diff --git "a/ferrimagnetic resonance/2.json" "b/ferrimagnetic resonance/2.json" new file mode 100644--- /dev/null +++ "b/ferrimagnetic resonance/2.json" @@ -0,0 +1 @@ +[ { "title": "2004.10475v2.Magnetic_correlations_in_polycrystalline___mathrm_Tb__0_15_Co__0_85___.pdf", "content": " \n \n 1 \n Magnetic correlations in polycrystalline Tb 0.15Co0.85 \nMathias Bersweiler1, Philipp Bender1, Inma Peral1, Lucas Eichenberger2, \nMichel Hehn2, Vincent Polewczyk3, Sebastian Mühlbauer4, and Andreas Michels1 \n1 Department of Physics and Materials Science, University of Luxembourg, 162A Avenue de la \nFaïencerie, L -1511 Luxembourg, Grand Duchy of Luxembourg \n2 Institut Jean Lamour (UMR CNRS 7198), Université de Lorraine, 54000 Nancy, France \n3 Istituto Officina dei Materiali (IOM) -CNR, Laboratorio TASC, 34149, Trieste, Italy \n4 Heinz Maier -Leibnitz Zentrum (MLZ), Technische Universität München, D -85748 Garching, \nGermany \n \nE-mail: mathias.bersweiler @uni.lu , andreas.michels@uni.lu \n \nSubmitted to Journ al of Physics D: Applied Physics \nReceived 5 February 2020 \nAccepted for publication 21 April 2020 \nAccepted Manuscript online 21 April 2020 \n \nDoi: https://doi.org/10.1088/1361 -6463/ab8b95 \nAbstract \nWe investigated a polycrystalline sample of the ferrimagnetic compound Tb 0.15Co0.85 by magnetometry and small -angle \nneutron scattering (SANS). The magnetization curve at 300 K is characteristic for soft ferrimagnets but at 5 K the hysteresis \nindicates the exi stence of magnetic domains. The magnetic SANS signal suggests that at 300 K the Tb and Co moments are \ncorrelated over large volumes within the micrometer -sized grain s with correlation lengths > 100 nm. At 5 K, however, the \nmagnetic SANS analysis reveals a reduced correlation length of around 4.5 nm, which indicates the formation of narrow \nmagnetic domains within the ferrimagnet with one d imension being in the nm range. We attribute the observed changes of the \ndomain structure to the tempera ture-dependence o f the magnetic properties of the Tb sublattice. \nKeywords: ferrimagnetism , magnetic domains , small -angle neutron scattering \n \n1. Introduction \nOver the last decades ferrimagnetic rare -earth transition -\nmetal alloys (RE -TM) raise d a lot of attention, since they are \ninteresting materials for fundamental research [1–4] and \npromising candidates for technological applications, such as \nmagneto -optical recording media [5], permanent magnets [6], \nor spintronic devices [7–10]. Previously, all -optical switching \nhas been observed in some selected RE -TM alloys [11–13], \nrendering them suitable candidates for optically -controlled \nmagnetic data storage devices. More recently, Mangin et \nal. [14] have demonstrated that all -optical helicity -dependent \nswitching can be extended to more complex, multilayered RE -\nTM systems containing for example HoFeC o, DyCo, or TbCo. \nMost of the studies are focused on amorphous RE-TM alloys, \nsince their fabrication is relatively easy and their magnetic properties can be straightforwardly controlled by changing the \nconcentrations, the nature of RE and TM, or the \ntemper ature [15,16] . It is well established that amorphous RE -\nTM alloys exhibit a noncollinear spin structure, the so -called \nsperimagnetic structure [17,18] , where the magnetic moments \nare frozen into random orientations. In contras t to the \namorphous alloys, in crystalline binary intermetallic \nferrimagnetic RE -TM alloys, it is assumed that the magnetic \nmoment of RE and TM are antiparallel coupled and form a \nferrimagnetic collinear arrangement [19–22]. \nThe goal of the present work is to investigate the structural \nand magnetic properties of a binary intermetallic Tb -Co \nferrimagnetic alloy, one of the most promising candidate \nsystem for the next generation of magnetic memories based on \nall-optical switching [14]. In particular, we study the \ntemperature dependence of the magnetic properties using \nconventional magnetometry combined with magnetic field - \n 2 \n dependent unpolarized small -angle neutron scattering \n(SANS). Magnetic SANS is a very powerful technique which \nprovides volume -averaged information about variations of the \nmagnetization vector field on a mesoscopic length scale of ~ \n1-300 nm [23,24] . This method has previously been applied to \nstudy the structures of magnetic nanoparticles [25–32], soft \nmagnetic nanocomposites [33,34] , proton domains [35–37], \nmagnetic steels [38–42], or Heusler -type alloys [43–46]. \nHere, we aim to estimate the temperature de pendence of the \nmagnetic correlation length in a polycrystalline bulk Tb-Co \nferrimagnetic alloy . \n2. Methods \nThe polycrystalline Tb 0.15Co0.85 sample has been prepared \nby arc melting under a high -purity argon atmosphere starting \nfrom stoichiometric quantitie s of the two high -purity elements \n(> 99.9% wt. % from Alfa Caesar). The mixture was melted \nin a water -cooled copper crucible and was not annealed after \nmelting. The sample was then ground manually, compacted to \na pellet, and enclosed in a silica tube under purified argon to \nprevent oxidation. The structural properties were determined \nby X -ray wide -angle diffraction of the powder using a Bruker \nD8 DISCOVER diffractometer with a Co -Kα radiation source. \nThe magnetic analysis was performed on a pellet using a \nPhysical Property Measurement System (PPMS) from \nQuantum Design (from 350 K to 5 K in applied magnetic \nfields up to 4 T). Thermomagnetization M(T) curves and \nhysteresis loops M(H) were recorded after cooling down the \nsample under a constant magnetic field of 4 T. The SANS \nexperiments were also performed on a circular pellet, in this \ncase with a diameter of 8 mm and a thickness of 1.2 ± 0.1 mm. \nThe neutron experiments were performed at the instrument \nSANS -1 at the Heinz -Maier -Leibnitz Zentrum (MLZ), \nGarching , Germany [47]. The measurements were done using \nan unpolarized incident neutron beam with a mean wavelength \nof λ = 4.51 Å and a wavelength broadening of Δλ/λ = 10 % \n(FWHM). The measurement s were conducted at room (300 K) \nand low temperature (5 K) and within a q-range of 0.06 nm-1 \n≤ q ≤ 3.0 nm-1. A magnetic field H0 was applied perpendicular \nto the incident neutron beam ( H0 ⊥ k0). Neutron data were \nrecorded at the maximum field available (4 T) and then in the \nremanent state (0 T). The neutron -data reduction (correction \nfor background scattering, sample transmission, and detector \nefficiency) was performed using the GRASP software \npackage [48]. \nIn the neutron data analysis (see below), the magnetic \nSANS cross section is discussed when the total (nuclear + \nmagnetic) SANS cross section at the highest field (near to \nsaturation) is subtracted from the total cr oss section at a lower \nfield. This procedure assumes that the nuclear SANS cross \nsection is independent of the applied magnetic field. \nTherefore, it is useful to explicitly display the total and the \npurely magnetic (difference) SANS cross sections. When t he applied magnetic field is perpendicular to the \nincident neutron beam ( H0 ⊥ k0), the elastic total (nuclear + \nmagnetic) unpolarized SANS cross section d Σ/dΩ and the \npurely magnetic SANS cross section d ΣM/dΩ are given as: \n \n𝑑𝛴\n𝑑𝛺(𝒒)=8𝜋3\n𝑉𝑏H2(𝑏H−2|𝑁̃|2+|𝑀̃𝑥|2+|𝑀̃𝑦|2cos2(𝜃)\n+|𝑀̃𝑧|2sin2(𝜃)\n−(𝑀̃𝑦𝑀̃𝑧∗+𝑀̃𝑦∗𝑀̃𝑧)sin(𝜃)cos(𝜃)) (1) \n \n𝑑𝛴𝑀\n𝑑𝛺(𝒒)=8𝜋3\n𝑉𝑏H2(𝛥|𝑀̃𝑥|2+𝛥|𝑀̃𝑦|2cos2(𝜃)\n+𝛥|𝑀̃𝑧|2sin2(𝜃)\n−𝛥(𝑀̃𝑦𝑀̃𝑧∗+𝑀̃𝑦∗𝑀̃𝑧)sin(𝜃)cos(𝜃)) (2) \n \nwhere V is the scattering volume, bH = 2.91 x 108 A-1m-1 relates \nthe atomic magnetic moment to the atomic magnetic scattering \nlength, 𝑁̃(𝒒) and 𝑴̃(𝒒)=[𝑀̃𝒙(𝒒),𝑀̃𝒚(𝒒),𝑀̃𝒛(𝒒)] represent \nthe Fourier transforms of the nuclear scattering length density \nN(r) and of the magnetization vector field M(r), respectively, \nθ specifies the angle between H0 and q q{0, sin ( θ), cos ( θ)} \nin the small -angle approximation, and the asterisks “*” denote \nthe complex conjugated quantities. For small -angle scattering \nthe component of the scattering vector along the incident \nneutron beam, here qx, is smaller than the other two \ncomponents, so that only correlations in the plane \nperpendicular to the incoming neutron beam are probed. The \n’s in equation (2) represent the difference between the \nFourier components at a certain applied fiel d and the highest \nfield of 4 T, which is subtracted in the data analysis. More \ndetails about the magnetic SANS technique can be found in \nRefs. [23,49] . \n3. Results \n3.1 XRD and SEM \nFigure 1 shows the X -ray diffraction results and displays a \nscanning electron microscopy (SEM) image s of the powder. \nThe SEM image s show that the primary particles (i.e., grains) \nare several μm in size. The XRD analysis confirms that \nTb0.15Co0.85 crystallizes in the hexagonal CaCu 5 structure -type \nwith the space group P6/mmm, indicat ing a pure single phase \nTbCo 5. This is expected from the hypothetical phase diagram \nof Tb xCo1-x and for a composition of x = 0.15 [50]. Moreover, \nthe XRD pattern exhibits no impurity peaks , which confirms \nthe high -quality synthesis of the Tb -Co alloy by arc melting. \nThe lattice -parameter values a and c were determined by the \nLe Bail fit method (LBF) implemented in the Fullprof \nsoftware [51].The values obtained from the XRD refinement \n(a ≈ 0.49 3 nm and c ≈ 0.40 1 nm) are consistent with the values \ntypically obtained in TbCo 5 alloy [50,52] . Furthermore, no \nadditional broadening of the diffraction peaks (apart from \ninstrumental broadening) are observed which verifies that the \ncrystallites are at least 100 nm in si ze. \n 3 \n \n3.2 Magnetometry \nFigure 2(a) shows the magnetization curves at 300 K and 5 \nK. At 300 K, the measured hysteresis loop is similar to that \nexpected for a soft polycrystalline ferrimagnet with randomly \ndistributed anisotropy axis. By cooling down to 5 K, the \nmagnetization curve significantly changes, namely , the \nmagnetization is strongly reduced over the whole field range \nas compared to 300 K, and the shape of the hysteresis is \ndistinctly different. \nThe characteristic shape of the hysteresis measured at 5 K \n(zoom in figure 2(b)) indicates that the reversal becomes \ndominated by the nucleation (i.e., the jump of M at small \nreversal fields) and propagation of magnetic domains (i.e., the \nshearing of the h ysteresis at intermediate fields). In fact, the \nmagnetization curve is qualitatively similar to that obtained in \nsynthetic antiferromagnetic magnetic systems whose field \nreversal behavior has been correlated to the collective \npropagation of magnetic stripe domains (see figure 3(a) in \nRef. [53]). \nThe temperature dependence of the total magnetization , \nmeasured under a cooling -field of 4 T, is displayed in figure \n2(c). By decreasing the temperature the total magnetization \ndecreases, as expected by considering the negative exchange \ncoupling between the Co and Tb sublattices (ferrimagnetic) \nand the temperature dependences of the Co and Tb magnetic \nmoments within the TbCo 5 crystal structure. As shown in \nRef. [54], in case of Tb -Co alloys having a TbCo 5 structure, \nthe magnetization of the Co sublatt ice remains roughly \nconstant over the temperature range 2 -400 K, whereas the \nmagnetization of the Tb sublattice increases significantly with \ndecreasing temperature, which consequently results in a \nreduction of the total magnetization. \n3.3 Magnetic Small -angle neutron scattering \nFigure s 3(a) and (b) display the two dimensional (2D) total \n(i.e., nuclear + magnetic) SANS cross sections at 300 K and at \n5 K, respectively , while figure 4 features the corresponding \n(over 2) azimuthally -averaged 1D SANS cross sections. As \ncan be seen, the total 2D SANS cross sections d Σ/dΩ are only \nweakly field -dependent and isotropic, which suggests the \ndominance of the isotropic nuclear scattering contribution. \nAccording to magnetometry (see figure 2(a)), the sample is \nnearly magnetically saturated at a field of 4 T for both \ntemperatures. Therefore, assuming a field -independent and \nisotropic nuclear SANS cross section d Σnuc/dΩ, the 1D sector \naverage of the total SANS cross section parallel to the applied \nfield ( q // H0) at 4 T is a good approximation for the nuclear \nSANS cross section d Σnuc/dΩ [compare equation (1)]. The in \nthis way estimated 1D d Σnuc/dΩ [red filled circles in figure 5] \nexhibit an asymptotic q-4 Porod behavior at the smallest \nmomentum transfers. This indicates scattering due to large -scale structures (e.g., grains or pores), which lie outside of the \nexperimentally accessible q-range (> 2 /qmin 100 nm). This \nis expected for the studied sample that consists of μm sized \ngrains (compare the results of the XRD anal ysis in figure 1). \nThe 2D magnetic SANS cross section d ΣM/dΩ in the \nremanent state at both temperatures [compare Eq. (2)] is \ndetermined by subtracting the 4 T data from the measurements \nat zero field . This data reduction procedure has already been \nused to extract the purely magnetic SANS cross section of \nnanoparticle systems [55,56] . The obtained 2D d ΣM/dΩ are \ndisplayed in figures 3(c) and (d) for 300 K and 5 K , \nrespectively. With reference to equation (2) it is reemphasize d \nthat the 2D magnetic cross sections d ΣM/dΩ contain in the \nsector perpendicular to the field (vertical sector) the difference \nof 𝑀̃𝑧(𝒒) at zero field and at 4 T. On the other hand, the sector \naverage parallel to the field contains the corresponding \ndifference s between the transverse magnetization Fourier \ncoefficients 𝑀̃𝑥(𝒒) and 𝑀̃𝑦(𝒒) at zero field and at 4 T. Thus, \nanalysis of this sector allows to access the transversal \nmagnetic correlation lengths in the remanent state. The 1D \nmagnetic cross sections at 300 K and 5 K, which are obtained \nby integration along the field direction over an angular range \nof ± 20°, are displayed in figure 5. At 300 K, the q-dependence \nof dΣM/dΩ is similar to that obtained for d Σnuc/dΩ ∝ q-4 (at the \nsmallest momentum transfers). This suggests the presenc e of \nlarge magnetic spin -correlation lengths ( lC > 100 nm), lying \noutside of the measured q-range. By contrast, at 5 K, a \ndeviation from the q-4 dependence can be discerned below 0.2 \nnm-1. This q-dependence can be described using a Lorentzian -\nsquared funct ion (blue solid line in figure 5) from which an \nestimate for the transversal magnetic correlation length of lC = \n4.5 ± 0.3 nm is obtained . As discussed by Hellman et al. [57], \na Lorentzian -squared term in magnetic SANS data may be \nattributed to meandering domain walls with lC being a measure \nfor the domain size. The magnetization data at 5 K ( figure \n2(a)) together with the estimated nanoscale transversal \ncorrelation length are compatible with this result. \n4. Discussion \nWe surmise that the formation of narrow magnetic domain s \nobserved at 5 K is connected to the temperature dependence \nof the magnetic anisotropy in TbC o5. The magnetic anisotropy \nof RE -TM systems is determined by the magnetic anisotropy \nof both sublattices (here the Tb and Co sublattice s). In \nRef. [54], it could be shown that the magnetic properties of the \nCo sublattice barely change within the temperature range 300 –\n5 K. Therefore, it can be assumed that the temperature -\ndependenc y of the magnetic properties of Tb Co5 is dominated \nby the one of the Tb sublattice . The magnetic anisotropy of the \nTb sublattice depends of the inter -sublattice exchange energy \nand on the easy-magnetization direction [58], which are both \nvery sensitive regarding temperature . The magnetization of \nthe Tb sublattice significantly increases with decreasing \ntemperature [54], so that it can be assumed that also the \n 4 \n magnetic anisotropy constant K increases accordingly [59]. \nAn increase of K favors the formation of narrow domain walls , \nsince the domain -wall width 𝛿w is proportional to 1√𝐾⁄ [60]. \nAs reported for Tb2Co17 [61], narrow domain walls are \ndifficult to move and thus may qualitatively explain the \nobservation of narrow domains at low temperature in TbCo 5. \nAt room temperature, on the other hand, the magnetic \nanisotropy of Tb is expected to be weak due to the increased \nthermal fluctuation of the magnetic moments of the Tb \nsublattice and , thus, 𝛿w is expected to become larger than at 5 \nK (the temperature dependence of the magnetic anisotropy in \nTbCo 5.1 suggests that K can vanish around 300 K [62]). \nTherefore, at 300 K, the system may rather favor a correlated \nsingle -domain structure within the grains. This feature \nqualitatively explains the soft ferrimagnet ic behavior of the \nmagnetization and the observation of a large spin -correlation \nlength by magnetic SANS , lying outside of the measured q-\nrange at room temperature. Further neutron studies, for \ninstance, magnetic -field-dependent polarized SANS and very \nsmall -angle neutron scattering (providing access to lower \nmomentum transfers) are required to shed light on the precise \nnature of the observed correlation lengths : In agreement with \nprevious neutron work [57], we interpreted the origin of the \ncorrelation lengths with the domain size, although it has to be \nconsidered that the correlation length could also be attributed \nto the domain walls. \n \n5. Conclusion \nTo summari ze, we employed magnetometry and \nunpolarized SANS to investigate the structural and magnetic \nproperties of polycrystalline samples of the ferrimagnetic \nalloy Tb 0.15Co0.85. The XRD analysis confirms the high quality \nof the synthesis with a single phase TbCo 5 as expected for this \ncomposition. The magnetometry results suggest a reversal of \nthe magnetization by rotation at 300 K, whereas at 5 K the \ncharacteristic shape of the hysteresis indicates the nucleation \nand propagation of magnetic domains. From the unpolarized \nSANS measurements, the purely m agnetic SANS cross \nsections in the remanent state were determined by subtracting \nthe scattering patterns measured at a large magnetic field of 4 \nT. The 1D magnetic SANS cross section parallel to the applied \nfield suggests that at 300 K both the Co and Tb m oments are \ncorrelated over large distances with correlation lengths of at \nleast 100 nm. At 5 K, on the other hand, analysis of the \nmagnetic SANS signal in terms of a Lorentzian -squared \nscattering function reveals a reduced correlation length of \naround 4.5 nm. 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For the analysis, the Le \nBail fit method (implemented in the Fullprof software) was \nused, considering the space group P/6mmm. The “*” indicate \nthe diffraction peaks coming from the Kβ radiation of the \nCobalt source. The bottom black solid line represents the \ndifference between the calculated and observed intensities. (b) \nSecondary electron scanning electron microscopy images of \nthe grain microstructure of our sample. Here the black color \ncorresponds to the carbon tape used for the discharging. \n \nFigure 2. (a) Magnetization curves measured in a field \nrange of ± 4 T at 300 K (black solid line) and 5 K (blue solid \nline). (b) Zoom of the magnetization curve measured at 5 K in \na field range of ± 1 T. The onset of nucleation and propagation \nof the magn etic domains are sketched by the arrows (1) and \n(2), respectively. (c) Temperature dependence of the total \nmagnetization under a fixed field of 4 T. \n \nFigure 3. (a) and (b) E xperimental two -dimensional (2D) \ntotal (nuclear + magnetic) unpolarized SANS cross sections \ndΣ/dΩ measured at 300 K and 5 K, respectively . (c) and (d) \nPurely magnetic 2D SANS cross sections d ΣM/dΩ measured \nat 300 K and 5 K, respectively . The purely magnetic 2D SANS \ncross section s in the remanent state were obtained by \nsubtracting the total scattering at the (near) saturation field of \n4 T from the data at H = 0 T. The applied magnetic field H0 is \nhorizontal in the plane of the detector ( H0 ⊥ k0). Note that the \ndΣ/dΩ and d ΣM/dΩ scales are plotted in polar coordinates ( q \nin nm-1, θ in degree, and the intensity in cts/exposure time). \n \nFigure 4. (a) and (b) Azimuthally -averaged 1D total SANS \ncross sections d Σ/dΩ as a function of the momentum transfer \nq and at selected applied -field values (see insets) (log -log \nscale) at 300 K and 5 K , respectively. The error bars of d Σ/dΩ \nare smaller than the data point size. \n \nFigure 5. Red filled circles: nuclear 1D SANS cross section \ndΣnuc/dΩ as a function of momentum transfer q. Color ed filled \nsquares: radially -averaged 1D magnetic SANS cross sections \ndΣM/dΩ along the field direction at 300 K (white filled \nsquares) and at 5 K (blue fi lled squares). Red dashed line: \npower law d Σnuc/dΩ ∝ q-4. Blue solid line: Lorentzian -squared \nfit of the transverse scattering contribution at 5 K to determin e \nthe magnetic transverse correlation length lC. The d Σnuc/dΩ \nwas determined by 10° horizontal sector averages ( q // H0) \nof the total d Σ/dΩ at an applied magnetic field of μ 0H0 = 4T \nand T = 300 K. The radially -averaged 1D magnetic SANS \ncross section was determined by 20° horizontal sector \naverages ( q // H0) of the 2D magnetic SANS cross section at the remanent state taken from figure 3. Note: the magnetic \nSANS cross section intensities at 300 K and 5 K have been \nrescale d to the nuclear 2D SANS cross section intensity for \nbetter comparison. (log -log scale)Figure 1 \n \n \n \n \n 8 \n Figure 2 \n \n \n \n \n 9 \n Figure 3 \n \n \n 10 \n Figure 4 \n \n \n \n 11 \n Figure 5 \n \n \n" }, { "title": "0808.2015v2.Co_resonant_enhancement_of_spin_torque_critical_currents_in_spin_valves_with_synthetic_ferrimagnet_free_layer.pdf", "content": "Co-resonant enhancement of spin-torque critical currents in spin-valves \nwith synthetic-ferrimagnet free-layer \n \nNeil Smith, Stefan Maat, Matthew J. Carey, Jeffrey R. Childress . \nSan Jose Research Center, Hitachi Globa l Storage Technologies, San Jose, CA 95135 \n \nIt is experimentally shown that the critical current for onset of spin-torque instability in \ncurrent-perpendicular-to-plane spin-valves can be strongly enhanced using \"synthetic \nferrimagnet\" free-layers of form FM 1/Ru/FM 2 (FM=ferrromagnet). However, this \nenhancement occurs for only one polarity of bi as current. A two-macrospin model is shown \nto reproduce the observations. The model sugge sts that this phenomenon is related to a \npolarity-dependent, spin-torque induced co-resonance betw een the two natural dynamic \nmodes of the FM 1/FM 2 couple. The resonance c ondition facilitates energy transfer out of the \nspin-torque destabilized mode into the othe r stable mode whose effective damping is \nactually enhanced by spin-torques, thereby delayi ng the onset of instability of this coupled \nsystem to larger critical currents. \n Spin-torque phenom ena, as m anifested in giant-m agnetores istive spin-v alves film stacks \nlithographically patterned in to ~100 nm nanopillars and driven with electrical cu rrents perpendicular to \nthe film plane have in recen t years been the activ e study of num erous theoretical and experim ental \npapers,1 both for their novel physics as well as potential applications for m agnetic m emory elem ents, \nmicrowave oscillato rs, and m agnetic recording read heads. In essen tially a ll of these s tudies, the \ndynam ically active m agnetic layer, or \"free layer\" of the spin-valve film stack, is either theoretically \nmodeled or experim entally fabricat ed as a single ferrom agnetic layer. This paper investigates, through \nboth experim ental m easurem ent and theoretical modeling, the novel spin -torque dynam ics of a \n\"synthetic-ferrim agnetic\" free-layer of the for m FM1/Ru/FM2, consis ting of two ferrom agnetic (FM) \nfilms of une qual thickness separated by a thin (0.8 nm ) Ru spacer which prom otes well-\nknown2FM 1MF t t >>\n2, strong antiparallel coupling betw een the two FM layers. C ompared to the sim ple free-layer \nsystem , the FM1/Ru/FM2 couple has two addition al spin-torque-pro ducing (Ru/FM) interfaces, and \nperm its (in the sim ple macrospin pi cture) two independent, non-degenerate, natural modes of oscillation. \nAs will be d iscussed below, these features can lead to a novel conditio n of spin-to rque-induced \"quasi-\nco-resonance\" of these two m odes which greatly im pacts the spin-torqu e-stab ility o f such devices, and \nwhich carries poten tially im portan t practical im plicatio ns f or the ir use in the af orem entioned \napplic ation s. \nThe presen t exper iments use m ultilayer f ilms of form AFM/PL/Ru/RL/Cu/FL1/Ru/FL2 (exc luding \nseed and cap layers). The first ferrom agnetic pi nned-layer (PL) is exchange-pinned to the \nantiferrom agnetic (AF M) layer, and is also strongly antiparallel-couple d to a second FM reference-layer \n(RL) across a thin Ru spacer. The PL and RL layers are clo sely m oment-m atched, for ming a \"synthetic-\nantiferromagnetic \" cou ple (as is c ommon practic e for such devices) which consequently does not \nrespond to a m odest external m agnetic fields. More uni que to the present structur es, the first free-layer \n(FL1) is also antiparallel-coupled to a second fr ee-layer (FL2), for ming the \"synthetic-ferrimagnetic-\nfree-lay er\" (SFM-FL) with shee t-film M-H behavior (at modest external fields) equivalent to a single \nFM f ilm of thickness . 2FL 1FL t t−\nA first set of experim ental m easurem ents, desc ribed in Fig. 1, uses Ni Fe f ree-lay ers with \n including a contro l with = 0, and two SFM-FL designs with = 2nm and 3nm . \nThe devices tested have been patterned into 75-nm circular pillars using E-beam lithography.2FL 1FL nm4 t t + =2FLt2FLt\n3 \nResistan ce (R-H) loops are m easured (at -5m V bias) in fields collin ear and transverse to \nthe IrMn pinning direction. All chosen devices have nonhysteretic, square (for )(xH )(yH\nxHR- kOe1≤xH ) and \nnear-symm etric (about ) . Accom panying each R-H data set ar e two loops, which 0=yHyHR-eIN-Fig. 1. (a) cartoon of devi ce geom etry. (b-d) R-H loops (as % δR/R) an d N-I e loops (as rm s power spectral \ndensity at 75 M Hz); for tFL2 = 0 (b) , 2nm (c), an d 3nm (d) . Spin-valve stack structure: \nIrMn(7)/Co Fe(3)a/Ru (0.7)/CoFe(3)/Cu(4)/ NiFe(4+ tFL2)/Ru(0 .7)/NiFe (tFL2); ( ) de notes film thickness in nm. \n \nmeasure narrow-band noise N vs electron curren t with constant applied fields of either \n or -600 Oe to align FL1 m agnetization either an tiparallel (A P) or para llel (P) with that of \nthe RL. Positiv e ele ctron curren t travels f rom RL to FL (Fig. 1a). Th e curr ent is driven by a 2-Hz \nsawtooth generator with s ync pulse triggering the 0.5- sec sw eep of a (zero-span ) spectrum analyzer. The \n loops are averaged over sweeps. The eI\nOe600+≅xH\neIN- 50≈ 0≅eI electronics n oise ( Hz nV/8.0~ ) is \nsubtracted out. \nThe techniqueeIN-4 measures the 1/ f-like noise as sociated with therm al perturbations of well-\nknown precessional m otion of a unid irectionally stable FL once sp in-torque instability begins.5. This \nonset is readily observed by the sh arp increase in noise above the /mAHz nV/03.0≈ residual \nelectronics noise for th ese devices . The \"critical currents\", , for this onset are found by \nsimple inspection. The were typically insensitive to few hundred Oe variations in . Ω≈11crit\neI\ncrit\neIxH\nFor all th icknesses of FL2, there is an observed AP-state negative critical poin t \nwhich was previously shown to be sp in-to rque-instab ility o f the RL/PLmA5.22crit\nAP -≈ −eI\n5,6. The SFM-FL devices alone \nshow an addition al positive critical p oint in the P-state, which is discussed further below. For the crit\nPeI+(a) \n0123\nvs. Hx\nvs. HytFL1=4 nm\ntFL2=0 nm\n(%)δR\nR\n0123\nvs. Hx\nvs. Hy(%)δR\nRtFL1=6 nm\ntFL2=2 nm\n-2 -1.5 -1 -0.5 0 0.5 1 1.5 20123\nH (kOe )vs. Hx\nvs. Hy(%)δR\nRtFL1=7 nm\ntFL2=3 nm00.2AFMx\nyzHa\nIe\nPL RL FL1 FL275 nm\n0.40.60.81\nnV\nHzPSD\nPAPH = ! 600 OextFL1=4 nm\ntFL2=0 nm\n00.20.8(b) \n0.40.61\ntFL1=6 nm\ntFL2=2 nmnV\nHzPSD\nPAP\nH = ! 600 Oex\n-6-5-4-3-2-1 012345600.20.8(c) \n0.40.61\nnV\nHzPSD\nI (mA)ePAPH = ! 600 OextFL1=7 nm\ntFL2=3 nm\n(d) 02FL=t control, the polarity asymmetry ratio is s imilar to earlier \nobservations,35.2 ) /() (1FL 1FLcrit\nAPcrit\nP - ≈ + −e e I I\n4-6 and is a known consequence of the intrinsic angular dependenc e of electrical transport in \nthese a ll-metallic dev ices. It is unre lated to th e unexpected polarity asym metry discussed below. \nFig. 2 shows a summary of si milar measurements for a second, modified film stack which \nincludes thin CoFe layers at the Cu/FL1 and FL1,2/Ru interfaces.eIN-\n7 Referenced to Ni 80Fe20 film s of equa l \nmom ent, the m agnetic thicknesses of FL1,2 ar e sim ilar to those shown in Fig. 1. \nFig. 2. P-state N-I e loops (rms po wer sp ectral d ensity); for spin-valve stack :: \nIrMn(7)/CoFe(3 )/Ru(0 .6)/CoFe(3 )/Cu(5)/CoFe(0.6)NiFe(4+ tFL2) /CoFe(0.2)/Ru(0.6)/CoFe(0.2)/NiFe(tFL2); \n( ) denotes film th ickness in nm.. In Ni80 Fe 20 equivalent Angstroms thickness, tFL1 = tFL2+45, tFL2 as indicated. \n \nFig. 3 summ arizes the experim ental results for for both stack structures, which includes 3 or 4 \ndevices for each value of , for which the data in F igs. 1 and 2 are representative. crit\neI\n2FLt\nFig. 3. Critical current vs. tFL2 for film stacks from Fig. 1 (a), and Fig. 2 (b). Solid circles (squares) for \nP-state (AP-sta te). Solid curves are m odel results as described in text, using η-coefficien ts as ind icated in \nfigures. Dashe d curves are for η2 = η3 = 0, but which exclude IePcrit > 0. \n -9-8-7-6-5-4-3-2-1012345678900.20.40.60.81\nnV\nHzPSD\nH = -600 Oex\nI (mA)eP-state\n00\n5515152525\n0 0.5 1 1.5 2 2.5 3 3.5-8-6-4-2024\n(nm)Iecrit\n(mA)\ntFL2P-state\nP AP η =0.25, η =0.75, η =0.75, η =0.61 1 2 3\nH = ! 600 OexAP-state\n0 0.5 1 1.5 2 2.5 3 3.5-10-4-2024(a) \n-8-6Iecrit\n(mA)\n(nm) tFL2P-state\nη =0.26, η =0.65, η =0.4, η =0.451P\n1AP\n2 3\nH = ! 600 OexAP-state\n(b) In both cases , there is a striking increase in the magnitude of with incre asing , which is \neven more dramatically demons trated for the second, Co Fe-int erfaced FL stack structure. In stark \ncontrast to these observations regarding , it is seen that shows rather little abso lute \nchange with . These tw o counter-intuitive resu lts agree well with analogous dV/dI measurem entscrit\nPeI−2FLt\ncrit\nPeI−crit\nAPeI+\n2FLt7 \nusing test devices (sam e film stack as in Fig. 2) of similar area but asymm etric non-circular geom etry. \nFor a theoretical insight into this phenom enon, we start w ith a sim ple two-m acrospin m odel for \nFL1/FL2 which tr eats the RL/PL as an iner t, spin-current polarizer. Magne tostatic, Ru-couplin g and \nZeem an term s for the system free energy E are taken to be \n] ) /() ( [ˆˆ ) /(\n2 12 1 ru2\n1,;21\n1FL 2FL1FL\nx s s x aku\nkjuu\njk ju s\nm tM tM mHH mHm VME\n+ −⋅ + =∑\n=mm\n (1) \nwhere are the unit m agnetization vectors fo r FL1,2, the set of tri-ind icied are \nmagnetostatic energy coefficients, is th e interfacial cou pling s trength, , \n is the app lied f ield, and is th e volum e of FL1. Slonczewski-typ e2,1ˆmzyxu\njkH,,\n2,1=\n=\nruJ1FL) /(ru ru tM J Hs ≡\nx H ˆa aH=1FLV8 spin-torques \n at the Cu/F L1, FL1/Ru, and Ru/FL2 interfaces are included as follows: mH ˆ ) (ST\n1FL × = VMs τ\n \nST\n1FL1FL ST STSTST STST\nˆ/ ) (/1) /()2/( ;ˆ ˆˆ ˆ )ˆ(ˆ\neff\n2,11 2 3 21 2 2 1 1 1\nj j s js e\nE VMVMIe H HH H\nHm Hmm Hmm x m H\n+ ∂∂ −=≡ × η=× η+ ±× η=\n=h (2) \n \nto form the tota l effective f ield . In (2), eff\n2,1=jH xˆ± refers to (P and AP-states), and the RLˆm 0=eI \nequilibrium state now defined to be x m ˆ ˆ01=, x m ˆ ˆ02 −= . The η-coef ficients will b e discuss ed below. \nThe additional (two) degrees of freedom of th e second m acrospin substantially com plicates the \nalgebraic description relative to the well-known 1- macrospin ca se.. As described previously,5,9 \nindividual, local coordinates where zyx′′′02 01ˆ ˆ ˆ mx m =′= are u sed to constru ct the f ollowin g matrix \nformulation of the linearized Gilbe rt equ ations of motion f or the two-dim ensional vectors \n: ) , (21 zj yj ,j m m ′ ′ == ′my z\nay z\naz yvu vkkv\nkvju\njuuj\njk j jvu\njkjkj\njk jkj\njk\nH H H H Htt H HH H H H H H HH H H H H HH HH HHH HH HHH HH HHH HH HHmm\nmH\nmm\nHtt\nDtt\nGtdtd\n22 22 7 8 21 3 711 11 5 6 3 512 ru 4 12 ru 3 2 18 33 7\n22\n4 33 3\n214 22 3\n12\n65\n11,eff\neff1 121\n, )/(,, ,,,)ˆ (1001,0110,0)( ) (\nSTST\nSTSTSTST\nSTST\n− + ≡ − ≡− + ≡ + ≡+ = + ≡ η+η±≡η′⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nη−η≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nηη −≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nη−η− −≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nη′η′−≡′∂∂\n∂∂\n∂′∂\n−δ ⋅ ≡δ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nγα\n≡ δ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−\nγ≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n′′≡′ =′⋅+′⋅ +\n∑\n′′ ′′\nt tt tt tt tt\nmHmmm mHmGD\n (3) \n \nwhere gyrom agnetic ratio Oe) Mrad/(sec19 - ≅γ , and 0>α is the Gilb ert dam ping param eter. In \n(2),vu\njkH′′⇔Ht\n is a tensor-m atrix f ormed from the 2D Cartesian tensors 44×jkHt\n given explicit in (3 ), \nand sim ilar for Dt\n and G. The expressions for assum e the symm etry . t\n7,5Hy\njkx\njkH H =\nThe natural modes for this system are nont rivial solutions of (3) of the for m . The 4 \nroots satisfy stet−∝′)(m\n2 2 1 1 , ω±σω±σ= i i s 0|) ( |det = + − GDHttt\ns , but are more generally found f rom the \neigenvalues of the matrix .1) (−+⋅GDHttt9 Only two of these roo ts, 2,1 2,1 2,1 ω+σ= i s , describe physically \ndistinct m odes. The spin-torque terms in (2) yield a nonreciprocal Ht\n (i.e., ), which can be \nshownuv\nkjvu\njk H H′′ ′′≠\n10 to perm it unstable modes with 0) Re( eI\n0crit\nP>eI\n4.03≥η . In \naddition, the m odel fails for the data in Fig. 3b at 0crit\nP>eI nm5.02FL=t , which m ay reflect excess \ndamping as , and/or a breakdown (due to finite transverse spin-diffusion length 02FL→t12) of the purely \ninterfacial f orm of spin-torque im plicitly assum ed in (2). \nIn the lim it z z y y m m m m J ′ ′ ′ ′ −→ → ⇒∞→1 2 1 2 ru , one m ay obtain from (3) the following \nanalytica l solutions f or : crit\neI\n∑\n=+− − + −≡η+ η−α\n⎥⎦⎤\n⎢⎣⎡\n−+→\n2\n1,2101 10 crit\nAP)or(P\n) ()1( )/ 1() or ( e2/) (\n1FL 2FLAP P1FL\n2FL 1FL2FL 1FL\nkjy\njkz\njkkj\nas\ne\nH H Ht t HH VM\nt tt tIh\n (4) \nwhich also a pplies when With .02FL=t2FL 1FL t t− fixed, (4) indicates (excluding any -dependence of \n) a scalin g, but equa lly so for eithe r or . The la tter sharp ly \ncontradicts experim ent. Further, (4) ex cludes the additional observed cases of when FLt\n0H1FL 2FL 1FL ) ( t t t ⋅ +crit\nPeI−crit\nAPeI+\n0crit\nP>eI.02FL>t The underlying physics behind the asymmetric , strong super linear (or weak) -dependence \nof (or ) would thus appear connect ed with the finiteness of . 2FLt\n0crit\nPeIruJ\nThis is f urther elucid ated in Fig.4. Com puted a s a continuous function of are critical currents \n, as well as natura l-mode param eters ruJ\n)(rucritJ Ie ) (21\n< >ω−ω ≡∆πf and ), max(2 1σσ ≡σ> evaluated at \n. In the case of negative in Figs. 4a ,b, the spin -torque term s radic ally alter th e \nnatural oscillation frequencies , even to the point of inducing a literal co-resonance , i.e., )(rucritJ I Ie e= 0crit\nP πf , between th e two m odes at a finite . Closely accom panying the co-resonance \nis a broad peak in both and ruJ\n)(rucrit\nPJ Ie− )(ruJ>σ , with a large maximum at . max\nru ruJ J≡\nThe are th e tem poral decay rate s (or line-wid ths) of the stable mode. Here, spin-torques can \nincrease the rate of energy loss from that m ode of osci llation well beyond that of intrinsic dam ping (e.g., \n at .). The th ird (dam ping) m atrix >σ\n1Gsec4−\n>≈σ 0=eI Dt\n in (3), along w ith nonreciprocal spin-torque \n0 02 04 06 08 1 12 1403691215\n01224364860\n(mA)−IecrittFL1=6.0 nm\ntFL2=1.5 n m\n(GHz)∆f\nor\nσ>P-state\nH = -600 Oex\n0 02 04 06 08 1 12 1(a) \n0510152025\nFig. 4. Mo deled values for critical cu rrent (solid line), ∆f (dashe d line), and σ> (dotted line) v s Jru , for \nparameter values ind icted in tex t and/or figure. η-coefficients sam e as use d in Fig. 3b. \n 401020304050\n(mA)−IecrittFL1=7.0 nm\ntFL2=2.5 nm\n(GHz)∆f\nor\nσ>P-state\nH = -600 Oex\n0 02 04 06 08 1 12 1(b) \n012345\n40510152025\n(mA)tFL1=7.0 nm\ntFL2=2.5 nm\n(GHz)∆f\nor\nσ>AP-state\n+IecritH = +600 O ex\n0 0.2 0.4 0.6 0.8 1 1.2 1.4048121620(c) \n01020304050\n(mA)−Iecrit\nJru(erg/cm )2tFL1=7.0 nm\ntFL2=2.5 nmP-state\nH =H = 600 O ek1 k2\n(GHz)∆f\nor\nσ>\n(d) contributions to Ht\n, imply that the two natura l modes are non-orthogonal, and are coupled both by spin-\ntorques and weakly by intrinsic dam ping. This dynam ic coupling allows energy tr ansfer between m odes, \nwhich is further strongly enhanced at and near the condition of co-resonance. However, this \nenhancem ent does not strictly require , but can occur under more general conditions of \"quasi -\nco-reson ance\" where , i.e., when the difference in the m odes' resonant frequencies is sm aller 0→∆f\n1 /<σ∆>f\nthan the ef fective line -width of the dam ped mode. This enhanced inter-m ode coupling provides >σ\nanother energy loss path (in addition to intrinsic dam ping) to counter the positive rate of work by spin-\ntorques on the destabilized m ode, delaying onset of spin-torque-instability and increasing . It is \nthus not surprising that the -depend ence of closely follows that of . Further, the \nbroad peaks of in Figs. 4a,b approxim ately coincide with the quasi-co-resonant condition \n. Finally, since , the broad tail of and the m onotonic increase \nwith of (e.g., com pare Figs. 4a and 4b) yields an explanation for the supe rlinear inc rease of \n with . crit\nPeI−\nruJcrit\nPeI− )(ruJ>σ\n)(rucrit\nPJ Ie−\n1 /<σ∆>fdevice\nrumax\nru J J < ) (max\nru ruJ J> σ>\n2FLtmax\nruJ\ncrit\nPeI−2FLt\nBy contras t, for positiv e modeled in Fig. 4c, the co-res onance condition does not occur, and \nthe spin -torque term s actua lly r educe below that of intrinsic dam ping ( ). \nAccordingly , shows little enhanc ement with the SFM -FL design, and the model add itionally \nindicates a moderate reduction in relative to th e fictitious c ase crit\nAPeI\n) (crit\neI>σ1Gsec4−≈\ncrit\nAPeI\ncrit\nAPeI 03,2= η (Fig. 3b). Finally, Fig. 4d \nshows results for a bi-stable SFM-FL with uniaxial anisotropy Oe600=kH in FL 1,2 replacing the \nextern al field (i.e., for t he term , k a H H→5Hk a H H −→ for in (3)). Although \nagain resembles in shape, there is no co-reson ance nor superlinear enhancem ent with of \n, which at is about 10% less than predicted by the model of Fig. 3b. \nThis m odeling resu lt is consis tent with 7H )(rucrit\nPJ Ie−\n)(ruJ>σ2FLt\ncrit\nPeI−2 device\nru erg/cm1≅ J 03,2= η\n0=aH dV/dI measurem ents7 on non-circular devices with \nmagnetostatic shape anisotropy. \nThe las t result in Fig. 4d clea rly indicates a connection between observable qu asi-co -reson ant \nenhancem ent of , and the presence o f an external field antipa rallel to (although all states in \nFig. 4 are m agnetostatically stable at crit\nPeI−2FLˆm\n0=eI with ). This situa tion would n aturally 2\nru erg/cm1.0>Joccur in practice for a current-per pendicular-to-plane giant-m agnetoresistive m agnetic read sensor, \nwhere the FL is conventionally stabilized by uni directional fields from abutted perm anent m agnet \nlayers.13 The increase in bias cu rrent (while m aintaini ng device stability) afforded by use of the SFM-FL \nthus has ready application for im proving sensor output signal for future read heads in hard disk drives.7 \n \nREFERENCES \n \n1 D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mat. 320, 1190 (2008) and m any references therein. \n2 S.S.P. Parkin, N. More and K.P. Roche, Phys. Rev. Lett., 64, 2304 (1990). \n3 D. Lacour, J.A. Katine, N. Sm ith, M.J. Carey and J.R. Childress, Appl. Phys. Lett. 85, 4681 (2004) \n4 N. Sm ith, J.A. Katine, J.R. Childress, M.J. Carey, IEEE Trans. Magn. 41, 2935,(2005). \n5 N. Sm ith, Phys. Rev. B 74, 026401 (2006) . \n6 J.R. Childr ess, M.J. Carey, S.I Kis elev, J.A. Katine, S.Ma at, and N. Sm ith, J. Appl. Phys. 99, 08S305 \n(2006). \n7 M.J. Carey, N. Sm ith, S. Maat, and J.R. Childress, arXiv:cond-m at/0808.2001. \n8 J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); 247, 324 (2002). \n9 N. Sm ith, J. Magn. Magn. Mater. (2008), doi: 10.1016/j.jm mm.2008.05.009 \n10 N. Sm ith, arXiv:cond-m at/0406486. \n11 J. Bass, and W . P. Pra tt Jr., J. Phys.: Conden. Matter 19, 183201, (2007) \n12 W. Chen, M. J. Rooks, N. Ruiz, J. Z. Sun, and A. D. Kent, Phys. Rev. B, 74, 144408 (2006). \n13 C. H. Tsang, R. E. Font ana, T. Lin, D. E. Heim . B. A. Gurney , M. L. W illiam s, IBM J. Res. Develpt. , \n42 103 (1998). \n " }, { "title": "0902.3109v1.Majority_spin_non_quasiparticle_states_in_half_metallic_ferrimagnet_Mn__2_VAl.pdf", "content": "arXiv:0902.3109v1 [cond-mat.mtrl-sci] 18 Feb 2009Majority-spin non-quasiparticle states in half-metallic ferrimagnet Mn 2VAl\nL. Chioncel,1,2E. Arrigoni,1M.I. Katsnelson,3and A.I. Lichtenstein4\n1Institute of Theoretical Physics, Graz University of Techn ology, A-8010 Graz, Austria\n2Faculty of Science, University of Oradea, RO-410087 Oradea , Romania\n3Institute for Molecules and Materials, Radboud University of Nijmegen, NL-6525 ED Nijmegen, The Netherlands\n4Institute of Theoretical Physics, University of Hamburg, D E-20355 Hamburg, Germany\nThe density of non-quasiparticle states in the ferrimagnet ic full-Heuslers Mn 2VAl alloy is cal-\nculated from first principles upon appropriate inclusion of correlations. In contrast to most half-\nmetallic compounds, this material displays an energy gap in the majority-spin spectrum. For this\nsituation, non-quasiparticle states are located below the Fermi level, and should be detectable by\nspin-polarizedphotoemission. Thisopensanewwaytostudy many-bodyeffectsinspintronic-related\nmaterials.\nHalf metals display a particular type of itinerant-\nelectron magnetism as well as unusual electronic prop-\nerties: they are metallic for one spin channel, and in-\nsulating or semiconducting for the opposite one [1, 2].\nElectronic structure calculations based on density func-\ntional theory offer an explanation for the half-metallicity\nbased on the interplay between the crystal structure, the\nvalence electron count, the covalent bonding, and the\nlarge exchange splitting in addition to symmetry con-\nstrains. The expected 100% spin polarization of half-\nmetals turned out to be an excellent motivation in de-\nveloping the field of spintronics both from a theoretical\nand an experimental point of view [2, 3]. In reality many\npotential half-metallic ferromagnets exhibit a dramatic\ndecrease of bulk spin polarization at temperatures well\nbelow their Curie temperature. In order to understand\nsuch a behavior from a theoretical point of view it is nec-\nessary to consider finite temperature many-body effects\n[2].\nAn important effect of dynamical electron correlations\ninhalf-metalsistheexistenceofnon-quasiparticle(NQP)\nstates [4, 5, 6]. These states contribute significantly\nin reducing the tunneling transport in heterostructures\ncontaining HMF [7, 8, 9, 10, 11], even in the presence\nof disorder. NQP states strongly influence the value\nand temperature dependence of the spin polarization in\nHMF [2, 6, 12], which is of primary interest for po-\ntential applications. These states originate from spin-\npolaron processes whereby the minority spin low-energy\nelectron excitations, which are forbidden for HMF in the\nsingle-particle picture, are possible as superpositions of\nmajority-spin electron excitations and virtual magnons\n[2, 4, 5, 6]. Recently we have applied the LDA+DMFT\n(local density approximation plus dynamical mean field\ntheory) method (for review of this approach, see Ref.13)\nto describe from first principles the non-quasiparticle\nstates in several half-metals [14, 15, 16, 17, 18]. Up to\nnow, our studies were restricted to half metals with a\ngap in the minority spin channel. In this situation NQP\nstates appear just abovethe Fermi level [6].\nOn the contrary, it was predicted that in half-metallic\nmaterials with a gap in the majority (say, “up”) spinchannel, NQP states should appear below the Fermi\nlevel [4, 5, 6]. This asymmetry can be understood in\nterms of electron-magnon scattering processes, as pre-\nsented in the followings.\nA well studied model which takes into account the in-\nteraction of charge carriers with local moments is the s-d\nexchange model. The interacting part of the Hamilto-\nnian is given by −I/summationtextSiσαβc†\niαciβ, where Iis thes-d\nexchange parameter, Sirepresents the localized spin op-\nerators,σαβare the Pauli matrices, and ciσare operators\nfor conduction electrons. The NQP picture turns out to\nbe essentially different for the two possible signs of the\ns−dexchange parameter.\nThe ground state of the system with I >0 (assum-\ning that the Fermi energy is smaller than the spin split-\nting 2IS) has maximum spin projection and, thus, the\nminority-electron band should be empty. For this case\n(I >0), NQP states in the minority spin gap develop\nas a superposition of the majority-electron states plus\nmagnon states, and of the minority-electron states, so\nthat the totalspin projection of the system is conserved.\nAs a result of this spin-polaronic effect, the minority-\nelectron density of states has a tail corresponding to\nthe virtual conduction electron spin-flip processes with\nmagnon emission. However, these virtual flips are im-\npossible below the Fermi energy EFdue to the Pauli\nprinciple (all majority-electron states are already occu-\npied and thus unavailable). Therefore, for the positive\ns-d exchange interaction the NQP states form above EF.\nContrary,fornegative Ithe minority-spinbandliesbe-\nlow the majority-spin one [2, 6]. Occupied minority-spin\nstates can be superposed with majority-electron states\nplus magnons, with conserved total spin projection, so\nthe NQP states occur below EF. At the same time, for\nI <0 the ferromagnetic ground state is non-saturated\nand thus zero-point magnon fluctuations are allowed. It\nis the fluctuations which are responsible for formation of\noccupied majority-electron NQP states there.\nFormally, the difference between I <0 and the previ-\nousI >0 cases, can be explained in terms of a particle-\nholetransformation c†\niσ→di¯σ, andciσ→d†\ni¯σ. Thismod-\nifies thes-dexchange Hamiltonian into I/summationtextSiσαβd†\niαdiβ.2\nIn other words, the Hamiltonian with I >0 for electrons\nis equivalent to that with I <0 for holes.\nThe above argument based on the s-dexchange model\ncan be generalized for arbitrary multi-band half-metallic\nelectronic structures [2, 6]. The conclusion remain un-\nchanged: for the case of minority-electron gap, NQP\nstates are situated above the Fermi energy, while for\nthe cases when the gap is present for majority-electrons,\nNQP states are formed below the Fermi energy.\nMost HMF materials have a gap in the minority spin\nchannel so that NQP states arise above the Fermi level.\nAs a consequence, these states cannot be studied by\nthe very well-developed and accurate technique of spin-\npolarized photoemission [19], which can only probe occu-\npied states. The spin-polarized Bremsstrahlung Isochro-\nmat Spectroscopy (BIS) probing unoccupied states [20]\nhas a much lower resolution. For this reason, HMF with\na gap in the majority-spin channel, and, consequently,\nNQP states in the occupied region of the spectrum, allow\nfor a detailed experimental analysis of these correlation-\ninduced states and are, therefore, potentially of great in-\nterest. Itisthepurposeofthepresentworktoperforman\nelectronic structure calculation based on a combination\nof the generalized-gradient approximation (GGA) and of\nDMFT for the half-metallic ferrimagnetic full-Heusler al-\nloy Mn 2VAl, which has a gap in the majority spin chan-\nnel. By appropriately taking into account effects due to\nelectronic correlations, we demonstrate explicitly the ex-\nistence of majority spin NQP states arising just below\nthe Fermi level, and study the temperature dependence\nof their spectral weight.\nIn full Heusler compounds with the formula X 2YZ, Mn\natomsusuallyoccupytheY-position,whilecompoundsin\nwhich Mn assumes the X-position Mn 2YZ, are very rare.\nThe prototype from the latter category is Mn 2VAl, for\nwhich a large number of theoretical and experimental in-\nvestigations have been made. Neutron diffraction experi-\nments [21] demonstrated the existence of a ferrimagnetic\nstate in which Mn has a magnetic moment of 1 .5±0.3µB\nandVmomentis −0.9µB. X-raydiffractionandmagneti-\nzationmeasurements[22] foundatotal magneticmoment\nof 1.94µBat 5K, close to the half-metallic value of 2 µB.\nThe Curie temperature of the sample was found to be\nabout 760 Kand the loss of half-metallic character was\nattributed to the small amount of disorder. Electronic-\nstructurecalculationsperformedbyIshida[23]withinthe\nlocal-density approximation (LDA), predict the ground\nstate of Mn 2VAl to be close to half-metallicity. Weht\nand Pickett [24] used the GGA for the exchange cor-\nrelation potential and showed that Mn 2VAl is a half-\nmetallic ferrimagnet with atomic moments in very good\nagreement with the experiment. Recent calculations of\nthe exchange parameters for Mn 2VAl [25] show a strong\nMn−Vexchange interaction that influence the ordering\nin the Mn sublattice. The estimated Curie temperatures\nare in good agreement with the experimental values [25].The intermixing between V and Al atoms in the Mn 2VAl\nalloyshowedthat a small degreeofdisorderdecreasesthe\nspin polarization at the Fermi level from its ideal 100%\nvalue, but the resulting alloy Mn 2V1−xAl1+xstill show\nan almost half-metallic behavior [26, 27].\nAccording to the ideal full Heusler ( L21) structure, the\nV atom occupy the (0 ,0,0) position, the Mn atoms are\nsituated at (1 /4,1/4,1/4)aand (3/4,3/4,3/4)a, and the\nAl at (1/2,1/2,1/2)a, wherea= 5.875˚Ais the lattice\nconstant of the Mn 2VAl compound. In our work, corre-\nlation effects in the valence V and Mn dorbitals are in-\ncluded via an on-site electron-electron interaction in the\nform1\n2/summationtext\ni{m,σ}Umm′m′′m′′′c†\nimσc†\nim′σ′cim′′′σ′cim′′σ. The\ninteraction is treated in the framework of dynamical\nmean field theory (DMFT) [13], with a spin-polarized T-\nmatrix Fluctuation Exchange (SPTF) type of impurity\nsolver [28]. Here, cimσ/c†\nimσdestroys/creates an elec-\ntron with spin σon orbital mon lattice site i. The\nCoulomb matrix elements Umm′m′′m′′′are expressed in\nthe usual way [29] in terms of three Kanamori parame-\ntersU,U′=U−2JandJ. Typical values for Coulomb\n(U= 2eV) and Stoner ( J= 0.93eV) parameters were\nused for Mn and V atoms. The above value of U is con-\nsiderablysmallerthanthebandwidthofMn 2VAl(7–8eV)\ntherefore the use of a perturbative SPTF-solver is justi-\nfied. In addition, the same solver was used to investi-\ngate spectroscopic properties of transition metals with\nremarkable results [30, 31, 32, 33, 34].\nSince the static contribution from correlations is al-\nready included in the local spin-density approximation\n(LSDA/GGA), so-called “double counted” terms must\nbe subtracted. To achieve this, we replace Σ σ(E) with\nΣσ(E)−Σσ(0) [35] in all equations of the DMFT pro-\ncedure [13]. Physically, this is related to the fact\nthat DMFT only adds dynamical correlations to the\nLSDA/GGA result. For this reason, it is believed that\nthis kind of double-counting subtraction “Σ(0)” is more\nappropriate for a DMFT treatment of metals than the\nalternative static “Hartree-Fock” (HF) subtraction [36].\nIn Fig. 1 we present the total density of states com-\nputed in GGA and GGA+DMFT, for T=200K. The\nGGA density of states displays a gap of about 0.4eV in\nthe majority spin channel in agreement with previous\ncalculations [24]. As expected from the s-d model calcu-\nlation, majority spin NQP states are visible just below\nthe Fermi level, with a peak around −0.25eV. In order\nto evaluate the spectral weight of these NQP states, we\nfit the low-energy density of states below the Fermi level\nin the majority channel with a Gaussian centered around\nthe peak position. The NQP spectral weight is then de-\nfined as the area below the Gaussian curve. The inset\nshows the NQP spectral weight for several temperatures\nup to 300K. It is interesting to note that within the com-\nputed temperature range 50 ≤T≤300 these values are\nalmost constant and considerably larger in comparison\nwith similar values for (NiFe)MnSb [16]. The data pre-3\n-3 -2 -1 0 1 2 3\nE-EF(eV)-12 -12-8 -8-4 -40 04 48 8DOS(states/eV/spin/unit cell)majority spinminority spinGGA\nDMFT\nGaussian fit\n0 100 200 300\nT(K)00.30.6spectral weight\nFIG. 1: (color online) Total density of states, computed\nwithin GGA (dashed/blue), and GGA+DMFT (full/red).\nThe Gaussian fit to the density of NQP states is shown as a\ndotted-dashed (black) line just below the Fermi level for th e\nmajority spin channel. The temperature dependent spectral\nweight of NQP states is displayed in the inset.\nsented in the inset can be extrapolated down to T= 0K,\nandaspectralweightof ≈0.544±0.018(states/Mn −d)is\nobtained. ThisdemonstratesthatNQPstatesarepresent\nalsoatT=0K,andobviouslytheyarenotcapturedbythe\nmean-field, GGA result. As it will be discussed below,\nNQP states predominantly consist of Mn-delectrons, so\nthe value for the integrated spectral weight (inset of Fig.\n1) only comes from Mn-dorbtials.\nThe atom resolved DOS is presented in Fig. 2. In\nGGA, the net magnetic moment per unit cell is 2 µBwith\nparallel Mn moments having values close to 1 .6µBand\noppositely oriented V moments close to −0.8µB. Be-\nlow the gap, the majority spin total DOS is mainly of\nMn character. The Mn and V moments have a strong\nt2gcharacter, and a small Al contribution to the mag-\nnetic moment is present. Most of the Vmajority spin\nstates lie above the gap, along with the Mn egstates.\nMinority-spin states below 0.5eV have roughly an equal\namounts of V(t2g) and Mn character. States around the\nFermi energy have a predominant Mn(t2g) character, in\nagreement with [24]. In contrast to the GGA results,\nthe many-body DMFT calculation (see Fig. 2) yields a\nsignificant DOS for the majority spin states just below\nthe Fermi level. These are the majority spin NQP states\ndiscussed above [12, 14, 15, 16, 17, 18]. As can be seen\nin Fig. 2 majority spin NQP states are predominantly\nofMn−3d↑character. Their spectral weight is quite-3-1.501.53\nUMn=2eV, JMn=0.9eV\nUV=2eV, JV=0.9eVGGA\nDMFT\n-5-4 -3 -2 -1 0 1 2 3 4\nE-EF(eV)-3-1.501.53DOS(states/eV)majority spin majority spinminority spin minority spinVMn\nFIG. 2: (color online) Atom resolved density of states, com-\nputed within the GGA (dashed/blue) and GGA+DMFT\n(full/red) approach, at T=200K. The majority spin NQP\nstates are visible in the Mn-3d↑DOS just below the Fermi\nlevel.\nsignificant (see inset of Fig. 1) so that accurate spin-\npolarized photoemission experiments should be able to\nidentify the existence of such states. In contrast, ma-\njority spin V(t 2g) states below the Fermi energy are not\nsignificantly changed. Above the Fermi level, the Mn(e g)\nand V(e g) states are pushed to higher energy, such that a\ngap is formed just above EF. In the minority spin chan-\nnel below EF, both V and Mn(t 2g) states are slightly\nmodified, while above EF, V(eg) states are shifted to\nhigher energies by 0 .5eV. Around the Fermi level, the\ndominant Mn−3d↓DOS is not significantly changed\nwith respect to the GGA values.\nThe applicability of the local DMFT approach to the\nproblem of the existence of NQP states has been dis-\ncussed in ref. [2] and [14]. It is essential to stress that\nthe accurate description of the magnon spectrum is not\nimportantfortheexistenceofnonquasiparticlestatesand\nfortheproperestimationoftheirspectralweight, but can\nbe important to describe an explicit shape of the density\nofstates“tail”inaveryclosevicinityoftheFermienergy.\nThe imaginary part of the atom and orbital resolved\nself-energies,forT=200KarepresentedinFig. 3. Forthe\nminority Mn- eg, V-egand V-t2g-orbitals we observe that\nthe imaginary part of the self-energy has a rather sym-\nmetric energy dependence around the Fermi level, with\na normal Fermi-liquid-type behavior −ImΣ↓\nMn/V(E)∝4\n-0.18-0.12-0.060.00\nMn(t2g) maj.\nMn(t2g) min.\n-1 0 1 2\nE-EF(eV)-0.18-0.12-0.060.00\nMn(eg) - maj.\nMn(eg) - nim.-0.18-0.12-0.060.00\nV(t2g) maj.\nV(t2g) min.\n-1 0 1 2-0.18-0.12-0.060.00\nV(eg) maj.\nV(eg) min.Σ Im Im Σ \nMn(t2g)\nIm Σ \nIm Σ Mn(eg)V(t2g)\nV(eg)Im \nIm Σ Σ \nIm \nΣ Im Σ \nFIG. 3: (color online) Imaginary part of the self-energies\nImΣσ\nMn/Vfort2g-orbitals on Mn (left upper panel) and V\n(right upper panel). The solid (red) line shows results for t he\nmajority( ↑) spins, while the dashed (blue) line for minority\nspins. The lower pannels show the corresponding ImΣσ\nMn/V\nfor theegorbitals.\n(E−EF)2. The majority spin −ImΣ↑\nMn/V(E) shows\na significant increase right below the Fermi level which\nis more pronounced for the t2g-orbitals. In addition, a\nslight kink is evidenced for an energy around -0.25eV.\nThe majority-state nonquasiparticles are visible in the\nmajority spin channel Fig. 2 at about the same energy.\nThese results shown in Fig. 3 suggests that many-body\neffects are stronger on Mn than on V sites. Therefore,\nNQP states are mainly determined by the Mn-datoms.\nThe behavior of the imaginary part of the self-energy\n(Fig. 3) and the Green function (Fig. 2) can be corre-\nlated with the analysis of the spin-resolved optical con-\nductivity. We have estimated the latter for different\ntemperatures whithin the GGA and GGA+DMFT ap-\nproaches using an approximation of constant matrix ele-\nments. Already at 50K the majority-spin optical spectra\nshows the appearance of a Drude peak signaling the clo-\nsure of the majority spin gap. With increasing temper-\natures, spectral weight is transfered towards the Drude\npeak contributing to the depolarization discussed below\nin Fig. 4.\nAs it was demonstrated previously [2, 4, 5, 6], the non-\nquasiparticlespectral weightin the density ofstates (Fig.\n2) is proportionalto the imaginarypart ofthe self-energy\n(Fig. 3), therefore it is determined by the quasiparticle\ndecay, which is the reason for the name of these states.\nFig. 4 displays the temperature dependence of the050100150200250300\nT(K)0.5 0.50.6 0.60.7 0.70.8 0.80.9 0.91 1MDMFT(T)/MGGAPDMFT(EF,T)P(EF,T)=N (EF,T) + N (EF,T)N (EF,T) - N (EF,T)\nMDMFT(T)/MGGA\nPDMFT(EF,T)\nFIG. 4: (color online) Temperature dependence of the spin\npolarization of conduction electrons (full/red) at the Fer mi\nlevelP(EF,T), and normalized magnetization M(T)/M(0)\n(dashed/black).\nmagnetization obtained directly from the GGA+DMFT\ncalculations and the spin polarization at the Fermi level,\nobtained using the relation P(EF,T) = (N↑(EF,T)−\nN↓(EF,T))/(N↑(EF,T)+N↓(EF,T)),Nσbeingtheden-\nsity of states. These results reflect a general trend\nvalid for half-metals in the presence of NQP states\n[2, 11, 14, 15, 16, 17, 18], namely that magnetization\nandpolarizationbehavedifferentlyasfunction oftemper-\nature. However, as can be seen in Fig. 4, this difference\nis not as sharp as in other HMF materials [12, 18].\nThe effect of disorder on half-metallicity was recently\ndiscussed in Mn 2V1−xAl1+xalloys, for −0.2≤x≤0.2\n[22, 26, 27]. The excess of both Al and V atoms ( x=\n0.1/−0.1 orx= 0.2/−0.2) has the effect of shrinking\nthe gap to zero, but with the Fermi level situated within\nthe gap. In addition, the Mn moment is not affected by\ndisorder and remains constant, in contrast to the V mo-\nment. Spin polarization is decreased by about 10%, with\nelectrons around the Fermi level having a dominant mi-\nnority spin character [26]. In contrast, many-body corre-\nlations have a much more dramatic effect. For non-zero\ntemperatures, all atomic magnetizations are decreased.\nFor instance, near room temperature ( T= 300K) the\nstrongest decrease occurs in V, for which the moment\ndrops almost by 47%, the Mn moment is reduced by\n32%, while the Al experiences just a small reduction by\n4%. As one can see from Fig. 4, already at 50 Kpolar-\nization drops to 75%, and is further decreased upon in-\ncreasing the temperature. As we discussed previously for5\nthe case of FeMnSb [16], many-body induced depolariza-\ntion is significantly stronger than the effect of disorder or\nof other spin-mixing mechanisms such as spin-orbit cou-\npling. This observation seems to hold also for the case of\nMn2VAl, although we can not exclude the fact that for\na larger degree of disorder, the material could possibly\ndepart from its almost half-metallic situation displayed\nfor small degree of substitution ( −0.2< x <0.2).\nInconclusion,inthispaperwehaveshownforaspecific\nmaterial that NQP states are also present in half-metals\nwith a gap in the majority spin channel, and appear\njust below the Fermi level, as predicted in model cal-\nculations [5]. In the case of Mn 2VAl, these states mainly\nconsist of Mn-3d↑electronsand have a considerablespec-\ntral weight. Although this material was reported to be a\nhalf-metal from electronic structure calculations [24], the\nexperimental evidence is not clear. Several reasons are\ninvoked such as existence of defects or the reduced sym-\nmetry at surface and interfaces. From a theoretical point\nofview, weshowthatcorrelation-inducedNQPstatessig-\nnificantly changethe majorityspin electronicstates, thus\nreducing the spin polarization at EF. The appearance of\nNQP states and its connection with tunneling magne-\ntoresistance was recently studied in Co 2MnSi-based tun-\nnel magnetic junction [12]. A great challenge would be\nto produce TMR junctions based on the ferri-magnetic\nMn2VAl. This would allow for a direct experimental in-\nvestigation of the existence of majority spin NQP states.\nPromisingcandidateHMFmaterialswithamajorityspin\ngap of similar magnitude as Mn 2VAl are the double per-\novskites Sr 2FeMO 6(M=Mo,Re) or Sr 2CrReO 6often as-\nsociated with collosal magnetoresistance behavior. In\nparticular the electronic structure of Sr 2CrReO6shows\na closure of its majority spin gap in the presence of spin-\norbit coupling [37], with states having a small spectral\nweight symmetrically distributed around the Fermi en-\nergy. Discrepancies between the experiment and theoret-\nical computations were explained based on possible anti-\nsite disorder [37]. We suggest that a significant density\nof NQP states could be present in the above perovskites\nas well. Work on these lines is in progress.\nL.C. and E.A. acknowledge financial support by the\nAustrian science fund under project nr. FWF P18505-\nN16. L.C. also acknowledge the financial support offered\nby Romanian Grant CNCSIS/ID672/2009. M.I.K. ac-\nknowledges financial support from FOM (The Nether-\nlands). A.I.L. acknowledge financial support from the\nDFG (Grants No. SFB 668-A3).\n[1] R. A. de Groot, F. M. Mueller, P. G. van Engen, and\nK. H. J. Buschow, Phys. Rev. Lett. 50, 2024 (1983).\n[2] M. I. Katsnelson, V. Y. Irkhin, L. Chioncel, A. I. Licht-\nenstein, and R. A. de Groot, Reviews of Modern Physics\n80, 315 (2008).[3] I. Zutic, J. Fabian, and S. D. Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[4] D. M. Edwards and J. A. 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Missert,3Mingzhong Wu,2and Andrew D.\nKent1,a)\n1)Center for Quantum Phenomena, Department of Physics, New York University,\nNew York, New York 10003, USA\n2)Department of Physics, Colorado State University, Fort Collins, Colorado 80523,\nUSA\n3)Sandia National Laboratories, Albuquerque, New Mexico 87185,\nUSA\n(Dated: 1 July 2019)\nWe present a study of the transport properties of thermally generated spin currents\nin an insulating ferrimagnetic-antiferromagnetic-ferrimagnetic trilayer over a wide\nrange of temperature. Spin currents generated by the spin Seebeck e\u000bect (SSE) in a\nyttrium iron garnet (YIG) YIG/NiO/YIG trilayer on a gadolinium gallium garnet\n(GGG) substrate were detected using the inverse spin Hall e\u000bect in Pt. By studying\nsamples with di\u000berent NiO thicknesses, the NiO spin di\u000busion length was deter-\nmined to be 4.2 nm at room temperature. Interestingly, below 30 K, the inverse\nspin Hall signals are associated with the GGG substrate. The \feld dependence of\nthe signal follows a Brillouin function for a S=7/2 spin (Gd3+) at low temperature.\nSharp changes in the SSE signal at low \felds are due to switching of the YIG mag-\nnetization. A broad peak in the SSE response was observed around 100 K, which\nwe associate with an increase in the spin-di\u000busion length in YIG. These observa-\ntions are important in understanding the generation and transport properties of\nspin currents through magnetic insulators and the role of a paramagnetic substrate\nin spin current generation.\nKeywords: spin current, spin transport, spin Seebeck e\u000bect, spin valve\nI. INTRODUCTION\nA spin current, or a \row of spin angular momentum, can be carried by conduction\nelectrons1,2or spin waves3,4. In a material with large spin-orbit coupling, like Pt, a spin\ncurrent can be converted into a measurable voltage by the inverse spin Hall e\u000bect (ISHE)5.\nSpin currents can be generated by the spin Hall e\u000bect (SHE)6{8, spin pumping9, or the spin\nSeebeck e\u000bect10{13. The spin Seebeck e\u000bect refers to the generation of spin currents when\na temperature gradient is applied to a magnetic material and has potential applications in\nconverting waste heat into electricity.\nA conventional spin valve consists of two ferromagnetic metals separated by a non-\nmagnetic metal14,15. Recently, a new spin valve structure based on an antiferromagnetic\ninsulator (AFI) sandwiched between two ferromagnetic insulators (FI) was proposed16. An\nAFI can conduct both up and down spins due to the degeneracy of its magnon spectrum at\nzero \feld. The predicted valve e\u000bect associated with thermally induced spin currents has\nbeen observed by controlling the relative orientations of Y 3Fe5O12(YIG) magnetization in a\nYIG/NiO/YIG structure17. YIG is a ferrimagnetic insulator with low magnetic dissipation,\nhighly e\u000ecient spin current generation18, and long-distance magnon transport19. Nickel\na)Electronic mail: andy.kent@nyu.eduarXiv:1906.12288v1 [cond-mat.mes-hall] 28 Jun 20192\nFIG. 1. A schematic of the sample and cross-sectional characterization of the sample by scanning\ntransmission electron microscopy energy-dispersive X-ray spectroscopy (STEM-EDS). (a) Sample\ngeometry showing the layers, the electrical contacts and the applied magnetic \feld. ~jcis the density\nof the charge current applied in the x-direction. Vxyis the voltage measured in the transverse\ndirection, and 'is the angle between the applied magnetic \feld and the current. GGG, YIG, NiO,\nand Pt are represented as purple, yellow, green, and grey, respectively. (b) Sample cross section\ncharacterized by STEM-EDS. GGG, YIG, NiO, and Pt layers are colored in blue, red, green, and\ndark gray, respectively.\nFIG. 2. Angular dependence and NiO thickness dependence of V2!\nxymeasured with an in-plane\nmagnetic \feld of 0 :4 T at room temperature. (a) Angular dependence of V2!\nxywith an AC density\nofjAC= 1:5\u00021010A=m2. \u0001V2!\nxyis extracted by \ftting the curve with a cosine function. (b) \u0001 V2!\nxy\nas a function of the NiO thickness. The curve is \ftted with V=V0e\u0000t=\u0015NiO, whereV0= 182 \u000644 nV\nand the spin di\u000busion length of NiO is \u0015NiO\u00194:2\u00061:1 nm.\nOxide (NiO) is an antiferromagnetic insulator used to decouple the two ferrimagnetic layers\nwhile conducting thermally generated spin currents.\nTo understand the generation, transmission, and detection of spin currents through a mul-\ntilayer consisting of di\u000berent magnetic insulators, transport measurements were performed\nin samples consisting of GGG(500 \u0016m)/YIG(20 nm)/NiO(t nm)/YIG(15 nm)/Pt(5 nm).\nHere GGG (Gd 3Ga5O12) is the standard substrate used to grow epitaxial YIG. Above the\nspin-glass transition temperature ( \u0018\u00000:18 K), GGG is paramagnetic with no long-range\nmagnetic order20,21. In addition, GGG has been shown to have a SSE, with a magnitude\ncomparable to the SSE that of YIG at low temperatures22.\nIn this article, room-temperature measurements were \frst performed to characterize the\nspin di\u000busion length of NiO. Then experiments were conducted over a broad range of tem-\nperature from 5 to 300 K. These revealed a strong enhancement of the SSE below 30 K that\noriginates from the GGG substrate. Further, \feld-dependent experiments show behavior3\nFIG. 3. Second harmonic response V2!\nxymeasured at several temperatures with an applied magnetic\n\feld\u00160H= 1:0 T for NiO thicknesses of 2.5, 5, and 10 nm. (a) Angular dependence of V2!\nxy\nmeasured from 5 to 300 K. Note that the angle 'is de\fned in Fig. 1(a). (b) \u0001 V2!\nxymeasured as a\nfunction of the temperature. Inset: a broad peak is observed around 100 K for the sample with a\n5 nm NiO thickness.\nassociated with switching of YIG magnetization and paramagnetism of GGG. Furthermore,\na broad peak in the SSE response around 100 K was observed, which may originate from\nthe temperature dependence of the spin di\u000busion length in YIG.\nII. SAMPLE FABRICATION AND MEASUREMENT TECHNIQUES\nThe sample was fabricated in the following way. First, a 20 nm YIG layer was grown\nepitaxially on a (111)-oriented GGG substrate (500 \u0016m) at room temperature and annealed\nin O 2at high-temperature23. An Ar plasma was used to clean the surface of the samples\nbefore depositing NiO via radio frequency (RF) sputtering in another chamber. Afterward,\na 15 nm YIG layer was grown on top with the same growth conditions of the \frst layer. Then\nthe sample was capped with a 5 nm Pt layer. For transport and SSE measurements, the\nPt was patterned into Hall bar structures using electron beam lithography and Ar plasma\netching. The Hall bar has a width of 4 \u0016m and the length between the two longitudinal\ncontacts is 130 \u0016m. An alternating current (AC) with a frequency of 953 Hz was used. As\nthe temperature gradient induced by the AC oscillates at twice the frequency, the second\nharmonic Hall voltage V2!\nxymeasured by a lock-in ampli\fer is proportional to the amplitude\nof the SSE-produced spin current24,25. Room-temperature measurements were performed\nwith a 0.4 T magnetic \feld applied in-plane. Temperature-dependent measurements are\ncarried out in the Quantum Design PPMS system also with an in-plane applied magnetic\n\feld.\nIII. EXPERIMENTAL RESULTS\nFigure 1(a) is a schematic of the GGG/YIG/NiO/YIG/Pt sample. A magnetic \feld is\napplied in-plane at an angle 'with respect to the current. The cross section of the sample\nis characterized by scanning transmission electron microscopy with energy-dispersive X-ray\nspectroscopy, shown in Fig. 1(b). Both the top and bottom YIG layers are crystalline, with\nthickness of 15 nm and 20 nm. NiO is polycrystalline, with a thickness of 5 nm for this\nsample (see Fig. S1 in the supplemental materials).\nFirst,V2!\nxywas measured as a function of 'for samples with NiO thicknesses of 2.5, 5,\n7.5, and 10 nm. V2!\nxyreaches a maximum at '= 180oand minimum at '= 0o. This is\nconsistent with the ISHE symmetry of the Hall voltage VISHE/~js\u0002^\u001b/rT\u0002^m/cos('),4\nwhere~jsis the spin current, ^ \u001bis the spin polarization direction, rTis the temperature\ngradient, and ^ mis a unit vector in the direction of magnetization. The angular dependence\nof theV2!\nxywas \ftted with a cosine function and the amplitude \u0001 V2!\nxyis plotted as a function\nof the NiO thickness (Fig. 2). \u0001 V2!\nxydecays rapidly as the NiO thickness increases and\nis \ftted to an exponentially decaying function V=V0e\u0000t=\u0015NiO. The characteristic spin\ndi\u000busion length of NiO is \u0015NiO\u00194:2\u00061:1 nm, close to what has been found in previous\nwork on YIG/NiO/Pt structures26.\nTo further understand the generation and transport of thermally generated spin currents\nthrough the heterostructure, the angular dependence of V2!\nxywas measured from 5 to 300 K\nwith an applied magnetic \feld of 1.0 T (Fig. 3(a)). The amplitude \u0001 V2!\nxyis extracted by the\nsame method discussed above and is plotted as a function of the temperature (Fig. 3(b)).\nFor the 5 nm thick NiO sample, as temperature decreases from 300 to 100 K, \u0001 V2!\nxyincreases\nsteadily from 150 to 271 nV. From 100 to 50 K, \u0001 V2!\nxyslightly decreases to 257 nV, forming\na broad peak around 100 K, shown in the inset of Fig. 3(b). However, as temperature\ndecreases below 30 K, \u0001 V2!\nxyincreases dramatically from 297 to 994 nV. The enhancement\nbelow 30 K was observed for all samples.\nAs has been previously noted, the SSE depends on the magnon population, the spin\ndi\u000busion length, and the interfacial spin-mixing conductance in the heterostructure. In\norder to understand the correlation between the SSE signal and the magnetization of the\nsamples, \feld-dependent measurements of V2!\nxywere performed in the sample with 2.5 nm\nthick NiO. Fig. 4(a) shows V2!\nxyas a function of the applied magnetic \feld between -5.0\nand 5.0 T with temperature ranging from 5 to 50 K. At 50 K, as the applied \feld goes\nfrom -5.0 to -1.0 T, V2!\nxy\u0019530 nV, almost independent of the applied \feld. As the applied\n\feld increases from -1.0 T to 20 mT, V2!\nxydecreases slowly to 108 nV. Then V2!\nxydrops\nsharply to -156 nV as the applied \feld increases from 20 to 100 mT. As the applied \feld\nincreases to 1.0 T, V2!\nxydecreases slowly to -540 nV and again is nearly constant thereafter.\nThe sharp switching steps observed in the V2!\nxy\u0000Hcurves around\u000650 mT occur at the\ncoercive \feld of the YIG, which is smaller than 50 mT at room temperature (see Fig. S2\nin the supplemental materials). Only one magnetization reversal can be identi\fed between\n-200 and 200 mT in the V2!\nxy\u0000Hcurves. As temperature decreases from 50 to 5 K, V2!\nxy\nincreases from 535 to 1970 nV, while the low-\feld step does not change signi\fcantly. A\nclear correlation between the V2!\nxyand the magnetization of a paramagnet can be seen by\ncomparing the V2!\nxy\u0000Hcurves with the Brillouin function of a S = +7/2 spin (Gd3+),\nshown in Fig. 4(b).\nIV. DISCUSSION\nThe SSE voltages decay rapidly as NiO thickness increases, as presented in Fig. 2. This\nindicates that spin currents were generated not only from the top YIG layer but also from\nthe bottom YIG or GGG layer. The NiO spin di\u000busion length is close to what has been\nfound before at room temperature in YIG/NiO/Pt structures26.\nA dramatic enhancement of the SSE voltages has been observed below 30 K, which\nis likely associated with GGG. The same enhancement has been observed in GGG(500\n\u0016m)/YIG(20 nm)/Pt(5 nm), shown in Fig. S3 in the supplemental materials. A previous\nstudy has shown that the spin current jsgenerated by paramagnetic SSE in GGG/Pt bilayer\nhas aT\u00001temperature dependence, associated with GGG susceptibility, which follows the\nCurie-Weiss law \u001f=C=(T\u0000\u0002CW), whereCis the Curie constant and \u0002 CWCurie-\nWeiss temperature22. At low temperatures, the GGG thermal conductivity kGGG has a\nT3temperature dependence. Therefore, the temperature gradient generated by a constant\npower isrT/1=kGGG/1=T\u00003. The resulting SSE voltage goes as VSSE/js\u0001rT/T\u00004.\nIn addition, a broad peak of the SSE signal observed around 100 K in the YIG/Pt structure\nsuggests that the spin di\u000busion length in YIG has a strong temperature dependence27. As\nspin currents generated and transmitted through YIG layers, the temperature-dependent5\nFIG. 4. (a) Field dependence of the V2!\nxymeasured between 5 and 50 K, with '= 0o. The sample\nis GGG (500 \u0016m)/YIG(40 nm)/NiO(2.5 nm)/YIG(20 nm)/Pt (5 nm). The magnetic \feld is swept\nbetween -5 and +5 T. The o\u000bset of V2!\nxyhas been removed. (b) Brillouin function of an S=+7/2\nspin.\nspin di\u000busion length in YIG would have a signi\fcant e\u000bect on the ISHE voltage generated\nin Pt. So the broad peak observed around 100 K in Fig. 3 may be associated with the\ntemperature dependence of spin di\u000busion length in YIG. However, further experiments\nare needed to understand how spin currents are transmitted through bulk GGG, NiO,\nGGG/YIG, YIG/NiO, and NiO/YIG interfaces at di\u000berent temperatures.\nComparing the \feld dependence of SSE voltages and the Brillouin function from 5 to 50\nK, it is clear that there is a contribution to SSE from GGG at low temperatures. At 5 K,\nthe SSE voltage follows the Brillouin function as the magnetic \feld swept from -5.0 to 5.0\nT. As temperature increases, the SSE voltages start deviating from the Brillouin function\n(Fig. 4 and Fig. S4). The underlying physics is not yet fully understood, since the role\nplayed by GGG, YIG, NiO and their corresponding interfaces vary with temperature.\nV. SUMMARY\nIn summary, the spin transport properties of an insulating trilayer based on two ferrimag-\nnetic insulators separated by a thin antiferromagnetic insulator were presented. The spin\ndi\u000busion length of NiO was found to be \u0015NiO'4:2 nm at room temperature. In addition,\na large increase of the SSE signal was observed below 30 K, revealing the dramatic e\u000bects\nof paramagnetic SSE from the GGG substrate. The \feld dependence of the SSE shows the\nswitching of YIG magnetization at low \feld as well as paramagnetic behavior associated\nwith GGG. Furthermore, the SSE voltages show a broad peak around 100 K, a feature\nthat may be related to the temperature dependence of spin di\u000busion length in YIG. This\nexperimental study provides information on how spins can be generated, transported and\ndetected in a heterostructure consisting of paramagnetic, ferrimagnetic and antiferromag-\nnetic insulators.\nSUPPLEMENTARY MATERIAL\nThe supplementary material provides the details of sample characterization by scanning\ntransmission electron microscopy (SEM), Vibrating Sample Magnetometer (VSM), trans-\nport measurements of a GGG/YIG/Pt sample, and \feld-dependent measurements of a\nGGG/YIG/NiO/YIG/Pt sample above 50 K.6\nACKNOWLEDGEMENTS\nThis work was supported partially by the MRSEC Program of the National Science\nFoundation under Award Number DMR-1420073. The instrumentation used in this research\nwas support in part by the Gordon and Betty Moore Foundations EPiQS Initiative through\nGrant GBMF4838 and in part by the National Science Foundation under award NSF-DMR-\n1531664. ADK received support from the National Science Foundation under Grant No.\nDMR-1610416. 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X 6(3), 031012 (2016)." }, { "title": "2012.14620v1.Strongly_modulated_ultrafast_demagnetization_and_magnetization_precession_dynamics_in_ferrimagnetic_Gdx_CoFe_1_x_alloys_via_3d_4f_intersublattice_exchange_coupling.pdf", "content": "1 \n Strongly modulated ultrafast demagnetization and magnetization precession dynamics \nin ferrimagnetic Gd x(CoFe) 1-x alloys via 3d-4f intersublattice exchange coupling \n \nY. Ren1, L. L. Zhang1, X. D. He1, G. J. Wu2, J. W. Gao1, P. Ran1, L. Z. Dan, T.Wang1, X. W. \nZhou1, Z. Liu1, J. Y. Xie3, Q. Y. Jin2, Zongzhi Zhang2 \n \n \nABSTRACT: \nManipulation of the intersublattice interaction strengh (\nTMREJ− ) in rare earth ( RE)-\ntransition metal (TM) alloys is a key issue to understand how efficien tly the laser-\ninduced angular momentum transfer s from 3 d to 4f spins and to have a better control of \nthe ultrafast spin dynamic s. In this work, the relationships between laser-induced \ndemagnetization process and the intersublattice 3d-4f interaction for the GdCoFe alloys \nwere systematically studied . The ultrafast two -stage demagnetization process could \nchange into a one-stage mode as the angular momen tum transferring channel between \n3d and 4 f spins is switched off , which could be modulated by \nTMREJ− . Furthermore , \nboth the effective g-factor and damping constant deduced by the subsequently laser-\ninduced magnetization precession process diverge at the angular momentum \ncompensation point based on the ferromagnetic resonance method with the LLG \nequations . The results provide an alternative way to efficiently manipulate the ultrafast \ndemagnetization time for practical applications. \n \n \n \n \n \n \n1 School of Physics and Astronomy and Yunnan Key Laboratory for Quantum Information, Yunnan University, \nKunming 650091 , China. 2 Department of Optical Science and Engineering, Fudan University, Shanghai 200433, \nChina . 3 Key Laboratory of LCR Materials and D evices of Yunnan Province, School of Materials and Energy, Yunnan \nUniversity, Kunming 650091, China. Correspondence and requests for materials should be addressed to Z.Z.Z. \n(email: zzzhang@fudan.edu.cn ) . 2 \n Introduction \n Femtosecond (fs) laser -induced spin dynamics of ferromagnetic materials provide s \nan effective way to manipulate spin orientations within the time regime of sub -\npicosecond (ps), which may have the potential applications in ultrafast magnetic writing \nand break through the limitation of magnetic recording writing speed (within ~ 100 \nps)1-3. It was known that the ferromagnetic transition metals (TM) usually present a \nsub-ps one -step demagnetization process upon laser excitation, followed by \nmagnetization recovery which may last for hundreds of ps4-7. To explain this \nphenomenon, Beaurepaire et al. proposed a phenomenological three temperature (3 -T) \nmodel, i.e. energy transfers among the electron, spin and lattice systems3. Recently , the \nlaser-induced spin dynamics of r are earth -transition metal (RE -TM) alloys have \nattracted great attention due to their potential applications for spin -orbit -torque driven \nmagnetization switching or all -optical switching8-17. With the antiferromagnetic ally- \ncoupled sublattices , the ultrafast demagnetization for RE-TM alloys always displays an \nabnormal two -step characteristic and a much longer decay time , as compared to that of \nTM18-22. It was predicted that the laser induced two -step demagnetization and long \ndecay time of RE-TM alloys arise from the 4 f electrons of RE metals, which are buried \nin the deep shell and far away from the Fermi surface23,24. It has been reported that the \ntypical decay time is 8 ps for Tb (orbital angular momentum L=3 ) and 40 ps for Gd \n(L=0), investigated b y time -resolved x -ray magnetic circular dichroism23. \nWhat is intriguing is that, the fs laser with photon energ ies of both 1.55 eV and 3.1 eV \ncould induce a continuous two -step demagnetization for Tb -TM alloys, but a \ndiscontinuous two -step demagnetization and a slightly faster recovery process for Gd -\nTM alloys18,22. As w e know, the ultrafast demagnetization of Gd couldn ’t be excited by \nthe fs laser because the 4 f spins of Gd ow n a binding energy of about 8.4 eV compared \nto that of ~2.4 eV for Tb25,26. Moreover , it is not available for the angular momentum of \nGd-TM alloys ( L=0 for Gd) transferring directly from the 4 f spins to lattice via spin \norbit coupling. It comes to the important issues that how efficient the laser induced \nangular momentum transferring is and how we can manipulate the fast demagnetization \nfor Gd -TM alloys. In particular f or GdCoFe films, the stron g 3d-5d6s-4f exchange 3 \n interaction s from intersublattices play important roles in the laser -induced ultrafast spin \ndynamics11,24,27 ,28. The intersublattice coupling strength JRE-TM, which could be \nmodulated by varying the Gd co ntent, might be a key to accelerating the laser -induced \nultrafast demagnetizatio n time. \nAnd to date, theoretical explanation of the two -step demagnetization occurring in \nthe time regime of sub -ps to tens of ps is still in debate for RE-TM alloys 18,19, 29. \nKoopmans et al. employed the spin -flip scattering theory of Elloitt -Yafe (EY) type to \nexplain the underlying mechanism , which is mainly decided by\nat cT/ , where\ncT and\nat\nare the Curie temperature and atomic magnetic moment, respectively19. A small\nat cT/\nof the RE content in RE -TM alloys would readily predict a two -step demagnetization \nphenomenon, which could be understood based on the phenomenological 4 -\ntemperature (4 -T) model with both 3 d and 4 f spins involved in the heat transferring \nprocess29. Apart f rom the unique laser -induced demagnetization process, the \nsubsequent magnetization precession and damping behaviors can also be influenced by \nthe RE dopants in RE -TM alloys. It has been shown that t he pr ecession frequency , \neffective gyromagnetic ratio and magnetic damping constant in Gd -TM alloys sharply \nincrease and diverg e around the angular momentum compensation point, which is RE \ncomposition dependent30,31. Nevertheless, the detailed dependences of precession and \ndamping properties on JRE-TM are still ambiguous , the underlying mechanisms are \ndeserved to explore for the future memory applications of fast switching. \nIn order to gain a comprehensive understanding of the laser -induced ultrafast \ndemagnetization and magnetization precession process , in this article we show a \ndetailed study in ferrimagnetic Gdx[FeCo] 1-x alloys by the time-resolved magneto -\noptical K err effect (TR-MOKE) technique, where t he intersublattice exchange coupling \nstrength JRE-TM are systematically modulated by varying the Gd composition of x. We \nfound that t he demagnetization processes of films with different x could be described \nas only one stage (stage I) or two stages (stages I and II), and the decay times of both \nstages could be greatly modulated by JRE-TM. In contrast, the precession frequency f, \neffecti ve g-factor geff and damping constant \n that determined from the magnetization 4 \n precession processes, are hardly affected by the coupling strength , except for the nearly \ncompensat ed Gdx[FeCo] 1-x samples. These results are not only helpful to gain a better \nunderstanding of the exchange coupling interaction in multi -sublattice ferrimagnetic \nalloys, but also provides an alternative way to achieving effective manipulation of \nmagnetization dynamics. \n \nResults and disc ussion: \nThe static magnetic properties of GdCoFe thin films. Fig. 1( a) show s the typical in-\nplane magnetic hysteresis loops of Gdx(CoF e)1-x films with various x. The magnetic \nhysteresis loops indicate that all the samples display in -plane magnetic anisotropy. With \nthe increase of Gd composition, the saturation magne tization Ms and magnetic \ncoerciv ity Hc display opposite nonmonotonic variation trends, as shown in Fig. 1(b). \nThe Ms value first decreases with increasing x, reaches almost zero around x = 0.22, and \nthen increases . Considering that the sign of Kerr signals of x = 0.22 is opposite to that \nof x = 0.24 , we can infer that the magnetic compensation point xcp is near x ~ 0.23 for \nour GdCoFe films, where the magnetizations of two sublattices (Gd and CoFe) are \nequal in magnitude and opposite in direction. More evidence could be seen from the \nexpected significant increase of Hc at x ~ 0.23, since the torque of the applied magnetic \nfield is inversely proportional to the net saturation magnetization of Ms32. \nFig. 1 (a) Typical in -plane magnetic hysteresis loops for Gd x(CoFe) 1-x films with var ious x. (b) \nThe saturation magnetization Ms and magnetic coercivity Hc as a function of Gd content x. \n5 10 15 20 25 30 35050100150200250300\n-800 -400 0 400 800-300-1500150300\n080160240320CoFe rich Gd rich\nGd content x [at.%]Hc (Oe) Ms (emu/cm3)\n x = 0.12\n x = 0.16\n x = 0.22\n x = 0.24\n x = 0.32M (emu/cm3)\nH (kOe)(b) (a)5 \n \nThe ultrafast demagnetization spectroscopy of GdCoFe thin films and theoretical \ncalculations. Figs. 2(a) and 2(b) illustrate the TR-MOKE measurement configurations \nof the ultrafast demagnetization and magnetization precession processes, with the \nmagnetic field applied either in -plane or tilted with a polar angle of 19 with respect to \nthe Gd x(CoF e)1-x thin film plane , respectively . Fig. 2(c) displays several typical \ndemagnetization curves under a saturated in -plane field of H = 6.0 kOe. Since the 4 f \nshell of Gd atoms is buried around 8.4 eV below EF, only s and d electrons near the \nFermi level participate in the optical excitation by the fs laser with the photon energy \nof 1.55 eV. The transient MOKE signal s of all the GdCoFe thin films significantly v ary \nwith the pump -probe delay time t, and the dynamic demagnetization process could be \ndivided into three types according to the composition of Gd . Type I of x=0.08~0.22 \n(CoFe rich) owns a rapid one-step demagnetization within sub -ps time range \n(defined as t1) and a subsequent rapid recovery within several ps. For the type II samples \nof x=0.24~0.28 (Gd rich) where the sign of the transient Kerr signal reverses, two kinds \nof two -step demagnetization behaviors occur. Similar to the SmFe and TbFe alloys18,33, \nthe MOKE signal of x=0.24 presents a continuously decay trend which lasts for several \ntens of ps. In contrast, for x=0.26 -0.28, after the initial fast demagnetization (t1), an \nintermediate recovery process ( tR) occurs, which is then followed by a much slow \ndemagnetization process (t2). As for the Gd -rich samples of type III ( x = 0.32), the \nvarying trend is similar to that of type I, but with an opposite Kerr signal sign and a \nmuch slower recovery process . \n To deeply understand the mechanism of ultrafast demagnetization for GdCoFe \nfilms, t he characteristic demagnetization times of t1 and t2, as well as the intermediate \nrecovery time tR were determined by using a bi-exponential fitting to the experimental \ndata. As shown in Fig. 2(d), t1 shows a nonmonotonic variation with increasing x. It is 6 \n as low as ~ 0.5 ps for the samples with a rather low or high Gd content, which could be \nascribed to the thermalization of 3d electrons . However, near the compensation \ncomposition of x=0.22 -0.28, t1 is relatively larger and reaches a maximum of 1.16 ps at \nx=0.24 (close to xcp). The decay time constant of Gd of 14 ps for 4 f spins and 0.8 ps for \n5d spins were reported , and the 3 d-5d6s-4f intersublattice exchange coupling could help \nthe 4 f spins participating in the initial fast demagnetization process24,34. So we consider \nthe observed larger t1 is resulting from the intersublattice exchange coupling, which \nprobably has a maximum strength at x = xcp. Meanwhile, tR shows a similar trend to that \nof t1 which could be ascribed to the longer demagnetization time and larger amplitude \nduring the initial fast demagnetization process. As for the decay time t2 that related to \nthe samples of type II, it increases linearly with the Gd composition x, following an \ninverse proportion to\nat cT/ , as predicted by the EY scattering theory. \n A schematic diagram of phenomenological explanation for the demagnetization \nprocess of GdCoFe thin films could be shown in Fig. 3(a). The conduction electrons \n \nFig. 2 TR-MOKE measurement configurations of (a) ultrafast demagnetization with H \napplied in -plane and (b) magnetization precession with a tilted H. (c) The typical ultrafast \ndemagnetization curves of Gdx(CoFe) 1-x films measured under a saturation in -plane \nmagnetic field of 6.0 kOe. (d) The Gd composition dependence of characteristic times of t1，\nt2 and tR. \n7 \n absorb the energy from the photons. At the beginning, the laser -induced angular \nmomentum transfers from the conduction electrons to 3 d spins, resulting in a rapid \ndemagnetization for all the samples. For the spin dynamics of type I, the 3 d spins of \nCoFe dominates the fast demagnetization, and the following rapid magnet ization \nrecovery could be ascribed to the electrons -phonons and 3 d spins - phonons coupling. \nFor type II s amples with a higher Gd content , the 5 d6s electrons serve as a heat-transfer \nchannel for transfer ring the the laser -induced angular momentum to 4 f spins , resulting \nin an increase in the 4 f spin temperature . The heat could also returned from the 4f spins \nto 3d via the intersublattice exchange couplin g, being responsible for the subsequent \nsecond demagnetization. It could be inferred that t he stronger \nTMREJ− is, the longer the \ninitial fast demagnetization time t1 will be , because t1 can be delayed by the excited 4f \nspins which participat e in the angular momentum transferring as well . \nBased on the above analysis regarding , the temporal evolution of \ndemagnetization are described by the 4 -T model with three different 3d-4f coupling \nconstan t GssGdCoFe . As shown in Fig. 3(a), the 3 d and 4 f spins are regarded as two \nseparate spin reservoirs since they have different responses to laser disturbances. By \nincluding the electronic temperature Te，lattice temperature 𝑇l, CoFe spin temperature \n𝑇sCoFe, and Gd spin temperature 𝑇sGd，the 4 -T model is repr esented by four coupled \ndifferential equations, which were solved based on the Runge -Kutta method18,29 \ndescrib ing heat flow between the different heat sources. The theoretically calculated \nresults are shown in Figs. 3(b-d), which agree well with the experimental data. For all \nthe cases, the value of T e gets to about 1020 K within ~100 fs, while the temperatures \nof Tl, TsCoFe and TsGd keep almost at the room temperature. After heat transfer s from \nelectrons to other heat baths, the values of Tl and TsCoFe increase while T e decreases. \nDepending on the different value s of GssGdCoFe, there are three kinds of varying trend s \nfor the CoFe spin temperature : (i) For GssGdCoFe=0 , TsCoFe increases with the delay \ntime t to the maximum of 415 K within ~4 ps, and then decreases with further increasing \nt. Eventually, the three heat baths reach a quasi -thermal equilibrium , i.e. Te, 8 \n Tl and TsCoFe get to a constant. In this case, TsGd does not involve in the temporal \nevolution of CoFe spins due to the closed heat transferring channel (without \nintersubl attice couplin g), which corresponds to the dynamic behaviors of both type I \nand III samples. (ii) For GssGdCoFe=0.75×1015W/m3K , the value of TsCoFe apidly \nincreases within t < 4 ps. In this time regime, T e greatly affects TsCoFevia the e lectron -\nspin coupling between electrons and CoFe spins . As 4 ps < t < 10 ps, accompanying \nwith the decrease of Te, TsCoFedrops as well , resulting in a n intermediate magnetization \nrecovery. Finally, as t > 10 ps, since the intersubl attice couplin g is strong enough , heat \ntransferring channel between the spins of Gd and CoFe open s, and then TsCoFe \nincreases again with the increas ed TsGd , resulting in the second demagnetization of \nCoFe with a long er time t2 . (iii) For GssGdCoFe=1.5×1015W/m3K, as the electron and \nlattice systems reach their quasi -thermal equilibrium at t~10 ps , both TsCoFeand TsGd \nFig. 3 (a) The Schematic diagram for the exchange coupling of the vario us heat baths in the \n4-T model. Numerically calculated curves for the temporal evolution of the heat -reservoir \ntemperatures ( Te , Tl ,TsCoFe, and TsGd) in the GdCoFe system using the 4-T model with \nGssGdCoFe=0 (b),GssGdCoFe=0.75×1015 W/m3K (c), GssGdCoFe=1.5×1015W/m3K (d). \n9 \n keep increasing until t~60 ps. This is because the large intersubl attice couplin g could \nimprove the heat transferring between 4f and 3d spins , thus accelerating the \ndemagnetization of Gd spins, which happens just a few ps after the CoFe spins . Such \nspin dynamics can be considered as a continuous two -stage demagnetization process , \nas compared to the discontinuous two -step demagnetization case. Apparently, since the \n3d-4f coupling constan t GssGdCoFe is comparable with \nTMREJ− , the laser -induced \ndynamic process of type II could be qualitative ly describe d with a n appropriate value \nof \nTMREJ− . According to our experimental and theoretical results, it gets to a conclusion \nthat the value of \nTMREJ− first increases with increasing the Gd composition x, and \nreaches the maximum at xcp, then falls with further increasing x. The obtained \ndependence of \nTMREJ− on x is quite different from the previous publications, which \neither reported a monotonic increas ing trend , or a constant value28,35. \n \nField-driven ultrafast demagnetization of GdCoFe thin films. To deeply explore the \nsecond step d emagnetization of type II, the transient Kerr signals as a function of delay \ntime were measured under various magnetic field H for the sample s of x = 0.24 and \n0.28, as shown in Figs.4 (a) and (b), respectively. The initial fast-step demagnetization \ntime t1 is field-independent and completes within ~1.5 ps , which is mainly related to the \nthermalization of 3 d spins and the influence of the partially involved 4 f spins on 3 d \nspins due to the strong inter-sublattice e xchange coupling24. Nevertheless, the dynamic \ncurves of different H become separated at t > 10 ps, which suggests that the second -\nstep demagnetizat ion process is apparently dependent on the applied field . In order to \nclarify this phenomenon , the characteristic time t2 of x = 0.24 and 0.28 and the ir relative \namplitude ratio of the second step demagnetization \n2M to the total demagnetization 10 \n \nTM, are shown in Figure s 4(c) and 4(d) as a function of H, respectively . For both \nsamples , t2 decreases linearly with increasing H, which is consistent with previous \nreports37. Accompanied with the decrease of t2, the ratio of \n2T/MM also shows a \ndownward trend , suggest ing that the stronger demagnetization can result in the longer \nt2. It suggests that magnetic field could drive the second demagnetization process, \nascribed to the acceleration of the growth and recovery of the reversed domains by \nexternal magnetic field36-38. The larger the magnetic field is, the faster the domains \nmove. As a result, the equilibrium of demagnetization will be achieved rapidly, resulting \nin a small t2. It could also be caused by the magnetic cooling effect, since the spin -\nlattice interaction can be effectively manipulated by changing the magnitude of the \napplied magnetic field, leading to the forced magnetization alignment in ultrafast time \nregime39. \n \nFig. 4 The laser-induced dem agnetization curves measured at various magnetic fields for \nthe films of x =0.24 (a) and 0.28 (b). The inset in (a) and (b) shows the enlarged area of \ndelay time 12 ps. ( c) and ( d) demonstrate the external field dependen ce of t2 and \n2T/MM\n for x =0.24 and 0.28 , respectively. Here \n2M and \nTM are the amplitudes \nof the second step demagnetization and the total demagnetization, respectively. \n \n11 \n The magnetization precession and damping dynamics of GdCoFe thin films. The \nmagnetization precession dynamics were further measured by applying various external \nfield H with a tilted angle of H = 19 with respect to the film plane, see Fig. 2(b). As \nwe know , the magnetization vector M is along the direction of effective field , which \nmainly includes H and the demagnetization field Hd. The variation of Hd by laser \nheating will give rise to the precession of M. Fig. 5(a) shows the typical transient Kerr \nsignals as a function of delay time measured at H = 6.0 kOe . Clearly , the precession \ncurves show great changes with increasing x. The precession behavior gradually \ndisappears a t x > 0.22, and reappears at x = 0.32. Note that, as shown in the inset for \n0.24\n x\n0.28, all the curves show a similar two -step decay with an intermediate \nrecovery . Especially, the inflection points of the transient Kerr signals at t ~14 ps for \nboth the demagnetization and precession curves of x = 0.28 are almost the same. It could \nbe inferred that the demagnetization of the type II dynamics ha s a significant effect on \nFig. 5(a) The transient precession Kerr signals of GdCoFe films with various Gd contents \nand the corresponding fitting lines for H = 6.0 kOe. The inset in (a) shows the short range \nKerr signal curves of x = 0.28 measured unde r the in -plane field (black triangles) and \ntilted field with a polar angle of 71 (red circles) . (b) The precession frequency f as a \nfunction of H (solid symbols) and the corresponding fitted lines (solid lines) . (c) The \neffective damping factor αeff as a function of H. The inset of (b) shows the polar coordinate \nsystem used for analys es of the TR -MOKE spectra . (d) and (e) show the Gd content \ndependence of αs and geff values determined from the correlation between f and H. \n \n \n \n12 \n the precession process in the time regim e of several tens of ps, since the \ndemagnetization is not completed in the time regime. \nFig. 5(b) illustrates the polar coordinate system used for analyses of the TR -\nMOKE spectra, and the f ield dependent precession frequency f curves for various x \nderived from the Kerr signal fitting by a damped sine function 40,41. According to the \nsimple relation of 𝛼eff = 1/2πfτ42, the corresponding effective magnetic damping factors \nare calculated and shown in Fig. 5(c). The value of 𝛼eff decrease s gradually with \nincreasing H, and presents a constant as H >10 kOe for all the samples with various x. \nThe extrinsic contribution of 𝛼eff from local distributions of magnetization/ magnetic \nanisotropy could be eliminated by applying a large external magnetic field43,44. \nTherefore, w e define the 𝛼eff value at H=12 kOe as the intrinsic damping term s. Figs. \n5(d) and (e) show composition dependences of the Gilbert damping parameter s and \nthe g -factor g eff, which are expected to diverge at the angular momentum compensation \npoint based on the ferromagnetic resonance mode with the LLG equations30,45-47. The \neffective gyromagnetic ratio \neff and Gilbert damping parameter s could be \nexpressed as follows 48. \nGdGd\nCoFeCoFeGd CoFe\neffT M T MT M T MT\n )( )()( )()(\n−−=\n (1), \nTMTM\nRERE T M T MTA )( )()( − =\n (2), \n)()( )(\n)(TAT M T M\nTGdGd\nGd\nCoFeCoFe\nCoFe\ns\n+\n=\n (3), \nWhere M , and \n are the net magnetic moment , gyromagnetic ratio and Gilbert \ndamping constant of RE and TM sublattices, respectively . A(T) is the net angular \nmomentum, A0 is a constant which is independent of temperature . It is known that the \ng factor is ~2 for Gd and 2.16 for CoFe, which means that the angular momentum \ncompensation point xAP is very close to the magnetization compensation po int of xCP ~ 13 \n 0.23. Because s and \necmge eff\neff2= are both inversely proportional to A(T), they \nincrease first ly and then fall with increasing x, showing a diverge nce around x ~ 0.23. \nThe largest s at x ~ 0.23 lead s to the elimination of magnetization precession. In \naddition , for the samples with x =0.24-0.26, the long two-step demagnetization process \nwhich lasts for several tens of ps may also be responsible for the abnormal \nmagnetization precession curves. \n \nDiscus sion: \nUsually, 800 nm fs laser with the photon energy of 1.55eV could only excite the \nmagneto -optical signal of GdCoFe from the Co(Fe) 3 d moments , since the binding \nenergy of 4f for Gd require photon energies about~ 8.4 eV. However, from the \nexperimental results, two -step demagnetization could be observed due to the \nparticipation of 4f spins of Gd. The exchange interactions of Gd -CoFe ( JRE-TM) \nintersublattice s play important roles in the laser -induced demagnetization dynamics . \nFor the CoFe -rich samples, the 3d moments of CoFe sublattices via ferromagnetic \ncoupling dominate the temperature dependence of the one-step demagnetization within \n1 ps. At x ~ xcp, the strong JRE-TM builds up an energy transfer channel for the 3d and 4 f \nspins. As a result, 4 f spins could not only participate the ultrafast demagnetization \nwithin 1 ps, but also drive the second step demagnetization persisting in several tens of \nps. It seems that 4 f spins delays the ultrafast demagnetization of 3 d spins. In other words, \nthe 3 d-5d6s-4f exchange interaction establishes a channel for heating up 4 f spins. As to \nthe Gd -rich samples with a high Gd content, the 4 f spins are no longer involved in the \nultrafast demagnetization. Since the JRE-TM sharply decreases due to the increase of the \namount of Gd atoms, resulting in the heat transferring channel switching off between \n3d and 4 f spins, so the ultrafast demagnetization is dominated by 3 d spins again. The \nvariation trend s of the first and secon d step decay time t1 and t2 with increasing x could \nbe well explained by the participation of 4 f spins via i ntersublattice 3 d-5d6s-4f \nexchange couplin g, and the spin-flip scattering theory combining with the 4 -T model , \nresepctively . 14 \n Moreover, we have also studied the magnetization precession processes of the \nGdCoFe thin films. The magnetization precession frequency f and damping constant\n\ncould be significant modulated by the variation of x. Both the effective g -factor geff and \nintrinsic damping constant\ns significantly increase and diverge at the angular \nmomentum compensation point, which agree well with the ferromagnetic resonance \nmodel. These results are helpful for better understand ing and controlling the magnetic \ndynamic behaviors of RE -TM systems with the antiferromagnetic coupling , which may \nfind potential applications for magnetic data storage and related spintronic devices. \n \n \nMethods: \nMagnetron sputtering of GdCoFe films on Si/SiO 2 substrates . A series of Si /SiO 2 \n/Ta (5nm) / Gdx (Co 0.8Fe0.2)1-x (20 nm) (GdCoFe) /Al (5nm) samples were fabricated by \nDC magnetron sputtering , with x varies from 0.08 to 0.32 . The base pressure of chamber \nwas 510-8 Torr and the Ar working pressure was 5.0 mTorr. The 5 nm thick Al capping \nlayer was used in all samples to prevent from oxidation. The relative composition of \nGdx(CoF e)1-x were controlled by adjusting the deposition power of Gd target while \nfixing the deposition power of CoFe target and measured by energy dispersive X -Ray \nspectroscopy. The magnetic properties were checked by both magneto -optical Kerr \neffect (MOKE) and vibrating sample magnetometer (VSM) . All the samples show an \namorphous phase analyzed by XRD with Cu K radiation. \n \nTime resolved Magneto -optical Kerr spectroscopy. The fs laser pulses (duration 100 \nfs, center photon energy1. 55 eV) from a Ti:sapphire laser oscillator (repetition rate \n1kHz) is used to excite the Gdx(CoF e)1-x alloy films. The experimental setup schematic \nis shown in Fig.1. T he laser -induced ultrafast demagnetization process was measured \nby TR -MOKE in the longitudinal geometry shown in Fig. 1(a) as a function of delay \ntime t between the pump and probe pulses , with an in -plane saturation field of 6.0 kOe \napplied for all the samples. In contrast, the magnetization precession and damping \nbehavior were obtained in a polar geometry at various magnetic fields. In order to set 15 \n the magnetization orientation away from the in -plane easy axis , the external magnetic \nfield H was applied at an angle of 71 with respect to the film normal , as shown in Fig. \n1(b). \n \nTheoretical simulations and calculations of the ultrafast demagnetization. The type \nI and III dynamic curves with only one -step demagnetization could be well fitted by the \nfollowing equation49-51 \n𝜃(𝑡)=𝜃0+𝐻(𝑡)[𝐵(1−𝑒−𝑡𝑡1⁄)+𝐶(1−𝑒−𝑡𝑡𝑅⁄)], (1) \nwhere 𝜃0 is the initial Kerr signal and 𝐻(𝑡) is the Heaviside step function . t1, tR, B \nand C a re the lifetimes and amplitudes of the initial demagnetization and the following \nrecovery, respectively. As for the type II dynamics , the two step demagnetization curves \ncould be described by 52 \n𝜃(𝑡)=𝜃0+𝐻(𝑡)[𝐵1(1−𝑒−𝑡𝑡1⁄)+𝐵2(1−𝑒−𝑡𝑡2⁄)]. (2) \nWhere 𝑡1(2) and 𝐵1(2) refer to the lifetimes and amplitudes of the demagnetization \nstage I and II, respectively. \nThe 4-T model is represented by four coupled differential equations that describe \nthe heat flow between four heat sources, which includes electrons, Co Fe spins, Gd spins \nand lattices. These equations can be written as follows: \nCe(Te)dTe\ndt=−Gel(Te−Tl)−GesCoFe(Te−TsCoFe)−GesGd(Te−TsGd)+P(t), (3) \nCl(Tl)dTl\ndt=−Gel(Tl−Te)−GlsCoFe(Tl−TsCoFe)−GlsGd(Tl−TsGd), (4) \nCsCoFe(TsCoFe)dTsCoFe\ndt=−GesCoFe(TsCoFe−Te)−GlsCoFe(TsCoFe−Tl)−GssGdCoFe(TsCoFe−TsGd), (5) \nCsGd(TsGd)dTsGd\ndt=−GesGd(TsGd−Te)−GlsGd(TsGd−Tl)−GssGdCoFe(TsGd−TsCoFe). (6) \nWhere Ce is the electron specific heat and Cl is the lattice specific heat. \nCsCoFe (CsGd) is the 3 d spins (4 f spins) contribution to the specific heat , respectively . \nGel , GesCoFe and GesGd are the electronic -lattice, electronic -spin interaction constants , \nrespectively . GlsCoFe(Gd) and GssGdCoFe are lattice -spin and 3d-4f spins interaction \nconstants , respectively . P(t) is the heat source term, which is only applied to the \nelectronic subsystem. We used a laser pulse with a full width at half maximum (FWHM) 16 \n of 100 femtoseconds (FWHM) and a peak power density of 3.5×105𝑊/𝑚3 . The \nspecific heat of the crystal lattice is taken as a constant. This approximation is available \nbecause our experiment is carried out at room temperature, and the lattice temperature \nis greater than the Debye temperature of the alloy. The specific heat of an electron is \nproportional to the temperature of the electron: 𝐶𝑒(𝑇𝑒)=𝛾𝑇𝑒 , where 𝛾=714𝐽/\n𝑚3𝐾2 .For CoFe sublattices, the following parameters are used: CsCoFe=0.13×\n105W/m3K，Gel=2×1017W/m3K，GesCoFe=0.13×1016W/m3K，GlsCoFe=0.48×\n1016W/m3K. These parameters of CoFe are close to the typical values of metals. For \nGd sublattices, the following parameters are used: CsGd=0.65×105W/m3K，CesGd=\n0.6×1015W/m3K，ClsGd=0.23×1015W/m3K. The 4 f spin of Gd and the 3 d spin of \nCoFe are arranged antiparallel through the 3 d-5d6s-4f exchange interaction. The \nRunge -Kutta method is used to solve the four coupled differential equations \nnumerically. \n \nThe formulas for the magnetization precession. The magnetization precession curves \ncould be fitted by using the following formula: \n𝜃𝑘=𝑎sdn(2𝜋𝑓𝑡+𝜑)exi (−𝑡/𝜏), (7) \nwhere a, f, and are the precession amplitude, frequency , life time, and initial phase , \nrespectively . \nThe precession frequencies are well fitted by the Kittel formula 40: \n𝑓=(𝛾/2𝜋)√𝐻1𝐻2, ( 8) \nwith 𝐻1=𝐻cos(𝜃𝑀−𝜃𝐻)−\nKeffH 𝑐𝑜𝑠2𝜃𝑀, (9) \n𝐻2=𝐻cos(𝜃𝑀−𝜃𝐻)−\nKeffH 𝑐𝑜𝑠2𝜃𝑀. (10) \nHere 𝛾 is the gyromagnetic ratio . \nKeffH\nsM4= represents the perpendicular \nuniaxial magnetic anisotropy field, where \nsM4 corresponds to the demagnetization \nfield. 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B 89, 214423 (2014). \n \n \n \n " }, { "title": "1012.5161v1.On_the_derivation_of_the_magnetocaloric_properties_in_ferrimagnetic_spinel_Mn3O4.pdf", "content": "On the derivation of the magnetocaloric properties in ferrimagnetic spinel \nMn 3O4 \nSubhash Thota, Francois Guillou, Vincent Ha rdy, Alexandre Wahl and Wilfrid Prellier* \nLaboratoire CRISMAT, CNRS UMR 6508, ENSICAEN, 6 Boulev ard du Maréchal Juin, \nF-14050 Caen Cedex, France \n \nJitendra Kumar \nMaterials Science Programme, Indian Inst itute of Technology Ka npur, Kanpur-208016, India \n \n \nAbstract \n \nLarge magnetocaloric effect has been observed in Mn 3O4 around its ferrimagnetic transition at \nTN = 42.75 K. Field-induced isothermal entropy changes ( ∆S) were derived from both magnetic \nand calorimetric techniques. The maximum | ∆S| and adiabatic temperature change ( ΔTad) at T N \nare 11 J kg-1 K-1 and 1.9 K, respectively, for a magnetic fi eld change of 20 kOe. Moreover, it is \nfound that the complex magnetic phase transitions taking place below T N produce additional \n⎯but smaller ⎯ features on ∆S(T). \n \n PACS number(s): 75.50.Gg, 75.25.+z, 75.30.Kz, 75.60.Ej \n \n*Corresponding author. Email: wilfrid.prellier@ensicaen.fr \n \n I. Introduction: \nResearch activity in the area of magneto-thermodynamics (MT) has received rapid \nimpetus in the recent past, mainly because of th e discovery of giant magnetic entropy changes in \nthe Gd 5(SixGe1−x)4 compounds.1, 2 Many efforts are underway to disc over materials showing large \nmagneto-caloric-effect (MCE) under moderate applied magnetic fields3, so that magnetic cooling \ntechnology may become a real ity in the near future.4,5 MCE is an importan t phenomena of MT \nwhich manifests itself as an is othermal magnetic entropy change ( ∆S) or an adiabatic \ntemperature change ( ∆Tad) when the magnetic material is exposed to a varying magnetic field.6 \nRefrigeration based on the MCE is advantageous in that it is an environmentally friendly and \nenergy efficient alternative to the co mmonly used vapor-cycle refrigeration.6 It was found that \ngiant MCE is exhibited by various in termetallic materials such as Gd 5(Si,Ge) 4 1,2, Mn(As,Sb) 7, \nLa(Fe,Si) 13 8,9, MnFe(P,As)10 , Heusler alloys 11,12 and some other compounds (see reviews Ref. \n3, 13-15 and references therein). In the case of tr ansition metal oxides, most of the MCE studies \nso far were focused on mixed-valenc y rare-earth based manganese oxides3,16, since some of them \nexhibit large | ∆S| values close to room temperat ure which are comparable with Gd.17 \nUp to now, no studies were devoted to the magnetocaloric properties of binary \nmanganese oxides (like MnO 2, Mn 2O3,, Mn 3O4 and Mn 5O8). This family of compounds combines \nseveral features favorable for applications; th ey are cheap, harmless and have a high chemical \nstability.18 Also, these oxides are potenti al candidates for i) rechargeable lithium batteries, ii) \ncatalysts, and iii) soft magnetic materials for transformers cores.18 Among all these \nantiferromagnetic oxides, the spinel Mn 3O4 is the only one which exhibits a long-range \nferrimagnetic ordering19, i.e. a transition at wh ich significant MCE can be expected. It belongs to \nthe class of normal-spinel (AB 2O4) structured compounds, in which the tetrahedral A sites are occupied by a divalent cation (Mn2+) while all the octahedral B si tes are occupied by trivalent \ncations (Mn3+). At 1445 K, this compound undergoes a tetragonal to cubic phase transition (c/a \n~1.16) resulting from a collective Ja hn-Teller distortion within the e g orbitals of Mn3+ (t2g3 eg1).20 \nBelow 42 K~T N, a complex succession of rearrangements takes place in the magnetic structure. \nDespite the longstanding history19-24, this issue is still the subject of intense research activity.25-27 \nAt the present time, the most widely accep ted picture of its magnetic ordering is 25-27: (i) At T N, \nMn 3O4 exhibits a transition from the paramagne tic state towards a non-collinear ferrimagnetism \nof Yafet-Kittel type.28 That means a triangular structure in which the Mn2+ spins order \nferromagnetically along [110] (cubic settings) while the Mn3+ are symmetrically canted around [-\n1 -1 0 ], leading to a net mome nt antiparallel to that of Mn2+; (ii) At T 1∼39 K this coplanar spin \nstructure lying in (1 -1 0) evol ves to a spiral-type ordering relate d to a conical di stribution of the \nMn3+ spins around [110] (this type of magnetic struct ure is incommensurate with the chemical \nlattice); (iii) Finally, at T 2∼33 K, the spin order rec overs a planar type stru cture. However, there \nis a doubling of the magnetic cell with respect to the chemical one, induced by a subdivision of \nthe Mn3+ into two sets. While the Mn2+ spins still order ferroma gnetically along [110], the Mn3+ \nspins exhibit a canting angle of ≈ 69° with respect to [-1 -1 0]. Very recently, Kim et al. reported \nthat this magnetic transition is accompanied by a structural transition from tetragonal to \nmonoclinic symmetry, which is highly sensitive to the strength and orientation of the external \nmagnetic field.27 \nIn this context, our motivation to investigate the magnetocaloric properties of Mn 3O4 is \ntwofold. First, from a fundamental point of view , this compound offers an opportunity to study \nthe MCE associated with a wide variety of ma gnetic or magnetostructural transitions. Second, \nfrom an application point of view, the existen ce of such a sequence of transitions could be expected to generate MCE over a wide temperatur e range. This aspect is a favorable condition to \nobtain large refrigeration capacit y. Furthermore, our investigati ons also provide a comparative \nstudy on the reliability of the calorimetric and magnetic methods which are commonly used to \nderive the isothermal entropy changes. \n \nII. Experimental details: \nOxalate based sol-gel process has been chosen to synthesize Mn 3O4 polycrystalline \nsample. The initial product is manganese oxalate dihydrate (MnC 2O4·2H 2O), prepared using \nmanganese acetate tetrahydrate [Mn(CH 3COO) 2·4H 2O] and oxalic acid [C 2H2O4·2H 2O] as \nprecursors with ethanol as a solvent. Thermal decomposition of MnC 2O4·2H 2O at 500°C for 2 h \nduration yields Mn 2O3 as a major product; further annealing of this sample at 1100°C for 4 h in \nair leads to the formation of Mn 3O4 powder. This powder is grounded in an agate mortar, pressed \ninto bars and finally sintered at 1100°C for 12h. The cr ystallinity and struct ure has been studied \nusing a Panalytical X’Pert Pr o diffractometer using Cu K α radiation, which indicate monophasic \nnature of this compound consis tent with the hausmannite Mn 3O4 reported in the literature. A \ndetailed study of the formation me chanism has been reported elsewhere.18, 29 A superconducting \nquantum interference device (SQUID) based ma gnetometer (Quantum Design, MPMS-XL5) has \nbeen used for the temperature dependent magnetization measurements M H(T) in low-field, while \nmagnetic isotherms M T(H) up to 90 kOe were recorded by m eans of an extraction technique in a \n“physical property measurement system” (PPMS, Quantum design). The heat-capacity curves \nC(T) (in H=0 and H=20 kOe) were recorded from 2 to 100 K by using a semi-adiabatic relaxation technique in the PPMS. To stress the comparison between the calorimetric and magnetic methods, the paper is focused on the data corresponding to a field change from zero to 20 kOe. Note also that H = 20 kOe is the typical value to be considered for potential applications \nusing permanent magnets. \nIII. Results and discussion: \nThe location of the above mentioned transitions (T N,T1,T2) on the magnetization and heat \ncapacity data are reported in the insets of Figure 1 and 2a , respectively. Apart from T N which has \na clear signature in both cases, the features associated to T 1 and T 2 are faint. To reveal them \nclearly, we display the temperature dependence of the first derivative of the magnetization curve \n(–dM/dT) recorded upon cooling in a small magne tic field of 200 Oe (inset of Figure 1) while \ninset of Figure 2a shows the temperature depe ndence of C/T in zero field. There is a good \nconsistency between these two results, both leading to T N ≈ 42.75 K, T 1 ≈ 39.75 K and T 2 ≈ \n34.25 K. The positions of these transitions are in line with those previously reported in the \nliterature.23,25-27 \nIn order to derive the magnetic entropy ch ange, we first used the common method based \non the Maxwell equation applied to a series of isothermal magnetization curves. These \nisothermal magnetization curves were recorded as follows: (i) Each of th ese curves was preceded \nby a zero-field-cooling, ensuring to start from a virgin magneti c state; (ii) The temperature \nspacing between two successive M T(H) curves was quite large out of the region of the transitions \n(δT=5K), and it was decreased to a small value inside it ( δT=2K); (iii) The data were recorded \nupon increasing the magnetic fiel d from 0 to a value of H Max. Two series were recorded, with \nHMax = 20 kOe and H Max = 90 kOe. The results of ΔS(T) for magnetic field changes equal or \nlower than 20 kOe were checked to be the sa me in both cases. Making use of the Maxwell \nequation with such data, the temperature dependence of ΔS associated to a magnetic field change \nfrom 0 to H is usually derived from the formula:6 ∫ ⎥⎦⎤\n⎢⎣⎡\n∂∂= ΔH\nHdHTHTMHTS\n0'')',(),( (1a) \nIt must be emphasized that this relationship assumes that M(T,H) can be regarded as a state \nfunction, i.e. taking a single value independent of the magneto thermal history. Accordingly, the \nMaxwell equation can be safely used around a sec ond-order transition, whereas it is in principle \ninapplicable to a first-order transition.30-42 In the latter case, the analysis of the magnetic data that \nis required to estimate ΔS is much more complex. In the recent years, various models have been \nproposed to address this issue wh ich is still intensively debated.31-33,35,37,38,40,42 \nIn the present case of Mn 3O4, the MCE primarily takes place around T N which is a second-order \ntransition. We thus derived ΔS(T) from the Maxwell equation, by using a slightly modified form \nof this equation which is more tractable in practice:15,41,43,44 \n∫∂∂= ΔH\ndHHTMTHTS\n0')',( ),( (1b). \nThe main panel of Figure 1 displays the - ΔS(T) curve obtained at H = 20 kOe, when considering \nthe M T(H) curves recorded upon field increasing. As expected for a ferrimagnetic transition, one \nobserves a large peak centered around T N, which reaches a maximum value of ∼ 7.5 Jkg-1K-1. \nThe overall shape of - ΔS(T) was found to be the same for larg e values of H, the main difference \nbeing the height of the peak which reaches ∼ 13 Jkg-1K-1 and ~ 18 Jkg-1K-1 for H = 50 and 90 \nkOe, respectively. Beside this main peak, the second prominent feature of - ΔS(T) in Figure 1 is \nthe presence of a crossing from positive to negative values as temperature is decreased below 20 \nK. Similar features were reported in various types of manganese oxides.45 However, it must kept \nin mind that artifacts can easily be ge nerated when using the Maxwell equation.30-42 Before addressing this issue more precisel y, let us consider the results of ΔS(T) obtained from the \ncalorimetric method. \nThis approach is based on the recording of heat capacity curves, C(T) measured in zero-\nfield and in the external magnetic field H. Assuming that the basi c relationship dS = (CdT)/T is \nalways obeyed (i.e, dealing with reversible tran sformations), the isothermal entropy change is \ndirectly computed from the following equation,46 \n∫ ⎥⎦⎤\n⎢⎣⎡− ∫ ⎥⎦⎤\n⎢⎣⎡= − = ΔT T\nH dTTTCdTTHTCTS TS HTS\n0 00)0,( ),()( )( ),( (2) \nThe C(T) curves in H = 0 and H = 20 kOe were measured upon warming after a zero-\nfield cooling down to the lowest reachable temperature, i.e. 2 K in the present case. Below 2K, \nthe C/T values are linearly extrapolated to zero at T = 0 K. The - ΔS(T) derived using the above \nprocess is depicted in Figure 2a, where the main feature is again the presence of a large peak \ncentered at T N. However two main differences emerge when comparing to the magnetic method. \nFirst, the height of the peak is substantia lly higher, reaching a maximum value of 11 J K-1 kg-1 \ninstead of 7.5 J K-1 kg-1. Second, the - ΔS(T) at low-temperatures monotonically reaches zero as \nthe temperature is decreased wit hout exhibiting any crossover to negative values. The first issue \nis a direct consequence of the limited reso lution of the magnetization method. Indeed, the ΔS \nvalue derived from Eq. (1) actually reflects an average over the temperature range T ± δT, where \nδT is the spacing betw een two consecutive M T(H) curves. In particular, one can check that the \nmaximum value derived from magnetization, ΔS (44K) ≈ -7.5 J kg-1 K-1, well corresponds to \n44.7 2/ )(44\n44−=⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛Δ∫+\n−T dTTST\nTC δδ\nδ J K-1 kg-1, where ΔSC(T) is the curve derived from heat capacity. \nOf course, reducing δT minimizes this effect, but this is limited by a growing uncertainty in the derivative of Eq. (1) when the M H(T) become too close to each othe r. In reality, the intrinsically \nhigher resolution of the calorimetric method makes it more suitable to derive the peak of ΔS(T) \nin case of a sharp transition, as presently found around T N. \nIn other respects, the absen ce of positive values in the ΔS(T) curves derived from C(T) at \nlow-temperatures led us to reconsider the results obtained from the magnetization method. \nActually, the question is not about the quality of magnetic data nor the Maxwell equation in \nitself, but it deals with a po ssible inappropriate use of it.6,35,36,46 For instance, it is now widely \nrecognized that such a type of problem is at the origin of a lot of huge ΔS values reported in the \nliterature, which are in fact artifacts.30-42 An important point sometimes forgotten is that the \nMaxwell equation is supposed to hold only in presence of a reversible magnetic behavior.6,35,36,46 \nHence, a first check about the legitimacy of usi ng the Maxwell equation is to compare the M(H) \ncurves recorded either upon fi eld-increasing (“Up” curve) or field-decreasing (“Down” curve). \nFigure 3 shows some of such magnetic cycles at various temperatures. The M(H) curves \nrecorded at temperatures higher th an ~ 30 K (see Figure 3a), i.e. in the temperature range of the \ntransitions, are found to be reversible (or almost reversible), authorizing the use of the Maxwell \nequation. In contrast, at lower temperature (see Figure 3b), a noticeable hysteresis is observed between the field increasing and field decreasing branches of M(H) curves. This behavior \nbecomes much more prominent at low temperatures (below ∼ 15 K). Therefore, the Maxwell \nequation is simply not valid to estimate the ΔS using Equation (1) at low-temperatures.\n Looking \nat Figure 3b, one can notice that the apparent “positive” values of ΔS(T) reflect a shift of the \nM(H) “Up” curves towards higher fields as the temperature is decreased. Meanwhile the \napplication of the Maxwell equation on the “Down” series of M(H) curves would just yield \nvanishing positive values of ΔS(T) in the same low-temperature regime. In fact, such a low-temperature behavior of the M(H) curves is mainly driven by the pronounced temperature \ndependence of the coercive field in Mn 3O4,19 a phenomenon which should not be confused with a \ngenuine MCE. \nFinally, a closer look at the - ΔS(T) curve derived from heat-capacity reveals two \nadditional features which are more clearly ev idenced in the logarithmic scale (Figure 2b). \nAround 40 K (close to T 1), one can observe a weak “kink”, while a small “dip” appears around \n33 K (near to T 2). The kink at T 1 can be regarded as an addi tional negative contribution of \nentropy change superimposed onto the low-temp erature side of the main peak around T N. Such a \nbehavior is well corresponding to a shoulder observed around T 1 in –dM/dT curve (see inset \nFigure 1). In terms of heat capacity , the above feature is also consistent with the fact that around \nT1, C(T, H=20kOe) lies below C(T, H=0) and has its maximum slightly shifted towards higher \ntemperature. At T 2, the anomaly looks different, since it ra ther appears as a “dip” superimposed \nonto the low-temperature wing of the main peak. It turns out that this feature comes from a \npeculiar influence of the magnetic field on the C(T) curve around T 2. As compared to zero-field, \nthe peak present on the C(T) curve in 20 kOe starts developing at a lower temperature ( ≈ 32.5 K \ninstead of ≈ 33.0 K) while its maximum is slightly shifted to higher temperature ( ≈ 34.3 K \ninstead of ≈ 34.0 K). The crossing between the C(T) cu rves in H=0 and H=20 kOe is close to ≈ \n33.6 K. In terms of MCE, such a field-induced modification in the prof ile of the transition \ninduces a dip in - ΔS(T), that is centered at the crossing temperature. Moreover, the height of \nsuch a peak is expected to be only a fraction of the entropy jump associated with the first-order \ntransition at T 2. By integrating the anomaly on the C/T(T) curves in zero or 2T, this entropy \njump is found to be ∼ 0.5 J K-1 kg-1, while the height of the dip-like anomaly on the ΔS(T) curve \nis indeed smaller, i.e. ∼ 0.1 J K-1 kg-1. With the calorimetric method, one can also obta in the second characte ristic quantity of \nthe MCE, i.e. the adiaba tic temperature change ΔTad. These values were derived from the \nentropy curves using the equation,46 \n[])( 00)( )( ) 0,(TS H ad STST H TT − = → Δ ( 3 ) \nThe resulting ΔTad(T) curve in the case of field change from 0 to 20 kOe is shown on \nFigure 4. Similarly to the ΔS(T) curve, the main feature in ΔTad(T) is a prominent peak located \nclose to T N. One can also observe a clear signature of the anomaly at T 2 in the ΔTad(T) plot. It \ncan be noted that the maximum ΔTad of 1.9 K for a moderate applied magnetic field of 20 kOe, is \nlarge enough to regard Mn 3O4 as a potential refrigerant materi al. Nevertheless, the width of the \nΔS(T) and ΔTad(T) curves are quite small, which woul d yield only a modest relative cooling \npower. The best magnetocaloric materials reported so far in the temperature range around 50 K \nbelong to the family of the Laves phases.3,15 Even though the performances of Mn 3O4 well \ncompare to those of these compounds showing second-order ferromagnetic transitions, slightly \nbetter results are observed in the case of first-order transitions. As a matter of fact, the best Laves \nphase ErCo 2 (TC = 37 K) yields a maximum ΔTad = 3 K for a field change of 20 kOe.3,15,47,48 \n \nIV. Conclusions: \nIn conclusion Mn 3O4 is found to exhibit a substant ial MCE around its ferrimagnetic \ntransition at T N (42.75 K). For a field change of 20 kOe, the peak values of ⎥ΔS⎥ and ⎥ΔTad⎥ are \n11 J K-1 kg-1 and 1.9 K, respectively. The vari ous transitions taking place from T N down to ∼ 33 \nK are also found to have signature s on the isothermal entropy cha nge. However, these effects are \nsmaller and do not really contribute to widen th e operating temperature range for applications in \nmagnetic refrigeration. This study also well illustrates the advantages of the calorimetric method as compared to \nthe widely used magnetic technique based on the Ma xwell equation: First, it is less prone to the \ngeneration of spurious features, as those we presently observed in Mn 3O4 owing to the \nappearance of magnetic irreversib ility at low temperatures ; S econd, the calorimetric method \nfacilitates a better temperature resolution which allows to detect the small anomalies and avoid \nunderestimating the maximum ΔS value in case of sharp peaks. \n \n \nV. 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Temperature dependence of the isothermal entropy change derived from the magnetic \nmethod, using the field-increasing branches of M( H) for a field variation from 0 to 20 kOe. The \ninset displays the derivative of the magnetizat ion curve measured in 200 Oe. The arrows show \nthe transitions taking place, from right to left, at T N, T1, and T 2. \n \nFigure 2. (a): Main panel: Temperature dependen ce of the isothermal entropy change for a field \nvariation of 20 kOe, where the “magnetic” ΔS(T) (open squares) is derived from the field \nincreasing branches of M(H), while the “calorimetric” ΔS(T) (filled circles) is derived from C(T) \ncurves recorded upon warming. The inset shows the temperature dependence of C/T in zero-field. (b): Zoomed view of the isothermal entropy change estimated from the heat capacity data, represented in a semi-logarithmic scale. The arro ws highlight the anomalies present, from right \nto left, at T\n1 and T 2. \n \nFigure 3. Isothermal magnetization curves reco rded either upon increasing (filled symbols) or \ndecreasing (empty symbols) the ma gnetic field. (a): High-T regime with curves at 30, 35, 40 and \n45 K; (b): Low-T regime with curves at 5, 10, 15 and 20 K. Figure 4. Temperature dependence of the adiabatic temperature change in Mn\n3O4, measured for a \nmagnetic field change of 20 kOe. " }, { "title": "1703.05220v1.Modeling_ultrafast_all_optical_switching_in_synthetic_ferrimagnets.pdf", "content": "arXiv:1703.05220v1 [cond-mat.mes-hall] 15 Mar 2017Modeling ultrafast all-optical switching in synthetic fer rimagnets\nS. Gerlach1,∗L. Oroszlany2, D. Hinzke1, S. Sievering1, S. Wienholdt1, L. Szunyogh3, and U. Nowak1\n1Fachbereich Physik, Universit¨ at Konstanz, D-78457 Konst anz, Germany\n2Department of Physics of Complex Systems, E¨ otv¨ os Univers ity,\nH-1117 Budapest, P´ azm´ any P´ eter s´ et´ any 1/A, Hungary an d\n3MTA-BME Condensed Matter Research Group and Department of T heoretical Physics,\nBudapest University of Technology and Economics, Budafoki ´ ut 8., HU-1111 Budapest, Hungary\n(Dated: September 13, 2018)\nBased on numerical simulations, we demonstrate thermally i nduced magnetic switching in syn-\nthetic ferrimagnets composed of multilayers of rare-earth and transition metals. Our findings show\nthat deterministic magnetization reversal occurs above a c ertain threshold temperature if the ratio\nof transition metal atoms to rare-earth atoms is sufficiently large. Surprisingly, the total thickness\nof the multilayer system has little effect on the occurence of switching. We further provide a simple\nargument to explain the temperature dependence of the rever sal process.\nI. INTRODUCTION\nThe demonstration of helicity-dependent all-optical\nmagnetization switching1,2was one of the most sur-\nprising findings in ultrafast magnetization dynamics.\nThe experiments showed that magnetization switching\nis possible solely triggered by a single laser pulse in\nthe sub-picosecond range avoiding any externally applied\nmagnetic field. These experiments were performed on\nGdFeCo, a rare-earth-based ferrimagnet where the rare-\nearth (RE) sublattice is antiferromagnetically coupled to\nthe transition metal (TM) sublattice. First attempts to\ndescribe these unexpected processes were based on the\nassumption that the circularly polarized laser pulse in-\nduces a strong magnetic field via the inverse Faraday ef-\nfect which determines the direction of the switching2,3.\nOver all, this process takes place on a time scale orders\nof magnitude shorter than today’s writing procedures\nin hard discs. This calls for applications in magnetic\ndata storage and alternative materials including alloys4,\nheterostructures5and synthetic ferrimagnets6are cur-\nrently investigated.\nThe discovery of thermally-induced all-optical switch-\ning using linearly polarized light7,8cast a new light on\nall-optical switching and called for more sophisticated\nmodels since this switching works without any external\nor optically induced magnetic field, which could define\nthe magnetization direction during its recovery after the\nultrafast quenching.\nIn simulations this switching was observed in an atom-\nistic spin model developed by Ostler et al.7–9. An\nattempt to explain the occurrence of the transient\nferromagnetic-like state (TFMLS) was given by Mentink\net al. who identified angular momentum transfer driven\nby the inter-sublattice exchange as the crucial process10.\nLater on the thermally induced switching was more quan-\ntitatively described by means of an orbital-resolved spin\nmodel, where the magnetic moments stemming from d-\nelectrons of the TM, the d-electrons of the RE and the\nf-electrons of the RE are distinguished11. Here, it was\nshown that the initial laser excitation brings the sublat-\ntices into a strong non-equilibrium after 1 ps. This hap-pens due to the different demagnetization times of the in-\ndividual sublattices12. On that time scale, electron and\nphonon temperatures are nearly equilibrated below TC\nagain, but the Fe sublattice is already completely demag-\nnetized while the Gd sublattice remains still rather or-\ndered. The remagnetization dynamics of Fe taking place\nsubsequently leads to a state where the Gd spins and the\nFe spins are aligned — the transient ferromagnetic-like\nstate. The TFMLS arises naturally as a consequence of a\nredistribution of the energy and angular momentum be-\ntween the different sublattices due to the maximization of\nentropy under the constraint of energy and angular mo-\nmentum conservation. These processes, which are driven\nvia the precession term of the Landau-Lifshitz-Gilbert\nequation, dominate on shorter times scales. The follow-\ning relaxation back to a ferrimagnetic equilibrium state\nis on a longer time scale, where dissipative processes are\nresponsible, and does not necessarily lead to a switched\nstate4,13. The details of this relaxation process depend\non the material properties as well as the experimental\nspecifications and are still under investigation.\nIn the following, we will explore the possibility of\nthermally-induced switching in synthetic ferrimagnets\ncomprised of bilayers of Fe and Gd. We use ab-initio\nmethods to estimate spin model parameters for Fe-Gd bi-\nlayers. The dynamic simulations of the spin model allows\nfor an investigation of the preconditions for thermally\ninduced switching. We find that deterministic magne-\ntization reversal occurs only above a certain threshold\ntemperature and in bilayers where the ratio of transition\nmetal atoms to rare-earth atoms is sufficiently large. Fi-\nnally, we find a simple explanation why the compensation\ntemperature is so important.\nII. MODEL\nOur aim is to model a synthetic ferrimagnet as a bi-\nlayer of two ferromagnets, Fe and Gd, with a negative\ncoupling between the two layers. For that we consider an\natomistic spin model where localized spins are arranged\non a simple-cubic lattice structure. Spins experience ex-2\nchange interactions with their nearest neighbors only and\ndipole-dipole interaction is neglected. The Heisenberg\nHamiltonian of the system studied reads\nH=−/summationdisplay\nNNJijSiSj−/summationdisplay\nidzS2\ni,z. (1)\nThe first term represents the Heisenberg exchange en-\nergy, where the exchange interaction is either between\nspins of the Fe layer, spins of the Gd layer, or, across the\ninterface, between Fe and Gd spins. This term contains,\ntherefore, three interactions, JFe−Fe,JGd−GdandJFe−Gd.\nThe second term represents a uniaxial anisotropy with\nanisotropy constant dz. The lateral dimensions of the\nmodel are 150 ×150 atoms with the layers stacked along\nthezaxis and with periodic boundary conditions in\ntransverse directions. The thicknesses of the Fe and Gd\nlayers are varied.\nAb-initio calculations have been performed in terms of\nthe fully relativistic screened Korringa–Kohn–Rostoker\n(KKR) method, designed, in particular, for layered sys-\ntems and surfaces14. The LSDA parametrization from\nRef. 15 was used. The strong correlation of the localized\n4f-states of the Gd atoms was treated within the frame-\nwork of the LSDA+U approach16as implemented within\nthe KKR method17. The calculations were carried out\nwith the commonly used U= 6.7 eV and J= 0.7 eV\nvalues of the Coulomb and exchange integrals16and the\ndouble counting term derived in Refs. 18 and 19 satis-\nfying the atomic limit for the LSDA total energy. The\nexchange constants have been obtained by means of the\nrelativistic torque method20.\nThe geometry used in the ab-initio calculations were\nbased on a heterostructure of six Fe layers between two\nsemi-infinite bulk Gd regions. For the hcp structure of\nGd bulk we used the experimental c/aratio of 1.5904\nand an optimized lattice constant a= 3.450˚A21. The\ninterlayer distance in the Fe region was chosen such that\nthe volume per Fe atom was identical to that in bcc Fe\nwith the experimental lattice constant of 2.867 ˚A. The\ndistance of the Fe planes to the nearest Gd planes was\ntaken to be the average of the Fe-Fe and Gd-Gd layer dis-\ntances. The interlayer exchange interactions obtained by\ntheab-initio procedure were then mapped to the simple\ncubic structure with nearest-neighbor interactions used\nin the spin-dynamics simulations.\nThe calculations outlined above result in exchange con-\nstants of the spin model above with the ratios JFe-Fe :\nJGd-Gd :JFe-Gd = 1 : 0 .286 : −0.388 which we use in\nthe subsequent dynamic simulations. The magnetic mo-\nments of Fe and Gd have the values of µTM= 1.92µB\nandµRE= 7.63µB, refering to the values found for bulk\nFeGd in Ref. 11. These values are close to those obtained\nin our KKR LSDA+U calculations. In addition, we used\nan anisotropy constant of dz= 0.2 meV favoring magne-\ntization along the zaxis.\nThe dynamics of the system is governed by the stochas-0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4\nTemperature in JFe/kB−3.0−2.5−2.0−1.5−1.0−0.50.00.51.0Magnetization Mz(T)/MFe\nz(0)\nTGd\nCTcomp TFe\nCMFe\nMGd\nM\nFIG. 1. (Color online) Temperature dependent equilibrium\nmagnetization for a bilayer with 3 monolayers of Fe and 2\nmonolayers of Gd. The magnetization of the element with\nthe stronger intra-layer exchange interaction, in this cas e Fe,\nclosely follows the shape of a ferromagnet with its Curie tem -\nperature TFe\nCclose to the Curie temperature of the coupled\nsystem. The element with the weaker intra-layer exchange\ninteraction, in this case Gd, exhibits magnetic ordering ab ove\nits own bulk Curie temperature, TGd\nC, due to the interaction\nwith the other sublattice. The chosen ratio of layer thick-\nnesses leads to a larger magnetic moment of the Gd layer at\nlow temperatures and, consequently, to a magnetic compen-\nsation point at a temperature Tcomp which is slightly above\nthe bulk Curie temperature of the Gd.\ntic Landau-Lifshitz-Gilbert equation of motion,\n(1 + α2)µi\nγ˙Si=−Si×[Hi+α(Si×Hi)],(2)\nwith the gyromagnetic ratio γand a dimensionless\nGilbert damping constant αthat describes the coupling\nto the heat-bath. In our simulations the damping con-\nstant is set to α= 0.0211,22. Thermal fluctuations are\nincluded as an additional white noise term ζiin the in-\nternal fields Hi=−∂H\n∂Si+ζi(t) with\n/angbracketleftζi(t)/angbracketright= 0,/angbracketleftζiη(0)ζjθ(t)/angbracketright=2kBT αµ i\nγδijδηθδ(t),(3)\nwhere i, jdenotes lattice sites and η, θare Cartesian com-\nponents. All algorithms we use are described in detail in\nRef. 23.\nIn Fig. 1we present the equilibrium properties of a bi-\nlayer consisting of 3 monolayers of Fe and 2 monolayers of\nGd. This bilayer behaves as a synthetic ferrimagnet with\na magnetic compensation point Tcomp at a temperature\nof about 40% of the Curie temperature Tc.\nIt is well-known that laser-induced demagnetization in\ntransition metals is several times faster than in rare-eart h\nelements24,25. Several approaches have been proposed to\nexplain this behavior, including electron-phonon scatter -\ning processes of the Elliott-Yafet type26and intra-atomic3\nenergy transfer within the electronic subsystem11. How-\never, based on the different demagnetization time scales,\nwe may assume that heating a multilayer system with\nincident laser light can lead to a situation where the TM\nlayer is completely demagnetized while the RE layers still\nretains a substantial net magnetization. We use this fact\nin the following and do not calculate the action of the\nlaser pulse on the spin system explicitly. Instead we focus\non the relaxation of the magnetization at constant tem-\nperature starting our simulations with a spin configura-\ntion where the Fe sublattice is completely demagnetized\n(spins are randomly oriented) while we vary the degree\nof magnetization of the Gd layer. This initial Gd magne-\ntization and the temperature will turn out to be crucial\nquantities for the understanding of thermally triggered\nswitching.\nIII. RESULTS\nFor a fixed layer configuration, we treat the initial\nRE magnetization remaining after the laser excitation\nand the temperature Tas the relevant parameters which\ndetermine the magnetization dynamics triggered by the\nlaser pulse. In Fig. 2three different possible scenarios are\nshown, switching (top), switching followed by switching\nback to the initial state (center), and no-switching (bot-\ntom). The chosen values for Tand Gd magnetization are\nindicated in Fig. 3. The switching scenario corresponds\nto the work of Radu et al.7, while the back-switching\nwas measured by Khorsand et al.27. Note, that in all\nthree cases the magnetization of the transition metal\nstarts towards negative values (while the original sign\nbefore demagnetization by the effect of the laser heating\nwould have been positive) so that a TFMLS is obtained.\nNote also, that the transverse components of the mag-\nnetization are usually not small — apart from the case\nof switching — which indicates a linear mechanism for\nthe case of successful switching but a more precessional\nprocess for the case of no-switching and back-switching.\nA systematic variation of the two parameters, temper-\nature and initial Gd magnetization, allows for the con-\nstruction of a switching diagram as shown in Fig. 3. It\nis ternary in the way that it provides information about\nthe relaxation process, with the three scenarios above\n(switching, back-switching, or no-switching) as shown in\nFig. 2. The diagram is constructed by taking into ac-\ncount the magnetization dynamics of the two species in-\nvolved for a single run during a time interval of 40 ps. We\nidentify three distinct regions, a connected no-switching\nregion (/squaresolid), a connected switching region ( /squaresolid), and a back-\nswitching region along the boundary of the other two re-\ngions (/squaresolid).\nFor an interpretation of this diagram we first note that\nfor very low values of initial RE magnetization one would\nexpect the system to randomly pick one of the two pos-\nsible equilibrium configurations, either switched or not.\nThis sort of statistical behavior is indeed evident from−1.0−0.50.00.51.0Magnetization Mz(T)/Mz(0) Fe\nGd\n−1.0−0.50.00.51.0Magnetization Mz(T)/Mz(0) Fe\nGd\n0 5 10 15 20\nTimetin ps−1.0−0.50.00.51.0Magnetization Mz(T)/Mz(0) Fe\nGd\nFIG. 2. (Color online) The three possible time relaxation\nscenarios for the sublattice magnetizations of a RE-TM lay-\nered system. (top) Magnetization switches with respect to\nthe initial configuration, (center) both sublattice magnet iza-\ntions change their signs twice, ending up back in the initial\nstate, (bottom) no-switching, i.e. the rare-earth layer ma gne-\ntization does not change sign. Also shown are the transverse\ncomponents (dotted and dashed lines). For the case of back-\nswitching and no-switching these components are of the orde r\nof the magnitude of the magnetization.\nthe isolated data points at the very left hand side of the\ndiagram. Furthermore one would expect that high val-\nues of MREwould prevent the system from switching at\nlow temperatures because the level of order in the RE\nlayer is too high to become demagnetized. This is indeed\nwhat we find from Fig. 3. The no-switching region is\nfound in the bottom right corner, corresponding to low\ntemperature and high MRE.\nSo far, the switching diagram meets our intuitive ex-4\n20 40 60 80 100\nInitial Gd magnetization in %0.00.10.20.30.40.50.6Final temperature in JFe/kBTcomp\nTGd\nC\nno-switchingswitching\nFIG. 3. (Color online) Switching in a 3:2 layer sample after\n40 ps. Color coding: /squaresolidMagnetization switches, /squaresolidmagneti-\nzation does not switch and returns to the initial state, /squaresolidboth\nmagnetizations change their sign twice, ending up in the ini -\ntial state (back-switching). The compensation temperatur e\nTcomp is indicated by a solid black line, the Curie tempera-\nture TGd\nCof the isolated Gd layer by a black dashed line. The\nthree points indicate the chosen values for the scenarios in\nFig. 2.\npectations. It is less obvious, however, why high tem-\nperatures (above Tcomp) lead to switching regardless of\nhow strongly the RE layers are demagnetized by the heat\npulse. Qualitatively, this can be understood keeping\nin mind the linear switching mechanism, which avoids\ntransverse magnetization components (see Fig. 2and\nRefs.2,28). Linear switching needs a high degree of spin\ndisorder in the system and, consequently elevated tem-\nperatures.\nMore quantitatively, the role of temperature can be\nunderstood via the equilibrium layer magnetizations as\nshown in Fig. 1, since those mark the final values for\nthe relaxation process. Let us assume that following the\nexcitation of the laser pulse the Fe layer is completely de-\nmagnetized while the Gd is only demagnetized by 50%.\nThe bilayer is far from equilibrium and a relaxation pro-\ncess will set in of which the details depend on the tem-\nperature. Each layer will relax towards its individual\nequilibrium value. We can identify three important tem-\nperature ranges:\n•T < TRE\nC: Both layers tend to increase their net\nmagnetization magnitudes towards higher values.\n•TRE\nC< T < TTM\nC: To reach equilibrium, the mag-\nnitude of the TM magnetization must still increase,\nwhile that of the RE must decrease.\n•T > TTM\nC: The system will completely demagne-\ntize.\nFor most TM-RE layer ratio Tcomp (if it exists) is\nhigher than TRE\nCand only the temperature range be-−1.0−0.50.00.51.0Magnetization Mz(T)/Mz(0)\nFe\nGd\n0.0 0 .5 1 .0 1 .5 2 .0\nTimetin ps−1.0−0.50.00.51.0Magnetization Mz(T)/Mz(0)\nFe\nGd\nFIG. 4. (Color online) Strong dissipation ( α= 1) in a 3:2\nlayer sample, implying no conservation of angular momen-\ntum. The relaxation does not lead to a TFMLS, neither at\nlow temperature where the Gd magnetization increases nor at\nhigh temperature where it decreases. (top: T < TGd\nC, bottom:\nTGd\nC< T < TFe\nC). Also shown are the transverse components\n(dotted and dashed lines) which remain small for both cases.\ntween TRE\nCandTTM\nCsupports the dynamics necessary for\nswitching — decreasing Gd magnetization and increasing\nFe magnetization. Consequently, the temperature must\nbe at least above the critical temperature of bulk Gd.\nAbove the compensation temperature switching is always\npossible. That means, if Tcomp is very low, switching can\nalso be done below TRE\nC.\nThe next question is, why the Fe layer magnetization\nstarts recovering towards negative magnetization which\nresults in a TFMLS. This is a consequence of angular mo-\nmentum conservation, as was already pointed out by sev-\neral authors10,11. While the Gd still demagnetizes (since\nthe temperature is above the bulk critical temperature\nof Gd), the dynamics of the Fe layer magnetization must\nchange into the other direction keeping the angular mo-\nmentum constant. The argument also explains that one\nneeds a certain initial Gd magnetization to start with.\nWithout initial Gd magnetization there is no angular mo-\nmentum reservoir to drag the Fe magnetization towards\nnegative values.\nAngular momentum conservation is not strictly ful-5\n0 1 2 3 4 5 6 7\nTimetin ps−0.8−0.6−0.4−0.20.00.2Magnetization Mz(T)/Mz(0)\nFe\nGd\nFIG. 5. (Color online) Layer resolved magnetization dynam-\nics for switching in a 3:2 layer sample. The Gd layer is de-\nmagnetized to 50% of its zero temperature value. The bold\nlines corresponds to the layers at the Gd-Fe interface and th e\ndashed lines shows the average value for Gd or Fe. The tem-\nperature is T= 0.5JFe/kB.\nfilled in the spin system. The relaxation part of the equa-\ntion of motion breaks this conservation on time scales\nwhich are determined by the value of the damping con-\nstant α. For low values of α, the precessional part of the\nequation of motion is much larger leading to dynamics\nwhich keeps total energy and total angular momentum\nconserved on shorter time scales. This is different when\nconsidering larger values of α. For comparison, we inves-\ntigate in Fig. 4the regime of strong dissipation ( α= 1).\nHere, the time scales of precessional dynamics is of the\nsame order as the time scale of relaxation and dissipative\neffects counteract the conservation of angular momentum\nin the system. Only if the damping constant αis suffi-\nciently small, angular momentum is almost conserved on\nshort time scales, along with the total energy, leading\nto a TFMLS as seen in Fig. 2. The figure also illus-\ntrates that — depending on the temperature — the Gd\nmagnetization might relax either towards higher or lower\nequilibrium values. The transition from dissipationless\ndynamics to the regime where damping effects dominate\nthe dynamics has previously been investigated11.\nIn the following we turn to the peculiarities of the lay-\nered system. The layer resolved magnetization in Fig. 5\nshows the importance of the interface layers for switch-\ning. For low damping, the angular momentum conserva-\ntion leads to a relaxation dynamics with an exchange of\nangular momentum between the still demagnetizing Gd\nlayer and the Fe layer, leading to a negative Fe magne-tization and, consequently, to a TFMLS. Because of the\nantiferromagnetic coupling along the Fe-Gd interface, the\nFe interface monolayer lags behind the other layers. Af-\nter some ps, however, the Fe magnetization has reached\nits new, negative equilibrium value, pushing the Gd via\nthe negative interface coupling towards positive values.\nHere, the dynamics is quicker at the interface as for the\nother Gd layer which is lagging behind.\nWe also simulated other layer thicknesses and ratio.\nWhile Fe-Gd bilayer with 20-40% Gd (for example 4:1,\n3:1 or 3:2 layer) turned out to have the correct thickness\nratio for switching (and having a magnetization compen-\nsation temperature) we found that the over-all thickness\nis less relevant. Successful switching can also be seen in\nmuch bigger samples (although on longer time scales), for\ninstance in a 30:20 Fe-Gd layer. The antiferromagnetic\ncoupling of the interface layers finally leads to a switch-\ning of all layers when the temperature of the heat bath\nexceeds Tcomp (which is almost the same as TGd\nCfor big\nsamples). Changing the ratio of Fe and Gd layer does\nnot change the switching behavior as long as the sam-\nple maintains a magnetization compensation tempera-\nture. Only the temperature range for switching increases\nwith decreasing percentage of Gd.\nIV. SUMMARY\nWe explored thermally induced magnetic switching in\nsynthetic ferrimagnets composed of a bilayer of rare-earth\nand transition metal on the basis of spin model simula-\ntions where the model parameters were calculated from\nfirst principles. Varying the temperature and the de-\ngree of initial rare-earth magnetization directly after th e\nlaser pulse one may find either, back-switching, or no-\nswitching. Deterministic magnetization reversal occurs\nabove a certain threshold temperature which is above the\nbulk Curie temperature of the rare-earth since only then\nthe magnetization of the rare-earth sublattice relaxes to-\nwards lower magnitude. The optimal ratio of transition\nmetal atoms to rare-earth atoms for successful switch-\ning has 20-40 % Gd layer while the total thickness of the\nmultilayer system only affects the time scale of switching.\nACKNOWLEDGMENTS\nThis work has been funded by the Center for Applied\nPhotonics at the University of Konstanz, the Deutsche\nForschungsgemeinschaft and by the Hungarian National\nScientific Research Fund (NKFIH) under project Nos.\nK115575 and K108676. O. L. acknowledges support from\nthe Janos Bolyai Scholarship of the Hungarian Academy\nof Sciences.\n∗stefan.gerlach@uni-konstanz.de1C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Physical Review6\nLetters 99, 047601 (2007).\n2K. Vahaplar, A. M. Kalashnikova, A. V. Kimel, D. Hinzke,\nU. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kiri-\nlyuk, and T. Rasing, Physical Review Letters 103, 117201\n(2009).\n3K. Vahaplar, A. M. Kalashnikova, A. V. Kimel,\nS. Gerlach, D. Hinzke, U. 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Nowak, European Physical Letter 86, 27006 (2009)." }, { "title": "2001.08701v2.Magnetization_plateaus_and_bipartite_entanglement_of_an_exactly_solved_spin_1_2_Ising_Heisenberg_orthogonal_dimer_chain.pdf", "content": "Magnetization plateaus and bipartite entanglement of an exactly solved\nspin-1 /2 Ising-Heisenberg orthogonal-dimer chain\nLucia G ´alisov ´aa,\u0003, Jozef Stre ˇckab, Taras Verkholyakc, Samuel Havadejb\naInstitute of Manufacturing Management, Faculty of Manufacturing Technologies with the seat in Preˇ sov, Technical University of Koˇ sice,\nBayerova 1, 080 01 Preˇ sov, Slovakia\nbInstitute of Physics, Faculty of Science, P . J. ˇSaf´ arik University, Park Angelinum 9, 040 01 Koˇ sice, Slovakia\ncInstitute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Svientsitskii Street 1, 790 11 L’viv, Ukraine\nAbstract\nSpin-1 /2 orthogonal-dimer chain composed of regularly alternating Ising and Heisenberg dimers is exactly solved in a presence\nof the magnetic field by the transfer-matrix method. It is shown that the ground-state phase diagram involves in total six di \u000berent\nphases. Besides the ferromagnetic phase with fully polarized spins one encounters the singlet antiferromagnetic and modulated\nantiferromagnetic phases manifested in zero-temperature magnetization curves as zero magnetization plateau, the frustrated fer-\nrimagnetic and singlet ferrimagnetic phases causing existence of an intermediate one-half magnetization plateau, and finally, the\nintriguing modulated ferrimagnetic phase with a translationally broken symmetry leading to an unconventional one-quarter mag-\nnetization plateau. The quantum character of individual ground states is quantified via the concurrence, which measures a strength\nof the bipartite entanglement within the pure and mixed states of the Heisenberg dimers at zero as well as nonzero temperatures.\nThe parameter region, where the bipartite entanglement may be in contrast to general expectations reinforced upon increasing of\ntemperature and /or magnetic field, is elucidated.\nKeywords: Ising-Heisenberg model, orthogonal-dimer chain, magnetization plateaus, bipartite entanglement\n1. Introduction\nThe gapped quantum ground states manifested in low-tem-\nperature magnetization curves as intermediate plateaus remain\nat a forefront of intense theoretical studies, because several in-\ntriguing fractional magnetization plateaus were experimentally\ndetected in high-field magnetization curves of the magnetic com-\npound SrCu 2(BO 3)2[1–5] providing a long-sought experimen-\ntal realization of the Shastry-Sutherland model [6, 7]. Despite\nsubstantial e \u000borts, the number and microscopic nature of inter-\nmediate magnetization plateaus of SrCu 2(BO 3)2still remain un-\nresolved issue due to extraordinary mathematical di \u000eculties re-\nlated to a theoretical modeling of the Shastry-Sutherland model\nat zero [8–13] as well as nonzero [14, 15] temperatures.\nBy contrast, the magnetization process of one-dimensional\ncounterpart of the Shastry-Sutherland model, which is com-\nmonly referred to either as the spin-1 /2 Heisenberg orthogonal-\ndimer or dimer-plaquette chain [16–18], is quite well estab-\nlished nowadays. Except three most pronounced magnetiza-\ntion plateaus at zero, one-quarter and one-half of the saturation\nmagnetization, one additionally encounters an infinite series of\nsmaller fractional magnetization plateaus at rational numbers\nn=(2n+2)=1=4;1=3;3=8;:::; 1=2 ranging in between one-\nquarter and one-half magnetization plateaus [19, 20]. Unfortu-\nnately, a respective magnetic compound that would enable an\n\u0003Corresponding author\nEmail address: galisova.lucia@gmail.com (Lucia G ´alisov ´a)experimental testing of this peculiar sequence of the fractional\nmagnetization plateaus is not available to date.\nRecent experimental discovery of the polymeric coordina-\ntion compound [Dy(hfac) 2(CH 3OH)] 2[Cu(dmg)(Hdmg)] 2[21,\n22], which will be further referred to as the polymeric chain\n[Dy 2Cu2]n, has a \u000borded a valuable experimental realization of\nthe spin-1 /2 Ising-Heisenberg orthogonal-dimer chain with a\nregular alternation of the highly anisotropic dimeric units of\nDy3+magnetic ions (Ising dimers) with the almost isotropic\ndimeric units of Cu2+magnetic ions. It is supposed that the\nmost dominant coupling in [Dy 2Cu2]nis by far an antiferromag-\nnetic interaction between the nearest-neighbouring Dy3+and\nCu2+magnetic ions, whereas the dinuclear entities of Dy3+ions\nand Cu2+ions are coupled presumably through much weaker\nferromagnetic interaction [22]. It is worthwhile to remark that\nthe spin-1 /2 Ising-Heisenberg orthogonal-dimer chain with a\nregularly alternating Ising and Heisenberg dimers arranged in\nan orthogonal fashion can be exactly solved by adapting the ap-\nproach elaborated in Refs. [23–25] for various versions of this\nintriguing one-dimensional quantum spin model.\nIn the present work we will introduce and rigorously solve\na spin-1 /2 Ising-Heisenberg orthogonal-dimer chain composed\nof regularly alternating Ising and Heisenberg dimers in a pres-\nence of the external magnetic field. Although the investigated\nquantum spin chain is somewhat oversimplified in that it does\nnot take into account two di \u000berent exchange pathways between\nDy3+and Cu2+magnetic ions existing within the polymeric\ncompound [Dy 2Cu2]n, we believe that this quantum spin chain\nPreprint submitted to Physica E June 2, 2020arXiv:2001.08701v2 [cond-mat.stat-mech] 31 May 2020may shed light on the nature of unconventional quantum ground\nstates invoked by the external magnetic field in a low-temperatu-\nre magnetization process of the polymeric complex [Dy 2Cu2]n.\nThe organization of this paper is as follows. In Section 2 we\nwill introduce and solve the spin-1 /2 Ising-Heisenberg orthogo-\nnal-dimer chain within the framework of the transfer-matrix\nmethod. Section 3 includes a comprehensive discussion of the\nmost interesting results obtained for the ground-state phase dia-\ngram, the magnetization process and the bipartite entanglement\nemergent within the Heisenberg dimers. The most important\nfindings and future outlooks are briefly mentioned in Section 4.\n2. Model and its exact solution\nIn the present paper, we will consider the quantum spin-1 =2\nIsing-Heisenberg orthogonal-dimer chain schematically depict-\ned in Fig. 1 and defined through the total Hamiltonian:\nˆH=JHNX\ni=1\u0000ˆS1;i\u0001ˆS2;i\u0001\n\u0001+J0\nINX\ni=1ˆ\u001bz\n1;iˆ\u001bz\n2;i\n+JINX\ni=1\u0002ˆSz\n1;i\u0000ˆ\u001bz\n1;i+ˆ\u001bz\n2;i\u0001+ˆSz\n2;i\u0000ˆ\u001bz\n1;i+1+ˆ\u001bz\n2;i+1\u0001\u0003\n\u0000hHNX\ni=1\u0000ˆSz\n1;i+ˆSz\n2;i\u0001\u0000hINX\ni=1\u0000ˆ\u001bz\n1;i+ˆ\u001bz\n2;i\u0001: (1)\nIn above,\u0000ˆS1;i\u0001ˆS2;i\u0001\n\u0001= \u0001(ˆSx\n1;iˆSx\n2;i+ˆSy\n1;iˆSy\n2;i)+ˆSz\n1;iˆSz\n2;i,ˆS\u000b\n1(2);i\n(\u000b=x;y;z) label the spatial components of the standard spin-\n1/2 operators corresponding to the Heisenberg spins forming\nhorizontal dimers, ˆ \u001bz\n1(2);iare the spatial components of the stan-\ndard spin-1 /2 operators related to the Ising spins forming ver-\ntical dimers, the parameter JHdenotes the XXZ Heisenberg\nintra-dimer interaction on horizontal bonds with the parameter\nof exchange anisotropy \u0001,J0\nIrepresents the Ising intra-dimer\ninteraction between the Ising spins on vertical bonds, and JI\ndenotes the Ising inter-dimer interaction between the nearest-\nneighboring Ising and Heisenberg spins. The last two terms hH\nandhIare Zeeman terms, which account for the magnetostatic\nenergy of the Heisenberg and Ising spins in an applied longitu-\ndinal magnetic field, respectively. Finally, Ndenotes the total\nnumber of the Heisenberg and Ising spin dimers and the peri-\nodic boundary conditions ˆ \u001bz\n1(2);N+1\u0011ˆ\u001bz\n1(2);1are assumed for the\nsake of simplicity.\nFor further convenience, the total Hamiltonian (1) of the\nspin-1=2 Ising-Heisenberg orthogonal-dimer chain can be alter-\nnatively rewritten as a sum of the six-spin cluster Hamiltonians\nschematically delimited in Fig. 1 by a dotted rectangle:\nˆH=NX\ni=1ˆHi; (2)\nˆHi=JH\u0000ˆS1;i\u0001ˆS2;i\u0001\n\u0001+J0\nI\n2\u0000ˆ\u001bz\n1;iˆ\u001bz\n2;i+ˆ\u001bz\n1;i+1ˆ\u001bz\n2;i+1\u0001\n+JI\u0002ˆSz\n1;i\u0000ˆ\u001bz\n1;i+ˆ\u001bz\n2;i\u0001+ˆSz\n2;i\u0000ˆ\u001bz\n1;i+1+ˆ\u001bz\n2;i+1\u0001\u0003\n\u0000hH\u0000ˆSz\n1;i+ˆSz\n2;i\u0001\u0000hI\n2\u0000ˆ\u001bz\n1;i+ˆ\u001bz\n2;i+ˆ\u001bz\n1;i+1+ˆ\u001bz\n2;i+1\u0001:(3)\n/s49/s44 /s105/s43/s49\n/s50/s44 /s105/s43/s49/s83\n/s49/s44 /s105/s43/s49/s83\n/s50/s44 /s105/s83\n/s49/s44 /s105/s49/s44 /s105\n/s50/s44 /s105/s83\n/s50/s44 /s105/s43/s49\n/s105/s116/s104/s32/s115/s105/s120/s45/s115/s112/s105/s110/s32/s99/s108/s117/s115/s116/s101/s114/s74\n/s72/s40 /s41\n/s74\n/s73/s39/s74\n/s73\n/s74\n/s73/s74\n/s73\n/s74\n/s73/s83\n/s50/s44 /s105/s45/s49Figure 1: The magnetic structure of the frustrated spin-1 /2 Ising-Heisenberg\northogonal-dimer chain. Green (red) balls denote lattice sites occupied by the\nIsing (Heisenberg) spins, thin green (thick red) lines correspond to the Ising\n(Heisenberg) intra-dimer bonds and dashed black lines illustrate the Ising inter-\ndimer bonds. The dotted rectangle represents the ith six-spin cluster described\nby the cluster Hamiltonian (3).\nIt is noteworthy that the ith six-spin cluster Hamiltonian (3) in-\nvolves the vertical Ising dimers from two adjacent unit cells,\nwhereas the factor 1 =2 emergent at the Ising coupling J0\nIand\nthe Zeeman term hIavoids a double counting of these two in-\nteraction terms being symmetrically split into two consecutive\ncluster Hamiltonians.\nIt is obvious from Eq. (3) that di \u000berent cluster Hamiltoni-\nans satisfy the standard commutation relation [ ˆHi;ˆHj]=0, and\nthus, the determination of eigenvalues of ˆHiis su\u000ecient to ex-\nactly resolve the considered spin-1 =2 Ising-Heisenberg orthogo-\nnal-dimer chain. The relevant calculation can be performed in\na matrix representation of the Hilbert subspace spanned over\nthe orthonormal basis of four available spin states correspond-\ning to the ith Heisenberg spin pair\bj\"\"i i=j\"i1;ij\"i2;i;j\"#i i=\nj\"i 1;ij#i 2;i;j#\"i i=j#i 1;ij\"i 2;i;j##i i=j#i 1;ij#i 2;i\t, where\nj\"i 1(2);iandj#i 1(2);idenote eigenvectors of the spin operator\nˆSz\n1(2);iwith the eigenvalues Sz\n1(2);i=1=2 and\u00001=2, respectively.\nAs a result, one obtains the following set of eigenvalues:\nEi;1=JH\n4+J0\nI\n2\u0000\u001bz\n1;i\u001bz\n2;i+\u001bz\n1;i+1\u001bz\n2;i+1\u0001\n+JI\u0000hI\n2\u0000\u001bz\n1;i+\u001bz\n2;i+\u001bz\n1;i+1+\u001bz\n2;i+1\u0001\u0000hH;(4a)\nEi;2=JH\n4+J0\nI\n2\u0000\u001bz\n1;i\u001bz\n2;i+\u001bz\n1;i+1\u001bz\n2;i+1\u0001\n\u0000JI+hI\n2\u0000\u001bz\n1;i+\u001bz\n2;i+\u001bz\n1;i+1+\u001bz\n2;i+1\u0001+hH;(4b)\nEi;3=\u0000JH\n4+J0\nI\n2\u0000\u001bz\n1;i\u001bz\n2;i+\u001bz\n1;i+1\u001bz\n2;i+1\u0001\n+1\n2q\nJ2\nI\u0000\u001bz\n1;i+\u001bz\n2;i\u0000\u001bz\n1;i+1\u0000\u001bz\n2;i+1\u00012+(JH\u0001)2\n\u0000hI\n2\u0000\u001bz\n1;i+\u001bz\n2;i+\u001bz\n1;i+1+\u001bz\n2;i+1\u0001; (4c)\nEi;4=\u0000JH\n4+J0\nI\n2\u0000\u001bz\n1;i\u001bz\n2;i+\u001bz\n1;i+1\u001bz\n2;i+1\u0001\n\u00001\n2q\nJ2\nI\u0000\u001bz\n1;i+\u001bz\n2;i\u0000\u001bz\n1;i+1\u0000\u001bz\n2;i+1\u00012+(JH\u0001)2\n\u0000hI\n2\u0000\u001bz\n1;i+\u001bz\n2;i+\u001bz\n1;i+1+\u001bz\n2;i+1\u0001; (4d)\n2and corresponding eigenvectors:\nj ii;1=j\"\"i i; (5a)\nj ii;2=j##i i; (5b)\nj ii;3=sin'ij\"#i i+cos'ij#\"i i; (5c)\nj ii;4=sin'ij\"#i i\u0000cos'ij#\"i i; (5d)\nwhere tan (2'i)=JH\u0001=\u0002JI\u0000\u001bz\n1;i+\u001bz\n2;i\u0000\u001bz\n1;i+1\u0000\u001bz\n2;i+1\u0001\u0003.\nHaving the full set of eigenvalues of the cluster Hamilto-\nnian (3), the partition function Zof the investigated quantum\nspin chain can be derived by applying the standard transfer-\nmatrix approach [26, 27]:\nZ=X\nf\u001b1;i;\u001b2;igNY\ni=1TrS1;i;S2;iexp\u0000\u0000\fHi\u0001=X\nf\u001b1;i;\u001b2;igNY\ni=14X\nj=1exp\u0000\u0000\fEi;j\u0001\n=X\nf\u001b1;i;\u001b2;igNY\ni=1T\u0000\u001bz\n1;i;\u001bz\n2;i;\u001bz\n1;i+1;\u001bz\n2;i+1\u0001=TrTN=4X\nj=1\u0015N\nj:(6)\nIn above,\f=1=(kBT) is the inverse temperature ( kBis the\nBoltzmann’s constant and Tis the absolute temperature), the\nsymbolP\nf\u001b1;i;\u001b2;igdenotes a summation over all possible config-\nurations of the Ising spins from all vertical bonds, the prod-\nuctQN\ni=1runs over all six-spin clusters visualized in Fig. 1\nand Tr S1;i;S2;istands for a trace over degrees of freedom of the\nith Heisenberg spin dimer. Apparently, the applied formalism\nenables one to express the partition function Zof the spin-\n1/2 Ising-Heisenberg orthogonal-dimer chain in terms of four\neigenvalues \u00151,\u00152,\u00153,\u00154of the 4\u00024 transfer matrix T, whose\nelements are formed by 16 Boltzmann’s weights corresponding\nto all available states of two adjacent Ising spin dimers from the\nith six-spin cluster (see Fig. 1) as defined by the formula:\nT\u0000\u001bz\n1;i;\u001bz\n2;i;\u001bz\n1;i+1;\u001bz\n2;i+1\u0001=2exp\"\n\u0000\fJ0\nI\n2\u0000\u001bz\n1;i\u001bz\n2;i+\u001bz\n1;i+1\u001bz\n2;i+1\u0001#\n\u0002exp\"\fJH\n4+\fhI\n2\u0000\u001bz\n1;i+\u001bz\n2;i+\u001bz\n1;i+1+\u001bz\n2;i+1\u0001#\n\u0002(\nexp\u0012\n\u0000\fJH\n2\u0013\ncosh\u0014\fJI\n2\u0000\u001bz\n1;i+\u001bz\n2;i+\u001bz\n1;i+1+\u001bz\n2;i+1\u0001\u0000\fhH\u0015\n+cosh\"\f\n2q\nJ2\nI\u0000\u001bz\n1;i+\u001bz\n2;i\u0000\u001bz\n1;i+1\u0000\u001bz\n2;i+1\u00012+(JH\u0001)2#)\n:(7)\nFrom the physical point of view, the transfer matrix (7) rep-\nresents the e \u000bective Boltzmann’s factor, which was obtained\nafter tracing out spin degrees of freedom of two Heisenberg\nspins from the ith horizontal dimer. Explicit expressions of the\ntransfer-matrix eigenvalues emerging in the final form of the\npartition function (6) are:\n\u00151=0; (8a)\n\u0015j=a\n3+2\n3sgn(q)ppcos26666641\n3tan\u000010BBBBB@p\np3\u0000q2\nq1CCCCCA+2\u0019(j\u00002)\n33777775\n(j=2;3;4);(8b)\nwhere:\na=A1+2A0+A\u00001;\np=a2\u00003(A1A\u00001+2A0A\u00001+2A0A1\u00002B2\n1\u00002B2\n\u00001\u0000B2\n0);q=a3\u00009a(A1A\u00001+2A0A\u00001+2A0A1\u00002B2\n1\u00002B2\n\u00001\u0000B2\n0)\n+27(A1A0A\u00001\u0000A\u00001B2\n1\u0000A1B2\n\u00001\u0000A0B2\n0+2B1B0B\u00001):\nThe coe \u000ecients AxandBx(x=\u00001;0;1) entering into the for-\nmula (8b) either directly, or through the parameters p,q, are\ngiven by:\nAx=2exp\"\fJH\n4+\fJ0\nI(\u00001)x\n4+\fhIx#\n\u0002\"\n2exp\u0012\n\u0000\fJH\n2\u0013\ncosh (\fJIx\u0000\fhH)+cosh \fJH\u0001\n2!#\n;\nBx=2exp266664\fJH\n4+\fJ0\nI(x2\u00001)\n4+\fhIx\n2377775\n\u0002(\nexp\u0012\n\u0000\fJH\n2\u0013\ncosh\u0012\fJIx\n2\u0000\fhH\u0013\n+cosh\u0014\f\n2q\nJ2\nI(x2\u00002)2+(JH\u0001)2\u0015)\n:\nAfter the explicit form of the transfer-matrix eigenvalues (8a)-\n(8b) is substituted into the final expression for the partition\nfunction (6) one obtains a crucial result of our calculations,\nfrom which the whole thermodynamics of the spin-1 /2 Ising-\nHeisenberg orthogonal-dimer chain directly follows. As a mat-\nter of fact, the Gibbs free energy Gper elementary unit cell can\nbe expressed in the thermodynamic limit N!1 in terms of\nthe largest transfer-matrix eigenvalue:\nG=\u0000kBTlim\nN!11\nNlnZ=\u0000kBTln(maxf\u00150;\u00151;\u00152;\u00153g):(9)\nOther important physical quantities, such as the local magne-\ntization mI=hˆ\u001bz\n1;i+ˆ\u001bz\n2;ii=2,mH=hˆSz\n1;i+ˆSz\n2;ii=2 per Ising\nand Heisenberg spin, respectively, the total magnetization m\nper lattice site, as well as the pair correlation functions cz\nII=\nhˆ\u001bz\n1;iˆ\u001bz\n2;ii,cxx(yy)\nHH=hˆSx\n1;iˆSx\n2;ii=hˆSy\n1;iˆSy\n2;ii,czz\nHH=hˆSz\n1;iˆSz\n2;iiand\nczz\nIH=hˆSz\n1;iˆ\u001bz\n1;ii=hˆSz\n1;iˆ\u001bz\n2;ii=hˆSz\n2;iˆ\u001bz\n1;i+1i=hˆSz\n2;iˆ\u001bz\n2;i+1i, which\nbring insight into the local order of the nearest-neighbour spins\ncan be subsequently obtained by means of the di \u000berential cal-\nculus:\nmI=\u00001\n2@G\n@hI;mH=\u00001\n2@G\n@hH;m=1\n2(mI+mH);(10)\nczz\nII=@G\n@J0\nI; cxx(yy)\nHH=1\n2JH@G\n@\u0001; (11)\nczz\nHH=@G\n@JH\u0000\u0001\nJH@G\n@\u0001;czz\nIH=1\n4@G\n@JI: (12)\nThe knowledge of rigorous results for the local magnetization\nmHand pair correlation functions cxx(yy)\nHHandczz\nHHcorresponding\nto the Heisenberg spin pairs gives the opportunity to rigorously\ncalculate an interesting physical quantity called concurrence ac-\ncording to the formula [28–31]:\nC=max8>>><>>>:0;4jcxx(yy)\nHHj\u00002s\u00121\n4+czz\nHH\u00132\n\u0000m2\nH9>>>=>>>;: (13)\nThe quantity (13) represents a feasible measure of bipartite\nentanglement of the Heisenberg spins forming the horizontal\ndimers at zero as well as nonzero temperatures.\n33. Results and discussion\nIn this section, we will discuss a diversity of zero-tempera-\nture spin arrangements, magnetization process and bipartite en-\ntanglement of the particular version of the quantum spin-1 =2\nIsing-Heisenberg orthogonal-dimer chain with the antiferromag-\nnetic exchange interactions JH>0,JI>0 and J0\nI>0. Without\nloss of generality, we will restrict our analysis to the special\ncase of the orthogonal-dimer chain with the isotropic Heisen-\nberg intra-dimer interaction (the case \u0001 =1), which exhibits all\ngeneric features of the more general quantum spin-1 =2 Ising-\nHeisenberg orthogonal-dimer chain with the anisotropic XXZ\nHeisenberg intra-dimer interaction with \u0001,1. To reduce a\nnumber of free parameters, we will also assume the same mag-\nnetic fields acting on the Ising and Heisenberg spins hI=hH=\nh, which corresponds to setting the same g-factors for these\nspins from the physical point of view.\n3.1. Ground-state phase diagram\nLet us start by analyzing possible zero-temperature spin ar-\nrangements of the considered quantum spin chain. By compar-\ning the eigenvalues (4a)–(4d) for all available configurations of\nthe Ising spins from ith and ( i+1)st vertical dimers one can\nidentify in total six di \u000berent ground states specified below:\n(i) The ferromagnetic (FM) phase – the unique classical phase\nwith the perfect ferromagnetic arrangement of all the Ising\nand Heisenberg spins:\njFMi=NY\ni=1\f\f\f\"\n\"E\ni\nj\"\"i i: (14)\nThe energy is: EFM=N\n4\u0000JH+J0\nI+4JI\u00008h\u0001;\n(ii) The frustrated ferrimagnetic (FI) phase – the macroscop-\nically degenerate (2N) ferrimagnetic phase with the anti-\nferromagnetic spin arrangement on the vertical Ising di-\nmers and the ferromagnetic spin arrangement on the hor-\nizontal Heisenberg dimers:\njFIi=NY\ni=1\f\f\f\"\n#E\ni\u0010\nor\f\f\f#\n\"E\ni\u0011\n\nj\"\"i i: (15)\nThe energy is: EFI=N\n4\u0000JH\u0000J0\nI\u00004h\u0001;\n(iii) The singlet ferrimagnetic (SFI) phase – the unique quan-\ntum phase with the ferromagnetic spin arrangement of the\nvertical Ising dimers and the fully entangled singlet state\nof the horizontal Heisenberg dimers:\njSFIi=NY\ni=1\f\f\f\"\n\"E\ni\n1p\n2\u0010\nj\"#i i\u0000j#\"i i\u0011\n: (16)\nThe energy is: ESFI=\u0000N\n4\u00003JH\u0000J0\nI+4h\u0001;(iv) The singlet antiferromagnetic (SAF) phase – the macro-\nscopically degenerate (2N) antiferromagnetic phase with\nthe perfect antiferromagnetic spin arrangement of the ver-\ntical Ising dimers and the fully entangled singlet state of\nthe horizontal Heisenberg dimers:\njSAFi=NY\ni=1\f\f\f\"\n#E\ni\u0010\nor\f\f\f#\n\"E\ni\u0011\n\n1p\n2\u0010\nj\"#i i\u0000j#\"i i\u0011\n:(17)\nThe energy is: ESAF=\u0000N\n4\u00003JH+J0\nI\u0001;\n(v) The modulated ferrimagnetic (MFI) phase – the macro-\nscopically degenerate (2N=2) phase characterized by a reg-\nular alternation of the ferromagnetically and antiferro-\nmagnetically ordered vertical Ising dimers and the sing-\nlet-like state of the horizontal Heisenberg dimers:\njMFIi=N=2Y\ni=1\f\f\f\"\n\"E\n2i\u00001\n\u0010\nsin'1j\"#i 2i\u00001\u0000cos'1j#\"i 2i\u00001\u0011\n\n\f\f\f\"\n#E\n2i\u0010\nor\f\f\f#\n\"E\n2i\u0011\n\n\u0010\ncos'1j\"#i 2i\u0000sin'1j#\"i 2i\u0011\n:(18)\nThe energy is: EMFI=\u0000N\n4\u0012\nJH+2q\nJ2\nI+J2\nH+2h\u0013\n;\n(vi) The modulated antiferromagnetic (MAF) phase – the two-\nfold degenerate phase characterized by a regular alterna-\ntion of two kinds of fully polarized vertical Ising dimers\nand the other singlet-like state of the horizontal Heisen-\nberg dimers:\njMAFi=N=2Y\ni=1\f\f\f\"\n\"E\n2i\u00001\n\u0010\nsin'2j\"#i 2i\u00001\u0000cos'2j#\"i 2i\u00001\u0011\n\n\f\f\f#\n#E\n2i\n\u0010\ncos'2j\"#i 2i\u0000sin'2j#\"i 2i\u0011\n:(19)\nThe energy is: EMAF=\u0000N\n4\u0012\nJH+2q\n4J2\nI+J2\nH\u0000J0\nI\u0013\n:\nNote that the two-site ket vectors with vertically written ar-\nrows in the eigenvectors (14)–(19) determine spin arrangements\nwithin the vertical Ising dimers, while the ones with horizon-\ntally written arrows determine spin arrangements within the\nhorizontal Heisenberg dimers. Up (down) arrow refers to the\nspin state 1 =2 (\u00001=2) in both kinds of ket vectors. The mix-\ning angles'1and'2in the last two eigenvectors (18) and (19),\nwhich determine a degree of quantum entanglement within the\nhorizontal Heisenberg spin dimers in the MAF and MFI phases,\nrespectively, are given by the relation tan (2'n)=JH=(nJI)\n(n=1;2).\nThe overall ground-state behavior of the investigated spin-\n1=2 Ising-Heisenberg orthogonal-dimer chain is depicted in\nFig. 2 in the parameter planes J0\nI=JI\u0000JH=JIforh=JI=0\n(panel a) and J0\nI=JI\u0000h=JIfor three representative values of the\ninteraction ratio JH=JI=1=4,p\n2=2, 1 (panels b-d). Black solid\nlines in the displayed phase diagrams indicate first-order (dis-\ncontinuous) phase transitions between the coexisting phases.\n4/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s83/s65/s70/s74\n/s72/s32/s47/s32/s74\n/s73\n/s74\n/s73/s39/s32 /s47/s32/s74\n/s73/s77/s65/s70\n/s97/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s83/s65/s70/s104/s32/s47/s32/s74\n/s73\n/s74\n/s73/s39/s32 /s47/s32/s74\n/s73/s70/s77\n/s77/s65/s70/s77/s70/s73/s83/s70/s73\n/s98/s70/s73\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s83/s65/s70/s104/s32/s47/s32/s74\n/s73\n/s74\n/s73/s39/s32 /s47/s32/s74\n/s73/s70/s73\n/s77/s65/s70/s77/s70/s73/s83/s70/s73\n/s99/s70/s77\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s83/s65/s70/s104/s32/s47/s32/s74\n/s73\n/s74\n/s73/s39/s32 /s47/s32/s74\n/s73/s70/s77\n/s70/s73\n/s77/s65/s70/s77/s70/s73/s83/s70/s73\n/s100/s48/s46/s48/s48/s48/s48/s46/s49/s48/s53/s48/s46/s50/s49/s48/s48/s46/s51/s49/s53/s48/s46/s52/s50/s48/s48/s46/s53/s50/s53/s48/s46/s54/s51/s48/s48/s46/s55/s51/s53/s48/s46/s56/s52/s48/s48/s46/s57/s52/s53/s49/s46/s48/s48/s48/s67Figure 2: The ground-state phase diagrams of the spin-1 =2 Ising-Heisenberg orthogonal-dimer chain in the J0\nI=JI\u0000JH=JIparameter plane by assuming zero magnetic\nfield h=JI=0 (panel a) and in the J0\nI=JI\u0000h=JIparameter plane for three representative values of the interaction ratio JH=JI=0:25 (panel b), JH=JI=p\n2=2\n(panel c) and JH=JI=1 (panel d). The figures are supplemented with a density plot of the concurrence Cmeasuring a bipartite entanglement within the horizontal\nHeisenberg dimers.\nThey were analytically calculated by comparing the ground-\nstate energies corresponding to the eigenvectors listed in\nEqs. (14)–(19). As one can see from Fig. 2a, only two quan-\ntum phases SAF and MAF emerge as possible ground states at\nzero magnetic field h=JI=0. The SAF phase is stable in the\nparameter region J0\nI=JI>p\n4+(JH=JI)2\u0000JH=JI, where the\npredominant intra-dimer coupling J0\nImaintains the antiparallel\nspin alignment of the vertical Ising dimers and the intra-dimer\ninteraction JHis strong enough to create the fully entangled\nsinglet-dimer state on all horizontal Heisenberg dimers. In the\nrest of the parameter space, the peculiar MAF phase with a two-\nfold broken translational symmetry due to a regular alternation\nof two kinds of fully polarized vertical Ising dimers and two\nalternating kinds of singlet-like states of the horizontal Heisen-\nberg dimers appears. It is noteworthy that the total magnetiza-\ntion is zero within both zero-field ground states SAF and MAF.\nThe situation becomes more complex after turning on the\nexternal magnetic field. Besides the SAF and MAF phases,\nthree ferrimagnetic phases SFI, MFI and FI with nonzero mag-\nnetization can be observed in addition to the fully polarized\nFM phase due to a mutual interplay between the applied mag-netic field hand the coupling constants JH,J0\nI,JI. It is obvious\nfrom Figs. 2b-d that the parameter regions corresponding to the\nquantum phases SAF and SFI (MAF) are gradually extended\n(reduced) upon increasing value of the interaction ratio JH=JI\npromoting existence of the singlet-dimer state on the horizon-\ntal bonds, while the classical FI and FM phases are contrarily\nshifted towards higher magnetic fields. As a result, both spin\narrangements inherent to SAF and SFI phases simultaneously\nappear in a zero-temperature magnetization process for mod-\nerate values of the interaction ratio J0\nI=JI2\u0000p\n4+(JH=JI)2\u0000\nJH=JI;2JH=JI\u0001after a relative strength of the Heisenberg intra-\ndimer coupling exceeds the value JH=JI=p\n2=2 (see Fig. 2d).\nLast but not least, the evolution of the parallelogram-shaped pa-\nrameter space corresponding to the MFI phase, which emerges\nat moderate values of the interaction ratio J0\nI=JIand the mag-\nnetic field h=JIfaithfully follows the trend of adjacent phases\nMAF, SFI, SAF, and FI: the phase boundaries MFI–SFI and\nSAF–MFI are gradually prolonged, while the ones MAF–MFI,\nMFI–FI are gradually shortened upon increasing of the interac-\ntion ratio JH=JI. For the particular value JH=JI=p\n2=2, the\nparameter region corresponding to the MFI phase has the shape\n5of a rhombus with the shorter diagonal parallel to the field-axis\n(see Fig. 2c).\n3.2. Magnetization process at zero and nonzero temperatures\nThe observed diversity of the ground states suggests vari-\nous magnetization scenarios at zero temperature. In fact, the\nzero-temperature magnetization curves of the studied spin-1 =2\nIsing-Heisenberg orthogonal-dimer chain may exhibit the zero\nplateau as well as intermediate magnetization plateaus at one-\nquarter, one-half and three-quarters of the saturation magnetiza-\ntion according to the Oshikawa-Yamanaka-A \u000feck rule [32, 33]\nas long as the period doubling of a magnetic unit cell is consid-\nered. In accordance with this rule, the plateau at zero magneti-\nzation is pertinent either to SAF or MAF ground state, the in-\ntermediate 1 /4-plateau corresponds to the MFI ground state, the\nintermediate 1 /2-plateau relates either to the SFI or FI ground\nstate, while the last possible intermediate 3 /4-plateau does not\nemerge in general. A comprehensive view of the situation is\nprovided by three-dimensional (3D) plots of the total magne-\ntization mnormalized with respect to its saturation value msat,\nwhich are depicted in Fig. 3 against the magnetic field h=JIand\nthe interaction ratio J0\nI=JIby assuming either zero temperature\nkBT=JI=0 (left panels) or su \u000eciently small but finite tempera-\nturekBT=JI=0:1 (right panels). For the sake of a comparison,\nthe interaction ratio JH=JIis fixed to the same values as used\nfor a construction of the ground-state phase diagrams depicted\nin Figs. 2b-d. Obviously, the zero-temperature magnetization\ncurves plotted in left panels of Fig. 3 reflect up to six di \u000berent\nsequences of the field-driven phase transitions depending on a\nmutual interplay between the intra- and inter-dimer coupling\nconstants JH,JI,J0\nI:\n(i) MAF!SFI!FM ,\n(ii) MAF!MFI!SFI!FM ,\n(iii) MAF!MFI!FI!FM ,\n(iv) SAF!MFI!FI!FM ,\n(v) SAF!FI!FM ,\n(vi) SAF!MFI!SFI!FM .\nIn agreement with the aforementioned ground-state analysis,\nthe sequences of the field-induced phase transitions MAF !\nSFI!FM, MAF!MFI!SFI!FM, SAF!MFI!FI!\nFM and SAF!FI!FM emerge in the zero-temperature mag-\nnetization curves for any value of the interaction ratio JH=JI,\nwhile the ones MAF !MFI!FI!FM and SAF!MFI\n!SFI!FM can be identified in the magnetization process\nonly for JH=JIp\n2=2, respectively (see\nFigs. 3a1 and 3c1). Of course, the actual magnetization plateaus\nand discontinuous magnetization jumps at the critical fields,\nwhich correspond to individual field-induced phase transitions,\ncan be detected merely at zero temperature, because any finite\ntemperature completely wipes a discontinuity in the magnetiza-\ntion and also diminishes the perfect plateaus from the respective\nisothermal magnetization curves (see right panels of Fig. 3).In general, the staircase character of the magnetization curves\nis gradually smoothing upon increasing of temperature until it\ncompletely vanishes due to su \u000eciently strong thermal fluctua-\ntions.\n3.3. Bipartite quantum entanglement\nTo gain a better insight into a degree of bipartite quantum\nentanglement emergent within the pure states of the horizontal\nHeisenberg spin dimers, the zero-temperature phase diagrams\ndisplayed in Fig. 2 are supplemented with the corresponding\ndensity plots of the concurrence calculated according to Eq. (13).\nAs expected, the concurrence Cbecomes non-zero in all quan-\ntum ground states SAF, MAF, SFI, and MFI, while it equals\nzero within two classical ground states FM and FI phases with a\nfull alignment of the horizontal Heisenberg dimers towards the\nmagnetic field [see the respective eigenvectors (14) and (15)].\nThe quantum phases SAF, SFI, MAF and MFI generally show\na di\u000berent strength of the bipartite quantum entanglement as\nevidenced by zero-temperature asymptotic values of the con-\ncurrence:\nCSAF=CSFI=1; CMAF=JH=JIp\n4+(JH=JI)2;\nCMFI=JH=JIp\n1+(JH=JI)2: (20)\nIt can be understood from Eq. (20) as well as the density plots\nshown in Fig. 2 that the Heisenberg dimers residing on the hor-\nizontal bonds are fully entangled only within the SAF and SFI\nground states. On the other hand, the strength of the bipar-\ntite quantum entanglement within the MAF and MFI ground\nstates basically depends on a relative strength of the Heisenberg\nand Ising intra-dimer interactions. More specifically, the higher\nvalue the interaction ratio JH=JItakes, the more strongly en-\ntangled the horizontal Heisenberg dimers are. It could be gen-\nerally concluded that the Heisenberg dimers generally display\na stronger quantum entanglement in the MFI phase than in the\nMAF phase when assuming the same value of the interaction\nratio JH=JI. In contrast to the ground states SAF and SFI with\na perfect quantum entanglement of the horizontal Heisenberg\ndimers, the perfect bipartite quantum entanglement cannot be\nreached neither in MAF nor in MFI phase for any finite value\nof the interaction ratio JH=JI, because the Heisenberg spin pairs\nreside in the outstanding singlet-like states instead of a perfect\nsinglet-dimer state [cf. the eigenvectors (18) and (19) with the\nones (16) and (17)].\n3.4. Bipartite thermal entanglement\nLast but not least, let us turn our attention to a detailed ex-\namination of the bipartite thermal entanglement, which refers\nto a bipartite entanglement emergent within the mixed states of\nthe horizontal Heisenberg dimers at nonzero temperatures. The\noverall picture of this issue can easily be created from density\nplots of the concurrence Calong with its magnetic-field and\ntemperature dependencies, which are depicted in Figs. 4–6 for\nthe particular value of the interaction ratio JH=JI=p\n2=2 and\nfour di \u000berent values of the interaction ratio J0\nI=JI=1=4;1;2\n6Figure 3: 3D plots of the total magnetization mreduced with respect to its saturation value msatas a function of the magnetic field h=JIand the interaction ratio J0\nI=JI\nfor three fixed values of the interaction ratio JH=JI=1=4 (panels a1, a2), JH=JI=p\n2=2 (panels b1, b2) and JH=JI=1 (panels c1, c2) at two di \u000berent temperatures\nkBT=JI=0 (left panels) and kBT=JI=0:1 (right panels). The curves of distinct colors refer to di \u000berent magnetization scenarios, which can be observed in the\nindividual 3D plots.\n7/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s83/s70/s73/s107\n/s66/s84/s32/s47/s32/s74\n/s73\n/s104/s32 /s47/s32/s74\n/s73/s77/s65/s70\n/s97/s70/s77\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s83/s65/s70/s107\n/s66/s84/s32/s47/s32/s74\n/s73\n/s104/s32 /s47/s32/s74\n/s73/s70/s77 /s77/s65/s70 /s77/s70/s73 /s83/s70/s73\n/s98\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s107\n/s66/s84/s32/s47/s32/s74\n/s73\n/s104/s32 /s47/s32/s74\n/s73/s83/s65/s70 /s77/s70/s73\n/s99/s70/s77 /s70/s73\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s48/s46/s48/s48/s48/s48/s46/s49/s48/s53/s48/s46/s50/s49/s48/s48/s46/s51/s49/s53/s48/s46/s52/s50/s48/s48/s46/s53/s50/s53/s48/s46/s54/s51/s48/s48/s46/s55/s51/s53/s48/s46/s56/s52/s48/s48/s46/s57/s52/s53/s49/s46/s48/s48/s48\n/s83/s65/s70/s107\n/s66/s84/s32/s47/s32/s74\n/s73\n/s104/s32 /s47/s32/s74\n/s73/s83/s65/s70\n/s100/s67\n/s70/s73Figure 4: The density plots of the concurrence Cin the h=JI\u0000kBT=JIparameter plane for the fixed values of the interaction ratio JH=JI=p\n2=2 and J0\nI=JI=1=4\n(panel a), J0\nI=JI=1 (panel b), J0\nI=JI=2 (panel c), J0\nI=JI=3 (panel d).\nand 3. As one could expect, the displayed data for the con-\ncurrence at low enough temperatures kBT=JI.0:05 faithfully\nresemble zero-temperature asymptotic values, which were dis-\ncussed in above by the ground-state analysis. For the inter-\naction ratios J0\nI=JI=1=4 and 1 the concurrence Cfirst in-\ncreases upon increasing of the magnetic field near the critical\nfields hc1=JI\u00190:7071 (for J0\nI=JI=1=4) and hc1=JI\u00190:3966,\nhc2=JI\u00191:0176 (for J0\nI=JI=1) due to a strengthening of the bi-\npartite entanglement within the horizontal dimers near the field-\ninduced phase transitions MAF !SFI and MAF!MFI,\nMFI!SFI, respectively. The second (third) field-induced\nphase transition SFI !FM, which can be found for both partic-\nular values of the interaction ratio J0\nI=JIat the same saturation\nfield hc2(3)=JI\u00191:7071 is responsible for a sudden drop of the\nconcurrence Cto zero, which confirms a breakdown of the bi-\npartite entanglement (see Figs. 4a,b and 5a,b). On the other\nhand, the bipartite entanglement of the horizontal Heisenberg\ndimers generally weakens for J0\nI=JI=2 along the whole mag-\nnetization process. In fact, the first rapid decrease of the con-\ncurrence observable around the critical field hc1=JI\u00190:4824 is\nattributable to the field-induced transition SAF !MFI, whilethe second abrupt decline of the concurrence associated with a\ncomplete breakdown of the concurrence emerges near the crit-\nical field hc2=JI\u00190:9319 of the field-driven phase transition\nMFI!FI (see Figs. 4c and 5c). Finally, the bipartite ther-\nmal entanglement disappears upon strengthening of the mag-\nnetic field according to the most standard scheme for the high-\nest value of the interaction ratio J0\nI=JI=3 illustrated in Figs. 4d\nand 5d. A breakdown of the concurrence already appears in a\nvicinity of the first critical field hc1=JI\u00190:7071, which corre-\nsponds to the field-induced phase transition from the quantum\nground state SAF to the classical one FI (see Fig. 4d and also\nFig. 5d).\nOf course, an increase in temperature causes a gradual\nsmoothing of abrupt changes of the concurrence observable at\nlow enough temperatures in a proximity of the critical fields, be-\ncause the bipartite entanglement between the Heisenberg spin\npairs is in general suppressed by thermal fluctuations above\nall quantum ground states (see Fig. 5). However, it should\nbe also mentioned that a small temperature rise may eventu-\nally invoke a gentle strengthening of the thermal entanglement.\nAs a matter of fact, the concurrence Cmay exhibit an out-\n8/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s107\n/s66/s84/s32 /s47/s32/s74\n/s73\n/s32/s32/s48/s46/s48/s53\n/s32/s32/s48/s46/s50/s48\n/s32/s32/s48/s46/s51/s53\n/s32/s32/s48/s46/s53/s48/s67\n/s104/s32 /s47/s32/s74\n/s73/s97/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s107\n/s66/s84/s32 /s47/s32/s74\n/s73\n/s32/s32/s48/s46/s48/s53\n/s32/s32/s48/s46/s50/s48\n/s32/s32/s48/s46/s51/s53\n/s32/s32/s48/s46/s53/s48\n/s83/s65/s70/s67\n/s104/s32 /s47/s32/s74\n/s73/s83/s70/s73\n/s98\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s107\n/s66/s84/s32 /s47/s32/s74\n/s73\n/s32/s32/s48/s46/s48/s53\n/s32/s32/s48/s46/s50/s48\n/s32/s32/s48/s46/s51/s53\n/s32/s32/s48/s46/s53/s48\n/s83/s65/s70/s67\n/s104/s32 /s47/s32/s74\n/s73/s99/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s107\n/s66/s84/s32 /s47/s32/s74\n/s73\n/s32/s32/s48/s46/s48/s53\n/s32/s32/s48/s46/s50/s48\n/s32/s32/s48/s46/s51/s53\n/s32/s32/s48/s46/s53/s48\n/s83/s65/s70/s67\n/s104/s32 /s47/s32/s74\n/s73/s100Figure 5: The magnetic-field dependencies of the concurrence Cfor the fixed values of the interaction ratio JH=JI=p\n2=2 and J0\nI=JI=1=4 (panel a), J0\nI=JI=1\n(panel b), J0\nI=JI=2 (panel c), J0\nI=JI=3 (panel d) by assuming four di \u000berent values of temperature kBT=JI.\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s32 /s104/s32 /s47/s32/s74\n/s73\n/s32/s32/s48/s46/s49\n/s32/s32/s48/s46/s54\n/s32/s32/s48/s46/s55/s48/s55/s49\n/s32/s32/s49/s46/s48\n/s32/s32/s49/s46/s55/s48/s55/s49\n/s32/s32/s49/s46/s56\n/s32/s32/s50/s46/s53/s67\n/s107\n/s66/s84/s32 /s47/s32/s74\n/s73/s97/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s32 /s104/s32 /s47/s32/s74\n/s73\n/s32/s32/s48/s46/s49\n/s32/s32/s48/s46/s54\n/s32/s32/s48/s46/s55/s48/s55/s49\n/s32/s32/s48/s46/s56\n/s32/s32/s49/s46/s53/s67\n/s107\n/s66/s84/s32 /s47/s32/s74\n/s73/s98\nFigure 6: The temperature dependencies of the concurrence Cfor the fixed values of the interaction ratio JH=JI=p\n2=2 and J0\nI=JI=0:25 (panel a), J0\nI=JI=3\n(panel b) by assuming a few di \u000berent values of the external magnetic field h=JI.\nstanding temperature-induced rise on account of thermal ex-\ncitations from a less entangled quantum ground state towardsa more entangled excited state (see for instance the low-field\nparts of solid red and dashed green curves corresponding to\n9the temperatures kBT=JI=0:05 and 0:2 in Figs. 5a,b and the\ndashed green curve for the magnetic field h=JI=0:6 in Fig. 6a.\nFurthermore, the density plots of the concurrence along with\nthe magnetic-field and temperature dependencies depicted in\nFigs. 4–6 clearly evidence that the thermal entanglement of the\nhorizontal Heisenberg dimers, although relatively weak, is also\ndetectable at nonzero temperatures above the classical FI and\nFM ground states. This peculiar finding can be repeatedly ex-\nplained in terms of a thermal activation of the entangled low-\nlying excited states related to some of the quantum phases SFI,\nMFI, MAF or SAF.\n4. Conclusion\nIn the present work we have introduced and exactly solved\na spin-1 /2 Ising-Heisenberg orthogonal-dimer chain, which is\ncomposed of regularly alternating Ising and Heisenberg dimers\nplaced in an external magnetic field. After tracing out spin\ndegrees of freedom of the Heisenberg dimers, the considered\nquantum spin chain has been rigorously treated by making use\nof the classical transfer-matrix approach. It is shown that the\nground-state phase diagram involves in total six di \u000berent ground\nstates. In addition to the classical ferromagnetic phase emergent\nabove the saturation field one also encounters two ground states\nwith zero total magnetization referred to as the singlet and mod-\nulated antiferromagnetic phases, two ground states with the to-\ntal magnetization equal to a half of the saturation value referred\nto as the frustrated ferrimagnetic phase and the singlet ferrimag-\nnetic phase and, finally, one peculiar ground state with the total\nmagnetization equal to a quarter of the saturation value referred\nto as the modulated ferrimagnetic phase. It is also evidenced\nthat the diversity of the ground states gives rise to six di \u000ber-\nent magnetization scenarios depending on a mutual interplay of\nthree considered coupling constants.\nA particular attention has been paid to quantification of the\nbipartite entanglement within the pure and mixed states of the\nhorizontal Heisenberg dimers at zero and nonzero temperatures\nwith the help of concurrence. Surprisingly, it turns out that\nthe bipartite entanglement may be reinforced by increasing of\ntemperature or even upon strengthening of the magnetic field,\nwhich is in contrast with general expectations. In addition, the\nbipartite thermal entanglement has been identified also above\ntwo classical phases: the ferromagnetic phase and the frustrated\nferrimagnetic phase. This unexpected finding can be ascribed to\nthermal excitations driving the investigated quantum spin chain\nfrom a pure classical ground state to a mixed state incorporating\nlow-lying excited state(s) closely connected to other quantum\nground states.\nAlthough the magnetic structure of the investigated spin-1 /2\nIsing-Heisenberg orthogonal-dimer chain was inspired by a het-\nerobimetallic backbone of the coordination polymer [Dy 2Cu2]n,\nthe substantial di \u000berence between the Land ´e g-factors of Dy3+\nand Cu2+magnetic ions ( gDy\u001920 vs. gCu\u00192:2) precludes a\nstraightforward comparison of the obtained theoretical results\nwith the available experimental data [21, 22]. The investiga-\ntion of the di \u000berence of the respective Land ´e g-factors alongwith anisotropy of the couplings constants is accordingly left as\nfuture task for our forthcoming study.\nAcknowledgment\nThis work was financially supported by the grant of The\nMinistry of Education, Science, Research and Sport of the Slo-\nvak Republic under the contract No. VEGA 1 /0105/20 and by\nthe grant of the Slovak Research and Development Agency un-\nder the contract No. APVV-16-0186.\nReferences\n[1] H. Kageyama, K. Yoshimura, R. Stern, N. V . Mushnikov, K. Onizuka, M.\nKato, K. Kosuge, C. P. Slichter, T. Goto, Y . Ueda, Phys. Rev. Lett. 82\n(1999) 3168.\n[2] K. Onizuka, H. Kageyama, Y . Narumi, K. Kindo, Y . Ueda, T. Goto, J.\nPhys. Soc. Jpn. 69(2000) 1016.\n[3] H. Kageyama, Y . Narumi, K. Kindo, K. Onizuka, Y . Ueda, T. Goto, J.\nAlloys Compd. 317–318(2001) 177.\n[4] S.E. Sebastian, N. Harrison, P. Sengupta, C.D. Batista, S. Francoual, E.\nPalm, T. Murphy, N. Marcano, H.A. Dabkowska, B.D. Gaulin, Proc. Natl.\nAcad. Sci. USA 105(2008) 20157.\n[5] Y . H. Matsuda, N. Abe, S. Takeyama, H. Kageyama, P. Corboz, A. Ho-\nnecker, S. R. Manmana, G. R. Foltin, K. P. Schmidt, F. Mila, Phys. Rev.\nLett. 111(2013) 137204.\n[6] B.S. Shastry, B. Sutherland, Physica B +C108(1981) 1069.\n[7] S. Miyahara, K. Ueda, Phys. Rev. Lett. 82(1999) 3701.\n[8] J. Dorier, K.P. Schmidt, F. Mila, Phys. Rev. Lett. 101(2008) 250402.\n[9] M. Nemec, G.R. Foltin, K.P. Schmidt, Phys. Rev. B 86(2012) 174425.\n[10] P. Corboz, F. Mila, Phys. Rev. Lett. 112(2014) 147203.\n[11] T. Verkholyak, J. Stre ˇcka, F. Mila, K.P. Schmidt, Phys. Rev. B 90(2014)\n134413.\n[12] G.R. Foltin, S.R. Manmana, K.P. Schmidt, Phys. Rev. B 90(2014)\n104404.\n[13] D.A. Schneider, K. Coester, F. Mila, K.P. Schmidt, Phys. Rev. B 93(2016)\n241107(R).\n[14] S. Wessel, I. Niesen, J. Stapmanns, B. Normand, F. Mila, P. Corboz, A.\nHonecker, Phys. Rev. B 98(2018) 174432.\n[15] A. Wietek, P. Corboz, S. Wessel, B. Normand, F. Mila, A. Honecker,\nPhys. Rev. Research 1(2019) 033038.\n[16] J. Richter, N.B. Ivanov, Czech. J. Phys. Suppl. 46(1996) 1919.\n[17] N.B. Ivanov, J. Richter, Phys. Lett. A 232(1997) 308.\n[18] J. Richter, N. B. Ivanov, J. Schulenburg, J. Phys.: Condens. Matter 10\n(1998) 3635.\n[19] J. Schulenburg, J. Richter, Phys. Rev. B 65(2002) 054420.\n[20] J. Schulenburg, J. Richter, Phys. Rev. B 66(2002) 134419.\n[21] S. Ueki, A. Okazawa, T. Ishida, T. Nogami, H. Nojiri, Polyhedron 26\n(2007) 1970.\n[22] A. Okazawa, T. Nogami, H. Nojiri, T. Ishida, Chem. Mater. 20(2008)\n3110.\n[23] H.G. Paulinelli, S. M. de Souza, O. Rojas, J. Phys.: Condens. Matter 25\n(2013) 306003.\n[24] T. Verkholyak, J. Stre ˇcka, Phys. Rev. B 88(2013) 134419.\n[25] T. Verkholyak, J. Stre ˇcka, Acta Phys. Polonica A 126(2014) 22.\n[26] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic\nPress, New York, 1982.\n[27] J. Stre ˇcka, M. Ja ˇsˇcur, Acta Physics Slovaca 65(2015) 235.\n[28] W. K. Wooters, Phys. Rev. Lett. 80(1998) 2245.\n[29] L. Amico, A. Osterloh, F. Plastina, R. Fazio, Phys. Rev. A 69(2004)\n022304.\n[30] L. Amico, R. Fazio, A. Osterloh, V . Vedral, Rev. Mod. Phys. 80(2008)\n517.\n[31] A. Osterloh, Int. J. Mod. Phys. B 27(2013) 1345018.\n[32] M. Oshikawa, M. Yamanaka, I. A \u000feck, Phys. Rev. Lett. 78(1997) 1984.\n[33] I. A \u000feck, Phys. Rev. B 37(1998) 5186.\n10" }, { "title": "2304.13698v2.Direct_observation_of_Néel_type_skyrmions_and_domain_walls_in_a_ferrimagnetic_DyCo__3__thin_film.pdf", "content": "Direct observation of Néel-type skyrmions and domain walls in a\nferrimagnetic DyCo 3thin film\nChen Luo,1, 2Kai Chen,1, 3,∗Victor Ukleev,1Sebastian Wintz,1, 4Markus\nWeigand,1, 4Radu-Marius Abrudan,1Karel Prokeš,5and Florin Radu1,†\n1Helmholtz-Zentrum-Berlin für Materialen und Energie,\nAlbert-Einstein-Straße 15, 12489 Berlin, Germany\n2Institute of Experimental Physics of Functional Spin Systems,\nTechnical University Munich, James-Franck-Straße 1,\n85748 Garching b. München, Germany\n3National Synchrotron Radiation Laboratory,\nUniversity of Science and Technology of China, Hefei, Anhui 230029, China\n4Max-Planck-Institut für Intelligente Systeme, 70569 Stuttgart, Germany\n5Helmholtz-Zentrum-Berlin für Materialen und Energie,\nHahn-Meitner Platz 1, D-14109 Berlin, Germany\n(Dated: August 11, 2023)\n1arXiv:2304.13698v2 [cond-mat.mtrl-sci] 10 Aug 2023Abstract\nIsolated magnetic skyrmions are stable, topologically protected spin textures that are at the fore-\nfront of research interests today due to their potential applications in information technology. A\ndistinct class of skyrmion hosts are rare earth - transition metal (RE-TM) ferrimagnetic materials.\nTo date, the nature and the control of basic traits of skyrmions in these materials are not fully\nunderstood. We show that for an archetypal ferrimagnetic material DyCo 3that exhibits a strong\nperpendicular anisotropy, the ferrimagnetic skyrmion size can be tuned by an external magnetic\nfield. Moreover, by taking advantage of the high spatial resolution of scanning transmission X-ray\nmicroscopy (STXM) and utilizing a large x-ray magnetic linear dichroism (XMLD) contrast that\noccurs naturally at the RE resonant edges, we resolve the nature of the magnetic domain walls of\nferrimagnetic skyrmions. We demonstrate that through this method one can easily discriminate\nbetween Bloch and Néel type domain walls for each individual skyrmion. For all isolated ferri-\nmagnetic skyrmions, we observe that the domain walls are of Néel-type. This key information is\ncorroborated with results of micromagnetic simulations and allows us to conclude on the nature of\nthe Dzyaloshinskii–Moriya interaction (DMI) which concurs to the stabilisation of skyrmions in this\nferrimagnetic system. Establishing that an intrinsic DMI occurs in RE-TM materials will also be\nbeneficial towards a deeper understanding of chiral spin texture control in ferrimagnetic materials.\n∗kaichen2021@ustc.edu.cn\n†florin.radu@helmholtz-berlin.de\n2INTRODUCTION\nMagnetic skyrmions are stable nanoscale whirls of magnetic spin textures [1–6]. Due to\ntheir topological stability, small size at the nanometer scale and controlled mobility under\nlow current densities, skyrmions hold the promise to impact significantly next-generation\ninformation storage technology [7–16]. Initially introduced in nuclear physics as soliton\nsolutions of non-linear field equations [1, 17], they hold now a distinct place in solid state\nphysics as well, following their theoretical prediction [18] and experimental observation [19].\nBroken inversion symmetry that is characteristic to certain crystal structures induces a non-\ncollinear coupling mechanism that contributes as an asymmetric term in the Hamiltonian\ndescribing the resulting magnetically chiral ground states [20]. Under certain conditions,\nthese chiral states concur in forming magnetic skyrmions as observed in archetypal cubic\nchiral crystals that exhibit a bulk Dzyaloshinskii–Moriya interaction (DMI) [19, 21–25].\nMoreover, symmetry breaking along with spin-orbit coupling present at magnetic interfaces\nlead to a weak interfacial DMI that contributes to the stabilisation of skyrmions observed\nin thin films [26–30] and multilayers [31–36].\nSkyrmion lattices that occur in single crystals fill a small pocket in the phase diagram for\ntemperatures that typically extends over few Kelvin [19, 37]. This is detrimental to applica-\ntions which require stability over a broad range around room temperature. The temperature\npocket can be eventually extended by reducing the dimensionality of the structures or by\nengineering the interfacial DMI of ferromagnetic thin films and multilayers [38]. Yet, caused\nby the skyrmion Hall effect [39, 40], the trajectories of these ferromagnetic topological units\nin devices are not straight, being deflected away by the Lorentz forces. To overcome this\nlimitation, ferrimagnetic materials are offering an advantage due to a versatile tunability of\ntheir magnetic properties.\nRare-earth-transition-metals (RE-TM) ferrimagnetic (FiM) alloys consist two anitferro-\nmagnetically coupled sublattices. At the compensation temperature (T comp), the magne-\ntizations of both sub-lattices are equal, leading to a vanishing net magnetization, just as\nfor an antiferromagnet. By the choice of the elemental composition and through tempera-\nture variation, their magnetic properties, including T comp, net magnetization and magnetic\nanisotropy, can be easily engineered, which makes FiM materials advantageous for spintron-\nics devices [41–43]. By selecting the RE element, two classes of FiM can be distinguished,\n3namely Gd-base FiM alloys that exhibit a weak perpendicular magnetic anisotropy (PMA)\ndue to the vanishing orbital moment of the RE, and Dy, Tb, Ho-based FiM alloys which\nhave a stronger PMA due to the large orbital moment of the RE. The latter category has\nthe potential to offer a higher stability of stored information, but the reports on skyrmions\nin these systems are scarce [44].\nBy contrast, for weak PMA FiM alloys, like FeGdCo, magnetic skyrmions have been\nrecently observed [45] and they can be controlled in microstructured devices [46, 47]. Since\nthen much research has been carried out to enable control and fuctionalization of ferrimag-\nnetic skyrmions: they have been observed in ferrimagnetic confined nanostructures [48];\ntopological spin memory is reported for Co/Gd multilayers exhibiting skyrmion stabil-\nity in fully compensated antiferromagnetically coupled heterostructure [49];observation of\nspin spirals and individual skyrmions in synthetic Pt/CoGd/Pt ferrimagnetic multilayers at\nroom-temperature has been achieved without the assistance of external magnetic fields [50];\nmagneto-transportmeasurementshaverevealedatopologicalcontributionresultingfromthe\noccurrence of an interfacial DMI in Ho/CoFeGd/ β−W multilayers [51]; evidence for chiral\nferrimagnetism in an ultrathin GdCo layer has been demonstrated through a combination\nof high-resolution Lorentz microscopy and XMCD [52]; Néel-type homochirality has been\nobserved over a large temperature range in Ta/Ir/Fe/GdFeCo/Pt multilayers using scanning\nelectron microscopy with polarization analysis [53]; and information on Néel versus Bloch\nDWs can be inferred by a tilt geometry with Lorentz transmission microscopy as shown for\na Mn 3Sn topological antiferromagnet [54]. However, a direct determination of the type of\nthe spin structures, namely Néel-type versus Bloch-type, was not experimentally reported,\nfor neither of the two FiM classes.\nAn unambiguous determination of the skyrmion type is crucial for understanding the sta-\nbilization mechanism of skyrmions in these materials. Indeed, one would expect stabilization\nof Néel-type skyrmions in a thin film system with an interfacial DMI induced by engineering\nof the spin-orbit coupling of the ferrimagnetic layer with neighboring heavy-metal (HM)\nlayers [8]. However, this approach usually requires stacking ultrathin magnetic and HM\nlayers into an asymmetric periodic multilayer in order to achieve a sizeable magnitude of\nDMI and, consequently, a small enough ( ∼100nm) skyrmion size [32]. On the other hand,\n\"bulk\" DMI stabilizing chiral Bloch-type skyrmions requires an intrinsic lack of inversion\nsymmetry within the material [20]. Alternatively, skyrmion bubbles having the same topo-\n4logical charge, but degenerate chirality can also be stabilized by dipolar interactions [55].\nFurthermore, dipolar interaction can compete with an interfacial DMI and change the spin\nrotation sense from Néel to Bloch type, or give rise to hybrid spin textures [56]. Therefore,\nunveiling the skyrmion type in FiM will shed light onto the microscopic spin Hamiltonian\nin this class of thin-film FiM alloys.\nIn this study, we report real space imaging of the magnetic structures in a FiM DyCo 3\nthin film by means of scanning transmission X-ray microscopy (STXM) utilizing both x-\nray magnetic circular dichroism (XMCD) and x-ray magnetic linear dichroism (XMLD)\ncontrast [57–61]. Using XMCD-STXM, we directly observe well-isolated FiM skyrmions\nand their transformation to maze-like domains as a function of the out-of-plane external\nfield. With XMLD-STXM, we demonstrate that these FiM skyrmions are Néel-type and\nthe maze-like domains also show a preference of Néel-type domain walls. We confirm our\nexperimental results to be consistent with micromagnetic simulations. Please note that the\nexperiments reported here are performed at low temperatures (26 K) which correspond to a\nnon-fully compensated magnetization state. The fully compensated magnetization for this\nferrimagnet occurs at a temperature that is well above the room temperature, therefore a\nvanishing skyrmion Hall effect is not addressed(see Supplementary Note S1.2).\nRESULTS AND DISCUSSION\nField dependence of Skyrmion size\nIn Fig. 1a we show the hysteresis loop in perpendicular geometry which was measured\nby SQUID magnetometry. The magnetic reversal of the DyCo 3film as a function of applied\nfield was investigated at 26 K using STXM. Because the maximum magnetic field for the\nSTXM experiments is limited to 260 mT which is not enough to saturate the sample at\nlow temperatures, we saturated the sample out-of-plane at room temperature prior to the\nSTXM measurements. After cooling down the sample (in a perpendicular magnetic field of\n+260 mT) to 26 K, the STXM images were obtained with the magnetic field sweeping from\n+260 mT to -260 mT. Figure 1b shows selected examples of XMCD-STXM areal images of\nthe FiM skyrmions acquired at different perpendicular magnetic fields. One can distinguish\ntheevolutionofwell-isolatedFiMskyrmionsasafunctionofthemagneticfield, rangingfrom\n5260 mT to 140 mT (within the field range that is marked by a grey rectangle in Fig. 1a). The\naverage density and radius of the skyrmions extracted from the STXM images as a function\noffieldareshowninFig.1candinFig.1d, respectively. Itincreasesfrom2.5to9.6skyrmions\nper square micrometer when the field is decreased from 260 mT to 140 mT, indicating that\nthe skyrmions can be created and annihilated by varying the field. At the same time, the\naverage skyrmion radius Rincreases from 45 nm to 65 nm, demonstrating that the skyrmion\nsize can be reduced and inflated with the external field. When the magnetic field decreases\nfurther, the skyrmions will merge into worm- and maze-like domain structures, as shown in\nthe insets of Fig. 1a.\nLateral imaging of skyrmions with XMCD contrast\nTo resolve the details of the FiM skyrmions, high resolution XMCD-STXM images were\nrecorded at the Co L 3and the Dy M 5edges at an external field of 140 mT, as shown in Fig. 2.\nThe left panels display the images of several single skyrmions, whereas in the right panels we\nshowtheirlineprofilesalongtheXandYaxes, whichrepresentthenormalizedperpendicular\nmagnetic moment of Dy ( MDy) and Co ( MCo) elements. One can easily observe that the\nmagnetic profiles of MDyandMCooverlap nicely, which clearly demonstrate the formation\nof FiM skyrmions with an antiparallel alignment of the Dy and Co moments. At this field,\nthe radius (FWHM) of the respective skyrmion is about 65 nm.\nLateral imaging of skyrmion domain walls with XMLD contrast\nTo identify the spin structure of the domain walls of the FiM skyrmions, XMLD was\nutilized by taking advantage of the strong linear dichroism of Dy at the M 5absorption edge.\nFigure 3a presents the demonstration of X-ray absorption spectroscopy (XAS) at the Dy M 5\nedgeforcircularleft(CL),circularright(CR),linearvertical(LV)andlinearhorizontal(LH)\npolarizations, respectively. The XMCD spectrum was obtained by taking the difference of\n(CR-CL), and the XMLD spectrum was obtained by taking the difference of (LV-LH). One\ncan see that the maximum XMLD signal appears enhanced at the second peak of the Dy\nM5edge whereas the maximum XMCD signal is located at the third peak. The intensity\nof the XMLD spectra at the Dy edge is sufficiently large to be exploited for the STXM\n6measurements [61]. Unlike XMCD which is sensitive to the magnetic moments collinear to\nthe x-ray propagation direction, XMLD is sensitive to the magnetic moments collinear to\nthe− →Evector of linearly polarized X-rays, which lies in the plane of the sample surface for\nour experimental geometry. To distinguish Néel-type and Bloch-type skyrmions from each\nother, we made simulations on how the two different types of skyrmions should look like in\nthe presence of circular and linearly polarized X-rays, which are shown in Fig. 3b and 3c.\nBy comparison, one can see that our experimental data match very well with the Néel-type\ncontrast, indicating that the FiM skyrmions in our DyCo 3thin film are Néel-type skyrmions.\nLateral imaging of domain walls of a maze domain state with XMLD contrast\nAfter successfully identifying Néel-type skyrmions utilizing the advantage of XMLD, we\nalso investigated the domain walls for maze-like domains using the same technique, as shown\ninFig.4(a-c). Onecaneasilyobservethatthedomainwallsatthetop/bottomsidesaremore\npronounced for LV, and that the domain walls at the left/right sides show more intensity\nfor LH, similar to the domain walls in skyrmions shown in Fig. 3d. Note that, one still\ncan see a weak contrast for some domain walls which do not follow this rule. This result\nindicates that the majority of the domain walls for maze-like domains are Néel-type with\na low mixing of Bloch-type. We also applied Fast Fourier Transform (FFT) to the STXM\nimages, which show different patterns for different polarizations (see Fig. 4(d-f). The FFT\nshows a ring pattern for circular polarization. For linear polarization, however, it shows\nan ellipse with long vertical axis for LV and an ellipse with long horizontal axis for LH,\nwhich represent the preference of Néel-type domain walls (compare also to the results of\nmicromagnetic simulations shown in the Supplementary Note S3 and Supplementary Note\nS4).\nMicromagnetic simulations\nMicromagnetic simulations were performed using the MuMax3 package [62] using mag-\nnetic parameters for a DyCo 3thin film deduced in a previous study [44] and in the present\nmagnetometry measurements that can be found in the Supplementary Information file (see\nSupplementary Note S1.1).\n7Figure 5 shows simulated magnetic structures as a function of the magnetic field applied\nperpendicular to the sample plane. The top row shows the color-coded three-dimensional\norientation of the magnetization, and the bottom row shows the in-plane component Mx. A\nmaze domain pattern shows up upon a relaxation of the random magnetization state at zero\nfield (Fig. 5a), featuring also some Néel-type skyrmions with opposite polarities. Skyrmions\nwith core magnetization parallel to the applied magnetic field collapse and disappear upon\nincreasing the field (Fig. 5b). Finally, at a higher field of µ0H= 530mT the maze domain\npattern evolves into a isolated skyrmion phase (Fig. 5c). This phase persists up to 810mT\nwhen the skyrmions collapse and the sample gets fully magnetized along the field. Bottom\npanels of Figs. 5a-d show the in-plane magnetization component within each cell of the\nsimulation. The contrast inversion from blue (left) to red (right) of each stable Néel-type\nskyrmion represents the chirality of the system given by the sign of the DMI constant, which\nis not picked up by the STXM experiment. Importantly, the simulation captures accurately\nthe impact of the domain wall type on the polarization-dependent STXM contrast. Circular\nandellipticshapesofFFTpatternsinFigs. 4d,e,fcorrespondverywelltotheonescalculated\nfrom the simulations (compare to Supplementary Figure S7) described in the Supplementary\nNote S3.\nInterestingly, the simulated skyrmion size is very tunable and can be changed by the\nexternal magnetic field by a factor of three from R= 32nm at 530 mT to R= 9nm\nat 810 mT. While the skyrmion size differs from the experimental value quite significantly,\nlowerDMIparametersdonotallowtostabilizepurelyNéel-typeskyrmionsbutratherhybrid\nones that carry Bloch caps at the surface (see Supplementary Note S5.3), being consistent\nwith the theory reported for magnetic multilayers [63]. If no DMI is assumed, the interplay\nbetween PMA and the stray field results in the formation of a bubble lattice with dominantly\nBloch-type domain walls (see Supplementary Note S5.1). Once a DMI term of sufficient\nstrength is introduced, the stability of the skyrmions is increased towards higher fields, and\nthe unique rotation sense of the domain walls gets defined. Note also that besides the DMI\nstrength, the saturation net magnetization and the magnetic anisotropy parameters play an\nimportant role for the skyrmion formation in this material (see Supplementary Figure S3\nand Supplementary Note S5.2).\nFor a discussion on the possible origin of a \"bulk\" DMI in this system see Ref. [44] and\nthe afferent Supplementary Note S6. Similar observations have recently been reported in\n8Fe3GeTe 2flakesof various thicknesswhere aninterplaybetween dipolarand DMinteractions\nresults in a complex history-dependent magnetic phase diagram of spin textures [64].\nCONCLUSIONS\nWe presented an experimental resolve of skyrmion traits in a ferrimagnetic DyCo 3thin\nfilmat26KusingSTXMimaging, utilizingbothXMCDandXMLDcontrast. WithXMCD-\nSTXM,themagneticstructuresinrealspacearerevealedasafunctionofdecreasingexternal\nfield. Well-isolated ferrimagntic skyrmions are observed between +260 mT and +140 mT,\nand the density as well as the radius of the skyrmions can be controlled by the external\nmagnetic field. When the magnetic field is further reduced, these skyrmions will merge\ninto maze-like domains, which matches very well with the results of magnetic simulations.\nUtilizing XMLD-STXM at the Dy M 5edge, we successfully identify the domain wall type\nof ferrimagnetic skyrmions to be of Néel-type. Moreover, the domain walls for the maze-like\ndomains are also investigated, revealing also a majority of Néel-type domain walls. Hence,\nwe are able to unambiguously conclude the interfacial-type symmetry [65] of DMI in DyCo 3\nthin film. Nevertheless, the origin of the strong DMI of this type remains an open question.\nThe technique of using XMLD contrast in the STXM measurements at the rare earth M\nedges provides a promising way to study complex spin textures in real-space, which is highly\nuseful for the characterization of skyrmions, chiral domain walls and various non-collinear\nmagnetic systems.\nMETHODS\nSample preparation and characterization\nThe ferrimagnetic DyCo 3film of 50 nm thickness was prepared by magnetron sputtering\nchamber (MAGSSY) at room temperature and in an argon atmosphere of 1.5×10−3mbar\nwith a base pressure of 5×10−9mbar. The stoichiometry of the DyCo 3alloy was controlled\nby varying the deposition rates of the Co and the Dy targets in a co-sputtering scheme.\nA Si 3N4membrane with a thickness of 100 nm was used as substrate for the soft X-ray\ntransmission measurements. A capping layer of 3 nm thick Ta was deposited on top of the\nsample surface to prevent surface oxidation. The magnetic properties of the sample have\n9been measured by SQUID magnetometry and by anomalous Hall effect (Tensormeter RTM1,\nHZDR Innovation, Germany), and they are described in the Supplementary Information file.\nX-ray measurements\nScanning transmission X-ray microscopy (STXM) measurements were performed at the\nMAXYMUS endstation at the Bessy II electron storage ring operated by the Helmholtz-\nZentrum Berlin für Materialien und Energie [66]. The X-ray beam was focused with a\nzone plate and an order selecting aperture on the transmissive sample in the presence of an\napplied out-of-plane magnetic field which was controlled by varying the arrangement of four\npermanent magnets. The STXM images were collected pixel by pixel using a piezoelectric\nsample stage at the Co L 3edge and the Dy M 5edge by exploiting the effects of x-ray\nmagnetic circular dichroism (XMCD) and x-ray magnetic linear dichroism (XMLD). The\nXMLD contrast represents an intensity map for LV (vertical axis in real space, parallel to\nthe sample surface) and LH (horizontal axis in real space, parallel to the sample surface)\norientations of the linear polarization axes. When the linear polarization is perpendicular to\nthe spin axis, the XAS intensity measured at the middle resonance peak of the M5 edge is\nlow (high in transmission), whereas for a parallel orientation of the linear polarization axis\nwith respect to the spin axis the intensity is high (low in transmission). (see for instance\nFigure S1, in Ref [61]). This makes the XMLD contrast easy to comprehend for the present\ntransmissiongeometryoffilmswithperpendicularmagneticanisotropy: achangeofintensity\nalong the linear direction shown on the LV, LH and L45 maps (see Supplementary Note\nS2) can be given only by Néel walls, whereas a change of intensity towards a direction\nperpendicular to the linear polarization direction can be given only by Bloch walls. Note\nthat the experimental XMLD maps shown in the manuscript are all logarithm of the raw\ndata images.\nThe XAS, XMLD and XMCD spectra for the Dy M 5edge (Fig. 3a) were performed at the\nDeimos beamline at synchrotron Soleil [67] in transmission mode using the same 50 nm thick\nDyCo 3sample grown on a Si 3N4membrane. The magnetic field of 2 T, which is much higher\nthanthesaturationfield, wasappliedalongthebeamduringtheXMCDmeasurementsusing\ncircular polarized X-rays and perpendicular to the beam during the XMLD measurements\nusing linear horizontal and linear vertical polarized X-rays.\n10Magnetic simulations\nMicromagnetic simulations were performed using MuMax3 [62]. The simulation was\nperformed on a three-dimensional grid 512×512×25voxels with the size of 2×2×2nm3.\nThe material parameters used for the simulation are given further below and can be found\nin the Supplementary Information file. A larger scale simulation with a grid of 1024×\n1024×25voxels was carried out for the simulation without DMI, in order to account for\nthe larger domain size. The computation was performed using a graphics processing unit\n(GPU) NVIDIA GeForce RTX 3080 Ti. The following material parameters were used:\nexchange stiffness Aex= 6 pJ m−1, saturation magnetization Ms= 600 kA m−1, and uniaxial\nanisotropy Ku= 130 kJ m−3. The interfacial-type DMI constant Dintwas tuned to obtain\nthe isolated field-induced Néel-type skyrmion phase without admixtures of maze domains.\nThe minimal value of DMI required for the purely Néel-type skyrmion stability was found to\nbeDint=0.0015 J m−2. It is remarkable that this value amounts about 8% of the exchange\nenergy of DyCo 3[68], in agreement with the suggestion of up to ∼20% of the isotropic\nexchange expected for DMI in disordered systems [69, 70] (see Supplementary Note S6).\nACKNOWLEDGMENTS\nWe thank the Helmholtz-Zentrum Berlin für Materialien und Energie for the allocation\nof synchrotron radiation beamtime (Proposal No. 212-10386). The authors acknowledge\nfinancial support by the German Federal Ministry for Education and Research (BMBF\nproject No. 05K19W061). F.R. acknowledges financial support by the German Research\nFoundation via Project No. SPP2137/RA 3570. We acknowledge the use of the Physical\nproperties laboratory, which is part of the CoreLab \"Quantum Materials\" operated by HZB.\nF.R. acknowledges insightful information provided by Dr. Eugen Weschke on the UE46\nundulator operation.\nCONTRIBUTIONS\nC.L., K.C. and F.R. conceived and designed the experiments. K.C. prepared the samples.\nC.L. and F.R. performed the STXM experiments with the help of S.W. and M.W., C.L. and\nK.C. analyzed the STXM data and prepared the figures. V.U. performed the micromagnetic\n11simulations. K.P. performed the magnetic characterization by SQUID. R.A., C.L., V.U.\nand F.R. performed the magneto-transport experiments. C.L., V.U. and F.R. wrote the\nmanuscript draft. All authors discussed the results and contributed to the manuscript.\n12FIG. 1. Hysteresis loop (measured by superconducting quantum interference device\n(SQUID)magnetometry)andscanningtransmissionX-raymicroscopy(STXM)images\nof ferrimagnetic (FiM) skyrmions at 26 K.\n(a)Out-of-planemagnetichysteresisloopmeasuredat26KbySQUIDmagnetometry. Theexternal\nmagnetic field µ0His expressed in units and sub-units of Tesla(T), with µ0being the vacuum\nmagnetic permeability and Hdenoting the magnetic field strength. (b) STXM images showing the\nFiM skyrmions at different perpendicular magnetic fields recorded at the Dy M 5edge and 26 K.\nThe average density (c) and radius (d) of the skyrmions as a function of external magnetic fields,\nwere extracted from the STXM images. The grey rectangle box in panel (a) represents the magnetic\nfield range where the FiM skyrmions are probed. The arrow represents the field sweeping direction.\nThe insets of panel (a) show the transition into maze-like domains. 13FIG. 2.X-ray magnetic circular dichroism imaging of isolated skyrmions.\n(a) Scanning transmission x-ray microscopy (STXM) images at the Co L 3edge and the Dy M 5edge\n(b), respectively. The color bar represents the normalized sublattice magnetization (M z) along the\nmagnetic field (and the x-ray beam) direction. (c, d) Line profiles across a skyrmion along the X\nand Y axes (see the dashed lines in panel a), representing the normalized perpendicular magnetic\nmoment for Dy and Co elements.\n14FIG.3.Resolveofthemagneticdomainwalltypeusingx-raymagneticlineardichroism.\n(a) x-ray absorption spectra (XAS) measured by circular left (CL), circular right (CR), linear\nhorizontal (LH) and linear vertical (LV) polarized X-rays, as well as x-ray magnetic linear dichroism\n(XMLD) and x-ray magnetic circular dichroism(XMCD) spectra at the Dy M 5edge at 4 K. The\nthree peaks of the Dy M 5edge are marked by dashed lines. Expected magnetic image contrast when\nusing circular, LH and LV polarized X-rays for Bloch-type (b) and Néel-type (c) skyrmions. (d)\nExperimental Scanning transmission X-ray microscopy (STXM) results. Here the STXM images\nfor LV and LH x-ray polarizations were obtained at the middle peak of the Dy M 5edge, and the\nSTXM image for circular polarized X-rays was measured at the third peak of the Dy M 5edge.\n15FIG. 4.Imaging of the magnetic domains and domain walls in a maze domain state\nand their corresponding Fast Fourier Transform.\n(a-c) Scanning transmission X-ray microscopy (STXM) images of the maze domains at -260 mT for\ncircular right (CR), linear horizontal (LH) and linear vertical (LV) polarized X-rays, respectively.\n(d-f) Fast Fourier Transform (FFT) of the top STXM images, with the dashed circles and ellipses\nas guide for the eye.\n16FIG. 5.Micromagnetic simulations demonstrating Néel-type skyrmion formation in the\npresence of a finite Dzyaloshinskii–Moriya interaction.\nTop row of (a,b,c,d) panels: Micromagnetic simulations of relaxed spin configurations for the DyCo 3\nfilm as a function of out-of-plane magnetic field. The external magnetic field µ0His expressed\nin sub-units of Tesla(T), with µ0being the vacuum magnetic permeability and Hdenoting the\nmagnetic field strength. The white and black colors represent the net normalized magnetization\nbeing parallel and antiparallel to the z-axis, respectively. The other colors represents the orientation\nof the in-plane component of the net magnetization within each cell as shown in the color wheel in\ntop panel (d). Bottom row of (a,b,c,d) panels: the magnetization component along the x-axis. The\ncolor bar shown in the bottom panel (d) represents the normalized net magnetization Mx.\n17SUPPLEMENTARY INFORMATION\nS1. MAGNETIC CHARACTERIZATION\nS1.1. SQUID magnetometry\nThe magnetic properties of the sample have been measured by SQUID magnetometry.\nThesamplehasbeencooleddownto2Kinamagneticfieldof2Teslaandmagnetichysteresis\nloops have been measured as a function of temperature, on warming. They are shown in the\nSupplementary Figure S1. The left axis displays the magnetization in kA/m as a function\nof an external magnetic field which was applied in a direction perpendicular to the sample\nsurface. At each temperature a field dependent magnetization was measured without the\nsample to correct for the eventual contributions from the sample holder itself. The absolute\nvalue of the magnetization is obtained by dividing the corrected SQUID raw response (emu)\non the volume of the layer which is area ×thickness, where the measured surface area was\nmeasured to be 6.53×10−6m2and the thickness of the film is 50×10−9m. The hysteresis\nloops exhibit a typical characteristic shape for films with perpendicular magnetic anisotropy,\nshowing the onset of the magnetization reversal at the nucleation field (the magnetic field\nwhere the global remagnetization initiates), followed by magnetic domains formation down\nto the annihilation field where the magnetization is fully reversed.\nThe nucleation field is extracted for each temperature and is shown in Figure S2. It\nexhibits a peculiar behavior: it has negative values at room temperature, changes sign at\nan intermediate temperature of about 230 K, increases in absolute values up to about 50\nK and decreases again, even to negative values, towards lower temperatures. This type of\nbehavior is expected for ferrimagnetic alloys due to the interplay between the anisotropy\nand the demagnetization energies.\nin Supplementary Figure S3 the saturation net magnetization extracted at the highest\napplied field is shown together with the magnetization at the nucleation field. The net\nmagnetization increases from room temperature towards lower temperatures in a monotonic\nway. Interestingly, the net magnetization at the nucleation field begins to deviate signifi-\ncantlyfromthesaturationmagnetizationatatemperatureofabout150K.Thistemperature\nrange (150 K to 2 K) can be considered as the phase diagram for the skyrmions formation.\nWith the magnetic parameters determined from the hysteresis loops, one can extract the\n18FIG. S1. Hysteresis loops measured by SQUID at different temperatures with field applied perpen-\ndicular to the sample’s surface.\nuniaxial magnetic anisotropy of the film as [71, 72]:\nKu=1\n2µM net(−Hn+ 4πM net) (1)\nwhere HNis the nucleation field, Mnetis the net magnetization at the nucleation field. The\nresults are shown in Supplementary Figure S4. We observe that the magnetic anisotropy\nchanges as function of temperature in an expected fashion. For ferrimagnetic films, it is\ndominated by the single ion anisotropy which further follows the character of the orbital\nmagnetic moment of the rare earth. The orbital magnetic moment was measured previously\nby soft x-ray spectroscopy [73]. It exhibits a non-monotonic increase below 200 K which\ncorrelates well with the temperature dependence of the anisotropy determined from the\nSQUID data.\n19FIG. S2. Temperature dependence of the nucleation field determined from data shown in Supple-\nmentary Figure S1\n20FIG. S3. Top panel: Saturation net magnetization extracted from the hysteresis loops at 2 Tesla\n(black down-triangles) and the magnetization at the nucleation field(blue up-triangles). Bottom\npanel: The difference between the saturation net magnetization and the magnetization at the\nnucleation field. This panel can be considered at a phase diagram for temperature range (0-150K)\nof skyrmions formation.21FIG. S4. Temperature dependence of the magnetic anisotropy calculated according to Supplemen-\ntary Equation 1.\n22S1.2. Electrical Transport\nThe characterization across the compensation magnetization was performed by magneto-\ntransport measurements as a function of temperature from 290 K to 460 K. The method\ninvolved here is spinning-current anomalous Hall magnetometry [74] which is implemented\nin a commercial device named Tensormeter (HZDR Innovation, Germany). This spinning-\ncurrent approach has the advantage that extrinsic parasitic contributions to the anomalous\nHall output signal are compensated dynamically which lifts the need of sample microstruc-\nturing.\nThe sample has been wire-bonded and connected with low resistance wires to the measur-\ning device. The transport measurements were performed under high vacuum( 3×10−8mbar)\nwith the Alice II instrument [25, 75]. The Hall resistance R xyof the sample was measured\nin a four-wire configuration using the Zero-Offset Hall preset of the Tensormeter. The tem-\nperature was calibrated by a temperature sensor mounted on the sample holder and the\nmagnetic field was applied perpendicular to the sample’s surface.\nIn Supplementary Figure S5a we show few selected hysteresis loops at temperatures be-\nlow and above the magnetization compensation. The hysteresis loops show an increased\ncoercive field close to the compensation temperature. Moreover, the sign of the hysteresis\nloops reverses as the temperature increases above the compensation. This effect reflects the\nsensitivity of the anomalous Hall resistance (R xy) whichin effectum is proportional to the\ndifference of the up and down electronic density of states at the Fermi energy. For 3d fer-\nromagnets this is proportional to the net magnetization. However, for RE-TM ferrimagnets\nalloys the 3d density of states at the Fermi energy are dominated by the TM element (in our\ncase Co), whereas the density of 4f-states of the RE ion (in our case Dy) are more localized\nbelow the Fermi energy(see Fig. 1c of [76]), therefore TM is contributing less significantly\nto the anomalous Hall resistivity.\nThe coercive field of each hysterysis loop was extracted and plotted in the Supplementary\nFigure S5b. At the divergence position, the net magnetization of the film vanishes. This\ncorresponds to magnetic compensation temperature Tcomp, which is for this sample 415 ±5\nK.\n23FIG. S5. The anomalous Hall effect (AHE) was measured at different temperatures. (a) The\nHall resistance R xyas a function of perpendicular magnetic field. The hysteresis loops change\nsign between 397.3 K and 426.1 K, which indicates the crossing of the magnetic compensation\ntemperature Tcomp. (b) Temperature dependence of the coercivity field µ0Hcexhibits a divergent\nbehavior near Tcompat≈415 K.\n24S2. STXM IMAGES FOR A 45 DEGREES ORIENTATION OF THE LINEAR PO-\nLARIZATION\nA 45◦rotation of the linear X-ray polarization was implemented to investigate the\nskyrmions for larger areas, as demonstratively shown in Supplementary Figure S6. By\ncomparing such a STXM image to those with LH and LV polarization, it can be seen that\nthe contrast of the domain walls also rotates by 45 degrees, which further confirms that the\nskyrmions are Néel-type.\nNote that in spite of a clear Néel character, the the skyrmion and domain walls shape\nexhibit distortions. Given that the atomic arrangements in amorphous alloys are not well-\ndefined, local variation of magnetic anisotropy and stiffness may impact on the homogeneity\nof the magnetic ground states. Also, local pinning at eventual defect sites may contribute\nto complex skyrmion shape distortions [77].\nFIG. S6. XMLD-STXM images of skyrmions at 140 mT and 30 K for LH and LV, and 45◦linear\nx-ray polarization.\nS3. MICROMAGNETIC SIMULATIONS: FAST FOURIER TRANSFORM OF\nTHE MAZE-DOMAIN STATE WITH NÉEL WALLS\nTheMz,Mx, and Mymagnetization components of the maze domain pattern resulting\nfrom the micromagnetic simulations are shown in Supplementary Figure S7, together with\ntheir Fast Fourier transform (FFT) patterns. The FFTs correspond well to the experimental\n25data shown in Figure 4(d-f) of the main manuscript. While the FFT of the Mzcomponent,\nwhich corresponds to the contrast mechanism to the circular light, results in a ring of\nintensity (Supplementary Figure S7a), the FFT patterns for the square of Mx,Myshow\narcs of intensity aligned in horizontal and vertical directions, respectively (Supplementary\nFigure S7b,c). On the other hand, the FFTs of the experimental data measured with linear\nhorizontal and vertical light polarizations (Figure 4e,f) appear as elongated ellipses because\nof a broader width distribution of the maze domains in the real sample, which is further\nconvoluted with the finite resolution of STXM images.\nFIG. S7. Micromagnetic simulations of the zero-field maze domain pattern for the DyCo 3film.\nMagnetization components (a) Mz, (b) Mx, and (c) Myare shown, respectively. First (from the\nleft) bottom panel shows the Fast Fourier transform (FFT) of the Mzcomponent while the other\ntwo bottom panels show the FFT of the square of the corresponding upper MxandMypatterns.\n26S4. MICROMAGNETICSIMULATIONS:FOURIERTRANSFORMOFTHEMAZE-\nDOMAIN STATE WITH BLOCH WALLS\nFFT of Mz,Mx,Mycomponents were also calculated for the micromagnetic simulations\nresults without taking a DMI into account (Supplementary Figure S8). While the ring-like\nintensity is clearly observed in the FFT of the Mzcomponent, FFTs for MxandMyare\nclearly rotated by 90◦compared to the previous case (Supplementary Figure S7). This\nclearly indicates the Bloch type character of the maze domain walls for a vanishing DMI in\ncontrast to the Néel type case when the DMI is present.\nFIG. S8. Micromagnetic simulations of the zero-field maze domain pattern for the DyCo 3film\nwithout taking DMI into account. Magnetization components (a) Mz, (b) Mx, and (c) Myare\nshown, respectively. First (from the left) bottom panel shows the Fast Fourier transform (FFT)\nof the Mzcomponent while the other two bottom panels show the FFT of the square of the\ncorresponding upper MxandMypatterns.\n27S5. MICROMAGNETIC SIMULATIONS: FEW MORE RELEVANT SHOWCASES\nS5.1. Case 1: Simulations for magnetic parameters corresponding to 26 K, without\nDMI\nThe magnetic field dependence of the maze domain pattern has been calculated also\nfor the case of a vanishing DMI, where the magnetic ground state is determined by the\ninterplay between dipolar interaction and uniaxial anisotropy only. Except for the DMI\nconstant, all the other parameters in the simulation are similar to the ones in the main text.\nSupplementary Figure S9 shows the simulated magnetization distributions at zero applied\nmagnetic field, 230mT, 340mT and 450mT. Relatively large ( ∼100nm) magnetic domains\nare observed at zero field, that evolve into worm-like stripes and, finally, into skyrmion\nbubbles as higher out-of-plane magnetic field is applied. The size of the bubbles originally\nformed at 340mT decreases to 45nm at 450mT and then the skyrmion state collapses into\na homogeneous one as the field is further increased. As a result, in the absence of DMI,\nmaze domain walls and field-induced skyrmions are clearly identified as Bloch-type, which\ncan be easily seen in the bottom panels of Supplementary Figure S9, where the intensity of\ntheMxcomponent changes from positive to negative at the top and bottom edges of the\nskyrmions. This is in contrast to the case of Néel type domain walls and skyrmions where\nthe intensity of these maps is rotated by 90 degrees (see main text).\nS5.2. Case 2: Simulations for the magnetic parameters corresponding to 200 K\nwith DMI\nMicromagnetic simulations for a saturation magnetization of Ms= 358 .5kA/m, an uni-\naxial anisotropy of Ku= 69 .8kJm−3and DMI equal to Dint= 0.0015Jm−2were carried\nout to mimic the sample behavior at 200K. The saturation net magnetization and magnetic\nanisotropy parameters were extracted from the magnetometry measurements. Here, large\nmagnetic domains (of the order of hundreds of nm) with Néel-type walls take place, evolving\ninto a very sparse skyrmion state ( ∼1per 1 µm2) at higher applied fields (Supplementary\nFigure S10). The simulation matches well with the STXM results which show similar large-\nscale textures at 200K [44] and absence of the skyrmion phase. We speculate that tuning\nthe DyCo composition in order to increase the anisotropy and saturation magnetization to-\n28FIG. S9. Micromagnetic simulations of the spin configurations for the DyCo 3film as a function\nof magnetic field without taking any DMI into account.The white and black represent the magne-\ntization parallel and anti-parallel to the z-axis. The color code represents the orientation of the\nin-plane component of the magnetization within each cell as shown in the color wheel in the panel\n(d). The bottom panels show the magnetization component along the x-axis. The color code of\nthe intensity of Mxis given in the bottom panel (d).\nwards room temperature can pave a way to stabilize technologically promising ferrimagnetic\ntopological spin textures in DyCo at 300K.\nInterestingly, a single anti-skyrmion has been spotted at an intermediate field range of\n150-180mT in the simulations (Supplementary Figure S10c), which is unexpected in the sys-\ntem with this type of DMI. Confirmation of this observation requires further computational\nand experimental studies.\nS5.3. Case 3: Simulations for the magnetic parameters corresponding to an inter-\nmediate DMI parameter at 26K\nAs we describe in the main text, the minimal value of the interfacial-type DMI con-\nstant required to stabilize Néel-type skyrmions in the micromagnetic simulations is found\natDint= 0.0015Jm−2, while lower values result in Bloch- or hybrid-type skyrmion textures.\n29FIG. S10. Micromagnetic simulations of the spin configurations for the DyCo 3film as a function\nof magnetic field with the MsandKuparameters corresponding to 200K. The black and white\ncontrast indicates the magnetization being parallel and anti-parallel to the z-axis. The color code\nrepresents the orientation of the in-plane component of the magnetization within each cell as shown\nin the color wheel in the top panel (d). The bottom panels show the magnetization component\nalong the x-axis. The color code of the intensity of Mxis given in the bottom panel (d).\nAn example of such a hybrid skyrmion is shown in Supplementary Figure S11. Here, the\nskyrmion winding changes from Bloch-type in the top layer of the film ( z= 1) to Néel-\ntype in the bottom layer ( z= 25) via an intermediate state in the middle. This behavior\ncorrespond well to predictions given by Lemesh and Beach in their analytical theory for\nmultilayers [63].\n30FIG. S11. Magnetic structure of hybrid skyrmions in DyCo 3film with an intermediate value of the\nDMI constant Dint= 0.00125Jm−2at 470mT as obtained from the micromagnetic simulations.\nThe panels show the single skyrmion structure in the top ( z= 1), middle ( z= 12) and bottom\n(z= 25) layers of the film, respectively.\n31S6. POSSIBLEORIGINOFA\"BULK\"DMIINTHEFERRIMAGNETICALLOYS\nOne main outcome of the resolve of the domain walls as Néel-type is that they require\na \"bulk\" DMI of an interfacial-type symmetry [65] to occur. In the absence of a DMI, the\ndomain walls are Bloch-type as revealed by the micromagnetic simulations. It is well known\nthat Ta or Pt buffer layers induce an interfacial DMI which has positive or negative sign,\nrespectively. However, a DMI that is localized at the interface only is too weak to stabilize\nNéel walls. For instance, in FM/Ta bilayers, the iDMI has an noticeable impact only for a\nFM layer thickness that is below 1 nm [78]. In our case, the ferrimagnetic layer is 50 nm\nthick, and therefore a DMI localized at the interface only, is not likely to stabilize skyrmions\nas observed experimentally. Instead a bulk DMI owes to be responsible for the skyrmions\nformation in this system [44].\nThe DyCo 3single crystal has a complex rhombohedral structure where Dy occupies two\nsites, whereas Co occupies three sites in the unit cell. According to Ref. [79], for one site the\nDy will align itself with the easy axis, whereas the Dy on the second site will prefer to align\nparallel to the c-axis of the crystal which is oriented perpendicular to the easy axis. As such,\na non-collinear spin arrangement is possible due to the frustrated site magnetic anisotropy\nof the rare earth ions. For the amorphous DyCo 3there is also strong experimental evidence\nfor the occurrence of a noncollinear spin state. In Ref. [80] an amourphous DyCo3 thick film\nhas been studied by Mössbauer spectroscopy. It has been observed that the Dy ion exhibit\na sperimagnetic arrangement, whereas the Co sublattice is ferromagnetically ordered. The\nabove experimental evidence for noncollinear behavior manifest intrinsically for both single\ncrystal and amourphous DyCo 3.\nCertainly, non-collinear interactions are yet not sufficient to lead to skyrmion formation.\nTwo more aspects play an important role for the formation of skyrmions in our DyCo3\nthin film, namely: an amorphous thin film naturally lacks a rotational and transnational\nsymmetry, (i. e. a lack of inversion symmetry that is similar to the B20 structures) which,\naccording to Ref. [70] may lead to a DMI that amounts up to about 10% of the isotropic\nexchange; and the dipolar interaction strength for thin ferrimagnetic films with perpendicu-\nlar magnetic anisotropy varies as a function of temperature(see section S1). 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Lett. 36, 1061\n(1976).\n38" }, { "title": "1705.09049v2.Fast_Vortex_Oscillations_in_a_Ferrimagnetic_Disk_near_the_Angular_Momentum_Compensation_Point.pdf", "content": "Fast Vortex Oscillations in a Ferrimagnetic Disk near the Angular Momentum\nCompensation Point\nSe Kwon Kim and Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n(Dated: October 15, 2018)\nWe theoretically study the oscillatory dynamics of a vortex core in a ferrimagnetic disk near its\nangular momentum compensation point, where the spin density vanishes but the magnetization is\n\fnite. Due to the \fnite magnetostatic energy, a ferrimagnetic disk of suitable geometry can support\na vortex as a ground state similar to a ferromagnetic disk. In the vicinity of the angular momentum\ncompensation point, the dynamics of the vortex resemble those of an antiferromagnetic vortex, which\nis described by equations of motion analogous to Newton's second law for the motion of particles.\nOwing to the antiferromagnetic nature of the dynamics, the vortex oscillation frequency can be an\norder of magnitude larger than the frequency of a ferromagnetic vortex, amounting to tens of GHz\nin common transition-metal based alloys. We show that the frequency can be controlled either by\napplying an external \feld or by changing the temperature. In particular, the latter property allows\nus to detect the angular momentum compensation temperature, at which the lowest eigenfrequency\nattains its maximum, by performing FMR measurements on the vortex disk. Our work proposes a\nferrimagnetic vortex disk as a tunable source of fast magnetic oscillations and a useful platform to\nstudy the properties of ferrimagnets.\nFerromagnets are used as platforms to produce GHz\nmagnetic oscillations, e.g., spin-torque and spin-Hall os-\ncillators [1], which can be used as microwave signal gen-\nerators. Among them, the core oscillation of a vortex in a\nmicrodisk [2, 3], which is a curling magnetization texture\nstabilized by magnetostatic energy, stands out because of\nits high coherence [4, 5]. This advantage allows it to be\nused in areas requiring highly precise oscillations such as\nbiomedicine [6]. Aside from the dynamics of an isolated\nvortex, coupled oscillations of vortices in an array of disks\nhave been studied as magnonic vortex crystals [7, 8]. The\noscillation can be driven in several ways such as with an\nexternal \feld [9, 10] or with an electric current [4, 5]. Its\ncharacteristic frequency is given by\nfFM\u0018\r\u00160Ms; (1)\nup to the geometric factor, where \ris the gyromagnetic\nratio,\u00160is the permeability constant, and Msis the sat-\nuration magnetization. The observed frequency ranges\nfrom several hundred MHz up to 2 GHz [3, 5, 10, 11]. See\nFig. 1 for some schematic illustrations of vortex disks.\nRecently, antiferromagnets have been gaining atten-\ntion as possible hosts of magnetic oscillations that\nare orders-of-magnitude faster than their ferromagnetic\ncounterparts [12, 13]. For example, a spin Hall oscillator\nbased on an antiferromagnet/heavy-metal heterostruc-\nture has been proposed as a possible THz signal gener-\nator [14]. However, the possibility of fast oscillations of\nan antiferromagnetic vortex has not been investigated be-\ncause, di\u000bering from ferromagnetic cases, antiferromag-\nnetic disks do not support a vortex as an equilibrium\nstate due to the absence of magnetostatic energy.\nHere, we show that a ferrimagnetic disk can host a sta-\nble vortex whose oscillations can be orders-of-magnitude\nfaster than those of ferromagnetic vortices. A ferrimag-\nnetic disk can harbor a vortex as a ground state due to\nthe \fnite magnetostatic energy [15], which con\fnes the\n(a)(b)p=1\nLRp=\u00001FIG. 1. Schematic illustrations of the magnetization of ferri-\nmagnetic disks harboring (a) a vortex with polarization p= 1\nand (b) a vortex with polarization p=\u00001. The thickness and\nthe radius of disks are denoted by LandR, respectively.\nvortex core to the center of the disk. We show that at\nthe angular momentum compensation point (CP), where\nthe spin density vanishes, the nature of the vortex-core\ndynamics is antiferromagnetic and in this way the core\nbehaves like a classical particle in a potential well [16].\nThe characteristic frequency of an oscillation of the vor-\ntex core at the CP, which will be derived below, is given\nby\nfAFM\u0018r\nJ\n~\u0003\rt\u00160Ms; (2)\nup to a dimensionless geometric factor. Here, Jis the\nantiferromagnetic exchange energy between neighboring\nmagnetic moments and \rtis the transverse gyromagnetic\nratio with respect to the direction of the magnetization.\nWe will show that this frequency is in the range of tens of\nGHz, e.g.,fAFM\u001930 GHz for the material parameters\nof CoTb [17]. As a fast-oscillating source, a ferrimagnetic\nvortex in a microdisk has two advantages over other pro-\nposals using antiferromagnets. First, it can be easily op-\nerated by an external magnetic \feld due to the \fnite mag-\nnetization. Secondly, since the spin density can be con-arXiv:1705.09049v2 [cond-mat.mes-hall] 30 Jun 20172\ntrolled by changing the temperature [18], the oscillation\nfrequency can be thermally tuned. In particular, the lat-\nter advantage allows us to detect the angular momentum\ncompensation temperature, where the lowest frequency\nwill exhibit its peak, by performing FMR measurements.\nThe high coherence of the vortex oscillation [4, 5] will\nyield a narrow FMR linewidth. A recent realization of\na vortex in an arti\fcial-ferrimagnet microdisk composed\nof Py/Gd superlattices [19] supports the experimental\nfeasibility of our proposal.\nLet us begin by deriving the equations of motion for a\nvortex in a ferrimagnetic disk. We will describe the low-\nenergy dynamics of a ferrimagnet using coarse-grained\nvariables over the constituent sublattices, following the\napproach taken by Andreev and Marchenko [20] and sub-\nsequently used in Refs. [21{23]. This approach is based\non approximate exchange symmetry and does not use any\nmodel representations of the state of the magnet, e.g.,\nthe number of sublattices as pointed in Ref. [20]. The\ncollinear order of a ferrimagnet much below the Curie\ntemperature can be described by the unit vector nde-\nnoting its direction. Along this direction, a ferrimag-\nnet has the net magnetization, M=Msn, and the net\nspin density, s=sn, which are related by the (longi-\ntudinal) gyrotropic coe\u000ecient \rasM=\rs. Here,Ms\nis the net saturation magnetization density. There is a\nclass of ferrimagnets such as rare-earth transition-metal\nalloys, which exhibit two distinct special values of tem-\nperature or chemical composition: the angular momen-\ntum CP, where the spin density vanishes, s= 0, and\nthe magnetization CP, where the magnetization vanishes,\nMs= 0 [15]. These two points are di\u000berent due to the\ndistinct gyromagnetic ratios of constituent sublattices.\nIn particular, at the angular momentum CP, these ferri-\nmagnets have the \fnite magnetization and the zero spin\ndensity [18], allowing the unconventional coexistence of\nmagnetostatic energy and antiferromagnetic dynamics.\nThe former stabilizes a vortex in the microdisk and the\nlatter yields fast oscillations of the vortex, as will be ex-\nplained below.\nThe dynamics of a generic collinear ferrimagnet can be\ndescribed by the following Lagrangian density [21, 22]:\nL=\u0000sa[n]\u0001_n+\u001a_n2=2\u0000U[n]; (3)\nwhere nis the unit vector in the direction of the local\nmagnetization, ais the vector potential for the magnetic\nmonopole, rn\u0002a=n, and\u001ais the e\u000bective inertia\ndensity. Here, the \frst term represents the Berry phase\nterm associated with the spin density; the second term\nrepresents the inertia of the dynamics of nwhich can\narise due to the relative canting of sublattice spins; the\nthird term is the free-energy density. This Lagrangian\nfor ferrimagnets can be interpreted as a hybrid of those\nfor ferromagnets and antiferromagnets. We model the\nfree-energy density in the absence of an external \feld by\nU=A(rn)2+\u00160H2\nms\n2: (4)Here, the \frst term is the exchange energy parametrized\nby the sti\u000bness coe\u000ecient A > 0; the second term is\nthe magnetostatic energy, where Hmsis the dipolar \feld\ninduced by the magnetization satisfying the equations\nr\u0001(Hms+M) = 0 and r\u0002Hms= 0. At the angular\nmomentum CP, s= 0, the kinetic part of the Lagrangian\nis antiferromagnetic, but the magnetostatic energy is \f-\nnite owing to the \fnite magnetization, Ms6= 0. Recog-\nnizing this peculiar coexistence of antiferromagnetic dy-\nnamics and magnetostatic energy motivated this work.\nThe energy dissipation associated with the dynamics can\nbe accounted for by considering the Rayleigh dissipation\nfunction,R=s\u000b_n2=2, which is half of the energy (den-\nsity) dissipation rate. The parameter s\u000bis reduced to\nthe product of the Gilbert damping constant [24] and\nthe spin density in the ferromagnetic limit, \u001a!0.\nThere is a length scale associated with the energy den-\nsityU, which is given by \u0015\u0011p\nA=\u0016 0M2s. It is referred to\nas the exchange length at which the magnetostatic energy\nis comparable to the exchange energy. When a dimension\nof the sample is larger than the exchange length, the mag-\nnetostatic energy dominates the total energy and gener-\nates nonuniform ground states. Let us consider a circular\nferrimagnetic disk of radius Rand thickness Las shown\nin Fig. 1. When the thickness is order of the exchange\nlength,L\u0018\u0015, and the aspect ratio of the thickness to\nthe radius is small enough, g\u0011L=R.0:5, the disk sup-\nports a vortex as a ground state [25]. For a quantitative\nestimate of the geometry of the physical system, let us\ntake the example of Co 1\u0000xTbx, which exhibits its angu-\nlar momentum CP at x\u00190:17. According to the exper-\nimental results in Ref. [17], the parameters are given by\nA\u00191:4\u000210\u000011J/m andMs\u0019105A/m at the angular\nmomentum CP, which yields the exchange length \u0015\u001930\nnm. Therefore, for example, a disk made of CoTb with\nthicknessL= 100 nm and radius R= 1000 nm would\nhave a vortex as a ground state.\nThe observed vortex states of magnetic disks\ncan be described by the following ansatz in\nthe spherical-coordinate representation, n =\n(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012):\u0012= (1\u0000p)\u0019=2 +\n2 arctan(p\nx2+y2=Rc) forp\nx2+y2< Rcand\n\u0012=\u0019=2 otherwise, and \u001e='+c\u0019=2, whereRc\nis the vortex-core radius that is on the order of the\nexchange length \u0015[26]. Here, p=\u00061 is the direction\nof the magnetization at the core, n=p^z, which is\nreferred to as the polarization; c=\u00061 corresponds to\nthe counter-clockwise (+) or clockwise ( \u0000) rotation of\nthe magnetization in the plane, which is referred to\nas the chirality. The low-energy dynamics of a vortex\ncan be described by the dynamics of its core position,\nR(t) = (X;Y ), with the aforementioned ansatz. The\nequations of motion for the position can be derived\nfrom the above Lagrangian and the Rayleigh dissipation\nfunction within the collective-coordinate approach,\nassuming a rigid magnetization pro\fle [22, 27]:\nMR+G_R\u0002^z+D_R=\u0000KR; (5)3\n0G/Gco12\u00001\u00002!/!AFM2!+!\u0000\n-2 -1 0 1 2121\nFIG. 2. The eigenfrequencies !\u0006=!AFM as functions of the gy-\nrotropic coe\u000ecient G=G co. The solid and dashed lines show\nthe eigenfrequencies for the counter-clockwise ( !+) and clock-\nwise (!\u0000) rotations of the core. See the main text for discus-\nsions.\nwhere\nM= 2\u0019\u001aLln(R=Rc) (6a)\nG= 2\u0019psL; (6b)\nD= 2\u0019s\u000bLln(R=Rc); (6c)\nK=\u00160LM2\ns[F1(L=R)\u0000(\u0015=R)2]: (6d)\nHere,Mrepresents the mass of the vortex, which orig-\ninates from the inertial term in the Lagrangian; G\nparametrize the gyrotropic force on the vortex, which\nis rooted in the spin Berry phase; Dparametrizes the\nviscous force on the vortex. The right-hand side is the\nrestoring force on the vortex, which has been obtained\nin Ref. [9] for ferromagnetic vortices. When the vortex\ncore is away from the center of the disk, magnetostatic\ncharges are created on the boundary. The corresponding\nmagnetostatic energy engenders a con\fning potential for\nthe vortex core, which is parametrized by K. The fac-\ntor [F1(L=R)\u0000(\u0015=R)2] is a dimensionless number that\nis on the order of 0.1 for L=R\u00180:1. The de\fnition of\nthe function F1(x), which is an increasing function of x,\ncan be found in Ref. [28]. Note that the gyrotropic co-\ne\u000ecient is proportional to the spin density, G/s, and\nthus it vanishes at the angular momentum CP.\nLet us study the excitation mode for the core dynam-\nics described by the above equation. To this end, it is\nconvenient to express the equations of motion in terms\nof the complex variable, \t \u0011X+iY:\nM\t\u0000iG_\t +D_\t =\u0000K\t: (7)\nWhen the damping constant is small, \u000b\u001c1, the main ef-\nfect of the viscous force is to broaden the linewidth of the\nexcitation spectrum, not to change the spectrum itself,\nand so we will neglect it henceforth by setting D= 0.\nThe oscillation frequencies of the monochromatic solu-\ntion, \t(t)/exp(i!t), are given by\n!\u0006=G\n2M\u0006s\u0012G\n2M\u00132\n+K\nM: (8)\nAt the angular momentum CP, where the gyrotropic\nforce vanishes G= 0, the eigenfrequencies are given by\nTTCP!!AFMH=0H=HcoH=\u0000HcoFIG. 3. Schematic illustrations of the lowest eigenfrequency\n!of the vortex-core oscillation as a function of temperature\nT, in the vicinity of the angular momentum CP ( TCP) sub-\njected to an external \feld, H=H^z. The vortex with po-\nlarizationp= 1 is considered. For CoTb disks, the maxi-\nmum eigenfrequency and the crossover \feld are estimated as\n!AFM\u00192\u0019\u000230 GHz and Hco\u00192 T, respectively. See the\nmain text for discussions.\n!\u0006=\u0006!AFM with!AFM\u0011p\nK=M , which is reduced\nto Eq. (2) if we recast it in terms of the microscopic pa-\nrameters using \u001a\u0018~2=Ja3withathe lattice constant.\nThe absence of any gyrotropic coupling between Xand\nYis one characteristic of the antiferromagnetic dynamics\nof two-dimensional solitons [29] such as vortices [16] and\nskyrmions [30]. Note that two circularly polarized modes\nare degenerate at the CP, where the spin con\fgurations\nrespect time-reversal symmetry. Far away from the an-\ngular momentum CP, where the gyrotropic force domi-\nnates the dynamic part in the equations of motion, the\nlowest eigenfrequencies are given by !=\u0000psgn(\r)!FM\nwith!FM\u0011K=jGj, which corresponds to the ferromag-\nnetic case [9]. The crossover between antiferromagnetic\nand ferromagnetic dynamics occurs when the two fre-\nquencies are comparable, !FM\u0018!AFM, corresponding\ntojGj\u0018Gco\u0011p\nMK. The above equation (8) for gen-\neral cases can be written as a function of the gyrotropic\ncoe\u000ecientG:!\u0006=!AFM =G=2Gco\u0006p\n(G=2Gco)2+ 1,\nwhich is shown in Fig. 2.\nLet us provide a numerical estimate for the antiferro-\nmagnetic oscillation at the angular momentum CP. The\nalloy Co 1\u0000xTbxat its CP, x\u001917, has the exchange-\nsti\u000bness coe\u000ecient, A\u00191:4\u000210\u000011J/m, and the lat-\ntice constant, a\u00190:4 nm, which yield the microscopic\nexchange constant J=Aa\u001935 meV [15]. By us-\ning the inertia, \u001a=~2=2Jza3[23, 31], where z= 6\nis the coordination number for three-dimensional bipar-\ntite lattices, and the additional parameters, L=R = 0:1\nandRc\u0019\u0015, we can estimate the eigenfrequency at\nthe CP:fAFM\u0011!AFM=2\u0019\u001930 GHz, which is one\norder-of-magnitude larger than the observed frequencies\nin ferromagnetic disks of several hundred MHz up to 2\nGHz [3, 5, 10, 11]. For example, for a cobalt disk of the\nsame shape, the ferromagnetic resonance frequency is cal-\nculated as fFM\u0011!FM=2\u0019\u0019700 MHz when using the\nsaturation magnetization Ms\u00191:2\u0002106A/m measured\nfor 30-nm-thick \flms [32].\nThe dependence of the eigenfrequency on the gy-\nrotropic coe\u000ecient can be used to infer the angular mo-4\nmentum CP. For example, when we measure the ferro-\nmagnetic resonance (FMR) frequency of the vortex os-\ncillation by varying the temperature across the angular\nmomentum CP denoted by TCP, the lowest resonance fre-\nquency should attain its maximum at TCP. In addition,\nsince the rotational directions of the core oscillation be-\nlow and above TCPare opposite, TCPcan be measured\nby detecting the change of the oscillation direction, which\ncan be probed by time-resolved scanning transmission X-\nray microscopy [8]. See Fig. 3 for schematic illustrations\nof the lowest oscillation frequency as a function of a tem-\nperature. These methods using a vortex oscillation in\na ferrimagnetic disk to determine TCPcan be an alter-\nnative to a recent proposal based on domain-wall speed\nmeasurements [33].\nWe now study the e\u000bects of an external \feld applied\nperpendicular to the disk. In Refs. [16], it has been shown\nthat the application of an external \feld can induce a gy-\nrotropic force on antiferromagnetic solitons. The same\nmechanism works for ferrimagnets. In the presence of an\nexternal \feld, H, the inertial term in the Lagrangian (3)\nis changed to \u001a(_n\u0000\rtn\u0002H)2=2 [20]. The \feld-induced\ndynamical term, \u0000\u001a\rt_n\u0001(n\u0002H), is linear in the time\nderivative of n, and thus can be interpreted as the e\u000bec-\ntive geometric phase associated with the magnetic dy-\nnamics. When the \feld is perpendicular to the disk,\nH=H^z, it gives rise to an extra force term in the\nequations of motion (5) for a vortex within the collec-\ntive coordinate approach [16]:\nMR+G_R\u0002^z+GH_R\u0002^z+D_R=\u0000KR;(9)\nwhere\nGH\u00112\u0019p\u001a\rtLH (10)\nis the contribution to the gyrotropic coe\u000ecient induced\nby an external \feld. Its origin can be understood by writ-\ning\u001a\rtHin the \feld-induced kinetic term as \u001fH=\rtby\nusing the relation, \u001a=\u001f=\r2\nt[23], where \u001fis the trans-\nverse magnetic susceptibility. This induced spin density\ncontributes to the gyrotropic coe\u000ecient G[Eq. (6b)]. The\ncrossover \feld corresponding to the crossover gyrotropic\ncoe\u000ecient is given by Hco=Gco=2\u0019\u001a\rtL, which is esti-\nmated to be Hco\u00192T for the aforementioned CoTb disk.\nThe \feld-induced gyrotropic force can be inferred by per-\nforming FMR measurements. See Fig. 3 for schematic\nillustrations [34].\nTo conclude, we have studied the oscillation dynamics\nof a vortex core in a ferrimagnetic disk. We have shownthat the oscillation frequency can be orders-of-magnitude\nlarger than that of a ferromagnetic vortex in the vicinity\nof the angular momentum CP, where the nature of the\ndynamics is antiferromagnetic due to the vanishing spin\ndensity. This is similar to the recently observed enhance-\nment of the ferrimagnetic domain-wall speed around the\nangular momentum CP [33]. The vortex oscillation fre-\nquency can be tuned by changing the temperature or by\napplying an external \feld. Our results exemplify the pos-\nsibility of using ferrimagnets to realize desired functional-\nities that have been di\u000ecult to achieve with conventional\nferro- and antiferromagnets. One possible research topic\nrelated to this is the dynamics of a vortex domain wall\nin a long ferrimagnetic strip, which can be faster than its\nferromagnetic counterpart in the vicinity of the angular\nmomentum CP and thus may realize a better racetrack\nmemory [35].\nWe made several approximations in this work to de-\nscribe the dynamics of a ferrimagnetic vortex. First, we\nobtained the mass of a vortex by assuming that its mag-\nnetization pro\fle is rigid. There can be contributions\nto the mass from deformations of the pro\fle as shown\nfor ferromagnetic magnetic bubbles [36]. Secondly, we\nneglected the magnetic crystalline anisotropy by assum-\ning that the magnetostatic energy dominates it. We re-\nmark that rare-earth transition-metal ferromagnetic al-\nloys are known to have uniaxial crystalline anisotropy.\nThe anisotropy type can be easy axis or easy plane, de-\npending on, e.g., the deposition parameters controlling\nthe structure of the \flm [37]. Thirdly, the derived equa-\ntions of motion for the dynamics of a vortex are valid to\nlinear order in the vortex-core displacement. There can\narise nonlinear e\u000bects beyond our description when the\namplitude of oscillations is su\u000eciently large. 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B 67,\n224432 (2003)." }, { "title": "2005.08914v3.Noncollinear_Magnetic_Modulation_of_Weyl_Nodes_in_Ferrimagnetic_Mn__3_Ga.pdf", "content": "Noncollinear Magnetic Modulation of Weyl Nodes in Ferrimagnetic Mn 3Ga\nCheng-Yi Huang,1, 2Hugo Aramberri,2,\u0003Hsin Lin,1and Nicholas Kioussis2,y\n1Institute of Physics, Academia Sinica, Taipei 11529, Taiwan\n2Department of Physics and Astronomy, California State University, Northridge, CA 91330-8268, USA\n(Dated: July 16, 2020)\nThe tetragonal ferrimagnetic Mn 3Ga exhibits a wide range of intriguing magnetic properties.\nHere, we report the emergence of topologically nontrivial nodal lines in the absence of spin orbit\ncoupling (SOC) which are protected by both mirror and C4zrotational symmetries. In the presence\nof SOC we demonstrate that the doubly degenerate nontrivial crossing points evolve into C4z-\nprotected Weyl nodes with chiral charge of \u00062. Furthermore, we have considered the experimentally\nreported noncollinear ferrimagnetic structure, where the magnetic moment of the Mn Iatom (on the\nMn-Ga plane) is tilted by an angle \u0012with respect to the crystallographic caxis. The evolution of the\nWeyl nodes with \u0012reveals that the double Weyl nodes split into a pair of charge-1 Weyl nodes whose\nseparation can be tuned by the magnetic orientation in the noncollinear ferrimagnetic structure.\nPACS numbers: 73.20.At, 75.50.Gg\nI. INTRODUCTION\nThe discovery of topological states of matter repre-\nsents a cornerstone of condensed-matter physics that may\naccelerate the development of quantum information and\nspintronics and pave the way to realize massless particles\nsuch as Dirac and Weyl fermions. A Weyl semimetal\n(WSM) is a topological semimetallic material hosting\ndoubly-degenerate gapless nodes near the Fermi level in\nthe three-dimensional (3D) momentum space1{4. The\nnodes correspond to e\u000bective magnetic monopoles or an-\ntimonopoles which carry nonvanishing positive and neg-\native chiral charge \u0006q. Typically, qtakes values of\u00061\ncorresponding to Weyl nodes, but is also possible to have\nintegers,q=\u00062;\u00063;::: for double Weyl nodes, etc.5\nThe Weyl nodes gives rise to surface states which form\nopen Fermi arcs rather than closed loops.\nCompared to their Dirac semimetal counterparts,\nWSMs require the breakdown either of inversion symme-\ntry or time reversal symmetry (TRS) to split each four-\nfold degenerate Dirac node into a pair of Weyl nodes. A\nnumber of WSMs that break inversion symmetry have\nbeen identi\fed in the past few years1{4. Moreover the\npresence of crystalline symmetries can further protect\nmultiple Weyl nodes with large chiral charge6{8. On the\nother hand, the discovery of their broken TRS counter-\nparts, which link the two worlds of topology and spintron-\nics, remains challenging and elusive6. Many potential\nTRS-breaking WSM have been proposed. Recently, three\ngroups have provided unambiguous and direct experi-\nmental con\frmation that Co 3Sn2S29,10, which becomes\na ferromagnet below 175 K, and Co 2MnGa, a room-\ntemperature ferromagnet11, are TRS-breaking WSMs.\nThe discovery of magnetic WSMs give rise to exotic quan-\ntum states ranging from quantum anomalous Hall e\u000bect\nto axion insulators3.\nAnother remarkable and highly promising class of\nmagnetic materials is the Heusler family12,13which in-\ncludes half metals,14ferromagnets, ferrimagnets, antifer-\nromagnets, and even topological insulators15,16and Weylsemimetals. In particular the ferrimagnetic and antifer-\nromagnetic compounds with antiparallel exchange cou-\npling, have recently garnered intense interest because of\nthe faster spin dynamics (in the terahertz range) com-\npared to the gigahertz-range magnetization dynamics of\ntheir ferromagnetic counterparts.17\nThe Mn 3X (X=Ga, Ge, Sn) Heusler compounds are\nconsidered prototypes with promising applications in the\narea of spintronics13,18. These compounds can be ex-\nperimentally stabilized in either the hexagonal DO 19\nstructure ( \u000fphase) or the tetragonal DO 22structure\n(\u001cphase)19. The high-temperature hexagonal crystal\nstructure is antiferromagnetic with a high N\u0013 eel temper-\nature (\u0018470 K) and a noncollinear triangular magnetic\nstructure. Recently, several experimental and theoretical\nstudies have demonstrated20{26the emergence of large\nanomalous Hall e\u000bect (AHE) in the noncollinear AFM\nhexagonal Mn 3X family, whose origin lies on the non-\nvanishing Berry curvature in momentum space. In addi-\ntion, ab initio calculations have revealed that these chiral\nAFM materials are topological Weyl semimetals22. On\nthe other hand, the low-temperature tetragonal phase,\nwhich can be obtained by annealing the hexagonal phase,\nis ferrimagnetic at room temperature and shows a unique\ncombination of magnetic and electronic properties, in-\ncluding low magnetization,27high uniaxial anisotropy,28\nhigh spin polarization ( \u001988%),29{31low Gilbert damp-\ning constant,29high Curie temperature,32and large volt-\nage controlled magnetic anisotropy e\u000eciency33. Interest-\ningly, neutron scattering experiments have reported34a\nnoncollinear ferrimagnetic magnetic structure in Mn 3Ga,\nwhere the magnetic moment orientation of the Mn atoms\non the Mn-Ga (001) plane is tilted by about 21 °with re-\nspect to the crystallographic caxis.\nThe objective of this work is to carry out \frst-\nprinciples electronic structure calculations to investigate\nthe emergence of topological nodal lines in the absence\nor presence of SOC in tetragonal ferrimagnetic Mn 3Ga.\nFurthermore, we present results of the e\u000bect of non-\ncollinear magnetism on the evolution of the Weyl nodes.arXiv:2005.08914v3 [cond-mat.mtrl-sci] 15 Jul 20202\nII. METHODOLOGY\nThe electronic structure calculations were carried out\nby means of \frst-principles spin-polarized collinear cal-\nculations within the density functional theory (DFT)\nframework as implemented in the VASP package35.\nThe Perdew-Burke-Ernzerhof36(PBE) implementation\nof the generalized gradient approximation (GGA) for\nthe exchange-correlation functional was employed. The\nplane-wave cuto\u000b energy was set to 400 eV, which was\nenough to yield well-converged results. The Brillouin\nzone (BZ) was sampled using a \u0000-centered mesh of\n10x10x10 k-points. The structure was allowed to relax\nuntil residual atomic forces became lower than 0.01 eV/ \u0017A\nand residual stresses became smaller than 0.01 GPa. The\nelectron-electron interactions are included, where indi-\ncated, within the GGA+U approach of Dudarev et al.37.\nIn this way, the electron correlations are taken into ac-\ncount through a single e\u000becive parameter Ue\u000b=U\u0000J.\nThe values of Ue\u000bfor the Mn I, MnIIand Ga are set to 2.6\neV, 0 and 0, respectively. Previous studies have shown\nthat these values yield lattice parameters closer to their\nexperimental values38. The spin-orbit coupling (SOC) of\nthe valence electrons is in turn included self-consistently\nusing the second-variation method employing the scalar-\nrelativistic eigenfunctions of the valence states39, as im-\nplemented in VASP. Then, DFT derived wave functions\nboth without and with SOC were in turn projected to\nWannier functions using the wannier90 package40.\nIn theDO22structure (I4/mmm space group) the\ntwo (001) antiferromagnetically-coupled Mn sublattices,\nshown in Fig. 1(a), consist of Mn Iatoms at the Wycko\u000b\npositions 2b (0,0,1/2) [Mn I-Ga (001) plane] and Mn II\natoms at the 4d (0,1/2,1/4) positions [Mn II-MnII(001)\nplane]. For the noncollinear calculation, where the mag-\nnetic moment of the Mn Iis rotated by an angle \u0019\u0000\u0012with\nrespect to the [001] direction, the angular dependence of\nthe Wannier Hamiltonian in the presence of SOC is de-\ntermined from,\nH(k;\u0012) =H0(k) +U(\u0012)Hex(k)Uy(\u0012): (1)\nHere,H0(k) is the TRS preserving Hamiltonian\nwith SOC, [ TH0(k)T\u00001=H0(\u0000k)],Hex(k) is the\nTRS-breaking exchange Hamiltonian, [ THex(k)T\u00001=\n\u0000Hex(\u0000k)],T=i\u001b2Kis the TRS operator, \u001b2acts on\nthe spin degrees of freedom, Kis complex conjugation,\nU(\u0012) =e\u0000i\u0019\u0000\u0012\n2\u001b2;MnIis the spin rotation operator, and\n\u001b2;MnIis theycomponent of Pauli matrix acting on the\nspin degrees of freedom of Mn I.\nIII. RESULTS AND DISCUSSION\nA. Nodal lines in the absence of SOC\nThe calculated lattice parameters a=b= 3.78 \u0017A and\nc= 7.08 \u0017A, are in good agreement with previous calcu-TABLE I: List of ab initio and experimental lattice constants\nand magnetic moments values for the collinear case. \u00162b\nz(\u00164d\nz)\ndenotes the z-component of the magnetic moment of the Mn I\n(Mn II) atom.\nMethod a( \u0017A) c( \u0017A)\u00162b\nz(\u0016B/Mn)\u00164d\nz(\u0016B/Mn)\nGGA 3.78 7.08 -2.83 2.30\nGGA+U 3.91 7.00 -3.76 2.45\nGGA+U+SOC 3.91 7.00 -3.87 2.51\nexperiment343.92 7.08 -3.07 2.08\nE-EF(eV) \nΓ X M Γ N\nMn IGa \nMn II\nbc\na(a) (b) \n(c) \nΓ\nXM\nNkz\nkx ky\nFIG. 1: (Color online) (a) The tetragonal cell of the DO 22\nferrimagnetic structure with [001] spin polarization. Arrows\ndenote the magnetic moments of Mn I(purple) and Mn II(red)\nsublattices which are coupled antiferromagnetically. (b) First\nBrillouin zone of the primitive cell shown in panel (a). (c)\nSpin-polarized band structure without SOC along the high\nsymmetry directions of the primitive cell, where the spin-up\n(spin-down) bands are denoted by blue (red).\nlations34,41,42, which are, however, lower than the exper-\nimental values of a=b= 3.92 \u0017A andc= 7.08 \u0017A(see\nTable I). The e\u000bect of U on the topology of the band\nstructure is discussed in Sec. III. Overall, our calculated\nGGA values of the magnetic moments of -2.83 \u0016Band\n2.30\u0016Bfor the Mn Iand MnIIatoms, respectively, are in\ngood agreement with previous DFT calculations34,41,42.\nFig. 1(c) shows the spin-polarized band structure of\nthe majority- (blue) and minority-spin (red) bands of\nMn3Ga without SOC and with collinear spins along the\nsymmetry lines of the Brillouin zone (BZ) of the prim-\nitive cell, shown in Fig. 1(b). For each spin channel,\nthe energy bands can be labeled by the eigenvalues of\nthe crystalline symmetry operator of a particular high3\n(a)\n(b)\nE-EF(eV)\nkz(Å-1)(-1,-1)(1,1)\nkz(Å-1)E-EF(eV)\nFIG. 2: (Color online) (a) Spin polarized band structure along\nthekz-axis (\u0000\u0000Msymmetry direction) without SOC, where\nthe blue (black) bands denote the spin-up states calculated\nfrom GGA (GGA+U). The two nontrivial crossing points, de-\nnoted with the blue dots, are labeled with the pair of eigenval-\nues, (\u00061,\u00061), of the mirror, M[110], and four-fold rotational,\nC4z, symmetries, respectively, which protect them. Red dots\ndenote the nontrivial crossing points when U is turned on.\n(b) 3D landscape of the nodal lines where the two blue dots\ndenote the two nontrivial crossing points in (a). The color\nbar represents the energy of the nodal points relative to the\nFermi energy.\nsymmetry direction. The band structure along the M-\n\u0000-M direction, shown in Fig. 2(a), features several band\ncrossings close to the Fermi level. Thus, throughout the\nremainder of the manuscript, we only focus on the cross-\ning points, marked by blue dots in Fig. 2 (a), between\nthe majority-spin bands along the kz(\u0000\u0000M) direction.\nThese points are protected by both a mirror re\rection\nsymmetry normal to the [110] direction, M[110], and a\nfour-fold rotational symmetry, C4z, and hence can be la-\nbeled by the pair of eigenvalues, ( \u00061,\u00061), ofM[110]and\nC4z, respectively. The e\u000bective k\u0001pmodel in the basis\nfj(1;1)i;j(\u00001;\u00001)igup to order of k2in the absence ofTABLE II: The C4z-protected Weyl fermion on kzaxis.\nuc(uv) denotes the eigenvalue of C4zin conduction (valence)\nband.Cdenotes chiral charge.\nkz(\u0017A\u00001)E-EF(meV)uc=uvCDispersion\nonkx-kyplane\n0.2811 229 -1 2 k2\n-0.2811 229 -1 -2 k2\nSOC can be straightforward derived and is given by\nHNL= (m1\u0000m2k2\nz)s3+a(k2\nx\u0000k2\ny)s1; (2)\nwherekis close to the \u0000 point, m1m2>0, thesi's are\nPauli matrices and the nodal lines lie on kz=\u0006q\nm1\nm2\nandkx=\u0006ky. We have tracked the nodal lines on\ntheM[110]-invariant plane. The other nodal lines on the\nM[1\u001610]-invariant plane were determined using the C4zro-\ntational symmetry. Fig. 2(b) shows the 3D landscape of\nnodal lines in momentum space. We \fnd that the nodal\nlines are topologically nontrivial characterized by the \u0019\nBerry phase43{45. The two blue points denote the non-\ntrivial crossing points as well as the intersecting points of\nnodal lines along the kzdirection in Fig. 2(b). Notably\nthe crossing points remain gapless and robust against a\ndistortion breaking either M[110]orC4z.\nB. Weyl Nodes in the Presence of SOC\nIn the presence of SOC, the symmetry conservation\ndepends on the magnetic orientation and the crystalline\nsymmetries. More speci\fcally the [001] collinear mag-\nnetic con\fguration is invariant under (1) inversion sym-\nmetry (P), (2) fourfold rotational symmetry about the\nz-axis (C4z) and (3) mirror re\rection symmetry normal\nto thezdirection (Mz). We next discuss the e\u000bect of\nmagnetization orientation (collinear versus noncollinear)\non the topological features of the band structure.\nWeyl Nodes in Collinear Ferrimagnetism| In the\npresence of SOC, the mirror symmetry M[110]is no longer\npreserved when the magentization of the collinear fer-\nrimagentic Mn 3Ga is along the [001] direction. Con-\nsequently, in general the nodal points in Fig. 2(b) are\ngapped out except for those crossing points along kz\nwhich are protected by the C4zrotational symmetry46.\nThus, for the band structure along the C4z-invariantkz-\naxis, shown in Fig. 3(a), we can identify the states by\nthe eigenvalues of C4zand locate the nontrivial crossing\npoints associated with di\u000berent eigenvalues. The non-\ntrivial crossing points, marked by red circles in Fig. 3(a),\nare Weyl nodes protected by C4zsymmetry, whose posi-\ntion alongkz, energy relative to E F, ratio of conduction\nto valence band C4zeigenvalues, uc=uv, chiral charge, C,\nand dispersion are summarized in Table II. Interestingly,\ntheC4z-protected Weyl fermion with uc=uv= -1 carries4\n+2 -2\n+2 \n-2kz(Å -1)\nky(Å -1)E-EF(eV) \nkz(Å -1)(a) (b) \n\u0001\u0002\u0003\u0004\n\u0005 \u0001\u0006\u0002\u0004\n\u0005\nFIG. 3: (Color online) (a) Band structure along the kz-axis (\u0000\u0000Msymmetry direction) with SOC, protected by C4zsymmetry.\nThe two Weyl points, denoted with red dots, have chiral charge of \u00062. Thee\u0000i\u0019\n4andei3\u0019\n4indicate the eigenvalues of C4zfor\nthe crossing bands, respectively. (b) Two Fermi arcs on the (100) surface where green (red) color denotes the spectral weight\nof the surface (bulk) states. Solid (hollow) white circle denotes positive (negative) chiral charge, and white arrows indicate the\nFermi arcs emerging from the Weyl nodes.\nchiral charge +2 and has quadratic dispersion on the kx-\nkyplane,6in sharp contrast to the double Weyl fermion\nwith fourfold degeneracy and linear dispersion1. Its other\nparity partner has opposite chiral charge of -2. Based on\nthe above analysis, the e\u000bective k\u0001pmodel in the basis\nof theC4zeigenstates,fje\u0000i\u0019\n4i;jei3\u0019\n4ig, up to order of k2\nfor theC4z-protected Weyl nodes can be straightforward\nadapted from Ref. 6, and reads\nHWP= (m1\u0000m2k2\nz)s3+(ak2\n++bk2\n\u0000)s++(ak2\n\u0000+bk2\n+)s\u0000;\n(3)\nwherekis around the \u0000 point, k\u0006=kx\u0006iky,s\u0006=\ns1\u0006is2,jaj6=jbj,m1m2>0 and the Weyl node positions\nare atkz=\u0006q\nm1\nm2. Interestingly, if a=b,HWPreduces\ntoHNLin Eq. (2), implying that the gap opening of the\nnodal lines would close. Fig. 3(b) displays the two Fermi\narcs on the (100) surface emerging from the two charge-2\nWeyl nodes.\nEvolution of Weyl Fermions in NonCollinear\nFerrimagnetism|\nNeutron scattering experiments have reported34a non-\ncollinear ferrimagnetic magnetic structure in the DO 22\nferrimagnetic Mn 3Ga structure, where there is a signi\f-\ncant in-plane magnetic moment, \u00162b\nx= 1.19\u0016Bcarried by\nthe MnIatoms [on the Mn-Ga (001) plane] leading to a\n21\u000etilt of the Mn Imoment from the crystallographic c\naxis [see Fig. 4(a)]. This noncollinear magnetic ordering\nspontaneously breaks both the C4zandMzsymmetry op-\nerations while only preserving P. Consequently, the C4z-\nprotected double Weyl fermion on the kzaxis for the case\nof collinear ferrimagntism splits into two charge-1 Weyl\nfermions which shift away from the kzaxis.\nIn order to investigate this scenario, we have stud-\nied the evolution of the Weyl points upon rotation of\nall magnetic moments of the Mn Iatoms at the Wyck-\no\u000b positions 2b with respect to the crystallographic z\naxis by the angle \u0012,\u00162b=\u00162b(\u0000sin\u0012^x+ cos\u0012^z), while\nkz(Å-1)\nΓ+2\n-2 -1\n-1-1 +1\n+1 \n+1+1\n-1\nMnIGa\nMnII (a) (b)\nM\nMz\nx θ\nμ2bFIG. 4: (Color online) (a) Noncollinear ferrimagnetic DO 22\nstructure of Mn 3Ga,34where the Mn Iatoms [on the Mn-Ga\n(001) plane] carry a substantial in-plane magnetic moment\nleading to a tilt of their moments from the crystallographic c\naxis. (b) Evolution of Weyl nodes in the 3D BZ as a function\nof tilt angle \u0012, where the red, green and blue circles denote the\nWeyl nodes at \u0012= 180 °, 170 °and 160 °,respectively. Dashed\narrows show the motion of Weyl points with decreasing \u0012. At\n\u0012=180 °, (collinear case) the two charge-2 Weyl nodes lie on\ntheC4z-protectedkz-axis. For\u00126=180 °each charge-2 Weyl\nnode splits into two charge-1 Weyl nodes which in turn move\naway from the kz-axis. The integer above each Weyl node\ndenotes the chiral charge.\n\fxing the direction of the Mn IImagnetic moments, as\nshown in Fig. 4(a). Here, \u0012= 180 °indicates the collinear\n(001) ferrimagnetism. Using the Wannier functions we\n\fnd that at \u0012= 160 °the magnitude of the calculated\nx-component of the magnetic moment of the Mn Iatoms\nis 0.94\u0016B/Mn in good agreement with the correspond-\ning experimental values of 1.19 \u0016B. Table III summarizes\nthe comparison of the values of the magnetic moments\nof the Mn Iatom between theory and experiment. The\nmagnetic moments from the rotated Wannier approach\nagree with DFT calculations well. Fig. 4(b) shows the5\nTABLE III: Comparison of the values of the magnetic mo-\nments for the Mn Iatoms for the noncollinear case from the-\nory and experiment. \u00162b\nz(\u00162b\nx) denotes the z(x)-component\nof the magnetic moment of Mn I.\nMethod \u00162b\nz(\u0016B/Mn)\u00162b\nx(\u0016B/Mn)\u0012(°)\nrotated Wannier -2.60 -0.94 160\nGGA+SOC -2.65 -0.95 160\nexperiment34-3.07 1.19 159\nevolution of the Weyl nodes as \u0012changes from 180 °to\n170 °and \fnally to 160 °. Initially, at \u0012= 180 °, the two\ncharge-2 Weyl nodes lie on the kz-axis. As\u0012decreases\neach charge-2 Weyl node splits into two charge-1 Weyl\nnodes which move away from the kz-axis, leading to the\nemergence of four charge-1 Weyl fermions in the case\nof noncollinear ferrimagnetism. Our electronic structure\ncalculations of the Fermi arcs on the (100) surface for \u0012\n= 160 °show that the noncollinear e\u000bect is small on the\nFermi arcs in Fig. 3(b), at least for small angle.\nIV. DISCUSSION\nIn this section we discuss the e\u000bect of electron-electron\ninteractions, U, on the equilibrium lattice constants,\nmagnetic moments, and the topology of the band struc-\nture for the collinear magnetic structure. As was alluded\nearlier we employed U=2.6 eV for Mn Iatoms and U=0\nfor the remaining atoms, which were found38to give a\nvalue for the alattice constant of 3.91 \u0017A in good agree-\nment with the experimental value. However, as shown in\nTable I, this is at the expense of a worse agreement for\nboth the lattice parameter cand thez-component of the\nmagnetic moment of the Mn2b(MnI) atoms. As shown\nin Fig. 2(a) in the absence of SOC the presence of U\nshifts the position of the Weyl nodes (red dots) to higher\nenergies and slightly towards the center of BZ. Moreover,\neven in the presence of SOC, the nodes are robust and re-\nmain gapless at about the same energies. Therefore, the\ncharge-2 Weyl nodes survive in the presence of electronic\ncorrelations. Moreover, since the e\u000bect of electronic cor-\nrelations can not gap out the charge-2 Weyl nodes, the\ncharge-1 Weyl nodes splitting of the charge-2 Weyl nodesshould be robust in the noncollinear magnetic structure\nas well.\nDue to the large shift of the Weyl nodes to higher en-\nergies induced by U, it will be challenging to observe the\nFermi arcs above the Fermi level in Fig. 3(b) employing\nangle-resolved photoemission spectroscopy (ARPES). On\nthe other hand, the time-resolved ARPES (trARPES),\na fast-growing and powerful technique to observe con-\nduction electron states up to hundreds meV above the\nFermi level47,48, may be a suitable platform to observe\nthe Fermi arcs above the Fermi level on the (100) sur-\nface in future experiments. Moreover, electron doping or\nalloying that preserves the C4zsymmetry, e.g., Mn 3Ge\nin the cubic structure18, can rise the chemical potential\nwhich may allow in turn the observation of these non-\ntrivial surface states emerging from the Weyl nodes.\nV. CONCLUSION\nIn summary, our ab initio electronic structure calcula-\ntions have shown that in the absence of SOC, nontrivial\nnodal lines emerge in collinear ferrimagnetic tetragonal\nMn3Ga. The nodal lines are protected by both mirror re-\n\rection symmetry normal to the [110] direction, M[110],\nand a four-fold rotational symmetry, C4z. 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Lett. 122, 167401\n(2019)." }, { "title": "2007.11214v3.Nuclear_Magnetic_Relaxation_Time_near_Compensation_Temperature_in_Ferrimagnetic_Insulator.pdf", "content": "Journal of the Physical Society of Japan FULL PAPERS\nNuclear Magnetic Relaxation Time near the Compensation Temperature in a\nFerrimagnetic Insulator\nMichiyasu Mori\nAdvanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 117-1195, Japan\nThe nuclear magnetic relaxation time T1in a ferrimagnetic insulators is calculated within the mean-field approxima-\ntion for the magnetic exchange interactions and the Raman process involving the hyperfine interaction. We find that the\nvalue of 1 /T1on one type of site increases rapidly near the compensation temperature T0, whereas that on the other type\nof site does not increase up to Curie temperature Tc. This is due to the fact that the soft-magnon bandwidth becomes\ncomparable to T0. An increase in 1 /T1below Tcis found also in another type ferrimagnet, which shows a hump structure\nin the temperature dependence of magnetization instead of compensation. Also in that case, we find the rapid increase in\n1/T1below Tc, even though the magnetization does not show compensation. The coexistence of soft and hard magnons\nleads to these remarkable properties of ferrimagnets.\n1. Introduction\nAferri- magnet is a kind of ferro- magnet, and it was\ntheoretically predicted by N ´eel.1–3)Soon afterward, the\nmagnetization compensation was observed in the LiFeCr\nspinel ferrite, for which the magnetization becomes zero at\nmagnetization-compensation temperature TMfar below the\nCurie temperature Tc.4)Such a ferrimagnet, called an N-type\nferrimagnet, also has been found in rare-earth iron garnets\n(RIGs).2, 5–11)The RIGs have been studied by many authors in\norder to apply their magnetization-compensation properites to\nmagneto-optical memories.12–14)\nThe dynamical aspects of ferrimagnetism were initially\nstudied using the electron spin resonance (ESR).15–23)The\nferrimagnetic resonance (FIR) di \u000bers from the ferromagnetic\nresonance (FMR) in having two branches. One gives the usual\nFMR, while the other, called the exchange frequency, is lo-\ncated higher in energy.21)It was di \u000ecult to measure the ex-\nchange frequency when it was first discovered, since its wave-\nlength is of the order of a tenth of a millimeter. However,\na singular behavior of the gyromagnetic ratio was observed\naround the angular momentum compensation temperature TA\nin a LiFeCr spinel ferrite.15, 16)On the lower branch, the e \u000bec-\ntive gyromagnetic ratio becomes small around TMand then\nincreases rapidly around TA.15–23)Theg-value of the upper\nbranch becomes small in a measurable range around TA.16)\nThe magnetization is the product of the Lande g-factor and\na total angular momentum. In general, hence, TMis di\u000berent\nfrom TA, when the orbital angular momentum is involved. In\ncontrast to the magnetization, the dynamics of a ferrimagnet\nbecome singular around TA.\nBecause the magnetization couples to a magnetic field,\nwhile the total angular momentum itself does not, it can be\ndi\u000ecult to measure TAdirectly using conventional methods.\nRecently, however, Imai et al. have successfully observed TA\nusing the Barnett e \u000bect.24–26)In a rotating frame, the rota-\ntion frequency couples to the angular momentum instead of\nto the magnetization, without any coupling constant. By spin-\nrotation coupling, a magnetization is induced through the an-\ngular momentum by mechanical rotation. This was originally\nstudied by Barnett,27)and it is now used to determine TAina RIG.24, 25)It has been reported that around TAthe magneti-\nzation reverse rapidly and that domain walls move fast.28–31)\nThose properties, which are advantageous for magnetic mem-\nories, are attributed to angular-momentum compensation.\nNuclear magnetic resonance (NMR) also is a powerful tool\nfor studying the magnetism of a broad range of materials.\nMagnetic excitations can be characterized by the nuclear mag-\nnetic relaxation time T1, which originates in the hyperfine in-\nteraction between electron and nucleus. For magnetic insu-\nlators, however, the origin of T1is not so obvious. If the sys-\ntem is isotropic and the nuclear and electron quantization axes\nare identical, the relaxation cannot be obtained within the lin-\nearized spin-wave approximation. Misalignment of the quan-\ntization axes–and /or the dipole-dipole interactions between an\nelectronic and a nuclear spins–induces relaxation through the\nRaman process.32–34)Interactions among magnons are also\nthe source of relaxation, e.g., through the three-magnon pro-\ncess.32–34)Those processes can be studied for both ferromag-\nnetic and antiferromagnetic insulators. Recently, Imai et al.\nhave reported an enhancement of the NMR signal around TA,\nwhich is closely related to domain wall motion.26)In contrast\nto ESR, NMR provides a site-selective measurement of mag-\nnetism. It is therefore interesting to study the dynamical as-\npects of magnetism site-by-site in a ferrimagnet. In addition,\na consistent understanding of ferrimagnetism among experi-\nmental methods–NMR, ESR, and neutron scattering–will be\nuseful.\nIn this paper, we study the nuclear magnetic relaxation\ntime in ferrimagnets. Section 2 explains the model Hamilto-\nnian and the approximation used. The nuclear magnetic relax-\nation time due to the Raman process is given in Sec. 3. Ad-\nditional changes due to orbital angular momentum are briefly\ndiscussed in Sec. 4. Below, Bohr magneton \u0016Band Planck\nconstant ~=h=2\u0019are set equal to 1 for brevity.\n2. Formalism: Magnons in Ferrimagnet\nWe will focus on a ferrimagnetic ”insulator,” which is sim-\nply called a ”ferrimagnet” below. The magnetic exchange in-\nteraction due to the Pauli principle and to the Coulomb in-\nteraction between electrons is the source of magnetism in a\nferrimagnet. Two sub-lattices with di \u000berent spin magnitudes\n1arXiv:2007.11214v3 [cond-mat.mtrl-sci] 29 Sep 2020J. Phys. Soc. Jpn. FULL PAPERS\nSA,SBcomprise the simplest model. The Hamiltonian is\ngiven by\nH=\u0000JAX\nhi;i0i~Si\u0001~Si0\u0000JBX\nhj;j0i~Sj\u0001~Sj0+JCX\nhi;ji~Si\u0001~Sj;(1)\nwith the spin operators ~Si(~Sj) on site i2A-sites ( j2B-sites).\nThe angular brackets h\u0001\u0001\u0001i denotes nearest neighbor sites. The\nmagnitudes of the magnetic exchange interactions JA,JB, and\nJCare assumed to be positive for brevity. First, we do not con-\nsider the orbital angular momentum ~Li. Hence, there is only\none compensation temperature T0; i.e., TM=TA\u0011T0. What\nis changed by ~Liwill be discussed in the last section. At T0,\nthe expectation values hSz\nAi\u0011MA>0 andhSz\nBi\u0011\u0000 MB<0\nsatisfy MA\u0000MB=0, where the bracket denotes the thermal\naverage. See also Appendix A. It is known that there are some\npossible cases of compensation. In the case considered above,\nboth sub-lattices have the same number of sites in a unit cell,\nas shown in the inset of Fig. 1. Another case has MAnA\u0000MBnB\n=0, for which the number nAof spins on sub-lattice Ais dif-\nferent from the number nBon sub-lattice B. As shown in Ap-\npendix B, those lattice structures have characteristic features\nin common. Hence, we consider the simplest case, shown in\nFig. 1 below.\nCompensation occurs at a finite temperature. To include the\ntemperature dependences of MAandMB, we adopt the mean-\nfield approximation and use the linearized spin-wave approx-\nimation around the mean-field solution. This is equivalent to\nTyablikov decoupling in the Green’s function method and is\na kind of random-phase approximation.35, 36)The mean-field\nsolution for JA=0.1,JB=1.0,JC=0.05, SA=1, and SB=1/2, is\nshown in Fig. 1, where T0=Tc\u00180.3 and the Curie temperature\nisTc\u00183.0.\nExpectation value of spin\nFig. 1. (Color online) The mean-field solution for JA=0.1, JB=1.0,\nJC=0.05, SA=1, and SB=1/2. The inset shows the lattice structure, which\nis three dimensional. The red (upward) and blue (downward) triangles denote\nMAandMB, respectively. The black circles are the sum of two expectation\nvalues, MA\u0000MB. The broken line indicates T0.\nThe Holstein-Primako \u000b(HP) bosons (magnons)–for which\nthe creation and annihilation operators are ay\ni;aionA-\nsublattice and by\nj;bjonB-sublattice–are given by, S\u0000\ni\u0018p2MAay\ni,S+\ni\u0018p2MAai,Sz\ni=MA\u0000ay\niai,S+\nj\u0018p2MBby\nj,S\u0000\nj\u0018p2MBbj,Sz\nj=by\njbj\u0000MB. Below, the spins are assumed to\nbe ordered in the z-direction. By the linearized approximation,\nthe action of the magnons is given by37)\nS=X\nq;i!n\by\" \u0000i!n 0\n0 i!n!\n+ \"1q\"\u0003\n3q\n\"3q\"2q!#\n\b;(2)\n\by\u0011\u0010\nay\nq(i!n);b\u0000q(\u0000i!n)\u0011\n; (3)\n\"1q\u0011zJCMB+zAJAMA(1\u0000\u0010Aq); (4)\n\"2q\u0011zJCMA+zBJBMB(1\u0000\u0010Bq); (5)\n\"3q\u0011JCp\nMAMBX\n\u0018eiq\u0001\u0018; (6)\n\u0010A(B)q\u00111\nzA(B)X\n\u0011cos(q\u0001\u0011); (7)\nWe use the boson operators aq(i!n) and b\u0000q(\u0000i!n) with mo-\nmentum q=(qx;qy;qz) and Matsubara frequency !n. The q-\nsummation is taken over the first Brillouin zone. The magnon\ndispersion relation depends on the connectivity of the sub-\nlattice, which gives the number of nearest-neighbor sites zA\nand zBon each sub-lattice, and the number zof nearest-\nneighbor sites between the two sub-lattices. In Eqs. (6) and\n(7),\u0018and\u0011mean the summation over the nearest-neighbor\nsites between the two sub-lattices and within each sublat-\ntice, respectively. From Eq. (2), the magnon Green’s functions\ng\u0017(q;i!n) are given by\ngA(q;i!n)\u0011haq(i!n)ay\nq(i!n)i\n=\u0000i!n+\"2q\u0010\ni!n\u0000E\u000bq\u0011\u0010\ni!n+E\fq\u0011; (8)\ngB(q;i!n)\u0011hbq(i!n)by\nq(i!n)i\n=\u0000i!n+\"1q\u0010\ni!n\u0000E\fq\u0011\u0010\ni!n+E\u000bq\u0011; (9)\nand the magnon dispersion relations E\u000bqandE\fqare given by\nE\u000bq=1\n22666664\u0010\n\"1q\u0000\"2q\u0011\n+r\u0010\n\"1q+\"2q\u00112\u00004\f\f\f\"3;q\f\f\f23777775; (10)\nE\fq=1\n22666664\u0000\u0010\n\"1q\u0000\"2q\u0011\n+r\u0010\n\"1q+\"2q\u00112\u00004\f\f\f\"3;q\f\f\f23777775:(11)\nThe small- Qapproximation for MA>MBleads to,\nE\u000bq\u0018CQ2; (12)\nE\fq\u001812JC(MA\u0000MB)+DQ2; (13)\nwhere CandDare constants given in Appendix B and Q\n\u0011q\nq2x+q2y+q2z. The mode E\u000bqis gapless, while E\fqhas an\n”optical gap,” Eg\u0011jE\u000bq=0\u0000E\fq=0j=12JC(MA\u0000MB), which\ndisappears at T0. The dispersion relations degenerate at the\ngamma point and increases linearly with Q, similar to an anti-\nferromagnet. Away from the gamma point, on the other hand,\nthe two dispersion relations deviate from each other:\nE\u000bq\u0018C1Q+C2Q2; (14)\nE\fq\u0018C1Q\u0000C2Q2; (15)\n2J. Phys. Soc. Jpn. FULL PAPERS\nwhere C1andC2are constants given in Appendix B. These\nQ-dependences are relevant to the temperature dependence of\nT1at low temperatures.\n3. Results: Nuclear Magnetic Relaxation due to the Ra-\nman Process\nIn this study, we consider the nuclear magnetic relaxation\ntime T1originating from the contact interaction between a nu-\ncleus and an electron\nHn\u0000el=1\n2X\n\u0017=A;B2666664\r\u0017X\nifi\u0017\u0010~Si\u0001~Ii\u00113777775; (16)\nwith the g-factor on\u0017-sites being given by \r\u0017and the nuclear\nspin~Iioni-site. If the system is isotropic and the quantiza-\ntion axes of the nucleus and electron are identical, we cannot\nobtain relaxation within the linearized approximation. Mis-\nalignment of the quantization axes–and /or the dipole-dipole\ninteractions between electronic and nuclear spins–will in-\nduce relaxation due to the Raman process.32–34)Interactions\namong magnons also cause relaxation, e.g., through the three\nmagnon-process.32–34)Below, we focus on the Raman process\ninduced by misalignment. This is su \u000ecient to enable us to\nfind some of the characteristics of T1near T0. The critical ex-\nponent of T1is beyond the scope of this study and will be\ndiscussed elsewhere. When the quantization axis of nucleus\ndeviates by an angle \u0012from that of the electron, Eq. (16) re-\nduces to the following component\nHz\nn\u0000el=1\n2X\n\u0017=A;B2666664\r\u0017X\nisin\u0012fi\u0017Sz\ni\u0000I+\ni+I\u0000\ni\u00013777775; (17)\nwhich are relevant for calculating T1. Assuming that \rA=\rB\n\u0011\rand the form factors fiAare constant, i.e., fiA=fiB\u0011f,\ntheT1on site\u0017=A;Bis given by\n1\nT1\u0017=FX\nqC\u0017(q;!0); (18)\nC\u0017(q;!0)=Z\ndt ei!0tD\nSz\n\u0017q(t)Sz\n\u0017;\u0000q(0)+Sz\n\u0017;\u0000q(t)Sz\n\u0017q(0)E\n;\n(19)\nwhereh\u0001\u0001\u0001i means the thermal average. The nuclear magnetic\nresonance energy is denoted by !0, and F=(\rfsin\u0012=2)2.\nUsing Eqs. (8) and (9), the spin-spin correlation function\nC\u0017(q;!0) is given by\nC\u0017(q;!0)=2\n1\u0000e!0=kBTIm\u0005R\n\u0017(q;!0); (20)\n\u0005\u0017(q;i!0)=kBTX\np;ng\u0017(p+q;i!n+i!0)g\u0017(p;i!n);(21)\nwhere Tis temperature, kBis the Boltzmann’s constant, and\nmomentum p=(px,py,pz) in the first Brillouin zone. The re-\ntarded function of \u0005\u0017(q;i!0) is denoted by \u0005R\n\u0017(q;!0). When\n!0is much smaller than kBT, the nuclear magnetic relaxation\ntime T1\u0017on site\u0017is given by\n1\nT1\u0017=2FX\nqlim\n!0!0kBT\n!0Im\u0005R\n\u0017(q;!0); (22)\n=2\u0019FX\np;q\u001a\nnB(E\u0017p)h\nnB(E\u0017p)+1i\u000bp\u000bq\n\u0001p\u0001q\u000e\u0010\nE\u0017p\u0000E\u0017q\u0011+nB(E\u0016p)h\nnB(E\u0016p)+1i\fp\fq\n\u0001p\u0001q\u000e\u0010\nE\u0016p\u0000E\u0016q\u0011\u001b\n; (23)\n\u000bp\n\u0001p=1\n2 \"1p+\"2p\n\u0001p+1!\n; (24)\n\fp\n\u0001p=1\n2 \"1p+\"2p\n\u0001p\u00001!\n; (25)\n\u0001p=r\u0010\n\"1p+\"2p\u00112\u00004\f\f\f\"3p\f\f\f2; (26)\nHere\u0016,\u0017, i.e.,\u0016=\u000bfor\u0017=\for\u0016=\ffor\u0017=\u000band the Bose\ndistribution function is denoted by nB(x)\u00111=[ex=(kBT)\u00001].\nNote that Eq. (23) can be checked by considering the ferro-\nand the antiferromagnet cases as discussed in Appendix C.\nUsing Eq. (23) and the mean-field solution shown in Fig. 1,\ntheT-dependence of 1 /T1\u0017can be calculated numerically as\nshown in Fig. 2 (a). Note that 1 =T1;Aincreases rapidly around\nFig. 2. (Color online) (a) The T-dependence of 1 =T1on the A-sites and B-\nsites (1 /T1Aand 1 /T1B) are plotted with upward triangles (red) and downward\ntriangles (blue), respectively. The mean-field solution for JA=0.1, JB=1.0,\nJC=0.05, SA=1, and SB=1/2 was used. The broken line indicates T0. See\nalso Fig. 1. (b) At low temperatures, T1Ais well fitted by T2(the dashed-\ndotted line) similar to the ferromagnet (See also Appendix C), whereas it is\ndeviated with increasing temperature as AT2+BT5(solid line). AandBare\nconstants.\nT=Tc\u00180.3, which corresponds to T0indicated by the bro-\nken line in Fig. 2. This contrasts sharply with 1 =T1B, which\ndiverges just below Tc. As shown in Fig. 2 (b), at low tem-\nperatures, T1Ais well fitted by T2, similar to the ferromagnet\n3J. Phys. Soc. Jpn. FULL PAPERS\n(See also Appendix C). With increasing temperature, on the\nother hand, it is fitted by AT2+BT5with constants AandB.\nThis is similar to the behavior of a ferromagnet, except for the\nfact that 1=T1Aincreases around T0instead of Tc.\nTo understand this behavior of 1 /T1around T0,E\u000bqandE\fq\nare plotted in Fig. 3 for (a) T=Tc=0.1 and (b) T=Tc=0.3 with\nqy=qz=0. At low temperatures, the T-dependence of 1 /T1\nFig. 3. (Color online) The dispersion relations E\u000bqandE\fqare plotted by\nupward triangles (red) and downward triangles (blue), respectively, for (a)\nT=Tc=0.1 and (b) T=Tc=0.3, with qy=qz=0. In the pre-set lattice struc-\nture, at q=(\u0019;\u0019= 2;0),E\u000bqis a maximum. This can be approximated by its\nvalue at q=(\u0019;0;0) due to the small value of JC. The low-energy region is\nenlarged and plotted in the insets.\nis determined by the Q2-dependences of E\u000bqandE\fqaround\nQ\u00180. See also the inset of Fig. 3 (a). At T=T0,Egbecomes\nzero, as shown in the inset of Fig. 3 (b), and both E\u000bqand\nE\fqbecome proportional to Qinstead of Q2around Q=0.\nNote that kBT=JBis shown by the broken line in Figs. 3 (a)\nand (b) as a measure of the temperature. In the pre-set lattice\nstructure, E\u000bqis a maximum at q=(\u0019;\u0019= 2;0). It can be ap-\nproximated by the value at q=(\u0019;0;0), since the small value\nofJC=0.05. We then find that at T=T0the bandwidth of\nE\u000bqbecomes comparable to kBT. This means that all states of\nE\u000bqcontribute to 1 /T1through nB(x) in the first term in Eq.\n(23), where the first term is dominant. This is the origin of\nthe rapid increase in 1 /T1AatT0. This is not accidental, due\nto the following points. The bandwidth of E\u000bqcan be very\nroughly estimated by \"1;q=(\u0019;0;0)\u0018(6JC+16JA)M0\u00188JA=0.8\nwith MA=MB\u0011M0\u00180.5. On the other hand, T0can be\nroughly estimated as T0\u0018zAJAXA=0.8. Since MAis soft and\ndecreases rapidly with T,T0is close to the Curie tempera-\nture of a system limited to the A-sublattice. Therefore, 1 /T1Arapidly increases around T0.\nSo far, we have discussed an N-type ferrimagnet.1, 3)An-\nother type of ferrimagnet–called P-type–shows a hump in the\ntemperature dependence of the magnetization instead of com-\npensation. Figure 4 is calculated from Eq. (1) for JA=0.5,\nJB=1.0,JC=0.2,SA=1/2, and SB=1. The lattice structure is\nthe same as the inset of Fig. 1. In a P-type ferrimagnet, the\nFig. 4. (Color online) The mean-field solution for a P-type ferrimagnet\nwith JA=0.5,JB=1.0,JC=0.2,SA=1/2, and SB=1.\nmagnetization does not show any singular behavior, such as\ncompensation, although it is composed of two di \u000berent sub-\nlattices, i.e., with soft and hard dispersion relations E\u000bqand\nE\fq. The value of 1 /T1\u0017in aP-type ferrimagnet is plotted in\nFig. 5 in the same way as for the N-type. We find that 1 /T1A\nFig. 5. (Color online) The relaxation rate 1 /T1in aP-type ferrimagnet. The\nT-dependences of 1 =T1on the A-sites and B-sites (1 /T1Aand 1 /T1B) are plot-\nted by upward triangles (red) and downward triangles (blue), respectively.\nThe mean-field solution for JA=0.5,JB=1.0,JC=0.2,SA=1/2, and SB=1 was\nused. The broken line is near the top of the hump structure.\nrapidly increases around T=Tc\u00180.4 close to the top of the\nhump structure, while 1 /T1Bincreases rapidly near Tc. It is\nnow straightforward to understand this behavior, since the A-\nsublattice is soft and the B-sublattice is hard. This is clearly\nshown by the dispersion relation in Figs. 6 for (a) T=Tc=0.1\n4J. Phys. Soc. Jpn. FULL PAPERS\nFig. 6. (Color online) The dispersion relations E\u000bqandE\fqare plotted by\nupward triangles (red) and downward triangles (blue), respectively, for (a)\nT=Tc=0.1 and (b) T=Tc=0.4, with qy=qz=0. The broken line indicates\nthe corresponding temperature kBT=JB.\nand (b) T=Tc=0.4. In each panel, kBT=JBis shown by the\nbroken line as measure of the temperature. At T=Tc=0.4, all\nofE\u000bqcontribute to 1 /T1A. Therefore, even in a P-type fer-\nrimagnet without compensation, we find a rapid increase of\n1/T1.\n4. Summary and Discussions\nWe have studied T1in a ferrimagnetic insulator that is in-\nduced by the Raman process involving the hyperfine interac-\ntion. To calculate 1 /T1, we adopted a Heisenberg model com-\nposed of two sublattices and used the linearized spin-wave ap-\nproximation around the mean-field solution. At T00\ncan be approximated as17–23)\n\n\u000b\u0018\rAhJz\nAi\u0000\rBhJz\nBi\nhJz\nAi\u0000hJz\nBiH\u0011\re\u000bH; (D\u00016)\n\n\f\u0018\u0015\u0010\nhJz\nAi\u0000hJz\nBi\u0011\n\u0000\rBhJz\nAi\u0000\rAhJz\nBi)\nhJz\nAi\u0000hJz\nBiH; (D\u00017)\nwith the e \u000bective gyromagnetic ratio \re\u000b. AtTA, note that the\ntwo frequencies become close each other, since \n\u000b\u0000\n\f=\n(\rA+\rB)H.\n1) L. N ´eel, Ann. Phys. 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G. de Gennes, Comptes Rendus 247, 1836 (1958).\n40) A. Szutuła, and J. Leciejewicz, in Handbook on the Physics and Chem-\nistry of Rare Earths vol. 12, eds. K. A. Gschneider, Jr. and L. Eyring\n(Elsevier, Amsterdam, 1989) p. 131.\n41) J. S. Plant, Journal of Physics C: Solid State Physics 10, 4805 (1977).\n42) Y . Nambu, J. Barker, Y . Okino, T. Kikkawa, Y . Shiomi, M. Enderle, T.\nWeber, B. Winn, M. Graves-Brook, J.M. Tranquada, T. Ziman, M. Fu-\njita, G.E.W. Bauer, E. Saitoh, K. Kakurai, Phys. Rev. Lett. 125, 027201\n(2020).\n8" }, { "title": "2308.12410v1.Magnetic_analogue_of_liquid_gas_phase_transition_of_water__case_study_of_a_spin_1_2_Ising_Heisenberg_model_on_a_diamond_decorated_square_lattice.pdf", "content": "1\nMAGNETIC ANALOGUE OF LIQUID-GAS PHASE TRANSITION OF WATER:\nCASE STUDY OF A SPIN-1/2 ISING-HEISENBERG MODEL ON A DIAMOND-\nDECORATED SQUARE LATTICE\nJ. Streˇ cka, jozef.strecka@upjs.sk, K. Karl’ov´ a, katarina.karlova@upjs.sk, Institute of Physics, Faculty of Science, P. J.\nˇSaf´ arik University, Park Angelinum 9, 04001 Koˇ sice, Slovakia, T. Verkholyak, werch@icmp.lviv.ua, Institute for Condensed\nMatter Physics, NASU, Svientsitskii Street 1, 790 11, L’viv, Ukraine, N. Caci, caci@physik.rwth-aachen.de, S. Wessel,\nwessel@physik.rwth-aachen.de, Institute for Theoretical Solid State Physics, JARA FIT, and JARA CSD, RWTH Aachen\nUniversity, 52056 Aachen, Germany and A. Honecker, andreas.honecker@cyu.fr, Laboratoire de Physique Th´ eorique et\nMod´ elisation, CNRS UMR 8089, CY Cergy Paris Universit´ e, Cergy-Pontoise, France\nINTRODUCTION\nPhase transitions of diverse physical systems have\ncaptured significant attention due to the occurrence of\nabrupt discontinuities or divergences in several physi-\ncal quantities, which consequently preclude the proper\ndefinition at the relevant phase transitions [1]. One\nof the most extensively studied physical substances is\nwater, whose phase diagram encompasses solid, liq-\nuid, and gaseous phases separated from each other\nby lines of discontinuous phase transitions. Among\nthese phase-transition lines, the most intriguing is the\nline of liquid-gas phase transitions, which starts at a\ntriple coexistence point of all three phases and ends\nat the critical point where a discontinuous phase tran-\nsition changes to a continuous one. A similar phase-\ntransition line has been recently found in the pressure-\ntemperature phase diagram of the quantum magnetic\nmaterial SrCu 2(BO 3)2, which provides an experimen-\ntal realization of the spin-1/2 Heisenberg model on the\nShastry-Sutherland lattice [2].\nThe line of discontinuous phase transitions termi-\nnating at an Ising critical point is not unique to the\nShastry-Sutherland model, but it may be encountered\nin other frustrated two-dimensional quantum spin sys-\ntems such as the spin-1/2 Heisenberg model on a\nfully frustrated bilayer [3–5], trilayer [6] and diamond-\ndecorated square lattice [7]. To capture thermal phase\ntransitions of the spin-1/2 Heisenberg model on frus-\ntrated two-dimensional lattices one has to resort to\nstate-of-the-art numerical calculations as for instance\nquantum Monte Carlo simulations in a dimer or trimer\nbasis in order avoid the notorious sign problem [8]. In-\nterestingly, it has been recently verified that the sim-\npler spin-1/2 Ising-Heisenberg model on the diamond-\ndecorated square lattice already captures the essential\nfeatures of thermal phase transitions of the spin-1/2\nHeisenberg model on the diamond-decorated square\nlattice [7,9].\nThe spin-1/2 Ising-Heisenberg diamond-decorated\nsquare lattice in a magnetic field can be rigorously\nmapped to an effective zero-field spin-1/2 Ising square\nlattice in a particular parameter subspace correspond-\ning to the thermal phase transitions [9]. From this per-\nspective, the spin-1/2 Ising-Heisenberg model on the\ndiamond-decorated square lattice represents a valu-\nable example in the realm of exactly solved lattice-\nstatistical models [10], which allows us to rigorously\nexamine the thermal phase transitions in the pres-\nence of an external magnetic field quite similarly as\nS 1,i,jJ2\nJ1S 2,i,jS 3,i,jS 4,i,jS 5,i,jFig. 1. Schematic illustration of the spin-1/2 Ising-\nHeisenberg model on a diamond-decorated square lattice.\nLarge (light blue) circles determine nodal lattice sites oc-\ncupied by the Ising spins S1,i,j, while small (violet) circles\ndetermine decorating lattice sites occupied by the Heisen-\nberg spins Sk,i,j(k= 2−5).\nfor Fisher’s superexchange antiferromagnet [11] and its\ndifferent variants [12–16].\nIn the present paper we provide a more compre-\nhensive understanding of the thermal phase transitions\nof the spin-1/2 Ising-Heisenberg diamond-decorated\nsquare lattice. Except temperature and magnetic-field\nvariations of the local and total magnetization, we will\nbring insight into the respective changes of the mag-\nnetic susceptibility and specific heat at the discontin-\nuous as well as continuous thermal phase transitions.\nThe paper is organized as follows. In the following\nsection we will briefly recall the definition of the stud-\nied quantum spin model and review the mapping to\nthe effective spin-1/2 Ising model on a square lattice.\nThe subsequent section is devoted to a detailed de-\nscription of basic magnetic and thermodynamic quan-\ntities (magnetization, susceptibility, and specific heat)\nin close vicinity of the thermal phase transitions. Fi-\nnally, the most important findings are summarized in\nthe concluding part.\nISING-HEISENBERG DIAMOND-DECORA-\nTED SQUARE LATTICE\nAt first, let us recall the definition of the spin-\n1/2 Ising-Heisenberg model on the diamond-decorated\nsquare lattice, which is schematically illustrated in2\nFig. 1 and mathematically given by the Hamiltonian\nˆH=J1LX\ni=1LX\nj=1h\u0010\nˆSz\n1,i,j+ˆSz\n1,i+1,j\u0011\u0010\nˆSz\n2,i,j+ˆSz\n3,i,j\u0011\n+\u0010\nˆSz\n1,i,j+ˆSz\n1,i,j+1\u0011\u0010\nˆSz\n4,i,j+ˆSz\n5,i,j\u0011i\n+J2LX\ni=1LX\nj=1\u0010\nˆS2,i,j·ˆS3,i,j+ˆS4,i,j·ˆS5,i,j\u0011\n−h5X\nk=1LX\ni=1LX\nj=1ˆSz\nk,i,j. (1)\nThe first term accounts for an anisotropic (Ising-type)\nexchange interaction J1between nearest-neighbor Ising\nand Heisenberg spins schematically shown in Fig. 1\nas large and small circles, respectively, while the sec-\nond term stands for the isotropic exchange interac-\ntion between nearest-neighbor Heisenberg spins. The\nlast term takes into account the Zeeman energy of the\nHeisenberg and Ising spins in an external magnetic field\nhandLdenotes the linear size of the considered two-\ndimensional lattice.\nIt has been verified in our previous paper [9] that\nthe partition function of the spin-1/2 Ising-Heisenberg\nmodel on the diamond-decorated square lattice given\nby the Hamiltonian (1) can be related via the general-\nized decoration-iteration transformation [17–19] to the\npartition function of the effective spin-1/2 Ising model\non a square lattice:\nZ(β, J1, J2, h) =A2NZeff(β, Jeff, heff), (2)\nwhich is defined through the effective Hamiltonian in-\nvolving temperature-dependent nearest-neighbor inter-\nactions Jeffand magnetic field heff:\nHeff=−JeffLX\ni=1LX\nj=1(Sz\n1,i,jSz\n1,i+1,j+Sz\n1,i,jSz\n1,i,j+1)\n−heffLX\ni=1LX\nj=1Sz\n1,i,j. (3)\nAn explicit form of the mapping parameters A,Jeff,\nandheffis given by Eqs. (7)–(10) in Ref. [9]. It follows\nfrom the exact mapping between the partition func-\ntions (2) that the spin-1/2 Ising-Heisenberg model on\nthe diamond-decorated square lattice becomes exactly\nsoluble in the particular parameter subspace with zero\neffective field heff= 0 due to the famous exact solution\nby Onsager [20], while in the parameter space with\nnonzero effective field heff̸= 0 one may obtain precise\nnumerical results by exploiting classical Monte Carlo\nsimulations.\nExcept for a trivial fully saturated paramagnetic\nphase, the spin-1/2 Ising-Heisenberg model on the\ndiamond-decorated square lattice displays two further\nground states, which can be classified as the classical\nferrimagnetic (FRI) phase:\n|FRI⟩=LY\ni,j=1|↓1,i,j⟩⊗|↑ 2,i,j↑3,i,j⟩⊗|↑ 4,i,j↑5,i,j⟩(4)and the quantum monomer-dimer (MD) phase:\n|MD⟩=LY\ni,j=1|↑1,i,j⟩ ⊗1√\n2(|↑2,i,j↓3,i,j⟩ − |↓ 2,i,j↑3,i,j⟩)\n⊗1√\n2(|↑4,i,j↓5,i,j⟩ − |↓ 4,i,j↑5,i,j⟩). (5)\nAt zero temperature the FRI and MD ground states\ncoexist along the phase boundary given by:\nhMD−FRI= 2(J2−J1), J 2∈[J1,3J1]. (6)\nThe ground-state phase boundary (6) represents a\nlower boundary for the wall of discontinuous (first-\norder) phase transitions between the FRI and MD\nphases, which are bounded from above by a line of con-\ntinuous (second-order) phase transitions from the uni-\nversality class of the two-dimensional Ising model [20]\nas dictated by the constraint βcJeff= 2 ln(1 +√\n2) and\nheff= 0.\nTHERMAL PHASE TRANSITIONS BE-\nTWEEN FERRIMAGNETIC AND MONO-\nMER-DIMER PHASES\nIn this section our particular attention will be fo-\ncused on typical signatures of discontinuous and con-\ntinuous thermal phase transitions between the FRI\nand MD phases. To this end, we will limit our fur-\nther analysis to the spin-1/2 Ising-Heisenberg model\non the diamond-decorated square lattice with the spe-\ncific value of the interaction ratio J2/J1= 1.5, which\ncaptures all typical features of both types of thermal\nphase transitions emergent in the close vicinity of the\ncoexistence point between the FRI and MD ground\nstates ( hMD−FRI/J1= 1 for J2/J1= 1.5).\nFigure 2 reports the finite-temperature phase dia-\ngram of the spin-1/2 Ising-Heisenberg model on the\ndiamond-like decorated lattice in the magnetic field\nversus temperature plane for this ratio. It can be seen\nfrom this figure that the line of discontinuous ther-\nmal phase transitions between the FRI and MD phases\n(blue broken line) actually starts from the relevant\ncoexistence point h/J1= 1 before it bends towards\nlower magnetic fields as temperature rises. This line\nof discontinuous thermal phase transitions finally ter-\nminates at the Ising critical point at h/J1= 0.9279\nandkBT/J 1= 0.2403 for J2/J1= 1.5, which can be\nascribed to a continuous thermal phase transition be-\ntween the FRI and MD phases (red circle).\nNext, let us illustrate a few typical features of\nthe magnetization and magnetic susceptibility when\nthe magnetic field drives the spin-1/2 Ising-Heisenberg\nmodel on the diamond-like decorated lattice across the\nthermal phase transition. For this purpose, a few typ-\nical magnetic-field dependencies of the local and total\nmagnetization are plotted in Fig. 3 for three selected\ntemperatures. The calculated values of the local mag-\nnetization of the Ising spins mI(red lines) and the local\nmagnetization of the Heisenberg spins mH(blue lines)3\n0.850.900.951.001.051.100.00.10.20.3M\nD FRI[0.9279; 0.2403]h\n / J1J2 / J1 = 1.5 \nkB T / J1\nFig. 2. Finite-temperature phase diagram in the magnetic\nfield versus temperature plane for the fixed value of the\ninteraction ratio J2/J1= 1.5. The blue broken curve dis-\nplays the line of discontinuous thermal phase transitions\nbetween the quantum monomer-dimer (MD) phase and the\nclassical ferrimagnetic (FRI) phase, which terminates at\nthe Ising critical point (red circle) with the coordinates\nh/J 1= 0.9279 and kBT/J 1= 0.2403.\nare indeed consistent with the presence of the MD and\nFRI phase for h/J1≲1 and h/J1≳1, respectively.\nAt the lowest temperature kBT/J 1= 0.2 one detects a\ndiscontinuous thermal phase transition from the MD\nphase towards the FRI phase accompanied with an\nabrupt magnetization jump [see Fig. 3(a)]. The dis-\ncontinuous magnetization jump gradually shrinks upon\nincreasing temperature until it completely vanishes in\nvicinity of the critical temperature kBT/J 1≈0.2403,\nwhere the magnetization discontinuity is replaced by\nan inflection point with a vertical (infinite) tangent [see\nFig. 3(b)]. The inflection point finally acquires a finite\nslope at the even higher temperature kBT/J 1= 0.3,\nwhere one finds a rather simple crossover from the\nMD phase towards the FRI phase. A similar crossover\ncan be observed around the saturation field h/J1= 4,\nwhere the local and total magnetization undergo a sub-\nstantial magnetic-field-driven change from the values\ntypical for the FRI phase towards their fully saturated\nvalues.\nAll aforementioned features of the isothermal mag-\nnetization curves of the spin-1/2 Ising-Heisenberg\ndiamond-decorated square lattice can be unambigu-\nously corroborated by the magnetic-field-induced\nchanges of the isothermal magnetic susceptibility pre-\nsented in Fig. 4. The finite cusp of the magnetic\nsusceptibility encountered at the lowest temperature\nkBT/J 1= 0.2 verifies the existence of a discontin-\nuous thermal phase transition between the MD and\nFRI phases [Fig. 4(a)]. On the other hand, there are\nstrong indications that in proximity of the critical tem-\nperature kBT/J 1≈0.2403 of the continuous thermal\nphase transition the magnetic susceptibility displays a\npronounced power-law divergence [Fig. 4(b)]. Last but\nnot least, the magnetic susceptibility exhibits a rather\n01 2 3 4 5 -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5m\nTmHm\nI MD FRIk\nB T / J1 = 0.2J\n2 / J1 = 1.5 \nmI , mH , mT(\na) h / J1\n01 2 3 4 5 -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5m\nTmHm\nI MD FRIk\nB T / J1 = 0.2403J\n2 / J1 = 1.5 \nmI , mH , mT(\nb) h / J1\n01 2 3 4 5 -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5m\nTmHm\nI k\nB T / J1 = 0.3J\n2 / J1 = 1.5 \nmI , mH , mT(\nc) h / J1Fig. 3. A few typical magnetic-field dependencies of the\nlocal and total magnetization for the fixed value of the in-\nteraction parameter J2/J1= 1.5 and three selected values\nof temperature: (a) kBT/J 1= 0.2; (b) kBT/J 1= 0.2403;\n(c)kBT/J 1= 0.3.\nsharp but rounded finite maximum at higher tempera-\ntures such as kBT/J 1= 0.3 [see the inset in Fig. 4(c)].\nIt is also worth mentioning that the magnetic suscepti-\nbility displays for arbitrary temperature another round\nmaximum located around h/J1≈4, which confirms a\nsimple crossover from the FRI phase towards the fully\nsaturated paramagnetic phase rather than a true phase\ntransition.\nIn the following we will examine in detail typical4\n01 2 3 4 5 0.00.20.40.60.81.0M\nD FRIkB T / J1 = 0.2 J\n2 / J1 = 1.5 \n/s61539T(\na) h / J1\n01 2 3 4 5 0.00.20.40.60.81.0M\nD FRIkB T / J1 = 0.2403 J\n2 / J1 = 1.5 \n/s61539T(\nb) h / J1\n0123450.00.20.40.60.81.0M\nD k\nB T / J1 = 0.3 J\n2 / J1 = 1.5 \n/s61539T(\nc) h / J10.70.80.91.01.10.00.20.40.60.81.0 \n \nFig. 4. A few typical magnetic-field dependencies of the\nisothermal magnetic susceptibility for the fixed value of the\ninteraction parameter J2/J1= 1.5 and three selected values\nof temperature: (a) kBT/J 1= 0.2; (b) kBT/J 1= 0.2403;\n(c)kBT/J 1= 0.3.\nfeatures of the magnetization and magnetic specific\nheat induced by temperature , when driving the spin-\n1/2 Ising-Heisenberg model on the diamond-decorated\nsquare lattice across the thermal phase transitions. To\nstart with, Fig. 5 depicts typical temperature varia-\ntions of the local and total magnetization for three\n0.100.150.200.250.300.350.40-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5m\nTm\nH mI M\nD FRIh / J1 = 0.95J2 / J1 = 1.5 \nmI , mH , mT(\na) kB T / J1\n0.10 .20 .30 .40 .5-0.4-0.3-0.2-0.10.00.10.20.30.40.5m\nHm\nTm\nIMD FRIh / J1 = 0.9279J2 / J1 = 1.5 \nmI , mH , mT(\nb) kB T / J1\n0.10 .20 .30 .40 .5-0.3-0.2-0.10.00.10.20.30.40.5m\nTmHm\nI MDh / J1 = 0.9J2 / J1 = 1.5 \nmI , mH , mT(\nc) kB T / J1Fig. 5. A few typical temperature variations of the local and\ntotal magnetization for the fixed value of the interaction\nratio J2/J1= 1.5 and three different magnetic fields: (a)\nh/J 1= 0.95; (b) h/J 1= 0.9279; (c) h/J 1= 0.9.\nselected values of the magnetic field. The discontinu-\nous nature of the local and total magnetization, ob-\nservable at the selected value of the magnetic field\nh/J1= 0.95, corroborates the presence of a discon-\ntinuous thermal phase transition between the FRI and\nMD phases [Fig. 5(a)]. If the magnetic field is fixed suf-\nficiently close to the critical value h/J1≈0.9279, the\nlocal and total magnetization show an inflection point\nwith a vertical tangent serving in evidence of the con-\ntinuous thermal phase transition between the FRI and5\n0.10 .20 .30 .40.00.51.01.52.0h\n / J1 = 0.95M\nD FRIJ2 / J1 = 1.5 \nC / N kB (\na) kB T / J1\n0.10 .20 .30 .40246810h\n / J1 = 0.9279M\nD FRIJ2 / J1 = 1.5 \nC / N kB (\nb) kB T / J1\n0.10 .20 .30 .40246810h\n / J1 = 0.9M\nDJ2 / J1 = 1.5 \nC / N kB (\nc) kB T / J1\nFig. 6. A few typical temperature variations of the mag-\nnetic specific heat for the fixed value of the interaction\nratio J2/J1= 1.5 and three different magnetic fields: (a)\nh/J 1= 0.95; (b) h/J 1= 0.9279; (c) h/J 1= 0.9.\nMD phases [Fig. 5(b)]. At even lower magnetic field\nh/J1= 0.9 the local and total magnetization display a\nsmooth continuous thermally-assisted change due to a\ncrossover from the MD phase towards the FRI phase.\nLet us conclude our discussion by a detailed anal-\nysis of the temperature dependencies of the magnetic\nspecific heat of the spin-1/2 Ising-Heisenberg model\non the diamond-decorated square lattice presented inFig. 6. The finite cusp of the magnetic specific heat seen\nin Fig. 6(a) around the temperature kBT/J 1≈0.216\nevidences the discontinuous thermal phase transition\nfor the relevant choice of the magnetic field h/J1=\n0.95. Contrary to this, the temperature dependence\nof the magnetic specific heat is quite reminiscent of\na steep logarithmic divergence if the magnetic field\nh/J1≈0.9279 is tuned sufficiently close to the contin-\nuous thermal phase transition [Fig. 6(b)]. Finally, the\nmagnetic specific heat may be completely free of any\nthermal phase transition as exemplified by the tem-\nperature dependence with a sizable but finite round\nmaximum shown in Fig. 6(c) for the magnetic field\nh/J1= 0.9.\nCONCLUSION\nIn the present article we have examined in detail\ndiscontinuous and continuous thermal phase transi-\ntions of the spin-1/2 Ising-Heisenberg model on the\ndiamond-decorated square lattice between the classical\nFRI phase and the quantum MD phase in the presence\nof an external magnetic field. It has been demonstrated\nthat the spin-1/2 Ising-Heisenberg diamond-decorated\nsquare lattice shows for a fixed value of the interac-\ntion ratio a line of discontinuous thermal phase transi-\ntions, which terminates at an Ising-type critical point\ninherent to a continuous thermal phase transition. It\ncould be thus concluded that the phase boundary be-\ntween the FRI and MD phases of the spin-1/2 Ising-\nHeisenberg model on the diamond-decorated square\nlattice is reminiscent of the liquid-gas phase boundary\nof water. In addition, the results for discontinuous and\ncontinuous thermal phase transitions of the spin-1/2\nIsing-Heisenberg diamond-decorated square lattice are\nexact as they can be directly descended from a rigorous\nmapping correspondence to an exactly solved spin-1/2\nIsing model on a square lattice with a temperature-\ndependent nearest-neighbor interaction Jeff̸= 0 and\nzero effective field heff= 0 [20]. Out of the param-\neter region corresponding to thermal phase transi-\ntions the spin-1/2 Ising-Heisenberg diamond-decorated\nsquare lattice can be rigorously mapped onto the spin-\n1/2 Ising model on a square lattice with a nonzero\nnearest-neighbor interaction Jeff̸= 0 and effective field\nheff̸= 0, which can be subsequently treated by classical\nMonte Carlo simulations.\nBy making use of extensive Monte Carlo simu-\nlations we have evidenced that the spin-1/2 Ising-\nHeisenberg model on the diamond-decorated square\nlattice displays a discontinuous magnetization jump\nwhen temperature or magnetic field drives the investi-\ngated spin system across a discontinuous thermal phase\ntransition. The magnetic susceptibility and specific\nheat consequently exhibit at the discontinuous thermal\nphase transitions finite cusps related to a discontinuous\nchange of both quantities. On the contrary, the mag-\nnetization of the spin-1/2 Ising-Heisenberg diamond-\ndecorated square lattice varies continuously when tem-6\nperature or magnetic-field changes drive the investi-\ngated spin system across a continuous thermal phase\ntransition. Under this condition, the magnetic suscep-\ntibility and specific heat display a strong divergence at\nthe continuous thermal phase transition.\nThe present accurate results are reminiscent of\nthe experimental findings, which recently reported un-\nconventional thermal phase transitions of the two-\ndimensional quantum magnet SrCu 2(BO 3)2[2]. In the\ncase of the diamond-decorated square lattice, the FRI-\nMD phase transition in the spin-1/2 Ising-Heisenberg\nmodel is not only qualitatively but even quantitatively\nclose to the Lieb-Mattis to MD phase transition in the\nfull spin-1/2 Heisenberg model [7,9]. The exact solution\nof the spin-1/2 Ising-Heisenberg model thus permits\nus to unveil, for example, a reentrance in the finite-\ntemperature transition line [9] that would be beyond\nthe accuracy of the numerical methods used to solve\nthe spin-1/2 Heisenberg model [7].\nIt is our hope that the present essentially exact re-\nsults will stimulate further exploration in this exciting\nresearch field.\nACKNOWLEDGEMENT\nThis work was funded by the Slovak Research\nand Development Agency and the French Ministry for\nEurope and Foreign Affairs, the French Ministry for\nHigher Education and Research under the ˇStef´ anik\nprogramme for Slovak-France bilateral projects under\ncontract Nos. SK-FR-19-0013/45125RC and SK-FR-\n22-0011/49880PG. J.S. and K.K. acknowledge the par-\ntial financial support by a grant of the Slovak Re-\nsearch and Development Agency under the contract\nNo. APVV-20-0150. We acknowledge support from\nthe Deutsche Forschungsgemeinschaft (DFG) through\nGrant No. WE/3649/4-2 of the FOR 1807 and RTG\n1995.\nREFERENCES\n1. C. Domb, M. S. Green: Phase Transitions and Criti-\ncal Phenomena: Volume 1 (Academic Press, London\n- 1972); ibid. Volume 2 (Academic Press, London -\n1972); ibid. Volume 3 (Academic Press, London -\n1976); ibid. Volume 4 (Academic Press, London -\n1977); ibid. Volume 5 (Academic Press, London -\n1983).\n2. J. Larrea Jim´ enez, S. P. G. Crone, E. Fogh,\nM. E. Zayed, R. Lortz, E. Pomjakushina, K. Con-\nder, A. M. L¨ auchli, L. Weber, S. Wessel, A. Ho-\nnecker, B. Normand, Ch. R¨ uegg, P. Corboz,\nH. M. Ronnow, F. Mila, Nature 592, 370 (2021).\n3. J. Stapmanns, P. Corboz, F. Mila, A. Honecker,\nB. Normand, S. Wessel, Phys. Rev. Lett. 121,\n127201 (2018).\n4. L. Weber, A. Y. D. Fache, F. Mila, S. Wessel, Phys.\nRev. B 106, 235128 (2022).\n5. Y. Fan, N. Xi, C. Liu, B. Normand, R. Yu, arXiv:\n2306.16288.6. L. Weber, A. Honecker, B. Normand, P. Corboz,\nF. Mila, S. Wessel, SciPost Phys. 12, 054 (2022).\n7. N. Caci, K. Karl’ov´ a, T. Verkholyak, J. Streˇ cka,\nS. Wessel, A. Honecker, Phys. Rev. B 107, 115143\n(2023).\n8. J. Gubernatis, N. Kawashima, P. Werner: Quan-\ntum Monte Carlo Methods: Algorithms for Lattice\nModels (Cambridge University Press, Cambridge -\n2016).\n9. J. Streˇ cka, K. Karl’ov´ a, T. Verkholyak, N. Caci,\nS. Wessel, A. Honecker, Phys. Rev. B 107, 134402\n(2023).\n10. R. J. Baxter: Exactly Solved Models in Statistical\nMechanics (Academic, London, 1982).\n11. M. E. Fisher, Proc. R. Soc. London A 254, 66\n(1960); ibid.256, 502 (1960).\n12. M. Hattori, H. Nakano, Prog. Theor. Phys. 40, 958\n(1968).\n13. H. Mashiyama, S. Nara, Phys. Rev. B 7, 3119\n(1973).\n14. W. T. Lu, F. Y. Wu, Phys. Rev. E 71, 046120\n(2005).\n15. L. ˇCanov´ a, M. Jaˇ sˇ cur, Condens. Matter Phys. 9, 47\n(2006).\n16. L. G´ alisov´ a, J. Phys.: Condens. Matter 28, 476005\n(2016).\n17. M. E. Fisher, Phys. Rev. 113, 969 (1959).\n18. O. Rojas, J. S. Valverde, S. M. de Souza, Physica\nA,388, 1419 (2009).\n19. J. Streˇ cka, Phys. Lett. A 374, 3718 (2010).\n20. L. Onsager, Phys. Rev. 65, 117 (1944)." }, { "title": "2208.00179v1.On_field_driven_domain_wall_motion_in_compensated_ferrimagnetic_nanowires.pdf", "content": "On field-driven domain wall motion in compensated ferrimagnetic nanowires\nK. Y . Jing,1X. Gong,1and X. R. Wang1, 2,\u0003\n1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong\n2HKUST Shenzhen Research Institute, Shenzhen 518057, China\n(Dated: August 2, 2022)\nThe fascinating high-speed field-driven domain wall (DW) motion along ferrimagnetic nanowires near the\nangular momentum compensation point (AMCP) is solved based on the generic ferrimagnetic dynamics. The\nphysics of the absences of precessional torque and infinite high Walker breakdown field at the AMCP is proved\nunder general conditions. Based on the energy conservation principle, an almost exact DW velocity formula,\nvalid beyond the Walker breakdown field, is obtained. Our results agree with all existing experiments and\nsimulations. This theory provides useful guidances to DW manipulation.\nIntroduction .— Magnetic domain wall (DW) dynamics in\nnanowires have attracted much attention for its rich physics\n[1, 2] and promising device applications such as racetrack\nmemories [3]. One critical issue in applications is the real-\nization of high stable DW speed under external forces such\nas magnetic fields and electrical currents. This requires a de-\nlay or removal of so-called Walker breakdown [4]. The en-\ndeavour of increasing DW speed leads to studying DW mo-\ntion in antiferromagnetic nanowires [5–7], and, very recently,\nto that in ferrimagnetic nanowires [8–19]. A ferrimagnet has\nat least two spin sublattices antiferromagnetically interacting\nwith each other. It has two special states called the angu-\nlar momentum compensation point (AMCP) at which the an-\ngular momenta of the two sublattices cancel each other and\nthe magnetization compensation point at which the magneti-\nzations cancel each other. One class of ferrimagnets is rare-\nearth-transition-metal alloys whose AMCP and magnetization\ncompensation point are di \u000berent in general and can be tuned\nby compositions, other than the temperature. Unlike an an-\ntiferromagnet, ferrimagnetic states can be manipulated by a\nmagnetic field, a spin transfer torque, and a spin-orbit torque.\nAlso, unlike a ferromagnet, the net magnetization of a ferri-\nmagnet can be very small but not zero, especially around an\nAMCP such that it is susceptible to the magnetic field with\nsmall Zeeman energy. One fascinating discovery is the very\nhigh DW speed of thousands meters per second in compen-\nsated ferrimagnetic (FiM) nanowires near the AMCP [9–11].\nHere we show that high DW speed near the AMCP is related\nto the absence of precessional torque and Walker breakdown\nphenomenon at the AMCP.\nAlthough FiM dynamics should be described by coupled\npartial di \u000berential equations for magnetizations on at least two\nantiferromagnetically coupled sublattices, existing theoreti-\ncal studies treat a ferrimagnet either as a ferromagnet whose\ndynamics follows Landau-Lifshitz-Gilbert (LLG) equation\n[10, 11] or an antiferromagnet with the N ´eel order governed\nby a second-order partial di \u000berential equation [9, 12, 16, 20–\n22]. DW dynamics is then obtained from converting the par-\ntial di \u000berential equations into ordinary di \u000berential equations\nfor the collective coordinates of DW center and DW-plane\ncanting angle [9, 12, 16, 20–22]. Indeed, existing theories\nhave enriched our understanding of DW dynamics in ferri-magnets in many aspects. However, there are some drawbacks\nin these approaches. These approaches fail to provide a quan-\ntitative explanation to both experiments and simulations since\nthey rely on the existence of a DW plane and a rigid body\nassumption for the Thiele equation [23]. It often needs to as-\nsume also certain DW structure such that the approaches are\ndi\u000ecult, if not impossible, to generalize to situations where\nthe assumptions are not valid such as for vortex DWs and\nDWs in chiral magnets. Furthermore, the physical picture be-\nhind the FiM DW motion is unclear in these approaches and\nan accurate description of the DW speed beyond the Walker\nbreakdown field is still challenging.\nIn this work, the origin of the high DW speed and absence\nof Walker breakdown field at the AMCP of a FiM nanowire\nare explained based on generic dynamics for coupled sublat-\ntice magnetizations of a ferrimagnet with a general Rayleigh\ndissipation. We show that a static DW between two domains\nwith di \u000berent energy densities does not exist. Spins in the\nDW must move in a field that creates such an energy density\ndi\u000berence. Moving spins must dissipate energy due to the in-\nevitable coupling between spins and its environment described\nby Gilbert damping in magnetization dynamics. The dissi-\npated energy must be compensated by the Zeeman energy re-\nleased from the DW propagation toward domain of the higher\nenergy density. At the AMCP, precessional torque vanishes\ndue to the zero angular momentum and the Walker breakdown\nfield become infinity, leading to the high DW speed. Further-\nmore, a universal relationship between DW speed and DW\nstructure is obtained, and an almost exact formula for high-\nfield DW velocity is derived.\nModel .— We consider a head-to-head (HH) DW in a FiM\nnanowire, whose easy axis is along the wire defined as the z-\naxis as shown in Fig. 1. M1andM2are the magnetizations\non two sublattices with M1andM2being their saturation mag-\nnetization. The total magnetic energy of the wire in the pres-\nence of a uniform magnetic field HisE=R\n\"d3xwith the\nenergy density of\n\"=JM1\u0001M2+X\ni=1;2h\nAi(rMi)2+fi(Mi)\u0000\u00160Mi\u0001Hi\n;(1)\nwhere J>0 is the antiferromagnetic interlattice-spin coupling\nconstant. Aiand fiare the ferromagnetic exchange sti \u000bnessarXiv:2208.00179v1 [cond-mat.mes-hall] 30 Jul 20222\nzxy\nM1\nM2DW\nRegion I Region II Region III\nFIG. 1. Schematic of a HH FiM DW in a nanowire. Region I and III\nare two uniform FiM domains, separated by a DW (region II) whose\nwidth is \u0001. DW structure can be very complicated. His the external\nfield. Colours denote the spin orientations: The red for spins along ˆ z\nand the light-blue for spins along \u0000ˆz.\nand anisotropic magnetic energy density for sublattice i(i=\n1;2).fiis assumed to have two equal minima at Mi=\u0006Miˆz.\nThe FiM magnetization dynamics is generically governed\nby the following equations [24, 25]\n1\n\r1@M1\n@t=\u0000M1\u0002 \nH1\u0000\u000b11\n\r1M1@M1\n@t\u0000\u000b12\n\r1M1@M2\n@t!\n1\n\r2@M2\n@t=\u0000M2\u0002 \nH2\u0000\u000b22\n\r2M2@M2\n@t\u0000\u000b21\n\r2M2@M1\n@t!\n;(2)\nwhereHi=\u0000\u0016\u00001\n0\u000eE=\u000eMiand\ri=gi\u0016B=~(i=1;2) are\nthe e\u000bective field and the gyromagnetic ratio for Mi, respec-\ntively. gi,\u0016B, and ~are the Land ´eg-factor of sublattice i\n(i=1;2), the Bohr magneton, and the Planck constant, re-\nspectively.\u000b11;\u000b22and\u000b12;\u000b21are intra-sublattice and inter-\nsublattice damping coe \u000ecients. We have\u000b12\n\r1M1=\u000b21\n\r2M2due to\nthe action-reaction law. si=Mi=\riis the spin density of sub-\nlattice i(i=1;2).\r1,\r2in a general ferrimagnet because of\nthe di \u000berence in Land ´eg-factors of sublattices. For example,\nin GdFeCo alloys, gGd'2,gFeCo'2:2 [9].\nResults .— We prove first that no static DW is allowed in\nthe presence of a magnetic field along the z\u0000direction, except\nat magnetization compensation point. If a static DW solution\nexists, the DW structure should satisfy equations Mi\u0002Hi=0\n(i=1;2). As illustrated in the Supplemental Materials [26], it\nimpliesMi(x;t) (i=1;2) satisfying following equation\n\t\n@\n26666664\" 1\u0000j=x;y;zX\ni=1;22Ai(rMi;j)\n(rMi;j)37777775\u0001d\u001b=const:(3)\nwhere@\nis any closed surface of the system, 1is the 3\u00023\nunit matrix, and\ndenotes the dyadic product. Eq. (3) cannot\nbe true for a DW with M1=M1ˆz;M2=\u0000M2ˆzon its left and\nM1=\u0000M1ˆz;M2=M2ˆzon its right as shown in Fig. 1, or\nvice versa, because it requires ( M1\u0000M2)H=0. Thus, a static\nDW can only exist either with H=0 or M1=M2. In other\nwords, a static DW cannot exist between two domains with\ndi\u000berent energy density. This result can also be understood\nfrom following argument: Assume Mi(x) is a static DW that\nseparate a left domain with a lower energy density \"1from\nthe right domain with a higher energy density \"2(> \" 1). The\nenergy change by shifting DW to the right by a distance L, i.e.Mi(x)!Mi(x+Lˆz), isLS(\"1\u0000\"2)<0, here Sis the cross\nsection area of the wire. The DW is not stable against a rigid\nshift to the right because this small change in spin structure\nalways lower the system energy. Thus a DW must vary with\ntime under a magnetic field.\nWhen Jis much larger than the Zeeman energy, M1and\nM2are always anti-parallel to each other. We define Me\u000b=\n(M1\u0000M2)m, wheremis the unit vector of M1. Thenm\nsatisfies the following equation\n(s1\u0000s2)@m\n@t=\u0000(M1\u0000M2)m\u0002He\u000b+\u000bm\u0002@m\n@t;(4)\nwhereHe\u000b=M1H1\u0000M2H2\nM1\u0000M2. In terms of m, the total energy\nisE[m]=Rh\nA(rm)2+f(m)\u0000\u00160(M1\u0000M2)m\u0001Hi\nd3x\nwith A=A1M2\n1+A2M2\n2, where ais the lattice constant. De-\nnote\u000b=\u000b11s1+\u000b22s2\u0000\u000b12s2\r2\n\r1\u0000\u000b21s1\r1\n\r2, the thermodynamic\nsecond law requires \u000b > 0 to ensure the Rayleigh dissipation\nfunctionalR=\u00160\u000b\n2R\u0010@m\n@t\u00112d3x[24, 25, 27] to be positive-\ndefinite. Equation (4) says that the change of spin angular\nmomentum (left-hand side) equals the net torque (right-hand\nside) that is the sum of a torque from an e \u000bective field on the\nnet magnetization ( M1\u0000M2,0) and a dissipative torque from\nthe motion ofm. At the AMCP, the dissipative torque cancels\nthe field torque.\nEquation (4) can be recast as an e \u000bective LLG equation\n[11, 20, 28–30]\n@m\n@t=\u0000\re\u000bm\u0002He\u000b+\u000be\u000bm\u0002@m\n@t; (5)\nwith an e \u000bective gyromagnetic ratio \re\u000b=jM1\u0000M2j=(s1\u0000s2)\nand an e \u000bective Gilbert damping \u000be\u000b=\u000b=(s1\u0000s2).\re\u000b\u000be\u000bis\nalways positive because a moving magnetization must dissi-\npate its energy to its environment (See Eq. (6) below). s1>s2\nands1\nHW. From Eqs. (8) and (9), we have\n¯v=\u000be\u000b\re\u000b\n2HS\u0010\n1+\u000b2\ne\u000b\u0011Zh\n(Hsin\u0012\u0000G)2+H2\ne\u000b;\u001ei\nd3x:(12)\nAverage DW velocity is (see the Supplemental Materials [26]\nfor detailed derivation),\n¯v=c1H+c1\n\u000b2\ne\u000b\u0012\nH\u0000q\nH2\u0000H2\nW\u0013\n(13)\nwhere c1=\u000be\u000b\re\u000b\n2S(1+\u000b2\ne\u000b)R\nsin2\u0012d3x=(M1\u0000M2)\u000b¯\u0001\n(s1\u0000s2)2+\u000b2is peaked at the\nAMCP. Equation (13) is exact under very sensible assump-\ntions, and all coe \u000ecients in Eq. (13) are fully determined by\nthe model parameters.\nEquation (13) predicts a negative di \u000berential DW mobility\nin the range of HWHW, we use MuMax3 [35] to nu-\nmerically solve Eq. (2) for a synthetic ferrimagnetic strip wire\nas shown in Fig. 1 that consist of two antiferromagnetically-\ncoupled ferromagnetic-layers of 1nm thick each. The strip\nsize is 16 nm\u00022 nm\u00021024 nm. The cell size in simulations\nis chosen to be 1 nm \u00021 nm\u00021 nm. To mimic a GdFeCo al-\nloy [9], the model parameters are J=1:2\u000210\u00004J\u0001A\u00002m\u00001,\nA1=9:8\u000210\u000024J\u0001m\u0001A\u00002,A2=1:23\u000210\u000023J\u0001m\u0001A\u00002,\nbiaxial anisotropy are considered for each sublattice, fi=\n\u0000Kz;i\nM2\niM2\ni;z+Ky;i\nM2\niM2\ni;y,i=1;2,Kz;1=Kz;2=0:65 MJ=m3,\n\u000b12=\u000b21=0.Ky;iand\u000bii(i=1;2) are used for simulat-\ning di \u000berent systems as labelled by Set 1-6 in Table I. The\ngyromagnetic ratios \r1=\r2=1:76\u00021011s\u00001T\u00001, the satura-\ntion magnetizations are M1=1010 kA=m,M2=900 kA=m.\nThe coupling field between two sublattices is of hundreds of\nTesla to guarantee collinearity of two spin sublattices. Dif-\nferent from a natural ferimagnet, inter-sublattice coupling is4\nData set Set 1 Set 2 Set 3 Set 4 Set 5 Set 6\nKy;1( MJ=m3) 0.05 0.035 0.02 0.1 0.1 0.1\nKy;2( MJ=m3) 0.05 0.035 0.02 0.1 0.1 0.1\n\u000b11 0.02 0.02 0.02 0.005 0.01 0.015\n\u000b22 0.02 0.02 0.02 0.005 0.01 0.015\n\u000be\u000b 0.3473 0.3473 0.3473 0.0868 0.1736 0.2605\nKy( MJ=m3) 0.1 0.07 0.04 0.2 0.2 0.2\n\u00160HW(T) 0.3157 0.2210 0.1263 0.1579 0.3157 0.4736\n¯\u0001(nm) 3.85 3.87 3.89 3.79 3.75 3.79\nc1(\u00160\u0001m\u0001s\u00001T\u00001) 210.00 211.13 212.34 57.48 111.37 162.69\nTABLE I. Ky;1,Ky;2,\u000b11, and\u000b22are model parameters. \u000be\u000b,Ky,\u00160HW,¯\u0001, and c1are computed quantities.\nalong the y-direction in our synthetic ferrimagnet. In the sim-\nulation, a DW is first created at the center of nanowire, then a\nuniform magnetic field is applied in the +ˆzdirection. The ve-\nlocity is obtained from the linear fit of time-evolution curve of\nthe DW center (where mz=0). For high fields above Walker\nbreakdown, the average velocities are obtained from data ac-\ncumulated for more than 4 velocity oscillating periods.\nWe consider six di \u000berent systems with various Ky;iand\u000bii\n(i=1;2). The detail values of the model parameters are given\nin Table I. Because of large speed di \u000berence, Fig. 2(a) plot ¯ v\nvs.Hfor three systems with the same \u000bii=0:02 and di \u000berent\nKy;i, label as Set 1, 2, 3. Figure 2(b) is the similar plots for\nthree systems with the same Ky;i=0:1 MJ=m3, but di \u000berent\n\u000bii, label as Set 4, 5, 6. The corresponding values of c1,\u000be\u000b,\nandHWcomputed from this theory are also given in Table\nI. The perfect agreements between the simulation results (the\nsymbols) and theoretical prediction (the solid curves) demon-\nstrate that Eq. (13) is almost exact.\nDiscussion and Conclusion .— Before conclusion, we\nwould like to make a few remarks. 1) The relationship be-\ntween the instantaneous DW speed and the DW structure is ex-\nact that explains why our high-field DW speed formula with-\nout any fitting parameters agree perfectly with simulation re-\nsults. 2) Since no collective-mode approximation is used, the\ntheory is applicable to all types of DWs. 3) High DW speed\nis a result of the absence of Walker breakdown field at the\nAMCP. This explains the observed high DW speed of more\nthan 1.5km /s at the AMCP although the mobility \u0016=(M1\u0000M2)\u0001\n\u000b\nforH0.25 T) that gradually annihilate the\nmaze-like domain pattern. The detection of a topological\nHall e\u000bect contribution indicates the existence of isolated\nskyrmions in broader \feld regimes at a lower temperature\nof 50 K. The magnetic microstructure of the skyrmions is\nfound to consist of a small core region of 40 nm and a sur-\nprisingly broad outer wall width of 50 nm. The skyrmion\npro\fle can be reproduced when considering the intrinsic\nmagnetic material parametes including a DMI constant\nof about 0.3 mJ/cm2, which is consistent with previous\nstudies on DyCo x\flms. The promising properties of fer-\nrimagnetic skyrmions in DyCo xalloy thin \flm systems in\ncombination with the possibility to easily tune their mag-\nnetic properties by varying their stoichiometry might be a\npromising route for skyrmions to be used in practical ap-\nplications in future spintronic devices. In particular, the\ndetrimental skyrmion Hall e\u000bect can be minimized when\nsetting the compensation temperature close to room tem-\nperature, which will possibly enable the realization of\nfunctional skyrmion devices that can be operated under\nambient conditions. Future systematic investigations of\nthe skyrmion pocket in the magnetic phase diagram for\nDyCo xand TM-RE alloys in general will reveal their full\npotential.\nThe authors acknowledge the \fnancial support for the\nPM2-VEKMAG beamline by the German Federal Min-\nistry for Education and Research (BMBF 05K10PC2,\n05K10WR1, 05K10KE1) and by HZB. A.P.-K. grate-\nfully acknowledges support from the DFG via Sonder-\nforschungsbereich (collaborative research center) SFB\n925 (subproject B3) and S. Rudor\u000b is acknowledged for\ntechnical support.SUPPLEMENTARY-METHODS\nThe samples were prepared by magnetron sputtering\n(MAGSSY chamber at BESSY) in an argon atmosphere\nof 1:5\u000210\u00003mbar with a base pressure of 5 \u000210\u00009mbar\nat a deposition temperature of 300 K. Si 3N4membranes\nwith a surface area of 5 \u00025 mm2and a thickness of 100 nm\nwere used as substrates for the soft x-ray transmission\nmeasurements including SAXRMS and STXM. A 3 nm\nthick Ta capping layer was grown on the DyCo 3layer to\nprevent surface oxidation.\nSAXRMS experiments have been performed at the\nVEKMAG end-station at the PM2 beamline, Helmholtz-\nZentrum Berlin (HZB). The di\u000bracted x-rays are col-\nlected on a Peltier-cooled square-shaped CCD detector\ncovering 2.1\u000eat the working distance of this study. The\nSAXRMS spectra (Fig. 1d) were retrieved by azimuthal\naveraging of the 2D patterns (Fig. 1c) after background\nsubtraction and masking of beamstop shadow and charge\nscattering streaks from the membrane edges. All intensi-\nties were normalized to the charge-scattering signal from\nthe membrane edges. The magnetic spectra were \ftted\nwith a split Pearson type VII distribution.\nElement-speci\fc STXM measurements were performed\nat MAXYMUS, beamline UE46 at HZB in the presence\nof an external magnetic \feld, H zparallel or antiparallel\nto the x-ray beam. 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Bagschik, et al., Employing soft x-ray resonant mag-\nnetic scattering to study domain sizes and anisotropy in\nCo/Pd multilayers, Phys. Rev. B 94, 134413 (2016).\n[67] N. Nagaosa, S. Onoda, A. H. MacDonald, & N. P. Ong,\nAnomalous Hall e\u000bect, Rev. Mod. Phys. 82, 1539-1592\n(2010)." }, { "title": "2106.10111v2.Atomistic_spin_model_of_single_pulse_toggle_switching_in_Mn__2_Ru__x_Ga_Heusler_alloys.pdf", "content": "Atomistic spin model of single pulse toggle switching in Mn 2RuxGa Heusler alloys\nF. Jakobs and U. Atxitia\u0003\nDahlem Center for Complex Quantum Systems and Fachbereich Physik,\nFreie Universität Berlin, 14195 Berlin, Germany\nSingle femtosecond pulse toggle switching of ferrimagnetic alloys is an essential building block\nfor ultrafast spintronics. Very different element-specific demagnetization dynamics is believed to\nbe a hard limit for switching in ferrimagnets. This suggests that ferrimagnets composed of two\nions of different nature, such as rare earth transition metal alloys, are necessary for switching.\nHowever, experimental observation of toggle switching in Mn 2RuxGa Heusler alloys, has contested\nthis limit since Mn ions are of the same nature. To shed some light into this question, we present an\natomistic spin model for the simulation of single pulse toggle switching of Mn 2RuxGa. The magnetic\nparameters entering in our model are extracted from previous experimental observations. We show\nthat our model is able to quantitatively reproduce measured magnetization dynamics of single pulse\ntoggle switching. We demonstrate that differently to previous understanding toggle switching in\nMn2RuxGa is possible even when both Mn sublattices demagnetization at very similar rate.\nSingle pulse femtosecond toggle switching in ferrimag-\nnets has attracted a lot of attention as a promising\nsolution for low energy, faster memory applications. [1–\n4]. It has already been demonstrated in micro and\nnanostructures[5,6], toswitchmagnetictunneljunctions\n[7], as passive component to induce switching in\nferromagnets[8]andevenusingpicosecondelectricpulses\n[9]. Toggle switching has been mostly found in one class\nof material, systems composed of transition metals rare-\nearth, e.g. GdFeCo [1] and TbFeCo [10] alloys, and\nGd/Co[11] and Tb/Co stacks [12]. Integrating all optical\nswitching with spintronics using this class of material\nhas been proposed[13], however, Mn 2RuxGa alloys, the\nsecond class of material showing toggle switching, is\nbettersuited[14]. WhileforGdFeCointenseresearchhas\nprovided a large amount of experimental data permitting\nthe validation of a number of theoretical models, for\nMn2RuxGa, there only exist a few experimental works\nshowing ultrafast magnetization dynamics and switching\n[14–17].\nThe Mn 2RuxGa Heusler alloy crystallises in the cubic\nspace group F 43mwith the magnetic Mn atoms on the\n4aand4csites (Fig. 1a). Spins at 4aand4csites are\ncoupledantiferromagnetically, whereastherespectiveMn\n4a- and 4csublattices are coupled ferromagnetically [18].\nThe Ga atoms appear at the 4bsites and Ru atoms at the\n4dsites of the lattice, and their magnetic contribution\ncan be neglected. Ultrafast magnetization dynamics of\nthe magneto-optically active Mn 4csublattice was first\nmeasured by Bonfiglio et al. [15]. In a subsequent\nwork, they concluded that Mn 4csublattice shows an\nultrafast demagnetization ( \u0018100s fs) followed by either\na secular equilibrium or by a fast remagnetization ( \u0018\n1ps). By using a phenomenological model based on\nthe so-called four temperature – electron, phonon, and\nspin temperatures of Mn 4aand 4c– these dynamics\nwere interpreted as a signature of strong exchange-\ndriven relaxation [16]. Spin temperature models however\nare unable to describe angular momentum transfer and\nFIG. 1. a) Crystal structure of Mn 2RuxGa with the two\nmagnetic Mn-sublattices represented as blue and red arrows.\nb) Schematic of the exchange constants J4a\u00004a,J4a\u00004cand\nJ4a\u00004c. c) Mn-specific atomic magnetic moments, symbols\ncorrespond to experimental data [19] and lines to the values\nused in the atomistic spin model.\nswitching.\nExperiments carried out by Davies et al. demonstrated\nthat switching is possible in Mn 2RuxGa, but only when\nthe initial temperature ( T0) lies below the magnetization\ncompensation temperature ( TM) [20]. Based upon a\nphenomenological model for the magnetization dynamics\nof two-sublattice magnets [21], the authors argued\nthat in order to link static thermodynamic properties\n(equilibrium TM) to highly non-equilibrium dynamics\n(switching), both sublattices should demagnetize at the\nsame rate, conserving the total angular momentum\nduring the whole process. This picture explains theirarXiv:2106.10111v2 [cond-mat.mtrl-sci] 12 Jan 20222\nobservation of the switching onset for a wide range of\nsystem parameters, including switching with picosecond-\nlong pulses. This interpretation collides with the\nphenomenology proposed by Bonfiglio et al. [16], which\nstates that both sublattices only demagnetize at the\nsame rate after \u00181.5 ps. Still, since data on the\nmagnetization dynamics of the switching process was\nmissing, the question remained open. An apparent\nstep forward along this direction was made by Banerjee\net al. [14] by measuring the dynamics of the Mn 4c\nsublattice when switching occurs. They were unable to\nobserve an explicit switching of the sign of the magneto-\noptical signal, however the dynamics of the Mn 4c\nsublattice behaved like those observed by Bonfiglio[16].\nThese observations differ strongly with the well-known\nelement specific signal switching measured in GdFeCo\n[22]. Recently, however, Banerjee and co-workers have\nclearlyobservedthedynamicsofmagnetizationswitching\nof the Mn 4csublattice [17], in clear disagreement to\ntheir own previous observations [14]. This disagreement\ncould be related to the different strength of the resetting\nmagnetic field, stronger in the latter. Similarly to\nthe previous works[14, 16, 20], the understanding of\nthe physics behind the switching process rested on\nphenomenological arguments. An attempt to describe\nswitching in Mn 2RuxGa using a first-principles model\nexists [23]. Although this model provides useful insights\non the potential origin of switching, due to its simplicity\nit is unable to describe the temperature dependence of\nthe switching condition ( T0< TM) observed by Davies\net al. [20] and Banerjee et al. [14]. A quantitative\nmodel able to account for thermodynamic aspects of\nthe magnetization switching dynamics in Mn 2RuxGa is\nnecessary but so far missing.\nIn this work we address this issue by presenting an\natomistic spin model to describe ultrafast magnetization\nand switching in Mn 2RuxGa alloys. Atomistic spin\nmodels have demonstrated themselves to be able to\ndescribe the dynamics of the magnetization switching in\ntransition metal rare-earth alloys[1, 24] and multilayers\n[25, 26]. Atomistic spin models are based on a\nsemiclassical spin Hamiltonian, which is defined by\nthe exchange and anisotropy constants as well as the\natomic magnetic moments (see Supplemental Material\nfor details). In a minimal model for Mn 2RuxGa,\none needs to determine at least three exchange\nconstants, J4a\u00004a,J4c\u00004candJ4a\u00004c(see Fig. 1 b)).\nWithin a Heisenberg spin model, the values of the\nexchange constants determine the critical temperature\nTc, and in ferrimagnets, besides Tc, the magnetization\ncompensationtemperature, TM, ifitexists. Thereported\nTcrange from Tc= 625K [18] (x= 0:68),Tc= 450K\n(x >0:5) [27],Tc= 550K (x= 0:7) [16]. We fix Tcto\nabout630Kwhichisattheupperendofreportedcritical\n200 400 600\nTemperature [K]0123Magnetization [ µB]\nCompensation\nTemperaturea) (b\n|Mn4a|\n|Mn4c|\nMn4a+4c\n0.6 0.8 1.0\nRuthenium Concentration x0100200300400\nCompensation Temperature [K]\nRoom Temperature\nSwitchingExperiment\nSimulationFIG. 2. Magnetic properties of Mn 2RuxGa alloys gained\nfrom atomistic spin model simulations. a) The equilibrium\nmagnetization for a Mn 2Ru0:68Ga alloy as function of\ntemperatureoftheMn 4asublatticeinred,theMn 4csublattice\nand the total magnetization Mn 4a+4c=jMn4aj\u0000jMn4c|. b)\ndisplays the magnetization compensation temperature TMas\nfunction of Ru-concentration, x. Black dots correspond to\nexperimental measurements [14] and red crosses from our\nmodel simulations. The red line is a linear fit of the simulated\nresults and is guidance to the eye.\ntemperatures. Here, we use values of the exchange\nconstants already determined experimentally [18], but\nslightly re-scaled such that our model reproduces the\nexperimental values of both TcandTM(Fig. 2). The\nanisotropy constant in Mn 2RuxGa has been reported to\nbesite-specific, with dz;4c= 1:1664\u000110\u000023Janddz;4a= 0\n[18], which we use in our model. We note that the role of\nthe anisotropy in the ultrafast magnetization dynamics\nis minimal since dz=J\u001c1and it is included to fix the\naverage magnetization towards the z-axis. Any choice of\nrelatively small values of the anisotropy constant would\nyield the same simulation results.\nLastly values for the site-specific Mn atomic magnetic\nmoments are needed. Based upon experimental data, we\nassume that the atomic moment of the Mn 4asublattice\nstays constant at \u0016s;4a= 2:88\u0016Bfor the range of Ru-\nconcentration studied here [19] (Fig. 1c)). Differently,\nthe Mn 4csites have been shown to have a stronger\ndependence on the Ru-concentration [19](Fig. 1c). Here\nweusevaluesof \u0016s;4c= 4:71\u0016B\u0001x(Fig. 1c))thatareboth\nclose to the experimentally measured values and provide\ngood results for TM(see Fig. 2). These values not only\ncompare well to those found in experiments[19, 28] but\nalso agree with the general trend found in first-principles\ncalculations [23, 29]. Simulations of our model are able\nto reproduce well the observation of an increasing TMfor\nan increasing Ru-concentration. For x<0:66, our model\nstarts to deviate from the experiments [14] (Fig. 2 b)).\nThe shared wisdom of single femtosecond pulse toggle\nswitching in ferrimagnets is based on experimental\nobservations in only one class of material, transition\nmetal rare-earth compounds. Despite their structural\nor morphological differences, in those materials the\nconditions for switching are based in the same working3\nprinciples, the rare-earth spin sublattice response to the\nlaser pulse is slower than that of the transition metal.\nThe physical reason behind this difference relies in the\ncore character of the highly localized 4 felectron spins in\nGd, plusthe absence of orbitalmagneticmoment( L= 0)\n[30]. The laser only excites them indirectly, by their\ncoupling to the 5d6s itinerant electrons. The transition\nmetal spins respond quickly to changes in the electronic\nstructure band due to the quick temperature rise.\nBased on this it has been argued for long that toggle\nswitching is only possible for two sublattice compounds\nwith components that demagnetize at different enough\nrates. Since data on site-specific dynamics is unavailable,\nit is unclear whether or not this criterion holds in\nMn2RuxGa. It is tempting to draw similarities to\nGdFeCo to explain switching in Mn 2RuxGa. For\nexample, since Mn 4aspins are localized, they could\nplay the role of the slow rare earth, while Mn 4c\nhave a more delocalized character, and thus play the\nrole of the transition metal. In this work we show\ndifferently, that Mn 2RuxGa switches even though its\ntwo sublattices show similar demagnetization times,\nunlocking different demagnetization times as a necessary\ncondition for switching. Another hard constrain for\nswitching Mn 2RuxGa is that it is only possible when\nthe initial temperature lies below TM[14, 20]. Davies et\nal. [20] found that this condition is robust and that can\nbe explained assuming that the dynamics is exchange-\ndriven, which has as a consequence that dM4a=dt=\n\u0000dM4c=dt. This condition also holds, when the coupling\nto the heat-bath is similar for both sublattices and they\ndemagnetize at similar rates. Here, we demonstrate that\nswitching is possible when the initial demagnetization\ndynamics is dominated by the coupling to the heat-bath\ninstead the exchange relaxation.\nWe use the atomistic stochastic-Landau-Lifshitz-\nGilbert equation (sLLG) [31] to describe the magne-\ntization dynamics of the Mn 2RuxGa alloys and the\ntwo-temperature model (TTM) to describe the electron\ntemperature Teland the phonon temperature Tph(see\nSupplemental Material (SM) for details) [32, 33]. The\nparameters defining the TTM for Mn 2RuxGa are not\nestablished yet. We use parameters similar to GdFeCo\nalloys, which are close to those used by Bonfiglio et\nal. within the 4TM [16]. The complete set of system\nparameter used in our model are summarized in table II\nof the SM. Within atomistic spin dynamics models, the\ndemagnetization time scales with \u001c\u0018\u0016at=(\u000bTc)[21, 34].\nIn two-sublattice magnets, the three parameters, \u000b;Tc\nand\u0016atare sublattice-specific. While Tcand\u0016atare\ndeterminedbyequilibriumpropertiesasdiscussedbefore,\nthe value of the damping parameter is related to the\ncoupling to the heat-bath. One method to estimate this\nvalue from experiments is to photo-excite magnetization\nprecession. These experiments have been conducted\nin Mn 2RuxGa and measured by time resolved Faraday\nFIG. 3. Comparison between the experimentally measured\nMn4cdynamics for a switching event (points) [17] and our\natomistic spin model simulations (lines).\neffect as a function of the applied field and temperature.\nFrom the decay of the precession the intrinsic damping\nparameter has been determined to have values smaller\nthan 0:02far fromTM[15]. We chose a damping value\n(\u000b4c= 0:01and\u000b4c= 0:013) that reproduces the\nultrafast magnetization dynamics of the Mn 4csublattice\nduring a switching event as measured by Banerjee et\nal. [17] and allows switching for x > 0:85similar to\nthe experimental findings of Banerjee et al. [14]. We\nnote, that Davies et al. [20] find switching already for\nx= 0:75which could be reproduced for a different choice\nof damping values (discussion in the SM). With this our\nmodel is able to quantitatively reproduce the recently\nmeasured demagnetization dynamics of the 4 clattice by\nBanerjee et al. (Fig. 3).\nHow does our model compare to the current understand-\ning of switching in Mn 2RuxGa? Previous works [16,\n17, 20] have suggested, that exchange-driven dynamics\ndominate the first step of the demagnetization and\nswitching process. This is assumed since the inter-\nlatticeantiferromagneticexchangeisstrongerthanin(for\nexample) transition metal rare earth alloys. Exchange-\nrelaxation stems from processes driving both sublattices\nto a mutual equilibrium. Thus, it is unlikely that\nexchange processes play a role on the first steps of the\ndynamics. Moreover, since exchange processes describe\ntransfer of angular momentum between sublattices, it\nis more likely that those processes dominate when the\nsublattice magnetic order is small rather than when its\nsaturated. Near equilibrium, relaxation by coupling to\nthe heat-bath dominates, while for situations of non-\nequilibrium and reduced magnetic order, the exchange-\nrelaxation dominates.\nFigure 4 shows the site-specific dynamics for three\ncharacteristic Ru-concentrations of Mn 2RuxGa forx=\n0:76(a),x= 0:86(b) andx= 0:96(c). These three cases4\n-3-2-1012mz[µB]a)\nMn2Ru0.76GaMn4a Mn4c\n-3-2-1012mz[µB]b)\nMn2Ru0.86Ga\n0 1 2 3 4 5\nTime in ps-3-2-1012mz[µB]c)\nMn2Ru0.96Ga-0.5-0.2500.250.5\nmz,total[µB]\nMz,total\n-0.5-0.2500.250.5\nmz,total[µB]\n-0.5-0.2500.250.5\nmz,total[µB]\nFIG. 4. Site-specific magnetization dynamics of the 4aand\n4cMn sites (red, blue; left axis) after a 100 fs laser pulse\nexcitation at t= 0(Gaussian pulse peak) for an increasing\nRuthenium concentration ( \u000b4c= 0:01,\u000b4a= 0:013). The\ntotal magnetization ( mz;total=mz;4a+mz;4c) is shown as a\nblack line (right axis). All laser parameters were the same for\nall three simulations. a) shows the no-switching scenario for a\nMn2Ru0:76Ga alloy. b) shows the non-deterministic scenario\nfor a Mn 2Ru0:86Ga alloy, with a prolonged demagnetization\nstate. c) shows the switching scenario for a Mn 2Ru0:96Ga\nalloy.\nrepresent alloys below/at/above the threshold of x=\n0:9, which defines the experimentally found switching\ncondition [14].\nA closer look to mz;totalin Fig. 4 indicates that\nup to the first picosecond mz;totalis not conserved,\nalthough it only changes slightly due to the similar\ndemagnetization dynamics of the Mn 4aand 4c\nsublattices (note, that the scale of the right y-axis is\nmuch smaller than the scale of the left axis). This\nmeans that relaxation by coupling to the heat-bath\ndominates. This relaxation can in turn be interpreted as\nexcitation of ferromagnetic (optical) magnons (following\nBanerjee and co-workers[17]). However, as the sublattice\nmagnetization reduces to small values ( \u00181 ps),\nthe value of mz;totalstays constant in time. This\nmeans that exchange-relaxation dominates the dynamics\n(conservation of total angular momentum). This can be\ninterpreted as excitation of antiferromagnetic (acoustic)\nmagnons. Total angular momentum is conserved for a\nrelativelylongperiodoftime(fewpicoseconds). Oncethe\nsystem has reached the exchange relaxation dominated\nregime (\u00191 ps) the interpretation of switching of Davies\nand co-workers remains valid. The exchange relaxation\nresults inj\u0001m4aj=j\u0001m4cj, but sincejm4a;0j0:5[27], whereas Ref. 16 reports\nTc= 550K forx= 0:7. Furthermore the T cmeasured in Ref. 27 features a peak around x= 0:4which could explain\nwhy our parameters only reproduce reproduce Tcompforx>0:68. Thus, a dependence of the exchange parameters on\nRu-concentration may therefore be possible for lower values of x. Since we are mostly interested in Ru-concentration\nfor which switching has been experimentally demonstrated ( x > 0:7), we restrict our study to Ru concentrations\nofx= 0:6\u00001:0. We note that we have increased J4a\u00004aby 10% to reproduce the magnetization compensation\ntemperature as reported in Ref. [14] while keeping T c= 630K. Using exchange constants with the ratios as stated in\nRef. 18 couldn’t reproduce the experimentally measured Mn 4cdynamics of C. Banerjee et al. [17] that are shown in\nFig. 3 (main text). However, the fact that there are reports of largely varying different Curie temperatures [16, 18, 27]\nand compensation temperature [14, 20] indicate that there seems to be a wide spectrum of valid exchange constants\nbetween different experiments, so that an adjustment of 10% of one of the parameters seems reasonable.\nTheuniaxial anisotropyisalso consideredtobesite-specific, inparticular, theanisotropyenergydensity K z;4c= 216\nkJm\u00003and K z;4a= 0kJm\u00003taken from Ref. [18]. Assuming a unit cell size of ( \u00190:6nm)3([36]) we obtain\ndz;4c= 1:1664\u000110\u000023J as on-site anisotropy and dz;4a= 0. The anisotropy is included to yield an alignment\nalong thez-axis after demagnetization or switching. Since it is much smaller the Heisenberg exchange it does not\nhave meaningful impact on the switching itself. Furthermore ref. 15 finds, based on XMCD experiments, different\ng-factors,g4a= 2:05andg4c= 2:00for the 4a- and 4c-sublattice. However since both \rand\u0016sare proportional to\ngithis does not enter the LLG.\nThe spin dynamics of this system are described by the atomistic stochastic-Landau-Lifshitz-Gilbert equation\n(sLLG) [31]\n(1 +\u000b2\ni)\u0016s;i\n\r@Si\n@t=\u0000(Si\u0002Hi)\u0000\u000bi(Si\u0002(Si\u0002Hi)): (2)\nWhere\rrepresents the gyromagnetic ratio and \u000biis a site-specific atomic damping parameter. Here, we also draw\non experimental observations to estimate the values of the damping parameters. Photo-excited spin precession was\nobserved by time resolved Faraday effect as a function of the applied field and temperature. From the decay of\nthe precession the intrinsic damping parameter was also determined to have values smaller than \u000b= 0:02, far from\ncompensation [15]. Here, we decided to use \u000b4c= 0:01and\u000b4c= 0:013since it reproduces experimental observations\nas discussed in the main text. The temperature dynamics are described using the two-temperature model (TTM)\nthat describes the electron temperature Teland the phonon temperature Tphvia a pair of two coupled differential\nequations [32, 33]:\nCel@Tel\n@t=\u0000gep(Tel\u0000Tph) +Pl(t) (3)\nCph@Tph\n@t= +gep(Tel\u0000Tph): (4)\nCelandCphrepresent the specific heat of the electron- and phonon system and Pl(t)describes the absorbed energy of\ntheelectronsystem, comingfromthelaser. Sinceexperimentaldataontheelectronandphonontemperaturedynamics8\nis missing, we use similar parameters to typical GdFeCo values for our TTM and similar ones to the experimental\nwork by Bonfiglio et al. (Ref. [16]). Table I provides an overview of the used TTM-parameters in comparison to\nRef. [16] and to parameters used in GdFeCo from Ref. 37 and Ref. 38.\nTABLE I. Two temperature model parameters comparison between GdFeCo and Mn 2RuxGa between different sources.\nTTM Unit Mn2RuxGaMn2RuxGa[16]GdFeCo[38] GdFeCo[37]\nCphJ/m3K3\u00021062.27\u00021063\u00021063\u0002106\ngphJ/m3Ks6\u000210178\u000210172\u0002101717\u00021017\n\rJ/m3K2350 484 714 700\nThe laser pulse is assumed to be Gaussian shaped with a FWHM of 100 fs. The electron temperature Telyielding\nfrom the TTM is used to scale the temperature effects in the spin system. This is done by including a Langevin\nthermostat, which adds an effective field-like stochastic term \u0010ito the effective field Hi=\u0010i(t)\u0000@H\n@Siwith white noise\nproperties [39]:\nh\u0010i(t)i= 0andh\u0010i(0)\u0010j(t)i= 2\u000bikBTel\u0016s;i\u000eij\u000e(t)=\r: (5)\nThe complete set of system parameter used in our model are summarized in table II.\nTABLE II. Table of the Heisenberg spin Hamiltonian parameters (left) and the two temperature model (TTM) (right).\nH Value Unit TTM Unit\nJ4a\u00004a 1:28\u000210\u000021[J]Cph 3\u0002106[J/Km3]\nJ4c\u00004c 4:0\u000210\u000022[J]Cel\rel\u0001Te[J/Km3]\nJ4a\u00004c\u00004:85\u000210\u000022[J]\rel350 [J/K2m3]\n\r 1:76\u000210\u000021[1\nTs]gep 6\u00021017[J/sKm3]\ndz 1:17\u000210\u000023[J]\n\u0016s;4a 2.88 [\u0016B]\n\u0016s;4c 4:71\u0001x [\u0016B]\n\u000b4a 0:013\n\u000b4c 0:01\nSwitching behaviour in dependence of Gilbert damping parameters\nFigure 6 shows the switching behavior of Mn 2RuxGa alloys as function of the Ru-concentration xand the absorbed\nlaser energy for different dampings \u000b4a. Red areas indicate switching behavior, blue marks non-switched simulations\nand grey areas indicate simulations with a prolonged transient ferromagnetic state or a demagnetized state.\nThe simulation was counted as switched, when the 4a-sublattice crossed mz= 0and reached a threshold of mz;4a<\n\u00000:12after 15 ps (starting at positive mzvalues). Otherwise it was counted as demagnetized if jmz;4aj<0:12, or\nremagnetized if mz;4a>= 0:12(see Fig. 4 for examples).\nThe damping \u000b4c= 0:01was kept constant while \u000b4awas varied from \u000b4a= 0:011(top) to\u000b4a= 0:013(middle)\nto\u000b4a= 0:015(bottom). Figure 6, shows clearly distinguish behaviors for the simulated alloys depending on the Ru-\nconcentration x. For low absorbed laser energies below 5:5\u0001108J/m3no switching occurs for all Ru concentrations\nconsidered here. This is due to the insufficient energy to temporarily demagnetize both sublattices. For all three\ncases, we find that for low Ruthenium concentrations below a damping-dependent threshold value the alloy does not\nswitch, independent of the laser energy. Only above that threshold we find deterministic switching. In the top panel\nwith\u000b4c= 0:01and\u000b4a= 0:011, we find the threshold Ru-concentration for switching to be around x\u00190:9\u00000:95.\nWhen the damping of the 4asublattice is increased to \u000b4a= 0:013the threshold moves to x\u00190:85\u00000:9, which\napproximatelycorrespondstotheswitchingthresholdfoundinRef.14. Finally, for \u000b4a= 0:015(bottom)theswitching\nthreshold decreases to x\u00190:8. Therefore we find, that by increasing the element specific damping discrepancy the\nswitching threshold moves towards lower Ruthenium concentrations. Our results compare best to the experiments\nby Banerjee and co-workers (threshold around x= 0:8\u00000:9) [14] when choosing \u000b4c= 0:01and\u000b4a= 0:013(Fig.\n6 middle). We note that in our model the only parameter that directly depends on xis\u0016s;4c, which impacts the9\n4567Energy [108J/m3]\nNo SwitchingSwitchingα4a= 0.011\n4567Energy [108J/m3]\nNo SwitchingSwitchingα4a= 0.013\n0.6 0.7 0.8 0.9 1.0\nRuthenium Concenentration x4567Energy [108J/m3]\nNo SwitchingSwitchingα4a= 0.015\nFIG. 6. Switching behavior as color of Mn 2RuxGa alloys as function of the Ruthenium concentration xand the absorbed\nlaser energy for different dampings \u000b4aof the 4a-sublattice. Red areas indicate switching behavior, blue marks areas without\nswitching and grey indicates a prolonged transient ferromagnetic state or a demagnetized state. The damping \u000b4c= 0:01of the\n4c-sublattice was kept constant while \u000b4awas varied from \u000b4a= 0:011(top) to\u000b4a= 0:013(middle) to \u000b4a= 0:015(bottom).10\nspeed of the 4c-sublattice. Our model shows that by increasing x,\u0016s;4calso increases and in turn the sublattice\ndemagnetization speed continuously slows down, up to the point where the demagnetization speed difference in the\n4a- and 4c-sublattice is large enough to enable switching behavior. The model also shows that this relatively different\ndemagnetization speed can also be controlled by the intrinsic site-dependent damping parameters, which influences\nthex-dependent threshold between switching and non switching behavior." }, { "title": "1907.07997v2.Electric_bias_controlled_switching_of_magnetization_of_ferrimagnetically_coupled_Mn_delta_layers_in_a_GaAs_AlGaAs_quantum_well.pdf", "content": "arXiv:1907.07997v2 [cond-mat.str-el] 12 Feb 2020Highlights\nElectric bias-controlled switching of magnetization of fe rrimagnet-\nically coupled Mn delta-layers in a GaAs-AlGaAs quantum wel l\nN. V. Agrinskaya, A. M. Kalashnikova, V. I. Kozub\n•We suggest a design of an artificial ferrimagnet consisting of two Mn -\ndoped delta-layers with different concentrations placed in a GaAs-\nAlGaAs quantum well and coupled via extra holes in the well.\n•We analyze a scenario of the magnetization switching in the artificial\nferrimagnet realized via heating the holes by a picosecond electric bia s\npulse which facilitates exchange scattering of the holes on Mn ions.Electric bias-controlled switching of magnetization of\nferrimagnetically coupled Mn delta-layers in a\nGaAs-AlGaAs quantum well\nN. V. Agrinskaya, A. M. Kalashnikova, V. I. Kozub\nIoffe Institute, 194021 St. Petersburg, Russia\nAbstract\nWe suggest a model of synthetic ferrimagnetic semiconductor str ucture based\non GaAs-AlGaAs quantum well doped by two Mn delta-layers. The cou pling\nbetween the delta-layers is mediated by extra holes, and can be swit ched\nbetween ferro- and antiferromagnetic one by gating the structu re. A proper\nchoice of Mn concentrations in the delta-layers and of local degree of dis-\norder enables fabrication of a ferrimagnetic structure supportin g ultrafast\nswitching of magnetization by short pulses of electric bias without an ex-\nternal magnetic field. The switching mechanism in the structure relie s on\nkinetic spin exchange between the two delta-layers which is mediated by ex-\nchange scattering of electric-pulse heated holes by magnetic ions w ithin the\nlayers. Owing to specific interplay between characteristics of the e xchange\nscattering, spin decay times, and the heat withdraw in the suggest ed syn-\nthetic ferrimagnetic semiconductor, the necessary parameters of electric-bias\npulse are within the technologically accessible range, and do not cont radict\ntypical thermal kinetics of semiconductor structures.\nKeywords: synthetic ferrimagnet, GaAs quantum well, ultrafast heating,\nultrafast switching of magnetization\nPACS: 75.50.Gg, 75.50.Pp, 75.50.Ss, 71.70.Gm, 75.78.-n\n1. Introduction\nRecently, a great attention [1] was attracted by experiments dem onstrat-\ning extremely fast ( ∼10−12s) magnetization reversal triggered by a single\nfemtosecond laser pulse in a ferrimagnetic metallic rare-earth (RE) - tran-\nsition metal (TM) alloy GdFeCo without external magnetic field [2, 3, 4 ].\nPreprint submitted to Journal of Magnetism and Magnetic Mat erials February 13, 2020Follow-up studies have suggested that laser-induced magnetizatio n switch-\ning can be also realized in other RE-TM alloys, as well as in a variety\nof the engineered ferrimagnetic structures, including exchange c oupled RE-\nTM multilayers and heterostructures comprised by two transition m etal lay-\ners antiferromagnetically coupled through a nonmagnetic metallic int erlayer\n[5, 6, 8]. Most importantly, experimental studies have demonstrat ed that the\nall-optical reversal of magnetization is not precessional and relies on subpi-\ncosecond quenching of the magnetizations [7] of RE and TM sublattic es [3].\nThe time-resolved X-ray [9, 10] and optical [11] experiments have u nveiled\nunconventional distinct dynamics of these sublattices leading to em ergence\nof a transient ferromagnetic-like state. Such nonequilibrium dynam ics is be-\nlieved to enable deterministic magnetization reversal, without any ne ed for\nany other stimulus defining the magnetization direction [4, 12].\nNaturally, microscopical mechanism underlying unconventional res ponse\nof magnetization of a ferrimagnetic metallic system to a femtosecon d laser\npulse is the subject of intense discussions nowadays. Several the oretical and\nmodelling approaches were employed to account for main features o f the all-\noptical switching. Atomistic and multiscale calculations based on a Lan dau-\nLifshitz-Bloch equation [13] for an ensemble of the exchange-coup led spins\nwere developed in Refs. [4, 14, 15, 16, 17] for describing the all-op tical rever-\nsal in single phase alloys, as well as in exchange-coupled multilayers [18 ], and\nto examine a feasibility of switching in ordered RE-TM alloys [17]. In [19] a\ncomprehensive phenomenological model based on the Onsager’s re lations [20]\nwas developed introducing an exchange-dominated regime of laser- induced\ndynamics in a ferrimagnet, which allows the reversal of magnetizatio n solely\ndue to the ultrafast heating. This work highlighted importance of th e an-\ngular momentum exchange between the sublattices of the ferrimag net. In\norder to gain insight into microscopical nature of the all-optical swit ching,\ndissipationless energy and angular momentum exchange between TM and\nRE sublattices mediated by 5 d-4fcoupling in RE ions has been explored in\n[21], and the exchange electron-electron scattering as the driving mechanism\nof the magnetization reversal was discussed in [22]. In [23] a genera l micro-\nscopic approach based on the rate equations was suggested for a ddressing\nthe problem of the angular momentum exchange between two noneq uivalent\nmagnetic sublattices in a metal. In the latter work, the exchange sc attering\nwas found to be the driving mechanism of the switching. Similarly, switc hing\nenabled by the exchange scattering was also considered in [24], with h owever,\nprincipally different model of a RE-TM system. Importantly, since a la ser\n2pulse serves as an ultrafast heating pulse only [4, 12], in [23] it was sug gested\nto realize electric-bias induced switching in a metallic ferrimagnetic str uc-\nture. Recently, the first report appeared on the experimental o bservation\nof the switching of the magnetization of the GdFeCo wire by a picosec ond\ncurrent pulse [25].\nAlong with unveiling a nature of unconventional dynamics of the spin s ys-\ntem, the goal of these theoretical studies is to provide recipes fo r novel sys-\ntems and alternative stimuli which enable ultrafast switching of magn etiza-\ntion under technologically accessible conditions [18, 17, 26]. Here we p ropose\na synthetic semiconducting ferrimagnet where the exchange scat tering-based\nmagnetization switching can be realized upon application of an electric -bias\npulse of a picosecond duration. The ferrimagnet is comprised by two fer-\nromagnetic Mn-doped delta-layers in a GaAs-AlGaAs quantum well (Q W).\nAntiferromagnetic coupling between the delta-layers is supported by the ex-\ntra holes in the QW, and can be controlled by proper gating the struc ture.\nFerrimagnetism of the whole structure is realized due to different Mn con-\ncentration in the ferromagnetic delta-layers. Using the general m icroscopic\napproach developed earlier in [23], we show that ferrimagnetic prope rties\nof this structure support switching of the net magnetization via tr ansient\nferromagnetic-like state in zero applied magnetic field. Occurrence of the\ntransient ferromagnetic-like state and consequent switching of t he net mag-\nnetization of the structure is enabled by different magnetizations a nd Curie\ntemperatures of the delta-layers, while the rates of exchange sc attering of\nfree carriers (holes) at the magnetic ions (Mn) are similar in both laye rs.\nThe latter is in contrast to the all-optical or electric-bias switching in RE-\nTM alloys, where different exchange scattering rates for the TM an d RE\nmetals were playing a decisive role. Further, we argue that semicond ucting\nproperties make the switching more energetically profitable than in m etals,\nand also ease constrains regarding short duration of the electric b ias pulse\nrequired for switching.\n2. A syntheticferrimagnet based ontwoMn delta-layers ina G aAs-\nAlGaAs QW\nAs a candidate structure for a synthetic ferrimagnet we consider a QW\nGaAs-GaAlAs containing two delta-layers of Mn with concentrations N1and\nN2. The delta-layers are separated by a distance D, which is comparable\nto the well width L∼15 nm (Fig.1). Recently a system comprised by a Mn\n3delta-layer in GaAs-based QW attracted attention [27, 28] as an a lternative\nto a bulk diluted ferromagnetic semiconductor GaMnAs [29]. These st udies\nimply that the most promising realization of such a ferromagnet is the one\nwith a delta-layer situated in the barrier in the vicinity of the QW. The\nferromagnetic ordering is then realized due to indirect exchange su pported\nby holes within the well. It was suggested that in such a way the holes w ithin\nthe well do not experience a disorder imposed by Mn delta-layer [27, 28].\nThis supposedly allows obtaining higher Curie temperatures TCdespite of\nthe tunneling exponent which is necessary to pay for the holes to co ntact Mn\nions.\nRecently an observation of ferromagnetism in delta-doped GaAs-A lGaAs\nQWs with Curie temperatures around 200 K at unusually small Mn dopin g\nlevels of∼5·1012cm−2was reported in [30, 31] The indirect exchange in the\ndelta-layer in this case was supported by the holes supplied by Mn ato ms\nthemselves. The lower concentration of Mn dopants lead to a decre ase of\ndisorder potential which favors higher TCdue to, in particular, a suppres-\nsion of concentration of Mn interstitials which are known to be compe nsating\ndefects. This fact was proved experimentally by demonstrating th at an in-\ncrease of Mn concentration leads to a suppression of ferromagne tism in the\nin delta-doped GaAs-AlGaAs QWs [30]. This result advocates attempt s to\nfabricate ferromagnetic structures by doping a region within a QW, which\nmagnetic properties are controlled by holes within the well. Then, one can\nalso fabricate twodelta-layers of Mn in the same QW. Importantly, in this\ncase the layers will be coupled to the same holes subsystem localized w ithin\nthe well.\nHere we consider a particular case when N2> N1, withN1being large\nenough to support ferromagnetism in the corresponding delta-lay er. The\nseparation distance Dbetween the delta-layers is much larger than the lo-\ncalization length for a Mn hole. Then the ferromagnetism of each delt a-layer\nis supported by own holes of Mn acceptors [30]. In this case the laye r with\nMn concentration N2is characterized by a lower value of coupling between\nthe ferromagnetic ions due to an increase of degree of disorder. T he rough\nestimate of the disorder effect on TC2can be given by TC2∝exp(−N−1/2\n2/l).\nHerelis the mean free path for the holes in the vicinity of the second layer,\nand it is somewhat less than the distance between the Mn atoms ���N−1/2\n2.\nNote that the condition l < N−1/2\n2 can be also reached by any additional\nintended contamination of the corresponding interface region. In any case\n4one can control the value of Tc2for a given value of N2> N1. Thus we\nconsider the system of two ferromagnetic layers 1 and 2 with Tc1> Tc2and\nthe saturation magnetizations Ms1< Ms2(Fig. 1).\nUnder the assumptions considered above, the delta-layers are no t ex-\nchange coupled. An exchange coupling of a RudermanKittelKasuyaY osida\n(RKKY) type between the layers can be realized by additional carrie rs sup-\nplied by doping of the barriers with shallow Be acceptors. Doping both\nbarriers with nearly the same surface concentrations N3/2 allows cancelling\nthe electric fields produced by the barrier acceptor layers within th e well.\nThus, these fields do not affect the bare well potential. The RKKY ex change\ncoupling between the delta-layers could be either ferromagnetic or antifer-\nromagnetic, depending on the band holes energies. We note that ad ditional\nholes within the well would improve ferromagnetic ordering in the delta -\nlayers via RKKY exchange coupling. We further note that a slightly hig her\ndoping level in the barrier closest to the second layer can impose add itional\ndisorder potential on this layer resulting from the screening the ad ditional\nbarrier charges by the holes within the second layer.\nHere we have to address the role of the second Hubbard band for t he Mn\nacceptors, since the Hubbard energy for them is at least twice lowe r than\nionization energy [32]. Thus one expects that the barrier dopants w ill ini-\ntially lead to partial filling of the upper Hubbard bands of the Mn accep tors.\nHowever, the complete filling of the upper Hubbard band correspon ds to\nadditional hole and thus additional positive charge per any site within the\ndelta-layer. The concentration of Mn-sites allowing ferromagnetic ordering\nstarts around 1013cm−2giving the intersite distances of order of 3 nm [31],\nand the complete filling of the upper Hubbard bands leads to large incr ease\nof the on-site hole energies resulting from coupling to the holes on th e neigh-\nboring sites (at distances at least of the order of distance to the n egatively\ncharged layers within the barriers). Such holes will be inevitably push ed to\nthe band states which are delocalized within the well. Indeed, any cha rged\nneighbor at the distance 3 nm gives additional energy of the order o f 30 meV\n∼300 K while the ionization energy for Mn is 100 meV. Correspondingly,\nonly relatively small part of the upper Hubbard band can be situated lower\nthan the bottom of the band resulting from lateral quantization. A t par-\ntial occupations of the upper Hubbard bands the states of these bands are\nexpected to be delocalized along the corresponding layer giving addit ional\nsupport to the ferromagnetic ordering within the layer. For larger concentra-\ntion of the additional holes within the well, N3−(N1+N2)> βN1(β << 1),\n5the presence of the upper Hubbard band can be neglected due to a presence\nof the band holes. As a result, the whole system can be considered a s the\nferromagnetically ordered delta-layers with the delocalized band ho les.\nConcentration of these holes of the order of 1019cm−3gives the total\nsurface concentration of the holes of the order of 1013cm−2[31], that is of\nthe order of the surface concentration within the ferromagnetic layers. In this\ncase the screening length is of the order of ∼2 nm for the holes with energy\n∼30 meV ( ∼300K). Such distance is smaller or, at least, comparable to the\nintersite distances within the layers. Thus, at corresponding conc entrations\nand the free holes energies we can consider the potentials of the do ubly\noccupied sites to be screened. This gives and additional argument t o neglect\na filling of the upper Hubbard band in our model.\nWe assume N1∼5·1012cm−2,N2∼1.5·1013cm−2,N3∼1013cm−2.\nFor the width of the well L∼1.5·10−6cm and the hole mass 10−27g one\nestimates the energy of the first quantization level of /planckover2pi12π2/2mL2∼2 meV,\nthe Fermi wavelength of the order of 10−6cm and concentration of the order\nof 1012cm−2. For concentration of the order of 1013cm−2distributed over\nthe well one has the interhole spacing ∼101/2·10−7cm that is around λF∼\n3 nm. It corresponds to occupation of the 3 dlevel of lateral quantization\nwhich gives oscillating wave function providing a possibility to have RKKY\nexchange coupling of any sign between the Mn delta-layers. Here we also\nnote that the Fermi energy of the band holes at the considered co ncentrations\nbecomes to be of the order or larger than the temperature of the system (200\n- 300 K). Thus, at the temperatures close to the critical ones, wh ich are also\naround 200 - 300 K, the situation is close to degeneration which facilit ates\nthe RKKY interaction.\nWe note that several attempts to realize and study exchange inte ractions\nbetween two Mn delta-layers imbedded into GaAs matrix were report ed (see\ne.g. [33, 34]). However the main attention in these studies has been p aid to\nthe role of the carriers supplied by Mn itself, and the interlayer dista nce was\nassumed to be small in comparison to localization length of the Mn holes\nin the GaAs host. This is in strong contrast to the structure consid ered in\nthe present work, where the additional holes are supplied by dopan ts within\nthe AlGaAs barriers. As a result the distance between Mn layers can be at\nleast of the order of several localization lengths, and the waveleng th of the\ndoped holes is comparable to the interlayer distance, and the situat ion is\nclearly corresponds to the RKKY interactions. Importantly, this in terlayer\nRKKY exchange coupling can be controlled by the gate situated in any of\n6the barriers. Indeed, the gate potential allows to control the co ncentration\nof the band holes within the well, which, in turn, affects the characte r of\ninterlayer coupling.\n3. Switching the magnetization of the delta-layers by elect ric-bias\npulse\nNow we consider an evolution of the magnetizations of the two delta-\nlayers with antiferromagnetic RKKY exchange between them in resp onse to\na short electric bias pulse of a duration of a few picoseconds. The ap plication\nof the electric bias pulse rapidly increases the temperature of the h oles in the\nQW. To treat the following evolution of the spin system we use the mod el\ndeveloped in [23]. In brief, it describes evolution of the occupation nu mbers\nof twodifferent ferromagnetic sublattices coupled antiferromagnetically via\ndelocalized carriers. The spin exchange between the localized ferro magnetic\nsubsystems is mediated by delocalized carriers which temperature is rapidly\nincreased above critical temperatures of both magnetic sublattic es. Either\nfemtosecond laser excitation or applying short electric bias pulse ca n serve\nfor the carriers heating. Spin exchange results in the switching of t he net\nmagnetization of the system without any additional stimuli, such as e xternal\nmagnetic field, circular polarization of light, or spin polarization of a cu rrent.\nThe switching of the net magnetization in this model relies on a delicate bal-\nance between the exchange scattering, spin relaxation, and coolin g times.\nImportantly, this model is not restricted to the case of RE-TM alloy s or het-\nerostructures, and is also applicable for the case of the structur es composed\nby two different transition metals.\nThe important difference between the system analyzed here and th e RE-\nTM metallic systems considered in [23], is that here two ferromagnet ic sub-\nsystems, i.e. two delta-layers, are comprised by the same Mn ions. T hus,\nthe exchange scattering of the holes on Mn ions is nearly equal for t he both\nsubsystems. We can estimate the characteristic times of this scat tering in the\nfollowing way. A 3D cross-section for Coulomb scattering of the mob ile holes\nby an unscreened ion is of ( e2/κε)2∼10−12cm2. 3D geometry is consid-\nered since the holes are delocalized in the direction normal to the well plane.\nHowever, the effect of nonlinear screening by the holes, in particula r by the\nones supplied by Mn atoms itself, diminish this estimate down to 10−13cm2\nwhich is controlled by the distance between Mn atoms. The cross-se ction for\nthe exchange scattering is expected to be less by a factor γ2, whereγcharac-\n7terizes the relative strength of the exchange potential, and can b e estimated\nto be of ∼0.3. Now we take into account that the inverse mean free path\nwith respect to the exchange scattering is given by a product of th is cross-\nsection and 3D concentration of Mn ions in the well ( ∼1019cm−3). This\ngives the value of 10−5cm for the mean free path. Correspondingly, for the\nholes velocity ∼107cm/s [38] the exchange scattering time can be estimated\nto be ofτex∼1 ps.\nAnother important difference between the spin dynamics in the cons id-\nered artificial ferrimagnet and in the RE-TM metallic alloy is that the sp in\nrelaxation times in the semiconductor GaAs appear to be considerab ly longer\nthan the exchange scattering times. In [35] the spin relaxation tim e for the\nholes in GaAs structures was found to be of ∼20 ps. This value is signifi-\ncantly larger than the spin relaxation time ∼0.1−1 ps in metals. Therefore,\nat the first stage of the electric-bias driven evolution of the spin ba lance in\nthe system, the effect of spin relaxation can be neglected. We note that this\nassumptions corresponds to the model suggested in [24], where th e two fer-\nromagnetic subsystems with different magnetizations and similar exc hange\nscattering times were considered. As we discussed above, such as sumptions\nare justified in the case of Mn layers within the semiconductor QW, wh ile\ntheir applicability to the metallic RE-TM alloy is arguable.\nRight after the application of the electric bias pulse, the temperatu re of\nholes in the QW is increased to the value T∗\nh, and the evolution of the spin\nsystem at this stage is reduced to redistribution of the total angu lar momen-\ntums(N2−N1), wheresis the spin of a hole, between the two ferromagnetic\nsubsystems. When the holes temperature reached T∗\nh, the resulting momenta\nsn1andsn2of the layers are then given by\nn1=(N2−N1)\nN2+N1N1, n 2=(N2−N1)\nN2+N1N2. (1)\nAs it is seen from Eqs. (1), the sign of the magnetization for the bot h sub-\nsystems now coincide since we consider the system with N2> N1. That is\nat this stage the mutual spin orientations in the two delta-layers co rrespond\nto a ferromagnetic-like state.[9]\nEvolution of delta-layers magnetizations from their equilibrium values to\nthe ones given by Eqs. (1) occurs on the time scale given by the estim ated\nexchange scattering time τex∼1 ps. We note that, in contrast to the evolution\nof the TM and RE magnetizations in the metallic RE-Tm alloys [23], the\nparticular time when the Ms1vanishes appear not to play crucial role, since\n8the spin relaxation times are much slower that the exchange scatte ring times.\nOnly if the spin relaxation is fast or if the temperature of the system remains\nelevated (i.e. no cooling is present) then the total angular moment, and,\ncorrespondingly, the total magnetization would finally vanish.\nTo describe the evolution of the system from the ferromagnetic-lik e state\nwe take into account the cooling down process. First, we consider a case\nwhen the dominant cooling mechanism is related to optical phonons wit h\na frequency ω0. This is relevant when the temperature of the holes is high\nenough. In this case the cooling time necessary to lower the temper ature\nof the holes from T∗\nh, down to some value Th, can be roughly estimated as\nt=τh−phkB(T∗\nh−Th)/(/planckover2pi1ω0), where the characteristic hole-phonon relaxation\ntime isτh−ph∼10−12s [39, 40].\nAt temperatures lower that /planckover2pi1ω0the further cooling is assisted by acous-\ntical phonons. To estimate the corresponding relaxation time we ta ke into\naccount that the holes are coupled to phonons with momentum equa l or less\nthan the holes momentum due to a momentum conservation. The hole s ve-\nlocities are of ∼107cm/s for temperature range in a vicinity of the Curie\ntemperatures of the layers, i.e. in the range of 200 - 300 K [38]. The wavevec-\ntors of corresponding phonons are less than 107cm−1. Thus, one concludes\nthat the phonon energies /planckover2pi1ωare about 10 times lower than the holes energies\nεh. In this case the energy relaxation time is larger by a factor ∼(εh//planckover2pi1ω)2\nthan the typical hole-phonon relaxation time τh−ph∼10−12s, provided that\nthe temperature This not much below the room temperature. Thus, we\nobtain characteristic time t=τh−ph(kBTh//planckover2pi1ω)2for the further energy relax-\nation.\nIn our estimates we assume that the heat withdrawal from the qua ntum\nwell is efficient enough, which can be ensured by the purity of the AlGa As\nbarriers and small mismatch of the elastic constants between GaAs and Al-\nGaAs. Thus, the phonon transport outside of the well can be cons idered\nas a ballistic one, and the characteristic time of the phonon escape c an be\nestimated as D/w , whereDis of the order of the thickness of the QW, and\nwis the sound velocity. Than the phonon escape time is of the order of\n10−12s which is comparable to the hole-phonon relaxation time τh−ph. The\nheat capacity of phonon subsystem is much larger than that of the holes\nsubsystem. For hole concentrations around 3 ·1012cm−2the corresponding\nfactor is around 104. Thus at time scale less than 10−8s the problem of heat\nwithdrawal can be neglected.\nSince the critical temperature for the first subsystem, Tc1is larger, then\n9that for the second subsystem ( Tc2), the temperature of the holes reaches the\nvalue ofTc1first upon cooling. Once the holes temperature is below Tc1the\nmagnetization of the first subsystem starts to restore accordin g to standard\nthermodynamical law as (see e.g. [36])\n∆Ms1∼Ms1/parenleftbiggTc1−Th\nTc1/parenrightbigg1/2\n, (2)\nwhere the This controlled by a process of cooling. Thus at a moment when\nN1/parenleftbiggTc1−Th\nTc1/parenrightbigg1/2\n∼(N2−N1)\nN2+N1N2 (3)\nthe magnetization of the layer 1 starts to dominate over the magne tization of\nthe layer 2 provided that it this moment the holes temperature is still larger\nthan the critical temperature Tc2.\nAt the final stage of the process, i.e. when the holes temperature is\nlowered down to the critical temperature Tc2of the layer 2, the magnetization\nof the latter starts to restore as well. The preferential direction for the layer\n2 magnetization is now set by the antiferromagnetic RKKY coupling to the\nmagnetization of the layer 1. Correspondingly, the net magnetizat ion of the\nsystem is switched at this stage.\n4. Conclusions\nWe suggested a model of a synthetic ferrimagnetic structure bas ed on\nthe Mn-doped GaAs-AlGaAs quantum well. It contains two Mn delta-la yers\nwithin the well with different Mn concentrations and different degree s of dis-\norder. Two symmetric layers of barrier acceptors are situated on different\nsides of the well in order to provide extra holes to the QW. These ext ra holes\nare essential since they participate in the RKKY coupling between th e delta-\nlayers. The sign of the RKKY coupling can be controlled by concentra tion of\nthese holes, by distance between Mn layers, and by gating the stru cture. As\na result, a structure of two antiferromagnetically coupled Mn delta -layers can\nbe fabricated with one layer possessing larger saturation magnetiz ation and\nlower critical temperature that the second layer. Using recently s uggested\ntheory for the switching of a ferrimagnet driven by the kinetic exch ange\nscattering, we show that the magnetization of such semiconducto r-based syn-\nthetic ferrimagnet can be switched by a short pulse of electric bias w ith no\n10additional external stimuli, e.g. external magnetic field, required t o set the\nmagnetization direction.\nWe note that the suggested switching scenario of the synthetic se miconductor-\nbased ferrimagne can be also realized by using short laser pulses inst ead of\nelectric bias ones, since the former are known to trigger ultrafast demag-\nnetization in GaMnAs [37]. It may be important from a point of view of\nintegration of Mn-doped GaAs-AlGaAs QW allowing toggle switching of\nmagnetization with the ultrashort semiconducor laser sources. Ga As and\nAlGaAs are among media for developing such lasers (for a review see, e.g.,\n[41] and more publications on this subject, e.g. [42]). Thus, one could en-\nvisage designing a device contained both a ultrafast laser source an d the\nmagnetic toggle switch ”on a single chip”. For the later to be realized, the\nCurie temperatures of the Mn-doped GaAs-AlGaAs QW should be abo ve\nroom temperature. Mn delta-doped GaAs possesses one of the hig hest tem-\nperatures among the III-V semiconductors reaching 250 K [27]. O n the other\nhand, among the bulk ferromagnetic III-V semiconductors, thos e doped with\nFe ions have the Curie temperature higher than (Ga,Mn)As [43]. Ther e-\nfore, it would be important to investigate if the III-V semiconducto r-based\nstructures with Fe delta-layers can be fabricated for the ultrafa st switching.\nPresently, however, there are no reports on successful increa se of the Curie\ntemperature by Fe delta-doping of a III-V semiconductor [44].\n5. 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Growth 511, 127 (2019).\n15Tc1Tc2(90\u000efor\u0011= 1:3.\nThe varying deformation originates from the fact that the di-\nrection of the Magnus force is determined by the sign of both\nthe topological charge Qand the net spin density \u001b.\nAs shown in Fig. 5(b-d), \u0011= 0:9(\u001b= +0:58A2s\u0001m\u00002),\nthe drift speed vyof the inner skyrmion is along the \u0000ydi-\nrection, while the Magnus force acting on the outer skyrmion\npoints up. Besides, the magnitudes of vxandvyare propor-\ntional to the size of skyrmions. As a consequence, < 90\u000e,\nand the components of tensors Das well as Iobviously vary\nwith time. On the contrary, when \u0011= 1:3(\u001b=\u00000:58\nA2s\u0001m\u00002), the drift speed vyof the inner skyrmion and the\nouter skyrmion are upward and downward respectively, which\ninduce a deformation angle larger than 90\u000e. As discussed\nFIG. 6. Comparison between the current-induced dynamics of (a)\nFM, (b) FiM ( \u0011= 0:9) and (c) AFM skyrmioniums. Insets show the\ntop view of a skyrmionium driven by the selected current density j=\n20MA\u0001cm\u00002att= 6 ns, which is indicated by the vertical dashed\nline. Symbols represent the results from numerical simulations, and\nthe dashed lines indicate the analytical results based on Eq. ( 7).\nabove, the skyrmions have no drift speed along the y-axis\nat the compensation point of the angular momentum, so that\nthe two skyrmions consisting of the FiM skyrmionium move\nalong thex-axis at different velocities leading to = 90\u000e. It\nis worth mentioning that, when \u0011= 1:1(\u001b= 0:0A2s\u0001m\u00002),\nthe components Dxy,IxxandIyyare still 0and do not\nvary with time. However, when \u00116=1:1, these quantities are6\nFM\nFiM25 MA· cm-2\n30 MA· cm-230 MA· cm-2\n30 MA· cm-227 m· s-1\n41 m· s-1\n61 m· s-1(a)\n(b)\n(c)\n(d)\nFIG. 7. Snapshots of (a-b) a FM skyrmion, (c) a FiM skyrmion and (d) a FiM skyrmionium during their motion driven by a spin current, where\nthe yellow dashed line represents the initial position of spin structures, and the gray lines stand for their trajectories. The time evolution of\nvelocities for these magnetic structures driven by a spin current with (e) j= 25 MA\u0001cm\u00002and (f-h)j= 30 MA\u0001cm\u00002. Here,\u0011= 0:9for a\ntypical uncompensated FiM system.\nnonzero, indicting the different deformations. In general,\nDxy<0(Ixx<0andIyy>0) corresponds to the defor-\nmation angle <90\u000e, whileDxy>0(Ixx>0andIyy<0)\nis related to the deformation angle >90\u000e. The final config-\nurations of these deformed FiM skyrmioniums at such current\ndensities are provided in Ref. [63].\nWe now compare the current-induced dynamics of FM,\nFiM and AFM skyrmioniums. In Fig. 6, we demonstrate\nthat the response of a skyrmionium to the driving current\nin the three magnetic systems. It is seen that a relatively\nsmall driving current density results in the distortion of a FM\nskyrmionium during its motion, while the AFM skyrmionium\nis the most difficult to be destroyed. The FiM skyrmionium\nis the intermediate one between the cases of FM and AFM\nskyrmioniums. Moreover, driven by the same current den-\nsity ofj= 20 MA\u0001cm\u00002and compared with FiM and AFM\nskyrmioniums, the FM skyrmionium is most fragile. Taking\nthe micro-structures of the three systems into account, the net\nangular momentum is completely canceled in the AFM sys-\ntem due to the two sublattices coupled antiferromagnetically\nwith the same magnetic moments and the same gyromagnetic\nratio. Different from the AFM system, there is a resultant\nangular momentum in the FiM system, where the two sub-\nlattices normally have different spin densities Mi=\ri. On the\nother hand, the net angular momentum of the FiM system is\nstill smaller than that of the FM system. Note that the Magnus\nforce depends on \u001bas shown in Eq. 5. From the point of view\nof applications, the FiM skyrmionium may be a good choice\nsince it not only suppresses the skyrmion Hall effect more ef-\nfectively than the FM skyrmionium, but also is easier to be\ndetected compared with the AFM one.\nHere, the current-induced dynamics of a FiM skyrmion-\nium is also compared with that of a FM skyrmion and a FiM\nskyrmion. During the motion driven by a spin current, the FM\nskyrmion shows an inevitable skyrmion Hall effect due to the\nMagnus force associated with the nonzero topological charge.In a confined nanotrack, as shown in Fig. 7(a), this Magnus\nforce is canceled by the repulsive force arisen from the bound-\nary so that the skyrmion moves along the edge with a steady\nspeed. However, for a large driving current density, the en-\nergy barrier of the edge is insufficient to confine a skyrmion\nin a nanotrack. Consequently, the skyrmion disappears at the\nedge [see Fig. 7(b)]. Considering that the net angular momen-\ntum of two sublattices are not completely cancelled, the FiM\nskyrmion shows the similar motion behaviors. From Figs. 7(c)\nand (d), it is seen that the steady speed of a FiM skyrmion is\nsmaller than that of a FiM skyrmionium driven by the same\ncurrent density. Moreover, the FiM skyrmionium moves along\nthe centerline of the nanotrack without drift speed as dis-\ncussed before. Figures 7(e-h) also show the time evolution\nof velocities for these spin configurations during their motion.\nFor the undamaged FM skyrmion and FiM skyrmion, the x-\ncomponent of the velocity begins with a constant, and then\nincreases as the skyrmion approaches the edge and eventually\nreaches a new steady value, while the y-component decreases\nto zero. However, the velocity of a skyrmionium remains un-\nchanged with a relatively large value. Therefore, compared\nwith skyrmions, the FiM skyrmionium as a carrier of informa-\ntion can effectively prevent the accumulation and annihilation\nof skyrmions at the edge, and potentially improve the access\nspeed of storage devices.\nIV . CONCLUSION\nIn conclusion, we have analytically and numerically inves-\ntigated the current-induced dynamics of a FiM skyrmionium\nin a nanotrack. Our results show that, at the angular momen-\ntum compensation point, the FiM skyrmionium is most robust\nto resist the deformation due to the zero intrinsic skyrmion\nHall effect, which is same as the case of an AFM skyrmio-\nnium. Nevertheless, the position of a FiM skyrmioniums is7\nobservable due to a nonzero magnetization. It is found that\nthe direction of distortion depends on the sign of the net an-\ngular momentum. Above the angular momentum compensa-\ntion point, the deformation angle is smaller than 90\u000e. How-\never, the deformation angle is larger than 90\u000ebelow the an-\ngular momentum compensation point. We have also demon-\nstrated the change in components of two tensors DandIfor a\nFiM skyrmionium during the motion to describe its deforma-\ntion. Furthermore, we have made a comparison between the\ncurrent-induced dynamics of FM, FiM and AFM skyrmion-\niums. The motion of a FiM skyrmionium is also compared\nwith that of FM and FiM skyrmions. Our results open a new\nfiled of the skyrmionium physics in the FiM system and could\nprovide guidelines for the design of future spintronic devices\nbased on ferrimagnets.\nACKNOWLEDGMENTS\nThis research was supported by Guangdong Special\nSupport Project (2019BT02X030), Shenzhen Fundamen-\ntal Research Fund (Grant No. JCYJ20210324120213037),\nShenzhen Peacock Group Plan (KQTD20180413181702403),\nPearl River Recruitment Program of Talents (2017GC010293)\nand National Natural Science Foundation of China\n(11974298, 61961136006). X.L.’s PhD study was financially\nsupported by the National Natural Science Foundation of\nChina (Grant No. 12004320). X.Z. was an International\nResearch Fellow of the Japan Society for the Promotion of\nScience (JSPS). X.Z. was supported by JSPS KAKENHI\n(Grant No. JP20F20363). J.X. acknowledges the support by\nthe National Natural Science Foundation of China (Grant No.\n12104327). M.E. acknowledges the support by the Grants-\nin-Aid for Scientific Research from JSPS KAKENHI (Grant\nNos. JP17K05490 and JP18H03676) and the support by\nCREST, JST (Grant Nos. JPMJCR16F1 and JPMJCR20T2).\nX.X.L. acknowledges the support by the Grants-in-Aid\nfor Scientific Research from JSPS KAKENHI (Grant Nos.\nJP20F20363 and JP21H01364).\nAPPENDIX A: ANALYTICAL DERIVATION OF THE\nMOTION EQUATIONS\nTo derive the motion equation of magnetization in a ferri-\nmagnetic system, we start from the Laudau-Lifshitz-Gilbert\n(LLG) equations with the spin-orbit torque for the two sublat-\ntices:\n_s1=\u0000\r1s1\u0002H1+\u000bs1\u0002_s1+\r1BD1s1\u0002(p\u0002s1);(A1)\n_s2=\u0000\r2s2\u0002H2+\u000bs2\u0002_s2+\r2BD2s2\u0002(p\u0002s2);(A2)\nwhere Hi =\u0000\u000e\"=(\u00160Mi\u000esi)andBDi =\n(\u0016B\u0012SHj)=(\rieMitz)are the effective fields associated\nwith various energies in the system and the damping-like\ntorque induced by the spin current, respectively. Multiplying\nthe Eqs. (A1) and (A2) by M1=\r1andM2=\r2, respectively,\nand then finding the addition and the subtraction of thesetwo equations, we obtain the following coupled equations of\nmotion:\n\u001a_n+\u001b_m=\u0000(m\u0002fn+n\u0002fm)\n+\u000b[\u001a(n\u0002_m+m\u0002_n) +\u001b(m\u0002_m+n\u0002_n)]\n+u1[m\u0002(p\u0002n) +n\u0002(p\u0002m)]\n+u2[n\u0002(p\u0002n) +m\u0002(p\u0002m)]; (A3)\n\u001a_m+\u001b_n=\u0000(m\u0002fm+n\u0002fn)\n+\u000b[\u001a(m\u0002_m+n\u0002_n) +\u001b(n\u0002_m+m\u0002_n)]\n+u1[n\u0002(p\u0002n) +m\u0002(p\u0002m)]\n+u2[m\u0002(p\u0002n) +n\u0002(p\u0002m)]: (A4)\nHere,\u001a=M1=\r1+M2=\r2,\u001b=M1=\r1\u0000M2=\r2,fm=\nf1+f2,fn=f1\u0000f2andu1=\f1+\f2,u2=\f1\u0000\f2with\n\fi=BDiMiandfi=HiMi. It is important to noted that\nthe rule for the linear combination of two variables is used in\nthis derivation, Ax+By= (1=2)[(A+B)(x+y) + (A\u0000\nB)(x\u0000y)]andAx\u0000By= (1=2)[(A+B)(x\u0000y) + (A\u0000\nB)(x+y)]. Considering the fact that jmj\u001cjnj\u00191for the\ncolinear ferrimagnets, and keeping leading-order terms, the\nabove equations are reduced as\n\u001a_n=fm\u0002n+\u000b(\u001an\u0002_m+\u001bn\u0002_n)\n+u2n\u0002(p\u0002n) +Tn\nnl; (A5)\n\u001a_m+\u001b_n=fm\u0002m+fn\u0002n+\u000b\u001an\u0002_n\n+u1n\u0002(p\u0002n) +Tm\nnl; (A6)\nwhere, Tn\nnl=u1n\u0002(p\u0002m)andTm\nnl=\u000b\u001bn\u0002_m+u2n\u0002\n(p\u0002m)are the weak nonlinear terms that will be discarded\nin the following derivation. The micromagnetic simulation of\nthe spin dynamics in this work is based on numerically solving\nEqs. (A5) and (A6).\nSubstituting fm=\u0000(1=\u00160)[\u0015m+L(@xn+@yn)]into\nEq. (A5), we obtain the total magnetization mthat depends\non the spatial Néel vector n.\nm=\u00160\n\u0015f\u0000\u001a(1 +\u000b2)n\u0002_n+\u000b[n\u0002fn\u0000u1n\u0002(p\u0002n)]\n\u0000u2p\u0002ng\u0000L\n\u0015(@xn+@yn); (A7)\nwhere the dissipation term \u000b[n\u0002fn\u0000u1n\u0002(p\u0002n)]\ncan be ignored. Rewriting the effective field fn=f\u0003\nn+\n(A\u0003=\u00160)(@xx+@yy+ 2@xy)n+ (L=\u0016 0)(@xm+@ym)with\nA\u0003=A=2and substituting mintofn\u0002n, we obtain\nfn\u0002n=f\u0003\nn\u0002n+A\n2\u00160(@xx+@yy+ 2@xy)n\u0002n+L\n\u00160\nh\u00160\n\u0015f\u0000\u001a( 1 +\u000b2)n\u0002(@x+@y)_n\u0000u2p\u0002(@x+@y)ng\n\u0000L\n\u0015(@xxn+@yyn+ 2@xyn)i\u0002n: (A8)\nSince\u0015= 4A=a2andL=p\n2A=a, we deduce that A=2 =\nL2=\u0015, and then the second and the last terms on the right side\nof the Eq.(A8) can be cancelled. We substitute mandfn\u0002n8\ninto Eq. (A6),\n\u00160\u001a\n\u0015f\u0000\u001a(1 +\u000b2)n\u0002n\u0000u2p\u0002_ng\u0000L\u001a\n\u0015(@x_n+@y_n)\n=\u0000\u001b_n+fm\u0002m+f\u0003\nn\u0002n+L\n\u0015f\u0000\u001a(1 +\u000b2)\nn\u0002(@x+@y)_n\u0000u2p\u0002(@x+@y)ng\u0002n\n+\u000b\u001an\u0002_n+u1n\u0002(p\u0002n): (A9)\nSupposing that (1 +\u000b2)\u00191and multiplying the Eq. (A9) by\nn, we obtain the following closed equation of the Néel vector\nn\n\u00160\u001a2\n\u0015n\u0002(n\u0002n) =\u001bn\u0002_n+\u000b\u001a_n+u1p\u0002n\n\u0000\u00160\u001au2\n\u0015n\u0002(p\u0002_n)\u0000n\u0002(f\u0003\nn\u0002n)\n+Lu2\n\u0015n\u0002f[p\u0002(@x+@y)n]\u0002ng: (A10)\nFor a centrosymmetric magnetic soliton, the order parameter\nnis described by both the position rand the azimuthal angle\n', i.e.,n(r;') = [ sin\u0012(r)cos\b(');sin\u0012(r)sin\b(');cos\u0012(r)].\nTaking the scalar product of Eq. (A10) with @in, and integrat-\ning over the space, we obtain the motion equation\nMR+\u001bG\u0002_R+\u000b\u001aD_R\u0000u1Ip=0; (A11)\nwhereM=\u00160\u001a2d=\u0015is the effective mass of the soliton, G=\n(0;0;G)is the topology-dependent gyrovector, DandIare\ntensors related to the damping term and the SOT, respectively.\nHere,\ndij=Z\n(@in\u0001@jn)dxdy =\u000eijd\n=\u000eij\u0019Z\n(\u00122\nr+sin2\u0012=r2)rdr; (A12)\nG=Z\n[n\u0001(@xn\u0002@yn)]dxdy = 2\u0019Z\nsin\u0012\u0012rdr; (A13)\nIij=Z\n(n\u0002@in)jdxdy; (A14)Z\n@in\u0001[n\u0002(p\u0002_n)]dxdy =0; (A15)\nZ\n@in\u0001[n\u0002(f\u0003\nn\u0002n)]dxdy =0; (A16)\nZ\n@in\u0001fn\u0002[p\u0002(@x+@y)n]\u0002ngdxdy = 0:(A17)\nThis term (n\u0002@in)jdenotes thej-component of the vector\n(n\u0002@in). Different from the gyrovector Gand the tensor\nD, the tensor Istrongly depends on the angle \b(')due to\nthe fact that\n(n\u0002@xn)x=\u0000sin\bcos'\u0012r+sin\u0012cos\u0012sin'cos\b\nr;(A18)\n(n\u0002@xn)y=cos\bcos'\u0012r+sin\u0012cos\u0012sin'sin\b\nr;(A19)\n(n\u0002@yn)x=\u0000sin\bsin'\u0012r\u0000sin\u0012cos\u0012cos'cos\b\nr;(A20)\n(n\u0002@yn)y=cos\bsin'\u0012r\u0000sin\u0012cos\u0012cos'sin\b\nr:(A21)\nIn general, the steady velocity is written as\n\u0014\nvx\nvy\u0015\n=u1\n\u000b2\u001a2d2+\u001b2G2\u0014\n\u000b\u001ad \u001bG\n\u0000\u001bG \u000b\u001ad\u0015\u0014\nIxxIxy\nIyxIyy\u0015\u0014\npx\npy\u0015\n:\n(A22)\nFor a Néel-type skyrmionium, \b =',Ixy=\u0000Iyx=I=\n\u0019R\n(r\u0012r+sin\u0012cos\u0012)drandIxx=Iyy= 0, above velocity\nequation becomes\n\u0014\nvx\nvy\u0015\nNéel=u1I\n\u000b\u001ad\u0014\npy\n\u0000px\u0015\n: (A23)\nSimilarly, for the Bloch-type skyrmionium, \b ='+\u0019=2,\nIxx=Iyy=\u0000IandIxy=Iyx= 0, the velocity is given by\n\u0014\nvx\nvy\u0015\nBloch=\u0000u1I\n\u000b\u001ad\u0014\npx\npy\u0015\n: (A24)\nTherefore, one can find that the velocity of a skyrmionium\ndepends on both the internal spin distribution and the polar-\nization direction of the spin current.\n[1] J. 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J. \nWalker1, Ignace Jarrige7, Sohrab Ismail -Beigi1,4, Charles Ahn1,4 \n \n1Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA \n2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama 35487 , USA \n3Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA \n4Department of Physics, Yale University, New Haven, Connecticut 06520, USA \n5Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA \n6Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, \nNew York 11973, USA \n7National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York, 11973, USA \n \nAbstract \nWe reveal in this study the fundamental low energy landscape in the ferrimagnetic Sr2CrReO 6 double \nperovskite and describe the underlying mechanisms responsible for the three low-energy excitations \nbelow 1.4 eV. Based on resonant inelastic x -ray scattering (RIXS ) and magnetic dynamics calculati ons, and \nexperiment s collected from both Sr2CrReO 6 powder s and epitaxially strained thin films, we reveal a strong \ncompetition between spin -orbit coupling, Hund’s coupling and the strain -induced tetragonal crystal field . \nWe also demonstrate that a spin -flip process is at the origin of the lowest excitation at 200 m eV, and we \nbring new insights into the predicted presence of orbital ordering in this material . We study the nature of \nthe magnons through a combination of ab initio and spin -wave theory calculations , and show that two \nnon-degenerate magnon bands exist and are dominated either by rhenium or chromium spins. The \nrhenium band is found to be flat at about 200meV ( ±25meV) through X -L-W-U high -symmetry points and \ndispersive to ward Γ. \nIntroduction \nMagnonics is an emerging alternative to modern electronics to carry and process information at high \nfrequency and low -energy consumption using magnetic degrees of freedom. This field of research focuses \non the use of spin waves or magn ons in magnetic materials. This type of collective excitation is not limited \nby Joule heating as it does not involve the transport of electrons , and allows the superposition of signals \nthrough multiplexing. While ferromagnets have been first considered for this application, the community \nhas recently turned to antiferromagnets for their higher frequencies of operation, the possibility of using \nleft- and right -handed wave s as an additional degree of freedom to carry information, and greater stability \nagainst parasitic magnetic field s. This stability results from the absence of net magnetization in 2 antiferromagnets, which provide s a relative immunity to external magnetic field [1] but creates challenges \nregarding excitation and control of spin waves. \nThe complex spin inte raction in ferrimagnets leads to original spin dynamic s that has been recently \nexploited to specifically overcome this obstacle [1–3]. While a promising ferrimagnet -based magnonic \nplatfor m ha s already been demonstrated using Y3Fe5O12 [4], other studies have revealed that unique spin \ncharacters can be realized in ferrimagnets due to the competition between different spins [1–3]. In rare \nearth –3d-transition metal ferrimagnetic compounds , fast field -driven antiferromagnetic spin dynamics \nare realized and field -driven domain wall mobility is remarkably enhanced at the angular momentum \ncompensatio n temperature (TA), where the net angular momentum vanishes [3]. At T A, both the left - and \nright -handed spin waves intersect, and chara cters of antiferromagnetic and ferrimagnetic spin wave \nmodes are observed [1]. \nRhenium -based double perovskites (DP), A2BReO 6 (A=Sr, Ca, Ba, and B=Cr, Fe) are ferrimagnetic materials \nwith a Curie temperature above room temperature [5]. They are strongly correlated materials, where the \nenergy scales of sp in-orbit coupling ( 𝝀), tetragonal crystal field (Δt) and electronic correlations (J H, U) \ncompete, leading to a diverse manifold of ground states and phenomena, such as magnetic ordering \nabove room temperature and an interplay between structural and elect ronic transitions [6]. Among the \nRe-based DPs, Sr 2CrReO 6 (SCRO) has the highest Curie temperature of T c=620 K [7] (508 K from [8]) and \nshows a large strain -dependent magnetocrystalline anisotropy (MCA) [9]. \nThrough a combination of state -of-the-art epitaxial growth of thin films, resonant x -ray scattering \nexperiments and calculations , and spin -dynamics simulations , we explore in this work the mechanisms \nresponsible for the fundamental low energy structure of SCRO and examine evidence for the presence of \nspin waves. We also bring additional insights into the presence of orbital ordering in this material [6]. The \nbest agreement between theory and experiment reveals a strong competition b etween spin -orbit \ncoupling, Hund’s coupling and strain -induced tetragonal crystal field . We show that two non -degenerate \nmagnon bands exist and are dominated by rhenium or chromium spins. The rhenium band is found to be \nflat at about 200meV ( ±25meV) throug h X-L-W-U high -symmetry points and dispersive toward Γ. \n \nMethod s \nThe three 90 nm (001) Sr 2CrReO 6 thin films characterized in this study were grown by off -axis magnetron \nsputtering on different substrates to achieve different strain states: -1% (compressive) on \n(LaAlO 3)0.3(Sr 2AlTaO 6)0.7 (LSAT), +1% (tensile) on a relaxed SrCr 0.5Nb 0.5O3 (SCNO) and 0% (unstrained) on \nSrTiO 3 (STO). No additio nal phases are detected through x -ray diffraction experiments, and t he quality of \nthe epitaxy is confirmed with the (103) reciprocal space map presented Figure S1 of the supplementary \nmaterial [10]. \nThe RIXS characterization was performed at the Advanced Photon Sour ce at the 27 -ID beamline. The \nincident photon energy was selected by a high -resolution mmE Si(440) monochromator near the Re -L3 \nedge (10.539 keV) and the final photon energy was analyzed with a 2 meter Si(119), leading to an overall \nresolution (Full Width at Half Maximum - FWHM) of 70 meV in this configuration. With an analyzer of 10 \ncm diameter 200 cm away from the sample, the momentum resolution in h, k and l is estimated to be \nlower than 0.07 r.l.u. in the pseudocubic system of coordinates. To enhance t he signal coming from the 3 films and to reduce the elastic background intensity, the incident beam grazes the sample surface (0.5 < \nθ < 9 degrees for thin films, depending on the vector q , and 1.2 degree for the powder) while the scattering \nangle 2θ was kep t close to 90 degrees in a horizontal scattering geometry. \nDensity functional theory (DFT) calculations are performed using projector augmented wave (PAW) \nmethod [11] implemented in the VASP code [12]. We use revised version of the generalized gradient \napproximation (GGA) proposed by Perdew et al. (PBEsol) [13]. Spin -orbit coupling (SOC) is not included, \nsince the effect of SOC is weak for SCRO from our previous study [6]. We use 500 eV of kinetic energy \ncutoff and 9 ×9×7 of k -point mesh. \nThe RIXS calculations were performed using EDRIXS, an open -source toolkit for simulating XAS and RIXS \nspectra based on exact diagonalization of model Hamiltonians. It is developed as part of the COMSCOPE \nproject in the Center for Computational Material Spectroscopy and Design, Brookhaven National \nLaboratory [14]. We use a single atomic multiplet model, considering a single Re5+ (d2) ion. We obtain the \ncrystal field splitting parameters from the Wannier function projections on Re d states [15], and use these \nparameters as a n initial guess. Then we explored a large phase space of parameters near the set of initial \nvalues . We fix the eg-t2g splitting (10Dq) of 3.23 eV from Wannier projection, since 10Dq is large and thus \nonly t 2g orbitals are important on the RIXS spectra below 1.5 eV. \nThe magnon spectrum are computed via DFT and spin -wave theory following Toth et. al [16]. The \nexchange coupling parameters J of a Heisenberg model are obtained from DFT, and we assume an \nisotropic magnetic interaction Jx=Jy=Jz. We compute the energy difference between the ferromagnetic (FM) \nand antiferromagnetic (AFM) configurations Δ𝐸=𝐸[AFM]−𝐸[FM]=∑𝐽𝑺𝑖∙𝑺𝑗 𝑖,𝑗 , where Si and Sj are \nspins of Cr ( S= 3/2) and Re ( S= 1/2) and obtain J = 33.58 meV. We then use this Heisenberg model to \ncompute the magnon spectrum via spin -wave theory. The main equations are given in the supplementary \nmaterial [10]. \nThe REXS experiments were carried out at the Advanced Photon Source at the 33 -ID beamline with a \n‘4S+2D’ six -circle diffractometer. The data are collected with a Pilatus solid state detector located about \n1 meter from the SCRO samples, which are cooled to a temperature of 15 K under a beryllium hemispheric \ndome. The expected reflection induced by the orbital -ordering (OO) at (0 0 ½) is reached with an incident \nangle θ i=4 degrees and measured in a fixed q mode as a function of energy in the 10.5 to 10.6 keV range, \nwith azimuthal angle φ defined as the angle between the <100> crystal axis of SCRO and the in -plane \ncomponent of the incident beam. The polarization sigma of the incident beam is fully in -plane. The \nfluorescence spectra are extracted from the backgr ound signal of the detector during the energy scans \nand the (0 0 ½) reflections are captured by a smaller region of interest and after background subtraction. \nThe energy -dependent REXS intensity of the OO -induced reflection is calculated from the DFT -deriv ed \natomic positions of Sr, Cr and O in a doubled SCRO unit cell (under tensile, compressive, or relaxed strain) , \nthe complex scattering factor extracted from the absorption (XAS) spectrum that result from the RIXS \ncalculations in the presence of orbital -ordering, and the Cromer -Mann coefficients . The intensity is then \nfinally corrected for self -absorption. See the supplementary material for more details [10]. \n \nResults 4 An overview of the energy scale s in the RIXS map of SCRO powder near the Re -L3 edge is presented in \nFigure 1. In addition to the elastic line at 0 eV, sharp excitations near 1 eV are observed (zone I), as well \nas broader features at 5, 8 and 9 -10 eV ( zone II, III, and IV, respectively). The resonance of some of these \nexcitations at specific energies of the incident light provides a signature of the electronic states involved \nin the intermediate |𝑛⟩ and final |𝑓⟩ steps of the RIXS process [17], as described schematically in Figure 1 \na). The excitations in zone I are enhanced when the incident photon energy is set to E i=10.534 keV, while \nzone II resonates at E i=10.538 keV. We attribute this difference of 4 eV to the splitting of the t2g and eg \nlevels of rhenium by an octahedral crystal field, similar to what is observed in another rhenium -based \ndouble perovskite Ba 2YReO 6 [18]. The broad zone III dominates the spectrum in the incident energy range \nbetween the resonances of zone I and II, and is identified as a charge transfer from the rhenium to the \noxygen ligand, whereas zone IV features a linear dependence with incident photon energy and is \nattributed to fluorescence. While the excitations in zone I only involve electronic transitions from and to \nthe t 2g levels of rhenium, namely dd intra -t2g, we identified the excitations in zone II as the result of \ntransitions from t 2g and to e g, namely dd t 2g-eg, although other excitations yielding broad spectra, like inter -\nband or charge -transfer excitations, cannot be excluded. A closer look at the spectral weight distribution \nin zone I reveals three components. The two higher energy pea ks at 0.55 and 0.9 eV are likely dd intra -t2g, \nfurther split by strain, spin -orbit coupling, or Hund’s coupling, but a feature as low as 0.2 eV suggests the \npresence of an additional relaxation channel such as bosonic excitations or the suspected orbital o rdering \n(OO) in SCRO , as discussed in a recent review on rhenium -based double perovskite s [6]. \n \nFigure 1: Incident photon energy versus energy loss RIXS mapping , near the L 3-edge of rhenium for a powder of SCRO at T=10 K. \nIn addition to the elastic line at 0 eV, four zones (I -IV) are identified. I) The lowest energy part of the spectrum (energy loss<1.4 \neV) inc ludes d -d transitions between the t 2g orbitals of rhenium, namely intra -t2g, and potential quasiparticle excitations. II) The \n 5 broader feature between 4 and 6 eV resonates 4 eV higher than the ones in zone I. Its spectral weight is identified as d-d transitions \nbetween the e g and t 2g orbitals. The strong signal located at 8 eV results from transitions between oxygen 2p and rhenium 5d \n(charge transfer) and the linearly incident -energy -dependent intensity in zone IV is induced by fluorescence. \nRecen t progress in the epitaxial thin film growth of SCRO on different substrates [9, 19, 20] allow s one to \ninvestigate the momentum dependence of these excitations and the role of strain on the low -energy \nexcitations belo w 1.4 eV (zone I). The data in Figures 2 , 3 and 4 have been collected from 90 nm thick \nSCRO thin films under tensile (+1%) and compressive strain ( -1%). \n \nFigure 2: RIXS spectra for different momentum transfer . a) Low -energy RIXS spe ctra (zone I in the map of Figure 1) collected from \nthe compressive SCRO thin film for different momentum transfer q expressed in the double perovskite cubic system (dp), at a \ntemperature of 17 K. The spectra are normalized by the integrated intensity betw een 0.05 and 1.4 eV. Three excitations are \nresolved at 0.2, 0.65 and 1 eV , in addition to the elastic line at 0 eV. The black curve represents the sum of four Voigt functions \nthat are used to fit the experimental data collected at the X point. b) Schematic unit cell of the fcc - SCRO in real space, and the \ncorresponding reciprocal space and Brillouin zone. The X marks repre sent the moment a that have been probed, with the same \ncolor code used for the spectrum. \nThree non -elastic features are clea rly resolved in F igure 2a) at 0.2, 0.65 and 1 eV for the SCRO film under \ncompressive strain and at a temperature of T=17 K. We do not observe significant dispersion of these \nexcitations for all measured momentum transfers, at L, X, and near U and W high -symmetry points . The Γ \npoint could not be measured because of the Bragg reflection that occurred at this coordinate (see figure \n2b), which would dominate the spectrum with a strong elastic peak centered at 0 eV and prevent the \nobservation of non -elastic feat ures below 1.4 eV. We also note that our RIXS measurements required a \ngrazing incidence condition of the incident photons to maximize the interaction of the light with the \nmaterial and improve the signal to noise ratio, which limits our capability to navig ate through reciprocal \nspace , specifically W and U. \n 6 The absence of dispersion in Figure 2, also observed in an unstrained (0% strain) SCRO film as shown in \nFigure S2 of the supplement ary material [10], is consistent with a local d-d intra -t2g mechanism but remains \nsurprising in this energy window below 0.5 eV. Similar low-energy excitation s have been reported in \npowder samples of Ca 2FeReO 6, Ba 2YReO 6, Ba 2FeReO 6 and in the iridate Sr 3CuIrO 6, but their origin is \nunclear. They have been ascribed to multi -phonon excitation s in the r henates [18] or impurities in iridates \n[21]. The former is unlikely at the high energy of the Re L3-edge because of the very short core -hole \nlifetime that reduces the RIXS cross -section for phonons [22–24]. \nNext, we delve deeper into the role of strain in SCRO . The direct comparison of the RIXS spectra of two \nfilms under opposite and equal strain, Figure 3 a) and b) , directly reflects the effect of a crystal -field sign \nchange on the electronic structure. The elastic line has been removed to better resolve the peak at 0.2 \neV. The three excitations discussed previously are observed for all temperatures in the 17 -300 K range \nand for both strain states. From compressive to tensile strain , the positions of the exci tations shift slightly \ntoward lower energies and the spectral weight is transferred from the feature near 1 eV to the one at 0.2 \neV. The three excitations are independent of temperature for the compressive SCRO film, but a \ntemperature dependence arises for T>100K when the strain is reversed to tensile, where the excitation at \n0.2 eV weakens in favor of the other two excitations. \n \nFigure 3: RIXS spectra collected at different temperature for a) compressive SCRO film at q=(6.4, 0.34, 6.8) and b) tensile SCRO \nfilm at q=(6, 0.34, 7.2) in the 17 -300 K ran ge. The elastic line at 0 eV has been removed for a better reading of the low energy \nexcitations in the experimental data , and the spectra are normalized by the integrated intensity betwe en 0.05 and 1.4 eV. c) and \n 7 d) Best-matched EDRIX calculations at 17 K in presence of an exchange magnetic field . The calculated intermediate and final state \nallowed us to identify t he peak near 0.2 eV as the result of a spin -flip process. The inset s show the electronic level splitting due to \nthe Hund’s coupling and strain -induced tetragonal crystal field (Δ t). \nA few scenarios can be considered to explain the emergence of these low -energy excitations. The first one \nis the effect of spin -orbit coupling, Hund ’s coupling and epitaxial strain. While spin -orbit coupling can \nfurther split the t 2g levels into a J eff=3/2 and 1/2 subset, this by itself is not sufficient to explain the three \nexcitations in the experimental data, but spin -orbit combined with a crystal field induced by the epitaxial \nstrain ( Δt) may result in three excitations. We investigated this idea using an atomic multiplet model solved \nby exact -diagonalization, as implemented in the EDRIXS software [14]. We systematically explored a range \nof values of the on -site Coulomb interaction (U), spin -orbit coupling ( 𝝀), Hund ’s coupling (J H) and \ntetragonal crystal field (∆ t) induced by the strain, in the range 0 ≤ 𝝀 ≤ 0.4, |∆ t| ≤ 0.5, 0.1 ≤ JH ≤ 0.6 and U=0, \n2 (unit of eV) , and typical results are presented Figure 4 a) and b ). No good match with the experimental \ndata was obtained for the two states of strain , especially for the lowest energy excitation, ruling out this \nfirst mechanism . \n \nFigure 4: RIXS calculations and experimental data for (a) compressive and (b)-(c) tensile strained SCRO at T=17 K. A large set of \ninput parameters, 0.2 ≤ 𝝀 ≤ 0.4 eV, -0.4 ≤ Δt ≤ 0.4 eV, 0 ≤ OO ≤ 0.2 eV, and U=0, 2 eV, are used to match the experimental data. \nThe best set is found to be U=2, 𝝀 = Δt =0.3 eV and J h=0.25 eV, although the quality of the fit is poor, especially for the excitation \nat 0.2 eV for tensile SCRO . The agre ement between experiment and calculation is not improved with OO as shown in (c). \nThe second scenario that we considered is based on the presence of orbital ordering. A recent theoretical \nstudy showed using DFT+U calculations that SCRO may stabilize at low temperatures in a new phase where \nthe Re d yz and d xz orbitals spontaneously order in a checkerboard fashion , which results in an energy gap \nformation [6]. The presence of a 0.2 eV gap has also been claimed in another experimental study based \non IR -absorption spectroscopy data [8] and might share a common origin with the 0.2 eV excitation \nobserved here in RIXS. Using the same atomic model, we mimicked the effect of OO on the energy levels \nof rhenium with a site -dependent splitting of the d yz and d xz orbitals via an additional OO energy \nparameter in the calculation , averaging the two Re sites involved in the checkerboard -like orbital ordering \ndescribed in reference [6]. We explored the range 0 ≤ OO ≤ 0.2 eV and obtain the lowest energy peak near \n0.2 eV with OO=0.2 eV , as shown in Figure 4 c). However , the relative magnitude of the peak is too small, \nindicating that OO may not be the origin of the lowest energy excitation near 0.2eV. \n \nBased on this result, we performed an additional experimental characterization on these SCRO thin films, \nwhich aimed at revealing the OO by Resonant Elastic X -ray Spectroscopy (REXS). Si nce OO slightly modifies \n 8 the absorption spectrum (XAS) of the two type s of rhenium involved in the orbital ordering, new Bragg \nreflections at specific reciprocal space coordinates should be observed. We note that due to the nature \nof the orbitals involved in the predicted OO (d yz and d xz), the intensities of the emergent Bra gg reflections \nnot only depend on the incident photon energy near the Re L 3 edge , but also on the photon polarization \ndirection relative to the orbitals as shown from the calculation Figure 5 a) . We experimentally probed \nthese new Bragg positions as a func tion of the photon energy near the Re L 3 edge and for different \npolarization direction s, and compared the results to REXS calculations for the tensile SCRO in Figure s 5 b). \nSame results are obtained for the compressive SCRO in the supplementary material Fi gure S 4 [10]. No \nadditional Bragg reflections are observed in this REXS study , from which we conclude that there is no long -\nrange OO in our SCRO . Impurities, defects, or strain gradient s in SCRO (see Figure S1c) may be responsible \nfor the absence of long -range order. \n \nFigure 5: Experimental and calculated energy scans at q=(0 0 ½) for the OO -induced reflections in tensile SCRO. a) Calculated \nintensity of the OO -induced r eflection at (0 0 ½) and an incident energy E i=10.565 keV, as a function of the azimuthal angle φ, for \ndifferent incident angle θi and for π, σ polarization. b) Experimental (solid lines) and calculated (dashed lines) energy -dependent \nintensity of the OO -induced reflection for σ polarization , θi=8.5 degrees and for different φ values . The simulation is calculated \nfrom the absorption spectra resulting from the EDRIXS calculation with OO for the tensile SCRO . The data are corrected for self -\nabsorption and nor malized by the Bragg reflection intensity at q=(0 0 2). \nA third scenario is the presence of magnons. While their dispersion has not been observed in Figure 2, \nthey remain a promising candidate in this low -energy part of the spectrum, especially in a ferrimagnetic \nmaterial such as SCRO . One way to distinguish magnons from charge excitations experimentally is to use \na more elaborate spectrometer to characterize the changes in the polarization between the incident and \noutgoing x -rays in RIXS [25]. Here, w e investigated theoretically the presence of an exchange magnetic \nfield in the range 0 ≤ M ≤ 0.1 eV in our atomic model. We also took into account the direction of this field \nby considering the strain -dependent m agnetic anisotropy in SCRO thin films [9]. The magnetic easy axis i s \nperpendicular (parallel) to the film surface when the strain is tensile (compressive) , which translates into \na field along the z -direction [0, 0, M] (in the xy -plane [𝑀/√2, 𝑀/√2, 0]) in the model. We obtained good \nagreement between theory and experimen t in this framework, as shown in Figure 3 c) and d) . We use \n𝝀=0.3, Jh=0.25, U=2, ∆t = −0.2, and M=[0, 0, 0.05] for tensile strain , and 𝝀=0.3, Jh=0.15, U=2, ∆t = 0.3, and \nM=[0.035, 0.035, 0] (unit of eV) for compressive strain. The discrepancy in the peak intensities is likely \ndue to the limitation of the single atomic model , which excludes any form of interaction between the Re \nand the surrounding ions. Each peak originates from many excitations that are close in energy and that \nare broadened by a cor e-hole lifetime of 0. 065 eV [26]. Using the calculated intermediate and final states, \n 9 we can identify the 0.2 eV excitation as a spin -flip p rocess for both compressive and tensile strain, which \nis allowed by symmetry [27]. For example, spin flip from 𝑑𝑥𝑧↓/𝑑𝑦𝑧↓ to 𝑑𝑥𝑧↑/𝑑𝑦𝑧↑ is allowed for \ncompressive strain. \nThe decrease in intensity observed for the low -energy peaks in tensile SCRO at high temperatures is still \nan open question. While it is known that bulk SCRO undergoes a transition from a tetragonal to a cubic \nphase at 2 60 K [7], this transition will be altered in the thin film due to the clamping effect of the substrate. \nTo estimate the effect of crystal field, we provide a summary of the effect of the tetragonal crystal field \nparameter |∆t| in Figure S3, and demonstrate that the lowest energy peak i s shifted towards lower energy \nlevels, resulting in a reduction in magnitude near 0.2eV after eliminating the elastic peak. However, the \npeak near 0.6eV is also shifted to ~0.4 eV, which is inconsistent with the experiment. Future studies are \nneeded to und erstand the temperature dependence of the low -energy peaks depending on the strain. \nWhile the physics of a spin -flip process can be understood by a single Re -atom model, it cannot describe \nthe dispersion of the spin excitation (magnon) in reciprocal space, which is typically large in \nantiferromagnetic cuprates or iridates [28–31], nor can it describe the spin -spin interaction between Cr \nand Re. We address this challenge by combining DFT calculations and spin -wave theory [16]. The exchange \ncoupling parameter s J of a nearest -neighbor Heisenberg model are obtained from DFT, and we assume \nan isotropic magnetic interaction Jx=Jy=Jz. We did not include SOC for DFT calculations because the effect \nof SOC is almost negligible [6]; due to the large magnetic moments and strong magnetic interaction in \nSCRO, the effect of SOC is quenched within DFT . We extract the nearest neighbor Heisenberg coupling J = \n33.58 meV (see method section) , and use the Heisenberg model to compute the magnon spectra. We \nobtain T C = 415 K, which is comparable to the experimental value of T C = 508 K from our fully ordered film \n[8]. The discrepancy between the spin -wave model and the experiment may be due to the next nearest \nneighbor interaction : previous theoretical work on Sr2CrOsO 6 predicts nearest (J1) and next nearest \nneighbor (J2) interaction parameters of J1=35meV and J 2/J1=0.4, and the predicted TC is 725 K that includes \nspin canting [32]. \nAs presented in Figure 6, there are two magnon bands whose energies are 0 and 110 meV at Γ, and 201 \nand 302 meV at X. Differing from typical antiferromagnetic materials like cuprates and iridates [30], which \nhave doubly degenerate magnon ban ds because of equal up - and down -spins magnitudes (| SA| = |SB|, \nwhere A and B are the two opposite -spin TM ions ) [33], SCRO is ferrimagnetic and the two lowest magnons \nbands are not degenerate (| SCr| > |SRe|). We also note that the dispersion is only about 25 meV for both \nbands along the X -L-W-U path but reaches 200 meV between Γ and any of these high -symmetry points. \nUnlike antiferromagnetic materials , the different TM ions with majority spin and minority spin can be \ndisentangled in the magnon band in SCRO . In Figure 6, we also present the relative weight of the Re spin s \nin magnon bands [16, 34, 35] (see supplementary material [10] for detail s). We find that the lower magnon \nband is Re -dominated and the upper one is Cr -dominated, and that this differentiation is largest at the k -\npoints far from Γ. Specifically, along the X -L-W-U path, the lowest magnon band at 200 meV is almost \nexclusively from rh enium. For this reason, and because of the large energy difference between the \nabsorption spectrum (XAS) of Cr and Re, only Re -dominated band s are observed in a RIXS experiment at \nthe Re L 3 edge at the wave vectors considered in our experiments . The energy of this experimental Re-\nband is about 200 meV along the X -L-W-U path , in agreement with our theory. This finding show s that \nRIXS can be used to disentangle the majority and minority spins from different TM ions such as 3d -4d or \n3d-5d ferrimagnetic double perovskites A 2BB’O 6 (A=Sr, Ca, B=Cr, Fe, B’=Mo, W, Re). 10 \n \nFigure 6: Magnon band structure. Two lowest, non -degenerate magnon bands. Color scale indicates the weight of Re spin involved \nin the magnon mode. Exchange parameters are obtained from the DFT and spin -wave calculation. The X markers represent the \nexperimentally measured k -points and their energies. \n \nConclusion \nWe identified in this study the origin of the three excitations observed in the RIXS spectra below 1.4 eV \nfor a Sr2CrReO 6 powder and thin films under different state s of strain. The strain mainly leads to a spectral \nweight redistribution below 1.4 eV and a temperature de pendence of the lowest energy peak at 0.2 eV \nfor tensile strain. Three scenarios were explored by means of RIXS calculations to understand the \nexperimental data, based on (i) a combination of spin -orbit coupling, Hund’s coupling and strain -induced \ntetragonal crystal field, (ii) an orbital ordering between the d yz and d xz orbitals of rhenium, as predic ted in \na recent theoretical study, and (iii) the presence of magnons in Sr2CrReO 6. A good match between theory \nand experiment was obtained in the second (ii) framework but subsequent REXS experiment experiments \nshow ed no trace of long -range OO -induced Brag g reflections, which may be due to strain gradients or the \npresence of defects in the strained Sr2CrReO 6 films. This leads us to consider scenario (iii). \nWe showed that a spin -flip process, mainly on the Re site, could also be responsible for the lowest 0.2 eV \nexcitations, and that the limited dispersive behavior observed in experiment is fully compatible with the \nmagnon calculations . The best fit of the RI XS spectra for the different strain states within the magnon \ntheory framework is obtained with similar value s of spin -orbit coupling, Hund’s coupling and strain -\ninduced tetragonal crystal field, which also emphasizes the competitive nature of the energy sc ale in \nSr2CrReO 6, which is likely responsible for the rich physics observed in this compound. \n \n 11 \n 12 ACKNOWLEDGMENTS \nWork at Yale was supported by the Air Force Office of Scientific Research (AFOSR) under Grant No. \nFA9550 -21-1-0173. \nWork at Brookhaven Nation al Laboratory was supported by the U.S. Department of Energy, Office of \nScience, Office of Basic Energy Sciences under Contract No. DE -SC0012704 . \nThis research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office \nof Scienc e user facility operated for the DOE Office of Science by Argonne National Laboratory under \nContract No. DE -AC02 -06CH11357. \nF.Y.Y. 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B 106, 134421 (2022). \n " }, { "title": "2207.10443v1.First_principles_insights_into_all_optical_spin_switching_in_the_half_metallic_Heusler_ferrimagnet_Mn__2_RuGa.pdf", "content": "arXiv:2207.10443v1 [cond-mat.mtrl-sci] 21 Jul 2022First-principles insights into all-optical spin switchin g in the\nhalf-metallic Heusler ferrimagnet Mn 2RuGa\nG. P. Zhang∗\nDepartment of Physics, Indiana State University, Terre Hau te, IN 47809, USA\nY. H. Bai\nOffice of Information Technology, Indiana State University, Terre Haute, IN 47809, USA\nM. S. Si\nSchool of Materials and Engineering,\nLanzhou University, Lanzhou 730000, China\nThomas F. George\nDepartments of Chemistry & Biochemistry and Physics & Astro nomy\nUniversity of Missouri-St. Louis, St. Louis, MO 63121, USA\n(Dated: July 22, 2022)\n1(July 22, 2022)Abstract\nAll-optical spin switching (AOS) represents a new frontier in magnetic storage technology –\nspin manipulation without a magnetic field, – but its underly ing working principle is not well\nunderstood. Many AOS ferrimagnets such as GdFeCo are amorph ous and renders the high-level\nfirst-principles study unfeasible. The crystalline half-m etallic Heusler Mn 2RuGa presents an op-\nportunity. Here we carry out hitherto the comprehensive den sity functional investigation into the\nmaterial properties of Mn 2RuGa, and introduce two concepts - the spin anchor site and th e optical\nactive site - as two pillars for AOS in ferrimagnets. In Mn 2RuGa, Mn(4 a) serves as the spin anchor\nsite, whose band structure is below the Fermi level and has a s trong spin moment, while Mn(4 c)\nis the optical active site whose band crosses the Fermi level . Our magneto-optical Kerr spectrum\nand band structure calculation jointly reveal that the deli cate competition between the Ru-4 dand\nGa-4pstates is responsible for the creation of these two sites. Th ese two sites found here not only\npresent a unified picture for both Mn 2RuGa and GdFeCo, but also open the door for the future\napplications. Specifically, we propose a Mn 2RuxGa-based magnetic tunnel junction where a single\nlaser pulse can control magnetoresistance.\nPACS numbers:\nKeywords:\n2(July 22, 2022)I. INTRODUCTION\nLaser-induced ultrafast demagnetization1changes the landscape of spin manipulation,\nwhere the laser field plays a central role in magnetism. All-optical spin switching (AOS)2is\na prime example, where a single laser pulse can turn spins from one dire ction to another,\nfree of an external magnetic field. As more and more materials are d iscovered3–5, a critical\nquestion on the horizon is what properties are essential to AOS. Ea rlier studies have focused\non magnetic orderings such as ferrimagnetic versus ferromagnet ic6, sample composition7,\ncompensation temperature8, magnetic domains9and others10, but most AOS materials are\namorphous and difficult to simulate within state-of-the-art density functional theory. This\ngreatly hampers the current effort to decipher the mystery of AO S at a microscopical level\nthat goes beyond the existing phenomenological understanding11.\nHeusler compounds represent a new opportunity12,13. Their properties can be systemat-\nically tailored, only subject to the structure stability. Different fro m rare-earth-transition\nmetals2,14, one has an empirical Slater-Pauling rule to predict spin moments15–22. Although\nthis rule is simple23, the actual synthesis of a desired material is a monumental task o f\ndecades in making24, because many materials are unstable experimentally. In 2002, Hor i\net al.25successfully synthesized various (Mn 1−xRux)3Ga alloys with x= 0.33−0.67 and\ndetermined the spin moment of 1.15 µBper formula. In 2014, Kurt et al.26demonstrated\nthat Ru can significantly reduce the spin moment in ferrimagnet Mn 2RuxGa. Because\none can tune composition x, Mn2RuxGa is likely to be a half-metal and fully-compensated\nferrimagnet27,28, with no stray field, ideal for spintronics13,29,30. Research has intensified\nimmediately21,31–35. Lenneet al.36found that the spin-orbit torque reaches 10−11Tm2/A in\nthe low-current limit. Banerjee et al.37reported that a single 200-fs/800-nm laser pulse can\ntoggle the spin from one direction to another in Mn 2RuGa within 2 ps or less. Just as found\nin GdFeCo2,11, for every consecutive pulse, the spin direction is switched. This dis covery37\ndemonstrates the extraordinary tunability of Heusler compounds , which now changes the\ntrajectory of AOS research38–40, with Mn 2RuGa as a crystalline model system where the\nfirst-principles investigation is now possible.\nIn this paper, we carry out the comprehensive first-principles den sity-functional study\nto pin down the material properties essential to all-optical-spin swit ching in ferrimagnet\nMn2RuGa. We introduce two concepts - the spin anchor site (SAS) and t he optical active\n3(July 22, 2022)site (OAS) as two essential pillars of AOS in ferrimagnets. SAS has a s trong spin moment,\nand in Mn 2RuGa it is the Mn(4 a) site. Our band structure reveals that the Mn(4 a)’s band\nis 0.5 eV below the Fermi level. By contrast, OAS has a smaller spin mome nt, easier to be\nswitched optically41, and in Mn 2RuGa it is the Mn(4 c) site. Its band is around the Fermi\nlevel and accessible to optical excitation42. The creation of SAS and OAS is the making\nof Ru and Ga. The Ru-4 delectrons set up the initial spin configuration with a strong\nspin moment concentrated on the distant Mn(4 c), but Ga tips this balance and reverses\nthe relative spin magnitude between Mn(4 a) and Mn(4 c). Although Ru and Ga are weakly\nmagnetic, their energy bands appear in the same energy window as t wo Mn atoms, which is\nmanifested in the magneto-optical Kerr spectrum. Guided by two e ssential sites, we can now\nunify Mn 2RuGa with GdFeCo, despite of their apparent structural differenc es, and extract\nthree essential properties for AOS. (i) A ferrimagnet must have a spin anchor site, i.e. Gd\nin GdFeCo and Mn(4 a) in Mn 2RuGa. (ii) It must have an optical active site, i.e. Fe in\nGdFeCo and Mn 2(4c) in Mn 2RuGa. (iii) Its spin anchor site and optical active site must\nbe antiferromagnetically coupled to minimize the potential energy ba rrier43. We propose a\nlaser-activated magnetic tunnel junction based on the same mate rial Mn 2RuxGa, but with\ndifferent compositions xwhich form optical activation, spin filtering and reference layers.\nThisdevice, ifsuccessful, representsanidealintegrationoffully- compensatedhalf-metallicity\nin spintronics into all-optical spin switching in femtomagnetism44,45.\nThe rest of the paper is arranged as follows. In Sec. II, we presen t our theoretical\nformalism. Section III is devoted to the results and discussion, whic h includes the crystal\nstructure, electronic band structure, ultrafast demagnetizat ion, and Kerr rotation angle.\nFinally, we conclude this paper in Sec. IV.\nII. THEORETICAL FORMALISM AND CALCULATION\nElement Mn lies in the middle of 3 dtransition metals, with a half-filled 3 dshell and\nzero orbital moment, just as Gd in the middle of 4 frare-earth metals. Mn is the only\n3dtransition metal element in inverse Heusler compounds, which is similar to a rare-earth\nelement46. Mn2RuGa crystallizes in an inverse XAHeusler structure21,22,31,35,46(see Fig.\n1(a)), where two manganese atoms, Mn 1and Mn 2, are situated at two distinct Wyckoff\npositions 4a(0,0,0) and 4c(1\n4,1\n4,1\n4), and are antiferromagnetically coupled. Ru and Ga sit at\n4(July 22, 2022)4d(3\n4,3\n4,3\n4) and4b(1\n2,1\n2,1\n2), respectively. The inverse Heusler XAstructure hastwo Mn atoms\nseparated by a vector (1\n4,1\n4,1\n4), while the L21structure by (1\n2,1\n2,1\n2)26,32,33. The experimental\nlattice constants in this nearly cubic material are a=b=c= 5.97˚A26. Viewing along the\ndiagonal direction, four atoms form chains Mn 1-Mn2-Ga-Ru-Mn 1···. Therefore, Mn 2RuGa\nloses both inversion and time reversal symmetries due to the antife rromagnetic coupling\nbetween two Mn atoms.\nWe employ the state-of-the art density functional theory and th e full-potential linearlized\naugmented plane wave (FLAPW), as implemented in the Wien2k code47. We first self-\nconsistently solve the Kohn-Sham equation\n/bracketleftBigg\n−¯h2∇2\n2me+VNe+VH+Vxc/bracketrightBigg\nψnk(r) =Enkψnk(r), (1)\nwhereψnk(r) is the wavefunction of band nat the crystal momentum kandEnkis its band\nenergy. The terms on the left are the kinetic energy operator, th e attraction between the\nnuclei and electrons, the Hartree term, and the exchange-corr elation48, respectively. The\nspin-orbit coupling is included using a second-variational method in th e same self-consistent\niteration.\nIII. RESULTS AND DISCUSSIONS\nA. Crystal structure\nOrderedHeusler alloyshavethreedistinctivekindsofstructures49: (1)normalfull-Heusler\nX2YZalloys with group symmetry L21, (2) half Heusler XYZcompounds with group\nsymmetryC1b, and (3) inverse Heusler X2YZalloys with group symmetry XA. (1) has the\nspace group No. 225. (2) and (3) have the same space group No. 2 16.L21has (8c) site,\nwhich is split into two different sites in XA46. However, over the years, various Wyckoff\npositions are adopted in the literature. For the XAstructure, Wollmann et al.46used a\ndifferent set of Wyckoff positions for Mn at 4 d(1\n4,1\n4,1\n4),Yat 4c(3\n4,3\n4,3\n4), Mn at 4b(1\n2,1\n2,1\n2),\nandZat 4a(0,0,0). So in their paper, their (4 a),(4b),(4c),(4d) positions have different\nmeanings from those in31. In order to convert Wollmann’s notation to the latter notation,\none has to shift the entire cell by 4 d(1\n4,1\n4,1\n4). For theL21structure, Wollmann et al.46also\nadopted different positions, which were again used in their review pap er13(see Table I).\n5(July 22, 2022)In Mn 2RuGa, several versions have also been used. It adopts an XAstructure. Kurt et\nal.26correctly assigned the space group symmetry L21to the full Heusler compound, but\ninappropriately assigned the same group to Mn 2RuGa, and so did Zic et al.32. Both Zic\net al.32and Fleischer et al.33had the correct notations for all the atoms, but their figure\nswitched the positions for Ru and Mn 2, where Ru(4 d) appears at position 4 c(1\n4,1\n4,1\n4) and\nMn2(4c) at 4d(3\n4,3\n4,3\n4). Since they never used the figure to characterize their experime ntal\ndata, this changedoes not affect their results. Wealso noticethat Bettoet al.50assignedC1b\ngroup symmetry to Mn 2RuGa, where two Mn atoms are at 4 a(0,0,0) and 4c(3/4,3/4,3/4)\nwhile Ru at 4 d(1/4,1/4,1/4) and Ga at 4 b(1/2,1/2,1/2). One can see from Table I that\nC1bhas no 4csite.\nGalanakis et al.31exchanged the positions for Ru and Ga, so Ru is at site 4 band Ga is\nat site 4d. Although in general such an exchange is allowed, they do not match the existing\nexperimental results42. For instance, only Mn 1(4a) has Ru as its neighbor. If we exchange\nthe positions for Ru and Ga, then Mn 2(4c) would have Ru as its neighbor.\nWe summarize those used Wyckoff positions in the same table, so the r eader can see\nthe difference. We adopt the common convention, as listed in the last line in Table I. This\nconvention matches the experimental results better33. In particular, the magneto-optics\nsignal agrees with the experimental one.\nB. Band structure\nWe choose a big kmesh of 44 ×44×44, with 11166 irreducible kpoints in the Brillouin\nzone. The product of the Muffin-tin radius RMTand the planewave cutoff is 7, where\nRMT(Mn1,Mn2,Ru) = 2.42 bohr, and RMT(Ga) = 2.28 bohr. We find that Mn 1has spin\nmoment of M4a= 3.17µB. We call Mn 1(4a) the spin anchor site, SAS, as it pins the\nmagnetic configuration, so the magnetic structure can be stabilize d and is immune to optical\nexcitation. Mn 2(4c) atoms form another spin sublattice with a smaller spin moment of\n−M4c=−2.31µB. The entire cell has the spin moment of 1.027 µB, in agreement with prior\nstudies21,31. Figure 1(a) shows the spatial valence spin density integrated fro m 2 eV below\nthe Fermi level for each atom, where the red(blue) color refers t o the majority(minority)\nspin. One can see the spin density is mainly localized on these two Mn ato ms, where Mn 1\nhas a larger spin in the spin up channel and Mn 2has the spin density in the spin down\n6(July 22, 2022)channel, so they are antiferromagnetically coupled.\nOur first finding is that the above spin configuration hinges on the de licate balance\nbetween Ru and Ga. Figure 1(b) shows their respective atomic ener gies. Ru’s 4 d75s1states\nare close to Mn’s unoccupied 5 d0states. Without Ga, when Mn and Ru form a solid, the\nspin moment on Ru increases by five times to 0.39 µBand is antiferromagnetically coupled\nto the Mn 1(4a)’s spin, but its spin is now ferromagnetically coupled to the distant Mn 2(4c),\nwhichisoppositetothenativeMn 2RuGa(seeTableII).Adding Gatipsthebalance, because\nGa’s 4p1is higher than Mn’s unoccupied 5 d0orbitals (see Fig. 1(b)), so Ga can transfer\nelectrons to Mn atoms more easily than Ru. We integrate the atom-r esolved density of\nstates around each sphere, and find that the number of the Ru 4 delectrons is 5.87, reduced\nby 1.13 with respect to its atomic 4 d7, while the 4 pelectron of Ga is 0.81, reduced by 0.2\nfrom 4p1. The total number of electrons within the Mn 1(4a) and Mn 2(4c) spheres are almost\nexactly the same, 6.04, but the number of 3 delectrons in each spin channel is very different.\nTable II shows that Mn 1(4a) has 4.09 3 delectrons in the majority channel and 1.01 in the\nminority channel, in contrast to Mn 2(4c) where 1.44 and 3.72 electrons are present. The\ntotal number of 3 delectron is still close to 5. Table II summarizes these results. In gen eral,\nthe orbital moment on Mn 1is small, around 0.025 µB, and that on Mn 2is slightly larger,\nreaching -0.046 µB, which is beneficial to the spin-orbit torque38,43, important for AOS37.\nFigure 2(a) shows the band structure, superimposed with the Mn 1-3dorbital character\nfrom its spin majority channel. The orbital characters are highlight ed by the circles, whose\nradius is proportional to the weight of the Mn 1-3dcharacter, and the lines are the actual\nband dispersion. Bands with a clear dominance of a single orbital are h ighlighted, and in\nthe figure,dz2anddx2−y2are denoted by z2andx2−y2for simplicity; and this is the same\nfor other orbitals. The entire set of detailed orbital characteriza tion is presented in51. We\nsee that the Mn 1’s occupied majority band centers around -0.6 eV below the Fermi lev elEF\n(horizontal dashed line), with a smaller contribution close to the Fer mi level. This feature\nis reflected in the 3 d-partial density of states (pDOS) in Fig. 2(b), where a small peak\nat the Fermi level is found, consistent with two prior studies21,31, indicative of structural\ninstability46. Figure 2(c) shows that the Mn 1spin minority band has a single dxz/dyzband,\nwhich crosses the Fermi level from the L to Γ, and then to X point, b ut this single band\ncrossing does not constitute a major contribution to the density o f states (DOS). Figure 2(d)\nshows the partial 3 ddensity of states at the Fermi level is very tiny but not zero. The o ther\n7(July 22, 2022)occupied minority dband is at -1.5 eV below the Fermi level. Because Mn 1(4a)’sdband is\naway from EFand has a small density of states around the Fermi level, optical ex citation\nat Mn 1is weak42.\nMn2(4c) is quite different from Mn 1(4a). Figure 2(f) shows that its majority bands cross\nthe Fermi level at multiple points, have mixed dcharacters, and are highly dispersive. Its\n3d-pDOS (Fig. 2(e)) has a larger peak at the Fermi level than Mn 1, quantitatively 1.80-1.81\nstates/eV for the former and 0.65-0.69 states/eV for the latter . This explains why Mn 2(4c)\nis more optically active than Mn 1(4a)42, where we call Mn 2(4c) the optical active site. In\nthe minority channel, Mn 2has a strong admixture of orbital characters (see Fig. 2(h)), an d\nits overall density of states at the Fermi level is also small (see Fig. 2(g)). We note that the\nminority band structure is very similar to that of Mn 3Ga16, and they both have a flat dz2\nband along the Γ-X direction.\nBefore we move on to ultrafast demagnetization, we must emphasiz e that the band struc-\nture is not solely contributed by these two Mn atoms. Both Ru and Ga significantly affect\nthe magnetic properties of Mn atoms. Thin lines in Figs. 2(e) and 2(g) are the Ru’s 4 d\npDOS for the spin majority and minority channels, respectively. One can see that the Ru- d\nmajority density of states follows the Mn 1(4a)’s pDOS (compare Figs. 2(b) and (e)), but its\nminority state follows the Mn 2(4c)’s pDOS (compare the thin and thick lines in Fig. 2(g)).\nThe split role from the same atom is remarkable.\nC. Ultrafast demagnetization\nThe circles in Fig. 3(a) are the experimental ultrafast demagnetiza tion42, and consist of\ntwo regions. Region I is from 0 to 0.26 ps, highlighted by the red arrow in Fig. 3(a), and\nregion II starts from 0.26 ps to 5 ps. This time separation of 0.26 ps is consistent with a\nprior study52. In region I, a sharp decrease in spin moment is observed, but in reg ion II\nthere is a peak. We can fit these two regions with the same equation,\n∆M(t)\nM=A/parenleftBiggM4ae−α4a(t−T)−M4ce−α4c(t−T)\nM4a−M4c/parenrightBigg\n−B, (2)\nwheretis the time. Ais necessary, since without it the laser field amplitude cannot enter t he\nequation.Bdetermines the net amount of demagnetization. Tsets the characteristic time\nfordemagnetizationorremagnetization. Sinceourspinmomentsar efixedbyourcalculation,\n8(July 22, 2022)we only have four fitting parameters for each region, where α4a(4c)is the demagnetization\nrate for site 4 a(4c). Table II shows that in region I, αis site-dependent, α4a= 4.5/ps and\nα4c= 2.8/ps, demonstrating that the larger the spin moment is, the larger αbecomes,\nα=cM,or,τM=1\ncM, (3)\nwherecis a constant. This equation is consistent with the empirical formula p roposed by\nKoopmans and coworkers53. Fromα, we find the demagnetization times τM(4a) = 222 fs,\nandτM(4c) = 357 fs. These intrinsic demagnetization times, called the H¨ ubner times10, are\nwell within the times for other transition and rare-earth metals: 58 .9 fs (Fe), 176 fs (Ni),\n363 fs [Gd(5 d)], 690 fs [Gd(4 f)]. An extreme point will appear if ∂/parenleftBig∆M(t)\nM/parenrightBig\n/∂t= 0, and the\nsecond-order time-derivative determines whether the extreme is a maximum or minimum,\n∂2/parenleftBig∆M(t)\nM/parenrightBig\n∂t2=Aα4cM4ce−α4c(t−T)(α4a−α4c). (4)\nIfα4a>α4c, weonlyhaveaminimumwhich explains thespinchangeinregionI. Inreg ionII,\nbothα4aandα4carereduced, but α4aisreducedmuch more, so α4a<α4c, whichcorresponds\nto a peak in region II. Table II shows that region II has α4a= 0.6/ps andα4a= 1.5/ps. The\ndemagnetization on the 4 asite slows down significantly.\nD. Kerr rotation angle\nUnderlying ultrafast demagnetization and subsequent all-optical s pin switching is the\nmagneto-optical property of Mn 2RuGa, which is characterized by the conductivity54in units\nof (Ωm)−1,\nσαβ(ω) =i¯he2\nm2\neV/summationdisplay\nk;m,nfnk−fmk\nEmk−Enk/angb∇acketleftnk|pα|mk/angb∇acket∇ight/angb∇acketleftmk|pβ|nk/angb∇acket∇ight\n(¯hω+iη)+(Enk−Emk), (5)\nwheremeis the electron mass, Vis the unit cell volume, fnkis the Fermi distribution\nfunction,Emkisthebandenergy ofstate |mk/angb∇acket∇ight,/angb∇acketleftnk|pα|mk/angb∇acket∇ightisthemomentum matrixelement\nbetween states |mk/angb∇acket∇ightand|nk/angb∇acket∇ight, andηis the damping parameter. The summation is over the\ncrystal momentum kand all the band states |mk/angb∇acket∇ightand|nk/angb∇acket∇ight, andωis the incident photon\nfrequency. Here αandβrefer to the directions, such as the xandydirections, not to be\nconfused with the above demagnetization rate. The anomalous Hall conductivity is just the\noff-diagonal term. In the limit of η,ω→0, the term behind the summation over kis the\n9(July 22, 2022)Berry curvature\nΩk,n\nα,β=/summationdisplay\nm/negationslash=n¯h2(fmk−fnk)/angb∇acketleftnk|vα|mk/angb∇acket∇ight/angb∇acketleftmk|vβ|nk/angb∇acket∇ight\n(Emk−Enk)2. (6)\nThe general expression given in Eq. 5 is better suited for metals with partial occupation\nthan the treatment with a separate sum over occupied and unoccu pied states55,56, though\nthe latter is faster. The intraband transition with n=mis included by replacing ( fnk−\nfmk)/(Emk−Enk)byitsderivative −∂fnk/∂Enk, whichis −β/2\ncoshβ(Enk−EF)+1, withoutresorting\nto more complicated numerics57. Hereβ= 1/(kBT), wherekBis the Boltzmann constant,\nTis the temperature, and EFis the Fermi energy. The Kerr effect is characterized by the\nKerr rotation θand ellipticity ǫ, in the small angle limit and with magnetization along the\nzaxis,\nθ+iǫ=−σxy\nσxx/radicalBig\n1+4πiσxx/ω, (7)\nwhereσxxmust be converted to 1/s. The SI version of/radicalBig\n1+4πiσxx/ωis/radicalBig\n1+iσxx/(ωǫ0).\nExperimentally, Fleischer et al.33measured the magneto-optical Kerr effect for a series of\nMn2RuxGa samples with compositions x= 0.61,0.62,0.69,0.83 and with thickness from 26\nto 81 nm. Figure 3(b) reproduces two sets of data from their supp lementary materials. One\ncan see that both the thickness and composition affect the Kerr ro tation angle. The thicker\nsamplehasalargerangle(comparedottedandlong-dashedlineswith x= 0.61,0.62),andthe\nangle peaks between 1.6-1.9 eV. Our theoretical Kerr angles with th ree different dampings\nare three solid lines with η= 0.8, 0.6 and 0.4 eV from the bottom to top, respectively. One\nnotices that the overall shape is similar to the experimental data, a nd the main peak is also\naround 2 eV, slightly higher than the experimental one, but a more d irect comparison is not\npossible since there are no experimental data at x= 1. The best agreement in terms of the\nKerr angle is obtained with η= 0.8 eV. The convergence of our spectrum is tested against\nthe mesh of 92 ×92×92, and there is no visual difference between this much bigger mesh\nand the one used in Fig. 3(b).\nWe can pinpoint the origin of the main peak by removing some atoms. We useη= 0.4\neV since it gives us more structures. First, we remove Mn 2(4c), without changing the lattice\nstructure and the rest of atoms, so we have Mn(4 a)RuGa. The solid line in Fig. 3(c) shows\nthat the Kerr rotation angle for the new Mn(4 a)RuGa is very different from the one in\nFig. 3(b), highlighting the fact that Mn 2(4c), not Mn 1(4a), contributes significantly to the\noverall signal. To verify this, we remove Mn 1(4a) but keep Mn 2(4c). The red long-dashed\n10 (July 22, 2022)line shows clearly thatthe overall shapeiswell reproduced, but the Kerr angleislarger. This\nconcludes that Mn 2(4c) is optically active and plays a decisive role in the magneto-optical\nresponse as OAS, consistent with the experiment42, but the role of Ru and Ga should not\nbe underestimated. Magnetically, they are silent and do not contrib ute to the spin moment\nsignificantly, but when we remove Ru, the spectrum changes comple tely (see the dotted\nline in Fig. 3(c). The same thing happens to the removal of Ga atom. T his reveals the\nsignificant contributions of Ru and Ga to the optical response of Mn 2RuGa. The discovery\nof two sites (spin anchor site and optical active site) found here ha s some resemblance to\nthe laser-induced intersite spin transfer58. In their system, Dewhurst et al.found that the\nspins of two Mn atoms are aligned and coupled ferromagnetically, not antiferromagnetically\ncoupled as found here. Additional calculations are necessary since their materials are not\nMn2RuGa. Mentink et al.59proposed a two-sublattice spin model where AOS is realized\nthrough the angular momentum exchange between sublattices. Bu t their did not reveal the\ndifferent roles played by two spin sublattices. Our mechanism makes a clear distinction\nbetween two sublattices, and thus ensures that two sublattices d o not compete optically and\nmagnetically.\nIV. CONCLUSION\nThrough Mn 2RuGa, our state-of-the-art first-principles density functional calculation es-\ntablishes two concepts: the spin anchor site and the optical active site as the key to AOS\nin ferrimagnets. The formation of SAS and OAS in Mn 2RuGa is accomplished by weakly\nmagnetic Ru and Ga atoms. In GdFeCo2Gd is SAS while Fe is OAS. Switching starts with\nOAS52; because the ferrimagnetic coupling between SAS and OAS is frustr ated and has a\nlower potential barrier to overcome if the spin moment is smaller60, SAS is dragged into\nthe opposite direction by OAS through the spin torque JSi×Sj38,41, to realize all-optical\nspin switching. Because the Heusler compounds have excellent tuna bility13,17,19,22,29,33, the\nfuture research can investigate the effect of the spin moments at SAS and OAS on the spin\nswitchability8. We envision an integrated device based on Mn 2RuGa as illustrated in Fig.\n1(c). All three parts of the device are made of the same materials b ut with different concen-\ntrationx. In the middle, xis close to 0.6, so we have a full-compensated half-metal, while\nat two ends, xis close to 1, whose spin is designed to be optically switched. This forms an\n11 (July 22, 2022)ideal magnetic tunnel junction that a light pulse can activate. A fut ure experimental test\nis necessary. We should add that experimentally, concentrations o f both Mn and Ru can\nbe already tuned in several different experimental groups. Chatt erjeeet al.35were able to\nadopt two concentrations x= 0.2,0.5 in Mn 2−xRu1+x, while Siewierska et al.34were able to\nindependently change xandyin Mn yRuxGa films. In fact, Banerjee et al.37already used\n13 samples with x= 0.5 up to 1.0 in Mn 2RuxGa.\nAcknowledgments\nThe authors appreciate the numerous communications with Dr. K. R ode (Dublin) and\nDr. K. Fleischer (Dublin). Dr. Fleischer provided the original experim ental data in text\nform, which is very convenient to plot, with a small correction to the thickness of their sam-\nples. G.P.Z. and Y.H.B. were supported by the U.S. Department of Ene rgy under Contract\nNo. DE-FG02-06ER46304. Part of the work was done on Indiana St ate University’s high\nperformance Quantum and Obsidian clusters. 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The full-Heusler compound has L2 1symmetry,\nthe half-Heusler one has C1bsymmetry, and the inverse Heusler compound has XAsymmetry.\nGroup symmetry Prototype X1 X2 Y Z Ref.\nL21(No. 225, Fm¯3m)Cu2MnAl8c(1\n4,1\n4,1\n4) share 4b(1\n2,1\n2,1\n2) 4a(0,0,0)46\nL21(No. 225, Fm¯3m) (0,0,0) (1\n2,1\n2,1\n2)(1\n4,1\n4,1\n4) (3\n4,3\n4,3\n4)61\nL21(No. 225, Fm¯3m)Cu2MnAl8c(1\n4,1\n4,1\n4) (3\n4,3\n4,3\n4)4a(0,0,0) 4b(1\n2,1\n2,1\n2)62\nC1b(No. 216, F¯43m)MgAgAs 4a(0,0,0) vacant 4b(1\n2,1\n2,1\n2) 4c(1\n4,1\n4,1\n4)62\nXA(No. 216, F¯43m)Li2AgSb4d(1\n4,1\n4,1\n4) 4b(1\n2,1\n2,1\n2)4c(3\n4,3\n4,3\n4) 4a(0,0,0)46\nXA(No. 216, F¯43m) 4a(0,0,0) 4c(3\n4,3\n4,3\n4)4b(1\n2,1\n2,1\n2) 4d(1\n4,1\n4,1\n4)31\nNo. 216 4a(0,0,0) 4c(1\n4,1\n4,1\n4)4d(3\n4,3\n4,3\n4) 4b(1\n2,1\n2,1\n2)63\nTABLE II: Spin and orbital moments of Mn 1, Mn2, Ru and Ga. The electron populations in their\nmajority and minority 3 dstates are listed as n3d,↑andn3d,↓. The demagnetization rate in regions\nI and II is denoted as αIandαII, respectively. In two artificial structures, Mn 2Ru and Mn 2Ga,\nonly the spin moments are given.\nElement Ms(µB)Mo(µB)n3d↑n3d↓αI(1/ps)αII(1/ps)Ms(µB)Ms(µB)\n(Mn2RuGa) (Mn 2RuGa) (Mn2Ru) (Mn 2Ga)\nMn1(4a) 3.17 0.025 4.09 1.01 4.5 0.6 2.80 3.35\nMn2(4c) -2.31 -0.046 1.44 3.72 2.8 1.5 -3.64 -3.27\nRu(4d) 0.076 -0.035 -0.39 NA\nGa(4b) 0.032 -0.000 NA -0.04\n18 (July 22, 2022)FIG. 1: (a) Structure of Mn 2RuGa and the spatial spin densities on the Mn 1(4a) (red, positive)\nand Mn 2(4c) (blue, negative). The Ru and Ga atoms have a small spin densi ty. (b) Atomic energy\nlevels of Ru, Mn and Ga. The energy splitting is due to the spin -orbit coupling in atoms. (c) Our\nproposed device has a junction structure and consists of thr ee layers of the same Mn 2RuxGa, but\nwith different composition x. The layer on the left is an optically active layer, the middl e is the\nspin filter, and the right layer is a spin reference layer. The magnetoresistance is controlled by\nlight.\n19 (July 22, 2022)W LΓXWK−2−1012Energy (eV)Mn1(4a)−2−1012Energy (eV)Mn1(4a)\nW LΓXWK−2−1012Mn2(4c)−2−1012Mn2(4c) 0 2 4 0 2 4\n0 2 4 0 2 4pDOS (states/eV)\n(a) (b)\n(c) (d)(e) (f)\n(g) (h)x2−y2\nz2\nz2\nxz,yzxz,yzxz,yz\nxyz2\nx2−y2\nz2\nxz,yzEF\nEFRuMn2\nRu\nMn2\nFIG. 2: (a) and (c) Orbital-resolved band structure with the 3dstate characters for the Mn 1(4a)\nspin-majority and spin-minority channels, respectively. (b) and (d) Partial density of states for the\nMn1spin-majority and spin-minority channels, respectively. Bands with clear orbital characters\nare denoted by their orbitals. The Fermi level is set at 0 eV (h orizontal dashed line). (f) and\n(h) Band structure with the 3 dstate characters for the Mn 2(4c) spin-majority and spin-minority\nchannels, respectively. (e) and (g) Partial density of stat es for the Mn 2’s 3d(thick lines) and Ru’s\n4d(thin lines) spin-majority and spin-minority, respective ly.\n20 (July 22, 2022)0 1 2 3 4 5\nEnergy (eV)−2−10123Kerr rotation (mrad) x=0.61, d=55 nm\nx=0.62, d=81 nm\n0 1 2 3 4 5\nEnergy (eV)−4−20246\nKerr rotation (mrad)Mn(4a)RuGa\nMn(4c)RuGa\nMn2Ga0 1 2 3 4 5 6Time (ps)\n−20−15−10−50∆M/M (%)Exp\nfitting1\nfitting2\n(a)\n(b) (c)II I\nη=0.4 eV\nη=0.6 eV\nη=0.8 eVη=0.4 eVlaser pulse\nFIG. 3: (a) Experimental demagnetization fitted by Eq. 2, wit h two sets of fitting parameters given\nin Table II, provides a crucial insight that demagnetizatio n rates at two Mn spin-sublattices change\nbetween region I (between 0 to 0.26 ps) and region II (between 0.26 ps to 5 ps). The experimental\ndata are extracted from Ref.42. The thick red curve is the laser pulse of duration 40 fs. (b) T he\nexperimental (dotted and dashed lines from Ref.33) and our theoretical Kerr rotation angles. The\nthree solid lines are our theoretical results with three diffe rent dampings η= 0.4,0.6,0.8 eV. (c)\nElement-resolved Kerr rotation angles when Mn 2(4c) (solid line), or Mn 1(4a) (dashed line), or Ru\n(dotted line) is removed separately. η= 0.4 eV is used.\n21 (July 22, 2022)" }, { "title": "2103.16872v1.Room_temperature_antiferromagnetic_resonance_and_inverse_spin_Hall_voltage_in_canted_antiferromagnets.pdf", "content": " \n1 \n Room temperature antiferromagnetic resonance and inverse spin -Hall \nvoltage in canted antiferromagnet s \nI. Boventer1, H. T. Simensen2, A. Anane1, M. Kläui2,3,4, A. Brataas2, R. Lebrun1 \n1. Unité Mixte de Physique CNRS, Thales, Univ. Paris -Sud, Université Paris -Saclay, Palaiseau 91767, France \n2. Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, \nTrondheim, Norway \n3. Institut für Physik, Johannes Gutenberg -Universität Mainz, D -55099, Mainz, Germany \n4. Graduate School of Excellence Materials Science in Mainz (MAINZ), Staudingerweg 9, D -55128, Mainz, Germany \n \nWe study theoreti cally and experimental ly the spin pumping signals induced by the resonance of \ncanted antiferromagnets with Dzyaloshinskii -Moriya interaction and demonstrate that they can \ngenerate easily observable inverse spin -Hall voltage s. Using a bilayer of hematite /heavy metal as \na model system, w e measure at room temperature the antiferromagnetic resonance and an \nassociated inverse spin -Hall voltage , as large as in collinear antiferromagnet s. As expected for \ncoherent spin -pumping , we observe that the sign of the inverse spin -Hall voltage provides direct \ninformation about the mode hand edness as deduced by comparing hematite, chromium oxide and \nthe ferrimagnet Yttrium -Iron Garnet . Our results open new means to generate and detect spin -\ncurrent s at terahertz frequencies by functionalizing antiferromagnets with low damping and \ncanted moment s. \n \nContemporary s pintronics , utiliz ing the electronic spin for information processing and microelectronics, \nis mostly based on ferromagneti c device architectures. In view of long term perspectives to enable \nenhanced data processing speeds and downscaling for on -chip information processing [1], spintronics \nwith antiferromagnets is a promising avenue [2]. Antiferromagnets exhibit the key advant age over \nferromagnets that their resonance frequency is enhanced by the exchange coupling of the sublattices , \nand thus generally in the terahertz regime [2,3] . In compensated antiferromagnets, the absence of a net \nmoment however strongly impede s simple access to their ultrafast dynamics , especially in thin films, \nand the development of ultra-fast antiferromagnet -based devices [4,5] . As a result, interfacial spin-\ntransport phenomena could provide new insights into the spin-relaxation processes and spin -dynamics \nin antiferromagnets [5–8]. \nExperimental access to the spin dynamics can be facilitated by spin to charge conversion mechanisms \nsuch as the spin pumping effect largely studied in ferromagnets [9,10] . The spin pumping effect \ngenerates alternating (ac) and continuous (dc) spin-currents from the spin precession of a magnetic \nmaterial into an adjacent conductor [11,12] . These spin -current s can be detected electrically by \nmeasuring a voltage via the inverse spin -Hall effect (ISHE) [12] or through the inverse Rashba -\nEdelstein effect [13]. Applied to antiferromagnets (AFMs) , these combined spintronic effects c ould \nprovide a direct and surface sensitive access to the magnetization dynamics in both bulk AFM and thin \nfilms. However, t heoretical [14,15] and recent experimental studies [16,17] showed that , in co llinear \nantiferromagnets , the amplitude of the pumped spin -currents scales with the dynamical sub -lattice \nsymmetry breaking. Thus , its amplitude is proportional between the ratio of the anisotropy field H A and \nthe exchange field H E. This ratio is of less than 0.1% in many compounds [5,6,15] , and only reaches 1 -\n2% in two known compounds MnF 2 [18] and FeF 2 [19] without applying large magnetic fields of t he \norder of the spin -flop field . This limit ation has until now largely restrained the investigat ion of spin-\npumping signals in antiferromagnets. \nIn parallel , route s to generate spin-pumping signals from noncol linear antiferromagnets have not been \nwidely considered yet. Early studies have reported that the bulk Dzyaloshinskii Moriya interaction \n(DMI) can reach a few Tesla in some antiferromagnets, and induce small canted moments both for easy \naxis (e.g. NiF 2 [20], CoF 2 [20] or most orthoferrites [21,22] ) and easy plane antiferromagnets (e.g. \nMnCO 3 [20] or hematite , α-Fe2O3, above the Morin transition [23]), or ch iral antiferromagnets (e.g. \nMn 3Sn [24]). \n2 \n In this Letter, we explore both theoretically and experimentally spin-pumping in non-collinear \nantiferromagnets with a DMI induced canting and capped with a heavy metal . First, w e demonstrate \ntheoretically that the AFM resonance generates a dc spin-pumping and an inverse spin -Hall voltage \nVISHE in the adjacent heavy metal, proportional to the ratio 𝐻𝐷/𝐻𝐸 (𝐻𝐷 : DMI field , 𝐻𝐸 : exchange field) . \nThis result is valid in both easy -axis and easy -plane canted AFMs . Note, that the zero-field mode \nfrequenc ies do not depend directly on the DMI field. We anticipate inverse spin -Hall voltages V ISHE > \n100 nV in many canted AFMs with 𝐻𝐷/𝐻𝐸���1−10 % [23,25] . In parallel, the ir zero-field mode \nfrequenc ies can range from tens to hundreds of gigahertz (depending on the exchange H E and anisotropy \nHA fields of the material s [23,25] ). Second, we experimentally study hematite (α-Fe2O3) capped with \nplatinum to confirm our theoretical predictions. Due to the low Gilbert damping and its residual \nanisotropy in the easy -plane phase [23,26,27] , we easily measure the resonance of the low frequency \nmode 𝑓− of hematite [6,23,28,29] and its associated VISHE. We report VISHE > 30 nV at 300 K as large \nas in the uniaxial antiferromagnet Chromium oxide (Cr 2O3) at low temperature s and only one order of \nmagnitude smaller than in the ferrimagnetic insulator YIG . Furthermore, a direct comparison between \nthe signs of the VISHE in the three compounds allows an identi fication of their respective mode \nhand edness. Altogether, o ur results highlight that c anted antiferromagnets embrace the rich dynamics \nof antiferromagnets , whilst keeping a net moment as in ferromagnets, and a larger temperature stability \nthan ferrimagnets with compensated angular momentum [30]. \n \nWe start b y deriving theoreticall y the spin-pumping response associated with the magnetization \ndynamics of both canted easy -axis and easy -plane antiferromagnets when put in contact with normal \nmetals. To this end , we model the magnetization dynamics in a macrospin approximation with two \nsublattices . We include the contributions from a DMI field 𝑯𝑫 and an external magnetic field 𝑯 \northogonal to the Néel order parameter . Both these fie lds contribute to the canted moments and, hence, \nto the emergence of a finite magnetic moment. The system’s free energy reads : \n \n𝐹=𝑀[𝐻𝐸(𝒎𝑨⋅𝒎𝑩)−𝐻𝐷𝒛̂⋅(𝒎𝑨×𝒎𝑩)+𝐻𝐴\n2(𝑚𝐴,𝑧2+𝑚𝐵,𝑧2)−𝐻𝑎\n2(𝑚𝐴,𝑦2+𝑚𝐵,𝑦2)−𝑯⋅(𝒎𝑨+\n𝒎𝑩)] (1), \n \nwhere γ denotes the gyromagnetic ratio, 𝒎𝑖 the unit vector of the sublattice magnetization i (𝑖∈{𝐴,𝐵}), \n𝑴𝑖=𝑀𝒎𝑖, and 𝑯=𝐻𝒙̂ is the externally applied static magnetic field, 𝐻𝐴 denotes the hard -axis \nanisotropy along the 𝑧 axis, and 𝐻𝑎 the easy -axis anisotropy within the 𝑥𝑦 plane . The above model can \nbe applied on both canted easy -axis (𝐻𝐴=0) and canted easy-plane (𝐻𝐴≫𝐻𝑎) AFMs . In these AFM s, \nthe eigen modes are non -degenerate with a low frequency mode 𝑓− and a high frequency mode 𝑓+: \n{𝑓−=(𝛾\n2𝜋)√2𝐻𝐸𝐻𝑎+𝐻(𝐻+𝐻𝐷) (2)\n𝑓+=(𝛾\n2𝜋)√2𝐻𝐸(𝐻𝑎+𝐻𝐴)+𝐻𝐷(𝐻+𝐻𝐷) (3). \nThe eigen modes are qualitatively similar in canted easy -plane and canted easy -axis AFMs (see \nSupplementary Material [31]). Hence, we can continue with a general ized model that describe s both \nsystems. This contrasts with the collinear easy-axis and easy -plane AFMs , in which the modes are \nprofoundly different and need separate treatment s [14–16,32,33] . The low frequency mode of canted \nAFM s is characterized by a right -handed elliptical precession of the magnetic moment (see Fig. 1. ( a)). \nThe high frequency mode is characterized by a linearly oscillating magnetization (see Suppl . \nMat. [31]). Note that , in the absence of an applied field , the gap of the low frequency mode [Eq. (2)] \ndepends only o n the exchange 𝐻𝐸 and easy -axis anisotropy 𝐻𝑎, and not on the DMI field 𝐻𝐷. This \nremarkable feature causes a low frequency mode in the THz range in canted easy -axis AFMs such as \nErFeO 3 [34]. In canted easy -plane AFM s where 𝐻𝑎 is often much smaller, the low frequency mode can \neven be found in the range of a few GHz, such as for hematite above the Morin temperature [23]. \n \nWe next derive an expression for the inverse spin Hall voltage 𝑉ISHE induced by spin pumping from the \nlow frequency mode into a n adjacent heavy metal layer. We consider excitation s by an oscillating \nmagnetic field hac (applied along the y axis) at the resonance frequency 𝑓_, which induces the precession \n3 \n of the magnetic moment s within the (𝑦𝑧) plane . The transverse 𝑉ISHE voltage (along y) is proportional \nto the dynamic magnetization amplitudes in both the 𝑦 and 𝑧 direction s. Following the approach of \nRef. [14], we establish the expression for the inverse spin -Hall voltage VISHE [see Suppl . Mat. [31] for \ndetails ]: \n \n𝑉ISHE =ħ𝜃N\n𝟖𝑑V \n𝑑N𝛾3(ℎ𝑎𝑐\n𝐻𝐸)2𝑄2(𝐻+𝐻𝐷)3\n4𝜋2𝑓−2 𝜆𝑒𝐺Rtan(𝑑𝑁2𝜆⁄ )\nℎ𝜎+2𝜆𝑒2𝐺Rcoth (𝑑𝑁𝜆⁄), (4) \nwhere 𝐺R denotes the real part of the spin mixing conductance per unit area of the interface, ℎac is the \namplitude of the excitation field, 𝑒 is the elementary charge, 𝑑V is the distance between the voltage \nleads, 𝑑N is the Pt layer ’s thickness, 𝜃N, λ, and 𝜎 are the thickness, spin diffusion length and the \nconductivity of Pt, respectively. 𝑄−=𝑓−\nΔ𝑓− correspond s to the material quality factor with Δ𝑓−=\nα(γ/π)𝐻𝐸 the linewidth of the low frequency resonance peak [28]. The expression of the \nantiferromagnetic linewidth and the damping values remain under strong debate in \nantiferromagnets [23,28,35,36] . However, r esonance measurements in insulating antiferromagnets \ngenerally show 𝑄− factor s in the range of 100 - 1000 [37–41]. We thus chose to evaluate the expecte d \ninverse spin -Hall voltages V ISHE for prototypical canted antiferromagnets such as orthoferrites with 𝑄− \nfactors of 500 , in the range of the existing reports [37,40,41] , and note that it can reach 0.5 μV (for hac \n= 1 mT, i.e. one ten th of the ac field used in Ref. [16]). For a given 𝑄− facto r, V ISHE scales with (𝐻+𝐻𝐷)3\n𝐻𝐸2𝑓−2. \nAs 𝑓_ scales with 𝐻, we anticipate a linear increase of the inverse spin -Hall voltage VISHE with the \napplied field (for H >> HD) which is opposite to the ferromagnetic case [42]. \nTable 1: Expected inverse spin Hall effect voltages V ISHE with their leading quantities H E, H A, H a and H D (taken at room \ntemperature) for each listed canted antiferromagnets . All voltages are calculated with a material quality factor 𝑄−=500 for \nthe different materials, an excitation field hac=1 mT, an external field H=0.2 T to ensure a mono -domainization , and a distance \nbetween the voltage lead s of 3 mm . 𝐺𝑅 is set to 6×1018, λ to 1.2 nm, 𝜃𝑁 to 0.1 taken from Refs. [26,42,48] . \nThe predicted inverse spin -Hall voltages presented in Table 1 are all above 10 nV which is in the \naccessible range of conventional voltage measurements [12,16,17,49] . Therefore, c anted \nantiferromagnets with DMI induced canting open very promising perspectives for examining the \nelectrical response of antiferromagnetic dynamics in the terahertz regime. \nTo confirm our predictions, we next experimentally investigate the antiferromagnetic resonance and \nthe generated inverse -spin Hall voltage in 500 μm thick bulk crystals of hematite covered [26] by a 3 \nnm thick Pt layer . We then compare the recorded inverse spin Hall effect voltage s to the one s of the \nferrimagnet YIG and the easy -axis antiferromagnet Cr2O3 [17] to also obtain further information such \nas the mode handedness as expected f or a coherent spin -pumping signal . \nIn the antiferromagnetic insulator hematite , the DMI field induce s a small canted moment ( ~ 3 \nemu/cm3) of the two sub -lattices above the Morin temperature (T M ~ 250 K) . For T > TM, its magnetic \nconfiguration corresponds to a canted easy-plane phase with a residual in -plane anisotropy H a such that \nit exhibits a low frequency mode in the gigahertz (GHz) range [23,28,29] . Therefore, w e can conduct \nour measurements using a state -of the art highly sensitive wideband resonance spectrometer (1 - 40 \nGHz) with a broadband coplanar waveguide ( c.f. Suppl. Mat. Fig . 1) and at room temperature. In Fig. \n1. (a-c), we present the dynamics and the frequency dispersion of the low frequency mode of hematite \nwhich has a gap around 15 GHz in agreement with previous reports [23,28] . Material HE (T) HA (T) Ha (T) HD (T) f- (THz) VISHE (nV) \nα-Fe2O3 [23,26,28] 1000 2.10-3 6.10-5 2.26 0.022 200 \nYFeO 3 [43,44] 640 / 0.06 14 0.25 711 \nErFeO 3 [45,46] 600 / 0.03 10 0.17 628 \nTmFeO 3 [30,31] 550 / 0.1 4 0.3 18 \n4 \n \nFigure 1: Magnetic resonance of the low frequency mode of hematite . (a) Illustration of the low frequency mode of hematite \nin the easy plane phase in presence of a DMI induced canting. (b) The frequency dispersion of the low frequency mode of \nhematite measured from 10 to 40 GHz . Using Eq. ( 2), we extract the parameters for the anisotropy field H a=6x10-5 T and \nHD=2.26 T. (c) Resonance curves for different values of the externally applied field. The data is normalized by the input power . \nCorrespondingly, we characterize the spin-pumping efficiency at the α -Fe2O3/Pt interface by \nmeasuring th e voltage generated at resonance in the top platinum layer using a lock-in technique . This \nallows us to val idate our model and investigate the spin dynamics of this non-collinear antiferromagnet . \nWe measure a voltage peak at resonance only in the transverse configuration (see inset of Fig. 2(a) ), \nand a clear sign reversal when we reverse the direction of the applied magnetic field. This sign inversion \ndemonstrate s the spin-pumping origin of the generated voltage . This result is to our knowledge the first \nevidence of spin -pumping from an easy -plane antiferromagnet and, more generally , at room \ntemperature for an antiferromagnet . \nFigure 2: Inverse spin Hall effect voltages V ISHE in the easy plane antiferromagnet hematite with a Dzyaloshinskii Moriya \ninduced canted moment. (a) ISHE v oltage m easurement at ±0.2 T in transverse configuration ( See inset for different signs of \nthe external field H. The antisymmetric signal shape indicates the recorded voltage to originate from spin-pumping . (b) VISHE \nfor different frequencies as a function of the external magnetic field . Colour coding of frequencies is consistent with Fig. 1 (c). \n(c) Dependence of VISHE peak as a function of the applied microwave power for a fixed excitation frequency of 31 GHz. In the \nmeasured range, t he ISHE voltage increases linearly as expected from Eq. (4). Inset shows the field dependency for the \ndifferent powers. \nAs for ferromagnets [12], we only measure a non -zero V ISHE in the transverse configuration when \nthe magnetic field is perpendicular to the voltage contact s as shown in Fig. 2 (a). In this configuration, \nthe dc spin accumulation 𝝈 generated at resonance must be parallel to the applied field and thus directed \nalong the canted net moment in order to generate a non -zero charge current 𝑱𝒄∝𝑱𝑺×𝝈, leading to \n \n5 \n VISHE. This result seem s at first contradictory to the spin -transport measurement s in easy-plane \nantiferromagnets where pairs of propagating magnons carry spin -angular momentum along the \ndirection of the antiferromagnet Néel order [29,50] . However, there is a key difference between the \nexcitation processes. For the spin -transport case, the current induced spin -accumulation generates pairs \nof correlated magnon modes with different wave vectors k, leading to an effective non -zero spin -angular \nmomentum along the Néel order. For the antiferromagnetic resonance case, the microwave magnetic \nfield excites only uniform oscillations (k= 0) which are linearly polarized in an easy-plane system . \nThus, only the dynamics of the canted moment , induced by the external magnetic field H or the DMI \nfield HD, can generate at resonance a non -zero dc spin-accumulation and thus a dc inverse spin -Hall \nvoltage . As shown in Fig. 2 (c) , we also observed a linear increase of V ISHE with the input power, \nconfirming that we are still in a linear regime of excitation. \n \nHaving detected the inverse spin -Hall volt age we need to determine the origin of the pumped spin \ncurrent . This origin is a key point in a long -standing debate in ferro - and ferrimagnet ic system s [51,52] \nand, more recently, also debated for antiferromagnets [16,17,53,54] . At resonance , oscillations of the \nexcited mode [14,15] or incoherent contribution s from thermal magnons (due to a resonance induced \nthermal gradient [55]) can both contribute to a spin-pumping signal . However, right (RH) - and left -\nhanded (LH) circularly polarized modes carry opposite angular momentum and should thus result in \ninverse spin -Hall voltage s with opposite signs [16,17] . Thus, o ne can obtain key information about the \nmode contributing to the spin -pumping signal s by analysing the sign of the inverse spin -Hall voltages. \nUsing broadband coplanar wa veguides up to 40 GHz , we can only access the low frequency RH \nmode above the Morin transition , and the LH mode below the Morin transition (see Supplemental) . \nHowever, we did not detect any V ISHE for the LH of hematite at all temperatures below the Morin \ntransition (See Supplemental [31]). This observation is consistent with a coherent spin -pumping signal \nassociated to the dynamical sub -lattice symmetry breaking [14], which is extremely small in the easy -\naxis phase of hematite ( HA/HE ≈ 10-6). To analyse its VISHE sign, we thus measure under the same \nconditions the inverse spin -Hall voltage s from the LH mode of the easy -axis antiferromagnet Cr 2O3 [17] \nand the RH mode of the ferrimagnet YIG [49]. To detect the LH mode of Cr 2O3, we perform \nmeasurements close to the spin -flop field to reduce the mode frequency below 40 GHz . We observe \nthat t he LH mode of Cr 2O3 and the RH modes of α-Fe2O3 and YIG show inverse spin -Hall voltages \nwith opposite signs as expected from a coherent spin-pumping model . In line with previous report s on \nCr2O3 [17], the VISHE of the LH mode disappears at high temperature (see Supplemental [31]) which is \nan indication of its coherent origin. Furthermore, the RH mode s of α-Fe2O3 and YIG generate inverse \nspin Hall voltages with opposite signs compared to the thermal spin-pumping contribution o f the RH \nmode of Cr 2O3 reported in Ref. [17]. Lastly, the inverse spin -Hall voltages have comparable amplitudes \nfor the LH mode of the easy -axis AFM Cr 2O3 and for the RH mode of the canted AFM α -Fe2O3, and \nare smaller than in YIG by less than an order of magnitude. This feature indicates that both col linear \nand nonco llinear antiferromagnets can efficiently generate spin -pumping whilst significantly enhancing \nthe operating frequency of spintronic devices. \nFigure 3. Inverse spin -Hall voltages recor ded at 31 GHz for (a) the left -handed AFM mode of the easy -axis antiferromagnet \nCr2O3 (b) the right -handed FM mode for the ferrimagnet YIG and (c) the low frequency (right -handed) mode of the canted \neasy-plane antiferromagnet α-Fe2O3 capped with a platinum layer . For an inverse spin -Hall voltage originating from \ncoherent spin pumping, a differen ce in hand edness of the mode’s polari sation is expected which induce s a change in the sign \nof VISHE. We confirm the correlation between the sign of the inverse spin -Hall voltage V ISHE and the mode handednes s since \nthe sign changes from (a) to (b) and is equal from (b) to (c) . Insets show the dispersion curves for the different modes. \n \n6 \n Illustration s show the orientation of the magnetic moments with respect to the external static field H. The thickness of the YIG \nfilm is 200 nm , and the two AFMs crystals are 500 μm thick . The measurements on YIG and α -Fe2O3 are performed at room \ntemperature and the ones on Cr 2O3 are performed at 30K (see Supplemental [31]). \nIn summary, we theoretically and experimentally demonstrate that noncollinear antiferromagnets \nwith DMI induced canting can generate sizeable spin-pumping signals and associated inverse spin -Hall \nvoltages . Using hematite as a room temperature model canted AFM, we measured a signal of VISHE > \n30 nV for the low frequ ency right -handed mode confirms our theoretical predictions . By compari ng \nwith the right -handed mode of YIG and the left -handed mode of Cr2O3, we confirm that the mode \nhanded ness determines the sign of the inverse spin -Hall voltage as expected for a coherent spin-\npumping signal . Consequently, c anted antiferromagnets allow for access ing the spin dynamics of the \nhitherto almost unexplored spin dynamics of noncollinear antiferromagnets. Such extension s not only \nbroaden the understanding o f the physics of the spin dynamics and of the relaxation processes for \nvarious classes of antiferromagnets but also represents a step further towards the realization of THz \napplications based on antiferromagnetic spintronics. \n7 \n \nAcknowledgement \nR.L., A.A. and M.K. acknowledge financial support from the Horizon 2020 Framework Programme of \nthe European Commission under FET -Open grant agreement no. 863155 (s -Nebula) . M.K. \nacknowledge s support from the Graduate School of Excellence Materials Science in Mainz ( MAINZ) \nDFG 266, the DAAD (Spintronics network, Project No. 57334897) . M.K. acknowledge s support from \nthe DFG project number 423441604 and additional support from SFB TRR 173 Spin+X (projects A01 \nand B02 # 268565370 ). H.T.S., A.B., M.K. were supported by th e Research Council of Norway through \nits Centres of Excellence funding scheme, project number 262633 “QuSpin”. 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B 93, 014425 (2016). \n " }, { "title": "2210.01441v1.Local_density_of_states_as_a_probe_for_tunneling_magnetoresistance_effect__application_to_ferrimagnetic_tunnel_junctions.pdf", "content": "Local density of states as a probe for tunneling magnetoresistance e\u000bect: application\nto ferrimagnetic tunnel junctions\nKatsuhiro Tanaka,1Takuya Nomoto,1and Ryotaro Arita1, 2\n1Research Center for Advanced Science and Technology,\nUniversity of Tokyo, Komaba, Meguro-ku, Tokyo 153-8904, Japan\n2Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan\n(Dated: October 5, 2022)\nWe investigate the tunneling magnetoresistance (TMR) e\u000bect using the lattice models which\ndescribe the magnetic tunnel junctions (MTJ). First, taking a conventional ferromagnetic MTJ as\nan example, we show that the product of the local density of states (LDOS) at the center of the\nbarrier traces the TMR e\u000bect qualitatively. The LDOS inside the barrier has the information on\nthe electrodes and the electron tunneling through the barrier, which enables us to easily evaluate\nthe tunneling conductance more precisely than the conventional Julliere's picture. We then apply\nthis method to the MTJs with collinear ferrimagnets and antiferromagnets. We \fnd that the\nTMR e\u000bect in the ferrimagnetic and antiferromagnetic MTJs changes depending on the interfacial\nmagnetic structures originating from the sublattice structure, which can also be captured by the\nLDOS. Our \fndings will reduce the computational cost for the qualitative evaluation of the TMR\ne\u000bect, and be useful for a broader search for the materials which work as the TMR devices showing\nhigh performance.\nI. INTRODUCTION\nUtilizing the close connection between the spin and\ncharge degrees of freedom of electrons in solids, spintron-\nics has developed various phenomena that are novel from\nthe viewpoint of fundamental physics and promising\nfor industrial use [1{5]. Among those, the tunneling\nmagnetoresistance (TMR) e\u000bect [6, 7] is one of the\nrepresentative phenomena in its wide application [8{12].\nThe TMR e\u000bect is observed in the magnetic tunnel junc-\ntion (MTJ), which consists of two magnetic electrodes\nand the insulating barrier in between. The electrons\ncan tunnel through the MTJ as a quantum mechanical\ncurrent, and the tunneling resistances become di\u000berent\nwhen the magnetic moments of the two electrodes\nalign parallelly or antiparallelly. The set of these two\nalignments with di\u000berent tunneling resistances corre-\nsponds to a bit taking a binary 0 or 1, which has been\nutilized to the magnetic head and the magnetic random\naccess memory devices for the storages and the readout.\nAs well as the theoretical approaches [13{16], large\nTMR ratios have been experimentally observed in the\nMTJs such as the Fe(Co)/Al 2O3/Fe(Co) [17, 18],\nFe(Co)(001)/MgO(001)/Fe(Co) [19, 20] and\nCoFeB/MgO/CoFeB systems [21, 22]. Ferromag-\nnetic Heusler compounds have also been utilized as the\nelectrodes thanks to their half-metalicity [23{26].\nWhile the main target of the spintronics was ferromag-\nnets, recent spintronics has been extended to antiferro-\nmagnets and ferrimagnets owing to their superiorities to\nferromagnets; the smaller stray \feld and the faster spin\ndynamics [27{35]. The antiferromagnetic version of the\nspintronic phenomena, e.g., the giant magnetoresistance\ne\u000bect [36{38] and the anomalous Hall e\u000bect [39{42], has\nbeen developed. Along with these advances, the TMR\ne\u000bect using antiferromagnets has also been intensivelyinvestigated [43{48]. While most of the studies have\nbeen theoretical attempts, experiments have also been\ndeveloped; the TMR e\u000bect is observed in the MTJ whose\ntwo electrodes are the ferromagnet and the ferrimagnetic\nHeusler compound [44]. However, for more practical ap-\nplication of the MTJs with antiferromagnets and ferri-\nmagnets to the devices, we should search for materials\nconstructing the MTJs which show a large TMR ratio,\nand handy methods for the search are required.\nIn this paper, we examine the TMR e\u000bect using the\nlattice models mimicking the MTJs whose electrodes are\nmade of collinear ferrimagnets, including the antiferro-\nmagnets. Motivated by the studies indicating that the\ninterfacial electronic structures a\u000bect the TMR e\u000bect\nand they can be probed by the local density of states\n(LDOS) [49{53], we particularly focus on the LDOS to\nanalyze the TMR e\u000bect. We \fnd that the product of the\nLDOS at the center of the barrier usually reproduces the\ntransmission properties qualitatively in the ferrimagnetic\nMTJs as well as the ferromagnetic ones. The LDOS has\nthe information both on the magnetic properties of elec-\ntrodes and on the tunneling electrons. Besides, from the\nphysics point of view, we show that multiple con\fgura-\ntions can be realized in the ferrimagnetic MTJs due to the\nsublattice structure for each of the parallel and antipar-\nallel magnetic con\fgurations. The resultant TMR e\u000bect\nchanges depending on the con\fgurations, which suggests\nthat the magnetic con\fgurations should be carefully ex-\namined when we deal with the ferrimagnetic MTJs.\nConsidering the above qualitative estimation in terms\nof the LDOS, we present a hierarchy for evaluating the\nTMR e\u000bect in Fig. 1. To quantitatively estimate the\nTMR e\u000bect, we have to calculate the conductance itself\nthrough the Landauer{B uttiker formula [54{57]. Tech-\nnically, this method can be applied to any system and\ngives us highly accurate results, whereas its numerical\ncost is often expensive, particularly in calculating fromarXiv:2210.01441v1 [cond-mat.mes-hall] 4 Oct 20222\nLocal density of states\nAppearance: \nbulk density of states / spin configurationTransport calculation:\nLandauer–Büttiker formulaCoverage / Computational cost\nHigh\nLow\nFIG. 1. Hierarchy for the evaluation of the tunneling mag-\nnetoresistance e\u000bect. The upper has a broader coverage and\ngives more quantitative results, with the higher calculation\ncost. Symbols in the right side, G(E),N\u001b(E;x; y), and\nN\u001b(E), denote the conductance, the local density of states,\nand the density of states of the bulk, respectively, which are\nthe quantities obtained in each calculation.\n\frst-principles. By contrast, the Julliere's picture, which\nclaims that the density of states of the bulk electrodes\ndetermines the e\u000eciency of the MTJ [6, 58], is a sim-\nple and convenient picture to predict the TMR e\u000bect.\nHowever, the picture is valid only for limited cases, and\ncurrently it is found that the electronic states of the\ntunneling electrons are signi\fcant rather than those of\nthe bulk electrodes. Our results can be placed between\nthese two methods. Calculating the LDOS of the MTJs\nis much easier than the Landauer{B uttiker calculation.\nAdditionally, the estimation from the LDOS can cover a\nbroader range of the MTJs with higher reliability than\nthe prediction from the electronic structures of the bulk\nelectrodes.\nThe remainder of this paper is organized as follows. In\nSec. II, we introduce the model describing the MTJ. We\nsimulate the TMR e\u000bect using the ferromagnetic elec-\ntrodes in Sec. III, and see how the LDOS works for pre-\ndicting the tunneling conductance. In Sec. IV, we calcu-\nlate the TMR e\u000bect in the ferrimagnetic and antiferro-\nmagnetic MTJs applying the prediction from the LDOS.\nWe discuss in Sec. V the hierarchy shown in Fig. 1 and the\ncorrespondence between our models and the MTJ with\nreal materials. Section VI is devoted to the summary and\nperspective of this study.\nII. MODEL AND METHOD\nWe construct the two-dimensional square lattice MTJ\nusing two semi-in\fnite lattices which work as electrodes\nand the barrier in between, which is schematically shown\nin Fig. 2. We treat the tight-binding Hamiltonian withthes{dcoupling on this system, which is given as\nH=H0+Ht+Hs\u0000d; (1)\nH0=X\ni\"ini; (2)\nHt=\u0000tX\nhi;ji;\u001b\u0010\ncy\ni;\u001bcj;\u001b+ h.c.\u0011\n; (3)\nHs{d=\u0000JX\ni2electrode(si\u0001\u001b)\u000b\fcy\ni;\u000bci;\f: (4)\nHere,cy\ni;\u001b(ci;\u001b) is the creation (annihilation) opera-\ntor of an electron with spin- \u001bon thei-th site, and\nni=P\n\u001bcy\ni;\u001bci;\u001bis the number operator. The on-site\npotential is denoted as \"i, and the electron hopping be-\ntween two sites is written as t. The summation in Htis\ntaken over the neighboring two sites, which is expressed\nbyhi;ji. The e\u000bect of the magnetism of the electrodes\nis introduced by the s{dcoupling,Hs{d, where the local-\nized spin moment, si, and the conducting electrons cou-\nple each other with the magnetic interaction constant, J.\nThe spin degrees of freedom of the conducting electrons\nare expressed by \u001b, which is the vector representation of\nthe 2\u00022 Pauli matrices. We take the two-dimensional\ncartesian coordinate, ( x;y), for the MTJ, where the x-\naxis is parallel to the conducting path which in\fnitely\nextends, and the y-axis is perpendicular to the conduct-\ning path (see Fig. 2(a)). The width of the barrier in\nthex-direction is L, and the width of the MTJ in the\ny-direction is W. The lattice constant is taken to be\nunity. We hereafter impose the open boundary condition\non they-direction, while we con\frmed that the periodic\nboundary condition does not change the overall results.\nThe calculations of the transmissions are performed\nusing the kwant package [59], in which the quantum\ntransport properties are computed based on the scatter-\ning theory and the Landauer{B uttiker formula.\nIII. TUNNELING MAGNETORESISTANCE\nEFFECT WITH FERROMAGNETIC\nELECTRODES\nLet us \frst recall the conventional TMR, namely the\nTMR using the ferromagnetic electrodes as shown in\nFig. 2(a). We set the localized spin moments as si=\nt\u00000 0 1\u0001\nfor all sites in the left electrode. For the\nsites in the right electrodes, we set si=t\u00000 0 1\u0001\nand\nt\u00000 0\u00001\u0001\nfor the parallel and antiparallel con\fgura-\ntions, respectively. The schematics of these two con\fg-\nurations are shown in Figs. 3(a) and 3(b). We set the\nhoppingtas a unit,t= 1. The on-site potential is set\nas\"i= 0 for the electrodes, and \"i= 10 for the barrier\nregion. The system size is set as L= 8 andW= 160.3\nelectrode\n(ferrimagnetic)electrode\n(ferrimagnetic)barrier\n(nonmagnetic)(a)\nelectrode\n(ferromagnetic)electrode\n(ferromagnetic)barrier\n(nonmagnetic)\n(b)\nFIG. 2. Schematics of the two-dimensional magnetic tun-\nnel junctions (MTJ) used in our calculations. (a) MTJ with\nthe ferromagnetic electrodes. (b) MTJ with the ferrimagnetic\nelectrodes. Arrows represent the localized spin moments on\nthe electrodes.\nA. Bulk properties\nBefore discussing the properties of the tunneling con-\nductance, we see the properties of the bulk ferromagnetic\nmetals used as the electrodes, namely, the energy bands\nand the density of states (DOS). The DOS is given as\nN\u001b(E) =R\nBZdk\u000e(E\u0000Ek;\u001b), whereEk;\u001bis the energy\nband with spin- \u001b. For the energy bands of the bulk elec-\ntrodes, we consider the two-dimensional square lattice\ndescribed byHgiven in Eq. (1). The energy bands are\nfound to be E\u0006\nk=\u00002t(coskx+ cosky)\u0006J. The DOS of\nthe ferromagnet is shown in Fig. 3(c) at J= 1, where the\nright-hand side and the left-hand side are the ones with\nthe up-spin and down-spin, respectively. By introducing\na \fniteJ, the energy band with up-spin gains the energy\n\u0000J, while that with down-spin shifts by + J.\nB. Transmission and local density of states\nIn Fig. 3(d), we show the TMR ratio de\fned as\nrt=TP\u0000TAP\nTP+TAP; (5)\non the plane of JandE. Here,TP/AP denotes the trans-\nmission for the parallel/antiparallel con\fguration. We\nnote that the de\fnition above is slightly di\u000berent from\nthe conventional optimistic/pessimistic ones for the rea-\nson on normalization. At J= 0, the whole system is\nnonmagnetic and has the degeneracy on the spin degrees\nof freedom, and thus TP=TAPholds at each energy.\nNamely,rtis zero. When we introduce a \fnite magnetic\ninteraction J, the degeneracy is lifted, and TPstarts totake a larger value than TAP; a \fnite TMR ratio is ob-\nserved. Due to the asymmetric structure of the barrier in\nthe energy, rtis also asymmetric with respect to E= 0.\nTheJ-dependence of TPandTAPis shown in Figs. 3(e)\nand 3(f). At E=\u00002, both ofTPandTAPdecrease when\nJincreases, and TAPreaches zero at J= 2 [60]. When\nE= 0,TPincreases with J, whileTAPdecreases to zero.\nTo better understand the transmission properties, we\nexamine the LDOS in addition to the bulk DOS. In the\nnaive Julliere's picture, the product of the bulk DOS,\ng(E), de\fned as\ng(E) =X\n\u001bNL;\u001b(E)NR;\u001b(E); (6)\ndescribes the transmission [6, 58], where NL/R;\u001b(E) is\nthe bulk DOS with spin- \u001b,N\u001b(E), of the left/right elec-\ntrodes. However, g(E) does not consider the barrier, and\nthus this picture holds only for the limited cases. In-\nstead, the LDOS has been utilized to capture the details\non the MTJs. In particular, it has been proposed that\nthe transmission can be described using the LDOS at the\ninterfaces of the MTJ. In fact, when the potential of the\nbarrier is high enough, the conductance derived by the\nKubo formula [14] is proportional to the product of the\nLDOS at the left and right interfaces, and to the expo-\nnential function, e\u0000\u0014L, representing the decay inside the\nbarrier [49, 50]. Here, \u0014stands for the decaying property,\nnamely, 1=\u0014means the spin di\u000busion length [61]. Hence,\nwe consider the product of the LDOS at the interfaces,\ngi(E) given as\ngi(E) =1\n2X\n\u001b\u0014\nN\u001b\u0012\nE; 1;W\n2\u0013\nN\u001b\u0012\nE;L;W\n2\u0013\n+N\u001b\u0012\nE; 1;W\n2+ 1\u0013\nN\u001b\u0012\nE;L;W\n2+ 1\u0013\u0015\n;(7)\nwhereN\u001b(E;x;y) (1\u0014x\u0014L, 1\u0014y\u0014W) is the LDOS\nof the barrier at ( x;y). Since we impose the open bound-\nary condition in the y-direction, we take the average over\nthe sites at y=W=2 andW=2 + 1 in Eqs. (7) and (8) to\nreduce the e\u000bects of the oscillation due to the boundary,\nand we implicitly assume that the spins do not \rip in-\nside the barrier [62]. However, in general, it is not easy\nto precisely evaluate the exponent \u0014; we should need the\ntransmission coe\u000ecients [63] in addition to \fnding the\nelectronic structures.\nAlternatively, we here utilize the LDOS at the center of\nthe barrier. Expecting that it contains both the details\nof the MTJ and the decay, we consider the product as\nfollows;\ngc(E) =1\n2X\n\u001b\u0014\nN\u001b\u0012\nE;L\n2;W\n2\u0013\nN\u001b\u0012\nE;L\n2+ 1;W\n2\u0013\n+N\u001b\u0012\nE;L\n2;W\n2+ 1\u0013\nN\u001b\u0012\nE;L\n2+ 1;W\n2+ 1\u0013\u0015\n:\n(8)4\n(b)(e)\n(h) (i)(f) (a)\n(c)\n00\n-6-4-2246\n0.6 -0.6\ndensity of states013\n0 2 1 002 1 0 2 1120246\n0123\n0 200.3\n000.5\n22Parallel\nAntiparallel(g)(d)\n-404\n2\n-2\n01\n01\n-404\n2\n-2\nFIG. 3. (a), (b) Schematic pictures of (a) the parallel and (b) antiparallel con\fguration of the magnetic tunnel junction (MTJ)\nwith ferromagnetic electrodes. (c) Spin-resolved density of states of the electrode with J= 1. Broken lines are the energies\nwhere we show the results in (e), (f), (h), (i). (d){(i) Results of the calculation on the ferromagnetic MTJ. (d) TMR ratio rt\n(Eq. (5)) on the plane of JandE. (e), (f) Transmissions as a function of the magnetic interaction, J, at (e) E=\u00002 and (f)\n0. (g) Ratio rc(Eq. (9)). (h){(i) Magnetic interaction, J, dependence of gc(E).\nIn this scheme, we only have to \fnd the electronic struc-\ntures to estimate the transmission. While we now con-\nsider the case with even- Land calculate the product of\nthe LDOS at x=L=2 andL=2 + 1, we should replace\nboth of them with the one at x= (L+1)=2 for the odd- L\ncase.\nSimilarly to the TMR ratio rt, we calculate the ratio,\nrc, de\fned as\nrc=gc;P(E)\u0000gc;AP(E)\ngc;P(E) +gc;AP(E); (9)\nwheregc;P(E) andgc;AP(E) aregc(E) for the parallel\nand antiparallel con\fgurations, respectively. Figure 3(g)\nshowsrcon the plane of JandE. We \fnd that rcquali-\ntatively reproduces the TMR ratio rtshown in Fig. 3(d).\nWe show the J-dependence of gc(E) in Figs. 3(h) and\n3(i), and that of g(E) in the insets. At E=\u00002,gc(E)\nreproduces the overall properties of the transmission,\nwhileg(E) increases with Jand does not reproduce the\ntransmission properties. When we increase the energy to\nE= 0, the bulk DOS with the up and down spins take the\nsame values, and the estimation in terms of g(E) would\nlead to the absence of the TMR. Even in this case, the\nproduct of the LDOS at the center of the barrier well re-\nproduces the transmission properties; the J-dependence\nofgc(E) is qualitatively the same as the one of the trans-\nmissions.\nWe remark that we also examine the J-dependence of\ngi(E) following the previous proposals [14, 49, 50]. We\n\fnd thatgc(E) better reproduce the transmission prop-erties than gi(E) when we compare these two quantities\n(see Appendix for the detailed results of gi(E)), which\nis due to the absence of the decay e\u000bect in gi(E).\nIV. TUNNELING MAGNETORESISTANCE\nEFFECT WITH\nFERRIMAGNETIC/ANTIFERROMAGNETIC\nELECTRODES\nNext, we examine the tunneling conductance of the\nMTJ with ferrimagnetic electrodes, including the anti-\nferromagnet, and apply the analysis by means of the\nLDOS. We here concentrate on the ferrimagnet with two-\nsublattices, A and B, on the square lattice. We assume\nthat the ferrimagnet has the G-type structure; all spins\nat the nearest-neighbors of the spins in the A-sublattice\nare in the B-sublattice, and vice versa. In the ferrimag-\nnetic MTJ, we de\fne the parallel and antiparallel con-\n\fgurations by the alignments of the spins on the same\nsublattice. Due to the two-sublattice structure, there\nare two pairs of the parallel{antiparallel con\fgurations\nwhose schematic views are shown in Figs. 4(a){4(d). In\nthe \frst one, which we call con\fguration-1, two local-\nized spins next to the barrier layer on the left and right\nelectrodes with the same y-coordinates are in the di\u000ber-\nent sublattices. The parallel and antiparallel arrange-\nments of con\fguration-1 is shown in Figs. 4(a) and 4(b),\nrespectively. In the second one, which we refer to as\ncon\fguration-2, those two spins are in the same sublat-5\n(a)\n(b)Configuration-1\nParallel\nAntiparallel\n(d)Configuration-2\nAntiparallel(c) Parallel\nFIG. 4. Two con\fgurations of the magnetic tunnel junc-\ntions (MTJ) using the ferrimagnetic electrodes, con\fguration-\n1 and 2. (a) Parallel and (b) antiparallel alignments of\ncon\fguration-1. (c) Parallel and (d) antiparallel alignments\nof con\fguration-2.\ntices, whose parallel and antiparallel arrangements are\nrespectively shown in Figs. 4(c) and 4(d). The param-\neters in the Hamiltonian are taken in common with the\nferromagnetic MTJ; \"i= 0 and 10tfor the electrodes and\nthe barrier respectively, and L= 8 andW= 160.\nA. Bulk properties\nWe \frst see the bulk properties of the ferrimagnetic\nelectrodes; we calculate the energy band and the DOS\nof the system described by the Hamiltonian Hon the\nsquare lattice. The spins of A- and B-sublattices are set\nassA=t\u00000 0sA\u0001\nandsB=t\u00000 0sB\u0001\n, respectively.\nSince the ferrimagnet has two-sublattices, there are two\nenergy bands, E\u0006\nk;\u001b, for each spin degrees of freedom \u001b.\n(a) (b)\ndensity of states6\n0\n-66\n0\n-6-1 0 1\ndensity of states-1 0 1-4-224\n-4-224FIG. 5. Spin-resolved density of states of (a) the two-\ndimensional ferrimagnet with ( sA; sB) = (1 :0;\u00000:5), and (b)\nthe antiferromagnet with ( sA; sB) = (1 :0;\u00001:0). For both\ncases, J= 1:0.\nThe energy bands are written as\nE\u0006\nk;\"=\u0000J(sA+sB)\u0006q\nJ2(sA+sB)2\u00004 (J2sAsB\u0000\r2\nk)\n2;\n(10)\nE\u0006\nk;#=+J(sA+sB)\u0006q\nJ2(sA+sB)2\u00004 (J2sAsB\u0000\r2\nk)\n2;\n(11)\nwhere\rk=\u00002t(coskx+ cosky). In Figs. 5(a) and 5(b),\nwe show examples of the DOS of the ferrimagnet with\n(sA;sB) = (1:0;\u00000:5), and that of the antiferromagnet\nwith (sA;sB) = (1:0;\u00001:0), respectively.\nB. Transmissions and local density of states\n1. Ferrimagnetic electrode\nWe discuss the transmission properties of the ferrimag-\nnetic MTJ. First, we investigate the system where we\n\fx the magnetization of two sublattices as ( sA;sB) =\n(1:0;\u00000:5). Figures 6(a) and 6(b) show the TMR ratio,\nrt, on the plane of JandEfor con\fguration-1 and 2,\nrespectively. The J-dependence of the transmissions for\ncon\fguration-1 and 2 at E= 3:6 is respectively shown\nin Figs. 6(c) and 6(d). Without the s{dcouplingJ,\nthe localized magnetic moment does not a\u000bect the trans-\nmission, and TPandTAPtake the same values for both\ncon\fguration-1 and 2. When a small Jis introduced in\ncon\fguration-1, TAPis larger than TP, andTPbecomes\nlarger than TAPatJ'1:5. Meanwhile, TPis larger than\nTAPat \fniteJin con\fguration-2.\nAs well as the ferromagnetic MTJ discussed in Sec. IV,\nwe focus on gc(E) de\fned by Eq. (8) and calculate rc\ngiven in Eq. (9). In Figs. 6(e) and 6(f), we plot rc\nfor con\fguration-1 and 2, respectively, which indicates\nthatgc(E) qualitatively traces the TMR property also\nin the ferrimagnetic MTJs. We plot the J-dependence\nofgc(E) in Figs. 6(g) and 6(h) for con\fguration-1 and6\n(b) (c) (d)\n(e)(a)Configuration-1 Configuration-2 Configuration-1 Configuration-2\n(f) (g) (h)0123\n0123\n2\n00\n2 11\n0 2 1 0 2 12\n00\n2 11000.08\n2\nParallel\nAntiparallel1\n0\n-1\n1\n0\n-1 -404\n2\n-2-404\n2\n-2\n-404\n2\n-2-404\n2\n-2\nFIG. 6. Results of the transmission calculation for the ferrimagnetic tunnel junction with ( sA; sB) = (1 :0;\u00000:5) for con\fguration-\n1 and 2. (a), (b) TMR ratio rt(Eq. (5)) on the plane of JandE. (c), (d) Transmissions at E= 3:6 with respect to J. (e), (f)\nRatio rc(Eq. (9)). (g), (h) Product of the local density of states at the center of the barrier, gc(E), as a function of J. Inset\nin (g) is the J-dependence of g(E).\n2 atE= 3:6, respectively, together with g(E) given in\nEq. (6) in the inset of Fig. 6(g). We can see that the\ntransmission and gc(E) similarly changes with J, while\ng(E) does not reproduce the transmission. Here the av-\nerage is meaningful also for capturing the two-sublattice\nmagnetic structure in the y-direction in the ferrimagnetic\nMTJ, whereas in the ferromagnetic MTJ the average over\ny=W=2 andW=2 + 1 is important to take the open\nboundary conditions into account. We note that gc(E)\ntraces even the reversal of the transmission occurring in\ncon\fguration-1, whereas g(E) orgi(E) de\fned by the\ninterfacial DOS does not (see Fig. 10 for details).\n2. Antiferromagnetic electrode\nNext we discuss the antiferromagnetic limit, ( sA;sB) =\n(1:0;\u00001:0). In this case, con\fguration-1 and 2 are\ndegenerate; the parallel and antiparallel alignments of\ncon\fguration-1 correspond to the antiparallel and par-\nallel alignments of con\fguration-2, respectively. Hence,\nhere we consider con\fguration-1 only and de\fne the par-\nallel and antiparallel con\fgurations by con\fguration-1.\nWe show the TMR ratio rt(Eq. (5)) in Fig. 7(a), and\ntheJ-dependence of the transmission at in Fig 7(b) at\nE= 1. AtE= 1, both of TPandTAPincrease with J\natJ.1 as shown in Fig. 7(b). At J&1,N\u001b(E) is zero,\nnamely, there is no incidence from the electrodes. Thus,\nthe transmission of each con\fguration becomes zero.\nFigures 7(c) represents the ratio rc(Eq. (9)). We con-\frm thatrchas a parameter dependence qualitatively\nthe same as the one of rt. In fact, the J-dependence of\ngc(E) atE= 1 shown in Fig. 7(d) well reproduce the\nJ-dependence of the transmissions shown in Fig. 7(b).\nIn the inset of Fig. 7(d) we show g(E) for the paral-\nlel and antiparallel con\fgurations, which are degenerate\nand do not predict a \fnite TMR e\u000bect. We note that\ngi(E) de\fned by the interfacial LDOS has a parameter\ndependence qualitatively di\u000berent from the one of the\ntransmissions (see Fig. 11 for details).\nV. DISCUSSION\nA. Hierarchy in estimating the transmission\nproperties\nWe presented a hierarchy in the evaluation of the trans-\nmission properties in Fig. 1. Here we discuss how the\nappearance, namely, the bulk DOS and the spin con\fg-\nurations, and the LDOS are related to the transmission\nproperties. For the ferrimagnetic tunnel junction with\n(sA;sB) = (1:0;\u00000:5) atJ= 1:85, we show the energy\ndependence of the transmissions and gc(E) in Fig. 8. Fig-\nures 8(a) and 8(b) are the results for con\fguration-1, and\nFigs. 8(c) and 8(d) are for con\fguration-2. When either\nof the two spin-states has a \fnite DOS, TPtakes a larger\nvalue than TAP, which supports the Julliere's picture.\nWhen both spin-states have \fnite DOS, shown as the\nshaded regions in Fig. 8, TAPis sometimes larger than7\n(a) (b)\n(c) (d)\n048048\n0 1 2 0 1 2001\n2Parallel\nAntiparallel1\n0\n-1\n1\n0\n-1 -404\n2\n-2-404\n2\n-2\nFIG. 7. Results of the transmission calculation of the an-\ntiferromagnetic tunnel junction with ( sA; sB) = (1 :0;\u00001:0).\n(a) TMR ratio rt(Eq. (5)) on the plane of JandE. (b)\nTransmissions for the parallel and antiparallel con\fgurations\natE= 1. (c) Ratio rc(Eq. (9)). (d) Magnetic coupling, J,\ndependence of gc(E) atE= 1. Inset is the J-dependence of\ng(E).\nTP, e.g., atE\u00183 for con\fguration-1 (Fig. 8(a)). This\nmeans that the conventional Julliere's description breaks\ndown in these regions. Instead, the spin con\fgurations\nat the interface usually gives very rough estimation of\nthe transmission. Let us focus on the spin con\fgurations\nat the interface. As shown in Figs. 4(a) and 4(b), for\ncon\fguration-1, the interfacial spins of the two electrodes\nwith the same y-coordinates align antiparallelly in the\nparallel arrangement, and those spins in the antiparallel\narrangement align parallelly. For the con\fguration-2, the\nparallel arrangement have the interfacial spins with op-\nposite directions, and the antiparallel arrangement have\nthe interfacial spins pointing the same directions, which\nis represented in Figs. 4(c) and 4(d). Since the spins\nunlikely to \rip through coherent tunneling, the trans-\nmissions become larger when the interfacial spins of two\nelectrodes align parallelly. In fact, in the case where two\nspin-states have nonzero DOS, TP T APholds in a broad\nE-region for con\fguration-2 (see Figs. 8(a) and 8(c)).\nFrom the interfacial spin con\fgurations, we can\nroughly predict the TMR properties in many cases. As\nshown in Fig. 8(a), however, TPbecomes larger than TAP\ncontrary to the prediction from the interfacial spins at\nE\u00183:7 in con\fguration-1 with the bulk DOS consist-\ning of both spin-states. Still in this case, gc(E) for the\nparallel con\fguration takes a larger value than that for\nthe antiparallel con\fguration. Furthermore, gc(E) gives\n(a) (b)\n(c) (d)\n0 0Configuration-1\nConfiguration-2\n6\n4\n2\n0\n4 2 1 2Parallel\nAntiparallel\n0 0 4 2 1 26\n4\n2\n0FIG. 8. Results of the transmission calculations for the fer-\nrimagnetic tunnel junction with ( sA; sB) = (1 :0;\u00000:5) at\nJ= 1:85 for (a), (b) con\fguration-1 and (c), (d) 2. (a),\n(c) Transmissions and (b), (d) gc(E) for the parallel and an-\ntiparallel alignments. Shaded regions represent the energies\nwhere both spin-states have \fnite DOS (see Fig. 5(a)).\nus the detailed information on the parameter dependence\nas shown in Figs. 8(b) and 8(d), while we can only know\nfrom the appearance whether the parallel or antiparal-\nlel con\fguration gives the larger transmission. To com-\npletely understand the transmission properties, of course\nwe should calculate transmission itself, but we expect\nthat the estimation in terms of the LDOS is enough as\nan initial way.\nB. Details of the magnetic tunnel junctions\nWe have considered the simplest cases, where the elec-\ntronic orbitals are isotropic and the barrier is structure-\nless. In reality, we should consider the details of the\nMTJs. In the Fe(001)/MgO(001)/Fe epitaxial MTJ, for\nexample, the \u0001 1-symmetry state with a large spin po-\nlarization has less decay, which dominantly contributes\nto the large TMR ratio [15, 16]. If we focus on this \u0001 1\nBloch state, we can apply our treatment and estimate the8\nTMR e\u000bect in terms of the LDOS. Besides, due to struc-\ntural disorders or hybridization of orbitals, the electronic\nand magnetic states at the interfaces may be modulated.\nWe can trace the e\u000bect of the modulation with gc(E),\nwhich is indicated by the results that gc(E) reproduces\nthe transmissions with each of two di\u000berent interfaces,\ncon\fguration-1 and 2, for the ferrimagnetic MTJs.\nIn the typical metals used in the ferrimagnetic spin-\ntronics such as GdFeCo [64{66] or Mn 2RuxGa [67], the\nvalences of the ions carrying the magnetic moments of\nthe di\u000berent sublattices are di\u000berent. This charge in-\nequivalence determines the interfacial structure to keep\nthe charge neutrality at the interface. Namely, if we\nalso take the charge degrees of freedom into account,\nwe can in principle design the interfacial magnetic struc-\nture. On the other hand, when we cannot control the\ninterfacial structures precisely, the averaged structure of\ncon\fguration-1 and 2 seems to be realized, which can be\nregarded as the ferromagnetic MTJ of the net magnetic\nmoments. We have numerically con\frmed that the Jul-\nliere's picture with the bulk DOS holds like ferromagnets\nin that case.\nFor antiferromagnetic MTJs, when we use the antifer-\nromagnets with the macroscopic time-reversal symmetry,\ncon\fguration-1 and 2 are not distinguished. The TMR\ne\u000bect then vanishes since there is a degeneracy on the\ntransmission between con\fguration-1 and 2 as mentioned\nin Sec. IV. By contrast, the antiferromagnets macroscop-\nically breaking the time-reversal symmetry separate the\nMTJs with con\fguration-1 and 2, which enables us to ob-\nserve a \fnite TMR e\u000bect in the antiferromagnetic MTJs.\nActually, the MTJs using such antiferromagnets have\nbeen theoretically proposed [46{48].\nVI. SUMMARY AND PERSPECTIVES\nIn summary, we have numerically studied the tunnel-\ning magnetoresistance (TMR) e\u000bect modelizing the mag-\nnetic tunnel junction (MTJ) consisting of the ferrimag-\nnetic electrodes as well as the well-known ferromagnetic\nones. To grasp the transmission properties, we have fo-\ncused on the local density of states. We have shown that\nthe transmission properties can be qualitatively repro-\nduced by the product of the local density of states at the\ncenter of the barrier. In the physical aspect, there can be\nmultiple con\fgurations for the ferrimagnetic MTJs owing\nto the sublattice structure of the electrodes. Those mul-\ntiple con\fgurations give the di\u000berent transmission prop-\nerties, and thus we should be careful for the magnetic\ncon\fgurations in the ferrimagnetic TMR.\nOur approach can be applied to more complicated\ncases where the detailed structures are taken into ac-\ncount. When one performs a more realistic TMR calcu-\nlation, the electronic structures and the wave-functions\nshould be obtained from \frst-principles. To calculate the\ntransmission by using \frst-principles wave-functions, the\nmethods such as the nonequilibrium Green's function for-\n(c) (b)(a)\n048\n0 2 1 002 1123\nParallel\nAntiparallel-4-2024\n01\n0 2 1FIG. 9. Results of the simulation for ferromagnetic tunnel\njunctions (Sec. III). See also Fig. 3 as a comparison. (a) Ratio\nri(Eq. (A.1)) on the plane of JandE. (b), (c) Product of the\ninterfacial local density of states gi(E) (Eq. (7)) with respect\ntoJat (b) E=\u00002 and (c) 0.\nmalism [68{70] or the scattering problem approach [71{\n74] are widely adopted. However, these methods usually\ndemand huge numerical costs, which probably has pre-\nvented us from exploring the MTJ using various materi-\nals. The calculation of the local density of states is much\nless costly, so that the approach with the local density\nof states will serve an easy means to search for the MTJ\nwith high e\u000eciency.\nACKNOWLEDGMENTS\nThis work was supported by JST-Mirai Program (JP-\nMJMI20A1), a Grant-in-Aid for Scienti\fc Research (No.\n21H04437, No. 21H04990, and No. 19H05825), and JST-\nPRESTO (JPMJPR20L7).\nAppendix: Product of the local density of states at\nthe interfaces\nIn the main text, we have discussed the similarities\nbetween the transmissions and the product of the local\ndensity of states (LDOS) at the center of the barrier re-\ngion. Here we discuss the J-dependence of the product\nof the LDOS at the interface [49, 50]. Similarly to rc\nde\fned in Eq. (9), we de\fne the ratio for the interfacial9\n(c) (d)(a) (b)Configuration-1 Configuration-2\n0 1 2024\n0 1 2024\nParallel\nAntiparallel-4-2024\n0 2 1-4-2024\n0 2 11\n0\n-1\nFIG. 10. Results of the calculation for ferrimagnetic tun-\nnel junctions with ( sA; sB) = (1 :0;\u00000:5) (Sec. IV B 1). See\nalso Fig. 6 as a comparison. (a), (b) Ratio ri(Eq. (A.1))\non the plane of JandEfor (a) con\fguration-1 and (b) 2.\n(c), (d) Product of the interfacial local density of states gi(E)\n(Eq. (7)) as a function of JatE= 3:6 for (c) con\fguration-1\nand (d) 2.\n(a) (b)\n01\n0 1 2Parallel\nAntiparallel\n-4-2024\n0 2 1-101\nFIG. 11. Results of the simulation for antiferromagnetic tun-\nnel junctions with ( sA; sB) = (1 :0;\u00001:0) (Sec. IV B 2). See\nalso Fig. 7 as a comparison. (a) Ratio ri(Eq. (A.1)) on the\nplane of JandE. (b) Product of the interfacial local density\nof states gi(E) (Eq. (7)) at E= 1 as a function of J.LDOS,ri, as\nri=gi;P(E)\u0000gi;AP(E)\ngi;P(E) +gi;AP(E): (A.1)\nHeregi;P/AP isgi(E) for the parallel/antiparallel con\fg-\nuration. We plot riin Fig. 9(a). We \fnd that rialso\ndescribes the transmission in the large- or small- Ere-\ngions where rt'1 owing to the absence of the bulk DOS\nof either of the spin-states. However, ridoes not repro-\nduce the intermediate- Eregion. Actually, as shown in\nFig. 9(b),gi(E) changes similarly to the transmission at\nE=\u00002 (see Fig. 3(e)), whereas the J-dependence of\ngi(E) shown in Fig. 9(c) largely deviates from that of the\ntransmission at E= 0 (Fig. 3(f)).\nIn Figs. 10(a) and 10(b), we present rifor the ferrimag-\nnetic MTJ with ( sA;sB) = (1:0;\u00000:5) for con\fguration-1\nand 2, respectively. Figures 10(c) and 10(d) are the J-\ndependence of gi(E) atE= 3:6. 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Rev.\nB74, 045429 (2006)." }, { "title": "1903.04293v1.High_Field_Anomalies_of_Equilibrium_and_Ultrafast_Magnetism_in_Rare_Earth_Transition_Metal_Ferrimagnets.pdf", "content": "1 \n High Field Anomalies of Equilibrium and Ultrafast Magnetism in Rare -Earth -Transition \nMetal Ferrimagnet s \nA. Pogrebna , 1 K. Prabhakara ,1 M. Davydova,2 J. Becker ,1,3 A. Tsukamoto,4 Th. Rasing ,1A. Kirilyuk1 \nA. K. Zvezdin,2 P. C. M. Christianen,3 and A.V. Kimel1 \n1 Radboud University, Institute of Molecules and Molecules, Nijmegen, The Netherlands \n2Prokhorov General Physics Institute of the Russian Acad emy of Sciences, Moscow, Russia \n3High Field Magnet Laboratory (HFML –EMFL) , Radboud University, Nijmegen, The Netherlands \n4College of Science and Technology, Nihon University, Chiba, Japan \nABSTRACT : Magneto -optical spectroscopy in fields up to 30 Tesla reveals anomalies in the equilibrium and ultrafast mag-\nnetic properties of the ferrimagnetic rare -earth -transition metal alloy TbFeCo. In particular, in the vicinity of the magneti-\nzation compensation temperature , each of the magnetizations of the antiferromagnetically coupled Tb and Fe Co sublattices \nshow triple hysteresis loops. Cont rary to state-of-the-art theory , which explains such loops by sample inhomogeneit ies, \nhere we show that the y are an intrinsic property of the rare -earth ferrimagnet s. Assuming that the rare-earth ions are par-\namagnet ic and have a non -zero orbital momentum in the ground state and, therefore, a large magnetic anisotropy , we are \nable to reproduce the experimentally observed behavior in equilibrium. The same theory is also able to describe the exper-\nimentally observed critical slowdown of the spin dynamics in the vicinity of the magnetization compensation tempera ture, \nemphasizing the role played by the orbital momentum in static and ultrafast magnetism of ferrimagnets. \nIntroduction \nThe need for ever faster and energy -efficient data storage and \ninformation processing is a strong motivation to search for un-\nconventional ways to control magnetism by means other than \nmagnetic field s [1–4]. Several successful realizations of mag-\nnetiz ation control with the help of an electric current [5,6], \nelectric field [2] and light [3] have been demonstrated . This \nhas become heavily debated topic in modern magnetism and \nrevealed a lack of understanding of the mechanisms that are \nresponsible for these phenomena [3,7]. It is clear, however, \nthat in all these cases the spin -orbit and the exchange spin -\nspin interactions play a decisive role. For instance, spin -orbit -\ntorques [8–12], multiferro icity [13,14] , opto - and photomag-\nnetic [3] phenomena cannot be understood wi thout taking \ninto account the spin-orbit interaction as well as orbital mo-\nmenta of the ground and excited states. The exchange inter-\naction , in turn, can be efficiently harnessed for spin manipu-\nlation in multi -sublattice spin systems or multilayers [15–20]. \nFerrimagnetic rare -earth intermetalli cs, and rare -earth -transi-\ntion metal (RE- TM) alloys in particular, are among the most \nstudied systems in fundamental and applied magnetism. For \nexample, unique functionalities of GdFeCo, GdFe, and GdCo \nalloys have been demonstrated in spintronics [21,22] and ul-\ntrafast magnetism [23–25]. The interplay between the ex-\nchange and the spin -orbit interaction in rare-earth ferrimag-\nnets facilitate electric field, current and optical control of \nspins . In particular, due to the antiferromagne tic coupling be-\ntween the non-equivalent magnetic sublattices in GdFeCo , it \nis possible to reverse its magnetization solely with a single \nfemtosecond laser pulse [26]. Anomalous hysteresis loop s and \ncritical slow ing down of laser -induced spin dynamics in high \nmagnetic field s were reported for GdFeCo [27], but a theory \nof such behavior has not been developed until now. Interest-\ningly , as the Gd ion in the alloy is in an S-ground state , its s ub-\nstitution with other rare -earth ions having non -zero orbital \nmomentum in the ground state must enhance the spin -orbit \ninteraction and thus open up fundamentally new opportuni-\nties in the field of spintronics and ultrafast magnetism. TbFeCo is an example of such a material, that, due to large \ncoercive fields above 10 T , is well suitable for high density \nmagnetic recording [28,2 9]. Although several attempts of \nmodeling the laser -induced spin dynamics in TbFeCo have \nbeen performed [30–32], not only spin dynamics, but even the \nstatic spin structure of unperturbed TbFeCo are poorly under-\nstood. Moreover, experimental studies of high -coercive -field \nmaterials are seriously hampered by the need for even higher \nmagnetic fields and thus require unique experimental instal-\nlations. \nHere we report the experimental observation of anomalous \nhysteresis loops of the magnetization s of the antiferromag-\nnetically coupled FeCo and Tb sublattices of ferrimagnetic \nTbFeCo in high magnetic fields. The loops appear to be very \nsensitive to temperature near the magnetization compensa-\ntion point, where the magnetizations of the two sublattices \ncancel each other . We show that the observed behavior can be \nexplained in the framework of an f-d ferrimagnet by taking \ninto account the orbital momentum and, as a result , the large \nmagnetic anisotropy of the rare -earth ions. In order to bring \nthe developed theory into an ultimate test , we experimentally \ninvestigate the magnetization dynamics in TbFeCo trigg ered \nby a femtosecond laser pulse in high magnetic fields and com-\npare the outcome of the experiment with the mode ling. It is \nsurprising that the theory developed to explain equilibrium \nmagnetic properties i s also able to predict the experimentally \nobserved critical slow ing down of the spin dynamics that was \nobserved in the experiment . \nThe paper is organized as follows . Section I describes static \nmagneto -optical measurements in which anomalous hystere-\nsis loops were observed. Next, in section II we introduce the \ntheoretical model aimed to describe the ground state of a fer-\nrimagnet with large rare -earth anisotropy. We plot magnetic \nfield -temperature ( H-T) phase diagram and expla in the equi-\nlibrium magnetism of TbFeCo using the developed theory . In \nsection III we present the experimental results on laser -in-\nduced ultrafast dynamics of TbFeCo in high magnetic field , 2 \n and theoretically explain the observed anomalies using the de-\nrived p hase diagram. We conclude the manuscript with a \nsummary, which emphasizes the simplicity, and at the same \ntime predictive power of the proposed theor y. Finally, we \nhighlight experimentally observed features whose explana-\ntion, being beyond the capabilities of the presented model, is \nthe next challenge in the physics of rare -earth alloys. \n \nI. Magneto -optical spectroscopy in high \nmagnetic fields \nThe s tudied material was an amorphous rare-earth -transi-\ntion metal alloy with stoichiometric composition \nTb22Fe68.2Co9.8. Without applied magnetic field, the antifer-\nromagnetic exchange coupling between the rare -earth and \nthe transition metal magnetic sublattices favor a collinear \nantiparallel alignment. The magnetizations of the transi-\ntion met al and the rare -earth sublat tices have different \ntemperature dependences so that at the magnetization \ncompensation temperature TM the net magnetization is \nzero, if no magnetic field is applied. In the studied sample \nTM=322 K. Below this temperature, T < T M, the net magnet-\nization is dominated by the RE sublattice. Above the com-\npensation point , the situation is the opposite , and the TM \nsublattice dominates the net magnetization . The strong in-\nter-sublattice 3d-4f exchange interaction between the Tb \nand FeCo magnetic moments defines t he Curie tempera-\nture, TC, which is around 700 K [33]. The studied sample is \na thin film structure with composition glass /SiN(5 nm)/RE -\nTM(20 nm)/SiN(60 nm). The TbFeCo magnetic layer has an \neasy-axis type of magnetic anis otropy, oriented perpendic-\nularly to the sample plane. \n \nFig. 1 (color online) Schematic of the experiment. The sam-\nple was insert ed into 37 T dissipative magnet . Dashed lines \n– pump and probe beams. \nThe e xperiments were performed at the High Field Magnet \nLaboratory (HFML) in Nijmegen . Magnetic field s up to 30 \nT were applied along the normal to the sample (see Fig. 1) . \nTo benefit from elemental specificity of the magneto -opti-\ncal phenomena in TbFeCo in the visible spectral \nrange [34], we performed the measurements of the polar \nmagneto -optical Kerr effect (MOKE) with light of two pho-\nton energies . In particular, for photon energy =1.55 eV \nthe effect is expected to probe the normal component of the magnetization of the FeCo sublattice, while light with \n=2.41 eV is mainly sensitive to the normal magnetization \ncomponent of Tb. \nFigure 2 shows the MOKE as a function of the applied mag-\nnetic field at different temperatures near the magnetiza-\ntion compensation temperature. Above TM = 322 K t he ob-\nserved behavior is anomalous as the loops have a triple \nshape and are very sensitive to the sample temperature . At \ntempe ratures below TM = 322 K, the anomalies are far less \npronounced, but sti ll visible ( see arrows in Fig. 2 ) \n \nFig. 2 (color online) Static MOKE data of t he TbFeCo sam-\nple measured at 2.41 eV (a) and 1.55 eV (b) photon energies \nat different temperatures from 267.0 K to 342. 5 K. A para-\nmagnetic background was subtracted from the measure-\nments. The magnetization compensation temperature TM is \naround 3 22 K. Black arrows correspond to the hysteresis \nedges around the first-order field -induced phase transitions \ndiscussed in section II , while a second -order phase transition \nis shown with a blue arrow. \nSimilar hysteresis loops were observed in rare -earth ferri-\nmagnetic alloys and intermetallics earlier and explained by \ninhomogeneities with the strongest ever reported ex-\nchange bias fields [35,36] . In particular, an application of \nthat theory to our case would imply that the exchange bias \nbetween the inhomogeneities reaches 10 T. However, it is \nalso known that hysteresis loop dependencies of the mag-\nnetization on temperature or field is a signature of a first \norder phase transition. Gradual changes of the magnetiza-\ntion upon a change of temperature or field are generally \nexplained as second -order phase transition s. Despite in-\ntense interest in rare-earth - transition metal alloys, their \nequilibrium prop erties in high magnetic field s and H-T \nphase diagram s near the compensation temperature, in \nparticular, are still unexplored. H-T phase diagram of uni-\naxial ferrimagnets has been studied theoretically ear-\nlier [37], under the assumption that only the TM-sublattice \nis anisotropic. In systems with different symmetry [38,39] \nit was found tha t the behavior of the phase diagram is \ngreatly affected by magnetic anisotropies of both sublat-\ntices. \n3 \n In order to reveal the origin of the observed anomalous \nhysteresis loops, we develop ed a theory of magnetism of \nTbFeCo in thermodynamic equilibrium and calculate d the \nH-T phase diagrams of this compound . \nMagnetic fields at which magnetization changes were sub-\ntracted from the static MOKE data (see Fig. 2) in a wide \nrange of temperatures are shown in phase diagram in Fig. \n3(a). The blue line data poin ts correspond to a second or-\nder phase transition, where the magnetization changes \ngradually. Black diamonds and associated black d ashed \nlines correspond to the edges of the experimental hystere-\nsis, where the magnetization changes abruptly. \nII. H-T phase diagram and anomalous hyste-\nresis loops \nSimilarly to Refs [37,40] , the calculations are based on \nanalysis of the free energy for a two -sublattice f-d (RE-TM) \nferrimagnet assuming that rare-earth ion is paramagnetic . \nWe take advantage of the fact that t he single -domain \nmodel work s especially well in the vicinity of the compen-\nsation point. This is because the domain size diverges at \nthe compensation point [41] as the magnetostatic energy \ndrops to zero when the magnetization \nfd M M M \nvanishes at T M. The free energy has the form: \n0- ( , ) ( , )effH\nd f a d f W h T d W M H M h M M\n, (1) \nwhere \neff dH H M is the effective magnetic field act-\ning upon the rare -earth ion, \n22\n22f d\na d f\ndfW K KMM Mn Mn\n is the uniaxial anisotropy \nenergy, with\n,dfKK being d- and f-sublattice anisotropy \nconstants , respectively , and \nn is an easy axis unit vector. \nThe magnetization function for rare -earth ions was taken \nin the following form: \n35\n1 2 3 ( , ) ( ) ( ) ( )f eff eff eff effM H T T H T H T H \n, (2) \nand it is directed along the effective field. Whereas for Gd \nthe f-sublattice magnetization is accurately described by \nthe Brillouin function, it doesn’t hold well for other rare -\nearth ions with non -zero orbital momentum in the ground \nstate [42], which is further compl icated by the amorphous \nnature of the alloy. Therefore, w e fit the sus ceptibilities \nnumerically, s tarting from the functions obtained by an \nexpansion of the Langevin function \n()eff\nTb eff\nBHLkT in series \nand numerically adapt ing them and the effective magnetic \nparameters to reproduce the experimentally observed \nfeatures. We present the calculations performed for the \nfollowing parameters: \nex dHM\n 80 T, TM = 322 K, TC = \n700 K [33]. Magnetic moments of the Tb and FeCo \nsublattices used in the calculations are \n10eff\nTb B and \n1.8eff\nFeCo B\n, respectively [31]. As it was shown earlier [38], the rare -earth anisotropy is one of the defining \nfactors for the character of the phase diagram near the \ncompensation point . In o ur model we assume that \n63 10fK\n erg/cc and \ndK = 0 (we assume the d-sublattice \nanisotropy to be much smaller than that of the rare-earth \nsublattice, as Fe ions are in the S-ground state). \n \nFig. 3 (color online) (a) Experimental and (b) theoretical \nmagnetic field - temperature phase diagram for TbFeCo . The \nlines in the experimental phase diagram are guides for the \neyes. The black points (dashed black curves) correspond to \nthe points stability loss, the red curve (TMP) is the first -order \ntransition line , and the blue points (blue curve BR) corre-\nspond to the second -order phase transition. (c) Schematics \nof the three phases AF 1, AF 2 and NC that are present in the \ndiagram . The states of the magnetization of both sublattices \nare shown with arrows, where the red arrow correspond s to \nthe FeCo magneti zation , and the green arrow correspond s \nto the Tb magnetization . \nUsing expression (1) for the thermodynamic potential we \nnumerically calculate the magnetic (H-T) phase diagram. \nThe ground states of the system are found by minimization \nof the thermodynamic potential w ith regard to the order \nparameter \nd , which denotes the polar angle of the FeCo \nmagnetization in the coordinate system with the z-axis \naligned along the external magnetic field. At the local \nminima one finds \n0\nd and \n2\n20\nd . The lines of \nstability loss, where \n2\n20\nd , are found for each phase. At \n4 \n the lines of the first -order phase transition two phases, \ncorresponding to solutions \n(1)\nd and \n(2)\nd , correspond to the \nlocal minima of the thermodynamic potential and the \ncondition \n(1) (2)\ndd is fulfilled. \nThe phase diagram in Fig. 3(a,b) shows three phases: two \nantiferromagnetic collinear phases AF 1 and AF 2 (rare -earth \nmagnetization is along the magnetic field below the com-\npensation temperature TM in phase AF 1, and opposite to it \nabove TM in phase AF 2) and one canted phase NC. The blue \nline BR represents the second -order phase transition be-\ntween phases AF 1 and NC and defines the spin -flop field de-\nnoted as HBR below the compensation point. Above the \ncompensation temperature the s pin-flop occurs discontin-\nuously, via a first-order phase transition ( see line RP in Fig. \n3 (a) ). The red line (TMP) corresponds to a first-order phase \ntransition between the collinear phases AF 1 and AF 2 along \nsegment TMR as well as between the phases AF 2 and NC \nalong the rest of the line , i.e. along segment RP. Each of the \ndashed curves corresponds to the stability loss of one of the \nphases. Following the markup in Fig. 3 (a), lines AA’, QQ’ \nand RB’ (we denote the corresponding fields HAA’, HQQ’ and \nHRB’, respectively ) are the lines of stability loss of phases \nAF 2, NC and AF 1, respectively. One might notice that the \nfirst-order phase transition TMRP goes to the outside of the \narea shown in the phase diagram. The point P is a tricritical \npoint at which the order of the phase transition between \ncollinear and noncollinear phases changes from first to sec-\nond. A number of unusual phenomena are expected to oc-\ncur in ferrimagnets near this point [43,44] , being an inter-\nesting subject for future studies. \nThe features of the magnetic phase diagram can be ob-\nserved experimentally by mea suring the dependence of the \nmagnetization on external magnetic field . In particular, de-\nducing magnetic fields corresponding to jumps in the ex-\nperimentally measured hysteresis loops, we define the \nfields and the temperatures at which collinear and non-col-\nlinear phases loose stability. In this way we plotted the ex-\nperimental phase diagram shown in Fig. 3(a). The structure \nof the theoretical phase diagram (see Fig. 3(b)) explains \nwell the behavior of the experimentally observed phase \ntransitions. To demonstrate the agreement of the theoret-\nical predictions with the experimental results , we also cal-\nculate the dependence of all possible equilibrium values of \nthe order parameter as a function of external magnetic \nfield \n()()i\ndH at fixed temperature . The obtained \nmagnetization plots explain the experimental results \nshown in Fig. 2. Above the compensation temperature, as \nillustrated in Fig. 4 (a), the theory reproduces the triple \nhysteresis loops observed in the experiment. The large cen-\ntral loop at lower fields encompasses the first -order phase \ntransition between the two collinear phases. Two loops \nthat appear at higher positive and negative fields are due \nto the two first -order phase transitions between collinear \nand non -collinear phases (from AF 2 to NC and vice versa , \nrespectively ). From Fig. 3 (a) one can see that the first -or-\nder transition, from which the additional loop at positive \nfield originate s, occurs at the line RP. The hysteresis \naround these first -order phase tra nsitions is defined by the position of the stability loss lines in the ( H-T) phase dia-\ngram and corresponds to the dashed lines in Fig. 3 (a). The \nloops disappear if the phase transition to the non -collinear \nphase NC becomes of second -order. At 290 K, i.e. below the \ncompensation point, we see that the second -order phase \ntransition occurs below the coercive field (see Fig. 4(b)). \n \nFig. 4 (color online) Calculated Tb magnetization curves (a) \nbelow and (b) above compensation temperature , at T = 290 \nK and T = 32 4 K, respectively . \nTherefore, our relatively simple theory is able to qualita-\ntively explain the observed anomalous hysteresis loops \nwithout involving inhomogeneities and huge exchange \nbias fields. The single -domain picture holds well near the \ncompensation point, where the domain size in the magnet \ndiverges. The triple hystere sis loops can be explained as a \nseries of first -order phase transitions. The quantitative dif-\nference between the theory and the experiment can be im-\nproved by taking into account such features of realistic \namorphous alloys as random single ion anisotropy , result-\ning in sperimagnetism [45]. \nTo test our theory further, it is interesting to check if the \ntheory can also explain anomalies in ultrafast magnetism \nof rare -earth -transition metal ferrimagnets . Some of these \nanomalies were seen in earlier experimental studies of ul-\ntrafast laser -induced spins dynamics in GdFeCo in the vi-\ncinity of the spin-flop transition [29], but n either theory \nnor simulations of the dicovered high field dynamics have \nbeen reported up to now. \nIII. Ultrafast m agnetism and critical slow-\ndown in high magnetic fields \nTo investigate the dependence of ultrafast spin dynamics \non bias temperature and high magnetic field , we per-\nformed time -resolved measurements of the polar mag-\nneto -optic al Kerr effect (tr -MOKE) using a pum p-probe \ntechnique , with 50 fs optical pulses generated by a 1-kHz \nTi:Al2O3 regenerative amplifier seeded with a Ti:Al2O3 os-\ncillator . The central photon energy of the pulses could be \ntuned with the help of an Optical Parametric Amplifier. Re-\nlying on the conventionally accepted approximation that \nthe main effect of optical pump pulse on a metallic magnet \nis ultrafast heating and r elying on conclusions of earlier \n5 \n studies [33] , we assumed that the laser -induced sp in dy-\nnamics is independent of the photon energy of the pump . \nTuning the photon energy of probe one can be sensitive to \nFeCo and Tb sublattices [34], respectively. Therefore, we \nperformed two types of pump -probe experiments. In order \nto monitor the laser -induced spin dynamics of the Tb-sub-\nlattice, the pump and probe photon energies were chosen \nat 1.55 eV and 2.48 eV , respectively . In order to monitor the \ndynamics of the FeCo -sublattice, the pump and probe pho-\nton energies were altered. The pump beam was focus ed on \nthe sample into a spot around 90 µm in diameter and the \ndiameter of the probe beam was smaller - around 30 µm. \nThe fl uence of the pump p ulses was 0.15 mJ/cm2, while the \nprobe fluence was kept around 1.5 µJ/cm2. All time -re-\nsolved measurements were performed at magnetic fields \noutside the hysteresis loops. \n \nFig. 5 (color online) Transient magneto -optical Kerr effect \nmeasured from TbFeCo at different magnetic fields. Traces \nmeasured with probe photon energy at 1.55 eV (FeCo sublat-\ntice) are shown by open orange squares . Experiments per-\nformed at photon energy 2.48 eV (Tb sublattice) are shown \nby filled green circles. In the left panel shown traces meas-\nured below the compensation point, T = 160 K (1.55 eV) and \nT = 220 K (2.48 eV). In the right panel shown traces meas-\nured above the compensation point , T = 350 K. The lines are \ncorresponding fits with functions , discussed in the text. \nThe results of tr -MOKE measurements on TbFeCo at vari-\nous magnetic fields and at diff erent probe photon energies \nbelow and above the compensation point are shown in left \nand right panels of Fig. 5, respectively. To analyze the magnetization dynamics, we will distin-\nguish two time -domains: (i) sub -10 ps longitudinal dynam-\nics i.e. demagnetiz ation of the magnetic sublattices and (ii) \nsub-100 ps transversal dynamics of the magnetic sublat-\ntices, i.e. tilt of the magnetization. The data shown in Fig. \n5 were fitted with a double -exponential function \n0 0 1exp( / ) exp( / )rise A t A t \n, where \n0 and \nrise are \nthe characteristic times of the ultrafast longitudinal and \ntransversal dynamics, respectively. A0 and A1 are the ampli-\ntudes. Assuming that the ultrafast demagnetization of both \nTb and FeCo sublattices is completed within a few ps [33], \nin the fit we set \n0 = 1.5 ps , while A0, A1,\nrise were taken as \nfitting parameters (see Supplementary II for details ). \nIn the collinear phase (Fig. 5 (e)) the dynamics is very fast \nand associated with the longitudinal demagnetization (i), \nwhich is in good agreement with previous re-\nports [27,33,46,47] . \nFigure 6 show s the field dependencies of the rise time \nrise\n, as deduced from the fit, below and above the compensa-\ntion temperature in the noncollinear phase . It is clearly \nseen that the dynamics slows down close to the spin -flop \ntransition . It is remarkable that below the compensation \ntemperature , decreasing the external magnetic field from \nHAA’+5 T to HAA’, which is close to the spin -flop field HBR at \nthat temperature , leads to a 400% increase of the rise time. \nA similar decrease of the field from HAA’+5 T to HAA’ above \nthe compensation temperature result s in a rather moder-\nate increase of \nrise by 25%. Finally, we find that above the \ncompensation temperature the magnetization of Tb and Fe \nhave clearly different dynamics with a slower response of \nthe Tb spins. \n \n \nFig. 6 (color online) (a, b) The rise time of the tr-MOKE sig-\nnal (see Fig. 5) below and above the compensation point, re-\nspectively . Orange circle s (open) and curves show charac-\nteristic times constants which correspond to the FeCo mag-\nnetic sublattice, while green circle s (filled) and curves corre-\nspond to the Tb magnetic sublattice. Grey dashed lines cor-\nrespond to the hysteresis edge around the magnetic field -in-\nduced first order phase transitions. \nTo assign the observed features of the presence and ab-\nsence of the critical slowing down to the peculiarities of the \nphase diagram, we simu late the ultrafast laser -induced \nmagnetization dynamics . We start with the corresponding \n6 \n magnetic structure in thermodynamic equilibrium and as-\nsume that a femtosecond laser pulse demagnetizes both \nsublattice s by 10% (see Suppl ementary II). The subsequent \ntransversal magnetization dynamics was modeled with the \nhelp of the Landau -Lifshitz -Gilbert ( LLG) equation . We \nshow that the observed dynamics in the canted phase can \nbe explained in the framework of coherent magnetization \nprecession of thermalized sublatt ices brought out of equi-\nlibrium by ultrafast demagnetization. After the demagnet-\nization, the spins of the sublattices will relax towards the \nequilibrium orientation via a heavily damped precession. \nIn the framework of the LLG equation , one can derive the \nout-of-equilibrium position of the magnetization right af-\nter the demagnetization [48]. We divide the magnetiza-\ntion dynamics into three time -domains, as earlier: after the \ninitial longitudinal demagnetization (i), the coherent rota-\ntion of spins further away from the initial equilibrium ori-\nentation occurs (ii). We f ind that a change in magnetiza-\ntion of any of the sublattice s of the order of one percent is \nalready enough to trigger the magnetization dynamics \nsimilar to that observed in the experiment. Using the \nframework described above w e derived analytical expres-\nsions for the rise time, corresponding to the strongly \ndamped dynamical regime as observed in the experiment: \n \n \n 2 2\n2 2arctanh / 2 / / 2\n/2ex r ex\npr\nrise\nex r \n\n \n , (3) \nwhere \n is the effective Gilbert damping constant for the \nferrimagnet, the exchange frequency \nex ex H , and the \nresonance frequency is defined as \n2 2 2\n002\ncos cos 2fd\nrKK mHH \n, where \ndf m M M\n and \n is the gyromagnetic ration of \nelectron . The angle \n0 is the angle between the external \nmagnetic field and the antiferromagnetic vector \ndfL M M\n and \nis the component of the magnetic \nsusceptibility perpendicular to the external magnetic field. \nWe assume the effective Gilbert damping constant, which \nis a function of the composition and temperature [49], to \nbe equal to 0.2 . An example of the calculated dynamics can \nbe found in Fig. S4 (see Supplementary). \nThe phase diagram predicts th at below the compensation \npoint the transition to the angular phase upon an increase \nof the external magnetic field occurs via a second -order \nphase transition (Fig. 3) . At the phase transition the fre-\nquency of the ferromagnetic resonance softens ,\n0r , \nand the dynamics of the order parameter slows critically \n(diverges to infinity) down as predicted by equation (3) \nand calculations shown in Fig. 7 (a). Similar slowing down \nis seen experimentally ( Fig. 6 (a)). \nFigure 7 (b) summarize s the calculated field dependenc y of \nthe response time above the compensation temperature, \nwhere the phase transition between collinear and angular phases is of first order. The slowing down at the phase tran-\nsitions is not critical , as it would be expected at a first-order \ntransition from the general theory of phase transi-\ntions [50]. Therefore , the experimental results reported in \nFig. 7 (b) agree well with the theoretically predicted behav-\nior based on the magnetic phase diagram. Note that exper-\nimental verification of the theoretically predicted features \nof first -order phase transitions is often obscured by such \nfactors as sample inhomogeneities . \n \nFig. 7 (color online) Dependence of the calculated rise time \non the external magnetic field . The calculations were per-\nformed below (a) and above (b) the co mpensation tempera-\nture at T 1 = 290 K and T2 = 324 K , respectively . The charac-\nteristic fields at the given temperatures are HBR (T1) = 7 T and \nHQQ’ (T2) = 8.7 T . \nFinally, our model could not reproduce the dramatic dif-\nference between the time scales for the sublattices of Tb \nand FeCo observed experimentally (Fig. 7 (b)) . We must \nnote that a difference in dynamics of the two sublattices at \nthe timescales of the order of 60 ps has been reported ear-\nlier [34], but despite several efforts of computation al stud-\nies [19,51] , the origin of such a behavior is still not under-\nstood . Different excitation times for TM and RE magnetic \nsublattices were previously observed in time -resolved X -\nray studies [15]. The mechanism was explained by a larger \nmagnetic moment per rare -earth ion in comparison to the \nmagnetic moment of transition metal ion s. However, re-\nported experiments were done in the collinear phase where \nthe rise time is on a 1 ps timescale, and the difference was \nobserved only on timescales where the electron -phonon \nsystem is still not thermalized. Distinct spin dy namics on \na time -scale 10 -100 ps must have a different origin. We sug-\ngest that a possible explanation of such a behavior can be \nsperimagneti sm reported for realistic TbFeCo alloys [45]. \nAs a matter of fact, m odeling spin dynamics of sperimag-\nnets is one of the challenges in modern computational \nmagnetism. \nConclusions \nWe performed experimental and theoretical studies of \nanomalous hysteresis loops of the magnetizations of the \nantiferromagnetically coupled FeCo and Tb sublattices \nof ferrimagnetic TbFeCo in high magnetic fields. Unlike \nprevious theories which ex plained such loops by ex-\nchange bias between the surface and bulk layers within \none fi lm, here we showed that such a loop can be an in-\ntrinsic feature a f-d ferrimagnet. By taking into account \n7 \n the orbital momentum that results in a large magnetic \nanisotropy of the rare -earth ions, we computationally \nexplored and defined the phase diagram of TbFeCo in \nH-T coordinates. In order to bring the developed theory \ninto an ultimate test, we experimentally investigate d \nthe magnetization dynamics in TbFeCo triggered by a \nfemtosecond laser pulse and compare d the outcome of \nthe experiment s with the modeling. It is surprising that \nthe theory developed to explain equilibrium magnetic \nproperties is also able to predict the experimentally ob-\nserved dynamics , including critical slow ing down of the \norder parameter in the vicinity of the magnetic c ompen-\nsation temperature. Finally , we note that above the \ncompensation temperature , we experimentally ob-\nserved clearly different dynamics of the magnetization \nof Tb and Fe sublattices. 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Rev. B \n87, 224417 (2013). \n " }, { "title": "2101.07300v1.Quantum_fluctuation_effects_on_the_ordered_Moments_in_a_two_dimensional_frustrated_ferrimagnet.pdf", "content": "arXiv:2101.07300v1 [cond-mat.str-el] 18 Jan 2021Quantum Fluctuation effects on the Ordered Moments in a two di mensional\nfrustrated Ferrimagnet\nKingshuk Majumdar∗\nDepartment of Physics, Grand Valley State University, Alle ndale, Michigan 49401, USA\nSubhendra D. Mahanti†\nDepartment of Physics and Astronomy, Michigan State Univer sity, East Lansing, Michigan 48824, USA\n(Dated: January 20, 2021)\nWe propose a novel two-dimensional frustrated quantum spin -1/2 anisotropic Heisenberg model\nwith alternating ferromagnetic and antiferromagnetic mag netic chains along one direction and anti-\nferromagnetic interactions along the other. The (mean-fiel d) ground state is ferrimagnetic in certain\nrange of the interaction space. Spin-wave theory analysis o f the reduction of ordered moments at in-\nequivalent spin sites and the instability of the spin waves s uggest a quantum phase transition which\nhas the characteristics of both the frustrated two-dimensi onal antiferromagnetic S=1/2 ( J1,J2)\nmodel and 1D S 1=1, S 2=1/2 quantum ferrimagnetic model.\nPACS numbers: 71.15.Mb, 75.10.Jm, 75.25.-j, 75.30.Et, 75. 40.Mg, 75.50.Ee, 73.43.Nq\nI. INTRODUCTION\nLow-dimensional quantum spin systems are excellent\nexamples to explore the physics of strongly interacting\nquantum many-body systems.1In addition to the inher-\nent quantum nature of the interacting elements (for ex-\nample localized spins with S=1/2), these systems pro-\nvide an array of choices where the effects of competing\ninteractions, non-equivalent nearest neighbor bonds, and\nfrustration on quantum fluctuations of the long range\norder parameter and on quantum phase transitions at\nT= 0K (no thermal fluctuations) can be explored. Al-\nthough extensive studies using different theoretical ap-\nproaches and using different spin models have been done\nover the last severaldecades we will first discuss two sim-\nple models relevant to our present work.2,3They are (i)\ntwo-dimensional (2D) antiferromagnetic S=1/2 Heisen-\nberg model on a square lattice with nearest (NN) and\nnext nearest neighbor (NNN) antiferromagnetic interac-\ntions(J1,J2)[ModelI]and(ii)one-dimensional(1D)spin\nchainconsistingofalternatingS 1=1andS 2=1/2spinsin-\nteracting antiferromagnetically [Model II]. The classical\ngroundstate (GS) ofmodel I in certain ( J1,J2) domain is\nlong range ordered (LRO) antiferromagnet whereas that\nofmodelIIisalongrangeorderedferrimagnet. Quantum\nspin fluctuations (QSF) dramatically affect the physical\nproperties of these systems, which we review briefly after\nfirst introducing a new model [Model III] below.\nWe propose a novel 2D Heisenberg model at T= 0K\nconsistingofonlyS=1/2spins,whichcombinestheessen-\ntial features of the two above models, extreme anisotropy\nof the NN bonds (some ferro and some antiferro) and\nfrustration. The classical ground state (discussed in de-\ntail later in the paper) is a four-sublattice ferrimagnet in\ncertain parameter space. Our focus in this paper is on\nthe stability of this ferrimagnetic ground state and effect\nof QSFs at T= 0K on the long range ordered sublattice\nmagnetizations.II. REVIEW OF MODELS I AND II\nA. Model I\nThe classical ground state of the 2D S=1/2 ( J1,J2)\nHeisenberg model on a square lattice depends on the\nfrustration parameter η=J2/J1.1,2Forη <0.5, the GS\nis a Ne´ el state with ordering wave vector ( π,π), similar\nto the unfrustrated case whereas for η >0.5 the GS is\nthe degenerate columnar antiferromagnetic state (CAF)\nwith ordering wave vectors ( π,0) and (0 ,π). There is a\nfirst-order phase transition from the Ne´ el state to CAF\nstate at η= 0.5. Effects of QSF on this phase transi-\ntion and other properties of this model have been inves-\ntigated using a large number of methods.4–13Here we\nreview the main results obtained within linear spin wave\ntheory (LSWT). Sublattice magnetization, mis reduced\nby QSF from its classical value of 0.5 to 0.303 at η= 0\nand then decreases monotonically with increasing ηand\napproaches zero at the first critical point ηc1= 0.38.\nSimilarly m= 0.37 atη= 1 and then steadily decreases\nto zero at the second critical point ηc2= 0.49. LSWT\nclearly indicates that QSF effects are enhanced in the\npresence of frustration. Also it suggests that in the re-\ngionηc1< η < η c2, the classical GSs are not stable. The\nnature of the GS (e.g. spin-liquid, valence bond) and low\nenergy excitations in this region have been extensively\nstudied during past several years and continue to be of\ngreat current interest.\nB. Model II\nThe second model deals with ferrimagnets. Ferrimag-\nnets are somewhere between ferromagnets and antifer-\nromagnets.14–25It is well known that for 1D quantum\nS=1/2 ferromagnet, the ground state is long-range or-\nderedandQSFsdonotreducethe classicalvalue of m. In2\ncontrast, in a 1D quantum S=1/2 antiferromagnet (AF),\nQSFscompletelydestroythe classicalLRO.Thequestion\nwhat happens forferrimagnetsdrew considerableinterest\nin the late 90’s and several interesting works were done\nusingasimpleisotropicNNantiferromagneticHeisenberg\nmodel with two types of spins, S 1= 1 and S 2= 1/2 in a\nmagnetic unit cell (MUC).17,18,20–26Following Refs. [17]\nand [21] we discuss some of the interesting physical prop-\nerties of this 1D system and point out how our proposed\n2D model differs from this.\nThe Hamiltonian of the 1D system is given by\nH=/summationdisplay\nn/bracketleftbig\nJ/parenleftbig\nS1n·S2n+S2n·S1n+1/parenrightbig\n−hSz\nTn/bracketrightbig\n,(1)\nwhereS1nandS2nare spin-1 and spin-1/2 operators\nrespectively in the nthunit cell (UC), effective field h(=\ngµBHwithggyro-magnetic ratio, µBBohr magneton,\nandHexternal magnetic field) and Sz\nTn=Sz\n1n+Sz\n2n.\nAccording to the Lieb-Mattis theorem27, forH= 0,\nthe GS is long range ordered as the system has total\nspinST=N/2, where Nis the number of UCs in GS,\n/angbracketleftSz\n1n/angbracketright= 1 and /angbracketleftSz\n2n/angbracketright= 0.5. The problem of looking at\nthe excitations is well suited for the LSWT approach.\nSince the elementary magnetic cell consists of two spins,\nLSWT predicts two types of magnons: a gapless “acous-\ntic” or “ferromagnetic” branch with Sz\nT=N/2−1, and\na gapped “optical” or “antiferromagnetic” branch with\nSz\nT=N/2+1. The optical magnon gap for this model\nhas been numerically found to be ∆ opt= 1.759J.23An\nintriguing property of this 1D quantum ferrimagnet is\nthat when one turns on the magnetic field H, the acous-\ntic branch opens up a gap but the optical gap decreases\nand at a critical value of the field Hc1this gap van-\nishes, the system then enters into a quantum spin liq-\nuid (QSL) phase, where the GS is dominated by QSFs\nwith spinon-like excitations.17,18,21With further increase\nin the strength ofthe field this QSL phasegoesinto asat-\nurated ferromagnetic phase.\nBrehmer et al. [17] calculated the sublattice magneti-\nzation for the S=1 sublattice ( mA) and found it to be (1-\nτ) withτ≈0.305. The sublattice magnetization of the\nS=1/2 sublattice can be calculated using their method\nand is found to be mB=−0.5+τ. The ordered moment\nof the S=1/2 sublattice is reduced by a factor of ∼2.5\ndue to QSF. There are two important points worth not-\ning here: (1) the total magnetization (ferromagnetic) per\nmagnetic unit cell is mA+mB= 0.5, the classical value\nand (2) QSF reduction of the S=1/2 sublattice is larger\nthan the 2D S=1/2 Heisenberg model for a square lat-\ntice where η∼0.2. Point (1) is consistent with the fact\nthat the ferromagneticlong rangeorder is not affected by\nQSF. Also mAandmBare independent of the magnetic\nfield.III. MODEL III\nAs mentioned in the beginning, we introduce a new 2D\nHeisenberg model which incorporates different aspects\nof the two models discussed above, anisotropic bonds\nand frustration. Also, instead of two types of spins and\nsingle exchange parameter, our model consists of only\nS=1/2 spins interacting with Heisenberg exchange cou-\nplings of different signs (both ferro and antiferro). The\nunit cell consists of four types of spins which we denote\nasS(µ)(µ= 1..4), it is a Bravais lattice. The lattice\nvectors for the four spins in a rectangular lattice with\nparameters ( a,b) along the xandydirections are given\nbyRiµ=ixaˆ x+iybˆ y+τµwhereτ1= (0,0),τ2=\n(0,b/2),τ3= (a/2,b/4) andτ4= (a/2,3b/4) (see Fig. 1).\nAs we will show, the ground state is ferrimagnetic in cer-\ntain range of exchange parameter space. Three spins\ncombine to form the S=3/2 sublattice. In contrast to\nthe 1D S=(3/2,1/2) model, where the magntitudes of\nthe spins in each sublattice are fixed, in our model, the\nS=3/2 sublattice can undergo amplitude fluctuations. In\nfact, the present model was inspired by recent inelastic\nneutron scattering experiments on a quasi 2D spin sys-\ntems containing Cu+\n2ions, Cu 2(OH)3Br.28However, in\nthis system the effect of orbital ordering of active mag-\nnetic orbitals driven by the ordering of the Br+ions on\nthe exchange parameters is such that the ground state is\nan antiferromagnet with eight spins per unit cell.\nThe Heisenberg spin Hamiltonian ( H) for model III is\ndivided into two parts, intra-chain ( H1) and inter-chain\n(H2):\nH=H1+H2, (2)\nwhere\nH1=−J1/summationdisplay\ni/bracketleftBig\nS(1)\ni·S(2)\ni+1\n2/parenleftBig\nS(1)\ni·S(2)\ni−bˆy+S(2)\ni·S(1)\ni+bˆy/parenrightBig/bracketrightBig\n+J2/summationdisplay\ni/bracketleftBig\nS(3)\ni·S(4)\ni+1\n2/parenleftBig\nS(3)\ni·S(4)\ni−bˆy+S(4)\ni·S(3)\ni+bˆy/parenrightBig/bracketrightBig\n,\n(3a)\nH2=1\n2J3/summationdisplay\ni/parenleftBig\nS(1)\ni+S(2)\ni/parenrightBig\n·/parenleftBig\nS(3)\ni+S(3)\ni−aˆx/parenrightBig\n+1\n2J4/summationdisplay\ni/bracketleftBig\nS(1)\ni·/parenleftBig\nS(4)\ni−bˆy+S(4)\ni−aˆx−bˆy/parenrightBig\n+S(2)\ni·/parenleftBig\nS(4)\ni+S(4)\ni−aˆx/parenrightBig/bracketrightBig\n+1\n2J3/summationdisplay\niS(3)\ni·/parenleftBig\nS(1)\ni+S(1)\ni+aˆx+S(2)\ni+S(2)\ni+aˆx/parenrightBig\n+1\n2J4/summationdisplay\niS(4)\ni·/parenleftBig\nS(2)\ni+S(2)\ni+aˆx+S(1)\ni+bˆy+S(1)\ni+aˆx+bˆy/parenrightBig\n.\n(3b)\nAll exchangeparameters Jµarepositive (see Fig. 1 foran\nillustrative long range ordered ferrimagnetic). We refer\nto this model as ( J1,J2,J3,J4) model.3\n1JJ\n2 J J4J\nJ4\n3S3J3J3\n3J11\nJ\nSS 2J4J4\nS4\nFIG. 1. (Color online) Classical ferrimagnetic ground stat e\nof 2D F-AF chain. The basic magnetic unit cell comprises\nof three up-spins S1,S2,S4and one down-spin S3. The in-\nteraction strengths J1,J2,J3are all positive and J4is the\nfrustrated bond.\nClassical Ground State: The basic model consists of\nalternating 1D ferro (strength J1= 1) and antiferro-\nmagnetic (strength J2=η2J1) S=1/2 chains (along the\ny-axis). The nearest chains interact with interaction\nstrengths J3(=η3J1) andJ4(=η4J1) which are an-\ntiferromagnetic. Before discussing the excitations and\nquantum spin fluctuations, we first consider the ground\nstate of our model when the spins are treated classically\n(mean field state). With J3=J4= 0, the ground state\n(G0) with broken global symmetry consists of decoupled\nalternating ordered F chains ( S1andS2spins) and AF\nchains (S3andS4spins). Due to the time reversal sym-\nmetry, the F chains can be either up or down (chosen\narbitrarily) and the AF chains can be in one of the two\nNe´ elstates. Thedegeneracyofthe G0is 22M, whereMis\nthenumberofF(orAF) chains. For J3>0andJ4= 0, if\nwe fix the orientation of one F chain, the nearest two AF\nchain orientations are fixed by the J3bond. The neigh-\nboring F chain orientations are then fixed. In this way,\nwe have the exact ground state Gas each bond takes its\nminimum energy value. When η3>0 andη4<0 (ferro-\nmagnetic), the system is not frustrated and the classical\nGS is a collinear ferrimagnetic state as shown in Fig. 1.\nHowever, for η3>0 andη4>0, spinS4is frustrated.\nFor weak frustration i.e. η4<< η3,Gis most likely the\nexact ground state and with increasing frustration ( J4)\nthe system will undergo a phase transition to a new state\nwhich may or may not be long range ordered. One ap-proach to attack the problem is to use the generalized\nLuttinger Tisza method [29] first proposed by Lyons and\nKaplan.30It turns out that for our Bravais lattice with\nfour-spin/unit cell system the calculations are quite dif-\nficult. So in the absence of the knowledge of the exact\nground state for large J4, we have used a different ap-\nproach. We study the local stability of Gwith increasing\nstrength of the frustrating bond ( J4). As we will show\nlater, depeding on the strength of J2/J1, there is a criti-\ncal value of J4/J3where the ground state Gis no longer\nlocally stable. Thus in our current analysis of the phase\ndiagram and excitations of the model using spin-wave\ntheory we use Gas the ground state.\nIV. SPIN-WAVE THEORY\nIt is well-known that spin-wave theory is best suited to\ntreat the dynamics of long range-ordered states in quan-\ntum spin system with large spin S. In the leading order\n(linear spin wave theory - LSWT), the excitations are\nmagnons. When magnon-magnon interaction effects are\nnegligible (for example for S >>1/2 and three dimen-\nsions), LSWT provides a very good description of the\nquantum spin fluctuation effects, one example being the\nreduction of the ordered moment in Heisenberg quantum\nantiferromagnets. However, for S=1/2 systems in 2D,\nmagnon-magnon interactions are not negligible and one\nmust incorporate higher order spin (1/S) corrections to\ntreat the system.6,7,31,32Even for these systems, LSWT\nprovides qualitatively correct physics. For example, for\n2D Heisenberg spin systems with nearest neighbor (NN)\nantiferromagnetic (AF) coupling [( J1,J2) model with no\nfrustration i.e. J2= 0] on a square lattice, the ordered\nmoment (average sublattice spin /angbracketleftSz/angbracketright) reduces due to\nQSF from 0.5 to 0.303 as given by LSWT.4,5When one\nincludes the higher order magnon-magnon interaction ef-\nfects using (1/S) expansion theory /angbracketleftSz/angbracketright= 0.307,7,32in-\ndicating that LSWT is very reasonable. For the gen-\neral (J1,J2) model, the effect of frustration is much more\nsubtle. Frustration tends to destabilize long range order.\nWith increase in the strength of frustration, /angbracketleftSz/angbracketright= 0\nat a critical value of J2=J2c. LSWT gives J2c= 0.38\nwhereas including the magnon-magnon interaction one\nfindsJ2c= 0.41,6,7againindicating the reasonablenessof\nLSWT in providinga measureof the QSF induced reduc-\ntion of the magnetization M. In a recent work (Ref. [12])\nresults for this model is obtained using a four-spin bond\noperator technique where it is found that /angbracketleftSz/angbracketright= 0.301\nforJ2= 0 and J2c= 0.36, which are close to the LSWT\nresults. We should mention here that all these method\nfail in the spin disordered state i.e. when J2> J2c.\nIn view of the abovediscussion, we opted to use LSWT\nto analyze the effect of QSF on the averagemagnetic mo-\nment and the critical strength of the frustration where\nthe ordered moments vanish. Unlike the ( J1,J2) model\n(two sublattice with same value of the ordered moment)\nour2D frustrated( J1,J2,J3,J4) model has a 4-sublattice4\nstructure as shown below and different sublattice mo-\nments are affected differently by QSF.\nFor our analysis we only consider the parameter space\n(η2,η3,η4) of the Hamiltonian H[Eq. (2)] where the GS\nis stable and is long range ordered collinear ferrimagnetic\nstate. The spin Hamiltonian in Eq. (3) is mapped onto a\nHamiltonian of interacting bosons by expressing the spin\noperators in terms of bosonic creation and annihilation\noperators a†,afor three “up” spins (spins 1, 2, and 4)\nandb†,bfor one “down’ spin (spin 3) using the standard\nHolstein-Primakoff representation33\nS+i\nin≈√\n2Sain, S−i\nin≈√\n2Sa†\nin, Szi\nin=S−a†\ninain,\nS+j\njn≈√\n2Sb†\njn, S−j\njn≈√\n2Sbjn, Szj\njn=−S+b†\njnbjn,\nand expand the Hamiltonian [Eq. (3)] perturbatively in\npowers of 1 /Skeeping terms only up to the quadratic\nterms. The resulting quadratic Hamiltonian is given as:\nH=Ecl+H0+···, (4)\nwhere\nEcl=−2J1NS2/bracketleftbig\n1+η2+2(η3−η4)/bracketrightbig\n(5)\nis the classical GS energy and\nH0= 2SJ1/summationdisplay\nk∈BZ/bracketleftBig\n(1+η3−η4)/parenleftBig\na(1)†\nka(1)\nk+a(2)†\nka(2)\nk/parenrightBig\n−γy/parenleftBig\na(1)\nka(2)†\nk+a(1)†\nka(2)\nk/parenrightBig\n+(η2−2η4)a(4)†\nka(4)\nk\n+(η2+2η3)b(3)†\n−kb(3)\n−k+η2γy/parenleftBig\nb(3)†\n−ka(4)†\nk+b(3)\n−ka(4)\nk/parenrightBig\n+η3γx/parenleftBig\neikyb/4a(1)†\nkb(3)†\n−k+e−ikyb/4a(2)†\nkb(3)†\n−k+h.c./parenrightBig\n+η4γx/parenleftBig\ne−ikyb/4a(1)†\nka(4)\nk+eikyb/4a(2)†\nka(4)\nk+h.c./parenrightBig/bracketrightBig\n(6)\nwithγx= cos(kxa/2) andγy= cos(kyb/2).\nIn the absence of inter-chain coupling ( η3=η4= 0)\nthemagnonspectrumcanbeobtainedusingthestandard\nBogoliubov transformations.34We find four modes for\neachky(−π/b < k y< π/b) independent of kx(−π/a <\nkx< π/a): two from the F-chains ( α-branches) and two\nfrom the AF-chains (one αand one β). The quadratic\nHamiltonian takes the following form:\nH0=/summationdisplay\nk∈BZ/bracketleftBig\nǫ(1)\nkα(1)†\nkα(1)\nk+ǫ(2)\nkα(2)†\nkα(2)\nk\n+ǫ(3)\nk/parenleftBig\nα(4)†\nkα(4)\nk+β(3)†\n−kβ(3)\n−k/parenrightBig/bracketrightBig\n+/summationdisplay\nk∈BZ/parenleftBig\nǫ(3)\nk−2J1S/parenrightBig\n,\n(7)\nwhere\nǫ(1,2)\nk= 2J1S[1∓γy], (8a)\nǫ(3)\nk= 2J2S/radicalBig\n1−γ2y= 2J2S|sin(kyb/2)|.(8b)\nThe last term in Eq. (7) are the LSWT corrections to\nthe classical ground state energy Eclin Eq. (5) for the\nspecial case η3=η4= 0.With inter-chain coupling (i.e. η3,η4>0), we have\nnot been able to find the analytical Bogoliubov trans-\nformations that transforms the bosonic spin operators to\nBogoliubov quasiparticle operators that diagonalize the\nHamiltonian H0[Eq. (6)]. For the special case kx=π/a\ni.e.γx= 0, we use the equation of motion method\n(see Appendix A) and obtain analytical solutions for the\nmagnon dispersion which are:\nǫ(1,2)\nk= 2J1S/bracketleftbig\n(1+η3−η4)±γy/bracketrightbig\n, (9a)\nǫ(3,4)\nk= 2J1S/vextendsingle/vextendsingle(η3+η4)±/radicalBig\n(η3−η4+η2)2−η2\n2γ2y/vextendsingle/vextendsingle.\n(9b)\nWhenη3=η4= 0 the above dispersions reduce to\nEq. (8) as expected.\nFor the general case we use an elegant method devel-\noped by Colpa to obtain both the eigenenergies (magnon\ndispersions) and eigenvectors (required for the calcula-\ntion of magnetization).35,36First we write the 8 ×8\nHamiltonian [Eq. (6)] in a symmetrized form:\nH0=J1S/summationdisplay\nk∈BZ8/summationdisplay\ni=1X(i)†\nkhkX(i)\nk\n−2J1SN[1+η2+2(η3−η4)],(10)\nwith the eigenvectors\nXk= [a(1)\nk,a(2)\nk,a(4)\nk,b(3)\nk,a(1)†\nk,a(2)†\nk,a(4)†\nk,b(3)†\nk]. The\nhermitian matrix hkis:\nhk=\nA1−B1C∗\n20 0 0 0 C1\n−B1A1C20 0 0 0 C∗\n1\nC2C∗\n2A20 0 0 0 B2\n0 0 0 A3C1C∗\n1B20\n0 0 0 C∗\n1A1−B1C20\n0 0 0 C1−B1A1C∗\n20\n0 0 0 B2C∗\n2C2A20\nC∗\n1C1B20 0 0 0 A3\n,(11)\nwhere the constants are given in Eqs. (A2).\nTheCholeskydecompositionhastobeappliedon hkto\nfindthecomplex Kmatrixthatfulfillsthecondition hk=\nK†K. However, the Cholesky decomposition only works\nif the matrix hkis positive definite (i.e. the eigenvalues\nare all positive).35In case the spectrum of the Hamilto-\nnianH0containszeromodes,onecanaddasmallpositive\nvalue to the diagonal of hkto make the matrix positive\n“definite”. We find that the criterion for the Cholesky\ndecomposition to work for all kisη4≤η2η3/(η2+2η3).\nAs an example, with η2= 3.0,η3= 0.4,η4≤η4c, where\nη4c= 0.316. Ifη4> η4cthe matrix hkis not positive\ndefinite and the procedure fails. As we discuss later, this\nis precisely the same condition for the stability of the fer-\nrimagnetic state. After obtaining the matrix K, we solve\nthe eigenvalue problem of the hermitian matrix KgK†,\nwheregis a diagonal paraunitary matrix with elements\ngii= diag(1 ,1,1,1,−1,−1,−1,−1). The resulting eigen-\nvectorsarethen arrangedin suchawaythat the first four\ndiagonal elements of the diagonalized L=U†KgK†U5\nmatrix are positive and the last four elements are nega-\ntive. The first four positivediagonalelements correspond\nto the magnon dispersions.\nTo calculate the sublattice magnetization miwe first\nconstruct the diagonal matrix, E=gLand then find the\ntransformation matrix T, which relates the boson modes\nXkwith the Bogoliubov modes αkviaXk=Tαk. The\nmatrixTis calculated using36:T=K−1UE1/2. mi=1,2,4\nof spins S1,S2,S4are positive but m3for spin S3is\nnegative. So we calculate the magnitude of mi=1−4for\neach of the four sublattices using\n|mi|= 0.5−|τi|. (12)\nwhereτiare the reduction caused by QSFs:\n|τi|=1\nN/summationdisplay\nk∈BZ/braceleftBig\nTkDT†\nk/bracerightBig\ni+4,i+4. (13)\nDis a diagonal matrix with [0 ,0,0,0,1,1,1,1] as the di-\nagonal elements. We again reiterate that the parameters\nη2,η3,η4are chosen such a way that the condition for the\nCholesky decomposition is satisfied, i.e. η4≤η4c.\nV. MAGNON DISPERSION AND SUBLATTICE\nMAGNETIZATION\nA. Magnon Dispersion\nEffects of inter-chain interaction on the magnon dis-\npersion is displayed in Fig. 2(a-e) where for illustration\nwe have chosen η2= 3,��3= 0.4 and the frustration pa-\nrameter η4is increased from 0.05 to 0.315. The disper-\nsion along ky(along the chains) is given for two values\nofkx:kx= 0 (top two panels) and kx=π/a(bottom\ntwo panels). Also for comparison we give the dispersions\nfor the non-interacting chains ( η3=η4= 0). Later we\nwill discuss the kxdependence for some special modes.\nAs expected, there are four magnon modes for each k.\nFor the non-interacting chains, there are two F-magnon\nmodes which are split (the lower mode ∼k2\nyfor small\nky) and two AF-magnons which are degenerate ( ∼kyfor\nsmallky). In the presence of couplings (discussed below)\nwe will (loosely) refer to these four modes as two F and\ntwo AF modes.\nFirst we consider the case kx=π/a(bottom two\npanels) where the hybridization between the F and AF\nmodes is absent (as γx= 0) - so the F and AF chains\ninteract only through effective fields. In this limit, we\nfind from Eq. (9a) and Eq. (9b) that the F-modes get\nrigidly shifted upwards by 2 J1S(η3−η4), the two degen-\nerate AF-modes are split by 4 JS(η3+η4), and both the\nmodes∼k2\ny. Atky= 0 the lower F-mode and the lower\nAF-mode are gapped, ∆ F(π/a,0) = 2J1S(η3−η4) and\n∆AF(π/a,0) = 2J1S[/radicalbig\n(η2+η3−η4)2−η2\n2−(η3+η4)].\nWhen the frustration parameter η4is increased towards\nη3, there is a critical value η4c=η2η3/(η2+ 2η3)< η3,11.5 22.5 301234ωk\n11.5 22.5 301234\n11.5 22.5 301234ωk\n11.5 22.5 301234\n11.5 22.5 3\nky (π/b)01234ωk\n11.5 22.5 3\nky (π/b)01234(a) η4=0.05 (b) η4=0.1\n(c) η4=0.2A. kx=0, η2=3.0, η3=0.4\n(d) η4=0.3\n(f) η2=3.0, η3=η4=0 (e) η4=0.315F\nFAF\nAFAF AFAF\nAFAF\nAF\nAF\nAFAFF\nFF\nF\nF\nF\nFF\nF\n11.5 22.5 301234ωk\n11.5 22.5 301234\n11.5 22.5 301234ωk\n11.5 22.5 301234\n11.5 22.5 3\nky (π/b)01234ωk\n11.5 22.5\nky (π/b)01234(a) η4=0.05 (b) η4=0.1\n(c) η4=0.2B. kx=π/a, η2=3.0, η3=0.4\n(d) η4=0.3\n(f) η2=3.0, η3=η4=0 (e) η4=0.315FFAFAF\nF\nFAFFFAFAF\nAF\nAFAF\nAF AF\nF\nFF\nF\nFAF\nFAF\nFIG. 2. Magnon dispersion of the ferrimagnetic state for A.\nkx= 0 and B. kx=π/a[Figs. (a-e)] with η2= 3.0,η3= 0.4.\nThe frustration parameter η4is varied from 0.05 (small frus-\ntration) to η4= 0.315. (f) Limiting case with no inter-\nchain coupling: the two AF-branches are degenerate, the F-\nbranches are gapped, and the lower F-branch vanishes at\nky= 2π/b(= 0). Note that due to 2 πperiodicity ky=\n[−π/b,π/b] = [π/b,3π/b].\nwhere ∆ AF(π/a,0) = 0 but ∆ F(π/a,0)>0. The fer-\nrimagnetic GS becomes locally unstable and the system\ntransits to a new ground state (For the parameter values\nwe have chosen η4c= 0.316 - this is also the place where\nCholesky decomposition fails because the matrix hkis\nnot positive definite). This is similar to the field induced\nquantum phase transition as a function of the external\nmagnetic field for the 1D quantum S 1= 1,S2= 1/2\nmodel discussed in the introduction.17,21Here the optic\nmode gap goes to zero at a critical field and the system\nundergoes a quantum phase transition from a ferrimag-\nnetic state to some other state. This phase transition oc-\ncurs in the range η3> η4> η4c. Fig. 3 shows a schematic\nphase diagram in the ( η4/η2,η3/η2) space. We also note\nthat for given η3andη4≤η3, the strength of the ex-\nchange in the AF chains η2should be greater than a\ncritical value η2c= 2η3η4/(η3−η4) for the ferrimagnetic\nstate to be stable.\nForkx= 0, the picture is qualitatively similar, but\nwith two fundamental differences resulting from hy-6\n0 0.2 0.4 0.60.8 1\nη3/η200.20.40.60.81η4/η2\nCollinear F-AF Ground StateCollinear F-AF Ground State\nFIG. 3. Phase diagram of H[Eq (2)]: normalized ˜ η4=η4/η2\nis plotted against normalized ˜ η3=η3/η2. The dashed lines\nare given by the equations ˜ η4= ˜η3/(1+2˜η3) (lower one) and\n˜η3= ˜η4/(1 + 2˜η4) (upper one). They are the boundaries of\nthe stability of the ferrimagnetic state. The solid thick li ne\n˜η4= ˜η3is most likely a critical line.\nbridization between ferro and antiferro chain excitations.\nFirst, the lower F-mode goes to zero when ky→0 as it\nshould for the Goldstone mode. However the dispersion\nfor large kydiffers qualitatively from the non-interacting\nchains. Second, hybridizationbetween the upper F-mode\nand the lower AF-mode opens up a hybridization gap\nat a finite kyand the size of the gap increases with\nη4. However, as for the ( kx,ky) = (π/a,0) the gap\n∆AF(π/a,0)→0 asη4→η4c. In fact ∆ AF(kx,0)→0\nfor all values of 0 ≤kx≤π/aforη4→η4c. In Fig. 4B we\nshow the kxdependence of ∆ AF(kx,0) for three different\nvalues of the frustration parameter η4. Also we show in\nFig. 4A the kxdependence of ∆ F(kx,0). This suggests\nthat the chains become dynamically decoupled and since\nthe decoupled AF chains are spin liquids without any\nlong range order, the system goes from an ordered state\nto a spin disordered state when η4> η4c. Exact calcula-\ntions will tell us about the precise nature of the ground\nstate for η4c< η4< η3.\n0 0.2 0.4 0.6 0.8 1\nkx (π/a)00.10.20.30.4∆F (kx,0)η4=0.05\nη4=0.2\nη4=0.315\n0 0.2 0.4 0.6 0.8 1\nkx (π/a)00.30.60.91.2∆AF(kx,0)η4=0.05\nη4=0.2\nη4=0.315A. B.\nFIG. 4. Gaps for F-mode (∆ F) and AF-mode (∆ AF) with\nincrease in kxforky= 0 with η2= 3.0,η3= 0.3 and three\ndifferent values of η4= 0.05,0.2,and 0.315.B. Sublattice Magnetization\nFollowing Colpa’s method we have calculated the sub-\nlattice magnetizations mifor the four sites. We have\nchecked that the sum of the reduction in the four sublat-\ntice moments due to quantum fluctuations,/summationtext4\ni=1τi= 0,\nwhich results in the total magnetic moment equal to\none as expected. This is equivalent to the results ob-\ntained for S 1= 1, S 2= 1/2 1D quantum ferrimag-\nnetic state for which the total magnetization/unit cell\nis equal to 0.5. Next we discuss the effect of frustra-\ntion on the quantum fluctuation induced reduction of\nthe long-range ordered moments for the four different\nspins of the unit cell. In the absence of interchain cou-\npling [Fig. 1], m1=/angbracketleftS1z/angbracketright=m2=/angbracketleftS2z/angbracketright= 0.5 and\nm3=/angbracketleftS3z/angbracketright=−m4=/angbracketleftS4z/angbracketright= 0 (due to quantum spin\nfluctuation in 1D AF). When we turn on η3, its effect\nis to produce an ordering field at the S3sites and or-\nder them in the direction opposite to the F-chain spins.\nThe intra AF chain interaction orders the S4spins par-\nallel to the F-chain spins, resulting in a 2D ferrimagnetic\nground state. If η2≪η3then the system will be more\n2D,m1=m2∼=0.5, andm3,m4will be non-zero with\nthe magnitude of m3larger than m4. On the other hand\nifη2≫η3, then intra-chain AF bonds will dominate,\nmaking the AF chains nearly decoupled and the LRO in\nthe AF chains will be small, m4≈ −m3≪0.5.\n0 0.1 0.2 0.3\nη400.10.20.30.40.5Sublattice Magnetization mi η2=3.0, η3=0.4m1=m2\n|m3|\nm4\nFIG. 5. Magnitude of sublattice magnetizations, mi, for\nη2= 3.0,η3= 0.4 as function of η4. Magnetizations for the\ntwo degenerate ( m1=m2) ferro-modes (solid) corresponding\nto spins 1 and 2 slowly increase as η4is increased. Magne-\ntizations m3,m4for spins 3 (dashed) and 4 (solid) decrease\ndue to the increase in quantum fluctuations with increase in\nη4. The ferrimagnetic ground state is stable in the parame-\nter space ( η2,η3,η4) as long as η3> η4andη4≤η4c, where\nη4c= 0.316 forη2= 3.0,η3= 0.4.\nIn Fig. 5, we show how the ordered moments change\nwith the increasing strength of the frustrated bond η4for\nspecificvaluesof η2= 3.0andη3= 0.4. Asη4approaches\nthe criticalvalue 0.316the magnetizationofthe AF chain\ndecreases but remains finite ( |m3| ∼0.07,|m4| ∼0.06)\njust before quantum phase transition to other ground\nstate within LSWT. This is in contrast to what happens7\nin the (J1,J2) model whereas J2approaches Jc(Jc1from\ntheNe´ elstateand Jc2fromtheCAFstate),thesublattice\nmagnetization goes to zero.\nFinally, in Fig. 6(a-b), we show the η2dependence of\nthemagnitudesofthefourorderparameters mi(i= 1..4)\nforη3= 0.4 for two fixed values of the frustrated inter-\nchain bond η4. For our assumed collinear ferrimagnetic\nground state η3> η4andη2> ηc\n2. Forη3= 0.4,η4= 0.1,\nthe critical value of ηc\n2is =0.27 and for η4= 0.2,η2c=\n0.80. For small η2i.eη2≪η3,m1=m2=|m3| ∼0.46\n1 2 3 4 56\nη200.10.20.30.40.5Sublattice Magnetization mi(a) η3=0.4, η4=0.1\nη2c=0.27m1=m2\n|m3|\nm4\n1 2 3 4 5 6\nη200.10.20.30.40.5Sublattice Magnetization mi(b) η3=0.4, η4=0.2m1=m2\nη2c=0.8|m3|\nm4\nFIG. 6. (a-b) Magnitude of sublattice magnetizations, mifor\nη3= 0.4,η4= 0.1 (Fig. a) and 0 .2 (Fig. b) as function of η2.\nThe ferrimagnetic ground state is stable for η2≥η2cwhere\nη2c= 0.27 forη4= 0.1 andηc\n2= 0.80 forη4= 0.2. Ferro\nmodes (solid) corresponding to spins 1 and 2 are degenerate\n(m1=m2). Magnetizations m3,m4for spins 3 (dashed) and\n4 (solid) decrease due to the increase in quantum fluctuation s\nwith increase in η2.\nandm4∼0.38, a reduction from 0.5 by 8% and 24%\nrespectively. The small antiferromagnetic coupling be-\ntween spins of the AF-chain induces a relatively large\nvalue of the moment at the site 4. When η2increases theQSF in the AF-chain reduces the moments at sites 3 and\n4. Notice that site 3 still has a larger moment (in magni-\ntude) than at site 4. For large η2values, say η2∼6, ferro\nchain spins have moments ∼0.495, whereas AF chain\nspins have moments of magnitude ∼0.14(>0) due to\nsmall stabilizing interchain coupling η3= 0.4 [Fig. 6(a)].\nIncreasing the strength of the frustrated bond η4essen-\ntially decouples the chains. For example with η4= 0.2,\natη2= 6.0 ferro chains have moments close to 0.5 and\nAF-chains have moments of magnitude 0 .08 [Fig. 6(b)].\nForη2< η2c, the system is most likely a spin liquid state\nwithout LRO.\nVI. CONCLUSIONS\nIn summary, we have proposed a 2D frustrated Heisen-\nberg model consisting of alternating 1D ferro ( J1) and\nantiferro( J2) chainswhich interact with alternatingfrus-\ntrated (J4) and unfrustrated( J3) bonds (strengths). The\nground state is a long range ordered ferrimagnetic state\nin certain region of the parameter space. Analysis using\nlinear spin wave theory suggests that the system under-\ngoesaquantumphasetransitiontoaquantumdisordered\nphasewith increasingstrengthof η4, similartothe classic\n2D (J1,J2) model. However in contrast to the ( J1,J2)\nmodel, the sublattice magnetizations of the AF chains\ndo not vanish at the critical value η4c, similar to the 1D\nS1= 1,S2= 1/2 model of a quantum ferrimagnet. The\nexact nature of the phase transition, the nature of the\nGS above η4c, and whether the order parameter vanishes\nat the transition should be explored by other theoretical\nand numerical techniques.\nVII. ACKNOWLEDGMENT\nSDM would like to thank Dr. Xianglin Ke for stimu-\nlating discussions.\nAppendix A: Equation of Motion Method\nWith inter-chain coupling (i.e. η3,η4>0), we are\nunable to find the the Bogoliubov transformations that\ndiagonalizes the Hamiltonian in Eq. (6). Thus we opt for\nanother way - the canonical equation of motion method\nto obtain the magnon dispersion.1The variouscommuta-\ntors that are needed for the canonical equation of motion\nmethod are:8\n/bracketleftbig\na(1)\nk,H0/2J1S/bracketrightbig\n=A1a(1)\nk−B1a(2)\nk+C1b(3)†\nk+C∗\n2a(4)\nk, (A1a)\n/bracketleftbig\na(2)\nk,H0/2J1S/bracketrightbig\n=A1a(2)\nk−B1a(1)\nk+C∗\n1b(3)†\nk+C2a(4)\nk, (A1b)\n/bracketleftbig\na(4)\nk,H0/2J1s/bracketrightbig\n=A2a(4)\nk+B2b(3)†\nk+C2a(1)\nk+C∗\n2a(2)\nk, (A1c)\n/bracketleftbig\nb(3)†\nk,H0/2J1s/bracketrightbig\n=−A3b(3)†\nk−B2a(4)\nk−C∗\n1a(1)\nk−C1a(2)\nk, (A1d)\n/bracketleftbig\na(1)†\nk,H0/2J1S/bracketrightbig\n=−A1a(1)†\nk+B1a(2)†\nk−C∗\n1b(3)\nk−C2a(4)†\nk, (A1e)\n/bracketleftbig\na(2)†\nk,H0/2J1S/bracketrightbig\n=−A1a(2)†\nk+B1a(1)†\nk−C1b(3)\nk−C∗\n2a(4)†\nk, (A1f)\n/bracketleftbig\na(4)†\nk,H0/2J1s/bracketrightbig\n=−A2a(4)†\nk−B2b(3)\nk−C∗\n2a(1)†\nk−C2a(2)†\nk, (A1g)\n/bracketleftbig\nb(3)\nk,H0/2J1s/bracketrightbig\n=A3b(3)\nk+B2a(4)†\nk+C1a(1)†\nk+C∗\n1a(2)†\nk, (A1h)\nwhere\nA1= (1+η3−η4), A2= (η2−2η4),(A2a)\nA3= (η2+2η3), B1=γy, B2=η2γy,(A2b)\nC1=η3γxeikyb/4, C2=η4γxeikyb/4.(A2c)\nWe notice that the first four commutators [Eqs. (A1a)\n- (A1d)] are decoupled from the second four commu-\ntators [Eqs. (A1e) - (A1h)]. With the basis vectors\nXk= (a(1)\nk,a(2)\nk,a(4)\nk,b(3)†\nk), thecanonicalequationofmo-\ntion can be deduced from the Hamiltonian in Eq. (6) in\nthe following way:\n/bracketleftBig\nX(i)\nk,H0\n2J1S/bracketrightBig\n=idX(i)\nk\ndt=gω(i)\nkX(i)\nk.(A3)In Eq. (A3) gis a 4×4 diagonal matrix with gii=\n(1,1,1,−1) in the diagonal elements. The eigenvalues,\nω(i)\nkare obtained by solving the determinant:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(A1−ω(i)\nk)−B1 C2 C∗\n1\n−B1(A1−ω(i)\nk)C∗\n2 C1\nC∗\n2 C2(A2−ω(i)\nk)B2\nC1 C∗\n1 B2(A3+ω(i)\nk)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.\n(A4)\nThe above determinant leads to a fourth-order polyno-\nmial:\nω4\nk+aω3\nk+bω2\nk+cωk+d= 0, (A5)\nwhere the coefficients are:\na=−2(1−2η4), (A6a)\nb= (1+η3−η4)2−4(1+η3−η4)(η3+η4)−(η2+2η3)(η2−2η4)−(1−η2\n2)γ2\ny+2(η2\n3−η2\n4)γ2\nx, (A6b)\nc= 2(1+η3−η4)2(η3+η4)+2(1+ η3−η4)(η2−2η4)(η2+2η3)−2η2\n2(1+η3−η4)γ2\ny−2(η3+η4)γ2\ny\n−2(1+η2+η3−3η4)η2\n3γ2\nx+2(1−η2−η3−η4)η2\n4γ2\nx−2(η2\n3−η2\n4)γ2\nxγ2\ny+4η2η3η4γ2\nxγ2\ny, (A6c)\nd=−(1+η3−η4)/bracketleftBig\n(1+η3−η4)(η2−2η4)(η2+2η3)−(1+η3−η4)η2\n2γ2\ny−2(η2−2η4)η2\n3γ2\nx\n−2(η2+2η3)η2\n4γ2\nx+4η2η3η4γ2\nxγ2\ny/bracketrightBig\n+(η2−2η4)(η2+2η3)γ2\ny+2(η2−2η4)η2\n3γ2\nxγ2\ny+2(η2+2η3)η2\n4γ2\nxγ2\ny\n−η2\n2γ4\ny−4η2\n3η2\n4γ2\nx−4η3η4γ2\nxγ2\ny(η2−η3η4). (A6d)\nThe other set of four boson operators\n(a(1)†\nk,a(2)†\nk,a(4)†\nk,b(3)\nk) lead to a similar fourth or-\nder polynomial equation, but the signs before the linear\nand cubic terms are negative. There is thus a ωk↔ −ωk\nsymmetry between the two sets of solutions. This fourth\norder polynomial [Eq. (A5)] has to be solved numerically.\nThe four real eigen-values can be positive or negative. If\nwe solve the fourth order polynomial associated with the\nother four boson operators we will get again four real\nsolutions which are negative of the solutions of Eq. (A5).\nFor the magnon frequencies we will consider only thefour positive solutions. 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Matter 27, 166002\n(2015)." }, { "title": "0707.2133v2.Martensitic_transition__ferrimagnetism_and_Fermi_surface_nesting_in_Mn_2NiGa.pdf", "content": "arXiv:0707.2133v2 [cond-mat.other] 15 Oct 2007Martensitic Transition, Ferrimagnetism and Fermi Surface\nNesting in Mn 2NiGa\nS. R. BARMAN1, S. BANIK1, A. K. SHUKLA1,\nC. KAMAL2, and APARNA CHAKRABARTI2\n1UGC-DAE Consortium for Scientific Research,\nIndore, 452017, Madhya Pradesh, India and\n2Raja Ramanna Centre for Advanced Technology,\nIndore, 452013, Madhya Pradesh, India\nAbstract\nPACS. 81.30.Kf - Martensitic transformations\nPACS. 71.20.Be - Electron density of states and band structu re of transition metals and alloys\nPACS. 71.18.+y - Fermi surface: calculations\nPACS. 71.15.Nc - Total energy calculations\nThe electronic structure of Mn 2NiGa has been studied using density functional theory and ph o-\ntoemission spectroscopy. The lower temperature tetragona l martensitic phase with c/a=1.25 is\nmore stable compared to the higher temperature austenitic p hase. Mn 2NiGa is ferrimagnetic in\nboth phases. The calculated valence band spectrum, the opti mized lattice constants and the mag-\nnetic moments are in good agreement with experiment. The maj ority-spin Fermi surface (FS)\nexpands in the martensitic phase, while the minority-spin F S shrinks. FS nesting indicates occur-\nrence of phonon softening and modulation in the martensitic phase.\nPACS numbers:\n1Introduction: Recentadvent ofmultiferroicshapememoryalloys(SMA)likeNi-Co-M n-In,\nNi-Mn-Ga that exhibit both ferroelastic and ferromagnetic proper ties has ushered a flurry\nof activity in this field1,2,3,4,5,6,7,8,9. In particular, Ni-Mn-Ga has generated immense interest\nbecause of very large strain (10%) in a moderate magnetic field ( ≈1 Tesla)3,4. Moreover,\nin Ni-Mn-Ga the actuation is much faster ( ≈2kHz) than conventional SMA5. However,\nNi2MnGa are brittle and so search for materials with better mechanical properties exhibit-\ningsimilarmagneticfieldinducedstrainisbeingactivelypursued10,11. Mn2NiGaisarecently\ndiscovered ferromagnetic SMA in the Ni-Mn-Ga family. It has Curie an d martensitic start\ntemperatures of 588 and 270 K, respectively11. Ferromagnetism in Mn 2NiGa is surprising\nbecause direct Mn-Mn interaction normally leads to antiferromagne tic alignment12,13. More-\nover, the origin of the martensitic transition involving a relatively larg e tetragonal distortion\n(c/a=1.21) has not been studied theoretically till date. Recently, a dens ity functional the-\nory (DFT) study on Mn 2NiGa shows a large enhancement of the density of states (DOS)\nnear the Fermi level ( EF) and quenching of Mn and Ni magnetic moments in the marten-\nsitic phase14. However, such large change in the magnetic moments or DOS has no t been\nobserved in any other SMA either from experiment9,15,16or theory8,17,18.\nThegeometryoftheFermisurface(FS)isresponsible foravariet yofphenomena likespin\norchargedensity waves, Kohnanomalies, Friedel oscillationsinmeta ls. IftheFShasparallel\nplanes, strong electronic response can occur at the wave vector that translates one parallel\nplane of the FS to the other. This wave vector is called the nesting ve ctor (n.v.). FS nesting\nhas been reported to cause softening of the transverse-acous tic (TA 2) phonon mode along\n[110] direction resulting in modulated pre-martensitic phase of SMA’s like Ni 2MnGa and\nNi-Ti19. Recently, an inelastic neutron scattering study on Ni 2MnGa showed the presence\nof charge density wave in the martensitic phase resulting from FS ne sting7. Thus, it is\nworthwhile to study the FS of Mn 2NiGa, particularly because the relatively large tetragonal\ndistortion is likely to modify the FS substantially.\nIn this work, a DFT study of the electronic structure of Mn 2NiGa using full poten-\ntial linearized augmented plane wave method (FPLAPW) is presented . The valence band\n(VB) spectrum, calculated from the theoretical DOS, is in agreeme nt with the ultra-violet\nphotoemission spectroscopy (UPS). We find that the total energ y (Etot) is lower in the\nmartensitic phase with a tetragonal distortion of c/a=1.25. We show that Mn 2NiGa is an\nitinerant ferrimagnet in both the martensitic and austenitic phases . The equilibrium lattice\n2constants and the magnetic moments are in agreement with x-ray d iffraction and magneti-\nzation data, respectively. The FS in the martensitic phase is drastic ally different from the\naustenitic phase. A highly nested hole-type majority-spin cuboidal FS sheet around the Γ\npoint appears in the martensitic phase that is absent in the austenit ic phase.\nMethodology: FirstprinciplesDFTcalculationswereperformedusingtheWIEN97co de20.\nGeneralized gradient approximation (GGA) for the exchange corre lation that accounts for\nthe density gradients was used21. An energy cut-off for the plane wave expansion of 16 Ry\nis used (RMTKmax= 9). The cut-off for charge density is Gmax= 14. The maximum l(lmax)\nfor the radial expansion is 10, and for the non-spherical part: lmax,ns=6. The muffin-tin\nradii are Ni: 2.1364, Mn: 2.2799, and Ga: 2.1364 a.u. The number of kpoints for self-\nconsistent field cycles in the irreducible Brilloiun zone is 256 and 484 in th e austenitic and\nmartensitic phase, respectively. The convergence criterion for Etotis 0.1 mRy, which implies\nthat accuracy of Etotis±0.34 meV/atom. The charge convergence is set to 0.001. FS has\nbeen calculated using XcrySDen22. Mn2NiGa ingot was prepared by arc furnace melting and\nannealing at 1100K9. It was characterized by x-ray diffraction (XRD), energy dispers ive\nanalysis of x-rays and differential scanning calorimetry16. Atomically clean specimen surface\nwas prepared by in situscraping using a diamond file and the chamber base pressure was\n6×10−11mbar. UPS was performed with a HeI (h ν=21.2 eV) photon source using electron\nenergy analyzer from Specs GmbH, Germany. The overall resolutio n was 120meV.\nMn2NiGa has a cubic L21structure in the austenitic phase that consists of four inter-\npenetrating f.c.c. lattices at (0,0,0), (0.25,0.25,0.25), (0.5,0.5, 0.5), and (0.75,0.75,0.75)\n(Fig. 1a)11,16. The structure of Mn 2NiGacan be better explained incomparison to Ni 2MnGa\nthat also has L21structure. In Ni 2MnGa, the Ni atoms are at (0.25,0.25,0.25) and\n(0.75,0.75,0.75), while Mn and Ga are at (0.5,0.5,0.5) and (0,0,0), r espectively and there\nis no direct Mn-Mn interaction, with Mn having eight Ni atoms as neare st neighbours. In\ncontrast, Mn 2NiGa has one Mn atom at (0.5,0.5,0.5) (referred to as MnII), while th e other\nMn atom (MnI) occupies the Ni atom position (0.75,0.75,0.75) of Ni 2MnGa. Thus, MnI\nand MnII occupy inequivalent sites in the unit cell, and there is a direct Mn-Mn interaction\nsince MnI and MnII are nearest neighbours. In the martensitic pha se, the XRD pattern for\nMn2NiGa has been indexed by a tetragonal unit cell with c/a=1.21 (Fig. 1b)11,16.\nTotal energy and magnetic moment calculation: To determine whether minimiza-\ntion ofEtotcauses the structural transition, we have calculated Etotfor both phases as a\n3function of the lattice parameters in the lowest energy magnetic st ate (discussion about the\nmagnetic state is given later). In the austenitic phase, Etotas a function of cell volume\n(V) exhibits a parabolic behaviour and the minimum (shown by arrow) det ermines the op-\ntimized lattice constant ( a=11.059 a.u.=5.85 ˚A) (Fig. 2a). The agreement is within 1% of\nthe experimental value of 5.9072 ˚A11. For the martensitic phase, in the first step, Etot(V)\nis calculated to obtain optimized V=1330 a.u.3at fixedc/a=1.21 (XRD value). Next,\nEtot(c/a) is calculated at V=1330 a.u.3. This gives the optimized c/ato be 1.25. In the\nfinal step, Etot(V) is calculated again with c/a=1.25 (Fig. 2a). Least square fitting of the\ndata8,23gives the Etotminimum at 1335.2 a.u.3(shown by arrow). From Fig. 2a, the Etot\nminimum in the martensitic phase is 6.8 meV/atom lower than the austen itic phase. This\ndemonstrates that the martensitic phase is stabilized through a siz able tetragonal distortion\n(c/a=1.25). The optimized lattice constants ( a=5.409 and c=6.762, ˚A) are within 2.1%\nand0.85%oftheexperimental latticeconstants a=5.5272 ˚A andc=6.7044 ˚A,respectively11.\nThus, the agreement of the lattice constants for both the phase s is satisfying, considering\nthat even forfree-electron-like non-magneticmetals therecould beabout 2%discrepancy be-\ntween experiment andGGAbased DFTtheory24. The decrease of Vby 1.2%is inagreement\nwith the experimental volume decrease of 0.64% in the martensitic ph ase11.\nThelowestenergymagneticstateisobtainedbyperforming Etotminimizationovervarious\npossible starting MnI and MnII magnetic moment combinations, as dis cussed in details in\nRef.25. For both austenitic and martensitic phase, the anti-parallel star ting spin (equal or\nunequal)configurationsofMnIandMnIIconvergetoaferrimagne ticstatethathasminimum\nEtot. We have used starting Mn magnetic moments for structure optimiz ation runs to be\n3µBfor both Mn atoms in anti-parallel orientation. However, when the s tarting MnI and\nMnII moments are parallel (equal or unequal), Etotconverges to different magnetic moments\nrelated to local minima at higher energies. For example, in the austen itic phase there are\nthree local minima25. Also in the martensitic phase, multiple local minima are obtained\nwith parallel starting moments of MnI and MnII. In particular, a loca l minimum that is 108\nmeV/atom higher in Etot, gives MnI and MnII moments to be 0.24 and 2.38 µB25. Thus,\none Mn moment is small, as has been reported in Ref.14. Our calculation based on the\nmagnetic moments reported in Ref.14converges at 193 meV/atom higher energy than the\nEtotminimum25. This gives an idea why the results from Ref.14are in disagreement with\nexperimental data, as discussed later.\n4The spin magnetic moment distribution in the martensitic phase clearly shows that it is\nferrimagnetic with MnI magnetic moment anti-parallel and smaller tha n MnII (Fig. 2b). Ni\nmoment is small and is parallel to MnII moment. For the martensitic (a ustenitic) phase,\nthe local spin magnetic moments are -2.21 (-2.43), 2.91 (3.2), 0.27 (0 .32), 0.01 (0.01) µB\nper formula unit ( µB/f.u.) for MnI, MnII, Ni, and Ga, respectively. The moment related\nto the interstitial charge is small (-0.04 µB). The total moment for the martensitic phase\n(1.01µB/f.u.) is 11% less than the austenitc phase (1.14 µB/f.u). The lowering of the mag-\nnetic moment in the martensitic phase has been reported by Liu et al.from magnetization\nstudies: 1.21 µB/f.u. (28.28 emu/g) and 1.29 µB/f.u. (30.3 emu/g) in the martensitic and\naustenitic phase, respectively11. Thus, the magnetic moment values and the trend that\nmagnetization is lower in the martensitic phase are in agreement with o ur calculations.\nDensity of states and photoemission spectroscopy: The stabilization of the tetrago-\nnally distorted martensitic phase in Ni 2MnGa has been related to band Jahn-Teller effect,\nwhere a DOS peak at EFin the cubic phase splits into two peaks below and above EFin\nthe tetragonal phase, resulting in a lowering of the total energy17. Splitting and shift of the\nDOS peaks just below EFhave also been observed in Ni 2.25Mn0.75Ga9. For Mn 2NiGa, the\ndifferences inthetotalDOSnear EFareinteresting: apeakat-0.1eVintheausteniticphase\nshifts to lower energy (-0.35 eV) and diminishes in intensity in the mart ensitic phase (both\npeaks indicated by arrows). The peak above EFat 0.35 eV (tick) does not shift but is en-\nhanced in intensity in the martensitic phase indicating a transfer of D OS from the occupied\ntotheunoccupiedstates. FromthepartialDOS(PDOS),itiscleart hatthepeaksat-0.1and\n-0.35eVarise primarily due to Ni 3 dand MnI3 dhybridization. The shift of the-0.1eV peak\nto lower energy in the martensitic phase results from enhanced Ni 3 d- MnI 3dhybridization\ncaused by decrease in Ni-MnIdistance from2.925 ˚A(austenitic) to 2.701 ˚A(martensitic) and\nis a possible reason for the stabilization of the martensitic phase. Th e DOS at EFis sub-\nstantially reduced in the martensitic phase (1.29 states/eVf.u.) com pared to the austenitic\nphase (3.39). Thus, decrease in electronic specific heat in the mart ensitic phase could be\nexpected.\nThe antiferromagnetic alignment of MnI and MnII spin moments can b e understood from\nthe 3dspin resolved PDOS (Fig. 3b). MnI 3 dminority-spin states appear below EFbetween\n-1to -3.5eV, whereas MnII 3 dmajority-spin states appear below EFwith two well separated\nhigh PDOS region around -1.5 and -2.7eV. MnI 3 dmajority-spin states appear primarily\n5aboveEFcentered around 0.7eV; while MnII 3 dminority-spin states appear above EF\nwith the main peak at 1.1eV and a smaller peak at 0.35eV. Thus, while the minority-spin\nstates are mostly excluded from the MnII 3 dshell, the majority-spin states are excluded\nfrom the MnI 3 dshell resulting in large but oppositely aligned moments. MnI and MnII a re\nnearest neighbors (n.n.) with n.n. distance of 2.549 (2.533) ˚A in the martensitic (austenitic)\nphase. The exchange pair interaction as a function of Mn-Mn separ ation was calculated by a\nHeisenberg-like model and an antiferromagnetic coupling at short in teratomic distances was\nfound that becomes ferromagnetic at larger distances12. Thus, direct Mn-Mn interaction\nat short interatomic distance is responsible for their opposite alignm ent12,13. The energy\nseparation between the centroid of the occupied and the unoccup ied spin states of opposite\npolarization gives an exchange splitting of 2.7eV (3.1eV) for MnI (MnI I) in the martensitic\nphase. In the austenitic phase, the exchange splittings are 2.8 and 3.6 eV for MnI and MnII,\nrespectively. Thus, the Stoner parameter (ratio of exchange sp litting and magnetic moment)\nis roughly about 1 eV/ µBin both phases, which is characteristic of itinerant magnetism.\nIt was shown for Mn excess Ni 2Mn1+xGa1−xthat the magnetic moments of Mn atom in\nGa site is equal but anti-parallel to the Mn atom at Mn site26. This would tend to suggest\nthat in Mn 2NiGa, the Mn moments would cancel and a small total moment might re sult\nfrom Ni. However, this does not happen and the difference of MnI an d MnII moments is key\nto the larger total moment ( ≈1µB). This originates fromthe stronger hybridization between\nthe majority-spin Ni and MnII 3 dstates in comparison to hybridization between Ni and MnI\n3dminority-spin states. Note that Ni and MnII are n.n. separated by 2.549 (2.533) ˚A in\nthe martensitic (austenitic) phase and stronger hybridization pulls down almost all the MnII\n3dmajority-spin states below EFresulting in strong spin polarization and larger moment.\nOn the contrary, hybridization between Ni and MnI 3 dminority-spin states is relatively\nweaker, distance being larger: 2.701 (2.925) ˚A in the martensitic (austenitic) phase, and\nthere are sizable MnI 3 dminority-spin states above EFincluding the 0.35 eV peak, resulting\nin smaller moment on MnI.\nPhotoemission spectroscopy is a direct probe of the DOS in the VB re gion. In Fig. 4, the\nmain peak of the UPS VB spectrum appears at -1.4 eV and the Fermi c ut-off is at 0 eV.\nIn order to calculate the VB spectrum, we note because of the ord er of magnitude larger\nphotoemission cross-sections of Ni 3 dand Mn 3 d(4.0 and 5.3 mega barns at h ν=21.2 eV,\nrespectively)27, these PDOS determine the shape of VB28. So, we have added the Ni and\n6Mn 3dPDOS in proportion to their cross-sections, multiplied by the Fermi f unction and\nbroadened by the instrumental Gaussian resolution and the life-tim e width related energy\ndependent Lorenzian to obtain the calculated VB (Fig. 4). This is a st andard procedure of\ncomparing the photoemission spectrum from a polycrystalline sample with the calculated\nDOS28,29. The position of the main peak at -1.4 eV and the ratio between the ma in peak\nand the intensity at EFare in good agreement with UPS VB spectrum. It is clear from\nFig. 4 that the main peak is dominated by Mn 3 d-Ni 3dhybridized states that have almost\nequal contribution. States near EFare dominated by Mn 3 dstates, and the MnI 3 din\nparticular.\nThe martensitic phase DOS from Ref.14, obtained by adding up the majority and\nminority-spin DOS from Fig. 5 of Ref.14, is in clear disagreement with our DOS (Fig. 3a).\nThis prompted us to calculate the VB spectrum from the PDOS of Ref .14following the same\nprocedure as discussed above and compare it with the experimenta l UPS VB. As shown in\nFig. 4, the calculated VB based on Ref.14is in obvious disagreement with UPS VB: no\nclear peak is observed in the former; a weak broad feature is prese nt at -2 eV and the in-\ntensity near EFis highest. This shows that the martensitic phase DOS reported in Re f.14\nis inconsistent with experiment. Moreover, the large change of loca l moments (austenitic\nMnI=-2.2, MnII=3.15, Ni 0.27 µBto martensitic MnII=Ni ≈0, MnI=1.4 µB) obtained in\nRef.14is physically unexpected25, since the MnI-MnII distance change by only 0.6% in the\nmartensitic phase. Thus, it is no wonder why the total moment repo rted in Ref.14is higher\nin the martensitic phase compared to the austenitic phase, in contr adiction to their own\nmagnetization data11,14.\nElectronic bands and Fermi surface: Austenitic phase majority spin states: We now\nturn to the discussion of the electronic bands and Fermi surface o f Mn2NiGa. The majority-\nspin bands in the austenitic phase show that band 29 forms electron pockets (Fig. 5b). The\ncorresponding FS, shown in Fig. 5d, is distorted prolate ellipsoidal in s hape and occurs\naround the Xpoint of the Brillouin zone (BZ) with the long axis along the Γ Xdirection.\nThe BZ is shown in Fig. 5a. The projection of the FS along Γ Xis a square (inset, Fig. 5d),\nwhich indicates that the FS nests onto itself with n.v. 0.44(1,0,0) and 0 .44(0,1,0), in units\nof 2π/a(=1 a.u.). The nested portion of the FS is a rhombus (shown by black lin es in\nFig. 5d) of area 0.052 a.u.2with an opening angle of about 15◦.\nMartensitic phase majority spin states: In the martensitic phase, the majority-spin FS\n7TABLE I: Nesting vectors for the Fermi surface of Mn 2NiGa, in units of 2 π/a(=1 a.u.).\nAustenitic phase Martensitic phase\nBandno. Majority spin Minority spin Majority spin Minority\nspin\n29 0.44(1,0,0),\n0.44(0,1,0)– 0.34(1,0,0),\n0.34(0,1,0)–\n28 – 0.31{1,0,0}0.75(1,1,0),\n0.75(1,-1,0),\n1.13(0,0,1)–\n27 – 0.4{1,0,0}– –\nexhibits interesting modification (Fig. 5e). The majority-spin band 2 9 related electron type\nFS is now connected as continuous pipes along (1,0,0) direction, but w ith varying cross-\nsection with flat parallel parts that nest onto each other (green/ pink sheet in Fig. 5e). The\nn.v. are 0.34(1,0,0) and 0.34 (0,1,0), and compared to the austenitic p hase the direction\nis same but the magnitude of the n.v.’s is reduced. Interestingly, a se cond majority-spin\nband (28) crosses EFthat results in a hole-type cuboid FS around the Γ point that has no\ncounterpart in the austenitic phase (blue sheet, Fig. 5e). Two mut ually perpendicular n.v.’s\n0.75(1,1,0) and 0.75(1,-1,0) are identified, along with a larger n.v. of 1.1 3(0,0,1). The n.v.’s\nalong the {1,0,0}, identified above, are not expected to contribute to phonon soft ening\nbecause these hardly contribute to the electron-phonon coupling matrix element19. On the\nother hand, the 0.75(1,1,0) and 0.75(1,-1,0) n.v.’s might be responsible for the softening\nof the TA 2[110] phonon resulting in a modulated martensitic phase. The differen t nesting\nvectors are shown in Table I.\nFrom Fig. 5d and e, the majority spin FS is clearly enlarged in the marte nsitic phase\ncompared to the austenitic phase. In the contrary, for the minor ity-spin states (Fig. 5f-i),\nthe FS clearly shrinks in the martensitic phase.\nAustenitic phase minority spin states: In the austenitic phase, minority spin band 27 is\nhole-type dispersing above (below) EFat 0.2ΓL(0.5LW) and generates distorted cubic FS,\nwhere one pair of diagonally opposite corners taper out (Fig. 5f). F S nesting is observed\nbetween the cube faces with n.v. 0.4 {1,0,0}, as shown by the yellow arrows. The second\n8sheet of the FS (band 28) is electron-like, consisting of multiply conn ected pipes of square\ncross-section (inset, Fig. 5h). The parallel surfaces of the pipes nest onto each other with a\nn.v. of 0.31 {1,0,0}a.u. and a nesting area of 0.16a.u.2\nMartensitic phase minority spin states: In the martensitic phase, the minority spin hole\ntype FS (band 27) has a flower-like shape with a perforation in the mid dle (Fig. 5g). The\nelectron type FS sheet shrink to disconnected pipes of varying diam eter (Fig. 5i). These\nminority-spin FS sheets (Fig. 5g,i) in the martensitic phase do not exh ibit nesting.\nConclusion: We observe FS nesting in the martensitic phase along [1,1,0] direction in the\nmajority-spin FS that might lead to the instability of the TA 2phonon mode in Mn 2NiGa.\nThe austenitic phase FS is drastically modified in the martensitic phase . The majority spin\nFS expands in the martensitic phase, while the minority-spin FS shrink s. We show that\nMn2NiGa is an itinerant ferrimagnet in both austenitic and martensitic pha se, and that the\nMnII or Ni moments do not become zero in the martensitic phase, re futing a recent work by\nLiuet al.14. The unequal spin magnetic moments in the two inequivalent Mn atoms (MnI\nand MnII) arise from the difference in the hybridization of the MnI 3 d-Ni 3dand MnII 3 d-\nNi 3dstates, which inturn isrelated to theinteratomic distances. We fur thermoreshow that\nin Mn 2NiGa a large tetragonal distortion ( c/a=1.25) decreases the total energy, stabilizing\nthe lower temperature martensitic phase. Mn 2NiGa would be an ideal system to study\ndifferent models of magnetization in metals since it has a simple L21structure and three\nsublatticemagnetizationwithparallel(betweenMnIIandNi)andant i-parallel(betweenMnI\nand MnII) magnetic moment alignment. Possibility of incommensurate magnetic phase or\ncharge density wave instabilities could be expected at low temperatu res due to presence\nof FS nesting and ferrimagnetism. Low temperature x-ray diffract ion might be able to\ndetect possible occurrence of a charge density wave state. Neut ron scattering, angle resolved\nphotoemission or Compton scattering experiments can verify the t heoretically predicted FS.\nIn fact, FS nesting, ferrimagnetism and large magnetoelastic coup ling makes Mn 2NiGa a\nhighly interesting material that has remained largely unexplored so f ar.\nWe thank K. KUNC and A. DE SARKAR for fruitful discussions. P. CHA DDAH, V.\nC. SAHNI, K. HORN, A. GUPTA and S. M. OAK are thanked for suppor t. Ramanna\nFellowship Research Grant and D.S.T.-Max Planck Partner Group Proj ect are thanked for\n9funding.\n1KAINUMA R. et al., Nature, 439(2006) 957.\n2TAKEUCHI I. et al., Nature Materials, 2(2003) 180.\n3SOZINOV A., LIKHACHEV A. A., LANSKA N., AND ULLAKKO K., Appl. Phys. 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B 62(2000) 1806.\n24PERDEW J. P. et al., Phys. Rev. B, 46(1992) 6671.\n25BARMAN S. R. AND CHAKRABARTI A., Phys. Rev. B (Comment, submitted ).\n26ENKOVAARA J., HECZKO O., AYUELA A., AND NIEMINEN R. M., Phys. Rev. B, 67\n(2003) 212405.\n27YEH J. J. AND LINDAU I., At. Data Nucl. Data Tables, 32(1985) 1.\n28CHAKRABARTIA. , BISWASC., BANIKS., DHAKAR.S., SHUKLAA.K. , ANDBARMAN\nS. R., Phys. Rev. B, 72(2005) 073103.\n29SARMA D. D. et al., Phys. Rev. Lett., 75(1995) 1126; FUJIMORI A. AND MINAMI A.,\nPhys. Rev. B, 30(1984) 957; BARMAN S. R. AND SARMA D. D., Phys. Rev. B, 51(1995)\n4007; BARMAN, S. R. et al., Phys. Rev. B 53(1996) 3746; BROWN D. et al., Phys. Rev. B\n57(1998) 1563.\n11Figure Captions\nFig. 1The structure of Mn 2NiGa in the (a) austenitic and (b) martensitic phase; the blue,\ngreen, red, and brown spheres represent Ni, MnI, MnII and Ga, r espectively.\nFig. 2 (a) The calculated total energies ( Etot) of Mn 2NiGa as a function of cell volume\nof the austenitic and martensitic phase. (b) Three dimensional plot of the spin magnetic\nmoment distribution (in unit of e ˚A−3) in the (110) plane in the martensitic phase, a contour\nplot is shown in the bottom.\nFig. 3(a) Comparison of total density of states (DOS) and Ni 3 dand Mn 3 dpartial DOS\nof Mn 2NiGa between the martensitic and austenitic phases (b) minority- an d majority-spin\ncomponents of the DOS in the martensitic phase.\nFig. 4UPS valence band (VB) spectrum of Mn 2NiGa in the martensitic phase compared\nwith theoretical VB spectrum calculated from the DOS in Fig. 3a. The contributions from\nthe Mn 3 dand the Ni 3 dpartial DOS are also shown. The spectra have been shifted along\nthe vertical axis for clarity of presentation.\nFig. 5 (a) The f.c.c. Brillouin zone showing the high symmetry directions. (b) Majority\nand(c)minority-spinenergybandsofMn 2NiGaintheausteniticphase. Majority-spinFermi\nsurface (FS) of the (d) austenitic phase compared to the (e) mar tensitic phase FS related to\nbands 28 and 29. Minority-spin austenitic phase FS related to (f) ba nd 27 and (h) band 28.\nInsets show the FS in a different orientation. Martensitic phase mino rity-spin FS related to\n(g) band 27 and (i) band 28. All the FS are shown in the repeated zon e scheme and yellow\narrows represent the nesting vectors. Black arrows relate the F S of the two phases.\n12This figure \"Fig1_mngth.gif\" is available in \"gif\"\n format from:\nhttp://arxiv.org/ps/0707.2133v2This figure \"Fig2m_mngth.gif\" is available in \"gif\"\n format from:\nhttp://arxiv.org/ps/0707.2133v2This figure \"Fig3_mngth.gif\" is available in \"gif\"\n format from:\nhttp://arxiv.org/ps/0707.2133v2This figure \"Fig4m_mngth.gif\" is available in \"gif\"\n format from:\nhttp://arxiv.org/ps/0707.2133v2This figure \"Fig5m_mng.gif\" is available in \"gif\"\n format from:\nhttp://arxiv.org/ps/0707.2133v2" }, { "title": "1101.3645v3.Mott_transition_and_ferrimagnetism_in_the_Hubbard_model_on_the_anisotropic_kagomé_lattice.pdf", "content": "arXiv:1101.3645v3 [cond-mat.str-el] 16 Apr 2013Mott transition and ferrimagnetism in the Hubbard model on t he anisotropic kagom´ e\nlattice\nA. Yamada1, K. Seki1, R. Eder2, and Y. Ohta1\n1Department of Physics, Chiba University, Chiba 263-8522, J apan\n2Karlsruhe Institut of Technology, Institut f¨ ur Festk¨ orp erphysik, 76021 Karlsruhe, Germany\n(Dated: November 12, 2018)\nMott transition and ferrimagnetism are studied in the Hubba rd model on the anisotropic kagom´ e\nlattice using the variational cluster approximation and th e phase diagram at zero temperature and\nhalf-fillingis analyzed. The ferrimagnetic phaserapidly g rows as thegeometric frustration is relaxed,\nand the Mott insulator phase disappears in moderately frust rated region, showing that the ferri-\nmagnetic fluctuations stemming from the relaxation of the ge ometric frustration is enhanced by the\nelectron correlations. In metallic phase, heavy fermion be havior is observed and mass enhancement\nfactor is computed. Enhancement of effective spatial anisot ropy by the electron correlations is also\nconfirmed in moderately frustrated region, and its effect on h eavy fermion behavior is examined.\nPACS numbers: 71.30.+h, 71.10.Fd, 71.27.+a\nEffect of geometric frustration is one of the most\nimportant subjects actively studied in the field of\nstrongly correlated electron systems. For instance,\nthe heavy fermion behavior in LiV 2O41,2with py-\nrochlore lattice structure, and the spin liquid states\nin the triangular-lattice organic materials κ-(BEDT-\nTTF)2X3–5and herbertsmithite ZnCu3(OH) 6Cl2with\nkagom´ e lattice structure6,7have attracted a lot of at-\ntentions.\nWhen spatial anisotropy is introduced in systems with\ngeometricfrustration,theinterplaybetweenthespinfluc-\ntuations and Mott transition appears as a new feature\nand provides unique phenomena which take place nei-\nther in the unfrustrated nor fully frustrated systems. A\nreentrant behavior of the Mott transition observed in the\nκ-(DEBT-TTY) 2Cu[N(CM) 2]Cl under pressure3,5is an\ninterestingexample realizedonan anisotropictriangular-\nlattice, where that behavior stems from the enhancement\nof the antiferromagnetic fluctuations due to the electron\ncorrelations.8,9\nAs for the kagom´ e lattice, which is a prototype of frus-\nFIG. 1: (Color online) (a) Anisotropic kagom´ e lattice.\nIn our lattice geometry, the three sites 1 ,2, and 3 form\nan equilateral triangle of the unit length and the dashed\nlines are along the xdirection. Inside the hexagon (dotted\nline) and square (dash-dotted line) are the 12- and 6-site\nclusters, respectively, which will be used in our analysis.\n(b) The first Brillouin zone of the anisotropic kagom´ e latti ce.trated systems, a fully frustrated case has been theoreti-\ncally studied in detail,10–13however the issues related to\nthe anisotropy have been considered only recently. The\nMott transition and magnetic properties near the transi-\ntion havebeen studied using the cellular dynamicalmean\nfield theory14, where the Mott transition point was an-\nalyzed and enhancement of spatial anisotropy and spin\ncorrelations were observed. Such enhancement may give\nrisetothe extensionoftheordered(ferrimagnetic)phase.\nTherefore, if the Mott transition itself persists without\nbeing veiled by the ferrimagnetic phase remains to be\nexamined. Also, the effect of the enhanced anisotropy on\nthe heavy fermion behavior is worth being studied.\nIn this paper, we investigate the ferrimagnetism and\nMott transition on the anisotropic kagom´ e lattice using\nthe variational cluster approximation (VCA)15–17, which\nis formulated based on a rigorous variational principle\nand exactly takes into account the short-range correla-\ntions. We study the phase diagram at zero temperature\nand half-filling. We show that, in moderately frustrated\nregion,theferrimagneticphaserapidlygrowsdowntothe\nmetal-insulator phase boundary, indicating that the spin\ncorrelations stemming from the relaxation of the frus-\ntration is enhanced by the electron correlations and the\nMott insulator (MI) phase disappears. In the metallic\nphase, heavy fermion behavior is observed and the mass\nenhancement of the quasiparticle is computed. Effective\nspatial anisotropy becomes also larger due to the elec-\ntroncorrelations,inagreementwiththepreviousstudy.14\nThis effect givesrise to an enhancement ofthe anisotropy\nof the effective masses of the quasiparticles.\nThe Hamiltonian of the Hubbard model on the\nanisotropic kagom´ e lattice (see Fig. 1) reads\nH=−/summationdisplay\ni,j,σtijc†\niσcjσ+U/summationdisplay\nini↑ni↓−µ/summationdisplay\ni,σniσ,(1)\nwheretij=tbetween the sites 1 and 2, 3 and tij=t′\nbetweenthesites2and3, Uistheon-siteCoulombrepul-\nsion, and µis the chemical potential. The annihilation\n(creation) operator for an electron at site iwith spin σis2\ndenoted as cjσ(c†\niσ) andniσ=c†\niσciσ. The system corre-\nsponds to the fully frustrated kagom´ e lattice at t′/t= 1,\nandfrustrationbecomesweakerwithdecreasing t′/t. The\nend memberat t′/t= 0is adecoratedsquarelattice. The\nenergy unit is set as t= 1 hereafter.\nWe use VCA15–17to examine the phase diagram and\nbehavior of the quasiparticles in the metallic phase at\nzero temperature. VCA is an extension of the cluster\nperturbationtheory15basedontheself-energy-functional\napproach.17This approach uses the rigorous variational\nprinciple δΩt[Σ]/δΣ = 0 for the thermodynamic grand-\npotential Ω twritten as a functional of the self-energy Σ\nΩt[Σ] =F[Σ]+Trln( −(G−1\n0−Σ)−1).(2)\nIn the above expression, F[Σ] is the Legendre transform\nof the Luttinger-Ward functional18and the index tde-\nnotes the explicit dependence of Ω ton all the one-body\noperators in the Hamiltonian. The stationary condition\nforΩt[Σ]leadstotheDyson’sequation. AllHamiltonians\nwith the same interaction part share the same functional\nform ofF[Σ], and using that property F[Σ] can be eval-\nuated from the exact solution of a simpler Hamiltonian\nH′, though the space of the self-energies where F[Σ] is\nevaluated is now restricted to that of H′. In VCA, one\nuses forH′a Hamiltonian formed ofclusters that are dis-\nconnected by removing hopping terms between identical\nclusters that tile the infinite lattice. A possible symme-\ntry breaking is investigated by including in H′the cor-\nresponding Weiss field that will be determined by mini-\nmizing the grand-potential Ω t. Rewriting F[Σ] in terms\nof the grand-potential Ω′≡Ω′\nt[Σ] and Green function\nG′−1≡G′\n0−1−Σ of the cluster Hamiltonian H′, the\ngrand-potential is expressed as\nΩt(t′) = Ω′−/integraldisplay\nCdω\n2πeδω/summationdisplay\nKlndet/parenleftbig\n1+(G−1\n0−G′\n0−1)G′/parenrightbig\n(3)\nand is now a function of t′. The functional trace has be-\ncome an integral over the diagonal variables (frequency\nandsuperlatticewavevectors)ofthelogarithmofadeter-\nminant over intra-cluster indices. The frequency integral\nis carried along the imaginary axis and δ→+0. The\nstationary solution of Ω t(t′) and the exact self-energy of\nH′at the stationary point, denoted as Σ∗, are the ap-\nproximate grand-potential and self-energy of Hin VCA,\nand physical quantities, such as expectation values of the\none-body operators, are calculated using the Green func-\ntionG0−1−Σ∗. In VCA, the restriction of the space of\nthe self-energies Σ into that of H′is the only approxi-\nmation involved and short-range correlations within the\ncluster are exactly taken into account by exactly solving\nH′.\nIn our analysis, the 6- and 12-site clusters in Fig. 1(a)\nare used to set up the cluster Hamiltonian H′. These\nclustershaveevennumberofsitessothatasingletground\nstate is possible. To study the ferrimagnetism, the Weissfield\nHF=hF/summationdisplay\nisign(i)(ni↑−ni↓) (4)\nwith sign( i) =−1 for the site 1 and sign( i) = 1 for\nthe sites 2 and 3, is also included. In the stationary\npoint search of Ω( µ′,hF), which we denote as the grand-\npotential per site, the Weiss field hFand the cluster\nchemical potential µ′are treated as the variational pa-\nrameters, where the latter should be included for the\nthermodynamic consistency.19During the search, the\nchemical potential of the system µis also adjusted so\nthat the electron density nis equal to 1 within 0.1%. In\ngeneral, a stationary solution with hF/ne}ationslash= 0 corresponding\nto the ferrimagnetic state and that with hF= 0 corre-\nspondingtothe paramagneticstateareobtained, and the\nground-state energies per site E= Ω+µnare compared\nto determine which solution (ferrimagnetic or paramag-\nnetic) is stable. The density of state\nD(ω) = lim\nη→0/integraldisplayd2k\n(2π)23/summationdisplay\nσ,a=1{−1\nπImGaσ(k,ω+iη)}(5)\nis also calculated to examine the gap. To be precise,\nD(ω) is calculated for η= 0.2, 0.1, and 0.05, andη→0\nlimit is evaluated by the standard extrapolation method.\nThe numerical error after this extrapolation is estimated\nto be of order 10−3, so the gap is identified as the re-\ngion ofωaroundω≃0 where the extrapolated D(ω) is\nless than 10−2. We also compute the ferrimagnetic order\nparameter per site\nM=3/summationdisplay\na=1(/an}bracketle{tna↑/an}bracketri}ht−/an}bracketle{tna↓/an}bracketri}ht)\nand the double occupancy per site\nDocc.=1\n33/summationdisplay\na=1/an}bracketle{tna↑na↓/an}bracketri}ht=dE\ndU\nwhere/an}bracketle{tnaσ/an}bracketri}htand/an}bracketle{tna↑na↓/an}bracketri}htare the expectation values of\nnaσandna↑na↓, respectively, with a=1, 2, and 3 being\nthe sites in Fig. 1(a).\nIn Fig. 2, we show the phase diagram at zero temper-\nature and half-filling obtained from this analysis using\nthe 12-site cluster. The results obtained using the 6-site\ncluster are also shown to quantitatively see the cluster\nsize dependence. The critical interaction strength UF\nseparating the ferrimagnetic and MI phases rapidly de-\ncreasesinthemoderatelyfrustratedregion t′= 0.5∼0.7,\nshowingthat the ferrimagneticfluctuations due to the re-\nlaxation of the geometric frustration is enhanced by the\nelectron correlations. (At t′= 0.75UF>20 for the 6-\nsite cluster.) In this region of t′, the ferrimagnetic phase\nis an insulator since there is a gap, and the transition\nbetween the ferrimagnetic and paramagnetic (including\nMI) phases is a level crossing (first order) because the3\n 0 5 10 15\n 0.5 0.6 0.7 0.8 0.9 1U\nt'Ferrimagnetic insulator Mott insulator\nParamagnetic metal\nFIG. 2: (Color online) Phase diagram of the Hubbard model\non the anisotropic kagom´ e lattice at zero temperature and\nhalf-filling as a function of t′andUobtained by VCA, where\nthe 12-site cluster is used (the crosses and circles). Lines are\nguides to the eye. The triangles and squares are the results\nobtained using the 6-site cluster. The crosses and triangle s\ncorrespond to the ferrimagnetic and paramagnetic transiti on\npoints and circles and squares are for the Mott transition\npoints.\nferrimagnetic solutions exist also U < U Feven though it\nis energetically disfavored there. The critical interaction\nstrength UMIseparating the MI and metallic phases is\nslightly smaller than the noninteracting band width W,\nwhereW= 6 att′= 1 and W= 4√\n2≃5.66 att′= 0.\nUMIdecreasesasthe geometricfrustrationisrelaxed, and\nthe slope becomes steeper in moderately frustrated re-\ngion. For the 12-site results, at t′= 0.5,UF= 4.0 while\nUMI= 4.1 so the MI phase has disappeared. Taking into\naccount the drastic growth of ferrimagnetic phase and\nthe fact that Wremains almost the same, the decrease\nofUMIaccording to the relaxation stems from the fer-\nrimagnetic fluctuations. As for the Mott transition, we\ncould not find out the Mott insulator and paramagnetic\nmetal coexisting region of Uat half-filling within our two\ncontrolling parameters µandµ′. Also as will be shown\nlater the Mott gap changes continuously as a function of\nU. Therefore, we could not find out an indication of the\ndiscontinuity at the Mott transition in this analysis. To\nsupplement this analysis, we show in Fig. 3 the double\noccupancy Docc.as a function of Ufor the 12-site clus-\nter, which also looks continuous at the transition point.\nIn Ref. 14 this transition is reported to be first order.\nFirst order Mott transitions are obtained in other models\nin the variational cluster approach with bath degrees of\nfreedom and treating the hybridization between the bath\nsites and cluster sites as a variational parameter.20,21In\nthese analyses, the coexisting metal and insulator solu-\ntions, leading to the first order transition, differ by the\nvalue of these hybridization parameters, and these situa-\ntions will be similar to the case of Ref. 14. Our analysis\ndoes not have bath degrees of freedom and technically\nthis will be the origin of the difference. It remains to be\nclarified which is the correct picture.\nNext we consider the cluster size dependence of our 0.08 0.1 0.12 0.14 0.16 0.18 0.2\n 3.5 4 4.5 5 5.5 6 6.5Docc.\nUt' = 1.0\nt' = 0.8\nt' = 0.6\nFIG. 3: (Color online) The double occupancy Docc.as a\nfunction of Ufort′= 1.0, 0.8, and 0 .6. The 12-site cluster\nisused. Thethreearrows indicatetheMott transitionpoint s.\n 0 2 4∆(a) t' = 1.0\n12 sites\n6 sites(b) t' = 0.8\n12 sites\n6 sites\n01\n0 2 4 6 8∆\nU(c) t' = 0.6\n12 sites∆\nM\n0 2 4 6 8 10 0 1\nM\nU(d) t' = 0.6\n6 sites∆\nM\nFIG. 4: (Color online) Mott gap ∆ as a function of Uat (a)\nt′= 1.0, (b)t′= 0.8, and (c), (d) t′= 0.6. These parameter\nregions correspond to the three vertical lines in Fig. 2. For\nt′= 0.6, order parameter Mis also included and the vertical\nlines separate the ferrimagnetic and MI phase.\nresults. In general, UMIis larger for larger clusters,\nsince the kinetic energyofthe cluster Hamiltoniancan be\nlarger for larger clusters. As for UF, when spin correla-\ntions are highly suppressed due to the frustration, cluster\nwavefunctions with small ferrimagneticfluctuations play\nan important role to examine near the true minimum of\nthe effective potential, so UFis smaller for larger clus-\nters. When the geometric frustration is moderate and\nspin correlations are not largely suppressed, the differ-\nence of cluster kinetic energies due to the cluster size be-\ncomes more important to determine the phase boundary,\nsoUFbecomes larger for larger clusters. Our result is\nconsistent with this general argument on the cluster size\ndependence. Quantitatively, UFis almost the same for\nthe 12- and 6-site clusters at t′= 0.6, andUFis smaller\nfor the 12-site clusters for t′>0.6. Relatively large dif-\nference of UFbetween the 12- and 6-site cluster results in\nstrongly frustrated region indicates strong suppression of\nthe spin correlations. The difference of UMIbetween the4\nFIG. 5: Spectral density at t′= 0.75 for (a), (b) U= 8\n(ferrimagnetic state), (c) U= 6 (MI state), and (d) U= 3\n(metallic state) along the dotted line in Fig. 1(b). The\nLorentzian broadening with η= 0.15tis used in all the\ncases. In (a) and (b), the solid lines are the mean-field\nSDW dispersion for the same values of U,t′andµ. In\n(d), the solid lines are the noninteracting band structure.\nIn(a), (b), and (c)thepeaks are scaled by5compared to(d).\n12-site and 6-site analysis is less than 20% of W. The\nbehavior of our UMIaccording to the relaxation of the\nfrustration is qualitatively consistent with the previous\nresults14, though our values for UMIare relatively small\ncompared to those in Ref. 14. At present the origin of\nthese discrepancies are not clear to us.\nIn Fig. 4 we show the Mott gap ∆ and ferrimagnetic\norderparameter Masfunctionsof Ufort′= 1.0,0.8, and\n0.6 corresponding to the three vertical lines in Fig. 2. ∆\nmonotonically decreases as Udecreases in all cases and\nMis always smaller for the 12-site cluster at t′= 0.6.\nThe gap ∆ Fin the ferrimagnetic phase is slightly larger\nthan ∆ if Uis the same. For example, for U= 10,\n∆ = 4.35 att′= 0.8 while ∆ F= 4.91 att′= 0.7 and\n∆F= 5.54 att′= 0.5 in the 12-site analysis.\nIn Fig. 5 we show the spectral weight function ρ(ω,k)\ncalculated using the 12-site cluster for solutions corre-\nsponding to (a), (b) the ferrimagnetic phase (up and\ndown spin parts are plotted separately), (c) the MI\nphase, and (d) the metallic phase at t′= 0.75. In (a) 1 1.1 1.2 1.3 r(a)\nt' = 0.6\nt' = 0.8\nt' = 1.0t' = 0.4\nt' = 0.5\n12\n m*/m(b) t' = 1.0 x\ny\n12\n0 1 2 3m*/m\nU(c) t' = 0.8x\ny\n0 1 2 312\n m*/m\nU(d) t'= 0.6x\ny\nFIG. 6: (Color online) (a) Ratio ras a function of Ufor\nt′= 1.0, 0.8, and 0.6. The two arrows indicate the value of r\nfor noninteracting band at t′= 0.5 (r= 1.24) and at t′= 0.4\n(r= 1.30). (b)∼(d) Mass enhancement factor m∗/min the\nxandydirections as functions of Ufor (b)t′= 1.0, (c)\nt′= 0.8, and (d) t′= 0.6. The lines are obtained using the\n12-site cluster and symbols (squares, triangles, and circl es)\nare the results with the 6-site cluster. In (a) the squares,\ntriangles, and circles correspond to t′= 1.0, 0.8, and 0 .6,\nrespectively. In (b) ∼(d) the triangles correspond to the x\ndirection while the squares correspond to the ydirection.\nand (b), the mean-field spin-density-wave (SDW) disper-\nsion is also included (solid lines) to see its general fea-\ntures, where M= 0.92 in the mean-field solution while\nM= 0.72 in VCA. In (d), the noninteracting band struc-\nture is also plotted (solid lines) for comparison. In (a)\nand (b), the dispersion is largely affected due to the elec-\ntron correlations compared to the mean-field solution.\nIn (c), the SDW dispersion disappears and the spectral\nfunction displaysa Mott gap acrossall wavevectors. The\ngap is smaller compared to (a) and (b) since Uis smaller.\nComparingwith (d), in (c) it looksthat the lowestenergy\nband in (d) is shifted downwards and the second band\nsplits into lower and upper Hubbard bands while the top\nflat band remains almostthe same. In (d), we notice that\nthe spectral function is consistent with the Fermi liquid\nstateandthe interactingbandsslightlyshrinktowardthe\nFermi surface, leading to heavy fermion behavior.\nTo study it in detail, we consider well below the MI\ntransition line where Fermi liquid natures are confirmed\nfrom the behavior of the spectral function and com-\npute the mass enhancement factor m∗/malong the x\nandydirections in kspace, where the xdirection cor-\nresponds to the direction of t′hopping in real space.\nNear the Fermi surface the position of the peak ( ω,k) =\n(ωF+δω,kF+δk) ofρ(ω,k) changes according to the re-\nlation|δω|= (kF/m∗)δkand the band mass m∗is calcu-\nlatedusingthisrelation. Wealsocomputetheratioofthe\nFermi momenta in the xandydirections, r=kyF/kxF.\nAst′decreases, the noninteracting Fermi surface slightly\nshrinks in the xdirection and slightly evolves in the y\ndirection10, soris a measure of the anisotropy includ-\ning the effect of the electron correlations. Even though5\nprecise values of these quantities may depend on the lat-\ntice geometry, we show in Fig. 6 randm∗/min thex\nandydirections as functions of Ufort′= 1.0, 0.8, and\n0.6 in our lattice geometry, to see general features about\nthe effect of the electron correlations on these quantities.\nThe lines are the results obtained using the 12-site clus-\nter and symbols are the results with the 6-site cluster.\nAtt′= 0.6,rrapidly grows around U≃3 for the 12-\nsite cluster. This tendency is also observed for the 6-site\ncluster, where rturns to grow around U≃1.5. The\nrapid growth of rindicates that the effective anisotropy\nis enhanced due to the electron correlations in moder-\nately frustrated region. In fact, for the 12-site cluster,\nthe value of ratt′= 0.6 andU= 3.5 is equal to that\nof noninteracting band at t′= 0.4. The analysis of the\neffective anisotropy was also done in Ref. 14 by consid-\nering the renormalization of the hoping parameters and\nour results are qualitatively consistent with their analy-\nsis. This enhancement of effective anisotropy indicates\nthat the spin fluctuations due to the relaxation of the\ngeometric frustration are also enhanced by the electron\ncorrelations. Within our analysis, this indication is con-\nsistent with the rapid growth of the ferrimagnetic phase\naboveUMI, and it is demonstrated by the analysis of\nthe spin correlations.14As is shown in Fig. 6(b) ∼(d), the\nheavy fermion behavior is observed in all cases for the\n12-site cluster. For the 6-site cluster sizable mass en-\nhancements are not observed and the 6-site cluster may\nnot be large enough for subtle analysis related to thespectral function. The growth of raffects also m∗/m\nsincekFenters into the calculation of m∗. In general,\nm∗is enhanced in the ydirection and suppressed in the\nxdirection since the Fermi surface shrinks in the xdirec-\ntion and evolves in the ydirection. This appears largely\nin moderately frustrated region due to the rapid growth\nofr, as is observed around U∼3.5 for the 12-site clus-\nter results in Fig. 6(d). Therefore the anisotropy of the\neffective masses is enhanced in moderately frustrated re-\ngion.\nIn summary we have investigated the ferrimagnetism\nand Mott transition on the anisotropic the kagom´ e lat-\ntice using VCA. The phase diagram at zero temperature\nand half-filling is determined. The ferrimagnetic phase\nrapidly grows in moderately frustrated region and the\nMI phase disappears there. In the metallic phase, heavy\nfermion behavior is studied and the mass enhancement\nis computed. Enhancement of spatial anisotropy due to\nthe electron correlations is also observed for moderately\nfrustrated region and its effect on the heavy fermion be-\nhavior is discussed. Thus, the interplay between the spin\ncorrelationsand Mott transitionis quantitatively studied\nabove and below the metal-insulator transition.\nOne of us (A.Y.) would like to thank N. Fukui,\nK. Kurasawa, H. Mikami, and H. Nakada for useful dis-\ncussions on numerical analysis. This work was supported\nin part by Kakenhi Grant No. 22540363of Japan. A part\nof computations was done at Research Center for Com-\nputational Science, Okazaki, Japan.\n1S. Kondo, D. C. Johnston, C. A. Swenson, F. Borsa1, A.\nV. Mahajan, L. L. Miller, T. Gu, A. I. Goldman, M. B.\nMaple, D. A. Gajewski, E. J. Freeman, N. R. Dilley, R. P.\nDickey, J. Merrin, K. Kojima, G. M. Luke, Y. J. Uemura,\nO. Chmaissem, and J. D. Jorgensen, Phys. Rev. Lett. 78,\n3729 (1997).\n2P. E. J¨ onsson, K. Takenaka, S. Niitaka, T. 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Tremblay, M.\nPotthoff, Eur. Phys. Lett. 85, 17002 (2009)." }, { "title": "1710.07779v1.Correlation_between_Compensation_Temperatures_of_Magnetization_and_Angular_Momentum_in_GdFeCo_Ferrimagnets.pdf", "content": " \n1 Correlation between Compensation Temperatures of Magnetization and \nAngular Momentum in GdFeCo Ferrimagnets \nYuushou Hirata1, Duck-Ho Kim1†, Takaya Okuno1, Tomoe Nishimura1, Dae-Yun Kim2, \nYasuhiro Futakawa3, Hiroki Yoshikawa3, Arata Tsukamoto3, Kab-Jin Kim4, Sug-Bong Choe2, \nand Teruo Ono1,5† \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 6 11-0011, Japan. \n2Department of Physics and Institute of Applied Physics, Seoul N ational University, Seoul \n08826, Republic of Korea. \n3College of Science and Technology, Nihon University, Funabashi, Chiba 274-8501, Japan. \n4Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon \n34141, Republic of Korea. \n5Center for Spintronics Research Network, Graduate School of Eng ineering Science, Osaka \nUniversity, Machikaneyama 1-3, Toyonaka, Osaka 560-8531, Japan. \n†Correspondence to: kim.duckho.23z@st.kyoto-u.ac.jp , ono@scl.kyoto-u.ac.jp \nDetermining the angular momentum compensation temperature of fe rrimagnets \nis an important step towards ferrimagnetic spintronics, but is not generally easy to \nachieve it experimentally. We propose a way to estimate the ang ular momentum \ncompensation temperature of ferrimagnets. We find a linear rela t i o n b e t w e e n t h e \ncompensation temperatures of the magnetization and angular mome ntum in GdFeCo \nferrimagnetic materials, which is proved by theoretically as we ll as experimentally. The \nlinearity comes from the power-l aw criticality and is governed by the Curie \ntemperature and the Landé g factors of the elements composing t he ferrimagnets. \n2 Therefore, measuring the magnetization compensation temperature and the Curie \ntemperature, which are easily ass essable experimentally, enable s to estimate the angular \nmomentum compensation temperature of ferrimagnets. Our study pr ovides efficient \navenues into an exciting world of ferrimagnetic spintronics. \nAntiferromagnets came into the spotlight in the last decade [1– 6] as a promising \nmaterial for spintronics devices because they exhibit fast magn etic dynamics and low \nsusceptibility to magnetic fields. These advantages originate f rom the antiferromagnetic \nordering in which the magnetic moments are compensated on an at o m i c s c a l e . T h i s a l s o \nimplies that it is difficult to efficiently manipulate antiferr omagnets using external magnetic \nfields, hindering the study of antiferromagnetic spin dynamics. However, recently, magnetic \nfield-controlled antiferromagnetic spin dynamics has been achie ved using ferrimagnets [7]. \nH e n c e , f e r r i m a g n e t s h a v e b e c o m e a p r o m i s i n g m a t e r i a l i n t h e e m e rging field of \nantiferromagnetic spintronics. \nFerrimagnetic materials comprise rare earth (RE) and transition m e t a l ( T M ) \ncompounds, wherein the spins of two inequivalent sublattices ar e coupled \nantiferromagnetically [8-10]. Because of the different Landé g factors of RE and TM \nelements, ferrimagnets exhibit compensation temperatures of mag netization and angular \nmomentum [11], at which the magnetizations (angular momenta) of the RE and TM \nsublattices have the same magnit ude but opposite directions. Co nsequently, the net \nmagnetization (angular momentum) is compensated. The compensati on temperatures have \nbeen studied experimentally and theoretically [7, 12–23]. In pa rticular, Kim et al . recently \nobserved fast field-driven domain wall (DW) motion in the vicin ity of the compensation \ntemperature of the angular momen tum [7]. This observation revea ls that ferrimagnets exhibit \nthe antiferromagnetic dynamics because of the zero net angular momentum a t the \n3 compensation temperature of the angular momentum, even though t hey have magnetic m\noments. Although remarkable efforts have been made theoreticall y and experimentally [7, \n12–23] in understanding the role of angular momentum compensati on in DW dynamics, it is \ndifficult to determine the angular momentum compensation temper ature because of the \nmethodological complexities, impeding the rapid development of this exciting research field \nas well as the fundamental understanding of the compensation te mperatures. \nI n t h i s l e t t e r , w e r e p o r t t h e c o r r e l a t i o n b e t w e e n t h e a n g u l a r m omentum and \nmagnetization compensation temper atures in ferrimagnetic GdFeCo a l l o y s . I t i s \nexperimentally demonstrated that the angular momentum compensat ion temperature is \ndirectly related to the magnetization compensation temperature, regardless of the sample \nstructures. The results show that there exists a strong correla tion between the two types of \ncompensation temperatures. We theoretically verified the correl ation on the basis of a simple \nmodeling technique. Moreover, the proposed approach is a novel method of determining the \nangular momentum compensation temperature. \nFor this study, we prepared six types of amorphous ferrimagnet GdFeCo films. Table \nI lists the detailed sample struc tures. The films were grown by co-sputtering, and the \ncompositions were estimated fr om the relative deposition rates of Gd and FeCo. The samples \nexhibit perpendicular magnetic anisotropy (PMA) with circular d omain expansion. As shown \nin Fig. 1( a), an e-beam lithography technique was applied to the structura l devices with a \nHall bar geometry in order to de tect the anomalous Hall resista nce. \nFirst, we characterized the magne tic properties of the GdFeCo s amples. Figure 1( b) \nshows the hysteresis loop of the GdFeCo microstrip. The anomalo us Hall resistance ܴୌ i s \nmeasured as a function of the perpendicular magnetic field ܪ௭ at room temperature. The \nclear, square, hysteresis loop shows that the GdFeCo samples ha ve strong PMA. The orange- \n4 colored arrow represents the magnetization switching field, whi c h i s r e f e r r e d t o a s t h e \ncoercive field ܪୡ. \nTo determine the magnetization compensation temperature ܶ, we measured ܴୌ b y \nsweeping ܪ௭ at each temperature [7, 18, 20, 23]. The magneto-transport pro perties are \ndominated by the FeCo moment because the energy of the 4f shell of Gd is located far below \nthe Fermi energy level [23]. A sign change in ܴୌ indicates a change in the direction of the \nFeCo moment. Using ܴୌ as a function of ܪ௭, we define the Hall resistance difference as \n∆ܴୌ≡ܴୌ൫ܪ,ୱୟ୲൯െܴୌ൫െܪ,ୱୟ୲൯ ( s e e F i g . 1 ( b)). Figure 2( a) shows the Hall resistance \ndifference ∆ܴୌ as a function of the temperature ܶ f o r S a m p l e I I , w h e r e ܪ,ୱୟ୲ and \nെܪ,ୱୟ୲ are the saturation fields with the condition ܪୡ൏ܪ,ୱୟ୲ (see inset of Fig. 1( b)). ∆ܴୌ \nto zero determines ܶൌ\t160 K, indicated by a blue dot. \nTo determine ܶ, we measured the field-driven DW speed as a function of the \ntemperature, as proposed elsewhere [7]. We first applied a suff iciently strong magnetic field \nwith a magnitude of –200 mT ( െܪ,ୱୟ୲) to saturate the magnetization along the –z \ndirection, and subsequently, a constant ܪ௭ for driving the DW. ܪ௭ i s s e l e c t e d l o w e r t h a n \nܪୡ, to eliminate the nucleation of the domain. Next, we applied a DC current ܫ௫ a l o n g t h e \nwire to detect the Hall signal (see red arrow in Fig. 1( a)), where ܫ௫ is sufficiently small to \nprevent spin torques and Joule heating effect [24–26]. We then injected a current pulse ܫ௬ \n(12 V, 100 ns) through the writing line (see blue arrow in Fig. 1(a)) to nucleate the reversed \ndomain, thereby creating two DWs in the wire. The created DW mo ves along the wire \nbecause of the presence of ܪ௭, and then, passes through the Hall bar; the DW arrival time \ncan be detected by monitoring the change in the Hall voltage us ing an oscilloscope. The DW \nspeed can be calculated from the arrival time and the distance traveled between the writing \nline and the Hall bar (400 µm). \n5 Figure 2( b) shows the DW speed ݒ as a function of the temperature ܶ a t ߤܪ௭ൌ\n\t80 mT for Sample II. This figure clearly shows that ݒ exhibits a peak at a certain \ntemperature (indicated by purple arrow). This tendency of ݒ with respect to ܶ is consistent \nwith the results given elsewhere [7]. Accordingly, ܶ can be determined, as shown in Fig. \n2(b). Here, the difference between ܶ and ܶ is defined as ∆\tܶሺ≡ܶെܶሻ, indicated by \nblack double arrows. \nFor a quantitative comparison, the values of ܶ are directly plotted with respect to \nܶ for all the samples, as shown in Fig. 3. It is interesting to note that all the values ሺܶ,ܶሻ \nlie on a single curve with linearity. The red line represents t he best linear fit with a slope of \n0.87 and a y-axis intercept of 101.1 K. From this result, we ex perimentally found that there \nexists a strong correlation between ܶ and ܶ for all the GdFeCo films. \nTo understand the correlation of ሺܶ,ܶሻ, we employ a theory based on a power-law \ncriticality, given that the variation in the magnetization as a function of the temperature can \nbe reasonably well approximated [27, 28, 29]. This function des cribes the temperature \ndependence of the magnetization, which can be expressed as ܯሺܶሻ~ሺܶେെܶሻఉ, where ܯ i s \nthe saturation magnetization, ܶେ is the Curie temperature, and ߚ is the critical exponent. \nAccordingly, the temperature dependencies of the magnetization f o r G d a n d F e C o c a n b e \nwritten as ܯୋୢሺܶሻൌܯୋୢሺ0ሻሺ1െܶ/ܶ େሻఉృౚ and ܯୣେ୭ሺܶሻൌܯୣେ୭ሺ0ሻሺ1െܶ/ܶ େሻఉూి, \nrespectively, where ߚୋୢ ( o r ߚୣେ୭) is the critical exponents of Gd (or FeCo) and ܯୋୢሺ0ሻ \n(or ܯୣେ୭ሺ0ሻ) is the saturation magnetization of Gd (or FeCo) at zero tempe rature, where \nߚୋୢߚୣେ୭ and ܯୋୢሺ0ሻܯୣେ୭ሺ0ሻ [27, 28]. The total saturation magnetization ܯ୲୭୲ୟ୪ \ncan be determined using the relation, ܯ୲୭୲ୟ୪ሺܶሻൌܯୣେ୭ሺ0ሻሺ1െܶ/ܶ େሻఉూిെܯୋୢሺ0ሻሺ1െ\nܶ/ܶେሻఉృౚ. As ܯ୲୭୲ୟ୪ൌ0 \tat ܶൌܶ, the following equation can be written. \n\t\t\t\t\t\t\tܶେെܶൌܶେሾܯୋୢሺ0ሻ/ܯୣେ୭ሺ0ሻሿଵ/ሺఉూిିఉృౚሻ.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \t\t\t\t\t\t\t\tሺ1ሻ \n6 Similarly, the total angular momentum ܣ୲୭୲ୟ୪ሺܶሻ can be given as ܣ୲୭୲ୟ୪ሺܶሻൌሺܯୣେ୭ሺ0ሻ/\nߛୣେ୭ሻሺ1െܶ/ܶ େሻఉూిെሺܯୋୢሺ0ሻ/ߛୋୢሻሺ1െܶ/ܶ େሻఉృౚ, where ߛୋୢ ( o r ߛୣେ୭ ) is the \ngyromagnetic ratio of Gd (or FeCo). The gyromagnetic ratio of G d (or FeCo) can be defined \nas ߛୋୢൌ݃ୋୢఓా\n ( o r ߛୣେ୭ൌ݃ୣେ୭ఓా\n\t), where ݃ୋୢ ( o r ݃ୣେ୭) is the Landé g factor of Gd \n(or FeCo), \tߤ is the Bohr magneton and is the reduced Plank’s constant. As ܣ୲୭୲ୟ୪ൌ0 \tat \nܶൌܶ, the following equation is obtained. \n\t\t\t\t\t\t\tܶେെܶൌܶେሾሺܯୋୢሺ0ሻ݃ୣେ୭ሻ/ሺܯୣେ୭ሺ0ሻ݃ୋୢሻሿଵ\nሺఉూిିఉృౚሻ.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tሺ2ሻ \nBy subtracting Eq. (1) from Eq. (2), we can write the relations hip between ܶ a n d ܶ a s \nfollows. \n\t\t\t\t\t\t\tܶൌܶܶେቈ1െሺ݃ୣେ୭/݃ୋୢሻଵ\nሺఉూిିఉృౚሻ൫ܯୋୢሺ0ሻ/ܯୣେ୭ሺ0ሻ൯ଵ\nሺఉూిିఉృౚሻ.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tሺ3ሻ \nBecause of the spin-orbit coupling of FeCo and zero orbital ang ular momentum of Gd, it is \nknown that ݃ୣେ୭ (~ 2.2) is slightly greater than ݃ୋୢ (~ 2) [30–32]. Consequently, we can \nexpect that ܶܶ due to the condition of ߚୋୢߚୣେ୭, ܯୋୢሺ0ሻܯୣେ୭ሺ0ሻ, and \n݃ୣେ୭݃ୋୢ [27, 28, 30–32]. From Eq. (3), the linearity of ሺܶ,ܶሻ can be easily \nunderstood. It is noteworthy that ܶ depends on ܶେ in Eq. (3), and therefore, ܶେ a f f e c t s \n∆ܶ .This results in a slight deviation from linearity. A standard scaling treatment is employed \nto examine the universal behaviors. By scaling Eq. (3) and divi ding it by ܶେ, we can obtain \nthe relation ܶ/ܶେൌܶ/ܶେߟ ,where \n\t\t\t\t\t\t\tߟ ≡ ቈ1െ ሺ݃ୣେ୭/݃ୋୢሻଵ\nሺఉూిିఉృౚሻ൫ܯୋୢሺ0ሻ/ܯୣେ୭ሺ0ሻ൯ଵ\nሺఉూిିఉృౚሻ.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tሺ4ሻ \nAs ߟ is decided by the given material parameters, Eq. (4) shows tha t ܶ/ܶେ i s d i r e c t l y \nproportional to ܶ/ܶେ. \n7 To confirm the above theoretical prediction, we measured ܶେ for each sample, as \nlisted in Table II by performing the temperature dependence of the ܴୌ [33]. Figure 4( a) \nshows the variation in ܶ/ܶେ with respect to ܶ/ܶେ. This relationship is clearly linear. The \nslope and y-axis intercept of the best linear fit are 0.99 ± 0. 06 and 0.19 ± 0.02, respectively, \nimplying that ߟ i s a p p r o x i m a t e l y c o n s t a n t f o r t h e s a m p l e s w i t h d i f f e r e n t ܶେ and ܶ. \nFigure 4( b) shows the results of ߟ for each sample. This result confirms that ߟ is invariant \n(= 0.19) irrespective of the sam ples. Therefore, the results pr ove the validity of the general \nassumption used in the theory. \nFor a better insight, we study ߟ for each parameter: ܯୋୢሺ0ሻ, ܯୣେ୭ሺ0ሻ, ߚୋୢ, and \nߚୣେ୭. Previous studies show that ܯୋୢሺ0ሻ/ܯୣେ୭ሺ0ሻ ranges from 1.1 to 1.2, and ߚୋୢ and \nߚୣେ୭ a r e 0 . 4 5 ൏ߚୣେ୭൏0.5 and 0.65 ൏ߚୋୢ൏0.7, respectively [27, 28]. Using these values, \nwe can numerically calculate ߟ . The blue dashed lines, shown in Fig. 4( b), indicate the \ncalculation results of the upper and lower limits of ߟ on the basis of the reported parameters \n[27, 28]. For the case of typical ranges, the figure shows that the experimental results are in \ngood agreement with the numerical calculation. Therefore, ߟ is approximately constant \nwithin the experimental accuracy. \nFrom the relation ܶ/ܶେൌܶ/ܶେߟ , we can estimate the angular momentum \ncompensation temperature. We denoted the estimated angular mome ntum compensation \ntemperature as ܶ∗, which can be calculated using the relation ܶܶߟେ. Here, we used ߟൌ\n\t0.19, from Fig. 4( b). To confirm the accordance with the measured angular momentum \ncompensation temperature ܶ, ܶ∗ was calculated for all the samples and is plotted with \nrespect to ܶ, as shown in Fig. 4( c). The solid red line ( ܶ∗ൌܶ) indicates a good conformity \nof the estimated angular momentum compensation temperature. Thi s observation proves that \nthe experimental inaccuracy in determining ܶ∗ i s w i t h i n a f e w K e l v i n , w h i c h c o u l d b e \n8 acceptable for estimating ܶ. In Fig. 4( c), the inaccuracy remains lower than approximately 5 \nK. \nFinally, it is worthwhile to discuss the general application of the estimation method \n(of the ܶ∗) to other ferrimagnetic materials, e.g., TbFeCo. Because the L andé g factors of RE \nand TM elements, ݃ୖ and ݃, are different for other elements, ߟ can be changed \ndepending on the type of ferrimagnetic material. If ߟ can be determined for a given RE–TM \nferrimagnet, ܶ∗ can be easily estimated for any ferrimagnetic material by dete rmining ܶୡ \nand ܶ. \nIn conclusion, we investigate the correlation between ܶ and ܶ i n G d F e C o \nferrimagnets. The results show a strong correlation between ܶ and ܶ, which can be \ndemonstrated experimentally and theoretically. Moreover, simple yet efficient method was \nemployed for estimating ܶ by measuring ܶେ and ܶ. Therefore, this observation will help \nin easily determining the angular momentum compensation tempera ture. Accordingly, the \nproposed scheme can be potentially applied to for ferrimagnet-b ased spintronics devices. \n \n9 References \n1. A. H. MacDonald and M. Tsoi, Phil. Trans. R. Soc. A 369, 3098 (2011). \n2. S. H. Yang, K.-S. Ryu, and S. Parkin, Nat. Nanotech. 10, 221 (2015). \n3. T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. N anotech. 11, 231 (2016). \n4. O. Gomonay, T. Jungwirth, and J. Sinova, Phys. Rev. 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Lett. 95, 047202 (2005). \n10 16. X. Jiang, L. Gao, J. Z. Sun, and S. S. P. Parkin, Phys. Rev . Lett. 97, 217202 (2006). \n17. J. Finley and L. Liu, Phys. Rev. Applied 6, 054001 (2016). \n18. W. Ham, S. Kim, D.-H. Kim, K.-J. Kim, T. Okuno, H. Yoshikaw a, A. Tsukamoto, T. \nMoriyama, and T. Ono, Appl. Phys. Lett. 110, 242405 (2017). \n19. K. Ueda, M. Mann, C.-F. Pai, A.-J. Tan, and G. S. Beach, Ap pl. Phys. Lett. 109, 232403 \n(2016). \n20. K. Ueda, M. Mann, P . W . P . de Brouwer, D. Bono, and G. S. D . Beach, Phys. Rev. B 96, \n064410 (2017). \n21. N. Roschewsky, T. Matsumura, S. Cheema, F. Hellman, T. Kato , S. Iwata, and S. \nSalahuddin, Appl. Phys. Lett. 109, 112403 (2016). \n22. R. Mishra, J. Yu, X. Qiu, M. Motapothula, T. Venkatesan, an d H. Yang, Phys. Rev. Lett. \n118, 167201 (2017). \n23. T. Okuno, K.-J. Kim, T. Tono, S. Kim, T. Moriyama, H. Yoshi kawa, A. Tsukamoto, and T. \nOno, Appl. Phys. Express 9, 073001 (2016). \n24. K.-J. Kim, J.-C. Lee, S.-B. Choe, and K.-H. Shin, Appl. Phy s. Lett. 92, 192509 (2008). \n25. D.-H. Kim, K.-W. Moon, S.-C. Yoo, B.-C. Min, K.-H. Shin, an d S.-B. Choe, IEEE Trans. \nMagn. 49(7), 3207 (2013). \n26. J.-C. Lee, K.-J. Kim, J. Ryu, K.-W. Moon, S.-J. Yun, G.-H. Gim, K.-S. Lee, K.-H. Shin, \nH.-W. Lee, and S.-B. Choe, Phys. Rev. Lett. 107, 067201 (2011). \n27. T. A. Ostler, et al., Phys. Rev. B 84, 024407 (2011). \n28. A. Kirilyuk, A. V. Kimel, and Th. Rasing, Rep. Prog. Phys. 76, 026501 (2013). \n11 29. S. Chikazumi, Physics of Ferromagnetism (Clarendon Press, O xford, 1997), p. 128. \n30. C. Kittel, Phys. Rev. 76, 743 (1949). \n31. G. G. Scott, Rev. Mod. Phys. 34, 102 (1962). \n32. B. I. Min and Y .-R. Jang, J. Phys. Condens. Matter 3, 5131 (1991). \n33. D. Chiba, S. Fukami, K. Shimamura, N. Ishiwata, K. Kobayash i and T. Ono, Nat. Mater. \n10, 853 (2011). \n \n \n12 Figure Captions \nF i g u r e 1 ( a ) Schematic of the GdFeCo microwire device, and (b) Anomalous Hall effect \nresistance ܴୌ as a function of the perpendicular magnetic field ߤܪ௭ at room temperature \n(300 K). The orange arrow indicates the coercive field ߤܪୡ and the black up–down arrow \nindicates the Hall resistance difference ∆ܴୌ≡ܴୌ൫ܪ,ୱୟ୲൯െܴୌ൫െܪ,ୱୟ୲൯, where ܪ,ୱୟ୲ \nand െܪ,ୱୟ୲ are the saturation fi elds with the condition ܪୡ൏ܪ,ୱୟ୲. \nFigure 2 (a) ∆ܴୌ as a function of the temperature ܶ for Sample II. The blue dot indicates \nthe magnetization compensation temperature ܶ, and (b) DW speed ݒ as a function of ܶ \nfor Sample II at ߤܪ௭ൌ\t80 mT. The blue dot indicates the magnetization compensation \ntemperature ܶ, and the purple arrow indicates the angular momentum compensat ion \ntemperature ܶ. \nFigure 3 \tܶ with respect to ܶ. The red line is the best linear fit. \nFigure 4 (a) ܶ/ܶୡ as a function of ܶ/ܶୡ (the red line is the best linear fit), (b) ߛ w i t h \nrespect to the sample number (the red line indicates ߛൌ \t0.19, and the blue dashed lines \nindicate the calculation of the upper and lower limits of ߛ based on the reported parameters), \nand (c) ܶ∗ with respect to ܶ (the red line is the best linear fit). \n \n13 Acknowledgements \nThis work was supported by JSPS KAKENHI (Grant Numbers 15H05702 , 26870300, \n26870304, 26103002, 26103004, 25220604, and 2604316). Collabora tive Research Program \no f t he I n s tit ut e f o r C he m ica l R e sea r ch , K yo to U n iv e r s i t y , an d R & D p r o j e ct fo r IC T Ke y \nTechnology of MEXT from the Japan Society for the Promotion of Science (JSPS). This \nwork was partly supported by The Cooperative Research Project P rogram of the Research \nInstitute of Electrical Communi cation, Tohoku University. D.H.K . was supported as an \nOverseas Researcher under Postdoctoral Fellowship of JSPS (Gran t Number P16314). D.Y.K. \nand S.B.C. were supported by a National Research Foundations of Korea (NRF) grant funded \nb y t h e M i n i s t r y o f S c i e n c e , I C T a n d F u t u r e P l a n n i n g o f K o r e a ( M SIP) \n(2015R1A2A1A05001698 and 2015M3D1A1070465). K.J.K. was supporte d by the National \nResearch Foundation of Korea (NRF) grant funded by the Korea Go vernment (MSIP) (No. \n2017R1C1B2009686, NRF-2016R1A5A1008184) and by the DGIST R&D Pr ogram of the \nMinistry of Science, ICT and Future Planning (17-BT-02). \nContributions of Authors \nD.H.K. conceptualized the work. D.H.K. and T.O. supervised the study. Y .F., H.Y., and A.T. \nprepared the films and Y.H. and T.O. made the devices. Y .H., D. H.K., T.O., T.N., and D.Y.K. \nconducted the experiments. D.H.K. and Y .H. performed the analys is, and D.H.K. modeled the \ndata. D.H.K., T.O., S.B.C., and K.J.K. wrote the manuscript. Al l authors discussed the results \nand commented on the manuscript. \n \n14 Table I. Summary of the sample structures \n Sample Structures \n۷\t܍ܔܘܕ܉܁ 5-nm SiN/20-nm Gd 23Fe67.4Co9.6/100-nm SiN/Si substrate \n۷۷\t܍ܔܘܕ܉܁ 5-nm SiN/30-nm Gd 23.5Fe66.9Co9.6/100-nm SiN/Si substrate \nSample III 5-nm SiN/1-nm Gd/ 5-nm Gd 23Fe67.4Co9.6/1-nm Gd/100-nm SiN/Si substrate \nSample IV 5-nm SiN/30-nm Gd 23Fe67.4Co9.6/5-nm Cu/5-nm SiN/Si substrate \nSample V 5-nm SiN/30-nm Gd 23.5Fe66.9Co9.6/5-nm SiN/Si substrate \nSample VI 5-nm SiN/20-nm Gd 23Fe67.3Co9.7/5-nm Pt/100-nm SiN/Si substrate \n \nTable II. Curie temperature ܶେ for each sample \n ( u n i t s : K ) \n Sample I Sample II Sample III Sample IV Sample V Sample VI \n 475 ± 10 450 ± 10 355 ± 10 412 ± 10 412 ± 10 441 ± 10 \n -0.4 -0.2 0.0 0.2 0.4-3.0-1.50.01.53.0RH []\n0H [T]RH0HC\nFigure 1\nV\nIyIx y\nxz\nba\nHzSiN\nGdFeCo\nBufferBuffer0 100 200 300-6-3036\nRH []\nT [K]\nFigure 20 100 200 30003006009001200\nv [m/s]\nT [K]Tba0 100 200 3000150300450\n TA [K]\nTM [K]\nFigure 30.0 0.2 0.4 0.6 0.80.00.20.40.60.81.0\n1234560.000.250.50b\nTA/TC\nTM/TCa\n \n\n# of Sample\nFigure 4150 200 250 300 350150200250300350\n T *\nA [K]\nTA [K]c" }, { "title": "0704.3139v2.Element_resolved_x_ray_ferrimagnetic_and_ferromagnetic_resonance_spectroscopy.pdf", "content": "Element-resolved x-ray ferrimagnetic and\nferromagnetic resonance spectroscopy\nG Boero, S Mouaziz, S Rusponi\nEcole Polytechnique F\u0013 ed\u0013 erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland\nP Bencok\nEuropean Synchrotron Radiation Facility (ESRF), F-38043 Grenoble, France\nF Nolting\nSwiss Light Source (SLS), Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland\nS Stepanow\nCentre d'Investigacions en Nanoci\u0012 encia i Nanotecnologia (CIN2-ICN), UAB Campus,\nE-08193 Bellaterra, Barcelona, Spain\nP Gambardella\nInstituci\u0013 o Catalana de Recerca i Estudis Avan\u0018 cats (ICREA)\nand Centre d'Investigacions en Nanoci\u0012 encia i Nanotecnologia (CIN2-ICN), UAB\nCampus, E-08193 Bellaterra, Barcelona, Spain\nE-mail: pietro.gambardella@icrea.es\nAbstract. We report on the measurement of element-speci\fc magnetic resonance\nspectra at gigahertz frequencies using x-ray magnetic circular dichroism (XMCD). We\ninvestigate the ferrimagnetic precession of Gd and Fe ions in Gd-substituted Yttrium\nIron Garnet, showing that the resonant \feld and linewidth of Gd precisely coincide\nwith Fe up to the nonlinear regime of parametric excitations. The opposite sign of\nthe Gd x-ray magnetic resonance signal with respect to Fe is consistent with dynamic\nantiferromagnetic alignment of the two ionic species. Further, we investigate a bilayer\nmetal \flm, Ni 80Fe20(5 nm)/Ni(50 nm), where the coupled resonance modes of Ni and\nNi80Fe20are separately resolved, revealing shifts in the resonance \felds of individual\nlayers but no mutual driving e\u000bects. Energy-dependent dynamic XMCD measurements\nare introduced, combining x-ray absorption and magnetic resonance spectroscopies.\nPACS numbers: 76.50.+g, 78.70.Dm, 78.20.Ls, 76.30.DaarXiv:0704.3139v2 [cond-mat.mtrl-sci] 22 Jan 2008Element-resolved XFMR 2\n1. Introduction\nRecent interest in magnetization dynamics has been fostered by progress in fast\nmagnetic recording and microwave technologies [1, 2]. Despite considerable e\u000borts,\nhowever, the description of magnetodynamics remains essentially phenomenological.\nInductive, magnetoresistive, and magneto-optical techniques solely measure the\nintegrated magnetic response of complex heterogeneous materials, typically magnetic\nalloys and multilayer structures, whose functionality depends on the interplay of several\nelements. The development of methods capable of elemental analysis constitutes\nan obvious advantage for investigating fundamental problems related to time- or\nfrequency-dependent magnetization phenomena. Examples include the dynamic\ncoupling of elemental moments in ferrites [3, 4, 5, 6], metallic alloys [7], and spin-valve\nheterostructures [8, 9], as well as spin-orbit induced damping e\u000bects attributed to the\npresence of high [5, 10, 11] and low [12] Z elements. Advances in this direction are\nmostly based on stroboscopic pump-probe experiments exploiting the element-resolving\npower of x-ray magnetic circular dichroism (XMCD) and the sub-ns bunch structure of\nsynchrotron radiation beams. Pulsed magnetic \felds in synchrony with x-ray photon\nbunches are usually employed to excite the reversal [8, 13] or the precessional motion\n[7] of the magnetization. More recently, continuous wave rf \felds have been applied to\nexcite resonant modes in trilayer metal \flms [14, 15] and microstructures [16, 17].\nWith respect to time-resolved measurements, techniques such as ferromagnetic\nresonance spectroscopy (FMR) o\u000ber an alternative and powerful way to gain insight\ninto the energy scales that govern magnetization dynamics. Frequency-domain methods\nthat allow to detect magnetic resonance using the core level absorption of circularly\npolarized x-rays have been developed independently by our group in the soft x-ray\nenergy range [18] and by Goulon et al. in the hard x-ray regime [19, 20]. These methods\nexploit the XMCD dependence on the scalar product M\u0001Pof the magnetization vector\nMand photon helicity Pto measure the time-invariant changes of the longitudinal\nmagnetization component \u0001 Mzas a function of microwave (MW) \feld B1and bias\n\feldB0. Microstrip resonators [18] and tunable cavities [21] have been employed to\ngenerate MW excitations together with di\u000berent detection schemes. In the hard x-ray\nregime, XMCD at the Kedge of transition metals relates purely to orbital magnetization\ncomponents; measurements at the Fe K-edge and Y L2;3edges by Goulon et al. provided\nevidence for the precession of the Fe orbital moments as well as induced Y spin moments\nin yttrium iron garnet (YIG) [19, 20].\nIn this article, we report on di\u000berent applications of soft x-ray MCD to FMR\nmeasurements and on a novel way to combine FMR and XMCD spectroscopy. Element-\nspeci\fc magnetic resonance spectra are measured on both magnetic oxides and metallic\nmultilayers. We show that ferrimagnetic resonance measurements of Gd-substituted\nYIG are consistent with the antiferromagnetic (AFM) alignment of Gd and Fe ions in the\nferromagnetic resonance mode of YIG in the non-linear regime, above the threshold for\nparametric spin wave excitations. Further, FMR spectra of coupled thin metal bilayersElement-resolved XFMR 3\nFigure 1. (a) Diagram of the experimental setup. (b) Close-up view of the resonator\nand photodiode situated between the poles of the electromagnet. Note that one of the\nmagnet poles and the photodiode have an opening to allow for the passage of x-rays.\nare separately resolved, allowing the investigation of interlayer dynamics in stacks of\nmagnetic layers. Finally, we show that the x-ray FMR (XFMR) signal measured at\nresonance as a function of photon energy yields dynamic XMCD spectra, which relate\nto the magnetic state of the atoms undergoing microwave absorption. The latter can\nbe combined with static XMCD spectra to derive information on the dynamics of the\norbital and spin magnetization components.\n2. Experimental\nA schematic diagram of the experimental setup is given in Fig. 1. A coplanar waveguide\n\u0015=2-resonator is used to generate a MW \feld B1\u00190:01 to 0.5 mT parallel to the sample\nsurface with input power 0 to 34 dBm at frequency !=2\u0019= 2:21 GHz. The resonator-\nsample assembly is positioned between the pole expansions of an electromagnet, which\nproduces a \feld 0 \u0014B0\u00140:8 T aligned perpendicular to the sample surface and parallel\nto the photon propagation direction. In the absence of MW \feld, Maligns with B0\nparallel to P, which is the geometry commonly employed in static XMCD measurements.\nIfB1is turned on, as B0matches the resonance \feld of the sample ( Br) the precessional\nmotion of Minduces a reduction of the longitudinal magnetization component Mzthat\ncan be measured as a steady-state e\u000bect in the frequency domain, i.e., without requiring\nsub-ns time resolution. Here, x-ray absorption spectra (XAS) corresponding to positive\n(P+) and negative (P\u0000) helicity are measured by recording the dc \ruorescence yield\n(FY) of the sample as a function of photon energy using a Si photodiode (Eurisys-\nCanberra, Ref. [22]). XMCD is de\fned as the di\u000berence spectrum P+-P\u0000(Fig. 2). The\nXFMR signal, either P+or P\u0000, is obtained by square-modulating the MW power source\nat relatively low frequency ( <100 kHz) and by measuring the corresponding amplitude\nof the ac FY photocurrent by means of a lock-in ampli\fer, as shown in Fig. 1 (a). We\nintroduce two methods to measure magnetic resonance using XMCD: the \frst, in analogy\nwith FMR spectroscopy, consists in recording the XFMR intensity during a sweep of B0\nacrossBr, \fxing the photon energy in correspondence of a static XMCD peak [18]. We\ndenote this type of measurements as XFMR B-scan , which e\u000bectively generate element-Element-resolved XFMR 4\nFigure 2. (a) One octant portion of the unit cell of GdIG, showing the AFM\nspin alignment of octahedral Fe (black circles), tetrahedral Fe (gray circles), and\ndodecahedral Gd sites (empty blue circles), from Ref. [23]. Oxygen ions have been\nomitted. (b) FY XAS spectra and corresponding XMCD of Fe and (c) Gd sites\nmeasured at room-temperature with B0= 0:21 T.\nspeci\fc longitudinal magnetic resonance spectra. The second method consists in taking\nthe sample at resonance by setting B0=Brand recording the XFMR as a function\nof photon energy. This, denoted as XFMR E-scan , is analogous to recording XAS and\nXMCD spectra, but corresponding to the precessional motion of Mrather than to a\nstatic situation. Examples of either type of measurements will be given later.\nTwo di\u000berent type of samples are employed in the present study: a\nrare earth substituted iron oxide and a metallic heterostructure, which were\nchosen in order to highlight the broad spectrum of materials where new insight\ncan be obtained by XFMR. A polished 30 \u0016m-thick slab of polycristalline\nGd 1Y2Fe5O12(Gd:YIG) with lateral dimensions 1 \u00022 mm2was selected to investigate\nferrimagnetic resonance in garnet systems composed of di\u000berent magnetic ions. An\nAl(10 nm)/Ni 80Fe20(5 nm)/Ni(50 nm)/Cr(5 nm) multilayer deposited on glass by e-\nbeam evaporation in high vacuum (1 \u000210\u00006mbar) was fabricated in order to address\nlayer-speci\fc resonance modes in metallic heterostructures. The x-ray spot size at the\nsample position was 0.1 mm long and 1 mm wide at full width half maximum, while\nthe coplanar resonator had a central conductor with a width of 1.5 mm and a length\nof 44 mm, thus ensuring that the MW excitation covers the whole area sampled by the\nx-ray beam. XAS and XFMR spectra were recorded at the L2;3edges of Fe and Ni,\nand at the M4;5edges of Gd. XAS spectra are normalized to the incident photon \rux\nmeasured by the photocurrent of an Au grid upstream from the sample, and are givenElement-resolved XFMR 5\nFigure 3. (a) Magnetization of a 30 \u0016m thick, 1\u00022 mm2wide Gd 1Y2Fe5O12slab\nmeasured by SQUID with applied \feld perpendicular to the sample plane at 300 K.\n(b) Magnetization vs. temperature of a 100 \u0016m thick Gd 1Y2Fe5O12slab \feld-cooled\nin a 3 mT \feld.\nin arbitrary units. Apart from normalization, the spectra are raw data; in particular, no\nenergy-dependent correction for self-absorption has been applied. As the signal-to-noise\nratio is proportional to the square root of the photocurrent [18], energy resolution has\nbeen sacri\fced to intensity by opening the exit slits of the beamline monochromator.\nThe e\u000bective energy resolution corresponds to about 1.2 and 3 eV at 700 and 1200 eV,\nrespectively, which results in signi\fcant broadening of the multiplet features of Fe and\nGd spectra in Gd:YIG, as shown in Fig. 2. This is not an essential problem for XFMR\nB-scans, but may limit the spectral resolution of E-scans; in the latter case, however,\nhigher resolution can be achieved simple by reducing the slit apertures while increasing\nthe averaging time to maintain a constant signal-to-noise ratio. Throughout the paper\nXFMRB-scans are given in pA, as measured by the FY photodiode. Simultaneously\nwith XFMR, the transverse part of the imaginary susceptibility \u001f00was measured, as in\nconventional FMR, by monitoring the power re\rected o\u000b the \u0015=2-resonator via a MW\nbridge and diode detector, as schematized in Fig. 1 (a). XFMR B-scans were measured\nat the ID08 beamline of the European Synchrotron Radiation Facility, while E-scans\nwere recorded at the SIM beamline of the Swiss Light Source; two undulators were\noperated in series with 99 \u00061 % circularly polarized beams in both type of measurements.\n3. Element-resolved XFMR spectra of Gd:YIG\nThe structure of Gd 1Y2Fe5O12(Gd:YIG) consists of three sublattices [Fig. 2 (a)]. Two\nof them, the octahedral and tetrahedral sites, contain Fe ions which are strongly AFM\ncoupled by superexchange. The third lattice, the dodecahedral sites, contains Gd and\ndiamagnetic Y ions [23]. While their mutual interaction is very weak, Gd ions couple\nAFM to tetrahedral Fe ions with a moderate exchange \feld of the order of 24 T (16 K)\n[24]. Such a system thus e\u000bectively behaves as a two-sublattice ferrimagnet, where\nthe Gd moments order spontaneously only at low temperature ( <50 K). Figure 3\n(a) shows the out-of-plane magnetization of Gd:YIG measured by superconducting\nquantum interference device magnetometry (SQUID) at room temperature. The curveElement-resolved XFMR 6\nFigure 4. FMR spectra of Gd:YIG measured by the re\rected power from the \u0015=2-\nresonator at 0 dBm using \feld and MW amplitude modulation (bottom and middle\ntraces, respectively). The top trace shows the high power (31 dBm) FMR for MW\namplitude modulation. B0is oriented perpendicular to the sample surface in all cases.\nis composed by a hard-axis ferromagnetic loop that saturates above 0.1 T, as expected\nfrom shape anisotropy considerations, and a linear term proportional to the applied \feld.\nThe latter is a common feature of rare-earth garnets and ascribed to the continuous\nrotation of MGdtowards MFewith increasing \feld, in accordance with N\u0012 eel's theory\nof ferrimagnetism. The temperature behavior of the magnetization, shown in Fig. 3\n(b), is characteristic of two AFM-coupled lattices with inequivalent magnetization.\nWhile for all rare-earth garnets the Curie temperature is associated to the pairing of Fe\nmoments and nearly independent on rare-earth composition [23, 25], the compensation\ntemperature depends sensibly on the rare-earth content. In Gd 3Fe5O12compensation\noccurs at 290 K [23]. Figure 3 (b) shows that the total magnetization of Gd 1Y2Fe5O12is\napproximately constant from 300 to 150 K; below this temperature magnetic order sets\nin throughout the Gd lattice, compensating the Fe magnetization at about 45 K. The\nXAS and XMCD spectra of Fe and Gd in Gd 1Y2Fe5O12recorded at room temperature\nwith applied \feld B0= 0:21 T are shown in Figs. 2 (b) and (c). The opposite sign of\ntheM5vsL3andM4vsL2intensity re\rects the static alignment of the resultant MGd\nagainst MFe.\nLinearization of the coupled equations of motion shows that two resonances can be\nexcited in a ferrimagnetic compound: the ferromagnetic mode, which is independent\nof the exchange \feld since the angle between MFeandMGddoes not vary during the\nprecession, and the high-frequency exchange mode, where the two sublattices precess\nout-of-phase but phase-locked to each other with non collinear magnetization vectors\n[3, 4, 26]. The \frst mode is the one accessible at relatively low \felds in usual FMR\nexperiments, as in our case, while the second one is situated at \felds of several\ntens of Teslas for frequencies in the MW range [27]. Neglecting magnetocrystalline\nanisotropy, the resonant \feld for uniform precession in the ferromagnetic mode is\ngiven byBr=!\n\r+\u00160Nz(MFe\u0000MGd) = 190 mT, where \ris the gyromagneticElement-resolved XFMR 7\nratio,Nz= 0:935 is the demagnetizing factor calculated for our geometry [28], and\n\u00160(MFe\u0000MGd) = 120\u00066 mT. Figure 4 shows the conventional FMR spectra of\nGd:YIG. Owing to the sample \fnite dimensions, the low power FMR shows a series of\nmagnetostatic modes with the principal one close to Br. The longer wavelength modes\nare resolved in the \feld-modulated spectrum (bottom trace) and appear as shoulders\nof the main peak in the MW-modulated spectrum (middle trace). For a sample 30 \u0016m\nthick with lateral dimensions of the order of 1 mm their separation corresponds to that\nexpected for magnetostatic forward volume wave modes with the excitation geometry\nof Fig. 1 [29, 30]. At high MW power (top trace) the FMR shifts to a lower \feld due to\nheating of the sample and related decrease of the resultant magnetization MFe\u0000MGd.\nMoreover, the FMR lineshape is signi\fcantly distorted due to e\u000bects such as foldover\nand nonlinear spin wave instabilities [31]. In such a regime, nonlinear terms in the\nLandau-Lifschitz equation of motion transfer energy from the uniform precession mode\ndriven by the external MW \feld to nonuniform magnon modes, which become unstable\nabove a critical \feld threshold [32]. These phenomena lead to saturation of the main\nresonance and precession angle together with excitation of spin waves above thermal\nvalues. Of relevance to the present discussion is the fact that nonlinear coupling terms\nescape conventional treatments of ferrimagnetic resonance, which reduce the dynamics\nof individual sublattices to that of a single macrospin (e.g., of amplitude MFe\u0000MGd\nfor Gd:YIG) [3, 4, 5, 6]. Moreover, the assumed equivalency of the equations of motion\nfor di\u000berent sublattices might not hold true when nonlinear phenomena are taken into\naccount. For example, substitution of foreign ions in a material where all equivalent\nlattice sites are occupied by identical ions, as in Gd:YIG, provides a site-dependent\nadditional scattering channel leading to spin wave excitations [33]. Element-resolved\nFMR spectra can thus put the macrospin concept to test, speci\fcally in the nonlinear\nregime where relatively large deviations \u0001 Mzmake the XFMR intensity easier to detect.\nFigure 5 compares the inductive FMR spectrum of Gd:YIG (a) with the XFMR\nP+-P\u0000intensity recorded at the Fe L2edge (b) and Gd M4edge (c) as a function of\nB0. Several comments are in order. First, we note that conventional FMR and XFMR\nspectra di\u000ber for obvious reasons, namely: (i) XFMR is a measure of \u0001 Mz, while\nFMR is proportional to the transverse dynamic magnetization component. Only if jMj\nis conserved the two measurements can be considered to be equivalent. (ii) XFMR is\nsurface-sensitive, with the same probing depth as FY XAS ( \u001820 nm at the Fe L2;3edges\n[34]) and probes a limited portion of the sample, while FMR averages over the whole\nsample volume. In Fig. 5 (a) the FMR lineshape is asymmetric and heavily saturated due\nto nonlinear e\u000bects that limit the FMR precession cone amplitude. The XFMR signal\nin (b), on the other hand, is composed of a broad resonant feature and a sharp peak\nlocated at about B0= 165 mT with linewidth \u0001 B= 1 mT. It may be observed that the\nintensity of both features is centered around the low-\feld rising edge of the FMR peak\nand does not follow the FMR intensity distribution. The origin of such di\u000berences lies\nin (i) and (ii); a detailed understanding of the XFMR vs. FMR lineshape, however, isElement-resolved XFMR 8\nFigure 5. (a) FMR spectrum of Gd:YIG measured simultaneously with the XFMR\ndata. (b) XFMR P+\u0000P\u0000intensity measured at the L2edge of Fe (723.8 eV) and (c)\nat theM4edge of Gd (1222 eV). The MW power is 31 dBm. The data are averaged\nover 40 sweeps of B0in the positive direction, with a sweep time of 80 s and lock-in\ntime constant of 100 ms.\npresently missing. To appreciate this point, we o\u000ber a number of consideration based on\nprevious FMR and XFMR studies of YIG. The sharp peak observed by XFMR denotes\na sudden increase of \u0001 Mz, whereMzis proportional to the total number of magnons\nin the system. De Loubens et al. , using magnetic resonance force microscopy on a\nsingle crystal YIG \flm, observed a dramatic increase of \u0001 Mzat the onset of the second\norder Suhl's instability threshold, which was attributed to the parametric excitation of\nlongitudinal spin waves with a low spin-lattice relaxation rate compared to the uniform\nmode [35, 36]. In this model, the total number of magnons is considered to be constant,\nwhile changes of Mzare attributed to a redistribution of their occupation number from\nmodes with relatively high to low relaxation rate, favoring larger precession angles [37].\nGoulon et al. , using XFMR on a single crystal Y 1:3La0:47Lu1:3Fe4:84O12\flm, also observed\na sharp decrease of Mzmeasured at the Fe K edge, taking place in correspondence with\nthe foldover critical \feld of the FMR spectrum [21]. They explained this e\u000bect by\nthe degeneracy of the uniform mode with long-wavelength longitudinal magnetostatic\nwaves caused by foldover in perpendicular FMR. In this regime, parametric excitation\nof coupled magnetostatic-magnetoelastic waves becomes possible [21], which may lead\nto an e\u000bective transfer of angular momentum to the lattice and therefore to a decrease\nofMz. This is substantially di\u000berent from the model proposed by De Loubens et al. ,Element-resolved XFMR 9\nFigure 6. Restricted range of (a) FMR and (b) XFMR spectra of Gd:YIG at the L2\nedge of Fe (723.8 eV) and M5edge of Gd (1191 eV) recorded with the parameters of\nFig. 5.\nas the total number of magnons needs not be conserved. The validity of either of these\nexplanations for the present measurements may be questioned due to the inhomogeneous\ncharacter of local magnetic \felds in polycrystalline samples, e.g., owing to magnetic\nanisotropy \ructuations or microstructure \raws, which results in broadened FMR lines.\nSpeci\fcally, if individual crystal grains went through resonance individually according\nto their orientation in the applied \feld and one would have to worry about strongly\ninhomogeneous resonance conditions; however, as the magnetocrystalline anisotropy\n\feld is more than a factor 10 smaller compared to the saturation magnetization in\nGd:YIG, dipolar coupling between di\u000berent grains predominates and resonance occurs\nas a collective phenomenon [38, 39]. The observation of di\u000berent magnetostatic modes\nin Fig. 4 supports this view, although a much smaller number of modes are resolved\ncompared to single crystal YIG \flms [21, 35]. The granular structure of the material\nand related local changes of the anisotropy \feld have also a well-known e\u000bect on the\ncritical \feld for parametric spin wave excitations, raising it up to 0.1-1 mT in YIG\n[40], and leading to a smooth onset of this e\u000bect rather than an abrupt threshold [41].\nThe saturation as well as the distorted shape of the FMR spectrum indicate that the\nconditions for foldover and parametric spin wave ampli\fcations are met at high power\nin Gd:YIG and likely contribute to the observed XFMR features. In general, however,\nwe cannot identify a unique origin for the XFMR peak nor exclude it to be related to\na mode localized at the vacuum-Gd:YIG interface, which would be selectively probed\nby XFMR and only weakly observed in the bulk FMR signal [see Fig. 6 (a)]. More\nmeasurements shall be performed to clarify this point.\nWe proceed now to compare the XFMR spectra of Fe and Gd, discussing whatElement-resolved XFMR 10\ntype of information may be derived on the relative motion and relaxation of dissimilar\nmagnetic moments in a bulk compound at resonance. Apart from the noise and a\nscaling factor, the Gd M4spectrum in Fig. 5 re\rects specularly the one measured at\nthe FeL2edge. The resonant \feld and linewidth derived from the Gd B-scan XFMR\nprecisely match those of Fe, but the XFMR intensity has opposite sign. This is even\nmore evident in the restricted range B-scan in Fig. 6 (b), where the Fe L2and GdM5\nspectra are reported; note that the relative sign of the Fe and Gd intensity depends on\nthe absorption edge, as for XMCD. Sign inversion of the XFMR at the Fe L2(L3) and Gd\nM4(M5) edges, consistent with that observed in the static XMCD [Figs. 2 (b) and (c)],\nreveals the coupled AFM dynamics of the Fe and Gd magnetic moments. Their relative\n\u0001Mz=Mdeviations can be quanti\fed in terms of the XFMR cross section, de\fned as\nthe ratio between the dynamic and static dichroism FY photocurrents \u001b=XFMR (E)\nXMCD (E),\nwhich depends on the x-ray photon energy Eas well as on the spin and orbital magnetic\nmoment precession in a way dictated by the XMCD sum rules [42]. At 31 dBm MW\npower, we have \u001bL2(Fe) = (2:0\u00060:2)\u000210\u00003and\u001bM4(Gd) = (1:7\u00060:2)\u000210\u00003. These\ndata, together with the above observations, are consistent with Fe and Gd maintaining\nrigid AFM alignment in nonlinear excitation modes (diagram in Fig. 6). We note that,\nin principle, the same result can be obtained for noncollinear MFeandMGdvectors\nprecessing on the cone shown in Fig. 6; however, in the noncollinear case, di\u000berent \rexing\nangles (\u001b) would be expected for Fe and Gd, given that the local exchange \felds acting\non the two ionic species are strongly dissimilar [3, 24, 27]. Full con\frmation of the type\nof AFM coupling would in any case require to measure the phase of the precessing Fe and\nGd moments, which may be retrieved only by time-resolved detection of the transverse\nmagnetization components [14, 15, 21]. Within the experimental error, XFMR data thus\nshow that the resonating longitudinal components of MFeandMGdhave opposite sign\nand equal relative deviations from static equilibrium up to the nonlinear regime of high-\npower MW excitations. This is consistent with collinear dynamic AFM alignment of\nMFeandMGdpredicted by the theory of ferrimagnetic resonance for uniform precession\nat low \felds, but extends into the nonlinear regime beyond the approximations usually\nmade in theoretical models [3, 4, 26] and at temperatures where thermal \ructuations\nstrongly a\u000bect magnetic order in the Gd lattice (Fig. 3). Further, the observation of\nequal Fe and Gd linewidths, within the experimental accuracy of the results reported\nin Fig. 6 (b), implies that the relaxation mechanisms of the Fe and Gd lattice can be\ndescribed by a common e\u000bective damping parameter, as also predicted by theory [4].\nEven though \u001b, and therefore \u0001 Mz, cannot be uniquely related to precessing\nmagnetic moments in the uniform mode due to the presence of nonlinear excitations,\nit is interesting to de\fne an e\u000bective precession angle related to \u0001 Mz=Mmeasured by\nXFMR. In doing so, one must take into account that \u001bis a photon energy-dependent\nparameter. In other words, considering that XAS involves 2 p!3d(3d!4f)\ntransitions for the Fe L2;3(GdM4;5) edges,\u001bdepends on the precession of both spin\nand orbital magnetic components of the d- (f-) projected density of states probed by\nphotons of energy E. This point has been discussed in detail by Goulon et al. inElement-resolved XFMR 11\nRef. [42], who have shown that the precession angles of the spin and orbital magnetic\ncomponents may be derived by combining \u001bL2and\u001bL3measurements and applying the\ndi\u000berential form of the XMCD sum rules. By assuming spin-only magnetic moments,\nthe relationship between \u001band the e\u000bective precession angle becomes extremely simple,\n\u001b= (1\u0000cos\u0012eff), yielding \u0012eff(Fe) = 3:6\u000e\u00060:2\u000eand\u0012eff(Gd) = 3:4\u000e\u00060:2\u000efor the\nmeasurements reported above. Even if the orbital magnetization of Gd and trivalent\nFe ions is usually very small, the extent to which orbital precession contributes to \u001b,\nin particular for Fe, remains to be determined. This matter touches on the interesting\nquestion of separately measuring the spin and orbital moment precession angles, which\nrequires either a comparison between Kedge andL2;3edges measurements recorded\nusing identical experimental conditions [42] or full XMFR E-scans over the entire L2;3\nregion. The latter possibility is further discussed in Sect. 5.\n4. Element-resolved XFMR spectra of metallic bilayers\nWe consider now the extension of XFMR to thin metallic \flms, and show that\nlayer-speci\fc magnetic resonance spectra of multilayer magnetic structures can be\nseparately resolved. This is of interest, e.g., to investigate interlayer coupling e\u000bects,\ndistinguish superposed spectra of layers with similar resonance \felds, and investigate\ncurrent induced precessional dynamics in spin-torque devices. Here we study a\nAl(10 nm)/Ni 80Fe20(5 nm)/Ni(50 nm)/Cr(5 nm) multilayer, where the thickness of the\ntwo magnetic \flms was adjusted so as to reduce Brof Ni 80Fe20to within range of our\nelectromagnet for perpendicular FMR.\nFigure 7 (a) shows the inductive FMR of the magnetic bilayer, where two resonances\nare observed at 530 and 740 mT. These are close but not equal to the resonances\nof individual Ni and Ni 80Fe20\flms, respectively, that were prepared with the same\nprocedure. The high \feld resonance peak, in particular, appears to be shifted by an\namount \u0001B=\u0000170 mT with respect to the resonance of an individual Ni 80Fe20layer,\nwhich is indicative of ferromagnetic exchange coupling at the Ni - Ni 80Fe20interface.\nThe elemental components of the two resonance peaks are straightforwardly resolved by\nXFMR, as shown in Fig. 7 (b). We observe that the low-\feld resonance originates from\nthe Ni layer alone, while the high-\feld one comprises both Ni and Fe components. In\nthe high-\feld resonance, the scaled Ni and Fe XFMR intensities coincide, implying\na common g-value and relaxation channel for the two elements, as expected for a\nferromagnetic alloy such as Ni 80Fe20[7]. We therefore conclude that, despite the\npresence of exchange coupling at the interface, mutual resonance-driving e\u000bects between\nperpendicularly-magnetized Ni and Ni 80Fe20layers are not signi\fcant. This result can be\nrationalized within the theoretical model developed by Cochran et al. for a thin overlayer\ncoupled to a thick magnetic substrate [43]. The model assumes that two ferromagnetic\nlayersAandBdeposited on top of each other are exchange coupled at their interface by\na surface energy per unit area of the form Eexc=\u0000JMA\u0001MB, whereJis the interface\ncoupling constant [44, 45]. In the two extreme limits of strong and zero coupling, theElement-resolved XFMR 12\nFigure 7. (a) FMR of Ni 80Fe20(5 nm)/Ni(50 nm) measured simultaneoulsy with (b)\nL2XFMR spectra of Fe and Ni at E= 722:2 and 871.7 eV, respectively. The MW\npower is 34 dBm.\nmagnetizations of the two layers precess locked together or independently of each other,\nrespectively. For small but \fnite J, mutual driving terms in the equations of motion\nbecome unimportant, with the overlayer responding to the driving MW radiation as if\nit were an isolated \flm subject to an e\u000bective anisotropy \feld of magnitude JMB=tA,\nwheretAdenotes the overlayer thickness and MBthe thick \flm magnetization [43]. This\nbehavior corresponds to the data reported in Fig. 7. From the shift \u0001 Bwe estimate\nJ= 2:1\u000210\u000015Vs/A andEexc\u00196\u000210\u00004J/m2. According to theory [43, 45], also\nthe resonance position of the thicker Ni layer should be down-shifted in the presence of\nferromagnetic interface coupling, namely by the amount JMA=tB. Indeed, with respect\nto a single 50 nm thick Ni layer in a Al(10 nm)/Ni(50 nm)/Cr(5 nm) stack, a shift\n\u0001B=\u000030 mT is observed, which yields J= 1:9\u000210\u000015Vs/A, consistently with the\nvalue reported above.\nCompared to the exchange energy of ferromagnetic metals, Eexcestimated from the\nresonance shifts turns out to be rather small for metallic \flms in direct contact with\neach other. Although this explains the absence of Ni 80Fe20(Ni) response upon excitation\nof the Ni (Ni 80Fe20) resonance, its origin could not be uniquely determined during the\npresent study. The magnitude of Eexcis known to be extremely sensitive to the quality of\nthe interface between magnetic materials. Roughness, as well as adsorption of impurities\nsigni\fcantly diminish the coupling strength. In high vacuum, the few seconds intervened\nbetween evaporation of the Ni and Ni 80Fe20\flms are su\u000ecient to deposit a monolayer-\nlike quantity of contaminants, which may strongly decrease the magnetization of the\ninterface metal layers. In vacuum conditions similar to ours, Ho\u000bmann et al. found\nEexc= 1:2\u000210\u00003J/m2for a double Ni/Ni 80Fe20/Ni interface [44]. Fully oxidizedElement-resolved XFMR 13\nFigure 8. Static Fe XMCD (solid line) of Ni 80Fe20(5 nm) and Fe XFMR E-scan\nmeasured at B0= 0:74 T (squares) and 0.70 T (dashed line). The MW power is\n34 dBm.\nNiO/Ni 80Fe20interfaces, on the other hand, have interfacial coupling energies as small\nas 2\u000210\u00005J/m2[46].\nFinally, we note that the smallest XFMR cross-section measured for Ni 80Fe20(5 nm)\ncorresponds to \u001bFe= 5\u000210\u00004, representing a very remarkable dichroism sensitivity in\nthe soft x-ray range, still susceptible of further improvements.\n5. Dynamic XMCD spectra\nSo far we have dealt with the information contained in XFMR B-scans. One of the main\npoints of XFMR, however, is that the measured intensity contains all the information\nderived from the x-ray absorption process, in particular that related to the unoccupied\n\fnal density of states of a given chemical species together with its spin and orbital\nmagnetization components. In other words, two powerful spectroscopical methods, x-\nray absorption and magnetic resonance, are combined together in XFMR. Here we show\nhow the information related to the electronic state of the atoms whose magnetization is\nprecessing can be practically retrieved by XFMR E-scans, i.e., by recording the XFMR\nintensity as a function of photon energy at B0=Br. Figure 8 shows the XFMR\nenergy-dependent intensity of Fe in the Ni 80Fe20layer measured on- and o\u000b-resonance,\ncompared with the static XMCD signal measured at the same \feld value. One can\nsee that, while the on-resonance XFMR displays a strong energy dependent intensity,\nthe XFMR measured o\u000b-resonance is zero within the noise, emphasizing the dynamic\norigin of the XFMR E-scan. Indeed, the latter can be considered as a dynamic XMCD\nspectrum, where the probed magnetization corresponds to that resonantly excited by\nthe MW \feld into uniform precession or other resonant modes selected by the choice of\nB0. Here, although the signal-to-noise ratio needs to be improved to reach quantitative\nconclusions, the overall similarity between the static and dynamic XMCD lineshape\nsuggests a similar orbital-to-spin ratio for the static and precessing magnetic moments\nof Fe.\nThis method eliminates the need to resort to the di\u000berential form of the XMCD\nsum rules to extract information on the precession dynamics of the spin and orbitalElement-resolved XFMR 14\nmagnetization components of the d-density of states introduced in Ref. [42]. By\nintegrating XFMR E-scans and XMCD spectra simultaneously measured, the standard\nXMCD sum rules [47, 48] can be applied, deriving information on the dynamic vs. static\ntotal orbital and spin magnetic moments. Assumptions made in applying the XMCD\nsum rules regarding integration cut o\u000bs, magnitude of the spin dipole moment, and\nisotropic absorption intensity [47, 48, 49] shall hold equally well (or badly) for XFMR\nE-scans and XMCD spectra, thus making their relative comparison most relevant. Two\ncaveats should be mentioned concerning this type of measurements. The \frst is the\nquantitative accuracy of the XMCD sum rules for soft x-ray absorption spectra measured\nin the FY mode, as discussed, e.g., in Ref. [50]. The second is the presence of strong self-\nabsorption e\u000bects for thick \flms and bulk samples, which alter the measured intensity\nof the most prominent XAS and XMCD features. Di\u000berent methods may be used to\nretrieve the true XAS absorption coe\u000ecients from FY data [51, 52]; a relative, qualitative\ncomparison of static and dynamic XMCD measurements is nonetheless always possible\nsince self-absorption a\u000bects them in the same way. Moreover, such e\u000bects may be\nneglected in ultrathin \flms and dilute samples, and entirely bypassed by measuring\nXFMR in a transmission geometry, with a signi\fcant additional gain of XAS intensity.\nRecently, XAS and XMCD spectra have been measured also by time-resolved pump-\nprobe methods, addressing the transfer of angular momentum from the spin and orbital\nmagnetic moments to the lattice in Fe/Gd multilayers [53] and polycrystalline Ni \flms\n[54]. Ultrafast heat transients produced by fs-laser pulses are used to pump electronic\nexcitations, inducing strong demagnetization e\u000bects and consequent transfer of angular\nmomentum from the magnetic system to the lattice. XMCD spectra recorded at \fxed\ndelay times allow to monitor the spin and orbital magnetic moments during this process.\nTime resolution is achieved either by temporally dispersing the intensity of x-ray photon\nbunches transmitted by the sample using a streak camera [53] or by employing fs x-\nray probe pulses produced by femtoslicing techniques [54], achieving resolutions of the\norder of 2 ps and 100 fs, respectively. \"Slower\" time-resolved schemes based on pulsed\nmagnetic \felds [7, 13] or continuous wave excitations [14, 15] as pump and x-ray photon\nbunches of\u001850\u0000100 ps duration as probe may also be employed to measure full XMCD\nspectra, although this, to our knowledge, has not yet been reported. With respect to\ntime-resolved methods, XFMR E-scans appear particularly suited to study stationary\nprecessional dynamics. The averaging time required to measure the Fe spectrum in\nFig. 8 amounts to about 1 hour. Improving the detection e\u000eciency using transmission\nrather than FY is expected to reduce this time further while leading to a better XFMR\nsignal-to-noise.\n6. Conclusions\nIn summary, we have shown that time-invariant x-ray magnetic dichroism and magnetic\nresonance spectroscopy at GHz frequency can be combined to yield element-resolved\nmagnetic resonance spectra as well as dynamic XMCD spectra, depending on whetherElement-resolved XFMR 15\nthe photon energy is kept constant while the applied magnetic \feld is varied or\nviceversa. We reported two case studies concerning a Gd 1Y2Fe5O12garnet and an\nAl(10 nm)/Ni 80Fe20(5 nm)/Ni(50 nm)/Cr(5 nm) metallic \flm. Antiferromagnetic\ncoupling at resonance between Fe and Gd sublattices in Gd:YIG has been resolved\nand shown to hold also in the nonlinear regime where the FMR response is heavily\nsaturated. The Fe and Gd XFMR linewidths coincide to within the experimental\naccuracy, supporting the notion of a common e\u000bective damping parameter for the two\nsublattices introduced in early theoretical treatments of ferrimagnetic resonance [4].\nThe Ni 80Fe20(5 nm)/Ni(50 nm) bilayer presents two resonance modes whose elemental\ncomponents have been separately identi\fed by XFMR. It was shown that while one\nlayer is excited the other is at rest, i.e., that interlayer driving e\u000bects are negligible\nfor moderate values of the interface exchange energy, as predicted by theory [43].\nFinally, the comparison between static and dynamic Fe XMCD lineshape in Ni 80Fe20\nsuggests a constant orbital-to-spin magnetic moment ratio for the steady and precessing\nmagnetization.\n7. 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Liu1,2 \n1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials \nand Institute for Advanced Materials, South China Academy of Advanced Optoelectronics, \nSouth China Normal University, Guangzhou 510006, China \n2Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China \n \n[Abstract] Magnetic s kyrmion s are magnetic texture s with topolog ical protection , which are \nexpected to be information carriers in future spintronic devices. In this work, we propose a \nscheme to implement hybrid magnetic skyrmion s (HMS) in ferrimagne ts, and we study \ntheoretically and numerically the dynamics of the HMS driven by spin-orbit torque . It is \nrevealed that the skyrmion Hall effect depends on the skyrmion helicity and the net angular \nmomentum ( δs), allowing the effective modulation of the HMS motion through tuning \nDzyaloshinskii -Moriya interaction and δs. Thus , the H all effect can be suppressed through \nselecting suitable materials to better control the HMS motion . Moreover, Magnus force for \nfinite δs suppresses the transverse motion and enhances the longitudinal propagation, resulting \nin the HMS dynamics in ferrimagnets faster than that in antiferromagnets. \n \nKeywords: ferrimagne ts, hybrid magnetic skyrmion, spin-orbit torque , spintronics \n \n \n*Email: qinmh@scnu.edu.cn I. Introduction \nMagnetic skyrmion s are localized spin texture s with a topological number [1–5], which \nhave been observed in a series of chiral magnets and heavy metal/ferromagnetic films where \nDzyaloshinskii -Moriya interaction (DMI) due to the broken inversion symmetry plays an \nimportant role in stabilizing the skyr mion s. Specifically, the bulk DMI in chiral magnets \nstabilize s Bloch skyrmion s (typical spin configuration is shown in Fig. 1(a)) [6], and the \ninterfacial DMI in films favors the formation of Néel skyrmion s (Fig. 1(b) ) [7]. Importantly, \nskyrmions are relatively steady with small s ize owing to the topological protection [8–12], and \nthey can be driven by electric current with a rather low density, making them exciting candidates \nfor high -density and low -power -consuming information storage devices . \nSpintronic devices based on skyrmion require precise modulation of the skyrmion s, while \nthe skyrmion Hall effect remains a key challenge [13–16]. In detail , when skyrmion is driven \nby spin -polarized current, it suffers a transverse Magnus force and deviates from the current \ndirection. As a result, the skyrmion can be driven out of the track under a high current , resulting \nin a loss of information. To overcome this deficiency , a number of so lutions including the use \nof antiferromagnets [7], boundary effect [17], and other similar spin structures [18] have been \nproposed. \nMost recently, hybrid magnetic skyrmion (HMS) in ferromagnets with a structure \ninterpolating between Néel and Bloch skyrmions stabilized by the hybrid DMI (a mixture of \ninterfacial and bulk DMIs) or by coupling with vortex structure s has been reported , which is \nrevealed to has an enhanced mobility and a reduced skyrmion Hall effect [19–22]. Interestingly, \nthe HMS dynamics driven by spin -orbit torque (SOT) significantly depends on the skyrmion \nhelicity, providing another parameter in modulating the Hall effect. For example, one can tune \nthe ratio of the interfacial and bulk DMIs through various methods such as applying \nvoltage [23], electric field [24] and strain [25,26] , and in turn modulates the skyrmion helicity \nand dynamics. As a matter of fact, the skyrmion Hall effect induced by SOT in ferromagnets \ncan be eliminated through delicate tuning the skyrmion helicity , while the strong stray field and \nrelatively slow spin dynamics are still disadvantages in applications . In antiferromagnets, on the other hand, the Magnus forces of two -sublattices induced by \nspin-orbit /spin -transfer torque on Néel or B loch skyrmion are well canceled out, and the \nskyrmion moves along the cu rrent direction [27,28] . Thus, besides zero stray field and fast spin \ndynamics, antiferromagnets also provide an important platform for achieving skyrmion long -\nrange transmission . However, effectively detection and modulation of antiferromagnetic \nskyrmion s are still challenging in practice due to zero net magnetization of the system . \nMoreover, in antiferromagnets, a strong Hall motion of HMS is also revealed when it is driven \nby SOT, which is different from the case of spin -transfer torque driven HMS motion with zero \nHall angle [29]. This phenomenon attributes to the fact that the effective SOT depending on the \nhelicity of t he HMS generally deviate s from the current direction , resulting in the HMS Hall \neffect . \nAlternatively , ferrimagnets are suggested to combine the benefits of ultra -high working \nfrequencies with ease of detecting and modulating [30–32], noting that ferrimagnet has a \nnonzero magnetic moment and ultrafast spin dynamics comparable to antiferromagnets around \nthe angular momentum compensation temperature ( TA). Importantly , the net angular \nmomentum δs can be adjusted through tuning temperature and ion doping, which has a \nsignificant effect on skyrmion dynamics [33–38]. For example, δs can effectively modulate the \nMagnus force induced by SOT and control the skyrmion Hall motion in ferrimag nets [18,32,39] . \nNaturally , it is expected that the joint action of δs and skyrmion helicity gives rise to \ninteresting skyrmion dynamics in ferrimagnets , which urgently deserves to be uncovered to \nprovide a clear scenario for skyrmion motion manipulation . On the one hand, this study \nuncovers new dynamic behavior, contributing to the development of spintronics. For example, \nour earlier work has revealed that δs determines the momentum onto the skyrmion in \nferrimagnets imposed by the injected ma gnons , and consequently affects the Hall angle [12]. In \nsome extent , δs probably has an important effect on the SOT-driven HMS dynamics in \nferrimagnets . On the other hand, the Hall effect of the HMS in ferrimagnets can be suppressed \nwhen the Magnus force depending on δs suppresses the transverse motion induced by the SOT. Thus, the HMS motion with zero Hall effect could be available for certain δs, which is \ninstructive for experiment and device design to achieve the straight skyrmion motion . \n In this work, we study the dynamics of HMS driven by SOT in ferrimagnets using Thiele \ntheory and numerical simulations based on solving the two coupled Landau -Lifshitz -Gilbert \n(LLG) equations . The dependence of the skyrmion Hall angle on δs and the helicity is \ninvestigated, which exhibits the gradual transition in Hall angle o n δs and the helicity. Thus, \nHall motion of the HMS can be suppressed through choosing suitable δs, allowing one to select \nsuitable materials to better control the skyrmion motion. Furthermore, the mobility of the HMS \nin ferrimagnets could be enhanced attr ibuting to Magnus force . \n \n \nII. Theory for SOT -driven HMS dynamic s in ferrimagnets \nIn this section, we study theoretically the HMS dynamics in ferrimagnets driven by SOT \nthrough deriv ing the equations of motion for a HMS based on Thiele theory . We consider a spin \nmodel composed of two unequal sublattices which are antiferromagnetically coupled , as shown \nin Fig. 1( d). The unit magnetization vectors of the two sublattices are m1 and m2, respectively. \nIn the continuum approximation [40], one introduce s the Néel vector n = (m1 − m2)/2 and the \nmagnetization vector m = (m1 + m2)/2 to deal with the dynamic equations of ferrimagnets . \nConsidering the SOT term, t he dynamics of vectors m and n can be described by the following \nequation , \n( )()() 2s m n p ss + =− + + + m n m f n f n n n m n\n, (1a) \n()()()ms ss =− + + n n f n n n m\n, (1b) \nwhere s = (s1 + s2) with the angular momentum densit ies of the two sublattices s1 and s2, δs = \n(s1 − s2), sα = (s1α1 + s2α2) with the damping constants α1 and α2, and α = (s1α1 + s2α2)/s is the \ndamping coefficient . β = ħjθSH / et is the coefficient of the SOT term, where ħ is the reduced \nPlanck constant, e is the electron charge , and t is the thickness of the magnetic layer . mp is the \npolarization vector of the spin polarized current , θSH is the spin Hall angle , and j is the current \ndensity. fn = − δU / δn and fm = −δU / δm denote the effective fields of n and m, respectively , with the free energy density U. Considering the fact that |m| << |n| in slowly evolving system, \nthe dynamic equation can be represented by n after neglecting the weak term s: \n()() 20s n p s − − + + =n n n n f n n n m n\n, (2) \nwhere ρ = s2/a is the inertia coefficient . \nOwing to the topological protection , the HMS behaves as a soliton or a rigid body that \nkeeps its shape unchanged during the propagation. For convenience, the HMS motion is \nintroduced into the dynamic equation in the form of collective coordinates R(t), n(r, t) = n(r − \nR(t)). Projecting the dynamic equation onto the HMS translational mode, the following Thiele \nequation of the HMS dynamics is obtained [41]: \n() 20sp M s IR − − + =R G R R m\n, (3) \nwhere M = −ρ∫(∂in ∙ ∂in)dxdy is the effective mass of the HMS, and G = (0, 0, 4πQ) is the \ngyromagnetic coupling vector with the topological charge Q. Here, Q, the viscous coefficient \n, and tensor I read \n()() 1/ 4 d dxy Q x y = n n n\n, (4a) \n()\n()d d 0\n0 d dii\njjxy\nxy = \nnn\nnn\n, (4b) \n()() sin cos dr I r r r r = +\n, (4c) \nwhere θ is the polar angle of the Néel vector in the spherical coordinate, and r is the radius in \nthe polar coordinate, as shown i n Fig. 1 (c). The tensor R(Θ) reads \n()sin cos\ncos sinR− =− − \n, (5) \nwhere the skyrmion helicity Θ = arctan( Db/Di) is determined by the bulk DMI magnitude (Db) \nand the interfacial DMI magnitude (Di), which could be altered between 0 and 2π. It is worth \nnoting that the tensor R(Θ) and SOT are directly coupled , allowing one to modulate the SOT \nacting on the HMS through tuning the HMS helicity. First, w e focus on the dynamics of the HMS at TA with δs = 0. In this case, the Magnus \nforce denoted by t he second term in Eq. (3) is elimin ated, and the acceleration\nR\n can be safely \nignored considering the s teady motion of the HMS. When the current is polarized along the x-\ndirection mp = ex, one obtains the velocity components of the HMS, \n2 sinxv I s =− \n, (6a) \n2 cosyv I s =− \n. (6b) \nIt is clearly shown that t he motion of the HMS depends on its helicity . Particularly , the \ndependence of vx/vy on the DMI coefficient s reads : \nn / / tabi xyv DD v= =\n, (7) \nthe same as that obtained through solving the LLG equation in the earlier report [29]. \nSubsequently , we derive the velocit y of the HMS in uncompensated ferrimagnets for a \nfinite δs, which is given by: \n( )( )2 2 2 2 28 cos 2 sin 16x s a sv I Q s Q s =− + +\n, (8a) \n( )( )2 2 2 2 28 sin 2 cos 16y s a sv I Q s Q s =− − + +\n. (8b) \nThus , the dependence of the velocity on both δs and Θ is clearly demonstrated, allowing one to \nmodulate the dynamics through tuning these parameters. Taking Θ = π/4 as an example, the \nMagnus force ~Qδs for negative δs enhances the longitudinal motion and suppresses the \ntransverse motion, resulting in the decrease of the Hall angle ~vx/vy. Subsequently, the HMS \nspeed v and Hall angle θH are derived , respectively, \n2 2 2 2 22\n16sIv\nQs\n=\n+\n, (9a) \n8 sin 2 cosarctan( )8 cos 2 sinsa\nH\nsaQs\nQs− + =+ \n. (9b) \nIt is demonstrated that v linearly depends on the current intensity and hardly be affected by the \nHMS helicity. Importantly , the helicity and δs effectively modulates the skyrmion Hall angle , \ndemonstrating the important role of the DMI magnitude s and δs in controlling the HMS dynamics in ferrimagnets . For tanΘ = sα/4πQδs, zero Hall angle is achieved, and the skyrmion \nmoves straightly along the current direction with a speed of 2βI/(16πQ2δs2 + sα22)1/2. \nInterestingly , sα decreases as δs increases, which may enhanc e the speed of the HMS . Thus, it \nis suggested theoretically that the Hall motion of the HMS in ferrimagnets can be completely \nsuppressed in accompany of the speed enhancement by tunin g δs, which definitely favors future \napplications. \nTo check these predictions, a comparison betw een the theoretical analysis with the \nnumerical simulations is indispensable. In Sec. III, we introduce numerical simulation s of the \nHeisenberg spin model , and then give the calculated results and discussion . \n \n \nIII. Numerical simulations and discussion \nIn this section, we first introduce the simulation method based on the standard Heisenberg \nmodel through solving the LLG equation, and then give the simulated and calculated results. A \nbrief discussion on potential application of the HMS is disc ussed at last. \n \nA. Model and simulation method \nThe micromagnetic simulations are performed based on the classical Heisenberg model , \nand t he model Hamiltonian is given by \n()() ()()2\n,,ˆ ˆAB\nABAB i j A i j B i j\n i,j i,j\nAB\ni ij i j b ij i j i\ni j i j iH J J J\nD z +D K z \n = + + \n+ + \n s s s s s s\nu s s u s s s\n, (10) \nwhere si is the normalized spin at lattice site i, JAB is the antiferromagnetic interlayer interaction \nbetween sublattice A and sublattice B, JA and JB are the ferromagnetic intra -sublattice coupling \nfor sublattice A and sublattice B between the nearest neighboring spins , respectively . DA \niand D\nB \nbare the interfac ial DMI and b ulk DMI coefficients for two sublattices , respectively, uij is the \nunit vector connecting two spins in sublattices , and K is the anisotropy constant . The coupling parameters are the same as those in the theoretical analysis. Actually, the \nstabilization of ferrimagnetic skyrmions have been experimentally reported in GdCo films [42]. \nHerein, we consider a GdCo/Co heterostructure to produce HMS, where the coupling between \nthe two magnetic layers can be modulated by tuning the thickness of the spacer [43]. The \ninterfacial DMI in Co laye r is induced by coupling with a heavy metal layer which generate s \nstrong spin-orbit coupling s [44], and SOT can be easily applied by injecting in -plane current \ninto the heavy metal layer [8,15] . Withou t loss of generality, we set the intra -sublattice \nexchange stiffness AGd-Gd = 5 pJ/m, ACo-Co = 5 pJ/m , the perpendicular magnetic anisotropy \nconstant K = 0.16 MJ/m3, the bilinear surface exchange coefficient σ = −1 mJ/m2, and t he DMI \ncoefficients Di and Db vary within a reasonable range. \nThen , the dynamics of the HMS is investigated by solving the LLG equation, \n() ,Mi i i\ni i eff i i i i p i\ni tt=− + + sss H s s m s\n, (11) \nwhere Heff,i = Mi−1H/si is the effective field with the magnetic moment Mi at site i, and the \ngyromagnetic ratio γi = giμB/ħ with the g-factors g1 = 2.2 and g2 = 2. We set the spin Hall angle \nθSH = 0.2 , and the damping constants α1 = α2 = 0.4. \nHere, the micromagnetic simulations are performed using the Object -Oriented \nMicroMagnetic Framework (OOMMF) with extended DMI module. We start from a discrete \nlattice with the size of 100 nm × 100 nm × 9 nm and cell size of 1 nm × 1 nm × 3 nm, and set \nthe time step to 10−13 s. The used magnetic moments Mi for nine different cases are shown in \nTable 1, correspond ing to nine different δs. \n \nB. Results and discussion \nIn Fig. 2(a), w e present the simulated and theoretical calculated vx/vy as function s of tan Θ \nat TA. The simulations coincide well with the theory, confirming the validity of the theory \nanalysis . A linear dependence of vx/vy on tanΘ is observed for mp = ex, the same as that in \nantiferromagnets [29]. Similarly, when the polarization vector of the injected spin current is \ntuned from ex to ey, the HMS moves along the direction perpendicular to the motion for mp = ex, and vy/vx linearly increases with tanΘ. Therefore, one can modulate the HMS motion through \ntuning t he helicity and spin polarization. \nSubsequently, the effect of the helicity on the speed of the HMS v is investigated, and the \ncorresponding results are shown in Fig. 2 (b) where present the simulated v as a function of \ncurrent density j for various Di/Db with a fixed \n22\nib D D D=+ . v linearl y increases with the \nincreasing j due to the enhancement of SOT , consistent with the earlier report. For a fixed j, the \nHMS moves at a highest speed for Di = Db, and speed down when Di and Db deviate from each \nother. This phenomenon can be understood from the following aspects. It is well noted that \nskyrmion size mainly depends on DMI and anisotropy . For two uncoupled magne tic layers, \nDMIs with a same magnitude stabilize the isolated skyrmion s with a same size , while the sizes \nof the skyrmions are different for DMIs with different magnitudes . Thus, for Di = Db, the \nskyrmions in two sublattices are well coupled due to the sam e skyrmion size. However, when \nDi deviates from Db, the interlayer antiferromagnetic coupling competes with the DMI stronger \nthan that for Di = Db, resulting in the increase of the internal energy and the reduction of the \nskyrmion mobility. \nIn some extent, this phenomenon is analogous to the barrel effect, i.e., the size and speed \nof the HMS are mainly determined on the smaller one of Di and Db. As a matter of fact, this \nqualitative explanation is confirmed in the calculated I/shown in the insert of Fig. 2(b) , noting \nthat I/ depends on the HMS structure and determines the dynamics . I/reaches its maximum \nfor Di = Db, and decreases with the deviation between Di and Db. Moreover, for a same deviation \nmagnitude, Di/Db = 1:4 or 4:1 as an example, the HMS have a same speed, as shown in our \nsimulations. \nSubsequently, the effect of δs on HMS dynamics in uncompensated ferrimagnets is \nnumerically investigated, and the corresponding results are shown in Fig. 3. Generally, during \nthe propagation of the HMS, Magnus force proportional to δs is generated via the gyrotropic \ncoupling , while the inertia coefficient ρ is decreased with the increase of δs. As a result, the \nspeed of the HMS slightly increases as δs increases, as shown in Fig. 3(a) where presents the simulated and calculated v as a function of δs for various j with mp = ex. Furthermore, the \nsimulations coincide well with the theory analysis for low current density (j ~ 6 × 1011 A/m2), \nwhile deviate s from the theory for large j. It is noted that a deformation of the HMS may occur \nduring the propagation with a high speed, attributing to the slight discrepancy between \nsimulations and theory. \nImportantly, a significant effect of δs on the HSM Hall motion is revealed, as shown in Fig . \n3(b) where gives the simulated velocity components vx and vy as functions of δs for various j. \nHere, without loss of generality, we consider the case of Db = Di, corresponding to the helicity \nof –3π/4. For δs = 0, vx equals to vy, and the HMS moves along the [1, 1] direction. vy increases \nwith the increase of δs, while vx less depends on δs. Thus, the Hall angle of the HMS could be \nmodulated through tuning δs. On the one hand, the Magnus force along the [-1, 1]/[1, -1] \ndirection is generated during the HMS propagation for positive/negative δs, which is enhanced \nas |δs| increases. Thus, the longitudinal/transverse motion vx/vy is suppressed/enhanced by \nMagnus force with the increase of δs. On the other hand, the parameter sα also decreases, and \nthe contribution of the dissipative term to vx/vy is enhanc ed. Moreover , the dissipative term \ncompetes with the Magnus force in determining vx, as demonstrated in Eq. 8, resulting in a \nrather stable vx for various δs. However , both the two terms contribute to vy, and vy extensively \nincreases with the increasing δs. As a result, the HMS Hall motion can be suppressed and even \neliminated by choosing suitable δs. \nSimilar effect of δs on the Hall motion of HMS with other helicity has been revealed, and \nthe results are summarized in Fig. 4 where gives the simulated Hall a ngle in the ( Di/Db, δs) \nparameter plane. It is clearly shown that both the HMS helicity and δs can be used in modulating \nthe Hall angle, and zero Hall angle can be achieved for these parameter values shown with the \ndashed line . For exampl e, the Bloch skyrmion with the helicity ~ −π/2 stabilized by Db exhibit s \na Hall motion, as shown in Fig. 5(a) where gives the trajectory of the skyrmion for δs = -3.1×10-\n7 J∙s/m3. The Hall motion is suppressed for introducing addition Di = Db /30 which stabilize the \nHMS with the helicity of -91.9° , and the HMS moves straightly along the x-direction, as shown \nin Fig. 5(b) . Figs. 5(c) and 5(d) gives the trajectories of the Bloch skyrmion and HMS for δs = 3.1×10-7 J∙s/m3, respectively, which also dem onstrate the suppression of Hall motion by \nintroducing suitable Di. Interestingly , the HMS moves faster than the case of negative δs, \nconsistent with the theory analysis in Eq. ( 9a). In details, the speed is enhanced up to ~ 20% due \nto the enlargement of the Magnus force , noting that Magnus force suppresses the transverse \nmotion and enhances the longitudinal propagation. Thus, ultrafast dynamics of HMS in \nferrimagnets is available in the absence of the Hall motion, confirming ag ain the advantages of \nferrimagnets in future spintronic applications. \n \nC. Possible applications \nSo far, this study unveils the important role of the skyrmion helicity and δs in modulating \nthe SOT-driven dynamics of the HMS in ferrimagnets , which is helpful in guiding future \nexperiments and device design. \nGenerally , most of the parameters chosen in this study are comparable to those in \nGdCo /Co heterostructure [42–44]. The current density is in the order of 1011 A/m2 which is the \ntypical magnitude in experiments [8,45,46] , and the DMI magnitudes are set within a \nreasonable range . Importantly, the ratio Di/Db has been proven to be a core control parameter \nin modulating the skyrmion Hall effect , allowing one to better control the HMS dynamics \nthrough tuning the DMI coefficients in ferrimagnets. In particular, the DMI can be easily \nadjusted through applying voltage, electric field, or strain. Of course, the theoretically revealed \nHMS dynamics in ferrimagnet s deserves to be checked in future experiments. \nA precise manipulation of skyrmion is import ant for implementing skyrmion -based logic \ndevices such as logic arrays. For the application of skyrmionic logic circuit, we show here a \nskyrmion diversion operation in a Y -junction , as depicted in Fig . 6. Here, a gate voltage is \napplied to tune the DMI [24], and the e lectrodes are set at the node s to control the interfac ial \nDMI in the local area. When the input signal is “0” with low potential , no additional DMI is \ngenerated at the node and the skyrmion will move along its original track , as shown in fig. 6(a). \nOne tunes the input voltage to “1” with high potential, a significant skyrmion Hall motion is \ninduced and the skyrmion move s to another track , as depicted in fig. 6(b). Moreover, e ach voltage node can be considered as a logical unit , and t he skyrmion -based programmable arrays \ncan be implemented by connecting nodes in a series. \n \n \nV . Conclusion \nIn conclusion , we have studied theoretically and numerically the dynamics of HMS in \nferrimagnets driven by SOT . The dependence of the skyrmion Hall angle on the net angle \nmomentum δs and the helicity is unveiled by numerical simulations, which demonstrates the \ngradual transition in Hall angle on δs and the helicity . Particularly, Hall motion of the HMS can \nbe eliminated through tuning δs, allowing one to select suitable materials to better control the \nskyrmion motion . Moreover, the HMS for finite δs may exhibit faster dynamics than that of \nantiferromagnets, attributing to the Magnus force which suppresses the transverse motion and \nenhances the longitudinal propagation. Thus, this work demonstrates the advantages of the \nferrimagnetic hybrid skyrmions for spintronic application, which is very meaningful for \nexperiments and device design. Acknowledgment \nWe sincerely appreciate the insightful discussions with Zhejunyu Jin and Xue Liang. The work \nis supported by the Natural Science Foundation of China (Grants No. U22A20117, No. \n51971096 , No. 92163210, and No. 51721001), the Guangdong Basic and Applied Basic \nResearch Foundation (Grant No. 2022A1515011727) , and Funding by Science and Technology \nProjects in Guangzhou (Grant No. 202201000008). References: \n[1] N. Nagaosa and Y. Tokura, Topological Properties and Dynamics of Magnetic Skyrmions , \nNat. Nanotechnol. 8, 899 (2013). \n[2] A. Fert, V. Cros, and J. Sampaio, Skyrmions on the Track , Nat. Nanotechnol. 8, 152 (2013). \n[3] X. Zhang, Y. Zhou, and M. Ezawa, High -Topological -Number Magnetic Skyrmions and \nTopologically Protected Dissipative Structure , Phys. Rev. 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Schematic spin configuration of (a) Bloch -skyrmion, (b) Néel -skyrmion and (c) hybrid \nskyrmion , and (d) sketch of a system with hybrid skyrmion where red and blue arrows represent \nthe magnetization direction of CoGd and Co/Pt alloy , respectively . \n \n \n \n \n \n \nFIG. 2. (a) The calculated (lines) and simulated ( symbols ) vx/vy as function s of the skyrmion \nhelicity tanΘ for various mp, and (b) the simulated HMS speed as a fu nction of the current \ndensity j for various DMI ratios . The insert shows the calculated I/ for j = 5 × 1011 A/m2 for \nvarious DMI ratios. \n \nFIG. 3. (a) The calculated (lines) and simulated (symbols) v, and (b) the simulated vx and vy of \nthe HMS as a function of δs for various j. \n \nFIG. 4. The simulated Hall angle of the HMS for mp = ex in the (Di/Db, δs) parameter plane. T he \nblack dashed line corresponds to the absence of the skyrmion Hall effect . \n \nFIG. 5. Trajectories of the skyrmion for (a) δs = −3.1 ×10-7 J∙s/m3 and Θ = −90º, and (b) δs = \n−3.1 ×10-7 J∙s/m3 and Θ = −91.9º, and (c) δs = 3.1 ×10-7 J∙s/m3 and Θ = −90º, and (d) δs = 3.1 \n×10-7 J∙s/m3 and Θ = −86.8º. The magnetic texture s in (a) and (c) are Bloch -type skyrmions, \nwhile those in (b) and (d) are HMS stabilized by the additional interfacial DMI. \n \nFIG. 6 . The propagation of HMS driven by SOT in the Y -junctions with the input of (a) low \npotential “0”, and (b) high potential “1”. Here, the interfacial DMI is tuned by the applied \nvoltage. \n" }, { "title": "1104.4032v1.Magnetic_properties_of_the_ferrimagnetic_cobaltite_CaBaCo4O7.pdf", "content": "arXiv:1104.4032v1 [cond-mat.str-el] 20 Apr 2011Magnetic properties of the ferrimagnetic cobaltite\nCaBaCo 4O7\nZhe Qua,∗, Langsheng Linga, Lei Zhanga, Li Pib,a, Yuheng Zhanga,b\naHigh Magnetic Field Laboratory, Chinese Academy of Science s,\nHefei, Anhui, 230031, China\nbHefei National Laboratory for Physical Sciences at the Micr oscale,\nUniversity of Science and Technology of China, Hefei, Anhui , 230026, China\nAbstract\nThe magnetic properties of the ferrimagnetic cobaltite CaBaCo 4O7are sys-\ntematically investigated. We find that the susceptibility exhibits a dow nward\ndeviation below ∼360 K, suggesting the occurrence of short range magnetic\ncorrelations at temperature well above TC. The effective moment is de-\ntermined to be 4.5 µB/f.u, which is consistent with that expected for the\nCo2+/Co3+high spin species. Using a criterion given by Banerjee [Phys.\nLett.12, 16 (1964)], we demonstrate that the paramagnetic to ferrimagn etic\ntransition in CaBaCo 4O7has a first order character.\nKeywords: A. magnetically ordered materials, D. phase transitions\n1. Introduction\nTransition metal oxides with geometry frustration have attracte d consid-\nerable interest over decades. [1, 2, 3, 4] They commonly exhibit th e per-\nsistence of strong spin fluctuations at low temperatures. As a res ult, the\nlong-range magnetic order is at least partially suppressed and vario us short\nrange correlated phases such as spin liquid, spin glass or spin ice deve lop.\nIn some cases, frustration can be partially or entirely released, eit her by\nstructural distortions that lift the ground-state degeneracy, or by the ”order-\nby-disorder” mechanism, [5] resulting in the establishment of a long- range\nmagnetic order.\n∗Corresponding author. Tel: +86-551-559-5640; Fax: +86-551- 559-1149.\nEmail address: zhequ@hmfl.ac.cn (Zhe Qu)\nPreprint submitted to Solid State Communications October 1 1, 2018Two well-known structural topology causing the presence of geom etry\nfrustrationaretwo-dimensionaltriangularlatticeandtwo-dimens ionalkagome\nlattice. Compositions whose structural motif embraces triangular or kagome\nlayers are of great interest as model systems and have been the f ocus of\nnumerous studies. In this respect, the recently discovered ”114 ” cobaltite\nCaBaCo 4O7[6, 7] is particularly interesting because its crystal structure is\nbuilt up of an alternate stacking of triangular or kagome layers form ed by the\nCoO4tetrahedra (shown in theinset to Fig. 1). There is very largedistor tion\nin the crystal, characterized by a strong buckling of the kagome lay ers. [6, 7]\nIn addition, it exhibits charge ordering, with Co2+sitting on two sites and\n”mixed valent” cobalt Co3+/Co2+Lsitting on two other sites. [7] Due to the\nlarge structural distortion and the charge ordering, the geomet ry frustration\nis lifted, resulting in a ferrimagnetic ground state at low temperatur es. [6, 7]\nAlthough significant progress has been made in understanding the m ag-\nnetic properties in CaBaCo 4O7, a few questions remain to be answered. For\nexample, does the system shows short-range magnetic correlatio ns above TC\nlike their ”114” cousins such as YBaCo 4O7? [8, 9] Why the obtained effec-\ntive moment differs significantly from the expected value in CaBaCo 4O7? [6]\nWhat’s the nature of the paramagnetic to ferrimagnetic transition ?\nTo address these questions, we systematically measured the magn etic\nproperties of CaBaCo 4O7. It is found that the susceptibility exhibits an\ndownward deviation below ∼360 K, suggesting the occurrence of short range\nmagnetic correlations at temperature well above TC. By extending the mag-\nnetization measurement up to 800 K, the effective moment is determ ined to\nbe 4.5µB/f.u through a Curie-Weiss analysis, which is consistent with that\nexpected for the Co2+/Co3+high spin species. Using a criterion given by\nBanerjee, [10] we demonstrate that the paramagnetic to ferrima gnetic phase\ntransition in CaBaCo 4O7is a first order one.\n2. Experiment\nPolycrystalline sample of CaBaCo 4O7was prepared by using the con-\nventional solid-state reaction method described in Ref. [6]. Stoich iometric\nproportionsofhighpurity CaCO 3, BaCO 3andCo 3O4were mixedandheated\nat 900oCin air to decarbonation. They are then pelletized, and then sin-\ntered at 1100oCin air for 12 hours and quenched to room temperature. The\nstructure and the phase purity of the samples were checked by po wder X-ray\ndiffraction (XRD) at room temperature. Magnetization measureme nts were\n2performed with a commercial superconducting quantum interfere nce device\n(SQUID) magnetometer (Quantum Design MPMS 7T-XL) and a Physic al\nPropertyMeasurement System(QuantumDesignPPMS-16T)equip ped with\na vibrating sample magnetometer (VSM).\n3. Results and Discussion\nFigure 1 displays the powder XRD pattern of CaBaCo 4O7at room tem-\nperature. Rietveld refinement [11, 12] of the XRD pattern confirm s that the\nsample is single phase with an orthorhombic structure ( Pbn21space group).\nThe lattice parameters are determined to be a= 6.2871 ˚A,b= 11.0106\n˚Aandc= 10.1945 ˚A, which agree well with previous reports within the\nexperimental error. [6, 7]\nThe temperature dependence of the magnetization M(T) between 2 K\nand 400 K under 0.1 T is shown in the upper panel of Fig. 2. They are me a-\nsured during field cooling sequence (FCC), during warming after field cooling\nsequence (FCW) andduring warming afterzerofield coolingsequenc e (ZFC),\nrespectively. All curve shows a rapid increase below ∼70 K, suggesting the\noccurrence of the transition into a magnetically ordered state. At 5 K, the\nsaturated magnetic moment is still relatively small, only ∼1.1µB/f.u.under\n14 T (see the lower panel of Fig. 2), agreeing with a ferrimagnetic gr ound\nstate. The Curie temperature, defined as the temperature corr esponding to\nthe maximum in the dM/dTcurve, is determined to be 60 K (see the inset to\nFig. 2). Below TC, we observe significant irreversibility between the magne-\ntization curve measured after ZFC and FCC histories. This is attribu ted to\nthe large coercive field compared to the applied field. [6, 13] As show n in the\nlower panel of Fig. 2, the coercive field of CaBaCo 4O7is about 2 T at 5 K,\nwhich is much larger than the applied field of 0.1 T. Therefore, the mag netic\ndomains will be locked in random direction during ZFC sequence while be\naligned to the same direction during FCC or FCW sequences, resulting in the\nlarge irreversibility below TC. All these results are consistent with previous\nreport, [6] confirming that our sample is of high quality.\nA close look on the temperature dependence of the magnetization r eveals\nmoreinformation. Theinset toFig. 3displays theenlargedviewofthe M(T)\ncurves between TCand 400 K. One can see that while the magnetization\ndecreases with increasing temperature above TCthe slope of the M(T) curve\ndoes not decrease monotonously as that expected for a pure par amagnetic\nstate where the Curie-Weiss law predicts χ∝C/(T−TCW) (HereCis\n3the Curie constant and TCWis the Curie-Weiss temperature). In order to\nunderstand this behavior, we extend the measurement of the M(T) during\nFCC sequence up to 800 K and perform the Curie-Weiss analysis. As s hown\nin Fig. 3, 1/ χshows an upward deviation from the linearity below ∼360 K.\nSince the deviation temperature is much higher than the Curie tempe rature,\nthis behavior could not be attributed to the critical enhancement o f the\nspin fluctuations due to the approach to the paramagnetic to ferr imagnetic\ntransition but suggests the occurrence of short range magnetic correlations.\nThe Curie-Weiss temperature TCWis determined to be ∼-890 K. This gives\nf=TCW/TC∼14.8 which means CaBaCo 4O7is strongly frustrated. The\neffective moment is determined to be ∼4.5µB/f.u., which agrees well with\nthevalueofa1:1combinationofCo2+/Co3+highspinspecies expected based\non the chemical formula. The Curie-Weiss temperature and the effe ctive\nmoment obtained here are different from previous report. [6] This should be\nunderstood because short-range magnetic correlations might ap pears in their\nfitting temperature region.\nIn order to obtain further information on the paramagnetic to fer rimag-\nnetic transition in CaBaCo 4O7, we use the criteria proposed by Banerjee to\ndetermine the order of this transition. By considering the essentia l similarity\nbetween the Landau-Lifshitz [14] and Bean-Rodbell [15] criteria, B anerjee\nshows that the slope of the H/MversusM2curves near the critical tem-\nperature can distinguish the first-order magnetic transition from the second\norder ones: a negative slope means the former and a positive slope m eans the\nlatter. [10] We then measured the initial isothermal magnetization c urves at\ntemperatures in the vicinity of the Curie temperature. Before eac h run, the\nsample is warmed up to 200 K and then cooled to the measuring temper ature\nunder zero field to ensure a perfect demagnetization of the sample . The data\nare summarized in the inset to Fig. 4. It is noted that the M(H) curve ex-\nhibitsapeculiar behaviorthatitsslopeshowsadecrease beforeanin crease at\nintermediate fields. This behavior was also observed in MnAs, where a first\norder transitionoccursatits Curietemperature andisused totes t Banerjee’s\ncriteria. [10, 15] We replotted the M(H) curves as H/Mvs.M2in Fig. 4.\nNegative slope is clearly observed between 64 and 70 K, which confirm s that\nthe paramagnetic to ferrimagnetic transition occurred in CaBaCo 4O7has a\nfirst order character according to the criterion.\n44. Conclusion\nIn conclusion, we systematically investigate the magnetic propertie s of\nCaBaCo 4O7. TheCurie-Weiss temperatureisdetermined tobe ∼-890Kand\nthe effective moment be ∼4.5µB/f.u.. The susceptibility shows downward\ndeviation from the Curie-Weiss law below ∼360 K, hinting that short range\nmagnetic correlations might occur at temperature much higher tha nTC=\n60 K. The paramagnetic to ferrimagnetic transition in CaBaCo 4O7is found\nto have the first order character.\n5. Acknowledgments\nThis work is financially supported by the National Key Basic Research of\nChina under Grant No. 2007CB925001 and 2010CB923403, and by N ational\nNatural Science Foundation of China under contract No. 1100419 8.\nReferences\n[1] A. P. Ramirez, Annu. Rev. Mater. Sci. 24, (1994) 453.\n[2] P. Schiffer and A. P. Ramirez, Comments Condens. Matter Phys. 18,\n(1996) 21.\n[3] J. E. Greedan, J. Alloys Compd. 408-412, (2006) 444.\n[4] R. Moessner and A. P. Ramirez, Phys. Today 59 (2), (2006) 24.\n[5] J. Villain, R. Bidaux, J. P. Carton, and R. Conte, J. Phys. (Franc e)41,\n(1980) 1263.\n[6] V. Caignaert, V. Pralong, A. Maignan, andB.Raveau, SolidState Com-\nmun.149, (2009) 453.\n[7] V. Caignaert, V. Pralong, V. Hardy, C. Ritter, and B. Raveau, P hys.\nRev. B81, (2010) 094417.\n[8] M. Soda, Y. Yasui, T. Moyoshi, M. Sato, N. Igawa, and K. Kakura i, J.\nPhys. Soc. Jpn. 75, (2006) 054707.\n[9] P. Manuel, L. C. Chapon, P. G. Radaelli, H. Zheng, and J. F. Mitche ll,\nPhys. Rev. Lett. 103, (2009) 054707.\n5[10] Phys. Lett. 12, (1964) 16; see also, for example, S. V. Vonsovskii, Mag-\nnetism(Wiley, New York, 1974), Vol. 2, Chap. 25.\n[11] A. C. Larson and R. B. Von Dreele, General Structure Analysis System\n(GSAS)(Los Alamos National Laboratory Report LAUR 86-748, 2000).\n[12] B. H. Toby, J. Appl. Cryst. 34, 2001 (210).\n[13] Z. R. Yang, S. Tan, and Y. H. Zhang, Appl. Phys. Lett. 79, (2001) 3645.\n[14] L. D. Landau, Zh. Eksp. Teor. Fiz. 7, (1937) 19; ibid. 7, (1937) 627;\nibid.E. M. Lifshitz, 11, (1941) 269; ibid.V. L. Ginzburg, 17, (1947) 833;\nS. V. Vonsovskii, Izv. Akad. Nauk SSSR, Ser. Fiz. 11, (1947) 485.\n[15] C. P. Bean and D. S. Rodbell, Phys. Rev. 126, (1962) 104.\n620 40 60 80 100 \n Intensity (arb. units) \n2θ (degree) Ba \nCa K Co \nT Co a\nc\nFigure 1: (Color online) Powder XRD patterns of CaBaCo 4O7. The solid curve is the\nbest fit from the Rietveld refinement using GSAS, with Rp= 11.68% and Rwp= 10.08%.\nThe vertical marks indicate the position of Bragg peaks and the bot tom curves show the\ndifference between the observed and calculated intensities. Inset shows the structure of\nCaBaCo 4O7viewed along baxis. K Co and T Co represent kagome layer and triangular\nlayer of CoO 4tetrahedra, respectively.\n70 100 200 300 0.0 0.2 0.4 0.6 0.8 \n-10 -5 0 5 10 -1.0 -0.5 0.0 0.5 1.0 H = 0.1 T \n ZFC \n FCC \n FCW \n M ( µb/f.u.) \nT (K) 40 60 80 100 \n dM/dT \nT (K) \n M ( µb/f.u.) \nH (T) T = 5 K \nFigure 2: (Color online) Upper panel: The magnetization as function o f the temperature\nunder an applied field of 0.1 T. inset shows the dM/dTas function of the temperature.\nLower panel: the magnetization as function of the field measured at 5 K.\n80 200 400 600 800 10 -2 10 -1 10 0\n1/ χ (T f.u./ µB)\nχ ( µB/f.u. T) \nT (K) 020 40 60 H = 0.1T \n100 200 300 400 2.0 2.5 3.0 3.5 \n M (10 -3 µb/f.u.) \nT (K) \nFigure 3: (Color online) The susceptibility and the reciprocal of the s usceptibility as\nfunction of temperature between 2 and 800 K under 0.1 T measured in FCC sequence.\nThe solid lines represent the Curie-Weiss fitting. Inset shows the en larged view of the\nM(T) curve to highlight the deviation from the Curie-Weiss fitting.\n0.0 0.2 0.4 0.6 010 20 \n50 K \n H/M (T f.u./ µB)\nM 2 ( µB2/f.u. 2)70 K \n0 1 2 3 4 50.0 0.2 0.4 0.6 \n \n70 K 50 K M ( µB /f.u.) \nH (T) \nFigure 4: Inset shows the initial isothermal magnetization curves a t temperatures in the\nvicinity of the Curie temperature TC= 60 K at an interval of 1 K. The main panel shows\nthese curves replotted as H/Mvs.M2.\n9" }, { "title": "2103.13433v2.Magnetism_and_Spin_Dynamics_in_Room_Temperature_van_der_Waals_Magnet_Fe__5_GeTe__2_.pdf", "content": "Magnetism and Spin Dynamics in\nRoom-Temperature van der Waals Magnet\nFe5GeTe 2\nLaith Alahmed,yBhuwan Nepal,zJuan Macy,{Wenkai Zheng,{Arjun Sapkota,z\nNicholas Jones,yAlessandro R. Mazza,xMatthew Brahlek,xWencan Jin,kMasoud\nMahjouri-Samani,ySteven S.L. Zhang,?Claudia Mewes,zLuis Balicas,{Tim\nMewes,zand Peng Li\u0003,y\nyDepartment of Electrical and Computer Engineering, Auburn University, Auburn, AL\n36849\nzDepartment of Physics, University of Alabama, Tuscaloosa, AL 35487\n{National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida\n32310\nxMaterials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,\nTN, 37831, USA\nkDepartment of Physics, Auburn University, Auburn, AL 36849\n?Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106\nE-mail: tmewes@ua.edu,balicas@magnet.fsu.edu,wzj0029@auburn.edu,pzl0047@auburn.edu\nAbstract\nTwo-dimensional (2D) van der Waals (vdWs) materials have gathered a lot of at-\ntention recently. However, the majority of these materials have Curie temperatures\n1arXiv:2103.13433v2 [cond-mat.mtrl-sci] 14 Sep 2021that are well below room temperature, making it challenging to incorporate them into\ndevice applications. In this work, we synthesized a room-temperature 2D magnetic\ncrystal Fe 5GeTe 2with a Curie temperature Tc= 332K, and studied its magnetic\nproperties by vibrating sample magnetometry (VSM) and broadband ferromagnetic\nresonance (FMR) spectroscopy. The experiments were performed with external mag-\nnetic fields applied along the c-axis ( Hkc) and the ab-plane ( Hkab), with temperatures\nranging from 300 K to 10 K. We have found a sizable Landé g-factor difference between\ntheHkcandHkabcases. In both cases, the Landé g-factor values deviated from g=\n2. This indicates a possible contribution of orbital angular momentum to the magnetic\nmoment but may also be caused by lack of saturation at FMR. The FMR measurements\nalso show a broadened linewidth at low temperatures; together with the VSM data, our\nmeasurements indicate that magnetic domains of different orientations develop within\nFe5GeTe 2in both HkabandHkccases, with a higher level of randomness in the Hkc\ncase, especially at lower temperatures. Our experiments highlight key information re-\ngarding the magnetic state and spin scattering processes in Fe 5GeTe 2, which promote\nthe understanding of magnetism in Fe 5GeTe 2, leading to implementations of Fe 5GeTe 2\nbased room-temperature spintronic devices.\nI. Introduction\nTheincreasinginterestinmagnetictwo-dimensional(2D)vanderWaals(vdWs)materials\nin recent years is warranted by their importance for fundamental studies of 2D magnetism,\nas well as potential applications for spintronic devices. Compared to three-dimensional (3D)\nmagnets, 2D magnetic materials exhibit exotic electrotransport, optical, and spin proper-\nties.1–3One of the biggest practical issues of most 2D vdWs magnetic materials is that they\ngenerally have a Curie temperature ( Tc) that is well below room temperature, making it\ndifficult to incorporate them into relevant devices.4–9For example, the Curie temperatures\nof 2D magnetic materials such as Cr(Si,Ge)Te 3(33 K and 61 K),4,5Cr(Br,I) 3(47 K and 61\n2K),6,7and Fe 3GeTe 2(220 K),8,9are all significantly below 300 K. This is due to their 2D\nnature, where the pair-exchange interaction is much weaker than in 3D magnets, as it is\nmostly mediated by neighboring magnetic atoms in the 2D plane. The interactions between\nthe magnetic layers across the vdWs gaps are much weaker in comparison.\nThe low Tcof the aforementioned 2D vdWs ferromagnets makes it impossible to use\ntheminroom-temperaturespintronicdevices. Furthermore, apotentialchallengeariseswhen\napproachingthe2DlimitofvdWscrystals, explainedbytheMermin-Wagnertheorem,10that\nintrinsic long-range magnetic order cannot be observed in the isotropic Heisenberg magnet\ndue to strong thermal fluctuations. This hinders the emergence of long-range magnetic\norder. However, uniaxial magnetic anisotropy may come to the rescue by stabilizing the\nferromagnetic order against finite thermal fluctuations. More specifically, in the 2D limit,\nit was shown theoretically that the Curie temperature is given by the uniaxial magnetic\nanisotropy constant K, and the spin-exchange interaction J, as follows:11\nTc\u00184\u0019J\n3ln(\u00192J=K)(1)\nAs the magnetic anisotropy in vdWs ferromagnets, set by the spin-orbit coupling, is much\nsmaller than the exchange interaction, Tcis low.12Extensive research efforts succeeded\nin engineering materials that could overcome these challenges. For example, Tccan be\nsignificantly raised to about room temperature by enhancing exchange interaction while\nkeeping the vdWs structure,12such as in the layered 2D Fe nGeTe 2(n\u00153).13,14This led to\nFe3GeTe 2withTcaround 220 K,9,15,16Fe4GeTe 2withTc= 270 K,12and Fe 5GeTe 2with T c\nranging from 260 - 310K, depending on the Fe content.14,17,18\nFerromagnetic resonance (FMR) spectroscopy is an important technique to study mag-\nnetization dynamics.19,20Several studies have analyzed the spin dynamics in 2D magnets by\nFMR,21–23which revealed their magnetocrystalline anisotropy dependence on temperature.\nIt is found that there is a discrepancy between the Landé g-factor along the c-axis and the\n3ab-plane directions in the 2D magnet Cr 2Ge2Te6.23,24However, the Landé g-factor is quite\nisotropic along different directions in another 2D magnet CrI 3.22These measurements were\nall performed on 2D magnets with low Curie temperatures (e.g. Tc= 61 K for CrI 3). With\nthe most promising room-temperature vdWs magnet arguably being Fe 5GeTe 2, we are in-\nterested in understanding its quasi-statis and dynamic magnetic properties, which can shed\nlight on its magnetic states and spin scattering mechanisms.\nWe first synthesize the vdWs magnet Fe 5GeTe 2, and then we study its magnetization\nproperties using both vibrating sample magnetometry (VSM) and ferromagnetic resonance\n(FMR) spectroscopy, in the temperature range of 300 K to 10 K. For FMR, a microwave field\nwas applied to the sample in addition to a quasistatic magnetic field, thus triggering spin\nprecession. At the resonance field Hresfor a given microwave frequency f, FMR oscillation\n(uniform-mode excitations of spin waves with k\u00190) occurs. The FMR spectroscopy has\nrevealed different Landé g-factors along the c-axis and the ab-plane in Fe 5GeTe 2, indicative\nof possible orbital momentum contribution to the magnetic moment. After examining the\ntemperature dependence of the FMR linewidth, we conclude that Fe 5GeTe 2maintains a\nferromagnetic phase with randomized magnetic domains at lower temperatures.\nII. Structure Characterization\nNominal Fe5GeTe 2crystals are grown using a mixture of precursor materials filled into\na quartz ampoule that is vacuumed and sealed with 1.9 mg/cm3of iodine as a transport\nagent. The mixture consists of pure elements of Fe:Ge:Te in the molar ratio of 6.2:1:2 (Fe:\n99:998%, powder, Alfa Aesar; Ge: 99:999%, 100 mesh, Alfa Aesar; Te: 99:999%, powder,\nAlfa Aesar). The excess Fe powder is to compensate for any possible Fe-site vacancies that\nmight occur during the growth.\nA standard MTI 2-zone model OTF-1200X furnace was employed, where the reactants or\nelemental precursors were placed in the high-temperature zone and the products were grown\n4in the low-temperature side. The ramping rate for both the hot (775\u000eC) and cold (700\u000eC)\nzones to their target temperatures was 1\u000eC/min. This temperature differential was held for\n14 days with the Fe5GeTe 2crystals being subsequently quenched in an ice bath.\nPrior to characterization, the excess iodine was removed through a bath and rinse cycle\nof acetone and isopropyl alcohol, respectively. Samples were either stored in a glove box\nwith high purity argon gas ( 99:99%) of 0.01 ppm O 2/H2O, or a desiccator under vacuum\nwith pressures ranging between 100-200 mTorr.\nThe crystalline structure characterization is measured on a Fe5GeTe 2bulk crystal with\nproperties shown in Figure 1. Figure 1a shows the crystal structure schematic of Fe 5GeTe 2.\nThe vdW-separated eight atomic-thick monolayers of two unit cells can be observed, where\nthe vdWs gaps exist between the Te atoms of neighboring unit cells. The light-blue circles\nlabeled Fe(1) represent the two possible occupation locations for the Fe(1) atoms, either\nabove or below a given Ge atom, with an occupation probability not exceeding 50%, as the\nFe-Ge bond becomes non-physical if both locations are occupied simultaneously.17\nThe X-ray diffraction (XRD) data collected for the experimental Fe 5GeTe 2sample crystal\nare shown in Figure 1b. The (00 l) reflections reveal the c-axis of the single crystal. The (00 l)\npeaks, where l=3n, reflect an ABCstacking sequence in the unit cell of the bulk crystal.\nThis is consistent with the rhombohedral lattice structure of the R3m (No. 166) space group,\nas previously reported.14,17\nThe rocking curves measured at (00 l) peak angles are shown in Figure 1c. The full-\nwidth-at-half-maximum values of the acquired curves, with values less than 0.06o, reflect the\nhigh level of crystallinity of the Fe 5GeTe 2samples. Figures 1d-f are Bragg reflection scans\nof different crystal planes (0 kl), (h0l), (hk0) from high-resolution XRD. They all show clear\nstreaks, confirming the high-quality of the single-crystal samples.\n5Figure 1: Crystal structure and x-ray diffraction (XRD) of single crystal Fe 5GeTe 2.a.\nSchematic of crystal structure of Fe 5GeTe 2.b. XRD 2\u0012/!scan showing (00 l) peaks. c.\nRocking curve scan of the peaks at 9\u000e, 27\u000e, 67\u000eshowing high crystallinity. d-f: Single\ncrystal XRD scan of Bragg reflections of different planes. d: (0kl) plane. e: (h0l) plane. f:\n(hk0) plane.\nIII. Quasi-Static Magnetization Properties\nQuasi-static magnetization properties were measured using VSM in a Quantum Design\nDynacool PPMS system. The measurements were carried out with the magnetic field applied\nalong both the c-axis (Hkc) and the ab-plane (Hkab) directions. To determine the Curie\ntemperature of the sample, we measured field-cooled curves, as well as heat capacity curves\nin the absence of a magnetic field.\nFigure 2a shows the results of the VSM magnetization versus field measurements of the\nFe5GeTe 2sample, for temperatures ranging from 1.8 K to 350 K, and for Hkc(dashed lines)\nandHkab(solid lines). The curves show an in-plane anisotropy due to strong demagnetizing\nfield along the c-axis. Possible spin-reorientation features, such as the ones observed in\nFe4GeTe 2,12are not observed in this sample. Our results are reasonable considering the fact\nthat the out-of-plane magnetocrystalline anisotropy in Fe 5GeTe 2crystals is very weak, i.e.,\n0.39 J/cm3.12,25Figure 2b shows the field-cooled (FC) curves for the Hkcand the Hkab\n6cases. The magnetization magnitude change between 100 K and 200 K in the Hkabcurve\nindicates possible magnetic phase transitions. This feature has been reported in previous\npublications.12,17,25The Curie temperature of the sample is determined to be Tc= 332 K\naccording to the transition points in the FC curves.\nFigure 2: Static magnetization of the Fe 5GeTe 2bulk single crystal. a. Temperature-\ndependent hysteresis loops at various temperatures for Hkc(dashed lines) and Hkab(solid\nlines). b. Field-cooled curves ( H= 50 Oe) for the Hkcand theHkabcases, respectively. c.\nHeat capacity as a function of temperature. The transition at 332 K and 110 K mark the\nCurie temperature and possible magnetic phase transition, respectively.\nHeatcapacitymeasurementswereusedtovalidatetheCurietemperatureestimation. The\nmeasurements were set to start from the highest temperature setpoint, T= 390 K, then the\ntemperature was gradually reduced to 1.8 K as the heat capacity data was collected. This\nprocedure guarantees an appropriate time constant is used to achieve more stable readings.\nThe measurement results are shown in Figure 2c. Two points of interest are highlighted on\n7the curve by two black arrows. The first is a transition at T= 332 K, which is consistent\nwithTcfrom the FC measurement. The second is the observation of a slope change around\nT= 110 K. The slope change indicates that Fe 5GeTe 2experiences some phase transition,\nand we will discuss it more extensively in the analysis of FMR linewidth in Section IV.\nIV. Magnetization Dynamics\nWe measured the FMR response of the Fe 5GeTe 2sample for Hkcand forHkab, at\ntemperatures varying from 10 K to 300 K. In our custom-built system, a coplanar waveguide\n(CPW) with impedance matched to 50 \nwas used to guide the microwave field to the\nsample. Thetestedmicrowavefrequenciesrangedfrom5GHzto40GHz. Foreachmicrowave\nfrequency, the magnetic field was swept from 15 kOe to zero. A microwave diode was used\nto convert the transmitted microwave signal to a dc voltage. To improve the signal-to-noise\nratio, we used a set of field-modulation coils supplemented by a lock-in amplifier to detect\nthe signal. Thus, the detected FMR response is identified as the derivative of the microwave\npower absorption.\nAs shown in Figure 3, we detected strong FMR responses at 300 K, demonstrating fer-\nromagnetism of Fe 5GeTe 2at room temperature. Figures 3a and 3b show the temperature\ndependence of the FMR profiles at 10 GHz and 20 GHz, for HkcandHkab, respectively.\nBesides the data points, the curves show fits to the derivative of a combination of symmetric\nand antisymmetric Lorentzian functions.19From the fits one can extract the resonance field\nHrand peak-to-peak linewidth \u0001Hppas shown in Figures 4 and 5. It is observed that the\nmagnitude of the FMR peaks decays with reducing temperature along the Hkcdirection.\nBelow 100 K, the FMR signal becomes undetectable in this orientation. We attribute this\nphenomenon to the broadening of the FMR resonance peaks. As illustrated in the later\ndiscussions in this section, the broadening and weakening FMR peaks indicate the devel-\nopment of an increasing number of magnetic domains with random orientations at lower\n8Figure 3: Ferromagnetic resonance (FMR) measurements of Fe 5GeTe 2single crystal. a.\nFMR profiles for Hkcat 200 K, 250 K, 280 K, and 300 K. b. FMR profiles for Hkabat 10\nK, 200 K, 250 K and 300 K.\ntemperatures.\nThe resonance frequencies fvs. the FMR resonance fields Hrat different temperatures\nare plotted in Figures 4a and 4b for the Hkcand for the Hkabcases, respectively. The\nfitting equation for the Hkcmeasurements is:26\nf=\r0(Hr\u00004\u0019Meff) (2)\nand the fitting equation for the Hkab-plane measurements is:26\nf=\r0p\n(Hr+ 4\u0019Meff)Hr (3)\nwherefis frequency, \r0is the reduced gyromagnetic ratio ( \r0=j\rj\n2\u0019), and 4\u0019Meffis the\neffective magnetization. The fitted curves are also presented in Figures 4a and 4b. By\nconstraining the combined fit such that Meffis the same along HkcandHkab, we obtain\ndifferent\r0and corresponding spectroscopic Landé g-factor values for these orientations as\nshown in Figure 4c. The left vertical axis shows \r0, and the right vertical axis shows the\nLandég-factor calculated using j\rj=g\u0016B\n\u0016h. Theg-factor exhibits a weak dependence on\ntemperature along both the ab-plane and the c-axis directions. However, it deviates from\n9Figure 4: Analysis of the FMR data of the Fe 5GeTe 2single crystal. a. Frequency fvs.\nresonance field Hrat 100 K, 200 K, 250 K, and 300 K for Hkc. The data points are fitted to\nEq. (2). b. Frequency fvs. resonance field Hrat 10 K, 200 K, 250 K, and 300 K for Hkab.\nThe data points are fitted to Eq. (3). c. Temperature dependence of the gyromagnetic\nratio\rand spectroscopic g-factor for the Hkc(red) and Hkab(black) cases, respectively.\nd. Temperature dependence of saturation magnetization 4 \u0019Msand effective magnetization\n4\u0019Mefffrom VSM and FMR measurements, respectively.\ng= 2: it is between 2.05 and 2.23 in the Hkabcase, while it is between 1.88 and 1.99 in\ntheHkccase. Thus, our data appears to indicate a sizable difference of the g-factor along\ndifferent directions in Fe 5GeTe 2.\nSimilar to Cr 2Ge2Te6,23the deviation of the g-factor from g= 2 may suggest an orbital\ncontribution to the magnetization due to spin-orbit coupling in Fe 5GeTe 2. It was found that\nstrong spin-orbit coupling results in nontrival Berry phase in Fe 3GeTe 2, another member in\nthe FenGeTe 2(n\u00153) family. In this case, the orbital character is formed by a mixture of 3D\norbitals from the Fe I–Fe I dumbbells and Fe II sites.27In Fe 5GeTe 2, the spin-orbit coupling\ncould be characterized by the dorbitals of Fe atoms and porbitals of Te atoms.23In addi-\n10tion, the anisotropy of the g-factor, which follows from that of the orbital moment, is also\nexpected physically: a small orbital moment arising from reduced crystalline symmetry may\n“lock” the large isotropic spin moment into its favorable lattice orientation through spin-orbit\ncoupling, giving rise to a sizable magnetic anisotropy. Therefore, it is likely that the orbital\nmoment is closely linked to the magnetocrystalline anisotropy in itinerant ferromagnets, as\nshown theoretically by Bruno et al.28for transition-metal monolayers. To form a clearer un-\nderstanding of the contribution of the orbital momentum, x-ray magnetic circular dichroism\n(XMCD) measurements are planned for future studies.\nHowever, an unsaturated magnetization at FMR resonance can also lead to an inac-\ncurate estimation of the gyromagnetic ratio.29Because the c-axis magnetization saturates\nat significantly large magnetic fields (e.g., H sat\u00196 kOe at 100 K), it is possible that the\nmagnetization was not fully saturated at FMR. To estimate to first order the influence of\nan unsaturated sample at resonance, we write 4\u0019Meff(H) = 4\u0019Meff,0+ 4\u0019pH, whereHis\nthe external field, 4\u0019Meff,0is the effective magnetization extrapolated to zero field, and 4\u0019p\nis the slope of 4\u0019Meffvs.Hcurve in the region where FMR occurs. Using this equation,\nequation (2) can be written as: f=\r0\nmeas(Hr\u00004\u0019Meff,meas ), with\r0\nmeas=\r0(1\u00004\u0019p)and\n4\u0019Meff,meas =4\u0019Meff\n1\u00004\u0019p. Using the VSM data of Figure 2a, we find that the magnetization can\nchange by 10% over the FMR resonance field range, which can possibly explain the differ-\nence of the g-factor values measured along the c-axis and the ab-plane directions. Thus, we\nconclude that while the g-factor of Fe 5GeTe 2may be anisotropic, the broad linewidth and\nslow saturation for Hkcprovide another possible explanation. Further clarification could be\nachieved using high field FMR, but this is beyond the scope of the current manuscript.\nThe fits also yield the effective magnetization 4\u0019Meffat 100 K, 200 K, 250 K and 300\nK. Figure 4d plots 4\u0019Meffand the saturation magnetization 4\u0019Msmeasured from FMR\nand VSM, respectively. One can see that there is a difference between 4\u0019Msand4\u0019Meff,\nrevealing an effective anisotropy field. As plotted in Figure 4d, Hk=4\u0019Ms-4\u0019Meffis\nnegative and shows a moderate easy-plane anisotropy field between 100 K and 300 K. The\n11effective anisotropy coefficient Keffis -4.81\u0002105erg/cm3at 100 K and reduces to -3.6 \u0002103\nerg/cm3at 300 K. The negative sign indicates an easy-plane anisotropy. This is consistent\nwith a previous report.25\nFigure 5: Characterization of FMR linewidth in Fe 5GeTe 2.a. Peak-to-peak linewidth \u0001Hpp\nvs. frequency for Hkab.b. Temperature dependence of effective damping parameter \u000befffor\nHkab.c.\u0001Hppvs. frequency for Hkc.\nNext, we analyze the FMR linewidth in order to gain insights on the spin scattering\nmechanisms in Fe 5GeTe 2. In Figures 5a and 5c, we plot the peak-to-peak linewidth \u0001Hpp\nvs. frequency at different temperatures measured for the Hkaband theHkccases, respec-\ntively. For an ideal magnetic thin film that is homogeneous and defect-free, the linewidth vs.\nfrequency plot reflects the intrinsic FMR damping; the Gilbert damping parameter \u000bcan be\nextracted from the slope. \u000busually increases or decreases with temperature as a consequence\nof interband or intraband scattering.30However, due to non-uniform magnetization states\nanddefectsinthesample, thelinewidthcanbebroadenedbyextrinsicscatteringmechanisms\nsuch as inhomogeneous line broadening \u0001H0and two-magnon scattering \u0001HTMS. Thus, the\nFMR linewidth \u0001Hppcan be expressed by the following form:31\n\u0001Hpp=2\u000beffp\n3j\rjf\n2\u0019+ \u0001H0+ \u0001HTMS (4)\nThrough the analysis of the data in Figure 5, we note our main observations as follows:\nFirst, the Fe 5GeTe 2linewidth is broadened by extrinsic scattering mechanisms such as \u0001H0\nand\u0001HTMS. The increment of \u0001H0is more pronounced when temperature reduces. Larger\n12\u0001H0at lower temperatures is a consequence of internal local moment distribution. Be-\ncause the Fe 5GeTe 2sample is single crystal that is defect-free, the larger H0indicates some\nintriguing magnetic orders. We will discuss this point further in the discussions below.\nSecond, as can be seen from Figure 5b, the effective damping parameter \u000beffchanges\nbetween 0.007 and 0.032 from 10 K to 300 K. The magnitude is similar to that of soft\n3D transition metal magnets such as Permalloy.32Because\u000beffis estimated from the Hkab\nmeasurement, it is likely that \u0001HTMSalso contributes to \u000beff. In this case, the extracted \u000beff\nprovides the upper boundary of intrinsic damping of Fe 5GeTe 2.\nWe now summarize all the measured data and provide our perspectives on the intriguing\nmagnetism in Fe 5GeTe 2: The hysteresis loops show that Fe 5GeTe 2is ferromagnetic with an\neasy-plane magnetic anisotropy at all the temperatures. The ab-plane FC curve in Figure 2b\nshows a sharp drop of magnetization from 200 K to 100 K and flattens upon further reducing\nT. These features indicate that magnetic domains of different orientations were developed\nwith decreasing T. The FMR data further support this hypothesis: the linewidth increases\nasTreduces along both the ab-plane and the c-axis. The disappearance of the FMR peaks\nbelow 100 K along the c-axis indicates a higher-level randomness of the magnetic domains\ncompared with the ab-plane. The features observed in our work are consistent with those\nreported in Ref.,25which proposes a complicated magnetism picture. Nevertheless, we show\nthat all the intriguing features in Fe 5GeTe 2can be explained by a ferromagnetic phase with\nrandomized magnetic domains at lower T.\nSummary and conclusion\nIn summary, we have synthesized a vdWs Fe 5GeTe 2crystal that showed a bulk Curie\ntemperature of 332 K. While the Curie temperature of Fe 5GeTe 2is expected to decrease\nwhen the 2D vdWs crystal is exfoliated into thin layers, the bulk value is still one of the\nhighest recorded for a 2D magnet until now, making it an attractive 2D option to be used\n13in spintronic devices. We used both VSM and FMR to study the magnetic properties of the\nFe5GeTe 2sample. The experiments were performed with external magnetic fields applied\nalong the c-axis and the ab-plane directions from 300 K to 10 K. We have focused on the\ntemperature and field dependences of the g-factor and spin scattering mechanisms. The\ng-factor values are differentiated by a sizable \u0001g= 0.3 between the HkcandHkabcases,\nindicative of considerable orbital momentum arising from spin-orbit coupling in Fe 5GeTe 2\nor fluctuation caused by lack of saturation at FMR. The FMR linewidth analysis reveals\nlow temperature-enhanced inhomogeneous line broadening, together with the VSM data,\nthey indicate a ferromagnetic phase with randomized magnetic domains at lower T. For\nfuture studies, it will be interesting to exploit Fe 5GeTe 2thin films for spin tranport and\nspin-to-charge conversion experiments at room temperature.\nAcknowledgement\nL.B. is supported by the US DOE, Basic Enery Sciences program through award DE-\nSC0002613. A portion of this work was performed at the National High Magnetic Field\nLaboratory, which is supported by the National Science Foundation Cooperative Agreement\nNo. DMR-1644779 and the State of Florida. M.B. is supported by the U.S. Department\nof Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering\nDivision. B.N. acknowledges support through NSF MEMONET grant #1939999, A.S. ac-\nknowledges support NSF-CAREER Award #1452670, and P.L. acknowledges the Auburn\nUniversity faculty start-up funding and valuable discussions with Prof. Mingzhong Wu, Prof.\nSatoru Emori, Prof. Wei Zhang, Dr. Chuanpu Liu, and Dr. Kaya Wei.\nReferences\n(1) Frisenda, R.; Niu, Y.; Gant, P.; Muñoz, M.; Castellanos-Gomez, A. Naturally occurring\nvan der Waals materials. npj 2D Materials and Applications 2020,4, 38.\n14(2) Mak, K. F.; Shan, J.; Ralph, D. C. Probing and controlling magnetic states in 2D\nlayered magnetic materials. Nature Reviews Physics 2019,1, 646–661.\n(3) Mitta, S. B.; Choi, M. S.; Nipane, A.; Ali, F.; Kim, C.; Teherani, J. T.; Hone, J.;\nYoo, W. J. Electrical characterization of 2D materials-based field-effect transistors. 2D\nMaterials 2020,8, 012002.\n(4) Casto, L. D.; Clune, A. J.; Yokosuk, M. O.; Musfeldt, J. L.; Williams, T. J.;\nZhuang, H. L.; Lin, M.-W.; Xiao, K.; Hennig, R. G.; Sales, B. 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Ferromagnetism Near Room Temperature in the\nCleavable van der Waals Crystal Fe5GeTe2. ACS Nano 2019,13, 4436–4442, PMID:\n30865426.\n16(18) May, A. F.; Bridges, C. A.; McGuire, M. A. Physical properties and thermal stability\nofFe5\u0000xGeTe 2single crystals. Phys. Rev. Materials 2019,3, 104401.\n(19) Oates, C.; Ogrin, F.; Lee, S.; Riedi, P.; Smith, G.; Thomson, T. High field ferromag-\nnetic resonance measurements of the anisotropy field of longitudinal recording thin-film\nmedia.Journal of Applied Physics 2002,91, 1417–1422.\n(20) Shaw, J. M.; Nembach, H. T.; Silva, T. J.; Boone, C. T. Precise determination of\nthe spectroscopic g-factor by use of broadband ferromagnetic resonance spectroscopy.\nJournal of Applied Physics 2013,114, 243906.\n(21) Lee, I.; Utermohlen, F. G.; Weber, D.; Hwang, K.; Zhang, C.; van Tol, J.; Gold-\nberger, J. E.; Trivedi, N.; Hammel, P. C. Fundamental Spin Interactions Underlying\nthe Magnetic Anisotropy in the Kitaev Ferromagnet CrI3.Phys. Rev. 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Scientific Reports 2016,6, 22890.\n18" }, { "title": "2206.13593v1.Bridging_atomistic_spin_dynamics_methods_and_phenomenological_models_of_single_pulse_ultrafast_switching_in_ferrimagnets.pdf", "content": "Bridging atomistic spin dynamics methods and phenomenological models of single pulse ultrafast\nswitching in ferrimagnets\nFlorian Jakobs and Unai Atxitia\nDahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit ¨at Berlin, 14195 Berlin, Germany\nWe bridge an essential knowledge gap on the understanding of all-optical ultrafast switching in ferrimagnets;\nnamely, the connection between atomistic spin dynamics methods and macroscopic phenomenological models.\nAll-optical switching of the magnetization occurs after the application of a single femtosecond laser pulse to\nspecific ferrimagnetic compounds. This strong excitation puts the involved degrees of freedom, electrons, lattice\nand spins out-of-equilibrium between each other. Atomistic spin models have quantitatively described all-\noptical switching in a wide range of experimental conditions, while having failed to provide a simple picture\nof the switching process. Phenomenological models are able to qualitatively describe the dynamics of the\nswitching process. However, a unified theoretical framework is missing that describes the element-specific spin\ndynamics as atomistic spin models with the simplicity of phenomenology. Here, we bridge this gap and present\nan element-specific macrospin dynamical model which fully agrees with atomistic spin dynamics simulations\nand symmetry considerations of the phenomenological models.\nI. INTRODUCTION\nSince its experimental discovery [1], the theoretical de-\nscription of laser induced all-optical switching (AOS) of the\nmagnetization in GdFeCo ferrimagnetic alloys has remained\na challenge. Despite intense experimental and theoretical re-\nsearch in the field [1–12], an established and unified picture\nof the process is still missing. Experimental findings are\nmostly compared or interpreted in terms of atomistic spin\ndynamics simulations [13–17], multisublattice spin dynam-\nics based on symmetry arguments [5, 18, 19], and based on\nthe Landau-Lifshitz-Bloch equation [20–22]. The main goal\nof the present work is the revision, extension and merging of\nthese approaches into a unified model.\nAtomistic spin dynamics (ASD) models have been used be-\nfore to quantitatively describe ultrafast dynamics in 3 dtransi-\ntion metals [23, 24] and 4 frare-earth ferromagnets [25, 26].\nThey have also been used in GdFeCo, to describe the equilib-\nrium thermal properties [13], the thermal character of AOS\n[4], the so-called transient ferromagnetic-like state [3], the\ndemonstration of spin-current-mediated rapid magnon local-\nisation and coalescence [27] and the possibility of AOS using\npicosecond-long laser pulses [16]. Results from atomistic spin\nmodels also compare qualitatively well to an analytical the-\nory based on the excitation of spin-wave exchange modes [8],\nprovide insights for optimal electron, phonon and magnetic\ncharacteristics for low energy switching [28] and predict max-\nimum repetition rate using two consecutive laser pulses [29].\nMore sophisticated, orbital-resolved atomistic models provide\ninsights on the role of the intra-exchange coupling between\n4fand 5 delectrons in the dynamics of GdFeCo alloys[14].\nAtomistic models can naturally describe switching in Gd/Fe\nmultilayers composed of very thin layers [30, 31]. Recent ob-\nservations [32, 33] of single pulse switching in Mn 2RuxGa\nalloys are also well-described by ASD methods [34]. De-\nspite the demonstrated success in modeling AOS, ASD sim-\nulation results are cumbersome to interpret without an ana-\nlytical model that unveils the role of the different processes\nand interactions during the switching process. This potential\nsemi-analytical model has to capture most of the features ofthe ASD simulations.\nSemi-phenomenological models describing switching al-\nready exist. A macroscopic theory for the description of the\ndynamics and relaxation of the macroscopic (sublattice) mag-\nnetization of ferromagnets and antiferromagnets was devel-\noped originally by Baryakhtar [9, 35]. An extension of such\nphenomenology to ferrimagnets in the context of ultrafast spin\ndynamics was introduced in Ref. [5]. At the ultrafast scale,\nmagnetization dynamics are dominated by atomic scale spin\nexcitations, these spin dynamics are driven by dissipative pro-\ncesses which in ferrimagnets are two-fold, relativistic and ex-\nchange driven. Relativistic processes allow for exchange of\nangular momentum between the spins and lattice degree of\nfreedom due to the presence of spin-orbit interaction connect-\ning them. Exchange processes can arise due to transport of\nspin angular momentum – spin and magnon transport – which\nis the only mean to exchange angular momentum in ferromag-\nnets. In multisublattice magnets another, different pathway\nopens, namely, local exchange of angular momentum. To ac-\ncount for such local exchange processes in ferrimagnets, the\nequation of motion for the magnetization dynamics proposed\nby Landau and Lifshitz [36] is enhanced by an exchange re-\nlaxation term [5, 9, 19, 37]. Within this macroscopic model,\nthe exchange relaxation dominates the dynamics when the\nmagnetic sublattices are driven into mutual non-equilibrium.\nQualitative agreement to experiments in two-sublattice mag-\nnets has been demonstrated [19], such as AOS in ferrimag-\nnetic GdFeCo using fs laser pulses [5] and ps laser pulses\n[38], AOS in Heusler semimetals Mn 2RuxGa [39], or element-\nspecific demagnetization of ferromagnetic NiFe alloys [18].\nQuantitative comparison of this model to neither experiments\nnor ASD simulations have been conducted so far. While the\narguments behind such phenomenology are robust, the range\nof applicability and the validity of the model parameters could\nbe questioned. For instance, the parameters defining the rel-\nativistic and exchange relaxation are assumed to be constant\nand of the same order. The magnetic free energy functional\nis calculated for near thermal equilibrium states. This implies\na relatively strong coupling to the heat-bath, while switching\nconditions are supposedly fulfilled when exchange relaxationarXiv:2206.13593v1 [cond-mat.mtrl-sci] 27 Jun 20222\nbetween sublattices dominates over the relaxation to the heat-\nbath.\nAn alternative macroscopic model directly derived from an\natomistic spin model has also been proposed. This model is\nbased in the Landau-Lifshitz-Bloch (LLB) equation of mo-\ntion [20, 40–43]. The LLB model for two-sublattice mag-\nnets [20, 42] has been used in the context of AOS in GdFeCo,\ne.g. the element-specific demagnetization rates compare well\nto experiment, and it predicts that near the magnetic phase\ntransition the otherwise slower Gd sublattice becomes faster\nthan Fe [22], as recently observed [44]. The LLB model has\nbeen demonstrated to provide accurate analytical expressions\nfor the temperature dependence of the relativistic relaxation\nparameter as well as for the non-equilibrium effective fields\nbelow and above the critical temperature [42]. Moreover, the\nLLB model also describes the transverse motion of the mag-\nnetization. This makes it the preferred model for computer\nsimulations of heat-assisted magnetic recording [45] and re-\nalistic description of all-optical switching [46], and ultrafast\nspintronics, such as domain wall motion [47, 48] or skyrmion\ncreation by ultrafast laser pulses [49]. So far the LLB model\nand Baryakhtar-like models have been considered as comple-\nmentary approaches. Here, we merge them into one unified\napproach.\nIn this work we address the issues discussed above by di-\nrectly comparing both phenomenological models to ASD sim-\nulations. We do so since ASD simulations have been al-\nready quantitatively compared to experiments in literature.\nWe find that quantitative comparison between ASD and both\nphenomenological models is partially possible for laser exci-\ntation producing small deviation from equilibrium. However,\nthose models hardly reproduce magnetic switching using the\nsame parameter values describing the relaxation of small per-\nturbations. Here, based upon those phenomenological mod-\nels, we propose a macroscopic model that compares precisely\nto the magnetization dynamics calculated using ASD simula-\ntions, including element-specific magnetization relaxation and\nswitching. This model bridges atomistic spin dynamics based\nmodels and previously proposed phenomenological models.\nNotably, it provides a deeper understanding to the parameters\nentering the phenomenological models and sheds some light\ninto the process of ultrafast switching in ferrimagnets.\nThe work is broken down in the following way: in Sec. II,\nwe present the atomistic spin model for the calculation of the\nmagnetic equilibrium properties and non-equilibrium dynam-\nics. The equilibrium properties are compared to a mean field\nmodel. We then provide atomistic calculations of the ultra-\nfast magnetization dynamics with input from the two temper-\nature model. These results are the basis for the comparison to\nthe phenomenological models presented in Sec. III. Firstly,\nwe present the Baryakhtar model and the Landau-Lifshitz-\nBloch model. Secondly, we compare the ultrafast magneti-\nzation dynamics calculated with those models to the atomistic\nspin dynamics results. Finally, in Sec. III C we present the\nunified phenomenological model, a hybrid model combining\nBaryakhtar and LLB models, and its comparison to atomistic\nspin dynamics.II. ATOMISTIC SPIN MODEL\nFerrimagnetic materials characterise by spontaneous mag-\nnetization as a resultant of two or more components of non-\nparallel magnetic moments [50]. Atomistic spin models based\non the Heisenberg Hamiltonian can be considered one of the\nsimplest microscopic models able to reproduce the equilib-\nrium properties of ferrimagnets. The spin system energy due\nto only the exchange interactions can be described by an ef-\nfective Heisenberg model:\nH=\u0000å\ni6=jJaSa;i\u0001Sa;j\u0000å\ni6=jJbSb;i\u0001Sb;j\u0000å\ni6=jJabSa;i\u0001Sb;j(1)\nwhere Ja(b)(ab)is the exchange constant between neighbor-\ning sites represented by two classical spin vectors SiandSj\n(jSj=1). Further, one can include magnetic anisotropy terms\nto Eq. (1) to set a preferential axis for the magnetization.\nHowever, since the anisotropy energy is relatively low it plays\na marginal role in the switching process. This makes for a\nsimpler Hamiltonian and a more direct comparison to the phe-\nnomenological models. To model a ferrimagnet, one needs to\nconsider two alternating sublattices of unequal and antiparal-\nlel moments, with three exchange coupling constants: ferro-\nmagnetic for each sublattice ( JaandJb) and a third for the an-\ntiferromagnetic interaction between them, Jab. For instance,\nGdFeCo alloys are composed of a transition metal FeCo and\na Gd rare-earth sublattices. We model the Fe and Co spins\nas only one magnetic sublattice, and we assume a common\natomic magnetic moment of mFeCo=1:94mB. In these alloys\nthe rare-earth impurities add localised 4 fspins to the sys-\ntem assumed to be, mGd=7:6mB. The amorphous nature of\nGdFeCo is modelled by using a simple cubic lattice model\nbut with random placements of Gd moments within the lattice\nto the desired concentration. The applicability of the Heisen-\nberg approximation relies on the stability of local moments\nunder rotation and at high temperature where Stoner excita-\ntions are generally weak [51]. It is assumed that the electronic\nproperties are temperature-independent in the range where the\nsystem is magnetically ordered.\nA. Atomistic spin dynamics\nEquilibrium and non-equilibrium element specific mag-\nnetic properties of a ferrimagnet are calculated using atomistic\nspin dynamics simulations which are based in the stochastic-\nLandau-Lifshitz-Gilbert equation (s-LLG) [52]\n(1+l2\ni)ms;i˙Si=\u0000gSi\u0002[Hi\u0000li(Si\u0002Hi)]; (2)\nwhere gis the gyromagnetic ratio, and liis the so-called\nphenomenological sublattice specific damping parameter. By\nincluding a Langevin thermostat the spin dynamics includ-\ning statistical – equilibrium and non-equilibrium thermo-\ndynamic properties can be obtained. An effective field-\nlike stochastic term ziis added to the effective field Hi=\nzi(t)\u0000¶H\n¶Si, with white noise properties [53]: hzi(t)i=\n0 andhzi(0)zj(t)i=2likBTms;idi jd(t)=g:The variance3\n−2−1.6−1.2−0.8−0.400.40.81.21.6\n0 100 200 300 400 500 600 700M(T)[µB]\ntemperature / KMnet\nFe\nGd\nFIG. 1. Equilibrium magnetization of a GdFeCo alloy for Gd concen-\ntration, xGd=25%. Element-specific normalized equilibrium mag-\nnetization and net equilibrium magnetization, M(T) =xGdmGdmGd\u0000\nxFemFemFe, where mGd(Fe)is the atomic magnetic moment of Gd(Fe).\nLines correspond to the mean-field approximation with renormalized\nexchange parameters. Symbols correspond to atomistic spin dynam-\nics simulations.\nof the Langevin noise is chosen such that the fluctuation-\ndissipation theorem is full filled.\nB. Mean-field approximation\nExact analytical expressions for the M(T)curve are cum-\nbersome to derive due to the many body character of the prob-\nlem. Here we resort the mean field approximation (MFA),\nalready used in previous works [8, 13, 54]. We note that to\nbe able to apply the MFA for the GdFeCo impurity model,\nand thus translation non-symmetric with respect to spin vari-\nables Si, we need to transform the Heisenberg Hamiltonian to\na symmetric one. We use the spin analogy of the virtual crys-\ntal approximation (VCA) to transform the disordered lattice\nHamiltonian Hto a symmetric VCA Hamiltonian HVCA.\nWithin the VCA we evaluate the effective sublattice exchange\nparameters, given by the sum of the exchange interactions of a\ngiven spin at a site riof sublattice iwith all other atoms of this\nsublattice. This involves weighting the exchange parameters\nby the relative composition, xi\u0011concentration species i[8],\nJi=å\nri;r0\niJ(ri;r0\ni)\u0011|{z}\nVCAxiJ(ri;r0\ni)intrasublattice (3)\nwhereas the intersublattice effective exchange reads\nJi j=å\nri;r0\nj=2AiJ(ri;r0\nj)\u0011|{z}\nVCAxiJ(ri;r0\nj)intersublattice (4)\nThus the VCA Hamiltonian reads\nHVCA=å\nj2AiJiSi\u0001Sj+å\nj=2AiJi jSi\u0001Sj (5)where Airepresent the magnetic sublattice of the spin Si. In\nthe exchange approximation we define the MFA field as\nmaHMFA\na=zaJaama+zabJabmb (6)\nThe element-specific equilibrium magnetization is calculated\nvia the self-consistent solution of ma=L(b maHMFA\na)and\nmb=L(b mbHMFA\nb).zaandzabcorrespond to the number\nof first nearest neighbours of type aandb, respectively. It\nis well-known that the MFA overestimates the value of the\ncritical temperature TC. However, a very good agreement be-\ntween ASD and MFA can be obtained by using a reduced\nvalue for the exchange parameters, even for multilattice mag-\nnets [54]. Figure 1 shows element-specific Ma=xamama(T)\nusing ASD simulations and renornalized MFA for xGd=25%.\nNet magnetization is also shown in Fig. 1, which is defined\nasM(T) =xGdmGdmGd\u0000xFemFemFe. The agreement between\nASD and MFA is good enough for all the temperature regions.\nWe observe the presence of compensation temperature TMat\nroom temperature for xGd=25% at which the thermally av-\nerage magnetization of both sublattices are equal but oppo-\nsite, so that the magnetization of the system is equal to zero\nM(TM) =0. The mapping of the atomistic spin model and the\ncorresponding mean-field approximation turns out to be nec-\nessary for a quantitative comparison to the phenomenological\nmodels, and thereby paramount for the unification of both pic-\ntures.\nC. Two Temperature Model\nSingle pulse all-optical switching has been demonstrated to\nbe a thermal process in ferrimagnetic GdFeCo alloys [4] and\nin Mn 2RuxGa Heusler semi-metals [32]. Ultrafast heating by\noptical or electric means are sufficient to achieve switching in\nspecific GdFeCo alloys [55]. Although the minimum achiev-\nable duration of the electric pulses are limited to picoseconds,\nthose are better suited for potential integration into applica-\ntions. Laser pulses can be as short as only a few femtoseconds,\nwhich permits to excite the electron system in timescales of\nthe order of the exchange interaction allowing for the inves-\ntigation of fundamental physics governing switching. In this\nwork, we center in excitation of the ferrimagnetic GdFeCo\nusing femtosecond laser pulses. When a metallic ferrimag-\nnetic thin film is subjected to a near infrared laser pulse, only\nthe electrons are accessible by the photon electric field. Ini-\ntially, the absorbed energy is barely transferred to the lattice\nand consequently the electron system heats up. The electron\nand phonon temperatures are decoupled for up to several pi-\ncoseconds until the electron-phonon interaction equilibrates\nthe two heat-baths. This phenomenology is well captured by\nthe so-called two-temperature model (2TM) [56, 57] which\ncan be written as two coupled differential equations:\nCel¶Tel\n¶t=\u0000gep\u0000\nTel\u0000Tph\u0001\n+Pl(t) (7)\nCph¶Tph\n¶t= +gep\u0000\nTel\u0000Tph\u0001\n: (8)4\nCel=gelTelwhere gel=6\u0002102J/m3K2, and Cph=3:8\u0002106\nJ/m3K represent the specific heat of the electron- and phonon\nsystem. The electron-phonon coupling is taken temperature\nindependent, Gep=7\u00021017J/m3K. Here, P(t)is a Gaussian\nshaped pulse with a duration of 55 fs. The exact values of the\nparameters entering the TTM in GdFeCo are still unknown.\nThe values we use here are close to the commonly used, e.g.\nRefs. [4, 8, 34].\nD. Ultrafast magnetization dynamics using ASD\nElement-specific magnetization dynamics induced by a\nfemtosecond laser pulse are calculated by combining the\natomistic s-LLG equation for the spin dynamics (Eq. (2)) and\nthe 2TM for the electron temperature (Eq. (7)). The electron\nsystem acts as heat-bath for the atomic spins. We consider a\nlattice with N=50\u000250\u000250 spins, and damping parameters,\nlGd=0:01=lFe. Figure (2) shows, for t<0, the dynam-\nics of the element-specific magnetization from an initial sat-\nurated state ( T=0 K), towards thermal equilibrium with the\nheat-bath which is set to T=300 K. The relaxation dynamics\nof Fe sublattice is faster than those of the Gd sublattice. This\ncomes out naturally as the element-specific dissipation of an-\ngular momentum scales as ˙ mz\u0018gl=ms, in Gd ( mGd=7:6mB)\nis slower than in Fe sublattice ( mGd=1:94mB). Once the mag-\nnetic system is in thermal equilibrium with the heat-bath, we\napply the laser pulse, t>0, which introduces energy into the\nelectron system and induces ultrafast magnetization dynam-\nics. To illustrate the switching and no switching dynamics we\nconsider two limiting cases, dynamics induced by low laser\npower, P0, and large laser power, 2 P0. The electron temper-\nature increases up and above the Curie temperature in time\nscales of a few hundreds of femtoseconds Fig. (2) (a). This re-\nflects in the magnetic system as a fast demagnetization of both\nFe and Gd sublattices. For relatively low laser power, P0, the\nmagnetization of both sublattices reduces while the electron\ntemperature remains relatively high. Once the electron tem-\nperature reduces and equalizes to the lattice temperature, the\nmagnetization recovers to the thermal state given by the heat-\nbath temperature, which is higher than initially ( T=300 K).\nThis is why the final magnetization value is smaller than the\ninitial one. For higher laser powers, 2 P0, the magnetization\nof both sublattices reduces quickly. The Fe sublattice faster\nthan the Gd one. Once the magnetization of the Fe sublattice\nhits zero, instead of remaining demagnetized, the magnetiza-\ntion starts to develop toward the opposite direction, while the\nmagnetization of the Gd sublattice is still in the process of de-\nmagnetization. During a couple of picoseconds, both sublat-\ntice magnetization are aligned along the same direction, simi-\nlar to a ferromagnet. Consequently, this non-equilibrium state\nhas been named the transient ferromagnetic-like state [3]. One\ncan observe in Fig. (2) (b) that the demagnetization rates of\nboth sublattices slow down when the Fe magnetization crosses\nzero. This change reveals the set in of a process driving the\nmagnetization dynamics different to the one driving the initial\ndemagnetization. It has been argued that at this point direct\nexchange of angular momentum between sublattices domi-\n30060090012001500\n−1−0.75−0.5−0.2500.250.50.751\n-10 -5 0 5 10 15relaxation tothermal stateswitchingno switchingT[K]electronlatticemztime [ps]FeCoGd(a)(b)\nFIG. 2. (a) Electron and lattice temperature dynamics for two laser\npulse power values, P0and 2 P0. Both electron and lattice tempera-\nture are kept constant, T=300K, for t<0. At t=0 a laser pulse is\napplied and the dynamics of the electron and lattice temperature heat\nup. The dynamics of those temperatures are theoretically described\nby the two-temperature model. (b) Element-specific magnetization\ndynamics induced by the heat profile at (a). The dynamics are calcu-\nlated using atomistic spin dynamics methods. For lower laser powers\nP0, the magnetization of both sublattices demagnetize rapidly and re-\nmagnetize towards the new equilibrium. For laser power 2 P0, the\nmagnetization of both sublattices demagnetizes and switches. After\nswitching they relax towards the thermal equilibrium state. GdFeCo\nalloys with xGd=25% are calculated.\nnates over processes of relativistic origin, which in turn dissi-\npate angular momentum into the heat-bath. Interestingly, soon\nafter switching, both sublattice magnetization rapidly relax to\nequilibrium indicating that relaxation into the heat-bath dom-\ninates the dynamics.\nIII. PHENOMENOLOGICAL MODELS\nDifferently to ASD simulations, phenomenological mod-\nels describe the element-specific magnetization dynamics by\nsolving two coupled equations of motion, one for each sub-\nlattice. In this work we aim at finding a phenomenological\nmodel that describes the same element-specific magnetization\ndynamics as those coming out from the ASD simulations (Fig.\n2). The starting point is the comparison of the ASD simula-\ntions to well-known phenomenological models. We show that\nthose models are unable to describe in a satisfactory way the\ndifferent element-specific magnetization dynamics studied in\nthe previous section and summarized in Fig. 2.5\nA. Baryakhtar model\nThe simplest model to describe element-specific magneti-\nzation dynamics and switching in ferrimagnets was proposed\nby Mentink and co-workers [5]. Longitudinal spin dynamics\nwas derived from Onsager’s relations\nma\ngadma\ndt=aB\namaHa+aB\ne(maHa\u0000mbHb) (9)\nmb\ngbdmb\ndt=aB\nbmbHb+aB\ne(mbHb\u0000maHa) (10)\nhere, aB\na;bstands for the relaxation parameter of relativistic\norigin, which dissipates angular momentum out of the spin\nsystem, and aB\nestands for the exchange relaxation parame-\nter and describes the rate of dissipation of angular momen-\ntum between sublattices. By construction exchange relaxation\nconserves the total angular momentum. We emphasize here\nthe difference in the notation between the atomic relaxation\nparameter, l, describing the dissipation of the atomic spins\nin ASD simulations and the macrospin relaxation parameter,\na, describing the dissipation of the whole magnetic sample.\nWithin this model, the values for aB\na;bandaB\neare unknown\nbut used as fitting parameters when compared to experiments.\nThe internal effective field Ha(b), acting on sublattice a(b)are\nderived from a non-equilibrium mean-field approximation,\nmaHa=\u0000b\u00001L\u00001(ma)+maHMFA\na (11)\nwhere, L\u00001(x)is the inverse Langevin function, b=1=kBT,\nwhere Trepresents the temperature of the heat-bath to which\nthe spin system is coupled to. At equilibrium, the effective\nfield is Ha=0, as ma=L(b maHMFA\na). The same arguments\napply for sublattice b. It turns out that by solving Eqs. (9)\nand (10) together with the 2TM, described in Eqs. (7) and\n(8), one obtains similar ultrafast magnetization dynamics as\nthose using ASD simulations (Fig. (2)). Element-specific\ndemagnetization [18] and switching dynamics [19] based on\nthis approach have been discussed thoughtfully before. On\nthose works, the values for the relaxation parameters, rela-\ntivistic and exchange, are taken constant and of the same or-\nder,aB\nFe\u0019aB\nGd\u0019aB\ne. We note that here aB\nadefines the rate\nof change of angular momentum ( mm=g). It differs from the\ndefinition of intrinsic damping parameters in ASD, which are\nrelated to the rate of change of the magnetization ( m). Sim-\nilarly to ASD methods though, within the Baryakhtar model\nthe observed fast dynamics of the Fe sublattice is related to a\nsmaller value of atomic magnetic moment.\nThe switching process within the Baryakhtar-like model\nis explained in the following manner. Since the Fe sublat-\ntice reacts faster than Gd to heating it is expected to remain\ncloser to thermal equilibrium with the heat-bath. This trans-\nlates into a smaller non-equilibrium effective field acting on\nFe than in Gd, HFe\u001cHGd, during the action of the laser pulse.\nFor strong enough pulses, the Fe magnetization rapidly re-\nduces, mFe\u00190, still HFeis small in comparison to HGd, in a\nway that the dynamics of Fe can be fairly approximated by\n˙mFe\u0019aB\neHGd. This drives the magnetization of Fe towards\n(a)\n(b)\n-1-0.500.51\n-10 -5 0 5 10laser powerP0laser power 2P0-1-0.500.51laser powerP0laser power 2P0mztime [ps]αBe/αBa=0αBe/αBa=0.3αBe/αBa=3mzFeGdFIG. 3. Element-specific magnetization dynamics of GdFeCo cal-\nculated using atomistic spin dynamics (symbols) and macroscopic\nBaryakhtar-like equation (solid lines) for two laser pulse power val-\nues, (a) P0and (b) 2 P0. Both electron and lattice temperature are\nkept constant, T=300 K, for t<0. At t=0 a laser pulse is ap-\nplied. In the Baryakhtar-like model the relativistic relaxation pa-\nrameters aBahave a value different to the Gilbert damping in ASD\nsimulations, (g=mFe)aB\nFe=0:005 and (g=mGd)aB\nGd=0:01. The ex-\nchange relaxation parameter is varied, aBe=aB\nFe=0;0:3 and 3. The\nrelaxation to thermal state ( t<0) is only well described for the Fe\nsublattice. (a) For P0, the laser induced dynamics is well described\nbyaBe=aB\nFe=0:1. (b) For 2 P0the demagnetization phase of both\nsublattices is relatively well described in comparison to ASD sim-\nulations. Switching is also possible, here one instance, for a value\naBe=aB\nFe=3.\nthe opposite direction. The field, HGdis defined by the en-\nergy of the system, HMFA\nGd(Eq. (6)) and aB\nefrom the cou-\npling between the Gd and the Fe sublattices. After switching,\nHFe\u0019HGdand relativistic relaxation processes dominate the\ndynamics and drive magnetization to complete the switching.\nThe question here is to what extent the non-equilibrium fields\nas given by Eq. (11) are accurate, and how are the relaxation\nparameters related to atomic damping parameters in ASD.\nSo far the connection between the relaxation parameters in\nthe ASD and Baryakhtar-like model is unknown. In ASD sim-\nulations shown in Fig. 2 we have used lFe=lGd=0:01 as\natomistic relaxation parameter. One would expect that the re-\nlaxation parameters in the atomistic and macroscopic models\nare related as la\u0019aB\na(ga=ma). In an attempt to find this cor-\nrespondence, we directly compare results from ASD simula-\ntions and Baryakhtar-like models for different values of aB\na\nandaB\nein Eqs. (9) and (10). We numerically solve Eqs.\n(9),(10), and (11) coupled to the 2TM with exactly the same\nparameters as for the ASD simulations. After exploring the\nresults of the Baryakhtar model for a range of values for aB\na\nandae, we find that for some values the agreement is good,\nas one observes in Fig. 3, however, it is not possible to find a\ngood match for all scenarios.6\nIn order to illustrate this, we first focus on the dynamics in-\nduced by the laser pulse with power P0(Fig. (3)(a)). We find\na good match for the laser induced magnetization dynamics\n(t>0 for (g=mFe)aFe=0:005 and (g=mGd)aGd=0:01, and\nfor values of exchange relaxation of up to aB\ne=aB\nFe=0:3. For\nvalues aB\ne=aB\nFe<0:3, thermal relaxation ( t<0) of the Fe is\nalso well described, however the relaxation of the Gd sublat-\ntice is significantly faster. For larger values of the exchange\nrelaxation aB\ne=aB\nFe=3, the dynamics of both sublatttices are\nsubstantially speed up and strongly disagree with ASD simu-\nlations.\nFor larger laser pulse power 2 P0the magnetization switches\nusing ASD simulations. We keep the same values for the re-\nlaxation parameters in Baryakhtar-like model as for P0, and\ncompare to the ASD simulations. For small values of aB\ne\n(Fig. (3)(b)), differently to the P0case (Fig. (3)(a)), the dy-\nnamics described by the Baryakhtar-like model is not only\nslower than those of ASD simulations but it hardly reproduces\nmagnetization switching. In order to reproduce switching, we\nneed to use larger values of the exchange relaxation parameter,\naB\ne=aB\nFe=3. These findings are in agreement with previous\nworks using Baryakhtar-like model where switching was re-\nproduced for comparable values of aB\ne. However, as we have\ndiscussed before, for those values of aB\ne, thermal relaxation\ndynamics ( t<0) is much faster than in ASD simulations.\nThis brings us to the question of how much understanding\nabout switching can we gain by using this bare Baryakhtar-\nlike model, are we missing something?\nB. The Landau-Lifshitz-Bloch model\nSince the Baryakhtar-like model is based on symmetry ar-\nguments, the macroscopic magnetization dynamics coming\nout from ASD simulations should also be described by that\nmodel with adequate expression for the relaxation parameters\nand non-equilibrium effective fields. The magnetization dy-\nnamics coming out from ASD simulations is well described\nby the LLB equation of motion.\ndma\ndt=Gk;a(ma\u0000m0;a); (12)\nwhere\nGk;a=2lag\nmakBT1\nxaL(xa)\nL0(xa); (13)\nwith xa=b maHMFA\na, where HMFA\na is given in Eq. (6), and\nm0;a=L(xa). The same equation applies to the second sublat-\nticeb. Here, the relaxation rate Gk;adepends non-linearly on\nthe non-equilibrium sublattice magnetization, ma(b), through\nthe parameter xa. We note that Eq. (12) can be expanded\naround equilibrium for small perturbations of the magnetiza-\ntion. By doing so, the relaxation rates and effective fields are\nexpressed in terms of equilibrium properties such as equilib-\nrium magnetization and zero-field susceptibilities [20]. In the\npresent work, however, we use the version in Eq. (12). Direct\ncomparison between ASD simulations and the LLB model\n-1-0.500.51\n-10 -5 0 5 10laser powerP0laser power 2P0-1-0.500.51laser powerP0laser power 2P0mztime [ps]αe/αa=0αe/αa=0.1αe/αa=1mzFeGd(a)\n(b)FIG. 4. Element-specific magnetization dynamics of GdFeCo calcu-\nlated using atomistic spin dynamics (symbols) and macroscopic LLB\nequation (solid lines) for two laser pulse power values, (a) P0and (b)\n2P0. For t<0, electron and lattice temperature are T=300K, and at\nt=0 a laser pulse is applied. The exchange relaxation parameter is\nvaried, ae=aa=0;0:1 and 1, where aa=0:01, and a=FeCo or Gd.\nThe initial relaxation dynamics is well described by ae=aa=0. (a)\nFor laser power P0, the element-specific dynamics is well-described\nforae=aa=0:1. (a) For ae=aa=1, exchange relaxation dominates\nand the element-specific dynamics are similar. (b) For laser power\n2P0, the switching dynamics is not described by the LLB model.\nof element-specific magnetization dynamics is possible and\nwith relatively good agreement. Importantly, since the LLB\nmodel is derived directly from the ASD microscopic model,\nthe damping parameters, la(b)in Eqs. (13) and (2) stand for\nthe same physics, the rate of angular momentum dissipation of\nthe atomic spins. Differently to the Baraykhtar model where\naB\na(b)is taken as a fitting parameter, within the LLB model the\nvalue of la(b)in Eq. (13) is the same as in the ASD simu-\nlations. A key difference between the Baryakhtar-like model\nand the LLB model is that in the latter an exchange relaxation\nterm is missing. In order to find a meeting point between these\nphenomenological models, we rewrite Eq. (12) in terms of a\ndamping term multiplied by an effective field,\ndma\ndt=2laL(xa)\nxag\nmama\u0000m0;a\nbL0(xa)=gaaHa; (14)\nwhere\naa=2laL(xa)\nxa: (15)\nDifferently to Baryakhtar-like model, in the LLB model, the\nrelaxation parameter strongly depends on temperature and\nnon-equilibrium sublattice magnetization through the thermal\nfield, xa=b maHMFA\na. At the same time, the non-equilibrium\nfields maHawithin the LLB and Baryakhtar-like models differ.7\nThe effective field in the LLB model is defined as\nmaHa=(ma\u0000m0;a)\nbL0(xa): (16)\nEquation (16) provides a microscopic description of the effec-\ntive field driving the magnetization dynamics in ferrimagnets,\nbased on the Heisenberg spin model (Eq. (1)). Under the as-\nsumption of small perturbations around the equilibrium both,\nLLB and Baryakhtar-like effective fields, simplify to Landau-\nlike expressions [19]. Equation (14) describes with a very\ngood degree of accuracy the relaxation of the angular mo-\nmentum via dissipation to the heat-bath, which corresponds\nto the relativistic term in Eqs. (9) and (10). Previously, it has\nbeen found that ASD simulations compare well to Eq. (14) for\ncoupling parameters of around la\u00190:1\u00001 [20, 42]. These\nvalues can be considered to correspond to the intermediate-to-\nhigh coupling regime. Direct comparison between ASD sim-\nulations and experiments of single pulse switching in GdFeCo\nhas suggested values of lFe\u00190:06 and lGd\u00190:01 [16]. In\nthe context of the present work we find that Eq. (14) describes\nrelatively well the thermal relaxation dynamics in direct com-\nparison to ASD simulations (Fig. (4)).\nIn order to account for the exchange relaxation in the LLB\nmodel, we follow the Baryakhtar-like model ((9) and (10)),\nand add an exchange relaxation term to Eq. (14),\ndma\ndt=gaaHa+gae\nma(maHa\u0000mbHb) (17)\nwhere aeis a phenomenological exchange relaxation param-\neter to be determined by comparison to ASD dynamics. The\ninclusion of the exchange relaxation (second term in r.h.s) in\nthe LLB improves the agreement to ASD simulations. With\nthis addition, the LLB model describes well thermal relax-\nation for small values of the ratio ae=aaas demonstrated in\nFig. 4. For large values of aethe LLB model is unable to\ndescribe thermal relaxation dynamics ( t<0 in Fig. 4(a) and\n(b)). For laser power P0(Fig. 4(a) ( t>0)) the magnetization\ndynamics is slightly slower using the LLB model than those\ngained by ASD simulations for ae=aa=0. For ae=aa=0:1,\nthe agreement is even better than without exchange relax-\nation. The agreement vanishes when the exchange relax-\nation is increased to ae=aa=1. Critically, when the laser\npower is increased from P0to 2P0, for which ASD simula-\ntions show ultrafast switching, the LLB model only shows\ndemagnetization-remagnetization of both sublattices. We find\nsome agreement on the demagnetization time scales when a\nquite large exchange relaxation is used, ae=aa=1. These\ndynamics are similar to those observed using the Baryakhtar-\nlike model for intermediate values of the exchange relaxation\nparameter (Fig. (3)). It has been demonstrated previously\nthat by including the transverse components of the equation\nof motion, switching is possible via a precessional path when\na canting between the magnetization of each sublattice exists\n[21]. Here, we restrict to purely longitudinal switching within\nthe LLB model.\n02468 1 0-1-0.500.51−10−50 5 1 0laser powerP0laser power 2P0time [ps]mztime [ps]FeGd(a)(b)FIG. 5. Element-specific magnetization dynamics of GdFeCo calcu-\nlated using atomistic spin dynamics (symbols) and the unified phe-\nnomenological model derived here, following Eq. (17) (solid lines)\nfor two laser pulse power values, (a) P0and (b) 2 P0. Both electron\nand lattice temperature are kept constant, T=300 K, for t<0. At\nt=0 a laser pulse is applied, the same as in Figure (2). GdFeCo\nalloys with xGd=25% are calculated.\nC. Unified phenomenological model\nSo far we have constructed a phenomenological model\nbased on the LLB and Baryakhtar-like models, the dynam-\nics is given by Eq. (17), the effective field by Eq. (16)\nand the relativistic relaxation parameter Eq. (15). We still\nneed an expression for the exchange relaxation parameter. We\nconstruct this expression starting with single species ferro-\nmagnets, where sublattices aandbrepresent the same spin\nlattice, hence exchange of angular momentum is non-local.\nTherefore, maHa\u0000mbHb=maHexa2\n0Dma, with a0represent-\ning the lattice constant. Hence, the rate of non-local angu-\nlar momentum transfer reads Gnon\u0000loc:\nex =aex(maHa\u0000mbHb) =\naa(A=Ma(T))Dma, where Ais the so-called micromagnetic\nexchange stiffness [58]. Ma(T) = (ma=ua)mais the magne-\ntization density at temperature T, where uais the unit cell\nvolume. Therefore, we find that aex=aa=(zma). By con-\nsidering that the exchange relaxation rate should conserve the\nsymmetry under the exchange of lattice index, aex(M1;M2) =\naex(M2;M1), we find that\naex=1\n2\u0012aa\nzabma+ab\nzbamb\u0013\n: (18)\nThis expression is the extension of the non-local exchange re-\nlaxation in ferromagnets to local exchange relaxation in ferri-\nmagnets. This explicit expression for the exchange relaxation\nparameter in Eq. (18) completes our unified model, which\nbridges the atomistic spin dynamics model and the Baryakhtar\nand LLB macroscopic models.\nWe find that the agreement between our unified phe-\nnomenological model and ASD simulations is excellent, see\nFig. (5)(a) and (b). Figure 5(a) shows that for t<0, the sub-\nlattice magnetization relaxation towards thermal equilibrium\nvalue is described with a high level of accuracy by our model.\nFort>0 and a relatively low laser power P0, the agreement\nis also excellent for the demagnetization and remagnetization\ndynamics. Figure 5(b) shows the comparison between the uni-\nfied model and ASD simulations of the switching dynamics.\nWe conclude that Eq. (17) for the sublattice magnetization8\ndynamics together with the Eq. (16) for the effective field\nand Eqs. (15) and (18) for the relaxation parameters, unify\nthe Barayakhtar and the LLB phenomenological models for\nsingle-pulse all-optical switching in ferrimagnets.\nIV . DISCUSSION AND CONCLUSION\nThe macroscopic model presented in this work solves some\nopen questions in the field of ultrafast magnetization dynam-\nics in ferrimagnets. For example, it answers the question of\nthe range of applicability and the validity of the parameters of\nthe Barayakhtar and LLB phenomenological models. In the\none hand, within our model, the relativistic relaxation param-\neters ( aa) are element-specific and strongly depend on both\nthe temperature and the non-equilibrium sublattice magneti-\nzation. The temperature and magnetization dependence of\nthe relativistic relaxation parameters are well described by the\nLLB model. In the other hand, the exchange relaxation pa-\nrameter ( aex) is cast in terms of the element specific relativis-\ntic relaxation parameters and sublattice magnetization. We\nhave demonstrated that in order to reproduce the ASD sim-\nulations results, the relaxation parameters in the Barayakhtar\nmodel have to be both temperature and magnetization depen-\ndent. The explicit expression of the exchange relaxation pa-\nrameter is the main result of the present work since it allows\nus to unify the Barayakhtar and LLB models. While for the\nBarayakhtar model aeis unconnected to aa, within our pro-\nposed model they are proportional to each other, ae\u0018aa=ma.\nThis relation is the key to bridge both ASD simulations and\nBarayakhtar and LLB models together. Additionally, we have\nalso demonstrated the validity of the non-equilibrium effective\nfields given in Eq. (16) as derived in the LLB model instead\nof the Barayakhtar model.\nSingle-pulse switching in ferrimagnets has been described\nbefore by the Baryakhtar model. A necessary condition for\nswitching is that the system transits from the relativistic relax-\nation regime to the so-called exchange-dominated relaxation\nregime. Although details of switching in such a regime have\nbeen already discussed in detail [5, 19], so far it has remained\nunknown how this transition could be described theoretically.\nOur model resolves this question. When the system is at equi-\nlibrium or weakly excited, the exchange-relaxation parameter\nfulfills, ae\u001caa. For strong excitation, such that the mag-\nnetic order of one sublattice reduces significantly, close to\nzero ma!0, the exchange relaxation will dominate the dy-namics since ae\u0018aa=ma\u001daa. From our model, one can\nderive universal criteria for switching in ferrimagnets, includ-\ning GdFeCo and Mn 2RuxGa [59].\nThe provided understanding is paramount for further re-\nsearch on material engineering, for example, to find alter-\nnative material classes showing all-optical switching. No-\ntably, our model predicts that the exchange relaxation term\nis enhanced as the number of neighbours reduces. This de-\npendence suggests that magnetic systems of lower dimen-\nsion, e.g. 2D magnets [60], could show a faster, more ef-\nficient switching than bulk materials. Further, the extension\nof our model to the micromagnetic level will allow to opti-\nmize switching conditions. The use of micromagnetic com-\nputational solvers permits for a realistic description of ultra-\nfast AOS processes in ferrimagnetic alloys, such as helicity-\nindependent and helicity-dependent AOS, where multidomain\nstates and thermal gradients play an important role in the pro-\ncess [46].\nTo summarize, in the present work we have presented a\nunified model for single-pulse all-optical switching in ferri-\nmagnets. Our model merges and improves previous semi-\nphenomenological models, the Landau-Lifshitz-Bloch model\nand Barayakhtar-like models. To verify the accuracy of the\nproposed model, we directly compare the laser induced mag-\nnetization dynamics to atomistic spin dynamics computer\nsimulations. Differently to previous models, our model has\nthe advantage that it can be directly compared to ASD simu-\nlations. Further, we have established the connection between\nASD and macroscopic equations of motion. Importantly, we\nprovide here the stepping stone for the construction of a mi-\ncromagnetic model valid for ferrimagnets including exchange\nrelaxation between sublattices. This is paramount for a ro-\nbust construction of a multiscale scheme of the switching pro-\ncess in which not only local magnetization dynamics is de-\nscribed but also magnetic domain nucleation and motion un-\nder strong non-equilibrium. Multiscale-based micromagnetic\nmodels will allow for the description of realistic sample sizes\nand describe recent spintronics phenomena using laser pulses,\ne.g. magnetic skyrmion creation/deletion with fs laser pulses,\nor domain-wall motion under dynamics thermal gradients.\nACKNOWLEDGMENTS\nThe authors acknowledge support from the Deutsche\nForschungsgemeinschaft through SFB/TRR 227 ”Ultrafast\nSpin Dynamics”, Project A08.\n[1] C. Stanciu, F. Hansteen, A. Kimel, A. Kirilyuk, A. Tsukamoto,\nA. Itoh, and T. Rasing, All-Optical Magnetic Recording with\nCircularly Polarized Light, Physical Review Letters 99, 047601\n(2007).\n[2] K. Vahaplar, A. Kalashnikova, A. V . Kimel, D. Hinzke,\nU. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk,\nand T. Rasing, Ultrafast Path for Optical Magnetization Rever-\nsal via a Strongly Nonequilibrium State, Physical Review Let-ters103, 117201 (2009).\n[3] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A.\nD¨urr, T. A. Ostler, J. 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San-\ntos, The magnetic genome of two-dimensional van der waals\nmaterials, ACS Nano 10.1021/acsnano.1c09150 (2022), pMID:\n35442017, https://doi.org/10.1021/acsnano.1c09150." }, { "title": "1107.5225v1.Magnetic_Behaviour_of_Disordered_Ising_Ferrimagnet_in_High_Magnetic_Field.pdf", "content": "arXiv:1107.5225v1 [cond-mat.mtrl-sci] 26 Jul 2011Magnetic Behaviour of Disordered Ising Ferrimagnet in High Magnetic Field\nSobhendu K.Ghatak\nDepartment of Physics\nRKMVivekananda University,Belur,Howrah-711202,India\nAbstract\nThe magnetic behaviour of a disordered ferrimagnetic syste mApB1−pwhere both AandBrepresent\nthe magnetic atoms with respective spin SA= 1/2 andSB= 1 in presence of high magnetic field is treated\ntheoretically.Assumingthemagneticinteraction canbede scribedthroughIsingHamiltonian theapproximate\nfree energy is obtained using the cluster-variational meth od.The field dependence of the magnetization\nis then obtained for different concentration pand exchange parameters ( JAA,JBBandJAB).Forp=\n0.5,the magnetization Min ferrimagnetic state and in absence of compensation tempe ratureTcmvanishes\natTC.Field induced reversal of Mis found at switching temperature TS(< TC) which is decreasing function\nof fieldH.A maximum in Mis found above TSand the maximum value of Mincreases with field.In\nferrimagnetic state Mincreases almost linearly at high Hregion. For system with large ferromagnetic\nJAA,the compensation temperature Tcmis increasing function of JBBandJAB.The decrease in compensation\ntemperature is linear at small field and tends to saturate at h igher field.The sharpness of the magnetization\nreversal is increased with H.For fully compensated state of the system with p= 2/3,the magnetization in\npresence of Halso exhibits switching behaviour at TS.Forp= 0.2 the field induced reversal of magnetization\noccurs more sharply.The orientational switching of the sub lattice magnetization MAandMBwith field\nincreases the Zeeman energy and is the origin of magnetizati on reversal at TS.\nPACS numbers: 05.50, +9;75.10Hk;75.10.-b\nKeywords :MixedSpinIsingmodel,Disorderedferrimagneticalloy,Rare-earth- transitionmetalalloy,Cluster-\nvariational method\n∗Corresponding author.\nE-mail address: skghatak@phy.iitkgp.ernet.in\n11 INTRODUCTION\nFerrimagnetic state in its simplest form is characterized by an oppos ing and unequal magnetization of two\nsublattices below a critical temperature Tc. The finite magnetization below Tcresults from unequal magnetic\nmoment of constituents metal ions of the material. In addition, diffe rences in rate of thermal demagnetization\nof sublattice magnetization can lead to complete cancelation at lower temperature -referred as compensation\ntemperature Tcmthat exists in number of ferrimagnetic system [1].The studies of ferr imagnetic materials are\nnormally centered around their importance in technical applications [2-4].In crystalline lattice the constituents\nmetal ions normally occupy respective sites.On the other hand, the site occupancy tends to be random in\ndisordered lattice. The relative composition of constituent metal io ns can also be varied over a wide range in\ndisordered (amorphous) state produced through rapidly quench ed method[6,7].The rare-earth -transition metal\nferrimagnetic alloys in amorphous state had been investigated for t heir potential in magneto-optical recording\n[8].The real alloy contains, apart from magnetic atom,glass former th at stabilizes the disordered state.The\namorphous alloy with two kinds of magnetic atom can be, to a first app roximation, considered as binary spin\nsystemwith twosublatticenetworks.Thefielddependenceofmagne tizationindisorderedferrimagneticmaterials\nhasbeenofrecentinterest,andthemagnetizationisfoundtobene arlylinearinfieldathighfieldregion[4,5]. The\ncompensation temperature is expected to be field dependent. In this context it is appropriate to examine\nthe field behaviour of the magnetization and the compensatio n temperature in ferrimagnetic\nstate, and is attempted here based on simple theoretical mod el.\nTheoreticalmodelfrequentlyutilizedtodescribethephasediagra mandthemagneticbehaviourofdisordered\nmagnetic alloy is disordered Ising model [9-13].The mixed spin system wit h two different spins whose interaction\nis Ising-like is considered as simple model for ferrimagnetic system.Diff erent theoretical methods like the mean-\nfield approximation [14],the effective field theories [15,16], the renorma lization-group calculations [17,18] and\nthe Monte-Carlo simulations [19,20] are used to get the phase diagra m and critical behaviour of Ising model\nwith spin S= 1/2 andS= 1 for two sublattices in ordered lattice. The model has also been st udied in different\ndecorated lattices [21,22] and it is predicted that the compensation temperature exists within a specific region\nofJAA−JBBplane where JAA(JBB) is intra-sublattice exchange interaction in A-(B) sublattice [22].\nIn this article the results of the field dependence of magnetization in ferrimagnetic state of mixed Ising spin\nsystemApB1−pwithSA= 1/2 andSB= 1 are presented.The approximate procedure as suggested by Og uchi\n[23] for pure Ising system and extended by Ghatak [13] for disorde red Ising system is utilized for evaluation\nof the free energy.The magnetization,compensation temperature and their field dependence are then obtained\nfrom the configuration averaged free energy.In sec.2 the model a nd the method of calculation are outlined and\nresults are presented in Sec.3.\n2 MODEL AND METHOD OF CALCULATION\nWe take a binary alloy ApBqof two magnetic atoms A and B with respective concentration pandq= 1−p. It\nis assumed that all magnetic interactions are localized and can be des cribed by Ising Hamiltonian\nH=−/summationdisplay\nijJijSizSjz−/summationdisplay\niHiSiz (1)\nWhereSizis the Ising spin at i−thsite and takes the value SA= 1/2 orSB= 1 depending upon the\noccupation of the site by AorBatom. These values are chosen to reduce the algebraic complexity .T he second\nterm is the Zeeman energy where the magnetic field Hi(expressed in dimension of energy) at i−thsite is in\n2z-direction. The nearest neighbour exchange interaction Jijtakes value JAA,JBBandJABfor the magnetic\nbond A-A, B-B and A-B respectively. For quenched disordered alloy the free energy F, given by [14]\nF=−kT[lnTrexp(−βH)]av (2)\nWhere [.....]avrepresents the average over all possible alloy configurations and β= 1/kT. The free energy\ncan be expressed as\nF=F0−kT[ln]av (3)\nThe quantity F0=−kT[lnTrexp(−βH0)]avrefers to the configuration averaged free energy for non-\ninteracting system described by the Hamiltonian H0.The symbol < ... >in second term of Frepresents the\nensemble average over the states of H0and the operator V=H−H0. The non-interacing Hamiltonian H0is\ntaken as\nH0=/summationdisplay\niIiSiz (4)\nHereIiis parametric field to be determined.\nForevaluationofthesecondtermofEq.(3)theapproximateproce dureusedearlierisadopted.Theapproximation\nisbasedonassumptionthatthesystemisbuiltoutofsmallindepende nt”buildingblock”.Theinteractionamong\nspins within the block are treated exactly and the rest of the intera ction is represented by field Ii.The field Ii\nis determined from minimization of the approximate free energy Fvthat can be written as [23,13]\nFv=F0−kTL/summationdisplay\nl=1[ln]av (5)\nWhereVlis thelthdivision of Vwhich is divided into Lnumber of blocks. With decrease of number\n(L) of division, Fvtends to exact value of free energy. With the increase in size of the block the algebraic\ncomplexity grows at faster rate compared to improvement of the r esult related to transition temperature of\nan Ising model.The building block consisting of 4-spins for Ising model le ads to the results equivalent to the\nBethe approximation. Here the same block is taken for the disorder ed mixed spin system. The possible atomic\nconfigurations for 4-spin block with A or B distributed at lattice point s are shown in Fig. 1. The respective\nprobability of occurrence of the configuration is noted below the re spective figure. The number of spin states\nin a given configuration depends on number of A and B atoms. The num ber varies from maximum 34for block\nwith allS= 1 atoms to minimum 24for all A-atom block. In this approximation the configuration - aver aged\ntrial free energy Fvcan be expressed as [24]\nFv=F0−(z/8β)[p4lnZ0+q4lnZ4+4p3qlnZ1+4pq3lnZ3+2p2q22(lnZ2+2lnZ22)] (6)\nwhere\nF0=−((1−z/2)/β)[pln(cosh(βIA/2))+qln(1+2cosh( βIB))] (7)\nand\nZ0= 2[X4\nAcosh2α1+4coshα1+2+X−4\nA] (8)\nZ4= 2X4\nBcosh4α2+8X2\nBcosh3α2+4(3+2 XB)cosh2α2\n+ 8(3+ X2\nB)coshα2+8X−1\nBB+9+2X−4\nB(9)\nZ1= 2X2\nAXCcosh(α2+3α1/2)+2X2\nAX−1\nCcosh(−α2+3α1/2)+2[2+ X−2\nAXC]cosh(α2+α1/2)\n+ 2[2+ X−2\nAX−1\nC]cosh(−α2+α1/2)+2[2+ X−2\nA]cosh(α1/2)+2X2\nAcosh(3α1/2) (10)\n3AA\nB\nAB B\nB BA BB BA\nB\nA\nB4p3q\n4pq3 q44p2q2\n2p2q2B BAA\nAAA\nAA\nJAA JBB JABp4\nFigure 1: Schematic representation of ’building block’ of four atoms . Solid, Dashed and Dotted lines represent\nrespectively JAA,JABandJBB. The probabilities of different configurations are given below the diag rams.\nZ3= 2X2\nBXCcosh(3α2+α1/2)+2X2\nBX−1\nCcosh(3α2−α1/2)\n+ 2[3+ X−2\nBXC+2X1/2\nC]cosh(α2+α1/2)+2[3+ X−2\nBX−1\nC+2X1/2\nC]cosh(α2−α1/2)\n+ 2[2XBX1/2\nC+XC]cosh(2α2+α1/2)+2[2XBX−1/2\nC+X−1\nC] cosh(2α2−α1/2)\n+ [6+4( XC+X−C)X−1\nB]cosh(2α1/2) (11)\nZ2= 2X2\nCcosh(2α2+α1)+2X−2\nCcosh(2α2−α1)+4cosh(2 α2)+6cosh( α1)\n+ 8cosh( α2)+6+4 XCcosh(α2+α1)+4X−1\nCcosh(α2−α1) (12)\nZ22= 2XAXBXCcosh(2α2+α1)+2XAXBX−1\nCcosh(2α2−α1)+4XBX−1\nAcosh(2α2)\n+ 2XA[1+2X−1\nB]cosh(α1)+4XAX1/2\nCcosh(α2+α1)+4XAX−1/2\nCcosh(α2−α1))\n+ 2[2X−1\nA(X−1/2\nC+X1/2\nC]cosh(α2)+2X−1\nA+2X−1\nAX−1\nB(X−1\nC+XC) (13)\nα1= [IA(1−2/z)+2HA/z]β,α2= [IB(1−2/z)+2HB/z]β (14)\nwhere z =number of nearest neighbours, XA= exp(βJAA/4) ,XB= exp(βJBB) andXC= exp(βJAB). In\nabove expressions of α’s the symbols H’s andI’s represent respectively the magnetic field and the variational\nparameters at respective site.\nThe equations for the variational parameters IAandIBare obtained from the minimization of the trial free\nenergyFv\ndFv/dIA=dFv/dIB= 0 (15)\nThis leads to the coupled equations for IAandIBas\n2tanh(βIA/2) =p3(A0/Z0)+4p2q(A1/Z1)+4q3(A3/Z3)+2pq2(A2/Z2)+2(A22/Z22) (16)\n8sinh(βIB)\n1+2cosh( βIB)=q3(B4/Z4)+4p3(B1/Z1)+4pq2(B3/Z3)+2p2q((B2/Z2)+2(B22/Z22)) (17)\n4whereAi= (2β/z)∂Zi\n∂HAandBi= (2β/z)∂Zi\n∂HB.The non-trivial self-consistent solutions of the equs. (16-17 )th en\nprovide the approximate free energy.The sub-network magnetiza tion per atom then becomes\nMA= 0.5tanh(βIA0/2) (18)\nMB= 2sinh( βIB0)/[1+2cosh( βIB0)] (19)\nand the total magnetization Mper atom\nM=pMA+qMB. (20)\nHereIA0andIB0are the self-consistent solution of the coupled equations (16) and (17). The finite values of\nIA0andIB0lead to spontaneous sub-network magnetization which appears be low the transition temperature\nTc[24].The numerical results for magnetization are presented for p= 0.5,2/3 and 0.2 by varying magnetic\nfield.The model parameters e.g kT, magnetic field Hand the exchange interactions are scaled in terms of\nstrongest exchange parameter and number of nearest neighbou r is taken as z= 8.The direction of the magnetic\nfield is assumed to be parallel to the magnetization of A-sub-lattice.\n3 RESULTS AND DISCUSSIONS\n3.1 Magnetization\n(i) Concentration p= 0.5\nWe first examine the ferrimagnetic behaviour of the system with equ al concentration of AandBand the\nexchange interaction between AandBis anti-ferromagnetic.It is also assumed that the exchange integra lJAB\nis stronger than the ferromagnetic exchange between A−AandB−B. To represent this case, a typical values\nJAB=−1.0,JAA= 0.2 andJBB= 0.1 for the exchange interactions are taken.The spontaneous magn etization\nMA,MBofA- andB- sublattice and the net magnetization Mdecrease smoothly with Tfrom their maximum\nvalue at T= 0 and vanish at critical temperature Tc(Fig.-2). At low T,Mvaries little from its maximum\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54 /s48/s46/s57 /s49/s46/s50 /s49/s46/s53/s45/s48/s46/s57/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54 /s48/s46/s57 /s49/s46/s50 /s49/s46/s53/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s98/s32\n/s32/s77\n/s66\n/s84/s47/s84\n/s67/s77\n/s65\n/s84/s47/s84\n/s67/s97\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s50/s53/s45/s48/s46/s50/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48\n/s84/s47/s84\n/s67/s77/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72\n/s32/s48/s46/s48\n/s32/s48/s46/s48/s53\n/s32/s48/s46/s49\n/s32/s48/s46/s51\n/s32/s48/s46/s53\n/s32/s48/s46/s56\n/s32/s49/s46/s48/s77\nFigure 2: a) Variation of net magnetization Mforp= 0.5 with reduced temperature T/TCfor different field H.\nb) that of sublattice magnetization MAandMBforH= 0 (dotted line) and 0 .5 (solid line). Value of exchange\nparameters are JAB=−1.0,JAA= 0.2 andJBB= 0.1.Inset Fig.: M/M0vsT/TCfor ordered AB(dashed\ncurve) and disordered (solid curve) case with same exchange para meters\nvalueM0=pMA+(1−p)MB=−0.25.There is no compensation point below TCfor this situation.In\npresence of a magnetic field along z-direction, the thermal d emagnetization becomes faster for\nT≤Tcand changes its sign at a particular temperature T=TS(< Tc)(Fig.2a) which depends\n5on the magnetic field.Above T > T S,the magnetization continues to increase until it reaches a\nmaximum at a temperature slightly higher than TS.With further increase in Ta slower decrease\nin the magnetization is found.As the field increases the reve rsal occurs at lower temperature\n,and so TS, termed as the switching temperature, decreases with the fie ld.The maximum value of\nthe magnetization ( ∆M) also becomes higher.This large change in behaviour of MnearTSis the\nresult of a different thermal behaviour of the sublattice mag netization in presence of a moderate\nfield.The results of MAandMBare presented in Fig. 2bforH= 0(dotted) and 0.5(solid) and\nshow that a simultaneous alteration of the orientation of su blattice magnetization with respect\nto the field direction occurs at TS.The switching of the orientation of the magnetization is th e\nresult of combined effects of the Zeeman and the exchange ener gies.As the magnetic moment of\nBis higher than that of A, the free energy in the switched state is lowered by increasing the m agnitude of\nthe Zeeman energy without any loss in the exchange energy.We also n ote that apart from the field,the inter-\nsublattice exchange parameters alter the switching temperature ,the sharpness of switching and the maximum\nvalue ofM.Inset Fig.2a shows thermal variation of the reduced magneti zation (dashed curve)of an\nordered ferrimagnet where two interpenetrating sublattic esAandBare antiferromagnetically\naligned.The corresponding variation of M/M0for disordered case is given by solid curve.The\nmagnetization at intermediate temperature interval in dis ordered system is reduced compared to\nthat of ordered state.Close to transition temperature the m agnetization are nearly equal in both\ncases.Thisis expected when the correlation length around TCis undisturbedby disorder.Similarity\nof the magnetization at low temperature is related to the abs ence of the spin wave excitation.The\nreversal of the magnetization in ordered system is found to o ccur at higher field.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s32\n/s72/s84\n/s83/s47/s32/s84\n/s67\n/s32/s77/s32/s47/s32/s77\n/s48\n/s97\n/s98/s77/s32/s47/s32/s73/s77\n/s48/s73\n/s72/s32/s32/s32/s32/s32/s32/s32/s84/s32/s47/s32/s84\n/s67\n/s32/s49/s46/s48\n/s32/s48/s46/s57\n/s32/s49/s46/s50\nFigure 3: a) Variation of ∆ M/|M0|andTS/TCwithHand b)Hdependence of magnetization M/M0at\nT/TC= 1,0.9 and 1.2 for same set of J’s as in Fig.2 and p= 0.5.\nIn Fig.3athe field variation of the reduced switching temperature TS/Tcand normalized ∆ M/|M0|are\ndepicted for range H= 0 to 1. In this range ∆ M/|M0|sharply increases at lower range of Hand tends to\nsaturate at higher region.On the other hand, the reduced switchin g temperature TS/Tcexhibits nearly linear\ndecrease at high Hregion.The field dependence of reduced M/|M0|is also shown in Fig.3 bforT/Tc= 1.2,1.0\nand 0.9.AtTc,Mgrows faster at low field ,and the growth rate slows down at higher fie ld. ForT < T c,M\nchanges sharply at certain field that depends on T. In high field region, a nearly linear dependence of MonH\nis found,and the slope becomes smaller at higher Tdue to more thermal fluctuation.\nNext we consider a system where the exchange parameters are su ch that the JAAis largest compared to\nothers. This is commonly the situation for the rare-earth transitio n metal alloy where the exchange integral\n6between the transition metal ions is much stronger than that betw een rare-earth and transition metal ions or\nbetween rare-earth ions.The moment of the rare-earth ( B) is larger than that of transition metal ion ( A).Again,\nthere are situations where JABis not very weak compared to JAA.In Fig.4 the magnetization M(4a) and the\nsublattice magnetization MAandMB(4b) are displayed for the exchange parameters JAA= 1.0,JAB=−0.5,\nandJBB= 0.05. The net magnetization Min the ferrimagnetic phase at H= 0 varies little for small\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s45/s48/s46/s57/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50\n/s98\n/s84/s47/s84\n/s67/s77\n/s65\n/s77\n/s66\n/s84/s47/s84\n/s67/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72\n/s32/s48/s46\n/s32/s48/s46/s49\n/s32/s48/s46/s50\n/s32/s48/s46/s51\n/s32/s48/s46/s52\n/s32/s48/s46/s53\n/s77\n/s32/s32\n/s97\nFigure 4: a) Variation of net magnetization Mforp= 0.5 with reduced temperature T/TCfor different field\nH.b) that of sublattice magnetization MAandMBH= 0 (dotted line) and 0 .4 (solid line). Value of exchange\nparameters are JAA= 1.0,JAB=−0.5,andJBB= 0.05\nT/TC.However,a larger variation is found at higher temperature due to fa ster demagnetization of MB.This is\nassociated with smaller inter- and intra-exchange JBBinteractions. For this set of parameters there is no com-\npensation temperature and Mvanishes at TC.In presence of a magnetic field Mchanges sign at a temperature\nT=TS(field induced switching temperature) and attains a maximum at a tem perature TMless than TC. The\nreversal of MatTSis the effect of higher Zeeman energy gain when MAandMBswitches their orientation\nwith respect to the magnetic field (Fig.4b). The field variation of the m aximum value of the magnetization\nwith respect to its maximum magnitude at zero field MM/|M0|,, is given in Fig.5.A sharp increase is found at\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48\n/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56\n/s77\n/s77/s47/s77\n/s48/s84\n/s77/s47/s84\n/s67\n/s72/s84\n/s83/s47/s84\n/s67\nFigure 5: Field variation of the maximum value of MM/|M0|,the normalized switching temperature TS/TCand\nthe temperature TM/TCwhereMis maximum.Other parameters are same as in Fig.4.\nlowHand tends to saturate at higher field region.With increase in HbothTMandTS(normalized by TC) are\nreduced.The variation is almost linear for TMwhereas nonlinearity is evident for TS.When the inter-sublattice\ninteraction is further reduced then the compensation temperatu re appears before the transition temperature\nTC.One such situation is described by the Fig.6 for a set of exchange JAA= 1.0,JAB=−0.1,andJBB= 0.05\nfordifferent field rangingfrom0 to0 .5.At lowtemperature Mis againdominated by Bsublattice magnetization,\n7/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s98/s32/s32/s77/s47/s73/s77\n/s48/s73\n/s72/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84/s47/s84\n/s67\n/s32/s32/s49/s46/s48\n/s32/s32/s49/s46/s50\n/s32/s32/s48/s46/s53/s49\n/s32/s32/s48/s46/s51/s77\n/s84/s47/s84\n/s67/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s72\n/s32/s48\n/s32/s48/s46/s48/s53\n/s32/s48/s46/s49\n/s32/s48/s46/s50\n/s32/s48/s46/s51\n/s32/s48/s46/s53\n/s97\nFigure 6: Variation of net magnetization Mforp= 0.5 with reduced temperature T/TCfor different field H\n(a),and the reduced magnetization M/|M0|with field H(b) for different values of T/TC.Values of exchange\nparameters are JAA= 1,0,JAB=−0.1,andJBB= 0.05\nchanges its sign at the compensation temperature TCMand finally vanishes at TC. At compensation point the\nmagnetization of whole system becomes zero due to equal and oppo site magnetization of two sub-lattices. A\nfaster demagnetization of sublattice Bcompared to Asublattice occur as both JAB=−0.1,andJBB= 0.05\nare much weaker compared to JAA.In presence of Hthe compensation occurs at lower temperature and a well\ndefined maximum appears (Fig.6 a).The temperature dependence is also highly asymmetric about temp erature\nwhere the maximum appears. For high field the reversal of Mis sharp around TCM.In Fig.6bthe field\ndependence of Mis shown for temperature T/TC= 1.2,1,0.51and0.3.The curve labeled 0.51corre-\nsponds to the result at compensation point TCM= 0.51TCwhenH= 0.The curve labeled 0.3is the\nresult when temperature is selected as T < T CM.The magnetization at low temperature T < T CM\nchanges its direction at faster rate.At TCM,Mgrows with high slope for small field,however the\nvariation of Mbecomes flattened at a higher field. Nearly linear variation i s observed at high\nfield although the rate of variation depends on temperature. The compensation temperature\nTCMatH= 0depends on exchange parameters JBBandJAB[24]. The compensation point in\nferrimagnetic state appears when the sublattice Bwith higher moment is thermally demagne-\ntized at a faster rate compared to that of other sublattice A.This happens when intra-sublattice\ninteraction JBBis weak.For given JAB,it is expected that higher value of ferromagnetic JBBresults\nhigherTCM.The dependence of TCMonJBBfor two values of JAB=−0.1 and−0.2 is displayed in\nFig.7a,andTCMis linearly dependent on JBB.ForJAB=−0.2 the compensation point exists when JBBis less\nthan 0.1.The dependence of TCMonHis given in Fig.7 bforJAA= 1.0 and a) JAB=−0.2 ,JBB= 0.02, b)\nJAB=−0.1,JBB= 0.02,and c) JAB=−0.1,JBB= 0.05,JAB= 0.1. For low H, a linear decrease in TCMis\nfound for all cases,and at higher field TCMtends to saturate.\n(i) Concentration p= 2/3\nThe concentration p= 2/3 corresponds to the situation where total magnetization is comple tely compensated\natT= 0 due to choice of spin values. When the inter-sublattice exchange JAB=−1.0 is larger than the\nintra-sublattice exchange ( JAA=JBB= 0.4)interactions the state with M= 0 persists up to T/TC= 0.2\n(Fig.8a).Within intermediate temperature interval ( 0 .2≤T < T C) andH= 0,Mbecomes finite but\nsmall and dominated by B-sublattice magnetization as thermal demagnetization effect is sam e for both sublat-\ntices.However,the presence of field changes the behavior of M. There is a sharp variation of Mfrom negative to\n8/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s98/s99/s32/s84\n/s99/s109\n/s72/s97/s98\n/s48/s46/s48/s49 /s48/s46/s48/s50 /s48/s46/s48/s51 /s48/s46/s48/s52 /s48/s46/s48/s53/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s84\n/s99/s109\n/s74\n/s66/s66/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s74\n/s65/s66\n/s32/s32/s32/s45/s48/s46/s50\n/s32/s32/s32/s45/s48/s46/s49/s97\nFigure 7: a)Dependence of the compensation temperature TCMwith exchange JBBfor two values of JAB\nand = 0.b) Field dependence of TCMfor three set of exchange parameters a) JAB=−0.2,JBB= 0.02,\nb)JAB=−0.1,JBB= 0.02,c)JAB=−0.1,JBB= 0.05 andp= 0.5\npositive value at switching temperature as Hincreases. With increases in field the switching temperature TSis\nlowered.The switching like behaviour of Mis due to simultaneous flipping of sublattice magnetization in order\nto maximize Zeeman energy and keeping antiferromagnetic alignment of the sublattice magnetizations.The field\ndependence at high field is also nearly linear as in earlier case of p= 0.5. We note that the magnitude of\nchange of the magnetization in this case much smaller compared to th e system p= 0.5. On the other hand\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57/s48/s46/s49/s50/s32\n/s84/s47/s84\n/s67/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72\n/s32/s32/s48\n/s32/s48/s46/s48/s53\n/s32/s48/s46/s49\n/s32/s48/s46/s50\n/s32/s48/s46/s51\n/s77/s97\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57\n/s77\n/s84/s47/s84\n/s67/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72\n/s32/s32/s48\n/s32/s48/s46/s48/s53\n/s32/s48/s46/s49\n/s32/s48/s46/s50\n/s32/s48/s46/s51/s98\nFigure 8: Variation of net magnetization Mforp= 2/3 with reduced temperature T/TCfor different field H.\nValue of exchange parameters are for a) JAA= 0.4,JAB=−1.0,andJBB= 0.4 and for b) JAA= 1.0,JAB=\n−0.5,andJBB= 0.2\nwhenJAA= 1.0 dominates overother exchangeinteractions ( JAB=−0.5,JBB−0.2) (Fig.8b) the magnetization\nalwaysremain parallelto H.This is due larger exchangeenergy in sublattice Acompared to that in Bsublattice,\nand the induce magnetization is dominated by change in MAfor all field.\n(i) Concentration p= 0.2\nThis refers to the situation where the concentration of ions with hig her values of magnetic moment is smaller\nand can simulate amorphous transition metal -rare-earth alloy with smaller concentration of rare-earth.Fig.9 a\ndisplays the net magnetization Mfor different field H= 0 to 0 .5 for exchange JAA= 1.0,JAB=−0.5,and\nJBB= 0.2.There is no compensation temperature for this set of parameter s.But the reversal occurs in presence\nof field.It is found that Mswitched very sharply from negative to positive value.Again the switc hing behaviour\nis associated with reversal of direction of MAandMBwith respect to the field direction at TS.\n9/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s32\n/s72/s84\n/s83/s47/s84\n/s67/s32/s32/s32/s32/s32/s32/s32/s32/s74\n/s65/s66\n/s32/s48/s46/s53\n/s32/s48/s46/s48/s53\n/s97/s98/s77\n/s84/s47/s84\n/s67/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32\n/s32/s48/s46/s48\n/s32/s48/s46/s48/s53\n/s32/s48/s46/s49\n/s32/s48/s46/s50\n/s32/s48/s46/s51\n/s32/s48/s46/s53\nFigure 9: Variation of net magnetization Mforp= 0.2 with reduced temperature T/TCfor different field\nH(a).Value of the exchange parameters are JAA= 1.0,JAB=−0.5,andJBB= 0.2.b)Field dependence of\nTS/TCfor two values of JAB.\nIn this work it is assumed that the magnetization of the sub-l atticeAcarrying smaller moment\nper site is aligned parallel to +z-axis and antiferromagnet ically to the magnetization of sub-lattice\nBwith higher moment.This leads to net magnetization is in neg ative z-direction.In real system\nthe direction of the magnetization is determined by an aniso tropy energy which often is repre-\nsented by anisotropic field acting on sub-lattice. So by addi ng small anisotropic field along +z\ndirection in sub-lattice Athe presented ferrimagnetic state is realized.The switchi ng behaviour\nof the magnetization in the ferrimagnetic state with magnet ic field is expected to follow when\nthe field is in opposite direction of the magnetization.The s witching of magnetization had been\nobserved in multilayer of rare-earth and transition metal[ 25,26] or alloy[26].The magnetization of\nrare-earth layers is aligned opposite to that of transition metal layers due to antiferromagnetic in-\nteraction at the interface of two layers.The magnetization reverses its direction in presence of the\nmagnetic field and exhibits complex hystersis.Although the se systems are different compared to\nsystem considered here however, it is worth noting the simil arity of global field behaviour of the\nmagnetization.We envisage possible application of the fiel d induced magnetization reversal.The\nswitching device that relied on the sign of the magnetizatio n is a possible area of application.At\na fixed field the switching will be induced by changing tempera ture through TS.The ferrimag-\nnetic system with higher concentration of ion carrying high magnetic moment would be more\nappropriate due to sharp nature of reversal (Fig.9).\n4 CONCLUSIONS\nA disordered ferrimagnetic alloy ( ApB1−p) with Ising like interaction between the spins ( SA= 1/2andSB= 1)\nof sub-lattices is treated using a cluster-variational method in pre sence of magnetic field.In this method the\ninteractions within the cluster of different configurations are trea ted exactly and the rest of the interaction\nis described by variational field which is obtained from minimization of fr ee energy. The results on the mag-\nnetization and compensation temperature are presented for p= 0.5,2/3 and 0.2 for different values of field\nand exchange parameters.In absence of a compensation tempera ture at zero field,a field induced magnetization\nreversal in ferrimagnetic state is found at a temperature -terme d as switching temperature.With increase in the\nmagnetic field the switching occurs at lower temperature and the ma gnetization reversal becomes sharper.At\n10switching temperature the free energy is gained by interchanging o rientation of sublattice magnetization with\nrespect to applied field.The compensation temperature at zero field increases with increase in intra-sublattice\nexchange interaction of Bsublattice.In presence of magnetic field the the compensation point appears at smaller\ntemperature.The magnetization in paramagnetic state varies almos t in linear fashion with Hin high field re-\ngion.The nearlylinear increasein magnetizationhas been observedin r are-earth- transition metal alloy[4,5].For\nfully compensated composition ( p= 2/3) the magnetization reverses at a switching temperature when int er-\nsublattice exchangedominatesoverothers.Oncontrarythe magn etizationpassesthroughamaximum at T < T C\nwhen the intra-sublattice exchange is largest.The magnetization re versal is found to be much sharper for system\nwith higher concentration of Bwith weak JAB.The effect of field induced magnetization reversal can be utilized\nfor switching device.\n5 Acknowledgement\nThe author gratefully acknowledges assistance from authority of RKMVivekananda University.It is also my\ngreat pleasure to acknowledge Prof.D.Sherrington for his help and e ncouragement during work on disordered\nspin system.\nReferences\n[1] L.Neel Ann.Phys,(Paris) 3137 (1948)\n[2] T. 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Phys. 67, 4432 (1990).\n[4] El-HagariM,Michor H,OzcanS,Giovamoni,MalarA,Heiba Z,KerschlP,Sch onhartM,BauerF,Grossinger\nR,Hilscher G,Freudenberg J,Rosner H,J.Phys.:Condens.Matter 184567 (2006)\n[5] R.Grossinger,M.Schonhart,P.Kerschl,S.OZcan,M.El-Hagary,J.Freude nberger and H.Michor\nJ.Phy.:Conference series 51139-142 (2006)\n[6] K.Moorjani and J.M.D.Coey Magnetic Glasses, Elsevier Publication Am sterdam (1984)\n[7] C.W.Chen Magnetism and Metallurgy of Soft Magnetic Materials,Dove r Publication,NewYork,(1986)\n[8] P.Choudhuri,J.J.Cuomo and R.J.Gambins Appl.Phys.Lett. 22,337 (1973)\n[9] M.Sajieddine,Oh.Baur,K.Cheriff,C.Dufour,G.Marchal and R.E.Camley Phy s.RevB49(1994) 8815,\nS.Honda and M.Nawate J.Magn.Magn.Mater. 136163 (1994)\n[10] D.Sherrington and S. Kirkpatrick Phys.Rev.Lett 341348 (1975)\n[11] M.F.Thorpe and A.R.McGurn Phys.Rev. B 202142 (1979)\n[12] T.Keneyoshi Phys.Rev. B 387688 (1986) and Solid State Comm. 93(1995) 691\n[13] S. K. 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B72(2005) 224404\n[23] A.Oguchi, Prog.Theor.Phys. 561442 (1976)\n[24] S.K.Ghatak,Int.J.Mod.Phy.B 222421 (2008)\n[25] S.Demirtas,R.E.CamleyandA.R.Koymen,Appl.Phys.Lett. 8720211(2005),R.E.Camley,W.Lohstroh,G.P.Felcher,\nN.Hosoito and H.Hashizume,J.Magn.Magn.Mater 28665 (2005)\n[26] S.Demirtas,M.R.Hosu,R.E.Camley,H.C.Mireles and A.R.Koymen,Phys.Rev.B 87184433 (2005)\n12" }, { "title": "1607.08689v1.A_rock_salt_type_Li_based_oxide__Li3Ni2RuO6__exhibiting_a_chaotic_ferrimagnetism_with_cluster_spin_glass_dynamics_and_thermally_frozen_charge_carriers.pdf", "content": "1 \n (Accepted for publication in Scientific Reports) \n \nA rock -salt-type Li-based oxide, Li 3Ni2RuO 6, exhibiting a chaotic ferrimagnetism with \ncluster spin-glass dynamics and thermally frozen charge carriers \n \nSanjay Kumar Upad hyay,1 Kartik K Iyer,1 S. Rayaprol2, P.L. Paulose ,1 and E.V. \nSampathkumaran1,* \n1Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India \n2UGC -DAE Consortium for Scientific Research, Mumbai Centre, R -5 Shed, BARC Campus, Trombay, \nMumbai – 400085, India \n \n*Corresponding author: sampath@mailhost.tifr.res.in \nThe area of research to discover new Li containing materials and to understand their physical \nproperties has been of constant interest due to applications potential for rechargeable batteries. \nHere, we present the results of magnetic investigations on a Li compound, Li 3Ni2RuO 6, which was \nbelieved to be a ferrimagnet below 80 K. While our neutron diffraction (ND) and isothermal \nmagnetization (M) data support ferrimagnetism, more detailed magnetic studies establish that this \nferri magnetic phase exhibits some f eatures similar to spin -glasses . In addition, we find another broad \nmagnetic anomaly around 40-55 K in magnetic susceptibility (χ), attributable to cluster spin-glass \nphenomenon . Gradual dominance of cluster spin-glass dynamics with a decrease of temperature ( T) \nand the apparent spread in freezing temperature suggest that the ferrimagnetism of this compound \nis a chaotic one. The absence of a unique freezing temperature for a crystalline material is interesting. \nIn addition, p yroelectric current (Ipyro) data reveal s a featu re in the range 40 -50 K , attributable \nto thermally s timulated depolarization current . We hope this finding motivates future work to \nexplore whether there is any intriguing correlation of such a feature with cluster spin-glass \ndynamics . We attribute these magnetic and electric dipole anomalies t o the crystallographic disorder, \nintrinsic to this compound. \n \nThe phenomenon of spin -glass ordering in which the magnetic moments are randomly frozen as the \ntemperature is lowered below a characteristic temperature (Tg) discovered several decades ago for magnetic \nimpurities in non -magnetic matrices , is commonly observed in many concentrated magnetic systems as \nwell1,2. Such a type of magnetic ordering in compounds is usuall y facilitated b y crystallographic order and \ncan also be triggered by geometrical frustration [see, for instance, Refs. 3-5]. Some materials exhibit what \nhas been known as ‘re -entrant spin -glass behavior6,7; in such materials , the one occurring at a higher \ntemperature is of a ferro /antiferromagnetic type, which can enter into a spin -glass reg ime with a lowering \nof temperature with a unique freezing temperature . A few ferromagnetic materials exhibiting spin -glass \ncharacteristics have also been labelle d ‘chaotic’ magnetic systems7. Evidences for multiple spin -glass \ntransitions are generally scarce barring some exceptions8-10, and in some systems of this kind9,10, \nferromagnetic cl usters behave like spin -glasses . In this article, we provide evidence for an interesting \nsituation in which one sees a gradual dominance of cluster spin-glass features, as though there is no unique \nfreezing temperature, with a decrease of temperature , for a crystalline material, viz., Li 3Ni2RuO 6, which \nwas believed to be a ferrimagnet (TC =80 K) , [Ref. 11 ]. Our results thus reveal that the ferrimagnetis m \nof this compound is not that simple . This conclusion is based on viewing together the results of ac and dc \nmagnetization and heat -capacity ( C) as well as neutron diffraction studies . Interestingly, pyroelectric \ncurrent reported here also exhibits an anomaly, which appears to arise from thermally frozen -in electric \ndipoles12-14, in the same T-range in which spin -glass -like features appear . But it is not clear to us at present 2 \n whether these electric dipole and magnetic phenomena are coupled. It may be stated that, following Ref. \n11, this compound was not paid much attention in the literature. \nThe monoclinic crystal structure (space group, C2/c) in which the compound forms is related to that \nof Li 2TiO 3-type rock-salt structure15. In this structure, three distinct positions for Li [Li1 ( 8f), Li2 ( 4d), \nand Li3 ( 4e)] and two different ( 4e) positions for Ti are possible. In the compound under investigation, it \nwas found that Li1 and Li2 positions are occupied by Li and Li3 is o ccupied by Ni. Ti1 site is occupied by \nNi and almost all Ti2 site is occupied by Ru. A fraction (<10%) of Ru and Ni go to Li1 and Li2 sites \nrespectively and Li in turn occupies majority Ni and Ru sites. If one ignores this disorder, the structure \nessentially consists of LiO 6 octahedra alternating with (Ni 2/3Ru1/3)O6 octahedral planes, running along c-\ndirection. It is however clear from the above discussions that there is a significant crystallographic disorder \nin this material. A view of the crystal structure of this compound, however ignoring disorder, along a -axis \nis shown in Supplementary Information (see Supplementary Fig. S1 online) . \n \nResults \nDc magnetization \nThe results of dc magnetic susceptibility as a function of T obtained in a magnetic field of (H=) 5 kOe \nare shown in Fig. 1 a. There is a gradual increase of χ with decreasing T below 300 K, which is cut off by \nan upturn below 100 K , which becomes sharper below about 80 K, due to the onset of magnetic ordering , \nfollowed by a peak around 50 K and finally a fall. Inverse χ exhibits a linear region in a narrow temperature \ninterval ( 225-300 K), below which there is a deviation from this high -temperature Curie -Weiss behavior, \nattributable to short -range magnetic correlations; the value of the effective moment obtained from the linear \nregion is about 5.3 μB per formula unit which is very close to that expected (5.57 μ B) for high -spin divalent \nNi (S= 1) and pentavalent Ru (S= 3/2). The value of the paramagnetic Curie -temperature is found to be \nabout -305 K. These findings are in agreement with those reported by Laha et al11 by measurements with 1 \nkOe. However, a further study with low -fields ( e.g., H= 100 Oe, see Fig. 1b), presented here, for zero-\nfield-cooled ( ZFC) and field -cooled (FC) conditions of specimen during measurements offer s an insight . \nWhile ZFC curve qualit atively resembles that obtained with 5 kOe, the FC curve deviates from this curve \nbelow about 100 K , without any downfall even in ZFC curve. χ(FC) continues to increase with a tendency \nto flatten , only below about 30 K , but not at the onset of magnetic ordering . There is a weak peak at about \n50 K in FC curve , coinciding with the peak temperatu re in ZFC curve . It should also be noted that there is \na shoulder near 30 K (where FC curve flattens) even in ZFC curve; the ZFC curve additionally shows a \nshould er near 80 K, which apparently gets smeared in the FC curve due to the steeper variation in this \ntemperature range. While irreversibility in ZFC -FC curves is a s ignature of spin -glass freezing, the \n“delayed” flattening of the FC curve with multiple featu res as described above already signals complex \nnature of magnetic ordering . \nWe have also me asured hysteresis loops at low fields (Fig. 1 c) and isothermal magn etization up to 140 \nkOe (Fig. 1 d) at selected temperatures. M(H) plots continue to increase without any evidence for saturation \nand thus the ferromagnetic state at high fields could not be obtained. This finding emphasizes that the \nmagnetic ordering can not be of a ferromagnetic -type. The hysteresis loops at 30 and 60 K to show a weak \nhysteres is, which is a characteristic feature of spin -glasses and ferrimagnetism . \n \nNeutron diffraction \nCrystallographic Structure : We have analyzed the crystallographic structure of this compound at \nroom temperature with the ND pattern recorded at 300K. The ND pattern was refined using a structural \nmodel given by L aha et al11. The observed pattern fits very well to this model . The occupancies for each \nsite obtained from Rietveld refinement is as follows: The Wyckoff site 8f is occupied by Li1 and Ru1 in \nthe ratio 92.5:7.5. There are three 4e sites, two of which are occupied by Li/Ni and Ni respectively, and the \nthird is occupied by Li/Ru. The first 4e site is occupied by Li3 and Ni1 in the ratio 87:13, and the second \n4e site is fully occupied by Ni2. The third 4e site is occupied by Ru:Li in 85:15 ratio. The 4d site is occupied \nby Li:Ni3 in the ratio 87:13. All the oxygen positions are fully occupied. In general, good fits were obtained 3 \n between calculated and observed ND patterns recorded at different tempera tures (300K, 150K, 100K, 65K, \n30K, 10K, and 3K). \nMagnetic features: In Fig. 2 we have shown the ND data along with Rietveld refinement profile for \nthree selected temperatures, 3, 65 and 150K. The first observation one can make here is that, with decreasin g \ntemperature, there is no additional or un -indexed Bragg peak. There is a distinct increase in peak int ensity \non entering magnetically ordered state (see Supplementary Fig. S2 online ). Fig. 3 shows the variation of \ncell parameters (including the monoclin ic angle β (in degree )). A decrease in temperature decreases the \noverall unit cell volume. \nSince the neutron diffraction patterns measured down to 3K do not show additional magnetic \nBragg peak, the magnetic ordering in this compound could be assumed to be ferrimagnetic with the \npropagation vector, k = (0 0 0), also taking note of the fact that isothermal magnetization at high fields is \nnot ferromagnetic -like (that is, absence of saturation) as mentioned earlier . Magnetic moments were refined \nfor temperatures well below TC only. Using BasIreps progr am of the Fullprof suite16,17, irreducible \nrepresentations and basis vectors were obtained for all the magnetic ions at different crystallographic sites. \nFor each magnetic ion, the 1 representation was sufficient to correctly represent the magnetic structure \nwith reasonable values of the magnetic mo ment. As per the cationic distribution, Ru1 is found at the site 8f \n(shared with Li1). However, Ru1 does not seem to possess a magnetic moment and hence was not \nconsidered for final refinement. Starting from the ND data measured at 3K, the magnetic moment s were \nrefined independently for Ni3 at site 4d, Ni1 and Ni2 at 4e sites and Ru2 at site 4e. This arrangement clearly \nshows that all moments at site 4e lie on the same plane and the moment on site 4d lie above and below this \nlayer. The coupling of magneti c moments between sites 4d and 4e is anti -parallel, thereby giving rise to \nferrimagne tic structure as shown in Fig. 4 . The refined magnetic moments for Ni and Ru at different sites \nare tabulated in Table 1 and shown pictorially in Fig. 5 . It can be clearly seen that among the magnetic ions, \nNi3 (at 4d site) exhibits negative moments, indicating that these moments are anti -parallel to the rest of the \nmagnetic ions (at site 4e) as clear ly seen in the Fig. 5 . It therefore appears that it is this antisite Ni w hich \nresults in net ferrimagne tism. We believe that non -monotonic variation of magnetic moments with \ntemperature could be genuine, considering complex features in the temperature dependence of dc magnetic \nsusceptibility. \nAc magnetic susceptibility \nFig. 6a show s real (χ) and imaginary (χ ) parts of ac χ. It is obvious that, following the upturn below \n100 K with lowering temperature, there is a peak in both these parts at 82 K for the frequency ( ν) = 1 Hz \nand this peak shifts towards higher T range wi th increasing ν, for instance by 2 K for 1333 Hz; apart from \nthis, the intensity of the peak also decreases w ith increasing ν. χ also exhibits a ν -depe ndent peak near 80 \nK, which is a characteristic feature of spin -glass freezing2. This implies that the ferrimagnetism could be a \nchaotic one, as proposed for a nother re-entrant ferromagnet long ago7. With a further lowering of \ntemp erature, a broad peak appears around 53 K in χ for ν= 1.3 Hz, which varies with frequency, with this \npeak -temperature increasing by about 1 K for 1333 Hz. A careful look at the left side of this χ peak \nsuggest s a weak change of slope near 40 K, as though there is a superposition o f at least two peaks below \n70 K, as though there is more than one characterist ic freezing temperature. In fact, this is more clearly \nreflected in χ (T), which peak s near 40 K for ν= 1.3 Hz. This peak shows an apparent upward shift by a \nfew degrees when measured with 1333 Hz. We also measured ac χ in the presence of a dc magnetic field \n(5 kOe) and the above -described features are completely suppressed with a dramatic reduction in the values \nand with overlapping -curves for different ν (see Fig. 6 a). This is a key support for spin-glass -like dynamics . \n \nHeat -capacity \nIn the inset of Fig. 6c, we show the plot of C(T) below 100 K and there is no evidence for any feature \ndown to 1.8 K that can be attributed to long range magnetic ordering. The absence of a feature at the onset \nof magnetic ordering (near 80 K) may be either due to the fact that rapidly varying large lattice contribution 4 \n around this temperature obscures the expected λ -anomaly. Crystallographic disorder also can contribute to \nsmearing the feature in the entire temperature range. Therefore, it is difficult to delineate the contributions \ndue to magnetic frustration, though this phenomenon also must play a role for the lack of C(T)anomaly. \n \nIsothermal remnant magnetization (MIRM) \nWe measured MIRM at three se lected temperatures, 1.8, 30, 65 and 125 K. The specimen was zero -field-\ncooled to desired temperature, and then a field of 5 kOe was applied. After waiting for some time, the field \nwas switched off, and then MIRM was measu red as a function of time ( t). We find that MIRM drops to \nnegligibly small values within seconds of reducing the field to zero at 125 K ; however, it decays slowly \nwith t at other te mperatures, as shown in Fig. 6b. These offer support to spin -glass dynamics. The curves \ncould be fitted to a stretch ed exponential form of the type18 MIRM (t) = M IRM(0)[1+Aexp( -t/τ)1-n), where A \nand n are constants and τ here is the relaxation time. It is found that the value of n falls in the range 0.5 – \n0.7. The values of relaxation times are rather large (e.g., 100 mins at 2 K and about 28 mins for 30 and 65 \nK). These values are in fact in agreement with th at reported for cluster spin-glasses18. \n \nWaiting time dependence of dc magnetization \nWe looked for aging effects [see, for instance, Ref. 19 ] in dc magnetization at two temperatures \n(30 K and 65 K) characterizing spin -glass phase. For this purpose we have followed ZFC and FC protocols \nas des cribed, for instance, in Ref. 20 . In ZFC proto col, we cooled the sample to the desired temperature, \nwaited for certain period of time, switched on a dc field of 100 Oe and measured the increase of M as a \nfunction of time. In FC protocol, the specimen was cooled in 100 Oe , and after waiting for certain period, \nthe decay of M was measured as a function of time after the field was switched off. Th e curves thus obtained \nfor two waiting times are s hown in Fig. 7. It is obvious from this figure that the curves for 3000 s are \ndisplace d with respe ct to those for a lower waiting time of 300 s. This is very distinct at 30 K for both ZFC \nand FC protocols, establishing spin -glass -like spin dynamics at such low temperatures. For 65 K, this \ndisplacement of curves is visible for ZFC protocol, but it is feeble for FC protocol, as though spin -glass -\nlike behavior tends to weaken with increasing temperature in the magnetically ordered state. Clearly aging \nphenomenon is present in this compound and demonstrates gradual nature of the variations in spin -glass \nfreezing with changing temperature . \n \n‘Memory effect’ in dc magnetization \nIn order to look for memory ef fect, we obtained χ(T) curves in different ways. In addition to ZFC curve \nin the presence of 100 Oe without a long wait at any temperature (which is a reference curve), we have \nobtained a ZFC curve after waiting at two temperatures 25 and 60 K for 3 hours each (and also for 6 hours \neach in another independent experiment). We obtained the difference bet ween these two curve s and plotted \nthe same as ΔM versus T in Fig. 6c. It is distinctly clear that there are clear ‘dips’ at these two temperatures \nin this plot. It is found that the intensity of the ‘dip’ is increase d for a wait of 6 hours, with respect t o tha t \nfor 3 hours. This is a signature of frustrated magn etic behavior, even in the ferri magnetic phase (just below \n80 K), as discussed for assemblies of nanoparticles with ferromagnetic core and antiferromagnetic shell21. \n \nComplex permittivity and pyroelectric current behavior \nComplex permittivity and pyroelectric current studies also reveal interesting behavior well below TC. \nDielectric constants ( ɛ) and the loss factor (tan ) are shown in Fig. 8a below 100 K for two selected \nfrequencies (1 and 100 kHz). Beyond 100 K, tan increases dramatically and therefore extrinsic \ncontributions tend to dom inate . The observation we would like to stress is that both ɛ and tan undergo a \ngradual increase with T from 1.8 K, with out any apparent peak or any other feature. Therefore, we rule out \nthe presence of any ferro electricity below 150 K in this compound. We have also measured \nmagnetocapacitance at various temperatures and the changes observed in ɛ for H= 140 kOe are 0.01 a nd \n0.1% at 2 and 25 K respectively . Therefore , magnetodi electric coupling is rather weak . 5 \n However, Ipyro as a function of T exhibits a distinct feature (measured with two poling electric fields \n100 and 200 V corresponding to 2.08 kV/cm and 4.16 kV/cm for the sample used ). That is, the plot (Fig. \n8b) shows a peak at about 40 K for a rate of warming of temperature (dT/dt ) of 2K /min for the poling by -\n2.08 kV/cm at 100 K. The peak gets reversed in sign when pole d by +100 V. The intensity of the peak \nincreases for 200 V, as shown in Fig. 8b. This finding mimics that expected for ferroelectricity. However, \nsince we do not find any anomaly in ɛ(T) (Fig. 8a), these peaks can n ot be attributed to ferroelectricity . To \ngather further support for this conclusion, we have performed pyroelectric current measurements for \ndifferent dT/dt . The results (see Fig. 8c) reveal that the peak in fact shifts to higher temperatures with \nincreasing dT/dt , for instance, to ~43 K an d ~46 K for 5 K/min and 8K/min respectively. Such a str ong \nvariation is not expected13 for ferroelectric transitions . We have also obtained the behavior of Ipyro in the \npresence of a dc magnetic field of 10 kOe and we find (se e Fig. 8 b) that the intensity of the peak in the plot \nis dra matically suppressed, as in the case of ac χ. \n \nDiscussion \nFrom the results presented above, it is clear that there appears to be a contradiction between the \nconclusion from neutron dif fraction results (suggesting well -defined magnetic structure, say, \nferrimagnetism ) and that from other bulk measurements (in particular, frequency dependence in ac \nsusceptibility, and aging, memory and ZFC -FC curves bifurcation behavior in dc magnetization, revealing \nspin-glass features ). Clear ly, the magnetism of this compound is very complex. The fact that the peaks in \nχ in the range 40 to 60 K are not cusp -like suggests that there is a spread in the freezing temperatures in \nthis T-range. This spread is consistent with various features noted aro und 30 - 50 K and 75 K in Fig. 1b. \nFor this reason, it is tempting to claim that this compound could be one of the rare examples for multiple \nspin-glass freezing phenomenon . It is not clear whether a relaxation phenomenon of ferrimagnetic structu re \nis operative. \n In order to understand the nature of magnetism better, w e have analyzed the ac χ results in terms of \nthe conventional power law, associated with the critical slowdown of relaxation time, τ/τ0 = (Tf/Tg − 1)−zν. \nHere, τ represents the observation time (1/2πν), τ0 is the microscopic relaxation time, Tg is the spin -glass \ntransition temperature, T f corresponds to freezing temperature for a given observation time and zv is the \ncritical exponent. For the feature around 40 K, we obtained Tg ≈28 K, zν ≈ 7.12 and τ0 ≈ 1.8×10−4 s. For \nthe one around 80 K, corresponding values are: ~80 K, ~2.6 and ~1.6 ×10−7 s. For a conventional spin \nglasses2, the zν value falls in the range ~4 -13, and τ 0 value ranges between 10−10 and 10−13 s. It is clear that \nthe values of τ 0 obtained are in general higher than that for the conventional spin glasses. But the deviation \nis highly pronounced for the feature around 20-40 K. Judged by t hese values, one can interpret22,23 that the \nferrimagnetic regions form cluster s exhibiting spin -glass -like inter-cluster dynamics, with the cluster -glass \nbehavior gradual ly strengthening with decreasing temperature. The parameters derived from isothermal \nremnant behavior also offer s support for cluster -glass behavior, as described earlier. Therefore, long-range \nmagnetic ordering is labelled ‘chaotic’ in this article. \nIt is i nteresting to see a feature in pyroelectric current in the glassy magnetic phase. This does not arise \nfrom ferroelectricity , as mentioned earlier. Therefore, an alternate explanation should be offered for the \nobservation of the peak in Ipyro. At this juncture, it may be recalled that such a dependence of the peak on \ndT/dt has been explained in terms of ‘thermally stimulated d epolarization current (TSDC)’12,14 in the past \nliterature. This phenomenon can be explained as follows: The mobile charge carriers , presumably \nintroduced by crystallographic defects due to intrinsic disorder described above, tend to organise \nthemselves to screen the applied electric field, and, with a lowering of temperature, these charge carriers \nget trapped randomly forming electric dipoles and persist for a very long time after removal of the electric \nfield. These carriers can be released thermally, which appears as a peak in Ipyro. The type of charge carriers \ntrapped determine the sign of Ipyro with respect to that of the electric field. For instance, positive Ipyro for a \nnegative electric field implies trapping of negative charges, whereas the same sign for both implies holes -\ntrapping. Thus, in DyMnO 3, such a p eak was attributed to holes14, whereas, in the case of yttrium iron \ngarnet13, electrons are responsible. Therefore, in the present case, considering the opposite sign of the \nelectric field and Ipyro, one can confidently state that the negative charges get trapped. 6 \n It is intriguing to note that the T-range over which this phenomenon occurs is essentially the same as \nthat of the spin -glass anomalies in ac χ. The observation that Ipyro feature is suppressed by a dc magnetic \nfield of 10 kOe as in the case of ac χ may signal some connection between these magnetic and elect ric \ndipole phenomena. However, it is an open question whether this is truly the case or whether it is just \naccidental. Intrinsic crystallographic disorder must be the root-cause of all these electric and magnetic \ndipole anomalies. \nIn short, the present results reveal that the new Li -based compound, Li3Ni2RuO 6, is not a simple \nferrimagnet, but is characterized by growing i nfluence of spin -glass dynamics with decreasing temperature \nin the magnetically ordered state. Due to strong chemical disorder, intrinsic to this compound, it appears \nthat spin -glass clusters form with apparently different freezing temperature s, rather tha n a single freezing \ntemperature. Thus, such multiple cluster -glass freezing temperature for a crystalline compound is not \ncommonly reported and we hope this compound would serve as an ideal system to model such a behavior . \nWe find pyroelectric anomalies attributable to thermally stimulated depolarization mechanism, presumably \ndue to trapping of negative char ge by crystallographic disorder, though the physical origin for possible \nconnection with cluster -glass dynamics is not obvious at present. Thus this compound exhibits interesting \nmagnetic and pyroelectric anomalies. \n \nMethods \nPolycrystalline sample was prepar ed as described by Laha et al11 by a solid state reaction route. \nRequired amounts of high purity (>99.9%) starting materials, Li 2CO 3, Ni oxalate (NiC 2O4.2H 2O), and \nRuO 2, as per the stoichiometry of the compound, were mixed thoroughly, heated at 673 K for 4 h, at 1073 \nK for 12 h, and subsequently at 1198 K for 12 h with intermediate grindings between these three stages of \nheating. X -ray diffraction (XRD) pattern (Cu K α) confirms single phase nature of the compound. The \nbackscattered electron images of scanning electron microscope (SEM) have been obtained to check the \nhomogeneity of the sample. We have also performed energy dispersive sca nning electron microscopic \nstudies to determine the composition, particularly for Ni and Ru, though it is not easy to obtain precise \ncomposition for low atomic number elements, Li and O, with the sensitivity of the SEM employed. \nT-dependent dc magnetizati on studies were carried out with the help of a commercial supeconducting \nquantum interference magnetometer (Quantum Design, USA) and ac χ study with different frequencies (ν= \n1.3, 13, 133, and 1333 Hz) with a ac field of 1 Oe was also carried out with the same magnetometer. Heat -\ncapacity studies were carried with a commercial Physical Properties Measurement s System (Quantum \nDesign, USA). T he same system was used to measure complex dielectric permittivity using an Agilent \nE4980 A LCR meter with a home -made sample holder with several frequencies (1 kHz to 100 kHz) and \nwith a bias voltage of 5 V; the same sample holder was used for pyroelectric studies with Keithley 6517B \nelectrometer by poling at 100 K with different electric fie lds. Unlike otherwise stated, all the measurements \nwere performed for zero -field-cooled condition ( ZFC from 300 K) of the specimen. \nNeutron diffraction measurements were carried out on polycrystalline samples on a focusing crystal \nbased powder diffractome ter (PD -3) at Dhruva reactor, Trombay24. Sample was filled in a vanadium can \nsubjected to temperature variation using a closed cycle refrigerator. Neutrons at a wavelength of 1.48Å \nwere used for the diffraction experiments. ND patterns were analyzed for n uclear (crystalline) and magnetic \nstructures by the Rietveld refinement method u sing the Fullprof program16,17,24 \n. \n \n 7 \n Table 1: The values of magnetic moments of Ni and Ru atoms at 4d and 4e sites at three \ntemperatures are tabulated. Ni1 and Ni2 are at 4e: (0 y ¼) with different y -positions and \nNi3 is at 4d: (¼ ¼ ½). Ru is also found at site 4e, with different y position. The variation \nin the y -position with respect to temperature is also shown in the table along with the \nmagnetic moments of Ni and Ru at these different sites. \nTemperature \n(K) Ni1@4e Ni2@4e Ni3@4d Ru2@4e \n (y = 0.0913) (y = 0.4230) (¼ ¼ ½) (y = 0.7423) \n3 0.925 0.849 -0.485 0.601 \n (y = 0.0906) (y = 0.4243) (¼ ¼ ½) (y = 0.7422) \n10 1.22 1.439 -0.131 0.821 \n (y = 0.0910) (y = 0.4223) (¼ ¼ ½) (y = 0.7412) \n30 1.11 1.22 -0.283 0.653 \n (y = 0.0962) (y = 0.4199) (¼ ¼ ½) (y = 0.7442) \n65 1.024 1.047 -0.137 1.199 \n \n \n 8 \n \nReferences \n1. Binder, K. & Young, A.P. Spin glasses: Experimental facts, theoretical concepts, and open \nquestions. Rev. Mod. Phys. 58, 801-976 (1986). \n2. Mydosh, J. A. Spin glasses: An experimental introduction (Taylor & Francis, 1993). \n3. Zvyagin, A. A. New physics in frustrated magnets: Spin ices, m onopoles, etc. Low Temp. \nPhys. 39, 901-922 (2013). \n4. Hardy, V. et al. Temperature and time dependence of the field -driven magnetization steps \nin Ca3Co2O6 single crystals. Phys. Rev. B 70, 064424 (2004). \n5. Sampathkumaran, E. V. & Niazi, A. Superparamagnetic -like ac susceptibility behavior in the \npartially disordered antiferromagnetic compound Ca3CoRhO 6. Phys. Rev. B 65, 180401(R) \n(2002). \n6. Verbeek, B. H., Nieuwenhuys, G. J., Stocker, H., & Mydosh, J. A. Evidence for a \nFerromagnet —Spin-Glass Transition in PdFeMn. Phys. Rev. Lett. 40, 586-589 (1978). \n7. Jonason, K., Mattsson, J. & Nordblad, P. Chaos in t he Ferromagnetic Phase of a Reentrant \nFerromagnet. Phys. Rev. Lett. 77, 2562 (1996). \n8. Wang, Y. T., Bai, H. Y., Pan, M. X., Zhao, D. Q. & Wang, W. H. Multiple spin -glass -like \nbehaviors in a Pr -based bulk metallic glass. Phys. Rev. B 74, 064422 (2006). \n9. Mao, J. et al. Evidence of two distinct dynamical freezing processes in single -layered \nperovskite La 0.7Sr1.3CoO 4. J. Phys.: Condens. Matter 23, 336001 (2011). \n10. Phan, M.H. et al. D. Magnetism and cluster -glass dynamics in geometrically frustrated \nLuFe 2O4. J. Appl. Phys. 105, 07E308 (2009). \n11. Laha, S. et al. New rock salt -related oxides Li 3M2RuO 6 (M=Co, Ni): Synthesis, structure, \nmagnetism and electrochemistry. J. Solid State Chem . 203, 160-165 (2013). \n12. Bucci, C., Fieschi, R. & Guide, G. Ionic thermocurrents in dielectrics . Phys. Rev. B. 148, 816-\n823 (1966). \n13. Kohara, Y., Yamasaki, Y., Onose, Y. & Tokura, Y. Excess -electron induced polarization and \nmagnetoelectric effect in yttrium iron garnet. Phys. Rev. B 82, 104419 (2010). \n14. Zou, T. et al. Excess -hole induced high temperature polarized state and its correlation with \nthe multiferroicity in single crystalline DyMnO 3. App. Phys. Lett. 105, 052906 (2014). \n15. Yu, C. L. et al. The structure of H 2TiO 3 — a short discussion on “Lithium recovery from salt \nlake brine by H 2TiO 3”. Dalton Trans. 44, 15721 -15724 (2015). \n16. Carvajal, J. R. Recent advances in magnetic structure determination neutron powder \ndiffraction. Physica B 192, 55-69 (1993). \n17. Rietveld, H. M. A Profile Refinement Method for Nuclear and Magnetic Structures. J.Appl. \nCryst. 2, 65-71 (1969). \n18. Xu, Q. et al. Magnetic interactions in BiFe 0.5Mn 0.5O3 films and BiFeO 3/BiMnO 3 superlattices. \nSci. Rep. 5, 9093 (2014). \n19. Shvartsman, V.V., Bedanta, S. et al. (Sr,Mn)TiO 3: A Magnetoelectric Multiglass. Phys. Rev. \nLett. 101, 165704 (2008). \n20. Bisht, V. & Rajeev , K. P. Memory and aging effects in NiO nanoparticles. J. Phys.: Condens. \nMatter 22, 016003 (2010). \n21. Vasilakaki, M. et al. Memory effects on the magnetic behavior of assemblies of nanoparticles \nwith ferromagnetic core/antiferromagnetic shell morphology. Phys. Rev. B 88, 140402 (R) \n(2013). \n22. Chakrabarty, T., Mahajan A. V. & Kundu, S., Cluster spin glass behavior in geometrically \nfrustrated Zn 3V3O8. J. Phys.: Condens. Matter 26, 405601 (2014). \n23. De, K., Thakur, M., Manna A. & Giri, S. Unusual glassy states in LaMn 0.5Fe0.5O3: Evidence \nof two distinct dynamical freezing processes. J. Appl. Phys. 99, 013908 (2006). 9 \n 24. Siruguri, V., Babu, P. D. , Gupta, M., Pimpale, A. V. & Goyal, P. S. A high resolution powder \ndiffractometer using focusing optics. Pramana 71, 1197 -1202 (2008). \n \nAuthor contributions \nS.K.U prepared the sample and characterized the same. He performed ac and dc magnetization \nmeasurements in association with P.L.P, carried out heat -capacity and dielectric and electric \npolarization studies along with K.K.I and analyzed the results. S.R. performed neutron diffraction \nstudies and analyzed this data. E.V.S. proposed the proble m, formulated the manuscript and finalized \nin consultation with other authors. \n \nCompeting financial interests \nThe authors declare no competing financial interests. \n \n \nFigure 1 | Magnetization data for Li 3Ni2RuO 6. Dc magnetic susceptibility as a function of temperature \nmeasured in a magnetic field of (a) 5 kOe, and (b) 100 Oe are plotted in (a) and (b) respectively. In (a), \ninverse susceptibility is also plotted with a line through the Curie -Weiss region. In (b) the curves obtained \nfor ZFC and FC conditions are shown Low-field h ysteresis loops at 30 and 60 K and isothermal \nmagnetization extended to high fields at 1.8, 30 and 60 K are also shown in (c) and (d) respectively. \n \nFigure 2 | Neutron diffraction patterns of Li 3Ni2RuO 6 measured at 150, 65 and 3K. The data for 3 and \n65K include a fitting for the magnetic structure, as described in the text. \n \n \nFigure 3 | Temperature dependence of different unit cell parameters obtained from the Rietveld \nrefinement of neutron diffraction patterns is plotted. Lines are drawn through the data points as a guide \nto the eyes. \n \nFigure 4 | The magnetic structure of Li 3Ni2RuO 6 at 3K. The arrow in red colour represents magnetic \nmoment of Ni3 (at 4d site) ions, whereas arrows in blue a nd cyan represent magnetic moments of Ni1 and \nNi2 (both at 4e site) respectively. The Ru2 (at 4e site) moment is shown in green colour. \n \nFigure 5 | The values magnetic moments at different sites in Li 3Ni2RuO 6 structure are plotted as a function \nof temperature. \n \nFigure 6 | (a) Real and imaginary parts of ac susceptibility measured with various frequencies (1.3, \n13, 133 and 1333 Hz) , (b) isothermal remnant magnetization at 1.8, 30 and 65 K, and (c) the \ndifference in magne tization curves, ΔM, obtained with and without waiting at 25 and 60 K for \nLi3Ni2RuO 6. In (a), t he arrows show the direction in which the peaks shift with increasing frequency and \nwith the omission of data points through the lines, In the inset of (c), he at-capacity as a function of \ntemperature is shown. The χ curves in (a) for 133 and 1333 Hz are shifted along y -axis (by 0.01 and 0.02 \nemu/mol respectively), for the sake of clarity. \n \nFigure 7 | Dc magnetization as a function of time (waiting time depen dence or aging experiments) for \nzero-field-cooled (ZFC) and field -cooled (FC) protocols as described in the text for Li 3Ni2RuO 6 for 30 \nand 65 K. \n \nFigure 8 | Temperature dependence of (a) dielectric constant ( ɛ) and loss factor (tan ) shown for two \nfrequencies (1 and 100 kHz), (b) pyroelectric current, Ipyro, for two poling fields, obtained by increasing the \ntemperature at the rate of 2K/min, and (c) Ipyro as a function of T for different rates of change of T, after \npoling with 2.08 V/cm, for Li 3Ni2RuO 6. In (b), the curve obtained in a field of 10 kOe is also included. 10 \n \n \n \n \n11 \n \n \n \n \n \n \n \n \n \n12 \n \n \n \n13 \n \n \n \n \n \n \n \n14 \n \n \n \n \n15 \n \n \n \n16 \n \n \n \n17 \n Supplementary Information \n \nA rock -salt-type Li-based oxide, Li 3Ni2RuO 6, exhibiting a chaotic ferrimagnetism with \ncluster spin -glass dynamics and thermally frozen charge carriers \n \nSanjay Kumar Upadhyay,1 Kartik K Iyer,1 S. Rayaprol2, P.L. Paulose,1 and E.V. \nSampathkumaran1,* \n1Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India \n2UGC -DAE Consortium for Scientific Research, Mumbai Centre, R -5 Shed, BARC Campus, Trombay, \nMumbai – 400085, India \n \n*Corresponding author: sampath@mailhost.tifr.res.in \n \n \nHere, we show crystal structure of Li3Ni2RuO 6 along the [100] direction (Fig. S1) and also compare raw \nneutron diffraction data for three different ranges in an expanded sca le (Fig. S2) at three temperatures. In \nFig. S1, we ignored crystallographic disorder discussed in the article, for the sake of simplicity. \nFig. S2 clearly brings out subtle variation s in the int ensity . \n \n \nSupplementar y Figure S1: Crystal structure of Li 3Ni2RuO 6 viewed along the [100] dir ection. \n \n18 \n \nSupplementary Figure S2: The raw data of the neutron diffraction patterns recorded at T = 3, 65 and 150K \nare plotted on the same scale for three different regions. \n \n" }, { "title": "0810.4871v1.Hybrid_resonant_phenomenon_in_a_metamaterial_structure_with_integrated_resonant_magnetic_material.pdf", "content": "arXiv:0810.4871v1 [cond-mat.mtrl-sci] 27 Oct 2008Hybrid resonant phenomenon in a metamaterial structure wit h integrated resonant\nmagnetic material\nJonah N. Gollub1, David R. Smith2, and Juan D. Baena3\n1Department of Physics, University of California, San Diego , CA, 92037 USA\n2Department of Electrical and Computer Engineering, Duke Un iversity, Durham, NC, 27708 USA and\n3Department of Physics, National University of Colombia, Bo got´ a, Colombia\n(Dated: August 6, 2018)\nWe explore the hybridization of fundamental material reson ances with the artificial resonances\nof metamaterials. A hybrid structure is presented in the wav eguide environment that consists of\na resonant magnetic material with a characteristic tuneabl e gyromagnetic response that is inte-\ngrated into a complementary split ring resonator (CSRR) met amaterial structure. The combined\nstructure exhibits a distinct hybrid resonance in which eac h natural resonance of the CSRR is split\ninto a lower and upper resonance that straddle the frequency for which the magnetic material’s\npermeability is zero. We provide an analytical understandi ng of this hybrid resonance and define\nan effective medium theory for the combined structure that de monstrates good agreement with nu-\nmerical electromagnetic simulations. The designed struct ure demonstrates the potential for using a\nferrimagnetic or ferromagnetic material as a means of creat ing a tunable metamaterial structure.\nINTRODUCTION\nMetamaterials encompass a class of artificial electro-\nmagnetic media that provide electromagnetic properties\nbeyond those available from natural materials [1, 2, 3].\nTheir unique electromagnetic properties are obtained by\nharnessing the resonant behavior of many periodic and\nsubwavelength composite structures. Unfortunately, the\nresonant nature of metamaterials also ensures that they\nare frequency dispersive and limited to a narrow fre-\nquency band of operation. One method of bypassing\nthis constraint is to design tunable metamaterials, which\nthough still dispersive, can be tuned to have the desired\npropertiesoverthefrequencyrangeofinterest. Atunable\nmetamaterial can be made by placing a material with a\ntunable electromagnetic parameter, permittivity or per-\nmeability, in a region of the metamaterial cell where the\nlocal fields are concentrated. Tuning the local fields will\nin turn affect the bulk electromagnetic response of the\nmetamaterial. This hasbeen demonstratedbytuning the\ncapacitance in the gap regionof metamaterials composed\nof split-ring resonators (SRRs) using ferroelectric mate-\nrials [4, 5]. In this paper we investigate a tunable meta-\nmaterial structure that incorporates a resonant magnetic\nmaterial, that in contrast to ferroelectric material, is it-\nselffrequencydispersiveoverthe rangeofoperation. The\ntwo dispersive systems, metamaterial structure and mag-\nnetic material, combine to exhibit a distinct hybrid res-\nonance for which we provide an analytical model and\ndemonstrate its good agreement with numerical results.\nGyromagnetic materials have a permeability tensor of\nthe form [6],\nµ=µ0\nµ1iµ20\n−iµ2µ10\n0 0 1\n, (1)\nwhereµ1andµ2are resonant functions of frequency. InMagnetic Layer \nDielectri c Layer E\nHkzI/2 I/2 \nH\n60 70 80 90 100 110 120 \n-2 2468\nFrequency (Ghz) Relative Parameters Real( ε) Imag( ε)\nReal( μ) Imag( μ)\n1st Resonance 2nd Resonance yx\nz\nFIG. 1: A diagram of a unit cell of the CSRR parallel plate\nwaveguide is shown. At resonance, current flows along the\nedges of the CSRR structure and produces a magnetic field\nthat is approximately perpendicular in the gap region. The\nextracted permeability and permittivity are shown for the f re-\nquency range that includes the first two resonances of the\nCSRR waveguide (without the influence of the magnetic ma-\nterial).\nthis study we consider a simplified “generic” resonant\nmagnetic material which is isotropic and does not in-\nclude the off-diagonal permeability component, µ2, in or-\nder to elucidate the underlying physics of the hybridiza-\ntion. Never-the-less, this structure is suggestive of how\nferrimagnetic or ferromagnetic materials might be used\nto make a tunable metamaterial structure. At the end\nof this letter we briefly discuss the potential for integrat-2\ning a real gyromagnetic material into the metamaterial\nstructure.\nMetamaterials with artificial magnetic properties have\nbeen constructed using SRR structures and exhibit an\napproximate magnetic Drude-Lorentz response that has\nbeen well documented in the literature [2]. A related\nresonant structure is called the complimentary split-ring\nresonator (CSRR) and has the “complementary” meta-\nsurface of a SRR (as shown in Fig. 1). The dual CSRR\nand SRR structures obey Babinet’s principle of equiva-\nlence with equal resonant responses [7, 8], but the CSRR\nis excited by a perpendicular electric field while the SRR\nstructure is excited by a perpendicular magnetic field.\nThe CSRR structure is well suited for implementation\nof the metamaterial concept in a waveguidegeometrybe-\ncausethe CSRR structures canbe etched into the ground\nplane of a waveguide and excited by an incident TEM\nwave. Just as for bulk metamaterial structures, it is pos-\nsible to define effective bulk parameters for a waveguide\nmetamaterialstructure[9]. ByBabinet’sequivalence,the\nCSRRs permittivity is the dual of the SRR structure and\nis given by\nǫ(ω) = 1−Fω2\nω2−ω2\n0+iωΓc, (2)\nwhereFis a constant, ω0is the resonant angular fre-\nquency, Γ cis the dissipation factor, and ωis the angular\nfrequency [3]. The resonant frequency ω0is related to\nthe inductance, L, and capacitance, C, of the structure\nby\nω0=1√\nLC. (3)\nTheinclusionofathinlayerofmagneticmaterialbelow\nthe CSRR structure as shown in Fig. 1 provides a means\nto influence the the local fields of the CSRR structure.\nWe considera magnetic materialwith relativepermeabil-\nity of the form,\nµr(ω) =(ξMs+ξH0−iωΓm)2−ω2\n(ξH0−iωΓm)(ξMs+ξH0−iωΓm)−ω2(4)\nwhereMsis the magnetic saturation of the material, H0\nis the magnetic bias field, Γ mis the damping constant,\nandξ=γµ0is a constant with γdefined as the gyromag-\nnetic ratio [10]. The CSRR resonance and magnetic ma-\nterial resonance interact to produce a hybrid resonance.\nThe mechanism of interaction can be understood by con-\nsideringthedynamicsofthecurrentflowwhenthe CSRR\nis resonating. An incident TEM wave in the waveguide\ndrives current symmetrically in and out of the CSRR\nstructure along the edges of the gap region as shown in\nFig. 1. The current is concentrated on either side of the\ngap of the CSRR structure but flows in opposite direc-\ntions creating a nearly perpendicular magnetic field in\nthe gap and the regions directly above and below thegap. Consequently, determining the inductance of the\nCSRR structure is analogous to analyzing the magnetic\ncapacitormodelofparallelcurrentsheetsfilledwith some\nvolume fraction q(to be determined) of frequency depen-\ndent magnetic material. Exploiting this simple model,\nthe inductance is found to be,\nL=µ0/parenleftbiggµr(ω)\nµr(ω)(1−q)+q/parenrightbigg\nggeom, (5)\nwhereµr(ω) is the relative permeability of the magnetic\nmaterial and ggeomis a constant with units of length\nthat is determined by the geometry of the CSRR struc-\nture. If the parallel current approximation were exact\nthenggeom= (1/2)wd/hwithwthe length of the gap, d\nthe width ofthe gap, and hthe height ofthe gap(the fac-\ntor of 1/2 follows from the parallel current flow into the\nCSRR structure). In general, the geometrical function\nis more complex but it can be extracted from numerical\nsimulations of the empty CSRR waveguide as we demon-\nstrate below. The capacitance of the CSRR structure\nfollows from the capacitance across the gap (above and\nbelow the metal-surface). It is given by, C=ǫ0fgeom,\nwhere again fgeomis a constant with units of length that\nis determined by the geometry of the CSRR structure\nand can be extracted from numerical simulations.\nInserting Eq. 5 into Eq. 3 provides an equation that\ncan be solved to determine the resonant frequency of the\nhybrid CSRR/magnetic structure,\nω′\n0=1/radicalBig\nµr(ω′\n0)\nµr(ω′\n0)(1−q)+qω0. (6)\nwhere,ω0= 1//radicalbig\nǫ0µ0fgeomggeom. We note that Eq. 6 re-\nduces to the resonantfrequency ofthe empty structure in\nthe case µr(ω) is of unity. As previously mentioned, the\nmagneticfieldgeneratedinthegapregionoftheCSRRat\nresonanceispredominatelyperpendicular(seeFig.1)and\ninteracts principally with the magnetic material through\nthex-component ofthepermeability. Infact, simulations\n(not shown here) have shown that variation of any of the\nother diagonal permeability components has no effect on\nthe response of the structure. We can insert the perme-\nability Eq. 4 into Eq. 6 and solve to get the new resonant\nfrequencyofthe hybridstructure. Eq.6istranscendental\nin nature, as a result of the magnetic materials frequency\ndependence, but it can be solved straightforwardly using\nnumerical methods.\nIn order to understand the dynamics of Eq. 6 it is in-\nstructive to plot the left and right side of the equation as\na function of frequency, as shown in Fig. 2(a). Note that\nthe solution to Eq. 6 is found at the intersection of these\ntwo lines, i.e. where the function is self-consistent. The\ncharacteristic phenomenon of the hybridization is seen\nto be a splitting of the CSRR’s resonance into a lower,\n1a, and upper, 1 b, hybrid resonance which straddle the3\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 60 65 70 \nBiasing Component H 0 (kG) Frequency (Ghz) HFSS Simulations \nAnalytic Solution Empty CSRR Resonance 0 20 40 60 80 100 120 20 40 60 80 100 120 Frequency (Ghz) \nFrequency (Ghz) 1st Resonance (CSRR) 2nd Resonance (CSRR) \nLine of equality RHS for 1st Resonance RHS for 2nd Resonance \n1a1b\n2a2b(a)\n(b)\nFIG. 2: (a) The left hand side (line of equality) and right\nhand side (RHS) of Eq. 6 are plotted for the 1st/2nd CSRR\nresonance with H0= 1.5 kG and characteristic parameters\nstated in the text. (b) The dispersion curve for the 1st or-\nder hybrid mode of the CSRR/magnetic material waveguide\nis calculated both analytically and through numerical elec tro-\nmagnetic simulations as a function of the biasing field, H0.\nfrequency at which the magnetic material’s permeabil-\nity is zero (above resonance—not the zero point at reso-\nnance). The hybridization is strongest when the natural\nresonance of the CSRR structure is near the zero perme-\nability point of the magnetic material. If one increasethe\nbiasing field of the magnetic material, such that its zero\npermeability frequency advances through the CSRR’s\nnatural resonance, a characteristic growth and shift of\nthe 1aresonanceis observedand then asubsequentdecay\nand shift of the 1 bresonance is observed. Hence, tuning\nthe magnetic material effectively tunes the response of\nthe metamaterial structure. Because a CSRR structure\ninherently exhibits multiple resonance (the first two res-\nonances of the CSRR structure are shown in Fig. 1) it\nis not only the fundamental resonance that is hybridized\nbut also the higher order resonances. However, these\nhigher order resonances are highly damped if they are\nfar away from the magnetic material’s zero permeability\nfrequency. In practice this means that if the zero per-\nmeability frequency of the magnetic material is aligned\nwith the fundamental CSRR structure resonance, then\ndepending on losses in the system, residual higher order\nhybrid resonances 2 a, ..., n amight be found near the\nfundamental, 1 a, hybrid resonance as shown in Fig. 2(a).\nIn contrast, the 1 b,2b, ..., n aresonances remain largely\nseparated. This suggests that the 1 bresonance—which\nis strongly resonant, isolated from the other hybrid reso-\nnances, and strongly tunable—maybe the most useful for\napplication.In order to investigate the hybridization of the CSRR\nand magnetic material we used Ansoft’s commercially\navailable electromagnetic finite element solver (HFSS).\nHFSS calculates the scattering matrix for a simulated\nstructure via a full wave analysis. It is sufficient to sim-\nulate one CSRR unit cell, as shown in Fig. 1, and apply\nappropriate boundary conditions to reproduce the char-\nacteristic response of a parallel plate waveguide struc-\nture. The simulated structure consisted of two parallel\nperfectly conducting plates and adjoining perpendicular\nperfect magnetic boundaries. The structure was excited\nthrough waveports on either sides of the waveguide with\na fundamental TEM mode. The CSRR structure was\ncut into the top waveguide plate and an exterior volume\nabove the structure was defined in order to accommo-\ndate electromagnetic fields emanating above the struc-\nture. Perfect magnetic boundaries were defined on the\nexterior volume’s surfaces perpendicular to the propa-\ngation mode while perfect electric boundaries were de-\nfined on the other surfaces including the top of the ex-\ntra boundary volume (note that the height of the extra\nvolume is chosen such that on the boundary the electro-\nmagnetic fields decay to near zero—and hence its speci-\nfication is irrelevant, but numerical convergence is faster\nby defining an electric boundary here).\nThe geometrical parameters of the CSRR were chosen\nto have a resonance near 64 Ghz. The unit cell was 1\nmm, the ring radius was 0.4 mm, the neck was 0.15 mm\nwide, and the notch width was 0.035 mm. The spac-\ning between the plates, which determines the strength\nof the CSRR resonance, was set to 0.4 mm. The top\nand bottom waveguide plates were 1 µm thick and had\nthe conductivity of copper ( σ= 5.6×107). For sim-\nplicity, the dielectric between the plates was set to have\na dielectric constant of ǫr= 1. The structure was first\nsimulated without the magnetic layer to determine the\nDrude-Lorentz fit parameters in Eq. 2. A parameter ex-\ntraction was performed on the S-parameters [11] of the\nsimulated structure to determine the effective permit-\ntivity and then a least square fit of the Drude-Lorentz\nfunction, Eq. 2, was used to determine the resonant fre-\nquency,ω0= 1//radicalbig\nǫ0µ0ggeomfgeom; the constant, F; and\nthe loss tangent, Γ c. In fact, CSRR structures (and\nmetamaterials in general) are not perfect Drude-Lorentz\nresonators as they exhibit spatial dispersion effects near\nresonance. These spatial dispersion effects were incor-\nporated into our fitting procedure [12] to determine the\nresonant frequency more accurately but then the simpler\nDrude-lorentz model was used in our analytic analysis\nwith good accuracy.\nOnce the empty CSRR structure was characterized,\nnumerical simulations incorporating the magnetic ma-\nterial were performed. In the interest of exploring the\npotential of incorporating high frequency (40-70 Ghz)\nthin film magnetic materials, such as hexagonal fer-\nrites [13], we considered a 1 µm thick layer of magnetic4Transmission (dB) 55 60 65 70 75 Frequency(Ghz) \n-20 -15 -10 -5 \n55 60 65 70 75 \n-20 -15 -10 -5 \n55 60 65 70 75 \n-20 -15 -10 -5 \n55 60 65 70 75 \n-20 -15 -10 -5 Bias = 0 kG \nBias = 3 kG Bias = 2 kG Bias = 1 kG HFSS Simulation \nAnalytic Solution 1a\n1a\n1a\n1a1b\n1b\n1b\n1b2a\n2a\n2a\n2a\nFIG. 3: The transmission results are shown for the HFSS\nelectromagnetic simulation and the analytic prediction us ing\neffective medium theory.\nmaterial with parameters (Eq. 4): ξ= 18 GhzkG−1,\nΓm= (0.70Ghz)/ωandξMs= 380Ghz. Thepermeabil-\nity was calculated in MATLAB and imported into HFSS\nas a data table of real values (Real[ µ]) and loss tangent\nvalues (Imag[ µ]/Real[µ]). As previously mentioned, it is\nonlythex-componentof µr(ω) thatcontributesto thein-\nteraction. Usingthefitted parametersextractedfromthe\nempty structure we solved the analytic equation, Eq. 6,\nfor the first hybrid mode, 1 a/1b, using a numerical root\nfinding method implemented in MATHEMATICA. The\nmagnetic filling fraction, q, was determined to have a\nvalue of 0.013 through comparison to the HFSS simula-\ntions results. A comparison of the analytic dispersion\ncurve calculated from Eq. 6 versus the numerical simu-\nlations performed with HFSS is shown in Fig. 2(b) and\ndemonstrates excellent agreement with variation of the\nbias field.\nThe transmission through the CSRR/magnetic mate-\nrialwaveguidecanbe calculatedanalyticallybyassigning\nbulk electromagnetic parameters to the waveguide struc-\nture. The effective bulk permittivity of the waveguide\nstructure can be calculated by inputing Eq. 6 into Eq. 2\nand then solving the Fresnel equations [11] to determine\nthe transmission and reflection of the structure. This\nwas done for the CSRR/magnetic waveguide structure\nand compared to HFSS simulations as shown in Fig. 3.\nIn the HFSS simulations a splitting of the second order\nCSRR resonance can also be seen. In the analytic ex-\npression we were able to reproduce this by applying our63 64 65 66 \nFrequency (Ghz) \nFrequency (Ghz) Relative Permeability ξMs=380 Ghz \nξMs=189 Ghz \nξMs=38 Ghz \n60 62 64 66 68 70 \n-5 510 20 30 40 50 60 70 \n-100 -50 50 100 Magnetic Material \nHybrid Structure zero point (a)\n(b)\nRelative Permittivity \nFIG. 4: (a) The real part of the permeability is shown for sev-\neral magnetization values of the magnetic material with the\nbiasfieldchosentohavethesame zeropermeability frequenc y.\n(b) The resulting hybrid permeability is shown.\ntechnique to the first (electric) and second (magnetic)\nresonance of the CSRR structure. The overall agreement\nis seen to correlate well over a range of biasing values.\nIt is remarkable to note that the width and strength\nof the hybrid resonances is primarily a function of the\nCSRR’s resonant properties. The properties of the mag-\nneticmaterial’sresonance(andfillingfraction q)predom-\ninantly determines the bandwidth over which the hybrid\nmode interaction exists. In Fig. 4 the permeability of the\nmagnetic material, Eq. 4, and the associated effective\npermeability of the hybrid structure are plotted for vari-\nous value of magnetization Ms. When the biasing field is\nchosen such that the zero point of the magnetic material\ncorrelates to the empty CSRR structure resonance, we\nsee that the relative permeability strength of the hybrid\nstructure is the same but that the splitting of the res-\nonance is smaller for smaller values of Msvalues. This\nsuggest that it may be possible to harness narrowly reso-\nnant magnetic materials, that cannot be utilized directly,\nby using this metamaterial approach (though only over\na narrow bandwidth). These potential materials include\nhexagonal ferrites and even antiferromagnetic materials\nsuch as MnF 2[14].\nIn summary, the hybrid resonance that results from\ncombining a resonant magnetic material and CSRR\nstructure results in a unique hybrid resonance which can\nbe harnessed to make tunable metamaterial structures.\nGyromagnetic materials have a resonant permeability\ntensor form similar to that considered here but with the\nadded complexity of off-diagonal resonant components.\nAs a result, the resonant response of the combined struc-\nture has a more complicated dependance on the orienta-\ntion of the biasing direction of the gyromagnetic mate-\nrial with respect to the geometry of the CSRR structure\n[15, 16]. Preliminary work suggest that if we limit our-5\nselves to using thin layers we should expect to see an\nanalogous phenomenon as demonstrated here. Though\ngyromagnetic materials have been used directly to make\ntunable microwave devices [17, 18, 19], using them in-\ndirectly in metamaterial structures has the potential ad-\nvantageofincreasingthe rangeofeffective materialprop-\nerties while at the same time reducing the amount of the\nmagnetic material needed in the structure and its asso-\nciated losses.\nThis research was supported by U.S. Army Research\nOffice DOA under Grant No. W911NF-04-1-0247. We\nalsoacknowledgesupportfromtheAirForceOfficeofSci-\nentific Research through a Multiply University Research\nInitiative under Contract No. FA9550-06-1-0279.\n[1] V. G. Veselago, Soviet Physics Uspekhi 10, 509 (1964).\n[2] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-\nNasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).\n[3] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J.\nStewart, IEEE MTT 47, 2075 (1999).\n[4] S. Lim, C. Caloz, and T. Itoh, IEEE Trans. on Microw.\nTheory. Tech. 52, 2678 (2004).\n[5] T. Hand and S. Cummer, J. of Appl. Phys. (2007).\n[6] C. Kittel, Introduction to Solid State Physics (Wiley &\nSons, Inc., 1996).[7] F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena,\nJ. Bonache, M. Beruete, R. Marqu´ es, F. Mart´ ın, and\nM. Sorolla, Phys. Rev. Lett. 93, 197401 (2004).\n[8] J. D. Baena, J. Bonache, F. mart´ ın, R. M. Sillero, F. Fal-\ncone, T. Lopetegi, M. A. G. Laso, J. Garc´ ıa-Garc´ ıa,\nI. Gil, M. F. Protillo, et al., IEEE MTT-S Digest 53\n(2005).\n[9] C. Caloz, C. C. Chang, and T. Itoh, J. Appl. Phys. 90,\n5483 (2001).\n[10] V. S. Liau, W. S. T. Wong, S. Ali, and E. Schloemann,\nIEEE MTT-S Digest p. 957 (1991).\n[11] D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis,\nPhys. Rev. B 65, 195104 (2002).\n[12] R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith,\nPhys. Rev. E 76, 26606 (2007).\n[13] H. L. Glass, Proc. IEEE 76, 151 (1988).\n[14] M. Lui, J. Drucker, A. R. King, J. P. Kotthaus, P. K.\nHansma, andV.Jaccarino, Phys.Rev.B 33, 7720(1986).\n[15] R. Marques, F. Mesa, and F. Medina, IEEE Microwave\nand Guided Wave Letters 10, 225 (2000).\n[16] M. Horno, F. L. Mesa, F. Medina, and R. Marques, IEEE\nTrans. Microwave Theory Tech. 38, 1059 (1990).\n[17] N. Cramer, D. Lucic, R. E. Camley, and Z. Celinski, J.\nAppl. Phys. 87, 6911 (2000).\n[18] R. J. Astalos and R. E. Camley, J. Appl. Phys. 83, 3744\n(1998).\n[19] R. E. Camley and D. L. Mills, J. Appl. Phys. 82, 3058\n(1997)." }, { "title": "1610.02550v1.Perpendicularly_magnetized_CoFeB_multilayers_with_tunable_interlayer_exchange_for_synthetic_ferrimagnets.pdf", "content": "arXiv:1610.02550v1 [cond-mat.mtrl-sci] 8 Oct 2016Perpendicularly magnetized CoFeB multilayers with tunabl e interlayer\nexchange for synthetic ferrimagnets\nP. Pirro,1,a)A. Hamadeh,1M. Lavanant-Jambert,1T. Meyer,2B. Tao,1E. Rosario,1Y. Lu,1M. Hehn,1S.\nMangin,1and S. Petit Watelot1\n1)Institut Jean Lamour, Université de Lorraine, UMR 7198 CNRS , 54506 Vandoeuvre-lès-Nancy,\nFrance\n2)Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern,\n67663 Kaiserslautern, Germany\n(Dated: 22 September 2018)\nA study of the multilayer system MgO/CoFeB(1.1nm)/Ta( t)/CoFeB(0.8nm)/MgO is presented, where the two\nCoFeB layers are separated by a Ta interlayer of varying thic knesst. The magnetization properties deduced\nfrom complementary techniques such as superconducting qua ntum interference magnetometry, ferromagnetic\nresonance frequency measurements and Brillouin light scat tering spectroscopy can be tuned by changing the\nTa thickness between t=0.25 nm, 0.5 nm and 0.75 nm. For t=0.5 nm, a ferromagnetic coupling is observed,\nwhereas for t=0.75 nm, the antiferromagnetic coupling need ed to construct a synthetic ferrimagnet is realized.\nIn the later case, the shape of magnetic domain walls between two ferrimagnetic alignments or between a\nferro- and a ferrimagnetic alignment is very different. This behavior can be interpreted as a result of the\nchange in dipolar as well as interlayer exchange energy and d omain wall pinning, which is an important\nconclusion for the realization of data storage devices base d on synthetic ferri- and antiferromagnets.\nI. INTRODUCTION\nThe demand to develop faster and more reliable mag-\nnetic memories with large data storage capabilities has\nlead to the proposition of the racetrack memory concept1,\nwhere data is stored in form of domain walls in magnetic\nnanowires. Since these domain walls can be moved via\nspin torque effects to the magneto-resistive readout ele-\nments, this concept allows for a three-dimensional data\nstorage device. However, dipolar effects like, e.g., the\nWalker breakdown limit the velocity and storage den-\nsity of magnetic domain walls in single layer wires. To\novercome this limitation, the use of synthetic ferrimag-\nnets (SFI) with perpendicularly-to-plane magnetized lay-\ners has proven to be promising2,3. Thus, the development\nand careful characterization of suitable SFI systems with\nlow magnetic damping to reach high domain wall speeds\nand high device efficiencies is of paramount interest.\nIn this context, we present a study of the mul-\ntilayer system MgO(2.5nm)/Co 40Fe40B20(1.1nm\n)/Ta(t)/Co 40Fe40B20(0.8nm)/MgO(2.5nm) (all nominal\nlayer thicknesses), where the two CoFeB layers are\nseparated by a Ta interlayer4,5of varying thickness\noft=0.25, 0.5 and 0.75 nm. This system is directly\ngrown on a GaAs substrate and can thus easily be\nintegrated into semiconductor based devices, e.g. spin\nLED structures6,8. In the investigated layer stack, the\nhybridization of the 3d orbitals of the transition metals\nwith the O2p orbitals of MgO provides the perpendicular\nmagnetic anisotropy (PMA). To establish PMA, thermal\nannealing is needed to crystalline the CoFeB starting\na)currently at: Fachbereich Physik and Landesforschungszen trum\nOPTIMAS, Technische Universität Kaiserslautern, 67663 Ka iser-\nslautern, Germanyfrom the CoFeB/MgO interface. During this process,\nthe diffusing boron is absorbed by the Ta interlayer. In\naddition, Ta provides an RKKY-like coupling5,9which\nis needed to create a SFI.\nUsing a superconducting quantum interference de-\nvice SQUID, magneto-optical Kerr effect microscopy\n(MOKE), inductive ferromagnetic resonance frequency\n(FMR) measurements and Brillouin light scattering spec-\ntroscopy (BLS), the magnetic properties of the systems\nhave been investigated. The experimental results are\nconsistently reproduced by a simple macrospin model.\nThe interlayer exchange coupling as well as the mag-\nnetic properties of the two individual CoFeB layers vary\nstrongly with the Ta thickness t. Fort=0.75 nm, a SFI\nwith an antiferromagnetic (AFM) exchange coupling is\nobserved and the nucleation of domain walls in between\nthe two ferrimagnetic configurations as well as in between\nthe ferri- and the ferromagnetic (FM) configuration are\nrealized.\nII. SAMPLE PREPARATION\nThe investigated multilayer stacks are deposited on\nGaAs substrates which are first desorbed in a MBE\nchamber at 300◦C by monitoring reflection high energy\nelectron diffraction patterns to remove the As capping\nlayer, which is used to passivate the surface of the sub-\nstrate. Then, the samples are transferred to the sputter-\ning chamber without breaking the vacuum to deposit the\nmultilayer at room temperature. The magnetization of\nthe samples prior to an annealing is in-plane, as usual for\nTa/CoFeB/MgO systems. To establish PMA, the sam-\nples were annealed at 250◦C for 3 mins in a rapid ther-\nmal annealing system (for details, see6. Prior studies of a\nsimilar system have show that a magnetically dead layer2\ncan be formed at the Ta/CoFeB interface7. According\nto6, the effective magnetic CoFeB layer thickness dis sig-\nnificantly reduced compared to the nominally deposited\nthickness if t≥0.5nm. The estimated values for dbased\non the findings in6are given in Table I.\nIII. SQUID MEASUREMENTS\nFigure 1 shows the magnetization along the field axis\nas a function of an applied out-of-plane (Fig.1 (a)-(c))\nand in-plane field (Fig.1 (d)-(e)) as obtained from SQUID\nmeasurements at 300 K. The sample with t=0.25 nm of\nTa shows an in-plane easy axis with a small coercive field\nof approximately 1 mT and a hysteresis similar to a single\nlayer, thus the two layers of CoFeB seem to be strongly\nFM coupled, either by strong RKKY-like exchange or by\na direct coupling due to a discontinuous Ta layer. The\nPMA of this system is not sufficient to overcome the de-\nmagnetization field and to consequently orient the mag-\nnetization out of plane. For t=0.5 nm, a single hysteresis\nloop with small coercivity for both field directions is ob-\nserved, indicating that either one of the layers has an\nin-plane easy axis and the other one an out-of-plane easy\naxis, or a multidomain state is occurring in remanence\n(2 mT coercivity field for both directions, not visible in\nFig. 1 for the in-plane field due to the scaling). Since\nthe hysteresis loops are both centered around zero field,\nwe conclude that the coupling between the layers is FM.\nFort=0.75 nm, a clear three-step hysteresis is observed\nfor the out-of-plane field (see also Fig. 3(a)), whereas for\na field in plane a continuous transition without any de-\ntectable hysteresis is revealed. Thus, in this case, both\nlayers have an out-of-plane easy axis and are AFM cou-\npled, which leads to the two additional hysteresis loops\ncentered around BRKKY=±20 mT.\nTo achieve a more quantitative characterization, the\nvalues of the saturation magnetization Msfrom the\nSQUID measurements have been extracted. To cal-\nculateMsfrom the measured magnetic moment per\narea, the effective magnetic thicknesses dgiven in Ta-\nble I have been used. To For t=0.25 nm, a saturation\nmagnetization averaged over both layers of Ms=1019\nkA/m is measured, whereas for t=0.5 nm, Ms= 1063\nkA/m is obtained. In the case of the AFM coupling\nfort=0.75 nm, assuming that the smaller magnetic mo-\nment can be attributed to the thinner layer, a satura-\ntion magnetization of M1\ns=1006 kA/m for the lower layer\nandM2\ns=973 kA/m for the upper layer (thus averaged\nMs=993 kA/m) can be deduced.\nIV. MACROSPIN MODELING\nTo access further material parameters, we model the\ntotal energy Etotof the system including Zeeman en-\nergyEZee, demagnetization energy EDM, uniaxial magne-\ntocrystalline anisotropy energy EAniwith the anisotropy-400 -200 0 200 400-1000-50005001000\n-100 -50 0 50 100 -40 -20 0 20 40(b) (c) (a)\nTa 0.75 nm Ta 0.25 nm Ta 0.5 nm\n-20 -10 0 10 20-1000-50005001000\n-600 -300 0 300 600 -600 -300 0 300 600(e) (d) (f)Mop(kA/m)\nBop(mT) Bop(mT) Bop(mT)\nMip(kA/m)\nBip(mT) Bip(mT) Bip(mT)Ta 0.75 nm Ta 0.25 nm Ta 0.5 nm\nFIG. 1. (color online) Magnetization as a function\nof the applied field along the out-of-plane plane direc-\ntion (a-c) and along the in-plane direction (d-f) for\nCoFeB(1.1 nm)/Ta( t)/CoFeB(0.8 nm) with t= 0.25nm (a,d),\nt= 0.5nm (b,e) and t= 0.75nm (c,f). (c) shows that for\nt=0.75 nm, the magnetization of both layers is out of plane\nwith a three step hysteresis loop indicating an AFM coupling\nwith a direct transition between the two ferrimagnetic stat es.\neasy axis ualong the out-of-plane direction and the in-\nterlayer exchange energy ERKKY between the two layers:\nEtot=i=1,2/summationdisplay\nlayer[EZee(Mi)+EDM(Mi)+EAni(Mi)]\n+ERKKY(M1,M2) (1)\nEZee(Mi) =−Mi·B·Vi (2)\nEDM(Mi) =−µ0\n2(Mi·u)2·Vi (3)\nEAni(Mi) =−(m·u)2·Ks\ni·S (4)\nERKKY(Mi) =−(m1·m2)·J·S (5)\nwith the normalized magnetization m=M\nMs, the out-of-\nplane unit vector u, the interlayer exchange coupling J\nand the volume Vi=S·di, which depends on the surface\nSand effective thickness diof the individual CoFeB layer.\nIn the case of AFM coupling ( t=0.75 nm), the switching\nfrom the ferromagnetic to the ferrimagnetic states takes\nplace at the field BRKKY, so we can conclude that at\nthis field EZee=ERKKY, which leads to the estimate of\nthe exchange constant J=−M2·BRKKY·d2≈ −0.011\nmJ/m2, which is on the same order of magnitude than J\nof Ref.5,9. Furthermore, we reconstructed the hard axis\nSQUID loops by numerically minimizing Etot(M1,M2)\nas a function of the applied field and subsequently ex-\ntracting the average magnetization along the field (blue3\ntd1d2M1M2Ks\n1Ks\n2J\n(nm) (nm) (kA/m) (mJ/m2)\n0.25 1.1 0.8101910190.54 0.54≥0.4\n0.50.975 0.675 106310630.690.565 0.045\n0.75 0.84 0.54 1006 9730.580.385 -0.011\nTABLE I. Overview of the obtained material parameters. The\nvalues for MandKsas well as the value for the interfacial\nexchange coupling Jfort=0.75 nm have been obtained from\nthe SQUID measurements, whereas Jin the case of t=0.25nm\nand 0.5 nm are estimates based on the FMR frequency mea-\nsurements. The effective magnetic thickness ddepends on\nthe Ta interlayer thickness t. If a distinction between the two\nlayers has not been possible, an average value is given.\ncurves in Fig. 1(a),(e) and (f)). From this reconstruc-\ntions, the magnetocrystalline surface anisotropy con-\nstantsKs\nihave been obtained which are listed in Table I.\nV. FMR EXPERIMENTS AND MODELING\nTo check the validity of the obtained material constants\nand to estimate the exchange coupling for the thicknesses\ntwith FM coupling, measurements of the frequencies of\nthe small angle precession eigenmodes have been per-\nformed (see Fig. 2). The external magnetic field has\nbeen applied along the in-plane axis and the eigenmode\nfrequencies have been obtained by standard FMR mea-\nsurements using a vector network analyzer (VNA). In\naddition, for the case of AFM coupling t=0.75 nm, the\nthermal spin-wave spectrum with a wave vector k→0\nequivalent to the ferromagnetic resonances has been ac-\nquired by Brillouin light scattering spectroscopy11,12.\n0 200 400 60005101520\n0 200 400 60005101520\n0 200 400 60005101520\nFMR\nModel\nBip(mT)FMR-VNA\nModel\nModel 1°FMR-VNA\nBLS\nModel\nModel 1°\n(c) (b) (a)\nBip(mT) Bip(mT)f(GHz)\nFIG. 2. (color online) Measurements of the ferromagnetic re s-\nonance frequencies (red dots) together with the eigenfrequ en-\ncies predicted by the macrospin model (blue and black lines)\nusing the material parameters presented in Table I obtained\nfrom SQUID. Black curves represent the macrospin model for\na perfect in-plane field whereas blue curves assume a small\nout-of-plane tilt of 1◦.\nTo compare the experimental results with our\nmacrospin model, we transform all energy expressions in\na new base given by the equilibrium magnetizations me\n1andme\n2. In this base, we calculate the effective fields\nBeff\ni=−1\nVi·Mi∇me\niEtot\ni(me\n1,me\n2) (6)\nand subsequently linearize the two coupled Landau-\nLifshitz equations:\ndme\n1\ndt=−γme\n1×Beff\n1,dme\n2\ndt=−γme\n2×Beff\n2 (7)\nwithme\nz→1 (external field axis along z) and solve the\nsystem numerically as an eigenvalue problem. The fre-\nquencies of two modes are obtained, which correspond\nto the in-phase and out-of-phase oscillations of the two\nlayers when assuming two identical layers.\nFort=0.25 nm (Fig. 2(a)), only one mode within the\nexperimental accessible range from 1 to 20 GHz has been\nfound, again indicating a strong coupling between the\nlayers. The blue curve shows the frequency evolution\npredicted by the macrospin model when using the pa-\nrameters obtained from the SQUID measurement with\nan exchange coupling J= 0.4mJ/m2. This is the min-\nimal value of Jneeded to shift the second mode above\n20 GHz and thus constitutes the lower boundary for the\ncoupling. A good agreement between the model and the\nexperiment is observed and the small deviation is proba-\nbly due to a residual misalignment of the applied fields.\nFort=0.5 nm (Fig. 2(b)), two modes with a frequency\nspacing of about 5 GHz have been observed. Using an\nexchange coupling of J=0.045 mJ/m2and the parame-\nters obtained from the SQUID, the macrospin model has\nbeen evaluated for a perfect in-plane alignment of the\nexternal field (dotted black curve) and for a small out-\nof-plane tilt of the external field of 1◦(continuous blue\nline). The tilt of the field influences especially the local\nfrequency minimum close to 200 mT. Since the trend for\nthe tilted field reproduces the experimentally obtained\nvalues more accurately, a slight misalignment from the\nin-plane axis during the FMR experiment can be con-\ncluded. No clear susceptibility peaks have been observed\nfor fields below the in-plane saturation field of 200 mT\nindicating an inhomogeneous magnetization state which\ncannot be described by the macrospin model.\nFort=0.75 nm (Fig. 2(c)), again two modes have been\nobserved. Here, with the macrospin model, we only use\nvalues obtained from the SQUID measurements. The\nmodel nicely reproduces the experimental trend if we al-\nlow for a 1◦misalignment of the external applied field\n(continuous blue curve). A perfect alignment of the field\nalong the in-plane direction would again lead to a pro-\nnounced frequency minimum, which has not been ob-\nserved (dotted black curve). Due to the smaller inter-\nlayer coupling J, the frequency gap which is occurring at\nabout 250 mT with an opening of about 800 MHz is much\nsmaller than in the t=0.5 nm case. Due to the compa-\nrably large frequency linewidth of more than 1 GHz, the\ngap could not be observed in the FMR and BLS exper-\niments. Since the frequency linewidth shows no system-\natic frequency dependence, the underlying broadening4\nmechanism is probably arising from an inhomogeneous\ndistribution of the material parameters.\nThe good general agreement between the experimen-\ntally obtained frequencies and the calculated eigenfre-\nquencies based on the material parameters of the SQUID\nmeasurements demonstrates that a comprehensive and\nself-consistent characterization of the system has been\nachieved.\nVI. MOKE MICROSCOPY\nIn the following, we will address the possible domain\nconfigurations in the case of the AFM coupling of the\nlayers (t=0.75 nm), since this configuration is especially\npromising for future high density storage elements2. The\nnucleation of DWs was studied experimentally in the\npresence of an out-of-plane magnetic field. Figure 3\npresents magneto-optical Kerr microscopy (MOKE) hys-\nteresis loops and images showing the nucleation of a do-\nmain in the different configurations of the two magneti-\nzations in the thinner upper and thicker lower magnetic\nlayer. In Fig. 3(a), the magnetic configurations are de-\npicted for the different levels of the MOKE signal by a\npair of arrows. Three jumps are observed in the hys-\nteresis loop, similar to the observations using SQUID.\nStarting from positive fields, the first step (marked by\n(b) in Fig. 3(a)) corresponds to the switching of the up-\nper thinner layer to minimize the interlayer exchange en-\nergy, which becomes more important than the Zeeman\nenergy at this field. Thus, a transition from the paral-\nlel (P) to the antiparallel (AP) state occurs. The sec-\nond step (marked by (c)) corresponds to the switching of\nboth layer’s magnetization, so a transition between the\ntwo AP states is taking place. Here, the Zeeman energy\nis reduced by aligning the residual magnetic moment of\nthe AP state along the external field, while the interlayer\nexchange energy is unaffected. With further decreasing\nfield, the Zeeman energy reaches again the value of the\ninterlayer exchange and a switching to the P state occurs\n(around -25 mT in Fig. 3(a)). In contrast to the SQUID\nmeasurement, a minor hysteresis loop around zero field\nwith a clear separation between the two AP states is ob-\ntained using MOKE. We attribute this behavior to the\ndifferent time scales of the measurements: MOKE is sig-\nnificantly faster, so the observed coercivity increases as\nthe system has less time to undergo a thermal activated\nswitching.\nFigures 3(b)-(e) present MOKE images after the nu-\ncleation of domains at different external fields around a\ndefect in the thin film. The nucleation takes place at\nthe transitions indicated in Fig. 3(a), whereas the fields\nwhere the domains are stabilized and imaged are indi-\ncated below the pictures. The MOKE contrast is deter-\nmined by the net magnetization direction of the domains\naveraged over the two layers. The orientation of the mag-\nnetization is again indicated by arrows.\nTwo magnetic domains are depicted in Figure 3(c),\n100 µm(d)\n (e)\n(c)\n24 mT(b)(a)\n(b)\n(c)\n(e)\n(d)\n-2.9 mT\n-24 mT 1.2 mTMOKE signal (a.u.)\nBop(mT)-50 -40 -30 -20 -10 0 10 20 30 40 50-0.04-0.020.000.020.04\nFIG. 3. (color online) (a) Hysteresis loop ob-\ntained from MOKE microscopy for AFM coupling in\nCoFeB(1.1nm)/Ta(0.75nm)/CoFeB(0.8nm). The arrows\nindicate the alignment of the two layers for the four differen t\nlevels of the MOKE signal. (b)-(e) MOKE microscopy images\nof different domain configurations which have been initially\nnucleated at the transition fields marked in (a) and imaged\nat the indicated field values.\neach of which includes MOKE contributions from the\nupper and lower magnetic layer. In the bright domain,\nthe magnetization is oriented down in the lower layer\nand up in the upper layer. In the darker surroundings,\nthe magnetic orientations of the two layers are both in-\nverted. Consequently, a domain wall is formed between\nthe two AP states which can be considered as a domain\nwall in a synthetic ferrimagnet. In Fig. 3(e), the role of\nthe domain around the defect and the surroundings is\njust inverted due to the different sign of the nucleation\nfield. For higher fields, the domain configurations shown\nin Fig.3(b) and (d) are stabilized in the center of the\nouter hysteresis loops and show both a domain with AP\nalignment of the two layers surrounded by an area with\nP alignment. Thus, depending on the external field, two\ndifferent kinds of domain walls can be nucleated in this\nsystem.\nFinally, it is clear that the shape of the domain de-\npends on the configuration between the upper and lower\nmagnetic layer. When the transition occurs between the5\nP and AP orientation (Fig. 3(b) and (d)), the domain\nwall shows a zigzag shape. For the transitions between\ntwo AP orientations of the layers (Fig. 3(c) and (e)),\nthe domain grows as a circle. The difference of these\nshapes might be attributed to a combination of different\neffects which all can lead to distinct domain wall dy-\nnamics for the two configurations: first, in the P to AP\ntransition, a domain wall is present only in the upper\nlayer. Thus, it could experience different pinning com-\npared to the domain wall between the two AP states,\nwhich is also present in the lower layer. In addition, the\ndipolar stray fields of a domain with P configuration are\nmuch stronger than those generated in the AP configu-\nration, which leads to different contributions of dipolar\nand interlayer exchange energy in the two cases.\nTo conclude, we have demonstrated that\nCoFeB/Ta/CoFeB multilayers grown on GaAs/MgO\npresent strong perpendicular magnetocrystalline\nanisotropy. The magnetization properties could be\ntuned by changing the Ta thickness. A 0.5 nm Ta\nthickness leads to a ferromagnetic coupling whereas a\n0.75 nm Ta thickness leads to a synthetic ferrimagnetic\nbilayer with an antiferromagnetic interlayer coupling.\nIn the later case, the shape of magnetic domain walls\nbetween two ferrimagnetic alignments and between a\nferromagnetic and a ferrimagnetic alignment is very\ndifferent, possibly due to the change in dipolar and\ninterlayer exchange energy as well as domain wall pin-\nning. Our results show that with the proper interlayer\nthickness, the CoFeB/Ta/CoFeB system is a promising\ncandidate for the realization of data storage devices\nbased on synthetic ferrimagnets.\nACKNOWLEDGMENTS\nThis work was supported by the ANR-NSF Project,\nANR-13-IS04-0008-01, COMAG by the ANR-LabcomProject LSTNM and by the Université de la Grande\nRegion (UniGR funded P. Pirro Post-Doc). Y. Lu also\nacknowledges the support by the joint French National\nResearch Agency (ANR)-National Natural Science Foun-\ndation of China (NSFC) SISTER project (Grants No.\nANR-11-IS10-0001 and No. NNSFC 61161130527) and\nENSEMBLE project (Grants No. ANR-14-0028-01 and\nNo. NNSFC 61411136001). Experiments were performed\nusing equipment from the TUBE - Daum funded by\nFEDER (EU), ANR, the Region Lorraine, and Grand\nNancy.\n1S. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008).\n2S.-H. Yang, K.-S. Ryu, and S. Parkin, Nature Nanotech 10, 221\n(2015).\n3M. Kuteifan, M.V. Lubarda, S. Fu, R. Chang, M.A. Escobar, S.\nMangin, E. Fullerton, and V. Lomakin, arXiv 1406, 3253 (2014 ).\n4C.-W. Cheng, T.-I. Cheng, C.H. Shiue, C.-L. Weng, Y.-C. Tsai ,\nand G. Chern, IEEE T Magn 49, 4433 (2013).\n5C.-W. Cheng, C.H. Shiue, T.-I. Cheng, and G. Chern, J. Appl.\nPhys.112, 033917 (2012).\n6B. S. Tao, P. Barate, J. Frougier,P. Renucci, B. Xu, A. Djeffal , H.\nJaffre‘s, J.-M. George, X. Marie, S. Petit-Watelot, S. Mangi n, X.\nF. Han, Z. G. Wang, and Y. Lu, Appl. Phys. Lett. 108, 152404\n(2016).\n7S. Y. Jang, C.-Y. You, S. H. Lim, and S. R. Lee, J. Appl. Phys.\n109, 013901 (2011).\n8S.H. Liang, T.T. Zhang, P. Barate, J. Frougier, M. Vidal, P.\nRenucci, B. Xu, H. Jaffres, J.M. George, X. Devaux, M. Hehn, X.\nMarie, S. Mangin, H.X. Yang, A. Hallal, M. Chshiev, T. Amand,\nH.F. Liu, D.P. Liu, X.F. Han, Z.G. Wang, and Y. Lu, Phys Rev\nB90, 085310 (2014).\n9S. S. P. Parkin, Phys. Rev. Lett. 67, 3598 (1991).\n10R.R. Gareev, V. Zbarsky, J. Landers, I. Soldatov, R. Schäfer ,\nM. Münzenberg, H. Wende, and P. Grünberg, Appl. Phys. Lett.\n106, 132408 (2015).\n11T. Sebastian, Y. Kawada, B. Obry, T. Brächer, P. Pirro, D.A.\nBozhko, A.A. Serga, H. Naganuma, M. Oogane, Y. Ando, and\nB. Hillebrands, J. Phys. D Appl. Phys. 48, 164015 (2015).\n12J.V. Harzer, B. Hillebrands, R.L. Stamps, G. Güntherodt, C. D.\nEngland, and C.M. Falco, J. Appl. Phys. 69, 2448 (1991)." }, { "title": "1810.00584v2.Stabilizing_Mechanism_for_Bose_Einstein_Condensation_of_Interacting_Magnons_in_Ferrimagnets_and_Ferromagnets.pdf", "content": "Stabilizing Mechanism for Bose-Einstein Condensation of Interacting Magnons\nin Ferrimagnets and Ferromagnets\nNaoya Arakawa\u0003\nDepartment of Physics, Toho University, Funabashi, Chiba, 274-8510, Japan\n(Dated: November 5, 2018)\nWe propose a stabilizing mechanism for the Bose-Einstein condensation (BEC) of interacting\nmagnons in ferrimagnets and ferromagnets. By studying the e\u000bects of the magnon-magnon interac-\ntion on the stability of the magnon BEC in a ferrimagnet and two ferromagnets, we show that the\nmagnon BEC remains stable even in the presence of the magnon-magnon interaction in the ferrimag-\nnet and ferromagnet with a sublattice structure, whereas it becomes unstable in the ferromagnet\nwithout a sublattice structure. This indicates that the existence of a sublattice structure is the\nkey to stabilizing the BEC of interacting magnons, and the di\u000berence between the spin alignments\nof a ferrimagnet and a ferromagnet is irrelevant. Our result can resolve a contradiction between\nexperiment and theory in the magnon BEC of yttrium iron garnet. Our theoretical framework may\nprovide a starting point for understanding the physics of the magnon BEC including the interaction\ne\u000bects.\nBose-Einstein condensation (BEC) has been exten-\nsively studied in various \felds of physics. The BEC is\na macroscopic occupation of the lowest-energy state for\nbosons [1]. This phenomenon was theoretically predicted\nin a gas of noninteracting bosons [2], and then it was ex-\nperimentally observed in dilute atomic gases [3{5]. This\nobservation opened up research of the BEC in atomic\nphysics [1]. Since the concept of the BEC is applicable\nto quasiparticles that obey Bose statistics, research of the\nBEC has been expanded, and it covers condensed-matter\nphysics, nuclear physics, and optical physics.\nThere is a critical problem with the magnon BEC.\nThe magnon BEC was experimentally observed in yt-\ntrium iron garnet (YIG), a three-dimensional ferrimag-\nnet [6{9]. However, a theory [10] showed that if low-\nenergy magnons of YIG are approximated by magnons of\na ferromagnet without a sublattice structure, the magnon\nBEC is unstable due to the attractive interaction between\nmagnons. Note \frst, that YIG is often treated as the\nferromagnet for simplicity of analyses [11, 12], second, in\ngeneral, the attractive interaction between bosons desta-\nbilizes the BEC [13, 14]. Thus the stabilizing mechanism\nfor the BEC of interacting magnons in a ferrimagnet re-\nmains unclear. To clarify it, we should understand the\ninteraction e\u000bects in a ferrimagnet. In addition, we need\nto understand the essential e\u000bects of the di\u000berences be-\ntween a ferrimagnet and the ferromagnet in order to un-\nderstand the reason for the contradiction between exper-\niment [6{9] and theory [10].\nIn this Letter, we study the interaction e\u000bects on the\nmagnon BEC in three magnets and propose a stabiliz-\ning mechanism. We use the Heisenberg Hamiltonian and\nconsider a ferrimagnet and two ferromagnets. By using\nthe Holstein-Primako\u000b transformation [15{17], we derive\nthe kinetic energy and interaction for magnons. Then,\nwe construct an e\u000bective theory to study the interaction\ne\u000bects on the magnon BEC in a similar way to the Bo-\ngoliubov theory [14, 18] for Bose particles. By combiningthe results for the three magnets, we show that the exis-\ntence of a sublattice structure, not the di\u000berence in the\nspin alignment, is the key to the stabilizing mechanism\nfor the BEC of interacting magnons. We also discuss the\ncorrespondence between our model and a more realistic\nmodel of YIG and several implications.\nWe use the Heisenberg Hamiltonian as a minimal\nmodel for ferrimagnets and ferromagnets. It is given by\nH= 2X\nhi;jiJijSi\u0001Sj; (1)\nwhereJijdenotes the Heisenberg exchange energy be-\ntween spins at nearest-neighbor sites, and Sidenotes the\nspin operator at site i.\nWe consider three cases. In the \frst case, we put\nJij=J,hSii=SAfori2A, andhSii=\u0000SBfori2B,\nwhereAandBdenoteAandBsublattices, respectively;\neach sublattice consists of N=2 sites. This case corre-\nsponds to a ferrimagnet with a two-sublattice structure\n[Fig. 1(a)]. In the second case, we put Jij=\u0000Jand\nhSii=Sfor alli's. In the third case, we put Jij=\u0000J,\nhSii=SAfori2A, andhSii=SBfori2B. The sec-\nond and third cases correspond to ferromagnets without\nsublattice and with a two-sublattice structure, respec-\ntively [Figs. 1(b) and 1(c)]. As we will show below, by\nstudying the BEC of interacting magnons in these three\ncases, we can clarify the stabilizing mechanism in a fer-\nrimagnet and the key to resolving the contradiction in\nthe magnon BEC of YIG. (We will focus mainly on the\nsign of the e\u000bective interaction between magnons and its\ne\u000bect on the stability of the magnon BEC.)\nWe begin with the \frst case of our model. We \frst\nderive the magnon Hamiltonian by using the Holstein-\nPrimako\u000b transformation [15{17]. After remarking on\nseveral properties in the BEC of noninteracting magnons,\nwe construct the e\u000bective theory for the BEC of interact-\ning magnons. By using this theory, we study the inter-\naction e\u000bects in the ferrimagnet.arXiv:1810.00584v2 [cond-mat.mes-hall] 2 Nov 20182\n\tB\n \tC\n \tD\nFIG. 1. Spin alignments on a plane of the cubic lattice in the three cases of our model; panels (a), (b), and (c) correspond\nto the \frst, second, and third cases, respectively. The direction and length of an arrow represent the direction and size of an\nordered spin. The ordered spins are ferrimagnetic in panel (a) and ferromagnetic in panels (b) and (c); sublattice degrees of\nfreedom are present in panels (a) and (c) and absent in panel (b).\nThe magnon Hamiltonian is obtained by applying the\nHolstein-Primako\u000b transformation to the spin Hamilto-\nnian. In general, low-energy excitations in a magnet can\nbe described well by magnons, bosonic quasiparticles [15{\n17, 19{23]. The magnon operators and the spin opera-\ntors are connected by the Holstein-Primako\u000b transforma-\ntion [15{17]. This transformation for our ferrimagnet is\nexpressed as follows:\nSz\ni=SA\u0000ay\niai; S\u0000\ni=p\n2SAay\nir\n1\u0000ay\niai\n2SA;(2)\nSz\nj=\u0000SB+by\njbj;S+\nj=p\n2SBby\njr\n1\u0000by\njbj\n2SB;(3)\nwherei2A,j2B,S\u0000\ni=Sx\ni\u0000iSy\ni= (S+\ni)y, and\nS+\nj=Sx\nj+iSy\nj= (S\u0000\nj)y;aianday\niare the operators of\nmagnons for the Asublattice, and bjandby\njare those for\ntheBsublattice. A substitution of Eqs. (2) and (3) into\nEq. (1) gives the magnon Hamiltonian.\nIn the magnon Hamiltonian, we consider the kinetic\nenergy terms and the dominant terms of the magnon-\nmagnon interaction. This is because our aim is to clarify\nhow the magnon-magnon interaction a\u000bects the magnon\nBEC, which is stabilized by the kinetic energy terms.\nSince the kinetic energy terms come from the quadratic\nterms of magnon operators and the dominant terms of the\ninteraction come from part of the quartic terms [16, 17],\nour magnon Hamiltonian is given by Hmag=Hnon+\nHint[24], where\nHnon= 2X\nqJ(0)(SBay\nqaq+SAby\nqbq)\n+2X\nqJ(q)p\nSASB(aqbq+ay\nqby\nq); (4)\nand\nHint=\u00002\nNP\nq;q0[J(0)ay\nqaqby\nq0bq0+J(q\u0000q0)ay\nqaq0by\nqbq0\n+J(q)pSASB(SAaqby\nq0bqbq0+SBbqay\nq0aq0aq)] + (H.c.):(5)\nWe have used ai=q\n2\nNP\nqeiq\u0001iaq,by\nj=q\n2\nNP\nqeiq\u0001jby\nq, andJ(q) =P\n\u000eJeiq\u0001\u000ewith\u000e, a\nvector to nearest neighbors.\nBefore formulating the e\u000bective theory for the BEC of\ninteracting magnons, we remark on several properties in\nthe BEC of noninteracting magnons in our ferrimagnet.\nTo see the properties, we diagonalize Hnonby using\n\u0012aq\nby\nq\u0013\n=\u0012cq\u0000sq\n\u0000sqcq\u0013\u0012\u000bq\n\fy\nq\u0013\n; (6)\nwherecq\u0011cosh\u0012qandsq\u0011sinh\u0012qsatisfy tanh 2 \u0012q=\n2pSASBJ(q)\n(SA+SB)J(0). After some algebra, we obtain\nHnon=X\nq\u000f\u000b(q)\u000by\nq\u000bq+X\nq\u000f\f(q)\fy\nq\fq; (7)\nwhere\u000f\u000b(q) = (SB\u0000SA)J(0) + \u0001\u000f(q) and\n\u000f\f(q) = (SA\u0000SB)J(0) + \u0001\u000f(q) with \u0001\u000f(q) =p\n(SA+SB)2J(0)2\u00004SASBJ(q)2; in Eq. (7) we have\nneglected the constant terms. Hereafter, we assume\nSA> SB; this does not lose generality. For SA> SB\n\u000f\u000b(0) = 0 is the lowest energy. Thus many magnons\noccupy theq= 0 state of the \u000bband in the BEC of non-\ninteracting magnons in the ferrimagnet for SA>SB. In\naddition, the low-energy excitations from the condensed\nstate are described by the \u000b-band magnons near q=0.\nWe now construct the e\u000bective theory for the BEC of\ninteracting magnons. To construct it as simple as possi-\nble, we utilize the properties in the BEC of noninteract-\ning magnons. As described above, in the ferrimagnet for\nSA> SBthe condensed state is the q=0state of the\n\u000bband and the low-energy noncondensed states are the\nsmall-qstates of the \u000bband. Thus we can reduce Hmag\nto an e\u000bective Hamiltonian He\u000b, which consists of the ki-\nnetic energy term of the \u000bband and the intraband terms\nof the magnon-magnon interaction for the \u000bband;He\u000b\nis given by He\u000b=H0+H0, whereH0is the \frst term of\nEq. (7), and H0is obtained by substituting Eq. (6) into\nEq. (5) and retaining the intraband terms. This He\u000bis\nsu\u000ecient for studying properties of the BEC of interact-\ning magnons at temperatures lower than a Curie tem-\nperature, because the dominant excitations come from3\nthe small-qmagnons in the \u000bband and the interband\nterms may be negligible in comparison with the intra-\nband terms. Then we can further simplify H0. Since its\nmain e\u000bects can be taken into account in the mean-\feld\napproximation, the leading term of H0is given by [24]\nH0=\u00004\nNX\nq;q0\u0000\u000b\u000b(q;q0)nq0\u000b\u000by\nq\u000bq; (8)\nwhere \u0000\u000b\u000b(q;q0) =J(0)(c2\nqs2\nq0+c2\nq0s2\nq) + 2J(q\u0000\nq0)cqsqcq0sq0\u0000J(q)pSASBcqsq(SAs2\nq0+SBc2\nq0)\u0000\nJ(q0)pSASBcq0sq0(SAs2\nq+SBc2\nq), andnq0\u000b=h\u000by\nq0\u000bq0i=\nn[\u000f\u000b(q0)] with the Bose distribution function n(\u000f). By\ncombining Eq. (8) with H0=P\nq\u000f\u000b(q)\u000by\nq\u000bq, we obtain\nHe\u000b=X\nq\u000f\u0003\n\u000b(q)\u000by\nq\u000bq; (9)\nwith\u000f\u0003\n\u000b(q) =\u000f\u000b(q)\u00004\nNP\nq0\u0000\u000b\u000b(q;q0)nq0\u000b.\nBy using the theory described by He\u000b, we study\nthe interaction e\u000bects on the stability of the magnon\nBEC. Since the magnon energy should be nonnegative,\nthe magnon BEC remains stable even for interacting\nmagnons as long as \u000f\u0003\n\u000b(0) is the lowest energy. This\nis realized if H0is the repulsive interaction. If H0is\nthe attractive interaction, the magnon BEC becomes un-\nstable. Thus we need to analyze the sign of \u0000 \u000b\u000b(q;q0)\nin Eq. (8). Since the dominant low-energy excitations\nare described by the \u000b-band magnons near q=0, we\nestimate \u0000 \u000b\u000b(q;q0) in Eq. (8) in the long-wavelength\nlimitsjqj;jq0j! 0. For a concrete simple example we\nperform this estimation in a three-dimensional case on\nthe cubic lattice. By expressing J(q) in a Taylor series\naroundjqj= 0 and retaining the leading correction, we\ngetJ(q)\u0019J(0)[1\u0000q2\n6]. Then, by using this expression\nand performing some calculations [24], we obtain the ex-\npression of \u0000 \u000b\u000b(q;q0) including the leading correction in\nthe long-wavelength limits. The derived expression is\n\u0000\u000b\u000b(q;q0)\u0019\u00002\n9J(0)q2q02(SASB)2\n(SA\u0000SB)4: (10)\nThe combination of Eqs. (10) and (8) shows that the\nleading term of the magnon-magnon interaction is re-\npulsive. Thus the magnon BEC remains stable in the\nferrimagnet even with the magnon-magnon interaction.\nThe above result di\u000bers from the stability of the\nmagnon BEC in the ferromagnet without a sublattice\nstructure. This can be seen by applying a similar\ntheory to the second case of our model and compar-\ning the result with the above result. The Holstein-\nPrimako\u000b transformation in the ferromagnet without\na sublattice structure is expressed as Sz\ni=S\u0000cy\nici,\nS\u0000\ni=cy\niq\n2S\u0000cy\nici, andS+\ni= (S\u0000\ni)yfor alli's;\nciandcy\niare the magnon operators. By using thistransformation and the Fourier transformations of the\nmagnon operators, such as ci=1p\nNP\nqeiq\u0001icq, we ob-\ntain the magnon Hamiltonian Hmag =Hnon+Hint,\nwhereHnon=P\nq\u000f(q)cy\nqcqwith\u000f(q) = 2S[J(0)\u0000\nJ(q)] andHint=\u00001\n2NP\nq;q0[J(0)cy\nqcqcy\nq0cq0+J(q\u0000\nq0)cy\nqcq0cy\nq0cq\u00002J(q)cy\nq0cqcy\nq0cq] + (H.c.). Then, by ap-\nplying the mean-\feld approximation to Hint, the lead-\ning term of the magnon-magnon interaction is reduced\ntoH0=\u00002\nNP\nq;q0\u0000(q;q0)nq0cy\nqcq, where \u0000(q;q0) =\nJ(0) +J(q\u0000q0)\u0000J(q)\u0000J(q0) andnq0\u0011n[\u000f(q0)]. Since\n\u0000(q;q0)\u00150, the magnon-magnon interaction becomes\nattractive. Thus the BEC of interacting magnons be-\ncomes unstable in the ferromagnet without a sublattice\nstructure.\nIn order to understand the key to causing the above\ndi\u000berence, we study the stability of the BEC of interact-\ning magnons in the third case of our model. As we can\nsee from Fig. 1, the di\u000berence between the third and \frst\ncases is about the spin alignment, and the di\u000berence be-\ntween the third and second cases is about the sublattice\nstructure. Thus, by comparing the result in the third case\nwith the result in the \frst or second case, we can deduce\nwhich of the two, the di\u000berences in the spin alignment\nand in the sublattice structure, causes the di\u000berence in\nthe stability of the BEC of interacting magnons.\nThe stability in the third case can be studied in a sim-\nilar way to that in the \frst case. In the third case, the\nHolstein-Primako\u000b transformation of Sifori2Ais the\nsame as Eq. (2), whereas that of Sjforj2Bis given\nbySz\nj=SB\u0000by\njbj,S\u0000\nj=p2SBby\njq\n1\u0000(by\njbj=2SB),\nandS+\nj= (S\u0000\nj)y; this di\u000berence arises from the di\u000ber-\nent alignment of the spins belonging to the Bsublattice.\nIn a similar way to the \frst case, we obtain the magnon\nHamiltonian Hmag=Hnon+Hint, whereHnonandHint\nare given by\nHnon= 2X\nqJ(0)(SBay\nqaq+SAby\nqbq)\n\u00002X\nqJ(q)p\nSASB(aqby\nq+ay\nqbq); (11)\nand\nHint=\u00002\nNX\nq;q0[J(0)ay\nqaqby\nq0bq0+J(q\u0000q0)ay\nqaq0by\nq0bq\n\u0000J(q)pSASB(SAay\nqby\nq0bq0bq+SBbqay\nqay\nq0aq0)] + (H.c.);(12)\nrespectively, with ai=q\n2\nNP\nqeiq\u0001iaqandbj=q\n2\nNP\nqeiq\u0001jbq. In addition, Hnoncan be diagonal-\nized by using aq=cq\u000bq\u0000sq\fqandbq=\u0000sq\u000bq+\ncq\fq, wherecq\u0011cosh\u0012qandsq\u0011sinh\u0012qsatisfy\ntanh 2\u0012q=\u00002pSASBJ(q)\n(SA+SB)J(0). The diagonalized Hnonis\nHnon=P\nq[\u000f\u000b(q)\u000by\nq\u000bq+\u000f\f(q)\fy\nq\fq] with\u000f\u000b(q) and\n\u000f\f(q), which are the same as those in the \frst case. Thus,4\nthe ferromagnet and ferrimagnet with the two-sublattice\nstructure have the same properties of the BEC of nonin-\nteracting magnons. Then we can construct the e\u000bective\ntheory for the BEC of interacting magnons in the third\ncase in a similar way. For SA> SB, in the third case,\nthe BEC of interacting magnons can be e\u000bectively de-\nscribed by He\u000b=P\nq\u000f\u0003\n\u000b(q)\u000by\nq\u000bqwith\u000f\u0003\n\u000b(q) =\u000f\u000b(q)\u0000\n4\nNP\nq0~\u0000\u000b\u000b(q;q0)nq0\u000b, where ~\u0000\u000b\u000b(q;q0) =J(0)(c2\nqs2\nq0+\nc2\nq0s2\nq) + 2J(q\u0000q0)cqsqcq0sq0+J(q)pSASBcqsq(SAs2\nq0+\nSBc2\nq0) +J(q0)pSASBcq0sq0(SAs2\nq+SBc2\nq). By estimating\n~\u0000\u000b\u000b(q;q0) in the long-wavelength limits in a similar way,\nwe obtain ~\u0000\u000b\u000b(q;q0)\u0019\u00002\n9J(0)q2q02(SASB)2\n(SA\u0000SB)4. Thus the\nBEC of interacting magnons is stable in the ferromagnet\nwith the two-sublattice structure.\nCombining the results in the three cases, we \fnd that\nthe di\u000berence between the interaction e\u000bects in the fer-\nrimagnet and in the ferromagnet without a sublattice\nstructure arises not from the di\u000berence in the spin align-\nment, but from the di\u000berence in the sublattice struc-\nture. This can resolve the contradiction between exper-\niment [6{9] and theory [10] because that theory uses a\nferromagnet without a sublattice structure. This also\nsuggests that the existence of a sublattice structure is\nthe key to stabilizing the BEC of interacting magnons in\nferrimagnets and ferromagnets. One possible experiment\nto test our mechanism is to measure the stability of the\nmagnon BEC in ferromagnets without and with a sublat-\ntice structure; a sublatttice structure, such as that shown\nin Fig. 1(c), can be realized, for example, by using two\ndi\u000berent magnetic ions.\nWe remark on the role of sublattice degrees of free-\ndom. As shown above, the magnon BEC remains stable\neven in the presence of the magnon-magnon interaction\nas long as a magnet has the sublattice degrees of free-\ndom. This remarkable property can hardly be expected\nfrom the properties of noninteracting magnons because in\nall the three cases, the low-energy properties can be de-\nscribed by a single magnon band. The magnon-magnon\ninteraction becomes repulsive only in the presence of the\nsublattice degrees of freedom because the magnons in dif-\nferent sublattices give the di\u000berent contributions to the\nintraband interaction for a single magnon band; the dif-\nferent contributions arise from the di\u000berent coe\u000ecients\nin the Bogoliubov transformation [e.g., see Eq. (6)].\nNext we discuss the correspondence between our model\nand a model derived in the \frst-principles study in\nYIG [25]. The latter is more complicated than our\nmodel because the magnetic primitive cell of YIG has\n20 Fe moments [26] and its spin Hamiltonian consists of\nthe Heisenberg exchange interactions for three nearest-\nneighbor pairs and six next-nearest-neighbor pairs [25].\nNote \frst, that all of the Fe ions are categorized into FeO\nand FeTions, Fe ions surrounded by an octahedron and a\ntetrahedron of O ions, respectively, and second, that YIGis a ferrimagnet due to the antiparallel spin alignments\nof the FeOand FeTions and the 2 : 3 ratio of the FeO\nand FeTions in the unit cell [27]. Although our model\ndoes not take into account all of the complex properties\nof YIG, our model can be regarded as a minimal model to\nstudy the stability of the BEC of interacting magnons in\nYIG. This is because of the following three facts: First,\nthe largest term in the spin Hamiltonian of YIG is the\nantiferromagnetic nearest-neighbor Heisenberg exchange\ninteraction between the FeOand FeTions and the others\nare at least an order of magnitude smaller. Second, the\nlow-energy magnons of YIG can be described by a single\nmagnon band around q=0. Third, the main e\u000bect of\nthe terms neglected in our theory is to modify the value\nof \u0000\u000b\u000b(q;q0) in Eq. (8). Since this modi\fcation may\nbe quantitative, our mechanism can qualitatively explain\nwhy the magnon BEC is stabilized in YIG.\nOur work has several implications. First, our results\nsuggest that a ferromagnet without a sublattice structure\nis inappropriate for describing the properties of interact-\ning magnons in ferrimagnets, such as YIG. This sugges-\ntion will be useful for future studies towards a compre-\nhensive understanding of magnon physics and spintron-\nics using magnons in YIG. Furthermore, it may be nec-\nessary to reconsider some results of YIG if the results\nare deduced by using a ferromagnet without a sublat-\ntice structure, in particular, the results depend on the\nsign of the magnon-magnon interaction. Our theoretical\nframework can then be used to study the BEC of interact-\ning magnons in other magnets as long as the low-energy\nmagnons can be described by a single magnon band. For\nthe magnets whose low-energy magnons have degeneracy,\nan extension of this framework enables us to study the\nBEC of interacting magnons. Thus our theory may pro-\nvide a starting point for understanding properties of the\nBEC of interacting magnons in various magnets.\nIn summary, we have studied the stability of the BEC\nof interacting magnons in a ferrimagnet and ferromag-\nnets, and we proposed the stabilizing mechanism. By\nadopting the Holstein-Primako\u000b transformation to the\nHeisenberg Hamiltonian, we have derived the magnon\nHamiltonian, which consists of the kinetic energy terms\nand the dominant terms of the magnon-magnon inter-\naction. We then construct the e\u000bective theory for the\nBEC of interacting magnons by utilizing the properties\nfor noninteracting magnons and the mean-\feld approx-\nimation. From the analyses using this theory, we have\ndeduced that in the ferrimagnet and ferromagnet with\nthe sublattice structure the magnon BEC remains stable\neven in the presence of the magnon-magnon interaction,\nwhereas it becomes unstable in the ferromagnet without\na sublattice. This result shows that the existence of a\nsublattice structure is the key to stabilizing the BEC of\ninteraction magnons, whereas the di\u000berence in the spin\nalignments is irrelevant. In addition, this result is consis-\ntent with the experimental results [6{9] of YIG and the5\ntheoretical result [10] of a ferromagnet without a sublat-\ntice structure.\n\u0003naoya.arakawa@sci.toho-u.ac.jp\n[1] C. J. Pethick and H. Smith, Bose-Einstein Condensa-\ntion in Dilute Gases (Cambridge University Press, Cam-\nbridge, England, 2002).\n[2] A. Einstein, Sitzungsberichte der Preussischen Akademie\nder Wissenschaften, Physikalisch-mathematische Klasse\n(1924) p.261; (1925) p.3.\n[3] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E.\nWieman, and E. A. Cornell, Science 269, 198 (1995).\n[4] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van\nDruten, D. S. Durfee, D. M. Kurn, and W. Ketterle,\nPhys. Rev. Lett. 75, 3969 (1995).\n[5] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G.\nHulet, Phys. Rev. Lett. 75, 1687 (1995).\n[6] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A.\nMelkov, A. A. Serga, B. Hillebrands, and A. N. Slavin,\nNature (London) 443, 430 (2006).\n[7] V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A.\nMelkov, and A. N. Slavin, Phys. Rev. Lett. 99, 037205\n(2007).\n[8] V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A.\nMelkov, and A. N. Slavin, Phys. Rev. Lett. 100, 047205\n(2008).\n[9] A. V. Chumak, G. A. Melkov, V. E. Demidov, O.\nDzyapko, V. L. Safonov, and S. O. Demokritov, Phys.\nRev. Lett. 102, 187205 (2009).[10] I. S. Tupitsyn, P. C. E. Stamp, and A. L. Burin, Phys.\nRev. Lett. 100, 257202 (2008).\n[11] V. Cherepanov, I. Kolokolov, and V. L'vov, Phys. Rep.\n229, 81 (1993).\n[12] J. Barker and G. E.W. Bauer, Phys. Rev. Lett. 117,\n217201 (2016).\n[13] A. L. Fetter and J. D. Walecka, Quantum Theory of\nMany-Particle Systems (Dover Publications, Inc., New\nYork, 2003).\n[14] A. A. Abrikosov, L. P. Gor'kov, and I. E. Dzyaloshinski,\nMethods of Quantum Field Theory in Statistical Physics\n(Dover Publications, Inc., New York, 1963).\n[15] T. Holstein and H. Primako\u000b, Phys. Rev. 58, 1098\n(1940).\n[16] T. Oguchi, Phys. Rev. 117, 117 (1960).\n[17] T. Nakamura and M. Bloch, Phys. Rev. 132, 2528 (1963).\n[18] N. N. Bogoliubov, Izv. Akad. Nauk SSSR, Ser. Fiz. 11,\n77 (1947).\n[19] F. Bloch, Z. Phys. 61, 206 (1930).\n[20] F. Dyson, Phys. Rev. 102, 1217 (1956).\n[21] R. Kubo, Phys. Rev. 87, 568 (1952).\n[22] A. B. Harris, D. Kumar, B. I. Halperin, and P. C. Ho-\nhenberg, Phys. Rev. B 3, 961 (1971).\n[23] E. Manousakis, Rev. Mod. Phys. 63, 1 (1991).\n[24] See Supplemental Material, which includes Refs. 16 and\n17, for the details of the derivations of Eqs. (4) and (5),\nEq. (8), and Eq. (10).\n[25] L.-S. Xie, G.-X. Jin, L. He, G. E. W. Bauer, J. Barker,\nand K. Xia, Phys. Rev. B 95, 014423 (2017).\n[26] A. Harris, Phys. Rev. 132, 2398 (1963).\n[27] F. Bertaut, F. Forrat, A. Herpin, and P. M\u0013 eriel, Comptes\nRendus Acad. Sci. 243, 898 (1956)." }, { "title": "0907.1348v1.Magnetic_hysteresis_in_a_molecular_Ising_ferrimagnet__Glauber_dynamics_approach.pdf", "content": "arXiv:0907.1348v1 [cond-mat.stat-mech] 8 Jul 2009Magnetic hysteresis in a molecular Ising ferrimagnet: Glau ber\ndynamics approach\nA. A. Bukharov1, A. S. Ovchinnikov1, N. V. Baranov1,2, K. Inoue3\n1Department of Physics, Ural State University, Ekaterinbur g, 620083 Russia\n2Institute of Metal Physics RAS and Ural State University, Eka terinburg, 620083, Russia\n3Institute for Advanced Materials Research,\nHiroshima University, Hiroshima, Japan\n(Dated: October 31, 2018)\nMotivated by recent experimental results reporting giant c oercive fields in Co(II)-\nbased molecular magnets we present a theory of hysteresis ph enomena based on the\nGlauber stochastic dynamics. Unusual form of hysteresis lo ops is similar to those\nof found in Co-based quasi-one-dimensional ferrimagnet Co PhOMe at low tempera-\ntures. Temperaturedependenceofthecoercive field hasacha racteristic formwithan\ninflection that may serve as an indicator of the Glauber dynam ics in real compounds.\nA relevance of the model for other Co-based molecular magnet s is discussed.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nOne of the remarkable phenomena found in molecular magnetic mater ials is the magnetic\nhysteresis similar to those observed in hard magnets.1,2In particularly, magnetic hysteresis\ncan be observed in the absence of a long-range magnetic order in ze ro-dimensional single-\nmolecule magnets (SMM)3and one-dimensional (1D) single chain magnets (SCM).4,5The\nhysteresis phenomena in SMM are affected by large easy-axis magne tic anisotropy and by\nthe weak intermolecular interactions.6Although a large easy-axis magnetic anisotropy can\nalso be of importance in quasi-1D magnets, an origin of magnetic hyst eresis in these systems\nis not yet well understood.7\nDiscovery of slow relaxation of magnetization in quasi-1D compound\nCo(hfac) 2[NIT(C 6H4p-OMe)] (or CoPhOMe)4evoked interest to the Glauber kinetic\nmodel suggested for relaxation dynamics in 1D Ising ferromagnet.8,9,10The source of a2\nstrong easy-axis anisotropy are Co(II) ions that enable to treat collective properties of this\ncompound by means of 1D Ising model. Recently, pronounced hyste resis loops with giant\ncoercivity were observed in another Co(II)-based materials, nam ely [Co(hfac) 2NIT-C 6H4-\nO-R]11, where R=(CH 2)3CH3, and [Co(hfac) 2]·BNO∗(BNO∗is chiral triplet bis(nitroxide)\nradical).12Despite the Glauber dynamics seems to be plausible, one should take in to\naccount that chains are packed in a 3D structure, and with a cooling 1D units undergo a\nphase transition into 3D magnetic array. It means that another ex planation of hysteresis\nphenomena related with domain wall (DW) dynamics comes into the play . High coercive\nfields reported in Refs.11,12at low temperatures, i.e. broad hysteresis loops, support the\nreasonings. Note that considerations of slow magnetic relaxation b elow 3D ordering in\nthe framework of the DW approach turns out to be fruitful for Mn (II)-based quasi-1D\nferrimagnets.13,14\nBeingguidedthismotivationwepresentatheoryofmagnetichyster esis forIsingferrimag-\nnetic chain calculated within the Glauber dynamics approach. An exte nsion of the Glauber\nmodel for higher spins (more than 1 /2) has been done in a modeling of photoinduced re-\nversible magnetization15,16. Despite equilibrium properties of mixed spin Ising systems have\nattracted a much of attention, a study of nonequilibrium aspects o f the model started only\nrecently.17In the paper we perform the mean field treatment of the Glauber-t ype stochastic\ndynamics of the mixed (3/2,1) ferrimagnetic chain by considering fluc tuations of local fields\nin the spirit of the generalized mean-field theory.18Our goal is to verify a validity of the\nmodel for real Co-based molecular ferrimagnetic chains. The choic e of the spin quantum\nnumbers is stipulated by a possible prototype of the model, the meta l-organic compound\n[Co(hfac) 2]·BNO∗studied recently by two of us, which embodies Co(II) ions with spin 3/ 2\nand the chiral triplet biradical ligands BNO∗with spin 1. However, the results we obtain\nmay be of interest for another Co-based quasi-1D magnets.\nThe paper is organized as follows. In Sec.II we obtain dynamic equatio ns of the Ising\nferrimagnetic ( S,σ) chain model derived through the Glauber stochastic dynamics in a p res-\nence of a time-varying magnetic field. In Sec. III we present result s of hysteresis behavior\nobtained via numerical calculations of these equations for the case S= 3/2,σ= 1. A depen-\ndence of shape of the hysteresis cycles on the ratio between the m agnetic field frequency and\nthe spin flip frequency is studied. In Sec. IV we confirm by a Monte Ca rlo (MC) method\nthat a hysteresis loops arise in a static magnetization process. Disc ussions are relegated to3\nthe Conclusion part.\nII. MODEL\nThe Hamiltonian of the Ising (3/2,1) ferrimagnetic chain is given by\nH=J/summationdisplay\ni(σiSi+σi+1Si)−H(t)/parenleftBigg/summationdisplay\niσi+/summationdisplay\niSi/parenrightBigg\n, (1)\nwhere the first term sums interactions between the nearest neigh bors with the spin variables\nσi= 0,±1 andSi=±1/2,±3/2, andJ >0 favors an antiferromagnetic coupling of the\nadjacent sites. The Zeeman term describes interaction of the spin s with an oscillating\nmagnetic field of the sinusoidal form\nH(t) =H0cos(ωt)\nwith a frequency ω. The system is in contact with an isothermal heat bath at given tem-\nperature T.\nThe relaxation of the interacting Ising system cannot be obtained f rom its Hamiltonian\nbecause it eliminates intrinsic spin dynamics (precession in the local fie ld). Nevertheless,\nrelaxation phenomena can be described by means of a phenomenolog ical equation which\nspecifies the transition rate from one spin configuration to anothe r. In general, we assume\nthat transition from one configuration to another involves changin g a single spin, i.e. ac-\ncording to Glauber dynamics,8,9,10at a rate of 1 /τtransitions per unit time.\nAccording to formalism of Ref.17the master equation for one sublattice can be written\nin suggestion that spins on another sublattice momentarily frozen, i.e.\nd\ndtP{S}(σ1,σ2,...,σ j,...,σ N)\n=/summationdisplay\nj/summationdisplay\nσ′\nj/ne}ationslash=σjωj(σ′\nj→σj)P{S}(σ1,σ2,...,σ′\nj,...,σ N)\n−/summationdisplay\nj\n/summationdisplay\nσ′\nj/ne}ationslash=σjωj(σj→σ′\nj)\nP{S}(σ1,σ2,...,σ j,...,σ N),(2)4\nd\ndtP{σ}(S1,S2,...,S j,...,S N)\n=/summationdisplay\nj/summationdisplay\nS′\nj/ne}ationslash=Sjωj(S′\nj→Sj)P{σ}(S1,S2,...,S′\nj,...,S N)\n−/summationdisplay\nj\n/summationdisplay\nS′\nj/ne}ationslash=Sjωj(Sj→S′\nj)\nP{σ}(S1,S2,...,S j,...,S N),(3)\nwhereP{S}(σ1,σ2,...σj,...,σ N)istheprobabilitythatthesystemhasthespinconfiguration\n{σ1, σ2, ... σ j, ..., σ N}in the first sublattice leaving the spins {S}of the second sublattice\nfixed,ωj(σ′\nj→σj) is the probability per unit time that the j-th spin changes from the value\nσ′\njtoσj. The similar notations hold for another sublattice.\nThe transition probabilities ωj(σ′\nj→σj) andωj(S′\nj→Sj) are imposed by the principle\nof detailed balance. Indeed, the principle requires the probabilities o f statesP{S}andP{σ}\nto be stationary at equilibrium\nωj(σj→σ′\nj)P{S}(σ1,σ2,...,σ j,...,σ N) =ωj(σ′\nj→σj)P{S}(σ1,σ2,...,σ′\nj,...,σ N),\nωj(Sj→S′\nj)P{σ}(S1,S2,...,S j,...,S N) =ωj(S′\nj→Sj)P{σ}(S1,S2,...,S′\nj,...,S N).\nProvided the probabilities P{S}andP{σ}are Boltzmann distributions, i.e.\nP{S}(σ1,σ2,...,σ′\nj,...,σ N)\nP{S}(σ1,σ2,...,σ j,...,σ N)=exp/bracketleftBig\n−β/braceleftBig\nJ/parenleftBig\nσ′\njSj−1+σ′\njSj/parenrightBig\n−Hσ′\nj/bracerightBig/bracketrightBig\nexp/bracketleftBig\n−β/braceleftBig\nJ/parenleftBig\nσjSj−1+σjSj/parenrightBig\n−Hσj/bracerightBig/bracketrightBig, (4)\nP{σ}(S1,S2,...,S′\nj,...,S N)\nP{σ}(S1,S2,...,S j,...,S N)=exp/bracketleftBig\n−β/braceleftBig\nJ/parenleftBig\nσjS′\nj+σj+1S′\nj/parenrightBig\n−HS′\nj/bracerightBig/bracketrightBig\nexp/bracketleftBig\n−β/braceleftBig\nJ/parenleftBig\nσjSj+σj+1Sj/parenrightBig\n−HSj/bracerightBig/bracketrightBig (5)\none get\nωj(σj→σ′\nj)\nωj(σ′\nj→σj)=exp[σ′\njyj]\nexp[σjyj],ωj(Sj→S′\nj)\nωj(S′\nj→Sj)=exp[ξjS′\nj]\nexp[ξjSj], (6)\nwhereyj=β/bracketleftbig\nH−J(Sj−1+Sj)/bracketrightbig\n, andξj=β/bracketleftbig\nH−J(σj+σj+1)/bracketrightbig\n,β= 1/T.\nIn Glauber dynamics the relationships (6) are resolved as follows\nωj(σj→σ′\nj) = Ωexp[σ′\njyj]/summationtext\nσ′′\njexp[σ′′\njyj], ω j(Sj→S′\nj) = Ωexp[ξjS′\nj]/summationtext\nS′′\njexp[ξjS′′\nj], (7)\nwhere Ω = 1 /τis a number of spin changes per unit time. Strictly speaking, Eqs.(7) a re\nrelevant for a equilibrium process with a constant magnetic field. Nev ertheless, it is reliable\nwhen the field sweep frequency is much less then that of spin transit ionsω≪Ω.5\nExpectation value of the σ-sublattice magnetization at the moment tis given by\n/an}bracketle{tσi/an}bracketri}ht=/summationdisplay\nσiσip(σi), (8)\nwhere\np(σi) =/summationdisplay\n{S}/summationdisplay\n{σ}′P{S}(σ1,...,σ i,...,σ N) (9)\nis the probability to find the σivalue. The sum with the prime runs over all σvariables\nexcept that of i-th site. Similar definitions are hold for another sublattice.\nThe probability satisfies the following dynamic equation\ndp(σi)\ndt=−Ωp(σi)+Ω/angbracketleftBigexp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]/angbracketrightBig\n, (10)\nwhich is obtained from Eq.(9) with an account of Eqs.(2-3) and (7). T he average in the\nright-hand side is determined as follows\n/angbracketleftBigexp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]/angbracketrightBig\n=/summationdisplay\n{S}/summationdisplay\n{σ}exp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]P{S}(σ1,...,σ i,...,σ N). (11)\nThe analogous equation is derived for another sublattice\ndp(Si)\ndt=−Ωp(Si)+Ω/angbracketleftBigexp[ξiSi]/summationtext\nS′\niexp[ξiS′\ni]/angbracketrightBig\n. (12)\nThe system of Eqs.(10,12) can be treated in the mean-field approxim ation (MFA)\n/angbracketleftBigexp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]/angbracketrightBig\n≈exp[σi/an}bracketle{tyi/an}bracketri}ht]/summationtext\nσ′\niexp[σ′\ni/an}bracketle{tyi/an}bracketri}ht], (13)\nthat yields\n\nd/an}bracketle{tσi/an}bracketri}ht\ndt=−Ω/bracketleftBig\n/an}bracketle{tσi/an}bracketri}ht−σBσ(σ/an}bracketle{tyi/an}bracketri}ht)/bracketrightBig\n,\nd/an}bracketle{tSi/an}bracketri}ht\ndt=−Ω/bracketleftBig\n/an}bracketle{tSi/an}bracketri}ht−SBS(S/an}bracketle{tξi/an}bracketri}ht)/bracketrightBig\n,(14)\nwhere the Brillouine function Bs(x) is introduced\nBs(x) =/parenleftBig\n1+1\n2s/parenrightBig\ncoth/bracketleftBig/parenleftBig\n1+1\n2s/parenrightBig\nx/bracketrightBig\n−1\n2scoth/bracketleftBigx\n2s/bracketrightBig\n.6\nFIG. 1: Hysteresis loops of magnetization per cell M=/an}bracketle{tσ/an}bracketri}ht+/an}bracketle{tS/an}bracketri}htat different temperatures θ.\nInset: the coercive field hcvs temperature.\nBy taking the uniform arrangement /an}bracketle{tσi/an}bracketri}ht=/an}bracketle{tσ/an}bracketri}ht,/an}bracketle{tSi/an}bracketri}ht=/an}bracketle{tS/an}bracketri}htand, as a consequence, /an}bracketle{tyj/an}bracketri}ht=\nβ[H(t)−2J/an}bracketle{tS/an}bracketri}ht],/an}bracketle{tξj/an}bracketri}ht=β[H(t)−2J/an}bracketle{tσ/an}bracketri}ht] one obtain eventually the dynamical equations\n\n\nd/an}bracketle{tσ/an}bracketri}ht\ndt=−Ω/bracketleftBig\n/an}bracketle{tσ/an}bracketri}ht−σBσ/parenleftbig\nσβ{H(t)−2J/an}bracketle{tS/an}bracketri}ht}/parenrightbig/bracketrightBig\n,\nd/an}bracketle{tS/an}bracketri}ht\ndt=−Ω/bracketleftBig\n/an}bracketle{tS/an}bracketri}ht−SBS/parenleftbig\nSβ{H(t)−2J/an}bracketle{tσ/an}bracketri}ht}/parenrightbig/bracketrightBig\n.(15)\nAt equilibrium ( d/dt→0) one recover the usual MFA equations from the system.\nIn one dimension the mean-field approximation is poor since the local fi eldsyiandξi\nfluctuate strongly from one site to another. To overcome partly t he drawback we use a\ngeneralization of the MFA approach like those used in Ref.18\nTake the following approximation for the averages (11)\n/angbracketleftBigexp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]/angbracketrightBig\n≈/summationdisplay\nSi−1,Siexp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]p(Si−1)p(Si), (16)7\n/angbracketleftBigexp[ξiSi]/summationtext\nS′\niexp[ξiS′\ni]/angbracketrightBig\n≈/summationdisplay\nσi,σi+1exp[ξiSi]/summationtext\nS′\niexp[ξiS′\ni]p(σi)p(σi+1), (17)\ni.e. the averages depend on the spin variables at the adjacent sites of another sublattice.\nAfter substitution Eqs.(16-17) into Eqs.(10,12) one obtain the dyn amic equations of the\ngeneralized mean-field approximation written through the probabilities\nd\ndtp(σi) =−Ω/bracketleftbigg\np(σi)−/summationdisplay\nSi−1,Siexp[σiyi]/summationtext\nσ′\niexp[σ′\niyi]/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nyi=β[H−J(Si−1+Si)]p(Si−1)p(Si)/bracketrightbigg\n,\nd\ndtp(Si) =−Ω/bracketleftbigg\np(Si)−/summationdisplay\nσi,σi+1exp[ξiSi]/summationtext\nS′\niexp[ξiS′\ni]/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nξi=β[H−J(σj+σj+1)]p(σi)p(σi+1)/bracketrightbigg\n.(18)\nBy determining a time evolution of these quantities one recover a time dependence of ex-\npectation values for the sublattice magnetizations according to Eq .(8).\nIII. NUMERICAL RESULTS\nLet us now discuss hysteresis phenomenon for (3 /2,1) ferrimagnetic Ising chain obtained\non the base of Eqs.(15) and (18). For numerical calculations it is con venient to rewrite the\nMFA equations in the form\nd/an}bracketle{tσ/an}bracketri}ht\nd˜τ=−/an}bracketle{tσ/an}bracketri}ht+B1([h−2/an}bracketle{tS/an}bracketri}ht]/θ),\nd/an}bracketle{tS/an}bracketri}ht\nd˜τ=−/an}bracketle{tS/an}bracketri}ht+(3/2)B3\n2(3[h−2/an}bracketle{tσ/an}bracketri}ht]/2θ).\nwhere ˜τ= Ωt, andh=H/J,θ=T/Jare reduced field and temperature, respectively.\nRegarding the system (18) there is a way to bring down a number of d ifferential equa-\ntions from 7 till 5 by using the normalizing condition for probabilities pand definition of\nobservables. Indeed, hold four differential equations for the pro babilities p(±1),p(±3/2)\nd\nd˜τp(±1) =−p(±1)+/summationdisplay\nSi−1,Siexp[±(h−Si−1−Si)/θ]/summationtext\nσ′\niexp[σ′\ni(h−Si−1−Si)/θ]p(Si−1)p(Si), (19)\nd\nd˜τp(±3/2) =−p(±3/2)+/summationdisplay\nσi,σi+1exp[±3(h−σj−σj+1)/(2θ)]/summationtext\nS′\niexp[S′\nih−σj−σj+1\nθ]p(σi)p(σi+1),(20)8\nand add the corresponding equation for the observable /an}bracketle{tS/an}bracketri}ht\nd/an}bracketle{tS/an}bracketri}ht\nd˜τ=−/an}bracketle{tS/an}bracketri}ht+/summationdisplay\nσi,σi+13\n2B3\n2/parenleftBig\n3[h−σj−σj+1]/2θ/parenrightBig\np(σi)p(σi+1),\nobtained from Eq.(18). The system is supplemented by the pure alge braic relations\np(0) = 1−p(−1)−p(1),\np(−1/2) =p(3/2)−2p(−3/2)−/an}bracketle{tS/an}bracketri}ht+1/2,\np(1/2) =/an}bracketle{tS/an}bracketri}ht−2p(3/2)+p(−3/2)+1/2\nthat yield the leaving probabilities.\nThe sums in the right-hand sides of Eqs.(19,20) are explicitly given by\n/summationdisplay\nSi−1,Siexp[σi(h−Si−1−Si)/θ]/summationtext\nσ′\niexp[σ′\ni(h−Si−1−Si)/θ]p(Si−1)p(Si) =exp[σi(h+3)/θ]\nZ1[(h+3)/θ]p(−3/2)2\n+2exp[σi(h+2)/θ]\nZ1[(h+2)/θ]p(−3/2)p(−1/2)+exp[σi(h+1)/θ]\nZ1[(h+1)/θ]/bracketleftbig\np(−1/2)2+2p(−3/2)p(1/2)/bracketrightbig\n+2exp[σih/θ]\nZ1[h/θ]/bracketleftbig\np(−3/2)p(3/2)+p(−1/2)p(1/2)/bracketrightbig\n+exp[σi(h−1)/θ]\nZ1[(h−1)/θ]/bracketleftbig\np(1/2)2+2p(−1/2)p(3/2)/bracketrightbig\n+2exp[σi(h−2)/θ]\nZ1[(h−2)/θ]p(1/2)p(3/2)+exp[σi(h−3)/θ]\nZ1[(h−3)/θ]p(3/2)2,\nfor the spin-1 sublattice and\n/summationdisplay\nσi,σi+1exp[Si(h−σj−σj+1)/θ]/summationtext\nS′\niexp[S′\ni(h−σj−σj+1)/θ]p(σi)p(σi+1) =exp[Si(h+2)/θ]\nZ3\n2[(h+2)/θ]p(−1)2\n+2exp[Si(h+1)/θ]\nZ3\n2[(h+1)/θ]p(−1)p(0)+exp[Sih/θ]\nZ3\n2[h/θ][p(0)2+2p(−1)p(1)]\n+2exp[Si(h−1)/θ]\nZ3\n2[(h−1)/θ]p(0)p(1)+exp[Si(h−2)/θ]\nZ3\n2[(h−2)/θ]p(1)2,\nfor the spin-3/2 one. Here,\nZ1(x) =/summationdisplay\nσ′\niexp[σ′\nix] = 2cosh( x)+1, Z 3\n2(x) =/summationdisplay\nS′\niexp[S′\nix] = 2cosh( x/2)+2cosh(3 x/2).\nThe results of such calculations performed within the Runge-Kutta method are presented\nin Figs.1-3 for the field frequency ω/Ω = 10−4(Ω = 1). Initial values were chosen to\ncorrespond either to total saturation or disorder in both sublatt ices. A stationary regime is9\napproximately reached after one full sweep. Fig.1 shows the evolut ion of hysteresis loops at\nvarioustemperatures. Itcanbeclearlyseenthattheareaofthe hysteresisloopmonotonically\ndecreases with increasing temperature. The hysteresis curves a re characterized by well-\ndefined steps, and for higher temperature ( θ >0.7) have the form with a loop in the middle\nclose to that of observed experimentally in Co(II)-based quasi-1D ferrimagnetic compound\nCoPhOMe at temperatures around /greaterorsimilar4.5 K.4The process of thermally activated spin flip\nformation reduces the effective intrinsic coercive field (see inset in F ig.1). Note that the\ntemperature dependence of the coercivity exhibits a behavior (a c urve with an inflection)\ndistinguished from that of predicted by a thermal activation theor y by Egami19used for\nhard magnetic materials.20WhenTtends to zero the value of the coercive field becomes\nequal to 2 J, i.e. it is determined by the exchange coupling (not by anisotropy) as expected\nfor Ising systems.\nThe both analytical approaches give qualitatively same results (see Fig.2). Note only\nthat a more detailed account of fluctuations within the generalized M FA squeezes the area\nof the hysteresis loops.\nTo elucidate physics standing for this hysteresis behavior, magnet izations of the both\nsublattices arepicked outin Fig.3. The results resemble a spin-orient ational phase transition\nin the Heisenberg antiferromagnet and are interpreted as follows. At strong fields both\nsublattices are polarized along the field. With its decreasing the bigge r spins retain their\ndirections while smaller spins change smoothly their alignment into oppo site to gain in an\nexchange energy. During further demagnetization process, whe nh <0, the bigger spins\nreorient to be again arranged along the field. This causes a sharp sp in reorientation in\nanother sublattice due to the exchange coupling which is stronger t han the corresponding\nZeeman interaction. The rapid change is accompanied by the side effe ct which stands out\nmore noticeably in the MFA calculations, namely, an appearance of ”w hiskers” in the entire\nhysteresis curve at low temperatures. The process is completed b y a gradual saturation\nof small-spin sublattice magnetization along the applied field. The back sweep goes on\nsimilarly.\nWealsofoundouthowthemagnetizationprocessdependsonthere altionshipbetween the\nfrequency Ω (a number of spin transitions per unit time) and the mag netic field frequency\nω. We plotted the corresponding curves in Fig.4 where two cases are p resented, namely,\nthe quasistatic regime ω/Ω = 10−4and the regime when the discrepancy between both the10\n-2-1 0 1 2\n-10 -5 0 5 10M\nhθ = 0.7\n-2-1 0 1 2\n-10 -5 0 5 10M\nhθ = 1\n-2-1 0 1 2\n-10 -5 0 5 10M\nhθ = 1.3\nFIG. 2: Comparison of hysteresis curves calculated within M FA (solid) and generalized MFA\n(dashed) approaches.\nfrequencies are not so drastic ω/Ω = 10−1. As well we show in the Figure the pure static\nmagnetization process which almost coincides with that of the quasis tatic regime. One see\nthat in the case ω/Ω = 10−1the hysteresis loop transforms into the narrow S-like form11\nFIG. 3: Hysteresis loops for the sublattice /an}bracketle{tσ/an}bracketri}ht(big dotted), /an}bracketle{tS/an}bracketri}ht(small dotted) and the entire M\n(solid) magnetizations ( θ= 0.4).\nsimilar to those found experimentally in the compound CoPhOMe at 2 .0 K and 3 .5 K.\nPhysically, the increasing the ratio ω/Ω at a fixed magnetic field frequency means that a\nspin dynamics governed by Ω slows down.\nIV. CONFIRMATION BY MONTE CARLO SIMULATIONS\nIn the previous section we have shown that in the quasistatic regime ω≪Ω the both\nanalytical models demonstrate an appearance of hysteresis loops of a peculiar form in the\nmiddle of magnetization curves. However, we note that spin fluctua tions taken into account\nin the framework the generalized MFA are strongly restricted since they hold a translational\nsymmetry of the chain, i.e. they are the same within an each elemenat ry cell. In order to\ncheck whether the hysteresis phenomena are stable against fluct uations in common case, we\nstudy the static magnetization process by a Monte Carlo method.\nWe apply standard importance sampling methods21,22to simulate the Hamiltonian given\nby Eq.(1). Periodic boundary conditions on N= 64, 256 chains were imposed and configu-12\n-2-1 0 1 2\n-10 -5 0 5 10M\nhθ = 0.2\na)\n!/Ω = 0\n!/Ω = 1/10000\n!/Ω = 1/10\n-2-1 0 1 2\n-10 -5 0 5 10M\nhθ = 0.2\nb)\n!/Ω = 0\n!/Ω = 1/10000\n!/Ω = 1/10\nFIG. 4: Evolution of magnetization curves for different frequ ency regimes ( θ= 0.2): (a) MFA; (b)\ngeneralized MFA13\nrations were generated by sequentially traversing the sublattices and making single-spin flip\nattempts. The flips are accepted or rejected according to a heat -bath algorithm. Our data\nwere generated with 104Monte Carlo steps per spin in the chain after 103warming steps\nper spin. We checked that the size effects do not substantially effec t the results (Fig. 5).\n-2-1 0 1 2\n-10 -5 0 5 10M\nhθ = 0.2\nN = 256\nN = 64\nFIG. 5: MC magnetization curves for different chain lengths.\nQualitatively, the same features of the hysteresis loops as obtaine d in MFA and the\ngeneralized MFA are also found in MC. Namely, the hysteresis loop of a n almost ideal\nrectangle form appears in the middle of the magnetization curve at lo w temperatues. The\ntemperature evolution of the hysteresis curves reproduces qua liatatively that of predicted\nby the analytical treatments, however, the MC hysteresis loop na rrows more rapidly with\nan increasing of temperature (Fig. 6).\nV. CONCLUSIONS\nWe analyse a magnetic hysteresis of Co-based quasi-1D ferrimagne tic magnets within\nthe model of the mixed spin Ising chain. By using a Glauber dynamics ap proach we build\nhysteresis loops that come up when a sinusoidal magnetic field is applie d. We found that the\nunusual shape of the calculated hysteresis cycles coincide with tho se found experimentally\nin the CoPhOMe ferrimagnet at low temperatures. However, anoth er Co-based molecular14\nchains11,12demonstrate a hysteresis behavior that is scarcely agreed with th e analytical\ntreatment, i.e. these materials behave similar to very hard magnets with a high coercive\nfield and broad hysteresis loops.\nIt is likely that a reason behind the hardness of the materials is interc hain interactions\nthat result in 3D ordering at low temperatures, and, as consequen ce, change a character\nof elementary excitations. Namely, random spin flips throughout th e Ising chain, that is a\nfeature of the Glauber dynamics, are substituted for spin flips in th e vicinity of domain walls\nseparating regions of opposite magnetizations that causes their d isplacements. Then the\ndemagnetization process is related with the thermally activated DW m otion. Similar ideas\nhave been recently argued in Ref.7, where the Mn(III)-based systems23,24were suggested to\nfind out a role of easy axis anisotropy in hysteresis phenomena.\n1R. Sessoli, D. Gateschi, A. Caneschi, M.A. Novak, Nature 365(1993) 365 .\n2L. Bogani, A. Vindigni, R. Sessoli, D. Gatteschi, J. Mater. C hem.18(2008) 4750.\n3R. Sessoli, H. L. Tsai, A. R. Schake, S. Wang, J. B. Vincent, K. Folting, D. Gatteschi, G.\nChristou and D. N. Hendrickson, J. Am. Chem. Soc. 115(1993) 1804.\n4A. Caneschi, D. Gatteschi, N. Lalioti, C. Sangregorio, R. Se ssoli, G. Venturi, A. Vindigni, A.\nRettori, M. G. Pini and M. A. Novak, Europhys. Lett. 58(2002) 771; Angew. Chem. Int. Ed.\n40(2001) 1760.\n5R. Clerac, H. Miyasaka, M. Yamashita and C. Coulon, J. Am. Che m. Soc.124(2002) 12837.\n6D. Gatteschi, R. Sessoli and J. Villain, Molecular Nanomagn ets, Oxford University Press, Ox-\nford, UK, 2006.\n7R. Sessoli, Angew. Chem. Int. Ed. 47(2008) 5508.\n8R. J. Glauber, J. Math. Phys. 4(1963) 294.\n9K. Kawasaki, Phys. Rev. B 145(1966) 224.\n10M. Suzuki, R. Kubo, J. Phys. Soc. Jpn 24(1968) 51.\n11N. Ishii, Y. Okamura, S. Chiba, T. Nogami, T. Ishida, J. Am. Ch em. Soc. 130(2008) 24.\n12Y. Numata, K. Inoue, N. Baranov, M. Kurmoo, and K. Kikuchi, J. Am. Chem. Soc. 129(2007)\n9902.\n13A.S. Ovchinnikov, I.G. Bostrem, V.E. Sinitsyn, A.S. Boyarc henkov, N.V. Baranov, K. Inoue,15\nPhys. Rev. B 74(2006) 174427.\n14E. Lhotel, D.B. Amabilino, C. Sporer, D. Caneau, J. Veciana, and C. Paulsen, Phys. Rev. B 77\n(2008) 064416.\n15M. Nishino, S. Miyashita, Phys. Rev. B 63(2001) 174404.\n16M. Nishino, K. Boukheddaden, S. Miyashita, F. Varret, Phys. Rev. B79(2005) 064452.\n17B. Deviren, M. Keskin, O. Canko, J. Mag. Magn. Mat. 321(2009) 458 and references therein.\n18E.Z. Meilikhov, JETP Letters 79(2004) 620.\n19T. Egami, phys. stat. sol. (a) 19(1973) 747.\n20F.T. Parker, Solid State Commun. 50(1984) 637; H. Kronm¨ uller, J. Mag. Magn. Mat. 7(1978)\n341.\n21G.M. Buend ´ia and R. Cardona, Phys. Rev. B 59, (1999) 6784.\n22M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statist ical Physics, OxfordUniversity\nPress, Oxford, UK, 1999.\n23K. Bernot, J. Luzon, R. Sessoli, A. Vindigni, J. Thion, S. Ric heter, D. Leclercq, J. Larionova,\nA. van der Lee, J. Am. Chem. Soc. 130(2008) 1619.\n24C. Coulon, H. Miyasaka, R. Cl´ erac, Struct. Bonding (Berlin )122(2006) 163.16\n-2-1 0 1 2\n-10 -5 0 5 10M\nhθ = 0.2\nMC\nGMFA \nMFA\n-2-1 0 1 2\n-10 -5 0 5 10M\nhθ = 0.8\nMC\nGMFA \nMFA\nFIG. 6: Temperature evolution of MC hysteresis loops in a com parison with the mean field (MF)\nand the generalized mean field (GMF) calculations ( N= 256)." }, { "title": "0805.3604v2.Frustration_Induced_Quantum_Phases_in_Mixed_Spin_Chain_with_Frustrated_Side_Chains.pdf", "content": "arXiv:0805.3604v2 [cond-mat.str-el] 12 Aug 2008Frustration Induced Quantum Phases in Mixed Spin Chain with Frustrated Side\nChains\nKazuo Hida1and Ken’ichi Takano2\n1Division of Material Science, Graduate School of Science an d Engineering,\nSaitama University, Saitama, Saitama, 338-8570, Japan\n2Toyota Technological Institute, Tenpaku-ku, Nagoya 468-8 511, Japan\n(Dated: October 26, 2018)\nA mixed Heisenberg spin chain with frustrated side chains is investigated by numerical and per-\nturbational calculations. A frustration-induced quantum partially polarized ferrimagnetic phase\nand a nonmagnetic spin quadrupolar phase are found adjacent to the conventional Lieb-Mattis type\nferrimagnetic phase or the nonmagnetic singlet cluster sol id phases. The partially polarized ferri-\nmagnetic phase has an incommensurate spin structure. Simil ar structures are commonly found in\nother frustration-induced partially polarized ferrimagn etic phases. Numerical results also suggest a\nseries of almost critical nonmagnetic ground states in a hig hly frustrated regime if the side chain\nspins weakly couple to the main chain.\nPACS numbers: 75.10.Jm, 75.10.Pq, 75.30.Et, 75.30.Kz\nI. INTRODUCTION\nThe interplay of frustration and quantum fluctua-\ntion has been extensively studied in a variety of low-\ndimensional quantum magnets. Even in one-dimensional\ncases, various exotic quantum phenomena such as spon-\ntaneous dimerization, [1] 1/3-plateau with spontaneous\ntrimerization, [2] and transition between quantum and\nclassical plateaus [3] are reported. On the other hand,\nthe mixed quantum spin chains also have a variety of\nground states ranging from quantum ferrimagnetism [4]\nto spin gap phases [5–8].\nRecently, it has been reported that frustration induces\na partiallypolarizedferrimagnetic(PPF) phase [9–12] in\nadditiontoaconventionalLieb-Mattistypeferrimagnetic\n(LMF) phase. The PPF phase appear when both frus-\ntration and quantum fluctuations are fairly strong. It is\nan interesting issue how general such a ferrimagnetismis.\nHence it is important to investigate the features of PPF\nphasesin variousspin systems. We arethen motivated to\nfind a PPF phase in other models and investigate them\nin detail.\nWe have introduced a spin chain with side chains in a\nprevious paper [8]. This spin chain has frustration ow-\ning to the interaction among spins in the main chain and\nthose of side chains. The frustration varies in strength\nand in feature with the variation of parameters in the\nmodel. Sincewefocusedonthespin-gapphasesin [8], we\nhave investigated the parameter regimes where frustra-\ntion is not strong enough to destroy the spin-gap phases.\nWe then found two spin-gap phases and explained them\nby singlet cluster solid (SCS) pictures.\nIn the present work, we examine this model in a highly\nfrustrated regime. This regime, in which the frustration\nplays a central role, is of interest in its own right, since\nthe model exhibits features very different from those in\nthe weak frustration regimes. We actually find clear nu-\nmerical evidences not only for the above mentioned fas-\ncinating PPF phase but also for the spin quadrupolar(QP) phase [13–19]. These phases are totally different\nfromthe conventionalphasessuch asspin gapphasesand\nthe LMF phase which can be realized even in the unfrus-\ntrated case. They will be investigated in detail in the\npresent paper. Also numerical data are obtained which\nsuggest the possible existence of an exotic almost critical\nnonmagnetic ground state in the regime where the cou-\nplings between the side chain and main chain spins are\nweak but strongly frustrated.\nThe transition from a ferrimagnetic phase to a non-\nmagnetic phase in the present model takes place because\nthe quantum fluctuation in the side-chains destroys the\nferrimagnetic long range order in the main chain. The\nmechanism of quantum destruction of ferromagnetism\nand ferrimagnetism has been less studied than that of\nantiferromagnetism; the latter has been extensively stud-\nied in relation with the high- Tcsuperconductivity. Re-\ncently, however, experiments have been reported on the\nnonmagnetic ground states in one- and two-dimensional\nmaterials [23, 24] with ferromagnetic nearest neighbour\nand antiferromagnetic next nearest neighbour couplings.\nTheoretical investigation has also been carried out for\ncorresponding models [25–28].\nIn an unfrustrated ferrimagnet, the spontaneous mag-\nnetizationisuniquelydetermined bytheLieb-Mattisthe-\norem [29]. This type of quantum ferrimagnetism has\nbeen investigated in detail [4]. As far as the frustration\nis weak, the spontaneous magnetization remains locked\nto this value [30]. This phase is the LMF phase [10].\nThe spontaneous magnetization in this phase is a sim-\nple fraction of the saturated magnetization. In contrast,\nthe spontaneous magnetization in the PPF phase con-\ntinuously varies with the parameter characterizing the\nstrength of frustration and is not a simple fraction of\nthe saturated magnetization. The PPF phase appears\nbetween the LMF phase and the nonmagnetic spin gap\nphases. This type of phase is first predicted in the pio-\nneering work of Sachev and Senthil [21] in the quantum\nrotor model. Bartosch, Kollar and Kopietz [22] proposed2\na possibility of ferromagnetic Luttinger liquid in an itin-\nerant one-dimensional Fermi system. The first explicit\nexample of quantum PPF phase induced by frustration\nin one-dimensional quantum spin systems was proposed\nby Ivanov and Richter [9] in a frustrated mixed spin lad-\nder. Similar phases are also found by Yoshikawa and\nMiyashita [10] in a uniform spin chain and by one of the\npresent authors in a trimerized zigzag chain [11, 12]. We\nproposeanotherexampleofthe PPFphasein the present\nmixed spin chain. The present example is substantially\ndifferent from previous ones, because it is accompanied\nby the destruction of the ferrimagnetic order in the main\nchain by the frustrated coupling to the quantum fluctu-\nation in the side chains.\nThe QP phase is well known for the spin-1 bilinear-\nbiquadratic chain between the Haldane and ferromag-\nnetic phases [13–19]. The exact Bethe ansatz solution\nis available if the coefficients of the bilinear and the bi-\nquadratic terms coincide with each other [13–15]. Re-\ncently, similar phases are found in the frustrated two-\ndimensional Heisenberg model with ferromagnetic near-\nest neighbour interaction and antiferromagnetic next\nnearest neighbour interaction [28]. In this paper we\nexplicitly show that our model reduces to the bilinear-\nbiquadratic chain in appropriate limiting cases. It is also\nargued that the QP phase should appear in a wide class\nof complex spin models between ferrimagnetic and spin\ngap phases.\nThis paper is organized as follows. In the next section,\nthe model Hamiltonian is presented. Various limiting\ncases are discussed using the perturbational approxima-\ntion from the strong coupling limit in Sec. III. The nu-\nmerically obtained ground state phases are explained in\nSec. IV. The properties of PPF phase are described in\ndetail in Sec. V. Section VI is devoted to summary and\ndiscussion.\nT(p)\nS (p)1S (p)2J J 2K1K2\n1\nFIG. 1: The quantum spin chain with side chains which we\nstudy in this paper. S1(p),S2(p) andT(p) are spins in the\np-th unit cell. These magnitudes are S1= 1,S2=1\n2, and\nT=1\n2, respectively.II. HAMILTONIAN\nWeconsiderthemixedHeisenbergspinchaindescribed\nby the Hamiltonian\nH=N/3/summationdisplay\np=1[J1S1(p)S2(p)+J2S2(p)S1(p+1)\n+K1S1(p)T(p)+K2S2(p)T(p)]. (1)\nwhereS2(p) andT(p) are the spin-1 /2 operators and\nS1(p) is the spin-1 ones in the p-th unit cell, as shown\nin Fig. 1. In what follows, we use nondimensional pa-\nrameters j=J2/J1,k=K1/J1andr=K2/(2K1), and\nthe unit of J1= 1. The parameter rcharacterizes the\nstrength of frustration. The total number of spin sites is\ndenoted by N, and then the number of unit cells is N/3.\nIn regime 0 ≤r/lessorsimilar0.5, where frustration is not strong,\nwe have found two types of nonmagnetic ground states\nand have explained them by SCS pictures [8]. When\nfrustrationbecomesstrong,thephasediagramdrastically\nchanges. In the present paper, we will investigate the\nstrongly frustrated case r/greaterorsimilar1 in detail.\nIII. LIMITING CASES\nA. Small jregime\nForj= 0, the chain is decoupled into an assembly of\n3-spin clusters described by the Hamiltonian\nHA(p) =S1(p)S2(p)+kS1(p)T(p)+2krS2(p)T(p).\n(2)\nAs discussed in Ref. [8], the cluster ground state is a\nsinglet state with energy\nEs=−k−1+kr\n2(3)\nfork < kc≡2\nr−1, and a triplet state with energy\nEt=−(1+k+2kr)−/radicalbig\n(1+k−4kr)2+8(k−1)2\n4\n(4)\nfork > kc. Therefore, the ground state of the chain for\nsmalljis a gapped local 3-spin singlet phase for k <\nkc. We call this phase as Gap I phase following Ref. [8].\nBecause this cluster ground state is gapped, it cannot\ngainenergywithinthefirstorderin jeveninthepresence\nofj.\nIn the 3-spin triplet ground state for k > kc, we define\na composed spin with magnitude 1 as ˆS(p)≡S1(p) +\nS2(p) +T(p) to describe low energy phenomena. The\ntotal Hamiltonian is written as H=HA\n0+HA\nintwith an3\nunperturbed part HA\n0=/summationtext\npHA(p) and an interaction\npart\nHA\nint=N/3/summationdisplay\np=1jS2(p)S1(p+1). (5)\nThe perturbation calculation up to the second order in\nthe interaction part HA\nintyields the following effective\nbilinear-biquadratic Hamiltonian for the composed spin\nˆS(p):\nHA\neff=N/3/summationdisplay\np=1/bracketleftBig\nJA\neffˆS(p)ˆS(p+1)+DA\neff(ˆS(p)ˆS(p+1))2/bracketrightBig\n,\n(6)\nwhere the effective interaction parameters consist of the\nfirst and the second order perturbation terms as JA\neff=\nJA(1)\neff+JA(2)\neffandDA\neff=DA(1)\neff+DA(2)\neff.\nIn the first orderperturbation, the effective interaction\nparameters coming from JA(1)\neffandDA(1)\neffare\nJA\neff≃ −jX(α), (7)\nDA\neff≃0, (8)\nwhereX(α) is given by\nX(α) =α√\n2(1+α2)2/parenleftbigg\n1−α\n2√\n2/parenrightbigg/parenleftbigg\n1+α2\n2/parenrightbigg\n(9)\nwith\nα=2√\n2(1−k)/radicalbig\n(k+1−4kr)2+8(k−1)2−1−k+4kr.(10)\nFor small but finite j, the energy of the LMF phase is\ngiven by Et+JA\neffper unit cell while the energy of the\n3-spin singlet (Gap I) phase is given by Eswith no first\norder correction in j. Therefore, comparing the energies\nof these two ground states, we find that the phase tran-\nsition between the gapped 3-spin singlet phase and the\nLMF phase takes place at\nk=kTF≡kc+(1−r)(2−2r+r2)(3−r)j\n2r(2−r)(3−4r+2r2).(11)\nThe effective coupling JA(1)\neffvanishes for k= 1. There-\nfore, in the neighbourhood of k= 1, the higher order\nterms come into play. Within the second order pertur-\nbation with respect to 1 −kandj, the effective coupling\nconstants are\nJA\neff≃j2\n8(r−1)(1+2r)−j(1−k)\n2(2r−1),(12)\nDA\neff≃j2\n8(r−1)(4r2−1). (13)\nThe effective model (6) has a variety of phases [19].\nWithin the present parameter regime, we find the Hal-\ndane phase for 0 < DA\neff< JA\neffwhich corresponds to Gap0 0.5 100.10.2\nGap II\n(Haldane)\nGap I\n(3 spin singlet)LMFQP\nkj r=1.2\nFIG. 2: Phase diagram for small jwithr= 1.2.\nII phase in Ref. [8], the QP phase for 0 < JA\neff< DA\neff\nand the LMF phase for JA\neff<0. However, in the orig-\ninal Hamiltonian (1), the LMF phase is limited by the\ntransition to the 3-spin singlet phase at JA\neff=Es−Et\nas discussed above. For JA\neff> Es−Et, the ground state\nis the 3-spin singlet (Gap I) phase.\nThusthe conditionsforeachphasein terms ofthe orig-\ninal parameters are summarized as follows: The ground-\nstate phase of the present model is (i) the 3-spin singlet\n(Gap I) phase for\n0< k < k TF, (14)\n(ii) the LMF phase for\nkTF< k < k FQ≡1−(2r−1)j\n4(r−1)(2r+1),(15)\n(iii) the QP phase for\nkFQ< k < k QH≡1−j\n2(2r+1),(16)\nand (iv) the Haldane (Gap II) phase for\nk > kQH. (17)\nThese phase boundaries are plotted on the k-jplane in\nFig. 2 for r= 1.2. It should be remarked that the\nQP phase in the present model is realized without bi-\nquadratic interaction in the original Hamiltonian (1).\nB. Large jRegime\nIn the large jlimit,S1(p) andS2(p−1) form an effec-\ntiveS= 1/2spinσ(p)≡S1(p)+S2(p−1). The effective\nHamiltonian is given by\nH=N/2/summationdisplay\np=1(Keff\n1σ(p)T(p)−Jeff\nFσ(p)σ(p+1)\n−Keff\nFσ(p+1)T(p)), (18)4\n.\nJFeffKFeffKeff: S=1/2T(p)\nσ(p)AF\nFerro\nFIG. 3: ∆-chain realized in the limit j >>1.\nwhich form a ∆-chain structure depicted in Fig. 3. The\neffective interactions are given by\nJeff\nF=4\n9, Keff\nF=2kr\n3, Keff\n1=4k\n3,(19)\nas argued in Ref. [8]. The detailed analysis of this model\nis reported in a separate paper [20]. Therefore, we only\nquotetheresultsandrewritethemintermsoftheoriginal\nmodel (1).\nTheferromagneticphaseofthe model(18)corresponds\nto the LMF phase in the original model (1). This phase\nis stable for\n0< k < k FQ=r−2\n3r(20)\neven in the limit of large j.\nFork > kFQ, the ground state is nonmagnetic. Never-\ntheless,therearestillseveraldifferentphases. Forlarge r,\nthe model (18) reduces to the S= 1 bilinear-biquadratic\nmodel and the QP phase appear for\nkFQ< k < k QH≡3r−4\n9r, (21)\nand Haldane phase appear for k > kQH.\nThe results of the numerical diagonalization calcula-\ntion for the model (18) is summarized in Fig. 4. There\nare the QP, the Haldane and the LMF phases, which we\ndiscussed above. In addition, numerical results suggests\nthattherepossiblyexistanarrowPPFphasebetweenthe\nLMF phase and the QP phase, and almost critical non-\nmagnetic ground states for small values of k. The PPF\nphase is so narrow that it cannot be represented in Fig.\n4. We speculate that the almost critical nonmagnetic\nphases are spin gap phases with extremely small energy\ngap with large scale resonating singlet cluster solid struc-\nture. Corresponding phases are also found for finite jas\ndescribed in the next section.\nC. Large rRegime\nWe examine the case of K2≫K1,J1(r,kr≫1) in\nthis subsection. In the the limit of r,kr→ ∞, the chain\nis decoupled into an assembly of 3-spin clusters, each of\nwhich described by the Hamiltonian\nHB(p) =jS1(p+1)S2(p)+2krS2(p)T(p).(22)0 0.1 0.2 0.301020\nr\nGap II\n(Haldane)\nkLMF\nQP\nFIG. 4: Phase diagram in large jlimit. The shaded region is\nthe almost critical nonmagnetic phase.\nThe eigenvalues of this 3-spin Hamiltonian are given by\nE(S= 2) =kr\n2+j\n2, (23)\nE(S= 1;g) =−j−2kr−/radicalbig\n16k2r2−8krj+9j2\n4,\n(24)\nE(S= 1;e) =−j−2kr+/radicalbig\n16k2r2−8krj+9j2\n4,\n(25)\nE(S= 0) =kr\n2−j. (26)\nThe ground states are the triplet states with energy\nE(S= 1;g). Itshouldbenotedthatthelowestexcitation\nenergy is of the order of kreven ifjis small. Therefore\nthe perturbation calculation from this limit is valid even\nfor small j. As a result, each cluster has an effective spin-\n1 degree of freedom ˜S(p)≡S1(p+1)+S2(p)+T(p). The\ntotal Hamiltonian is written as H=HB\n0+HB\nintwith an\nunperturbed part HB\n0=/summationtext\npHB(p) and an interaction\npart\nHB\nint=N/3/summationdisplay\np=1(S2(p)+kT(p))S1(p).(27)\nWe can write down the effective Hamiltonian for ˜S(p) up\nto the second order in HB\nintin the form,\nHB\neff=N/3/summationdisplay\np=1/bracketleftBig\nJB\neff˜S(p)˜S(p+1)+DB\neff(˜S(p)˜S(p+1))2/bracketrightBig\n.\n(28)\nWithin the first order in HB\nint, the effective interaction\nparameters are given as\nJB\neff≃ −(X(α)+kX(−α)), (29)\nDB\neff≃0, (30)5\n0 5 10−0.6−0.4−0.20\nj/rGap II\n(Haldane)r∆k\nQP\nLMFr∆kQH\nr∆kFQ\nFIG. 5: Phase diagram on the j/r-r∆kplane where ∆ k=\nk−kc(j/r).\nwhereX(α) is given by Eq. (9) with\nα=j−4kr+/radicalbig\n16k2r2−8krj+9j2\n2√\n2j.(31)\nThe effective exchange constant JB\neffvanishes up to the\nfirst order in HB\nintifjandksatisfy the relation\nj\nr=8k(k2−1)\n(3k−1)(3k−5). (32)\nWe denote the value of kwhich satisfies this relation by\nkc(j/r) as a function of j/r. The terms of O(r−1) come\ninto play for k≃kc(j/r), and constitute the bilinear-\nbiquadratic form (28) with JB\neff∼O(k−kc,r−1) and\nDB\neff∼O(r−1).\nWe do not explicitly present the second order expres-\nsion for JB\neffandDB\neff, since we numerically carried out\nthe summation over the intermediate states in the sec-\nond order perturbation calculation. The LMF-QP phase\nboundary kFQis determined by setting JB\neff= 0. Be-\ncause the correction terms are of O(r−1), deviations\n∆kFQ≡kFQ−kc(j/r) and ∆ kQH≡kQH−kc(j/r)\nscale with 1 /rfor fixed j/r. Figure 5 shows the j/r-\ndependence of r∆kFQandr∆kQH. Thephaseboundaries\nforr= 10 determined by the present approximation are\nshown in Fig. 6.\nThecalculationin thissubsectionsuggeststhat theQP\nphase found in the small- jlimit (IIIA) and that in the\nlarge-jlimit (IIIB) form a single phase, although it is\nexplicitly demonstrated only in the large rlimit.\nIV. NUMERICAL GROUND STATE PHASE\nDIAGRAM\nForsmall r, therearetwotypesofnonmagneticground\nstates and the Gaussian transition occurs between them\nas described in Ref. [8]. The perturbational approaches\nin III, however, predict the presence of the LMF and\nthe QP phases in addition to the conventional spin gap\nphases.\nWe start with the case of r= 2, where the frustration\nisfairlystrong. Thegroundstatephasediagramisshown0 0.5 102040\nj\nkLMFGap II\n(Haldane)r=10\nQP\nFIG. 6: Phase diagram for r= 10 using the approximation in\nsubsection IIIC.\n0 0.5 10123\njr=2\nLMFPPF\nGap II\n(Haldane)\nkN=18\nN=24\nQP\nFIG. 7: Ground state phase diagram of (1) for r= 2.0. In\nthis and following figures, phase boundaries determined fro m\nthe numerical diagonalization data for N= 18 and 24 are\nshown. The solid lines are guide for eye. The broken lines are\nthe results of the small- japproximation in IIIA.\nin Fig. 7. The phase boundaries calculated by using the\nnumerical diagonalization data for N= 18 and 24 are\nshown. Between the Haldane-like Gap II phase and the\nLMFphasewithspontaneousmagnetization M0=Ms/2,\nthere appear a QP phase, which is identified by the low-\nest excitation with total spin 2, [16, 17] for 0 .66/lessorsimilark≤1\nas expected from the perturbational calculation. For\nk/greaterorsimilar0.5, only the data for N= 18 are shown consid-\nering the quasi-trimerized nature of the QP phase. In\nspite of the limited system size, the phase boundary co-\nincides well with the perturbational results for small j\nas depicted by the broken lines. However, the QP phase\nvanishes when kdecreasesand jincreases. Instead, there\nappears a PPF phase with the spontaneous magnetiza-\ntion of intermediate values between Ms/2 and 0. The\ndetailed properties of this phase is discussed separately\nin the next section.6\n0 0.5 10123\nGap II\n(Haldane)Gap\n I\nGap ILMFPPF\nkjr=1.5\nN=18\nN=24\nQP\nFIG. 8: Ground state phase diagram of (1) for r= 1.5.\n0 0.5 10123\nGap IGap II\n(Haldane)\nPPF\nLMF Gap Ij\nkr=1.3\nN=18\nN=24\nQP\nFIG. 9: Ground state phase diagram of (1) for r= 1.3..\nForr <2, the 3-spin singlet Gap I phase appears for\n0< k < k c. As shown in the phase diagrams Fig. 8 for\nr= 1.5, Fig. 9 for r= 1.3 and Fig. 10 for r= 1.2,\nboth the QP phase and the PPF phase shrink to regions\naroundk∼1 with the decrease of r. Atr= 1, the 3-spin\nsinglet phase extends up to k= 1 for small j, and both\nthe QP and the PPF phases vanish.\nThe phase boundary between the Gap I and the\nGap II phases is determined by the twisted boundary\nmethod [31, 32]. In Fig. 8 for r= 1.5, the Gap I phase\nconsists of two separate regions which have the same\nparity under the twisted boundary condition. However,\nthese two regions merge with the decrease of ras shown\nin Fig. 10 for r= 1.2. Therefore we conclude these two\nregions belong to a single phase.0 0.50123\nGap II\n(Haldane)\nGap IPPF\nLMFQPN=18\nN=24r=1.2\nkj\nFIG. 10: Ground state phase diagram of (1) for r= 1.2.\nFor small k, we find the region in which the singlet-\ntriplet excitation gap ∆ Ebehaves almost critically. Fig-\nure11(a)showsthe j-dependence ofthe scaledgap N∆E\nforr= 1.3 andk= 0.12 forN= 12,18 and 24. Al-\nthough the boundary of such region cannot be precisely\ndetermined, they roughly correspond to the shaded re-\ngions in Figs. 8, 9 and 10. Applying the twist boundary\nmethod, [31, 32] we calculate the ground state energies\nE+andE−with spin inversion parities + and −, respec-\ntively. Then we find that the spin inversion parity of the\nground state, or the sign of E+−E−, changes several\ntimes with the variation of jas shown in Fig. 11(b). In\nthis region, the difference E+−E−is extremely small\n(typically less than O(10−3) forN= 24), and becomes\neven smaller with the decrease of k. This behavior is\nmost prominent for j∼1 as shown in Figs. 9 and 10,\nunless this regime is covered by the ferrimagnetic phase\nas in Figs. 7 and 8.\nThe critical values of jat which the parity changes\nare shown by the dotted lines in Figs. 8, 9 and 10 for\neach system size. They depend sensitively on the system\nsize. Due to the limitation of the system size, we cannot\nconclude whether these lines correspond to some phase\ntransitions in the thermodynamic limit. However, for\nlargej, these lines are expected to be continuously con-\nnected to the similar lines of the effective model (18) in\nthe corresponding regime (the shaded region in Fig. 4).\nThe numerically estimated values of the central charge\ncof the effective model (18) on these lines suggest that\nthey are Gaussian transition lines with c= 1 among spin\ngap phases with extremely small gap and large scale sin-\nglet clusters [20]. Therefore it is likely that these lines\nin the present model are also similar Gaussian transition\nlines. However, considering the large ambiguity in the\nestimation of cin Ref. [20], other possibilities cannot be\nruled out. The elucidation of the nature of the ground7\n.\n0 2 400.20.4\njN∆E r=1.3 k=0.12\nperiodic boundary\nN=12\nN=18\nN=24(a)\n0 2 4−0.00200.0020.004E+−E−\nN=12\nN=18r=1.3 k=0.12\ntwisted boundary\nN=24\nj(b)\nFIG. 11: (a) The j-dependence of thescaled singlet-triplet en-\nergy gap N∆Ewith periodic boundary condition and (b) the\nenergy difference between the different parity ground states\nE+−E−with twisted boundary condition. The parameters\narer= 1.3,k= 0.12 andN= 12,18 and 24. The triangles\nindicate the values of jwhere the ground state parity changes\nunder the twisted boundary condition for N= 24.\nstate in this regime is left for future studies.\nV. PARTIALLY POLARIZED\nFERRIMAGNETIC PHASE\nA. Numerical Results\nTo clarify properties of the PPF phase, we calculated\nthe spontaneous magnetization M0of the ground state\nby the density matrix renormalization group (DMRG)\nmethod. In Fig. 12, we show M0as a function of j\nforr= 1.3 andk= 0.5 withN= 72. For small j,\nthe ground state is in the Gap I phase with M0= 0.\nWhenjincreases, the LMF phase sets in where M0=\nMs/2. The slight deviation of M0/Msfrom 0.5 in this\nphaseisduetotheboundaryeffectinevitablefortheopen\nboundary DMRG. With further increase of j, the ground\nstate enters into the PPF phase where the spontaneous\nmagnetization gradually decreases down to zero.\nTypical magnetization curve calculated by the DMRG0 100.20.4M0/Ms\njr=1.3 k=0.5 N=72\nFIG. 12: The j-dependence of the spontaneous magnetization\nfork= 0.5 andr= 1.3. The results are calculated in the\nN= 72 system by the DMRG method. The magnetization is\nnormalized by the saturated magnetization Ms≡2N/3.\n0 2 400.51\nM/Ms\nHj=1.7 k=0.4 r=1.5\nN=192\n0 0.05 0.100.20.4M/Ms\nH\nFIG. 13: Magnetization curve in the PPF phase with k=\n0.4,j= 1.7 andr= 1.5 forN= 192 calculated by the DMRG\nmethod.\nis presented in Fig. 13 for r= 1.5,j= 1.7 andk= 0.4.\nThe magnetization increases continuously from the zero\nfield value in the PPF phase in contrast to the LMF\nphase where the magnetization is quantized to the zero\nfield value up to a finite critical field [4]. This implies\nthat the magnetic excitation is gapless in the PPF phase\nas in the previously reported systems [9, 21, 22].\nThe local magnetization profile /angbracketleftSz(p)/angbracketrightcalculated by\nthe DMRG is plotted against pin Fig. 14 for r= 1.5,j=\n1.7 andk= 0.4. In addition to the period 2 oscillation,\nan incommensurate modulation is clearly observed in the\nlocal magnetization.\nSimilar behaviors are found in other frustration in-\nduced PPF phases in the spin-1/2 period-3 chain with\nnext nearest neighbor interaction [11] and in the model\nof Ref. [10]. We expect these features are common as-\npects of the frustration induced quantum PPF phase.8\n0 100−0.500.51\nS1\nS2Tr=1.5 j=1.7 k=0.4\nN=192\np\nFIG. 14: Local magnetization profile in PPF phase with k=\n0.4,j= 1.7 andr= 1.5 forN= 192 calculated by the DMRG\nmethod.\nφ\nχ\nS1T\nS2θ\nFIG. 15: Classical planar spin configuration in a triangle.\nB. Classical Picture\nTo understand the physical picture of the PPF phase,\nwe consider the classical limit of the Hamiltonian (1).\nSince the J2-bonds are not frustrated, the relative angles\nbetween the spins S1(p),S2(p) andT(p), which form a\ntriangle, are not affected by jin the absence of magnetic\nfield. The ground state of the whole chain can be con-\nstructed by arranging the triangles so that S1(p) and\nS2(p−1) are antiparallel. The classical ground state en-\nergyE△\nGof a 3-spin cluster consisting of the spins S1(p),\n0 1 2024\nPPFLMFr\nk\nFIG. 16: Classical phase diagram on the k-rplane..\nS1(1)T(1)\nS2(1) S1(2) S2(2) S1(3)T(3)\nS2(3)T(2)(a)\nS1(1)T(1)\nS2(1) S1(2) S2(2) S1(3)T(3)\nS2(3)T(2)(b)\nFIG. 17: Examples of classical groundstate configurations ( a)\nwith finite magnetization and (b) with vanishing magnetiza-\ntion. The inner arrows indicate the direction of the rotatio n\nof spins along each triangle.\n0 5 1000.51r=1.5 j=1.7\nHM/Ms\nk=0.2\nk=1.0\nk=2.0\nFIG. 18: Classical magnetization curves for k= 0.2,1.0 and\n2.0 with j= 1.7 andr= 1.5.\nS2(p) andT(p) is given by\nE△\nG(φ,θ,χ) =k\n2cos(φ−θ)+kr\n2cos(φ−χ)+1\n2cos(χ−θ),\n(33)\nassuming a planar configuration. We confine ourselves\nto the planar configuration, because nonplanar configu-\nrations have higher energy. The angles θ,φandχdenote\nthe polar angle of S1(p),T(p) andS2(p) measured from\nz-axis as shown in Fig. 15, respectively. By minimiz-\ningE△\nG, we find that a stable noncollinear configuration\nwithin the triangle is realized in the region\n/vextendsingle/vextendsingle/vextendsingle/vextendsingler−1\nr/vextendsingle/vextendsingle/vextendsingle/vextendsingle< k r+1\nr,\n|⇑↑↓/angbracketright0< k is \nrequired to see possible cu rrent-induced switching in the W(3)/Co- Tb(10) device, but a current this \nlarge will typically destroy our samples. To achieve DL-SOT switching, we further reduce the \nthickness of deposited Co-Tb from 10 nm to 3.5 nm. In this thin Co-Tb case, the coercive field of \nCo-Tb is reduced to 10OecH≈ , which is beneficial for observi ng magnetization switching due to a \nlower depinning field. In Fig. 6(d), we show a representative current-induced DL-SOT switching curve \nfrom a W(4)/Co-Tb(3.5) Hall-bar de vice with lateral dimensions of 10μm6 0μm × . The critical 12 \n switching current is of ~ 1m A , from which the critical switching current density is estimated to be \n10 21.4 10 A/mcJ≈× . This number is even smaller than the case of W/Co-Fe-B, mainly due to the \nsmaller cH and magnetization eff\nFMsMt of thinner Co-Tb layer. Over all, our results indicate that \ncurrent-induced DL-SOT in W/Co-Tb heterostru cture is an efficient mechanism to induce \nmagnetization dynamics therein. Although previous st udy had shown that the field-like component of \nSOT could also possibly exist in a TM/Co-Tb system [32], the magnitude is much smaller than its \ndampinglike counterpart. Therefore, we conclude that the magnetizatio n switching we observe here is \nmainly due to current-induced DL-S OT from the SHE of amorphous W. \n \nV . CONCLUSION \n To summarize, we show that current-induced DL-SOT efficiencies DLξ from both W/Co-Fe-B \n(TM/ferromagnetic) and W/Co-Tb (TM/ferrimagnetic) heterostructures depend on the microstructure \nof W buffer layer. Thr ough hysteresis loop shift measurements, we estimate \nthin(amorphous)0.116 0.144DLξ ≈− for magnetic heterostructures with thin, amorphous W, while \nthick(crystalline)0.026 0.030DLξ ≈− for heterostructures with thick, crystalline W. By comparing results from \nboth systems, we find the spin transparency factor W/Co-Tb W/Co-Fe-B\nint intTT ≤ . We further demonstrate \ncurrent-induced DL-SOT switching in both W/Co-Fe-B and W/Co-Tb heterostructure systems. 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Rohart, E. Jue, V . Cros, and A. Fert, Europhys. Lett. 100, 57002 (2012). \n[42] T. Y . Chen, C. T. Wu, H. W. Yen, and C. F. Pai, Phys. Rev. B 96, 104434 (2017). 15 \n [43] J. H. Han, A. Richardella, S. A. Siddiqui, J. Fi nley, N. Samarth, and L. Q. Liu, Phys. Rev. Lett. \n119, 077702 (2017). \n \n 16 \n \nFigure 1. Out-of-plane hysteresis loops of (a ) W(4)/Co-Fe-B(1.4)/Hf(0.5)/MgO(2) and (b) \nW(14)/Co-Fe-B(1.4)/Hf(0.5)/MgO(2) magnetic hete rostructures. Cross s ection HR-TEM imaging \nresults from (c) W(4)/Co-Fe-B(1.4)/Hf(0.5)/MgO( 2) and (d) W(14)/Co-Fe-B(1.4)/Hf(0.5)/MgO(2) \nmagnetic heterostructures. The subpanels are the diffractograms derived by reduced fast Fourier \ntransformation (FFT) from the regi ons of interests (white boxes). \n \n \n17 \n \nFigure 2. (a) Schematic illustration of anomalous Ha ll voltage measurement. (b) Representative shifted \nHall voltage loops from a W(4)/Co-Fe-B(1 .4) sample with different DC currents DCI and an in-plane \nbias field xH=600 Oe. (c,d) Switching fields swH of W(4)/Co-Fe-B(1.4) and W(15)/Co-Fe-B(1.4) \nsamples for down-to-up (red circles) and up-to-down (blue triangles) switching pr ocesses as functions \nof DCI, with xH=600 Oe and 1500 Oe, respectively. effzH (black squares) represent the center of \nHall voltage loops. The solid li nes represent lin ear fits to effzH data. \n18 \n \nFigure 3. (a) Out-of-plane coercive field cH of the Co-Fe-B layer, (b) Hall-bar device resistance, (c) \ninverse of the Hall-bar device resistance, and (d) the magnitude of DL-SOT efficiency DLξ of \nW/Co-Fe-B magnetic heterostructures as functions of W thickness (Wt). L and w in (c) stand for length \nand width of the Hall-bar device channel, respectivel y. The solid line and dashed line in (c) represent \nlinear fits to W4nm t≤ and W4nm t> data, respectively. The red solid line and blue dashed line in \n(d) represent fits to a spin diffusion model for W4nm t≤ and W4nm t> data, respectively. \n \n19 \n \nFigure 4. (a) Schematic illustration of cu rrent-induced SOT switching measurement. swI represents \nthe amplitude of injected current pulse. The applied current pulse duration is 50 ms. (b) A \nrepresentative current-induced SO T switching result from a W(4)/Co-Fe-B(1.4) Hall-bar sample under \nin-plane bias field 80OexH= . The black arrows represent the sweeping directions of applied current \npulse swI. \n \n20 \n \nFigure 5. (a) Representative out-of-plane hysteresis loop of a W(3)/Co-Tb(10) heterostructure. (b) \nCross section HR-TEM imaging result from a W( 3)/Co-Tb(10) sample. (c,d) Switching fields swH of \nW(3)/Co-Tb(10) and W(10)/Co-Tb( 10) samples for down-to-up (red circles) and up-to-down (blue \ntriangles) switching processes as functions of DCI, both with xH=1000 Oe. effzH (black squares) \nrepresent the cente r of Hall voltage loops. The solid lines represent linear fits to effzH data. \n \n21 \n \nFigure 6. (a) Out-of-p lane coercive field cH of the Co-Tb layer and (b) the magnitude of DL-SOT \nefficiency DLξ of W(Wt)/Co-Tb(10) magnetic heterostructur es as functions of W thickness (Wt). \nThe red solid line and blue dashed line in (b) represent fits to a spin diffusion model for W4nm t≤ \nand W4nm t> data, respectively. (c) Out-of-plane hysteresis loop of a W(4)/Co-Tb(3.5) \nheterostructure. (d) Current-indu ced SOT switching curve of a W(4)/Co-Tb(3.5) Hall-bar sample under \nin-plane bias field 800OexH= . The black arrows represent the sweeping directions of applied \ncurrent pulse swI. The dashed lines serve as guide to the eye. \n" }, { "title": "2307.00475v1.Giant_coercivity__resistivity_upturn__and_anomalous_Hall_effect_in_ferrimagnetic_FeTb.pdf", "content": "1 \n Giant coercivity , resistivity upturn , and anomalous Hall effect in ferrimagnetic FeTb \nLijun Zhu,*1,2 Lujun Zhu3, Qianbiao Liu,1 Xin Lin1,2 \n1. State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of \nSciences, Beijing 100083, China \n2. College of Materials Science and Opto -Electronic Technology, University of Chinese Academy of Sciences, Beijing \n100049, China \n3. College of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China \nEmail: *ljzhu@semi.ac.cn \nAbstract: Despite the blooming interest , the transition -metal rar e-earth ferrimagnets have not been comprehensive ly \nunders tood in terms of their coercivity and transport properties. Here, w e report a systematic study of the magnetic and \ntransport properties of ferrimagnetic FeTb alloy by varying the layer thickness and temperature. The FeTb is tuned from the \nTb- dominated regime to the Fe-dominate d regime via the layer thickness , without varying the compos ition. The coercivity \nclosely follows the 1/cos θH scaling (where θH is the polar angle of the external magnetic field ) and increases quasi -\nexponentially upon cooling (exceeding 90 kOe at low temperatures) , revealing that the nature of the coercivity is the \nthermally -assisted domain wall depinning field. The resistivity exhibits a quasi -linear upturn upon cooling possibly due to \nthermal vibrations of the structure factor of the amorphous alloy . The existing scaling laws of the anomalous Hall effect in \nthe literature break down for the amorphous FeTb that are either Fe- or Tb -dominated. These findings should advance the \nunderstanding of the transition -metal -rare-earth ferrimagnets and the associated ferrimagnetic phenomen a in spintronics . \n \nI. Introduction \n \nFerrimagnetic materials (FIMs) , which have two \nantiferromagnetically coupled sublattices, are of \nconsiderable interest in the field of spintronics [1-5]. FIMs \nare potentially advantageous for dense magnetic recording \napplications [6] because of their tunable magnetism , less \nsensitivity to stray magnetic fields than ferromagnets (FMs) , \nand easier and fast detection than antiferromagnets (AFs) . \nMoreover , FIMs are an exotic platform to study the interplay \nof spin -orbit physics and AF coupling as a function of the \ndegree of magnetic compensation. Several striking spin -\norbit -coupling (SOC) phenomena have been demonstrated to \narise from the magnetization compensation within the bulk \nof the FIMs, such as large magnetic domain wall velocities \nnear compensation [3-5], strong compensation -dependen ce \nand sign reversal of bulk spin-orbit torques (SOTs) [7], strong \nvariation of the “interfacial” SOTs with t he relative spin \nrelaxation rates within the bulk of FIMs [8,9] , lack of \ncurrent -driven magnetization switching at full magnetization \ncompensation [10-12]. However, the spin -mixing \nconductance of the interfaces of metallic FIMs is insensitive \nto temperature and magnetic compensation or the areal \ndensity of the magnetic moment of the interface [8], in \ncontrast to the insulat ing FIM case [13-14]. \nAn i n-depth understanding of the ferrimagnetic \nphenomen a in magnetic heterostructures requires insights \ninto the mechanisms of the coercivity (Hc), the electron \nmomentum scattering , and the anomalous Hall effect (AHE) \nof the FIMs . Note that t he coercivity of a perpendicular \nmagnetization represents the switching barrier to overcome \nby the driving magnetic field or SOT [15-17], while electron \nmomentum scattering affects the generation [18-20] and \nrelaxation of spin current via SOC [8,21,22] . The AHE \ntypically functions as the indicator of t he magnetization \norientation in a variety of experiments (e.g., harmonic Hall \nvoltages [7-9] and magnetization switching [14-\n15,19, 23,24]). However, the magneti zation of transition -metal rare -earth FIMs arises from two competing sublattices \nthat have been suggested to contribute to the AHE in distinct \nmanners (i.e., the 3d states of the transition metal governed \nthe transport properties , while the 4 f states of Tb were less \ninvolved [23,25,26]). The scaling behavior of the AHE in \nsuch transition -metal rare -earth FIMs thus become s \nstimulating open question s. \nIn this work , we systematically examine t he magnetic \nand transport properties of FeTb, a representative FIM, as a \nfunction of layer thickness and temperature. We show that \nthe coercivity can be rather high and increase quasi -\nexponentially upon cooling due to the nature of the \nthermally -assisted domain wall depinning field . The large \nlongitudinal resistivity (ρxx) increases quasi -linearly upon \ncooling . The anomalous Hall resistivity ( ρAH) cannot be \ndescribed by the existing scaling laws [27-29]. \n \nII. Samples and magnetization \n \nFor this work, we deposit four FeTb films with different \nthicknesses (t = 8 nm, 16 nm, 32 nm, and 48 nm) on Si/SiO 2 \nsubstrates by co -sputtering at room temperature. Each \nsample is capped by a MgO ( 2 nm)/Ta ( 2 nm) bilayer that is \nfully oxidized upon exposur e to the atmosphere. The \ncomposition of all the FeTb layers is Fe0.55Tb0.45 in volume \npercentage as calibrated using the deposition rate s of the Fe \nand the Tb. This composition correspon ds to a Tb/Fe atomic \nratio of ≈ 0.3 (as calculated using the atomic volumes of bulk \nFe and Tb crystals) , which is consistent with the energy \ndispersive spectr oscopy (EDS) result s (0.32± 0.01) in Figs. \n1(a) and 1(b). Such FeTb films are amorphous and \nhomogeneous as indicated by scanning trans mission electron \nmicroscopy and electron energy loss spectra results of the \nsamples prepared by the same sputtering tool using similar \nparameters within the same few days [7]. 2 \n \nFig. 1. (a) Energy dispersive spectra collected using a Bruker \nEDS tool and (b) the Tb/Fe atomic ratio calculated using the \nTb L α and Fe K α EDS spectra for the FeTb films with \ndifferent thicknesses, revealing that the composition is the \nsame for the FeTb films studied in this work. \n \nThe saturation magnetization (Ms) for each FeTb film, \nthe sum contribution of the Fe and the Tb sublattices ( Fig. \n2(a)), is measured using a superconducting quantum \ninterference device (SQUID). Figure 2(b) shows the \ntemperature profile of the saturation magnetization of the \nFeTb films with dif ferent thicknesses. Ms for the 8 nm FeT b \nvaries non -monotonically with temperature and peaks at 250 \nK. For the 16 nm FeTb, Ms increases monotonically from \nbeing negligibly small at temperatures below 50 K to ≈60 \nemu/cm3 as the temperature approaches 300 K. In contrast, \nfor both the 32 nm and 48 nm FeTb, Ms decreases first slowly \nand then more rapidly as the temperature increases. The \ndiverse temperature profile s of the saturation magnetization \nof the FeTb films impl y a strong tu ning of the magnetization \ncompensation configuration by thickness. As plotted in Fig. \n2(b), the magnetization exhibits full compensation at the \n“compensation thickness” of 16 nm at 5 K and of ≈20 nm at \n300 K. Thus, the magnetization compensation of the FeTb \nalloy is not only a function of substrate [7], composition \n[7,30,31 ], and strain [ 32] but also of temperature and \nthickness. \nThe films are then patterned into 5×60 μm2 Hall bars for \nelectrical measurements using a physical properties \nmeasurement system (PPMS -9T). From the AHE \nmeasurements (see Figs. 2(c) and 2(d)), the FeTb films have \nfairly square hysteresis loops for the t ransverse resistivity \n(ρxy), implying good magnetic uniformity of these films. The \npolarity of the hysteresis loop is opposite for the 8 nm and \n16 nm FeTb samples compared to that for the 32 nm and 48 nm ones, suggesting that the films are Tb -dominated at 8 nm \nand 16 nm but Fe -domin ated at 32 nm and 48 nm. Such \nstriking thickness dependences of the magnetization and the \ntransverse resistivity are interesting observations and worth \nfuture investigation . While the unambiguous identification \nof the exact mechanism is beyond the scope of this paper, we \nspeculate that the striking thickness dependence of the \nmagnetic and transport properties might be related to some \nhidden short -range ordering within the amorphous films . \n \nIII. Giant, strongly temperature -dependent coercivity \n \nFigure 3(a) shows the out -of-plane coercivity and the \neffective perpendicular magnetic anisotropy field ( Hk) at 300 \nK. Here, the out -of-plane coercivity is estimated from the \nswitching of the transverse resistivity by a perpendicular \nmagnetic field ( Hz, Fig. 2(d)), while Hk is estimated from the \nparabolic scaling of ρxy with the in-plane magnetic field (Hxy) \ndue to tilting of the magnetization ( Fig. 3(b)), i.e., \n ρxy = ρAH cos(arcsin( Hxy/Hk)) ≈ ρAH [1-1/2(Hxy/Hk)2]. (1) \nBoth Hc and Hk vary as a function of the layer thickness and \ntend to increase upon approaching the “compensation \nthickness”. As shown in Fig. 3(c) and 3(d), Hc follows a \n1/cos θH scaling (θH is the polar angle of the driving magnetic \nfield Hxz) and is typically much smaller than the \nperpendicular magnetic anisotropy field at θH = 0o. This is i n \ncontrast to the case of a perpendicular macrospin , for which \nthe coercivity varies as \nHc = H k (cos2/3θH + sin2/3θH)-3/2, (2) \nand is equal to Hk at θH = 0o and 90 o. As plotted in Fig. 3(e), \nthe out-of-plane coercivity of each FeTb film increases upon \ncooling in a quasi -exponential manner . These observations \nconsistently reveal that the coercivity of the FeTb represents \nthe thermally -assisted depinning field of the magnetic \ndomain walls rather than the bulk magnetic anisotropy field. \nThis is general ly the case for magnetic systems in which the \nreversed domain nucleation and domain wall propagation \n(with the energy barrier of the domain wall pinning field ) \nrequire less energy than coherent rotation (with the energy \nbarrier of Hk) [17]. \nWe also note that the out -of-plane field required to \nswitch these films, i.e., the coercivity , exceeds 90 kOe at \ntemperatures below 175 K for the 16 nm FeTb and at \ntemperature s below 25 K for the other three samples. The \nrapid increase of the coercivity upon cooling is distinct from \nthe enhancement of coercivity at the magnetization \ncompensation points [ 1-5,7, 33] because it occurs in the \nwhole temperature region, includ ing the temperatures at \nwhich the magnetization is very hi gh (Fig. 2(b) ). The giant \ncoercivity and square hysteresis loops ( Fig. 3(f)) may make \nthese FeTb interesting hard magnets for some specific \nspintronic applications [34]. \n3 \n \nFig. 2. (a) Schematic depict of the ferrimagnetism and the anomalous Hall effect in the FeTb. (b) Dependence of the saturation \nmagnetization on temperature , (c) Dependence of the saturation magnetization on the thickness at 5 K and 300 K, a nd (d) \nTransverse resistivity at 300 K for the FeTb with different layer thicknesses. \n \nFig. 3. (a) Perpendicular coercivity and perpendicular magnetic anisotropy field and (b) Parabolic scaling of the transverse \nresistivity with in -plane magnetic field for FeTb with different thicknesses (300 K). (c) Dependence on the polar angle ( θH) \nof the room -temper ature coercivity. (d) Transverse resistivity hysteresis loops for the 8 nm FeTb measured at θH = 0o and 85o \nand 300 K. (e) Dependence on the temperature of the perpendicular coercivity (θH = 90o) of the FeTb films. (f) Transverse \nresistivity hysteresis loop at 25 K for the 32 nm FeTb, displaying a giant coercivity of 65 kOe . \n4 \n IV. Resistivity upturn \nResistivity or electron momentum scattering is also a \nkey property of a spintronic material. For ins tance, e lectron \nmomentum scattering affects spin-dependent scattering , the \ngeneration and relaxation of spin current via SOC . To provide \ninsight into the electron momentum scattering mechanism, \nwe measure the resistivity of the FeTb samples as a function \nof temperature (Fig. 4(a)). ρxx varies between 210 µΩ cm and \n250 µΩ cm. In analog y to the magnetic properties and the \nanomalous Hall resistivity (see below) , ρxx shows also an \ninteresting non -monotonic thickness dependence at each \nfixed temperature due to some exotic mechan ism yet to know . \nHere, i nterfacial scattering is unlikely to play any significant \nrole in the determination of ρxx of these thick , resistive FeTb \nbecause they should have a very short mean -free path. \nMore interestingly, ρxx shows a quasi -linear upturn upon \ncooling for each sample . Similar resistivity upturn has also \nbeen observed in 200 nm thick FeTb films [35]. In general, a \nresistivity upturn can arise from weak localization, hopping \nconductance, orbital one -channel Kondo effect, orbital two -\nchannel Kondo effect, electron -electron scattering, magnetic \nBrillouin zone scattering , or scattering of electrons by \nthermal vibration of structure factor . However, none of these \nmechanisms appear to explain the resistivity upturn of these \nFeTb films. First, w eak localization , which diminishes under \nan external magnetic field or a strong internal exchange field, \nis not expected in the ferrimagnetic FeTb that have giant \nperpendicular magnetic anisotropy (Fig. 3(a)). Hopping \nconductance is known to occur in Mott -Anderson insulators \nwith extremely high resistivity (e.g., 106-109 µΩ cm for \nquasicrystal AlPdRe [36], amorphous GeTe, and GeSb 2Te4 \nannealed at 150 oC [37], 1011-1017 µΩ cm for the Pt -SiO 2 \ngranular film with Pt concentration of 0.11 [38,39]) but not \nin metals like FeTb with several orders of magnitude lower \nresistivity . The absence of hopping conductance is reaffirmed \nby the lack of a T-1/4 scaling (so-called Mott’s law [40]) in the \nresistivity ( Fig. 4(b)). The orbital one -channel Kondo effect \n[41], if important, should increase the resistivity as a function \nof ln T, which is not the case for the FeTb ( Fig. 4(c)). The \nresistivity upturn due to e lectron -electron interaction would \nfollow a T1/2 scaling at low temperatures [42], which is not \nconsistent w ith the evident deviation from the T1/2 scaling at \ntemperatures below 150 K ( Fig. 4(d)). \nThe orbital two -channel Kondo effect [41,43-45] is also \nless likely to explain the resis tivity upturn in the amorphous \nFeTb films. This is because it would imply a Kondo \ntemperature of >300 K (below which the T1/2 scaling emerges) \nand a deviation temperature of 150 K (below which the \nresistivity deviates from the T1/2 scaling), both of which are \nsurprisingly high. Note that the Kondo temperature is only 23 \nK for the L10-MnAl [42] and 14.5 K for L10-MnGa films [45], \na few K for glasslike ThAsSe [43] and Cu point contacts [41], \nwhile the deviation temperature is typically below 1 K for all \nthe previously studied two -channel Kondo systems [41-43]. \nAnother possible mechanism for a resistivity upturn is \nthe magnetic Brillouin zone scattering (the periodic potentials \ndue to antiferromagnetic alignment of the magnetic \nsublattices can produce an additional magnetic Brillouin zone, of smaller volume in k-space than the ordinary lattice \npotential, whose planes further incise and contort the Fermi \nsurface [47]). While this possibility cannot be quantitatively \ntested due to a lack of knowledge about the exact functional \ndependence on temperature, magnetic Brillouin zone \nscattering should be weak in the amorphous FeTb which has \nno long -range periodicity in the crystalline and magnetic \nlattices. The resistivity upturn is also absent in epitaxial \nferrimagnets of Mn 1.5Ga [28] and Mn 2Ga [48] in which the \ntwo Mn sublattices are also AF coupled. \nAfter we have excluded any important role of weak \nlocalization, hopping conductance, orbital one -channel \nKondo effect, orbital two -channel Kondo effect, electron -\nelectron scattering, and magnetic Brillouin zone scattering , \nscattering of electrons by thermal vibration s of the structure \nfactor [49,50] is left as the most likely mechanism for the \nincrease of resistivity with decreasing temperature over a \nwide range of temperature in our amorphous FeTb films. \nThermal vibration s of the structure factor have been reported \nto explain the resistivity upturn in many liquid transition \nmetals and metallic glass alloys [35,49,50]. Note that such \nthermal vibration s of the structure factor in disordered alloys \nare distinct from the phonon scattering that increases t he \nresistivity with increasing temperature in ordered crystalline \nmaterials [ 27-29,44 ]. Future theoretical calculations of the \nstructure factor as a function of temperature would be \ninformative for a more quantitatively understanding of the \nresistivity of the FeTb samples, which is, however, beyond \nthe scope of this article. \n \nFig. 4. Resistivities ( ρxx) of the FeTb films plotted as a \nfunction of (a) temperature T, (b) T (in log plot), (c) T-1/4, and \n(d) T1/2. In (b) -(d) the resistivity data for the 8 nm FeTb is \nmultiplied by 1.05 for clarity. In (b) the l og plot of ρFeTb as a \nfunction of T-1/4 indicates a lack of Mott’s law for hopping \nconduction [38], the latter predicts ln ρxx to vary linearly with \nT-1/4. In (d) the straight lines represent the best linear fits to \nthe data in the high-temperature regime. \n5 \n \nFig. 5. Scaling of the anomalous Hall effect. (a) Dependence on the temperature of ρAH of the FeTb films with different \nthicknesses. (b) ρAH vs ρxx2 and (c) ρAH vs ρxx for the FeTb films and the control Mn 1.5Ga sample. The solid straight lines in \n(b) represent f its of the data to Eq. (3) and the solid curves in (c) represent the fits of the data to Eq. (4) . \n \nV. Scaling of the s trong Anomalous Hall effect \n \nWe now discuss the scaling of the anomalous Hall \nresistivity ( ρAH) with the longitudinal resistivity ( ρxx). The \nscaling analysis is interesting as it can disentangle the \nintrinsic and extrinsic contributions of the anomalous Hall \nresistivity of magnetic materials in which the electron \nscattering is dominated by impurity and phonon scattering. In \nthat case, ρAH of a given sample is simply a linear function of \nρxx2, i.e., \n ρAH = αρxx0 + β0 ρxx02 +bρxx2, (3) \nwhere α, β0, ρxx0, and b are constant for a given sample and a \n= αρxx0 + β0 ρxx02 goes to zero when the residual resistivity ρxx0 \n(due to static impurity scattering at low temperatures) is zero. \nEquation (3) describes the AHE scaling of epitaxial FIM \nMn 1.5Ga [28] and some other 3 d ferromagnets. Hou et al. [29] \nalso proposed a multivariable scaling relation for the AHE in \nmagnetic materials in very high -conductivity regime by \nassuming two major competing scattering sources: i.e. \n ρAH = αρxx0 + β0 ρxx02 + γ0 ρxx0 ρxxT + β1ρxxT2, (4) \nwhere ρxx0 is also the residual resistivity and ρxxT = ρxx-ρxx0 is \nresistivity due to dynamic phonon scattering at high \ntemperatures , α, β0, γ0, and β1 are fitting parameters. Equation \n(4) describes well the AHE scaling in epitaxial ferromagnetic \nFe [27] grown by molecular -beam epitaxy. For convenience, \nwe rewrite Equation (4) as \nρAH = αρxx0 + (γ 0-2 β1) ρxx0ρxx +( β0+ β1-γ0) ρxx02 + β1ρxx 2, (5) \nWe note that Eq. (3) -Eq. ( 5) predict that ρAH is a monotonic \nfunction of ρxx and scales smoothly to zero at zero ρxx (ρxx0 = \nρxxT=0). In Fig. 5(a), we plot the values of ρAH for the FIM FeTb \nas a function of temperature. While the anomalous Hall \nresistivities of the Fe -dominated and Tb -dominated films are \nof opposite signs, the magnitude increases monotonically, by \n50%, for each sample. A similar increase of ρAH with \ntemperature has also been reported in ferromagnetic MnAl \nwith orbital two -channel Kondo effect [51] and is distinct \nfrom that of FMs (e.g., Fe [27], Co [52], Ni [53], and FePt \n[54]) and FIMs (MnGa [28]) in which the electron scattering \nis dominated by impurity scattering and p honon scattering. \nMore surprisingly, ρAH of the FeTb is not a linear function of \nρxx2 and even does not have an obvious monotonic scaling \ntowards zero as ρxx decreases. This observation suggest s the \nbreakdown of the conventional AHE scaling for the dirty \nmetal of amorphous FeTb ferrimagnets . This breakdown is \nnot a general case for FIMs as the Mn 1.5Ga with AF -coupled \nMn sublattices does follow Eq. (3) (see Fig. 4(b) ). \nWe show i n Fig. 5(c) that the anomalous Hall resistivity \nphenomenally foll ows the law \n ρAH = α +βρxx + γρxx2, ( 6) \nwhere α, β, and γ are non-zero constant s. Equation (6) \npredicts a peak and decay in ρAH at very high resistivities \nwhich might be consistent with the expectation that ρAH \nshould reduce to wards zero in the limit of infinite ρxx \n(insulators). Note that we have tested that the data in Fig. 5(c) \ncannot be fit by a monotonically varying exponential, \nlogarithm, hyperbola, and othe r functions. The underlying \nphysics and the precise application regime of the new scaling , \nEq. ( 6), require theoretical and experimental investigations in \nthe future and is beyond the scope of this work. \nFinally, we mention that the anomalous Hall resistivity \nof the FeTb is giant compared to that of the 3 d magnets Fe \n[27], Co [52], Ni [53], Co40Fe40B20 [55], Mn 1.5Ga [28], MnAl \n6 \n [51], and Mn 3Ge [56] (Fig. 6(a)) due to the large anomalous \nHall angle ( ρAH/ρxx, see Fig. 6(b)) and the high resistivity. As \nplotted in Fig. 6(c), the anomalous Hall conductivity of the \nFeTb is also stronger than that of MnAl, MnGa, Mn 3Ge, and \nCoFeB with significantly higher longitudinal conductivities. \nSuch giant anomalous Hall effect is highly preferred for \nsensor applications . \n \nFig. 6. Dependence on the longitudinal resistivity of (a) the \nanomalous Hall resistivity and (b) the anomalous Hall angle \nof representative magnetic films . (c) The anomalous Hall \nconductivity of the same materials plotted as a function of the \nlongitudinal conductivity. \n \nConclusion \nWe have presented a systematic study of the magnetic and \ntransport prop erties of the ferrimagnetic FeTb alloy by \nvarying the layer thickness and temperature. The FeTb is \ntuned from the Tb -dominated regime to the Fe -dominated \nregime simply via the increase of the layer thickness , without \nvarying the composition . For each of th e studied FeTb \nsamples, the coercivity closely follows the 1/cos θH scaling \n(where θH is the polar angle of the external magnetic field ) \nand increases quasi -exponentially upon cooling and exceeds \n90 kOe below a certain low temperature, revealing that the \nnature of the coercivity is thermally -assisted domain wall \ndepinning field. The resistivity increases quasi -linearly with \ntemperature upon cooling likely due to thermal fluctuations of the structure fact or of the amorphous FeTb . The anomalous \nHall resistivities of both Fe - or Tb -dominated FeTb layers \nthat are in the dirty limit cannot be described by any of the \nexisting AHE scaling laws proposed in the literature. These \nexotic findings should advance the understanding of the \nmagnetic and transport behaviors of the transition -metal -rare-\nearth ferrimagnets. \n \nThe authors thank Changmin Xiong for help with PPMS \nmeasurements. 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Awad,1Chunlei Wang,2Mohammad Zahedinejad,1Nilamani\nBehera,1Himanshu Fulara,1Roman Khymyn,1Afshin Houshang,1Jonas Weissenrieder,2and J. \u0017Akerman1,b)\n1)Physics Department, University of Gothenburg, 412 96 Gothenburg, Sweden.\n2)Department of Applied Physics, KTH Royal Institute of Technology, 106 91 Stockholm,\nSweden\nFerromagnetic materials dominate as the magnetically active element in spintronic devices, but come with\ndrawbacks such as large stray \felds, and low operational frequencies. Compensated ferrimagnets provide an\nalternative as they combine the ultrafast magnetization dynamics of antiferromagnets with a ferromagnet-like\nspin-orbit-torque (SOT) behavior. However to use ferrimagnets in spintronic devices their advantageous prop-\nerties must be retained also in ultrathin \flms ( t<10 nm). In this study, ferrimagnetic Gd x(Fe87:5Co12:5)1\u0000x\nthin \flms in the thickness range t= 2{20 nm were grown on high resistance Si(100) substrates and studied\nusing broadband ferromagnetic resonance measurements at room temperature. By tuning their stoichiometry,\na nearly compensated behavior is observed in 2 nm Gd x(Fe87:5Co12:5)1\u0000xultrathin \flms for the \frst time,\nwith an e\u000bective magnetization of Me\u000b= 0.02 T and a low e\u000bective Gilbert damping constant of \u000b= 0.0078,\ncomparable to the lowest values reported so far in 30 nm \flms. These results show great promise for the\ndevelopment of ultrafast and energy e\u000ecient ferrimagnetic spintronic devices.\nI. INTRODUCTION\nSpintronic devices utilize the spin degree of freedom for\ndata storage, information processing, and sensing1,2with\ncommercial applications such as hard drives, magnetic\nrandom access memories, and sensors. Besides conven-\ntional memory applications based on quasi-static opera-\ntion of magnetic tunnel junctions, high frequency spin-\ntronic oscillators3,4have recently been demonstrated for\nanalog computing applications such as bio-inspired neu-\nromorphic computing5,6, logic operations, energy har-\nvesting and Ising Machines.7For the \frst time, such oscil-\nlators are now used in commercial magnetic hard drives\nto facilitate writing to the disc.8The key challenges in\ndeveloping such devices is to \fnd material combinations\nwhich allow for fast operation, low-power consumption,\nnon-volatility, and high endurance. Due to their nat-\nural spin polarization and easy manipulation, ferromag-\nnetic materials (FM) dominate as active elements in these\ndevices.4However, FMs come with drawbacks such as:\n(i) large magnetic stray \felds a\u000becting the operation of\nneighbouring devices; (ii) limited scalability of magnetic\nbits in memory devices; (iii) the operating frequency of\nspin-based oscillators limited by ferromagnetic resonance\nfrequency, and (iv) slow synchronization of such oscilla-\ntors. These shortcomings drive researchers to \fnd more\nsuitable materials for future spintronic devices.\nVery recently, the interest in antiferromagnetic (AFM)\nspintronics9{11increased rapidly, as AFM materials have\nno stray \felds and can o\u000ber ultrafast spin dynamics, in-\ncluding AFM resonance frequencies in the THz region.\nIt was theoretically shown that such high-frequency ex-\ncitations are possible to achieve without any applied\nmagnetic \feld by injecting spin currents into AFM\na)Electronic mail: lakhan.bainsla@physics.gu.se\nb)Electronic mail: johan.akerman@physics.gu.sematerials.12{15Experiments have since demonstrated\npossible THz writing/reading capabilities.16However,\nthe absence of a net magnetic moment in AFMs leads\nto di\u000eculties in the read-out of the spin dynamics, in-\ncluding any microwave output signal from the AFM\noscillators.13{15\nA possible solution is presented by ferrimagnets\n(FiMs), which combine the properties of FMs and AFMs.\nFiMs posses magnetic sub-lattices in the same way as\nAFMs do, but their sub-lattices are inequivalent. The\nmagnetic sub-lattices in FiMs often consist of di\u000berent\nmagnetic ions, such as rare earth (e.g. Gd) and transi-\ntion metal (e.g. Fe, Co) alloys (RE-TM) such as CoGd,\nand as a result, a large residual magnetization remains\ndespite the two opposing sub-magnetizations. The tem-\nperature dependence of RE and TM sub-magnetizations\nin FiM can be quite di\u000berent which result in magneti-\nzations that can increase, and even change sign, with\ntemperature17,18, in stark contrast to the non-monotonic\ndecreasing temperature dependence for FMs and AFMs.\nSimilar e\u000bects could also be seen by varying the com-\nposition of ferrimagnetic alloys instead of changing the\ntemperature.19In addition, the di\u000berent properties of the\ntwo magnetic sub-lattices also results in two compen-\nsation points, namely the magnetization compensation\npointTmand the angular compensation point Ta. AtTm,\nthe two magnetic sub-lattices cancel each other, which re-\nsults in a zero net magnetic moment, while at T a, their\nnet angular momentum vanishes, as in AFMs. Therefore,\natTa, FiMs can have a near-THz resonance as in AFMs,\nwhile still having a net magnetic moment which can lead\nto strong read-out signals, including e\u000ecient microwave\nsignal output from FiM-based oscillators20, as well as ef-\n\fcient control and excitation. FiMs also show high spin\npolarization which also make them suitable candidate for\ne\u000ecient magnetic tunnel junctions.21\nDue to these unique properties, research in FiMs for\nspintronic applications is intensifying22, focusing mainly\non RE-TM based systems such as CoTb23, CoGd24, andarXiv:2111.08768v1 [cond-mat.mtrl-sci] 16 Nov 20212\nFigure 1. (a) Schematic illustration of the coplanar waveguide (CPW), the thin \flm sample and its orientation, the directions\nof the applied magnetic \feld H, the microwave \feld hrf, and the e\u000bective magnetic \feld He\u000bduring FMR measurements.\nInset shows the \flm stack. (b) FMR response (derivative of the FMR absorption) for a 10 nm Gd 12:5Fe76:1Co11:4\flm (S2)\nrecorded at di\u000berent frequencies and \ftted (solid lines) to Eq. 1. While FMR curves were recorded at 1 GHz frequency intervals\nthroughout this study, \fgure (b) only shows curves with \u0001 f= 2 GHz for clarity.\nGdFeCo25and Mn 3\u0000xPtxGa26,27based Heusler alloy.\nAmong these, GdFeCo has been studied the most with\ndemonstrations of fast domain wall motion28and ultra-\nfast spin dynamics17nearTa, large spin-orbit torques\nand their sign reversal,25,29low magnetic damping in\nthick 30 nm \flms,30and sub-picosecond magnetization\nreversal,31to name a few. What is missing, however, is a\ndemonstration that these unique material properties per-\nsist down to much thinner \flms, which will ultimately be\nneeded if FiMs are to be used in spin-Hall nano oscillators\n(SHNOs).4\nIn the present study, we systematically study the\ngrowth and functional properties of ultrathin ferrimag-\nnetic Gd x(Fe87:5Co12:5)1\u0000xthin \flms [referred to as\nGdx(FeCo) 1\u0000xhereafter]. GdFeCo thin \flms in the\nthickness range of 2{20 nm were grown on high resis-\ntance silicon (HR-Si) substrate. The atomic composi-\ntion of Gd x(FeCo) 1\u0000xwas controlled using co-sputtering\nand determined using inductively coupled plasma optical\nemission spectroscopy (ICP-OES). The magnetic prop-\nerties and Gilbert damping were studied using broad-\nband ferromagnetic resonance (FMR) measurements. We\nalso demonstrate ultra low Gilbert damping for 2 nm\nGdFeCo, near the compensation point of Gd x(FeCo) 1\u0000x.\nThese results paves the way for integration of FiMs into\nvarious spintronic devices and applications.\nII. RESULTS AND DISCUSSION\nThe growth conditions for GdFeCo were \frst optimized\nby growing four 10 nm thick Gd 12:5Fe76:1Co11:4\flms on\nHR-Si (100) substrates using di\u000berent MgO seed layer\nthicknesses: 0 nm (S1), 6 nm (S2), 10 nm (S3 & S4);in S4, the seed was annealed at 600C for 1 hour prior\nto GdFeCo deposition to check the e\u000bect of MgO crys-\ntallinity. MgO was chosen as seed since it is insulating\nand therefore will not contribute any spin sinking to the\nmagnetic damping.32\nA. Seed layer dependence on 10nm thick\nGd12:5Fe76:1Co11:4\flms\nFurther details of the growth conditions are given in\nthe experimental section. FMR measurements, on 6 \u00023\nmm2rectangular pieces cut from these \flms, were then\nperformed using a NanOsc PhaseFMR-40 FMR Spec-\ntrometer. The sample orientation on the coplanar waveg-\nuide (CPW), together with the directions of the applied\n\feld, the microwave excitation \feld hrf, and the e\u000bec-\ntive magnetic \feld He\u000b, are shown in Fig. 1(a). Typical\n(derivative) FMR absorption spectra obtained for S2 are\nshown in \fgure 1(b) together with \fts to a sum of sym-\nmetric and anti-symmetric Lorentzian derivatives:33\ndP\ndH(H) =\u00008C1\u0001H(H\u0000HR)\n[\u0001H2+ 4(H\u0000HR)2]2+2C2(\u0001H2\u00004(H\u0000HR)2)\n[\u0001H2+ 4(H\u0000HR)2]2\n(1)\nwhereHR, \u0001H,C1, andC2represent the resonance \feld,\nthe full width at half maximum (FWHM) of the FMR ab-\nsorption, and the symmetric and anti-symmetric \ftting\nparameters of the Lorentzian derivatives, respectively.\nThe extracted values of HRvs.fare shown in \fgure\n2 (b) together with \fts to Kittel's equation:34\nf=\r\u00160\n2\u0019q\n(HR\u0000Hk)(HR\u0000Hk+Meff) (2)3\nFigure 2. (a) Seed layer dependence of frequency vs resonance \feld of the 10 nm thick Gd 12:5Fe76:1Co11:4\flms, here solid\nsymbols and solid lines are the experimental data points and \ftting with equation (2), respectively. (b) Resonance linewidth\n(\u0001H)vs.frequency of the 10 nm thick Gd 12:5Fe76:1Co11:4\flms, here solid symbols and solid lines are the experimental data\npoints and \ftting with equation (3), respectively. The e\u000bective Gilbert damping constant values of all the samples are given in\n\fgure 2 (b). The black and violet dotted lines in \fgure 2(b) shows the \ftting of equation (3) in low and high frequency regions,\nrespectively.\nwhere,\r,HkandMe\u000bare the gyromagnetic ratio, the\nin-plane magnetic anisotropy \feld, and the e\u000bective mag-\nnetization of the sample, respectively, all allowed to be\nfree \ftting parameters. Values for \randHkonly showed\nminor variation between the four samples, with \r=2\u0019=\n29.4-30.0 GHz/T and Hk= 66-104 Oe. Me\u000bvaried more\nstrongly, with values of 0.79, 1.19, 0.71 and 0.76 T ob-\ntained for S1, S2, S3 and S4, respectively.\nThe e\u000bective Gilbert damping constant \u000bcan then be\nobtained from \fts of \u0001 Hvs.fto:35\n\u0001H= \u0001H0+4\u0019\u000bf\n\r\u00160(3)\nwhere the o\u000bset \u0001 H0represents the inhomogeneous\nbroadening. Equation (3) is well \ftted to the experimen-\ntal values, using \u0001 H0and\u000bas adjustable \ftting param-\neters for all the four samples, as shown in the \fgure 2(b).\n\u0001H0= 2{4 mT is essentially sample independent within\nthe measurement accuracy. In contrast, the obtained val-\nues of\u000bvary quite strongly and are given inside \fgure\n2(b). The GdFeCo grown with 6 nm MgO seed layer (S2)\nclearly shows the lowest value of \u000b= 0:0055, although\nthis might be a\u000bected by the slight non-linear behavior\naround 10 to 15 GHz. However, when only the high-\feld\ndata is \ftted, the extracted damping of \u000b= 0.0076 is\nstill the lowest and at all frequencies the linewidth of S2\nlies well below all the other samples. As damping is one\nof the most important parameters for spintronic devices,\nwe hence chose the growth conditions of S2 for all subse-\nquent \flms in this study.B. Thickness dependence on Gd 12:5Fe76:1Co11:4\flms\nAfter optimizing the growth conditions for\nGd12:5Fe76:1Co11:4, the thickness dependence of the\n\flms was studied with the same composition using the\ngrowth conditions of sample S2. The FMR linewidth\n\u0001Hvs. f is shown in \fgure 3(a) and exhibits a relatively\nstrong dependence on thickness. It is noteworthy\nthat the 4 nm \flm shows the narrowest linewidth at\nall frequencies, clearly demonstrating that very low\ndamping can be achieved also in ultra-thin GdFeCo.\nThe extracted Me\u000band\u000bare shown vs.thickness in\n\fgure 3(b), both showing a strong thickness dependence.\nDamping as low as \u000b= 0:0055 is obtained for the 10\nnm thick \flms. If only the high-\feld portion of the data\nis \ftted, the extracted damping increases to 0.0076,\nwhich is still about an order of magnitude lower than\nany literature value on 10 or 30 nm \flms.19,36Both\nthe 10 and 20 nm \flms showed a minor nonlinearity in\n\u0001Hvs.fdata and were therefore analysed by \ftting\nthe data in both the low and the high \feld regions\nseparately, as shown by the dotted lines in \fgure 3(a).\nThe\u000bvalue for the 20 nm \flm increased slightly from\n0.0098 to 0.0109 if only high \feld data is used for\nanalysis. The relatively higher damping for the 20 nm\n\flm might be due to the radiative damping mechanism\nwhich increases proportionally with magnetic layer\nthickness.37We conclude that 2 nm ultrathin \flms can\nindeed be grown with reasonably low damping. Since\nthe damping is strongly thickness dependent in this\nregime, the optimum thickness for devices may likely be\nfound in the 2{4 nm range.4\nFigure 3. (a) FMR linewidth \u0001 Hvs.ffor four Gd 12:5Fe76:1Co11:4\flms with di\u000berent thicknesses, together with linear \fts to\nequation (3). The dotted lines show \fts for the 20 nm \flm in its low and high frequency regions, respectively. (b) E\u000bective\nmagnetization and e\u000bective Gilbert damping constant vs.thickness; lines are guides to the eye.\nC. Composition dependence on 2nm thick \flms\nTo \fnally investigate whether we can achieve a com-\npensated ferrimagnetic behavior also in ultra-thin \flms,\nwe grew 2 nm Gd x(FeCo) 1\u0000x\flms in the composition\nrange 12{27 at.% Gd. The \flms were characterized using\nFMR spectrometry as described above and the extracted\nresults are shown in \fgure 4.\nThe extracted Me\u000band\u000bfollow a similar trend as re-\nported earlier for one order of magnitude thicker GdFeCo\n\flms characterized using an all-optical pump-probe tech-\nnique.17We \frst note that we can indeed reach an es-\nsentially fully compensated antiferromagnetic behavior in\ntwo \flms around a composition of 25 at.% Gd. We have\nmarked this compensation point with xmand a dashed\nline in \fgure 4 (c). Both \flms show very low damping of\n0.0078 and 0.009 respectively. However, just below this\ncomposition, the damping shows a peak, which is con-\nsistent with an angular compensation point, which we\ndenote byxa. It is noteworthy that the extracted damp-\ning value of \u000b= 0.0142 is still more than an order of\nmagnitude lower than \u000b= 0.45 of 30 nm \flms measured\nusing FMR spectrometry19and\u000b= 0.20 of 20 nm \flms\nmeasured using an optical pump-probe technique.17\nIII. CONCLUSION\nIn view of the potential application of compensated\nferrimagnets to spintronic devices, we prepared ferri-\nmagnetic thin \flms of Gd x(FeCo) 1\u0000xon high resistance\nSi(100) substrates and studied them using the FMR mea-\nsurements. Their growth conditions were optimized us-\ning 10 nm thick Gd 12:5Fe76:1Co11:4\flms, after which\nthickness dependent studies were done on the same com-\nposition in the thickness range of 2{20 nm. Composi-\ntion dependence studies were \fnally done on 2 nm thick\nGdx(FeCo) 1\u0000x\flms and an essentially compensated fer-rimagnetic behavior was observed for the \frst time in\nultrathin 2 nm \flms. The angular momentum compensa-\ntion and magnetic compensation points observed in this\nwork are very close to those reported earlier on much\nthicker \flms in the literature. A record low \u000bvalue of\nabout 0.0078 is obtained near the magnetic compensa-\ntion point, which is an order of magnitude lower than\nthe values reported in the literature using similar analysis\nmethods. The observation of compensated ferrimagnetic\nbehavior in ultrathin \flms together with very low value\nof\u000bare promising results for the future development of\nultrafast and energy e\u000ecient ferrimagnetic spintronic de-\nvices.\nEXPERIMENTAL SECTION\nA. Thin \flms growth and composition analysis\nAll the samples were prepared on high resistivity\nSi(100) substrates using a magnetron sputtering sys-\ntem with a base pressure of less than 2 \u000210\u00008torr.\nThin \flms of Gd x(FeCo) 1\u0000xwere deposited using the\nco-sputtering of high purity (more than 99.95%) Gd\nand Fe 87:5Co12:5targets, and composition analysis\nwas done using the inductively coupled plasma mass\nspectroscopy (ICP-MS). Thin \flms stacking structure\nof Si(100)/MgO(t)/Gd 12:5Fe76:1Co11:4(10)/SiO 2(4)\nwere used for seed layer dependence studies, here,\nthe number in the bracket is the thickness of the\nlayer in nm, where t=0, 6 and 10 nm. Four sam-\nples, namely S1 to S4 were prepared to obtain the\nbest conditions to grow Gd 12:5Fe76:1Co11:4(10) \flms.\nFor S1, Gd 12:5Fe76:1Co11:4(10) was grown directly\nover HR-Si (100) substrates, while in both S2 and\nS3 Gd 12:5Fe76:1Co11:4were grown with MgO seed\nlayer of 6 and 10 nm, respectively. All the lay-\ners in S1-S3 were grown at room temperature and5\nFigure 4. (a) Frequency vs.resonance \feld and (b) resonance linewidth vs.frequency, of 2 nm thick Gd x(FeCo) 1\u0000x\flms as\na function of Gd content in atomic %. (c) E\u000bective magnetization and e\u000bective Gilbert damping constant vs.Gd content.\nSolid symbols represent the values obtained by \ftting the experimental FMR data in (a) and (b) using the equation (2) and\n(3), respectively; solid lines in (c) are guides to the eye. xaand xmshow the angular and magnetic compensation points,\nrespectively, obtained from the literature17,19.\nno further heat treatment was given to them. In\nS4, 10 nm MgO seed layer were grown over HR\nSi(100) substrates at RT and followed by a in-situ\npost-annealing at 600C for 1 hour, and after that\nGd12:5Fe76:1Co11:4were deposited. The stacking struc-\nture of Si(100)/MgO(6)/Gd 12:5Fe76:1Co11:4(m)/SiO 2(4)\nwere used for thickness dependence studies, where\nm is the thickness of Gd 12:5Fe76:1Co11:4layer,\nand varied from 2 to 20 nm. For composi-\ntion dependence studies, stacking structure of\nSi(100)/MgO(6)/Gd x(FeCo) 1\u0000x(2)/SiO 2(4) were used,\nwhere xvaried from 12.5 to 26.7. The composition of\nGdx(FeCo) 1\u0000x\flms was varied by changing the sput-\ntering rate of Fe 87:5Co12:5target, while keeping the Gd\nsputtering rate \fxed for most \flms. All the samples for\nthickness dependence and composition dependence were\ngrown at room temperature and no post-annealing was\nused. Layer thicknesses were determined by estimating\nthe growth rate using the Dektak pro\fler on more than\n100 nm thick \flms.B. Inductively coupled plasma mass spectroscopy\n(ICP-MS) measurements\nThe elemental composition (Co, Fe, and Gd) of the\nthin \flm samples was determined by inductively coupled\nplasma optical emission spectroscopy (ICP-OES) using a\nThermo Fisher Scienti\fc iCAP 6000 Series spectrometer.\nEach thin \flm sample was exhaustively extracted in 5 mL\nHNO3 (65%, Supelco, Merck KgaA, Sigma-Aldrich) for\na duration of 30 min. 5 mL ultrapure MilliQ-water (18\nM\ncm) was added to the solution and the extract was al-\nlowed to rest for 30 minutes. The extract was transferred\nto a 100 mL volumetric \rask. The extracted sample was\nthen rinsed for several cycles in ultrapure water. The\nwater used for rinsing was transferred to the same volu-\nmetric \rask. The extract was diluted to 100 mL for ICP\nanalysis. ICP check standards were prepared from stan-\ndard solutions (Co and Fe: Merck, Germany; Ga: Accu-\nstandard, USA). The relative standard deviation (from\nthree individual injections) were within 1%.6\nTable I. The obtained values of e\u000bective Gilbert damping constant \u000bat room temperature (RT) in this work and comparison\nwith the lowest values reported so far in the literature at RT and also at their respective angular momentum compensation\n(Ta) and magnetic compensation (T m) points.\nFilm composition Film thickness \u000b Measurement technique Analysis method Reference\nGd23:5Fe68:9Co7:6 30 \u00180.45 (at RT) FMR Kittel's FMR19\n\u00180.35 (at RT) Pump-probe\nGd22Fe74:6Co3:4 20 \u00180.21 (at T a) Pump-probe -do-17\n\u00180.13 (at T m)\nGd25Fe65:6Co9:4 10 \u00180.07 (at RT) Spin torque FMR -do-36\n\u00190.01 (at RT) Spin torque FMR Ferrimagnetc resonance\nGd23:5Fe66:9Co9:6 30 0.0072 (at RT) Domain wall (DW) Field driven DW30\nmotion mobility\nGd12:5Fe76:1Co11:4 10 0.0055 Broadband FMR Kittel's FMR This work\n0.0076 (HF data) -do- -do- This work\nGd12:5Fe76:1Co11:4 4 0.0064 -do- -do- This work\nGd12:5Fe76:1Co11:4 2 0.0101 -do- -do- This work\nGd23:4Fe67:0Co9:6 2 0.0141 -do- -do- This work\nGd24:4Fe66:1Co9:5 2 0.0078 -do- -do- This work\nC. Ferromagnetic resonance (FMR) measurements\nRectangular pieces of about 6 \u00023 mm2were cut from\nthe blanket \flms and broadband FMR spectroscopy was\nperformed using a NanOsc Phase FMR (40 GHz) system\nwith a co-planar waveguide for microwave \feld excita-\ntion. Microwave excitation \felds hrfwith frequencies up\nto 30 GHz were applied in the \flm plane, and perpendic-\nular to the applied in-plane dc magnetic \feld H. All the\nFMR measurements were performed at the room tem-\nperature. The schematic of FMR measurement setup is\nshown in 1(a), and further details about the measure-\nments are given in Section 2 (results and discussions).\nSUPPORTING INFORMATION\nSupporting Information is available from the Wiley\nOnline Library or from the corresponding author.ACKNOWLEDGEMENTS\nLakhan Bainsla thanks MSCA - European Commission\nfor Marie Curie Individual Fellowship (MSCA-IF Grant\nNo. 896307). This work was also partially supported\nby the Swedish Research Council (VR Grant No. 2016-\n05980) and the Horizon 2020 research and innovation\nprogramme (ERC Advanced Grant No. 835068 \"TOP-\nSPIN\").\nCONFLICT OF INTEREST\nThe authors declare no con\rict of interest.\nAUTHOR CONTRIBUTIONS\nL.B. and J. \u0017A. planned the study. L.B. grew the \flms,\nperformed the FMR measurements and analysed the ob-\ntained FMR data. J.W. helped with ICP-MS measure-\nments and analysis. L.B. wrote the original draft of the\npaper. J. \u0017A. coordinated and supervised the work. All\nauthors contributed to the data analysis and co-wrote\nthe manuscript.7\nDATA AVAILABILITY STATEMENT\nThe data that support the \fndings of this study are\navailable from the corresponding author on reasonable\nrequest.\nREFERENCES\n1S. Wolf, D. Awschalom, R. Buhrman, J. Daughton, v. S. von\nMoln\u0013 ar, M. Roukes, A. Y. Chtchelkanova, and D. 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Mandal1\n1)Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, India\n2)Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU,\nUK\n(Dated: 19 November 2021)\nWe have studied magnetodielectric and spin-lattice coupling in CoNb 2O6single crystals. Magnetostriction\nand magnetodielectric experiments are performed at temperatures in and above anti\u000beromagnetic phase of\nquasi 1D Ising spin chain CoNb 2O6. Field induced magnetic transitions are clearly re\rected in magnetodi-\nelectric measurement as well as magnetostriction measurement also. Two sharp anomalies are found around\nthe critical \felds of antiferromagnetic to ferrimagnetic transition and ferrimagnetic to saturated paramag-\nnetic transition in both magnetodielectric and magnetostriction experiments. High \feld anomaly is more\npronounced for magnetodielectric response and magnetostriction also. So, in CoNb 2O6, spins are strongly\ncoupled with lattice as well as charges also.\nI. INTRODUCTION\nIn recent years geometrically frustrated triangular lat-\ntice systems have attracted immense interest due to\nits di\u000berent kind of magnetic phase transitions and de-\ngenerate ground states. Several triangular lattice sys-\ntems also exhibit multiferroic behavior.1Geometrical\nfrustration plays a key role to produce magnetodielec-\ntric coupling. Dielectric constant measurements in pres-\nence of magnetic \feld can probe the coupling between\ncharges and spins in insulating systems. Ising spin\nchain CoV 2O6,2Ca3Co2O63{5with triangular network\ndisplay magnetodielectric coupling at low temperature.\nQuasi-one-dimensional Ising spin chain CoNb 2O6is a\nvery good example of frustrated triangular lattice sys-\ntem which exhibits several interesting features like meta-\nmagnetic transition, quantum criticality behavior etc.\nRecently, quantum phase transition in transverse \feld\nhas been experimentally evidenced in CoNb 2O6.6A E 8\nsymmetry has been experimentally observed near the\nquantum critical point of Ising ferromagnet CoNb 2O6.\nAt low temperature, this system also exhibits var-\nious degenerate magnetically ordered states such as\nfourfold-degenerate antiferromagnetic (AF) phase, \feld-\ninduced threefold-degenerate ferrimagnetic (FR) phase,\nsinusoidally amplitude-modulated incommensurate (IC)\nphase, con\frmed by Neutron di\u000braction study.7{9In\nCoNb 2O6system, Co2+ions form zigzag chains along\nc-direction and they are arranged into isosceles trian-\ngular geometry in the a\u0000bplane. At low tempera-\ntures, Co spins orient along two di\u000berent easy axes in\nthe nearly a\u0000cplane with a 31\u000ecanting angle from\nthecaxis. Intrachain interaction is ferromagnetic in\nnature and chains are weakly coupled by antiferromag-\nnetic interaction. In this paper, we have performed meg-\nnetodielctric measurement to evidence the coupling be-\ntween electrical charges and magnetic moments. In ad-\ndition, we have done magnetostriction measurements to\nprobe coupling between spin and lattice. In some sys-\ntems, spins are simultaneously coupled with both latticeand charges. For example, in EuTiO 3, magnetostriction\nmeasurement exhibits several similarities with the \feld\ndependence of the dielectric constant.10The correlation\nbetween spin-phonon coupling and dielectric constant has\nbeen observed in TbFe 3(BO 3)4.11In this paper we have\nstudied and compared magnetic, magnetodielectric, mag-\nnetothermal properties of CoNb 2O6.\nII. EXPERIMENTAL DETAILS\nSingle crystal of CoNb 2O6was grown by the travel-\ning solvent \roating zone method.12Laue XRD was per-\nformed to determine crystal axes and crystal was cut\nalong di\u000berent crystallographic planes according to ex-\nperiment. Laue di\u000braction patterns are illustrated in\nFigure 1. A rectangular shaped piece of single crystal\nwas used for dielectric measurements. Two parallel faces\nof the crystal were covered with silver paint in order to\napply an electric \feld perpendicular to the chains. Here\nelectric \feld was applied along aaxis where as magnetic\n\feld was applied along the easy axis direction cso that\nthe~E?~Hcondition was always ful\flled. Magnetostric-\ntion measurements were done by capacitive method using\na miniature tilted-plates dilatometer with applied \feld\nparallel to caxis. The capacitance measurements were\nperformed using a commercial AH2700A ultra-precision\ncapacitance bridge. The magnetic measurements were\ndone in SQUID-VSM (Quantum Design). The speci\fc\nheat measurements were done using a physical property\nmeasurement system (Quantum Design) by conventional\nrelaxation time method.\nIII. MAGNETIZATION MEASUREMENTS\nTemperature dependence of magnetization along c-axis\nin zero \feld cool(ZFC) and \feld cool(FC) conditions is\nplotted in Figure 2(a). At low temperature, M vs. T\ncurve shows two successive transitions below 3 and 2 K.arXiv:1609.02048v1 [cond-mat.str-el] 7 Sep 20162\nFIG. 1. Laue di\u000braction patterns of (100), (010) and (001)\nplanes.\nZFC and FC of M(T) do not show any bifurcation down\nto 1.8 K. M(T) curve exhibits slope change around 3\nK (T 1) due to transition from paramagnetic to incom-\nmensurate phase. Another transition occurs below 2 K\n(T2) where a sharp drop in magnetization has been ob-\nserved due transition from IC to AFM phase. In inset\nof Figure 2(a), temperature dependence of speci\fc heat\n(Cp(T)) is plotted at zero \feld. A very sharp peak has\nbeen observed at 2.9 K in Cp(T) due to PM-IC tran-\nsition. Field dependence of magnetization along c-axis\nfor some selected temperatures both above and below\nT1and T 2are plotted in Figure 2(b). In inset of Fig-\nure 2(b), a closer view of M(H) at 1.8 K is given which\nexhibits multiple magnetization plateaux due to \feld in-\nduced magnetic phase transitions, similar to previously\nobserved data.13M is very small up to 200 Oe then it\nshows step like jump at \frst critical \feld Hc1and obtains\n1/3 of saturation magnetization value in a certain \feld\nrange. This step like increase in magnetization can be ex-\nplained from magnetic phase diagram by S. Kobayashi.14\nThe saturation magnetization value is consistent with a\nCo2+moment of about 3 \u0016B. At 1.8 K, the system re-\nmains at AFM phase below 200 Oe, then it enters to fer-\nrimagnetic phase via an incommensurate phase with in-\ncreasing \feld. So this \feld induced AFM to ferrimagnetic\nphase transition is re\rected in sharp step-like increase in\nM(H) curve around 200 Oe. Another increase in M(H)\naround 3.8 kOe (second critical \feld Hc2) is observed due\n12 3 4 5 0123450\n4 8 1 2012340\n1 2 3 4 5 6 7 010203040500\n120.00 .30 .60 .9c (emu/mol Oe)T\n (K) zfc \nfcH|| c100 Oea\n)C\np (JK-1mol-1)T\n (K)H\n|| cM (emu/g)H\n (T) 1.8K \n3K \n5K \n9K \n14Kb)H\nc1H (T) M (mB/f.u.) 1.8 KH c2FIG. 2. (a) Plots of \u001f(T) with zero \feld cool and \feld cool\ncondition for CoNb 2O6at 100 Oe \feld applied along caxis.\nInset:Speci\fc heat ( Cp) versus temperature plot for CoNb 2O6\nat zero \feld. (b) Isothermal magnetization at some selected\ntemperatures when \feld is applied along caxis. Inset shows\nthe closer view of M-H curve at 1.8 K.\nto \feld induced transition from ferrimagnetic state to sat-\nurated PM state. With increasing temperature step-like\nincrease in M(H) curve gradually disappears. Just above\n3 K, magnetization linearly increases with H and then\nsaturates. Further increase of temperature makes the M\nalmost linear with H.\nIV. DIELECTRIC CONSTANT MEASUREMENTS\nThe isothermal dielectric constant measurements per-\nformed with a frequency 1 kHz, as a function of external\nmagnetic-\feld at some selected temperatures are plot-3\n-0.9-0.6-0.30.00.30.60.9-0.020-0.015-0.010-0.0050.0000.0050\n.00 .20 .40 .60 .81 .0-0.020-0.015-0.010-0.0050.0000.005De'/e'0 %H\n (T) 1 \n2 \n3 \n4 \n51.5 KH || ca\n)b\n)H|| cDe'/e'0 %H\n (T) 1.5 K \n1.6 K \n2 K \n2.3 K \n4.3 K \n5.5 K\nFIG. 3. (a) Five segment curves for relative change of dielec-\ntric constant as a function of magnetic \feld are plotted at 1.5\nK when magnetic \feld applied along caxis. (b) Plot of rel-\native change of dielectric constant as a function of magnetic\n\feld for some selected temperatures when magnetic \feld is\napplied along caxis.\nted in Figure 3. Here external magnetic \feld is applied\nalongcaxis and electric \feld is applied along aaxis.\nActually, Figure 3 shows percentage of relative change\nin dielectric constant (\u0001 \"0/\"00=[\"0(H)-\"00]/\"00), where\n\"00is the dielectric constant of the sample in absence\nof magnetic \feld. Sharp anomalies are found in isother-\nmal relative change of dielectric constant at 1.5 K in \fve\nsegment curve, shown in Figure 3(a). With increasing\n\feld \u0001\"0/\"00% remains constant initially and then ex-\nhibit a sharp negative peak around Hc1, then it shows\nalmost a constant positive plateau region in a certain\n\feld range. With further application of magnetic \feld,\n\u0001\"0/\"00% displays a sharp step-like jump around Hc2andhysteresis has also been found here. Then, \u0001 \"0/\"00% de-\ncreases very slowly with increasing H. Depending on \feld\nstrength, \u0001 \"0/\"00% obtains positive value as well as neg-\native value. With increasing temperature, these anoma-\nlies gradually disappear, shown in Figure 3(b). At 4.3\nK, sharp peak-like feature around Hc1is totally disap-\npeared but \u0001 \"0/\"00% becomes positive in a certain \feld\nrange though the plateau-like behavior is disappeared.\nNo positive region has been found with further increase\nof temperature where dielectric constant monotonically\ndecreases with increasing \feld. At 5.5 K, where system\nis aboveT1, peak around Hc1and step-like jump around\nHc2in dielectric constant disappear and it shows almost\nlinear dependence with \feld. According to the magnetic\nphase diagram with \feld applied along c-axis, drawn from\nneutron di\u000braction data, CoNb 2O6undergoes multiple\n\feld induced magnetic transitions at 1.5 K.14In low \feld\nregion, below T 2, CoNb 2O6exhibits successive \feld in-\nduced antiferromagnetic (AF) to incommensurate (IC)\nthen IC to ferrimagnetic (FR) transitions around 225 Oe\nand 395 Oe respectively. Dielectric constant shows two\nsuccessive slope changes in low \feld region which can be\ninterpreted by these AF to FR transition via intermediate\nIC phase. Above a certain critical \feld, this system en-\nters to \feld induced FR state and remains in this state up\nto 3.2 T where corresponding dielectric constant exhibits\na plateau-like feature obtaining a certain positive value.\nObserved sharp step-like jump in \u0001 \"0/\"00% around 0.33\nT may be related to transition from \feld induced FR to\nsaturated PM state. Field induced magnetic transitions\nare re\rected in \feld dependent dielectric constant mea-\nsurements also. So it is clear that dielectric constant is\nmagnetically coupled in this system.\nV. THERMAL EXPANSION MEASUREMENTS\nWe have also performed magnetostriction measure-\nments at some selected temperatures. Figure 4(a) shows\nmagnetostriction \u0001 L(H)/L0=[L(H)-L0]/L0, whereL0is\nthe length of the sample in absence of magnetic \feld, at\n1.6 K for \feld increasing and decreasing conditions. Simi-\nlar to dielectric constant measurement, magnetostriction\nat 1.6 K also exhibits two anomalies around Hc1and\nHc2. Hysteresis has also been found here. \u0001 L(H)/L0\nshows a weak cusplike anomaly around 500 Oe where a\npeak type feature has been found in dielectric constant\nmeasurement. A very pronounced peak has been found\nin \u0001L(H)/L0around 3500 Oe where dielectric constant\nexhibits a very sharp step-like jump almost near about\nthis \feld. The very sharp peak around 3500 Oe is as-\nsociated with the transition from the ferrimagnetic state\nto a saturated PM high-\feld phase. Apart from this,\nmagnetostriction measurement shows an interesting be-\nhavior. \u0001L(H)/L0obtains positive value as well as neg-\native value. Similar kind of behavior is also observed\nin magnetostriction of EuTiO 3where it shows a sign\nchange with increasing magnetic \feld10. With increasing4\n0.00 .20 .40 .60 .81 .0-3-2-101230\n.00 .40 .81 .21 .62 .0-3-2-101234561.6 KDL/L0(10- 5)H\n (T)H|| ca)b\n)H\n|| cDL/L0(10- 5)H\n (T) 1.6 K \n2.1 K \n2.9 K \n4 K \n5.5 K\nFIG. 4. (a)Magnetostriction, \u0001 L(H)=L0at 1.6 K for both\n\feld increasing and decreasing condition when magnetic \feld\napplied along caxis. (b) Magnetostriction, \u0001 L(H)=L0, at\nseveral temperatures both above and below T1andT2for\nsome selected temperatures when magnetic \feld is applied\nalongcaxis.\n\feld \u0001L(H)/L0remains negative up to 4700 Oe then it\nbecomes positive and increases linearly with \feld above\n6000 Oe. Magnetostriction curves for some selective tem-\nperatures both below and above T1andT2are shown in\nFigure 4(b). At 2.1 K, weak anomaly around Hc1dis-\nappears but sharp peak around Hc2has been observed.\nWith increasing temperature, height of the peak around\n3500 Oe decreases gradually and disappears above 4 K.\nAt 5.5 K, \u0001 L(H)/L0exhibits very small value up to\n6000 Oe and then it increases monotonically with in-\ncreasing \feld but it remains always positive throughoutthis region. Particularly at low temperature, \feld in-\nduced metamagnetic transitions are re\rected in magne-\ntostriction measurements also which suggests that spins\nare strongly coupled with lattice in this system.\nVI. SUMMARY\nWe have carried out magnetostriction and magne-\ntodielectric measurements on single crystalline CoNb 2O6\nsample at low temperature. The samples are well char-\nacterized by magnetization and speci\fc heat measure-\nments. We have related \feld dependence of dielec-\ntric constant and thermal expansion measurements with\nmagnetic phase diagram by neutron di\u000braction and com-\npared with magnetization. Multiple phase transitions ob-\nserved by neutron di\u000braction data are clearly re\rected\nin \feld dependence of dielectric constant and magne-\ntostriction measurements. Field dependence of the di-\nelectric constant display several similarities with magne-\ntostriction measurements in CoNb 2O6. Both \feld depen-\ndence of dielectric constant and magnetostriction exhibit\ntwo anomalies around two critical \felds of metamagnetic\ntransitions and obtain positive as well as negative value.\nSo it can be concluded that spin-lattice coupling plays a\nkey role and spins are magnetically coupled with charges\nin this system.\n1S. Seki, Y. Onose, and Y. Tokura, Phys. Rev. Lett. 101, 067204\n(2008).\n2K. Singh, A. Maignan, D. Pelloquin, O. Perez, and Ch. Simon,\nJ. Mater. Chem. 22, 6436 (2012).\n3N. Bellido, Ch. Simon, and A. Maignan, Phys. Rev. B 77, 054430\n(2008).\n4N. Bellido, Ch. Simon, and A. Maignan, J. Magn. Magn. Mater.\n321, 1770 (2009).\n5T. Basu, K. K. Iyer, K. Singh, and E. V. Sampathkumaran, Sci.\nRep. 3, 3104 (2013).\n6R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D.\nPrabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer,\nScience 327 177 (2010).\n7S. Mitsuda, S. Kobayashi, K. Aga, H. Katagiri, H. Yoshizawa,\nM. Ishikawa, K. Miyatani, and K. Kohn, J. Phys. Soc. Jpn. 64,\n2325 (1995).\n8S. Kobayashi, S. Mitsuda, M. Ishikawa, K. Miyatani, and K.\nKohn, Phys. Rev. B 60, 3331 (1999).\n9C. Heid, H. Weitzel, P. Burlet, M. Bonnet, W. Gonschorek, T.\nVogt, J. Norwig, and H. Fuess, J. Magn. Magn. Mater. 151, 123\n(1995).\n10P. G. Reuvekamp, R. K. Kremer, J. K ohler, and A. Bussmann-\nHolder, Phys. Rev. B 90, 094420 (2014).\n11U. Adem, L. Wang, D. Fausti, W. Schottenhamel, P. H. M. van\nLoosdrecht, A. Vasiliev, L. N. Bezmaternykh, B. B uchner, C.\nHess, and R. Klingeler, Phys. Rev. B 82, 064406 (2010).\n12D. Prabhakaran, F. R. Wondre, A. T. Boothroyd, J. Cryst.\nGrowth 250, 72 (2003).\n13I. Maartense, I. Yaeger, B.M. Wanklyn, Solid State Commun. 21\n93 (1977).\n14S. Kobayashi, S. Mitsuda, and K. Prokes, Phys. Rev. B 63,\n024415 (2000)." }, { "title": "2109.00169v1.Epitaxial_Integration_of_a_Perpendicularly_Magnetized_Ferrimagnetic_Metal_on_a_Ferroelectric_oxide_for_Electric_Field_Control.pdf", "content": "Epitaxial Integration of a Perpendicularly Magnetized Ferrimagnet ic \nMetal on a Ferroelectric oxide for Electric -Field Control \nXin Zhang, Pei-Xin Qin, Ze -Xin Feng, Han Yan, Xiao -Ning Wang, Xiao -Rong Zhou, Hao-\nJiang Wu, Hong -Yu Chen, Zi -Ang Meng, Zhi-Qi Liu* \nSchool of Materials Science and Engineering, Beihang University, Beijing 100191, China \nEmail: zhiqi@buaa.edu.cn \n \n \nAbstract : Ferrimagnets , which contain the advantages of both ferromagnets (detectable \nmoments) and antiferromagnets (ultrafast spin dynamics), have recently attracted great \nattention. Here we report the optimization of epitaxial growth of a tetragonal \nperpendicularly magnetized ferrimagnet Mn 2Ga on MgO . Electrical transport, magnetic \nproperties and the anomalous Hall effect (AHE) were systematically studied. \nFurthermore, we successfully integrated high-quality epitaxial ferrimagnetic Mn 2Ga \nthin films onto ferroelectric PMN -PT single crystals with a MgO bu ffer layer. It was \nfound that the AHE of such a ferrimagnet can be effectively modulated by a small \nelectric field over a large temperature range in a nonvolatile manner. This work thus \ndemonstrates the great potential of ferrimagnets for developing high -density and low -\npower spintronic device s. \n \nKeywords : ferroelectric oxides; ferrimagnetic metals; PMN -PT; Mn 2Ga; anomalous \nHall effect \n 1. Introduction \nContemporary mass storage for data centers predominantly relies on hard disk drives that are \nbased on perpendicularly magnetized ferromagnetic granular films , the spin states of which can \nbe easily manipulated by external magnetic fields generated by current coils due to remarkable \nmacroscopic moments. However, as limited by the characteristic GHz spin dynamics, \nferromagnetic materials could hardly be utilized for the static random -access memory \ntechnology and the in -memory computing, which needs a sub -ns response speed. Similar to \nferromagnets, ferrimagnets possess large magnetic moments. Besides, they exhibit ultrahigh \nspin dynamics of THz as a result of antiferromagnetic exchange coupling [1-6] akin to \nantiferromagnets. Hence, ferrimagnets are rising star materials for new -generation sub-ns \ninformation devices [7-12]. \nTetragonal D0 22 Mn 2Ga with a = 3.905 Å and c = 7.193 Å , a classical ferrimagnet with a high \nCurie temperature of ~710 K, exhibits large magnetic anisotropy along its [001] \ncrystallographic orientation. (001) -oriented single -crystalline or textured films are thus \nperpendicularly magnetized. In addition, D0 22 Mn2Ga is of high spin polarization at the Fermi \nlevel and low Gilbert damping constant [13], both of which are useful for spin valves, \nperpendicular magnetic tunnel junctions [14], narrow -band terahertz emission from coherently \nexcited spin precession [15], and high -density spin -transfer -torque magnetoresistive random \naccess memories (STT -MRAM) [16]. \nHowever, the information writing (corresponding to the spin state manipulation) of Mn 2Ga-\nbased spintronic devices such as spin val ves and STT -MRAM mostly depend on electric al-\ncurrent -generated magnetic fields or electric al currents, which generates significant Joule \nheating and hence results in high power consumption. Alternatively, if one can effectively \ncontrol the spin state of Mn 2Ga with a non -electrical -current manner, the energy for writing a \nbit could be substantially lowered. \nAn electric field applied onto a conductor yields an electrical current. In contrast, for highly \ninsulating ferroelectric oxide materials, the application of an electric field generates a negligible \ncurrent. Instead, it induces piezoelectric strain [1-7,17-23]. Assuming (001) -oriented D0 22 \nMn 2Ga could be epitaxially integrated onto ferroele ctric oxides, electric -field-generated strain could harness the spin state of Mn 2Ga as the electronic states of solid -state materials are all \nsensitive to the periodic lattice. \nNevertheless, epitaxial growth of intermetallic Mn 2Ga on a ferroelectric oxide is quite \nchallenging. The key reasons are: (1) highly -ordered epitaxial films need high thermal energy \nto let the atoms diffuse into their equilibrium sites during thin film growth and therefore high \ngrowth temperatures are requ ired; (2) intermetallic alloys are of strong chemical activity, which \ncan easily be oxidized or form secondary alloys at high temperatures while landing on \nferroelectric oxide substrates ; (3) lattice mismatch between intermetallic alloys and \nferroelectric oxides could break epitaxial growth. Accordingly , a proper oxide substrate or an \noxide buffer layer with robust high -temperature stability and low oxygen diffusion coefficient \ncould be crucial for realizing epitaxial growth of Mn 2Ga on ferroelectrics. In this letter, we \nreport the optimal epitaxial growth of D0 22 Mn 2Ga on ferroelectric PMN -PT. Furthermore, \nelectric -field control of its AHE has been achieved over a large temperature range, which paves \nthe way for low -power Mn 2Ga spintroni c device applications. \n2. Experimental \nMgO, a transparent insulating oxide with a large bandgap of 7.8 eV, shows excellent high -\ntemperature stability with a melting point of 2800°C. It has a cubic lattice with a = 4.212 Å , \nwhich is close to the in -plane lattice constant of D0 22 Mn 2Ga. Therefore, Mn 2Ga films were \nfirstly grown on single -crystalline (001) -oriented MgO substrates using the magnetron \nsputtering technique at different temperatures ranging from 30 to 550°C . The base pressure and \nthe D.C. sputtering power were 7.5×10-9 Torr and 60 W, respectively. During the deposition, \nthe Ar pressure was kept at 3mTorr. With X -ray reflectometry, t he deposition rate was \ndetermined to be 35 Å/ min and the total thickness was first kept at 100 nm , which was later \nchanged to 50 and 30 nm f or thickness dependence studies . The crystal structure of Mn 2Ga thin \nfilms was measured by X -ray diffraction (XRD). Magnetic characterization and electrical \nmeasurements were performed by a Quantum Design VersaLab with a vibrating sample \nmagnetometer option . The standard linear four -probe method and the Hall geometr y were used \nfor longitudinal and Hall resistance measurements, respectively. \n3. Results and discussion Figure 1a shows single -crystal XRD spectra of Mn 2Ga films grown on MgO substrates \nfabricated at different growth temperatures TG. For TG below 450°C, no thin -film peaks are seen, \nimplying disordered polycrystalline films. At 450°C, the (002) peak and a weak (001) peak of \nD0 22 Mn 2Ga show up. With increasing the growth temperature to 550°C, both the (001) and \n(002) peaks of D0 22 Mn 2Ga become sharpe r, indicating enhanced crystallinity and chemical \nordering. The growth temperature could not be further raised. That is because at higher \ntemperatures the surface energies of intermetallic Mn 2Ga and ferroelectric oxide PMN -PT are \nof large different, due to the non -wetting issue, Mn 2Ga could not form continuous thin films \nbut only separate islands. \nThe metallicity, which could be examined by the normalized resistivity relative to room -\ntemperature values, is demonstrated in Fig. 1b. When the substrate temperature is lower than \n450°C, Mn 2Ga/MgO films are semiconducting, which is contrary to its bulk met allic behavior. \nThis suggests the existence of a large degree of chemical disorder including grain boundaries . \nThe films fabricated at 450 and 550°C are metallic and the metallicity is improved with \nenhancing substrate temperature. All these electrical transport results are consistent with the \nXRD spectra shown in Fig. 1a . \nThe out -of-plane and in -plane magnetic moments versus magnetic fields ( M-H) are mea sured \nat 50 and 300 K for single -crystalline Mn 2Ga/MgO film s (Fig. 1c , f). Overall, both films exhibit \nthe feature of perpendicularly magnetized anisotropy. The room -temperature saturation \nmagnetization of Mn 2Ga/MgO films are ~290 and ~335 emu/cc for growth temperature of 450 \nand 550°C, respectively, which are comparable with the magnetization values of previously \nreported ferrimagnetic Mn 2Ga films [24]. Similar to the XRD and electrical resistivity data, the \nincrease of the growth temperature improves the squareness of the out -of-plane M-H loop, \nwhich is favorable for perpendicular spintronic device applications. Therefore, 550°C is the \noptimized growth t emperature for high -quality epitaxial and perpendicularly magnetized \nferrimagnetic Mn 2Ga films. \nIn addition, the room -temperature anisotropy field 0Hk is determined to be ~10 T for the \n550°C -fabricated Mn 2Ga film, which corresponds to a uniaxial magnetocrystalline anisotropy energy Ku of ~1.68 MJ/m3, which is of the same order with the previously reported highest Ku \n(~2.17 MJ/m3) [25] for molecular -beam -epitaxy -fabricated ferrimagnetic Mn-Ga films. For real \napplications, the duration of information storage needs to be more than 10 years, which requires \nthe ratio of magnetocrystalline energy of a bit KuV greater than 40 times room -tempe rature \nthermal energy 40kBT ~ 1.67×10-19 J. Considering a bit cell consisting of a single Mn 2Ga layer \nwith a cubic shape, the critical bit size could accordingly be reduced to ~ 4.6 nm. This thus \nimplies great potential of our optimized Mn 2Ga ferrimagnetic thin films with perpendicular \nmagnetized anisotropy for high -density data storage. \nLongitudinal magnetotransport properties of Mn 2Ga thin films fabricated at various \ntemperatures were examined for out -of-plane magnetic fields . It was found that for TG < 300 °C, \nthe magnetoresistance (MR) effect is rather weak and almost not affected by the growth \ntemperature . Figure 2 shows the MR curves collected at different temperatures ranging from 50 \nto 300 K for Mn 2Ga thin films with TG > 30°C. For TG = 150°C (Fig. 2a ), the MR above 50 K \nis positive, which implie s that the orbital scattering due to the Lorent z force is dominant. \nHowever, the MR turns into negative for 50 K, suggesting the important role of magnetic \nmoments of Mn. For TG = 300°C (Fig. 2 b), the room -temperature MR is negligible while the \nlow-temperature MR curves are interestingly linear. It is worth noticing that the positive MR at \n150 K is the largest, reminiscent of the maximal magnetotransport properties at ~200 K for \nnoncollinear antiferromagnets Mn 3Sn [26] and Mn 3Ge [4]. For crystallized epitaxial Mn 2Ga \nfilms , the butterfly -shape hysteresis MR curves (Fig. 2c , d) are clearly seen, characteristic of \nlong-range ferrimagnetic/ferromagnetic order. In addition, the positive orbital scattering is more \nsignificant at low temperatures, leading to suppressed negative MR. \nSystematic transverse magnetotransport properties, i.e., the Hall effect, of the Mn 2Ga/MgO \nfilms deposited below 450°C are demonstrated in Fig. 3. Similarly, the Hall curves for TG = 30 \nand 150°C (Fig. 3a , b) are comparable and are linear above 50 K . At 50 K, the Hall effect \nbecomes nonlinear , signature of the magnetic -moment -related AHE, which is consistent with \nthe negative MR at 50 K in Fig. 2a . For TG = 300°C, the AHE is obvious below 200 K , indicating \nthe formation of magnetic order. The Hall effect of epitaxial Mn 2Ga films is shown in Fig. 4a , b. The g eneral shape of the Hall \ncurves is in excellent agreement with that of the M-H loops. Therefore, the Hall effect could \nserve as a sensitive electrical probe to magnetic p roperties of perpendicularly magnetized \nMn 2Ga. Detailed scaling law analysis [7] (Fig. 4c) on the optimized film reveals that for low \nlongitudinal resistivity range 280 < ρxx < 330 Ωcm, ρxy ∝ ρxx2, the Berry curvature is the \ndominant origin for the AHE , which is a pseudo magnetic field in momentum space and \ndetermined by the topological bands interaction of Bloch electrons [27,28] . While for the large \nresistivity region with ρxx > 330 Ωcm, skew scattering becomes more important for generating \na transverse Hall voltage , leading to ρxy ∝ ρxx [29]. The excellent perpendicular magnetic \nanisotropy remains in thinner films such as 50 and 30 nm (Fig. 5) , which, in turn, lead to \nsignificantly enhanced anomalous Hall resistance. The much larger anomalous Hall resistance \nin thinner Mn 2Ga films could facilitate the electrical read -out for memory devices. \nBased on the experimental results mentioned above, 30-nm-thick Mn 2Ga films were further \nepitaxially integrated onto (001) -oriented PMN -PT ferroelectric oxides with a 25-nm-thick \nMgO buffer layer so as to manipulate its AHE or magneti sm by electric -field-induced \npiezoelectric strain [1-6,17-23,30] . The MgO buffer layer s were grown by a pulsed laser \ndeposition system at 400°C and post -annealed at 600°C for 1 h, which was utilized to prevent \nPb element in PMN -PT substrate s from diffusing into the chamber and Mn 2Ga film s to form \nsecondary alloy s at high temperature s [18,31] . As shown in Fig. 6a, the XRD spectrum \ncontain ing the (002) peak of the MgO buffer layer and the (001) and (002) peaks of the Mn 2Ga \nthin film indicates the epitaxial growth of the Mn 2Ga on the MgO buffer layer. The f ield-\ndependent Hall signals of an epitaxial Mn 2Ga/MgO/PMN -PT heterostructure (Fig. 6b) at \ndifferent temperatures are in consistence with that of Mn 2Ga/MgO in Fig . 4b, which suggest s \nthe excellent perpendicular magnetic anisotropy of the epitaxially integrated Mn 2Ga films on \nferroelectric PMN -PT. \nTo explore the effect of piezoelectric strain on the AHE in the Mn 2Ga/MgO/PMN -PT \nheterostructure , an electric field EG of -5 kV/cm was perpendicularly applied across the PMN -\nPT substrate (Fig. 7a) to pole the ferroelectric substrate at room temperature. To examine any \npossible variation of the AHE, the Hall curves were re -measured after electric poling of the \nPMN -PT. It turns out that under such an electric -field excitation, the AHE is enhanced for all the temperatures (Fig. 7b-g). The relative electric -field-induced nonvolatile modulation of the \nzero-field anomalous Hall resistance is extracted and plotted in Fig. 7h, which reaches ~ 16% \nbelow 200 K and ~ 14% at 300 K. \nTo further confir m the nonvolatile nature of the electric -field-induced piezoelectric strain in \nPMN -PT, the room -temperature electric -field dependent longitudinal resistance of the Mn 2Ga \nfilm in the Mn 2Ga/MgO/PMN -PT heterostructure is measured with the linear four -probe \ngeometry (Fig. 8a). As shown in Fig. 8b, the positive and negative peaks in perpendicular gating \ncurrent through the MgO buffer layer and PMN -PT clearly exhibit the reversible ferroelectric \npolarization switching feature. Correspondingly, the electric -field-dependent longitudinal \nresistance (Fig. 8c) shows an asymmetric and nonvolatile butterfly loop, which is similar to \nwhat we obtained in previous measur ements [3,23] . \nEmpir ically , the AHE in ferromagnetic materials is closely related to magnetization. Motivated \nby this understanding, we examined the out -of-plane magnetization change of the Mn 2Ga film \nfor the Mn 2Ga/MgO/PMN -PT heterostructure upon electric -field poling (Fig. 9a) of the \nferroelectric substrate PMN -PT. As shown in Fig. 9b , c, the perpendicular magnetization has \nbeen changed substantially. At 50 K, the electric -field poling of the PMN -PT alter the saturation \nmagnetization of Mn 2Ga alters from ~360 to ~423 emu/cc (Fig. 9b) , which corresponding to an \n~17.5% magnetization enhancement, similar to the anomalous Hall resistance increase in Fig. \n7b. For 300 K, the out -of-plane magnetization changes from ~335 to ~382 emu/cc (Fig. 9c) , \nwell consistent with the ~14% anomal ous Hall resistance variation in Fig. 7g. Thus, these \nexperimental results clearly illustrate that the piezoelectric -strain -induced anomalous Hall \neffect modulation is predominantly caused by the strain -induced magnetization variation. For \nferrimagnetic ma terials with two opposite unequal sublattices, the enlargement of the net \nmagnetization would likely pertain to the weakening of the compensation of two sublattices in \nterms of the spin rotation, reminiscent of the scenario of noncollinear antiferromagnetic spin \nstructure modulation by piezoelectric strain as theoretically described by Lukashev et al. [32]. \n4. Conclusions \nIn conclusion, we have fabricated epitaxial ferrimagnetic Mn 2Ga thin films with perpendicular \nmagnetic anisotropy on MgO substrates. The mechanisms of the AHE were unveiled for different longitudinal resistivit y ranges . When MgO is used as buffer layer, Mn 2Ga thin films \nwith perpendicular anisotropy have been successfully integra ted onto ferroelectric PMN -PT \nsubstrates which is useful for utilizing ferrimagnetic material s in high-density spintronic \ndevices and could enable the fabrication of other exotic epitaxial heterostructures with \nferrimagnets interfacing with some novel mate rials [33-45]. Via the defects engineering in thin \nfilms [46], the spin structure and the AHE of ferrimagnetic Mn 2Ga could further be modulated \nto realize the topological Hall effect . More importantly , the AHE of ferrimagnetic Mn 2Ga films \nis largely modulated by the electric -field-induced piezoelectric strain , which paves the way for \nmagnetic -field-free low -power ferrimagnetic spintronic device applications. \n \nAcknowledgements: Zhi-Qi Liu acknowledges financial support from National Natur al \nScience Foundation of China (NSFC Grant Nos: 52121001, 51822101, 51861135104 & \n51771009). \n References \n[1] Feng Z, Yan H, Wang X, Guo H, Qin P, Zhou X, Chen Z, Wang H, Jiao Z, Leng Z, Hu Z, Zhang \nX, Wu H, Chen H, Wang J, Zhang T, Jiang C, Liu Z. Nonvolatile electric control of the anomalous \nHall effect in an ultrathin magnetic metal. Adv Electron Mater. 2019;6(2):1901084. \n[2] Wang X, Feng Z, Qin P, Yan H, Zhou X, Guo H, Leng Z, Chen W, Jia Q, Hu Z, Wu H, Zhang X, \nJiang C, Liu Z. Integration of the noncollinear antiferromagnetic metal Mn 3Sn onto ferroelectric \noxides for electric -field control. 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Phys Rev B. 2021;104(6):064428. \n Figure 1 \n \nFig. 1 a X-ray diffraction pattern s of 100-nm-thick Mn 2Ga/MgO heterostructure s grow n at different \ntemperatures ; b Temperature -dependent normalized resistivity of Mn 2Ga thin films deposited at different \ntemperatures ; c-f The magnetization curves of Mn 2Ga samples deposited at 450 and 550 °C are measured \nat 50 and 300 K, respectively \n \nFigure 2 \n \nFig. 2 Magnetoresistance of 100-nm-thick Mn 2Ga/MgO films fabricated at different growth \ntemperatures TG. a TG = 150 °C; b TG = 300 °C; c TG = 450 °C; f TG = 550 °C \n \nFigure 3 \n \nFig. 3 Hall effect for 100-nm-thick Mn 2Ga/MgO films fabricated below 450 °C. a TG = 30°C; b TG \n= 150 °C; c TG = 300 °C \n \nFigure 4 \n \nFig. 4 a Hall effect for a 100-nm-thick Mn 2Ga/MgO film fabricated at TG = 450 °C; b Hall effect for a \n100-nm-thick Mn 2Ga/MgO film fabricated at TG = 550 °C; c Scaling law analysis for the Mn 2Ga film \nfabricated at TG = 550 °C \n \nFigure 5 \n \nFig. 5 Hall effect of Mn 2Ga/MgO film s with smaller thickness fabricated at TG = 550 °C. a 50 nm ; \nb 30 nm \nFigure 6 \n \nFig. 6 a X-ray diffraction pattern of a Mn 2Ga/MgO/PMN -PT heterostructure ; b Hall effect \nmeasurement s of the Mn 2Ga/MgO/PMN -PT heterostructure at different temperatures ranging from 50 to \n400 K \n \nFigure 7 \n \n \nFig. 7 a Schematic of the perpendicular electric -field modulation of the AHE for the Mn 2Ga(30 \nnm)/MgO(25 nm)/PMN -PT heterostructure ; b-g AHE curves for the Mn 2Ga film before and after electric \npoling at various temperatures ; h Relative zero -field Hall resistance modulation as a function of \ntemperature \n \nFigure 8 \n \n \nFig. 8 a Schematic of the perpendicular electric -field modulation of the longitudinal resistance of Mn 2Ga \nin a Mn 2Ga/MgO/PMN -PT heterostructure ; b Room -temperature gating current as a function of \nperpendicular electric field ; c Room -temperature longitudinal resistance of the Mn 2Ga film versus \nelectric field. \n \nFigure 9 \n \n \nFig. 9 a Schematic of the out-of-plane magnetization measurements upon electric -field poling of the \nferroelectric PMN -PT substrate for the Mn 2Ga(30 nm)/MgO(25 nm)/PMN -PT heterostructure ; b Out-of-\nplane magnetization of the Mn 2Ga film at 50 K before and after electric -field poling ; c Out-of-plane \nmagnetization of the Mn 2Ga film at 300 K before and after electric -field poling \n" }, { "title": "0905.1036v1.Towards_the_theory_of_ferrimagnetism_II.pdf", "content": "arXiv:0905.1036v1 [cond-mat.str-el] 7 May 2009Towards the theory of ferrimagnetism II\nNaoum Karchev\nDepartment of Physics, University of Sofia, 1126 Sofia, Bulga ria\nThe present paper is a sequel to the paper by Karchev (2008 J.P hys.:Condens.Matter 20325219).\nA two-sublattice ferrimagnet, with spin- s1operators S1iat the sublattice Asite and spin- s2oper-\natorsS2iat the sublattice Bsite, is considered. Renormalized spin-wave theory, which accounts\nfor the magnon-magnon interaction, and its extension are de veloped to describe the two ferrimag-\nnetic phases (0 ,T∗) and (T∗,TN) in the system, and to calculate the magnetization as a funct ion of\ntemperature.\nThe influence of the parameters in the theory on the character istic temperatures TNandT∗is\nstudied. It is shown that, increasing the inter-sublattice exchange interaction, the ratio TN/T∗>1\ndecreases approaching one, and above some critical value of the exchange constant there is only\none phase TN=T∗, and the magnetization-temperature curve has the typical C urie-Weiss profile.\nWhen the intra-exchange constant of sublattice with strong er intra-exchange interaction increases\ntheNe`eltemperature increases while T∗remains unchanged. Finally, when the magnetic order of\nthe sublattice with smaller magnetic order decreases, T∗decreases. The theoretical predictions are\nutilize to interpret the experimentally measured magnetiz ation-temperature curves.\nPACS numbers: 75.50.Bb, 75.30.Ds, 75.60.Ej, 75.50.-y\nI. INTRODUCTION\nThe present paper is a sequel to the paper [1]. A two-\nsublattice ferrimagnet, with spin- s1operators S1iat the\nsublattice Asite and spin- s2operators S2iat the sub-\nlatticeBsite. The true magnons of a two-spin system\nare transversal fluctuations of the total magnetization\nwhich includes both the magnetization of the sublattice\nAandBspins. The magnon excitation is a complicate\nmixture of the transversal fluctuations of the sublattice\nAandBspins. As a result the magnons’ fluctuations\nsuppress, in different way, the magnetic orders on the\ndifferent sublattices and one obtains two phases. At low\ntemperature (0 ,T∗) the magnetic orders of the Aand\nBspins contribute to the magnetization of the system,\nwhile at the high temperature ( T∗,TN) the magnetiza-\ntion of the spins with a weaker intra-sublattice exchange\nis suppressed by magnon fluctuations, and only the spins\nwiththestrongerintra-sublatticeexchangehavenon-zero\nspontaneous magnetization.\nRenormalizedspin-wavetheory, which accountsforthe\nmagnon-magnon interaction, and its extension are de-\nveloped to describe the two ferrimagnetic phases in the\nsystem and to calculate the magnetization as a function\nof temperature. It is impossible to require the theoret-\nically calculated N´eeltemperature and magnetization-\ntemperaturecurvestobe in exactaccordancewith exper-\nimental results. The models are idealized, and they do\nnotconsidermanyimportanteffects: phononmodes, sev-\neral types of disorder, Coulomb interaction, etc. Because\nof this it is important to formulate theoretical criteria for\nadequacy of the method of calculation. In my opinion\nthe calculations should be in accordance with Mermin-\nWagner theorem [2]. It claims that in two dimensions\nthere is not spontaneous magnetization at non-zero tem-\nperature. Hence, thecriticaltemperatureshouldbeequal\nto zero. It is well known that the Monte Carlo methodof calculation does not satisfy this criteria, and ”weak\nz-coupling” 3D system is used to mimic a 2D layer. It is\ndifficult within Dynamical Mean-Field Theory (DMFT)\nto make a difference between two dimensional and three\ndimensional systems. DMFT is a good approximation\nwhen the dimensionality goes to infinity. The present\nmethods of calculation, being approximate, capture the\nbasic physical features and satisfy the Mermin-Wagner\ntheorem.\nThere is an important difference between N´eeltheory\n[3] and the results in the present paper. N´eel’s calcula-\ntions predict a temperature TNat which both the sublat-\nticeAandBmagnetizations become equal to zero and\nT∗is a temperature at which the magnetic moment has\na maximum.\nThe influence of the parameters in the theory on the\ncharacteristic temperatures TNandT∗is studied. It\nis shown that, increasing the inter-sublattice exchange\ninteraction, the ratio TN/T∗>1 decreases approach-\ning one, and above some critical value of the exchange\nconstant there is only one phase TN=T∗, and the\nmagnetization-temperature curve has the typical Curie-\nWeiss profile. When the intra-exchange constant of the\nsublattice with stronger intra-exchange interaction in-\ncreases the Ne`eltemperature increases while T∗remains\nunchanged. Finally, when the magnetic order of the\nsublattice with smaller magnetic order decreases, T∗de-\ncreases.\nTo compare the theoretical results and the experimen-\ntalmagnetization-temperaturecurvesonehas, firstofall,\nto interpretadequatelythe measurements. The magnetic\nmoments in some materials are close to ”spin only” value\n2µBSand the sublattice spins s1ands2can be obtained\nfrom the experimental curves. As an example I consider\nthesulpho-spinel MnCr 2S4−xSex[4]. Onthe otherhand\nthere are ferrimagnets with strong spin-orbital interac-\ntion. Itisconvenient,inthatcase,toconsider jjcoupling2\nwithJA=LA+SAandJB=LB+SB. As an example\nI consider the vanadium spinel MnV2O4[5, 6, 7, 8].\nThe paper is organized as follows. In Sec. II the\nmodel is presented and a renormalized spin-wave theory\nisworkedouttocalculatethemagnetization-temperature\ncurves for different parameters of the model. The influ-\nence of the theory parameters on the N´eelandT∗tem-\nperatures is studied in Sec. III. I consider three cases:\ni) when the inter-sublattice exchange constant increases\nandallthe otherparametersarefixed, ii) oneoftheintra-\nsublattice parameters is changed and iii) when one of the\nspins decreases. Applications and analyzes of experimen-\ntal magnetization-temperature curves are given in Sec.\nIV. A summary in Sec. V concludes the paper.\nII. SPIN-WAVE THEORY\nA. Renormalized spin-wave (RSW) theory\nThe Hamiltonian of the system is\nH=−J1/summationdisplay\n≪ij≫AS1i·S1j−J2/summationdisplay\n≪ij≫BS2i·S2j\n+J/summationdisplay\n/angbracketleftij/angbracketrightS1i·S2j (1)\nwhere the sums are over all sites of a three-dimensional\ncubic lattice: ∝angbracketlefti,j∝angbracketrightdenotes the sum over the nearest\nneighbors, ≪i,j≫Adenotes the sum over the sites of\nthe A sublattice, ≪i,j≫Bdenotes the sum over the\nsites of the B sublattice. The first two terms describe\nthe ferromagnetic Heisenberg intra-sublattice exchange\nJ1>0,J2>0, while the third term describes the inter-\nsublattice exchangewhichisantiferromagnetic J >0. To\nstudy a theory with the Hamiltonian Eq.(1) it is conve-\nnient to introduce Holstein-Primakoff representation for\nthe spin operators\nS+\n1j=S1\n1j+iS2\n1j=/radicalBig\n2s1−a+\njajaj\nS−\n1j=S1\n1j−iS2\n1j=a+\nj/radicalBig\n2s1−a+\njaj(2)\nS3\n1j=s1−a+\njaj\nwhen the sites jare from sublattice Aand\nS+\n2j=S1\n2j+iS2\n2j=−b+\nj/radicalBig\n2s2−b+\njbj\nS−\n2j=S1\n2j−iS2\n2j=−/radicalBig\n2s2−b+\njbjbj(3)\nS3\n2j=−s2+b+\njbj\nwhen the sites jare from sublattice B. The operators\na+\nj, ajandb+\nj, bjsatisfytheBosecommutationrelations.\nIn terms of the Bose operators and keeping only the\nquadratic and quartic terms, the effective Hamiltonian\nEq.(1) adopts the form\nH=H2+H4 (4)where\nH2=s1J1/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\n+s2J2/summationdisplay\n≪ij≫B/parenleftbig\nb+\nibi+b+\njbj−b+\njbi−b+\nibj/parenrightbig\n(5)\n+J/summationdisplay\n/angbracketleftij/angbracketright/bracketleftbig\ns1b+\njbj+s2a+\niai−√s1s2/parenleftbig\na+\nib+\nj+aibj/parenrightbig/bracketrightbig\nH4=1\n4J1/summationdisplay\n≪ij≫A/bracketleftbig\na+\nia+\nj(ai−aj)2+(a+\ni−a+\nj)2aiaj/bracketrightbig\n+1\n4J2/summationdisplay\n≪ij≫B/bracketleftbig\nb+\nib+\nj(bi−bj)2+(b+\ni−b+\nj)2bibj/bracketrightbig\n+1\n4J/summationdisplay\n/angbracketleftij/angbracketright/bracketleftbigg/radicalbiggs1\ns2/parenleftbig\naib+\njbjbj+a+\nib+\njb+\njbj/parenrightbig\n(6)\n+/radicalbiggs2\ns1/parenleftbig\na+\niaiaibj+a+\nia+\niaib+\nj/parenrightbig\n−4a+\niaib+\njbj/bracketrightbigg\nand the terms without operators are dropped.\nThe next step is to represent the Hamiltonian in the\nHartree-Fock approximation\nH≈HHF=Hcl+Hq (7)\nwhere\nHcl= 12NJ1s2\n1(u1−1)2+12NJ2s2\n2(u2−1)2\n+ 6NJs1s2(u−1)2, (8)\nH2=s1J1u1/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\n+s2J2u2/summationdisplay\n≪ij≫B/parenleftbig\nb+\nibi+b+\njbj−b+\njbi−b+\nibj/parenrightbig\n(9)\n+Ju/summationdisplay\n/angbracketleftij/angbracketright/bracketleftbig\ns1b+\njbj+s2a+\niai−√s1s2/parenleftbig\na+\nib+\nj+aibj/parenrightbig/bracketrightbig\nandN=NA=NBis the number of sites on a sublattice.\nEquation (9) shows that the Hartree-Fock parameters\nu1, u2andurenormalize the intra-exchange constants\nJ1, J2and the inter-exchange constant J, respectively.\nIt is convenient to rewrite the Hamiltonian in momen-\ntum space representation\nHq=/summationdisplay\nk∈Br/bracketleftbig\nεa\nka+\nkak+εb\nkb+\nkbk−γk/parenleftbig\na+\nkb+\nk+bkak/parenrightbig/bracketrightbig\n,\n(10)\nwhere the wave vector kruns over the reduced first Bril-\nlouinzone Brofacubiclattice. Thedispersionsaregiven\nby equalities\nεa\nk= 4s1J1u1εk+ 6s2Ju\nεb\nk= 4s2J2u2εk+ 6s1J u (11)\nγk= 2J u√s1s2(coskx+cosky+coskz)3\nwith\nεk= 6−cos(kx+ky)−cos(kx−ky)−cos(kx+kz)\n−cos(kx−kz)−cos(ky+kz)−cos(ky−kz).(12)\nTo diagonalize the Hamiltonian one introduces new Bose\nfieldsαk, α+\nk, βk, β+\nkby means of the transformation\nak=ukαk+vkβ+\nka+\nk=ukα+\nk+vkβk\n(13)\nbk=ukβk+vkα+\nkb+\nk=ukβ+\nk+vkαk\nwhere the coefficients of the transformation ukandvk\nare real function of the wave vector k\nuk=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nεa\nk+εb\nk/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk+ 1\n\n(14)\nvk=sign(γk)/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nεa\nk+εb\nk/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−1\n\nThe transformed Hamiltonian adopts the form\nHq=/summationdisplay\nk∈Br/parenleftBig\nEα\nkα+\nkαk+Eβ\nkβ+\nkβk+E0\nk/parenrightBig\n,(15)\nwith new dispersions\nEα\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−εb\nk+εa\nk/bracketrightbigg\n(16)\nEβ\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk+εb\nk−εa\nk/bracketrightbigg\nand vacuum energy\nE0\nk=1\n2/bracketleftbigg/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk−εb\nk−εa\nk/bracketrightbigg\n(17)\nFor positivevalues ofthe Hartree-Fockparametersand\nall values of k∈Br, the dispersions are nonnegative\nEα\nk≥0, Eβ\nk≥0. For definiteness I choose s1> s2.\nWith these parameters, the αkboson is the long-range\n(magnon) excitation in the two-spin system with Eα\nk∝\nρk2, near the zero wavevector, while the βkboson is a\ngapped excitation.\nTo obtain the system ofequations for the Hartree-Fock\nparameters we consider the free energy of a system with\nHamiltonian HHFequations (8) and (15)\nF= 12J1s2\n1(u1−1)2+12J2s2\n2(u2−1)2\n+ 6Js1s2(u−1)2+1\nN/summationdisplay\nk∈BrE0\nk (18)\n+1\nβN/summationdisplay\nk∈Br/bracketleftBig\nln/parenleftBig\n1−e−βEα\nk/parenrightBig\n+ ln/parenleftBig\n1−e−βEβ\nk/parenrightBig/bracketrightBig\n.whereβ= 1/Tis the inverse temperature. Then the\nthree equations\n∂F/∂u1= 0, ∂F/∂u2= 0, ∂F/∂u= 0 (19)\nadopt the form (see the appendix)\nu1= 1−1\n6s11\nN/summationdisplay\nk∈Brεk/bracketleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/bracketrightBig\nu2= 1−1\n6s21\nN/summationdisplay\nk∈Brεk/bracketleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/bracketrightBig\nu= 1−1\nN/summationdisplay\nk∈Br/bracketleftbigg1\n2s1/parenleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/parenrightBig\n(20)\n+1\n2s2/parenleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/parenrightBig\n−2\n3Ju/parenleftBig\n1+nα\nk+nβ\nk/parenrightBig(coskx+cosky+coskz)2\n/radicalBig\n(εa\nk+εb\nk)2−4γ2\nk\n\nwherenα\nkandnβ\nkare the Bose functions of αandβ\nexcitations. The Hartree-Fock parameters, the solution\nof the system of equations (20), are positive functions\nofT/J,u1(T/J)>0, u2(T/J)>0 andu(T/J)>0.\nUtilizing these functions, one can calculate the sponta-\nneous magnetization of the system, which is a sum of\nthe spontaneous magnetization on the two sublattices\nM=MA+MB, where\nMA=< S3\n1j> j is from sublattice A\n(21)\nMB=< S3\n2j> j is from sublattice B\nIn terms of the Bose functions of the αandβexcitations\nthey adopt the form\nMA=s1−1\nN/summationdisplay\nk∈Br/bracketleftBig\nu2\nknα\nk+v2\nknβ\nk+v2\nk/bracketrightBig\n(22)\nMB=−s2+1\nN/summationdisplay\nk∈Br/bracketleftBig\nv2\nknα\nk+u2\nknβ\nk+v2\nk/bracketrightBig\nThemagnonexcitation- αkin the effectivetheoryequa-\ntion(15)-isacomplicatedmixtureofthetransversalfluc-\ntuations of the AandBspins. As a result the magnons’\nfluctuationssuppressinadifferentwaythemagnetization\non sublattices AandB. Quantitatively this depends on\nthe coefficients ukandvkin equations (22). At char-\nacteristic temperature T∗spontaneous magnetization on\nsublattice Bbecomes equal to zero, while spontaneous\nmagnetization on sublattice Bis still nonzero. Above\nT∗the system of equations (20) has no solution and one\nhas to modify the spin-wave theory. The magnetization\ndepends on the dimensionless temperature T/Jand di-\nmensionless parameters s1, s2, J1/JandJ2/J. For pa-\nrameters s1= 1.5, s2= 1, J1/J= 0.94 andJ2/J= 0.01\nthe functions MA(T/J) andMB(T/J) are depicted in\nfigure 1. The upper (blue) line is the sublattice Amag-\nnetization, the bottom (red) line is the sublattice Bmag-\nnetization.4\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52\n/s45/s49/s44/s48/s45/s48/s44/s53/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53\n/s32/s77/s65\n/s77/s66/s84/s47/s74/s115\n/s49/s61/s49/s46/s53/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s115\n/s50/s61/s49\n/s74\n/s49/s47/s74/s61/s48/s46/s57/s52/s32/s32/s32/s32/s32/s32/s74\n/s50/s47/s74/s61/s48/s46/s48/s49/s83/s80/s79/s78/s84/s65/s78/s69/s79/s85/s83/s32/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78/s84/s42\n/s47/s74\nFIG. 1: (color online)The spontaneous magnetization MA-\nupper (blue) line and MB-bottom (red) line as a function of\nT/Jfor parameters s1= 1.5, s2= 1, J1/J= 0.94 and\nJ2/J= 0.01.T∗is the temperature at which sublattice B\nmagnetization becomes equal to zero\nB. Modified RSW theory\nOnce suppressed, the sublattice Bmagnetization can-\nnot be restored increasing the temperature above T*. To\nformulate this mathematically we modify the spin-wave\ntheory using the idea of a description of the paramag-\nnetic phase of 2D ferromagnets ( T >0) by means of\nmodified spin-wave theory [10, 11] and its generalization\n[1]. We consider a two-sublattice system and, to enforce\nthe magnetization on the two sublattices to be equal to\nzero in paramagnetic phase, we introduce two parame-\ntersλAandλB[1]. The new Hamiltonian is obtained\nfrom the old one equation (1) by adding two new terms:\nˆH=H−/summationdisplay\ni∈Aλ1S3\n1i+/summationdisplay\ni∈Bλ2S3\n2i(23)\nIn momentum space the new Hamiltonian adopts the\nform\nˆH=/summationdisplay\nk∈Br/bracketleftbig\nˆεa\nka+\nkak+ ˆεb\nkb+\nkbk−γk(bkak+b+\nka+\nk)/bracketrightbig\n(24)\nwhere the new dispersions are\nˆεa\nk=εa\nk+λ1,ˆεb\nk=εb\nk+λ2.(25)\nUtilizing the same transformation equations (13) with\nparameters\nˆuk=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nˆεa\nk+ ˆεb\nk/radicalBig\n(ˆεa\nk+ ˆεb\nk)2−4γ2\nk+ 1\n\n(26)\nˆvk=sign(γk)/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nˆεa\nk+ ˆεb\nk/radicalBig\n(ˆεa\nk+ ˆεb\nk)2−4γ2\nk−1\none obtains the Hamiltonian in diagonal forma\nˆH=/summationdisplay\nk∈Br/parenleftBig\nˆEα\nkα+\nkαk+ˆEβ\nkβ+\nkβk+ˆE0\nk/parenrightBig\n,(27)\nwhere\nˆEα\nk=1\n2/bracketleftbigg/radicalBig\n(ˆεa\nk+ ˆεb\nk)2−4γ2\nk−ˆεb\nk+ ˆεa\nk/bracketrightbigg\nˆEβ\nk=1\n2/bracketleftbigg/radicalBig\n(ˆεa\nk+ ˆεb\nk)2−4γ2\nk+ ˆεb\nk−ˆεa\nk/bracketrightbigg\n(28)\nˆE0\nk=1\n2/bracketleftbigg/radicalBig\n(ˆεa\nk+ ˆεb\nk)2−4γ2\nk−ˆεb\nk−ˆεa\nk/bracketrightbigg\nIt is convenient to represent the parameters λ1andλ2\nin the form\nλ1= 6Jus2(µ1−1), λ2= 6Jus1(µ2−1).(29)\nIn terms ofthe new parameters µ1andµ2the dispersions\nˆεa\nkand ˆεb\nkadopt the form\nˆεa\nk= 4s1J1u1εk+ 6s2J uµ1\n(30)\nˆεb\nk= 4s2J2u2εk+ 6s1J uµ2\nThey are positive (ˆ εa\nk>0, ˆεb\nk>0) for all values of the\nwavevector k, if the parameters µ1andµ2are positive\n(µ1>0, µ2>0). The dispersions Eq.(28) are well de-\nfined if square-roots in equations (28) are well defined.\nThis is true if\nµ1µ2≥1. (31)\nTheβkexcitation is gapped ( Eβ\nk>0) for all values of\nparameters µ1andµ2which satisfy equation (31). The\nαexcitation is gapped if µ1µ2>1, but in the particular\ncase\nµ1µ2= 1 (32)\nˆEα\n0= 0, and near the zero wavevector\nˆEα\nk≈ˆρk2(33)\nwith spin-stiffness constant\nˆρ=8(s2\n2J2u2µ1+s2\n1J1u1µ2) + 2s1s2Ju\n(s1µ2−s2µ1)(34)\nIn the particular case equation (32) αkboson is the long-\nrange excitation (magnon) in the system.\nWeintroducedtheparameters λ1andλ2(µ1,µ2)toen-\nforce the sublattice AandBspontaneousmagnetizations\nto be equal to zero in the paramagnetic phase. We find\nouttheparameters µ1andµ2, aswellastheHartree-Fock\nparameters, as functions of temperature, solving the sys-\ntem of five equations, equations (20) and the equations\nMA=MB= 0, where the spontaneous magnetizations5\nhave the same representation as equations (22) but with\ncoefficients ˆ uk,ˆvk, and dispersions ˆEα\nk,ˆEβ\nkin the ex-\npressions for the Bose functions. The numerical calcula-\ntions show that for high enough temperature µ1µ2>1.\nWhen the temperature decreases the product µ1µ2de-\ncreases, remaining larger than one. The temperature at\nwhich the product becomes equal to one ( µ1µ2= 1) is\ntheN´eeltemperature. Below TN, the spectrum contains\nlong-range(magnon) excitations, thereupon µ1µ2= 1. It\nis convenientto representthe parametersin the following\nway:\nµ1=µ, µ 2= 1/µ. (35)\nIn the ordered phase magnon excitations are the ori-\ngin of the suppression of the magnetization. Near the\nzero temperature their contribution is small and at zero\ntemperature sublattice AandBspontaneous magneti-\nzation reach their saturation. On increasing the tem-\nperature magnon fluctuations suppress the sublattice A\nmagnetization and sublattice Bmagnetization in differ-\nent ways. At T∗the sublattice Bspontaneous magne-\ntization becomes equal to zero. Increasing the tempera-\nture above T∗, the sublattice Bmagnetization should be\nzero. This is why we impose the condition MB(T) = 0\nifT > T∗. For temperatures above T∗, the parameter µ\nand the Hartree-Fock parameters are solution of a sys-\ntem of four equations, equations (20) and the equation\nMB= 0. The Hartree-Fockparameters, as a functions of\ntemperature T/J, are depicted in figure 2 for parameters\ns1= 1.5, s2= 1, J1/J= 0.94 andJ2/J= 0.01. The\nvertical dotted (green) line corresponds to T∗/J.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s48/s44/s48/s48/s44/s49/s48/s44/s50/s48/s44/s51/s48/s44/s52/s48/s44/s53/s48/s44/s54/s48/s44/s55/s48/s44/s56/s48/s44/s57/s49/s44/s48/s49/s44/s49\n/s117\n/s49\n/s117\n/s117\n/s50\n/s84/s47/s74/s72/s65/s82/s84/s82/s69/s69/s45/s70/s48/s67/s75/s32/s80/s65/s82/s65/s77/s69/s84/s69/s82/s83\nFIG. 2: (color online)Hartree-Fock parameters u1,u2andu\nas a function of T/Jfors1= 1.5, s2= 1, J1/J= 0.94 and\nJ2/J= 0.01. The vertical dotted (green) line corresponds to\nT∗/J\nThe function µ(T/J) is depicted in figure 3 for the\nsame parameters.\nWe utilize the obtained function µ(T),u1(T),u2(T),\nu(T) to calculate the spontaneous magnetization as a/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48\n/s84/s42\n/s47/s74/s84\n/s78/s47/s74\nFIG. 3: (color online) µ(T/J) for parameters s1= 1.5, s2=\n1, J1/J= 0.94andJ2/J= 0.01. Theverticaldotted(green)\nline corresponds to T∗/J, while (red) dashed lines to TN/J\nandµ(TN/J).\nfunction of the temperature. Above T∗, the magneti-\nzation of the system is equal to the sublattice Amagne-\ntization. For the same parameters as above the functions\nMA(T/J) andMB(T/J) are depicted in figure 4a. The\nupper (blue) line is the sublattice Amagnetization, the\nbottom (red) line is the sublattice Bmagnetization. The\ntotal magnetization M=MA+MBis depicted in fig-\nure 4b.\nIII.TNANDT∗DEPENDENCE ON MODEL’S\nPARAMETERS\nThe existence of two ferromagnetic phases (0 ,T∗) and\n(T∗,TN) is a generic feature of two spin systems. The\ncharacteristic temperatures TNandT∗strongly depend\non the parameters of the model. Intuitively, it is clear\nthat, if the inter-exchange is much stronger than intra-\nexchanges, the ferromagnetic order sets in simultane-\nously on both sublattices. This is not true, if inter-\nexchange is not so strong. To demonstrate this I study\na system with sublattice Aspins1= 1.5, and sublat-\nticeBspins2= 1. For parameters J1/J= 0.5 and\nJ2/J= 0.005 the magnetization-temperature curve is\ndepicted in FIG.5 curve ”c”. The ratio of the charac-\nteristic temperatures equals TN/T∗= 1.722. Increasing\nthe inter-exchange coupling, J1/J= 0.3,J2/J= 0.003\n(curve ”b”), the ratio decreases TN/T∗= 1.229, and\nabove some critical value of the inter-exchange constant\nJ1/J= 0.05,J2/J= 0.0005N´eel’stemperature becomes\nequal to T∗. There is only one ferromagnetic phase, and\nmagnetization-temperature curve ”a” is a typical Curie-\nWeiss curve. Despite this the system does not describe\nferromagnet, because the spin wave excitations are su-\nperposition of the sublattice AandBspin excitations.6\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53\n/s45/s49/s44/s48/s45/s48/s44/s56/s45/s48/s44/s54/s45/s48/s44/s52/s45/s48/s44/s50/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s49/s44/s50/s49/s44/s52/s49/s44/s54\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s49/s44/s50/s49/s44/s52/s49/s44/s54/s32/s77/s65\n/s77/s66/s84/s47/s74\n/s115\n/s49/s61/s49/s46/s53/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s115\n/s50/s61/s49\n/s74\n/s49/s47/s74/s61/s48/s46/s57/s52/s32/s32/s32/s32/s32/s32/s74\n/s50/s47/s74/s61/s48/s46/s48/s49/s83/s80/s79/s78/s84/s65/s78/s69/s79/s85/s83/s32/s77 /s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78/s97\n/s84/s47/s74 /s84/s42\n/s47/s74/s98\n/s77/s65\n/s43/s77/s66\nFIG. 4: (color online)a) The sublattice Aspontaneous mag-\nnetization MA-upper (blue) line and sublattice Bsponta-\nneous magnetization MB-bottom (red) line as a function of\nT/Jfor parameters s1= 1.5, s2= 1, J1/J= 0.94 and\nJ2/J= 0.01.\nb) The total spontaneous magnetization MA+MB.T∗/J-\nvertical dotted (green) line\nNext, I consider a system with sublattice Aspins1=\n1.5, and sublattice Bspins2= 1. The ratio of sublat-\nticeBexchange constant J2and inter-exchange constant\nJis fixedj2=J2/J= 0.01, while the ratio j1=J1/J\nvaries. When the sublattice Aexchange constant J1in-\ncreasesj1=J1/J= 0.64,0.84,0.94, the magnetization-\ntemperature curve at temperatures below T∗does not\nchange. There is no visible difference between T∗tem-\nperatures for the three values of the parameter J1/J.\nThe difference appears when the temperature is above\nT∗. Increasing sublattice Aexchange constat increases\ntheN´eeltemperature. The three curves are depicted in\nfigure 6.\nFinally, I consider three systems with equal exchange\nconstants J1/J= 0.4,J2/J= 0.004 and sublattice A\nspins1= 4, but with three different sublattice Bspins\n(figure 7). The calculations show that decreasing the\nsublattice Bspindecreases T∗temperature, increasesthe\nmaximum of magnetization at T∗and zero temperature/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53/s50/s44/s48/s50/s44/s53/s51/s44/s48\n/s97\n/s98/s99\n/s84/s47/s74\n/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78\n/s91\n/s93\nFIG. 5: (color online) The magnetization 2 MA+ 2MBas\na function of T/Jfors1= 1.5 ands2= 1, curve a:J1/J=\n0.05,J2/J= 0.0005, curve b:J1/J= 0.3,J2/J= 0.003,\ncurvec:J1/J= 0.5,J2/J= 0.005.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53/s50/s44/s48/s50/s44/s53/s51/s44/s48/s106\n/s49/s61/s48/s46/s57/s52/s32/s32/s84\n/s78/s47/s84/s42/s61/s51/s46/s48/s48/s52\n/s106\n/s49/s61/s48/s46/s56/s52/s32/s32/s84\n/s78/s47/s84/s42/s61/s50/s46/s54/s57/s50\n/s106\n/s49/s61/s48/s46/s54/s52/s32/s32/s84\n/s78/s47/s84/s42/s61/s50/s46/s48/s56/s57\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s106\n/s50/s61/s48/s46/s48/s49\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s115\n/s49/s61/s49/s46/s53/s32/s32/s115\n/s50/s61/s49\n/s84/s47/s74/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78 /s91 /s93\nFIG. 6: (color online) The magnetization 2 MA+ 2MBas a\nfunction of T/Jfors1= 1.5,s2= 1,j2=J2/J= 0.01 and\nthree values of the parameter j1=J1/J;j1= 0.94 (black)\nsquares, j1= 0.84 (red) circles, j1= 0.64 (blue) triangles\nmagnetization.7\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48/s48/s49/s50/s51/s52/s53/s54/s55/s56\n/s84/s47/s74\n/s84\n/s99/s42\n/s47/s74/s84\n/s98/s42\n/s47/s74/s84\n/s97/s42\n/s47/s74/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78 /s91\n/s66/s93/s97\n/s98\n/s99\nFIG. 7: (color online) The magnetization 2 MA+ 2MBas\nfunction of T/JforJ1/J= 0.4,J2/J= 0.004,s1= 4, and\ns2= 0.5-curvea(green), s2= 1.5-curveb(red),s2= 2.5-\ncurvec(black)\nIV. THEORY AND EXPERIMENT\nA. Sulpho-spinel MnCr 2S4−xSex\nThe sulpho-spinel MnCr 2S4−xSexhas been investi-\ngated by measurements of the magnetization at 15 .3kOe\nas a function of temperature (figure 94 in [4]). The max-\nimum in the magnetization versus temperature curve,\nwhich is typical of MnCr 2S4(x= 0), increases when\nxincrease, and disappears at x= 0.5. TheN´eeltem-\nperature decreases from 74 Katx= 0 to 56 Katx= 2.\nThe authors’ conclusion is that the observed change of\nthe magnetic properties is attributed to a decrease of the\nstrength of the negative Mn2+−Cr3+superexchange\ninteraction with increasing Seconcentration.\nWe obtained, see figure 5, that the maximum of the\nmagnetization is at T∗. Above T∗the magnetization of\nthe system is equal to the magnetization of sublattice A\nspins. If we extrapolate this curvebelow T∗down to zero\ntemperature we will obtain a value close to 2 s1µB, where\ns1is the spin of the sublattice Aspin operators. The\nexperimentalfigures[4]showthatextrapolationsgiveone\nand the same result for all values of x. One can accept\nthe fact that the Seconcentration do not influence over\nthe value of sublattice Aspin and s1= 1.5.\nBelowT∗the magnetization is a sum of sublattice\nAandBmagnetization. Hence, the magnetization at\nzero temperature is equal to 2( s1−s2)µB. Therefore,\none can determine the sublattice Bspins2. The re-\nsults of the theoretical calculations of magnetization, in\nBohr magnetons, are depicted in figure 8 for parameters\ns1= 1.5, J1/J= 0.47, J2/J= 0.001 and s2= 1-curve\na(black); s2= 0.7-curveb(red), and s2= 0.4-curvec(blue). The temperature and magnetization axis are\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s49/s44/s50/s49/s44/s52/s49/s44/s54/s49/s44/s56/s50/s44/s48/s50/s44/s50/s50/s44/s52/s50/s44/s54/s50/s44/s56/s51/s44/s48/s51/s44/s50/s51/s44/s52/s51/s44/s54\n/s97/s98/s99\n/s84/s47/s74/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78 /s91\n/s66/s93\nFIG. 8: (color online) The magnetization 2 MA+ 2MBas\nfunction of T/JforJ1/J= 0.47,J2/J= 0.001,s1= 1.5,\nands2= 1-curve a(black), s2= 0.7-curveb(red),s2= 0.4-\ncurvec(blue)\nchosen in accordance with experimental figure. Compar-\ning figure 94 in [4] and figure 7 in the present paper, one\nconcludesthatthe effectivesublattice Bspins2decreases\nwith increasing Se concentration, and this is the origin\nof the anomalous temperature variation of magnetiza-\ntion. The figure 8 shows that the present calculations\ncapture the essential features of the system; increasing\ntheSeconcentration (decreasing s2) leads to a decrease\nofN´eeltemperature, T∗temperature decreases too, and\nthe maximum of the magnetization increases. Compar-\ning the figure 8 in the present paper and figure 5 in [1]\none realizes the importance of the present method of cal-\nculation for adequate reproducingthe characteristictem-\nperatures TN,T∗, and the shape of the magnetization-\ntemperature curves.\nB. Vanadium spinel MnV2O4\nThe spinel MnV2O4is a two-sublattice ferrimagnet,\nwithsite Aoccupiedbythe Mn2+ion, whichisinthe3 d5\nhigh-spin configuration with quenched orbital angular\nmomentum, which can be regarded as a simple s= 5/2\nspin. The B site is occupied by the V3+ion, which takes\nthe 3d2high-spin configuration in the triply degenerate\nt2gorbital and has orbital degrees of freedom. The mea-\nsurements show that the setting in of the magnetic order\nis atN´eeltemperature TN= 56.5K[5] and that the\nmagnetization has a maximum near T∗= 53.5K. Be-\nlowthistemperaturethemagnetizationsharplydecreases\nand goes to zero when temperature approaches zero.8\nWe consider a system which obtains its magnetic prop-\nertiesfrom MnandVmagneticmoments. Becauseofthe\nstrongspin-orbitalinteractionit is convenienttoconsider\njjcoupling with JA=SAandJB=LB+SB. The sub-\nlatticeAtotal angular momentum is jA=sA= 5/2,\nwhile the sublattice Btotal angular momentum is jB=\nlB+sB, withlB= 3, and sB= 1 [5]. Then the g-factor\nfor the sublattice AisgA= 2, and the atomic value of\nthegBisgB=5\n4. The sublattice Amagnetic order is\nantiparallel to the sublattice Bone and the saturated\nmagnetization is σ= 25\n2−5\n44 = 0, in agreement with\nthe experimental finding that the magnetization goes to\nzero when the temperature approaches zero. The Hamil-\ntonian of the system is\nH=−κA/summationdisplay\n≪ij≫AJA\ni·JA\nj−κB/summationdisplay\n≪ij≫BJB\ni·JB\nj\n+κ/summationdisplay\n/angbracketleftij/angbracketrightJA\ni·JB\nj (36)\nThe first two terms describe the ferromagnetic Heisen-\nberg intra-sublattice exchange κA>0,κB>0, while\nthe third term describes the inter-sublattice exchange\nwhich is antiferromagnetic κ >0. To proceed we use\nthe Holstein-Primakoff representation of the total angu-\nlar momentum vectors JA\nj(a+\nj,aj) andJB\nj(b+\nj, bj), where\na+\nj, ajandb+\nj, bjare Bose fields, and repeat the calcula-\ntions from sections II and III. The magnetization of the\nsystemgAMA+gBMBas a function of the tempera-\nture is depicted in figure 9 for parameters κA/κ= 0.45\nandκB/κ= 0.001. The parameters are chosen so that\nthe calculations to reproduce the experimental value of\nthe ratio TN/T∗.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53 /s54/s48 /s54/s53 /s55/s48/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53/s50/s44/s48/s50/s44/s53/s51/s44/s48/s51/s44/s53\n/s84/s47\n/s84/s42/s47/s77/s65/s71/s78/s69/s84/s73/s90/s65/s84/s73/s79/s78 /s91\n/s66 /s93/s65 /s47 /s61/s48/s46/s52/s53\n/s66/s47 /s61/s48/s46/s48/s48/s49\n/s84\n/s78/s47/s84/s42/s61/s49/s46/s48/s56/s53\nFIG. 9: (color online)The magnetization gAMA+gBMBas\na function of T/κfor parameters κA/κ= 0.45 andκB/κ=\n0.001.\nThe profile of the magnetization-temperature curve is\nin a very good agrement with the experimental zero-\nfield cooling (ZFC) magnetization curves [6, 7]. Theanomaloustemperaturedependence ofthe magnetization\nis reproduced, but there is an important difference be-\ntween the interpretation of the experimental results in\n[5, 6, 7, 8, 9], and the present theoretical results. In\nthe experimental papers TNis the temperature at which\nboth the MnandVmagnetization become equal to zero.\nThe present theory predicts two phases: at low temper-\natures (0 ,T∗) sublattice Mnmagnetization and sublat-\nticeVmagnetization contribute to the magnetization of\nthe system, while at high temperatures ( T∗,TN) only\nMnions have non-zero spontaneous magnetization. The\nvanadium sublattice magnetization set in at T∗, and ev-\nidence for this is the abrupt decrease of magnetization\nbelowT∗, which also indicates that the magnetic order\nof vanadium electrons is anti-parallel to the order of Mn\nelectrons.\nFor samples cooled in a field (FC magnetization) the\nfield leads to formation of a single domain and, in addi-\ntion, increases the chaotic order of the spontaneous mag-\nnetization of the vanadium sublattice, which is antipar-\nallel to it. As a result the average value of the vana-\ndium magnetic order decreases and does not compensate\ntheMnmagnetic order. The magnetization curves de-\npend on the applied field, and do not go to zero. For\na larger field the (FC) curve increases when tempera-\nture decreases below N´eeltemperature . It has a max-\nimum at the same temperature T∗< TNas the ZFC\nmagnetization, and a minimum at T∗\n1< T∗. Below T∗\n1\nthe magnetization increases monotonically when temper-\nature approaches zero.\nTheexperimentswith samplescooledinfield (FCmag-\nnetization) providea new opportunity to clarify the mag-\nnetism of the manganese vanadium oxide spinel. The\napplied field is antiparallel with vanadium magnetic mo-\nment and strongly effect it. On the other hand, the ex-\nperiments show that there is no difference between ZFC\nand FC magnetization curveswhen the temperature runs\nover the interval ( T∗,TN) [6, 7]. They begin to diverge\nwhen the temperature is below T∗. This is in accor-\ndance with the theoretical prediction that the vanadium\nmagnetic moment does not contribute the magnetization\nwhenT > T∗andT∗is the temperature at which the\nvanadium ions start to contribute the magnetization of\nthe system. Because of the strong field, the two vana-\ndium bands are split and the magnetic moment of one of\nthet2gelectrons is reoriented to be parallel with the field\nand magnetic orderof the Mnelectrons. The description\nof this case is more complicate and requires three mag-\nnetic orders to be involved. When T∗< T < T Nonly\nMnions have non zero spontaneous magnetization. At\nT∗vanadium magnetic orderantiparallelto the magnetic\norder ofMnsets in and partially compensates it. Below\nT∗\n1the reoriented electron gives contribution, which ex-\nplains the increasing of the magnetization of the system\nwhen the temperature approaches zero. A series of ex-\nperiments with different applied field could be decisive\nfor the confirmation or rejection of the T∗transition. In-\ncreasing the applied field one expects increasing of T∗\n19\nand when the field is strong enough, so that all vana-\ndium electrons are reoriented, an anomalous increasing\nof magnetization below T∗would be obtained as within\nthe ferromagnetic phase of UGe2[12].\nV. SUMMARY\nIn summary, I have worked out a renormalized spin-\nwave theory and its extension to describe the two phases\n(0,T∗) and (T∗,TN) of a two sublattice ferrimagnet.\nComparing the figure 4 in the present paper and figure 4\nin [1] and figure 8 in the present paper and figure 5 in [1]\nonebecomesawareoftherelevanceofthepresentcalcula-\ntions for the accurate reproduction of the basic features\nof the system near the characteristic temperatures TN\nandT∗.\nThe present theory of ferrimagnetism permits to con-\nsider more complicate systems such as CeCrSb 3com-\npound [13] or the spinel Fe3O4which are two sublattice\nferrimagnets but with three spins.\nVI. ACKNOWLEDGMENTS\nThis work was partly supported by a Grant-in-Aid\nDO02-264/18.12.08 from NSF-Bulgaria.\nAPPENDIX A\nTo make more transparent the derivation of the equa-\ntions for the Hartree-Fock parameters Eq.(20) I consider\nthe first term (the sublattice Aterm) in the Hamiltonian\nof the magnon-magnon interaction Eq.(6). To write this\nterm in the Hartree-Fock approximation one represents\nthe product of two Bose operators in the form\na+\niaj=a+\niaj−< a+\niaj>+< a+\niaj>(A1)\nand neglects all terms ( a+\niaj−< a+\niaj>)2in the four\nmagnon interaction Hamiltonian. The result is\n1\n2a+\niaja+\niai≈ −< a+\niaj>< a+\niai>\n+< a+\niaj> a+\niai+a+\niaj< a+\niai>\n1\n2a+\njaia+\njaj≈ −< a+\njai>< a+\njaj>\n+< a+\njai> a+\njaj+a+\njai< a+\njaj>\n1\n2a+\njaja+\niaj≈ −< a+\njaj>< a+\niaj> (A2)\n+< a+\njaj> a+\niaj+a+\njaj< a+\nraj>a+\niaia+\njaj≈ −< a+\niai>< a+\njaj>\n+< a+\niai> a+\njaj+a+\niai< a+\njaj>\n−< a+\niaj>< a+\njai>\n+< a+\niaj> a+\njai+a+\njai< a+\niaj>\nThe Hartree-Fockapproximationofthe sublattice Apart\nof the Hamiltonian of magnon-magnon interaction reads\n1\n4J1/summationdisplay\n≪ij≫A/bracketleftbig\na+\nia+\nj(ai−aj)2+(a+\ni−a+\nj)2aiaj/bracketrightbig\n≈12NJ1s2\n1(u1−1)2(A3)\n+J1s1(u1−1)/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\nwhere the Hartree-Fock parameter u1is defined by the\nequation\nu1= 1−1\n6s11\nN/summationdisplay\nk∈Brek< a+\nkak>(A4)\nCombining the sublattice Apart of the Hamiltonian\nEq.(5) (the first term) and Eq.(A3) one obtaines the\nHartree-Fock approximation for the sublattice Apart of\nthe Hamiltonian\nHA≈12NJ1s2\n1(u1−1)2(A5)\n+J1s1u1/summationdisplay\n≪ij≫A/parenleftbig\na+\niai+a+\njaj−a+\njai−a+\niaj/parenrightbig\nIn the same way one obtains the Hartree-Fock approx-\nimation of the sublattice Band inter sublattices parts\nof the Hamiltonian. The result is the HHFHamiltonian\nEqs.(7,8,9).\nTo calculate the thermal average < a+\nkak>, in the\nEq.(A4), one utilizes the Hamiltonian HHF. Therefor,\nthe matrix element depends on the Hartree-Fock param-\neters, andequation(A4)isoneoftheselfconsistentequa-\ntions for these parameters.\nThe matrix element can be represented in terms of\nαk(α+\nk) andβk(β+\nk) Eq.(13)\n< a+\nkak>=u2\nknα\nk+v2\nknβ\nk+v2\nk(A6)\nwherenα\nk=< α+\nkαk>, nβ\nk=< β+\nkβk>are the Bose\nfunctions of αandβexcitations. Substituting the ther-\nmal average in Eq.(A4) with Eq.(A6), one obtains that\nequation (A4) is exactly the first equation of the system\nEq.(20) which in turn is obtained from the first of the\nequations (19).\n[1] N.Karchev, J.Phys.:Condens.Matter, 20, 325219 (2008). [2] N. D. Mermin and H. Wagner, Phys. Rev. Let t.17, 113310\n(1966).\n[3] L.N´eel, Ann. Phys., Paris, 3, 137 (1948).\n[4] P. P. Van Stapele, in Handbook of Magnetic Materials ,\nVolume3, 603, EditedbyE.P Wohlfarth, (North-Holland\nPublishing Company, 1982).\n[5] K. Adachi, T. Suzuki, K. Kato, K. Osaka, M. Takata,\nand T. Katsufuji, Phys. Rev. Lett., 95, 197202 (2005).\n[6] V. O. Garlea, R. Jin, D. Mandrus, B. Roessli, Q. Huang,\nM. Miller, A. J. Schultz, and S. E. Nagler, Phys. Rev.\nLett.,100, 066404 (2008).\n[7] S.-H.Baek, K.-Y Choi, A.P.Reyes, P.L. Kuhns,\nN.J.Curro, V.Ramanchandran, N.S. Dalal, H. D.\nZhou, and C.R. Wiebe, J. Phys.:Condens.Matter, 20,135218 (2008).\n[8] H. D. Zhou, J. Lu, and C. R. Wiebe, Phys. Rev., B 76,\n174403 (2007).\n[9] Vincent Hardy, Yohan Br´ eard, and Christine Martin,\nPhys. Rev. B 78, 024406 (2008).\n[10] M.Takahashi, Prog. Theor. Physics Supplement 87, 233\n(1986).\n[11] M.Takahashi, Phys. Rev. Lett. 58, 168 (1987).\n[12] C. Pfleiderer and A. D. Huxley, Phys. Rev. Lett., 89,\n147005 (2002).\n[13] D. D. Jackson, S. K. McCall, A. B. Karki, and D. P.\nYoung, Phys. Rev. B 76, 064408 (2007)." }, { "title": "1508.01427v1.Large_spin_wave_bullet_in_a_ferrimagnetic_insulator_driven_by_spin_Hall_effect.pdf", "content": "arXiv:1508.01427v1 [cond-mat.mes-hall] 6 Aug 2015Large spin-wave bullet in a ferrimagnetic insulator driven by spin Hall effect\nM. B. Jungfleisch,1,∗W. Zhang,1J. Sklenar,1,2J. Ding,1W. Jiang,1H. Chang,3\nF. Y. Fradin,1J. E. Pearson,1J. B. Ketterson,2V. Novosad,1M. Wu,3and A. Hoffmann1\n1Materials Science Division, Argonne National Laboratory, Argonne IL 60439, USA\n2Department of Physics and Astronomy, Northwestern Univers ity, Evanston IL 60208, USA\n3Department of Physics, Colorado State University, Fort Col lins CO 80523, USA\n(Dated: June 25, 2021)\nDue to its transverse nature, spin Hall effects (SHE) provide the possibility to excite and detect\nspin currents and magnetization dynamics even in magnetic i nsulators. Magnetic insulators are out-\nstanding materials for the investigation of nonlinear phen omena and for novel low power spintronics\napplications because of their extremely low Gilbert dampin g. Here, we report on the direct imaging\nof electrically driven spin-torque ferromagnetic resonan ce (ST-FMR) in the ferrimagnetic insulator\nY3Fe5O12based on the excitation and detection by SHEs. The driven spi n dynamics in Y 3Fe5O12\nis directly imaged by spatially-resolved microfocused Bri llouin light scattering (BLS) spectroscopy.\nPreviously, ST-FMR experiments assumed a uniform precessi on across the sample, which is not\nvalid in our measurements. A strong spin-wave localization in the center of the sample is observed\nindicating the formation of a nonlinear, self-localized sp in-wave ‘bullet’.\nMagneticmemoryandlogicdevicesrelyontheefficient\nmanipulation of the orientation of their magnetization\nusing low power1,2. Recently, there has been revitalized\ninterest in the ferrimagnetic insulator yttrium iron gar-\nnet (YIG, Y 3Fe5O12) motivated by the discovery of spin-\ntronic effects by combining this material and heavy met-\nals such as Pt3–7. Its extremely small magnetic damping\nenables low power data transmission and processing on\nthe basis of magnons, the elementary quanta of magnetic\nexcitations.3–5,8–12. In addition the low damping YIG\nalso enables nonlinear phenomena where the superposi-\ntion principle breaks down11. Previous work reported on\nthe formation of spin-wave caustics13, Bose Einstein con-\ndensationofmagnons14andnonlinearmode conversion15\nto name only a few. Recently, it has become possible to\ngrow nanometer-thick YIG films, which allow the prepa-\nration of micro- and nanostructured devices5,8,9,12,16.\nTherefore,the studyofnonlinearspin dynamicsin minia-\nturized YIG systems has only just begun.\nIndependent of the progress of the YIG film growth,\nthe development in employing spin-orbit interaction in\nheavy metals17,18and their alloys19in contact with a\nferromagnet (FM) has flourished. The SHE20,21can be\nused for the generation of strong current-driven torques\non the magnetization in the FM layer. The resultant\nspin current can drive spin-torque ferromagnetic reso-\nnance (ST-FMR) in bilayers consisting of ferromagnetic\nand nonmagnetic metals and be detected by a hom-\ndyne mixing of the microwavesignal with the anisotropic\nmagnetoresistance22. Recent theories propose that ST-\nFMR can be extended to insulating FM/normal metal\nbilayers. Here, the detection of magnetization precession\noccurs by spin pumping and a rectification of the spin\nHall magnetoresistance23,24. We showed recently that\nthis rectification process is indeed possible in YIG/Pt\nbilayers25. All previousanalysisof electric measurements\nassume uniform precession across the sample22,26. In or-\nder to validate this assumption it is highly desirable to\nimageaccurrent-driven spin dynamics spatially-resolvedand frequency-resolved. These investigations provide\ninteresting insights in the underlying physics, such as\nwhether bulk or edge modes are preferably excited by\nST-FMR or nonlinear spin dynamics may occur.\nIn this letter, we show experimentally the excitation of\nspin dynamics in microstructured magnetic insulators by\nthe SHE of an adjacent heavy metal and observe the for-\nmation of a nonlinear, self-localized spin-wave intensity\nin the center of the sample27–29. The magnetization dy-\nnamics in a nanometer-thick YIG layer is driven simulta-\nneouslybythe Oerstedfield andaspin torqueoriginating\nfrom a spin current generated by the SHE of an attached\nPt layer. The dynamics is detected in two complemen-\ntary ways: (1) Electrically, by a rectification mechanism\n(a) (b)\n(c) (d)\n-2.0-1.5-1.0-0.50.00.5DC voltage VDC (µV) \n12001000800600\nMagnetic field H (Oe) Data\n Total fit\n SMR\n Spin pumping1.5\n1.0\n0.5\n0.0DC voltage VDC (µV) \n12001000800600\nMagnetic field H (Oe)\nFIG. 1. (a) Schematic of the ST-FMR experimental setup (b)\nST-FMR mechanism in the YIG/Pt bilayer. The alternat-\ningrfcurrent drives an Oersted field hrfexerting a field-like\ntorqueτHon the magnetization M. At the same time a oscil-\nlatorytransverse spinaccumulationat theYIG/Ptinterfac e is\ngenerated by the SHE which results in a damping-like torque\nτSTT. (c) and (d) Typical dcvoltage spectra recorded at in-\nplane angles of φ= 30◦andφ= 240◦andP= +10 dBm.2\n-4-2024 VDC (µV)\n-1000 -500 0500 1000\n Magnetic field H (Oe)+15 dBm\n+12 dBm\n+10 dBmf = 4 GHz(a)\n(b)\n-3-2-10123SMR voltage VSMR (µV)\n250 200 150 10050 0\nIn-plane angle φ (°)-400-2000200400Spin-pumping\nvoltage VSP (nV)-3-2-1VDC (µV)subsidiary \n mode main\nmode\nFIG. 2. (a) Typical VDCspectra at a constant frequency\nf= 4 GHz for various applied microwave powers. The in-\nset shows the resonance peak at P= +15 dBm. Two modes\nare detected. (b) In-plane angular dependence of the SMR,\nVSMR, and of the spin-pumping contribution, VSP, to the dc\nvoltage. The solid lines represent fits ∝cosφsin 2φ.\nof the spin Hall magnetoresistance (SMR)30–32as well as\nby spin pumping3–5,33–35and (2) Optically, by spatially-\nresolved Brillouin light scattering (BLS) microscopy36.\nThe experimental findings are further validated by mi-\ncromagnetic simulations37.\nYIG(40 nm)/Pt bilayers were fabricated by in-situ\nmagnetron sputtering under high-purity argon atmo-\nsphere on single crystal gadolinium gallium garnet\n(GGG, Gd 3Ga5O12) substrates of 500 µm thickness with\n(111) orientation16. For the electrical measurements a\nPt thickness of 2 nm was used, while for the optical in-\nvestigations the thickness was 5 nm in order to minimize\nthe influence of additional heating effects by the laser.\nIn a subsequent fabrication process, stripes in the shape\nof 30×5µm2(electrical measurements) and 5 ×5µm2\n(optical measurements) were patterned by photolithog-\nraphy and ion milling5. A coplanar waveguide (CPW)\nmade of Ti/Au (3 nm/120 nm) was structured on top of\nthe bar allowing the signal line to serve as a lead for the\nYIG/Pt bar as illustrated in Fig. 1(a). In this ST-FMR\nconfiguration a bias-T is utilized to allow for simultane-\nous transmission of a microwave signal with dcvoltage\ndetection via lock-in technique across the Pt. For this\npurpose the amplitude of the rfcurrent is modulated at\n3 kHz. We use a BLS microscope with a spatial reso-\nlution of 250 nm, where the laser spot is focused onto\nthe sample and the frequency shift of the back reflected\nlight is analyzed by a multi-pass tandem Fabry P´ erot\ninterferometer36. The detected BLS intensity is propor-\ntional to the square of the dynamic magnetization, i.e.,\nthe spin-wave intensity.In order to excite a dynamic response by ST-FMR in\ntheYIGsystema rfsignalispassedthroughthePtlayer.\nThe magnetization dynamics is governed by a modified\nLandau-Lifshitz-Gilbert equation23,24:\ndM\ndt=−|γ|M×He���+αM×dM\ndt+|γ|/planckover2pi1\n2eMsdFJs,(1)\nwhereγis the gyromagnetic ratio, Heff=hrf+HD+H\nis the effective magnetic field including the microwave\nmagnetic field hrf, demagnetization fields HD, and the\nbias magnetic field H.αis the Gilbert damping param-\neter [the second term describes the damping torque τα,\nFig.1(b)] and Jsis a transverse spin current at the inter-\nface generated by the SHE from the alternating charge\ncurrent in the Pt layer23,24:\nJs=Re(g↑↓\neff)\neM×(M×µs)+Im(g↑↓\neff)\neM×µs.(2)\nHere,g↑↓\neffis the effective spin-mixing conductance and µs\nis the spin accumulation at the YIG/Pt interface. The\nfirstterminEq.( 2)describesananti-damping-liketorque\nτSTTand the second term is a field-like torque τH. As\nillustrated in Fig. 1(b) and described by Eq. ( 1) the mag-\nnetizationis drivenbythe independent torquetermscon-\ntaininghrfandJs.\nFirst, we describe the electrical characterization of the\nYIG/Pt bars by means of ST-FMR. Figure 1(c) and\n(d) illustrate typical dcvoltage spectra; exemplarily, we\nshow spectra recorded at in-plane angles of φ= 30◦and\nφ= 240◦, with applied rfpowerP= +10 dBm. A sig-\nnalisobservedwhen thesystemisdrivenresonantly. The\ndata is analyzed using the model proposed by Chiba et\nal. (supplementary information)23,24. According to the\nmodel, two signals contribute to the dcvoltage: (1) Spin\npumping which manifests in a symmetric contribution to\nthe Lorentzian lineshape. (2) Spin Hall magnetoresis-\ntance which is a superimposed symmetric and antisym-\nmetric Lorentzian curve [Fig. 1(c,d)].\nFIG. 3. Color-coded dispersion relation measured by BLS\nmicroscopy. The laser spot was focused onto the center of\nthe sample while the rffrequency as well as magnetic field\nwere varied. As for the electrical measurements two modes\nare detected by BLS. The inset shows corresponding field de-\npendence of the resonance measured by electrical means.3\nFIG. 4. Spatially-resolved BLS map of the 5 ×5µm2large\nYIG/Pt sample. The magnetic field H= 665 Oe is applied\natφ∼45◦. (a) - (d) Driving microwave frequency increases\nfrom 3.7 GHz to 3.85 GHz, microwave power P= +17 dBm.\nFigure2(a) illustrates dcvoltage spectra at a fixed\nmicrowave frequency f= 4 GHz for three different ap-\nplied powers. The offset is due to the longitudinal spin\nSeebeck effect6,7(see supplementary information) and\ndoes not affect the conclusions drawn from the resonance\nsignal7. The inset in Fig. 2(a) shows the resonance peak\natP= +15 dBm. Clearly, a less intense, secondary\nmode in addition to the main mode is detected. Accord-\ning to the Chiba model23,24thedcvoltage signal can be\ndeconvoluted into a spin-pumping and a SMR contribu-\ntion as also shown in Fig. 1(c) and (d). To analyze the\ndata employing the model we use a spin-mixing conduc-\ntance ofg↑↓\neff= 3.36×1014Ω−1m−2and a spin-Hall angle\nofγSHE= 0.0938. A fit to the angular-dependent data\nyields a phasedifference between Oersted field and the ac\ncurrent of δ= 64±5◦[see Fig. 1(c,d)]. Figure 2(b) shows\nthe angular dependences of the fitted spin-pumping and\nthe SMR signals. The model predicts the same angular\ndependent behavior ∝cosφsin2φfor spin pumping and\nSMR. As seen in Fig. 2(b), we find a good agreement be-\ntweentheory(solidlines)andexperimentforbothcurves.\nPlease note that we observe a small, non-vanishing volt-\nage at angles φ=n·90◦,n∈N, where the model sug-\ngests zero voltage23–25. In this angular range the model\nbreaks down and the experimental data cannot be fitted\n(see supplementary information).\nIn the following we compare the electrical measure-\nmentswiththeresultsobtainedbyBLSimaging. Theop-\ntical measurements wereperformed on YIG(40 nm)/Pt(5\nnm) bars having a lateral size of 5 ×5µm2. The ex-\nternal magnetic field is applied at an angle of φ∼45◦\nwhere the dcvoltage detection is maximized [Fig. 2(b)].\nFigure3shows the dispersion relation measured by BLS\nin a false color-coded image where red indicates a highspin-wave intensity and the blue area shows the ab-\nsence of spin waves. The measured dispersion is in\nagreement with the electrical measurements as shown in\nthe inset: As the field increases the resonance shifts to\nhigher frequencies as is expected from the Kittel equa-\ntion,f=|γ|\n2π/radicalbig\nH(H+4πMeff), where Meffis the effec-\ntive magnetization.\nAs is apparent from Fig. 2(a) and Fig. 3magnetization\ndynamics can be excited in a certain bandwidth around\nthe resonance which is determined by the specific de-\nvice characteristics. Furthermore, both figures (electrical\nand optical detection) suggest that there is an additional\nmode below the main mode. At first, one might identify\nthis mode as an edge mode39,40. However, this is not the\ncase as it will be discussed below.\nIn order to spatially map the spin-wave intensity, the\napplied magnetic field is kept fixed at H= 665 Oe. Fig-\nure4illustrates the experimental observations in false\ncolor-coded images. At an excitation frequency below\nthe resonance frequency, e.g., f= 3.7 GHz no magneti-\nzation dynamics is detected [Fig. 4(a)]. As the frequency\nincreases the system is driven resonantly and a strong\nspin-wave intensity is observed from f= 3.725 GHz\ntof= 3.8 GHz, Fig. 4(b,c). Increasing the frequency\neven further results in a diminished signal, Fig. 4(d) for\nf= 3.85 GHz. At even larger frequencies no magnetiza-\ntion dynamics is detected as it is also apparent from the\ndispersion illustrated in Fig. 3. In conventional electrical\nST-FMR measurements, a uniform spin-wave intensity\ndistribution across the lateral sample dimensions is as-\nsumed. However, as our experimental results show, this\nassumption is not fulfilled: A strong spin-wave signal is\nlocalized in the center of the YIG/Pt bar. It is desir-\nable to experimentally investigate at what minimum ex-\ncitation power the formation of the localization occurs.\nHowever, in the investigated range of powers we always\nobserve a localization in the center of the sample (see\nsupplementary information). For rfpowers of less than\n+11 dBm the signal is below our noise-floor.\nIn spite of this experimental limitation, we also car-\nried out micromagnetic simulations in order to gain fur-\nther insight into the underlying magnetization dynamics.\nThe simulations confirmed qualitatively the experimen-\ntal observations as is depicted in Fig. 5: Two modes can\nbe identified in the simulations, Fig. 5(a). In the low\npower regime, which is not accessible experimentally, we\nfind that the spatial magnetization distribution of the\nmain mode is almost uniform and the less intense sub-\nsidiary mode is localized at the edges ( hrf= 0.25 Oe,\nnot shown). With increasing rfpower, the spatial dis-\ntributions of both modes transform and at a threshold\nofhrf≈1 Oe a localization of both modes in the center\nof the sample is observed. Figure 5(b,c) show the corre-\nsponding spatial dynamic magnetization distributions at\nhrf= 5 Oe and agreewell with the experimental findings,\nFig.4.\nThis spatial profile can be understood asthe formation\nof a nonlinear, self-localized ‘ bullet’-like spin-wave inten-4\n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 Intensity (a.u.) \n3.8 3.7 3.6 3.5 3.4 3.3 \nFrequency f (GHz)(a) \nmain mode subsidiary mode \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 Normalized integrated \n BLS intensity (a.u.) 50 40 30 20 \nrf power P (mW) 1.0 \n0.8 \n0.6 \n0.4 \n0.2 Normalized integrated \nintensity simulation (a.u.) \n20 15 10 5 0rf magnetic field h rf (Oe) (d) \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 (b) \n (c) subsidiary mode main mode \n1 µm \nFIG. 5. Micromagnetic simulations: (a) The spectrum reveal s\ntwo modes. Spatially-resolved magnetization distributio n of\nthe main mode, (b), and the less intense, subsidiary mode,\n(c). (d) The normalized integrated BLS intensity saturates\nat high excitation powers P, which is validated by micromag-\nnetic simulations at large driving rfmagnetic fields hrf.\nsity caused by nonlinear cross coupling between eigen-\nmodes in the system15. This process is mainly deter-\nmined by nonlinear spin-wave damping which transfers\nenergy from the initially excited ferromagnetic resonance\ninto other spin-wave modes rather than into the crys-\ntalline lattice15. To check this assumption, we plot-\nted in Fig. 5(d) the normalized integrated BLS intensity\nas well as the integrated spatial magnetization distribu-\ntion as a function of the applied microwave power and\ntherfmagnetic field, respectively. Both integrated sig-\nnals demonstrate a nonlinear behavior and saturate at\nhigh powers/microwave magnetic fields. This observa-\ntion is a direct manifestation of nonlinear damping: en-\nergy is absorbed by the ferromagnetic resonance and re-\ndistributedtosecondaryspin-wavemodesmoreandmore\neffectively15.\nUntil now, ST-FMR experiments assumed a uni-\nform magnetization precession22–24,26. However, as our\nspatially-resolved BLS results demonstrate and con-\nfirmed by micromagnetic simulations, the driven lateral\nspin-wave intensity distribution in insulating FMs devi-ates from this simple model at higher excitation powers\nwhich are common in ST-FMR measurements. The for-\nmation of a localized spin-wave mode was not considered\nin previous ST-FMR experiments neither in metals nor\nin insulators. Our findings have direct consequences on\nthe analysis and interpretation of ST-FMR experiments.\nThe precession amplitude is not uniform across the sam-\nple implying that the effective spin-mixing conductance\ng↑↓\neffis actually an average over the sample cross sec-\ntion. In areas where the precession amplitude is large,\ng↑↓\neffis underestimated, whereas it is over estimated in\nlow-intensityareas. This also complicates the determina-\ntion of the spin-Hall angle from ST-FMR measurements.\nMicromagnetic simulations show phase inhomogeneity,\nspecifically aroundthe perimeter ofthe mode. The phase\ninhomogeneity tends to equally lag and lead the main\nuniform phase of the center mode; effectively the phase\ninhomogeneity then leads to no significant change to the\nlineshape. However,assuming the phaseat the perimeter\nto be uniformly leading the bulk phase results in a cor-\nrectionto the lineshape that is still negligiblebecause the\neffective areaand amplitude where the phase is deviating\nis significantly smaller than the bulk area. Nevertheless,\nin general the issue of inhomogeneous phase distribution\nmay complicate the analysis of electrical ST-FMR spec-\ntra, especially in smaller samples.\nInconclusion,wedemonstratedthattheconceptofST-\nFMR can be extended to magnetic insulators where the\nformation of a nonlinear, self-localized spin-wave inten-\nsity driven by an accurrent was observed. We adopted\nan electrically-driven ST-FMR excitation and detection\nscheme in magnetic insulator (YIG)/heavy normal metal\n(Pt)bilayersthatwasoriginallydevelopedforall-metallic\nsystems. A dcvoltage in YIG/Pt bilayers was observed\nunder resonance condition by a SMR-mediated spin-\ntorque diode effect in agreement with theoretical pre-\ndictions. Spatially-resolved BLS microscopy revealed a\nstrong ‘bullet’-like spin-wave localization in the center\nof the sample due to nonlinear cross coupling of eigen-\nmodes in the system. Since the observed electrical signal\nis sufficiently large and the signal-to-noise ratio is rea-\nsonably good, down-scaling of sample dimensions to the\nnanometer-scale is feasible.\nACKNOWLEDGMENTS\nWe thank Stephen Wu for assistance with ion milling.\nThe work at Argonne was supported by the U.S. De-\npartment of Energy, Office of Science, Materials Science\nand Engineering Division. Lithography was carried out\nat the Center for Nanoscale Materials, an Office of Sci-\nence user facility, which is supported by DOE, Office of\nScience, Basic Energy Science under Contract No. DE-\nAC02-06CH11357. The work at Colorado State Univer-\nsity was supported by the U. S. Army Research Office\n(W911NF-14-1-0501),theU.S.NationalScienceFounda-\ntion (ECCS-1231598),C-SPIN(one ofthe SRCSTARnet5\nCenters sponsored by MARCO and DARPA), and the U. S. Departme nt of Energy (DE-SC0012670).\n∗jungfleisch@anl.gov\n1D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2008).\n2A. 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Alff1,∗\n1Technische Universit¨ at Darmstadt, Petersenstr. 23, 6428 7 Darmstadt, Germany\n2Royal Institute of Technology(KTH), Brinellv¨ agen 23, 100 44 Stockholm, Sweden\n3ACRHEM, University of Hyderabad, Hyderabad 500 046, India\n4European Synchrotron Radiation Facility (ESRF),\n6 Rue Jules Horowitz, BP 220, 38043 Grenoble Cedex 9, France\n(Dated: received 29 April 2009)\nWe haveinvestigated spin andorbital magnetic momentsof th eRe5dionin thedoubleperovskites\nA2FeReO 6(A= Ba, Sr, Ca) by X-ray magnetic circular dichroism (XMCD) at t he ReL2,3edges.\nIn these ferrimagnetic compounds an unusually large negati ve spin and positive orbital magnetic\nmoment at the Re atoms was detected. The presence of a finite sp in magnetic moment in a ’non-\nmagnetic’ double perovskite as observed in the double perov skite Sr 2ScReO 6proves that Re has\nalso a small, but finite intrinsic magnetic moment. We further show for the examples of Ba and Ca\nthat the usually neglected alkaline earth ions undoubtedly also contribute to the magnetism in the\nferrimagnetic double perovskites.\nPACS numbers: 75.25.+z, 75.30.-m, 75.50.-y\nI. INTRODUCTION\nOrdered double perovskites of the composition\nA2MNO6(withAan alkaline earth, Ma magnetic tran-\nsition metal ion, and Na non-magnetic ion) have come\nagain into the focus of research because of their inter-\nesting magnetic properties. First, in Sr 2FeMoO 6a large\nroom-temperature magnetoresistance was observed [1].\nSecond, within the group of ferrimagnetic double per-\novskites materials with higher Curie-temperatures, TC,\nthan in the simple perovskites (e.g. doped manganites)\ncan be obtained. At the moment the highest TCvalues\nhave been reported for Sr 2CrReO 6(TC≈635K) [2, 3, 4]\nand Sr 2CrOsO 6(TC≈725K) [5, 6, 7]. Third, the mech-\nanism leading to magnetic coupling is believed to be as-\nsociated with a strong tendency to a half-metallic nature\nofthe chargecarriersatthe Fermilevel[8, 9, 10]. Clearly,\nthese materials are interesting candidates for spintronic\napplications [11], in particular when having in mind fully\nepitaxial structures based on perovskite materials.\nRecently, Majewski et al.and Sikora et al.have\nproposed a simple scaling law between the Curie-\ntemperature and the induced magnetic moment at the\nnon-magnetic site in the double perovskite structure\n[4, 12, 13]. Philipp et al.have discussed that a high\nCurie-temperature is associated with a tolerance factor\nclose to one for the corresponding crystal [14]. The only\nexception for this rule is found in the series A2FeReO 6\n(A= Ba, Sr, Ca). In this particular FeRe-system it\nis the strongly monoclinically distorted Ca-based com-\npound having an anomally high TC, namely about 540K\n[15, 16, 17] (comparing to about 400K for Sr 2FeReO 6\n[16] and 325K for Ba 2FeReO 6[18, 19]). The dimen-\n∗Electronic address: alff@oxide.tu-darmstadt.desionless tolerance factor, f, inA2FeReO 6whose devia-\ntion from unity implies structural distortion varies from\naboutf= 1.057 forA= Ba over f= 0.997 forA=\nSr tof= 0.943 for A= Ca [14]. In general, the\nBa-based ferrimagnetic double perovskites are close to\na structural transition into a hexagonal lattice where\nferro(i)magnetism is not allowed for symmetry reasons;\nthe Sr-based compounds are always close to a perfect cu-\nbic structure with maximal TC, and the Ca-based double\nperovskites are orthorhombically or monoclinically dis-\ntorted, with still a large but - due to the reduced ex-\nchange - clearly reduced ferrimagnetic transition temper-\nature. The exceptional large TCof Ca2FeReO 6is accom-\npanied by an insulating state at low temperatures, in\ncontrast to Sr 2FeReO 6or even the similarly monoclini-\ncally distorted Ca 2FeMoO 6which both are metallic [20].\nThe metal-insulator transition in Ca 2FeReO 6has been\nreported to occur between 100 and 150K [15, 16, 20].\nThis behavior has been attributed to strongly enhanced\nelectron-electron correlations on the Re site due to a re-\nduced transfer integral between Fe and Re corresponding\nto an extremely large effective Coulomb repulsion, Ueff,\nofabout 4eV on both ions [20]. This, however, is in some\ncontradiction to the observed high Curie-temperature,\nwhich is believed to be a consequence of a kinetic energy\ngain due to the hybridization of the Fe 3 dand Re 5 d t2g-\norbitals. The prediction that a decreased band-filling is\nfavorable for TC[21], which could be used to conciliate a\nhighTCwith a reduced Re-Re overlap, has turned out to\nbe not valid: an increased band-filling actually leads to\na strong TCenhancement for both the FeMo-system [22]\nand the CrW-system [23]. Within the kinetically driven\nexchangemodel[8, 9, 10], theincreaseof TCismorenatu-\nrallyexplained asaconsequenceofincreasedband-filling.\nNote, that the casesof Ca 2FeReO 6and Sr 2CrOsO 6, both\nbeing insulating and having a high TCat the same time,\narecompletelydifferent: InthecaseofSr 2CrOsO 6having2\nTable I: Summary of sample properties from x-ray diffrac-\ntion at 300K (calculated by Rietfeld-refinement) and SQUID\nmagnetometry.\nmaterial TCsymm. lattice antisites\n(K) (˚A) (%)\nBa2FeReO 6317Fm3ma= 8.0571(2) 0.9\nSr2FeReO 6418Fm3ma= 7.8752(4) 2.6\na= 5.3992(2)\nCa2FeReO 6556P21/nb= 5.5269(2) 3.6\nc= 7.6826(3)\na= 5.6760(2)\nSr2ScReO 6-P21/nb= 5.6534(2) 7\nc= 7.9862(3)\nonlyatinyrhombohedraldistortion,theOs5 dt2gbandis\ncompletelyfilled, whileforCa 2FeReO 6itisthestructural\ndistortion that drives the metal-insulator transition. Re-\ncently, it was suggested that in double perovskites with\nheavy ions as Re a large orbital contribution to the mag-\nnetic moment leads to an enhanced total magnetization\nabovetheintegervaluethatisexpectedforahalf-metallic\nmaterial [25]. This elsewhere predicted and calculated\n[26] strong influence of spin-orbit coupling leads to a\nquasi-half metallicity which still is from the view point of\napplications in spintronics very high (above 90%). An-\nother point of interest is the possibility of an intrinsic\nenhancement of the Re spin magnetic moment due to\nthe peculiar Re5+state in the ferrimagnetic double per-\novskites. In this study, we present the XMCD analysis\nof the system A2FeReO 6(A= Ba, Sr, Ca), compare the\nexperimental data to theoretical predictions calculated\nwithin the full-potential linear muffin-tin orbital method\n(FP-LMTO) [27] with included spin-orbit coupling, and\ncomplete the so far suggested scaling law [4] by using the\nidentical method to extract separately spin and orbital\nmagnetic moments. Furthermore, we search for a con-\ntribution of the alkaline earth element to the magnetic\nbehavior, and also look for an intrinsic Re moment in\na suited double perovskite compound with Mbeing a\nnon-magnetic ion: Sr 2ScReO 6.\nII. EXPERIMENTAL\nA summary of the sample properties is given in Ta-\nble I. All values where a comparison can be made to\nliterature values are in good agreement with these data\n[2, 15]. Note that the small amount of antisite disorder\ndoes not affect our results. The XMCD measurementson\nthe ReL2,3edges were performed at the European Syn-\nchrotron Radiation Facility (ESRF) at beam line ID12\n[29]. The spectra were recorded within the total fluores-\ncenceyield detection mode. The XMCDspectrawereob-\ntained as direct difference between consecutive XANES/s49/s48/s53/s53/s48 /s49/s48/s54/s48/s48 /s49/s49/s57/s53/s48 /s49/s50/s48/s48/s48/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48\n/s82/s101/s32 /s76\n/s50/s32/s101/s100/s103/s101/s82/s101/s32 /s76\n/s51/s32/s101/s100/s103/s101\n/s32/s67/s97\n/s50/s70/s101/s82/s101/s79\n/s54/s54/s32/s84/s101/s115/s108/s97/s44/s32/s49/s48/s32/s75/s101/s108/s118/s105/s110/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s66/s97\n/s50/s70/s101/s82/s101/s79\n/s54\n/s32/s83/s114\n/s50/s70/s101/s82/s101/s79\n/s54\nFigure 1: XMCD spectra for A2FeReO 6(A=Ba, Sr, Ca).\nscans (X-ray Absorption Near Edge Spectrum) recorded\nwith opposite helicities of the incoming x-ray beam. To\nensure that the XMCD spectra are free from any exper-\nimental artefacts the data were collected for both direc-\ntions of the applied magnetic field of 6T (parallel and\nantiparallel to the x-ray beam). The degree of circular\npolarization of the monochromatic x-ray beam was 98%.\nThe measurements were performed at about 10K for all\nsamples ( T≪TC), if not indicated otherwise. Since the\nsamples measured in backscattering geometry were very\nthick, the spectra were first normalized to the edge jump\nof unity and then corrected from self-absorption effects.\nThe edge jump intensity ratio L3/L2was then normal-\nized to 2.19/1 [30]. This is different from the statistical\n2:1branching ratiodue to the difference in the radial ma-\ntrix elements of the 2 p1/2to 5d(L2) and 2p3/2to 5d(L3)\ntransitions. The XMCD measurement as a function of\napplied field suggests that our samples are closer to sat-\nuration at 6T as is concluded by de Teresa at al.[25]\nfrom high-field SQUID measurements. This issue has to\nbe clarified in future by high-field XMCD measurements.\nIII. RESULTS AND DISCUSSION\nIn this paper, the XANES spectra themselves are\nnot further discussed. As shown in Fig. 1, for FeRe-\ncompounds at both absorption edges we find a rather\nintense XMCD signal. This is a clear evidence for the\nexistence of a magnetic moment at the Re 5 dshell. For\nall three compounds, the XMCD spectra at the L2edge\nare largest (as expected for m= 1 orbitals) and similar\nin shape. In Ca 2FeReO 6the size of the XMCD signal\nis by a factor of 2 smaller compared to the two other\nFeRe compounds. At the L3edge, the Ca-based double\nperovskite again stands out by a pronounced peak with\nnegative XMCD signal which is absent for Sr 2FeReO 6\nand Ba 2FeReO 6. The data at the L3edge look slightly\ndifferent in amplitude as compared to previously pub-\nlished data [13]. However, the data are consistent in that\nthe integrated XMCD intensity at the L3edge is nega-\ntive only in the case of Ca 2FeReO 6. In this sense, all\ndata support the unusual behavior of Ca 2FeReO 6, which\ncannot only be attributed to the different ionic size of the3\nTable II: Measured (exp., normalized to 5K) and calculated\n(th., calculated within the generalized gradient approxim a-\ntion including spin-orbit coupling (GGA+SO)) magnetic mo-\nments at the Re site for different double perovskites at about\n10K. For a detailed discussion of the applied band-structur e\ncalculation see e.g. [26, 28]. Calculation in [31] is GGA with\nspin-orbit coupling. The number of d-holes was taken from\nthe band-structure calculation. In our case this number was\naround 5 .3. The error of the measured values is estimated as\n2.5%.\nmaterial mS(µB/f.u.)mL(µB/f.u.)|mL/mS|\nexp. Ba 2FeReO 6 −0.56 0 .15 0 .27\nSr2FeReO 6 −0.74 0 .21 0 .28\nCa2FeReO 6 −0.47 0 .16 0 .34\nSr2CrReO 6[4] −0.68 0 .25 0 .37\nSr2ScReO 6(80K) 0 .013 −0.002 0 .15\nth. Ba 2FeReO 6 -0.65 0.19 0.29\nSr2FeReO 6 -0.68 0.15 0.22\nSr2FeReO 6[31] -0.85 0.23 0.27\nSr2CrReO 6[26] -0.85 0.18 0.21\n/s45/s48/s46/s48/s48/s52/s45/s48/s46/s48/s48/s50/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50/s48/s46/s48/s48/s52\n/s49/s48/s53/s50/s53 /s49/s48/s53/s53/s48 /s49/s48/s53/s55/s53 /s49/s49/s57/s53/s48 /s49/s50/s48/s48/s48/s48/s46/s48/s48/s46/s57/s49/s46/s56\n/s54/s32/s84/s101/s115/s108/s97/s44/s32/s56/s48/s32/s75/s101/s108/s118/s105/s110\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41/s88/s65/s78/s69/s83/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s83/s114\n/s50/s83/s99/s82/s101/s79\n/s54/s82/s101 /s32/s76\n/s50\n/s32/s82/s101 /s32/s76\n/s51\nFigure 2: XANES and derived XMCD spectra at the Re L2\nandL3edges of Sr 2ScReO 6.\nAsite ions.\nIn Fig. 2 we show XANES and XMCD spectra for the\ncompound Sr 2ScReO 6. This compound is important be-\ncause the absence of any free electrons at Sc3+which\nhas a 3d0configuration will lead to a complete break-\ndown of the inducedmagnetic moment at the Re site.\nThis compound therefore allows the measurement of the\nintrinsic magnetic moment of Re5+(also in contrast to\nRe6+compounds as Sr 2MgReO 6). Previously, Kato et\nal.have calculated from a Curie-Weiss fit to the suscep-\ntibility aneffective magneticmomentofRe in Sr 2ScReO 6\nof about 1.1 µB/f.u., as expected within the ionic picture\n[32]. In contrast, our data show the existence of a much\nsmaller, but finite intrinsic moment at the Re site, in-\ndicating an increased tendency to magnetic ordering of\nRe5+. Since this moment is present above the antiferro-\nmagnetic transition temperature, it is not related to spin\nglass behavior. The spin magnetic moment is about 50\ntimes smaller than corresponding induced moments on/s45/s48/s46/s48/s48/s50/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50\n/s53/s50/s53/s48 /s53/s50/s55/s53 /s53/s54/s50/s53 /s53/s54/s53/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s54/s32/s84/s101/s115/s108/s97/s44/s32/s49/s48/s32/s75/s101/s108/s118/s105/s110\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41/s88/s65/s78/s69/s83/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s66/s97\n/s50/s70/s101/s82/s101/s79\n/s54/s97/s41\n/s66/s97/s32 /s76\n/s50\n/s66/s97/s32 /s76\n/s51\n/s52/s48/s51/s53 /s52/s48/s52/s48 /s52/s48/s52/s53 /s52/s48/s53/s48 /s52/s48/s53/s53 /s52/s48/s54/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s45/s48/s46/s48/s48/s56/s45/s48/s46/s48/s48/s54/s45/s48/s46/s48/s48/s52/s45/s48/s46/s48/s48/s50/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50\n/s54/s32/s84/s101/s115/s108/s97/s44/s32/s49/s48/s32/s75/s101/s108/s118/s105/s110/s67/s97\n/s50/s70/s101/s82/s101/s79\n/s54\n/s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41/s88/s65/s78/s69/s83/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s32/s40/s97/s46/s117/s46/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s67/s97/s32 /s75/s98/s41\n/s32\nFigure 3: XANES and derived XMCD spectra at the a) Ba\nL2andL3edges of Ba 2FeReO 6and b) at the Ca Kedge of\nCa2FeReO 6.\nRe5+, and the orbital magnetic moments even by a fac-\ntor of 100. However, due to the high sensitivity of the\nset-up at ESRF, one can unambiguously prove the exis-\ntence of this moment. In contrast to the opposite sign of\ntheinducedmagneticmomentwithrespecttotheapplied\nfield, the spin magnetic moment at the Re in Sr 2ScReO 6\nis aligned with the field. This is expected because the\nkinetic exchange via fully polarized spin down is not at\nwork. This intrinsic moment of Re5+therefore has to be\nconsidered as an indicator of the tendency to unusually\nhigh magnetization of Re based double perovskites.\nAs a last point, we address magnetism in the earth al-\nkaline ions itself, which usually are completely neglected\nin the magnetic scenario. The XANES and XMCD spec-\ntra of the Ba L2andL3edges of Ba 2FeReO 6and of the\nCaK-edge of Ca 2FeReO 6are shown in Fig. 3. The 5 d\nspin magnetic moment (calculated with 9 as the number\nofd-holes corresponding to the band-structure calcula-\ntion) of Ba is µS=−0.0065 and the 5 dorbital magnetic\nmoment µL=−0.0013 (both in µB/f.u.),|µL/µS| ≈0.2.\nThe theoretical predictions calculated as described else-\nwhere [26, 28] are µS=−0.0084 and µL=−0.0014\nwhich is in fair agreement with our experimental data.\nFor Ca 2FeReO 6we can only qualitatively say that a fi-\nnite magnetic moment is observed, because the K-edge\nprobes only the 4 porbital magnetism. Since the Ledges\nare experimentally not accessible, a quantative analysis\ncannotbe done. Theobservationofa magneticallypolar-\nized density of states gives clear evidence for a magnetic\ninteraction of the earth alkaline ions with the other ions.\nThe magnetic contribution of Ba in this case is a fac-\ntor of 2 smaller than the contribution of the intrinsic Re\nmoment. Naturally, one expects that the magnetic con-4\ntribution increases with ionic size due to the increased\nexchange with the neighboring ions. The clear orbital\ncontribution in Ba 2FeReO 6is not unexpected due to the\nheavy ionic mass. Our data provide a test for a detailed\ntheoretical study of the magnetism in the double per-\novskites, and underlines the importance of taking spin-\norbit coupling into account. Note, that for example in\nCrO2, where the importance of oxygen in the magnetic\nmechanism is undoubted, comparable values of spin and\norbital moments of the oxygen ion have been measured\n[33] as compared to our results on Ba in Ba 2FeReO 6.\nIn Table II we summarize our results for the spin and\norbital magnetic moments at the Re site as derived from\nthe XMCD measurements by applying the standard sum\nrules [34, 35] and compare them to theoretical values.\nAlso, the ratio |mL/mS|is calculated, since this quan-\ntity is not affected by possible uncertainties in the calcu-\nlated number of holes. In general, the calculated dataare\nin surprisingly good agreement with the measured data.\nOne of the main reasons certainly is, that spin-orbit cou-\npling is taken into account from the beginning. Note,\nthat in the hard x-ray range the sum rules apply with\nhigh validity due to the large spin-orbit splitting of the\ncore level.\nLetusfinallydiscussagainourdataforthe threeFeRe-\nbased compounds. Our data are in good qualitative and\nquantitative agreement with literature data with one ex-\nception: Ca 2FeReO 6. While Sikora et al.[13] find, that\nthe spin magnetic moment of Re in Ca 2FeReO 6scales\nwith the high TC, in our case it has the lowest spin mag-\nnetic moment, letting Ca 2FeReO 6stand out from the\nscaling law [4, 12, 13] which so far holds in all other\ncases. This behavior is certainly more natural, since oneexpects that a reduced exchange will also lead to a re-\nduced spin magnetic moment on the Re site. Note that\nthe ratio of orbital and spin magnetic moments are con-\nsistent with the previous data. As suggested previously\nby Kato et al.[16], a Re t2gorbital ordered state or its\nglass-state analog associated with the monoclinic lattice\ndistortion occurs, pointing out the importance of corre-\nlation effects in this compound. Recently, Sikora et al.\n[36] proposed a scenario with a complex competition be-\ntween two phases with different electronic and crystallo-\ngraphic structure. Our data give further indication that\nCa2FeReO 6is exceptional among the double perovskites\ndue to the strong octahedral-site distortions.\nIV. SUMMARY\nIn summary, we have elucidated the Re magnetic mo-\nments in the FeRe-based series of double perovskites as a\nfunction of the earth alkaline ion, confirming the excep-\ntional position of Ca 2FeReO 6. We have measureda finite\nintrinsic magneticmomentattheRe5+siteinSr 2ScReO 6\nindicating the tendency to enhanced magnetic moments\nobserved in Re based double perovskites. Furthermore,\nfor the first time we were able to measure by XMCD the\nmagnetic moments directly at the alkaline earth site it-\nself. Our result shows that the usually neglected Ca and\nBa ions play a role in the magnetic scenario of the kinet-\nically driven exchange model, comparable in size to the\nrole of oxygen.\nThis work was supported by the ESRF (HE-\n2114/2115/2379).\n[1] Kobayashi K I, Kimura T, Sawada H, Terakura K and\nTokura Y 1998 Nature395677\n[2] Kato H, Okuda T, Okimoto Y, Tomioka Y, Takenoya\nY, Ohkubo A, Kawasaki M and Tokura Y 2002\nAppl. Phys. Lett. 81328\n[3] AsanoH, KozukaN, Tsuzuki A and Matsui M 2004 Appl.\nPhys. Lett. 85263\n[4] Majewski P, Gepr¨ ags S, Sanganas O, Opel M, Gross R,\nWilhelm F, Rogalev A and Alff L 2005 Appl. Phys. Lett.\n87202503\n[5] Krockenberger Y, Mogare K, Reehuis M, Tovar M,\nJansen M, Vaitheeswaran G, Kanchana V, Bultmark F,\nDelin A, Wilhelm F, Rogalev A, Winkler A and Alff L\n2007Phys. Rev. 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B 79220402(R)" }, { "title": "1810.01158v1.Magnetocrystalline_anisotropy_and_exchange_probed_by_high_field_anomalous_Hall_effect_in_fully_compensated_half_metallic_Mn2RuxGa_thin_films.pdf", "content": "arXiv:1810.01158v1 [cond-mat.mtrl-sci] 2 Oct 2018Magnetocrystalline anisotropy and exchange probed by high -field anomalous Hall\neffect in fully-compensated half-metallic Mn 2RuxGa thin films\nCiar´ an Fowley,1,∗Karsten Rode,2Yong-Chang Lau,2Naganivetha Thiyagarajah,2Davide\nBetto,2Kiril Borisov,2Gwena¨ el Atcheson,2Erik Kampert,3Zhaosheng Wang,3,†Ye Yuan,1\nShengqiang Zhou,1J¨ urgen Lindner,1Plamen Stamenov,2J.M.D. Coey,2and Alina Maria Deac1\n1Institute of Ion Beam Physics and Materials Research,\nHelmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstra ße 400, 01328 Dresden, Germany\n2AMBER and School of Physics, Trinity College Dublin, Dublin 2, Ireland\n3Hochfeld-Magnetlabor Dresden (HLD-EMFL), Helmholtz-Zent rum Dresden - Rossendorf,\nBautzner Landstraße 400, 01328 Dresden, Germany\n(Dated: October 3, 2018)\nMagnetotransport is investigated in thin films of the half-m etallic ferrimagnet Mn 2RuxGa in\npulsed magnetic fields of up to 58T. A non-vanishing Hall sign al is observed over a broad tem-\nperature range, spanning the compensation temperature (15 5K), where the net magnetic moment\nis strictly zero, the anomalous Hall conductivity is 6673 Ω−1m−1and the coercivity exceeds 9T.\nMolecular field modelling is used to determine the intra- and inter-sublattice exchange constants\nand from the spin-flop transition we infer the anisotropy of t he electrically active sublattice to be\n216kJm−3and predict the magnetic resonances frequencies. Exchange and anisotropy are compa-\nrable and hard-axis applied magnetic fields result in a tilti ng of the magnetic moments from their\ncollinear ground state. Our analysis is applicable to colli near ferrimagnetic half-metal systems.\nPhySH: Ferrimagnetism, Magnetotransport, Half-metals, Anomalous Hall effect, Magnetic anisotropy, Exchange\ninteraction\nThin films with ultra-high magnetic anisotropy fields\nexhibit magnetic resonances in the range of hundreds of\nGHz [1–3] which is promising for future telecommunica-\ntions applications. Spin-transfer driven nano-oscillators\n(STNOs) working on the principle of angular momentum\ntransferfromaspin-polarisedcurrenttoasmallmagnetic\nelement[4,5], haveachievedoutputpowersofseveral µW\nand frequency tuneabilities of ∼GHzmA−1[6, 7], useful\nfor wireless data transmission [8]. Output frequencies of\nSTNOs based on standard transition-metal based ferro-\nmagnets, such as CoFeB, or cubic Heulser alloys such as\nCo2Fe0.4Mn0.6Si are in the low GHz range [9–13].\nCertain Heusler alloys [14, 15] are a suitable choice\nfor achieving much higher output frequencies, aimed at\nenabling communication networks beyond 5G [16]. The\nMn3−xGa family contains two Mn sublattices which are\nantiferromagnetically coupled in a ferrimagnetic struc-\nture [14]. They have low net magnetization, Mnet, and\nhigh effective magnetic anisotropy, Keff, with anisotropy\nfields of µ0HK= 2Keff/Mnetexceeding 18T [17, 18],\nwhich results in resonance frequencies two orders of mag-\nnitude higher [1, 2] than Co-Fe-B.Furthermore, the mag-\nneticpropertiesoftheseferrimagneticalloyscanbetuned\neasily with composition [19–21]. Mn 3−xGa films have\nshown tuneable resonance frequencies between 200GHz\nto 360GHz by variation of the alloy stoichiometry and\nmagnetic anisotropy field [2].\nHere we focus on the fully-compensated half-metallic\nHeusler compound, Mn 2RuxGa (MRG) [20–25]. Filmsof MRG were first shown experimentally [20] and subse-\nquently confirmed by DFT calculations [25] to exhibit a\nspin gap at EF. The material crystallises in the cubic\nspace group, F¯43m. Mn on the 4 aand 4csites are anti-\nferromagnetically coupled, while those on the same sites\nare ferromagnetically coupled. The crystal structure is\nFIG.1: (a)CrystalstructureofMn 2RuxGa, themagneticmo-\nments of the Mn 4aand Mn 4care aligned antiparallel. (b) and\n(c), a two-sublattice macrospin model used to explain the ob -\nserved temperature and field dependences of electronic tran s-\nport in the presence and absence of an applied field µ0Hz,\nrespectively. Two key points of the model are: below (above)\nTcompthe moment of the Mn 4cis parallel (antiparallel) to\nMnet; and, in the absence of an applied field the sublattice\nmoments donot change their orientation uponcrossing Tcomp.2\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s32/s49/s48/s32/s75\n/s32/s49/s49/s48/s32/s75\n/s32/s49/s56/s48/s32/s75/s100\n/s120/s121/s47/s100 /s181\n/s48/s72\n/s122/s32/s40/s97/s46/s117/s46/s41\n/s181\n/s48/s72\n/s122/s32/s40/s84/s41/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s32/s49/s55/s53/s32/s75\n/s32/s49/s54/s53/s32/s75\n/s32/s49/s51/s53/s32/s75\n/s32/s49/s50/s53/s32/s75\n/s32/s49/s49/s48/s32/s75/s77/s82/s32/s40/s37/s41\n/s181\n/s48/s72\n/s122/s32/s40/s84/s41/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s45/s49/s48/s107/s48/s49/s48/s107/s50/s48/s107/s51/s48/s107/s52/s48/s107/s53/s48/s107/s54/s48/s107/s55/s48/s107/s120/s121/s32/s40/s79/s104/s109/s46/s109/s41/s45/s49\n/s181\n/s48/s72\n/s122/s32/s40/s84/s41/s32/s49/s55/s53/s32/s75\n/s32/s49/s54/s53/s32/s75\n/s32/s49/s51/s53/s32/s75\n/s32/s49/s50/s53/s32/s75\n/s32/s49/s49/s48/s32/s75\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s52/s54/s56/s49/s48/s181\n/s48/s72\n/s99/s32/s40/s84/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s52/s53/s53/s48\n/s181\n/s48/s72\n/s115/s102/s32/s40/s84/s41/s45/s52/s53 /s45/s51/s48 /s45/s49/s53 /s48 /s49/s53 /s51/s48 /s52/s53/s45/s50/s48/s107/s48/s50/s48/s107/s52/s48/s107/s54/s48/s107/s56/s48/s107/s49/s48/s48/s107 /s120/s121/s32/s40/s79/s104/s109/s46/s109/s41/s45/s49\n/s48/s72\n/s122/s32/s40/s84/s41/s32/s50/s50/s48/s32/s75\n/s32/s49/s56/s48/s32/s75\n/s32/s49/s53/s48/s32/s75\n/s32/s49/s49/s48/s32/s75\n/s32/s56/s48/s32/s75\n/s32/s53/s48/s32/s75\n/s32/s49/s48/s32/s75\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s49/s48/s107/s45/s56/s107/s45/s54/s107/s45/s52/s107/s45/s50/s107/s48/s50/s107/s52/s107/s54/s107/s56/s107/s49/s48/s107\n/s32/s47/s32 /s32/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s115/s119/s101/s101/s112/s115/s32/s40/s48/s32/s84/s41\n/s32/s43/s32/s32/s53/s56/s32/s84/s32\n/s32/s43/s32/s54/s46/s53/s32/s84/s32/s120/s121/s32/s40/s79/s104/s109/s46/s109/s41/s45/s49\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s40/s102/s41/s40/s100/s41/s40/s99/s41/s40/s98/s41/s40/s97/s41\n/s40/s101/s41\nFIG. 2: (a) AHC loops up to 6 .5T for Mn 2Ru0.61Ga around the compensation temperature (155K). Loops are off set vertically\nfor clarity. (b) Magnetoresistance loops recorded at the sa me time as the data in (a). Loops are offset vertically for clar ity. (c)\nAHC loops up to 58T, where the spin-flop transition is indicat ed by the grey arrows. The linear slope is due to the ordinary\nHall effect. Loops are offset vertically for clarity. (d) Deri vative of the selected data in (c) clearly highlighting the s pin-flop.\n(e)µ0Hc(black circles) and µ0Hsf(red squares) as a function of temperature. The divergence o f the coercivity is expected at\nTcompsince with Mnet= 0 and Keff∝negationslash= 0. (f) Temperature dependence of the remanent Hall conduct ivity when saturated at\n10K in negative (solid line) and positive (dashed line) appl ied field. The black open (closed) circles record the remanen t Hall\nresistivity after the application of 6 .5T (58T).\nshown in figure 1 (a). The Ga is on the 4 bsites and Ru\noccupies a fraction of the 4 dsites [20]. We will discuss\nMn on the 4 aand 4csites by referring to the Mn 4aand\nMn4csublattices. By changing the Ru concentration, the\nmagnetic properties of the Mn 4csublattice are altered,\nwhile those of the Mn 4asublattice remain relatively con-\nstant [21]. Thin films grown on MgO have an out-of-\nplane magnetic easy axis due to biaxial strain induced by\nthe substrate during growth [23]. Unlike the uncompen-\nsated tetragonal D022Mn3−xGa family of alloys, MRG\nhas a compensation temperature, Tcomp, where there isno net magnetization [20, 21]. Nonetheless, there is non-\nvanishing tunnel magnetoresistance [22], spin Hall angle\n[23] and magneto-optical Kerr effect [24], which all arise\nfrom the Mn 4csublattice. The occupied electronic states\noriginating from the Mn 4asublattice lie below the spin\ngap [25].\nThe electrical transport on MRG reported to date [22,\n23] can be explained using the model shown in figure\n1 (b) and (c) where the direction of spin polarisation\nis governed by the direction of the Mn 4csublattice and\nnot Mn 4aorMnet. Here we make use of the dominant3\ninfluence of a single sublattice on the electron transport\nto study the magnetism of a compensated half-metal at\ncompensation, and evaluate the exchange and anisotropy\nenergies.\nWe measure magnetotransport, especially the anoma-\nlous Hall effect in the temperature range 10K to 300K\nin magnetic fields up to 58T. The anomalous Hall con-\nductivity (AHC) of a metallic ferromagnetic film, σxy, is\nproportional to the out-of-plane component of magneti-\nzation,Mz, which is defined as Mcosθwhereθis the an-\ngle between the z-axis and the magnetization, M[26]. In\nferrimagnets, however, the AHC will depend on the band\nstructure at the Fermi level, EF, so when the material is\nhalf-metallic, one expects σxy∝MslcosθMsl, whereMsl\nis the magnetization of the sublattice that dominates the\ntransport.\nMn2RuxGa layers of varying composition, x= 0.55,\n0.61 and 0.70, were deposited on MgO substrates in a\nfully automated Shamrocksputtering system. The thick-\nness of the films, ≈27nm, was determined by X-ray re-\nflectivity. Hallcrossesofwidth100 µmandlength900 µm\nwere patterned using direct-laser-write lithography, Ar+\nion milling and lift-off. The Hall bars were contacted\nwith Cr 5nm / Au 125nm pads.\nA Lakeshore Hall system was used to measure the\nlongitudinal ( ρxx) and transverse ( ρxy) resistivities from\n10K to 300K in out-of-plane fields up to 6 .5T. The\nAHC,σxy=ρxy/ρ2\nxx[27, 28], is obtained from the raw\ndata. In-plane, µ0Hx, and out-of-plane, µ0Hz, pulsed\nmagnetic fields of up to 58T were applied at the Dres-\ndenHighMagneticFieldLaboratoryatselectedtempera-\nturesbetween10Kand220K. WefocusonMn 2Ru0.61Ga\nwithTcomp≈155K. All three compositions were found\nto have compensation temperatures between 100K and\n300K, and exhibit similar properties.\nAHC loops versus µ0HzaroundTcompare shown in\nfigure 2 (a). At all temperatures, MRG exhibits strong\nperpendicular magnetic anisotropy. The reversal of the\nsign ofσxybetween 135K and 165K indicates a rever-\nsal of the spin polarisation at EFwith respect to the\napplied field direction, as expected on crossing Tcomp.\nThe coercivity, µ0Hc, varies from 3T to 6T between\n110K and 175K. The longitudinal magnetoresistance,\nρxx(H)/ρxx(0) shown in figure 2 (b) is small ( <1%), as\nexpected fora half-metal[29]. Pulsedfield measurements\ninfigure2(c)showthat, closeto Tcomp,µ0Hcexceeds9T\nand that MRG exhibits a spin-flop transition at higher\nfields, indicated in the figure by the grey arrows. The\nderivative of selected curves of σxyversus applied field,\nfigure 2 (d), show clearly the spin-flop field especially at\nlower temperatures. We note that the longitudinal mag-\nnetoresistance up to 58T also does not exceed 1% (not\nshown). The divergence in coercivity (black circles in\nfigure 2 (e)) is expected at Tcompbecause the anisotropy\nfield in uniaxial magnets is µ0HK= 2Keff/Mnet, where\nKeffis the effective anisotropyenergy and Mnetis the netmagnetization. The anisotropy field is an upper limit on\ncoercivity. The temperature dependence of the spin-flop\nfield,µ0Hsf, is also plotted in figure 2 (e) (red squares).\nThe solid (dashed) line in figure 2 (f), traces the\ntemperature dependence of σxywhen the sample is ini-\ntially saturated in a field of −6.5T (6.5T) at 10K\nand allowed to warm up in zero-applied magnetic field.\nThe spontaneous Hall conductivity, σxy, decreases from\n7859Ω−1m−1to5290Ω−1m−1anddoesnotchangesign\nfor either of the zero-field temperature scans. The rema-\nnent value of σxyafter the application of 6 .5T (58T) is\nplotted with open (solid) symbols. The combined data\nestablish that, in MRG films, neither the AHE nor the\nAHCareproportionalto Mnet. Theydepend onthemag-\nnetization of the sublattice that gives rise to σxy. While\nsimilar behaviour is well documented for the anomalous\nHall effect in rare-earth – transition-metal (RE-TM) fer-\nrimagnets, where both RE and TM elements contribute\nto the transport [30–32], in MRG both magnetic sub-\nlattices are composed of Mn which has been confirmed\nto have the same electronic configuration, 3 d5[21]. If\nboth sublattices contributed equally to the effect, the\nsum should fall to zero at Tcomp.\nWe refer to the model presented in figure 1 (b) and\n(c) to explain the behaviour shown in figure 2 (f). Fig-\nure 1 (b) shows the Mn 4aand Mn 4csublattice moments\nand the net magnetic moment in the case of an applied\nfield,µ0Hz, along the easy-axis of MRG. Below Tcomp,\nthe Mn 4cmoment (green arrow) outweighs that of Mn 4a\n(blue arrow), and Mnet(orange arrow) is parallel to the\nMn4csublattice. At Tcomp,Mnetis zero but the direc-\ntions of the sublattice moments have not changed with\nrespect to µ0Hz. AboveTcomp,µ0Hzcauses a reversal of\nMnet(provided it exceeds µ0Hc). Here, the Mn 4asublat-\ntice has a larger moment than Mn 4candMnetwill be in\nthe samedirection asthe Mn 4amoment. Due to the anti-\nferromagnetic alignment of both sublattices the moment\non Mn 4cis parallel (antiparallel) to µ0Hzbelow (above)\nTcomp.\nIn the absence of an applied field (figure 1 (c)), the\ndirection of Mnetwill reverse crossing Tcompdue to the\ndifferent temperature dependences of the sublattice mo-\nments. However, the sublattice moments only change in\nmagnitude, and not direction. The uniaxial anisotropy\nprovided by the slight substrate-induced distortion of the\ncubic cell [20] provides directional stability along the z-\naxis. Therefore, crossing Tcompin the absence of applied\nfield, we expect no change in sign of σxy, nor should\nit vanish. The Mn 4csublattice dominates the electron\ntransport and determines spin direction of the available\nstates at EF, while the Mn 4astates form the spin-gap.\nThe results of a molecular field model [33] based on\ntwo sublattices are presented in figure 3. The molecular\nfield,Hi, experienced by each sublattice is given by:\nHi\n4a=n4a−4aM4a+n4a−4cM4c+H(1)4\nHi\n4c=n4a−4cM4a+n4c−4cM4c+H (2)\nwheren4a−4aandn4c−4care the intra-layer exchange\nconstants and n4a−4cis the inter-layer exchange con-\nstant.M4aandM4care the magnetizations of the 4 a\nand 4csublattices. His the externally applied mag-\nnetic field. The moments within the Mn 4aand Mn 4csub-\nlattices are ferromagnetically coupled and hence n4a−4a\nandn4c−4care both positive. The two sublattices couple\nantiferromagnetically and therefore n4a−4cis negative.\nThe equations are solved numerically for both tempera-\nture and applied field dependences to obtain the projec-\ntion of both sublattice magnetizations along the z-axis,\nMz−α=Mαcosθα, whereα= 4a,4c. In the absence of\nan applied field, θ= 0, therefore Mz−αreduces simply\ntoMα.\nThe model parameters are given in table I. Based on\npreviousXMCDmeasurements[21]aswellasDFTcalcu-\nlations [25] we take values of547kAm−1and 585kAm−1\nfor the magnetizations on the 4 aand 4csublattice, re-\nspectively. The values of n4a−4a,n4c−4candn4a−4c\nare fit to reproduce Tcompand the Curie temperature,\nTC. The temperature dependences of Mz−4a(blue line),\nMz−4c(green line) and Mnet(orange line) with n4a−4a=\n1150,n4c−4c= 400 and n4a−4c=−485 are shown in fig-\nure 3 (a). In orderto numericallyobtain the temperature\ndependence in zero applied field a strong field of 60T is\nusedtosetthedirectionof Mnetandthenreducedtozero,\nso the sublattice moments reverse at Tcomp= 155K as in\nthe experiment. TCis625K. Mnetvariesfrom38kAm−1\nat 10K to a maximum of 97kAm−1at 512K, close to\nTC.\nFigure 3 (b) shows the measured AHC (circles), along\nwith|Mz−4c|(green line) from the molecular field model.\nIt can be seen clearly that σxyfollows the temperature\ndependence of Mz−4cbelowTcompand not Mnet. As\na further step, we plot |Mnet|from the molecular field\nmodel (orange line) with |σxy(T)−σxy−comp|(triangles).\nAsσxyis proportional only to M4cand at compensa-\nTABLE I: Initial parameters input to the molecular field\nmodel according to equations 1 and 2. M4a,M4candK4a,\nK4care the magnetizations and uniaxial anisotropies on the\n4a, 4csublattices. n4a−4aandn4c−4care the intralayer ex-\nchange constants. n4a−4cis the interlayer exchange constant.\nDerived parameters are outputs of the molecular field model.\nInitial parameters\nM4a(0K) 547kAm−1n4a−4a 1150\nM4c(0K) 585kAm−1n4c−4c 400\nK4a 0kJm−3n4a−4c -485\nK4c 216kJm−3\nDerived parameters\nMnet(10K) 38kAm−1TC 625K\nMnet(max.) 97kAm−1Tcomp 155K/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s56/s48/s46/s48/s49/s48/s48/s46/s48/s49/s50/s48/s46/s48/s49/s52/s48/s46/s48/s49/s54/s48/s46/s48/s49/s56\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s120/s121/s47/s77\n/s110/s101/s116\n/s40/s79/s104/s109/s46/s109/s46/s65/s47/s109/s41/s45/s49\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s120/s121/s47/s77\n/s122/s32/s40/s79/s104/s109/s46/s109/s46/s65/s47/s109/s41/s45/s49\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32\n/s120/s121/s47/s77\n/s122/s45/s52 /s99/s32 /s32/s76/s105/s110/s101/s32/s102/s105/s116/s32\n/s120/s121/s47/s77\n/s122/s45/s52 /s99\n/s32\n/s120/s121/s47/s77\n/s122/s45/s52 /s97/s32 /s32/s76/s105/s110/s101/s32/s102/s105/s116/s32\n/s120/s121/s47/s77\n/s122/s45/s52 /s97/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48 /s55/s48/s48/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s32/s77\n/s122 /s45/s52 /s97\n/s32/s77\n/s122 /s45/s52 /s99\n/s32/s124 /s77\n/s110/s101/s116/s124 /s77\n/s122/s32/s40/s77/s65/s47/s109/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48 /s55/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s32/s124 /s77\n/s122/s45/s52 /s99/s124\n/s32/s124 /s77\n/s110/s101/s116/s124/s124/s77\n/s122/s124/s32/s40/s77/s65/s47/s109/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s84\n/s99/s111/s109/s112\n/s48/s50/s107/s52/s107/s54/s107/s56/s107\n/s32\n/s120/s121\n/s32/s124\n/s120/s121/s40/s84 /s41/s32/s45/s32\n/s120/s121 /s45/s99/s111/s109 /s112/s124\n/s32\n/s120/s121/s32/s32/s40/s79/s104/s109/s46/s109/s41/s45/s49\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s52/s48/s46/s54/s32/s124 /s77\n/s122 /s45/s52 /s97/s124\n/s32/s124 /s77\n/s122 /s45/s52 /s99/s124/s124/s77\n/s122/s124/s32/s40/s77/s65/s47/s109/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s54/s107/s56/s107/s32\n/s120/s121\n/s120/s121/s32/s32/s40/s79/s104/s109/s46/s109/s41/s45/s49\n/s40/s99/s41/s40/s97/s41\n/s40/s98/s41\nFIG. 3: (a) Temperature dependence of Mz−4aandMz−4c\nandMnet. The data is obtained from numerical integration\nwith an applied field to set the direction of the net magneti-\nzation and then reduced to zero, therefore the magnetizatio n\nreverses at Tcomp.Tcompis 155K and the Curie temperature\nis 625K. (b) σxy,|σxy(T)−σxy−comp|,|Mz−4c|, and|Mnet|as\nafunction of temperature. Inset: σxyplotted with Mz−4aand\nMz−4cas a function of temperature, to show that σxydoes\nindeed follow Mz−4cand not Mz−4a. (c) Ratio of σxy/Mz−4c\nandσxy/Mz−4a(dotted lines) over the experimentally mea-\nsuredtemperature rangecomplete withlinear fits(solid lin es).\nThe ratio is almost constant with no significant linear back-\nground slope showing that σxy∝Mz−4c. The inset shows the\nclear divergence of σxy/MnetatTcomp.\ntionM4c=M4a, subtracting the value of σxyatTcomp\n(|σxy(T)−σxy−comp|) gives an approximate indication of5\n/s48 /s49/s53 /s51/s48 /s52/s53/s45/s54/s107/s45/s52/s107/s45/s50/s107/s48/s50/s107/s52/s107/s54/s107/s120/s121/s32/s40/s79/s104/s109/s46/s109/s41/s45/s49\n/s181\n/s48/s72\n/s105/s32/s40/s84/s41/s32/s47/s32 /s32/s181\n/s48/s72\n/s122\n/s32/s47/s32 /s32/s181\n/s48/s72\n/s120\n/s50/s50/s48/s32/s75/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s77\n/s122/s45 /s52 /s99/s32/s40/s77/s65/s47/s109/s41\nFIG. 4: Comparison of experimental data (solid lines) and\nmolecular fieldmodel (dashed lines) at 220K for fields applie d\nalongµ0Hzandµ0Hx.µ0Hsfis observed in both cases at\n26T.\nhowMnetbehaves with temperature. Even though this\nignorestheweak M4atemperaturedependence, thetrend\nofMnetfollows|σxy(T)−σxy−comp|, showingthat σxyis a\nreflection of M4cand notMnet. The inset in figure 3 (b)\nshows both Mz−4aandMz−4cwith the experimentally\nobtained σxy, and shows that σxymore closely follows\nMz−4c. Figure 3 (c) shows the ratio of σxytoMz−4c\n(green dotted line), Mz−4a(blue dotted line) and Mnet\n(inset). Linear fitting of σxy/Mz−4c(solid green line),\nandσxy/Mz−4a(solid blue line) show that σxy/Mz−4c\nremains constant over the measured temperature range,\nand is equal to 0 .0136Ω−1m−1A−1m, similar to what\nhas been reported for other itinerant ferromagnetic sys-\ntems [34, 35]. The linear slope for σxy/Mz−4aand the\ndivergence of σxy/Mnetshows that σxyreflects neither of\nthese two quantities.\nA recent study has shown via ab initio calculations\nthat this must be the case for a fully-compensated half-\nmetallic ferrimagnetic system [36] although previous re-\nports on bulk films found ρxy, and hence, σxyfalling to\nzero atTcomp[37].\nFor the evaluation of the magnetic anisotropy we use\nthe initial low-field change of σxyversusµ0Hxand ex-\ntrapolate to zero and obtain K4c(not shown). The val-\nues obtained vary from 100kJm−3to 250kJm−3over\nthe entire data range. We also calculate the anisotropy\ndirectly from the spin-flop transition Hsf=/radicalbig\n2HKHex\n4c,\nwhereHKis thesublattice anisotropyfield and Hex\n4cis the\nexchange field, the first term in Eqn. 2. The anisotropy\nfield,HK, is related to the sublattice anisotropy energy\nK4c.\nA comparison between the experiment and the model\nat 220K for both µ0Hzandµ0Hxis shown in figure\n4. The solid lines plot the experimentally obtained σxy,\nwhile the dashed lines plot Mz−4cfrom the model. The\nspin-flop field is observed in both cases at µ0Hz= 26T.\nFor the case of µ0Hx, it can first be seen that the 4 c\nmoment does not saturate along the field as one wouldexpect [18, 38]. It initially decreases but then returns\nto a saturated value in both the experimental data and\nthe model. This behaviour is due to the fact that in\nMRG the exchange and anisotropy energies are compa-\nrable and weak. If the exchange coupling is strong then\nthe net magnetic moment could be saturated along µ0Hx\nas both sublattices can remain antiparallel up to the\nanisotropy field µ0HK= 2(K4a+K4c)/(M4a+M4c) =\n2Keff/Mnet. If the exchange coupling is weak then both\nsublattice moments will tilt from their antiparallel align-\nment, breaking exchange, before the net magnetic mo-\nment can be saturated along µ0Hxat the appropriate\nsublattice anisotropy field µ0HK= 2Ksl/Msl, sl = 4a,4c.\nThe model and experiment disagree slightly on the\ntemperature dependence of HsfbelowTcomp. Bet-\nter agreement can be obtained by using much higher\nanisotropyenergiesofoppositesign: K4a=−1.5MJm−3\nandK4c= 1.7MJm−3. This has the effect of increasing\n(decreasing) Hsfabove (below) Tcomp. While this im-\nproves the match between σxyversusµ0HzbelowTcomp,\nit worsens the match of σxyversusµ0Hx, at all tem-\nperatures. This and the slight discrepancies between\nthe model and experiment when a low value of K4c\nis used (Fig. 4) indicate that additional anisotropies,\nlikely cubic, in MRG, as well as anti-symmetric exchange\n(Dzyaloshinskii-Moriya interaction) should be taken into\naccount.\nWe have shown that the uniaxial molecular field model\nreproduces the main characteristics of the experimental\ndata and we confirm the relationship σxy∝M4ccosθM4c.\nKnowing HKandHex\n4cwe can predict the frequencies\nof the anisotropy, fanis=γµ0HK, and the exchange,\nfexch=γµ0/radicalbig\n2HKHex\n4c=γµ0Hsf, magnetic resonance\nmodes, where γ= 28.02GHzT−1[39]. At 220K, µ0Hsf\n= 26T and µ0Hex\n4c=n4a−4cM4a= 294T, therefore\nµ0HK= 1.15T and the resonances are fanis= 32GHz\nandfexch= 729GHz.\nIn conclusion, σxyfor fully-compensated half-metallic\nferrimagnetic alloys follows the relevant sublattice mag-\nnetization, MslcosθMsl, and not MnetcosθMnet. High-\nfield magnetotransport and molecular field modelling al-\nlows the determination of the anisotropy and exchange\nconstants provided the half-metallic material is collinear.\nMn2RuxGa behaves magnetically as an antiferromagnet\nand electrically as a highly spin polarised ferromagnet; it\nis capable of operation in the THz regime and its trans-\nport behaviour is governed by the Mn 4csublattice. The\nimmediate, technologically relevant, implication of these\nresultsis that spin-transfertorqueeffects in compensated\nferrimagnetic half-metals will be governed by single sub-\nlattice.6\nACKNOWLEDGEMENTS\nThis project has received funding from the Euro-\npean Union’s Horizon 2020 research and innovation pro-\ngramme under grant agreement No 737038 (TRAN-\nSPIRE). This work is supported by the Helmholtz Young\nInvestigator Initiative Grant No. VH-N6-1048. 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The Heusler compounds Mn 2TiZ(Z= Al, Ga, In, Si, Ge, Sn, P, As,\nSb) are of large interest due to their potential ferrimagnet ic properties and high\nspin polarization. Here, we present calculations of the str uctural and magnetic\nproperties of these materials. Their magnetic moment follo ws the Slater-Pauling\nrulem=NV−24. None of them is actually a perfect half-metallic ferrima gnet,\nbut some exhibit more than 90% spin polarization and Curie te mperatures well\nabove room temperature. The exchange interactions are comp lex, direct and\nindirect exchange contributions are identified. The Curie t emperature scales with\nthe total magnetic moment, and it has a positive pressure dep endence. The role\nof theZelement is investigated: it influences the properties of the compounds\nmainly via its valence electron number and its atomic radius , which determines\nthe lattice parameter. Based on these results, Mn 2TiSi, Mn 2TiGe, and Mn 2TiSn\nare proposed as candidates for spintronic applications.Ferrimagnetism in Mn 2TiZ Heusler compounds 2\n1. Introduction\nA very interesting class of Heusler compounds that has received co nsiderable\ntheoretical, but only few experimental attention to date, are the half-metallic\nferrimagnetsMn 2YZ,whereY=V,Cr, Mn, Fe, Co, Ni, Cuand ZisagroupIII,IV, or\nV element [1, 2, 3, 4, 5, 6, 7, 8, 9]. Half-metallic compounds are chara cterized by a gap\nfor either the spin-down or the spin-up density of states (DOS) at the Fermi energy,\nso that an electric current has purely up or down electrons. This pr operty makes\nthem highly interesting for applications in spintronics. A half-metallic f errimagnet\nhas advantages over the well-known half-metallic ferromagnets: d ue to the internal\nspin compensation it has rather low magnetic moment, while the Curie t emperature\nremains fairly high. A low magnetic moment gives rise to low stray fields, which is\ndesiredforspintronics,asisahighCurietemperatureandthusago odthermalstability\nof the compound [10]. The most prominent compound out of this class is Mn2VAl,\nwhich has been studied thoroughly by experiment and theory [11, 1 2, 13, 14, 15].\nTogether with numerous other compounds in the Mn 2VZseries it has been predicted\nto be a half-metallic ferrimagnet [1, 16]. Its low magnetic moment of ab out 2µBper\nformula unit (f.u.) and the high Curie temperature of 760K make it a pr omising\ncompound for spintronics [13]. Several other materials classes ha ve been proposed\nto be half-metallic ferrimagnets, e.g., Cr 0.75Mn0.25Se and Cr 0.75Mn0.25Te in the zinc\nblende structure [17], or Cr antisites in CrAs, CrSb, CrSe, and CrTe , having the zinc\nblende structure [18].\nIdeally, an electrode material for spintronics would be a half-metal with zero\nnet moment. This can not be achieved with antiferromagnets becau se of the spin-\nrotational symmetry (resulting in zero polarization), but well chos en half-metallic\nferrimagnets can be tuned to zero moment. This property is also kn own as half-\nmetallic antiferromagnetism, and has been first predicted for Mn an d In doped\nFeVSb [19]. Among others, La 2VMnO 6and related double perovskites [20] and\ncertain diluted magnetic semiconductors have been later predicted to be half-\nmetallic antiferromagnets as well [21]. Finally, the ferrimagnetic Heus ler compounds\nMn2VAl and Mn 2VSi have been proposed as a starting point for doping with Co\nto achieve the full compensation [23]. However, it should be noted th at the half-\nmetallic antiferromagnetism is limited to zero temperature and a small macroscopic\nnet moment is expected at elevated temperature—in particular nea r the Curie\ntemperature—because of the inequivalent magnetic sublattices [22 ].\nFollowing the Slater-Pauling rule connecting the magnetic moment mand the\nnumber of valence electrons NVviam=NV−24 in the half-metallic Heusler\ncompounds [24], it is expected to find another series of ferrimagnet ic half-metals\nin the Mn 2TiZsystem with −3 to−1µB/f.u. The negative moment indicates\nthat the half-metallic gap would appear for the majority states. Th ese compounds\ncould—if they are half-metals—provide another series of potential electrodes for\nspin-dependent applications and could also become a starting point f or half-metallic\nantiferromagnetism.\nIn this paper, we discuss ab initio calculations of the properties of the\n(hypothetical)Mn 2TiZcompounds, crystallizedintheL2 1structure. Noexperimental\ndata are available for this system, and only Mn 2TiAl has been studied theoretically\nbefore [25]. However, it is expected that parts of this series will exis t in the L2 1\nstructure, seeing that Mn 2VAl and Mn 2VGa, as well as parts of the Co 2TiZseries\nhave been prepared [26, 27, 28].Ferrimagnetism in Mn 2TiZ Heusler compounds 3\n2. Calculational approach\nThe calculations presented in this study were performed within two d ifferent density\nfunctional theory-based band structure codes: the full-poten tial linearized augmented\nplane waves (FLAPW) package Elk [29] and the full-potential Korrin ga-Kohn-\nRostoker Munich SPRKKR [30] package. Although both methods are in principle\nequivalent for crystalline systems, there are subtle differences as sociated with their\nnumerical implementations, and thus it is worth to compare both met hods on the\nrather complex intermetallic system Mn 2TiZ.\nElk was used to determine the theoretical lattice parameters and t he total energy\ndifferences between ferrimagnetic and nonmagnetic states. Thes e calculations were\ncarried out on a 12 ×12×12kpoint mesh (72 points in the irreducible wedge of the\nBrillouin zone). The muffin-tin radii of all atoms were set to 2.0 a.u. to a void overlaps\nat small lattice parameters. The equilibrium lattice parameters awere determined\nusing a third-degree polynomial fit to the total energies. To obtain accurate magnetic\nmoments and densities of states, the calculations were performed at the equilibrium\nlattice parameter using a 16 ×16×16k-mesh (145 points in the irreducible wedge)\nand nearly touching muffin-tin spheres.\nThe SPRKKR calculations were performed on the theoretical equilibr ium lattice\nparametersdeterminedwithElk. Thecalculationswerecarriedoutin thefull-potential\nmode with an angular momentum cutoff of lmax= 3 on a 22 ×22×22kpoint mesh\n(289 points in the irreducible wedge of the Brillouin zone). Both the fu ll potential\nas well as the increased angular momentum cutoff are necessary to ensure accurate\nresults. The DOS were calculated on a denser mesh of 1145 kpoints with 0.5mRy\nadded as the imaginary part to the energy.\nThe exchange-correlation potential was modeled within the genera lized gradient\napproximationofPerdew,Burke,andErnzerhofinbothschemes[ 31]. Thecalculations\nwere converged to about 0.1 meV. All calculations were carried out in the scalar-\nrelativisticrepresentationofthe valencestates, thusneglecting the spin-orbitcoupling.\nSPRKKR allows to calculate the Heisenberg exchange coupling parame tersJij\nwithin a real-space approach using an expression proposed by Liech tenstein et al.\n[32]. Using the Jijthe Curie temperatures were calculated within the mean field\napproximation (MFA). For a single-lattice system the Curie tempera ture is given\nwithin the MFA by\n3\n2kBTMFA\nC=J0=/summationdisplay\njJ0j. (1)\nIn a multi-sublattice system—as, e.g., the Heusler compounds with fo ur sublattices—\none has to solve the coupled equations\n3\n2kBTMFA\nC/angbracketlefteµ/angbracketright=/summationdisplay\nνJµν\n0/angbracketlefteν/angbracketright (2)\nJµν\n0 =/summationdisplay\nR/negationslash=0Jµν\n0R,\nwhere/angbracketlefteν/angbracketrightis the average zcomponent of the unit vector eν\nRpointing in the direction\nof the magnetic moment at site ( ν,R). The coupled equations can be rewritten as an\neigenvalue problem:\n(Θ−TI)E= 0 (3)\n3\n2kBΘµν=Jµν\n0Ferrimagnetism in Mn 2TiZ Heusler compounds 4\n/s65/s115\n/s68/s69/s32/s61/s32/s48/s46/s48/s49/s52/s32/s101/s86/s71/s101\n/s68/s69/s32/s61/s32/s48/s46/s49/s57/s32/s101/s86/s50/s46/s53\n/s50/s46/s48\n/s49/s46/s53\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s71/s97\n/s68/s69/s32/s61/s32/s48/s46/s53/s50/s32/s101/s86/s80\n/s68/s69/s32/s61/s32/s48/s46/s48/s48/s52/s52/s32/s101/s86/s83/s105\n/s68/s69/s32/s61/s32/s48/s46/s49/s55/s32/s101/s86/s50/s46/s53\n/s50/s46/s48\n/s49/s46/s53\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48/s65/s108\n/s68/s69/s32/s61/s32/s48/s46/s53/s48/s32/s101/s86\n/s54/s46/s52/s48 /s54/s46/s49/s48 /s53/s46/s56/s48 /s53/s46/s53/s48/s83/s98\n/s68/s69/s32/s61/s32/s48/s46/s48/s51/s57/s32/s101/s86\n/s54/s46/s52/s48 /s54/s46/s49/s48 /s53/s46/s56/s48 /s53/s46/s53/s48\n/s108/s97/s116/s116/s105/s99/s101/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s32/s97/s32/s40/s197/s41/s83/s110\n/s68/s69/s32/s61/s32/s48/s46/s50/s53/s32/s101/s86/s50/s46/s53\n/s50/s46/s48\n/s49/s46/s53\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48\n/s54/s46/s52/s48 /s54/s46/s49/s48 /s53/s46/s56/s48 /s53/s46/s53/s48/s73/s110\n/s68/s69/s32/s61/s32/s48/s46/s55/s48/s32/s101/s86\nFigure 1. Total energies of the investigated compounds in dependence of their\nlattice parameters. The results for the ferrimagnetic and t he non-magnetic states\nare represented with + and ×, respectively.\nwith a unit matrix Iand the vector Eν=/angbracketlefteν/angbracketright. The largest eigenvalueof the Θmatrix\ngives the Curie temperature [16, 33]. In order to separate the two Mn lattices, the\ncalculations were run in F ¯43m space group, in which the Mn atoms are not equivalent\nby symmetry. The R-summation in Eq. (2) was taken to a radius of Rmax= 3.0a,\nwhich has been shown to be sufficient for half-metallic Heusler compou nds [34, 35].\n3. Results\n3.1. Energy minimization and lattice parameters\nThree types of magnetic configurations were tested: ferro-, fe rri-, and nonmagnetic.\nIt was found for all compounds that the ferromagnetic configura tions were unstable\nand converged into the ferrimagnetic state. Fig. 1 displays the tot al energies of\nthe ferrimagnetic and the nonmagnetic configurations in dependen ce on the lattice\nparameters a. We find that the ferrimagnetic state has always lower energy than the\nnon-magnetic state; the difference in total energy reduces with in creasing number of\nvalence electrons, but it increases within the groups with the atomic number. The\nlattice parameters follow roughly a linear dependence on the atomic r adius of the Z\nelement with the correlation coefficient of r= 0.92 (Fig. 2 (a)). Some compounds\nshow a strong asymmetry of the total energy curve in the ferrima gnetic configuration\nand even kinks in the curves for very large a. This is caused by a steep increase of the\nmagnetic moments for increasing awhich causes a stronger binding. However, thisFerrimagnetism in Mn 2TiZ Heusler compounds 5\nTable 1. Results of the ground state properties calculations with El k and\nSPRKKR. The total magnetic moments are given in µBper formula unit, the\natomic magnetic moments are given in µBper atom. The SPRKKR results for\nMn2TiAs were obtained with a= 5.95˚A (see text).\nElk SPRKKR\nMn2TiZa(˚A)m m MnmTiP (%) m m MnmTiP (%)\nAl 5.96 2.98 1.83 -0.57 21 2.98 1.76 -0.49 82\nGa 5.95 2.95 1.84 -0.60 45 2.97 1.77 -0.53 79\nIn 6.23 3.17 2.17 -0.86 7 3.08 1.98 -0.82 32\nSi 5.78 1.98 1.16 -0.31 94 1.98 1.13 -0.26 87\nGe 5.87 1.97 1.20 -0.37 94 1.97 1.16 -0.33 89\nSn 6.14 1.97 1.32 -0.51 97 2.00 1.25 -0.48 93\nP 5.68 0.30 0.18 -0.05 -3 — — — —\nAs 5.82 0.94 0.59 -0.20 84 0.97 0.61 -0.22 58\nSb 6.07 0.97 0.65 -0.25 88 0.98 0.62 -0.24 79\neffect is never strong enough to shift the equilibrium lattice paramet er to such a high-\nmstate. The equilibrium lattice parameters are summarized in Table 1. T ypically we\nfind the equilibrium lattice parameters of Heusler compounds obtaine d with Elk to be\naccurate within ±0.5% compared to experiment.\n3.2. Magnetic moments and densities of states\nThe results discussed in this subsection are summarized in Table 1 and Fig. 3.\n3.2.1. Mn 2TiAl, Mn 2TiGa, Mn 2TiInFrom the rule m=NV−24 we expect to find\na magnetic moment of 3 µB/f.u.for these compounds. The FLAPW calculations show\nsmall deviations from this rule, indicating that the compounds are no t perfect half-\nmetals. This is confirmed by the DOS, which show spin polarizations at t he Fermi\nlevel below 50%, and in particular only 7% for Mn 2TiIn, where the magnetic moment\nis enhanced to 3 .17µB/f.u.. This arises from the large lattice parameter and the fact\nthat all three compounds do not form a gap in the DOS. The Fermi lev el for Mn 2TiAl\nand Mn 2TiGa is in a region with low DOS for both spin channels (see insets in Fig.\n3), but both of them have a very large empty spin-down DOS right ab oveEF. Small\nvariations of the lattice parameter would thus lead to strong variat ions of the spin\npolarization.\nThe calculations performed with SPRKKR reproduce the magnetic mo ments\nobtained in Elk very well. Although the total moments are practically e qual, a larger\ndeviation is found for the atom-resolved moments. The Fermi ener gy is found at\nslightly different positions in the DOS, and the detailed structures ob served in Elk\naroundEFare less pronounced, especially the dip in the spin-down states at EF. This\nleads to significantly higher spin polarization values in SPRKKR. Howeve r, the trend\nthat Mn 2TiIn has the lowest polarization within this group is reproduced.Ferrimagnetism in Mn 2TiZ Heusler compounds 6\n/s48/s46/s55/s53\n/s48/s46/s55/s48\n/s48/s46/s54/s53\n/s48/s46/s54/s48\n/s48/s46/s53/s53/s109/s77/s110/s32/s47/s32/s109\n/s54/s46/s50 /s54/s46/s49 /s54/s46/s48 /s53/s46/s57 /s53/s46/s56 /s53/s46/s55\n/s108/s97/s116/s116/s105/s99/s101/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s32/s97/s32/s40/s197/s41/s45/s48/s46/s51/s48/s45/s48/s46/s50/s53/s45/s48/s46/s50/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s109/s84/s105/s32/s47/s32/s109/s80/s73/s110/s83/s110/s83/s98\n/s83/s105/s65/s115\n/s71/s101/s71/s97\n/s65/s108\n/s40/s98/s41/s54/s46/s51\n/s54/s46/s50\n/s54/s46/s49\n/s54/s46/s48\n/s53/s46/s57\n/s53/s46/s56\n/s53/s46/s55\n/s53/s46/s54/s108/s97/s116/s116/s105/s99/s101/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s32/s97/s32/s40/s197/s41\n/s49/s54/s48 /s49/s52/s48 /s49/s50/s48 /s49/s48/s48\n/s90/s32/s97/s116/s111/s109/s105/s99/s32/s114/s97/s100/s105/s117/s115/s32/s40/s112/s109/s41/s80/s83/s105/s71/s97\n/s71/s101\n/s65/s115/s83/s110\n/s65/s108/s83/s98/s73/s110\n/s40/s97/s41\nFigure 2. (a): Dependence of the lattice parameter aon the atomic radius of\ntheZelement. (b): Normalized magnetic moments of Mn and Ti in dep endence\nof the lattice parameter.\n3.2.2. Mn 2TiSi, Mn 2TiGe, Mn 2TiSnAccording to the “rule of 24” a total magnetic\nmoment of 2 µB/f.u.is expected. Again, small deviations from this rule are observed;\nall moments are lower by about 1.5%. In Elk, the three compounds ar e found to\nform a half-metallic gap in the spin-up states slightly above EF. The gap onset above\nEF(width) is 0.16eV (0.49eV) for Si, 0.24eV (0.25eV) for Ge, and 0.19eV ( 0.01eV)\nfor Sn. Nevertheless, the spin polarization is above 90% in these calc ulations. The\nstructure of the DOS around EFleads to a stable spin polarization and magnetic\nmoment upon isotropic lattice compression or expansion. For this se ries, having the\nsame valence electron counts and nearly half-metallic DOS, one can o bserve clearly\na narrowing of the bands, i.e., the DOS are contracted towards EF, while the Fermi\nlevel itself moves upwards. This is directly associated with the gradu ally increasing\nlattice parameter in this series, which reduces the overlap of the 3d orbitals and\ntherebyreducesthe itinerancyofthe system. An increasedlocaliz ationofthe electrons\nprovides also an explanation for the increasing atomic magnetic mome nts along this\nseries. Similar behavior has been observed earlier for Co 2MnZ, withZ=Si, Ge, Sn\n[36, 37] and Ni 2MnSn [38]. In the first case the Mn moment is increased and the Co\nmoment is lowered along the series, keeping the total moment intege r. Calculations\non Co2MnSi with increased lattice parameter reproduced this behavior. I n the second\ncase, the pressure dependence of the moments was studied. Und er increasing pressure,\ni.e., with reduced lattice parameter, both the Ni and the Mn moment d ecrease, and\nthus the total moment decreases. However, Ni 2MnSn is not a half-metal, hence the\ntotal moment is not restricted to an integer value. Consequently, both observations\non quite different ferromagnetic Heusler compounds are in accord w ith our case of\n(nearly) half-metallic ferrimagnetic Heusler compounds.\nWe note, that the magnetic moments and DOS from SPRKKR are in ver y good\nagreement with the ones obtained from Elk. However, the Fermi lev el is found at a\nlower position, giving rise to the slightly reduced polarization values.\n3.2.3. Mn 2TiP, Mn 2TiAs, Mn 2TiSbIn these cases a total magnetic moment of only\n1��B/f.u.is expected. Because of the very small lattice parameter of Mn 2TiP, itsFerrimagnetism in Mn 2TiZ Heusler compounds 7\n/s49/s48\n/s53\n/s48\n/s45/s53\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s83/s98/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s65/s115/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s80\n/s45/s56/s45/s52/s48/s52/s56\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s83/s110/s45/s52/s48/s52\n/s71/s101/s45/s52/s48/s52\n/s83/s105\n/s49/s48\n/s53\n/s48\n/s45/s53\n/s45/s49/s48/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s32/s47/s32/s101/s86/s32/s47/s32/s117/s110/s105/s116/s32/s99/s101/s108/s108/s41/s45/s48/s46/s49 /s48/s46/s49\n/s71/s97\n/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s73/s110/s56\n/s52\n/s48\n/s45/s52/s45/s48/s46/s49 /s48/s46/s49\n/s65/s108\nFigure 3. Densities of states calculated with Elk. The spin-up DOS is p ointing\nup, the spin-down DOS is pointing down. The insets for Al and G a show the\nregion around the Fermi energy.\nspin-splitting is small with only 0 .3µB/f.u.in the Elk calculation. The situation of\nMn2TiAs and Mn 2TiSb is similar to that of Mn 2TiSi and Mn 2TiGe. A spin-up gap\nis formed above the Fermi level with onset (width) of 0.29eV (0.53eV ) for As and\n0.19eV (0.44eV) for Sb. Though not being half-metallic, both compou nds have spin\npolarizations of more than 80%.\nFinally, the magnetic moments of Mn 2TiSb in SPRKKR agree very well with\nthose obtained with Elk. But again, the Fermi level is lower and the sp in polarization\nis reduced. ForMn 2TiP and Mn 2TiAs the situation is quite different. They can not be\nconverged into ferrimagnetic states at the equilibrium lattice param eters determined\nby Elk; instead, they are found to be nonmagnetic. This is caused by the tiny\nenergy difference between the ferrimagnetic and the nonmagnetic configuration, which\nleads to a numerical instability of the ferrimagnetic state. By increa sing the lattice\nparameterofMn 2TiAsbyabout2%to5.95 ˚A,theseparationisincreasedartificiallyto\nabout 30meV/f.u. and the calculation convergesinto the ferrimagn etic state. Because\nof this, the properties obtained with SPRKKR for this compound hav e to be taken\nwith care: in all other cases the individual atomic moments are slightly lower in\nSPRKKR than those from Elk; here instead, larger moments are fou nd. However, the\nsame procedure can not be applied to Mn 2TiP, within a reasonable range of latticeFerrimagnetism in Mn 2TiZ Heusler compounds 8\nparameters.\n3.2.4. General remarks It is worth to note that the magnetic moments of the Z\ncomponent are always below 0.06 µBand that they are always parallel to the Ti\nmoment. In detail, the values are Al 0.044 µB, Ga 0.052 µB, In 0.058 µB, Si 0.034 µB,\nGe 0.035 µB, Sn 0.034 µB, P 0.0062 µB, As 0.018 µB, and Sb 0.017 µB.\nAnother property worth noting is the fact that the ratios mMn/mandmTi/m\nfollow a linear dependence (with correlationcoefficients of r≈0.9 in both casesfor the\nElkdata)onthelatticeparameter(andhencetheinteratomicdista nces)independently\non theZtype, see Fig. 2 (b). As mentioned above, with increasing lattice par ameter\nthe itinerantcharacterofthesystemisreducedandlocalizesthem omentsgraduallyon\nthe atoms. Therefore, the influence of the Zcomponent in Mn 2TiZis twofold. First,\nit determines the lattice parameter of the compound and following fr om that, the\ndegree of electron localization. And second, the total magnetic mo ment is determined\nvia the number of electrons supplied, if the lattice parameterdoes n ot exceed a certain\nrange (which is not the case for P and In).\n3.3. Exchange interactions and Curie temperatures\nThe exchange interactions are investigated here for Mn 2TiGa, Mn 2TiGe, and\nMn2TiSb, which are representative compounds for their respective Zgroup. Fig.\n4 (a) displays the Jijcalculated for the intra-sublattice interaction Mn1(2)-Mn1(2)and\nthe inter-sublattice interactions Mn1(2)-Mn2(1)and Mn-Ti of the three compounds.\nAll other interactions are very small and can be neglected for the f ollowing discussion.\nIn all three cases it is clear that the Mn1(2)-Mn2(1)inter-sublattice interaction\nprovides the largest contribution to the exchange. Further, the nearest neighbor\ninteraction of Mn-Ti is always negative, hence all compounds are fe rrimagnets. All\ninteractions are mostly confined within a radius of 1 .5a. Apart from these similarities,\nthere are many interesting differences.\nFirst, we discuss the details of the dominating inter-sublattice inter action Mn1(2)-\nMn2(1). The first and second nearest neighbors provide a large, positive e xchange.\nThe second nearest neighbors have two different values of Jij. This is a feature that\nis not observed in frozen-magnon calculations (see, e.g. [16]), beca use the Fourier\ntransform that is necessary to obtain the exchange parameters involves a spherical\naveraging. Instead, with the real-space approach used here we o bserve a difference for\nMn atoms with a Ti atom or a Zatom in between. We found larger values on the\nMn atoms mediated via Ti and lower values on the Zmediated ones. The nearest Mn\nneighborshaveadistanceofabout2.95 ˚A,andthe exchangeisapparentlyindirect. For\ndirect exchange, one would expect a scaling with the magnetic momen ts, which is not\nobserved here. It rather oscillates with the sp electron number. A similar result has\nbeen obtained earlier on other half and full Heusler compounds [39]. The ratio of the\nnearest and second nearest neighbor coupling is significantly reduc ed with increasing\nelectron concentration, and the nearest neighbor interaction do minates in Mn 2TiSb.\nThe antiferromagnetic Mn-Ti interaction is only significant for the n earest\nneighbors. Accordingly, the interaction between Mn and Ti, which ha ve a distance of\nabout 2.55 ˚A is essentially given by direct exchange coupling and the scaling with th e\nTi moment corroborates this assumption.\nThe intra-sublattice interaction of Mn1(2)-Mn1(2)exhibits a notable oscillatory\nbehavior. In the two cases with odd valence electron number it is pos itive for theFerrimagnetism in Mn 2TiZ Heusler compounds 9\n/s51\n/s50\n/s49\n/s48\n/s45/s49 /s77/s110/s50/s84/s105/s83/s98/s32/s77/s110/s49/s40/s50/s41/s45/s77/s110/s49/s40/s50/s41\n/s32/s77/s110/s49/s40/s50/s41/s45/s77/s110/s50/s40/s49/s41\n/s32/s77/s110/s45/s84/s105/s54\n/s52\n/s50\n/s48\n/s45/s50 /s77/s110/s50/s84/s105/s71/s101/s71/s101/s32/s109/s101/s100/s105/s97/s116/s101/s100/s84/s105/s32/s109/s101/s100/s105/s97/s116/s101/s100 /s56\n/s52\n/s48\n/s45/s52/s74/s105/s106/s32/s40/s109/s101/s86/s41\n/s77/s110/s50/s84/s105/s71/s97\n/s54\n/s52\n/s50\n/s48\n/s45/s50\n/s45/s52/s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53 /s48/s46/s48/s77/s110/s50/s84/s105/s71/s101\n/s97/s32/s61/s32/s54/s46/s48/s55/s32/s197/s54\n/s52\n/s50\n/s48\n/s45/s50\n/s45/s52/s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53 /s48/s46/s48\n/s114/s32/s47/s32/s97/s77/s110/s50/s84/s105/s71/s101\n/s97/s32/s61/s32/s53/s46/s56/s55/s32/s197/s54\n/s52\n/s50\n/s48\n/s45/s50\n/s45/s52/s74/s105/s106/s32/s40/s109/s101/s86/s41\n/s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53 /s48/s46/s48/s77/s110/s50/s84/s105/s71/s101\n/s97/s32/s61/s32/s53/s46/s54/s55/s32/s197/s40/s97/s41\n/s40/s98/s41\nFigure 4. Heisenberg exchange parameters Jijin dependence on the normalized\ndistance r/a. (a):Jijfor Mn 2TiGa, Mn 2TiGe, Mn 2TiSb for their respective\nequilibrium lattice parameters. (b): Jijfor Mn 2TiGe with different lattice\nparameters. Note the different scales of the vertical axes in the top row.\nnearest neighbors, negative for the second, and again positive fo r the third nearest\nneighbors. For Mn 2TiGe with its even electron count the first two neighbors have\nnegative and the third neighbor has positive interaction. So in the lat ter case, the\ntotal Mn-Mn intra-sublattice interaction is effectively antiferroma gnetic.\nIn order to study the dependence of Jijon the lattice parameter as a possible\nexplanation for the differences discussed above, additional calcula tions on Mn 2TiGe\nhave been performed with lattice parameters of (5 .87±0.2)˚A. This compound was\nchosen because of the wide (pseudo-)gap for the spin-up states , which warrants a\nstable total magnetic moment and minimal band structure effects o ver the range of a\nused here.\nThe resultsfrom these calculationsare givenin Fig. 4 (b). Obviously, the changes\nherearerathersubtleandcannotaccountforthethe largediffer encesdiscussedabove.\nHowever, we note a reduction of the nearest neighbor Mn1(2)-Mn2(1)interaction and\nof the Ti mediated second nearest Mn1(2)-Mn2(1)neighbor. Meanwhile, the Mn-Ti\ninteraction increases, in agreement with increased Mn and Ti momen ts.\nThe strong confinement of the exchange interactions to a sphere with a radius of\nabout 1.5ais reflected in the Curie temperature calculated as a function of the cluster\nradius which is nearly converged at r/greaterorsimilar1.5a, see Fig. 5 (a). At larger radii a weak\noscillation of TMFA\nCis observed, indicating long-ranged RKKY-like behaviour.\nA deeper discussion of the exchange interaction is beyond the scop e of this\npaper. However, it was recently shown for numerous half and full H eusler compounds\nthat various exchange mechanisms—such as RKKY, superexchang e and Anderson s-d\nmixing—contribute to the indirect exchange interactions [39].Ferrimagnetism in Mn 2TiZ Heusler compounds 10\n/s56/s48/s48\n/s55/s48/s48\n/s54/s48/s48\n/s53/s48/s48\n/s52/s48/s48\n/s51/s48/s48\n/s50/s48/s48\n/s49/s48/s48\n/s48/s84/s67/s77/s70/s65/s32/s40/s75/s41\n/s51/s46/s48 /s50/s46/s53 /s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53 /s48/s46/s48\n/s114/s32/s47/s32/s97/s32/s65/s108 /s32/s83/s105\n/s32/s71/s97 /s32/s71/s101 /s32/s65/s115\n/s32/s73/s110 /s32/s83/s110 /s32/s83/s98\n/s40/s97/s41/s56/s48\n/s55/s48\n/s54/s48\n/s53/s48\n/s52/s48\n/s51/s48\n/s50/s48\n/s49/s48\n/s48\n/s45/s49/s48\n/s45/s50/s48\n/s45/s51/s48\n/s45/s52/s48\n/s45/s53/s48/s74/s48/s32/s40/s109/s101/s86/s41\n/s65/s108/s71/s97 /s73/s110/s83/s105/s71/s101/s83/s110 /s80/s65/s115/s83/s98/s32/s77/s110/s49/s40/s50/s41/s45/s77/s110/s49/s40/s50/s41\n/s32/s77/s110/s49/s40/s50/s41/s45/s77/s110/s50/s40/s49/s41\n/s32/s77/s110/s45/s84/s105\n/s40/s98/s41\nFigure 5. (a): The Curie temperature TMFA\nCin dependence on the normalized\ncluster radius r/ataken into the summation. (b): R-summed exchange coupling\nparameters J0.\nThe relevant contributions to the J0matrix in Eq. (2) are displayed in Fig. 5\n(b). In agreement with the previous discussion it is found that the in ter-sublattice\ninteraction Mn1(2)-Mn2(1)provides the largest contribution, followed by the Mn-Ti\ninteraction, which can become as large as the Mn1(2)-Mn2(1)interaction in Mn 2TiIn.\nThe intra-sublattice interaction Mn1(2)-Mn1(2)is generally weak, positive for Al, Ga,\nIn, and negative for Si, Ge, Sn. All other inter- and intra-sublattic e contributions are\nbelow 1meV. A negative intra-sublattice contribution means that th e interaction acts\nagainstthe ferromagneticorderonthis latticeandthusreducest heCurietemperature.\nTo estimate the accuracyofour method for the Curie temperatur e determination,\nwe calculated the Curie temperatures of some Heusler compounds a t their respective\nexperimental lattice parameters. The calculated (experimental) v alues are: Co 2MnSi\n1049K (985K)[40], Co 2TiSn 383K (355K)[41], Mn 2VAl 605K (760K)[13] and\nMn2VGa 560K (783K)[26]. Further values, obtained using the same meth od, can\nbe found in Ref. [35]. For the Co-based ferromagnetic compounds, the calculated\nmean-field values are in good agreement with experiment. However, in the case of the\ntwo ferrimagnetic Mn-based compounds, the MFA Curie temperatu re is about 25%\nlower than the experimental one.\nTable 2summarizesourcalculated Curietemperatures. Theyarewe llaboveroom\ntemperature for the compounds with 21 and 22 valence electrons, but considerably\nlower for Mn 2TiAs and Mn 2TiSb. The Curie temperature scales roughly linear\nTable 2. Curie temperatures TMFA\nCcalculated in the mean-field approximation.\nMn2TiZ Al Ga In Si Ge Sn P As Sb\nTMFA\nC(K) 665 663 630 424 398 354 — 132 156Ferrimagnetism in Mn 2TiZ Heusler compounds 11\n/s49/s46/s53\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48\n/s45/s48/s46/s53/s109/s32/s40/s109/s66/s41\n/s54/s46/s48/s55 /s53/s46/s57/s55 /s53/s46/s56/s55 /s53/s46/s55/s55 /s53/s46/s54/s55\n/s108/s97/s116/s116/s105/s99/s101/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s32/s97/s32/s40/s197/s41/s40/s99/s41\n/s32/s109/s77/s110\n/s32/s109/s84/s105/s52/s50/s48\n/s52/s48/s48\n/s51/s56/s48\n/s51/s54/s48\n/s51/s52/s48/s84/s67/s77/s70/s65/s32/s40/s75/s41/s40/s98/s41/s54/s48\n/s52/s48\n/s50/s48\n/s48\n/s45/s50/s48/s74/s48/s32/s40/s109/s101/s86/s41/s40/s97/s41\n/s32/s77/s110/s49/s40/s50/s41/s45/s77/s110/s49/s40/s50/s41\n/s32/s77/s110/s49/s40/s50/s41/s45/s77/s110/s50/s40/s49/s41\n/s32/s77/s110/s45/s84/s105\nFigure 6. Dependence of J0(a),TMFA\nC(b) and magnetic moments (c) on the\nlattice parameter in Mn 2TiGe. Markers in (b) are the same as in (a). Magnetic\nmoments in (d) are mMn(⊓ ⊔) andmTi(△).\nwith the total magnetic moment. Within one group, the Curie temper atures are\ncomparable, though a trend to decrease with increasing atomic num ber of the Z\ncomponent is clear for 21 and 22 valence electrons.\nThe Curie temperatures of Mn 2TiAl, Mn 2TiGa and Mn 2TiIn are quite similar.\nThe slightly reduced TMFA\nCof Mn 2TiIn is caused by the steep reduction of the\nMn1(2)-Mn2(1)interaction. On the other hand, a simultaneous increase of the\nMn-Ti interaction stabilizes TMFA\nCat a still high level. In the series Mn 2TiSi –\nMn2TiGe – Mn 2TiSn the Mn1(2)-Mn2(1)decreases, but here the increase of the Mn-Ti\ninteraction can not compensate this and hence the Curie temperat ure decreases. In\nany case, the Mn1(2)-Mn2(1)interaction provides the dominant contribution to TMFA\nC,\nonly in Mn 2TiIn the Mn-Ti interaction is dominant. The significantly lower Curie\ntemperature of Mn 2TiAs with respect to Mn 2TiSb can be attributed to the artificially\nincreased lattice parameter used in the calculation.\nThe dependence of the exchange parameters and TMFA\nCon the lattice constant\nwas studied above for Mn 2TiGe. The corresponding terms of the J0matrix, the\nCurie temperature and the magnetic moments are presented in Fig. 6 (a)-(c). A\ndecrease of the Mn1(2)-Mn2(1)interaction and simultaneously of TMFA\nCwith increasing\nais observed, although both mMnandmTiincrease. Obviously, the individual\nmoments play only a minor role in the exchange and the interatomic dist ances are\nmoreimportant. The Mn-TiaswellastheMn1(2)-Mn1(2)interactionsbecomestrongerFerrimagnetism in Mn 2TiZ Heusler compounds 12\nwith increasing a, but they nearly compensate each other. In agreement with a dire ct\nexchange coupling, the Mn-Ti interaction scales with the magnetic m oments. The\nchanges in J0reproduce very well the changes observed in Fig. 5 (b) for the Si – Ge\n– Sn series.\nPut in terms of a pressure dependence, we observe d TC/dp >0, i.e., the\nCurie temperature increases with increasing pressure. Kanomata et al.proposed\nan empirical interaction curve for Ni 2MnZand Pd 2MnZfull Heusler compounds that\nsuggestes d TC/dp >0 for these compounds [42]. The origin of this behavior is\nattributed to the Mn-Mn distance and the indirect exchange betwe en the Mn atoms,\nwhich fully carry the magnetism of the compounds. Hence, all other interactions\ncan be neglected. A numerical confirmation by first principles of this interaction\ncurve was given recently [38]. For half-metallic Heusler compounds of type Co 2YZ\nK¨ ubleret al.analyzed the dependence of TCon the valence electron number, which\nis approximately linear, and scales thus with the total magnetic mome nt [43]. Further\nit was also proposed for Co 2MnZcompounds to have d TC/dp >0, although the Co\natom participates significantly in the exchange interactions [37]. Exp erimentally this\ndependence on the lattice parameter was even observed for the C o2TiZseries (with Z\n= Si, Ge, Sn), where the Ti atoms have nearly vanishing magnetic mom ent [27].\nInterestingly, the magnetic moments of Mn and Ti in Mn 2TiGe vary within the\nsamerangeasthemomentsfordifferentcompoundsshowninFig. 2( b), whilethe total\nmoment remains fixed at 2 µB/f.u. These findings demonstrate the stronginfluence of\nthe lattice parameter, while the details of the electronic structure of theZelement are\nless important. Consequently, the Zelement influences the properties of the Mn 2TiZ\ncompound mainly via its number of valence electrons and its atomic rad ius, which\ndetermines the equilibrium lattice parameter.\n4. Conclusion\nOur results suggest that the Mn 2TiZHeusler compound series with Z= Al, Ga,\nIn, Si, Ge, Sn, P, As, Sb, can exhibit ferrimagnetism in accordance w ith the rule\nm=NV−24. Most of the compounds have large spin polarization and a spin-up gap\nforms above the Fermi energy. The Curie temperatures calculate d within the mean-\nfield approximation indicate that the compounds with 21 and 22 valenc e electrons will\nbe ferrimagnetic at room temperature. A thorough understandin g of the influence of\ntheZcomponent on the properties of the compounds has been establish ed on the\nbasis of ab initio band structure and exchange coupling calculations. It was found\nthat the pressure dependence of TCis positive, in agreement with ferromagentic full\nHeusler compounds. Because of their large and stable spin polarizat ions and their\nhigh Curie temperatures we propose in particular Mn 2TiSi, Mn 2TiGe, and Mn 2TiSn\nas candidates for spintronic applications.\nAcknowledgements\nThis work has been supported by the German Bundesministerium f¨ u r Bildung und\nForschung (BMBF) under contract number 13N9910. Helpful disc ussions with Prof.\nAndrei Postnikov are acknowledged.Ferrimagnetism in Mn 2TiZ Heusler compounds 13\nReferences\n[1]¨Ozdo˜ gan K, Galanakis I, S ¸a¸ sioglu E and Akta¸ s B 2006 J. Phys.: Condens. Matter 182905\n[2] Fujii S, Okada M, Ishida S and Asano S 2008 J. Phys. Soc. Jpn 77074702\n[3] Luo H, Zhu Z, Liu G, Xu S, Wu G, Liu H, Qu J and Li Y 2008 J. Magn. Magn. Mater. 320421\n[4] Wurmehl S, Kandpal H C, Fecher G H and Felser C 2006 J. Phys.: Condens. Matter 186171\n[5] Luo H Z, Zhang H W, Zhu Z Y, Ma L, Xu S F, Wu G H, Zhu X X, Jiang C B a nd Xu H B 2008\nJ. Appl. 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B 76024414" }, { "title": "1006.5531v2.Magnetostatics_of_synthetic_ferrimagnet_elements.pdf", "content": "arXiv:1006.5531v2 [cond-mat.mtrl-sci] 11 Jun 2011Magnetostatics of synthetic ferrimagnet elements\nOlivier Frucharta,∗, Bernard Diényb\naInstitut NÉEL, CNRS & Université Joseph Fourier – BP166 – F-38 042 Grenoble Cedex 9 – France\nbSPINTEC (UMR8191 CEA/CNRS/UJF/G-INP), CEA Grenoble, INAC, 3805 4 Grenoble Cedex 9,\nFrance\nAbstract\nWe calculate the magnetostatic energy of synthetic ferrima gnet (SyF) elements, con-\nsisting of two thin ferromagnetic layers coupled antiferro magnetically, e.g. through\nRKKY coupling. Uniform magnetization is assumed in each lay er. Exact formulas as\nwell as approximate yet accurate ones are provided. These ma y be used to evaluate\nvarious quantities of SyF such as shape-induced coercivity and thermal stability, like\ndemagnetizing coefficients are used in single elements.\nSynthetic antiferromagnets (SAF, resp. ferrimagnets, SyF )[1,2] consist of two thin\nferromagnetic films of moments of same (resp. different) magn itude, strongly coupled\nantiferromagnetically thanks to the RKKY interaction thro ugh an ultrathin spacer layer,\ntypically Ru 0.6−0.9nmthick[3]. Hereon we consider only the case of in-plane magne-\ntized layers. SyFs are widely used to provide spin-polarize d layers displaying an overall\nweak moment. One benefit is to minimize cross-talk of neighbo ring (e.g.memory bits) or\nstacked ( e.g.in a spin-valve) elements through stray-field coupling[ 1,2], such as in Mag-\nnetic Random Access Memory (MRAM)[ 4]. SyFs are also used to decrease the Zeeman\ncoupling with external fields, e.g.to increase coercivity in reference layers[ 5], decrease\neffects of the Oersted field in magneto-resistive or spin-tor que oscillator pillars, or more\nrecently boost the current-induced domain-wall propagati on speed in nanostripes[ 6,7].\n∗Corresponding author\nEmail address: Olivier.Fruchart@grenoble.cnrs.fr (Olivier Fruchart)\nPreprint submitted to Elsevier December 2, 2018In practice SyFs are used as elements of finite lateral size. I t has been shown[ 8] and\nit is widely used [ 9,10] that for flat and magnetically soft nanomagnets of lateral s ize\nsmaller than a few hundreds of nanometers, the macrospin app roximation (uniform mag-\nnetization) is largely correct. In this framework the coerc ive field equals the anisotropy\nfield2K/µ0Msand the energy barrier KV(Vis the volume of the dot) preventing spon-\ntaneous magnetization reversal equals the magnitude of ani sotropy of the total magnetic\nenergyE, to which all the physics therefore boils down. Elongated do ts are often used to\ninduce or contribute to an easy axis of magnetization and an e nergy barrier ∆E, based on\ndipolar energy. Dipolar energy is a quadratic form and thus i t is fully determined by its\nvalue along the two main in-plane axes. For single elements ∆Ed=KdV∆N with∆N\nthe difference between the two in-plane demagnetizing coeffic ients, and Kd= (1/2)µ0M2\ns.\nAnalytical formulas have been known for a long time to evalua te the mutual energy\nof an arbitrary set of prisms[ 11]. However while simple expressions for Nand thus ∆Ed\nhave been described, displayed and discussed for single-la yer flat elements[ 11,12], the\nanalytical expressions and the evaluation of Edin SyFs have not been discussed in detail\nso far. Instead the studies requiring estimation of the dipo lar energy in SyF, mainly\npertaining to MRAM cells[ 9,10,13], have in the best case made use of an effective so-\ncalled attenuation coe���cient with respect to self-energy[ 10], which requires a numerical\nevaluation[ 13]. The meaning and scaling laws of this attenuation coefficien t have never\nbeen discussed in detail, hiding the physics at play. As ther mal stability, coercivity and\ntoggle switching fields[ 9,10,14] depend crucially on the interlayer magnetostatic cou-\npling, it is desirable to have a simple yet accurate analytic al expression for interlayer\ndipolar fields. In this manuscript we report exact analytica l expressions for the magne-\ntostatics of SyFs uniformly-magnetized in each sub-layer. From the numerical evaluation\nwe discuss the physics at play, while from the analytical for mulas we propose an approx-\nimate yet accurate scaling law for their straightforward ta ble-top evaluation.\nWe first consider SyF prisms and name F1 and F2 the two ferromag netic layers (Fig-\nure1), with magnetization aligned along z. This covers the case of both finite-size\nprisms as well as infinitely-long stripes with a rectangular cross-section. We apply for-\nmulas expressing the interaction between two parallel char ged surfaces[ 11], and adopt\nthe convenient notation of Fijkfunctions, the i,jandk-fold indefinite integrals along x,\n2Figure 1: Geometry and notations of a prismatic SyF element c omprising two ferromagnetic layers F1\nandF2.\nyandzof the Green’s function F000= 1/r[15]. The only such function needed here is\nF220=1\n2[x(v−w)Lx+y(u−w)Ly]−xyPz+1\n6r(3w−r2) (1)\nwithu=x2,v=y2,w=z2,r=√u+v+w,Lx= (1/2)ln[(r+x)/(r−x)]etc,\nPx=xarctan(yz/xr)etc, and Lx= 0andPx= 0forx= 0etc.\nThe integrated magnetostatic energy of a single prismatic e lement of thickness tis:\nEd=2Kd\nπ/summationdisplay\nδa,δt,δc∈{0,1}(−1)δa+δt+δcF220(aδa,tδt,cδc) (2)\nwhich normalized to Kdyields the demagnetizing coefficient Nz. It can be verified that\nEq. (2) coincides with the explicit formula already known[ 12]. The magnetostatic energy\nof a prismatic SyF element may be calculated using the same fo rmalism , may be written\nas:\nEd=Kd,1Nz(a,t1,c)V1+Kd,2Nz(a,t2,c)V2\n+2/radicalbig\nKd,1Kd,2Nm(a,t1,s,t2,c)/radicalbig\nV1V2 (3)\nwithNm(a,t1,s,t2,c) =1\nπa√t1t2c/summationtext\nδ1,δ2,δa,δc∈{0,1}(−1)δ1+δ2+δa+δc×F220(aδa,s+t1δ1+\nt2δ2,cδc)is a mutual magnetostatic coefficient with a negative value, a ndVi=atic(resp.\nKd,i) is the volume (resp. dipolar constant) of each single prism i. This equation of dipo-\nlar energy is a quadratic form of M1andM2, generalizing the definition of demagnetizing\ncoefficients.\n3Figure 2: Magnetostatic energy of a SyF with c= 2a= 100nm ,M1,2= 106A/m,s= 0.7nm. (a) Sum\nof the energies of two prisms without mutual interaction, an d when embedded in the SyF geometry. t1\nis kept constant at 2.5nm, whilet2is varied. (b) Energy of the general SyF. The curved lines are those\nof minimum energy for either constant t1ort2. The thick horizontal dotted line highlights the path for\nthe SyF curve shown in (a).\nFigure 2(a) shows Edupon building a SyF via the progressive thickness increase o f F2\nabove F1, considering or not the interaction between the two layers. In the latter case the\nenergy increase nearly scales with t2\n2, which is understandable because it is a self-energy\n(inF2alone). In the coupled case (a SyF) Edretains like for the uncoupled case an overall\nclose-to-parabolic convex shape as can be verified with fitti ng, however with an initial\nnegative slope. This can be understood as for low t2the extra edge charges induced by\nan infinitesimal increase δt2mainly feel the stabilizing stray field arising from F1, whil e\nfor large t2they feel more the nearby charges induced by F2 itself. Notic e that, contrary\nto what could be a first guess, the minimum of Ed(t2)occurs before the compensation of\nmoment ( t1=t2). This stems from the same argument as above, which is that ma gnetic\ncharges at an edge of F2 are closer to another than to the charg es on the nearby edge of\nF1, thus for an identical amount δt2contribute more to Ed.\nFigure 2(b) shows the full plot of Ed(t1,t2)fors= 0.7nm. The above arguments\nappear general. From this figure let us outline three take-aw ay messages. 1. For a given\nt1the minimum of Edof a SyF is found for t2/greaterorsimilart1/2. 2. At this minimum Edis reduced\nby only≈20−30%with respect to a single-layer element of thickness t1considered alone.\n3.Edroughly regains the value of the single layer at the moment co mpensation point\n(t1=t2).\n4This sheds light on results previously noticed empirically , however whose origin and\ngenerality had not been highlighted. Wiese et al. reported that the effective dipolar\nfield anisotropy of a SyF basically scales with the inverse net moment [16],i.e.like the\ninverse strength of Zeeman energy. This suggests that ∆Edis essentially independent of\nthe imbalance of moment, which goes against the widespread b elief that magnetostatic\nenergy nearly vanishes upon moment compensation. Our resul ts clarify and quantify\nthis: the dipolar energy does not differ more than 20-30 % from that of a single layer\nfort2/lessorsimilart1(Figure 2a, dotted line). Saito et al. also reported that the thermal stability\nofCo90Fe10[3]/Ru[0.95]/Co90Fe10[5]is similar to that of Co90Fe10[3]. As explained in\nthe introduction, we recall that thermal stability is deter mined by the energy barrier\nalong the hard axis direction, with respect to the easy axis d irection. In the case of\nanisotropy arising from dipolar energy and an elongated sha pe of the element, this bar-\nrier can be evaluated straightforwardly by calculating onc eEdalong the short edge of\nthe dot, and second along the longedge of the dot. Doing this we explain the findings of\nSaitoet al.., whereas a reduction of 50%would be expected on the basis of compensated\nmoments (the numbers in brackets are thicknesses in nanomet ers). Our calculations may\nalso be applicable to the cross-over of vortex versus single domain in flat disks[ 17] or\nvortex versus transverse domain walls in stripes[ 18], whose scaling law t×a= Cte may\nbe derived qualitatively by equaling the energy of a vortex ∼tand that of a single-\ndomain∼a2t(t/a)(hereV=a2tis the volume, and t/athe demagnetizing coefficient).\nInterestingly Tezuka et al. noticed that there is an optimum ferromagnetic film thick-\nness at which SyAF can obtain a single-domain structure . This minimum (related to\na minimum of demagnetization energy) is found for an imbalanced thickness in good\nquantitative agreement with Figure 2b.\nWith a view to promote the use of accurate magnetostatics for SyF while eliminating\nthe need for numerical evaluation, we derived approximate y et highly accurate expressions\nforEd. Figure 3a shows that to a very good approximation, Edis proportional to the\nwidth of the element (along x) and is independent of its length (along z). This is already\naccurate for a single-layer ( t2= 0), and is very accurate close to the compensation\nt1=t2because edges then behave as lines of dipoles, whose stray fie ld quickly decays\nwith distance ( ∼1/r2). ThusEdboils down to a single line integral along its edge:\n5Energy(10 J)-19\nDot length (nm)200 400 600 800 1000456\n200 400 600 800 1000Energy(10 J)-18\n0123456\nDot length (nm)(a) (b)\nFigure 3: (a) Energy of a single layer (full symbols) and SyF ( open symbols) as a function of dot length,\ni.e.alongz, whilea= 100nm . (b) Energy of a single layer as a function of width (along x, open\nsymbols), and length (along z, full symbols, same curve as in a), while the other in-plane d imension is\nkept constant at 100nm . The lines are linear fits. For both plots the parameters are: Mi= 106A/m,\nt1=t2= 2.5nm,s= 0.7nm.\nEd=Eλ/contintegraldisplay\n(m.n)2ds (4)\n=Eλ/contintegraldisplay|dx|/radicalbig\n1+(∂xf)2(5)\n=Eλ/integraldisplay2π\n0(rsinθ−∂θrcosθ)2\n/radicalbig\n(∂θr)2+r2dθ (6)\nEq. (4) is the general expression, expressed in the following two l ines in cartesian and\npolar coordinates (Figure 4a).Eλis the density of magnetostatic energy per unit length\nof edge, a concept once discussed in the case of single layers [19]. Equations ( 4-6) apply to\nan arbitrary shape of perimeter (not simply rectangles for p risms) by considering the in-\nplane angle ϕbetween magnetization and the normal to the edge. It can be ve rified that\nfor a SAF we have, with an accuracy better than 10%for geometrical parameters relevant\nfor practical cases, i.e.t1,2in the range of 2−10nm andsintherangeof 0.5−1nm:\nEλ≈(1/2)Kdt2(7)\nThe meaning of Eq. ( 7) is straightforward: due to the short range of interaction\n6\u0001\u0002\u0003\u0004\u0005\u0006\u0007 \b\t\n\u0002\u000b\f\r\n\u000e\u000f\f\u0010\u0011\u0012\u0006\n\u0013\u0014\u0014\u0005\u0015\u0006\t\u000e\t\u0016\u0017\u0002\r\u000b\u0014\t\u0001\u0011\u0003\nFigure 4: (a) notations for the calculation of edge energy (b ) integrated magnetostatic energy for various\nshapes. Eis the elliptical integral of the second kind. See text for th e definition of Eλ.\nbetween dipolar lines, the density of dipolar energy is non- zero only in the vicinity of\nthe edges, with a lateral range t. Thus a volume t2is concerned with a line density\nof energy of the order of Kd. Expressions for non-compensated cases (including single\nelements) may also be evaluated. This provides us with analy tical expressions for the\nmagnetostatics of SyFs for the most usual shapes (Figure 4b).\nA scaling law sometimes used as a first guess is based on the poi nt dipole approxima-\ntion. In this framework the energy gained by coupling F1andF2would roughly scale\nwithKda4/t, resulting from two point moments Msa2tinteracting like 1/t3(fors≪t,\nand assuming lateral dimensions of the order of a). The scaling arising from our exact\ncalculation is aEλ∼Kdat2(Eq. ( 7) and Figure 4a). The point dipole approximation is\nthus clearly incorrect with an extra scaling (a/t)3(see Figure 4a) which largely overesti-\nmates the dipolar coupling. This is a general argument for an y flat element, where dipolar\nfields are short-ranged[ 20] and thus the point-dipole approximation is clearly incorr ect.\nTo conclude we derived exact formulas for the magnetostatic s of prism SyF, and sim-\nple yet accurate forms for SyFs of arbitrary shapes. These si mple forms may be used\nstraightforwardly to derive scaling laws for all aspects of SyF physics pertaining with\n7dipolar energy such as thermal stability, coercivity and an isotropy field. Notice that sim-\nilar to the case of single flat elements edge roughness is liab le to reduce significantly dipo-\nlar energy[ 21,22,23], so that the theoretical predictions need to be considered as upper\nbounds to the experimental values. The non-uniformity of ma gnetization is not expected\nto have a significant impact for lateral sizes below a few hund reds of nanometers[ 8].\nWe acknowledge useful discussions with Y. Henry, IPCMS-Str asbourg.\nReferences\n[1] D. Heim, S. S. P. Parkin, US patent 5,465,185, 1995. Magne toresistive spin valve sensor with\nimproved pinned ferromagnetic layer and magnetic recordin g system using the sensor.\n[2] H. Van den Berg, US patent 5,686,838, 1997. Magnetoresis tive sensor having at least a layer system\nand a plurality of measuring contacts disposed thereon, and a method of producing the sensor.\n[3] S. S. P. 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(200 9).\n9" }, { "title": "2006.12553v3.Insights_into_nature_of_a_magnetization_plateau_of_3_d__4_f__coordination_polymer__Dy__2_Cu__2____n__from_a_spin_1_2_Ising_Heisenberg_orthogonal_dimer_chain.pdf", "content": "Condensed Matter Physics, 2020, Vol. 23, No 4, 43708: 1–11\nDOI: 10.5488/CMP.23.43708\nhttp://www.icmp.lviv.ua/journal\nInsights into nature of a magnetization plateau\nof 3𝒅-4𝒇coordination polymer [Dy 2Cu2]𝒏from\na spin-1/2 Ising-Heisenberg orthogonal-dimer chain\nJ. Strečka1, L. Gálisová2, T. Verkholyak3\n1Department of Theoretical Physics and Astrophysics, Faculty of Science, P. J. Šafárik University,\nPark Angelinum 9, 040 01 Košice, Slovakia\n2Institute of Manufacturing Management, Faculty of Manufacturing Technologies with the seat in Prešov,\nTechnical University of Košice, Bayerova 1, 080 01 Prešov, Slovakia\n3Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,\n1 Svientsitskii St., 79011 Lviv, Ukraine\nReceived June 22, 2020, in final form August 13, 2020\nThe ground state and magnetization process of an exactly solved spin- 12Ising-Heisenberg orthogonal-dimer\nchain with two different gyromagnetic factors of the Ising and Heisenberg spins are investigated in detail. It\nis shown that the investigated quantum spin chain exhibits up to seven possible ground states depending\non a mutual interplay of the magnetic field, intra- and inter-dimer coupling constants. More specifically, the\nfrustrated and modulated quantum antiferromagnetic phases are responsible in zero-temperature magneti-\nzation curves for a zero magnetization plateau. The intermediate 1/11- and 5/11-plateaus emerge due to the\nfrustrated and modulated quantum ferrimagnetic phases, while the intermediate 9/11- and 10/11-plateaus\ncan be attributed to the quantum and classical ferrimagnetic phases. It is conjectured that the magnetiza-\ntion plateau experimentally observed in a high-field magnetization curve of 3 𝑑-4𝑓heterobimetallic coordi-\nnation polymer [{Dy(hfac) 2(CH3OH)} 2{Cu(dmg)(Hdmg)} 2]𝑛(H2dmg=dimethylglyoxime; Hhfac =1,1,1,5,5,5-\nhexafluoropentane-2,4-dione) could be attributed to the classical and quantum ferrimagnetic phases.\nKey words: Ising-Heisenberg orthogonal-dimer chain, magnetization plateau, 3 𝑑-4𝑓coordination polymer\n1. Introduction\nThe frustrated spin-1\n2Heisenberg orthogonal-dimer (or dimer-plaquette) chain [1–4] has attracted\nconsiderable attention as it represents one-dimensional counterpart of the famous Shastry-Sutherland\nmodel [5], which is widely studied by virtue of elucidation a peculiar sequence of fractional plateaus\nexperimentally observed in low-temperature magnetization curves of SrCu 2(BO 3)2[6] and rare-earth\ntetraborides RB 4[7]. It has been argued by Schulenburg and Richter [8, 9] that a zero-temperature\nmagnetization curve of the spin-1\n2Heisenberg orthogonal-dimer chain displays an infinite series of\nfractional magnetization plateaus at rational numbers𝑛\n2𝑛¸2=1\n41\n31\n2, whereas the lowermost and\nuppermost plateaus from this series are the widest ones. Another interesting feature of the spin-1\n2\nHeisenberg orthogonal-dimer chain lies in its belonging to a prominent class of flat-band models, for\nwhich low-temperature thermodynamics can be elaborated by making use of an effective lattice-gas\ndescription as extensively discussed by Derzhko and coworkers [10, 11].\nRegrettably,thermodynamicpropertiesofquantumHeisenbergspinmodelsareingeneralinaccessi-\nblebyexactcalculationsatnonzerotemperature.ReplacementofsomeofthequantumHeisenbergspins\nby the classical Ising ones paves the way to an exact solution of the analogous Ising-Heisenberg models\nbyemployingexactmappingtransformationsandthetransfer-matrixmethod[12–14].Forinstanceafew\nexactly solved versions of the spin-1\n2Ising-Heisenberg orthogonal-dimer chain [15–19], to be further\nThis work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution\nof this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.43708-1arXiv:2006.12553v3 [cond-mat.str-el] 18 Jan 2021J. Strečka, L. Gálisová, T. Verkholyak\nFigure 1.(Colour online) (a) A crystal structure of 3 𝑑-4𝑓coordination polymer [Dy 2Cu2]𝑛(see the text\nforafullchemicalformula)adaptedaccordingtocrystallographicdatareportedinreference[21].Large\ncyanballsdeterminecrystallographicpositionsofDy3¸magneticions,whilesmallgreenballsstandfor\ncrystallographic positions of Cu2¸magnetic ions (a coloured scheme for atom labeling is presented in\nthelegend);(b)ThemagneticstructureofthecorrespondingIH-ODC,inwhichDy3¸magneticionsare\ntreated as the Ising spins while Cu2¸magnetic ions are treated as the Heisenberg spins. The coupling\nconstants𝐽I,𝐽0\nIand𝐽HareassignedtotheIsinginter-dimerinteractionbetweenDy3¸andCu2¸magnetic\nions (solid lines), the Ising intra-dimer interaction between Dy3¸magnetic ions (dashed lines) and the\nHeisenberg intra-dimer interaction between Cu2¸magnetic ions (dotted lines), respectively.\nabbreviated as IH-ODC, brought a deeper insight into the magnetization process [15, 17, 19], magne-\ntocaloric effect [15, 17], low-temperature thermodynamics [16–18] and bipartite thermal entanglement\n[16, 19] of this frustrated quantum spin chain. Besides, it was demonstrated that the exact solution for\nthe IH-ODC may serve as a useful starting point for the many-body perturbation treatment of the fully\nHeisenberg counterpart model within a more advanced strong-coupling approach [20].\nAlthough we are currently not aware of any experimental realization of the spin-1\n2Heisen-\nberg orthogonal-dimer chain, it surprisingly turns out that 3 𝑑-4𝑓heterobimetallic coordination poly-\nmer [{Dy(hfac) 2(CH 3OH)} 2{Cu(dmg)(Hdmg)} 2]𝑛(H2dmg=dimethylglyoxime; Hhfac =1,1,1,5,5,5-\nhexafluoropentane-2,4-dione) [21], hereafter abbreviated as [Dy 2Cu2]𝑛, provides an experimental real-\nizationoftheIH-ODC.Infact,thepolymericcompound[Dy 2Cu2]𝑛displaysapeculiarone-dimensional\narchitecturewithregularlyalternatingdimericunitsofDy3¸-Dy3¸andCu2¸-Cu2¸magneticionsstacked\nin an orthogonal fashion with respect to each other as illustrated in figure 1 (a) [21]. It should be\npointed out, moreover, that Dy3¸magnetic ion represents Kramers ion with the ground-state multiplet\n6H152, which is subjected to a relatively strong crystal-field splitting into eight well-separated Kramers\ndoublets[22,23].Inthisregard,themagneticbehaviourofDy3¸magneticionwiththetotalangularmo-\nmentum𝐽=152andtheassociated 𝑔-factor𝑔𝐽=43canbeapproximatedatlowenoughtemperatures\nbytheclassicalIsingspinwiththeeffectivegyromagneticfactor 𝑔Dy=20whenneglectingtheadmixture\nof all excited Kramers dublets [22, 23]. Getting back to a magnetic structure of the polymeric complex\n[Dy 2Cu2]𝑛,whichisschematicallydrawninfigure1(b),theverticaldimerofDy3¸-Dy3¸magneticions\nmay be approximated by a couple of the Ising spins, while the horizontal dimer of Cu2¸-Cu2¸magnetic\nions may be approximated by a couple of the Heisenberg spins.\nThe organization of this article is as follows. In section 2 we describe the studied IH-ODC and\nrecall basic steps of its exact analytical solution. The ground state and magnetization process of the\ninvestigated quantum spin chain are theoretically studied in section 3. The available experimental data\nfor the high-field magnetization curve of the polymeric compound [Dy 2Cu2]𝑛are interpreted by virtue\noftheIH-ODCinsection4.Finally,severalconclusionsandfutureoutlooksarementionedinsection5.\n43708-2Nature of a magnetization plateau of 3 𝑑-4𝑓coordination polymer (Dy 2Cu2)𝑛\n2. Spin-1\n2Ising-Heisenberg orthogonal-dimer chain\nLet us start by introducing the IH-ODC in a magnetic field through the following Hamiltonian [see\nfigure 1 (b) for a schematic illustration]:\nˆH=𝐽H𝑁∑︁\n𝑖=1ˆS1𝑖\u0001ˆS2𝑖¸𝐽0\nI𝑁∑︁\n𝑖=1ˆ𝜎𝑧\n1𝑖ˆ𝜎𝑧\n2𝑖¸𝐽I𝑁∑︁\n𝑖=1\u0002ˆ𝑆𝑧\n1𝑖\u0000ˆ𝜎𝑧\n1𝑖¸ˆ𝜎𝑧\n2𝑖\u0001¸ˆ𝑆𝑧\n2𝑖\u0000ˆ𝜎𝑧\n1𝑖¸1¸ˆ𝜎𝑧\n2𝑖¸1\u0001\u0003\n\u0000𝑔H𝜇B𝐵𝑁∑︁\n𝑖=1\u0000ˆ𝑆𝑧\n1𝑖¸ˆ𝑆𝑧\n2𝑖\u0001\u0000𝑔I𝜇B𝐵𝑁∑︁\n𝑖=1\u0000ˆ𝜎𝑧\n1𝑖¸ˆ𝜎𝑧\n2𝑖\u0001 (2.1)\nwhere ˆS1¹2º𝑖\u0011 ¹ˆ𝑆𝑥\n1¹2º𝑖ˆ𝑆𝑦\n1¹2º𝑖ˆ𝑆𝑧\n1¹2º𝑖ºdenote standard spin-1\n2operators ascribed to Cu2¸magnetic\nions approximated by the notion of quantum Heisenberg spins and ˆ𝜎𝑧\n1¹2º𝑖refer to the𝑧-component of\nthe standard spin-1\n2operators ascribed to Dy3¸magnetic ions approximated by the notion of classical\nIsing spins. The coupling constants 𝐽Hand𝐽0\nIdetermine a strength of the Heisenberg and Ising intra-\ndimer interactions within the horizontal Cu2¸-Cu2¸and vertical Dy3¸-Dy3¸dimers, respectively, while\nthe coupling constant 𝐽Idetermines a strength of the Ising inter-dimer interaction between the nearest-\nneighbour Cu2¸and Dy3¸magnetic ions. The Zeeman’s terms ℎH=𝑔H𝜇B𝐵andℎI=𝑔I𝜇B𝐵take into\naccount a magnetostatic energy of magnetic moments relating to the Heisenberg and Ising spins in\npresenceoftheexternalmagneticfield 𝐵,whichdifferduetodifferentgyromagneticfactors 𝑔Hand𝑔Iof\nCu2¸and Dy3¸magnetic ions, respectively.\nIt is worthwhile to remark that the partition function, Gibbs free energy and magnetization of the\nIH-ODC defined through the Hamiltonian (2.1) was exactly calculated under the periodic boundary\nconditions ˆ𝜎𝑧\n1¹2º𝑁¸1\u0011ˆ𝜎𝑧\n1¹2º1inourprecedingpaper[19].Thecalculationprocedureusedtakesadvan-\ntage of splitting the total Hamiltonian (2.1) into commuting six-spin cluster Hamiltonians involving one\nhorizontalCu2¸-Cu2¸dimerandtwoenclosingverticalDy3¸-Dy3¸dimers,whichallowastraightforward\nfactorizationofthepartitionfunctionintoaproductoftherespectiveBoltzmannfactors.Aftertracingout\nspin degrees of freedom of the horizontal Cu2¸-Cu2¸dimer (Heisenberg dimer), the partition function\nis in fact expressed in terms of four-by-four transfer matrix depending on spin states of two adjacent\nvertical Dy3¸-Dy3¸dimers (Ising dimers) and the whole magnetothermodynamics can be elaborated\nby making use of the transfer-matrix method (the readers interested in further calculation details are\nreferredtoreference[19]).However,allnumericalresultspresentedinreference[19]wererestrictedjust\ntotheparticularcase ℎH=ℎI,whichcorrespondstosettingthesameLandé 𝑔-factorsforCu2¸andDy3¸\nmagnetic ions which is contrary to the expected (typical) values of the gyromagnetic factors 𝑔H\u00192for\nCu2¸magnetic ions and 𝑔I\u001920for Dy3¸magnetic ions. Therefore, in the present article we adapt the\nexact solution for the IH-ODC reported in reference [19] in order to investigate the effect of different\n‘localmagneticfields’ ℎH≠ℎIarisingfromthedifferenceofthegyromagneticfactorsofCu2¸andDy3¸\nmagnetic ions 𝑔H≠𝑔I.\n3. Theoretical results\nIn this section we examine in detail the ground state and magnetization process of the IH-ODC by\nassuming the gyromagnetic factors 𝑔H=2and𝑔I=20, which are close to typical values of the Landé\n𝑔-factors for Cu2¸and Dy3¸magnetic ions, respectively. To reduce the number of free parameters, a\nsizeofthecouplingconstant 𝐽I¡0correspondingtotheantiferromagneticIsinginter-dimerinteraction\nbetween Dy3¸-Cu2¸magnetic ions serves as an energy unit when defining a relative strength of the\nHeisenberg intra-dimer interaction 𝐽H𝐽Iwithin the horizontal dimers, the Ising intra-dimer interaction\n𝐽0\nI𝐽Iwithin the vertical dimers and the magnetic field 𝜇B𝐵𝐽I.\n43708-3J. Strečka, L. Gálisová, T. Verkholyak\n3.1. Ground state\nThe IH-ODC with the gyromagnetic factors 𝑔H=2and𝑔I=20may display, in presence of the\nmagnetic field, up to seven different ground states depending on a mutual interplay of the coupling\nconstants𝐽H𝐽I,𝐽0\nI𝐽Iand the magnetic field 𝜇B𝐵𝐽I. The typical ground-state phase diagrams are\nreportedinfigure2inthe 𝐽H𝐽I\u0000𝜇B𝐵𝐽Iparameterplaneforfourrepresentativevaluesoftheinteraction\nratio𝐽0\nI𝐽I=\u000005,05,20and25. It is quite evident from figure 2 (a) that the ground-state phase\ndiagram for the particular case with the ferromagnetic Ising intra-dimer coupling 𝐽0\nI𝐽I=\u000005involves\njustfourdifferentgroundstates,morespecifically,theclassicalferrimagneticphase jIiwithanantiparallel\nspin arrangement of the Ising and Heisenberg dimers characterized through the following eigenvector\nand corresponding eigenenergy\njIi=𝑁Ö\n𝑖=1\f\f\f\"\n\"E\n𝑖\nj##i𝑖 𝐸 I=𝑁\n4\u0000𝐽H¸𝐽0\nI\u00004𝐽I\u00004ℎI¸4ℎH\u0001 (3.1)\nthemodulatedquantumantiferromagneticphase jIIiwithalternatingcharacteroftheIsingdimersanda\nsinglet-like state of the Heisenberg dimers\njIIi=𝑁2Ö\n𝑖=1\f\f\f\"\n\"E\n2𝑖\u00001\n\u0010\nsin𝜑2j\"#i 2𝑖\u00001\u0000cos𝜑2j#\"i 2𝑖\u00001\u0011\n\n\f\f#\n#E\n2𝑖\n\u0010\ncos𝜑2j\"#i 2𝑖\u0000sin𝜑2j#\"i 2𝑖\u0011\n\n𝐸II=\u0000𝑁\n4\u0012\n𝐽H¸2√︃\n4𝐽2\nI¸𝐽2\nH\u0000𝐽0\nI\u0013\n 𝜑 2=1\n2arctan\u0012𝐽H\n2𝐽I\u0013\n (3.2)\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48\n/s40/s97/s41/s57/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73\n/s40/s49/s56\n/s66/s41\n/s48/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73/s73\n/s40/s48\n/s66/s41/s49/s48/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73/s73/s73\n/s40/s50/s48\n/s66/s41/s115/s97/s116/s117/s114/s97/s116/s105/s111/s110/s124/s73/s86\n/s40/s50/s50\n/s66/s41\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48\n/s124/s86/s73/s66/s66 /s32/s47/s32/s74\n/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32\n/s66/s66 /s32/s47 /s32/s74\n/s73\n/s74\n/s72/s32 /s47/s32/s74\n/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s74\n/s72/s32 /s47/s32/s74\n/s73/s40/s98/s41/s40/s49/s48\n/s66/s41/s57/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73\n/s40/s49/s56\n/s66/s41\n/s48/s45/s112/s108/s97/s116/s101/s97/s117\n/s40/s48\n/s66/s41/s124/s73/s73/s49/s48/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73/s73/s73\n/s40/s50/s48\n/s66/s41/s115/s97/s116/s117/s114/s97/s116/s105/s111/s110/s124/s73/s86\n/s40/s50/s50\n/s66/s41\n/s124/s86\n/s40/s48\n/s66/s41\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48\n/s57/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s66/s66 /s32/s47/s32/s74\n/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32\n/s66/s66 /s32/s47 /s32/s74\n/s73\n/s74\n/s72/s32 /s47/s32/s74\n/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s74\n/s72/s32 /s47/s32/s74\n/s73/s40/s99/s41/s124/s73\n/s40/s49/s56\n/s66/s41/s115/s97/s116/s117/s114/s97/s116/s105/s111/s110/s124/s73/s86\n/s40/s50/s50\n/s66/s41\n/s49/s48/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73/s73/s73\n/s40/s50/s48\n/s66/s41\n/s48/s45/s112/s108/s97/s116/s101/s97/s117/s124/s86\n/s40/s48\n/s66/s41/s83/s65/s70/s53/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s86/s73\n/s40/s49/s48\n/s66/s41\n/s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48\n/s83/s65/s70\n/s40/s100/s41/s57/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73\n/s40/s49/s56\n/s66/s41/s115/s97/s116/s117/s114/s97/s116/s105/s111/s110/s124/s73/s86\n/s40/s50/s50\n/s66/s41\n/s49/s48/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s73/s73/s73\n/s40/s50/s48\n/s66/s41\n/s48/s45/s112/s108/s97/s116/s101/s97/s117/s124/s86\n/s40/s48\n/s66/s41/s83/s65/s70/s53/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s86/s73\n/s40/s49/s48\n/s66/s41\n/s83/s65/s70/s49/s47/s49/s49/s45/s112/s108/s97/s116/s101/s97/s117/s124/s86/s73/s73\n/s40/s50\n/s66/s41\nFigure 2.Asemi-logarithmicplotoftheground-statephasediagramoftheIH-ODCwiththegyromagnetic\nfactors𝑔H=2and𝑔I=20in the𝐽H𝐽I\u0000𝜇B𝐵𝐽Iplane for four representative values of the interaction\nratio: (a)𝐽0\nI𝐽I=\u000005; (b)𝐽0\nI𝐽I=05; (c)𝐽0\nI𝐽I=20; (d)𝐽0\nI𝐽I=25. The numbers in round brackets\ndetermine the total magnetization in units of Bohr magneton 𝜇Band the fractions represent its relative\nsize with respect to the saturation magnetization.\n43708-4Nature of a magnetization plateau of 3 𝑑-4𝑓coordination polymer (Dy 2Cu2)𝑛\nthequantumferrimagneticphase jIIIiwithfullypolarizedIsingdimersandaperfectsinglet-dimerstate\nof the Heisenberg dimers\njIIIi=𝑁Ö\n𝑖=1\f\f\f\"\n\"E\n𝑖\n1p\n2¹j\"#i𝑖\u0000j#\"i𝑖º 𝐸 III=\u0000𝑁\n4\u00003𝐽H\u0000𝐽0\nI¸4ℎI\u0001 (3.3)\nand finally, the saturated paramagnetic phase jIViwith a fully polarized nature of the Ising as well as\nHeisenberg dimers\njIVi=𝑁Ö\n𝑖=1\f\f\f\"\n\"E\n𝑖\nj\"\"i𝑖 𝐸 IV=𝑁\n4\u0000𝐽H¸𝐽0\nI¸4𝐽I\u00004ℎI\u00004ℎH\u0001 (3.4)\nNote that all eigenvectors are written as a tensor product over states of the vertical Ising dimers (former\nstate vector) and the horizontal Heisenberg dimers (the latter state vector), respectively. At low enough\nmagnetic fields, the classical ferrimagnetic phase jIidominates in the parameter region with the fer-\nromagnetic Heisenberg intra-dimer coupling 𝐽H0, while the modulated quantum antiferromagnetic\nphasejIIiand the quantum ferrimagnetic phase jIIIidominate in the parameter region with the antifer-\nromagnetic Heisenberg intra-dimer coupling 𝐽H¡0. Of course, the saturated paramagnetic phase jIVi\nrepresentstheactualgroundstateathighenoughmagneticfieldsregardlessofwhethertheferromagnetic\nor antiferromagnetic Heisenberg intra-dimer coupling is considered.\nOn the other hand, the ground-state phase diagram of the IH-ODC with a relatively weak antiferro-\nmagnetic Ising intra-dimer coupling additionally involves the other two ground states [see figure 2 (b)\nfor𝐽0\nI𝐽I=05], which could be identified as the frustrated quantum antiferromagnetic phase jViwith\na two-fold degenerate antiferromagnetic state of the Ising dimers and a perfect singlet-dimer state of the\nHeisenberg dimers\njVi=𝑁Ö\n𝑖=1\f\f\f\"\n#E\n𝑖\u0010\nor\f\f#\n\"E\n𝑖\u0011\n\n1p\n2¹j\"#i𝑖\u0000j#\"i𝑖º 𝐸V=\u0000𝑁\n4\u00003𝐽H¸𝐽0\nI\u0001 (3.5)\nand the highly degenerate modulated quantum ferrimagnetic phase jVIiwith alternating ferro-\nantiferromagnetic character of the Ising dimers and a singlet-like state of the Heisenberg dimers\njVIi=𝑁2Ö\n𝑖=1\f\f\f\"\n\"E\n2𝑖\u00001\n\u0010\nsin𝜑6j\"#i 2𝑖\u00001\u0000cos𝜑6j#\"i 2𝑖\u00001\u0011\n\n\f\f\"\n#E\n2𝑖\u0010\nor\f\f#\n\"E\n2𝑖\u0011\n\n\u0010\ncos𝜑6j\"#i 2𝑖\u0000sin𝜑6j#\"i 2𝑖\u0011\n\n𝐸VI=\u0000𝑁\n4\u0012\n𝐽H¸2√︃\n𝐽2\nI¸𝐽2\nH¸2ℎI\u0013\n 𝜑 6=1\n2arctan\u0012𝐽H\n𝐽I\u0013\n (3.6)\nItisquiteclearfromfigure2(b)thattheground-statephasediagramhasnotchangedintheparameterspace\nwith the ferromagnetic Heisenberg intra-dimer coupling 𝐽H0, while two novel ground states emerge\natlow(uptomoderate)magneticfieldsintheparameterspacewithsufficientlystrongantiferromagnetic\nHeisenberg intra-dimer coupling 𝐽H¡0. It is noteworthy that the frustrated quantum antiferromagnetic\nphasejVisuppresses the modulated quantum antiferromagnetic phase jIIiupon increasing the relative\nstrengthoftheantiferromagneticIsingintra-dimercoupling 𝐽0\nI𝐽Iuntilthislattergroundstatecompletely\nvanishes from the ground-state phase diagram as evidenced by figure 2 (c) for 𝐽0\nI𝐽I=20. Last but not\nleast, the sufficiently strong antiferromagnetic Ising intra-dimer coupling 𝐽0\nI𝐽I¡2may additionally\ncauseanupriseofthenovelgroundstatealsointheparameterspacewiththeferromagneticHeisenberg\nintra-dimercoupling 𝐽H0andsmallenoughmagneticfields,whichcouldbeidentifiedasthefrustrated\nferrimagneticphase jVIIiwithatwo-folddegenerateantiferromagneticstateoftheIsingdimersandfully\npolarized Heisenberg dimers given by\njVIIi=𝑁Ö\n𝑖=1\f\f\f\"\n#E\n𝑖\u0010\nor\f\f#\n\"E\n𝑖\u0011\n\nj\"\"i𝑖 𝐸 VII=𝑁\n4\u0000𝐽H\u0000𝐽0\nI\u00004ℎH\u0001 (3.7)\n43708-5J. Strečka, L. Gálisová, T. Verkholyak\n3.2. Zero- and low-temperature magnetization curves\nThe gapped ground states (3.1)–(3.7) should be manifested in zero- and low-temperature magneti-\nzation curves of the IH-ODC as intermediate plateaus emergent at fractional values of the saturation\nmagnetization.Byconsideringspecificvaluesofthegyromagneticfactors 𝑔H=2and𝑔I=20oneshould\naccordinglydetectazeromagnetizationplateauinastabilityregionofthemodulatedquantumantiferro-\nmagneticphasejIIiandthefrustratedquantumantiferromagneticphase jVigivenbyequations(3.2)and\n(3.5),theintermediate1/11-plateaumayemergeduetothefrustratedferrimagneticphase jVIIigivenby\nequation (3.7), the intermediate 5/11-plateau can be ascribed to the modulated quantum ferrimagnetic\nphasejVIigiven by equation (3.6), the intermediate 9/11-plateau corresponds to the classical ferrimag-\nneticphasejIigivenbyequation(3.1)andfinally,theintermediate10/11-plateaurelatestothequantum\nferrimagneticphase jIIIigivenbyequation(3.3).Fromthisperspective,theIH-ODCexhibitsasubstan-\ntial diversity of the magnetization curves with nine possible magnetization scenarios as exemplified in\nfigure 3.\nInagreementwiththereportedground-statephasediagrams,onefindsthreedifferentmagnetization\nscenarios with either a single field-driven phase transition jIi\u0000jIVi[figure 3 (a)], three field-induced\nphase transitionsjIIi\u0000jIi\u0000jIIIi\u0000jIVi[figure 3 (b)] or two field-driven phase transitions jIIi\u0000jIIIi\u0000jIVi\n/s49/s69/s45/s51 /s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s66 /s32/s66/s32 /s47/s32/s74\n/s73/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32\n/s66 /s32/s66/s32 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/s49 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s74\n/s73/s39/s32 /s47/s32/s74\n/s73/s32/s61/s32/s50/s46/s53\n/s74\n/s72/s32 /s47/s32/s74\n/s73/s32/s61/s32/s48/s46/s53\n/s40/s105/s41/s49/s48/s47/s49/s49/s57/s47/s49/s49\n/s53/s47/s49/s49\nFigure 3.(Colouronline)Asemi-logarithmicplotofallrepresentativeisothermalmagnetizationcurvesof\ntheIH-ODCwiththegyromagneticfactors 𝑔H=2and𝑔I=20calculatedatthreedifferenttemperatures\n𝑘B𝑇𝐽I=0(red solid lines), 0.05 (blue dashed lines) and 0.15 (green dotted lines). Different panels\ndemonstrate a diversity of the magnetization process depending basically on a choice of the coupling\nconstants𝐽H𝐽Iand𝐽0\nI𝐽Iquoted in the respective panels.\n43708-6Nature of a magnetization plateau of 3 𝑑-4𝑓coordination polymer (Dy 2Cu2)𝑛\n[figure3(c)]ontheassumptionthattheIsingintra-dimercouplingisferromagnetic 𝐽0\nI0.Theparticular\ncase with a weak antiferromagnetic Ising intra-dimer coupling 𝐽0\nI𝐽I&0displays the other three types\nof magnetization processes due to the presence of the phases jViand/orjVIi: one type involves a\nsequence of four field-driven phase transitions jIIi\u0000jVIi\u0000jIi\u0000jIIIi\u0000jIVi[figure 3 (d)] and the other two\ntypes include a sequence of three field-induced phase transitions jIIi\u0000jVIi\u0000jIIIi\u0000jIVi[figure 3 (e)]\norjVi\u0000jVIi\u0000jIIIi\u0000jIVi[figure 3 (f)], respectively. Finally, the specific case with a sufficiently strong\nantiferromagneticIsingintra-dimercoupling 𝐽0\nI𝐽I\u001d0mayexhibitthreeothermagnetizationscenarios,\nwhichconsecutivelyincludeasequenceoftwofield-drivenphasetransitions jVIIi\u0000jIi\u0000jIVi[figure3(g)],\nfive field-induced phase transitions jVi\u0000jVIIi\u0000jVIi\u0000jIi\u0000jIIIi\u0000jIVi[figure 3 (h)] or four field-induced\nphase transitionsjVi\u0000jVIi\u0000jIi\u0000jIIIi\u0000jIVi[figure 3 (i)], respectively.\nNow,afewcommentsareinorderasfarasathermalstabilityoftheindividualmagnetizationplateaus\nisconcerned.Itisquiteobviousfromfigure3thatsomeintermediateplateausarequiterobustwithrespect\ntothermalfluctuationsandtheycanbeclearlydiscernedintherespectiveisothermalmagnetizationcurves\natlow(𝑘B𝑇𝐽I=005)orevenatmoderate( 𝑘B𝑇𝐽I=015)temperatures.Thisistypicallythecaseforthe\nzero magnetization plateau and for the intermediate 9/11- and 10/11-plateaus emergent at higher values\nofmagnetization.Ontheotherhand,theintermediate1/11-and5/11-plateausemergentatsmallervalues\nofthemagnetizationaretypicallysubjectedtoaconsiderablesmoothinguponincreasingthetemperature\nandtheycannotbeclearlydiscernedintherespectiveisothermalmagnetizationcurvesevenatarelatively\nlow temperature ( 𝑘B𝑇𝐽I=005). This striking finding can be related to a substantial difference of the\nenergygapoftheindividualgroundstatesascribedtotherelevantmagnetizationplateaus.Asamatterof\nfact, the gapped ground states, for instance the classical and quantum ferrimagnetic phases jIiandjIIIi,\nduetotheiruniquenaturepossessahighenergygapinanexcitationspectrum.Onthecontrary,thehighly\ndegenerate ground states, such as the frustrated ferrimagnetic phase jVIIiand the modulated quantum\nferrimagnetic phase jVIi, possess only a tiny energy gap and hence, they are subjected to a substantial\nthermal smoothing owing to their macroscopic degeneracies.\n4. Magnetization curve of the polymeric compound [Dy 2Cu2]𝒏\nIn this part we compare available experimental data for the magnetization curve of the 3 𝑑-4𝑓\nheterobimetallic coordination polymer [Dy 2Cu2]𝑛measured in pulsed magnetic fields up to 10 T at\ntemperature 𝑇=05K [21] with a relevant theoretical prediction based on the IH-ODC. It should be\nmentioned that the magnetization data recorded in pulsed magnetic fields show a small hysteresis in\na low-field region 𝐵.05T, which was ignored for simplicity because it is beyond the scope of the\nintroduced IH-ODC. It actually follows from the inset of figure 4 (a) of reference [21] that the magnetic\nhysteresis basically depends on a field scan rate and it may be thus attributed to a quantum tunneling of\nthemagnetizationtypicallyobservedinsingle-chainmagnetsduetolevelcrossingamongexcitedstates.\nItisworthnoticingthatallfivemodelparametersoftheIH-ODCdefinedthroughtheHamiltonian(2.1)\nwere supposed to be free fitting parameters in order to get the best theoretical fit of the experimental\nmagnetization data (see figure 4 for a comparison). The best theoretical fit was accordingly obtained for\nthefollowingsetofthemodelparameters: 𝐽I𝑘B=802K,𝐽0\nI𝑘B=1735K,𝐽H𝑘B=173K,𝑔H=228\nand𝑔I=1854.\nThe reported value of the gyromagnetic factor 𝑔H=228is quite typical for Cu2¸magnetic ions,\nwhile the other reported value 𝑔I=1854is only by a few percent (cca. 7%) lower than the value\n𝑔Dy=20theoretically expected for the effective Ising-spin description of Dy3¸magnetic ions with the\ntotal angular momentum 𝐽=152and the respective 𝑔-factor𝑔𝐽=43[22, 23]. It should be pointed\nout, moreover, that the values of the Ising inter- and intra-dimer coupling constants 𝐽Iand𝐽0\nIshould be\nalso rescaled by the factors of 15 and 225 in order to get true values of the coupling constants between\nDy3¸-Cu2¸and Dy3¸-Dy3¸magnetic ions when passing away from the effective Ising-spin description\nof Dy3¸magnetic ions. The actual values of the three considered coupling constants consequently read:\n𝐽Dy-Cu𝑘B=053K,𝐽Dy-Dy𝑘B=008K and𝐽Cu-Cu𝑘B=173K, whereas the predicted values of the\ncouplingconstants 𝐽Dy-Cuand𝐽Dy-Dyfallintoareasonablerangeforthecouplingconstantsbetween3 𝑑-4𝑓\nand 4𝑓-4𝑓magnetic ions, respectively [22, 23]. Note, furthermore, that the former exchange constant is\ncomparable withthe meanvalue ofthe exchangecoupling 𝐽Dy-Cu𝑘B=048K, whichcan becalculated\n43708-7J. Strečka, L. Gálisová, T. Verkholyak\nFigure 4. (Colour online) A comparison between the magnetization curve of the polymeric compound\n[Dy2Cu2]𝑛measured in pulsed magnetic fields up to 10 T at temperature 𝑇=05K (a black solid line\nwithopencirclesadaptedfromreference[21])andthebesttheoreticalfitobtainedbyusingoftheIH-ODC\nwith the following fitting set of the parameters: 𝐽I𝑘B=802K,𝐽0\nI𝑘B=1735K,𝐽H𝑘B=173K,\n𝑔H=228and𝑔I=1854. The theoretical results for the isothermal magnetization curves are also\npresented for very low temperature 𝑇=005K (light blue dotted line) and slightly higher temperature\n𝑇=10K(reddashedline)inadditiontothetemperature 𝑇=05K(greensolidline)correspondingto\nthe displayed experimental data.\nfromthecouplingconstants 𝐽A𝑘B=0895Kand𝐽B𝑘B=0061Kassignedpreviouslytotwodifferent\nexchangepathwaysbetweenDy3¸-Cu2¸magneticionswithinthesimplifiedtetranuclearmodel[21].Itis\nworthwhile to remark that the reported value for the coupling constant 𝐽Cu-Cu𝑘B=173K is relatively\nsmallwithrespecttotheshortestdistancebetweenCu2¸-Cu2¸magneticions,butthissmallvaluecanbe\nattributed to a rather inefficient exchange pathway involving a double-oxygen bridge between magnetic\n(3𝑑𝑥2-𝑦2) and nonmagnetic ( 3𝑑𝑧2) orbitals of Cu2¸magnetic ions in an elongated square-pyramidal\nenvironment [see figure 1 (a)].\nThe predicted values of the coupling constants of the IH-ODC are consistent with the following\nvalues of the interaction ratio 𝐽0\nI𝐽I\u001922and𝐽H𝐽I\u001902, which should cause a magnetization sce-\nnario quite analogous to that shown in figure 3 (i) with a sequence of four field-driven phase transitions\njVi\u0000jVIi\u0000jIi\u0000jIIIi\u0000jIVi. A low-field part of the magnetization curve should accordingly display two\nnarrow plateaus at zero and approximately 5/11 of the saturation magnetization (note that the gyro-\nmagnetic factors slightly deviate from the ideal values 𝑔H=20and𝑔I=200), which are, however,\ncompletely smeared out by thermal fluctuations at temperatures as low as 𝑇&05K due to tiny energy\ngaps of the phases jViandjVIi. As a matter of fact, the zero magnetization plateau corresponding\nto the frustrated quantum antiferromagnetic phase jViis restricted to the magnetic fields smaller than\n017T, while the 5/11-plateau related to the modulated quantum ferrimagnetic phase jVIiis limited to\nthe magnetic-field range 017\u0000028T.\nOntheotherhand,thehigh-fieldpartofthemagnetizationcurveshouldexhibittwowiderintermediate\nplateaus roughly at 9/11- and 10/11 of the saturation magnetization. The 9/11-plateau pertinent to the\nclassical ferrimagnetic phase jIiis stable within the magnetic-field range 028\u0000412T, while the 10/11-\nplateau ascribed to the quantum ferrimagnetic phase jIIIiis stable within the magnetic-field range\n412\u0000637T. In this regard, it seems quite puzzling that the field-driven transition between the classical\nand quantum ferrimagnetic phases cannot be evidently seen from the magnetization data recorded for\nthe polymeric compound [Dy 2Cu2]𝑛at a relatively low temperature 𝑇=05K as both ground states jIi\nandjIIIiwithasubstantialenergygapshouldbequiteresistantwithrespecttoathermalsmoothing.Itis\nplausibletoconjecturetwopossiblereasonsforthediscrepancybetweentheexperimentalmagnetization\ndatarecordedattemperature 𝑇=05Kandtherespectivetheoreticalfit:eitherthesampleofthepolymeric\ncompound[Dy 2Cu2]𝑛wasnotkeptduringthemagnetizationprocessunderaperfectisothermalcondition\n43708-8Nature of a magnetization plateau of 3 𝑑-4𝑓coordination polymer (Dy 2Cu2)𝑛\nsince small heating (e.g. due to the magnetocaloric effect) could resolve this discrepancy as evidenced\nby a theoretical curve calculated for a slightly higher temperature 𝑇=10K, or the discrepancy is of\nintrinsic origin and it comes from two different exchange pathways between Dy3¸-Cu2¸magnetic ions\nneglected within the investigated quantum spin chain.\n5. Conclusion\nIn the present article we have scrupulously investigated diverse ground-state phase diagrams and\nmagnetization curves of the IH-ODC by assuming two different gyromagnetic factors of the Ising\nand Heisenberg spins. The investigated quantum spin chain including two different Landé 𝑔-factors\nwas inspired by the polymeric coordination compound [Dy 2Cu2]𝑛, whose magnetic structure shows a\npeculiar one-dimensional architecture with regularly alternating dimeric units of Dy3¸-Dy3¸and Cu2¸-\nCu2¸magnetic ions stacked in an orthogonal fashion [21]. The vertical dimer of Dy3¸-Dy3¸magnetic\nionswasapproximatedbyacoupleoftheIsingspins,whilethehorizontaldimerofCu2¸-Cu2¸magnetic\nions was approximated by a couple of the Heisenberg spins.\nIt has been found that the IH-ODC exhibits a rich variety of the classical and quantum ground\nstates, which, besides the fully saturated paramagnetic phase emergent at sufficiently high magnetic\nfields,involvesixmoregroundstates:thefrustratedandmodulatedquantumantiferromagneticphase,the\nfrustratedandmodulatedquantumferrimagneticphase,aswellas,thequantumandclassicalferrimagnetic\nphase. These remarkable ground states are responsible for the presence of magnetization plateaus in\nzero- and low-temperature magnetization curves, which are manifested at 0, 1/11, 5/11, 9/11 and/or\n10/11ofthesaturationmagnetization.Astabilityoftheintermediatemagnetizationplateauswithrespect\nto thermal fluctuations was investigated in detail. It has been verified that the rather narrow 1/11-\nand 5/11-plateaus ascribed to the frustrated and modulated quantum ferrimagnetic phases with a high\nmacroscopicdegeneracyandatinyenergygapareeasilydestroyeduponasmallincreaseoftemperature,\nwhile the relatively wide 9/11- and 10/11-plateaus ascribed to the nondegenerate classical and quantum\nferrimagneticphaseswitharobustenergygaparequiteresistantwithrespecttothermalfluctuationsand\nmay be also discernible at relatively low (up to moderate) temperatures.\nConsequently, we have successfully applied the IH-ODC with two different gyromagnetic factors\nof the Ising and Heisenberg spins for a theoretical modelling of the high-field magnetization data\nrecordedpreviouslyforthepolymericcoordinationcompound[Dy 2Cu2]𝑛atasufficientlylowtemperature\n𝑇=05K [21]. The respective low-temperature magnetization curve evidently displays intermediate\nmagnetizationplateau(s),whichstartsatarelativelyhighvalueofthemagnetizationbeingapproximately\n78%ofthesaturatedvalue.ThebesttheoreticalfitoftheavailableexperimentaldatabasedontheIH-ODC\nsuggeststhatthemagnetizationplateau(s)observedexperimentallycouldbeascribedtotheclassicaland\nquantum ferrimagnetic phases given by equations (3.1) and (3.3) inherent to the intermediate 9/11- and\n10/11-plateau, respectively. It should be mentioned, however, that the magnetization data measured for\nthepolymericcomplex[Dy 2Cu2]𝑛atlowenoughtemperature 𝑇=05Kdonotprovideanyexperimental\nevidencefortherelevantfield-inducedphasetransitionbetweentheclassicalandquantumferrimagnetic\nphase [21]. This discrepancy could be resolved either by a small deviation from a perfect isothermal\ncondition during the experimental measurement caused by heating of the sample (e.g., due to the\nmagnetocaloriceffect)oritmayrelatetotheoversimplifiednatureoftheintroducedIH-ODCneglecting\ntwo structurally inequivalent exchange pathways between Dy3¸-Cu2¸magnetic ions. In our future work\nwe plan to examine the magnetization process of the IH-ODC taking into consideration two different\nexchange constants between Dy3¸-Cu2¸magnetic ions in order to clarify this issue.\nAcknowledgements\nTheauthorsaredeeplyindebtedtoOlegDerzhkoforalotofinsightfulandinspiringscientificdiscus-\nsions, from which they have substantially benefited over their whole careers. This work was financially\nsupportedbySlovakResearchandDevelopmentAgencyprovidedunderthecontractNo.APVV-16-0186\n43708-9J. Strečka, L. Gálisová, T. Verkholyak\nand by The Ministry of Education, Science, Research and Sport of the Slovak Republic provided under\nthe grant No. VEGA 1/0105/20.\nReferences\n1. Ivanov N.B., Richter J., Phys. Lett. A, 1997, 232, 308–312, doi:10.1016/S0375-9601(97)00374-5.\n2. Richter J., Ivanov N.B., Schulenburg J., J. Phys.: Condens. Matter, 1998, 10, 3635–3649,\ndoi:10.1088/0953-8984/10/16/015.\n3. Koga A., Okunishi K., Kawakami N., Phys. Rev. B, 2000, 62, 5558–5563, doi:10.1103/PhysRevB.62.5558.\n4. Miyahara Sh., In: Introduction to Frustrated Magnetism, Lacroix C., Mendels Ph., Mila F. (Eds.), Springer\nSeries in Solid-State Sciences, Vol. 164, Springer-Verlag, Berlin, Heidelberg, 2011, 513–536.\n5. Shastry B.S., Sutherland B., Physica B ¸C, 1981, 108, 1069–1070, doi:10.1016/0378-4363(81)90838-X.\n6. Matsuda Y.H., Abe N., Takeyama S., Kageyama H., Corboz P., Honecker A., Manmana S.R., Foltin G.R.,\nSchmidt K.P., Mila F., Phys. Rev. Lett., 2013, 111, 137204, doi:10.1103/PhysRevLett.111.137204.\n7. Gabáni S., Flachbart K., Siemensmeyer K., Mori T., J. Alloys Compd., 2020, 821, 153201,\ndoi:10.1016/j.jallcom.2019.153201.\n8. Schulenburg J., Richter J., Phys. Rev. B, 2002, 65, 054420, doi:10.1103/PhysRevB.65.054420.\n9. Schulenburg J., Richter J., Phys. Rev. B, 2002, 66, 134419, doi:10.1103/PhysRevB.66.134419.\n10. Derzhko O., Richter J., Eur. Phys. J. B, 2006, 52, 23–36, doi:10.1140/epjb/e2006-00273-y.\n11. Derzhko O., Richter J., Maksymenko M., Int. J. Mod. Phys. B, 2015, 29, 1530007,\ndoi:10.1142/S0217979215300078.\n12. Fisher M.E., Phys. Rev., 1959, 113, 969–981, doi:10.1103/PhysRev.113.969.\n13. Rojas O., Valverde J.S., de Souza S.M., Physica A, 2009, 388, 1419–1430, doi:10.1016/j.physa.2008.12.063.\n14. Strečka J., Phys. Lett. A, 2010, 374, 3718–3722, doi:10.1016/j.physleta.2010.07.030.\n15. Ohanyan V., Honecker A., Phys. Rev. B, 2012, 86, 054412, doi:10.1103/PhysRevB.86.054412.\n16. Paulinelli H.G., de Souza S.M., Rojas O., J. Phys.: Condens. Matter, 2013, 25, 306003,\ndoi:10.1088/0953-8984/25/30/306003.\n17. Verkholyak T., Strečka J., Phys. Rev. B, 2013, 88, 134419, doi:10.1103/PhysRevB.88.134419.\n18. Verkholyak T., Strečka J., Acta Phys. Pol. A, 2014, 126, 22–23, doi:10.12693/APhysPolA.126.22.\n19. Gálisová L., Strečka J., Verkholyak T., Havadej S., Physica E, 2021, 125, 114089,\ndoi:10.1016/j.physe.2020.114089.\n20. Verkholyak T., Strečka J., Phys. Rev. B, 2016, 94, 144410, doi:10.1103/PhysRevB.94.144410.\n21. Okazawa A., Nogami T., Nojiri H., Ishida T., Chem. Mater., 2008, 20, 3110–3119, doi:10.1021/cm703530n.\n22. De Jongh L.J., Miedema A.R., Adv. Phys., 1974, 23, 1–260, doi:10.1080/00018739700101558.\n23. Jensen J., Mackintosh A.R., Rare Earth Magnetism, Oxford University Press, Oxford, 1991.\n43708-10Nature of a magnetization plateau of 3 𝑑-4𝑓coordination polymer (Dy 2Cu2)𝑛\nПояснення природи плато намагнiчення 3d-4f\nкоординацiйного полiмера [Dy 2Cu2]𝒏на основi спiн-1/2\nортогонально-димерного ланцюжка Iзiнґа-Гайзенберґа\nЙ. Стречка1, Л. Ґалiсова2, Т. Верхоляк3\n1Iнститут фiзики, Факультет природничих наук, Унiверситет iменi П. Й. Шафарика, парк Ангелiнум 9,\nКошицi 04001, Словаччина\n2Iнститут управлiння виробництвом, Факультет виробничих технологiй у Прешовi,\nТехнiчний унiверситет в Кошицях, вул. Баєрова 1, Прешов 08001, Словаччина\n3Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна\nДетально дослiджено основний стан i процес намагнiчення точно-розв’язного спiн- 12ортогонально-\nдимерного ланцюжка Iзiнґа-Гайзенберґа з двома рiзними гiромагнiтними факторами Iзiнґових та Гай-\nзенберґових спiнiв. Показано, що до��лiджений квантовий спiновий ланцюжок виявляє до семи можли-\nвих основних станiв в залежностi вiд взаємної дiї магнiтного поля, констант внутрiшньо- i мiж-димерної\nвзаємодiї. А саме, фрустрована та модульована антиферомагнiтнi фази вiдповiдальнi за нульове пла-\nто намагнiчення при нульовiй температурi, промiжнi 1/11- та 5/11-плато виникають у фрустрованiй\nта модульованiй квантових ферiмагнiтних фазах, в той час як промiжнi 9/11- та 10/11-плато можна\nвiднести до квантової i класичної ферiмагнiтних фаз. Запропоновано, що плато намагнiчення експе-\nриментально спостереженi при високих полях у 3d-4f гетеробiметалiчному координацiйному полiмерi\n[{Dy(hfac) 2(CH3OH)} 2{Cu(dmg)(Hdmg)} 2]𝑛(H2dmg = dimethylglyoxime; Hhfac = 1,1,1,5,5,5-hexafluoropentane-\n2,4-dione) можна вiднести до класичної та квантової феромагнiтних фаз.\nКлючовi слова: ортогонально-димерний ланцюжок Iзiнґа-Гайзенберґа, плато намагнiчення, 3d-4f\nкоординацiйний полiмер\n43708-11" }, { "title": "1812.00714v1.Mn2V0_5Co0_5Z__Z__Ga__Al__Heusler_alloys__Fully_compensated_ferrimagnets_with_high_Tc_and_compensation_temperature.pdf", "content": "1 \n Mn2V0.5Co0.5Z (Z= Ga, Al) Heusler alloys: Fully compensated \nferrimagnets with high T c and compensation temperature \n \nP V Midhunlal1, J Arout Chelvane2, D Prabhu3, Raghavan Gopalan3, and Harish Kumar N 1* \n1. Department of Physics, Indian Institute of Technology-Madras, Chennai-600036, India. \n2. Defense Metallurgical Research Laboratory, Kanchanbagh (PO), Hyderabad-500058, India. \n3. International Advanced Research Centre for Powder Metallurgy and New Materials (ARCI), \n Chennai-600113, India. \n*nhk@iitm.ac.in \n \nAbstract: High T C fully compensated ferrimagnets are potential candidates for spin \ntransfer torque based spintronic devices. We report the structural and magnetic properties of \nhigh T C fully compensated ferrimagnets Mn 2V0.5Co0.5Z (Z=Ga, Al) in the melt-spun ribbon \nand arc melted bulk form. While the parent alloys Mn 2YZ (Y=V, Co; Z= Ga, Al) exhibits a \nmagnetic moment value around 2 f.u., the Mn 2V0.5Co0.5Ga alloy exhibits room \ntemperature nearly fully compensated moment value of 0.09 and 0.13 f.u. in the bulk and \nribbon form respectively. For Mn 2V0.5Co0.5Al this turned out to be 0.04 and 0.08 f.u. In \nContrast to the bulk sample’s Néel P-type ferrimagnetic behaviour, ribbon samples exhibit \nNéel N-type ferrimagnetic characteristic with a high compensation temperature of 420 K for \nZ=Ga and 275 K for Z=Al. The observed T C values are more than 640 K for all samples. The \ndifferences in the magnetic properties of arc melted and melt-spun alloys indicates that even a \nslight variation in stoichiometry and sample preparation method can influence the physical \nproperties of a compensated system. \n1. Introduction \n Half-metallic antiferromagnets (HMAFs) or more precisely half-metallic fully \ncompensated ferrimagnets (HMFCFis) are a new class of magnetic materials which exhibits \nzero macroscopic moment with high spin polarization at the Fermi level. These materials \ndiffer from the conventional antiferromagnets in such a way that the different magnetic sub-\nlattices are chemically or crystallographically inequivalent and the compensation occurs in a \nwide range of temperature. Also, the characteristic temperature is the Curie temperature (T C) \nand not the Néel temperature as in the case of conventional antiferromagnets 1. This 2 \n interesting class of materials are expected to be utilized as tips in Spin-polarized Scanning \nTunneling Microscopes (SP-STM) and electrode material in Spin Torque Transfer (STT) \nbased Magnetic Tunnel Junctions (MTJs) which would be the building blocks of future \nMagnetic Random Access Memory (MRAM)2,3. Among the various HMAFs which includes \ncertain double perovskites, dilute magnetic semiconductors and superlattices 4,5, Heusler \nalloys have got much interest due to their magnetic moment tunability and high T C. The \nSlater-Pauling relation which expresses the total magnetic moment (M t) to the total number \nof valence electron per unit cell (Z t) can be utilized to design HMFCFis by varying the \nvalence electron number. As per the rule, M t= Zt-24 for full-Heusler alloys (general formula \nX2YZ, X & Y are transition metal and Z is a main group element) and M t= Zt-18 for half-\nHeusler alloys (chemical formula XYZ), zero magnetic moment state is expected for full-\nHeusler alloys with 24 valence electrons and half-Heusler alloys with 18 valence electrons6. \nThe ab initio studies carried out by I.Galanakis et al. have shown a way of achieving half-\nmetallic fully compensated ferrimagnetism in Mn 2VAl and Mn 2VSi Heusler alloys by \nsubstituting Co at the Mn site7. Motivated by this interesting theoretical observation, \nresearchers have carried out experimental investigation on the structural and magnetic \nproperties of (Mn 2-xCox)VZ (Z = Ga, Al) Heusler alloys8,9. Even though the compensation \nwas achieved in these alloys, the T C has drastically come down from more than 700 K to \nbelow the room temperature as the compensation point approaches (x=1) which is a \ndrawback as far as the applications are concerned. At the same time, our recent investigation \non the Co substitution at the V site of Mn 2VZ (Z= Ga, Al) alloys could sort out the issue of \ndecreasing T C. Mn2V1-xCoxZ (Z = Ga, Al) alloys exhibited near zero moment state when \nx=0.5 by preserving high T C (more than 650 K)10. Here in this report we investigate the \nstructural and magnetic properties of high T C fully compensated ferrimagnetic Mn 2V0.5Co0.5Z \n(Z = Ga, Al) ribbons with high compensation temperature. The properties are compared with \nthat of the bulk samples and the interesting deviations in the compensation behaviour are \ndiscussed. \n2. Experimental details \n Mn2V0.5Co0.5Z (Z = Ga, Al) bulk samples were prepared by arc melting the individual \nelements with high purity. Initially, titanium was melted several times to absorb any leftover \noxygen in the chamber. Samples were melted several times after flipping to ensure \nhomogeneity. Samples were sealed in an evacuated quartz tube for annealing. \nMn2V0.5Co0.5Ga was annealed at 1073 K and Mn 2V0.5Co0.5Al was annealed at 673 K for three 3 \n \ndays followed by furnace cooling. Mn 2V0.5Co0.5Z (Z = Ga, Al) ribbon samples were prepared \nfrom the arc melted ingots by vacuum induction melting followed by melt spinning method. \nThe molten alloys were ejected over rotating copper wheel rotating at 1000 rpm. Flakes with \n30 - 40 µm thickness and approximate length of 10 mm and breadth of 2 mm were obtained. \nThe structural characterization was carried out using Rigaku SmartLab high-resolution X-ray \ndiffractometer with Cu-K α radiation. The compositions and microscopic images of the \nribbons were recorded using FEI-InspectF Scanning Electron Microscope (SEM). The \nmagnetic measurements in the temperature range 5- 300 K were carried out using Quantum \nDesign MPMS 3 SQUID VSM and high-temperature magnetic measurements (300–850 K) \nwere carried out using Microsense Vibrating Sample Magnetometer, Model EZ9. \n3. Results & Discussion \n3.1 Structural properties \n Fig. 1(a) - (f) shows the cross-sectional and surface SEM images of the ribbon samples. \nSurfaces morphology appeared to be different for the two sides for both the alloys (one side is \nsmooth and the other side is rough). This is expected as one of the surfaces touches the wheel \nand the other surface exposed to the inert atmosphere during the melt spinning technique. \n \n \n \n \n \n \n \n \n Fig. 1(a) Cross-sectional SEM images of Mn 2V0.5Co0.5Ga and (b) Mn 2V0.5Co0.5Al ribbons. \n(c) & (d) surface images (both sides) of Mn 2V0.5Co0.5Ga and (e) & (f) Mn 2V0.5Co0.5Al \nribbons. 4 \n \nFig. 2(a) and (b) shows the X-ray diffraction (XRD) patterns of Mn 2V0.5Co0.5Z ribbon and \nbulk samples. There was not much difference in the XRD pattern of bulk and corresponding \nribbon samples. The (111) and (200) Superlattice reflections were absent in the case of Z=Ga \nsamples while Z=Al samples exhibited these reflections with low intensity. The absence/low \nintensity could be due to the similar atomic scattering factors of x-ray for the atoms which are \nin the same period in the periodic table. Here it is to be noted that as per the earlier reports, \nthe parent Mn 2VGa alloy has not exhibited any superlattice reflection in the XRD pattern and \nthe neutron diffraction pattern has shown the superlattice peaks with huge intensity11,12. It is \nreported that Mn 2VZ alloys crystallize in the cubic L21 structure (space group:225, Fm𝟑ഥm) \nand Mn 2CoZ alloys crystallize in the cubic Xa structure (space group:216, F𝟒ഥ3m)12–14 . As the \nparent alloys/end members possess different crystal structure, it is difficult to predict the \ncrystal structure of Mn 2V0.5Co0.5Z samples. Bearing this in mind, Rietveld refinement of the \nXRD patterns were carried out by using FullProf software15. assuming both L21 and Xa \nstructures for all samples. Interestingly both the structures have given a similar fit with nearly \nthe same lattice parameter and χ2 values. To have a better understanding, the XRD patterns \nwere simulated for both the structures and it was evident that the patterns are \nindistinguishable. Since the lattice parameters are same for both the structures, refined pattern \nassuming Xa structure is shown in Fig. 2 (a) and (b). The obtained lattice parameters are \n5.872 and 5.857 Å for Z=Ga and Al ribbon respectively which are close to the bulk values. \nThe estimated composition, lattice parameter and other magnetic parameters are shown in \ntable I. \n Fig. 2(a) Refined X-ray diffraction patterns of Mn 2V0.5Co0.5Z ribbon samples and (b) Bulk \nsamples. 5 \n \n3.2 Magnetic properties \n Our earlier report has investigated the detailed magnetic properties of the Mn 2V1-xCoxZ \n(Z = Ga, Al) bulk alloys. The obtained moment values for the end members in the series are \n1.80, 2.05, 1.83 and 2.06 f.u. for Mn 2VGa, Mn 2CoGa, Mn 2VAl and Mn 2CoAl \nrespectively10. Here we focus on the x=0.5 member in the bulk and ribbon forms, which \nexhibits total magnetic moment compensation with high T C. The isothermal magnetization \ncurves at 5 K and 300 K for the ribbons and bulk samples with x=0.5 are shown in Fig. 3 (a)-\n(d). The nearly compensated magnetic moment values obtained through Honda plots of M-H \ncurves at 5 K are 0.23 and 0.10 f.u. for Z= Ga and Al ribbon respectively. This is slightly \nhigher than the magnetization values of the corresponding bulk samples which are 0.1 and \n0.06 f.u. The magnetic parameters are shown in table I. The moment was found to \ndecrease with increase in temperature for the ribbon samples and an increment was observed \nfor the bulk samples as shown in Fig. 3. Moment values at 300 K are 0.09 and 0.04 f.u. for \nZ= Ga and Al ribbons respectively. This indicates that even though magnetic moment \ncompensation happens in both ribbon and bulk samples, the effect of temperature on the \nmagnetic properties are different. This is more evident from the temperature variation of \nmagnetization (M-T) measured in the range 5- 850 K as shown in Fig. 4 (a)-(d). Since the \nlow-temperature M-T curves (5-300 K) and high-temperature M-T curves (300-850 K) are \nrecorded using different instruments, combined normalized curves are shown in the figures. Alloy Type EDS Composition Lattice \nparameter \n( Å ) M (f.u.) \nTC \n(K) 5 K 300 K \nMn2V0.5Co0.5Ga Ribbon Mn 2.09V0.51Co0.52Ga0.87 5.872 0.23 0.09 672 \nBulk Mn 2.08V0.52Co0.47Ga0.91 5.878 0.10 0.13 706 \nMn2V0.5Co0.5Al Ribbon Mn 2.03V0.51Co0.52Al0.91 5.857 0.10 0.04 641 \nBulk Mn 2.02V0.48Co0.50Al1.00 5.825 0.06 0.08 659 \nTable I: Structural and magnetic parameters of Mn 2V0.5Co0.5Z alloys in the ribbon and bulk forms 6 \n \nThe low temperature M-T curves measured at 100 Oe field and in the range 5-300 K are also \nshown in the insets of Fig. 4 (a)-(d). \n \n \n \n \n \n \n Fig. 3(a) isothermal magnetization curves of Mn 2V0.5Co0.5Ga and (b) \nMn2V0.5Co0.5Al ribbons measured at 5 and 300 K. (c) & (d) isothermal \nmagnetization curves of Mn 2V0.5Co0.5Ga and Mn 2V0.5Co0.5Al bulk samples. 7 \n \n \n \nThe M-T curves for the ribbons show different behaviour compared to the bulk samples. For \nthe ribbons, magnetic moment was found to decrease with increase in temperature from a \nnon-zero value and nearly full compensation occurs around 420 K for Z=Ga and 275 K for \nZ=Al. This behaviour is in agreement with the decrease in moment observed in the M-H \ncurves recorded at different temperatures. By increasing the temperature further, \nferrimagnetic to paramagnetic transition has been occurred and the T C values were found to \nbe 672 K and 641 K for Z=Ga and Z= Al ribbons respectively. This is slightly less compared \nto the bulk values of 706 K and 659 K for Z=Ga and Z=Al respectively. The M-T curves \nclearly show the presence of a full compensation temperature which was absent in the case of \nbulk samples. A similar behaviour is observed in the compensated ferrimagnet Mn 1.5FeV0.5Al Fig. 4(a) M-T curves of Mn 2V0.5Co0.5Ga and (b) Mn 2V0.5Co0.5Al ribbons \nmeasured at 100 Oe field (c) & (d) M-T curves of Mn 2V0.5Co0.5Ga and \nMn2V0.5Co0.5Al bulk samples. Insets shows the low temperature ZFC and FC \nM-T curves. 8 \n (bulk arc melted sample)16,17. It is to be noted that the compensation temperature for \nMn1.5FeV0.5Al bulk alloy was 127 K (which has been tuned up to 226 K by varying the \nstoichiometry) which is much lower than the compensation temperature of Mn 2V0.5Co0.5Ga \nribbon reported in this paper (420 K). In the earlier reported Mn 1.5FeV0.5Al system, three \ndifferent kinds of M-T behaviour was observed, when the stoichiometry was varied. The first \ncase is the fully compensated ferrimagnetic state where a zero moment is observed near 0 K \nand it is maintained up to a certain temperature (50 K in the case of Mn 1.5FeV0.5Al) followed \nby an increase in magnetization and then transition from ferrimagnetic state to paramagnetic \nstate. The authors could also simulate this M-T curve using a molecular field model for a two \nsub-lattice ferrimagnet. The second case is the overcompensated ferrimagnetic state where \nthe compensation temperature was shifted to a certain temperature (308 K) by choosing an \nappropriate stoichiometry for the parent alloy. Here the magnetization is non-zero near 0 K \nand then decreases with increase in temperature, reaching a fully compensated state and then \nfollow similar behaviour as in the first case. This is Néel N-type ferrimagnetic behaviour. In \nthe third case which is known as the Néel P-type ferrimagnetic behaviour, full compensation \ndoes not occur in the entire temperature range and the moment keeps on increasing with the \nincrease in temperature18. Now comparing these three M-T characteristics, it is clear that the \nMn2V0.5Co0.5Z ribbon samples fall in the second category which is Néel N-type ferrimagnet. \nBut the decrease in the magnetization up to the full compensation temperature is not as fast as \nthat of Mn 1.5FeV0.5Al alloy indicating different exchange coupling strength for the different \nmagnetic sublattices. As far as the bulk samples are concerned, a full compensation point was \nnot observed as in the case of ribbon samples. An increase in the magnetic moment was \nobserved till the T C indicating that the bulk samples could be Néel P-type ferrimagnets. The \nTC was found to be 706 K and 659 K for Z=Ga and Al respectively. It is to be noted that even \nthough the moment is varying with the temperature, the magnitudes show that the samples \nare in almost a near compensated state at least up to 420 K for the ribbon and 300 K for the \nbulk samples. It is expected that a slight variation in the composition could highly affect the \nmagnetic sub-lattices ordering of a fully compensated system which would give a different \nmagnetic response with the temperature as in the case of Mn 1.5FeV0.5Al 17. In addition to this, \nthe sudden quenching of molten liquid would have affected the magnetic sub-lattices ordering \nof ribbon samples (quenching rate is around 104 K/s). Mn 2V0.5Co0.5Ga ribbon and bulk \nsamples exhibit a negligibly small transition around 50 K. Unlike the Z=Ga samples, Z=Al \nsamples (ribbon and bulk) has not shown any transition in the low-temperature regime 9 \n indicating that the presence of small second magnetic phase is not influencing the \ntemperature dependent compensation in Z=Ga sample. \n4 Conclusions \n The magnetic properties of fully compensated ferrimagnets Mn 2V0.5Co0.5Z (Z=Ga, Al) \nshow distinctly different magnetic properties in their ribbon and bulk form. While the ribbon \nsamples exhibit Néel N-type ferrimagnetic behaviour with high compensation temperature, \nthe bulk samples exhibit Néel P-type ferrimagnetic behaviour without any full compensation \ntemperature. Even though there exists a temperature dependent moment variation in the \nsamples, an overall nearly fully compensated state is preserved at least up to 420 K for the \nribbons and 300 K for the bulk samples. These materials having zero moment, high T C and \nwide temperature range of compensation would be attractive for the future spintronic devices \nutilizing fully compensated ferrimagnets. \n \nReferences \n1 S. Wurmehl, H.C. Kandpal, G.H. Fecher, and C. Felser, J. Phys. Condens. Matter 18, 6171 \n(2006). \n2 R.A. de Groot, Phys. B Phys. Condens. Matter 172, 45 (1991). \n3 B. Balke, G.H. Fecher, J. Winterlik, and C. Felser, Appl. Phys. Lett. 90, 1 (2007). \n4 W.E. Pickett, Phys. Rev. B - Condens. Matter Mater. Phys. 57, 10613 (1998). \n5 X. Hu, Adv. 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Chen, Y.X. Li, G. Xiao, and G.H. Wu, 1 (2008). \n15 J. Rodríguez-Carvajal, Phys. B Condens. Matter 192, 55 (1993). \n16 R. Stinshoff, A.K. Nayak, G.H. Fecher, B. Balke, S. Ouardi, Y. Skourski, T. Nakamura, \nand C. Felser, Phys. Rev. B 95, (2017). \n17 R. Stinshoff, G.H. Fecher, S. Chadov, A.K. Nayak, B. Balke, S. Ouardi, T. Nakamura, and \nC. Felser, AIP Adv. 7, (2017). \n18 L. Neel, Science, 174, 985 (1971). \n \n \n \n " }, { "title": "2109.06033v1.Chiral_Cavity_Quantum_Electrodynamics.pdf", "content": "Chiral Cavity Quantum Electrodynamics\nJohn Clai Owens,1, 2Margaret G. Panetta,1Brendan Saxberg,1Gabrielle Roberts,1Srivatsan\nChakram,3Ruichao Ma,4Andrei Vrajitoarea,1Jonathan Simon,1and David Schuster1\n1James Franck Institute and Department of Physics, University of Chicago, Chicago, IL 60637, USA\n2Thomas J. Watson, Sr., Laboratory of Applied Physics and Kavli Nanoscience Institute,\nCalifornia Institute of Technology, Pasadena, California 91125, USA\n3Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA\n4Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA\n(Dated: September 14, 2021)\nCavity quantum electrodynamics, which explores the granularity of light by coupling a resonator\nto a nonlinear emitter [1], has played a foundational role in the development of modern quantum\ninformation science and technology. In parallel, the \feld of condensed matter physics has been\nrevolutionized by the discovery of underlying topological robustness in the face of disorder [2{4],\noften arising from the breaking of time-reversal symmetry, as in the case of the quantum Hall\ne\u000bect. In this work, we explore for the \frst time cavity quantum electrodynamics of a transmon\nqubit in the topological vacuum of a Harper-Hofstadter topological lattice [5]. To achieve this,\nwe assemble a square lattice of niobium superconducting resonators [6] and break time-reversal\nsymmetry by introducing ferrimagnets [7] before coupling the system to a single transmon qubit.\nWe spectroscopically resolve the individual bulk and edge modes of this lattice, detect vacuum-\nstimulated Rabi oscillations between the excited transmon and each mode, and thereby measure\nthe synthetic-vacuum-induced Lamb shift of the transmon. Finally, we demonstrate the ability to\nemploy the transmon to count individual photons [8] within each mode of the topological band\nstructure. This work opens the \feld of chiral quantum optics experiment [9], suggesting new routes\nto topological many-body physics [10, 11] and o\u000bering unique approaches to backscatter-resilient\nquantum communication.\nI. INTRODUCTION\nMaterials made of light are a new frontier in quantum\nmany-body physics [12]; relying upon non-linear emitters\nto generate strong photon-photon interactions and ultra-\nlow-loss meta-materials to manipulate the properties of\nthe individual photons, this \feld explores the interface\nof condensed matter physics and quantum optics whilst\nsimultaneously producing novel devices for manipulating\nlight [13, 14]. Recent progress in imbuing photons with\ntopological properties [15], wherein the photons undergo\ncircular time-reversal-breaking orbits, promises opportu-\nnities to explore photonic analogs of such solid-state phe-\nnomena as the (fractional) quantum Hall e\u000bect [2, 3],\nAbrikosov lattices [16], and topological insulators [4].\nIn electronic materials, the circular electron orbits re-\nsult from magnetic or spin-orbit couplings [4]. Unlike\nelectrons, photons are charge-neutral objects and so do\nnot directly couple to magnetic \felds. There is thus an\ne\u000bort to generate synthetic magnetic \felds for photons\nand more generally to explore ideas of topological quan-\ntum matter in synthetic photonic platforms. Signi\fcant\nprogress in this arena has been made in both optical-\nand microwave- topological photonics: in silicon photon-\nics [17, 18] and optics [19, 20], synthetic gauge \felds have\nbeen achieved while maintaining time-reversal symme-\ntry by encoding a pseudo-spin in either the polarization\nor spatial mode. In RF and microwave meta-materials,\nboth time-reversal-symmetric [21, 22] and time-reversal-\nsymmetry-broken models have been explored, with the\nT-breaking induced either by coupling the light to fer-rimagnets in magnetic \felds [7, 23] or by Floquet engi-\nneering [24].\nTo mediate interactions between photons, a nonlinear\nemitter or ensemble of nonlinear emitters must be in-\ntroduced into the system [25]. This has been realized\nfor optical photons by coupling them to Rydberg-dressed\natoms, providing the \frst assembly of two-photon Laugh-\nlin states of light [26]. In quantum circuits, a 3-site lat-\ntice of parametrically-coupled transmon qubits enabled\nobservation of chiral orbits of photons/holes [24], and a\n1\u00028 lattice of transmons enabled exploration of Mott\nphysics [27]. In nanophotonics, a topological interface\nenabled helical information transfer between a pair of\nquantum dots [28].\nIn this work, we demonstrate a scalable architecture for\nprobing interacting topological physics with light. Build-\ning on prior room-temperature work [7], we demonstrate\na 5\u00025 array of superconducting resonators that acts as\na quarter \rux ( \u000b=1\n4) Hofstadter lattice [5], exhibiting\ntopological bulk and edge modes for the photons that\nreside within it. We couple a single transmon qubit to\nthe edge of this system, and enter, for the \frst time,\nthe regime of strong-coupling cavity quantum electrody-\nnamics for a highly nonlinear emitter interacting with the\nspectrally resolved modes of a topological band structure.\nIn Section II, we introduce our superconducting topo-\nlogical lattice architecture compatible with transmon\nqubits. We then characterize its properties both spectro-\nscopically and spatially in Section III. In Section IV, we\ncouple a single transmon qubit to the lattice, employing\nit to detect and manipulate individual photons in bulkarXiv:2109.06033v1 [cond-mat.mes-hall] 9 Sep 20212\na b c\nN\nSReadout\ncavity\nQubit drive\n+ readoutLattice\ndrive\ngrgl\nFIG. 1. Elements of chiral cavity quantum electrody-\nnamics. a, The apparatus consists of a 5 \u00025,\u000b=1\n4Hof-\nstadter lattice [5] of resonators in which microwave photons\npropagate as charged particles in a magnetic \feld, coupled to\na single qubit on the edge that is sensitive to the precise num-\nber of photons and their energies. Each site, implemented as a\ncoaxial resonator milled into a block of niobium [6], exhibits a\nresonance frequency !0determined by the length of a central\npost, and a nearest neighbor tunneling rate Jdetermined by\nthe size of a machined coupling hole. The synthetic magnetic\n\feld manifests as an Aharonov-Bohm \rux \u0019=2 when photons\nhop around minimal closed loops (green), generated by the\nspatial structure of the resonator modes: each 2-site by 2-site\nplaquette includes one lattice site (red) that exhibits a px+ipy\norbital, while the other three sites exhibit sorbitals [7, 10].\nThe additional site (blue) on the system edge serves as read-\nout cavity into which transmons may be inserted. b,px+ipy\nsites instead contain three posts and thus support three mi-\ncrowave modes ( s,px\u0006ipy). Because our Hofstadter lattice\nemploys only the px+ipymode, we must isolate it: the smode\nis tuned away by the electromagnetic coupling between posts,\nwhile a Yttrium-Iron-Garnet (YIG) ferrimagnet (black) cou-\nples primarily to the px\u0000ipymode (due to the orientation of\nthe B-\feld of the red/blue bar magnet), thereby detuning it in\nenergy and isolating the px+ipymode. c,A transmon qubit\nis inserted into a gap between readout (left) and lattice (right)\ncavities on a sapphire carrier (turquoise), and couples to the\ntwo cavities with Rabi frequencies grandglrespectively. An\nSMA connector (gold) allows direct microwave probing of this\nreadout cavity and thus the transmon.\nand edge modes of the lattice and to measure the Lamb\nshift of this synthetic vacuum. In Section V we conclude,\nexploring the opportunities opened by this platform.\nII. A SUPERCONDUCTING HOFSTADTER\nLATTICE FOR MICROWAVE PHOTONS\nIn vacuum, photons are neither (i) massive, nor (ii)\ncharged, nor (iii) con\fned to two dimensions, the crucial\ningredients for quantum Hall physics [2]. To realize these\nessentials, we follow the road-map laid out in [10]: mi-\ncrowave photons are trapped in a 2D array of microwave\nresonators, and thereby con\fned to two transverse di-\nmensions and imbued with an e\u000bective mass due to the\n\fnite tunneling rate between the resonators. Rather\nthan attempting to actually imbue photons with electric\ncharge, we note that when electrons are con\fned to alattice, the entire impact of a magnetic \feld on their dy-\nnamics is encompassed by the Aharonov-Bohm-like phase\nthat they acquire when tunneling around closed trajec-\ntories. We engineer this \\Peierls phase\" via the spatial\nstructure of the on-site lattice orbitals.\nFig. 1a shows the square Hofstadter lattice that we\nhave developed for this work. Each square in the diagram\nis a lattice site, implemented as a resonator of frequency\n!0\u00192\u0019\u00029 GHz, tunnel-coupled to its nearest-neighbors\nwithJ= 2\u0019\u000218 MHz (see SI C). Sites with counter-\nclockwise red arrows exhibit modes with spatial structure\npx+ipy, while all other sites have s-like modes. The phase\nwinding in a px+ipysite causes photons tunneling in/out\nfrom di\u000berent directions to acquire a phase \u001e=\u000e\u0012, where\n\u000e\u0012is the angle between input and output directions [7].\nThis ensures that when photons tunnel around a closed\nloop enclosing nplaquettes, they pick up an Aharonov-\nBohm-like phase \u001eloop=n\u0019\n2. Such a tight-binding model\nwith a \rux per plaquette of \u0019=2 is called a \\quarter \rux\nHofstadter lattice\" [5].\nTo avoid seam losses and thus achieve the highest qual-\nity factors, the lattice structure is machined from a sin-\ngle block of high-purity (RRR=300) niobium and cooled\nto 30 mK to reduce loss and eliminate blackbody pho-\ntons (see SI A). Sites are realized as coaxial resonators,\nwhile tunneling between adjacent sites is achieved via\nhole-coupling through the back side: as in [29], the cou-\npling holes are sub-wavelength and thus do not lead to\nleakage out of the structure. s-orbital sites are imple-\nmented as single post resonators, while px+ipysites are\nrealized with three posts in the same resonator, coupled\nto a Yttrium-Iron-Garnet (YIG) ferrimagnet that ener-\ngetically isolates the px+ipymode from the (vestigial)\npx\u0000ipyandsmodes (see Fig. 1b, SI Fig. S3). The T-\nsymmetry of the ferrimagnet is broken with the \u00180:2\nTesla magnetic \feld of ?1:6 mm N52 rare-earth magnets\nplaced outside of the cavity, as close to the YIG as pos-\nsible to minimize quenching of superconductivity (see SI\nFig. S4).\nA single \fxed-frequency transmon qubit on a sapphire\nwafer is inserted between the top-left resonator in the\n5\u00025 square lattice and the adjacent readout resonator\n(see Fig. 1a); a zoom-in of this setup is shown in Fig. 1c.\nIII. PROBING THE TOPOLOGICAL LATTICE\nWe \frst characterize the mode structure of the topolog-\nical lattice itself in the linear regime, prior to introducing\nthe transmon nonlinearity. Fig. 2a shows the anticipated\nenergy spectrum of a semi-in\fnite strip \u000b= 1=4 Hofs-\ntadter lattice with four bands and topologically protected\nedge channels living below the top band and above the\nbottom band. In a \fnite system, these continuum bands\nand edge channels fragment into individual modes satis-\nfying the boundary conditions. Fig. 2b shows the mea-\nsured response of the lattice when probed spatially both\nwithin the bulk (left) and on the edge (right), with the3\nFIG. 2. A superconducting Chern circuit. The central ingredient of a chiral cavity QED platform is a long-lived,\nspectrally-isolated chiral (unidirectional) mode to couple to a real or synthetic atom. For our experiments this mode is a\nquantized edge-excitation of a synthetic Hall system realized in a \u000b= 1=4 Hofstadter square lattice [5]. The numerically-\ncomputed band structure of our implementation of this model is depicted in a,for an in\fnite strip geometry. The top and\nbottom bands each exhibit a Chern number C=\u00001, while the middle two bands, which touch at Dirac points, have a total\nChern number C= +2; chiral edge channels exist above the bottom band and below the top band, as anticipated from the bulk-\nboundary correspondence [4]. b,shows microwave transmission spectra measured through our actual 5 \u00025 lattice, where both\nthe bulk bands and chiral edges manifest as well-resolved resonant modes due to the \fnite system size. \\Bulk\" measurements\nare performed by exciting and measuring at two distinct sites on the interior of the lattice, while \\edge\" measurements employ\ntwo sites on the lattice perimeter. In c,we measure the spatial structure of the modes identi\fed with arrows in band observe\nthat, as expected, the mode residing predominantly in the interior of the lattice is located energetically within a lattice band,\nwhile the one localized to the edge resides within an energy gap (see SI F for measurement details). In d,we excite a single\nedge site at energies within the upper (red) and lower (blue) bulk gaps, and observe the response of the resulting traveling\nexcitation as a function of time averaged over the full perimeter ( main panels ) and vs. site index around the system edge\n(inset panels ). The insets demonstrate that upper and lower edge channels have opposite chiralities and re\rect the numerous\norbits of the pulse before it damps away. The ability of a photon to undergo numerous round trips prior to decay is equivalent\nto spectroscopic resolution of the individual edge modes.\nenergies aligned to Fig. 2a. It is clear that the bulk spec-\ntrum exhibits modes within the bands, while the edge\nspectrum exhibits modes within the bandgaps. We fur-\nther validate that the modes we have identi\fed as \\bulk\"\nand \\edge\" modes reside in the correct spatial location\nby exciting modes identi\fed with arrows in Fig. 2b and\nperforming full microscopy of their spatial structure in\nFig. 2c.\nTo demonstrate that the edge channels are indeed long-\nlived and chiral (handed), we abruptly excite the system\nat an edge site within each of the two bulk energy gaps\nin Fig. 2d (see SI F). By monitoring the edge-averaged\nresponse as the excitation repeatedly orbits the lattice\nperimeter, we determine that the excitation can circle\nthe full lattice perimeter >20 times prior to decay (see\nFig. 2d). In the insets to Fig. 2d, we probe in both space\nandtime, and observe that the excitations move in op-posite directions in the upper and lower band gaps, as\nanticipated from the bulk-boundary correspondence [4].\nIV. COUPLING A QUANTUM EMITTER TO\nTHE TOPOLOGICAL LATTICE\nTo explore quantum nonlinear dynamics in the topo-\nlogical lattice we couple it to a transmon qubit (see Fig. 1\nand SI G) which acts as a quantized nonlinear emitter\nwhose properties change with each photon that it ab-\nsorbs. Unlike traditional cavity and circuit QED exper-\niments in which a nonlinear emitter couples to a single\nmode of an isolated resonator, here the transmon couples\nto all modes of the topological lattice. In what follows\nwe will induce a controlled resonant interaction between\nthe transmon and individual lattice modes, investigating4\nthe resulting strong-coupling physics.\nThejgi$jeitransition of the transmon ( !q\u00192\u0019\u00027:8\nGHz) is detuned from the lattice spectrum ( !lattice\u0019\n2\u0019\u00029 GHz) by \u0001\u00192\u0019\u00021:2 GHz. We bring the trans-\nmon controllably into resonance with individual lattice\nmodes via the dressing scheme in Fig. 3a inset (see Meth-\nods and ref. [30] for details); this dressing also gives us\ncomplete control over the magnitude of the qubit-lattice\nsite coupling.\nIn Fig. 3a we tune the excited transmon into reso-\nnance with individual lattice modes and observe vacuum-\nstimulated Rabi oscillations (see SI I) of a quantized exci-\ntation between the transmon and the mode. Comparing\nwith the predicted band structure shown in Fig. 3b, we\nsee that the transmon couples e\u000eciently to both bulk and\nedge modes of the lattice, despite being physically located\non the edge. This is because the lattice is only 5 sites\nacross, comparable to the magnetic length lB\u00181=\u000b= 4\nsites, so the lattice site coupled to the transmon has sub-\nstantial participation in both bulk and edge modes; fur-\nthermore, the system is su\u000eciently small that the number\nof bulk sites is comparable to the number of edge sites,\nso all modes have approximately the same \\volume.\"\nTo unequivocally demonstrate strong coupling between\nthe transmon and a single lattice mode, we examine a\nsingle frequency slice of Fig. 3a versus evolution time.\nFig. 3c shows such a slice and demonstrates high-contrast\noscillations that take several Rabi cycles to damp out, as\nis required for strong light-matter coupling. For sim-\nplicity, we choose our dressed coupling strength to be\nless than the lattice mode spacing; stronger dressing to\nexplore simultaneous coupling to multiple lattice modes\nopens the realm of super-strong-coupling physics [31, 32],\nwhere the qubit launches wavepackets localized to smaller\nthan the system size.\nWhen a qubit is tuned towards resonance with a sin-\ngle cavity mode it experiences level repulsion [33] and\nthen an avoided crossing at degeneracy. The situation is\nmore complex for a qubit coupled to a full lattice, where\none must account for interactions with alllattice modes,\nboth resonant and non-resonant. In total these couplings\nproduce the resonant oscillations observed in Fig. 3c plus\na frequency-dependent shift due to level repulsion from\no\u000b-resonant lattice modes, which may be understood as a\nLamb shift from coupling to the structured vacuum [34].\nWe quantify this Lamb shift by comparing the frequen-\ncies of the modes observed in linear lattice spectroscopy,\nas in Fig. 2a but with the transmon present (see SI H), to\nthose observed in chevron spectroscopy in Fig. 3a. These\ndata are shown in the lower inset to Fig. 3a. When the\nqubit is tuned near the low-frequency edge of the lat-\ntice spectrum it experiences a downward shift from all of\nthe modes above it, and when it is tuned near the upper\nedge of the lattice, it experiences a corresponding upward\nshift. These two extremes smoothly interpolate into one\nanother as modes move from one side of the qubit to\nthe other. There is also a near-constant Stark shift of\n\u00183.5 MHz arising from the classical dressing tone. Toour knowledge, this is the \frst measurement of the Lamb\nshift of a qubit in a synthetic lattice vacuum.\nFinally, we demonstrate the ability to count photons\nwithin an individual lattice mode. If the transmon were\ncoupled to a single lattice site and not to the full lattice,\neach photon in that site would shift the qubit jgi$jei\ntransition by 2 \u001f, where\u001f\u0019g2\nl\n\u0001\u0002\u000bq\n\u0001+\u000bq\u00192\u0019\u00025 MHz, and\n\u000bqis the transmon anharmonicity. This photon-number-\ndependent shift, and thus the intra-cavity photon num-\nber, can be measured by performing qubit spectroscopy\ndetected through the readout cavity (see SI J). When the\ntransmon is coupled to a lattice rather than an isolated\ncavity, the \u001fshift is diluted by the increased volume of\nthe modes. In Fig. 3d, we inject a coherent state into\nthe highlighted mode in Fig. 3a and then perform qubit\nspectroscopy to count the number of photons within the\nmode. The observed spectrum corresponds to a coherent\nstate with \u0016 n\u00191:4, with the individual photon occu-\npancies clearly resolved. Indeed, when we perform this\nexperiment as a function of the amplitude of the coher-\nent excitation pulse (Fig. 3d, inset), we \fnd a continuous\nevolution from vacuum into a Poisson distribution over\nthe \frst six Fock states.\nV. OUTLOOK\nIn this work we have demonstrated a photonic materi-\nals platform that combines synthetic magnetic \felds for\nlattice-trapped photons with a single emitter. This has\nenabled us to explore interactions between the individual\nmodes of a topological system and the non-linear excita-\ntion spectrum of the emitter, entering for the \frst time\nthe realm of fully-granular chiral cavity QED and thus\ndemonstrating the ability to count and manipulate indi-\nvidual photons in each mode of the lattice. We anticipate\nthat coupling a transmon to a longer edge would enable\nqubit-mediated photon-induced deformation of the edge\nchannel (in the \\super strong\" coupling limit of the edge\nchannel [31, 35]), as well as universal quantum compu-\ntation via time-bin-encoding [36] or blockade engineer-\ning [37]. Introduction of a qubit to the bulk of this sys-\ntem would allow investigation of the shell-structure of a\nLandau-photon polariton [11], a precursor to Laughlin\nstates. Addition of a second qubit on the edge would al-\nlow chiral, back-scattering-immune quantum communi-\ncation between the qubits [9]. Scaling up to one qubit\nper site will enable dissipative stabilization [38{41] of\nfractional Chern states of light [10] and thereby provide\na clean platform for creating anyons and probing their\nstatistics [42].\nVI. METHODS\nThe transmon qubit (see SI B) has a jgi$jeitran-\nsition frequency of !q= 2\u0019\u00027:8 GHz, compared with5\nQubita\nc b\nQubit LatticeEnergyClassical driveCavity drive\namplitude\nBulk Edge Bulk Bulk Edge\nFIG. 3. Quantum nonlinear dynamics on a chiral lattice. When a transmon qubit is coupled to the edge of the topological\nlattice, many of the properties of the (nonlinear) qubit are transferred to the (linear) modes of the lattice. In a,we prepare\nthe qubit in its second excited state jfi(seetop inset ), and drive it with a classical tone (see Methods), thereby scanning\nthe energy of resulting dressed excited state j~eithrough the lattice band structure. The qubit can then coherently exchange\na single photon with the individual lattice modes. The resulting multimode chevron pattern exhibits fast, low-amplitude Rabi\noscillations when the qubit is detuned from the lattice modes, and slower, high-contrast Rabi oscillations on resonance with\neach lattice mode. The vacuum Rabi coupling to each mode is determined by the wavefunction overlap of that mode with\nthe qubit site (see SI J); for this reason all edge modes exhibit fast Rabi oscillations, but many of the bulk modes exhibit\nslower oscillations. The bottom inset shows the Lamb shift of the transmon due to the topological vacuum of the lattice,\nmeasured by comparing the frequency shift of the modes between the chevrons and the linear spectroscopy like Fig. 2. There is\nan additional overall Stark shift of all chevron modes from the strong classical drive. The gray and white at bottom highlight\nthe locations of the bands and gaps. b,When the qubitj~eistate is tuned to resonance with a particular lattice mode (blue\nline in a), vacuum-stimulated Rabi oscillations between qubit and cavity are apparent, demonstrating strong coupling cavity\nQED where information exchange to a single chiral mode is faster than all decay processes. c,To count photons in a particular\nedge mode, in this case the highlighted mode in a, we directly excite that mode with a coherent pulse and detect the number\nof photons it contains as a discrete shift of the qubit frequency (as probed through its readout cavity, see SI G) resulting from\nthe photon-number-dependent dispersive shift of the qubit frequency. The line is a multi-Lorentzian \ft, and the individual\nLorentzians for each photon number are shown in gray. Inset, by measuring the qubit excitation spectrum as a function\nof chiral-mode excitation power, we observe a response transitioning from vacuum (single high-frequency resonance) to the\nexpected Poisson distribution (appearance of multiple lower-frequency resonances). The uncertainties are smaller than the\ndata points in bandc(see SI L).6\nthe lattice spectrum centered on 8 :9 GHz and spanning\n\u000650MHz (due to lattice tunneling J= 2\u0019\u000218MHz). In\nthe presence of the applied magnetic \felds, the jgi$jei\ntransition of the transmon exhibits a T1= 2:9\u0016s and\nT2= 3:9\u0016s (see SI G). The anharmonicity of the trans-\nmon is\u000bq= 2\u0019\u0002346 MHz. The transmon is coupled to\nits readout cavity ( !r= 2\u0019\u000210:6 GHz,\u0014r= 2\u0019\u0002500\nkHz) withgr= 2\u0019\u000266 MHz. The transmon is coupled to\nthe (1;1) site of the lattice with gl= 2\u0019\u0002111MHz. The s-\nlike lattice sites have a linewidth of \u0014s= 2\u0019\u00025kHz, while\nthepx+ipy-like sites have a linewidth \u0014px+ipy= 2\u0019\u000250\nkHz. The lattice sites themselves are tuned with \u00061 MHz\naccuracy for the s-sites and\u00063 MHz accuracy for the\npx+ipysites.\nWe perform linear spectroscopy of the lattice with\ndipole antennas inserted into each lattice site and use\ncryogenic switches to choose which sites we excite/probe\n(see SI C).\nWe perform nonlinear spectroscopy by exciting the\ntransmon through its readout cavity. The transmon is\n\fxed-frequency to avoid unnecessary dephasing from sen-\nsitivity to the magnetic \felds applied to the YIG spheres.\nAs a consequence, we \\tune\" the transmon to various lat-\ntice modes by dressing through the readout cavity [30]:\nwe prepare the transmon in the second excited ( jfi) state\nand then provide a detuned drive on the jfi$jeitran-\nsition to create a dressed j~ei\u0019jfi\u0000\n\u0001jeistate at any\nenergy in the vicinity of the lattice band structure, with\na dipole moment for coupling to the lattice which is\nrescaled by the ratio of the dressing Rabi frequency to\nthe detuning from the jfi$jeitransition. The result-\ning vacuum-stimulated jf;0i $ jg;1kiRabi frequency\ngk\u0019gl\u0002\n\u0001\u0002hxtransmonj ki. Here \n is the Rabi frequency\nof the dressing tone on the jfi$jeitransition of the\ntransmon andhxtransmonj kiis the participation within\nmodekof the lattice site where the transmon resides.\nThe dressing scheme may alternatively be understood\nas a 2-photon Rabi process, where the jf;0i je;0i\ntransition is stimulated by the classical drive, and the\nje;0i jg;1kitransition is stimulated by the vacuum\n\feld of mode k.For the qubit measurements, the the lattice is tuned\nto a center frequency of 2 \u0019\u00028:9 GHz, corresponding to\na dressing frequency of 2 \u0019\u00026:35 GHz\u000650 MHz. Note\nthat with the additional signi\fcant \fgure, the jgi$jei\ntransition has a frequency of 2 \u0019\u00027:75 GHz.\nVII. ACKNOWLEDGEMENTS\nThis work was supported primarily by ARO MURI\nW911NF-15-1-0397 and AFOSR MURI FA9550-19-1-\n0399. This work was also supported by the University\nof Chicago Materials Research Science and Engineering\nCenter, which is funded by National Science Foundation\nunder award number DMR-1420709. J.O., M.P., and\nG.R. acknowledge support from the NSF GRFP. We ac-\nknowledge Andrew Oriani for providing a rapidly cycling\nrefrigerator for cryogenic lattice calibration.\nVIII. AUTHOR CONTRIBUTIONS\nThe experiments were designed by R.M. J.O., D.S. and\nJ.S. The apparatus was built by J.O., R.M., and B.S. J.O.\nand M.P. collected the data, and all authors analyzed the\ndata and contributed to the manuscript.\nIX. AUTHOR INFORMATION\nThe authors declare no competing \fnancial interests.\nCorrespondence and requests for materials should be ad-\ndressed to D.S. (dis@uchicago.edu).\nX. DATA AVAILABILITY\nThe experimental data presented in this manuscript is\navailable from the corresponding author upon request.\n[1] Walther, H., Varcoe, B. 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Soviet Physics JETP 30, 1068{1075\n(1970).\n[44] Cochran, J. F. & Mapother, D. Superconducting transi-\ntion in aluminum. Physical Review 111, 132{142 (1958).\n[45] Nigg, S. E. et al. Black-box superconducting circuit quan-\ntization. Physical Review Letters 108, 240502 (2012).\n[46] Reed, M. et al. High-\fdelity readout in circuit quantum\nelectrodynamics using the Jaynes-Cummings nonlinear-\nity.Physical Review Letters 105, 173601 (2010).8\nSupplement A: Cryogenic Setup\nThe cryogenic setup employed for measuring the lat-\ntice coupled to the qubit is shown in Fig. S1 and is similar\nto the setup used to measure the lattice prior to the in-\ntroduction of the qubit. The lattice is mounted on the\nmixing chamber (MXC) plate of a Bluefors LD-250 dilu-\ntion refrigerator at \u001831 mK. The lattice sites depicted\nin blue are connected to a 10-way cryogenic switch which\nis in turn connected to a circulator so that these sites\ncan all connect to either an input line or an output line.\nSites in green are connected to only input lines and thus\ncan only be excited, not measured. The site (1 ;1) is\nconnected separately to a circulator so that it can be\nmeasured and excited independently of the blue lattice\nsites. The remaining six sites in the 5x5 lattice are not\nconnected to a either an input line or output line. The\nqubit readout resonator is also connected to a circula-\ntor so that re\rection measurements can be performed to\nmeasure the state of the qubit. The qubit is excited o\u000b-\nresonantly through the readout resonator. Each output\nline is \fltered by an eccosorb \flter to suppress accidental\nhigh-frequency qubit excitation.\nFor the measurements performed without a qubit (see\nFig. 2), two cryogenic switches were used within the\nfridge, enabling direct probing of 20 sites, while a 21st\nsite was independently connected to a circulator. The\nonly sites not measured were the 4 chiral cavities, whose\nmodes are more localized at the bottom of the cavity,\nmaking coupling to them via a dipole antenna di\u000ecult\nwithout spoiling the quality factor.\nA Keysight PNA-X N5242 is used to perform lattice\nspectroscopy. For the qubit measurements, a Keysight\narbitrary waveform generator (M8195A, 64 GSa/s) is\nused to synthesize a local oscillator signal near the\nqubit frequency, while Berkeley Nucleonics 845-M mi-\ncrowave synthesizers provide separate local oscillator sig-\nnals near lattice and dressing frequencies. The local oscil-\nlators are then I/Q modulated by Keysight PXIe AWGs\n(M3202A, multichannel, 1GS/s) to generate the indi-\nvidual qubit drive, qubit readout, and dressing pulses.\nThe qubit drive and readout pulses are combined out-\nside the fridge and sent to the readout resonator (Fig. S2\nand Fig. S1). The re\rected readout signal is routed\nto the output line via circulators and ampli\fed with a\nHEMT ampli\fer at 4 K and additional room temperature\nampli\fers (Miteq AFS3-00101200-22-10P-4, Minicircuits\nZX60-123LN-S+). The signal is then demodulated using\nan IQ mixer and recorded using a fast digitizer (Keysight\nM3102A, 500 MSa/s). A schematic layout of the room-\ntemperature components of the experimental setup can\nbe found in Fig. S2.\nSupplement B: Transmon Fabrication\nThe transmon qubit is made of aluminum deposited\non a \u001c50:8mm, 430\u0016m thick C-place (0001) sapphire\nMXC plate ~31 mK Still 4 K 50 K 300 K\n......\n30 dB20 dB 30 dB\n20 dB\nEccosorb\nfilterQuinstar \n4-12 GHZ\nisolator \nRadiall\nR574F32005\nRF switchLNF 8-12 GHz\nsingle junction\ncirculatorQuinstar\ndouble junction\ncirculatorXMA SS cryo \nattenuator\nXMA gold-plated \ncopper threaded\ncryo attenuator4-12 GHz\nHEMT\nSMA \nfeedthrough\nLines to 8 \nlattice sitesLines to 10 \nlattice sites\nLatticeTransmon\nQubit drive \nand readout\nLattice site driveReadout\nMiteq amplifierMinicircuits \nZX60-123LN-S+30 dBResonator + qubit driveReadout\n30 dB 20 dBSingle lattice site driveReadout\n20 dB20 dB20 dB20 dBFIG. S1. Partial diagram of cryogenic setup. Ex-\nperiments involving the qubit were performed in a Bluefors\nLD-250 dilution refrigerator operating at approximately 31\nmK. The antennas coupled to some lattice sites (highlighted\nin blue) are accessible via an input and readout line connected\nby a circulator, allowing re\rection measurements on individ-\nual sites. Other antennas coupled to lattice sites (highlighted\nin green) are accessible via input line only.\nwafer. The wafer was annealed at 1200\u000eC for 1.5 hours.\nThe wafer was then cleaned with toluene, acetone, iso-\npropanol, and DI water in an ultrasonic bath immedi-\nately prior to junction deposition. The transmon was\ndeposited in a single layer. The mask for deposition\nwas de\fned by electron-beam lithograthy using a Raith\nEBPG5000 Plus E-Beam Writer. The mask was a bi-\nlayer of resist made of a stack of MMA and PMMA. The\ncapacitor pads were rectangles with dimensions 50 mi-\ncrons by 1100 microns and were written with the 40 nA\nbeam. The junctions were patterned Manhattan-style\nusing a \fner 1 nA beam. In the intermediate region that\nconnects the junction to the capacitor a beam of 4 nA was\nused. Before writing, a 50 nm layer of Au was thermally\nevaporated onto the wafer in order to provide adequate\ngrounding the the electron beam. After writing the re-\nsist, the resist stack was developed for 1.5 minutes in a9\nQ\nRFLO\nIBNC 645 AWG\nTrigger\nTrigger Trigger\nKeysight M3202A \n1Gs/s AWGKeysight M3201A\n500 Ms/s AWGKeysight M3102A \n500 Ms/s digitizer\nSR 445A\npreamplifierLO LOKeysight PXIe Chassis\nMinicircuits \nZFSC-2-10G+ \nsplitter\nMarki MLIQ218L \n1851 IQ mixer\nHP 33320H \nmanual atten.Minicircuits\nZX60-832-N-S+\namplifier\n3 dB SMA atten.20 dB SMA atten.\nDigital atten. \n0-30 dB \n+ insertion lossMinicircuits \nVLFX-80+ \nlowpass filter\nMicro Lambda \nMLFI-1017 tunable \nYIG bandpass filter LOLattice site drive\nReadoutResonator + \nqubit driveSingle lattice site drive\nMinicircuits \nZVE-3W-83+ high\npower amplifierQ\nRFLO\nI Q\nRFLO\nI\nFIG. S2. Schematic of room-temperature experimen-\ntal setup. In any given measurement, one of the three read-\nout lines pictured in Fig. S1 is selected for use. Drives on\nlattice sites, important for the pulsed measurements with se-\nquences detailed in Fig. S9, are either passed directly into\nthe refrigerator after I/Q modulation or, in the case of strong\ndressing tones, are \fltered by a YIG (Yttrium Iron Garnet)\ntunable bandpass \flter and ampli\fed before use.\nsolution of 3 parts IPA and 1 part DI water chilled on a\n6\u000eC cold plate. The junction was then deposited using\nelectron-beam evaporation in three steps. First, 80 nm\nof aluminum was deposited at an angle of 40\u000erelative\nto the plane of the wafer and parallel to a \fnger of the\njunction. Next, the junction was oxidized in 20 mbar of\nhigh purity oxygen for 10 minutes. Last, a 45 nm layer\nof aluminum was deposited at the same evaporation an-\ngle of 40\u000ebut orthogonal to the \frst evaporation. After\nevaporation, the rest of resist as well as the aluminum\nattached to the resist were removed via lift-o\u000b in an 80\u000e\nC solution of PG remover for 3 hours. The wafer was\nthereafter diced into the dimensions that \ft the lattice\nsetup.Supplement C: Lattice Fabrication\nThe lattice is machined from a solid block of niobium,\nwhich exhibits a low Hc1to allow magnetic \felds to pen-\netrate the material, and a high Hc2to ensure that the\n\feld does not quench the superconductivity [43]. High\nniobium purity is not required since the magnetic \feld\nused to bias the YIG spheres already limits the quality\nfactor of the cavities. The resonators are arranged in a\n5\u00025 square lattice, which is the minimum lattice size\nthat supports a clear distinction between bulk and edge.\nWhile the edge channels predominantly reside in sites on\nthe lattice perimeter, they have some participation in the\nsites one removed from the edge, exponentially decaying\ninto the bulk. Each lattice site consists of a\u0015\n4coaxial\nresonator described in detail in our earlier work [7]. The\nfrequency of the lattice site is inversely dependent on the\nlength of the post in the center of the cavity. By adding\nindium foil to the end of a post the site frequency can\nbe tuned to a precision better than 1 MHz. In order to\nmaximize the quality factor of the resonator, the depth\nof the cavity is set so that the evanescent decay from the\npost mode is much less than the residual resistive loss\nof the superconductor. The niobium lattice is mounted\nin a copper box that is screwed into the niobium in or-\nder to adequately thermalize the niobium to the mixing\nchamber plate of the dilution refrigerator in which it is\nplaced. Antennas are mounted onto the lid of the copper\nbox so that a single antenna protrudes into each lattice\nsite from the top of the cavity. The length of the antenna\nsets the coupling quality factor of each lattice site. For\nthese measurements, each lattice site is weakly coupled to\nits antenna so that the total quality factor of the lattice\nis maximized.\nSpecial care is required to maintain cryo- and qubit\ncompatibility with the ferrimagnetic elements and their\nassociated B-\felds. The YIG spheres are located physi-\ncally inside of their lattice cavities in order to couple to\nthe microwave modes, so magnetic \feld must be routed\nto the YIG spheres through bulk superconductor. We\nmake the cavities out of the type II superconductor nio-\nbium so that magnetic \felds can penetrate it while it is\nin the vortex superconducting state. To create the bias\n\feld, we place a 1.6 mm diameter permanent neodymium\ncylindrical magnet in a hole outside of the cavity but di-\nrectly underneath each YIG sphere, leaving a 0.3 mm\nthick layer of niobium between the magnet and the in-\nside of the cavity and the YIG (Fig. 1c). The proximity\nof the magnet to the YIG sphere achieves a bias \feld\nof\u00180:2 T on the YIG sphere, while the small size of\nthe magnet minimizes the amount of \feld that passes\nthrough the posts of the cavity and the amount of nor-\nmal or vortex state niobium in areas with large current\n\row. This bias \feld achieves splittings between the two\nchiral rotating modes of the YIG cavity of 2 \u0019\u0002200 MHz\nwhile retaining cavity quality factors of 2 \u0002105. Trans-\nmission through the two chiral modes of the YIG cavity\nis shown in Fig. S3.10\nCavity quality factors were measured after machining\nthe lattice, inserting the ferrites (YIG spheres) to relevant\ncavities, and applying magnetic \felds. After machining\nbut before applying any surface treatment, lattice sites\nhad quality factors of \u00182\u0002106. Introducing the YIG\nsphere to the cavity does not degrade the Q, so long as\nno additional materials are added to hold the YIG sphere\nin place. After applying the magnetic \feld in the \fnal\ncon\fguration, cavity quality factors dropped to \u00182\u0002\n105, suggesting that the limiting loss factor of the cavity\nmodes is the resistive losses in the normal regions created\nby the magnetic \feld piercing the superconductor.\nThe cavities are coupled together via holes milled into\nthe back side of the niobium block. These holes open\npaths between lattice sites that allow the Wannier func-\ntions of the lattice sites to overlap with their neighbors,\ncreating coupling between sites. These coupler holes cre-\nate additional coupling via a virtual coupling mechanism,\nin which the couplers act as higher frequency resonators.\nIn some (non-cryogenic) lattices we added a screw that\ncould tune the frequency of the coupler lower, allowing\nus to achieve greater couplings between lattice sites (up\nto 150 MHz). For the cryogenic lattice design we reduced\nthe tunneling between sites to J= 2\u0019\u000218 MHz. This was\ndone because we wanted to preserve the quality factor of\nthe lattice modes by using less magnetic \feld, which in\nturn reduced the amount by which we broke time-reversal\nsymmetry. In order to increase both the quality factor of\nthe modes and the tunneling ratio, more e\u000bective meth-\nods of funneling magnetic \feld are required.\nThree readout cavities for qubits are machined on the\nedge of the lattice, though for the results shared in this\nletter only a single qubit was introduced to one such cav-\nity. These cavities are designed to be much higher in fre-\nquency (\u001810:5 GHz) than the main lattice so that they\ndo not interact directly with it. The qubit is mounted so\nthat its capacitor pads act as antennas that can directly\ncouple the qubit both to the readout cavity and to the\nlattice modes (see Fig. 1d). The antenna that couples to\nthe readout cavity is inserted into the bottom of this cav-\nity so that it can attain a low coupling Q (20,000), while\nbeing short enough that the modes introduced by the\nantenna are much higher in frequency than the readout\ncavity.\nSupplement D: Cryogenic Chiral Lattice Site\nCharacterization\nA major innovation of this work was its design of\na way to break time-reversal symmetry in the lattice\nwhile maintaining low-loss modes and compatibility with\nqubits that are sensitive to magnetic \felds. In prior work\nwe designed interactions between 3D microwave photons\nand a DC magnetic \feld in a room temperature alu-\nminum lattice [7, 10]. This kind of interaction is me-\ndiated via a ferrite sphere (made of Yttrium Iron Gar-\nnet, or YIG) placed inside a lattice cavity. A DC mag-netic \feld is applied to the YIG sphere through the cav-\nity, tuning magnon modes of the sphere into resonance\nwith the cavity modes. The hybridization of the chiral\nmagnon modes with the 3D cavity modes results in time-\nreversal-symmetry-broken cavity modes. Lattice cavities\npopulated with YIG spheres are engineered to host two\ndegenerate modes. One of the modes' magnetic \felds\nat the center of the cavity precesses clockwise, and the\nother precesses counter-clockwise. This opposite chirality\nof the modes creates a di\u000berence in the coupling strength\nbetween the two cavity modes and the chiral YIG mode.\nThe mode that precesses with the same chirality as the\nYIG sphere couples more strongly to the chiral YIG mode\nthan the mode with the opposite chirality. In Fig. S3, we\nshow the mode frequencies of the cavity as a function\nof an applied uniform magnetic \feld on a cavity with a\nYIG sphere inside. At low \felds, the two cavity modes\nare nearly degenerate, but as the bias magnetic \feld is\nincreased, the YIG mode comes into resonance with the\ncavity and splits the cavity modes by up to 400 MHz.\nThe color of plotted data indicates the phase acquired\nduring transmission through the cavity via ports that\nare placed 45\u000eapart with respect to the cavity center.\nTransmission through one of the chiral modes results in\na phase accumulation of\u0019\n2radians, while the other chi-\nral mode acquires a phase of3\u0019\n2=\u0000\u0019\n2. The observed\nphase is twice the physical angle between the two ports\nbecause we are comparing S21andS12to remove global\nreciprocal phases accrued in cables.\nIn practice, routing the magnetic \feld to the ferrite\nposes a challenge when using superconducting lattices,\nas magnetic \felds either are repelled by superconductors\nor induce normal regions that greatly increase the cav-\nity loss. In Fig. S4 we show the e\u000bects of an increasing\nmagnetic \feld applied to a niobium cavity. The \feld is\nuniform, applied externally at 12 Kelvin (K) before the\ncavity superconducts. For each data point the cavity is\nwarmed to 12 K before the \feld is changed. At the \felds\nused to bias the YIG sphere to the cavity frequency (350\nmT as shown in Fig. S3), the quality factor of the cav-\nity would be as low as 104. Additionally, the Josephson\njunction in the qubit is made out of aluminum, a type I\nsuperconductor with a low Hc1[44]. Simply applying an\nexternal magnetic \feld would adversely e\u000bect both qubit\nand cavity lifetimes.\nTo minimize the magnetic \felds permeating the sys-\ntem, we generate a local magnetic \feld in the vicin-\nity of the YIG spheres with small Neodymium magnets\n(\u00181:5mm diameter). In order to achieve the required\n\feld at each ferrite, we insert a magnet into a small hole\nmilled into the back the cavity (shown in Fig. 1) directly\nbeneath the ferrite. This minimizes the amount of nio-\nbium which sees its superconductivity quenched by the\nB-\feld, as the strongest part of the magnetic \feld is lo-\ncalized to the area between the ferrite and the magnet:\nindeed, the bulk of the modal surface currents \rows in\nthe posts of the cavity, locations where the B-\feld has\ndecayed substantially. As shown in Fig. S5, we generate11\nFIG. S3. Splittingpx\u0006ipymodes in a B-\feld. Chiral\ncavity mode frequencies are measured as a function of mag-\nnetic \feld applied to the YIG sphere. For each slice in the\ny-direction, the cavity is warmed above the TCof niobium to\n12 K, so that the magnetic \feld on the YIG sphere can be\nchanged. The cavity is then cooled to 2 K and the magnetic\n\rux is locked in place by the superconducting transition of\nthe cavity. Transmission is then measured between two an-\ntennas placed 45\u000eapart (with respect to the center of the\ncavity) so that the phase accumulated in transmission can be\nmeasured. In this plot, we take the di\u000berence between trans-\nmission from antenna 1 to 2 and transmission from antenna\n2 to 1 to isolate the non-reciprocal phase shift in the cavity.\nThis plot shows both the magnitude and phase of transmis-\nsion; the color shows the phase while the brightness shows the\nmagnitude. The two cavity modes split in frequency when the\nmagnetic \feld is applied and the phase shifts of the modes are\n\u000090\u000eand +90\u000edue to the opposite chirality of the modes.\nsu\u000ecient \feld to break the degeneracy between the two\nchiral cavity modes by 200 MHz, whilst maintaining a\nquality factor of 200,000. The splitting between these two\nmodes is a measure of how strongly time-reversal sym-\nmetry is broken. This splitting also limits the maximum\ntunneling rate in the lattice, as the Hofstadter model as-\nsumes one orbital per lattice site, requiring the tunneling\nenergy to remain small enough to avoid coupling to the\ncounter-rotating orbitals. The ratio between the tunnel-\ning rate and the loss rate in our cavities is then a measure\nof how fast the dynamics are compared to the loss rate,\nwhich is an important benchmark for the system that de-\ntermines how far the photons move within their lifetimes.\nIn this work, the ratio between tunneling and loss rates\nis18MHz\n50kHz\u0019400.\na\nb\nFIG. S4. Quality factor of chiral cavities. a, Quality\nfactor of a three post cavity as a function of magnetic \feld.\nFor every measurement, the cavity is warmed above niobium's\nsuperconducting transition of 9 K to 12 K so that the mag-\nnetic \feld can be changed. The cavity then is cooled to 2 K\nwithin the magnetic \feld and the quality factor is measured.\nThe cavity quality factor decreases to 30000 at the \feld that\ntunes the YIG to the cavity resonance. b,Quality factor of\na three post cavity with a YIG sphere as a function of tem-\nperature while the magnetic \feld is held at zero. The YIG\nsphere lowers the quality factor of the cavity but the quality\nfactors still reach \u00182 million.\n/uni0000001c/uni00000011/uni00000014 /uni0000001c/uni00000011/uni00000015 /uni0000001c/uni00000011/uni00000016 /uni0000001c/uni00000011/uni00000017 /uni0000001c/uni00000011/uni00000018\n/uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000000b/uni0000002a/uni0000002b/uni0000005d/uni0000000c/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000035/uni00000048/uni00000049/uni0000004f/uni00000048/uni00000046/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni0000000b/uni00000044/uni00000055/uni00000045/uni00000011/uni0000000c\nFIG. S5. Chiral cavity spectrum with a permanent\nmagnet. The magnet is inserted into a hole bored from the\nbottom of the cavity so that the magnet sits below the YIG\nsphere with\u00180.5 mm of niobium between the magnet and the\nYIG sphere. The ensemble is then cooled to 2 K. The magnet\ngenerates enough B-\feld to break the degeneracy of the two\nchiral modes ( px\u0006ipy) by\u0018200 MHz while still maintaining\na quality factor of 2 \u0002105.12\na\nb\nFIG. S6. Quality factors of lattice modes from edge\nand bulk spectra. Internal (Qi) and coupling ( Qc) quality\nfactors are extracted from \fts to modes in the lattice spectra\ndisplayed in Fig. 2b in the main text. Taken on the bare lat-\ntice prior to addition of the transmon qubit, these microwave\ntransmission spectra were measured between a pair of lattice\nbulk sites (extracted Qs are represented by triangles in the\nplots) and between a pair of edge sites (extracted Qs are rep-\nresented by circles in the plots). Expected locations of bulk\nbands, based on mode counting, are highlighted in gray and\nare intended to serve as a guide for the eye. Note that not\nall 25 lattice eigenmodes are apparent in the spectra and thus\nnot all 25 eigenmodes are apparent in this plot. In a,coupling\nquality factors Qcrange between\u00194.9e5 and 1.3e9. Towards\nthe center of the band, some lattice eigenmodes with higher\nbulk participation fractions display relatively higher Qcval-\nues. In b,internal quality factors Qirange between 6 :7\u0002104\nand 3:9\u0002105. In the gaps between gray bulk bands and in the\nvery center of the spectrum, lattice eigenmodes with higher\nedge participation fractions display relatively higher Qival-\nues, consistent with expectations that these modes su\u000ber less\nloss due to their smaller relative participation in the lossier\nbulk sites equipped with YIG spheres.\nSupplement E: Lattice Disorder Characterization\nand Compensation\nDisorder in the lattice site frequencies must be con-\ntrolled to a level below other energy scales of the sys-\ntem Hamiltonian (tunneling, particle-particle interac-tions, and magnetic \feld interactions). Lattice sites are\ntuned to degeneracy at room temperature, though the\ndi\u000berent types of lattice sites (single post, three post\nYIG-coupled) change frequency di\u000berently as the lattice\nsites cool to 20 mK. We \frst measure the change in fre-\nquency from cooling for the di\u000berent lattice sites ( \u001823\nMHz for single post cavities and \u001840 MHz for cavities\nwith YIG spheres) and then adjust for the di\u000berential\nat room temperature by adding indium to the top of\nthe cavity posts, which e\u000bectively lengthens the cavity\npost and decreases the cavity frequency. Because indium\nis malleable and superconducting, it attaches easily to\nthe cavity posts and does not decrease the cavity quality\nfactors. After modifying post lengths with indium, we\nadjust each cavity frequency at 1K until the disorder in\nlattice site frequencies is less than \u00061 MHz for the single\npost cavities and \u00063 MHz for the YIG cavities. We tune\nthe lattice to !l\u00192\u0019\u00028:9 GHz for the measurements\nwithout a qubit.\nSupplement F: Probing Lattice Dynamics and\nSpectra\nIn order to measure the linear response of the lattice,\nwe insert dipole antennas into each site and connect them\nto a cryogenic switching network which is routed through\ncirculators to enable performance of re\rection measure-\nments on most sites of the lattice (see SI A for details\non wiring and connectivity). To characterize the lattice,\nwe \frst measure re\rection on each site. Fig. 2d shows\na pulse propagating on the edge of the lattice when the\nlattice edge site (1,1) is excited with a 80 ns Gaussian\npulse at the frequency of the green mode in Fig. 2c. The\nround trip time for an edge pulse is \u0018120 ns compared\nto the decay time of \u00183\u0016s. The direction the pulse trav-\nels is set by the direction of the magnetic \feld and which\nband gap is excited. The two band gaps support edge\nchannels of opposite chiralities.\nSupplement G: Transmon Characterization\nThe transmon chip is clamped in a copper holder that\nis then mounted on the side of the niobium lattice. In-\ndium foil is added to the surface of the copper holder\nthat clamps the qubit in order to better thermalize the\nsapphire chip on which the qubit is printed. There is\none measurement port in the system strongly coupled to\nthe readout cavity. All of the qubit measurements are\nperformed via re\rection measurements on the readout\ncavity.\nThe Hamiltonian of the 25 lattice sites and a readout13\nresonator coupled to the qubit is the following:\nH\n~=X\np=q;r\u0010\n!p^ay\np^ap\u0000\u000bp\n2^ay2\np^a2\np\u0011\n\u00002\u001fr^ay\nq^aq^ay\nr^ar\n+Nmodes =25X\nl=1\u0010\n!l^ay\nl^al\u0000\u000bl\n2^ay2\nl^a2\nl\u00002\u001fl^ay\nl^al^ay\nq^aq\u0011\n+X\nl6=m\u00002\u001flm^ay\nl^al^ay\nm^am;\n(S1)\nwhere!q=!geis theg$etransition frequency\nof the transmon, \u000bqis the transmon anharmonicity (so\n!ef=!q\u0000\u000b),!ris the bare readout resonator fre-\nquency,\u000bris the self-Kerr shift of the readout resonator,\n2\u001fris the qubit-readout dispersive shift, !lare the lattice\nmode frequencies, 2 \u001flare the qubit-lattice mode disper-\nsive shifts,\u000blare the self-Kerr shifts of the lattice modes,\nand\u001flmare the cross-Kerr shifts of the lattice modes.\nInitial qubit characterization was done with the sin-\ngle lattice site to which the qubit was coupled tuned to\nthe lattice frequency ( \u00189 GHz) while the other 24 sites\nwere temporarily blocked o\u000b using screws lowered from\nthe lid of the lattice until they contacted the post of the\nlattice site. This was done so that the simpler system of\nthe qubit, readout resonator, and lattice resonator could\nbe analyzed in isolation. Importantly, in this con\fgura-\ntion the e\u000bects of the YIG-biasing magnets on the qubit\nlifetime could be directly measured by characterizing the\nqubit with and without the magnets present. This direct\ncomparison would be di\u000ecult to make with every lattice\nsite tuned to the lattice frequency since the magnets are\nrequired to achieve this tuning. We compare the mea-\nsurements of the qubit with no magnets in the lattice\nto measurements in which the magnets were added to\nthe lattice in Table I. The introduction of the magnets\nresulted in the lifetime of the qubit dropping to 3 \u0016s, ap-\nproximately commensurate with the lifetimes of the cav-\nity modes shown in Fig. 2. This suggests that improving\nthe funneling of the magnetic \feld will be necessary to\nfurther increase the qubit lifetime.\nNext we characterized the qubit with every lattice site\ntuned to the lattice frequency and magnets added to the\nchiral site, so that the lattice was in the con\fguration\ndescribed in Fig. 2, but with the qubit coupled to a sin-\ngle site on the corner of the lattice. Table II summarizes\nrelevant parameters of this system and compares them\nto parameters found when only the single lattice site was\ntuned to the lattice frequency. When the lattice is tuned\ninto resonance, the modes delocalize over the lattice, de-\ncreasing the electric \feld near the qubit antenna which in\nturn decreases the coupling to the qubit. To tune the lat-\ntice into resonance, the lattice cavities were temporarily\ndecoupled and their bare frequencies individually tuned\nto 2\u0019\u00028:901 GHz. After the inter-site tunneling was re-\nstored in the lattice, the lattice modes' frequencies split\ninto the chiral band structure described in Fig. 2. Qubit\nParameterNo MagnetMagnet!!/2$(GHz)7.8157.814%\"(μs)9.7±0.32.86±0.07%#(μs)6.9±0.23.9±.1Temperature (mK)<200<200 TABLE I. Parameters of qubit coupled to a single lat-\ntice mode with and without a magnet in the lattice.\nAdding the magnet decreases the lifetime of the qubit to 3 \u0016s,\na timescale similar to the lifetime of the cavity modes after\nthe \feld is added.\nparameters were measured using standard techniques.\nThe qubit frequency was measured through Ramsey spec-\ntroscopy.\u001frwas measured by \u0019-pulsing the qubit and\nmeasuring the shift in the readout resonator frequency.\nThe\u001flshifts were measured by driving the cavity with a\nweak coherent tone and measuring the photon-number-\ndependent splitting in the qubit frequency. The self-Kerr\n\u000blof the modes were calculated from the measured dis-\npersive shifts \u001fland transmon anharmonicity \u000bq, using\nthe relation 2 \u001fl=p\u000bq\u000bl[45].\nSupplement H: Spectroscopy of the Lattice Modes\nCoupled to the Qubit\nFor these measurements, transmission is measured\nanalogously to Fig. 2b, but with the qubit introduced\ninto the lattice. Fig. S7a compares transmission between\ntwo edge sites (sites (1 ;4) and (3;1)) and two bulk sites\n(site (3;3) and (2;3)) at high powers, where the qubit is\nsaturated and thus e\u000bectively decoupled from the lattice,\ncausing the lattice modes to appear at their bare frequen-\ncies [46]. Similar to the spectra in the lattice without the\nqubit, the edge-edge transmission has modes located in\nthe large gaps between the bulk modes. These measure-\nments have a large background that was not present in\nthe measurements in Fig. 2, suggesting that some direct\ncoupling between the input and output lines was acquired\nwhile setting up the measurements with the qubit.\nThe lower frequency bulk band gap hosts three distinct\nedge modes between 8 :86 GHz and 8 :88 GHz, consistent\nwith simulation of a 5 \u00025 quarter-\rux Hofstadter model\n(see Fig. S8), while the higher frequency bulk band gap\nis seen to host only two distinct edge modes between\n2\u0019\u00028:92 and 2\u0019\u00028:94 GHz. The third edge mode\nassociated with this band gap is predicted to be much\ncloser to the bulk modes as seen in Fig. S8. The top\nand bottom bulk bands are expected to have only four\nmodes, so the peak doublet near 2 \u0019\u00028:94 GHz is likely\nto be a hybridization of one bulk mode and one edge14\nParameterHamiltonian NotationValueReadout frequency!$/2$10.5835 GHzReadout linewidth&$/2$500 kHzQubit frequency!!/2$7.815 GHzQubit anharmonicity'/2$346 MHzBare lattice frequency!%/2$8.901 GHzReadout dispersive shift($/2$113 kHzSingle lattice site dispersive shift(&'/2$5.3 MHzCoupled lattice mode dispersive shift (mode at 8.8719 GHz)(%…)/2$(0.45 MHz)Single lattice site self Kerr)&'/2$81 kHzCoupled lattice mode self Kerr (mode at 8.8719 GHz)!%…)/2$(0.6 kHz)\nTABLE II. Parameters of qubit/lattice/readout res-\nonator system. Terms designated \\single lattice site\" are\nmeasured when only the lattice site to which the qubit is di-\nrectly coupled is tuned to the lattice frequency while the rest\nof the lattice sites are far detuned. The bare lattice frequency\nis the target frequency to which each lattice site is tuned be-\nfore the couplings to adjacent lattice sites are enabled. Values\nin parentheses are example quantities for the lattice mode in-\nvestigated in Fig. 3b and 3c.\nmode that are pushed closer in frequency than in the\ndisorder-free model. This trend of the top band being\ncompressed is consistent with the data taken in Fig. 2\nand is likely due to the presence of the opposite-chirality\nmodes in the chiral cavities. These modes are located\n2\u0019\u0002140\u00002\u0019\u0002200 MHz higher in frequency than the\nbare lattice frequency and are strongly coupled to the\nneighboring sites ( \u00182\u0019\u000220 MHz). The presence of these\nmodes can shift the higher frequency modes of the lattice\nby\u00182\u0019\u00024 MHz and compress the upper band gap.\nFig. S7b shows the same edge-edge transmission as probe\npower is varied. When the power is lowered, the lattice\nmodes acquire a dispersive shift from the qubit that is\nproportional to the lattice corner site's participation in\nthe mode.\nEdge modes of the lattice experience power dependent\nshifts of 2\u0019\u0002(1\u00182) MHz while modes constrained to the\nbulk of the lattice shift by much less than the linewidth\nof the mode.We calculate the eigenmodes and eigenvectors of an\nideal 5\u00025 Hofstadter Hamiltonian (see Fig. S8), at a \rux\nof\u0019\n2per plaquette. We see excellent qualitative agree-\nment of the eigenvalues of the modes with the data shown\nin Fig. 2. For a system this small there are only four\nmodes located in each bulk band gap for a total of eight\n\\edge\" modes. However, two of the edge modes in each\ngap are much closer to the bulk band frequency, which\ncauses the eigenvectors of the these modes to have more\nparticipation in the bulk. In Fig. 2, these two edge modes\nthat are isolated further from the edge mode have a\nstronger response in edge-edge transmission, while trans-\nmission through the modes located closer to the bulk\nbands starts to decrease in comparison. This transition\nfrom bulk to edge mode at the band edge is seen in larger\nsystems as well. For future experiments where we will use\nthe chiral channel to transport quantum states, photons\nmust be transferred into a superposition of edge modes in\norder to create a localized traveling wave packet. These\ntwo modes located closest to the center of the band gap\nwould be ideal modes to create a traveling single photon\nstate, since they are isolated from the bulk and are thus\nmore robust to disorder.\nSupplement I: Structure of Pulsed Measurements\nwith Qubit\nThe transmon qubit is controlled and read out disper-\nsively via drive and readout tones applied to its read-\nout resonator, shown enclosed in the left side of the blue\nboxes in Fig. 1a and enclosed in red in Fig. S9a. To gen-\nerate number splitting data like that shown in Fig. 3c, we\npopulate relevant eigenmodes with photons by providing\nlong and weak drive tones at these modes' frequencies\nthrough an antenna weakly coupled to the corner site of\nthe lattice most proximate to, and directly coupled to,\nthe qubit. While this weak drive is still being supplied\nto the lattice corner site, we excite the qubit from its\nground state by supplying its readout resonator with a\nlong Gaussian drive pulse ( \u001b= 1400 ns) that has a frac-\ntion of the amplitude it would take to fully place the qubit\nin its \frst excited state. We then measure the degree to\nwhich the qubit is excited from its ground state with\na readout tone applied at a range of frequencies in the\nvicinity of the original (zero-lattice-photon) qubit transi-\ntion. In this way we can characterize the photon-number-\ndependent shift of the qubit resonance, which depends on\nphotonic population in each lattice eigenmode that has\nenough edge participation to couple notably to the qubit.\nThis pulse sequence is diagrammed in Fig. S9a.\nTo generate Rabi oscillations between the qubit and\nlattice eigenmodes, as demonstrated in Fig. 3a and\nFig. 3b, we supply a sequential set of \u0019pulses at the\nqubit transition frequencies !geand!efto prepare the\nqubit in itsjfistate. We then supply a square pulse\nof varying length (0 \u00002000 ns) to the qubit's readout\nresonator to tune the qubit into resonance with targeted15\nSaturated bulk\nSaturated edge\nLow power edge\n8.84 8.86 8.88 8.90 8.92 8.94 8.96-0.006-0.004-0.0020.0000.002\nFrequency(GHz)Transmission(magnitude)\nBands Gap: edge modes Bands Gap: edge modes Bandsa\nb\nFIG. S7. Transmission spectroscopy through the lattice coupled to a qubit. Ina,the blue plot and the light red\nplot compare the transmission between two bulk cavities (site (3 ;3) and (2;3)) and two edge cavities (sites (1 ;4) and (3;1)) at\nhigh powers so that the qubit is saturated and the lattice is measured while e\u000bectively decoupled from the qubit. As predicted,\ntwo large gaps exist in the bulk-bulk transmission and modes in the edge-edge transmission reside in these gaps. As shown\nin Fig. S8, there are four edge modes per band, but two of the edge modes are quite close to the bulk band and share some\ncharacteristics with the bulk modes. The dark red plot in bis the edge-edge transmission taken at lower power so that the\nmodes are shifted by their coupling to the qubit. The modes shift in energy relative to the spectra at higher powers, showing\nthat they are strongly coupled to the qubit.\nFIG. S8. Simulations of the eigenmodes of the 5\u00025quarter-\rux Hofstadter Hamiltonian. The eigenvectors of\nthe \frst 13 modes of the lattice are calculated and plotted versus these modes' frequencies. Only these 13 modes are shown\nbecause they represent all unique eigenvectors of the system, as the modes are symmetric around the middle (13th) mode. In\na system of this size, there are two modes that are localized strongly to the edge of the system, and two modes located in the\ngap but close enough to the bulk bands that they have more participation in the bulk lattice sites. The 13th mode is located\nwhere the two middle bulk bands touch and maintains its large delocalization for a lattice of any size. Each mode's color scale\nis normalized to the site in that mode with the largest participation.16\nt\n19.9 μs 5.6 μs 2.1 μs\nt\n0-2,000 ns 2,100 ns 252.6 ns148 ns 148 ns24.6 μsb\ncLattice photon number splitting experiment\nQubit/lattice mode swap experimentQubit drive \n+ readoutLattice drive a\nFIG. S9. Pulse sequences used for Fig. 3. Ina,Locations\nof qubit drive, readout tone, and drive on the lattice corner\nsite are shown in a schematic cross-section of the lattice edge.\nThe transmon chip is represented in yellow, while the readout\ncavity is boxed in red and the lattice corner site is boxed in\nblue. b,To measure the photon-number-dependent shift in\nthe qubit resonance resulting from photons populating the\nlattice eigenmodes, we apply a very long (24.6 \u0016s) weak drive\n(blue pulse) at lattice eigenmode frequencies to the corner site\nof the lattice which is directly coupled to the qubit. After 19.9\n\u0016s, we apply a weak Gaussian pulse of \u001b= 1400 ns (yellow)\nat!qand then read out (purple pulse) the qubit absorption\nafter the lattice eigenmode drive is completed. c,To generate\nRabi oscillations between qubit and a swath of the lattice\neigenspectrum, we prepare the qubit in its jfistate with two\n\u0019pulses at!q=!ge(yellow) and !ef(red). We drive dressed\nvacuum Rabi oscillations with a square pulse of varying length\nat!f0$g1= (!ge+!ef\u0000!l), apply a\u0019pulse at!ef(red)\nto enable readout on the jgi$jeitransition, and read out\n(purple).\nlattice eigenmodes and drive oscillations between qubit\nand lattice. We \fnally supply an additional \u0019pulse at\n!efbefore reading out the qubit state. This pulse se-\nquence is diagrammed in Fig. S9b.\nSupplement J: Mode Dependence of Dispersive Shift\nThe transmon qubit is directly coupled to a single cor-\nner site of the lattice, as depicted at top left of Fig. 1a\nand in Fig. 1c. If all other sites in the 5 \u00025 lattice are\ndetuned from this resonator, each photon in the corner\nsite induces a shift of the qubit jgi$jeitransition of\n2\u001f, with\u001fLS\u0019g2\nl\n\u0001\u0002\u000bq\n\u0001+\u000bq\u00192\u0019\u00025:3 MHz. When all\nother lattice sites are tuned to resonance and hybridize\nforming the band structure, the dispersive shift is diluted\nbetween the modes of the band structure. Lattice mode\nlexperiences a dispersive shift 2 \u001fl= 2\u001fLS\u0002jhuljLSij2,\n/uni0000001b/uni00000011/uni0000001b/uni00000019 /uni0000001b/uni00000011/uni0000001b/uni0000001b /uni0000001b/uni00000011/uni0000001c/uni00000013 /uni0000001b/uni00000011/uni0000001c/uni00000015 /uni0000001b/uni00000011/uni0000001c/uni00000017\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|l|/uni00000003/uni0000000b/uni00000030/uni0000002b/uni0000005d/uni0000000c\n/uni00000037/uni0000004b/uni00000048/uni00000052/uni00000055/uni0000005c\n/uni00000027/uni00000044/uni00000057/uni00000044FIG. S10. Mode-dependent dispersive shift of qubit.\nTheoretical predictions of the dispersive shift between the\nqubit and each lattice eigenmode are compared to measured\nvalues from number splitting data. An example plot of num-\nber splitting due to population of a particular lattice eigen-\nmode is seen in inset c.of Fig. 3. Theory is based on the\nmeasured (single-site) \u001fLSscaled by the spectral weight of\neach lattice mode at the lattice corner site which is coupled\nto the qubit. Only modes present in the chevron spectrum of\nthe lattice pictured in Fig. 3 are included. The separation be-\ntween 0- and 1-photon peaks when mode lis driven is de\fned\nas 2\u001fl. Expected regions of the bulk bands, estimated from\nmode indices, are highlighted in gray. Error bars are derived\nfrom \fts to number splitting data.\nwherejuliis the wavefunction of mode l, andjLSiis\nthe wavefunction of a photon localized on the corner site\ncoupled to the transmon. Note thatP\nl\u001fl=\u001fLS.\nThe mode employed in Fig. 3c, with index 7 in Fig. S8,\nexhibits a shift per photon of 2 \u001f7where\u001f7= 2\u0019\u0002\n0:45 MHz, extracted from the frequency di\u000berence be-\ntween zero- and one-photon resonances of Fig. 3c. This\nmode has a wavefunction overlap with the corner site of\njhu7jLSij2= 0:09, and thus we anticipate a shift of 2 \u001f7=\n2\u001fLS\u0002jhu7jLSij2= 2\u00022\u0019\u00025:3 MHz\u00020:09\u00192\u00022\u0019\u0002480\nkHz, in agreement with the measured \u001f7= 2\u0019\u0002450 kHz.\nIn Fig. S10 we compare, for each mode, the predicted\nshift with the shift extracted from the observed split-\ntings between zero- and one-photon peaks of additional\nnumber splitting measurements.\nSupplement K: Numerical Modeling of the\njg0i$jf1iOscillations in Fig. 3b\nTo provide a \ft for the stimulated vacuum-Rabi os-\ncillations in the data plotted in Fig. 3b, we numerically\nsimulate a model for a transmon qubit coupled to two\nmodes through the jg0i$jf1idriven process. We take\ninto consideration the transmon population in the jgi,\njei, andjfistates along with the photon population in\na selected pair of lattice eigenmodes over a timescale of\n2\u0016s. We write the Hamiltonian of the qubit coupled to17\nour selected subset of the lattice eigenspectrum in the\nrotating frame of the classical drive\nH=~= (!q\u0000!d) ^ay\nq^aq\u0000\u000bq\n2^ay\nq^ay\nq^aq^aq\n+X\nl=7;8(!l\u0000!d) ^ay\nl^al+gl\u0010\n^ay\nq^ay\nq^al+ ^aq^aq^ay\nl\u0011\n:(S1)\nThe classical drive induces a resonant qubit-mode cou-\npling at!d= 2!q\u0000\u000bq\u0000!lfor each mode l2f7;8g. The\ndynamics is simulated by incorporating the photon loss\nin the transmon (\u0000) and lattice modes ( \u0014) and solving\nthe master equation in the standard Lindblad form\n_\u001a=\u0000i\n~[H;\u001a] +D[p\u0014(^a7+ ^a8)]\u001a+D[p\n\u0000^aq]\u001a; (S2)\nusing the usual de\fnition for the damping superoperator\nD[^L]\u001a=^L\u001a^Ly\u00001\n2f^Ly^L;\u001ag. In our numerics, we restrict\nthe transmon and lattice modes to accommodate up to\nfour excitations each and further truncate the Hilbert\nspace to manifolds that have a maximum of four total\nexcitations in the combined system. Initializing the sys-\ntem with the qubit excited to its jfistate and the lattice\nmodes empty of photons, we calculate the qubit excitedstate populations P(jfi) andP(jei) during the exchange\ndynamics, along with photon populations ^ nl= ^ay\nl^ain the\ntwo lattice eigenmodes. The simulation results are dis-\nplayed in Fig. S11. Further simulation results, plotted for\na range of detunings of drive frequency from band cen-\nter, are shown alongside data in Fig. S12. We \fnd very\ngood agreement with the data in Fig. 3b if the measured\ntransmon excited state population includes population\nin thejeistate explained by the jfi!jeidecay during\nthe dynamics. Additionally, simulation results illustrate\nthat it is possible to selectively drive exchange dynamics\nbetween the qubit and primarily a single lattice mode by\nselecting a drive so that the dressed state j~eiis resonant\nwith the targeted mode.\nSupplement L: Estimating Uncertainties for Fig. 3\nWe estimate the errors for Fig. 3b & c as the stan-\ndard deviation of signal-free regions of the datasets. For\nFig. 3b, the data comes from the slice at t= 0 of\nFig. 3a. For Fig. 3c, the data comes from the zero-\ndrive-power slice of Fig. 3c, inset, for readout detunings\n\u000e2[\u00005:5;\u00001:5] MHz.18\na\nb\nFIG. S11. Simulated qubit excited state population for Fig. 3b. Simulated populations P(jfi),P(jei), andP(jgi) in\nthe transmon qubit's jfi,jei, andjgistates are plotted in time. In a,we retrieve a \ft (yellow) to the measured data (blue\npoints) plotted in Fig. 3b and taken at the frequency of the dressed j~eistate singled out by blue line in Fig. 3a. To achieve this\n\ft we represent the qubit excited state population as P(jfi) +P(jei). Photonic occupancies n7(green) and n8(purple) of the\neigenmodes selected for simulation are also plotted. It can be seen that the occupancy n7of the lattice eigenmode resonant with\nthe dressedj~eistate oscillates qualitatively out of phase with the qubit excited state population P(jfi) +P(jei), consistent\nwith a swap of excitation between qubit and mode. In b,we retrieve a \ft (yellow) to the measured data (red points) taken\nat the frequency of the dressed j~eistate close to resonance with a nearby lattice eigenmode to the one targeted in a. The\nfrequency of the dressed j~eistate at which this data was taken is singled out by a red line in Fig. S12a.19\nc d\nf eb a\nFIG. S12. Simulated observables for qubit/lattice eigenmode swap experiment Ina,a subset of Fig. 3a at a narrower\nrange of dressedj~eistate frequencies is overlaid with vertical lines which point out the frequencies at which data in Fig. S11a\n(blue line) and Fig. S11b (red line) were taken. b,Simulated qubit excited state population, rendered as P(jfi) +P(jei), is\nplotted over a range of drive detunings from band center shared by the data in a. Note that this simulation is performed for\na system of a qubit coupled to two lattice eigenmodes, and does not incorporate contributions from coupling to other regions\nof the lattice eigenspectrum. c,SimulatedP(jei) is plotted for the range of drive detunings from band center used in a.d,\nSimulatedP(jfi) is plotted for the range of drive detunings from band center used in a.e,Simulated occupancy n7=h^ay\n7^a7i\nof the seventh lattice eigenmode is plotted for the range of drive detunings from band center used in a. The drive frequency\nat which the qubit dressed j~eistate is brought near resonance with this lattice eigenmode is overlaid with a blue vertical line\nina. Note that this eigenmode is the edge mode for which swap data is plotted in Fig. 3b and for which \u001fl=\u001f7and!l=!7\nare quoted in Table II. f,Simulated occupancy n8=h^ay\n8^a8iof the eighth lattice eigenmode is plotted for the range of drive\ndetunings from band center used in a. The drive frequency with which the qubit dressed j~eistate is brought near resonance with\nthis lattice eigenmode is overlaid with a red vertical line in a. Together, eandfillustrate that under the selected simulation\nparameters, it is possible to drive oscillations in the photonic occupancy of each of the pair of lattice eigenmodes separately by\nselecting di\u000berent drive frequencies." }, { "title": "1808.06690v1.Tunable_Magnonic_Thermal_Hall_Effect_in_Skyrmion_Crystal_Phases_of_Ferrimagnets.pdf", "content": "Tunable Magnonic Thermal Hall E\u000bect in Skyrmion Crystal Phases of Ferrimagnets\nSe Kwon Kim,1, 2Kouki Nakata,3Daniel Loss,4, 5and Yaroslav Tserkovnyak1\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA\n3Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan\n4Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: August 22, 2018)\nWe theoretically study the thermal Hall e\u000bect by magnons in skyrmion crystal phases of ferri-\nmagnets in the vicinity of the angular momentum compensation point (CP). To this end, we start\nby deriving the equation of motion for magnons in the background of an arbitrary equilibrium spin\ntexture, which gives rise to the \fctitious electromagnetic \feld for magnons. As the net spin density\nvaries, the resultant equation of motion interpolates between the relativistic Klein-Gordon equation\nat CP and the nonrelativistic Schr odinger-like equation away from it. In skyrmion crystal phases,\nthe right- and the left-circularly polarized magnons with respect to the order parameter are shown\nto form the Landau levels separately within the uniform skyrmion-density approximation. For an\nexperimental proposal, we predict that the magnonic thermal Hall conductivity changes its sign\nwhen the ferrimagnet is tuned across CP, providing a way to control heat \rux in spin-caloritronic\ndevices on the one hand and a feasible way to detect CP of ferrimagnets on the other hand.\nIntroduction. |Magnetic skyrmions are swirling spin\ntextures, which are characterized by the topological\nskyrmion number de\fned in terms of the real-space spin\ncon\fguration [1]. Their topological characteristic in\ru-\nences not only the dynamics of themselves, e.g., by en-\ngendering the Magnus force, but also the dynamics of\nelectrons moving through them [2]. In particular, when\nferromagnetic skyrmions form a crystal lattice, electrons\nwhose spin follows the local spin texture adiabatically ex-\nperience the Lorentz force due to the \fctitious magnetic\n\feld proportional to the skyrmion density and thereby\nexhibit the so-called topological Hall e\u000bect [3]. Recently,\nthere has been a growing interest in skyrmion crystals in\nantiferromagnets and their electronic transport proper-\nties [4, 5] because of their fundamental di\u000berences from\nferromagnetic counterparts as well as technological ap-\nplications for THz-speed magnetic devices [6].\nMagnons, which are quanta of spin waves [7], can\ntransport information and exhibit topological phenom-\nena similarly to electrons. Their potential ability to\nrealize devices based on insulating magnets, which are\nfree from drawbacks of conventional electronics such as\nsigni\fcant energy loss due to Ohmic heating, has led\nto the emergence of magnon-based spintronics [8]. In\nskyrmion crystal phases of ferromagnets, magnons have\nbeen shown to experience the \fctitious magnetic \feld\nby keeping their spin antiparallel to the local spin tex-\nture [9]. As a result, magnons form the approximate\nLandau levels with the \fnite Berry curvature [10], caus-\ning the thermal Hall e\u000bect [11{13]. However, the magnon\nbands and their transport properties in antiferromagnetic\nskyrmion crystals have not been studied.\nIn this Letter, we \fll this gap by investigating a\ntheoretically more general problem: The dynamics of\nmagnons in the presence of skyrmion crystals in ferrimag-\nferrimagnet in skyrmion crystal phasexyrT\njQs=0s<0s>0FIG. 1. Schematic illustration of the magnonic heat \rux\njQthrough a ferrimagnet in its skyrmion crystal phase sub-\njected to a temperature gradient rT. The colored small ar-\nrows depict a single skyrmion texture of the order parameter\nn. Magnons can exhibit the thermal Hall e\u000bect since the\nskyrmion crystal gives rise to the \fctitious magnetic \feld,\nwhich magnons of left and right circular polarization (with\nrespect to the order parameter) experience as if they carry\nthe positive and the negative charge, respectively. The in-\nduced transverse heat \rux changes its direction as the net\nspin density salong nvaries across 0.\nnets exhibiting the angular momentum compensation\npoint (CP) [14], at which the net spin density vanishes,\nbut the magnetization can be \fnite. One class of such\nferrimagnets is rare-earth transition-metal alloys such as\nGdFeCo or CoGd whose net spin density can be tuned\nby varying either temperature [15] or chemical composi-\ntion [16]. To this end, we start by deriving the equation\nof motion for magnons in the presence of an arbitrary\nspin texture, which includes the \fctitious electromag-\nnetic \feld. The obtained equation of motion is reduced\nto the nonrelativistic Schr odinger-like equation for ferro-\nmagnetic magnons away from CP and to the relativistic\nKlein-Gordon equation for antiferromagnetic magnons at\nCP, interpolating between the dynamics of ferromagnets\nand that of antiferromagnets as previously shown for thearXiv:1808.06690v1 [cond-mat.mes-hall] 20 Aug 20182\ndynamics of domain walls and skyrmions [17, 18]. In the\npresence of a skyrmion crystal, two species of magnons\nwith right and left circular polarization (with respect to\nthe order parameter) will be shown to experience the \fc-\ntitious magnetic \felds of opposite directions and form\nthe Landau levels separately, realizing two-dimensional\nmagnonic topological insulators [19]. As an experimen-\ntal proposal, we will show that the thermal Hall conduc-\ntivity changes its sign as the ferrimagnet is tuned across\nCP. See Fig. 1 for a schematic illustration. One promis-\ning platform is o\u000bered by GdFeCo \flms, where isolated\nskyrmions have been observed [20] and the antiferromag-\nnetic domain-wall dynamics has been demonstrated at\nCP [17]. The proposal provides not only a feasible way\nto control the direction of the thermal \rux, which can\nbe useful in spin caloritronics [21], but also a thermal\ntransport measurement for determining CP, which can\ncomplement the other methods based on magnetic reso-\nnances [15, 22] or domain-wall speed measurements [17].\nGeneral formalism. |Our model system is a collinear\nferrimagnet, whose potential energy density is given by\nU[n] =A(rn)2=2 +Dn\u0001(r\u0002n)\u0000h\u0001n;(1)\nwhere n(r;t) is the three-dimensional unit vector repre-\nsenting the direction of the magnetic order. Here, the\n\frst term is the exchange energy; the second term is\nthe Dzyaloshinskii-Moriya interaction (DMI), which ex-\nists when the inversion symmetry is broken [23]; the last\nterm represents the Zeeman coupling between the exter-\nnal \feld hand the magnetization along the direction of\nthe order parameter. Here, we are neglecting the other\nterms such as the dipolar interaction by following the\nprevious literature on chiral magnets [24]. With a suit-\nable choice of the coe\u000ecient values, the ground state\nis a skyrmion crystal [24], which will be the phase of\nour main interest for later discussions. The equilibrium\norder-parameter con\fguration will be denoted by n0.\nThe dynamics of the order parameter nof the ferrimag-\nnet can be described by the following Landau-Lifshitz-\nlike equation [17, 18, 25]:\ns_n+\u001an\u0002n=n\u0002\u0000\nAr2n\u00002Dr\u0002n+h\u0001\n;(2)\nwheresis the equilibrium spin density and \u001a\nparametrizes the inertia associated with the dynamics\nof the order parameter. The left-hand side is the time\nderivative of the net spin density, s=sn+\u001an\u0002_n, the\nformer and the latter of which are the longitudinal and\nthe transverse component of the spin density with re-\nspect to the order parameter, respectively [26]. Conven-\ntional ferromagnets and antiferromagnets have only the\n\frst and the second term, respectively, on the left-hand\nside. The parameter of focus in this work is the spin den-\nsitys, which can be varied across zero. We are interested\nin the change of the magnon bands as a function of it.\nTo obtain the equation of motion for a magnon, which\nis a quantum of small-amplitude \ructuations from theequilibrium state, we use the local coordinate system\nn0, where the equilibrium state is in the positive zdi-\nrection, n0\n0\u0011^z[11, 27]. The transformation can be\nimplemented by a three dimensional rotation matrix R\nsatisfying n0=Rn0\n0. We will use one explicit realiza-\ntion of it in this work: R= exp(\u001e0Lz) exp(\u00120Ly) for\nn0= (sin\u00120cos\u001e0;sin\u00120sin\u001e0;cos\u00120) whereLyandLz\nare the generators of the rotations about the yand thez\naxis, respectively [28].\nThe equation of motion for magnons can be obtained\nfrom Eq. (2) to linear order in the \ructuation \u000en0\u0011n0\u0000\n^z=n0\nx^x+n0\ny^y. Since the equation is second order in\ntime derivative, there are two types of solutions. It is\nconvenient to represent the two monochromatic solutions\nwith the complex \felds: +=n0\nx\u0000in0\ny/exp(\u0000i\u000ft=~) for\nright-circularly polarized magnons and \u0000=n0\nx+in0\ny/\nexp(\u0000i\u000ft=~) for left-circularly polarized magnons where\n\u000fis the magnon energy. We will refer the former and the\nlatter to the positive-chirality ( q= 1) and the negative-\nchirality (q=\u00001) solutions, respectively. The equation\nof motion for a magnon of chirality qis given by\n\u0000qs\u0012\ni@t\u0000q\u001e\n~\u0013\n q+\u001a\u0012\ni@t\u0000q\u001e\n~\u00132\n q=A\u0012r\ni\u0000qa\n~\u00132\n q;\n(3)\nwhich is our \frst main result. See Supplemental Ma-\nterial for its derivation [29]. Here, \u001eis the texture-\ninduced scalar potential given by \u001e=~(R\u00001@tR)12=\n\u0000~cos\u00120@t\u001e0, whereM12represents a corresponding\nmatrix element of M. The vector potential consists of\ntwo contributions [11]: a=at+ad, where the \frst term\nis from the exchange energy, at\ni=\u0000~(R\u00001@iR)12=\n~cos\u00120@i\u001e0, and the second term is from the DMI,\nad=\u0000(~D=A)n0. The texture-induced \fctitious elec-\ntric and magnetic \felds are given by\net\ni=\u0000@i\u001e\u0000@tat\ni=~n0\u0001(@tn0\u0002@in0); (4)\nbt\ni=\u000fijk@jat\nk= (~=2)\u000fijkn0\u0001(@kn0\u0002@jn0);(5)\nin the Einstein summation convention. The obtained\n\felds are identical to those for electrons [2, 30] and\nmagnons [11, 27] in ferromagnets. The DMI-induced\nvector potential adgives rise to another contribution\nto the \fctitious \felds, ed=\u0000@tad= (~D=A)@tn0and\nbd=\u0000(~D=A)r\u0002n0[11]. Note that the chirality\nq=\u00061 of a magnon serves as its charge with respect\nto the electromagnetic \felds, which can be understood\nas follows. Since the positive- and negative-chirality\nmagnons carry spin whose directions are locked paral-\nlel and antiparallel with respect to the background spin\ntexture n0[31], they pick up the Berry phase with the op-\nposite signs and experience the opposite \fctitious electro-\nmagnetic \felds [32]. During the derivation, we neglected\nthe second and higher order terms in \u001eandaand the\nterm proportional to the external \feld, by focusing on\nhigh-energy magnons whose wavelength is much smaller3\nthan the spatial extension of the texture and whose ki-\nnetic energy dominates the Zeeman energy, which we will\nrefer to as the exchange approximation [33].\nAt CP, the equilibrium spin density vanishes, s= 0,\nand the nature of the dynamics becomes antiferromag-\nnetic. The equation of motion is then reduced to the fol-\nlowing Klein-Gordon equation [34] which describes the\ndynamics of a relativistic particle with charge qin the\npresence of an electromagnetic \feld:\n(i~@t\u0000q\u001e)2 q=c2\u0012~\nir\u0000qa\u00132\n q; (6)\nwherec\u0011p\nA=\u001a is the characteristic speed that is the\nmagnon speed in the absence of electromagnetic \felds.\nThis equation describing the dynamics of magnons in\nantiferromagnets moving through a general spin texture\nhas not been derived before except for a special case of a\none-dimensional domain wall [31, 34].\nWhen su\u000eciently distant from CP, a ferrimagnet has\nenough spin density sto neglect the inertial term /\u001a\nin Eq. (2) for the low-energy dynamics. The equation of\nmotion (3) for ferrimagnetic magnons is then reduced to\nthat for ferromagnetic magnons [11, 27]:\n\u0000sgn(qs)i~@t q=\"\n1\n2m\u0012~\nir\u0000qa\u00132\n\u0000sgn(s)\u001e#\n q;\n(7)\nwith the e\u000bective mass m=~jsj=2A, which resembles the\nSchr odinger equation for a nonrelativistic charged parti-\ncle subjected to an electromagnetic \feld.\nIt is instructive to discuss the solutions to Eq. (3) in\nthe absence of the scalar and the vector potentials \u001e= 0\nanda=0, which are given by the plane-wave solutions,\n q/exp(ik\u0001r\u0000i\u000ft=~) [35]. The energy-momentum\nrelation is given by\n\u000fq(k) =p\n(mc2)2+~2c2k2+ sgn(qs)mc2: (8)\nThe solution to Eq. (6) for an antiferromagnetic magnon\nis given by the high-kinetic energy limit, i.e., ~jkj\u001dmc:\n\u000f\u0006\u0019~cjkj. The two solutions are degenerate at the\nlevel of approximation taken in Eq. (6) where the time\nreversal symmetry is respected by having vanishing spin\ndensitys= 0. The lower-energy solution to Eq. (7) for a\nferromagnetic magnon is given by the low-kinetic energy\nlimit, i.e., ~jkj\u001cmc:\u000fq\u0019~2k2=2mwith the chirality\nq=\u0000sgn(s). Note that spin of the low-energy magnons\nis locked antiparallel to the direction of the background\nspin density sn0. Here, the momentum scale governing\nthe separation between a nonrelativistic and a relativistic\nregime is given by mc=~jsj=2pA\u001a.\nMagnon in a skyrmion crystal. |Now, let us apply the\nabove formalism to one speci\fc example: Magnons in a\nskyrmion crystal of a quasi-two-dimensional ferrimagnet.\nWe will assume that the skyrmion crystal is static, for\nn=0n=1n=2n=3n=0n=1n=2n=3\nsl/2⇢c✏l/~c\n012\u00001\u000021p3p5p7q=\u0000q=+q=+q=-FIG. 2. The plot of the Landau levels [Eq. (11)] of magnon\nbands in ferrimagnetic skyrmion crystals in terms of the\nrescaled energy \u000fl=~cand the rescaled spin density sl=2\u001ac.\nThe solid gold and the dashed blue lines represent the right-\ncircularly polarized ( q= +) and the left-circularly polarized\n(q=\u0000) magnon bands, respectively.\nwhich the \fctitious electric \feld vanishes. A skyrmion is\ncharacterized by its integer skyrmion number [36]:\nQ=1\n4\u0019Z\ndxdyn0\u0001(@xn0\u0002@yn0)\u0011Z\ndxdy\u001asky;(9)\ncounting how many times the order parameter n0wraps\nthe unit sphere. Under suitable conditions, skyrmions\nwith the de\fnite skyrmion number can crystalize in the\ntriangular lattice as observed in several ferromagnetic\nmaterials [2], giving rise to the \fnite skyrmion number\ndensity per unit area, which we denote by \u001asky. The\nassociated \fctitious magnetic \feld [Eq. (5)] is given by\nbt=\u00004\u0019~\u001asky^z: (10)\nThe spatial pro\fle of the magnetic \feld depends on the\ndetailed values of material parameters, making it cum-\nbersome to take into account analytically. Therefore,\nbelow, we will account for its e\u000bects by spatially aver-\naging it: b=\u00004\u0019~h\u001askyi^z. The corresponding mag-\nnetic length is given by l=p\n~=jbzj= 1=p\n4\u0019jh\u001askyij,\nwhich is proportional to the distance between neighbor-\ning skyrmions. The DMI-induced contribution bdvan-\nishes after spatial averaging. In addition, we will assume\nthe negative skyrmion density \u001asky<0 and thus bz>0\nwithout loss of generality for subsequent discussions.\nTo solve Eq. (3), we adopt the known results for the\nLandau levels of a nonrelativistic charged particle sub-\njected to a uniform magnetic \feld [29, 37]. Plugging the\nmonochromatic function, q(r;t) = exp(\u0000i\u000ft=~) nn0(r)\ninto Eq. (3), where nn0(r) is the known eigenfunction of\nthe right-hand side of Eq. (3) for the nth Landau level ( n0\nis the index for states within each Landau level), yields\nthe following solutions:\n\u000fq\nn=p\n(mc2)2+~c2bz(2n+ 1) + sgn( qs)mc2:(11)4\n012\u00001\u0000201\u00001sl/2⇢c+rT-jQs<0s>0+rT-jQxy(a)(b)(c)-+2\u00002¯yx(⇥102)\n-2 -1 0 1 2-200-1000100200\n-200-1000100200\nFIG. 3. (a) The plot of the rescaled thermal Hall conduc-\ntivity \u0016\u0014yx\u0011(2\u0019~t=k2\nBT)\u0014yx, which parametrizies the ratio\nof the induced transverse heat \rux jy\nQto the applied lon-\ngitudinal temperature gradient @xTfor the ferrimagnet \flm\nof thickness t. (b) and (c) the schematic illustrations of the\npositive-chirality (+) and the negative-chirality ( \u0000) magnon\nmotions subjected to a temperature gradient rT= (@xT)^x\nand the resultant transverse energy \rux jQ=jy\nQ^y.\nThese magnon bands in a ferrimagnetic skyrmion crystal\nwithin the approximation of the uniform skyrmion den-\nsity is our second main result. The number of states\nin one Landau level is given by the total number of\nthe \fctitious magnetic \rux quanta through the plane,\nwhich is twice the total number of skyrmions in the sys-\ntem. The massless relativistic limit is given by \u000f\u0006\nn\u0019\ncp\n~bz(2n+ 1), which agrees with the known result for\nthe Klein-Gordon equation [38]. The lower band in the\nmassive limit, mc2\u001dcp\n~bz(2n+ 1), is reduced to the\nsolution for nonrelativistic particles: \u000fn\u0019(~bz=2m)(2n+\n1). The solution can be cast into the dimensionless form\nin terms of the rescaled energy \u0018\u0011\u000fl=~cand the rescaled\nspin density \u0010\u0011sl=2\u001ac:\u0018\u0006\nn=p\n\u00102+ 2n+ 1\u0006\u0010. See\nFig. 2 for the plot. The solid gold and the dashed blue\nline represent the right-circularly polarized ( q= +) and\nthe left-circularly polarized ( q=\u0000) magnon bands, re-\nspectively. The xaxis of Fig. 2 can be swept through the\norigin by varying the temperature across CP, the physical\nimplication of which is discussed below.\nTunable thermal Hall e\u000bect. |Each \rat Landau level of\nmagnons [11, 12] has the Chern number \u00170=\u0000q, which\nis the integral of the uniform Berry curvature de\fned in\nterms of the magnonic wavefunction over the Brillouin\nzone [29]. Magnons with the \fnite Berry curvature can\ngive rise to the thermal Hall e\u000bect [39, 40], a phenomenon\nof the generation of the transverse energy \rux jy\nQupon\nthe application of the longitudinal temperature gradient\n@xT:jy\nQ=\u0000\u0014yx@xT, which is quanti\fed by the thermal\nHall conductivity \u0014yx[41]. Within the linear response\ntheory, the thermal Hall conductivity for our case is given\nby\u0014yx= (k2\nBT=2\u0019~t)P\nn[c2(\u001aB(\u000f\u0000\nn))\u0000c2(\u001aB(\u000f+\nn))],wheretis the thickness of the ferrimagnet, c2(x) =\n(1+x)(log(1+x\u00001))2\u0000(logx)2\u00002Li2(\u0000x), Li 2(z) is the\npolylogarithm function, and \u001aB(\u000f) = [exp(\u000f=kBT)\u00001]\u00001\nis the Bose-Einstein distribution [39]. Figure 3(a) shows\nthe plot of the rescaled thermal Hall conductivity \u0016 \u0014yx\u0011\n(2\u0019~t=k2\nBT)\u0014yxas a function of the rescaled spin den-\nsity\u0010=sl=2\u001ac. The plot is drawn with the following\nvalue for the ratio of the characteristic relativistic energy\nscale to the temperature: ~cl\u00001=kBT\u00180:03, which is ob-\ntained from the magnon speed c\u0018104m/s calculated for\nGdFeCo [18], the magnetic length l= 1=p\n4\u0019jh\u001askyij\u0018\n10 nm for the inter-skyrmion distance \u001850 nm of tri-\nangular lattice of skyrmions observed in chiral ferromag-\nnets [42], and the temperature T= 70 K. The induced\ntransverse heat \rux changes its direction as the net spin\ndensity varies across zero, at which magnons of two chi-\nralities are degenerate and thus the thermal Hall e\u000bect is\nabsent similarly to antiferromagnets [19]. When the spin\ndensity is negative, for example, there are more positive-\nchirality magnons than the others, causing the negative\nthermal Hall conductivity as shown in Fig. 3(b). Here,\nwe remark that, although the numerical results shown in\nFig. 3(a) are obtained within the approximation of the\nuniform \fctitious magnetic \feld, the sign change of \u0014yx\nats= 0 is the generic property of Eq. (3) under the time\nreversal and thus does not rely on the approximation.\nNext, let us estimate the change of the thermal Hall\nconductivity \u0001 \u0014yxas the net spin density svaries by\n\u0001s\u00185\u000210\u00008J\u0001s/m3, which can be achieved by changing\nthe temperature by \u0001 T= 10 K around CP of GdFeCo\n\flms according to the numerical results in Ref. [17]. Here,\nwe assume that all the parameters except the spin den-\nsity are constant. By using the inertia \u001a\u0018~2=Jd3ob-\ntained with the Heisenberg exchange energy J\u00185 meV\nand the lattice constant d\u00180:5 nm, the above \u0001 syields\n\u0001\u0014yx\u00180:05 W/K\u0001m for 50-nm thick \flms, which is com-\nparable to the large thermal Hall conductivities observed\nin frustrated magnets [43].\nDiscussion. |By investigating the magnon bands in\na skyrmion crystal of a ferrimagnet in the vicinity of\nCP, we have shown that, under certain approxima-\ntions, the positive-chirality and the negative-chirality\nmagnons form Landau levels separately by experiencing\nthe skyrmion-induced \fctitious magnetic \feld, leading\nto the identi\fcation of ferrimagnets in their skyrmion-\ncrystal phases as magnonic topological insulators. We\nhave predicted that the sign of the resultant thermal Hall\nconductivity changes its sign across CP. We mention here\nthat the Berry curvature of the magnon bands will cause\nthe spin Nernst e\u000bect [21, 44], a phenomenon of the gen-\neration of a transverse spin current by a longitudinal tem-\nperature gradient, in addition to the thermal Hall e\u000bect.\nThe former, respecting the time reversal symmetry, will\nbe \fnite even at CP unlike the latter disappearing at CP.\nS.K.K. acknowledges the enlightening discussions with\nOleg Tchernyshyov and Ari Turner. S.K.K. and Y.T.5\nwere supported by the Army Research O\u000ece under Con-\ntract No. W911NF-14-1-0016. K.N. was supported\nby Leading Initiative for Excellent Young Researchers,\nMEXT, Japan. D.L. was supported by the Swiss Na-\ntional Science Foundation, the NCCR QSIT, and JSPS\nKAKENHI Grant Numbers 16K05411.\n[1] A. N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP\n68, 101 (1989); A. Bogdanov and A. Hubert, J. Magn.\nMagn. Mater. 138, 255 (1994).\n[2] N. Nagaosa and Y. Tokura, Nat. 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Lett. 117, 217202 (2016).7\nSupplemental Material: Tunable Magnonic Thermal Hall E\u000bect in Skyrmion Crystal Phases of\nFerrimagnets\nSe Kwon Kim,1;2Kouki Nakata,3Daniel Loss,4;5and Yaroslav Tserkovnyak1\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA\n3Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan\n4Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: August 20, 2018)\nThis Supplemental Material contains the derivation of the equations of motion for magnons and the\nsummary of the known results for the Landau levels of a charged particle.\nTHE DERIVATION OF THE EMERGENT ELECTROMAGNETIC FIELDS.\nWe use the three-dimensional rotation matrix Rthat transforms the zaxis to the equilibrium con\fguration\nn0:n0=Rn0\n0with n0\n0\u0011^z. One explicit form for Ris given byR= exp(\u001e0Lz) exp(\u00120Ly) for n0=\n(sin\u00120cos\u001e0;sin\u00120sin\u001e0;cos\u00120) whereLx;Ly, andLzare the generators of the three-dimensional rotations about\nthex;y; andzaxis given by\nLx=0\n@0 0 0\n0 0\u00001\n0 1 01\nA;Ly=0\n@0 0 1\n0 0 0\n\u00001 0 01\nA;Lz=0\n@0\u00001 0\n1 0 0\n0 0 01\nA: (S1)\nNote that they can be de\fned in terms of the Levi-Civita symbol: ( La)bc=\u0000\u000fabc. The derivatives of n=Rn0can\nthen be expressed in terms of the covariant derivatives of n0as follows:\nR\u00001@\u0016n= (@\u0016+R\u00001@\u0016R)n0\u0011(@\u0016+At\n\u0016)n0; (S2)\nwhere\u0016= 0 denotes the temporal derivative and \u0016= 1;2, and 3 denote the spatial derivatives with respect to the\ncoordinates x;y, andz, respectively. The e\u000bects of the background spin texture on top of which a magnon lives are\ncaptured by the matrices At\n\u0016, which is skew-symmetric due to the orthonormality of the rotation matrix R.\nExplicitly, we have the following transformation of each derivative term in the equations of motion by keeping the\nterms that include the derivative of the order parameter n0at least once:\n@\u0016n=R(@\u0016+At\n\u0016)n0; (S3)\n@2\n\u0016n=R(@\u0016+At\n\u0016)2n0: (S4)\nThe transformation of the DMI is as follows:\nr\u0002n=RRT(r\u0002n) (S5)\n=R\u0002\nRT(r\u0002Rn0)\u0003\n: (S6)\nThe factor inside the square bracket is given by\n\u0002\nRT(r\u0002Rn0)\u0003\na=RT\nac\u000fcde@dRefn0\nf (S7)\n=\u0002\nAd\nb@bn0\u0003\na; (S8)\nwhere\nAd\nb=RTLbR: (S9)\nIn terms of the n0, the Landau-Lifshitz-Gilbert Eq. (3) is given by\ns(@t+At\n0)n0+\u001an0\u0002(@t+At\n0)2n0=n0\u0002A\u0000\n@i+At\ni\u0000(D=A)Ad\ni\u00012n0; (S10)8\nwhen we keep the terms that include the temporal or spatial derivative of n0at least once by focusing on high-energy\nmagnons. For small deviations from the equilibrium, n0\u0019^z+n0\nx^x+n0\ny^ywithjn0\nxj;jn0\nyj\u001c1, we obtain the following\nequation of motion for n+\u0011n0\nx\u0000in0\ny:\n\u0000is(@t+i\u001e=~)n+\u0000\u001a(@t+i\u001e=~)2n+=\u0000A\u0002\n@i\u0000i(at\ni+ad\ni)=~\u00032n+; (S11)\nwhere the scalar potential \u001e, the vector potential atfrom the spin texture, the vector potential adfrom the DMI are\ngiven by\n\u001e\u0011~(At\n0)12; at\ni\u0011\u0000~(At\ni)12; ad\ni\u0011(~D=A)(Ad\ni)12: (S12)\nOne set of explicit expressions of them is given by \u001e=\u0000~cos\u00120@t\u001e0,at=~cos\u00120r\u001e0, and ad=\u0000(~D=A)n0.\nTHE DYNAMICS OF A NONRELATIVISTIC CHARGED PARTICLE IN THE PRESENCE OF A\nUNIFORM MAGNETIC FIELD\nIn the presence of a strong perpendicular magnetic \feld, the bands of a charged particle in two-dimensional systems\nform the Landau levels. Here, we summarize the known results for the Landau levels of a nonrelativistic charged parti-\ncle in the presence of a uniform magnetic \feld [37], which has been adopted for a ferromagnetic magnon previously [11].\nThe pertinent Schr odinger equation is given by\ni~@t =\"\n1\n2m\u0012~\nir\u0000qa\u00132#\n ; (S13)\nwhereq=\u00061 parametrizes the charge of the particle. The external magnetic \feld bz= (r\u0002a)\u0001^zperpendicular\nto the plane is characterized by the magnetic length, which is given by l=p\n~=jbzj. The corresponding cyclotron\nfrequency is given by !c=~=ml2.\nThe solution to Eq. (S13) is then given by (r;t) = nn0(r) exp(\u0000i\u000fnt=~), with the energy of the nth Landau level\n(n= 0;1;2;\u0001\u0001\u0001)\n\u000fn=~!c(n+ 1=2) = ( ~jbzj=2m)(2n+ 1); (S14)\nand the spatial part nn0(r) is the eigenfunction of the right-hand side of Eq. (S13) with the eigenvalue \u000fn. See\nRef. [37] for the explicit construction of the spatial part nn0. The number of each Landau level's states, which are\nindexed by the integer n0in nn0(r), is given by the total number of magnetic \rux quanta through the plane. The\nsemiclassical equations of motion for the wavepacket localized at the position rand the momentum kare given by\n_r=1\n~@\u000fn(k)\n@k\u0000_k\u0002\nn(k) (S15)\n~_k=\u0000rU(r); (S16)\nwhere \nis the Berry curvature de\fned in terms of the magnetic Bloch wavefunctions [39, 40]. For \rat Landau levels,\nthe Berry curvature is uniform, and the Chern number, de\fned as the integral of the z-component of the Berry\ncurvature, is determined by the product of the charge and the direction of the magnetic \feld: \u0017n=R\nBZdkxdky\nn\u0001\n^z=2\u0019=\u0000sgn(qbz)." }, { "title": "2001.01042v2.Observation_of_spin_motive_force_in_ferrimagnetic_GdFeCo_alloy_films.pdf", "content": " \n1 Observation of s pin-motive force in ferrimagnetic GdFeCo alloy films \n \nShun Fukuda1, Hiroyuki Awano1 and Kenji Tanabe1* \n1Toyota Technological Institute, Nagoya, 468 -8511, Japan \n*electronic mail: tanabe@toyota -ti.ac.jp \n \nAbstract \nNon-uniform magnetic structure s produce emergent electromagnetic phenomena such as the \ntopological Hall effect and the spin -motive force (SMF). The experimental reports on the SMF, \nhowever, are very few an d the relationship between the SMF and material parameters is still unclear. \nIn this study, we investigated the SMF in ferrimagnetic GdFeCo alloy films using the spin -torque -\ninduced ferromagnetic resonance method and clarified the relationship . The amplitude of the detected \nSMF becomes larger than that of the transition metal alloy FeCo by the Gd doping and reaches the \nmaximum near a Gd composition of the boundary between in -plane and perpendicular ly magnetized \nfilms . According to the analytical calc ulation, t he enhancement is related to the trajectory of the \nmagnetization precession . Moreover, we find that the SMF induced by the magnetic resonance is \ninversely proportional to the square of the damping constant. \n2 Recently, non-uniform magnetic structures such as magnetic domain wall1-2, magnetic vortex3-5, \nanti-vortex6-8, and skyrmion9-13 have attracted much attention in both fundamental and applied physics. \nThe magnetic domain wall and skyrmion, which can be controlled by a current, have been s tudied \ntoward practical application for future memory devices1,2,11. Moreover, t he non -uniform magnetic \nstructure induces effective electric and magnetic fields for conduction electrons . The effective \nmagnetic field induced by the skyrmion s has been detected as a Hall effect, which is termed the \ntopological Hall effect12-16. The effective electric field induced by their dynamics is termed the spin -\nmotive force (SMF)17-30. These phenomena are of great interest as the emergent electromagnetism in \nfundamental physics . \nThe f irst obse rvation of the SMF had been reported by Yang and Beach et al. in 200 921. They \naccurately control led magnetic -field-driven domain wall motion in NiFe alloy wires and detected the \nSMF induced by the motion using the unique lock -in method . After that, several studies on the SMF \nwere also reported by other groups22-24,29,30. The theoretical studies22,26 reported by Yamane et al. \nsuggest that the SMF induced by the magnetic resonance stro ngly depends on the damping constant \nand the SM F induced by the magnetic domain wall motion increases with increasing magnetic \nanisotropy. In all the experiment al studies, however, transition metal alloy s such as NiFe alloy21-24,30 \nand related oxides such as the magnetite Fe3O429 were used and the relation ship between material \nparameters and the SMF is still unclear . Furthermore, the detected voltages are almost 0.1 - 1 V and \nwe ha ve few guidelines for the material development owing to the enhancement of the SMF . \nHere, we investigate the SMF in GdFeCo alloy film s and report the influence of Gd on the SMF . \nThe GdFeCo alloy31-37 is a ferr imagnetic material, where magnetic moment of FeCo is antiparallel to \nthat of Gd. We can fabricate RE-rich films, where net magnetization is parallel to the moment of the \nrare-earth metal (Gd) , and TM -rich films, where the transition metal (FeCo) mainly contributes net \nmagnetization, by controlling the composition of the alloy. The boundary composition betwee n the \nRE-rich and the TM -rich alloys is termed the magnetization compensation point (MCP , 𝑥=𝑥𝑐). The \nGdFeCo alloy has large perpendicular magnetic anisotropy near the MCP . Moreover, the Gilbert \ndamping constant in GdFeCo strongly depends on the composition. Therefore, GdFeCo is one of the \nsuitable materials in studies on the relationship between the SMF and some material parameters such \nas the saturation magnetization, the demagnetizing field, the composition , the magnetic anisotropy , \nand the damping constant . \nThe Si3N4(10 nm)/ Gdx(Fe82Co28)1-x(16 nm)/Pt(10 nm) strips were fabricated by a magnetron \nsputtering, electron -beam lithography, and a lift -off process. The GdFeCo alloy layer was deposited \nby a co-sputtering method from Gd and FeCo targets and the composition of the alloy can be changed \nby optimization of the cathode power. The composition of all the samples was checked by an energy \ndispersive X -ray spectrometry . The Pt layer under the GdFeCo layer is used as an electrode for rf \ncurrents that induces spin torques and(/or) rf Ampère magnetic fields, which is quite similar to an \n3 experimental method of the spin -torque ferromagnetic resonance ( ST-FMR). The widths of the strips \nare changed along a longitudinal dire ction of the strips and the structures are not rectangular , but \ntrapezoidal. \nWe measured the SMF by using the unique ST-FMR method proposed by Nagata et al .29 in Fig. 1(a). \nWhen an rf-current is injected into a bilayer of Pt/GdFeCo , a magnetic resonance is excited by the rf \nspin current and the Ampère field induced by the rf current in the Pt layer38. Since the rf current density \n𝑗𝑒 is inversely proportional to the shrinking width of the strip 𝑤, the cone angle of the magnetization \nprecession is changed along the longitudinal direction of the strips. Th us, the excited magnetic \nresonance become s non -uniform dynamics and is expected to induce the SMF along the longitudinal \ndirection . The SMF was measured by using a lock -in and an amplitude modulation method s. The \nmeasured dc voltage involve s the SMF and the other contribution that originat es from the rectification \neffect of the anisotropic magnetoresistance and the rf current. Although the rectification effe ct van ishes \nwhen the external magnetic field is parallel to the longitudinal direction of the strips, it appears even \nif the field slightly differs from the longitudinal direction of the strips . The SMF is detected as a \nsymmetrical component centered at zero magnetic field. On the other hand, the rectification voltage \ninvolves both symmetric and anti -symmetric components centered at the resonant magnetic field . In \nparticular , both of the contributions are anti -symmetric at zero magnetic field s38. Therefore, we \nextracted the SMF from the measu red dc voltage using the symmetry for the magnetic field. \nFigure 1(b) shows the measured typical dc voltage, whose shape is similar to the Lorentz function. \nWe extract the SMF by using a fitting function of \n 𝑉=𝑉0+𝑉1{𝑆𝐻(𝐻0,𝐻𝑅)+𝑆𝐻(𝐻0,−𝐻𝑅)}+𝑉2{𝑆𝐻(𝐻0,𝐻𝑅)−𝑆𝐻(𝐻0,−𝐻𝑅)}\n+𝑉3{𝐴𝐻(𝐻0,𝐻𝑅)+𝐴𝐻(𝐻0,−𝐻𝑅)}, (1) \nwhere 𝑆𝐻(𝐻0,𝐻𝑅)=Δ𝐻2/((𝐻0−𝐻𝑅)2+Δ𝐻2) and 𝐴𝐻(𝐻0,𝐻𝑅)=(𝐻0−𝐻𝑅)Δ𝐻/((𝐻0−\n𝐻𝑅)2+Δ𝐻2). 𝐻0 is the external magnetic field , 𝐻𝑅 is the resonant magnetic field and Δ𝐻 is the \nfull width at half maximum (FWHM). The second term has the same symmetry as the SMF . The sign \nof the SMF does not depend on the sign of the magne tic field, but is determined by the arrangement \nof the magnetic precession regions. The third and fourth terms attribute to the rectification effe ct in \nFig. 1(c) . The fitting provides information on the resonant frequency , the FWHM, the signal amplitude \nof the SMF, and the rectification effe ct. \nFigure s 2(a-b) show the signal amplitude 𝑉1 as a function of the right and left width of the strip. \nAlthough the signal of almo st 100 nV is detected in the trapezoidal strips, the signal disappears in \nthe rectangular strip, 𝑤𝑟=𝑤𝑙=100 μm. The sign of the detected signal depends on the shrinking \nsides of the strip. Note that the se tendency cannot be expressed by any rectification effects in ST -FMR \nstudies. Moreover, t he SMF is represented as \n \n𝑉1∝1\n𝑤𝑙2−1\n𝑤𝑟2 (2) \n4 from the calculation (as shown below) . The solid curves in Fig s. 2(a-b) indicate the fitting curves of \nthe Eq. (2), which is roughly consistent with the experimental results. Hence, these results reveal that \n𝑉1 is the SMF. \nFigure 2(c) shows the Gd -composition dependence of the SMF 𝑉1. The deposited GdFeCo film has \nthe MCP at about 𝑥𝑐=0.23 (as shown below). Hence, it is TM -rich when 𝑥<0.23 and RE-rich \nwhen 𝑥>0.23. The GdFeCo film is in-plane magnetized from 𝑥=0.0 to 𝑥=0.17 and in 𝑥>\n0.27 owing to the large demagnetizing fields, which is proportional to the saturation magnetization. \nThe alloy films at 𝑥=0.18,0.27 , which were perpendicularly magnetized , were measured by \ncontrolling the magnetization direction via the large in -plane magnetic field . Except for 𝑥=0.12, the \ndetected voltage surprisingly increases with increasing Gd composition as 𝑥<0.17, which indicates \nthat the SMF can become larger than that in alloys that consist of only transition metals such as the \nFeCo alloy via the Gd doping. Moreover, we notice that the SMFs in both the TM -rich and the RE -\nrich films increase near the Gd composition s of the boundary between in -plane and perpendicularly \nmagnetized films. \nTo understand the enhancement of the SMF via the Gd doping as 𝑥<0.17, we model the motion \nof the magnetic moment in GdFeCo by assuming that net magnetic moment that consist s of Gd and \nFeCo is a single magnetic moment. The assumption can be reasonable in the case that the Gd \ncomposition is far away fro m an angular momentum com pensation point (ACP) , which is the \ncomposition that the angular momenta of FeCo and Gd cancel out39. The used Landau –Lifshi tz–\nGilbert (LLG) equation is represented as \n 𝑑𝒏\n𝑑𝑡=−𝛾𝒏×𝑯eff+𝛼𝒏×𝑑𝒏\n𝑑𝑡+ℏ\n2𝑒𝜇0𝑀𝑠𝑡𝐽𝑠(𝒏×𝒔)×𝒏. (3) \nHere, 𝒏 is a normalized magnetization vector, 𝛾 is the gyromagnetic ratio, 𝛼 is the Gilbert \ndamping constant , 𝜇0 is the permeability, 𝑀𝑠 is the saturation magnetization of GdFeCo , 𝒔 is the \nspin magnetic moment and 𝑡 is the thickness of the GdFeCo layer. The xyz -coordinate direction is \ndefined as the xyz arrows in Fig. 1. 𝑯eff=(ℎ𝑥−(4𝜋𝑀𝑠𝑁𝑦+𝐴𝑦)𝑛𝑦𝐻0)𝑇 , where ℎ𝑥 is the \nAmpère field induced by the rf current , 𝑁𝑦 is the demagnetizing coefficient, 𝐴𝑦 is the anisotropic \nconstant , and 𝐻0 is the external magnetic field. Since the GdFeCo film is perpendicularly magnetized \nnear the MCP (𝑥=𝑥𝑐), we add only the perpendicular magnetic anisotropy to the LLG equation . \nWhen 𝛼≪1, |𝑛𝑥|,|𝑛𝑦|≪1, and |ℎ𝑥|≪|𝐻0|, \n \n{ 𝑛𝑥={𝐴𝐻𝜔(𝛾ℎ𝑥)+𝑆𝐻𝜔𝑅(𝛽𝐽𝑠)}+𝑖{−𝑆𝐻𝜔(𝛾ℎ𝑥)+𝐴𝐻𝜔𝑅(𝛽𝐽𝑠)}\n𝛼𝜔𝑅𝛾(2𝐻𝑅+𝑘)\n𝑛𝑦={−𝑆𝐻𝜔(𝛾ℎ𝑥)+𝐴𝐻𝜔𝑅(𝛽𝐽𝑠)}+𝑖{−𝐴𝐻𝜔(𝛾���𝑥)−𝑆𝐻𝜔𝑅(𝛽𝐽𝑠)}\n𝛼𝜔𝛾(2𝐻𝑅+𝑘), (4) \nwhere 𝑘=4𝜋𝑀𝑠𝑁𝑦+𝐴𝑦 , 𝜔=𝛾√𝐻𝑅(𝐻𝑅+𝑘) is an angular frequency of the rf current, 𝜔𝑅=\n𝛾𝐻𝑅 , and 𝛽=ℏ‖𝒔‖2𝑒𝜇0𝑀𝑠𝑡 ⁄ . By substituting 𝑁𝑦=1 and 𝐴𝑦=−𝐻𝑦𝑎𝑛𝑖 , we can obtain the \n5 relationship between the r esonant field and the frequency 𝑓, \n \n𝑓=𝛾\n2𝜋√𝐻𝑅(𝐻𝑅+4𝜋𝑀𝑠−𝐻𝑦𝑎𝑛𝑖). (5) \n𝑆𝐻 and 𝐴𝐻 are the Lorentz and the anti-Lorentz functions in the Eq. (1) , respectively. These FWHM s \nΔ𝐻 are proportional to the frequency, \n \nΔ𝐻=2𝜋𝛼\n𝛾𝑓. (6) \nAssuming that ℎ𝑥=ℎ𝑒𝑖𝜔𝑡 and 𝐽𝑠=𝑗𝑒𝑖𝜔𝑡, we can obtain \n \n{ 𝑛𝑥=√𝑆𝐻√(𝜔𝛾ℎ)2+(𝜔𝑅𝛽𝑗)2\n𝛼𝛾(2𝐻𝑅+𝑘)1\n𝜔𝑅cos(𝜔𝑡+𝛿)\n𝑛𝑦=√𝑆𝐻√(𝜔𝛾ℎ)2+(𝜔𝑅𝛽𝑗)2\n𝛼𝛾(2𝐻𝑅+𝑘)1\n𝜔sin(𝜔𝑡+𝛿), \ntan𝛿= 𝐴𝐻𝜔𝑅𝛽𝑗− 𝑆𝐻𝜔𝛾ℎ\n 𝑆𝐻𝜔𝑅𝛽𝑗+ 𝐴𝐻𝜔𝛾ℎ (7) \nwhich reveal that the magnetization trajectory is not a circular orbit, but an elliptical orbit when \n4𝜋𝑀𝑠−𝐻𝑦𝑎𝑛𝑖≠0. In particular, its ellipticity mainly depend s on the magnetic anisotropy and the \ndemagnetizing field and is independent of the ratio of the rf spin current to the rf Ampère field. On the \nother hand, the phase of the trajectory is strongly influenced by the ratio of the rf spin current to the rf \nAmpèr e field. \nFigures 3(a-c) show 𝐻𝑅 and 𝛥𝐻, and 𝑉1 as a function of the frequency of the rf current. 4𝜋𝑀𝑠−\n𝐻𝑦𝑎𝑛𝑖 is estimated by using the fitting function of the Eq. (5). Since the anisotrop y field is parallel to \nthe demagnetizing field, 4𝜋𝑀𝑠−𝐻𝑦𝑎𝑛𝑖 is regarded as a single parameter. T he saturation \nmagnetization 4𝜋𝑀𝑠 is estimated from the M-H curve in Fig. 3(d). The damping constant 𝛼 is \nobtained from a linear fitting of the Eq. (6) including the extrinsic damping constant40. \nThe elect ric field induced by the SMF is represented as \n \n𝐸𝑧=−ℏ\n2𝑒𝒏̃∙(𝜕𝒏̃\n𝜕𝑡×𝜕𝒏̃\n𝜕𝑧), (8) \nwhere ℏ is the Plank constant , 𝑒 is the elementary charge , and 𝒏̃ is the direction of the internal \nexchange field in a ferromagnetic material20, which corresponds to the direction of the magnetization , \n𝒏. The SMF is calculated by 𝑉𝑆𝑀𝐹=∫𝐸𝑧𝑑𝑧. By substituting the Eq. (7) into the Eq. (8), \n \n𝑉𝑆𝑀𝐹=𝑆𝐻ℏ𝐼2\n4𝛼2𝛾𝑒𝑡2(1\n𝑤𝑙2−1\n𝑤𝑟2)(𝐻𝑅+𝑘)(𝛾ℎ̃)2+𝐻𝑅(𝛽𝑗̃)2\n(2𝐻𝑅+𝑘)2, (9) \nwhere, 𝐼 is the rf current , ℎ̃=ℎ/𝑗𝑒 , and 𝑗̃=𝑗/𝑗𝑒 because both ℎ and 𝑗 are proportional to the \ncurrent density, 𝑗𝑒 . The equation reveals that t he SMF should be the Lorentz function and be \nproportional to 𝑤𝑙−2−𝑤𝑟−2. Since the SMF is inversely proportional to 𝛼2, we evaluate the SMF \nas the prod uct of the SMF and the square of the damping constant, 𝛼2𝑉𝑆𝑀𝐹. Besides , the SMF depends \n6 on two unknown parameters, which indicates the contributions of the rf Ampère field and the rf spin \ncurrent , and we could not analyze the experimental data using the equation. Hence, w e roughly assume \n𝛾ℎ̃≅𝛽𝑗̃ from the obtained fitting parameters of 𝑉2 and 𝑉3 in the Eq. (1). Then, 𝛼2𝑉𝑆𝑀𝐹 is \nrepresented by \n \n𝛼2𝑉𝑆𝑀𝐹∝{𝛾2(4𝜋𝑀𝑠−𝐻𝑦𝑎𝑛𝑖)2+4𝜔2}−1\n2, (10) \nwhich means that the SMF decreases with increasing frequency and increases with decreasing \n4𝜋𝑀𝑠−𝐻𝑦𝑎𝑛𝑖. When the equation is applied to the frequency dependence of the SMF in Fig. 3(c) , the \nfitting curve is good agreement with the decreasing tendency in Fig. 3(c), which su ggests the validity \nof the approximation. \nFigures 4(a -c) show the Gd-composition dependences of 𝛼, 4𝜋𝑀𝑠, 4𝜋𝑀𝑠−𝐻𝑦𝑎𝑛𝑖, and 𝛼2𝑉1. The \ndependence of 4𝜋𝑀𝑠, which is obtained from the M -H curve, shows that the MCP is near 𝑥=0.23 \nand 4𝜋𝑀𝑠−𝐻𝑦𝑎𝑛𝑖 decreases with increasing Gd -composition toward the boundary composition \nbetween the in -plane and perpendicularly magnetized films . The damping constant trends to increase \nnear the MCP, which is also consistent with the previous report32,39. The t heoretical suggestion32 \nindicates that the damping constant diversifies to infinity at the ACP, which is near the MCP owing to \nslight differences between Fe, Co, and Gd g-factors . Though the ACP is measured by the Barnett effect \nmeasurement technique40,41, we estimate the ACP at 𝑥=0.21 using g factors 𝑔Gd=2.0 and \n𝑔FeCo=2.2. The SMF is evaluated as 𝛼2𝑉1 in Fig. 4(c ) to take into account the influence of the \ndamping constant . As a results, 𝛼2𝑉1 increases with increasing Gd composition up to 𝑥=0.16 , \nwhich is close to the boundary composition , and decreases with increasing Gd composition after that. \nThe results reveal that t he suppression of 𝑉1 at 𝑥=0.12 in Fig. 2(c) stems from the apparent effect \ndue to the increase in the damping constant. The cur ve in Fig. 4(c), which is a guide by eye, is \napproximately drawn by using the Eq. (10). Since the curve overlaps with the experimental data, the \nenhancement via the Gd doping can be roughly expressed by the competition between the \ndemagnetizing field and t he magnetic anisotropy field. In other word s, when the trajectory of the \nmagnetization precession becomes circular , the output of the SMF is maxim ized. \nWe have investigated the SMF induced in the ferrimagnetic GdFeCo alloy films by using the unique \nspin-torque ferromagnetic resonance method . The amplitude of the SMF , which strongly depends on \nthe Gd composition , is maxim ized at the boundary composition between the in -plane and the \nperpendicular ly magnetized films, whi ch shows that the trajectory of the magnetization precession is \ncrucial in order to enhance the SMF. Since the SMF induced by the magnetic resonance is inverse ly \nproportional to the square of the damping constant, ferromagnetic materials with low er damping \nconstants are desired to enhance the SMF . \n7 We would like to thank Mr . A. Takahashi and Dr. S. Sumi for collaboration at an early stage of this \nwork. 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Chudo, M . Ono, K . Harii, M . Matsuo, Y . Ohnuma, S. Maekawa, E. Saitoh, \nAppl. Phys. Lett. 114, 162402 (2019) . \n11 Figure 1 \n \nFig. 1. (a) Schematic illustration of the measurement setup. (b) The detected typical dc \nvoltage as a function of the external magnetic field. The blue curve represents the fitted \ndata. The power of the rf current is 10 dBm. The Fe 82Co18 alloy is used. (c) Each \ncomponent of the fitting function. The data in the upper figure ha ve the same symmetry as \nthe SMF. On the other hand, the data in the lower figure ha ve the same symmetry as the \nsignals of the rectificatio n effect . \n \n12 Figure 2 \n \nFig. 2. (a-b) The detected voltages, 𝑉1, as a function of the widths of the left and the right \nedges of the ferromagnetic strips . The frequency and the power of the rf current are 6 GHz \nand 10 dBm, respectively. 𝑤𝑟 and 𝑤𝑙 are defined in the insets. (c) The Gd composition \ndependence of the detected voltages, 𝑉1. The frequency and the power of the rf current are \n6 GHz and 10 dBm, respectively . The width of the left edge is 100 m and the width of the \nright edge is 20 m. \n \n \n13 Figure 3 \n \nFig. 3. (a -c) The resonant field, the FWHM , and the SMF as a function of the frequency of \nthe rf curr ent. The power of the rf current is 10 dBm.The width of the left edge is 100 μm \nand the width of the right edge is 20 μm. (d) The magnetization -field curve. The \ncomposition of the used alloy is 𝑥=0.16. The magnetic field direction is in -plane. \n \n14 Figure 4 \n \nFig. 4. (a) The Gilbert damping constant as a fu nction of the Gd composition. (b ) The \nsaturation magnetization 4π𝑀𝑠 and 4π𝑀𝑠−𝐻𝑦 as a function of the Gd composition. ( c) \n𝛼2𝑉1 as a function of the Gd composition. The frequency and the power of the rf current \nare 6 GHz and 10 dBm, respectively. The blue curve is a guide for the eye , which is written \nby using the Eq. (10). \n" }, { "title": "0705.3731v1.Ferrimagnetism_and_antiferromagnetism_in_half_metallic_Heusler_alloys.pdf", "content": "arXiv:0705.3731v1 [cond-mat.mtrl-sci] 25 May 2007pss header willbe provided by the publisher\nFerrimagnetism and antiferromagnetism in half-metallic\nHeusler alloys\nIosifGalanakis∗1,Kemal ¨Ozdo˘gan2,ErsoyS ¸as ¸ıo ˘glu3,4,andBekir Aktas ¸2\n1Department of Materials Science, Universityof Patras,GR- 26504 Patra,Greece\n2Department of Physics,Gebze Instituteof Technology, Gebz e, 41400, Kocaeli, Turkey\n3Institutf¨ ur Festk¨ orperforschung, Forschungszentrum J ¨ ulich, D-52425 J¨ ulich, Germany\n4FatihUniversity, PhysicsDepartment, 34500, B¨ uy¨ ukc ¸ek mece,˙Istanbul,Turkey\nReceived 15November 2003, revised30 November 2003, accept ed 2December 2003\nPublishedonline 3December 2003\nPACS75.47.Np, 75.50.Cc, 75.30.Et\nHalf-metallic Heusler alloys are among the most promising m aterials for future applications in spintronic\ndevices. Although most Heusler alloys are ferromagnets, fe rrimagnetic or antiferromagnetic (also called\nfully-compensated ferrimagnetic) alloys would be more des irable for applications due tothe lower external\nfields. Ferrimagnetism can be either found in perfect Heusle r compounds or achieved through the creation\nof defects inferromagnetic Heusler alloys.\nCopyrightlinewillbeprovidedbythe publisher\nIntroduction : The family of the ferromagnetic Heusler alloys, e.g. NiMnSb or Co2MnSi, have been\nextensively studied during the last years due to their poten tial applications in magnetoelectronic devices\n[1]. Their main advantage with respect to other half-metall ic systems is their structural similarity with\nthe binary semiconductors and their high Curie temperature s. First principles calculations have been ex-\ntensively employed to study their electronic and magnetic p roperties (see Refs. [2, 3, 4] and references\ntherein). One of the most important features of these alloys is the Slater-Pauling behavior of their total\nspin magnetic moment which is given simply as a function of th e number of valence electrons in the unit\ncell [5, 6]. Authors have studied in the recent years several aspects of these half-metallic alloys like the\nproperties of surfaces [7, 8, 9] and interfaces with semicon ductors [10, 11], the quaternary [12, 13], the\norbitalmagnetism[14,15],theeffectofdopinganddisorde r[16,17],theexchangeconstants[18]andthe\nmagneto-opticalproperties[19].\nHalf-metallicferrimagneticHeusler alloyslike Mn 2VAl, where Mn andV atomshaveantiparallel spin\nmoments, are of particular interest since they create small er external magnetic fields and thus lead to\nsmaller energy losses [20, 21]. In the extreme case like Cr 2MnSb (alloys with 24 valence electrons) Cr\nandMnspinmomentscanceleachotherandthecompoundsarena medasfully-compensatedhalf-metallic\nferrimagnetsorsimplyhalf-metallicantiferromagnets[2 2]. Defectsinthesealloysshowaveryinteresting\nbehavior. When we substitute Co atomswith Cr(Mn)in the Co 2Cr(Mn)Al andCo 2Cr(Mn)Sicompounds,\nthe impurity atoms couple antiferromagnetically with the o ther transition metal atoms lowering the total\nspin moment [23, 24]. Co and Fe impurities in Mn 2VAl and Mn 2VSi ferrimagnetic alloys have spin\nmoments antiparallel to Mn and thus the total spin moment rea ches closer to the zero value [25]. In this\nmanuscript we will complete these studies presenting resul ts for the Cr(Mn) impurites in ferromagnetic\nCo2Mn(Cr)Al and Co 2Mn(Cr)Si alloys, the case of V and Cr impurities in ferrimagn etic Mn 2VAl and\nMn2VSi alloys and finally the 24-valence half-metallic antifer romagnetic alloys Cr 2FeZ (Z= Si, Ge, Sn)\nusingthefull–potentialnonorthogonallocal–orbitalban dstructurescheme(FPLO)[26].\n∗Corresponding author: e-mail: galanakis@upatras.gr , Phone+30 2610 969925, Fax +30 2610 969368\npss datawillbe providedby thepublisher2 I.Galanakis, K. ¨Ozdo˘ gan, E.S ¸as ¸ıo˘ glu, and B.Aktas ¸:\nTable 1 Total and atom-resolved spin magnetic moments in µBfor the [X 1−xX∗\nx]2YZ compounds. The atom-\nresolved spin moments have been scaled toone atom. We donot p resent the moments of the Z sp-atom since they are\nnegligible withrespect tothe X andY transition-metal atom s.\nCompound mComX∗mYmTotalCompound mMnmX∗mVmTotal\n[Co0.9Cr0.1]2MnAl 0.72 -2.57 2.79 3.43 [Mn0.9V0.1]2VAl -1.68 -1.00 1.03 -2.13\n[Co0.9Cr0.1]2MnSi 0.97 -1.55 3.03 4.40 [Mn0.9V0.1]2VSi -1.17 -1.10 0.92 -1.33\n[Co0.9Mn0.1]2CrAl 0.74 -1.75 1.70 2.60 [Mn0.9Cr0.1]2VAl -1.66 -1.8 1.12 -2.20\n[Co0.9Mn0.1]2CrSi 0.94 -0.95 2.14 3.60 [Mn0.9Cr0.1]2VSi -1.06 -1.58 0.96 -1.20\n[Co1−xCrx]2MnZ and [Co 1−xMnx]2CrZ (Z= Al, Si) alloys : In the first part of our study we will\nconcentrate on the case of defects in the strong ferromagnet ichalf-metallic Co 2CrZ and Co 2MnZ (Z= Al\norSi) Heusler alloys. InRef. [23]we havestudiedthe case of Cr defectsinCo 2CrAl andCo 2CrSi alloys;\nCr atoms substitute Co atoms at the perfect X sites. We expand ed this study in Ref. [24] studying the\ncase of Mn impurities in Co 2MnAl and Co 2MnSi compounds. In all cases the impurity atoms had spin\nmoments antiparallel to the other transition metal atoms lo wering the total spin moment while keeping\nthe half-metallic character of the parent compounds. We hav e expanded this study now considering Mn\nimpurities in Co 2CrAl and Co 2CrSi and Cr impurities in Co 2MnAl and Co 2MnSi and present both the\ntotalandatom-resolvedDOS’sinFig. 1. Aswasthecasealsoi nRefs. [23,24]allcompoundswithdefects\npresent the half-metallic behaviorand only in the case of [C o0.9Cr0.1]2MnSi alloy is the width of the gap\nslightly smaller with respect to the perfect Co 2MnSi alloy (not shown here). The total spin moments(see\nTable 1) are considerably smaller than the spin moments of th e perfect compounds: 5 µBfor Co2MnSi,\n4µBfor Co 2MnAl and Co 2CrSi, and 3 µBfor Co 2CrAl. Thus these kind of defected alloys could be\nvaluable for realistic applications. As for the compounds i n the previous paragraph we have performed\ncalculationsforseveralconcentrationsbut resultsaresi milar totheconcentrationwhichwepresent.\n[Mn1−xVx]2VZ and [Mn 1−xCrx]2VZ (Z= Al, Si) alloys : In Ref. [25] we have shown that when we\nsubstitute Co or Fe for Mn in the ferrimagnetic hafl-metallic Mn2VAl and Mn 2VSi alloys, the Co and\nFe impurity atoms have spin moments parallel to the V atoms at the Y site and antiparallel to the Mn\natoms. Thus the negative total spin moment of the alloys beco mes smaller in magnitude and for specific\nconcentrationofdefectswe geta half-metallicantiferrom agnet. We shownowresultswhenwe use V and\nCrasimpurityatomswhichhavelowervalencethanMn. Ascanb eseeninFig. 2,alloyswithCrimpurities\nalmost keep unaltered the gap in the spin-up band for both Al a nd Si-based compounds. Contrary to Cr\natoms, the lighter vanadium shows a smaller exchangesplitt ing and the Fermi level falls within a spin-up\npickof theV DOS completelydestroyingthe half-metallicit y. We shouldalso notethat we presentresults\nonly for a concentration of defects of x=0.1. We have performed calculations also for x=0.025,0.05 and\n0.2butthebehaviorofthecompoundsissimilar tothecase wh ichwepresent.\nEvenmoreinterestingisthebehaviorofthespinmomentspre sentedinTable1. InthecaseofCoandFe\nimpurities,duetotheirpositivespinmoment,thetotalspi nmomentwasapproachingzeroasweincreased\nthe concentration. Contraryto these chemical elements, V a nd Cr hybridizeless with the neighboringMn\natomswhichshowamoreatomiclike behaviorandthusincreas etheabsolutevalueoftheirspinmagnetic\nmoment. V or Cr atomshave smaller spin momentsthan Mn atomsb ut they are in low concentrationand\nas a result the total spin momentincreasesin magnitudefrom the -2µBof the perfectMn 2VAl and the -1\nµBof the perfect Mn 2VSi (note that the total spin momentsare negativesince thes e alloys have less than\n24 valence electrons according to the Slater-Pauling rule [ 6]). Thus such a doping with Cr and V would\nhavenoadvantagesovertheperfectMn 2VAlandMn 2VSi compoundsforrealisticapplications.\nc/circlecopyrt2003WILEY-VCH VerlagGmbH&Co. KGaA, Weinheimpss header willbe provided by the publisher 3\n-6-3036[Co0.9Cr0.1]2MnAl\n[Co0.9Cr0.1]2MnSi\n-4 -2 0\nE (eV)-6-3036[Co0.9Mn0.1]2CrAl\n[Co0.9Mn0.1]2CrSi\n-202\n-4 -2 0\nE (eV)-202\n-4 -2 0-202\n-202DOS (states/eV)Co Cr\nCr Z Co MnMn Z\nFig. 1(Color online) Total and atom-resolved DOS\nfor Cr impurities in Co 2MnAl and Co 2MnSi alloys\n(upper panel) and Mn impurities in Co 2CrAl and\nCo2CrSicompounds (lower panel).-6-3036[Mn0.9V0.1]2VAl\n[Mn0.9V0.1]2VSi\n-4 -2 0\nE (eV)-6-3036[Mn0.9Cr0.1]2VAl\n[Mn0.9Cr0.1]2VSi\n-202\n-4 -2 0\nE (eV)-202\n-4 -2 0-202\n-202DOS (states/eV)Mn Vimp\nV ZMn CrV Z\nFig. 2(Color online) Total and atom-resolved DOS\nfor V (upper panel) and Cr (lower panel) impurities\nin the case of Mn 2VAl and Mn 2VSi alloys. DOS’s\nhave beenscaled toone atom.\n-6-3036DOS (states/eV)Fe\nCr\ntotal\n-4 -2 0\nE (eV)-6-3036\n-4 -2 0\nE (eV)Cr2FeSi\nCr2FeGe Cr2FeSn\nFig. 3Total, Cr- and Fe-resolved DOS for the\nCr2FeZalloyswhereZisSi,GeorSnandforalattice\nconstant of 6.2 ˚A.-6-3036DOS (states/eV)Fe\nCr\ntotal\n-4 -2 0\nE (eV)-4 -2 0\nE (eV)-6-3036a=6.0 Å a=6.2 Å \na=6.1 Å a=6.3 ÅCr2FeSi\nFig. 4(Color online) Total, Cr- and Fe-resolved\nDOS for the Cr 2FeSialloyand for four different val-\nues of the latticeconstant.\nCr2FeZ(Z=Si,Ge,Sn)alloys: MotivatedbyourresultsontheCr 2MnZalloys[22],wedecidedtostudy\nalso another family of 24-valence electrons compounds cont aining Fe instead of Mn: the Cr 2FeZ alloys\nwhereZ is Si, Ge orSn. In Fig. 3 we present the total andthe Cr a ndFe-resolveddensityof states (DOS)\nfor a lattice constant of 6.2 ˚A for all three alloys. For this lattice constant all the comp oundsunder study\npresent a region of low DOS in the spin-up band and the Fermi le vel falls within this region. Contrary\nto the compounds containing Mn, where the spin-up occupied s tates are mainly of Cr character and the\noccupied spin-downstates are of mainly Mn character, for th e Fe-based alloys both the spin-up and spin-\ndown occupied states exhibit a more mixed character. The ele ctronic structure is more complicated than\nthe Mn-based alloys and as a result when we slightly vary the l attice constant the band-structure changes\nc/circlecopyrt2003WILEY-VCH VerlagGmbH&Co. KGaA, Weinheim4 I.Galanakis, K. ¨Ozdo˘ gan, E.S ¸as ¸ıo˘ glu, and B.Aktas ¸:\nin a way that the the region of low DOS is completely destroyed . This is illustrated in Fig. 4 where we\npresenttheDOSforCr 2FeSiforfourvaluesofthelatticeconstant. Crspin-upstat esbelowtheFermilevel\nand the Cr spin-up states just above the Fermi level move one t owards the other completely destroying\nthe regionof high spin-polarizationuponcompression. Lat tice constants largerthan 6.3 ˚A are unlikelyto\nbe achieved experimentally. The reason is the different hyb ridization between the transition-metal atoms\nwhenwesubstituteFeforMn. Thesmallerexchangesplitting oftheFeatomsclosesthegapandthisleads\nalsotoadifferentdistributionoftheCrchargewhichnomor eshowsalargeband-gapasfortheMn-based\nalloys. Finally we shouldalso discussthe spinmagneticmom ents. Fe atomsforthe lattice constantof6.2\n˚A possess a moment slightly smaller than -3 µBand each Cr atom has a spin moment of around 1.5 µB\nresultingin the almost vanishingtotal spin momentrequest edby theSlater=Paulingbehaviorforthe half-\nmetalswith 24valenceelectrons[6]. Overallthe Cr 2FeZ alloyswith 24-valenceelectronsarenot suitable\nfor realistic application, contrary to the Cr 2MnZ compoundssince half-metallicity is very fragile in the ir\ncase (e.g. calculations suggest that the region of low spin- up DOS persists for Cr 2FeSi only between 6.2\nand6.3˚A).\nSummary : We havepresentedfirst-principlescalculationsonseveral Heusleralloys. Cr(Mn)impurites\noccupying Co sites in Co 2Mn(Cr)Al and Co 2Mn(Cr)Si alloys couple antiferromagnetically to the other\ntransition metal atoms resulting in ferrimagnetic compoun ds with lower total spin moments similarly to\nour previous studies on these alloys [23, 24]. V and Cr impuri ties occupying Mn sites in ferrimagnetic\nMn2VAl and Mn 2VSi alloys present spin moments parallel to the Mn atoms cont rary to the behavior of\nthe Fe and Co impurities in these alloys and they lead to large r absolute values of the total spin moments\n[25]. Finally,rhe24-valenceelectronalloysCr 2FeZ(Z=Si,Ge,Sn)showaregionoflowdensityofstates\ninsteadoftherealgappresentedbytheisovalentCr 2MnZ(Z=P,As, Sb,Bi)[22]whichpersistsonlyfora\nverynarrowrangeoflattice constants.\nReferences\n[1] Half-metallic alloys: fundamentals and applications, Eds.: I. Galanakis and P. H. Dederichs, Lecture notes in\nPhysics vol.676 (BerlinHeidelberg: Springer 2005).\n[2] I.Galanakis, Ph. Mavropoulos, and P.H. Dederichs, J.Ph ys. D:Appl. Phys. 39, 765 (2006).\n[3] I.Galanakis andPh. Mavropoulos, J.Phys.: Condens. Mat ter inpress [preprint: cond-mat/0610827].\n[4] P.H.Dederichs, I. Galanakis, andPh. Mavropoulos, J.El ectronMicroscopy 54, i53(2005).\n[5] I.Galanakis, P.H.Dederichs, andN. Papanikolaou, Phys .Rev. B66, 134428 (2002).\n[6] I.Galanakis, P.H.Dederichs, andN. Papanikolaou, Phys .Rev. B66, 174429 (2002).\n[7] I.Galanakis, J. Phys.: Condens. Matter 14, 6329 (2002).\n[8] I.Galanakis, J. Magn. Magn. Mater. 288, 411 (2005).\n[9] M. Leˇ zai´ c,I. Galanakis, G.Bihlmayer, andS.Bl¨ ugel, J.Phys.: Condens. Matter 17, 3121 (2005).\n[10] I.Galanakis, J. Phys.: Condens. Matter 16, 8007 (2004).\n[11] I.Galanakis, M. Leˇ zai´ c, G.Bihlmayer, andS.Bl¨ ugel , Phys.Rev. B 71, 214431 (2005).\n[12] I.Galanakis, J. Phys.: Condens. Matter 16, 3089 (2004).\n[13] K.¨Ozdo˘ gan, B. Aktas ¸,I. Galanakis, E.S ¸as ¸ıo˘ glu, J.Appl. Phys.101, 073910 (2007).\n[14] I.Galanakis, Phys. Rev. B 71, 012413 (2005).\n[15] Ph.Mavropoulos, I.Galanakis, V. Popescu, and P.H.Ded erichs, J. Phys.: Condens. Matter 16, S5759 (2004).\n[16] I.Galanakis, K. ¨Ozdo˘ gan, B.Aktas ¸, andE.S ¸as ¸ıo˘ glu, Appl. Phys.Lett. 89, 042502 (2006).\n[17] K.¨Ozdo˘ gan, E.S ¸as ¸ıo˘ glu, B.Aktas ¸,and I.Galanakis, Phys .Rev. B74, 172412 (2006).\n[18] E.S ¸as ¸ıo˘ glu, L.M. Sandratskii,P.Bruno, and I.Gala nakis, Phys.Rev. B 72, 184415 (2005).\n[19] I.Galanakis, S.Ostanin, M. Alouani, H.Dreyss´ e, and J .M. Wills,Phys. Rev. B 61, 599 (2000).\n[20] K.¨Ozdo˘ gan, I. Galanakis, E.S ¸as ¸ıo˘ glu, andB.Aktas ¸,J. Ph ys.: Condens. Matter 18, 2905 (2006).\n[21] E.S ¸as ¸ıo˘ glu, L.M. Sandratskii,and P.Bruno, J.Phys .: Condens. Matter 17, 995 (2005).\n[22] I.Galanakis, K. ¨Ozdo˘ gan, E.S ¸as ¸ıo˘ glu, andB.Aktas ¸,Phys.Rev. B 75, 172405 (2007).\n[23] K.¨Ozdo˘ gan, I. Galanakis, E.S ¸as ¸ıo˘ glu, andB.Aktas ¸,Phys .Stat.Sol.(RRL) 1, 95(2007).\n[24] K.¨Ozdo˘ gan, I. Galanakis, E.S ¸as ¸ıo˘ glu, andB.Aktas ¸,Sol. St.Commun. 142, 492 (2007).\n[25] I.Galanakis, K. ¨Ozdo˘ gan, E.S ¸as ¸ıo˘ glu, andB.Aktas ¸,Phys.Rev. B 75, 092407 (2007).\n[26] K.KoepernikandH.Eschrig,Phys.Rev.B 59,1743(1999); K.Koepernik, B.Velicky,R.Hayn, andH.Eschr ig,\nPhys.Rev. B 58, 6944 (1998).\nc/circlecopyrt2003WILEY-VCH VerlagGmbH&Co. KGaA, Weinheim" }, { "title": "1712.08204v1.Exchange_torque_induced_excitation_of_perpendicular_standing_spin_waves_in_nanometer_thick_YIG_films.pdf", "content": "Exchange-torque-induced excitation of\nperpendicular standing spin waves in\nnanometer-thick YIG films\nHuajun Qin1,*, Sampo J. H ¨am¨al¨ainen1, and Sebastiaan van Dijken1,*\n1NanoSpin, Department of Applied Physics, Aalto University School of Science, P .O. Box15100, FI-00076 Aalto,\nFinland\n*huajun.qin@aalto.fi, sebastiaan.van.dijken@aalto.fi\nABSTRACT\nSpin waves in ferrimagnetic yttrium iron garnet (YIG) films with ultralow magnetic damping are relevant for magnon-based\nspintronics and low-power wave-like computing. The excitation frequency of spin waves in YIG is rather low in weak external\nmagnetic fields because of its small saturation magnetization, which limits the potential of YIG films for high-frequency\napplications. Here, we demonstrate how exchange-coupling to a CoFeB film enables efficient excitation of high-frequency\nperpendicular standing spin waves (PSSWs) in nanometer-thick (80 nm and 295 nm) YIG films using uniform microwave\nmagnetic fields. In the 295-nm-thick YIG film, we measure intense PSSW modes up to 10th order. Strong hybridization between\nthe PSSW modes and the ferromagnetic resonance mode of CoFeB leads to characteristic anti-crossing behavior in broadband\nspin-wave spectra. A dynamic exchange torque at the YIG/CoFeB interface explains the excitation of PSSWs. The localized\ntorque originates from exchange coupling between two dissimilar magnetization precessions in the YIG and CoFeB layers.\nAs a consequence, spin waves are emitted from the YIG/CoFeB interface and PSSWs form when their wave vector matches\nthe perpendicular confinement condition. PSSWs are not excited when the exchange coupling between YIG and CoFeB is\nsuppressed by a Ta spacer layer. Micromagnetic simulations confirm the exchange-torque mechanism.\nIntroduction\nMagnonics aims at the use of spin waves for the processing, storage, and transmission of information1–6. With the smallest\ndamping parameter of all magnetic materials, ferrimagnetic YIG has attracted considerable interest. Several building blocks\nfor spin-wave-based technologies have been realized using YIG magnonics, including magnonic crystals6–8, logic gates9,\ntransistors10, and multiplexers11. In these experiments, coplanar waveguides (CPWs) or microstrip antennas are typically used\nto excite propagating magnetostatic spin waves. The frequency of these spin waves depends on their wave vector ( k), the\nsaturation magnetization of YIG ( Ms), and the external magnetic field ( Hext). Because the saturation magnetization of YIG is\nsmall and the wave vector is limited by the width of the antenna signal line, the spin-wave frequency is only 1\u00002GHz in weak\nmagnetic fields12, 13. Higher frequencies can be attained by the excitation of magnetostatic spin wave modes with larger wave\nvector using a grating coupler14, at the expense of emission efficiency.\nAnother spin-wave mode that can be excited in a magnetic film is the PSSW. The wave vector of this confined mode is\napproximated by k=pp=d, where dis the film thickness and pis the order number. Thus in nanometer-thick magnetic films,\nthe wave vectors of PSSW modes are large and their frequency is high. The formation of a PSSW requires a nonuniform\nexcitation across the magnetic film thickness. Laser pulses15–17, microwave magnetic fields from a miniaturized antenna18, and\neddy-current shielding in conducting films19have been used to excite PSSWs. In these experiments, the excitation field is\nnonuniform and both odd and even PSSW modes are measured. Uniform microwave magnetic fields can also excite PSSWs\nif the magnetization of the film is pinned at one or both of its interfaces20–26. Symmetrical pinning only induces odd PSSW\nmodes, whereas both odd and even modes should be detected if the magnetization is pinned at one of the interfaces. Most work\non PSSWs has focussed on metallic ferromagnetic materials such as Co/Py19, Py23–26, and CoFeB27, 28. In YIG, Klingler et al.\nused PSSWs to extract the exchange constant18and Navabi et al. demonstrated the excitation of a 1st order PSSW mode in a\n100-nm-thick YIG film on top of an undulating substrate28.\nHere, we report on efficient excitation of PSSWs in nanometer-thick YIG films. The excitation mechanism is based on\nexchange coupling between the YIG film and a CoFeB layer. We show that forced magnetization precessions in YIG and CoFeB,\ndriven by an approximately uniform CPW field, induce a dynamic exchange torque at the interface when the precessions are\ndissimilar. Consequently, the emission of spin waves into YIG is most efficient if the dynamic exchange torque is maximized\nnear the ferromagnetic resonance (FMR) frequency of either YIG or CoFeB. Because the PSSW dispersion relations cross thearXiv:1712.08204v1 [cond-mat.mtrl-sci] 21 Dec 2017FMR curve of the CoFeB layer, PSSWs with high order numbers are efficiently excited in YIG at high frequencies.\nResults\nSpin-wave spectra of YIG, CoFeB, and YIG/CoFeB films\nSingle-crystal ferrimagnetic YIG films with a thickness of 80 nm and 295 nm were grown on (111)-oriented Gd 3Ga5O12(GGG)\nsubstrates using pulsed laser deposition (PLD). To measure broadband spin-wave spectra, we placed the films face-down onto a\nCPW with a 50 mm-wide signal line. A microwave current provided by a vector network analyzer (VNA) was used to generate\na microwave magnetic field around the CPW. The main excitation strength of the CPW was at wave vector k\u00190. We recorded\nabsorption spectra in transmission by measuring the real part of the S12scattering parameter. The experiments were performed\nwith an external magnetic bias along the CPW. A schematic of the measurement geometry is shown in Fig. 1a.\nWe first discuss the spin-wave spectrum of a single 295-nm-thick YIG film. Figure 1b shows the absorption at a magnetic\nbias field of 20 mT (dashed blue line). Data as a function of magnetic field are shown in Fig. 1c. Obviously, only one\nspin-wave mode is excited in the film. The mode corresponds to uniform magnetization precession in YIG, i.e., the FMR mode.\nHigher-order PSSW modes are not detected in this sample, as expected from the near-uniform microwave magnetic field and\nthe absence of magnetization pinning at the interfaces. After characterization, we deposited a 50-nm-thick CoFeB layer onto\nthe same YIG film using magnetron sputtering. Figure 1b (solid orange line) and Fig. 1e show the spin-wave spectrum of this\nsample. Now, a large number of spin-wave modes are measured. The lowest-frequency mode, labeled as p= 0, corresponds to\nFMR in YIG (compare the blue and orange lines in Fig. 1b or the spectra in Figs. 1c and 1e). The higher-order modes ( p= 1, 2,\n...) are PSSWs in YIG (see next Section for details). The excitation of both odd and even modes implies that exchange coupling\nat the YIG/CoFeB interface produces an asymmetric pinning configuration. Finally, we note that a FMR mode is also excited in\nCoFeB ( p= 0 (CoFeB)). To support this conclusion, we show the spin-wave spectrum of a single 50-nm-thick CoFeB film on\nGGG in Fig. 1b (dashed green line) and in Fig. 1d.\nThe data in Fig. 1 clearly indicate that the excitation of PSSWs in YIG is particularly efficient near the FMR of the CoFeB\nfilm. This point is further exemplified by the spin-wave spectrum of Fig. 1f, demonstrating the formation of PSSW modes\nwith order numbers up to p= 10 at high frequencies. High-order PSSWs are only visible if their frequency approaches that of\nthep= 0 mode in CoFeB. Exchange coupling at the YIG/CoFeB interface results in mode hybridization and characteristic\nanti-crossing behavior. Consequently, the frequency of the CoFeB FMR mode in the YIG/CoFeB bilayer is slightly shifted with\nrespect to the same resonance in the single CoFeB film (orange and green curves in Fig. 1b).\nIn the following, we first analyze the PSSW modes in YIG/CoFeB using a phenomenological model. Next, we assess their\nintensity and linewidth. Results for YIG/CoFeB bilayers with a 80-nm-thick YIG film are subsequently discussed. Finally, we\nelucidate the origin of efficient PSSW excitation in exchange-coupled YIG/CoFeB bilayers using control experiments with a\nnonmagnetic spacer layer and micromagnetic simulations.\nAnalysis of PSSW mode dispersions\nTo investigate the dependence of the PSSW resonance frequency on external bias field, we first extract experimental data for p=\n0 ... 6 from the spin-wave spectra in Fig. 1. The results are plotted as symbols in Fig. 2a. We derive the saturation magnetization\nof our YIG film by fitting the frequency dependence of the p= 0 mode to the Kittel formula29:f=gm0=2pp\nHext(Hext+Ms).\nUsing g=2p= 28 GHz/T, we obtain a good fit for Ms=192kA/m. This magnetization value compares well to previous results\non YIG films30, 31. Next, we fit the PSSW modes ( p= 1 ... 6) to the following dispersion relation23, 27, 28:\nfPSSW =gm0\n2ps\u0014\nHext+2Aex\nm0Ms\u0010pp\nd\u00112\u0015\u0014\nHext+2Aex\nm0Ms\u0010pp\nd\u00112\n+Ms\u0015\n; (1)\nwhere Aexis the exchange constant. By inserting g=2p= 28 GHz/T, Ms= 192 kA/m, d= 295 nm, and p=1\u00006, we obtain\ngood fits to all PSSW modes using Aex=3:1 pJ/m (lines in Fig. 2a). This value also agrees with literature18.\nThe agreement between our experimental data and the model confirms that the higher-order resonances in the spin-wave\nspectra of the YIG/CoFeB bilayer correspond to PSSW modes in YIG. The observation that integer numbers of pand the actual\nthickness of the YIG film in Eq. 1 provide excellent fits to all dispersion curves over a large bias-field range signifies strong\nmagnetization pinning at the YIG/CoFeB interface. Weak pinning at the boundary of the YIG film would require the use of\np\u0000Dpin the fitting formula23, where correction factors Dp= 0 and Dp= 0 would correspond to full and zero magnetization\npinning, respectively. In our YIG/CoFeB bilayer, short-range exchange coupling between the two magnetic films provides a\nstrongly pinned interface.\n2/9Intensity and linewidth of PSSW resonances\nDamping of PSSW modes in magnetic films can have different origins. Besides intrinsic damping, eddy-current damping (in\nmetallic films), and radiative damping caused by inductive coupling between the sample and the microwave antenna can also\ncontribute26. To assess the damping of PSSWs in our YIG/CoFeB bilayer, we plot the full width at half maximum (FWHM)\nlinewidth of the p= 1 ... 4 modes relative to that of the p= 0 mode (Fig. 2c). The frequency evolution of this data was obtained\nfrom the spin-wave spectra in Fig. 1e for magnetic fields ranging from 0 to 30 mT. The linewidths of all PSSW modes are large\nat low frequencies. The broad resonances in YIG are caused by hybridization with the higher-loss FMR mode in CoFeB. As the\nfrequency increases, the frequency gap between the PSSWs in YIG and the FMR mode in CoFeB becomes larger. Once the\ntwo modes decouple, the linewidths of the PSSW modes decrease. In the decoupled state, the PSSW linewidths are similar to\nthat of the p= 0 mode in YIG, independent of frequency. Since eddy-current damping can be omitted in insulating YIG and\nradiative damping would cause the linewidth to increase with frequency26, the data in Fig. 2c suggest that damping of PSSWs\nis dominated by intrinsic material parameters.\nFigure 2d shows the relative intensity of the same PSSW modes. The dashed lines indicate the frequency where\nFWHM p/FWHM p=0= 1 in Fig. 2c, which we use as an indicator for dehybridization between the PSSWs in YIG and\nthe FMR mode in CoFeB. For the pure PSSW modes beyond this critical frequency, we still measure high intensities. The\nintensities of the p= 1 and p= 2 modes are up to 50% of the p= 0 resonance and this value drops to about 30% for p= 3 and p\n= 4. The large intensities of the PSSW modes demonstrate a highly efficient excitation mechanism.\nTuning of PSSW modes in YIG/CoFeB bilayers\nThe frequency of a PSSW depends on the wave vector of the confined mode and the external magnetic bias field. Since\nk=pp=d, the frequency of a PSSW could be enhanced by a reduction of the film thickness d. For an efficient excitation\nmethod, this would enable high-frequency spin waves in YIG at small magnetic fields. To test this prospect, we prepared a\n80 nm YIG/50 nm CoFeB bilayer. The spin-wave spectrum of this sample is shown in Fig. 3. In addition to the FMR modes\nin YIG and CoFeB, the first two PSSW modes are also measured. Anti-crossing behavior between the p= 1 mode and the\nCoFeB resonance and an increase of the PSSW intensity near the anti-crossing frequency are again apparent. Compared to the\n295-nm-thick YIG film, the PSSWs in thinner YIG are shifted up in frequency. At a moderate magnetic bias field of 20 mT,\nthe increase of frequency amounts to about 3 GHz for p= 1 and 8 GHz for p= 2. The data of Fig. 3 thus confirm that PSSW\nmodes are efficiently excited at high frequencies if the thickness of YIG is reduced.\nPSSW excitation mechanism\nWe explain the excitation of PSSWs in YIG/CoFeB bilayers by a dynamic exchange torque at the interface. The uniform\nmicrowave excitation field from the CPW induces forced magnetization precessions in both magnetic layers. If the amplitudes\nof these precessions are different, a dynamic exchange torque is generated, causing the emission of spin waves from the\ninterface. The efficiency of this excitation mechanism depends on the strength of the dynamic exchange torque, which is\nmaximized at the FMR frequency of YIG and CoFeB. While spin waves are emitted from the interface over a broad frequency\nrange, PSSWs only form when the wave vector of the excited spin waves matches the perpendicular confinement condition\n(k=pp=d) of the YIG film. Our experiments support this scenario. PSSWs are only measured after the YIG film is covered by\na CoFeB layer and the PSSW resonances are most intense if the induced precession of magnetization is large in one of the two\nlayers, i.e., near the FMR of YIG or CoFeB (see Figs. 1b,e,f and Fig. 3a).\nTo confirm the crucial role of exchange coupling at the YIG/CoFeB interface, we prepared a 295 nm YIG/10 nm Ta/50\nnm CoFeB trilayer on GGG. The spin-wave spectrum of this sample is shown in Fig. 4. As expected, no PSSW modes are\nmeasured in this case. The two resonances in the spectrum are identical to those in Figs. 1c and 1d and, thus, correspond to\nthe FMR mode in the YIG and CoFeB film, respectively. The Kittel formula fits the experimental data for Ms= 192 kA/m\n(YIG) and Ms= 1280 kA/m (CoFeB). The results of Fig. 4 demonstrate that the elimination of exchange coupling between\nmagnetization precessions in YIG and CoFeB by the Ta spacer layer destroys the driving force behind PSSW excitation. This\nalso implies that dipolar coupling between YIG and CoFeB is insignificant.\nWe performed micromagnetic simulations in MuMax332to further study the microscopic origin of PSSWs in YIG/CoFeB\nbilayers. In the simulations, we considered a 295-nm-thick YIG film and a 50-nm-thick CoFeB layer. The structure was\ndiscretized using finite-difference cells of size x= 54 nm, y= 54 nm and z= 2.7 nm, as schematically shown in the inset of\nFig. 5a. Two-dimensional periodic boundary conditions were applied in the film plane to mimic an infinite bilayer. We used\nthe following input parameters: Ms= 192 kA/m (YIG), Ms= 1280 kA/m (CoFeB), Aex= 3.1 pJ/m (YIG), and Aex= 16 pJ/m\n(CoFeB). The damping constant was set to 0.005 for both magnetic films. For YIG, this relatively large value was selected to\nlimit the computation time. Spin waves in the bilayer were excited by an uniform 3 mT sinc-function-type magnetic field pulse\nor a sinusoidal ac magnetic field (see Methods). The excitation field was along xand a magnetic bias field was aligned along y.\nFor comparison, we also performed micromagnetic simulations for a structure where the YIG and CoFeB films are separated by\na 10-nm-thick nonmagnetic spacer, to mimic the response of a YIG/Ta/CoFeB trilayer.\n3/9The top panel of Fig. 5a shows simulated spin-wave spectra for the YIG/CoFeB bilayer (solid orange line) and the\nYIG/Ta/CoFeB trilayer (dashed green line). The simulations were performed with a magnetic bias field of 30 mT. For\ncomparison, we plotted the measured spectra of these samples in the bottom panel of Fig. 5a. The simulations reproduce\nthe excitation of PSSW modes in the YIG film of the YIG/CoFeB bilayer ( p= 1 ... 7) and the absence of these modes in\nYIG/Ta/CoFeB. The simulated PSSW frequencies are in good agreement with the experiments, except for frequencies near\nthe CoFeB FMR mode. This discrepancy is attributed to stronger hybridization between the PSSWs and the CoFeB FMR\nmode in the simulations, caused by stronger exchange coupling at a perfectly flat interface. Mode hybridization also shifts the\nFMR mode in CoFeB. Results for YIG/Ta/CoFeB confirm this view. For this decoupled structure, the frequency of spin-wave\nresonances are the same in the simulations and experiments (dashed green curves in Fig. 5a). We note that different parameters\nare plotted in the simulated and measured spectra. In the simulations, the intensity of the resonances is proportional to the\namplitude of magnetization precession. The intensity of the modes in the experiments, on the other hand, are determined by\ninduction-related absorption of a microwave current in the CPW. For constant magnetization precession, the absorption signal\nwould increase with frequency. As a result, the relative intensity of the CoFeB resonance at higher frequency is larger in the\nlower panel of Fig. 5a. Simulated spin-wave spectra as a function of magnetic bias field for both structures are shown in Figs.\n5b and 5c.\nWe now focus on the spatial distribution of spin-wave modes in the YIG and CoFeB films. Figure 6a and 6b show simulation\nresults for YIG/CoFeB and YIG/Ta/CoFeB, respectively. Magnetization precession in YIG at the FMR frequency is reduced\nnear the CoFeB interface. This effect, which is absent in the YIG/Ta/CoFeB structure, signifies strong exchange coupling to the\nCoFeB layer. The PSSWs in YIG/CoFeB are nearly symmetric and strongly confined to the YIG film for the non-hybridized\nmodes. The number of nodes corresponds to order parameter p. Hybridization between the PSSW modes in YIG and the FMR\nmode in CoFeB at higher frequencies also induces a significant magnetization precession in the CoFeB layer.\nThe simulated time evolution of PSSW mode formation in the YIG/CoFeB bilayer is depicted in Figs. 6c and 6d. Here, we\nfocus on p= 4 at an excitation frequency of 4.9 GHz. The simulations illustrate how the magnetization responds to the onset\nof a spatially uniform sinusoidal ac magnetic field at t= 0 s. Just after the excitation field is switched on, spin waves with a\nwavelength of l\u0019150 nm ( l=2d=p) are emitted from the YIG/CoFeB interface. This excitation is triggered by a dynamic\nexchange torque originating from dissimilar magnetization precessions in the YIG and CoFeB layers. The emitted spin waves\npropagate along the thickness direction of the YIG film and reflect at the GGG/YIG interface. At the selected frequency of 4.9\nGHz, the forward and backward propagating spin waves interfere constructively. As a result, a p= 4 PSSW is formed. The\nlarge-amplitude PSSW is fully established after t\u00196 ns.\nThe simulated time evolution of magnetization dynamics in the YIG/Ta/CoFeB trilayers is shown in Figs. 6e and 6f. As\ndiscussed previously, this structure does not support the excitation of PSSWs. Instead, the ac magnetic field induces uniform\nsmall-amplitude precessions of magnetization in the YIG and CoFeB layers. Because of different precession amplitudes, a\ntime-dependent divergence of magnetization emerges at the location of the Ta insertion layer. This divergence is the source\nof the dynamic exchange torque in structures where the magnetization of YIG and CoFeB are directly coupled by interface\nexchange interactions.\nFinally, we discuss the off-resonance time evolution of magnetization dynamics in the YIG/CoFeB bilayer (Figs. 6g and\n6h). We consider an excitation frequency of 4.5 GHz, thus, in between the frequencies of the p= 3 and p= 4 PSSW modes (see\nFig. 6a). Under these circumstances, spin waves are again emitted from the YIG/CoFeB interface by the dynamic exchange\ntorque. However, since the condition for constructive spin-wave interference along the film thickness of YIG is not fulfilled,\ntheir amplitude is not amplified and a PSSW does not form.\nIn summary, we have demonstrated an efficient method for the excitation of PSSWs in nanometer-thick YIG films. The\nmethod relies on direct exchange coupling between the YIG film and a CoFeB top layer. The application of an uniform\nmicrowave magnetic field produces a strong dynamic exchange torque at the YIG/CoFeB interface. This results in short-\nwavelength spin-wave emission. A PSSW is excited if one of the perpendicular confinement conditions is met. Our findings\nopen up a new route towards the excitation high-frequency spin waves in YIG. The results can also be generalized to other\nexchange-coupled systems. The excitation of intense PSSWs with large order numbers requires crossings between their\ndispersion relations and the FMR mode in a second, exchange-coupled magnetic layer. This situation is attained if the saturation\nmagnetization is smallest in the PSSW carrying film.\nMethods\nSample fabrication\nWe grew YIG films with a thickness of 80 nm and 295 nm on single-crystal GGG(111) substrates using PLD. The GGG\nsubstrates were ultrasonically cleaned in acetone and isopropanol before loading into the PLD vacuum chamber. We degassed\nthe substrates at 550\u000eC for 15 minutes. After this, oxygen was inserted into the chamber. After setting the oxygen pressure\nto 0.13 mbar, we increased the temperature to 800\u000eC at a 5\u000eC per minute rate. The YIG films were deposited under these\n4/9conditions from a stoichiometric target. We used an excimer laser with a pulse repetition rate of 2 Hz and a laser fluence of\n1.8 J/cm2. After film growth, we annealed the YIG films at 730\u000eC in an oxygen environment of 13 mbar. The annealing time\nwas 10 minutes. This was followed by a cool down to room temperature at a rate of \u00003\u000eC per minute. The deposition process\nresulted in single-crystal YIG films, as confirmed by X-ray diffraction. The composition of CoFeB was 40%Co, 40%Fe, and\n20%B. The CoFeB and Ta layers were grown by magnetron sputtering at room temperature.\nSpin-wave spectroscopy\nWe recorded spin-wave absorption spectra in transmission by measuring the real part of the S12scattering parameter. To\nenhance contrast, a reference spectrum taken at larger magnetic field or frequency was subtracted from the measurement data.\nThe setup consisted of a two-port VNA and a quadruple electromagnet probing station. The CPW with a 50 mm-wide signal\nline and two 800 mm-wide ground lines was patterned on a GaAs substrate. The gap between the signal and ground lines was\n30mm. The CPW was designed to provided a k\u00190excitation field in the plane of the YIG film. During broadband spin-wave\nspectroscopy measurements, the sample was placed face-down onto the CPW.\nMicromagnetic simulations\nWe performed micromagnetic simulations using open-source GPU-accelerated MuMax3 software. A 6900\u00026900\u0002345nm3\nCoFeB/YIG bilayer structure was discretized into 54\u000254\u00022:7nm3cells and two-dimensional periodic boundary conditions\nwere applied in the film plane. We abruptly changed the magnetic parameters at the YIG/CoFeB interface and used the\nharmonic mean value of the exchange constants in YIG and CoFeB to simulate the interface exchange coupling. The system\nwas initialized by an external magnetic field along the yaxis followed by relaxation to the ground state. After this, a spatially\nuniform 3 mT sinc-function-type magnetic field pulse with a cut-off frequency of 20 GHz was applied along the xaxis. The\nmagnetic field pulse excited all spin-wave modes up-to the cut-off frequency with uniform excitation power (Fig. 5). To study\nthe spatial dependence of magnetization dynamics (Fig. 6), the system was driven be a sinusoidal ac magnetic field with an\namplitude of 3 mT. In these simulations, the time evolution of the perpendicular magnetization component ( mz) was recorded for\n50 ns in 3 ps time steps along the thickness direction of the system at the center of the simulation mesh. The spatially-resolved\nintensity was obtained by applying a Fourier imaging technique where the time evolution of mzwas Fourier-transformed on a\ncell-by-cell basis.\nReferences\n1.Kruglyak, V . V ., Demokritov, S. O. & Grundler, D. Magnonics. J. Phys. D: Appl. Phys. 43, 264001 (2010).\n2.Serga, A. A., Chumak, A. V . & Hillebrands, B. YIG magnonics. J. Phys. 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A. & Hillebrands, B. Magnon transistor for all-magnon data processing. Nat. Commun. 5, 4700\n(2014).\n11.Davies, C. S. et al. Field-controlled phase-rectified magnonic multiplexer. IEEE Trans. Magn. 51, 1–4 (2015).\n12.Yu, H. et al. Magnetic thin-film insulator with ultra-low spin wave damping for coherent nanomagnonics. Sci. Rep. 4,\n6848 (2014).\n13.Collet, M. et al. Spin-wave propagation in ultra-thin yig based waveguides. Appl. Phys. Lett. 110, 092408 (2017).\n14.Yu, H. et al. Approaching soft x-ray wavelengths in nanomagnet-based microwave technology. Nat. Commun. 7, 11255\n(2016).\n15.Demokritov, S. O., Hillebrands, B. & Slavin, A. N. Brillouin light scattering studies of confined spin waves: linear and\nnonlinear confinement. Phys. Reports 348, 441–489 (2001).\n5/916.Busse, F., Mansurova, M., Lenk, B., von der Ehe, M. & M ¨unzenberg, M. A scenario for magnonic spin-wave traps. Sci.\nRep. 5, 12824 (2015).\n17.Razdolski, I. et al. Nanoscale interface confinement of ultrafast spin transfer torque driving non-uniform spin dynamics.\nNat. Commun. 8, 15007 (2017).\n18.Klingler, S. et al. Measurements of the exchange stiffness of yig films using broadband ferromagnetic resonance techniques.\nJ. Phys. D: Appl. Phys. 48, 015001 (2015).\n19.Kennewell, K. J. et al. Magnetization pinning at a py/co interface measured using broadband inductive magnetometry. J.\nAppl. Phys. 108, 073917 (2010). DOI 10.1063/1.3488618.\n20.Kittel, C. Excitation of spin waves in a ferromagnet by a uniform rf field. Phys. Rev. 110, 1295–1297 (1958).\n21.Soohoo, R. F. General exchange boundary condition and surface anisotropy energy of a ferromagnet. Phys. Rev. 131,\n594–601 (1963).\n22.Wigen, P. E., Kooi, C. F., Shanabarger, M. R. & Rossing, T. D. Dynamic pinning in thin-film spin-wave resonance. Phys.\nRev. Lett. 9, 206–208 (1962).\n23.Gui, Y . S., Mecking, N. & Hu, C. M. Quantized spin excitations in a ferromagnetic microstrip from microwave photovoltage\nmeasurements. Phys. Rev. Lett. 98, 217603 (2007).\n24.Khivintsev, Y . V . et al. Spin wave resonance excitation in ferromagnetic films using planar waveguide structures. J. Appl.\nPhys. 108, 023907 (2010).\n25.Magaraggia, R. et al. Exchange anisotropy pinning of a standing spin-wave mode. Phys. Rev. B 83, 054405 (2011).\n26.Schoen, M. A. W., Shaw, J. M., Nembach, H. T., Weiler, M. & Silva, T. J. Radiative damping in waveguide-based\nferromagnetic resonance measured via analysis of perpendicular standing spin waves in sputtered permalloy films. Phys.\nRev. B 92, 184417 (2015).\n27.Conca, A. et al. Annealing influence on the gilbert damping parameter and the exchange constant of cofeb thin films. Appl.\nPhys. Lett. 104, 182407 (2014).\n28.Navabi, A. et al. Efficient excitation of high-frequency exchange-dominated spin waves in periodic ferromagnetic structures.\nPhys. Rev. Appl. 7, 034027 (2017).\n29.Kittel, C. On the theory of ferromagnetic resonance absorption. Phys. Rev. 73, 155–161 (1948).\n30.Howe, B. M. et al. Pseudomorphic yttrium iron garnet thin films with low damping and inhomogeneous linewidth\nbroadening. IEEE Magn. Lett. 6, 1–4 (2015).\n31.Sokolov, N. S. et al. Thin yttrium iron garnet films grown by pulsed laser deposition: Crystal structure, static, and dynamic\nmagnetic properties. J. Appl. Phys. 119, 023903 (2016).\n32.Vansteenkiste, A. et al. The design and verification of mumax3. AIP Adv. 4, 107133 (2014).\nAcknowledgements\nThis work was supported by the European Research Council (Grant No. ERC-2012-StG 307502-E-CONTROL). S.J.H.\nacknowledges financial support from the V ¨ais¨al¨a Foundation. Lithography was performed at the Micronova Nanofabrication\nCentre, supported by Aalto University.\nAuthor contributions statement\nH.J.Q., S.J.H., and S.v.D. designed and initiated research. H.J.Q. fabricated the samples and conducted the measurements.\nH.J.Q. and S.J.H. performed the micromagnetic simulations. S.v.D. supervised the project. H.J.Q. and S.v.D. wrote the\nmanuscript, with input from S.J.H.\nAdditional information\nCompeting financial interests: The authors declare that they have no competing interests.\n6/9Figure 1. (a) Schematic of the measurement geometry (not to scale). Spin-wave spectra are obtained by placing the sample\nface-down on a CPW. A microwave current is injected into the CPW signal line using a VNA. This produces a nearly uniform\nspin-wave excitation field. Spin-wave absorption is measured in transmission using scattering parameter S12. The external\nmagnetic bias field is oriented parallel to the CPW. (b) Spin-wave spectra for a single YIG film (dashed blue line), a single\nCoFeB film (dashed green line), and a YIG/CoFeB bilayer (solid orange line), measured with an external magnetic bias field of\n20 mT. The YIG and CoFeB films are 295 nm and 50 nm thick. The FMR modes in YIG and CoFeB ( p= 0) and higher order\nPSSW modes in YIG ( p= 1 ... 6) are labeled. (c)-(e) Spin-wave spectra of the same samples as a function of magnetic bias\nfield: (c) YIG, (d) CoFeB, (e) YIG/CoFeB. (f) YIG/CoFeB spin-wave spectrum at higher frequency and larger magnetic bias\nfield, demonstrating the excitation of PSSWs with large order numbers.\n-150 -100 -50 0 50 100 15024681012\n0, 123456Frequency (GHz)\nMagnetic field (mT )p\n20 40 60 80 100 12081012141618202 4 6 8 10fPSSW (GHz)\nkPSSW (rad/ □m)Order number pa\ncb\nd\n1 2 3 4 50.00.51.0\np = 3 p = 4 p = 2I_p / I_p = 0\nFrequenc y (GHz)p = 1\n1 2 3 4 5012345\np = 4 p = 3 p = 2\nFrequenc y (GHz)FWHM_p / FWHM_p = 0 p = 1\nFigure 2. (a) Frequency of the FMR and PSSW modes in a 295-nm-thick YIG film as a function of magnetic bias field. The\nsymbols are extracted from the experimental spin-wave spectrum in Fig. 1e. The lines are fits to the data using the Kittel\nformula ( p= 0) and the PSSW dispersion relation (Eq. 1, p= 1 ... 6). (b) PSSW mode frequency as a function of wave vector\nfor a magnetic bias field of 185 mT. The solid and empty symbols denote experimental data and calculated values (Eq. 1),\nrespectively. (c),(d) FWHM linewidth and intensity of the PSSW resonances as a function of frequency. The properties of the p\n= 1 ... 4 modes are normalized to those of the FMR mode ( p= 0).\n7/9Figure 3. (a) Spin-wave spectrum of a 80 nm YIG/50 nm CoFeB bilayer. (b) Extracted frequency of the p= 0 ... 2 modes in\nYIG, demonstrating an up-shift in PSSW frequency compared to data for the 295-nm-thick YIG film (Fig. 2a).\nFigure 4. (a),(b) Spin-wave spectrum for a 295-nm-thick YIG film and a 50-nm-thick CoFeB layer, separated by 10 nm of Ta.\nOnly the FMR modes in YIG and CoFeB are measured in this case. The lines in (b) are fits to the two resonances using the\nKittel formula.\nFigure 5. (a) Simulated (top) and measured (bottom) spin-wave spectra for a 295 nm YIG/50 nm CoFeB bilayer (solid orange\nline) and a 295 nm YIG/10 nm Ta/50 nm CoFeB trilayer (dashed green line). The magnetic bias field is 30 mT. The inset\nillustrates the simulation geometry. (b),(c) Simulated spectra for the same structures as a function of magnetic bias field.\n8/9Figure 6. (a),(b) Simulated spatial distribution of the FMR and PSSW modes in (a) a 295 nm YIG/50 nm CoFeB bilayer and\n(b) a 295 nm YIG/10 nm Ta/50 nm CoFeB trilayer. (c)-(f) Simulated time evolution of magnetization dynamics in (c),(d) a 295\nnm YIG/50 nm CoFeB bilayer and (e),(f) a 295 nm YIG/10 nm Ta/50 nm CoFeB trilayer. The excitation frequency in these\nsimulations is 4.9 GHz, which corresponds to the frequency of the p= 4 PSSW mode. (g),(h) Simulated time evolution of\nmagnetization dynamics in a 295 nm YIG/50 nm CoFeB bilayer at an off-resonance frequency of 4.5 GHz.\n9/9" }, { "title": "2303.10979v2.Anapole__chiral_and_orbital_states_in_Mn3Si2Te6.pdf", "content": "Anapole, chiral and orbital states in Mn 3Si 2Te 6 \nS. W. Lovesey 1, 2, 3 \n1 ISIS Facility, STFC, Didcot, Oxfordshire OX11 0QX, UK \n2 Diamond Light Source, Harwell Science and Innovatio n Campus, Didcot, Oxfordshire \nOX11 0DE, UK \n3 Department of Physics, Oxford University, Oxford OX 1 3PU, UK \nAbstract The ferrimagnet Mn 3Si 2Te 6 attracts attention because of a recently discovere d \ncolossal magnetoresistance (CMR) with unique magnet ic field properties. An improved \nmagnetic structure for the material has emerged fro m a neutron diffraction study linked to \nunderstanding the CMR. A deeper theoretical investi gation of the magnetic structure has now \nrevealed anapole, chiral and orbital states of mang anese ions not previously mentioned. \nMoreover, it is shown that existence of these state s in the low temperature form of Mn 3Si 2Te 6, \nwith a magnetic field applied, can be tested by neu tron and resonant x-ray diffraction. \nI. INTRODUCTION \n Silicon manganese telluride (Mn 3Si 2Te 6) is semiconducting and ferrimagnetic below a \ntemperature ≈ 78 K [1]. A trigonal structure for the compound wa s established many years \nbefore recent measurements of its colossal magnetor esistance (CMR) and an improved \nferrimagnetic structure [2, 3, 4, 5]. The nature of CMR in Mn 3Si 2Te 6 sets it aside from other \nCMR materials. Basal plane conductivity drastically increases (a factor 10 7) when an external \nmagnetic field is applied along the magnetic hard a xis normal to the plane, and there is only a \nmodest change to the conductivity when the field is applied parallel to an easy axis in the plane \n[6, 7, 8]. \n The theoretical study reported here paves the way for numerous tests of the improved \nmagnetic structure of Mn 3Si 2Te 6. A tried and tested structure is required for mean ingful \ninterpretations of experimental data and a consensu s understanding of the unusual magnetic \nproperties [5]. Tests can result in a refinement or rejection of the defining magnetic space \ngroup, which is a standard procedure in establishin g a reliable space group for a chemical \nstructure. The study takes the form of calculations informed by magnetic symmetry of \nscattering amplitudes for beams of neutrons and x-r ays. Bragg diffraction of these radiations \nunveils anapole, chiral and orbital states of a cry stal. Calculations predict all these states for \nMn 3Si 2Te 6 in an applied magnetic field, none of which have b een mentioned in the literature, \nto the best of our knowledge. Anapoles (a Dirac dip ole depicted in Fig. 1) appear in diffraction \namplitudes of neutrons, and x-rays tuned to an atom ic resonance [9 - 12]. Unlike loopy entities \n[8, 13], axial and Dirac atomic multipoles have spe cific discrete symmetries. Tuning the energy \nof x-rays to an atomic resonance has two obvious be nefits in diffraction experiments. In the \nfirst place, there is a welcome enhancement of Brag g spot intensities and, secondly, spots are \nelement specific [14, 15]. Another attribute of res onant x-ray Bragg diffraction is realized with \ncircularly polarized x-rays that detect charge-like and magnetic chiral states [16, 17, 18]. \nAbsorption of x rays at the K edge exposes orbital states alone [19], while spin and orbital \ndegrees of freedom contribute to Bragg spots enhanc ed by absorption at L edges, for example [20]. Some iridates and ceramic superconductors har bour Dirac multipoles that have been \nshown to diffract neutrons [21]. Notably, the exist ence of Dirac quadrupoles in the pseudo-gap \nphases of cuprate superconductors YBCO and Hg1201 a ccount for magnetic Bragg diffraction \npatterns [22, 23]. In the present theoretical study of neutron and x-ray diffraction by Mn 3Si 2Te 6 \nparticular attention is given to weak magnetic Brag g spots that do not appear in the diffraction \npattern of the parent structure. The intensity of o ne such basis-forbidden Bragg spot for \nMn 3Si 2Te 6 at a temperature ≈ 5 K as a function of an applied magnetic field has been reported \n[5]. By their nature, basis-forbidden Bragg spots r eveal details of an electronic structure. Best \nknown, perhaps, is Templeton-Templeton (T & T) scat tering due to departures from spherical \nsymmetry in atomic charge distributions [24, 25]. A ccording to Neumann's Principle, such \ndepartures delineate the local symmetry of resonant ions [26, 27, 28]. As we mentioned, \nanalogous effects occur in magnetic diffraction tha t is the principal subject of our study. The \ntwo independent Mn ions in Mn 3Si 2Te 6 have different magnetic properties on account of th eir \ndifferent environments. \n The nominal atomic configuration of a manganese io n in Mn 3Si 2Te 6 is 3d 5 (Mn 2+). A \npure configuration 3d 5 allows axial magnetic dipoles alone, with spin = 5 /2 and no orbital \nangular momentum. Silicon manganese telluride shows a surprisingly large magnetic \nanisotropy. In simulations of its magnetic structur e, the anisotropy field originates from an \nanisotropy in the Mn orbital moments, and concomita nt orbital angular momentum [3]. An \nadmixture of states with opposite parities creates Dirac multipoles, e.g., d and p atomic orbitals, \nand informative model calculations for V and Cu ion s in V 2O3 and CuO, respectively, appear \nin Refs. [11, 29, 30]. By the same token, simulatio ns of parity-odd electric dipole- electric \nquadrupole (E1-E2) x-ray scattering amplitudes depe nd on an admixture of states with opposite \nparities [31]. Anion and cation states determine t he hybridization matrix that is proportional \nto the wavefunction overlap of metal and Te holes, and Te p-like states penetrate Mn 4p \norbitals. Indeed, in the vicinity of a Mn ion Te p- wave functions can be expanded in the basis \nof the Mn orbitals [32]. In the same vein, axial ma gnetic quadrupoles, formed by a correlation \nof anapole and orbital degrees of freedom, are expe cted to be different from zero. They are \nallowed in magnetic neutron scattering when the ato mic wave function is drawn from two, or \nmore, j-manifolds. Such is the case in the j eff = 1/2 state of Ir ions (Ir 4+, 5d 5), often exploited in \nstudies of Sr 2IrO 4, which is an admixture of manifolds with total ang ular momenta j = 3/2 and \nj = 5/2 [21]. \nII. MAGNETIC PROPERTIES \n The parent structure of Mn 3Si 2Te 6 is trigonal. It belongs to the space group P 3\u00021c (No. \n163.79, BNS [33]) with Mn1 and Mn2 in Wyckoff sites 4f and 2c, respectively [2, 3]. \nFerrimagnetic order depicted in Fig. 2 develops bel ow ≈ 78 K without changes to the lattice \nstructure [3]. Recently, the monoclinic magnetic sp ace group C2 '/c ' (No. 15.89, cell choice 1 \n[33]) with unique axis b was assigned to the magnet ic order [5]. It belongs to the magnetic \ncrystal class 2'/m' that is centrosymmetric, and pe rmits ferromagnetism and a piezomagnetic \neffect, while a linear magnetoelectric effect is no t allowed. The magnetic space group allows a \nferromagnetic component within the (ac)-plane. This implies that application of a magnetic field along the c-axis, or any other direction with in the (ac)-plane, does not change the C2 '/c ' \nmagnetic symmetry. A bulk magnetic moment is shown to be due to axial multipoles alone. \n Mn1 and Mn2 use Wyckoff positions 8f (0, 1/3, ≈ 0) and 4e (0, 1/3, 1/4), respectively, \nin C2 '/c '. Sites do not contain centres of inversion symmetr y and both Mn ions are permitted \naxial (parity signature σπ = +1) and polar (Dirac, σπ = −1) magnetic multipoles. Monoclinic and \nhexagonal structures are related by a = − ah, b = ah + 2bh = (0, ao√3, 0) and c = − ch, with \nhexagonal vectors ah = ( ao, 0, 0), bh = ( 1 ̸ 2) ( −ao, ao√3, 0) and ch = (0, 0, co) [5]. Unit cell lengths \na ≈ 7.029 Å, b ≈ 12.175 Å, c ≈ 14.255 Å, and all cell angles = 90 o. Miller indices for the \nmonoclinic and hexagonal structures are denoted ( h, k, l) and (H o, K o, L o), respectively, with h \n= −Ho, k = H o + 2K o, l = −Lo. Atomic multipoles are calculated in orthogonal co ordinates ( ξ, η, \nζ) derived from a, b, c, with unique axis b and ηη ηη parallel. \n For our atomic description of electronic propertie s, Mn ions are assigned spherical \nmultipoles OKQ with integer rank K and projections Q in the inter val − K ≤ Q ≤ K. Cartesian \nand spherical components of a dipole n = (x, y, z), for example, are related by x = (n −1 − n+1)/ √2, \ny = i(n−1 + n+1)/ √2, z = n 0. A complex conjugate is defined as OKQ* = ( −1) Q OK−Q, and a \nphase convention OKQ = [ OKQ' + iOKQ'' ] for real and imaginary parts labelled by single \nand double primes, respectively. In which case, the diagonal multipole OK0 is purely real. \nAngular brackets ... denote the time-average, or expectation value, of the enclosed tensor \noperator, i.e., Mn multipoles feature in the electr onic ground state of Mn 3Si 2Te 6. \n Sites 8f occupied by Mn1 have no symmetry. They di ffer in this respect from sites 4e \nwith symmetry 2 η', i.e., an anti-dyad parallel to the unique axis b of the monoclinic cell. In \nconsequence, a multipole OKQ2 for Mn2 obeys the identity σθ (−1) K + Q OK−Q2 = OKQ2, \nwhere the time signature σθ = −1 for a magnetic multipole. An alternative version of the identity \nfor Mn2 multipoles is σθ (−1) K OKQ2* = OKQ2. Thus, components of Mn2 magnetic dipoles \nparallel to the unique axis b are not permitted. \n An electronic structure factor, \n ΨKQ = [exp( iκκ κκ•d) OKQd], (1) \ndetermines neutron and x-ray diffraction patterns f or a particular space group [20]. In the \npresent study, the implied sum in Eq. (1) is over p ositions d of Mn ions in a unit cell. The \nreflection vector κκ κκ with integer Miller indices referred to the monocli nic structure. There are \nthree independent Te ions using Wyckoff positions 8 f, and they possess general coordinates \nunlike Mn1. Tellurium nuclei contribute to neutron scattering for Miller indices that allow the \nscalar component of the electronic structure factor (K = 0, Q = 0) to be different from zero for \nσθ = +1 (non-magnetic) and σπ = +1 (axial). Examination of Ψ00(8f) with these auxiliary \nconditions reveals that Ψ00(8f) = 0 for ( h, 0, l) with even h and odd l, and (0, k, l) with even k \nand odd l; these are (basis-forbidden) reflections. Conventi onal neutron polarization analysis \nmeasures the ratio of magnetic and nuclear scatteri ng amplitudes at basis-allowed, or core, \nBragg reflections. In the case of x-rays, Thomson s cattering from all ions is absent at forbidden \nreflections. For Mn2 we find a structure factor, \n ΨKQ(2) = OKQ2 exp( iπl/2) [1 + (−1) h + k] [exp( iϕ) + σπ (−1) l exp( −iϕ)], (2) \nwith ϕ = (2 πky) and y = 1/3 [5]. Bulk properties are found by se tting h = k = l = 0. The \ncorresponding structure factor can be different fro m zero for axial multipoles ( σπ = +1) and \ninclude dipoles (K = 1). C-centring is responsible for the reflection condition even ( h + k). As \nwe shall see, Mn1 multipoles share the latter condi tion and, also, those for a bulk magnetic \nmoment. Multipoles with σθ (−1) K = + 1 are purely real, i.e., magnetic multipoles with odd rank \nare purely real. In the present case of magnetic sc attering, ΨKQ(2) for a reflection ( h, k, l) is \npurely real for odd K, and even (odd) l and σπ (−1) l = + 1 ( −1). Conventional neutron \npolarization analysis is available when magnetic an d nuclear scattering have a like phase [35, \n36, 37]. \n Specifically, reflections ( h, 0, l) are absent for σπ (−1) l = − 1. Bragg spots labelled by \neven h and odd l are caused by Dirac multipoles denoted GKQ. Site symmetry allows anapoles \nG1ξ and G1ζ, cf. Fig. 1. Likewise, axial magnetic dipoles T1ξ and T1ζ cause diffraction \nfor (h, 0, l) with even h, l, since site symmetry for Mn2 does not include the parity signature. \n Turning to Mn1, \n ΨKQ(1) = [1 + (−1) h + k] [exp( iϕ) + σπ exp( −iϕ)] \n × [OKQ1 + σθ (−1) K + Q (−1) l OK−Q1], (3) \nwith ϕ = (2 πk/3). An alternative form of the the third factor = [OKQ1 + σθ (−1) K + l OKQ1*]. \nA significant difference between Mn1 and Mn2 is tha t the former ion is permitted dipoles \nparallel to the unique axis b. \nIII. NEUTRON DIFFRACTION \n A magnetic scattering amplitude Q⊥ generates an intensity [| Q⊥1|2 + |Q⊥2|2] of \nunpolarized neutrons, where subscripts 1 and 2 deno te independent ions Mn1 and Mn2. In more \ndetail, Q⊥(±) = [ e × (Q(±) × e)] with a unit vector e = κ/κ, and superscripts refer to axial ( +) \nand Dirac ( −) multipoles. The intermediate amplitude Q(+) is proportional to the axial \nmagnetic moment µµ µµ in the forward direction of scattering, with Q(+) = µµ µµ/2 for κ = 0, while \nthe dipole in Q(−) can be related to anapoles [30, 38 - 42]. The magn etic content of a Bragg \nspot with overlapping nuclear and magnetic amplitud es can be detected by spin-flip scattering \n[22, 23, 43]. A fraction ∝ {(1/2) (1 + P2) | Q⊥|2 − |P • Q⊥|2} of neutrons participate in events \nthat change (flip) the neutron spin orientation, wh ere P is the primary polarization. Taking P • \nP = 1 yields a standard spin-flip signal [43], \n \n(SF) = {| Q⊥|2 − |P • Q⊥|2}. (4) \n \nIn what follows, we show expressions for Q, or Q⊥, appropriate to Bragg spots that expose \nintriguing facets of the magnetic structure assigne d to Mn 3Si 2Te 6 by the monoclinic magnetic \nspace group C2 '/c ' [5]. \n Magnetic multipoles in neutron diffraction depend on the magnitude of the reflection \nvector, κ. An axial dipole T1 contains standard radial integrals j0(κ) and j2(κ) shown in \nFig. 3, with j0(0) = 1 and j2(0) = 0. An approximation to the transition-metal dipo le, \n \n T1 ≈ (µµ µµ/3) [ j0(κ) + j2(κ) (g − 2)/g], (5) \nis often used [39, 40]. Here, the magnetic moment µµ µµ = g S and the orbital moment L = (g \n− 2) S. The coefficient of L is approximate, while T1 = (1/3) 2S + L for κ → 0 is an \nexact result. This is the extent of the analysis us ed by Ye et al . to interpret their Mn 3Si 2Te 6 \ndiffraction pattern [5]. Subtraction of patterns ga thered above and below the magnetic ordering \ntemperature (200 K and 5 K) provided an estimate of the magnetic content. Polarization \nanalysis offers greater sensitivity to magnetic con tributions. Higher order multipoles in the \naxial neutron scattering amplitude depend on the el ectronic position operator n. The equivalent \noperator [( S × n ) n] for T2 shows that actually the quadrupole measures the cor relation between \nthe spin anapole ( S × nn nn) and orbital degrees of freedom [21, 41]. The quadru pole T2 is \nproportional to j2(κ) in Fig. 3. \nThe Dirac dipole d depends on three radial integrals. We use [30, 41] , \n d = (1/2) [ i(g 1) n + 3 (h 1) S × nn nn − (j 0) ΩΩ ΩΩ]. (6) \nRadial integrals (g 1) and (j 0) shown in Fig. 3 diverge in the forward direction of scattering ( κ \n→ 0). Not so for (h 1) shown in Fig. 3, which also carries the κ-dependence of the Dirac \nquadrupole H2 ∝ [(h 1) {S ⊗ n}2] observed in neutron diffraction from high-T c compounds \nHg1201 and YBCO [22, 23]. Returning to Eq. (6), ΩΩ ΩΩ = [ L × n × n × n × n − n × × × × L ] is an orbital \nanapole (toroidal dipole) depicted in Fig. 1. We re tain dipoles and quadrupoles in the following \nneutron diffraction amplitudes. \nA. Forbidden reflections \n \n Ye et al . report the intensity of the Bragg spot (0, 2, −1) for a sample temperature of 5 \nK as a function of a magnetic field applied along t he crystal c-axis; Fig 3a in Ref. [5]. In the \nabsence of multipoles other than dipoles Q1(+) = (0, Qη1(+), 0) with Qη1(+) = [12 cos(2 πk/3) \nT1η1] for (0, k, l) with even k and odd l. Eq. (5) provides an estimate of the axial dipole. \nHowever, it is likely that quadrupoles T2 proportional to j2(κ) contribute to the scattering \namplitudes. In which case, Qξ1(+) ∝ [e η eζ (T2+21' + √(3/2) T201)], Qζ1(+) ∝ [e η eζ T2+11'] \nand a contribution proportional to [e ζ2 T2+11'] is subtracted from Qη1(+). Dirac multipoles are \nalso allowed. Corresponding results for Mn2 ions ar e, Q2(+) ∝ [6 sin(2 πk/3) ( T1ξ2, 0, T1ζ2)] \nwith quadrupoles T2+12'' and T2+22'' as corrections to dipoles in the (ac)-plane and Qη2(+) ∝ \n[e η eζ T2+22'' ]. Reflections of the type ( h, 0, l) with even h and odd l are more selective than \n(0, k, l) with even k and odd l, since Mn1 Dirac multipoles and Mn2 axial multipol es are not \npermitted for k = 0. \n For Mn1 and ( h, 0, l) with even h and odd l we find Q⊥ξ1(+) = Q⊥ζ1(+) = 0 and, Q⊥η1(+) ≈ 8 [(3/2) T1η1+ √3{e ξ eζ (T2+21' − √(3/2) T201) + (e ξ2 − eζ2) T2+11'}]. (7) \nUnit vectors are, \n e ξ = h/[ h2 + (rl)2]1/2 , e ζ = rl/[ h2 + (rl)2]1/2 with r = ( a/c) ≈ 0.493. (8) \nWe find Q⊥ξ2(−) = Q⊥ζ2(−) = 0 and, \n Q⊥η2(−) ≈ 4 (−1) m [e ξ dζ2 − eζ dξ2 + (3/ √5) (e ζ H2+12'' − eξ H2+22'' )]. (9) \nwith Miller index l = (2 m + 1). Axial and Dirac contributions in Eqs. (7) and (9) are \ndistinguished by different dependences on the refle ction vector. Both amplitudes are purely \nreal. However, conventional neutron polarization an alysis is not available in the absence of \nnuclear scattering [35, 36, 37]. \nB. Allowed reflections \n Axial amplitudes for Mn1 and reflections ( h, k, l) with even h + k, l are, \n Qξ1(+) ≈ (3/2) T1ξ1 − √3[e ξ eζ T2+21'' + (e η2 − eζ2) T2+11''], \n Qη1(+) ≈ √3 eη [eξ T2+11'' + eζ T2+21''], (10) \n Qξ1(+) ≈ (3/2) T1ζ1 − √3[e ξ eζ T2+11'' + (e η2 − eξ2) T2+21'']. \nDipole components in the ξ-ζ plane contribute. A common factor 8 cos(2 πk/3) is omitted in \nEq. (10). Recall that e ξ ∝ h, e η ∝ k and e ζ ∝ l. Dirac Mn1 amplitudes have a common factor [8 \nsin(2 πk/3)], and they are zero for the special case ( h, 0, l). We find, \n Q⊥ξ1(−) ≈ − eη [dζ1 + (3/ √5) {(1 − 2 e ξ2) H2+21'' + 2 e ξ eζ H2+11''}], \n Q⊥η1(−) ≈ eξ dζ1 − eζ dξ1 + (3/ ���5) (1 − 2 e η2) [e ζ H2+11'' − eξ H2+21''], \n Q⊥ζ1(−) ≈ eη [dξ1 + (3/ √5) {(1 − 2 e ζ2) H2+11'' + 2 e ξ eζ H2+21''}]. (11) \nAmplitudes for Mn1 and Mn2 are similar, because Mil ler index l is even. \nIV. RESONANT X-RAY DIFFRACTION \n The Bragg angle θ in Fig. 4 for the reflection ( h, 0, l) is determined by, \n sin( θ) = ( λ/2 a) [ h2 + (r l)2]1/2 , (12) \nwith a wavelength λ ≈ (12.4/E) Å, and the resonance energy E in keV. Man ganese absorption \nedges used in an electric dipole- electric dipole ( E1-E1) scattering event have energies E ≈ \n6.537 keV for the K edge (1s → 4p), and L 2 ≈ 0.649 keV and L 3 ≈ 0.638 keV (2p → 3d). \nEnhancement at the K edge reveals Mn orbital states alone [19]. Previous studies using resonant \nx-ray diffraction at the K-edge of 3d-transition me tal compounds include V 2O3, α-Fe 2O3 and \nNiO [11, 15, 45, 46]. Spin and orbital states are r evealed at L edges and their contributions to \nabsorption satisfy useful sum-rules [47]. The space group forbidden Bragg spot (0, 0, 1) can be \naccessed at L edges, while many Bragg spots indexed by ( h, 0, l) with even h are available \nusing the K edge. Radial matrix elements for Mn (3d5) calculated from Cowan’s program [44] are (1s|R|4p)/a o = − 0.00354 and (2p|R|3d)/(1s|R|4p) = 58.25, where a o is the Bohr radius. Dirac \nmultipoles contribute to an E1-E2 event allowed by 3d-4p mixing (1s → 3d, 1s → 4p), and the \nMn E2 radial integral (1s|R 2|3d)/ a o2 = 0.00095. An E1-E2 event manifests itself as a pre -edge \nfeature to K edge absorption spectra [48, 49]. \n \n Four states of polarization in the primary ( σ, π) and secondary ( σ', π') x-ray beams are \ndepicted in Fig. 4. The scattering amplitude for pr imary σ and secondary π' is denoted ( π'σ), \nfor example. Intensity of a Bragg spot in the rotat ed channel of polarization is proportional to \n|( π'σ)| 2, and likewise for unrotated channels. Proportional ity factors include radial integrals, \nand E1-E1 and E2-E2 amplitudes for the K edge are p roportional to (1s|R|4p) 2 and \n[(1s|R 2|3d)/λ]2, respectively. Reported amplitudes are functions o f the angle ψ that measures \nrotation of the illuminated crystal about the refle ction vector and they are derived from \nuniversal expressions [20, 50]. \n It follows from the structure factor Eq. (3) for M n1 that diffraction indexed ( h, 0, l) \nproceeds by a parity-even absorption event E1-E1, o r E2-E2, with σθ (−1) K = +1 [20, 50]. On \nthe other hand, selecting forbidden reflections wit h even h and odd l means diffraction by Mn2 \nproceeds by a parity-odd event E1-E2. Consider firs t Mn1, an E1-E1 event and odd l. The \ncorresponding electronic structure factor ΨKQ(1) is purely imaginary. An axial dipole t1η1 \ncontributes to both ( π'π) and ( π'σ), and not ( σ'σ) [20, 50]. For the E1-E1 rotated channel, \n ( π'σ) = cos( θ) cos( ψ) ( i/√2) t1η1 − sin( θ) cos(2 ψ) [ α t2+11'' + β t2+21'' ] \n + cos( θ) sin( ψ) [β t2+11'' − α t2+21'' ]. (13) \nThe reciprocal lattice vector b* and the η-axis are antiparallel at the start of an azimuthal angle \nscan ( ψ = 0). In Eq. (13), α = cos( χ), β = sin( χ) with, \n cos( χ)= h/[ h2 + (r l)2]1/2 , (14) \n and we omit a common factor = 8. Schmitt et al . [51] set out attributes of ( π'σ) that make it \nattractive to measurements. Quadrupoles t2Q1'' in Eq. (13) are time-even (non-magnetic) and \naccount for T & T scattering, i.e., nominally weak scattering created by angular anisotropy in \nthe electronic charge distribution of the resonant ion [24, 25]. \n Intensity picked out by circular polarization in t he primary photon beam = P 2ϒ where \n[16, 17, 18, 52], \n \n ϒ = {( σ'π)*( σ'σ) + (π'π)*( π'σ)} '' . (15) \n \nThe Stokes parameter P 2 (a purely real pseudoscalar) measures helicity in the primary x-ray \nbeam. Since intensity is a true scalar, ϒ and P 2 must possess identical discrete symmetries, \nspecifically, both scalars are time-even and parity -odd. The signature is extracted from \nobserved Bragg spot intensities by subtraction of i ntensities measured with left- and right-\nhanded primary x-rays, or x-rays with opposite heli cities, namely, ± P2. From Eq. (15), ions Mn1 observed at ( h, 0, l) with even h and odd l using an E1-E1 absorption event possess a \nchiral signature, \n ϒ = − (1/ √2) t1η1 cos( θ) sin( ψ) {sin(2 θ) sin( ψ) [β t2+11'' − α t2+21'' ] \n + 2 [1 − (cos( θ) sin( ψ)) 2] [ α t2+11'' + β t2+21'' ]}, (16) \napart from a factor = 64. A non-zero ϒ relies on interference between a magnetic dipole t1η1 \nparallel to the unique axis b and charge-like quadr upoles t2Q1'' . Estimates of the two \nquadrupoles can be inferred from contributions to ϒ that are even and odd functions of ψ. \n Continuing with ( h, 0, l) with even h and odd l, Mn2 ions diffract Dirac multipoles \nincluding anapoles G1ξ2 and G1ζ2. For an E1-E2 event, \n ( σ'σ) ≈ (4 i/5) √3 ( −1) m cos( θ) sin( ψ) [ β G1ξ2 − α G1ζ2]. (17) \nat the level of anapoles. A complete result for ( σ'σ) appears in an Appendix. Notably, a simple \ndependence on the azimuthal angle sin( ψ) extends to quadrupoles in the unrotated amplitude . \nThe corresponding chiral signature is zero, because all four diffraction amplitudes in Eq. (15) \nare purely imaginary. Indeed, ( σ'σ) and ( π'π) have identical structures for an E1-E2 event \nwhereas they differ by a dipole contribution for an E1-E1 event [50]. \nV. CONCLUSIONS \n In summary, we have shown through calculations inf ormed by magnetic symmetry that \nferrimagnetic Mn 3Si 2Te 6 likely possesses many unusual properties, in additi on to a recently \ndiscovered unique CMR [6, 7]. Our revelations are d erived from a monoclinic space group \npreviously proposed on the basis of an investigatio n using magnetic neutron diffraction [5]. \nManganese ions use two symmetry independent sites, as in the trigonal paramagnetic structure \n[1]. Beyond axial magnetic dipoles, depicted in Fig . 2, are magnetic quadrupoles and anapoles \nthat are possibly visible in an analysis of high-re solution diffraction patterns. By way of an \nexample, the intensity of a Bragg spot with a refle ction vector κ ≈ 1.122 Å −1 is reported in Ref. \n[5]. Contributions to the neutron scattering amplit ude from spin and orbital anapoles are \naccompanied by atomic form factors (h 1) and (j 0) with values ≈ 10% of the axial form factor \nj0 for this reflection vector, as can be seen by insp ection of Fig. 3. In consequence, the spin \nanapole might account for ≈ 30% of the neutron scattering amplitude Eq. (6). M agnetic \nquadrupoles have form factors j2 that peak above ≈ 7 Å −1. \n A nominally weak Bragg spot, forbidden by reflecti on conditions for the parent \nstructure of Mn 3Si 2Te 6, has been measured as a function of a magnetic fie ld [5]. The standard \nmethod to estimate the magnetic content of diffract ion patterns is to analyse the difference \nbetween two measured above and below the magnetic o rdering temperature. Polarization \nanalysis in neutron diffraction provides greater se nsitivity to the magnetic content of Bragg \nspots with overlapping magnetic and nuclear content s permitted by reflection conditions of the \nparent structure [21, 35-37, 43]. In presenting our results for future neutron and resonant x-ray \ndiffraction experiments, we pay particular attentio n to weak magnetic Bragg spots. In x-ray diffraction by non-magnetic materials they arise fr om Templeton-Templeton scattering created \nfrom angular anisotropy in electronic charge distri butions [24, 25]. \n Magnetic multipoles revealed by resonant x-ray dif fraction do not contain atomic form \nfactors [14, 15, 20]. Axial and Dirac multipoles co ntribute to different diffraction amplitudes \nat different resonance energies. Orbital degrees of freedom in the Mn p-type valence state are \nexposed through signal enhancement by an (parity-ev en) electric dipole - electric dipole (E1-\nE1) event at the Mn K edge [19, 20]. Absorption at an L edge samples the Mn d-type valence \nstate. Whereas, Dirac multipoles are exposed in an (parity-odd) electric dipole - electric \nquadrupole (E1-E2) event using a pre-edge feature t o the Mn K edge [48, 49]. Amplitudes in \nSection IV are functions of rotation of the Mn 3Si 2Te 6 sample about the reflection vector, and \nazimuthal-angle displays are different for differen t absorption events. Notably, the more \nabundant Mn ions possess an E1-E1 chiral signature Eq. (16) not found for the second type of \nMn ions for the same diffraction conditions. \nACKNOWLEDGEMENT A theoretical study of diffraction patterns for Mn 3Si 2Te 6 was \nproposed by Dr P. Manuel. Dr D. D. Khalyavin and Dr K. S. Knight assisted with applications \nof crystal and magnetic symmetries. Dr V. Scagnoli contributed Fig. 1. Professor G. van der \nLaan produced Fig. 3, and all cited estimates of ra dial matrix elements for x-ray diffraction. \nAPPENDIX \nMn2, reflection ( h, 0, l) even h and l = (2 m + 1), a common factor 4, and an E1-E2 absorption \nevent. The abbreviations α = cos( χ), β = sin( χ), with cos( χ)= h/[ h2 + (r l)2]1/2 with r = ( a/c) ≈ \n0.493. \n ( σ'σ) = (4 i/5) √3 ( −1) m cos( θ) sin( ψ) { α [− G1ζ2 + (1/3) √10 G2+22''] \n + β [G1ξ2 − (1/3) √10 G2+12''] + √(2/3) α [1 − 5 α2 cos 2(ψ)] G302 \n + (1/3) √2 β [1 − 15 α2 cos 2(ψ)] G3+12' + (2/3) √5 α [1 − 3 (1 + β2) cos 2(ψ)] G3+22' \n − √(10/3) β [α2 + (1 − 4 α2) cos 2(ψ)] G3+32'}. (A1) \nEvidently, octupole contributions to ( σ'σ) modify the simple sin( ψ) azimuthal-angle \ndependence that hallmarks anapoles and quadrupoles. \nReferences \n[1] R. Rimet, C. Schlenker and H. Vincent, J. Magn. Magn. Mater. 25 , 7 (1981). \n[2] H. Vincent, D. Leroux, D. Bijaoui, R. Rimet and C. Schlenker, J. 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E. von Neumann, Vorlesungenüber die Theorie Elastizität der festen Körper und des \n Lichtäthers (Leipzig: Teubner, 1885). \n[27] J. F. Nye, Physical Properties of Crystals (Oxford University Press, London, 1964). \n[28] A. P. Cracknell, Magnetism in Crystalline Materials (Pergamon Press, Oxford, 1975). \n[29] S. W. Lovesey and E. Balcar, J. Phys. Soc. Jpn . 82 , 021008 (2013). \n[30] S. W. Lovesey, J. Phys. Condens. Matter 26 , 356001 (2014). [31] O. Bun ău, A. Y. Ramos and Y. Joly, International Tables for Crystallography Vol. I: \n (The Netherlands: Springer, 2021). \n[32] J. Haase et al ., Phys. Rev. B 69 , 094504 (2004). \n[33] We use the BNS setting of magnetic space group s, see Bilbao Crystallographic server, \n http://www.cryst.ehu.es. \n[34] MAGNDATA, http://webbdcrista1.ehu.es/magndata. \n[35] H. A. Alperin, P. J. Brown, R. Nathans, and S. J. Pickart, Phys. Rev. Lett. 8, 237 (1962). \n[36] R. M. Moon, T. Riste and W. C. Koehler, Phys. Rev. 181 , 920 (1969). \n[37] P. J. 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Wang, ibid . 70 , 694 (1993); P. Carra, H. König, \nB. T. Thole, and M. Altarelli, Physica B 192 , 182 (1993). \n[48] D. Cabaret et al ., Phys. Chem. Chem. Phys. 12 , 5619 (2010). \n[49] D. F. Leto and T. A. Jackson, Inorg. Chem. 53 , 6179 (2014). \n[50] V. Scagnoli and S. W. Lovesey, Phys. Rev. B 79 , 035111 (2009). \n[51] A. T. Schmitt et al ., Optica 8, 56 (2021). \n[52] J. Fernández-Rodríguez, S. W. Lovesey and J. A . Blanco, Phys. Rev. B 77 , 094441 \n (2008). \n \n \n \n \n \n \n \n \n \n \nFIG. 1. Depiction of a toroidal dipole, also known as an anapole. Figure prepared by V. \nScagnoli. \n \nFIG. 2. Mn axial dipoles in Mn 3Si 2Te 6 depicted in an hexagonal setting. Si ions in blue and \nMnTe 6 units.After A. F. May et al . [3]. Reproduced from MAGNDATA [34]. \nFIG. 3. Radial integrals for Mn 2+ (3d 5) displayed as a function of the magnitude of the \nreflection vector κ = 4 πs with s = sin( θ)/ λ (Å −1), Bragg angle θ and neutron wavelength λ. \nAlso, a dimensionless variable w = 3a oκ where a o is the Bohr radius. Blue and purple lines \nare standard radial integrals j0(κ) and j2(κ) that occur in the axial dipole Eq. (5). Red, green \nand blue curves are radial integrals in the polar d ipole Eq. (6). Two integrals (g 1) and (j 0) \ndiverge in the forward direction of scattering, and quantities w(g 1) and w(j 0) are displayed for \nthis reason . Calculations, performed with Cowan's atomic code [44], and figure made by G. \nvan der Laan. \n \nFIG. 4. Primary ( σ, π) and secondary ( σ', π') states of polarization. Corresponding wavevector s \nq and q' subtend an angle 2 θ specified in Eq. (12). The Bragg condition for dif fraction is met \nwhen q − q' coincides with the reflection vector indexed ( h, k, l). Monoclinic crystal vectors a, \nb, and c that define local axes ( ξ, η, ζ) and the depicted Cartesian (x, y, z) coincide in the \nnominal setting of the crystal where the azimuthal angle ψ = 0. \n" }, { "title": "1212.6032v1.Magnetization_process_in_the_exactly_solved_spin_1_2_Ising_Heisenberg_model_on_decorated_Bethe_lattices.pdf", "content": "arXiv:1212.6032v1 [cond-mat.stat-mech] 25 Dec 2012Condensed Matter Physics, 2012, Vol. 15, No 4, 43003: 1–10\nDOI: 10.5488/CMP.15.43003\nhttp://www.icmp.lviv.ua/journal\nMagnetizationprocessintheexactlysolvedspin-1/2\nIsing-HeisenbergmodelondecoratedBethelattices\nJ. Strečka1, C. Ekiz2\n1Department of Theoretical Physics and Astrophysics, Facul ty of Science, P.J. Šafárik University,\nPark Angelinum 9, 040 01 Košice, Slovak Republic\n2Department of Physics, Faculty of Science, Adnan Menderes U niversity, 090 10 Aydın, Turkey\nReceived June 22, 2012, in final form August 10, 2012\nThe spin-1/2 Ising-Heisenberg model on diamond-like decor ated Bethe lattices is exactly solved in the pres-\nence of the longitudinalmagnetic field by combiningthe deco ration-iterationmapping transformation with the\nmethodofexactrecursionrelations.Inparticular,thegro undstateandlow-temperaturemagnetizationprocess\nof the ferrimagnetic version of the considered model is inve stigated in detail. Three different magnetization\nscenarios with up to two consecutive fractional magnetizat ion plateaus were found, whereas the intermedi-\nate magnetization plateau may either correspond to the clas sical ferrimagnetic spin arrangement and/or the\nfield-inducedquantum ferrimagnetic spin ordering without any classical counterpart.\nKeywords: Ising-Heisenberg model, Bethe lattice, exact results, mag netization plateau\nPACS:05.50.+q, 75.10.-b, 75.10.Jm, 75.10.Kt 75.40.Cx, 75.60.E j\n1.Introduction\nLow-dimensionalquantumspinsystemshaveattractedmucha ttentionoverthepastfewdecades,\nsincetheyexhibitalotofstrikingquantumphenomenainclu dingfractionalmagnetizationplateaus,\nspin-Peierlsdimerization,unconventionalspin-liquidg roundstates,ormanyotherpeculiarvalence-\nbond-solidgroundstatessuchastheHaldanephase[1,2].It isworthnotingthatthemostremarkable\nexperimentalfindingsreportedforlow-dimensionalspinsy stemsweremostlysatisfactorilyinterpreted\nwiththehelpofquantumHeisenbergmodelanditsvariousext ensions.Fromthetheoreticalpointof\nview,anexacttreatmentofthequantumHeisenbergmodelrem ainsanunresolvedproblemmainlydue\ntosubstantialmathematicaldi fficulties,whicharisefromanoncommutabilityofspinoperat orsinvolved\nintherelevantHamiltonian.However,thismathematicalco mplexitycanbeavoidedbyconsideringsim-\nplerIsing-Heisenbergmodels,whichdescribehybridclass ical-quantumspinsystemsconstitutedbothby\nthe‘classical ’IsingaswellasthequantumHeisenbergspins.ThehybridIsi ng-Heisenbergmodelscanbe\nexactlytreatedbymakinguseofgeneralizedmappingtransf ormations,whichwereoriginallyintroduced\nbySyozi[3,4]andlaterongeneralizedbyFisher[5],Rojase tal.[6,7]andoneofthepresentauthors[8].\nInthiswork,thegeneralizeddecoration-iterationtransf ormationiscombinedwiththemethodofex-\nactrecursionrelationsinordertoobtainexactresultsfor thespin-1\n2Ising-Heisenbergmodelondiamond-\nlikedecoratedBethelatticesinthepresenceofthelongitu dinalmagnetic field.Itshouldbenotedthatthe\napplieddecoration-iterationtransformationestablishe sarigorousmappingequivalencebetweenthein-\nvestigatedmodelsystemandthespin-1\n2IsingmodelonacorrespondingsimpleBethelatticewiththe\neffectivenearest-neighbourinteractionandtheeffectiv emagnetic field.Owingtothisprecisemapping\ncorrespondence,exactresultsforthespin-1\n2Ising-Heisenbergmodelonthediamond-likedecoratedBeth e\nlatticescanbesubsequentlyextractedfromtherelevantex actsolutionofthespin-1\n2Isingmodelonasim-\npleBethelatticebymeansofthemethodofexactrecursionre lations[9 –11].\nTheorganizationofthispaperisasfollows.Insection2,th edetaileddescriptionoftheinvestigated\nmodelsystemispresentedtogetherwiththebasicstepsofit sexactsolution.Themostinterestingre-\n©J. Strečka, C. Ekiz, 2012 43003-1J. Strečka, C. Ekiz\nsultsarethenpresentedanddiscussedinsection3.Inparti cular,ourattentionisfocusedontheground\nstateandlow-temperaturemagnetizationprocessofthefer rimagneticversionoftheconsideredmodel.\nFinally,someconcludingremarksaredrawninsection4.\n2.Ising-HeisenbergmodelondecoratedBethelattices\nLetusintroducethespin-1\n2Ising-Heisenbergmodelonadiamond-likedecoratedBethel attice,which\nisschematicallyillustratedontheleft-hand-sideof figure1ontheparticularexampleoftheunderlying\nBethelatticewiththecoordinationnumber q=3.Inthis figure,thefullcircleslabellatticepositionsof\ntheIsingspins µ=1\n2,whiletheemptycirclesmarklatticepositionsoftheHeise nbergspins S=1\n2.One\nmayinferfrom figure1thatthemagneticstructureoftheinvestigatedmodel isformedbytheIsingspins\nplacedatlatticesitesofadeepinteriorofin finiteCayleytree(Bethelattice),whicharelinkedtogether\nthroughtheHeisenbergspinpairsplacedin-betweeneachco upleoftheIsingspins.ThetotalHamiltonian\nofthespin-1\n2Ising-Heisenbergmodelondiamond-likedecoratedBethela tticesreads\nH=− JHN q/2/summationdisplay\n(k,l)/bracketleftbig\n∆/parenleftbig\nSx\nkSx\nl+Sy\nkSy\nl/parenrightbig\n+Sz\nkSz\nl/bracketrightbig\n−JI2N q/summationdisplay\n(k,i)Sz\nkµz\ni−HAN/summationdisplay\ni=1µz\ni−HBN q/summationdisplay\nk=1Sz\nk.(1)\nHere, Sα\nk(α=x,y,z)andµz\nirepresentspatialcomponentsofthespin-1\n2operator,theparameter JHde-\nnotestheXXZinteractionbetweenthenearest-neighbourHe isenbergspins,theparameter ∆controlsa\nspatialanisotropyinthisinteractionbetweentheeasy-ax is(∆<1)andeasy-plane (∆>1)regime,and\ntheparameter JImarkstheIsinginteractionbetweenthenearest-neighbour HeisenbergandIsingspins,\nrespectively.Furthermore,twoZeeman ’sterms HAand HBdeterminethemagnetostaticenergyofthe\nIsingandHeisenbergspinsinalongitudinalmagnetic field.\nDIT\nSk1\nSk2/c109k1\n/c109k2/c109k1\n/c109k2Jeff\nq=3JI\nJH( )/c68\nFigure1. The spin-1\n2Ising-Heisenberg model on the diamond-like decorated Beth e lattice (figure on the\nleft) and its exact mapping via the decoration-iteration tr ansformation (DIT) onto the spin-1\n2Ising model\non a simple Bethe lattice (figure on the right). The full (empt y) circles denote lattice positions of the Ising\n(Heisenberg) spins, the ellipse demarcates the elementary diamond-shaped spin cluster described by the\nkth bond Hamiltonian (3).\nItisquiteevidentfrom figure1thateachpairofHeisenbergspinsissurroundedbyone coupleof\ntheIsingspinslocatedatlatticesitesofasimpleBethelat ticeandhence,themodelunderconsideration\ncanalternativelybeviewedastheBethelatticeofIsingspi nswhose( fictitious)bondsaredecoratedina\ndiamond-likefashionbytwoquantumHeisenbergspins.Invi ewoffurthermanipulations,itis,therefore,\nofpracticalimportancetorewritethetotalHamiltonian(1 )asasumofbondHamiltonians\nH=N q/2/summationdisplay\nk=1Hk, (2)\nwhereasthebondHamiltonian Hkinvolvesalltheinteractiontermsbelongingtothe kthdiamond-\n43003-2Spin-1/2 Ising-Heisenberg model in the external magnetic fi eld\nshapedclusterspeci ficallydelimitedin figure1byanellipse\nHk= − JH/bracketleftbig\n∆/parenleftbig\nSx\nk1Sx\nk2+Sy\nk1Sy\nk2/parenrightbig\n+Sz\nk1Sz\nk2/bracketrightbig\n−JI/parenleftbig\nSz\nk1+Sz\nk2/parenrightbig/parenleftbig\nµz\nk1+µz\nk2/parenrightbig\n−HB/parenleftbig\nSz\nk1+Sz\nk2/parenrightbig\n−HA\nq/parenleftbig\nµz\nk1+µz\nk2/parenrightbig\n. (3)\nOwingtothevalidityofthecommutationrelationshipbetwe endifferentbondHamiltonians [Hi,Hj]=\n0,thepartitionfunctioncanbepartiallyfactorizedintoap roductofbondpartitionfunctions\nZIHM=/summationdisplay\n{µi}N q/2/productdisplay\nk=1Trkexp(−βHk)=/summationdisplay\n{µi}N q/2/productdisplay\nk=1Zk, (4)\nwhere β=1/(kBT),kBistheBoltzmann ’sconstantand Tistheabsolutetemperature.ThesymbolTr k\ndenotesatraceoverdegreesoffreedomoftwoHeisenbergspi nsfromthe kthdiamond-shapedcluster\nandthesummation/summationtext\n{µi}runsoverallpossiblecon figurationsoftheIsingspins.Thebondpartition\nfunction Zkcanbeevaluatedinthemoststraightforwardwaybyadirectd iagonalizationofthebond\nHamiltonian(3)withintheparticularsubspaceofthe kthHeisenbergspinpairandemployingatrace\ninvarianceofthebondpartitionfunctionwithrespecttoau nitarytransformation.Afterexecutingthis\nprocedureonegainstheresultantexpression,whichimplie sapossibilityofapplyingthegeneralized\ndecoration-iterationtransformation[5 –8]\nZk/parenleftbig\nµz\nk1,µz\nk2/parenrightbig\n= 2exp/bracketleftbiggβHA\nq/parenleftbig\nµz\nk1+µz\nk2/parenrightbig/bracketrightbigg/braceleftbigg\nexp/parenleftbiggβJH\n4/parenrightbigg\ncosh/bracketleftbig\nβJI/parenleftbig\nµz\nk1+µz\nk2/parenrightbig\n+βHB/bracketrightbig\n+exp/parenleftbigg\n−βJH\n4/parenrightbigg\ncosh/parenleftbiggβJH∆\n2/parenrightbigg/bracerightbigg\n=Aexp/bracketleftbigg\nβJeffµz\nk1µz\nk2+βHeff\nq/parenleftbig\nµz\nk1+µz\nk2/parenrightbig/bracketrightbigg\n.(5)\nConsideringfouravailablecombinationsofspinstatesoft woIsingspins µz\nk1andµz\nk2,onegetsfromthe\ntransformationformula(5)threeindependentequationsth atunambiguouslydeterminethemapping\nparameters A,Jeffand Heff\nA=2/parenleftbig\nV+V−V2\n0/parenrightbig1/4,βJeff=ln/parenleftBigg\nV+V−\nV2\n0/parenrightBigg\n,βHeff=βHA+q\n2ln/parenleftbiggV+\nV−/parenrightbigg\n, (6)\nwhichareforsimplicityde finedthroughthefunctions V±and V0\nV±=exp/parenleftbiggβJH\n4/parenrightbigg\ncosh/parenleftbig\nβJI±βHB/parenrightbig\n+exp/parenleftbigg\n−βJH\n4/parenrightbigg\ncosh/parenleftbiggβJH∆\n2/parenrightbigg\n,\nV0=exp/parenleftbiggβJH\n4/parenrightbigg\ncosh/parenleftbig\nβHB/parenrightbig\n+exp/parenleftbigg\n−βJH\n4/parenrightbigg\ncosh/parenleftbiggβJH∆\n2/parenrightbigg\n. (7)\nAtthisstage,thedirectsubstitutionofthealgebraicmapp ingtransformation(5)intothefactorizedform\nofthepartitionfunction(4)leadstoarigorousmappingrel ationship\nZIHM(β,JI,JH,∆,HA,HB,q)=AN q\n2ZIM(β,Jeff,Heff,q), (8)\nwhichconnectsthepartitionfunction ZIHMofthespin-1\n2Ising-Heisenbergmodelonthediamond-like\ndecoratedBethelatticewiththepartitionfunction ZIMofthespin-1\n2Isingmodelonacorresponding\nsimple(undecorated)Bethelatticeschematicallyillustr atedontheright-hand-sideof figure1andmath-\nematicallygivenbytheHamiltonian\nHIM=− JeffN q/2/summationdisplay\n(i,j)µz\niµz\nj−HeffN/summationdisplay\ni=1µz\ni. (9)\nApparently,themappingparameters Jeffand Heffgivenbyequations(6) –(7)determinetheeffective\nnearest-neighbourinteractionandtheeffectivemagnetic fieldofthecorrespondingspin-1\n2Isingmodel\n43003-3J. Strečka, C. Ekiz\nonthesimpleBethelattice,whilethemappingparameter Aisjustasimplemultiplicativefactorinthe\nestablishedmappingrelation(8)betweenbothpartitionfu nctions.\nNow,otherphysicalquantitiesofourparticularinterestf ollowquitestraightforwardly.Forinstance,\nwiththehelpofequation(8),oneeasily findsasimilarmappingrelationbetweenthefreeenergy FIHMof\nthespin-1\n2Ising-Heisenbergmodelonthediamond-likedecoratedBeth elatticeandthefreeenergy FIM\noftheequivalentspin-1\n2IsingmodelonasimpleBethelattice\nFIHM=−kBTlnZIHM=FIM−N qk BT\n2lnA. (10)\nConsequently,thesingle-sitesublatticemagnetizationo ftheIsingspinscanbecalculatedbydifferentiat-\ningthefreeenergy(10)withrespecttotherelevantmagneti cfieldHA\nmA=−1\nN∂FIHM\n∂HA=−1\nN/parenleftbigg∂FIM\n∂βHeff/parenrightbigg∂βHeff\n∂HA=mIM(β,Jeff,Heff). (11)\nAccordingtoequation(11),thesublatticemagnetizationo ftheIsingspinsinthespin-1\n2Ising-Heisenberg\nmodelonthediamond-likedecoratedBethelatticeisequalt othemagnetizationofthecorresponding\nspin-1\n2IsingmodelonthesimpleBethelatticewiththeeffectivene arest-neighbourinteraction Jeffand\ntheeffectivemagnetic fieldHeffgivenby(6) –(7).Asimilarcalculationprocedurecanalsobeperformed\nforobtainingthesingle-sitesublatticemagnetizationof theHeisenbergspins,whichcanbeforconve-\nnienceexpressedintermsofthemagnetization mIMandthenearest-neighbourpaircorrelation εIMof\ntheequivalentspin-1\n2IsingmodelonthesimpleBethelattice\nmB= −1\nN q∂FIHM\n∂HB=1\n2∂lnA\n∂βHB−/parenleftbigg1\nN q∂FIM\n∂βJeff/parenrightbigg∂βJeff\n∂HB−/parenleftbigg1\nN q∂FIM\n∂βHeff/parenrightbigg∂βHeff\n∂HB\n=1\n8/parenleftbiggW+\nV+−W−\nV−+2W0\nV0/parenrightbigg\n+εIM\n2/parenleftbiggW+\nV+−W−\nV−−2W0\nV0/parenrightbigg\n+mIM\n2/parenleftbiggW+\nV++W−\nV−/parenrightbigg\n.(12)\nThenewlyde finedfunctions W±and W0aregivenby\nW±=exp/parenleftbiggβJH\n4/parenrightbigg\nsinh/parenleftbig\nβJI±βHB/parenrightbig\n, W0=exp/parenleftbiggβJH\n4/parenrightbigg\nsinh/parenleftbig\nβHB/parenrightbig\n. (13)\nTocompleteourexactcalculationofbothsublatticemagnet izations,itisnowsu fficienttosubsti-\ntuteintothederivedformulas(11) –(12)therelevantexactresultsforthemagnetizationandne arest-\nneighbourspin-spincorrelationofthecorrespondingspin -1\n2IsingmodelonthesimpleBethelatticewith\ntheeffectivenearest-neighbourinteraction Jeffandtheeffectivemagnetic fieldHeffgivenby(6) –(7).The\nsublatticemagnetizationandspin-spincorrelationfunct ionofthespin-1/2Isingmodelontheundeco-\nratedBethelatticecanberigorouslyfoundwithintheframe workofexactrecursionrelations[9 –13].If\nthesimpleBethelattice(see figure1, figureontheright)is ‘cut’atacentralsitewiththespin µk1,itwill\ndisintegrateinto qidenticalbranchesandthepartitionfunctionofthesystem willtaketheform\nZ=/summationdisplay\nµk1exp(βHeffµk1)/bracketleftbig\ngn(µk1)/bracketrightbigq, (14)\nwhere gn(µk1)isthepartitionfunctionofaseparatebranch\ngn(µk1)=/summationdisplay\nµk2exp(βJeffµk1µk2+βHeffµk2)/bracketleftbig\ngn−1(µk2)/bracketrightbigq−1. (15)\nByusingof(15),onecaneasilyobtainarecursionrelations hipforthevariable xn=gn(−1/2)\ngn(+1/2)\nxn=exp/parenleftBig\n−βJeff\n4+βHeff\n2/parenrightBig\n+exp/parenleftBigβJeff\n4−βHeff\n2/parenrightBig\nxq−1\nn−1\nexp/parenleftBigβJeff\n4+βHeff\n2/parenrightBig\n+exp/parenleftBig\n−βJeff\n4−βHeff\n2/parenrightBig\nxq−1\nn−1. (16)\n43003-4Spin-1/2 Ising-Heisenberg model in the external magnetic fi eld\nEventhoughtheparameter xndoesnothaveadirectphysicalsense,itplaysacrucialrole indetermining\nthecanonicalensembleaveragesofallphysicalquantities inthelimit n→ ∞.Forinstance,oneeas-\nilyobtainsthefollowingexpressionsforthemagnetizatio nandnearest-neighbourspin-spincorrelation\nfunctionofthespin-1/2IsingmodelontheBethelattice\nmIM=1\n2exp/parenleftbig\nβHeff/parenrightbig\n−xq\nexp/parenleftbig\nβHeff/parenrightbig\n+xq,\nεIM=1\n4exp/parenleftBigβJeff\n4+βHeff/parenrightBig\n−2exp/parenleftBig\n−βJeff\n4/parenrightBig\nxq−1+exp/parenleftBigβJeff\n4−βHeff/parenrightBig\nx2q−2\nexp/parenleftBigβJeff\n4+βHeff/parenrightBig\n+2exp/parenleftBig\n−βJeff\n4/parenrightBig\nxq−1+exp/parenleftBigβJeff\n4−βHeff/parenrightBig\nx2q−2.(17)\nwhichcanbebothexpressedthroughastable fixedpoint x=limn→∞xnoftherecurrencerelation(16).\n3.Resultsanddiscussion\n/s99/s50/s99/s49/s83/s80/s80\n/s67/s70/s80/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s81/s70/s80/s72/s32/s61/s32/s74/s72/s40/s45/s49/s41/s47/s50/s32/s43/s32/s124/s74/s73/s124\n/s72/s32/s61/s32/s113/s91/s74/s72/s32/s40/s45/s49/s41/s47/s50/s32/s45/s32/s124/s74/s73/s124/s93/s47/s40/s113/s45/s50/s41/s72/s32 /s61/s32 /s113/s124/s74\n/s73/s124/s32\n/s72/s32\nFigure2. The general ground-state phase diagram in\nthe∆−Hplane.Inthispart,letusproceedtoadiscussionof\nthemostinterestingresultsobtainedforthefer-\nrimagneticversionofthespin-1\n2Ising-Heisenberg\nmodelonthediamond-likedecoratedBethelat-\nticewiththeferromagneticHeisenberginterac-\ntion JH>0andtheantiferromagneticIsingin-\nteraction JI<0,which,atsu fficientlylow fields,\nwill favour the antiparallel alignment between\nthe nearest-neighbouring Ising and Heisenberg\nspins,respectively.Itisworthwhiletoremarkthat\nthecriticalbehaviouroftheconsideredmodelin\nthe absence of the external magnetic field has\nbeeninvestigatedinsomedetailinourprevious\nwork[14]andhence,theeffectofanon-zeromag-\nneticfieldwillbeatthemainfocusofourresearch\ninterest.Toreducethetotalnumberoffreepa-\nrameters,themostnotablefeaturesofthemagnetizationpr ocesswillbeillustratedforaspeci ficchoice\nH≡HA=HB,whichcoincideswithsettingLandég-factorsoftheIsinga ndHeisenbergspinsequalto\neachother.\nFirst,letuscommentonpossiblespinarrangementsemergin gatzerotemperature.Owingtothe\nvalidityofthecommutationrelationshipbetweenthediffe rentclusterHamiltonians,theground-state\nspinarrangementscaneasilybeobtainedbysearchingforth elowest-energyeigenstateofthecluster\nHamiltonian(3).Theground-statephasediagramdisplayed infigure2impliestheexistenceofthree\ndifferentgroundstates,whichcanbethoroughlycharacter izedbythefollowingeigenvectors\n|CFP〉 =N/productdisplay\nk=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleµz\nk=−1\n2/angbracketrightbiggN q/2/productdisplay\nk=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk1=1\n2/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk2=1\n2/angbracketrightbigg\n,\n|QFP〉 =N/productdisplay\nk=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleµz\nk=sgn(H)1\n2/angbracketrightbiggN q/2/productdisplay\nk=11/rad〉callow\n2/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk1=1\n2/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk2=−1\n2/angbracketrightbigg\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk1=−1\n2/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk2=1\n2/angbracketrightbigg/parenrightbigg\n,\n|SPP〉 =N/productdisplay\nk=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleµz\nk=1\n2/angbracketrightbiggN q/2/productdisplay\nk=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk1=1\n2/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleSz\nk2=1\n2/angbracketrightbigg\n. (18)\nAscouldbeexpected,twogroundstatescorrespondtoclassi calspinarrangementswithaperfectparal-\nlelandantiparallelalignmentsbetweenthenearest-neigh bourIsingandHeisenbergspinstobefurther\nreferredtoastheclassicalferrimagneticphase(CFP)andt hesaturatedparamagneticphase(SPP),respec-\ntively.Apartfromthoserathertrivialphases,onemayalso detectamorespectacularquantumfrustrated\nphase(QFP)withapeculiarspinfrustrationoftheIsingspi nsstemmingfromaquantumentanglement\n43003-5J. Strečka, C. Ekiz\noftheHeisenbergspinpairs.Asamatteroffact,theemergen tquantumsuperpositionoftwopossible\nantiferromagneticstatesoftheHeisenbergspinpairsisre sponsibleinQFPforacompleterandomness\noftheIsingspinsatazeromagnetic fieldasconvincinglyevidencedinourpreviousstudy[14].Du eto\nthespinfrustration,alltheIsingspinstendtoalignintot heexternal- fielddirectionforarbitrarybut\nnon-zeromagnetic fieldand,consequently,astriking quantumferrimagneticphase developsfromQFP\nwithafullpolarizationoftheIsingspinsandthenon-magne ticnatureoftheHeisenbergspinpairs.The\nexistenceofQFPaloneseemstobeaquitegeneralfeatureoft heIsing-Heisenbergmodels,whereamu-\ntualcompetitionbetweentheeasy-axisIsinginteractiona ndtheeasy-planeHeisenberginteractiontakes\nplace[15,16].Furthermore,allphasetransitionsbetween threeavailablegroundstatesareofthe first\norderandtheirexplicitformisgivenin figure2alongthedepictedphaseboundaries.\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48\n/s113 /s32/s61/s32/s51\n/s40/s97 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32 /s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s48/s53\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s40/s98 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s52\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s40/s99 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s54\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s40/s100 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s50\nFigure3. (Color online) The total and sublattice magnetizations as a function of the external magnetic\nfield for the spin-1\n2Ising-Heisenberg model withthecoordinationnumber q=3,theinteractionratio\nJH/|JI|=1.0,theexchangeanisotropy ∆=1.0andfourdifferenttemperatures.\nNow,letusillustratetypicalmagnetizationscenariosasd isplayedin figures3 –5forthespin-1\n2Ising-\nHeisenbergmodelonthediamond-likedecoratedBethelatti cewiththecoordinationnumber q=3,the\nspecificvalueoftheinteractionratio JH/|JI| =1.0,threedifferentvaluesoftheexchangeanisotropy\n∆andseveraltemperatures.Itisworthwhiletoremarkthatth etotalsingle-sitemagnetization mT≡/parenleftbig\nmA+qm B/parenrightbig\n/(1+q)isalsoplottedin figures3 –5inadditiontobothsublatticemagnetizations mAand\nmBoftheIsingandHeisenbergspins,respectively.Iftheexch angeanisotropyisselectedbelowits first\ncriticalvalue ∆<∆c1≡1+2|JI|/JH,then,oneencountersarathertypicalmagnetizationcurve reflecting\nthefield-inducedtransitionfromCFPtoSPPasshownin figure3.Itisquiteclearthattheintermedi-\natemagnetizationplateauobservedatahalfofthesaturati onmagnetizationindeedcorrespondstothe\nclassicalferrimagneticspinarrangementinherenttoCFPa ndthemagnetizationplateaugraduallydi-\nminishesuponincreasingthetemperature.Themostsigni ficantchangesinthedisplayedmagnetization\ncurveevidentlyoccurifthetemperatureisselectedslight lyabovethecriticaltemperatureofCFP(note\nthat kBTc/|JI|≈0.5for∆=1).Eventhoughbothsublatticemagnetizationsalreadystar tfromzerointhis\nparticularcase,theyobviouslytendtowardstypicalmagne tizationvaluesforCFPstillbearingevidence\nofanintermediatemagnetizationplateauatmoderate fieldsandtemperatures[see figure3(c)].\n43003-6Spin-1/2 Ising-Heisenberg model in the external magnetic fi eld\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s40/s97 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/s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s50\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s40/s99 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s51\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s40/s100 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32/s109\n/s66/s32\n/s109\n/s65/s32\n/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32\n/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s50\nFigure4.(Coloronline)Thetotalandsublatticemagnetizationsasa functionoftheexternalmagnetic\nfieldforthespin-1\n2Ising-Heisenbergmodelwiththecoordinationnumber q=3,theinteractionratio\nJH/|JI|=1.0,theexchangeanisotropy ∆=4.0andfourdifferenttemperatures.\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s40/s97 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32\n/s109\n/s66/s32/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s48/s53\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s40/s98 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32\n/s109\n/s66/s32/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s51\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s40/s99 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32\n/s109\n/s66/s32/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s48/s46/s54\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s40/s100 /s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s32 /s47/s32 /s124/s74\n/s73/s124/s109\n/s84/s32\n/s109\n/s66/s32/s109\n/s65/s32\n/s113 /s32/s61/s32/s51/s109\n/s65/s32/s44/s32/s109\n/s66 /s32/s32/s44\n/s32/s109\n/s84/s32/s32/s74\n/s72/s32/s32/s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s48/s107\n/s66/s32/s84/s32 /s47/s32 /s124/s74\n/s73/s124/s32/s61/s32/s49/s46/s50\nFigure5.(Coloronline)Thetotalandsublatticemagnetizationsasa functionoftheexternalmagnetic\nfieldforthespin-1\n2Ising-Heisenbergmodelwiththecoordinationnumber q=3,theinteractionratio\nJH/|JI|=1.0,theexchangeanisotropy ∆=6.0andfourdifferenttemperatures.\n43003-7J. Strečka, C. Ekiz\nHowever,themostinterestingmagnetizationprocesscanbe foundiftheexchangeanisotropyisse-\nlectedfromtheinterval ∆∈(∆c1,∆c2)with∆c2≡1+2(q−1)|JI|/JH.Underthiscondition,atlowenough\ntemperatures,thetotalmagnetizationexhibitstwosucces sivefractionalmagnetizationplateausatone\nquarterandonehalfofthesaturationmagnetization[see figures4(a) –(b)],whichendupattwodifferent\nfield-inducedtransitionsfromQFPtoCFPand,respectively, fromCFPtoSPP.Thelowermagnetization\nplateauatonequarterofthesaturationmagnetizationgive saclearevidenceofQFP,becausethetotal\nmagnetizationstartsfromzeroanditbecomesnon-zeromain lyduetothe field-inducedalignmentof\nthefrustratedIsingspins.Moreover,itisquiteinteresti ngtoobservefrom figure4thattheformer field-\ninducedtransitionbetweenQFPandCFPismuchsharperatagi ventemperaturethanthelatter field-\ninducedtransitionbetweenCFPandSPP.Ofcourse,therelev antmagnetizationcurvebecomessmoother\nuponincreasingtemperatureuntilbothmagnetizationplat eauscompletelydisappearfromthemagneti-\nzationprocessaboveacertaintemperature( kBT/|JI|≈0.5for∆=4.0).\nLastbutnotleast,themagnetizationcurvewithoutthehigh erintermediatemagnetizationplateau\natahalfofthesaturationmagnetizationcanbedetectedwhe nevertheexchangeanisotropyexceedsits\nsecondcriticalvalue ∆>∆c2.Inagreementwiththeground-statephasediagramshownin figure2,the\nlow-temperaturemagnetizationcurvedisplaysadirect field-inducedtransitionfromQFPtowardsSPP\nwithoutpassingthroughanothermagnetizationplateauCFP .Forillustration,themagnetizationscenario\nofthistypeisdepictedin figure5.Itisworthnotingthatthe field-inducedpolarizationoftheHeisen-\nbergspins,whichappearsinthevicinityofthesaturation ���eld,maycause,atmoderatetemperatures,\natransientloweringofthesublatticemagnetizationofthe Isingspinsasitcanbeclearlyseenin fig-\nures5(b) –(c).Thepartialloweringofthesublatticemagnetizationo ftheIsingspinscanbeattributed\ntoaspinreorientationoftheHeisenbergspinstowardsthee xternal- fielddirectionandatendencyof\nthenearest-neighbourIsingandHeisenbergspinstoaligna ntiparallelwithrespecttoeachotherdueto\ntheantiferromagneticinteractionin-betweenthem.Furth ermore, figure5(d)showsaninterestingcross-\ningofbothsublatticemagnetizations,whichoccursatsu fficientlyhightemperaturesonaccountofthe\nantiferromagneticcorrelationsbetweenthenearest-neig hbourIsingandHeisenbergspins.\nFigure6.(Coloronline)Acolormapofthetotalmagnetizationasafun ctionofthedimensionlesstemper-\natureandexternalmagnetic fieldforthespin-1\n2Ising-Heisenbergmodelonthediamond-likedecorated\nBethelatticewiththecoordinationnumber q=3,theinteractionratio JH/|JI|=1.0andthreedifferent\nvaluesoftheexchangeanisotropy:(a) ∆=1.0;(b)∆=4.0;(c)∆=6.0.\nLetusconcludeouranalysisofthemagnetizationprocessby fewcommentsonacolormapofthetotal\nmagnetizationdepictedin figure6asafunctionoftemperatureandexternalmagnetic field.Accordingto\nauniquecolormaplabellingusedin figure6,twofractionalvaluesofthetotalmagnetization mT=0.125\nand 0.25thatcorrespondtotheintermediatemagnetizationplateau sassociatedwiththeappearanceof\nQFPandCFParedisplayedbycyanandgreencolor,respective ly.Ascouldbeexpected,thequiteextensive\ngreenregionin figure6(a)indicatesaratherwidemagnetizationplateauata halfofthesaturation\nmagnetizationemergingforrelativelyweakexchangeaniso tropies∆<∆c1,whilethewidecyanregion\ninfigure6(c)impliestheexistenceofarelativelyrobustmagne tizationplateauatonequarterofthe\nsaturationmagnetizationforstrongenoughexchangeaniso tropies∆>∆c2.Hence,iffollowsthatthe\nmoststrikingmagnetizationpro filewithtwosuccessiveintermediatemagnetizationplateau smightbe\nindeedexpectedfortheintermediateexchangeanisotropie s∆∈(∆c1,∆c2).Infact, figure6(b)serves\n43003-8Spin-1/2 Ising-Heisenberg model in the external magnetic fi eld\ninevidenceofthepresenceofbothintermediatemagnetizat ionplateaus,whicharegraduallysmudged\nbythermal fluctuationsastemperatureincreases.Interestingly,ittu rnsoutthatthelowerfractional\nmagnetizationplateaupertinenttoQFPdiminishesmuchmor esteadilywithanincreasingtemperature\nincomparisonwiththehigherfractionalmagnetizationpla teaupertinenttoCFP,whichseemstobemuch\nmoreresistantagainstthermal fluctuations.\n4.Conclusion\nThepresentworkdealswiththespin-1\n2Ising-Heisenbergmodelondiamond-likedecoratedBethela t-\nticesinthepresenceofthelongitudinalmagnetic field.Exactsolutionfortheinvestigatedmodelhas\nbeenobtainedbycombiningthedecoration-iterationmappi ngtransformationwiththemethodofexact\nrecursionrelations.Theformertransformationmethodmak esitpossibletoestablisharigorousmapping\nrelationshipwiththeequivalentspin-1\n2IsingmodelonasimpleBethelattice,whichissubsequently ex-\nactlytreatedwithintheframeworkofthelattermethodbase donexactrecursionrelations.Exactresults\nforthepartitionfunction,Gibbsfreeenergy,totalandbot hsublatticemagnetizationswerederivedby\nmakinguseofthisrigorousapproach.\nOurparticularattentionwasfocusedonexploringthegroun dstateandlow-temperaturemagnetiza-\ntionprocessoftheferrimagneticversionofthemodelconsi dered.Themostinteresting findingstemming\nfromourpresentstudyisanexactevidenceofaratherdivers emagnetizationprocess.Asamatteroffact,\nwehavedemonstratedthreedifferentmagnetizationscenar ioswithuptotwodifferentfractionalmagne-\ntizationplateaus,whereastheintermediatemagnetizatio nplateaumayeithercorrespondtotheclassical\nferrimagneticspinarrangementand/orthequantumferrima gneticspinorderingwithoutanyclassical\ncounterpart.Theoriginofthestrikingquantumferrimagne ticphaseliesinapeculiarspinfrustrationof\ntheIsingspins,whichcomesfromthenonmagneticnatureoft heHeisenbergspinpairsgovernedbythe\nsymmetricquantumsuperpositionoftheirtwointrinsicall yantiferromagneticspinstates.\nAcknowledgements\nThisworkwassupportedbytheScienti ficGrantAgencyofMinistryofEducationofSlovakRepublic\nundertheVEGAGrantNo.1/0234/12andbyERDFEU(EuropeanUn ionEuropeanregionaldevelopment\nfund)grantunderthecontractITMS26220120005(activity3 .2.).\nReferences\n1. MillerJ.S.,DrillonM.,Magnetism:MoleculestoMateria ls,Wiley,Weinheim,2001.\n2. LacroixC.,MendelsP.,MilaF.,IntroductiontoFrustrat edMagnetism,Springer,Berlin,2011.\n3. SyoziI.,Prog.Theor.Phys.,1951, 6,341;doi:10.1143/PTP.5.341.\n4. SyoziI.,In:PhaseTransitionandCriticalPhenomena,Vo l.1,ed.byDombC.,GreenM.S.,AcademicPress,New\nYork,1972,pp.269 –329.\n5. FisherM.E.,Phys.Rev.,1959, 113,969;doi:10.1103/PhysRev.113.969.\n6. RojasO.,ValverdeJ.S.,deSouzaS.M.,PhysicaA,2009, 388,1419;doi:10.1016/j.physa.2008.12.063.\n7. RojasO.,deSouzaS.M.,J.Phys.A:Math.Theor.,2011, 44,245001;doi:10.1088/1751-8113/44/24/245001.\n8. StrečkaJ.,Phys.Lett.A,2010, 374,3718;doi:10.1016/j.physleta.2010.07.030.\n9. BaxterR.J.,ExactlySolvedModelsinStatisticalMechan ics,Academic,NewYork,1982.\n10. ThompsonC.J.,J.Stat.Phys.,1982, 27,441;doi:10.1007/BF01011085.\n11. MukamelD.,Phys.Lett.,1974, 50A,339;doi:10.1016/0375-9601(74)90050-4.\n12. OhanyanV.R.,AnanikyanL.N.,AnanikianN.S.,PhysicaA ,2007,377,501;doi:10.1016/j.physa.2006.11.034.\n13. IzmailianN.Sh.,HuChin-Kun,PhysicaA,1998, 254,198;doi:10.1016/S0378-4371(98)00193-9.\n14. StrečkaJ.,EkizC.,ActaPhys.Pol.A,2010, 118,725.\n15. StrečkaJ.,Ja ščurM.,ActaPhys.Slovaca,2006, 56,65.\n16. EkizC.,StrečkaJ.,Ja ščurM.,J.Magn.Magn.Mater.,2011, 323,493;doi:10.1016/j.jmmm.2010.10.001.\n43003-9J. Strečka, C. Ekiz\nПроцеснамагнiченостiвточнорозв’язнiйспiн-1/2моделi\nIзинга-ГайзенберганадекорованихграткахБете\nЙ.Стречка1,С.Екiз2\n1Природничийфакультет ,Унiверситет iм.П.Й.Шафарика ,Кошiце,Словацькареспубл iка\n2Природничийфакультет ,Унiверситет iм.АднанаМендереса ,Айдин090 10,Туреччина\nСпiн-1/2модель Iзинга-Гайзенберганаромбопод iбнiйдекорован iйгратц iБетерозв ’язаноточноупри -\nсутност iпоздовжньогомагн iтногополя ,поєднуючидекорац iйно-iтерацiйнеперетвореннязметодомто -\nчнихрекурсивнихсп iввiдношень .Зокрема ,детальнодосл iдженоосновнийстан iнизькотемпературне\nнамагн iченняферимагн iтноїверс iїрозглянутоїмодел i.Знайденотрир iзнiсценарiїнамагн iченостiзщо-\nнайбiльшедвомапосл iдовнимидробовимиплато ,депром iжнеплатонамагн iченостiможев iдповiдати\nкласичномуферимагн iтномусп iновомувпорядкуваннюта /абоiндукованомуполемквантовомуферима -\nгнiтномусп iновомувпорядкуванню ,якенемаєжодногокласичногоаналога .\nКлючовiслова:модель Iзинга-Гайзенберга ,граткаБете ,точнiрезультати ,платонамагн iченостi\n43003-10" }, { "title": "1308.0203v1.Exchange_relaxation_as_the_mechanism_of_ultrafast_spin_reorientation_in_two_sublattice_ferrimagnets.pdf", "content": "arXiv:1308.0203v1 [cond-mat.str-el] 1 Aug 2013Exchange relaxation as the mechanism of ultrafast spin reor ientation in two-sublattice\nferrimagnets.\nV. G. Baryakhtar,1V. I. Butrim,2and B. A. Ivanov1,3,∗\n1Institute of Magnetism, 03142 Kiev, Ukraine\n2Taurida National V.I. Vernadsky University, 95007 Simfero pol, Ukraine\n3National Taras Shevchenko University of Kiev, 03127 Kiev, U kraine\nIn the exchange approximation, an exact solution is obtaine d for the sublattice magnetizations\nevolution in a two-sublattice ferrimagnet. Nonlinear regi mes of spin dynamics are found that include\nboth the longitudinal and precessional evolution of the sub lattice magnetizations, with the account\ntaken of the exchange relaxation. In particular, those regi mes describe the spin switching observed\nin the GdFeCo alloy under the influence of a femtosecond laser pulse.\nPACS numbers: 75.10.Hk, 78.47.J-, 05.45.-a\nMagnetic materials have various applications in\nmodern electronics and informatics, but probably the\nmost important research direction is still the creation\nof information storage and processing systems. The\nchallenge of designing magnetic devices with ever\nincreasing information density and recording speed\nrequires solving certain fundamental problems of the\nmagnetism dynamics. The possibility to manipulate the\nmagnetization by means of femtosecond laser pulses\nopens wide opportunities in this direction. This field\nhas been incepted by the work [1], where a fast (within\na time shorter than a picosecond) reduction of nickel\nmagnetization after the exposure to a 100 femtosecond\nlaser pulse has been observed, as well as the subsequent\nrelaxation of the magnetization with a characteristic\ntime of the order of picoseconds. The authors explained\nthe initial drop in the magnetization either by an\nextremely rapid heating of the sample above the Curie\npoint, see review [2], or by spin-dependent super-diffusive\nelectron transfer in the laser-excited metal [3]. Further\nwork in this area followed for various materials, and\nunexpected and rather unusual effects were discovered.\nIn the ferrimagnetic rare earth and transition metal\nalloy GdFeCo, a femtosecond pulse lead, in the first\nstage, to a similar spin reduction (i.e., the reduction of\nthe magnetization of sublattices) as for nickel, but the\nsubsequent evolution turned out to be fundamentally\ndifferent. Instead of a simple relaxation to the initial\nvalue, within about the same time (a few picoseconds),\nboth sublattice magnetizations changed their signs, i.e.,\na switching of the net magnetic moment took place [4],\nand during this picosecond-scale evolution there occurred\nana priori energetically unfavorable state with parallel\nsublattice moments. Such a magnetization switching\neffect is of a threshold type, and is observed only for\nsufficiently strong pulses. It has been detected in films\nas well as in microparticles [5] and nanoparticles [6],\nboth for ferromagnets with and without a compensation\npoint [5]. There is also a way of “selective” switching:\ndue to the magnetic dichroism, the absorbed energy ofa circularly polarized pulse depends on the direction of\nthe magnetic moment of the particles, and a pulse of\ncertain polarization would only switch the moments of\nthe particles which are in a matching state [7]. All that\nmakes possible to create a purely optically-controlled\nmagnetic memory with a picosecond recording speed.\nAlthough an analytical explanation of this effect is\nhighly desirable, The theoretical description has been\nperformed only by means of numerical simulation [4, 5].\nIt has been found that the change of the sublattice\nmagnetization lengths S1=|S1|иS2=|S2|, i.e. a\nlongitudinal spin evolution, is crucially important for th is\nphenomenon [5, 8]. The magnetization length is formed\nby the exchange interaction, and all the salient features\n(particularly, picosecond-scale characteristic evoluti on\ntimes, and the fact that the effect persists even in\nmagnetic fields up to 300 KOe) point out to the\nimportance of the exchange-dominated evolution [5].\nThe Landau-Lifshitz (LL) equation [9], with the\nstandard relaxation terms [9, 10] preserves the\nmagnetization length. The problem of the correct\nstructure of the relaxation terms in the LL equation,\nincluding the question of a purely exchange relaxation,\nwas previously considered by one of the present authors\n[11, 12]. It was shown that the longitudinal spin evolution\narises naturally when the general equations describing\nthe magnetization dynamics of ferromagnets [11] and\nantiferromagnets [12] are constructed, but has certain\nlimitations. Because of the obvious symmetry of the\nexchange interaction, it can not lead to a change (in\nparticular, relaxation) of the total spin of the system.\nTherefore, the evolution of the magnetization length of\na simple ferromagnet is reduced to a diffusion process\n(that is generally nonlinear), and is absent in the\nhomogeneous case which we are interested in, see the\ndetailed analysis in [11]-[13]. However, for a magnet with\ntwo sublattices, the situation is different, and a purely\nexchange relaxation is possible even for a homogeneous\ndynamics [12].\nThese ideas were used in [8] for a qualitative2\ndescription of the experimental data. Since the duration\nof the laser pulse used (less than 100 fs) is much shorter\nthan the characteristic evolution time, the analysis\ncan be performed by considering the dynamics of the\nmagnetization outside of the time interval of the pulse.\nIn doing so, a highly non-equilibrium state created by\nthe pulse plays the role of the initial condition for the\nequations describing the magnetization dynamics. The\nfollowing scheme has been proposed: the light pulse\ntransfers the system into a non-equilibrium state, in\nwhich, however, the direction of the spin sublattices is the\nsame as in the initial state. The system evolves further\nunder the influence of a faster exchange relaxation,\nfollowing along the straight line S1+S2=S1(0) +\nS2(0) = const in the(S1,S2)plane, see Fig. 1 of Ref.\n[8]. The analysis showed that the evolution of the system\nquickly leads to a state of partial equilibrium, which\ncorresponds to the spin values differing from the initial\nonesS1(0),S2(0)not only in the magnitude, but in the\nsign as well. The further evolution is due to the slower\nrelativistic relaxation, and the system goes to one of\nthe two equivalent states of complete equilibrium. In\na wide range of the initial values, consistent with the\nexperiment, the final state after the two-step process\ndiffers from the initial one only by the signs of S1and\nS2, which explains the effect of spin switching. However,\nRef.[8] studied a purely longitudinal dynamics, that is, it\nwas assumed that the vectors S1andS2remain collinear\nto their initial values.\nIn the present work, the exchange evolution of\nsublattice spin vectors of a ferrite is investigated in\na general way, without the assumption of collinearity.\nWe have found nonlinear regimes of spin dynamics,\nincluding both longitudinal and precessional evolution\nof the sublattice spins. It is shown that in the case of\na strong deviation from equilibrium an instability of the\nlongitudinal dynamics is possible, in which the amplitude\nof the precession increases due to the transfer of the\nenergy associated with the nonequilibrium character of\nlength of the antiferromagnetism vector L=S1−S2into\nthe deviation of Lfrom its equilibrium direction that is\ncollinear to the total magnetization M=S1+S2.\nThe LL equations for a two-sublattice magnet, with\npurely exchange relaxation terms can be written as\n/planckover2pi1∂S1\n∂t= [S1,H1]+λ(H1−H2)−λ1∇2H1,\n/planckover2pi1∂S2\n∂t= [S2,H2]−λ(H1−H2)−λ2∇2H2,(1)\nwhereS1,S2are the sublattice spins, H1,2=\n−δw/δS1,2are the effective fields for the sublattices, and\nw=w{S1,S2)}is the non-equilibrium thermodynamic\npotential per elementary cell, written as a functional of\nthe sublattice spin density. In what follows we set the\nPlanck constant to unity, and it will only be recovered\nin some final results. The relaxation terms can bewritten in the form δQ/δH1,2, whereQis the dissipative\nfunction, dw/dt−= 2Q, whose density in the exchange\napproximation is given by the following expression [11]:\n2Q=λ(H1−H2)2+λ1(∇H1)2+λ2(∇H2)2.\nHereafter, we will discuss only the homogeneous\ndynamics, and the terms containing λ1,2, which\ndetermine the spin diffusion, will be neglected.\nHere a general remark is in order, regarding the\nequations of motion of the magnetization. For the LL\nequation, both dynamic and dissipative terms (including\nthe standard relaxation term of the relativistic nature as\nwell as the exchange terms such as those in Eq. (1)) are\nchosen to be linear in the components of the effective\nfield. This approach is consistent with the Onsager\nprinciple, see [11]. However, the linearity of equations\nin the effective field does not limit the applicability\nof these equations to the linear approximation . For a\nmagnetically ordered state, a significant nonlinearity\nis present in the expression for the nonequilibrium\nthermodynamic potential, which determines well-known\nnon-linear properties of the LL equation. The presence\nof this non-linearity, reflecting the properties of the\nsystem, makes this approach very natural and reasonable.\nOf course, it is possible to consider generalizations of\nthese equations including the terms nonlinear in the\ncomponents of the effective field, but we do not know\nany examples where such a generalization would lead to\nnew physical effects.\nIn the homogeneous case and in the exchange\napproximation, the relaxation for two-sublattice magnets\nis actually determined by a single parameter λ. This is\neasy to understand by noticing that Eqs. (1) preserve\nthe total spin M, which is the consequence of the\nexchange approximation. We remark that the SU(2)\nexchange symmetry does not exclude the change of\nlength as well as the direction of the antiferromagnetism\nvectorL. Thus, we come to the conclusion that the\ninter-sublattice exchange plays the dominating role in\nrelaxation (in contrast to the independent relaxation of\nevery sublattice, as it comes out when the relaxation\nterm is taken in the Gilbert form), which is supported\nby recent experiments [14].\nNaturally, (1) describes only the relaxation to a\npartially equilibrium state, which corresponds to a\nminimum of the thermodynamic potential at fixed (and,\ngenerally, non-equilibrium) M. The value of λcan\nbe found from the damping decrement γlinof small-\namplitude Loscillations, which in the framework of\n(1) is determined by the formula γlin=λJ12(¯S1−\n|¯S2|)2/¯S1|¯S2|, where ¯S1,¯S2are the equilibrium spin\nvalues. It is important that the damping of optical spin\nwaves, connected to the transversal oscillations of the\nantiferromagnetism vector L, and the relaxation of the\nlength of Lare both determined by the same constant\nλ. First, this allows one to establish the value of λ3\nfrom independent measurements, and second, one can use\nthe known results of microscopic calculations of magnon\ndamping [15] to estimate it, which yields λ∝T4.\nIn what follows, our starting point will be the following\nexpression for the thermodynamic potential of a two-\nsublattice ferrite with purely exchange symmetry, written\nfor the homogeneous case as a function of the sublattice\nspins:\nw(S1,S2) =f1(S2\n1)+f2(S2\n2)+J12S1S2, (2)\nwhereS2\n1,2=S2\n1,2, and the exact form of the functions\nf1andf2is not yet specified. It is clear that the terms\ncontaining f1,f2do not contribute to the dynamical part\nof (1), and [S1,2,H1,2]→ ±J12[S1,S2]. It is convenient\nto pass to the equations for irreducible vectors Mand\nL. The equation for Myields∂M/∂t= 0, and for L\none obtains the closed-form vector equation ∂L/∂t=\nJ12[M,L] + 2λeHL,HL=−∂w/∂L. Let us choose\nthezaxis along the constant vector M=Mez. In the\nconvenient notation L=Ll,l2= 1those equations take\nthe form\n∂L\n∂t= 2λe(lHL)−2λe∂w\n∂L,\n∂l\n∂t=J12[M,l]+2λe\nL[HL−l(lHL)], (3)\nwhere the dissipative term in the equation for lresembles\nthe Landau-Lifshitz one. Equations for Landl, with the\naccount taken of the specific form of the thermodynamic\npotential can be also cast in the following convenient\nform:\n∂L\n∂t=−2λe(J12L−∂f1\n∂S1+∂f2\n∂S2),\n∂l\n∂t=J12[M,l]+λe\nL/parenleftbigg∂f1\n∂S2\n1−∂f2\n∂S2\n2/parenrightbigg\n[M−l(lM)].(4)\nHaving written lx+ily= sinθexp(iϕ), lz= cosθ, it is\neasy to show that ϕ=ωt,/planckover2pi1ω=J12M, and at θ∝ne}ationslash= 0,π\nvectorlprecesses with a constant frequency ω, and the\nprecession amplitude Lsinθchanges with time because\nof the dissipation. It is interesting that for small Ma\n“slowdown” of this precession takes place. Thus, nonlinear\noscillations of arbitrary (not small) amplitude have the\nform of a precession of Laround the constant vector M,\nwith the frequency /planckover2pi1ω=J12M:\nL=Lzez+L⊥(excosωt+eysinωt),\nLz=Lcosθ, L⊥=Lsinθ,\nwhere the quantities Lz(t),L⊥(t)exhibit a dissipative\nevolution. It is convenient to write down the equations in\nL,θvariables:\n∂L\n∂t= 2λL(J12−∂f1\n∂S2\n1−∂f2\n∂S2\n2)+2λ(∂f2\n∂S2\n2−∂f1\n∂S2\n1)Mcosθ,\n∂θ\n∂t=−M2λ\nL(∂f2\n∂S2\n2−∂f1\n∂S2\n1)sinθ. (5)For the sake of simplicity and physical clarity let us\ntakef1,2in the form of the Landau expansion of the\nform\nf1=J1\n4(S2\n1−S2\n0)2, f2=J2S2\n2\n2. (6)\nHere we assume that the second sublattice consists\nof paramagnetic rare-earth ions, f2is determined by\nthe spin entropy, and J2is of the order of the\ntemperature T. The parameter S0=S0(T)formally\ncoincides with the equilibrium value of the iron sublattice\nmagnetization if one neglects its interaction with the\nrare-earth sublattice. Using (5), one obtains simple close d\nformulae for the equilibrium values of the sublattice\nspins,¯S1=/radicalBig\nS2\n1,0+J2\n12/J1J2and¯S2=−J12¯S1/J2,\nwhile the equations can be written in the form\nt0∂L\n∂t=f(L,θ), t0∂θ\n∂t=g(L,θ), t0=4/planckover2pi1\nλJ1,(7)\nгдеf(L,θ) =−L3−3L2Mcosθ+AL+B,\ng(L,θ) =−Msinθ\nL(4J2\nJ1+4S2\n0,1−L2−2LMcosθ−M2)\nA=−M2(1+2cos2θ)−4J2\nJ1+4S2\n0,1+8J12\nJ1,\nB=Mcosθ(4J2\nJ1+4S2\n0,1−M2).\nIt is worth noting that the evolution of Lz(t),L⊥(t)\noccurs on a naturally emerging universal time scale t0=\n4/planckover2pi1/λJ1, which is larger than the “purely exchange” time\ntex∼/planckover2pi1/J1∼t0/λsince the relaxation constant λis\nsmall. For not very small values of M∼1and not too\nweak inter-sublattice interaction J12∼J1, this time scale\nis also larger than the precession period of vector L.\nProceed further to the analysis of the evolution of Lz\nandL⊥. It is clear that all singular points occur at θ=\n0,π, and their positions are determined by zeros of the\nfunction f(L,θ)atsinθ= 0. The condition f(L,sinθ=\n0) = 0 can be represented as a cubic equation in Lcosθ=\n±L(it is convenient to assume that L >0, andθvaries\nin the range 0≤θ≤π). AtM= 0the three roots\nareL1,3cosθ=±√\nAandL2= 0, so it is clear that at\nsufficiently small M≤Mcthere will also be three real\nroots. A simple analysis shows that L=L1, θ= 0(or\nLz=L1>0,L1=√\nAatM= 0)corresponds to the\nequilibrium position (a stable node), L=L3, θ=π,\ni.e.,Lz=−L3<0corresponds to a saddle point, and\nthe unstable node lies at L=L2. ForM≤Mcone has\nL1< L3, and the unstable node will correspond to a\nnegative value of Lz=L2cosθ <0. AtM=MctheL2\nandL3roots merge, and for M > M cthe system has\nonly one singular point at Lcosθ=L1>0.\nIt is important to note that for all values of Mthe\nsystem (7) has another solution Lz=Lz(t),L⊥(t) = 0,4\n/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s45/s50/s46/s48 /s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s76 \n/s122 /s76 /s32\nРис. 1: The evolution of L⊥andLz, calculated numerically\nforJ1=J2= 2J12andS0= 1atM= 0.4and shown as\na phase portrait. Singular points are shown with circles, an d\nthe separatrix is shown as a dotted line.\nwhich corresponds to a purely longitudinal evolution,\nbut the physical sense of this solution is very different.\nAtM > M c, for all initial conditions ( L(0)< L1or\nL(0)> L1), the value of Lztends to its equilibrium\nvalueL1for this class of solutions. Numerical analysis\nshows that in his case the evolution remains close to the\npurely longitudinal one even if the direction of Ldeviates\nfrom the equilibrium. The only exception is for large\ndeviations, when L(0)∼ −L1; in this case the length of\nLis already close to the equilibrium, and a rotation of L\nbecomes favorable. At extremely small Mthe evolution\nis degenerate: θchanges much slower than L, and the\nphase portrait in the (L⊥,Lz)plane consists of radial\nstraight line intervals θ= const and of parts of the circle\nL=L1≈√\nA. For finite M < M cthe situation is much\nmore interesting: in this case one also has a solution of\nthe form Lz=Lz(t), L⊥(t) = 0 , but with the initial\nconditions −L3< Lz(0)<−L2, i.e., between the saddle\npoint and the unstable node, the longitudinal evolution\ntakes the system away from the equilibrium. This is\nillustrated in Fig. 1, which shows the phase portrait of\nthe system in (L⊥,Lz)plane, calculated numerically for\nJ1=J2= 2J12andS0= 1atM= 0.4< Mc(at those\nparameter values one has Mc= 4/(3√\n3)≃0.77).\nThus, the exact solution of the full system of\nequations of motion for sublattice spins in the exchange\napproximation shows the existence of two qualitatively\ndifferent regimes. The character of the evolution is mainly\ndetermined by the initial value of the magnetization M,\nwhich is conserved in the exchange approximation. For\nlargeM > M c, as well as for all Mand the initial\nvalueLz(0)>0, a longitudinal relaxation occurs, as\nstudied previously in [8]. For M < M c, approximately\nthe same behavior is also retained for negative Lz(0),provided that Lz(0)>−L2. In all those cases, there is\na special solution of the form Lz=Lz(t)which leads to\nthe equilibrium, and even for a nonzero (but small) value\nof the transversal initial deviation L⊥(0)the value of\nL⊥remains small in the process of relaxation. However,\nthe situation is changed dramatically, if the initial value\nenters the region of the unstable node situated around\nLz≃ −L2(Lz≃0.83in Fig. 1). As seen in Fig. 1,\nin the vicinity of this point and to its left, even small\ninitial values of L⊥increase with time. In this case, in\na wide range of the initial conditions all trajectories in\nthe(Lz,L⊥)plane tend to the separatrix which connects\nthe saddle point and the unstable node; the values of\nL⊥are not small at the separatrix. In this way, strongly\nnonlinear evolution regimes with L⊥∼Lz∼1become\npossible. The solution with L⊥∝ne}ationslash= 0atM∝ne}ationslash= 0is of\nthe precession type, i.e., for the initial condition with\nLz<−L2approaching equilibrium is accompanied by\nthe growth of the precession amplitude of L, at the\nconstant precession frequency /planckover2pi1ω=J12M, so that L=\nLzez+L⊥(excosωt+eysinωt). It should be remarked\nthat the experimentally observed time dependence of the\nsublattice magnetizations in the time interval between\n0.5and3ps shows some non-monotonic behavior at the\nbackground of a smooth magnetization change, which\nresembles oscillations with the period of about 0.3ps, see\nFig. 2 of Ref. [4]. The results of numerical simulation of\nthis process, reported in the same work [4], did not show\nsuch a behavior, but oscillations were found in recent\nnumerical studies [16].\nTaking into account the transversal spin deviations\nin the process of evolution may be important for\nunderstanding the recent experiment on TbFeCo alloy\n[17]. An obvious difference between this material and\nGdFeCo is the presence of a strong easy-axis anisotropy,\nbut it is clear that such anisotropy should not affect\na purely longitudinal evolution. Despite that, spin\nswitching characteristic for GdFeCo was not observed in\nTbFeCo, although the initial reduction of the sublattice\nmagnetizations was roughly the same as in the GdFeCo\nexperiment. Of course, there could be other reasons\nfor such a different behavior, e.g., the presence of\nunquenched orbital moment of Tb, but the detailed\nanalysis of this problem is beyond the scope of the present\npaper.\nThis work is partly supported by the joint Grant\n0113U001823 of the Russian Foundation for Fundamental\nResearch and the Presidium of the National Academy of\nScience of Ukraine, and by the joint Grant Φ53.2/045 of\nthe Russian and Ukrainian Foundations for Fundamental\nResearch.\n∗Electronic address: bivanov@i.com.ua5\n[1] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. 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B 56, 619 (1997).\n[14] V. L´ opez-Flores, N. Bergeard, V. Halt´ e et. al, Phys. R ev.\nB87, 214412 (2013).\n[15] V. N. Krivoruchko and D. A. Yablonskii, Zh. Eksp.\nTheor. Fiz. 74, 2268 (1978) [Sov. Phys. JETP 47, 1179\n(1978) ]; Fiz. Tverd. Tela 21, 1502 (1979) [in Russian].\n[16] U. Atxitia, T. Ostler, J. Barker, R. F. L. Evans, R. W.\nChantrell, and O. Chubykalo-Fesenko, Phys. Rev. B 87,\n224417 (2013).\n[17] A. R. Khorsand, M. Savoini, A. Kirilyuk, A.V. Kimel, A.\nTsukamoto, A. Itoh, and Th. Rasing, Phys. Rev. Lett.\n110, 107205 (2013)." }, { "title": "1807.08650v1.Tilted_and_type_III_Dirac_cones_emerging_from_flat_bands_in_photonic_orbital_graphene.pdf", "content": "1 \n Tilted and type-III Dirac cones emerging from flat bands in photonic \norbital graphene \nM. Milicevic 1, G. Montambaux 2, T. Ozawa 3, I. Sagnes 1, A. Lemaître 1, L. Le Gratiet 1, A. Harouri 1, \nJ. Bloch 1, A. Amo 4* \n1Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Université Paris-Saclay, C2N-\nMarcoussis, 91460 Marcoussis, France \n2Laboratoire de Physique des Solides, CNRS, Univ. Paris -Sud, Université Paris-Saclay, 91405 Orsay Cedex, \nFrance \n3 Interdisciplinary Theoretical and Mathematical Scie nces Program (iTHEMS), RIKEN, Wako, Saitama 351-0198, \nJapan \n4Univ. Lille, CNRS, UMR 8523 – PhLAM – Physique des Lase rs Atomes et Molécules, F-59000 Lille, France \n*alberto.amo-garcia@univ-lille.fr \n \nThe extraordinary electronic properties of Dirac ma terials, the two-dimensional partners \nof Weyl semimetals, arise from the linear crossings in their band structure. When the \ndispersion around the Dirac points is tilted, the e mergence of intricate transport \nphenomena has been predicted, such as modified Klei n tunnelling, intrinsic anomalous \nHall effects and ferrimagnetism. However, Dirac mat erials are rare, particularly with \ntilted Dirac cones. Recently, artificial materials whose building blocks present orbital \ndegrees of freedom have appeared as promising candi dates for the engineering of \nexotic Dirac dispersions. Here we take advantage of the orbital structure of photonic \nresonators arranged in a honeycomb lattice to imple ment photonic lattices with semi-\nDirac, tilted and, most interestingly, type-III Dir ac cones that combine flat and linear \ndispersions. The tilted cones emerge from the touch ing of a flat and a parabolic band \nwith a non-trivial topological charge. These result s open the way to the synthesis of \norbital Dirac matter with unconventional transport properties and, in combination with \npolariton nonlinearities, to the study of topologic al and Dirac superfluids in photonic \nlattices. \n 2 \n The extraordinary transport properties of Dirac mat erials arise from the spinor nature of their \nelectronic wavefunctions and from the linear disper sion around Dirac and Weyl points. In two-\ndimensions, Klein tunnelling, weak antilocalisation , unconventional Landau levels or bulk pseudo-\nconfinement appear as some of their most remarkable features 1,2 . Standard Dirac cones, like those \npresent in graphene and other two-dimensional mater ials, have rotational symmetry about the Dirac \nquasi-momentum. Their topological properties make t hem particularly robust to deformations of the \nlattice: Dirac cones always appear in pairs, each o f them characterised by a topological charge 3, and in \nthe presence of time-reversal and inversion symmetr y, they can only be annihilated via their merging \nwith a Dirac point of opposite charge 4–9. \nDirac cones can be classified according to the geom etry of their Fermi surface. The cylindrically \nsymmetric Dirac cones described above belong to the family of type-I Dirac cones. They are \ncharacterised by a closed Fermi surface eventually becoming a single point at the band crossing, where \nthe density of states vanishes (Fig. 1a). However, they are not the only kind of linear band crossings \nthat can be found in Dirac materials. The general H amiltonian describing a Dirac cone in two \ndimensions can be expressed as 10 \u0001\u0002\u0003\u0004\u0005\u0006\u0007\b\t\n\t\u000b\u0007\b\f\n\f\r\u000e\b\u000b\u0007\t\n\t\u000e\u000f\u000b\u0007\f\n\f\u000e\u0010, where \n\t,\f is the \nwave vector measured from the Dirac point; \u0007\b\t,\u0007\b\f,\u0007\t and \u0007\f represent effective velocities; \u000e\b is the \n2\u00132 identity matrix and \u000e\u000f,\u0010\u0005\u0014\u0015,\u0016⋅\u0018, where \u0014\u0015,\u0016 are suitably chosen orthogonal unit vectors and \n\u0018\u0005\u0006\u000e\t,\u000e\f,\u000e\u0019\r is the vector of Pauli matrices. The eigenenergies of this Hamiltonian form two bands: \n\u001a\u001b\u0002\u001c\u0004\u0005\u0007\b\t\n\t\u000b\u0007\b\f\n\f\u001b\u001d\u0002\u0007\t\n\t\u0004\u001e\u000b\u0006\u0007\f\n\f\r\u001e. (1) \nIf both coefficients \u0007\b\t and \u0007\b\f are equal to zero we obtain the energy spectrum of a type-I Dirac cone \nwith Fermi velocities \u0007\t and \u0007\f. If any of the \u0007\b\t,\u0007\b\f coefficients is non zero, then the Dirac cone is \ntilted (Fig. 1(b)). In two dimensional materials, t his kind of tilted Dirac dispersion has been predic ted \nto appear in quinoid-type 10 and hydrogenated graphene 11 , and it has been indirectly evidenced in the \norganic semiconductor 10,12,13 α-(BEDT-TTF) 2I3. The interest of materials presenting this dispers ion \n \nFigure 1. Types of Dirac dispersions in two dimensio ns . (top) Dispersions together with the zero energy pla ne (grey); \n(bottom, in red) zero-energy Fermi surface: a Standard, type- I Dirac cones characterised by a linear dispersion in all directions \nin k-space and a point-like Fermi surface. b Type-I tilted Dirac cone. c Type-III Dirac point (critically tilted), combinin g a flat \nband along a line and linear dispersions. Its Fermi surface is a line. d Type-II Dirac cone. \n3 \n resides in its non-isotropic transport properties, which can be used for valley filtering in p−n junc8 ons 14 \nor for the generation of photocurrent 15 . \nWhen the tilt parameter \u0007 \u001f\b≡\u001d\u0002\u0007\b\t\u0007\t⁄ \u0004\u001e\u000b\u0006\u0007\b\f\u0007\f⁄\r\u001e is larger than 1, a type-II Dirac point 10,16 is \nformed (Fig. 1d). Its Fermi surface is no longer a point but two crossing lines, and the density of st ates \nat the energy of the Dirac point becomes finite 16 . They have been recently observed in the form of \nFermi arcs in three-dimensional semimetals 17–19 . A particularly interesting situation takes place at the \ntransition between type-I and type-II Dirac cones, that is, when the tilt parameter \u0007 \u001f\b\u00051. In this case, \nthe cone is critically tilted, with a flat band alo ng one direction (Fig. 1c). Because of its distinct Fermi \nsurface, a single line, and its diverging density o f states, this kind of dispersion has been labelled type-\nIII Dirac cone 20,21 . While most of their electric and magnetic propert ies are still to be unveiled, they \nhave been predicted to greatly enhance the supercon ducting gap in Weyl semimetals 22 , and provide a \nnew platform for the study of correlated phases wit h a flat band 23 . \nType-III Dirac points have not been yet reported ex perimentally, and tilted type-I and type-II have be en \nchallenging to synthetize 13,17,19 because they require materials whose constituent a toms are arranged \nin lattices with intricate electron hoppings 10,12,16 . Existing proposals rely on the engineering of nex t-\nnearest neighbours tunnelling, difficult to find in natural materials and to implement in electronic \nmetamaterials. Artificial photonic lattices represe nt an opportunity to explore the physics of \nunconventional Dirac points thanks to the at-will c ontrol of onsite energies and hoppings 24 . Current \nschemes are based on the design of long distant cou pling of photons in lattices of resonators 25,26 ; \nevidence of type-II Weyl points has been recently r eported in microwave metamaterials 27,28 and via \nconical diffraction in laser written waveguides wit h elaborate couplings 29 . \nIn this article we propose and demonstrate experime ntally a new method to implement tilted and \nsemi-Dirac cones for photons, and provide the first experimental observation of type-III Dirac cones. \nWe employ #\t, #\f orbital bands in a honeycomb lattice of polariton micropillars, which arise from the \nnearest-neighbour hopping of photons confined in th e first excited modes of each resonator of the \nlattice 30,31 . In this analogue system, the band structure is di rectly accessible in photoluminescence \nexperiments. The orbital bands with symmetric hoppi ngs contain a flat band that touches a parabolic \nband. When asymmetry in the hopping is introduced, which simulates uniaxial strain in solid-state \ngraphene, the flat-parabolic band touching evolves into tilted and type-III Dirac cones. The richness of \nthis multi-band system allows, in addition, the obs ervation of semi-Dirac cones which combine \nmassless and massive dispersions. By analysing thei r topological charge, i.e. the winding of the \nHamiltonian around each Dirac point, we show that t he semi-Dirac, tilted and type-III Dirac cones \nemerge as a consequence of topological Lifshitz tra nsitions induced by strain in the orbital bands. Th e \npresent realisation shows the potential of orbital bands to engineer the properties of Dirac matter. \nPhotonic orbital lattice \nThe photonic platform we employ is a honeycomb latt ice of coupled micropillars. The lattice is etched \nfrom a semiconductor planar microcavity made out of two AlGaAs Bragg mirrors that confine photons \nin the vertical direction, and twelve GaAs quantum wells embedded in the spacer between the mirrors \n(see Sec. I of Supplementary Information). At 10 K, the temperature of our experiments, the confined \nphotons and the quantum well excitons are in the st rong coupling regime and form polaritons, light-\nmatter hybrid quasi-particles. Each micropillar, wi th a diameter of 2.75 µm presents an additional 4 \n lateral confinement due to the index of refraction contrast between the semiconductor and air \n(Fig. 2a). The photonic spectrum of an individual m icropillar is thus discrete with the lowest energy s-\nmode being cylindrically symmetric, and the first e xcited modes formed by two degenerate #\t,#\f \norbitals with lobes oriented 90º from each other (F ig. 2b) 31 . In the honeycomb lattice, the micropillars \noverlap (centre-to-centre distance $\u00052.4 µm) enabling the hopping of photo ns between adjacent \nsites (Fig. 2d). \nThe coupling of s-modes results in two bands with a spectrum and eigenmodes very similar to those \nof electrons in graphene, shown in the low energy p art of Fig. 2e and studied in previous works 30,32 . \nHere we concentrate on the orbital bands which aris e from the coupling of p-modes (high-energy set \nof bands in Fig. 2e). Orbitals oriented along the l ink between adjacent pillars ( #\f in the example of \nFig. 2c) present a coupling '( much stronger than '), the coupling of orbitals oriented perpendicular to \nthe link ( #\t in Fig. 2c). \nFigure 3a shows the angle resolved photoluminescenc e of the p-bands when exciting the lattice at its \ncentre with a non-resonant continuous wave laser at 745 nm, focused on a 3 µm diameter spot. The \npower of the laser is 6 mW, well below the threshol d for any nonlinear effect. The laser creates \nelectrons and holes in the quantum wells that relax down through phonon scattering to form \npolaritons distributed over the different bands of the lattice. When polaritons recombine via the \nescape of a photon out of the microcavity, photons are emitted with an in-plane momentum and a \nfrequency that correspond to those of the original polaritons within the lattice of resonators. An ang le \nand energy resolved measurement employing an imagin g spectrometer coupled to a CCD allows \nreconstructing the dispersion relation (see Supplem entary Information). To avoid destructive \ninterference effects along high symmetric crystallo graphic directions, characteristic of bi-partite \nlattices 33 , we record the emission as a function of *\t for *\f\u00054+/3$ , passing through the ., Γ, .′ \npoints in the second Brillouin zone (dashed line in Fig. 3e). Four bands are observed: the lowest one is \nflat, while the two central bands present two type- I Dirac points at K and K’ and touch the flat band at \nthe Γ point 30 . This is in good agreement with a nearest-neighbou r tight-binding calculation 34 assuming \n')\u00050 (white solid lines, see Supplementary Information) . In the experiments, the uppermost band Figure 2. Honeycomb polariton lattice with orbital bands . Scanning electron microscopy (SEM) image of a singl e micropillar, \nthe elementary building block of the honeycomb latt ice. b Characteristic emission spectrum of a single microp illar, showing \ns, p and d discrete modes. c Scheme of the coupling of #\t and #\f orbitals in the honeycomb lattice with hoppings '(≫'). d\nImage of a honeycomb lattice of micropillars with h omogeneous hoppings ( '(\u0005'′(, \t4\u0005 1, see inset ). The circles pinpoint \nthe upper surface of the micropillars in two adjace nt hexagons; the white arrows indicate the directio n along which the \ncoupling '′( is modified. e Experimental photoluminescence of the lattice show ing s- and p-bands. \n5 \n deviates from a flat band due to the coupling to hi gher energy bands (arising from photonic d-orbitals \nin the micropillars). \n \nFigure 3. Tilted Dirac cones in orbital graphene un der strain. The central panels a-d show the measured polariton \nphotoluminescence intensity as a function of *\t for different values of 4 (the colour scale of each panel has been \nindependently normalised to its maximum value). The cut is done for *\f\u00052+/\u0002$5\u000b$ 2⁄ \u0004, dashed line in e. $\u00052.40 µm \nfor the unstrained coupling, while for the strained coupling $′ is 2.40 µm, 2.60 µm, 2.70 µm and 2.72 µm, respectively, in a-\nd. \u001a\b\u00051.5687 eV and photon-exciton detuning :\u0005; 10 meV at the energy minimum of the p-bands for pane ls a, b. d; \n\u001a\b\u00051.5780 meV and :\u0005;2 meV for c. <'(\u0005;0.90 meV for panel a, ;0.85 meV for b and d, and ;0.65 for c, which are \nobtained by fitting the measured spectra with a tig ht-binding Hamiltonian (fits are shown in white lin es). The left inset of c\nshows the semi-Dirac dispersion measured along *\f for the value of *\t marked by a dashed line in c. The left column, depicts \nthe calculated tight-binding bands for energies clo se to \u001a\b (orange dashed rectangle in a) and the winding of the \nwavefunctions around the Dirac points. The right co lumn shows the tight- binding bands in the region at the bottom of the \nbands (green dashed rectangle in a) together with the winding of the wavefunctions. e Sk etch of the Brillouin zones in \nmomentum space. f Green dots: absolute value of \u0007\t and \u0007\b\t extracted from fits of Eq. 1 to the two experimental Dirac cones \nvisible at low energies in b-d (see Supplementary Information). The plotted value is the average of |\u0007\t| and |\u0007\b\t| extracted \nfrom the two Dirac points. Solid lines: tight-bindi ng result. \n6 \n Tilted and semi-Dirac cones \nThe dispersion of the p-bands can be modified by in troducing an artificial uniaxial strain in the latt ice. \nTo do so, we change the centre-to-centre distance $′ between the micropillars whose link is oriented \nalong the ? direction. This is equivalent to modifying '′(, defined in the inset of Fig. 2d, while keeping \nthe other two couplings '( constant 35 . The central panels of Fig. 3 show the spectra of four different \nlattices with decreasing strain parameters 4≡' ′('(⁄ . The value of 4 is extracted from the fit of the \ntight-binding model (white lines) to the experiment al dispersions. Let us first focus on the type-I Di rac \ncones in the central region (orange rectangle). The left panels depict the calculated dispersion aroun d \n\u001a\b obtained from the tight-binding model. Decreasing beta, that is, emulating the stretching of the \nlattice, brings the Dirac cones closer together in the @ direction (Fig. 3a-b) until they merge at 4\u0005 0.5 \nin a single band touching (Fig. 3c). For 4 < 0.5 a gap is opened (Fig. 3d and Supplementary Fig. S4 ). \nThis is a topological Lifshitz transition in which two Dirac cones with opposite topological charge me rge \nand annihilate, predicted for standard s-band graph ene 4,36 and reported in photonic 7,8 and atomic 5,6 \nhoneycomb lattices and in black phosphorus 9. At the merging point ( 4 = 0.5 , Fig. 3c), we provide the \ndirect observation of a semi-Dirac dispersion, with the touching of two parabolic bands along the *\t \ndirection and a linear dispersion along *\f (shown in the left inset of Fig. 3c). \nThe Dirac points that we have just analysed do not pres ent any tilt: \u0007\b\t= \u0007 \b\f= 0 for any value of the \nstrain. Tilted Dirac cones become apparent when ana lysing the evolution of the touching between the \nquadratic and flat bands (green dashed square in Fi g. 3a) as a function of strain. For decreasing valu es \nof 4, the flat band evolves into a dispersive band with negative effective mass at the Γ point, and the \nband touching divides into two Dirac points that mo ve away from each other in the *\t direction 37 \n(Fig. 3b-e). Remarkably, they are tilted as indicat ed by the angle bisector in green dashed lines in \nFig. 3c. A fit of Eq. 1 to the experimental dispers ions close to the Dirac points reveals the evolutio n of \n\u0007\b\t and \u0007\t as a function of the strain, as depicted in Fig. 3 f. The measured values of the tilt ( \u0007\b\t) agree \nwell with those expected from the tight-binding Ham iltonian shown as solid lines in Fig. 3f (see \nSupplementary Information). \nEngineering type-III Dirac cones \nCritically tilted type-III Dirac cones with \u0007 \u001f\b=\u001d\u0002\u0007\b\t\u0007\t⁄ \u0004\u001e+\u0006\u0007\b\f\u0007\f⁄\r\u001e= 1 can be engineered in our \nsystem when instead of expanding the lattice ( 4 < 1 ), it is compressed ( 4 > 1 ). This is shown in Fig. 4 \nfor 4 = 1.5 : in the direction *\f (Fig. 4c), parallel to that along which $′ is reduced, two new Dirac \npoints emerge from the flat-parabolic band touching . As a reference, Fig. 4a shows the case of 4 = 1 \nalong *\f. The most striking feature of the new Dirac points is that they show the crossing of a flat band \nwith zero group velocity and a linear band with fin ite group velocity (see Fig. 4c, dashed rectangle). \nThis is precisely the signature of a type-III Dirac cone, and it implies \u0007\b\f= −\u0007 \f. From the experimental \nphotoluminescence of Fig 4c we measure C\u0007\b\fC=\u00020.29 ± 0.03 \u0004$̅'( and C\u0007\fC=\u00020.35 ± 0.03 \u0004$̅'( \n(with $̅=\u001eEFGE\nH), which is in good agreement with the tight bindin g prediction ( C\u0007\b\fC=C\u0007\fC=\n0.37$' (, see Supplementary Information). Along the perpend icular direction ( *\t, Fig. 4d), the type-III \ncones present a symmetric linear crossing, and we m easure |\u0007\b\t|= \u00020.00 ± 0.04\u0004$' (, |\u0007\t|=\n\u00020.46 ± 0.04 \u0004$'(, in agreement with the tight-binding Hamiltonian ( 0 and 0.48$' (, respectively), \nwhose dispersion is shown in Fig. 4e. The degenerac y of the flat band results in a divergent density o f \nstates at the energy of the type-III Dirac point. 7 \n Topological invariants of tilted and type-III Dirac cones \nThe emergence of unconventional Dirac cones when ap plying a uniaxial deformation to the orbital \nlattice can be understood from topological argument s. The analysis of the topological charge \nassociated to each cone, that is, the winding of th e Hamiltonian around each Dirac point, provides a \nclear picture of their birth and evolution. The vic inity of the band touching point around the Γ point \nfor 4 = 1 can be described by an effective 2\u00132 Hamiltonian obtained from the projection of the ful l \n4\u00134 Hamiltonian on the subspace of the two lowest ener gy bands 38 , resulting in the following \neffective Hamiltonian: \n\u0001\u0002\u0003\u0004= −H\n\u001e'(\u000e\b+H\nI'(J\n\t\u001e\n\t\n\f\n\n\t\n\f \n\f\u001eK≡ −H\n\u001e'(\u000e\b+ L\u0002\u0003\u0004⋅ \u0018. (2) \nMatrix elements coupling the high and low energy ma nifolds are treated in a second order \nperturbation theory 38 . In this form, the winding of L\u0002\u0003\u0004 can be calculated as a winding vector 36 : M =\n\u00022+\u0004NO∮Q\u0002\u0003\u0004\u0013$Q\u0002\u0003\u0004,\t where Q\u0002\u0003\u0004=L\u0002\u0003\u0004\n|L\u0002\u0003\u0004|, and the integral is performed along a closed line in \nmomentum space that encircles the band touching poi nt (see Supplementary Information). The \nmodulus of M is always an integer, thus providing a topological charge to the touching point, and it \nhas a value R =2 in this case. Figure 4. Type-III Dirac cones. a Measured photoluminescence intensity for 4 = 1 ($ = $′\u00052.60 µm) along *\f for *\t=\n−2+√3$⁄ (line 1 in b). \u001a\b=1.5687 eV and <'(= −0.85 meV. b Emission in momentum space at \u001a\b for 4 = 1, showing \nthe usual Dirac points at the high symmetry points in recipro cal space. Crosses indicate the position of the Dir ac points \nreported in c. c Zoom of the low energy section of the measured spe ctrum for a lattice with 4 = 1.5 ($ =2.60 µm, $′\u0005\n2.40 µm) showing the emergence of a type-III Dirac cone c ombining flat and dispersive bands. d Measured dispersion along \nthe *\t direction for *\f= −0.5\t\u00132+ 3$⁄ (line 2 in b), crossing the Dirac point in the dashed rectangle in c. e Dispersion \nobtained from the tight-binding model zoomed in the momentum space region marked with a dashed rectang le in c and d. \nIn a, c, d, the white lines show the tight-binding model. \n8 \n We can extend this analysis to the emerging Dirac c ones when 4 ≠ 1 . Following the same procedure, \nthe tight-binding Hamiltonian can be reduced to an effective 2 × 2 matrix close to the considered Dirac \ncone. For 4 < 1 , the energy of the Dirac cones in the lower part o f the spectrum is \u001aU= \u001a \b−\nO\n\u001eV3 + 64\u001e'(. Taking \u001aU as the origin of energies, the effective Hamiltoni an near a Dirac point reads: \n\u0001\u0002\u0003\u0004= \u0007 cos Z \n \t\u000e\b+ \u0007\n \t\u000e[+ \u0007 cos Z \n \f\u000e\t, (3) \nwhere \u0007 =√H\n\\V|1 − 4\u001e|$'(, the angle Z ∈ ^0, + 2 ⁄_ is defined as tan Z = V1 − 4\u001e√34 c (with Z →\n+ − Z for the other Dirac point), and \u000e[= \u0018 ⋅ \u0014e, with \u0014e= − sin Z \u0014 g+ cos Z \u0014 h, where \u0014i,g,h are \nCartesian unit vectors. Analogously, for 4 > 1 , around a type-III Dirac point ( \u001aU= \u001a \b−H\n\u001e'(), the \nreduced Hamiltonian reads: \n\u0001\u0002\u0003\u0004= \u0007 cos j \n \f\u000e\b+ \u0007\n \t\u000ek− \u0007 cos j \n \f\u000e\u0019, (4) \nwhere tan j = V4\u001e− 1 V4 − 4\u001ec and \u000ek= \u0018 ⋅ \u0014k, with \u0014k= cos j \u0014 i− sin j \u0014 g (for the other \nDirac point: j → + − j ). By comparing \u0001\u0002\u0003\u0004 in Eqs. 3 and 4 with the generalised Dirac Hamilto nian, \nwe can directly extract the Dirac effective velocit ies as a function of 4 (shown as solid lines in Fig. 3f \nfor \u0007\b\t and \u0007\t) as well as the tilt parameter \u0007 \u001f\b: \n |\u0007\b\t| C\u0007\b\fC |\u0007\t| C\u0007\fC \u0007\u001f\b \n4<1 \u0007cos Z 0 \u0007 \u0007cos Z cos Z \n4>1 0 \u0007cos j \u0007 \u0007cos j 1 \n \nTo compute the winding of the Dirac cones, Hamilton ians 3 and 4 can be rearranged similarly to the \nrightmost hand side of Eq. 2, with a term in the fo rm L\u0002\u0003\u0004⋅ \u0018. For 4 < 1 (Hamiltonian 3), the winding \nof L\u0002\u0003\u0004 around each of the Dirac cones is R = 1 , as indicated in the right panels of Fig. 3. This is also \nthe case for the type-III Dirac cones for 4 > 1 (winding of Hamiltonian 4). A prominent feature of these \nHamiltonians is that the vector L\u0002\u0003\u0004 winds on a plane that depends on the deformation 4, namely on \nthe plane \u0002\u0014i, \u0014e\u0004 for 4 < 1 , and \u0006\u0014k, \u0014\u0019\r for 4 > 1 (see Supplementary Information). \nThe analysis of the windings sheds light on the mec hanisms behind the creation of Dirac cones starting \nfrom a flat band touching a dispersive band. The si ngle band touching at 4 = 1 is described by a \nwinding R = 2 . When 4 ≶ 1 the band touching evolves into a pair of Dirac con es (tilted or type-III) \nwith winding R = 1 . This is an illustration of one of the two possibl e scenarios for Dirac merging in \ntwo dimension, characteristic of, for example, bila yer graphene 37 . Remarkably, the orbital p-bands \nshow the two possible universal scenarios for the m erging of Dirac cones in two dimensions 39 : (i) two \nDirac cones with R = 1 emerge from a point with R = 2 , observed in the lower part of the spectra \nof Fig. 3; (ii) two Dirac cones with opposite windi ng R = ±1 merge in a semi-Dirac cone with R =\n 0, reported in the central part of the spectra in Fi g. 3 (a 2 × 2 effective Hamiltonian analysis can also \nbe done for the Dirac points at \u001a\b). \nThe photonic realisation here reported demonstrates the flexibility of orbital bands to implement \nunconventional Dirac points. This is an asset for t he engineering of photonic materials that combine \ndifferent types of Dirac dispersions, an promising configuration for the study of analogue black holes \nin photonics 21,40 . Moreover, our experiments provide a recipe for th e implementation of Dirac cones 9 \n in solid state materials: the touching of a flat an d a dispersive band with winding R = 2 evolves into \ntwo Dirac cones in the presence of strain. This beh aviour has been predicted for other lattice \ngeometries 36 , and it presents a natural playground to investiga te the transition between different \ntopological phases when particle interactions are p resent 41,42 or when time reversal symmetry is \nbroken. Polaritons are particularly well suited to study these scenarios: thanks to their excitonic \ncomponent they present significant repulsive intera ctions in the high density regime 43 and they are \nsensitive to external magnetic fields, allowing the implementation of quantum Hall phases 44,45 . \nPolariton orbital bands of the kind here reported o pen exciting perspectives for the study of topologi cal \nlasers 46 and of Dirac superfluids 47 . \nAcknowledgements . This work was supported by the ERC grant Honeypol, the EU-FET Proactive grant \nAQuS, the French National Research Agency (ANR) pro ject Quantum Fluids of Light (ANR-16-CE30-\n0021) and the Labex CEMPI (ANR-11-LABX-0007) and Na noSaclay (ICQOQS, Grant No. ANR-10-LABX-\n0035), the French RENATECH network, the CPER Photon ics for Society P4S, and the Métropole \nEuropéenne de Lille via the project TFlight. T.O. a cknowledges support from the Interdisciplinary \nTheoretical and Mathematical Sciences Program (iTHE MS) at RIKEN \nReferences \n1. Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic prope rties of graphene. \nRev. Mod. Phys. 81, 109–162 (2009). \n2. Zhao, Y. et al. Creating and probing electron whispering-gallery m odes in graphene. Science 348, 672–675 (2015). \n3. Bernevig, B. A. & Hughes, T. L. Topological Insulators and Topological Superconduct ors . (Princeton University Press, \n2013). \n4. 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Paris -Sud, Université Paris-Saclay, 91405 Orsay Cedex, \nFrance \n3 Interdisciplinary Theoretical and Mathematical Scie nces Program (iTHEMS), RIKEN, Wako, Saitama 351-0198, \nJapan \n4Univ. Lille, CNRS, UMR 8523 – PhLAM – Physique des Lase rs Atomes et Molécules, F-59000 Lille, France \n \nContent : \nI.- Sample description and experimental set-up .... ................................................... .................... 11 \nII.- Tight-binding Hamiltonian and extraction of m .................................................. ..................... 13 \nIII.- Gap opening at m = n. op .................................................. ................................................... 15 \nIV.- Tomography of tilted Dirac cones ............. ................................................... ......................... 16 \nV.- Measurement and calculation of the effective Di rac velocities .................................... ........... 17 \nVI.- Hamiltonian reduction and winding around the b and touching points ............................... .... 18 \n \nI.- Sample description and experimental set-up \nThe sample used in the experiments is a semiconduct or microcavity grown by molecular beam epitaxy. \nTwo Bragg mirrors made of 28 (top) and 40 (bottom) pairs of q 4⁄ alternating layers of \nGa 0.05 Al 0.95 As/Ga 0.80 Al 0.20 As embed a q 2⁄ Ga 0.80 Al 0.20 As cavity. q = 775 nm is the emission/absorption \nwavelength of free excitons of 12 GaAs, 7 nm wide q uantum wells grown in groups of four in the three \ncentral maxima of the electromagnetic field of the cavity. At 10 K, the temperature of the experiments , \nthe excitonic and photonic resonances are in the st rong coupling regime, with a Rabi splitting of \n15 meV. The experiments are performed at a photon-e xciton detuning of -10 meV (measured at the \nenergy of the lowest energy flat band), except for the data shown in Fig. 3c, for which the detuning i s \n-2 meV. \nTo fabricate the honeycomb lattices, the as- grown planar structure is subject to e-beam lithogr aphy \nand Inductively Coupled Plasma etching down to the GaAs substrate. Each micropillar in the lattice has \na diameter of 2.75 µm, and the centre-to-centre distance varies between 2.4 and 2.72 µm, ensuring \nthe overlap between adjacent micropillars. The pola riton lifetime measured in a similar unetched 12 \n microcavity is of the order of 30 ps. In the etched structures, the lateral defects induced during the \nmicrofabrication process reduce the lifetime to abo ut 3-5 ps. \nExperiments are performed by exciting the centre of each lattice with a continuous wave laser focalise d \nin a spot of 3 µm in diameter (full width at half maximum). The wav elength of the laser is 745 nm \n(1.6642 eV), corresponding to an energy about 100 m eV above the lower polariton s- and p-bands. The \nlaser injects electron and holes that relax down in coherently to form polaritons that populate the \nlower bands of the lattice. The photons that escape out of the sample conserve the energy and in-\nplane momentum of polaritons in the lattice. The la tter is related to the angle of emission following \nthe expression: *∥=s\n 1 they are: \n\u001aU= \u001a \b−H\n\u001e'(, cos . \f=\u001eN\n0, .\t= 0, (E6b) \nIn Figs. 3 and 4 of the main text, the experimental value of 4 is obtained from fits of the spectrum of \nthe tight-binding model to the measured dispersions , in which ')= 0 and '(, 4 and the onsite energy \n\u001a\b are the fitting parameters. \n 15 \n III.- Gap opening at m = n.op \nFigure 3c of the main text shows the formation of a semi-Dirac cone at \u001a = \u001a \b for 4 =0.5. If 4 is \nfurther reduced to 0.45, a gap opens. To further ev idence the gap opening, Fig. S3 shows the spectrum \nobtained from Figs. 3c and d at the momentum of the Dirac point ( *\t= −3\u00132+/3√3$) for 4 =0.5 \nand 4 =0.45 . In the latter case a splitting is apparent at \u001a\b, corresponding to the gap opening. \n \n \n \nFigure S3. Spectrum at gap opening. a, b Photoluminescence spectra taken at the momentum po sition of the semi- Dirac \ncone *\t= −3\u00132+/3√3$, *\f=2+/\u0002$5\u000b$ 2⁄ \u0004, for 4 =0.5 (a) and 4 =0.45 (b), corresponding to Fig. 3c and d, \nrespectively. In b, a splitting is apparent in the circled area. \n16 \n IV.- Tomography of tilted Dirac cones \nFigure 3 of the main text shows the emergence of ti lted Dirac cones from the flat band – dispersive \nband touching when 4 < 1. The tilt is clearly visible in the *\t direction, resulting in a non-zero value \nof \u0007\b\t (as defined in Eq. 1). Figure S4 shows the dispers ion across one of the newly created Dirac cones \nin the *\f direction for 4 =0.5. The white solid lines depict the eigenvalues calc ulated by diagonalising \nHamiltonian E2. As it is apparent from the measured spectra and the tight-binding bands, the newly \ncreated cones are not tilted along the *\f direction, resulting in \u0007\b\f0. \n \n \nFigure S4. Tomography of tilted Dirac cones. a Measured dispersion along the *\t direction for *\f=4+/\u0002$5\u000b$ 2⁄ \u0004 (line 1 \nin panel c) for 4 =0.5 (same as Fig. 3c). b Measured dispersion along the *\f direction for *\t=2+/3√3$, traversing the \ntilted Dirac cone (green dashed line in a, line 2 in c) . c Sketch of the different Brillouin zones in momentum space. The orange \nhexagons show the Brillouin zones as defined for 4 = 1. The crosses indicate the position of the emergent tilted Dirac cones. \nd Zoom of the tight- binding eigenenergies in the region close to the lo west energy Dirac points. Parameters are the same a s \nfor Fig. 3c. \n17 \n V.- Measurement and calculation of the effective Di rac velocities \nFigure 3f of the main text shows the effective Dira c velocities \u0007\b\t and \u0007\t measured from the measured \ndispersions along the *\t direction plotted in Fig. 3. To measure these velo cities, we first extract the \ndispersion around the Dirac cones from the maximum photoluminescence intensity as a function of \nenergy and momentum. For the case of 4 =0.45 , the data points are shown in Fig. S5b. A linear f it \nallows measuring the slope of the dispersion. By co mparison with Eq. 1 of the main text, the two \nmeasured slopes correspond to |\u0007\b\t|−|\u0007\t| and |\u0007\b\t|+|\u0007\t|. From that, we extract |\u0007\b\t| and |\u0007\t|. We \nrepeat the procedure for the symmetric Dirac point and we take the average of their absolute value. \nNote that for type-I Dirac cones, \u0007\b\t= \u0007 \b\f=0, and the absolute value of the two slopes should b e \nidentical and equal to \u0007\t. To test this hypothesis, we have measured the slo pe around the K’ type-I \nDirac cone at \u001a\b in Fig. 3a at positive momenta. After converting t he slopes to effective Weyl velocities \n(Eq. 1 of the main text) we obtain |\u0007\b\t|=0.07\u001b0.03 and |\u0007\t|=0.72\u001b0.03 in units of $'(. \nFrom the tight-binding Hamiltonian, the effective D irac velocities can be obtained from the derivative \nalong *\t and *\f of the eigenvalues of E3-E5 at the position of the emergent Dirac cones (E6), resulting \nin the analytical expressions for \t4A 1 (normalised to $'(, where we have assumed that in the \nexperiment $ $′): \n|\u0007\b\t|=H\n\\\u001dON\nOG\u001e, C\u0007\b\fC=0, |\u0007\t|=√H\n\\V1 − 4\u001e, C\u0007\fC=VHNH\n\\. (E7) \nThe analytical expressions for |\u0007\b\t| and |\u0007\t| are shown in solid lines in Figs. S5 and 3f. \nFor 1 A 4 A2 we have (same units as above): \n|\u0007\b\t|=0, C\u0007\b\fC=C\u0007\fC=O\n\\V\u00024\u001e−1\u0004\u00024 − 4\u001e\u0004, |\u0007\t|=√H\n\\V4\u001e−1. (E8) \nNote that the fact that C\u0007\b\fC=C\u0007\fC (with opposite sign: \t\u0007\b\f\u0005;\u0007\f) for 4 > 1 is the signature of the \ntype-III Dirac cone, and it results in a flat band along the *\f direction. \nFigure S5. a Measured dispersion along the *\t direction for *\f=2+/\u0002$5\u000b$ 2⁄ \u0004 for 4 =0.45 (same as Fig. 3d). b Extracted \ndata points from the Dirac point marked in a dashed rectangle in a. From linear fits to the points we obtain |\u0007\b\t| and |\u0007\t|, \nplotted in Fig. 3f of the main text. \n18 \n \nVI.- Hamiltonian reduction and winding around the b and touching points \nThe vicinity of touching point between two bands ca n be described by an effective 2\u00132 Hamiltonian \nobtained from the projection of the full 4\u00134 Hamiltonian on the subspace of the two lowest ener gy \nbands 3. The reduced Hamiltonian can be expressed in the g eneric form: \n\u0001\u0002\u0003\u0004= x \u0002\u0003\u0004\u000e\b+ L\u0002\u0003\u0004⋅ \u0018 = x \u0002\u0003\u0004\u000e\b+ \u000f\u0002\u0003\u0004⋅ \u000e\u000f+ \u0010\u0002\u0003\u0004⋅ \u000e\u0010, (E9) \nwhere \u000e\b is the 2\u00132 identity matrix and \u000e\u000f,\u0010\u0005\u0014\u0015,\u0016⋅\u0018, with \u0014\u0015,\u0016 unit vectors and \u0018 = \u0006\u000e\t,\u000e\f,\u000e\u0019\r \nis the vector of Pauli matrices. Without loss of ge nerality and for simplicity, we consider the case o f \n\u0014\u0015⋅ \u0014\u0016=0, which implies \u000e\u000f,\u000e\u0010\u00050. \nThe winding vector of this Hamiltonian around the b and touching point is 4: \nM =O\n\u001e∮Q\u0002\u0003\u0004× $Q \u0002\u0003\u0004=\u0014\n\u001e∮¡¢\u0002\u0003\u0004£¡¤\u0002\u0003\u0004N¡ ¤\u0002\u0003\u0004£¡¢\u0002\u0003\u0004\n¡¢\u0002\u0003\u0004G¡¤\u0002\u0003\u0004¥\u0003 , (E10) \nwhere \u0014= \u0014 \u0015× \u0014 \u0016 and \tQ\u0002\u0003\u0004\u0005L\u0002\u0003\u0004\n|L\u0002\u0003\u0004|. The integral is performed along a closed line in momentum \nspace that encircles the band touching point. The m odulus of M can only take integer values. \nThe effective 2\u00132 Hamiltonian described in the main text close to th e flat-parabolic touching at the \nΓ point for 4 = 1 takes the form: \n\u0001\u0002\u0003\u0004= −H\n\u001e'(\u000e\b+H\nI'(J\n\t\u001e\n\t\n\f\n\n\t\n\f \n\f\u001eK≡ −H\n\u001e'(\u000e\b+ L\u0002\u0003\u0004⋅ \u0018, (E11) \nand it is characterized by the winding vector M = +2\t\u0014h. \nFor 4 < 1, close to the emerging Dirac cone, it reads: \nFigure S5. \u0007\b\t and \u0007\t as a function of 4 calculated from the derivative of the tight- binding spectrum at the position of the \nemergent Dirac cones. \n19 \n \u0001\u0002\u0003\u0004= \u0007 cos Z \n \t\u000e\b+ \u0007\n \t\u000e[+ \u0007 cos Z \n \f\u000e\t, (E12) \nwhere \u0007 =√H\n\\V|1 − 4\u001e|$'(, the angle Z ∈ ^0, + 2 ⁄_ is defined as tan Z = V1 − 4\u001e√34 c (with Z →\n+ − Z for the other Dirac cone), and \u000e[= \u0018 ⋅ \u0014e, with \u0014e= − sin Z \u0014 g+ cos Z \u0014 h, where \u0014i,g,h are \nCartesian unit vectors. \nAnalogously, for 4 > 1 , around one of the type-III Dirac points, the redu ced Hamiltonian reads: \n\u0001\u0002\u0003\u0004= \u0007 cos j \n \f\u000e\b+ \u0007\n \t\u000ek− \u0007 cos j \n \f\u000e\u0019, (E13) \nwhere tan j = V4\u001e− 1 V4 − 4\u001ec and \u000ek= \u0018 ⋅ \u0014k, with \u0014k= cos j \u0014 i− sin j \u0014 g (for the other \nDirac point: j → + − j ). \nBy comparing the form of Eqs. E12 and E13 with the generalised Dirac Hamiltonian presented in the \nmain text and its eigenvalues in Eq. 1, we can writ e the effective velocities and tilt parameter \u0007 \u001f\b as a \nfunction of \u0007, Z and j: \n |\u0007\b\t| C\u0007\b\fC |\u0007\t| C\u0007\fC \u0007\u001f\b=¦\u00070@\n\u0007@2\n+J\u00070?\n\u0007?K2\n \n4<1 \u0007cos Z 0 \u0007 \u0007cos Z cos Z \n4>1 0 \u0007cos j \u0007 \u0007cos j 1 \n \nIf \u0007, Z and j are explicitly expressed in terms of 4 we obtain Eqs. E7 and E9, in agreement with the fu ll \ntight-binding calculation. \nThe modulus of the winding vector for Hamiltonian E 11 is R = 2 , while for each of the tilted ( 4 < 1 ) \nand type-III Dirac cones ( 4 > 1\u0004 , the winding is R = 1 . As stated in the main text, a prominent feature \nof these Hamiltonians is that vector L\u0002\u0003\u0004 winds on a plane whose orientation depends on the \ndeformation 4. Namely, for 4 < 1 the pseudo-field L\u0002\u0003\u0004 resides on the plane \u0002\u0014i, \u0014e\u0004 and the \nwinding vector points in the direction M\n|M|= \u00020, cos Z, ± sin Z\u0004 . The winding vector plane thus rotates \n \nFigure S6. Scheme of the rotation of the winding plane as a f unction of 4 for the two Dirac cones (red and blue lines) emerging \nfrom the parabolic-flat band touching. \n20 \n as a function of 4. As each of the two emergent Diract cones is descr ibed by an angle Z of opposite \nsign, the winding plane turns in opposite direction s for each Dirac cone when 4 goes to zero. This is \nillustrated by the red and blue circles in Fig. S6, which show, respectively, the winding plane of eac h of \nthe two Dirac cones. \nFor 4 > 1, the vector L\u0002\u0003\u0004 resides on the plane \u0006\u0014k,\u0014h\r and the winding vector points in the \ndirection \tM\n|M|\u0005\u0002\u001bsinj,cosj,0\u0004 . Note that at 4 =2 (bottom-right panel in Fig. S6), the two \nemergent Dirac cones are described by a winding vec tor residing in the same plane but pointing in \nopposite directions. Therefore, the winding around each Dirac cone has opposite sign: R = + 1 and \nR = − 1. At 4 =2, the Dirac cones merge at the M-point and annihila te, and a gap opens for 4 >2. \nThis situation is shown in Fig. S7, and it could no t be observed experimentally because the engineered \nhoppings in the designed structure were limited to 4 = 1.5. This kind of merging preceded by a \nrotation of the winding plane has been recently dis cussed in detail in the context of the Mielke latti ce \nunder strain 4. \n1. Jacqmin, T. et al. Direct Observation of Dirac Cones and a Flatband i n a Honeycomb Lattice for Polaritons. Phys. Rev. \nLett. 112, 116402 (2014). \n2. Wu, C. & Das Sarma, S. p x , y -orbital counterpa rt of graphene: Cold atoms in the honeycomb optical lattice. Phys. \nRev. B 77, 235107 (2008). \n3. Cohen- Tannoudji, C., Grynberg, J. & Dupont-Roc, G. Atom-Photon Interactions: Basic Process and Appilca tions . (Wiley, \n1992). \n4. Montambaux, G., Lim, L.-K., Fuchs, J.-N. & Piécho n, F. Winding vector: how to annihilate two Dirac p oints with the \nsame charge. arXiv:1804.00781 (2018). \nFigure S7. Tight binding spectra for different values of 4 > 1 at *\t=0. At 4 =2 the type- III Dirac cones merge and annihilate \nresulting in a gap opening for 4 >2. \n" }, { "title": "2307.15246v1.Analysis_of_magneto_optical_Kerr_spectra_of_ferrimagnetic_Mn__4_N.pdf", "content": "Analysis of magneto-optical Kerr spectra of ferrimagnetic Mn 4N\nJ. Zemen1\nFaculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 160 00, Prague 6,\nCzech Republic\n(Dated: 31 July 2023)\nSimulations of magneto-optical Kerr effect in biaxially strained Mn 4N are performed using density functional theory\nand linear response theory. We consider three ferrimagnetic phases, two collinear and one noncollinear, which have\nbeen corroborated separately by earlier studies. The simulated spectra are compared to magneto-optical data avail-\nable in recent literature. A collinear ferrimagnetic phase with a small saturation magentization, a large perpendicular\nanisotropy, and Curie temperature above 700 K is found to be consistent with the measured spectra. We hypothesise\nthat an admixture of the noncollinear phase, which could explain the lower than predicted net moment and magnetic\nanisotropy observed experimentally, is also present.\nTraditionally, ferrimagnets have been applied in magneto-\noptical recording,1–3where a laser beam is used to increase the\ntemperature close to the Curie point so that the magnetization\ncan be reversed by a small applied magnetic field. Ferrimag-\nnetic alloys containing rare earth and transition metals, such as\nGdFeCo and TbCo, with large perpendicular anisotropy near\nroom temperature have show exceptional performance in this\narea.4\nRecently, ferrimagnetic materials have attracted much at-\ntention for applications in high-density magnetic random ac-\ncess memories (MRAM) and logic devices, which can com-\nbine high perpendicular magnetic anisotropy (PMA) with\nsaturation magnetization much lower than in typical fer-\nromagnets and with Curie temperature well above room\ntemperature.5–8\nFerrimagnets usually consist of two magnetic sublattices\nwith anfiterromagnetic coupling between them. This antifer-\nromagnetic exchange interaction allows for fast spin dynamics\nwhich has motivated the extensive research in antiferromag-\nnetic spintronics.9–11The absence of net magnetic moment\nin antifferomagnetic materials prevents dipolar coupling be-\ntween closely packed memory arrays. However, at the same\ntime, it precludes control of the magnetic domain structure by\nexternal magnetic filed. The small but finite net magnetizaiton\nof ferrimagnetic materials alleviates this problem. Moreover,\nclose to a compensation point the reduced net spin polariza-\ntion makes the state of a ferrimagentic layer in, e.g., a mag-\nnetic tunnel junction (MTJ), more susceptible to spin-transfer\ntorque (STT).12–14\nThe compensation of magnetic sublattices is commonly\nachieved in metallic alloys containing rare earths men-\ntioned above. However, one can avoid the reliance on\nrare earth elements by turning to metallic antiperovskite ni-\ntrides some of which have collinear or non-collinear ferri-\nmagnetic structure.15–17Mn-based antiperovskiet nitrides are\na broad family of materials hosting a range of phenom-\nena including magneto-transport,18–20magneto-caloric,21–23\nor magneto-optical properties tunable by chemical composi-\ntion or lattice strain.24–26We predicted that Mn 3GaN, which\nis a widely studied member of this family with a fully com-\npensated triangular antifferomagntic ground state (cubic lat-\ntice), can develop a collinear ferrimagnetic (FIM) phase when\napplying compressive biaxial strain at room temperature.27\nSubsequently, we used magneto-optical Kerr spectroscopy toanalyse the magnetic structure of Mn 3NiN thin film which is\na closely related antiperovskite with triangular antiferromag-\nnetic ground state. The measured data are consistent with the\npresence of a collinear FIM phase at room temperature.28\nMn 4N is another member of the antiperovskte family. Its\nmagnetic structure is even more complicated than that of\nMn 3NiN as Ni on the 1 asite is replaced by another Mn with\na large magnetic moment. Mn 4N has received much atten-\ntion in recent years mainly due to a large perpendicular mag-\nnetic anisotropy (PMA ≈105J/m3) and ultrafast response to\nexternal field.29–38In addition, the saturation magnetization,\nMsis relatively low as demonstrated by numerous studies of\nMn 4N thin films listed in Table I. Therefore, low critical STT-\nswitching current density, Jc∝αMstHk(where αis the damp-\ning constant, tis the magentic layer thickness, and Hkis the\nanisotropy field proportional to the PMA energy density, Ku)\nis expected.39Moreover, the shape anisotropy is negligible\ndue to low Msso the magnetic anisotropy is dominated by\nthe magnetocrystalline contribution.39\nIn order to utilize the potential of Mn 4N for spintronic ap-\nplications such as MTJs or skyrmionic devices,40it is crucial\nto attain thorough understanding of the microscopic origin\nof the large PMA combined with low magnetization and to\nbe able to grow thin films with well defined magnetic struc-\nture on substrates compatible with CMOS technology. (So\nfar, superior magnetic properties including sharp magnetiza-\ntion switching have been observed on STO substrates,41which\ncomplicates integration due to the requirement of post-growth\nannealing at high temperatures to achieve crystallinity of the\nSTO barrier.)\nIn studies listed in Table I, Mn 4N has been deposited\nsince 2014 on a range of substrates with different lattice mis-\nmatches, e.g., MgO, SrTiO 3(STO), and LaAlO 3(LAO) with\nmismatch of approximately −6%,−0.1%, and +2%, respec-\ntively, assuming (001) surfaces and lattice constant of cubic\nMn 4N equal to 0.3865 nm.47It has been observed using x-\nray diffraction that the films have c/a≈0.98−0.99, where\naandcare the in-plane and out-of-plane lattice constant, re-\nspectively, despite the different magnitude and sign of lattice\nmismatch.\nTherefore, experimental studies that keep track of c/aand\nPMA generally conclude that the origin of PMA in Mn 4N\nfilms is the tetragonal distortion.29,30,33,42Moreover, studies\nthat include ab initio simulations have associated the ob-arXiv:2307.15246v1 [cond-mat.mtrl-sci] 28 Jul 20232\nTABLE I. Comparison of Mn 4N films with thickness ( t), and mea-\nsured magnetic anisotropy energy (K u) and saturation magnetization\n(Ms).\nSubstrate c/at[nm] method Ku[MJ/m3]Ms[kA/m]\nMgO300.987 35 PLD 0.16 157\nMgO290.99 26 MBE 0.22 145\nMgO420.99 100 Sputtering 0.88 110\nSTO310.99 25 MBE 0.1 80\nMgO430.983 9 MBE 0.18 127\nMgO440.991 30 MBE 0.075 100\nMgO41- 10 MBE 0.11 118\nSTO41- 10 MBE 0.11 105\nSTO32- 10 MBE 0.11 71\nMgO330.993 18 MBE 0.06 63\nSTO330.989 17 MBE 0.126 73\nLAO330.998 19 MBE 0.045 53\nMgO390.99 30 Sputtering 0.1 80\nMgO450.99 28 Sputtering 0.17 156\nGlass360.993 45 Sputtering 0.022 36\nGlass460.988 48 Sputtering 0.073 99\nMgO/VN380.987 28 Sputtering 0.043 85\nserved PMA with a collinear ferrimagnetic phase, so called\nFIM Bphase, with total energy minimum at c/a=0.98.31,39\nCollinear ferrimagnetic phases FIM Aand FIM Bshown in\nFig. 1 were revealed by neutron diffraction experiments as\nearly as 1962.47Ito et al.31and Isogami et al.39demonstrated\nbyab inito simulations that FIM Bhas a significantly lower to-\ntal energy than FIM Ain the range c/a∈(0.96−1.1), in agree-\nment with our simulations shown in Fig. 1(d). Both theoretical\nstudies suggest that this intrinsic tetragonal phase with large\nPMA explains why c/a∈(0.98−0.99)has been reported in\nMn 4N films epitaxially grown on different substrates regard-\nless of the film thickness and lattice mismatch.\nHowever, negligible dependence of PMA on c/awas pre-\ndicted by Isogami et al.39which is in disagreement with PMA\nmeasured by Hirose et al.33in Mn 4N on MgO, STO, and\nLAO. Moreover, the theoretical studies mentioned so far31,39\nhave not considered any noncollinear magnetic phases de-\nspite the fact that a noncollinear ferrimagnetic phase (ncFIM)\nwas identified by neutron diffraction in bulk Mn 4N in 1979.17\nMagnetic moments of Mn atoms in this \"umbrella-like\" struc-\nture shown in Fig.1(a) do not mutually compensate as in cubic\nMn 3NiN or Mn 3GaN even though the lattice has cubic sym-\nmetry with space group Pm¯3m.17This is due to the fact that\n1acorner site is occupied by Mn with a large local magnetic\nmoment, m1a. The moments in face-center 3 cpositions, m3c,\nare tilted out of the (111) planes (where the Mn atoms form\na kagome lattice) by approximately 20◦to have a component\nalong the [111] axis, antiparallel to m1a. The ncFIM struc-\nture was confirmed computationally by Uhl et al.48and more\nrecently by Zhang et al.49including the net moment along\nthe [111] axis, mnet=1.1µB. The same team also proposed\nrelated noncollinear ferrimagnetic phases in Mn 4N film on\nMgO45based on ncFIM and the coplanar triangular antifer-\nromatic structure of Mn-based antiperovskite nitrides.\nFIG. 1. Comparison of ferrimagnetic structures considered: (a-c)\nunit cells with magnetic moments and chemical identities: Mn on 3 c\nsites in red, Mn on 1 asites in blue, N in green, and (d) corresponding\ncalculated total energies vs c/aratio.\nHere we compare the collinear and noncollinear FIM\nphases of strained Mn 4N based on total energy and Magneto-\noptical Kerr effect (MOKE) calculated using Density Func-\ntional Theory (DFT) and linear response theory. This is pri-\nmarily motivated by a recent study of MOKE in 23 nm thick\nMn 4N films sputtered on MgO substrate.50The authors inter-\npret their MOKE spectra based on projected densities of states\n(PDOS) simulated by Isogami et al.39and conclude that: \"The\nfine structures observed in Kerr rotation could be attributed to\na superposition of different magnetic phases from the dom-\ninant ferrimagnetism in Mn 4N, although theoretical calcula-\ntions may be necessary for further interpretation.\" Moreover,\nan in-plane component of magnetization has been detected in\none Mn 4N film deposited on MgO by MBE,51and the authors\nexplained it by presence of a FIM phase with magnetization\nalong the [111] axis.17,52\nTherefore, we simulate the MOKE spectra in this work\nfor the three main magnetic structures FIM A, FIM B, and nc-\nFIM. We employ noncollinear spin polarized DFT following\nthe approach of Ref. [53] and our earlier work.28We use\nthe projector augmented wave method as implemented in the\nV ASP code54with with generalized gradient approximation\n(GGA) parameterized by Perdew–Burke–Ernzerhof.55Our re-\nsults were obtained using a 500 eV energy cutoff and a 23 ×\n23 × 23 k-mesh (for a unit cell with 5 atoms) to ensure conver-\ngence (in agreement with numerical settings of Ref. [39]). The\nvalence configurations of manganese and nitrogen are 3 d64s1\nand 2 s22p3, respectively.3\nTABLE II. Comparison of magnetic space groups and corresponding\nanomalous Hall conductivity tensor (AHC) for each pahse.\nncFIM FIM A, FIM B\nSpace group 123, P4/mmm 123, P4/mmm\nMag. space group 12.62, C2’/m’ 123.345, P4/mm’m’\nAHC, σα,β\n0 σxyσxz\n−σxy 0σxz\n−σxz−σxz0\n\n0σxy0\n−σxy0 0\n0 0 0\n\nTaking into account our recent study of MOKE spectra\nin the closely related antiperovskite Mn 3NiN,28and earlier\nMOKE studies of some collinear antiferromagnets such as\nCuMnAs,56we modify the intra-atomic Coulomb interaction\nwithin GGA through the rotationally invariant approach to\nGGA+U as proposed by Dudarev et al.57We explore values\nof U from 0.7 eV (refined in Ref. [28]) to 2.2 eV on the Mn-\n3dorbital. (All data shown here in figures were simulated\nusing U = 0.7 eV .) This repulsion lifts the unoccupied man-\nganese d-states further away from the Fermi level, resulting in\na blueshift in the optical and magneto-optical responses which\nimproves the agreement with the available measured data.50\nThe unit cells and corresponding total energies as functions\nof the c/aratio are shown in Fig. 1. By analysing available x-\nray data of films listed in Table I we noticed a range of values\nof Poisson’s ratio, ν. Therefore, for each c/aratio, we calcu-\nlate the total energy assuming ν=0.3345as well as a=a0,\nwhere a0=0.389 nm is the equilibrium lattice parameter for\nncFIM from our DFT simulations. This is close to the exper-\nimental value a0=0.3865 nm and to an earlier DFT calcula-\ntion, a0=0.382 nm.49We note that our conclusion is inde-\npendent of this choice: The energy minimum for FIM Aphase\nobtained at c/a>1 is more than 0.2 eV higher than the energy\nminimum of FIM Batc/a=0.98, in agreement with earlier\nDFT studies,31,39which suggest that the Mn-Mn direct AFM\ninteraction might be stabilizing the FIM Bstructure. However,\nthe ground state of ncFIM phase ( c/a=1) is another 30 meV\nlower than the energy minimum of FIM B. (This energy differ-\nence is much bigger than 4 meV between ncFIM and a cubic\ncollinear FIM phase with m1a(m3c) parallel (antiparallel) to\nthe [111] axis predicted by Zhang et al.49)\nTherefore, it is conceivable that ncFIM phase with c/a∈\n(0.99−1.0)coexists with FIM Bphase in films where the lat-\ntice mismatch with the substrate does not induce a large biax-\nial strain thanks to dislocations30,31or a \"dead layer\"39im-\nmediately above the interface. Areas of the film with less\ndislocations (better epitaxy) are then more likely to stabilise\nthe tetragonal FIM Bphase. This suggestion further elabo-\nrates on the aforementioned explanation why Mn 4N films with\nc/a≈0.99 have been reported on various substrates regard-\nless of the film thickness and lattice mismatch. Furthermore,\na mixture of FIM Band ncFIM phases could explain the lower-\nthan-predicted M sand PMA as we will discuss below.\nMOKE spectra offer a valuable insight into the magnetic\nstructure of thin films so we take this opportunity to com-\npare spectra simulated for all there ferrimagnetic phases to\nthe measured data.50As in our recent study of MOKE spec-\ntra in Mn 3NiN,28we note that MOKE is an optical coun-\nFIG. 2. Simulated MOKE spectra: (a) Kerr rotation and (b) Kerr\nellipticity; for two values of c/aand U = 0.7 eV (without smoothing).\nterpart of the Anomalous Hall Effect (AHE). Both MOKE\nand the intrinsic contribution to Anomalous Hall Conductiv-\nity (AHC), σα,β, originate (within linear response thoery) in\nnon-vanishing integral of Berry curvature Ωn,α,β(k)over the\nBrillouin zone:\nσα,β=−e2\n¯hZdk\n2π3∑\nn(occ. )f[εn(k)−µ]Ωn,α,β(k),\nΩn,α,β(k) =−2I∑\nm̸=n⟨km|να(k)|kn⟩⟨kn|νβ(k)|km⟩\n[εkn−εkm]2,(1)\nwhere f[εn(k)−µ]denotes the Fermi distribution function\nwith εn(k)andµbeing the energy eigenvalues of occupied\n(unoccupied) Bloch band, n, and the Fermi energy, respec-\ntively, and where να(k)corresponds to the velocity operator\nin Cartesian coordinates.\nThe Kerr angle ( θK) and ellipticity ( ηK) in case of polar-\nMOKE geometry depend on AHC as follows28:\nθK+iηK=−σxy\nσxxp\n1+i(4π/ω)σxx, (2)4\nwhere ωis the photon energy and we assume that the z-axis\n(parallel to the incident beam) is perpendicular to the film sur-\nface.\nThe presence of MOKE and AHE can be determined by an-\nalyzing the transformation properties of the Berry curvature\nunder all symmetry operations of a particular magnetic space\ngroup. In Table II we list the space groups of the three FIM\nphases of Mn 4N (subject to tetragonal distortion) obtained by\nFINDSYM software.58,59The last row of Table II presents\nthe form of the AHC tensor in linear response regime ob-\ntained using software Symmetr60considering both set of sym-\nmetry operations and the spin-orbit coupling. We note that\nboth collinear FIM phases share the same form of AHC ten-\nsor with one independent nonzero element, σx,y, inducing the\npolar-MOKE. The AHC tensor of ncFIM has two independent\nnonzero elements, σx,yandσx,z=σy,zas in case of strained\nMn 3NiN28. Cubic Mn 4N would have a magnetic space group\n166.101, R ¯3m’ and σx,y=σx,z=σy,z. The listed forms of the\nAHC tensor are determined by the symmetry of the structure\nrather than the net magnetization so they would not change\n(AHE and MOKE would not vanish) even if the net magnetic\nmoment vanished, i.e., if full compensation of the ferrimag-\nnet was achieved, which is desirable when seeking ultrafast\nspintronic devices.40,49\nFig 2 presents the main result of the work. It shows Kerr\nrotation and Kerr ellipticity as a function of energy, ω∈\n(1−7)eV . Our model does not include the intraband con-\ntribution which dominates below 1 eV , hence the choice of\nenergy interval. We calculated the spectra for several ratios\nc/a∈(0.97−1.03)but the dependence on tetragonal distor-\ntion appears to be much smaller than the differences between\nthe three FIM phases so we plot only spectra for c/a=0.99\nand 1.01. The spectra in this figure include fine features as we\nuse small Gaussian smearing, σ=0.01 eV , to treat the partial\noccupancies in k-space integration. This approach would cor-\nrespond to experimental data measured in a crystalline film\nwith few defects at low temperature. However, the avail-\nable MOKE data were measured in sputtered films at ambient\ntemperature.50Therefore, we include also Fig. 4 where the\nfine features are smoothed out.\nIn order to interpret our spectra based on features of the\nband structure, we plot the projected DOS for all three phases\nin Fig. 3. We resolve PDOS only for Mn 3d orbitals as\nthe other contributions are small and too far from the Fermi\nenergy to play an important role in the visible magneto-\noptical response. Mn 1, Mn 2, and Mn 3occupy the 3 csites\nwith cartesian coordinates (0.0, 0.5, 0.5), (0.5, 0.0, 0.5), and\n(0.5, 0.5, 0.0), respectively, whereas Mn 4occupies the 1 asite\nwith coordinates (0, 0, 0) as shown in Fig. 1. We note that\nour PDOS for FIM Bphase is in agreement with Fig. 7(a) of\nRef. [39].\nFigure 3(a) for FIM Ashows PDOS with one dominant tran-\nsition indicated by a grey arrow between a peak in occupied\nstates of Mn 1−3and a peak in excited states of Mn 4. This tran-\nsition described by energy difference, dE≈2 eV , corresponds\nto the sharp peak in magneto-optical response at ω≈2 eV .\nFigure 3(b) for FIM Bshows PDOS with two dominant tran-\nsition indicated again by arrows described by dE≈2 eV and\nFIG. 3. Comparison of Projected Density of States (PDOS) for FIM A\n(a), FIM B(b), and ncFIM (c) phases with c/a=0.99 and U = 0.7 eV .\nGrey arrows indicate potentially dominant excitations.\n3 eV which correspond to a dip and a peak in Kerr rotation,\nrespectively.\nFigure 3(c) for ncFIM shows PDOS lacking prominent\npeaks around 2 eV below the Fermi level, which are present\nin case of FIM B. The peak in PDOS of Mn 4present in FIM A\naround 0.5 eV above Fermi energy is shifted to 1 eV in ncFIM.\nSuch band structure results in absence of prominent peaks in\nthe magneto-optical response at photon energies below 4 eV .\nWe conclude that the predicted spectral features can be inter-\npreted based on transitions from Mn-3d orbitals on 3 csites to\nMn-3 dorbitals on 1 asite and that the three FIM phases should\nrelatively easily distinguishable even after smearing their dis-\ntinctive features by effects such as lattice defects or elevated\ntemperature.\nIn order to compare our data to MOKE spectra measured\nby Sakaguchi et al.,50we interpolate our curves plotted in\nFig. 2 using the UnivariateSpline function from Python Scipy\npackage with smoothing parameter set to 0.0008 on a linear\ngrid with 200 points. The smeared spectra for c/a=0.99\nare shown in Fig. 4. Our Kerr rotation, φ(ω), and elliptic-\nity,η(ω), can be compared directly to Fig. 5(a) and (b) of5\nFIG. 4. Simulated MOKE spectra: (a) Kerr rotation and (b) Kerr\nellipticity; for c/a=0.99, U = 0.7 eV , and smoothing of DFT data in\nFig. 2.\nRef. [50], respectively, where the boron content in Mn 4N is\nzero.\nFirstly, we note the the amplitude of the measured spectra\nis approximately 0.1 degree which is an order of magnitude\nlarger than in the related noncollinear Mn-based systems such\nas Mn 3NiN28and Mn Sn.61The amplitude of the simulated\nspectra is only a factor of two larger which is an unexpectedly\ngood agreement indicating high quality of the Mn 4N film.\nSecondly, the measured φ(ω)is positive below 1.5 eV , has a\ndip around 2.2 eV , and seems to come back to positive values\nabove 3 eV , where the studied interval ends. Such trend is\nobserved only in data simulated for FIM Bphase, although the\ncrossing points, φ(ω) =0, are shifted to lower energies (1\nand 2.6 eV instead of the measured 1.5 and 3 eV). A shift of\noptical and magneto-optical responses is commonly achieved\nin DFT studied by increasing the Coulomb repulsion on sites\nwith more localized states (here the Mn-3 dstates). Therefore,\nwe simulated the spectra for a range of Hubbard parameters,\nU, from U=0.7 eV (shown here) to U=2.2 eV . Spectra with\nU≈2.2 eV have crossing points very close to 1.5 and 3 eV , as\nin the experiment. However, we believe that this value of Uistoo high compared to Mn 3NiN28and CuMnAs56where U=\n0.7 and U=1.7 eV was used, respectively, for Mn-3 dorbitals\nto predict MOKE spectra in semiquantitative agreement with\nmeasurement.\nThirdly, we speculate that the crossing point, φ(ω) =0, at\n2.6 eV for FIM Bcould be shifted to higher energy by a as-\nsuming an admixture of ncFIM, which is negative throughout\nthe energy interval. Such superposition of spectra would also\nshift the crossing point at 1 eV to lower energies in contrast to\nexperiment. However, our predictions are less reliable below\n1.5 eV as our model does not include the intraband contribu-\ntions (the Drude peak).\nFinally, we check the agreement in case of Kerr ellipticity,\nη(ω), which is Kramers-Kronig-related to φ(ω). Fig. 5(b)\nof Ref. [50] shows a monotonous decrease of η(ω)to zero\naround 3 eV . As expected, FIM Bis the only phase that shows\nsuch trend in the simulated spectrum but the crossing point is\nagain shifted to lower energy.\nTo complete the analysis and to formulate our hypothe-\nsis about dominant FIM Bphase with an admixture of nc-\nFIM phase, we discuss Figure 5 which shows the magneto-\ncrystalline anisotropy profile and the component of magneti-\nzation perpendicular to the film (direction of the applied field),\nMz, for the two relevant phases. In Fig. 5(a) the total energy\nis plotted as a function of angle θbetween the net magneti-\nzation and the [001] axis (perpendicular to film). The insets\nshow the orientation of the local moments of FIM Bphase for\nnet magnetization pointing along [001], [110], and [00 1]. This\nchoice is relevant for the polar-MOKE experiment, where the\nsample is measured in magnetic field applied parallel and an-\ntiparallel to the [001] axis and the two spectra are subtracted\nto eliminate nonlinear MOKE effects.50The film has to un-\ndergo a rotation of magnetic moments driven by the reversal\nof the perpendicular applied field between the two measure-\nments. So it has to overcome the energy barrier of the in-plane\norientation, θ=π/2, which is dE=1.4 meV per formula\nunit or 3.78 MJ/m3for the FIM Batc/a=0.99. This PMA\nis in perfect agreement with earlier calculations.39However,\nthe value is much larger than experimental PMA ≈0.1 MJ/m3\nas listed in Table I. We cannot compare PMA directly in case\nof Sakaguchi et al.50as they give only the anisotropy field,\nHk=1.5 T, which is a typical value on Mn 4on MgO but\nwe do not know the saturation magnetization and the size of\nthe applied field. So we have to assume that the sample was\nfully aligned with the applied field during MOKE measure-\nment. However, the discrepancy between the predicted and\nmeasured PMA remains an open question which complicates\nthe interpretation of MOKE spectra.\nThe energy barrier is lower for ncFIM at c/a=0.99,dE=\n0.67 eV/f.u. or 1.81 MJ/m3, which speaks in favour of our hy-\npothesis of ncFIM and FIM Bcoexistence. However, the spin\nreorientation mechanism becomes much more complicated in\ncase of ncFIM. There are 8 variants of this phase with the net\nmagnetization pointing parallel or antiparallel to the 4 body-\ndiagonals, in perfect analogy to Mn 3NiN.20The applied field\ncan align the net magnetization with the [001] axis but the\nrotation of the moments to the opposite field orientation can\ngo through different energy minima and energy barriers de-6\npending on the local conditions in the film. We have carried\nout an extensive DFT study of the total energy landscape as a\nfunction of coherent rotations of the 4 local moments.\nWe considered all rotations that belong to the Pm¯3mspace\ngroup of the cubic perovskite lattice: 2-fold and 4-fold ro-\ntations about the main axes, 3-fold rotations about the body\ndiagonals, and 2-fold rotations about the side diagonals. De-\ntails about our findings will be summarised elsewhere. Here\nwe show an example of a rotation between two energy minima\n(from [111] to [ ¯11¯1]) which incurs the lowest energy barrier,\ndE=0.67 eV/f.u. mentioned above. This rotation is a simul-\ntaneous rotation by π/2 about the [010] axis, by π/2 about\nthe [001] axis and by [2 π/3] about [ ¯11¯1] axis. (An intuitive\nsimple rotation of magnetization from [111] to [11 ¯1] about the\n[¯110] axis would not restore the ground state magnetic struc-\nture.) The dash-dotted line in Fig. 5 consists of: (i) a simple\nrotation about the [ ¯110] axis from a state with net magnetiza-\ntion along [001] to the ground state along [111] denoted by\nθ0, (ii) the simultaneous rotation to [ ¯11¯1] denoted by π−θ0),\n(iii) a simple rotation about [110] to a state with net magneti-\nzation along [00 ¯1]. All four significant states are depicted as\ninsets in Fig. 5(b). Notably, the structure in the first inset is of\nthe same type (and has the same direction of magnetization)\nas the noncollinear structure proposed for Mn 4N in Fig. 2(d)\nof Ref. [45].\nAs illustrated by Fig. 5, a film in ncFIM phase with cu-\nbic lattice or with slight tetragonal distortion, c/a∈(0.99,1),\ncan be in a multi-domain state even at saturation (when the net\nmagnetization is fully aligned with applied field) and the mag-\nnetization reversal can follow different paths for each domain.\nInvestigation of magnetic domain wall (WD) propagation in\ncase of ncFIM is beyond the scope of this study but we believe\nthat the availability of more energy minima and lower energy\nbarriers between them (compared to FIM B) would facilitate\nDW propagation, thereby lowering the effective anisotropy\nfield to values observed experimentally.\nOur hypothesis would have implications for the observed\nsaturation magnetization, Ms, so we include Fig. 5(b) to show\nthe out-of-plane magnetization component, M z. Isogami et\nal.39predict 180 mT for FIM Band observe 110 mT exper-\nimentally. They are able to attribute the discrepancy to a\ndead layer at the interface with substrate, and nitrogen de-\nficiency and top surface oxidation. Here we suggest that\nthe lower Mscould be due to the admixture of ncFIM with\nMs=0.727µb/f.u. = 143 mT. However, we admit that Zhang\net al.49predict Ms=1.24µB/f.u. = 244 mT for ncFIM using\nDFT. Our M sis lower due to the use of U = 0.7 eV which\nleads to larger local moments on 3 csites and consequently\nmore compensation of m1a.\nIn summary, we simulated the MOKE spectrum of strained\nMn 4N using DFT+U assuming three ferrimagnetic phases dis-\ncovered by earlier neutron diffraction as well as theoretical\nstudies. We compared our results to polar-MOKE spectra\nmeasured by Sakaguchi et al.50in Mn 4N films on MgO sub-\nstrate. We found that the key features of the simulated spec-\ntra are consistent with the measured spectrum only in case\nof the FIM Bphase. The agreement of the simulated Kerr ro-\ntation could be further improved if a fraction of the ncFIM\nFIG. 5. Comparison of FIM Band ncFIM phases based on: (a) total\nenergy and (b) magnetization projection, M z; angle θdescribes the\nswitching from magnetization along the [001] axis to the opposite\ndirection. The insets show the magnetic structure at significant points\nalong the selected switching paths.\nphase was added to the dominant FIM Bphase. At the same\ntime, the admixture of ncFIM could explain the lower PMA\nand M sobserved experimentally. We believe that our analysis\nwill motivate further MOKE studies where the applied field\ncan be inverted along a chosen path, using a vector magnet\nin particular. This could shed more light on the ncFIM phase\npreferred at lower tensile strains and enable sub-nanosecond\nspin dynamics at room temperature.\nACKNOWLEDGMENTS\nWe acknowledge fruitful discussions with Freya Johnson,\nLesley F. Cohen, Martin Veis, and Jakub Železný. 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We show\nthat the interband magnon drag enhances \u001bmand reduces \u0014m, whereas its total e\u000bects on Smare\nsmall. This drag results from the interband momentum transfer induced by the magnon-magnon\ninteractions. We also show that the higher-energy band magnons contribute to Sm,\u001bm, and\u0014m\neven for temperatures smaller than the energy di\u000berence between the two bands.\nI. INTRODUCTION\nMagnon transport is the key to understanding spin-\ntronics and spin-caloritronics phenomena of magnetic in-\nsulators1{3. For example, a magnon spin current is vital\nfor the spin Seebeck e\u000bect2,4{7. Magnon transport is im-\nportant also for other relevant phenomena8{13.\nThere are two key issues about magnon transport in\nferrimagnetic insulators. One is about multiband e\u000bects.\nYttrium iron garnet (YIG) is a ferrimagnetic insulator\nused in various spintronics or spin-caloritronics phenom-\nena1{3,8{12. Its magnons have been often approximated\nas those of a ferromagnet. However, a study using its\nrealistic model14showed that not only the lowest-energy\nband magnons, which could be approximated as those of\na ferromagnet, but also the second-lowest-energy band\nmagnons should be considered except for su\u000eciently low\ntemperatures. Since the experiments using YIG are per-\nformed typically at room temperature1{3,8,9,11,12, it is\nnecessary to clarify the e\u000bects of the higher-energy band\nmagnons on the magnon transport. The other is about\ninteraction e\u000bects. The magnon-magnon interactions are\nusually neglected. However, their e\u000bects may be drastic\nin a ferrimagnet because they can induce the interband\nmomentum transfer, which is expected to cause an in-\nterband magnon drag by analogy with various drag phe-\nnomena15{40. Nevertheless, it remains unclear how the\nmagnon-magnon interactions a\u000bect the magnon trans-\nport.\nIn this paper, we provide the \frst step towards re-\nsolving the above issues and propose a new drag phe-\nnomenon, the interband magnon drag. We derive three\ntransport coe\u000ecients of interacting magnons for a two-\nsublattice ferrimagnet and numerically evaluate their\ntemperature dependences. We show that the interband\nmagnon drag enhances a magnon conductivity and re-\nduces a magnon thermal conductivity, whereas its total\ne\u000bects on a spin-Seebeck coe\u000ecient are small. We also\nshow that the higher-energy band magnons contribute to\nthese transport coe\u000ecients even for temperatures lower\nthan the energy splitting of the two bands.\n\"#\nYZ[FIG. 1. Our ferrimagnetic insulator. The up or down arrows\nrepresent the spins on the AorBsublattice, respectively. The\nx,y, andzaxes are also shown.\nII. MODEL\nOur ferrimagnetic insulator is described by\nH= 2JX\nhi;jiSi\u0001Sj\u0000hN=2X\ni=1Sz\ni\u0000hN=2X\nj=1Sz\nj; (1)\nwhere the \frst term is the Heisenberg exchange interac-\ntion between nearest-neighbor spins, and the others are\nthe Zeeman energy of a weak magnetic \feld ( jhj\u001cJ).\n(The ground-state magnetization is aligned parallel to\nthe magnetic \feld.) We have disregarded the dipolar in-\nteraction and the magnetic anisotropy, which are usually\nmuch smaller than J14,41. For concreteness, we consider\na two-sublattice ferrimagnet on the body-centered cubic\nlattice (Fig. 1); i's andj's in Eq. (1) are site indices\nof theAandBsublattice, respectively. There are N=2\nsites per sublattice. Our model can be regarded as a\nminimal model of ferrimagnetic insulators because a fer-\nrimagnetic state, the spin alignments of which are given\nbySi=t(0 0SA) for alli's and Sj=t(0 0\u0000SB) for all\nj's, is stabilized for J >0 with the weak magnetic \feld.\nWe set ~= 1,kB= 1, anda= 1, where ais the lattice\nconstant.\nTo describe magnons of our ferrimagnetic insulator,\nwe rewrite Eq. (1) by using the Holstein-Primako\u000b\nmethod42. By applying the Holstein-Primako\u000b transfor-arXiv:2205.03058v1 [cond-mat.mes-hall] 6 May 20222\nmation43{45to Eq. (1) and using a 1 =Sexpansion43,44,46\nand the Fourier transformation of magnon operators, we\ncan write Eq. (1) in the form\nH=HKE+Hint: (2)\nHereHKErepresents the kinetic energy of magnons,\nHKE=X\nq\u0000\nay\nqbq\u0001\u0012\u000fAA\u000fAB(q)\n\u000fAB(q)\u000fBB\u0013 \naq\nby\nq!\n;(3)\nwhere\u000fAA= 2J0SB+h,\u000fAB(q) = 2pSASBJq,\u000fBB=\n2J0SA\u0000h, andJq= 8Jcosqx\n2cosqy\n2cosqz\n2;Hintrepre-\nsents the leading terms of magnon-magnon interactions,\nHint=\u00001\nNX\nq1;q2;q2;q4\u000eq1+q2;q3+q4(2Jq1\u0000q3ay\nq1aq3by\nq4bq2\n+r\nSA\nSBJq1aq1by\nq2bq3bq4+r\nSB\nSAJq1bq1ay\nq2aq3aq4) + (H.c.):\n(4)\nWe can also express HKEas a two-band Hamiltonian by\nusing the Bogoliubov transformation43{45:\nHKE=X\nq[\u000f\u000b(q)\u000by\nq\u000bq+\u000f\f(q)\fq\fy\nq]; (5)\nwhere\u000f\u000b(q) =h+J0(SB\u0000SA) + \u0001\u000fq,\n\u000f\f(q) =\u0000h+J0(SA\u0000SB) + \u0001\u000fq, and\n\u0001\u000fq=q\nJ2\n0(SA+SB)2\u00004SASBJ2q. ForSA> SB\nwe have\u000f\u000b(q)<\u000f\f(q). Note that the Bogoliubov trans-\nformation is given by aq= (Uq)A\u000b\u000bq+ (Uq)A\f\fy\nqand\nby\nq= (Uq)B\u000b\u000bq+ (Uq)B\f\fy\nq, where (Uq)A\u000b= (Uq)B\f=\ncosh\u0012q, (Uq)A\f= (Uq)B\u000b=\u0000sinh\u0012q, and these hyper-\nbolic functions satisfy cosh 2 \u0012q= [J0(SA+SB)]=\u0001\u000fq\nand sinh 2\u0012q= (2pSASBJq)=\u0001\u000fq. Then, by using the\nBogoliubov transformation, we can decompose Hintinto\nthe intraband and the interband components47. Because\nof these properties, our model is a minimal model to\nstudy the two key issues explained above.\nIII. DERIVATIONS OF TRANSPORT\nCOEFFICIENTS\nWe consider three transport coe\u000ecients: a spin-\nSeebeck coe\u000ecient Sm, a magnon conductivity \u001bm, and\na magnon thermal conductivity \u0014m. They are given by\nSm=L12,\u001bm=L11, and\u0014m=L22, whereL\u0016\u0011's are\nde\fned as\njS=L11ES+L12\u0010\n\u0000rT\nT\u0011\n; (6)\njQ=L21ES+L22\u0010\n\u0000rT\nT\u0011\n: (7)\nHerejSandjQare magnon spin and heat, respectively,\ncurrent densities, ESis a nonthermal external \feld, andrTis a temperature gradient. (Note that one of the\npossible choices of ESis a magnetic-\feld gradient48.)\nL21=L12holds owing to the Onsager reciprocal the-\norem. It should be noted that although \u0014mis generally\ngiven by\u0014m=L22\u0000L21L12\nL11, our de\fnition \u0014m=L22is\nsu\u000ecient to describe the thermal magnon transport at\nlow temperatures at which the magnon picture is valid\nbecause the L22gives the leading temperature depen-\ndence. Since a magnon chemical potential is zero in equi-\nlibrium, jQ=jE, where jEis a magnon energy current\ndensity. Hereafter we focus on the magnon transport\nwithESor (\u0000rT=T) applied along the xaxis.\nWe express L\u0016\u0011's in terms of the correlation functions\nusing the linear-response theory23,49{54. First,L12is\ngiven by\nL12= lim\n!!0\bR\n12(!)\u0000\bR\n12(0)\ni!; (8)\nwhere \bR\n12(!) = \b 12(i\nn!!+i\u000e) (\u000e= 0+),\n\b12(i\nn) =ZT\u00001\n0d\u001cei\nn\u001c1\nNhT\u001cJx\nS(\u001c)Jx\nEi; (9)\nand \nn= 2\u0019Tn (n > 0). HereT\u001cis the time-ordering\noperator51, andJx\nSandJx\nEare spin and energy, respec-\ntively, current operators. They are obtained from the\ncontinuity equations55{57(see Appendix A); the results\nare\nJx\nS=\u0000X\nqX\nl;l0=A;Bvx\nll0(q)xy\nqlxql0; (10)\nJx\nE=X\nqX\nl;l0=A;Bex\nll0(q)xy\nqlxql0; (11)\nwherevx\nll0(q) = (1\u0000\u000el;l0)@\u000fAB(q)\n@qx,xqA=aq,xqB=by\nq,\nex\nBB(q) =\u0000ex\nAA(q) =\u000fAB(q)@\u000fAB(q)\n@qx, andex\nAB(q) =\nex\nBA(q) =1\n2(\u000fAA\u0000\u000fBB)@\u000fAB(q)\n@qx. In deriving Eqs. (10)\nand (11), we have omitted the corrections due to Hint\nbecause they may be negligible23. Then we can obtain\nL11by replacing Jx\nEin \b 12(i\nn) byJx\nS, andL22by re-\nplacingJx\nS(\u001c) in \b 12(i\nn) byJx\nE(\u001c). Thus the derivation\nofL12is enough in obtaining L\u0016\u0017's. In addition, since\nwe can derive L12in a similar way to the derivations of\nelectron transport coe\u000ecients23,33,50,54,58, we explain its\nmain points below. (Note that the Bose-Einstein con-\ndensation of magnons is absent in our situation.)\nBy substituting Eqs. (10) and (11) into Eq. (9) and\nperforming some calculations (for the details see Ap-\npendix B), we obtain\nL12=L0\n12+L0\n12: (12)\nFirst,L0\n12, the noninteracting L12, is given by (see Ap-\npendix B)\nL0\n12=1\n\u0019NX\nqX\n\u0017;\u00170=\u000b;\fvx\n\u00170\u0017(q)ex\n\u0017\u00170(q)I(I)\n\u0017\u00170(q); (13)3\nwherevx\n\u00170\u0017(q) =P\nl;l0=A;Bvx\nll0(q)(Uq)l\u00170(Uq)l0\u0017,\nex\n\u0017\u00170(q) =P\nl;l0=A;Bex\nll0(q)(Uq)l\u0017(Uq)l0\u00170, and\nI(I)\n\u0017\u00170(q) =Z1\n\u00001dz@n(z)\n@zImGR\n\u0017(q;z)ImGR\n\u00170(q;z):(14)\nHeren(z) = (ez=T\u00001)\u00001,GR\n\u000b(q;z) = [z\u0000\u000f\u000b(q) +i\r]\u00001,\nGR\n\f(q;z) =\u0000[z+\u000f\f(q) +i\r]\u00001, and\ris the magnon\ndamping. Next, L0\n12, the leading correction due to the\n\frst-order perturbation of Hint, is given by (see Appendix\nB)\nL0\n12=1\n\u00192N2X\nq;q0X\n\u00171\u00172;\u00173;\u00174vx\n\u00171\u00172(q)ex\n\u00173\u00174(q0)V\u00171\u00172\u00173\u00174(q;q0)\n\u0002[I(I)\n\u00171\u00172(q)I(II)\n\u00173\u00174(q0) +I(II)\n\u00171\u00172(q)I(I)\n\u00173\u00174(q0)]; (15)\nwhere\nI(II)\n\u0017\u00170(q) =Z1\n\u00001dzn(z)Im[GR\n\u0017(q;z)GR\n\u00170(q;z)];(16)\nV\u00171\u00172\u00173\u00174(q;q0) = 4Jq\u0000q0P\nl(Uq)l\u00171(Uq)\u0016l\u00172(Uq0)\u0016l\u00173(Uq0)l\u00174,\nand\u0016lisBorAforl=AorB, respectively. Then we\nobtain\nL11=L0\n11+L0\n11; L22=L0\n22+L0\n22; (17)\nwhereL0\n11,L0\n11,L0\n22, andL0\n22are obtained by replacing\nex\n\u0017\u00170(q) in Eq. (13) by\u0000vx\n\u0017\u00170(q),ex\n\u00173\u00174(q0) in Eq. (15) by\n\u0000vx\n\u00173\u00174(q0),vx\n\u00170\u0017(q) in Eq. (13) by\u0000ex\n\u00170\u0017(q), andvx\n\u00171\u00172(q)\nin Eq. (15) by\u0000ex\n\u00171\u00172(q), respectively.\nSince we suppose that the magnon lifetime \u001c= (2\r)\u00001\nis long enough to regard magnons as quasiparticles, we\nrewrite Eqs. (13) and (15) by taking the limit \u001c!1 .\nFirst, Eq. (13) reduces to\nL0\n12\u0018L0\n12\u000b+L0\n12\f; (18)\nwhere\nL0\n12\u0017\u00181\nNX\nqvx\n\u0017\u0017(q)ex\n\u0017\u0017(q)\u001c@n[\u000f\u0017(q)]\n@\u000f\u0017(q): (19)\n(The detailed derivation is described in Appendix C.)\nThis expression is consistent with that obtained in the\nBoltzmann theory with the relaxation-time approxima-\ntion59. Equation (18) shows that L0\n12\u0019L0\n12\u000bat\nsu\u000eciently low temperatures for SA> SBowing to\n@n[\u000f\u000b(q)]\n@\u000f\u000b(q)\u001d@n[\u000f\f(q)]\n@\u000f\f(q). Similarly, we obtain\nL0\n11\u0018L0\n11\u000b+L0\n11\f; L0\n22\u0018L0\n22\u000b+L0\n22\f; (20)\nwhereL0\n11\u0017andL0\n22\u0017are obtained by replacing ex\n\u0017\u0017(q) in\nEq. (19) by\u0000vx\n\u0017\u0017(q) and by replacing vx\n\u0017\u0017(q) by\u0000ex\n\u0017\u0017(q),\nrespectively. Then, as we show in Appendix C, Eq. (15)\nreduces to\nL0\n12\u0018L0\n12-intra +L0\n12-inter1 +L0\n12-inter2; (21)whereL0\n12-intra is the correction due to the intraband in-\nteractions,\nL0\n12-intra =L0\n12-intra-\u000b+L0\n12-intra-\f; (22)\nL0\n12-intra-\u0017=\u00002\nN2X\nq;q0vx\n\u0017\u0017(q)ex\n\u0017\u0017(q0)\u001cV\u0017\u0017\u0017\u0017(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)@n[\u000f\u0017(q0)]\n@\u000f\u0017(q0); (23)\nandL0\n12-inter1 andL0\n12-inter2 are the corrections due to the\ninterband interactions,\nL0\n12-inter1 =\u00002\nN2X\nq;q0vx\n\u000b\u000b(q)ex\n\f\f(q0)\u001cV\u000b\u000b\f\f(q;q0)\n\u0002@n[\u000f\u000b(q)]\n@\u000f\u000b(q)@n[\u000f\f(q0)]\n@\u000f\f(q0)\n\u00002\nN2X\nq;q0vx\n\f\f(q)ex\n\u000b\u000b(q0)\u001cV\f\f\u000b\u000b(q;q0)\n\u0002@n[\u000f\f(q)]\n@\u000f\f(q)@n[\u000f\u000b(q0)]\n@\u000f\u000b(q0); (24)\nL0\n12-inter2 =L0\n12-inter2-\u000b+L0\n12-inter2-\f\n= (L0\nE\u000b+L0\nS\u000b) + (L0\nE\f+L0\nS\f); (25)\nL0\nE\u0017=2\nN2X\nq;q0vx\n\u0017\u0017(q)ex\n\u000b\f(q0)\u001cV\u0017\u0017\u000b\f(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)n[\u000f\u000b(q0)]\u0000n[\u0000\u000f\f(q0)]\n\u000f\u000b(q0) +\u000f\f(q0);(26)\nL0\nS\u0017=2\nN2X\nq;q0vx\n\u000b\f(q)ex\n\u0017\u0017(q0)\u001cV\u000b\f\u0017\u0017(q;q0)\n\u0002n[\u000f\u000b(q)]\u0000n[\u0000\u000f\f(q)]\n\u000f\u000b(q) +\u000f\f(q)@n[\u000f\u0017(q0)]\n@\u000f\u0017(q0):(27)\nHere theV\u00171\u00172\u00173\u00174(q;q0)'s are given by\nV\u0017\u0017\u0017\u0017(q;q0) =V\u000b\u000b\f\f(q;q0) =V\f\f\u000b\u000b(q;q0)\n= 2Jq\u0000q0sinh 2\u0012qsinh 2\u0012q0; (28)\nV\u0017\u0017\u000b\f(q;q0) =V\u000b\f\u0017\u0017(q0;q)\n=\u00002Jq\u0000q0sinh 2\u0012qcosh 2\u0012q0: (29)\nEquation (24) shows that the interband components\nof the magnon-magnon interactions cause the energy-\ncurrent-drag correction and the spin-current-drag cor-\nrection, which are, in the case for SA> SB, the \frst\nand the second term, respectively, of Eq. (24). Further-\nmore, Eqs. (26) and (27) show that other interband com-\nponents cause the energy-current-drag corrections L0\nE\u0017's\nand the spin-current-drag corrections L0\nS\u0017's. Since these\ninterband components cause the interband momentum\ntransfer,L0\n12-inter1 andL0\n12-inter2 are the corrections due\nto the interband magnon drag. The similar corrections\nare obtained for L0\n11andL0\n22:\nL0\n11\u0018L0\n11-intra +L0\n11-inter1 +L0\n11-inter2; (30)\nL0\n22\u0018L0\n22-intra +L0\n22-inter1 +L0\n22-inter2; (31)4\nwhereL0\n11-intra andL0\n22-intra are the corrections due to\nthe intraband interactions,\nL0\n11-intra =L0\n11-intra-\u000b+L0\n11-intra-\f; (32)\nL0\n11-intra-\u0017=2\nN2X\nq;q0vx\n\u0017\u0017(q)vx\n\u0017\u0017(q0)\u001cV\u0017\u0017\u0017\u0017(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)@n[\u000f\u0017(q0)]\n@\u000f\u0017(q0); (33)\nL0\n22-intra =L0\n22-intra-\u000b+L0\n22-intra-\f; (34)\nL0\n22-intra-\u0017=2\nN2X\nq;q0ex\n\u0017\u0017(q)ex\n\u0017\u0017(q0)\u001cV\u0017\u0017\u0017\u0017(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)@n[\u000f\u0017(q0)]\n@\u000f\u0017(q0); (35)\nandL0\n11-inter1 ,L0\n11-inter2 ,L0\n22-inter1 , andL0\n22-inter2 are the\ncorrections due to the interband interactions,\nL0\n11-inter1 =4\nN2X\nq;q0vx\n\u000b\u000b(q)vx\n\f\f(q0)\u001cV\u000b\u000b\f\f(q;q0)\n\u0002@n[\u000f\u000b(q)]\n@\u000f\u000b(q)@n[\u000f\f(q0)]\n@\u000f\f(q0); (36)\nL0\n11-inter2 =L0\n11-inter2-\u000b+L0\n11-inter2-\f; (37)\nL0\n11-inter2-\u0017=\u00004\nN2X\nq;q0vx\n\u0017\u0017(q)vx\n\u000b\f(q0)\u001cV\u0017\u0017\u000b\f(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)n[\u000f\u000b(q0)]\u0000n[\u0000\u000f\f(q0)]\n\u000f\u000b(q0) +\u000f\f(q0);(38)\nL0\n22-inter1 =4\nN2X\nq;q0ex\n\u000b\u000b(q)ex\n\f\f(q0)\u001cV\u000b\u000b\f\f(q;q0)\n\u0002@n[\u000f\u000b(q)]\n@\u000f\u000b(q)@n[\u000f\f(q0)]\n@\u000f\f(q0); (39)\nL0\n22-inter2 =L0\n22-inter2-\u000b+L0\n22-inter2-\f; (40)\nL0\n22-inter2-\u0017=\u00004\nN2X\nq;q0ex\n\u0017\u0017(q)ex\n\u000b\f(q0)\u001cV\u0017\u0017\u000b\f(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)n[\u000f\u000b(q0)]\u0000n[\u0000\u000f\f(q0)]\n\u000f\u000b(q0) +\u000f\f(q0):(41)\nAs well as L0\n12-inter1 andL0\n12-inter2 ,L0\n11-inter1 ,L0\n11-inter2 ,\nL0\n22-inter1 , andL0\n22-inter2 are the interband magnon drag\ncorrections.\nIV. NUMERICAL RESULTS\nWe numerically evaluate Sm,\u001bm, and\u0014m. We setJ=\n1,h= 0:02J, and (SA;SB) = (3\n2;1).SA:SB= 3 : 2 is\nconsistent with a ratio of FeTto FeOsites in the unit cell\nof YIG41. The reason why ( SA;SB) = (3\n2;1) is considered\nis that the transition temperature derived in a mean-\feld\napproximation in this case with J= 3 meV at h= 0\n[i.e.,Tc= (16=3)JSA(SB+ 1)\u0018557 K] is close to the\nCurie temperature of YIG, TC. To perform the momen-\ntum summations numerically, we divide the \frst Brillouinzone into a Nq-point mesh and set Nq= 243(=N=2) (for\nmore details, see Appendix D). The temperature range\nis chosen to be 0 < T\u001410J(\u00180:6Tc) because a previ-\nous study60showed that the magnon theory in which the\nmagnon-magnon interactions are considered in the \frst-\norder perturbation theory can reproduce the perpendicu-\nlar spin susceptibility of MnF 2up to about 0 :6TN, where\nTNis the N\u0013 eel temperature. For simplicity, we deter-\nmine\u001cby\u001c\u00001=\r0+\r1T+\r2T2, where\r0= 10\u00002J,\n\r1= 10\u00004, and\r2= 10\u00003. (The results shown below\nremain qualitatively unchanged at h= 0:08Jand 0:16J,\nas shown in Appendix E.)\nWe begin with the temperature dependence of Sm.\nFigure 2(a) shows that in the range of 0 < T\u00142J\nL12\u0019L0\n12\u000bholds, whereas for T\u00153Jthe contribu-\ntion fromL0\n12\fis non-negligible. For example, at T= 6J\nwe haveL0\n12=L0\n12\u000b\u00180:7. This result indicates that the\nhigher-energy band magnons contribute to Smeven for\nT < [\u000f\f(q)\u0000\u000f\u000b(q)] = 7:96J. This may be surprising\nbecause their contributions are believed to be negligi-\nble at such temperatures. Then, Fig. 2(a) shows that\nthe magnitude of Smis enhanced by the intraband cor-\nrectionL0\n12-intra [=L(a)\n12\u0000L0\n12], whereas it is reduced by\nthe interband corrections L0\n12-inter2 [=L(b)\n12\u0000L(a)\n12] and\nL0\n12-inter1 [=L0\n12+L0\n12\u0000L(b)\n12] (Table I). Among these cor-\nrections,L0\n12-intra gives the largest contribution. (As we\nwill see below, this contrasts with the result of L11orL22,\nfor which the largest contribution comes from L0\n11-inter2\norL0\n22-inter2 , respectively.) The reason why the inter-\nband magnon drag corrections L0\n12-inter2 andL0\n12-inter1 are\nsmall is that the energy-current-drag contributions and\nspin-current-drag contributions [e.g., L0\nE\u000bandL0\nS\u000bin Eq.\n(25)] are opposite in sign and are nearly canceled out.\nFigure 2(a) also shows L0\n12+L0\n12\u0019L0\n12. These results\nsuggest that the total e\u000bects of the interband magnon\ndrag onSmare small.\nWe turn to \u001bmand\u0014m. Their temperature depen-\ndences are shown in Figs. 2(b) and 2(c). First, we see the\n\f-band magnons contribute to L11forT\u00154Jand toL22\nforT\u00153J. This result is similar to that of L12and indi-\ncates that the multiband e\u000bects are signi\fcant also for \u001bm\nand\u0014m. The largest e\u000bects on L22are due to the prop-\nerty thatex\n\u0017\u0017(q) includes\u000f\u0017(q) [more precisely, ex\n\u000b\u000b(q) =\nvx\n\u000b\u000b(q)\u000f\u000b(q) andex\n\f\f(q) =\u0000vx\n\f\f(q)\u000f\f(q)]. Then, Figs.\n2(b) and 2(c) show that \u001bmis enhanced by L0\n11-intra ,\nL0\n11-inter2 , andL0\n11-inter1 , and that \u0014mis enhanced by\nL0\n22-intra and reduced by L0\n22-inter2 andL0\n22-inter1 (Table I).\n[Note thatL0\n\u0016\u0011-intra =L(a)\n\u0016\u0011\u0000L0\n\u0016\u0011,L0\n\u0016\u0011-inter2 =L(b)\n\u0016\u0011\u0000L(a)\n\u0016\u0011,\nandL0\n\u0016\u0011-inter1 =L0\n\u0016\u0011+L0\n\u0016\u0011\u0000L(b)\n\u0016\u0011.] In contrast to L0\n12, the\nlargest contributions to L0\n11andL0\n22come fromL0\n11-inter2\nandL0\n22-inter2 , respectively. Since L0\n11-inter2 ,L0\n11-inter1 ,\nL0\n22-inter2 , andL0\n22-inter1 are the interband magnon drag\ncorrections, the above results suggest that the interband\nmagnon drag enhances \u001bmand reduces \u0014m. This implies\nthat the interband magnon drag could be used to enhance\nthe spin current and to reduce the energy current. Since\nthis drag results from the interband momentum transfer5\n\tC\n \tB\n \tD\n−20−15−10−5 0\n 0 2 4 6 8 10h = 0.02JSm (arb. unit)\nT/JL0\n12 α\nL0\n12 \nL(a)\n12 \nL(b)\n12 \nL0\n12 +L’12 \n 0 1 2 3 4 5\n 0 2 4 6 8 10h = 0.02J\nσm (arb. unit)\nT/JL0\n11 α\nL0\n11 \nL(a)\n11 \nL(b)\n11 \nL0\n11 +L’11 \n 0 50 100 150 200 250\n 0 2 4 6 8 10h = 0.02Jκm (arb. unit)\nT/JL0\n22 α\nL0\n22 \nL(a)\n22 \nL(b)\n22 \nL0\n22 +L’22 \nFIG. 2. The temperature dependences of (a) Sm(=L12), (b)\u001bm(=L11), and (c) \u0014m(=L22) for (SA;SB) = (3\n2;1) at\nh= 0:02J.L(a)\n\u0016\u0011andL(b)\n\u0016\u0011are de\fned as L(a)\n\u0016\u0011=L0\n\u0016\u0011+L0\n\u0016\u0011-intra andL(b)\n\u0016\u0011=L0\n\u0016\u0011+L0\n\u0016\u0011-intra +L0\n\u0016\u0011-inter2 , respectively. Note that\nL0\n\u0016\u0011=L0\n\u0016\u0011\u000b+L0\n\u0016\u0011\fandL0\n\u0016\u0011=L0\n\u0016\u0011-intra +L0\n\u0016\u0011-inter1 +L0\n\u0016\u0011-inter2 . ForSm, theL0\n12\fis non-negligible for T\u00153Jand the largest\nterm of the drag terms is L0\n12-intra , which enhances jSmj. For\u001bm, theL0\n11\fis non-negligible for T\u00154Jand the largest term of\nthe drag terms is L0\n11-inter2 , which enhances \u001bm. For\u0014m, theL0\n22\fis non-negligible for T\u00153Jand the largest term of the drag\nterms isL0\n22-inter2 , which reduces \u0014m. The e\u000bects of the other drag terms are summarized in Table I.\nTABLE I. The e\u000bects of the drag terms on L12(=Sm),L11(=\u001bm), andL22(=\u0014m).jL12jis enhanced by L0\n12-intra and reduced\nbyL0\n12-inter1 andL0\n12-inter2 .L11is enhanced by L0\n11-intra ,L11-inter1 , andL0\n11-inter2 .L22is enhanced by L0\n22-intra and reduced by\nL0\n22-inter1 andL0\n22-inter2 .\nTransport coe\u000ecient Intra term Inter1 term Inter2 term\njL12j Enhanced Reduced Reduced\nL11 Enhanced Enhanced Enhanced\nL22 Enhanced Reduced Reduced\ninduced by the magnon-magnon interactions, its e\u000bects\ncould be controlled by changing the band splitting energy\nconsiderably via external \felds. (Such control is mean-\ningful if and only if the magnon picture remains valid.)\nNote that for ferrimagnetic insulators the e\u000bects of the\nweak magnetic \feld on the band splitting energy are neg-\nligible because this energy for h= 0 is of the order of J.\n(The actual analysis about the possibility of controlling\nthe interband magnon drag is a future problem.)\nV. DISCUSSIONS\nWe discuss the validity of our theory. Since Hint\ncould be treated as perturbation except near TC, we be-\nlieve our theory is appropriate for describing the magnon\ntransport for T < T C. It may be suitable to treat the\nmagnon-magnon interactions in the Holstein-Primako\u000b\nmethod because the unphysical processes that can ap-\npear in aS= 1=2 ferromagnet61are absent in our case.\nThen the e\u000bects of the magnon-phonon interactions may\nnot change the results qualitatively. First, since the\ninteraction-induced magnon polaron occurs only at sev-\neral values of h62, its e\u000bect can be avoided. Another e\u000bect\nis to cause the temperature dependence of \u001c63,64, and it\ncould be approximately considered as the temperature-\ndependent \u001c. Although the phonon-drag contributions\nmight change Sm21, experimental results59suggest thatsuch contributions are small or negligible.\nWe make a short comment about the relation between\nour theory and the Boltzmann theory. Our theory is\nbased on a method of Green's functions, which can de-\nscribe the e\u000bects of the damping and the vertex correc-\ntions appropriately. In principle, these e\u000bects can be\ndescribed also in the Boltzmann theory if the collision\nintegral is treated appropriately65. However, in many\nanalyses using the Boltzmann theory, the collision inte-\ngral is evaluated in the relaxation-time approximation,\nin which the vertex corrections are completely omitted.\nSince our interband magnon drag comes from the ver-\ntex corrections due to the \frst-order perturbation of the\nquartic terms, the similar result might be obtained also\nin the Boltzmann theory if the interband components of\nthe collision integral are treated appropriately.\nWe remark on the implications of our results. First,\nour interband magnon drag is distinct from a magnon\ndrag in metals. For the latter, magnons drag an elec-\ntron charge current via the second-order perturbation\nof asd-type exchange interaction25. Second, the inter-\nband magnon drag is possible in various ferrimagnetic\ninsulators and other magnetic systems, such as anti-\nferromagnets47,56,66and spiral magnets57,67. Note that\nthe possible ferrimagnetic insulators include not only\nYIG, but also some spinel ferrites, such as CoFe 2O4and\nNiFe 2O468,69. Third, our theory can be extended to\nphonons and photons. Thus it may be useful for study-6\ning transport phenomena of various interacting bosons.\nFourth, our results will stimulate further studies of YIG.\nFor example, the reduction in jSmjdue to the multiband\ne\u000bect could improve the di\u000berences between the voltages\nobserved in the spin-Seebeck e\u000bect and obtained in the\nBoltzmann theory of the ferromagnet59at high temper-\natures because the voltage is proportional to Sm.\nVI. CONCLUSION\nWe have studied Sm,\u001bm, and\u0014mof interacting\nmagnons in the minimal model of ferrimagnetic insu-\nlators. We derived them by using the linear-response\ntheory and treating the magnon-magnon interactions as\nperturbation. We showed that some interband compo-\nnents of the magnon-magnon interactions give the cor-\nrections to these transport coe\u000ecients. These correc-\ntions are due to the interband magnon drag, which is\ndistinct from the magnon drag in metals. Then we nu-\nmerically calculated the temperature dependences of Sm,\n\u001bm, and\u0014mfor (SA;SB) = (3\n2;1) andh= 0:02J. We\nshowed that the total e\u000bects of the interband magnon\ndrag onSmbecome small, whereas it enhances \u001bmand\nreduces\u0014m. The latter result may suggest that the in-\nterband magnon drag could be used to enhance the spin\ncurrent and reduce the energy current. For Sm, the in-\nterband corrections become small because they lead to\nthe energy-current-drag contributions and spin-current-\ndrag contributions, which are opposite in sign and are\nnearly canceled out. We also showed that the contribu-\ntions from the higher-energy band magnons to Sm,\u001bm,\nand\u0014mare non-negligible even for temperatures lower\nthan the band splitting. This result indicates the impor-\ntance of the multiband e\u000bects.\nACKNOWLEDGMENTS\nThis work was supported by JSPS KAKENHI Grants\nNo. JP19K14664 and JP22K03532. The author also ac-\nknowledges support from JST CREST Grant No. JP-\nMJCR1901.\nAppendix A: Derivations of Eqs. (10) and (11)\nWe explain the details of the derivations of Jx\nSand\nJx\nE, Eqs. (10) and (11). As described in the main text,\nthey are obtained from the continuity equations. Such a\nderivation is explained, for example, in Ref. 55.\nWe begin with the derivation of Jx\nS. (Note that the fol-\nlowing derivation, which is applicable to collinear mag-\nnets, can be extended to noncollinear magnets.) We sup-\npose that the zcomponent of a spin angular momentum,Sz\nm, satis\fes\ndSz\nm\ndt+r\u0001j(S)\nm= 0; (A1)\nwhere j(S)\nmis a spin current operator at site m. Using\nthis equation, we have\nd\ndt\u0010X\nmRmSz\nm\u0011\n=\u0000X\nmRmr\u0001j(S)\nm\n=X\nmj(S)\nm=J(S)\nl: (A2)\nHerelisAorBwhen the sumP\nmtakes over sites on\ntheAor theBsublattice, respectively. In deriving the\nsecond equal in Eq. (A2) we have omitted the surface\ncontributions. Jx\nSis given by the xcomponent of JS,\nwhere\nJS=J(S)\nA+J(S)\nB: (A3)\nCombining Eq. (A2) with the Heisenberg equation of\nmotion, we obtain\nJ(S)\nl=iX\nmRm\u0002\nH;Sz\nm\u0003\n; (A4)\nwhereHis the Hamiltonian of the system considered.\nThen, since we focus on the magnon system described\nbyH=HKE+Hint, whereHKEandHintare given in\nthe main text, and treat Hintas perturbation, we replace\nHin Eq. (A4) by HKEandSz\nmin Eq. (A4) either by\nSA\u0000ay\nmamforl=Aor by\u0000SB+by\nmbmforl=B; as a\nresult, we obtain\nJ(S)\nA=iX\nhi;jiX\nmRm\u0002\nh0\nij;SA\u0000ay\nmam\u0003\n; (A5)\nJ(S)\nB=iX\nhi;jiX\nmRm\u0002\nh0\nij;\u0000SB+by\nmbm\u0003\n; (A6)\nwhereHKE=P\nhi;jih0\nijandh0\nij= (2JSB+\u000ei;jh)ay\niai+\n(2JSA\u0000\u000ei;jh)by\njbj+2JpSASB(ay\niby\nj+aibj). Note that the\nreplacement of HbyHKEmay be suitable because the\ncorrections due to Hintare next-leading terms; and that\nthe replacement of Sz\nmbySA\u0000ay\nmamor by\u0000SB+by\nmbm\ncorresponds to the Holstein-Primako\u000b transformation of\nthe ferrimagnet. After some algebra, we can write Eqs.\n(A5) and (A6) as follows:\nJ(S)\nA=\u0000i2Jp\nSASBX\nhi;jiRi(aibj\u0000ay\niby\nj); (A7)\nJ(S)\nB=i2Jp\nSASBX\nhi;jiRj(aibj\u0000ay\niby\nj): (A8)\nCombining these equations with Eq. (A3), we have\nJS=\u0000i2Jp\nSASBX\nhi;ji(Ri\u0000Rj)(aibj\u0000ay\niby\nj):(A9)7\nThen, by using the Fourier coe\u000ecients of the magnon\noperators,\nai=r\n2\nNX\nqaqeiq\u0001Ri; by\nj=r\n2\nNX\nqby\nqeiq\u0001Rj;(A10)\nwe can rewrite Eq. (A9) as follows:\nJS=\u00002Jp\nSASBX\nq@Jq\n@q(aqbq+ay\nqby\nq)\n=\u0000X\nq@\u000fAB(q)\n@q(xy\nqBxqA+xy\nqAxqB); (A11)\nwhereJq=JPz\nj=1eiq\u0001(Ri\u0000Rj)= 8Jcosqx\n2cosqy\n2cosqz\n2,\n\u000fAB(q) = 2JpSASBJq,xqA=aq, andxqB=by\nq. Note\nthatzis the number of nearest-neighbor sites ( z= 8).\nThexcomponent of Eq. (A11) gives Eq. (10).\nIn a similar way we can obtain the expression of Jx\nE.\n(The following derivation is similar to that for an anti-\nferromagnet56.) First, we suppose that the Hamiltonian\nat sitem,hm, satis\fes\ndhm\ndt+r\u0001j(E)\nm= 0; (A12)\nwhere j(E)\nmis an energy current operator at site m. Be-\ncause of this relation, the energy current operator JEcan\nbe determined from\nJE=J(E)\nA+J(E)\nB; (A13)\nwhere J(E)\nlis given by\nJ(E)\nl=iX\nm;nRn\u0002\nhm;hn\u0003\n; (A14)\nthe sumP\nmtake over sites on the Aor theBsublattice,\nand the sumP\nntake over sites on sublattice l. Then, to\ncalculate the commutator in Eq. (A14), we consider the\ncontributions only from HKEand neglect the corrections\ndue toHint, as in the derivation of J(S)\nl. As a result, hm\nform2Ais given by\nh0\nmA= (2SBzJ+h)ay\nmam+p\nSASBX\njJmj(ambj+ay\nmby\nj);\n(A15)\nand that for m2Bis given by\nh0\nmB= (2SAzJ\u0000h)by\nmbm+p\nSASBX\niJim(aibm+ay\niby\nm):\n(A16)\nHerem2AorBmeans that mis on theAorBsublat-\ntice, respectively, and Jij=Jji=Jfor nearest-neighbor\nsitesiandj. Note thatPN=2\ni=1h0\niA+PN=2\nj=1h0\njB=HKE. In\nour de\fnition, the energy current operator includes theconribution from the Zeeman energy [see Eq. (A14){\n(A16)]. Combining Eqs. (A15) and (A16) with Eqs.\n(A13) and (A14), we have\nJE=iX\nm;nRn\u0002\nh0\nmA;h0\nnA\u0003\n+iX\nm;nRn\u0002\nh0\nmB;h0\nnB\u0003\n+iX\nm;nRn\u0002\nh0\nmA;h0\nnB\u0003\n+iX\nm;nRn\u0002\nh0\nmB;h0\nnA\u0003\n:(A17)\nThen we can calculate the commutators in Eq. (A17) by\nusing the commutation relations of the magnon opera-\ntors and the identities [ AB;C ] =A[B;C] + [A;C]Band\n[A;BC ] = [A;B]C+B[A;C]; the results are\n\u0002\nh0\nmA;h0\nnA\u0003\n=SASBX\njJmjJnj(ay\nnam\u0000ay\nman);(A18)\n\u0002\nh0\nmB;h0\nnB\u0003\n=SASBX\niJimJin(bmby\nn\u0000bnby\nm);(A19)\n\u0002\nh0\nmA;h0\nnB\u0003\n=SASBX\njJmnJmj(bjby\nn\u0000bnby\nj)\n+ [2Jz(SA\u0000SB)\u00002h]p\nSASBJmn\n\u0002(ambn\u0000ay\nmby\nn)\n+SASBX\niJmnJni(ay\niam\u0000ay\nmai);(A20)\n\u0002\nh0\nmB;h0\nnA\u0003\n=SASBX\njJmnJnj(bmby\nj\u0000bjby\nm)\n+ [\u00002Jz(SA\u0000SB) + 2h]p\nSASBJnm\n\u0002(anbm\u0000ay\nnby\nm)\n+SASBX\niJnmJmi(ay\nnai\u0000ay\nian):(A21)\nBy substituting these equations into Eq. (A17) and per-\nforming some calculations, we obtain\nJE= 2iX\nm;n;j(Rn\u0000Rm)SASBJnjJjmay\nnam\n\u00002iX\nm;n;i(Rn\u0000Rm)SASBJniJimbnby\nm\n+iX\nm;n(Rn\u0000Rm)[2Jz(SA\u0000SB)\u00002h]p\nSASB\n\u0002Jmn(ambn\u0000ay\nmby\nn): (A22)\nAs in the derivation of JS, we can rewrite Eq. (A22)\nby using the Fourier coe\u000ecients of the magnon operators\n[Eq. (A10)]; as a result, we have\nJE=\u0000X\nq2p\nSASBJq2p\nSASB@Jq\n@q(ay\nqaq\u0000bqby\nq)\n\u0000[J0(SA\u0000SB)\u0000h]2p\nSASB@Jq\n@q(aqbq+ay\nqby\nq):\n(A23)8\nSince\u000fAA= 2J0SB+h,\u000fBB= 2J0SA\u0000h, and\u000fAB(q) =\n2pSASBJq, we can write Eq. (A23) as follows:\nJE=\u0000X\nq\u000fAB(q)@\u000fAB(q)\n@q(ay\nqaq\u0000bqby\nq)\n+X\nq1\n2(\u000fAA\u0000\u000fBB)@\u000fAB(q)\n@q(aqbq+ay\nqby\nq)\n=X\nqX\nl;l0=A;Bell0(q)xy\nqlxql0; (A24)\nwhere eAA(q) =\u0000eBB(q) =\u0000\u000fAB(q)@\u000fAB(q)\n@qand\neAB(q) =eBA(q) =1\n2(\u000fAA\u0000\u000fBB)@\u000fAB(q)\n@q. Equation\n(A24) for the xcomponent is Eq. (11).\nAppendix B: Derivations of Eqs. (13) and (15)\nWe derive Eqs. (13) and (15). As described in the\nmain text, their derivations can be done in a simi-\nlar way to the derivations of electron transport coe\u000e-\ncients23,50,54,58: the transport coe\u000ecients can be derived\nby using a method of Green's functions51. We \frst de-\nriveL0\n12, the noninteracting L12, and then derive L0\n12,\nthe leading correction to L12due to the \frst-order per-\nturbation of Hint.\nFirst, we derive L0\n12, Eq. (13). Substituting Eqs. (10)\nand (11) into Eq. (9), we have\n\b12(i\nn) =\u00001\nNX\nq;q0X\nl1;l2;l3;l4=A;Bvx\nl1l2(q)ex\nl3l4(q0)\n\u0002ZT\u00001\n0d\u001cei\nn\u001chT\u001cxy\nql1(\u001c)xql2(\u001c)xy\nq0l3xq0l4i\n=\u00001\nNX\nq;q0X\nl1;l2;l3;l4=A;Bvx\nl1l2(q)ex\nl3l4(q0)\n\u0002G(II)\nl1l2l3l4(q;q0;i\nn); (B1)\nwhere \nn= 2\u0019Tn withn > 0. (Note that the nand\nmused in this section are di\u000berent from those used in\nAppendix A.) Equation (B1) provides a starting point\nto deriveL0\n12andL0\n12. To derive L0\n12, we calculate\nG(II)\nl1l2l3l4(q;q0;i\nn) in the absence of Hintby using Wick's\ntheorem51; the result is\nG(II)\nl1l2l3l4(q;q0;i\nn) =\u000eq;q0TX\nmGl2l3(q;i\nn+i\nm)\n\u0002Gl4l1(q;i\nm); (B2)\nwhereGll0(q;i\nm) is the magnon Green's function in thesublattice basis with \n m= 2\u0019Tm and an integer m,\nGll0(q;i\nm) =\u0000ZT\u00001\n0d\u001cei\nm\u001chT\u001cxql(\u001c)xy\nql0i:(B3)\nThen the magnon operators in the sublattice basis, xql\nandxy\nql, are connected with those in the band basis, xq\u0017\nandxy\nq\u0017, through the Bogoliubov transformation,\nxql=X\n\u0017=\u000b;\f(Uq)l\u0017xq\u0017; (B4)\nwherexq\u000b=\u000bq,xq\f=\fy\nq, (Uq)A\u000b= (Uq)B\f= cosh\u0012q,\nand (Uq)A\f= (Uq)B\u000b=\u0000sinh\u0012q; as described in the\nmain text, these hyperbolic functions satisfy cosh 2 \u0012q=\nJ0(SA+SB)\n\u0001\u000fqand sinh 2\u0012q=2pSASBJq\n\u0001\u000fq. ThusGll0(q;i\nm)\nis related to the magnon Green's function in the band\nbasis,G\u0017(q;i\nm):\nGll0(q;i\nm) =X\n\u0017=\u000b;\f(Uq)l\u0017(Uq)l0\u0017G\u0017(q;i\nm);(B5)\nwhere\nG\u000b(q;i\nm) =1\ni\nm\u0000\u000f\u000b(q); G\f(q;i\nm) =\u00001\ni\nm+\u000f\f(q):\n(B6)\nCombining Eq. (B5) with Eqs. (B2) and (B1), we have\n\b12(i\nn) =\u00001\nNX\nqX\n\u0017;\u00170=\u000b;\fvx\n\u00170\u0017(q)ex\n\u0017\u00170(q)\n\u0002TX\nmG\u0017(q;i\nn+m)G\u00170(q;i\nm)\n=\u00001\nNX\nqX\n\u0017;\u00170=\u000b;\fvx\n\u00170\u0017(q)ex\n\u0017\u00170(q)G(II)\n\u0017\u00170(q;i\nn);\n(B7)\nwhere\nvx\n\u00170\u0017(q) =X\nl1;l2=A;Bvx\nl1l2(q)(Uq)l1\u00170(Uq)l2\u0017; (B8)\nex\n\u0017\u00170(q) =X\nl3;l4=A;Bex\nl3l4(q)(Uq)l3\u0017(Uq)l4\u00170: (B9)\nThen we can rewrite G(II)\n\u0017\u00170(q;i\nn) in Eq. (B7) as follows:\nG(II)\n\u0017\u00170(q;i\nn) =Z\nCdz\n2\u0019in(z)G\u0017(q;i\nn+z)G\u00170(q;z)\n+T[G\u0017(q;i\nn)G\u00170(q;0) +G\u0017(q;0)G\u00170(q;\u0000i\nn)];\n(B10)\nwheren(z) is the Bose distribution function, n(z) =\n(ez=T\u00001)\u00001, and C is one of the contours shown in Fig.\n3. Using Eqs. (B10) and (B6), we obtain9\n\tB\n \tC\n \tD\n \tE\nFIG. 3. The contours used for the integrations in (a) G(II)\n\u000b\u000b(q;i\nn), (b)G(II)\n\u000b\f(q;i\nn), (c)G(II)\n\f\u000b(q;i\nn), and (d)G(II)\n\f\f(q;i\nn).\nThe horizontal dashed lines correspond to Im z=\u0000\nn.\nG(II)\n\u0017\u00170(q;i\nn) =Z1\n\u00001dz\n2\u0019in(z)n\nGR\n\u0017(q;z+i\nn)[GR\n\u00170(q;z)\u0000GA\n\u00170(q;z)] + [GR\n\u0017(q;z)\u0000GA\n\u0017(q;z)]GA\n\u00170(q;z\u0000i\nn)o\n;(B11)\nwhereGR\n\u0017(q;z) is the retarded magnon Green's function,\nGR\n\u000b(q;z) =1\nz\u0000\u000f\u000b(q) +i\r; GR\n\f(q;z) =\u00001\nz+\u000f\f(q) +i\r; (B12)\nGA\n\u0017(q;z) is the advanced one, and \ris the magnon damping. By combining Eq. (B11) with Eq. (B7) and performing\nthe analytic continuation i\nn!!+i\u000ewith\u000e= 0+, we have\n\bR\n12(!) = \b 12(i\nn!!+i\u000e) =\u00001\nNX\nqX\n\u0017;\u00170=\u000b;\fvx\n\u00170\u0017(q)ex\n\u0017\u00170(q)Z1\n\u00001dz\n2\u0019in(z)\n\u0002n\nGR\n\u0017(q;z+!)[GR\n\u00170(q;z)\u0000GA\n\u00170(q;z)] + [GR\n\u0017(q;z)\u0000GA\n\u0017(q;z)]GA\n\u00170(q;z\u0000!)o\n:\n(B13)\nBy usingG(z+!) =G(z) +!@G(z)\n@z+O(!2) and performing the partial integration, we obtain\nL0\n12= lim\n!!0\bR\n12(!)\u0000\bR\n12(0)\ni!\n=\u00001\n4NX\nqX\n\u0017;\u00170=\u000b;\fvx\n\u00170\u0017(q)ex\n\u0017\u00170(q)Z1\n\u00001dz\n\u0019@n(z)\n@zh\nGR\n\u0017(q;z)GR\n\u00170(q;z)\u00002GR\n\u0017(q;z)GA\n\u00170(q;z) +GA\n\u0017(q;z)GA\n\u00170(q;z)i\n=1\nNX\nqX\n\u0017;\u00170=\u000b;\fvx\n\u00170\u0017(q)ex\n\u0017\u00170(q)Z1\n\u00001dz\n\u0019@n(z)\n@zImGR\n\u0017(q;z)ImGR\n\u00170(q;z): (B14)\nIn deriving this equation we have used the symmetry relations vx\n\u00170\u0017(q) =vx\n\u0017\u00170(q) andex\n\u0017\u00170(q) =ex\n\u00170\u0017(q). Equation\n(B14) is Eq. (13).\nNext, we derive L0\n12, Eq. (15). By using Eq. (B1), we can write the correction due to the \frst-order perturbation\nofHintas follows:\n\u0001\b12(i\nn) = +1\nNX\nq;q0X\nl1;l2;l3;l4=A;Bvx\nl1l2(q)ex\nl3l4(q0)ZT\u00001\n0d\u001cei\nn\u001cZT\u00001\n0d\u001c1hT\u001cxy\nql1(\u001c)xql2(\u001c)xy\nq0l3xq0l4Hint(\u001c1)i:(B15)\n[Note that Hinthas been de\fned in Eq. (4).] By using Wick's theorem51, we can calculate\nhT\u001cxy\nql1(\u001c)xql2(\u001c)xy\nq0l3xq0l4Hint(\u001c1)i; the result is\nhT\u001cxy\nql1(\u001c)xql2(\u001c)xy\nq0l3xq0l4Hint(\u001c1)i=\u00001\nNX\nl5;l6;l7;l8=A;BVl5l6l7l8(q;q0)Gl5l1(q;\u001c1\u0000\u001c)Gl2l6(q;\u001c\u0000\u001c1)\n\u0002Gl7l3(q0;\u001c1)Gl4l8(q0;\u0000\u001c1); (B16)10\nwhereGll0(q;\u001c) =TP\nme\u0000i\nm\u001cGll0(q;i\nm),\nVl5l6l7l8(q;q0) =8\n>>>>>>>>>>><\n>>>>>>>>>>>:4J0 (l5=l6=l;l7=l8=\u0016l);\n4Jq\u0000q0 (l5=l8=l;l6=l7=\u0016l);\n2Jq0q\nSA\nSB(l5=l6=B;l7=l;l8=\u0016l);\n2Jqq\nSA\nSB(l5=l;l6=\u0016l;l7=l8=B);\n2Jq0q\nSB\nSA(l5=l6=A;l7=l;l8=\u0016l);\n2Jqq\nSB\nSA(l5=l;l6=\u0016l;l7=l8=A);(B17)\nand\u0016l=BorAforl=AorB, respectively. Then, by substituting Eq. (B16) into Eq. (B15) and carrying out the\nintegrations, we obtain\n\u0001\b12(i\nn) =\u00001\nN2X\nq;q0X\nl1;l2;\u0001\u0001\u0001;l8=A;Bvx\nl1l2(q)ex\nl3l4(q0)Vl5l6l7l8(q;q0)T2X\nm;m0Gl5l1(q;i\nm)Gl2l6(q;i\nn+m)\n\u0002Gl7l3(q0;i\nn+m0)Gl4l8(q0;i\nm0): (B18)\nFurthermore, we can rewrite this equation by using the Bogoliubov transformation [i.e., Eq. (B4)]; the result is\n\u0001\b12(i\nn) =\u00001\nN2X\nq;q0X\n\u00171;\u00172;\u00173;\u00174=\u000b;\fvx\n\u00171\u00172(q)ex\n\u00173\u00174(q0)V\u00171\u00172\u00173\u00174(q;q0)\u0001G(II)\n\u00171\u00172\u00173\u00174(q;q0;i\nn); (B19)\nwhere\nV\u00171\u00172\u00173\u00174(q;q0) =X\nl5;l6;l7;l8=A;BVl5l6l7l8(q;q0)(Uq)l5\u00171(Uq)l6\u00172(Uq0)l7\u00173(Uq0)l8\u00174; (B20)\n\u0001G(II)\n\u00171\u00172\u00173\u00174(q;q0;i\nn) =T2X\nm;m0G\u00171(q;i\nm)G\u00172(q;i\nn+m)G\u00173(q0;i\nn+m0)G\u00174(q0;i\nm0): (B21)\nSincevx\n\u00171\u00172(q) andex\n\u00173\u00174(q0) are odd functions in term of qxandq0\nx, respectively, and G\u0017(q;i\nm)'s are even functions,\nthe \fnite terms of V\u00171\u00172\u00173\u00174(q;q0) in Eq. (B19), i.e., the terms which are \fnite even after carrying outP\nq;q0, come\nonly fromVABBA (q;q0) =VBAAB (q;q0) = 4Jq\u0000q0[Eq. (B17)]; because of this property, we can replace Eq. (B20) by\nV\u00171\u00172\u00173\u00174(q;q0) =X\nl=A;B4Jq\u0000q0(Uq)l\u00171(Uq)\u0016l\u00172(Uq0)\u0016l\u00173(Uq0)l\u00174: (B22)\nThen, as in G(II)\n\u0017\u00170(q;i\nn) [Eq. (B10)], we can replace the sums in Eq. (B21) by the corresponding integrals:\n\u0001G(II)\n\u00171\u00172\u00173\u00174(q;q0;i\nn) =hZ\nCdz\n2\u0019in(z)G\u00171(q;z)G\u00172(q;z+i\nn) +AihZ\nC0dz0\n2\u0019in(z0)G\u00173(q0;z0+i\nn)G\u00174(q0;z0) +A0i\n=G(II)\n\u00172\u00171(q;i\nn)G(II)\n\u00173\u00174(q0;i\nn); (B23)\nwhereA=T[G\u00171(q;0)G\u00172(q;i\nn)+G\u00171(q;\u0000i\nn)G\u00172(q;0)],A0=T[G\u00173(q0;i\nn)G\u00174(q0;0)+G\u00173(q0;0)G\u00174(q0;\u0000i\nn)],\nandCorC0is one of the contours shown in Fig. 3. By substituting Eq. (B11) into Eq. (B23) and performing the\nanalytic continuation i\nn!!+i\u000e(\u000e= 0+), we have\n\u0001\bR\n12(!) = \u0001\b 12(i\nn!!+i\u000e)\n=\u00001\nN2X\nq;q0X\n\u00171;\u00172;\u00173;\u00174=\u000b;\fvx\n\u00171\u00172(q)ex\n\u00173\u00174(q0)V\u00171\u00172\u00173\u00174(q;q0)\n\u0002Z1\n\u00001dz\n2\u0019in(z)n\n[GR\n\u00171(q;z)\u0000GA\n\u00171(q;z)]GR\n\u00172(q;z+!) +GA\n\u00171(q;z\u0000!)[GR\n\u00172(q;z)\u0000GA\n\u00172(q;z)]o\n\u0002Z1\n\u00001dz0\n2\u0019in(z0)n\nGR\n\u00173(q0;z0+!)[GR\n\u00174(q0;z0)\u0000GA\n\u00174(q0;z0)] + [GR\n\u00173(q0;z0)\u0000GA\n\u00173(q0;z0)]GA\n\u00174(q0;z0\u0000!)o\n:(B24)11\nThen, by performing the calculations similar to the derivation of Eq. (B14), we obtain\nL0\n12= lim\n!!0\u0001\bR\n12(!)\u0000\u0001\bR\n12(0)\ni!\n=1\n4\u00192iN2X\nq;q0X\n\u00171;\u00172;\u00173;\u00174=\u000b;\fvx\n\u00171\u00172(q)ex\n\u00173\u00174(q0)V\u00171\u00172\u00173\u00174(q;q0)h\nF(I)\n\u00171\u00172(q)F(II)\n\u00173\u00174(q0) +F(II)\n\u00171\u00172(q)F(I)\n\u00173\u00174(q0)i\n;(B25)\nwhere\nF(I)\n\u0017\u00170(q) =\u00001\n2Z1\n\u00001dz@n(z)\n@zh\nGR\n\u0017(q;z)GR\n\u00170(q;z) +GA\n\u0017(q;z)GA\n\u00170(q;z)\u00002GA\n\u0017(q;z)GR\n\u00170(q;z)i\n= 2Z1\n\u00001dz@n(z)\n@zImGR\n\u0017(q;z)ImGR\n\u00170(q;z); (B26)\nF(II)\n\u0017\u00170(q0) =Z1\n\u00001dz0n(z0)h\nGR\n\u0017(q0;z0)GR\n\u00170(q0;z0)\u0000GA\n\u0017(q0;z0)GA\n\u00170(q0;z0)i\n= 2iZ1\n\u00001dz0n(z0)h\nReGR\n\u0017(q0;z0)ImGR\n\u00170(q0;z0) + ImGR\n\u0017(q0;z0)ReGR\n\u00170(q0;z0)i\n: (B27)\nA combination of Eqs. (B26), (B27), and (B25) gives Eq. (15).\nAppendix C: Derivations of Eqs. (18), (19),\n(21){(27)\nWe explain the details of the derivations of Eqs. (18),\n(19), (21){(27). These equations are obtained by deriving\nthe expressions of L0\n12andL0\n12in the limit \u001c!1 , where\n\u001c= (2\r)\u00001is the magnon lifetime.\nFirst, we derive Eqs. (18) and (19). Using Eq. (B12),\nwe have\nImGR\n\u000b(q;z) =\u0000\r\n[z\u0000\u000f\u000b(q)]2+\r2; (C1)\nImGR\n\f(q;z) =\r\n[z+\u000f\f(q)]2+\r2: (C2)\nSince\u001c! 1 corresponds to \r!0, we can express\nI(I)\n\u0017\u00170(q) [i.e., Eq. (14)] in this limit as follows:\nI(I)\n\u000b\u000b(q)\u0018@n[\u000f\u000b(q)]\n@\u000f\u000b(q)Z1\n\u00001dz\r2\nf[z\u0000\u000f\u000b(q)]2+\r2g2\n=\u0019\n2\r@n[\u000f\u000b(q)]\n@\u000f\u000b(q); (C3)\nI(I)\n\f\f(q)\u0018\u0019\n2\r@n[\u000f\f(q)]\n@\u000f\f(q); (C4)\nI(I)\n\u000b\f(q) =I(I)\n\f\u000b(q)\u00180: (C5)Combining these equations with Eq. (13), we have\nL0\n12\u0018L0\n12\u000b+L0\n12\f; (C6)\nL0\n12\u0017=1\nNX\nqvx\n\u0017\u0017(q)ex\n\u0017\u0017(q)@n[\u000f\u0017(q)]\n@\u000f\u0017(q)\u001c: (C7)\nThese are Eqs. (18) and (19).\nNext, we derive Eqs. (21){(27). Since L0\n12is given by\nEq. (15), the remaining task is to derive the expression of\nI(II)\n\u0017\u00170(q) in the limit \u001c!1 . By performing the similar\ncalculations to the derivations of Eqs. (C3){(C5), we\nobtain12\nZ1\n\u00001dzn(z)ReGR\n\u000b(q;z)ImGR\n\u000b(q;z) =\u0000\rZ1\n\u00001dzn(z)z\u0000\u000f\u000b(q)\nf[z\u0000\u000f\u000b(q)]2+\r2g2\n=\u0000\rZ1\n\u00001dzn(z)@\n@zn\n\u00001\n21\n[z\u0000\u000f\u000b(q)]2+\r2o\n\u0018\u0000\u0019\n2@n[\u000f\u000b(q)]\n@\u000f\u000b(q); (C8)\nZ1\n\u00001dzn(z)ReGR\n\u000b(q;z)ImGR\n\f(q;z) =\rZ1\n\u00001dzn(z)z\u0000\u000f\u000b(q)\nf[z\u0000\u000f\u000b(q)]2+\r2gf[z+\u000f\f(q)]2+\r2g\u0018\u0000\u0019n[\u0000\u000f\f(q)]\n\u000f\u000b(q) +\u000f\f(q);\n(C9)\nZ1\n\u00001dzn(z)ReGR\n\f(q;z)ImGR\n\u000b(q;z) =\rZ1\n\u00001dzn(z)z+\u000f\f(q)\nf[z+\u000f\f(q)]2+\r2gf[z\u0000\u000f\u000b(q)]2+\r2g\u0018\u0019n[\u000f\u000b(q)]\n\u000f\u000b(q) +\u000f\f(q);(C10)\nZ1\n\u00001dzn(z)ReGR\n\f(q;z)ImGR\n\f(q;z) =\u0000\rZ1\n\u00001dzn(z)z+\u000f\f(q)\nf[z+\u000f\f(q)]2+\r2g2\n=\u0000\rZ1\n\u00001dzn(z)@\n@zn\n\u00001\n21\n[z+\u000f\f(q)]2+\r2o\n\u0018\u0000\u0019\n2@n[\u000f\f(q)]\n@\u000f\f(q): (C11)\nBy combining these equations with Eq. (16), we can\nexpressI(II)\n\u0017\u00170(q) in the limit \u001c!1 as follows:\nI(II)\n\u000b\u000b(q)\u0018\u0000\u0019@n[\u000f\u000b(q)]\n@\u000f\u000b(q); (C12)\nI(II)\n\f\f(q)\u0018\u0000\u0019@n[\u000f\f(q)]\n@\u000f\f(q); (C13)\nI(II)\n\u000b\f(q) =I(II)\n\f\u000b(q)\u0018\u0019n[\u000f\u000b(q)]\u0000n[\u0000\u000f\f(q)]\n\u000f\u000b(q) +\u000f\f(q):(C14)\nSubstituting these equations and Eqs. (C3){(C5) into\nEq. (15), we obtain\nL0\n12\u0018L0\n12-intra +L0\n12-inter1 +L0\n12-inter2; (C15)\nwhere\nL0\n12-intra =X\n\u0017=\u000b;\fL0\n12-intra-\u0017; (C16)\nL0\n12-intra-\u0017=\u00002\nN2X\nq;q0vx\n\u0017\u0017(q)ex\n\u0017\u0017(q0)\u001cV\u0017\u0017\u0017\u0017(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)@n[\u000f\u0017(q0)]\n@\u000f\u0017(q0); (C17)\nL0\n12-inter1 =X\n\u0017=\u000b;\fn\n\u00002\nN2X\nq;q0vx\n\u0017\u0017(q)ex\n\u0016\u0017\u0016\u0017(q0)\u001cV\u0017\u0017\u0016\u0017\u0016\u0017(q;q0)\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)@n[\u000f\u0016\u0017(q0)]\n@\u000f\u0016\u0017(q0)o\n; (C18)and\nL0\n12-inter2 =X\n\u0017=\u000b;\f(L0\nE\u0017+L0\nS\u0017); (C19)\nL0\nE\u0017=2\nN2X\nq;q0vx\n\u0017\u0017(q)ex\n\u000b\f(q0)V\u0017\u0017\u000b\f(q;q0)\u001c\n\u0002@n[\u000f\u0017(q)]\n@\u000f\u0017(q)n[\u000f\u000b(q0)]\u0000n[\u0000\u000f\f(q0)]\n\u000f\u000b(q0) +\u000f\f(q0);(C20)\nL0\nS\u0017=2\nN2X\nq;q0vx\n\u000b\f(q)ex\n\u0017\u0017(q0)V\u000b\f\u0017\u0017(q;q0)\u001c\n\u0002n[\u000f\u000b(q)]\u0000n[\u0000\u000f\f(q)]\n\u000f\u000b(q) +\u000f\f(q)@n[\u000f\u0017(q0)]\n@\u000f\u0017(q0):(C21)\nIn Eq. 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Goennenwein1;2;4\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2 and\n4Nanosystems Initiative Munich, Schellingstra\u0019e 4, D-80799 M unchen, Germany\n(Dated: October 17, 2018)\nWe experimentally investigate magnon-polaritons, arising in ferrimagnetic resonance experiments\nin a microwave cavity with a tuneable quality factor. To his end, we simultaneously measure the\nelectrically detected spin pumping signal and microwave re\rection (the ferrimagnetic resonance sig-\nnal) of a yttrium iron garnet (YIG) / platinum (Pt) bilayer in the microwave cavity. The coupling\nstrength of the fundamental magnetic resonance mode and the cavity is determined from the mi-\ncrowave re\rection data. All features of the magnetic resonance spectra predicted by \frst principle\ncalculations and an input-output formalism agree with our experimental observations. By changing\nthe decay rate of the cavity at constant magnon-photon coupling rate, we experimentally tune in\nand out of the strong coupling regime and successfully model the corresponding change of the spin\npumping signal. Furthermore, we observe the coupling and spin pumping of several spin wave modes\nand provide a quantitative analysis of their coupling rates to the cavity.\nI. INTRODUCTION\nMotivated by the vision of hybrid quantum information\nsystems combining the fast manipulation rates of super-\nconducting qubits and the long coherence times of spin\nensembles, strong spin-photon coupling is a major goal\nof quantum information memory applications. Coher-\nent information exchange between microwave cavity pho-\ntons and a spin ensemble was initially demonstrated for\nparamagnetic systems1{3, but only recently has this con-\ncept been transferred to magnetically ordered systems,\nwhere coupling rates of hundreds of megahertz can be\nachieved.4{7Utilizing the \rexibility of exchange coupled\nmagnetically ordered systems, more complex architec-\ntures involving multiple magnetic elements have already\nbeen developed8,9. Additionally, magnetically ordered\nsystems allow to study classical strong coupling physics\neven at room temperatures.5{10\nMoreover, a key advantage of magnetically ordered sys-\ntems over their paramagnetic counterparts { which has\nyet to be fully explored { is the ability to probe mag-\nnetic excitations electrically through spin pumping and\nthe inverse spin Hall e\u000bect. Spin pumping, in general, re-\nlies on ferromagnet-normal metal (FM/NM) heterostruc-\ntures and has been demonstrated for a wide variety of ma-\nterial combinations11. Under resonant absorption of mi-\ncrowaves, the precessing magnetisation in the ferromag-\nnet sources a spin current into the normal metal, where it\nis converted into a charge current via the inverse spin Hall\ne\u000bect. This spin Hall charge current is then detected.\nIn ferromagnetic insulator (FMI)-based FMI/NM het-\nerostructures, charge current signals from the recti\fca-\ntion of the microwave electric \feld are very small12, lead-\ning to a dominant spin pumping/spin Hall signal. This\nhas led to much research on FMI/NM heterostructures, of\nwhich the Yttrium Iron Garnet (YIG)/Platinum(Pt) bi-\nlayers we use are a prime example. Spin pumping is a well\nunderstood e\u000bect for weak photon-magnon coupling11,13,i.e. for situations where the decay rates of the cavity\nand the magnetic system are larger than the photon-\nmagnon coupling strength. However, the large spin den-\nsity of YIG and the resulting large e\u000bective coupling\nstrength allows one to reach the strong coupling regime\nalso in typical spin pumping experiments. The exper-\nimental observation14and theoretical treatment15,16of\nspin pumping in a strongly coupled magnon-photon sys-\ntem has only recently been performed. These results sug-\ngest that combining spin pumping and strong magnon-\nphoton coupling may enable the transmission and elec-\ntrical read out of quantum states in ferromagnets using\na hybrid architecture. Experiments directly linking spin\npumping in the weak and strong coupling regime are,\nhowever, still missing. Such experiments are one impor-\ntant step towards understanding the functional principle\nand key requirements for such a hybrid architecture.\nIn this paper, we present a systematic study of the\nmagnon-photon coupling in magnetic resonance exper-\niments in a YIG/Pt bilayer mounted in a commer-\ncially available EPR cavity. We measure both the mi-\ncrowave re\rection spectra and the electrically detected\nspin pumping signal in the system. The tuneable cavity\nquality allows us to systematically move in and out of the\nstrong coupling regime. Measurements with high mag-\nnetic \feld and frequency resolution allow us to clearly\nobserve the coupling of spin wave modes with the hy-\nbridized cavity{fundamental FMR mode. We explore a\ndi\u000berent approach as recently used by Zhang et al.7: In\nour setup, instead of tuning the cavity frequency we tune\nits decay rate while the e\u000bective magnon-photon coupling\nrate and the magnon decay rate stay constant. We thus\nachieve a transition from the strongly coupled regime\nwhere the decay rates of spin and cavity system are both\nconsiderably smaller than the e\u000bective coupling rate, to\nthe weakly coupled regime where the cavity decay rate\nis much higher than the magnon-photon coupling rate.\nThis regime is also called the regime of magnetically in-arXiv:1601.05681v1 [cond-mat.mtrl-sci] 21 Jan 20162\nduced transparency (MIT)7.\nThis paper is organized as follows: In Sec. II we re-\nview the general theory of the coupled magnon-photon\nsystem and the main features of spin pumping in the case\nof strong coupling. In Sec. III we describe the experi-\nmental details of recording the microwave re\rection of\nthe system as a function of frequency and applied mag-\nnetic \feld while simultaneously recording the DC spin\npumping voltage across the Pt. Finally in Sec. IV we\npresent our observation of strong coupling between the\ncavity mode and both the fundamental magnetic reso-\nnance and standing spin wave modes. We also demon-\nstrate the transition from strong to weak coupling by\ntuning the cavity line width and discuss the di\u000berence\nin the experimental spin pumping signature in both the\nstrong and weak regimes.\nII. THEORY\nA. Photon-Magnon Dispersion\nConventionally, ferromagnetic resonance (FMR) is\nmodeled in terms of the Landau-Lifshitz-Gilbert (LLG)\nequation which describes the dynamics of a magnetic mo-\nment in the presence of a magnetic \feld. In a static\nmagnetic \feld H0, the magnetic moment will precess\nwith the Larmor frequency !s. In detail, !sdepends\non the static \feld strength and on its orientation due to\nanisotropy17. This precessional motion can be resonantly\nexcited by a time varying microwave magnetic \feld H1\nwith a frequency close to !s. To observe spin pumping\nin FM/NM heterostuctures, the \feld H0should be ap-\nplied perpendicular to the surface normal (i.e. in the\ninterface plane)11,13,18,19. In this case, the FMR disper-\nsion (in the absence of crystalline magnetic anisotropy)\nis!s=\r\u00160p\nH0(H0+Ms)20. Here,Msis the mate-\nrial speci\fc saturation magnetization, \ris the material\nspeci\fc gyromagnetic ratio and \u00160is the vacuum perme-\nability. In the limit H0\u001dMsthe resonance frequency\nis thus linear in magnetic \feld. Contrary to the spin\nresonance frequency !s, the resonance frequency !cof a\nmacroscopic cavity is determined by geometrical and di-\nelectric parameters only and therefore does not depend\non the magnetic \feld. However, since the magnonic and\nthe photonic mode interact in resonance, we expect mod-\ni\fcations to the pure FMR and pure cavity dispersions.\nTo be speci\fc, we will observe an anticrossing of the\nFMR and the cavity dispersion for a su\u000eciently strong\nmagnon-photon coupling.\nTo describe the coupling between the cavity mode\nand the spin excitation the quantum mechanical Tavis-\nCummings model21,22and classical \frst principles15ap-\nproaches using the input-output formalism5have success-\nfully been used. For the dipolar interaction assumed\nin the models, the single spin-single photon coupling\nstrengthg0is proportional to the vacuum microwave\nmagnetic \feld H0\n1and the dipole moment mof the spin.In the scope of the Tavis-Cummings model, it has been\nshown that the collective coupling strength ge\u000bto an en-\nsemble of spins is proportional to the square root of the\nnumber of polarized spins for the coupling to the vac-\ncum microwave magnetic \feld. In a classical theory, Cao\net al.15derived that thisp\nNbehaviour prevails also for\nthe magnon-photon coupling in magnetically ordered sys-\ntems. Here, the total magnetization and thus the \flling\nfactor of the ferromagnetic material in the cavity, can be\nused as a measure for the total number of spins.\nThe characteristic \fngerprint of strong coupling is the\nformation of an observable anti-crossing of the cavity and\nthe spin dispersion relation close to resonance. Note,\nthat the presence of stong coupling and the accompanied\nvisible anti-crossing of the dispersion relations requires\nthat the e\u000bective coupling ge\u000bexceeds the loss rates of\nthe spins (\rs) and the cavity ( \u0014i+\u0014e). Experimentally,\nwe tune the spin resonance frequency !sacross the cavity\nresonance frequency !cvia an externally applied static\nmagnetic \feld. The coupled system can most simply be\nmodelled in the vicinity of the resonance frequency using\ntwo coupled harmonic oscillators, where the resonance\nfrequency is5:\n!\u0006=!c+\u0001\n2\u00061\n2q\n\u00012+ 4g2\ne\u000b(1)\nHere, \u0001 =\r(\u00160H0\u0000\u00160Hres) is the spin-cavity detuning\nwithHresstatisfying the spin resonance condition for a\ngiven cavtiy frequency !c.\nIn ferromagnetic \flms, additional magnetic modes, so-\ncalled perpendicular standing spin waves modes, appear\ndue to magnetic boundary conditions. For the condition\nwhere the magnetization is pinned at least at one sur-\nface of the \flm (and in the absence of any anisotropies\nor magnetic gradients) the magnon spectrum can easily\nbe calculated20. The di\u000berence of the resonance \feld of\nthenth-mode from the fundamental mode Hn\nres\u0000H1\nresis\nproportional to n2. Cao et al.15also calculated the ex-\npected coupling strength for di\u000berent modes and found\nthat the coupling decreases with increasing mode number\nasge\u000b/1=n. This can be understood when considering\nthe microwave mode pro\fles and the fact that the spa-\ntial mode pro\fle of the microwave \feld H0\n1in a cavity\nis typically homogeneous and in phase throughout the\nthickness of the (thin \flm) sample. Therefore only every\nsecond mode can be excited and the e\u000bective magnetiza-\ntion to which the microwave can couple to is reduced to\nM\nn.\nB. Spin pumping and strong coupling\nSpin pumping in ferromagnet/normal metal bilayers\nin the weak coupling regime is well understood11,13,19:\nAn additional mechanism which damps the magnetiza-\ntion precession becomes available by spin pumping, as\nthe precessing magnetisation is driving a spin current\ninto the adjacent normal metal.13In electrically detected3\nVNA 9-10 GHZ\nLNA\n1 ADC\nA B\nelectro magnetfeed\nlineDC lines\nYIG sample\nsample holderDC meas. lines\ncavity\ngeffκi κe γsp\nγiH0\ncryostatA-B\nFIG. 1. Block diagram of the experimental setup and sample\nmounting. (Inset) Schematic of the coupling scheme illus-\ntrating cavity decay due to intrinsic losses viz. losses to the\nfeedline\u0014c=\u0014i+\u0014e, spin system decay consisting of intrinsic\ndamping and spin pumping damping \ri=\rs+\rspas well the\ncollective coupling rate ge\u000b\nspin pumping, this spin current is then converted into a\ncharge current via the inverse spin Hall e\u000bect (ISHE). For\nelectrical open circuit conditions, one thus obtains a volt-\nage which scales as11,19VSP/g\"#\u0015SDtanhtN\n2\u0015SDsin2\u0012.\nIt, thus, contains information on the spin mixing con-\nductanceg\"#, the spin di\u000busion length \u0015SD, the mag-\nnetization precession cone angle \u0012and depends on the\nthickness of the normal metal layer tN. The maximal\nprecession cone angle \u0012and thus the maximal expected\nspin pumping voltage depends on the microwave power\nbut also on the coupling strength between cavity and spin\nsystem. For strong coupling, the cone angle is expected\nto be reduced as compared to the weak coupling case due\nto the hybridized nature of the excitation at its maximal\nintensity.\nThe other contributions in the equation for VSPare\nmaterial constants: The spin mixing conductance g\"#de-\nscribes the the transparency of the ferromagnet/normal\nmetal interface und limits the spin pumping e\u000eciency\ngenerally; the spin di\u000busion length \u0015SDin conjunction\nwith the normal metal thickness tNaccounts for a spin\naccumulation in the normal metal and reduces the spin\npumping e\u000eciency if tN/\u0015SD.\nIII. EXPERIMENTAL DETAILS\nA. Sample preparation\nIn our experiments we used YIG/Pt heterostruc-\ntures grown by liquid phase epitaxy on (111)-oriented\nGadolinum Gallium Garnet (GGG) substrates. The YIG\flm thickness was 2 :8µm. In order to produce a high\nquality interface between YIG and Pt, and thus a large\nspin mixing conductance g\"#, we followed the work of\nJung\reisch et al.23and \frst treated the YIG surface\nby piranha etching for 5 minutes in ambient conditions.\nThereafter, the sample was annealed at 500\u000eC for 40 min-\nutes in an oxygen atmosphere of 25 µbar. Under high\nvacuum, it was then transferred into an Electron Beam\nEvaporation (EVAP) chamber where 5 nm Pt was de-\nposited. The exact Pt thickness was determined using\nX-Ray re\rectometry. However, we note that for our anal-\nysis the Pt layer thickness is of minor importance as it\nwas consistently larger than the spin di\u000busion length \u0015SD\nof Pt such that the Pt layer simply serves as a perfect spin\nsink.\nIn order to achieve collective strong coupling between\nmagnons and cavity photons, the number of magnetic\nmoments must be su\u000eciently high. Therefore, we diced\nthe sample into several pieces of di\u000berent lateral dimen-\nsions. Magnetic resonance experiments in the strong cou-\npling regime showed that thep\nNscaling of the coupling\nstrength discussed in Section II is indeed obeyed upon\ncomparing samples with di\u000berent volume and thus dif-\nferent total magnetic moment. In the following, we will\nfocus on a sample with lateral dimensions 2 mm \u00023 mm\nwhich, with the e\u000bective spin density \u001aS= 2:1\u00021022\u0016B\ncm3\nof iron atoms in YIG24, contains on the order of 4 \u00021017\nspins. Finally, the sample was mounted on a PCB sample\ncarrier and wire bonded as depicted in the inset of Fig. 1.\nThe carrier itself was mounted on a sample rod which al-\nlowed the sample to be accurately positioned in the elec-\ntrical \feld node of a Bruker Flexline MD5 dielectric ring\ncavity in an Oxford Instruments CF935 gas \row cryo-\nstat. Shielded DC cabling allowed for the measurement\nof the ISHE voltage. The detailed design blueprints of\nthe sample rod and chip carrier can be retrieved online25.\nB. Experimental setup\nThe Bruker cavity exhibits a TE 011mode with an elec-\ntric \feld node at the sample position. Its quality factor\nQ=!=\u0001!FWHM\nc (\u0001!FWHM\nc being the full width half\nmaximum line width of the cavity) is dominated by the\ndissipative losses in the dielectric and its \fnite electrical\nresistance ( \u0014i) as well as radiation back into the cavity\nfeed line (\u0014c). By changing the cavity's coupling ratio to\nthe feed line, unloaded coupled quality factors Qcfrom 0\nto 8000 can be achieved. This allows tuning in and out\nof the strong coupling regime easily. Using the gas \row\ncryostat, di\u000berent temperatures can be stabilized. All the\nfollowing experiments have, however, been performed at\nroom temperature.\nTo measure ferromagnetic resonance (FMR) the cavity\nwas connected to the port of an Agilent N5242A vector\nnetwork analyzer (VNA). The driving power of 15 dBm\nexcites at maximum on the order of NPh= 1:3\u00021014pho-\ntons in the cavity which is considerably smaller than the4\nnumber of spins in the sample (4 \u00021017). In this case,\nthe theory presented in Sec. II is well justi\fed26. The\nfrequency dependent cavity re\rection S11was measured\nwhile sweeping the external \feld \u00160Hthat is created by\na water cooled electromagnet. The IF bandwidth was\nchosen to be 100 Hz which leads to a frequency sweep\ntime of approximately 2 s for each magnetic \feld step. A\ncalibration of the microwave leads up to the resonator's\nSMA connector was performed. The calibration did not\ninclude the feed line inside the resonator mount, which\ngave rise to a background signal in the re\rection param-\neter. However, by utilizing the full complex S-parameter\nfor the background subtraction with the inverse map-\nping technique outlined by Petersan and Anlage27and\na subsequent Lorentzian \ft to the magnitude, a reliable\nmeasurement of Qis still possible, even for a completely\nuncalibrated setup. We note that even though standing\nwaves in the mirowave feed line will not appear in the\ncalibrated re\rection measurement they will still change\nthe total power in the cavity and therefore may compli-\ncate the electrically detected DC spin pumping signal.\nUncalibrated measurements did not show sharp feed line\nresonances in the frequency range studied here but only\nsmooth oscillations with an amplitude of less than 1 dB\nand there was no correlation in the DC signal resolved.\nIn order to \ft the data and as it improves clarity, we only\ndiscuss calibrated measurements in the following.\nThe DC voltage from the sample was measured along\nthe cavity axis and thus perpendicular to the external\nmagnetic \feld and the sample normal. It was ampli-\n\fed with a di\u000berential voltage ampli\fer model 560 from\nStanford Research Systems. The ampli\fer was operated\nin its low noise (4 nV/p\nHz) mode and set to a gain of\n2\u0002104. The analog high-pass \flter of the ampli\fer was\ndisabled, however, a low-pass \flter with a 6dB roll-o\u000b at\n1 kHz was employed. Limiting the bandwidth of the am-\npli\fer by \fltering is required in order to achieve a good\nsignal-to-noise ratio. Care has, however, to be taken as\nthe lineshape may be quickly distorted by inappropriate\nsettings and thus the signature of spin pumping might be\nmasked. High-pass \fltering can easily lead to a dispersive\nlike contribution to the signal, whereas low-pass \fltering\nwill give rise to asymmetric line shapes depending on the\nratio of IF bandwidth and low-pass frequency. We made\nsure that no such distortions contribute to the presented\nmeasurements. The ampli\fed voltage signal was \fnally\nrecorded using the auxiliary input of the VNA simulta-\nneously with the cavity re\rection S11.\nIV. RESULTS AND DISCUSSION\nWe \frst focus on the case of the so-called critical cou-\npling of the feed line to the cavity in which most FMR\nexperiments are conducted. In this case, the internal loss\nrate of the cavity equals the loss rate to the feed line and\nthe quality factor is Qc=Qinternal=2. Note that inserting\na sample and holder into the cavity will reduce the cavity\n275 267 259\nStatic magnetic0H(mT)\n259 267 275\n 180 0180\nV9.509.559.609.659.709.759.80\n0.20.40.6 240 0 240\n9.559.609.659.709.759.80|S |11 VDC(µV)\n57911=nFrequency (GHz)(a)\n(b) 2geff/2π = 64 MHzFIG. 2. (a)Re\rection parameter S11recorded while sweeping\nthe magnetic \feld. Strong coupling of the collective spin ex-\ncitations is indicated by a clear anticrossing, spin wave modes\nto the low \feld side of the main resonance are visible. Black\nnumbers indicated the spin wave mode number. (b)Simul-\ntaneously recorded DC voltage. Fundamental and spin wave\nmodes are visible where the latter couple less strongly and\nthus pump spin current more e\u000eciently. Insets: Detail of\nn=5 spin wave mode including the dispersion relation of the\nstrong coupling between the fundamental FMR mode and the\ncavity as solid red line and the anti-crossing of this hybrid and\nthe spin wave mode (#5) as white lines.\nQby an amount which depends on the sample and holder\ndetails such as conductivity and dielectric losses. Based\non our measured loaded Qc= 706, the cavity decay rate\nis calculated to be \u0014c=2\u0019=!r\n2\u0019=2Qc= 6:8 MHz.\nStrong coupling of the magnon and cavity system man-\nifests itself in a characteristic anti-crossing of the (mag-\nnetic \feld independent) cavity resonance frequency and\nthe magnon dispersion that is (approximately) linear in\nmagnetic \feld. This anti-crossing corresponding to two\ndistinct peaks in a line cut at the resonance \feld, are im-\nmediately visible in the re\rection spectrum in Fig. 2. The\nminimal splitting gives the collective coupling strength\nge\u000b=2\u0019= 31:8 MHz of the fundamental mode to the cav-\nity. Taking into account the number of spins in the\nsample, the single spin coupling rate is on the order of\ng0=2\u0019= 0:1 Hz which is in agreement with experiments\non paramagnetic systems28.\nIn our setup, even the coupling of higher order spin\nwave modes to the cavity can be resolved. We number\nthe spin waves as noted in Fig. 2 taking into account\nthat with an uniform driving \feld only odd modes can\nbe excited. Analysis of the resonance position of the spin\nwave modes reveals that Hn\nres\u0000H1\nresin our sample is\nproportional to nrather than n2. This indicates a non-\nsquare like potential well. Similarly, complicated mode\nsplittings have been reported in literature29. The low-\nest order spin wave mode that can be easily observed in\nour setup is shown in the inset of Fig. 2 (a) in detail.5\nIt exhibits the largest e\u000bective coupling (3 MHz) of all\nspin wave modes. The red and white lines in Fig. 2 (a)\ncorrespond to the harmonic-oscillator model (Eqn. 1) for\nthe fundamental mode and the lowest order spin wave\nmode, respectively. As the spin wave couples to an al-\nready hybridized sytem, we superimposed the dispersion\n!c=!r(B) of the hybridized system of fundamental\nmode and unperturbed cavity as the \"cavity\" mode in\nthe modelling of the spin wave mode couplings.\nIn order to quantify the coupling strength of the higher\norder modes which only interact weakly with the hy-\nbridized cavity{fundamental FMR mode, we follow the\napproach of Herskind et al.30. For each \feld, we \ft\na Lorentzian to the magnitude of the cavity absorp-\ntion. From this \ft we extract the resonance frequency\n!cand the half width half maximum of the absorption\n\u0001!which, in the weakly coupled spin waves reads as30\n\u0001!= \u0001!c+ge\u000b\rs=\u0000\n\r2\ns+ \u00012\u0001\n:\nThe coupling of the spin waves to the already hy-\nbridized cavity resonance decreases with the order of the\nmode. This can be understood by taking into account\nthat the e\u000bective magnetization to which the homoge-\nneous microwave \feld can couple decreases with increas-\ning mode number. The extracted values, gn=7;9;11;13=\n[3:65;2:49;1:64;1:16] MHz, match accurately with the ex-\npected1\nndependence of the coupling strength15.\nWe attribute the pronounced feature that is seen to\nthe right of the anti-crossing to an unidenti\fed spin\nwave mode. A similar feature was found in other\nexperiments14and has been interpreted in the same man-\nner. In our data, we can clearly distinguish between the\nfundamental mode and this additional mode { simply\nby remembering that the relative intensity and coupling\nstrength is expected to be higher for the fundamental\nmode. Possible origins for the additional mode are an in-\nhomogeneous sample or a gradient in the magnetic prop-\nerties across the \flm thickness31. This would be consis-\ntent with the unusual spin wave mode splitting. Lastly,\nwe note that the recorded signal in the re\rection parame-\nter is completely symmetric upon magnetic \feld reversal.\nThe simultaneously recorded DC voltage is shown in\nFig. 2 (b). Contrary to the re\rection parameter, the volt-\nage signal reverses sign on reversing \u00160H0. The lineshape\nthat we record for all modes is completely symmetric as\nfar as they can be clearly distinguished from each other.\nWe thus conclude that we observe a signal purely caused\nby spin pumping and not by any rectifying e\u000bect. In a\nFMI/NM bilayer ( \u001aYIG\u001510 G\n m)32recti\fcation can\nonly arize from a change of the spin Hall magnetoresis-\ntance (SMR) in the normal metal. According to model\ncalculations12this e\u000bect is negligible for the system we\ninvestigate because of the small magnitude of the SMR\ne\u000bect (<0:1%). This notion is further corroborated by\nthe fact that the change in lineshape expected for recti\f-\ncation type signals is not visible in our data. Apart from\nthe spin wave modes which are clearly resolved in the\nDC voltage signal, we can also clearly see the electricallydetected spin pumping voltage originating from the hy-\nbridized system of cavity and fundamental FMR mode\n(the main anti-crossing). The hybridized cavity eigen-\nmodes can, however, pump spin current into the normal\nmetal only very ine\u000eciently and thus the DC voltage we\nobserve is very low.\nThe upper panels of Fig. 3 show the change in cav-\nity re\rection as we gradually increase the coupling of the\ncavity to the feed line and thus increase the cavity decay\nrate. Starting from the critically coupled case (inter-\nnal cavity losses are equal to losses into the feedline) in\nthe left panel to a highly overcoupled cavity (losses into\nthe cavity feed line dominate the cavity's decay rate) in\nthe right panel, we clearly see an increase in the cavity\nlinewidth up to the point were the unperturbed cavity\nis no longer recognizable. Correspondingly, the cavity\ndecay rate increases from left to right and, in turn, the\nmicrowave magnetic \feld strength H1in the cavity de-\ncreases. For the already weakly coupled spin wave modes\nthe spin pumping voltage decreases with decreasing mi-\ncrowave magnetic \feld strength H1resp. available mi-\ncrowave power(indicated by the higher S11parameter)\nin the cavity. The DC spin pumping voltage amplitude\ncorresponding to the fundamental mode (lower panels of\nFig. 3) does, however, not decrease for lower Q-factors\nbut stays approximately constant. Considering that the\nabsorbed power of the cavity-spin system stays approxi-\nmately constant when changing the cavity decay rate as\ncan easily be seen in the line cuts in the upper panels of\nFig. 3 this behaviour can also be understood.\nThe best measure of the true magnon spectrum and\nline widths of the spin system can be extracted from\nthe highly overcoupled case (right panels of Fig. 3 and\nFig. 4). There, the magnon-photon coupling is negligi-\nble compared to the cavity loss rate and therefore, the\nmagnon-cavity mode hybridization does not distort the\nline shape. A mode that strongly couples with the cav-\nity, on the contrary, can vanish completely in the \fxed-\nfrequency spectrum. We \fnally note that we observe\nthe described anti-crossing due to the magnon-photon\ncoupling and thus the distortion of the lines in a \fxed-\nfrequency experiment (with the cavity tuned to high Q,\nas usually done in cavity-based FMR experiments) al-\nready for sample volumes as small as V= 2\u000210\u00003mm3\nin the case of YIG ( MS= 140 kA m\u00001). These sample\nvolumes are easily achieved for LPE grown samples and\nwe note that in most cavity FMR experiments33the ef-\nfects of the coupling need to be taken into account in\norder to yield accurate results especially when automatic\nfrequency control is employed.\nV. CONCLUSIONS\nIn summary, we presented systematic measurements\nof spin pumping in di\u000berent regimes of the magnon-\nphoton coupling strength. For the fundamental mode of\na YIG/Pt bilayer, strong coupling with an e\u000bective cou-6\n0.0 0.4 0.8\n260 265 270 275 280\n 260 265 270 275 280\nStatic magnetic field µ0H0 (mT)\n260 265 270 275 280\n −250 0250\nVDC (µV)99.559.609.659.709.759.Frequency (GHz)0.16 0.32 0.48 0.64|S11|\n−240 −800 80 240\n9.559.609.659.709.759.80\nVDC (µV)\n|S11|\nFIG. 3. Increasing the coupling of the cavity to the feed line (from left to right) increases the cavity loss rate \u0014cand thus line\nwidth \u0001!=2\u0019. This enables experimental control of the transition between strong and weak coupling. The line cuts at positive\n\feld (intense colors) and negative \feld (pale colors) again con\frm the symmetry, V(\u0000H0) =\u0000V(H0) andS11(\u0000H0) =S11(H0)\nand also show the merging of the two dispersion curves during the strong/weak transition.\n−4\n−8\n−12|S11| (dB)weak coupling\nstrong coupling\n255 260 265 270 275\nMagnetic Field (mT)050100150VDC(µV)(a)\n(b)\nFIG. 4. Line cuts of (a) re\rection parameter and (b) DC volt-\nage at the resonator frequency !c(H0= 0). In the strongly\ncoupled magnon-photon case (green lines), the fundamental\nmode vanishes as opposed to the weakly coupled case where\nthe magnon spectrum is accurately reproduced\npling strength of ge\u000b=2\u0019= 31:8 MHz has been achieved\nat room temperature in a standard EPR cavity. The\ncharacteristics of the coupled magnon-photon system \ft\nwell to the established theory and are consistent with\nrecent results. Simultaneously, we recorded the electri-\ncally detected spin pumping signal of the fundamental\nmode. We were able to tune the system from the strongto the weak coupling regime by changing the cavity's de-\ncay rate. The evolution of the spin pumping signal of the\nfundamental mode has been analyzed qualitatively and\nfollows the predictions of Lotze16: In the strongly cou-\npled magnon-photon system the spin pumping e\u000eciency\nis reduced as the precession cone angle is smaller than in\nthe weakly coupled case. Additionally, we were able to\nobserve coupling and electrically detected spin pumping\nof several spin wave modes with distinctly di\u000berent cou-\npling strengths and observe for the \frst time their 1 =n\ndependence predicted by Cao et al.15. Furthermore, we\ndirectly demonstrated the implications of strong coupling\non \fxed-frequency FMR experiments. We conclude that\nsmall sample volumes or an highly overcoupled cavity are\nmandatory for a qualitatively and quantitatively correct\nevaluation of the magnon spectrum.\nACKNOWLEDGEMENTS\nWe thank Christoph Zollitsch and Johannes Lotze\nfor many valuable discussions and Michaela Lammel for\nassistance in the sample preparation. M. Harder ac-\nknowledges support from the NSERC MSFSS program.\nWe gratefully acknowledge funding via the priority pro-\ngramme Spin Caloric Transport (spinCAT) of Deutsche\nForschungsgemeinschaft (Project GO 944/4), SFB 631\nC3 and the priority programm SPP 1601 (HU 1896/2-1).\n1D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo,\nL. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B.Buckley, D. D. Awschalom, and R. J. Schoelkopf, Physical7\nReview Letters 105, 140501 (2010).\n2Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques,\nD. Zheng, A. Dr\u0013 eau, J.-F. Roch, A. Au\u000beves, F. Jelezko,\nJ. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve,\nPhysical Review Letters 105, 140502 (2010).\n3C. W. Zollitsch, K. Mueller, D. P. Franke, S. T. B. Goen-\nnenwein, M. S. Brandt, R. Gross, and H. Huebl, Applied\nPhysics Letters 107, 142105 (2015).\n4O. O. Soykal and M. E. Flatt\u0013 e, Phys. Rev. Lett. 104,\n077202 (2010).\n5H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. 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Venimadhav, R. Chandra,\nC. Mitra ·U. Kumar*\nReceived: date / Accepted: date\nAbstract Inthisreport,themagneticbehaviourof NiCr2O4bulkandnanopar-\nticle samples under different applied magnetic field has been investigat ed ex-\ntensively. Nanoparticles of NiCr2O4were obtained by mechanical milling of\npolycrystalline powder prepared by polyol method. FC-ZFC measur ement of\nbulk at different applied magnetic field has revealed the existence of a ferri-\nmagnetic transition around 66K followed by an antiferromagnetic tr ansition\nclose to 30K. However, its nano counterpart has shown remarkab le change in\nmagnetic properties - a suppression of ferrimagnetic transition ac companied\nby strengthening low temperature magnetic phase and observatio n of a new\ntransition at 90K ( TP), which is weakly magnetic in nature. The frequency\ndependent ac susceptibility data of nanoparticle have been fitted t o the well\nknown de Almedia-Thouless equation and a H2/3dependence of the low tem-\nperature peak is observed with a resulting zero field freezing tempe rature (T0\nf)\nequal to 10.1K. Further, the dynamical behaviour near freezing t emperature\nhasbeenanalysedintermsofcriticalbehaviourandthe obtainedfit ted param-\neters values being as τ0(relaxation time constant) = 3 .6X10−6s,T0\nf= 8.7K\nandzν= 11.1. Moreover, Vogel-Fulcher law has been used to understand the\nH. Singh, T. Chakraborty, K. Srikanth, C. Mitra, U. Kumar\nIndian Institute of Science Education and Research (IISER) Kolkata, Mohanpur Campus,\nPO: BCKV Campus Main Office, Mohanpur 741252, Nadia, West Beng al, India.\nTel.: +91-33-25873121\nFax: +91-33-25873020\nE-mail: udayphy@iiserkol.ac.in\nT. Ono\nDepartment of Physical Science, Osaka Prefecture Universi ty, Gakuen-cho 1-1, Naka-ku,\nSakai, Osaka 599-8531, Japan.\nA. Venimadhav\nCryogenic Engineering Centre, Indian Institute of Technol ogy, Kharagpur-721302, India.\nR. Chandra\nNano Science Laboratory, Institute Instrumentation Centr e and Centre of Nanotechnology,\nIndian Institute of Technology Roorkee, Roorkee 247667, Ut tarakhand, India.2 H. Singh\nnature of freezing transition and the parameter after fitting are obtained as\nEa/kB= 58.9K,τ0= 5.22×10−8andT0= 8.03K.Finally,thespin-glassphase\nis concluded. Moreover, in contrast to bulk, the H2/3dependence of freezing\ntemperature of nanoparticle sample (75h) does support the 2D su rface like\nspin glass nature.\nKeywords Nanoparticles ·Spin Glass ·Antiferromagnetism\nPACS75.75.Fk ·75.50.Lk ·75.47.Lx\n1 Introduction\nOne of the interesting family of magnetic materials is spinel oxides, ma inly\nrepresented by the formula AB2O4[1]. Here the tetrahedral ‘A’ sites were\noccupied by divalent ion and octahedral ‘B’ site were equally occupied by di-\nvalent and trivalent ions. Among normal spinel compounds, nickel c hromite\nhas recently received considerable interest because of its classific ation into dy-\nnamically spin frustrated systems, based upon the realization of re sonance\nlike magnetic excitation observed in neutron scattering experiment [2]. How-\never, despite the presence of pyrochlore lattice which is formed by Cr3+ion,\nthe occurrence of Jahn-Teller distortion nullify the possibility of NiCr2O4to\nbe considered as geometrically frustrated magnet. A number of po tential ap-\nplication (light or heat sensitive micromechanical device, catalytic ma terials,\ngas sensors) has provided technological recognition to this mater ial [3,4]. Sin-\ngle crystals of NiCr2O4has been extensively studied from the perspective of\nneutron scattering [2], heat capacity [3], magnetodielectric [5] and m agnetic\n[6] measurements. However, magnetic studies of nanoparticles of this com-\npound have not been reported so far. Below a critical physical dime nsion (in\nnanometer range) magnetic nanoparticles become single domain as c ompared\nto the normal multi-domain structure of the bulk counterpart. Th ese single\ndomain nanoparticle are quite interesting owing to unusual phenome non they\nexhibit like superparamagnetism [7], quantum tunnelling of magnetizat ion [8]\nand large coercivities [9]. The magnetic studies investigated have sho wn the\noccurrenceoftwotransitiontemperatures TCandTS, correspondingto the on-\nsetofferrimagnetic(longitudinal) componentand spiral(transve rse)antiferro-\nmagnetic component. Recently , Tomiyasu et. al.has reported a new magnetic\nstructure in which for transverse and longitudinal components, B -sites were\nsorted into two sublattices [6]. NiCr2O4undergoes structural transition from\ncubic to tetragonal structure below 310K owing to Jahn-Teller dist ortion. The\nc\n(c−a)ratio is 4% at 4.2K [10]. However,Ishibashi et. al.has confirmed another\nstructural transition to orthorhombic structure at 65K and is co rrelated with\nthe onset of ferrimagnetic ordering [11]. In the present study, we reported the\nmagnetic study of NiCr2O4nanoparticles prepared by mechanical milling of\npolycrystallinepowder.FC-ZFCmeasurementweredoneat0.005an d0.1Tap-\nplied magnetic field to investigate the effect of magnetic field on the be haviour\nof the nanoparticle samples. The observed magnetic transition are in good2D spin-glass phase in NiCr2O4nanoparticles 3\nagreement with the earlier reported results. However, a new tran sition (TP),\nat 90Khasbeen reportedfor the firsttime in bulk NiCr2O4. In high field mea-\nsurement this transition disappears. But in nanoparticle samples, s oftening of\nTCwith decrease in particle size was found, whereas TPis distinctly visible.\nThe decrease in magnetization values in the magnetic nanoparticle sy stem in\ncomparison with its bulk phase is well known but still the subject matt er of\ndebate. This could be seen in the present study. The reason for th is reduction\nin magnetization could be associated with canted spin arrangement o r spin\ndisorder on the surface or finite size effect [12,13,14]. The result o f these phe-\nnomenon leads to superparamagnetism or spin-glass phases. We ha ve explored\nthe low temperature regime (down to 0.4K) as a function of magnetic field to\ninvestigate in detail the magnetic behaviour of the low temperature anomaly\n(TS). Frequency dependent ac susceptibility was also carried out to re veal any\nfrequency dependent magnetic phase. A 2D spin-glass like behaviou r has been\nestablishedfromdeAlmeida-Thouless(AT) analysiswhichisfurthers upported\nbyfrequency dependent acsusceptibility study. Finally, wetried to understand\nthe mechanism responsible for the existence of2D spin glasslike phas e and the\nsurface spin disorder is concluded as the reason behind such an obs ervation.\n2 Experimental Details\nNiCr2O4bulk was prepared by decomposition of NiCr2O4obtained by re-\nactingNiCl2.6H2Owith (NH4)2.Cr2O7. In a typical preparation, 10 mmol\nofNiCl2.6H2Oand 10 mmol of ( NH4)2Cr2O7were mixed in 40 mL distilled\nwater. To this mixture, 90 mmol of ethylene glycol mixed in 40 mL of dis -\ntilled water was added. The resulting solution was stirred for 12 hour s after\nwhich the solution was heated at 60C to evaporate water and obtain a brown\ncoloured gel which on further heating results in a greenish-black po wder pre-\ncursor. The greenish-black powder was heated at 350C to remove ethylene\nglycol and ammonium chloride which was finally heated at 1200C for 72 h ours\nto obtain NiCr2O4bulk crystalline form. This bulk powder form was used\nfor the synthesis of different nano size NiCr2O4sample using high energy ball\nmill machine Fritsch PlanetaryMono Mill Pulverisette 6. For this purpo se, the\nbulk powder sample was kept in an 80 ml agate bowl with 10 mm agate ba ll\nin 1: 8 sample and ball weight ratio and milled up to different milling time.\nAll the nano size samples were annealed at 400C for 4 hours and coole d very\nslowly to room temperature (40C/h) to reduce the possible existing strain in\nthe nano size sample. The phase purity and crystal structure (fr om 320 K\nto 10 K) of all the powder samples were studied by using Rigakus Smar tLab\nX-ray diffractometer using Cu K lines in parallel beam geometry mode.\nThe X-ray diffraction (XRD) pattern of all the samples were scanne d from\n15 to 70 at the step angle of 0.02 and anode power 9 kW. To know the e xact\nparticle size distribution, the samples were subjected to transmiss ion electron\nmicroscopy study and micrographs were collected under 200 kV ano de volt-\nage using model. A set of three sample 0 (Bulk), 33, 75h were prepar ed and4 H. Singh\nsubsequently studied. Zero field cool (ZFC) and field cool (FC) tem perature\ndependent magnetic moment measurement was done at 0.005 and 0.1 T field\nusing Magnetic Property Measurement System (MPMS). A systema tic mini-\nmization of the trapped magnetic field in the superconducting coil of MPMS,\nis usually performed before commencing any measurement. To stud y the low\ntemperature transition, ZFC measurement up to very low tempera ture of0.4K\nwere carried out at applied magnetic field of 0.005, 0.01, 0.05 and 0.1T. These\nlow temperature measurements were performed at Osaka, Japan . The fre-\nquency dependent ac susceptibility curves as a function of temper ature were\nalso collected at 7, 73 and 143 Hz in the MPMS at small oscillating magnet ic\nfield of 1Oe amplitude.\n3 Results and Discussion\nThe room temperature XRD pattern of bulk, 33h, and 75h milled samp les are\nshown in Fig. 1. The diffraction pattern of the bulk NiCr2O4powder sample\nis identical to the earlier reported results [5] confirming the phase purity of\nthe sample. The profile fitting for bulk diffraction patterns was carr ied out\nusing Fullprof Suite version 2009 and structural parameters were determined.\nAbove 310 K, the bulk NiCr2O4exits in cubic spinel structure with space\ngroup Fd(-3m) and lattice parameter 8.3194 ˚A. However, it adopts a tetrago-\nnal cubic spinel structure with space group I 41/amd and lattice pa rameters\nof a = b = 5.8369 ˚Aand c = 8.4312 ˚Abelow 310 K. The splitting of Bragg\npeaks in Fig. 1 can be seen as an indication of cubic tetragonal struc ture and\nis because of Jahn-Teller distortion. These lattice parameters for cubic and\ntetragonal structure are in good agreement with earlier reporte d values [5,6].\nA clear and systematic base broadening in the XRD pattern of 75h an d 33h\nsamples compared to bulk one can be seen in Fig. 1 which is associated t o\ndecreasing particle size effect. The dominant peak at nearly 36 [(211 )d plane]\nis taken under consideration for particle size determination using st andard\nDebye-Scherrer equation. The average particle size of 33h and 75 h ball milled\nsamples was 14.5 and 22.15 nm respectively. To confirm these particle sizes,\nthe 75h sample were subjected to TEM study as shown in Fig. 2. Distin ctly\nvisible lattice fringes confirms the high quality of the sample as is usually\nobserved in single crystalline nanostructures.\nFC-ZFC measurement of bulk and nanoparticle samples were perfor med at\ndifferent applied magnetic field (0.005, 0.01, 0.1 and 2T). For 0.005T ap plied\nmagneticfield, in caseofbulk sample, the observedtransitionsat67 Kand 29K\ncorresponding to TCandTSis well reproducible and is close to the reported\nresults in this system [6] as shown in Fig. 3. We have zeroed the magne tic\nfield of the superconducting magnet of MPMS by following a standard zeroing\nprotocol to ensure that the field was zero at the time of cooling. In terestingly,\nthe ZFC measurement at 0.005T of this system has shown a new tran sition at\n90K (named as TPhere) as shown in the inset of Fig. 3. ZFC measurement\nat 0.005T has not been reported so far in the literature for this sys tem. We2D spin-glass phase in NiCr2O4nanoparticles 5\nFig. 1 X-ray pattern of bulk and nanoparticle samples prepared wit h milling time of 33\nand 75h, shown along with bulk sample.\narguethat there is development of a weakmagnetic phase in the sys tem at this\ntransition temperature, however, the real nature of interactio n might be more\ncomplex. Study performed by Ishibashi et. al., has shown that the high tem-\nperature magnetic transition is occurring simultaneously with the st ructural\ntransition from tetragonal to orthorhombic at TC[11]. The temperature de-\npendent XRD measurement (not shown here) performed over bulk sample did\nnot revealed any structural transition happening at 90K. Hence, we verified\nthat this new transition at TPis solely magnetic.\nThe variation of temperature dependent magnetic behaviour at 0.0 05T ap-\nplied magnetic field for all the samples is shown in Fig. 3. The decrease in\nmagnetization at TCis observed, with decrease in particle size. This soften-\ning ofTCindicates the breaking of spin-lattice coupling owing to particle size\nreduction. The presence of spin-lattice coupling is common among ch romium\nbased spinels. The weakening of spin-lattice coupling is also supporte d by a\nshift in of spiral AFM ordering at TSin our samples towards lower temper-\nature, as the particle size decreases and can be seen in the insets o f Fig. 3\n[15]. The decrease in particle size destroys the long range AFM corre lation,\nresulting in a shift in TStowards lower temperatures [16]. As the particle\nsize decreases, a reduction in the magnetization value is observed. Lowering of\nmagnetization can be explained through increased surface spin can ting, spin\ndisorder or finite size effect at nanoparticle surface [17]. All the sam ples were\nalso investigated at 0.01T magnetic field as shown in Fig. 4. Apart from in-\ncreasein magnetization value no significanteffect ofmagnetic field is o bserved.\nHowever, the new transition TPcould not be observed distinctly. This could\nbe due to the formationof ferrimagneticspin clusters which is overc omeby the\napplication of higher fields causing a disappearance of TPat higher fields. On\nthe other hand, TPis clearly visible in the nanoparticlesamples. The exchange\ncoupling between magnetic ions ( Ni2+andCr3+) is directly proportional to\nCurie-Weiss temperature ( θCW) and can be expressed by relation,6 H. Singh\nFig. 2TEM image of the 75h milled sample revealing the lattice grat ing lines. In the inset\nis the particle size distribution of the same sample with ave rage particle size close to 12nm .\nJ=A|θCW| (1)\nwhere,A=3kB\nZS(S+1),kBis the Boltzmann constant, Z is the number of\nnearest neighbour interaction and S is the total spin. The high temp erature\nregion (T >150K) of ZFC magnetization data at 0.005 and 0.01T was fitted\nwith the Curie-Weiss law and subsequently θCWvalue was calculated for both\nbulk and nanoparticles. For 0.005T field value, the θCWfor bulk, 33 Hrs and\n75h samples are found to be −346.6±2,−295.05±2 and−238.26±2K re-\nspectively. However, for 0.01T field value, it varies as −607.23±2,−295.4±2\nand−236.3±2K. The negative value of θCWindicates the presence of antifer-\nromagnetic correlation interaction. One can see that for both 0.00 5 and 0.01T\nfield,θCWdecreases with decrease in particle size. Now, according to eq. 1,\nwith decreasein θCW, the exchangeinteractionamongmagnetic ionsdecreases\nand this signifies the weakening of TCin our nanoparticle samples. According\nto mean field theory the frustration parameter is defined as f = θCW/TN. For\nbulk, ‘f’ is nearly 20 and for 75h nanoparticle sample it is 23.6. The gene ral\ncondition for the existence of geometrically frustrated magnets, the ground\nstate should be either antiferromagnetic or spin-glass in nature [18 ].\nTo investigate the low temperature behaviour of TS, magnetization mea-\nsurement down to 0.4K at different applied dc magnetic fields were car ried\nout as shown in Fig. 5. With increase in applied dc magnetic field, the pea k\ntemperature ( TS) is found to be shifted to lower temperatures which is a char-\nacteristic feature of a spin-glass. In such a situation, the peak te mperature\n(TS) is identified as the freezing temperature represented as Tfin the ensuing\nanalysis. The small kinks seen in the main panel of Fig. 5, in low tempera ture\nregion (T <6K) are may be due to experimental errors. We fit the observed\nfreezing temperature to the power law relation which is well known as De\nAlmeidaThouless (AT) equation [19], given by2D spin-glass phase in NiCr2O4nanoparticles 7\n/s48/s46/s48/s48/s48/s48/s48/s46/s48/s48/s52/s53/s48/s46/s48/s48/s57/s48/s48/s46/s48/s49/s51/s53/s48/s46/s48/s48/s48/s48/s48/s48/s46/s48/s48/s48/s50/s57/s48/s46/s48/s48/s48/s53/s56/s48/s46/s48/s48/s48/s56/s55\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s46/s48/s48/s48/s48/s46/s48/s51/s57/s48/s46/s48/s55/s56/s48/s46/s49/s49/s55/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s49/s46/s53/s51/s46/s48/s52/s46/s53\n/s32/s32/s77/s32 /s40/s109\n/s66/s47/s70/s46/s85/s46 /s41/s32/s88/s32/s49/s48/s45/s52\n/s84/s32/s40/s75/s41/s84\n/s83\n/s84\n/s80/s90/s70/s67 /s32\n/s84\n/s67\n/s51/s51/s104/s32/s77/s32 /s40\n/s66/s47/s70/s46/s85/s46 /s41\n/s32/s32\n/s84\n/s83/s84\n/s67/s84\n/s80/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s32\n/s32 /s84/s32/s40/s75/s41/s77/s32 /s40\n/s66/s47/s70/s46/s85/s46 /s41/s32/s88/s32/s49/s48/s45/s52\n/s32/s32\n/s90/s70/s67\n/s84\n/s83\n/s84\n/s67/s84\n/s80/s32\n/s84/s32/s40/s75/s41\n/s32/s55/s53/s104/s32\n/s84\n/s83\n/s84\n/s67/s84\n/s80\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s48/s50/s52/s54/s56\n/s32/s32\n/s84/s32/s40/s75/s41/s77/s32 /s40\n/s66/s47/s70/s46/s85/s46 /s41/s32 /s88/s32/s49/s48/s45/s51\n/s32/s32\n/s90/s70/s67\n/s84\n/s83/s84\n/s67\n/s84\n/s80\n/s32/s32\n/s66/s85/s76/s75/s32\n/s84\n/s83\n/s84\n/s67/s84\n/s80\nFig. 3 Magnetization FC-ZFC measurement of 75h, 33h and bulk NiCr2O4sample in\n0.005T cooling field. Inset shows the ZFC of respective sampl es in 0.005T cooling field. All\nthe three transitions can be clearly seen.\nHAT(Tf)\n∆J∝/parenleftBigg\n1−Tf\nT0\nf/parenrightBigg3/2\n(2)\nandis shown in the inset ofFig. 5. Here, ∆Jis the width ofthe distribution\nof exchange interaction and T0\nfis the freezing temperature at zero magnetic\nfield. The fitting parameter we used is T0\nf. The zero field freezing point was\nobtained by extrapolating the AT line to the temperature axis and is f ound\nto beT0\nf=10.1K as shown in the inset of Fig. 5. The error in determining\nfreezing temperatures is very small (barely visible) and is evaluated by the\ntemperature step size for the measurement of ∼0.2K. It can be seen that\nthe freezing temperature Tf, corresponding to which the magnetization value\nMFC−MZFC=∆Mbecomes non-zero (Fig. 3, 4) indicating the onset of\nfreezing temperature, decreases (low temperature shift) with in crease in ap-\nplied dc magnetic field (H). It shows a Tf∝H2/3dependence, which reflects\n2D surface like spin-glass behaviour. It is worth mentioning here tha t, in bulk\nthe system reflects 3D ferrimagnetic behaviour and in nanoparticle regime\nit reflects a 2D spin-glass like behaviour with Tf= 10.1K. This 2D spin-\nglass like nature is associated with the surface spin character of th eNiCr2O4\nnanoparticles. However, core of the nanoparticle is still ferrimagn etic in na-\nture. In this regard, the temperature dependent response of m agnetization\nfrom bulk to nanoparticle could be more informative (Fig. 3, 4). The w eak-\nening of ferrimagnetic transitions ( TC) and strengthening of low temperature\npeak (TN/Tf) could be seen from bulk to nanoparticle regime with the evolu-\ntion of a new peak ( TP∼90K) towards higher temperature side. Obviously,\nthe surface (shell) spins have dominating role/effect compared to b ulk (core)8 H. Singh\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s48/s52/s48/s46/s48/s48/s56/s48/s46/s48/s49/s50/s48/s46/s48/s49/s54/s48/s46/s48/s50/s48\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57\n/s84\n/s67/s84\n/s83\n/s32/s32\n/s84/s32/s40/s75/s41/s77/s32/s40\n/s66/s47/s70/s46/s85/s46/s41\n/s32/s32\n/s90/s70/s67/s32\n/s84/s32/s40/s75/s41/s77/s32 /s40\n/s66 /s47/s70/s46/s85/s46 /s41\n/s32/s32\n/s66/s85/s76/s75/s32\n/s84\n/s83/s84\n/s67/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s48/s46/s48/s48/s49/s48/s46/s48/s48/s50/s48/s46/s48/s48/s51/s48/s46/s48/s48/s52/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56\n/s84\n/s80/s84\n/s67/s32\n/s32/s84/s32/s40/s75/s41/s77/s32/s40\n/s66/s47/s70/s46/s85/s46/s41\n/s32/s32\n/s90/s70/s67\n/s84\n/s83\n/s84\n/s80\n/s32/s32 /s32/s32\n/s51/s51/s104/s32\n/s84\n/s83/s84\n/s67/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s48/s46/s48/s48/s49/s48/s46/s48/s48/s50/s48/s46/s48/s48/s51/s48/s46/s48/s48/s52/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55\n/s84\n/s67/s32\n/s32/s77/s32/s40\n/s66/s47/s70/s46/s85/s46/s41\n/s84/s32/s40/s75/s41\n/s32/s32\n/s32/s90/s70/s67\n/s84\n/s83\n/s84\n/s80\n/s32/s32 /s32/s55/s53/s104/s32\n/s84\n/s83\n/s84\n/s67/s84\n/s80\nFig. 4FC-ZFC magnetization measurement of 75h, 33h and bulk NiCr2O4sample in 0.1T\ncooling field. Inset shows the ZFC of respective samples. All the three transitions can be\nclearly seen.\nspins in the nanoparticles which leads to the conclusion that ferrimag netically\nordered spins are quite less in number in comparasion to the surface spins.\nThe origin and nature of the TPwill be discussed latter in the letter.\nThe presence of spin-glass like phase in the reported system might b e due\nto superparamagnetism and AT line analysis alone is not sufficient. Hen ce it is\nnecessarytoimplementanotherexperimentalevidencetoestablis htheclaimof\nspin-glassphaseinoursystem.Thus,thefrequencydependenta csusceptibility\nanalysis was done to establish this claim. Fig. 6, shows the real part o f the ac\nsusceptibility ( χ′) as a function of temperature measured at frequencies 7, 73,\nand143Hz.Forthismeasurement,thesamplewasfirstcooleddown to4Kfrom\n300K. Then a probing ac magnetic field of amplitude 1 Oe was applied for the\nac susceptibility measurement. The dc biasing field was set to zero du ring data\naccusation.The freezingtemperature( Tf) isidentified asthe peak in the curve\nand found to be shifted towards high temperature with increase in f requency.\nIt is believed that superparamagnetism (SPM) can also give rise to pe aks in ac\n‘χ’ measurementsaccompaniedby frequencydependent shift in pea k (blocking\ntemperaturein SPM)positions canbe seen,similartothe resultssho wninFig.\n6. The two magnetic phases (SPM and spin-glass) can be distinguishe d by the\nempirical quantity ∆Tf/[Tflog(f)], with values varying from 0.004-0.018 for\nspin-glass to as large as 0.3 for SPM [20]. Here, ∆Tf=Tf−T0\nf, is the shift in\nfreezing temperature for spin-glass phase from T0\nf. For 75h sample, the value\nof the empirical quantity is found to be 0.05 which is close to the spin-g lass\nphase. To understand the dynamical behaviour of the spin glass ph ase near\nfreezing temperature, two different theoretical approaches ha ve been adopted.\nThe first approach assumes that a phase transition takes place at the freezing2D spin-glass phase in NiCr2O4nanoparticles 9\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48 /s50/s50/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48\n/s55/s46/s53 /s56/s46/s48 /s56/s46/s53 /s57/s46/s48 /s57/s46/s53 /s49/s48/s46/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s32/s32/s72/s50/s47/s51\n/s32/s40/s79/s101/s32/s50/s47/s51\n/s41\n/s84\n/s102/s32/s40/s75/s41/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s32/s65/s84/s45/s76/s105/s110/s101/s32/s70/s105/s116\n/s84\n/s102/s48/s77 /s32/s40/s101/s109 /s117/s47/s77 /s111/s108/s101/s41\n/s32/s32\n/s84/s32/s40/s75/s41/s32/s48/s46/s48/s48/s53/s84\n/s32/s48/s46/s48/s49/s84\n/s32/s48/s46/s48/s53/s84\n/s32/s48/s46/s49/s84\nFig. 5ZFC curves for 75h sample, measured at different cooling field of 0.005, 0.01, 0.05,\n0.1T. The inset shows the field field dependence of freezing te mperature and the solid line\nrepresents a fit with De- Almedia Thouless equation.\ntemperature in the vicinity of which a critical behaviour in the temper ature\ndependence ofrelaxationtime ( τ) isexpected. This is knownascriticalslowing\ndown model. In this model, the critical relaxation time, defined as τ= 1/f,\nnear the transition point is related to the correlation length (Tf\nT0\nf−1) of the\nspins and is expressed in the form of a power law as given below [21,22, 23].\nThe freezing temperatures obtained from Fig. 6 were fitted with th e power\nlaw equation,\nτ=τ0/parenleftBigg\nTf\nTf\n0−1/parenrightBigg−zν\n, (3)\nwhere,τ=1/f, f is the frequency of the ac χmeasurement, τ0is the relax-\nation time constant normally lying in the range of 10−9to 10−13sec andzνis\nthe critical exponent. The fitting shown in the inset (a) of Fig. 6, ha s yielded\nparameter values as τ0= 3.6×10−6,T0\nf= 8.66K and zν=11.1. Here, the\nvalue of relaxation time constant τ0= 3.6×10−6is larger at least by a fac-\ntor of 103than the normal value for spin glass phase. The reason for this is\nnot understood. However, the value of T0\nf(= 8.7K) is close to the T0\nfvalue\n10.1K obtained from AT line analysis. The obtained value of critical exp onent\n‘zν’ (11.1K) matches well with the typical value of ‘ zν’ for spin-glass system\nranging from 4 to 12 [20,24], suggesting a spin-glass ground state in NiCr2O4\nnanoparticles.\nThe second approach assumes that the freezing phase transition is a non-\nequilibrium phenomenon and the dynamical properties of spin glass ph ase can\nbe explored by Vogel-Fulcher law. This law takes into account the inte racting\nproperty of spin-glass clusters. The Vogel-Fulcher law is expresse d as,10 H. Singh\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54/s48/s46/s50/s48/s48/s46/s50/s52/s48/s46/s50/s56/s48/s46/s51/s50\n/s45/s53 /s45/s52 /s45/s51 /s45/s50/s49/s50/s46/s48/s49/s50/s46/s50/s49/s50/s46/s52/s49/s50/s46/s54/s49/s50/s46/s56/s49/s51/s46/s48/s69/s120/s112/s46/s32/s100/s97/s116/s97\n/s32/s70/s105/s116\n/s32/s32\n/s32/s32/s40/s98/s41\n/s108/s110/s32/s40/s49/s47/s102/s41/s84\n/s102/s32/s40/s75/s41/s49/s50/s46/s48 /s49/s50/s46/s51 /s49/s50/s46/s54 /s49/s50/s46/s57/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57/s48/s46/s49/s50/s48/s46/s49/s53\n/s32/s32/s40/s115/s101/s99/s46/s41\n/s84\n/s102/s32/s40/s75/s41/s32/s69/s120/s112/s46/s32/s100/s97/s116/s97\n/s32/s70/s105/s116/s40/s97/s41\n/s32/s32/s39/s32/s88/s32/s49/s48/s45/s51\n/s32/s40/s101/s109/s117/s47/s103/s41\n/s84/s32/s40/s75/s41/s32/s55/s72/s122\n/s32/s55/s51/s72/s122\n/s32/s49/s52/s51/s72/s122\nFig. 6 A plot of AC susceptibility as a function of temperature meas ured at different\nfrequencies of 7, 73 and 143 Hz for 75h sample. The inset (a) an d (b) shows the fitting as\nper eq. 3 and 4.\nτ=τ0exp/bracketleftbiggEa\nkB(Tf−T0)/bracketrightbigg\n, (4)\nhere,Earepresents the energy barrier or activation energy and T0is the\nphenomenological parameter describing interaction between clust ers. The plot\nofTfvs. ln(1/f) is plotted together with the fitted curve given in eq. 4 an d is\nshown in the inset (b) of Fig. 6. The fitted parameters are Ea/kB= 58.9K,\nτ0= 5.22×10−8andT0= 8.03K. The non-negative value of T0= 8.03K\nindicates the presence of interacting spin-glass clusters. Moreov er, forT0= 0,\neq.4reducestoArrheniuslawgivenby τ=τ0exp(Ea/kBTf).Inthissituation,\ntheinter-clusterinteractionisnegligible.Inviewofthisrelation,the fitteddata\ngive unreasonably low value of τ0(∼10−40sec), which seems much faster than\nthe atomic relaxation of spins ( ∼10−14sec) which is physically not possible.\nNow our basic concern is to understand the mechanism responsible f or the\ngeneration of spin glass like phase in NiCr2O4nanoparticle which shows a\ndifferent behaviour as compared to its bulk counterpart. The 2D na ture of\nthe spin glass phase established using AT-line fitting indicates a surfa ce effect.\nThis may be associated with either spin canting or surface spin disord er. Spin\ncanting exists due to two competing magnetic exchanges (one is isot ropic and\nthe other is asymmetric in nature). The marginal weakening of ferr imagnetic\ntransition in 75h sample is an indication of weakening of isotropic excha nge.\nFurther, the bulk NiCr2O4does not show glassy feature in our measurements\nconfirming existing reports to the best of our knowledge. Therefo re, we ruled\nout spin canting as the reason for existing surface effect in NiCr2O4nanopar-\nticle. Actually, surface effects result from the lack of translationa lsymmetry at\nthe boundaries of the particle because of the lower coordination nu mber there\nand the existence of broken magnetic exchange bond which leads to surface2D spin-glass phase in NiCr2O4nanoparticles 11\nspin disorder and frustration [27]. Another important effect is the fi nite size\neffect which is observed in nanoparticle sample. This effect originates from the\ncut off of some characteristic length due to the purely geometric co nstraint on\nfinite volume. It results in superparamagnetic behaviour, which is no t present\nin our sample (already pointed out in earlier discussion). Finally, it is obs erved\nthat nanoparticle sample has higher value of mean field frustration p arame-\nter ‘f’ (23.6) as compare to its bulk sample (f=20). Also, weakening o fTCin\nnanoparticlesampleindicatesabsenceoflongrangeferrimagnetico rderingand\nin fact exhibits spin-glass behaviour. This fulfils the general conditio n for the\nsystem to be geometrically frustrated magnet (GFM). Hence, the NiCr2O4\nnanoparticle sample might be retaining geometrical frustration.\n4 Conclusion\nFrom the entire study, the following points can be concluded: 1. The nanopar-\nticle ofNiCr2O4shows 2D spin glass like character at low temperature with\nTf= 10.1K. Vogel-Fulcher law analysis confirms the existence of interac ting\nspin clusters.\n2. The surface spin disorder seems to be the reason for the existe nce of 2D\nspin glass feature.\n3. The decrease in magnetization value of NiCr2O4nanoparticle compared to\nits bulk counterpart may be due to the surface spin disorder.\n4. The existence of new transition ( TP) may be due to the presence of weak\nferrimagnetic spin clusters.\n5. TheNiCr2O4nanoparticles may be considered as a GFM due to higher\nvalue of ‘f’ parameter, absence of long range ordering and the pre sence of spin\nglass ground state.\n5 Acknowledgement\nWewouldliketothankMinistryofHumanResourceandDevelopment(M HRD),\nGovt. of India, for funding.\nReferences\n1. S. Krupika, and P. Novak, ( Ferromagnetic Materials ), edited by E. P. Wolfarth, Vol 3,\np.189. North-Holand, Amsterdam (1982).\n2. K. Tomiyasu, H. Hiraka, K. Ohoyama, and K. Yamada, Resonan ce-Like Magnetic Exci-\ntations in Spinel Ferrimagnets FeCr2O4andNiCr2O4Observed by Neutron Scattering,\nJ. Phys. Soc. Jpn., 77, 12 (2008).\n3. S. Klemme, and J. C. 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Sharma, S. K. Kulshreshtha, and G. K. Dey, Dynamics\nof spin clusters in amorphous Fe2O3, Phys. Rev. B, 72, 174408 (2005).\n24. S. Bedanta, and W. Kleemann, Supermagnetism, J. Phys. D: Appl. Phys., 42, 013001\n(2009).\n25. D. N. H. Nam, R. Mathieu, P. Nordblad, N. V. Khiem, and N. X. Phuc, Spin-glass\ndynamics of La0.95Sr0.05CoO3, Phys. Rev. B, 62, 8989 (2000).\n26. K.Gunnarsson,P.Svedlindh, P.Nordblad,L.Lundgren,H .Aruga,andA.Ito, Dynamics\nof an Ising Spin-Glass in the Vicinity of the Spin-Glass Temp erature, Phys. Rev. Lett.,\n61, 754 (1988).\n27. X. Batlle, and A. Labarta, Finite-size effects in fine part icles: magnetic and transport\nproperties, J. Phys. D: Appl. Phys., 35, R15-R42 (2002)." }, { "title": "2007.08908v3.Coherent_coupling_between_multiple_ferrimagnetic_spheres_and_a_microwave_cavity_in_the_quantum_limit.pdf", "content": "Coherent coupling between multiple ferrimagnetic spheres\nand a microwave cavity at millikelvin temperatures\nN. Crescini,1, 2,\u0003C. Braggio,2, 3G. Carugno,2, 3A. Ortolan,1and G. Ruoso1\n1INFN-Laboratori Nazionali di Legnaro, Viale dell'Universit\u0012 a 2, 35020 Legnaro (PD), Italy\n2Dipartimento di Fisica e Astronomia, Via Marzolo 8, 35131 Padova, Italy\n3INFN-Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy\n(Dated: March 7, 2022)\nThe spin resonance of electrons can be coupled to a microwave cavity mode to obtain a photon-\nmagnon hybrid system. These quantum systems are widely studied for both fundamental physics\nand technological quantum applications. In this article, the behavior of a large number of ferrimag-\nnetic spheres coupled to a single cavity is put under test. We use second-quantization modeling\nof harmonic oscillators to theoretically describe our experimental setup and understand the in\ru-\nence of several parameters. The magnon-polariton dispersion relation is used to characterize the\nsystem, with a particular focus on the vacuum Rabi mode splitting due to multiple spheres. We\ncombine the results obtained with simple hybrid systems to analyze the behavior of a more complex\none, and show that it can be devised in such a way to minimize the degrees of freedom needed to\ncompletely describe it. By studying single-sphere coupling two possible size-e\u000bects related to the\nsample diameter have been identi\fed, while multiple-spheres con\fgurations reveal how to upscale\nthe system. This characterization is useful for the implementation of an axion-to-electromagnetic\n\feld transducer in a ferromagnetic haloscope for dark matter searches. Our dedicated setup, con-\nsisting in ten 2 mm-diameter YIG spheres coupled to a copper microwave cavity, is used for this aim\nand studied at mK temperatures. Moreover, we show that novel applications of optimally-controlled\nhybrid systems can be foreseen for setups embedding a large number of samples.\nI. INTRODUCTION\nIn the fruitful and renowned \feld of light-matter\ninteraction1, the study of hybrid quantum systems based\non magnonics gave outstanding results2,3. Magnons,\nquanta of spin excitations, can coherently couple to pho-\ntons through a magnetic dipole interaction4,5. The elec-\ntron spin resonance of a magnetic material can thus be\ncoupled to the rf magnetic \feld of a microwave cavity\nmode, to form a photon-magnon hybrid system (HS)6{9\nat GHz frequencies. This kind of HSs are employed\nin \felds including, but not limited to, the develop-\nment of quantum computers10, quantum networks11,12,\nand quantum sensing13. In these areas, the coherent\ncoupling of spin ensembles, microwave cavities6{9,14{16\nand superconducting quantum circuits17{19is used to\ncreate quantum memories20,21, to convert optical pho-\ntons to microwaves22and vice versa23{25, and to de-\ntect single quanta of magnetization26. In spintronics,\nmagnons27are suitable carriers of spin information, and\nhave several advantages over electrical currents used in\nmodern electronics. Furthermore, the investigation of\nnon-Hermitian quantum mechanics28is currently pur-\nsued with HSs. Exceptional points are the signature\nof non-Hermitian physics29,30, and their observation was\nrecently proposed31, and experimentally veri\fed32in\nphoton-magnon HSs. These results were followed by the\ndemonstration of exceptional surfaces33, and the measure\nof a third order exceptional point has been proposed in\nsystems of multiple ferrimagnetic spheres34. Besides the\nstudy of non-Hermitian quantum mechanics, di\u000berent ap-\nplications can be envisioned for these points, for example\nas sensitive probes of magnetic \felds35. These and manyother studies36{40maintain that photon-magnon HSs are\nan outstanding testbed for the study of many physical\nphenomena.\nThe study of HSs comprising a large amount of\nmagnetic samples has been explored to create gradient\nmemories9, or to detect faint pseudo-magnetic \felds41.\nIn precision physics, rf spin-magnetometers based on HSs\nare used as axion haloscopes under the name of ferro-\nmagnetic haloscopes41,42(FH). In these devices, a large\nnumber of spins increases the sensitivity to magnetic os-\ncillations induced by dark matter axions. As the axion\ncouples to magnons43in a FH the HS acts as an axionic-\nto-electromagnetic \feld transducer.\nThis work reports on the coherent coupling of ten\n2.1 mm-diameter ferrimagnetic spheres to a cylindrical\ncopper cavity, with the aim of implementing the result-\ning HS in a FH. Our results, however, go beyond this\nsole purpose, and investigate the dynamics of such large\nand complex systems, which are promising for multiple\nusages9,39,44{47. In our setup, the strong coupling regime\nis largely reached already with a single sphere, thus we\ncan study the scaling without qualitatively changing the\nbehavior of the HS. We singularly tested spheres with\ndiameters ranging from 0.5 mm to 2.5 mm and observed\ntwo di\u000berent size-e\u000bects, a\u000becting the g-factor and the\nsphere's zero-\feld splitting. By choosing a diameter of\n2.1 mm, we focus on the coherent coupling of multiple\nspheres. The multi-samples design is preferred to a sin-\ngle rod of magnetic material for practical reasons, like\nthe material availability and geometric demagnetization.\nThe in\ruence of the dimension and relative distance of\nthe samples are consistently accounted for in a model\nwhich precisely reproduces the experimental data. OurarXiv:2007.08908v3 [quant-ph] 3 Mar 20222\nstudy thus demonstrates that it is feasible to scale up a\nHS while preserving its optimal control.\nIn Sec. II we detail the model used to describe the ex-\nperiment. Sec. III illustrates the setup, and is further di-\nvided into two parts: the \frst (III A) explains the room\ntemperature tests used to understand the behavior of sin-\ngle spheres, while the second (III B) reports on the cou-\npling of many multiple spheres to the cavity, and on the\nroom temperature and ultra cryogenic tests of the result-\ning HS. The last part of the paper, Sec. IV, is dedicated\nto the possible improvements of the setup, in terms of\nincreasing the quantity of material coupled to the cavity,\nand to a summary of the obtained results.\nII. SECOND-QUANTIZATION MODEL\nThis section deals with the model used to study the\nHS dynamics, which essentially consists in a system of\ncoupled harmonic oscillators in the second-quantization\nformalism48,49that e\u000eciently describes the number of\n(quasi-)particle in a state by using creation and annihi-\nlation operators to add or remove a quanta. Similar ap-\nproaches have been used to study the interaction of mul-\ntiple qubits in a cavity50, and test the Tavis-Cummings\nmodel51for a low number of two-level systems. Since\nwe are interested in describing resonant quanta, raising\nand lowering operators share the same algebra of a col-\nlection of interacting quantum harmonic oscillators. Letus consider Nphoton modes X=fx1;x2;:::;x Ng, and\nMmagnon modes Y=fy1;y2;:::;y Mg, and label with\n!andgtheir frequencies and couplings, respectively. Ev-\nery modeXcan couple to any of the modes Y, so the\nHamiltonian of the system in the rotating wave approxi-\nmation reads\nH=~X\nx2X!xxyx+~X\ny2Y!yyyy\n+X\nx2XX\ny2Ygxy(xyy+yyx)\n+X\ni6=jgyiyj(yy\niyj+yy\njyi);(1)\nwhere~is the reduced Planck constant, and xy,yy(x,\ny) are the creation (annihilation) operators of the corre-\nsponding mode quanta. Our interest lies in the evolution\nof the mean values of xandyoperators that can be cal-\nculated by Heisenberg-Langevin equations. The e\u000bects of\ndissipations in a resonant system due to its coupling with\na thermal reservoir can be easily taken into account by\nadding an imaginary part to the mode frequencies which\ncorrespond to their linewidths \rs. Thus the equations of\nmotion for the HS evolution can be recast as a \frst or-\nder system of di\u000berential equations (see Appendix A for\nthe complete derivation), and its solution can be readily\nfound in Fourier space. The associated HS matrix reads\nH=0\nBBBBBBBBBBBBBBBB@!x1\u0000i\rx1=2 0 ::: 0\n0!x2\u0000i\rx2=2::: 0\n............\n0 0 ::: ! xN\u0000i\rxN=2gx1y1gx1y2::: g x1yM\ngx2y1gx2y2::: g x2yM............\ngxNy1gxNy2::: g xNyM\ngx1y1gx2y1::: g xNy1\ngx1y2gx2y2::: g xNy2\n............\ngx1yMgx2yM::: g xNyM!y1\u0000i\ry1=2gy2y1::: g y1yM\ngy1y2!y2\u0000i\ry2=2::: g y2yM............\ngy1yMgy2yM::: ! yM\u0000i\ryM=21\nCCCCCCCCCCCCCCCCA:(2)\nThe top-left block stands for the cavity modes, which are\nconsidered uncoupled to each other, as typically their\nresonant frequency di\u000berence is much larger than their\nlinewidths. The lower-right block shows the material\nmagnetic modes, which may auto-interact mainly thanks\nto the dipole coupling between di\u000berent spheres (see Sec-\ntion III). The o\u000b-diagonal blocks are the couplings be-\ntween the cavity and magnetic modes. The matrix in\nEq. (2) completely describes the system dynamics, and\ncan be used to calculate the dispersion relation of themagnon-polariton.\nIn our study we variate the static magnetic \feld to\nunderstand how it a\u000bects the transmission function of\nthe HS. The spin precession frequency of a free elec-\ntron in a generic magnetic \feld Bis!0(B) =\rB, with\n\r= (2\u0019)28 GHz=T. This basic element can be used in\nthe theoretical model to account for a variation of the\nmagnetostatic mode frequencies !Y. In particular, we\nwrite the frequencies of the Ymodes as\n!Y(B) =\r(B+bY); (3)3\nwhere the zero \feld shift \rbYis a constant depending on\nthe magnetostatic mode under consideration. In Eq. (3)\nwe choose a linear dependence between !and the ap-\nplied static \feld, but it is straightforward to include\nnon-linear magnetic \feld dependencies if any are present.\nThe poles of the cavity magnon-polariton dispersion rela-\ntion can be found by solving the characteristic equation\ndet(K(!;!Y)) = 0, where\nK(!;!Y) =!IN+M\u0000H(!Y); (4)\nandIN+Mis the identity matrix of dimension N+M.\nHere!can be seen as the frequency of a monochromatic\nprobe signal. Experimental spectra are collected through\ntwo antennas coupled to the resonant cavity. In princi-\nple, we should consider the transmission of all the cavity\nmodes (see Appendix A), but, as the model does not ac-\ncount for the antenna coupling strength, we can restrict\nto one mode and neglect other transmission channels. To\ncalculate the transmission spectra of the HS, we can con-\nsider the monochromatic system excitation coupled to\nthe \frst cavity mode x1\nsx1(!;!Y) =\rx1!2\n2j(K\u00001(!;!Y)\u0001^e1)1j2; (5)\nwhere ^e1is the unit vector of x1. Note that the param-\neters ofHcan be determined experimentally by \ftting\nthe transmission spectra to Eq. (5).\nIn the following, to indicate a particular HS con\fgu-\nration we will write the Hamiltonian of the system, and\ntake for granted that to get the dispersion relation one\nfollows Eq. (4) and (5).\nTo reduce the complexity of the HS, and understand\nhow the di\u000berent couplings a\u000bect its dynamics, we re-\nduce the number of considered modes to four: two cavity\nmodescandd, and two magnetic modes mandn. The\nmodec\u0011x1corresponds to the one coupled to the an-\ntennas, and m\u0011y1is the Kittel mode with frequency\n!m. The resulting Hamiltonian is\nH4=0\nBB@!c\u0000i\n2\rc 0gcmgcn\n0!d\u0000i\n2\rdgdmgdn\ngcmgdm!m\u0000i\n2\rmgmn\ngcngdngmn!n\u0000i\n2\rn1\nCCA:\n(6)\nThe resonant frequency of the cavity modes are set\nto!c= (2\u0019)10:65 GHz and !d= (2\u0019)10:9 GHz, and\nthe linewidths to \rc=\rd= (2\u0019)1:0 MHz, while\nthe linewidths of the magnon modes \rm=\rn=\n(2\u0019)2:0 MHz. The e\u000bects of the di\u000berent terms are stud-\nied by \frst setting to zero all the couplings but gcm, set to\n(2\u0019)90 MHz to be close to the value measured with a sin-\ngle 2.1 mm diameter sphere. Then, the other couplings\nare selectively turned on one by one, and the resulting\ndispersion relations are compared in Fig. 1.\nPlot 1(a) is a usual anticrossing curve, where the cou-\npling 2gcmis the frequency splitting of the two HS modes\nwhen!=!m. This HS is the starting point for all the\nfollowing considerations.\n(a) (b)\n(c) (d)c-m c-m-d\nn-c-m c-m-n\nFIG. 1: Plots of the transmission spectra sx1(!;! 0) illustrat-\ning the dispersion relations of the magnon-polariton systems.\nThe color scale is in logarithmic arbitrary units, where blue\ncorresponds to low and light green to high transmission. The\ndash between the labels of two modes indicates that the two\nare coupled. Upper plots: normal anticrossing curve (a), and\ncoupling of the Kittel mode with the two cavity modes cand\nd(b). Lower plots: (c) c-mode coupled to two magnetostatic\nmodesmandn, withbn6= 0; (d)c-mode coupled to m, cou-\npled ton, withbn= 0 andgmn= 25 MHz. One can obtain\nthe same plot with gmn= 0 andbn\u00180. See text for further\ndetails.\nThe interaction of the Kittel magnetic mode mwith\ntwo cavity modes c;dis considered in Fig. 1(b), where we\nset equal couplings gcm=gdm, producing a combination\nof two anticrossing curves. The second cavity mode d,\nat the frequency !d= (2\u0019)10:9 GHz, is not coupled to\nthe antennas, as is shown by Eq. (5). For this reason no\ntransmission happens at !dif not mediated by the m\nmode.\nFig. 1(c) shows a single cavity mode ccoupled to two\nindependent magnetostatic modes, mandn. The fre-\nquency of the mode nis obtained by shifting !mof\n\rbn= (2\u0019)0:4 GHz [see Eq. (3)], corresponding to a \feld\nbias of about 15 mT. This o\u000bset \feld may describe higher\norder modes52,53or two spheres coupled to the same\nmode but through slightly di\u000berent \felds. In fact, as we\nshall see in the following Section, a size e\u000bect related to\nthe diameter of the spheres54{58is the presence of a bias\n\feld. The coupling of this mode to the cavity mode is\narbitrarily set to gcn= (2\u0019)25 MHz to roughly resemble\nthe coupling to higher order modes53.\nThe dispersion relation (d) of Fig. 1 is obtained by\nadding a magnon-magnon coupling gmn= (2\u0019)25 MHz\nbetween the magnetostatic mode nand the Kittel mode\nm, while keeping nuncoupled from the cavity mode c. A\nthird hybrid mode appears, its resonance frequency lies4\nsampleE-\feld\nhigh\nlowc d\nFIG. 2: Section of the cavity body geometrical shape, and\npro\fle of the canddmodes. The sample lies on a maximum\nof the rf magnetic \feld for both modes.\nbetween the two ones of the c-msystem, and dispersively\nshifts them. It is interesting to note that a similar result\ncan be obtained by using a small detuning bnin the pre-\nviousn-c-mcon\fguration but with no gmncoupling and\nsuch that\rm\u001c\rbn<2p\ng2cm+g2cn. This means that an\nidentical dispersion relation results from hybridizing, un-\nder the same static \feld, a cavity and two spheres with\na di\u000berent o\u000bset \feld bY. The central mode is usually\nreferred to as dark mode, which was studied e. g. for\nqubits50and for magnons9,59. The fact that the e\u000bect of\na magnon-magnon coupling results similar to the one of\nan-c-msystem with a small bias bnis further discussed\nand experimentally explored in Sec. III B.\nWhenMmagnon modes are degenerate (i. e. have the\nsame resonant frequency) they couple with a cavity mode\nas a single oscillator, and the vacuum Rabi splitting, the\nfrequency di\u000berence of the two hybrid modes, scales as\nthe root of the sum of the squared couplings. In this case\nthe two-modes system c-mholds. Ideally, the addition\nof more spheres gives the same result, and the splitting\nscales as the square root of the total number of spins.\nHowever, a large increment of the number of spheres in\na single cavity forces us to face the dynamics outlined\nin Fig. 1. These e\u000bects can be combined to obtain more\nconvoluted dispersion relations, which can be explained\nthanks to the understanding of their basic elements pro-\nvided by this Section.\nIII. EXPERIMENTAL RESULTS\nThe photonic resonators of our setup are copper cavi-\nties with a cylindrical body and cone-shaped end caps60.\nThe TM110 mode of a perfectly cylindrical cavity is de-\ngenerate for rotations about its axis, complicating the\ncoupling to the sample. To lift the degeneracy, the sym-\nmetry of the cavity body is broken by two \rat walls\nwhich reduce its diameter, as shown in Fig. 2. As a conse-\nquence, the two polarizations have frequencies which dif-\nfers about 200 MHz, and we identify the two correspond-\ning modes with the canddappearing in Eq. (6). Themagnetic material is Yttrium Iron Garnet (Y 3O5Fe12), a\nferrimagnetic insulator61{64widely known due to its ap-\nplications in microwave electronics65, mainly related to\nits narrow linewidth66{68. The values of resonance fre-\nquencies, linewidths and couplings used in the previous\nSection were chosen to resemble the measured ones of\ncopper cavities and of YIG.\nWe tested several spheres of di\u000berent diameter to infer\nthe maximum size that can be used and characterized\nwithout taking into account perturbative e\u000bects. Except\nfor 0.5 mm and 1 mm-diameter spheres, the others were\nmanufactured on site from a large YIG single crystal.\nAfter cutting the crystal into small cubes, spheres are\nobtained by grinding them with the procedure described\nin Appendix B. In the measurement setup, a fused silica\npipe and PTFE cups hold the spheres in a position close\nto the cavity center and let them rotate freely, guarantee-\ning alignment of their easy axis to the external magnetic\n\feld. Such condition is necessary in order to coherently\nmaximize the coupling strength as prescribed in [41]. In\nthe following tests this condition is always ful\flled, and\nin Sections III A and III B it is analyzed and understood.\nThe magnetic \feld is provided by a superconducting\nmagnet, and care is taken to place the mounted spheres\nat its center: within a cylindrical region of 0.5 cm diame-\nter and 7 cm height the homogeneity of the \feld is better\nthan 7 ppm, guaranteeing no inhomogeneous broadening\nof the electron spin resonance. The linearity and ab-\nsence of o\u000bsets for the relation between the power supply\ncurrent and the magnetic \feld has been veri\fed using a\nparamagnetic BDPA sample, whose g-factor is remark-\nably close to the one of the free electron. Further details\non the experimental apparatus can be found in Appendix\nC.\nThe aim of this analysis is to build an optimally-\ncontrolled HS which embeds the largest possible quantity\nof magnetic material. The measurements in Sec. III A\nand in the \frst part of Sec. III B are obtained at room\ntemperature, while the test of the haloscope transducer\nwas performed at T= 90 mK. At this temperature, quan-\ntum e\u000bects prevail over thermal ones, as kBT <~!,\nwherekBis the Boltzman constant.\nA. Single sphere coupling\nIn the \frst study, single spheres of diameters rang-\ning from 0.5 mm to 2.5 mm have been mounted in two\ndi\u000berent cavities. The TM110 mode frequency is 14.09\nGHz for the \frst cavity and 10.65 GHz for the second,\nwith wavelengths of the rf radiation of 21 mm and 28 mm,\nrespectively. Such values allow us to test the system\nin a regime where the diameter-to-wavelength ratio is\nlarger than 1/10, and dimensional e\u000bects should come\ninto play54.\nFig. 3 shows some results in the form of anticrossing\ncurves. We have veri\fed that the hybridization scales\nas the square root of sample volume, con\frming that all5\n(a) 0.5 mm (b) 1.5 mm\n(c) 2.5 mm (d) 2.5 mm\nFIG. 3: Anticrossing curve of spheres with di\u000berent diameter.\n(a) and (b) show the dispersion relation of single spheres of\ndi\u000berent diameters coupled to the same cavity mode, the cou-\npling strength is obtained by the \ft (dashed lines), and scales\nas expected. (c) and (d) are the anticrossings of the same\nsphere coupled to the TM110 mode of two di\u000berent cavities,\nand the dashed line is the lower frequency hybrid mode. In\nplot (d) one can see a discrepancy between the theory and\nthe result, possibly due to the fact that the sphere diameter\nis larger than one tenth of the microwave \feld wavelength.\nthe spins are behaving coherently, in agreement with the\nTavis-Cummings model. The same behavior is expected\nwhen the number of spheres is increased.\nOne can note that the dispersion relation of Fig. 3(c)\nclosely resembles the one of Fig. 1(b), as the coupling\nstrength between the 2.5 mm sphere and the 10.65 GHz\nmode of the cavity is large enough to involve both the\ncanddpolarizations of this TM110 mode. As expected\nfrom the symmetry of the system, the coupling strengths\ngcmandgdmare about the same. In Fig. 1(b) the cou-\npling to the dmode is much weaker because of the char-\nacteristic of the antennas, discussed in Section II.\nA \frst dimensional e\u000bect is observed when the largest\ndiameter YIG sphere is coupled to the higher frequency\ncavity, i.e. the one with smaller wavelength. It consists\nin an apparent variation of the g-factor, as was \frst ev-\nidenced in [55] for various ferrites. The 2.5 mm sphere's\nelectrons, placed in the 14 GHz cavity, exhibit a gyro-\nmagnetic ratio of 26 GHz/T and not of 28 GHz/T, as re-\nported in Fig. 3(d), which corresponds to an apparent\ng-factor of 1.86. In the \fgure, the dashed line shows\nthe expected behavior of the hybrid mode's resonant fre-\nquency for g= 2, which signi\fcantly di\u000bers from the\nmeasured spectra. The e\u000bect is not present if the same\nsphere is coupled to the lower frequency cavity, as shown\nin Fig. 3(c), were the dashed line is superimposed to the\nmeasured hybrid frequency. The variation of the g-factor\nbmFIG. 4: Relation between the sphere's diameter and the full\nhybridization \feld Bfh, measured with the 14 GHz cavity. The\npositive o\u000bset follows from the zero-\feld splitting of the YIG.\nwas observed only when the sample size exceeds one tenth\nof the radiation \feld wavelength, in agreement with the-\noretical suggestions54.\nA second size e\u000bect has also been evidenced in our\nmeasurements, and it relates the dimension of the ferrite\nto its resonant frequency. Such e\u000bect was \frst observed\nin the \ffties55and described ten years later55,57,58, and\nstates that larger spheres need higher magnetic \felds to\nreach the same resonance frequency. Here, Bfhis the ap-\nplied magnetic \feld value that realizes full hybridization,\nmeasured through the current \rowing in the magnet. In\nperfectly spherical samples, the related theory55,57,58in-\ntroduces a diameter dependent o\u000bset frequency\nbm(\u001e) =!c=\r\u0000Bfh=\u00002\u00192M0\n45\u00152m(5 +\u000fr)\u001e2+b0(7)\nwhereM0'178 mT and \u000frare the YIG saturation\nmagnetization and relative dielectric constant, \u0015mis the\nwavelength associated to the Larmor frequency !m,\u001eis\nthe sphere's diameter, and b0is the zero-diameter o\u000bset.\nFig. 4 shows the measured values for our set of spheres,\ntogether with a linear \ft of the data. Using Eq. (7), the\nmagnetic \feld o\u000bset for null sphere diameter is found to\nbeb0= 14\u00069 mT, compatible with the zero-\feld splitting\ncalculations performed for YIG69. The resulting slope is\n6:2 mT=mm2, from which we extract a value of \u000fr'30,\nwhich di\u000bers by a factor about 2 from the one found in\nthe literature \u000fr'1570,71. It is true that other measure-\nments of the YIG permittivity yielded values higher72or\nlower73than 15, however this discrepancy could be ex-\nplained by some other diameter-dependent size-e\u000bect, for\nwhich indications may already be present in the X-band\nmeasurements of a previous work55. Furthermore, as we\ndo not have a precise control over the sample shape, it is\nnot possible to exclude that our spheres are a\u000bected by\na size-dependent ellipticity that reduces the goodness of\nEq. (7).\nWhilst the second size-e\u000bect is not expected to in\ru-\nence the magnon-to-photon transduction, it is not clear6\nwhether the variation of the gyromagnetic ratio does. In\nthe forthcoming measurements, to avoid this issue, we\nconservatively choose to engineer our HS-based trans-\nducer employing spheres almost with the same diameter,\nnamely\u001e'2:1 mm.\nB. Multiple spheres acting as one\nTo move forward in the realization of the transducer,\none needs to grasp the coupling of two spheres before\nscaling up the system to an arbitrarily large number of\nsamples. We focus on understanding the system disper-\nsion relation in the light of the interactions shown in\nFig. 1, to maintain the optimal control of the HS, and\ndemonstrate its applicability as an axion to electromag-\nnetic transducer.\nFirst of all, as it is clear from Fig. 4, two spheres with\ndi\u000berent diameters will exhibit di\u000berent Larmor frequen-\ncies even when subjected to the same magnetic \feld. An\nHS with two such spheres would have a dispersion re-\nlation as the one described in Fig. 1(d), i.e. equivalent\nto the situation of two samples with di\u000berent zero \feld\nsplitting and with the presence of a dark mode. It follows\nthat, to preserve the control of the HS and the coherent\ncoupling of a large number of spheres, the samples' di-\nameter should be as akin as possible.\nSecondly, the chance of a magnon bouncing through\nmagnetic modes before being converted to photons has\nto be minimum, otherwise it will result in a non-negligible\nsignal loss. Magnons in di\u000berent samples are assumed to\ninteract with each other through a dipole-like potential,\nhence we studied the sphere-sphere interaction by varying\nthe distance between two YIGs as shown in Fig. 5, which\ncan be compared with the theoretical results of Fig. 1(d).\nThe spheres are placed along the cavity axis in both rf\nand dc uniform magnetic \felds. Being the dc \feld aligned\nwith the cavity axis, when this is turned on all the spheres\nwill have their easy axis along the same line, i.e. the\ncavity axis again. Along this direction, the transmission\nspectra are independent of the sphere position provided\nthat this is kept in a central range of about 5 cm length.\nIn our measurements, we will not care about the ab-\nsolute positions of the spheres, albeit in their relative\nseparation, which, as we will see, is strongly in\ruenc-\ning the results. Following the plots of Fig. 5, in (a) and\n(b) we obtain the usual hybridization with one sphere\nby turning on the static \feld. Then, in (c), a second\nsphere is introduced in the system with less than 0.5 mm\nof spacing from the \frst one: the vacuum Rabi splitting\nincreases and extra peaks appear between the two hy-\nbrid modes. By increasing the separation between the\nspheres to 4 mm \u00182\u001e, we obtain the plot (d) showing\na reduced amplitude of the peaks which are not hybrid\nmodes and a slightly smaller Rabi splitting, now closer to\n\u0018p\n2 times the single sphere case. Relying on the model,\ndescribed essentially by Figure 1(d), the plots 5(c) and\nB=0\nB=0.5 T\nB=0.5 T\nB=0.5 T(a)\n(d)(c)(b)YIG65 dBFIG. 5: Coherent coupling of two spheres, and e\u000bect of their\nrelative distance. On the left side of the plots four schematic\ndrawings show the YIGs (black sphere) position inside the\npipe (blue). The pipe is placed on the cavity axis, parallel to\nthe static magnetic \feld. Plots (a) and (b) show the single\nsphere con\fguration without and with magnetic \feld, that\nproduces the hybrid modes indicated by the small vertical\narrows. In (c) one sphere is placed close to the \frst one, with\nthe e\u000bect of increasing the coupling, i.e. the hybrid mode\nseparation, but also of generating a number of dark modes\nin the intermediate frequency interval. It is evident from (d)\nthat the strength of such modes is reduced by tens of dB\nthanks to the distance between the two YIGs, in agreement\nwith what is expected from the model.\n(d) are evidencing the e\u000bects of a magnon-magnon cou-\npling for close spheres: an increase of the splitting due\nto dispersive shifts of the hybrid resonances and a high\ntransmission through the dark modes. These observa-\ntions bolster the assumption that the sphere-sphere cou-\npling is dipole-like, as it depends on their distance and,\nspeci\fcally, it is much weakened when the samples are\nseparated by more than their diameter.\nThe scaling of the system to a larger number of spheres,\nnecessary to realize a more sensitive ferromagnetic halo-\nscope, is straightforward once the behavior of the two-\nYIGs HS is understood. The conditions considered to\nengineer such an enhanced HS are summarized as fol-\nlows: (i) the single sphere diameter is smaller than 1/10\nof\u0015m; (ii) the spheres are free to rotate and align with the\nexternal \feld; (iii) each bare sphere is far in the strong\ncoupling regime with the cavity mode, and is positioned\non the cavity axis; (iv) all the spheres must have the same\ndiameter; (v) the relative distance between the spheres\nshould be about 2 \u001e. Following these constraints we \flled\nthe volume having uniform magnetic \felds, rf of the cav-\nity and dc of the magnet, with ten 2.1 mm diameter YIG\nspheres. Each one placed along a cylindrical fused silica\npipe and separated by thin PTFE caps (see Fig. 7 for a\nsketch and a picture of the system).\nWe now note that the experimental errors of the mea-\nsurements are mostly systematic, and possibly due to\nvariations of the samples positioning in the di\u000berent tests.7\nFIG. 6: Ten spheres HS: simulated (left, where blue-to-light green corresponds to low-to-high transmission) and measured\n(right, bright colors correspond to higher transmission) anticrossing dispersion relation, obtained with the model based on H4\nand a room temperature transmission measurement, respectively. The parameters used to obtained the left anticrossing curve\n!\u0000are detailed in the text.\nAs a consequence, the uncertainties in determining the\neigenfrequencies of the HS are not related to the fre-\nquency resolution, and we estimate them by confronting\nrepeated measurements of the same HS con\fguration,\nwhich are found to be consistent within an 8% error.\nThe dispersion relations of Fig. 6, comparing the the-\nory to the experiment, demonstrate that the resulting HS\ncan be completely described by means of our model based\non the Hamiltonian H4in Eq. (6). The theoretical spec-\ntrum is calculated considering only an external coupling\nto the cavity mode cat 10.65 GHz (see Section II). The\nexperimental spectrum in Fig. 6 (right) was collected\nwith the use of a movable antenna allowing for a good\ncoupling with mode c, and, for symmetry reasons, a much\nweaker one with mode d. Our model yields a sound agree-\nment with the measurements for what concerns mode fre-\nquencies and their relative transmission amplitude. For\nexample, the measured transmission of the dmode is ex-\ntremely low, and the one of the dark mode almost van-\nishes at higher frequencies. In the experimental plot, a\n\ft of the lower frequency hybrid mode !\u0000is also shown\nas a dashed line. The analysis of the anticrossing curve\nshows a Rabi splitting of 528 MHz, correctly scaling as\nthe square root of the number of spheres from the single\nsphere value of about 180 MHz. The resonant frequen-\ncies of the cavity modes are the ones reported in Sec-\ntion II, and remaining parameters are gdm= 264 MHz\nandgmn= 28 MHz. The local discrepancies between the\ntwo plots (i. e. the anticrossing of !\u0000around 0.4 T) can\nbe imputed to spurious magnon modes. This mode pre-\nserves a good antenna output, and is less a\u000bected by the\npresence of other resonances. In fact, the closer cavity\nmodes are the TM11 z, forz= 1;2;:::, which lie at fre-\nquencies higher than !c, making!\u0000well isolated. This\nmode originates from the coherent coupling of all the ten\nYIG spheres, and has an adequate coupling to the dipole\nantenna used to extract the signal, and hence its signal\ntransduction is e\u000ecient.This system, with minor modi\fcations for thermaliza-\ntion (see Appendix C), was then operated at ultra cryo-\ngenic temperatures. The anticrossing pattern obtained\nat 90 mK is reported in Fig. 7(b) and closely resembles\nthe one of Fig. 6, with the increase of the coupling gcm\nas expected from the higher value of the YIG saturation\nmagnetization at such temperatures. The single sphere\nsplitting increases by about 17%74and becomes 210 MHz.\nA \ft with the model based on H4on the experimental\ndispersion relation gives 2 gcm= (2\u0019)638 MHz, to be com-\npared with the value 210p\n10 MHz = 664 MHz. Within\nthe experimental uncertainties, the ten YIG spheres are\ncoherently coupled to the cavity mode, as they e\u000bectively\nreact as a single oscillator. Furthermore, the HS behaves\nas expected also at temperatures where quantum e\u000bects\ndominate over thermal \ructuations7.\nSome other traits of the HS dynamics can be seized by\nlooking at the dispersion relations in Fig. 6 and Fig. 7(b):\nthe coupling of the magnetic m-mode with the d-mode\nof the cavity, and the existence of a dark mode related\nto a magnetic mode n. Evidence for these facts comes\nby looking at the two models described in Fig. 1(b) and\nFig. 1(d), respectively. The explanation for the dark\nmode is not straightforward. In addition to the Kit-\ntel mode of uniform precession, a ferromagnetic sphere\nhas several higher order magnetostatic modes. Following\nthe mode classi\fcation of Walker62, we identify the uni-\nform mode m= (1;1;0), which is degenerate with the\n(4;3;0) one. The latter could be the n-mode producing\nthe dark mode, since sphericity is not perfect in our home\nmade samples, and mandncould actually be slightly de-\ntuned from each other62. However, we are actually not\nfavoring such explanations since single spheres dispersion\ncurves only show standard behavior. The presence of a\nmodencould also be due to the use of spheres with dif-\nferent diameters, as it was discussed in Section III (see\nFig. 4). Also for this case, we believe this is not the cor-\nrect explanation. In fact, by performing measurements8\nPTFE supports\nMicrowave cavity\nAntennas\nYIG spheres\nNbTi magnet\n(a)\n(b) (c)\nFIG. 7: (a) Picture (left) and scheme (right) of the HS tested\nat a temperature of 90 mK and used in a FH75. See text and\nAppendix C for a detailed description of the apparatus. (b)\nAnticrossing curve obtained at 90 mK with the !\u0000frequency\nevidenced with a dashed line, and (c) tuning of the hybrid\nmode frequency !\u0000with a \feld variation of 3 mT.\nwith two spheres we have veri\fed that, for our set, co-\nherent interaction with all the available spins is not real-\nized only when the di\u000berences in sphere diameters satisfy\n\rbm(\u001e)\u001d\rm(see discussion of Fig. 1(d) and in Section\nII). Since all the YIGs used for the ultra cryogenic mea-\nsurement have much close diameters, we believe that the\ndark mode in our anticrossing curves of Fig. 6 and 7(b)\nresults from some residual magnon-magnon coupling. A\nway to certify this could be testing the presence of the\ndark mode with almost identical spheres. In Fig. 7(b) are\npresent two low frequency modes which are absent in the\nroom temperature anticrossing of Fig. 6. This unidenti-\n\fed cavity resonances are visible also at 300 K, but their\nfrequency shifts with the temperature. Even if they ap-\npear in the spectra, they are not coupled to magnon\nmodes and so does not interfere with the FH operation.\nAs the apparatus behavior is consistently described bythe model based on H4, we can now use it to \fnd the\nmaximal transduction bandwidth of the realized set-up\nfor an input on the magnetic mode mand readout on\nthe cavity mode cwith matched antenna. This is the\ngeneral situation of a ferrimagnetic haloscope described\nin the introduction, where a pseudo-magnetic \feld due\nto a dark matter axion excites a magnetic resonance in\nthe material, producing an rf power on the coupled cav-\nity mode. By tuning the Larmor frequency !mthrough\nthe magnetic \feld it is possible to search for dark matter\n\felds oscillating at various frequencies !\u0000, considering\na transduction e\u000eciency given by the relative change of\nmagnon-to-photon conversion probability. We can de-\n\fne the detection bandwidth as the range of frequencies\nfor which transduction e\u000eciency is within 3 dB from the\nmaximum; we mention that the veri\fcation of this ap-\nproach validity will be presented in a forthcoming work76.\nFor the apparatus described by Fig. 7(b) this bandwidth\nis about 400 MHz. In Fig. 7(c) a close up of the tuning\nobtained with a variation of 3 mT of the magnetic \feld is\nshown. The possibility to achieve large and simple tun-\nability for the input frequency is a major advantage of a\nferromagnetic haloscope75, and can be used to simplify\nmany HS-based applications2,40.\nIV. FURTHER DEVELOPMENTS AND\nCONCLUSIONS\nDetection sensitivity of FH is directly related to the\namount of sensitive material. To improve sensitivity,\nwhile keeping the same central working frequency, longer\ncylindrical cavities holding more spheres can be used, and\no\u000b axis loading of the spheres can be implemented. In or-\nder to coherently use all the available spins, each sphere\nmust be strongly coupled to the cavity mode76. This\nresults in the production of ultra strong vacuum Rabi\nsplitting for the complete system, with the drawback of\npossible interference with other cavity modes or between\nhigher order magnetic modes. To avoid such problems,\nwe leverage the fact that as long as the single sphere\ncoupling is larger than the losses the transduction ef-\n\fciency is preserved. Hence, magnetic spheres can be\nplaced in a region of lower rf magnetic \feld of the cavity\nmode, thereby reducing the single sphere coupling to a\nfew times the magnon or cavity linewidth, with the re-\nsult of a smaller total splitting. Moreover, some prelim-\ninary measurements that we have performed show that\nmagnon-magnon interaction is kept small when two or\nmore spheres are placed in a plane perpendicular to the\ncavity axis, thus allowing a small gap separation between\nthe spheres in such direction. It is then possible to fore-\nsee a single microwave cavity with a volume of active\nmaterial one order of magnitude larger than the one em-\nployed in this article, just by \flling the cavity with planes\nof spheres, each plane separated by the required distance\nto avoid mutual interaction. If carefully devised, these\nfuture HSs may still be described by a few oscillators9\nmodels, preserving their optimal control, and remarkably\nincreasing their possible applications.\nIn conclusion, we outlined a model composed of an ar-\nbitrary number of photon and magnon oscillators. The\nanalysis was limited to the experimental case of two cav-\nity modes and two magnetic modes, and the various cou-\nplings among them were separately studied. Afterwards,\nwe make use of a \rexible HS to experimentally reproduce\nthe theoretical results and understand the constraints to\ntake into account when devising a HS with a considerable\nmaterial quantity. The outcome is a recipe that we use to\nbuild the largest optimally-controlled HS to date, which\nwe test at room and at millikelvin temperature. This de-\nvice is used as axion-to-electromagnetic \feld transducer\nin a FH75, paving the way to the use of these devices\nto search for Dark Matter with a cosmologically relevant\nsensitivity. Our results benchmark a starting point for\nfuture large HS designs, and demonstrates that the de-grees of freedom of such a complex system can be reduced\nto an acceptable number.\nAcknowledgments\nThe authors would like to deeply acknowledge Enrico\nBerto for the help in the fabrication of the spheres, and\nRiccardo Barbieri for the initial formulation of the model.\nWe are thankful to Andrea Benato, Fulvio Calaon and\nMario Tessaro for the work on the cryogenics and elec-\ntronics of the setup, and to David Alesini and Mario Zago\nfor the cavity design and realization. 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Journal of Applied Physics , 29, 03 1958.\n69Zhang Zheng-Wu, Wu Ping-Feng, Yang Jian-Hua, and\nZhou Kang-Wei. Absorption spectrum and zero-\feld split-\nting of y 3fe5o12.Phys. Rev. B , 48:16407{16409, Dec 1993.\n70Jerzy Krupka, Stephen A Gabelich, Krzysztof Derza-\nkowski, and Brian M Pierce. Comparison of split post di-\nelectric resonator and ferrite disc resonator techniques for\nmicrowave permittivity measurements of polycrystalline\nyttrium iron garnet. Measurement Science and Technol-\nogy, 10(11):1004{1008, sep 1999.\n71M. N. Afsar, K. M. Lee, Yong Wang, and K. Kocharyan.\nMeasurements of complex permittivity and permeability of\ncommon ferrimagnets at millimeter waves. IEEE Transac-\ntions on Magnetics , 40(4):2826{2828, 2004.\n72A. M. Hofmeister and K. R. Campbell. Infrared spec-\ntroscopy of yttrium aluminum, yttrium gallium, and yt-\ntrium iron garnets. Journal of Applied Physics , 72(2):638{\n646, 1992.\n73Takanori Tsutaoka, Teruhiro Kasagi, and Kenichi\nHatakeyama. Permeability spectra of yttrium iron garnet\nand its granular composite materials under dc magnetic\n\feld. Journal of Applied Physics , 110(5):053909, 2011.\n74Elmer E. Anderson. Molecular \feld model and the magne-\ntization of yig. Phys. Rev. , 134:A1581{A1585, Jun 1964.\n75N. Crescini, D. Alesini, C. Braggio, G. Carugno,\nD. D'Agostino, D. Di Gioacchino, P. Falferi, U. Gam-bardella, C. Gatti, G. Iannone, C. Ligi, A. Lombardi,\nA. Ortolan, R. Pengo, G. Ruoso, and L. Ta\u000barello. Axion\nsearch with a quantum-limited ferromagnetic haloscope.\nPhys. Rev. Lett. , 124:171801, May 2020.\n76N. Crescini et al. Studying the dynamics of photon-magnon\nhybrid systems with ultrafast laser pulses.\n77H. Maier-Flaig, S. Klingler, C. Dubs, O. Surzhenko,\nR. Gross, M. Weiler, H. Huebl, and S. T. B. Goennenwein.\nTemperature-dependent magnetic damping of yttrium iron\ngarnet spheres. Phys. Rev. B , 95:214423, Jun 2017.\nAppendix A: Langevin equations and the e\u000bective\nHamiltonian\nThe dynamics of the system can be described by the\nquantum Langevin equations\n_x=\u0000i(!x\u0000i\rx)x\u0000iX\ny2Ygxy\n2y\n_y=\u0000i(!y\u0000i\ry)y\u0000iX\nx2Xgxy\n2x\u0000iX\ny\u00036=ygyy\u0003\n2y\u0003;(A1)\nwhich are valid for every cavity mode x2Xand magnon\nmodey2Y, and where iis the imaginary unit and\ny\u00032Y. For the sake of this work it su\u000eces considering\nonly the intrinsic decay rates of the cavity modes, as they\ndominate over the ones introduced by external couplings\nif we assume that the setup's antennas are weakly cou-\npled. Eq.s (A1) may be recast in their matrix form as\n_W=\u0000iHW; (A2)\nwhereW=fX;Yg=fx1;x2;:::;x N;y1;y2;:::;y Mg,\nandHis the matrix reported in Eq. (2) and used through-\nout this work, in particular to determine sx1(!;!m) in\nEq. (5). Here we note that it is experimentally challeng-\ning to have an antenna coupled to a single mode, mak-\ning our calculated transmission spectra an incomplete de-\nscription of the apparatus. Nevertheless, as is detailed in\nSection III and Fig. 2, this approximation holds thanks to\nthe geometric con\fguration of the two considered cavity\nmodes. We stress that one can get a more realistic de-\nscription of the HS spectra by summing over the possible\ntransmissions due to external couplings to other modes\nas\nstot(!;!m) =X\nx2X\u0011xsx(!;!m); (A3)\nwhere\u0011xis a constant accounting for the coupling\nstrength between the antenna and the di\u000berent cavity\nmodes. This is an e\u000bective approximation which resists\nif all the external couplings are weak, and do not in\ruence\nthe linewidths of the modes. An even better description\nconsists in the addition of a mode-dependent dissipation\nin Langevin equations to account for the coupling of the\nantennas, see for example Ref. [34], which naturally yields\nthe increase of the modes' decay rate.12\nFIG. 8: The instrument used to produce the spheres (left).\nImprovement of linewidth with subsequent steps for a batch\nof six spheres (right).\nAppendix B: Spheres production\nSuper polished YIG spheres show the sharpest mag-\nnetic resonance linewidth66{68. Material purity and sur-\nface roughness are the two key elements to be cured\nin order to obtain the best linewidth values at all\ntemperatures77. Our spheres have been produced on site\nto have better control of all the relevant parameters. We\nbuy large single crystal high purity YIG cylinders, nor-\nmally a few cm length and 5 to 7 mm diameter, and\ncut them into 3 mm size cubes. Each cube then follows\na grinding procedure to obtain spheres of about 2.1 mm\ndiameter. The grinder is based on a high frequency rotat-\ning plate with replaceable silicon carbide (SiC) sandpa-\nper foils where the YIG piece get abraded while tumbling\ninside a plastic holder (see the left plot in Fig. 8). Start-\ning from a P800 SiC paper, \fner and \fner grit size are\nused in sequence up to the \fnal P4000. The duration\nof each step has been optimized after several trials. The\nlast step consists of four passages with P4000 virgin SiC\npaper, each lasting half an hour. At the end of this pro-\ncess, we get spheres with the nominal diameter within a\nfew percent and linewidths of about 2 MHz. A \fnal 24\nhours polishing with Alumina-based suspension on a low\nfrequency rotating system results in a linewidth slightly\nabove 1.3 MHz at room temperature (see the right plot\nof Fig. 8).\nAppendix C: Hot-bore superconducting magnet and\nmillikelvin temperature hybrid system\nThe setup used for testing the single and multiple-\nspheres con\fgurations includes a superconducting mag-\nnet, described in Section III, with a hot bore in which\nthe HS is inserted. This custom setup was built to test\nseveral HSs per-day, which is incompatible with multiple\nrefrigeration cycles. The cavity is mounted on a plastic\nsupport that aligns its axis with the one of the magnet,\nand a PTFE support holds the pipe which positions the\nYIG spheres on this same axis.\nAt milli-Kelvin temperatures, the cavity is anchored\nto the mixing chamber of a dilution refrigerator with two\nlarge copper bars. Its temperature is monitored witha thermometer attached to the cavity body. The setup\nmust ensure a proper thermalization of the YIG spheres,\nwhich for this reason are contained in a pipe \flled with\ngaseous helium. The preparation of the fused silica pipe\nis done at room temperature as follows. A vacuum sys-\ntem is designed in such a way to remove the air from the\npipe, which is then immersed in a 1 bar helium controlled\natmosphere and closed. In this way the pipe is \flled with\nhelium, and can be sealed by using a copper plug glued\nwith Stycast. First the sealing is tested without the sam-\nples by measuring the frequency shift of the TM110 mode\nof the cavity-pipe system with and without helium. The\nfrequency is measured with the helium-\flled pipe, which\nis then immersed in liquid nitrogen and again placed in\nthe cavity. Re-measuring the same frequency excludes\nthe presence of leaks. The plug used to close the pipe\nis eventually anchored to the cavity body with a copper\nrod, granting a good thermalization of the whole system." }, { "title": "1101.2303v1.Gigantic_terahertz_magnetochromism_via_electromagnons_in_hexaferrite_magnet_Ba__2_Mg__2_Fe___12__O___22__.pdf", "content": "arXiv:1101.2303v1 [cond-mat.str-el] 12 Jan 2011Gigantic terahertz magnetochromism via electromagnons in\nhexaferrite magnet Ba 2Mg2Fe12O22\nN. Kida,1,∗S. Kumakura,1,2S. Ishiwata,3,†Y. Taguchi,3and Y. Tokura1,2,3\n1Multiferroics Project (MF), ERATO,\nJapan Science and Technology Agency (JST),\nc/o Department of Applied Physics,\nThe University of Tokyo, 7-3-1 Hongo,\nBunkyo-ku, Tokyo 113-8656, Japan\n2Department of Applied Physics, The University of Tokyo,\n7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\n3Cross-Correlated Materials Research Group (CMRG)\nand Correlated Electron Research Group (CERG),\nASI, RIKEN, 2-1 Hirosawa, Wako, 351-0198, Japan\n(Dated: December 1, 2018)\nEffects of temperature (6–225 K) and magnetic field (0–7 T) on th e low-energy (1.2–5 meV)\nelectrodynamics of the electromagnon, the magnetic resona nce driven by the light electric field,\nhave been investigated for a hexaferrite magnet Ba 2Mg2Fe12O22by using terahertz time-domain\nspectroscopy. We find the gigantic terahertz magnetochromi sm via electromagnons; the mag-\nnetochromic change, as defined by the difference of the absorpt ion intensity with and without\nmagnetic field, exceeds 500% even at 0.6 T. The results arise f rom the fact that the spectral in-\ntensity of the electromagnon critically depends on the magn etic structure. With changing the\nconical spin structures in terms of the conical angle θfrom the proper screw ( θ= 0◦) to the ferri-\nmagnetic ( θ= 90◦) through the conical spin-ordered phases (0◦< θ <90◦) by external magnetic\nfields, we identify the maximal magnetochromism around θ≈45◦. On the contrary, there is no\nremarkable signature of the electromagnon in the proper scr ew and spin-collinear (ferrimagnetic)\nphases, clearly indicating the important role of the conica l spin order to produce the magnetically-\ncontrollable electromagnons. The possible origin of this e lectromagnon is argued in terms of the\nexchange-striction mechanism.\nPACS numbers:\n1I. INTRODUCTION\nThe control of light signal with the magnetic state or conversely th e light sensing of\nthe magnetic state is a key issue in the contemporary information te chnology, as realized\nin many magneto-optical devices and elements. Most of the magnet o-optical functions ex-\nploit the change of light polarization with the magnetic state, usually g enerating the light\npolarization rotation (Kerr or Faraday rotation). On the other ha nd, the magnetic-field\nchange of optical absorption itself, i.e., magnetochromism, is usually very small in spite of\nthe potential demands in spin-electronic science and technology. T his is reasonable since\nthe electronic resonance is usually tied to the magnetic state only th rough spin-orbit inter-\naction in the magnetic medium. Some exceptions are found for the ma terials endowed with\ninherent strong charge-spin coupling and magnetically controllable c ompeting orders, such\nas spin-crossover complexes1, colossal-magnetoresistive oxides2, and low-dimensional molec-\nular magnets3. Even for these materials, however, a high magnetic field of severa l tesla is\nnecessary to cause the gigantic magnetochromism.\nA more promising candidate to host the gigantic magnetochromism is t he multiferroic\ncompound, where the electricity and magnetism is inherently coupled4,5. Such a coupling\nis recently evidenced for a variety of oxide compounds, as typified b y the magnetic field\ncontrol of ferroelectricity in TbMnO 36and by the electric field control of the magnetiza-\ntion in GdFeO 37. As a consequence of such a strong magnetoelectric response, a number\nof multiferroics may have the potential to show the magnetic reson ance in the dielectric\nconstant ( ǫ) spectrum, in addition to (or rather than) the magnetic permeabilit y spectrum.\nThis collective mode, now termed electromagnon, becomes electrica lly active and thus can\nbe excited by the light Evector. This is contrasted by the case of the conventional mag-\nnetic resonance (the k= 0 magnon) driven by the light Hvector. Indeed, recent optical\nexperiments at terahertz frequencies have revealed the emerge nce of the electromagnon in\na family of multiferroics8,9;RMnO3(R= rare earth ions)10,RMn2O511, Ba2Mg2Fe12O2212,\nCuFe1−xGaxO213, and Ba 2CoGe2O714. As the origin of the electromagnons observed in anti-\nferromagnets with non-collinear spin configurations such as RMnO3, the exchange-striction\nmechanism is proposed15,16: The non-collinear spins ( SiandSjon the neighboring i-th and\nj-th sites) can produce the local polarization ∆ Pijin response to Ecomponent of light, as\ngiven by ∆ Pij∝∆Si·Sj. In fact, the spectral shape of the electromagnon in RMnO3can\n2be reproduced by this mechanism17. As only the scalar product Si·Sjon the underlying\nlattice is relevant, the electromagnon should be insensitive to the dir ection of the spiral spin\nplane, as already confirmed by the experiments applying the magnet ic field on DyMnO 318.\nAmong several magnetoelectrics to host electromagnons, a hexa ferrite, Ba 2Mg2Fe12O22\ninvestigated here, occupies a unique position. Recent combined stu dies of terahertz time-\ndomain spectroscopy and inelastic neutron scattering have revea led the emergence of the\nelectromagnon around 0.7 THz (corresponding to 2.8 meV in photon e nergy)12; the elec-\ntromagnon becomes active in the conical spin ordered phase when t he lightEvector is set\nparallel to the propagation vector along [001] [see Fig. 1(d)].\nAt room temperature, Ba 2Mg2Fe12O22shows the ferrimagnetic order composed of two\nmagnetic sublattice blocks, LandS[Fig. 1(b)] with the opposite directions of the in-plane\nmagnetizations. Below 195 K, it becomes the proper screw structu re, in which the in-plane\nLandSspins rotate in proceeding along [001] [Fig. 1(c)]25. Finally, below 50 K, it forms the\nlongitudinalconicalspinstructureasthein-plane LandSspinsdeclinetoward[001]withthe\nconical angle θ[Fig. 1(d)]. Noticeably, in magnetic fields along [001], Ba 2Mg2Fe12O22yields\nthelargesaturationmagneticmoment( ∼8µBperf.u.; f.u.,formulaunit). Thisenablesusto\neasilymodifytheelectromagnoninherenttotheconicalspinphase[s eetheinsetofFig. 4(a)],\nas compared to other multiferroics with the antiferromagnetic ord er (no net spontaneous\nmagnetization). Another remarkable characteristic of Ba 2Mg2Fe12O22is the emergence of\nthe ferroelectricity by controlling the spin cone with magnetic field19,20: Transverse conical\nstructure, as realized by declining the cone axis toward [100] [see th e inset of Fig. 6(a)], can\nproduce the ferroelectricity. In such a transverse spiral or con ical magnet, the spontaneous\npolarization stems from the spin-current21,22or inverse Dzyaloshinskii-Moriya mechanism23,\nas expressed by /vectorP=/summationtext\naij/vector eij×(/vectorSi×/vectorSj); here/vectorSi,/vector eij,aijare the spin at i-th site, the unit\nvector connecting i- andj-th sites, and the coupling constant, respectively. This scenario is\nwell proved also for this compound by the detailed analysis of the mag netic structure24. The\napplicationofexternalmagneticfieldishighlyefficient forcontrolof θ;θcanbechangedfrom\n90◦to 0◦. Therefore, the transverse conical structure, where the scr ew axis is perpendicular\nto the propagation vector, can host the ferroelectricity along [12 0]. According to recent\nneutron scattering experiments, θwas found to increase from nearly 0◦at 50 K to 20◦at\n25 K24. From the viewpoint of the magnetochromism, the magnetically-flex ible conical spin\nstructure is expected to produce the dramatic modification of the electromagnon spectrum.\n3For this purpose, we perform the terahertz time-domain spectro scopy and fully map out\nthe spectrum of the electromagnon in a variety of the spin-ordere d phases of Ba 2Mg2Fe12O22\nby tuning the magnetic field (up to 7 T). The preliminary experiments in the zero and low\nmagnetic field (up to 0.2 T) have identified the existence of electroma gnon12. Here we reveal\nthe gigantic magnetochromism via electromagnons at terahertz fr equencies with the fine\ncontrol of the conical spin order. Furthermore, we could determ ine the magnetoelectric\nphase diagram of Ba 2Mg2Fe12O22by conversely using the electromagnon spectra as the\nprobe.\nThe format of this paper is as follows: In Sec. II we describe the exp erimental setup for\nterahertz time-domain spectroscopy combined with a split-type su perconducting magnet.\nSection III is devoted to the systematic optical investigations of t he electromagnons in\na variety of spin-ordered phases tuned by magnetic fields (from th e proper screw to the\nferrimagnetic through the conical spin-ordered phases), in which we reveal the important\nrole of the conical spin order to produce the electromagnon activit y by the measurements in\nparaelectric longitudinal (Sec. III A) and ferroelectric transver se (Sec. III B) conical spin-\nordered phases. On the basis of the results of Sec. III, we discus s in Sec. IV the possible\norigin of the electromagnon activity in terms of the symmetric-exch ange mechanism. The\nconclusion is given in Sec. V.\nII. METHODS\nThe single-crystalline samples were prepared by a flux method as des cribed in Ref. 19,\nand their magnetic and ferroelectric characteristics were all cons istent with previous results.\nThe obtained samples were mechanically polished to the thickness of 7 0– 300µm (thickness\nof the sample is different in the measurements at respective temper atures so as to gain the\nsignal-to-noise ratio).\nWe used the terahertz time-domain spectroscopy in a transmission geometry to directly\nestimate the complex dielectric constant spectra. Our experiment al scheme using the ZnTe\ncrystal as a terahertz emitter was reported in Ref. 18. In the pr esent experiments, to\nfurther access the low-energy electrodynamics of the electroma gnon, especially down to 0.3\nTHz(1.2meVinphotonenergy), weusedthephotoswitching sampling technique withalow-\ntemperature-grown GaAs (LT-GaAs) as a terahertz emitter, as schematically shown in Fig.\n42. The femtosecond laser pulses delivered from the mode-locked Ti:s apphire laser (center\nwavelength of800nm; pulsewidth of100fs; repetitionrateof82MH z) were impinged onthe\nphotoswitching device made on the LT-GaAs with dipole-antenna. Th e radiated terahertz\npulse was collimated by a pair of off-axis paraboloidal mirrors and focu sed on the another\nphotoswitching device made on the LT-GaAs; the induced photocur rent was monitored in\ntime-domain by changing the arrival time of the trigger femtosecon d laser pulse. In this\nsetup, we inserted the split-type superconducting magnet (up to 7 T) in between off-axis\nparaboloidal mirrors. To eliminate the contribution of the conventio nal magneto-optical\neffects, we adopted the Voigt geometry, where the light kvector is perpendicular to the\ndirection of the magnetic field. The polarizationof theradiated tera hertz pulse was carefully\ntuned by a wire-grid polarizer.\nWith use of the fast Fourier transformation of the measured tera hertz pulse in time do-\nmain, we obtained the amplitude and phase spectra with and without t he sample. From the\nobtained complex transmission, we numerically estimated the complex refractive constants\nn(ω) =/radicalbig\nǫ(ω)µ(ω), where ǫandµare complex dielectric constants and complex magnetic\npermeability, respectively. As we previously confirmed on the basis o f the measurements\nof the light polarization dependence (both the electric-field and mag netic-field components\nof light) using crystal plates with different crystal orientations12, the contribution of µ(ω)\nton(ω) was negligible in the measured frequency range, namely µ(ω)≈1. In this paper,\ntherefore, we used ǫ(ω) as a quantity. Further details of our estimation procedure were\nreported in Ref. 18.\nIII. RESULTS\nA. Effect of magnetic fields on electromagnons in the paraelectric phase\nFirst, we examine the temperature evolution of the electromagnon in zero magnetic field.\nThe light Evector was set parallel to [001] to induce the electromagnon. In th e proper screw\nphase in the temperature range of 190–50 K, there is no signature of electromagnons in the\nimaginary part of the dielectric constant ( ǫ2) spectra, as shown in Figs. 3(b) to 3(d). Below\n50 K, where the longitudinal conical spin order evolves [Fig. 1(d)], th e sharp resonance can\nbe discerned in ǫ2spectrum and its peak magnitude reaches about 6, as exemplified by the\n5ǫ2spectrum at 6 K [Fig. 3(a)]. This absorption was assigned to the elect romagnon, inherent\nto the longitudinal conical spin ordered phase, on the basis of the c ombined measurements\nof the complete set of the light-polarization dependence as well as o f the magnetic excitation\nspectrum with use of the inelastic neutron scattering12.\nNext, we applied the magnetic field along [001] at 6 K so as to control θof the longi-\ntudinal conical structure. In the experiments in magnetic fields H, we adopted the Voigt\ngeometry to exclude the contribution of the conventional magnet o-optical effects such as\nFaraday rotation. Owing to the ferrimagnetic nature of Ba 2Mg2Fe12O22, the magnetization\nalong [001] increases with magnetic field and yields a large magnetic mom ent about ∼8\nµB/f.u. at 7 T [Fig. 4(a)]. The saturated magnitude of Mis consistent with the anticipated\nferrimagnetic order, where LandSsublattice blocks were perfectly directed along [001], as\nschematically shown in the inset of Fig. 4(a). Therefore, we could ro ughly estimate θas a\nfunction of magnetic field, as given by θ= sin−1(M/Ms) [Fig. 4(a)]. At 6 K in zero magnetic\nfield,θwas estimated to be 20◦, which is consistent with the measured value by the recent\nneutron scattering experiments24. About 3 T, we can induce the ferrimagnetic phase as θ\nnearly reaches about 90◦. In magnetic fields, there is a remarkable change of the electro-\nmagnon ǫ2spectra [Fig. 3(a)]; this is viewed as the gigantic terahertz magneto chromism.\nWith increasing magnetic field, the electromagnon grows in spectral intensity and reaches\nthe maximum around 1 T, as can be seen in Fig. 3(a). The relative chan ge of the magnitude\nofǫ2at 0.76 THz with and without magnetic field was estimated to be ∼300% in 0.6 T.\nWith further increasing magnetic field, the intensity of the electrom agnon decreases and the\npeak position shifts to the lower energy. The electromagnon reson ance disappears above 3\nT in the ferrimagnetic (all spin collinear) phase.\nFurther noticeable magnetochromism can be seen in the ǫ2spectra at 64 K on the verge\nof the transition from the proper screw to longitudinal conical spin structures [Fig. 3(b)].\nThe tendency that the intensity of the electromagnon increases w ith increasing magnetic\nfield is similar to the case at 6 K [Fig. 3(a)], except for the behavior belo w 0.5 T. In zero\nmagnetic field, the proper screw state is stable, hence no electrom agnon is observed. Even in\nweak magnetic fields below 0.5 T, we can induce the electromagnon, via the evolution of the\nlongitudinal conical spinorder, asidentified inthe ǫ2spectrum around0.1T. Therefore, near\nthistemperature, themagnetochromismisstronglyenhanced; th erelativeabsorptionchange\nyields a gigantic value, ∼500% at 0.6 THz even at 0.6 T. With increasing temperature, the\n6value of the relative change tends to decrease, since the applicatio n of magnetic field can\nhardly induce the longitudinal conical spin phase, as exemplified by th eǫ2spectra at 90 K\n[Fig. 3(c)]. Finally, at 164 K, ǫ2spectrum is found to be independent of magnetic field ( <7\nT) [Fig. 3(d)].\nTo quantify this trend, we show in Fig. 4(b) the contour plot of the s pectral weight (SW),\nas defined by the integration of the optical conductivity (= ǫ0ǫ2ω,ǫ0being a permittivity of\nvacuum) between 0.4 THz to 1 THz, in the plane of temperature and m agnetic field. Open\ncircles represent the measured points. The scale color bar is shown at the right side of the\nfigure; for example, red region indicates the maximum value of the SW . This contour plot\nclearly reflects the magnetic ( H∝bardbl[001]) phase diagram of Ba 2Mg2Fe12O22. In the restricted\narea of the phase diagram, the longitudinal conical spin ordered ph ase is identified with the\nenhanced electromagnon spectral intensity dependent on θ. The present experiments with\nuse of the electromagnon as a probe can provide the full-range pha se diagram including\nthe thermal and magnetic-field variations among the proper screw , conical, and collinear\nstructures.\nB. Effect of magnetic fields on electromagnons in the ferroelectric phase\nIn Ba2Mg2Fe12O22, we can also induce the transverse conical structure by applying t he\nmagnetic field along [100]. In this phase, the ferroelectricity emerge s along [120], i.e., along\nthe direction perpendicular to both the magnetic q-vector ( ∝bardbl[001]) and the conical axis\n(∝bardbl[100]), as is evidenced by the measurements of the spontaneous po larization [Fig. 6(a)]19.\nFigure 5 presents the ǫ2spectra at respective temperatures in magnetic fields ( ∝bardbl[100]) up\nto 7 T. In zero magnetic field, a signature of the electromagnon at 0 .7 THz is discerned\nonly in the longitudinal conical spin phase, e.g., at 5 K [Fig. 5(a)]. When t he magnetic\nfield was applied along [100] at 5 K, the magnetization steeply increase s around 0.2 T [Fig.\n6(a)]. Accordingly, thespontaneouspolarizationemergesandrea chesthemaximumabout80\nµC/m2. These are all consistent with the transformation from the longitu dinal to transverse\nconical spin orders, which was directly confirmed by the recent pola rized neutron scattering\nexperiments24. The similar magnetochromism is observed in the electromagnon spec tra to\ncase ofH∝bardbl[001] (Fig. 3); ǫ2increases first with increasing magnetic field and reaches the\nmaximum at around 1 T, and finally, the electromagnon resonance dis appears about 3 T, at\n7which the transverse conical spin order transforms to the ferrim agnetic order as evidenced\nby the reduction of the spontaneous polarization [Fig. 6(a)]. Thus, the emergence of the\nelectromagnon is only limited to the conical spin (transverse conical forH∝bardbl[100], and\nhence ferroelectric in this case) phase. In turn, we uniquely deter mine the magnetic phase\ndiagram with use of the electromagnon spectral intensity as a prob e. We show in Fig. 6(b)\nthe contour plot of the SW (the integrated optical conductivity fr om 0.3 THz to 1.2 THz)\nin the plane of temperature and magnetic fields. Thus, we can identif y the region, where\nSW shows the enhancement in the conical spin arrangement, as colo red in red and yellow.\nThe solid lines are the guides to indicate the ferroelectric phase in the phase diagram, as\ninferred from the recent neutron scattering experiments24,26, which is consistent with our\npresent data. On the basis of the such mapping of the electromagn on activity in terms of\ntemperature and magnetic field, we confirm again that the conical s pin order can act as an\neffective source of the electromagnons, whose intensity critically d epends on θ.\nIV. DISCUSSION\nAlthough the detailed theoretical study is needed, the most plausib le scenario to explain\nthe above behaviors of the electromagnon is to consider the symme tric exchange-striction\nmechanism. As extensively discussed in RMnO3as the model case15,16,17, the non-collinear\nspins with the symmetric (Heisenberg type) exchange interaction ( JSi·Sj) causes the elec-\ntromagnon activity through the relationship, ∆ Pij∝∆Si·Sj, when the local electric\ndipole is tied to the i−jbond. According to the detailed magnetic structure analysis\nof Ba2Mg2Fe12O22by neutron scattering25,LandSsublattice blocks are composed of the\nstacking of Fe ions along [001], as labeled by “1”, “4”, “5”, “8”, “11” , “12”, “11′”, “8′”, “5′”,\n“4′”, and “1′” [Fig. 7(a)]. Figure 7(b) shows the reported spin arrangement at r espective\nFe sites within the plane in the ferrimagnetic phase at 294 K25. In this phase, the spins are\nall collinear. Therefore, ∆ Siis perpendicular to Sjand thus ∆ Pijin response to Eω∝bardbl[001]\nis always zero, as schematically shown in the lower panel of Fig. 7(b). On the other hand,\nthe spins become non-collinear in the proper screw spin phase below 1 95 K. The spin ar-\nrangement within the plane in this phase is schematically shown in Fig. 7( c); the turn angle\nof 59.6◦at 77 K was taken from Ref. 25. The electromagnon activity would be possible in\nthis phase [see the lower panel of Fig. 7(c)]. However, this mechanis m cannot explain the\n8present experimental results, where there is no remarkable signa ture of the electromagnon\nin the proper screw phase [Figs. 4(b) and 6(b)]. This is partly due to t he fact that the\nin-plane J(scaled with the transition temperature ∼195 K) is large compared to the mea-\nsured photon energy range ( ≤6 meV). In this proper screw spin phase of Ba 2Mg2Fe12O22,\nLandSspins lie only within the plane [Fig. 7(c)]. On the contrary, spins decline t oward\n[001] in the longitudinal conical ordered phase below 50 K, where the electromagnon shows\nup. The spin component projected on to (120) plane, emphasized a t Fe “4”, “5”, and “8”\nsites, is schematically shown in Fig. 7(d); the upper, middle, and lower panels stand for\nthe situations, where θis 90◦, 45◦, and 0◦, respectively. In the case for 0◦< θ <90◦, there\nemerges another route to ∆ Pijas ∆Siis not perpendicular to Sj. For example, we can\nconsider ∆ Pproduced around Fe “5” site while simply assuming S4= (−S[100],0,−S[001])\nandS8= (S[100],0,S[001])27, the situation for θ= 45◦is schematically shown in the middle\npanel of Fig. 7(d). We then obtain non-zero component of ∆ Palong [001], being propor-\ntional to 2∆ S5S[001]. In this simple consideration, the intensity of the electromagnon sh ould\nbe scaled with sin2θand reach the maximum at θ= 45◦[middle panel of Fig. 7(d)]. In\nthe longitudinal conical spin phase at 6 K (Fig. 4), for example, the o bserved SW of the\nelectromagnon reaches the maximum around 1 T, which nearly corre sponds to θ= 45◦. In\nthe both ferrimagnetic phases induced by magnetic field along [001] o r [100], electromagnons\nbecome inactive (Figs. 4 and 6) as all the spins become collinear within t he projection of\n(120) plane [upper panel of Fig. 7(d)], being consistent with this mec hanism. Contrary to\nthe origin of the ferroelectricity, i.e. the antisymmetric exchange in teraction described by\nvector spin chirality Si×Sj, the symmetric exchange interaction ( Si·Sj) acting along [001]\nis likely the dominant source of the observed electromagnons. Uniqu ely, such a symmetric\nexchange interaction depends on θ. Therefore, the magnetic field control of θranging from\n0◦and 90◦can produce the gigantic terahertz magnetochromism via electrom agnons as we\nobserved here.\nV. CONCLUSION\nIn conclusion, we performed the terahertz time-domain spectros copy for a variety of the\nmagnetically ordered phases of the hexaferrite Ba 2Mg2Fe12O22, while tuning temperature\nand magnetic field. The electromagnon appears only in the conical sp in ordered phase but\n9irrespectiveofthelongitudinalortransverseconicalform,whilen oelectromagnonisobserved\nin the proper screw nor ferrimagnetic phase. The intensity of the e lectromagnon is found to\nbe highly sensitive to the conical angle tuned by external magnetic fi elds. Such a sensitivity\ncan provide the unique opportunity to determine the magnetoelect ric phase diagram in a\nwide range of temperature and magnetic field. From the technical v iewpoint, we clarified\nthat the electromagnon can induce the gigantic magnetochromism a t terahertz frequencies;\nthe estimated relative change of ǫ2(∼400%) is remarkably large even in a magnetic field\nas low as 0.6 T. The important implication of the present work is that th e source of the\nelectromagnon is not limited to multiferroics; electromagnon resona nce emerges irrespective\nof the presence of the ferroelectricity. There are a variety of no n-collinear magnets at room\ntemperature, which are candidates to potentially host the gigantic magnetochromism via\nelectromagnons.\nAcknowledgments\nWe thank S. Miyahara and T. Arima for enlightening discussion. 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Momozawa, Y. Yamaguchi, and M. Mita, J. Phys. Soc. Jpn. 55,1350 (1986).\n26H. Sagayama, K.Taniguchi, N. Abe, T. Arima, Y. Nishikawa, S. Yano, Y. Kousaka, J. Akimitsu,\nM. Matsuura, and K. Hirota, Phys. Rev. B 80,180419(R) (2009).\n27S8andS4belong to LandSsublattice blocks, respectively. Therefore, to be precise , its magni-\ntude should be different. This also produces the non-zero comp onent along [100]. Even in this\ncase, the present discussion is valid in the context that the non-zero component along [001]\nemerges only in the conical spin ordered phase.\n12FIG. 1: (color online) Crystal and spin structures of Ba 2Mg2Fe12O22in zero magnetic field.\nSchematic illustrations of (a) hexagonal structure and spi n structures in (b) ferrimagnetic, (c)\nproper screw, and (d) longitudinal conical spin ordered pha ses. The conical angle θis defined in\n(d). The unit cell can be conveniently divided by two sublatt ice blocks; spinel Sand hexagonal\nLsublattice blocks along [001]. In the conical spin ordered p hase shown in (d), electromagnon\nbecomes active when the light Evector is set parallel to [001].\n13FIG. 2: (color online) Schematic illustration of experimen tal setup for terahertz time-domain spec-\ntroscopy combined with split-type superconducting magnet . We employed the photoconducting\nsampling technique with low-temperature-grown GaAs (LT-G aAs) as terahertz emitter and de-\ntector. The magnetic field was applied up to 7 T in Voigt geomet ry to prevent the conventional\nmagneto-optical effect.\n140 0.5 1 1.5 010 20 30 40 50 60 \nFrequency (THz) ε2\n0 T 1.5 T 2 T 3 T 4 T 5 T 7 T \n6 T \n 0.05 T 0.1 T 0.15 T 0.2 T 0.3 T 0.4 T 0.6 T 0.8 T 1 T \n0 0.5 1 1.5 010 20 30 40 50 60 \nFrequency (THz) 0 T 0.05 T 0.1 T 0.15 T 0.2 T 0.3 T 0.6 T 0.8 T 0.9 T 1 T 1.3 T 1.6 T 2 T 3 T 4 T 5 T 7 T \n0.5 T \n0 0.5 1 1.5 010 20 30 40 50 60 \nFrequency (THz) 0 T 0.1 T 0.2 T 0.4 T 0.6 T 0.8 T 1 T 1.3 T 1.6 T 2 T 2.5 T 3 T 4 T 5 T 6 T 7 T \n0 0.5 1 1.5 010 20 30 40 50 60 \nFrequency (THz) 0 T 0.2 T 0.5 T 1 T 1.5 T 2 T 3 T 4 T 5 T 6 T 7 T (a) 6 K (b) 64 K (c) 90 K (d) 164 K H||[001] \nFIG. 3: (color online) Gigantic magnetochromism via electr omagnons in the paraelectric phases\nof Ba2Mg2Fe12O22. Imaginary part of the dielectric constant spectra ( ǫ2) in magnetic fields up to\n7 T, measured at (a) 6 K originally in the longitudinal conica l spin ordered phase, at (b) 64 K\nnear the conical to proper screw transition temperature, at (c) 90 K and at (d) 164 K originally in\nthe proper screw phase. Respective ǫ2spectra are arbitrarily off-set for clarity; the off-set revel i s\nindicated by the horizontal solid line. The light E-vector was set parallel to [001] and the magnetic\nfieldHwas applied along [001] in Voigt geometry.\n150 1 2 3 4 5 6 7050 100 150 200 250 \nMagnetic Field (T) Temperature (K) \n2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.5 H≠0 H=0 H≥3 T Conical Ferri.\nHDC \n0 1 2 3 4 5 6 702468\n045 90 M[001] ( µB/f.u.) θ (deg.) \n6 K (a) \n(b) \nθSpectral Weight (arb. units) \nFIG. 4: (color online) Magnetic phase diagram of Ba 2Mg2Fe12O22, as determined by the signature\nof the electromagnon. The magnetic field Hwas applied along [001] to finely control the conical\nangleθof the longitudinal conical structure, as schematically sh own in the inset of (a). (a)\nMagnetization along [001] M[001]and estimate θas a function of magnetic field, measured at 6 K.\n(b) Contour map of the spectral weight (SW) of the electromag non in terms of temperature and\nmagnetic field. The measured points are indicated by open cir cles. The magnitude of SW (see text\nfor the definition) is shown with the scale color bar. The unit of SW is arbitrary.\n160 0.5 1 1.5 010 20 30 40 50 60 \nFrequency (THz) 0 T 0.2 T 0.5 T 1 T 2 T 3 T 4 T 6 T 7 T \n0 0.5 1 1.5 010 20 30 40 50 60 \nFrequency (THz) ε2\n0 T 0.05 T 0.1 T 0.3 T 0.6 T 0.8 T 1 T 1.2 T 1.4 T 1.8 T 2.6 T 3 T 3.5 T 4 T 5 T 6 T 7 T \n0 0.5 1 1.5 010 20 30 40 50 60 \nFrequency (THz) 0 T 0.1 T 0.2 T 0.4 T 0.6 T 0.8 T 1 T 1.3 T 1.4 T 1.6 T 2 T 2.5 T 3 T 4 T 5 T 6 T 7 T \n0 0.5 1 1.5 010 20 30 40 50 60 \nFrequency (THz) 0 T 0.06 T 0.1 T 0.13 T 0.2 T 0.3 T 0.4 T 0.6 T 0.8 T 1.3 T 1.6 T 2 T 2.5 T 3 T 3.5 T 4 T 5 T 6 T 7 T (a) 5.5 K (b) 60.3 K (c) 91 K (d) 164 K H||[100] \nFIG. 5: (color online) Gigantic magnetochromism via electr omagnons in the ferroelectric phase of\nBa2Mg2Fe12O22. Imaginary part of the dielectric constant spectra ( ǫ2) in magnetic fields up to 7\nT measured at (a) 5.5 K originally in the conical spin ordered phase, at (b) 60.3 K near the conical\nto proper screw transition temperature, at (c) 91 K and at (d) 164 K originally in the proper screw\nphase. Respective ǫ2spectra are arbitrarily off-set for clarity; the off-set revel i s indicated by the\nhorizontal solid line. The light E-vector was set parallel to [001] and the magnetic field Hwas\napplied along [100] in Voigt geometry.\n170 1 2 3 4 5 6 7050 100 150 200 250 \nMagnetic Field (T) Temperature (K) \n3.00 4.20 5.40 6.60 7.80 9.00 10.2 11.4 12.6 13.8 15.0 0 1 2 3 4 5 6 702468\n020 40 60 80 100 Polarization ( µC/m 2)M[100] ( µB/f.u.) \n5 K H≠0H=0H≥4 T Conical Ferri. \nH(a) \n(b) Spectral Weight (arb. units) \nFIG. 6: (color online) Magnetoelectric phase diagram of Ba 2Mg2Fe12O22, as determined by the sig-\nnatureof the electromagnon. Themagnetic field Hwas applied along [100] to inducethe transverse\nconical spin phase, as schematically shown in the inset of (a ). In this phase, the ferroelectricity\nemergesalong[120], perpendiculartoboth[100] and[001] ( seealsoFig. 1). (a) Magnetization along\n[100]M[100]and spontaneous polarization along [120] as a function of ma gnetic field, measured at\n5 K. (b) Contour map of the spectral weight (SW) of the electro magnon in terms of temperature\nand magnetic field. The measured points are indicated by open circles. The magnitude of SW (see\ntext for the definition) is shown with the scale color bar. The unit of SW is arbitrary. The solid\nline represents the region, where the transverse conical sp in structure was identified by the neutron\nscattering experiments24. In this region, the spontaneous polarization emerges, as s hown in (a).18FIG. 7: (color online) Schematic illustrations of the spin a rrangement in Ba 2Mg2Fe12O22. (a) Ac-\ncording to the previous neutron scattering experiments25,LandSsublattice blocks were composed\nof Fe ions, which are labeled by “1”, “4”, “5”, “8”, “11”, “12” , “11′”, “8′”, “5′”, “4′”, and “1′”. (b)\nIn the ferrimagnetic phase at 294 K, the spins at respective s ites are collinear with a turn angle of\n0◦; the reported data were taken from Ref. 25. (c) In the proper s crew phase at 77 K, the spins are\nnon-collinear with a turn angle of 59.6◦25. (d) Possible source of the electromagnons we observed\nhere in the conical spin ordered phase. Within the framework of the symmetric-exchange induced\nstriction, electromagnon becomes active when ∆ Siis not perpendicular to Sj. We focus on the Fe\n“4”, “5”, and “8” sites. In the conical phase, the components of the spins projected on to (120)\nplane can give rise to the electromagnon activity, as schema tically shown in (d). In this case, the\nelectromagnon intensity depends on the conical angle θand shows the maximum at θof 45◦. This\ncan be experimentally realized in magnetic fields. Furtherm ore, there is no chance to produce the\nelectromagnon activity with θof 0◦or 90◦. These are all consistent with present experiments.\n19" }, { "title": "2105.08088v2.Fluctuation_induced_ferrimagnetism_in_sublattice_imbalanced_antiferromagnets_with_application_to_SrCu__2__BO__3____2__under_pressure.pdf", "content": "Fluctuation-induced ferrimagnetism in sublattice-imbalanced antiferromagnets\nwith application to SrCu 2(BO 3)2under pressure\nPedro M. C^ onsoli, Max Fornoville, and Matthias Vojta\nInstitut f ur Theoretische Physik and W urzburg-Dresden Cluster of Excellence ct.qmat,\nTechnische Universit at Dresden, 01062 Dresden, Germany\n(Dated: August 16, 2021)\nWe show that a collinear Heisenberg antiferromagnet, whose sublattice symmetry is broken at the\nHamiltonian level, becomes a \ructuation-induced ferrimagnet at any \fnite temperature Tbelow the\nN\u0013 eel temperature TN. We demonstrate this using a layered variant of a square-lattice J1-J2model.\nLinear spin-wave theory is used to determine the low-temperature behavior of the uniform magne-\ntization, and non-linear corrections are argued to yield a temperature-induced qualitative change\nof the magnon spectrum. We then consider a layered Shastry-Sutherland model, describing a frus-\ntrated arrangement of orthogonal dimers. This model displays an antiferromagnetic phase for large\nintra-dimer couplings. A lattice distortion which breaks the glide symmetry between the two types\nof dimers corresponds to broken sublattice symmetry and hence gives rise to ferrimagnetism. Given\nindications that such a distortion is present in the material SrCu 2(BO 3)2under hydrostatic pressure,\nwe suggest the existence of a \ructuation-induced ferrimagnetic phase in pressurized SrCu 2(BO 3)2.\nWe predict a non-monotonic behavior of the uniform magnetization as function of temperature.\nI. INTRODUCTION\nThe \feld of quantum magnetism harbors a wealth of\nfascinating phenomena which are driven by \ructuations\n[1]. These include quantum spin liquids [2, 3] { stable\nstates of matter devoid of symmetry-breaking order {,\nseveral types of unconventional quantum phase transi-\ntions [4], as well as a variety of symmetry-breaking states\nstabilized by \ructuations. A large class of the latter are\ndescribed as \\order by disorder\", a mechanism where a\nsubset of states is selected from a classically degener-\nate manifold by either quantum or thermal \ructuations\n[5]. Order by disorder is prominent in strongly frustrated\nmagnets, one important example being the easy-plane\npyrochlore antiferromagnet where an ordered state is cho-\nsen from a one-parameter degenerate manifold [6].\nAmong the various frustrated spin systems, the\nShastry-Sutherland model [7] plays a prominent role. It\ndescribes a planar Heisenberg model of coupled pairs of\nspins 1=2 with a particular orthogonal-dimer structure.\nIts ground-state phase diagram features a dimer-singlet\nstate, a symmetry-breaking plaquette-singlet state, and\na N\u0013 eel antiferromagnet as function of increasing ratio\nof inter-dimer to intra-dimer couplings, x=J0=J[8{\n10]. Very recently, a narrow quantum spin-liquid phase\nhas been proposed in addition [11]. Local moments ar-\nranged on the Shastry-Sutherland lattice appear in a\nnumber of compounds, the most prominent one being\nthe spin-1=2 Mott insulator SrCu 2(BO 3)2[8, 12]. Re-\nmarkably, hydrostatic pressure can be used to tune x\nin SrCu 2(BO 3)2, and signatures of magnetic transitions\nhave been detected around 1 :8 GPa [13{19] and 4 :5 GPa\n[14], with various experimental aspects being under ac-\ntive debate [18, 20]. NMR experiments [13] yield evidence\nfor two distinct Cu sites in the intermediate phase, sug-\ngesting two types of inequivalent dimers. Antiferromag-\nnetic (AF) order has been detected by neutron di\u000bractionin the high-pressure phase [21].\nIn this paper we discuss the phenomenon of\n\ructuation-induced ferrimagnetism in antiferromagnets,\nand we propose that SrCu 2(BO 3)2at high pressure is in\nfact a ferrimagnet. Ferrimagnetism refers to states which\ndisplay both staggered and uniform magnetizations, and\nit commonly occurs in systems with two di\u000berent types\nof magnetic ions with unequal spin sizes [22]. Here, we\nidentify a distinct mechanism for ferrimagnetism: In a\nsystem with equal-sized spins which displays N\u0013 eel anti-\nferromagnetism in the ground state, a uniform magne-\ntization is induced at \fnite temperature solely by \ruc-\ntuation e\u000bects. More precisely, we show that thermal\n\ructuations generically produce a \fnite magnetization\nin a Heisenberg antiferromagnet once the Z2symmetry\nbetween the two sublattices is broken at the Hamiltonian\nlevel. Remarkably, quantum \ructuations do not produce\na \fnite magnetization at T= 0 due to spin conserva-\ntion, such that the uniform magnetization becomes a\nnon-monotonic function of temperature, as illustrated in\nFig. 1. We exemplify this in a layered toy model consist-\ning of two interpenetrating square-lattice ferromagnets,\nfor which we employ spin-wave theory to calculate the\ntemperature-induced magnetization. In addition, simple\nTmtot\nTN00~T 4~(TNT) \nFIG. 1: Qualitative temperature dependence of the\n\ructuation-induced uniform magnetization in sublattice-\nimbalanced Heisenberg antiferromagnets, with the critical ex-\nponent\f= 0:37 [23, 24] in d= 3 dimensions.arXiv:2105.08088v2 [cond-mat.str-el] 13 Aug 20212\nLandau theory is used to analyze the behavior near the\nN\u0013 eel temperature TN. We then consider a layered ver-\nsion of the Shastry-Sutherland model, as appropriate for\nthe material SrCu 2(BO 3)2. We predict the existence of a\nuniform magnetization in its orthorhombic high-pressure\nphase and provide a rough estimate for its amplitude.\nThe remainder of the paper is organized as follows:\nIn Sec. II we introduce the toy model and demonstrate\nthe phenomenon of \ructuation-induced ferrimagnetism.\nWe also discuss temperature-induced corrections to spin-\nwave spectrum and link them to general hydrodynam-\nics. Sec. III is devoted to the layered Shastry-Sutherland\nmodel appropriate for SrCu 2(BO 3)2where we provide\nquantitative results of relevance for its high-pressure\nphase. A discussion and outlook close the paper.\nII. FERRIMAGNETISM FROM THERMAL\nFLUCTUATIONS: TOY MODEL\nIn this section we utilize a simple toy model to discuss\nthe emergence of ferrimagnetism from thermal \ructua-\ntions in antiferromagnets with broken sublattice symme-\ntry. We also connect the results to general aspects from\nLandau theory and hydrodynamics.\nA. Model\nOur model is constructed from a bipartite square-\nlattice Heisenberg model with nearest-neighbor AF cou-\nplingJbetween spins S, displaying collinear N\u0013 eel order.\nTheZ2symmetry between the two sublattices is broken\nby adding second-neighbor couplings which are di\u000berent\nfor the two sublattices AandB; we label them J0\naandJ0\nb\nand choose them to be ferromagnetic in order to stabilize\nN\u0013 eel antiferromagnetism. Finally, we add a (small) fer-\nromagnetic inter-layer interaction J?such that magnetic\norder also appears at \fnite temperatures. The model is\ndepicted in Fig. 2, its Hamiltonian reads\nH=JX\nhijim~Si;m\u0001~Sj;m\u0000J?X\nim~Si;m\u0001~Si;m+1\n\u0000J0\naX\nhhii02Aii~Si;m\u0001~Si0;m\u0000J0\nbX\nhhjj02Bii~Sj;m\u0001~Sj0;m (1)\nwherei;jdenote in-plane lattice coordinates, mis the\nlayer index, andhijiandhhii0iidenote pairs of \frst and\nsecond neighbors, respectively. The model displays a\nglobal SU(2) spin symmetry. For J0\na6=J0\nbit features\na two-site unit cell, and we will set the lattice constant\nof the underlying square lattice to unity.\nVarious limiting cases are of interest: On one hand,\nforJ\u001dJ0\na;bwe have an antiferromagnet in which J0\na6=\nJ0\nbinduces weak sublattice symmetry breaking. On the\nother hand, J0\na;b\u001dJcorresponds to two inequivalent\nferromagnets on the two sublattices which are weakly\ncoupled byJsuch that global collinear AF order emerges.\nFIG. 2: Layered square-lattice Heisenberg antiferromagnet\nwith two inequivalent second-neighbor couplings J0\na(green)\nandJ0\nb(red). Thermal \ructuations induce a \fnite uniform\nmagnetization for 0 0. The mode dispersions are illustrated in Fig. 3\nfor parameter sets with (a) J\u001dJ0\na;band (b)J\u001cJ0\na;b.\nNote that !~k\u0006is symmetric with respect to rotations\naround the kzaxis up toO\u0000\nk2\u0001\n. Moreover, the quadratic\nterm in Eq. (4) vanishes when J0\na=J0\nb, i.e., for equivalent\nsublattices. We also see a vanishing of the quadratic term\nforkk= 0 because the interlayer coupling J?does not\ndistinguish sublattice AfromB. Hence, spin waves with\nzero in-plane momentum do not experience the sublattice\nsymmetry breaking.\nIt is instructive to discuss the limit J0\na;b\u001dJ, which, as\nmentioned earlier, describes two inequivalent and weakly\ncoupled ferromagnetic subsystems. In this setting, de-\ncreasingJshould restrict the linear portion of the spec-\ntrum to smaller and smaller values of kas the Gold-\nstone modes approach the quadratic shape expected for\n(a)\n012345\n(b)024680.000.010.02FIG. 3: Spin-wave dispersion for the toy model (1) along\na path in the Brillouin zone and parameters (a) J= 100,\nJ0\na= 10,J?= 2 and (b) J= 0:1,J0\na= 10,J?= 1 in units\nofJ0\nb, both with \u0011= 1. Blue (red) curves correspond to !~k+\n(!~k\u0000), respectively. The inset in (b) shows a zoom into the\nlow-energy part of the dispersion near ~k= (0;0;0).\ndecoupled ferromagnets. One can track this transforma-\ntion by computing the nonzero wavenumber k\u0003(\u0012), with\ntan\u0012=kk=kz, at which the magnitudes of the linear and\nquadratic terms in Eq. (4) become equal. A simple cal-\nculation yields\nk\u0003(\u0012) =2p\n2J\njJ0a\u0000J0\nbjsin\u0012\u0012\nJ0\na+J0\nb+J+J?\ntan2\u0012\u00131=2\n;(5)\nwhich con\frms our expectation: For \fxed J0\na6=J0\nband\nJ?,k\u0003indeed decreases with J.\nInspecting the Bogoliubov coe\u000ecients, explicitly listed\nin Eq. (A8), shows that the two modes have di\u000berent\nweights on the two sublattices once J0\na6=J0\nb. More specif-\nically, forJ0\na> J0\nbthe mode + (\u0000) is primarily located\non sublattice A(B).\nWe note that the low-energy behavior of the mode dis-\npersion is qualitatively di\u000berent for \u00116= 1, and we will\nget back to this in Sec. II F below. This section will also\ndiscuss corrections to the mode dispersion beyond linear\nspin-wave theory.4\nD. Uniform magnetization at low temperatures\nSpin-wave theory can be used to calculate \ructuation\ncorrections to the sublattice magnetizations via a 1 =S\nexpansion. The next-to-leading order result, obtained\nfrom linear spin-wave theory (see Appendix A for de-\ntails), reads\nmA=S\u0000m0(T) +1\nNX\n~k\u0010\nn~k\u0000\nBE\u0000n~k+\nBE\u0011\n;\nmB=\u0000\u0011S+m0(T) +1\nNX\n~k\u0010\nn~k\u0000\nBE\u0000n~k+\nBE\u0011\n: (6)\nHere,n~k\u0006\nBE= 1=(e!~k\u0006=T\u00001) is the Bose-Einstein distri-\nbution function where we have set Boltzmann's constant\nto unity,kB= 1, and\nm0(T) =X\n~kF~k\nN\u0010\n1 +n~k\u0000\nBE+n~k+\nBE\u0011\n\u00001\n2; (7)\nwithF~kbeing a temperature-independent coe\u000ecient\nspeci\fed in Eq. (A9). One can see that the quantum cor-\nrections,m0(T= 0), are equal on both sublattices, lead-\ning to vanishing total magnetization at T= 0 for\u0011= 1,\nas announced in Sec. II B above. The thermal correc-\ntions, however, are di\u000berent because the non-degenerate\nspin-wave modes experience di\u000berent thermal occupa-\ntions. As a result, we obtain a uniform magnetization\nper site,\nmtot=(1\u0000\u0011)\n2S+1\nNX\n~k\u0010\nn~k\u0000\nBE\u0000n~k+\nBE\u0011\n; (8)\nwhich is \fnite for non-zero temperature even if the spin\nsizes on the two sublattices are equal, \u0011= 1. This is a\ncentral result of this paper.\nWe expect these expressions to yield reliable results\nfor low temperatures where the occupation of the spin-\nwaves modes remains small to validate the approximation\nof noninteracting magnons. This corresponds to having\nsmall 1=Scorrections in Eq. (6) in comparison to the\nleading-order term.\nThe \ructuation-induced uniform magnetization\nemerges at order S0in the spin-wave expansion. Fig. 4\ndepicts its temperature dependence, together with the\nthermal corrections to the sublattice magnetizations,\nat a \fxed ratio J0\na=J0\nb= 10 when (a) J\u001dJ0\na;band (b)\nJ\u001cJ0\na;b. In both cases, the low-temperature corrections\ntomAandmBscale asT2, a feature which is also\nobserved in antiferromagnets without broken sublattice\nsymmetry [25]. This can be easily rationalized by power\ncounting: For linearly dispersing modes, F~kscales\nas 1=k, and hence the leading contribution to the T\ndependence of the integral in Eq. (7) has the formR\ndkkd\u00002nBE(k=T) and scales as Td\u00001.\nIn this low-temperature regime, the only accessible ex-\ncited states lie within the energy range where the magnon\n(a)\n10010110210310-1010-810-610-410-2100\n(b)\n10-210-110010110210-910-710-510-310-1101FIG. 4: Fluctuation-induced uniform magnetization,\nmtot, as function of temperature, together with the \fnite-\ntemperature corrections to the sublattice magnetizations,\n\u0001mA;B=mA;B(T)\u0000mA;B(0). The coupling constants were\nset to (a)J= 100,J0\na= 10,J?= 2 and (b) J= 0:1,J0\na= 10,\nJ?= 1 in units of J0\nb. In both plots, the horizontal grid\nline marks the value 1 =2, whereas the vertical grid line in (b)\nindicates the temperature T=JS.\nbranches are nearly degenerate, so that a di\u000berence in\nthe occupation of the spin-wave modes emerges as a sub-\nleading e\u000bect in T. Expanding the dispersions in next-\nto-leading order in kand noting that the diverging F~k\nfactor is absent from Eq. (8), we \fnd that the uniform\nmagnetization scales as T4, see Appendix A.\nIn Fig. 4(a), we see that for J\u001dJ0\na;bthis low-Tpower\nlaw continues well beyond the point where T=S matches\nthe smallest coupling in the system, J0\nb, and extends up to\ntemperatures T\u0018JS, beyond which spin-wave theory is\nno longer valid. The reason is that strong Jyields both\nlarge spin-wave velocities at small k(4)and large val-\nues ofk\u0003(\u0012) (5), such that a signi\fcant di\u000berence in the\noccupations of the spin-wave modes only appears at rela-\ntively high energies. As a result, mtotonly reaches values\nof 10\u00003at the temperature where thermal corrections to\nmA;Bbecome signi\fcant, i.e., of order 10\u00001. The oppo-\nsite limit,J\u001cJ0\na;b, leads to a markedly di\u000berent behav-\nior, shown in Fig. 4(b). The now weak coupling Jstabi-\nlizes approximately sublattice-symmetric AF order only\nat very low temperatures. Thus, once T&JS, sublattice\nBbecomes much more susceptible to \ructuations than5\n(a)10-510-410-310-210-1\n(b)\n123456789100.000.010.020.030.04\nFIG. 5: Fluctuation-induced magnetization as in Fig. 4, but\nnow as function of J0\na=J0\nbat \fxed \fnite temperature. Param-\neters were set to (a) T=S = 50,J= 100,J?= 2 and (b)\nT=S = 1,J= 0:1,J?= 1 all in units of J0\nb, and are such that\nthe results with the ratio J0\na=J0\nb= 10 correspond to data in\nFig. 4 at the speci\fed temperatures.\nsublatticeA, which explains why mB(mA) starts grow-\ning faster (slower) than T2around the vertical dashed\nline. The deviation from the low-temperature scaling is\nthen responsible for enhancing the uniform magnetiza-\ntion, which becomes as large as 5 \u000110\u00002when thermal\ncorrections to mBreach 10\u00001.\nFig. 5 illustrates how the same quantities in Fig. 4 vary\nwith the ratio J0\na=J0\nbat \fxedT=(J0\nbS) when (a) J\u001dJ0\na;b\nand (b)J\u001cJ0\na;b. The uniform magnetization vanishes\nin the limit J0\na=J0\nb!1, which corresponds to restoring\nsublattice symmetry, while mtot/(J0\na=J0\nb\u00001) for small\nimbalance. However, the increase in mtotsaturates at a\npoint where J0\na=J0\nbbecomes so large that the spin-wave\nvelocities in Eq. (4) start growing at the same rate that\nk\u0003(\u0012) decreases in Eq. (5). Put more simply, saturation\noccurs when increasing J0\na=J0\nbonly leads to further split-\nting of the spin-wave bands at energies larger than T=S.\nThe main di\u000berence between Figs. 5(a) and (b) lies in the\norder of magnitude of the \ructuation-induced e\u000bects: At\nJ0\na=J0\nb= 10,mtotis two orders of magnitude larger in\npanel (b) compared to panel (a). Once again, this di\u000ber-\nence follows from the fact that the strong AF coupling J\nbetween the sublattices in (a) balances how \ructuationsact on each of them, therefore diminishing the resulting\nuniform magnetization.\nE. Uniform magnetization at elevated\ntemperatures\nTo access elevated temperatures where spin-wave the-\nory is no longer reliable, we resort to arguments from\nLandau theory. The sublattice-imbalanced antiferromag-\nnet { equivalent to a ferrimagnet { is described by two\norder parameters, a staggered magnetization and a uni-\nform magnetization, which are linearly coupled [26]. As\na result, there is a single transition upon cooling at TN\nfrom the paramagnet to the ferrimagnet, and both order\nparameters are zero (non-zero) above (below) TN, respec-\ntively. The linear coupling also implies that the onset of\nthe uniform magnetization below TNis identical to that\nof the staggered magnetization, hence mtot/(TN\u0000T)\f\nwhere\fis the order-parameter exponent [27]. For the\nclassical phase transition at hand, the universality class\nfor the Heisenberg magnet remains O(3) independent of\nwhether the transition is into an antiferromagnetic or fer-\nrimagnetic state. Hence, \f= 0:37 ind= 3 space dimen-\nsions [23, 24]. Together with the low-temperature result\nmtot/T4, we conclude that the uniform magnetization\ndisplays a non-monotonic temperature dependence as il-\nlustrated in Fig. 1.\nAgain, it is instructive to discuss the limit J\u001cJ0\na;b.\nForJ0\na>J0\nb, the transition at TNconcerns primarily the\nonset of ferromagnetism on the A sublattice. Weak J\nproduces a small opposite magnetization on the Bsub-\nlattice, resulting in a collinear ferrimagnet. (Note that\nthe energy gained by maintaining the collinearity of the\nglobal magnetic order is extensive, whereas the entropy\nassociated with directional \ructuations of sublattice B\nis only intensive.) In the limit of large sublattice im-\nbalance,J0\na\u001dJ;J0\nb, there is hence a window of tem-\nperatures below TNwhere the uniform magnetization is\nlarge,mtot\u0019mA=2 asmB\u001cmA. In this limit, it is also\neasy to see that the sign of the uniform magnetization is\nthe same at low Tand close to TN: It is the sublattice\nwith weaker magnetism that experiences stronger ther-\nmal \ructuations, such that mtotaligns with the magne-\ntization of the more strongly ordered sublattice, i.e., the\nAsublattice if J0\na>J0\nb.\nF. Hydrodynamic modes and corrections to the\nspin-wave spectrum\nFor a broader picture, we connect our \fndings\nto general hydrodynamic considerations. A collinear\ntwo-sublattice antiferromagnet, spontaneously breaking\nSU(2) symmetry, is characterized by a non-conserved or-\nder parameter and displays two linearly dispersing Gold-\nstone modes which are degenerate in the long-wavelength\nlimit. This is in agreement with the spin-wave result (4)6\nfor\u0011= 1.\nA ferrimagnet, in contrast, has in addition a conserved\norder parameter, namely uniform magnetization mtot.\nAs a result, it features a single quadratically dispersing\nGoldstone mode [28, 29]. This can be nicely seen in the\nexplicit spin-wave expressions (2) for \u00116= 1: Here !~k\u0000\nis gapless and quadratic in kwhereas!~k+, though also\nquadratic, exhibits a gap given by 4 j\u0011\u00001jJS.\nTogether, this implies that the low-energy spectrum of\nthe model (1) must change qualitatively when going from\nT= 0 toT >0: The system turns from an antiferromag-\nnet to a ferrimagnet, such that one of the T= 0 Gold-\nstone modes must acquire a temperature-induced gap,\nand the other one must change its dispersion from linear\nto quadratic at small k. This change can be captured by\nnon-linear spin-wave theory, i.e., has the form of 1 =Scor-\nrections at \fnite T. While it is straightforward to write\ndown the quartic terms in the spin-wave Hamiltonian,\nanalyzing all terms at \fnite temperature turns out to be\nrather laborious, and therefore we refrain from doing so.\nHowever, as these corrections are suppressed as T!0,\nthey have no in\ruence on the leading low-temperature\nbehavior of the uniform magnetization, mtot/T4.\nIII. FERRIMAGNETISM IN A LAYERED\nDISTORTED SHASTRY-SUTHERLAND MODEL\nAfter having established that sublattice-imbalanced\nantiferromagnets generically display \ructuation-induced\nferrimagnetism, we now turn to an experimentally rele-\nvant example, namely the Shastry-Sutherland lattice as\nrealized in the compound SrCu 2(BO 3)2.\nA. Model and symmetries\nOur starting point is the Heisenberg model on\nthe Shastry-Sutherland lattice, consisting of orthogonal\ndimers of spins 1 =2 with intra-dimer coupling Jand inter-\ndimer coupling J0, Fig. 6(a). This model features a four-\nsite unit cell, containing two dimers, and displays, in ad-\ndition to mirror symmetries along the dimer axes, a non-\nsymmorphic glide symmetry which maps the two types\nof dimers into each other.\nThe phase diagram of the Shastry-Sutherland model\nhas been determined numerically [9, 10], Fig. 6(b): It\ncontains a paramagnetic dimer phase for x=J0=J <\n0:675, a bipartite N\u0013 eel antiferromagnet for x > 0:765,\nand a plaquette-ordered singlet paramagnet, the so-called\nempty-plaquette phase, in between [10]. In addition, a\nvery recent numerical study [11] has proposed that a\ngapless quantum spin-liquid phase is realized in a nar-\nrow range, 0 :79< x < 0:82, intervening between the\nplaquette-singlet and AF phases [30].\nThe ferrimagnetism discussed in this paper appears\nupon breaking the glide symmetry, such that two dif-\nferent types of (mutually parallel) dimers emerge. Such\n(a)\n(b)\ndimer\nsingletplaquette\nsingletNéel\nAFSL?FIG. 6: (a) Shastry-Sutherland model with intra-dimer cou-\nplingsJa;band inter-dimer coupling J0. The dashed lines\nindicate the unit cell. (b) Ground-state phase diagram of the\nS= 1=2 Shastry-Sutherland model with Ja=Jb, as reported\nin Ref. 10. An intermediate spin-liquid (SL) phase (shaded)\n[30] has been recently proposed in Ref. 11. In the present\nwork, the focus is on the antiferromagnetic phase at large\nJ0=Jshown in red.\nFIG. 7: Layered Shastry-Sutherland model (9) used to de-\nscribe SrCu 2(BO 3)2. An orthorhombic distortion is assumed\nto generate di\u000berent intra-dimer couplings Ja,Jb.\nsymmetry breaking corresponds to an orthorhombic dis-\ntortion where half of the intra-dimer bonds elongate and\nthe other half contract [31, 32]. In the AF state, each of\nthe intra-dimer couplings acts on one AF sublattice only,\nsuch that the symmetry between the two sublattices is\nbroken. In the following we will therefore consider a dis-\ntorted Shastry-Sutherland model with intra-dimer cou-7\nplingsJa;band inter-dimer coupling J0. To meaningfully\ndiscuss magnetic order at \fnite temperature, we work\nwith a layered version of the model. Guided by the struc-\nture of SrCu 2(BO 3)2, we consider a stacking of the layers\nsuch that orthogonal dimers are on top of each other, and\ninclude a (small) antiferromagnetic Heisenberg interlayer\ncouplingJ?which pairwise connects vertically stacked\ndimers [33]. The model, illustrated in Fig. 7, is described\nby the Hamiltonian\nH=JaX\nhhij2Aiim~Si;m\u0001~Sj;m+JbX\nhhij2Biim~Si;m\u0001~Sj;m\n+J0X\nhiji~Si;m\u0001~Sj;m\n+J?X\nhhijiim(~Si;m+~Sj;m)\u0001(~Si;m+1+~Sj;m+1) (9)\nwhere each term in the last sum represents four couplings\nbetween the spins of neighboring dimers in zdirection,\nand we consider spins of general size S.\nThe ground states of the single-layer version of the dis-\ntorted Shastry-Sutherland model (9) with S= 1=2 have\nbeen studied in Refs. 31, 32. While all phases of the orig-\ninal Shastry-Sutherland model appear stable against a\nsmall dimer imbalance, the main \fnding of Ref. 31 is the\nexistence of a Haldane-like phase for strongly imbalanced\ndimers and weak inter-dimer coupling J0. This phase is\ndominated by one-dimensional correlations; it is adiabat-\nically connected to a so-called full-plaquette phase and\nhas been argued [32] to be a candidate for the interme-\ndiate phase observed experimentally in SrCu 2(BO 3)2.\nB. Spin-wave theory in the antiferromagnetic phase\nAs announced, we are interested in antiferromagnets\nwith broken sublattice symmetry. Hence we focus on the\nphysics of the model (9) in the regime of larger x=J0=J,\nwhere one encounters a clear connection to the toy model\ndiscussed in Sec. II: Both systems display a collinear anti-\nferromagnetic classical ground state with two sublattices,\neach of which experiences an independent internal cou-\npling. Therefore, the two models share the same mecha-\nnism for breaking sublattice symmetry.\nAs in Sec. II, we perform a spin-wave calculation to\ndetermine its properties in a 1 =Sexpansion both at zero\nand \fnite temperature. For the standard 2D Shastry-\nSutherland model, the linear-spin-wave theory descrip-\ntion of the AF phase breaks down for x < 1, signaling\na transition to a di\u000berent phase at this level of the ap-\nproximation. Hence, we work with parameter sets corre-\nsponding to x&1.\nIn the AF phase of model (9), the symmetry-broken\nstate features four sites per unit cell, such that the Bo-\ngoliubov transformation can only be performed numer-\nically. For convenience, we employ an in-plane coordi-\nnate system corresponding to the square lattice shown in\n012345FIG. 8: Spin-wave dispersion of the distorted Shastry-\nSutherland model (9) along a path in the Brillouin zone for\nparameters Ja= 0:97,Jb= 0:9 andJ?= 0:2 in units of J0.\nFig. 6(a), such that the basis vectors of the (magnetic)\nunit cell are given by a1= (2a;0;0),a2= (0;2a;0), and\na3= (a;a;c ) whereaandcare the in-plane and out-of-\nplane lattice constants which we set to unity in the fol-\nlowing. The relevant details of the calculation are given\nin Appendix B; here we summarize the key results.\nThe spin-wave spectrum along an exemplary path in\nthe BZ is illustrated in Fig. 8. Of the four spin-wave\nmodes, two are gapped at small momenta, while the\ntwo others are linearly dispersing Goldstone modes. As\nwith the toy model, these modes are degenerate only\nforJa=Jb, i.e. when the sublattice symmetry is pre-\nserved; for Ja6=Jbthey share the same velocity, but\ndi\u000ber at quadratic order except for kk= 0. Notably,\nthe Goldstone-mode velocity is highly anisotropic, e.g., it\nis di\u000berent even for di\u000berent in-plane directions because\nJa6=Jbleaves only mirror symmetries intact.\nC. Ferrimagnetism\nThe qualitative arguments for \ructuation-induced fer-\nrimagnetism brought forward in Sec. II apply unchanged\nto the sublattice-imbalanced antiferromagnet of the\nShastry-Sutherland model. Our numerical evaluation of\n1=Scorrections to the magnetizations on the individual\nsites of the unit cell, as detailed in Appendix B, con-\n\frms this expectation. The quantum corrections are\nequal on all sites, resulting in a vanishing total magne-\ntization at T= 0. In contrast, the thermal corrections\nare di\u000berent on the A(up) andB(down) sublattices,\nwhile they are pairwise equal on the two unit-cell sites\nbelonging to the AandBsublattice, respectively. As\nbefore, the thermal corrections to the sublattice mag-8\n(a)10-1210-1010-810-610-410-2100\n(b)10-210-110010-1010-810-610-410-2100\nFIG. 9: Uniform magnetization, mtot, and thermal correc-\ntions, \u0001mA;B, to the sublattice magnetization of the distorted\nShastry-Sutherland model, with parameters (a) Ja= 0:9,\nJb= 0:8,J?= 0:1 and (b)Ja= 0:97,Jb= 0:9,J?= 0:01 in\nunits ofJ0.\nnetizations, \u0001 mA(T) and \u0001mB(T), scale proportional\ntoT2at low temperature. The uniform magnetization,\nmtot= [mA(T) +mB(T)]=2, scales as T4because two\nmode dispersions di\u000ber at quadratic order only.\nNumerical results illustrating the variation of the mag-\nnetization with temperature are shown in Fig. 9. The\nparameters in panels (a) and (b) correspond to weaker\n(stronger) \ructuation corrections, driven both by the\ndi\u000berentJ?and byJa;bbeing further away (closer) to\nthe critical value Ja;b=J0within spin-wave theory.\nConsequently, the uniform magnetization is much larger\nin (b) compared to (a) at the same temperature, even\nthough the ratio Ja=Jbis similar in both cases. While\nthe extreme limit of two weakly coupled ferromagnets,\ndiscussed for the toy model, cannot be realized in the\nShastry-Sutherland model, the magnetization neverthe-\nless can get as large as 5 \u000110\u00003at the temperature where\nthe largest \u0001 mis 10\u00001.\nFig. 10 depicts the e\u000bect of varying the sublattice im-\nbalance at a \fxed temperature while keeping J0as the\nlargest coupling in the system. Di\u000berently from the toy\nmodel, the thermal corrections are now larger on the\nstrongly coupled sublattice, since the AF couplings Ja;b\nare frustrated. Still, the uniform magnetization shows\nthe same trend as in Fig. 5, growing with increasing sub-\n1 2 3 4 5 6 7 8 9100.000.050.100.150.20FIG. 10: Same quantities as in Fig. 9, but now as a function\nofJa=Jbat \fxed temperature T=J0Sand withJa= 0:9\n(and varying Jb) andJ?= 0:1 in units of J0. The ratio\nJa=Jb= 1:125 reproduces the data in Fig. 9(a) at the speci\fed\ntemperature.\nlattice imbalance until it saturates at large Ja=Jb. For\nSrCu 2(BO 3)2, small orthorhombic distortion likely im-\nplies thatJa=Jbremains close to unity.\nD. Application to SrCu(BO 3)2under pressure\nSrCu 2(BO 3)2assumes a tetragonal structure at ambi-\nent pressure and low temperatures, where its magnetic\nproperties are in very good agreement with those of the\ntwo-dimensional Shastry-Sutherland model in the small-\nxdimer phase. The magnetic couplings have been esti-\nmated to be J\u001985 K andJ0\u001954 K [8, 33].\nHigh-pressure studies of SrCu 2(BO 3)2detected various\nsignatures of pressure-driven phase transitions. In par-\nticular, indications for a di\u000berent, but still paramagnetic,\nphase were found above 2 GPa [13], and this transition\nwas later located more precisely to be around 1 :8 GPa\n[14{19]. While it is natural to assume that this para-\nmagnetic phase represents the empty-plaquette phase of\nthe Shastry-Sutherland model, both the NMR results of\nRef. 13 and the neutron scattering results of Ref. 16\nappear to be incompatible with this idea: The empty-\nplaquette phase displays equivalent magnetic sites and\nC4symmetry while the NMR data indicate the existence\nof two inequivalent magnetic sites. To resolve this con-\ntradiction, it has been argued [32] that an orthorhombic\ndistortion, stabilizing a di\u000berent plaquette phase in the\nintermediate regime, is most compatible with the NMR\n[13] and neutron scattering [16] data.\nAt higher pressures, a structural transition to a mon-\noclinic structure occurs around 4 :5 GPa [14], and AF or-\nder with a rather high N\u0013 eel temperature of 120 K has\nbeen detected at 5 :5 GPa via neutron scattering [21]. It\nhas been suggested, but not clari\fed beyond doubt, that\nthis magnetic order in fact emerges around 4 GPa before\nthe structural transition [20]. In addition, a recent low-9\ntemperature thermodynamic study [18] found indications\nfor a previously undetected AF state below 4 K occurring\nbetween 3 and 4 :2 GPa. It has been suggested [18] that it\nis this low-temperature AF state which should be inter-\npreted as the genuine AF state of the Shastry-Sutherland\nmodel, given that the higher-pressure monoclinic sys-\ntem no longer features the orthogonal dimers character-\nistic of the Shastry-Sutherland model. The same study\nalso presented evidence for an additional phase transi-\ntion occurring above 4 :2 GP at 8 K and proposed that\nthis transition is related to the existence of yet another\nlow-temperature magnetic state, which is likely to dis-\nplay AF order as well. However, a full characterization\nof this phase is still lacking. Apparently, more work is\nneeded to discern the fascinating high-pressure physics of\nSrCu 2(BO 3)2.\nFor our purpose, we focus on the fact that pressure-\ninduced structural distortions lead to inequivalent\ndimers; this likely applies to all pressures larger than\n2 GPa [13, 21, 32]. AF order in each layer will thus be\nsublattice-imbalanced because of the broken glide sym-\nmetry. Achieving a \fnite uniform magnetization then\nrelies on the uniform magnetization in adjacent layers\nbeing parallel. Our model in Fig. 7 assumes the estab-\nlished layer stacking, with orthogonal dimers on top of\neach other [21, 33], an antiferromagnetic interlayer cou-\npling [18, 21, 33], and an orthorhombic distortion of the\ntetragonal structure as proposed in Ref. 32. Together,\nthis yields a macroscopic uniform magnetization, which\nwe can estimate to reach up to 5 \u000110\u00003\u0016Bper Cu atom\nat its temperature maximum, i.e., slightly below the N\u0013 eel\ntemperature, see Fig. 1. It would be very interesting to\ntest this prediction in future high-pressure magnetization\nmeasurements. For such an experiment, one needs to\nkeep in mind that ferrimagnets generically form magne-\ntization domains much like ferromagnets [22]; detecting\na uniform magnetization might therefore require cooling\nin an applied \feld.\nWe note that the type of dimer distortion suggested to\nexist at 5:5 GPa at elevated Tin Ref. 21, see their Figs. 5\nand 6, would lead to a vanishing total magnetization in-\nstead, because a strong up-spin dimer in one layer would\ncouple to a strong (instead of weak) down-spin dimer in\nthe next layer. The resulting \fnite magnetization per\nlayer may still be detectable as a surface e\u000bect.\nFinally, we comment on the e\u000bect of small\nDzyaloshinskii-Moriya (DM) interactions which break\nSU(2) symmetry at the Hamiltonian level and are known\nto be present in SrCu 2(BO 3)2[21, 34]. In the AF phase\nof interest here, DM interactions may lead to a small,\nbut \fnite, uniform magnetization at T= 0 and also alter\nits leading low-temperature corrections as the spin-wave\nspectrum may acquire a gap at T= 0. Nonetheless, the\nconclusion that inequivalent sublattices experience dif-\nferent thermal \ructuations remains, and for small DM\ninteractions the non-monotonic temperature dependence\nof the magnetization will be preserved.IV. CONCLUSIONS AND OUTLOOK\nIn summary, we have shown that collinear N\u0013 eel anti-\nferromagnets, whose Z2symmetry between the two sub-\nlattices is broken in the Hamiltonian, become ferrimag-\nnets at \fnite temperature. Interestingly, this is an e\u000bect\ndriven by thermal \ructuations but not by quantum \ruc-\ntuations, constituting an interesting example where both\ntypes of \ructuations produce di\u000berent physics { in con-\ntrast to many instances of order by disorder where ther-\nmal and quantum \ructuations lead to very similar state\nselection [5]. We have proposed that such a \ructuation-\ninduced ferrimagnetic phase is realized in SrCu 2(BO 3)2\nunder high pressure, where a lattice distortion breaks the\nglide symmetry of the Shastry-Sutherland lattice.\nOur work suggests a number of future directions: First,\nit is conceivable that similar thermal \ructuation ef-\nfects occur in antiferromagnets with more complicated\nground-state spin structures. Second, while we have ar-\ngued that the antiferromagnetic phase with mtot= 0\nis a stable state of matter at T= 0 despite sublat-\ntice imbalance, it is interesting to ask whether quantum\n\ructuations can generate additional, more non-trivial,\nzero-temperature phases in sublattice-imbalanced anti-\nferromagnets. Finally, considering the same phenomenol-\nogy in the presence of charge carriers will lead to a\n\ructuation-induced anomalous Hall e\u000bect.\nAcknowledgments\nWe thank L. Janssen and E. Andrade for discus-\nsions and collaborations on related work. Financial sup-\nport from the Deutsche Forschungsgemeinschaft through\nSFB 1143 (project-id 247310070) and the W urzburg-\nDresden Cluster of Excellence on Complexity and Topol-\nogy in Quantum Matter { ct.qmat (EXC 2147, project-id\n390858490) is gratefully acknowledged.\nAppendix A: Spin-wave calculations for toy model\nThe starting point for our analysis of the toy model\nproposed in Sec. II was to expand the Hamiltonian in\nEq. (1) in powers of 1 =p\nSaround a collinear N\u0013 eel state.\nSince we allow the two magnetic sublattices to have, at\nleast in principle, spins of unequal sizes, the Holstein-\nPrimako\u000b transformation reads\nSz\ni=S\u0000ay\niai;\nS+\ni=q\n2S\u0000ay\niaiai=p\n2Sai+O\u0010\n1=p\nS\u0011\n;\nS\u0000\ni=ay\niq\n2S\u0000ay\niai=p\n2Say\ni+O\u0010\n1=p\nS\u0011\n;(A1)10\nfor sitesilocated on sublattice Aand\nSz\ni=\u0000\u0011S+by\nibi\nS+\ni=by\niq\n2\u0011S\u0000by\nibi=p\n2\u0011Sby\ni+O\u0010\n1=p\n\u0011S\u0011\n;\nS\u0000\ni=q\n2\u0011S\u0000by\nibibi=p\n2\u0011Sbi+O\u0010\n1=p\n\u0011S\u0011\n;(A2)\nfor sites belonging to sublattice B. As usual, ay\niandby\ni\nare bosonic creation operators with corresponding anni-\nhilation operators aiandbi.\nAfter substituting Eqs. (A1) and (A2) into the Hamil-\ntonian, applying a Fourier transform to the bosonic op-\nerators, and neglecting terms beyond O(S), one arrives\nat a quadratic Hamiltonian of the form\nHLSW=S2Ecl+S\n2X\n~kh\n\ty\n~kM~k\t~k\u0000\u0000\nA~k+B~k\u0001i\n;(A3)\nwhere\nS2Ecl=\u0000NS2\"\n2\u0011J+J0\na+\u00112J0\nb+\u0000\n1 +\u00112\u0001\n2J?#\n(A4)\nis the classical ground-state energy for a system with N\nsites, \t~k=\u0010\na~k;b~k;ay\n\u0000~k;by\n\u0000~k\u0011T\n, and\nM~k=0\nB@A~k0 0C~k\n0B~kC~k0\n0C~kA~k0\nC~k0 0B~k1\nCA (A5)\nwith\nA~k= 4\u0000\n\u0011J+J0\na\u0018~k\u0001\n+ 2J?(1\u0000coskz);\nB~k= 4\u0000\nJ+\u0011J0\nb\u0018~k\u0001\n+ 2\u0011J?(1\u0000coskz);\nC~k= 4\u00111=2J\r~k: (A6)\nThe linear spin-wave Hamiltonian in Eq. (A3) can be\ndiagonalized by means of a Bogoliubov transformation\na~k=u~k\u000b~k\u0000v~k\fy\n\u0000~k; b~k=u~k\f~k\u0000v~k\u000by\n\u0000~k(A7)\nwith coe\u000ecients\nu~k= sgn\u0000\nC~k\u0001r\nF~k+ 1\n2; v~k=r\nF~k\u00001\n2(A8)\ngiven in terms of the function\nF~k=A~k+B~kq\u0000\nA~k+B~k\u00012\u00004C2\n~k: (A9)\nWhen expressed in terms of the Bogoliubov operators,\nEq. (A3) takes on the diagonal form\nHLSW=S2Ecl+1\n2X\n~kh\n!~k\u0000+!~k+\u0000S\u0000\nA~k+B~k\u0001i\n+X\n~k\u0010\n!~k+\u000by\n~k\u000b~k+!~k\u0000\fy\n~k\f~k\u0011\n: (A10)The resulting mode dispersions !~k\u0006are speci\fed in\nEq. (4) of the main text, with the abbreviations be-\ning related to the terms de\fned in Eq. (A6) as P~k=\n(A~k+B~k)=2,R~k= (A~k\u0000B~k)=2, andQ~k=C~k.\nWe can then use the previous relations to derive\nthe sublattice magnetizations to next-to-leading order in\n1=S. For sublattice A, we have\nmA=2\nNX\ni2AhSz\nii=S\u00002\nNX\n~kD\nay\n~ka~kE\n=S\u00002\nNX\n~k\u0010\nu2\n~kD\n\u000by\n~k\u000b~kE\n+v2\n~kD\n1 +\fy\n~k\f~kE\u0011\n=S\u00001\nNX\n~kh\u0000\nF~k+ 1\u0001\nn~k+\nBE+\u0000\nF~k\u00001\u0001\u0010\n1 +n~k\u0000\nBE\u0011i\n;\n(A11)\nwhich is equal to the \frst expression in Eq. (6). One\nrecovers the second expression straightforwardly after a\nsimilar sequence of steps for sublattice B.\nThe low-temperature behavior of the uniform magneti-\nzation can in fact be predicted analytically by expanding\nthe spin-wave dispersion up to quadratic order in kand\nretaining the leading contribution in Tto Eq. (8). Let us\nfocus on the case \u0011= 1 for concreteness. After rewriting\nEq. (4) in the form\n!~k\u0006\u0019!1(\u0012)k\u0006!2(\u0012)k2; (A12)\nwe \fnd\nn~k\u0000\nBE\u0000n~k+\nBE\u00192e\u0000\f!1ksinh\u0000\n\f!2k2\u0001\n1\u00002e\u0000\f!1kcosh (\f!2k2) +e\u00002\f!1k\n\u00192e\u0000\f!1k\f!2k2: (A13)\nThe approximation made in the last step of (A13) is\njusti\fed by the fact that we are dealing with momenta\nk.1=(\f!1), and hence \f!2k2.!2=\u0000\n\f!2\n1\u0001\n\u001c1. We then\nsubstitute Eq. (A13) into Eq. (8) to obtain\nmtot/\fZ\u0019\n0d\u0012sin\u0012!2(\u0012)Z1\n0dkk4e\u0000\f!1(\u0012)k\n/jJ0\na\u0000J0\nbj\n(\fS)4J5Z\u0019\n0d\u0012sin\u0012\u0014!2(\u0012)\njJ0a\u0000J0\nbjS\u0015\u0014JS\n!1(\u0012)\u00155\n:\n(A14)\nSince the integral in the last line above is expressed solely\nin terms of dimensionless quantities, we conclude that in\nthe low-temperature limit\nmtot/jJ0\na\u0000J0\nbj\nJ\u0012T\nJS\u00134\n; (A15)\nwhich is precisely what our numerical results indicate.\nAppendix B: Spin-wave calculations for\nShastry-Sutherland model\nThe spin-wave calculations for the Shastry-Sutherland\nmodel are based on a Holstein-Primako\u000b representation11\nof the spin operators as above. Since the unit cell consists\nof four spins which exhibit collinear N\u0013 eel order in the\nclassical limit, we assign the transformations given by\nEqs. (A1) and (A2) to two spins each, with spin size S\non all sublattices. Here, the spins on sublattice A(B)\ncorrespond to Holstein-Primako\u000b operators aiandci(bi\nanddi), respectively, see Fig. 6(a).\nAs noted in the main text, for calculation purposes\nwe choose a coordinate system where the Shastry-\nSutherland plane is located on a square lattice along the\nJ0bonds. Moreover, the planes are stacked such that\ninequivalent dimers are positioned on top of each other.\nWith lattice constants set to unity, this yields real-space\nbasis vectors a1= (2;0;0),a2= (0;2;0),a3= (1;1;1)\nand reciprocal-space basis vectors b1=\u0019(1;0;\u00001),b2=\n\u0019(0;1;\u00001),b3= 2\u0019(0;0;1).\nSubstitution of the Holstein-Primako\u000b operators into\nthe Hamiltonian given by Eq. (9) and a subsequent\nFourier transform yields the quadratic Hamiltonian\nHLSW=S2Ecl+S\n2X\n~kh\n\ty\n~kM~k\t~k\u00002B~ki\n(B1)\nwhere\nS2Ecl=NS2\u0014Ja+Jb\n4\u00002J0\u00002J?\u0015\n(B2)\nis again the classical ground state energy for a system\nconsisting of Nspins, \t~k=\u0010\na~k;c~k;by\n\u0000~k;dy\n\u0000~k\u0011T\nand\nM~k=0\nBB@A~kC~kE~kF~k\nC\u0003\n~kA~kF\u0003\n~kE\u0003\n~k\nE\u0003\n~kF~kB~kD~k\nF\u0003\n~kE~kD\u0003\n~kB~k1\nCCA(B3)\nwhere\u0003denotes the complex conjugate and\nA~k= 4J0\u0000Ja+ 4J?;\nB~k= 4J0\u0000Jb+ 4J?;\nC~k=Jaei(\u0000kx+ky);\nD~k=Jbei(kx+ky);\nE~k= 2J0cosky+J?(ei(\u0000kx+kz)+ei(\u0000kx\u0000kz));\nF~k= 2J0coskx+J?(ei(ky+kz)+ei(ky\u0000kz)): (B4)\nThe Hamiltonian in Eq. (B1) can now be diagonalized\nby means of a generalized Bogoliubov transformation [35]\n\t~k=T(~k)\b~k: (B5)with \b~k=\u0010\n\u000b~k;\r~k;\fy\n~k;\u000ey\n~k\u0011T\nbeing the normal mode\nspinor. The columns of T(~k) correspond to the eigen-\nvectors of \u0006 M~kwith\n\u0006 =\u0012\nI0\n0\u0000I\u0013\n(B6)\nwhere Idenotes the 2\u00022 unit matrix. Solving the eigen-\nvalue problem for \u0006 M~kyields two positive eigenvalues\n\u0015~k1;2and two negative eigenvalues \u0015~k3;4. The normal\nmodes of the Hamiltonian are then given by\n!~k1;2=S\u0015~k1;2;\n!~k3;4=\u0000S\u0015~k3;4: (B7)\nImportantly, the positive and negative eigenvalues are\nnotof pairwise equal magnitude. With Eqs. (B5) and\n(B7) the linear spin-wave Hamiltonian then takes the di-\nagonal form\nHLSW=S2Ecl+X\n~k\u0014!~k3+!~k4\n2\u0000SB~k\u0015\n+X\n~kh\n!~k1\u000by\n~k\u000b~k+!~k2\ry\n~k\r~k+!~k3\fy\n~k\f~k+!~k4\u000ey\n~k\u000e~ki\n:\n(B8)\nThe sublattice magnetization can then be derived from\nthe Holstein-Primako\u000b operators using the respective Bo-\ngoliubov coe\u000ecients and Eq. (B5). The magnetizations\non theaandcsites, both belonging to sublattice A, are\nequal and read\nmA=2\nNX\ni2AhSz\nii=S\u00004\nNX\n~khay\n~ka~ki\n=S\u00004\nNX\n~k\u0010\njT11(~k)j2h\u000by\n~k\u000b~ki+jT12(~k)j2h\ry\n~k\r~ki\n+jT13(~k)j2h1 +\fy\n~k\f~ki+jT14(~k)j2h1 +\u000ey\n~k\u000e~ki\u0011\n;(B9)\nfrom which the zero-temperature magnetization mA(T=\n0) and its temperature correction can be calculated; ex-\npressions for the Bsublattice follow analogously.\n[1] C. Lacroix, P. Mendels, and F. Mila (Eds.), Introduction\nto Frustrated Magnetism , Springer, Heidelberg (2011).\n[2] L. Savary and L. Balents, Rep. Prog. Phys. 80, 016502(2017).\n[3] Y. Zhou, K. Kanoda, and T.-K. Ng, Rev. Mod. Phys. 89,\n025003 (2017).12\n[4] M. Vojta, Rep. Prog. Phys. 81, 064501 (2018).\n[5] J. Villain, R. Bidaux, J.-P. Carton, and R. Conte, J.\nPhys. France 41, 1263 (1980).\n[6] M. E. Zhitomirsky, M. V. Gvozdikova, P. C. W.\nHoldsworth, and R. Moessner, Phys. Rev. Lett. 109,\n077204 (2012).\n[7] B. S. 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Results have been obtained for a YIG sphere (d = 5 mm , 𝑀𝑆 = 140 \nkA/m ). Solid lines refer to the unshielded sphere, and dotted lines refer to the shielded sphere with 𝐷𝑐 = 10 mm. \nb) Q-factors as a function of magnetic bias obtained with the electrodynamic TDE . For the shielded structure , it \nhas been assumed that the shield is perfectly conducting . \n \nFig.4. a) Frequency offset characteristics of selected modes solved with the electrodynamic TDE for a shielded \nYIG sphere (d = 5 mm, 𝐷𝑐 = 10 mm , 𝑀𝑆 = 140 kA/m ). b) Plots of the electric field distribution for four TE10p \nmodes. Plots of the electric field were obtained at 𝐻0𝑟=2.5 for the TE101 mode and at 𝐻0𝑟=5 for the remaining \nthree TE10p modes. \nThe Q-factor of a few TE modes is presented in Fig. 3b. For the unshielded sphere, the total Q-\nfactor accounts for radiation and magnetic losses, while for the shielded structure, only magnetic losses \nare present assuming that the cavity is perfectly conducting. It can be noticed that the electrodynamic \nTDE allows to predict the frequency above which radiation losses exceed magnetic losses [21], and this \nfrequency increases with the elevation mode order, n. \n \n3.3 Electrodynamic analysis of factors influencing mode spacing \nFrequency offsets obtained for TE n01 modes, 𝑛=1…3, are shown in Fig. 5 for two diameters of the \nsample, while Fig. 6 presents the frequency spacing between the two modes 𝛥𝑤21=𝑤2−𝑤1. In \nqualitative accordance with Eq. (5), the reson ance frequencies of TE n01 modes decrease with increasing \nsample diameter d and permittivity 𝜀𝑓. As demonstrated in Fig. 5, increasing permittivity of the \nsurrounding medium, 𝜀𝑑, has a similar effect. The increase of either of the model parameters 𝐻0𝑟, d, \n𝜀𝑓, 𝜀𝑑 lowers each consecutive TE n01 mode frequency to a lesser degree, leading to the increase of 𝛥𝑤𝑛1 \nmode spacings. Similarly , as it is seen in Fig.6, the frequency spacing increases with the increase of the \ndiameter of the metal enclosure, 𝐷𝑐. It was shown in [13] that the spacing between the modes is \nessentially independent of crystalline orientation. However, in general, since crystalline orientation \ninfluences the internal bias [7], it can influence the mode spacing, but it will be practically noticeable \nonly for large anisotropy values. \n \nFig.5. Differences between normalized frequencies and the normalized static magnetic field for TE 101, TE 201 and \nTE301 modes as a function of the normalized static magnetic field for two different permittivity values of medium \nsurrounding ferromagnetic samples ( 𝜀𝑑) for sample s having diameter equal to a) d =0.5 mm, and b) d =1.0 mm. \n \nFig.6. Differences bet ween normalized frequencies of TE 201 (w2) and TE 101 (w1) modes as functions of \nnormalized static magnetic field for various diameters ( Dc) of metal enclosures. Ms = 140 kA/m, εf = 16, εd = \n1 were assumed. a) sample diameter d =0.5 mm, b) sample diameter d =1.0 mm. \n \n4. Experiments \n \nExperiments have been performed in the setup shown in Fig. 7. The single -crystal YIG sphere (d = 0.5 \nmm) in a PTFE tube is inserted between the orthogonally oriented coupling loops connected to a vector \nnetwork analyzer (VNA), which is employed to measure the scattering matrix coefficient ( S21). The \nloops surrounding the sample are soldered to the metal enclosure made of copper ( Dc = 2.5 mm). An \nelectromagnet was used as a source of static magnetic bias. Fig. 8 shows an exemplary spectrum of \nferromagnetic modes with TE 101, TE 201 and TE 301 modes annotated . Such spectra were recorded for \nseveral static magnetic bias values . \nAs is postulated in this paper, considering variations of the frequency spacing between the TE n0p \nmodes using the electrodynamic TDE , it is possible to improve the accuracy of the determination of 𝑀𝑠. \nThe procedure to obtain Ms from the set of measured resonant frequencies fk, fl obtained for different \nexternal magnetic bias values Hkext is as follows: \n1. As a starting point, t ake an initial value of Ms based on knowledge of the material or determine \nMs using Eq. (6) from fk, fl measured at low static bias assuming magnetostatic values of wk and \nwl. \n2. Compute wk and wl for each Hkext using the electrodynamic TDE. \n3. Compute Ms for each Hkext using Eq. (6). \n4. Repeat the procedure starting from the average value of Ms (Hext) until convergence. \n \n \nFig.7. Schematic of experimental setups – top view (left) and side view (right) . \n \nTable I shows measured frequency spacing for the spectrum shown in Fig. 8 and the \ncorresponding Ms determined with the aid of both TDEs . It is seen that the normalized frequenc y \ndifferences obtained employing the electrodynamic model are smaller than their magnetostatic \ncounterparts , which confirms theoretical results shown in Fig. 6a. Values of Ms obtained with the aid of \nthe electrodynamic and magnetostatic TDEs using Eq. (6) well agree with the nominal value provided \nby the manufacturer (𝑀𝑆 =139. 2 kA/m) if one considers the spacing between the modes TE 301 and TE 201. \nHowever, t he magnetostatic model gives rise to about 7 % of error in 𝑀𝑆 if TE 201 and TE 101 modes are \ncavityco eranalyzed d ue to the erroneous prediction of Δw21 at the given magnetic bias . The use of the latter pair of \nmodes has the practical advantage that it is easier to couple to the TE 201 than to the TE 301 mode. \n \nFig.8. Spectrum of | S21| in the setup shown in Fig. 7 for a YIG sphere (d = 0.5 mm , 𝑀𝑆 =139.2 kA/m ) at H0r ≈ \n0.65. \n \nTable I. Saturation magnetization determined for a YIG sample ( d = 0.5 mm) in metal enclosure ( Dc = 2.5 mm) \nfrom m easure d resonance frequencies and the corresponding computed frequency offsets. The static magnetic \nbias was such that H0r ≈ 0.65. \nk \nf \n(MHz) 𝛥𝑤𝑘,𝑘−1 \nmagnetoststic 𝛥𝑤𝑘,𝑘−1 \nelectrodynamic Fletcher’s \nmode \nlabelling Electrodynamic \nTEn0p mode \ndesignation 𝑀𝑠 \n(kA/m) \nelectrodynamic 𝑀𝑠 \n(kA/m) \nmagnetostatic \n1 4809 1 1 0 1 0 1 \n2 5113 0.066667 0.062096 2 2 0 2 0 1 139.0 4 129.51 \n3 5253 0.028571 0.028568 3 3 0 3 0 1 139.1 8 139.24 \n \n \nFig.9. Saturation magnetization , Ms, determined from the frequency offset between TE 101 and TE 201 modes as a \nfunction of magnetic bias obtained for a) a single -crystal doped YIG (d = 0.483 mm , 𝛥𝐻≈ 1 Oe at 10 GHz) , and \nb) polycrystalline calcium vanadium ferrite (d = 1.4 mm , 𝛥𝐻≈ 4 Oe at 10 GHz). The larger relative error s in Fig. \n9b as compared to Fig. 9a arise because multiple modes are excited in the larger sphere and they influence each \nother’s resonance frequency due to moderate mode Q-factors. \n \nFig.9 shows the saturation magnetization determined from the frequency offsets between TE 101 \nand TE 201 modes measured in a broad range of magnetic bias for two additional ferromagnetic samples . \nThe first sample (Fig. 9a) is made of single -crystal Ga-doped YIG having d = 0.483 mm and \n𝛥𝐻≈1 Oe at 10 GHz . The second sample (Fig. 9b) is made of polycrystalline calcium vanadium ferrite \nhaving d = 1.4 mm and 𝛥𝐻≈4 Oe at 10 GHz. Samples were measured in the experimental setup shown \nin Fig. 7. For the smaller sample, Ms obtained with the proposed electrodynamic approach amounted to \nca. 66.6 kA/m within a 2% relative error , which is lower than the nominal value of 69.6 kA/m declared \nby Vendor #1. Calculat ing Ms using the magnetostatic approach leads to an overestimation of Ms by up \nto ca. 10% compared to the electrodynamic approach . For the larger calcium vanadium sample having \nd = 1.4 mm , computed values of Δw21 and measured frequency offsets between the corresponding modes \nincrease with H0r by as much as 50% in the measurement range . As a result , the saturation magnetization \ndetermined with Eq. (6) with the aid of the electrodynamic TDE remains constant within an experimental \nerror of a few per cent (see Fig. 9b) and is in close agreement with the nominal value of 147.2 kA/m \ndeclared by Vendor #2. On the contrary, t he saturation magnetization determined from the magnetostatic \nTDE, where 𝛥𝑤21=2\n5−1\n3=𝑐𝑜𝑛𝑠𝑡 , increases up to about 50% above the nominal value at H0r = 3. \n \n \n5. Discussion \n \nAnalysis of the methods applicable to the measurement of Ms using Eq. (6) available in the literature \n[15] reveal s that experimental errors can reach even +58% if the magnetostatic approach is used. The \npresented experiments , thus, confi rm the superiority of the electrodynamic approach as the error in the \nmeasurement of Ms predominantly depends on the accuracy of the theoretical assessment of 𝑤𝑘 and 𝑤𝑙. \nContrary to experiments, these coefficients obtained with the magnetostatic model are constant with H0r \nand other parameters. It should be mentioned that the scattering theory ( see Eq. (5)) allows to \napproximate the resonance frequency changes of the mode of uniform precession as function of sample \nsize and its permittivity in an infinitely large metal enclosure and up to d/ ≈ 0.1 [12]. However, this \ntheory is not strictly applicable for the higher order modes. Therefore , the electrodynamic model is a \nnatural choice for accurate computations of 𝑤𝑘−𝑤𝑘−1 and, consequently, accurate determination of \nMs. The use of the electrodynamic TDE instead o f Walker -Fletcher’s TDE allows to remove many \npractical sources of error present in the determination of Ms with Eq. (6) and extend the measurement \nrange to higher static bias fields. \n6. Conclusions \n \nResults of our computations of TE n0p mode s in shielded and unshielded ferromagnetic spheres \nemploying a rigorous electrodynamic approach confirm that these modes can be treated as ordinary \nelectromagnetic resonances in a gyrotropic, dispersive and lossy magnetic medium . The magnetostatic \nmodel of these resonances allows to analyze only some of their properties, most notably resonance \nfrequencies for sufficiently small samples. Most of the dominant resonances that exist in small \nferromagnetic spheres below the frequency of FMR are related either to the negative effective \npermittivity value (𝜇𝑟) for circularly polarized [ n,n,0] (also known as TE n01 using electrodynamic \nnotation ) and [ n,n-1,0] modes or to the negative value of the diagonal component 𝜇 of permeability \ntensor. The presented electrodynamic theory allows for accurate measurements of the basic parameters \nof magnetic materials including, but not limited to, the saturation magnetization. \nAcknowledgement \nThis work was partially supported by t he TEAM -TECH 2016 -1/3 Project entitled “High -precision \ntechniques of millimeter and sub -THz band characterization of materials for microelectronics” o erated \nwithin the Foundation for Polish Science TEAM TECH Program co -financed by the European Regional \nDevelopment Fund, Operational Program Smart Growth 2014 -2020 . \nThis work was partially supported by “Project PROM - International Scholarship Exchange for PhD \nStudents and Academic Staff” , financed from the European Social Fund under the Operational Program \nKnowledge Education Development, non -competitive project entitled International Scholarship \nExchange for PhD Students and Academic Staff, contract number: POWR.03.03.00 -00-PN13/18. This work was partiall y su orted by “The ARC Centre of Excellence for Engineered Quantum \nSystems”, roject grant number CE170100009. \nThis work was partially supported by the National Science Centre, project registration number \n2018/31/B/ST7/04006. \n \nAppendix A \n \nPermeability tensor \nThe ferromagnetic resonance (FMR) phenomenon (FMR) can be quantitatively described with a permeability \ntensor 𝜇̿ derived from the Landau -Lifshitz equation [6]. In a presence of uniform static magnetic field magnetizing \nferrite material along z-axis of Cartesian or cylindrical coordinate system the permeability tensor takes the form \n(1a). When the static magnetic field is sufficiently strong to saturate gyromagnetic medium, then 𝜇||=1 and \npermeability tensor is known as the Polder tensor [5]. \n𝜇̿= 𝜇0[𝜇𝑗𝜅0\n−𝑗𝜅𝜇0\n00𝜇||] (1a) \nThe diagonal and the off -diagonal relative components of the Polder tensor take the following form [6]: \n𝜇=1+𝐻0𝑟 + 𝑗 𝛼 𝑤̂\n𝐻0𝑟2 − 𝑤̂2 + 2 𝑗 𝛼 𝐻0𝑟 𝑤̂ (2a) \n𝜅=𝑤̂\n𝐻0𝑟2 − 𝑤̂2 + 2 𝑗 𝛼 𝐻0𝑟 𝑤̂ (3a) \nwhere: 𝐻0𝑟=𝐻0/𝑀𝑆 , 𝑤̂ =𝑓̂ /𝑓𝑚, 𝑓𝑚=𝛾𝑀𝑆, 𝐻0 is the static magnetic field inside the sample (the internal static \nmagnetic field), 𝑀𝑆 is the saturation magnetization, 𝛼 is a Gilbert damping factor, and 𝑓̂ is the complex frequency. \n \nThe natural frequency of ferromagnetic resonance 𝑓𝑚 is defined as 𝑓𝑚=𝛾𝑒𝑓𝑓𝑀𝑆 where 𝑀𝑆 is the \nsaturation magnetization of ferrite material, and 𝛾𝑒𝑓𝑓 is the effective gyromagnetic ratio. For free electron \ngyromagnetic ratio is known with high precision from cyclotron measurements 𝛾𝑒=|𝑒|\n2𝑚𝑒𝑔𝑒≈ 35.21719 \nMHz/(kA/m), and g-factor 𝑔𝑒=2.002319 , while for real ferrite materials g-factor values are de termined from \nexperiments. g-factor values for typical microwave ferrites are close to 𝑔𝑒 so it is often assumed that 𝛾𝑒𝑓𝑓=𝛾𝑒 = \n𝛾. \nThe imaginary part of frequency describes time dependence of electromagnetic fields (transient solutions) \nfor a micr owave resonator containing lossy medium. Instead to the Gilbert damping factor the relaxation time, 𝜏=\n1/(𝛼𝛾𝐻0), and the ferromagnetic resonance linewidth, ∆𝐻=2𝛼𝐻0=2/(𝛾𝜏), are often used to describe the losses \nin ferromagnetic material. \nFor circularly polarized electromagnetic fields permeability of ferromagnetic material can be expressed \nas (4). \n𝜇𝑟,𝑙=𝜇± 𝜅 (4a) A positive (negative) sign in Equation (4 a) corresponds to the clockwise (counter -clockwise) polarization of the \nmagnetic field. \nFor lossless medium formulas (2 a), (3a) and (4 a) reduce to \n𝜇=1+𝐻0𝑟 \n𝐻0𝑟2 − 𝑤2 (2b) \n𝜅=𝑤\n𝐻0𝑟2 − 𝑤2 (3b) \n𝜇𝑟=1+1 \n𝐻0𝑟 − 𝑤 (4b) \nFerromagnetic resonance frequency 𝑓𝐹𝑀𝑅 is defined as the frequency for which the denominators in expressions \n(2a) and (3a) and (4a) vanish, which takes place when 𝑓𝐹𝑀𝑅=𝛾𝐻0 or 𝑤=𝐻0𝑟. \n \nAppendix B \n \nTranscendental equations of electrodynami cs \nFor the TE n0p modes of free oscillations of an isotropic sphere having permeability 𝜇𝑟, relative complex \npermittivity 𝜀𝑓, and immersed in dielectric medium having permittivity 𝜀𝑑 , as in Fig.1a, the TDE can be written \nas follows [3]: \n{𝑛𝐽𝑛+1\n2(𝑘𝑅1)−𝑘𝐽𝑛−1\n2(𝑘𝑅1)}𝐻𝑛+1\n2(2)(𝑘0𝑅1)−𝜇𝑟{𝑛𝐻𝑛+1\n2(2)(𝑘0𝑅1)−𝑘0𝐻𝑛−1\n2(2)(𝑘0𝑅1)}𝐽𝑛+1\n2(𝑘𝑅1)=0 (6a) \nn, p are elevation (related to in Fig.1) and radial mode indices, respectively, c is the speed of the EM wave in a \nvacuum, J (H) are Bessel (Hankel) functions. The radial mode index p denotes consecutive roots of (6a) while the \nazimuthal mode index m has no impact on the resonance frequency, so it is assumed to be m = 0. \nFor the TE n0p modes of free oscillations of an isotropic sphere having permeability 𝜇𝑟, relative complex \npermittivity 𝜀𝑓, and immersed in dielectric medium having permittivity 𝜀𝑑 , as in Fig.1 b, the TDE can be written \nas follows [ 8]: \ndet(𝑊(𝜔))=0 (7a) \nwhere: 𝑊=[𝑤11𝑤12𝑤13\n𝑤21𝑤22𝑤23\n0𝑤32𝑤33] \n𝑤11=𝜇𝑟(𝑘1𝑅1)1\n2𝐽𝑛+1\n2(𝑘1𝑅1) \n𝑤12=−(𝑘2𝑅1)1\n2𝐽𝑛+1\n2(𝑘2𝑅1) \n𝑤13=−(𝑘2𝑅1)1\n2𝑌𝑛+1\n2(𝑘2𝑅1) \n𝑤21=(𝑘1𝑅1)3\n2[𝐽𝑛−1\n2(𝑘1𝑅1)−𝑛\n𝑘1𝑅1𝐽𝑛+1\n2(𝑘1𝑅1)] \n𝑤22=−(𝑘2𝑅1)3\n2[𝐽𝑛−1\n2(𝑘2𝑅1)−𝑛\n𝑘2𝑅1𝐽𝑛+1\n2(𝑘2𝑅1)] \n𝑤23=−(𝑘2𝑅1)3\n2[𝑌𝑛−1\n2(𝑘2𝑅1)−𝑛\n𝑘2𝑅1𝑌𝑛+1\n2(𝑘2𝑅1)] 𝑤32=(𝑘2𝑅2)1\n2𝐽𝑛+1\n2 (𝑘2𝑅2) \n𝑤33=(𝑘2𝑅2)1\n2𝑌𝑛+1\n2(𝑘2𝑅2) \n \nwhere: \n𝑘1=𝜔(𝜀𝑓𝜇𝑟)0.5/𝑐, 𝑘2=𝜔(𝜀𝑑)0.5/𝑐 \nY – Bessel functions of the second kind . \n \nWhen equations (6a) or (7a) are solved with respect to the complex angular frequency 𝜔̂, the Q-factors can be \ndetermined as 𝑄=𝜔′/2𝜔′′. \n \nReferences \n[1] Baden Fuller, Ferrites at Microwave Frequencies . Institution of Engineering and Technology, 1987. \n[2] L. R. Walker, “Resonant Modes of Ferromagnetic S heroids,” J. Appl. Phys. , vol. 29, no. 3, pp. 318–\n323, Mar. 1958. \n[3] J. Kru ka, B. Salski, P. Ko yt, and W. Gwarek, “Electrodynamic study of YIG filters and resonators,” \nSci. Rep. , vol. 6, pp. 1 –9, 2016. \n[4] R. L. White, “Use of Magnetostatic Modes as a Research Tool,” J. Appl. Phys. , vol. 31, no. 5 , pp. S86–\nS94, May 1960. \n[5] D. Polder, “On the theory of ferromagnetic resonance,” Physica , 1949. \n[6] A. Gurevich, Ferrites at microwave frequencies . Consultants Bureau Enterprises Inc., 1963. \n[7] J. Krupka, P. Aleshkevych, B. Salski, P. Kopyt, and A. Pac ewicz, “Ferromagnetic Resonance Revised – \nElectrodynamic A roach,” Sci. Rep. , vol. 7, no. 1, p. 5750, Dec. 2017. \n[8] P. Fletcher, I. H. Solt, and R. Bell, “Identification of the Magnetostatic Modes of Ferrimagnetic \nResonant S heres,” Phys. 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Magid, “A Microwave Technique for the Analysis and Determination of Saturation Magnetization of \nNarrow -Linewidth Ferrimagnetic Malerials,” IEEE Trans. Instrum. Meas. , vol. IM -13, no. 4, pp. 329 –\n335, Dec. 1964. \n[16] D. N. Bose, S. R. Borgaonkar, and T. S . Vedavathy, “Measurement of magnetic ro erties of single \ncrystal YIG by non -resonant method,” Bull. Mater. Sci. , vol. 1, no. 2, pp. 121 –128, Oct. 1979. \n[17] I. H. Solt, “Tem erature De endence of YIG Magnetization,” J. Appl. Phys. , vol. 33, no. 3, pp. 11 89–1191, Mar. 1962. \n[18] P. Röschmann and H. Dötsch, “Pro erties of Magnetostatic Modes in Ferrimagnetic S heroids,” Phys. \nStatus Solidi , vol. 82, no. 1, pp. 11 –57, Jul. 1977. \n[19] P. C. Fletcher and R. O. Bell, “Ferrimagnetic resonance modes in s heres,” J. Appl. Phys. , 1959. \n[20] J. Bourhill, N. Kostylev, M. Goryachev, D. L. Creedon, and M. E. Tobar, “Ultrahigh coo erativity \ninteractions between magnons and resonant hotons in a YIG s here,” Phys. Rev. B , 2016. \n[21] J. Kru ka, P. Aleshkevych, B. Salski, P. Ko yt, and J. Hartnett, “Magnetic and Electric Solid -State \nPlasmon S herical Resonators,” Plasmonics , 2018. \n \n " }, { "title": "2001.08037v1.The_dynamics_of_a_domain_wall_in_ferrimagnets_driven_by_spin_transfer_torque.pdf", "content": "The dynamics of a domain wall in ferrimagnets driven by spin-transfer torque\nDong-Hyun Kim,1Duck-Ho Kim,2Kab-Jin Kim,3Kyoung-Woong\nMoon,4Seungmo Yang,4Kyung-Jin Lee,1, 5, 6and Se Kwon Kim7\n1Department of Semiconductor Systems Engineering,\nKorea University, Seoul 02841, Republic of Korea\n2Center for Spintronics, Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea\n3Department of Physics, KAIST, Daejeon 34141, Republic of Korea\n4Quantum Technology Institute, Korea Research Institute of Standards and Science, Daejeon 34113, Republic of Korea\n5Department of Materials Science and Engineering,\nKorea University, Seoul 02841, Republic of Korea\n6KU-KIST Graduate School of Converging Science and Technology,\nKorea University, Seoul 02841, Republic of Korea\n7Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA\n(Dated: January 23, 2020)\nThe spin-transfer-torque-driven (STT-driven) dynamics of a domain wall in an easy-axis rare-\nearth transition-metal ferrimagnet is investigated theoretically and numerically in the vicinity of\nthe angular momentum compensation point TA, where the net spin density vanishes. The partic-\nular focus is given on the unusual interaction of the antiferromagnetic dynamics of a ferrimagnetic\ndomain wall and the adiabatic component of STT, which is absent in antiferromagnets but exists\nin the ferrimagnets due to the dominant coupling of conduction electrons to transition-metal spins.\nSpeci\fcally, we \frst show that the STT-induced domain-wall velocity changes its sign across TA\ndue to the sign change of the net spin density, giving rise to a phenomenon unique to ferrimagnets\nthat can be used to characterize TAelectrically. It is also shown that the frequency of the STT-\ninduced domain-wall precession exhibits its maximum at TAand it can approach the spin-wave\ngap at su\u000eciently high currents. Lastly, we report a numerical observation that, as the current\ndensity increases, the domain-wall velocity starts to deviate from the linear-response result, calling\nfor a more comprehensive theory for the domain-wall dynamics in ferrimagnets driven by a strong\ncurrent.\nI. INTRODUCTION\nSpintronics is the \feld, in which electrons' spin is uti-\nlized in addition to charge for the advancement of infor-\nmation processing technology beyond the conventional\ncharge-based electronics, and, therefore, the interaction\nbetween charge and spin has been one of the central top-\nics in the \feld. In particular, the e\u000bect of a charge cur-\nrent on the magnetic dynamics, which is described as the\nspin-transfer torque (STT), has been intensively studied\nfor metallic ferromagnets since the \frst theoretical pre-\ndictions in 1996 [1, 2]. One of the major practical utilities\nof STT in spintronics is to drive a magnetic domain wall,\na topological defect between two uniform domains, that\ncan be used to realize racetrack memory [3]. Also, fun-\ndamental research on STT-induced domain-wall motion\nhas been allowing us to strengthen our understanding of\nspin-charge interaction in ferromagnets [4{7].\nDeparting from ferromagnets consisting of parallel\nspins, there is another class of magnets called antiferro-\nmagnets, where neighboring spins are antiparallel. An-\ntiferromagnets have been attracting great attention in\nspintronics, particularly for the last decade, owing to\ntheir much faster dynamics than ferromagnets and the\nensuing promise for ultrafast spintronic devices [8]. How-\never, the research on antiferromagnetic dynamics has\nbeen impeded by di\u000eculties in controlling and detecting\nthem due to, e.g., the absence of magnetization. For this\nreason, the previous research on STT in antiferromagnetshas been mostly theoretical [9{14].\nThere have been recent developments in understand-\ning and utilizing antiferromagnetic dynamics enabled by\nan emerging class of magnets called ferrimagnets, which\nconsist of two or more inequivalent magnetic sublattices\ncoupled antiferromagnetically [15]. These ferrimagnets,\nwhich are typi\fed by rare-earth transition-metal (RE-\nTM) ferrimagnetic alloys such as GdCo or GdFeCo, can\nexhibit antiferromagnetic dynamics owing to the anti-\nferromagnetic coupling of constituent spins, and, at the\nsame time, can be controlled and detected easily due to\nsmall, but \fnite magnetization caused by imperfect can-\ncellation of neighboring magnetic moments. This unique\ncombination of antiferromagnetic dynamics and \fnite\nmagnetization has recently allowed for achieving fast\ndomain-wall motion [16{19] and magnetization switch-\ning [20{22] in various ferrimagnets. In particular, Okuno\net al. [23] recently reported an experimental study of\nSTT in ferrimagnets through current-assisted \feld-driven\ndomain-wall motion, introducing ferrimagnets as a useful\nplatform to investigate STT in magnets with antiferro-\nmagnetic coupling [24].\nIn this work, we theoretically and numerically study\nthe domain-wall dynamics in RE-TM ferrimagnets driven\nby STT, which has not been explored yet. One of the\nunique features of ferrimagnets that are absent in ferro-\nmagnets and antiferromagnets is that their spin density\ncan be continuously tuned across zero by changing tem-\nperature or chemical composition. The temperature atarXiv:2001.08037v1 [cond-mat.mes-hall] 22 Jan 20202\n(a)(b)e\u0000\nAAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgxbAbBQUvAS8eI5oHJGuYnXSSIbOzy8ysEJZ8ghcPinj1i7z5N06SPWhiQUNR1U13VxALro3rfju5ldW19Y38ZmFre2d3r7h/0NBRohjWWSQi1QqoRsEl1g03AluxQhoGApvB6GbqN59QaR7JBzOO0Q/pQPI+Z9RY6R4fz7rFklt2ZyDLxMtICTLUusWvTi9iSYjSMEG1bntubPyUKsOZwEmhk2iMKRvRAbYtlTRE7aezUyfkxCo90o+ULWnITP09kdJQ63EY2M6QmqFe9Kbif147Mf0rP+UyTgxKNl/UTwQxEZn+TXpcITNibAllittbCRtSRZmx6RRsCN7iy8ukUSl75+XK3UWpep3FkYcjOIZT8OASqnALNagDgwE8wyu8OcJ5cd6dj3lrzslmDuEPnM8f5k2NhQ==Domain-wall motionDomain-wall motione\u0000\nAAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgxbAbBQUvAS8eI5oHJGuYnXSSIbOzy8ysEJZ8ghcPinj1i7z5N06SPWhiQUNR1U13VxALro3rfju5ldW19Y38ZmFre2d3r7h/0NBRohjWWSQi1QqoRsEl1g03AluxQhoGApvB6GbqN59QaR7JBzOO0Q/pQPI+Z9RY6R4fz7rFklt2ZyDLxMtICTLUusWvTi9iSYjSMEG1bntubPyUKsOZwEmhk2iMKRvRAbYtlTRE7aezUyfkxCo90o+ULWnITP09kdJQ63EY2M6QmqFe9Kbif147Mf0rP+UyTgxKNl/UTwQxEZn+TXpcITNibAllittbCRtSRZmx6RRsCN7iy8ukUSl75+XK3UWpep3FkYcjOIZT8OASqnALNagDgwE8wyu8OcJ5cd6dj3lrzslmDuEPnM8f5k2NhQ==\nFIG. 1. Schematic illustration of a domain-wall motion driven\nby the adiabatic STT in a RE-TM ferrimagnet. The red and\nthe blue arrows in the gray box represent spins of TM and RE\nelements, respectively. When a conduction electron (denoted\nbye\u0000) traverses the domain wall from the left to the right,\nits spin denoted by the red arrow follows the TM spin direc-\ntion adiabatically. After passing through the domain wall,\nthe change of electron's spin is transferred to the domain wall\nvia the adiabatic STT. (a) When TM spins are larger than\nRE spins (corresponding to T > T A), the net spin direction\nis given by the TM spin direction. The transfer of up-spin\nfrom conduction electrons to the spin texture expands the\nleft domain with net-spin up, pushing the domain wall to the\nright. (b) When RE spins are larger than TM spins (corre-\nsponding to T < T A), the net spin direction is given by the\nRE spin direction. The transfer of up-spin from conduction\nelectrons drive the domain wall to the left by expanding the\nright domain with net-spin up.\nwhich the net spin density vanishes is called the angu-\nlar momentum compensation point TA[25, 26], and it\no\u000bers one of the situations where the advantage of fer-\nrimagnets is most prominent: their dynamics is purely\nantiferromagnetic due to the vanishing spin density and\nthus is shown to be the fastest [16{18]. Therefore, in\nour study on STT-driven domain-wall dynamics, we will\nfocus on ferrimagnets in the vicinity of the angular mo-\nmentum compensation point TA.\nSTT of ferrimagnets that describes the e\u000bect of a cur-\nrent on a spatially varying spin texture is similar to STT\nof ferromagnets due to the dominant coupling of conduc-\ntion electrons' spin to one of multiple sublattices of ferri-\nmagnets, as have been invoked in Ref. [27], where the\nSTT-driven dynamics are studied for two-dimensional\nspin textures called skyrmions in ferrimagnets. It con-\nsists of the reactive and the dissipative components,\nwhich are also referred to as the adiabatic and the nonadi-\nabatic STT. The adiabatic STT, which is even under time\nreversal, describes the angular momentum transfer from\nconduction electrons to the background spin texture such\nas a domain wall when electrons' spin follows the local\nspin texture adiabatically. The nonadiabatic STT cap-\ntures the angular momentum transfer via the other pro-\ncesses deviating from the aforementioned adiabatic pro-\ncess, e.g., mistracking between conduction electrons' spin\nand the spin texture. While both terms exist in ferro-\nmagnets, antiferromagnets are lack of the adiabatic STT\nsince electrons' spin cannot follow atomically-changing\nstaggered spins adiabatically [11, 28]. STT of RE-TM\nferrimagnets possesses both terms akin to ferromagnets\nsince conduction electrons are known to interact mostlywith TM magnetic moments and thus are spin-polarized\nfollowing TM's magnetization [29, 30]. See Fig. 1 for the\nillustration. Therefore, in the vicinity of angular momen-\ntum compensation point, where the nature of dynamics is\nantiferromagnetic but STT possesses the adiabatic term,\nferrimagnets are expected to exhibit a unique phenomena\nthat can occur neither in ferromagnets nor in antiferro-\nmagnets, which we reveal theoretically and numerically\nin this work through the domain-wall dynamics.\nThis paper is organized as follows. In Sec. II, we study\nthe STT-driven domain-wall dynamics in ferrimagnets\nby varying the temperature across the angular momen-\ntum compensation point TA. Speci\fcally, by studying\nthe dependence of the velocity and the angular velocity\nof a domain wall on the net spin density and the current\nwithin the linear response, we show that the domain-wall\nvelocity changes its sign across TA(see Fig. 1 for the il-\nlustration), o\u000bering an electrical way to identify TAthat\nis known to be di\u000ecult to characterize. In addition, the\nangular velocity is shown to exhibit its maximum at TA,\nwhere most of the transferred angular momentum from\nconduction electrons is used for rotating the domain wall\nby accumulating a nonequilibrium spin density inside it.\nThe theoretical result based on the collective coordinate\napproach is supported by atomistic spin simulations. In\nSec. III, we numerically study the STT-driven dynamics\nof a domain wall exactly at TAby applying large currents\nto go beyond the linear-response regime. As the current\nincreases, the domain-wall angular velocity is shown to\nsaturate to the spin-wave gap, which is caused by the\nincrease of the domain-wall width. In addition, at large\ncurrents, we observe that the domain-wall velocity de-\nviates from what is predicted from the linear-response\ntheory, showing a limitation of the linear-response the-\nory for the domain-wall dynamics at high biases. We\nconclude the paper by providing a summary and future\noutlook in Sec. IV.\nII. STT-DRIVEN DOMAIN-WALL DYNAMICS\nWITHIN THE LINEAR RESPONSE\nIn this section, we study the dynamics of a domain\nwall in ferrimagnets driven by STT within the linear re-\nsponse, i.e., at su\u000eciently small currents. For concrete-\nness, we consider RE-TM ferrimagnets which are com-\nposed of two antiferromagnetically-coupled sublattices of\nRE spins and TM spins.\nA. Theory\nThe Landau-Lifshitz-Gilbert-like equation for a RE-\nTM ferrimagnet with STT is given by [23, 27, 31{35]\n\u000es_n\u0000\u000bsn\u0002_n\u0000\u001an\u0002n\n=\u0000n\u0002he\u000b+P(J\u0001r)n\u0000\fPn\u0002(J\u0001r)n;(1)3\nto linear order in the bias, a charge current density\nJ=J^x, where nis the unit vector along the magne-\ntization direction of RE elements, \u000esis the equilibrium\nspin density along \u0000n(i.e., along the spin direction of\nRE elements), \u000b>0 is the Gilbert damping constant, s\nis the sum of the spin densities of the two sublattices, \u001a\nis the moment of inertia representing antiferromagnetic\ndynamics of n[36],he\u000b\u0011\u0000\u000eU=\u000enis the e\u000bective \feld\nconjugate to n, andU[n] is the potential energy. The\nlast two terms on the right-hand side are the adiabatic\nand the nonadiabatic STT terms, where Pis the spin\nconversion factor given by P= (~=2e)(\u001b\"\u0000\u001b#)=(\u001b\"+\u001b#)\n(with the electron charge e>0) which characterizes the\npolarization of the spin-dependent conductivity \u001bs(s=\"\nor#with\"chosen along\u0000n), and\fis the dimensionless\nparameter characterizing the nonadiabatic torque term.\nNote that there exists the adiabatic component of STT\nsince the charge current can be spin-polarized according\nto one of two sublattices, which is in contrast with STT\nfor antiferromagnets where the adiabatic component is\nabsent [11, 28].\nWe consider a quasi-one-dimensional magnet with uni-\naxial anisotropy described by the following potential en-\nergy:\nU=Z\ndV(An02\u0000Km2\nz)=2; (2)\nwhich has been used to describe the domain-wall motion\nin magnets with perpendicular magnetic anisotropy and\nnegligible in-plane anisotropy (see, e.g., Refs. [17, 37{\n39]). Here, Ais the exchange coe\u000ecient, Kis the easy-\naxis anisotropy coe\u000ecient, and0represents the deriva-\ntive with respect to the spatial coordinate x. Here, we\nassume that the magnetic order is uniform in the yz\nplane. A domain wall is a topological soliton connecting\ntwo ground states n=\u0006^z. Its low-energy dynamics is\nknown to be well described by two collective coordinates,\npositionX(t) and angle \b( t), via the so-called Walker\nansatz [40]: n(x;t) =fsech((x\u0000X)=\u0015) cos \b;sech((x\u0000\nX)=\u0015) sin \b;\u0000tanh((x\u0000X)=\u0015)g, where\u0015represents the\ndomain-wall width. In equilibrium, the domain-wall\nwidth is given by \u00150=p\nA=K determined by the compe-\ntition between the exchange energy and the anisotropy.\nBy employing the collective-coordinate approach [5,\n41, 42], we obtain the equations of motion for Xand\n\b, which are given by\n\u000es\u0015_\b +\u000bs_X+\u001aX=\u0000\fPJ; (3)\n\u000es_X\u0000\u000bs\u0015_\b\u0000\u001a\u0015\b =\u0000PJ: (4)\nThe steady-state velocity Vand the angular velocity \nare then given by, respectively,\n_X!V=\u0000PJ(\u000es+\u000b\fs)\n\u000e2s+ (\u000bs)2; (5)\nand\n_\b!\n =PJ\n\u00150\u000bs\u0000\f\u000es\n\u000e2s+ (\u000bs)2: (6)This is our main analytical result: The domain-wall ve-\nlocityVand the angular velocity \n as a function of the\nnet spin density \u000es, which can be controlled in RE-TM\nferrimagnets by changing temperature or chemical com-\nposition.\nLet us discuss the obtained results for speci\fc cases.\nFirst, when the net spin density is su\u000eciently large, i.e.,\nwhen the temperature is su\u000eciently away from TA, the\ndomain-wall velocity can be approximated by\nV\u0019\u0000PJ=\u000e s;forj\u000esj\u001d\u000bs ; (7)\nwhile assumingj\fj\u001c1. This can be understood as the\nresult of the angular-momentum transfer /PJfrom con-\nduction electrons to the domain wall via the adiabatic\nSTT. The absorption of the transferred angular momen-\ntum translates into the expansion of one of the two do-\nmains at the velocity V[1, 2]. Note that the direction\nof the domain-wall motion changes when the sign of the\nnet spin density changes, i.e., across TA. The net spin\ndensity\u000esis de\fned with respect to \u0000n, and thus, in\nour domain-wall ansatz, the net spin densities of the left\n(x!\u00001 ) and the right ( x!+1) domains are given\nby +\u000esand\u0000\u000es, respectively, polarized in the + zdirec-\ntion. Therefore, for the given angular momentum trans-\nfer from conduction electrons, whether the left domain\nor the right domain expands is determined by the sign of\nthe net spin density. See Fig. 1 for the schematic illustra-\ntion. We would like to remark here that the analogous\nresult of the reversal of the domain-wall motion has been\nreported in the theoretical study of the spin-wave-driven\nferrimagnetic domain-wall motion [43].\nSecond, when the net spin density vanishes \u000es= 0, i.e.,\natTA, the domain-wall velocity is reduced to\nV=\u0000\fPJ=\u000bs; for\u000es= 0: (8)\nThis reproduces the known result for the STT-induced\ndomain-wall motion in antiferromagnets [11], where the\ndomain wall is driven by the nonadiabatic STT /\f.\nAlso, for\u000es= 0, the angular velocity is reduced to\n\n =PJ=\u000b\u0015 0s;for\u000es= 0: (9)\nThis can be understood as follows. Conduction electrons\ntransfer angular momentum at the rate /PJto the\ndomain wall by passing through it. When the net spin\ndensity is \fnite \u000es6= 0, the domain wall can absorb the\ntransferred angular momentum by moving, i.e., by ex-\npanding one of the two domains. However, when the net\nspin density vanishes \u000es= 0, the transferred angular mo-\nmentum cannot be absorbed by the domain-wall motion\nand thus it is accumulated inside the domain wall. This\nnonequilibrium spin density exerts the e\u000bective magnetic\n\feld [44], which in turn rotates the magnetic order inside\nthe domain wall at the angular velocity \n /PJ. The\nsteady-state amount of the nonequilibrium spin density\nand the corresponding precession frequency \n are deter-\nmined by balancing the spin dissipation rate caused by\nthe precession/\u000bs\n and the generation rate of the4\n(a)(b)•2(a) -DW velocity as a function of current density•2(b) -DW angular velocity as a function of current온도범위1~5 에서의Figure 입니다. 𝛽=0.0010.5𝛼일때의계산입니다.-Fig. 2(b) 를Eq. (6) 으로fitting 하였습니다.0246810-2-1012 1 2 3 (TA) 4 5Velocity (km/s)Current density (1011 A/m2)02468100400800120016002000 1 2 3 (TA) 4 5Angular velocity (109 rad/s)\nCurrent density (1011 A/m2)•2(a) -DW velocity as a function of current density•2(b) -DW angular velocity as a function of current온도범위1~5 에서의Figure 입니다. 𝛽=0.0010.5𝛼일때의계산입니다.-Fig. 2(b) 를Eq. (6) 으로fitting 하였습니다.0246810-2-1012 1 2 3 (TA) 4 5Velocity (km/s)Current density (1011 A/m2)02468100400800120016002000 1 2 3 (TA) 4 5Angular velocity (109 rad/s)\nCurrent density (1011 A/m2)\nFIG. 2. (a) The domain-wall velocity and (b) the domain-wall\nangular velocity as a function of the current density J\u00141012\nA/m2at various temperatures shown in Table I. The symbols\nare numerical results. The lines show the analytical results for\nthe velocity V[Eq. (5)] and the angular velocity \n [Eq. (6)]\nobtained within the linear response.\nnonequilibrium spin density /PJ. The similar phe-\nnomenon has been observed numerically and explained\ntheoretically in the dynamics of a domain wall in an anti-\nferromagnet driven by a circularly-polarized magnon cur-\nrent [37, 38].\nB. Simulation\nTo con\frm the obtained analytical results, we perform\nnumerical simulations by solving the following coupled\natomistic LLG equations for RE-TM ferrimagnets [16,\n43, 45]:\n@Ak\n@t=\u0000\rreAk\u0002Hk\ne\u000b,A+\u000breAk\u0002@Ak\n@t\n\u0000bre@Ak\n@x\u0000\frebreAk\u0002@Ak\n@x;\n@Bk\n@t=\u0000\rtmBk\u0002Hk\ne\u000b,B+\u000btmBk\u0002@Bk\n@t\n\u0000btm@Bk\n@x\u0000\ftmbtmBk\u0002@Bk\n@x;(10)\nwhere AkandBkare the normalized spins at the kth\nsites in RE and TM sublattices, respectively, Hk\ne\u000b,A =\n(1=\u0016re)\u0001@H=@ AkandHk\ne\u000b,B = (1=\u0016tm)\u0001@H=@ Bkare\nthe e\u000bective magnetic \felds, \u0016reand\u0016tmare the local\nmagnetic moments, \u000breand\u000btmare the Gilbert damp-\ning constants, \rreand\rtmare the gyromagnetic ratios,\nMreandMtmare the saturation magnetizations, bre=\n\u0000gre\u0016BPreJ=(2eMre) andbtm=\u0000gtm\u0016BPtmJ=(2eMtm)\nare the STT parameters [46], Jis the charge current den-\nsity,eis the electron charge, greandgtmare the g-factors,\nPreandPtmare the dimensionless spin polarizations, and\n\freand\ftmare the dimensionless nonadiabatic STT pa-\nrameters. Here, His the discrete Hamiltonian given by\nH=AsimP\nk(Ak\u0001Bk+Bk\u0001Ak+1)\u0000KsimP\nk[(Ak\u0001^z)2+\n(Bk\u0001^z)2].\nFor the sample geometry, we considered 3200 \u0002100\u00020:4\nnm3with cell size 0 :4\u0002100\u00020:4 nm3. Correspond-\ningly, the lattice constant in the xdirection is given\n(a)(b)•3(a) –DW velocity as a function of temperature •3(b) –DW angular velocity as a function of temperature-4-2024-200-1000100200Velocity (m/s)ds (10-8 Js/m3)Current density (1010 A/m2) 1 2 3 4 5 6 7 8 9-4-2024060120180240300Current density (1010 A/m2) 1 2 3 4 5 6 7 8 9Angular velocity (109 rad/s)\nds (10-8 Js/m3)•3(a) –DW velocity as a function of temperature •3(b) –DW angular velocity as a function of temperature-4-2024-200-1000100200Velocity (m/s)ds (10-8 Js/m3)Current density (1010 A/m2) 1 2 3 4 5 6 7 8 9-4-2024060120180240300Current density (1010 A/m2) 1 2 3 4 5 6 7 8 9Angular velocity (109 rad/s)\nds (10-8 Js/m3)FIG. 3. (a) The domain-wall velocity and (b) the domain-wall\nangular velocity as functions of the spin density \u000esat various\ncurrent densities. The symbols are numerical results. The\nlines show the analytical results: the velocity V[Eq. (5)] and\nthe angular velocity \n [Eq. (6)] within the linear response.\nNote the sign change of the velocity across \u000es= 0 and the\nmaximum of the angular velocity at \u000es= 0.\nbyd= 0:4 nm. The used material parameters are\nAsim= 7:5 meV,Ksim= 0:4 meV,\u000btm=\u000bre= 0:002,\nand the gyromagnetic ratios \rre= 1:76\u00021011rad/s\u0001T\nand\rtm= 1:936\u00021011rad/s\u0001T (corresponding to the\ng-factorsgtm= 2:2 andgre= 2 [26]). The used mag-\nnetic moments for \fve di\u000berent cases are shown in Ta-\nble I. For STT parameters, a current in RE-TM ferrimag-\nnets is known to interact mostly with TM magnetic mo-\nments [29, 30]. Therefore, in this work, we consider the\nsimplest case where the current interacts only with TM\nelements. Accordingly, we use the following STT param-\neters:Pre= 0;Ptm= 0:3, and\ftm= 0:001. Correspond-\ning parameters in the continuum model [Eq. (2)] are given\nbyA= 4Asim=dandK= 4Ksim=d3. In addition, we\nhave the following relations: n= (Ak\u0000Bk)=jAk\u0000Bkj,\nsre=Mre=\rre,stm=Mtm=\rtm,\u000es=sre\u0000stm,s=\nsre+stm,\u000b= (\u000bresre+\u000btmstm)=s,PJ=s(bre\u0000btm)=2,\nand\f= (\frebre+\ftmbtm)=2PJ.\nFigure 2(a) and (b) show the domain-wall velocity V\nand the angular velocity \n as a function of the current\ndensityJ\u00141\u00021012A/m2for various values of the net\nspin density \u000esshown in Table I. The analytical results\nshown as lines and the numerical results shown as sym-\nbols agree well for the current densities J.5\u00021011\nA/m2. Figure 3(a) and (b) show the velocity and the an-\ngular velocity as a function of the net spin density \u000esfor\nvarious current densities. Two main features predicted\nIndex 1 23 (TA)4 5\nMre(kA/m) 1020 1010 1000 990 980\nMtm(kA/m) 1130 1115 1100 1085 1070\n\u000es(10\u00008J\u0001s/m3)-4.13 -2.07 0 2.07 4.13\nTABLE I. The values of the magnetic moments MreandMtm\nfor transition-metal and rare-earth elements, respectively, and\nthe net spin density \u000esused in atomistic spin simulations. In-\ndex 3 represents the angular momentum compensation point\nTA.5\n(a)(b)012340.00.51.01.52.02.53.0b 0 0.0005 0.001Angular velocity (1012 rad/s)\nCurrent density (1012 A/m2)012341234567b 0 0.0005 0.001DW width (nm)Current density (1012 A/m2)At 𝑇𝐴•4(a) –DW angular velocity as a function of current density•4(b) –DW width as a function of current density012340.00.51.01.52.02.53.0b 0 0.0005 0.001Angular velocity (1012 rad/s)\nCurrent density (1012 A/m2)012341234567b 0 0.0005 0.001DW width (nm)Current density (1012 A/m2)At 𝑇𝐴•4(a) –DW angular velocity as a function of current density•4(b) –DW width as a function of current density\nFIG. 4. (a) The domain-wall angular velocity and (b) the\ndomain-wall width as functions of the current density for\nthree di\u000berent values of the nonadiabatic STT parameter\n\f= 0;0:0005;0:001 at the angular momentum compensa-\ntion point\u000es= 0. The symbols are numerical results. The\nsolid lines show the analytical results: the angular velocity \n[Eq. (12)] and the width \u0015[Eq. (13)].\nby the theory, the sign change of the domain-wall veloc-\nity across\u000es= 0 and the maximum angular velocity \nat\u000es= 0, are demonstrated in the numerical results.\nIII. STT-INDUCED DOMAIN-WALL\nDYNAMICS AT TA\nIn this section, we study the STT-induced dynamics of\na domain wall exactly at TA, where the net spin density\nvanishes and thus the nature of the magnetic dynamics\nis purely antiferromagnetic, by applying a charge-current\ndensity up to 4\u00021012A/m2to look for novel phenomena.\nA. Angular velocity of a domain wall\nLet us \frst discuss the numerical results on the\ndomain-wall angular velocity from atomistic spin simula-\ntions performed with three di\u000berent values of the nona-\ndiabatic STT parameter \f= 0;0:0005, and 0 :001. Fig-\nure 4(a) shows the angular velocity _\b as a function of the\ncurrent density. The angular velocity increases linearly\nas the current increases for the small current density as\npredicted by Eq. (6), but it deviates from the equation\nfor high current densities by showing the saturation.\nThis observed saturation of the angular velocity can\nbe understood as the e\u000bect of the change of the domain-\nwall width as follows. The width of the static domain\nwall is given by \u00150=p\nA=K determined by the competi-\ntion between the exchange energy /Aand the easy-axis\nanisotropy/K. When the domain wall is precessing uni-\nformly at the angular velocity \n in the laboratory frame,\nthe e\u000bective easy-axis anisotropy in the spin frame ro-\ntating at the domain-wall angular velocity \n is given by\nKe\u000b=K\u0000\u001a\n2as shown in Ref. [38]: the uniform spin\nrotation about the zaxis in the laboratory frame gives\nrise to the e\u000bective magnetic \feld along the zaxis in the\n(a)(b)01234-30-150Velocity (m/s)Current density (1012 A/m2)b 0 0.0005 0.001•5(a) -DW velocity as a function of current density. (low current density)•5(b) -DW velocity as a function of current density. (high current density)0.00.10.20.30.40.5-2.5-2.0-1.5-1.0-0.50.0b 0 0.0005 0.001Velocity (m/s)Current density (1012 A/m2)01234-30-150Velocity (m/s)Current density (1012 A/m2)b 0 0.0005 0.001•5(a) -DW velocity as a function of current density. (low current density)•5(b) -DW velocity as a function of current density. (high current density)0.00.10.20.30.40.5-2.5-2.0-1.5-1.0-0.50.0b 0 0.0005 0.001Velocity (m/s)Current density (1012 A/m2)FIG. 5. (a) and (b) The domain-wall velocity as a function of\nthe current density for three di\u000berent values of the nonadia-\nbatic STT parameter \f= 0;0:0005;0:001 at the angular mo-\nmentum compensation point \u000es= 0. The symbols are simula-\ntion results. The sold lines show the analytical results for the\ndomain-wall velocity Vwithin the linear response [Eq. (5)].\nrotating spin frame of reference (which is analogous to the\ncentrifugal force in a rotating frame of reference), which\nin turn decreases the easy-axis anisotropy as known for\nmagnets with antiferromagnetic coupling [36]. Therefore,\nthe width of the domain wall rotating at the angular ve-\nlocity \n is given by\n\u0015=\u00150p\n1\u0000(\n=!0)2; (11)\nwhere!0\u0011p\nK=\u001a is the spin-wave gap at TA[45]. By\nsolving Eq. (6) with \u00150replaced by \u0015[Eq. (13)] for \n, we\nobtain the domain-wall precession frequency as a func-\ntion of the current density:\n\n =PJ=\u000b\u0015 0sp\n1 + (PJ=\u000b\u0015 0s!0)2; (12)\nand the domain-wall width:\n\u0015=\u00150p\n1 + (PJ=\u000b\u0015 0s!0)2: (13)\nThe obtained expression for the angular velocity \n[Eq. (12)] is reduced to the linear-response result [Eq. (6)]\nwhen the quadratic e\u000bects in the current density Jis ne-\nglected. Note that the angular velocity converges to the\nspin-wave gap !0\u00193\u00021012rad/s as the current den-\nsity increases, but can never exceed it. The sold lines in\nFigs. 4(a) and (b) show the analytical solutions for the\nangular velocity \n [Eq. (12)] and width \u0015[Eq. (13)], re-\nspectively for several values of \f. They agree well with\nthe simulation results shown as the symbols.\nB. Velocity of a domain wall\nLet us now turn to the STT-induced translation mo-\ntion of a domain wall at large currents. Figure 5(a) shows\nthe domain-wall velocity Vas a function of the current\ndensity up to 0 :5\u00021012A/m2. For relatively small cur-\nrentsJ\u00140:2\u00021012A/m2, the simulation results shown6\nas symbols are well explained by the linear-response an-\nalytical result V=\u0000\fPJ=\u000bs [Eq. (5)] shown as solid\nlines. However, Fig. 5(b), where the current density as\nlarge as 4\u00021012A/m2is applied, shows that the domain-\nwall velocity starts to deviate signi\fcantly from Eq. (5)\nfor the current density J&1\u00021012A/m2. This de-\nviation is not due to the current-induced change of the\ndomain-wall width since V=\u0000\fPJ=\u000bs does not depend\non\u0015. There are two notable features. First, even when\nthe nonadiabatic torque is absent \f= 0, the domain\nwall exhibits a \fnite velocity at high current densities,\nwhich disagrees with the known results for antiferromag-\nnetic domain-wall motion obtained within the linear re-\nsponse [11]. Secondly, as the current density increases,\nthe domain-wall velocities corresponding to three di\u000ber-\nent values of \fappear to converge on the one univer-\nsal line, suggesting that it is not the nonadiabatic STT\n/\fPJ but the adiabatic STT /PJthat plays a main\nrole in the observed domain-wall velocity at high current\ndensities. Our numerical result demonstrates a limita-\ntion of the linear-response theory for the STT-induced\ndomain-wall motion at high currents. We leave a theoret-\nical understanding of the observed domain-wall velocity\nat higher currents as a future research topic.\nIV. DISCUSSION\nTo sum up, we have studied the STT-induced dynam-\nics of a domain wall in ferrimagnets theoretically and\nnumerically. The domain-wall velocity changes it sign\nacrossTAdue to the sign change of the net spin density,\ngiving rise to a phenomenon unique to ferrimagnets that\ncannot be found in ferromagnets and antiferromagnets.\nThe angular velocity of a domain wall is shown to exhibit\nits maximum at TA, which can be understood as the e\u000bect\nof the STT-induced accumulation of the nonequilibrium\nspin density inside the domain wall. At higher currents,\nwe have found numerically that the domain-wall velocity\ncan signi\fcantly deviate from the linear-response result,\ncalling for the development of a more general theory for\nthe dynamics of a domain wall subjected to strong cur-\nrents.\nIn this work, we have focused on the e\u000bects of STT on\nthe dynamics of a domain wall in ferrimagnets. The re-\nciprocal e\u000bects of a spin texture on a current are known\nto give rise to intriguing phenomena in ferromagnetssuch as the generation of electromotive force by domain-\nwall precession [47{50] and the topological Hall e\u000bect in\nskyrmion crystal phases [51{53]. The corresponding ef-\nfects in ferrimagnets would be worth being investigated\nin the future. In addition, our understanding of STT in\nantiferromagnetically-coupled magnetic systems can be\nadvanced further by pursuing the microscopic theory for\nthe spin-charge interaction in ferrimagnets as has been\ndone for ferromagnets within the Stoner model or the\ns-d model for itinerant ferromagnetism [34]. More gen-\nerally, we envision that the research on the spin dynam-\nics as well as the spin-charge interaction in ferrimagnets\nwill lead us to more uni\fed understanding of magnetic\nphenomena spanning various types of magnetic materi-\nals including ferromagnets and antiferromagnets as two\nspecial cases, and also it will facilitate the advancement\nof ferrimagnetic spintronics aiming at easily-controllable\nhigh-speed devices by combining the advantages of ferro-\nmagnetic and antiferromagnetic devices.\nACKNOWLEDGMENTS\nD.H.K was supported by the National Research Coun-\ncil of Science & Technology (NST) Research Fellowship\nfor Young Scientist of the National Research Council of\nScience & Technology (NST), the POSCO Science Fel-\nlowship of POSCO TJ Park Foundation, the Korea In-\nstitute of Science and Technology (KIST) institutional\nprogram (No. 2E29410 and 2E30600), and the National\nResearch Council of Science & Technology (NST) grant\n(No. CAP-16-01-KIST) funded by the Korea govern-\nment (Ministry of Science and ICT). K.J.K. was sup-\nported by the National Research Foundation of Korea\n(NRF) grant funded by the Korean government (MSIP)\n(Grant No. NRF-2016R1A5A1008184). K.W.M. and\nS.Y. were supported by the National Research Founda-\ntion of Korea (NRF-2019M3F3A1A02072478), the Na-\ntional Research Council of Science & Technology (NST)\ngrant (No. CAP-16-01-KIST) by the Korea govern-\nment (MSIP), the Future Materials Discovery Program\nthrough the National Research Foundation of Korea\n(No. 2015M3D1A1070467). K.J.L. was supported by\nthe KIST Institutional Program (No. 2V05750). S.K.K.\nwas supported by Young Investigator Grant (YIG) from\nKorean-American Scientists and Engineers Association\n(KSEA) and Research Council Grant URC-19-090 of the\nUniversity of Missouri.\n[1] J. Slonczewski, Current-driven excitation of magnetic\nmultilayers, J. Magn. Magn. Mater. 159, L1 (1996).\n[2] L. Berger, Emission of spin waves by a magnetic mul-\ntilayer traversed by a current, Phys. Rev. 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B 90, 094408 (2014)." }, { "title": "2102.11004v1.A_review_of_modelling_in_ferrimagnetic_spintronics.pdf", "content": "A review of modelling in ferrimagnetic spintronics\nJoseph Barker1, 2,\u0003and Unai Atxitia3, 4,y\n1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom\n2Bragg Centre for Materials Research, University of Leeds, Leeds LS2 9JT, United Kingdom\n3Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany\n4Dahlem Center for Complex Quantum System, Berlin, Arnimallee 14, 14195 Berlin, Germany\nIn this review we introduce computer modelling and simulation techniques which are used for ferrimagnetic\nmaterials. We focus on models where thermal effects are accounted for, atomistic spin dynamics and finite tem-\nperature macrospin approaches. We survey the literature of two of the most commonly modelled ferrimagnets\nin the field of spintronics–the amorphous alloy GdFeCo and the magnetic insulator yttrium iron garnet. We\nlook at how generic models and material specific models have been applied to predict and understand spintronic\nexperiments, focusing on the fields of ultrafast magnetisation dynamics, spincaloritronics and magnetic textures\ndynamics and give an outlook for modelling in ferrimagnetic spintronics\nI. INTRODUCTION\nIn the field of spintronics1the goal is to use the spin de-\ngree of freedom to create new devices which are superior to\nelectronic devices in some aspect such as energy consump-\ntion. It also provides insights into the fundamental physics of\nspin conversion with electrons and phonons2. For example,\ncreating memory devices which can be controlled with elec-\ntric or spin currents but with better durability and lower power\nconsumption than electronic memories. There is also a push\nto develop new types of devices for specialised roles such as\nmachine learning where the additional degrees of freedom in\nmagnetic materials can be exploited3.\nFerromagnets provide us with a single magnetisation vec-\ntor which can be manipulated using external stimuli such as\nmagnetic fields and spin transfer from electrical currents. The\ninteraction of the magnetisation vector with the spin quantisa-\ntion axis of electrons gives rise to a range of physical effects\nsuch as giant magnetoresistance4–6, spin Hall7–11and Rashba-\nEdelstein effects12,13, which can be used to create devices or\nas fundamental probes in experiments. For quite some time\nferromagnetic behaviour has been the focus of studies in spin-\ntronics. Whilst ferrimagnetic materials have been used it is\ngenerally because of a desirable material property, such as\na low magnetic damping or high magneto-optical coupling.\nThe ferrimagnetic nature was incidental and often ignored in\ntheory and simulations due to their minor role in the macro-\nscopic magnetic properties of interest by the time. As re-\nsearch progressed it became clear that some results could only\nbe explained due to the unique characteristics of ferrimagnets\nwhich occur with two or more coupled but anti-aligned mag-\nnetisation vectors. It has long been known that ferrimagnets\nhave special points in their phase diagram such as the angu-\nlar momentum and magnetisation compensation points. For\na two sublattice ferrimagnet with antiparallel sublattice mag-\nnetisationsM1(T)andM2(T), the magnetisation compensa-\ntion point is when M1(T) =M2(T)(see Fig. 1(a)). There is\nzero net magnetisation because of the cancellation of the two\nantiparallel sublattices. If the two sublattices also have differ-\nent gyromagnetic ratios \r1and\r2, then the angular momen-\ntum compensation temperature is different from the magneti-\nsation compensation point, being at M1(T)=\r1=M2(T)=\r2as shown in Fig. 1(b).\nRecently, researchers have become interested in using\nthe compensation points to tune the macroscopic proper-\nties of ferrimagnets to impart some of the desirable quali-\nties of both ferromagnets–a measurable net magnetisation–\nand antiferromagnets–high frequency dynamics. The hope is\nthat the versatile spin couplings present in ferrimagnets can be\ntechnologically exploited in devices for ultrafast spintronics14,\nwith the potential to overcome the gigahertz frequency limi-\ntations of ferromagnetic based technologies and realise tera-\nhertz spintronics15. Ferrimagnetic insulators may also enable\nlow power devices which work by pure spin currents16. Con-\nsequently, a large and ever increasing range of ferrimagnetic\nmaterials are now used to investigate and push forward the\nfield of spintronics.\nComputer modelling of ferrimagnets aims to describe the\nmacroscopic magnetic behaviour, which can be observed in\nexperiments, by considering the microscopic or multiple sub-\nlattice variables which are difficult to isolate experimentally.\nModelling ferrimagnets involves many of the similar hurdles\nas modelling antiferromagnets. There are multiple connected\nsublattices and antiferromagnetic coupling between them can\nlead to complex phase diagrams and magnetisation dynamics.\nEspecially important in modelling ferrimagnets is the correct\naccounting of the thermodynamic properties as temperature\ntunes the system across the compensation points. Simulations\nare also frequently used in non-equilibrium physics where ex-\nperimental results can be noisy or of limited resolution and\nanalytic theory can be difficult. In this Review we focus on\nthe two major numerical modelling approaches; atomistic spin\ndynamics (ASD)17–21and the Landau-Lifshitz-Bloch (LLB)\nequation22–27which are able to produce accurate descriptions\nof both the dynamics and thermodynamics of ferrimagnets.\nThese computational methods are used for different length\nand timescale limits. Atomistic spin-dynamics simulations\nare restricted by computational costs to lateral dimensions of\n10-100 nm. Timescales are limited to tens of nanoseconds.\nThe simulation of mesoscopic spin textures such as magnetic\ndomains is extremely computationally expensive. For com-\nparison with experiments much larger sample sizes are often\nnecessary. This is usually tackled with micromagnetic ap-\nproaches. Conventional micromagnetics, rests on a general-arXiv:2102.11004v1 [cond-mat.mtrl-sci] 22 Feb 20212\nFIG. 1. Diagram of the (a) magnetisation compensation and (b) an-\ngular momentum compensation points in a two sublattice ferrimag-\nnet.M1andM2are the magnetisations of the antiparallel sublattices,\nwhich have different gyromagnetic ratios \r1and\r2respectively. \re\u000b\nis the effective gyromagnetic ratio defined in Eq. (6). If \r1=\r2\nthen the magnetisation and angular momentum compensation tem-\nperatures coincide.\nisation of the Landau-Lifshitz equation, which is a zero tem-\nperature approach and cannot be used when the temperature\nof the spin system varies in space and time. The primary fi-\nnite temperature micromagnetic approach is the LLB equation\nof motion.\nWe proceed in Section II by introducing the basic theory\nand principles behind the common methods and how they\napply to ferrimagnets. In Section III we explain how two\nmaterials–GdFeCo and yttrium iron garnet–have been mod-\nelled and why these two materials are so focused on in spin-\ntronics. In Section IV we survey the literature where these\nmethods and material models have played an important role\neither in elucidating the understanding of experiments or pre-\ndictions of ferrimagnetic behaviour. We conclude in Section V\nwith an outlook of the future directions for modelling in ferri-\nmagnetic spintronics.\nII. MODELLING TECHNIQUES\nAt a constant temperature and away from the magnetisa-\ntion or angular momentum compensation points ferrimagnets\nbehave like ferromagnets. Ferrimagnets show a net magneti-sation and the low frequency dynamics can be described with\nequations analogous to a ferromagnet. A considerable amount\nof modelling of ferrimagnetic materials has used the conven-\ntional micromagnetism approach which assumes smooth spa-\ntial variation of the net magnetisation. Either by considering\nthe material simply as a ferromagnet, an effective medium ap-\nproach which adjusts the parameters to mimic a given temper-\nature, or by coupling multiple magnetisation vectors in a cell\nor close proximity. These approaches describe many experi-\nmental situations well28,29, but when temperature becomes a\ndynamic variable or higher energy modes play a role in dy-\nnamics or thermodynamics, more sophisticated approaches\nare needed.\nA. Atomistic Spin Dynamics (ASD)\nAtomic spin models are a natural formalism for ferrimag-\nnetism because they can explicitly account for the different\nmagnetic moments at the atomic scale within the material.\nThey are often complemented with statistical physics meth-\nods because with large ensembles of spins, thermodynamic\nproperties can be calculated. A Heisenberg Hamiltonian de-\nscribes the energetics of the system owning to the interactions\nbetween the classical ‘spin’ vectors. The word spin is used\ncolloquially in this field and means the classical local mag-\nnetic moment ascribed to an atomic site. Additional terms of\nrelativistic origin or due to coupling to external stimulus can\nbe added to the Hamiltonian according to the material being\nstudied. A typical example Hamiltonian is\nH=\u00001\n2X\ni6=jJijSi\u0001Sj\u0000X\nidzS2\nz;i\u0000X\ni\u0016iB\u0001Si:(1)\nwhere the first term is the Heisenberg exchange with Jijis the\nelectronic exchange energy (the 1=2is for the double counting\nin the sum) and Siis a classical spin vector of unit length.\nThe second term is a uniaxial anisotropy in the z-direction\nof energydzrepresenting magneto-crystalline effects and the\nthird term is an external magnetic field Bmeasured in Tesla\nand\u0016iis the size of the magnetic moment of the i-th spin in\nBohr magneton.\nThe exchange parameters quantify the interaction energy\nbetween the spins upon small rotation from the ground state,\nas demonstrated by Liechestein et al.30. The exchange inter-\naction here can arise from direct exchange of the overlap of\nelectronic orbitals, typical in transition metals, or indirect such\nas super-exchange mediated by a third (non-magnetic) ion as\nis common in oxides. Ferrimagnets can be described by con-\nsidering this Hamiltonian with two or more distinct types of\nspin. Two possibilities are i) when there the spin moments\nof different atoms have different magnitudes, for example a\ncombination of transition metal and rare earth elements as in\nthe alloy GdFeCo ii) in a crystal where the same element ex-\nists in different environments which are antiferromagnetically\ncoupled. The magnetic garnets are a typical example of this,\nwhere Fe3+ions exist in both octahedral and tetrahedral en-\nvironments in a 2:3 ratio with antiferromagnetic coupling be-\ntween the two sublattices. In material specific models the ex-3\nchange parameters Jijare usually parametrised either from\nneutron scattering measurements of the magnon spectrum or\ndirect calculation of the electronic structure from first princi-\nples methods. Often the exchange interaction contains contri-\nbutions beyond the nearest neighbours which leads to complex\nmagnon band structures.\nClassical atomistic spin models are often solved using\nMonte Carlo techniques31or atomistic spin dynamics–which\nprovides dynamical quantities of interest. The equation of\nmotion for the spin dynamics is the Landau-Lifshitz-Gilbert\n(LLG) equation:\n@Si\n@t=\u0000j\rij\n(1 +\u00152\ni)[(Si\u0002Hi)\u0000\u0015i(Si\u0002(Si\u0002Hi))]:\n(2)\n\u0015iis the (dimensionless) local intrinsic damping assigned to\nthei-th site,\ri=gis the gyromagnetic ratio on the i-th site in\nradians per second per Tesla. Hi=\u0000(1=\u0016i)(@H=@Si)is the\neffective field local to a site iderived from the Hamiltonian.\nThe LLG equation is usually solved numerically for large\ncollections of coupled spins in a manner similar to molecu-\nlar dynamics for atomic motion. Besides the antiferromag-\nnetic interactions between different sublattices, ferrimagnets\nare modelled by using distinct parameters on different atomic\nlattice sites. Elements can have magnetic moments of differ-\ning values, different gyromagnetic ratios or exchange interac-\ntions. This atomic scale detail is something not possible in\nmicromagnetic approaches based on the continuum approxi-\nmation.\nTemperature is included in this dynamical model following\nBrown32by adding a stochastic field \u0010ito the effective field\nHi=\u0010i(t)\u0000(1=\u0016i)(@H=@Si). Equation (2) now becomes\na Langevin equation. Conventionally the classical fluctuation\ndissipation theorem is used (as this is a classical spin model)\nand the statistical properties of the stochastic field are there-\nfore a white noise with correlations33:\nh\u0010i;\u000b(t)i= 0 andh\u0010i;\u000b(0)\u0010j;\f(t)i=2\u0015ikBT\u000eij\u000e\u000b\f\u000e(t)\n\u0016i\ri:\n(3)\nwherekBTis the thermal energy provided by the heat bath\nand\u000b,\fare Cartesian components. Within ASD spin-spin\ninteractions occur naturally due to the coupled system of LLG\nequations and hence their effect is present without explicit\ninclusion–something which often limits analytic thermody-\nnamic models. The magnon-magnon interactions become sig-\nnificant for thermodynamics at temperatures above approxi-\nmately half the Curie temperature ( TC). Recently advances\nhave been made in the thermostating to use the quantum fluc-\ntuation dissipation theorem instead34,35. This is essentially the\nsemi-classical limit often used in analytic theory where spin\nmoments are considered in the classical limit but the quantum\n(Planck) distribution is used as the thermal distribution for the\nmagnons. The use of quantum statistics enables quantitative\ncalculations of thermodynamics even at low temperatures34,36.\nThe thermostated spin dynamics gives a canonical ensem-\nble from which thermodynamics can be calculated. The total\nmagnetisation at a given time is M(t) =P\ni\u0016s;iSi(t)and\na thermodynamic average can be calculated from a long timeseries in equilibrium. For ferrimagnets it is often useful to\ncalculate the sublattice magnetisation, something which only\na few experimental methods can probe.\nM\u0017(t) =X\nj\u0016s;jSj(t) (4)\nwherejare the indices belonging to the \u0017-th sublattice.\nResolving quantities by sublattice links into the Landau-\nLifshitz-Bloch method (described below in Section II D)\nwhich requires these thermodynamic quantities. The thermal\nspin fluctuations cause the system to sample broad regions of\nthe free energy and so although the anisotropy, dz, and ex-\nchange energy, Jij, in the Hamiltonian are independent of\ntemperature, the macroscopic anisotropy (often denoted K)\nand the macroscopic exchange stiffness (often A) are temper-\nature dependent and can be calculated with ASD37. These are\nalso key input parameters for macroscopic formalisms such\nas the micromagnetic and LLB methods. The statistical ther-\nmodynamic aspects of ASD means it can be used to calculate\nphase diagrams. The accuracy exceeds what is possible with\nmean field approaches which are known to significantly over\nestimate critical points due to the absence of magnon-magnon\ninteractions. For a set of material specific parameters, these\nmicroscopic details produce the thermodynamic angular mo-\nmentum and magnetisation compensation points which dis-\ntinguish ferrimagnets from ferro and antiferromagnets. Non-\ncollinear phases such as spin flop which occur in high mag-\nnetic fields can be modelled and even non-equilibrium and\ndynamical states38.\nThe damping parameter in ASD describes the rate of en-\nergy and angular momentum dissipation to the heat bath. It\nrepresents the electron and phonon systems. Detailed descrip-\ntions of the heat-bath dynamics are rarely39,40introduced or\nmodelled in detail. The coupling to electron and phonon sys-\ntems are “direct” damping pathways41involving charge carri-\ners or the lattice, with spin-orbit coupling being paramount\nfor the transfer of angular momentum from the spin space\nto the lattice. The atomic damping parameter appearing in\nEq. (2) is different from the macroscopic damping parameter–\nthe relaxation rate of the total magnetisation–as measured in\nan experiment. The macroscopic damping includes further\ncontributions from “indirect” mechanisms–magnon-magnon\ninteractions–which are internal to the spin system. This re-\nsults in an increase in the damping of the macroscopic mag-\nnetisation with increasing temperature.\nASD is used for both generic modelling as well as mate-\nrial specific modelling. Generic models include many of the\ncomplex effects which are heavily approximated in analytic\ntheory, such as magnon-magnon interactions and non-linear\ndynamics, but without being tied to a single material. This is\nwell exampled in the field of spincaloritronics (Section IV B)\nwhich studies the coupling between spin and heat–an ideal\nfield for the application of ASD.4\nB. Macrospin models\nMuch of the basic theory of ferrimagnets comes from a sim-\nple macrospin model of two coupled magnetisation vectors.\nThis provides simple equations for effective bulk properties\nwithin the limit that the two macrospins are rigidly coupled,\nassuming a strong inter-sublattice exchange interaction.\nIn ferrimagnets the macroscopic damping, \u000be\u000b–defined as\nthe rate of relaxation of transverse magnetisation oscillations–\ndiffers qualitatively from the microscopic damping parame-\nters. The reason is the coupled dynamics of both sublattices.\nThe collective oscillations, described as one effective ferro-\nmagnet, relax at a rate which is a combination of the damping\nparameters from different sublattices, weighted by their mag-\nnetisation. This rate of relaxation is what is experimentally\nobservable42. In the macrospin rigid coupling approximation\nand not too close to the magnetisation compensation point the\neffective damping of the net magnetisation is\n\u000be\u000b(T) =\u000b1(T)M1(T)=\r1+\u000b2(T)M2(T)=\r2\nM1(T)=\r1\u0000M2(T)=\r2: (5)\nAt low temperatures (much below the Curie temperature,\nbefore the magnon-magnon interactions become significant),\nin a first step, \u000b1(T) =\u00151and\u000b2(T) =\u00152can be taken\nas constant. The effective damping scales inversely to the\nnet magnetisation M1\u0000M2, which has a more pronounced\ntemperature dependence than in ferromagnets, with the strik-\ning difference that as the system approaches the magnetisation\ncompensation temperature, M1\u0000M2!0, the macroscopic\ndamping (Eq. (5)) starts to diverge. Divergence of the damp-\ning parameter is unphysical, and points to the failure of the\neffective macrospin approach to describe ferromagnetic mag-\nnetisation dynamics.\nThe local gyromagnetic ratio \rican also be site dependent\nin a ferrimagnet leading to an angular momentum compen-\nsation point where the sublattices no longer precess. For the\nmacrospin model this is\n\re\u000b(T) =M1(T)\u0000M2(T)\nM1(T)=\r1\u0000M2(T)=\r2: (6)\nThe frequency of the lowest energy mode (often called the\nferromagnetic mode), when neglecting damping is\n!(T) =\re\u000b(T)H: (7)\nApproaching the magnetisation compensation point !!0.\nAs demonstrated in Figure 1, at the angular momentum com-\npensation point M1(T)=\r1\u0000M2(T)=\r2= 0so\re\u000band!di-\nverge. When the inclusion of the damping contribution is used\nin the calculations, != (\re\u000b=1 +\u000b2\ne\u000b)H. This correction re-\nmoves the divergence of the frequency at the compensation\npoint.\nC. Conventional micromagnetism\nLength scales covered by atomistic spin dynamics simu-\nlations are limited to only several tens of nanometers of lat-eral size and macrospin approaches fail at describing mag-\nnetic textures, such as domain walls, vortices or skyrmions\n(Section IV C) which are a key aspect of many spintronic de-\nvice functionalities. To model spintronic phenomena happen-\ning at the micrometer scale, such as magnetic domain motion,\none needs to use the so-called micromagnetic models. Mi-\ncromagnetics has become an pivotal tool for not only for the\ninterpretation of experimental results but also for the design\nof spintronics devices and predicting novel effects. Origi-\nnally micromagnetics was developed as a purely theoretical\ntool for the study of magnetisation processes, however nowa-\ndays computer software using finite difference or finite ele-\nment methods to solve the equations for different geometries\nis common43–46.\nConventional micromagnetism was originally formulated\nwithin the ’ferromagnetic’ framework. The basic underlying\nassumption of micromagnetism is that neighbouring atomic\nspins can be considered parallel over a certain scale given\nby the ratio between the anisotropy and exchange interaction.\nThe magnetisation is therefore treated as a continuous field\nwhere at each point of the continuum space rthe magnetisa-\ntion is defined by one vector, m(r). Ferrimagnetism was first\nintroduced in micromagnetics by assuming that the opposing\nsublattices are locked exactly antiparallel and so the coupled\nequations of motion can be re-written with a single Landau-\nLifshitz-Gilbert equation of motion47using the effective pa-\nrameters for the damping (Eq. (5)) and gyromagnetic ratio\n(Eq. (6)). This is sometimes called an effective medium ap-\nproach. It is useful in ferrimagnets with very strong exchange\ncoupling between the sublattices and away from the compen-\nsation points which diverge in the effective parameter equa-\ntions. Micromagnetism for ferrimagnets with explicit consid-\neration of at least two sublattices is recent48–50. The assump-\ntion of a smoothly changing magnetisation only holds in the\nspin subspace corresponding to each magnetic sublattice. At\nthe local level, the spins are aligned antiparallel. Similarly to\nantiferromagnets, this causes an ‘exchange enhancement’ of\nthe spin dynamics. Exchange enhancement effects are impor-\ntant since allow for the speed up of the dynamics, for exam-\nple the motion of domain walls at relativistic velocities–near\nto the maximum magnon velocity of the medium51. Other\nmagnetic textures, such as skyrmions, in ferrimagnets are also\nof great fundamental and technological interest. Micromag-\nnetic models allow the investigation of low frequency dynam-\nics of such magnetic textures at the device scale. However,\nconventional micromagnetism is unable to describe thermal\neffects. These are crucial in fields such as ultrafast spin dy-\nnamics and spincaloritronics and possibly for realistic device\nmodelling due to contact heating. Solutions to the shortcom-\ning of thermodynamic considerations have been proposed for\nexample by rescaling the micromagnetic parameters to their\nvalues of at specific temperatures. Stochastic fields have been\nalso added to the equations of motion, however since the finite\nsize of the computational cells means that the full range of\nspin waves excitations can not be described. At intermediate-\nto-high temperatures the contribution of those excitations to\nthe thermodynamic properties becomes large and they cannot\nbe neglected. An elegant solution is the use of the Landau-5\nLifshitz-Bloch equation as a basis for micromagnetism, first\nderived for ferromagnets22,24and later on for ferrimagnets26.\nD. Landau-Lifshitz-Bloch (LLB) model\nTo include thermal effects into the equation of motion of\nmagnetisation one has to either include a noise term, simi-\nlar to atomistic spin dynamics methods (Section II A), or to\nexpand the equation of motion to take care of the effect of\ntemperature on a mesoscopic level. This leads to the so-\ncalled Landau-Lifshitz-Bloch equation. The LLB equation\ndescribes magnetisation dynamics (rather than spin dynam-\nics) at finite temperatures. It can be considered as an ex-\ntension of already established micromagnetic methods with a\ncomparable numerical effort. Standard micromagnetic mod-\nels are strictly zero temperature formalisms—the length of the\nmagnetisation is fixed. In contrast, the LLB equation admits\nthe relaxation of the magnetisation length. The higher order\nspin-spin interactions are accounted for in the derivation, pro-\nducing an equation in which parameters have a temperature\ndependence. These parameters can be calculated within the\nmean-field approximation (MFA), higher order methods using\nGreen’s functions52, atomistic spin dynamics, or parametrised\nfrom experimental measurements. This makes the LLB equa-\ntion well suited for modelling scenarios where temperature\nalso changes dynamically, such as the field of ultrafast mag-\nnetisation dynamics. Because of the increasing interest in an-\ntiferromagnetic and ferrimagnetic materials the LLB equation\nof motion for two sublattice magnets was recently derived26\nand we briefly introduce it here.\nThe LLB equation in a two sublattice ferrimagnet is ele-\nment specific, and for each sublattice averaged magnetisation\nat each site, mi;\u0017=hSi;\u0017i,(\u0017=1,2), hence, its value is lim-\nited to 1. Moreover, in a first step it is common to assume a\nmagnetically homogeneous state, mi;\u0017!m\u0017for all lattice\nsites at the same sublattice \u0017, the dynamics equation reads as\nfollows\n1\n\r\u0017dm\u0017\ndt=\u0000[m\u0017\u0002He\u000b;\u0017]\u0000\u000b\u0017\n?\nm2\u0017[m\u0017\u0002[m\u0017\u0002He\u000b;\u0017]]\n+\u000b\u0017\nk\nm2\u0017(m\u0017\u0001He\u000b;\u0017)m\u0017: (8)\nThe relaxation of the magnetisation within the LLB equation\ndepends on the longitudinal and transverse damping parame-\nters,\u000b\u0017\nkand\u000b\u0017\n?, and the effective field, He\u000b. The damping\nand other input parameters for the two sublattice LLB equa-\ntion are element specific. When the gyromagnetic ratio, \r\u0017,\nis different for each sublattice, owing to the particular relation\nbetween orbital and spin degrees of freedom at each sublat-\ntice, the total angular momentum compensation point differs\nfrom the magnetisation compensation point. Below TC, the\ndamping parameters \u000b\u0017\nkand\u000b\u0017\n?are\n\u000b\u0017\nk=2kBT\u0015\u0017\neJ0;\u0017; \u000b\u0017\n?=\u0015\u0017 \n1\u0000kBT\neJ0;\u0017!\n: (9)\nwhereeJ0;\u0017=J0;\u0017+jJ0;\u0017\u0014jme;\u0014=me;\u0017,J0;\u0017is the intra-\nsublattice exchange and J0;\u0017\u0014is the inter-sublattice exchange(note the sign of the second term does not depend on the\nsign of the inter-sublattice exchange interaction), me;\u0017is the\nlength of the magnetisation at equilibrium for a given tem-\nperature. Above TCthe longitudinal and transverse damp-\ning parameters are equal and coincide with the expression22\nfor the classical LLB equation of a ferromagnet above TC,\n\u000b\u0017\n?=\u000b\u0017\nk= 2\u0015\u0016T=3TC, and the equation of motion reduces\nto Bloch equation. In Eqs. (9), the intrinsic damping parame-\nters\u0015\u0017depend on the particularities of the spin dissipation at\nthe atomic level as discussed above in Section II A. They can\nbe the same or different for each sublattice. The effective field\nHe\u000b;\u0017for sublattice \u0017is defined as\nHe\u000b;\u0017=H+HA;\u0017+J0;\u0017\u0014\n\u0016\u0017\u0005\u0014\n+\u00141\n2\u0003\u0017\u0017\u0012m2\n\u0017\nm2e;\u0017\u00001\u0013\n\u00001\n2\u0003\u0017\u0014\u0012\u001c2\n\u0014\n\u001c2e;\u0014\u00001\u0013\u0015\nm\u0017;(10)\nwhere H is an applied field, \u0005\u0017 =\n\u0000[m\u0014\u0002[m\u0014\u0002m\u0017]]=m2\n\u0014is transverse to m\u0014, and \u001c\u0017\nis the component of m\u0017parallel to m\u0014, in other words\n\u001c\u0017=m\u0014(m\u0017\u0001m\u0014)=m2\n\u0014, where\u00146=\u0017. The anisotropy\nfield,HA;\u0017is related to the zero-field transverse susceptibility\nor directly to the uniaxial anisotropy. The temperature depen-\ndence of the parameters defining the longitudinal dynamics in\nEq. (10) is\n\u0003\u0017\u0017=1\ne\u001fk;\u0017\u0012\n1 +J0;\u0017\u0014\n\u00160;\u0017e\u001fk;\u0014\u0013\n;\u0003\u0017\u0014=J0;\u0017\u0014\n\u00160;\u0014me;\u0014\nme;\u0017:\n(11)\nWithin the MFA, the equilibrium magnetisation\nof each sublattice can be obtained via the self-\nconsistent solution of the Curie-Weiss equations\nme;\u0017 = L ((J0;\u0017me;\u0017+jJ0;\u0017\u0014jme;\u0014)=kBT), (Lis the\nLangevin function) and the sublattice dependent longitudinal\nsusceptibilities derived directly from them, f\u001f\u0017\nk=@m\u0017=@H26.\nFor temperatures above TC, one can make use of the sym-\nmetry around TC,e\u001f(\u000f) = 2e\u001f(\u0000\u000f), where\u000f= 1\u0000T=TC\nis small26. These parameters can be calculated directly\nfrom MFA, gained from ASD simulations, or measured\nexperimentally.\nE. Comparisons of ASD and LLB approaches\nThere have been several studies comparing the ASD and\nferromagnetic LLB approaches to show consistency26,53,54.\nSince the LLB model describes both the longitudinal and\ntransverse relaxation of the magnetisation, two different types\nof comparison have been conducted so far. In the first case,\ntransverse relaxation, which can also be described by the con-\nventional micromagnetism, the LLB model provides also the\ntemperature dependence of the damping parameters, magneti-\nsation and effective fields. In the second case, longitudinal\nrelaxation, this is described by the last term in Eq. (8), and\naccounts for the dynamics of the magnetisation length. For\nexample, when the temperature of the heat-bath changes, the\nmagnetisation length also changes. This cannot occur in con-\nventional micromagnetic methods.6\nIn transverse relaxation numerical experiments, the main\ndifference between ferromagnets and ferrimagnets is found\naround the magnetisation and angular momentum compensa-\ntion temperatures. As already described by Eqs. (5) and (7),\neffective damping and frequency strongly depends on the net\nmagnetisation and angular momentum, which in turn depend\non temperature. Effective damping also depends on sublattice\ndamping parameters, which are assumed to be constant in con-\nventional micromagnetism, however within the LLB approach\nthey depend on temperature. As highlighted above, features\nsuch as the temperature dependence of damping should obey\nEq. (5). This emerges naturally in ASD but it is important\nthe LLB also reproduces this physics. Direct comparison be-\ntween ASD simulations and LLB predictions were made in\norder to validate the temperature dependence of the parame-\nters involved in the transverse dynamics. Schlickeiser et al.\nperformed this ASD-LLB comparison53. They showed good\nagreement between ASD and LLB results as shown in Fig. 2.\nTheir work makes it clear that the effective damping of ferri-\nmagnets is higher than predicted by macroscopic approxima-\ntions with a constant intrinsic damping parameter. As TCis\napproached, spin-spin interactions increase and so the intrin-\nsic damping increases. The simulations also provide access to\nnot only the effective Gilbert-like damping but also the damp-\ning of individual modes in the magnon spectrum. The ex-\nchange modes have a higher effective damping and a greater\ntemperature dependence. These modes can play an important\npart in non-equilibrium and dynamical processes55.\nIn longitudinal relaxation simulation, one aims to inves-\ntigate the time scales for the magnetisation length to reach\na new equilibrium when the temperature of the heat-bath is\nchanged suddenly. A longitudinal relaxation term was al-\nready proposed by Baryaktar as a direct generalisation of the\nLandau-Lifshitz equation56. However, the theory does not\nanswer what happens with phenomenological relaxation pa-\nrameters. Later on Garanin solved this issue with the LLB\nequation22, and Atxitia et al. extended Garanin’s ideas to\nferrimagnets26. In their work, Atxitia et al. already con-\nducted a systematic ASD-LLB comparison of the element-\nspecific longitudinal relaxation dynamics in a GdFeCo alloy.\nThis study was restricted to relatively large intrinsic damp-\ning parameters, \u0015\u0017= 0:1and temperatures below TC. Later,\nNieves et al.57extended the comparison to all temperatures.\nThe comparison between ASD simulations and LLB predic-\ntion is very good. This good agreement opens the door to use\nthe LLB model to derive analytical expression of the charac-\nteristic time scales of the longitudinal relaxation for situations\nwhere the deviations from equilibrium are relatively small. In\nferrimagnets, the main characteristic of the longitudinal relax-\nation is that the time scales are element specific. For example,\nin GdFeCo, Gd relaxes slower than Fe. This is rooted in their\ndifferent atomic magnetic moments. Interestingly, fluctua-\ntions act differently on Gd and Fe sublattices since the alloys\nare typical>70% Fe the Gd moments are like impurities in-\nside this Fe ferromagnet, therefore Gd behaves as a polarised\nparamagnet. As a consequence, whilst magnetic fluctuations\nmake Fe magnetisation dynamics experience critical slowing\ndown near the Curie temperature, in Gd they do not, and the\nFIG. 2. Comparison of ASD (points) and LLB (red and blue lines)\ncalculations of the effective frequency and damping of the ferromag-\nnetic (blue) and exchange (red) modes in a ferrimagnet. The black\ndashed line is the analytic solution when the temperature dependence\nof the microscopic damping parameters is not included. (Reprinted\nfigure with permission from [F. Schlickeiser et al., Phys. Rev. B, 86,\n214416 (2012)] Copyright (2012) by the American Physical Society.)\nGd sublattice relaxes faster than Fe at this point. This be-\nhaviour was predicted by the LLB model and it has recently\nbeen observed in femtosecond element-resolved experiments\nin GdFeCo58.\nIII. MATERIALS\nFerrimagnetic materials are everywhere, many of mag-\nnetic materials used in research are some kind of ferrimag-\nnet. Actually the majority of “ferromagnetic” insulators are\nferrimagnets59. There are many ferrimagnets with high poten-\ntial for technological applications. However two ferrimagnets\nhave been of sustained interest in spintronics ; GdFeCo metal-\nlic alloys for ultrafast spintronics applications, and YIG, for\ninsulator spintronics. From the computer modelling perspec-\ntive, ASD is ideal for the modelling of those two ferrimagnets.\nFirst, GdFeCo is most commonly grown as an amorphous al-\nloy, which makes it almost impossible for first principle cal-7\nculation of macroscopic properties but it is a problem well\nsuited for ASD and LLB models. Second, YIG presents a\ncomplex spin structure of the unit cell, and potential novel\nspintronic properties emerging from the atomic interactions\nbetween them can only be handled with atomistic models.\nA. Amorphous GdFeCo\nGdFeCo has long been used for magneto-optical based ex-\nperiments but is now finding use in other spintronics areas\ndue to its tuneable nature60–62. It is an amorphous alloy of\nGd, Fe and Co where the content of each can be adjusted in\nthe growth. In the Gd with partially filled 4f-shells a mag-\nnetic moment of\u00187\u0016Barises from the localised electrons\ndue to their coupling according to Hund’s rules. The valence\nband electrons are spin-polarised by the 4f electrons and give\nrise to an additional 5d6s atomic magnetic moment contribu-\ntion of around 0:5\u0016B. They are strongly exchange coupled to\nthe 4f electrons via a very strong intra-atomic exchange inter-\naction of the order of 200 meV , thus one can fairly consider\nboth kind of spins as one locked magnetic moment for most of\nthe physics studied here. The Gd moment is therefore much\nlarger than the Fe and Co moments, but in alloys with only\n15-30 % of Gd the net magnetisation can be balanced forming\nthe compensation points even though there are much fewer\nGd moments. Being able to continuously adjust the composi-\ntion allows fine tuning of the macroscopic quantities such as\nsaturation magnetisation and the temperature of the magneti-\nsation and angular momentum compensation points. In ex-\nperiments, the Gd or FeCo composition are often varied in a\nsystematic set of samples alongside the temperature or applied\nfield. This has lead to important discoveries such as field free\noptical switching38but makes the parameter space to model\nmuch larger55,63\nAlthough GdFeCo is amorphous it has generally been mod-\nelled on a lattice with random site occupancies64. The ex-\nchange interactions are parametrised by nearest neighbour\nvalues chosen to reproduce the magnetisation compensation\nand Curie temperatures63. While the number of neighbours of\neach site is constant (due to the lattice) the number of specific\nFe or Gd neighbours varies from site to site as would be ex-\npected in an amorphous system. Modelling the system as a\ntruly amorphous arrangement of atoms is eminently possible\nbut requires parameters such as the atomic radial distribution\nfunction and the distance dependent exchange which are un-\nknown and hard to obtain. The lattice based random site mod-\nelling is therefore a crossover between attempts at quantitative\nmodelling and toy models, where salient macroscopic features\nare reproduced using a somewhat generic base model.\nOstler et al. performed the first extensive ASD modelling\nof GdFeCo for different compositions, parametrised from a\nseries of experimental measurements63. Compensation points\nas well as critical temperatures were well reproduced by the\nmodel, showing an excellent agreement with both experimen-\ntal and mean field modelling (Fig. 3). They also parametri-\ncally change the Fe-Gd exchange coupling and calculate the\nthermal relaxation time of the element-specific magnetisation\nFIG. 3. The Cuire temperature ( TC- blue squares) and magnetisa-\ntion compensation temperature ( TM- blue circles) of the amorphous\nGdx(FeCo) 1\u0000xASD model as a function of the fractional Gd con-\ntent. Red triangles are experimental measurements of the magneti-\nsation compensation temperature (Reprinted figure with permission\nfrom [T.A. Ostler et al., Phys. Rev. B, 84, 024407 (2011)] Copyright\n(2011) by the American Physical Society.)\nlength, showing that stronger coupling leads to a faster ther-\nmalisation. Since then, the ASD model for GdFeCo proposed\nby Ostler et al. has been used for the description of ultrafast\nall-optical switching with great success, see Section IV A for\nfurther details.\nB. Garnets\nYttrium Iron Garnet (YIG)65is one of the main ferrimag-\nnetic materials used in insulator spintronics research. It’s very\nlow Gilbert damping, \u000b\u00181\u000210\u00005means that spin waves and\nmagnons have very long lifetimes. Recently it has become a\nplatform for spincaloritronics66research where magnon spin\nand heat currents are passed through the YIG16. Many other\ngarnets can be formed by substitution, for example the yttrium\nis easily substituted by other rare-earth elements such as Gd,\nTb or Er and the Fe can be substituted by gallium to reduce\nthe magnetisation and introduce compensation points.\nMacroscopically the magnetic properties of YIG appear\nsimilar to a ferromagnet. Most analytic theories approximates\nYIG as a simple one magnon band ferromagnet. In fact YIG\nis a complex ferrimagnet with the primitive cell containing\n20 Fe atoms. Much effort has been applied in building de-\ntailed, quantitative models of YIG and numerical modelling\nhas played a role in understanding spintronic effects in YIG\nand other garnets such as gadolinium iron garnet (GdIG) be-\nyond the ferromagnetic approximations.\nThe complexity of the crystal structures and magnetic spec-\ntra of ferrimagnetic insulators has meant that parametrising\natomistic models can be difficult. It is common to infer\nHeisenberg exchange parameters from inelastic neutron scat-\ntering measurements of the magnon spectrum. Linear spin8\nwave models are fitted to the measured dispersion67. In these\ncomplex magnets only a few of the magnon modes can be\nidentified in the neutron scattering making the fitting non-\nunique. For example YIG must contain 20 magnon modes but\nneutron measurements see only four or five modes clearly68.\nGd is also present in many ferrimagnets, particularly those\nused in magneto-optical based spintronics, as well as some\ninsulating garnets. But it provides a particular difficulty for\nneutron scattering because its natural isotope has one of the\nlargest neutron capture cross sections. Electronic structure\ncalculations are therefore being increasingly used to establish\nthe exchange interactions in material specific models.\nThe ferrimagnetic insulators like YIG have large electronic\nband gaps making the most common density functional the-\nory (DFT) methods poorly suited for calculations due to the\nsignificant Coulomb interaction. Attempts have been made\nto calculate the magnetic properties of ferrimagnetic garnets\nusing DFT+U but these could not simultaneously provide the\ncorrect electronic and magnetic properties69–72.\nRecently more advanced methods such as quasiparticle\nself-consistent GW73have shown great promise for these ma-\nterials with excellent results obtained for YIG74. This enables\na truly ‘ab initio’ parametrisation of atomic spin models, with-\nout even the choice of a Hubbard ‘U’. Progress in this area\nmay allow accurate modelling of specific ferrimagnetic insu-\nlators even where it is difficult to measure the exchange inter-\nactions experimentally.\nRecent advances in ASD calculations have introduced\nquantum statistics into the thermostat34,35. This means the\nthermal magnons obey Planck rather than Rayleigh-Jeans\nstatistics. The result is that quantitative calculations of ther-\nmodynamic properties can be made across almost the en-\ntire temperature range from zero Kelvin to the Curie point.\nThe method has been verified for complex ferrimagnets such\nas YIG with excellent agreement for the temperature depen-\ndence of magnetisation, magnon heat capacity34and magnon\nspectrum75. Excellent agreement was found. The magnon\nheat capacity in particular is an important parameter for ther-\nmal and spin transport76but can only be measured below\nabout 10 K in experiments due to the overwhelming contri-\nbution of the phonons77. Classical equipartition overestimates\nits value orders of magnitude and using quantum statistic al-\nlowed accurate calculations of the room temperature value34.\nThe ASD calculations showed that at room temperature the\npresence of the higher terahertz modes means the magnon heat\ncapacity is an order of magnitude larger than otherwise pre-\ndicted when assuming a simple single magnon band model78.\nIV . RESEARCH DOMAINS\nA. Ultrafast magnetisation dynamics\nThe field of ultrafast magnetisation dynamics began with\nthe discovery of the sub-picosecond magnetic response of\nNickel to femtosecond duration optical laser pulses79. The\nuse of femtosecond laser pulses has enabled the ultra-\nfast manipulation of the magnetic order, from femtoseconddemagnetisation79–81to ultrafast spin currents82,83and switch-\ning of magnetic polarity38,64,84–86. Later, alternative rapid ex-\ncitations have also been shown to drive magnetisation dynam-\nics, including recent demonstrations using picosecond electric\nand spin currents87,88.\nFerrimagnetic materials play a central role in this field.\nThe most prominent material is amorphous GdFeCo alloys\n(see Sec. III A) since it has long been the only material\nshowing the unusual deterministic heat-induced magnetisa-\ntion switching38. Recently, another class of ferrimagnet,\nMn2RuxGa89, has shown the same all thermal switching. The\ntheoretical description of laser induced all-optical switching\n(AOS)84of the magnetisation in GdFeCo ferrimagnetic alloys\nhas remained a challenge55,63,64,84,90–97Experimental findings\nare mostly compared or interpreted in terms of atomistic spin\ndynamics simulations55,63,98,99and the Landau-Lifshitz-Bloch\nequation26,93,100.\nLaser pulses can be as short as just a few femtoseconds,\nwhich can excite the electron system on timescales of the or-\nder of the exchange interaction allowing the investigation of\nthe fundamental physics governing switching. When a metal-\nlic ferrimagnetic thin film is subjected to a near infrared laser\npulse, only the electrons are accessible to the photon electric\nfield. Initially, the absorbed energy is barely transferred to the\nlattice and consequently the electron system heats up. The\nelectron and phonon temperatures, TelandTph, are decou-\npled for up to several picoseconds until the electron-phonon\ninteraction equilibrates the two heat-baths. This phenomenol-\nogy is well captured by the so-called two-temperature model\n(2TM)101,102which can be written as two coupled differential\nequations:\nCel@Tel\n@t=\u0000gep(Tel\u0000Tph) +Pl(t) (12)\nCph@Tph\n@t= +gep(Tel\u0000Tph): (13)\nThe parameters entering Eqs. (12) and (13) are material de-\npendent. For instance, for GdFeCo alloys, Cel=\relTelwhere\n\rel= 6\u0002102J/m3K2, andCph= 3:8\u0002106J/m3K rep-\nresent the specific heat of the electron- and phonon system.\nThe electron-phonon coupling, gep= 7\u00021017J/m3K. Here,\nPl(t) =P0exp(\u0000t2=t2\np)represents the absorbed energy by\nthe electron system, coming from the laser. The laser duration\nistp.\nASD simulations coupled to the 2TM predicted and ex-\nperiments subsequently confirmed, that the heat loaded by a\nfemtosecond laser pulse into these particular systems is suffi-\ncient to toggle switch the magnetisation polarity38as shown\nin the simulation results of Fig. 4. Despite the Gd and FeCo\nspin sublattices being coupled antiferromagnetically, during\nthe switching process, and for only a few picoseconds, the\nGd and FeCo spins show a parallel alignment, dubbed a ‘tran-\nsient ferromagnetic-like state’. These insights were only pos-\nsible thanks to element-specific time resolved femtosecond\nx-ray magnetic circular dichroism experiments conducted by\nRadu and co-workers64. This class of magnetic switching of-\nfers great promise for future applications as picosecond writ-\ning times have already been demonstrated in micro90and9\nFIG. 4. (a) Electron temperature in the two temperature model simu-\nlating the thermal laser pulses. (b) The sublattice magnetisation dy-\nnamics from ASD, Fe (dashed blue line) and Gd (solid red line). The\nsublattices switch deterministically with only a thermal pulse and no\napplied magnetic field. (c) Net magnetisation as a function of time\nthrough the switching events. (Reprinted figure from [T.A. Ostler et\nal., Nat. Commun., 3, 666 (2012)])\nnanoscale magnetic dots103. Recent experiments have brought\nthis closer to more conventional spintronics paradigms, show-\ning that besides using femtosecond laser pulses, GdFeCo\nnanostructures can be switched by the heat provided by pi-\ncosecond electric currents87and picosecond optical pulses104.\nThanks to ASD simulations it has been recently shown that\nswitching using femtosecond to picosecond heating follows\nthe same switching path.105\nFull micromagnetic models based in the LLB equation have\nbeen able to predict the ultrafast antiferromagnetic skyrmions\ncreation after ultrafast demagnetisation46. Micromagnetic\nsimulations based on the LLB have been also used for the\ndescription of all-optical switching of the magnetisation using\nmultiple-shot technique in cross devices made of synthetic fer-\nrimagnets composed of rare-earth free materials106. Switching\nbehaviours can be controlled by the Curie temperature of each\nlayer in the bilayer, which in turn is controlled by their thick-\nness.\nB. Spincaloritronics\nThe field of spincaloritronics studies the physical ef-\nfects coming out from the coupling between spin and\nheat66. Spincaloritronics phenomena include the spin Seebeck\neffect107–109, spin Peltier effect110,111, spin Nernst effect112, in\nmetallic, semiconductor, non-magnetic and magnetic insula-\ntors. Magnetic insulators are extensively used here because\nthey allow the study of purely magnon driven effects with-\nout charge transport convoluting results. Ideally, experiments\nwould study the magnon transport in ferromagnets before\nbranching into ferri- and antiferromagnets but, as mentioned\nbefore, most “ferromagnetic” magnetic insulators are in-fact\nferrimagnets. Many theoretical works in this field neglect thecomplications associated to the existence of at least two an-\ntiparallel sublattices and approximate the materials as a ferro-\nmagnet by assuming a single magnon band with a ~!\u0018k2\ndispersion. To explore what such approximations may miss\nin a true ferrimagnetic material ASD modelling has been used\nextensively.\nThe magnonic spin Seebeck effect in a ferrimagnet–the re-\nsponse of the magnetisation under a temperature gradient–has\nbeen studied using ASD simulations considering the simpli-\nfied situation of a temperature step instead of a gradient. In\nthis context, ASD modelling of a generic two sublattice fer-\nrimagnet in a temperature step provided information about\nthe spatial distribution of sublattice magnetisation113. In this\nnumerical experiment, a stationary non-equilibrium magnon\naccumulation arises and the shape corresponds to the deriva-\ntive of the equilibrium magnetisation @mz=@T in the absence\nof temperature step. In ferromagnets, @mz=@T < 0, but in\nferrimagnets showing compensation points, the sign of it can\nchange, hence, the existence of a temperature at which mag-\nnetisation accumulation vanishes at the step site.\nModelling in this field has also drawn attention to the\nmagnon polarisation in ferrimagnets68,75,114. Here polarisation\nis related to the clockwise versus anticlockwise rotation of\nthe spins or alternatively whether a magnon mode carries +~\nor\u0000~spin angular momentum. In a ferromagnet, magnons\nhave a single circular polarisation (or elliptical with dipolar\ninteractions) corresponding to the anticlockwise rotation of\nthe magnetic moments in a field. In a uniaxial antiferromag-\nnet there are both anticlockwise and clockwise polarisations\nbut the magnon modes degenerate so the polarisation cannot\neasily be measured. In a ferrimagnet the anticlockwise and\nclockwise polarisations also exist due to the opposing sublat-\ntices, but the exchange field between the sublattices splits the\nmodes so that clockwise magnons are higher in energy than\nanticlockwise.\nExperimental measurements and ASD modelling of the\nspin Seebeck effect in Gadolinium iron garnet (GdIG)\nshowed that magnon polarisation has observable effects in\nspintronics115. Two changes of sign in the spin Seebeck effect\nwere found as the temperature is increased. One is expected\nacross the magnetisation compensation point as the sublattices\nreverse in the applied field across this point. The change in\nsign of the spin Seebeck effect at low temperatures does not\ncorrespond to any macroscopic changes of the ferrimagnet.\nTheory and ASD modelling showed that the changing ther-\nmal occupation of magnon modes with different polarisation\ncauses the sign change at low temperature.\nAlthough the polarisation of magnons has been described\nby theory from the early understanding of magnons it had\nnever been directly measured, possibly due to the lack of ap-\nparent physical consequence until recently. Polarised inelas-\ntic neutron scattering was used to measure the polarisation of\nmagnons for the first time75, supported by quantitative ASD\nmodelling. The scattering cross section in the required exper-\nimental geometry is very small making the experiments dif-\nficult. The first attempt was only partially successful which\nwas identified because of the prior calculation of the polarised\nneutron scattering cross section using ASD modelling. A sec-10\nFIG. 5. Upper (c) experimental measurements of the magnetic chi-\nral part of polarised inelastic neutron scattering from YIG. Lower\n(d) ASD calculations of the polarised inelastic neutron scattering\ncross section from a parametrised model of YIG. The ASD results\nhave been convoluted with the experimental measurement resolution.\n(Reprinted figure with permission from [Y . Nambu et al., Phys. Rev.\nLett., 125, 027201 (2020)] Copyright (2020) by the American Phys-\nical Society.)\nond measurement was successful and very good agreement\nwas found between the ASD calculations and the experimen-\ntal measurements as shown in Fig. 5. The polarisation of dif-\nferent magnon modes may be useful in spintronics applica-\ntions as the magnon transport appears to be sensitive to the po-\nlarisation in experiments116. Theoretical suggestions of how\nto use the polarisation are also being made117.\nThe high frequency magnon modes of the YIG has\nmotivated experiments to excite these high frequency\nmodes118–121. Maehrlein et al.118resonantly excited long-\nwavelength THz phonons in YIG using THz laser pulses.\nChanges in the magnetisation were measured via the magneto-\noptical Faraday effect with an optical probe pulse. A reduction\nin the sublattice magnetisation was observed on a picosecond\ntimescale118. The change in magnetisation persists for mi-\ncroseconds before recovering. The excitation of the infra-red\nactive THz phonons predominantly makes the light oxygen\nions to move, displacing them from their equilibrium posi-\ntions. It is assumed to cause fluctuations in the superexchange\nbetween Fe atoms due to the changes in the bond distances\nand angles, shown pictorially in Fig. 6A. To confirm the ex-\nperimental findings ASD calculations were performed with\na modified Heisenberg exchange which included dynamical\nstochastic fluctuations of the intersublattice exchange param-\neters. From the model it was determined that the net magneti-\nsation remains constant, even though the sublattice magneti-\nFIG. 6. (A) Superexchange model of the inter-sublattice coupling in\nYIG. The oxygen displacement ( u(t)) caused by phonon excitations\nat 20 THz causes fluctuations in the superexchange Jad. (B) ASD\nmodel with stochastic exchange for 0.5 ps. The change in magneti-\nsation of each sublattice is shown in blue and green respectively. The\ntotal change in magnetisation (black line) is zero. (Reprinted figure\nfrom [Maehrlein et al., Sci. Adv., 4, eaar5164 (2018)])\nsation is decreasing (Fig. 6B). This is because the isotropic\nexchange interaction conserves the the spin angular momen-\ntum.\nThe fields of ultrafast magnetisation dynamics and\nspincaloritronics sometimes combine and effects such as\nthe spin Seebeck effect are probed on the sub-picosecond\ntimescale to understand the timescales of the fundamental pro-\ncesses. Seifert et al. used a femtosecond laser to suddenly\nheat the Pt of a YIG-Pt bilayer119. The YIG is transparent\nto the laser and so there is a significant temperature differ-\nence between the hot electrons in the Pt and the magnons and\nphonons of the YIG. The aim was to understand how quickly\nthe spin current across the YIG-Pt interface is established due\nto electron-magnon scattering. The spin current (measured\nfrom the THz emission due to the current formed by the in-\nverse spin Hall effect) emerges almost instantaneously and\npeaks within 500 femtoseconds of the laser excitation. A dy-\nnamical theory was created which required parameters such as\nthe frequency dependent spin susceptibility of the YIG. This11\nwas calculated using ASD with a detailed model of YIG and\nshowed that the magnetic system can respond instantly due to\nthe lack of inertia. The quantitative theory allowed an esti-\nmate of the interfacial sd-exchange coupling and spin mixing\nconductance in good agreement with other values in the liter-\nature.\nC. Magnetic textures\nThe motion of spin textures such as domain walls and\nskyrmions is a large area of study because of proposed de-\nvice concepts using the textures for information storage and\nprocessing. Much work has been done on ferromagnetic tex-\ntures but ferrimagnets are relatively unexplored beyond treat-\ning a ferrimagnet as a ferromagnet. The main difference be-\ntween ferro and ferrimagnets that can be expected is in how\nthe dynamics of textures changes due to the presence of the\nangular momentum and magnetisation compensation points.\nIt is expected that at the angular compensation point, fer-\nrimagnets behave like antiferromagnets. Antiferromagnetic\ntextures can move at high velocities due to the compensation\nof torques acting at each sublattice and responsible of the so-\ncalled Walker breakdown common for ferromagnets. Analyti-\ncal theory based on existing macrospin approaches have been\nused to predict and interpret domain wall dynamics under ex-\nternal fieldH122. The velocity, vDWand precession, \nDW\nof the ferrimagnetic domain wall of width \u0001DWcan be ex-\npressed in terms of effective damping (Eq. (5) and gyromag-\nnetic ratio (Eq. (6):\nvDW= \u0001 DW\u000be\u000b\n1 +\u000b2\ne\u000b\re\u000bH; \n DW=1\n1 +\u000b2\ne\u000b\re\u000bH:\n(14)\nStill, thermal effects, such as motion of magnetic textures\nunder thermal gradients require computational models. ASD\nmodelling has been useful in this area. Donges et al. per-\nformed an extensive study of how domain walls in ferrimag-\nnets move when thermal gradients are applied123. In ferro- and\nantiferromagnets, domain walls always move towards hotter\nregions124–126. In ferrimagnets they found the same behaviour\nbelow the Walker breakdown–where domain walls behave like\nan effective ferromagnetic DW under thermal gradient. But\nabove the Walker breakdown the domain walls can move to-\nwards colder regions if the temperature is less than the angular\nmomentum compensation point as shown in Fig. 7. Their re-\nsults explain anomalies in experimental measurements of do-\nmain wall motion in ultrafast laser experiments on GdFeCo\nwhere domain walls have been observed to move away from\nthe heated region127–the opposite of previous expectations.\nThis highlights the need for ASD modelling of ferrimagnets\nas these features cannot be seen in ferromagnetic models such\nas Eqs. (14) and are difficult to isolate in experiments from\nmany other possible effects. They also find a ‘torque com-\npensation point’ where the ferrimagnetic domain wall moves\nsimilarly to an antiferromagnet domain wall with inertia free\nmotion and no Walker breakdown.\nIn the magnetic skyrmion field there is an effort to remove\nor control the skyrmion Hall effect–the transverse motion of askyrmion when it is pushed by a directional stimulus. In an-\ntiferromagnets it is known that skyrmions have no hall effect\ndue to the exact cancellation of the Magnus force from the\ntwo sublattices114,128. Recent experiments in GdFeCo have\ndemonstrated that the skyrmion Hall effect can also be zero in\nferrimagnets at the angular momentum compensation point129\n. The experimental evidence comes from MOKE imaging and\nwere compared with ASD simulations showing a good agree-\nment confirming the absence of a skyrmion Hall effect when\nthe next spin density is zero.\nV . SUMMARY AND OUTLOOK\nThe modelling techniques presented in this review have al-\nready played a central role in uncovering material properties\nand physical phenomena in ferrimagnetic spintronics which\nare absent in the commonly used effective ‘ferromagnetic’\nmodels. Even with the rise of antiferromagntic spintronics130,\nwe believe ferrimagnetic spintronics will grow due to the\nmanipulations that can be induced by use of the compensa-\ntion points. One of the most important aspects of computer\nmodelling is the close collaboration with experimental part-\nners. This allows not only to interpret experimental results\nbut also to validate theories and predict new physical phe-\nnomena. Paradigmatic examples have been covered in this\nreview. For example, in the field of ultrafast dynamics, mod-\nelling techniques have been pivotal to the interpretation of the\ntransient ferromagnetic-like state in GdFeCo, validation of the\nfemtosecond formation of the Spin Seebeck effect in YIG,\nand the prediction of thermally-induced ultrafast magnetisa-\ntion switching in GdFeCo. One characteristic of ferrimagnets\nis that interesting physics appears when the energy input is\nof the same order as the antiferromagnetic coupling between\ndistinct sublattices. For example, by changing temperature\n(thermal energy input), magnetisation and angular momentum\ncompensation point can be crossed at which some dynami-\ncal properties are similar to antiferromagnets. Lately, models\nwhich include thermal effects have dominated this field. Im-\nprovements in the modelling of temperature by inclusion of\nquantum statistics are likely to become more prevalent and\nprovide more predictive weight to multiscale simulations. Ef-\nforts to directly model the heat bath subsystems are also likely\nto increase as there is much interest in the strong coupling be-\ntween systems such as phonons and magnons. As space and\ntimescales are pushed further models at the atomic level must\nbe further developed beyond the localised rigid spin approx-\nimation. There is also a notable absence of models for ele-\nments with significant orbital moments which may be required\nfor rare-earth elements which are found in many ferrimagnets.\nGoing forward we anticipate ferrimagnets to be ever more\ninvestigated for spintronic applications. Ferrimagnets are the\ndoor for fast control of magnetic functionalities as given by\ntheir antiferromagnetic character while conserving the advan-\ntage of ferromagnets in terms of measuring and controlling\nby conventional means such as magnetic fields. Recent ex-\namples, include the field of skyrmionics and spin-orbit torque\nswitching. Ferrimagnetic insulators in particular are likely to12\nFIG. 7. Velocity, precession and tilting of ferrimagnetic domain walls (DW) under thermal gradients from ASD simulations. (a) Biaxial\nanisotropy ferrimagnetic DWs dynamics can be described as an effective ferromagnet, velocity scales with temperature gradient. (b) Tilting\nof the DW vanishes at the torque compensation temperature Ttat which the DW dynamics shows similar characteristics to antiferromagnets,\nnamely, quasi-inertia-free motion and the absence of Walker breakdown. (c) Uniaxial anisotropy. Above Walker breakdown, the ferrimagnetic\nDW can show the opposite, counter intuitive behaviour of moving toward the cold end. Below the compensation temperature the wall moves\ntoward the cold end, whereas above it toward the hot end. This behaviour is driven by angular momentum transfer and therefore strongly\nrelated to the angular momentum compensation temperature. (e) The inset shows the temperature dependence of the non-adiabaticy parameter\n\fe\u000b, defined as the ratio between adiabatic and non-adiabatic torque created by a spin current. (Reprinted figure from [Donges et al., Phys.\nRev. Research, 2, 013293 (2020)])\ncontinue to be used widely in experiments as they provide an\nimportant spin source for fundamental and device inspired ex-\nperiments which attempt to modify spin current transmission.\nRecently, practical realisation of relativistic kinematics in iso-\nlated magnetic solitons has been demonstrated51in ferrimag-\nnets, establishing an experimental framework to study rela-\ntivistic solitonic physics.\nSo far ferrimagnets have hardly featured in some popular\ntopics such as topological magnetism where the additional\nsymmetries allowed by a two sublattice magnet with a net\nmagnetisation may be of interest. A similarly open avenue\nto be explored is the investigation of two-dimensional ferri-\nmagnets. A promising route to realise this class of 2D ferri-magnets could be by exploiting the highly tunable superlattice\npatterns in twisted (or moiré) bilayers of atomically thin ma-\nterials. 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Nan-\notechnol. 11, 231 (2016)." }, { "title": "2305.02971v1.Effective_rectification_of_THz_electromagnetic_fields_in_a_ferrimagnetic_iron_garnet.pdf", "content": "arXiv:2305.02971v1 [cond-mat.mtrl-sci] 4 May 2023Effective rectification of THz electromagnetic fields in a fer rimagnetic iron garnet\nT.G.H. Blank,1,2E.A. Mashkovich,3K.A. Grishunin,1C. Schippers,2\nM.V. Logunov,4B. Koopmans,2A.K. Zvezdin,5and A.V. Kimel1\n1Radboud University, Institute for Molecules and Materials , 6525 AJ Nijmegen, the Netherlands.\n2Department of Applied Physics, Eindhoven University of Tec hnology,\nP.O. Box 513, Eindhoven 5600 MB, the Netherlands.\n3University of Cologne, Institute of Physics II, Cologne D-5 0937, Germany.\n4Kotel’nikov Institute of Radioengineering and Electronic s, 125009 Moscow, Russia.\n5Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991 Moscow, Russia.\n(Dated: May 5, 2023)\nIt is found that single-cycle THz electromagnetic fields effic iently excite a GHz spin resonance\nmode in ferrimagnetic Tm 3Fe5O12, despite the near absence of GHz spectral components in the\nexciting THz pulse. By analyzing how the efficiency of excitat ion depends on the orientation and\nstrength of the THz electric field, we show that it can be expla ined in terms of the nonlinear THz\ninverse Cotton-Mouton effect. Here, the THz electric field ge ts effectively rectified and acts on the\nferrimagnetic spins as a uni-polar effective magnetic field p ulse. This interpretation is confirmed by\na theoretical model based on the phenomenological analysis of the effective magnetic field, combined\nwith the equations of motion derived from the effective Lagra ngian for a ferrimagnet. Moreover,\nby using the outcome of two-dimensional THz spectroscopy, w e conjecture a quantum-mechanical\ninterpretation of the observed effect in terms of stimulated Raman scattering of THz photons by\nthe crystal-field split f-f electronic transitions of Tm3+.\nI. INTRODUCTION\nThe development of ultrafast magnetism opened up\na research field [ 1], which explores new regimes of spin\ndynamics triggered in ferro, ferri, and antiferromagnetic\nmaterials by stimuli much shorter than the time required\nto reach thermodynamic equilibrium ( ∼100 ps). Con-\nsequently, in such regimes of spin dynamics, the con-\nventional approximations of equilibrium thermodynam-\nics to describe magnetic phenomena fail, and the result-\ning spin motion is often counter-intuitive. For instance,\neven though Curie’s principle [ 2] of equilibrium thermo-\ndynamics predicts that “the symmetry of the causes are\nto be seen in the effects” and thus solely heating cannot\nresult in magnetization reversal, it was shown that ultra-\nfast heating induced by femto or picosecond laser pulses\nor picosecond electrical pulses is able to reverse the mag-\nnetization in ferrimagnetic materials [ 3–6]. Similarly, in\nequilibrium thermodynamics, the fastest and the least\ndissipative route of magnetization reorientation seem to\nbe always mutually excluding. However, this seemingly\nimpossible exclusive combination was achieved by em-\nploying the effect of photo-induced magnetic anisotropy\nby a femtosecond laser-pulse, demonstrating the simul-\ntaneously record-fast and least-dissipative writing of a\nmagnetic bit [ 7]. Naturally, this counter-intuitive but\nvery appealing regime of spin dynamics attracted the at-\ntention of researchers in applied magnetism, including\nthe fields of spintronics, magnonics, and magnetic data\nstorage. This interest, in turn, fuels the search for ever\nnew ultrashort stimuli to enable more ultrafast and even\nless dissipative writing of magnetic bits.\nWhile femto and picosecond laser pulses in the near-\ninfrared and the visible spectral range are the most pop-\nular stimuli in ultrafast magnetism, it was realized thatnearly single-cycle THz electromagnetic pulses [ 8], con-\nsisting of photons with a thousand times smaller energy,\ncan affect the spins in magnetic media in a more energy\nefficient non-thermal way. In particular, it was shown\nthat the magnetic component of such THz pulses can\ndirectly couple to spins via Zeeman torque [ 9]. At the\nsame time, such an approach is associated with a sig-\nnificant disadvantage. The time integral of the electro-\nmagnetic field of the THz pulses, as in the case of any\nother freely propagating electromagnetic wave in a neu-\ntral medium, is strictly zero and the net effect of such\na stimulus on spins is thus questionable. But recently,\nit was shown that THz electric fields can be effectively\nrectified and thus become much more efficient in control-\nling spins [ 10,11]. Such rectification of THz fields has so\nfar only been demonstrated in canted antiferromagnetic\nmedia [12]. Here, we fill the gap and demonstrate that a\nsimilar mechanism of effective field rectification can also\nbe realized in ferrimagnets.\nIn this article, we show that a single-cycle THz pulse\nin ferrimagnetic Tm 3Fe5O12is able to excite not only a\nTHz [13], but also GHz mode of spin resonance. The ex-\ncitation ofthe lattermode is surprisingasits frequencyis\nnearly absent in the spectrum of the THz pulse. By an-\nalyzing how the amplitude of the GHz mode depends on\nthe orientation and the strength of the THz electromag-\nnetic fields, we rule out heating and show that the fields\nare effectively rectified and that the GHz spin resonance\nis excited by an effective magnetic field generated in the\nmedium duetothe inverseCotton-Moutoneffect (ICME)\n[14,15]. These findings are supported by a phenomeno-\nlogicalanalysisofthe rectified fieldin combinationwith a\nLagrangian model of spin dynamics [ 16], which all fit the\nexperimental observations. Quantum mechanically, the\nprocess can be described in terms of stimulated Raman2\nscattering [ 17], where a first photon with frequency ω1\nbrings the electron to the excited state, while a second\nphoton with frequency ω2stimulates fast recombination\nto the Stokes-shifted ground state accompanied by an\nemission of a magnon with frequency ω=ω1−ω2. By\nemploying two-dimensional (2D) THz spectroscopy [ 18],\nwe show that the excited state is characterized by a life-\ntime shorter than 1 ps. This fact practically excludes\nthat the GHz mode is excited via the second THz mode\nof spin resonance.\nThe paper is organized as follows. Section IIprovides\ndetails about the sample and describes the experimental\nsetup. Section IIIpresents the main experimental find-\nings. These findings are supported by the theory pre-\nsented in section IV, where we employed a Lagrangian\napproachto describe the magnetizationdynamics, driven\nby an effective rectified magnetic field due to the ICME.\nWe suggest a possible microscopic mechanism of the lat-\nterinthediscussionofsection V,supportedbyadditional\nmeasurements using 2D THz spectroscopy. Our conclu-\nsions are summarized in the final section VI.\nII. SAMPLE AND EXPERIMENTAL SETUP\nThe particular ferrimagnet that we examined was a\n19µm thick film of bismuth and gallium substituted\nthulium iron garnet (TmIG) Tm 2BiFe4.2Ga0.8O12grown\nby liquid phase epitaxy on a 500 µm gadolinium gallium\ngarnet (Gd 3Fe5O12) substrate with (111) orientation.\nThe garnet structure has eight formula units per unit\ncell and has space-group symmetry Ia 3d (point group\nOh) [19]. In the parent compound, Tm 3Fe5O12, two in\nfive Fe3+ions can be found in octahedral surroundings\nof O2−and the other three in a tetrahedral environment\n[20]. The spins of the tetrahedrally surrounded Fe3+ions\ncouple antiferromagnetically to the spins of those in the\noctahedral environment. The two Fe3+magnetic sublat-\ntices have different magnetizations, this difference results\nin a net magnetization MFe. The spins of Tm3+couple\nantiferromagnetically with respect to MFe, resulting in\na net magnetization MTm. Because the exchange inter-\naction between the iron sublattices is large compared to\nany other exchange interaction between the three sub-\nlattices (Tm-Tm and Tm-Fe), it is sufficient to treat\nthe two iron sublattices as one [ 21,22]. Therefore, al-\nthough Tm 3Fe5O12is, in reality, a three-sublattice ferri-\nmagnet, it is effectively treated as a more conventional\ntwo-sublattice Tm-Fe ferrimagnet with net magnetiza-\ntionM=MFe+MTmand N´ eel vector L=MFe−MTm.\nIn the garnet studied here, a part of the Tm3+ions\nwere substituted by Bi3+to enhance the magneto-optical\nFaraday effect [ 23–25]. Moreover, some Fe3+ions on\nthe tetrahedral sites [ 26] were partly substituted with\nnon-magnetic Ga3+to reduce MFeand thus ensure that\na possible ultrafast spin reorientation is not hampered\nby the need to facilitate an ultrafast exchange of large\nangular momentum between the lattice and the spinsystem. Moreover, a sufficiently large Ga3+dilution\nshould grant a magnetization compensation temperature\nto Tm 3Fe5O12that usually has no compensation point\nduetothesmallmagneticmomentofTm3+[24], butsuch\na temperature was not observed in the present composi-\ntion. Finally, both the parent compound as well as the\nsubstituted versionsdisplayuniaxialmagneticanisotropy\nalong the out-of-plane [111] axis [ 24], as was confirmed\nby magneto-optical measurements [ 13].\nWe performed a pump and probe experiment on\nthe sample by employing single-cycle THz-pulses gener-\nated by tilted-pulse-front optical rectification in LiNbO 3\n[8,27], yielding pulses with a peak electric field strength\nup to∼1 MV/cm in focus as calibrated by electro-optic\nsampling in a slab of GaP-(110) (see Fig. 1(b)). The\nTHz pulses were brought to temporal and spatial over-\nlap with low-intensity near-infrared (NIR) probe pulses\nwith a central wavelength of 800 nm and pulse duration\nof 100 fs. The experimental scheme and coordinate sys-\ntem are depicted schematically in Fig. 1(a), indicating\nthe THz pump and NIR probe polarization angles αand\nβdefined with respect to the y-axis. The polarizations\nwere controlled by a set of wire-grid polarizers for the\nTHz pump and a half wave-plate for the probe pulse.\nThe probe pulse transmitted through the sample and its\nTHz-induced rotation in polarization was mapped by a\ncombination of a Wollaston prism and a set of balanced\nphotodiodes. Bymeasuringthe polarizationrotationasa\nfunction of the time-retardation tbetween the pump and\nprobe pulses, we traced the THz-induced spin dynamics.\nFIG.1. (a)Schematicillustration oftheexperimental sche me.\n(b) The calibrated waveform of the THz pulse measured by\nelectro-optic sampling in GaP. The THz polarization is ini-\ntially along the y-axis, but it could be rotated using wire-grid\npolarizers. (c) Typical THz-induced transient of the probe -\npolarization rotation, measured at T= 6 K with an applied\nexternal field of 110 mT, pump polarization α=−45◦and\nprobe polarization β= 0◦. The data at long timescales has\nbeen multiplied by a factor of 5 for visibility.3\nOur previous results showed that the magneto-optical\nsignal of the transmitted probe pulse originates exclu-\nsively from out-of-plane magnetization components [ 13].\nTherefore it can be expected that no dynamic magneto-\noptical signal due to magnetization precession will be de-\ntected when the equilibrium magnetization is out of the\nplane. Tothis end, a magnetic field wasapplied predomi-\nnantlyintheplaneinordertosaturatethemagnetization\nin the plane, slightly titled at a small angle δ∼3◦. The\ntilt was required to be able to excite THz spin dynamics\nfor every polarization of the THz pulse (see Ref. [ 13]),\nas is explained in more detail in the conclusions of Ap-\npendixA. Finally, XRD analysis of the sample (see Sup-\nplemental Material [ 28]) confirmed its [111] orientation.\nMoreover, the analysis provided us with the orientation\nof the crystallographic axes with respect to the experi-\nmental coordinate system x∝bardbl[112] andy∝bardbl[110].\nIII. RESULTS\nFigure1(c) shows a typical dynamical polarization ro-\ntation transient. In our previous article [ 13], we showed\nthat the ultrafast THz dynamics in the first 50 picosec-\nonds can be attributed to the ferrimagnetic THz Kaplan-\nKittel exchange mode [ 29]. Due to the unequal g-factors\nof the Tm and Fe sublattices, the Zeeman torque (or\nmagnetic dipole interaction) acts differently on each sub-\nlattice, rendering a relatively efficient resonant excita-\ntion of the mode. Moreover, the bismuth-substitution\nyields a strong magneto-optical Faraday effect, which\nmade detection with a good signal-to-noise ratio possi-\nble. In addition to the previously reported THz mode,\nthe transients also reveal oscillations at a much lower\n(GHz) frequency. So low frequencies are typical for the\nferromagnetic resonance (FMR) mode in a ferrimagnet,\nas was also predicted by the theory of Kaplan and Kittel.\nGiven that the duration of our THz pulse ∼1 ps is much\nshorter than the period of the mode ∼50 ps, a resonant\nexcitation by the THz magnetic field similar to what was\nseen with the exchange mode is unlikely. Instead, the\nexcitation must be impulsive in nature. In order to re-\nveal the excitation mechanism of this supposedly FMR\nmode, we measured the dynamics as a function of the\npump polarization angle α, the probe polarization an-\ngleβ, the strength of the THz pump electric field, the\nstrength of the external magnetic field as well as sample\ntemperature.\nA. Pump and probe polarization\nThe measured dynamics are strongly dependent on\nthe polarization angle αof the THz pump pulse (see\nFig2(a)), which rules out heating as the dominant mech-\nanism. The optimal excitation occurs for α=±45◦(see\nFig.2(b)). The extrema are slightly asymmetric, such\nthat the excitation is actually stronger for α=−45◦.\n/s70/s97/s114/s97/s100/s97/s121/s32/s114/s111/s116/s97/s116/s105/s111/s110/s32/s40/s109/s100/s101/s103/s41\n/s84/s105/s109 /s101 /s32/s40/s112/s115/s41/s40/s97/s41\n/s97 /s32/s40/s100/s101/s103/s41\n/s113\n/s70/s32/s40/s109/s100/s101/s103/s41\n/s70/s70/s84/s32/s97/s109/s112/s108/s46/s32/s40/s110/s111/s114/s109/s46/s41\n/s84/s72/s122/s32/s112/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32 /s97 /s32/s40/s100/s101/s103/s41/s70/s77/s82\n/s70/s105/s116\n/s84/s105/s109 /s101 /s32/s40/s112/s115/s41/s48 /s49/s56/s48/s98 /s40/s100/s101/s103/s41/s40/s99/s41/s40/s98/s41\n/s97/s32 /s61/s32/s45/s52/s53/s111\nFIG. 2. (a) THz induced dynamics as a function of the pump\npolarization angle α, measured at a temperature of 6 K and\nexternal field of 110 mT. The data have been fitted by sine\nfunctions shown by the solid black lines. Two maxima at\napproximately ±45◦can be observed, with nearly opposite\nphases. (b) Normalized peak Fourier amplitudes of the mode\nfor different α, where positive and negative values indicate the\ndifferent phases of the dynamics. (c) THz-induced dynamics\nmeasured at the optimal THz polarization α=−45◦and\nfieldBext= 130 mT, for various angles of the initial probe\npolarization β, showing no variations.\nThe dynamics for negative and positive αare approxi-\nmately in opposite phases. These facts imply that the\nexcitation mechanism is π-periodic with respect to α.\nThis is in contrast with the 2 πperiodicity of the exci-\ntation of the exchange mode due to the Zeeman torque\n[13]. Instead, the results are more comparable to the\nnonlinear excitation of the quasi-ferromagnetic mode in\nantiferromagnetic FeBO 3, where the largest signals were\nalso found when the electric (or magnetic) field of the\npump pulses was polarized at ±45◦from the net magne-\ntization [ 12].\nNext, itcanbeseeninFig. 2(c)thatthesignaldoesnot\ndepend on the initial probe polarization angle β. This\nimplies that the observed dynamics are a result of the\nmagneto-optical Faraday effect, to which only the out-\nof-plane magnetization contributes θF(t)∝Mz(t) [13].\nTherefore, starting from Fig. 2, we referred to the in-\nduced polarization rotation as Faraday rotation θF(t).\nB. THz amplitude\nFigure3shows the FFT spectrum of the dynamics\n(red) measured at the optimal THz polarization angle\nα=−45◦, as well as the spectrum of the THz pump-\npulse (blue). The sharp red peak that corresponds to the\nGHz (FMR) mode lies completely out of the spectrum\nof the exciting THz pulse. This fact indicates that the4\n/s32/s84/s72/s122/s40/s97/s41 /s40/s98/s41\nFIG. 3. (a) Typical Fourier spectrum of the signal (red) and\nthe exciting THz pulse (blue). The sharp peak in the signal\nspectrum that corresponds to the FMR mode falls out of the\nTHz spectrum. (b) Peak FFT amplitude of the FMR mode\nas a function of THz electric field amplitude, measured at a\ntemperature of 6 K, THz polarization α=−45◦andBext=\n110 mT. When fitting the data with a power law dependence\nof the form Eγ\nTHzfor variable γ, the data is best fitted with a\nquadratic dependence γ= 2.\nexcitation is most likely nonlinear. This nonlinearity was\nconfirmed by measuring the peak Fourier amplitude of\nthe mode as a function ofthe strength of the THz electric\nfieldETHz(see Fig. 3(b)). The data could only be fitted\naccurately with a quadratic dependence E2\nTHz.\nC. External magnetic field and temperature\nThe assignmentofthe exactoriginofthe FMR mode is\nnot straightforward, because TmIG is, in reality, a three-\nsublattice ferrimagnet. This means that possible spin\nresonances are associated not only with the spins of Fe3+\nbut also with those of the Tm3+ions. To reveal the ori-\nginofthe mode, westudied the frequencyasafunction of\nthe external magnetic field and temperature. Figure 4(a)\nshows that small changes in the external magnetic field\nhave an enormous impact on the frequency of the mode.\nIt can be seen in Fig. 4(b) that the frequency initially\ndecays, but for larger fields >25 mT it increases approx-\nimately linearly. Theoretically, the linear slope can be\nrelated to the g-factor of the FMR mode [ 30]. In the\ncase of FMR involving only iron, which has a g-factor of\ng≈2,theslopeshouldamountto28GHz/T.However,in\nour case, the slope is approximately ∼90 GHz/T, which\nallows us to give an estimate of the effective g-factorgeff\nis about ˆ geff≈6.4.\nThe fact that the effective g-factor is so much larger\nthan that of iron, tells us that the thulium sublattice\nmust be involved. Let gTmandgFebe theg-factors of\nthe individual Tm and Fe sublattices, respectively. The\ntheory of Kittel predicts that the effective g-factor, in\nthat case, is given by [ 31]:\ngeff(T) =MFe(T)−MTm(T)\ng−1\nFeMFe(T)−g−1\nTmMTm(T).(1)\nThe effective g-factor can thus have a strong temper-ature dependence in the vicinity of the angular mo-\nmentum compensation point, i.e. when g−1\nFeMFe(T) =\ng−1\nTmMTm(T). Otherwise, no strong temperature depen-\ndenceoftheeffective g-factorisexpected. Hereweshould\nnote that the magnetization compensation point in this\nsample either does not exist or is slightly above 0 K.\nFigure4(c) shows that the frequency at a fixed mag-\nnetic field does drop significantly when increasing the\ntemperature. But at the same time, it can be seen in\nFig.4(d) that the slope of the frequency as a function\nof the field does not decrease as a function of tempera-\nture. It means that the decay in frequency observed in\nFig.4(c) is a result of a temperature-dependent effective\nfield such as magnetic anisotropy, and not of a decreas-\ning effective g-factor. In Fig. 4(e), we show the effective\ng-factors estimated from the slopes of the linear fits in\nFig.4(d). It reveals that the effective g-factor has a pe-\nculiar temperature dependence and remains to be large.\nLarge values of the effective g-factor have also been ob-\nserved in a Tm 3Fe5O12compound with La, Ca, and Ge\nsubstitutions [ 32]. Here, the result could be fitted by\nassuming that the Tm angular momentum is quenched,\ni.e.gTm= 0. It is therefore likely that the peculiar tem-\nperature dependence of geffis greatly influenced by the\ndilution of Bi and Ga. Other factors might also play a\nrole, such as the potential noncollinear arrangement of\nthe Tm spins [ 33,34] that were observed in these materi-\nals, which raises interesting questions for further studies.\nIn any case, the fact that the g-factor remains well above\nthat ofthe free electron( g= 2) in the entire temperature\nrangefrom 6 to 170K implies that the thulium sublattice\nis involved in the FMR mode.\nFinally, we measured how the dynamics depend on a\nchangein the polarityofthe externalmagnetic field. Fig-\nure5shows that for the excitation maxima α=±45◦,\nthe phase of the dynamics is independent of the change\noffield polarity ±Bext. This result is in contrastwith the\nexcitationoftheTHzmodebyZeemantorquereportedin\nRef. [13], where the detected magneto-optical transients\ndid reverse their phase upon a change of polarity of the\nexternalmagnetic field. This fact againconfirms that the\ndominatingmechanism ofexcitation ofthe FMR mode in\nthe studied TmIG is not due to the linear coupling of the\nTHz magnetic field to the spins via the Zeeman torque.\nOnly around α= 0◦, one can distinguish a very weak sig-\nnal that changes sign upon the field reversal. This signal\ncould be interpreted to be due to the linear excitation\nmechanism, but it is clear that this mechanism is quite\ninefficient and negligible in comparison with nonlinear\nexcitation.\nTo summarize, the obtained experimental dependen-\ncies reveal that THz electromagnetic pulse excites GHz\noscillations of the magneto-optical Faraday effect in\nthulium iron garnet, which probes the out-of-plane pro-\njection of the magnetization dynamics Mz(t). The exci-\ntation is π-periodic with respect to the THz pump polar-\nizationαand reaches a maximal efficiency for α=±45◦.\nThe dependency of the amplitude of the oscillations5\nFIG. 4. (a) THz-induced transients for several strengths of the applied magnetic field, measured at T= 6 K and α=−45◦.\nPart (b) shows the extracted frequencies, where the width of the error bars equals the FWHM of the fitted Gaussian in the\nFFT spectrum. The slope of the linear part of the curve provid es a reasonable estimate ˆ gefffor the effective g-factorgeff. (c)\nExtracted frequency as a function of temperature for a fixed m agnetic field of 130 mT, where the bars again depict the FWHM.\n(d) The frequency measured at three different external magne tic fields, for various temperatures, where the blurred area s depict\nthe 95% confidence bands of the fitted line. (e) The resulting e stimations of the effective g-factor based on the fitted slope in\n(d), the bars denote the standard error.\nclearly reveals that the mechanism of the excitation is\nnonlinear with respect to the strength of the THz field.\nThe dependency of the frequency of the oscillations on\nthe applied external magnetic field implies that the oscil-\nlations must be assigned to a spin resonance in the com-\npound. The unusually large effective g-factor deduced\nfrom the measurements and the typical GHz frequency\nsuggests that the oscillationsare associated with the low-\nfrequency FMR mode in the system of two macro spins\nformed by the magnetizations of iron MFeand thulium\nMTmsublattices. To further support this interpretation,\n/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48 /s52/s48/s48 /s52/s53/s48/s48/s50/s48/s52/s48/s54/s48/s70/s97/s114/s97/s100/s97/s121/s32/s114/s111/s116/s97/s116/s105/s111/s110/s32/s40/s109/s100/s101/s103/s41\n/s84/s105/s109 /s101 /s32/s40/s112/s115/s41/s97 /s32/s61/s32/s52/s53/s111\n/s97 /s32/s61/s32/s48/s111\n/s97 /s32/s61/s32/s45/s49/s53/s111\n/s97 /s32/s61/s32/s45/s51/s48/s111\n/s97 /s32/s61/s32/s45/s52/s53/s111\n/s97 /s32/s61/s32/s45/s54/s48/s111\n/s97 /s32/s61/s32/s45/s57/s48/s111/s43/s32/s49/s51/s48/s32/s109 /s84 /s45/s32/s49/s51/s48/s32/s109 /s84\nFIG. 5. THz-induced dynamics measured at T= 6 K, for\nseveralαand two opposite field polarities Bext=±130 mT.\nFor the angles αwhere the excitation is strong, the phase of\nthe mode remains unaltered under field reversal. However,\nit appears that the dynamics measured around the minimum\nα= 0◦is affected by the polarity of the field.we propose a theoretical model described in the next sec-\ntion.\nIV. THEORY\nThissectionisstructuredasfollows: insubsection IVA\nwe derived an expression for the FMR frequency. In the\nderivation, we employed an effective ferrimagnetic La-\ngrangian that was simplified to describe dynamics corre-\nsponding to the FMR mode only. A complete analysis\nwith the total effective Lagrangian is presented in Ap-\npendixA. From the derivations, it becomes clear that\nthe spin dynamics can be triggered if THz light acts as\nan effectively rectified magnetic field. Hence, in sub-\nsectionIVB, we phenomenologically describe the rec-\ntification in terms of the inverse Cotton-Mouton effect.\nWe derived the form of the effective rectified magnetic\nfield based on point-group symmetry. Finally, in subsec-\ntionIVC, we combine the Lagrangian with the ICME\ninteraction potential to obtain equations for the THz-\ninduced motion of the spins.\nA. Frequency of FMR\nLetθ,ϕbe the polar and azimuthal angles of\nthe net magnetization M=MFe+MTm≡\nm(sinθcosϕ,sinθsinϕ,cosθ), where m=MFe−MTm.\nThe total effective Lagrangian density describing magne-\ntization dynamicsofatwo-sublatticeferrimagnetis given\nin Appendix A. The equations of motion derived from6\nthis Lagrangian possess two eigenmodes, corresponding\nto the THz exchangemode and the GHz FMR mode [ 13].\nWhen considering only the FMR mode, the description\ncan be simplified using the fact that the involved mag-\nnetizations remain antiparallel during the dynamics [ 30],\nwhile for the exchange mode, the sublattices become mu-\ntually canted. Thewaytoimpose onthe Lagrangianthat\nwe only want to consider the FMR mode is, therefore, to\nenforce the sublattices to remain antiparallel. This can\nbe done by letting the antiferromagnetic exchange cou-\npling parameter λ, which defines the exchange energy\nUex=−λMFe·MTm, go to infinity λ→ −∞. Thisfer-\nromagnetic approximation does not influence the FMR\nmode and its frequency (see Appendix A). In this case,\nthe effective Lagrangian reduces to the low-frequency ef-\nfective Lagrangian density LFM:\nLFM=−m\nγeff˙ϕcosθ−U(θ,ϕ). (2)\nHere,γeffis the effective gyromagnetic ratio γeff≡\ngeffµB//planckover2pi1andU(θ,ϕ) is the static potential energy den-\nsity, which contains uniaxial anisotropy and Zeeman in-\nteraction with the external magnetic field:\nU(θ,ϕ) =−Ku(M·ˆz)2\nm2−M·Hext,(3)\nwhereKu>0 is the uniaxial anisotropy constant and\nHext= (Hx,0,Hz) the external magnetic field (in Tesla).\nSimilar to the experiment, we let the external magnetic\nfield be slightly tilted away from the sample plane, i.e.\nHz/Hx= tanδ(see Fig. 1). The ground state angles θ0,\nϕ0can be found by minimization of ( 3). Afterwards, the\nequationsofmotionthat aregivenbythe Euler-Lagrange\nequations can be linearized around the ground-state an-\nglesθ=θ0+θl,ϕ=ϕ0+ϕlwithθl,ϕl≪1:\n0 =d\ndt∂L\n∂˙θ−∂L\n∂θ≈ −m\nγeff˙ϕlsinθ0+U′′\nθ(θ0,ϕ0)θl,\n0 =d\ndt∂L\n∂˙ϕ−∂L\n∂ϕ≈m\nγeff˙θlsinθ0+U′′\nϕ(θ0,ϕ0)ϕl.(4)\nwhere we introduced the notation U′′\nθ(θ0,ϕ0)≡\n∂2\n∂θ2U(θ,ϕ)/vextendsingle/vextendsingle\nθ=θ0,ϕ=ϕ0. The FMR frequency is given by\nthe eigenfrequency of these coupled equations, which we\nseparately derived for a purely in-plane field and tilted\nfield configuration.\n1. In-plane field ( δ= 0◦)\nWhen ignoring the tilting of the external magnetic\nfield, the static energy potential becomes:\nU(θ,ϕ) =−Kucos2θ−mHxsinθcosϕ.(5)\nMinimization w.r.t. ϕyieldsϕ0= 0 for Hx>0 and\nϕ0=πforHx<0. For the minimization w.r.t. θ, weneed to consider two regimes: when the applied external\nmagnetic field is greater or smaller than the anisotropy\nfieldHa≡2Ku/m:\nθ0=/braceleftigg\nsin−1Hx\nHafor|Hx|< Ha,\nπ/2 for |Hx| ≥Ha.(6)\nUsing these ground-state angles, the eigenfrequency of\nthe equations of motion ( 4) can be found:\nωFM=/braceleftigg\nγeff/radicalbig\nH2a−H2xfor|Hx|< Ha,\nγeff/radicalbig\nHx(Hx−Ha) for|Hx| ≥Ha.(7)\nThe solution for this case ( δ= 0◦) is plotted in Fig. 6,\nwhich shows that the frequency drops to zero when\n|Hext| →Ha, and afterward approaches a linear trend.\nWe do observe the linear increase of the frequency in the\nexperiment, but we do not see such a significant drop at\nlow fields. Therefore, we need to include the small tilt of\nthe external magnetic field δ∝negationslash= 0◦.\n2. Tilted field ( δ/negationslash= 0◦)\nIn the case of a tilted magnetic field, the static poten-\ntial energy is given by:\nU(θ,ϕ) =−Kucos2θ−mHxsinθcosϕ−mHzcosθ.(8)\nAgain we have that ϕ0= 0 for Hx>0 andϕ0=π\nforHx<0, which means Hxcosϕ=|Hx|and we can\nminimize ( 8) only with respect to θ:\n1\nmU′\nθ(θ) = cosθ(Hasinθ−|Hx|)+Hzsinθ= 0.(9)\nTo be able to solve this equation, we treat the out-of-\nplane field as a perturbation with respect to the case of\n/s48 /s49 /s50 /s51 /s52/s48/s49/s50/s51/s52/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s103\n/s101/s102/s102/s72\n/s97/s41\n/s72\n/s101/s120/s116/s32/s47/s32/s72\n/s97/s32/s32/s32/s100/s32/s61/s32 /s48/s111\n/s32\n/s32/s100/s32/s61/s32 /s51/s111\n/s32/s100/s32/s61/s32 /s54/s111\n/s100\n/s100\n/s101/s120/s116/s32 \n/s101/s102/s102/s32 \nFIG. 6. Theoretical curves for the FMR frequency as a func-\ntion of the external magnetic field, for different tilting ang les\nδ.7\nan in-plane field. Therefore, we substitute θ=θ0−ǫ\nin Eq. (9) withθ0as in the in-plane case (Eq. ( 6)) and\nassumed ǫ≪1. We retained maximally cubic terms ∼\nǫ3and solved the third-degree polynomial for ǫ(Hx,Hz).The solutions gives the new ground-state angle θ′\n0=θ0−\nǫ(Hx,Hz). From here, we again linearized the Euler-\nLagrangeequations( 4) andcalculatedexpressionsforthe\nFMR resonance mode as presented in Eq. ( 10).\nωFM=\n\nγeff/radicalbigg\nH2a−H2x+/parenleftBig7\n2H2x−2H2a/parenrightBig\nǫ2+/parenleftBig\n3Hx(H2a−H2x)1\n2+HzHx/parenrightBig\nǫ+Hz(H2a−H2x)1\n2\n/radicalbigg\n1−ǫ(H2a/H2x−1)1\n2−1\n2ǫ2for|Hx|< Ha,\nγeff\n(1−1\n2ǫ2)1\n2/radicalig\nHx/bracketleftbig\n(Hx−Ha)+Hzǫ+(2Ha−1\n2Hx)ǫ2/bracketrightbig\nfor|Hx| ≥Ha.(10)\nWe solved the third-degree polynomial of ǫnumerically\nfor different external magnetic fields and tilt angles and\ncalculated the corresponding resonance frequencies. The\nresultisplotted in Fig. 6, whichshowsthat wecanobtain\na better agreement to our data of Fig. 4when account-\ning for the tilted field. We also see that the minimal\nfrequency occurs when the external magnetic field equals\ntheuniaxialanisotropyfield. Lookingbackatexperimen-\ntal data from Fig. 4(b), we see that the dip occurs at ap-\nproximately 25 mT which therefore allows us to estimate\nthe size of the uniaxial anisotropy field Ha≈25 mT.\nHowever, when inserting this value into Eq. 10, our the-\nory predicts ωFM=γeffHaat zero external field which\namounts to a frequency of (2 π)−1ωFM≈2.24 GHz, while\nexperimentally we observe ∼14 GHz (see Fig. 4(b)).\nThis unaccounted frequency offset of about 12 GHz sug-\ngests that we should include some effective biasing field\nwithamagnitudeofabout130mT.Atthispoint,wehave\nno definite answer to where this field stems from. Shape\nanisotropy should play a role given the clear domain pat-\nterns seen in this sample without an external magnetic\nfield [12]. Alternatively, the so-called “double umbrella\nstructure” - a noncollinear arrangement of the thulium\nions seen at helium temperatures in TmIG using neutron\ndiffraction [ 33,34] - might be involved. Also, strain in\nthe sample induced by the Gd 3Fe5O12substrate could be\nthe sourceofthe missingfield [ 35]. Fortunately, thisopen\nquestion does not obstruct the theoretical treatment of\nthe rectified effective magnetic field by the ICME which\nwill be held in the coming section.\nB. Energy considerations for the ICME\nInthissection, wederiveaneffectiveinteractionpoten-\ntialstartingfromthemostbasicprinciplesoflight-matter\ninteraction. This interaction potential will then enter the\nLagrangianin the next section. When an oscillating elec-\ntric field of light E(t) enters a non-absorbing medium,\nit interacts with the medium by inducing electric polar-\nization. The change in interaction energy density dW\ndue to the increase of electric polarization d Pis givenbydW=−E·dP. Here we ignored higher-order multi-\npole contributions as well as magnetic dipole interactions\nbecause they are expected to be weak. In the linear op-\ntical approximation, the amount of induced polarization\nis linear to the applied electric field Pi=χijǫ0Ej, with\nǫ0the dielectric permittivity of vacuum and χijthe elec-\ntric susceptibility tensor. Therefore, the total interaction\npotential energy density W, after integration, becomes:\nW=−1\n2χijǫ0EiEj. (11)\nThe electric susceptibility tensor is related to the dielec-\ntric permittivity ǫij= (1+χij)ǫ0. The presence of static\nmagnetization induces magneto-optical birefringence in\nthe medium, and modifies this dielectric permittivity\n[36]:\nδǫij=kijkMk+ξijklMkMl+···,(12)\nwhere the third-rank antisymmetric axial tensor kijkand\nfourth-rank symmetric polar gijkltensor describe the\nwell-known Faraday and Cotton-Mouton effects , respec-\ntively. The relevant interaction potential that describes\nthe interaction of light and magnetization can then be\nfound by substituting ǫ0(δχij) = (δǫij) from Eq. ( 12) in\n(14) to obtain [ 10]:\nW=−1\n2kijkEiEjMk−1\n2ξijklEiEjMkMl.(13)\nOn one hand, magnetization may therefore induce bire-\nfringence in the medium and modify the properties of\nlight via the well-known Faraday and Cotton-Mouton ef-\nfects. On the other hand, inversely, the presence of light\nmay induce magnetization. That is, by thermodynamics,\nthe electric field oflight acts asan effective magnetic field\nHeff=−δW\nδM[37]. The appearance of an effective field\noriginating from the two terms in Eq. ( 13) are known as\nthe inverse Faraday effect (IFE) and ICME, respectively.\nTo obtain an expression for the effective fields, we\nmake a Fourier expansion of the wave and assume it\nis monochromatic (the extension to non-monochromatic\nlight is straightforward) E(t) = Re[E(ω)exp(−iωt)] =8\n1\n2[E(ω)exp(−iωt)+E∗(ω)exp(iωt)], where E(ω) is the\n“Jones vector”. The Fourier components of the polariza-\ntionareagainrelatedtothoseofthe electricfield Pi(ω) =\nǫ0˜χij(ω)Ej(ω) where ˜χij(ω) is the optical susceptibility.\nBecause we consider a non-absorbingmedium, the tensor\n˜χij(ω) is required to be Hermitian ˜ χ∗\nij(ω) = ˜χji(ω) [38].\nUsing this fact, the interaction potential ( 14) becomes\n[39]:\nW=−1\n4ǫ0˜χijE∗\niEj+h.f., (14)\nwhere we neglect the high-frequency ( h.f.) terms as they\naverageout on the relevant timescales [ 38,40]. Similarly,\nEq. (13) becomes:\nW=−1\n4kijkE∗\niEjMk−1\n4ξijklE∗\niEjMkMl(15)\nBy Neumann’s principle [ 41], the tensors kijkandξijkl\nshould be invariant under the crystallographic point\ngroup operations. In an isotropic or cubic medium, kijk\nis an antisymmetric imaginary tensor with only a sin-\ngle nonzero tensor component kxyz=kzxy=kyzx=\n−kxzy=−kyxz=−kzyx=−ik[40,42,43], and the\ncorresponding effective field is:\nHIFE=ik\n4E(ω)×E∗(ω). (16)\nTherefore, a circularly polarized light pulse generates a\nrectified effective magnetic field with opposite directions\nforleft/right-handedpolarizedlight[ 40,42,44,45]. How-\never, for linearly polarized light, the effective field is zero.\nGiven that ourTHz pulses are linearlypolarized, we only\nconsider the inverse Cotton-Mouton field [ 15]:\n(HICME)l=ξijkl\n2E∗\ni(ω)Ej(ω)Mk.(17)\nUsing the crystallographic point group Oh, we can find\ntheminimalexpressionforthetensor ˆξ. Thenonzeroten-\nsor components in the standard cubic coordinate system\nx∝bardbl[100],y∝bardbl[010],z∝bardbl[001] are tabulated in Ref. [ 43],\nwhere it can be seen that the tensor has only two inde-\npendent components ξxxxxandξxxyy. We transformed\nthe tensor ξijklto our experimental coordinate system\n(see Fig. 1), where z∝bardbl[111],x∝bardbl[112] andy∝bardbl[110]. In\nthis coordinate system, the tensor can be expressed in\nVoight notation by a 6 ×6 matrix ˜ξ:\n˜ξ=\n3ξ1ξ1ξ1+ξ20−√\n2ξ20\nξ13ξ1ξ1+ξ20√\n2ξ20\nξ1+ξ2ξ1+ξ23ξ1−ξ20 0 0\n0 0 0 ξ1+ξ20√\n2ξ2\n−√\n2ξ2√\n2ξ20 0 ξ1+ξ20\n0 0 0√\n2ξ20ξ1\n\nwhereξ1≡ξxxxx\n6+ξxxyy\n2andξ2=ξxxxx\n6−ξxxyy\n2. Thisgives the ICME interaction energy from Eq. ( 13):\nWICME=−ξ1\n2/parenleftbig\n2(ExMx+EyMy)2+E2\nTHzm2/parenrightbig\n−ξ2\n2/parenleftig\nE2\nTHzM2\nz+2√\n2MxMz(E2\ny−E2\nx)\n+4√\n2ExEyMyMz/parenrightig\n.(18)\nwhere we used that m=/summationtext\ni/radicalbig\nM2\niconstant. Note that\nthe latter acts as a holonomic constraint on the system,\nmaking the expression for the ICME field as in Eq. ( 17)\na bit naive because it presumes no constraints on the\nvariables Mi. Although such a constraint complicates a\nNewtonian approach to describe the influence of HICME\nonM, in the Lagrangian approach treated in the next\nsubsection the problem is naturally circumvented.\nThe Newtonian approach is anyhow insightful and we\nbriefly treat it here before going back to the Lagrangian\napproach. We can obtain the approximate ICME field\nby assuming that |Hext| ≥Hawhile ignoring the tilt\nMz≪Mxsuch that M=Mxˆx. In that case, Mxcan\nbe regarded as constant (only changing sign for ±Hext),\nwhileMy,zare two independent variables. Furthermore,\nthe THz electric field ETHz(t) lies in the xyplane, and\ntherefore ETHz(t) =ETHz(t)(sinα,cosα,0). Then, an\napproximate expression for the inverse Cotton-Mouton\nfield derived from Eq. ( 13) is given by:\nHICME=E2\nTHzMx\n0\nξ1sin2α√\n2ξ2cos2α\n.(19)\nGiven that the THz pulse duration of about tTHz∼1 ps\nismuchshorterthanthatoftheperiodoftheFMRmode,\nwe could treat the effect of this field on magnetization M\nas an instantaneous (impulsive) torque τ=M×HICME,\nwhich triggers dynamics that can be described using the\nLandau-Lifshitz equationdM\ndt=−γM×H, with the ini-\ntial condition M(t= 0) = (Mx,0,0):\n˙M(t= 0) =γM2\nx\n0√\n2ξ2cos2α\n−ξ1sin2α.\n/integraldisplaytTHz\n0/bracketleftbiggd\ndtE2\nTHz/bracketrightbigg\ndt,\n(20)\nwhereETHz(t) is the THz pulse electric field. It can be\neasily seen that the torque fulfills the quadratic depen-\ndence with respect to the THz field amplitude ∼E2\nTHz.\nMoreover, in order to have maximal torque at α=±45◦,\nwe find out that ξ1must be dominant over ξ2, i.e.ξ1≪\nξ2. At the same time, it can be shown that the latter is\nalso required for the invariance of the phase of the ob-\nserveddynamicsunderfield reversalasweshowedexperi-\nmentally in Fig. 5. This can be derived fromthe fact that\nwe are only sensitive to the z-projection of the magneti-\nzation, while the precession of the magnetization around\nthe external magnetic field is always right-handed. If the\ntermξ2would have been dominant, magnetization pre-\ncession would have been launched with a z−projection9\nthat has an exactly opposite phase for ±Hext, unlike the\nexperiment. This invarianceofthe phaseon fieldpolarity\nwhenξ1≪ξ2can be seen directly in the coming section\nafter we solved the Lagrangian equations of motion.\nC. Lagrangian equations of motion driven by the\nICME\nTo include light-matter interaction in the Lagrangian,\nwe introduce the interaction energy f(θ,ϕ):\nLFM=−m\nγeffcosθ˙ϕ−U(θ,ϕ)+f(θ,ϕ),(21)\nwhich contains Zeeman interaction with a dynamic THz\nmagnetic field h(t) as well as the ICME of a THz electric\nfieldE(t) that is derived from the interaction potential\nin Eq. (18):\nf(θ,ϕ) =msinθ(hxcosϕ+hysinϕ)\n+m2ξ1sin2θ/bracketleftbig\nE2\nxcos2ϕ+2ExEycosϕsinϕ+E2\nysin2ϕ/bracketrightbig\n+m2ξ2\n2/bracketleftig\nE2cos2θ+2√\n2sinθcosθcosϕ/parenleftbig\nE2\ny−E2\nx/parenrightbig\n+4√\n2ExEysinθcosθsinϕ/bracketrightig\n. (22)\nNote that for the full Lagrangian (see Appendix A), the\ndrivingenergycontainsseveralothertermsthatareperti-\nnent to ferrimagnets (and absent for ferromagnets). The\nequations of motion derived from the Euler-Lagrange\nequations, after linearization, are given by:\nU′′\nθ(θ0,ϕ0)θl−sinθ0m\nγeff˙ϕl=fθ(t),\nU′′\nϕ(θ0,ϕ0)ϕl+sinθ0m\nγeff˙θl=fϕ(t),(23)\nwhere the generalized force terms fνare defined for ν=\nθ,ϕand are evaluated around the ground-state angles:\nfν(t) =−/bracketleftbiggd\ndt/parenleftbigg∂f\n∂˙ν/parenrightbigg\n−∂f\n∂ν/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nν=ν0=∂f\n∂ν/vextendsingle/vextendsingle/vextendsingle\nν=ν0,(24)\nThese generalized forces can be interpreted as torque.\nWhen ignoring the small canting angle δ, we obtain the\nfollowing driving terms evaluated for the case |Hx| ≥Ha:\nfθ(t) =−√\n2m2ξ2E2\nTHzcosϕ0cos2α,\nfϕ(t) =mhycosϕ0+m2ξ1E2\nTHzcos2ϕ0sin2α.(25)\nThe driving term fθ(t) proportional to ξ2is maximal\nwhenα= 0◦,90◦, in contrast with the experiment.\nTherefore, we again conclude that ξ2≪ξ1and essen-\ntially ignore this term. Next, as expected, it can be seen\ninfϕ(t) that the THz magnetic field component hy(t),\nwhich is perpendicular to the equilibrium magnetization,\ncould drive the precession. However, this term scales\nlinearly with the THz amplitude, which is also not thedominant mechanism in the experiment. Theoretically,\nthis term is also negligible as we will treat the excita-\ntion as instantaneous in which such linear driving terms\ndisappear because/integraltexttTHz\n0h(t)dt= 0. Therefore, we only\nconsider the ICME term in fϕthat is proportional to ξ1.\nExperimentally, we are only sensitive to the z-\ncomponent of the magnetization Mz(t) =mcosθ(t)≈\n−mθl(t)(for|Hx| ≥Hainwhichcase θ0=π/2). Inother\nwords, we only detect dynamics of θl(t). The equation\nof motion for θl(t), found by differentiation and mutual\nsubstitution of Eqns. ( 23), is given by:\n¨θl+ω2\nFMθl=mγeffξ1/bracketleftbiggd\ndtE2\nTHz/bracketrightbigg\ncos2ϕ0sin2α.(26)\nWe treat the excitation as instantaneous (see “photonic\nimpact” [ 46]), in which case the excited dynamics is fully\ndetermined by the initial condition θl(0) = 0 and:\n˙θl(0) =mγeffξ1cos2ϕ0sin2α/integraldisplaytTHz\n0/bracketleftbiggd\ndtE2\nTHz/bracketrightbigg\ndt.(27)\nAsϕ0= 0 for Hx>0 andϕ0=πforHx<0, we\nhave that cos2ϕ0= 1 in both cases such that a change in\nfield polarity does not change the phase of the dynamical\nout-of-plane magnetization ˙Mz(0)≈ −m˙θl(0), just as we\nobserved in the experiment (see Fig. 5). Moreover, the\nexcitation is maximal and has a mutually opposite sign\nforα=±45◦, and scales quadratically with the THz\nfield. Therefore, all experimentally observed features re-\ngarding the excitation are captured by this Lagrangian\napproach.\nV. 2D SPECTROSCOPY AND DISCUSSION\nThe predictions made by the phenomenological treat-\nmentoftheICMEfittheexperimentaldataverywell, but\nit does not yet clarify the underlying microscopic mecha-\nnism. The effect is conventionally seen as a consequence\nof impulsive stimulated Raman scattering (ISRS) involv-\ning either resonant or off-resonant excitation of a certain\nelectronic transition [ 17,42,47–51]. It is thus interesting\nif we are able to assign a certain transition that medi-\nates the ISRS process. Here, it is important to take into\naccount the possibility of the recently discovered mecha-\nnism of magnonic Raman scattering [ 52], where the GHz\nq-FM resonance in FeBO 3was excited through resonant\nRaman scattering of the q-AF THz resonance. A notable\nsimilarity between FeBO 3and the current experiment is\nthat the maxima of excitation both occur when the THz\npolarization is α=±45◦from the net magnetization.\nBut at the same time, the materials are largely differ-\nent since FeBO 3is an antiferromagnet with N´ eel vector\nL⊥M, while collinear ferrimagnets such as TmIG have\nL∝bardblM. In any case, if the exchange mode mediates\nthe excitation of the FMR mode through magnonic Ra-\nman scattering, it should become visible in 2D THz spec-\ntroscopy [ 52]. In this technique, instead of only one THz10\npulse, two THz pulses are applied at a mutual delay τ.\nThis allows us to track the Faraday rotation signal of the\nprobe pulse when both THz pulses arepresent θF,12(t,τ),\nbutalsothesignals θF,1(t,τ) andθF,2(t,τ)obtainedwhen\nonly oneofthe twopulses excited the sample. From here,\nwe obtained the nonlinear signal:\nθF,NL(t,τ)≡θF,12(t,τ)−θF,1(t,τ)−θF,2(t,τ).(28)\nIf the exchange mode mediates the excitation of the\nFMR mode, this should become apparent as periodicity\nin the excitation efficiency of the FMR mode with a pe-\nriod defined by the frequency of the exchange mode. We\nperformed this experiment and the result of θF,12(t,τ) is\nshown in Fig. 7. The measured signal does not show any\nperiodicity of the mode, and neither did we observea sig-\nnal in the nonlinear part calculated with Eq. ( 28). The\nonly logical explanation for the absence of a nonlinear\nsignal in 2D spectroscopy is that the nonlinear excita-\ntion is mediated by a state with a short lifetime <1 ps.\nIn this case, no information will be carried over between\ntwosubsequentTHzpulseswhichareseparatedbyatime\nlonger than their own pulse duration of ∼1 ps. In other\nwords, the second pulse delayed at a time τ >1 ps will\nnever be able to experience the presence of the first THz\npulse, meaningthatsuperpositionholdsandnononlinear\nsignal (see Eq. 28) will come up. Therefore, the THz ex-\nchange mode does not mediate the ISRS mechanism and\nis not responsible for the excitation of the FMR mode.\nFigure8illustrates the mechanism of ISRS via a\nshort-lived excited state. The process could occur com-\npletely off-resonantly via electronic or phononic transi-\ntions which lie beyond the THz excitation spectrum, in\nFIG. 7. Result of θF,12(t,τ) measured in 2D THz spec-\ntroscopy. Fast modulations shortly after the pulse arrival as-\nsociated with the exchange mode can be observed, as well as\nfaint contours which correspond to the FMR mode at longer\ntimescales. However, no modulations in the excitation effi-\nciency as a function of τof the latter mode were found.\nFIG. 8. Schematic illustration of ISRSfrom a short-lived el ec-\ntronic excited state, which is believed to be the microscopi c\nmechanism for the ICME.\nwhich case the excited state is virtual. However, a reso-\nnant excitation via a state that lies within the THz pulse\nspectrum is expected to be more likely. The only candi-\ndates for such an excited state are the crystal-field split\ntransitions of the Tm3+ground state. The dodecahe-\ndral crystal-field environment of the Tm3+(4f12) ions,\nwith local symmetry described by the dihedral point-\ngroup D 2, splits its ground-state multiplet3H6(J= 6)\ninto 2J+ 1 = 13 Stark levels. The energy level dia-\ngram for substituted Tm3+ions in dodecahedral D 2sites\nin Y3Al5O12was earlier obtained from experimental ab-\nsorption and emission spectra in [ 53,54] and more re-\ncently it was studied in [ 55]. The lowest electric-dipole-\nallowed transition has an energy of 27 cm-1correspond-\ning to about 0 .809 THz. This energy is in the vicinity of\na spectral feature we actually observed in this material\n[13], but which we couldn’t assign. Such a crystal-field\nsplit transition can be very short-lived and effectively act\nas an effective virtual electronic state from which light\ncan scatter. Moreover, since there are no other transi-\ntions in our spectrum, we conjecture by deduction that\nISRS of the lowest crystal-field split state of thulium is\nthe responsible microscopic mechanism of excitation (see\nFig.8for a schematic illustration of ISRS). In order to\nsubstantiate this hypothesis, one must develop a micro-\nscopic theory that takes into account both the Tm-Fe ex-\nchange interaction and the electronic structure of Tm3+.\nThis problem is beyond the scope of this article.\nVI. CONCLUSION\nWe showed that a single-cycle THz pulse is able to ex-\ncite a GHz magnon in ferrimagnetic TmIG. The experi-\nmental dependencies reveal that the excitation is a result\nof ICME, where the THz field of light becomes effectively\nrectified to generate a unipolar magnetic field pulse. The\nresultsaresupportedbytheequationsofmotionobtained\nfrom an effective ferrimagnetic Lagrangian, using a phe-\nnomenological expression for the rectified field. We dis-\ncussed the possible microscopic picture of ICME by con-11\nsidering the mechanism of ISRS. 2D spectroscopy ruled\nout magnon-magnonscattering, similar to what occurred\nin FeBO 3[52], to be responsible. Instead, we conjectured\nthat the effect is more similar to that in TmFeO 3[10],\nand is based on light-induced scattering from the crystal-\nfield split electronic states of Tm3+.\nOur results demonstrate that nonlinear THz optomag-\nneticeffectsdononotonlyplayaroleinantiferromagnets\nbut also in ferrimagnetic materials. In general, this non-\nlinearity facilitates a channel of energy transfer from the\nelectric field of light to the magnetic spin system. Such a\nchannel recently enabled coherent steering of spins overa\npotentialbarrierinantiferromagneticTmFeO 3[11]. Sim-\nilarly, our results, therefore, open a way for future data-\nwriting, spintronics, and magnonics applications basedon ferrimagnets.\nACKNOWLEDGMENTS\nThe authors thank Sergey Semin and Chris Berkhout\nfor their technical support. The work was supported by\nthe Dutch Research Council (NWO). The authors de-\nclare that this work has been published as a result of\npeer-to-peer scientific collaboration between researchers.\nThe provided affiliations represent the actual addresses\nof the authors in agreement with their digital identifier\n(ORCID) and cannot be considered as a formal collabo-\nration between the aforementioned institutions.\nAppendix A: Total Effective Lagrangian\nThe total effective Lagrangian density of a two-sublattice ferrima gnet in the vicinity of the compensation tem-\nperature can be written in terms of the polar and azimuthal angles o f the net magnetization M=MFe+MTm≡\nm(sinθcosϕ,sinθsinϕ,cosθ) [13,16]:\nLeff=χ⊥\n2\n/parenleftigg˙θ\nγ+Hxsinϕ/parenrightigg2\n+/parenleftbigg/parenleftbigg˙ϕ\nγ−Hz/parenrightbigg\nsinθ+Hxcosθcosϕ/parenrightbigg2\n−m˙ϕ\nγeffcosθ−U(θ,ϕ)+¯f(θ,ϕ),(A1)\nwhereχ⊥≡1/|λ|withλ <0 the exchange coupling parameter that defines the exchange ene rgy density Uex=\n−λMFe·MTm, andγ≡(MFe+MTm)/(MFe/γFe+MTm/γTm). The static potential Ucontains Zeeman interaction\nwith the external magnetic field and magnetic anisotropy as defined in Eq. (3). The interaction energy density ¯f(θ,ϕ)\nis equal to:\n¯f(θ,ϕ) =χ⊥\n2/bracketleftigg\ncos2θ(hxcosϕ+hysinϕ)2+2/parenleftbigg/parenleftbigg˙ϕ\nγ−Hz/parenrightbigg\nsinθ+Hxcosθcosϕ/parenrightbigg\ncosθ(hxcosϕ+hysinϕ)\n+2/parenleftigg˙θ\nγ+Hxsinϕ/parenrightigg\n(hxsinϕ−hycosϕ)+(hxsinϕ−hycosϕ)2/bracketrightigg\n+msinθ(hxcosϕ+hysinϕ)\n+m2ξ1sin2θ/bracketleftbig\nE2\nxcos2ϕ+2ExEycosϕsinϕ+E2\nysin2ϕ/bracketrightbig\n+m2ξ2\n2/bracketleftig\nE2cos2θ+2√\n2sinθcosθcosϕ/parenleftbig\nE2\ny−E2\nx/parenrightbig\n+4√\n2ExEysinθcosθsinϕ/bracketrightig(A2)\nDamping can be included through the Rayleigh function R=αM\n2γ/parenleftig\n˙θ+sin2θ˙ϕ/parenrightig\n, whereM ≡MFe+MTm. The\ngeneral equations of motion are defined by the Euler-Lagrange eq uations:\nd\ndt∂Leff\n∂˙θ−∂Leff\n∂θ+∂R\n∂˙θ= 0,d\ndt∂Leff\n∂˙ϕ−∂Leff\n∂ϕ+∂R\n∂˙ϕ= 0. (A3)\nThe result can be linearized around the ground-state angles θ=θ0+θl,ϕ=ϕ0+ϕl,θl,ϕl≪1 (where θ0,ϕ0are\nas in Sec IVA). This gives the linearized equations of motion:\n¨θl+ζ˙θl+γ2\nχ⊥U′′\nθ(θ0)θl−sinθ0γ2\nγeff|λ|m˙ϕl=γ2\nχ⊥¯fθ(t),¨ϕl+ζ˙ϕl+γ2U′′\nϕ(θ0,ϕ0)\nχ⊥sinθ0ϕl+γ2\nγeff|λ|m˙θl=γ2\nχ⊥¯fϕ(t)\nsinθ0,\n(A4)\nwhereζ=α¯γM\nχ⊥and the driving terms can be calculated from the interaction energy ¯ffrom (A2) using equation ( 24).\nThese equations have two eigenfrequencies, one corresponding t o the exchange mode ωex≈ |λ|(γTmMFe−γFeMTm),12\nand one corresponding to the FMR mode whose frequency is given by Eq. (10). The driving terms after linearization,\ntaking only the leading contributions within the first order of ǫinto account, are given by:\n¯fθ(t) =χ⊥γ−1˙hycosϕ0+√\n2m2ξ2cosϕ0/parenleftbig\nE2\nx−E2\ny/parenrightbig\n+ǫ/parenleftbig\nmhxcosϕ0+2m2ξ1E2\nxcos2ϕ0−m2ξ2E(t)2/parenrightbig\n¯fϕ(t) =/parenleftbig\nmhycosϕ0+2m2ξ1ExEycos2ϕ0/parenrightbig\n+ǫ/parenleftig\n−χ⊥γ−1˙hxcosϕ0+2√\n2m2ξ2ExEycosϕ0/parenrightig(A5)\nwhere we also ignored the terms −ǫχ⊥cos2ϕ0(h2\nx+2Hxhx) in¯fθ(t) and−ǫχ⊥Hzhycosϕ0in¯fϕ(t) because they are\nsmall. The expressions ( A5) display a weak field-derivative Zeeman torque ∝ǫ˙hx(whereǫ≪1) when the THz\nmagnetic field is along the x-axis (˙hy= 0,˙hx∝negationslash= 0), being even zero when there is no tilt of the field ǫ= 0. On the\nother hand, there is a strong field-derivative torque ∝˙hywhen the THz magnetic field is aligned along the y-axis,\nwhich is perpendicular to the equilibrium spin-direction and thus M. 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Takanashi, Observation of inverse spin Hall effect\nin ferromagnetic FePt alloys using spin Seebeck effect,\nApplied Physics Letters 107, 092401 (2015) .arXiv:2305.02971v1 [cond-mat.mtrl-sci] 4 May 2023Supplemental Material for “Effective rectification of THz\nelectromagnetic fields in a ferrimagnetic iron garnet”\nT.G.H. Blank,1,2E.A. Mashkovich,3K.A. Grishunin,1C. Schippers,2\nM.V. Logunov,4B. Koopmans,2A.K. Zvezdin,5and A.V. Kimel1\n1Radboud University, Institute for Molecules and Materials , 6525 AJ Nijmegen, the Netherlands.\n2Department of Applied Physics, Eindhoven University of Tec hnology,\nP.O. Box 513, Eindhoven 5600 MB, the Netherlands.\n3University of Cologne, Institute of Physics II, Cologne D-5 0937, Germany.\n4Kotel’nikov Institute of Radioengineering and Electronic s, 125009 Moscow, Russia.\n5Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991 Moscow, Russia.\n(Dated: May 5, 2023)\nXRD ANALYSIS\nWe performed X-ray diffraction (XRD) analysis on the sample to obta in the in-plane crystallo-\ngraphic orientation. Fig. S1(a) depicts the experimental setup. The scattering plane of the s ample\ncan be tilted at an angle ψ, and can be rotated around its normal axis by an angle φ. Because cubic\ncrystals are self-dual, we can identify the Miller planes ( hkl) to be simply the planes perpendicular\nto the vector [ hkl].\nThe data for φ= 0◦, andψ= 0◦confirms the [111] orientation of the sample [ ?]. From the\nillustrations in Fig. S1(c), it can be seen that after tilting the sample at ψ= 54.74◦(angle between\n[111] and [100]), it should be possible to let the scattering plane coincid e with one of the base\nvector planes (100), (010) or (001) by rotating the sample over a certain angle φ. Therefore, we\nputψ= 54.74◦and set the radiation and detection angle at 2 θ= 58.86◦corresponding to a (800)\npeak to recognize a (100) plane. By scanning the rotation angle φ, we obtained three candidates\natφ= 0◦,33.3◦,66.6◦(see Fig. S1(d)). Due to the three-fold symmetric [111] axis, the peaks at\nφ+n·2π/3 are equivalent. The data in Fig. S1(e) establishes that only φ= 33.3◦(and equivalent\nangles) coincides with a (100) plane. For additional confirmation, we note that by tilting the sample\natψ= 19.47◦, the same rotation φ= 33.3◦should correspond to a (211) plane. This is confirmed\nin Fig.S1(f). Therefore, we conclude that the crystallographic axes in our experiment are oriented\nlike the dotted arrows in Fig. S1(g).\nNote that although the peaks at φ+n·2π/3 withn= 0,1,2 should be equivalent, it turns out\nthat the equivalent axes for φ= 33.3◦were rather φ= 150◦andφ= 270◦, meaning that the first\nrotation is slightly shifted. Similar results were obtained in Ref. [ ?] and were explained as a result\nof strain between the thulium iron garnet film and the Gd 3Fe5O12substrate.\n1Figure S1: (a) Schematic illustration of the angles involved in the XRD analysis of the sample, where x,\nyandzcoincide with the experimental axes from Fig. 1(a) of the mai n article. (b) XRD data for the\nsample plane, which confirms the [111]orientation of the sample [ ?]. (c) Illustration of the cubic crystal\naxes and the two-dimensional projection seen from the [111]direction. (d) Scattering intensity at\n2θ= 58.86◦((800)) for different sample rotations φ, where the sample tilt is set at ψ= 54.74◦which is\nequal to the angle between the [111]and[100]axes. This data suggests three possible candidates\nφ= 0◦,33.3◦,66.6◦which could coincide with a (100)plane. (e) Angular scattering intensity\nmeasurements for a tilting angle of ψ= 54.74◦and for the three different candidate angles φ. The data at\nφ= 33.3◦shows what is typical for a (100)plane. In (f), we perform similar measurements, but for a\ntilting angle ψ= 19.47◦, corresponding to the angle between the [111]and[112]axes. The result confirms\nthatφ= 33.3◦corresponds to a (211)plane. (g) The results allowed us to conclude that the project ions of\nthe cubic axes are rotated at an angle of 33.3◦compared to the drawing in (c), meaning that the\nexperimental xandyaxis correspond to the [112]and[110]axes, respectively.\n2" }, { "title": "1907.04540v1.Temperature_dependence_of_magnetic_resonance_in_ferrimagnetic_GdFeCo_alloys.pdf", "content": "1 \n Temperature dependence of magnetic resonance in ferrimagnetic \nGdFeCo alloys \nTakaya Okuno1, Se Kwon Kim2,3†, Takahiro Moriyama1, Duck-Ho Kim1, Hayato \nMizuno1,4, Tetsuya Ikebuchi1, Yuushou Hirata1, Hiroki Yoshikawa5, Arata Tsukamoto5, \nKab-Jin Kim6, Yoichi Shiota1, Kyung -Jin Lee7,8, Teruo Ono1,9† \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 611 -0011, Japan \n2Department of Physics and Astronomy, University of California Los Angeles, California \n90095, USA \n3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, \nUSA \n4Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581 , Japan \n5College of Science and Technology, Nihon University, Funabashi, Chiba 274 -8501, Japan \n6Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon \n34141, Republic of Korea \n7Department of Materials Science & Engineering, Korea University, Seoul 02841, \nRepublic of Korea \n8KU-KIST Graduate School of Converging Science and Technology, Korea University, \nSeoul 02841, Republic of Korea \n9Center for Spintronics Research Network (CSRN), Graduate School of Engineering \nScience, Osaka University, Osaka 560 -8531, Japan \n†E-mail: kimsek@missouri.edu , ono@scl.kyoto -u.ac.jp \n \n 2 \n We provide a macroscopic theory and experimental results for magnetic resonances of \nantiferromagnetically -coupled ferrimagnets. Our theory, which interpolates the dynami cs of \nantiferromag nets and ferromagnets smoothly, can describe ferrimagnetic resonance s across \nthe angular momentum compensation point . We also present experimental results for spin-\ntorque induced ferrimag netic resonance at several temperatures. The spectral analysis based \non our theory reveals that the Gilbert damping parameter, which has been considered to be \nstrongly temperature dependent, is insensitive to temperature. We env ision that our work \nwill facilitate further investigation of ferrimagnetic dynamics by providing a theoretical \nframework suitable for a broad range of temperatures . \n \n 3 \n Antiferromagnets have been gaining much attention in spintronics because of their \npotential utility for high -speed ultra -dense spintronic devices.1-4) Due to the antiparallel \nalignment of adjacent spins , their dynamics is different from that of ferromagnets.5) One \nemerging material platform for studying antiferromagnetic dynamics is \nantiferromagnetically -coupled ferrimagnets ,6-11) for which we can use conventional \ntechniques for ferromagnets owing to small but finite magnetizations . Indeed, r ecent \nexperiments in such ferrimagnets have found that both field -driven and current -driven \ndomain -wall dynamics are fastest at the angular momentum compensation point 𝑇A where \nthe magnetic dynamics are antiferromagnetic .12-15) However, the magnetic resonance \nphenomenon of ferrimagnets (FiMR) has not been fully clarified so far because of \ninsufficient experimental investigations. In the literature, Stanciu et al . have studied the \nlaser-induced precession and its decay to equilibrium in ferrimagnets and concluded that the \neffective Gilbert damping parameter 𝛼 , which governs the dissipation rate of angular \nmomentum, is strongly temperature dependent and increases signif icantly at 𝑇A.6) However, \nsome recent studies have provided a new perspective on 𝛼 of ferrimagnets by full y \nconsidering the antiferromagnetic dynamics in ferrimagnets . Kamra et al. have theoretically \nrevealed that the temperature dependence of FiMR occurs because of the temperature \ndependence of magnetic dynamics , not because of temperature dependence of 𝛼.16) Kim et \nal. have also reported the temperature -insensitive 𝛼 of ferrimagnets through the DW motion \nexperiment s.17) In this paper, we provide an additional evidence of temperature -insensitive \n𝛼 of ferrimagnets by performing the FiMR experiment analyzed by our macroscopic FiMR \ntheory. \nFirst, we derive the equations for FiMR in a ferrimagnet consisting of two \nantiferromagnetically -coupled sublattices. Throughout the manuscript, we will focus on the \nregime where the ferrimagnet is away from the magnetization compensation temperature 𝑇M \nso that th e magnetization is finite and well defined. Our experiments are also performed well \nwithin the considered regime as detailed below. To this end, we expand the L andau -Lifshitz -\nGilbert -like (LLG -like) equation for ferrimagnet films12,18-20) at the uniform ground state \nalong the positive in-plane z directio n to linear order in the small fluctuations |𝑛𝑥|,|𝑛𝑦|≪\n1, where the unit vector 𝒏 represents the N éel order parameter . The resultant equations are \ngiven by \n𝑠net𝑛̇𝑥−𝛼𝑠total𝑛̇𝑦−𝜌𝑛̈𝑦=𝑀𝐻ext𝑛𝑦, (1) 4 \n \n𝑠net𝑛̇𝑦+𝛼𝑠total 𝑛̇𝑥+𝜌𝑛̈𝑥=−𝑀(𝐻ext+𝐻ani)𝑛𝑥, (2) \n \nwhere 𝑠net=𝑠1−𝑠2 is the net spin density of two sublattices 𝑠1>0 and 𝑠2>0, 𝛼 is the \nGilbert damping parameter , 𝑠total =𝑠1+𝑠2 is the sum of the magnitudes of the two spin \ndensities , 𝜌>0 is the moment of inertia for the dynamics (which is inversely proportional \nto the microscopic exchange field between the two sublattices and describes the \nantiferromagnetic dynamics of the magnet ),2) 𝐻ext is the external field along the z direction, \n𝐻ani is the effective anisotropy field along the x direction perpendicular to the film \n(including the effect of the demagnetizing field), and 𝑀 is the magnetization . Here, we are \nneglecting the terms that are quadratic or higher order in 𝐻ext and the time derivative of the \norder parameter. The damping term is added by considering the Rayleigh dissipation \nfunction 𝑅=𝛼𝑠total ∫𝑑𝑉 𝒏̇2/2, which is the half of the ener gy dissipation rate through the \nmagnetic dynamics.20) Note that the Rayleigh function is defined in terms of 𝑠total, not in \nterms of 𝑠net, so that it is well defined even in the vicinity of 𝑇A where 𝑠net vanishes.17) \nTo the zeroth -order in the damping paramete r 𝛼 , the resonance frequencies for \nmonochromatic solutions to the above equations are given by \n𝑓±2\n=𝑠net2+𝜌𝑀(2𝐻ext+𝐻ani)±√𝑠net4+2𝜌𝑀(𝑠net)2(2𝐻ext+𝐻ani)+𝜌2𝑀2𝐻ani2\n8𝜋2𝜌2, (3) \n \nwhere 𝑓+ and 𝑓− are the frequencies for higher and lower resonance frequencies for the given \nfield. Far away from 𝑇A, where the net spin density |𝑠net| is suf ficiently large, Eq. (1) and \nthe corresponding dynamics are dominated by the first -order time derivative term and thus \nwe can neglect the second -order term by setting 𝜌=0 . In that ferromagnetic limit, the \nexpression for the lower frequency is reduced to that for the ferromagnet resonance \nfrequency :21) \n 𝑓FiM= 𝑀\n2𝜋|𝑠net|√𝐻ext(𝐻ext+𝐻ani). (4) \nNote that 𝑀/|𝑠net| is the effective gyromagnetic ratio 𝛾eff of the ferrimagnets. As the \ntemperature approaches 𝑇A , the net spin density | 𝑠net| decreases and thus the resonance \nfrequency is expected to increase. However, this formula cannot be used in the vicinity of 5 \n 𝑇A, where 𝑠net vanishes and thus the second -order term cannot be neglected. Exactly at 𝑇A, \nthe net spin density vanishes 𝑠net=0, which reduces the obtained resonance frequencies \n[Eq. (3)] to \n 𝑓+= 1\n2𝜋√𝑀(𝐻ext+𝐻ani)\n𝜌, 𝑓−= 1\n2𝜋√𝑀𝐻ext\n𝜌. (5) \nInclusion of the second -order time derivative term ∝𝜌 in the LLG -like equations [Eq. ( 1) \nand Eq. ( 2)] is necessary to obtain finite resonance frequencies at 𝑇A; otherwise, the LLG -\nlike equations lack in the reactive dynamic term ∝𝑠net at 𝑇A and becom e unable to describe \nthe ferrimagnetic dynamics properly therein. \n Since our experimental results, which are presented below, are performed away from \n𝑇A, let us derive the resonance linewidth for ferrimagnets in the ferromagnetic regime. When \nwe include the Gilbert damping term, the resultant linewidth of ferrimagnets ∆𝐻 (half -\nwidth -half-maximum) is given by \n∆𝐻≈2𝜋𝛼\n𝛾eff 𝑠total\n|𝑠net| 𝑓FiM. (6) \n \nTherefore 𝛼 in ferrimagnet is given by \n𝛼FiM≈(𝛾eff\n2𝜋)|𝑠net|\n𝑠total (∆𝐻\n𝑓FiM). (7) \nNote that b oth 𝑠total and 𝑠net appear in the lin ewidth expression because 1) the energy \ndissipation rate is proportional to 𝑠total since two lattices contribute additively and 2) the \nresonance frequency is inversely proportional to 𝑠net. On the other hand, in conventional \nexpressions for ferromagnetic resonance, the two spin -density parameters are assumed to be \nidentical , 𝑠total =𝑠net, and the corresponding expre ssion 𝛼FM≈(𝛾eff\n2𝜋)(∆𝐻\n𝑓FiM) was used to \nanalyze the magnetic resonance of ferrimagnets in the previous reports.6,7) Below, these two \nexpressions for the Gilbert damping parameters, 𝛼FiM and 𝛼FM, will be compared based on \nour experimental results. \nWe experimentally investigate d the FiMR in the GdFeCo compounds by using the \nhomodyne technique22-24) as shown in Fig 1 . For this study, we used a 5-nm SiN/10 -nm \nGd25.0Fe65.6Co9.4/5-nm Pt/100 -nm SiN /Si substrate film. The film was patterned into a 10 -\nµm-wide and 10 -µm-long strip pattern structure using optical lithography and Ar ion milling. \nA coplanar waveguide made of 100 -nm Au /5-nm Ti were de posited at the ends of the strip. 6 \n The measurements were performed by sweeping an external magnetic field ����ext at a fixed \nrf current 𝐼rf (frequency 𝑓=4−18 GHz). 𝐻ext was applied in-plane 45° away from the \nlong axis of the strip. \nFigure 2a shows the FiMR spectra at several temperatures 𝑇 between 220 K and 295 \nK. Although a single peak was clearly observed at 295 K, a second peak was also observed \nat 𝐻ext≈50 mT when 𝑇 is lower than 240 K. Note that the spontaneous magnetization lies \nin the sample plane at 𝑇=295 K while it becomes perpendicular to the plane when 𝑇≤\n240 K . Thus, the two resonan ce peaks when 𝑇≤240 K originat e from the magnetic \nresonance of perpendicular (𝐻ext≈50 mT ) and in -plane (higher field) magnetization s, \nrespectively . Here we focus on the resonan ce peak originating from in -plane magnetization, \nso we cut off the low -field regime to exclude the resonance peak from perpendicular \nmagnetization and fit those spectra in Fig. 2a by the combination of symmetric and anti -\nsymmetric Lorentzian function s, from which the resonan ce parameters are obtained .22,23) \nFigure s 2b and 2c show the resonance frequency 𝑓res as a function of the resonance \nfield 𝐻res and the spectral linewidth ∆𝐻 (half -width -half-maximum) as a function of 𝑓res, \nrespectively . Firstly, we analyze th ese data using the conventional expressions of \nferromagnetic resonance,21,25) \n𝑓res=𝑔eff𝜇B\nℎ√𝐻res(𝐻res+𝐻ani), (8) \n \n∆𝐻=𝛼FM\n(𝑔eff𝜇Bℎ⁄)𝑓res+∆𝐻0. (9) \n \nHere, 𝑔eff is the effective Landé g-factor, 𝜇B is the Bohr magneton, ℎ is the Planck ’s \nconst ant, 𝐻ani is the effe ctive anisotropy field including the demagnetization field, 𝛼FM is \nthe effective Gilbert damping parameter defined as in Ref. 6, and ∆𝐻0 is a frequency -\nindependent linewidth known as the inhomogeneous broadening, which originates from \nmagnetic non -uniformity.25) Equation (8) can be matched with Eq. (4) once we identify \n𝑔eff𝜇B/ℏ as the effective gyrom agnetic ratio 𝑀/|𝑠net| (ℏ=ℎ2𝜋⁄ is the reduced Planck ’s \nconstant ) and 𝐻res as 𝐻ext . The 𝐻res vs 𝑓res shown in Fig. 2b are well fitted b y Eq. (8), \nindicated by the solid lines, and 𝑔eff and 𝐻ani are obtained as the fitting parameters. Figures \n3a and 3b show 𝑔eff and 𝐻ani as a function of 𝑇, respectively. It is found that 𝑔eff remarkably \nincreases as 𝑇 decreases. Since the 𝑇A of the device is estimated to be 160 K (see below the 7 \n estimation method) , the result shows that 𝑔eff increases as 𝑇 approaches 𝑇A. Note that the \ndrastic decrease in 𝐻ani with decreasing 𝑇 (Fig. 3b) is attributed to the change in magnetic \nanisotropy from in -plane (295 K) to perpendicular (220 K) direction as mentioned above. \nThe 𝑓res vs ∆𝐻 show n in Fig. 2c are well fitted by Eq. (9), indicated by the solid lines, and \n𝛼FM and ∆𝐻0 are obtained as the fitting parameters. Figures 3c and 3d show 𝛼FM and ∆𝐻0 \nas a function of 𝑇, respectively. It is found that 𝛼FM increases significantly as 𝑇 decreases, \ni.e. as 𝑇 approaches 𝑇A. The 𝑇 dependences of 𝑔eff and 𝛼FM are in good agreement with the \nprevious paper s.6,7,2 7) According to the previous paper s,6,7) the 𝑇 dependences of 𝑔eff and \n𝛼FM are understood in terms of that of the net angular momentum 𝑠net; both 𝑔eff𝜇Bℏ⁄=\n𝑀net 𝑠net⁄ and 𝛼FM [from Eq. (9)] are ill -defined at 𝑇A where 𝑠net vanishes , which makes \nthe theory based on ferromagnets invalid therein . However, as shown in the discussion of \nour theory for FiMR , by defining the Gilbert damping parameter in the Rayleigh dissipation \nfunction 𝑅=𝛼𝑠total ∫𝑑𝑉 𝒏̇2/2 in such a way that the damping parameter is always well -\ndefined, the resonance frequency and the li newidth of FiMR can be described properly \nacross 𝑇A . In order to test whether our theory can explain the experimental results, we \nanalyze those data in Figs. 2b and 2c based on our theory. \n As mentioned in the theory part, the ferrimagnetic resonance frequency in the \nferromagnetic limit is reduced to the conventional ferromagnetic case, while the spectral \nlinewidth is modified by including the additional term 𝑠net 𝑠total⁄ . Therefore, the Gilb ert \ndamping parameter [Eq. (7)] in our theory [Eq. (1) and Eq. (2)] for the dynamics of \nferrimagnets can be obtained by the following expression: \n𝛼FiM=𝛼FM|𝑠net\n𝑠total|. (10) \nTo obtain 𝛼FiM from Eq. (7) and Fig. 2c, 𝑠net 𝑠total⁄ needs to be acquired. Although the net \nspin density 𝑠net is easy to obtain from the effective gyromagnetic ratio, the total spin density \n𝑠total is not straightforward to obtain. To solve this problem , we perform the following \nanalysis . The effective net gyromagnetic ratio satisfies the following equation;6,7) \n𝑔eff𝜇B\nℏ=𝑀net\n𝑠net=𝑀FeCo −𝑀Gd\n𝑀FeCo\n(𝑔FeCo 𝜇Bℏ⁄)−𝑀Gd\n(𝑔Gd𝜇Bℏ⁄). (11) \n \nHere, 𝑀FeCo (𝑀Gd) is the magnetizations of transition metal (rare -earth metal), and 𝑔FeCo \n(𝑔Gd) is the Landé g-factor of transition metal (rare -earth metal) sublattice. 𝑔eff is shown in 8 \n Fig. 3a, 𝑔FeCo and 𝑔Gd are obtained from literature (𝑔FeCo ~2.2 and 𝑔Gd~2.0 ).28-30) Two \nquantities can be measured directly : 𝑀net is independently measured by SQUID as shown \nin Fig. 4a and 𝑠net can be obtained from the effective gyromagnetic ratio when the \nferrimagnet is well within the ferromagnetic regime. With the measured values of 𝑀net = \n𝑀FeCo −𝑀Gd and 𝑠net= 𝑀FeCo\n(𝑔FeCo 𝜇Bℏ⁄)−𝑀Gd\n(𝑔Gd𝜇Bℏ⁄), we can obtain the magnetizations of two \nsublattices, 𝑀FeCo and 𝑀Gd , and also the spin densities of two sublattices, 𝑠FeCo and 𝑠Gd . \nFrom these results, we can obtain the total spin density 𝑠total =𝑠FeCo +𝑠Gd. Figures 4a and \n4b show 𝑀FeCo and 𝑀Gd, and 𝑠net=𝑠FeCo −𝑠Gd and 𝑠total =𝑠FeCo +𝑠Gd as a function of \n𝑇, respectively . Note that the 𝑇M (110 K ) determined by SQUID ( Fig. 4a ) and the 𝑇A (160 \nK) roughly estimated from the 𝑇 dependence of 𝑠net (Fig. 4b) are clearly different,12,31) \nwhich supports the validity of this analysis. Finally, by substituting 𝑠net and 𝑠total into Eq. \n(10), the damping parameter 𝛼FiM is obtained as shown in Fig. 4c. It can be clearly seen that \n𝛼FiM(≈0.01) is insensitive to 𝑇, in sharp contrast to 𝛼FM which significantly increases as 𝑇 \napproaches 𝑇A. Note that Eq. (10) is valid only in the ferromagnetic limit and, therefore, it \nis necessary to confirm that the measured temperature range ( 220 – 295 K) is in the deep \nferromagne tic regime. This would be guaranteed by the facts that 1) Fig. 4b shows \n𝑇A~160 K , which is far below the lowest 𝑇 in our measurement s (220 K ), and 2) the \nresonance frequency at 𝑇A is expected to be similar to or larger than about 50 GHz under \n300 mT ,6) which is much larger than the experimentally obtained resonance frequency at 220 \nK (12 GHz under 300 mT). \nThe observation that 𝛼FiM is insensitive to 𝑇 indicates that the 𝑇 dependence of the \nspectral linewidth in FiMR is attributed to the 𝑇 dependence of the net spin density 𝑠net \ninstead of that of the effective Gilbert damping parameter. This conclusion is consistent with \nsome recent papers,16,17) but it is in sharp contrast to the interpretation of the previous \nreport s,6,7) where the 𝑇 dependence of the spectral linewidth in FiMR was attributed to the \nchange of the effective Gilbert damping parameter. Our results provide an additional and \nclear evidence that properly defined Gilbert damping parameter 𝛼FiM of ferrimagnets is \ninsensitive to temperature, supporting the validity of these papers .16,17) Here, we would like \nto mention that, e ven though Fig. 4c is an evidence for the temperature -insensitive 𝛼FiM of \nferrimagnets, it lacks the information of 𝛼FiM in the vicinity of 𝑇A. However, obtaining 𝛼FiM \nin the vicinity of 𝑇A based on FiMR experiment s is challenging because 1) experimental \nobservation of ferrima gnetic resonance at 𝑇A which was measured to be larger than 5 0 GHz 9 \n for certa in ferrimagnets6) is expected to be difficult with homodyne detection technique (40 \nGHz at maximum in our measurement system) and 2) obtaining the necessary parameter \n𝑠total is difficult because the net spin density 𝑠net cannot be obtained from the effective \ngyromagnetic ratio in the vicinity of 𝑇A. Therefore, we believe that Fig. 4c serves as a good \nexperimental evidence to conclude that 𝛼FiM of ferrimagnets is insensitive to temperature. \nIn conclusion, we have provided the macroscopic theoretical description of \nferrimagnetic resonance and experimental results that support it. Our theory shows that the \nresonance frequency and the spectral linewidth of ferrimagnetic resonance can be described \nwell across the angular momentum compensation point , by adding the antiferromagnetic -\nlike inertial term to the equations of motion and by defining the Gilbert damping parameter \nproperly through the Rayleigh dissi pation function. Moreover, we performed the spin -torque \ninduced ferrimagnetic resonances at various temperatures and successfully observed that the \nresonance frequency and the linewidth depend on temperature. By analyzing the spectrum \nbased on our theory, we found that the Gilbert damping parameter in ferrimagnets is \ninsensitive to temperature, which has been considered to be strongly temperature -dependent. \nOur work introduces a new framework for studying ferrimagnetic resonance that allows us \nto interpret the ferrimagnetic dynamics for a wide range of temperatures . \n \nAcknowledgments \nThis work was supported by the JSPS KAKENHI (Grants No. 15H05702, No. 17H04924 , \nNo. 17H05181 , No. 26103002, and No. 26103004), Collaborative Research Program of the \nInstitute for Chemical Research, Kyoto University, and R & D project for ICT Key \nTechnology of MEXT from the Japan Society for the Promotion of Science (JSPS). This \nwork was partly supported by The Cooperative Research Project Program of the Researc h \nInstitute of Electrical Communication, Tohoku University. S. K. K. w as supported by the \nstartup fund at the University of Missouri . D. H. 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The schematic illustration of the device and the measurement setup . The direction \nof the external magnetic field 𝐻ext and the AC current 𝐼rf are indicated. 𝐻ext was applied \nin-plane 45° away from the long axis of the strip . \n \nFig. 2. (a) The ferrimagnetic resonance spectra as a function of the external magnetic field \n𝐻ext at several temperatures from 220 -295 K. The emerging peak at 𝐻ext≈50 mT below \n240 K is attributed to the out -of-plane resonan ce peak and are neglected in this study. (b) \nThe resonance frequency 𝑓res as a function of the resonance magnetic field 𝐻res. The solid \nlines are the fitting results by Eq. (8). (c) The spectral linewidth ∆𝐻 as a function of 𝑓res. \nThe solid lines are the fitting results by Eq. (9). \n \nFig. 3. Resonance parameters as a function of temperature extracted by the fitting in Fig. \n(2). (a) The effective Landé g-factor 𝑔eff. (b) The effective anisotropy field 𝐻ani. (c) The \neffective Gilbert damping parameter 𝛼FM. (d) The f requency -independent linewidth ∆𝐻0. \n \nFig. 4. (a) The net magnetization 𝑀net and the magnetizations of two sublattices 𝑀FeCo and \n𝑀Gd as function s of temperature . (b) The net spin density 𝑠net, the spin densities of two \nsublattices 𝑠FeCo and 𝑠Gd, and the sum of the magnitudes of the two spin densit ies 𝑠total as \nfunction s of temperature . (c) The effective Gilbert damping parameter 𝛼FM and the \nproperly defined Gilbert damping parameter of ferrimagnets 𝛼FiM as function s of \ntemperature . \n \n \n \n 13 \n \nFig. 1 . \n \n14 \n Fig. 2. \n \n0 5 10 15 200204060 295 K\n 260 K\n 240 K\n 220 KDH [mT]\nfres [GHz]数式 y = a + b*x\n重み 機械的\n残差平方和 6.89093\nピ ア ソ ン の r 0.99995\n--\n補正Ｒ 二乗 0.99989\n値 標準誤差\nΔH切片 38.04028 0.57739\n傾き 18.31533 0.07106\n数式 y = a + b*x\n重み 機械的\n残差平方和 3.53715\nピアソンの r 0.99996\n--\n補正Ｒ 二乗 0.9999\n値 標準誤差\nΔH切片 47.96301 0.69263\n傾き 21.8284 0.08274\n数式 y = a + b*x\n重み 機械的\n残差平方和 2.37794\nピ ア ソ ン の r 0.99992\n--\n補正Ｒ 二乗 0.9998\n値 標準誤差\nΔH切片 53.44711 1.23634\n傾き 24.77597 0.13079\n0 100 200 300 400 50005101520\n 295 K\n 260 K\n 240 K\n 220 Kfres [GHz]\nHres [mT]\n050100150200\n f = 4 GHz\n 6 GHz\n 10 GHz\n 14 GHz\n 18 GHzT = 295 K\n050100150200\nT = 280 K\n050100150200\nT = 260 K\n050100150200\nT = 240 K\n050100150200\nT = 230 K\n0 100 200 300 400 500 600 700 800050100150200V [mV]\nHext [mT]T = 220 K\n(a) \n(c) \n(b) 15 \n Fig. 3. \n \n \n220 240 260 280 3004681012DH0 [mT]\nT [K]\n220 240 260 280 300-120-80-400Hani [mT]\nT [K]\n220 240 260 280 3000.050.100.150.20\naFM\nT [K]\n220 240 260 280 3002.02.53.03.54.04.5geff\nT [K]\n(a) \n(c) \n(b) \n(d) 16 \n Fig. 4. \n \n \n \n100 150 200 250 30001234Spin density [ ´10-6 J s/m3]\nT [K] snet\n sFeCo\n sGd\n stotal\nTA~160K\n0 50 100 150 200 250 300-101234Magnetization [ ´105 A/m]\nT [K] Mnet (by SQUID)\n MFeCo\n MGd\nTM » 110 K\n150 200 250 3000.000.050.100.150.20\na\nT [K] aFM\n aFiM\nTA~160K\n(a) \n(c) \n(b) " }, { "title": "2311.07350v1.Magnetoresistive_detection_of_perpendicular_switching_in_a_magnetic_insulator.pdf", "content": " \n1 \n Magnetoresistive detection of perpendicular switching in \na magnetic insulator \nSilvia Damerio1,*, Achintya Sunil2, M. Mehraeen2, Steven S. -L. Zhang2 and Can O . Avci1,* \n1Institut de Ciència de Materials de Barcelona, Campus de la UAB, Bellaterra, 08193, Spain \n2Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106, USA \n \n \nSpintronics offers promising routes for efficient memory, logic, and computing technologies. \nThe central challenge in spintronics is electrically manipulating and detecting magnetic states \nin devices. The electrical control of magnetization via spin -orbit torques is effective in both \nconducting and insulating magnetic layers . However, the electrical readout of magnetization in \nthe latter is inherently difficult, limiting its use in practical applications. Here, we demonstrate \nmagnetoresistive detection of perpendicular magnetization reversal in an electrically insulating \nferrimagnet, terbium iron garnet (TbIG). To do so, w e use TbIG|Cu|TbCo , where TbCo is a \nconducting ferr imagnet and serves as the reference layer, and Cu is a nonmagnetic spacer . \nCurrent injection through Cu|TbCo allows us to detect the magnetization reversal o f TbIG with \na simple resistance readout during an external magnetic field sweep. By examining the effect of \nmeasurement temperature, TbCo composition, and Cu thickness on the sign and amplitude of \nthe magnetoresistance, we conclude that the spin -dependent electron scattering at the TbIG|Cu \ninterface is the underlying cause. Technologically -feasible magnetoresistive detection of \nperpendicular switching in a ferrimagnetic garnet is a breakthrough, as it opens broad avenues \nfor novel insulating spintr onic devi ces and concepts . \nFerrimagnetic garnets (FMGs) are ubiquitous in solid -state physics. They host a wide variety of useful \nproperties , such as ultralow damping, high magnon density, tunable magnetic anisotropy and \nsaturation magnetization, and highly ordered single -crystal structures with sharp surfaces and \ninterfaces.1 Furthermore, single -crystal epitaxial FMG s can be grown from a few micrometers down \n2 \n to a few nanometers of thickness via physical and chemical deposition techniques on suitable \nsubstrates and, yet, retain bulk -like properties. As a result, FMGs have served as an excellent material \ntestbed for a multitude of spintronic phenomena in t he past few years.2–8 More recently, there has \nbeen a growing effort in exploiting FMGs as active components in magnonic integrated circuits9 and \nspintronic memory devices.10 Researc h into the latter has been particularly fruitful owing to the \ndevelopment of ultrathin FMGs with perpendicular magnetic anisotropy (PMA)11,12 and the highly \nefficient control of their magnetization vector by current -induced spin -orbit torques.8,10,13 –17 \nWhile the above advances are promising for the integration of FMGs in future computing \ntechnologies, the electrical detection of magnetization in this material family is still a critical \nchallenge. The spin Seebeck, local and non -local spin Hall magnetoresistance, and thermal spin drag \neffects have been utilized for the electrical detection of magnetization in FMGs.3–5,18 However, all of \nthese phenomena rely on voltages induced by inverse spin Hall effect in an adj acent nonmagnetic \nmetal, such as Pt, hence fundamentally limiting the output signal amplitudes and materials that can \nbe used for the purpose. Electrical detection of the perpendicular magnetization vector in FMGs \npossessing PMA, crucial for domain wall an d skyrmion -based racetrack devices, and for downscaling \nthe lateral size of memory cells, is particularly strenuous. Thus far, the anomalous Hall effect (AHE) \ndriven by interfacial spin -dependent scattering4,19–21 is used as the sole electrical method for \nmagnetization vector detection in PMA FMGs,10,22 yielding minute output signals , nonetheless in a \ntechnologically -prohibitive four-contact Hall cross device geometry . \nMagnetization vector detection in conventional spintronic devices is realized by tunnel and giant \nmagnetoresistances (GMR).23–26 The latter, discovered in the 1980s, has led to miniaturized GMR \nsensors, revolutionizing both the magnetic non -volatile data stora ge and field sensing technologies.27 \nWhat lies at the core of the GMR is the spin valve effect, arising from the spin -dependent electron \nscattering at the bulk and interfaces of ferromagnetic materials.28 GMR spin valves typically consist \nof conducting magnetic layers separated by nonmagnetic metal spacers. The electrical resistance of \n3 \n such systems exhibits a large asymmetry between the parallel and antiparallel alignment of the \nmagnetizations of adjacent magnetic layers , which serve as the reference and free layers , respectively . \nGMR can then be used to effectively probe the magnetization reversal or rotation of the free layer \nwith respect to the reference, also in a current -in-plane (CIP) geometry, leading to simple two -\nterminal devices. To date, FMGs have been excluded from such device concepts even though the \nGMR -based sensing of FMGs’ magnetization vector would be highly desirable. Few e arlier reports \nhave characterized the spin valve effect at other magnetic insulator |metal interfaces , but these were \nlimited to in -plane magnetization systems and cryogenic temperature measurements.29–31 While the \npresence of the spin valve effect is allowed at an FMG |metal interface in principle , its experimental \nobservation remains elusive. \nIn this work, we show a robust spin valve effect at the interface of a typical FMG, terbium iron garnet \n(Tb 3Fe5O12, TbIG ), enabling us to detect 180º perpendicular magnetization reversal via simple \nresistance measurements at room temperature. We achieve such spin valve behavior in a \nTbIG|Cu|TbCo trilayer ( see Fig. 1a) where the TbIG is engineered to be the free layer wherea s TbCo \nacts as the reference magnetic layer. We observe abrupt longitudinal resistance changes during a \nswept magnetic field upon crossing the coercive fields of the respective layers, analogous to the CIP -\nGMR, albeit with a smaller amplitude with respect to a fully metallic stack. The effect of measurement \ntemperature, TbCo layer composition, and spacer layer thickness on the sign and amplitude of the \nmagnetoresistance collectively pinpoint the spin valve effect at the TbIG|Cu interface as the \npredominant origin of the observed phenomenon. Theoretical calculations based on layer -resolved \nBoltzmann transport equations are in excellent agreement with our experimental findings and further \nconsolidat e the presumed origin of the magnetoresistance effect. The eff icient reading of the \nmagnetiza tion vector in FMGs in a two-terminal CIP geometry reported here provides a new platform \nfor novel spintronic device concepts using magnetic insulators as active components. \nLayer composition and magnetic characterization \n4 \n We deposited TbIG(25)|Cu(1 -5)|TbxCo1-x(8)|Pt(3) (thicknesses in nm, subscript x denotes atomic %) \nstacks and other reference layers by means of magnetron sputtering, and patterned them into standard \nHall bar devices (see Fig. 1b and Methods). Here, TbIG and TbxCo1-x (TbCo for short, x = 0.3 unless \nspecified otherwise) are used as the soft and hard magnetic layers, respectively, due to the lower \n(higher) coercivity of the former (latter). We preferred a Tb -based ferrimagnetic metallic alloy over \nan elemental t ransition metal such as Co, since the former exhibits a large and tunable bulk PMA \nover a broad range of composition, thickness, and temperature, independently of the under/over layer, \nthus eliminating the laborious engineering of the interfacial PMA in th e latter.32–34 The thickness of \nthe Cu spacer is varied between 1 and 5 nm and Cu is chosen due to its high spin conductivity when \ninterfaced with FMGs and TbCo.35,36 Finally, Pt is used as capping to avoid oxidation of TbCo and it \nhas no active role in the spin transp ort experiments reported in this paper, therefore will not be \nmentioned in the remainder of the text. \nThe magnetization behavior of TbIG and TbCo were examined by polar magneto -optic Kerr effect \n(MOKE, Fig. 1c) and AHE (Fig. 1d) measurements with a swept out-of-plane field ( Hz). In the former, \nwe observe d a clear hysteresis loop typical of PMA systems with 100% remanence, and large \ncoercivity Hc ~35 mT. A closer look -up to the lower field region reveals minor signal jumps at ~±10 \nmT. The inner loop measurement focusing on this region (see inset – dark green curve ) indicates a \nsecond magnetic signal with a much lower amplitude, superimposed to the larger hysteresis loop. The \nHc value of this inner loop corresponds to the one measured f or TbIG prior to the Cu|TbCo deposition \n(see inset - light green curve) , hence this signal corresponds to TbIG . The similar coercivity of the \nTbIG before and after deposition of the Cu|TbCo indicates negligible exchange coupling between the \ntwo magnetic la yers (see Supplementary Information). The Hall resistance ( RH) measurement shows \nthe hysteresis loop originating from the AHE in TbCo with Hc ~35 mT, consistent with the MOKE \nresult. However, an inner loop measurement similar to Fig. 1c - inset does not sh ow any signal related \nto the TbIG reversal. Because of the absence of current flow through TbIG, the AHE signal cannot \n5 \n be generated in this layer and the interfacial AHE is also inoperative due to the negligible spin Hall \neffect in Cu . These measurements clearly demonstrate that the magnetization vector of TbIG in the \nabove stack can only be probed optically but not electrically with standard Hall measurements. \nMagnetoresistive detection of perpendicular switching \nWe then examine the lon gitudinal resistance behavior of the above stack and compare it with \nreference material systems. The GMR in magnetic transition -metal multilayers is known to originate \nfrom the spin -dependent scattering in the spin -split d bands,37,38 typically leading to an increase of the \nresistance when the magnetization ( M) of consecutive layers is antiparallel compared to the parallel \ncase. Figure 2a shows the resistance behavior of a fully metallic Co|Cu|TbCo PMA spin valve device \nduring an Hz sweep. We observe two reversal events at low and high fields, corresponding to the \nswitching of Co and TbCo, respectively, resulting in two distinct resistance levels in the parallel and \nantiparallel configurations, expectedly of the CIP -GMR. The magnitude of the GMR is 4.9%, \ncomparable to previous reports on similar systems.36,39 Next, we measure Cu/TbCo deposited on a \nnon-magnetic gadolinium gallium garnet ( Gd 3Ga5O12, GGG) substrate (Fig. 2b). Here, the spin valve \nbehavior is not observed. Instead, the resistance displays a field -dependent variation resembling a \nbowtie shape . This unconventional resistance behavior was previously reported and attributed to the \nmagnon magnetoresistance40 and/or magnetoresistance due to the sperimagnetism of TbCo .41 This \neffect requires a conducting magnetic layer and exhibits a quasi -linear positive or negative slope \ndepending on the relative direct ion of M and Hz. \nFigure 2c shows the magnetoresistance of the TbIG|Cu|TbCo trilayer. The resistance clearly \nundergoes four distinct switching events corresponding to the perpendicular switching of TbIG and \nTbCo , superimposed to the quasi -linear background magnetoresistance described above. Initially, at \nlarge positive fields, the magnetizations of TbIG and TbCo are parallel and aligned with Hz. Upon \nreducing Hz, the resistance increases linearly until ~ -10 mT at which field the magnetization of TbIG \nreverses from up to down and a sudden resistance increase occurs. As Hz is reduced further , the \n6 \n resistance increase continues, until the coercivity of TbCo is reached at ~ -35 mT, where the resistance \ndrops sharply , followed by further gradual Hz-dependent decrease . A reciprocal sequence is observed \nwhen Hz is swept from the negative to positive direction. To better illustrate the GMR -like behavior, \nin Fig. 2d, we plot the magnetoresistance after subtraction of the back ground due to the unusual \nmagnetoresistance of TbCo not related to the GMR effect. Finally, Fig. 2e shows the minor loops, \ncorresponding to the reversal of M of TbIG, when the TbCo layer is pre -set in the up (+z) and down \n(-z) direction. In these measureme nts, we observe the hysteretic resistance behavior in both cases with \nan opposite sign due to the reversal of the reference layer. Based on the amplitude of the jump in Fig. \n2e, we estimate the relative change of resistance R/R0 =1.4x10-5, where R0 is the resistance at zero \napplied field. \nThe signals reported in Fig. 2c-e are consistent with a positive GMR in TbIG|Cu|TbCo. The sign of \nthe GMR in all -metallic systems is dictated by the spin -dependent scattering asymmetry at the bulk \nand interfaces of the m agnetic layers. In TbCo, the spin -polarized conduction is mostly dominated by \nthe s-d electrons of Co, hence the Co magnetization dictates the bulk and interface contribution to the \nGMR as well as the sign of the AHE. The positive AHE in Fig. 1d and the positive GMR in Fig. 2a \nestablish that the net magnetization in TbCo is parallel to the Co sublattice in the TbIG|Cu|TbCo \nstack. Likewise, independent measurements reported in our earlier study ( see Ref.[12]) and \nSupplementary Information show that the net magnetization in TbIG is Fe sublattice dominated. Since \nTbIG is an insulator, only the TbIG/Cu interface contributes to the GMR. Therefore, the positive \nGMR in TbIG|Cu| TbCo demonstrates that the sign of the spin -dependent scattering asymmetry at \nboth the TbIG|Cu and Cu|TbCo interfaces is the same and positive wit h respect to the standard \nconvention. \nTemperature -dependence of the magnetoresistance \nTo examine further the effect of the relative orientation of the ferrimagnetic sublattices on the sign, \nand the influence of the measurement temperature on the amplitude of R/R0, we studied the \n7 \n magnetoresistance in TbIG|Cu|TbCo as a function of temperature (T), across the magnetic \ncompensation temperature ( TM) of TbIG. In Fig. 3a, we plot representative data during a Hz sweep \nmeasured at T = 225 K and 175 K, i.e., above and below the TM of TbIG, respectively. We found TM \nof TbIG ~200 K for this specific sample (see Supplementary Information). Because of the linear \nmagnetoresistance background from TbCo th e overall shape of the signal becomes intricate for T ≤ \n200 K . Nevertheless, we observe that the resistance change s due to TbIG reversal (marked with black \narrows) is upward for T = 225 K and downward for T = 175 K , indicating a clear sign change of \nΔR/R 0. This is further highlighted in Fig. 3b with the minor loop measurements corresponding to the \nmagnetization reversal of TbIG when the TbCo layer is fixed along the up (+z) direction. Above TM \n(T = 225 K) we observe a negative hysteresis loop (similar to the room temperature measurement \nreported in Fig. 2e) whereas below TM (T = 175 K) the loop becomes positive. \nIn Fig. 3c we plot ΔR/R 0 in the entire temperature range tested in this study. The sign reversal of \nΔR/R 0 at TM of TbIG confirms that the spin -dependent reflection at the TbIG|Cu interface, similarly \nto the interfacial AHE42, is not governed by the net magnetization, but instead , by the orientation of \nthe Fe sublattice. In spite of the sign change at T ~ 200 K, the absolute value of ΔR/R 0 increases \nsmoothly and monotonically (open dots in Fig. 3c) with the decreasing temperature. This is expected \nfrom the strong temperature dependence of the GMR effect due to the increased spin conductivity of \nCu at lower temperatures.43 \nTbCo composition and Cu spacer thickness dependence \nThe amplitude and sign of the GMR can be modulated by the TbCo composition and Cu spacer \nthickness. To investigate the former, we measured ΔR/R 0 in TbIG|Cu(2)|Tb xCo1-x(8nm) layers with x \nranging between 0.3 and 0.65 and plotted in Fig. 4 – left. We note that for x < 0.3 the films were \nmagnetized in -plane and for x > 0.65 they were paramagnetic (see Supplementary Information). We \nfind that ΔR/R 0 strongly depends on the Tb content, displaying a rapid decrease going from x = 0.3 to \n0.5. When the Tb amount reaches x = 0.55 , the magnetization becomes Tb sublattice dominated and \n8 \n ΔR/R 0 becomes negative (i.e. R is lower when the magnetization of TbIG and Tb xCo1-x are \nantiparallel). This is analogous to the sign reversal observed upon crossing the TM of TbIG reported \nin Fig. 3c, but obtained at room temperature with a modification of the Tb xCo1-x composition instead. \nOverall, |ΔR/R 0| undergoes a ~22 -fold decrease when the Tb concentration is increased from 0.3 to \n0.6, and finally vanishes at x = 0.65. This result indicates the essential role played by Co in the spin -\npolarized current generation and spin -dependent scattering in the bulk and interfaces of TbCo. \nFigure 4 – right reports the effect of the Cu spacer thickness ( t) on ΔR/R 0 in TbIG|Cu( t)|TbCo layers. \nThe largest effect is found for Cu(2 nm). Increasing t results in a decrease of ΔR/R 0, as expected due \nto the increased current shunting and reduced spin coherence.43 On the other hand, for a Cu thickness \nof 1 nm, a slightly lower ΔR/R 0 is observed with respect to Cu(2 nm), whereas the effect is, in \nprinciple, expected to be enhanced. We associate this behavior to the lower uniformity of such thin \nCu layer, causing additional resistance, not participating in the magnetoresistance and negatively \naffecting the spin -dependent properties and interface quality with the TbCo layer. Indeed, we observe \na correlation between the Cu thickness and the coercivity and PMA of TbCo, indicating a sharper \ninterface when the Cu thickness is increased, promoting a higher quality TbCo. This might be the \nreason why the magnetoresistance is still substantial despite larger current shunting when the Cu \nspacer is increased to 5 nm. The strong dependence of the magnetic properties of TbCo alloys on the \nunderlayer material, thickness and interface quality is well known .44 Additionally, discrepancies in \nthe spin -dependent properties of the TbIG|Cu interface for TbIG grown on different dates on different \nsubstrates could influence the data presented in Fig. 4 – right. Although the essential features and \ntrends expected of the thickness dependence of ΔR/R 0 are observed, the data in Fig. 4 – right cannot \nreflect a universal behavior. Therefore, we believe that a more systematic study of the influence of \nthe spacer layer thickness on ΔR/R 0 should be conducted by continuously varying the Cu thickness \non a single substrate to ensure the unifo rmity of the TbIG properties, which falls outside our technical \ncapabilities. Finally, to confirm the relevance of the spacer layer with low spin -orbit coupling (high \n9 \n spin c onductivity ) on the magnetoresistance, we replaced Cu with 2 nm of Pt which exhibit s large \nspin-orbit coupling . We did not detect any spin valve effect within our experimental detection limit. \nThis negative result is anticipated due to the very short spin diffusion length of Pt (typically <2 nm) , \ndephasing nearly the entire spin current generated in the TbCo and at the Pt/TbCo interface .36 \nFurthermore, the absence of spin valve signal with the Pt spacer excludes the magnetic proximity \neffect at the TbIG |metal interface as the potential origin of the magnetoresistance, since Pt is much \nmore suscept ible to proximity magnetism than Cu. We n ote that ΔR/R 0 values of all the materials \nexamined in this work can be found in Table S1 of the Supplementary Information. \nBoltzmann Transport Calculations \nWe calculated the magnetization -dependent resistivity of T bIG|Cu|TbCo trilayer by solving the \nfollowing linearized Boltzmann transport equation with a layer -by-layer approach :38,45 \n 𝑣𝑧𝜕𝑔\n𝜕𝑧−𝑒𝐸\n𝑚𝜕𝑓0\n𝜕𝑣𝑥=−𝑔\n𝜏 (1) \nHere 𝑚 and 𝜏 are the effective mass and momentum relaxation time and mass of conduction \nelectrons, 𝑓0 is the equilibrium electron distribution fun ction, 𝑔(𝒗,𝑧) is the deviation from f0 induced \nby the external electric field with amplitude E applied along x. Full translation symmetry is assumed \nin the x -y plane (i.e., the layer plane) so that the spatial dependence of 𝑔 only occurs in the thickne ss \ndirection parallel to the z -axis. To further simplify our calculations, we assumed the same effective \nmass, relaxation time, and Fermi energy 𝜀𝐹 for the conducting Cu and TbCo layers, which would be \nsufficient for the purpose of an order of magnitude estimation of the ΔR/R 0 ratio of the trilayer system. \nThe calculation can be extended straightforwardly to involve different values of the above -ment ioned \nmaterials parameters without essential change to the formulation. \nIn order to capture the interfacial spin -dependent scattering, we divide 𝑔 into four additive \ncomponents depending on the orientation of an electron’s spin moment with respective to the \n10 \n magnetization of a magnetic layer and the sign of vz (i.e., the z -component of the electronic group \nvelocity). The general solutions of Eq. (1) can be written as \n 𝑔±↑(↓)(𝒗,𝑧)=𝑒𝐸𝜏\n𝑚𝜕𝑓0\n𝜕𝑣𝑥[1+𝐶±↑(↓)𝑒𝑥𝑝(∓𝑧\n𝜏|𝑣𝑧|)], (2) \nwhere the “ +” and “ −” signs denote electrons with 𝑣𝑧 being positive and negative , respectively, the \narrow ↑(↓) characterizes orientation of spin parallel (antiparallel) to the local magnetization of the \nmagnetic layer in question. The general solutions take the same form for electrons in the Cu and TbCo \nlayers, except for the coefficients, 𝐶±↑(↓), which need to b e determined by boundary conditions. \nTo calculate the ΔR/R 0 ratio, scatterings at three interfaces of the trilayer need to be taken into account \nthrough the boundary conditions, namely, 1) the spin -dependent reflection at the out surface of the \nTbCo layer , 2) the specular transmission and reflection at the interface between the metallic TbCo|Cu \ninterface, and 3) the spin -dependent specular reflection at the Cu|TbIG interface. In addition, when \nthe magnetizations of the TbCo and TbIG layers are antiparallel , we also take into account the change \nof spin quantization axis in the middle of the Cu layer. It is reasonable to assume specular reflection \nat the metallic interface is negligible and scattering at the outer surface of the TbCo layer is completely \ndiffu sive. And near the magnetic interface between the Cu and TbIG layer, the portions of electron \nfluxes that are specularly reflected from the interface are, in principle, also spin dependent ,21 as “spin -\nup” and “spin -down” electrons see different energy barriers effectively due to the exchange coupling \n(𝐽𝑒𝑥) between them and the magnetization of the TbIG layer even though the magnetic layer is \ninsulating. \nBy inserting the general solutions of 𝑔±↑(↓) for each layer into these boundary conditions, the \nunknowns in Eqs. (2) can be determined, allowing us to further evaluate the spatially -averaged \nlongi tudinal conductivity of the ith layer via 𝜎(𝑖)=1\n𝑑𝑖𝐸∫𝑑𝑧∫𝑑3𝒗𝑣𝑥(𝑔↑+𝑔↓), where 𝑑𝑖 is the \nthickness of the ith layer. The total resistivities for parallel and antiparallel magnetization \nconfigurations , denoted by 𝜌↑↑ and 𝜌↑↓respectively , are obtained by inverting the corresponding \n11 \n conductivity tensors. Finally, the ΔR/R 0 ratio is evaluated via ∆𝑅𝑅0⁄ =(𝜌↑↓ −𝜌↑↑)𝜌↑↑⁄. For the \nTbCo|Cu interface, the transmission coefficients are taken to be 𝑇↑=0.5 and 𝑇↓=0.95 (note that \nthe roughness of the interface can be characterized phenomenologically by the spin -dependent \ndiffusive scattering parameter defined as 𝐷↑(↓)=1−𝑇↑(↓).45 For 𝐽𝑒𝑥~0.01 eV and the averaged \nenergy barrie r of the insulator 𝑉𝑏~12 eV,2,21 the reflection coefficients are estimated to be 𝑅↑=\n0.4995 and 𝑅↓=0.5005 . This spin asymmetry of electron scatterings at the two magnetic interfaces \ngives rise to a ΔR/R 0 ratio of 6.2x10-5, close to the experimental value of 7.0x10-5 measured at 10 K. \nDetailed calculation s are provided in the Supplementary Information. \nConclusions \nIn summary, we demonstrate a simple magnetoresistive detection of perpendicular magnetization \nreversal in an insulating ferrimagnetic material TbIG. The detection relies on current -in-plane \nmagnetoresistance measurements in a TbIG|Cu|TbCo tril ayer system where the conducting \nferrimagnet TbCo is used as a reference magnetic layer and spin polarizer. The material and \ntemperature dependence of the magnetoresistance and theoretical calculations collectively pinpoint \nthe spin -valve effect at the TbI G|Cu interface as the underlying cause of the observed phenomenon. \nThese results will open a new chapter in the field of insulating spintronics as they will stimulate \nresearch into a wide spectrum of FMG, spacer and spin polarizer layer combinations, and e nable a \nwhole new range of device ideas and architectures based on magnetic insulators. While the effect is \nrelatively small for any microelectronic applications as of yet, it can be enhanced by orders of \nmagnitude by material s and device engineering , and support from theory . \nOn the materials side, ferromagnetic materials with higher spin polarization such as Heusler alloys \nwith an optimized thickness could provide a pathway to increase the magnetoresistance. To obtain \neven larger gains, the spacer layer c an be chosen from those having a very long spin coherence length \nand promoting better spin mixing conductance with the insulating and conducting magnetic layers \nforming the device. Potential candidates include two -dimensional materials such as graphene, \n12 \n transition metal dichalcogenides (TiSe 2, MoS 2, etc.), conducting oxides (SrVO 3, etc.) and other light \nmetals (Cr, Mn, etc.). Our work will also stimulate theoretical efforts , as the most suitable materials \ncould be determined by using relevant first -principl es calculations. \nOn the device side, the magnetores istive reading reported here will enable non -volatile binary \nmemory cells where a magnetic insulator could be used as an active component instead of \nconventional magnetic conductors. In such devices, usin g a magnetic insulator would bring about a \nseries of advantages such as higher structural stability, broader magnetic tunability , ultrafast \nswitching times, and low power consumption , among others. It is also possible to use the \nmagnetoresistance output to identify and characterize the skyrmions and domain walls in an insulating \nracetrack memory and study their field/current -driven dynamics in real time . Insulating domain wall - \nand sky rmion -based devices enabled by magnetoresistive reading could pave the way for novel \nanalog -like memory concept s that can be used in neuromorphic computing. \nMethods \nSamples preparation \n25 nm -thick TbIG thin films were deposite d on GGG(111) substrates by radio frequency (r.f.) \nsputtering at 800 ºC from a stoichiometric target. The deposition rate was ∼0.4 nm/min at the applied \npower of 150W in 3mTorr of a mixture of Ar and O 2 with a ratio of 30:2 and the base pressure in \nthe chamber was below 7x10-8 Torr. Detailed characterization and optimization procedures of our \nTbIG films are described in Ref.[12]. To fabricate the six -terminal Hall bar devices, the continuous \nTbIG films were covered in photoresist and patterned by laser -writer optical lithography. Finally, the \nmetallic stack described further below was deposited and lift-off was performed. The Hall bar \ndimensions are l=30 μm for the current line length, l/4 its width and l/10 the Hall branch width. \nThe metallic stack consisted of M( t)|Tb xCo1-x(8 nm)|Pt(3 nm) multilayers deposited by d.c. magnetron \nsputtering at room temperature in 3 mTorr Ar. The Tb xCo1-x alloys were obtained by co -sputtering \n13 \n pure Co and Tb targets and the relative atomic concentration of the two elements was controlled by \nthe relative sputtering power. The thickness t of the metal (M) spacer layer varied between 1 and 5 \nnm. The deposition rates were as follows: 0.13514 nm/s at 100 W for Cu, 0.104 nm/s at 200 W for \nCo, 0.082 nm/s t 50 W for Tb, 0.186 nm/s at 50 W for Pt and 0.057 nm/s at 200 W for Ti. The \nfollowing referen ce samples were also prepared: Cu(2 nm)|Tb 0.3Co0.7(8 nm)|Pt(3 nm) on GGG(111) \nsubstrate, Ti(3 nm)|Pt(3 nm)|Co(1 nm)|Cu(2 nm)|Tb 0.3Co0.7(8 nm)|Pt(3 nm) on Si, Ti(3 nm)|Pt(3 \nnm)|Co(1 nm)|Cu(2 nm)|Tb 0.65Co0.35(8 nm)|Pt(3 nm) on Si and Pt(1.5 nm)|Cu(1.5 nm)|Tb 0.3Co0.7(8 \nnm)|Pt(3 nm) on TbIG. \nMagnetic and Electrical Characterization \nThe magnetic hysteresis and anisotropy axis of the films were examined using a home -built magneto -\noptic Kerr effect (MOKE) setup in a polar geometry with a 532 nm wavelength gr een laser . The room \ntemperature magnetoresistance measurements were performed by recording the resistance ( R) as a \nfunction of H between the extremes of the current line using a Keithley DMM6500 digital multimeter \nand DC test current of 1 mA. For t he temp erature dependence of the magnetoresistance , selected \nsamples were inserted inside a physical property measurement system and measured in the range of \n10–300K with an AC probing current of 1 mA (root mean square ) and frequency 𝜔/2𝜋 = 999Hz. \nThe Hall effect measurements were performed at room temperature using a Zurich Instruments MFLI \ndigital lock -in amplifier. An AC probing curren t of amplitude 0.5 -1mA (root mean square) and \nfrequency 𝜔/2𝜋 = 1092Hz was sent across the current line and the first h armonic voltage ( R𝐻) was \nmeasured across the Hall arms. \nData availability \nAll the data supporting the findings of this study are available upon request from the corresponding \nauthor. \nCode availability \n14 \n The computer code used for data analysis is available upon request from the corresponding author. \nReferences \n1. Yang, Y., Liu, T., Bi, L. & Deng, L. Recent advances in development of magnetic garnet \nthin films for applications in spintronics and photonics. J. Allo ys Compd. 860, 158235 \n(2021). \n2. Kajiwara, Y. et al. Transmission of electrical signals by spin -wave interconversion in a \nmagnetic insulator. Nature 464, 262 –266 (2010). \n3. Uchida, K. I. et al. Observation of longitudinal spin -Seebeck effect in magnetic insulators. \nAppl. Phys. Lett. 97, 172505 (2010). \n4. Nakayama, H. et al. 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B 103, 014421 (2021). \n42. Rosenberg, E. R. et al. Magnetism and spin transport in rare -earth -rich epitaxial terbium and \neuropium iron garnet films. Phys. Rev. Mater. 2, 094405 (2018). \n43. Tripathy, D., Adeyeye, A. O. & Shannigrahi, S. Effect of spacer layer thickness on the \nmagnetic and magnetotransport propertie s of Fe3 O4/Cu/Ni80 Fe20 spin valve structures. \nPhys. Rev. B 75, 022403 (2007). \n44. Tolley, R. et al. Generation and manipulation of domain walls using a thermal gradient in a \nferrimagnetic TbCo wire. Appl. Phys. Lett. 106, 242403 (2015). \n45. Camle y, R. E. & Barna, J. Theory of giant magnetoresistance effects in magnetic layered \nstructures with antiferromagnetic coupling. Phys. Rev. Lett. 63, 664 (1989). \n \nAcknowledgments \nS.D. and C.O.A. acknowledge funding from the European Research Council (ERC) under the \nEuropean Union’s Horizon 2020 research and innovation program (project MAGNEPIC, grant \n18 \n agreement No. 949052). C. O. A. acknowledges funding from the Spanish Ministry of Science and \nInnovation through grant reference no. CNS2022 -136060. Work by A.S., M.M. and S.S. -L.Z was \nsupported by the College of Arts and Sciences, Case Western Reserve University. Authors thank M. \nFettizio for the insightful discussions. \n \nAuthor Informa tion \nContributions \nC.O.A. conceived the idea and supervised the study. S.D. designed and prepared the samples, carried \nout the measurements and analyzed the data. A.S. , M.M. and S. S.-L.Z. constructed the theoretical \nframework. S.D. and C.O.A. wrote the manuscript. All authors discussed the results and commented \non the manuscript. \nCorresponding author \nCorrespondence to Silvia Damerio ( sdamerio@icmab.es ) and Can Onur Avci (cavci@icmab.es) \n \nCompeting Interests \nThe au thors declare no competing interests. \n \n19 \n Figures and Tables \n \nFig.1 Illustration of the spin valve effect in TbIG|Cu|TbCo, device schematics, and optical and \nelectrical characterization . a Schematic representation of the spin valve structure in the high and \nlow resistance states at room -temperature. The red and blue arrows indicate the direction of the \nmagnetization ( M) and corresponding spin s of the majority and minority carriers. Cyan sph eres \nrepresent conduction electrons undergoing lower and higher scattering events in the parallel and \nantiparallel magnetic configurations, respectively. b Schematic representation of the device with the \ngeometry used for the electrical measurements. c Plot of the polar MOKE signal of TbIG|Cu|TbCo. \nThe inset shows the minor loops corresponding to the switching of TbIG prior (light green) and after \n(dark green) the deposition of TbCo. d Transverse Hall resistance ( RH) in TbIG|Cu|TbCo as a function \n-50 0 50-1.0-0.50.00.51.0\n-15 015-0.030.000.03MOKE (a.u.)\nHZ (mT)\n-50 0 50-0.8-0.40.00.40.8RH ()\nHZ (mT)a\nbc\ndi\niTbCo\nTbIGCu\nTbCo\nTbIGCuM\nM\nM\nMParallel state -Low Resistance\nAntiparallel state -High Resistance\nz\nx y\nRz\nx y\nRHie-\ne- \n20 \n of out-of-plane field ( HZ), corresponding to the AHE in TbCo. No fingerprints of TbIG switching are \nobserved in the inner loop (gray). \n \nFig.2 Magnetoresistance in TbIG|Cu|TbCo and other reference layers . a Resistance ( R) as a \nfunction of the out-of-plane fie ld (HZ) in an all -metallic Co(1)|Cu(2)| TbCo (8) trilayer. b in \nGGG(subs.)|Cu(2)| TbCo (8) and c in TbIG(25)|Cu(2)| TbCo (8). The latter clearly shows a spin valve \nsignal similar to the one observed in the all -metallic trilayer. d Plot of the resistance from panel c \nafter removal of the unconventional magnetoresistance contribution from TbCo (see text for more \ndetails). The red and blue arrows indicate the direction of M in TbIG and TbCo , respectively. e Inner \nloops showing the switching of TbIG when the direction of TbCo is fixed up ( red) or down ( blue) \nwith respect to the film normal. \n-100 0 100-0.010.000.010.020.03R-R0 ()\nHZ (mT)\nTbCo\nCo\nTbCo\nTbIG\nTbCo\nGGG\n-200 0 200360365370375380R ()\nHZ (mT)\n-50 0 50160.23160.24R ()\nHZ (mT)\n-100 0 100942.65942.70R ()\nHZ (mT)a b c\nd e\n-20 -10 0 10 20-0.010.000.010.02R-R0 ()\nHZ (mT) TbCo up TbCo down \n21 \n \nFig.3 Temperature dependence of the magnetoresistance in TbIG|Cu| TbCo . a Plot of the \nresistance ( R) as a function of out -of-plane field ( HZ) for 2 selected temperatures above (red ) and \nbelow (blue) the compensation temperature ( TM) of TbIG. The black arrows indicate the resistance \njump corresponding to TbIG magnetization reversal . b Plot of the inner loops showing the switching \nof TbIG when M of TbCo is fixed out -of-plane for two selected temperatures above (red) and below \n(blue) TM. The linear background has been subtracted. c Plot of the temperature ( T) dependence of \nthe magnetoresistance (close dots) and its absolute value (open dots). \n0.00.51.01.5\n \n0.60 0.55 0.35Cu(2)/TbxCo1-xCu(t)/Tb0.3Co0.7\n5 2 3 1\nx t (nm)R/R0 x10-5\n0.4 0.3 0.5 0.65\n \nFig.4 Amplitude of the magnetoresistance as a function of Tb xCo 1-x composition and Cu spacer \nthickness . Left: plot of ΔR/R 0 in TbIG(25)|Cu(2)|Tb xCo1-x(8) alloy as a function of Tb content ( x) \nfrom 0.3 to 0.65. We note that ΔR/R 0 undergoes a sign change for x > 0.5 and it is below our detection \n-200 -100 0 100 200R-R0\nHZ (mT)0.01 \n-200 0 200R-R0 \nHZ (mT)0.05 175K225Ka\n0 100 200 300-8-4048\nT (K)R/R0 x10-5\n225K\n175Kb c\nTM \n22 \n limit for x = 0.65. The dotted lines represent the absolute value of the GMR amplitude for visual \nillustration of the monotonic decay of ΔR/R 0. Right: plot of ΔR/R 0 in TbIG(25)/Cu(t)/Tb 0.3Co0.7(8) as \na function of Cu spacer thickness ( t) from 1 to 5 nm. \n \n23 \n Supplementary Information for “Magnetoresistive \ndetection of perpendicular switching in a magnetic \ninsulator” \nSilvia Damerio1,*, Achintya Sunil2, M. Mehraeen2, Steven S. -L. Zhang2, and Can O. Avci1,* \n1Institut de Ciència de Materials de Barcelona, Campus de la UAB, Bellaterra, 08193, Spain \n2Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106, USA \n \nSI1 – Exchange bias in TbIG|Cu|TbCo spin valves \nWe found that the magnetization of the magnetic layers in the spin valves of this study is \ncoupled via a small exchange bias for the Cu thickness ≤2 nm. This interaction appears as a \nlateral shift towards negative (positive) values of the hystereis loops of the “soft” TbIG when \nthe magnetization of Tb 0.3Co0.7 is fixed in the up (down) direction (see Fig. 2e of the main \ntext). We estimated the shift ( ΔH) as a function of the thickness of the Cu spacer layer and \nplot the trend in Fig. S1. Due to the low amount of samples available we could not perform \na detailed statistical analysis of these values, but the results seem to indicate that the exchange \nbias decreases upon increasing spacer layer thickness. Because the measured exchange bias \nis much lower than the coercivity o f TbIG, we consider it negligible in this study. \n0 2 4 6036H (mT)\nCu t (nm)\n \nFig.S1 Exchange bias in TbIG|Cu|TbCo spin valves. a Plot of the difference of the \ncoercive field of TbIG ( ΔH), measured from the R vs H loops, when the magnetization of \nTbCo is fixed in the up and down direction as a function of the Cu spacer layer thickness ( t). \n \nSI2 – Magnetic and magneto -transport properties of Tb xCo1-x alloys \nFerrimagnets are a class of magnets with unbalanced antiparallel -aligned sublattice moments, \nwhich results in a finite, albeit small, magnetization. In RE -transition metal alloys and \nmultilayers, the dominant sublattice (i.e. the sublattice whose moment is parallel to the net \nmagnetization) varies with the stoichiometry and temperature. Measurement of the \nanomalous Hall effect (AHE) provides information on the dominating sublattice, as the sign \nof the AHE coefficient is opposite for RE and transition -metals . \n24 \n \nBefore utilizing Tb xCo1-x in the fabrication of spin valve devices we characterized the \nmagnetic and magneto -transport properties of the alloy as a function of Tb concentration. To \nthis end, we grew 8 nm thick Tb xCo1-x films by co -sputtering on Si substrates with 3 nm Ti \nbuffer and capping layers. The samples were patterned into Hall -bar devices ( with sizes l=30 \nμm for the current line length, l/4 its width and l/10 the Hall branch width) by standard optical \nlithography and lift -off. Figure S2a shows the plot of the Hall resistance ( RH) as a function \nof out -of-plane field ( HZ) of the Tb xCo1-x films below the compensation composition. Here, \nPMA is only achieved above 25% of Tb due to the shape anisotropy that tends to orient the \nmagnetization in -plane when the films are patterned into micron -sized devices. Here, HC \nincreases from 65 mT at x=0.25 to 750 mT for x=0.5. Comparable values are observed in the \nspin valves of the main text (see Table S1) with a small difference due to the different buffer \nlayer (Cu instead of Ti). Between x=0.50 and x=0.55 the compensation composition of the \nTbxCo1-x alloy is reached and the sign of the AHE reverses, as shown in Fig. S2b. Upon \nfurther increasing Tb concentration, HC starts to decrease again, reaching 20 mT at x=0.65. \nThis allows us to conclude that Tb xCo1-x films are Co -dominated up to x=0.5 and Tb -\ndominated above x=0.55. \n \nFig.S2 Characterization of the AHE in Tb xCo1-x alloys as a function of Tb content. Plot \nof the Hall resistance ( RH) as a function of HZ for patterned Tb xCo1-x films with a x between \n0.25 and 0.5 and b x between 0.55 and 0.65. \n \nSI3 – Ferrimagnetic sublattices and sign of the magnetoresistance \nThe giant magnetoresistance (GMR) provides information on the relative orientation of the \nmoment of the transition metal sublattice in two neighboring layers, as it originates from the \nscattering of d -conduction electrons with the transition metal moments. Figure S3a shows \nthe AHE of a Co(1)|Cu(2)|Tb 0.3Co0.7(8) (thickness in nm) reference spin valve. As it can be \nseen, here the AHE coefficient is positive for both Tb 0.3Co0.7 and Co layers, indicating that \nthe magnetization ( M) of Tb 0.3Co0.7 is parallel to the Co sublattice. Consistently, the GMR \n(Fig. S3b also Fig. 2a of the main text) is positive, indicating that in both layers the Co \n-1000 -500 0 500 1000-0.4-0.20.00.20.4RH (a.u.)\nHZ (mT) x=0.25\n x=0.30\n x=0.35\n x=0.40\n x=0.50\n-500 0 500-0.4-0.20.00.20.4RH (a.u.)\nHZ (mT) x=0.55\n x=0.60\n x=0.65a b \n25 \n sublattice, which dominates the transport properties, is aligned with the magnetic field ( HZ). \nFigures S3c-d show the measurements of a reference Co(1)|Cu(2)|Tb 0.65Co0.35(8) spin valve. \nHere, due to the large amount of Tb, the alloy becomes RE -dominated and thus both the AHE \nand the GMR change sign. Figure S3e-f show the Hall resistance ( RH) as a function of HZ in \na TbIG(25)|Pt(1.5)|Cu(1.5)|Tb 0.3Co0.7(8) spin valve. In this case, the insertion of a thin (1.5 \nnm) layer of Pt in contact with the TbIG is necessary to read its magnetization direction via \nthe spin Hall magnetoresistance (SMR) effect. Here, we observe a major hysteresis loop ( Fig. \nS3e) corresponding to the AHE of Tb 0.3Co0.7 with positive AHE coefficient, and a minor loop \n(Fig. S3f) corresponding to the SMR -AHE of TbIG. The latter is negative, indicating that \nTbIG is above its compensation temperature a nd thus its magnetization is parallel to the \nmoment of the tetragonal Fe3+ sublattice (notice that 3d metal -dominated TbIG and TbCo \nhave opposite sign of the AHE coefficient). In this type of spin -valve the GMR is positive, \nas shown in Fig.2c of the main t ext. On the other hand, Fig. S3g shows the negative AHE in \na TbIG(25)|Cu(2)|Tb 0.55Co0.45(8) spin valve, which displays the negative GMR of Fig. S3h, \nas the TbIG and TbCo are respectively 3d metal and RE dominated. Here, the TbCo coercive \nfield is significa ntly higher, being closer to the magnetic compensation composition. These \ndata confirm that the transport properties in ferrimagnetic insulator|spacer|metal spin valves \nare dominated by the transition metal sublattice in both layers and show that the spin -\ndependent reflection coefficient at the TbIG|Cu interface has the same sign as that of the \nwell-known Co|Cu interface. \n \nTable S1 Amplitude of the magnetoresistance in different TbIG|spacer|magnetic metal \ntrilayers at room temperature. \n \nMetallic layer \n(t nm) Spacer layer \n(t nm) HC oxide \n(mT) HC metal \n(mT) R/R 0 \nTb0.25Co0.75 (8) Cu (2) 7 In-plane < detection \nTb0.3Co0.7 (8) Cu (2) 10 35 1.44E -5 \nTb0.35Co0.65 (8) Cu (2) 10 55 7.74E -6 \nTb0.4Co0.6 (8) Cu (2) 7 150 3.16E -6 \nTb0.5Co0.5 (8) Cu (2) 6 800 2.42E -6 \nTb0.55Co0.45 (8) Cu (2) 9 350 -2.0E-6 \nTb0.6Co0.4 (8) Cu (2) 10 65 -6.3E-7 \nTb0.65Co0.35 (8) Cu (2) 6 20 < detection \nTb0.3Co0.7(8) Cu (1) 50 0 1.01E -5 \nTb0.3Co0.7 (8) Cu (3) 10 48 7.57E -6 \nTb0.3Co0.7 (8) Cu (5) 10 40 7.74E -6 \nTb0.3Co0.7 (8) Pt (2) 15 73 < detection \n \n26 \n \nFig.S3 AHE and GMR across the ferrimagnetic compensation. Plot of a the Hall \nresistance ( RH) and b the longitudinal resistance ( R) as a function of out -of-plane magnetic \nfield ( HZ) of a Co(1)|Cu(2)|Tb 0.3Co0.7(8) reference spin valve. Plot of c RH and d R as a \nfunction of HZ of a Co(1)|Cu(2)|Tb 0.65Co0.35(8) reference spin valve. e-f Plot of RH as a \nfunction of HZ of a TbIG(25)|Pt(1.5)|Cu(1.5)|Tb 0.3Co0.7(8) (Co-dominated) and spin valve. \nPlot of g RH and h R as a function of HZ of a TbIG(25)|Cu(2)|Tb 0.55Co0.45(8) (Tb -dominated) \nspin valve. \n \n \n−200 0 200−0.6−0.4−0.20.00.20.4RH ()\nHZ (mT)\n-200 0 200360365370375380R ()\nHZ (mT)\n-50 0 50-0.4-0.20.00.20.4RH ()\nHZ (mT)\n-50 0 50493494R ()\nHZ (mT)\n-80 0 80-0.4-0.20.00.20.40.6RH ()\nHZ (mT)\n-10 -5 0 5 10-0.050.000.05RH (m)\nHZ (mT) inner\n-1000 -500 0 500 1000-0.6-0.4-0.20.00.20.40.6RH ()\nHZ (mT)\n−400 0 4001122.601122.611122.62R ()\nHZ (mT)a b\nc d\ne f\ng h \n27 \n SI4 – Temperature dependence \nThe magnetic compensation temperature of TbIG can be determined by measuring the \ntemperature dependence of the Anomalous Hall effect (AHE) in TbIG|Pt heterostructure, as \nshown in Ref.[12]and Fig. S4a. Here we observe a sign reversal of the AHE at approximately \n200 K, which coincides with the maximum coercivity (Fig. S4b), due to the reduced Zeeman \nenergy on the vanishingly small net magnetization. Simil arly, from the magnetoresistance \nmeasurements of a TbIG|Cu|TbCo spin valve shown in Fig. S4c, we can determine the \nmagnetic compensation of TbIG as the point at which the sign of the GMR reverses (see Fig. \n3c of the main text) and the coercive field diverg es. Above ~200K, the resistance ( R) displays \nan upward jump when the magnetization of the TbIG layers reverses and a downwards jump \nin correspondence of the coercivity of TbCo. The amplitude of the second jump is larger than \nthe first, as it is given by th e sum of both the GMR effect and the MMR effect, which also \nleads to a decrease of R. Below the TbIG compensation ( TM ~200K) the GMR is negative, \nthus the resistance becomes lower when the magnetization of TbIG reverses. In this case, due \nto the GMR effect , we expect an increase of R at the TbCo coercivity. However, here we \nobserve a second jump of R towards lower values. This is due to the sum of the GMR and \nMMR contributions, the latter being larger than the former, and thus resulting in a second \ndecrease of R. Figure S4d summarizes the temperature dependence of the coercivity of the \nTbCo and TbIG layers inferred from the magnetoresistance measurement of Fig. S4c. While \nthe Hc of both layers increases gradually as the temperature is reduced, due to the stronger \nexchange interaction between the sublattices and enhanced PMA, the HC of TbIG shows an \nadditional peak at TM = 200 K. The HC vs T trend for the TbIG layer obtained from AHE (Fig. \nS4b) and magnetoresistance (Fig. S4d) measurements is essentially the same. \n28 \n \nFig.S4 Temperature dependence. a Plot of the Hall resistance ( RH) and as a function of out -\nof-plane magnetic field ( HZ) of a TbIG/Pt(4) bilayer measured between 170 and 250 K . b \nPlot of the coercive filed ( HC) of TbIG as a function of T inferred from the AHE measurement \nof panel a. c Plot of the Resistance ( R) and as a function of HZ of a TbIG(25)|Cu(2)|TbCo(8) \nspin valve measured between 10 and 300 K . d Plot of HC of Tb IG (red) and TbCo (blue) as \na function of T inferred from the magnetoresistance measurement of panel c. \n \n \n-15000 0 1500010K100K\n50K300K\n275K\n225K250K\n150K\n125K175KR-R0 \nHZ (Oe)0.5 200K\n-800 -400 0 400 800180K\n170K190KRH-R0 \nHZ (mT) 250K\n 240K\n 230K\n 220K\n 210K\n 200K2 m\n180 200 220 2400200400600TMHC (mT)\nT (K)\n0 100 200 30005001000TMHC (mT)\nT (K) TbCo\n TbIGa b\nc d \n29 \n SI5 – Theoretical model \nWe compute the conductivity of the trilayer by solving the following linearized Boltzmann \ntransport equation with a layer -by-layer approach: \n 𝑣𝑧𝜕𝑔\n𝜕𝑧−𝑒𝐸\n𝑚𝜕𝑓0\n𝜕𝑣𝑥=−𝑔\n𝜏 (S1) \nHere 𝑚 and 𝜏 are the effective mass and momentum relaxation time of co nduction electrons, \n𝑓0 is the equilibrium electron distribution function, 𝑔(𝒗,𝑧) is the deviation from f0 induced \nby the external electric field with amplitude E applied along x. Full translation symmetry is \nassumed in the x -y plane (i.e., the layer plane) so that the spatial dependence of 𝑔 only occurs \nin the thickness direction parallel to the z -axis. To further simplify our calculations, we \nassumed the same effective mass, re laxation time, and Fermi energy 𝜀𝐹 for the conducting \nCu and TbCo layers, which would be sufficient for the purpose of an order of magnitude \nestimation of the MR ratio of the trilayer system. The calculation can be extended \nstraightforwardly to involve d ifferent values of the above -mentioned materials parameters \nwithout essential change to the formulation. \n \nFig.S5 Coordinate system for calculating the magnetoconductivity in the \nTbCo|Cu|TbIG trilayer. The TbCo and TbIG layers are denoted as Layer 1 and Layer 4, \nrespectively, the Cu spacer is divided into two parts with equal thickness (a) by a dotted line \nwhere change of spin quantization axis occurs when the magnetizations of the two magnetic \nlayers a re antiparallel. \n \nIn order to capture the interfacial spin -dependent scattering, we divide 𝑔 into four additive \ncomponents depending on the orientation of an electron’s spin moment with respective to the \nmagnetization of a magnetic layer and the sign of 𝑣𝑧 (i.e., the z -component of the electronic \ngroup velocity). The general solutions of Eq. (1) for each layer (as sketched in Fig. S5) can \nbe written as \n 𝑔±,↑(↓)(𝑖)(𝒗,𝑧)=𝑒𝐸𝜏\n𝑚𝜕𝑓0\n𝜕𝑣𝑥[1+𝐶±,↑(↓)(𝑖)𝑒𝑥𝑝 (∓𝑧\n𝜏|𝑣𝑧|)], (S2) \n \n30 \n where the “ +” and “ −” signs denote electrons with positive and negative 𝑣𝑧 components \nrespectively, the arrow ↑(↓) characterizes orientation of spin parallel (antiparallel) to the \nlocal magnetization of the magn etic layer in question, and the superscripts are the layer \nindices. \nThe twelve integration constants, 𝐶±,↑(↓)(𝑖) (𝑖=1,2,3) in Eqs. (S2) are determined by the \nfollowing twelve boundary conditions at the interfaces between Layer 1, 2, and 3. For the \nease of notation, we will suppress the group -velocity variable of 𝑔±,↑(↓)(𝑖). \n Reflection of electrons at the outer surface of the TbCo layer ( 𝑧=𝑏): \n 𝑔−,↑(1)(𝑧=𝑏)=𝑅↑(𝑏)𝑔+,↑(1)(𝑧=𝑏) (S3-1) \n 𝑔−,↓(1)(𝑧=𝑏)=𝑅↓(𝑏)𝑔+,↓(1)(𝑧=𝑏) (S3-2) \nwhere 𝑅↑(↓)(𝑏) is the reflection parameter for spin -up (spin -down) electrons respectively. The \nvalue of 𝑅↑(↓)(𝑏) ranges from 0 to 1 with “0” corresponds to fully diffusive reflection and “1” \ncorresponds to specular reflection. \n Spin-dependent electron reflection and transmission at the interface ( 𝑧=𝑎) between \nthe TbCo and Cu l ayers: \n 𝑔+,↑(1)(𝑧=𝑎)=𝑇↑(𝑎)𝑔+,↑(2)(𝑧=𝑎)+𝑅↑(𝑎)𝑔−,↑(1)(𝑧=𝑎) (S3-3) \n 𝑔+,↓(1)(𝑧=𝑎)=𝑇↓(𝑎)𝑔+,↓(2)(𝑧=𝑎)+𝑅↓(𝑎)𝑔−,↓(1)(𝑧=𝑎) (S3-4) \n 𝑔−,↑(2)(𝑧=𝑎)=𝑇↑(𝑎)𝑔−,↑(1)(𝑧=𝑎)+𝑅↑(𝑎)𝑔+,↑(2)(𝑧=𝑎) (S3-5) \n 𝑔−,↓(2)(𝑧=𝑎)=𝑇↓(𝑎)𝑔−,↓(1)(𝑧=𝑎)+𝑅↓(𝑎)𝑔+,↓(2)(𝑧=𝑎) (S3-6) \nwhere 𝑅↑(↓)(𝑎) and 𝑇↑(↓)(𝑎) are, respectively, the reflection and transmission parameters for spin -\nup (spin -down) electrons at the interface 𝑧=𝑎. \n Change of spin quantization axis in the middle of the Cu layer ( 𝑧=0): \n 𝑔+,↑(2)(𝑧=0)=𝑇↑↑(𝜃𝑚)𝑔+,↑(3)(𝑧=0)+𝑇↑↓(𝜃𝑚)𝑔+,↓(3)(𝑧=0) (S3-7) \n 𝑔+,↓(2)(𝑧=0)=𝑇↓↓(𝜃𝑚)𝑔+,↓(3)(𝑧=0)+𝑇↓↑(𝜃𝑚)𝑔+,↑(3)(𝑧=0) (S3-8) \n 𝑔−,↑(3)(𝑧=0)=𝑇↑↑(𝜃𝑚)𝑔−,↑(2)(𝑧=0)+𝑇↑↓(𝜃𝑚)𝑔−,↓(2)(𝑧=0) (S3-9) \n 𝑔−,↓(3)(𝑧=0)=𝑇↓↓(𝜃𝑚)𝑔−,↓(2)(𝑧=0)+𝑇↓↑(𝜃𝑚)𝑔−,↑(2)(𝑧=0) (S3-10) \nwhere 𝑇↑↑(𝜃𝑚)=𝑇↓↓(𝜃𝑚)=cos (2𝜃𝑚) and 𝑇↑↓(𝜃𝑚)=𝑇↓↑(𝜃𝑚)=sin (2𝜃𝑚) with 𝜃𝑚 the \nangle between the magnetizations of the TbCo and the TbIG layers. \n Reflection of electrons at the interface between the Cu and TbIG layers ( 𝑧=−𝑎): \n 𝑔+,↑(3)(𝑧=−𝑎)=𝑅↑(−𝑎)𝑔−,↑(3)(𝑧=−𝑎) (S3-11) \n 𝑔+,↓(3)(𝑧=−𝑎)=𝑅↓(−𝑎)𝑔−,↓(3)(𝑧=−𝑎) (S3-12) \nPlacing Eqs. (S2) into the above boundary conditions, one can determine the twelve unknowns, with \nwhich the spatially -averaged total conductivity of the trilayer structure can be calculated via \n31 \n 𝜎=−𝑒\n(2𝜋)3∑1\n𝑑𝑖∫𝑑𝑧∫𝑑3𝒌𝑣𝑥(𝑔↑(𝑖)+𝑔↓(𝑖))/𝐸3\n𝑖=1 (S4) \nwhere 𝑔↑(↓)(𝑖)=𝑔+,↑(↓)(𝑖) for 𝑣𝑧>0 and 𝑔↑(↓)(𝑖)=𝑔−,↑(↓)(𝑖) for 𝑣𝑧<0. The longitudinal resistivity, \n𝜌, can be obtained by inverting the conductivity tensor. And finally, the magnetoresistance \nratio is obtained by \n 𝑀𝑅 =𝜌𝐴𝑃−𝜌𝑃\n𝜌𝑃 (S5) \nwhere 𝜌𝐴𝑃 and 𝜌𝑃 are the resistivities for the magnetizations of the two magnetic layers in \nthe antiparallel ( 𝜃𝑚=𝜋) and parallel ( 𝜃𝑚=0) configurations, respectively. In Fig. S6, we \nplot the magnetoresistance ratio as a function of applied magnetic field for a trilayer TbIG(25 \nnm)|Cu(2 nm)|TbCo (8 nm), with the materials parameters given in Table S2. The calculated \nmagnetoresistance ratio is 0.0062%, comparable to the value obse rved experimentally. \n \n \nFig. S6 MR ratio as a function of the magnetic field applied perpendicular to the layer \nplane. The blue (red) curve denotes the change of the MR with field sweep from +1500 Oe \nto -1500 Oe (from -1500 Oe to +1500 Oe). The red and blue arrows indicate, respectively, \nthe magnetizations of the TbCo and TbIG layers, respectively. The coercive fields of TbIG \nand TbCo are taken to be 150 Oe and 500 Oe, respectively. And the materials used in the \ncalculation are listed in the table below. \n \n \n \n32 \n Table S 2 Reflection and Transmission parameters (Refs.[21],[45]) \n \nSymbols Value Parameter \n𝑅↑(𝑏) 0 Reflection at x = b for spin -up electrons \n𝑅↓(𝑏) 0 Reflection at x = b for spin -down electrons \n𝑅↑(𝑎) 0 Reflection at x = a for spin -up electrons \n𝑅↓(𝑎) 0 Reflection at x = a for spin -down electrons \n𝑇↑(𝑎) 0.5 Transmission at x = a for spin -up electrons \n𝑇↓(𝑎) 0.95 Transmission at x = a for spin -down electrons \n𝑇↑↑(𝜃𝑚) 0 for 𝜃𝑚=𝜋, \n 1 for 𝜃𝑚=0 Spin-conserved transmission at x = 0 for spin-up electrons \n𝑇↓↓(𝜃𝑚) 0 for 𝜃𝑚=𝜋, \n 1 for 𝜃𝑚=0 Spin-conserved transmission at x = 0 for spin -down electrons \n𝑇↑↓(𝜃𝑚) 1 for 𝜃𝑚=𝜋, \n 0 for 𝜃𝑚=0 Spin-flip transmission (from spin -up to spin -down) at x = 0 \n𝑇↓↑(𝜃𝑚) 1 for 𝜃𝑚=𝜋, \n 0 for 𝜃𝑚=0 Spin-flip transmission (from spin -down to spin -up) at x = 0 \n𝑅↑(−𝑎) 0.4995 Reflection coefficient at x = −a for spin -up electrons \n𝑅↓(−𝑎) 0.5005 Reflection coefficient at x = −a for spin -down electrons \n \n \n " }, { "title": "2105.06102v2.Thermal_instability_in_a_ferrimagnetic_resonator_strongly_coupled_to_a_loop_gap_microwave_cavity.pdf", "content": "arXiv:2105.06102v2 [cond-mat.str-el] 6 Aug 2021Thermal instability in a ferrimagnetic resonator strongly coupled to a loop-gap\nmicrowave cavity\nCijy Mathai,1Oleg Shtempluck,1and Eyal Buks1\n1Andrew and Erna Viterbi Department of Electrical Engineeri ng, Technion, Haifa 32000 Israel\n(Dated: August 9, 2021)\nWe study nonlinear response of a ferrimagnetic sphere reson ator (FSR) strongly coupled to a\nmicrowave loop gap resonator (LGR). The measured response i n the regime of weak nonlinearity\nallows the extraction of the FSR Kerr coefficient and its cubic damping rate. We find that there is\na certain range of driving parameters in which the system exh ibits instability. In that range, self-\nsustained modulation of the reflected power off the system is g enerated. The instability is attributed\nto absorption-induced heating of the FSR above its Curie tem perature.\nI. INTRODUCTION\nFerromagnetic and ferrimagnetic resonators [1–3] are\nwidely employed in a variety of microwave (MW) de-\nvices, including narrow band oscillators [4], filters [5],\nand parametric amplifiers [6]. These resonators exhibit a\nvariety of intriguing physical effects [7], including Bose-\nEinstein condensation [8] and magneto-optical coupling\n[9–12]. Here we study a strongly coupled hybrid system\ncomposed of a loop gap resonator (LGR) integrated with\na ferrimagnetic sphere resonator (FSR) made of yttrium\niron garnet (YIG) [13, 14]. We focus on the regime of\nnonlinearresponse. InsectionIIIbelowweexploretheef-\nfect onnonlineardampingin the regionofrelativelyweak\nmicrowavedriving. Aninstability, whichisobservedwith\na much stronger driving, is reported in section IV below,\nand a theoretical model, which attributes the instability\nto a driving-induced heating, is presented.\nMany nonlinear dynamical effects have been observed\nbefore in FSRs, including auto-oscillations [15, 16], opti-\ncal cooling [17], frequency mixing [18, 19] and bistabil-\nity [20–24]. The Suhl instability (of both first and sec-\nond orders)hasbeen observedwith transversemicrowave\ndriving, whereas parallel pumping instability has been\nobserved with longitudinal driving [25]. Applications of\nnonlinearity for quantum data processing have been ex-\nplored in [26–33].\nHeating a YIG sphere from room temperature to 400K\nby microwave driving having power of 450mW has been\nreported in [34]. At a Curie temperature given by Tc=\n560 K, YIG undergoes a phase transition between an\nordered ferrimagnetic state (FS) and a disordered para-\nmagnetic state (PS). Thermal instability was observed\nin a cavity magneto-mechanical system [35]. Microwave\noscillations induced by injecting spin-polarized current\n[36] into a magnetic-multilayer structure have been re-\nported in [37]. Self-excited oscillations induced by ohmic\nheating in a Y 3Fe5O12/Pt bilayer nanowire have been\ninvestigated in [38]. Imaging of heating induced by the\nspin Peltier effect has been demonstrated in [39].\nFIG. 1: FSR-LGR coupling: (a) A sketch of the FSR made\nof YIG having radius of Rs= 1mm that is integrated inside\nthe aluminum cylindrical LGR having gap width of 0 .3mm.\nThe sphere is held by ceramic ferrules (CFs). A sapphire\nwafer (labeled as S) is inserted into the gap to increase the\ncapacitance. (b) The numerically calculated magnetic field\nenergy density distribution (normalized with respect to th e\nmaximum value) corresponding to driving at the resonance\nfrequency ωe/(2π) = 3.3GHz. (c) A VNA reflectivity |S11|2\nmeasurement as a function of magnon frequency ωs(propor-\ntional to the externally applied magnetic field). The cou-\nplingcoefficient geffis extractedfrom thetheoretical fit(white\ndashed lines) following Eq. (2).\nII. LOOP GAP RESONATOR\nWith relatively low input power, the main mecha-\nnisms responsible for FSR nonlinear response are mag-\nnetic anisotropy [40] and exchange interaction[13]. Con-\nsider a MW cavity mode having angular frequency ωe\nand an integrated FSR having radius Rs. It is assumed\nthat the applied static magnetic field Hsis parallelto the\neasy axis. In the Holstein-Primakoff approximation [41]\n(which assumes that magnetization is nearly saturated),\nthe Hamiltonian of the system HDis expressedas [21, 42]\n/planckover2pi1−1HD=ωeNe+ωsNs+KMN2\ns\n+geff/parenleftbig\nA†\neAs+AeA†\ns/parenrightbig\n,\n(1)\nwhereNe=A†\neAe(Ns=A†\nsAs) is a cavity mode (FSR\nKittel mode) number operator, ωs=γgHsis the Kit-\ntel mode angular frequency, γg/2π= 27.98 GHz T−1is2\nthe gyromagnetic ratio, KM=/planckover2pi1γ2\ngKc1//parenleftbig\nVsM2\ns/parenrightbig\nis the\nanisotropy-induced Kerr frequency, Kc1is the first-order\nanisotropy constant, Vs= 4πR3\ns/3 is the volume of the\nsphere,Msis the saturation magnetization, and geffis\nthe cavity-FSR coupling coefficient. For YIG at room\ntemperature, Ms= 140 kA /m andKc1=−610 J/m3,\nhenceKM=−2.4×10−8Hz×(Rs/(100µm))−3.\nIn the linear regime, where the Kerr nonlinearity can\nbe disregarded, the Hamiltonian HD(1) can be diago-\nnalized. The angular frequencies ω±of the two hybrid\nphoton-magnon eigen modes are given by [43]\nω±=ωe+ωs\n2±/radicalBigg/parenleftbiggωe−ωs\n2/parenrightbigg2\n+g2\neff.(2)\nBoth angular frequencies ω±are positive provided that\ngeff<√ωsωe. Note that the super-radiance Dicke in-\nstability occurs in the ultra-strong coupling region where\ngeff>√ωsωe[44]. In the rotating wave approximation\n(RWA) the Kerrcoefficients K±ofthe hybridmodes hav-\ning angular frequencies ω±are given by Eqs. ( A9) and\n(A10) of appendix A [see Eq. ( A8)].\nIn the current experiment, we explore the response for\na wide range of the MW input powers Pp. We find that\nthe response is well described by the Hamiltonian HD\nprovided that Ppis sufficiently small. However, with suf-\nficiently high Pp, the FSR temperature Tmay exceed\nthe Curietemperature Tcdueto MWabsorption-induced\nheating. We study the response of the FSR-LGR system\nto an injected monochromatic pump tone having a fre-\nquency close to resonance. The off reflected power is\nmeasured using a spectrum analyzer (SA). We find that\nthere is a certain zone in the pump frequency - pump am-\nplitude plane, in which the resonator exhibits limit-cycle\n(LC) response resulting in self-sustained modulation of\nthe reflected power. The observed LC is attributed to\nthermal instability (TI) [45].\nA MW cavity made of an LGR allows achieving a\nrelatively large coupling coefficient geff[46, 47]. The\nMW LGR schematically shown in Fig. 1(a), is made\nof a hollow concentric aluminium tube having an in-\nner and outer radii of RLGR= 1.7mm and 3mm, re-\nspectively, and a height of HLGR= 12mm. A sapphire\nstrip of 260 µm thickness has been inserted into the gap\nin order to increase its capacitance, which in turn re-\nduces the frequency feof the LGR fundamental mode\n[fe=ωe/(2π) = 3.3GHz with sapphire] [48]. An FSR\nmade of YIG having radius of Rs= 1mm is held by two\nferrules inside the LGR. The static magnetic field Hsis\napplied perpendicularly to the LGR axis. The LGR-FSR\ncoupled system has been encapsulated in a metallic rect-\nangular shield made of aluminum. The cavity is weakly\ncoupled to a loop antenna (LA).\nThe numerically calculated magnetic energy density\ndistribution corresponding to the LGR fundamental\nmode is shown in Fig. 1(b). The calculated density is\nhomogeneous ( ≃95%) over the FSR volume, and it is\nwell confined inside the LGR inner volume. Note thatfor our device, the LGR inner volume, which is given\nbyπR2\nLGRHLGR, is 4 orders of magnitude smaller than\nthe volume λ3\ne, whereλe=c/feis the free space wave-\nlength corresponding to the LGR frequency fe, andcis\nthe speed of light in vacuum. Consequently, the coupling\ncoefficient geffcan be made much larger than typical val-\nues obtained with the commonly employed rectangular\ncavities [28], for which the mode volume commonly has\nthe same order of magnitude as λ3\ne.\nBased on Eq. (2) of Ref. [28], together with the eval-\nuated energy density shown in Fig. 1(b), the calcu-\nlated value of the coupling coefficient is found to be\ngeff= 176MHz for the LGR fundamental mode of fre-\nquencyfe= 3.3GHz. Alternatively, geffcan be extracted\nfrom measurements of MW reflection coefficient |S11|2\nas a function of the Kittel mode frequency ωs/(2π) and\ndriving frequency ωNA/(2π). Fitting |S11|2, which is\nmeasured at temperature of 3K using a vector network\nanalyzer (VNA), with Eq. ( 2) [see Fig. 1(c)] yields the\nvaluegeff= 200MHz, which is pretty much close to the\nvalueobtainedfromsimulation. Note that geffisonlyone\norder of magnitude smaller than the threshold value cor-\nresponding to the super-radiance Dicke instability [44].\nIII. KERR COEFFICIENT AND NONLINEAR\nDAMPING\nCavity driving having amplitude Ω pand angular fre-\nquencyωpis taken into account by adding a term given\nby/planckover2pi1Ωp/parenleftbig\nA†\nee−iωpt+Aee−iωpt/parenrightbig\nto the Hamiltonian HD\n(1). Steady state solution of the driven system was cal-\nculated in Ref. [40] for the case where damping is taken\nintoaccountto firstorderonly. Forthat casethe solution\nis found by solving a cubic equation for the FSR dimen-\nsionless energy Es=/angbracketleftNs/angbracketright[given by Eq. (36) of [40]]. We\nfind, however,thatthe calculatedsteadystate yieldsonly\na moderate agreement with experimental data. Better\nagreement can be obtained by taking into account non-\nlinear damping to cubic order [49]. In this approach the\ncubic equation for Esbecomes\n/parenleftbig\nδ′2\ns+γ′2\ns/parenrightbig\nEs=η|Ωp|2, (3)\nwhereδ′\ns=δs−ηδe+ 2KMEs,δs=ωs−ωpand\nδe=ωe−ωpare driving detuning angular frequencies,\nη=g2\neff//parenleftbig\nδ2\ne+γ2\ne/parenrightbig\n,γe=γ1e+γ2ewithγ1e(γ2e) be-\ning the external (intrinsic) cavity damping rate, γ′\ns=\nγs+ηγe+γ3sEs,γsis the FSR linear damping rate and\nγ3sis the FSR cubic nonlinear damping coefficient. Note\nthat|Ωp|2is proportional to the driving power Ppin-\njected into the LA. Note also that when nonlinear damp-\ning is disregarded (i.e. when γ3s= 0) Eq. ( 3) becomes\nidentical to Eq. (36) of [40].\nVNA measurements of the reflection coefficient |S11|2\nfor three different values of Ppare shown in Fig. 2(a-c).\nFor the data presented in both Fig. 2and Fig. 3, the\nradius of the FSR is Rs= 0.1mm. The theoretical fit3\nFIG. 2: Reflection coefficient |S11|2in dB units for three values of MW input power Pp. Panels (a), (b), and (c) present the\nexperimental data corresponding to MW input powers Ppof -20 dBm, -5 dBm, and +10 dBm, respectively. The second row\n[panels (c), (d), and (e)] shows the corresponding theoreti cal fits that are obtained from Eq. (3). The theoretical fit par ameters\nareγ2e= 1.5 MHz,γe= 4 MHz, γs= 1 MHz, KM= 6.325 nHz, δe= 35MHz , and γ3s= 0.001 nHz. To obtain a proper fit,\nNsandgeffare taken as variable values varying as a function of Pp. ForPp=−20 dBm, −5 dBm, and 10 dBm, Nsvalues are\ntaken as 1 ×1019m−3, 5×1019m−3and 8×1019m−3, andgeffvalues are taken as 14MHz , 14 MHz and 12MHz, respectively.\nshown in Fig. 2(d-f) is based on the cubic equation ( 3),\nwhich allows the calculation of the dimensionless energy\nEs, and on Eq. (3) of Ref. [28], which evaluates the\nreflectioncoefficient |S11|2asafunctionof Es. Thevalues\nof parameters assumed for the calculations are listed in\nthe caption of Fig. 2. Note the driving-induced blue\nshift observed in the magnetic resonance frequency [see\nFig.2(a-c)]. This shift cannot be accurately reproduced\ntheoretically when nonlinear damping is disregarded.\nIV. THERMAL INSTABILITY\nFurther insight can be gained by measuring the spec-\ntral density ISAof the signal reflected off the LA using a\nSA (see Fig. 3). We find that for Pp> Pc= 42.5 dBm,\nand for sufficiently small detuning from resonance, the\nmeasured spectral density ISAcontains equally-spaced\nside-bands (SB) on both sides of the driving frequency\nfp=ωp/(2π) [see Fig. 3(a)]. We measure the SB spac-\ning frequency ωSM/(2π) as a function of the driving fre-\nquencyfpand driving power Pp[see Fig. 3(c)].\nThe observed equally spaced SBs are attributed to\na thermal instability mechanism that is discussed in\nRef. [45]. The phase transition occurring at the Curie\ntemperature Tcbetween the FS and the PS gives rise\nto a sharp change in the resonance modes of the hy-\nbrid cavity-FSR system. Consider the case where the\nfrequency of the externally applied driving is tuned very\nclose to the frequency of one the hybrid system modes.\nWith sufficiently high driving amplitude the temperatureTof the FSR may exceeds the Curie temperature Tcdue\nto driving-induced heating. For that case no steady state\nwithT < T c(i.e. FS) exists. The transition from the FS\ntothePSoccurringat Tcisexpectedtogiverisetoareso-\nnance frequency shift. Consequently the driving-induced\nheating is expected to abruptly drop down, since above\nTcthe frequency detuning between the continuous wave\nexternal driving and the resonance frequency becomes\nlarger (in absolute value). Consider the case where the\nreduced heating gives rise to a temperature drop below\nT < T c. For this case, a steady state with T > T c(i.e.\nPS) also becomes impossible. In the region where no\nsteady state is possible, the temperature is expected to\noscillate around Tc. The frequency of temperature oscil-\nlation can be determined from the spacing between the\nmeasured SBs.\nFor the measurements presented in Fig. 3, the driving\nangularfrequency ωpistunedcloseto ω+. Theanalysisis\ngreatly simplified by disregarding the other hybrid eigen\nmode having angular frequency ω−. This approximation\nis applicable in the strong coupling regime, for which the\nresonances having angular frequencies ω±do not overlap\n[see Eq. ( 2)]. In this approach the FSR-cavity system is\ntreated as a single mode having angular frequency ω+=\n2π×3.32GHz, and Kerr coefficient K+=KMsin4(θg/2)\n[see Eq. ( A9)]. The mode damping rate γ+= 30 MHz is\nexpressed as γ+=γ1++γ2+, whereγ1+is the coupling\ncoefficient between the driven mode and the LA, and γ2+\nis the mode intrinsic damping rate (note that γ1+=γ2+\nfor critical coupling).\nTo account for the observed SB, we consider the ef-4\nFIG. 3: Thermal instability. (a) Spectral density ISAof the\nsignal reflected off the LA, as a function of the detuning fre-\nquencyfd, for the driving frequency fp= 3.2224 GHz and\nnormalized driving power Pp/Pc= 1.288 specified by the\nblack cross overlaid in (c). (b) Spectral density ISAin dB as a\nfunction of the driving frequency fpand detuning frequency\nfdforPp/Pc= 1.7 [indicated by the overlaid horizontal\ndashed line in (c)]. (c) The SB spacing frequency ωSM/(2π)\nin MHz as a function of driving frequency fpand normal-\nized driving power Pp/Pc. The overlaid blue (red) dashed\nline represents the threshold condition EF=EcF(EP=\nEcP). The following values are assumed for the calculations\nω+F/2π= 3.317GHz, ω+P/2π= 3.314GHz, γ+F= 1.3×γ+P,\nσF/wTF= 2.6×σP/wTP, (K+F/γ+F)(wTF/σF) = 0.5 and\nK+P= 0.\nfect of driving-induced heating on the FSR magnetic or-\ndering. The externally applied driving gives rise to a\nheating power Qgiven by Q= 2ℏω+γ2+|B|2, whereB\nis the complex amplitude of the driven mode (note that\nnonlinear damping is disregarded here). It is assumed\nthat the FSR temperature Tis uniform, and that the\ncooling power due to the coupling between the FSR and\nits environment at a base temperature of T0is given by\nH(T−T0), where His the heat transfer coefficient. The\nthermal heat capacity of the FSR is denoted by C. It is\nassumed that all the parameters characterizing the mode\nabruptly change at a critical temperature given by Tc. In\nthe adiabatic (diabatic) region, the mode linear damping\nrateγ+is much smaller (larger) than the thermal decay\nrateH/C.\nIn dimensionless form, system’s time evolution is gov-\nFIG. 4: Limit cycle. (a) Numerical integration of the equa-\ntions of motion (4) and (5) is performed with the following\nparameters Im( wF−wP) =−0.1, Re(wF) =−1, Re(wP) =\n−1.5,σF= 0.01,σP= 0.02, and wTF=wTP= 0.01. The\nvalues of driving detuning frequency Im( wF) and driving am-\nplitudew1=w1F=w1Pare indicated by the black cross in\n(b). The LC is shown in (a) as a closed curve in the complex\nBplane, in (c) as a periodic function of Θ −1 vs. the normal-\nized time τ, and in (d) as a periodic function |B|2vs.τ. The\nplane of driving frequency and driving amplitude is shown in\n(b). No steady state solution exists in the region between th e\nblue and red curves (labeled as A).\nerned by [45]\n˙B=wB−w1, (4)\n˙Θ =σ|B|2−wTΘ. (5)\nOverdot denotes a derivative with respect to a di-\nmensionless time τ, which is related to the time t\nbyτ=γ0t, where γ0is a constant rate. The di-\nmensionless complex frequency wis given by w=/parenleftBig\ni/parenleftBig\nωp−ω+−K+|B|2/parenrightBig\n−γ+/parenrightBig\n/γ0, the dimensionless\ndriving amplitude w1is given by w1=iγ−1\n0√2γ1+Ωp,\nthe dimensionless temperature Θ is given by Θ =\n(T−T0)/(Tc−T0), the dimensionless heating coeffi-\ncientσis given by σ= 2ℏω+γ2+γ−1\n0C−1(Tc−T0)−1,\nand the dimensionless thermal rate wTis given by wT=\n(H/C)/γ0.\nThe normalized parameters w,w1,σandwTare as-\nsumed to have a step function dependence on the tem-\nperature. Below (above ) the critical temperature Tc, i.e.\nfor Θ<1 (Θ>1), they take the values wF,w1F,σF\nandwTF(wP,w1P,σPandwTP), respectively. A steady\nstate (i.e. time independent) solution below (above) the\ncritical temperature Tc, i.e. in the region Θ <1 (Θ>1),\nis possible provided that EF< EcF(EP> EcP), where\nEF=|w1F/wF|2andEcF=wTF/σF(EP=|w1P/wP|2\nandEcP=wTP/σP) [see Eqs. ( 4) and (5) and Fig. 4(b)].\nNote that both EFandEPrepresent steady state values5\nof Eq. (4) for|B|2, whereas both EcFandEcPrepresent\nvalues of |B|2, for which Θ = 1 is a steady state value of\nEq. (5).\nHeat can be removed from the FSR by radiation, ex-\nchange with the surrounding air, and exchange with\nthe supporting ferrules, which hold the FSR inside the\nLGR. The contributions to the total heat transfer co-\nefficient Hdue to radiation, air and the ferrules are\ndenoted by hradSs,hairSsandHfer, respectively, where\nSs= 4πR2\nsistheFSRsurfacearea. Thecoefficient hradis\nroughly given by hrad≃αYIGσSB/parenleftbig\nT4\nc−T4\n0/parenrightbig\n/(Tc−T0),\nwhereαYIGis the averaged FSR absorption coefficient\nin the spectral band corresponding to room tempera-\ntureT0≃300K radiation (wavelength λ≃10µm),\nσSB=π2k4\nB//parenleftbig\n60/planckover2pi13c2/parenrightbig\nis the Stefan-Boltzmann constant,\nkBis the Boltzmann’s constant, /planckover2pi1is Plank’s constant,\nandTc= 560K is the YIG Curie temperature. The ab-\nsorption coefficient value αYIG≃10−1[50] yields hrad≃\n2Wm−2K−1. For ambient temperature and pressure\nhair≃15Wm−2K−1, hence ( hrad+hair)Ss(Tc−T0)≃\n0.6mW for a FSR having radius Rs= 0.1mm. In the\nregion where SB are observed the induced heating power\nappliedtotheFSRisabout3ordersofmagnitudeslarger,\nhenceH≃Hfer, i.e. both radiation and air have negligi-\nbly small contributions, and thus heat is mainly removed\nby the ferrules.\nThe thermal heat capacity of a FSR having radius\nRs= 0.1mm and volume Vs= 4πR3\ns/3 is given by\nC= 2.9×106JK−1m−3×Vs= 1.2×10−5JK−1[51],\nhence the thermal decay rate is roughly given by H/C≃\n320Hz×(Qc/W)((Tc−T0)/(260 K))−1, whereQcisthe\nheating power applied to the FSR, for which the steady\nstate temperature is Tc. Hence for the current device\n(H/C)/γ+≃10−5, and thus the diabatic approximation\nis applicable.\nA typical limit cycle (LC) in the diabatic regime is\nshown in Fig. 4. The LC is calculated by numerically\nintegrating the equations of motion ( 4) and (5). The\nblue (red) cross shown in Fig. 4(a) indicates the steady\nstate value w1/wofBcorresponding to the FS (PS), i.e.\nfor Θ<1 (Θ>1), and the blue (red) circle represents\nthe relation |B|2=EcF(|B|2=EcP). In the plane of\ndriving frequency and driving amplitude, which is shown\nin Fig.4(b), the blue and red curves are derived from the\nrelations EF=EcFandEP=EcP, respectively. In the\nregion labeled as A, no steady state solution to Eqs. ( 4)\nand (5) exists. The LC period time τLCcan be calculated\nby integrating Eqs. ( 4) and (5) over a single period. In\nthe diabatic limit, one finds that τP≃ |wP|−1+|wF|−1.\nThe measured value of LC frequency roughly agrees with\nthis theoretical estimation.\nV. SUMMARY\nIn summary, we demonstrate that relatively large cou-\npling coefficient geffcan be obtained by employing anLGR having mode volume much smaller than λ3\ne. The\nresponse of the system in the weak nonlinear regime al-\nlows the extraction of the Kerr coefficient KMand the\ncubic nonlinear damping rate γ3s. An instability is re-\nvealed by driving the system with a relatively high input\npower. Above the instability threshold the response of\nthe system to an externally applied monochromatic driv-\ning exhibits self-modulation. The instability, which is\nattributed to driving-induced heating, occurs in a region\nwhere the response has no steady state value. Further\nstudy will be devoted to developing sensors that exploit\nthis instability for performance enhancement.\nVI. ACKNOWLEDGMENTS\nThis work was supported by the Israeli science founda-\ntion, the Israeli ministry of science, and by the Technion\nsecurity research foundation.\nAppendix A: Rotating wave approximation\nThe Hamiltonian ( 1) can be expressed as\n/planckover2pi1−1HD=/parenleftbig\nA†\neA†\ns/parenrightbig\nM/parenleftbigg\nAe\nAs/parenrightbigg\n+KMN2\ns,(A1)\nwhere the 2 ×2 matrix Mis given by\nM=/parenleftbigg\nωegeff\ngeffωs/parenrightbigg\n. (A2)\nThe eigenvalues ω±of the matrix Mare given by ω±=\nωm±/radicalbig\nω2\nd+g2\neff[see Eq. ( 2)], where ωm= (ωe+ωs)/2\nandωd= (ωe−ωs)/2. The matrix Mcan be expressed\nas\nM=ωm/parenleftbigg\n1 0\n0 1/parenrightbigg\n+/radicalBig\nω2\nd+g2\neff/parenleftbigg\ncosθsinθ\nsinθ−cosθ/parenrightbigg\n,\n(A3)\nwhere\ntanθ=geff\nωd. 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Chubykalo-Fesenko2\n1Department of Physics, University of York, Heslington, York YO10 5DD United Kingdom and\n2Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain\n(Dated: May 31, 2022)\nUltrafast laser-induced magnetic switching in rare earth-transition metal ferrimagnetic alloys has\nrecently been reported to occur by ultrafast heating alone. Using atomistic simulations and a\nferrimagnetic Landau-Lifshitz-Bloch formalism, we demonstrate that for switching to occur it is\nnecessary that angular momentum is transferred from the longitudinal to transverse magnetization\ncomponents in the transition metal. This dynamical path leads to the transfer of the angular\nmomentum to the rare earth metal and magnetization switching with subsequent ultrafast precession\ncaused by the inter-sublattice exchange \feld on the atomic scale.\nThe behavior of magnetization dynamics triggered by\nan ultrafast laser stimulus is a topic of intense research\ninterest in both fundamental and applied magnetism [1].\nA range of studies using ultrafast laser pulses have shown\nvery di\u000berent timescales of demagnetization for di\u000berent\nmaterials; from 100 fs in Ni [2] to 100 ps in Gd [3]. Any\npotential applications utilizing such a mechanism would\nrequire, not only ultrafast demagnetization, but also con-\ntrolled magnetization switching.\nMagnetization reversal induced by an ultrafast laser\npulse has been reported in the ferrimagnet GdFeCo, to-\ngether with a rich variety of phenomena [4{8]. Several\nhypotheses have been put forward to explain the ob-\nserved magnetization switching: crossing of the angu-\nlar momentum compensation point [4], the Inverse Fara-\nday E\u000bect [5], and its combination with ultrafast heat-\ning [6]. It has been shown that the rare earth (RE)\nresponds more slowly to the laser pulse than the tran-\nsition metal (TM) [7], even though the sublattices are\nstrongly exchange coupled. Intriguingly, Radu et al. [7]\nshow experimentally and theoretically the existence of a\ntransient ferromagnetic-like state, whereby the two sub-\nlattices align against their exchange interaction, existing\nfor a few hundred femtoseconds. Recently [8], the atom-\nistic model outlined in [7, 8] predicted the phenomenon\nof magnetization reversal induced by heat alone, in the\nabsence of any external \feld; a prediction veri\fed experi-\nmentally. This remarkable result opens many interesting\npossibilities in terms of ultrafast magnetization reversal\nand potential areas of practical exploitation, however a\ncomplete theoretical understanding of this e\u000bect is cur-\nrently missing.\nIn magnets consisting of more than one magnetic\nspecies, excitation of the spins on a time scale compa-\nrable with that of the inter-sublattice exchange takes the\nsublattices out of equilibrium with each other. It is in\nthis regime where the thermally driven switching of fer-\nrimagnetic GdFeCo occurs. A recent study by Mentink\net al. [10] proposed an explanation of the process using a\nphenomenological model of the magnetization dynamics,\nwhich assumes the additive character of two relaxation\nmechanisms: one governed by the inter-sublattice ex-change and another by the relativistic contribution (cou-\npling to external degrees of freedom). The model is based\non the physically plausible argument that the switching\nis driven by angular momentum transfer in the exchange-\ndominated regime. However, the assumption of a linear\npath to reversal allows the angular momentum transfer\nto occur through longitudinal components only, since the\nperpendicular components are neglected. Additionally,\nthe dynamical equation in Ref. [10] was derived from\nthe Onsager principle, generally valid for small devia-\ntions from the equilibrium only. Thus far, a complete\nexplanation of the heat driven, ultrafast reversal process\nremains illusive.\nIn this Letter we demonstrate that the switching of\nmagnetization in a ferrimagnet after femtosecond heat-\ning is due to the transfer of angular momentum from the\nlongitudinal to the transverse magnetization components\nin the TM and consequent transfer of the angular mo-\nmentum through perpendicular components to the RE.\nWe present a general formalism, leading to a macroscopic\ndynamical equation for a ferrimagnet. This is in the form\nof a Landau-Lifshitz-Bloch (LLB) equation, in which, un-\nlike the phenomenological model of Ref. 10, the two re-\nlaxation mechanisms are not additive. Our theory gives\nthe non-equilibrium conditions necessary for this angu-\nlar momentum transfer to happen and thus to produce\nthe precessional rather than linear reversal as suggested\nin Ref. 10. These predictions are supported by calcula-\ntions using an atomistic model based on the Heisenberg\nexchange Hamiltonian with Langevin dynamics.\nIn the absence of any external stimulus, the energet-\nics of the atomistic spin model are described purely by\nexchange interactions, given by the spin Hamiltonian:\nH=\u0000X\nj0:\nHk\nT'\u0000TRm0\nR\nme\nRfor the case before the heat pulse is re-\nFIG. 3. (a) Precession of sublattice magnetizations around\nthe exchange \feld of each other in the macroscopic (LLB) de-\nscription. After the action of an ultrafast laser pulse the large\namplitude of the TM precession causes it to cross mz= 0, and\nfor su\u000eciently large angular momentum transfer, the angle\nbetween sublattices becomes small. After cooling the dom-\ninance of the TM sublattice forces the RE to realign along\nthe opposite direction, completing the switching process. (b)\nTrajectories of the parallel and transverse magnetization com-\nponents for TM calculated via atomistic simulations of the\nHeisenberg model (1) at di\u000berent maximum pulse tempera-\nturesTmax= 1000;1200;1250;1300;1350 and 1400 K. After\nthe pulse, the temperature is removed, this moment is indi-\ncated by small circles.\nmoved (mT>me;T= 0) and because after the heat pulse\nis gone the system cools down Hk\nT'[\u0000TT\u0000\u0000TR]=2>0\nwithmT\u001cme;T(T). The LLB equation for the TM is\nreduced to the following system of equations:\ndm2\nT\ndt= 2j\rTj\u000bk\nTHk\nTm2\nT;\nd\u001a\ndt=\u00002h\n\u000b?\nT\nTp\n1\u0000\u001a=m2\nT\u0000j\rTj\u000bk\nTHk\nTi\n\u001a(5)\nwhere\u001a= (mt\nT)2= (mx\nT)2+ (my\nT)2is the TM trans-\nverse magnetization component, \n T=m0\nRj\rTjjJ0;TRj=\u0016T\nis the precessional frequency of the anti-ferromagnetic\nexchange mode.\nThe trajectory \u001a= 0 corresponds to a linear dynamical\nmode. The standard analysis of the dynamical system\n(5) shows that for Hk\nT>0 andmz\nT< \u000b?\nT\nT=(j\rTjHk\nT)\nthis trajectory becomes unstable. Before the end of the\npulse it is equivalent to mT>(\u000b?\nT=\u000bk\nT)me;Twhich is also\neasily satis\fed, taking into account that \u000b?\nT> \u000bk\nT, see\nRef. [14]. The physical interpretation is that in this\ncase very small perturbations from \u001a= 0 will not be\ndamped but will lead to the development of a perpendic-\nular magnetization component, as is indeed observed by\nthe atomistic simulations Fig. 3(b), in which we use the\natomistic model and apply heat pulses of di\u000berent tem-\nperatures to drive the system into di\u000berent states. The\natomistic simulations clearly con\frm the development of\nthe perpendicular component.\nHowever, the dynamical system (5) alone does not de-\nscribe the reversal due to the assumption of the static4\nRE magnetization. In the same approximation, the LLB\nequation for the RE reads:\ndmx(y)\nR\ndt=\u0006\nRmy(x)\nT\u0000\u000b?\nR\nm0\nR\nRmx(y)\nT\u0000j\rRj\u000bk\nRHk\nRmx(y)\nR\n(6)\nwhere the upper sign corresponds to the equation\nformx\nRand the lower sign for the my\nRone, \nR=\nzqm0\nRj\rRjjJTRj=\u0016RandHk\nRis the RE longitudinal \feld.\nEquation (6) shows that the perpendicular motion of the\nTM triggers the corresponding precessional motion of the\nRE via the angular momentum transfer (the \frst two\nterms of Eq. (2), i.e. via perpendicular components)\nwith the same frequency \n T, but di\u000berent amplitude,\nsee Fig. 3(a). During this dynamical process in some\ntime interval the RE and TM magnetization have both\nthe same sign of the z-component, forming the transient\nferromagnetic-like state seen experimentally [7]. Note\nthat the subsequent precession has a frequency which is\nproportional to the exchange \feld and thus is extremely\nfast. The motion of the TM around RE direction and\nvice versa occurs during and after the ferromagnetic-like\nstate until the system has relaxed to equilibrium.\nAn outstanding question is whether the magnetiza-\ntion precession, a central part of the process, can be\nobserved experimentally on a macroscopic sample. We\nshould recall that in non-equilibrium at high tempera-\ntures the correlation between atomic sites is weak, thus\nwe cannot expect the precession to occur with the same\nphase in the whole sample; an e\u000bect which would make\nthe precession macroscopically unobservable. To demon-\nstrate the e\u000bect we present in Fig. 4 the results of atom-\nistic switching simulations in GdFeCo for di\u000berent system\nsizes (Tmax= 2000 K). In Fig. 4 we observe that for small\nsystem sizes transverse oscillations with the frequency of\nan exchange mode are visible, consistent with the predic-\ntion of our analytical model. However, in large system\nsizes of the order of (20 nm)3it is averaged out, con-\nsistent with the excitation of localized exchange modes\nwith random phase. Note that the same e\u000bect happens\nfor very high temperatures where the observed magne-\ntization trajectory appears close to linear; although we\nstress again the importance of a small perpendicular com-\nponent to initiate the magnetization reversal, which will\noccur on a local level as demonstrated by Fig. 4.\nIn conclusion, the LLB equation for a ferrimagnet de-\nscribes the mutual relaxation of sublattices which oc-\ncurs simultaneously under internal damping and inter-\nsublattice exchange. This model allows us to present a\nsimple picture of the magnetization reversal of GdFeCo\nin response to an ultrafast heat pulse alone. The physical\norigin of this e\u000bect is revealed within the LLB equation\nas a dynamical reversal path resulting from the insta-\nbility of the linear motion. To trigger the reversal path\na small perpendicular component is necessary. In prac-\ntice this will arise from random \ructuations of the mag-\n-0.8-0.6-0.4-0.20.00.20.40.60.8\n 0 5 10 15 20 25 30mx\nTime [ps](3nm)3\n(9nm)3\n(20nm)3FIG. 4. Atomistic modeling of the system size dependence\nof the transverse magnetization components of the TM un-\nder ultrafast switching, showing cancelation of the localized\ntransverse magnetization components arising from exchange\nprecession for larger system sizes. The time t= 0 corresponds\nto the end of the laser pulse.\nnetization at elevated temperatures. The perpendicular\ncomponent grows in time resulting in ultrafast magne-\ntization precession in the inter-sublattice exchange \feld,\nalso observed in atomistic simulations for small system\nsizes. The switching is initiated by the TM which ar-\nrives at zero magnetization faster than the RE and re-\nsponds dynamically to its exchange \feld. Thus, the non-\nequivalence of the two sub-lattices is an essential part of\nthe process. Switching into the transient ferromagnetic\nstate occurs due to large-amplitude precessional motion\nof the TM in the exchange \feld from the RE and a slow\ndynamics of RE.\nThis work was supported by the European Commu-\nnity's Seventh Framework Programme (FP7/2007-2013)\nunder grant agreements NMP3-SL-2008-214469 (Ultra-\nMagnetron) N 214810 (FANTOMAS), NNP3-SL-2012-\n281043 (FEMTOSPIN) and the Spanish Ministry of Sci-\nence and Innovation under the grant FIS2010-20979-C02-\n02.\n[1] J. St ohr, and H. C. Siegmann, Magnetism: from Funda-\nmentals to Nanoscale Dynamics (Springer, Berlin, 2006).\n[2] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot,\nPhys. Rev. Lett. 76, 4250 (1996).\n[3] M. Wietstruk, A. Melnikov, C. Stamm, T. Kachel, N.\nPontius, M. Sultan, C. Gahl, M. Weinelt, H. A. D urr, and\nU. Bovensiepen Phys. Rev. Lett., 106, 127401 (2011),\n[4] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, Th. Rasing, Phys. Rev. Lett. 99,\n047601 (2007).\n[5] F. Hansteen, A. Kimel, A. Kirilyuk and Th. Rasing,\nPhys. Rev. Lett. 95, 047402 (2005).\n[6] K. Vahaplar, A. M. Kalashnikova, A. V. Kimel, D.\nHinzke, U. Nowak, R. Chantrell, A. Tsukamoto, A.\nItoh, A. Kirilyuk and Th. Rasing, Phys. Rev. Lett. 103,5\n117201 (2009).\n[7] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius,\nH. A. D urr, T. A. Ostler, J. Barker, R. F. L. Evans, R.\nW. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, Th.\nRasing and A. V. Kimel, Nature, 472, 205 (2011).\n[8] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell,\nU. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui, L.\nLe Guyader, E. Mengotti, L. J. Heyderman, F. Nolting,\nA. Tsukamoto, A. Itoh, D. Afanasiev, B. A. Ivanov, A.\nM. Kalashnikova, K. Vahaplar, J. Mentink, A. Kirilyuk,\nTh. Rasing, and A. V. Kimel, Nature Commun. 3, 666\n(2012).\n[9] T. A. Ostler, R. F. L.Evans, R. W. Chantrell, U. Atxitia,\nO. Chubykalo-Fesenko, I. Radu, R. Abduran, F. Radu,\nA. Tsukamoto, A. Itoh, A. Kirilyuk, Th. Tasing and A.\nKimel, Phys. Rev. B. 84, 024407 (2011).\n[10] J. H. Mentink, J. Hellsvik, D. V. Afanasiev, B. A. Ivanov,A. Kirilyuk, A. V. Kimel, O. Eriksson, M. I. Katsnelson,\nand Th. Rasing, Phys. Rev. Lett. 108057202 (2012).\n[11] U. Atxitia, O. Chubykalo-Fesenko, J. Walowski, A. Mann\nand M. M unzenberg, Phys. Rev. B 81, 174401 (2010).\n[12] M. Sultan, U. Atxitia, A. Melnikov, O. Chubykalo-\nFesenko and U. Bovensiepen, Phys.Rev.B 85, 184407\n(2012)\n[13] D. Garanin, Phys. Rev. B 55, 3050 (1997).\n[14] U. Atxitia, P. Nieves and O. Chubykalo-Fesenko, Phys.\nRev. B 86, 104414 (2012)\n[15] O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell and\nD. Garanin, Phys. Rev. B 74, 094436 (2006).\n[16] R. F. L. Evans, D. Hinzke, U. Atxitia, U. Nowak, R. W.\nChantrell and O. Chubykalo-Fesenko, Phys. Rev. B 85,\n014433 (2012)\n[17] U. Atxitia and O. Chubykalo-Fesenko, Phys. Rev. B 84,\n144414 (2011)" }, { "title": "1506.07028v1.Design_of_compensated_ferrimagnetic_Heusler_alloys_for_giant_tunable_exchange_bias.pdf", "content": "arXiv:1506.07028v1 [cond-mat.str-el] 23 Jun 2015Design of compensated ferrimagnetic Heusler alloys for\ngianttunableexchangebias\nAjayaK.Nayak1∗,MichaelNicklas1,StanislavChadov1,PanchananaKhuntia1,ChandraShekhar1,\nAdel Kalache,1, Michael Baenitz1, YuriiSkourski2, Veerendra K. Guduru3, Alessandro Puri3, Uli\nZeitler3, J.M. D. Coey4&ClaudiaFelser1∗\n1Max Planck Institutefor Chemical Physicsof Solids, N¨ othn itzerStr. 40, D-01187 Dresden, Ger-\nmany\n2Dresden High Magnetic Field Laboratory (HLD), Helmholtz-Z entrum Dresden-Rossendorf, D-\n01328Dresden, Germany\n3High Field Magnet Laboratory, Institute for Molecules and M aterials, Radboud University Ni-\njmegen,Toernooiveld7, NL-6525ED Nijmegen,TheNetherlan ds\n4School ofPhysicsand CRANN, TrinityCollege, Dublin2, Irel and\nThe discovery ofmaterials withimprovedfunctionality can beaccelerated by rational mate-\nrial design.1Heusler compounds with tunable magnetic sublattices allow to implement this\nconcepttoachievenovelmagneticproperties.2Here,wehavedesignedafamilyofHeusleral-\nloyswithacompensatedferrimagneticstate. Inthevicinit yofthecompensationcomposition\nin Mn-Pt-Ga, a giant exchange bias (EB) of more than 3 T and a si milarly large coerciv-\nity are established. The large exchange anisotropy origina tes from the exchange interaction\nbetween the compensated host and ferrimagnetic clusters th at arise from intrinsic anti-site\ndisorder. We demonstrate the applicability of our design co ncept on a second material, Mn-\n1Fe-Ga, witha magnetic transition aboveroom temperature, e xemplifying the universality of\nthe concept and the feasibility of room-temperature applic ations. Our study points to a new\ndirection for novel magneto-electronic devices. At the sam e time it suggests a new route for\nrealizing rare-earth free exchange-biased hard magnets, w here the second quadrant magne-\ntizationcanbe stabilizedby the exchangebias.\nExchange bias corresponds to a shift of the hysteresisloop o f a ferromagnet along the mag-\nnetic field axis due to interfacial exchange coupling with an adjacent antiferromagnet.3,4It is em-\nployed for a variety of technological applications3–10, including magnetoresistive read heads and\nsensors. Oneofthemostimportantusesofexchangebiasis in spin-valvedevices,where an artifi-\ncialantiferromagnetexchange-biasedbyanormalantiferr omagnetallowsmagnetoresistancetobe\ndeveloped in a small field, stabilizing the magnetic state ag ainst repeated cycling.9Although the\noriginofexchangebiasisstillasubjectofdiscussion,its phenomenologyhasvariouslybeeninter-\npreted in terms of rough ferromagnetic (FM)- antiferromagn etic (AFM) interfaces11,12, a domain\nstatemodel13anduncompensatedinterfacialspins14,15. AlthoughbothFM andAFM subsystems\nare inseparable parts of an EB system, it is the latter that de termines the magnitude of the EB in\nthe system.15–17Therefore, it is important to search for an appropriate magn etically compensated\nmaterial to observe a maximum effect. Besides the EB phenome non, magnetically compensated\nmaterials are proposed to play a vital role in antiferromagn etic spintronics18,19and all-optical\nswitchingdevices20. Singlespinsuperconductivityhasevenbeenpredictedina spin-compensated\nhalf-metal.21Inthisletterwepresentadesignschemeforamagnetically- compensatedferrimagnet\nand demonstrate the relevance of this compensated magnetic state in realizing a giant exchange\n2biasand alarge coercivity.\nHeuslercompounds,whichprovidethematerialbaseforourd esignscheme,arewellknown\nfor their multi-functional properties, such as the giant ma gnetocaloric effect22, the field-induced\nshape-memory effect23and the topological insulating property24. The story of their success is\nbased onthefact thatnew materialscan bedesignedintheflex iblestructureoftheHeuslerfamily\non the basis of simple rules taking into account the position of the atoms, the number of valence\nelectrons,thedegreeofatomicdisorderandthestrengthof theexchangeinteractions2. Oneofthe\nfamous examples of valence electron count is the robustness of the Slater-Pauling rule, typically\nconnectedtohalf-metallicity,intheL2 1cubicHeuslercompounds.25Togetherwiththefactthatin\nHeusler compounds Mn develops a strong localized magnetic m oment, it can be used as a guide\nto search for compensated ferrimagnets in the Heusler famil y with 24 valence electrons. On the\nother hand, this provides us only a preliminary guide, since the compensated ferrimagnetic state\ntypically cannot be combined with half-metallicity. The cl assical example is Mn 3Ga26, which\nis predicted to exhibit a half-metallic state in the L2 1cubic phase, but the cubic phase does not\nform in the bulk due to a strong electronic instability leadi ng to a tetragonal strain breaking both\nthe compensated and half metallic states. However, either a half-metallic or a compensated state\ncan berecoveredbychemicalsubstitution.27Here, wefocusonthecompensatedmagneticstatein\nHeuslermaterials.\nCombining Mn 3Ga with another material of opposite net magnetization, Mn 2PtGa, forms\nan excellent starting point to design a zero-magnetization structure based on a compensated fer-\n3rimagnetic state. Mn 2PtGa consists of two non-equivalent types of Mn, one in the Mn -Ga and\nanother in the Mn-Pt planes of the inverse tetragonal struct ure (space group I-4m2)28. The Mn\nsitting in Mn-Ga planes possesses a higher magnetic moment d ue to its more localized nature.\nIt couples antiferromagnetically to the Mn in Mn-Pt planes. This configuration results in a net\nuncompensated magnetization of about 0.5 µB/f.u.arising from the larger moment of the Mn in\nMn-Ga planes.28On the other hand, the complete replacement of Pt by Mn to form Mn3Ga(space\ngroupI4/mmm)resultsinoneMnintheMn-GaandtwoMnintheMn-Mnplanes(f ormerMn-Pt\nplane). Thus, Mn 3Ga displaysa net uncompensated magnetizationof about 1 µB/f.u.of opposite\nsign to that in Mn 2PtGa.26,29Combining these two materials then suggests that we can crea te a\nfully compensated magnet with a compensation point for a par ticular Mn/Pt ratio. This design\nscheme is schematically depicted in Fig. 1. From first-princ iples calculations, it follows that the\ncritical composition with the zero magnetization is achiev ed in the solid solution Mn 3−xPtxGa at\na Pt content of about x0= 0.59, which is in good agreement with the experimental findings. O n\noptimizing the Mn/Pt ratio in Mn 3−xPtxGa, we always find a small lack of compensation in the\nmaterial, due to the formation of FM clusters by anti-site di sorder. This leads to an exceptionally\nlarge bulk EB and a large coercivity. In contrast to an artific ial antiferromagnet, which is a thin\nfilmstructurecomposedoftwoferromagneticlayersseparat edbyacouplinglayer9,herewecom-\nbine two isostructural ferrimagnetic compounds Mn 3Ga and Mn 2PtGa to obtain an intrinsically\nanisotropiccompensatedmagneticstateon an atomicscalei na bulkmaterial.\nIn order to characterize the magneticproperties of Mn 3−xPtxGa we have measured the tem-\nperature dependence of the low field magnetization, M(T). We find a systematic increase in the\n4ferrimagnetic N´ eel temperature ( TN) with increasing Mn content as shown in Fig. 2. The irre-\nversibility between the ZFC and FC curves reflects the appear ance of coercivity. We suggest that\nFM clusters embedded in the compensated host are the source o f this irreversibility. NMR mea-\nsurementsconfirmthattheseclustersoriginatefromrandom swapsbetweenPtintheMn-Ptplanes\nand Mn in the Mn-Ga planes (Supplementary Fig. 2). The irreve rsibility between ZFC and FC\nM(T)curves increases with increasing magnetic field demonstrat ing that cooling in higher fields\nhelpstheFM clusterstogrow insize(SupplementaryFig. 3).\nTheZFCmagnetizationat5K showsan unsaturatedbehaviorfo rmagneticfieldsupto14T\n(inset of Fig. 2), therefore, higherfields are required in or der to gain a better understandingof the\nmagnetic state of the system. Pulsed-field magnetization me asurements at 4.2 K up to a field of\n60TareshowninFig. 3a. For x= 0.5,0.6,and0.7thehysteresisloopscloseatafieldof 35−40T.\nThe non-saturating magnetization up to 60 T reflects the stre ngth of the dominant intersublattice\nMn-Mn exchange in the polycrystalline compensated ferrima gnetic host. The hysteresis reveals\nthat there is a FM component of 0.1−0.3µBin the compensated host, which may be compared\nwith its ferromagnetic collinear saturation of about 6µB/f.u.. We further obtain coercive fields of\nµ0HC= 2.2, 3.6, and 3.0 T at 4.2 K for x= 0.5, 0.6 and 0.7, respectively. These values are\ncomparable to the maximum HCobserved in rare-earth-based hard magnetic bulk materials . To\ninvestigatetheexchangeinteractionbetweentheFMcluste rsandAFMhost,wehavemeasuredFC\nhysteresisloops. TheFChysteresisloopstakenat4.2Kinad c-fieldof ±32TaredepictedinFig.\n3b. Theloopsfor x= 0.5,0.6,and0.7displayalargeshiftinthenegativefielddirec tionindicating\nthe existence of a large unidirectional anisotropy and a lar ge exchange bias. The hysteresis loops\n5close in a field of only 20−25T for all samples. Interestingly, the FC hysteresis loops re corded\nafter field cooling in 15 T and 25 T follow a similar path. This i mplies that a cooling field of\nµ0HCF= 15T is sufficientto saturatetheFM momentsand alarger cooling field does notfurther\nchange the magnetic state of the sample (inset of Fig. 3). For all samples the FC M(H)loops\npassalmostthroughtheorigininthethirdquadrant,result ingnearlysamecoercivityandexchange\nbias field. The EB increases monotonically for µ0HCFup to 10 T and then saturates for further\nincreasing coolingfields. It may beworthwhileto mentionth atfield of10 T is nearly sufficientto\nsaturatetheexchangebias,but thefull32 T isrequired tofu lly closetheloop.\nInMn3−xPtxGa,weobtainamaximumEBof HEB= 3.3Tforx= 0.6. Thisvalueisamong\nthe largest EB reported so far. The dependence of the EB and co ercivity on the Pt concentration\nmeasured after field cooling the sample in 15 T is displayed in the inset of Fig. 4a. Both, the EB\nfield (HEB) and the coercive field ( HC) exhibit a clear maximum around x= 0.6. We observe\na small increase in HCfor the FC loops in comparison to the ZFC loops, as generally f ound for\nmaterials showing an EB effect.4,5,30To demonstrate the temperature dependence of the EB we\nhavemeasuredhysteresisloopsatdifferenttemperaturesa fterfieldcoolingin5TforMn 2.4Pt0.6Ga\nand Mn 2.5Pt0.5Ga. The EBshowsa monotonicdecrease with increasing temper aturethatvanishes\naroundTN, asshowninFig. 4a.\nTheextremelylargeEBatlowtemperaturesintheMn-Pt-GaHe usleralloysisagoodstarting\npoint for achieving large EB in compensated materials. Ther efore, to establish a comprehensive\npicture we applied our design concept to another Heusler sys tem in the Mn-Ga family, Mn-Fe-\n6Ga, where the ordering temperature of the system can be varie d over a wide range to above room\ntemperature by tuning the composition (Supplementary Fig. 7-Fig. 14). Furthermore, Mn 2FeGa\nis nearly in a compensated magnetic state, whereas, Fe 2MnGa is ferromagnetic. By adjusting\nthe composition we can achieve a material with ferromagneti c clusters in an almost compensated\nferrimagnetic host and a N´ eel temperature aboveroom tempe rature. The temperature dependence\nofthe EBfor Mn 1.5Fe1.5Ga and Mn 1.8FeGa are shownin Fig. 4b. Mn 1.5Fe1.5Gaexhibitsan EB of\n1.1 T at 2 K, which is observableup to room temperature. ForMn 1.8FeGa we find an EB of 1.2 T\nand 0.04T at 2K and 300K, respectively.\nOurexperimentsshowthatmagneticcompensationofthehost isanimportantrequirementto\nachievealargeexchangebias. Wenowintroduceamodeltoexp laintheseobservations. Assuming\nferromagnetic clusters in the compensated ferrimagnetic h ost that are coherent with the host and\ncoupledtoitviaexchangebondsattheinterface, wecanelab oratetheconditionsfortheexchange\nbias. We assume that the FM clusters and their surroundings a re part of a single crystalline grain\nwith a uniform orientation of the uniaxial magnetocrystall ineanisotropy. By applying an external\nmagnetic field which is large enough to rotate the magnetizat ion in the ferromagnetic clusters,\nbut small enough not to rotate the compensated ferrimagneti c host, we find that the EB is well\ndescribed by\nH≈12JABSASB[1−cos(θK−θB)]·L2n2/3+KBsin2(θK−θB)·L3n\nδSA·cosθK·NA+SBcosθB·L3n. (1)\n7whereSA(B)are the amplitudes of the atomic magnetic moments in the host (cluster), δSA\naccounts for the imperfect magnetic compensation in the hos t;NA(B),ABare the number of atoms\ninside the host (cluster) and at the cluster/host interface , respectively; JABis the exchange cou-\npling constant at the cluster/host interface. KA,B>0are the anisotropy constants for the host\nand the cluster, θK,A,Bare the polar angles referring to the orientation of the anis otropy axis and\nmagnetizationoftheclusterand ofthehost,respectively.\nIn case of small FM clusters ( L3n≪NA) the key role is played by the compensationof the\nhost. TheEBfield can beexpressedas,\nH(M)≈α\nM+β, (2)\nwhereMis the total magnetization; αandβare parameters depending on the interface\nexchange-coupling,theparticularformoftheinterfaces, andthenumber,distributionandmagnetic\nmoments of the embedded ferromagnetic clusters. The theore tical estimate agrees well with the\nexperimentaldatathatshowanearlyinverserelationshipb etweenthe HEBandmagnetization(see\nFig. 1a).\nTo estimatethemaximumexchange-bias field Eq. 1 can be expre ssed in terms ofthe sizeof\ntheferromagneticcluster( L),\n8Hmax[T] = 0.21·(1/L[˚A])·105+0.54, (3)\nwhere the first term appears from the interface exchange coup ling and the second, from the\nmagnetic anisotropy (Supplementary). In order to achieve t he exchange bias fields of 3T, the\ncharacteristicFM clustersizemustnotexceed L∼104˚A.\nA key experimental observation is that at low temperature th e exchange bias is equal to the\ncoercivityin all samples. This can be explainedby consider ing additionalconditionsfor themag-\nnetization switching. For simplicity, we consider only tho se grains which are oriented differently\nfrom the polar angle, θK=π/2. Such grains exhibit a sharp rectangular-like hysteresis, in which\nthe coercivity corresponds to a regime of fast domain-wall m otion. This rapid process is the tran-\nsition between two states: one, comprising the domain wall ( induced by the competition between\nthe strong negative magnetic field and the positive exchange pinning at the host/cluster interface)\nand the new one, which is magnetically uniform. The eliminat ion of the domain wall occurs for\nthesmallexternalfieldvalues( H <0)at whichtheenergies ofthesetwostatesbecomecompara-\nble. Thedetailedtheoreticalinterpretationofourfinding scanbefoundinsupplementarymaterial.\nFrom this model we estimate that H+\nCis only about -1 mT, where the cluster size Lis the lead-\ning contribution. Taking into account that the measured hys teresis is the statistical sum over all\norientations, H+\nCwillhavean evensmalleramplitude.\nTo conclude, we have established a new approach to design com pensated ferrimagnetic\n9Heusler alloys. A small lack of compensation allows us to ach ieve an extremely large EB and\nmatching coercivity. A simple model allows us to quantify th e experimental results that show\nan inverserelationship between the total magnetization an d magnitudeof the exchange-bias field.\nThe present finding provides new insight into the mechanism o f EB and suggests exploring sim-\nilar coherent materials that can be implemented in thin film. Furthermore, the achievement of a\ncoercivity of morethan 3 T based on exchange anisotropy prop oses a new approach to permanent\nmagnet design. Permanent magnets are expected to deliver a l arge flux density. Their working\npoint,stabilizedbycoercivity,fallsnaturallyinthesec ondquadrantofthehysteresisloop,because\nofthedemagnetizingfield. Thepresentworkoffersanaltern ativeapproach-tostabilizethesecond\nquadrant magnetization by exchange bias. However, to reali zable in practical applications it must\nbecoupledwithalargermagnetizationinordertoproduceau sefulfluxdensity. Anon-percolating\nferromagnetic fraction approaching 30%, rather than aroun d 3% in Mn-Pt-Ga and less than 12%\nin Mn-Fe-Ga, is needed. The balance between the volume of FM c lusters and compensated host\nneeds to be optimized to achieve a large exchange bias as well as a large magnetization. Hence,\nthe present finding is a good starting point for designing rar e-earth free hard magnets comprising\nof two hard magnetic phases, which is different form the conv entional exchage spring approach.\nOur design concept together with the flexibility and huge num ber of materials crystallizing in the\nHeusler structure indicate the high potential in designing new materials with a large EB and a\nlarge coercivity at room temperature. Finally, the large EB and coercivity in the nearly compen-\nsatedHeusleralloysmaysignificantlyexpandtheimportanc eofthesematerialsinthephysicsand\nmaterialsciencecommunityduetohigh potentialforantife rromagneticspintronics.\n101. Eberhart, M. E. &Clougherty, D. P. Looking for design in materials design. Na t. Mater. 3,\n659-661(2004).\n2. Graf,T.,Felser,C. &Parkin,S.S.P.SimplerulesfortheunderstandingofHeusle rcompounds.\nProg. Solid StateChem. 39,1 (2011).\n3. Meiklejohn, W. H. &Bean, C. P. New Magnetic Anisotropy. Phys. 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Tetragonal phase of epitaxial room-tem perature antiferromagnet CuMnAs.\nNatureCommunications 4, 2322(2013).\n19. Soh, Y. &Kummamuru, R. K. Spintronics in antiferromagnets. Phil. Tr ans. R. Soc. A 369,\n3646-3657(2011).\n1220. Kimel,A.V.,Kirilyuk,A.,Tsvetkov,A.,Pisarev,R.V. &Rasing,Th.Laser-induced ultrafast\nspinreorientationin theantiferromagnet TmFeO 3. Nature429,850-853(2004)\n21. Pickett, W. E.SingleSpinSuperconductivity.Phys.Rev .Lett.77,3185-3188(1996).\n22. Krenke,T.,etal.Inversemagnetocaloriceffectinferr omagneticNi-Mn-Snalloys.Nat.Mater.\n4, 450-454(2005).\n23. Kainuma, R. et al. Magnetic-field-induced shape recover y by reverse phase transformation.\nNature439, 957-960(2006).\n24. Chadov,S.etal.Tunablemultifunctionaltopologicali nsulatorsinternaryHeuslercompounds.\nNat. Mater. 9, 541-545(2010).\n25. Wurmehl, S., Kandpal, H. C., Fecher, G. H. &Felser, C. Valence electron rules for prediction\nof half-metallic compensated-ferrimagnetic behaviour of Heusler compounds with complete\nspinpolarization.J. Phys.: Condens. Matter 18, 6171-6181(2006).\n26. Chadov, S., Kiss, J., &Felser, C. Improving Spin-Transport by Disorder. Adv. Func . Mater.\n23, 832-838(2013).\n27. Kurt, H. et al. Cubic Mn 2Ga Thin Films: Crossing the Spin Gap with Ruthenium. Phys. Re v.\nLett.112,027201(2014).\n28. Nayak, A. K., Nicklas, M., Chadov, S., Shekhar, C., Skour ski, Y., Winterlik, J. &Felser,\nC. Large Zero-Field Cooled Exchange-Bias in Bulk Mn 2PtGa. Phys. Rev. Lett. 110, 127204\n(2013).\n1329. Rode, K. et al. Site-specific order and magnetism in tetra gonal Mn 3Ga thin films. Phys. Rev.\nB87184429(2013).\n30. Leighton, C., Nogu´ es, J., J¨ onsson- ˚Akerman, B. J. &Schuller I. K. Coercivity Enhancement\nin Exchange Biased Systems Driven by Interfacial Magnetic F rustration. Phys. Rev. Lett. 84,\n3466-3469(2000).\nAcknowledgements: We thank J.A.Mydoshand Erik Kampert for valuablediscussio nson\nthe present work. This work was financially supported by the D eutsche Forschungsgemeinschaft\nDFG (Projects No. TP 1.2-A and No. 2.3-A of Research Unit FOR 1 464 ASPIMATT) and by\nthe ERC Advanced Grant No. (291472) ”Idea Heusler”. The expe riments at the High Magnetic\nFieldLaboratoryDresden(HLD)andHighFieldMagnetLabora toryNijmegenweresponsoredby\nEuro-MagNETIIundertheEuropeanUnionContract No. 228043 .\nAuthor Contributions Allauthorscontributedsubstantiallyto thiswork.\nAuthor Informations The authors declare no competing financial interests. Corre spon-\ndence and requests for materials should be addressed to C.F. (felser@cpfs.mpg.de) or A.K.N.\n(nayak@cpfs.mpg.de).\n14Figure 1 Design of a compensated magnetic state. a , A theoretical modeling showing\nthat thePt content controls theamount of Mn-spins pointing down by varyingtheuncom-\npensated magnetization, M, from1.1µBfor Mn 3Ga to0.5µB, in the opposite direction,\nfor Mn 2PtGa, thereby, going through the compensation point ( M= 0) at a Pt content of\nx= 0.59(upper panel). Approaching compensation maximizes HEB(lower panel): solid\ncurve corresponds to the model calculation deduced from the ab-initiocalculated mag-\nnetic moments, closed circles corresponds to the experimen t.b, Magnetic moments of\nmanganese in two different sublatices for the tetragonal Mn 3Ga (see the crystal struc-\nture), one Mn in Mn-Ga plane with a localized moment ≈3.1µB/f.u.(spin up, red arrow)\nand two Mn in Mn-Mn planes with a larger sum moment ≈4.2µB/f.u.(spin down, blue\narrow). TheMnintheMn-Gaplanecouplesantiferromagnetic ally totheMnintheMn-Mn\nplanes resulting in a net moment of 1.1µB/f.u.with spin down in the present illustration.\nc, Perfect magnetic compensation is expected in Mn 2.41Pt0.59Ga.d, Magnetic moment of\nmanganese in two sublatices for Mn 2PtGa (see the crystal structure), one Mn in Mn-Ga\nplanewithalargerlocalizedmoment(spinup,redarrow)and oneMninMn-Ptplanewith\na smaller moment (spin down, blue arrow). This configuration displays a net moment of\naround0.5µB/f.u.with spin up. Therefore, on going from Mn 3Ga to Mn 2PtGa, a mag-\nneticcompensation isencountered withequal andoppositel y aligned Mnmomentsinthe\nMn-Ga and Mn-Pt planes. e, A small anti-site disorder can bring about the formation FM\nclusters inside the compensated host. The exchange interac tion between the FM clus-\nters and the compensated host establishes a strong exchange anisotropy in the system\n15leading totheobservation of anexchange bias.\nFigure 2 Compensated ferrimagnetic state in Mn-Pt-Ga. M(T)for Mn 3−xPtxGa (x=0.2,\n0.5 and 0.6) measured in 0.01 T. Open symbols represent the ze ro-field cooled (ZFC)\ncurves and closed symbols the field cooled (FC) curves. In the ZFC mode, the sample\nwas initially cooled to 2 K and the data were taken upon increa sing the temperature in\nappliedmagneticfield. IntheFCmode,thedatawerecollecte dwhilecoolinginfield. The\ninset shows the zero field cooled M(H)hysteresis loop for Mn 2.4P0.6Ga measured at 2 K\ninafield of ±14 T.\nFigure 3 Hysteresis loops in Mn-Pt-Ga. a ,M(H)isotherms measured up to 60 T at\n4.2K.Thedataforsuccessivesamplesareshiftedby 0.3µBalongthemagnetizationaxis\nforbetter clarity. b,Fieldcooled(FC) M(H)loops measuredupto ±32Tforx= 0.5,0.6,\nand 0.7 after field cooling the samples in µ0HCF= 15T (closed symbols) and 25 T (open\nsymbols). The loop for x= 0.7is shifted by −0.3µBand forx= 0.5by+0.3µBalong the\nmagnetizationaxis. Theinsetshowsthedependence ofexcha nge-bias field( HEB)onthe\ncoolingfield( HCF).HEBiscalculatedusing HEB=−(H+\nC+H−\nC)/2,whereH+\nCandH−\nCare\nthelower and upper cut-offfields at which themagnetization becomes zero.\nFigure 4 Exchange bias and coercive fields for Mn-Pt-Ga and Mn-Fe-Ga. a, Tempera-\nture dependence of the EB for Mn 2.4Pt0.6Ga and Mn 2.5Pt0.5Ga. The inset of ashows the\ncoercive field HC(circles) and HEB(squares) as a function of the Pt concentration xin\nMn3−xPtxGa.b, Temperature dependence of theEB for Mn 1.5Fe1.5GaandMn 1.8FeGa.\n16/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s45/s49/s50 /s45/s56 /s45/s52 /s48 /s52 /s56 /s49/s50/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s72 /s61/s48/s46/s48/s49/s32/s84\n/s32/s120 /s61/s48/s46/s54\n/s32/s120 /s61/s48/s46/s53\n/s32/s120 /s61/s48/s46/s50/s77/s110\n/s51 /s45/s120/s80/s116\n/s120/s71/s97\n/s215/s48/s46/s54/s54/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41 /s215/s49/s48/s45/s52\n/s32\n/s84 /s32/s40/s75/s41/s215/s49/s46/s51/s77/s110\n/s50/s46/s52/s80/s116\n/s48/s46/s54/s71/s97\n/s72 /s32/s40/s84/s41/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s32/s32\n/s84 /s61/s53/s32/s75/s90/s70/s67/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50\n/s45/s51/s48 /s45/s49/s53 /s48 /s49/s53 /s51/s48/s45/s48/s46/s57/s45/s48/s46/s54/s45/s48/s46/s51/s48/s48/s46/s51/s48/s46/s54\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s49/s50/s51/s120 /s61/s48/s46/s53\n/s120 /s61/s48/s46/s54\n/s120 /s61/s48/s46/s55/s32\n/s32\n/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41/s84 /s61/s52/s46/s50/s32/s75/s97\n/s32/s48/s72 /s32/s40/s84/s41\n/s32/s98\n/s77/s110\n/s50/s46/s52/s80/s116\n/s48/s46/s54/s71/s97\n/s32/s32/s72\n/s69/s66/s32/s40/s84/s41\n/s72\n/s67/s70/s32/s40/s84/s41/s84/s61/s52/s46/s50/s32/s75/s48 /s52/s48 /s56/s48 /s49/s50/s48 /s49/s54/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50\n/s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56/s48/s49/s50/s51/s52/s48/s72\n/s69/s66/s40/s84/s41\n/s32/s77/s110\n/s50/s46/s53/s80/s116\n/s48/s46/s53/s71/s97\n/s32/s77/s110\n/s50/s46/s52/s80/s116\n/s48/s46/s54/s71/s97\n/s32\n/s84 /s32/s40/s75/s41/s48/s72\n/s67/s70/s61/s53/s32/s84/s97\n/s48/s72\n/s67/s70/s61/s53/s32/s84/s32/s77/s110\n/s49/s46/s53/s70/s101\n/s49/s46/s53/s71/s97\n/s32/s77/s110\n/s49/s46/s56/s70/s101/s71/s97\n/s32\n/s32/s98\n/s48/s72\n/s67/s70/s61/s49/s53/s32/s84/s84 /s61/s52/s46/s50/s32/s75\n/s77/s110\n/s51/s45 /s120/s80/s116\n/s120/s71/s97/s32/s72\n/s69/s66\n/s32/s72\n/s67/s48/s72 /s40/s84/s41\n/s32/s32\n/s120" }, { "title": "0711.3002v1.Probing_phase_coexistence_and_stabilization_of_the_spin_ordered_ferrimagnetic_state_by_Calcium_addition_in_the_YBa__1_x_Ca__x_Co__2_O__5_5__layered_cobaltites_using_neutron_diffraction.pdf", "content": "arXiv:0711.3002v1 [cond-mat.str-el] 19 Nov 2007Probing phase coexistence and stabilization of the spin–or dered ferrimagnetic state by\nCalcium addition in the Y(Ba 1−xCax)Co2O5.5layered cobaltites using neutron\ndiffraction\nG. Aurelio,∗J. Curiale,†and R. D. S´ anchez‡\nComisi´ on Nacional de Energ´ ıa At´ omica – Centro At´ omico B ariloche,\nAv. Bustillo 9500, 8400 S. C. de Bariloche, RN, Argentina\nG. J. Cuello\nInstitut Laue Langevin, BP 156, F-38042 Grenoble Cedex 9, Fr ance\n(Dated: November 30, 2018)\nIn this article we study the effects of a partial substitution of Ba with the smaller cation Ca in\nthe layered cobaltites YBaCo 2O5+δforδ≈0.5. Neutron thermodiffractograms are reported for\nthe compounds YBa 0.95Ca0.05Co2O5.5(xCa= 0.05) and YBa 0.90Ca0.10Co2O5.5(xCa= 0.10) in the\ntemperature range 20 K ≤T≤300 K, as well as high resolution neutron diffraction experim ents at\nselectedtemperatures for thesamples xCa= 0.05,xCa= 0.10 andtheparentcompound xCa= 0. We\nhave found the magnetic properties to be strongly affected by the cationic substitution. Although\nthe “122” perovskite structure seems unaffected by Ca additi on, the magnetic arrangements of\nCo ions are drastically modified: the antiferromagnetic (AF M) long–range order is destroyed, and\na ferrimagnetic phase with spin state order is stabilized be lowT∼290 K. For the sample with\nxCa= 0.05 a fraction of AFM phase coexists with the ferrimagnetic on e belowT∼190 K, whereas\nforxCa= 0.10 the AFM order is completely lost. The systematic refinemen t of the whole series\nhas allowed for a better understanding of the observed low–t emperature diffraction patterns of the\nparent compound, YBaCo 2O5.5, which had not yet been clarified. A two–phase scenario is pro posed\nfor thexCa= 0compoundwhichis compatible withthe phasecoexistence o bserved inthe xCa= 0.05\nsample.\nI. INTRODUCTION\nDuring the last decade, cobaltites have gained in-\ncreased attention. A great effort is being made to clarify\nand systematize the extremely rich variety of phenom-\nena they exhibit. Initially, they were expected to show\nsimilar properties to other perovskite–family members,\nsuch as manganites and cuprates,1–5but soon it was\nfound that they present additional tunable features, as\nthe cobalt spin state, that add to their complexity but\nalso make them even more fascinating and challenging.\nAmong cobaltites, the layered compounds RBaCo2O5+δ\n(Rbeing a rare earth) are currently being intensively\nstudied.3–6The oxygen content in these compounds can\nbe modified in a wide range (0 ≤δ≤0.9 depending\non theRcation and the synthesis conditions),6which in\nturn controls the mixed valence state of Co ions. Several\nfactors strongly influence the physical properties of these\ncobaltites: the non–stoichiometry, the Rcation size, the\nvacancies structural order, and — as we will show in the\npresent work— also the structural disorder introduced\nby doping the Ba site with small quantities of smaller\ncations with the same valence state.\nFrom a structural point of view, RBaCo2O5+δis\nformed by a stacking sequence of [CoO 2]–[BaO]–[CoO 2]–\n[ROδ]planesalongthe c−axis,2,7theusuallycalled“112”\nstructure derived form the aPxaPx2aPcell, being aP\nthe perovskite unit cell constant. The symmetry may\nbe tetragonal or orthorhombic, depending on the oxygen\ncontent and the Rcation. The oxygen vacancies have\na strong tendency to become ordered, which results inseveral superstructures.8\nOf particular interest is the case δ= 0.5, for which Co\nis expected to be completely in the +3 valence state. In\nthis case, a particular order of oxygen vacancies leads to\nthe “122” superstructure, consisting of an ordered array\nof 50% Co atoms in octahedral oxygen coordination and\n50% in a pyramidal environment. This, in turn, favors a\nmetal–insulator (MI) transition just above room temper-\nature (the TMIdepends again on the Rcation) which can\nbefoundonlyfor δvaluesverycloseto0.5.4Whendoping\nwith holes (Co4+,δ >0.5) these compounds behave as\nmetals above the TMItransition, but when doping with\nelectrons (Co2+,δ <0.5), these do not seem to partic-\nipate in charge transport, which has been explained in\nterms of a spin blockade.9There has arisen a big contro-\nversy regarding the physical phenomena which occur at\nTMI. Regardless of the Rcation, cobaltites with δ∼0.5\nall show a jump in resistivity and a concomitant lattice\ndistortion with a sudden volume collapse. The distortion\nis associated with specific changes in the Co–O distances\nin pyramids and octahedra, and became the subject of\ndifferent interpretations. A possible driving force for the\nMI transitionhasbeen proposedto be aspin state transi-\ntionfromthe Colow–spinstate(LS: t6\n2ge0\ng)toahigh–spin\nstate (HS: t4\n2ge2\ng) occurringonly at the octahedral sites.10\nFor the particular cases of R= Pr11and Gd,10,12this\nhypothesis would also be supported by a change in the\nslope of the inversesusceptibility curveat TMI, which has\nbeen analyzed in terms of the Curie–Weiss model. Fur-\nther support to this scenario was given by Maignan et\nal.9forR= Ho based on thermoelectric measurements,2\nshowing that the spin blockade mechanism is fully com-\npatible with this picture. Other authors have proposed\nthat at2g−eghybridization is enhanced in the metal-\nlic phase by the lattice distortion, such that the metallic\nor insulating behavior would be determined by the in-\ntersite mixing of the itinerant 3d electrons between the\noctahedral and pyramidal sites.13ForR= Tb the dis-\ntortions of pyramids and octahedra were interpreted as a\nd3x2−r2/d3y2−r2orbital ordering transition accompanied\nby a intermediate–spin (IS: t5\n2ge1\ng)to HS spin state transi-\ntion.7Furthermore, Pomjakushina et al.14proposed that\nthe observed volume collapse at the transition tempera-\nture and the existence of an isotopic effect are indicative\nof a charge delocalization breaking the orbital order of\nthe insulating phase, which could be compatible with a\nspin state switch, but occurring in pyramids, not in octa-\nhedra. It seems obvious that this issue is far from being\nclarified and some effort must be made to systematize\nthe study of the MI transition in cobaltites.\nA second controversy, closely related to the one men-\ntioned in the previous paragraph, concerns the low tem-\nperature ordering of the magnetic moments at the Co\nsites. Again, the feature which seems to be common to\nallRcobaltites is the existence of a spontaneous mag-\nnetization in a more or less narrow temperature range,\ndepending on R, below room temperature. Above this\nrange they are paramagnetic, and below this range they\ntransformtoanantiferromagnetic(AFM) state. Itisnow\nwell established that the origin of the spontaneous mag-\nnetization is not a ferromagnetic order, but among the\ntwo remaining possibilities, i.e., a ferrimagnetic phase or\na canted AFM phase, there are various different mod-\nels which have been proposed. Some of these models\ninvolve a so–called spin state ordering (SSO) in which\nnot only the spin state may be different between Co\natoms located at pyramids and Co atoms located at oc-\ntahedra, but also among the pyramidal15or the octahe-\ndral sites16,17a SSO may arise leading to a doubling of\nthea−axis in the unit cell. Indeed, a theoretical work\nby Khomskii and L¨ ow18showed that such spin super-\nstructures can be energetically favorable. These mod-\nels would correspond to a ferrimagnetic phase. On the\nother hand, the proposers of canted–AFM models argue\nthat there are no structural evidences for the doubling\nof thea−axis, and adopt the canted models which also\nexplain the neutron diffraction data, with a doubling of\nthea−axis just in the magnetic cell.11,19However, some\ncare must be taken when comparing all these experimen-\ntal data. For instance, there may be no evidence of a\n“222”superstructure in cobaltites with R= Gd and Pr11\nbut the case might be different for other lanthanides. In\nfact, some studies using NMR techniques showed that\nforR= Y, there are four non–equivalent Co sites at low\ntemperature,21and forR= Eu there are three,22which\nis compatible with the SSO scenario. It should be em-\nphasized, too, that the “222” superstructure is very hard\nto detect from diffraction measurements unless an ex-\nceptionally high signal–to–noise ratio is attained. Usingtransmission geometry, Chernenkov et al.23have shown\nthat the superstructure can indeed be observed in sin-\ngle crystals with R= Gd using X–ray diffraction. In all\ncases, there seems to be consensus on the IS character\nof pyramidal Co atoms,5,10,24–26although the spin state\n—or spin states— at octahedral sites remains uncertain\nor may, at least, depend on the Rsize.\nMoststudies oflayeredcobaltiteswereconducted for R\namongthelanthanides,butthecompoundwith R=Y3+,\nwhich is a small, non–magnetic ion, is a good candi-\ndate to isolate the intrinsic properties of Co and ex-\nplore the small– Rregion of the phase diagram. It is\nnow well documented, for instance, that the MI tran-\nsition temperature decreases with the Rsize. To gain\nmore insight into the possible role of disorder, we have\nintroduced a second source of distortion, by substitut-\ning the Ba–site with Ca, which has a smaller atomic\nradius. In addition, it has recently been postulated on\nthe basis of density–functional theory calculations, that\na smaller cation substitution in the Ba–site of small lan-\nthanide cobaltites could be a promising compound to\nexhibit enhanced giant magnetoresistance properties.27\nThe present work is aimed at characterizing and corre-\nlating the Ba–substituted compounds when compared to\nthe parent YBaCo 2O5.5cobaltite. We have performed a\nstructural characterization using neutron powder diffrac-\ntion (NPD) to study the interplay between the structures\nand their magnetic order, and correlate this information\nwith our previous magnetic studies.28,29\nII. EXPERIMENTAL METHODS\nThree polycrystalline samples were prepared by\nsolid–state reaction. High–purity powders of Y 2O3,\nBaCO 3, CaCO 3and Co 3O4were mixed at stoichio-\nmetric weights to prepare the compounds YBaCo 2O5+δ\n(xCa= 0), YBa 0.95Ca0.05Co2O5.5(xCa= 0.05) and\nYBa0.90Ca0.10Co2O5.5(xCa= 0.10). After a de–\ncarbonationprocessat1173Kfor18h, themixtureswere\npressed into pellets and annealed. The samples were an-\nnealed together during 25 h at 1273 K and slowly cooled\nat 1 K/min in oxygen flow. After a regrinding of the re-\nsulting pellets, the compression and annealing at 1273 K\nin oxygen processes were repeated. A single batch was\nused for all the samples to guarantee identical synthesis\nconditions, which resulted in samples of about 1.5 g.\nThe oxygen content in our samples has been deter-\nmined by refinement of our NPD data. In addition, we\nhave compared the macroscopic magnetization and resis-\ntivity of our xCa= 0 sample with a very detailed study\nof the parent compound YBaCo 2O5+δearly reported by\nAkahoshi and Ueda30. Their work presents the existing\ncorrelation between oxygen content and magnetic and\ntransport properties. In particular, the magnetization\ncurve for our sample (Fig. 5) reveals an excellent quan-\ntitative agreement with their results for δ= 0.5, and a\nclear disagreement outside the range 0 .44< δ≤0.52.3\nMoreover, the resistivity measurements in our samples29\nshow the characteristic sharp jump of the MI transition,\nwhich has been shown to occur only for 0 .45< δ≤0.65\nbut only to be sharp for δ≃0.54. These limits give\nus confidence in the refined values from our NDP data.\nThe global oxygen contents refined independently from 8\nhigh resolutiondiffractograms,correspondingto different\nsamples and temperatures —always below 350 K— were\nin mutual agreement within experimental error, yield-\ning an average value of δ= 0.46±0.02. In the fol-\nlowing we shall refer to the samples using the notation\nY(Ba,Ca)Co 2O5.5.\nNeutron thermodiffraction data were collected on the\nhigh–intensity two–axis diffractometer D20 located at\nthe High Flux Reactor of Institute Laue–Langevin ILL,\nGrenoble, France. Samples with xCa= 0.05 andxCa=\n0.10werecooledin astandardorangecryostatfromroom\ntemperature down to 20 K, and diffraction patterns were\nthen collected every two minutes at a warming rate of\n1 K/min from 20 K to 320 K. A wavelength of ∼2.41˚A\nwas used to highlight the magnetic diffraction and was\ncalibrated using a Silicon sample.\nIn addition, high–resolution NPD data were collected\nat diffractometer Super–D2B of ILL for samples with\nxCa= 0,0.05 and 0.10. A wavelength of ∼1.594˚A was\nused to collect patterns at selected temperatures for ap-\nproximately 3 h. It is worth noting that the volume of\nsampleavailablewasnotasmuchastheidealforthiskind\nof experiment, so we looked for a compromise between\nthe collection time, the available beamtime, and our ca-\npabilities for preparing all the samples in a single batch.\nThe NPD patterns were processed with the full–pattern\nanalysis Rietveld method, using the program FULLPROF\n31for refining the crystal and magnetic structures.\nIII. RESULTS\nA. Description of structures and refinement\nstrategy\nThe room temperature structures of the parent com-\npound YBaCo 2O5+δwere first reported by Akahoshi and\nUeda,30who showed that for δ= 0.5 there may form\ntwo competing structures. One of them is orthorhombic,\nand corresponds to the space group Pmmm having the\n“122” superstructure characteristic of similar cobaltites\nwithδ= 0.5.2,3,7,24A schematic representation of this\nphase is shown in Fig. 1. The vacancies order consists of\nalternating[CoO 6]octahedrachainsalongthe c−axisand\ncorner–sharing[CoO 5] pyramidsalongthe b−axis,result-\ning in alternating octahedral and pyramidal layers in the\na−cplane. This produces a doubling of the cell along\ntheb−axis, with a unit cell aPx2aPx2aP. The second\nstructure that may stabilize in this system for δ= 0.5\n(and other values as well) has a tetragonal symmetry\nand no doubling of the b−axis,i.e., no ordering between\npyramids and octahedra. In this case, the space groupBa YOCo \nabc\n(a) (b) \n(c) (d) \nFIG. 1: (Color online) (a) The structure of YBaCo 2O5.5. Half\nCo atoms are in square–pyramidal coordination and the other\nhalf in octahedral coordination. (b) Magnetic model adopte d\nfor the SSO ferrimagnetic phase after Ref. 17. The nuclear\n“122” unit cell and the magnetic “222” unit cell are indicate d,\nand light atoms correspond to octahedral coordination. In\nthisschematicrepresentation, onlyCoandOatomsare shown\nfor clarity. (c) Magnetic model adopted for the AFM– O2\nphase after Ref. 17. (d) Magnetic model adopted AFM– O1\nphase.\nisP4/mmmand the unit cell aPxaPx2aP. Recently,\nFrontera et al.6have shown that, although the order of\nvacancies may not always be perfectly achieved, the or-\nder between the R–cation layer and the Ba layer is well\nestablished and there is no mixing between them. Nei-\nther are there significant oxygen vacancies in the [BaO]\nlayers. The “122” structure admits a certain degree of\ndisorder, consisting of misplaced pyramids or octahedra,\nbut keeping the long–range “122” order.\nIn the present refinements, the Ca cations were ran-\ndomly introduced in the structure at the crystallographic\nsiteoccupied byBa, in appropriateproportions. We have\nfound no evidence of Ca segregation nor the formation of\nadditional phases, so we believe that Ca has been suc-\ncessfully incorporated into the cobaltites structure. The\nstrategyforthe Rietveldrefinement wasasfollows. First,\nthe high resolution data from D2B were refined to ob-\ntain an accurate nuclear structure for each sample. The\nraw data coming from the Super–D2B detector were pro-\ncessed using the LAMP software32to obtain two sets of\ndata: one of them having a better angle resolution at\nthe expense of losing some neutron counts, the other one\nhaving all neutron counts collapsed into a single diffrac-\ntogram. The first set was used to determine the lattice4\nparameters, while the second set was used to refine the\natomic positions and temperature factors, and both sets\nwere iteratively refined until convergence to the struc-\nture. The magnetic structures were also included in the\nrefinements. The models we have used will be discussed\nin the following sections. At a second step, the struc-\ntural data obtained were used to refine sequentially the\nneutron thermodiffractogramsobtained at D20. Temper-\nature scans where divided into different ranges according\nto the structural and magnetic order, and for each range\nthe atomic positions and occupations obtained at D2B\nwere kept fixed, while lattice parameters, temperature\nfactors and magnetic moments were allowed to vary.\nThe objective of this work is to focus on the role\nof Ca addition to the parent compound YBaCo 2O5.5.\nOur results will show that there is a clear logical se-\nquence between the three samples studied, corresponding\ntoxCa= 0,xCa= 0.05 andxCa= 0.10. Surprisingly, the\nsample with greater Ca content, xCa= 0.10, turned out\nto be the simplest one, and as Ca is removed the com-\nplexity increases, resulting in a quite complicated tem-\nperature evolution of the parent compound. This fact\nprobably explains why this compound has not yet been\nfully reported in such detail as other Rcobaltites, except\nfor the structural study by Akahoshi and Ueda,8and a\nrecent neutron diffraction study by Khalyavin et al.33fo-\ncusing on the ferrimagnetic to paramagnetic transition.\nFor the above reasons, we will present our results follow-\ning the decreasing Ca sequence.\nB. Sample with xCa= 0.10\nIn Fig. 2 we present two different sections of the pro-\njected thermodiffractograms for the sample xCa= 0.10.\nFig. 2(a) corresponds to the low–angle range, in which\nmost reflections are of magnetic nature, and disappear\nsimultaneously at ∼295 K on warming from 20 K to\n300 K. In Fig. 2(b) we focus on the 2 θrange where the\nBragg reflections (2 0 0) and (0 4 0) clearly show a dis-\ntortion occurring at room temperature.\nThe high resolution data were refined using a nuclear\nphase with the “122” structure, as described in sec-\ntion IIIA. We do not discard the possibility of the actual\nstructure being “222”, with a doubling of the a−axis and\nfour different crystallographicsites for the Co atoms,17,33\nin line with the magnetic model adopted. However, as we\nare interested in the temperature evolution of rather low\nresolutiondatafromD20,andgiventhecomplexityofthe\nother samples, we have decided to refine the whole series\nwith the averaged “122” structure. Moreover, following\nFrontera et al.11we have fixed the zcoordinate of the oc-\ntahedral Co site to 1/4, in orderto obtain centrosymmet-\nric octahedra. With these assumptions, we reduced the\nnumber of free parameters which is critical when dealing\nwith multiphasic systems. We have nevertheless allowed\nfor disorder between pyramids and octahedra, by refin-\ning the occupation of apical oxygen sites (site 1gmostlyTABLE I: Structural parameters refined from the high reso-\nlution D2B data for the compound YBa 0.90Ca0.10Co2O5.5at\nT= 70 K, 230 K and 350 K. Atomic fractional coordinates\ncorrespond to space group Pmmm in the following Wyck-\noff positions: Y ( 2p)=(1\n2,y,1\n2); Ba, Ca ( 2o)=(1\n2,y,0); CoOct\n(2r)=(0,1\n2,1\n4); CoPyr ( 2q)= (0,0,z); O1 (1a)=(0,0,0); O2\n(1e)=(0,1\n2,0); O3 ( 1g)=(0,1\n2,1\n2); O3’ ( 1c)=(0,0,1\n2); O4\n(2s)=(1\n2,0,z); O5 (2t)=(1\n2,1\n2,z); O6 (4u)=(0,y,z).\nT= 70 K T= 230 K T= 350 K\nY y0.2714(5) 0.2717(5) 0.2684(5)\nBa, Ca y0.253(1) 0.254(1) 0.248(1)\nCoPyr z0.260(1) 0.260(1) 0.262(1)\nO4 z0.3112(7) 0.3108(7) 0.3107(7)\nO5 z0.276(1) 0.276(1) 0.271(1)\nO6 y0.2481(8) 0.2479(8) 0.2416(8)\nO6 z0.2954(6) 0.2947(5) 0.2980(5)\nO3 Occ 0.94(2) 0.93(2) 0.89(2)\nO3’ Occ 0.0 0.0 0.02(2)\na(˚A) 3.8423(1) 3.8438(1) 3.8254(1)\nb(˚A) 7.7947(2) 7.8118(2) 7.8503(2)\nc(˚A) 7.4835(2) 7.4965(2) 7.5217(2)\nRB 7.4 7.4 7.1\nRmag 13.6 14.7\nχ27.4 5.8 5.2\noccupied, and site 1cmostly unoccupied in the Pmmm\nspace group). We remark that even for different degrees\nof vacancies disorder, the total occupation of sites always\nsummed up to the sameoxygencontentin all samplesbe-\nlow 350 K. In Table I we present the details of the refined\nstructure for the sample with xCa= 0.10 from D2B data\ncollected at 70 K, 230 K and 348 K.\nThe model we have adopted for refining the mag-\nnetic phase is the SSO ferrimagnetic model proposed by\nKhalyavin,17,33which leads to the best agreement with\nour NPDdata, macroscopicmagnetization data and high\ntemperature susceptibility data.28,29It is schematized in\nFig. 1(b). Although the model involves Co atoms at half\nthe octahedral sites being in low–spin state, and there-\nfore having a magnetic moment equal to zero, after a\nfirst step in the refinement we allowed this site to adopt\na non–zero magnetic moment, which resulted in a small\nvalue compatiblewith the fact that the apical oxygensite\n1gis not completely occupied, and therefore some octa-\nhedra are, in fact, misplaced pyramids.11In Fig. 3(a)\nwe present the evolution with temperature of the lattice\nparameters refined in the Pmmm “122” structure. The\nsequential D20 results are presented together with the\nresults from D2B at the studied temperatures. The char-\nacteristic structural distortion occurring at TMI∼295 K\nis clearlyobserved. We havealreadyreportedthat in this\nsystem, the MI transition occurs almost simultaneously\nwith the paramagnetic–ferrimagnetic transition,28which\nseems to be only a coincidence. Therefore, no further5\nanomaly is observed in the lattice parameters down to\n20 K, as there is in this sample no further magnetic tran-\nsition. In Fig. 4(a) we present the magnetic moment of\nCo atoms in each crystallographicenvironment. We have\nnot included in the figure the small magnetic moment of\nmisplaced pyramids, which remains always less than 0.4\nµB.\nC. Sample with xCa= 0.05\nOur preliminary X–ray diffraction pattern of the as–\nsynthesized sample xCa= 0.05 at room temperature re-\nsulted identical to xCa= 0.10with just a slight difference\nin the lattice parameters, indicating that the room tem-\nperaturestructure is the same in both samples. However,\nthe macroscopic magnetization data in Fig. 5 show that\natlowtemperature theybehavedifferently. Below200K,\non cooling, the magnetization of the xCa= 0.05 sample\nstarts dropping but the sample retains a net magnetiza-\ntion down to 5 K, in contrast with the parent compound\nxCa= 0 which shows an AFM behavior. Moreover, the\nbig hysteresis between the data collected on cooling and\nwarming in the xCa= 0.05 sample suggests that there is\nacompetitionbetweentwostates. TheNPDexperiments\nreveal the nature of these two states.\nIn Fig. 6 (c) we present a projection of the thermod-\n T[K] T[K] \n2θ2θ(a) \n(b) \nFIG. 2: (Color online) Projection of two selected sections\nof the thermodiffractograms corresponding to sample with\nxCa= 0.10. (a) 16◦<2θ <44◦and (b) 75◦<2θ <80◦\nshowing the (2 0 0) and (0 4 0) Bragg reflections of the “122”\nstructure. Data were collected at D20 with λ∼2.41˚A be-\ntween 20 K and 320 K.0 50 100 150 200 250 300 350 7.45 7.50 7.65 7.70 7.75 7.80 7.85 \n(a) b1\n2a1\nc1\n Lattice Parameters [Å] \nT [K] \n0 50 100 150 200 250 300 350 7.45 7.50 7.65 7.70 7.75 7.80 7.85 \n(b) b2b12a2\n2a1\nc2c1\n Lattice Parameters [Å] \nT [K] \n0 50 100 150 200 250 300 350 7.45 7.50 7.65 7.70 7.75 7.80 7.85 \n(c) cb\n2a Akahoshi & Ueda [ref. 8] \nb2b1\n2a2\n2a1\nc1=c2\n Lattice Parameters [Å] \nT [K] \nFIG. 3: (Color online) Thermal evolution of the lattice pa-\nrameters 2 a,bandcfor theO1 (dark symbols) and O2 (light\nsymbols) phases in samples with xCa= 0.10 (a),xCa= 0.05\n(b) and xCa= 0 (c), determined from data collected at D20\nand D2B. In (c) the data from D2B (solid symbols) are com-\npared with data reported by Akahoshi and Ueda,8plotted\nusing diamonds.\niffractograms collected at D20 for sample xCa= 0.05. In\n(a) and (b) we show the temperature evolution of the in-\ntensity of the magnetic reflections (0 1 0) and (1/2 1 1)\nrespectively, indexed in terms ofthe “122”nuclear phase.\nThese reflections correspond to the ferrimagnetic order\ndiscussed in the previous subsection. Figure 6 (a) and6\n0 50 100 150 200 250 300 0.0 0.5 1.0 1.5 2.0 2.5 3.0 \n Octahedral site 2 \n Pyramidal site \n(a) \n m [ µB/Co] \nT [K] \n0 50 100 150 200 250 300 0.0 0.5 1.0 1.5 2.0 2.5 3.0 \n(b) \n m [ µB/Co] \nT [K] \nFIG. 4: (Color online) Temperature dependence of the mag-\nnetic moments in pyramids (square symbols) and half the oc-\ntahedra (triangular symbols) for the samples xCa= 0.10 (a)\nandxCa= 0.05 (b), obtained from the Rietveld refinements\nof neutron data collected at D20. Moments were refined along\n[100].\n(b) compare the intensities of these reflections in sam-\nplesxCa= 0.05 andxCa= 0.10. We can observe that\nthey both behavealmostidentically from200K to320K,\nbut below 200 K the ferrimagnetic reflections are lower\nin thexCa= 0.05 sample. Moreover, at 200 K additional\nfeatures are evidenced in the thermodiffractograms. The\nsmall arrows in Fig. 6 (c) mark two reflections which ap-\npear only below 200 K. They are indicative of the pres-\nence of a second —magnetically ordered— phase which\nmay be indexed with a further doubling of the cparame-\nter as reported by various authors in the AFM region of\nlayered cobaltites.11,15–17,19But here we also observe si-\nmultaneouschangesin the nuclearstructureand not only\na magnetic phase separation, or gradual magnetic transi-\ntion to a different magnetic arrangement.11,15,16,20This\nis illustrated, for instance, by a peak arising at 200 K\nin 2θ∼77.6◦lying in between the (2 0 0) and (0 4 0)\nreflections of the “122” nuclear phase. In Fig. 7 we show\nthe evolution with temperature of the collected intensity\nat 2θ∼77.6◦, for this sample as well as for the sample\nwithxCa= 0.10 for comparison. When warming, there\nis a sudden drop at 200 K, to a value corresponding to0 50 100 150 200 250 300 012345678910 \n0 50 100 150 200 250 300 012345678910 \n0 50 100 150 200 250 300 012345678910 \nxCa =0.10 \nxCa =0.05 \nFCW FCW \nFCC FCC \nxCa =0 \n \n M [emu/g] \nT [K] \n \nFIG. 5: (Color online) Low–field magnetization as a func-\ntion of temperature for samples with xCa= 0,xCa= 0.05\nandxCa= 0.10. Empty symbols represent the magnetization\nmeasured on cooling under a magnetic field of 5 kOe (FCC),\nand solid symbols represent the magnetization subsequentl y\nmeasured on warming (FCW).\nthe overlap of the neighboring (2 0 0) and (0 4 0) reflec-\ntions, and a further drop to background values above the\nMI transition, when these reflectionsaresuddenly shifted\napart by the distortion. We first evaluated the possibil-\nity of this behavior being a result of a distortion of the\n“122” nuclear phase at 200 K. Such hypothesis gave no\nsatisfactory results when trying to refine simultaneously\nthe D2B and D20 data at 70 K. Additional attempts to\nmodel the 70◦<2θ <80◦region using Gaussian peaks\nof just one orthorhombic phase, even allowing for unreal-\nistically wide peaks, could not reproduce the triple–peak\nshape observed in the D20 spectra (inset in Fig. 7).\nWe also considered other structural models to account\nfor the low temperature data coming from D2B and D20.\nNeither the PccanorPmmaspace groups proposed by\nPlakhty et al.15and Khalyavin,33together with the re-\nspective magnetic models proposed in their work, gave\nsatisfactory results for the simultaneous refinement of all\nour data.\nAnother possible explanation is the presence of a sec-\nond structural phase. This is suggested by the poor\nresults obtained refining the D2B data using the same\n(single-phase)modelasforthe xCa= 0.10sampleat70K\n(Pmmmspace group), as well as with other single-phase\nmodels (Pcca,Pmma). Based on the D20 data, we con-\nsidered a possible second phase with a strong tetragonal\ndistortion, which would not be unreasonable considering\nthat a two-phase mixture of orthorhombic and tetrago-\nnal phases for YBaCo 2O5.5had already been reported\nby Akahoshi and Ueda,30both phases occurring compet-\nitively. It is also worth noting that in the same batch as\nthe present samples, we have also synthesized a series of\nsamples where Barium is replaced by Strontium, a sub-7\nT[K] Intensity [arb. units] x=0.10 \nx=0.05 \nT[K] Intensity [arb. units] x=0.10 \nx=0.05 \n2θT[K] (a) (b) \n(c) \nFIG. 6: (Color online) Thermal evolution of the intensity of Bragg reflections (0 1 0)(a) and (1/2 1 1)(b) indexed in terms o f\nthe “122” phase for samples with xCa= 0.05 (triangles) and xCa= 0.10 (diamonds) from data collected at D20. (c) Two–\ndimensional projection of the neutronthermodiffractogram s collected at D20 for sample xCa= 0.05 in the range 16◦<2θ <44◦.\nThe small arrows indicate theonset ofmagnetic reflections f rom theAFM “224” phase. The longarrows between graphs indic ate\nthe position in the thermodiffractogram of the two reflection s whose intensity is plotted in (a) and (b).\nstitution which clearly favors a tetragonal phase in which\nthe position of the (2 0 0) Bragg reflection is almost ex-\nactly coincident with this new peak in the xCa= 0.05\nsample.34Consequently, we tried to refine both the D2B\nand D20 sets of data using a mixture of the “122” phase\nand a tetragonal ( P4/mmm) phase. This gave no satis-\nfactoryresultseither, andmoreover,the proposedtetrag-\nonal symmetry could not account for the observed mag-\nnetic supercell.\nWe finally proceeded to adopt for the second phase an\northorhombic ( Pmmm) cell with the constraint b= 2a,\nand at a further step we allowed the bandalattice pa-\nrameters to vary independently. The diffractogram at\n70 K was therefore refined with one orthorhombic phase\nwith ferrimagnetic order ( O1 phase) plus a second or-\nthorhombic phase ( O2) with AFM order which is con-\nsistent with a “224” supercell. This finally gave muchmore satisfactory results for the refinement. The mag-\nnetic peaks arising below 200 K, although weak, could\nbe accounted for using the AFM model proposed by\nKhalyavin,17schematized in Fig. 1(c), with the con-\nstraint of a single magnitude for the magnetic moment of\nall Co atoms in pyramidal positions, and once again (as\nin the ferrimagnetic SSO model), two possible spins for\noctahedral Co. At 230 K, the O1 phase with ferrimag-\nnetic order was enough to refine the D2B diffractogram.\nIn TableII we presentthe details ofthe refined structures\nfor the sample with xCa= 0.05 from D2B data collected\nat 70 K and 230 K. Figure 8 shows the Rietveld refine-\nment (solid line) of the high resolution data (symbols)\ncollected at 70 K. The four sets of Bragg reflections indi-\ncated at the bottom by vertical bars correspond to each\nof the above mentioned phases. The difference pattern\nbetween observed and calculated data is also shown.8\nIntensity [arb. units] \nT[K] 2θxCa = 0.05 \nxCa = 0.10 \n T\n[K] \n2θ\nT\n[K] \nFIG. 7: (Color online) Thermal evolution of the intensity\nin the thermodiffractograms at 2 θ∼77.6◦, in the position\nbetween the (2 0 0) and (0 4 0) Bragg reflections of the\n“122” phase for the sample with xCa= 0.05 (diamonds) and\nxCa= 0.10 (squares). The insets show the three–dimensional\nthermodiffractograms for the relevant 2 θrange.\nThe presented scenario for the evolution with temper-\nature of the xCa= 0.05 sample yields consistent results\namong the D2B and D20 data, as well as it accounts\nfor other experimental facts. When cooling from room\ntemperature, theintensityoftheferrimagneticreflections\nstarts dropping because there is a second phase develop-\ning in the sample, so that the volumefraction ofthe ferri-\nmagneticphaseisreduced. Inaddition, theobservedhys-\nteresis among the FCC and FCW magnetization curves\nin Fig. 5 are also indicative of a possible phase separa-\ntion, as well as the hysteresis in the resistivity curves\npresented previously.29The presence of a second nuclear\nphase has become more evident when performing ther-\nmodiffractograms with λ= 2.52˚A. This scenario would\nbe very difficult to infer just from D2B data collected at\nlower wavelengths and at isolated temperatures, due to\npeak overlap and to a lack of perspective of the continu-\nous thermal evolution of the sample.\nThe evolution with temperature of the lattice param-\neters refined in the O1 andO2 phases is shown in\nFig. 3(b). The sequential D20 results are presented to-\ngether with the results from D2B at the studied temper-\natures. For the O1 phase, the distortion at TMI∼295K\nis again observed as in the sample with xCa= 0.10.\nFor the O2 phase, we observe a tetragonal distortion\nabove 170 K: above that temperature the a2andb2lat-\ntice parameterscouldonlybe refinedusing the constraint\n2a2=b2. The volume per atom of both phases is prac-\ntically the same, although it is observed that the lat-\ntice parameter relation is different (2 a1< b1, 2a2> b2).\nIt would be interesting to stabilize the O2 phase to get\nmore detailed information of its structural properties.34\nInFig.4(b)wepresentthemagneticmomentofCoatomsTABLE II: Structural parameters refined from the high reso-\nlution D2B data for the compound YBa 0.95Ca0.05Co2O5.5at\nT= 70 K and 230 K. Two sets of atomic fractional coordi-\nnates are given for the O1 andO2 phases, which correspond\nto space group Pmmm in the following Wyckoff positions: Y\n(2p)=(1\n2,y,1\n2); Ba, Ca ( 2o)=(1\n2,y,0); CoOct ( 2r)=(0,1\n2,1\n4);\nCoPyr ( 2q)= (0,0,z); O1 (1a)=(0,0,0); O2 ( 1e)=(0,1\n2,0);\nO3 (1g)=(0,1\n2,1\n2); O3’ (1c)=(0,0,1\n2); O4 (2s)=(1\n2,0,z); O5\n(2t)=(1\n2,1\n2,z); O6 (4u)=(0,y,z).\nT= 70 K T= 230 K\nO1 O2 O1\nY y0.2757(8) 0.264(2) 0.2731(5)\nBa, Ca y0.254(1) 0.233(3) 0.254(1)\nCoPyr z0.262(2) 0.267(2) 0.261(1)\nO4 z0.317(1) 0.291(5) 0.3125(7)\nO5 z0.261(2) 0.287(5) 0.274(1)\nO6 y0.2460(9) 0.243(2) 0.2473(7)\nO6 z0.2941(6) 0.301(1) 0.296(1)\nO3 Occ 1.0 0.84(6) 0.92(2)\nO3’ Occ 0.0 0.0 0.00(2)\na(˚A) 3.8460(2) 3.8845(3) 3.8468(1)\nb(˚A) 7.7887(4) 7.7099(6) 7.8075(2)\nc(˚A) 7.4827(5) 7.4819(7) 7.4959(2)\nf(%) 64(4) 36(3) 100\nRB 7.0 7.0 7.3\nRmag 15 32 16\nχ24.48\nin the ferrimagnetic phase for each crystallographic en-\nvironment. We have not included in the figure the small\nmagnetic moment of misplaced pyramids, which remains\nalwayslessthan 0.5 µB. Unfortunately, the qualityofour\nD20 data and the two-phase scenario do not allow for a\nconfident determination of the magnetic moments in the\nO2 AFM phase. These were constrained to be aligned\nalong [100] following the model described above and to\nadopt values similar to those in the O1 phase, in order\nto obtain reliable phase fractions which were comparable\nto the values refined from D2B data.\nFigure 9 shows the net spontaneous magnetization of\nsamples with xCa= 0.05 andxCa= 0.10, obtained\nasM=µCo·fO1, whereµCorepresents the net mag-\nnetic moment per Co atom in the ferrimagnetic phase\n(=µCoOct/4) andfO1is the refined phase fraction of\ntheO1 phase. These results can be compared with the\nmacroscopic determination of the magnetization of the\nsamples as a function of temperature (Fig. 5), always\nconsidering these were collected under an applied field of\n5 kOe. The overall similarity between the curves is in\nexcellent agreement.9\n2θ(deg) Intensity[arb. units]\nFIG.8: (Color online)Rietveldrefinementfor thesamplewit h\nxCa= 0.05 from data collected at D2B at T= 70 K. Vertical\nbars at the bottom indicate Bragg reflections from the phases\nincluded in the refinement: the nuclear phases O1,O2 and\nthe magnetic phases SSO “222” and AFM “224”.\nD. Sample with xCa= 0\nWe finally turn to the parent compound. This sample\nwas only studied in the high resolution instrument, so we\ncannot present continuous temperature scans in the low–\ntemperature range as in the other samples. We collected\nthree diffractograms at T= 70 K, 273 K and 348 K. A\nsimilar study on this system has been very recently re-\nported,33focused on the paramagnetic to ferrimagnetic\ntransition, and a verygood agreementis found. It should\nbe emphasized that those authors have refined the pat-\nterns using the expanded “222” cell for the nuclear phase\nin the ferrimagnetic region in the Pmmaspace group, an\nhypothesis which —as we mentioned before— we do not\ndiscard but prefer to use the “122” Pmmm cell to be\nconsistent along the whole series. For the lowest temper-\nature, however, those authors did not present any refine-\nment of their data. The intriguing fact that the “222”\nmagneticreflectionsreappearedat190Kafterhavingdis-\nappeared at 265 K was left unexplained. In the present\nwork, we show that our diffractogramat 70 K can be sat-\nisfactorily refined in the framework of the analysis pre-\nsented for the xCa= 0.05 sample. Therefore, the nuclear\ndiffraction was accounted for by using two orthorhombic\nphases,O1 andO2, and the whole set of magnetic reflec-\ntions could then be assigned to each of these phases. As\ninxCa= 0.05, theO2 phase presents an AFM ordering\nwith a “224” magnetic cell. The O1 phase, on the other\nhand, cannot keep its ferrimagnetic SSO order because0 50 100 150 200 250 300 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 \nxCa =0.10 \nxCa =0.05 \n MH=0 [ µB/Co] \nT [K] \nFIG. 9: (Color online) Spontaneous magnetization of the fer -\nrimagnetic phase in samples with xCa= 0.05 andxCa= 0.10,\nobtained from our NPD refinements as M=µCo·fO1,\nwhereµCorepresents the net magnetic moment per Co atom\n(=µCoOct/4) andfO1is the phase fraction of the O1 phase.\nthis would not be compatible with the macroscopic mag-\nnetization measurements. In addition, a close inspection\nof the magnetic peaks reveals that not all the reflections\nfrom the 273 K ferrimagnetic phase are present, but that\ncontributionstointensityrelatedtoFMplanesareabsent\nat 70 K. Therefore, we have used a second AFM model\nfor theO1 phase at 70 K, with a “222” magnetic cell,\nand a G–type–like ordering, although we have allowed\nfor different magnetic moment values in pyramids and\noctahedra. The model is schematized in Fig. 1(d). This\nscenariois also supported by our data on a different sam-\nple, substituted with 5 % Sr ( Y(Ba 0.95Sr0.05)Co2O5.5)\nfor which we have also performed neutron thermodiffrac-\ntion scans and apparently behaves almost the same as\nthe parent compound. These results will be published in\na separate paper.34\nIn Table III we present the details of the refined struc-\ntures for the sample with xCa= 0 from D2B data. The\ndiffractogram at 70 K was refined with the O1+O2 mix-\nture, plus two AFM phases associated to each of them,\none having a “224” supercell and the other one with a\n“222” supercell. At 230 K, only the O1 phase with fer-\nrimagnetic order was enough to refine the diffractogram,\nwhereas at 348 K just the nuclear O1 phase was refined.\nIt is worth noting that the parent compound seems to\nhave a higher degree of misplaced octahedra when com-\npared to the Ca–substituted samples. This is evidenced\nby the non–zero occupation of the O3’ site, which cor-\nresponds to the empty apical oxygen position of pyra-\nmids. At the highest temperature, however, there is a\nslight rearrangement of vacancies among the O3 and O3’\nsites. When comparing the low temperature phases O1\nandO2 in the xCa= 0 and xCa= 0.05 samples, we10\nTABLE III: Structural parameters refined from the high reso-\nlutionD2BdatafortheparentcompoundYBaCo 2O5.5atT=\n70 K, 273 K and 348 K . Two sets of atomic fractional coordi-\nnates are given for the O1 andO2 phases, which correspond\nto space group Pmmm in the following Wyckoff positions: Y\n(2p)=(1\n2,y,1\n2); Ba, Ca ( 2o)=(1\n2,y,0); CoOct ( 2r)=(0,1\n2,1\n4);\nCoPyr ( 2q)= (0,0,z); O1 (1a)=(0,0,0); O2 ( 1e)=(0,1\n2,0);\nO3 (1g)=(0,1\n2,1\n2); O3’ (1c)=(0,0,1\n2); O4 (2s)=(1\n2,0,z); O5\n(2t)=(1\n2,1\n2,z); O6 (4u)=(0,y,z).\nT= 70 K T= 273 K T= 348 K\nO1 O2 O1 O1\nY y0.2809(9) 0.271(1) 0.2745(6) 0.2692(5)\nBa y0.255(2) 0.238(2) 0.254(1) 0.244(1)\nCoP z0.251(3) 0.270(3) 0.261(1) 0.260(1)\nO4 z0.321(2) 0.300(2) 0.3127(9) 0.3126(7)\nO5 z0.251(2) 0.276(3) 0.274(1) 0.267(1)\nO6 y0.2449(9) 0.243(1) 0.2495(9) 0.2437(8)\nO6 z0.299(1) 0.300(1) 0.2935(7) 0.2984(5)\nO3Occ 1.0 0.67(6) 0.86(3) 0.81(2)\nO3’Occ 0.0 0.18(4) 0.05(2) 0.12(2)\na(˚A) 3.8515(2) 3.8819(2) 3.8496(1) 3.8221(1)\nb(˚A) 7.7785(5) 7.7156(4) 7.8085(2) 7.8581(3)\nc(˚A) 7.4859(6) 7.4845(6) 7.5032(2) 7.5250(2)\nf(%) 42(3) 58(3) 100 100\nRB 7.7 7.5 8.1 6.4\nχ23.98 3.97 3.53\nobserve that the O1 phase seems to prefer a more per-\nfect order of pyramids and octahedra, while the excess\noxygen vacancies accommodate in the O2 phase. There\nis also consistency among the structural parameters in\nboth samples for the O1 andO2 phases. The results\nfor the refined lattice parameters at the three temper-\natures studied are shown in Fig. 3(c). Our data are\ncompared with those published by Akahoshi and Ueda\n(diamonds).30The difference at 70 K is due to the use\nof one or two phases to refine the data. We have found\nno way of indexing the whole set of magnetic reflections\nbased on a single nuclear structure. It has been shown\nin other Rcobaltites that there could be two coexisting\nmagnetic arrangementson a single nuclearstructure,11,16however, the evidence found in the xCa= 0.05 thermod-\niffractograms, and the consistency obtained in the whole\nseries when adopting such a phase separationmodel, give\nus confidence in the proposed scenario. The complexity\nof the systems seems to be related to the small size of the\nRcation.\nIV. CONCLUDING REMARKS\nAlthough the layered cobaltites RBaCo2O5.5have re-\nceived great attention in the past five years, much of\nits behavior remains still controversial and unclear. In\nthis paper, an attempt is made to get some insight into\nthe role of cationic disorder by substituting the Ba–site,\na topic that has not yet been investigated to the best\nof our knowledge. Interestingly, the systematics of this\nsubstitution led us to clarify and to propose a model\nthat describes the behavior at low temperature of the\nundoped parent compound. Even though Ca addition\ndoes not lead to severe structural distortions, it has nev-\nertheless dramatic effects on the magnetic arrangement\nand stability of the ferrimagnetic phase on detriment of\nthe AFM long–rangeorder. Our results open up the pos-\nsibility of studying Ca–doped cobaltites in order to iso-\nlate the intrinsic properties of the “122” ferrimagnetic\nphase in monophasic samples, avoiding spurious effects\nin the analysis of macroscopic properties. Further work\nis in progress to investigate the role of different cations\nsubstitution and the systematics of the Ba–site disorder\neffects.\nAcknowledgments\nThis work is part of a research project supported by\nAgencia Nacional de Promoci´ on Cient´ ıfica y Tecnol´ ogica\n(Argentina), under grant PICT 17-21372 and 20144,\nby CONICET (Argentina) under grant PIP 5250/05\nand 5657/05, and by SECTyP, Universidad Nacional de\nCuyo. JC acknowledges a fellowship from CNEA and\nCONICET. 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Yarem-\nchenkoandV.V.Kharton, Phys.Rev.B 75, 134407 (2007).\n34Aurelio et al., in preparation.0 50 100 150 200 250 300 350 7.45 7.50 7.65 7.70 7.75 7.80 7.85 \n(a) b1\n2a1\nc1\n Lattice Parameters [Å] \nT [K] 0 50 100 150 200 250 300 350 7.45 7.50 7.65 7.70 7.75 7.80 7.85 \n(c) cb\n2a Akahoshi & Ueda [ref. 8] \nb2b1\n2a2\n2a1\nc1=c2\n Lattice Parameters [Å] \nT [K] 0 50 100 150 200 250 300 0.0 0.5 1.0 1.5 2.0 2.5 3.0 \n Octahedral site 2 \n Pyramidal site \n(a) \n m [ µB/Co] \nT [K] 0 50 100 150 200 250 300 0.0 0.5 1.0 1.5 2.0 2.5 3.0 \n(b) \n m [ µB/Co] \nT [K] " }, { "title": "1807.06884v2.Controlled_anisotropic_dynamics_of_tightly_bound_skyrmions_in_a_synthetic_ferrimagnet_due_to_skyrmion_deformation_mediated_by_induced_uniaxial_in_plane_anisotropy.pdf", "content": "Controlled anisotropic dynamics of tightly bound skyrmions in a synthetic ferrimagnet due to\nskyrmion-deformation mediated by induced uniaxial in-plane anisotropy\nP. E. Roy,1,\u0003Rub´en M. Otxoa,1, 2and C. Moutafis3\n1Hitachi Cambridge Laboratory, J. J. Thomson Avenue, CB3 OHE, Cambridge, United Kingdom\n2Donostia International Physics Center, Paseo Manuel de Lardizabal 4, Donostia-San Sebastian 20018, Spain\n3School of Computer Science, University of Manchester, Manchester M13 9PL, United Kingdom\n(Dated: July 20, 2018)\nWe study speed and skew deflection-angle dependence on skyrmion deformations of a tightly bound\ntwo-skyrmion state in a synthetic ferrimagnet. We condsider here, an in-plane uniaxial magnetocrystalline\nanisotropy-term in order to induce lateral shape distortions and an overall size modulation of the skyrmions due\nto a reduction of the effective out-of-plane anisotropy, thus affecting the skyrmion speed, skew-deflection and\ninducing anisotropy in these quantities with respect to the driving current-angle. Because of frustrated dipolar\ninteractions in a synthetic ferrimagnet, sizeable skyrmion deformations can be induced with relatively small\ninduced anisotropy constants and thus a wide range of tuneability can be achieved. We also show analytically,\nthat a consequence of the skyrmion deformation can, under certain conditions cause a skyrmion deflection with\nrespect to driving-current angles, unrelated to the topological charge. Results are analyzed by a combination\nof micromagnetic simulations and a compound particle description within the Thiele-formalism from which an\nover-all mobility tensor is constructed. This work offers an additional path towards in-situ tuning of skyrmion\ndynamics.\nI. INTRODUCTION\nMagnetic skyrmions have been extolled as candidates for\nthe constituent information carriers in spintronic devices such\nas racetrack memories and logic circuits1–6. Their advantage\nover conventional domain walls are that they are less sensi-\ntive to edge defects and can be driven at much lower current\ndensities. However, similar to the Hall-effect for an elec-\ntrically charged particle, a magnetic skyrmion viewed as a\nquasi-particle endowed with a topological charge, exhibits a\ndeflection known as the skyrmion Hall-effect7,8, resulting in a\nskew deflection known as skyrmion Hall-angle, \u0002Sk. This can\nlead to detrimental annihilation at device-boundaries at high\nenough driving amplitudes. In order to overcome this, there\nhave been proposals suggesting the usage of both intrinsic9–11\nand synthetic antiferromagnets12,13(SAFs) with identical but\noppositely magnetized sublattices, whereby the direction of\nthe gyroforce on a skyrmion in one sublattice is equal but op-\nposite in direction for the skyrmion on the other sublattice9.\nThis is because, by virtue of satisfying the antiferromagnetic\ncoupling they have opposite topological charge. The two de-\nflection forces then fully cancel out and the compound object\nmoves without deflection. An advantage of antiferromagnetic\nsystems is the robustness against external field perturbations.\nHowever this means that the manipulation and detection of an-\ntiferromagnetic textures is generally a difficult task. Devices\nbased on single layer ferromagnets as the functional compo-\nnent have the advantage of easy detection schemes but are\nsenstitive to external fields. However in ferrimagnetic sys-\ntems, a low net moment is present, offering reasonable de-\ntectibility and at the same time a good degree of robustness\nagainst disturbing stray-fields. A net moment though, means\na finite skyrmion-Hall angle (even if it is much reduced com-\npared to a single layer ferromagnetic system). An additional\nbenefit of synthetic systems is the large tuneability of mate-\nrial properties, but the conclusions extolled herein should bevalid also for instrinsic ferrimagnets. Apart from the desire\nto achieve some degree of control over \u0002Sk, another impor-\ntant dynamical property to tune and/or enhance is the speed\nof skyrmion propagation. Enhancement is desired due to in-\ncreased operating frequency of a device. Tuning in general\nwould find usage wherever the skyrmionic device is a sub-\nset of a bigger multifunctional device, whereby timescales\nneed to be matched at different points of device functional-\nity in space. In order to modulate the dynamical properties\nof a skyrmionic magnetic device, one may consider tailor-\ning of material properties during fabrication and/or affecting\nthe ready-made device by external means during operation.\nWe refer to the former as intrinsic and the latter as extrinsic\ntuning, respectively. In particular, extrinsic tuning offers ma-\nnipulation on the fly. Different approaches have previously\nbeen presented for dynamical control such as: by mismatch-\ning the saturation magnetization of the constituent ferromag-\nnetic (FM) layers in an SFIM14, spatially uniform modulation\nof the perpendicular anisotropy15, perpendicular anisotropy\ngradients16,17and radial magnetic field gradients18. In such\na scenario (for a given topological charge) the skyrmion ra-\ndius is isotropically varied and thus its dynamical behaviour is\nisotropic in the plane of propagation with respect to the driv-\ning current-angle. There have recently been some very in-\nteresting works considering the effect of inducing anisotropic\nDzyaloshinskii-Moriya interaction in single layer FMs as a\ncontrol-knob for skyrmion dynamics (speed and skew-angle),\noffering also due to skyrmion-shape distortion anisotropic dy-\nnamical behaviour with respect to the direction of the driving\nexcitation mechanism,19,20. The anisotropic dynamics adds an\nadditional degree of freedom in tuning the skyrmion dynami-\ncal behaviour.\nIn this work, we focus on synthetic ferrimagnets (SFIMs)\nfor reasons mentioned in the preceeding paragraph and con-\nsider here tuning the dynamics due to deformation of a bound\ntwo-skyrmion texture via only an induced uniaxial in-plane\nanisotropy term, achievable by e.g. an inverse magnetostric-arXiv:1807.06884v2 [cond-mat.mes-hall] 19 Jul 20182\ntive effect whereby a mechanical uniaxial stress induces a uni-\naxial contribution to the magnetocrystalline energy21. The\ndriving mechanism is provided by spin-Hall-effect induced\ntorques. By utilizing a synthetic ferrimagnet, we are able\nto induce large skyrmion deformations for relatively small\nvalues of the induced uniaxial in-plane anisotropy constants\nand predict a wide tuneability of the bound skyrmion skew-\ndeflection, speed and degree of anisotropy of the said quan-\ntities with respect to the driving current-angle. In addition,\nit is shown, that, given a deviation from circularly shaped\nskyrmions, that for driving current angles different from mul-\ntiples of\u0019=2, there is a contribution to the skyrmion deflection\naway from the driving-current direction which is independent\nof the topological charge.\nII. METHODS\nA. Thiele approach and the effective skyrmion mobility tensor\nThe system under consideration is schematically depicted\nin Fig. (1), whereby two FM layers (FM 1and FM 2) are sepa-\nrated by a Ru spacer, magnetically coupled via antiferromag-\nnetic (RKKY-type) and dipolar interactions. Each FM layer\nis also coupled to a heavy metal (HM) which promotes inter-\nfacial Dzyaloshinskii-Moriya (IDMI)-interaction. Each FM\nlayer contains one skyrmion. The two skyrmions are consid-\nered to be strongly antiferromagnetically coupled, i.e. tightly\nbound such that they move as one compound unit without in-\nternal nor relative dynamics. The means of propulsion of this\ncompound skyrmion is supplied by torques exerted by a spin-\naccummulation due to the Spin-Hall-Effect (SHE). The SHE\nis produced by passing a current Ithrough the HMs, generat-\ning current-densities J1andJ2in HM 1and HM 2, respectively\n(see Fig. 1 (a)). We consider an arbitrary in-plane current-\ndirection. Thus we define a global coordinate system and a\nprimed/local system, whereby the current direction defines the\nrotation of primed coordinates with respect to global coordi-\nnates, as shown in Fig. (1 (b)).\nIn order to analytically describe the dynamics of the\nskyrmions, we turn to the Thiele equation22with Spin-Hall\nforces2,7,8. We consider the tightly bound skyrmions in dy-\nnamic equilibrium and sum the forces to zero according to14:\nX\nL\u00160dLML\n\rLf\u0000QL[^z\u0002v]\u0000\u000bLDDDL\u0001v+ 4\u0019b0\nLIIIL\u0001JLg=0:\n(1)\nThe subscript L= 1;2denotes the FM layer num-\nber (Fig. 1 (a)), \rLis the gyromagnetic ratio, \u00160,\nthe permeability in vacuum, MLthe saturation magne-\ntization and dLthe layer thickness. The first term\nin Eq. (1) represents the gyroforce with topological\ncharge23QL=1\n4\u0019RR^mL\u0001[@x^mL\u0002@y^mL]dxdy (the\ntopological charge being a component of the gyrovector\nGLthrough the relation GL= [0 0\u00004\u0019QL]), with\n^mL=ML=ML. The second term constitutes a dissipa-\ntive drag force with an associated dissipation tensor DDDL=\n4\u0019h(Dii)L(Dij)L\n(Dji)L(Djj)Li\n, whose elements are given by (Dij)L=\nFIG. 1. (a) Geometry and constituent layers of the stack consid-\nered in this work. (b): Top view of the global ( xy) and primed/local\n(x0y0) coordinate systems. Here JLis the current density in the L:th\nFM layer whose direction is at an angle \u0012Jwith respect to the global\nsystem. \u0002Skof the bound skyrmion is defined as the angle of devi-\nation from the direction of the current density. Double arrows sig-\nnify the considered directions of induced uniaxial anisotropies (with\nanisotropy constants KxandKydepending on whether the easy di-\nrection is induced along xory, respectively). The velocity compo-\nnents of the effective skyrmion in the global coordinate system are\ndesignatedvxandvy, whereas the speed is denoted as v.\n1\n4\u0019RR\n[@i^mL\u0001@j^mL]dxdy2,22. The last term is the force ex-\nerted by the SHE-induced torque with b0\nL=\rL~\u001eLoL\n\u001602jejMLdL24and\n[Iqr]L=\u00001\n4\u0019RR\n[@q^mL\u0002^mL]s\u000fsrdxdy2,25;q;r2 fx;yg\nand\u000fsris the Levi-Civita symbol. Further, ~is Planck’s re-\nduced constant, \u001eLthe intrinsic spin-Hall angle and eis an\nelectron’s charge. The unit magnitude factor oLtakes into ac-\ncount the effect of HM/FM stacking order, on the direction\nof the resulting spin-accummulation. As HM 1is under FM 1\nand HM 2is above FM 2o1ando2have opposite sign. We set\nhere,o1= 1 ando2=\u00001. For brevity we assign a tensor\nSSSL=b0\nLIIIL. Note that the units of SSSis [m3/As] (velocity per\nunit current density) and as such may be viewed as a SHE-\nrelated mobility tensor in the absence of other forces. We re-\niterate, that the underlying assumptions in stating the problem\nby Eq. (1), are that the internal structures of the skyrmions are\nrigid and that the antiferromagnetic coupling between them is\nstrong enough such that they move together without any rela-\ntive dynamics. Now, intrinsic tuning of \u0002Skand speed of the\ntightly bound system would be to consider a mismatch of the\nsaturation magnetization, gyromagnetic ratio or thickness be-\ntween the two FM layers. This is more easily seen by rewrit-\ning Eq. (1) as\u0000Qe[^z\u0002v]\u0000DDDe\u0001v+wSSS1\u0001J1+SSS2\u0001J2=0,\nwhereQe=wQ1+Q2andDDDe=w\u000b1DDD1+\u000b2DDD2, with\nw=\r2d1M1\n\r1d2M2. Further, we write the current density J2in terms\nofJ1and make the reasonable assumption that J1andJ2\npoint in the same direction such that J2=ksJ1, whereks\nis a real scaling factor. Thus the inclusion of different mag-\nnitudes of current densities in the top and bottom FMs is re-\ntained. Now, we can define; SSSe=wSSS1+ksSSS2. Therefore,\nhere, the SHE-mobility of the compound particle is a function\nof the ratio of current-density amplitudes in the FM layers.\nThe Thiele equation for the single compound particle reads:\n\u0000Qe[^z\u0002v]\u0000DDDe\u0001v+SSSe\u0001J1=0: (2)\nViewing the system in terms of this compound particle de-3\nscription with effective properties, which henceforth is called\neffective skyrmion, it is easily seen that we can modulate Qe,\nDDDeandSSSeby varyingwand thus affect the dynamical prop-\nerties such as speed and \u0002Skof the effective skyrmion. How-\never, a path towards in-situ modulation of the skyrmion dy-\nnamics is rather to utilize the fact that the elements of the ten-\nsors in Eq. (1) and thus in Eq. (2) are determined by the ge-\nometry of the texture15. Provided a suitable material system\nis available, one route towards in-situ manipulation is by an\ninduced in-plane anisotropy in order to deform the magnetic\ntexture (induced by e.g. the inverse magnetostrictive effect,\nwhereby an imposed mechanical stress induces a magnetic\nuniaxial anisotropy). For the type of skyrmion deformation\nconsidered here (expansions along principal axes), we assume\nDe\nxy=De\nyx= 0 andSe\nxy=Se\nyx= 0 (which was verified\nwhen evaluating the tensor components from micromagneti-\ncally obtained magnetization distributions). From Eq. (2) the\nvelocity components are then vx=QeSyyJ1y+De\nyySxxJ1x\n(Qe)2+DexxDeyyand\nvy=De\nxxSyyJ1y\u0000QeSxxJ1x\n(Qe)2+DexxDeyy. Thus we can write the velocity\nof the effective skyrmion in terms of an over-all effective mo-\nbility tensor \u0016\u0016\u0016e. Remembering that the current density in the\nfirst FM is to be used and setting J=J1, we may write:\nvi=\u0016e\nijJj (3)\n\u0016\u0016\u0016e=2\n4De\nyySe\nxx\n(Qe)2+DexxDeyyQeSe\nyy\n(Qe)2+DexxDeyy\n\u0000QeSe\nxx\n(Qe)2+DexxDeyyDe\nxxSe\nyy\n(Qe)2+DexxDeyy3\n5 (4)\nEqs. (3-4) constitute the velocity components of the com-\npound particle with respect to the global coordinate system\n(see Fig. 1(b)) and as such, determining the \u0002Skfrom the\nratiovy=vxwould give an angle with respect to the global x-\naxis. Thus in order to study the \u0002Sk-dependence on current\ndirection we must define it with respect to a rotated coordinate\nsystem defined by the current direction. If we denote the angle\nof the uniform current density with respect to the global x-axis\nby\u0012J, andJ=J[cos (\u0012J) sin (\u0012J)]T, then we can express\nvxandvyin terms of the primed coordinate system as vx0=\ncos(\u0012J)vx+sin(\u0012J)vyandvy0=\u0000sin(\u0012J)vx+cos(\u0012J)vy.\nConsequently, \u0002Sk=atan\u0010vy0\nvx0\u0011\n.\nB. Micromagnetic technique\nIn order to calculate the mobility tensor under various states\nwith different induced in-plane anisotropies, we use equi-\nlibrium configurations determined by micromagnetic simula-\ntions. To stabilize the antiferromagnetic alignment between a\nskyrmion in FM 1and another skyrmion in FM 2, it is neces-\nsary for them to have the same handedness13. The handedness\nis determined by the sign of the IDMI. The sign itself, depends\non; i) intrinsic material properties and ii) whether the HM is\ncoupled below or above a FM, since opposite stacking order\nwill change the sign of the IDMI26,27. For the former, we de-\nnote the intrinsic IDMI strengths by DLfor a given layer L.\nThe latter contribution is taken into account for by a multi-\nplicative factor fLwhere we set f1= +1 andf2=\u00001.Whenf1D1=f2D2then textures in both FMs will have the\nsame handedness13. Thus for the stack considered here, HMs\nneed to be chosen such that D2=\u0000D113. Furthermore, in order\nfor the AFM coupled skyrmions to move in the same direction\nwhen current is passed through both HMs, the intrinsic Spin-\nHall angles, \u001eLof the two HMs need to be opposite in sign\n(\u001e2=\u0000\u001e1)13. One possible HM material pair could be e.g. Pt\nand W13. The total energy density considered of a given FM\nlayerLis:\n\u000fL=AL3X\ni=1jrmi;Lj2+\u001b\ntRu(1\u0000^mL=1\u0001^mL=2)\n+fLDL[mz;L(r\u0001^mL)\u0000(^mL\u0001r)mz;L]\n+K?\nU;L\u0010\n1\u0000(^mL\u0001^z)2\u0011\n+Kk\nU;L\u0010\n1\u0000\u0000^mL\u0001^uk\u00012\u0011\n\u00001\n2\u00160ML^mL\u0001Hm;L:(5)\nIn Eq. (5), ALare the intra-layer exchange stiffness con-\nstants,\u001bthe inter-layer exchange coupling constant (here\nantiferromagnetic) between the layers, tRuis the thickness\nof the Ru spacer, K?\nU;Lare out-of-plane uniaxial magne-\ntocrystalline anisotropy constants, Kk\nU;Lare in-plane uniax-\nial magnetocrystalline anisotropy constants with easy direc-\ntions ^uk. We shall henceforth use the denomination Kxif\n^uk=^xto describe induced in-plane anisotropy along the\nx-direction and Kyif^uk=^yfor they-direction. It is\nalso to be understood that in any state of induced anisotropy\nwe consider, Kk\nU;1=Kk\nU;2. Further, Hm;L is the mag-\nnetostatic field, is evaluated in the the entire computational\ndomain. The interaction terms in Eq. (5) of the main\ntext give rise to an effective field Heff;L=\u00001\n\u00160ML@\u000fL\n@^mLact-\ning on the magnetization. In particular the IDMI field is\nHIDMI;L=\u00002fLDL\n\u00160ML[(r\u0001^mL)^z\u0000rmz;L]28,29and the inter-\nlayer exchange field at a site idue to interaction with a site\njisHRKKY;i=2\u001b\ntRu\u00160Mi^mj. We consider here that this RKKY-\nlike interaction occurs between the computational cells sep-\narated purely along the ^z-axis. Due to the IDMI, Neumann\nboundary conditions on the lateral surfaces of the FMs are\n@^mL\n@^nL=\u0000fLD1;2\n2AL(^z\u0002^nL)\u0002^mL, where ^nLis the unit nor-\nmal vector to the lateral surfaces28,29. For surfaces normal to\nthe plane, homogeneous Neumann conditions are used. The\nmagnetostatic field, Hm;L is computed as the spatial convo-\nlution between a demagnetizing tensor30and the magnetiza-\ntion distributions by Fast Fourier Transform techniques31. In\nthe magnetostatic field computation, for the near-field, we use\nanalytical formulae for the demagnetizing tensor by Newell\nand Dunlop30and for the far field, at relative cell distances\nlarger than 40, the point dipole approximation is used. The\ndynamics of the system is modelled by solving the Landau-\nLifshitz-Gilbert (LLG) equation, with added Spin-Hall-Effect\ntorques28:\n@^mL\n@t=\u0000\rL^mL\u0002Heff;L+\u000bL^mL\u0002@^mL\n@t\n\u0000jb0\nLJLj[^mL\u0002(^mL\u0002^pL)];(6)4\nwhere\rL=2.21\u0002105m/As is the gyromagnetic ratio, \u000bLis\nthe Gilbert damping, b0\nL=~\rL\u001eL\n\u001602eMLdL24with\u001eLbeing the\nintrinsic Spin-Hall angles, ethe electron charge, dLare the\nFM layers’ thicknesses, JLare the current density amplitudes\nand^pLare the directions of the spin-accumulation acting on\nlayerLdue to the SHE at the FM/HM interfaces. The spin-\naccumulation due to a current density along direction ^jL(unit\nvector) present in HM Lis^pL=sgn\u001eL\u0010\n^jL\u0002^nHM-FM;L\u0011\n,\nwhere ^nHM-FM;Lis the unit normal vector directed from a HM\ntowards a FM layer28. For computational simplicity, Eq. (6)\nis cast into explicit form and solved by a fifth order Runge-\nKutta integration scheme32. The lateral space considered is a\nrectangular domain of dimensions 1200 \u0002768 nm2whereas\nthe comprising FM layers and spacer layer all have a thick-\nness of 0.8 nm. The domain is discretized into 1.5 \u00021.5\u00020.8\nnm3cells. Material parameters considered are the following:\nA1=A2=20 pJ/m,M1=0.6 MA/m,M2=0.75 MA/m,\nK?\nU;1=K?\nU;2=0.6 MJ/m3,\r1=\r2=2.21\u0002105m/As,\n\u000b1=\u000b2=0.1,D1=2.8 mJ/m2,D1=\u00002.5 mJ/m2,\n\u001e1=0.1,\u001e2=\u00000.1,\u001b=\u00000.5 mJ/m2andKk\nU;1=Kk\nU;2\nis varied in the range [0,0.09] MJ/m3. For determining static\nequilibrium configurations, initial conditions close to that of\ntwo antiferromagnetically coupled skyrmions were imposed\nand the system let to freely ring down until a convergence was\nreached in the whole computational domain (each layer satis-\nfying1\nMLj^mL\u0002Heff;Lj\u001410\u00006). In the calculation of Qe,\nDeandSefrom the static equilibrium distributions, contribu-\ntions from magnetization-canting at and close to the bound-\naries were excluded by removing 45 computational cells into\nthe computational domain along all lateral normals. For all\ncurrent-driven dynamical calculation we consider J16=J2,\nspecificallyJ1=1011andJ2= 2\u00021011A/m2( i.e.ks= 2).\nFor the evaluation of \u0002Skfrom the micromagnetic simula-\ntions, skyrmion positions versus time in each FM layer were\ntracked by the moments of topological density23. Simulations\nfor each case of induced anisotropy were run until steady\nstate motion of the skyrmion pair was achieved. Care was\ntaken to only extract data for positions sufficiently far away\nfrom the boundaries in order to exclude repulsive boundary-\ninteractions. From the extracted longitudinal and transverse\nspeeds,vxandvythe speedvwas obtained and \u0002Skwas ex-\ntracted from the velocities expressed in the primed coordinate\nsystem according to the preceding section.\nIII. RESULTS AND DISCUSSION\nIn order to quantify the effect of an in-plane anisotropy,\nequilibrium configurations were computed for a range of in-\nduced in-plane anisotropy states. A note however, is in or-\nder for our choice of the particular degree of ferrimagnetism,\ni.e. the choice of M2in relation to M1. Firstly, it is found\nthat for a given in-plane anisotropy constant, the larger the\nratio ofM2toM1, the larger was the resulting skyrmion\ndeformation. Secondly, although the stability of the bound\nskyrmion state ranges over a wide set of M2=M1-ratios in\nour system, 1\u0014M2=M1\u00141:5, the range of applied in-plane anisotropy, whereby the bound skyrmion state is stable\nreduces asM2=M1increases. We found M2=M1=1.25 to\nbe a good compromise between the applicable range of im-\nposed in-plane anisotropies and the degree of skyrmion defor-\nmation (which in turn means range of dynamical tuneability).\nA qualitative argument for the trend of increased skyrmion\ndeformability with increasing M2=M1-ratios can be formed\nby considering that the dipolar interaction between FM 1and\nFM2is frustrated, except in the region where the magnetiza-\ntion lies predominately in plane (within the skyrmion width).\nWhen a uniaxial anisotropy is imposed, the dipolar energy can\nbe somewhat lowered by tilting more moments towards the\nplane. The projection on the plane of this tilting should of\ncourse be in the direction of the induced in-plane anisotropy.\nThus, the higher the dipolar frustration is (meaning as the ra-\ntioM2=M1increases), the higher is the motivation to increase\nthe in-plane portion of the skyrmion state. This would trans-\nlate to a higher degree of skyrmion deformation for a given\nvalue of the in-plane uniaxial anisotropy. Fig. 2 shows the\nequilibrium configurations of the bound skyrmion state for a\nsubset of induced in-plane anisotropies. The effect of a Kxor\nKycan be stated by the following two observations: (i): The\nskyrmions elongate preferentially along the direction perpen-\ndicular to the induced anisotropy, i.e. become elliptical with\nthe major axis perpendicular to the induced anisotropy direc-\ntion. (ii): the over-all skyrmion-size increases with increasing\nin-plane anisotropy. Point (i) is a logical consequence of the\nsystem increasing its number of moments pointing by rotat-\ning the in-plane portion of the skyrmion towards the direction\nof induced anisotropy. Point (ii) is also to be expected, as by\nincluding an in-plane anisotropy, the out-of-plane anisotropy\nis in effect reduced. Test calculations were performed ver-\nifying that the skyrmion size increases as the perpendicular\nanisotropy decreases. Another contribution to skyrmion size-\nenhancement was touched upon above, in terms of lowering\nthe dipolar frustration. If the texture was circularly symmetric\nwith only the radius varying we should expect that the diag-\nonal elements of the effective mobility tensor (Eq. 4) to be\nequal and the absolute values of the off-diagonal elements to\nbe equal, because then the diagonal elements of DDDeare equal\nand the same would apply to SSSe, whileQedoes not depend\non the spin-profile of the skyrmion and thus remains constant\n[compare to circularly symmetric skyrmions in single layer\nFMs7]. In such a case, modulation of values of the tensor el-\nements will indeed alter the speed and magnitude of \u0002Sk, but\nthe dynamics will be isotropic in the plane, i.e. no matter in\nwhich direction the driving current flows, the speed and \u0002Sk\ndo not change. This could be achieved as has been proposed\nin other works; by modulating the perpendicular anisotropy\nconstant14. The situation changes drastically if the skyrmion\nis deformed. Immediately we can expect that the elements of\nthe diagonal tensors DDDeandSSSediffer in magnitude. This is\nshown in Fig. 3, whereby the dependence of DDDe,QeandSSSe,\non induced in-plane anisotropies along xandy-directions are\nshown. The elements were computed by using the micromag-\nnetically obtained configurations with parameters described\nin the caption. It was verified through direct calculations, that\nDe\nxy=De\nyx= 0andSe\nxy=Se\nyx= 0.5\nFIG. 2. (Colour online) Skyrmion mz-profiles along two cuts, XX and YY for various states of induced Kk\nU;L. Below each profile is shown\nthe corresponding vector-plots of the magnetization distributions in the bottom (L=1) and top (L=2) layers with colorcoding corresponding\ntomz; red = +1 and dark blue= -1. The circles with arrows in the magnetization distribution corresponding to Kx=Ky= 0 indicate\nthe in-plane components of the skyrmions. The direction of induced anisotropy is indicated by thick double arrows in the vector-plots : (a):\nKx=Ky=0. (b):Kx=0.05 MJ/m3. (c):Kx=0.09 MJ/m3. (d):Ky=0.05 MJ/m3. (e):Ky=0.09 MJ/m3.\nFIG. 3. (Colour online). The non-zero tensor elements of DDDe,\nSSSeand the effective charge Qeas a function of induced in-plane\nanisotropies along xandy-directions, obtained from micromagnet-\nically computed configurations. Parameters used are: \r1=\r2=\n2.21\u0002105m/As,\u001e1=0.1,\u001e2=\u00000.1,o1=1,o2=\u00001,M1=0.6\nMA/m,M2=0.75 MA/m, \u000b1=\u000b2=0.1 andd1=d2=0.8 nm.\nThe direction of uniaxial in-plane anisotropy is indicated in each plot\nby double arrows. (a): De\nxx,De\nyyandQevs.Kx. (b): Se\nxx,Se\nyyvs.\nKx. (c):De\nxx,De\nyyandQevs.Kyand (d): Se\nxx,Se\nyyvs.Ky.\nWe can see the above discussion of the expectations ver-\nified, that for zero in-plane anisotropy, the elements of the\ndiagonal tensors DDDeandSSSeare equal, signifying circularskyrmion shape. Let us look at Fig. 3 (a) and (b). As Kxin-\ncreases there is a splitting in the dependencies of De\nxxandDe\nyy\nin (a) and the same for Se\nxxandSe\nyyin (b). The charge Qeis as\nexpected unaffected by the presence of the anisotropy-induced\ntexture deformation. As Kxincreases, the skyrmions elongate\nmore alongy(Fig. 2 (b) and (c)). The effect on the drag force\nis then according to Fig. (3 (a)) a rapid increase along the di-\nrection of the elliptically shaped skyrmion’s minor axis and a\nslower increase along the axis of elongation (major axis). If\nwe think of the drag as a resistance to ��ow then we should\nexpect a larger drag in the direction of propagation with the\nlarger frontal area. Conversely along a direction whereby the\nobject is more stream-lined-shaped, a relatively smaller drag\nis to be expected. This view is consistent with the observa-\ntion made herein concerning sharp increase in De\nxxasKxin-\ncreases. The reason for a noticeable increase also in the De\nyy-\nelement is that the over-all size of the skyrmion also increases\nwith increasing Kx. In terms of Se\nxxandSe\nyy(Fig. 3 (b))\nwhich constitute the spin-Hall effect mobility, the largest in-\ncrease is for Se\nxxas the number of moments along xincreases\nboth as a result of satisfying the direction of anisotropy. The\nreasons for an increase of Se\nyyis similarly to the discussion on\nthe drag force, due to an increase in skyrmion width, mean-\ning an increase also in the number of moments along the y-\ndirection. Since the SHE-induced spin-accummulation is per-\npendicular to the current direction, then we should expect that\ne.g. given a current fixed along x, the speed enhancement\nis greater for an induced anisotropy Kxthan for an induced\nKy. The situation is reversed if the current is injected along y.\nThe same arguments can be applied to the case of an induced\nanisotropyKyalongy. This is shown in Figs. 3 (c) and (d),\nwhereby the situation is reversed with respect to Figs. 3 (a)6\nand (b), because the skyrmion preferential elongation is along\nx. We now evaluate the over-all mobility tensor using DDDe,SSSe\nandQe. In order to be as general as possible we consider the\ncase whereby J16=J2and set here ks=2. Evaluating the\nexpression in Eq. (4) and plotting the tensor elements versus\ninduced in-plane anisotropy along both xandy-directions,\nKxandKy, respectively, we obtain the results shown in Fig.\n4.\nFIG. 4. (Colour online) Effective mobility-tensor elements as a\nfunction of induced in-plane anisotropy. Here, ks=2. (a):\u0016e\nxxand\n\u0016e\nyyvs.Kx. (b):\u0016e\nxyand\u0016e\nyxvs.Kx. (c):\u0016e\nxxand\u0016e\nyyvs.Ky.\n(d):\u0016e\nxyand\u0016e\nyxvs.Ky.\nAs can be seen in Figs. 4 (a) and (c), both mobilities \u0016e\nxx\nand\u0016e\nyyincrease with increasing induced anisotropy. Therate of increase of these elements is greatest in the direction\nof the induced anisotropy. In terms of the off-diagonal ele-\nments, there is also a difference in their behaviour in terms of\ntheir rate of change with respect to the direction of induced\nanisotropy, see Figs. 4 (b), (d). These two behaviours mean\nthat the speed vand skyrmion Hall-angle \u0002Skare tuneable\nby the in-plane anisotropy and that we can have anisotropy in\nthis tuneability by two means; either by, for a given current\ndirection, induce the anisotropy along different directions, or\nfor a fixed induced anisotropy-direction, vary the angle of the\ndriving current. In what manner vand\u0002Skchanges with in-\ncreased magnitude of the induced anisotropy boils down to the\nrelative rate of change and magnitudes of the tensor elements\nin Fig. 3 as the induced anisotropy is varied. The speed for\na given induced anisotropy gets its largest contribution from\nthe diagonal elements of \u0016\u0016\u0016e. In conjunction with the relative\nrate of growth of these elements being large we can expect\nthat for a given fixed current-direction, vwill always increase\nwith increasing magnitude of in-plane anisotropy. In addi-\ntion, it will become clear that the source of effective skyrmion\ndeflection away from the driving current direction can in prin-\nciple stem from two sources (except for current-directions ex-\nactly alongxory-directions): one due to a finite topologi-\ncal charge (the dominant contribution) and the second comes\npurely from lateral shape-distortions of the skyrmions. In fact,\neven for a situation whereby Qe=0 such as in a perfectly bal-\nanced SAF, there can be a deflection away from the driving\ncurrent-angle for \u0012Jdifferent from multiples of \u0019=2. We shall\nnow address these issues in an orderly fashion. Although\nlengthy, it is instructive to write out the full expressions for\n\u0002Skandvgiven an arbitrary current direction. Recalling that\nJ=J[cos (\u0012J) sin (\u0012J)]Twith\u0002Skdefined with respect to\nthe primed coordinate system and vbeing a magnitude which\nis easily stated using the global system as starting point (as the\nmagnitude will be the same in both coordinate systems), one\narrives at:\nv=Jr\nsin (2\u0012J)\u0002\n\u0016exx\u0016exy+\u0016eyx\u0016eyy\u0003\n+ cos2(\u0012J)h\n(\u0016exx)2+\u0000\n\u0016eyx\u00012i\n+ sin2(\u0012J)h\u0000\n\u0016exy\u00012+\u0000\n\u0016eyy\u00012i\n(7)\n\u0002Sk= arctan(1\n2sin (2\u0012J)\u0002\n\u0016e\nyy\u0000\u0016e\nxx\u0003\n+\u0016e\nyxcos2(\u0012J)\u0000\u0016e\nxysin2(\u0012J)\n1\n2sin (2\u0012J)\u0002\n\u0016exy+\u0016eyx\u0003\n+\u0016exxcos2(\u0012J) +\u0016eyysin2(\u0012J))\n(8)\nLet us start with tuneability of vand\u0002Skby inducedKxand\nKyfor a fixed\u0012J: Consider, as an example the case \u0012J=0\n(i.e. current injected purely along the x-direction). Then, from\nEq. (7) and Eq. (4), v(\u0012J= 0) =Jq\n(\u0016exx)2+ (\u0016eyx)2=\nJSe\nxx\n(Qe)2+DexxDeyyq\n(Dyy)2+ (Qe)2. From this expression, it is\nthen clear that the speed will have a different dependency on\nKx(withKy= 0) than onKy(withKx= 0), based on the\nindividual behaviour of the individual tensor elements on the\ninduced anisotropy. Now, for \u0002Sk, then from Eq. (8) and Eq.(4),\u0002Sk(\u0012J= 0) = arctanhvy0(\u0012J=0)\nvx0(\u0012J=0)i\n= arctanh\u0016e\nyx\n\u0016exxi\n=\narctanh\n\u0000Qe\nDeyyi\n, which is the standard result for the skyrmion-\nHall angle [Ref] as a special case of \u0012J= 0, except that\nwe specifically state here the De\nyy-element in the denomina-\ntor. We can intuitively predict the behaviour of \u0002Sk. Let us\nconsider again \u0012J=0 (current flowing along x) and an in-\nduced anisotropy Kx; We know from Fig. 3, that with in-\ncreasingKx, the largest change in dissipation occurs for the\nDe\nxxelement, but De\nyyincreases as well with the result that7\nFIG. 5. (Colour online) speed, vand\u0002Skvs.Kxwith current\ninjected along x(\u0012J= 0). The parameters used are the same as be-\nfore. (a):vvs.Kk(KxandKy) predicted by the effective skyrmion\napproach and computed by full dynamic micromagnetic simulations.\nThe direction of induced in-plane anisotropy is indicated by double\narrows in the legends. (b): \u0002Skvs.Kk(KxandKy).\nthe magnitude of \u0002Skis expected to decrease at some rate.\nNow, keeping the same scenario except now we impose an\nanisotropyKy, we know that the rate of increase of De\nyyis\neven greater with increasing anisotropy Ky. The expectation\nthen of the change in magnitude of \u0002Skshould be expected to\nbe greater for induced anisotropy along ywhen driving with\n\u0012J=0. To illustrate this simple analysis, vand\u0002eas a\nfunction ofKxandKyfor the current-angle \u0012J= 0 is com-\nputed and shown in Fig. 5. We have also compared the ef-\nfective skyrmion approach to full dynamical micromagnetic\nsimulations with an excellent agreement, thus validating the\napproach. The overall range of tuneability is significant and\nis anisotropic with respect to induced-anisotropy direction. In\na real situation we could envision the device mounted on a\npiezoelectric stressor that can transmit strain in two ortogonal\ndirections (one at a time) in order to induce and discriminate\ndifferent speeds. We now address the general case \u0012Jis al-\nlowed to vary whereby anisotropic behaviour of speed and\ndeflection angle are expected to be present. In addition, we\npoint out, that in general, as long as \u0012J6=n\u0019=2(wherenis\nan integer) and the effective skyrmion radius is not isotropic\n(deviation from circular shape) , \u0002Skwill contain a contri-\nbution originating from only the anisotropic deformation of\nthe skyrmion, i.e. independent of the topological charge. In\naddition the speed may also be non-isotropic with respect to\ndriving-current direction. In order to see this, we briefly di-\ngress on this point as it may be of consequence for exper-\niments whereby there is a sizeable inverse magnetostrictive\neffect causing a deviation from circularly shaped skyrmions.\nIf we impose Qe= 0 (in practice this means balancing the\nFM layers such that w=1), such that all off-diagonal el-\nements in Eq. (4) are zero, then From Eqs. (7) and (8),\nv(Qe= 0) =Jq\n(\u0016exx)2cos2(\u0012J) +\u0000\n\u0016eyy\u00012sin2(\u0012J)and\n\u0002Sk(Qe= 0) = arctan\u0014\n1\n2sin(2\u0012J)[\u0016e\nyy\u0000\u0016e\nxx]\n(\u0016exx)2cos2(\u0012J)+(\u0016eyy)2sin2(\u0012J)\u0015\n. It\nis now clear that, provided \u0016e\nxx6=\u0016e\nyy, there will be a finite\n\u0002Skfor all current-angles \u0012J6=n\u0019\n2in the absence of a net\ntopological charge. In terms of the speed, we can also see\nthat the role of a topological charge on vis two-fold; it affects\nboth magnitude and shifting the value of \u0012Jwherebyvhasits maximum or minimum; with Qe= 0,vexhibits maxima\nand minima at \u0012J=n\u0019\n2, but forQe6= 0, thesin (2\u0012J)term\nin Eq. (7) imposes a shift away from n\u0019\n2. We now return to\nour original easily deformable system and make predictions\nfor a case whereby for fixed values of Kx(withKy= 0), we\nsweep the injected current angle \u0012J. The results for the de-\npendence of both vand\u0002Skare shown in Fig. 6. Results are\nalso compared for the highest value of Kxto full dynamic mi-\ncromagnetic simulations as a verification-step of the effective\nskyrmion prediction, with a very good match. The amplitudes\nof oscillations in the dynamical behaviours are quite signifi-\ncant and should be easily detectable in an experiment. Apart\nfrom the additional degree of freedom in terms of modulating\nthe dynamics, this anisotropic behaviour could be used to de-\ntect the possible presence of strain induced anisotropy in the\nsystem and thus of skyrmion deformation.\nFIG. 6. Effective skyrmion prediction of (a): vvs.\u0012Jfor different\nKxand (b): \u0002Skvs.\u0012Jfor different Kx. Solid lines correspond to\nthe effective skyrmion approach and ���lled circles are results from\nfull dynamical micromagnetic simulations. The value of induced\nanisotropyKx(whose direction in the plane is also indicated by the\ndouble arrows) corresponding to each curve is shown by a number in\nunits of MJ/m3.\nIV . CONCLUSIONS\nIn conclusion, we have shown, by combined micromagnetic\nsimulations and an effective skyrmion analytical model, that\nwe can effectively modulate both speed and skyrmion Hall-\nangle of tightly antiferromagnetically bound skyrmions by in-\nduced in-plane anisotropies. The cause of the said modula-\ntions stem from a deformation of the skyrmion-texture (going\nfrom circular to elliptical shapes). As a further consequence,\nwe showed that this introduces dynamical anisotropy in the\nplane of skyrmion propagation with respect to driving-current\ninjection-angle. In addition, we have shown, given a devia-\ntion from circularly shaped skyrmions, that for driving current\nangles\u0012J6=n\u0019\n2, there is a contribution to the skyrmion de-\nflection away from the driving-current direction independent\nof the topological charge, i.e. even for a perfectly balanced8\nSAF. This may be of consequence for SAF devices whereby\nskyrmions operate on relatively large areas, causing a build-\nup of in time of deviation from the intended target position.Finally, if the uniaxial anisotropies can be induced by me-\nchanical stress, it can possibly lead to less complex device\nstructures as compared to other proposed schemes.\n\u0003per24.ac.uk\n1A, Fert, V . 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Flan-\nnery, Numerical Recipes: The Art of Scientific Computing,\nCambridge University Press, Cambridge, England 1988)." }, { "title": "1909.05809v2.Single_pulse_all_optical_toggle_switching_of_magnetization_without_Gd__The_example_of_Mn2RuxGa.pdf", "content": "1 Single pulse all -optical toggle switching of magnetization without Gd: The \nexample of Mn 2RuxGa \nC. Banerjee, N. Teichert, K. Siewierska, Z. Gercsi, G. Atcheson, P. Stamenov, \nK. Rode, J. M. D. Coey and J. Besbas* \n1 CRANN, AMBER and School of Physics, Trinity College, Dublin 2, Ireland \nEnergy -efficient control of magnetization without the help of a magnetic field is a key \ngoal of spintronics1,2. Purely heat -induced single -pulse all -optical toggle switching has \nbeen demonstrated, but so far only in Gd ba sed amorphous ferrimagnet films3–6. In this \nwork, we demonstrate toggle switching in the half-metallic compensated ferrimagnetic \nHeusler alloys Mn 2Ru xGa, which have two crystallographically -inequivalent Mn \nsublattices7. Moreover, we observe the switching at room temperature in samples that are \nimmune to external magnetic fields in excess of 1 T, provided they exhibit compensation \nabove room temperature. Observation s of the effect in compensated ferrimagnets without \nGd challenges our understanding of all -optical switching. The dynamic behavior \nindicates that Mn 2Ru xGa switches in 2 ps or less. Our findings widen the basis for fast \noptical switching of magnetization and break new ground for engineered materials that \ncan be u sed for nonvolatile ultrafast switches using ultrashort pulses of light. \nKeywords: Single -pulse toggle switching; Half -metallic ferrimag nets; compensation \ntemperature; Mn 2Ru xGa; picosecond spin dynamics; \n*besbasj@tcd.ie \n \n 2 Driven by the demands for high speed, low cost and high -density magnetic recording, \nresearch in spintronics has always sought insight into new classes of magnetic materials and \ndevices that show efficient and reproducible magnetization switching. In this respect, interest \nin the magneti c properties of antiferromagnetically coupled sub -lattice systems has gained \nmomentum in the last decade. The total or partial cancellation of the moments makes these \nsystems insensitive to stray magnetic fields, and the interaction between the sublattice moments \nintroduces phenomena that are absent in conventional ferromagnets, opening new opportunities \nfor magnetic recording and information processing2,8–10. \nAn efficient way of controlling magnetism is to use ultrashort laser pulses1,11. XMCD \ninvestigations in 2011 by Radu et al.3 of the dynamics of the Gd and Fe atomic moments in a \nthin layer of amorphous ferrimagnetic Gd 25Fe65.6Co9.4 after a 50 fs laser pulse , revealed a \ntransient parallel alignment of the moments that was the precursor of swi tching. They were \nfollowed by the discovery of single -pulse all -optical toggle switching of the magnetization in \nthe same material by Ostler et al.12. A general basis for fast all -optical switching in multi -\nsublattice magnets was then proposed by Mentink et al.13. Amorphous Gd x(Fe,Co) 100-x with x \n≈ 25 is a metallic ferrimagnet with localized 4 f-shell magnetic moments on the Gd sublattice \nand delocalized 3 d-band moments on the Fe -Co sublattice. Upon excitation by a femtosecond \nlaser pulse, the Fe -Co undergoes sub -picosecond demagnetization leading to practically \ncomplete loss of the ordered 3 d shell magnetization, an effect that had been originally observed \nin ferromagnetic nickel14. Concomitantly, the Gd atoms experience a slower loss of m agnetic \nalignment , with partial transfer of angular momentum from the Gd f shell to the FeCo d shell3, \nentail ing a transient parallel alignment of the moments of the demagnetizing Gd and the re-\nmagnetiz ing FeCo that ultimately leads to magnetization toggle switching on a picosecond \ntimescale3,12. As the suggested mechanism for single pulse all optical switching (SP-AOS ) \nrelies on an ultrafast interplay between two inequivalent spin sublattic es, one with a slower \nresponse to the laser (the Gd 4 f electrons ) and the other with a faster one (the Fe 3 d electrons ), \nsubsequent researches on SP -AOS concentrated on rare -earth based ferrimagnets15. In these \nswitching measurements, it is useful to distinguish the very short timescale on whi ch the future \ndirection of t he n et magnetization is decided, and the longer timescale necessary for the \nequilibrium magnetization to be established and respond to an external magnetic field. In \npractice helicity -independent SP -AOS has only been demonstrated in ferrimagnetic 3 Gdx(Fe,Co )100-x thin films3, Gdx(Fe,Co) 100-x spin valves5 and in synthetic Gd/Co ferrimagnets4; \nIt has not been seen in other rare -earth based ferrimagnet s such as amorphous Tb 27Co7316,17 \nwhere the 4 f electrons experience strong spin -orbit coupling. Its thermal ori gin is established \nby the independence of the effect on the polarization and helicity of the light3,12, and the \nequivalent effect produced by pulses of hot electrons18. A related phenomenon has been \nreported in ferrimagnetic TbFeCo, however, under specific structural condition s19 and in \nferromagnetic Pt/Co/Pt structures, when the laser spot size matches that of the ferromagnetic \ndomains20. A different type of single -pulse, non -thermal, non -toggle switching has been \nreported with linearly -polarized light in insulating Co -doped yttrium iron garnet21. \n In this work, we present a new, rare -earth -free, ferrimagnet that exhibits SP -AOS where \nthe two sublattices should not according to the prevalent thermodynamical models13 have \ndrastically different response time to laser excitation. We report all -optical toggle switching in \nthe ferrimagnetic Heusler alloys Mn 2RuxGa (MRG)7 where both magnetic sublattices are \ncomposed of manganese, and establish MRG as a versatile alternative to Gd x(FeCo) 1-x for SP -\nAOS applications. In MRG, the Mn atoms occupy two inequivalent sub -lattices at Wyckoff \npositions 4 a and 4 c in the cubic 𝐹4´3𝑚 structure (See supplementary Fig. S1 a), with \nantiferromagnetic intersublattice coupling7. At low temperature the magnetization of the \nMn(4 c) sublattice is dominant, but as temperature increases the magnetization of the Mn(4 c) \nfalls faster than that of the Mn(4 a) sublattice, leading to a compensation temperature Tcomp \nwhere the two are equal and opposite as the coercivity tends to diverge when t he net \nmagnetization crosses zero22. The value of Tcomp can be varied by changing the Ru \nconcentration x, so it is possible to make MRG peculiarly insensitive to external magnetic fields \nby decreasing its magnetisation23. The electronic structures of the t wo sublattices are different. \nMRG has a spin gap ~1 eV close to the Fermi energy, which led to its identification as the first \nexample of a half -metallic ferrimagnet7; the Mn(4 c) electrons have a high spin -polarized \ndensity of states whereas that of the Mn (4a) electrons is much lower. The unusual electronic \nstructure accounts for an anomalous Hall effect (AHE) that is greater than those seen in \ncommon ferromagnets23 and a strong magneto -optical Kerr effect (MOKE), even when the net \nmagnetization vanishes at Tcomp23,24, because both AHE and MOKE probe mainly the spin -4 polarized conduction band associated with Mn in the 4c position. Domains can be directly \nimaged in the Kerr microscope, regardless of the net magnetization 25. \nIn our experiments, we investigated SP-AOS in 19 MRG thin films having different Ru \ncontents with Tcomp above or below room temperature (RT). The films are deposited on MgO \n(100) substrates, which leads to a slight tetragonal distortion of the cubic XA -type structure \n(from space group 216, 𝐹4´3𝑚 to space group 119, I 4´m2), which is responsible for the \nperpendicular magnetic anisotropy of the MRG films. Optical pulses of 800 nm waveleng th \nand about 200 fs duration were generated by a mode -locked Ti -sapphire laser seeding a 1 kHz \namplifier. Fi gure 1 displays the result s of irradiating a Mn 2Ru1.0Ga film by a single 200 fs pulse \nwith a Gaussian intensity profile, as observed by ex situ Kerr microscopy. Here the light or \ndark contrast indicates an orientation of the Mn(4 c) sublattice into or out of the plane. For either \ninitial magnetization direction, a single laser pulse of sufficient intensity will switch the \nmagnetization d irection in the irradiated area (The elliptical shape of the switched domain is \ncaused by astigmatism of the focusing lens). Pulses where the average energy density is sub -\nthreshold leave the magnetization unchanged , except just at the centre, where the intensity \nexceeds threshold (Fig 1a) . The whole irradiated spot is switched at 1.5 µJ (Fig 1b) , but at 3 µJ \na multidomain pattern appears in the center of the irradiated zone (Fig 1c) , where the \ntemperature of the film has transiently exceeded the Curie temperature of the sample (~500 K)7 \nleading to re -magnetization in sub -micron domains close in size to the resolu tion of the Kerr \nmicroscope. They are much smaller than t he ~ 100 µm domains normally observed at room \ntemperature during the reversal process after saturating the magnetization25. It is established \nthat such temperatures can be reached in equilibrium betw een the lattice and spin system in the \nvery first picoseconds following optical excitation in transition metal compounds14, after which \nthe system re -magnetizes randomly in the stray field during the cool down. The multidomain \npattern is directly surrounde d by a ring -shaped switched domain, which shows that SP -AOS \ninvolves a significant transient demagnetization. The variation of the size of the switched \ndomain area with increasing pulse energy has been employed to calculate the threshold fluence \nfor switch ing (See Supplementary Information VI). Interestingly, we never observed SP -AOS \nin any MRG film having Tcomp below RT (see Supplementary Information V). We verified that \nthe observed sequence of switching originates solely from laser induced heating, by re peating 5 the experiment with circularly polarized laser pulses of opposite helicities as well as different \ndirections of linear polarization with respect to the MRG crystallographic directions (not \nshown) . The SP -AOS occurred identically in all cases, which eliminates the possibili ty of any \ncontribution from magnetic circular dichroism26 or from transient spin -orbit torques generated \nby the electric field. On further increasing the laser power to 5 μJ, the center of the irradiated \nspot on the film is ablated. \nFigure 2 depicts the results of the irradiation with 1 to 5 successive laser pulses on \nMn 2Ru1.0Ga. The panels show different regions that were subjected to the given numbers of \nshots. Consistently, the irradiation by a series of laser pulses leads to a toggling of the direction \nof the magnetization, which was investigated for up to 12 consecutive pulses. \nMRG possesses a low net magnetization and a high anisotropy field. Therefore, the \ncoercive field of the films usually exceeds 0.2 T and can reach values as high as 10 T23 if the \ntemperature is very close to Tcomp. It is interesting to see whether a highly -coercive sample can \nbe switched by light at RT. Figure 3 shows the toggling of magnetization following a sequence \nof pulses in a film of Mn 2Ru0.75Ga with coercivity exceeding 1 T. That sample could not be \nsaturated in our electromagnet and it was therefore measured in its virgin state, which is \ncharacterized by a distribution of magnetic domains with a predominance of magnetization \ndirected toward the substrate. Toggling of each of the individual domains by the light pulse is \nobserved even though the sample is insensitive to an external magnetic field of 1 T. The \nthreshold fluence for this sample was approximately one third of that of Mn 2Ru1.0Ga. Th e SP-\nAOS observed in samples with compensation temperature close to RT is particularly important \nfor three reasons: 1) the threshold fluence required for switching is small, potentially enabling \nenergy -efficient applications in the future27; 2) the coerciv ity of MRG diverges close to Tcomp, \nwhich makes the magnetic state impervious to external magnetic fields; 3) switching of micron -\nsized domains is possible, which are much smaller than the laser spot size. \nNext we turn to the dynamics of the excitation and reversal processes. The magnetism \nin MRG originates from the 3 d moments of Mn(4 a) and Mn(4 c) sublattices which are \nantiferromagnetically coupled. The femtosecond laser pulse is expected to disrupt the inter -site \nexchange (< 0.1 eV) and rapidly destroy th e magnetic order; while the intra -atomic, on -site \nexchange that depends on stronger Coulomb interactions (3 – 5 eV) should not be completely 6 destroyed. The aftermath of the pulse therefore involves re -establishment of magnetic order \nfrom the atomic moments , which in a ferrimagnet c ould include effects of angular momentum \ntransfer between sites. To investigate this possibility, we have studied the magnetization \ndynamics using time -resolved polar MOKE (TR -MOKE) in the two -color collinear pump \nprobe geometry. In this part of the study, we compare two samples, Mn 2Ru1.0Ga and \nMn 2Ru0.65Ga. They have Tcomp of 390 K and 165 K respectively and their coercive fields at \nroom temperature are similar (~ 460 mT), as shown in the optically measured hysteresis loops \nin Fig. 4a. The loops have opposite signs, as expected , because the Mn(4 c) sublattice, which \ngives the dominant contribution to the MOKE signal, aligns parallel to the applied field below \nTcomp and antiparallel above. Intense laser pulses of wavelength 800 nm wer e used as the pump \nbeam to excite the magnetization dynamics, and the Mn(4c) sublattice magnetization was \nsubsequently probed in a stroboscopic manner using weaker 400 nm pulses. A field of 500 mT \nwas applied perpendicular to the films to ensure an identic al initial state before each pump \npulse. Figure 4b shows the TR-MOKE signal for different pump fluences for Mn 2Ru0.65Ga, \nwhich does not switch because Tcomp is below RT. Following the laser excitation, the transient \nMOKE signal shows a step -like change, ca used by the ultrafast destruction of the magnetic \norder as the electron temperature shoots up. Subsequently, the magnetization regains its initia l \nstate in tens or hundreds of ps, depending on laser fluence. This behavior is typical of \nferromagnetic metals . It should be noted here that even though the MOKE response indicates \nfull demagneti zation of the Mn(4 c) electrons, this does not mean we have transiently exceeded \nthe Curie temperature in the lattice. At a fluence of 1 5.1 mJ cm-2, a trace of magnetic order \nmight still persist in the magneto -optically silent Mn (4a) sublattice, to ensure that when the \nsystem re -magneti zes, it does so uniformly. At greater fluence s we observe a multi -domain \nstate after laser irradiation, indicating t hat the system has been thermally demagnetized when \nthe electrons re -establish thermal equilibrium with a lattice that is now above Tc. \nOn changing to Mn 2Ru1.0Ga, we f ind a strong dependence of the TR -MOKE signal on \nlaser fluence, which is quite different below and above the threshold fluence Fth for SP -AOS \n(See Fig. 4c 1 and 4c 2 respectively). Below threshold , the behavior is like that of Mn 2Ru0.65Ga \nat similar fluence; the recovery takes about 10 ps, and an increase in the fast demagnetization \nsignal is observed with increasing fluence (Fig 4 d). Upon crossing the fluence threshold \nindicated by the yellow bar, the following new features appear in the signal: i) an increase in 7 the pump fluence now leads to a decrease in the demagne tization amplitude at zero delay (Fig. \n4c3), contrary to the previous case and to Gdx(FeCo )1-x28. The behavior of the demagnetization \namplitude with increas ing pump fluence is seen in Fig. 4d , where it fall s away from full \ndemagnetization above Fth; ii) As the system relaxes, the signal undergoes a rapid partial \nrecovery within 2 ps (See Fig. 4c 3), after which it reverses, at the point marked by the arrow. \nThis anomaly , which has not been reported in Gdx(FeCo )1-x, appears in the sample \ncompensating above room temperature and only at fluences where toggle switching is observed. \nThere, at 2 ps, the film must have switched, in the short -time sense mentioned in the \nintroduction , because at longer times the Mn(4 c) sublattice continues to re -magnetize in a \ndirection opposite to the one it had originally, before being turned back after 50 ps by the weak \napplied field and gradually recovering over the course of several hundred picoseconds. \nExtrapolating the initial slope to negative saturation gives a switchi ng tim e, in the longer sense, \nof about 200 ps. The slope from 2 – 50 ps is comparable , but opposite in sign to that of the first \nsample (Fig 2b) and the timescale for recovery at high fluence is similar. \nAnother sample with Tcomp = 250 K, excited at 210 K or 230 K was found to behave \nsimilarly29, and the time for thermalization of the electrons and the lattice is deduced there from \na 4-temperature model to be 2 ps. Furthermore, Bonfiglio et al.29 have shown that magnetic \norder is already beginning to be re-established in MRG within 1 ps, permitting efficient \nexchange scattering and transfer of angular momentum from one sublattice to the other, even \nat extremely short timescales. We speculate that for MRG most of the demagneti zation is due \nto ex change of a ngular momentum between the sublattices, but a quite different wavelength of \nthe probe pulse would be needed to reveal the behavior of the 4 a sublattice and see whether \nthe transient parallel alignment of the two subattice moments that is seen in XMCD in \nGdx(FeCo )1-x3 is also present in MRG. The question of where the local memory of the \nmagnetization is stored just after the electrons are demagnetized by the laser pulse is germane \nto the explanation of the behavior shown in Fig 1 both below and above the sw itching threshold \nFth. If not in a substantially slower demagnetization rate for the silent 4 a sublattice, then the \nlocal angular momentum might be transient ly parked in optical phonon modes, or the nuclear \nspins. 8 In summary, we have demonstrated single -pulse all -optical thermal switching in less \nthan 2 ps in films of the half -metallic compensated Heusler ferrimagn et Mn 2RuxGa, where both \nmagnetic sublattices are composed of manganese atoms, occupying different crystallographic \nsites. These results extend the scope of the phenomenon beyond a limited range of amorphous \nGdx(Fe,Co) 100-x alloys with x ≈ 25, where the mag netic sublattices are defined chemically . A \ncomparison of the two systems is provided in Supplemental Information VIII. The Heusler \nalloys are a huge family, with an established body of knowledge about their magnetic and \nelectronic properties that will all ow us to advance our understanding of the SP -AOS \nphenomenon and design materials that can be the basis of future nonvolatile opto -magnetic \nswitches. Beyond the newly -demonstrated quality of MRG as an opto -magnetic material, its \nlarge intrinsic spin -orbit t orque, which relies on the absence of inversion symmetry of the \nMn(4 c) sublattice opens prospects for new multifunctionality30,31. Therefore, MRG and its \nchemically tailored successors offer the prospects of both new insights into condensed matter \non a fem tosecond timescale and new technological prospects taking advantage of ultra -fast \ncontrol of the magnetic state without any reliance on a magnetic field. \n9 Methods: MRG films with different Ru content were grown on MgO(001) substrates at 350◦C \nby DC magnetron sputtering in a Shamrock system with a base pressure of 2×10−8 Torr. They \nwere co -sputtered from Ru and stoichiometric Mn 2Ga targets. The Ru concentration was \ncontrolled by varying the Mn 2Ga target plasma power while fixing the Ru power. The samp les \nwere then capped with a protective layer of 2 nm of Al 2O3. \nFemtosecond laser pulses were generated by Ti -sapphire laser seeding a 1 kHz \namplifier with a Q -switched cavity. Their central wavelength was 800 nm and the pulse \nduration was about 200 fs. The amplifier can be operated in continuous mode where a train of \npulses is generated at a repetition rate of 1 kHz or in single pulse mode where the emission of \none single pulse can be externally triggered. In some cases, 400 nm laser pulses were obtained \nby second harmonic generation in a β-BaB 2O4 crystal. \n Prior to laser irradiation, the films were saturated at room temperature in the 1 T \nperpendicular magnetic field of our Evico Kerr microscope. Different areas of the films were \nthen irradiated with severa l linearly polarized laser pulses of different powers, followed by ex -\nsitu imaging of the final results in the Kerr microscope. For the imaging, a polarized beam is \nfocused onto the sample using a microscope objective. A rotation in the polarization due to \nKerr effect occurs in the reflected beam, which then passes through an analyzer, before \nreaching the camera. In order to increase the contrast, we keep the axis of the analyzer few \ndegrees away from cross position in both directions and acquire two images . Subsequently, the \nimages are subtracted to extract the final image. \n For the dynamic measurements, the laser beam with wavelength 800 nm was split into \na pump beam and a frequency -doubled probe beam at 400 nm. The intensity of the probe was \n10 kept low. Bot h pump and probe were linearly polarized and collinear. The spot sizes were \nmeasured to be about 150 and 70 µm respectively. The dynamical magneto -optic Kerr rotation \nwas measured using a balanced detection scheme and acquired using a lock -in amplifier and a \nmechanical chopper at 500 Hz in the pump beam. The pump/probe delay was varied using a \nmechanical delay line. \n \n11 Figure 1| Single -pulse all -optical switching (SP -AOS) in Mn 2Ru 1.0Ga. A film uniformly -\nmagnetized out of the plane (top) or into the plane (bottom) is irradiated by a single 800 nm \npulse focused onto a spot 150µm x 100µm. The pulse energy in a is 0.9 µJ, which only exceeds \nthe switching threshold in a small region at the centre where the fluence is hi ghest . The 1.5µJ \npulse in b switches the whole irradiated area, whereas the 3µJ pulse in c heats the film at t he \ncentre of the spot above the Curie temperature, produci ng a fine multidomain pattern. The most \nintense pulses (5 µJ) lead to ablation of the film. \nFigure 2| Toggling of the magnetization in Mn 2Ru 1.0Ga. Magnetization pattern as a \nfunction of the number of applied pulses . Pulse energy was 1.2 µJ. \nFigure 3| Toggling of magnetization in a high -coercivity Mn 2Ru 0.75Ga film. Repeated \ntoggling of the m icron -scale domain pattern of a virgin -state sample is observed with \nrepeated pulses. There was a net imbalance of domains pointing in and out of the plane. \nFigure 4| Time Resolved Magnetization Dynamics in Mn 2Ru xGa. a, Hysteresis loops \nmeasured by MOKE in Mn 2Ru0.65Ga and Mn 2Ru1.0Ga, which have compensation temperature \nbelow and above RT respectively. b, Transient Kerr signals of Mn 2Ru0.65Ga for different pump \nfluences. The variation of the Kerr signal is normalized to the total Kerr rotation at RT. c1, \nTransient Kerr signal of Mn 2Ru1.0Ga for fluences below the switching threshold ; c2, similar \ndata including fluences above threshold ; c3, A zoom of c2 in a shorter time window; the anomaly \nmarked by the arrow is discussed in the text. d, Variation of the demagnetization amplitude at \nzero delay for different pump fluences. The yellow shaded region indicates the threshold \nfluence for switching. The solid line is a guide to eye. \n \n12 References \n1. Kimel, A. V. & Li, M. Writing magnetic memory wit h ultrashort light pulses. Nat. \nRev. 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Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics. \nNat. Nanotechnol. 11, 231 –241 (2016). \n9. Baltz, V. et al. Antiferromagnetic spintronics. Rev. Mod. Phys. 90, 015005 (2018). \n10. Němec, P., Fiebig, M., Kampfrath, T. & Kimel, A. V. Antiferromagnetic opto -\nspintronics. Nat. Phys. 14, 229 –241 (2018). \n11. Kirilyuk, A., Kimel, A. V. & Rasing, Th. Ultrafast optical manipulation of magnetic \norder. Rev. Mod. Phys. 82, 2731 –2784 (2010). \n12. Ostler, T. A. et al. Ultrafast heating as a s ufficient stimulus for magnetization reversal \nin a ferrimagnet. Nat. Commun. 3, 666 (2012). \n13. Mentink, J. H. et al. Ultrafast Spin Dynamics in Multisublattice Magnets. Phys. Rev. \nLett. 108, 057202 (2012). \n14. Beaurepaire, E., Merle, J. -C., Daunois, A. & Bigot, J. -Y. Ultrafast Spin Dynamics in \nFerromagnetic Nickel. Phys. Rev. Lett. 76, 4250 –4253 (1996). \n15. Wienholdt, S., Hinzke, D., Carva, K., Oppeneer, P. M. & Nowak, U. Orbital -resolved \nspin model for thermal magnetization switching in rare -earth -based f errimagnets. Phys. Rev. \nB 88, 020406(R) (2013). \n16. El Hadri, M. S. et al. Two types of all -optical magnetization switching mechanisms \nusing femtosecond laser pulses. Phys. Rev. B 94, 064412 (2016). \n13 17. A. R. Khorsand et al. Element -Specific Probing of Ult rafast Spin Dynamics in \nMultisublattice Magnets with Visible Light. Phys. Rev. Lett. 110, 107205 (2013). \n18. Xu, Y. et al. Ultrafast Magnetization Manipulation Using Single Femtosecond Light \nand Hot -Electron Pulses. Adv. Mater. 29, 1703474 (2017). \n19. Liu, T.-M. et al. Nanoscale Confinement of All -Optical Magnetic Switching in \nTbFeCo - Competition with Nanoscale Heterogeneity. Nano Lett. 15, 6862 –6868. \n20. Vomir , M., Albrecht, M. & Bigot, J. -Y. Single shot all optical switching of intrinsic \nmicron size magnetic domains of Pt/Co/Pt ferromagnetic stack. Appl. Phys. Lett. 111, 242404 \n(2017). \n21. Stupakiewicz, A., Szerenos, K., Afanasiev, D., Kirilyuk, A. & Kimel, A. V. Ultrafast \nnonthermal photo -magnetic recording in a transparent medium. Nature 542, 71 (2017). \n22. Betto, D. et al. Site-specific magnetism of half -metallic Mn 2Rux thin films determined \nby x-ray absorption spectroscopy. Phys. Rev. B 91, 094410 (2015). \n23. Thiyagarajah, N. et al. Giant spontaneous Hall effect in zero -moment Mn 2RuxGa. \nAppl. Phys. Lett. 106, 122402 (2015). \n24. Fleisher, K. et al. Magneto -optic Kerr effect in a spin -polarized zero moment \nferrimagnet. Phys. Rev. B 98, 134445 (2018). \n25. Siew ierska, K. E., Teichert, N., Schäffer, R. & Coey, J. M. D. Imaging Domains in a \nZero -Moment Half Metal. IEEE Trans. Magn. 55, 2600104 (2019). \n26. Khorsand, A. R. et al. Role of Magnetic Circular Dichroism in All -Optical Magnetic \nRecording. Phys. Rev. Lett. 108, 127205 (2012). \n27. Barker, J. et al. Two-magnon bound state causes ultrafast thermally induced \nmagnetisation switching. Sci. Rep. 3, (2013). \n28. Stanciu, C. D. et al. Subpicosecond Magnetization Reversal across Ferrimagnetic \nCompensation Points. Phys . Rev. Lett. 99, 217204 (2007). \n29. Bonfiglio, G. et al. Sub-picosecond exchange -relaxation in the compensated \nferrimagnet Mn 2RuxGa. arXiv 2003.01420 , (2020). \n30. Lenne, S. et al. Giant spin -orbit torque in a single ferrimagnetic metal layer. arXiv \n1903.0 4432 (2019). \n31. Finley, J., Lee, C. -H., Huang, P. Y. & Liu, L. Spin -Orbit Torque Switching in a \nNearly Compensated Heusler Ferrimagnet. Adv. Mater. 31, 1805361 (2019). \n \n \n14 \n \nAuthor contributions: \nC. B., J. B . and J. M. D. C. designed the project. Experimental work was done by C. B., N. T. \nand J. B. Growth and characterization of the samples were carried out by G. A. and K. S. \nNumerical simulations were performed by Z. G. and J. B. ; C. B., J. B., P. S., J. M. D. C and \nK. R. interpreted the data. Al l authors discussed the results. C. B., J. B. , K. R. and J. M. D. C. \nwrote the paper. \n \nAcknowledgements: \nThis project has received funding from Science Foundation Ireland through contracts \n16/IA/4534 ZEMS and 12/RC/2278 AMBER and from the European Union’s FET-Open \nresearch programme under grant agreement No 737038. C.B. is grateful to Irish Research \nCouncil for her post -doctoral fellowship. N. T. would like to acknowledge funding from the \nEuropean Union’s Horizon 2020 research and innovation programme unde r the Marie \nSkłodowska -Curie EDGE grant agreement No 713567. \n \nThe authors declare no competing financial interest. \n \n \n \n \n15 \n \n \nFigure 1| \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n16 \n \nFigure 2| \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n17 \n \n \nFigure 3| \n \n \n \n \n \n \n \n \n \n \n \n \n50 μmMnet \n18 \n \nFigure 4| \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n4 8 12 160.250.500.751.00\nFluence (mJ.cm-2)Demag Ampl. \n0 10 20 30-0.8-0.40.0DMz/ Mz\nDelay (ps)\n0102030405060-0.8-0.40.0DMz/ Mz\n 4.3 mJ.cm -2\n 6.4 mJ.cm -2\n 7.5 mJ.cm -2\nDelay (ps)\n050200 400 600-0.9-0.6-0.30.0\n 7.5 mJ.cm -2\n 15.1 mJ.cm -2\n 17.2 mJ.cm- 2DMz/ Mz\nDelay (ps)b\n-1.2 -0.6 0.0 0.6 1.2-0.80.00.81.6 Mn2Ru0.65Ga\n Mn2Ru1.0GaKerr Signal (Norm.)\nm0 H (T)a\nd c3c1c2 a\n050200 400 6000.00.40.8DMz/ Mz\nDelay (ps) 7.5 mJ.cm-2\n 15.1 mJ.cm-2\n 17.2 mJ.cm-2 \n19 Supplementary Information \nSingle pulse all -optical toggle switching of magnetization without Gd: The example of \nMn 2Ru xGa \nC. Banerjee, N. Teichert, K. Siewierska, Z. Gercsi, G. Atcheson, P. Stamenov, \nK. Rode, J. M. D. Coey and J. Besbas* \n1 CRANN, AMBER and School of Physics, Trinity College, Dublin 2, Ireland \n*besbasj@tcd.ie \n \nI. STRUCTURAL AND MAGNETIC CHARACTERIZATION OF Mn 2Ru1.0Ga \n \n \n \n \n \n \n \n \n \nFigure S1: (a) Diagram of the inverted Heusler (XA) crystal unit cell of typical MRG film. \nCharacterisation of Mn 2Ru1.0Ga (MRG): ( b) X ray reflectivity pattern of the MRG thin film. \nFitting gives a thickness of 42. 8 nm and a density of 8.2 g . cm-3, (c) X ray diffraction pattern \nof MRG thin film on MgO (001) substrate. ( d) Reciprocal space map of MgO (113) peak and \nMRG (204) peak with lattice parameters calculated with respect to th e MRG unit cell . \n \n1 2 3 4100102104106Intensity\n2(o)\n30 40 50 60 70100102104106\nMRG (004)MgO (002)Intensity\n2 (o)MRG (002)a b\nc\n d \n20 MRG crystallises in an inverted Heusler (XA) structure of space group F -43m with two \ncrystallographically inequivalent magnetic Mn atoms at Wyckoff positions 4 a and 4 c and Ga \nand Ru atoms occupy the 4 b and 4 d positions, respectively, as shown in Fig. S1 (a ). Note the \nMn(4c) sublattice is non -centrosymmetric. Due to the biaxial non -volume c onserving strain of \nthe MgO substrate the unit cell of MRG is tetragonally distorted and hence the space group is \nreduced to I -42m. X-ray data on the Mn 2Ru1.0Ga film are shown in Fig. S1 (b), (c) and (d ). The \nX-ray reflectivity ( XRR) pattern shown in Fig. S1 (b ) has been fitted using X'Pert Reflectivity \nsoftware and the film thickness was calculated to be 42. 8 nm. The X -ray diffraction (XRD) \npattern in Fig. S1 (c) exhibits (002) and (004) reflections from the MRG, together with peaks \nfrom the MgO substrate. The c -parameter calculated from the (004) reflection is 604.7 pm. A \nreciprocal space map (RSM) of the MRG film in Fig. S1 (d ) confirms the c -parameter obtained \nfrom XRD, and shows a distribution of a -parameters around the central value of 595.8 pm, \nwhich corresponds to that of MgO. This demonstrates how substrate strain induces a ~ 1% \ntetragonal elongation of the MRG unit cell since the c/a ratio is 1.01, giving rise to the \nperpendicular magnetic anisotropy found in all MRG films . \n \n \n \n \n \n \n \n \n \n \n \n21 II. DEPENDENCE OF MAGNETIZATION ON APPLIED MAGNETIC FIELD FOR \nMn 2Ru0.65Ga AND Mn 2Ru1.0Ga \n \n \n \n \n \n \nFigure S2 : Magnetization versus field applied in plane and out of plane at 300 K (a) for the \nMn 2Ru0.65Ga film and (b) for the Mn 2Ru1.0Ga film. \nThe dependence of magnetization on the field applied out of plane and in plane at 300 K relative \nto the surface of the thin film was measured for Mn 2Ru0.65Ga and Mn 2Ru1.0Ga films (See Fig. \nS2), which have Tcomp below and above RT respectively. From the curves, t he saturation \nmagnetization s are 40 and 65 kA . m-1, respectively. Interestingly, we observe a soft component \nin the out-of-plane, as well as the in-plane magnetization , which originates partly from the non -\ncollinearity of the two exchange -coupled antiferromagnetically aligned Mn magnetic \nsublattices. \n \n \n \n \n \n \n \n \n-4 -2 0 2 4-40040Magnetisation (kA m-1)\nm0H (T) Out of Plane\n In Plane\n-4 -2 0 2 4-80-4004080Magnetisation (kA m-1)\nm0H (T) Out of Plane\n In Planea b \n22 III. MEASUREMENT OF OPTICAL AND TRANSPORT MAGNETOMETRY OF \nMn 2Ru1.0Ga \n \n \n \n \n \n \nFigure S3 : Measured hysteresis loops of Mn 2Ru1.0Ga via (a) Kerr microscopy and (b) \nanomalous Hall effect. \nWe have measured the magnetic hysteresis loop s using polar Kerr effect ( = 630 nm) and \nanomalous Hall effect in the perpendicular Mn 2Ru1.0Ga fi lm, which are compared in Fig. S3 . \nIn contrast to the two-step switching observed in the S QUID loop (See Fig. S 2 (b)), here the \nsquare hysteresis loops exhibit straightforward single -step switching , with an average \nswitching field of 460 mT in both cases. Whereas the SQUID probes the net magnetic moment, \nthe difference of the nearly -equal sublattice contributions, the optical and electrical transport, \non the other hand, relies on the distribution of spin-polarized electrons at or near the Fermi \nenergy, which in MRG reflects the 4c sublattice magnetization. Consequently, the hysteresis \nshown in Fig. S3 as well as the domain images presented here reflects the local magnetization \nstate of the 4c sublattice. It is therefore possible to explore the local magnetization state even \nat compensation. \n \n \n \n \n \n-0.8 -0.4 0.0 0.4 0.8-0.8-0.40.00.40.8Rxy ()\nm0 H (T)\n-0.8 -0.4 0.0 0.4 0.80255075\nm0 H (T)Kerr signal (a.u.)a b \n23 IV. DETERMINATION OF Tcomp FOR Mn 2Ru1.0Ga \n \n \n \n \n \n \n \nFigure S4: Optically measured hysteresis loops for Mn 2Ru1.0Ga at different temperatures (a) \nand the corresponding variation of coercive field (b). \n \nFigure S4 (a) presents the hysteresis loops measured by polar Kerr effect in Mn 2Ru1.0Ga at \ndifferent temperatures. On approaching Tcomp, the net moment falls, which increases the \nanisotropy field and coercivity , until they diverge at Tcomp. In addition, as the optical \nmeasurement s probes the Mn (4c) sublattice only, a change in the sign of the hysteresis loop is \nseen upon crossing Tcomp. The variation of the coercive field as a function of temperature is \nshown in Fig. S4b , from which Tcomp for this sample is estimated to be ~ 390 K. \n \n \n \n \n \n \n \n300 350 400 4500.00.20.40.6Coercivity (T)\nTemperature (K)\n-0.8 -0.4 0.0 0.4 0.8Kerr Signal (a.u.)\nField (T) 292 K\n 303 K\n 313 K\n 424 K\n 431 K\n 454 K\n 523 Ka b \n24 V. RESULT OF PULSE ENERGY DEPENDENT MEASUREMENT IN MRG \nSAMPLE WHERE AOS WAS NOT OBSERVED \n \n \n \n \n \nFigure S5 : Typical Kerr microscope images of MRG after single laser pulse of various pulse \nenergies were irradiated on different positions of the surface. The results show a multidomain \npattern has formed. \n \nIn the main text, single -pulse all-optical switching ( SP-AOS) is presented for MRG samples \nhaving Tcomp above room temperature. No toggling was not observed for the MRG samples \nhaving Tcomp below room temperature. Figure S 5 shows typical Kerr micrographs after the \npulse , for different pulse energies. The images reveal the presence of multidomain state with \nan onset at ~ 1 µJ. This is different to the Mn 2Ru1.0Ga results, where a ring of switched domain \nwas observed around t he thermally demagnetized region. \n \n \n \n \n \n \n \n \n1.5 µJ 2.0 µJ 2.5 µJ\n3.0 µJ 3.5 µJ\n 4.0 µJ\n1.0 µJ\n50 μm \n25 VI. DETERMINATION OF THE SPOT SIZE AND THRSHOLD FLUENCE FOR \nSWITCHING \n \n \n \n \n \n \n \nFigure S 6: Switched domain size as a function of pulse energy for Mn 2Ru1.0Ga. The red solid \nline is a fit to Eqn. 1. \nAs mentioned in the main text, we have employ ed the growth of the switched domain size with \nincreasing pulse energy for Mn 2Ru1.0Ga to calculate the laser spot size as well as the threshold \nfluence for switching using the Liu methodS1. This method exploits the fact that at the edge of \nthe magnetic contrast, the excitation fluence is equal to the threshold fluence. By assuming a \nGaussian pulse -shape , the switched area can be determined as, \n 𝐴𝑆=𝐴0𝑙𝑛(𝐸\n𝐸𝑡ℎ) ………(1) \n where AS is the switched area corresponding to pulse energy E, A0 is the laser spot area and Eth \nis the threshold pulse energy. Figure S 6 presents the variation of the switched area as a function \nof pulse energy, from which the spot area and the threshold pulse energy are extracted by fitting \nwith Eqn. 1 to be 18600 µm2 and 1.4 µJ respectively. The threshold fluence is then calculated \nby di viding the threshold pulse energy with the laser -spot area ,which yields ~7.5 mJ/cm2. \n \n[S1] Liu , J. M. Simple technique for measurements of pulsed Gaussian -beam spot sizes. Opt. \nLett. 7, 196 (1982 ). \n2.0 2.4 2.8 3.2 3.6 4.08121620Spot Size(103 mm2)\nPulse Energy (10-6 J) \n26 VII. SP-AOS MEASUREMENT ON Mn 2Ru0.92Ga USING LASER PULSES OF \nWAVELENGTH 400 NM \n \n \n \n \nFigure S7 : Single pulse all optical switching measurement performed on Mn 2Ru0.92Ga using \nlaser pulses of wavelength 400 nm and pulse energy 0.9 µJ. The number of laser pulses the \nregion is exposed to is labelled in each imag e. \n \nIn order to substantiate the thermal origin of SP -AOS in MRG, we examined the response of \nits magnetization to the laser pulses of different wavelength and polarization. In Fig. S 7 the \nresponse of Mn 2Ru0.92Ga is shown for laser light of wavelength 400 nm and up to five laser \npulses. The results are in line with the observation with 800 nm (See Fig. 2 in main text ). \nEssentially, a decrease in the excitation wavelength decreases the laser spot size, thereby \nincreasing th e thermal gradient across it. This directly affects the magnetization profile of the \nirradiated region and the switching is observed in a relatively narrow ring around the thermally \ndemagnetized area, as compared to the 800 nm case. \n \n \n \n \n \n \n \n1 shot 2 shots 3 shots 4 shots 5 shots\n50 μm\n \n27 VIII. COMPARISON OF TEMPERATURE DEPENDENCE OF MAGNETIZATION \nIN MRG AND Gd FeCo \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S8: Representation of the sublatttice magnetizations with respect to temperature \ncalculated in the mean field approach for Mn 2Ru1.0Ga and Gd(FeCo)3. \n \n \n \n \n \n \n \n \n \n \n \nMn 2Ru 1.0Ga \n Gd(FeCo) 3 \n \n28 TABLE S1: Outcome of single -pulse excitation in various MRG samples. In some cases, where \nsimilar behaviour was found for films with similar composition, only one entry is shown. \nValues of Tcomp shown in bl ack are experimentally measured , while the others (shown in blue) \nare interpolated. \nSample Tcomp (K) Coercive Field (mT) Switching Observed? \nMn 2Ru0.5Ga 75 150 No \nMn 2Ru0.55Ga 80 170 No \nMn 2Ru0.60Ga 130 260 No \nMn 2Ru0.62Ga 145 350 No \nMn 2Ru0.63Ga 160 370 No \nMn 2Ru0.65Ga 165 440 No \nMn 2Ru0.7Ga 245 740 No \nMn 2Ru0.9Ga 310 > 1000 Yes \nMn 2Ru0.92Ga 315 > 1000 Yes \nMn 2Ru0.93Ga 320 > 1000 Yes \nMn 2Ru0.94Ga 325 > 1000 Yes \nMn 2Ru0.95Ga 375 600 Yes \nMn 2Ru1.0Ga 390 480 Yes \n \n29 TABLE S2: MAGNETIC PROPERTIES OF Mn 2RuxGa AND Gd(FeCo )3 \nMagnetic properties of Mn 2RuxGa and Gd(FeCo )3 (i.e. GdFeCo) . Tc and Tcomp are respectiv ely \nthe Curie and compensation temperatures. Mi is the magnetization of the sublattice i and Mnet \nis the net magnetization of the system. τi are the characteristi c demagnetization times for \nsublattices i = 4a, 4c in MRG and i = Gd, FeCo in Gd(FeCo) 3. τe-l is the characteristic time \nassociated to the energy transfer between the hot electronic system and the lattice. \n Mn 2Ru xGa Gd(FeCo) 3 \nStructure Cubic Heusler XA Amorphous \nTc (K) 5001 (Mn 2RuGa) \n5006 (Gd 22Fe9.8Co68.2) \n \nTcomp (K) 3901,2 (Mn 2Ru1.0Ga) \n2505 (Gd 25Fe65.6Co9.4) \n \nM4a/FeCo (kA . m-1) 7901 (Mn 2Ru1.0Ga) \n5503 (Mn 2Ru0.61Ga) 6407 (Gd 25Fe65Co10) \nM4c/Gd (kA . m-1) 8601 (Mn 2Ru1.0Ga) \n5903 (Mn 2Ru0.61Ga) 10007 (Gd 25Fe65Co10) \nMnet (kA . m-1) 702 (0 K) (Mn 2Ru1.0Ga) \n403 (0 K)(Mn 2Ru0.61Ga) 3607 (0 K) (Gd 25Fe65Co10) \nτ4a/Gd (ps) 8.04 (Mn 2Ru0.7Ga) \n0.435 (Gd 25Fe65.6Co9.4) \n \nτ4c/ FeCo (ps) 0.5 (Mn 2Ru1.0Ga), \n0.5/8.04 (Mn 2Ru0.7Ga) 0.15 (Gd 25Fe65.6Co9.4) \n4.0/1506 (Gd 22Fe9.8Co68.2) \n \n τe-l (ps) 2.04 (Mn 2Ru0.7Ga) 1.55 (Gd 25Fe65.6Co9.4) \n5.06 (Gd 22Fe9.8Co68.2) \n1 From static MOKE vs Temperature measurements. \n2 From SQUID/XRD measurements. \n3 Fowley et al. Phys. Rev. B 98, 220406(R) (2018). \n4 Bonfiglio et al. ArXiv, 2003.01420 (2020). \n5 Radu et al. Nature 472, 205 -209 (2011). \n6 Mekonnen et al. Phys. Rev. B 87, 180406(R) (2013). \n7 Estimated for atomic spins µGd =7.6 µB and µFe/Co =1.6 µB. \n " }, { "title": "1209.3965v1.Nontrivial_ferrimagnetism_of_the_Heisenberg_model_on_the_Union_Jack_strip_lattice.pdf", "content": "arXiv:1209.3965v1 [cond-mat.str-el] 18 Sep 2012Nontrivial ferrimagnetism of the Heisenberg model on the Un ion\nJack strip lattice\nTokuro Shimokawa∗and Hiroki Nakano\nGraduate School of Material Science,\nUniversity of Hyogo, Kamigori, Hyogo 678-1297, Japan\n(Received 31 May 2012)\nAbstract\nWe study the ground-state properties of the S= 1/2 antiferromagnetic Heisenberg model on\nthe Union Jack strip lattice by using the exact-diagonaliza tion and density matrix renormaliza-\ntion group methods. We confirm a region of the intermediate-m agnetization state between the\nN´ eel-like spin liquid state and the conventional ferrimag netic state of Lieb-Mattis type. In the\nintermediate-state, we find that the spontaneous magnetiza tion changes gradually with respect to\nthe strength of the inner interaction. In addition, the loca l magnetization clearly shows an in-\ncommensurate modulation with long-distance periodicity i n the intermediate-magnetization state.\nThese characteristic behaviors lead to the conclusion that the intermediate-magnetization state is\nthe non-Lieb-Mattis ferrimagnetic one. We also discuss the relationship between the ground-state\nproperties of the S= 1/2 antiferromagnetic Heisenberg model on the original Union Jack lattice\nand those on our strip lattice.\nPACS numbers: 75.10.Jm, 75.30.Kz\nKeywords: quantum spin system, frustration, ferrimagnetism, D MRG, exact diagonalization\n∗Electronic address: t.shimokaw@gmail.com\n0I. INTRODUCTION\nFerrimagnetism is a fundamental phenomenon in the field of magnetis m. The most\nfamous type of ferrimagnetism is called Lieb-Mattis (LM) one[1–6]. Fo r example, this fer-\nrimagnetism appears in the ground state of the ( s,S)=(1/2, 1) mixed spin chain with\nnearest-neighbor antiferromagnetic interaction. In this system , the occurrence of the LM\nferrimagnetism originates from the situation that two different spin s are arranged alter-\nnately in a line owing to the AF interaction. In the LM ferrimagnetic sta te, the spontaneous\nmagnetization occurs and the magnitude is fixed to a simple fraction o f the saturated mag-\nnetization. As in the case of this mixed spin chain, not only the magnet ic properties but\nalso the occurrence mechanism of the LM ferrimagnetism are well kn own since this type of\nferrimagnetism has been studied extensively. Especially, the ferrim agnetism in the quantum\nHeisenberg spin model on the bipartite lattice without frustration is well understood within\nthe Marshall-Lieb-Mattis (MLM) theorem[1, 2].\nOn the other hand, a new type of ferrimagnetism that is clearly differ ent from the LM\nferrimagnetism has been found in the ground state of several one -dimensional frustrated\nHeisenberg spin systems[7–13]. The spontaneous magnetization in t his new type of ferri-\nmagnetism changes gradually with respect to the strength of frus tration. In addition, the\nincommensurate modulation with long-distance periodicity in local mag netizations is ob-\nserved as a characteristic quantum behavior of the new type of fe rrimagnetism. Hereafter,\nwe call the new type of ferrimagnetism non-Lieb-Mattis (NLM) type . The mechanism of the\noccurrence of the NLM ferrimagnetism have not yet been clarified in contrast to the case of\nthe LM ferrimagnetism.\nHistorically, some candidates of the NLM ferrimagnetism among the 2 D systems were\nalready reported. For examples, there are the mixed-spin J1-J2Heisenberg model on the\nsquare lattice[14] and the S= 1/2 Heisenberg model on the Union Jack lattice of Fig. 1(a)\n[15–18]. These 2D frustrated systems have the intermediate grou nd-state, namely “canted-\nferrimagnetic state” as described in Fig. 2, in which the spontaneou s magnetization is\nchanged when the inner interaction of the system is varied. It has n ot been, however, in-\nvestigated whether the incommensurate modulation with long-dista nce periodicity exists or\nnot in the local magnetization of the intermediate-magnetization st ate owing to the diffi-\nculty of treating these 2D frustrated systems numerically and the oretically. Therefore, the\n1'JH\u000f\u0001\u0012\tB\n 'JH\u000f\u0001\u0012\tC\n \n\"\n\"` $\n$` # %+\u0012\n+\u0013 +\u0012\n+\u0013\nFIG. 1: (Color online) Structures of the lattices: the Union Jack lattice (a), the Union Jack strip\nlattice (b). An S= 1/2 spin is located at each site denoted by a black circle. Antif erromagnetic\nbondsJ1(bold straight line) and J2(dashed line) are represented. Sublattices in a unit cell of\nlattice (b) are represented by A, A′, B, C, C′, and D.\nrelationships between the intermediate-magnetization states of t hese 2D frustrated systems\nand the NLM ferrimagnetic state are still unclear.\nUnder such circumstances, quite recently, the S= 1/2 antiferromagnetic Heisenberg\nmodel on the spatially anisotropic kagome lattice was studied[19]. In t his model, the\nintermediate-magnetization states exist between the LM ferrimag netic state and the non-\nmagnetic one[20–24]. It was reported that the local magnetization in these intermediate-\nstate shows large dependence on the position of the sites although it is difficult to judge\nclearly whether the incommensurate modulation with long-distance p eriodicity is present or\nabsent. In addition, the S= 1/2 Heisenberg models on the quasi-one-dimensional kagome\nstrip lattices were studied[25, 26]. These strip lattices share the sa me lattice structure in\ntheir inner part with the spatially anisotropic kagome lattice. The loca l magnetizations in\nthe intermediate-state clearly show incommensurate modulations w ith long-distance peri-\nodicity irrespective of the strip width. Therefore, these results s trongly suggest that the\nintermediate-magnetization states not only of the kagome strip lat tices but also of the orig-\ninal kagome lattice are the NLM ferrimagnetism.\nThese kagome results motivate us to investigate the ground-stat e properties of the quasi-\none-dimensional strip model whose lattice structure is common to t he part of the 2D lattice\n2/FFMʅ $BOUFE\u000eGFSSJ 4FNJTUSJQFE\u000eGFSSJ\n+\u0013\u0010+ \u0012П\nП\nЋ\u0012Ћ\u0013\nFIG. 2: (Color online) Ground-state phase diagram of the S= 1/2 Heisenberg model on the\nUnion Jack lattice depicted Fig. 1(a). Here, black and white circles represent up-spin and down-\nspin respectively.\nknown as the other candidates of the NLM ferrimagnetism. In this s tudy, we treat the\nS= 1/2 Heisenberg model on the Union Jack strip lattice depicted in Fig. 1(b ). This strip\nlattice share the same lattice structure in the inner part with the or iginal Union Jack lattice\ndepicted in Fig. 1(a). Our numerical calculations lead to the conclusio n that the NLM\nferrimagnetic phase appears in the ground-state of the Union Jac k strip model of Fig. 1(b).\nWe also discuss the relationship between the ground-state proper ties of the present strip\nmodel and those of the original 2D model.\nII. MODEL\nThe Hamiltonian of the S= 1/2 antiferromagnetic Heisenberg model on the Union Jack\nstrip lattice depicted in Fig. 1(b) is given by\nH=J1/summationdisplay\ni[Si,A·Si,C+Si,B·Si,D+Si,A′·Si,C′+Si,A·Si,B+Si,B·Si,A′\n+Si,C·Si,D+Si,D·Si,C′+Si,C·Si+1,A+Si,D·Si+1,B+Si,C′·Si+1,A′]\n+J2/summationdisplay\ni[Si,A·Si,D+Si,A′·Si,D+Si,D·Si+1,A+Si,D·Si+1,A′], (1)\nwhereSi,ξis anS= 1/2 spin operator at ξ-sublattice site in i-th unit cell. Positions\nof the six sublattices in a unit cell are denoted by A, A′, B, C, C′and D in Fig. 1(b).\nWe fixed J1= 1 hereafter as a energy scale. In what follows, we examine the reg ion of\n30≤J2/J1≤3.5 in the present study. Note that the number of total spin sites is d enoted\nbyN; thus, the number of unit cells is N/6.\nLet us introduce here the ground-state phase diagram of the S= 1/2 Heisenberg model\non the original Union Jack lattice of Fig. 1(a). At small J2/J1, one can see immediately that\nthe antiferromagnetic N´ eel order is observed since this model co rresponds to the S= 1/2\nantiferromagnetic Heisenberg one on the simple square lattice in the limit ofJ2/J1= 0.\nWhen the J2/J1is increased, the intermediate-magnetization state appears. A va riational\nanalysis for the classical model revealed that the spin configuratio ns in this intermediate\nregion are defined by the angle of cant ϕas illustrated in Fig. 2[18]. Therefore, this\nintermediate-state is called canted-ferrimagnetic one. The phase transition between the N´ eel\nandcanted-ferrimagneticphasesoccursat α1≡J2/J1∼0.84fromtheviewpointofthespin-\nwave theory[15, 16]. On the other hand, it was reported that this p hase transition occurs\natα1∼0.65 by using the series expansion (SE)[17] and coupled cluster metho d (CCM)[18]\ntechniques. It was also discussed the possibility that the semistripe d-ferrimagnetic state as\nillustrated in Fig. 2 appears at very large values of J2/J1(α2≡J2/J1≈125)[18].\nIn what follows, we examine the ground-state phase diagram of the S= 1/2 Heisenberg\nmodel on the Union Jack strip lattice depicted in Fig. 1(b) and compar e the results of the\noriginal 2D lattice with those of the our strip lattice.\nIII. NUMERICAL METHODS\nWe employ two reliable numerical methods: the exact diagonalization ( ED) method and\nthe density matrix renormalization group (DMRG) method[27, 28]. Th e ED method can\nbe used to obtain precise physical quantities for finite-size cluster s. This method does not\nsuffer from the limitation of the shape of the clusters. It is applicable even to systems with\nfrustration, in contrast to the quantum Monte Carlo (QMC) metho d coming across the\nso-called negative-sign problem for systems with frustration. The disadvantage of the ED\nmethod is the limitation that available system sizes are very small. Thus , we should pay\ncareful attention to finite-size effects in quantities obtained from this method.\nOn the other hand, the DMRG method is very powerful when a syste m is (quasi-)one-\ndimensional under the open-boundarycondition. The methodcan t reat much larger systems\n40 1 2 3\nJ2/J 100.20.4M/M sN=24, periodic\nN=48, open\nN=96, open\nN=144, open'JH\u000f\u0001\u0014\tB\n 'JH\u000f\u0001\u0014\tC\n \n0 10 20\nStotz–50050 E nergy J2/J 1=0.1 \nJ2/J 1=1.5 \nJ2/J 1=3.5 \nFIG. 3: (Color online) (a) Dependences of the lowest energy o nSz\ntot. Results of J2/J1= 0.1, 1.5\nand 3.5 for the system size of N= 48 are presented. Arrows indicate the values of the spontan eous\nmagnetization Min eachJ2/J1. (b)J2/J1-dependence of M/Msobtained from ED calculations for\nN= 24 (black cross) under the periodic-boundary condition an d DMRG calculations for N= 48\n(red triangle), 96 (blue square) and 144 (green pentagon) un der the open-boundary condition.\nthan the ED method. Note that the applicability of the DMRG method is irrespective of\nwhether or not systems include frustrations. In the present res earch, we use the “finite-\nsystem” DMRG method. Note that we carefully choose the maximum n umber of retained\nstates (MS) and the number of sweeps ( SW) in our DMRG calculations.\nIV. RESULTS\nIn this section, we present our numerical results in the ground-st ate of the S= 1/2\nHeisenberg model on the Union Jack strip lattice of Fig. 1(b). First, let us explain the way\nto obtain the spontaneous magnetization Min the ground state of the quantum system with\nisotropic interactions. We calculate the lowest energy E(J2/J1,Sz\ntot,N), where Sz\ntotis the\nz-component of the total spin. For example, the energies for each Sz\ntotin the three cases of\nJ2/J1are shown in Fig. 3(a). In this figure, the results of the DMRG calcula tions with the\nMS= 700 and SW= 15 are presented when the systems size is N= 48 for J2/J1=0.1, 1.5,\n3.5. The spontaneous magnetization M(J2/J1,N) is determined as the highest Sz\ntotamong\n5'JH\u000f\u0001\u0015\tB\n 'JH\u000f\u0001\u0015\tC\n \nFIG. 4: (Color online) (a) Spin configuration in the N´ eel-li ke spin liquid phase. (b) Spin configu-\nration in the LM ferrimagnetic phase of M/Ms= 1/3, where this configuration is obtained from\nthe numerical results of the local magnetization shown in Fi g. 5(b).\nthose at the lowest common energy [see arrows in Fig. 3(a)].\nOur results of the J2/J1-dependence of the M/Msare shown in Fig. 3(b), where Ms\nmeans saturated magnetization value, namely, Ms=N/2. In the limit of J2/J1= 0, this\nUnion Jack strip model depicted in Fig. 1(b) is reduced to the S= 1/2 antiferromagnetic\nHeisenberg model on the three-leg ladder as known the typical sys tem of the gappless spin-\nliquid ground states[29, 30]. According to the study of the S= 1/2 frustrated three-leg\nspin ladder[31], it is expected that the N´ eel-like spin liquid phase occur s in the ground-state\nwhen the strength of the J2/J1is small but finite, where the schematic spin configuration in\nthe N´ eel-like spin liquid state is illustrated in Fig. 4(a). Indeed, our nu merical calculations\nlead to the same conclusions that the nonmagnetic phase of M/Ms= 0 appears in the region\nwhereJ2/J1is relatively small.\nFor larger J2/J1, on the other hand, the magnetic phases with M/Ms/negationslash= 0 appears in\nthe ground state. Careful observation enables us to find that th ere are two magnetic phases\nin the thermodynamic limit; one is the intermediate magnetic phase of 0 < M/M s<1/3\nand the other is the phase of M/Ms= 1/3. It should be noted here that the phase of\nM/Ms= (1\n3−2\nN) which is found only under the open-boundary condition merges with the\nphase of M/Ms= 1/3 in the thermodynamic limit of N→ ∞since the value of M/Ms\nbecomes gradually larger and approaches the value of M/Ms= 1/3. This change due to the\nincrease of the system size comes from finite-size effect. It is impor tant that we successfully\nobserve the intermediate-magnetization phase where the sponta neous magnetization M/Ms\nchanges continuously with respect to the strength of J2/J1.\nNext, we calculate the local magnetization /angbracketleftSz\ni,ξ/angbracketrightto investigate the spin configurations in\n6'JH\u000f\u0001\u0016\tB\n 'JH\u000f\u0001\u0016\tC\n \n10 20 \ni–0.2 00.20.4N=144 J 2/J 1=3.5 M=24\nC’ D C A ’ B A\n10 20 \ni–0.2 00.20.4N=144 J 2/J 1=1.8 M=17\nC’ D C A ’ B A\nFIG. 5: (Color online) Local magnetization /angbracketleftSz\ni,ξ/angbracketrightat each sublattice ξ. Panels (a) and(b) areresults\nforJ2/J1=1.8 and 3.5 respectively. These results are obtained from o ur DMRG calculations for\nN= 144 (i=1,2,···, 24).\nthese two magnetic states, where /angbracketleftA/angbracketrightdenotes the expectation value of the physical quantity\nAandSz\ni,ξis thez-component of Si,ξ. Figure 5 depicts our results for a system size N= 144\non the lattice depicted in Fig. 1(b) under the open-boundary condit ion; Fig. 5(a) and Fig.\n5(b) correspond tothe case of J2/J1=1.8and3.5 respectively; we use greeninverted triangle\nforξ=A, blue pentagon for ξ= A′, red circle for ξ= B, black cross for ξ=C, aqua triangle\nforξ= C′, and purple square for ξ=D. In Fig. 5(a), we find clearly incommensurate\nmodulations with long-distance periodicity in the behavior of the local magnetization at\ntheB-sublattice sites. Therefore, we conclude that the intermediate- magnetization phase\nof 0< M/M s<1/3 is the NLM ferrimagnetic one. On the other hand, in Fig. 5(b),\nwe observe the uniform behavior of upward-direction spins at subla ttice-sites B, C, C′and\nD and downward-direction spins at sublattice A and A′as illustrated in Fig. 4(b). The\nspin configuration in the phase of M/Ms= 1/3 can be understood from the viewpoint of the\nMLMtheorembecausethepresent stripmodelcorrespondstoth eS= 1/2antiferromagnetic\nHeisenberg model on the diamond chain in the limit of J2/J1=∞. Therefore, it is naturally\nlead to the conclusion that the phase of M/Ms= 1/3 is the LM ferrimagnetic one.\nFinally, we discuss the relationship between the ground-state prop erties of the original\nUnion Jack model depicted Fig. 1(a) and those of the strip model de picted in Fig. 1(b).\n7It is confirmed that the schematic spin configuration in the N´ eel-like spin liquid state de-\npicted in Fig. 4(a) is consistent to that in the N´ eel state depicted in Fig. 2. We also\nconfirm that the schematic spin configuration depicted in Fig. 4(b) a grees completely with\nthat in the semistriped-ferrimagnetic state of the original Union Ja ck model as shown in\nFig. 2. In addition, there exists the intermediate-magnetization st ate where the sponta-\nneous magnetization is gradually changed in the ground-state of th e both models although\nthe incommensurate modulation with long-distance periodicity has no t been confirmed in\nthe case of 2D Union Jack model. Therefore, one finds that the the ground-state phase\ndiagram of the S= 1/2 antiferromagnetic Heisenberg model on the Union Jack strip lat-\ntice is qualitatively consistent to that of the S= 1/2 antiferromagnetic Heisenberg model\non the original Union Jack lattice. The intermediate-magnetization s tate of the original\nUnion Jack lattice have been understood from the view point of the c lassical configuration,\nnamely ”canted-state”. However, our numerical results of the U nion Jack strip model leads\nto the possibility that the intermediate-state of the original Union J ack lattice is also the\nNLM ferrimagnetic one whose characteristic behavior in the local ma gnetization originates\nfrom pure quantum effects. The future studies are desirable to co nfirm the presence of the\nincommensurate modulation in the canted-state of the 2D Union Jac k model.\nV. CONCLUSIONS\nWe have studied the ground-stateproperties of the S= 1/2 antiferromagnetic Heisenberg\nmodel on the Union Jack strip lattice depicted in Fig. 1(b) by the ED an d DMRG meth-\nods. Our numerical calculations have revealed that the intermediat e-magnetization state\noccurs between the N´ eel-like spin liquid state corresponding to the N´ eel state of the original\n2D model and the LM ferrimagnetic state which agrees with the semis triped-ferrimagnetic\nstate of the original 2D model. In this intermediate-magnetization s tate of this strip model,\nthe spontaneous magnetization changes gradually with respect to the strength of the inner\ninteraction. We have also found the existence of the incommensura te modulation with long-\ndistance periodicity of the local magnetization. From the finds of th ese characteristic be-\nhavior, it has concluded that the intermediate state of this strip mo del is the NLM ferrimag-\nnetism. These results naturally lead to the expectation that the int ermediate-magnetization\nstate of the original model is also the NLM ferrimagnetic one.\n8Acknowledgments\nWe wish to thank Professor T. Sakai for fruitful discussions. One of the authors (T. S.)\nacknowledgesthefinancialsupportfromtheMotizukiFundofYuk awaMemorialFoundation.\nThis work was partly supported by Grants-in-Aid (Nos. 20340096, 23340109, 23540388,\nand 24540348) from the Ministry of Education, Culture, Sports, S cience and Technology of\nJapan. 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Lecheminant, B. Bernu, C. Lhuillier, L. Pierre and P. Sindzingre, Phys. Rev. B 56, 2521\n(1997).\n[21] Ch. Waldtmann, H.-U. Everts, B. Bernu, C. Lhuillier, P. Sindzingre, P. Lecheminant, and L.\nPierre, Eur. Phys. J. B 2, 501 (1998).\n[22] K. Hida, J. Phys. Soc. Jpn. 70, 3673 (2001).\n[23] H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 79, 053707 (2010).\n[24] H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 80, 053704 (2011).\n[25] T. Shimokawa and H. Nakano, J. Phys.: Conf. Ser. 320, 012007 (2011).\n[26] T. Shimokawa and H. Nakano, to be published in J. Phys. So c. Jpn.\n[27] S. R. White, Phys. Rev. Lett. 69, 2863 (1992).\n[28] S. R. White, Phys. Rev. B 48, 10345 (1993).\n[29] E. Dagotto and T. M. Rice, Science 271618 (1996).\n[30] Azuma M, Hiroi Z, Takano M, Ishida K and Kitaoka Y, Phys. R ev. Lett. 73, 3463 (1994).\n[31] S. Abe, T. Sakai, K. Okamoto and K. Tsutsui, J. Phys.: Con f. Ser.320, 012015 (2011).\n[32] A. F. Albuquerque, F. Alet, P. Corboz, P. Dayal, A. Feigu in, L. Gamper, E. Gull, S. Gurtler,\nA. Honecker, R. Igarashi, M. Korner, A. Kozhevnikov, A. Lauc hli, S. R. Manmana, M. Mat-\nsumoto, I. P. McCulloch, F. Michel, R. M. Noack, G. Pawlowski , L. Pollet, T. Pruschke, U.\nSchollwock, S. Todo, S. Trebst, M. Troyer, P. Werner, S. Wess el, J. Magn. Magn. Mater. 310,\n1187 (2007) (see also http://alps.comp-phys.org).\n10" }, { "title": "2202.08115v1.Unusual_ferrimagnetic_ground_state_in_rhenium_ferrite.pdf", "content": "Unusual Ferrimagnetic Ground State in Rhenium Ferrite\nM. Hussein N. Assadi\u0003\nSchool of Materials Science and Engineering, University of New South Wales, Sydney NSW 2052, Australia.\nMarco Fronzi\nCollege of Engineering, Shibaura Institute of Technology, Toyosu, Koto City, Tokyo 135{8548, Japan.\nDorian A. H. Hanaor\nFachgebiet Keramische Werksto\u000be, Technische Universit at Berlin, 10623 Berlin, Germany.\n(Dated: 2022)\nThrough comprehensive density functional calculations, we predict the stability of a rhenium-\nbased ferrite, ReFe 2O4, in a distorted spinel-based structure. In ReFe 2O4, all Re and half of the Fe\nions occupy the octahedral sites while the remaining Fe ions occupy the tetrahedral sites. All Re\nions are predicted to be at a +4 oxidation state with a low spin con\fguration ( S= 3=2), while all\nFe ions are predicted to be at a +2 oxidation state with a high spin state con\fguration ( S= 2).\nMagnetically, ReFe 2O4adopts an unconventional ferrimagnetic state in which the magnetic moment\nof Re opposes the magnetic moments of both tetrahedral and octahedral Fe ions. The spin-orbit\ncoupling is found to cause a slight spin canting of \u00181:5\u000e. The predicted magnetic ground state\nis unlike the magnetic alignment usually observed in ferrites, where the tetrahedral cations oppose\nthe spin of the octahedral cations. Given that the density of states analysis predicts a half-metallic\ncharacter driven by the presence of Re t2gstates at the Fermi level, this compound shows promise\ntowards potential spintronics applications.\nKeywords: Rhenium ferrite, unconventional magnetism, ferrimagnetism, spin canting, density functional\ntheory, spin-orbit coupling\nINTRODUCTION\nSearching for and investigating exotic magnetic phases\ndeepens our fundamental knowledge of complex func-\ntional materials and opens new horizons for novel ap-\nplications [1]. Ferrites are among the most studied mag-\nnetic materials, with broad applications in spintronics\n[2], magnetic data storage [3], magnetically recoverable\ncatalysts [4-7], and microwave guides [8]. The utility\nof ferrites stems from their high magnetic saturation,\nhigh Curie temperatures and controllable coercivity. Fer-\nrites, most commonly synthesised by reactions of Fe 2O3\nwith a smaller proportion of other metal oxides, encom-\npass a wide range of chemical compositions, stoichiome-\ntries, and crystal structures. However, they mostly crys-\ntallise into cubic spinel structures, distorted spinels with\nlower symmetry, [9] or hexagonal structures [10]. Fer-\nrite cations are situated between octahedral and tetra-\nhedral voids, created through the oxygens' closed packed\narrangement [11]. Alongside Fe, cation sites may be oc-\ncupied by Sr, Ba, Pb, or transition metals (TMs). Gen-\nerally, ferrites are ferrimagnetic where the spin of the\ntetrahedral cations opposes but does not cancel the spin\nof the octahedral cations [12]. The second cation's type\nand its abundance determine the magnetic hardness and\nsaturation of the resultant ferrite, enabling the design of\nmaterials with desirable magnetic phase transition tem-\nperatures and magnetic coercivity [13].\nTypically, in complex materials containing multiple\nmagnetic cations, the magnetic ground state is stabilisedthrough competing ferromagnetic and antiferromagnetic\nsuperexchange interactions, resulting in frustrated sys-\ntems with multiple magnetic phase transitions. In fer-\nrites with only fourth row 3d TM ions, these competi-\ntions simply stabilise the ferrimagnetic state where the\ncations on the octahedral site align antiparallel to the\ncations on the tetrahedral sites [12]. However, in other\nclasses of complex oxides, such as double perovskites,\nmagnetic interactions between 3d and heavier 4d and\n5d elements has demonstrated substantially more com-\nplex magnetic behaviour [14,15], often deviating from the\nrules of thumb established by Goodenough and Kanamori\n[16,17], where exotic magnetic behaviours are often the\nresult of the strong spin-orbit coupling, structural distor-\ntions and higher bond covalency between heavier TM ions\nand oxygen. A detailed review of Goodenough-Kanamori\nrules can be found in the literature [18,19].\nWith all these exciting developments in perovskite\nmagnetism, one wonders if there are any similar coun-\nterparts in ferrites. The idea of harnessing heavier 4d\nand 5d TM ion ferrites to control anisotropy and magne-\ntostriction in ferrite was proposed by Hansen and Krish-\nnan in 1977 [20]. However, since then, this idea has not\nattracted the attention it deserves. In the present work,\nwe demonstrate the stability of a rhenium-based ferrite\nReFe 2O4and discusses its unconventional ferrimagnetic\nground state through comprehensive density functional\ncalculations.arXiv:2202.08115v1 [cond-mat.mtrl-sci] 16 Feb 20222\nCOMPUTATIONAL SETTINGS\nSpin-polarised collinear and noncollinear density func-\ntional calculations were performed with VASP code\n[21,22], using the projector augmented wave method\n(PAW) [23] and the Perdew{Burke{Ernzerhof (PBE)\nexchange-correlation functional [24,25]. To improve the\nelectronic band description, adequate intra-atomic inter-\naction terms ( Ue\u000b), based on the Liechtenstein et al. ap-\nproach [26], were added to the Fe 3d electrons. The Uand\nJparameters were 3.5 eV and 0.5 eV, respectively, result-\ning in an e\u000bective U(Ue\u000b) of 3 eV. Comparable values\nwere reported to improve the band description accuracy\nof ferrites [27,28]. More speci\fcally a Ue\u000bvalue of 3 eV\nis necessary to adequately describe the Fe 3d electrons in\noxides [29]. Furthermore, our comprehensive test demon-\nstrated the adequacy of these values (Fig. S1). Accu-\nrate electronic localisation through the GGA+ Uformal-\nism is essential for obtaining reliable structures as the\natomic forces are sensitive to magnetic moments borne\non cations [30]. The energy cut-o\u000b was set at 650 eV.\nThe precision key for the rest of the parameters was set\nACCURATE . The noncollinear calculations were initi-\nated with the WAVECAR \fles calculated with the spin-\npolarised collinear method to facilitate convergence.\nTo simulate the ReFe 2O4structure, as shown in Fig.\n1, two Fe ions in the primitive magnetite cell, with the\nchemical formula Fe 6O8[31], were substituted with Re\nions. As shown in Fig. 1a{j, we considered all possible\nrhenium placement scenarios, and for each scenario, we\nexamined various spin alignments, searching for the most\nstable structure. Substitution at the octahedral sites con-\nsistently resulted in lower total energy, so we further in-\nvestigated all plausible spin alignments in ReFe 2O4with\nonly octahedral Re. A dense 7 \u00027\u00027 k-point mesh, gen-\nerated with the Monkhorst-Pack scheme of \u00180:015\u0017A\u00001\nspacing, consisting of 172 irreducible sampling points in\nthe Brillouin zone, was used for geometry optimisation.\nFor geometry optimisation, the internal coordinates and\nthe lattice parameters were relaxed to energy and force\nthresholds smaller than 10\u00006eV and 0.02 eV \u0017A\u00001, re-\nspectively. No symmetry restriction was applied in ge-\nometry optimisation to allow relaxation to lower symme-\ntry, should it be more stable. It is well-known that even\nthe simplest of the spinel ferrites, Fe 3O4, although cu-\nbic (Fd\u00163m) at room temperature, transforms to a lower\nsymmetry monoclinic structure ( P21=c) below\u0018125 K,\nthrough a Verwey phase transition [32].\nRESULTS AND DISCUSSION\nTwo cation types based on coordination exist in the\nspinel ferrite structure: one that is tetrahedrally coordi-\nnated and another that is octahedrally coordinated withoxygen ions. The \frst type represents one-third, while\nthe second represents two-thirds of the total cations in\nthe crystal. We examined three possible rhenium place-\nments to identify the most stable position of the Re ions\nin ReFe 2O4. First, both Re ions were placed at the tetra-\nhedral sites while all Fe ions were left at the octahedral\nsite, which is the typical cationic distribution in spinels.\nSecondly, one Re ion was placed at the tetrahedral site,\nand the other Re ion was placed at the octahedral site,\na con\fguration usually referred to as an intermediate\nspinel. Lastly, both Fe ions were placed at tetrahedral\nsites, while the octahedral sites were equally occupied\nwith Re and Fe, which is referred to as an inverse spinel.\nWe investigated all these cationic distributions by cal-\nculating the total energy for two possible ferrimagnetic\nand ferromagnetic spin alignments. In the ferromagnetic\nstructure, all cations' spin was set parallel, while in the\nferrimagnetic structure, the tetrahedral cations' spin was\nset antiparallel to the spin of the octahedral cations. As\nshown in Fig. 1a and b, when Re ions are located at\ntetrahedral sites, the total energy is relatively high re-\ngardless of the direction of the Re spin. However, the\nferrimagnetic state is still more stable than the ferro-\nmagnetic one. For intermediate rhenium occupancy, as\nin Fig. 1c, d, the total energies for both spin alignments\nwere slightly higher than the previous case of complete\ntetrahedral Re occupation.\nWhen Re ions are located at octahedral sites (Fig. 1e{\nj), the compound's total energy generally decreased sig-\nni\fcantly, indicating greater stability. For instance, the\naforementioned ferrimagnetic spin alignment of Fig. 1e\nwas more stable than the counterparts with tetrahedral\nRe (Fig. 1a) and the mixed Re (Fig. 1c) con\fgurations\nby approximately 2 eV per unitcell. The stability of the\noctahedral Re warranted further investigations of other\npossible spin alignments of ReFe 2O4with octahedrally\ncoordinated rhenium. Accordingly, we calculated all dif-\nferent possible ferrimagnetic spin alignments for this fer-\nrite compound with octahedral Re [see con\fgurations (g)\nand (h)]. In particular, con\fguration (h) was the most\nstable among all. In this ferrimagnetic con\fguration,\nthe Re ions' spin direction is antiparallel to the spins of\nboth octahedral Fe and tetrahedral Fe ions. In this case,\nthe spins of the octahedral Fe and tetrahedral Fe ions\nwere parallel. Con\fguration (e), with similar spin align-\nment to a conventional ferrimagnetic inverse spinel such\nas magnetite, was higher in total energy relative to con-\n\fguration (h) by 0.4446 eV/u.c. (u.c. is unitcell). Con-\n\fguration (g), representing the other possible realisation\nof the ferrimagnetic alignment, also had higher total en-\nergy than con\fguration (h) by 2.1606 eV/u.c. Likewise,\nthe ferromagnetic state in con\fguration (f) had higher\ntotal energy of 1.1251 eV/u.c.\nTo unambiguously con\frm the stability of con\fgura-\ntion (h), we further calculated the total energy of con-\n\fguration (i), which is quite similar to con\fguration (h)3\nexcept that the spins of the two Re ions were set an-\ntiparallel to examine the strength of the magnetic cou-\npling among Re ions. We also examined con\fguration\n(j), which is antiferromagnetic; that is, every cation has\nan antiparallel spin to the one adjacent. Both con\fgu-\nrations (i) and (j) had total energies higher than that of\ncon\fguration (h) by more than 1 eV/u.c. The stability\nof con\fguration (h) relative to con\fgurations (i) and (j)\ndemonstrates that pairs of Re Oct, Fe Octand Fe Tetions\nhave a strong tendency towards parallel spin alignment\namong themselves.\nFor the most stable placement of Re which can be\nexpressed as Fe Tet(ReFe) OctO4, con\fguration (h), which\nrepresents the magnetic ground state, is remarkably sta-\nble as \ripping to even the second most stable spin con-\n\fguration (e) has an energy cost of 0.4446 eV/u.c. This\nlevel of stability is likely to correspond to an ambient\nCurie temperature in ReFe 2O4. This prediction is based\non a comparison with CoFe 2O4's stability margin. A\nspin-\rip from ferrimagnetic to ferromagnetic order in\nCoFe 2O4costs 0.536 eV/u.c. [33], resulting in a Curie\ntemperature ( TC) of 793 K in bulk cobalt ferrite [34].\nFurthermore, the mean-\feld approximation can estimate\nthe Curie temperature based on the magnetic exchange\nbetween Fe and Re sublattice systems as follows [35]:\n3\n2kBTC=X\ni6=jJij; (1)\nin whichkBis the Boltzmann constant and Jijis the\npair exchange coupling parameter between sites iand\nj. Assuming the nearest neighbour interaction between\nRe and Fe to be the most signi\fcant, Equation 1 yields\nTC= 734:9 K, which is close to the comparison made\nearlier.\nGiven that Re is a relatively heavier sixth-row tran-\nsition element, we anticipate that the role of spin-orbit\ncoupling (SOC) is potentially signi\fcant in ReFe 2O4[36].\nTo examine the signi\fcance of SOC, we re-optimised the\nmost stable con\fguration (h) with spin-orbit coupling\ntaken into consideration. The structural relaxation was\nminor as none of the internal coordinates and the lattice\nparameters changed by more than \u00181%. However, the\ntotal energy was lowered to \u0000107:7975 eV/u.c., indicat-\ning that SOC accounts for \u00180:67% of the total energy.\nSimilarly, SOC is not anticipated to change the magnetic\nalignment in ReFe 2O4as other competing con\fgurations\nare also estimated to have their total energy lowered by\napproximately the same amount when SOC is considered\n(Table S1). SOC brings about magnetic noncollinearity\nby coupling the spin to the orbital degrees of freedom, of\nwhich the latter depends on the lattice environment. The\nnoncollinear magnetic alignment of ReFe 2O4is shown in\nFig. 2a. More precisely, the spins of the Re ions formed a\ntight angle of 1 :329\u000ebetween each other, i.e., nearly par-\nallel; and the net spin of the Re ions formed an angle of177:593\u000ewith the net spin of the Fe ions, i.e., nearly an-\ntiparallel. It can be seen that the degree of noncollinear-\nity is relatively small as the spin directions are to a great\nextent similar to that of the collinear alignment of con-\n\fguration (h). The net magnetisation of the whole com-\npound was calculated to be 6.090 \u0016B/f.u. (f.u. stands for\nformula unit), which is approximately 1.5 times larger\nthan that of the saturation magnetic moment of mag-\nnetite [37]. A more qualitative description of the e\u000bect\nof theUe\u000bchoice on the total magnetisation and the local\nmagnetisation of the Re and Fe ions in this con\fguration\nis given in Table S2.\nThe density of states (DOS) in ReFe 2O4, calculated\nconsidering SOC, are presented in Fig. 3. Here, DOS is\nprojected onto the axes of an orthogonal frame of which\nthezaxis is parallel to the cpdirection of the ReFe 2O4\nprimitive cell. First, we notice that the DOS magnitude\nalong thexandyaxes is\u00183% of the DOS magnitude\nalong thezaxis, indicating a slight deviation from linear-\nity and corroborating the magnetic moment orientations\nobtained in Fig. 2a. Along the zaxis, (Fig. 3c), we can\nsee that both Fe Tetand Fe Octare in high-spin and +2\noxidisation states as one spin channel, comprising of \fve\nelectrons per Fe, for both Fe Tetand Fe Oct, is fully occu-\npied while the other spin channel is only partly occupied.\nTherefore, the electronic con\fguration for Fe2+\nTetise2\"t3\n2\n\"e1#, while for Fe2+\nOct, the electronic con\fguration is t3\n2g\n\"e2\ng\"t1\n2g#. Moreover, Re is in low-spin state with +4\noxidation. According to its partial DOS, Re has a fully\noccupiedt2g\"states, which are immediately followed by\nits emptyt2g#states. The Fermi level crosses the tail\noft2g\"states, giving rise to half-metallic conduction, as\nno states are available at the Fermi level with opposing\nnet spin direction. Moreover, because of the larger crys-\ntal \feld acting on Re's 5d states, Re's empty egstates\nare located at\u00184 eV above the Fermi level (Fig. S2).\nThe ReFe 2O4half-metallicity can be utilised for magneto-\nresistive response [38,39] or near-perfect spin-polarised\ncurrent injection [40,41].\nAs shown in Fig. 3, the Fe 3d states are spread through\nthe conduction band, hybridising extensively with O,\nwhile Re 5d states are mainly concentrated within 2\neV below the Fermi level. Nonetheless, in the region of\n\u00002< E\u0000EFermi<0, the spin-down Re and Fe states\nand O 2p states hybridise together, facilitating the mag-\nnetic superexchange interactions that stabilise the mag-\nnetic ground state. Furthermore, the net magnetic mo-\nments borne on all cations, as shown in Fig. 2, are smaller\nthan ideal ions. For high spin Fe2+(3d6), either in tetra-\nhedral or octahedral coordination, the magnetic moment\nshould have been 4 \u0016B. For octahedral Re4+(5d3), the\nmagnetic moment should have been 3 \u0016B. However, since\nFe{O or Re{O bonds are not purely ionic but possess a\ndegree of covalency, the magnetisation of transition metal\nions is expected to be lower than the purely ionic values\n[42]. The reduction in magnetisation is more profound4\n(b)\nEt = –103.7775 eV (FM)\n(f)\nEt = –105.9444 eV (FM)\n(j)\nEt = –104.2201 eV (AFM)ap\nbpcp\nEt = –105.9039 eV\n(i)\nFe\nRe\nO\n(e)\nEt = –106.6249 eV (FiM-a) Et = –107.0695 eV (FiM-c)\n(h)\nOctTet\n(g)\nEt = –104.9089 eV (FiM-b)\nOctTetOctTet(d)\nEt = –103.5364 eV (FM)\nOctTet(c)\nEt = –103.7854 eV (FiM) Et = –104.6162 eV (FiM)\n(a)\nTet\nOct\nAll starting at\nap = bp = cp = 5.961 Å\nαp = βp = γp = 60º\n( )Fd¯3m\nFig. 1. The spin con\fgurations used for determining the magnetic ground state of ReFe 2O4. In con\fgurations a\nandb, Re ions are at the tetrahedral sites. In candd, one Re ion is located at the tetrahedral site while the other\nis located at the octahedral site. In e,f,g,h,i, and j, both Re ions are located at the octahedral sites. The density\nfunctional total energy ( Et) of each con\fguration is also shown. FM, FiM, and AFM refer to ferromagnetic,\nferrimagnetic and antiferromagnetic, respectively.\n∠ReOct–O–ReOct: 86.626º, 86.634º∠FeOct–O–FeOct: 93.52º, 97.13º∠FeOct–O–ReOct: 91.29º, 91.32º, 98.81º, 98.84º∠ReOct–O–FeTet: 123.55º, 124.98º, 128.61º, 134.50º∠FeOct–O–FeTet: 111.94º, 112.10º, 114.02º, 114.05º, 127.39º, 127.42º\nccacbc(b)ReOct4+\nFeTet2+\nap = 6.243, bp = 6.198, cp = 6.243 Åαp = 60.24º, βp = 54.43º, γp = 60.24º\nac = 5.710 , bc = 6.198, cc = 6.243 Åαc = 119.76º, βc = 117.21º, γc = 90.00ºFeOct2+\nReOct4+FeOct2+FeTet2+\nap\nbpcp3.549 μB3.549 μB\n3.614 μB3.614 μB1.280 μB\n1.092 μB(a)\nFeTet2+\nFeOct2+\nFe\nRe\nO\nFig. 2. aThe optimised primitive cell of ReFe 2O4with SOC taken into account, along with the magnetic moments\nborne on all cations. bThe conventional representation of the optimised ReFe 2O4, which has a triclinic symmetry.\nInb, all possible bond angles between cations are also presented. The corresponding yellow marks show a\nrepresentative of each given angle. The lattice parameters shown are either indexed with porc, indicating the\nprimitive and conventional cell dimensions, respectively.5\nin heavier transition metal ions are their bonds are more\ncovalent [33]. A more quantitative description of the elec-\ntronic localisation function is provided in Fig. S3.\nAs shown in Fig. 2a, the primitive cell of ReFe 2O4\nis substantially transformed by geometry optimisation.\nThe lattice parameters of the optimised primitive cell do\nnot conform to the high symmetry of the initial structure\nas its lattice parameters asymmetrically changed and its\nvolume expanded. The initial structure volume, which\nwas based on magnetite, was 155.225 \u0017A3, while the opti-\nmised structure had a volume of 163.025 \u0017A3. The ionic\nradius of Fe can explain the expansion of the volume\nupon Re substitution at the tetrahedral site. The radius\nof high-spin Fe2+\nTetin ReFe 2O4is 0.63 \u0017A. In magnetite,\nthe tetrahedral site is occupied by Fe3+with a smaller\nradius of 0.49 \u0017A. The Re4+radius in octahedral coordi-\nnation is 0.63 \u0017A which is quite close to the replaced Fe3+\nradius (0.65 \u0017A) and is not expected to be a substantial\ndrive in the structural transformation.\nWe examined the optimised ReFe 2O4primitive cell's\nsymmetry to investigate the magnetic exchange among\nall cations. A triclinic symmetry was detected through\nthe FINDSYM symmetry detection algorithm [43] with a\ntight tolerance of 0.00001 \u0017A for lattice parameters (CIF\nprovided in the supplementary information). The con-\nventional cell with triclinic symmetry is shown in Fig. 2b.\nDetecting all possible magnetic exchanges that stabilise\nthe predicted ground state magnetism|con\fguration\n(h) re-optimised with SOC considered{is easier in the\nsymmetry-imposed structure. The TM{O{TM bond an-\ngles for this structure are all listed in Fig. 2. The TM{\nO{TM bond angles between cations on tetrahedral and\noctahedral sites are obtuse and thus dominate the mag-\nnetic exchange interactions, as the superexchange inter-\naction magnitude is proportional to cos2(6TM{O{TM).\nThe higher total energy of con\fguration (g) indicates the\nsuperexchange between Re4+\nOctand Fe2+\nTetis antiferromag-\nnetic, while the higher total energy of con\fguration (e)\nindicates that the superexchange between Fe2+\nOctand Fe2+\nTet\nis ferromagnetic. The TM{O{TM bonds among tetrahe-\ndral sites are all nearly right angles, indicating a minimal\norbital overlap favouring weaker ferromagnetic superex-\nchange. The strength of this ferromagnetic exchange can\nbe estimated from the total energy of con\fgurations (g)\nand (h) of Fig. 1, showing that setting adjacent cations\nto antiferromagnetic coupling raises the total energy.\nFinally, we examine the stability of ReFe 2O4in com-\npeting metallic and oxide phases. The compound's for-\nmation enthalpy (\u0001 Hmetallic ) was calculated relative to\nthe metallic Re (hexagonal paramagnetic) and Fe (body-\ncentred ferromagnetic) phases, and gaseous O 2was cal-\nculated as\n\u0001Hmetallic =Et(ReFe 2O4)\u0000Et(Re)\u00002Et(Fe)\u00002Et(O2):\n(2)\nHere,Etis the density functional total energy. \u0001 Hmetallic\n-0.40.00.4 \nO 2p \nFeTet 3d \nFeOct 3d \nReOct 4dE\n − EFermi (eV)(a) ReFe2O4 (x)-\n0.30.00.3(\nc) ReFe2O4 (z)(b) ReFe2O4 (y)-\n8- 6- 4- 20 2 -303DOS (eV−1)D OS (eV−1)t\n2gt\n2gt2g & egt\n2ge & t2et2g & ege & t2DOS (eV−1)Fig. 3. Partial density of states of ReFe 2O4at its most\nstable magnetic state [con\fguration ( h) of Fig. 1],\ncalculated with spin-orbital coupling considered. The\ndensity of states is projected along an orthogonal frame\nhaving the x,y, andzaxes. Thezaxis of this frame\ncoincides along the lattice parameter cpof the primitive\nlattice parameter, shown in the lower row of Fig. 1.\nwas found to be\u00005:1924 eV/f.u. The formation enthalpy\nrelative to the competing oxide phase ( \u0001 Hoxide) was\ncalculated as\n\u0001Hoxide =Et(ReFe 2O4)\u0000Et(ReO 2){2Et(FeO):(3)\nHere, ReO 2was the most stable Re4+oxide in orthorhom-\nbic structure (materials project identi\fer mp-7228 [44]),\nand FeO was the most stable Fe2+oxide in monoclinic\nstructure (materials project identi\fer mp-1279742 [44]).\n\u0001Hoxide was found to be\u00001:1903 eV/f.u. Given the neg-\native \u0001Hvalues, we can conclude that ReFe 2O4is stable\nagainst decomposition to oxides of its constituent ele-\nments and Re4+and Fe2+. For the future synthesis of\nReFe 2O4, one can draw inspiration from the recently de-\nveloped green fabrication methods for ferrites [45].\nCONCLUSIONS\nUsing density functional calculations, considering spin-\norbit coupling, we predict that a Re-based ferrite\nReFe 2O4is stable in a distorted spinel structure with\nreduced triclinic symmetry ( P\u00161), and adopts an uncon-\nventional magnetic ordering in where Re spin opposes\nthe spin of both Fe Tetand Fe Oct, while Fe Tetand Fe Oct6\nhave parallel spin alignment among themselves. The net\nmagnetic moment of this compound is evaluated at 6.090\n\u0016B=f:u:which is about 1.5 times greater than that of\nmagnetite. The magnetic ground state is remarkably sta-\nble as \ripping any spin incurs an energetic cost of at least\n0.2223 eV/f.u., which is likely to correspond to an am-\nbient Curie temperature. The compound is predicted to\nbe half-metallic, which implies that this compound may\nbe useful towards applications in spintronics where the\nspin polarisation of conduction electrons is desired.\nCONFLICTS OF INTEREST\nThe authors declare that there is no con\rict of interest.\nACKNOWLEDGMENTS\nThe authors gratefully acknowledge the funding of this\nproject by computing time provided by the Paderborn\nCenter for Parallel Computing (PC2).\nVERSION OF RECORD\nM. Hussein N. Assadi, Marco\nFronzi and Dorian A. H. Hanaor\nUnusual ferrimagnetic ground state in rhenium ferrite\nEur. Phys. J. Plus (2022) 137, 21. https:\n//doi.org/10.1140/epjp/s13360-021-02277-z\nREFERENCES\n[1] A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Pre-\njbeanu, B. Dieny, P. Pirro, and B. Hillebrands, J. Magn.\nMagn. Mater. 509, 166711 (2020). https://doi.org/\n10.1016/j.jmmm.2020.166711\n[2] R. K. Kotnala and J. Shah, in Handbook of Mag-\nnetic Materials, edited by K. H. J. 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Li, Nanoscale\nHoriz. 4 (2), 434 (2019). http://dx.doi.org/10.1039/\nC8NH00278A" }, { "title": "0807.4153v1.Frustration_induced_quantum_phase_transitions_in_a_quasi_one_dimensional_ferrimagnet__Hard_core_boson_map_and_the_Ton_ks_Girardeau_limit.pdf", "content": "arXiv:0807.4153v1 [cond-mat.str-el] 25 Jul 2008Frustration-induced quantum phase transitions in a quasi- one-dimensional\nferrimagnet: Hard-core boson map and the Tonks-Girardeau l imit\nR. R. Montenegro-Filho∗and M. D. Coutinho-Filho†\nLaborat´ orio de F´ ısica Te´ orica e Computacional, Departam ento de F´ ısica,\nUniversidade Federal de Pernambuco, 50670-901, Recife-PE , Brazil\nAbstract\nWe provide evidence of a superfluid-insulator transition (SIT) of ma gnons in a quasi-one-dimensional\nquantum ferrimagnetwith isotropic competing antiferromagneticspin interactions. This SIT occurs be tween\ntwo distinct ferrimagneticphasesdue to the frustration-induced closing ofthe gapto amagnon excitation. It\nthuscausesacoherentsuperpositionofsinglet andtripletstates at latticeunit cellsandapower-lawdecayon\nthe staggered spin correlation function along the transverse dire ction to the spontaneous magnetization. A\nhard-core boson map suggests that asymptotically close to the SI T the magnons attain the Tonks-Girardeau\nlimit. The quantized nature of the condensed singlets is observed be fore a first-order transition to a singlet\nmagnetic spiral phase accompanied by critical antiferromagnetic o rdering. In the limit of strong frustration,\nthe system undergoes a decoupling transition to an isolated gapped two-leg ladder and a critical single linear\nchain.\nPACS numbers: 75.10.Pq,75.10.Jm,75.40.Mg,75.30.Kz,75. 50.Gg\n∗Electronic address: rene@df.ufpe.br\n†Electronic address: mdcf@ufpe.br\n1I. INTRODUCTION\nRecently, several experimental and theoretical studies in dicate that, under very special condi-\ntions, magnons [1, 2, 3] and polaritons [4] undergo Bose-Ein stein condensation (BEC) in two- and\nthree-dimensional materials. In magnetic systems, BEC of m agnons can be driven by an applied\nmagnetic field ( h) (Ref. [1]), by varying the external pressure [2], or by micr owave pumping [3] In\n1D gapped antiferromagnets, e. g., spin-1 chains [5] and sin gle spin-1/2 two-leg ladders [6], the gap\nto the magnon excitation closes at a critical value ( hc) of the field and the magnetization increases\nas (h−hc)1/2. Although, stricto sensu , there is no BEC of magnons in these 1D systems, it is very\nappealing to describe the transition in terms of the condens ation of the uniform component of the\nmagnetization along the applied field [5]. In fact, rigorous results [7] on low dimensional ( D≤2)\nuniform interacting boson systems preclude the occurrence of BEC in finite temperature ( T). In\n2D systems phase fluctuations have mainly a thermal origin, s o that only the T= 0 condensate\nsurvives, with superfluid behavior persisting up to the Kost erlitz-Thouless temperature. In con-\ntrast, in 1D boson systems phase fluctuations have a quantum o rigin and there is no BEC, even at\nT= 0, but superfluidity is expected [7]. However, in finitesystems the scenario is more complex,\nsince in real confined systems [7, 8] one may be dealing with me tastable states.\nIn this work we introduce an isotropic Heisenberg spin Hamiltonian with two competing anti-\nferromagnetic (AF) exchange couplings [ J1(≡1) andJ] exhibiting a continuous quantum phase\ntransition at a critical value Jc1which, we argue, is a superfluid-insulator transition (SIT) of\nmagnons associated with the creation of a coherent superpos ition of singlet and triplet states at\nlattice unit cells. For J= 0, the model shares its phenomenology and unit cell topolog y with\nquasi-one-dimensional ferrimagnetic compounds [9], such as the line of trimer clusters present in\ncopper phosphates [10], and the organic ferrimagnet PNNBNO (Ref. [11]). On the theoretical side,\nseveral features of the ferrimagnetic phase have been studi ed through Hubbard [12], t−J(Ref.\n[13]) and Heisenberg [14] models, including magnetic excit ations [15, 16] and the occurrence of\nnew phases induced by hole doping of the electronic band [17] . Also, the physical properties of the\ncompoundCu 3(CO3)2(OH)2were successfully explained [18] by the distorted diamond c hain model\n[19], which is a system with three spin 1/2 magnetic sites per unit cell and coupling parameters\nsuch that the ferrimagnetic state is frustrated.\nNumericalresultshavebeenobtainedforfiniteclustersthr oughDensity MatrixRenormalization\nGroup (DMRG) (Refs. [20, 21]) using open boundary condition s and exact diagonalization (ED)\nusing periodic boundary conditions and Lanczos algorithm.\n2(a)B1\nB2A(b)\nJc1= 0.342 Jt= 0.445 0.5\nJ00.20.40.60.81Sg / SLMDMRG (Nc = 33)\nED (Nc = 10)\nHCB Model(c)F1 PhaseF2 Phase\nFIG. 1: (a) Illustration of the A and B sublattices (circles) and AF sp in couplings which favor (full lines)\nand destabilize (dashed lines) the LM ferrimagnetic GS: J1(≡1) andJ, respectively. (b) Illustration of the\nLM ferrimagnetic GS. (c) Results (see text) for Sg/SLM; dashed and dotted lines are guides to the eye.\nThe paper is organized as follows: in Sec. II we introduce the model Hamiltonian and analyze\nthe magnetic correlations of the competing phases close to J=Jc1. In Sec. III we define a\nhard-core boson model (HCB model), which is used to describe the main characteristics of the\nmagnon SIT at J=Jc1, in particular, the Tonks-Girardeau limit. Further, in Sec . IV we discuss\nthe singlet magnetic spiral phase accompanied by critical a ntiferromagnetic ordering, which sets\nin after a first-order transition at J=Jt, as well as the decoupling transition, at J=Jc2, to an\nisolated gapped two-leg ladder and a critical single linear chain. Finally, a summary of the results\nis presented in Sec. V.\nII. MODEL HAMILTONIAN AND ORDERED PHASES\nThe model Hamiltonian reads:\nH=Nc/summationdisplay\nl=1/summationdisplay\nα=1,2Al·(Bαl+Bα,l−1)+J(/summationdisplay\nlAl·Al+1\n+B1l·B2l+/summationdisplay\nα=1,2Bα,l·Bα,l+1), (1)\n30.34 0.36 0.38 0.4\nJ12FA(q=0)\nFA(q=π)(a)\nJc = 0.342\n0.34 0.36 0.38 0.4\nJ1234FB(q=0)\nFB(q=π)(b)\nJc= 0.342\n(c)\nFIG. 2: DMRG results for the magnetic structure factor, FX(q), atq= 0 and q=πfor (a)X= A and (b)\nX= B (B 1or B2) spins in a chain with Nc= 33; dashed lines are guides to the eye. (c) Illustration of the\nF2 phase.\nas sketched in Fig. 1(a). In Eq. (1), Al,B1landB2ldenote spin 1/2 operators at sites A l, B1l\nand B2lof the unit cell l, respectively, and Ncis the number of unit cells. For J= 0 the model\n(namedAB2chainordiagonal ladder ) is bipartite and the Lieb-Mattis (LM) theorem [22] predict s\na ground state (GS) total spin\nSg=|NA−NB|\n2=Nc\n2≡SLM, (2)\nwhereNA(NB) is the number of A (B 1and B2) sites. The GS spin pattern is represented in Fig.\n1(b). In Fig. 1(c) we report data for Sg/SLMas a function of Jusing DMRG ( Nc= 33) and ED\n(Nc= 10). Although the LM theorem is not applicable for J/negationslash= 0, the ferrimagnetic phase ( F1\nphase) is robust up to J≈0.342≡Jc1(Ref. [23]), beyond which Sgsteadily decreases ( F2 phase )\nbefore a first order transition to a phase with Sg= 0 (apart from finite size effects) at J≈0.445.\nIn order to characterize the F2 phase, we have calculated the magnetic structure factor,\nFX(q) =Nc/summationdisplay\nlCX(l)eiql, (3)\nwithq= 2πn/(Nc−1),n= 0,1,...,Nc−1, where CX(l) is the two-point correlation function\nbetween spins separated by lunit cells at sites X=A,B1andB2. We first noticed that the A\nspins remain ferromagnetically ordered as the critical poi ntJc1= 0.342 is crossed, although the\n40 0.01 0.02 0.03 0.04 0.05 0.06\n1 / (Nc+1)00.010.020.03\nm2\nAL(0)\nm2\nAL(π)\nm2\nAT(0)\nm2\nAT(π)(a)\n0 0.01 0.02 0.03 0.04 0.05 0.06\n1 / Nc00.020.030.045\nm2\nBL(0)\nm2\nBL(π)\nm2\nBT(0)\nm2\nBT(π)(b)\n1 10 100\nl10-610-510-410-310-2(-1)lCBT(l)\nDMRG (Nc=121)\na0 / l0.93(c)\nFIG. 3: DMRG results for the square of the longitudinal (L) and tra nsverse (T) order parameters at the\nspin se ctor Sz=SgandJ= 0.395 for (a) A and (b) B (B 1or B2) spins (full lines are polynomial fittings).\n(c) DMRG results for the transverse staggered correlation func tionCBTπ(l).\nmagnitude of the peak at q= 0 decreases for J > Jc1, as displayed in Fig. 2(a), while no peak\nis observed at q=π. The B i(i= 1 or 2) spins also remain ferromagnetically ordered (peak a t\nq= 0), with similar J-dependence, as shown in Fig. 2(b). However, an extra peak at q=π\ndevelops after the transition, which indicates the occurre nce of a period-2 modulation in the spin\npattern for J/greaterorsimilarJc1. Further, the average value of the correlation function /angbracketleftB1l·B2l/angbracketright, which\namounts to ≈0.25 (triplet state) in the F1 phase, steadily decreases after the transition at Jc1.\nThesefindingssuggest that theF2 phasewould display a cante d configuration, as illustrated in Fig.\n2(c). However, to check whether these features are robust in the thermodynamic limit, we have\nstudied the finite size scaling behavior of the transverse (T ) and longitudinal (L) order parameters\nin the F2 phase:\nm2\nX(L,T)(q) =FX(L,T)(q)\nNc, (4)\nforq= 0 (uniform component) and q=π(staggered component), in the subspace of maximum to-\ntal spinz-component ( Sz=Sg). The correlations are studied at J= 0.395, for which Sg=SLM/2,\nand the results are shown in Fig. 3. We confirmed that in the (ex trapolated) thermodynamic\n5limit the spins at sites A and B are ferromagnetically ordere d, as indicated by m2\nXL(q= 0)/negationslash= 0 in\nFigs. 3(a) and (b). Further, since the A and B net magnetizati ons are oppositely oriented, the F2\nphase is ferrimagnetic. The values of m2\nAL(q=π),m2\nBL(q=π) andm2\nBT(q= 0) nullifies linearly\nwith system size, which evidences short-range correlation s. On the other hand, the best fitting to\nthe data for m2\nBT(q=π) presents a nonlinear dependence with the inverse of the sys tem size and\nalso nullifies in the thermodynamic limit. This behavior ind icates that the staggered correlation\nfunction of the spins at sites B along the transverse directi on to the spontaneous magnetization,\nCBTπ(l), exhibits a power-law decay, as explicitly confirmed in Fig . 3(c). We thus conclude that\nforJc1< J <0.445 the GS is also ferrimagnetic but with critical correlati ons along the transverse\ndirection to the spontaneous magnetization ( F2 phase ).\nNext we focus on the effect of Jon the magnetic excitations. For J= 0 the Hamiltonian\nexhibits three magnon modes [15, 16]. One is AF, i. e., the spi n is raised by one unit with respect\nto the GS total spin, while the other two are ferromagnetic, a ssociated with the lowering of the\nGS total spin by one unit. The AF gapped dispersive mode is res ponsible for a quantized plateau\nin the magnetization curve as function of hand should also exhibit condensation, as suggested\nby the numerical data in Ref. [16]. One of the ferromagnetic m agnons is the gapless dispersive\nGoldstone mode, while the other is a flat mode and is the releva nt excitation for the transition\natJ=Jc1. To understand some nontrivial features of this excitation , we must comment on the\nsymmetry properties of the model. For J= 0 the Hamiltonian is invariant under the exchange\nof the B sites at the samecell. This symmetry implies that spins at these B sites can be found\nonly in singlet or triplet states (mutually exclusive possi bilities); in the GS only triplets are found.\nThe relevant magnon is a localized gapped mode which induces the formation of a singlet pair in\none cell, as illustrated in Fig. 4(a). For J/negationslash= 0, this local symmetry is explicitly broken and the\nspins at these B sites can be found in a coherent superpositio n of singlet and triplet states. In Fig.\n4(b), data using ED for the magnon band ( q= 2πn/Nc,n= 0,1,...,Nc−1) is displayed for various\nvalues of Jbefore the transition point. For J= 0 the band is flat with a gap ∆ 0≈1.0004. By\nincreasing J, the bandwidth increases and the gap to the GS lowers, closin g at the wave vector\nq=πat the transition point.\n6(a)\n00.25 0.5 0.75 1\nq / π00.51Magnon Band\nED (Nc=10)\nHCB Model(b)\nJc0.445 0.6 0.7 0.8\nJ05101520\nNS\nDMRG (Nc= 33)\nHCB Model(c)\n1 10 20 30 33 40\nl00.10.2<ηl> = 0.25 - < B1l. B2l>\n0.340\n0.345\n0.350\nHCB (NS = 2)\nHCB (NS = 4)(d)\nFIG. 4: (a) Illustration of the relevant magnon excitation for J= 0: ellipse indicates a local singlet state.\n(b) Magnon band for J= 0.00,0.05,0.10,0.15,0.20,0.25,0.30 and 0 .34, from top to bottom. (c) NSas a\nfunction of J. Full lines are the HCB model predictions in the TG limit and dashed lines a re guides to the\neye.\n7III. HARD-CORE BOSON MODEL, SUPERFLUID-INSULATOR TRANSIT ION AND\nTHE TONKS-GIRARDEAU LIMIT\nThe GS total number of singlets is given by\nNS=Nc/summationdisplay\nl=1/angbracketleftηl/angbracketright, (5)\nwith singlet density /angbracketleftηl/angbracketright=/angbracketlefts†\nlsl/angbracketright, where\ns†\nl≡1√\n2(B†\n1l,↑B†\n2l,↓−B†\n1l,↓B†\n2l,↑) (6)\nis the creation operator of a singlet pair at cell landB†\nil,σis the creation operator of an electron\nwith spin σat the B i(i= 1,2) site of cell l. In fact, it is easy to show that\n/angbracketleftηl/angbracketright=1\n4−/angbracketleftB1l·B2l/angbracketright, (7)\nsoNS= 0 forJ= 0. In Fig. 4(c) we observe that NSstarts to increase in steps of unity after\nJ=Jc1, indicating the quantized nature of the condensing singlet s.\nWe now examine the nature of the quantum critical point at J=Jc1. For this purpose we split\nthe Hamiltonian of Eq. (1) in three terms: the first favors fer rimagnetism,\nHAB=/summationdisplay\nlAl·(Sl+Sl−1), (8)\nwhereSl=B1l+B2l; the second one favors AF ordering between A spins, i. e.,\nHA=J/summationdisplay\nlAl·Al+1, (9)\nand shall play no significant role in our analysis; the last te rm, also unfavorable to ferrimagnetism,\nis a two-leg ladder Hamiltonian connecting spins at sites B1andB2(discarding a constant factor)\n[24, 25]:\nHB=J\n2/parenleftBigg/summationdisplay\nlS2\nl+/summationdisplay\nlSl·Sl+1+/summationdisplay\nlDl·Dl+1/parenrightBigg\n, (10)\nwhereDl=B1l−B2l. We represent the Hamiltonian in a basis with two states for e ach pair\nB1landB2l: the singlet and the triplet component in the magnetization direction. In addition,\nwe define the vacuum of the HCB model as the state with this trip let component in each cell.\nWe now study the GS energy when a number NSof singlet pairs is added to the vacuum. For\nJ= 0, the energy cost of a singlet pair is ∆ 0(the gap to the flat mode); thus, for NSsinglets, the\n8contribution from HABisNS∆0. The first term in HBis diagonal and will add a factor of −JNS;\nthe second causes a repulsion between singlets and adds also an extra factor of −JNS; finally,\nthe last term in HBintroduces the singlet itinerancy. Grouping these contrib utions, we arise to a\nmodel of hard-core bosons with nearest-neighbor repulsion :\nHS= (∆0−2J)NS+J\n2/summationdisplay\nlηlηl+1+J\n2/summationdisplay\nl(s†\nlsl+1+h.c.). (11)\nWeremarkthatthehard-corebosoninteraction isimpliedby thealgebraofthesingletoperators:\n[sl,s†\nl]+= 1; and (12)\n[sl,sm]−= 0 forl/negationslash=m. (13)\nBefore the transition, the single magnon dispersion relati on,\nωq(J) = ∆0−2J+Jcosq, (14)\nagrees well with the numerical data for q≈π, as can be seen in Fig. 4(b). The resulting critical\npoint:ωq=π(Jc1,S) = 0, i. e.,\nJc1,S=∆0\n3≈0.333, (15)\nis in excellent agreement with the numerical prediction Jc1= 0.342. Moreover, the closing of the\nmagnon gap is also in excellent agreement with the predictio n\n∆J=ω(q=π) = 3(Jc1,S−J) (16)\nand with the expected linear vanishing of the Mott gap [26]: zν= 1, where z= 2 and ν= 1/2 are\nthe correlation length and dynamic critical exponents, res pectively (see below).\nAfter the transition and in the highly diluted limit (NS\nNc≡η→0), the energy of NShard-core\nbosons in 1D is well approximated by the energy of NSfree spinless fermions [5]. Through this\nmap, the energy density reads:\nEGS(J) =EGS(J)\nNc=/integraldisplaykF\n−kFdk\n2π[ǫk(J)−µF] (17)\n≈3(Jc1,S−J)η+Jπ2η3\n6, (18)\nwherekF=πηand\nǫk(J)−µF=ωk+π(J)≈ −3(J−Jc1,S)+Jk2\n2. (19)\n90.36 0.38Jc\nJ-10EGS\nDMRG (Nc= 33)\nHCB Model(a)\n00.050.10.150.20.25η0.50.751\nKDMRG (Nc = 33)\nDMRG (Nc = 65)\nHCB Model\n1− 4η(b)\nFIG. 5: (a) Groundstate energy, EGS, relativeto the energyofthe LM state. (b) LuttingerLiquid expon ent,\nK, as function of η. Full lines are the HCB model predictions in the TG limit and dashed lines a re guides\nto the eye.\nNotice that the Fermi chemical potential satisfies the Tonks -Girardeau (TG) limit [7, 27, 28]\n(1D Bose gas of impenetrable particles), corresponding to a n infinitely high repulsive potential in\nthe Lieb-Liniger solution [29] of the δ-function 1D Bose gas:\nµF=ǫF(J) =π2Jη2\n2, (20)\nwhereJ−1is the fermion mass, /planckover2pi1≡1, andηis the density of singlets for J≥Jc1,Sderived from\nthe equilibrium condition ∂ηEGS(J) = 0:\nη=/radicalbig\n6(J−Jc1,S)\nπ√\nJ,η→0, (21)\nmuch in analogy with the 1D field-induced transition. Furthe r, in Fig. 4(d) we display the good\nagreement between thenumerical estimate for the density /angbracketleftηl/angbracketrightof a two (four) particle state, NS= 2\n(NS= 4), in an open system and the HCB model in a continuum space gi ven by [24]\n/angbracketleftη(l)/angbracketright=2\nNc−1NS/summationdisplay\nn=1sin2(knl), (22)\nwithkn= 1,...,πNS\nNc−1. Also, as shown in Figs. 1(c), 4(c) and 5(a), the HCB model pre dictions for\nSg\nSLM= 1−2η, (23)\nηandEGS(J), respectively, are very close to the numerical data for J/greaterorsimilarJc1≈Jc1,S.\nOn the other hand, using the Luttinger liquid description [3 0, 31] for our highly diluted HCB\n10model we have the following general relations for the sound v elocitycand the compressibility κ:\nc=πJη\nK; (24)\n1\nη2κ=πc\nK, (25)\nwhereKis the Luttinger parameter governing the decay of the correl ation functions. However,\nsince\n1\nη2κ=d2EGS\ndη2=π2Jη, (26)\nit implies that K= 1; thus c=πJη=JkF, in accord with the TG limit [7, 28]. Further, taking\nηas the order parameter of the SIT, Eq. (21) implies β= 1/2, while η2κdiverges with a critical\nexponent α=γ= 1/2, in agreement with the scaling and hyperscaling relations [26]:\nα+2β+γ= 2, (27)\n2−α=ν(d+z), (28)\nrespectively, assuring that the SIT is in the free spinless g as universality class [32].\nIn an interacting Bose gas [33], K= 1/2 is the separatrix between systems dominated by\nsuperfluid fluctuations, K >1/2, from those dominated by charge density fluctuations, K <1/2,\n(in our magnetic model spin fluctuations prevail). Affleck and collaborators [34] have succeeded in\ntaking into account corrections from interactions between pairs of dilute magnons parametrized by\na scattering length, a, thus implying that\nK= 1−2am+O(m2), (29)\nwheremis the field-induced magnetization for the S= 1 chain, with a≈ −2. The predicted\nincrease of Kwithmwas confirmed by numerical calculations [34]. This parametr ization can also\nbe implemented in our problem. In fact, in Fig. 5(b) we show th atK= 1−4η, witha≈2,\nfits quite well the data for the Luttinger liquid parameter in the highly diluted regime. Kwas\ncalculated using DMRG and assuming\nCBTπ∼a0\nl1\n2K. (30)\nIV. SPIRAL CORRELATIONS, WEAKLY COUPLED AF CHAINS, AND LADD ER-\nCHAIN DECOUPLING\nWe now turn our attention to the transition point Jt≈0.445, which marks the onset of a singlet\nphase, as can be seen in Fig. 1(c), characterized by non-quan tized values of NS, as shown in Fig.\n114(c). On the other hand, from the Hamiltonians in Eqs. (8)-(1 0), we can infer that for J >>1 the\nsystem should decompose into a linear chain (A sites) and an i sotropic two-leg ladder system (B 1\nand B2sites); see Fig. 1(b). The linear chain is known to be gapless with critical spin correlations\n(power-law decay), while the two-leg ladder is gapped with e xponentially decaying correlations. In\nwhat follows we discuss the complex phase diagram in the regi onJ > Jt.\nInitially, we display in Fig. 6 the magnetic structure facto rsFA(q) andFBi(q), withi= 1 or 2,\nas well as FS(q), which is associated to the magnetic structureof the compo site spin Sl=B1l+B2l.\nIn Fig. 6(a) we see that FA(q) peaks at q= 0 forJ= 0.44, i. e., the system remains in the F2\nphase and the A spins are ferromagnetically ordered. For J= 0.45, a sharp peak in a spiral wave-\nvectorqmaxis observed. The peak broadens and qmaxincreases with increasing J. ForJ= 0.56 we\nnotice the emergence of a commensurate AF peak, coexisting w ith the spiral one, particularly for\nJ≥0.60, as seen in Figs. 6(a) and 6(b). On the other hand, we observ e in Fig. 6(c) the presence of\ntwo peaks in FBi(q) forJ= 0.44: theq= 0 peak associated with the ferromagnetic ordering of the\nBisites in the F2 phase, and the q=πpeak related to the critical staggered transverse correlat ion\nat the same phase. Likewise, for J= 0.45, a spiral peak is observed at the same wave-vector qmax\nofFA(q). Further, notice in Fig. 6(d) that the magnitude of the AF pe ak drops in the interval\n0.96≤J <1.00.\nIn order to develop a physical meaning of the above referred d ata, we first point out that the\ncoupling between spins at A and B sites occurs through the com position Sl=B1l+B2l, as can\nbe seen in Eq. (8). Further, as the singlet component of Slis magnetically inert, only its triplet\ncomponents affect the magnetic ordering at the A sites. In fact , as shown in Figs. 6(e) and 6(f),\nshort-range spiral ordering is observed in the magnetic str ucture of Slup toJ≈1.00. However,\nsince the peak is weak and broad for J/greaterorsimilar0.6, its feature is overcomed by the AF one in the data\nof Figs. 6 [(a)-(d)]. In the sequence, we focus on the AF order ing observed for J/greaterorsimilar0.6 and study\nhow the system approaches the ladder-chain decoupling.\nIn Fig. 7(a), we present the staggered AF correlation functi on between A-spins as J→1. As\nobserved, itsbehavioriswelldescribedbythatfoundinasi nglelinearchain, whichisasymptotically\ngiven by\nC(l)∼(−1)l\nl, (31)\napartfromlogarithmiccorrections[30]. Asimilarbehavio risobservedinFig. 7(b)forthestaggered\nAF correlation between Bi-spins up to J= 0.88, a value beyond which the shape of the curve is\nvisibly changed. In order to understand this dramatic behav ior, we recall that in a two-leg ladder\n120 1 2 3 q0.70.80.91FA(q)J = 0.44\nJ = 0.45\nJ = 0.48\nJ = 0.52\nJ = 0.56\nJ = 0.60\nJ = 0.64(a)\n0 1 2 3 q12\nFA(q)J = 0.68\nJ = 0.80\nJ = 0.92\nJ = 0.96\nJ = 1.00(b)\n0 1 2 3 q0.60.811.2FBi(q)J = 0.44\nJ = 0.45\nJ = 0.48\nJ = 0.52\nJ = 0.56\nJ = 0.60\nJ = 0.64(c)\n2.6 2.8 3 q0.81.21.6\nFBi(q)J = 0.68\nJ = 0.80\nJ = 0.92\nJ = 0.96\nJ = 1.00(d)\n0 1 2 3 q00.511.52FS(q)(e)\n0.440.60 0.96\nJ00.501\n0.250.75\nqmax / π(f)\nFIG. 6: Magnetic structure factor FX(q) for the Aspins [(a) and (b)], Nc= 32, and for the Bispins\n[(c) and (d) with i= 1 or 2], Nc= 33, for the indicated values of J. (e) Magnetic structure factor for\nthe composition Sl=B1l+B2landJ= 0.44,0.45,0.48,0.52,0.56,0.60,0.64,0.68,0.80,0.92,0.96 and 1 .00,\nfrom top to bottom at q= 0. (f) The value of the wave-vector for which the peak at the mag netic structure\nfactor exhibited in (e) is observed. Dashed lines are guides to the ey e.\nsystem the asymptotic form of the correlation is given by [35 ]\nC(l)∼(−1)le−l/ξ\nl1/2, (32)\nwhereξ(≈3.2,see Ref.[36]) defines the correlation length, associated wi th the gapped spin liquid\n131 10 30\n l10-410-2(-1)lCA(l)\n~ 1 / l\nJ = 1.00\nJ = 0.96\nJ = 0.88\nJ = 0.80\nJ = 0.72\nJ = 0.64(a)\n1 10 30\n l10-410-310-210-1\n(-1)lCBi(l)\n~ 1 / l\nJ = 0.64\nJ = 0.72\nJ = 0.80\nJ = 0.88\nJ = 0.92\nJ = 0.96\nJ = 1.00(b)\n1 10 20 30\n l10-610-410-2(-1)lCBi(l)\n~ 1 / l\nLadder(c)\n1 1.2 1.4 1.61.8 2\nJ3.2681012\nξNc = 33\nNc = 65\nFitting \nLadder (d)\nFIG. 7: Staggered correlation functions between (a) A spins, Nc= 32, and (b) Bi(withi= 1 or 2)\nspins,Nc= 33, for the indicated values of J. In (a) and (b) solid lines indicate the asymptotic be-\nhavior for a single chain. (c) Staggered correlation functions betw eenBi(withi= 1 or 2) spins for\nJ= 0.88,0.92,0.96,1.0,1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9 and 2.0, circles-dash from top to bottom and\nNc= 33. For comparison we plot the behavior for a single chain (solid line) a nd for a two-leg ladder with\n32 rungs. (d) Correlation length ξas a function of J: solid line indicates the fitting of the data for Nc= 65\nto Eq. (33) with ggiven by Eq. (40); dashed line indicates the value of ξ(≈3.2) for a two-leg ladder.\nstate of this system. Indeed, as displayed in Fig. 7(c) the st aggered correlations CBiasymptotically\napproaches the correlation in a two-leg ladder system. In Fi g. 7(d) we present the behavior of ξ\nas a function of JforNc= 33 and Nc= 65. These data were obtained by a proper fitting of CBi\nin the interval l0< l <(Nc/2): starting from J= 2 and taking l0≈6 (about twice the value of\nξof a two-leg ladder), we find ξ; twice this value of ξwas used as input ( l0= 2ξ) for the next\nchosen value of J, and so on. Moreover, we have obtained a good fitting to these d ata by using the\ntwo-loop analytic form of the O(3) non-linear sigma model (N LSM) correlation length in (1+1)\ndimension [37]:\nξ=ae2π\ng/parenleftbigg\n1+2π\ng/parenrightbigg−1\n, (33)\nwhereais a constant and gis the NLSM coupling. Further, we assume (see below) that the\n14coupling gis the one suitable to the anisotropic quantum Heisenberg tw o-leg ladder to the NLSM\n[38]:\ng= 2κ/radicalBigg\n1+J⊥\n2J/bardbl, (34)\nwhereJ⊥(J/bardbl) is the exchange coupling between spins at the same rung (leg ) andκis a constant\nthat depends on the choice of the lattice regularization.\nIn order to justify Eq. (34) for g, we consider a mapping of the model Hamiltonian, Eq. (1), to\nthe Hamiltonian of an isolated two-leg ladder by eliminatin g the spin degrees of freedom associated\nwiththeA sites. Themappingis performed, inasemiclassica l manner, by thefollowing assumption\nonHAB[Eq. (8)]:\nHAB→HAB=γ/summationdisplay\nlAl·Sl, (35)\nwhereγis an effective coupling constant. This amounts to reduce the A -B coupling to spins within\nthe same unit cell, and cell-cell interactions are taken int o account through the effective coupling\nγ. We now write: ( Al+Sl)2=A2\nl+S2\nl+2Al·Sl, withS2\nl=B2\n1l+B2\n2l+2B1l·B2l; since within\na unit cell ( Al+Sl)2≈(1/2)2in an AF phase, and dropping constant terms, HABcan be written\nas\nHAB=−γ/summationdisplay\nlB1l·B2l. (36)\nSince correlations between spins at Asites does not play a significant role close to the transition ,\nwe discard the term HA, and, finally, obtain the following anisotropic two-leg ladder Hamiltonian:\nH→HaL=HAB+HB, (37)\nwhere the exchange couplings are given by\nJ⊥=J−γ, (38)\nJ/bardbl=J. (39)\nSubstituting Eqs. (38) and (39) into Eq. (34), we find the effect ive NLSM coupling:\ng=κ/radicalbigg\n6(J−Jc2)\nJ, (40)\nwhereJc2=γ/3. We have fitted the data in Fig. 7(d) to Eq. (33), with ggiven by Eq. (40), and\na,κandJc2as fitting parameters. The obtained value of a(=2.7) is such that ξ→3.1 asJ→ ∞,\n150.3420 1 2\nJ00.20.40.6η\nDMRG (Nc= 33)\nHCB Model\nFIG. 8: Density of singlets as function of J.\nwhich agrees with the expected value for an isolated isotropic two-leg ladder ( ≈3.2), while κ= 4.5\nandJc2= 0.91, in agreement with the correlation function behavior sho wn in Fig. 7(b).\nFinally, inFig. 8wedisplaytheveryinteresting behavioro fthedensity ofsinglets, η, as function\nofJ. It is clear that the effect of the A-spins and singlet-singlet interaction is relevant only for\nJt/lessorsimilarJ/lessorsimilarJc2, otherwise the solution, Eq. (21), for low density of single ts can be extended to the\nregion of low density of triplets above Jc2(strongly coupling limit), where correlations between\nB-spins are exponentially small [see Eq. (32)]. In fact, the asymptotic value predicted by Eq. (21),\ni. e.,η=√\n6/π≈0.78, compares well with the numerical one: ≈0.71.\nV. SUMMARY AND CONCLUSIONS\nJc1JtLieb−Mattis \nFerrimagnetism\nJc2+\nAntiferromagnetic\nCorrelationsSpiralFerrimagnetism\n+\nCritical\nAntiferromagnetic\nTransverse Correlations\nJ0DecouplingLadder−Chain\nFIG. 9: Schematic representation of the phase diagram.\nIn this work we have derived the rich phase diagram of a three- leg spin Hamiltonian related\nto quasi-one-dimensional ferrimagnets, as function of a fr ustration parameter Jwhich destabilizes\n16the ferrimagnetic phase. In Fig. 9 we present an illustratio n of the obtained phase diagram,\nwhich displays two critical points, Jc1≈0.342 and Jc2≈0.91, and a first order transition point\natJt≈0.445. Through DMRG, exact diagonalization and a hard-core bo son model, we have\ncharacterized the transition at Jc1as an insulator-superfluid transition of magnons (built fro m\nthe coherent superposition of singlet and triplet states be tween B sites at lattice unit cells), with\na well defined Tonks-Girardeau limit in the high diluted regi me. Ferrimagnetism with critical\nstaggered correlations in a direction transverse to the spo ntaneous magnetization is observed for\nJc1< J < J c2. Further, for Jc1< J < J tthe number of singlets in the lattice is quantized,\nwhile above the first order transition at J=Jtthis quantity is a continuous one. Also, in the\nintervalJt< J < J c2the magnetic structure factor displays a singlet phase with incommensurate\n(q/negationslash= 0 and π) spiral and AF peaks. However, the spiral peak broads and the AF peak is the salient\nfeature as Jincreases within this phase. At J=Jc2a remarkable gapped two-leg ladder / critical\nsingle-linear chain decoupling transition occurs, charac terized by an essential singularity in the\ncorrelation length as predicted by the NLSM through a mappin g of our model onto an anisotropic\nquantum Heisenberg two-leg ladder. For J≫Jc2the ladder approaches the isotropic limit (full\ndecoupling), while the linear chain remains critical.\nIn summary, our reported results clearly reveal that frustr ated quasi-one-dimensional magnets\nare quite remarkable systems to study magnon condensation, including the crossover to coupled\nladder systems of higher dimensionality [39] and related ch allenging phenomena [40], as well as\nfrustration-driven quantum decoupling transition in ladd er systems.\nVI. ACKNOWLEDGMENTS\nWe acknowledge useful discussions with A. S. F. Ten´ orio and E. P. Raposo. This work was\nsupported by CNPq, Finep, FACEPE and CAPES (Brazilian agenc ies).\n[1] M. Jaime et al., Phys.Rev. Lett 93, 087203(2004); T. 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Affleck, Phys. Rev.\nB72, 132414 (2005).\n[35] D. G. Shelton, A. A. Nersesyan, A. M. Tsvelik, Phys. Rev. B 53, 8521 (1996).\n[36] S. R. White, R. M. Noack, and D. J. Scalapino, Phys. Rev. Lett. 73, 886 (1994).\n[37] E. Br´ ezin and J. Zinn-Justin, Phys. Rev. B 14, 3110 (1976); S. H. Shenker and J. Tobochnik, Phys.\nRev. B22, 4462 (1980).\n[38] D. S´ en´ echal,Phys.Rev. B 52, 15319(1995); G. Sierra, J.Phys.A 29, 3299(1996); G. Sierra, in Strongly\nCorrelated Magnetic and Superconducting Systems , Lecture Notes in Physics Vol. 478, edited by G.\nSierra and M. A. Mart´ ın-Delgado (Springer-Verlag, Berlin, 1997) ( cond-mat/9610057); S. Dell’Aringa,\nE. Ercolessi, G. Morandi, P. Pieri, and M. Roncaglia, Phys. Rev. Lett .78, 2457 (1997).\n[39] E. Orignac, R. Citro, and T. Giamarchi, Phys. Rev. B 75, 140403(R) (2007).\n[40] S. E. Sebastian et al., Nature 441, 617 (2006); P. A. Sharma, N. Kawashima, and I. R. Fisher, Natur e\n(London) 441, 617 (2006).\n19" }, { "title": "1211.0123v1.Spin_Seebeck_effect_in_antiferromagnets_and_compensated_ferrimagnets.pdf", "content": "arXiv:1211.0123v1 [cond-mat.mtrl-sci] 1 Nov 2012Spin Seebeck effect in antiferromagnets and compensated fer rimagnets\nYuichi Ohnuma,1,∗Hiroto Adachi,2,3Eiji Saitoh,1,2,3,4and Sadamichi Maekawa2,3\n1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n2Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan\n3CREST, Japan Science and Technology Agency, Sanbancho, Tok yo 102-0075, Japan\n4WPI Research Center, Advanced Institute for Material Resea rch, Tohoku University, Sendai 980-8577, Japan\n(Dated: August 22, 2018)\nWe theoretically investigate the spin Seebeck effect (SSE) i n antiferromagnets and ferrimagnets,\nand show that the SSE vanishes in antiferromagnets but survi ves in ferrimagnets even at the magne-\ntization compensation point despite the absence of its satu ration magnetization. The non-vanishing\nSSE in ferrimagnets stems from two non-degenerate magnons. We demonstrate that the magnitude\nof the SSE in ferrimagnets is unchanged across the magnetiza tion compensation point.\nPACS numbers: 85.75.-d, 72.25.Mk, 75.30.Ds\nI. INTRODUCTION\nMuch attention is now focused on the thermal effects\nin spintronics, and the emergent research field of spin\ncaloritronics is rapidly developing.1,2One of the most\nimportant issues in spin caloritronics is the spin Seebeck\neffect (SSE).3The SSE is the mechanism by which a\nspin voltage is generated from a temperature gradient\nin a magnetic material over a macroscopic scale of sev-\neral millimeters.4Because the spin voltage is a potential\nfor electron spins to drive spin currents, this spin voltage\ninjects a pure spin current, i.e., a spin polarized current\nwhich is unaccompanied by a charge current, from the\nferromagnet into an attached nonmagnetic metal. The\ninverse spin Hall effect (ISHE)5,6converts the injected\nspin current into a transverse electric voltage and hence\nthe SSE is electrically detectable. Since its discovery in\n2008,this phenomenonhasdrawnmuchinterestasasim-\nple way of generating pure spin currents that are needed\nfor future spin-based technology,7,8and the recent obser-\nvation of the giant SSE in InSb9has attracted a consid-\nerable attention.\nThe SSE has been observed in various ferro-\nmagnetic materials ranging from metallic ferromag-\nnets, Ni 81Fe193and Co 2MnSi,10to semiconduct-\ning ferromagnet (Ga,Mn)As,11,12to insulating mag-\nnets LaY 2Fe5O1213and (Mn,Zn)Fe 2O4.14Although\nLaY2Fe5O12and (Mn,Zn)Fe 2O4are classified into fer-\nrimagnets in a rigorous terminology, the current under-\nstanding of the SSE in these systems relies on a model-\ningasferromagnets15,16becausethe low-energymagnetic\npropertiesrelevant to the SSE arewell described by a fer-\nromagnet modeling owing to the large gap between the\nacoustic and optical magnons. These observations have\nestablishedtheSSEasauniversalaspectofferromagnets.\nBesides ferromagnets, ferrimagnets and antiferromag-\nnets are known as prototypes of magnetic materials.17A\nferrimagnet is an ordered spin system in which two sub-\nlattice magnetizations point in the opposite directions,\nand an antiferromagnet is classified as a special case of\na ferrimagnet for which both sublattices have equal sat-uration magnetizations. Recently, there has been an on-\ngoing attempt to develop antiferromagnetic metal spin-\ntronics, and several experimental18and theoretical19–21\nwork are already in progress. Regarding ferrimagnets,\nthe intriguing characteristics of ferrimagnetic ordering\nare now drawing considerable attention22,23in develop-\ning a ultrafast magnetization manipulation technique.\nTherefore, it is quite natural to ask whether the SSE\ncan be observed in antiferromagnets and ferrimagnets.\nIn this paper, we address the issue of observing the\nSSE in antiferromagnets and ferrimagnets. Especially,\nwe focus on the SSE in ferrimagnets with magnetization\ncompensation. A certain class of ferrimagnets are known\nto possess a magnetization compensation temperature\nTM(angular-momentum compensation temperature TA),\nat which the two sublattice magnetizations (spins) have\nthe same magnitudes but opposite directions, leading to\nnet-zero saturation magnetization (spin angular momen-\ntum).24–28We show that two non-degenerate magnons\ngive rise to the non-vanishing SSE at TMorTAdespite\nthe absence ofnet saturationmagnetizationortotal spin.\nAlso, we show that for a uniaxial antiferromagnet the\nSSE vanishes because the thermal spin injection by the\ntwodegeneratemagnonsisperfectlycompensated. More-\nover, the SSE in an easy-plane antiferromagnet is shown\ntodisappearbecauseinthis instanceneithermagnoncar-\nries spins.\nThis paper is organized as follows. In Sec. II, we in-\nvestigate the SSE in uniaxial antiferromagnets as well as\nferrimagnets with magnetization compensation. Next, in\nSec. III we discuss the SSE in easy-plane antiferromag-\nnets. Finally, in Sec. IV we summarize and discuss our\nresults.\nII. SPIN SEEBECK EFFECT IN UNIAXIAL\nANTIFERROMAGNETS AND FERRIMAGNETS\nAs a general model of ferrimagnets and antiferromag-\nnets, we consider the following Hamiltonian defined on a2\nFIG. 1: (Color online) Schematic view of a hybrid structure\ncomposed of a nonmagnetic metal ( N) and a ferrimagnet ( F)\nwith two sublattices AandB.\nlattice composed of two sublattices AandB,29\nHF=−JA/summationdisplay\n/angbracketlefti,i′/angbracketright∈ASA,i·SA,i′−JB/summationdisplay\n/angbracketleftj,j′/angbracketright∈BSB,j·SB,j′\n+JAB/summationdisplay\n/angbracketlefti∈A,j∈B/angbracketrightSA,i·SB,j+δHA+δHB,(1)\nwhereJAandJB(JAB) are the nearest-neighbor intra-\nsublattice (inter-sublattice) exchange integrals, and /angbracketleft,/angbracketright\nspecifies nearest neighbor bonding (see Fig. 1). The last\ntwo terms in Eq. (1) for sublattice L=A,Bare given\nbyδHL=/summationtext\ni∈L[gLµ0H0·SL,i−DL\n2(/hatwidez·SL,i)2], where\nµ0is the Bohr magneton, H0=−H0/hatwidezis the external\nmagnetic field, gLandDLare the effective g-factor and\nthe anisotropy constant for sublattice L.\nFirst, we use the spin-waveapproximationto diagonal-\nize Eq. (1). Following standard procedures30using the\nlinear Holstein-Primakoff transformation for spin opera-\ntorsS±\nL,i=Sx\nL,i±iSy\nL,i(L=A,B), the Hamiltonian (1)\nis diagonalized to be\nHF=/planckover2pi1/summationdisplay\nq/parenleftBig\nω+\nqα†\nqαq+ω−\nqβ†\nqβq/parenrightBig\n, (2)\nwhereω±\nq=1\n2/radicalBig\n(ǫAq+ǫBq)2−4η2q±(ǫA\nq−ǫB\nq), and the\nprecise forms of ǫA\nq,ǫB\nq, andηqare given by ǫA\nq=\n2z0JASA[1−γq] +z0JABSB+ (gAµ0H0+DASA) and\nǫB\nq= 2z0JBSB[1−γq]+z0JABSA+(−gBµ0H0+DBSB).\nHere,γq=z−1\n0/summationtext\nδeiq·δis defined by the sum over\nz0nearest neighbors of the original lattice, and ηq=\nJAB√SASB/summationtext\nδ′eiq·δ′is defined by the sum over z0near-\nest neighbors of the sublattice AorB. In this paper,\nwe assume a cubic lattice for simplicity. In Eq. (2),\nthe magnon operators αqandβqare defined by the\nBogoliubov transformation31aq=u+\nqαq+u−\nqβ†\nqand\nbq=u−\nqα†\nq+u+\nqβq, where and aqandbqare the\nFourier transforms of operators ai= (2SA)−1\n2S+\nA,iand\nbi= (2SB)−1\n2S−\nB,iwithSA=|SA|andSB=|SB|, and\nu+\nq2−u−\nq2= 1.\nFIG. 2: (Color online) Spin-wave spectra ( H0= 0) with q\nalong the [111] direction calculated from Eq. (2) using pa-\nrameters for (a) a uniaxial antiferromagnet NiO, and (b) a\ncompensated ferrimagnet Er 3Fe5O12. The wavevector qis\nmeasured in units of the inverse of the nearest-neighbor dis -\ntance.\nIn Fig. 2, the spin-wave spectra ( H0= 0) calculated\nfrom Eq. (2) for a uniaxial antiferromagnet NiO and a\ncompensated ferrimagnet Er 3Fe5O12are plotted. For\nNiO, we use JAB= 6.3 meV ( JA=JB= 0),D=\n0.1 meV,SA=SB= 0.92,28,32whereas for Er 3Fe5O12,\nwe assign the net spin of the rare-earth ions (the fer-\nric ions) to SA(SB) on a model cubic lattice, and we\nsetJA= 0 meV, JB= 0.68 meV, JAB= 0.19 meV,\nSA= 4.2,SB= 2.5,gA= 1.4,gB= 2.0DA= 3.5×\n10−3meV, and DB= 3.0×10−4meV to reproduce the\nN´ eel temperature TN´ eel= 556 K and the magnetization-\ncompensation temperature TM= 83 K.28,33As is well\nknown, the two antiferromagnetic magnons are degener-\nate ifH0= 0, whereas the two ferrimagnetic magnons\nare non-degenerate because of the inequivalence of the\ntwo sublattices.\nWe discuss now the SSE in uniaxial antiferromagnets\nand ferrimagnets modeled by Eq. (1). Note that a uniax-\nial antiferromagnet can be modeled as a special case of\na ferrimagnet. We consider a model shown in Fig. 1, in\nwhich a ferrimagnet ( F) and a nonmagnetic metal ( N)\nare interacting weakly through the s-dexchange interac-\ntion at the interface. We assume that the ferrimagnet F\nhas a local temperature TF, and the nonmagnetic metal\nNhas a local temperature TN. We analyze the SSE in\nthe longitudinal configuration34by employing the linear-\nresponse formulation of the SSE in a ferromagnet de-\nveloped in Refs. 15 and 35. The s-dinteraction at the\ninterface is modeled by\nHsd=/summationdisplay\ni,j∈F/N-interface/parenleftBig\nJA\nsdσi·SA,i+JB\nsdσj·SB,j/parenrightBig\n,(3)\nwhere, for sublattice L=A,B,JL\nsdis thes-dexchange\ninteraction at the F/Ninterface, σiis the itinerant spin\ndensity operator in N. The total Hamiltonian of the\nsystem,H, is then given by\nH=HF+HN+Hsd, (4)3\nFIG. 3: (Color online) Feynman diagram representing two\nprocesses relevanttotheSSEinuniaxialantiferromagnets and\nferrimagnets. (a) Spin current injected by αqmagnons ( I+\ns).\n(b) Spin current injected by βqmagnons ( I−\ns). The signs of\nI+\nsandI−\nsare opposite. The solid and wavy lines represent\nmagnon and itinerant spin-density propagators, respectiv ely.\nwhereHNis the single-particle Hamiltonian of the con-\nduction electrons in N(see, e.g., Eq. (31) in Ref. 36).\nThe central quantity that characterizes the SSE is the\nspin current Isinjected into N, because it is proportional\nto the experimentally detectable electric field EISHEvia\nISHE:5,6\nEISHE=θSHρJs×σ, (5)\nwhereθSHandρare respectively the spin-Hall angle and\nthe resistivity of N, andJs= (Is/Aint)/hatwidexis the spin-\ncurrent density across the F/Ninterface having a con-\ntact area Aint. Following Refs. 15 and 35, we calculate\nIsas the rate of change of the spin accumulation in N,\ni.e.,Is=/summationtext\ni∈N/angbracketleft∂tσz\ni/angbracketrightwhere/angbracketleft···/angbracketrightdenotes the statisti-\ncal average. What is special in the present calculation\nis that we need to express the s-dinteraction [Eq. (3)]\nin terms of the αqandβqoperators [“ ±” branches in\nEq. (2)], because these are the magnon operators in F.\nFollowing procedures presented in Appendix A, the spin\ncurrent injected in Nis expressed as\nIs=−2√\n2√NNNF/planckover2pi1Re/summationdisplay\nk,q/integraldisplay\nω/bracketleftBig\nJ+\nsd(k,q)AK\nk,q(ω)\n+J−\nsd(k,q)BK\nk,q(ω)/bracketrightBig\n, (6)\nwhereAK(BK) is the Keldysh component of the inter-\nface correlation function between magnon operator αq\n(β†\nq) and the itinerant spin-density operator σ−\nk(see Ap-\npendix A), and we have introduced the shorthand no-\ntation/integraltext\nω=/integraltext∞\n−∞dω\n2π. HereJ±\nsd(k,q) is the effective s-d\ninteraction written in terms of magnon operators, and\nthe precise definition is given in the Appendix A.\nWe perform the perturbative approach in term of the\ns-dinteraction at the interfaceto evaluate Eq. (6). Then,\nthe spin current Isis given by the two diagrams shown\nin Fig. 3, and accordingly, Ishas two terms:\nIs=I+\ns+I−\ns, (7)\nFIG. 4: (Color online) Temperature dependence of the SSE\nsignalIs[red, Eq. (7)], saturation magnetization Ms[blue,\nEq. (9)], and total angular momentum Stot[green, Eq. (10)],\ncalculated for a compensated ferrimagnet Er 3Fe5O12using\nthesameparametersasinFig.2(b). Thecase fora Mspinned\nby the anisotropy field is shown; the data is normalized by its\nvalue at T/TN´ eel= 0.1. Inset: The case for a Mspinned by\nthe external magnetic field is shown.\nwhereI±\ns, representing the contribution from the ±\nbranch, is expressed by\nI±\ns=±/summationdisplay\nk,q8Nint[[|J±\nsd(k,q)|2]]\nNNNF/planckover2pi12/integraldisplay\nωImχR(k,ω)\n×ImGR\n±(q,ω)/bracketleftBig\ncoth(/planckover2pi1ω\n2kBTN)−coth(/planckover2pi1ω\n2kBTF)/bracketrightBig\n.(8)\nHereNintis the number of localized spins at the F/N\ninterface, NN(NF) is the number of lattice sites in N\n(sublattice sites in F), [[|J±\nsd(k,q)|2]] =SA(JA\nsdu±\nq)2+\nSB(JB\nsdu∓\nq)2. In Eq. (8), χR(k,ω) =χN/(1 +λ2\nNk2−\niωτsf) where χN,λN,τsfare respectively the param-\nagnetic susceptibility, the spin-diffusion length, and the\nspin-flip relaxation time in N, andGR\n±(q,ω) = 1/(ω−\nω±\nq+ iα±ω) where α±is the damping parameter in F.\nNote that the signs of the spin current injected by the\nαqmagnons ( I+\ns) and that by the βqmagnons ( I−\ns) are\nopposite.\nWe first consider the SSE in a uniaxial antiferromag-\nnet. As is depicted in Fig. 2 (a), the two magnons in a\nuniaxialantiferromagnetaredegenerateif H0= 0. More-\nover,owingtothe equivalenceofsublattices AandB, the\ns-dexchange interactions at the interface for these two\nsublattices are the same ( JA\nsd=JB\nsd). From these condi-\ntions we obtain |I+\ns|=|I−\ns|resulting in a null SSE due\nto Eq. (7), i.e., Is= 0. Thus, the SSE vanishes in a\nuniaxial antiferromagnet under a negligibly small exter-\nnal magnetic field because of the perfect compensation\nof the spin injection by the two degenerate magnons.\nWe next consider the SSE in a ferrimagnet close to\nthe magnetization compensation point, in which the two4\nmagnons are no longer degenerate. Figure 4 shows the\ntemperature dependence of the SSE signal Is(T) calcu-\nlated from Eqs. (7) and (8) for a compensated ferrimag-\nnet Er 3Fe5O12by using the same parameters as in Fig. 2\n(b). In Fig. 4 we also plot the saturation magnetization\nMs=µ0/parenleftBiggA\nNF/summationdisplay\ni∈A/angbracketleftSz\nA,i/angbracketright+gB\nNF/summationdisplay\nj∈B/angbracketleftSz\nB,j/angbracketright/parenrightBig\n(9)\nto determine the magnetization compensation point de-\nfined by Ms(TM) = 0. In addition, we plot the total\nangular momentum\nStot=/angbracketleftSz/angbracketright (10)\ntodeterminetheangular-momentumcompensationpoint\ndefined by Stot(TA) = 0. Here, Szis thez-component of\nthe total spin S=SA+SB, i.e.,\nSz=1\nNF/summationdisplay\ni∈ASz\nA,i+1\nNF/summationdisplay\nj∈BSz\nB,j.(11)\nClearly we see that the SSE signal is unchanged across\nboth compensation points, either TM≈0.15TN´ eelor\nTA≈0.32TN´ eel. We performed the same calculation\nfor several different choices of parameters, and confirmed\nthat the SSE is unchanged across TMandTA.\nIII. SPIN SEEBECK EFFECT IN EASY-PLANE\nANTIFERROMAGNETS\nIn this section, we show that the SSE in easy-plane an-\ntiferromagnets vanishes under a zero magnetic field be-\ncause neither of magnons carries spins in easy-plane an-\ntiferromagnets. We consider the following Hamiltonian\nfor easy-plane antiferromagnets:37\nHeAF=J/summationdisplay\n/angbracketlefti∈A,j∈B/angbracketrightSA\ni·SB\nj\n+/summationdisplay\nL=A,B/summationdisplay\ni∈L[gµ0H0·SL\ni−D\n2(/hatwidez·SL\ni)2] (12)\nwhereJisthenearest-neighborexchangeintegrals, H0=\nH0/hatwidexis the external magnetic field, and gis the g-factor\nandD <0 is the anisotropyconstant which selects the x-\nyplaneasaneasyplane. Notethattheexternalmagnetic\nfield is applied along the xaxis, and we assume SA/bardbl/hatwidez\nandSB/bardbl −/hatwidezwhenH0= 0. Following Ref. 37, we in-\ntroduce the linear Holstein-Primakoff transformation by\nchoosing the direction of each canted sublattice spin in\nthe ground state as a spin quantizing axis. Performing a\nπ/4 rotation to the operators to separate the mixing of\nthe two spin operators and using the Bogoliubov trans-\nformation, Eq. (12) is diagonalized to be\nHeAF=/planckover2pi1/summationdisplay\nq/parenleftBig\nε+\nqξ†\nqξq+ε−\nqζ†\nqζq/parenrightBig\n,(13)whereε±\nq=/radicalBig\n(A±q+2B±q)(A±q−2B±q),A±\nq=\n2z0JScos2θ+gµ0H0sinθ+|D|S∓z0JS(cos2θ−1)γq,\nB±\nq=∓z0JS(cos2θ−1)γq− |D|S/2, and γq=\nz−1\n0/summationtext\nδeiq·δis defined by the sum over z0nearest neigh-\nbors. In the above equation, θis the canted angle of\nthe sublattice magnetization, and the magnon operators\nξqandζqare defined by the Bogoliubov transformation\n1√\n2/parenleftbig\naq+b−q/parenrightbig\n=uqξq+vqξ†\n−qand1√\n2/parenleftbig\naq−b−q/parenrightbig\n=\nxqζq+yqζ†\n−q, whereu2\nq−v2\nq= 1 and x2\nq−y2\nq= 1 are\nrealcoefficients, and aqandbqarethe Fouriertransforms\nof Holstein-Primakoff operators aiandbi. As is seen in\nFig. 3 of Ref. 37, the two magnons in the easy-plane an-\ntiferromagnet are not degenerate even when H0= 0.\nNow we discuss the SSE in an easy-plane antiferro-\nmagnet modeled by Eq. (12) by using the same proce-\ndure as in the previous section. As before, we consider\na system in which an easy-plane antiferromagnet ( eAF)\nhaving local temperature TeAFand a nonmagnetic metal\n(N) having local temperature TNare interacting weakly\nthrough the s-dexchange interaction at the interface. In\nthe absence of an external magnetic field, a direct calcu-\nlation shows that the spin current Isinjected into Nis\nidentically zero, i.e.,\nIs= 0. (14)\nThis is understood by investigating the z-component of\nthe total spin S=SA+SB[Eq. (11)]. In the case\nof a uniaxial antiferromagnet discussed in the previous\nsection, the expectation value of Szis given by\n/angbracketleftSz/angbracketright=/parenleftBig\nSA−1\nNF/summationdisplay\nq/angbracketleftα†\nqαq/angbracketright/parenrightBig\n−/parenleftBig\nSB−1\nNF/summationdisplay\nq/angbracketleftβ†\nqβq/angbracketright/parenrightBig\n,\n(15)\nwhereαqandβqare the magnon operators defined in\nSec. II. Equation (15) means that the αqmagnons car-\nries spin one while βqmagnon carries spin minus one in\na uniaxial antiferromagnet. On the other hand, the ex-\npectation value of Szin an easy-plane antiferromagnet\nunder discussion is calculated to be identically zero, i.e.,\n/angbracketleftSz/angbracketright= 0. (16)\nEquation (16) means that magnons in an easy-plane an-\ntiferromagnet are similar to a linearly-polarized photon\nand hence neither of magnons carries spins if H0= 0.\nIn the presence of a finite external magnetic field\nH0=H0/hatwidexwith a nonzero canted angle θ, however, the\nx-component of Sbecomes nonzero. In this situation,\nthe spin current Isinjected into Nis shown to have the\npolarization along the xaxis, and its magnitude is given\nby\nIs=gµ0H0\nJz0/parenleftbig\nI+\ns+I−\ns/parenrightbig\n, (17)5\nwhere\nI±\ns=4J2\nsdNint\nNNNF/planckover2pi12/summationdisplay\nk,q/integraldisplay\nωImχR(k,ω)ImFR\n±(q,ω)\n×/bracketleftBig\ncoth(/planckover2pi1ω\n2kBTN)−coth(/planckover2pi1ω\n2kBTeAF)/bracketrightBig\n.(18)\nHere,FR\n±(q,ω) = 1/(ω−ε±\nq+iα±ω) is the retarded com-\nponent of the magnon propagator with α±is the damp-\ning parameter. Note that the signs of I+\nsandI−\nsare the\nsame, in contrast to the case of a uniaxial antiferromag-\nnet.\nFrom Eqs. (14) and (17), we conclude that the SSE\nvanishes in an easy-plane antiferromagnet if H0= 0.\nIV. DISCUSSION AND CONCLUSION\nThe main result of this paper is that the SSE in an-\ntiferromagnets vanishes, whereas the SSE in ferrimag-\nnets persists and is insensitive to either magnetization or\nangular-momentum compensation effects. The interpre-\ntationis asfollows. Forthe SSEtooccur, the existenceof\nthe transverse fluctuations of the total spin, i.e., Sx\ntotand\nSy\ntot, is needed. For a ferrimagnet at TMorTA, fluctua-\ntionsofSx,ydonot vanishevenwhen Stot= 0orMs= 0.\nTherefore, ferrimagnetic magnons can always generate\nthe SSE. Only for a uniaxial antiferromagnet, where the\ntwo magnons are degenerate, the SSE from the two de-\ngenerate magnons with the opposite sense compensates\nperfectly. Note that neither magnon in an easy-plane\nantiferromagnet carries spins.\nOur conclusion is not modified by considering the\nphonon-drag contribution to the SSE38because, as dis-\ncussed in Refs. 39 and 40, phonon drag can be taken\ninto account by replacing TFandTNin Eq. (8) with an\neffective magnon temperature T∗\nFand effective spin accu-\nmulationtemperature T∗\nN. We alsonotethat themagnon\nexcitations are well defined even at TA. The presumed\ndivergence of the magnon damping parameter at TA41\ndoes not exist when we recall the condition justifying the\nexpansion used in Ref. 42, where such an effect mani-\nfests itself as an enhancement of the damping parameter\nwithout any divergence (see Appendix B). Note that the\nmagnitude of magnon damping has less effect on the lon-\ngitudinal SSE, although it has a large influence on the\ntransverse SSE.\nTo conclude, we have theoretically investigated the\nSSE in antiferromagnets and ferrimagnets, and shown\nthat the SSE vanishes in antiferromagnets whereas it\npersists at either the magnetization or the angular-\nmomentum compensation points of ferrimagnets, despite\nthe absence of its saturation magnetization or total spin.\nBecause a fringing field by saturation magnetization is\nsuppressed at the magnetization compensation point,\nthis phenomenon can be useful for constructing a pure\nspin current device which is free from crosstalk of the\nfringing field.Acknowledgments\nWe are grateful to K. Uchida. This work is was finan-\ncially supported by a Grant-in-Aid from MEXT, Japan,\nand a Fundamental Research Grants from CREST-JST,\nPRESTO-JST, Japan.\nAppendix A: Linear-response expression of\nmagnon-driven spin injection in ferrimagnets\nIn this Appendix, we derive Eq. (6) in the main text.\nWe consider a system described by the Hamiltonian (4),\nand calculate the spin current Is=/summationtext\ni∈N/angbracketleft∂tσz\ni/angbracketright. We\nuse the momentum representation of σz\niand calculate\nthe quantity Is=√NN/angbracketleft∂tσz\nk0/angbracketrightk0→0. The Heisenberg\nequation of motion for σz\nk0gives\n∂tσz\nk0=i\n/planckover2pi1/summationdisplay\nk,q/bracketleftBig√8SAJA\nsd(k,q)√NFNN(u+\nqα−\nq+u−\nqβ+\nq)σ−\nk\n+√8SBJB\nsd(k,q)√NFNN(u+\nqβ+\nq+u−\nqα−\nq)σ−\nk/bracketrightBig\n+h.c.,(A1)\nwhere σ±\nk=1\n2(σx\nk±iσy\nk),JL\nsd(k,q) =/summationtext\ni∈N/F(L)JL\nsdei(k−q)·rifor sublattice L=A,B,\nand we have used the relation [ σz\nk,σ±\nk′] =±2√NNσ±\nk+k′.\nTaking the statistical average of the above quantity, the\nspin current is calculated to be\nIs(t) =−4√\n2√NNNF/planckover2pi1Re/summationdisplay\nk,q/bracketleftBig\nJ+\nsd(k,q)A<\nk,q(t,t)\n+J−\nsd(k,q)B<\nk,q(t,t)/bracketrightBig\n, (A2)\nwhereJ±\nsd(k,q) =JA\nsd(k,q)√SAu±\nq+JB\nsd(k,q)√SBu±\nq.\nHere,A<\nk,q(t,t′) =−i/angbracketleftαq(t′)σ−\nk(t)/angbracketrightandB<\nk,q(t,t′) =\n−i/angbracketleftβ†\nq(t′)σ−\nk(t)/angbracketrightmeasure the interface correlation func-\ntions between the magnon operators ( αqandβq) and\nspin-density operator σ−\nk. In the steady state the inter-\nface correlations A<\nk,q(t,t′) andB<\nk,q(t,t′) depends only\non the time difference t−t′. Introducing the frequency\nrepresentation A<\nk,q(t,t′) =/integraltext∞\n−∞dω\n2πA<\nk,q(ω)e−iω(t−t′)as\nwell as using the relationship A<=1\n2[AK−AR+AA],\nwe finally obtain Eq. (6) in the main text.\nAppendix B: Magnon damping near compensation\npoints\nIn this Appendix, we calculate temperature depen-\ndence of magnon damping parameter close to the com-\npensation points and show that the magnon excitation\nis well defined even at compensation points without any\ndivergences. We begin with two Landau Lifshitz Gilbert\nequations for sublattice L=A,B:41\ndML\ndt=−γLML×HL+αL\nMs,LML×dML\ndt,(B1)6\nFIG. 5: (Color online) Temperature dependence of the\neffective magnon damping parameter αeff[red, Eqs. (B8)\nand (B9)], saturation magnetization Ms[blue, Eq. (9)], and\ntotal angular momentum Stot[green, Eq. (10)], calculated for\na compensated ferrimagnet Gd 23Fe74.6Co3.4. The data is nor-\nmalized by its value at T/TN´ eel= 0.1.\nwhereMLis the sublattice magnetization with its mag-\nnitude given by ML,HLis the effective magnetic field,\nγL=gLµ0//planckover2pi1is the gyromagnetic ratio, and αLis the\nGilbertdampingparameter. Theeffectivefieldsaregiven\nbyHA=H0+Han\nA−λMBandHB=H0+Han\nB−λMA,\nwhereH0=H0/hatwidezisexternalmagneticfield, Han\nA=Han\nA/hatwidez\nandHan\nB=−Han\nB/hatwidezare the anisotropy fields, and λMA\nandλMBare the inter-sublattice exchange fields with\nλ=z0JAB/(gAgBµ2\n0). Because we here focus on the\nuniform mode, the intra-sublattice exchange couplings\nλA=z0JA/(gAµ0)2andλB=z0JB/(gBµ0)2are dis-\ncarded in Eq. (B1).\nBelow the magnetization compensation point we set\nMA=MA/hatwidez+mAandMB=−MB/hatwidez+mB, such that\nthe effective fields can be written as\nHA= (H0+Han\nA+λMB)/hatwidez−λmB,(B2)\nHB= (H0−Han\nB−λMA)/hatwidez−λmA.(B3)\nIntroducing the representation Eeff\nA=−(H0+Han\nA+\nλMB) andEeff\nB=−(H0−Han\nB−λMA), and linearizing\nwithrespectto mAandmB, theLandau-Lifshitz-Gilbert\nequations are transformed to be\ndmA\ndt=/hatwidez×/bracketleftBig\nγA(λMAmB−Eeff\nAmA)+αAdmA\ndt/bracketrightBig\n,\n(B4)\ndmB\ndt=−/hatwidez×/bracketleftBig\nγB(λMBmA+Eeff\nBmB)+αBdmB\ndt/bracketrightBig\n.\n(B5)We introduce m±=mx±imyand substitute m+\nL(t) =\nm+\nLe−iωtinto Eqs. (B4) and (B4). Then we obtain\n(−iω−αAω+iγAEeff\nA)m+\nA−iλγAMAm+\nB= 0,\n(B6)\n(−iω+αBω+iγBEeff\nB)m+\nB+iλγBMBm+\nA= 0.\n(B7)\nThe eigenfrequency ωis determined by the equation:\n(ω−iαAω−γAEeff\nA)(ω+iαBω−γBEeff\nB)\n+λ2γAγBMAMB= 0. (B8)\nAbove the magnetization compensation temperature,\nwe setMA=−MA/hatwidez+mAandMB=MB/hatwidez+mB\nbecause we consider a situation in which the satura-\ntion magnetization is pinned by an external magnetic\nfield. This situation can be analyzed by rewriting Eeff\nA=\n−(H0−Han\nA−λMB) andEeff\nB=−(H0+Han\nA+λMA) as\nwell as reversingthe signs of αAandαB. We numerically\nsolve Eq. (B8) by setting\nω=ω0+iαeffω0. (B9)\nFigure 5 shows the temperature dependence of the\neffective Gilbert damping parameter αeffthe lower fre-\nquency mode, calculated for a compensated ferrimagnet\nGd22Fe70Co8.41,43,44We assign Asublattice for Gd ions\nandBsublattice for Fe ions, and neglect Co ions for\nsimplicity. We set SA= 3.85,SB= 3.5,gA= 2.0,\ngB= 2.05,H0= 0.04 T,Han\nA= 0.0 T,Han\nB= 0.02 T,\nαA= 0.004,αB= 0.0039. The saturation magnetization\nand the total spin are calculated by using the mean field\napproximation by using JAB= 0.28 meV, JA= 0 meV,\nandJB= 0.34 meV to reproduece TN´ eel= 500 K. 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B 73,\n220402(R) (2006).\n42See a paragraph containing Eq. (22) in R. K. Wangsness,\nPhys. Rev. 91, 1085 (1953).\n43R. C. Taylor and A. Gangulee, J. Appl. Phys, 48, 358\n(1977).\n44P. Hansen, C. Clausen, G. Much, M. Rosenkranz, and K.\nWitter, J. Appl. Phys, 66, 756 (1989)." }, { "title": "2312.00367v2.Compensated_Ferrimagnets_with_Colossal_Spin_Splitting_in_Organic_Compounds.pdf", "content": "Compensated Ferrimagnets with Colossal Spin Splitting in Organic Compounds\nTaiki Kawamura1, Kazuyoshi Yoshimi2, Kenichiro Hashimoto3, Akito Kobayashi1, and Takahiro Misawa2∗\n1Department of Physics, Nagoya University, Nagoya, Aichi 464-8602, Japan\n2Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan\n3Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan\nThe study of the magnetic order has recently been invigorated by the discovery of exotic collinear\nantiferromagnets with time-reversal symmetry breaking. Examples include altermagnetism and\ncompensated ferrimagnets, which show spin splittings of the electronic band structures even at zero\nnet magnetization, leading to several unique transport phenomena, notably spin-current generation.\nAltermagnets demonstrate anisotropic spin splitting, such as d-wave, in momentum space, whereas\ncompensated ferrimagnets exhibit isotropic spin splitting. However, methods to realize compensated\nferrimagnets are limited. Here, we demonstrate a method to realize a fully compensated ferrimagnet\nwith isotropic spin splitting utilizing the dimer structures inherent in organic compounds. Moreover,\nbased on ab initio calculations, we find that this ferrimagnet can be realized in the recently discovered\norganic compound (EDO-TTF-I) 2ClO 4. Our findings provide an unprecedented strategy for using\nthe dimer degrees of freedom in organic compounds to realize fully compensated ferrimagnets with\ncolossal spin splitting.\nIntroduction.— Collinear antiferromagnets have tradi-\ntionally been viewed as conventional magnetic orderings\nthat lack unique phenomena such as spin current gener-\nation. However, recent theoretical advances have identi-\nfied exotic collinear antiferromagnets with time-reversal\nbreaking, notably altermagnets [1–3] and compensated\nferrimagnets [4]. These magnetic states exhibit spin\nsplitting in their electronic band structures even with-\nout net magnetization. Because spin splitting may drive\nspin-dependent novel transport phenomena and uncon-\nventional superconducting phases, these distinctive anti-\nferromagnets have attracted considerable attention.\nSeveral materials have been proposed as candidates\nfor altermagnetism [3, 5–13]. For example, κ-ET-\ntype organic compounds [6] and transition metal oxide\nRuO 2[8, 10, 14] exhibit altermagnetism with anisotropic\nspin splitting in electronic band structures. These mate-\nrials are expected to exhibit spin-dependent transport\nand anomalous Hall effects owing to anisotropic spin\nsplitting. In contrast, compensated ferrimagnets offer\nisotropic spin splitting, increasing efficiency for spin-\ncurrent generation. The concept of metallic compen-\nsated ferrimagnetism (half-metallic antiferromagnetism)\nwas introduced by van Leuken and de Groot [15]. Since\nthen, several candidate materials have been suggested us-\ningab initio calculations [16, 17]. More recently, mono-\nlayer MnF 2was proposed as an insulating compensated\nferrimagnet [18]. Since the magnetic moment of compen-\nsated insulating ferrimagnets is strictly zero due to the\nLuttinger theorem [4], small perturbations do not change\nthe compensation condition. In addition, compensated\nferrimagnets have lower crystal symmetry than altermag-\nnets [4]. Therefore, compensated ferrimagnets have ad-\nvantages over altermagnets, leading to various potential\napplications, such as thin-film synthesis. However, the\nnumber of compensated ferrimagnets discovered experi-\nmentally is limited [19–22]. To harness the compensated\n(a)\nA\nA’B\nB’t1s\nt2\n 0 0.5 1up spin\ndown spin\nk/2π(b)\n-3-2-1 0 1 2 3 4\ns s\ns s sFIG. 1. (a) Schematic of a one-dimensional model showing\nthe compensated ferrimagnetism. The broken lines show a\nunit cell. The inter-dimer hopping integral s, the intra-dimer\nhopping integrals t1andt2are represented by the horizontal\nthin black lines, the vertical thick red line, and the vertical\nthin blue line, respectively. The up (down) spin polariza-\ntions are described by the yellow upward (purple downward)\narrows. (b) Band structures of the Hamiltonian defined in\nEq. (2). We take t1= 1.0,t2= 0.6,s= 0.6,δ= 0.2, and\n∆ = 1 .5. For comparison, we also show the band structures\nof the equivalent dimer case with the broken curves, whose\nparameters are given by t1=t2= 1.0,s= 0.6,δ= 0.0, and\n∆ = 1 .5.\nferrimagnets, a simple method to realize them is neces-\nsary.\nIn this Letter, we present a path towards fully com-\npensated ferrimagnets with colossal spin splitting us-\ning typical dimer structures in organic compounds. Us-\ning a simple one-dimensional model, we demonstrate\nthat a collinear antiferromagnetic order with inequiva-\nlent dimers can induce fully compensated ferrimagnets.\nFurthermore, we find that the recently discovered or-\nganic compound (EDO-TTF-I) 2ClO 4[EDO-TTF-I=4,5-\nethylenedioxy-4′-iodotetrathiafulvalene] [23] can realize\nthis mechanism based on ab initio calculations. The ex-\nperiments demonstrate that (EDO-TTF-I) 2ClO 4under-arXiv:2312.00367v2 [cond-mat.mtrl-sci] 21 Mar 20242\ngoes a structural transition with anionic ordering at 190\nK. Below the structural transition, unit-cell doubling oc-\ncurs with the extended unit cell containing two inequiv-\nalent dimers. By deriving and solving the ab initio effec-\ntive Hamiltonian for the low-temperature phase of (EDO-\nTTF-I) 2ClO 4, we find that the ground state is a collinear\nantiferromagnet with isotropic spin splitting, i.e., a fully\ncompensated ferrimagnet.\nSimple model.— First, to show the key idea for real-\nizing the compensated ferrimagnets, we consider a one-\ndimensional model whose unit cell contains two inequiv-\nalent dimers. A schematic of the model is illustrated in\nFig. 1 (a). The inequivalence between the two dimers\nis characterized using difference between the intra-dimer\nhoppings t1andt2and chemical potential difference δ.\nWe also consider the collinear dimer antiferromagnetic\n(DAF) state, where the up-(down-)spin electrons are lo-\ncated on A and A′(B and B′). This DAF state is not in-\nvariant to any combination of time reversal with transla-\ntion/rotation operations because of dimer inequivalence.\nThus, isotropic spin splitting is expected in this dimer\ncollinear DAF state.\nTo determine the mechanism of spin splitting in this\nmodel, we consider the following tight-binding Hamilto-\nnian for the DAF state:\nH=X\nk,σc†\nkσHσ(k)ckσ, (1)\nHσ(k) =\nσ∆ t1 A(k) 0\nt1 σ∆ 0 A(k)\nA(k)∗0 −σ∆ +δ t 2\n0 A(k)∗t2 −σ∆ +δ\n,\n(2)\nwhere c†\nkσ= (c†\nAkσ, c†\nA′kσ, c†\nBkσ, c†\nB′kσ),A(k) = s(1 +\ne−ik) and ∆ denotes the gap induced by the DAF or-\nder. The spin index σtakes +1 and −1 for the up- and\ndown-spins, respectively. The eigenvalues of this Hamil-\ntonian are given by\nE0,σ,±(k) =δ/2+t+±[(σ∆ +t−−δ/2)2+ 2s2C(k)]1/2,\n(3)\nE1,σ,±(k) =δ/2−t+±[(σ∆−t−−δ/2)2+ 2s2C(k)]1/2,\n(4)\nwhere t±= (t1±t2)/2 and C(k) = (1+cos k). Thus, spin\nsplitting of the bands is induced by t−= (t1−t2)/2 and δ,\ni.e., inequivalence of the dimers. From these expressions,\nit is evident that the differences in intra-dimer hoppings\nhave the similar gap-opening effect as the differences in\nthe chemical potentials. Because t−andδare indepen-\ndent of the wave number, spin splitting is isotropic.\nFigure 1(b) shows the electronic band structures of\nthe Hamiltonian defined in Eq. (2) for a typical param-\neter set. We also plot the band structures when the two\ndimers are equivalent ( t1=t2andδ= 0). As expected,\nFIG. 2. (a) Crystal structure of (EDO-TTF-I) 2ClO 4atT=\n100 K and the real-space distribution of maximally localized\nWannier functions (MLWFs) drawn by VESTA [24]. Anions\nClO 4are represented by balls and rods. Colors of anions rep-\nresent their orientations. Due to the different configurations\nof anions around the dimers, two dimers (A-A′and B-B′) be-\ncome inequivalent. Black lines show the unit cell including\nfour molecules. (b) Energy band structure obtained by the\nDFT calculation (red lines) and the MLWFs (blue circles) for\nthe paramagnetic states. The Fermi energy is set to zero.\nisotropic spin splitting occurs in the band structures. At\ncommensurate filling (i.e., three-quarter, half, and quar-\nter filling), the DAF state is insulating, and the net mag-\nnetization is zero because the number of up- and down-\nbands below the Fermi energy are the same. Thus, the\nDAF state at the commensurate filling is the fully com-\npensated ferrimagnets with isotropic spin splitting.\nCrystal and electronic structures of (EDO-TTF-\nI)2ClO 4.— First, we summarize the crystal structure\nof (EDO-TTF-I) 2ClO 4. (EDO-TTF-I) 2ClO 4consists of\nEDO-TTF-I molecule with +1 /2 charge (3 /4 filling) and\nanion ClO 4layers. Above 190 K, the unit cell contains\ntwo EDO-TTF-I molecules and space inversion symme-\ntry is macroscopically protected because of the random\norientation of ClO 4. By lowering the temperature, the\nstructural phase transition with anion ordering occurs at\napproximately T= 190 K, which induces unit-cell dou-\nbling, as evidenced by the X-ray analysis. As shown in\nFig. 2 (a), each unit cell contains four molecules in the\nlow-temperature phase. A and A′(B and B′) molecules\nin a unit cell form a dimer, referred to as dimer I (dimer\nII). The inversion centers are at the centers of each\ndimer. The inequivalence of the dimers I and II can\nalso be understood from the partial density of states\n(PDOS), which are shown in the Supplemental Mate-\nrial [25]. Because dimers I and II are inequivalent af-\nter the structural phase transition, (EDO-TTF-I) 2ClO 4\nwill exhibit compensated ferrimagnetism if appropriate\nantiferromagnetic order occurs. We note the related or-\nganic compound (EDO-TTF-I) 2PF6[26] has similar crys-\ntal structures; however, all the dimers are equivalent.3\nThis indicates that the ordering of ClO 4plays an essen-\ntial role in realizing inequivalent dimers.\nNext, we summarize the low-temperature electronic\nstructure of (EDO-TTF-I) 2ClO 4. Although the resis-\ntivity exhibits semimetallic behavior immediately after\nthe structural phase transition, a metal-insulator phase\ntransition occurs at approximately T= 95 K [23]. It\nis confirmed that the metal-insulator transition does not\naccompany a structural change. As explained later, ab\ninitio analysis suggests that the DAF ordering is the ori-\ngin of the insulating phase although the origin of the\nmetal-insulator transition is not experimentally clarified\nyet.\nAb initio effective Hamiltonian for (EDO-TTF-\nI)2ClO 4.—To investigate the electronic structures of the\nlow-temperature phase in (EDO-TTF-I) 2ClO 4, we derive\ntheab initio low-energy effective Hamiltonian, based on\nband structures obtained by the density functional the-\nory (DFT). We use Quantum ESPRESSO [27] to ob-\ntain the DFT bands. In this study, we employ the opti-\nmized norm-conserving Vanderbilt pseudopotentials and\nplane-wave basis sets [28, 29]. The exchange-correlation\nfunctional used in this study is the generalized gradient\napproximation proposed by Perdew, Burke, and Ernzer-\nhof [30]. The cut-off energy of the wave functions and\ncharge densities are set to be 80 Ry and 320 Ry, respec-\ntively. During the self-consistent loop process, a 5 ×5×3\nuniform k-point grid and Methfessel-Paxton smearing\nmethod are used [31]. We use the crystal structure data\nat 100 K [23]. We perform structural optimization for\nhydrogen atoms and use the optimized structure in the\nfollowing analyses. In the DFT calculations, we only con-\nsider the paramagnetic solutions. Figure 2 (b) shows the\nobtained energy band structures of (EDO-TTF-I) 2ClO 4.\nThe four bands around the Fermi level, which are isolated\nfrom the other bands, mainly consist of the highest occu-\npied molecular orbitals of the four EDO-TTF-I molecules\nin the unit cell (A,A′,B, and B′). We select these four\nbands as the low-energy degrees of freedom and use them\nto construct the maximally localized Wannier functions\n(MLWFs) using RESPACK [32]. Isosurfaces of the ML-\nWFs are shown in Fig.2 (a). In addition, we confirm\nthat the bands interpolated by the MLWFs accurately\nreproduce the DFT band structures (Fig.2 (b)).\nAfter constructing MLWFs, we derive low-energy ef-\nfective Hamiltonian, which is given by\nHEDO=H0+Hint,\nH0=X\ni,j,α,β,σtiαjβc†\niασcjβσ,\nHint=X\ni,αUiαniα↑niα↓+1\n2X\ni,j,α,βViαjβNiαNjβ,(5)\nwhere c†\niασ(ciασ) is a creation (annihilation) operator\nfor an electron in the i-th unit cell with orbitals α= A,A′, B, B′and spin σ. The number operators are de-\nfined as niασ=c†\niασciασandNiα=niα↑+niα↓. The\ntransfer integrals tiαjβare evaluated using the MLWFs.\nWe also evaluate the screened Coulomb interactions Uiα\nandViαjβusing the constrained random phase approxi-\nmation [33, 34] implemented in RESPACK [32]. We set\nthe cutoff energy of the polarization function to 5 .0 Ry.\nDetails of the transfer integrals and interaction parame-\nters are summarized in the Supplemental Material [25].\nWe note that the difference in the intra-dimer hoppings\nt1andt2and the existence of the site potential δindi-\ncate the inequivalence of the dimers. In actual calcula-\ntions, we subtract a constant value ∆ DDFfrom onsite and\noffsite Coulomb interactions to consider the interlayer\nscreening [35, 36]. Following previous studies [36–39],\nwe employ ∆ DDF= 0.2 eV. We confirm that the results\nare insensitive to ∆ DDF. We also perform an electron-\nhole transformation to reduce the numerical cost.\nmany-variable variational Monte Carlo (mVMC)\nanalysis.— The effective Hamiltonian is solved using the\nmVMC method [40, 41], which can take into account\nquantum fluctuations and spatial correlations seriously.\nThe trial wave function used in this study is given by\n|ψ⟩=PGPJLS|ϕpair⟩, (6)\nPG= exp\"X\nigini↑ni↓#\n, (7)\nPJ= exp\n1\n2X\ni̸=jvijNiNj\n, (8)\n|ϕpair⟩=\nNsiteX\ni,jfijc†\ni↑c†\nj↓\nNe/2\n|0⟩, (9)\nwhere PG,PJ, and LSare the Gutzwiller factor [42],\nlong-range Jastrow factor [43, 44], and total spin projec-\ntor [45, 46], respectively [41, 47]. NeandNsiteindicate\nthe number of electrons and sites, respectively. We im-\npose a 1 ×4 sublattice structure on variational parame-\nters. We use the Hartree-Fock approximation results as\nthe initial fijvalues. The spin-singlet projection ( S= 0)\nis used in ground-state calculations for La=Lb≤8,\nwhile it is not used for La=Lb≥10 to reduce the nu-\nmerical costs. We confirm that the spin projection does\nnot largely affect physical quantities, such as the spin\nstructure factors. All variational parameters are simul-\ntaneously optimized using the stochastic reconfiguration\nmethod [48].\nAs detailed in the Supplementary Materials, the\nHartree-Fock approximation shows that the DAF state\n(Fig.3 (a)) and antiferromagnetic states with charge or-\ndering (AF+CO) are ground state candidates. Within\nthe Hartree-Fock approximation, the ground state of the\neffective Hamiltonian is the AF+CO state. However, us-\ning the mVMC method, we find that the DAF state be-4\nFIG. 3. (a) Schematic of the DAF state and representative\nhopping integrals ( t1-t6) in the conducting layer of (EDO-\nTTF-I) 2ClO 4. (b) Spin structure factor obtained by the\nmVMC method for La=Lb= 12. Sharp peaks appear at\nq= (0, π/2),(0,3π/2), which correspond to the DAF state.\n(c) Size dependence of the peak value of the spin structure\nfactors. (d) Doping dependence of the chemical potential.\ncomes the ground state of the effective Hamiltonian. We\nalso find that the AF+CO state converges to the DAF\nstate after optimization, even when using the variational\nparameters of AF+CO state as the initial state. This\nresult indicates that correlation effects beyond the mean-\nfield approximation are important in stabilizing the DAF\nstate.\nTo examine the existence of long-range antiferromag-\nnetic order, we calculate the spin-correlation functions of\nthe ground state, defined as\nS(q) =1\n(Nsite)2X\ni,j⟨Si·Sj⟩eiq·(ri−rj), (10)\nwhere the original lattice structure is mapped to the\nequivalent La×4Lbsquare lattice for simplicity. Fig-\nure 3 (b) shows S(q) in the momentum space, with sharp\nBragg peaks at ( qx, qy)=(0, π/2) and (0 ,3π/2). As shown\nin Fig. 3 (c), we confirm that the peak values of S(q) re-\nmain finite in the thermodynamic limit. We also confirm\nthat charge densities are uniform and there is no signa-\nture of a charge-ordered state. These results show that\nthe ground state of the effective Hamiltonian is the DAF\nstate. In addition, we calculate the charge gap ∆ c, given\nby ∆ c=µ(Ne+ 1)−µ(Ne−1), where the chemical po-\ntential is defined as µ(Ne+ 1) = [ E(Ne+ 2)−E(Ne)]/2.\nFigure 3 (d) shows the doping rate γd=Ne/Nsite−1.5\ndependence of the chemical potential. From this plot, we\nestimate the charge gap [∆ c=µ(Ne+ 1)−µ(Ne−1)] as\n∆c∼0.4 eV. This result indicates that the ground state\nFIG. 4. (a) Density of state (DOS) of the low-energy effective\nHamiltonian (EDO-TTF-I) 2ClO 4for the DAF state obtained\nby the Hartree-Fock approximation. DOS for up (down) spin\nis described by the red (blue) lines. (b) Band dispersions of\nthe DAF state obtained by the Hartree-Fock approximation.\nThe red (blue) surfaces describe up-spin (down-spin) band\ndispersions.\nof (EDO-TTF-I) 2ClO 4is the DAF insulator.\nSpin splitting.— Based on the results obtained using\nthe mVMC method, we analyze spin splitting in the DAF\nstate using the Hartree-Fock approximation (for more de-\ntails, see Ref. [25]). We assume the DAF order and scale\nthe interaction parameters to reproduce the charge gap\n∆c∼0.4 eV obtained from the mVMC calculations. The\nscaling ratio λ, which monotonically scales the onsite and\noffsite Coulomb interactions, is estimated to be λ= 0.7.\nThe details of the Hartree-Fock calculations are provided\nin the Supplemental Material.\nUsing the Hartree-Fock approximation, we calculate\nthe DOS defined as Dσ(ω)=(πLaLb)−1P\nk,nIm(ω−iη−\nEk,n,σ+µ)−1, where µandηare the chemical poten-\ntial and the smearing factor, respectively. Ek,n,σdenotes\nthen-th eigenvalue of the mean-field Hamiltonian at mo-\nmentum k. We set η= 0.002 eV. As shown in Fig. 4 (a),\nspin splitting occurs in the DAF order ( D↑(ω)̸=D↓(ω)).\nThe electronic band dispersions in Fig. 4(b) also show\nisotropic spin splitting over the entire Brillouin zone.\nThis demonstrates that (EDO-TTF-I) 2ClO 4can be fully\ncompensated ferrimagnets if a DAF order occurs.\nHere, we analyze the origin of spin splitting in (EDO-\nTTF-I) 2ClO 4. As in the case of the simple model,\nt−= (t1−t2)/2 and δcan induce spin splitting. From\ntheab initio calculations, we find that t−= 0.036 eV\nis comparable to δ= 0.047 eV. Thus, spin splitting in\n(EDO-TTF-I) 2ClO 4is induced by both t−andδ. One\nmight think that finite δwould make the total magneti-\nzation finite; however, a charge gap guarantees that the\ntotal magnetization is robust against perturbations. In\nthis case, the total magnetization is zero at δ= 0 and\nremains zero even if δis added, provided that δis signif-\nicantly smaller than the charge gap.\nSummary and discussion.— In this Letter, we present\na simple method to realize fully compensated ferrimag-\nnetism using the dimer degrees of freedom, which are5\ntypical of organic compounds. Using a simple model,\nwe demonstrate that the inequivalence of the two dimers\nand DAF order can induce fully compensated ferrimag-\nnets at commensurate filling. Furthermore, ab initio cal-\nculations suggest that the ground state of (EDO-TTF-\nI)2ClO 4is a DAF insulator with inequivalent dimers. As\na result, the DAF order induces isotropic spin splitting\nin the electronic band structure. Our study shows that\nthe key to realizing compensated ferrimagnetism lies in\ninequivalent dimer structures induced by anion ordering.\nThis finding offers an unanticipated direction in materi-\nals design, where exotic magnetism can be achieved by\nselecting and modifying anions to exhibit anion order-\ning. The discovery of compensated ferrimagnetism in\ninequivalent dimer structures, as well as the potential for\nmaterials design using anion ordering, demonstrates that\norganic compounds offer a versatile platform for realiz-\ning exotic magnetism. An intriguing future issue would\nbe to examine the doping effects in the DAF insulating\nstate. Because the lowest unoccupied band is fully po-\nlarized and its DOS is large (Fig.4 (a)), unconventional\nsuperconductivity, such as triplet superconductivity, is\nexpected [49]. Further experimental and theoretical in-\nvestigations in this direction are desirable.\nThe authors thank Y. Nakano, M. Ishikawa, H.\nYamochi, and A. Otsuka for the fruitful discussions. This\nwork was financially supported by Grants-in-Aid for Sci-\nentific Research (KAKENHI) (Grant Nos. 23H03818,\n23KJ1065, 15K05166, 22K18683 21H01793), Grant-in-\nAid for Scientific Research for Transformative Research\nAreas (A) “Condensed Conjugation” (No. JP20H05869)\nfrom Japan Society for the Promotion of Science (JSPS),\nand JST SPRING (Grant No. JPMJSP2125). The com-\nputations were performed using the facilities at the Su-\npercomputer Center, Institute for Solid State Physics,\nUniversity of Tokyo.6\nSupplemental Material for “Compen-\nsated Ferrimagnets with Colossal Spin\nSplitting in Organic Compounds”\nPARAMETERS IN AB INITIO LOW-ENERGY\nEFFECTIVE HAMILTONIANS\nWe calculate the electronic band structures by\nthe density functional theory (DFT) using Quantum\nESPRESSO [27] and evaluate the transfer integrals\nby the maximally localized Wannier functions (ML-\nWFs). Then, we evaluate Coulomb interaction by the\nconstraint random phase approximation (cRPA) using\nRESPACK [32]. The values of the transfer integrals\nlarger than 0 .020 eV and the Coulomb interactions are\nshown in Table I. We show schematic illustrations of the\ntransfer integrals and the Coulomb interactions in Fig. 5.\nAll input and output files are uploaded to the ISSP data\nrepository [50].\nTransfer integrals [eV] Coulomb interactions [eV]\nδ 0.047 UA=UA′ 2.094\nUB=UB′ 2.076\nt1 0.252 V1 0.903\nt2 0.179 V2 0.884\nt3 0.128 V3 0.880\nt4 0.112 V4 0.700\nt5 0.084 V5 0.681\nt6 0.058 V6 0.614\nt7 - V7 0.659\nt8 - V8 0.628\nTABLE I. Transfer integrals and screened Coulomb interac-\ntions of the ab initio low-energy effective Hamiltonians for\n(EDO-TTF-I) 2ClO 4.\nFIG. 5. Schematic of the transfer integrals and the off-site\nCoulomb interactions in the conduction layer.7\nPARTIAL DENSITIES OF STATES\nTo see the inequivalence of the dimers, we calculate\nthe partial densities of states (PDOS) using the one-body\npart of the low-energy effective Hamiltonians for (EDO-\nTTF-I) 2ClO 4in the momentum space, which is given by\nH0\nσ(k) =X\nkX\n∆r,α,βt∆r,αβeik·∆rc†\nk,α,σck,β,σ\n=X\nkX\n∆r,α,βH0\nαβ,σ(k)c†\nk,α,σck,β,σ.(11)\nHere, t∆r,αβis the transfer integrals obtained by\nRESPACK and ∆ rrepresents the translational vector.\nThe Hamiltonian H0\nσ(k) satisfies the following eigenvalue\nequation:\nH0\nσ(k)|k, n, σ⟩=En,σ(k)|k, n, σ⟩, (12)\n|k, n, σ⟩=\ndA,n,σ(k)\ndA′,n,σ(k)\ndB,n,σ(k)\ndB′,n,σ(k)\n, (13)\nwhere En,σ(k) and|k, n, σ⟩are the eigenvalue and eigen-\nvectors of H0. The band and spin indices are represented\nbynandσ, respectively. Using En,σ(k) and dα,n,σ(k),\nthe PDOS Dα(ω) can be calculated as\nDα(ω) =1\nπLaLbX\nk,n,σIm|dα,n,σ(k)|2\nω−iη−Ek,n,σ+µ,(14)\nwhere ηtakes the positive infinitesimal value and µis the\nchemical potential determined to set the electron num-\nber to 6. We set η= 0.002 eV in the numerical calcu-\nlation. Figure 6 shows the PDOS Dα(ω). We consider\nthe paramagnetic state in this calculation. The inequiva-\nlence in DA(ω) and DB(ω) indicates the inequivalence of\nthe dimer I and II. We note that the relations DA(ω) =\nDA′(ω) and DB(ω) =DB′(ω) are satisfied due to space-\ninversion symmetry.\nFIG. 6. PDOS obtained by the tight-binding model. Inequiv-\nalence of the dimers appears in the PDOS ( DA(ω) ̸=DB(ω)).8\nDETAILS OF THE HARTREE-FOCK\nAPPROXIMATION\nTo examine the ground state candidates of (EDO-\nTTF-I) 2ClO 4, we perform the Hartree-Fock (HF) ap-\nproximation to the ab initio low-energy effective Hamil-\ntonians defined in Eq. (5) in the main text. We use the\nunrestricted HF code implemented in mVMC [41]. Us-\ning the results of HF calculations, we generate the initial\nvariational parameters for the mVMC method. In this\nstudy, we examine the ordered states with q= 0, i.e.,\nwe consider the symmetry-broken states within the unit\ncell. As illustrated in Fig. 7, we consider five initial\nstates: (i) the paramagnetic (PM) state, (ii) the ferro-\nmagnetic (FM) state, (iii) the dimer antiferromagnetic\n(DAF) state, (iv) the AF state, (v) the AF state with\ncharge ordering (AF+CO). To examine the correlation ef-\nfects, we introduce the parameter λ, which monotonically\nscales the Coulomb interactions as ˜Uiα≡λ(Uiα−∆DDF)\nand˜Viαjβ≡λ(Viαjβ−∆DDF). Here, ∆ DDFdenotes the\nconstant shift that takes into account the screening ef-\nfects between conduction layers [35, 36].\nFigures 8 (a) and (b) show the charge density\nnC\ni\u000b\n=\n⟨ni↑+ni↓⟩and the spin density\nnS\ni\u000b\n=⟨ni↑−ni↓⟩at the\nith (=A, A′, B, B′) site in the ground states obtained\nby the HF approximation for ∆ DDF = 0.2 eV, respec-\ntively. We find that the PM state [ ⟨nC\nA(B)⟩=⟨nC\nA′(B′)⟩\nand⟨nS\ni⟩= 0] is the ground state below λ∼0.3. In\n0.3≲λ≲0.5, DAF state [ ⟨nC\nA(B)⟩=⟨nC\nA′(B′)⟩and\n⟨nS\nA⟩=⟨nS\nA′⟩=−⟨nS\nB⟩=−⟨nS\nB′⟩] becomes the ground\nstate. For λ≳0.5, the AF state with the CO [ ⟨nC\nA(B′)⟩>\n⟨nC\nA′(B)⟩and⟨nS\nA′(B′)⟩>⟨nS\nA(B)⟩] becomes the ground\nstate. Figure 8 (c) shows the λ-∆DDF phase diagram.\nSince amplitudes of the on-site and off-site Coulomb in-\nteractions become small with increasing ∆ DDF, the phase\nboundary between the DAF state and the AF+CO state\nslightly shifts to large λregion by increasing ∆ DDF.9\nFIG. 7. Schematic illustrations of initial states used in the HF approximation. The up and down arrows indicate the spin-up\nand spin-down states, respectively. The orange and blue ellipses represent the dimer states constructed by A-A′molecules and\nB-B′molecules, respectively. The green circles represent the charge-rich sites.10\nFIG. 8. λdependence of (a) the charge density and (b) the spin density. (c) λ-∆DDF phase diagram obtained by the HF\napproximation.11\nMVMC ANALYSIS OF THE GROUND-STATE\nPHASE DIAGRAM\nBased on the results obtained by the HF calcula-\ntions, we investigate the ground states using the mVMC\nmethod, which can treat correlation effects more ac-\ncurately. Following previous studies [36–39], we take\n∆DDF= 0.20 eV. Figure 9(a) shows the phase diagram\nas a function of λ. By increasing λ, the phase transi-\ntion between the DAF state and the PM state occurs\naround λ= 0.5. Above λ∼2.2, the AF+CO state be-\ncomes the ground state due to the off-site Coulomb inter-\nactions. Figure 9(b) shows the energy difference between\nthe PM state (the AF+CO state) and the DAF state,\ni.e., ∆ E1=EPM−EDAF(∆E2=EAF+CO −EDAF) as\na function of λforLa=Lb= 6 lattice. The AF+CO\nstate is a quasi-stable state for λ≳1.4 and becomes the\nground state for λ≳2.2. We note that the AF+CO state\nis not stabilized even when we select the AF+CO state\nas an initial state for λ≲1.2. Figure 10 shows the spin\nand charge density structure factors of the PM, the DAF,\nand the AF+CO states. The charge structure factor is\ndefined by\nN(q) =1\n(Nsite)2X\ni,j\n(Ni−¯N)·(Nj−¯N)\u000b\neiq·(ri−rj),\n(15)\n¯N=1\nNsiteX\ni⟨Ni⟩. (16)\nThe ordering wave vectors q= (0, π/2),(0,3π/2) in S(q)\n(Fig. 10 (b)) correspond to the DAF state. Meanwhile,\nthe ordering wave vectors q= (0, π/2),(0,3π/2) in S(q)\nandq= (0, π) inN(q) (Fig. 10(c) and (f)) correspond\nto the AF+CO state.12\nFIG. 9. (a) The ground state phase diagram as a function of λobtained by the mVMC calculation. (b) λ-dependencies of the\nenergy difference between the PM state and the DAF state (∆ E1) and the one between the AF+CO state and the DAF state\n(∆E2).13\nFIG. 10. Spin structure factors S(q) [charge structure factors N(q) ] of the PM state ( λ= 0.3), the DAF state ( λ= 1), and\nthe AF+CO state ( λ= 2.4) in (a), (b), and (c) [(d), (e), and (f)], respectively.14\nMEAN-FILED HAMILTONIAN IN THE\nMOMENTUM SPACE\nTo see the spin splitting of the DAF states, we calcu-\nlate band dispersions and DOS using the one-body Green\nfunctions obtained by the HF approximation. Here, we\nsetλ= 0.7, which reproduces the charge gap estimated\nby the mVMC method. By performing the Fourier trans-\nformation for the mean-field Hamiltonian in the real\nspace, we obtain the mean-field Hamiltonian in the mo-\nmentum space, which is given by\nH=X\nkX\n∆r,α,β,σt∆r,αβeik·∆rc†\nk,α,σck,β,σ\n+X\nkX\nα,σUα⟨nα,¯σ⟩c†\nk,α,σck,α,σ\n+X\nkX\n∆r,α,β,σV∆r,αβ⟨Nβ⟩c†\nk,α,σck,α,σ\n−X\nkX\n∆r,α,β,σV∆r,αβD\nc†\nr0+∆r,β,σcr0,α,σE\neik·∆rc†\nk,α,σck,β,σ.\n(17)\nHere, ∆ rdenotes the translational vector and ¯ σ=−σ.\nOff-diagonal one-body Green functions in the real space\nare represented by ⟨c†\nr0+∆r,β,σcr0,α,σ⟩. We take r0=0as\nthe representative coordinate of r0because r0is an ar-\nbitrary coordinate due to translational symmetry. Using\nthe mean-field Hamiltonians in the momentum space, we\ncalculate the band dispersions and the DOS. All values of\nthe one-body Green functions are uploaded to the ISSP\ndata repository [50].\n∗tmisawa@issp.u-tokyo.ac.jp\n[1] L. ˇSmejkal, A. H. MacDonald, J. Sinova, S. Nakatsuji,\nand T. 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Mazin, Notes on altermagnetism and superconduc-\ntivity, arXiv:2203.05000 .\n[50] https://isspns-gitlab.issp.u-tokyo.ac.jp/\nk-yoshimi/edottf ." }, { "title": "1403.0656v1.Off_Resonant_Manipulation_of_Spins_in_Diamond_via_Precessing_Magnetization_of_a_Proximal_Ferromagnet.pdf", "content": "O\u000b-Resonant Manipulation of Spins in Diamond via Precessing Magnetization of a\nProximal Ferromagnet\nC. S. Wolfe*,1V. P. Bhallamudi*,1H. L. Wang,1C. H. Du,1S.\nManuilov,1A. J. Berger,1R. Adur,1F. Y. Yang,1and P. C. Hammel1,\u0003\n1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA\n(Dated: September 29, 2018)\nWe report the manipulation of nitrogen vacancy (NV) spins in diamond when nearby ferrimagnetic\ninsulator, yttrium iron garnet, is driven into precession. The change in NV spin polarization, as\nmeasured by changes in photoluminescence, is comparable in magnitude to that from conventional\noptically detected magnetic resonance, but relies on a distinct mechanism as it occurs at a microwave\nfrequency far removed from the magnetic resonance frequency of the NV spin. This observation\npresents a new approach to transferring ferromagnetic spin information into a paramagnet and\nthen transducing the response into a robust optical signal. It also opens new avenues for studying\nferromagnetism and spin transport at the nanoscale.\nPACS numbers: 72.25.Mk, 75.76.+j, 75.78.-n, 75.78.-n, 75.78.-n\nKeywords: NV, magnetization dynamics, FMR, spin transport, YIG\nUnderstanding the transport of spin and energy be-\ntween dissimilar materials is a topic of intense current\ninterest re\recting both the scienti\fc richness of the topic\nas well as it technological potential[1{6]. Metal/metal in-\nterfaces have been extensively studied, and to a lesser de-\ngree metal/semiconductor and metal/insulator systems.\nHowever, the transfer of angular momentum between two\ninsulating materials has been more challenging to study\ndue to the lack of suitable detection methods.\nNV centers in wide band-gap insulating diamond pro-\nvide an exceptional platform for performing spin-based\nmeasurements. The paramagnetic NV center is optically\nactive, and its photoluminescence (PL) is dependent on\nthe relative occupation of the lowest lying electronic spin\nstate of the defect center [7, 8]. This enables optical mea-\nsurement of NV-center spin state with excellent sensitiv-\nity, making optically detected magnetic resonance of NV\ncenters an area of intense research activity [9{21]. How-\never, work done thus far relies on manipulation of NV\nspins using magnetic resonance.\nHere we present experimental evidence that the NV\ncenter state can be modi\fed non-resonantly ( i.e.,by ir-\nradiation with microwave magnetic \felds at frequencies\nfar from NV center Larmor frequencies) by coupling to\nthe dynamics of a proximal ferromagnetic insulator. This\nchange can be detected as a change in the NV center\nPL, as is done for conventional microwave-driven reso-\nnant spin manipulations. This new e\u000bect promises valu-\nable insights into interactions between spins in adjacent\ndissimilar materials. This is a particular case in which\nboth materials are insulating and the interaction is ef-\nfective over long distances ( >300 nm). More generally,\n* C. Wolfe and V. Bhallamudi contributed equally to this work.\nFIG. 1. Experimental Schematic : The sample is a 20 nm\nthick single crystal YIG \flm with nanodiamonds dispersed on\ntop with a thickness of about 500 nm. To apply microwave\n\felds to the sample a silver microwire is patterned on the YIG.\nGreen laser light is focused onto nanodiamonds near the wire,\nand the intensity of the resulting photoluminescence from the\nNV centers is measured. Inset is an SEM image of dispersed\nnanodiamonds.\nthis provides a novel method for manipulating NV center\nspins and could enable sensitive spatially resolved imag-\ning of ferromagnetic phenomena by means atomic scale\nNV centers [9, 12, 15].\nYttrium Iron Garnet (YIG), Y 3Fe5O12, was chosen for\nthis experiment as a well known ferrimagnetic insulator\nwith exceptionally low damping [22, 23]. Here an epitax-\nial YIG \flm, 20nm thick, was grown on a gadolinium\ngallium garnet (GGG) (111) substrate by o\u000b-axis sput-\ntering [24{26]. Continuous wave microwave \felds are ap-\nplied to the sample by means of a 300 nm thick and 30 \u0016m\nwide silver microstrip line patterned on top of the YIG.arXiv:1403.0656v1 [cond-mat.mes-hall] 4 Mar 20142\nNanodiamonds, 50-200 nm in size (as shown by SEM im-\nage analysis) and containing up to a few thousand NV\ncenters each, were dispersed on top of a lithographically\nde\fned microstrip line as shown in Fig. 1. AFM mea-\nsurement indicates that the nanodiamonds form a 500\nnm thick \flm. Photoluminescence is excited in the NV\ncenters using a 532 nm laser beam and is collected by a\nphotodiode. A lock-in measurement is performed on the\nphotodiode signal by modulating the amplitude of the\napplied microwave \feld.\nThe lock-in measurement of the resulting modulation\nof the PL intensity is presented in Fig. 2 as a function\nof an applied in-plane magnetic \feld and the applied mi-\ncrowave frequency. Data for a control sample with nan-\nodiamonds on a GGG substrate without YIG is shown\nin Fig. 2 (a). We observe the intrinsic and well-known\nmagnetic resonances of the NV center ground and excited\nstates, starting at 2.87 GHz and 1.43 GHz respectively.\nShown in Fig. 2 (b) is the same data overlaid with the\ntheoretically expected resonance conditions for the NV\ncenters, which are obtained by solving the NV hamilto-\nnian in the presence of magnetic \feld parallel and per-\npendicular to the NV axis (see supplementary informa-\ntion of [9]). We also see several features in the PL below\n1.25 GHz (Fig. 2 (a)) that are not related to the normal\nground- and excited-state resonances of the NV centers.\nThese will be discussed in a forthcoming publication.\nThe data obtained from the nanodiamonds on top of\nthe YIG \flm is shown in Fig. 2 (c). The key di\u000berence be-\ntween Fig. 2 (a) and (c) is the feature in (c) that extends\nup from the lower left-hand corner. This is highlighted in\nFig. 2 (d) where the data from (c) is overlaid with a solid\nblue curve showing the YIG ferromagnetic resonance con-\ndition. The blue dots show the YIG resonance condition\nas measured by re\rected microwave power. These data\nare \ft (blue curve) using the equation for the uniform\nFMR mode in a thin \flm [27], and we obtain a magneti-\nzation,\u00160Ms, of 183 mT (see supplementary information\nfor more information). As can be seen, the intensity of\nthe PL from the NV centers strongly changes precisely\nwhen the YIG FMR is excited.\nWe have considered the e\u000bect of heating, caused by the\nFMR absorption in YIG, on the PL of NV centers. Sev-\neral control experiments and estimates of the possible\ne\u000bect render this potential explanation highly unlikely.\nMore details can be found in the supplementary infor-\nmation.\nThere are two key points to note about the FMR-\ninduced feature in the PL signal. First, it is seen at fre-\nquencies and \felds well separated from the NV center's\nown resonance conditions. This is in clear contrast to\nthe quantum computing and magnetometry techniques\nbeing developed, where the spin-state of NV centers is\ncoherently manipulated by microwave \felds meeting the\nmagnetic resonance conditions [9{20]. Instead, we see a\nchange in the PL that correlates to the excitation of the\n3020100\nMagnetic Field (mT)\n3020100\nMagnetic Field (mT)4.0\n3.5\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5Microwave Frequency (GHz)\n4.0\n3.5\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5Microwave Frequency (GHz)\n(a) (b)\n(c) (d) NV || H \n NV /s94 H \n YIG data\n Fit\nOn YIGOn GGG\n1.00.50.0\nChange in NV PL (a.u.)FIG. 2. NV PL Data while YIG undergoes FMR : (a)\nRaw data showing the change in the intensity of the PL from\nthe NV centers in nanodiamonds dispersed on top of GGG,\nwith no YIG, as a function of microwave frequency and mag-\nnetic \feld. (b) The same data overlaid with theoretical reso-\nnance conditions for the NV centers. The black dashed lines\nshow the resonance condition for an NV center with the mag-\nnetic \feld parallel to the NV axis for the ground and excited\nstates. The grey crosses show the resonance condition for an\nNV center with the magnetic \feld perpendicular to the NV\naxis. (c) Similar data from nanodiamonds dispersed on the\nYIG sample with the distinct feature corresponding to the\nYIG FMR condition. (d) The data from (c) with the FMR\npeaks measured using re\rected microwave power (blue dots).\nAlso shown is a \ft (blue line) to the calculated dispersion re-\nlation for YIG \flm FMR with the magnetic \feld in plane (see\nmain text).3\nFIG. 3. NV center PL Data on YIG and on Wire\nAbove YIG : (a) Data taken with the laser spot focused on\ntop of the \u0018300 nm thick patterned microwire. The data\nshows that the signal corresponding to the YIG FMR is still\npresent, though reduced. The coupling must extend at least\n300 nm through the wire. (b) Data from the nanodiamonds\ndirectly on top of the YIG for comparison. (c) Line cuts of\nthe data in (a) and (b) at 16.5 mT. The FMR-induced peak is\nreduced by 3 times relative to the NV excited state peak for\nthe case of nanodiamonds on top of the microstrip, compared\nto the the case directly on YIG.\nYIG magnetization into precession by means of ferro-\nmagnetic resonance. It is remarkable that excitations at\nenergies as much as three to six times smaller than any\nNV center resonance have such a large e\u000bect on the NV\ncenter spin state. Second, the FMR-induced feature is\ncomparable in amplitude to the intrinsic NV resonances.\nThe large amplitude of the signal implies that a signif-\nicant number of NV centers in our laser spot must be\ncontributing to the signal. This suggests that since the\nnanodiamond \flm is 500 nm thick, the coupling must\nbe either long range (extending hundreds of nanometers)\nor that spin transport by means of spin di\u000busion plays a\nrole [28].\nTo probe the spatial extent of the coupling, we re-\npeated the measurement with our laser spot focused on\nthe nanodiamonds on top of the microwire, where the\nnanodiamonds are separated from the YIG by more than300 nm. This data can be seen in Fig. 3 (a) and compared\nto the signal when the diamond is directly on top of the\nYIG in Fig. 3 (b). Linecuts (at 16.5 mT), presented in\nFig. 3 (c), show that the FMR-induced feature is reduced\nbut clearly persists even when the nanodiamonds are not\nin direct contact with the YIG.\nWhile a clear explanation for the e\u000bect is not forthcom-\ning from our experimental results, we can make a few\nobservations. First, the insulating nature of both ma-\nterials rules out long range carrier-mediated transport\nof spins. Second, one of the unique aspects of this ex-\nperiment is that both the YIG magnetization and the\nNV center spins are out of equilibrium when the YIG\nis on FMR: YIG due to its resonance and the NV cen-\nters due to the hyperpolarizing action of the laser ex-\ncitation [29, 30]. This is in contrast to other systems\nwhere angular momentum is transferred between spin\nsub-systems, e.g. dynamic nuclear polarization or spin\npumping [1, 2, 31], where spins in one spin sub-system\nrelax by transferring polarization to another spin sub-\nsystem. The transfer of angular momentum from the\nYIG to the NV centers could result in relaxation towards\nequilibrium of both the spin systems, and thus be highly\ndesirable for the overall system.\nThis phenomenon o\u000bers the opportunity of probing\nspin transport in the absence of conductors and employ-\ning an all-optical readout. These advantages dramat-\nically reduce the potential confounding e\u000bects encoun-\ntered in studies of spin transport in metallic systems.\nThus this system o\u000bers an attractive and powerful ap-\nproach to better understanding microscopic details of re-\nlaxation and spin transport in magnetic heterostructures.\nExperiments are underway on other ferromagnets and\nstructures to gain more insight on this e\u000bect and how\nangular momentum is transferred between spin systems.\nIn summary, we have shown that the NV center spin\nstate can be manipulated by a dynamic coupling to the\nmagnetization of YIG. The availability of a wide selec-\ntion of ferromagnetic materials and structures (e.g., satu-\nration magnetization and anisotropies) potentially o\u000bers\na high degree of control in manipulating the NV spin\nstate. The potential for ultra high resolution imaging\nof ferromagnetic phenomena using individual NV centers\ncan have a signi\fcant impact as well. It should in\ru-\nence the \felds of spintronics and quantum information\nby combining the sensitivity of the NV center and the\ntunability and scope of ferromagnetism.\nThe authors wish to thank Yaroslav Tserkovnyak for\nuseful discussions. Funding for this research was provided\nby the Center for Emergent Materials at the Ohio State\nUniversity, an NSF MRSEC (Award Number DMR-\n0820414, ARO (Award number W911NF-12-1-0587) and\nDOE (Award number DE-FG02-03ER46054).4\n\u0003hammel@physics.osu.edu\n[1] C. P. Slichter, Principles of Magnetic Resonance\n(Springer-Verlag, New York, 1989).\n[2] F. Meier and B. P. Zakharchenya, Optical Orientation\n(North-Holland, Amsterdam, 1984).\n[3] P. C. Hammel, M. L. Roukes, Y. Hu, T. J. Gramila,\nT. Mamiya, and R. C. Richardson, Phys. Rev. Lett. 51,\n2124 (1983).\n[4] F. Pulizzi, ed., Nat. Mater. (Insight Issue:Spintronics) ,\nVol. 11 (Nature Publishing Group, 2012).\n[5] I. \u0014Zuti\u0013 c, J. Fabian, and S. D. Sarma, Rev. Mod. 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Lett. 88, 117601 (2002)." }, { "title": "1607.04312v1.Room_temperature_polarization_in_the_ferrimagnetic_Ga2_xFexO3_ceramics.pdf", "content": "Room temperature polarization in the ferrimagnetic Ga 2−xFe xO3ceramics\nB. Kundys, F. Roulland, C. Lefèvre, C. Mény, A. Thomasson, N. Viart \nInstitut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), UMR 7504 CNRS-UdS, 67034 Strasbourg, France \nAbstract\nThe effect of the Fe/Ga ratio on the magnetic and electric properties of the multiferroic Ga2−xFe xO3 compound has been studied in order to \ndetermine the composition range exhibiting magnetic and electric orders coexistence and their critical temperatures. A magnetoelectric phase \ndiagram, showing the evolution of both the Néel magnetic ordering temperature TN and the electric ordering temperature Tc, versus the iron content \nhas been established for 0.9 ≤ x ≤ 1.4. While the ferrimagnetic Néel temperature increases with the iron content, the electric ordering temperature \nshows an opposite trend. The electric polarization has been found to exist far above room temperature for the x = 1.1 composition which shows \nthe highest observed electric ordering temperature of Tc ≈ 580 K. The x = 1.3 and 1.4 compounds are ferrimagnetic–electric relaxors with both \nproperties coexisting at room temperature.\nKeywords: Dielectric permittivity; Polarization; Ferrimagnetism; Magnetoelectric phase diagram\n1. Introduction\nThe coexistence of magnetic and electric orders at the same\ntemperature and pressure regions present high research inter-\nest due to the potential to design technologically importantcross-functionalities in these materials. Such functionalities caninclude, butare not limited to, the electric field-controlled mag-\nnetization or magnetic field-controlled polarization.\n1–3There\nare however very few materials that present such propertiesat room temperature.\n4,5Until recently the only material show-\ning unambiguously both ferroelectricity and magnetoelectricityat room temperature was BiFeO\n3;6–8this explains why this\nmaterial has been the focus point of most of the researcheson multifunctional materials. Lately, convincing evidences ofthe coexistence of room temperature electric and magneticorders and room temperature magnetoelectric effect have beenobserved in a few other compounds such as solid solutionphases between lead iron based perovskites, and in particularthe PbFe\n0.5Ta0.5O3–PbZr 0.53Ti0.47O3(PFT–PZT)9,10and the\nPbFe 0.5Nb0.5O3–PbZr 0.53Ti0.47O3(PFN–PZT)11systems, the\ncomplex hexaferrite Sr 3Co2Fe24O41,12oreven /H9252-NaFeO 2.13\nThe preparation of these compounds is however rather tricky andthere is still room for improvement of their electric polarization,net magnetization and magnetoelectric effect.The Ga\n2−xFe xO3(GFO) compound, is known since\nthe 60s to present interesting pyroelectric, ferrimagneticand magnetoelectric properties near room temperature.\n14\nIt crystallizes in the orthorhombic space group Pc2 1n,\nwith a= 0.87512 ±0.00008 nm, b= 0.93993 ±0.00003 nm and\nc= 0.50806 ±0.00002 nm (Fig. 1).15This structure is based on\nan ABAC double hexagonal close-packed stacking of oxygens,and strongly differs from the usual perovskite structure observedfor most of the other multiferroic materials. The cations aredistributed among four cationic sites Ga1, Ga2, Fe1, and Fe2.\nGa1 and Fe1 are antiferromagnetically coupled to Ga2 and\nFe2. If the Fe\n3+cations only occupied the Fe1 and Fe2 sites, and\nthe Ga3+only the Ga1 and Ga2 one, the compound would be\nstrictly antiferromagnetic for x= 1.0. In fact a cationic site disor-\nder, observed by Arima et al.17by neutron diffraction, allows the\ncompound to exhibit a non-negligible net resulting magnetiza-tion (0.7 /H9262\nB/Fe for x= 1.0). The magnetic ordering temperature\nis relatively high in this family of compounds and is above\nroom temperature for x≥1.3. Although GaFeO 3was already\nreported to show magnetic field dependent polarization14,17the2 B. Kundys et al. \nFig. 1. Projection of the crystal structure of Ga 2−xFexO3along the caxis (space\ngroup Pc2 1n) produced using the VESTA16crystallographic software.\nanswer to the question whether it is electrically polar at room\ntemperature and in zero magnetic field remains unclear. A phasetransition to electrically polar state may be confirmed by a dipolereorientation induced maximum in the temperature variation ofthe dielectric permittivity. However the temperature variations ofthe dielectric properties reported so farfor pure GFO compounds\nconcern exclusively the x= 1.0 compound and the temperature\ndependence of the permittivity shows no maximum, indicatingthe absence of the transition to electrically polar state.\n18–21The\nonly evidence of a maximum in the temperature variation ofthe dielectric permittivity wasfound for a Mn-doped GaFeO\n3\nsample.22Recently, a partially reversible polarization of ca.\n0.3/H9262C/cm2washowever evidenced in polycrystalline GaFeO 3\nthrough pyroelectric measurements.23The reported ferroelec-\ntric Curie temperature (T Cf)wasbelow room temperature (ca.\n100 K). A similar value of the polarization, ca.0.2/H9262C/cm2,\nbutthis time fully reversible and at room temperature, has\nbeen observed on thin films, both for the Ga 0.6Fe1.4O3:Mg24\nand GaFeO 325compositions. It must be noted that a polar-\nization of ca.1/H9262C/cm2has also been measured on thin films\nof the isostructural compound /H9255-Fe 2O3at room temperature.26\nThe awaited polarization in GaFeO 3is however two orders of\nmagnitude bigger than the measured ones, with a value of ca.\n25/H9262C/cm2, as evaluated by Stoeffler using a simple point charge\nmodel.\nThere is therefore a need to, first, clarify the electric behavior\nof GFO compounds and, then, study the influence of the ironcontent xon the temperature of the prospective electric order. In\nthis work, we attend to the electric characterization of a series ofGa\n2−xFe xO3polycrystalline compounds (0.9 ≤x≤1.4) through\nboth the study of the variation of the dielectric constant withtemperature and pyroelectric current measurements.\n2. Materials and methods\nIn our experiment the Ga\n2−xFe xO3polycrystalline samples\n(0.9≤x≤1.4) were prepared viaan optimized ceramic pro-\ncess already published elsewhere.27High purity commercial\npowders of /H9251-Fe 2O3(Prolabo >99%) and Ga 2O3(Alfa Aesar99.999%) were first ball-milled in a Teflon jar in an optimized\ndispersive environment. The resulting slurries are subsequentlydried and the powders obtained are mixed with an organic binder\n(Rhodoviol) to be uniaxially pressed under 60 bars into 15 mmdiameter and 0.6 mm height disk shaped pellets using a hydraulic\npress. The green pellets were then sintered at the tempera-ture necessary for the formation of the desired Ga\n2−xFe xO3,\nphase exempt of any parasitic phase, which depends upon theFe content.\n27Forthe electric measurements, the temperature\nvariation wasperformed in an Instec cryostat with a possibility\nto heat up to 873 K. Dielectric measurements were performedin vacuum at different frequencies using an Agilent LCR meter.The samples were first taken to the highest available temperature(ca.600 K) and the dielectric constant wasthen measured upon\ncooling them down to 150 K. Pyroelectric current measurementswere performed with a Keithley electrometer upon cooling thesamples in a small electric field of 0.05 V/0.6 mm. The electricpolarization wasdeduced from those pyroelectric measurements\nthrough a time integration method. Magnetic measurementswere performed using a superconducting quantum interferencedevice magnetometer (SQUID MPMS XL, Quantum Design).\n3. Results and discussion\nThe temperature dependence of the dielectric permittivity\nwas measured for various compounds of the Ga\n2−xFe xO3fam-\nily (with 0.9 ≤x≤1.4). The results are shown in Fig. 2(a), and\nthe evolution of the dielectric losses with the Fe content arepresented in Fig. 2(b). A maximum in the temperature depend-\nent permittivity is clearly seen for compositions above x= 1.0,\nindicating thus the existence of an electric polarization. The posi-tion of this maximum varies from 570 K for the x= 1.1 sample to\nabout 400 K for the x= 1.3 sample. It is clear that both the electric\nordering temperature (T\nc) and dielectric losses of the GFO com-\npounds strongly depend on the Fe/Ga ratio. The maximum in thedielectric permittivity curve is within an experimentally reach-able range of temperatures for x≥1.1,butit strongly smoothens\nwith increasing x.\nThe Ga\n2−xFe xO3,x= 1.1 compound shows the most impor-\ntant dielectric anomaly. The temperature position of the observedpeak in the dielectric permittivity and dielectric loss is frequencyindependent, and allows determining a transition temperature of582 K (Fig. 3(a)). The ratio between the slopes of the 1/ε\n/primeversus\ntemperature curve above and below Tcis larger than 2 (Fig. 3(b)),\nthus indicating a first-order transition to the paraelectric stateabove 582 K.\nPyroelectric current measurements also reveal an anomaly\nnear this temperature for x= 1.1 (Fig. 4). The current integration\nwith respect to time method gives a rather large polarizationvalue of ca.33/H9262C/cm\n2, existing at room temperature. Although\npolarization loops have been measured in thin films of this mate-rial by different teams, including ours,\n24,28it has to be noted\nthat our attempts to measure ferroelectric loops in the bulk com-pound, at different temperatures between 150 and 500 K, and inelectric fields up to 60 kV/m, were unsuccessful. There is there-fore no evidence of ferroelectricity of the material in its bulkform. Although the reported value of 33 /H9262C/cm\n2has an onlyB. Kundys et al. 3\nFig. 2. (a) Temperature variation of the reduced dielectric permittivity measured at 100 kHz for various Ga 2−xFexO3compounds (0.9 ≤x≤1.4) and (b) dielectric\nloss evolution with the Fe content in Ga 2−xFexO3compounds at room temperature.\nFig. 3. (a) Frequency dispersion of the dielectric permittivity and dielectric loss (inset) as a function of temperature, (b) temperature dependence of 1/ε/primebelow and\nabove the electric ordering temperature Tc, for GFO x= 1.1.\nindicative character as it wasinduced by a specific electric field,\nit nevertheless, together with a peak in the dielectric permittiv-\nity, confirms the existence of an important electric polarizationstate in the sample, with a phase transition at 580 K. The I(T)\ncurve, measured under an applied electric field of 0.05 V/0.6 mm(Fig. 4) shows a zero current value in the temperature range of160–450 K. This therefore excludes any artifact due to a domi-nant leakage contribution. At the electric ordering temperature,the small applied electric field allows orienting the polariza-tion from random paraelectric state to polar state. It also has tobe noted that the observed polarization value perfectly matches\nFig. 4. Electric polarization (left) deduced from pyroelectric current (right)\nintegration with respect to time, for GFO (x = 1.1).with the value of 25 /H9262C/cm2calculated by Stoeffler29using first\nprinciple methods.\nFor the Ga 2−xFe xO3,x= 1.3 and 1.4 compounds, a relaxor-\nlike behavior is observed, where the temperature for which the\nmaximum permittivity is observed depends on the measurementfrequenc y(Fig. 5).\nSuch a behavior is correlated with an important increase of the\ndielectric losses with the increasing iron content (Fig. 2(b)). Thegeneral trend in the T\ncevolution can be deduced by comparing\nthe dielectric anomalies at one chosen frequency (Fig. 2(a)): itclearly decreases with increasing x.\nThe thermomagnetic curves of the samples were measured\nunder an applied field of 75 Oe, high enough to increase themagnetic response while still keeping the samples well belowmagnetic saturation. The evolution of the Néel temperature of thesamples versus their iron content is given in Fig. 6. As awaited,\n17\nthe magnetic properties are strongly dependent on the iron con-tent of the samples: the higher the iron content, the higher boththe magnetization and the magnetic transition temperature. Themost interesting results are ascribed to the two samples of high-est iron contents (x = 1.3 and 1.4) as their T\nNvalues are higher\nthan room temperature (T N= 330 K for x= 1.3 and TN= 400 K\nfor x= 1.4). The Ga 2−xFe xO3system belongs to type 1 multifer-\nroic compounds in which magnetic and electric orders occur atdifferent temperatures.4 B. Kundys et al.\nFig. 5. Temperature variation of the dielectric permittivity for (a) Ga 0.7Fe1.3O3and (b) Ga 0.6Fe1.4O3at different frequencies.\nFig. 6. M(T) dependences for the different GFO compounds (0.9 ≤x≤1.4)\nunder an applied field of 75 Oe.\nFig. 7. Magnetoelectric phase diagram of Ga 2−xFexO3compounds with x\nbetween 0.9 and 1.4.\nFig. 7 finally summarizes the magnetoelectric phase diagram\nfor Ga 2−xFe xO3compounds, showing the evolution of the Néel\nmagnetic ordering temperature TN, together with the electric\nordering temperature Tc,versus the iron content.\n4. Conclusion\nIn summary, the temperature variation (150 K < T< 600 K)\nof the dielectric properties of Ga 2−xFe xO3, 0.9≤x≤1.4, com-\npounds shows a maximum for the x> 1 compositions. The\nabsence of previous documented studies for compositions other\nthan x= 1 explains why this phenomenon had not been observed\nbefore. The composition x= 1.1 shows a polar state with anordering temperature of about 580 K and a polarization of33/H9262C/cm\n2very close to the value awaited from theoretical cal-\nculations. Compositions with x≥1.3 present a relaxor behavior,\nwith high dielectric losses. The temperature at which the maxi-mum of permittivity is observed decreases with increasing ironcontent. The dependence of the electric and magnetic propertiesof this system upon the Fe content is interestingly opposite, sinceT\nNincreases with x.Forsamples with x≥1.3, the coexistence of\nboth electric and magnetic polarizations in a wide temperaturerange including room temperature is possible. Further studiesin this area should focus on the optimal xvalues for which the\ndielectric relaxor behavior is transformed into a classical onewith low dielectric dissipation, allowing a large magnetizationand a large polarization near room temperature.\nAcknowledgments\nThis work was done with the financial support from the\ninternational ANR DFG Chemistry project GALIMEO #2011-\nINTB-1006-01. The authors are grateful to Dr. Ingrid CA ˜NERO\nINFANTE for fruitful discussions.\nReferences\n1.Binek C, Doudin B. Magnetoelectronics with magnetoelectrics. J Phys:\nCondens Matter 2005;17:L39–44.\n2.Bibes M, Barthelemy A. Multiferroics: towards a magnetoelectric memory.\nNat Mater 2008;7:425–6.\n3.Scott JF. Applications of magnetoelectrics. J Mater Chem 2012;22:4567–74.\n4.Hill NA. Why are there so few magnetic ferroelectrics? J Phys Chem B\n2000;104:6694–709.\n5.Scott JF. Room-temperature multiferroic magnetoelectrics. 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J Eur Ceram Soc\n2013;33:1029–35.\n28.Mukherjee S, Roy A, Auluck S, Prasad R, Gupta R, Garg A. Room temper-ature nanoscale ferroelectricity in magnetoelectric GaFeO\n3epitaxial thin\nfilms. Phys Rev Lett 2013;111:087601.\n29.Stoeffler D. First principles study of the electric polarization and of itsswitching in the multiferroic GaFeO\n3system. J Phys: Condens Matter\n2012;24:185502." }, { "title": "1712.04972v2.High_temperature_terahertz_optical_diode_effect_without_magnetic_order_in_polar_FeZnMo__3_O__8_.pdf", "content": "arXiv:1712.04972v2 [cond-mat.str-el] 6 Jan 2018High-temperature terahertz optical diode effect without ma gnetic order in polar\nFeZnMo 3O8\nShukai Yu,1Bin Gao,2Jae Wook Kim,2Sang-Wook Cheong,2Michael\nK.L. Man,3Julien Mad´ eo,3Keshav M. Dani,3and Diyar Talbayev1,∗\n1Department of Physics and Engineering Physics,\nTulane University, 6400 Freret St., New Orleans, LA 70118, U SA\n2Rutgers Center for Emergent Materials and Department of Phy sics and Astronomy, Piscataway, NJ 08854, USA\n3Femtosecond Spectroscopy Unit, Okinawa Institute of Scien ce and Technology Graduate University\n(Dated: August 7, 2021)\nWe present a terahertz spectroscopic study of polar ferrima gnet FeZnMo 3O8. Our main finding\nis a giant high-temperature optical diode effect, or nonreci procal directional dichroism, where the\ntransmitted light intensity in one direction is over 100 tim es lower than intensity transmitted in the\nopposite direction. The effect takes place in the paramagnet ic phase with no long-range magnetic\norder in the crystal, which contrasts sharply with all exist ing reports of the terahertz optical diode\neffect in other magnetoelectric materials, where the long-r ange magnetic ordering is a necessary\nprerequisite. In FeZnMo 3O8, the effect occurs resonantly with a strong magnetic dipole a ctive\ntransition centered at 1.27 THz and assigned as electron spi n resonance between the eigenstates of\nthe single-ion anisotropy Hamiltonian. We propose that the optical diode effect in paramagnetic\nFeZnMo 3O8is driven by signle-ion terms in magnetoelectric free energ y.\nPACS numbers:\nMultiferroic materials that combine ferroelectricity\nwith magnetismhavebeen thesourceoffascinatingphys-\nical phenomena and functionalities1,2. One example is\nthe terahertz (THz) nonreciprocal directional dichroism\nthatwasrecentlydiscoveredinmultiferroics3,4. Theterm\nrefers to the difference in absorption coefficient for lin-\nearly polarized light waves traveling in opposite direc-\ntions. A material can be transparent for light traveling\nin one direction and completely opaque when the same\nlight wave travels in the opposite direction. By analogy\nwith the semiconductor diode, we will use the term opti-\ncal diode effect (ODE) for the directional dichroism.\nODE functionality can potentially find applications\nin THz optical isolators. Practical ODE devices would\nneed to display the effect close to room temperature,\nwhile most of the reported ODE observations until\nnow occured in low-temperature magnetically ordered\nstates3–13. ODE exists at room temperature in ferroelec-\ntric antiferromagnet BiFeO 3, but its magnitude is rather\nsmall14. In this letter, we report the observation of a\ngiant high-temperature THz ODE in polar ferrimagnet\nFeZnMo 3O8. We demonstrate complete suppression of\nabsorption for one direction of traveling light, while the\nabsorption for the other direction remains very strong,\nresulting in 100-fold difference in transmitted light in-\ntensity between the two directions. Most remarkably,\nthe strong ODE persists in the high-temperature para-\nmagnetic state of FeZnMo 3O8without long-range mag-\nnetic order up to 110 K, which is in stark contrast with\nall previous reports of THz ODE where the presence\nof magnetic order is a necessity. The ODE and re-\nlatedopticalmagnetoelectriceffectsinroom-temperature\nparamagnetic state were first reported by Rikken et\nal. at visible wavelengths, were the strength of the ef-\nfect was many orders of magnitude smaller15–17. TheODE in FeZnMo 3O8is resonant with a 1.27 THz elec-\ntron spin transition between the eigenstates of single-ion\nanisotropy Hamiltonian, which is the dominant energy\nscale. We suggest that the high-temperature giant ODE\nin FeZnMo 3O8results from single-ion terms in magne-\ntoelectric interaction, which can open a new direction in\nthe quest for stronger ODE in other magnetoelectrics.\nHigh quality monodomain crystals of FeZnMo 3O8\nwere prepared by chemical vapor transport method18.\nTwo platelet-shaped crystals were selected, one with the\npolarcaxis perpendicular to the platelet, the other with\nthecaxis in the platelet plane. THz time domain spec-\ntroscopy (TDS) in high magnetic field (up to 16 T) was\nused to study the low-energy excitations19.\nFeZnMo 3O8is derived from the parent material\nFe2Mo3O8by susbtituting Fe with Zn. Both materials\nadopt a polar space group P63mc18,20–22. In the par-\nent, Fe ions occupy octahedrally and tetrahedrally oxy-\ngen coordinated sites, while Zn ions preferentially pop-\nulate the tetrahedral sites in the substituted variant21.\nWith equal amounts of Fe and Zn in FeZnMo 3O8, ex-\nchange interactions are heavily diluted. No signature of\nlong range magnetic order is found down to TC= 14\nK23, where the ferrimagnetic ground state appears with\nantiparallel alignment of spins on octahedral and tetra-\nhedral sites. M(H) curves along the caxis show a very\nnarrow hysteresis loop with coercivity HC= 0.06 T and\nsaturation magnetic moment of 3.9 µB/f.u. The M(H)\nmeasurement perpendicular to the caxis does not show\nsaturation up to 7 T, indicating a very strong magnetic\nanisotropy24.\nFigures 1(a,b) show the recorded absorption and re-\nfractive index spectra at low temperature (12 K and 14\nK) for different polarizations of the incident THz wave.\nPolarization analysis shows that the 1.27 THz resonance2\n/s48/s50/s48/s52/s48/s54/s48/s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48 /s49/s46/s50/s53 /s49/s46/s53/s48 /s49/s46/s55/s53 /s50/s46/s48/s48\n/s32/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s65/s98/s115/s111/s114/s98/s97/s110/s99/s101/s32/s40/s99/s109/s45/s49\n/s41\n/s40/s97/s41\n/s50/s46/s56/s51/s46/s48/s51/s46/s50/s51/s46/s52/s51/s46/s54/s51/s46/s56/s52/s46/s48\n/s40/s98/s41/s32/s49/s50/s32/s75/s44/s32/s32 /s101/s40 /s41/s32/s32/s124/s32 /s32/s99 /s44/s32/s32 /s104/s40 /s41/s32/s32/s124/s32 /s32/s99\n/s32/s49/s52/s32/s75/s44/s32/s32 /s101/s40 /s41/s32/s124/s124/s32 /s99 /s44/s32/s32 /s104/s40 /s41/s32/s32/s124/s32 /s32/s99\n/s32/s49/s52/s32/s75/s44/s32/s32 /s101/s40 /s41/s32/s32/s124/s32 /s32/s99 /s44/s32/s32 /s104/s40 /s41/s32/s124/s124/s32 /s99/s32/s82/s101/s102/s114/s97/s99/s116/s105/s118/s101/s32/s105/s110/s100/s101/s120\n/s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48 /s49/s46/s50/s53 /s49/s46/s53/s48 /s49/s46/s55/s53 /s50/s46/s48/s48/s48/s50/s48/s52/s48/s54/s48/s32/s49/s50/s32/s75/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s54/s48/s32/s75\n/s32/s50/s48/s32/s75/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s55/s48/s32/s75\n/s32/s51/s48/s32/s75/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s56/s48/s32/s75\n/s32/s52/s48/s32/s75/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s49/s48/s48/s32/s75\n/s32/s53/s48/s32/s75\n/s40/s99/s41/s32/s65/s98/s115/s111/s114/s98/s97/s110/s99/s101/s32/s40/s99/s109/s45/s49\n/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s101/s40 /s41/s32/s32/s124/s32 /s32/s99 /s44/s32/s32 /s104/s40 /s41/s32/s32/s124/s32 /s32/s99\nFIG. 1: (Color online) (a,b) Absorption and refractive inde x\nin zero applied magnetic field ( H= 0) for different polariza-\ntions of the incident THz wave. (c) Temperature dependence\nof absorption.\nis magnetic-dipole active, as the absorption disappears\ncompletely when the THz magnetic field h(ω) is ori-\nented along the polar cdirection. Cooling the crystal\nto 4 K into the ferrimagnetic state does not apprecia-\nbly change the THz absorption spectra. A similar res-\nonance near 1.4 THz was observed by Kurimaji et al.25\nin (Fe0.6Zn0.4)2Mo3O8, along with weaker resonances at\n2.3 and 2.5 THz. The weak 2.3 and 2.5 THz resonances\nare outside the frequency window of our zero-field THz\nTDS spectrometer. Kurumaji et al. suggest that the\n1.4 THz mode in (Fe 0.6Zn0.4)2Mo3O8may be related to\nlocal single-site spin transitions on Fe sites25. We show\nthat the 1.27 THz resonance in FeZnMo 3O8is indeed\nan electron spin resonance between the eigenstates of the\nsingle-ion anisotropy Hamiltonian on Fe2+cites.\nFigure 1(c) shows the temperature dependence of the\n1.27 THz absorption, which persists almost up to 100\nK temperature. The strong absorption in the paramag-\nnetic state servesevidence that the magnetic-dipoletran-\nsitionat1.27THzcannotberelatedtoacollectiveexcita-\ntion associated with an ordered magnetic state. Figure 2\nshows the magnetic field dependence of the THz trans-\nmission spectra with oscillating THz fields e(ω) andh(ω)\nboth perpendicular to the caxis. The static magnetic\nfieldHis applied both along and perpendicular to the\ncaxis. The frequency window of the high-magnetic-field\nTHz TDS setup includes frequencies up to 3 THz, albeit\nwith a narrow ”dark” band between 2.2-2.5 THz due to/s48/s46/s49/s49/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s32/s82/s101/s108/s97/s116/s105/s118/s101/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41\n/s32/s48/s32/s84\n/s32/s49/s32/s84\n/s32/s52/s32/s84\n/s32/s56/s32/s84\n/s32/s49/s50/s32/s84\n/s32/s49/s54/s32/s84/s40/s97/s41\n/s101/s40 /s41/s32/s32/s124/s32 /s32/s99 /s44/s32/s32 /s104/s40 /s41/s32/s32/s124/s32 /s32/s99/s72 /s32/s124/s124/s32 /s99\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s49/s49\n/s101/s40 /s41/s32/s32/s124/s32 /s32/s99 /s44/s32/s32 /s104/s40 /s41/s32/s32/s124/s32 /s32/s99/s72 /s32/s32/s124/s32 /s32/s99/s40/s99/s41/s32\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s32/s48/s32/s84\n/s32/s53/s32/s84\n/s32/s56/s32/s84\n/s32/s49/s50/s32/s84\n/s32/s49/s54/s32/s84\nFIG. 2: (Color online) Magnetic field dependence of transmit -\nted intensity at T= 4 K. (a) Magnetic field Happlied along\nthecaxis in Faraday measurement geometry. (b) Magnetic\nfieldHapplied perpendicular to the caxis in Voigt measure-\nment geometry.\nabsorption in polyethylene lenses that are part of the\nsetup. This dark band is blanked out in Fig 2. In high\nmagnetic field, our setup does not allow the collection\nof a free space reference spectrum, instead of which we\nused a reference spectrum that is an averageof spectraat\ndifferent recorded magnetic fields26. Due to the strong\nshift of the 1.27 resonance, such reference allows us to\nisolate and measure only the magnetic field-dependent\nabsorption. In Fig. 2, the relative transmission is the\nTHz intensity at a fixed magnetic field divided by the in-\ntensity of above-defined reference spectrum. Clearly, the\nposition of the 1.27 THz resonance shifts with magnetic\nfield, and the shift depends on the direction of applied\nfieldH. At high H, we also find and additional field-\ndependent absorption in the 2.5-3.0 THz range (Fig. 2).\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s72 /s32/s32/s124/s124/s32 /s99\n/s72 /s32/s32/s32/s124/s32 /s32/s99/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s84/s32/s61/s32/s52/s32/s75\nFIG.3: (Color online)Magnetic fielddependenceofresonanc e\nfrequencies at T= 4 K.\nFigure 3 summarizes the measured magnetic field de-3\npendence of the observed resonance frequencies. The\nmain 1.27 THz resonance shifts up linearly when His\napplied along the cdirection; the resonance shifts up\nquadraticallywhen Hisapplied perpendicularto c. This\nbehavior can be described well by the single-ion Hamil-\ntonian\nH=−DSz2+gµBH·S, (1)\nwhereS= 2 is the spin on Fe2+ions,D >0 is the\neasy axis anisotropyconstant, and His the applied mag-\nnetic field. In zero field, the oscillating THz field h(ω)\ndrives the transitions between the eigenstates of the un-\nperturbed part H=−DSz2with energy /planckover2pi1ω= 3D. For\nmagnetic field applied along zandxdirections (parallel\nand perpendicular to the caxis, respectively), we expect\nthe following resonance frequency shifts\n/planckover2pi1ω= 3D+gµBHz, (2)\n/planckover2pi1ω= 3D+2(gµBHx)2\n3D, (3)\nwhich were computed under the assumptions of an\nisotropic gfactor and of small frequency shifts compared\nto 3D. The solid green lines in Fig. 3 are frequencies\ncomputed using Eqs. (2) and (3) with g= 4.50. The\nagreement with data is very good. The measured tem-\nperature and magnetic field dependence of the 1.27 THz\nmode lead to the conclusion that it arises from the elec-\ntron spin resonance transitions between the eigenstates\nof single-ion Hamiltonian (1). In the paramagnetic state,\nthe transisions occur on individual Fe ions. In the fer-\nrimagnetic state, the transitions correspond to the pre-\ncession of the macroscopic magnetization of the majority\nFe spins on octahderal sites with single-ion anisotropy as\nthe dominant energy scale.\nThe high-energytransitionsseen in Figs. 2and 3 above\n2.5 THz are likely similarin originto the 2.3and 2.5THz\ntransitions observed in (Fe 0.6Zn0.4)2Mo3O825.\nWe now focus on the ODE associated with the 1.27\nTHz resonance in FeZnMo 3O8. Two distinct mecha-\nnismshavebeenestablishedforODE.Oneismagnetochi-\nral dichroism found in magnetic and chiral materials4,27.\nThe other is toroidal dichroism, for which the mate-\nrial must possess simultaneous electric polarization P\nand magnetization M3,28,29. The ODE happens for\nlight propagating along and opposite the toroidal vector\nT=P×M24. The THz wave must be polarized with\nits electric field e(ω) alongPand with its magnetic field\nh(ω) along magnetization Mwhile propagating along T.\nIn FeZnMo 3O8, the polarization Pis along the caxis.\nFigure4(a)demostratestheODEinFeZnMo 3O8inmag-\nnetic field Happlied perpendicular to the caxis at 4 K.\nThe THz wave is polarized along the caxis (e(ω)/bardblP)\nand is traveling along the P×Hdirection. ODE is\ndetected by reversing the direction of H. Figure 4(a)\ndisplays the intensity of transmitted THz wave at the\nresonance frequency for positive and negative magnetic\nfields. For both positive and negative fields, the reso-\nnance frequency shifts according to Eq. (3). However,together with the frequency shift we also find that the in-\ntensity of the resonance changes. For positive fields, the\nresonance is enhanced, for negative fields, the resonance\nis suppressed. In negative 8 T field, the resonant absorp-\ntion practically disappears; the difference in transmitted\nintensity between positive and negative 8 T field reaches\na factor of 100. Very clearly, this is a demonstration of a\ngiant ODE.\n/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s69/s45/s53/s49/s69/s45/s52/s49/s69/s45/s51/s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56 /s50/s46/s48\n/s84/s61/s52/s32/s75/s65/s98/s115/s111/s114/s98/s116/s105/s111/s110/s32 /s32/s40/s99/s109/s45/s49\n/s41\n/s32/s48/s32/s84\n/s32/s52/s32/s84\n/s32/s56/s32/s84\n/s32/s49/s50/s32/s84\n/s32/s49/s54/s32/s84\n/s40/s98/s41/s84/s61/s52/s32/s75\n/s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56 /s50/s46/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s32/s52/s32/s75\n/s32/s53/s48/s32/s75\n/s32/s55/s48/s32/s75\n/s32/s57/s48/s32/s75\n/s32/s49/s49/s48/s32/s75/s32/s91 /s40/s56/s32/s84/s41/s32/s45/s32 /s40/s45/s56/s32/s84/s41/s93/s32/s32/s40/s99/s109/s45/s49\n/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s40/s99/s41/s32/s48/s32/s84\n/s32/s50/s32/s84\n/s32/s52/s32/s84\n/s32/s54/s32/s84\n/s32/s56/s32/s84/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s84/s72/s122/s41/s84/s114/s97/s110/s115/s109/s105/s116/s116/s101/s100\n/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s40/s97/s41/s101/s40 /s41/s32/s124/s124/s32 /s99 /s44/s32/s32 /s72 /s32/s124/s124/s32 /s104/s40 /s41/s32/s32/s124/s32 /s32/s99\nFIG. 4: (Color online) (a,b) Optical diode effect and absorp-\ntion coefficient for positive and negative magnetic field H.\nSolid lines show absorption coefficient for positive magneti c\nfield. Open circles of the same color show the absorption\ncoefficient for negative magnetic field. (c) Temperature de-\npendence of the optical diode effect.\nTo quantitatively compare ODE in FeZnMo 3O8to\nother materials, we compute the absorption coefficient\nfor positive and negative magnetic fields, Fig. 4(b). The\ndifferenceinabsorptioncoefficientreaches∆ α= 72cm−1\nbetween positive and negative magnetic fields in the 8-\n12 T range. This is significantly higher than most re-\nported values of ODE3–6,8–14, with the only higher value\n∆α∼400 cm−1cited for Gd 0.5Tb0.5MnO37. Figure 4(c)\nshows the temperature dependence of ∆ αforH=±8\nT. ODE and ∆ αremain significant up to 110 K in\nthe paramagnetic state of FeZnMo 3O8, when no long\nrange magnetic order is present in the crystal. This\nobservation drastically contrasts with all previous re-\nports, where THz ODE occurs in a magnetically ordered\nstate. In the room-temperature paramagnetic state, the\nODE was first observed by Rikken et al. at visible\nwavelengths15. To quantitatively compare our THz ODE\nwith the observations of Rikken, we compute the quan-4\ntity (α(H)−α(−H))/(α(H)+α(−H))/2Hand find that\nit exceeds 5 ×10−2T−1in FeZnMo 3O8at 4 K temper-\nature. In the work of Rikken et al., this quantity was\nmeasured to be 2 .5×10−5T−1with ∆α= 1.2×10−4\ncm−1. By both measures, our THz ODE is many orders\nof magnitude stronger.\nWhat is the origin of the high-temperature ODE in\nFeZnMo 3O8? In addition to the presence of the toroidal\nmoment T=P×Min the crystal, another prerequisite\nfor ODE is nonzero dynamic magnetoelectric susceptibil-\nityχme\nxz(ω)28,29such that\nImχme\nxz(ω) =/summationdisplay\nnπcµ0\n2/planckover2pi1NV(/angbracketleft0|hx|n/angbracketright/angbracketleftn|ez|0/angbracketright+\n/angbracketleft0|ez|n/angbracketright/angbracketleftn|hx|0/angbracketright)δ(ω−ωn).(4)\nHere,/angbracketleft0|hx|n/angbracketrightand/angbracketleft0|ez|n/angbracketrightare magnetic and electric\ndipole matrix elements between ground and excited spin\nstates of the Fe2+ions. The first, magnetic dipole matrix\nelements is clearly not zero, as evidenced by the strong\nmagnetic absorption when h(ω) is perpendicular to the\ncaxis. We must conclude the existence of the second,\nelectric dipole matrix element, from the observed strong\nODE. We propose that such electric dipole matrix ele-\nments could exist on tetrahedrallycoordinated Fe2+sites\nwithnoinversionsymmetryduetospindependentmetal-\nligand hybridization30,31. Such interaction could induce\nan electric dipole at the spin site, as well as electric-\nfield driven spin transitions. The spin dependent metal-\nligand hybridization was invoked to explain the ODE in\nBa2CoGe2O7associated with the spin resonance due to\nanisotropy gap excitation28. In contrast to our presentresults, the ODE in Ba 2CoGe2O7disappeared above the\nmagnetic ordering temperature TN= 6.7 K3. Further\nwork is needed to clarify the details that govern the\nmagnetoelectric susceptibility χme\nxz(ω). For example, in\nFeZnMo 3O8composition with equal amount of Fe and\nZn, where the tetrahedral sites are preferentially occu-\npied by Zn and the octahedral sites are preferentially\noccupied by Fe, the 1.27 THz resonance would need to\nget some of its strength from tetrahedral sites to give\nrise toχme\nxz(ω). Alternatively, octahedral sites may need\nto experience local inversion symmetry breaking to allow\nthe same.\nTo summarize, we have reported a giant high-\ntemperature ODE (110 K) in polar paramagnetic\nFeZnMo 3O8without long-range magnetic order, which\nis fundamentally different all from prior reports of THz\nODE. The ODE in FeZnMo 3O8happens at the fre-\nquency of the strong electron spin resonance assigned as\nthe single-ion anisotropy gap excitation. We have pro-\nposed that necessary dynamic magnetoelectric suscepti-\nbilityχme\nxz(ω) alsoresults from single-ion magnetoelectric\ninteractions30,31. Our experimental results demonstrate\nthat single-site magnetic and magnetoelectric interac-\ntions can provide a new avenue in the search for high-\ntemperature THz ODE in other magnetoelectric materi-\nals.\nWe acknowledge fruitful discussions with Valery\nKiryukhin and Andrei Sirenko. The work at Tulane Uni-\nversity was supported by the NSF Award No. DMR-\n1554866. The work at Rutgers University was sup-\nported by the DOE under Grant No. DOE: DE-FG02-\n07ER46382.\n∗Electronic address: dtalbayev@gmail.com\n1S.-W. Cheong and M. Mostovoy, Nature Mat. 6, 13 (2007).\n2R. Ramesh and N. Spaldin, Nature Mat. 6, 21 (2007).\n3I. K´ ezsm´ arki, N. Kida, H. Murakawa,\nS. Bord´ acs, Y. Onose, and Y. Tokura,\nPhys. Rev. Lett. 106, 057403 (2011), URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.106.05 7403.\n4S. Bordacs, I. K´ ezsm´ arki, S. Seki, and\nY. Tokura, Nat. Comm. 5, 4583 (2014), URL\nhttp://dx.doi.org/10.1038/ncomms5583 .\n5Y. Takahashi, R. Shimano, Y. Kaneko, H. Murakawa, and\nY. 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B 89, 195145 (2014), URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.89.195145 .\n30C. Jia, S. Onoda, N. Nagaosa, and J. H.\nHan, Phys. Rev. B 74, 224444 (2006), URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.74.224444 .\n31T. Arima, Journal of the Physical So-\nciety of Japan 76, 073702 (2007),\nhttp://dx.doi.org/10.1143/JPSJ.76.073702, URL\nhttp://dx.doi.org/10.1143/JPSJ.76.073702 ." }, { "title": "2208.00756v2.Effect_of_magnetism_and_phonons_on_localized_carriers_in_the_ferrimagnetic_kagome_metals_GdMn__6_Sn__6__and_TbMn__6_Sn__6_.pdf", "content": "Effect of magnetism and phonons on localized carriers in the ferrimagnetic kagome\nmetals GdMn 6Sn6and TbMn 6Sn6\nM. Wenzel,1,∗A. A. Tsirlin,2, 3O. Iakutkina,1Q. Yin,4H.C. Lei,4M. Dressel,1and E. Uykur1, 5\n11. Physikalisches Institut, Universit¨ at Stuttgart, 70569 Stuttgart, Germany\n2Felix Bloch Institute for Solid-State Physics, Leipzig University, 04103 Leipzig, Germany\n3Experimental Physics VI, Center for Electronic Correlations and Magnetism,\nInstitute of Physics, University of Augsburg, 86135 Augsburg, Germany\n4Laboratory for Neutron Scattering, and Beijing Key Laboratory of Optoelectronic Functional Materials MicroNano Devices,\nDepartment of Physics, Renmin University of China, Beijing 100872, China\n5Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and Materials Research, 01328 Dresden, Germany\n(Dated: December 22, 2022)\nKagome metals possess peculiar optical spectra consisting of contributions from free charge carri-\ners in a Drude-type response, localized carriers seen as a strongly temperature-dependent localization\npeak, and, in some cases, phonons displaying strong anomalies. The rare-earth kagome metal series,\nRMn6Sn6, provides a marvelous playground to study the electronic properties of kagome metals in\nthe presence of variable magnetic order. Here, we report temperature-dependent reflectivity studies\non two members of the RMn6Sn6family, GdMn 6Sn6(in-plane ferrimagnet) and TbMn 6Sn6(out-\nof-plane ferrimagnet), in a broad energy range (50 - 18000 cm−1, equivalent to 6.2 meV - 2.23 eV)\ndown to 10 K. At high temperatures, a phonon mode at approximately 160 cm−1is observed, which\nbecomes screened out in TbMn 6Sn6below ∼150 K as the localization peak linearly passes through\nthe mode. In GdMn 6Sn6, the disappearance of the phonon is accompanied by the onset of saturation\nof the peak position, suggesting an unusual interplay between the two features.\nProposed by Syˆ ozi in 1951, the kagome lattice quickly\nbecame popular among both theoretical and experimen-\ntal physicists due to its unique magnetic and electronic\nproperties [1, 2]. Consisting of spatially separated metal-\nlic kagome planes, kagome metals are model compounds\nfor studying strong electronic correlations, magnetism,\nand topologically non-trivial states [3]. Here, the itiner-\nant carriers give rise to the peculiar kagome electronic\nband structure hosting dispersionless flat bands, saddle\npoints, as well as linearly dispersing Dirac bands [4–9].\nThe ternary rare-earth series, RMn6Sn6(R= Sc, Y,\nGd-Lu), opens new ways to investigate the influence of\nmagnetism on the electronic properties of kagome met-\nals and hence, distinguish between magnetic-driven and\nkagome layer-driven properties. While these compounds\nhave been studied extensively over the last three decades\nregarding their unusual magnetic structure, they recently\ngained attention in the framework of kagome metals [10–\n12]. These compounds crystallize in the P6/mmm space\ngroup featuring spatially decoupled magnetic Mn-kagome\nplanes stacked along the c-axis, which are stabilized by\nSn1 atoms. Within one unit cell, the kagome layers are\nseparated by non-magnetic Sn2 atoms forming a hon-\neycomb lattice, while RSn3 layers separate the kagome\nplanes from one unit cell to another, as sketched in\nFigs. 1(a) and 1(b). The underlying magnetic structure\nstrongly depends on the rare-earth element separating\nthe layers, resulting in a large variety of ferrimagnetic ( R\n= Gd, Tb, Dy, Ho) and antiferromagnetic ( R= Sc, Y,\nEr, Tm, Yb, Lu) ground states across the series [10, 13].\n∗maxim.wenzel@pi1.physik.uni-stuttgart.deAngle-resolved photoemission spectroscopy (ARPES)\nand Landau level measurements reveal the signatures\nof the kagome lattice, including topologically non-\ntrivial Dirac bands and flat bands in these materi-\nals [7, 9, 12, 14]. Comprising spin-polarized Mn 3 d\nstates with a strong intrinsic spin-orbit coupling, these\ntwo-dimensional kagome bands exhibit non-trivial Chern\nnumbers [6, 7, 15] giving rise to an intrinsic anomalous\nHall effect [16–21]. While the different magnetic struc-\ntures do not seem to affect the main band dispersions\nnear the Fermi energy EF, significantly, a gap at the\nDirac points has been proposed only for the ferrimag-\nnetic systems [12, 22–24]. Moreover, this Chern gap, as\nwell as the energy of the Dirac points ED, can be tuned\nwith the rare-earth element [22]. Here, the number of\nunpaired 4 felectrons of the rare-earth element plays an\nimportant role as a coupling between the 4 fand the 3 d\nelectrons is observed.\nThe key implications of these topological features lie in\nunusual transport properties that crucially rely on charge\ncarriers and their dynamics [12, 27–29]. Especially the ef-\nfect of magnetism is one of the central issues [30]. There-\nfore, here, we study these dynamics and their depen-\ndence on the magnetic order with temperature-dependent\nbroadband Fourier transform infrared spectroscopy stud-\nies on RMn6Sn6systems, namely on GdMn 6Sn6and\nTbMn 6Sn6. While both systems possess an almost iden-\ntical crystal structure and a ferrimagnetic ground state\nbelow room temperature, in the former one, the spins are\naligned within the kagome plane, whereas in the Tb com-\npound, a perpendicular alignment to the kagome layers is\nreported [10, 13, 31–33]. This was confirmed prior to our\noptical study by dc transport and magnetic susceptibil-\nity measurements shown in Figs. 1(c) and 1(d). We fur-arXiv:2208.00756v2 [cond-mat.str-el] 21 Dec 20222\n0.00.40.81.21.61\n01 001 000100000.00.40.81.21.60\n1 0020030001234565.05.56.06.57.0E\n II ab 10 K \n50 K \n100 K \n150 K \n175 K(\nf) σ1 (104Ω-1cm-1)(\ne) \nFrequency (cm-1) 200 K \n225 K \n250 K \n275 K \n300 KTbMn6Sn6 σ1 (104Ω-1cm-1) \nE II abGdMn6Sn6(c)G dMn6Sn6 \nTemperature (K)(d)T bMn6Sn6 χ (µB/f.u.)H\n = 0.1 TH\n II abTr0\n50100150ρ\n (µΩcm)3\n003300.30.60.9dρ/dTT\n (K)H = 0.1 TH\n II ab χ (µB/f.u.)0\n50100150ρ\n (µΩcm)0.010 .11 Energy (eV)\nFIG. 1. (a) and (b) Crystal and magnetic structure below 300 K of GdMn 6Sn6and TbMn 6Sn6, respectively [13, 25]. (c) and (d)\nMagnetic susceptibility and dc resistivity curves measured in the ab-plane. The Curie temperature of both systems lies above\nthe measured temperature range; however, a spin reorientation from the basal plane near to the c-axis around Tr∼310 K\nis visible for TbMn 6Sn6. For GdMn 6Sn6, no anomalies are observed in the measured temperature range. Open circles are\nthe dc resistivity values obtained from the Hagen-Rubens fits of the optical measurements as explained in the Supplemental\nMaterial [26]. (e) and (f) Temperature-dependent in-plane optical conductivity with the dotted lines being the Hagen-Rubens\nextrapolation to low energies.\nther performed density functional theory plus Hubbard\nU(DFT+ U) calculations to evaluate the electronic struc-\ntures, revealing the correlated character of the RMn6Sn6\nseries. Due to localization effects, the optical response\nof the charge carriers splits into the conventional Drude\npart and a prominent low-energy peak. This peak shows\na clear dependence on the magnetic order and underlies\nthe magnetic tunability of this compound family.\nFigures 1(e) and 1(f) display the temperature-\ndependent real part of the in-plane optical conductiv-\nity of GdMn 6Sn6and TbMn 6Sn6, respectively. At first\nglance, the spectra are remarkably similar and resem-\nble the spectrum of the ferromagnetic Fe 3Sn2[34, 35].\nConsistent with the metallic nature of these compounds,\na very narrow Drude component is observed at low en-\nergies, which becomes even sharper upon cooling. For\nGdMn 6Sn6, only the tail of this feature is visible even at\n300 K. Two step-like absorption features can be identified\nin the otherwise relatively flat conductivity at high ener-\ngies. Very similar steps were interpreted as the signature\nof two-dimensional Dirac fermions in Fe 3Sn2. In addi-\ntion to the sharp Drude component and interband tran-\nsitions, a phonon mode around 160 cm−1is observed.\nFurthermore, we have realized that the low-energy dy-\nnamics cannot be reproduced only with a single Drude\ncomponent, but an additional peak-like absorption fea-\nture is required as shown in Fig. 2 (a) and (b) for thedata at 300 K. With this peak showing a strong red-shift\nupon cooling, it puts the RMn6Sn6series on common\nground with other kagome metals and clearly separates\nthis feature from other low-energy transitions, which are\ninterband in nature [34, 36–38].\nA closer look at the low-energy regime reveals sub-\nstantial differences between the two ferrimagnetic com-\npounds. Figures 2 (b) and 2(d) show the temper-\nature evolution of this so-called localization peak in\nGdMn 6Sn6and TbMn 6Sn6after subtracting the fitted\nDrude, phonon, and interband contributions from the\nexperimental optical conductivity. Not only is the lo-\ncalization peak more pronounced in the in-plane ferri-\nmagnetic system GdMn 6Sn6, but the peak position sat-\nurates at low temperatures, as shown in Fig. 2(a). In\ncontrast, a linear red-shift over the whole temperature\nrange is observed in TbMn 6Sn6[see Fig. 2(c)]. Hence,\nthe peak moves out of the measured range at low tem-\nperatures, and its position has to be estimated from its\nhigh-frequency tail, as well as by considering the linear\nbehavior of the shift at higher temperatures, leading to\nincreasing error bars of the fits.\nVisually, the temperature evolution of the peak posi-\ntion in GdMn 6Sn6looks strikingly similar to the behavior\nin Fe 3Sn2. For the latter, a possible coupling between the\nlocalization peak and the underlying magnetic structure\nis discussed since the linear scaling breaks down after a3\n024681\n0100100010000024681\n01 001 0001 00000.00.20.40.60.81.01.20.00.20.40.60.81.01.20\n100200300(\nf)(e)(\nd) Peak Position (cm-1)G dMn6Sn6ω\nPhonon(c)0\n1 002003000100200300T\nbMn6Sn6TbMn6Sn6 Peak Position (cm-1)T\nemperature (K)ωPhonon \nσ1 (103Ω-1cm-1) \n300 K200 K100 K10 KGdMn6Sn60.010.11 Energy (eV) \n σ1 (103Ω-1cm-1)F\nrequency (cm-1)10 K1\n00 K2\n00 K3\n00 K(b) \nFrequency (cm-1) experiment \ntotal fit \nDrude \nlocalization \nphonon \ninterband \nσ1 (104Ω-1cm-1)3\n00 KT\nbMn6Sn6 σ1 (104Ω-1cm-1) \nG\ndMn6Sn6(a)3\n00 K0.010 .11 Energy (eV)1\n01 0010000.70.80.91.0ReflectivityF\nrequency (cm-1)1\n01 0010000.70.80.91.0ReflectivityF\nrequency (cm-1)\nFIG. 2. (a) and (b) Decomposed optical conductivity at 300 K, consisting of a Drude component (purple), a localization peak\n(blue), a phonon mode (green), and several interband transitions (orange). The insets show the total fit to the measured\nreflectivity. Details on the fitting process as well as the decomposed spectra at lower temperatures can be found in the\nSupplemental Material [26]. (c) and (d) Temperature dependence of the localization peak position. The red dashed line marks\nthe phonon mode, while the red arrow indicates the temperature where the mode disappears. (e) and (f) Temperature evolution\nof the localization peak, obtained by subtracting the fitted Drude, phonon mode, and interband contributions from the spectra.\nThe solid lines are the Fratini model fits to the total experimental conductivity as described in the Supplemental Material [26].\nreorientation of the Fe-spins at 120 K [34, 39]. Addition-\nally, the shape of the peak transforms into a sharp Fano\nresonance. The saturation as observed in GdMn 6Sn6was\nalso reported in the non-magnetic KV 3Sb5, suggesting\nthat the origin of this effect may be other than mag-\nnetic. Additionally, no change of the in-plane ferrimag-\nnetic structure of GdMn 6Sn6is reported below room tem-\nperature; hence, the primary cause for the observed satu-\nration must be something else. Nevertheless, a common-\nality between the two magnetic systems is the in-plane\ndirection of the magnetic moments in both Fe 3Sn2below\nits spin-reorientation transition and GdMn 6Sn6.\nOne plausible explanation for the observed saturation\nuniting magnetic and non-magnetic kagome metals is the\ninvolvement of a phonon mode. Indeed, phonons and\ntheir importance for the electronic structure of kagome\nmetals have been studied in multiple compounds. In the\nAV3Sb5family, phonons are discussed to be the driv-\ning force behind the charge-density-wave formation and\nthe low-temperature superconductivity [40, 41]. Opti-\ncal measurements revealed strong phonon anomalies as-\nsociated with a coupling of the phonon modes to the\nelectronic background in KV 3Sb5and RbV 3Sb5[36, 37].\nFurthermore, a strong interplay between phonons and\nfermionic degrees of freedom was revealed by scanningtunneling microscopy (STM) studies of paramagnetic\nCoSn [42].\nDFT calculations, shown in the Supplemental Mate-\nrial [26], reveal a total number of nine IR-active phonon\nmodes in each compound. Four of these modes have the\nA2usymmetry involving out-of-plane atomic displace-\nments and hence, cannot be detected by our in-plane\nmeasurements. While in highly metallic systems phonon\nmodes are often too weak to be detected and/or screened\nby the free carriers, our measurements were able to cap-\nture a prominent E 1umode around 160 cm−1at room\ntemperature. At low temperatures, this mode disap-\npears in both compounds. At first glance, this anomalous\nbehavior might be explained by a structural distortion;\nhowever, low-temperature XRD studies report almost no\nchanges in the crystal structure of RMn6Sn6down to 2 K\n[13, 25]. Hence, an interplay between the phonon mode\nand the localization peak has to be considered as a possi-\nble scenario, not least because both features are located\naround the same energy range.\nFor a further comparison of the two features, the posi-\ntion of the phonon mode is marked with the red dashed\nline in Figs. 2(a) and 2(c), while the red arrow points\nat the temperature at which the phonon mode disap-\npears in each compound. In TbMn 6Sn6, the phonon4\nmode disappears as soon as the localization peak passes\nthrough it, suggesting that the localization peak screens\nout the phonon mode. On the other hand, a more com-\nplex relationship between the two features is observed in\nGdMn 6Sn6. Here, the phonon mode shows an enhance-\nment and a slight broadening as the localization peak\npasses through it, and is retained even below the cross-\ning over a narrow temperature range. Eventually, the\nmode disappears around the temperature where the po-\nsition of the localization peak saturates. This behavior\nsuggests an unusual coupling between the phonon mode\nand the localization peak in GdMn 6Sn6. Based on the ob-\nservation that the strong localization peak anomalies ap-\npear in the in-plane ferromagnetic system, one plausible\nexplanation would be a magneto-elastic coupling to the\nin-plane infrared-active phonon mode. Additionally, the\nrare-earth element could directly influence the phonon\nmode and hence its interplay with the localization peak.\nUltimately, an interplay with some other bosonic ex-\ncitations such as magnons, for instance, could as well\nlead to the distinct behavior of the localization peak in\nGdMn 6Sn6compared to TbMn 6Sn6. Indeed, magnon\nbands extending to energies up to ∼100 meV have\nbeen reported in several members of the RMn6Sn6family\n[43, 44].\nThe presence of a red-shifting localization peak is a\ncommon occurrence in systems with slow electron dy-\nnamics, such as organic conductors, cuprates, and man-\nganites [45, 46], many of them being strongly correlated\nmaterials. Hence, we now turn to analyzing the elec-\ntronic correlations in the RMn6Sn6series. Figures 3(a)\nand 3(b) show the comparison between the experimental\nand the calculated optical conductivities using DFT tak-\ning into account the different magnetic structures. For\nall calculations, a Hubbard UR= 10 eV was added\nto the rare-earth element with the DFT+ Umethod us-\ning the double-counting correction in the fully localized\nlimit to treat the strongly correlated 4 felectrons [9, 47–\n49]. In the case of GdMn 6Sn6, a good agreement with\nthe experiment is found, while for TbMn 6Sn6, the low-\nenergy spectral weight cannot be reproduced with this\nmethod. The agreement is improved by adding a Hub-\nbardUMn= 0.4 eV to the Mn-atoms. Another possibility\nis shifting the Fermi energy down by 47 meV; however,\nthis requires removing one electron from the structure,\nwhich is hard to reconcile with the system.\nAlthough with different adjustments, one can bring the\ncalculations to the experiment’s level, in either case, the\nenergy of the calculated conductivity needs to be rescaled\nby a factor of 2.5 in GdMn 6Sn6(2 in TbMn 6Sn6). A very\nsimilar scaling factor was previously reported for ARPES\nmeasurements of GdMn 6Sn6[9]. This suggests that these\nsystems are clearly beyond DFT, and electronic correla-\ntions therein can not be fully treated on the mean-field\nDFT+ Ulevel.\nWe further observed the step-like absorption features,\ncombined with the relatively flat optical conductivity, as\nthe potential signatures of the Dirac points in these sys-\n0.00 .20 .40 .60 .81 .00123456 \nσ1 (103Ω-1cm-1) experiment at 10 K \nstoichiometricG\ndMn6Sn6(\nb)(\nc)1001 0001 00000123456 \nFrequency (cm-1) σ1 (103Ω-1cm-1) \n experiment at 10 K \nstoichiometric \nµ = - 47 meV \nUMn = 0.4 eVT\nbMn6Sn6(a) \nS\nWDrude / SWbandReMn6Sn6KV3Sb5RbV3Sb5CsV3Sb5correlationsC\no3Sn2S2ZrSiSeZrSiSW\nTe20.11 Energy (eV)FIG. 3. (a) and (b) Experimental interband transitions along\nwith the DFT+ Ucalculated optical conductivity. For all cal-\nculations a Hubbard UR= 10 eV was added to the rare-earth\nelement. Furthermore, the energy scale of the calculated con-\nductivity is rescaled for a better comparison with the experi-\nment. (c) Correlation scaling for different kagome metals and\nother topological materials taken from ref. [50].\ntems. Considering that there are two Dirac points, one\nabove and one below the Fermi energy (see Supplemen-\ntal Material [26]), one would expect these step-like ab-\nsorption features to appear [34]. This interpretation be-\ncomes even more tempting when the energies of the steps\nare compared with the ARPES measurements. However,\nconsidering the relatively high energy range of these fea-\ntures and the significant number of bands crossing the\nFermi energy, the step-like absorption is most likely just\na cumulative effect of different contributions; hence, one\nshould be careful in its assignment. On the other hand,\nabsorption features at lower energies ( ω < 1000 cm−1)\ncan be related to transitions between bands very close\nto the Fermi energy, most probably involving transitions\nbetween the saddle points nearby the Mpoint, as shown\nin our band structure calculations in the Supplemental\nMaterial [26].5\nAlthough the RMn6Sn6series lies beyond the limits\nof the DFT+ Umethods presented here, the calcula-\ntions can be used for an initial assessment of the cor-\nrelation strength. As proposed previously for different\ncompounds, including cuprates, iron pnictides, and topo-\nlogically nontrivial Dirac systems [50, 51], the ratio of\nthe spectral weight of the mobile carriers from the ex-\nperiment and the DFT calculations can be used as a\ngauge of electronic correlations. Here, SW Drude /SW band\nis close to 1 for uncorrelated materials, while the ra-\ntio becomes zero for Mott insulators showing the most\ncorrelated behavior. Figure 3(c) depicts this scaling\nfor the AV3Sb5series and topological semimetals taken\nfrom refs. [36, 50]. From the calculations, we can deter-\nmine a rough value of SW Drude /SW band≈0.2, point-\ning towards much stronger correlations in comparison\nwith the AV3Sb5series and other kagome metals re-\nported to date. Moreover, no significant difference be-\ntween GdMn 6Sn6and TbMn 6Sn6is observed, whereas\nthe correlation strength changes drastically between dif-\nferent members of the AV3Sb5family.\nIn summary, we establish the correlated nature of ferri-\nmagnetic kagome metals of the RMn6Sn6family and un-\ncover partial localization of charge carriers manifested by\nthe prominent low-energy peak in the optical conductiv-\nity. The temperature evolution of this peak is sensitive to\ndetails of the magnetic order. While in TbMn 6Sn6, the\nlocalization peak red-shifts linearly through the whole\ntemperature range upon cooling and screens out thephonon mode at ∼160 cm−1, it displays different char-\nacteristics in GdMn 6Sn6. Here, the peak is more pro-\nnounced, while its position saturates at low tempera-\ntures. This dissimilar behavior indicates a major dif-\nference in low-energy degrees of freedom that damp elec-\ntron dynamics and, consequently, should affect transport\nproperties at low temperatures. Both compounds dis-\nplay a strongly correlated character, as a good agreement\nwith the experimental interband transitions is only found\nafter rescaling the energy of the calculated optical con-\nductivity, and the experimental Drude spectral weight is\ndrastically lower than the DFT prediction.\nThe authors acknowledge the fruitful discussion with\nSimone Fratini, and technical support by Gabriele Un-\ntereiner. We also thank Falk Lissner and Rainer Niewa\nfor the XRD measurements. H.C.L. was supported\nby National Key R&D Program of China (Grant No.\n2018YFE0202600), the Beijing Natural Science Foun-\ndation (Grant No. Z200005), the Fundamental Re-\nsearch Funds for the Central Universities and Research\nFunds of Renmin University of China (RUC) (Grant\nNos. 18XNLG14, 19XNLG13, and 19XNLG17), and\nthe Beijing National Laboratory for Condensed Matter\nPhysics. The work has been supported by the Deutsche\nForschungsgemeinschaft (DFG) via Grants No. UY63/2-\n1, No. DR228/48-1, and No. DR228/51-1. E. U. ac-\nknowledges the European Social Fund and the Baden-\nW¨ urttemberg Stiftung for the financial support of this\nresearch project by the Eliteprogramme.\n[1] I. Syˆ ozi, Statistics of Kagom´ e Lattice, Progress of Theo-\nretical Physics 6, 306 (1951).\n[2] M. Mekata, Kagome: The Story of the Basketweave Lat-\ntice, Physics Today 56, 12 (2003).\n[3] D. F. Liu, A. J. Liang, E. K. Liu, Q. N. Xu, Y. W.\nLi, C. Chen, D. Pei, W. J. Shi, S. K. Mo, P. Dudin,\nT. Kim, C. 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Sofo, Linear optical properties\nof solids within the full-potential linearized augmented\nplanewave method, Comput. Phys. Commun. 175, 1\n(2006).1\nSupplemental Material for ”Effect of magnetism and phonons on localized carriers in\nferrimagnetic kagome metals GdMn 6Sn6and TbMn 6Sn6”\nM. Wenzel, A. A. Tsirlin, O. Iakutkina, Q. Yin, H.C. Lei, M. Dressel, and E. Uykur\nI. CRYSTAL GROWTH\nSingle crystals of GdMn 6Sn6and TbMn 6Sn6were grown by the Sn flux method with Gd/Tb : Mn : Sn = 1 : 6 : 20\nmolar ratio. Gd/Tb (ingots), Mn (pieces) and Sn (grains) were put into an alumina crucible and sealed in a quartz\nampule under partial argon atmosphere. The sealed quartz ampule was heated up to 1373 K and kept there for 20 h\nto ensure the homogeneity of melt. After that, for GdMn 6Sn6, the temperature was rapidly cooled down to 1023 K\nfor 20 h and subsequently cooling down to 873 K at 2 K/h. For TbMn 6Sn6, the temperature was cooled down directly\nto 873 K with the rate of 5 K/h. Finally, the ampules were taken out of furnace and the single crystals were separated\nfrom the flux by a centrifuge.\nII. EXPERIMENTAL DETAILS\nPrior to our optical study, we carried out four-point dc resistivity and magnetic susceptibility measurements within\ntheab-plane to monitor possible magnetic transitions and confirm the stoichiometry. For the magnetic susceptibility\nmeasurements, a small magnetic field of H= 0.1 T was applied. The obtained data agrees well with the literature\nand confirms the spin reorientation in TbMn 6Sn6around 310 K from the basal plane near to the c-axis [S1]. For\nGdMn 6Sn6, all magnetic transitions are above the measured temperature range; hence, we observed no anomalies in\nour data [S2].\nFreshly cleaved samples with the dimensions of 2 x 2 mm2surface area and thickness of about 100 µm were used for\nthe optical study. Here, temperature-dependent reflectivity measurements were performed in the ab-plane covering\na broad frequency range from 50 to 18000 cm−1(6.2 meV - 2.23 eV) down to 10 K, as shown in Fig. S1. For the\nhigh-energy range ( ω > 600 cm−1) a Bruker Vertex 80v spectrometer with an incorporated Hyperion IR microscope\nwas used, while the low-energy range was measured with a Bruker IFS113v spectrometer and a custom-built cryostat.\nFreshly evaporated gold mirrors served as reference in these measurements. The absolute value of the reflectivity was\nobtained by an in-situ gold-overcoating technique in the far-infrared range, as described in ref. [S3]\nConsidering the metallic nature of the samples, we used Hagen-Rubens extrapolation below 50 cm−1, while x-ray\nscattering functions were utilized for the high-energy range to extrapolate the data [S4]. The optical conductivity is\nthen calculated from the measured reflectivity by standard Kramers-Kronig analysis.\n101 001 0001 00000.50.60.70.80.91.01\n01 001 0001 0000 Reflectivity \n Frequency (cm-1) 10 K \n50 K \n100 K \n150 K \n175 K \n200 K \n225 K \n250 K \n275 K \n300 K \nG\ndMn6Sn6E II ab(b) \nFrequency (cm-1) \nT\nbMn6Sn6E II ab(a)0.010 .11 Energy (eV)0\n.010 .11 Energy (eV)\nFIG. S1. Temperature-dependent reflectivity over a broad frequency range (50 to 18000 cm−1) measured in the ab-plane. The\ndotted lines are the Hagen-Rubens extrapolations.arXiv:2208.00756v2 [cond-mat.str-el] 21 Dec 20222\nIII. DECOMPOSITION OF OPTICAL SPECTRA\nDifferent contributions to the total optical conductivity were modeled with the Drude-Lorentz approach. With ε∞\nbeing the high-energy contributions to the real part of the dielectric permittivity, the dielectric function [˜ ε=ε1+iε2]\nis expressed as\n˜ε(ω) =ε∞−ω2\np,Drude\nω2+iω/τ Drude+/summationdisplay\njΩ2\nj\nω2\n0,j−ω2−iωγj. (S1)\nHere,ωp,Drude and 1/τDrude are the plasma frequency and the scattering rate of the itinerant carriers, respectively.\nThe parameters ω0,j, Ωj, andγjdescribe the resonance frequency, width, and the strength of the jthexcitation,\nrespectively.\nFollowing the approach of previous optical studies of kagome metals, we base our analysis of the localization peak\non the displaced Drude formalism proposed in 2014 by Fratini et al. [S5]. Here, possible localization effects, due to\ninteractions of charge carriers with low-energy degrees of freedom, such as phonons, electric or magnetic fluctuations,\nare considered by modifying the classical Drude response with an additional backscattering of the electrons. This\n101 001 0001 00000.00.40.81.21\n01 001 0001 00000.00.40.81.20.00.40.81.20\n.00.40.81.20\n.00.40.81.20.00.40.81.2T\n = 10 K σ1 (104Ω-1cm-1) \n \nFrequency (cm-1)GdMn6Sn6G\ndMn6Sn6T = 100 KT = 200 KG\ndMn6Sn6 σ1 (104Ω-1cm-1) \n F\nrequency (cm-1)T = 10 KT = 200 KT\n = 100 K1\n01001000100000.40.60.81.0ReflectivityF\nrequency (cm-1)101001000100000.40.60.81.0ReflectivityF\nrequency (cm-1) \n σ1 (104Ω-1cm-1)(a)(\nb)(\nc)(d)(\ne)(\nf) σ1 (104Ω-1cm-1) σ1 (104Ω-1cm-1)TbMn6Sn6T\nbMn6Sn6T\nbMn6Sn6 σ1 (104Ω-1cm-1)1\n01001000100000.40.60.81.0ReflectivityF\nrequency (cm-1)1\n01001000100000.40.60.81.0ReflectivityF\nrequency (cm-1)101001000100000.40.60.81.0ReflectivityF\nrequency (cm-1)101001000100000.40.60.81.0ReflectivityF\nrequency (cm-1)0.010 .11 Energy (eV)0\n.010 .11 Energy (eV)\nFIG. S2. Decomposed optical conductivity at 200 K, 100 K, and 10 K, consisting of a Drude component (purple), a localization\npeak (blue), a phonon mode (green), and several interband transitions (orange) modeled with the Drude-Lorentz approach.\nThe insets show the total fit to the measured in-plane reflectivity.3\nLorentzian 1 GdMn 6Sn6 TbMn 6Sn6\nT(K) ∆ ε ω 0(cm−1)γ(cm−1) ∆ε ω 0(cm−1)γ(cm−1)\n10 172.705 432.556 651.473 909.009 332.377 682.769\n50 148.140 432.556 619.329 850.895 332.377 758.926\n100 156.009 432.556 660.067 760.626 332.377 899.605\n150 198.806 432.556 853.54 865.581 332.377 807.755\n175 189.058 432.556 901.554 - - -\n200 202.635 432.556 767.635 1024.99 332.377 1041.35\n225 215.642 432.556 1034.66 1050.23 332.377 982.126\n250 192.204 432.556 1164.59 1180.61 332.377 1006.25\n275 163.49 432.556 1105.97 1265.02 332.377 1002.79\n300 206.121 432.556 1215.73 1251.79 332.377 1049.3\nLorentzian 2 GdMn 6Sn6 TbMn 6Sn6\nT(K) ∆ ε ω 0(cm−1)γ(cm−1) ∆ε ω 0(cm−1)γ(cm−1)\n10 95.7128 1698.15 2554.68 69.8246 1705.89 1924.48\n50 99.9937 1698.15 2788.11 72.9802 1705.89 1985.85\n100 100.325 1698.15 2776.85 76.0712 1705.89 2027.15\n150 107.838 1698.15 2994.56 88.1694 1705.89 2441.02\n175 110.115 1698.15 3048.64 - - -\n200 113.452 1698.15 3017.55 98.043 1705.89 2775.34\n225 108.198 1698.15 3016.64 103.049 1705.89 2831.12\n250 126.412 1698.15 3437.89 93.1684 1705.89 2754.83\n275 127.789 1698.15 3432.88 96.9267 1705.89 2883.27\n300 121.526 1698.15 3432.53 100.372 1705.89 3007.04\nLorentzian 3 GdMn 6Sn6 TbMn 6Sn6\nT(K) ∆ ε ω 0(cm−1)γ(cm−1) ∆ε ω 0(cm−1)γ(cm−1)\n10 50.7642 6479.28 11939 74.6754 6338.93 12808.7\n50 47.1342 6479.28 11939 75.5724 6338.93 12808.7\n100 49.0074 6479.28 11939 76.3284 6338.93 12808.7\n150 50.8744 6479.28 11939 76.336 6338.93 12808.7\n175 50.9762 6479.28 11939 - - -\n200 53.1189 6479.28 11939 77.1148 6338.93 12808.7\n225 53.6501 6479.28 11939 78.7435 6338.93 12808.7\n250 52.5772 6479.28 11939 76.814 6338.93 12808.7\n275 53.1029 6479.28 11939 76.8908 6338.93 12808.7\n300 55.8173 6479.28 11939 73.3722 6338.93 12808.7\nLorentzian 4 GdMn 6Sn6 TbMn 6Sn6\nT(K) ∆ ε ω 0(cm−1)γ(cm−1) ∆ε ω 0(cm−1)γ(cm−1)\n10 18.9776 31246.3 80843.9 19.2266 27418.1 61229.7\n50 19.2876 31246.3 80843.9 19.0763 27418.1 61229.7\n100 19.3669 31246.3 80843.9 18.9045 27418.1 61229.7\n150 19.1541 31246.3 80843.9 19.0954 27418.1 61229.7\n175 19.0385 31246.3 80843.9 - -\n200 19.0366 31246.3 80843.9 18.7211 27418.1 61229.7\n225 18.8462 31246.3 80843.9 18.9102 27418.1 61229.7\n250 19.0347 31246.3 80843.9 18.7916 27418.1 61229.7\n275 19.029 31246.3 80843.9 18.7916 27418.1 61229.7\n300 19.2212 31246.3 80843.9 19.0186 27418.1 61229.7\nTABLE I. Fit parameters of the total number of four Lorentzians used to model the interband optical transitions in GdMn 6Sn6\nand TbMn 6Sn6.\nleads to a shift of the zero-frequency response to a finite value:\n˜σlocalization (ω) =C\nτb−τtanh{/planckover2pi1ω\n2kBT}\n/planckover2pi1ω·Re/braceleftbigg1\n1−iωτ−1\n1−iωτb/bracerightbigg\n. (S2)\nHere,Cis a constant, /planckover2pi1is the reduced Planck constant, kBthe Boltzmann constant, τbthe backscattering time, and4\n0100200300040801200\n10020030002004006000\n1002003000204060800\n1002003000100200300400500 \nTemperature (K) 1/τDrude (cm-1) \n Temperature (K) \n 1/τlocalization (cm-1)1\n/τ1\n/τb0\n20406080100ρ\n (µΩcm)(d)( c)T\nbMn6Sn6 1/τDrude (cm-1)T\nemperature (K)GdMn6Sn60\n4080120(b)ρ\n (µΩcm)(a) \nTemperature (K)1/τlocalization (cm-1) \n 1\n/τ1\n/τb\nFIG. S3. Elastic scattering rate, 1/ τ(blue) and backscattering rate, 1/ τb(red) of the Fratini model fits. Additionally, the\nelastic scattering rate of the Drude contribution (green), overlaid with the dc resistivity (orange), is given.\nτthe elastic scattering time of the standard Drude model.\nThe total dielectric permittivity takes the form\n˜ε(ω) = ˜εDrude (ω) + ˜εLorentz (ω) + ˜εlocalization (ω). (S3)\nThe complex optical conductivity [˜ σ=σ1+iσ2] is then calculated as\n˜σ(ω) =−iω[˜ε(ω)−ε∞]/4π. (S4)\nFig. S2 shows the decomposed optical conductivity at various temperatures. The spectra were fitted in a consistent\nway for all temperatures using one Drude contribution (purple), a total number of four Lorentzians (see Table I for the\nparameters) to describe the interband optical transitions (orange), a sharp Lorentzian for the phonon mode (green),\nas well as the Fratini model to describe the localization peak (blue). At 300 K the localization peak is only weakly\npronounced and additionally screened by low-energy interband transitions in TbMn 6Sn6. On the other hand, the\npeak is clearly visible by the eye in the spectrum of GdMn 6Sn6due to the absence of strong low-energy interband\nabsorptions and the sharper Drude contribution.\nIn Fig. S3, we show the elastic scattering rate and the backscattering rate obtained from the Fratini model fits to\nthe optical spectra, as well as the scattering rate of the classical Drude model. When overlaying the Drude scattering\nrate with the dc resistivity, a remarkably similar temperature evolution is found in TbMn 6Sn6, indicating that the dc\ntransport is governed by the free electrons. On the other hand, a clear deviation of this behavior above ∼200 K is\nobserved in GdMn 6Sn6. Considering the akin temperature dependence of the resistivity to the elastic scattering rate\nof the localization peak at high temperatures, this signals a significant contribution of the incoherent carriers to the\ndc transport in GdMn 6Sn6.\nIV. PHONON CALCULATIONS\nPhonon calculations were performed on the density-functional theory (DFT) level in VASP [S6, S7] using the refined\nstructural parameters given in Table II and the Perdew-Burke-Ernzerhof (PBE) flavor of the exchange-correlation\npotential [S8]. Spin-orbit coupling was included, and different directions of the magnetic moment were chosen.\nFerromagnetic order was introduced for Mn atoms, whereas f-electrons of Gd and Tb were placed into the core, and\nonly a small residual magnetic moment due to d-electrons appeared on these atoms. This simplification was necessary\nin order to achieve good convergence of total energies and forces, as required in phonon calculations. The 8 ×8×4\nk-mesh was used.\nFrequencies of Γ-point phonons were obtained from the built-in procedure with frozen atomic displacements of\n0.015 ˚A. Fig. S4 (a) and (b) depict the calculated IR-active phonon modes of GdMn 6Sn6and TbMn 6Sn6. In both\ncompounds, a total number of nine IR-active modes are expected, which do not significantly vary in frequency with\nchanges in the direction of the magnetic moments. Four of these are A 2uc-axis modes (dashed lines) and hence,\ncannot be observed in our in-plane measurements. The remaining five modes are E 1umodes (solid lines) involving\nin-plane atomic displacements. However, the appearance of phonon modes in reflectivity spectra strongly depends\non the intensity of the phonon mode, especially for highly metallic samples, as in the case of the ReMn6Sn6series.\nHence, it is possible that only the E 1umode around 160 cm−1is strong enough to be captured by our measurements.5\nGdMn 6Sn6\na=b= 5.5399(2) ˚A,c= 9.0318(5) ˚A\nV= 240.054(18) ˚A3\nP6/mmm\nλ= 0.71073 ˚A\nΘmin= 0.41◦, Θmax= 27.48◦\n−7≤h≤7,−7≤k≤7,−11≤l≤11\nRint= 0.0656\nAtomx/a y/b z/c U iso(˚A2)\nGd 0 0 0.5 0.01120(34)\nMn 0.5 0 0.25224(13) 0.01074(34)\nSn11\n32\n30 0.01147(33)\nSn21\n32\n30.5 0.01024(33)\nSn3 0 0 0.16206(11) 0.01184(33)TbMn 6Sn6\na=b= 5.5305(2) ˚A,c= 9.0223(5) ˚A\nV= 238.988(18) ˚A3\nP6/mmm\nλ= 0.71073 ˚A\nΘmin= 0.41◦, Θmax= 27.48◦\n−6≤h≤7,−7≤k≤7,−11≤l≤11\nRint= 0.0787\nAtomx/a y/b z/c U iso(˚A2)\nTb 0 0 0.5 0.01182(41)\nMn 0.5 0 0.25244(16) 0.01129(42)\nSn11\n32\n30 0.01155(40)\nSn21\n32\n30.5 0.01078(40)\nSn3 0 0 0.16276(13) 0.01211(40)\nTABLE II. Details of data collection and refined structural parameters for GdMn 6Sn6(left) and TbMn 6Sn6(right).\nThis mode can be represented with a sharp Lorentzian,\nσ1(ω) =∆εω2ω2\n0γ\n4π[(ω2−ω2\n0)2+γ2ω2]. (S5)\nHere, ∆εstands for the intensity, ω0for the resonance frequency, and γfor the linewidth. Consistent with the\nhardening of the lattice, we observe a slight blue shift of the mode upon cooling in both compounds. In GdMn 6Sn6, a\nsignificant enhancement of intensity and a slight broadening of the mode are observed as the localization peak crosses\nthe respective phonon mode. On the other hand, no such changes are observed in TbMn 6Sn6. Here, both the intensity\nas well as the linewidth stay constant within the error bars of our fits.\nV. CALCULATION OF THE OPTICAL CONDUCTIVITY\nDFT calculations of the band structure and optical conductivity were performed in the Wien2K code [S9] using\nthe same PBE functional [S8]. Spin-orbit coupling was included in all calculations. For a realistic implementation\nof the magnetic structures, the [100]-direction of the magnetic moments was chosen for GdMn 6Sn6, while the [001]-\ndirection was set for TbMn 6Sn6. Additionally, an antiferromagnetic coupling between the Mn- and rare-earth-atoms\nwas implemented. Moreover, a Hubbard UGd/Tb = 10 eV was added to the 4 fshell of the rare-earth element using\nthe DFT+Umethod with the FLL (fully localized limit) double-counting correction to push the minority 4 fstates to\nenergies well above the Fermi level. DFT calculations were converged on the 15 ×15×4k-mesh. Optical conductivity\nwas calculated within Wien2K [S10] on a denser 26 ×26×14k-mesh.\nFig. S5 shows the calculated band structures along high-symmetry paths of the first Brillouin zone. Both compounds\npossess flat bands around 0.5 eV and saddle points nearby the Mpoint. The Dirac points nearby Kare marked by\ncircles, and their energies are noted in Table III. In the case of a two-dimensional Dirac point, the optical conductivity\nis supposed to show a sharp Drude component along with a step-like onset at 2 |ED|, followed by a frequency-\nindependent behavior. Hence, the interpretation of the observed steps in the optical conductivity as the signature of\ntwo-dimensional Dirac fermions is very tempting. The obtained Dirac cone energies from our experiment are noted in\nTable III. A direct comparison with our calculations reveals a remarkable agreement of the determined energies. On\nthe other hand, a comparison with ARPES studies shows a larger deviation of the energies of the second Dirac point.\nHowever, it should be noted that ARPES only probes the states below the Fermi energy leading to less accurately\ndetermined values.\nDespite the good agreement between the experiment and our calculations, the step-like absorption features shown\nin Fig. S6 should be interpreted cautiously. A closer look at the calculated bandstructure reveals the large number\nof bands crossing the Fermi energy in these compounds. Thus, the multi-band nature of the ReMn6Sn6series should\nnot be disregarded.\nFor TbMn 6Sn6, the accuracy of the calculated optical conductivity in comparison with the experiment increases\nwhen either shifting the Fermi energy down by 47 meV or adding a Hubbard UMn= 0.4 eV to the Mn-atoms. However,6\n20022525027530002040602\n002252502753001571581591601612\n00225250275300468101214608 01 001 201 401 601 802 002 202 40 \nGdMn6Sn6 \nTbMn6Sn6 ΔεT\nemperature (K)(c) \n ω0 (cm-1)T\nemperature (K)(d) \n γ (cm-1)T\nemperature (K)(e) \n G\ndMn6Sn6(a)E1uA\n2u0.0100 .0150 .0200 .025Energy (eV)T\nbMn6Sn6 IR-active phonon modesF\nrequency (cm-1)(b)\nFIG. S4. (a) and (b) Calculated IR-active phonon frequencies of GdMn 6Sn6and TbMn 6Sn6. The solid lines represent the\nin-plane E 1umodes while dashed lines mark A 2umodes involving atomic displacements along the c-axis.(c)-(e) Fit parameters\nof the observed phonon mode in the optical spectra corresponding to the E 1umode marked by the red area in (a) and (b).\n-1.0-0.50.00.51.0-\n1.0-0.50.00.51.0 \nE - EF (eV)GdMn6Sn6/s61511\n M K /s61511 initial \nUMn = 0.4 eV/s61511\n M K /s61511E - EF (eV) \nTbMn6Sn6\nFIG. S5. DFT+ Uband structures for GdMn 6Sn6(left) and TbMn 6Sn6(right) shown along high-symmetry paths of the first\nBrillouin zone. The observed Dirac points at the Kpoint are marked with circles and their energies are noted in Table III.\nin both cases, the energy of the first Dirac point shifts above the Fermi level, which is not expected from ARPES\nstudies on the ReMn6Sn6series.7\n40008 00012000160000123456784\n0008 0001200016000012345678G\ndMn6Sn6 experiment at 10 K \nfit \nsteps σ1 (103Ω-1cm-1)F\nrequency (cm-1)TbMn6Sn6(b) σ1 (103Ω-1cm-1) \nF\nrequency (cm-1)(a)0.40 .81 .21 .62 .0Energy (eV)0\n.40 .81 .21 .62 .0Energy (eV)\nFIG. S6. Experimental optical conductivity after subtracting the localization peak and the low-energy interband transitions.\nThe remaining spectra resemble the optical conductivity of two-dimensional Dirac fermions. The steps at 2 |ED|are highlighted\nwith dots.\noptical study calculations ARPES estimates\nED1(meV)ED2(meV)ED1(meV)ED2(meV)ED1(meV)ED2(meV)\nGdMn 6Sn6 63 291 - 42 233 - 42 [S11] 170 [S12]\nTbMn 6Sn6 65 298 - 41 239 not reported 130 [S13]\nTABLE III. Energies of the Dirac points obtained from the optical study at T= 10 K, the DFT+ Ucalculations, and estimates\nfrom ARPES measurements.\n[S1] D. C. Jones, S. Das, H. Bhandari, X. Liu, P. Siegfried, M. P. Ghimire, S. S. Tsirkin, I. I. Mazin, and N. J. Ghimire,\nOrigin of spin reorientation and intrinsic anomalous Hall effect in the kagome ferrimagnet TbMn 6Sn6, arXiv:2203.17246.\n[S2] D. Gorbunov, M. Kuz’min, K. Uhl´ ıˇ rov´ a, M. ˇZ´ aˇ cek, M. Richter, Y. Skourski, and A. Andreev, Magnetic properties of a\ngdmn6sn6 single crystal, Journal of Alloys and Compounds 519, 47 (2012).\n[S3] C. C. Homes, M. Reedyk, D. A. Cradles, and T. Timusk, Technique for measuring the reflectance of irregular,\nsubmillimeter-sized samples, Appl. Opt. 32, 2976 (1993).\n[S4] D. B. Tanner, Use of x-ray scattering functions in Kramers-Kronig analysis of reflectance, Phys. Rev. B 91, 035123 (2015).\n[S5] S. Fratini, S. Ciuchi, and D. Mayou, Phenomenological model for charge dynamics and optical response of disordered\nsystems: Application to organic semiconductors, Phys. Rev. B 89, 235201 (2014).\n[S6] G. Kresse and J. Furthm¨ uller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a\nplane-wave basis set, Computational Materials Science 6, 15 (1996).\n[S7] G. Kresse and J. Furthm¨ uller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis\nset, Phys. Rev. B 54, 11169 (1996).\n[S8] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett. 77, 3865\n(1996).\n[S9] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, J. Luitz, R. Laskowski, F. Tran, and L. Marks, WIEN2k, An Augmented\nPlane Wave + Local Orbitals Program for Calculating Crystal Properties (Karlheinz Schwarz, Techn. Universit¨ at Wien,\nAustria), 2018. ISBN 3-9501031-1-2.\n[S10] C. Ambrosch-Draxl and J. Sofo, Linear optical properties of solids within the full-potential linearized augmented planewave\nmethod, Comput. Phys. Commun. 175, 1 (2006).\n[S11] Z. Liu, N. Zhao, M. Li, Q. Yin, Q. Wang, Z. Liu, D. Shen, Y. Huang, H. Lei, K. Liu, and S. Wang, Electronic correlation\neffects in the kagome magnet GdMn 6Sn6, Phys. Rev. B 104, 115122 (2021).\n[S12] W. Ma, X. Xu, J.-X. Yin, H. Yang, H. Zhou, Z.-J. Cheng, Y. Huang, Z. Qu, F. Wang, M. Z. Hasan, and S. Jia, Rare\nEarth Engineering in RMn6Sn6(R= Gd−Tm, Lu) Topological Kagome Magnets, Phys. Rev. Lett. 126, 246602 (2021).\n[S13] J.-X. Yin, W. Ma, T. A. Cochran, X. Xu, S. S. Zhang, H.-J. Tien, N. Shumiya, G. Cheng, K. Jiang, B. Lian, Z. Song,\nG. Chang, I. Belopolski, D. Multer, M. Litskevich, Z.-J. Cheng, X. P. Yang, B. Swidler, H. Zhou, H. Lin, T. Neupert,\nZ. Wang, N. Yao, T.-R. Chang, S. Jia, and M. Zahid Hasan, Quantum-limit Chern topological magnetism in TbMn 6Sn6,8\nNature 583, 533 (2020)." }, { "title": "2209.11562v1.Magnetostatics_of_Room_Temperature_Compensated_Co_Gd_Co_Gd_based_Synthetic_Ferrimagnets.pdf", "content": "arXiv:2209.11562v1 [cond-mat.mes-hall] 23 Sep 2022Magnetostatics of Room Temperature Compensated\nCo/Gd/Co/Gd-based Synthetic Ferrimagnets\nThomas J. Kools,1Marnix C. van Gurp,1Bert Koopmans,1and Reinoud Lavrijsen1\nDepartment of Applied Physics, Eindhoven University of Tec hnology\nP . O. Box 513, 5600 MB Eindhoven, The Netherlands\n(*Electronic mail: t.j.kools@tue.nl)\n(Dated: 26 September 2022)\nFlexibility for interface engineering, and access to all-o ptical switching of the magnetization, make synthetic ferr i-\nmagnets an interesting candidate for advanced opto-spintr onic devices. Moreover, due to their layered structure and\ndisordered interfaces they also bear promise for the emergi ng field of graded magnetic materials. The fastest and most\nefficient spin-orbit torque driven manipulation of the magn etic order in this material system generally takes place at\ncompensation. Here, we present a systematic experimental a nd modeling study of the conditions for magnetization\ncompensation and perpendicular magnetic anisotropy in the synthetic ferrimagnetic Co/Gd/Co/Gd system. A model\nbased on partial intermixing at the Co/Gd interfaces of this system has been developed which explains the experiments\nwell, and provides a new tool to understand its magnetic char acteristics. More specifically, this work provides new\ninsight in the decay of the Co proximity-induced magnetizat ion in the Gd, and the role the capping layer plays in the\nGd magnetization.\nThe ever expanding rate of data generation and consump-\ntion propels research into new material systems to use for\nprocessing and storage of information. Therefore, one ma-\njor challenge of contemporary research in spintronics is to de-\nvelop material systems of which the magnetization can be ma-\nnipulated both time- and energy-efficiently. 3d-4f ferrima g-\nnetic material systems, like GdFeCo and CoTb alloys, and\nmultilayers based on a combination of these metals, are attr ac-\ntive due to their antiferromagnetically coupled sublattic es1–7.\nThese materials aim to combine favorable properties of thei r\nferromagnetic and antiferromagnetic counterparts, and be ar\npromise for the emerging field of graded magnetism8. They\nhave garnered a great amount of attention from the sci-\nentific community due to their access to single-pulse all-\noptical switching (AOS) of the magnetization5,9–11, efficient\nspin-orbit torque (SOT)-driven manipulation of the mag-\nnetic order12–16and exchange torque driven current induced-\ndomain wall motion (CIDWM) with velocities over 1000\nm/s2,3,7. Hence, these developments push the search for ma-\nterial platforms for domain wall-based memory in advanced\nsolid state devices like racetrack memory17–19. Interestingly,\nthe combination of AOS and efficient CIDWM in this mate-\nrial system is also very promising to bridge the gap between\nphotonics and spintronics20–23.\nCo/Gd-based synthetic ferrimagnetic bilayers, where the\n3d and 4f-material are grown as discrete layers, have a few\ndistinct advantages over 3d-4f alloys. The layered structu re\nof these synthetic ferrimagnets allows for easier adaption to\nwafer scale production. Also, contrary to alloys, a much\nwider composition range between the 3d and 4f-metal ex-\nhibits AOS24,25. Combined with the increased access to in-\nterfacial engineering, this leads to more flexibility and tu n-\nability of its magnetic properties. Moreover, the Pt/Co/Gd\ntrilayer displays strong interfacial spintronic effects, such as\nperpendicular magnetic anisotropy (PMA), the spin-Hall ef -\nfect, and the interfacial Dzyaloshinskii–Moriya interact ion, all\nimportant aspects for applications based on efficient domai nwall motion19,26–28. Despite these favorable properties, the\nengineering relevance of the Co/Gd bilayer system has been\nlimited due to the absence of both magnetization and angu-\nlar momentum compensation, where the two magnetic sub-\nlattices cancel each other, at room temperature. For it is we ll\nknown that CIDWM2,3,21and in general SOT-driven ferrimag-\nnetic spin dynamics4,12–14are most effective close to the an-\ngular momentum or magnetization compensation point.\nIn this work, we therefore investigate the conditions for\ncompensation in Co/Gd/Co/Gd, which we from now on\ndub the quadlayer system. Compared to the Co/Gd bi-\nlayer, we double the magnetic volume of the Co while\ntripling the number of Co/Gd interfaces where magnetiza-\ntion is induced in the Gd through direct exchange with\nthe Co11,19. This is expected to enhance the contribution\nof the Gd to the net magnetic moment, while still main-\ntaining PMA. The samples nominally consist of stacks of\nTaN(4 nm)/Pt(4)/Co(0.6)/Gd( tGd1)/Co( tCo2)/Gd( tGd2)/TaN(4)\nas schematically drawn in Fig.2c, which were grown on\nSi/SiO 2substrates through magnetron sputtering in a cham-\nber with a typical base pressure of 5 ×10−9mBar. The first\nsample is fabricated using wedge sputtering in order to con-\nfirm that compensation is achieved. Specifically, in the first\nsample the middle Gd thickness tGd1is varied between 0 and\n1.5 nm over a few mm (see inset Fig. 1a), whereas tCo2and\ntGd2are constant and set to 0.7 and 1.5 nm, respectively.\nThe magnetic properties of this wedge were investigated\nby the polar magneto-optic Kerr effect (pMOKE), where we\nare only sensitive to out-of-plane (OOP) components of the\nCo magnetization, as Gd does not contribute appreciably to\nthe pMOKE signal at our used wavelength of 658 nm29. We\nscan the sample locally using a focused laser spot. At mag-\nnetic compensation (e.g. from a Co-dominated to a Gd dom-\ninated region or vice-versa) two effects are expected: a di-\nvergence of the coercivity and a sign change in the pMOKE\nsignal. The former can be observed in Fig. 1a, where the\ncoercivity extracted from hysteresis loops measured acros s2\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s32/s77/s79/s75/s69/s32/s115/s105/s103/s110/s97/s108\n/s32/s67/s111/s101/s114/s99/s105/s118/s105/s116/s121 \n/s71/s100/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41/s77/s79/s75/s69/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s46/s117/s46/s41\n/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48\n/s32/s67/s111/s101/s114/s99/s105/s118/s105/s116/s121/s32/s40/s109/s84/s41Co Gd Co Gd (a)\n/s45/s51/s48/s48 /s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s45/s49/s48/s49/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s77/s79/s75/s69/s32/s115/s105/s103/s110/s97/s108\n/s109\n/s48/s72 /s32/s40/s109/s84/s41/s32/s116\n/s71/s100/s61/s48/s46/s56/s51/s32/s110/s109\n/s32/s116\n/s71/s100/s61/s48/s46/s57/s52/s32/s110/s109(b)Right axisLeft axis\nFIG. 1. Polar MOKE characterization of a Co/Gd/Co/Gd sample\nwhere the middle Gd layer is wedged between 0 and 1.5 nm. a):\nRemanent polar MOKE signal normalized by its value at tGd1= 0 nm\n(black), and coercive field (red) as a function of Gd layer thi ckness.\nb): Two sample hysteresis loops measured by polar MOKE at a Co\n(black) and Gd-dominated magnetic composition (red) on the wedge\nin a).\nthe wedge is plotted in red. The divergence follows from\nthe inefficiency of the Zeeman interaction in a compensated\nsystem. This divergence coincides with a change in sign of\nthe remanent pMOKE signal (Kerr rotation, normalized to its\nvalue at tGd1= 0 nm) which is plotted in black in Fig. 1a.\nTo understand this sign change, we must consider that in the\nGd-dominated regime the Zeeman energy dictates that the Gd\nmagnetization aligns with the magnetic field. The measured\nCo-magnetization will consequently align antiparallel to the\nfield, leading to the change in sign of the pMOKE signal. The\nchange in the hysteresis is illustrated in Fig. 1b, where the\nblack and red loops are measured in the Co ( tGd1= 0.83 nm)\nand Gd-dominated magnetic regime ( tGd1= 0.94 nm), respec-\ntively. The 100% remanence observed indicates the PMA in\nthis sample.\nIn order to obtain information on the tunability of the com-\npensation point and PMA, as a low net magnetization would\nimply a large effective anisotropy, we use orthogonal doubl e\nwedge samples. In Fig. 2a we illustrate the Co/Gd/Co/Gd\ndouble wedge sample structure. After deposition of the first/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s67/s111/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\n/s71/s100/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41Gd \nCo \nGd \nCo (a) (b)\nmCo >m Gd \nmGd >m Co \nIn plane\nM\nz\n0Co 1\nGd 1 Co 2 \nGd 2 \nEd\nz\n0Co 1 Co 2 \nGd 2 \nTaN Pt TaN \na0\na0\na0\nGd 1 \na0 a0a0Co 1Gd 1Co 2Gd 2(c) (d)\n(e)λ0MCo0 \nMGd0\nFIG. 2. a): Schematic of the double wedge samples under inves -\ntigation in this report. b): Polar MOKE scan of a double wedge\nCo(0.6 nm)/Gd(x-axis)/Co(y-axis)/Gd sample(1.5). The co mpensa-\ntion boundary is indicated by the green line. c): Schematic i llus-\ntration of the model used to describe the compensation and SR T-\nboundary for the magnetostatic phase diagrams. Magnetic la yers are\nmodelled with the inclusion of an intermixing region with a w idth\na0. d) and e) respectively, illustrate the magnetization and s hape\nanisotropy energy as modelled throughout the four magnetic layers.\nGd wedge, the sample is rotated by 90 degrees and the Co\nwedge is deposited. After the sample is saturated with an\nOOP magnetic field of 1 T we scan the sample surface and de-\ntermine the remanence from the pMOKE signal at each point\nwhen no magnetic field is applied. Using this method allows\nus to scan the full parameter space of nominal layer thick-\nnesses of the middle two layers in a single sample. A typical\nresulting diagram of the remanent pMOKE signal is shown in\nFig. 2b for a sample where tGd1=0−3 nm and tCo2=0−2\nnm, keeping the top Gd thickness at tGd2= 1.5 nm at which\nwe anticipate, based on earlier work, the proximity-induce d\nmagnetization in the Gd to be saturated11,19. In the diagram,\nwe can distinguish between three basic states. The red and\ndark blue regions indicate stack compositions where the mag -\nnetization points OOP, with the Co or Gd magnetization being\ndominant, respectively. The light blue region indicates st ack\ncompositions where the magnetization points in-plane (IP) ;\nthis is above the spin reorientation transition (SRT), wher e the\ninterfacial PMA is not sufficient to keep the full stack OOP.\nThese three regions define two major transitions of interest :\nthe compensation boundary (red to dark blue) and the SRT\nboundary (red/dark blue to light blue).\nTo obtain a quantitative understanding of the shapes of\nthese boundaries, a model has been developed to simulta-\nneously describe the compensation boundary and the SRT\nboundary. Furthermore, the model will be used to get insight\nin the basic properties of the proximity induced magnetizat ion3\nin the Gd and level of intermixing, a quantity that has not bee n\ninvestigated using these double wedged samples. We set out\nto model the net magnetization, which is zero at the compen-\nsation boundary, as well as the magnetostatic free energy of\nthe anisotropy, which is zero at the SRT boundary. One of\nthe main assumptions of the model relies on the experimental\nobservation that the interface between Co and Gd thin films\nare intermixed30–32. The assumed Co and Gd concentration\nas a function of thickness is illustrated in Fig. 2c. The lay-\ners are modelled by means of the four magnetic layers in our\nCo/Gd/Co/Gd structure, with each layer assumed to be sepa-\nrated by an intermixing region with a constant and identical\nwidth of a0.\nIn order to find the magnetization compensation point, we\nthen describe the net magnetization of this multilayer stru c-\nture, which vanishes at compensation. We use typical assump -\ntions for the magnetization profile of the Co/Gd bilayer to de -\nscribe the magnetization in our Co/Gd/Co/Gd system, which\nare illustrated in Fig. 2d11,19. The Co magnetization is crudely\nassumed constant throughout the nominal thickness of the Co\nlayer, with a value MCo0, giving:\nMCo1=MCo0, (1)\nand\nMCo2=MCo0Feq,Co, (2)\nwhere, in order to implement the intermixing regions into th e\nchange of the magnetization with layer thickness, we empir-\nically define the continuous function Feq,Co(see Sup. A for\ndetails). It describes the transition to an equilibrium mag netic\nstate with middle Co layer thickness caused both by the effec t\nof intermixing on the magnetization, as well as percolative be-\nhavior. The latter of which describes the minimum thickness\nneeded to stabilize a coherent ferromagnetic state.\nIn contrast to the Co magnetization, the magnetization in\nthe Gd layers is mainly induced at the interface with the Co\nlayer11,19. Therefore, the magnetization as a function of the\ndistance to the Co/Gd interface z∗will be described by an ex-\nponentially decaying profile, which is typical to describe m ag-\nnetization induced at an interface between a ferromagnet an d\na non-magnetic metal33–35, given by:\nMGd1=MGd0(exp/parenleftbigg\n−z∗\nλ0/parenrightbigg\n+exp/parenleftbiggz∗−tGd1\nλ0/parenrightbigg\n)Feq,CoFeq,Gd,\n(3)\nand\nMGd2=MGd0exp(−z∗/λ0)Feq,CoFeq,Gd, (4)\nwhere MGd0is the magnitude of the magnetization at the in-\nterface, λ0is the characteristic decay length of the magneti-\nzation, and Feq,Gdis a similar empirical function to Feq,Co, de-\nscribing the development of the Gd magnetization with mid-\ndle Gd layer thickness (see Sup. A). Note that the effective\nexponential decay constant λ0is influenced by many parame-\nters, like surface roughness, the actual degree of intermix ing,local ratios between Co and Gd atoms and the actual decay\nof the magnetization induced in the Gd, and should hence be\ninterpreted as an effective parameter describing the colle ctive\nbehavior of all these effects. The resulting total magnetic mo-\nment per unit area mtotcan then be extracted by integrating the\nmagnetizations over the respective layer thicknesses:\nmtot=4\n∑\ni=1/integraldisplayti\n0Midz∗. (5)\nNext, the SRT-condition needs to be implemented. There\nare two main contributions to the effective anisotropy: the in-\nterfacial anisotropy energy and demagnetization energy. T he\nmagnetocrystalline anisotropy energy per unit area KSdue to\nthe Co/Pt interface is assumed constant. The free energy den -\nsity per unit area due to the shape anisotropy Edis schemati-\ncally plotted in Fig. 2e. It is calculated by treating the sys tem\nas a continuous magnetic system and integrating the typical\nexpression for the volume demagnetization energy density o f\na thin film with OOP magnetization MS:E∗\nd=1\n2µ0M2\nS, where\nµ0is the magnetic permeability of vacuum. In order to ac-\ncount for the smaller demagnetizing field in the intermixing\nregions where the magnetization is inherently lower, we sub -\ntract the demagnetization energy of the intermixing region ,\nwith a characteristic width of a0, from the total demagnetiza-\ntion energy leading to the following expression for the tota l\narea-normalized demagnetization energy Ed:\nEd=4\n∑\ni=1/integraldisplayti\n01\n2µ0M2\nidz∗\n−/parenleftbigg/integraldisplaya0/2\n01\n2µ0M2\nCo1dz∗+Feq,mix/integraldisplaya0\n01\n2µ0M2\nGd1dz∗\n+Feq,mix/integraldisplaya0\n01\n2µ0M2\nCo2dz∗\n+/integraldisplaya0/2\n01\n2µ0M2\nGd2dz∗/parenrightbigg\n, (6)\nwhere Feq,mixis an identical empirical function to Feq,Gdand\nFeq,Coused to describe the onset and saturation of the inter-\nmixing regions in the Gd layer upon changing the layer thick-\nness (see Sup. A). The resulting total free energy density pe r\nunit area is Etotcan then be calculated by adding EdandKS\ntogether.\nWe will use this model for the magnetization and anisotropy\nenergy to test our physical understanding and make an esti-\nmate of the (effective) physical parameters underpinning t hese\nsystems by fitting it to the measured phase diagrams, again\nconsidering the SRT-boundary and compensation boundary to\nbe at Etot= 0 and mtot= 0, respectively. To test the quantita-\ntive applicability of our model to these Co/Gd/Co/Gd system s,\nthree double-wedged samples are considered: Co(0.6)/Gd(0 -\n3)/Co(0-2)/Gd( tGd2) with tGd2= 0.7, 1.0 and 1.5 nm, respec-\ntively, where the thicknesses chosen are expected to probe\ndifferent degrees of decay of the induced magnetization in t he\nGd. The phase diagrams measured on the three samples are\nshown in Fig. 3 a, b and c for tGd2= 0.7, 1.0 and 1.5 nm,4\n/s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s108\n/s48/s32/s40/s110/s109/s41\n/s116\n/s71/s100/s50/s32/s40/s110/s109/s41/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s67/s111/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\n/s71/s100/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s67/s111/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\n/s71/s100/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s67/s111/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\n/s71/s100/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41(a) (b) (c)\ntCo1= 0.6 nm \ntGd2 = 0.7 nm tCo1 = 0.6 nm\ntGd2 = 1.0 nmtCo1 = 0.6 nm\ntGd2 = 1.5 nm\n(d) (e) (f)\n/s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s97\n/s48/s32/s40/s110/s109/s41\n/s116\n/s71/s100/s50/s32/s40/s110/s109/s41/s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s77\n/s71/s100/s48/s32/s40/s77/s65/s47/s109/s41\n/s116\n/s71/s100/s50/s32/s40/s110/s109/s41\nFIG. 3. Magnetostatic phase diagrams with model fits of the ma gnetization (orange) and demagnetization energy (green) f ortGd2=a): 0.7 nm,\nb): 1.0 nm, and c): 1.5 nm. The main magnetic parameters extra cted from the fitting procedure for the three phase diagrams: d): Magnetization\nof Gd at the Co/Gd interface MGd0, e): Magnetization decay length λ0, and f): intermixing region width a0.\nrespectively, where we can immediately observe that with in -\ncreasing top Gd thickness the area of the Gd-dominated regio n\nincreases.\nBefore fitting the model to explain the features of these\nphase diagrams, we experimentally characterize the interf a-\ncial anisotropy strength due to the Co/Pt interface KSto be\n1.22 mJ/m2(see Appendix B), and set the Co saturation mag-\nnetization MCo0equal to the bulk magnetization of Co at 1.4\nMA/m. The other parameters in the model are left uncon-\nstrained. In Fig. 3a,b,c we give the resulting fits for mtot=0\nand Ed=0 in orange and green, respectively. We find a\ngood correspondence between the model and the experiment.\nSpecifically, The curvature of the magnetization profile, an d\nthe corresponding magnetostatic energy balance character ized\nby the peaked shape, are both generally well described. Par-\nticularly for the samples with tGd2= 0.7 and 1 nm the cor-\nrespondence is good across the whole phase diagrams. For\ntGd2= 1.5 nm in Fig. 3c, the model correspondence on the\nSRT-boundary becomes worse for tGd1>1.5 nm. It is not yet\nbeen unequivocally established what causes this differenc e be-\ntween experiment and theory. We speculate that it might be\ndue to finite size effects affecting the Curie temperature in the\nGd and hence the total amount of magnetization induced be-\nyond what is currently implemented in the model. Based on\nthe model we can attribute the peak in effective anisotropy\nin the phase diagrams to the first ∼1 nm of Gd contribut-ing mainly to the intermixing regions, leading to the initia l\nincrease, after which pure Gd is found, which decreases the\neffective anisotropy, leading to the decline from ∼1 nm on-\nwards. Moreover, the change in curvature of the compensatio n\nboundary for tCo<0.5 nm between the three samples can now\nbe attributed to the percolation limit approach of the Co lay er\nleading to either one or three interfaces which induce a net\nmagnetization in the Gd layers.\nTo also illustrate the quantitative value of the model, we\nwill now discuss the magnetic parameters extracted from the se\nfits. In particular, the parameters fixing the hitherto unkno wn\nGd magnetization profile λ0,MGd0, and a0are of interest here.\nThese parameters are plotted for the three fitting procedure s in\nFig. 3d, e and f, respectively. All other parameters found in\nthe fitting procedure are listed in appendix C. The extracted\nGd interfacial magnetization of about 1.3 MA/m is compa-\nrable with values found in earlier work11,19,26. Next, the λ0\nvalues in the order of 1 nm suggest that the magnetization pro -\nfile extends well beyond the first monolayer affected by direc t\nexchange with the Co layer. This observation is further cor-\nroborated when considering the difference between identic al\nphase diagrams capped with Ta and TaN (see Appendix D).\nThere we observe that the Gd- dominated OOP regime (dark\nblue in earlier diagrams) in the phase diagram extends all th e\nway to Co-thicknesses of 0 nm in the TaN-capped sample,\nwhereas in Ta-capped samples of otherwise identical compo-5\nsition a minimum middle Co thickness is always required to\nreach the Gd-dominated regime, indicating an overall reduc -\ntion in the total magnetization of the Gd. We postulate that\nthis difference is caused by magnetization quenching in the\nGd due to intermixing between the capping layer and the Gd,\na process that will likely be more severe for atomic Ta than fo r\ncovalently bound TaN36. Finally, we will discuss the resulting\nvalues for a0(Fig. 3f). Regarding the growth of Gd on Co\nand vice versa, earlier work demonstrated that the interfac es\nbetween multilayers of Co and Gd are disordered30,31,37, and\nthat the exact growth and intermixing dynamics also depend\non the order of growth of the two layers30,38. This indicates\nthat the found intermixing region width a0of around 0.8 nm\nis in line with earlier work. A reasonable comparison can be\nmade to the [Pt/Co/Gd]-multilayers investigated by Nishim ura\net al., where using transmission electron microscopy inves ti-\ngations similar typical intermixing region widths were fou nd\nas we find from fitting our model here, i.e., in the 0.5-1 nm\nrange32.\nIn conclusion, we have experimentally demonstrated mag-\nnetic compensation in the synthetic ferrimagnetic quadlay er\nCo/Gd/Co/Gd system. It is found that compensation can be\neffectively tuned by layer thickness. We also demonstrated\nthe utility of orthogonally wedged samples to characterize the\nnominal thickness parameter space in order to investigate t he\nmagnetostatics of these systems and consequently find stack\ncompositions with favorable magnetic properties. Finally , a\ncrude model for the net magnetization and PMA was devel-\noped which described the experiments well, providing an ef-\nfective framework to discuss the magnetostatics in these co m-\npensated multilayered ferrimagnetic systems with PMA. We\nnote that this is probably an oversimplified model to describ e\nthe real intermixing profiles, e.g. the constant Co magneti-\nzation with thickness. It however provides a good qualitati ve\nframework to build more detailed models which will require\nrefinement of the assumed magnetization profiles using high-\nresolution depth sensitive magnetometry. This work improv es\nthe understanding of basic magnetostatic properties and gi ves\ninsight in the more fundamental aspects of the design and\nphysics of these promising and flexible multilayer systems.\nACKNOWLEDGMENTS\nThis work was part of the research program Foundation for\nFundamental Research on Matter (FOM) and Gravitation pro-\ngram “Research Center for Integrated Nanophotonics,” whic h\nare financed by the Dutch Research Council (NWO). This\nwork was suported by the Hedrik Casimir Institute.\nAUTHOR DELCARATIONS\nConflict of Interest\nThe authors have no conflicts to discloseAuthor Contriubtions\nThomas J. Kools: Conceptualization (equal); Investigation\n(lead); Methodology (lead); Writing – original draft (lead );\nWriting – review and editing (lead). Bert Koopmans: Con-\nceptualization (equal); Funding acquisition (equal); Sup er-\nvision (equal); Writing – review and editing (supporting).\nReinoud Lavrijsen: Conceptualization (equal); Funding ac-\nquisition (equal); Supervision (equal); Writing – review a nd\nediting (supporting).\nDATA AVAILABILITY\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.6\n/s45/s49/s53/s48/s48 /s45/s49/s48/s48/s48 /s45/s53/s48/s48 /s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s77/s65/s47/s109/s41\n/s109\n/s48/s72 /s32/s40/s109/s84/s41\nFIG. 4. In-plane SQUID characterization of the magnetic mom ent of\na Ta(4 nm)/Pt(4)/Co(1)/TaN(4) as a function of in-plane fiel d. The\nred dot indicates the extracted anisotropy field.\nAppendix A: Percolation functions\nIn order to implement the intermixing regions into the\nchange of the magnetization with layer thickness, we empir-\nically define continuous functions Feq,GdandFeq,Codescrib-\ning the transition to an equilibrium state with respective l ayer\nthickness. These describe both percolative behavior; the m in-\nimum thickness needed to stabilize a coherent ferromagneti c\nstate, and the effect of intermixing on particularly the Gd m ag-\nnetization:\nFeq,Gd(tGd1)=erf((tGd1−t0,Gd)/LGd1)+1\n2, (A1a)\nFeq,Co(tCo2)=erf((tCo2−t0,Co)/LCo2)+1\n2, (A1b)\nwhere t0,Gd1andt0,Co2, and LGd1andLCo2, are parameters\ndefining the critical thickness and characteristic width of the\npercolation, respectively.\nFeq,mixis an identical empirical function to those presented\nin Eq. A1 used to describe the onset of the intermixing region s\nin the Gd layer upon changing the layer thickness:\nFeq,mix(tGd1)=erf((tGd1−t0,mix)/Lmix)+1\n2. (A2)\nAppendix B: Characterization Ks\nIn order to estimate the anisotropy constant KS, we per-\nformed IP VSM-SQUID measurements (see Fig. 4) on a\nTa(4)/Pt(4)/Co(1)/TaN(4) sample. The resulting hard-axi s re-\nsponse to the IP field is typical of a sample with PMA like\nthis multilayer. We find an anisotropy field of 900 mT. To es-\ntimate the corresponding KS, we use a simple Stoner-Wolfarththeory, considering three contributions to the magnetosta tic\nfree energy: Zeeman energy Ez, interfacial anisotropy from\nthe Co/Pt interface EK, and the shape anisotropy Es. The re-\nsulting total energy Etotis then given by the sum of these three\ncontributions:\nEtot=1\n2µ0M2\nScos2(θ)+sin(θ)/parenleftbigg\n−µ0HM S+KSsin(θ)\nt/parenrightbigg\n,\n(B1)\nwhere µ0is the permeability of vacuum, MSis the saturation\nmagnetization, θis the angle between the magnetization and\nthe thin film sample normal, His the applied magnetic field,\nandtis the thickness of the magnetic layer. By minimiz-\ningEtotwith respect to θand setting θ=90◦we find the\nanisotropy field Ha:\nHa=1\n2µ0t/parenleftbig\nHM S+M2\nS/parenrightbig\n. (B2)\nForMS=1.4 MA/m (SQUID), t=1 nm and Ha=900 mT\n(SQUID), we find KS=1.22 mJ/m2.\nAppendix C: Fitting parameters\nThe parameters for the best fit of the model described in\nthe main text to the double wedge Co/Gd/Co/Gd samples as\nshown in Figs. 3a, 3b and 3c are described in this section.\nTables I, II and III show these fitting parameters for top Gd\nthickness tGd2=0.7 nm, tGd2=1.0 nm, tGd2=1.5 nm, re-\nspectively.\nTABLE I. Summary of the fitting parameters found to best descr ibe\nthe compensation boundary and boundary from OOP to IP magne-\ntization in the double wedge with top Gd thickness tGd2=0.7 nm\n(see Fig. 3a). For MS,CoandKSwe choose 1.4 MA/m and 1.22\nmJ/m2as explained in the main text and appendix B. Errors repre-\nsent 95% confidence intervals extracted from the fitting proc edure.\nParameter Value Error Unit\nMGd1 1.2 0.2 MA/m\nt0,Co 0.21 0.01 nm\nt0,Gd1 1.2 0.2 nm\nt0,mix 0.47 0.01 nm\nLCo 0.43 0.03 nm\nLGd1 0.83 0.02 nm\nLmix 1.3 0.1 nm\nλ0 0.97 0.16 nm\na0 0.66 0.01 nm\nAppendix D: Comparison phase diagram capping layers\nIn Fig.5 a and b the magnetostatic phase diagrams of a\nCo(0.6)/Gd(x)/Co(0.7)/Gd(1.5) stack with a 4 nm thick cap-\nping layer of TaN and Ta are plotted, respectively. The\nmost important difference to note here is that the region\nwhere the magnetization is OOP and the Gd-contribution is7\nTABLE II. Summary of the fitting parameters found to best de-\nscribe the compensation boundary and boundary from OOP to IP\nmagnetization in the double wedge with top Gd thickness tGd2=\n1.0 nm (see Fig. 3b). For MS,CoandKSwe choose 1.4 MA/m\nand 1.22 mJ/m2as explained in the main text and appendix B. Er-\nrors represent 95% confidence intervals extracted from the fi tting\nprocedure.\nParameter Value Error Unit\nMGd1 1.29 0.17 MA/m\nt0,Co 0.13 0.01 nm\nt0,Gd1 0.72 0.15 nm\nt0,mix 0.39 0.01 nm\nLCo 0.63 0.02 nm\nLGd1 0.56 0.06 nm\nLmix 0.27 0.01 nm\nλ0 1.3 0.2 nm\na0 0.83 0.02 nm\nTABLE III. Summary of the fitting parameters found to best de-\nscribe the compensation boundary and boundary from OOP to IP\nmagnetization in the double wedge with top Gd thickness tGd2=\n1.5 nm (see Fig. 3c). For MS,CoandKSwe choose 1.4 MA/m\nand 1.22 mJ/m2as explained in the main text and appendix B. Er-\nrors represent 95% confidence intervals extracted from the fi tting\nprocedure.\nParameter Value Unit\nMGd1 1.57 0.16 MA/m\nt0,Co 0.10 0.02 nm\nt0,Gd1 0.66 0.21 nm\nt0,mix 0.43 0.01 nm\nLCo 1.14 0.03 nm\nLGd1 3.1 1.1 nm\nLmix 0.28 0.01 nm\nλ0 1.50 0.35 nm\na0 0.90 0.03 nm\ndominant (dark blue) extends all the way to zero Co thick-\nness for the TaN cap. In contrast, the Ta-capped sample\nmagnetization only becomes dominated by the Gd magne-\ntization for a minimum Co thickness of about 0.4 nm. In\nFig. 5 a and b we plot magnetostatic phase diagrams for a\nTa(4)/Pt(4)/Co(0.6)/Gd(x)/Co(0.7)/Gd(1.5) with a TaN an d Ta\ncapping layer, respectively.\nBIBLIOGRAPHY\n1S. K. Kim, G. S. D. Beach, K.-J. Lee, T. Ono, T. Rasing, and H. Ya ng, “Fer-\nrimagnetic spintronics,” Nature Materials , vol. 21, pp. 24–34, Jan 2022.\n2K.-J. Kim, S. K. Kim, Y . Hirata, S.-H. Oh, T. Tono, D.-H. Kim, T . Okuno,\nW. S. Ham, S. Kim, G. Go, Y . Tserkovnyak, A. Tsukamoto, T. Mori yama,\nK.-J. Lee, and T. 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Brennan,\n“Structural characterization of multilayers using x-ray d iffraction,” MRS\nProceedings , vol. 239, p. 475, 1991." }, { "title": "1406.0523v1.Electronic_and_magnetic_properties_of__1_1_1__oriented_CoCr2O4_epitaxial_thin_film.pdf", "content": "Electronic and magnetic properties of (1 1 1)-oriented CoCr 2O4epitaxial thin \flm\nXiaoran Liu,1,a)M. Kareev,1Yanwei Cao,1Jian Liu,2, 3S. Middey,1D. Meyers,1J. W.\nFreeland,4and J. Chakhalian1\n1)Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701,\nUSA\n2)Department of Physics, University of California, Berkeley, California 94720,\nUSA\n3)Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley,\nCalifornia 94720, USA\n4)Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439,\nUSA\nWe report on the fabrication of high quality (1 1 1)-oriented ferrimagnetic normal\nspinel CoCr 2O4epitaxial thin \flms on single crystal Al 2O3substrates. The struc-\ntural, electronic and magnetic properties were characterized by in-siture\rection high\nenergy electron di\u000braction, atomic force microscopy, X-ray di\u000braction, X-ray photoe-\nmission spectroscopy, SQUID magnetometry and element resolved resonant X-ray\nmagnetic scattering. The comprehensive characterization reveals that no disorder in\nthe cation distribution or multivalency issue is present in the samples. As a result,\nKagom\u0013eand triangular layers are naturally formed via this speci\fc growth approach.\nThese \fndings o\u000ber a pathway to fabricate two dimensional Kagom\u0013 eheterostructures\nwith novel quantum many-body phenomena by means of geometrical design.\na)Electronic mail: xxl030@email.uark.edu\n1arXiv:1406.0523v1 [cond-mat.str-el] 2 Jun 2014In the past few years, two dimensional (2D) Kagom\u0013 elattices have attracted tremen-\ndous interest in the pursuit of novel quantum phenomena. Many intriguing exotic quantum\nstates on this lattice have been predicted by theoretical calculations including topological\ninsulators1,2, spin liquids3,4, kinetic ferromagnetism5, spin Hall e\u000bect6, anomalous Hall e\u000bect\nof light7, and chiral superconducting state8. However, the synthesis of a 2D Kagom\u0013 elattice\nis a quite challenging work. Several metal-organic hybrid compounds have been synthesized\nby chemical methods and possess Kagom\u0013 elattice but structural disorder and distortions re-\nmained a persistent hinderance9,10. There is, however, an alternative approach to realize this\nsituation. Recently, epitaxial thin \flms and heterostructures have been grown along the (1 1\n1) direction and this new control parameter which is referred as geometrical engineering, has\ncultivated many novel systems with emergent properties11{13. Following the same approach,\nit was pointed out that Kagom\u0013 elattice can be obtained by arti\fcially controlling the thin\n\flm growth of a spinel type oxide (general formula AB 2O4) along the (1 1 1) orientation14.\nAmong many choices of spinel oxides, CoCr 2O4(CCO) is a prototypical candidate which\nhas been widely studied to exhibit interesting physical phenomena in the bulk materials\nincluding conical spin states15{17, induced multiferroic behavior at low temperature18,19, and\nan unconventional magneto-structural phase transition in high magnetic \felds20. CCO is a\nnormal spinel oxide in which all of the Co2+occupies the tetrahedral sites while Cr3+resides\nwithin in the octahedral sites. The electronic con\fgurations of Co2+and Cr3+are 3 d7and\n3d3respectively, both having S = 3/215{17,21. When viewed along the (1 1 1) direction, CCO\nconsists of alternating stacked multi-layers with Cr3+ions composing a Kagom\u0013 e(K) lattice\nand a triangular (T) lattice while the Co2+ions composing two triangular (T') ones, forming\nthe stacking sequence [ \u0001\u0001\u0001T'/K/T'/T \u0001\u0001\u0001]. The crystal structure of the CCO conventional\nunit cell is depicted in Fig. 1(a). The 2D Kagom\u0013 elattice plane is shown in Fig. 1(b)\nwith the typical Kagom\u0013 eunit cell outlined by yellow lines. To date, despite the conceptual\nattractiveness of such geometrical design, the epitaxial growth of (1 1 1) oriented spinel\noxide thin \flms presents serious challenge due to the reported cation distribution disorder\nand mixture of various transition metal oxidation states22{25.\nIn this Letter, we report on the layer-by-layer growth of high quality (1 1 1)-oriented\nCCO thin \flms on Al 2O3(AlO) (0 0 0 1) single crystal substrates by the pulsed laser depo-\nsition technique. The crystallinity, surface morphology, \flm thickness and atomic structure\nare studied by in-situ high pressure re\rection-high-energy-electron-di\u000braction (RHEED),\n2atomic force microscopy (AFM), X-ray di\u000braction (XRD) and X-ray re\rectivity (XRR).\nElectronic and magnetic properties are extensively characterized by in-situ X-ray photoe-\nmission spectroscopy (XPS), SQUID magnetometry and synchrotron-based X-ray resonant\nmagnetic scattering (XRMS). The combined data con\frm that the (1 1 1) grown CCO sam-\nples maintain a normal spinel structure with no cation distribution disorder or multivalency\nissue.\nCCO thin \flms were fabricated under a partial pressure of 5 mTorr of oxygen by the\npulsed laser interval deposition method26using a KrF excimer laser operating at \u0015= 248\nnm. During the deposition, the substrate was kept at 800\u000eC. The laser's intensity and\npulse rate were 2 J/cm2and 18 Hz, respectively. Samples were annealed at the growth\ncondition for 10 min and then cooled down to room temperature. Fig. 1(c) displays the\nRHEED pattern of the substrate before the deposition. As seen, the specular spot together\nwith the (-1, -1) and (1, 1) spots on the Laue circle and the strongly developed Kikuchi\nlines testify the smooth morphology of the AlO (0 0 0 1) substrate27. During the deposition,\nthe recovery of the intensity of the specular re\rection spot and the occurrence of the half\norder spots indicate the 2D epitaxial growth of the \flm. After annealing and cooling down\nto room temperature, RHEED patterns of the resultant sample are shown in Fig. 1(d).\nThe distinct spots from specular and o\u000b specular re\rections with the well developed streak\npatterns con\frm excellent CCO thin \flm crystallinity and \rat terraces. In addition, smooth\nsurface morphology is corroborated by the AFM imaging shown in Fig. 1(e); the obtained\naverage surface roughness is below 60 pm for a 1 \u0016m by 1\u0016m scan.\nIt is of critical importance to ensuring the (1 1 1) growth direction is maintained through-\nout the sample and determining the \flm thickness, thus we performed the X-ray di\u000braction\n2\u0012-!scans on CCO thin \flms as well as on the pure AlO substrate, together with a X-ray\nre\rectivity scans on the \flm using Cu K \u000bradiation. As shown in Fig. 2, the expected CCO\n(2 2 2), (3 3 3) and (4 4 4) Bragg re\rections are clearly seen, which testify that the sample\nis (1 1 1) orientated. The lattice constant calculated from the peak position is 8.33 \u0017A, which\nis in excellent agreement with the bulk value15. Based on the XRD pattern, no impurity\nphases are detected. In addition, the inset in Fig 2 displays the X-ray re\rectivity, yielding a\n\flm thickness of about 25 nm according to the period of the thickness fringes that provides\nadditional evidence for the sample \ratness.\nDue to the well-known persistent multivalency problem in the thin \flm growth of spinel\n3oxides, the proper valency of the Cr and Co ions must be con\frmed. Fig. 3(a) and 3(b)\nshow XPS measurements taken on Cr and Co 2p core levels. As seen, Cr has two peaks at\nabout 577 eV and 587 eV which correspond to the 2p 3=2and 2p 1=2peaks indicating Cr is in\nthe trivalent state.28,29The 2p 3=2and 2p 1=2peaks of Co are at 781 eV and 797 eV indicating\nCo is in the divalent state28,29. In addition, no extra peaks from Cr2+, Cr4+or Co3+are\nobserved, which excludes the possibility of a multivalent charge state.\nTo further elucidate that the CCO thin \flm maintains the normal spinel structure without\ncation disorder, XRMS experiments were carried out at beamline 4-ID-C of the Advanced\nPhoton Source (APS) in Argonne National Laboratory. Left (I+) and right (I\u0000) polarized soft\nx-rays with an incident angle of 15\u000ewere tuned to the Ledges of Cr and Co and recorded in\nscattering mode30. In the XRMS mode, the di\u000berence between I+and I\u0000near the absorption\nedge represents the contribution to the scattering amplitude from uncompensated magnetic\nmoments of a speci\fc chemical element31, while the sum I++ I\u0000is connected to the charge\nstate32. Figures 3(c) and 3(d) show the total re\rectivity intensity (I++ I\u0000) of both Cr\nand Co as a function of incident photon energy. As seen, the lineshapes of each element\nmeasured at 15 K and 80 K are almost unchanged. The positions of the main absorption\nedges L3andL2together with the satellite peaks are almost identical to the previous reported\nstudy33, which implies all of the Cr3+are in the octahedral sites while all the Co2+sit in\nthe tetrahedral sites.\nNext we discuss the magnetic properties of the CCO thin \flm. To this end, temperature\ndependent magnetization was measured while cooling in an applied magnetic \feld of 0.2 T by\nSQUID. As seen in Fig. 4(a), the ferrimagnetic phase transition occurs at a T caround 95 K.\nAs the temperature further decreases, the collinear to incommensurate spiral ferrimagnetic\nphase transition17takes place at T s\u001922 K. Element-speci\fc spin alignments were further\ninvestigated by XRMS spectra on Cr and Co L2;3absorption edges at 15 K and 80 K in\ndi\u000berent applied \felds from 5 T to 0.1 T. All of the XRMS data have been normalized\nby using the corresponding total re\rectivity intensity. As presented in Fig. 4(b) and 4(c),\nboth Cr and Co exhibit signi\fcant XRMS signal in the vicinity of their absorption edges.\nThe maximal XRMS signal at Cr L3edge is about +20% while that of Co is around -80%,\nwhich indicate strong ferromagnetic ordering of the moments on each type of metal ions,\nrespectively. Note, the sign of the maximal XRMS signal is opposite for Cr and Co due to\nthe ferrimagnetic nature of this material, which implies the overall spin orientation of Cr is\n4antiparallel to those of the Co ions. In Fig. 4(d), the magnitudes of the maximal XRMS\nvalues of Cr and Co at 15 K and 80 K are plotted as a function of applied magnetic \feld\nshowing a saturation \feld of approximately 3 T. This is much larger than the reported bulk\nvalues17,18,29which are typically less than 0.6 T. This di\u000berence is attributed to the fact that\nin the bulk the magnetic \feld is commonly applied along the CCO [0 0 1] direction which\nis the spin easy axis.\nIn summary, we have successfully fabricated high quality (1 1 1)-oriented normal spinel\nCCO thin \flms on AlO (0 0 0 1) substrate. The structural, electronic and magnetic prop-\nerties are investigated by a combination of RHEED, XRD, XPS, XRMS and SQUID mea-\nsurements. No disorder of the cation distribution or multivalency is observed. Magnetic\nmeasurements con\frm the ferrimagnetic behavior. Since all the trivalent Cr are situated\nin the octahedral sites, they compose magnetic Kagom\u0013 eplanes naturally layered along this\norientation. The presented results pave a way for fabricating 2D-isolated Kagom\u0013 ehet-\nerostructures in which a plethora of novel quantum phenomena are expected.\nThe authors acknowledge M. Hawkridge for the assistances on the XRR measurement. JC\ndeeply acknowledges numerous fruitful discussions with D. Khomskii and G. Fiete. J.C. was\nsupported by the DOD-ARO under Grant No. 0402-17291. Work at the Advanced Photon\nSource, Argonne is supported by the U.S. DOE under Grant No. DEAC0206CH11357.\nREFERENCES\n1H. M. Guo, and M. Franz, Phys. Rev. B 80, 113102(2009)\n2X. Hu, A. R uegg, and G. A. Fiete, Phys. Rev. B 86, 235141(2012)\n3T. H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Broholm, and\nY. Lee, Nature 492, 406-410(2012)\n4M. Punk, D. Chowdhury, and S. Sachdev, Nat. Phys. 10, 289-293(2014)\n5F. Pollmann, P. Fulde, and K. Shtengel, Phys. Rev. Lett. 100, 136404(2008)\n6G. Liu, P. Zhang, Z. Wang, and S. Li, Phys. Rev. Lett. 79, 035323(2009)\n7A. Petrescu, A. A. Houck, and K. Le Hur, Phys. Rev. A 86, 053804(2012)\n8S. Yu and J. Li, Phys. Rev. B 85, 144402(2012)\n9Y. Liu, V. Ch. Kravtsov, D. A. Beauchamp, J. F. Eubank, and M. Eddaoudi, J. Am.\nChem. Soc. 127, 7266-7267(2005)\n510E. A. Nytko, J. S. 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Davis, J. F. Moulder, and G. E. Muilenberg, Handbook\nof X-ray Photoelectron Spectroscopy (Perkin-Elmer, Eden Prairie, 1979)\n29S. Lei, L. Liu, C. Wang, X. Shen, C. Wang, D. Guo, S. Zeng, B. Cheng, Y. Xiao, and L.\nZhou, CrystEngComm 16, 277(2014)\n30J. W. Freeland, J. J. Kavich, K. E. Gray, L. Ozyuzer, H. Zheng, J. F. Mitchell, M. P.\nWarusawithana, P. Ryan, X. Zhai, R. H. Kodama, and J. N. Eckstein, J. Phys: Condens.\nMatter 19, 315210(2007)\n31C. Kao, J. B. Hastings, E. D. Johnson, D. P. Siddons, G. C. Smith, and G. A. Prinz, Phys.\nRev. Lett. 65, 373(1990)\n32D. R. Lee, and S. K. Sinha, Phys. Rev. B 68, 224409(2003)\n33R. V. Chopdekar, M. Liberati, Y. Takamura, L. F. Kourkoutis, J. S. Bettinger, B. B.\nNelson-cheeseman, E. Arenholz, A. Doran, A. Scholl, D. A. Muller, and Y. Suzuki, J.\nMagn. Magn. Mater. 322, 2915-2921(2010)\n71 µm1 µm1 nmRa ≈ 60 pm\n(b)\n(c)\n(-1, -1)(1, 1)(0, 0)\n(a)\nCoCrO[001]\n(-1, -1)(1, 1)(0, 0)(-1/2, -1/2)(1/2, 1/2)(d)(e)FIG. 1. (a) Schematic crystal structures of the CCO unit cell. Cr3+ions along the (1 1 1)\norientation compose a Kagom\u0013 elattice plane. (b) Detailed 2D Kagom\u0013 elattice plane. The typical\ncorner-shared triangular structure is outlined by yellow lines. (c) RHEED patterns of AlO sub-\nstrate. (d) RHEED patterns of CCO thin \flms after cooling down to room temperature. Note,\nthe incident electron beam of the RHEED is \fxed along the [1 \u00161 0 0] direction of the substrate.\n(e) AFM image of the \flm surface after growth.\n8CCO (222)CCO (444)CCO (333)Intensity (a.u.)2θ (deg.)100101102103104105\n858075706560555045403530 CCO on AlO AlO substrate\n2θ (deg.)Reflectivity (a.u.)103104105106107\n4.03.02.01.0FIG. 2. X-ray di\u000braction of CCO thin \flms and the AlO substrate. Film peaks are labeled on the\ngraph. The sharp peaks belong to the AlO (0 0 0 1) substrate. Note, the lattice constant obtained\nis 8.34 \u0017A, which equals the bulk value. The inset is the X-ray re\rectivity data of the same sample.\nFilm thickness calculated according to the Kiessig fringes is about 25 nm.\n9Co 2pIntensity (a.u.)Binding Energy (eV)(b)2p1/22p3/284807672x103 805800795790785780775Binding Energy (eV)2p1/2Cr 2p(a)Intensity (a.u.)2p3/264605652x103 595590585580575570\nPhoton Energy (eV)L3L2Cr L edge(c)Scattering (a.u.)2.01.51.00.50.0595590585580575570 15K 80K(d)L3L2Co L edgeScattering (a.u.)\nPhoton Energy (eV)2.01.51.00.50.0800795790785780775 15K 80KFIG. 3. (a)-(b) Core level XPS data (Mg anode) of (a) Cr 2p and (b) Co 2p. (c)-(d) X-ray\nscattering spectra measured at 15 K and 80 K on the Ledge of (c) Cr and (d) Co.\n10Cr 15 K\nEnergy (eV)XRMS (%)(c)20151050-5595590585580575570 5T 3T 1T 0.1T(b)XRMS (%)Energy (eV)Co 15 K-80-60-40-20020\n800795790785780775 5T 3T 1T 0.1T(d)\nMagnetic Field (T)|XRMS| (%)806040200543210 Co 15 K Co 80 K Cr 15 K Cr 80 KTemperature (K)Moment (×10-6 emu)Tc = 95 KTs = 22 K(a)543210x10-6 120100806040204.554.504.45x10-6 353025201510FIG. 4. (a) Temperature-dependent magnetization curves of CCO in an applied \feld of 0.2 T\nalong the [1 \u00161 0 0] direction of the substrate. The inset is a magni\fed plot in the vicinity of the\nsecond transition point. (b)-(c) XRMS data on the L edge of (b) Co and (c) Cr measured with\ndi\u000berent applied magnetic \felds at 15 K. The color series of red, green, blue and pink stands for\n\feld strength of 5 T, 3 T, 1 T and 0.1 T, respectively. (d) Absolute values of Co and Cr XRMS\nmain peaks as a function of applied \feld and temperature.\n11" }, { "title": "2010.03979v1.Static_magnetic_proximity_effects_and_spin_Hall_magnetoresistance_in_Pt_Y___3__Fe___5__O___12___and_inverted_Y___3__Fe___5__O___12___Pt_bilayers.pdf", "content": "Static magnetic proximity e\u000bects and spin Hall magnetoresistance in Pt/Y 3Fe5O12and\ninverted Y 3Fe5O12/Pt bilayers\nStephan Gepr ags,1,\u0003Christoph Klewe,2Sibylle Meyer,1Dominik Graulich,3Felix Schade,1Marc Schneider,1\nSonia Francoual,4Stephen P. Collins,5Katharina Ollefs,6,yFabrice Wilhelm,6Andrei Rogalev,6Yves\nJoly,7Sebastian T.B. Goennenwein,8Matthias Opel,1,zTimo Kuschel,3and Rudolf Gross1, 9, 10\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germanyx\n2Advanced Light Source, Lawrence Berkeley National Laboratory, California 94720, USA\n3Center for Spinelectronic Materials and Devices,\nDepartment of Physics, Bielefeld University, 33615 Bielefeld, Germany\n4Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany\n5Diamond Light Source Ltd, Harwell Sci & Innovat Campus, Didcot OX11 0DE, Oxon, England\n6European Synchrotron Radiation Facility (ESRF), 38043 Grenoble Cedex 9, France\n7University Grenoble Alpes, CNRS, Grenoble INP, Institut N\u0013 eel, 38000 Grenoble, France\n8Technische Universit at Dresden, 01069 Dresden, Germany\n9Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n10Munich Center for Quantum Science and Technology (MCQST), 80799 M unchen, Germany\n(Dated: October 9, 2020)\nThe magnetic state of heavy metal Pt thin \flms in proximity to the ferrimagnetic insulator\nY3Fe5O12has been investigated systematically by means of x-ray magnetic circular dichroism and x-\nray resonant magnetic re\rectivity measurements combined with angle-dependent magnetotransport\nstudies. To reveal intermixing e\u000bects as the possible cause for induced magnetic moments in Pt, we\ncompare thin \flm heterostructures with di\u000berent order of the layer stacking and di\u000berent interface\nproperties. For standard Pt layers on Y 3Fe5O12thin \flms, we do not detect any static magnetic\npolarization in Pt. These samples show an angle-dependent magnetoresistance behavior, which\nis consistent with the established spin Hall magnetoresistance. In contrast, for the inverted layer\nsequence, Y 3Fe5O12thin \flms grown on Pt layers, Pt displays a \fnite induced magnetic moment\ncomparable to that of all-metallic Pt/Fe bilayers. This magnetic moment is found to originate\nfrom \fnite intermixing at the Y 3Fe5O12/Pt interface. As a consequence, we found a complex\nangle-dependent magnetoresistance indicating a superposition of the spin Hall and the anisotropic\nmagnetoresistance in these type of samples. Both e\u000bects can be disentangled from each other due\nto their di\u000berent angle dependence and their characteristic temperature evolution.\nI. INTRODUCTION\nThe understanding of spin phenomena in condensed\nmatter is of great importance in the \felds of spin-\ntronics and spin caloritronics.1,2In this context, heavy\nmetal/ferromagnetic insulator (HM/FMI) heterostruc-\ntures provide a unique platform for the generation and\ndetection of pure spin currents utilizing the (inverse) spin\nHall e\u000bect (SHE),3spin pumping,4or the spin Seebeck\ne\u000bect.5In these HM/FMI heterostructures, the transport\nof spin angular momentum across the HM/FMI interface\nis of key importance. Obviously, the magnetic and struc-\ntural properties at the interface between the HM layer\nand the FMI play a crucial role for the interfacial ex-\nchange interaction between the HM conduction electrons\nand the localized ions in the FMI.6{11In this respect,\npossible static magnetic moments in the HM layer at the\ninterface to the FMI induced by magnetic proximity ef-\nfects (MPE) came into the focus of research,12{17since\nHMs, such as Pt or Pd, are typically close to the Stoner\ncriterion for ferromagnetism and exhibit proximity in-\nduced \fnite magnetic moments in contact to ferromag-\nnetic metals (FMMs).18{20\nThe MPE can be attributed to a direct exchange inter-\naction between the magnetic elements in the ferromagnetand the conduction electrons of the HM, which is deter-\nmined by the overlap of their wave functions re\recting\nthe short range nature of the MPE.20,21While the ap-\npearance of a \fnite magnetic polarization in HMs in con-\ntact to FMMs is unquestionable,20the MPE in HM/FMI\nheterostructures is controversially discussed. In the pro-\ntotype Pt/Y 3Fe5O12(YIG) structure, Lu and coworkers\nreported an induced ferromagnetic moment of 0.054 \u0016B\nper Pt atom at room temperature by element-selective x-\nray magnetic circular dichroism (XMCD) measurements\nat the PtL2;3edges, which even increases to 0.076 \u0016B\nper Pt atom at 20 K.13However, the signi\fcant ferro-\nmagnetic Pt moment at room temperature could not be\nveri\fed by other groups using Pt or Pd as the HM layer\nand YIG or ferrites AFe2O4(A=Fe, Ni, Co, Mn) as the\nFMI.9,14,22{24In addition, no indication of a \fnite mag-\nnetic polarization in Pt grown on FMIs could be found\nby x-ray resonant magnetic re\rectivity (XRMR) mea-\nsurements, which is a direct element-speci\fc measure of\nthe spin polarization at the interface. The MPE has been\nexcluded by XRMR for Pt on top of NiFe 2O4deposited\nby chemical vapor deposition16and by sputtering.25Re-\ncently, however, a \feld induced magnetic polarization in\nPt on YIG was found at low temperature and high mag-\nnetic \felds by XMCD, mainly caused by the paramag-arXiv:2010.03979v1 [cond-mat.mtrl-sci] 8 Oct 20202\nnetic nature of Pt.17Furthermore, a strong XMCD sig-\nnal was observed in Fe 3O4/Pt/Fe 3O4epitaxial trilayers\nmainly caused by Fe-Pt interdi\u000busion and Fe-Pt alloying\ndue to the deposition of Pt at high temperature.26\nThe di\u000berence between the MPE in HM/FMM and\nHM/FMI structures seems to be consistent with the situ-\nation of superconducting thin \flms on FMM or FMI: the\nmagnetic moment of Cu in YBa 2Cu3O7is \fnite when in\nproximity to the FMM La 2=3Ca1=3MnO 3, but below the\ndetection limit on the FMI LaMnO 3.27\nThe presence or absence of MPEs in HM/FMI het-\nerostructures has direct consequences for the understand-\ning of spin current experiments. Additional magnetore-\nsistance (MR) or magneto-Seebeck and Nernst e\u000bects\nwill occur in spintronic and spin caloritronic experiments\nin the presence of static magnetic moments in the HM\nlayer.28,29As an example, the MR found in HM/FMI\nheterostructures was attributed to a magnetic-proximity\nMR based on the conventional anisotropic magnetoresis-\ntance (AMR).12,13,30\nNakayama et al.31and Althammer et al. ,32however,\ndemonstrated that the longitudinal resistivity of the HM\nlayer reaches its maximum value, if the magnetization of\nthe underlying FMI is either aligned along the current\ndirection jor the normal nof the thin \flm plane,\nwhile a minimum value was detected, when aligning the\nmagnetization in the \flm plane perpendicular to j. This\ncharacteristic angle dependence is inconsistent with the\nconventional AMR of a polycrystalline HM layer32and\nis, instead, explained by an interplay of charge and spin\ncurrents at the interface between the FMI and the HM\nlayer via the (inverse) SHE.33This so-called spin Hall\nmagnetoresistance (SMR) was further experimentally\ncon\frmed in a variety of HM/FMI heterostructures\nsuch as Pt/YIG,31,32,34{38Ta/YIG,35, Pt/Gd 3Fe5O12,39\nPt/Fe 3O4,32Pt/NiFe 2O4,32,40Pt/CoFe 2O4,41and\nPt/Cu 2OSeO 342as well as using antiferromagnetic\ninsulators NiO43{46, Cr 2O3,47,48and\u000b-Fe2O3.46,49{51\nThe exchange of spin angular momentum as the under-\nlying mechanism of the SMR is further con\frmed by\nPt/YIG/Pt trilayer structures52,53and non-local trans-\nport experiments in Pt/YIG bilayer nanostructures.54{56\nAs discussed by Kikkawa and coworkers,17the pres-\nence of a \fnite magnetic polarization in the HM caused\nby MPEs could be related to defects at the interface be-\ntween the FMI and the HM layer, such as interdi\u000busion\nof ions and alloying, thin amorphous layers, vacancies\nor free magnetic elements at the surface, demonstrating\nthat the quality of the interface is of crucial importance.\nThis is also con\frmed by Vasili et al. .9They showed that\nthe presence or absence of a MPE in Pt depends on the\nPt growth conditions, leading to a possible interfacial re-\nconstruction.\nTo get more insight into the origin of MPEs in\nHM/FMI heterostructures and their correlation to\nMR e\u000bects, we systematically investigate Pt/YIG het-\nerostructures with di\u000berent order of the layer stacking\nand di\u000berent interface properties using element-selectivex-ray and angle-resolved magnetotransport studies. We\n\fnd no indication for any MPE in standard Pt/YIG\nheterostructures with clean, in-situ grown Pt/YIG in-\nterfaces. In these samples, the existing temperature-\ndependent MR in the Pt layer can be explained within\nthe framework of the SMR model. In contrast, in in-\nverted YIG/Pt heterostructures, we clearly detect a \f-\nnite XMCD as well as a XRMR signal demonstrating\ninduced magnetic moments in the Pt layer most likely\ncaused by intermixing e\u000bects at the YIG/Pt interface. In\nthose samples, we \fnd a superposition of the SMR and\nthe AMR, which can be disentangled from each other\ndue to their characteristic angle and temperature depen-\ndence. We further investigated aging e\u000bects of the Pt\nlayers utilizing x-ray absorption near edge spectroscopy\nand magnetotransport measurements. While the white\nline intensity of the Pt L2;3edges in the inverted YIG/Pt\nheterostructures stays nearly una\u000bected over time, a clear\nincrease is observed in standard Pt/YIG bilayers indi-\ncating oxidation e\u000bects. However, the SMR amplitude\nchanges only marginally with time.\nII. SAMPLE FABRICATION AND\nCHARACTERIZATION\nA. Thin \flm deposition\nA series of Pt/YIG bilayer heterostructures on (111)-\noriented, single crystalline Y 3Al5O12(YAG) substrates\nwas fabricated in-situ in an ultra-high vacuum system.57\nThe YIG thin \flms were deposited by pulsed laser de-\nposition (PLD)58from a stoichiometric, polycrystalline\ntarget using a KrF excimer laser ( \u0015= 248 nm) with a\n\ruence of 2 J/cm2and a repetition rate of 10 Hz. The\nPt layers were deposited via electron-beam evaporation\nin ultra-high vacuum at room temperature with a depo-\nsition rate of 0 :4\u0017A/s. To probe possible intermixing ef-\nfects at the Pt/YIG interface, we fabricated two di\u000berent\ntypes of bilayer samples: The \frst one consists of a YIG\nthin \flm deposited in O 2atmosphere at 500\u000eC capped\nin-situ without breaking the vacuum with an approxi-\nmately 2 nm thick Pt layer (\\standard\" Pt/YIG//YAG\nbilayer).14,32For the second type of bilayers, we \frst de-\nposited a polycrystalline 10 nm thin Pt \flm on a YAG\nsubstrate and subsequently, in-situ , the YIG thin \flm on\ntop in an Ar atmosphere at 450\u000eC to suppress oxidation\nof the Pt layer (\\inverted\" YIG/Pt//YAG bilayer).38\nWe expect a clean and sharp interface between the\nmetallic, polycrystalline Pt thin \flm and the insulat-\ning YIG layer for the standard Pt/YIG//YAG bilayer\nsamples,14since the electron-beam evaporation method\nis associated with low, thermal kinetic energies of the\nPt particles. Hence, a vanishingly small intermixing at\nthe YIG/Pt interface is expected for this stacking se-\nquence. For the inverted YIG/Pt//YAG bilayer sam-\nples, however, the situation is di\u000berent. Here, Pt is\npartly incorporated into the YIG thin \flm and vice versa,3\n40°45°50°55°100101102103104105I(cps)\n2θ-4°0° 4°12I(cps)\n∆ω(c)\nPt (111)\nYIG (444)\nYAG (444)Pt (111)40°45°50°55°100101102103104105I(cps)\n2θ-0.04°0.00°0.04°102030I(cps)\n∆ω(a)\nPt (111)\nYIG (444)\nYAG (444)YIG (444)\n0.1 0.2 0.310-710-610-510-410-310-210-1100I/I0\nq(Å-1)experiment\nsimulation(b)\nYIGPt\nYAG 52 nm2 nm\n0.1 0.2 0.310-510-410-310-210-1100I/I0\nq(Å-1)experiment\nsimulation(d)\nPtYIG\nYAG 11 nm17 nm\nFIG. 1. Structural properties of (a),(b) a standard\nPt/YIG//YAG bilayer (black lines) and (c),(d) an inverted\nYIG/Pt//YAG bilayer (blue lines) measured at 300 K. The\nout-of-plane high-resolution x-ray !-2\u0012di\u000braction scans are\nshown in (a),(c). The expected 2 \u0012-positions of the Pt (111)\nand YIG (444) re\rections are indicated by dashed verti-\ncal lines. The insets display the rocking curves around the\nYIG (444) and the Pt (111) re\rections, respectively. (b),(d)\nNon-magnetic x-ray re\rectivity scans plotted against the scat-\ntering vector q. From the simulations (red lines), we deter-\nmine the thickness and roughness of the layers. The bilayer\nstacks with the layer thickness of the respective samples are\nsketched in the insets.\ndue to the high kinetic energies of the atoms and ions\nin the laser plume during the PLD-process of YIG.59,60\nOn the other hand, a partial oxidation of the Pt thin\n\flm of the standard Pt/YIG//YAG bilayer samples over\ntime is expected. This is not the case for the inverted\nYIG/Pt//YAG bilayer samples, since here the thin Pt\nlayer is covered by the thick YIG layer.\nIn the following, we will restrict our discussion to the\ndata of two representative standard and inverted bilayer\nsamples.B. Structural characterization\nThe samples were characterized with respect to their\nstructural properties using high-resolution x-ray di\u000brac-\ntometry (HR-XRD) in a four-circle di\u000bractometer with\nmonochromatic Cu K\u000b1radiation with a wavelength of\n0.15406 nm. The !-2\u0012scans of the bilayers reveal no sec-\nondary crystalline phases (cf. Figs. 1(a),(c)). However,\nwhile YIG was found to be crystalline in the standard\nPt/YIG//YAG bilayer with a fully relaxed crystal struc-\nture exhibiting a lattice constant of 1.238 nm,61no re-\n\rection of YIG was found in the inverted YIG/Pt//YAG\nbilayer indicating a polycrystalline growth of YIG on\nPt. For the standard Pt/YIG//YAG bilayer, the rock-\ning curves around the YIG (444) re\rection display a full\nwidth at half maximum (FWHM) of about 0 :03\u000e(cf. in-\nset in Fig. 1(a)), indicating a low mosaic spread of the\nYIG thin \flms despite the 3% lattice mismatch between\nYIG and the YAG substrate. On the other hand, we do\nnot detect any Pt-related re\rection, indicating a poly-\ncrystalline structure of the Pt thin \flm. The situation\nis di\u000berent for the inverted YIG/Pt//YAG bilayer sam-\nples. The!-2\u0012scan displays a weak Pt (111) re\rection,\nwhich can be attributed to a (111)-textured Pt layer with\na high mosaic spread as revealed by the large FWHM of\nabout 4\u000eof the rocking curve (cf. inset in Fig. 1(c)). The\ntextured structure arises when heating the Pt thin \flm\nto 450\u000eC in Ar atmosphere before the deposition of the\nYIG layer. However, the Pt (111) re\rection is found at a\nhigher 2\u0012-angle (2\u0012= 40:21\u000e) compared to Pt thin \flms\nfabricated from the same deposition chamber on various\nmagnetic materials or the literature value of 2 \u0012= 39:755\u000e\nfor bulk Pt.62This might be attributed to a \fnite inter-\nmixing of Pt most likely with Fe rather than oxidation\nof Pt.63,64\nThe thickness as well as an estimation of the rough-\nness of the respective layers were determined by non-\nmagnetic x-ray re\rectivity (XRR) at the Diamond Light\nSource (DLS) at beamline I16 and at the Deutsches\nElektronen-Synchrotron (DESY) at beamline P09 with\na photon energy of 11566 eV (cf. Figs. 1(b),(d)). From\nsimulations to the experimental data using the recursive\nParratt algorithm,65a N\u0013 evot-Croce roughness model,66\nand layers with individual refractive indices n= 1\u0000\n\u000e+i\f(\u000e: dispersion, \f: absorption), we obtain (1 :7\u0006\n0:1) nm (Pt) and (52 :0\u00060:1) nm (YIG) for the stan-\ndard Pt/YIG//YAG bilayer (cf. Fig. 1(b)). The inter-\nface roughness as well as the Pt surface roughness are\nfound to be 0.2 nm and 1.5 nm, respectively. Since the\nPt and the YIG fringes are not very pronounced, a pre-\ncise evaluation of the exact structural and optical pa-\nrameters is challenging resulting in uncertainties of the\ndetermined values/signi\fcant error bars. Furthermore,\nas the Pt roughness is in the same range as the total Pt\nlayer thickness, it might in\ruence the resistivity of the\nPt layer by surface roughness induced scattering.67For\nthe inverted YIG/Pt//YAG bilayer sample, we obtain\nlayer thicknesses of YIG and Pt of (17 :4\u00060:5) nm and4\n-4-20 2 4-100-50050100M(kA/m)\nµ0H(T)-4-20 2 4-100-50050100M(kA/m)\nµ0H(T)300 K 300 K(a) (b)\nYIGPt\nYAG 52 nm2 nm PtYIG\nYAG 11 nm17 nm\nFIG. 2. Magnetization curves of (a) a standard\nPt/YIG//YAG bilayer (black symbols) and (b) an inverted\nYIG/Pt//YAG bilayer (blue symbols) recorded at 300 K with\nthe magnetic \feld applied parallel to the \flm plane. The dia-\nmagnetic contributions from the YAG substrates have been\nsubtracted.\n(11:2\u00060:3) nm, respectively (cf. Fig. 1(d)). Here, the\ninterface roughness was found to be 0.7 nm and the YIG\nsurface roughness 1.5 nm.\nTaken together, the standard Pt/YIG//YAG bilayer\nsamples are composed of an epitaxially grown YIG thin\n\flm covered with a polycrystalline Pt thin \flm, whereas\nthe inverted YIG/Pt//YAG bilayer samples are consist-\ning of a polycrystalline YIG thin \flm grown on top of a\nPt layer exhibiting a (111)-textured structure with pos-\nsible \fnite intermixing e\u000bects at the YIG interface.\nC. Magnetic characterization\nThe in-plane magnetic properties of the bilayer sam-\nples were studied by superconducting quantum interfer-\nence device (SQUID) magnetometry (cf. Fig. 2). At room\ntemperature, we obtain similar magnetization curves for\nboth sample types with a saturation magnetization of\naroundMs= (110\u00065) kA/m. This value is in agree-\nment with our previous results,14but slightly lower than\nthe bulk value of MYIG\ns = 143 kA/m.68While for the\nstandard Pt/YIG//YAG bilayer sample type, this might\nbe caused by interdi\u000busion of Al from the YAG sub-\nstrate into the \frst monolayers of the YIG thin \flm,69\nthe reduced saturation magnetization of the inverted\nYIG/Pt//YAG sample is most likely also a result of in-\nterdi\u000busion at the YIG/Pt interface. This assumption is\nsupported by the di\u000berence in coercive \felds of 2 mT and\n40 mT for the standard Pt/YIG//YAG and the inverted\nYIG/Pt//YAG sample, respectively. Furthermore, a sec-\nond magnetically hard phase seems to be present in the\ninverted sample revealed by the additional hysteresis visi-\nble at high magnetic \felds between 0.5 T and 2 T. This in-\ndicates that interdi\u000busion at the YIG/Pt interface mightresult in a magnetically hard in-plane component similar\nto FePt.70\nIII. ELEMENT-SELECTIVE MAGNETIC\nPROPERTIES\nTo investigate the magnetic properties of the Pt lay-\ners in our bilayer samples, we take advantage of ad-\nvanced, element-selective, synchrotron-based techniques\nusing hard x-rays.71,72The x-ray absorption near edge\nspectra (XANES) and XMCD measurements were per-\nformed at the European Synchrotron Radiation Facility\n(ESRF) at the beamline ID12 using the total \ruorescence\nyield detection mode.73The XRMR measurements were\ncarried out at DLS (beamline I16) and DESY (beamline\nP09).\nA. X-ray absorption near edge spectra\nThe XANES were recorded around the Pt L3\n(11564 eV) and L2edges (13273 eV) with right and left\ncircularly polarized light under positive and negative\nmagnetic \felds of \u00060:6 T and\u00060:9 T. An electromagnet\nallowed to \rip the \feld direction at each energy value of\nthe incoming photons. The incident angle of the x-rays\nwas set between 3\u000eand 5\u000ewith respect to the surface\nplane, while the external magnetic \feld was aligned par-\nallel to the incident beam. Several XANES were recorded\nto increase the signal-to-noise ratio and normalized to an\nL3edge jump of unity and an L2edge jump of 0 :45 ac-\ncording to Mattheiss & Dietz.75\nThe XANES from a standard Pt/YIG//YAG (solid\nblack line) and an inverted YIG/Pt//YAG bilayer (solid\nblue line) recorded shortly after the fabrication of the\nsamples are shown in Figs. 3(a) and (b), respectively. We\nclearly observe the Pt L3and PtL2absorption edges\nfor both sample types. Both XANES also display ex-\ntended x-ray absorption \fne structure (EXAFS) wig-\ngles at around 11588 eV and 13300 eV (cf. vertical ar-\nrows in Figs. 3(a) and (b)). However, these EXAFS\nwiggles are shifted to lower energies in case of the in-\nverted YIG/Pt//YAG bilayer sample. To obtain more\ninformation about the valence state and chemical envi-\nronment of the absorbing Pt atoms, we calculated the\nXANES using the FDMNES code76,77in a Density Func-\ntional Theory (DFT) full potential, relativistic approach\nincluding spin-orbit coupling (Fig. 4). This code is ex-\ntensively used to simulate XANES and resonant x-ray\nscattering spectra. As obvious from Fig. 4(a), the calcu-\nlated XANES of Pt (full black line) is in good agreement\nwith the experimentally obtained XANES of our stan-\ndard Pt/YIG//YAG bilayer (open black symbols). Not\nonly the Pt L3and PtL2absorption edges but also the\nEXAFS wiggles are clearly reproduced. In case of the\ninverted YIG/Pt//YAG bilayer (see Fig. 4(b)), the mea-\nsured XANES (open blue symbols) show clear di\u000berences5\n11550 11600 13300 133500.00.51.01.5\n-1.5-1.0-0.50.00.51.01.5\n**norm.XANES(arb.units)\nphotonenergy(eV)*\nXMCD(%)(a)\nRT\n46 K\n0 50010001.21.41.6PtO1.36\ntime(days)norm.I(a.u.)\nPtPt L3Pt L2\n11550 11600 13300 133500.00.51.01.5\n**norm.XANES(arb.units)\nphotonenergy(eV)-4-3-2-10123\nXMCD(%)(b)\n46 KRT46 K\nRT\n0 50010001.21.41.6\ntime(days)norm.I(a.u.)\nPtO1.36\nPtPt L3Pt L2YIGPt\nYAG 52 nm2 nm\nPtYIG\nYAG 11 nm17 nm\nFIG. 3. Normalized Pt L2andL3XANES (full lines, left\naxis) and XMCD spectra (dashed lines, right axis) of (a) a\nstandard Pt/YIG//YAG bilayer (black) and (b) an inverted\nYIG/Pt//YAG bilayer (blue) measured at room temperature\n(RT) and 46 K. The magnetic \feld of \u00060:6 T was applied par-\nallel to the incoming x-rays under an angle of 3...5\u000eto the\nsample surface. Di\u000braction peaks are marked by asterisks ( ?).\nBoth samples display EXAFS wiggles at around 11588 eV and\n13300 eV, marked by the vertical arrows. The inset shows the\nnormalized XANES Pt L3white line intensity as a function\nof the time after the sample fabrication. For comparison, the\nvalues for Pt and PtO 1:36from Ref. 74 are indicated by dashed\nhorizontal lines.\nto the simulated Pt XANES (black full line) in the en-\nergy range between 11576 eV and 11595 eV ( L3) as well\nas between 13285 eV and 13304 eV ( L2), i.e. around the\n\frst EXAFS wiggle directly after the L3andL2absorp-\ntion edges (cf. Fig. 4(b)). In particular, an increased\nintensity of the EXAFS wiggles at around 11588 eV and\n13300 eV can be observed and furthermore the EXAFS\nwiggles are shifted to lower energies. To explain this, we\nconsidered a \fnite interdi\u000busion at the YIG/Pt interface\neither leading to regions with a PtFe alloy or a \fnite Pt-\ndoping of YIG. In the latter case, the calculated XANES\nof Pt on the di\u000berent sites of the YIG crystal structure\nreveal completely di\u000berent energy dependencies of the x-\nray absorbed intensity (cf. dashed, dashed-dotted, dotted\nlines in Fig. 4(b)). However, the calculated XANES ofPtFe (blue full line in Fig. 4(b)) agrees fairly well with\nthe experimental results of the inverted YIG/Pt//YAG\nbilayer reproducing the higher intensity as well as the en-\nergy shift of the EXAFS wiggles. This is in agreement\nwith experiments on Pt alloys78and indicates that a \f-\nnite interdi\u000busion at the YIG/Pt interface of the inverted\nYIG/Pt//YAG bilayer took place during the deposition\nleading to regions, where Pt is in direct contact to Fe.\nThe white line intensities at the absorption edges rep-\nresent another known sensitive measure for the valence\nstate as well as chemical environment of Pt, since it is re-\nlated to the 5 delectron vacancies.79For example, PtO 1:6\ndisplays an L3white line intensity of 2.20, PtO 1:36of\n1.50, and metallic Pt of 1.25, with respect to the edge\njump.74Furthermore, a small reduction of the white line\nintensity to around 1.20 is found for FePt, which is at-\ntributed to electronic hybrid states as a result of the al-\nloying of Pt with Fe.80We observe values of 1.29 and\n1.26 in our standard and inverted bilayer sample types,\nrespectively. These values are very close to our previous\nobservations14and the one reported for metallic Pt.74For\nPt on di\u000berently prepared NiFe 2O4\flms, slightly larger\nwhite line intensities between 1.33 and 1.35 have been\nfound,16,25while for Pt/CoFe 2O4the white line inten-\nsity was 1.24.22However, Lu et al. reported a larger\nwhite line intensity of 1.45 relative to the edge jump\nin a 1.5 nm thin Pt layer on YIG.13This discrepancy\ncompared to our results and to the literature values is\neven more pronounced for the white line intensity at\nthe PtL2edge. For Pt metal an intensity of 0 :7981is\nexpected, which is in nice agreement with our results\n(cf. Fig. 3(a)), but in contrast to the data reported by\nLuet al. . They found a value larger than 1.0.13One pos-\nsible reason for an enhanced white line intensity might be\naging of the Pt surface. For the Pt L3white lines, we ob-\nserve an increase in intensity to 1.57 within \u0018900 days\nafter the thin \flm deposition for the standard bilayer\nsample type, whereas the value for the inverted one stays\nconstant over time (cf. insets in Fig. 3(a),(b)). This can\nbe understood as partial oxidation of Pt when exposed\nto ambient atmosphere, which is highly possible for the\nstandard Pt/YIG//YAG bilayer, but suppressed by the\nYIG capping layer in the inverted YIG/Pt//YAG bilayer.\nRecently, possible aging e\u000bects have also been reported\nfor Pt(3.2 nm)/Fe(9.1 nm) bilayers, where the magnitude\nof the induced magnetic Pt moment decreased by about\n30% within half a year.16,82\nB. X-ray magnetic circular dichroism\nThe XMCD spectra were calculated as the direct di\u000ber-\nence between consecutive XANES recorded with opposite\nx-ray helicity or magnetic \feld direction. The following\nXMCD results, therefore, give access only to the projec-\ntion of the magnetization of Pt on the external magnetic\n\feld, i.e. the k-vector of the incoming x-ray beam.\nWhile the XANES of the Pt layer of both sample types6\n11550 11600 13300 133500.00.51.01.5\nXANESPt/YIG\nPtcalculationsnorm.XANES(arb.units)\nphotonenergy(eV)Pt L3\nYIGPt\nYAG(a)Pt L2\n11550 11600 13300 133500.00.51.01.5\nXANESYIG/Pt\nPtcalculations\nPtFecalculations\nPtonYc-site\nPtonFed-site\nPtonFea-sitenorm.XANES(arb.units)\nphotonenergy(eV)(b)Pt L3Pt L2\nPtYIG\nYAG\nFIG. 4. Calculated XANES around the L2- andL3-edge us-\ning the FDMNES code.77(a) Calculated XANES of Pt (full\nblack line) and experimental XANES (open symbols) of a\nstandard Pt/YIG//YAG bilayer. (b) Calculated XANES of\nPtFe (full blue line) and Pt-doped YIG with Pt replacing Y\non thec-site (dashed blue line), substituting Fe on the d-site\n(dashed dotted blue line), as well as on the a-site (dotted blue\nline) of the garnet crystal structure. For comparison, the cal-\nculated XANES of Pt (full black line) is also shown. The\nexperimental XANES of an inverted YIG/Pt//YAG bilayer\nis depicted with open symbols. Since the calculations are in\nphotoelectron energies with zero being the Fermi energy, the\ncalculated XANES were shifted in energy by around 11571 eV\n(L3-edge) and around 13283 eV ( L2-edge), respectively, for\ncomparison with the experimental data. The vertical arrows\nindicate again the \frst EXAFS wiggles.\nare almost identical, their XMCD spectra are di\u000berent\n(cf. dashed lines in Figs. 3(a) and (b)). For the standard\nPt/YIG//YAG bilayer, we do not observe a \fnite XMCD\nsignal at both Pt L2andL3edges down to a noise level of\n<0:1% with respect to the edge jump at room temper-\nature (RT) (cf. black dashed line in Fig. 3(a)) and 46 K\n(cf. grey dotted line in Fig. 3(a)). Therefore, we do not\n\fnd any indication for a \fnite induced magnetic moment\nin Pt on YIG due to MPEs. This supports our previ-\nous results from a comprehensive XMCD study of three\ndi\u000berent standard Pt/YIG bilayer samples with di\u000berent\nthicknesses (3 nm, 7 nm, and 10 nm) of the Pt top layer,from which we identi\fed an upper limit of a possible\ninduced moment of (0 :003\u00060:001)\u0016Bper Pt.14Further-\nmore, it is in agreement with the data reported recently\nby di\u000berent groups.9,22{24However, these results are in\ncontrast to the \fnite induced ferromagnetic magnetic mo-\nment of 0.054 \u0016Bin Pt detected by XMCD in Pt/YIG\nbilayers reported by Lu and coworkers.13We note that\nthe noise level of our data is at least 10 times lower than\ntheir XMCD signal of 1%.\nIn the inverted YIG/Pt//YAG bilayer sample, how-\never, we detect a \fnite XMCD signal at both Pt L3and\nL2edges (cf. Fig. 3(b)). The maxima of the XMCD signal\nare located at slightly lower energies than the maximum\nof the XANES in accordance with literature.83To quan-\ntify the induced magnetic moment of the Pt atoms aver-\naged over the Pt \flm thickness, we apply magneto-optic\nsum rules neglecting the magnetic dipole term due to the\ncubic symmetry and the polycrystalline nature of the Pt\nthin \flm.84,85To this end, we \frst determine the number\nof holes following the method proposed by Ref. 86 using\nthe Au x-ray absorption white line intensity published in\nRef. 81 as reference. We \fnd a number of holes per Pt\natom of 1.87, which is slightly larger than reported for\nmetallic Pt.81With this value, we obtain a spin moment\nofms= 0:058\u0016B/Pt and an orbital magnetic moment of\nml= 0:011\u0016B/Pt, resulting in a ratio of ml=ms= 0:188\nat room temperature. Thus, msis of the same order as\nthe total induced magnetic moment of 0 :032\u0016B/Pt ob-\ntained earlier from an all-metallic polycrystalline Pt/Fe\nreference sample with a 10 nm thin Pt top layer.14This\nhighmsvalue suggests that a large fraction of Pt atoms\nis in direct proximity to Fe atoms, indicating a high level\nof intermixing at the YIG/Pt interface in the inverted\nYIG/Pt//YAG bilayer sample. The XMCD signal even\nincreases at lower temperature (cf. dotted blue line in\nFig. 3(b)). At 46 K, we \fnd a spin and angular moment\nofms= 0:160\u0016B/Pt andml= 0:016\u0016B/Pt, respectively,\nwhich is almost three times larger than at room temper-\nature.\nC. X-ray resonant magnetic re\rectivity\nWe further con\frm the XMCD results by XRMR mea-\nsurements recorded at room temperature at the Pt L3\nedge at a \fxed photon energy close below the XANES\nmaximum.16,25,82An external magnetic \feld of \u00160H=\n85 mT was applied in the scattering plane parallel to\nthe sample surface by a four-coil electromagnet. Cir-\ncularly polarized light was used for the measurements\nwith a degree of polarization of (81 \u00065)% at DLS (stan-\ndard Pt/YIG//YAG sample) and of (99 \u00061)% at DESY\n(inverted YIG/Pt//YAG bilayer sample). The magnetic\n\feld direction was \ripped ( \u0006H) at DLS for each value of\nthe scattering vector q, while the re\rected intensity I\u0006\nwas detected and the x-ray polarization (left) was kept\nconstant. At DESY, the magnetic \feld stayed constant\n(+H), while the x-ray helicity was \ripped (left/right) for7\neach value of q, thus obtaining the re\rected intensity I\u0006.\nThe XRMR asymmetry ratio \u0001 I= (I+\u0000I\u0000)=(I++I\u0000)\nis simulated with ReMagX72using the structural pa-\nrameters from the non-magnetic XRR analysis (cf. sec-\ntion II B) and additional magneto-optic depth pro\fles,\nwhich describe the magneto-optic parameters \u0001 \u000eand \u0001\f\nvertical to the layer stack. Slightly di\u000berent energy cali-\nbrations of the beamlines lead to slight variations of the\nwhiteline energies. Therefore, di\u000berent \u0001 \f=\u0001\u000eratios for\nanalyzing the XRMR data have been chosen by either\ncomparing the XAS to the ab initio calculations used\nin Ref. 16 or by identifying the energy with maximal\ndichroic e\u000bect and vanishing \u0001 \u000e. We use a \u0001 \f=\u0001\u000era-\ntio of 7.3 for \ftting the DLS data and \u0001 \u000e= 0 for the\nDESY data (cf. Ref. 87). A detailed description of the\n\ftting procedure and more information on the shape of\nthe magneto-optic depth pro\fles can be found in Ref. 82.\nNote that \u0001 \u000eand \u0001\fare proportional to the magnetic\nPt moment and can be converted from magneto-optic\nconstant into a magnetic moment per Pt atom using\nthe conversion factor theoretically calculated in Ref. 16\nfor smooth standard Pt/FMM bilayers as con\frmed by\nXMCD.88However, this factor has to be handled with\ncare when analyzing the inverted YIG/Pt//YAG sam-\nple with probable interdi\u000busion at the YIG-Pt interface.\nTherefore, the focus of the XRMR analysis for the in-\nverted YIG/Pt//YAG sample is the absence or presence\nof the MPE rather than a rigorous quantitative evalua-\ntion.\nThe XRMR responses of the two investigated sample\ntypes are quite di\u000berent as presented in Fig. 5. The\nexperimental asymmetry ratio \u0001 Iof the standard sam-\nple type (black symbols in Fig. 5(a)) does not exhibit\nany oscillations. In comparison, we simulate a theoret-\nical asymmetry ratio of a Pt/YIG//YAG sample with\nthe same structural and optical parameters, but using\nthe magneto-optic depth pro\fle shown in the inset of\nFig. 5(a). The 1.3 nm e\u000bective layer thickness of the spin-\npolarized Pt is chosen according to the spin-polarized\ne\u000bective layer thickness found for Pt/Fe bilayers.16,25,82\nOur simulation yields a long-range oscillation (red line in\nFig. 5(a)) caused by the small Pt thickness of 1.7 nm. The\nexperimental data clearly do not show such an oscillation\nor any other non-zero oscillating feature aside from noise\n\ructuations. This is in accordance with XRMR mea-\nsurements on Pt/NiFe 2O416,25and con\frms the XMCD\nresults discussed in section III B.\nUsing the simulated maximum magneto-optic change\nin Pt as well as the conversion factor determined in\nRef. 16 and taking into account the degree of light polar-\nization, we calculate an upper limit of 0.002 \u0016Bper spin-\npolarized Pt atom within the 1.3 nm thick spin-polarized\nvolume. If the whole Pt \flm of 1.7 nm (spin-polarized\nplus non-polarized part) was taken into account, the\nupper limit would be 0 :002\u0016B\u0002(1.3 nm/1.7 nm) =\n0:0015\u0016Bper Pt atom, which improves our sensitivity\nlimit by a factor of two with respect to the XMCD results.\nThis advantage of XRMR is even more pronounced for\n-4-3-2-1012x10-2\nexperiment\nbackgroundXRMRasymmetry∆I0.1 0.2 0.3 0.4-0.4-0.3-0.2-0.10.00.10.2x10-2\nexperiment\nsimulationXRMRasymmetry∆I\nq(Å-1)\n5007I(a.u.)\nf(a.u.)(b) room temperature\nPtYIG\nYAG 11 nm17 nm(a) room temperature\nPt L3-edge\nYIGPt\nYAG 52 nm2 nm\nbackground\nYIG/Pt\n0.1 0.2 0.3 0.4-4-3-2-1012x10-2\nexperiment\nsimulationXRMRasymmetry∆I\nq(Å-1)(c) room temperature\n0 20∆β\n∆β(10-9)YIG\nPt1.9 nm0 20∆β\n∆β(10-10)Pt\nYIG1.3 nmFIG. 5. XRMR asymmetry ratios \u0001 Iat the PtL3edge of\n(a) a standard Pt/YIG//YAG bilayer (black symbols) and\n(b),(c) an inverted YIG/Pt//YAG bilayer (blue symbols) be-\ntween XRR curves measured under the same conditions as in\nFig. 1(b),(d), but with an applied magnetic \feld of \u000685 mT at\nroom temperature. (a) The asymmetry ratio of the standard\nbilayer does not show any XRMR response within experimen-\ntal error. The simulation (red line) using the magneto-optic\ndepth pro\fle (inset) gives an upper limit of 0.002 \u0016Bper spin-\npolarized Pt atom. (b) Raw data of the inverted bilayer shows\nclear oscillations in the asymmetry ratio \u0001 I. A Fourier analy-\nsis (inset) identi\fes two main contributions: oscillations from\nthe YIG/Pt (green) and a background at very low frequen-\ncies (red), which was extracted and back-transformed (dashed\nred line). (c) Asymmetry ratio after subtraction of the back-\nground contribution from (b). The simulation (red line) is\nbased on the magneto-optic depth pro\fle shown in the inset.8\nthicker Pt \flms, since XRMR is directly sensitive to the\nspin-polarized Pt, while XMCD is a\u000bected by both spin-\npolarized and non-polarized Pt, as discussed in Refs. 16\nand 82.\nIn contrast to the standard sample type, the XRMR\nasymmetry ratio \u0001 Iof the inverted YIG/Pt//YAG sam-\nple (cf. blue symbols in Fig. 5(b),(c)) shows clear oscil-\nlations due to magnetic Pt. The absolute asymmetry\nratio reaches values of up to 3.5%, which is the same or-\nder of magnitude as found for Pt on ferromagnetic met-\nals such as Pt/Fe16,25or Pt/Ni 33Fe67and Pt/Ni 81Fe19\n(permalloy).82By \ftting the raw XRMR asymmetry we\nare able to reproduce the high frequency oscillations from\nthe magnetized Pt. However, we observe a systematic\ndeviation between the simulation and the asymmetry ra-\ntio. A Fourier analysis of the XRMR asymmetry (in-\nset of Fig. 5(b)) reveals an additional low frequency\nbackground (red) on top of the magnetic response from\nYIG/Pt (green) that cannot be reproduced within our\nmodel. The background was separated from the other\nfrequencies and back-transformed (cf. red dashed line in\nFig. 5(b)). This long-range background, which was not\nobserved in previous measurements, will be investigated\nin future studies.\nIn order to model the XRMR asymmetry ratio based\non the structural and optical parameters of the XRR re-\nsults, the long-range background is subtracted as shown\nin Fig. 5(c). Using the magneto-optical depth pro\fle\ngiven in the inset of Fig. 5(c), we obtain a spin-polarized\nPt layer with an e\u000bective thickness of 1.9 nm and a mag-\nnetic moment of 0.09 \u0016Bper spin-polarized Pt atom,\nwhich is located near the YIG-Pt interface. Taking the\nwhole Pt thickness of 11.2 nm into account, we obtain\n0.09\u0016B\u00021.9 nm / 11.2 nm = 0.015 \u0016Bper Pt atom, which\nis in the same order of magnitude as the obtained mag-\nnetic moment by the XMCD experiment.\nThe XRMR results of the inverted YIG/Pt//YAG\nbilayer are not as comparable to XMCD as the re-\nsults that have been obtained for Pt grown on Fe and\nCoFe with almost perfect quantitative agreement be-\ntween XRMR and XMCD ( \u00062%).88One reason could\nbe the poorer \ft quality of the asymmetry ratio for\nthe inverted YIG/Pt//YAG bilayer compared to metal-\nlic bilayers. Still, the \ft result presented here re\rects a\nclear overall global minimum as tested by goodness-of-\ft\nspace mappings which have proven e\u000bective in previous\nXRMR studies.89,90In addition, the origin of the oscillat-\ning background observed for the inverted YIG/Pt//YAG\nbilayer is unknown and has not been observed previously\nfor metallic bilayers, but could a\u000bect the quantitative\nXRMR results as well. Moreover, the high roughness of\nthe YIG-Pt interface together with the possible interdif-\nfusion at this interface is not captured by the ab initio\ncalculations that form the basis of the \u0001 \fto\u0016Bper\nPt atom conversion factor.16In spite of these uncertain-\nties and the discrepancy between the quantitative XRMR\nand XMCD results, we clearly observe spin-polarized Pt\nat the YIG-Pt interface of the inverted YIG/Pt//YAGsample.\nIV. MAGNETOTRANSPORT\nMEASUREMENTS\nTo investigate the impact of the \fnite induced mag-\nnetic Pt-moment of the inverted YIG/Pt//YAG bilayer\nsample on the SMR, the thin \flm bilayer samples were\npatterned into Hall bar mesa structures (width 80 \u0016m and\ncontact separation 800 \u0016m) via photolithography and Ar\nion milling. The samples were mounted on a rotatable\nsample holder, which is placed in a superconducting mag-\nnet cryostat. We applied a current Ic= 100\u0016A in the Pt\nlayer with the current direction jk[121] of the YIG thin\n\flm and measured the longitudinal voltage Vlongvia a DC\ncurrent reversal technique,39while rotating the magnetic\n\feldHat a constant magnitude of \u00160H= 7 T. This value\nis well above the saturation \feld of YIG (cf. Fig. 2), en-\nsuring that the magnetization Mof the YIG layer is sat-\nurated along the magnetic \feld direction H. We rotated\nthe magnetic \feld Hin three di\u000berent rotation planes:\n(i) in the \flm plane with angle \u000b(ip), (ii) out of the \flm\nplane perpendicular to the current direction jwith angle\n\f(oopj), and (iii) out of the \flm plane perpendicular\nto the transverse direction twith angle \r(oopt) (cf. in-\nsets in Fig. 6(a)-(c)). From the measured longitudinal\nvoltage, we calculated the longitudinal resistivity \u001along\nand normalized it to the respective value \u001a0=\u001along;Hkt.\n\u001a0di\u000bers signi\fcantly for both sample types. While for\nthe inverted YIG/Pt//YAG bilayer \u001a= 2:71\u000210\u00007\nm\nat 300 K is in perfect agreement with previous reports,37\nan increased resistivity of \u001a= 1:19\u000210\u00006\nm at 300 K\nis found for the standard Pt/YIG//YAG bilayer sample\ntype. This can be attributed to size as well as rough-\nness e\u000bects, and can be described in a modi\fed Fuchs-\nSondheimer theory.91{93Moreover, the higher resistivity\nof the standard bilayer sample type further indicates a\npartially oxidized Pt layer.\nThe angle-dependent magnetoresistance (ADMR) of\nthe standard Pt/YIG//YAG as well as the inverted\nYIG/Pt//YAG bilayer sample types carried out in the\nthree rotation planes (ip, oopj, oopt) are shown in Fig. 6.\nThe ADMR of the standard bilayer is consistent with the\nSMR model as discussed in detail in Refs. 32 and 33.\nWithin this model, the modulation of \u001alongas a func-\ntion of the magnetization direction m=M=Msof the\nYIG layer re\rects the interplay between charge and spin\ncurrents at the interface between Pt and YIG via the\n(inverse) SHE. A charge current density Jcin the con-\nducting Pt layer induces a spin current density Jsper-\npendicular to the spin polarization \u001bandJcvia the SHE.\nThis results in a local spin accumulation at the Pt/YIG\ninterface if \u001bis collinear to m, which induces a di\u000bu-\nsive spin current back\row Jback\nscompensating Js. If\u001b\nis non-collinear to m, a spin transfer torque is exerted\non the magnetic moments, reducing the spin accumula-\ntion. This results in an additional dissipation channel for9\nYAG (111)PtYIG\nn\nt\njh\nβ\nYAG (111)PtYIGn\nt\nj\nγh\nYAG (111)PtYIG\nn\nt\njhα\nh || jh || th || - jh || - t h || nh || - th || - nh || t h || nh || jh || - nh || - j\n(c)\n0° 90° 180°270°360°-1012310-3(ρlong/ρ0)-1\nα0° 90° 180°270°360°\nβ0° 90° 180°270°360°\nγ(d) (e) (f)\n250 K\n110 K\n10 K110 K\n10 K250 K10 K\n110 K\n250 KPtYIG\nYAG 11 nm17 nm0° 90° 180°270°360°0.00.10.20.30.40.50.60.70.810-3(ρlong/ρ0)-1\nα0° 90° 180°270°360°\nβ0° 90° 180°270°360°\nγ(a) (b)250 K\n110 K\n10 K\n10 K250 K\n110 K\n10 K110 K250 K\nYIGPt\nYAG 52 nm2 nm\nFIG. 6. Normalized longitudinal resistivity \u001alongof (a),(b),(c) a standard Pt/YIG//YAG bilayer sample (black symbols)\nand (d),(e),(f) an inverted YIG/Pt//YAG bilayer sample (blue symbols) recorded at 250 K, 110 K, and 10 K in an external\nmagnetic \feld of \u00160H= 7 T. The magnetic \feld His rotated (a),(d) in-plane (ip-rotation), (b),(e) out-of-plane perpendicular\nto the current direction j(oopj-rotation), and (c),(f) out-of-plane perpendicular to the transverse direction t(oopt-rotation),\nas illustrated above. The lines are \fts to the data using cos2-functions (cf. Eqs. (1) and (2)). All curves are normalized to\n\u001a0=\u001along ;Hkt.\ncharge transport in the Pt layer leading to an increase of\nthe Pt resistivity.31,32The modulation of the component\nof the Pt resistivity tensor \u001aalong the current direction j,\ncoinciding with the longitudinal resistivity \u001along, is given\nby33\n\u001aSMR\nlong=\u001aSMR\n0+\u001aSMR\n1\u0002\n1\u0000m2\nt\u0003\n; (1)\nwhere\u001aSMR\n0 is approximately equal to the normal resis-tivity of the Pt layer.33\u001aSMR\n1 represents the SMR coe\u000e-\ncient with\u001aSMR\n1\u001c\u001aSMR\n0, andmtdenotes the projection\nofmont. Here we assume that the SMR amplitudes\n\u001aSMR\n1;1and\u001aSMR\n1;2of the two strongly antiferromagnetically\ncoupled Fe sublattices in the collinear ferrimagnetic state\nof YIG are equal: \u001aSMR\n1=\u001aSMR\n1;1=\u001aSMR\n1;2.39,44,46\nEquation 1 results in a cos2\u000band cos2\fdependence\nof\u001along, when rotating the magnetic \feld Hin the ip-10\nand oopj-plane, while no ADMR of \u001alongis expected for\nmagnetic \feld rotations in the oopt-plane.32This is fun-\ndamentally di\u000berent to the anisotropic magnetoresistance\n(AMR) of a polycrystalline ferromagnetic metal layer. In\nthis case\u001aAMR\nlong is described by the well-known expression\n\u001aAMR\nlong =\u001aAMR\n0 +\u001aAMR\n1m2\nj; (2)\nwhere\u001aAMR\n0 is given by the resistivity perpendicular to\nm(\u001a?) and\u001aAMR\n1 by the di\u000berence of the resistivities\nparallel (\u001ak) and perpendicular ( \u001a?) tomas\u001aAMR\n1 =\n\u001ak\u0000\u001a?.94We therefore expect an angle dependence of\n\u001aAMR\nlong, when rotating the magnetic \feld in the ip- and\noopt-plane.\nThe above equations are only valid for polycrystalline\nheavy-metal layers as it is the case for the standard\nPt/YIG//YAG bilayer samples. For crystalline mate-\nrials, the symmetry of the crystal has to be taken into\naccount for the calculation of the resistivity tensor \u001a.95{97\nFor the inverted YIG/Pt//YAG bilayer sample types, we\nfound that the Pt layer is weakly textured along the [111]-\ndirection (see Fig. 1(b)). However, since the FWHM of\nthe rocking curve along the Pt (111) re\rection is more\nthan 4\u000e, we neglect any contribution from the crystal\nsymmetry to \u001aalso in the inverted Pt/YIG//YAG bi-\nlayer sample type in the following discussion.\nAs shown in Figs. 6(a)-(c), the standard\nPt/YIG//YAG bilayer sample displays a cos2\u000band\ncos2\fangle-dependence of \u001along at all temperatures\nwhen rotating the magnetic \feld in the ip- and oopj-\nplane (cf. Figs. 6(a),(b)). However, while no ADMR\nsignal above the noise level of 4 \u000210\u00006is visible for\n250 K and 110 K in the oopt rotation plane, a small\nbut \fnite ADMR is observed at 10 K. We \ft our data\naccording to Eq. (1) using cos2functions (cf. solid\nlines in Fig. 6(a)-(c)). The obtained SMR amplitudes\n\u0001\u001a=\u001a0from the ip-, oopj-, and oopt-measurements are\ndepicted in Fig. 7(a). We observe a decrease of the SMR\namplitudes with decreasing temperature T, which is in\nagreement with our previous report.37However, since\nthe SMR depends on the spin di\u000busion length, the spin\nHall angle as well as the spin mixing conductance,33\ndiverse reports of the T-dependence of the SMR can be\nfound in literature due to di\u000berent T-dependencies of\nthese physical quantities in di\u000berent Pt/YIG samples.\nIn particular, the T-dependence of the SMR di\u000bers sig-\nni\fcantly for in-situ37andex-situ36fabricated Pt/YIG\nbilayers as well as for samples with Ar+-ion cleaning98,99\nor chemical etching100of the YIG layer prior to the\nPt deposition. This demonstrates that both intrinsic\nand extrinsic (phonon and impurity) scattering play an\nimportant role for the SMR.101,102TheT-dependence\nof the SMR can thus be regarded as a hallmark of the\nquality of Pt/YIG samples.\nAs obvious from Fig. 6(c) and Fig. 7(a), we observe\na \fnite ADMR for temperatures T < 50 K, while ro-\ntating the magnetic \feld in the oopt-plane. This be-\nhavior can be attributed to a MR in our Pt thin \flms,\nsince we also found a \fnite ADMR of similar magnitude\n0 50 1001502002503000.00.51.01.52.02.53.03.5\nMRip\nMRoopt10-3∆ρ/ρ0\nT(K)(b) YIG/Pt//YAG0 50 1001502002503000.00.10.20.30.40.50.60.70.8\n10 20 3005MRip\nMRoopj\nMRoopt10-3∆ρ/ρ0\nT(K) 10-5∆ρ/ρ0\nT(K)(a) Pt/YIG//YAG\nPt//YAGFIG. 7. MR amplitudes obtained from ADMR measurements\nof (a) a standard Pt/YIG//YAG bilayer sample (black sym-\nbols) and (b) an inverted YIG/Pt//YAG bilayer sample (blue\nsymbols) carried out at \u00160H= 7 T in di\u000berent rotation planes\n(ip, oopj, oopt) of the magnetic \feld. The inset shows the MR\namplitudes of a Pt//YAG reference sample.\nat low temperatures in Pt//YAG reference samples (see\ninset of Fig. 7(a)). This MR results from an increase\nof the resistance with magnetic \felds along the normal\nof the thin \flm ( n-direction), and is most likely caused\nby weak antilocalization e\u000bects.98,103,104Since no MR is\nobserved in ip-ADMR measurements of the Pt//YAG\nreference sample (square open symbols in the inset of\nFig. 7(a)), the \fnite MR in oopt-ADMR measurements\nof the Pt/YIG//YAG sample cannot be related to a con-\nventional AMR or MPE.105It causes a di\u000berence of the\nMR amplitude recorded in ip- and oopj-ADMR measure-\nments with MR ip150 K, this contribution dominates and\na single cos2\f-dependence is visible. Figure 8 clearly\ndemonstrates that the angle-dependence of \u001alongcan not\nbe simply described by Eqs. (1) and (2). Instead, a morecomplex resistance network has to be taken into account\nwhich requires a more detailed knowledge of the local\nmicrostructure of the Pt thin \flm.\nV. CONCLUSION\nIn summary, we showed that in heavy metal\n(HM)/ferromagnetic insulator (FMI) bilayer thin \flm\nsamples consisting of the HM Pt and the FMI YIG the\nappearance of magnetic proximity e\u000bects in the HM Pt\ncrucially depends on the quality of the Pt/YIG inter-\nface. On standard Pt/YIG bilayer samples with a clean\nand sharp interface, we do not observe any indication of\nan induced magnetic moment in the Pt layer in x-ray\nmagnetic circular dichroism (XMCD)14as well as x-ray\nresonant magnetic re\rectivity experiments (XRMR). In\nthese samples, the observed magnetoresistance can be\nexplained solely within the spin Hall magnetoresistance\ntheory. In contrast, in inverted YIG/Pt bilayer samples,\na \fnite induced magnetic moment of up to 0 :058\u0016B/Pt\ncan be found by XMCD and XRMR at room tempera-\nture, which increases at lower temperatures. This \fnite\nmoment, which often is attributed to a magnetic prox-\nimity e\u000bect at a perfect Pt/YIG interface, is shown to\noriginate from a \fnite interdi\u000busion at this interface due\nto the deposition of YIG on Pt, associated with both\nan elevated temperature and high kinetic energy of the\natoms/ions impinging on the surface during the PLD pro-\ncess. In those samples, the spin Hall magnetoresistance\nis superimposed by an induced magnetic moment based\nanisotropic magnetoresistance, which becomes dominant\nat low temperatures. This demonstrates that a com-\nbined temperature-dependent x-ray and magnetotrans-\nport study is essential to con\frm or exclude any magnetic\nproximity e\u000bects in HM/FMI heterostructures.\nACKNOWLEDGMENTS\nThis work was supported by the European Synchrotron\nRadiation Facility (ESRF) via HE-3784, HC-1500, HC-\n1783, HC-2058, and HC-3268 as well as the Deutsche\nForschungsgemeinschaft (DFG) via SPP 1538 (Projects\nNo. GO 944/4-1 and No. KU 3271/1-1). We acknowledge\nDiamond Light Source (Didcot, UK) for time on Beam-\nline I16 under Proposal MT12772-1 and DESY (Ham-\nburg, Germany), a member of the Helmholtz Association\nHGF, for the provision of experimental facilities. Parts\nof this research were carried out at beamline P09 at PE-\nTRA III. 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Materials 2, 011401 (2018)." }, { "title": "1308.1441v1.Loop_Liquid_in_an_Ising_Spin_Kondo_Lattice_Model_on_a_Kagome_Lattice.pdf", "content": "arXiv:1308.1441v1 [cond-mat.str-el] 6 Aug 2013APS/123-QED\nLoop Liquid in an Ising-Spin Kondo Lattice Model on a Kagome L attice\nHiroaki Ishizuka1and Yukitoshi Motome1\n1Department of Applied Physics, University of Tokyo, Hongo, 7-3-1, Bunkyo, Tokyo 113-8656, Japan\n(Dated: July 20, 2018)\nPhase diagram of an Ising-spin Kondolattice model on a kagom e lattice is investigated bya Monte\nCarlo simulation. We find that the system exhibits a peculiar ferrimagnetic state at a finite tem-\nperature, in which each triangle is in a two-up one-down spin configuration but the spin correlation\ndoes not develop any superstructure. We call this state the l oop liquid, as it is characterized by the\nemergent degree of freedom, self-avoiding up-spin loops. W e elucidate that the system shows phase\ntransitions from the loop liquid to ferrimagnetically orde red states and a crossover to a partially\nferromagnetic state by changing the electron density and te mperature. These can be viewed as\ncrystallization and cohesion of the loops, respectively. W e demonstrate that the loop formation is\nobserved in the optical conductivity as a characteristic re sonant peak.\nPACS numbers: 75.10.Kt, 71.10.Fd, 05.10.Ln\nThe interplay between charge and spin degrees of free-\ndom in electrons has long been studied as one of the cen-\ntral problems in condensed matter physics. In particular,\nitinerant electron systems coupled to localized spins are\nfundamental models in the study of such interplay in cor-\nrelated electrons. In these systems, effective spin-spin in-\nteractions mediated by itinerant electrons play a crucial\nrole in the magnetism. For instance, spin-glass behav-\nior in metallic alloys with doped magnetic ions is driven\nby the so-called Ruderman-Kittel-Kasuya-Yosida inter-\nactions1–3. Another example is metallic ferromagnetic\n(FM) behavior in perovskite manganese oxides, which is\nunderstood by the double-exchange (DE) interaction4,5.\nOn the other hand, the interplay between charge and\nspin also triggers significant changes of the electronic\nstructure and transport properties. In the perovskite\nmagnanese oxides, the competition between the FM DE\nand antiferromagnetic (AFM) super-exchange interac-\ntions plays an important role in the colossal magneto-\nresistance6.\nThe interest in the spin-charge coupling has recently\nbeen extended to frustrated systems, in which geometri-\ncal frustration provides an additional degree of freedom\nfor controlling the system. In these systems, a simple\nAFM ordering of localized spins is suppressed by geo-\nmetrical frustration, and instead, a disordered spin state\nwith strong local correlations is expected to be realized\nat a sufficiently low temperature ( T). Such a “liquid-\nlike”spinstateisanticipatedtohavecharacteristiceffects\non the coupled itinerant electrons. Indeed, the impor-\ntance ofcharacteristicnoncoplanarspin textures was dis-\ncussed for an unconventional anomalous Hall effect and\npeculiar metallic behavior observed in Nd 2Mo2O77and\nPr2Ir2O78,9.\nStimulated by such interesting experiments, many the-\noretical studies have been done recently. For instance,\ncharacteristic metal-insulator transitions were found in\nextendedFalicov-Kimballmodelswithlocalconstrainton\nthe spatial configuration of localized particles10. A simi-\nlar problem was also studied for the model with localized\nspins11. Meanwhile, a noncoplanar spin correlation inan Ising-spin Kondo lattice model on a pyrochlore lat-\ntice was studied for explaining the resistivity minimum\nin Pr2Ir2O712,13. A similar noncoplanar correlation on\na kagome lattice was shown to induce a charge gap and\nquantum anomalous Hall effect14,15. These results indi-\ncate that liquid-like states can be realized in frustrated\nspin-charge coupled systems and their peculiar spin cor-\nrelations significantly affect the electronic and transport\nproperties.\nIn this study, to explore such liquid states and ex-\notic electronic properties, we investigate a Kondo lattice\nmodelwithIsinglocalizedmomentsonatwo-dimensional\nkagome lattice. By using a Monte Carlo (MC) method,\nwe show that the model exhibits a locally-correlatedspin\nstate with a fractional magnetic moment. We call this\nferrimagnetic (FR) liquid-like state the loop liquid, as it\ncan be viewed as a soup of self-avoiding up-spin loops\nmixed with isolated down spins [see Fig. 1(b)]. Although\nthe electronicstructure waspreviouslystudied by assum-\ning similar loop liquids11, our results provide a convinc-\ning evidence of the thermodynamic stability of the loop\nliquid for the first time to our knowledge. The obtained\nphase diagram includes FM, partially FM, q= 0 and√\n3×√\n3 FR states in addition to the loop liquid; the\nphasetransitionsandcrossoverareinterpretedascrystal-\nlization and cohesion in terms of the loops, respectively.\nWe also demonstrate that the optical conductivity devel-\nops characteristic peaks corresponding to the formation\nof the self-avoiding loops.\nWe consider a single-band Kondo lattice model on a\nkagome lattice with localized Ising spin moments. The\nHamiltonian is given by\nH=−t/summationdisplay\n/angbracketlefti,j/angbracketright,σ(c†\niσcjσ+H.c.)+ J/summationdisplay\niσz\niSi.(1)\nThe first term represents hopping of itinerant electrons,\nwhereciσ(c†\niσ) is the annihilation (creation) operator of\nan itinerant electron with spin σ=↑,↓at theith site,\nandtis the transfer integral. The sum /angbracketlefti,j/angbracketrightis taken over\nnearest-neighbor (NN) sites on the kagome lattice. The\nsecond term is the on-site interaction between localized2\n(b) (c)\n(d)\nFIG. 1. (color online). (a) Phase diagram of the model in\nEq. (1) at J= 6 obtained by the Monte Carlo simulation.\nThe symbols shows the critical temperatures Tcfor magnetic\nstates: ferromagnetic (FM), partially ferromagnetic (PFM ),\nloop liquid (LL), q= 0 ferrimagnetic ( q= 0), and√\n3×√\n3\nferrimagnetic (√\n3×√\n3) states. Tcfor the√\n3×√\n3 state at\nn= 8/9 is shown by the diamond, while the upper limit for Tc\nforq= 0 state n= 0.83 is shown by the downward triangle,\nwhich is given by the temperature we reached with Ns= 82\ncalculations. The squares (circles) show Tcdetermined from\nthe Binder analysis of m(P), and the upward triangles show\nTcdetermined by the system-size extrapolation of the peak of\nχm. The curve connecting the symbols is a guide for the eyes.\nThe strip at the bottom is the ground state phase diagram\nobtained by the variational calculation for three magnetic or-\nders, FM, q= 0, and√\n3×√\n3. PS is the phase separation\nbetween the neighboring two phases. The schematic pictures\nof the magnetic states are given for (b) LL, (c) q= 0, and (d)√\n3×√\n3. The bold lines denote the loops connecting up-spin\nsites and the dots show down-spin sites.\nspins and itinerant electrons, where σz\ni=c†\ni↑ci↑−c†\ni↓ci↓\nrepresents the zcomponent of the itinerant electron spin\nandSi=±1 denotes the localized Ising spin at the ith\nsite;Jis the coupling constant (the sign of Jdoes not\nmatter in the present model). Hereafter, we take t= 1\nas the unit of energy, the lattice constant a= 1, the\nBoltzmann constant kB= 1, and e2/has the unit of\nconductance.\nThermodynamic properties of the model in Eq. (1) are\nstudied by a MC simulation which is widely used for sim-\nilar models16. The calculations were conducted up to the\nsystem size N= 3×NswithNs= 92under the periodic\nboundary conditions ( Nsis the number of three-site unit\ncells). To deal with the freezing of MC sampling, some0.00.20.40.60.81.0\n0.560.640.720.800.88(b)m\nχm [/40]P\nNS= 72\nNS= 82\nNS= 92NS= 62\nn\n(e) S(k)/NS\nkxky\n00\n2π2π0.00.10.20.30.40.5T(a) n=0.52\nn=0.64\nn=0.76\nn= 8/9n=0.83n=0.86m\n(c) S(k)/NS\nkxky\n00\n2π2π0.00.20.40.6(d) S(k)/NS\nkxky\n00\n2π2π0.000.050.100.150.00.20.40.60.81.0\n0.1 0.03\nFIG. 2. (color online). (a) MC results for Tdependences\nofmat different n. The data at n= 8/9 are calculated for\nNs= 92, while the others for Ns= 82. (b)ndependences\nofm,χm, andPatT= 0.03 forNs= 62, 72, 82, and 92.\nThe MC results of S(k)/Nsare shown for (c) n= 0.65, (d)\nn= 0.84, and (e) n= 8/9 atT= 0.03 andNs= 92.\nof the low- Tdata were calculated starting from a mixed\ninitialspinconfigurationoflow- Torderedandhigh- Tdis-\nordered states17. The thermal averages were calculated\nfor typically 15000-80000MC steps after 5000-18000MC\nsteps for thermalization. In addition, the ground state\nphase diagram is also obtained by comparing the energy\nof the dominant phases found in the MC simulation18.\nFigure 1(a) shows the phase diagram obtained by the\nMC simulation at J= 6 while varying electron density\nn=/summationtext\niσ/angbracketleftc†\niσciσ/angbracketright/N. As lowering T, the system exhibits\na phase transition with developing a net magnetization\nm=/radicalbig\n/angbracketleft(/summationtext\niSi/N)2/angbracketright.Tdependences of mare shown in\nFig. 2(a). In the low density region for n/lessorsimilar0.56,map-\nproaches its saturated value 1 in the low- Tlimit, namely,\nthe system exhibits a fully-polarized FM order. This\nphase is connected to the FM phase in the the large J\nregion, which is induced by the DE mechanism4,5. While\nincreasing n, the low- Tvalue ofmdecreases from 1 and\ncontinuously becomes smaller as nbecomes larger, as\nshown in Figs. 2(a) and 2(b). At the same time, the\nprobability to find a two-up one-down spin configuration\nin each triangle, P=/radicalbig\n/angbracketleft(/summationtext\nν3pν/2N)2/angbracketright, increases con-\ntinuously from zero [Fig. 2(b)]; here, pν= 1(−1) for two-\nup one-down (one-up two-down) and otherwise pν= 0,\nand the sum is over all triangles. The spin structure\nfactorS(k) for the same sublattice is featureless except\nfor the peak at k=0, as shown in Fig. 2(c); here,\nS(k) =1\nNs/summationtext\ni,j∈α/angbracketleftSiSj/angbracketrightexp(ik·rij), where rijis the\nvector from ith tojth site, and the sum is taken for the\nsitesi,jin the same sublattice α. We call this region\nwith the reduced mthe partially ferromagnetic (PFM)\nphase19.\nIn the region of 0 .8/lessorsimilarn <8/9, however, the low- T\nvalue of mbecomes almost independent of n, and satu-\nratestoafractionalvalue m= 1/3, asshownin Fig.2(a).3\nIn this region, most of the triangles on the kagome lat-\ntice are in two-up one-down spin configurations, namely,\nP≃1 [Fig. 2(b)]. As shown in Fig. 2(d), S(k) does not\nshowanysharppeak except forthe one at k=0, indicat-\ning that this state has no superstructure. Hence, this FR\nstate is a peculiar Coulombic state subject to the two-up\none-downlocalconstraint, in a similarsensetothe two-in\ntwo-out state in spin ice20,21. The spin state is composed\nof the emergent degrees of freedom, self-avoiding up-spin\nloops and isolated down-spins, as schematically shown in\nFig. 1(b). Hence, we call this Coulombic state the loop\nliquid (LL).\nAn interesting observation here is that the change be-\ntween the FM, PFM, and LL states is smooth and there\nis no sign of phase transition. Both mandPchanges\ncontinuously without showing any singularity, and the\nmagnetic susceptibility χmshows only a broad hump,\nas shown in Fig. 2(b). This indicates that the change\nfrom FM to LL is a crossover and not a phase transition.\nSuch behavior is understood from the symmetry point of\nview. In the LL state, though mis nonzero, the system\nstill remains disordered and preserves all the symmetries\nof the lattice; the situation is unchanged from the FM\nand PFM states. As a consequence, these phases are\nsmoothly connected by the crossover.\nOn the other hand, as decreasing Tor as further in-\ncreasing n, the LL state exhibits phase transitions with\nshowing a magnetic long-range order (LRO). In our MC\nsimulation, we identify two different transitions; one is\nthe transition to the state with q= 0 LRO of the two-up\none-down spin configurations [Fig. 1(c)], and the other\nto the state with√\n3×√\n3 LRO [Fig. 1(d)]. The for-\nmer is observed while decreasing Tatn∼0.83, and the\nlatter is found by increasing nto a commensurate filling\nn= 8/9.S(k)forthelatterstateisshowninFig.2(e). In\nthe corresponding density regions, the two phases are ob-\ntained in the variational calculation for the ground state,\nas shown in Fig. 1(a). These two LRO states are viewed\nascrystalphasesoftheemergentloopsinthetwoextreme\ncases; the former is a periodic array of one-dimensional\nchains, while the latterthe shortestsix-sitehexagons. In-\nterestingly, the peculiar LL state extends in the density\nregion between these two crystal phases.\nLet us closely lookat the formationofLL and the crys-\ntallization of loops. Figure 3 shows the MC results of T\ndependences of magnetic properties at n= 0.83. The re-\nsult in Fig. 3(a) shows the increase of mwith saturation\nto 1/3 and a divergent peak of χmatT∼0.05. At the\nsame time, as shown in Fig. 3(b), Pshows saturation\nto 1 and its susceptibility χPshows a peak, indicating\nthat most of the triangles become two-up one-down be-\nlowT∼0.05. The Binder parameters22formandP,\ngmandgP, respectively, are shown in Fig. 3(d). Both\nshow a crossing of the results for different sizes, indicat-\ning the transition is of second order. The critical tem-\nperatures determined from the two independent Binder\nanalyses show good accordance; Tc= 0.051(4). On the\nother hand, a rapid increase of S(k=0)/Nsto 1 is ob-0.00.20.40.60.81.0\n0.02 0.04 0.06 0.08\n0.700.750.800.850.900.951.00\n0.0400.0440.0480.0520.0560.750.800.850.900.951.001.05T(b)P\nχP[/300]\nT(d) gm\ngP\nT(c)\nS(k=0)/NST(a)\nmχm[/50]\n0.00.20.40.60.8\n0.02 0.04 0.06 0.08\n0.00.20.40.60.81.0\n0.02 0.04 0.06 0.08NS= 62\nNS= 72\nNS= 82NS= 52NS= 42\nNS= 62\nNS= 72\nNS= 82NS= 52NS= 42\nNS= 62\nNS= 72\nNS= 82NS= 52NS= 42NS= 62\nNS= 72\nNS= 82NS= 52NS= 42\nFIG. 3. (color online). MC results for (a) mandχm, (b)P\nandχP, (c)S(k=0)/Ns, and (d) gmandgPforNs= 42, 52,\n62, 72, and 82and atn= 0.83.\nservedin Ns= 42and 62, as shown in Fig. 3(c); the onset\nTdecreasesforlarger Nsalthoughtheresultsshowstrong\nfinite size effects with different behavior for even and odd\nNs. This suggests a phase transition to the q= 0 ordered\nstate at a lower Tthan 0.028; this is consistent with the\nground state obtained by the variational calculation, as\nshown in Fig. 1(a). Although the precise estimate of the\ncritical temperature is difficult within the present calcu-\nlation, these results indicate that successive phase transi-\ntions from PM to LL and LL to q= 0 FR state take place\natn= 0.83. The former corresponds to the formation of\nloops, and the latter their crystallization.\nNow we discuss the electronic and transport properties\nof the itinerant electrons. Figure 4 shows the result of\noptical conductivity σ(ω). First, to extract the effect of\ncharacteristic spin correlations in the LL state, we cal-\nculateσ(ω) by taking simple average over different spin\npatternsin the ideal LL manifold (all the trianglessatisfy\nthe two-up one-down local constraint). The calculations\nweredone byusingthe Kubo formulafor 24different spin\npatterns. Figure 4(a) is the result of σ(ω) calculated at\nn= 0.843 for various J. All the results show a sharp\npeak at ω=ωp∼1.0-1.2, which shifts to lower ωfor\nlargerJ.\nThe characteristic peak comes from the transition pro-\ncess between two localized states in the six-site loops. In\nthe limit of J→ ∞, electrons are confined in the loops or\nat isolated sites11; the contribution to σ(ω) comes only\nfrom the transition process between the electronic states\nin the same loop. Hence, sharp peaks appear in σ(ω)\ncorresponding to the discrete energy levels in the finite\nlength loops. In the current kagome case, the most dom-\ninant loops are the shortest ones with the length of six\nsites. In the six-site loops, the energy difference between\nthe unoccupied and occupied levels at this filling (the\nhighest and second highest levels) is 1. Hence, we expect\na sharp peak at ωp= 1 in the limit of J→ ∞. For large4σ(ω)(a)\nJ= 161210 8\n 6\n 4\n 2\nω\n0.00.10.20.30.40.50.60.7\n0.00.51.01.52.02.53.03.54.0σ(ω)\n0.00.20.40.6\n-6-5-4-3-2-1012DOS\nε(b)\nn=0.862\n0.840\n0.812\n0.792\n0.732\n0.618\n0.572\nω0.00.10.20.30.40.50.60.7\n0.00.51.01.52.02.53.03.54.01 / Jω\n0.91.01.11.21.3\n0.00.10.20.30.40.50.6\nFIG. 4. (color online). Optical conductivity σ(ω) calculated\n(a) by simple average over LL configurations while varying J\natn= 0.843 for a 22supercell of N= 3×122sites, and (b)\nby MC simulation while varying natJ= 6 for a 42supercell\nofN= 3×62sites at T= 0.04. The scattering rate in\nthe Kubo formula is taken as τ−1= 0.01. The typical error\nbars are shown at ω= 0.5. The inset in (a) shows the peak\nposition of σ(ω) atω∼1. The dotted line shows the fitting\nbyω= 0.995+0.558/J−0.155/J2. The inset in (b) is DOS\natn= 0.862. The Fermi level is set at ε= 0.\nbut finite J, the second order perturbation in terms of\nthe hopping between up and down spin sites shifts the\nsecond highest eigenenergy to a lower energy. On the\nother hand, this perturbation process does not affect the\nhighest eigenenergy. Hence, it is expected that the peak\nshifts to a higher ωas decreasing J; the asymptotic be-\nhaviorat J→ ∞isexpected tobe ωp= 1+O(1/J). This\nis confirmed by the fitting shown in the inset of Fig. 4(a).\nInterestingly, the peak persists in the weak Jregion\nwhere the exchange splitting 2 Jis comparable or smallerthan the bare bandwidth 6 tand the above perturbative\nargument appears to be no longer valid. In a recent\nstudy on a metal-insulator transition caused by corre-\nlated potentials, a LL-type local correlation induces a\nmetal-insulator transition at a considerably smaller po-\ntential than the bandwidth by confining the electrons in\ntheloops10. Thepersistingresonantpeakin σ(ω)islikely\nto be the consequence of this confinement.\nEmergence of the characteristic peak is also observed\nin the thermodynamic average obtained by the MC sim-\nulation. Figure 4(b) shows the MC result of σ(ω) while\nvaryingnatT= 0.04andJ= 6. With increasing nfrom\ntheFMregion,thepeakat ω∼1developsintheLLstate\nforn/greaterorsimilar0.8. The inset in Fig. 4(b) shows the density of\nstates (DOS) for itinerant electrons (lower half of two\nsplit bands) at n= 0.862. The result clearly shows the\npresence of two sharp peaks below and above the Fermi\nlevel set at ε= 0; the energy difference is about 1 .1,\nwhich well corresponds to the peak in σ(ω) in the main\npanel of Fig. 4(b).\nTo summarize, we studied an Ising-spin Kondo lattice\nmodel on a kagome lattice with focusing on the emer-\ngent magnetic states and their electronic properties. By\nusing an unbiased Monte Carlo simulation, we presented\nthat the loop-liquid state emerges in the finite tempera-\nture region, in addition to ferromagnetic, q= 0 ferrimag-\nnetic, and√\n3×√\n3 ferrimagnetic states. The loop liquid\nis a Coulombic ferrimagnetic state, characterized by the\nemergent up-spin loops originating from the two-up one-\ndown local spin configurations. The phase diagram is\nunderstood in terms of the emergent loops as crystalliza-\ntion and cohesion of the dense liquid of the loops. We\nalso showed that the loop-liquid formation is observed in\ncharacteristicpeaksintheopticalconductivity. Recently,\nthe spin-charge coupling in frustrated magnets has been\nrevealed to exhibit rich physics, both in magnetic and\ntransport properties. We hope that our finding of yet an-\nother emergent state would further stimulate the study\nof these systems.\nThe authors thank N. Furukawa, L. D. C. Jaubert,\nand K. Penc for fruitful discussions. H.I. is supported\nby Grant-in-Aid for JSPS Fellows. This research was\nsupported by KAKENHI (No. 22540372 and 24340076),\ntheStrategicProgramsforInnovativeResearch(SPIRE),\nMEXT, and the Computational Materials Science Initia-\ntive (CMSI), Japan.\n1M. A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954).\n2T. Kasuya, Prog. Theor. Phys. 16, 45 (1956).\n3K. Yosida, Phys. Rev. 106, 893 (1957).\n4C. Zener, Phys. Rev. 82, 403 (1951).\n5P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675\n(1955).\n6Colossal Magnetoresistive Oxides , ed. by Y. Tokura (CRC\nPress, 2000).\n7Y. Taguchi, Y. Ohhara, H. Yoshizawa, N. Nagaosa, and Y.Tokura, Science 291, 2573 (2001).\n8S.Nakatsuji, Y.Machida, Y.Maeno, T. Tayama, T. Sakak-\nibara, J. van Duijn, L. Balicas, J. N. Millican, R. T.\nMacaluso, and J. Y. Chan, Phys. Rev. Lett. 96087204\n(2006).\n9Y.Machida, S.Nakatsuji, Y.Maeno, T. Tayama, T. Sakak-\nibara, and S. Onoda, Phys. Rev. Lett. 98057203 (2007).\n10H. Ishizuka, M. Udagawa, and Y. Motome, Phys. Rev. B\n83, 125101 (2011).5\n11L. D. C. Jaubert, S. Piatecki, M. Haque, and R. Moessner,\nPhys. Rev. B 85, 054425 (2012).\n12M. Udagawa, H. Ishizuka, and Y. Motome, Phys. Rev.\nLett.108, 066406 (2012).\n13G.-W. Chern, S. Maiti, R. M. Fernandes, and P. Wolfle,\npreprint (arXiv:1210.3289).\n14H. Ishizuka and Y. Motome, Phys. Rev. B 87, 081105(R)\n(2013).\n15G.-W. Chern, A. Rahmani, I. Martin, and C. D. Batista,\npreprint (arXiv:1212.3617).\n16S. Yunoki, J. Hu, A. L. Malvezzi, A. Moreo, N. Furukawa,\nand E. Dagotto, Phys. Rev. Lett. 80, 845 (1998).17Y. Ozeki, K. Kasono, N. Ito, and S. Miyashita, Physica A\n321, 271 (2003).\n18For details, e.g., see Appendix of H. Ishizuka and Y. Mo-\ntome, Phys. Rev. B 87, 155156 (2013).\n19At the lowest T, the PFM phase is presumably taken over\nby a phase separation between FM and q= 0 FR states\n[see Fig. 1(a)].\n20M. J. Harris, S. T. Bramwell, D. F. McMorrow, T. Zeiske,\nand K. W. Godfrey, Phys. Rev. Lett. 79, 2554 (1997).\n21A. P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan,\nand B. S. Shastry, Nature (London) 399, 333 (1999).\n22K. Binder, Z. Phys. B 43, 119 (1981)." }, { "title": "1808.08466v1.Twisted_magnetization_states_and_inhomogeneous_resonance_modes_in_a_Fe_Gd_ferrimagnetic_multilayer.pdf", "content": "arXiv:1808.08466v1 [cond-mat.mtrl-sci] 25 Aug 2018Twisted magnetization states and inhomogeneous resonance modes\nin a Fe/Gd ferrimagnetic multilayer\nA.B. Drovosekova,∗, A.O. Savitskya,b, D.I. Kholina, N.M. Kreinesa, V .V . Proglyadoc, M.V . Ryabukhinac, E.A. Kravtsovc,d,\nV .V . Ustinovc,d\naP .L. Kapitza Institute for Physical Problems RAS, 119334 Mo scow, Russia\nbInstitute of Solid State Physics RAS, 142432 Chernogolovka , Moscow region, Russia\ncM.N. Mikheev Institute of Metal Physics UB RAS, 620137 Ekate rinburg, Russia\ndUral Federal University, 620002 Ekaterinburg, Russia\nAbstract\nStatic and dynamic magnetic properties of a ferrimagnetic [ Fe(35Å)/Gd(50Å)] 12superlattice were investigated in a wide 4 −300 K\ntemperature range using magneto-optical Kerr e ffect (MOKE) and ferromagnetic resonance (FMR) techniques. T he multilayer\nstructure was sputtered on a transparent glass substrate wh ich made it possible to perform MOKE measurements on both Fe a nd Gd\nterminated sides of the superlattice. These experiments al lowed us to detect a transition between field-aligned and can ted magnetic\nstates on both sides of the film and to distinguish between the bulk and surface twisted phases of the superlattice. As a res ult, the\nexperimental H−Tmagnetic phase diagram of the system was obtained. FMR studi es at frequencies 7 −36 GHz demonstrated\na complex evolution of absorption spectra as temperature de creased from room down to 4 K. Two spectral branches were dete cted\nin the sample. Theoretical simulations show that the observ ed spectral branches correspond to di fferent types of inhomogeneous\nresonance modes in the multilayer with non-uniform magneti zation precession inside Gd layers.\nKeywords: Fe/Gd multilayer, ferrimagnetics, magnetic properties, ferr omagnetic resonance\nPACS: 68.65.Ac, 75.70.Cn, 75.50.Gg, 76.50. +g\n1. Introduction\nLayered structures based on transition (TM) and rare-earth\n(RE) ferromagnetic (FM) metals, like Fe /Gd, are model ferri-\nmagnetic systems demonstrating a rich magnetic phase diagr am\nwith complex types of magnetic ordering [1, 2]. Due to an an-\ntiferromagnetic (AFM) coupling at Fe-Gd interfaces and ess en-\ntially different Curie temperatures of Fe and Gd (for bulk ma-\nterials, TFe\nC=1043 K and TGd\nC=293 K) a so-called ”compen-\nsation point” Tcompcan exist in the system. At T=Tcompmag-\nnetic moments of Fe and Gd layers are equal to each other and\nthe total magnetization of the system vanishes. Below Tcomp,\nthe magnetic moment in Gd subsystem exceeds that in Fe sub-\nsystem, while above Tcomp, opposite situation takes place. As a\nresult, in weak fields applied in the film plane, a collinear ma g-\nnetic phase is realized with Fe magnetization vector orient ed\nparallel (at T>Tcomp) or antiparallel to the field direction (at\nTHb) the bulk\ntwisted state is realized. Fig. 1 represents schematically the cor-\nresponding magnetization distributions calculated for di fferent\nfield values at T>Tcomp [11]. It is important to note that the\nsurface twist phase arises at the outermost layer of the supe rlat-\ntice when its magnetization is directed opposite to the appl ied\nfield. Thus, the surface twist phase arises on Gd-terminated\nside of the superlattice at T>Tcompand on Fe-terminated side\nof the superlattice at THb\nFigure 1: Different types of magnetization vector distribution in a Fe /Gd superlattice at T>Tcomp (calculated using the mean-field model [11]).\nmakes it difficult to perform detailed studies of stability regions\nof bulk and surface twisted phases as a function of temperatu re\nand magnetic field.\nMagneto-optical Kerr e ffect (MOKE) is a relatively simple\nand sensitive method to obtain direct information about the sur-\nface magnetic state of the multilayer. The penetration dept h of\nvisible light into metal is about ∼100 Å which is comparable\nwith typical thickness of individual layers in the superlat tice.\nThus, MOKE signal provides information about magnetizatio n\nin several upper layers of the superlattice. Hahn et al. [13] used\nMOKE to study surface magnetic states in a [Fe /Gd] 15struc-\nture. Since the samples were sputtered on non-transparent S i\nsubstrates, authors compared MOKE signals from superlatti ces\nterminated by Fe and Gd layers. The di fference of the MOKE\ncurves for two samples was explained by the surface magnetic\ntwist arising in case of the surface layer magnetization ori ented\nopposite to the field direction.\nIn our previous work [11] we studied static magnetization\ncurves of a [Fe/Gd] 12multilayer. Comparing the experimental\ndata with mean-field calculations, we found indications of fi eld-\ndependent phase transitions between field-aligned, surfac e- and\nbulk twisted states. However, the static magnetometry prov ides\nonly the net magnetic moment of the entire multilayer and the\nsurface effects are manifested too weakly. In this work we use\nMOKE to obtain more precise knowledge on the surface mag-\nnetic states in the superlattice. The investigated [Fe /Gd] 12mul-\ntilayer is grown on a transparent glass substrate which allo ws\ndirect probing magnetic states on both sides of the structur e. As\na result, we determine the stability regions of bulk and surf ace\ntwisted states in the superlattice, depending on temperatu re and\nmagnetic field. The experimental phase diagram is compared\nwith calculations based on the mean-field model [11].\nStudies of magnetization dynamics in RE /TM systems at-\ntract attention due to a recent idea to use such materials for re-\nalization of ultrafast magnetic switching, promising for p oten-\ntial applications in magnetic storage devices [14–16]. A nu m-ber of works were devoted to investigations of ferromagneti c\nresonance (FMR) in TM /Gd multilayers [17–26]. Room tem-\nperature studies [17–20] demonstrated the importance of sp in\npumping into RE metal to explain a large FMR line width in\nTM/RE systems. Several groups reported about the e ffect of\nline broadening and shift of the absorption peak to lower fiel ds\nat cooling the system below room temperature. Such behaviou r\nwas observed for Co /Gd [21, 22], Py/Gd [23], Fe/Gd [11] and\nFe/Cr/Gd [24, 25] multilayers.\nIn most of the cited works only one ”high-temperature” res-\nonance peak was detected. This peak became much weaker or\neven disappeared as temperature decreased below TGd\nCwhich\neffect was explained by non-local damping mechanisms in the\nsystem [11, 23]. In a short letter [26], Svalov et al. reported\nabout experimental observation of a second absorption peak be-\nlowTGd\nCin a Co/Gd multilayer. Similar behaviour was observed\nin our previous works for the Fe /Gd system [11]. Theoretical\nsimulations showed that the observed absorption peaks corr e-\nsponded to different types of inhomogeneous resonance modes\nin the multilayer. In this work we perform more detailed in-\nvestigation of temperature evolution of the resonance spec tra\nin the Fe/Gd superlattice. In contrast to the work [11], here\nwe pay special attention to the transformation of the spectr a in\nthe vicinity of TGd\nC. In particular, we note that the behaviour\nof the high-temperature resonance peak is strongly depende nt\non the pumping frequency. To explain this result and identif y\nthe observed resonance modes, the experimental data are com -\npared with model calculations based on Landau-Lifshitz equ a-\ntions describing magnetization dynamics in the system.\n2. Sample and experimental details\nThe [Fe(35 Å)/Gd(50 Å)] 12superlattice was prepared on a\nglass substrate using high vacuum magnetron sputtering tec h-\nnique. Two chromium layers with thickness 50 Å and 30 Å\nserved as buffer and cap layers respectively. X-ray di ffraction\n2/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s45/s50/s45/s49/s48/s49/s50\n/s72\n/s98/s50/s49\n/s50 /s49/s102/s105/s108/s109/s32/s115/s117/s114/s102/s46 /s115/s117/s98/s115/s116/s114/s46/s77/s79/s75/s69/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s46/s117/s46/s41\n/s72 /s32/s40/s107/s79/s101/s41/s116/s119/s105/s115/s116/s101/s100/s32/s115/s116/s97/s116/s101\n/s115/s117/s114/s102/s97/s99/s101/s32/s116/s119/s105/s115/s116\n/s70/s101/s45/s97/s108/s105/s103/s110/s101/s100/s32/s115/s116/s97/s116/s101/s115/s97/s109/s112/s108/s101/s77/s79/s75/s69/s32/s103/s101/s111/s109/s101/s116/s114/s105/s101/s115/s58\n/s72\n/s115\nFigure 2: MOKE curves measured at 155 K from two sides of the fil m: 1) from\nthe glass substrate side (Fe-terminated side of the superla ttice) and 2) from the\nfilm surface (Gd-terminated side of the superlattice). Comp aring the curves,\ndifferent types of magnetic ordering can be identified.\nstudies performed in [11] demonstrated well-defined layere d\nstructure of the sample with interfacial root mean square ro ugh-\nness of about 1–2 atomic monolayers.\nMagnetic properties of the multilayer were studied using\nMOKE and FMR techniques in the 4 −300 K temperature range\nin magnetic fields up to 10 kOe applied in the film plane.\nLongitudinal MOKE studies of the surface magnetization\nwere performed on both sides of the film, using a 635 nm semi-\nconductor laser. In our experimental geometry the MOKE sig-\nnal was proportional to the component of magnetization para llel\nto the applied field.\nFMR measurements were carried out using a conventional\nfield-sweep technique on a laboratory developed transmissi on\ntype spectrometer at di fferent frequencies in the range 7 −36 GHz.\n3. Results and discussion\n3.1. Magneto-optical Kerr e ffect\nStatic magnetometry of the investigated sample performed\nin [11] showed that Gd layers had reduced Curie temperature,\nTGd\nC≈200 K, comparing with the bulk value 293 K. The system\ndemonstrated the compensation point at Tcomp≈90 K.\nTesting MOKE experiments on Fe and Gd thin films showed\nthat both Fe and Gd layers should contribute to the total Kerr ef-\nfect for the combined Fe /Gd layered system. Under our exper-\nimental conditions, the MOKE signal from Gd is comparable\nwith that from Fe (about two times smaller at low temperature )\nbut has opposite sign. Thus, we expect di fferent signs of MOKE\nfor Gd- and Fe-aligned states in the investigated multilaye r./s45/s52 /s45/s50 /s48 /s50 /s52 /s45/s52 /s45/s50 /s48 /s50 /s52/s77/s79/s75/s69/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s46/s117/s46/s41\n/s72 /s32/s40/s107/s79/s101/s41/s50/s57/s52/s32/s75\n/s49/s53/s53/s32/s75\n/s49/s50/s51/s32/s75\n/s57/s56/s32/s75\n/s56/s57/s32/s75\n/s55/s49/s32/s75\n/s50/s48/s32/s75\n/s52/s46/s53/s32/s75/s40/s97/s41\n/s84\n/s99/s111/s109/s112/s40/s98/s41\n/s120\n/s32/s50/s120\n/s32/s50\n/s72 /s32/s40/s107/s79/s101/s41\n/s32/s50/s57/s55/s32/s75\n/s49/s53/s54/s32/s75\n/s49/s50/s48/s32/s75\n/s49/s48/s48/s32/s75\n/s57/s48/s32/s75\n/s55/s48/s32/s75\n/s51/s48/s32/s75\n/s52/s46/s53/s32/s75/s120\n/s32/s50\nFigure 3: MOKE curves obtained at di fferent temperatures on Gd-terminated\n(a) and Fe-terminated (b) sides of the superlattice. Black a rrows show transi-\ntions from field-aligned to canted state of the surface magne tization.\nFig. 2 shows the experimental MOKE hysteresis loops mea-\nsured at T=155 K from two sides of the superlattice. For both\ncurves, a flat part in the region of weak fields means that the\nmagnetic moment of the outermost layer remains collinear to\nthe external field. Positive sign of the MOKE signal at H>0\nindicates the Fe-aligned state. At some higher field the MOKE\nsignal decreases, indicating that the magnetization of the outer-\nmost layer begins to rotate. Note that on Gd-terminated side this\nrotation starts in weaker field ( H=Hs) than on Fe-terminated\nside ( H=Hb). Thus, we can conclude that in magnetic fields\nHsHb, a transforma-\ntion to the bulk twisted phase occurs.\nSimilar analysis of the MOKE curves was performed for\ndifferent temperatures in the range 4–300 K (see Fig. 3) and the\nresulting phase diagram of the system was obtained (Fig. 4). At\nT>TGd\nCwe observe simple rectangular hysteresis loops with-\nout any signs of possible phase transitions. At lower temper a-\ntures the shape of the MOKE curves changes. The compensa-\ntion point Tcomp≈90 K can be clearly detected as temperature\nwhere an inversion of the hysteresis loop occurs (Fig. 3), i. e.\ndifferent orientation of Fe magnetization is realized in weak\nfields above and below Tcomp. It is also clearly seen that at\n3/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s50/s52/s54/s56\n/s71/s100/s45/s116/s101/s114/s109/s105/s110/s97/s116/s101/s100/s32/s115/s105/s100/s101\n/s70/s101/s45/s116/s101/s114/s109/s105/s110/s97/s116/s101/s100/s32/s115/s105/s100/s101\n/s83/s117/s114/s102/s97/s99/s101\n/s116/s119/s105/s115/s116/s72 /s32/s40/s107/s79/s101/s41\n/s84 /s32/s40/s75/s41/s77/s97/s103/s110/s101/s116/s105/s99/s32/s116/s119/s105/s115/s116\n/s115/s116/s97/s116/s101\n/s70/s101/s32/s45\n/s97/s108/s105/s103/s110/s101/s100\n/s71/s100/s45/s97/s108/s105/s103/s110/s101/s100/s84\n/s99/s111/s109/s112/s84\n/s67/s71/s100\nFigure 4: Resulting H−Tphase diagram of the investigated Fe /Gd superlattice.\nPoints are obtained from MOKE data on two sides of the multila yer. Lines are\ncalculations within the mean-field approach [11]. The dashe d line corresponds\nto a situation when Gd magnetization vanishes in the middle o f Gd layer.\nT>Tcomp the rotation of magnetization starts in weaker fields\non Gd-terminated side of the superlattice. On the contrary, at\nT0.5, are shown). The model predicts\nthe existence of two spectral branches with di fferent types of\nmagnetic precession inside Gd layers (Fig. 6c). The HT-mode\nhas a gap in the spectrum at low temperatures and corresponds\nto strongly non-uniform precession inside Gd layers. The LT -\nmode is quasi-uniform. Its frequency vanishes at H=Hb, i.e.\nat phase transition from field-aligned to twisted magnetic s tate.\nIn general, the behaviour of calculated curves f(H) repeats\nqualitatively the experimental dependencies, except the t emper-\nature region≈200−225 K (i.e. slightly above TGd\nC) and in weak\nmagnetic fields H/lessorsimilar1.5 kOe. Above T=225 K the model pre-\ndicts the crossing of two spectral branches. One branch with in-\n4/s48 /s50 /s52 /s54 /s56 /s49/s48 /s48 /s50 /s52 /s54 /s56 /s49/s48 /s48 /s49 /s50 /s51 /s52/s102/s32/s61/s32/s55/s46/s54/s53/s32/s71/s72/s122/s50/s52/s56/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s46/s117/s46/s41\n/s72 /s32/s40/s107/s79/s101/s41/s50/s55/s56\n/s53/s48/s55/s48/s49/s48/s56/s49/s50/s54/s49/s51/s56/s49/s54/s55/s49/s56/s48/s49/s57/s49/s50/s49/s48/s50/s50/s57/s97/s41\n/s84 /s32/s40/s75/s41/s58/s84 /s32/s40/s75/s41/s58/s32/s98/s41\n/s102/s32/s61/s32/s49/s55/s46/s50/s32/s71/s72/s122\n/s72 /s32/s40/s107/s79/s101/s41/s50/s50/s49/s50/s51/s55/s50/s56/s53\n/s50/s52/s52/s55/s55/s49/s49/s48/s53/s49/s51/s52/s49/s54/s49/s49/s56/s48/s50/s53/s48\n/s49/s57/s56/s84 /s32/s40/s75/s41/s58/s99/s41\n/s32\n/s72 /s32/s40/s107/s79/s101/s41/s102/s32/s61/s32/s50/s53/s46/s57/s32/s71/s72/s122/s50/s50/s49/s50/s51/s53/s50/s56/s53\n/s51/s48/s53/s57/s49/s48/s57/s49/s51/s48/s49/s52/s57/s49/s55/s48/s49/s56/s48/s49/s57/s48/s50/s48/s48/s50/s49/s49\nFigure 5: Experimental resonance spectra at di fferent temperatures (shown in the plot) obtained at f=7.65 GHz (a), f=17.2 GHz (b), and f=25.9 GHz (c).\n/s48 /s49 /s50 /s51 /s52 /s53 /s54/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s48 /s49 /s50 /s51 /s52 /s53 /s54/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s48 /s50/s53 /s53/s48/s102/s32/s40/s71/s72/s122/s41\n/s72 /s32/s40/s107/s79/s101/s41/s49/s56/s48 /s50/s48/s48 /s50/s50/s53\n/s50/s57/s53\n/s49/s52/s48 /s56/s53/s49/s56/s48/s72/s84/s32/s109/s111/s100/s101\n/s76/s84/s32/s109/s111/s100/s101/s97/s41\n/s76/s84/s32/s109/s111/s100/s101/s72/s84/s32/s109/s111/s100/s101\n/s50/s50/s48\n/s50/s57/s53/s50/s48/s48\n/s50/s50/s53/s49/s56/s48\n/s56/s53\n/s49/s52/s48\n/s49/s56/s48\n/s72 /s32/s40/s107/s79/s101/s41/s102/s32/s40/s71/s72/s122/s41/s98/s41\n/s72\n/s98/s56/s53/s32/s75/s70/s101/s80/s114/s101/s99/s101/s115/s115/s105/s111/s110/s32/s112/s114/s111/s102/s105/s108/s101/s32/s99/s41\n/s70/s101\n/s49/s52/s48/s32/s75/s76/s84/s32/s109/s111/s100/s101/s44/s32/s49/s53/s32/s71/s72/s122\n/s71/s100/s32/s71/s100\n/s49/s56/s48/s32/s75/s50/s57/s53/s32/s75/s72/s84/s32/s109/s111/s100/s101/s44/s32/s51/s53/s32/s71/s72/s122\n/s68/s101/s112/s116/s104/s32/s40/s197/s41/s70/s101 /s70/s101\nFigure 6: Experimental (a) and calculated (b) frequency vsfield dependencies at di fferent temperatures (shown in the plots in Kelvins) and examp les of calculated\ndepth profiles of magnetization precession in the Gd layer fo r LT and HT modes (c).\n5/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s50/s52/s54/s56/s49/s48\n/s55/s46/s54/s53/s32/s71/s72/s122/s49/s55/s46/s50/s32/s71/s72/s122/s50/s53/s46/s57/s32/s71/s72/s122/s72\n/s114/s101/s115/s32/s40/s107/s79/s101/s41\n/s84 /s32/s40/s75/s41/s51/s53/s46/s55/s32/s71/s72/s122/s51/s53/s46/s55/s32/s71/s72/s122\n/s50/s53/s46/s57/s32/s71/s72/s122\n/s49/s55/s46/s50/s32/s71/s72/s122\n/s55/s46/s54/s53/s32/s71/s72/s122\n/s84\n/s99/s111/s109/s112/s81/s32/s62/s32/s53\n/s51/s32/s60/s32/s81/s32/s60/s32/s53\n/s81/s32/s60/s32/s51/s81/s117/s97/s108/s105/s116/s121/s32/s102/s97/s99/s116/s111/s114/s58\n/s72/s84/s45/s109/s111/s100/s101/s76/s84/s45/s109/s111/s100/s101\nFigure 7: Temperature dependencies of the resonance field at different frequen-\ncies. Points are experimental data, lines are calculations . Solid, dashed, and\ndotted lines correspond to di fferent Q-factor of resonance modes.\ncreasing dependence f(H) corresponds to preferable precession\nof Fe layers. This branch has large Q-factor and is observed e x-\nperimentally. The second branch with decreasing dependenc e\nf(H) corresponds to preferable precession of inner part of Gd\nlayers. This branch has small Q-factor and is not observed ex -\nperimentally. Below T=225 K the model predicts the repul-\nsion of these two crossing modes. As a consequence, a gap in\nthe spectrum opens. Experimentally, however, such a gap ari ses\nonly at T/lessorsimilar180 K (Fig. 6a,b).\nDespite this discrepancy, Fig. 6b helps to understand di ffer-\nent behaviour of the HT peak at frequencies below and above\nf≈12 GHz, i.e different direction of the line shift at cooling\nthe system below room temperature (Fig. 5). The critical val ue\n12 GHz corresponds to the frequency where the e ffect of modes\nrepulsion arises.\nFig. 7 shows the resulting experimental and calculated tem-\nperature dependencies of the resonance fields Hres(T) at dif-\nferent frequencies. It can be seen that the experimental and\ntheoretical curves demonstrate not only qualitative but al so a\ncertain quantitative agreement. The noticeable discrepan cy ob-\nserved for HT-mode at 17.2 GHz below T≈230 K is con-\nnected with the above-discussed inadequate description of the\nmode-repulsion region.It is interesting to note that at low frequency ( f=7.65 GHz)\nthe model predicts the existence of minimum in the Hres(T) de-\npendence for the LT-mode. This minimum is connected with\nthe fact that the LT-mode frequency vanishes at H=Hb(i.e.\nHres→Hbwhen f→0). Since Hbturns to zero at Tcomp,\nwe could expect the minimum of Hres(T) at this temperature.\nExperimentally, however, we did not manage to detect the ab-\nsorption line below Tcomp. The reason for this can be the large\ndamping of the corresponding resonance mode. Indeed, our\ncalculations show that the Q-factor of the LT-mode increase s\nbelow Tcompatf=7.65 GHz (see Fig. 7).\nTo summarize, we achieved a reasonable agreement between\nthe experiment and model calculations. The model describes\nmany features of the experimental spectra and helps to ident ify\nthe types of the observed resonance modes. The main discrep-\nancy between the experiment and model arises in the vicinity\nofTGd\nCwhere the calculated spectra are very sensitive to mag-\nnetic parameters of the system and can be strongly influenced\nby structural inhomogeneities of the real superlattice.\n4. Conclusion\nIn this work we demonstrated the realization of non-colline -\nar magnetic states and inhomogeneous magnetization dynami cs\nin a Fe/Gd artificial layered ferrimagnet. We have shown that\nboth static and dynamic properties of the system are describ ed\ntaking into account essentially non-uniform magnetizatio n dis-\ntribution inside Gd layers.\nUsing the magneto-optical Kerr e ffect, we defined the re-\ngions of stability for surface and bulk twisted states of the in-\nvestigated multilayer. The resulting experimental H−Tphase\ndiagram is in a good agreement with calculations based on the\nmean-field model.\nFerromagnetic resonance spectra obtained in this work re-\nveal a complex temperature evolution with two spectral bran -\nches that can not be explained in terms of uniform magnetic\nprecession within the superlattice. The performed theoret ical\nsimulations of magnetization dynamics in the system show th at\nthe observed resonance modes correspond to di fferent types of\ninhomogeneous precession inside Gd layers.\nIn the end we would like to emphasize that the nanostruc-\ntured ferrimagnets provide possibility to study such compl ex\nmagnetic phenomena under easily achievable experimental c on-\nditions: in magnetic fields up to 1 T and at microwave frequen-\ncies. The traditional ferrimagnetic crystals would requir e mag-\nnetic fields and frequencies that are several orders of magni tude\nlarger. In this respect, the artificial structures can be con sid-\nered as suitable model objects for experimental investigat ions\nof non-collinear magnetic phases and inhomogeneous magnet i-\nzation dynamics in ferrimagnets.\nAcknowledgments\nThe work is partially supported by the Russian Founda-\ntion for Basic Research (grants No. 16-02-00061, No. 18-37-\n00182), by the Ministry of Education and Science of the Rus-\n6sian Federation (grant No. 14-Z-50.31.0025), and by the Ba-\nsic Research Program of the Presidium of Russian Academy of\nSciences.\nResearch in Ekaterinburg was performed in terms of the\nState assignment of Federal Agency of Scientific Organizati ons\nof the Russian Federation (theme “Spin” No. AAAA-A18-188\n020290104-2).\nAppendix A. FMR frequency of a strongly coupled layered\nferrimagnet\nLet us consider two FM layers with di fferent magnetic mo-\nmentsµ1>µ 2. We suppose that these layers are strongly AFM\ncoupled (the exchange energy is infinity). In this case, the m ag-\nnetic field Happlied in the film plane aligns µ1andµ2parallel\nand antiparallel to the field direction respectively. Consi dering\nZeeman and demagnetizing energy of both layers, the total en -\nergy of the system can be written as\nE=−H/parenleftbigµ1+µ2/parenrightbig+2π/parenleftbigµ1·z/parenrightbig2\nV1+/parenleftbigµ2·z/parenrightbig2\nV2,\nwhere zis a unit vector normal to the film plane, V1andV2\nare volumes of layers. Taking into account that −µ2⇈µ1, the\nenergy expression can be rewritten in the form\nE=−Hµ+2πµ2\n1V/V1+µ2\n2V/V2\n(µ1−µ2)2·(µ·z)2\nV,\nwhereµ=µ1+µ2andV=V1+V2. Now it has the form of\nmagnetic energy for a single FM film with modified demagne-\ntizing factor. Thus, the FMR frequency of the system is define d\nby modified Kittel’s formula\nω=γeff/radicalbig\nH(H+4πMeff), (A.1)\nwhere\n4πMeff=4πµ2\n1/V1+µ2\n2/V2\nµ1−µ2, (A.2)\nandγeffis a net gyromagnetic ratio of two coupled layers [27]\nγeff=µ1−µ2\nµ1/γ1−µ2/γ2, (A.3)\nwhereγ1andγ2are gyromagnetic ratios of individual layers. If\nγ1≈γ2, Eqs. (A.1), (A.2) predict increasing FMR frequency\nwhenµ2is increasing. This behaviour is opposite to the case\nof amorphous or crystal ferrimagnetic film when the e ffective\ndemagnetizing field is defined by simple expression 4 πMeff=\n4π(M1−M2), where M1,2are magnetizations of FM sublattices\n[27]. In this situation FMR frequency is decreasing with M2\nincrease.\nIt is important to note that the approximation (A.1)–(A.3)\nis valid only when the exchange fields Hex,iacting on layers\ni=1,2 are much stronger than the corresponding demagnetiz-\ning fields Hex,i≫4πMiand the external field is far below the\ntransition to the canted state: H≪|Hex,1−Hex,2|[28].References\n[1] R. E. Camley, “Thermal Properties of Magnetic Multilaye rs and Nanos-\ntructures: Applications to Static and Dynamic Behavior” in Magnetism\nof Surfaces, Interfaces, and Nanoscale Materials , Handbook of Surface\nScience, V ol. 5, edited by R. E. Camley, Z. Celinski, and R. L. Stamps\n(Elsevier, North-Holland, 2015).\n[2] R. E. Camley and R. L. Stamps, J. Phys. Cond. Mat. 5, 3727 (1993).\n[3] R. E. Camley, Phys. Rev. B 353608 (1987).\n[4] R. E. Camley and D. R. Tilley, Phys. Rev. B 37, 3413 (1988).\n[5] R. E. Camley, Phys. Rev. B 39, 12316 (1989).\n[6] M. Sajieddine, Ph. Bauer, K. Cherifi, C. Dufour, G. 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Guti´ errez, and D. S. Schmool, Ch in. Phys.\nLett. 18, 973 (2001).\n[27] R. K. Wangsness, Phys. Rev. 91, 1085 (1953).\n[28] A. G. Gurevich, Magnetic Resonance in Ferrites and Anti ferromagnets\n(Nauka, Moscow, 1973) [in Russian].\n7" }, { "title": "2309.16168v1.Magnetism_and_magnetocaloric_properties_of_Co___1_x__Mn__x_Cr__2_O__4_.pdf", "content": "Magnetism and magnetocaloric properties of Co 1−xMnxCr2O4\nJoya A. Cooley,1,∗Gregor Dairaghi,2, 3Guy C. Moore,4, 5Matthew K. Horton,4, 5\nEmily C. Schueller,6, 7Kristin A. Persson,4, 8and Ram Seshadri6, 7, 9\n1Department of Chemistry and Biochemistry, California State University, Fullerton, California 92834, United States\n2Department of Physics, Carleton College, Northfield, Minnesota 55057, United States\n3Current Affiliation: Applied Physics Graduate Program,\nNorthwestern University, Evanston, Illinois 60208, United States\n4Department of Materials Science and Engineering, University of California Berkeley, Berkeley, California 94720, United States\n5Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States\n6Materials Research Laboratory, University of California, Santa Barbara, California 93106, United States\n7Materials Department, University of California, Santa Barbara, California 93106, United States\n8Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States\n9Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, United States\nCo1−xMnxCr2O4crystallizes as a normal spinel in the cubic Fd3mspace group, and the end mem-\nbers have been reported to display a region of collinear ferrimagnetism as well as a low-temperature\nspin-spiral state with variable coherence lengths from 3 nm to 10 nm in polycrystalline samples. Here,\nwe present the synthesis of the entire solid solution, and data showing that the ferrimagnetic or-\ndering temperature as well as the spin-spiral lock-in temperature are tunable with the Co/Mn ratio.\nThe peak magnetocaloric entropy change was determined to be ∆SM=−5.63 J kg−1K−1in an ap-\nplied magnetic field change of ∆H= 0 T to 5 T for the Mn end-member at the ferrimagnetic ordering\ntemperature. Using density functional theory (DFT), we explore the shortcomings of the magnetic de-\nformation proxy to identify trends in ∆SMacross composition in this spinel system, and explore future\nextensions of theory to address these discrepancies.\nINTRODUCTION\nIn the search for alternative refrigeration technologies,\nmagnetic refrigeration has emerged as an environmen-\ntally friendly and more efficient alternative to conven-\ntional vapor-compression refrigeration [1]. Magnetic re-\nfrigeration uses the magnetocaloric effect (MCE) by al-\nternating between states of high and low entropy (low\nand high magnetization, respectively) to impart a re-\nversible temperature change. Since the discovery of the\ngiant magnetocaloric effect in Gd 5Si2Ge2[2], research\nhas focused on methods of understanding which mate-\nrials may exhibit a large magnetocaloric effect. Satura-\ntion magnetization ( Msat) is typically used as an indica-\ntor for the magnitude of the magnetocaloric effect, and\nhigh magnetization compounds have been thought to be\nwell-performing magnetocalorics.\nRecent analysis has shown Msatis not always the most\neffective predictor of which materials may exhibit fa-\nvorable magnetocaloric properties, and Msatshows poor\ncorrelation with the MCE as quantified by the magnetic\nentropy change, ∆SM[3]. As such, there has been an\nincrease in interest in systems whose lattice and spin\ndegrees of freedom are strongly coupled, especially in\nmagnetocaloric materials. It has been proposed that this\ncoupling of spin and lattice – termed magnetostructural\ncoupling – is important to understanding, and even pre-\ndicting, the magnitude of the magnetocaloric effect in\nferromagnets [3].\n∗jcooley@fullerton.eduFor many decades, the MCE has been used to reach\ncryogenic (mK and µK) temperatures [4–6]. Recently,\nthere has been great interest in magnetocalorics for\nroom-temperature magnetic refrigeration [7]. This ne-\ncessitates magnetic ordering temperatures near room\ntemperature such that the maximum ∆SM(and, there-\nfore, MCE)t will be near room temperature. However,\nother technologies exist that would require excellent\nmagnetocaloric performance in more intermediate tem-\nperature ranges below the boiling point of liquid nitro-\ngen ( e.g.2 K to 70 K). This includes staged/cascaded cy-\ncles necessary to cover the wide temperature range re-\nquired for hydrogen liquefaction [8], which boils near\n20 K. While some effort has been made toward study-\ning inexpensive materials like chalcogenide spinels [9],\ncurrently much of the research into magnetocalorics vi-\nable for hydrogen liquefaction involves expensive rare-\nearth elements [10–12], stemming partially from the fact\nthat these often exhibit high Msatvalues. Additionally, a\nsegmented series of materials with gradually changing\ntransition temperatures would be beneficial to reach hy-\ndrogen liquefaction temperatures [13]. Here, we look to\nan earth-abundant transition metal oxide solid solution –\nCo1−xMnxCr2O4– with gradually varying transition tem-\nperatures magnets as a more cost-effective alternative to\nrare-earth magnetic materials.\nWe investigate how compositional changes and slight\nstructural changes affect the magnetism and MCE as\nquantified by the magnetic entropy change ∆SMin the\nspinel solid solution Co 1−xMnxCr2O4. This series of com-\npounds is geometrically frustrated stemming partially\nfrom the 3D-pyrochlore sublattice of the Cr3+and par-\ntially from the diamond sublattice formed by cations inarXiv:2309.16168v1 [cond-mat.mtrl-sci] 28 Sep 20232\nthe Mn/Co site. Frustrated spin systems often possess in-\nteresting and exotic ground states [14, 15], such as the\nspiral spin textures found in the Mn [16] and Co [17, 18]\nchromite spinels, and have been linked to good magne-\ntocaloric performance [19]. According to a 2014 study\nby Dey et al., MnCr 2O4exhibits some degree of magne-\ntostructural coupling and shows magnetoelastic transi-\ntions at both the N ´eel temperature and the spin spiral\nlock-in temperature [20], and its geometric frustration\nlikely plays a part in coupling the spin and lattice [21].\nAs the first study of the magnetocaloric properties of this\nsolid solution, we find that the Mn-rich members exhibit\na high ∆SMof up to −5.67 J kg−1K−1for a field change\nof 0 T to 5 T.\nI. MATERIALS AND METHODS\nA. Synthesis\nPolycrystalline powders of the solid solution series\nCo1−xMnxCr2O4were prepared using solid-state syn-\nthesis. Precursor materials CoC 2O4•2H2O, MnO, and\nCr2O3were used as received. For clarity, herein samples\nwill be referred to according to their nominal xvalues\n(i.e.x= 0.00, 0.25, 0.50, 0.75, and 1.00). CoCr 2O4was\nsynthesized following literature procedure [17] but re-\ngrinding and annealing was not found to improve phase\npurity so samples were heated at 800 °C for 24 hours,\nthen at 1000 °C for 24 hours without removing from the\nfurnace, then the furnace was allowed to cool. Samples\nx= 0.25 and 0.50 were also synthesized using this proce-\ndure. However, due to initial phase separation into two\ndifferent spinel phases with slightly different unit cell\nparameters, samples x= 0.25 and 0.50 were quenched\nfrom 1000 °C. MnCr 2O4was also synthesized according\nto literature [22], and x= 0.75 also followed this proce-\ndure.\nB. Structural characterization\nHigh-resolution synchrotron powder X-ray diffraction\n(XRD) data were collected at room temperature on all\nsamples using beamline 11-BM at the Advanced Pho-\nton Source (APS), Argonne National Laboratory. For\nroom temperature scans, powderized samples were\nloaded into 0.8 mm diameter Kapton capillaries with\neach end sealed with clay and measured for 6.7 minute\nscans. x= 0.00, 0.75, and 1.00 were measured with\nλ= 0.457850 ˚A and x= 0.25 and 0.50 were measured\nat another time with λ= 0.457845 ˚A. Rietveld refine-\nments of data were performed using TOPAS [23]. All\nsamples were fit with size and strain parameters, except\nthe complex peak shape of x= 0.25 which required a 2 θ-\ndependent split Pearson VII peak shape to describe varia-\ntion in peak asymmetry fully. Structures were visualized\nusing VESTA-3 [24].C. Magnetic measurements\nMagnetic properties were measured on 5 mg to 10 mg\nof powder loaded into capillaries and measured a Quan-\ntum Design MPMS3 equipped with a vibrating sample\nmagnetometer (VSM). Zero field– and field–cooled mag-\nnetization ( M) vs temperature ( T) measurements were\ntaken upon warming at a rate of 5 K min−1. In order to\ndetermine ∆SM,Mversus Tmeasurements were taking\non cooling (using a rate of 5 K min−1) at various fields\nfrom H= 0.1 T to H= 5 T. Temperature derivatives of\ntheM(T)s were calculated using Tikhonov regulariza-\ntion [25], and then integrals with field were calculated\nusing the trapezoid method to obtain ∆SM. Data were\nanalyzed using the magentro.py code, and more details\nof this procedure have previously been reported [26].\nD. First-principles calculations\nThe crystal structures corresponding to the x= 0and\nx= 1 endpoints of the spinel system Mn xCo1−xCr2O4\n(CoCr 2O4and MnCr 2O4, respectively) were obtained\nfrom the Materials Project database[27]. To identify the\nenergetically stable collinear magnetic ordering of these\nspinel structures, an enumeration of the possible mag-\nnetic orderings and evaluation of their respective ener-\ngies was performed using the atomate workflow [28].\nDespite the experimental results in support of both\nCoCr 2O4and MnCr 2O4exhibiting a spin-spiral ground\nstate [17, 20, 29–32], we performed collinear ferrimag-\nnetic calculations. The justification for this approach is\nbased on experiments that show that the spin-spiral tran-\nsition temperature is significantly lower than the N ´eel\ntemperature, TN. Furthermore, we are interested in the\nferrimagnetic (FiM) to paramagnetic (PM) phase transi-\ntion, which yields the greatest change in entropy. Based\non the experimentally reported values of the spin-spiral\ntransition temperatures for CoCr 2O4and MnCr 2O4, the\norder-disorder phase transition has been experimentally\ndetermined to involve a predominantly collinear FiM to\nPM transition.\nUsing the endpoint spinel structures, a set of possi-\nble structures of each composition Mn xCo1−xCr2O4were\nenumerated based on their crystallographic symmetries,\nup to a supercell size of two times the formula unit using\nthepymatgen interface with enumlib [33–36].\nFollowing the generation of intermediate crystal struc-\ntures for x=0.25, 0.5, and 0.75, we generalized the\ncollinear FiM ordering that would be expected for the\nspinel system endpoints, with a net spin-up moment on\nthe Cr atoms, and spin-down moment on the Mn and\nCo atoms. This collinear FiM ordering was used in\nspin-polarized calculations for computing the deforma-\ntion proxy, ΣM, and saturation magnetization, M0\nnet.3\nFIG. 1. A view of one unit cell of the spinel structure type,\nACr2O4(A= Co, Mn) where the Asite is tetrahedrally coor-\ndinated by O and corner shared, and the Cr site is octahedrally\ncoordinated and edge- and corner shared.\nII. RESULTS AND DISCUSSION\nA. Structure and Phase Purity\nThe spinel structure type is pictured in Figure 1 where\nCo/Mn atoms are in green corner-shared A-site tetrahe-\ndra, Cr atoms are in deep blue edge-and corner-shared\nB-site octahedra, and O atoms are in tangerine at the\nvertices of the tetrahedra and octahedra. The normal\nspinel structure type exists such that divalent cations are\nin tetrahedral A-sites and trivalent cations are in octa-\nhedral B-sites. Many spinel compositions have the abil-\nity to form partially- or fully-inverted spinels such that\nA-site atoms oxidize to trivalent and occupy the B-site,\nandB-site atoms reduce to divalent and occupy the A-\nsite. Chromite spinels, however, are a unique case. Since\ntrivalent Cr is d3, it is more stable in an octahedral crys-\ntal field splitting arrangement, where all spins are in the\nlowest energy, triply-degenerate t2gorbitals. Trivalent\nCr in a tetrahedral crystal field would necessarily have\ntwo low-energy spins in the doubly degenerate egorbital\nand one high energy spin in the t2gorbitals, destabiliz-\ning this arrangement relative to the octahedral arrange-\nment. Thus, in chromite spinels we can be relatively cer-\ntain that Co and Mn only occupy the tetrahedral A-sites\nwhile Cr only occupies the octahedral B-site.\nRietveld refinement results of synchrotron powder x-\nray diffraction for samples in this study are shown in Fig-\nure 2. The left panel shows that each sample fits to a sin-\ngle spinel phase at the resolution of the instrument. The\nright panel shows a close view of the evolution in peak\nposition of the (311) peak as a function of x: a mono-TABLE I. Crystallographic Data for Co 1−xMnxCr2O4.\nx a(˚A) u R wp Cr2O3(%)\n0.00 8.33304(4) 0.26125(9) 14.46 –\n0.25 8.34771(4) 0.25949(1) 10.81 –\n0.50 8.3851(4) 0.26043(6) 12.149 0.81(1)\n0.75 8.41180(2) 0.26241(7) 13.72 0.54(1)\n1.00 8.43853(1) 0.26385(6) 12.82 1.40(2)\ntonic decrease in Qcorresponding to an increase in unit\ncell parameter. We note that samples with x= 0.25 and\nx= 0.50 show slightly more broadening than other sam-\nples. These samples did require quenching from max-\nimum temperature in order to yield single phase sam-\nples, and we note that there could be two spinel phases\nwith slightly different unit cell parameters, yet even the\nextremely high resolution of beamline 11-BM at the Ad-\nvanced Photon Source ( <1.4×10−4∆Q/Q) is not able\nto resolve two different phases, so we proceed as if each\nsample is single phase. There are no crystalline fer-\nromagnetic impurities present in the samples, however\nx= 0.50, 0.75, and 1.00 contain a small ( <1.5 wt%)\namount of unreacted Cr 2O3, which is antiferromagnetic\nat room temperature [37], that could not be eliminated\nby further reaction.\nFurther results of Rietveld refinements, including the\nweighted profile R-value Rwp, are tabulated Table I. The\ncell parameter a(also depicted in Figure 3) and oxygen\nposition uwere determined during Rietveld refinement.\nThe ionic radii of tetrahedrally coordinated Co2+and\nMn2+are 0.72 ˚A and 0.80 ˚A [38], respectively, so it fol-\nlows that increasing Mn composition should increase the\nlattice parameter throughout the solid solution. The an-\nion in spinel structures – here, O – is located on the crys-\ntallographic equipoint 32eand variation in this position,\nrepresented as the value u, reflects how the structure\nchanges when accommodating different sizes of cations.\nWhen u= 0.25, the anions are in ideal cubic closest-\npacking arrangement with perfect CrO 6octahedra; in-\ncreases in uabove the ideal value of 0.25 indicate the\nsize of the tetrahedron is increasing, and the octahedron\nis shrinking and undergoing a trigonal compression [39].\nTable I shows that the uparameter overall increases with\nincreasing Mn substitution, shifting it higher than the\nideal value, and implying the tetrahedral site size in-\ncreases across the solid solution, in good agreement with\nthe unit cell parameter increase and larger ionic radius\nof Mn2+.\nFigure 3 shows that the lattice parameter aof the se-\nries increases with increasing xand follows V ´egard’s law\nas indicated by the dashed line (created from the liter-\nature lattice parameters of the end members CoCr 2O4\n[40] and MnCr 2O4[41]). This is in good agreement with\nthe larger ionic radius of tetrahedral Mn2+as compared\nto Co2+. Each data point is also in good agreement with\nthe V ´egard line indicating that this series forms a com-4\nFIG. 2. The results of Rietveld refinement of synchrotron X-\nray diffraction data measured for each sample show the spinel\nstructure type. The data are black circles, the overlaying line\nrepresents the model based on the structure, and the line be-\nlow each pattern indicates the difference between the data and\nmodel. A close view of the (311) peak (right) indicates a com-\nplete solid solution at the resolution of the instrument.\nplete solid solution.\nB. Magnetic Properties\nBoth the Co and Mn end members of the solid solu-\ntion undergo a paramagnetic to ferrimagnetic transition\nand, after a region of collinear ferrimagnetic order, have\na complex spiral ground state at low temperatures. In\nFigure 4, which shows temperature-dependent magneti-\nzation for each composition, the transition from the para-\nmagnetic state to the collinear ferrimagnetic state can be\nseen to decrease as the Mn composition is increased. This\nis important from an engineering standpoint as tunabil-\nity and control of transition temperature, and thus peak\n∆SM(discussed later), in this range is especially valu-\nable to cascaded magnetic refrigeration for hydrogen liq-\nuefaction.\nThe high temperature (300 K to 390 K) inverse sus-\nceptibility of each sample was fit to a linear regres-\nsion and the equation of the resulting line was used\nto extract parameters from the Curie-Weiss equation,\nχ=C/(χ−ΘCW) – namely, the effective paramagnetic\nmoment µeff, Curie constant C, and Curie-Weiss inter-\nFIG. 3. a) Lattice parameters from Rietveld refinement of syn-\nchrotron X-ray diffraction data for each sample show that each\nfollows the V ´egard law (dashed line) as expected for a solid\nsolution.\nFIG. 4. Temperature-dependent magnetization data reveal an\nincrease in magnetization and decrease in ordering tempera-\nture as Mn content is increased.\ncept ΘCW. These values are presented in Table II. The\nspin-only formula for µeffis generally taken as valid for\n3dtransition metal ions, however it only strictly applies\nto specific cases. In reality, the orbital contribution is\nnot totally quenched and spin-orbit coupling plays a role\nin determining µeff.d1−d4transition metal ions have\na spin orbit coupling constant smaller than zero, and\ntheir µeffvalues tend to be smaller than what is calcu-\nlated by the spin-only formula. d6−d9transition met-5\nals have a spin-orbit coupling constant larger than zero\nand tend to have µefflarger than calculated by spin-only\nformula. The samples in this series show a stochastic\ntrend with composition concerning how they compare\nto the spin-only and unquenched µeffvalues. Co2+is\nd7and we would expect compounds rich in Co2+to de-\nviate higher than the spin-only value; conversely Mn2+\nisd4and we would expect that Mn-rich samples devi-\nate lower than the spin-only value. Values of µefffromthe Co- and Mn-rich ends of the solid solution are both\nfar above unquenched estimates and far below spin-only\nestimates with no real trend to speak of. This may be\ndue to not fitting completely in the paramagnetic regime\nfor each of these samples as, due to instrumentation and\nsample limits, only measurement to 390 K is practical.\nAsxincreases, the magnitude of ΘCWdecreases sug-\ngesting that the number of dominant antiferromagnetic\nCo/Mn interactions with Cr decreases as Mn content is\nincreased, and the decrease is overall monotonic.\nTABLE II. Results from fitting inverse magnetic susceptibility data of Co 1−xMnxCr2O4to a linear regression to extract parameters\nin the Curie-Weiss equation. TCis assumed to be the peak of the ∆SMcurve. Estimated moments (spin-only and unquenched) are\ncalculated using: µeff=p\n2µ2\nCr+ (1−x)(µCo)2+x(µMn)2. Individual moments, such as µMnare calculated as µeff=gp\nS(S+ 1)\nusing an isotropic Land ´e g factor of 2.\nµeff(µB)\nx C( [µB/ f.u.] T−1K ) measured spin-only unquenched ΘCW(K) TC(K)\n0.00 7.83 7.9 6.7 7.5 −613 97.9\n0.25 7.00 7.5 7.1 7.7 −471 88.7\n0.50 6.96 7.5 7.4 7.8 −369 74.2\n0.75 7.82 7.9 7.7 7.9 −337 58.8\n1.00 7.53 7.8 8.1 8.1 −271 43.7\nTABLE III. Results from fitting inverse magnetic susceptibility data of Co 1−xMnxCr2O4to Equation 4 using\nscipy.optimize.curve_fit .TCvalues are reproduced from Table II. We hypothesize that the “ A” sublattice corresponds\nto tetrahedral sites in the spinel structure occupied by Mn and Co species, and the “ B” sublattice corresponds to octahedral sites\noccupied by Cr atoms.\nx|ΘCW|TC CB CA CA+CB λBB λAA λAB\n(K) ( [µB/ f.u.] T−1K ) ( [µB/ f.u.]−1T )\n0.00 97.4 97.9 0.95±0.01 9.53 ±0.11 10.48 66.5±0.2 86.2 ±0.4 127.6 ±0.6\n0.25 89.6 88.7 0.89±0.01 7.85 ±0.07 8.74 67.1±0.3 72.0 ±0.4 118.4 ±0.6\n0.50 78.7 74.2 6.76±0.02 1.20 ±0.00 7.96 71.7±0.1 -37.7 ±0.0 48.2 ±0.0\n0.75 65.8 58.8 8.66±0.03 0.57 ±0.00 9.23 60.2±0.1 -113.0 ±0.0 15.1 ±0.1\n1.00 45.4 43.7 8.37±0.11 2.54 ±0.01 10.91 73.6±0.4 1.3 ±0.1 38.9 ±0.0\nA plot of C/(χ|ΘCW|) +sgn(ΘCW)vsT/ΘCWcol-\nlapses all high temperature susceptibility data as shown\nin Figure 5. The dashed straight line intersecting the ori-\ngin corresponds to ideal Curie-Weiss behavior; since the\nhigh temperature data fit well onto this line, this indi-\ncates that the fits in the high temperature regime are\nvalid. Plotting the susceptibility data in this manner al-\nlows us to understand the nature of dominant magnetic\nexchange interactions [42, 43]. Since all show negative\ndeviations from the ideal Curie-Weiss line, this indicates\nuncompensated antiferromagnetism manifesting as ferri-\nmagnetism.\nFrom inspecting Table II, it is clear that |ΘCW|severely\noverestimates TC. This is not entirely surprising, be-\ncause this disagreement is expected for more than oneanti-parallel sublattice [44]. We explore the extension\nof Weiss mean field theory to two magnetic sublattices in\nSection V A. By fitting Equation 4 to 1/χdata versus tem-\nperature, we were able to achieve agreement between\n|ΘCW|andTCto within a few Kelvin, as reported in Ta-\nble III. This Curie-Weiss temperature, ΘCW, is defined as\n|ΘCW|= max {ηi[−W]}, as explained in Section V A.\nThe effectiveness of this theoretical extension is ex-\nemplified in the agreement of the model to 1/χat all\ntemperatures greater than TC, as shown in Figure 6.\nThe parameters of Equation 4 are reported in Table\nIII, with their associated uncertainty values output by\nscipy.optimize.curve_fit . The “ A” and “ B” labels in-\ndicate a separate magnetic sublattice. Based on the an-\nticipated low temperature ferrimagnetic ordering of this6\nFIG. 5. Scaled inverse susceptibility data as a function of scaled\ntemperature as described by equation are shown. The dashed\nline indicates simplified (ferromagnetic) Curie-Weiss paramag-\nnetism. Negative deviations in all samples reflect uncompen-\nsated interactions and suggest ferrimagnetic interactions.\nFIG. 6. The inverse susceptibility is plotted versus temperature\nfor each composition, xin Co 1−xMnxCr2O4. The lighter mark-\ners indicate the model equation fit to the data, Equation 4, for\ntwo magnetic sublattices. The fitted parameters of this equa-\ntion are supplied in Table III.\nspinel system, these lattices should each correspond to\neither tetrahedral sites, occupied by Mn or Co atoms, or\noctahedral sites, occupied by Cr atoms. Observing that\nλBBremains relatively constant versus Mn composition\nxcompared to the other values of λ, we anticipate that\ntheB-sublattice corresponds to the octahedral (Cr) sub-\nlattice, and the A-sublattice to the tetrahedral (Mn, Co)\nFIG. 7. Field-dependent isothermal magnetization of\nCo1−xMnxCr2O4were taken at 30 K for all samples to avoid\nthe spiral magnetic ordering region and reflect the magnetiza-\ntion in the collinear ferrimagnetic regime. Increasing Mn con-\ntent increases the saturation magnetization.\nTABLE IV. Spontaneous ( M0) and Saturation Magnetization\n(Msat) Data for Co 1−xMnxCr2O4.\nx M 0(µB/f.u.)aMsat(µB/f.u., 2 T) Msat(µB/f.u., 5 T)\n0.00 – 0.14 0.21\n0.25 0.32 0.38 0.44\n0.50 0.60 0.68 0.72\n0.75 0.88 0.92 0.99\n1.00 0.97 1.06 1.17\naExtrapolated values are from 30 K isotherms for each sam-\nple\nsublattice.\nFigure 7 shows field-dependent magnetization data of\neach sample at 30 K. This temperature was chosen to\navoid the spiral magnetic ordering region and reflect the\nmagnetization of in the collinear ferrimagnetic region in\neach sample. The x= 0.00 end has considerable coerciv-\nity as compared to the other compositions, and this coer-\ncivity has been seen to increase with decreasing temper-\nature (from 75 K to 3 K) [45]. Since none of the samples\nfully saturates at the highest applied field of 7 T, Table IV\nshows spontaneous magnetization ( M0) as extrapolated\nfrom Arrott-Belov plots (which could not be determined\nforx= 0.00) as well as saturation magnetization ( Msat)\nvalues at fields of 2 T and 5 T. Overall, this table shows\nthat magnetization values increase as Mn content is in-\ncreased. Also, none of the individual magnetization val-\nues are very large, again indicating that Msatis not al-\nways a useful indicator of magnetocaloric viability, dis-\ncussed further below.7\nFIG. 8. (a) Calculated ∆SMcurves for each sample for field\nchanges of 0 T to 2 T (dashed lines) and 0 T to 5 T (solid lines)\nin the species shows an increasing peak ∆SMand decreasing\npeak temperature with increasing Mn content. (b) Increasing\nRC with increasing Mn content indicates that the overall per-\nformance of the magnetocaloric increases with Mn content.\nC. Magnetocaloric Properties\nThe magnitude of the magnetocaloric effect can be de-\nrived from temperature-dependent magnetization mea-\nsurements at varying fields which are then derived and\nintegrated over field to yield ∆SM, depicted in Fig-\nure 8(a). The absolute value of ∆SMincreases as Mn\ncontent increases, and the temperature occurrence of\nthe peak value decreases as expected from previously\ndiscussed temperature-dependent magnetization mea-\nsurements. This series spans a wide range of peak\n∆SMvalues, from −1.67 J kg−1K−1forx= 0.00 to\n−5.63 J kg−1K−1forx= 1.00. It is difficult to visu-\nally determine whether or not ∆SMpeaks broaden or\nsharpen throughout the series, and this can affect overall\nmagnetocaloric performance. Thus we employ a more\nmathematically rigorous method of understanding mag-\nnetocaloric performance, the refrigerant capacity. This\ninvolves integrating the area under the ∆SMcurve using\nthe full width at half maximum of each curve as temper-\nature limits. These results are shown in Figure 8(b) and\nindicate that the overall magnetocaloric performance in\nthe series does improve as Mn content is increased since\nthe refrigerant capacity increases nearly 5-fold through-\nout the series.\nOften, Msatis used as a proxy for understanding the\nmagnitude of the magnetocaloric effect, however in this\ncase it is clear that other factors – such as magne-\ntostructural coupling – are at play. For example, the\nsaturation magnetization (Figure 7) of the x= 0.75 andx= 1.00 samples near 7 T is approximately 1.12 µBand\n1.25µB, respectively. However, the difference in their\npeak ∆SMvalues nearly doubles from 3.51 J kg−1K−1to\n5.67 J kg−1K−1, respectively, indicating that in this case\nMsatis not an appropriate metric by which to understand\nmagnetocaloric performance as quantified by ∆SM.\nD. First-Principles Calculations\nThe experimental measurements of ∆SMare com-\npared to first-principles computational proxies for the\n∆SMfigure of merit for magnetocaloric materials. Bo-\ncarsly et al. [3] and others have demonstrated the pre-\ndictive power of computational magnetic deformation\nproxy, ΣM[46]. The “magnetic deformation proxy” pro-\nvides a computationally inexpensive indicator of a large\nchange in magnetocaloric isothermal change in entropy,\n∆SMabove a threshold of ΣM>1.5%.ΣMis a mea-\nsure of the deformation strain between the magnetic and\nnonmagnetic structures of the material. The reason for\nthe strong correlation between ∆SMandΣMcan be ex-\nplained, in a general sense, by the role that coupled mag-\nnetic and structural degrees of freedom play in promot-\ning phase transitions with a large change in entropy, and\neven latent heat in the case of a first-order magnetostruc-\ntural phase transition.\nIn this study, the magnetic deformation proxy ΣMwas\ncomputed for a representative set of structures for each\nMn composition xin Mn xCo1−xCr2O4. In addition to\nΣM, previous studies have identified that the total mag-\nnetization at T= 0 K, M0\nnet, calculated from DFT, also\ncorrelates well with ∆SM, although to a lesser degree\nthan ΣMfor the materials database analyzed in the orig-\ninal study that tested the correlation of ∆SMwith differ-\nent first-principles indicators [3]. In this context, M0\nnet\nis defined as the “net magnetization” within the unit\ncell and can be computed from the difference between\nspin-up and spin-down electron densities produced by\nDFT calculations. Bocarsly and others have shown that\nfor a representative sample of ferrimagnetic and anti-\nferromagnetic materials, the product of the deforma-\ntion proxy with the saturation magnetization, M0\nnetΣM,\nin many cases provides a greater indication of a large\nchange in entropy, ∆SM, compared to either M0\nnetorΣM\nindividually [47].\nFigure 9(a) includes the distribution of ΣMvalues at\neach composition. Due to the large spread of values as-\nsociated with a different crystal structure, we are unable\nto make a conclusion regarding the trend of the defor-\nmation values versus composition. The reported stan-\ndard deviation of energy values was found to be no more\nthan 0.4 meV, normalized by the number of atoms in the\nunit cell. The maximum difference in average energy\nper atom output by VASP was 1 meV for each ferrimag-\nnetic configuration for both CoCr 2O4and MnCr 2O4. This\nsmall difference in energy values may be indicative of the\nfrustrated nature of magnetism that has been confirmed8\nFIG. 9. (a) The deformation proxy, ΣM, versus Mn composi-\ntion. (b) The computed net magnetization M0\nnet(µBper f.u.)\nscaled by the deformation proxy, M0\nnetΣM, versus Mn com-\nposition. (c) The experimentally measured saturation magne-\ntization Msat(µBper f.u.) scaled by the deformation proxy,\nMsatΣM, versus Mn composition. Colored data-points indicate\nthe individual calculations for each enumerated structure. Each\nblack X indicates the mean deformation value at each compo-\nsition. Over x=0.25, 0.5, and 0.75, the maximum standard\ndeviation in energy per atom values at each composition com-\nputed by VASP was 0.4 meV.\nexperimentally in in CoCr 2O4and MnCr 2O4[17, 20, 29–\n32] and theoretically using the classical theory of mag-\nnetic ground-states [48, 49].\nIn addition to ΣM, Figure 9(b) shows the trend of the\nproduct of the computed net magnetization and the de-\nformation proxy M0\nnetΣMversus composition. M0\nnetis\nthe net magnetization of the simulation cell output by\nVASP. These M0\nnetvalues are in units of Bohr magnetons\n(µB), normalized by the formula unit (f.u.). In this case,\nthere is a clear downward trend, which would be ex-\npected for the collinear configuration that we chose to\ncompute. M0\nnetΣMdecreases with x, which is opposite\nto the trend that was observed from the experimental\nmeasurements.\nCompared to the DFT derived M0\nnetΣM, Figure 9(c)\nprovides a plot of the experimentally measured Msat\nscaled by ΣMat each composition. These Msatvalues are\nalso reported in Bohr magnetons, normalized by the for-\nmula unit. The upward trend with composition is due to\nthe dominant behavior of the experimentally measured\nMsatvalues, which increase with Mn composition.\nFrom the results of the computed ΣMandM0\nnetΣM\nversus Mn composition, it is clear that there is a large\ndisagreement with the experimentally observed behav-ior. This is possibly due to the fact that ΣMsimply quan-\ntifies the degree of magnetostructural coupling in a ma-\nterial, without treating the lattice and spin contributions\nto∆Sof the phase transition explicitly. The opposite\ntrend of M0\nnetwith the experimental saturation magneti-\nzation, Msat, is stark, but can be described by the incon-\nsistencies between MsatandM0\nnet. The two are strictly\ncomparable only at zero temperature. Even then, M0\nnet\nis calculated under zero applied field, where Msatis not,\nby definition. The saturation magnetization and effec-\ntive moment µeffare inherently temperature-dependent\nquantities. For example, the µeffis derived from the high\ntemperature paramagnetic decay of the susceptibility ac-\ncording to the Curie-Weiss law. For this reason, Monte\nCarlo methods or a mean-field description is necessary in\norder to connect the DFT derived parameters to the ex-\nperimentally measured thermodynamic quantities such\nasMsatandµeff. In addition, it has been shown that M0\nnet\ndoesn’t correlate well with ∆SMfor a nonzero applied\nmagnetic field [3].\nIn future studies of this spinel system, thermodynamic\nquantities versus temperature will be computed using a\nspin-lattice coupling Hamiltonian with parameters de-\nrived from density functional theory calculations. This\nmodel will allow for the quantification of the change in\nentropy due to structural, magnetic, and their coupling\nusing Monte Carlo methods that directly quantify spin\nand phonon contributions to ∆SM.\nIII. CONCLUSION\nWe have studied the solid solution Co 1−xMnxCr2O4as\na candidate for magnetocaloric applications by synthesiz-\ning using standard solid-state synthesis. Synchrotron X-\nray diffraction measurements revealed a complete solid\nsolution of spinel samples. Magnetic measurements be-\ntween 2 K and 390 K show the spiral spin-state at low\ntemperatures transitioning to collinear ferrimagnetism\nat moderate temperatures then to paramagnetism at\nhigh temperatures, and the transition temperatures for\neach phase decrease monotonically as Mn content is in-\ncreased. The maximum magnetic entropy change is also\nfound to increase monotonically, from −1.67 J kg−1K−1\nfor the Co end member to −5.63 J kg−1K−1for the Mn\nend member (for a field change of 0 T to 5 T). Overall,\nthe tunability of this series and robust peak ∆SMvalues\nin the range of 40 K to 75 K make this series attractive for\ncascaded hydrogen liquefaction systems relying on active\nmagnetic refrigeration.\nThe effect of variation in Mn/Co composition on max-\nimum magnetic entropy difference, ∆SM, cannot be ex-\nplained by DFT-computed magnetic deformation proxy\nvalues, ΣM[3], but are more closely related to the trends\ninMsatacross composition. This is notthe case for\nM0\nnetfrom DFT, therefore, we suggest that there are cru-\ncial thermodynamic mechanisms that underlie the wide\nrange of ∆SMvalues across the series. We argue that9\nthis finite-temperature behavior cannot be resolved at\nthe level of DFT alone [50], but requires finite tempera-\nture modeling approaches, such as Monte Carlo, or even\nWeiss mean-field models.\nIV. ACKNOWLEDGEMENTS\nThis work and facilities employed here were supported\nby the National Science Foundation through the MR-\nSEC Program NSF DMR 1720256 (IRG-1). The UCSB\nMRSEC is a member of the Materials Research Facil-\nities Network (www.mrfn.org). Use of the Advanced\nPhoton Source at Argonne National Laboratory was sup-\nported by the U. S. Department of Energy, Office of Sci-\nence, Office of Basic Energy Sciences, under Contract No.\nDE-AC02-06CH11357. G.M. acknowledges support from\nthe Department of Energy Computational Science Gradu-\nate Fellowship (DOE CSGF) under grant DE-SC0020347.\nM.K.H acknowledges support by the U.S. Department of\nEnergy, Office of Science, Office of Basic Energy Sciences,\nMaterials Sciences and Engineering Division under Con-\ntract No. DE-AC02-05-CH11231 (Materials Project pro-\ngram KC23MP).\nV. APPENDIX\nA. Curie-Weiss Law for a Ferrimagnet\nThe Weiss molecular field theory for magnetism [51]\nis based on a mean-field approximation, and thereforeneglects fluctuations that influence behavior below and\nnear to the critical temperature. However, Curie-Weiss\ntheory is useful for studying the behavior of magnets at\ntemperatures above their paramagnetic transition tem-\nperature. This theory was originally formulated for\nthe ferromagnetic to paramagnetic order-disorder phase\ntransition, however, it is possible to generalize this theory\nto the study of ferrimagnetism and antiferromagnetism,\nby treating the magnetic configuration as an antiparallel\ncoupled arrangement of parallel (ferromagnetic) lattices.\nThis first section follows the reasoning presented in the\ntext of Kittel [44].\nFor the sake of clarity, we will start from the system of\nequations for the Weiss molecular field coupling between\nthe magnetization of A and B sublattices in a ferrimag-\nnet, assuming a negative - AFM exchange between A and\nB sites,\nMA=CA\nT(B0−λAAMA−λABMB)\nMB=CB\nT(B0−λABMA−λBBMB) (1)\nwhere MA&MBandCA&CBare the net magneti-\nzation and Curie-Weiss constants of the A and B sublat-\ntices, respectively. B0represents the applied field, and\nλAA,λBB, &λABare the Weiss field constants. These\nWeiss field constants ( λ’s) capture the net exchange in-\nteraction within and between the two sublattices. The\nlatter of which is symmetric under both directions of the\nexchange (A →B and vice versa). Equations 1 can be\nstated in matrix-vector form as Equation 2.\n(T·I+W)\"\nMA\nMB#\n= \nT\"\n1 0\n0 1#\n+\"\nλAACAλABCA\nλABCBλBBCB#!\"\nMA\nMB#\n=B0\"\nCA\nCB#\n(2)\nBy inspecting Equation 2, we see that for B0= 0, a\nnonzero solution for MA&MBexists only if [44]\ndet{T·I+W}= 0 (3)\nTherefore, the critical temperature of the system will cor-\nrespond to the opposite sign of an eigenvalue of W,\nspecifically TC= max {ηi[−W]}, where ηi[A]are the\neigenvalues of a matrix A. From Equation 2, we arrive\nat a generalized Curie-Weiss law, for T > T C\nχ=∂(Pn\ni=1Mi)\n∂B0=1T[T·I+W]−1C (4)\nFor the two lattice case ( n= 2),Pn\ni=1Mi=MA+MB,\n1T= [1 1] , andCT= [CACB].1. Uncertainty quantification\nIn order to fit Equation 4 to experimental data,\nwe used scipy.optimize.curve_fit [52]. This op-\ntimization routine utilizes the Jacobian of the objec-\ntive function in the minimization procedure, as well as\nfor calculating the propagation of uncertainty. If one\ndoes not specify the jacobian argument explicitly, then\nscipy.optimize.curve_fit computes derivatives using\nfinite differences (FD). If we use the default FD scheme,\nthe uncertainty values of the parameters are estimated to\nbe at least 104. However, we find improvement if we sup-\nply the analytical derivative presented below in Equation10\n5,\n∂\n∂φχ=−1TW−1\nT∂W\n∂φW−1\nTC+1TW−1\nT∂C\n∂φ,\nwhere WT= [T·I+W]. (5)\nIn this expression, φrepresents a parameter of the model\nfunction, φ∈ {CA, CB, λAA, λBB, λAB}. If we provide\nthe analytical Jacobian, all uncertainties fell below 1.\nThis improvement is likely due to numerical artifacts that\narise from using the FD scheme near to the singularity at\nthe critical temperature.\n2. Simplified exchange\nWe can simplify the Equation 2 by setting λAA=\nλBB= 0, and letting λ=λAB. This simplification yields\nthe following result for susceptibility above the critical\ntemperature TC[44]:\nχ=∂(MA+MB)\n∂B0=(CA+CB)T−2λCACB\nT2−T2\nC(6)\nAnother consequence of this solution is that TC=\n��√CACB[44].\n3. High temperature limit\nContinuing from Equation 6, we can obtain the following\nfactorization\nχ=∂(MA+MB)\n∂B0\n=CA+CB\nT+TC·T−2λCACB\nCA+CB\nT−TC(7)\n=CA+CB\nT+TC·T−ρ·TC\nT−TC(8)\nwhere ρis the ratio between the geometric and arith-\nmetic mean between CAandCB,\nρ=C1/2\nAC1/2\nB\n1\n2(CA+CB). (9)\nTherefore, ρ= 1ifCA=CB. If we examine the high-\ntemperature limit in which T >> T C, we can make the\nfollowing simplification\nlim\nT→∞T−ρ·TC\nT−TC= 1 (10)\nUnder approximation, we arrive at a more compact form\nof the CW law for a ferrimagnet\nχ≈CA+CB\nT+TC, (11)which is applicable for temperatures significantly larger\nthan the critical temperature. An illustration of this is\nshown in Figure 10.\nFIG. 10. Comparison between the Curie-Weiss (CW) expres-\nsions for χ, where the full expression (Equation 6) is plotted\nagainst the approximate high-temperature CW law (Equation\n11). The parameters, λ,CA, and CBare arbitrary dimension-\nless constants that are chosen to justify this approximation.\n4. Relation to effective moment\nThe effective moment can be expressed in terms of the\nCurie-Weiss law using Weiss mean-field theory [44]\nχ=Np2µ2\nb\n3kB(T−TC)=Nµ eff2\n3kB(T−TC)=C\nT−TC\nµeff2=3kB\nNC (12)\nNis the number of atoms per unit volume [44].\nCombining Equations 11 and 12, we arrive at the fol-\nlowing relationship between the “net” effective moment\nand the effective moment for each sublattice (A & B):\nC≈CA+CB=Naµ2\na\n3kB+Nbµ2\nb\n3kB\nµeff2≈Na\nNµ2\na+Nb\nNµ2\nb (13)\nThis relationship allows us to approximate, at T >\nTC, the effective magnetic moment of a sample from a\nweighted sum of the µeffof the constituent magnetic sub-\nlattices.11\n5. 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Bright, et al. , Nature methods 17, 261\n(2020)." }, { "title": "2012.10493v1.Rare_earth_free_ferrimagnetic_Mn4N_sub_20_nm_thin_films_as_high_temperature_spintronic_material.pdf", "content": "Rare -earth -free ferrimagnetic Mn 4N sub-20 nm thin films as high-temperature spintronic \nmaterial \nW. Zhou1,a), C.T. Ma1, T.Q. Hartnett2, P.V. Balachandran2, S.J. Poon1,a) \n \n1 Physics D epartment, University of Virginia , Charlottesville , Virginia,22903 , USA \n2 Material Science Engineering , University of Virginia, Charlottesville, Virginia,22903, \nUSA \n \nFerrimagnetic alloy thin films that exhibit perpendicular (out -of-plane) magnetic anisotropy (PMA) with \nlow saturation magnetization, such as GdCo and Mn 4N, were predicted to be favorable for hosting small \nNéel skyrmions for room temperature applications. Due to the exponential decay of interfacial \nDzyaloshinskii -Moriya interaction ( DMI ) and the limited range of spin -orbit -torques, which can be used \nto drive skyrmion motion , the thickness of the ferrimagnetic layer has to be small, preferably under 20 \nnm. W hile there are examples of sub -20 nm, rare earth -transition metal (RE-TM), ferrimagnetic thin \nfilms fabricated by sputter deposition , to date rare-earth -free sub-20 nm Mn 4N films with PMA have \nonly been reported to be achieved by molecular beam epitaxy, which is not suitable for massive \nproduction. Here we report the successful thermal growth of sub -20 nm Mn 4N films with PMA at 400 -\n450 °C substrate temperatures on MgO substrates by reactive sputtering. The Mn 4N films were achieved \nby reducing the surface roughness of MgO substrate through a high -temperature vacuum annealing \nprocess . The optimal films showed low saturation magnetization (M s = 43 emu/cc), low magnetic \nanisotropy energy (0.7 Merg/cc), and a remanent magnetization to saturation magnetization ratio \n(Mr/Ms) near 1 at room temperature . Preliminary ab -initio density functional theory (DFT) calculations \nhave confirmed the ferrimagnetic ground state of Mn 4N grown on MgO. The magnetic properties, along \nwith the high thermal stability of Mn 4N thin films in comparison with RE -TM thin films , provide the \nplatform for future studies of practical s kyrmion -based spintronic materials. \n \n \n \n \n \n \na) Authors to whom correspondence should be addressed : wz8he@virginia.edu and \nsjp9x@virginia.edu \n 2 \n As industry rapidly transition s to using big data in their operations and decision making, there is an \nurgent need to develop technologies that accommodate the increasing requirements of high -density data \nstorage 1. One promising candidate that has received increasing attention is using magnetic skyrmions as \ninformation carriers and manipulating them with current by spin -orbit torque (SOT) for logic or memory \noperations 2,3 (e.g. racetrack memories). Magnetic skyrmions are swirling spin configurations of \nneighbor atoms in a magnetic mate rial with topologic protection. The size of the skyrmion could be as \nsmall as a few nanometers 4 and manipulation of skyrmions could be energy efficient 5,6 in high -density \ndata storage applications 7,8. To date, most RT skyrmions have been discovered in f erromagnetic (FM) \nmultilayer stacks 7-10. The use of ferromagnets as skyrmions have certain shortcomings. For instance, \nferromagnets suffer from large stray fields and large saturation magnetization (M s), which makes the \nregion of parameter space for small skyrmion less accessible , especially at room temperature (RT) 11. \nUnlike ferromagnets, the ferrimagnetic counterparts have small stray fields and the low Ms, which \nmakes them suitable for hosting small RT skyrmions 11. This makes ferrimagnet a promising candidate \nmaterial for high -density data storage applications. One prototypical example is the amorphous GdCo \nferrimagnetic thin film. Small (10 –30 nm) room temperature skyrmions have been reported in Pt/GdCo \n(6 nm) /TaO x 12. While the amorphous GdCo ferrimagnet shows promising characteristics relative to the \nferromagnets, it suffers from poor thermal stability. It has been shown that in amorphous GdCo, PMA is \nlost after annealing at 300 -400 °C 13, the temperature range applied in complementary metal –oxide –\nsemiconductor ( CMOS ) fabrication 14. Since Néel skyrmions can only exist in a heterostructure with \nPMA, the poor thermal stability of RE -TM films will have a deleterious effect in both the fabrication \nand performance of amorphous RE -TM in devices that leverage skyrmions for storage technology. One \nof the potential solutions to this problem is to explore crystalline ferrimagnets with PMA synthesized by \ndeposition at high -temperature near 400 °C, thus ensuring compatibility with conventional CMOS \nprocessing . 3 \n In addition to the intrinsic effects that we have discussed thus far, film thickness is an important \nextrinsic effect that impacts the overall device performance. The importance of film thickness applies to \nboth ferro - and ferrimagnets. It is now well -established that the interfacial Dzyaloshinskii -Moriya \ninteraction (DMI ), which stabilizes the magnetic skyrmions in multilayers and heterostructures, decays \nexponentially with increasing film thickness 15-17. Further, the SOT scales inversely with the thickness 18. \nThis necessitates the growth of thin film magnets with sub -20 nm thickness, for the realization of small \nskyrmions in practical applications 12,19,20. \nAnti-perovskite Mn 4N has been known as a crystalline, rare-earth -free ferrimagnetic material. In \nMn 4N, the Mn -atoms have a face -centered cubic structure with one N -atom at the body center. A \nschematic of the unit cell is shown in Figure 1. The Mn -atoms at the corner and the face center have \ninequivalent magne tic moments and are ferrimagnetically coupled 21. Although the easy axis of bulk \nMn 4N is along the [111] direction, PMA is repeatedly and reproducibly observed in crystalline Mn 4N \nfilms 22-28. As a result, ferrimagnetic Mn 4N thin films have also attracted increasing interest in \nspintronics applications. Compared to the amorphous RE -TM ferrimagnetic alloys, the Mn 4N system has \nbetter thermal stability for two main reasons. First, in most thin film studies, it has been shown that the \nanti-perovskite crystal s tructure is formed at 400 -450 °C. No loss of PMA is reported in thin films after \nannealing, which is an encouraging outcome. Second, there is no known evidence for any structural \nphase transformation on cooling to room temperature. Thus, we interpret that the anti -perovskite crystal \nstructure is tolerant of high -temperature device fabrication processes (unlike the GdCo amorphous \nalloys). As noted earlier, the emergence of PMA is an important magnetic property for its use in \nspintronic applications. One of t he plausible reasons for the PMA in Mn 4N thin films could be attributed \nto the deviation of the out -of-plane lattice constant, c, to the in -plane lattice constant, a, (c/a) ratio from \n1, 22-28 due to the in -plane epitaxial strain. A recent study also showe d that the anisotropy energy is 4 \n correlated to the c/a ratio 27. We note that further studies are warranted to understand the interplay of \nmagnetic and strain effects on the PMA and this is beyond the scope of this paper. \nSeveral groups have grown crystalline Mn 4N epitaxial films (30 -100 nm) on MgO, SrTiO 3, or \nLaAlO 3 substrates by magnetron sputtering 22,23, molecular beam epitaxy (MBE) 24-27, and pulsed laser \ndeposition (PLD) 28, and they reported similar c/a ratios of ~0.99. The reported uniaxial magnetic \nanisotropy constant (K u) and M s of those Mn 4N films were about 0.5 –1 Merg/cc and 50 –100 emu/cc, \nrespectively 22-28, comparable with the data observed for amorphous GdCo thin films (K u ~ 0.25 Merg/cc \nand M s ~ 50 emu/cc) 12. \nTo date, however, only a few groups have grown sub -20 nm Mn 4N thin film s with good magnetic \nhysteresis (M(H)) loops, which has remanent magnetization M r to M s ratio larger than 0.5 (M r/Ms > 0.5), \nby MBE 26,27. For those reported Mn 4N film by sputtering , the thickness es were 30 n m 22 to 100 nm 23, \nsome of them even up to hundr eds nanometers 29,30. Similar results on sputter deposited sub -20 nm \nMn 4N thin film have not been reported. C ompared to MBE, sputter deposition is a more widely adopted \nmethod in CMOS technology . In this work, we have epitaxially grown sub-20 nm Mn 4N thin film s on \nMgO (001) substrate with PMA by reactive sputtering . The effect of MgO substrate morphology on the \nquality of Mn 4N film is studied. We then compare the magnetic properties of the Mn 4N films on MgO \nsubstrate from experiment and first principles -based density functional theory (DFT) calculations. The \nmain contribution of this paper lies in the demonstration of the growth of high magnetic quality sub -20 \nnm crystalline Mn 4N films on MgO subs trate using reactive sputtering that is more promising for scale -\nup production. \n 5 \n \nFigure 1. Mn 4N crystal structure \n \nMn 4N thin films with nominal thickness es of ~15 nm and 10 nm were deposited on MgO(001) \n5x5x0.5 mm substrate s by reactive rf -sputtering at 400 °C and 450 °C substrate temperature s and a base \npressure of 7x10-8 Torr. The flow rates of Ar and N 2 gases were controlled by a mass flow meter and we \nmaintained a flow rate Ar:N 2 ratio of 93:7. A 3 nm Pt capping layer was deposited on Mn 4N layer at \nroom temperature to prevent oxidization. Before loading the MgO (001) substrate into the vacuum \nchamber, it was wet -cleaned with 2% diluted Hellmanex III alkaline detergent, acetone, and \nisopropanol, sealed in a vacuum tube with pressure 30 mTorr . The MgO substrate was baked at high \ntemperatures 1,000 °C or 1,100 °C for 4 hours . Then, the substrates were annealed inside the chamber at \n500 0C for one hour to remove the surface contaminations. The Mn target was pre -sputtered with A r gas \nfor 20 minutes to remove the surface oxide. Surface roughness of the MgO substrate was measured \nusing atomic force microscopy (AFM). The deposition rate was measured by X -ray reflect ometry \n(XRR). The film compositions were determined with X -ray phot oelectron spectroscopy (XPS). The film \nthickness was verified by XRR with an XRR simulation/calculation program (Rigaku, GXRR) 31. \n The epitaxial growth of Mn 4N crystal layer was demonstrated using X -ray diffraction (XRD) with \nCu-Kα radiation. The magnetic properties of the samples were measured at room temperature with \nvibrating sample magnetometry (VSM). The diamagnetic component of the substrate was deduced from \nthe slope of raw M -H curves at large H region and subtracted from the raw data. K u was calculated from \neffective anisotropy K with Eq. (1) and (2) \n6 \n 𝐾𝑢=𝐾+2𝜋𝑀𝑠2 (1) \n𝐾=∫(𝑀𝑒𝑎𝑠𝑦(𝐻)−𝑀ℎ𝑎𝑟𝑑(H))𝑑𝐻∞\n0 (2) \nWhere M easy and M hard are the magnetization in out -of-plane applied field and in -plane applied field, \nrespectively. \nAb-initio electronic structure calculations were carried out in the Density Functional Theory (DFT) \nframework using the plane -wave pseudopotential Quantum ESPRESSO code 32, 33. Core and valence \nelectrons were treated using the ultrasoft pseudopotential method 34,35. The exchange -correlation \nfunctionals were described using the Perdew -Burke -Ernzerhof parameterization of the genera lized \ngradient approximation modified for solids (PBEsol) 36. The plane -wave cutoff energy was set to 60 Ry \nand a Γ -centered Monkhorst -Pack k-point mesh of 12x12x12 was used to sample the Brillouin Zone 37. \nLattice parameters were fixed to experimentally m easured value ( 𝑐/𝑎 = 0.987) and Ms was calculated \nusing the formula, 𝑀𝑠=|𝜇𝑡𝑜𝑡𝑎𝑙|\n𝑉, where |𝜇𝑡𝑜𝑡𝑎𝑙| is the absolute value of the total magnetization from DFT \ncalculations given in Bohr magnetons and 𝑉 is the unit cell volume 38. Self-consistent spin -polarized \ncalculations were performed for collinear spin structures with no spin -orbit coupling term in the \nHamiltonian. The Mn -atoms (Mn I) located in the cell corners with coordinates (0, 0, 0) were \nferromagnetically coupled to Mn -atoms (Mn IIa) located at (0.5, 0.5, 0). Both Mn I - and Mn IIa-atoms \nwere anti -ferromagnetically coupled to the Mn -atoms (Mn IIb) in the face centers with coordinates (0, \n0.5, 0.5) and (0.5, 0, 0.5). Since the total atomic magnetic moments at all three Mn -sites do not cancel \neach other out (Mn I = 3.47 μ B, Mn IIa = 0.75 μ B and Mn IIb = -2.36 μ B,), the ground state is a \nferrimagnet. \nTable I lists the five samples (S1 -S5) that were examined in this study . We begin by investigating \ndifferent substrate annealing temperatures to understand the impact of the MgO surface on epitaxial \ngrowth of Mn 4N. The S1 -S3 were deposited at 400 °C, and S4 and S5 were deposited at a higher 7 \n temperatu re of 450 °C. Film thickness was determined by XRR measurement. In the case of S2 -S5, the \nMgO substrates were pre -baked before loading into the sample deposition chamber. \nTABLE I. List of Mn 4N thin films on MgO substrates \nSample No. MgO a nnealing \nTemperature MgO a nnealing \nTime Film t hickness Deposition temperature \nS1 No annealing (RT) -- 16.6 ± 0.6 nm 400 °C \nS2 1000 °C 4 hours 16.8 ± 0.5nm 400 °C \nS3 1100 °C 4 hours 16.8 ± 0.5 nm 400 °C \nS4 1100 °C 4 hours 16.8 ± 0.5 nm 450 °C \nS5 1100 °C 4 hours 11.5 ± 0.6 nm 450 °C \n \nThe film thicknesses measured by the XRR technique indicated that the thicknesses of all five thin \nfilm samples were less than 20 nm (see Table I). Analysis of XRR results is shown in Supplementary \nFigure 1. S1 -S4 had a thickness of 16.8 ± 0.5nm, whereas S5 had a thickness of 11.5 ± 0.6 nm. Figure \n2(a) shows the normalized M -H curves with out -of-plane applied magnetic field for 1 6.8 nm Mn 4N film \non different heat -treated MgO substrates (S1 -S3). The M -H curves show a strong dependence on the \nannealing temperature. All out -of-plane hysteresis loops w ere open, indicating PMA. The substrate \nannealing temperature of S3 ( 1100 °C) is higher than that of S2 (1000 °C), whereas S1 is unannealed. \nThe squareness of the M(H) hysteresis loop improves from S1 to S2 to S3. Figure 2(b) shows the \ndependence of the substrate annealing temperature on M r/Ms. The M r/Ms ratio increased from 0.47 to \n0.72 as the annealing temperature increased from RT to 1100 °C . The increment of M r/Ms is attributed \nto the improved epitaxial growth of the thin film. High -temperature annealing not only decomposes the \nMg(OH) 2 and MgCO 3 contaminants , it also reconstructs the surface of the MgO substrate 39,40. During \nthis process, the substrate forms an atomically smooth surface 39,40, which is beneficial for epitaxial \ngrowth 41. The difference between S 1 and S3 is mainly attributed to the different substrate annealing 8 \n temperatures. Thin films annealed at 1100 °C produced a smoother surface than 1000 °C. The surface \nmorphology as characterized by the atomic force microscopy revealed an average root -mean -square \n(rms) roughness of 0.592 nm, 0.308 nm, 0.278 nm for the as -received, 1000 °C annealed and 1100 °C \nannealed thin films, respectively (See Supplement ary Figure 2). \nTo explore further improvement, we increased the deposition temperature to 450 °C, while retaining \nthe substrate annealing temperature and time at 1100 °C and 4 hours . It has been shown that the \nsquareness of the Mn 4N film can be improved by increasing deposition temperature 29 .We found that \nthe Mr/Ms ratio of these samples , S4 and S5 , increased further to near 1 , as shown in Figure 2(c). The \ndifference between S4 and S5 is in the film thickness: the S4 and S5 thin films were 16.8 ± 0.5 nm nm \nand 11.5 ± 0.6 nm thick, respectively (see Table I). S5, t he thinnest film, kept a high Mr/Ms ratio ~0.8, \nwhich is an encouraging outcome. \nFigure 2(d) shows the out -of-plane and in -plane M -H curves for the S4 film. The M s and K u for the \nS4 film is 43 ± 1 emu/cc and 0. 70 Merg/cc, respectively . All five samples had similar M s (40-60 \nemu/cc). Although the M s of the se films are smaller than the reported Ms (~100 emu/cc ) in thick er films \n(30-100 nm) 22-25,28 , they are comparable with that of the MBE -grown sub-20 nm Mn 4N film on MgO, \n50-80 emu/cc 26,27. The low er Ms of sub -20 nm Mn 4N films may be due to surface oxidation 25. \n 9 \n \n \nFigure 2 (a) Normalized out -of-plane M(H) loops of 16.8 nm Mn 4N film s deposited at 400 °C on MgO \nsubstrates annealed at different temperatures ; S1 (red) was unannealed, S2 (green) was annealed at 1000 \n°C, and S3 (blue) was annealed at 1100 °C, (b) M r/Ms ratio of 16.8 nm Mn 4N film (S1-S3) deposited at \n400 °C on MgO substrates, which have been annealed at different temperatures, (c) Normalized out -of-\nplane M(H) loops of 16.8 nm (S4, cyan ) and 11.5 nm (S5 , brown ) thick Mn 4N film s deposited at 450 °C. \non MgO substrates, which have been annealed at 1100 °C , (d) out-of-plane and in -plane M(H) loops of \nthe S 4 Mn 4N film. \n \nFigure 3 shows the out -of-plane 2θ -θ XRD profile (a) and φ scan (b) of S3. Besides the MgO \nsubstrate peaks, only Mn 4N (002) peak is observed in the 2θ -θ XRD profile, which indicates the Mn 4N \n(00l) orientation is parallel to the MgO (00l). In the XRD 360o φ scan, both Mn 4N and MgO shows four \npeaks with 90o interval, which corresponding to (202), (022), (-202), and (0 -22). The overlap of Mn 4N \npeaks and MgO peaks in φ -scan confirm their epitaxial relat ionship to be MgO (001) [100]// Mn 4N \n(001) [100]. The out -of-plane lattice constant c and in -plane lattice constant a deduced from the (002) \nand (202) diffraction peaks are 0.386 nm and 0.391 nm , respectively. Therefore, the c/a ratio was \ncalculated to be 0.987, which is close to the value reported earlier 22-28. \n10 \n \nFigure 3 . (a) 2θ- θ profile of S3 and MgO substrate annealed at 1100 °C. (b)φ scan of Mn 4N (101) and \nMgO (101) peaks. \n \nBased on the experimental lattice constant, DFT predicts an Ms value of 153 emu/cc, which is larger \nthan our experiment results. However, DFT is a zero -temperature electronic structure calculation. Li et \nal have shown that 𝑀𝑠 of the bulk Mn 4N would decrease as a function of increasing measurement \ntemperature 40. DFT also does not consider the possible effects of the mixing layer at the interface as \nwell as defects . Future investigation will address the temperature dependence of saturation \nmagnetization and magnetic anisotropy energy. \nWe have grown sub-20 nm ferrimagnetic Mn 4N epitaxial thin film s with high M r/Ms ratios at 400 -\n450 °C substrate temperatures on MgO substrates by reactive sputtering . The quality of the epitaxial \ngrowth was optimized by ex-situ pre-annealing of MgO substrates at temperatures as high as 1100 °C. \nThe annealing was found to reduce the surface roughness of the MgO substrate s at the atomic level, \nwhich was found to improve the quality of the out -of-plane magnetization hysteresis loops. The present \nresults established Mn 4N thin film as a thermally stable ferrimagnet for further investigation as \npromising skyrmion -based spintronic material. \n \n \n11 \n Suppl ementary material \nSee su pplementary material for the XRR results of the film and the AFM results of the MgO \nsubstrate s surface. \nAcknowledgment \nThis work was supported by the DARPA Topological Excitations in Electronics (TEE) program \n(grant D18AP00009). The content of the information does not necessarily reflect the position or the \npolicy of the Government, and no official endorsement should be inferred. Approved for public release; \ndistribution is unlimited. \nData Availability Statements \nThe data that support the findings o f this study are available from the corresponding author upon \nreasonable request. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 12 \n Reference: \n \n1 A. Fert, F . N. V. 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XRR curve of the S3(a) and S5(b) . the blue line is the raw data , and the red line is the fitting line. \n \nNotes on Supplementary Figure 1. \n \n15 \n X-ray reflectivity (XRR) was performed on the MgO /Mn 4N/Pt samples to confirm that the Mn 4N layer \nis thinner than 20 nm. The blue line is the raw data collected with a Cu -Kα source (λ = 1.5406 Å) and \nthe red line is a fitting line produced by the software Rigaku -GXRR31. Supplementary Figure 1(a) shows \nthe XRR curve and fitting line of S3. The fit gave a Mn 4N film thickness of 16.8 ± 0.5nm (nominal \nthickness was 15 nm) and a Pt capping layer thickness of 3.4 ± 0.6 nm nm. Supplementary Figure 1(b) \nshows the XRR curve and fitting line of S5. The fit gave a Mn 4N film thickness of 11.5 ± 0.6 nm \n(nominal thickne ss was 10 nm) and a Pt capping layer thickness of 4.2 ± 0.5 nm . The discrepancy \nbetween the XRR fitting model and the measured data at small angle (< 0.8) is because these angles are \nsmaller than the critical angle (approx. 0.8 degree) for total reflecti on. \n \nSupplementary Figure 2. AFM surface morphology of pure MgO (100) substrate surface (a) as received, and after annealing \nfor 4 hours at (b) 1000 °C, (c) 1100 °C. The lower panels (d-f) show the corresponding line scans of the AFM topographies. \n \nNotes on Supplementary Figure 2. \n \nAFM was performed on the MgO substrates, which have been annealed at different temperatures, to \ncheck the surface morphology of the MgO (001) substrates. Supplementary Figure 2 (a), (b), (c) shows \nthe AFM images of as -received MgO substrate, annealed at 1000 °C, and annealed at 1100 °C, \nrespectively. Supplementary Figure 2 (d), (e), (f) shows the corresponding line scans of the AFM \ntopographies. It is clear that as -received MgO has the roughest surface. However, from the image s, it is \nhard to tell the difference between the 1000 °C annealed MgO substrate surface (Supplementary Figure \n2(b), (e)) and the 1100 °C annealed MgO substrate surface (Supplementary Figure 2(c), (f)). Gwyddion \n43 was employed to analyze the roughness, whi ch gave the 1000 °C annealed MgO substrate surface rms \nroughness of 0.308 nm and the 1100 °C annealed MgO substrate surface rms roughness of 0.278 nm. \nFor comparison, the surface rms roughness of the as -received substrate was 0.592 nm. \n \n \n \n \n \n \n \n" }, { "title": "0711.1051v1.Low_energy_structure_of_the_intertwining_double_chain_ferrimagnets_A_3_Cu_3__PO_4___4___A_Ca_Sr_Pb_.pdf", "content": "arXiv:0711.1051v1 [cond-mat.str-el] 7 Nov 2007Low-energy structure of the intertwining double-chain fer rimagnets\nA3Cu3(PO4)4(A=Ca,Sr,Pb)†\nShoji Yamamoto and Jun Ohara\nDepartment of Physics, Hokkaido University, Sapporo 060-0 810, Japan\n(Dated: 11 March 2007)\nMotivated by the homometallic intertwining double-chain f errimagnets A3Cu3(PO4)4(A=\nCa,Sr,Pb), we investigate the low-energy structure of their model Hamiltonian H=/summationtext\nn[J1(Sn:1+\nSn:3) +J2(Sn+1:1+Sn−1:3)]·Sn:2, whereSn:lstands for the Cu2+ion spin labeled lin thenth\ntrimer unit, with particular emphasis on the range of bond al ternation 0 < J2/J1<1. Although\nthe spin-wave theory, whether up to O(S1) or up to O(S0), claims that there exists a flat band in\nthe excitation spectrum regardless of bond alternation, a p erturbational treatment as well as the\nexact diagonalization of the Hamiltonian reveals its weak b ut nonvanishing momentum dispersion\nunlessJ2=J1orJ2= 0. Quantum Monte Carlo calculations of the static structur e factor further\nconvince us of the low-lying excitation mechanism, elucida ting similarities and differences between\nthe present system and alternating-spin linear-chain ferr imagnets.\nPACS numbers: 75.10.Jm, 75.50.Gg, 75.40.Cx\nI. INTRODUCTION\nIt is a long-standing and still challenging theme in\nmaterials science to design molecular systems ordering\nferromagnetically.1The naivest idea of ferromagnetically\ncoupling nearest-neighbor magnetic centers leads to the\nhighest spin multiplicity but critically depends on some\nstructural parameters which are hard to handle chemi-\ncally. An alternative solution to highly magnetic ground\nstates consists of aligning molecular bricks so as to ob-\ntainanonzeroresultantspininthegroundstateandthen\ncoupling the chains again in a ferromagnetic fashion. A\nvariety of quasi-one-dimensional ferrimagnets were thus\nsynthesized and not a few of them have been attracting\ntheoretical as well as experimental interest.\nBimetallic chain compounds are early examples and\namong others is MnCu(pbaOH)(H2O)3(pbaOH =\n2-hydroxy-1 ,3-propylenebis(oxamato) = C 7H6N2O7),2\nwhich retains the long-range ferromagnetic order on\nthe scale of the crystal lattice. Replacing the Mn2+\nions by Fe2+, Co2+, and Ni2+ions, Kahn and co-\nworkers further synthesized a series of isomorphous\ncompounds,3which stimulated extensive chemical ex-\nplorations of heterometallic chain magnets4,5and sys-\nFIG. 1: Cu2+trimeric chains in A3Cu3(PO4)4. The strongly\ncoupled Cu2+trimer consists of a central square planar\nCu2+(1) ion (black circle) and two pyramidal Cu2+(2) ions\n(gray circles) bridged by oxygen ions (open circles).\n†Phys. Rev. B 76, 014409 (2007)tematic theoretical investigations of alternating-spin\nchains.6,7,8,9,10,11,12,13,14In an attempt to obtain sub-\nstantially larger couplings between neighboring mag-\nnetic centers and possibly attain transitions to three-\ndimensional order at higher temperatures, Caneschi et\nal.15made a distinct attempt to bring into interac-\ntion metal ions and stable organic radicals. The repre-\nsentative materials of general formula Mn(hfac)2NIT-R\n(hfac = hexafluoroacetylacetonate = C 5H2O2F6;\nNIT-R= nitronyl nitroxide radical = C 7H12N2O2-R\nwithR= CH 3,C2H5,C3H5,C6H5) indeed exhibit anti-\nferromagneticintrachaininteractionsrangingfrom200to\n330cm−1. The metal-radical hybrid strategy, combined\nwith fabrication of novel polyradicals,16yielded various\npolymerized heterospin chain compounds.17,18\nHomometallicferrimagnetismisalsorealizable19,20but\nits mechanism is often more subtle, essentially depend-\ning on the structural features of the system. Coron-\nadoet al.21,22pioneeringly synthesized chain-structured\ncompounds of such kind, M2(EDTA)(H 2O)4·2H2O\n(M= Ni,Co; EDTA = ethylenediamminetetraacetate =\nC10N2O8), whose ferrimagnetic behavior originates from\nthe alternating gfactors and is therefore faint. Ho-\nmometallic chain compounds of more pronouncedly fer-\nrimagnetic aspect23,24,25,26were not obtained until an-\nother decade had passed, where particular topologies\nwere elaborately imposed on the intrachain exchange\ninteractions. A series of compounds, M(R-py)2(N3)2\n(M= Cu,Mn;R-py = pyridinic ligand = C 5H4N-R\nwithR= Cl,CH3,···), consists of bond-polymerized ho-\nmometallic chains, where the neighboringmetal ion spins\nare bridged by versatile azido ligands and are coupled to\neach other ferromagnetically or antiferromagnetically.\nThe homometallic intertwining double-chain com-\npoundsA3Cu3(PO4)4(A= Ca,Sr,Pb),27,28,29which are\nillustrated in Fig. 1, aretopologicalferrimagnets30in the\nstrict sense. Their hybrid analogs Ca 3−xSrxCu3(PO4)4\n(0≤x≤3)28were also fabricated in an attempt to\ntune the antiferromagnetic bridges between the Cu(1)2\nand Cu(2) sites, labeled J1andJ2, and possibly ex-\nplore how paramagnetic spins grow into bulk ferrimag-\nnets. The magnetic centers without single ion anisotropy\nand the simple crystalline structure without any organic\nligand will contribute toward revealing intrinsic features\nof one-dimensional ferrimagnetic phenomena. Thus mo-\ntivated, various experiments have been performed on\nthese copper phosphates in recent years, including high-\nfield magnetization,31specific-heat,32inelastic neutron-\nscattering,33nuclear spin-lattice relaxation-time,34and\nelectron-spin-resonance35measurements.\nIt is therefore unfortunate that theoretical investiga-\ntions of this system still stay in their early stage.30,36,37\nIndeed there exists a field-theoretical study38deserving\nspecial mention, but the authors restricted their argu-\nment to the particular case of J1=J2taking a main\ninterest in realizing organic ferromagnetism. A recent\nnumerical diagonalization study39is also a fine guide\nto this system, but the authors still devoted themselves\nto clarifying the electronic correlation effect on unsatu-\nrated ferromagnetism rather than geometrically modify-\ning this unique bipartite lattice, starting from a model\nof the Hubbard type. An introduction of bond alterna-\ntionδ≡J2/J1/ne}ationslash= 1 to this system will not only con-\ntribute toward understanding the magnetic properties of\nA3Cu3(PO4)428,30,33,34but also illuminate the character-\nisticoftheuniform point δ= 1. We arethusled toreport\nthe whole excitation mechanism of homogeneous-spin\nintertwining double-chain ferrimagnets, employing both\nanalytical and numerical tools. According to the spin-\nwave theory, there exist three modes of elementary exci-\ntation, two of which exhibit parallel dispersion relations,\nwhile the rest of which is of no dispersion, regardless\nof bond alternation. However, the exact-diagonalization\nand perturbational calculations disprove the spin-wave\nscenario that the low-lying excitation spectrum remains\nqualitatively unchanged with varying δ. Indeed there\nexist local excitations which are rigorously immobile at\nδ= 1, but they can be itinerant with δmoving away from\nunity. Except for the two particular points δ= 1 and\nδ= 0, corresponding to a plaquette chain and decoupled\ntrimers, respectively, there is no flat band in the exci-\ntation spectrum of the homogeneous-spin trimeric chain .\nWefurtherinquireintothermalexcitationsbasedonsuch\nan energy spectrum. Calculating the static structure fac-\ntor as a function of temperature for an alternating-spin\nlinear-chainferrimagnetaswellasforthe presentsystem,\nwe show what are the universal ferrimagnetic features\nand how they vary with decreasing δ.\nII. PLAQUETTE CHAINS\nThe Hamiltonian of our interest is represented as\nH ≡ H 1+H2=N/summationdisplay\nn=1/bracketleftbig\nJ1(Sn:1+Sn:3)·Sn:2\n+J2(Sn+1:1+Sn−1:3)·Sn:2/bracketrightbig\n, (1)whereSn:lsymbolizes the Cu2+ion spin (S=1\n2) labeled\nlin thenth trimer unit (see Fig. 1) and the intratrimer\n(J1) and intertrimer ( J2) exchange interactions, denoted\nbyH1andH2, respectively, are defined as 0 ≤J2≤J1.\nFirst we take a look at the particular point of J2/J1≡\nδ= 1, where the model reads a plaquette chain, bearing\nsome analogy with a linear chain of alternating spins 1\nand1\n2.\nIntroducing bosonic operators through the Holstein-\nPrimakoff transformation\nS+\nn:1=/radicalBig\n2S−a†\nn:1an:1an:1, Sz\nn:1=S−a†\nn:1an:1,\nS+\nn:2=a†\nn:2/radicalBig\n2S−a†\nn:2an:2, Sz\nn:2=a†\nn:2an:2−S,\nS+\nn:3=/radicalBig\n2S−a†\nn:3an:3an:3, Sz\nn:3=S−a†\nn:3an:3,\n(2)\ndefining their Fourier transforms as\nak:l=1√\nN/summationdisplay\nnei(−1)lk(n+l/2−1)an:l,(3)\nwith the lattice constant set equal to unity, and further\nprocessing them via the Bogoliubov transformation\nα†\nk:−=ψ−1(k)a†\nk:1+ψ−2(k)ak:2+ψ−3(k)a†\nk:3,\nα†\nk:0=ψ01(k)a†\nk:1+ψ02(k)ak:2+ψ03(k)a†\nk:3,\nα†\nk:+=ψ+1(k)ak:1+ψ+2(k)a†\nk:2+ψ+3(k)ak:3,(4)\nwe reach a spin-wave Hamiltonian,\nH=Eg+/summationdisplay\nλ=∓,0ωλ(k)α†\nk:λαk:λ, (5)\nFIG. 2: Dispersion relations of the elementary excitations\nin the spin-1\n2plaquette chain, two (circles and diamonds) of\nwhich reduce the ground-state magnetization and are thus\nof ferromagnetic character, while the rest (squares) of whi ch\nenhances the ground-state magnetization and is thus of anti -\nferromagnetic character. The exact-diagonalization resu lts at\nN= 4, 6, and 8 are presented by symbols of small, middle,\nand large sizes, respectively, whereas the up-to- O(S1) linear\n(LSW) and up-to- O(S0) interacting (ISW) spin-wave calcu-\nlations are given by dotted and solid lines, respectively.3\nFIG. 3: (Color online) Dispersion relations of the elementa ry excitations in the spin-1\n2trimeric chain with varying δ. The\nfirst-order perturbational calculations, together with th e up-to-O(S1) linear spin-wave findings, are given in the upper five,\nwhereas the second-order perturbational calculations, to gether with the up-to- O(S0) interacting spin-wave findings, are given\nin the lower five. The exact-diagonalization results at N= 4, 6, and 8 are presented in both upper and lower panels by sym bols\nof small, middle, and large sizes, respectively\nwithEg=/summationtext\ni=2,1,0E(i)\ngandωλ(k) =/summationtext\ni=1,0ω(i)\nλ(k),\nwhereE(2)\ng=−2S2(J1+J2)Nis the classical ground-\nstate energy, while E(i)\ngandω(i)\nλ(k) (i= 1,0,···) are the\nO(Si) quantum corrections to the ground-state energy\nand the dispersion relation of mode λ, respectively. Here\nwe have discarded the O(S−1) terms. There are sev-\neral ways40,41,42,43,44of treating the quartic interactions.\nWhen we diagonalize the one-body terms and then take\naccount of the two-body terms perturbationally,45the\nspin-wave energies read\nE(1)\ng\nJ1N=S(1+δ)\n2N/summationdisplay\nk/bracketleftbig\nω(k)−3/bracketrightbig\n, (6)\nE(0)\ng\nJ1N=1+δ\n2(Γ2−1)−2δ\n1+δ\n×(3Γ2+2Λ2−5ΓΛ−3Γ+3Λ), (7)\nω(1)\n∓(k)\nJ1=S(1+δ)\n2/bracketleftbig\nω(k)∓1/bracketrightbig\n,\nω(1)\n0(k)\nJ1=S(1+δ), (8)\nω(0)\n∓(k)\nJ1=1+δ\n2/bracketleftbig\nΓΓ(k)∓Γ/bracketrightbig\n−δ\n1+δ\n×/bracketleftbig\n6ΓΓ(k)−5ΓΛ(k)−5ΛΓ(k)+4ΛΛ(k)\n−3Γ(k)+3Λ(k)∓Γ±Λ/bracketrightbig\n,ω(0)\n0(k)\nJ1=−1+δ\n2(Γ−1)−2δ\n1+δ(Γ−Λ),(9)\nand their eigenvectors are given by\nψ∓1(k) =ψ∗\n∓3(k) =2(e±ik/2+δe∓ik/2)\n(1+δ)/radicalBig\n2ω(k)/bracketleftbig\n3∓ω(k)/bracketrightbig,\nψ∓2(k) =/radicalBigg\n3∓ω(k)\n2ω(k), ψ01(k) =1√\n2,\nψ02(k) = 0, ψ03(k) =−e−ik/2+δeik/2\n√\n2(eik/2+δe−ik/2),(10)\nwhere\nω(k) =/radicalBigg\n1+32δ\n(1+δ)2sin2k\n2, (11)\nΓ=1\nN/summationdisplay\nkΓ(k) =1\nN/summationdisplay\nk1\nω(k),\nΛ=1\nN/summationdisplay\nkΛ(k) =1\nN/summationdisplay\nkcosk\nω(k).(12)\nFigure 2 shows the thus-calculated spin-wave ex-\ncitation modes together with the exact eigenvalues.\nFree spin waves well describe the ferromagnetic modes4\nω−(k) andω0(k), while higher-order quantum correc-\ntions play an essential role in reproducing the antifer-\nromagnetic mode ω+(k). TheO(S0) quantum correc-\ntions significantly improve fully delocalized magnetic ex-\ncitations in general,18,41,46but the standard Holstein-\nPrimakoff magnon series expansion seems not to work\nwell for highly localized excitations. The dispersive\nbranchesω∓(k) are nothing but the elementary excita-\ntion modes of spin-alternating linear-chain Heisenberg\nferrimagnets.47They are parallel within the spin-wave\ntheory, but their difference ω+(k)−ω−(k) is momentum\ndependent in fact. On the other hand, the flat band\nω0(k) arises from further excitation degrees of freedom\nin the present system. When J1=J2, the Hamiltonian\n(1) reads\nH=J1N/summationdisplay\nn=1(Sn:2+Sn+1:2)·Tn:3;n+1:1,(13)\nwith composite spins Tn:3;n+1:1≡Sn:3+Sn+1:1, each\nlying diagonally across an elementary palquette. Since\nthe Hamiltonian (13) commutes with T2\nn:3;n+1:1≡\nTn:3;n+1:1(Tn:3;n+1:1+ 1), we have good quantum num-\nbersTn:3;n+1:1, each taking either 0 or 1. Therefore, the\nplaquette-chain Hamiltonian is block-diagonalizedby the\nset of numbers {Tn:3;n+1:1;n= 1,2,···,N}48as well as\nby the total magnetization/summationtextN\nn=1(Sz\nn:2+Tz\nn:3;n+1:1)≡\nM. The Hilbert space of/summationtextN\nn=1(Tn:3;n+1:1)2/2 =/summationtextN\nn=1(Sn:3·Sn+1:1+ 3/4)≡ N=Ncorresponds to\nthe ferrimagnetic chain of alternating spins 1 and1\n2\nand consequently we here have exactly the same disper-\nsion relations46of elementary excitations. The Hilbert\nspace of N=N−1 andM=N/2−1 consists\nofNsubspaces labeled {T1:3;2:1,T2:3;3:1,···,TN:3;1:1}=\n{0,1,···,1},{1,0,1,···,1},···,{1,···,1,0}, and they\nall givethe sameset ofeigenvalues, forming Nflat bands.\nWe find the lowest one in Fig. 2.\nThus and Thus, the spin- Splaquette chain turns out a\ncombination of the alternating-spin-(2 S,S) linear chain\nand extra excitation degrees of freedom within the com-\nFIG. 4: Probability of two spin1\n2’s constructing a spin 1 in\nthe ground state of the spin-1\n2trimeric chain of N= 64 with\nvaryingδ, wherePn:3;n+1:1≡T2\nn:3;n+1:1/2 =Sn:3·Sn+1:1+\n3/4 andPn:1;n:3≡T2\nn:1;n:3/2 =Sn:1·Sn:3+3/4 are estimated\nby a quantum Monte Carlo method.posite spins Tn:3;n+1:1. All the composite spins are sat-\nurated in the ground state, T2\nn:3;n+1:1= 2S(2S+ 1),\nand therefore, their excitations are necessarily of ferro-\nmagnetic aspect. The ferromagnetic excitations of local\ncharacter are well understandable within the spin-wave\ndescription. Equations (4) and (10) show that a†\nn:1and\na†\nn:3, creating bosonic excitations on the Cu2+(2) sites,\nindeed participate in the construction of α†\nk:0, but any of\na†\nn:2, creating bosonic excitations on the Cu2+(1) sites,\ndoes not. Without mediation of bridging spins Sn:2, any\nintraplaquette excitation is never movable. Then what\nmay happen with δmoving away from unity? The spin-\nwave theory, whether up to O(S1) or up toO(S0), pre-\ndicts that the ferromagnetic and antiferromagnetic exci-\ntation modes ω∓(k) are still parallel and the extra ferro-\nmagnetic excitation mode between them, ω0(k), remains\ndispersionless. Let us verify the true scenario.\nIII. BOND-ALTERNATING TRIMERIC\nCHAINS\nWe demonstrate in Fig. 3 several schemes of calcu-\nlating low-lying excitation modes for the spin-1\n2bond-\nalternating trimeric chain. In spite of the persistent flat\nband within the spin-wave theory, the exact diagonal-\nization reveals that it can be dispersive with varying δ.\nWhenδ/ne}ationslash= 1, the Hamiltonian (1) does not commute\nwithT2\nn:3;n+1:1. Now that there is a certain probability\nof composite spins Tn:3;n+1:1being singlet even in the\nground state, the gapped ferromagnetic excitation mode\nω0(k) is no more describable as their individual triplet-\nto-singlet flips. At δ= 0, any excitation is localized\nwithinatrimerof Sn:1,Sn:2,andSn:3, andtheexcitation\nspectrum degenerates into three flat bands, ω−(k)≡0,\nω0(k) =J1, andω+(k) = 3J1/2. Figure 3 shows that\nthe middle branch of them connects with the flat band\natδ= 1. Without J2, the Hamiltonian (1) is reduced to\nH=H1=J1N/summationdisplay\nn=1Sn:2·Tn:1;n:3, (14)\nwith intratrimer composite spins Tn:1;n:3≡Sn:1+Sn:3\nand thus commutes with T2\nn:1;n:3. The plaquette chain\n(13) and the decoupled trimers (14) both exhibit a flat\nband due to gapped ferromagnetic excitations, but their\nways of constructing local immobile excitations are dif-\nferent from each other. Triplet-to-singlet [(4 S+1)-fold-\nmultiplet-breakingingeneral]flipsofintraplaquettecom-\nposite spins Tn:3;n+1:1are the elementary excitations in\nthe former, while those of intratrimer composite spins\nTn:1;n:3are the elementary excitations in the latter. Fig-\nure 4 shows how such composite spins behave in the\nground state with varying δ. Neither Tn:3;n+1:1nor\nTn:1;n:3form complete triplets at 0 < δ <1, due to\nnonvanishing off-diagonal matrix elements /an}bracketle{tTn:3;n+1:1=\n1|H|Tn:3;n+1:1= 0/an}bracketri}htand/an}bracketle{tTn:1;n:3= 1|H|Tn:1;n:3= 0/an}bracketri}ht.5\nAt the two particular points δ= 1 andδ= 0, only\nthe Cu2+(2) ion spins Sn:1andSn:3constitute the\ngapped ferromagneticexcitationmode, but otherwisethe\nCu2+(1) ion spins Sn:2also contribute to that. Without\ninterconnecting spins Sn:2, any excitation is immobile,\nwhereas with their mediation, all the local excitations\ncan be itinerant and the resultant bands are dispersive.\nPerturbational calculations support such a scenario.\nWith increasing couplings J2between isolated trimers,\nthe energy dispersion relations grow as follows:\nEg\nJ1N=−1−δ\n9−869\n2430δ2+O(δ3), (15)\nω−(k)\nJ1=4\n9δ(1−cosk)+δ2\n2430(929−474\n×cosk−455cos2k)+O(δ3), (16)\nω0(k)\nJ1= 1−0.38970δ+δ2(0.32099−0.26736\n×cosk+0.04212cos2k)+O(δ3), (17)\nω+(k)\nJ1=3\n2+δ\n18(7−12cosk)+δ2\n810(346−100\n×cosk−109cos2k)+O(δ3). (18)\nEquations (16)-(18) are also drawn in Fig. 3. The first-\norder perturbation points out that not only ω∓(k) them-\nselves but also their difference should be dispersive, but\nit cannot reveal nonvanishing momentum dependence of\nω0(k). We cannot reproduce the dispersive middle band\nuntil we take account of the second-order perturbation.\nThe lowest ferromagnetic and antiferromagnetic excita-\ntions of decoupled trimers (14), gapless and gapped by\n3J1/2 from the ground state, respectively, are both N-\nfold degenerate and are expressed as\n|E∓(m)/an}bracketri}ht=|−(1±3)J1/4;1/2∓1/an}bracketri}htm\n⊗\nn/negationslash=m|−J1;1/2/an}bracketri}htn(m= 1,2,···N),(19)\nwhiletheirgappedferromagneticexcitationsatanenergy\ncost ofJ1areN2-fold degenerate and are expressed as\n|E0(m,m′)/an}bracketri}ht=δmm′|0;−1/2/an}bracketri}htm⊗\nn/negationslash=m|−J1;1/2/an}bracketri}htn\n+(1−δmm′)|0;1/2/an}bracketri}htm⊗|−J1;−1/2/an}bracketri}htm′\n⊗\nn/negationslash=m,m′|−J1;1/2/an}bracketri}htn(m,m′= 1,2,···N),(20)\nin terms of the eigenstates of an isolated trimer\n|Sn:1,Sn:2,Sn:3/an}bracketri}ht,\n|−J1;1/2/an}bracketri}htn=1√\n6(|↑↑↓/an}bracketri}ht−2|↑↓↑/an}bracketri}ht+|↓↑↑/an}bracketri}ht),\n|−J1;−1/2/an}bracketri}htn=1√\n6/parenleftbig\n|↓↓↑/an}bracketri}ht−2|↓↑↓/an}bracketri}ht+|↑↓↓/an}bracketri}ht/parenrightbig\n,\n|0;1/2/an}bracketri}htn=1√\n2/parenleftbig\n|↑↑↓/an}bracketri}ht−|↓↑↑/an}bracketri}ht/parenrightbig\n,\n|0;−1/2/an}bracketri}htn=1√\n2/parenleftbig\n|↓↓↑/an}bracketri}ht−|↑↓↓/an}bracketri}ht/parenrightbig\n,|J1/2;3/2/an}bracketri}htn=|↑↑↑/an}bracketri}ht,\n|J1/2;1/2/an}bracketri}htn=1√\n3/parenleftbig\n|↑↑↓/an}bracketri}ht+|↑↓↑/an}bracketri}ht+|↓↑↑/an}bracketri}ht/parenrightbig\n,\n|J1/2;−1/2/an}bracketri}htn=1√\n3/parenleftbig\n|↑↓↓/an}bracketri}ht+|↓↑↓/an}bracketri}ht+|↓↓↑/an}bracketri}ht/parenrightbig\n,\n|J1/2;−3/2/an}bracketri}htn=|↓↓↓/an}bracketri}ht. (21)\nWith perturbational interactions H2turned on, the\nN-fold degeneracy of the eigenvalue −[N−3(1∓\n1)/4]J1=/an}bracketle{tE∓(m)|H1|E∓(m)/an}bracketri}htis completely lifted,\nwhereas the N2-fold degenerate eigenvalue −(N−1)J1=\n/an}bracketle{tE0(m,m′)|H1|E0(m,m′)/an}bracketri}htonly splits into Nflat bands\nwithin the first-order corrections. The second-order cor-\nrections are necessary for reproducing the dispersion re-\nlation ofω0(k). In this context we may be reminded that\nHonecker and L¨ auchli49pioneeringly investigated anal-\nogous but frustrated Cu2+trimeric chains. The gap-\nless ferromagnetic excitation mode (16) is indeed derived\nfrom their effective Hamiltonian under strong trimeriza-\ntionδ≪1.\nIV. SUMMARY AND DISCUSSION\nWe have investigated the low-energystructure of inter-\ntwining double-chain ferrimagnets composed of homoge-\nneous spins with particular emphasis on the gapped fer-\nromagnetic excitation mode. While there exist a macro-\nscopic number of flat bands50in the excitation spectrum\natδ= 1 andδ= 0, which signify uncorrelated excita-\ntions of local spin-2 Smultiplets in any case, they become\ndispersive as soon as δmoves away from these particular\npoints. Pair excitations of corner spins Sn:3andSn+1:1\nare elementary in plaquette chains of δ= 1, while those\nofSn:1andSn:3are elementary in decoupled trimers of\nδ= 0, both of which are completely immobile without\nany mediation of joint spins Sn:2. The spin-wave theory\nsuccessfully characterizes the plaquette chain but fails\nto find arising contribution of Sn:2to gapped ferromag-\nnetic excitations with bond alternation. Such a mislead-\ning prediction has been corrected by further numerical\nand analytical investigations.\nThe spin-Splaquette chain thus shares whole the na-\nture of the alternating-spin-(2 S,S) linear chain and fur-\nther exhibits ferromagnetic excitations of its own. All\nthe findings but the flat band in Fig. 2 are indeed ex-\nactlythesameaswehaveintheferrimagneticHeisenberg\nchain of alternating spins 1 and1\n2.45Even though the\nhomogeneous-spin plaquette chain and the alternating-\nspinlinearchainareequivalentintheirgroundstates, the\nformer demonstrates its extra excitation degrees of free-\ndom and deviates fromthe latter with increasingtemper-\nature. In order to illuminate similarities and differences\nbetween them, we show in Fig. 5 quantum Monte Carlo\ncalculations of their static structure factors\nS(q) =1\nN/summationdisplay\nn,l,n′,l′eiq(xn:l−xn′:l′)Sz\nn:lSz\nn′:l′,(22)6\nFIG. 5: (Color online) Quantum Monte Carlo calculations of t he static structure factor S(q), with the distance between\nneighboring spins in the chain direction set equal to unity, as a function of temperature. The whole view and enlargement s at\nq= 0 and q=πfor the spin-1\n2plaquette chain of N= 64 (a) and the alternating-spin-(1 ,1\n2) linear chain of N= 64 (b). The\nferromagnetic [ S(0)] and antiferromagnetic [ S(π)] peaks are observed in more detail (c), where the common asy mptotic values\nin the high-temperature limit, 3 /4 and 11 /12 for the the spin-1\n2plaquette chain and the alternating-spin-(1 ,1\n2) linear chain,\nrespectively, are indicated with arrows. Thermal averages of the projection Pn:3;n+1:1≡T2\nn:3;n+1:1/2 =Sn:3·Sn+1:1+ 3/4\nin the spin-1\n2plaquette chain are also shown for reference, where the asym ptotic value in the high-temperature limit, 3 /4, is\nindicated with an arrow.\nas functions of temperature, where the chain-directional\ncoordinates xn:lare given in the unit of neighboring-spin\nspacing. The pronounced peaks at q= 0 andq=π\nreflect the ferromagnetic and antiferromagnetic double\nexcitation mechanism in common. Without any field\napplied,S(0) andS(π) are, respectively, the uniform\nFIG. 6: (Color online) Quantum Monte Carlo calculations of\nthe static structure factor S(q), with the distance between\nneighboring spins in the chain direction set equal to unity, as\na function of bond alternation and temperature for the spin-1\n2\ntrimeric chain of N= 64.and the staggered susceptibilities multiplied by temper-\nature. With decreasing temperature, they both diverge\nas 1/T.38,45,51With increasing temperature, they both\napproach the paramagnetic value/summationtext\nlSn:l(Sn:l+1)/3 but\nbehave differently at intermediate temperatures. A mini-\nmumofS(0)asafunctionoftemperatureischaracteristic\nof ferrimagnets.6,7,18,25,30,52,53,54S(0) monotonically de-\ncreases and increases with increasing temperature in fer-\nromagnets and antiferromagnets, respectively.14Though\nthe thermal as well as quantum behaviors of the spin-\nSplaquette chain and the alternating-spin-(2 S,S) lin-\near chain are very much alike, yet there grows a differ-\nence between them with pair excitations of intraplaque-\ntte spins Sn:3andSn+1:1from their highest multiplets.\nThe ferromagnetic and antiferromagnetic structures of\nS(q) less survive increasing temperature in the spin- S\nplaquette chain than in the alternating-spin-(2 S,S) lin-\near chain. The larger Sthe major difference in S(q) as\nlimT→∞[S(2S,S)(q)−S(S,S,S)(q)] = 2S2/3. Alternating-\nspin-(2S,S) ferrimagnetic chains behave like combina-\ntionsofspin- Sferromagneticandspin-(2 S)antiferromag-\nnetic chains,14while such a simple magnetic sum rule\nis not available to intertwining double-chain ferrimag-\nnets of our interest. Additional intraplaquette antifer-\nromagnetic interactions induce incommensurate peaks in\nS(q),55making corner spins Sn:3andSn+1:1frustrated.\nOnceδmoves away from unity, the homogeneous-\nspin trimeric chain never more shares any feature of the7\nalternating-spin chain. Figure 6 presents S(q) with vary-\ningδand analyzes its features at q= 0 andq=πin\nparticular. At low temperatures, S(0) andS(π) both\ndecline with decreasing δ, but they still diverge as 1 /T\nunlessδ= 0.30At high temperatures, S(π) remains de-\ncreasing, whereas S(0) turns increasing, with decreasing\nδ. Decoupled trimers are nothing more than paramag-\nnets and their structure factor is given as\nS(q) =3\n4−2\n3eJ1/kBT−e−J1/2kBT\neJ1/kBT+1+2e−J1/2kBTcosq\n+1\n6eJ1/kBT−3+2e−J1/2kBT\neJ1/kBT+1+2e−J1/2kBTcos2q,(23)\nwhich is also drawn in Fig. 6 with solid lines. Equa-\ntion (23) at q= 0 reads as the effective Curie law for a\ntrimer entity, where the Curie constant varies from 1 /4,\nattributable to the ground-state doublet, to 3 /4, simply\ncoming from free spin1\n2’s, with increasing temperature.\nArising intertrimer couplings J2immediately pronounce\na quadratic dispersion relation of the ferromagnetic exci-\ntations at small momenta and their further increasecosts\nthe antiferromagnetic excitations higher energy. That is\nwhy growing global correlations enhance and reduce the\nuniform susceptibility-temperature product at low and\nhigh temperatures, respectively.\nWeak but nonvanishing dispersion of the gapped fer-\nromagnetic excitation mode is the most remarkable find-\nings of ours and is the very characteristic of intertwining\ndouble-chain ferrimagnets. As the existent compounds\nA3Cu3(PO4)4have all been reported to exhibit ratherstrong bond alternation δ<∼0.1,30,32,33,34,35it may be\nhard to detect the dispersion relation ω0(k) there. Ni\nanalogs, if available, will present an energy structure of\nthe same type on an enlarged energy scale. Highly lo-\ncalized excitations in fully exchange-coupled bulk mag-\nnets may either arise from an accidental arrangement of\nexchange couplings or come out of a particular lattice\nstructure of geometric aspect. 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Singh1,y\n1Indian Institute of Science Education and Research Bhopal, Bhopal, 462066, India\n2School of Physics and Materials Science, Thapar Institute of Engineering and Technology, Patiala 147004, India\n3Laboratoire CRISMAT, UMR 6508 CNRS ENSICAEN,\n6 bd du Marechal Juin, 14050 Caen Cedex 4, France\nGeometrically frustrated structures combined with competing exchange interactions that have dif-\nferent magnitudes are known ingredients for achieving exotic properties. Herein, we studied detailed\nstructural, magnetic, thermal (specific heat), magneto-dielectric, and magnetic exchange bias prop-\nerties of a mixed 3d - 4d spinel oxide with composition CoFeRhO 4. Detailed magnetization, heat\ncapacity, and neutron powder diffraction studies (NPD) highlight long-range ferrimagnetic order-\ning with an onset at 355 K. The magnetic structure is established using a ferrimagnetic model\n(collinear-type) that has a propagation vector k = 0, 0, 0. The magneto-dielectric effect appears\nbelow the magnetic ordering temperature, and the exchange bias (EB) effect is observed in field\ncooled (FC) conditions below 355 K. The magneto-dielectric coupling in CoFeRhO 4originates due\nto the frustration in the structure, collinear ferrimagnetic ordering, and uncompensated magnetic\nmoments. The unidirectional anisotropy resulting from the uncompensated magnetic moments\ncauses the room-temperature exchange bias effect. Remarkably, the appearance of technologically\nimportant properties (ferromagnetism, magnetodielectric effect, and EB) at room temperature in\nCoFeRhO 4indicates its potential use in sensors or spintronics.\nI. INTRODUCTION\nSpinel oxides are an intriguing class of materials, not\nonly for their potentiality for a wide range of applications\nbut also because of a variety of new and exciting physics\n(such as frustrated magnetism, multiferroic properties,\norbital glass system, spintronics applications, and spin-\norbital liquids) that continues to arise from the strong\ninteractions among spin, orbital, and structural degrees\nof freedom [1–14]. They have a unique structure with\na general formula of AB 2O4, where A and B are metal\nions (Fig.1). This structure comprises an array of metal\ncations in octahedral and tetrahedral coordination, sur-\nroundedbyoxygens, creatingtwosetsofmagneticsublat-\ntices [15]. B cations generally form a pyrochlore-like lat-\ntice by residing in the octahedral sites and originate frus-\ntrated magnetic interactions [16–19]. On the other hand,\nthe metal A ions occupy the tetrahedral sites (eightfold)\nand construct a diamond lattice [21–25]. This bipar-\ntite lattice can be interpreted as two face-centered inter-\npenetrated cubic (fcc) sublattices. These sublattices are\nshifted diagonally by one-quarter. Variation of magnetic\nand nonmagnetic cations on the tetrahedral A sites and\ntheoctahedralBsitescanoriginatecomplexmagneticin-\nteractions by affecting the magnitudes of superexchange\ninteractions (J AA, JBB, and JAB). Therefore, exotic\nproperties (and states) such as a spiral spin liquid phase\n[26], unique, glassy magnetic behavior [22, 27], and spin-\norbital liquids [28, 29] emerge in spinel materials. The\n\u0003soumarik@thapar.edu\nyrpsingh@iiserb.ac.inpyrochlore lattice is a fertile playground for theoretical\nandexperimental researchtoexplore newphysics. Atthe\nsame time, the bipartite diamond lattice (in spinel sys-\ntems) is a fruitful platform for realizing exotic quantum\nbehavior. In particular, the bipartite nature of the dia-\nmond lattice can be useful in designing the recently pro-\nposed 3D topological paramagnetism for the frustrated S\n= 1 diamond lattice [22, 23].\nIn addition to geometrical frustration and competing\nexchange interactions having different magnitudes, spin-\norbit coupling (SOC) is a known ingredient that favors\nmore exotic spin order and dynamics. In this connection,\n4d and 5d elements containing oxides attracted remark-\nable research interest in recent times. Being spatially\nmore extended (4d/5d orbitals), the on-site Coulomb re-\npulsion energy (U) is smaller in 4d and 5d elements than\ntheir 3d analogues. However, in the spinel oxide family,\nonly one 5d-containing material has been reported so far.\nThe iridium-containing spinel oxide with composition\nCu[Ir 1:5Cu0:5]O4highlights a highly frustrated magnetic\nstate [30]. Among 4d-rhodium-based diamond lattice\nspinel oxides, in diamond-lattice Heisenberg antiferro-\nmagnet CoRh 2O4, a combined experimental and theoret-\nical work showed that the S = 3/2 spins are unfrustrated\nand display static and dynamic properties [21]. Tetrago-\nnallydistortedCuRh 2O4showsanincommensuratemag-\nnetic order for the S = 1/2 spins and the presence of siz-\nable quantum effects [21]. A spin-orbit entangled param-\nagnetic state is suggested in NiRh 2O4[22, 31, 32]. Stud-\nies on magnetically diluted Cu 1\u0000xZnxRh2O4highlight\nspin transition triggered by an enhancement of preced-\ning spin fluctuations [33]. At the same time, it shows\nthe suppression of orbital order on the octahedral sitesarXiv:2304.13983v1 [cond-mat.mtrl-sci] 27 Apr 20232\nthrough the percolative manner. In general, geometri-\ncally frustrated structures with 4d/5d elements (SOC)\nare ideal for realizing exotic phenomena. Therefore, ex-\nploringnewfrustratedstructures(geometricalfrustration\nand competing magnetic interactions) coupled with SOC\nis essential.\nIn this article, we present the detailed structural (us-\ning X-ray powder diffraction, neutron powder diffrac-\ntion, and electron microscopy), magnetic (magnetiza-\ntion), thermal (specific heat), and magneto-dielectric\nstudies on a mixed 3d - 4d spinel oxide with composition\nCoFeRhO 4. In this material, non-magnetic Rh3+cations\noccupy octahedral B sites; however, they can be a source\nof SOC in the structure. An insulating and ferrimagnetic\nground state is observed near room temperature (T C=\n355 K). Detailed magnetic and magneto-dielectric mea-\nsurements highlight the room-temperature exchange bias\neffect and the magnetodielectric effect in this material.\nII. EXPERIMENTAL DETAILS\nSynthesis. We have used the standard solid-state re-\naction route to synthesize the polycrystalline CoFeRhO 4\nmaterials. Stoichiometric amounts of Co 3O4(99.9%)\nFe2O3(99.999%) and Rh (99.9%) metal powders were\nused for the synthesis. The stoichiometric amounts of\nthe starting materials were mixed with a mortar pestle.\nThe materials were heated several times and the final\nsintering was performed at 1473 K for 36 h.\nX-ray Powder Diffraction and Neutron Diffrac-\ntion.X-ray diffraction (XRD) using the powder form of\nthe material was collected at ambient temperature (RT)\nusing a PANalytical diffractometer (Cu-K \u000b,\u0015= 1.54056\nÅ). Neutron diffraction (NPD) data for the powdered\nCoFeRhO 4samplewerecollectedatvarioustemperatures\n(12 K, 100 K and 250 K) in JEEP II, Kjeller Reactor,\nNorway (\u0015= 1.5538 Å). We have performed the Ri-\netveld refinements of the diffraction patterns (XRD and\nNPD) using the fullProf suite software.\nElectron Microscopy. We have performed electron\nmicroscopy measurements (high-angle annular dark-field\nscanning transmission electron microscopy (HAADF-\nSTEM) and electron diffraction) to explore the crystal\nstructure of CoFeRhO 4. HAADF-STEM and electron\ndiffraction were carried out using a JEOL ARM-200F\ncold FEG probe image aberration corrected 200 kV mi-\ncroscope. The instrument is also equipped with a solid-\nangle CENTURIO EDX detector.\nMagnetism and Specific Heat. Direct current\nmagnetization (temperature and magnetic field depen-\ndent) measurements were done in a Quantum Design su-\nperconducting quantum interference device (MPMS 3).\nMagnetic measurements were conducted in field-cooled\n(FC) and zero-field-cooled (ZFC) modes. Specific heat\nwas measured (2 K - 400 K) in a physical property mea-\nsurement system (PPMS, Quantum Design) without a\nb\naFIG. 1. crystal structure for the CoFeRhO 4. Green spheres\nrepresenttheFe/Rhcations(OctahedralBsitesintheAB 2O4\nstructure), BluespheresrepresenttheCocations(Tetrahedral\nA sites), and red spheres represent oxygen.\nmagnetic field.\nDielectric Measurements. The temperature and\nmagnetic field-dependent dielectric measurements were\nperformed employing an LCR meter (Agilent 4284A) and\na sample insert for a PPMS (Quantum Design).\nIII. RESULTS AND DISCUSSION\nPreliminary Rietveld refinement uses the room-\ntemperature XRD pattern of CoFeRhO 4. It shows\nthat the sample crystallizes in a cubic structure with a\nspace group Fd3m, which is isostructural with CoFe 2O4\n[34, 35]. However, a small amount of Rh is detected\n(main peak at 2 \u0012'41 degrees, Fig. S1 in the sup-\nporting information (SI)) [20] in the RT XRD pattern\nfor CoFeRhO 4. Further, to explore the detailed nuclear\nand magnetic structure, we have collected the NPD pat-\nterns at 12 K, 100 K, and 250 K. Therefore, we will use\nthe results of the NPD Rietveld refinements to describe\nthe crystal structure of CoFeRhO 4. Similar to the XRD\npattern, a small amount of Rh is also detected in the\nNPD patterns. Fig.2 illustrates the plot of the NPD\nRietveld refinement for the pattern collected at 12 K.\nThe plots of the NPD Rietveld refinements for the pat-\nterns collected at different temperatures are provided in\nthe supporting information (Figure S2 in SI) [20]. The\nstructural parameters obtained from the 12 K - NPD re-\nfinement for CoFeRhO 4are summarized in Table 1. The\ncrystal structure for CoFeRhO 4is highlighted in Fig.1.\nRh atoms and most of the Fe cations occupy the octa-\nhedrally coordinated B site (16d (0.5, 0.5, 0.5)) of the\nstructure. Co-cations occupy the eightfold tetrahedral A\nsites (8a (0.125, 0.125, 0.125)). Oxygen atoms occupy\nthe 32e (x, x, x) Wycoff positions. The refinement of3\n3000\n2000\n1000\n0Intensity (a. u.)\n100 80 60 40 20\n2θ (º) Obs\n Calc\n Diff.\n Bragg Position900\n450\n0\n191817\n[111]300\n200\n100\n313029 12 K\n 100 K\n 250 K\n[220]\nFIG. 2. Rietveld refinement plot of the Neutron powder\ndiffraction (NPD) pattern collected at 12 K for CoFeRhO 4.\nThe lower ticks highlight the magnetic peak positions (k = 0,\n0, 0). Insets in the figure show the enhancement of the [111]\nand [220] NPD peaks with a lowering of the temperature.\nthe oxygen occupancy indicates that there is no devia-\ntion from the full occupancy. Rh shows a small deviation\nfromfulloccupancy (Occupancy=0.98(2)). Fig.3shows\nthe STEM- HAADF image along the [011] direction for\nCoFeRhO 4. The observed STEM images agree with the\nexpected crystal structure of the spinel material. The\nlattice parameter obtained from STEM images (8.4 Å)\nmatched the values found from the NPD (Table 1) and\nRT-XRD refinements.\nThestructural(Rietveld)refinementofthe250KNPD\nhighlights the existence of magnetic peaks at that tem-\nperature. These magnetic ordering-related peaks show\nan increase in intensity with decreasing temperature (in-\nset in Fig.2). The position of all magnetic peaks for\nCoFeRhO 4coincides with the allowed nuclear reflections.\nTherefore, we describe the magnetic structure using the\npropagation vector k = (0,0,0). For the long-range mag-\nnetic order, the best fit to the NPD magnetic reflections\nis achieved using a collinear ferrimagnetic ordering model\n(Fig. 4). A similar magnetic ordering scheme was pro-\nposed for the CoFe 2O4material [34]. The absence of a\n(200) magnetic peak discards the non-collinear arrange-\nment of the spins. In general, any long-range spin cant-\ning would result in the appearance of the (200) magnetic\nBragg peak. We do not observe the (200) peak at 21.27\n\u000e((400) peak is observed at 2\u0012= 43.43\u000e).\nMagnetic moments in different sublattices (M (T d)\n- tetrahedral A sites, M (Oct) - octahedral B sites)\nobtained from refinements are listed in Table 1. The\nresultant magnetic moment per formula unit obtained\nfrom the NPD refinements is 0.6 (1) \u0016Bat T = 12\nK. However, a discrepancy is observed between the\nmagnetic moments’ experimental and theoretical values\nin the tetrahedral and octahedral sites. Theoretically\ncalculated magnetic moments in different sublattices\nFIG. 3. HAADF image and crystal structure along the [011]\nzone axis for CoFeRhO 4, collected at room temperature. Fe,\nRh, and Co cations are highlighted in the figure. The corre-\nsponding electron diffraction pattern is shown in the inset.\nc\nb\na\nFIG. 4. The observed magnetic structure (k = 0, 0, 0)\nfor CoFeRhO 4. A collinear ferrimagnetic ordering scheme is\nobserved in the NPD refinements.\nTABLE I. Structural parameters for Ba 2ScRuO 6at RT ex-\ntracted from Rietveld refinement of powder XRD diffraction\ndata\nSpace group Fd3m,a = b = c = 8.4207 (1) Å\nRP= 6.98, R WP= 9.01,\u001f2= 1.03,R Bragg= 2.79\nRMag= 5.29, M (T d) = 3.4 (1) \u0016B, M (Oct.) = 4.1 (1) \u0016B\nAtom Wyck. Pos. x y z Occu. B iso\nCo(T d) 8a 0.125 0.125 0.125 0.88 (2) 0.1 (1)\nFe(T d) 8a 0.125 0.125 0.125 0.12(2) 0.1(1)\nCo(Oct.) 16d 0.5 0.5 0.5 0.065(5)0.11(3)\nFe(Oct.) 16d 0.5 0.5 0.5 0.44(1) 0.11(3)\nRh(Oct.) 16d 0.5 0.5 0.5 0.49(1) 0.11(3)\nO 32e 0.249(1)0.249(1)0.249(1) 1 0.10(3)\nare MA=0:88\u00023:5 + 0:12\u00025= 3.66\u0016Band MB=\n0:87\u00025 + 0:13\u00023:5)= 4.81\u0016B, calculated with M Fe3+\n= 5\u0016Band MCo2+= 3.5\u0016B. Reduced magnetic4\n(a)\n(b)\nFIG. 5. (a) Temperature-dependent magnetization (M -\nT) and (b) magnetic field variation of the magnetization (M\n- H) for CoFeRhO 4. Ferrimagnetic behavior is observed for\nCoFeRhO 4.\nmoments could result from local disorder (and/or local\ncanting) of spins. The complex cationic distribution\nand the existence of nonmagnetic Rh in the structure\ncan originate competing magnetic interactions and can\ncreate nonuniform spin canting. A similar reduced\nmagnetic moment due to the local spin canting is\nalso observed for the CoFe 2O4and Ti-doped CoFe 2O4\nmaterials [34, 36]. However, a collinear ferrimagnetic\nlong-range magnetic ordering scheme is suggested for\nboth of these materials.\nThe FC and ZFC magnetic moment (M-T) tempera-\nture variation measured at 0.1 T for CoFeRhO 4is pre-\nsented in 5. The M-T shows a sharp upturn at 355 K,\nhighlighting the onset of the ferrimagnetic transition at\nthat temperature. Upon further lowering the temper-\nature bifurcation between FC and ZFC magnetization\n(MFCand MZFC) is visible at 250 K. Below 250 K,\nMFCshows an increasing trend with decreasing temper-ature. Several competing magnetic interactions exist in\nCoFeRhO 4. Thecomplexcrystalstructure(diamondand\npyrochlore lattice) and the distribution of magnetic and\nnonmagnetic cations in two different sublattices can cre-\nate competing magnetic interactions and frustration. For\nexample,A-sitemagneticcationsaresurroundedby12B-\nsite magnetic and non-magnetic first neighbours, which\ncanbeFe3+orCo2+orRh3+; anyB-cationissurrounded\nby 6 A and 6 B-sited cations. Therefore, the bifurcation\nand the rise in M FCat a lower temperature could be due\nto the frustration in the spinel structure. However, the\nferrimagnetictransition inCoFeRhO 4ismuchlowerthan\nin CoFe 2O4. In CoFe 2O4, the collinear ferrimagnetic\ntransition is observed at 800 K with a strong intersub-\nlattice AFM superexchange interaction (J AB= - 12.39\nkB) [34]. Introducing non-magnetic Rh into the struc-\nture could dilute the magnetic interactions by decreasing\nthe strength of inter- and intra-sublattice superexchange\ninteractions and, therefore, can reduce the magnetic or-\ndering temperature. Figure 5(b) shows the magnetiza-\ntion as a function of the magnetic field (M - H) at var-\nious temperatures measured in the ZFC mode. Above\nthe magnetic transition temperature (at 400 K), a linear\nparamagnetic type M-H behavior is observed. However,\nat 350 K, the M-H loop illustrates non-linearity at low\nmagnetic fields, indicating theappearance of themagnet-\nicallyorderedstate. Aswefurtherlowerthetemperature,\nsaturated ferrimagnetic-type M - H loops are observed.\nThe saturation magnetization and coercive field (H C) en-\nhance with decreasing the temperature.\nFigure 6 highlights the temperature variation of to-\ntal specific heat for CoFeRhO 4. Magnetic ordering usu-\nally involves an entropy change, resulting in specific heat\nanomalies. The C Pvs T plot shows an anomaly at 355\nK, indicating a true phase transition. This also confirms\nthe magnetic ordering temperature for CoFeRhO 4. The\nlow-temperature part of the specific heat (2-20 K) can be\nwell represented by eqn. 1:\nCP=\rT+\fT3; (1)\nwhere\r= Sommerfeld coefficient and \f= lattice con-\ntributions to the specific heat. The obtained \r= 0.015\nJmol\u00001K\u00002. However, the total specific heat at higher\ntemperatures contains both the phonon and magnetic\nparts (CP=Cph+Cm). A combined Einstein-Debye\nmodel can generally estimate the total phonon contri-\nbution in the specific heat. Therefore, to estimate the\nmagnetic contributions ( Cm) in the specific heat, we fit\nthe data with the Einstein-Debye model [37]:\nCph= 9Rx\u00003\nDZxD\n0x4ex\n(ex\u00001)2dx+R2X\ni=1aix2\nEiex\nEi\n(ex\nEi\u00001)2\n(2)\nWhere xD=\u0012D/T and x Ei=\u0012Ei/T and, Debye tem-\nperature = \u0012Dand\u0012Ei= Einstein temperature. R =5\n150\n100\n50\n0C (J/mole-K)\n400 300 200 100 0\nT (K) Cexp\n fit (eqn. 1)\n fit (eqn. 2)\n6\n4\n2\n0C (J/mole-K)\n20100\nT (K)\nFIG. 6. Temperature variation of the specific heat (C) for\nCoFeRhO 4. An anomaly due to the magnetic ordering is ob-\nserved at 355 K. The green line highlights the fitted curve\nusing Eqn.2. Low-temperature C is also fitted using eqn.1\nand shown in the inset of the figure.\n8.31 J/mol K, and a i= degree of freedom for each Ein-\nstein mode. The fitted curve is shown in figure 6. The\nbest fit is obtained with \u0012D= 632 K,\u0012E1= 177 K and\n\u0012E2= 740 K. Then the contribution related to the mag-\nnetic ordering (C m) in the specific heat is calculated by\nsubtracting C phfrom the measured total specific heat\ndata. Figure 7 highlights the C m- T plot, showing a\nsharp maximum at T = 355 K. The entropy change due\nto the magnetic ordering can be calculated by using the\nfollowing formula:\nSm(T) =ZT\n0Cm(T)\nTdT (3)\nTheSm(T)vs T plot is shown in figure 7. Magnetic\nentropy increases with increasing temperature and shows\na saturation value of '4.4 J mole\u00001K\u00001(above 355 K),\nand this is less than the theoretically estimated magnetic\nentropy for CoFeRhO 4(Sm(T)= Rln (2S+ 1)= 26.41 J\nmole\u00001K\u00001, Co2+adopts the e g4t2g3electronic config-\nuration with S = 3/2, Fe3+adopts the e g2t2g3electronic\nconfiguration with S = 5/2). In general, the frustra-\ntion of the magnetic cations near and above the magnetic\ntransition temperature can reduce the entropy contribu-\ntiontothemagneticordering. InCoFeRhO 4, geometrical\nfrustration in the structure (diamond and pyrochlore lat-\ntice) and the distribution of magnetic and non-magnetic\ncations in two different sublattices could cause frustra-\ntionofmagneticcations. Thereducedmagneticmoments\nin the neutron powder diffraction experiments point to\nspins’ local disorder (and/or local canting). Therefore,\nthe reduced entropy related to magnetic ordering can be\nattributed to the frustration of the magnetic cations [38].\nOxides geometrical frustration and ferrimagnetic\ntransition often result in exciting magnetoelec-\ntric/magnetodielectric coupling. Geometric frustration\n12\n10\n8\n6\n4\n2\n0Cm (J/mole-K)\n350 300 250 200\nT (K)5\n4\n3\n2\n1\n0Sm (J/mole-K)\nCoefficient values ± one standard deviation\nA=0.02189 ± 0.00177\nB=0.00012499 ± 1.41e-006FIG. 7. Temperature variation of the magnetic component\nof the specific heat (C) (left axis) and calculated magnetic\nentropy (right axis) for CoFeRhO 4\n(local spin canting) in a magnetic system is proven to be\na key ingredient for magneto(di)electric coupling. For\nexample, the triangular Ising lattice Ca 3Co2O6shows\nmagneto dielectric coupling below the ferrimagnetic\nordering temperature (24 K) [39]. Haldane spin-chain\nsystem Dy 2BaNiO 5shows a magneto-(di)electric effect\nbelow the long-range ordering temperature (58 K)\n[40]. In spinels, the magnetodielectric effect is observed\nfrom the beginning of the ferrimagnetic ordering in\nCoCr 2O4(TC= 96 K) [41, 42] and NiCr 2O4[43].\nMnCr 2O4shows magneto dielectric effect below the\nferrimagnetic ordering temperature (43 K) [44]. How-\never, the magneto-(di)electric effect in ferrimagnetic\nmaterials near room temperature is rare. The near-\nroom-temperature magnetodielectric effect is important\ndue to its possible applications in spintronics devices,\nmagnetically accessible ferroelectric random-access\nmemories, and communication technology. Figure 8\nshows the temperature variation of the real part of the\ndielectric constant ( \u000fr) for CoFeRhO 4. The dielectric\nloss (tan\u000e) is highlighted in the inset of Figure 8. The ex-\ntremely low values of dielectric loss (tan \u000e) highlight the\ninsulating behaviour of CoFeRhO 4. It also excludes the\npossibility of extrinsic Maxwell-Wagner-like behaviour\nappearing because of the leakage currents. However,\nthe frequency dispersion of \u000frand tan\u000estarted above\n320 K, indicating the growing Maxwell-Wagner-type\nrelaxation or other sources of conductivity around 320\nK. Nonetheless, the dielectric constant is intrinsic below\n320 K. In addition to a small hump near the magnetic\nordering temperature (335 K), two anomalies at 220 K\nand 50 K are observed in the \u000fr- T plots. In particular,\nthe magnetisation measurements also observed similar\nanomalies at 220 K and 50 K.\nAs observed from our nuclear and magnetic structure\nrefinements, heat capacity, and magnetization studies,\nCoFeRhO 4hosts an exciting combination of complex6\n575\n570\n565\n560\n555 εr\n400 300 200 100\n T (K) 5 kHz\n 10 kHz\n 20 kHz\n 50 kHz\n 80 kHz\n0.12\n0.08\n0.04\n0.00 tanδ\n300 150\n T (K)\nFIG. 8. Temperature variation of the real part of the dielec-\ntric constant ( \u000fr) at several fixed frequencies for CoFeRhO 4.\nThe inset in the figure shows the dependence of the dielectric\nloss as a temperature function.\ncrystal structure (diamond and pyrochlore lattice) and\ncompeting magnetic interactions. The reduced magnetic\nmoments observed in the NPD refinements highlight the\npossibility of local spin canting. Therefore, the geomet-\nrical frustration, competing magnetic interactions, and\nthe local spin canting can originate magnetic frustra-\ntion in the structure. The magnetic frustration could be\nwhy these anomalies are observed below the long-range\nmagnetic ordering. To understand the effect of the mag-\nnetic field on the dielectric constant, we have performed\nmagnetic field dependence measurements of the dielectric\nconstant. Figure 9 shows the temperature variation of \u000fr\nmeasured in the presence of different magnetic fields. In-\nterestingly, the dielectric constant increases in the pres-\nence of a magnetic field below the magnetic ordering\ntemperature, and a positive magnetodielectrictance ef-\nfect is observed. The starting of the magneto(di)electric\neffect below the long-range ferrimagnetic ordering tem-\nperature indicates that the observed magneto(di)electric\neffect is related to the magnetic ordering of CoFeRhO 4.\nThe change in the dielectric data with the magnetic field\nis most pronounced near 220 K. The magneto-dielectric\nconstant ( \u0001\u000fr= [\u000fr(H)\u0000\u000fr(H= 0)]=\u000fr(H= 0)) and its\ndependence on the magnetic field are shown in the inset\nof figure 9. As discussed, the magnetic frustration in the\nstructure (geometric frustration, competing magnetic in-\nteractions, and local spin canting) could be the origin of\nthe increase in \u000frwith the magnetic field at that temper-\nature.\nIn recent times, complex magnetic materials (for\ninstance, ferrimagnetic systems Mn 3\u0000xPtxGa [45],\nBa2Fe1:12Os0:88O6[46] and SrFe 0:15Co0:85O2:62) [47], in\nparticular, those having a magnetic transition temper-\nature near room temperature have attracted significant\ninterest in exploring the exchange bias effect (EB). The\nEB effect shifts the isothermal magnetization loop, be-\ncomingasymmetricandshiftingalongthefieldaxis. This\n575\n570\n565\n560 εr\n40035030025020015010050\n T (K) H \n 0 T\n 5 T\n 14 T\n0.6\n0.4\n0.2\n0.0 Δεr (%)\n1050-5-10\n H (T) 300 K\n 220 K\n 100 KFIG. 9. Temperature dependent dielectric constant (real\npart,\u000fr) at different magnetic fields (H = 0 T, 5 T, and 14\nT) for CoFeRhO 4. The inset highlights the magnetic field\nvariation of the dielectric constant ( \u0001\u000fr= [\u000fr(H)\u0000\u000fr(H=\n0)]=\u000fr(H= 0)) at different temperatures.\neffecthasimportanttechnologicalapplications, suchasin\nthe development of magnetic sensors, magnetic recording\nread heads [48], random access memories [49], and other\nspintronic devices [50, 51]. Figure 10 highlights the mag-\nnetic field dependence of magnetization at 100 K, mea-\nsured in FC mode (cooling field, HFC= 3 T). It shows a\nclear shift towards the left field (negative magnetic field).\nThe EB field (H EB) can be calculated from the shift of\nthe hysteresis loop using the following equation\nHEB=\u0000(HC(L)+HC(R))=2 (4)\n(HC(L)andHC(R)are the intercepts with the positive\n(right) and negative (left) field axis. The calculated ex-\nchange bias field H EB= 65 Oe at 100 K. The exchange\nbias field is almost constant with varying cooling fields\n(HFC, ranging from 1 T to 7 T). To investigate the tem-\nperature evolution of the exchange bias effect and to ex-\nplore whether the exchange bias property in CoFeRhO 4\nis correlated with magnetic ordering, we have measured\nthe temperature variation of the EB field. The material\nwas cooled to the measuring temperatures in a magnetic\nfield of 3 T. Then the M-H loops were measured between\n\u00060.5 T. Fig. 11 highlights the temperature evolution of\nthe EB field for CoFeRhO 4. Interestingly, the EB effect\nemerges just below the long-range ferrimagnetic order-\ning (350 K); however, with decreasing temperature, H EB\nshows almost constant behavior down to 10 K. In ferri-\nmagnetic systems, the presence of uncompensated mag-\nnetic moments in a compensated AFM host is proven to\nbe very effective in designing the exchange bias effect (for\ninstance, Mn 3\u0000xPtxGa [45], Ba 2Fe1:12Os0:88O6) [46].\nIn this spinel structure, there is an uncompensated\nmagnetic moment due to the distribution of two differ-\nent magnetic cations (Co2+and Fe3+) in two different\nsublattices. In CoFeRhO 4, as observed from the neutron\ndiffraction refinements (table 1), below the magnetic or-7\n-30-20-100102030 M (emu/g)\n-0.4 -0.2 0.0 0.2 0.4\nH (T)-404M (emu/g)\n-400-2000200400\n H (Oe) 100 K\nHFC = 3 T\nFIG. 10. M-Hloop of CoFeRhO 4at 100 K measured in\nfield cooled (FC) mode ( HFC= 3 T). The inset highlights\nthe enlarged central part of the FC M-Hloop. A shift along\nthe left field axis is highlighted here.\nderingtemperature, themagneticmomentofBsublattice\n(M (Oct)) dominates over the M (T d). Under FC condi-\ntions, themagneticfieldorientsthenetmagneticmoment\nof the sublattices along the field direction. Now, as we\ngradually reverse the direction of the magnetic field, the\nmagnetic moments of the B sublattice (M (Oct)) do not\neasily reverse its direction as they are antiferromagneti-\ncally coupled with the A-sublattice (M (T d)). This pin-\nning effect of irreversible uncompensated spins (Octahe-\ndral B sublattice) develops the unidirectional anisotropy,\nand therefore, exchange bias is observed in CoFeRhO 4.\nIV. CONCLUSION\nTo summarize, we synthesized and examined the\ndetailed structural, magnetic, thermal (specific heat),\nmagneto-dielectric, and magnetic exchange bias proper-\nties of CoFeRhO 4, a mixed 3d-4d spinel oxide. We used\nvarious diffraction techniques, such as RT-XRD, NPD,\nRT-electron diffraction, and STEM, to explore the ma-\nterial’s structural details. The material crystallizes in a\ncubic structure with a space group of Fd3m. Our mea-\nsurements show a long-range ferrimagnetic ordering with\nan onset at 355 K, which is explained by a collinear fer-\nrimagnetic ordering model with k = [0, 0, 0]. However,\nwe observe reduced magnetic moments in the refinement,\npossibly due to the spins, local disorder, local canting, or\nfrustration. Magnetic frustration can also reduce mag-\nnetic entropy, as reflected in the specific heat measure-\nment. The magnetic entropy increases with temperature\nandreachesasaturationvalueof4.4Jmole\u00001K\u00001(above\n355 K), but it is less than the theoretically estimated\nvalue. Furthermore, our dielectric and FC magnetiza-\ntion measurements show the appearance of two techno-\nlogically significant phenomena near room temperature:\nthe magnetodielectric effect and the exchange bias ef-\n65\n52\n39\n26\n13\n0HEB(Oe)\n400 300 200 100 0\nT (K)HFC = 3 TFIG. 11. Temperature variation of the exchange bias field\n(HEB) for CoFeRhO 4. The exchange bias effect started to\nappear below the long-range ferrimagnetic ordering (350 K).\nfect. These effects appear below the magnetic ordering\ntemperature (355 K) and can originate from magnetic\nfrustration, collinear ferrimagnetic ordering, and uncom-\npensated magnetic moments. The uncompensated mag-\nnetic moments create unidirectional anisotropy, result-\ning in an exchange bias effect. Importantly, the pres-\nence of room temperature ferrimagnetism, EB, and the\nmagneto(di)electric effect demonstrates the potential of\nCoFeRhO 4in developing materials for sensors or spin-\ntronics applications operating at room temperature.\nV. ACKNOWLEDGMENTS\nR. P. S. acknowledges the Science and Engineering\nResearch Board (SERB), Government of India, for the\nCRG/2019/001028 Core Research Grant. S. M. ac-\nknowledges the SERB, Government of India, for the\nSRG/2021/001993 Start-up Research Grant.\n[1] V. W. J. Verhoeven, F. M. Mulder, and I. M. 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Oppeneer6, \nJoseph Barker9, Tobias Kampfrath1,2* \n \n1. Fritz Haber Institute of the Max Planck Society, Faradayweg 4 -6, 14195 Berlin, Germany \n2. Department of Physics, Freie Universität Be rlin, Arnimallee 14, 14195 Berlin, Germany \n3. Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, \nMax-Born -Straße 2A, 12489 Berlin, Germany \n4. Helmholtz -Zentrum Berlin für Materialien und Energie, Albert -Einstein -Straße 15, 12489 Berlin, Germany \n5. Institute for Optics and Atomic Physics, Technical University Berlin, \nHardenbergstraße 36, 10623 Berlin, Germany \n6. Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden \n7. Helmholtz -Zentrum Dresden -Rossendorf, Bautzner Landstr. 400, 01328 Dresden, Germany \n8. Ioffe Institute, 26 Polytechnicheskaya , 194021 St. Petersburg, Russia \n9. Institute for Materials Research, Tohoku University, Sendai 980 -8577, Japan \n# Present address: Department of Chemistry, Columbia University, \n3000 Broadway, New York, NY 10027, USA \n* Corresponding author s. Email: radu@mbi -berlin.de, tobias.kampfrath@fu -berlin.de \n \n \nABSTRACT \nTo gain contro l over magnetic order on ultrafast time scales, a fundamental understanding of the \nway electron spins interact with the surrounding crystal lattice is required . However, \nmeasurement and analysis even of basic collective processes such as spin -phonon equili bration \nhave remained challenging. Here, we directly probe the flow of energy and angular momentum in \nthe model insulating ferrimagnet yttrium iron garnet . Following ultrafast resonant lattice \nexcitation , we observe that magnetic order reduces on distinct time scales of 1 ps and 100 ns. \nTemperature -dependent measurements, a spin-coupling analysis and simulations show that the \ntwo dynamics directly reflect two stages of spin -lattice equilibration. On the 1 -ps scale, spins and \nphonons reach quasi -equilibrium in terms of energy through phonon -induced modulation of the \nexchange interaction. This mechanism leads to identical demagnetization of the ferrimagnet’s two \nspin-sublattices and a novel ferrimagnetic state of increased temperature yet unchanged total \nmagne tization . Finally, on the much slower, 100 -ns scale, the excess of spin angular momentum \nis released to the crystal lattice, resulting in full equilibrium. Our findings are relevant for all \ninsulating ferrimagnets and indicate that spin manipulation by pho nons, including the spin \nSeebeck effect, can be extended to antiferromagnets and into the terahertz frequency range. \n INTRODUCTION \nIn solids, vibrations of the crystal lattice have a significant impact on the orbital dynamics of the \nelectrons ( Fig. 1A). They strongly modify properties such as electrical conductivity and may even \ncause insulator -to-metal transitions1. Likewise, the interplay between phonons and electron spins \n(Fig. 1A) is relevant for equally drastic phenomena including colossal magnetores istance, \nfemtosecond magnetization control2,3,4,5,6 and the spin Seebeck effect7,8. Recently, ultrafast optical \ntechniques have provided new insights into the ultrafast coherent coupling of individual phonon \nand spin modes9,10,11,12,13,14. \nDespite this progres s, the microscopic origins of spin -phonon interaction s remain an intriguing \nproblem. Even the equilibration of crystal lattice and electron spins , arguably the conceptually \nsimplest collective process, is far from being understood. This notion is highlight ed in the model \nsystem yttrium iron garnet Y 3Fe5O12 (YIG), which is one of the best -studied magnetically ordered \nsolids15,16 and ubiquitous in the field of magnonics8. Only estimates of the time constant of spin-\nphonon equilibration exist, and they extend over as many as 6 orders of magnitude from ~1 µs \n(Ref. 17) to ~250 ps (Refs. 18 and 19) and down to ~1 ps (Ref. 20). Here, we introduce an \napproach based on resonant phonon excitation that allows us to directly probe the interaction s \nbetween the crystal lattice and electron spins over multiple time scales. \nExperiment. A schematic of our terahertz (THz) -pump magnetooptic -probe experiment is shown \nin Fig. 1A and detailed in the Materials and Methods section and Fig. S1A. An incident, intense, \nultrashort THz pump pulse21 (photon energy of ~0.1 eV, duration of ~25 0 fs, see Fig. S1B) \nselectively excites the crystal lattice by resonantly driving infrared -active transverse -optical (TO) \nphonons. The impact on the sample’s magne tic order is monitored from femtoseconds to \nmilliseconds by measuring the magnetooptic Faraday rotation of a time -delayed probe pulse (see \nFigs. 1A and S1A). In addition , we measure isotropic changes in the optical transmittance of the \nsample on ultrafast time scales . \nAs samples, we ch oose model systems for spin -wave dynamics in magnetic insulators7,8,16,22,23: \npure YIG and bismuth/gallium -substituted YIG (BiGa:YIG) . In these ferrimagnets, magnetic Fe3+ \nions at a - and d -sites in the unit cell ( Fig. 1B) comprise two inequivalent, ferromagnetic \nsublattices with magnetization Ma and Md, respectively, which couple antiferromagne tically. The \n2:3 ratio of a - to d-sites results in a nonzero net magnetization Ma+Md below the Curie \ntemperature TC. The Faraday rotation θ of the probe pulse is determined by24 \nθ=a aMa + adMd (1) \nwhere aa and ad are the sublattice magnetooptic coefficient s. Compared to YIG, the Faraday effect \nof BiGa:YIG is enhanced by about one order of magnitude and of opposite s ign24 (see Materials \nand Methods and Fig. S1C). \nIn both materials, we expect that the analysis of the pump -induced dynamics is simplified due to \nthe following featur es. First, the sublattice magnetizations Ma and Md almost entirely arise from \nthe spin rather the orbital angular momentum of the electrons . At room temperature, the g-factor \nof YIG differs by only 0.13% from that of the free electron25. Second, the sizeable electronic band \ngap (2.85 eV for YIG) impli es that the electron ic orbital degrees of freedom remain in their \nground state, thereby significantly reducing the complexity of the pump -induced processes. \nFigure 2 A displays the absorptance spectrum of a 15 µm thick BiGa:YIG film from 10 to 45 THz \nwhere absorption is known to be due solely to infrared -active phonons26. Our ab initio \ncalculations show that the pump pulse centered around 21 THz (red spectrum in Fig. 2A) \npredominantly excites long -wavelength TO( ) lattice normal modes characterized by an \nasymmetric Fe -O stretch vibration (see Figs. 1B, S4 and Materials and Methods ). RESULTS AND DISCUSSION \nUltrafast spin dynamics. Figure 2 A shows the relative pump -induced change Δθ/θ0=θ(t)/θ0−1 in \nthe Faraday rota tion as a function of the delay t since excitation . Here, θ0=θ(−2 ps) refers to the \nequilibrium case. When pumping off the TO( ) phonon resonances , at around 38 THz (blue pump \nspectrum in Fig. 2 A), a relatively small signal is found (blue curve in Fig. 2B). In marked \ncontrast, we witness a response more than one order of magnitude stronger for resonant phonon \nexcitation around 20 THz (red pump spectrum in Fig. 2A) : an ultrafast single -exponential drop of \nthe Faraday signal with a time constant as short as fast=1.6 ps (red curve in Fig. 2B). On much \nlonger time scales, an additional exponential reduction of θ with a time constant of slow=90 ns is \nfound (Fig. 2C). Recovery back to the initial state occurs over about 1 ms (Fig. S2D). \nSince the pump -probe sign al grows linearly with the pump fluence (inset of Fig. 2C), excitation is \ndominated by one -photon absorption, whereas strong -field effects such as field or impact \nionization are negligible27. This notion is further supported by Fig. 2 A which demonstrates th at \nthe sample absorptance and transient Faraday rotation are found to depend on the pump frequency \nin a very similar way. \nWe emphasize that almost identical dynamics are observed when a pure YIG sample instead of \nthe BiGa:YIG film is used (see Fig. S2B-D). Additional control experiments confirm that, as soon \nas the pump pulse has left the sample, the transient Faraday signal Δθ(t) reliably reflects the \ndynamics of the sublattice magnetizations Ma and Md with time -independent coefficients aa and ad \nin Eq. (1) (see Materials and Methods and Fig. S3). \nFigure 2B also shows the relative optical transmittance change of the BiGa:YIG sample for the \ncase of resonant pumping . The signal starts with a peak -like feature whose wi dth coincides with \nthat of the THz pump pulse. Already for delays t>1 ps, the signal is almost constant, changing by \nless than 10% in the subsequent evolution . Such relaxation to a quasi -steady state is substantially \nfaster than seen for the transient Fara day rotation which still doubles its value at t=1 ps in the \nfollowing 5 ps (Fig. 2B). Furthermore, the transmittance signal is found to be independent of the \nsample magnetization (Fig. S3C). These features suggest that the transmittance change \npredominantl y monitors the redistribution dynamics of the pump -deposited energy in the crystal \nlattice . \nAlthough the relative signal changes are small and still in the perturbative regime , our results in \nFig. 2 show a proof of concept that resonant phonon excitation provides an ultrafast manipulation \nof magnetic order . It only involves the crystal lattice and electron spins, yet no electron ic orbital \ndegrees of freedom (Fig. 1 A). The picosecond spin dynamics observed here (Fig. 2B) are \nunexpected because they are five orders of magnitude faster than the spin coherence lifetime \n(>0.1 µs) of YIG, which is known to be one of the longest amongst magnetically ordered \nmaterials8,15. \nTemperature dependence. To characterize the transient states established on the fast \n(fast=1.6 ps) and slow ( slow=90 ns) time scale, respectively, we increase the sample \ntemperature T0 from 300 K to 420 K. Simultaneously, we measure the equilibrium Faraday \nrotation θ0 as well as its pump -induced change Δθ at ultrashort ( t=10 ps, Fig. 2B) and long delays \n(t=1 µs, Fig. 2C) after phonon excitation. \nThe resulting θ0 versus T0 has the typical shape15 of a ferrimagne t’s static magnetization curve \n(Fig. 3 A). Its slope ∂θ0/∂T0 steepens with rising T0 until the transition into the paramagnetic phase \noccurs at the Curie temperature TC=398 K of BiGa:YIG . In contrast, the pump -induced Faraday \nsignal at t=1 µs (Fig. 3B) incr eases with T0 and reaches a maximum right below TC, reminiscent \nof the derivative of the static curve ( Fig. 3A). Indeed, we find that Δθ(1 µs) versus T0 closely \nfollows (∂θ0/∂T0)ΔT (Fig. 3B). Here, ΔT=0.39 K is the increase in equilibrium temperature as calculated from the energy density deposited by the 1 µJ pump pulse (see Materials and \nMethods ). The good agreement of both curves shows that ~1 µs after pumping, the BiGa:YIG \nfilm is in full thermodynamic equilibrium characterized by temperature T0+ΔT. \nRema rkably, Fig. 3B reveals that the changes in the Faraday signal at 10 ps are systematically yet \nnonuniformly smaller than at 1 µs, in agreement with Figs. 2B,C. While for temperatures \nT0<380 K, Δθ(10 ps) amounts to roughly 80% of Δ θ(1 µs), this ratio decrea ses strongly when T0 \nis increased . For example, slightly above 380 K (where both curves reach their maxima) and \nslightly below 400 K, Δθ(10 ps) has reduced to only 50% and 20% of Δ θ(1 µs), respectively. \nConsequently , Δθ(10 ps) vs T0 in Fig. 3B cannot be ma de to agree with Δ θ(1 µs) vs T0 by a \nsimple global rescaling of the Δ θ(10 ps) values. In particular, the shape of Δ θ(1 µs) vs T0 agrees \nmuch better with a suitably scaled derivative ∂θ0/∂T0 of the static Faraday rotation (black curve in \nFig. 3B) than Δθ(10 ps) vs T0. Therefore, at time t=10 ps after pump excitation , the spin system is \nin a state that is significantly different from the equilibrium state found at t=1 µs. \nAnalysis of spin couplings. To understand the microscopic mechanism driving the ultrafas t \nchange in magnetic order following resonant phonon excitation (Fig. 2B), we note that solids \nexhibit only three fundamental spin couplings. They can be understood as effective magnetic \nfields exerting torques on spins . In the following , we discuss all of them in terms of their \ncapability to modify the sublattice magnetizations Ma and/or Md on ultrafast time scales. \nFirst, spin -orbit (SO) coupling is often considered to dominate the ultrafast demagnetization of \nlaser -excited ferromagnetic metals in the ab sence of transport28. In our experiment, however, we \nfind identical ultrafast dynamics for pure YIG and BiGa:YIG (see Fig. S2B), even though the Bi \nions are known to increase SO coupling . This notion is manifested in the magnetooptic effects \nwhich are enha nced by more than one order of magnitude24. Therefore, SO coupling plays a \nnegligible role in the ultrafast spin dynamics seen in Fig. 2B. \nThe same conclusion can be drawn for the second type of coupling, spin -spin magnetic -dipole \n(SSMD) interaction, which has a comparable strength to SO coupling in YIG15. We note that SO \nand SSMD coupling also account for magnetic anisotropy , which can undergo strong photo -\ninduced c hanges in the garnet YIG:Co (Ref. 29). Ultrafast laser -induced modification of the \nanisotropy field of a YIG:Co film was even shown to drive precessional magnetization \nswitching29. The observed time constant of this process (~20 ps) is, however, still more than one \norder of magnitude longer than the ultrafast dynamics (fast=1.6 ps) observed here . In addition, we \ndo not observe any sign of coherent precession which is typical of anisotropy changes in the \nperturbative regime. Consequently, neither SO nor SSMD coupling can explain the spin dynamics \nobserved on the fast scale here. \nWe finally consider the only remaining fundamental spin coupling, isotropic exchange, which is \nthe strongest spin interaction in most magnets2. In YIG, it is responsible for the ferrimagnetic \norder a nd the high magnon frequencies extending to more than 20 THz at room temperature15,30. \nTherefore, exchange interaction may well account for the picosecond spin dynamics observed \nhere (Fig. 2B). Importantly, sinc e it conserves the total spin angular momentum, the \nmagnetization changes of a - and d -sublattice cancel each other. Despite ΔMa+ΔMd=0, the \nresulting change in Faraday rotation is nonzero because the magnetooptic coefficients aa and ad of \nYIG differ signifi cantly24 (see Eq. (1)). In summary, our analysis of the fundamental spin \ncouplings shows that only isotropic exchange interaction is capable of explaining the picosecon d \nspin dynamics seen in our experiment. \nModel. We suggest the following scenario of exchange -mediated spin -phonon coupling: the \nresonantly excited TO( ) phonons decay into other modes significantly faster than the time \nconstant fast=1.6 ps of the measured ultrafast spin dyna mics26, thereby heating up the crystal lattice. The assumption of ultrafast phonon r elaxation is consistent with our transient \ntransmittance data (Fig. 2B) which indicate that the phonons reach an approximately stationary \nstate within less than 1 ps. As the heat capacity of the crystal lattice of YIG is two orders of \nmagnitude higher than that of the spin system31, the lattice acts akin to a bath whose temperature \nis suddenly increased from T0 to T0+ΔT. \nAs a consequence of the ultrafast lattice heating , the O2- ions, which are the lightest, will undergo \nadditional random deflection Δu(t) (Fig. 4A). This perturbation modulates the superexchange32 of \nadjacent a -Fe3+ and d -Fe3+ spins and the associated coupling constant Jad by16,33,34 \nΔJad(t)=(∂ Jad/∂u) Δu(t). (2) \nWe now put this model to test by conduct ing calculations of both dynamic and equilibrium states \nand by their comparison to our experimental observations on the ultrafast ( fast, Fig. 2B) and the \nslower time scale (slow, Fig. 2C). \nDynamics on the fast scale . To determine the ultrafast rate of change of Ma und Md, atomistic \nspin-dynamics simulations based on ~106 coupled spi n equations of motion are performed30 (see \nMaterials and Methods ). We include time -dependent exchange parameters through Eq. (2) \nwhere Δu(t) is assumed to be random wit h a variance given by the pump -induced temperature \nincrease ΔT of the ionic lattice. \nSimulation results are shown in Fig. 4B. At times t<0, the magnetizations of both sublattices \nfluctuate around their constant mean s Ma0 and Md0. However, when fluctuations ΔJad(t) of the \nexchange parameter are switched on at t=0, Ma decreases linearly with time, until ΔJad(t) is \nswitched off. We find the opposite behavior for ΔMd. The sum curve ΔMa+ΔMd has a more than \none order of magnitude smaller amplitude . It arises from the much weaker random field s of the \nbath due to spin couplings other than isotropic exchange (see Materials and Methods ). \nFigure 4B demonstrates that thermal modulation of Jad induces demagnetization of the two s pin-\nsublattices by the same amount and, th us, transfers energy into the spin system. \nThe slope of the simulated ΔMa(t)/Md0 (Fig. 4B) can directly be compared to the initial slope of \nthe pump -probe signal Δθ(t)/θ0 (Fig. 2B), which amounts to approximately −1% ps-1. \nAgreement between experiment and theory is obtained by choosing ∂Jad/∂u~10 Jad Å-1 (see \nMaterials and Methods ). This value and the theory of superexchange32 imply that the overlap \nintegral of the ground -state wavefunctions of adjacent O2- and Fe3+ ions changes by 100% wh en \ntheir distance changes by ~0. 4 Å. As the distance between the two ions is one order of magnitude \nsmaller than the lattice constant a of YIG15, we estimate ∂u/∂a~0.1 and obtain \n∂Jad/∂a~(∂Jad/∂u)(∂u/∂a)~1 Jad Å-1. This result is in excellent agreeme nt with recent ab initio \ncalculations34 of the exchange -coupling constants as a function of a, which yielded \n∂Jad/∂a~0.7 Jad Å-1. Therefore, phonon -modulated exchange coupling successfully and \nconsistently expl ains the rate (∂θ/∂t)/θ0 at which magnetic order of YIG is observed to reduce \nimmediately after resonant phonon excitation . \nAccording to our model, ∂Ma/∂t, ∂Md/∂t and, thus, ∂ θ/∂t (Eq. (1)) are proportional to t he \ndifference between lattice and spin temperature (Eq. (10)), in agreement with the linear pump -\nfluence dependence of the transient Faraday rotation θ seen in the experiment (inset of Fig. 2C). \nConsequently, the measured exponential decay of θ to a constant value (Fig. 2B) indicate s that \nthe electron spins approach the lattice temperature T0+ΔT with the time constant fast=1.6 ps. \nNote that the subsequent slower dynamics evolve with a time constant of slow=90 ns. We \nconclude that for all pump -probe delays t≪slow, the transient chan ges in Ma and Md arise solely \nfrom inter-sublattice exchange coupling . As this interaction conserves the total spin angular \nmomentum, the dynamics are under the constraint Ma+Md=0 throughout th at time window . Dynamics on the slow scale. As indicated by our temperature -dependent measurements (Fig. 3), \nthe garnet film approaches full equilibrium at temperature T0+ΔT with the time constant slow. The \ntotal magnetization Ma+Md of this s tate is lower than for t≪slow where the change Ma+Md is \nstill zero . Thus, transfer of angular momentum from the spin system to the lattice has certainly \noccurred for times t~slow and above . \nThe only microscopic spin couplings capable of changing the t otal spin and, thus, magnetization \nMa+Md are SO and SSMD coupling. If SO coupling were dominant, one would expect a \ndifference in the spin dynamics in pure YIG and in BiGa:YIG in which SO coupling is enhanced \nby partial Bi substitution . However, the signal -to-noise ratio of the Faraday signal seen on the \nslow scale of pure YIG (Fig. S2B) does currently not allow us to draw a conclusion about which \nof the two couplings is dominant . \nConstrained spin state. As discussed above, t he spin system is under the constraint of constant \nmagnetization Ma+Md=Ma0+Md0 for times t≪slow whereas it is in full, unconstrained equilibrium \nfor t≫slow (Fig. 2C). Assuming that for t≫fast the spins and crystal lattice are characterized by a \nsingle temperature T0+ΔT, both the constrained and unconstrained spin state can be fully \ndescribed by equilibrium statistical physics (see Materials and Methods ). \nUsing a mean -field approximation, we obtain the change (∂Ml/∂T0)ΔT in the sublattice \nmagnetizations vs T0 for heating with and without the constraint of conserved Ma+Md (Fig. 5A). \nInterestingly, the constrained ∂Ma/∂T0 of minority s pin-sublattice a (green line in Fig. 5A) follows \nclosely its unconstrained counterpart. In contrast, the constrained ∂ Md/∂T0 of majority s pin-\nsublattice d (blue line in Fig. 5A) is systematically smaller t han the unconstrained ∂ Md/∂T0 \nbecause sublattice d can only demagnetize as much as sublattice a (ΔMd=−ΔMa). In particular, t he \ndifference between constrained and unconstrained ∂ Md/∂T0 increases considerably when the \ntemperature T0 approache s the Curie poin t (Fig. 5A). We emphasize that this behavior is in \nexcellent qual itative agreem ent with the measured Faraday signals Δ θ(10 ps) and Δθ(1 µs) vs T0 \n(Fig. 3B). \nIt is interesting to note that in the equilibrium formalism used here, the constant total \nmagnetiza tion is reinforced by a Lagrange multiplier which can be interpreted as a virtual \nhomogeneous magnetic field (see Eq. (13)). One could also term this field “spin pressure”, in \nanalogy to the pressure that is requir ed to keep a gas of particles in a bottle of constant volume \nwhile the gas temperature is increased. In our experiment, the “spin pressure” builds up on the \ntime scale given by fast and is released by angular -momentum transfer to the crystal lattice on the \ntime scale slow. \nPicture of spin -phonon equilibration. We summarize that our dynamic and equilibrium \ncalculations are fully consistent with the time scale s, fluence dependenc e, temperature \ndependence and magnitude of the transient Faraday rotation found in the experiment. This \nagreement leads us to the following picture of the flow of energy and angular momentum of \nphonon -pumped YIG (Fig. 5B): [1] the pump pulse excites zone -center TO() phonons which \n[2] decay within the pump duration, thereby increasing the crystal -lattice temperature. The \nadditional thermal modulation of a -d-exchange induces [3a] transfer of angular momentum \nbetween a - and d -spin-sublattices and implies [3b] energy t ransfer from the phonon to the spin \nsystem with the time constant fast=1.6 ps (Fig. 2B). The excess magnetization of this constrained \nstate decays by [4] transfer of angular momentum and energy between crystal lattice and spins \nwith the time constant slow=90 ns, resulting in f ull, unconstrained equilibrium (Fig. 2C). The \nangular -momentum transfer is mediated by SO and/or SSMD interactions22 which do not \nconserve Ma+Md. \n CONCLUSION \nWe employ a THz -pump magnetoo ptic-probe experiment to investigate spin-lattice equilibration \nfrom the femtosecond to the microsecond time scale. Combined with a spin-coupling analysis and \natomistic spin -dynamics simulations, o ur results reveal that the speed of spin -phonon relaxation in \nferrimagnetic insulators critically depends on whether one refers to energy or angular momentum . \nOn a picosecond scale, spins and phonons reach a quasi -equilibrium through phonon -modulated \nexchange interaction. It induces redistribution of energy among phonons and spins , whereas \nangular momentum is only transferred between spins. On a much longer time scale in the \nnanosecond range, the additional transfer of angular momentum between spins and crystal lattice \nresults in full equilibration. The quasi -equilibrium persists on the intermediate time scale and can \nbe understood as a transient spin state with elevated temperature yet unchanged net \nmagnetization. It could be considered as a realization of magnon populations with nonzero \nchemical potential which w ere recently introduced in the theory of magnon transport by the spin \nSeeb eck effect19. \nTransfer of angular momentum between spin -sublattices is also a key process in magnetic \nswitching of ferrimagnetic metalli c alloys by optical femtosecond laser pulses35,36,37,38. This \nprocess is considered to follow the preferential ultrafast demagnetization of one of the two spin-\nsublattices . Note that the flow of angular momentum can proceed through various mechanisms \nsuch as d irect exchange torque36 or nonlocal processes38 including the transport of spin -polarized \nelectrons. Our results reveal a new mechanism for manipulating the exchang e interaction on \nultrafast timescales and highlight the important role phonons can play in the direct exchange \ndynamics between spin-sublattices. \nIn terms of applications, our results suggest that resonant phonon excitation is a new pathway to \nthe prepar ation of spin states with increased temperature yet unchanged total magnetization. This \nroute may be particul arly interesting for the manipulation of antiferromagnetic order2, where spin \nangular momentum is inher ently conserved. Finally, the ultrafast spin -lattice coupling of YIG \nimplies that the magnon temperature follows the phonon temperature with a delay on the order of \nonly 1 ps, thereby shifting the cutoff frequency of the bulk spin Seebeck effect7 to the THz range. \n \n MATERIALS AND METHODS \nSamples . Our iron garnet films are grown by liquid -phase epitaxy on substrates of gadolinium \ngallium garnet Gd 3Ga5O12 (GGG). To prevent pump absorption by the substrate, several iron-\ngarnet films are transferred on diamond windows after mechanically removing the GGG. Test \nmeasurements confirm that GGG pump absorption is irrelevant to the ultrafast dynamics. \nWe study two types of garnet samples: pure yttrium iron garnet Y 3Fe5O12 (YIG) and \nbismuth/gallium -substituted iron garnet Bi xY3xGayFe5−yO60 (x=1.53, y=1.33, BiGa:YIG) . They \nhave, respectively, a Curie temperature of 545 K and 398 K, and in -plane and out -of-plane \nmagnetic anisotropy. Film thicknesses cover a range from 7 to 20 µm. Chemical composition of \nthe garnet films is determ ined by X -ray fluorescence measurements. Substitution of Y by Bi in \nthe BiGa:YIG sample enhances the magnetooptic Faraday effect by one order of magnitude24. \nTypical hysteresis loops obtained by measuring the s tatic Faraday rotation of the probe beam are \nshown in Fig. S1C. \nTHz spectra of the sample absorptance (which equals one minus the reflected and transmitted \npower, both normalized to the incident power) are obtained by combined reflection and \ntransmission m easurements with a broadband THz time -domain spectrometer. Figure 2A shows \nthat maximum absorptance occurs at 21 THz, which is 2 THz above the highest -frequency TO ()-\nphonon resonance26. According to Ref. 26, absorption in this frequency range solely arises from \nTO phonons, while crystal -field and charge -transfer -gap transitions are located at much higher \nfrequencies24. \nExperimental design. A THz pump pulse is used to resonantly excite long -wavelength TO \nphonons of the iron-garnet film under study . The instantaneous magnetic state of the sample is \ndetermined by a subsequent optical probe pulse measuring the magnetooptic Fa raday effect. In \nthis way, spin dynamics are monitored over a large range of pump -probe delays ranging from \nfemtoseconds to milliseconds. Details of our setup are shown in Fig. S1. \nTo generate intense THz pump pulses, we employ an amplified Ti:sapphire las er system \ndelivering pulses (energy of 15 mJ, center wavelength of 800 nm, duration of 40 fs, repetition rate \nof 1 kHz) to drive two optical parametric amplifiers that, in turn, generate intense infrared pulses \n(pulse energy of 1.5 mJ and 1.2 mJ, center wa velength of 1280 nm and 1400 nm, duration of 50 fs \nand 50 fs, respectively). By difference -frequency mixing of the two infrared pulses in a nonlinear -\noptical GaSe crystal21 (thickness of 1 mm), we obtain intens e THz pulses (tunable from 15 to \n60 THz, energy of typically 3 to 20 µJ, pulse duration around 250 fs) with stable carrier -envelope \nphase. For sample excitation, the THz pulse is focused onto the sample surface to a spot with \ndiameter 180 µm. Its transient electric field \nEt is measured by electrooptic sampling (see \nbelow). The pulse spectrum is obtained by Fourier transformation of \nEt and shown in Fig. 2A. \nLow-noise probe pulses (8 fs, 800 nm, 0. 5 nJ) are derived from the pulse train of the Ti:sapphire \nseed laser (repetition rate of \nrep80MHz f ). They traverse the sample collinearly with the pump \nafter a delay \nt. To measu re the probe’s transient polarization rotation \n Δt , we employ a \npolarimeter consisting of a Wollaston -prism polarizer followed by two fast photodiodes. \nMeasurement of the probe ellipticity \n Δt is accomplished by putting a quarter -wave plate \nbefore the Wollaston prism. The resulting photodiode current is a train of temporally isolated \nelectrical pulses (duration of 5 ns each, pulse -to-pulse distance of \nrep1/ 12.5nsf ), which is \nsampled using a computer -controlled fast analog -digital converter card39. In this way, we are able \nto measure signals at delays of \nrep τ/t j f (with integer \nj ) between −100 ns and ~1 ms within a single pump shot and with a delay spacing of \nrep1/ 12.5nsf . The ultrashort offset \nτ is set \nfrom −2 ps to 200 ps by a variable mechanical delay stage. \nPump -probe signal traces \nt are taken for the sample magnetization saturated in opposite \ndirections using an external magnetic field \nextB of 15 mT, respectively. To measure samples \nwith in -plane or out -of-plane anisotropy, the magnetic field is oriented 45° with respect to the \nincident pump and probe beams. To vary the sample temperature, the sample is mounted on a \nresistive heater. The sample temperature is measured with two thermocouples and an imaging \ninfrared thermometer, all of them yielding consistent temperature values. To measure the THz \nelectric field, we substitute the sample by a GaSe crystal and measure the transient ellipticity \n Δt E t\n induced by the electrooptic effect21. \nOptionally, our setup also permits measurement of the transient isotropic transmittanc e of the \nsample . For this purpose, we remove all polarization -sensitive components such as the Wollaston \nprism and wave plate and detect the probe -pulse energy with one of the two fast photodiodes of \nthe polarimeter. Based on the small attenuation length26 (~1 µm) of the resonant pump pulse into \nthe iron -garnet sample, we estimate that the temporal blurring of the pump -probe signals due to \npump -probe velocity mismatch is smaller than 10 fs and, therefore, neglig ible. \nSignal analysis. Up to first order in magnetization, the Faraday rotation \n (and likewise the \nellipticity \n ) of the probe’s outgoing polarization is a weighted average24 of the magnetization of \nthe two s pin-sublattices \na,dl , that is, \n.lla M d b \n (3) \nHere, \nla are the local magnetooptic coupling coefficients (Verdet constants) of YIG, \npr llMeM\n is the magnetization of sublattice \nl projected on the propagation -direction unit \nvector \npre of the probe, \nd is the sa mple thickness, and \nb is an offset arising from any \nmagnetization -independent optical anisotropy. The angular brackets \n denote averaging over the \nprobed sample volume. \nIn equilibr ium, the \nla are constant throughout the probed magnetic volume, \nl l l la M a M . \nThe impact of the pump pulse makes all quantities in Eq. (3) time-dependent, resulting in \nrelatively small changes \n0 t of the signal \n0 2 ps measured 2 ps before arrival of the \npump pulse. According to Eq. (3), the nonmagnetic offset \nb cancels by calculating the \nasymmetric part \n 00ΔΔΔ Δ Δ .2l l l l t a M M a \n (4) \nThe first term on the right -hand side scales with a weighted average of the sublattice \nmagnetizations along \npre , which is the quantity we are intereste d in. The second term, however, \nmakes a contribution independent of the \nΔlM and arises from a possible pump -induced \nmodulation \nΔla of the Verdet constants. Figure S3 shows that the second term is negligible and \nthat \nΔ reflects the true magnetization dynamics, \n0 ΔΔllaM , once the pump pulse has \nleft the sample. Excitation profile and heating . Since the absorption length of the THz pump pulse26 (~1 µm) is \nsmaller than the sample thickness (~10 µm), pump excitation is inhomogeneous along the \nz -axis \nnormal to the film plane. Because the pump -induced signal \n Δt depends linearly on the \nabsorbed pump fluence \nabsF (inset of Fig. 2C), the pump -induced change in the local Faraday \nrotation \n Δ,zt is proportional to the pump energy density \nabswz absorbed in the vicinity of \nthe plane at \nz. In other words, one has \nabs abs\n0 d d\nF z w z and \nabs Δ,z t C t w z where \nCt\n captures the temporal dynamics . As a consequence, the total pump -induced signal \n abs\n0Δ Δ , d Δ , .d\nt z t d z z t C t F \n (5) \nis independent of the shape of the absorption profile \nabswz . Therefore, without loss of \ngenerality, we can consider the absorption profile to be homogeneous with \nabs abs / w F d . Our \nargumentation is valid as long as transport of heat from the YIG film into the substrat e is \nnegligible. Figure S2D shows this assumption is fulfil led for pump -probe delays up to at least \n1 µs. \nAssuming the sample has reached (local) thermal equilibrium at \n1µst , we estimate the average \ntemperature increa se \n Δ 1 µsT of the BiGa:YIG film by dividing \nabsF by the pumped sample \nvolume [ d(pump diameter)2\n/4 ], by the mass density and by the specific heat capa city31 of \nYIG. We obtain a value of \n Δ 1 µs 0.39 KT for an incident pump energy of 1 µJ, which can be \ninterpreted as the temperature increase of a homogeneously excited YIG film. \nWe checked effects due to accumulative heating of the sample by reducing the repetition rate of \nthe pump pulses from 1000 Hz to 500 Hz by means of a mechanical chopper. The magnitude of \npump -induced signal that persists after 1 ms is found to be smal ler than 10% of the pump -induced \nsignal at \n~ 1µst . It corresponds to a static homogeneous temperature increase of the probing \nvolume of only ~39 mK, which has a negligible influence on our results, including the \ntemperature dependence s een in Fig. 3B. \nYIG ab initio calculations. Our ab initio phonon calculations are performed using the finite -\ndisplacement method implemented in the open -source package phonopy40. The electronic -\nstructure calculation and equilibrium geometry search of YIG ar e performed within the Density -\nFunctional Theory (DFT) framework, using the Vienna Ab-initio Simulation Package41 (VASP) \nand adopting the generalized gradient approximation42 of the exchange -correlation functional. \nOur calculations and resulting phonon dispe rsion relations are detailed in the Supplementary \nMaterial . \nFrom the element -resolved vibrational density of states (Fig. S4), we find that a continuum of \nmodes from ~7 to 20 THz contributes to the displacement of the O2- ion with approximately \nfrequency -independent weight. This result implies that the vibration of the O2- ion has a mean \nfrequency of \nOΩ / 2π 14 THz and a correlation time of \nO10 fs . \nDynamic spin model. Our consideration of all possible spin-coupling mechanisms ( SO, SSMD, \nexchange, see main text) leads us to the view that the ultrafast regime of the spin dynamics of \nYIG following phonon excitation arises from isotropic exchange interaction. More precisely, the \nthermal vibrations of the ionic lattice modulate the a -d-exchange interaction (Fig. 4A), thereby transferring spin angular momentum from the a - to d-spin-sublattice and quenching each \nsublattice magnetization at the same rate. \nTo test this hypothesis, we calculate both the rate of spin -angular -momentum transfer between \nsublattices (based on numerical simulations) and the temperature dependence of the sublattice \nmagnetizations (based on an analytical treatment ). Starting point is the Heisenberg spin \nHamiltonian of YIG15, \nB extˆ ˆ ˆ ˆ ,1\n2jj j j j\nll l l l\nll jj ljH J g \n\n S S B S\n (6) \nwhere \nˆj\nl\nS is the spin -angular -momentum operator of the total electron spin of an Fe3+ ion (spin \nquantum number \n5 / 2S ) located at site \nj in spin-sublatti ce \nl. Orbital contributions to the \nmagnetic moments are negligibly small25. In YIG, there are two different crystallographic sites \na,dl\n for the Fe3+ ions which are, respectively, surrounded by six O2- ions (octahedral sites) and \nfour O2- ions (tetrahedral sites). The exchange constants have been determined to be as follows15: \nif for given \nl , \nl, the \nj and \nj represent next neighbors, then \njj\nll llJJ\n . Otherwise, \n0jj\nllJ\n . \nThus, the exchange part of the Hamiltonian is fully determined by the three constants \naaJ , \nddJ \nand \nad daJJ . \nFollowing Eq. (2), the phonon -induced modulation of the exchange interaction is modeled as \nvariation \n ad adΔΔu J t J u t\n (7) \nof the exchang e coupling constant between \nal - and \ndl -spin-sublattices. Here, \n/u u , \nand \nΔu is the pump -heat-induced deflection of the O2- ion mediating the superexchange of \nadjacent a - and d -type Fe3+ ions, as depicted in Fig. 4A. This assumption is justified becau se \nadJ \npredominantly involves the a -Fe3+ and d -Fe3+ ions and the O2- ion in between. Since the O2- ion is \nthe by far lightest ion in the YIG unit cell, its motion is expected to modulate the a -d \nsuperexchange most st rongly. \nAtomistic spin -dynamics simulations. To calculate the rate of spin -angular -momentum transfer \nbetween sublattices, we conduct numerical simulations based on Eqs. (6) and (7). For the \nimplementation, we assume an ensemble of classical spins and replace each spin operator \nˆ\niS by \n1iSS s\n. Here, \nis is a vector of length one , and the compound index \nlj is summarized in \none index \ni . Likewise, the Hamilton operator \nˆH turns into the Hamilton function \nH . For the \nexchange constants \naaJ , \nddJ and \nadJ , we use the values 0.33 meV , 1.15 meV and −3.43 meV, \nrespectively15. \nBy applying Langevin theory, a stochastic Landau -Lifshitz -Gilbert -type equation is derived for \neach spin30,43,44, \n eff , eff , 2.\n1i\nt i i i i i i i\nii\n s s B s s B\n (8) \nwhere \ni is the gyromagnetic constant of a spin with magnetic moment \niis . Damping due to \ncoupling to non -spin degrees of freedom (the so -called bath ) is quantified by \n52 10i . The \neffective field \neff ,iB acting on \nis has three contributions: a deterministic part due to the exchan ge fields \n00\nexch, /ii H Bs of the adjacent spins for temporally constant \n0\nijJ and two stochastic \ncomponents \nitξ and \nexch,Δit B that arise from coupling of the spins to the bath. The first field \nitξ\n transfers spin angular momentum and energy between the spin system and the bath having \ntemperature \n0T . According to the fluctuation -dissipation theorem, in equilibrium, the fluctuations \nitξ\n are balanced by a friction force, the second, Gilbert -type term in Eq. (8), thereby driving \nthe spin system into an equilibrium state with temperature \n0T (Refs. 30, 43, 44). The time \nconstant of this relaxation process is set by the damping parameters \ni and found to be irrelevant \non the picosecond scale considered here. \nThe second stochastic field \nexch,ΔΔi ij j\njJ Bs arises from the phonon -induced modulation of the \nexchange i nteraction (Eq. (7)). It is distinctly di fferent from \nitξ as it causes energy transfer into \nthe spin system, but leaves the total spin angular momentum unchanged. Since the noise field at \ntemperature \n0T is already accounted for by \nitξ , the variance of \n ΔijJt scales with the \ntemperature increase \nΔT of the crystal lattice induced by the pump pulse. More precisely, owing \nto the equipartition theorem, the pump -induced deflection \nΔu of the O2- ion obeys \n2 2 2 2\nO O O O BΩ Δ Ω Δ Δm u m u k T \n, where \nOΩ / 2π 14 THz is the mean O2--ion vibration \nfrequency (see above). \nTo determine the initial rate of phonon -driven angular -momentum transfer betwe en the s pin-\nsublattices, phonons and spins are assumed to be thermalized at temperatures \n0ΔTT and \n0T , \nrespectively (see main text) . Following our ab initio result s (see above), \n Δut is assumed to \nhave a correlation function with a width of \nO10 fs , which is a ccordingly modeled as \n B\nO 2\nOOΔΔ Δ .ΩkTu t u t t tm \n (9) \nThe equation of motion (Eq. (8)) is integrated numerically for \n3564 2.6 10N unit cells at \nequilibrium temperature \n0300 KT . At ea ch time step \nt, we calculate the two sublattice \nmagnetizations \n1/j\nll\njS S N Ms with \na,dl . In our numerical implementation, the \npump-induced thermal noise of the exchange constants (Eqs. (7) and (9)) can be switched on or \noff over arbitrary time intervals and with variable strength \n2\nadΔuJT . \nResults of our atomistic simulations are shown in Figs. 4B and S5. We find the demagnetization \nrate of each sublattice scales according to \n2 d0\nad/\nΔtl\nu fMMJT\n (10) \nwith constant \n16 2 2 1 12 10 m J K psf . On the other hand, our experiment shows that the \nFaraday signal \nΔ and, thus, \n0/t l lMM scale with the pump fluence (see inset of Fig. 2C) \nand, thus, the initial \nΔT as well. In other words, we have 00/ Δ/\nΔΔt l l tMMgTT (11) \nwhere \n3 1 11.9 10 K psg has been determined from the initial slope of \n00 Δ / Δ /llMM \n(Fig. 2B) seen in our experimental results. Comparison of Eqs. (10) and (11) yields the estimate \nad ad / 10uJ g f J \nÅ -1. \nAccording to the theory of superexchange32, \nadJ is proportional to the fourth power of the overlap \nintegral of the Fe3+ and the O2- ion. Therefore, upon changing the distance of the two ions, the \noverlap integral undergoes relative changes on the order of 2.5 Å -1. \nMean -field sublattice magnetization. Our previous res ults strongly indicate that spins and \nphonons reach quasi -equilibrium for times t≫fast following phonon excitation. As exchange \ninteraction leaves the total spin angular momentum of the electrons unchanged, this thermal state \nis constrained by the boundar y condition of conserved total spin angular momentum. Such \nconstraints are ubiquitous in thermostatics (such as for a gas contained in a closed bottle) and can \nbe treated by equilibrium statistical physics45. \nOur goal is to determine the magnetization of t he d- and a-spin-sublattice of YIG in thermal \nequilibrium for two cases: the unconstrained (UC) situation without boundary condition and the \nconstrained (C) situation with the boundary condition of constant total spin angular momentum, \nconstant. ˆj\nl\nljS\n (12) \nIn both cases, we need to calculate the statistical operator \nˆD of the system, from which the \nexpectation value \n Tˆˆ ˆr A DA of an ob servable \nˆA (such as a sublattice magnetization) can be \nderived. In thermal equilibrium and for both the UC and C case, the statistical operator is given \nby45 \nˆ ˆˆ1expj\nl\nljDHZ pS\n (13) \nwhere \nB1/kT with \nBk being the Boltzmann constant. The partition function \nZ is determined \nby the normalization condition \nTr ˆ1D . In the UC case, we simply set \n0p in Eq. (13), thereby \nresulting in the standard canonical statist ical operator45. In the C case, the constraint of conserved \nspin angular momentum (Eq. (12)) has to be fulfilled by proper choice of the Lagrange \nmultiplier \np in Eq. (13). Interestingly, \np can be interpreted as a virtual homogeneous magnetic \nfield that is adjusted such to reinforce the boundary condition of Eq. (12). It is analogous to the \nchemical potential that keeps the number of particles in a giv en macroscopic system constant. \nThe general Heisenberg spin Hamiltonian \nˆH of Eq. (6) is too complex for an analytical \ncalculation of \nˆD. Consequently, we apply the mean -field appro ximation32, \nˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ j j j j j j j j\nl l l l l l l l \n S S S S S S S S\n, omit the number \nˆˆjj\nll\nSS , and allow for alignment \nexclu sively along the \nz -axis unit vector \nze , \n,ˆ ˆ ˆj j j\nl l z z l z SS S e e , as set by the external \nmagnetic field \next ext zBBe . In addition, we assume the spins order homogeneously on each \nsublattice \nl, that is, \n0 ˆˆj\nllSS (sublattice approximation32). By evaluating the \njj\nllJ\n of YIG (see \nabove), we finally obtain the mean -field (MF) Hamiltonian 0\nMF B ext1\n2ˆ ˆ ˆ ˆjj\nll ll l l l\nll j ljH J N S S g B S \n (14) \nwhere the \nllJ are defined as above, and denotes the number of the next neig hbors a site in \nsublattice \nl possesses in sublattice \n'l. For YIG, one has \naa8 N , \ndd4 N , \nad6 N and \nda4 N \n(Ref. 15). We can now write the statistical operator of Eq. (13) in a very compact form as \nMF1exp ˆ ˆj\nll\nljD F SZ\n (15) \nwhere the effective field \n0\nB ext1\n2ˆ\nl ll ll l\nlF J N S g B p \n \n (16) \ncaptures the mean field (first and second term) and the scalar virtual field \nz ppe imposed by \nthe constraint of Eq. (12). By taking advantage of the fact that all spin operators in Eq. (15) \ncommutate with each other, we obtain the expectation value \n00 1Tr exp . ˆˆ\nl l l\nlS F SF \n (17) \nfor any Fe3+ spin of sublattice \nl. Evaluation of this equation leads to an implicit relationship for \n0ˆ\nlS\n, \n0, ˆ\nl S lS S SF \n (18) \nwhere \n5 / 2S and \nS\n is the Brillouin function (Ref. 32). The magnetization \nlM of sublattice \nl \nis proportional to \n0ˆ\nllS where \nl is the number of \nl -Fe3+ ions per unit cell with \nd12 and \na8\n. \nFor the UC case (\n0p ), Eq. (18) is a system of two coupled equations and solved numerically, \nthereby yielding the unconstrained \nU0\nCˆ\nlS for each temperature \n0T . For the C case, the boundary \ncondition of Eq. (12), rewritten as \nC0\na C0\nd d a const ˆ ant, ˆSS \n (19) \nadds a third equation to Eq. (18) which determines \np. Note that in our experiment, we start from \nan UC equilibrium given by the \nU0\nCˆ\nlS at temperature \n0T . Subsequently, the pump pulse \nincreases \n0T by a small amount \nΔT while keeping the total spin angular momentum constant. \nSince we are only interested in small relative changes of the \n0ˆ\nlS , we linearize Eqs. (18) and (19) \nin terms of \nΔT and solve analytically for the small changes \n0\nC Δˆ\nlS . \nTo be consistent with our atomistic spin -dynamics model (see above), we choose the ratio of the \ncoupling parameters \nadJ , \naaJ and \nddJ to be identical to the values given in Ref. 15, but rescaled \nby a factor of 0.36 to fit the experimentally observed critical temperature of \nC398K T . For \ncomparison with our experiment, we finally determine the Farada y rotation resulting from the \nllNcalculated sublattice magnetizations \nlM (see Eq. (1)). The results of this procedure are shown and \ndiscussed in Figs. 5A and S6. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nACKNOWLEGMENT \n \nGeneral : We thank G.E.W. Bauer and S.M. Rezende for stimulating discussions . \nFunding : \nT.K. acknowledges funding through the grants European Researc h Council H2020 CoG \n“TERAMAG” (grant no. 681917) and H2020 FET -OPEN project “ASPIN ” (grant no. 766566) \nand DFG SFB/TRR 227 “Spin Dynamics” (project A05) . I.R. acknowledges funding through \nGerman Federal Ministry of Education and Research (BMBF) grant 05K16BCA “Femto -THz-X.” \nP.M. and P.M.O. acknowledge financi al support from the Swedish Research Council (VR), the K. \nand A. Wallenberg Foundation (grant no. 2015.0060), and the Swedish National Infrastructure for \nComputing. M.G. thanks for funding through BMBF grant no. 05K10KEB. R.V.P. acknowledges \nfinancial support from the Russian Scientific Foundation (grant no. 16 -12-10456). M.W. \nacknowledges funding by the Max Planck Society. \nAuthor contributions : I.R. conceived the original experiment idea. S.F.M ., I.R. and T.K. \ndesigned the experiment. S.F.M. built the setup , performed measurement s and analyzed data . I.R., \nA.M.K. and R.V.P. prepared the samples, which were characterized by S.F.M., I.R., A.M.K., A.P. \nand M.G. The th eoretical model was developed by T.K. and S.F.M., with contributions from J.B., \nP.M., A.P. and P.M.O. Atomistic spin -dynamics simulations were conducted by J.B. Equilibrium \nmagnetization curves were calculated by S.F.M. and T.K. Ab initio calculations of electronic and \nvibrational properties were performed by P.M. and P.M.O. The manuscript was written by T.K., \nJ.B. and S.F.M. , with contributions from A.P. , P.M.O., P.M. , M.W. and I.R . All authors \ncontributed to the discussion s of the results and commented on the manuscript . \n \n FIGURES \n \n \n \nFig. 1. Ultrafast probing of spin -phonon interaction s. (A) Experimental principle. A THz \npump pulse resonantly and exclusively excites optical phonons of a ferrimagnet. The impact \non the sample magnetization is monitored by the Faraday rotation θ of a subsequent \nfemtosecond probe pulse. By using an electric insulator, the electron ic orbital degrees of \nfreedom remain unexcited (see red crosses). ( B) Part of the unit cell of ferrimagnetic YIG. \nMagnetic Fe3+ ions at tetrahedral d -sites and octahedra l a-sites comprise, respectively, the \nmajority and minority spin-sublattice of the ferrimagnet. The pump pulse resonantly excites a \nTO(Γ) optical phonon associated with a Fe -O stretch vibration at the tetrahedral d -site. \n \n \n \n \nFig. 2. Ultrafast phonon -induced dynamics of magnetic order. (A) THz absorptance of the \nBiGa:YIG film (black solid line) . Pump intensity spectra are either resonant (red) or non-resonant \n(blue) with the TO( Γ) phonon absorption band. Open circles show the pump -induced Faraday \nsignal 10 ps after sample excitation as a function of the pump pulse center frequency. ( B) Pump -\ninduced change Δθ in Faraday rotation for resonant and off -resonant pumping on ultrafast and \n(C) microsecond time scales normalized to the equilibrium Faraday signal θ0=θ(−2 ps). The \nincident fluence is 10 mJ cm-2. Panel (B) also shows the isotropic transient change in the sample \ntransmittance for resonant pumping (thin black solid line) . Dashed lines in panels (B) and (C) are \nsingle -exponential fits with time constants of fast=1.6 ps and slow=90 ns, respectively. The inset \nof panel (C) displays the ultrafast Faraday signal Δθ(10 ps) as a function of the incident pump -\npulse fluence. Data are taken at a temperature of 296 K. \n \n \n \n \n \n \n \nFig. 3. Tw o regimes of spin -lattic e equilibration. (A) Equilibrium Faraday rotation \nθ0=θ(−2 ps) vs ambient temperature T0 along with a fit to an analytical function (thin solid \nline). ( B) Pump -induced change Δθ in the Faraday rotation at t=10 ps (red symbols) and 1 µs \n(blue) after pump -pulse arrival. The black curve is the change (∂θ0/∂T0)ΔT in the Faraday \nrotation expected from the increase ΔT of the sample temperature due to heating by the pump \npulse. θ0(T0) is taken from panel (A) (thin solid line), and ΔT=0.39 K is calculated from the \nabsorbed pump energy and the heat capacity of the excited volume. \n \n \n \n1\n0.8\n0.6\n0.4\n0.2\n00(T0)0(296 K)\n420 400 380 360 340 320\nTemperature T0 (K)Equilibrium\nFaraday signal 0(T0)TcA\nB\n12\n10\n8\n6\n4\n2\n0Faraday rotation 0(296 K) (10-3)\n420 400 380 360 340 320\nTemperature T0 (K) at 1 µs\n at 10 ps\n \n (0T0) T,\n T 0.39 K \n \n \nFig. 4. Atomistic spin -dynamics simulations . (A) Schematic of our model of ultrafast spin -\nphonon coupling. Thermal motion of the O2- ion modulates the superexchange constant Jad of \nthe adjacent a -Fe3+ and d -Fe3+ spins, thereby enabling transfer of spin angular momentum \nbetween a - and d -spin-sublattices. (B) Evolution of a - and d -spin-sublattice magnetizations \nas obtained by atomistic spin -dynamics simulations. From 0 to 0.5 ps, ther mal modulation of \nthe exchange constant Jad is switched on (orange square ). The variance of th e Jad fluctuation \nis proportional to the difference ΔT between crystal -lattice and spin temperature. To obtain \nagreement of the slope of ΔMa(t) with that found in the experiment directly after pump \nexcitation ( −0.1% ps-1, see Fig. 2B) where Δ T=0.39 K (see Fig. 3B), an approximately three \ntimes smaller ∂Jad/∂u of ~10 Jad Å-1 than used here has to be chosen. \n \n \n \n \nFig. 5. Constrained state and spin-phonon equilibra tion in YIG. (A) Calculated change in \nsublattice magnetization Ma and Md per increase of temperature T0, without and with the \nconstraint of constant spin angular momentum. The constrained and unconstrained ∂Md/∂T0 \ncurves exhibit good qualitative agreement with the measured Faraday signals Δθ(10 ps) and \nΔθ(1 µs) vs T0 shown in Fig. 3B. (B) Schematic of spin -phonon equilibration in YIG. \n[1] Pump -excited TO(Γ ) phonons [2] increase the population of other lattice modes. The \nincreased thermal modulation of the a-d-exchange by ΔJad(t) leads to [3a] transfer of angular \nmomentum between a - and d -spin-sublattices , accompanied by [3b] energy transfer from the \nphonon to the spin system on the time scale fast=1.6 ps. The resulting state is constrained by \nΔMa+ΔMd=0 and decays by [4] transfer of angular momentum and energy bet ween crystal \nlattice and electron spins on the slow=90 ns scale. 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In a broad sense, BMNs are \nnanoparticles of iron -containing materials that have sufficient magnetic susceptibility for movement in \nthe magnetic field of laboratory magnets10. These nanoparticles arecalled biogenic because their \nbiosynthesis is programmed at the genet ic level41-43. The main impetus for the study of biogenic \nferrimagnetism was the ability of m agnetotactic bacteria to move along the lines of force of the Earth’s \nmagnetic field (i.e. magnetotaxis)1,2. Subsequently, the idea of magnetotaxis was developed into the \nidea of magnetoreception of animals (i.e. their ability to orient in the geomagnetic field) due to the \npresence of BMN9,33. \nBut when examining organs and tissues for the presence of BMN, these magnetic nanoparticles \nwere found not only in those organs and tissues that may be responsible for the orientation of animals \nin the Earth’s external magnetic field (brain, beak of migratory birds, lateral line and ethmoid bone of \nmigratory fish), but also in a number of other organs, both migrat ory and non-migratory \norganisms8,14,19,22,24 ,25,28-30,33,37 -39,44-50 (Fig. 1, Table 1) . \n \n \nFig. 1. The organs of migratory fish, in which the presence of BMNs is experimentally shown: \nethmoid bone22,45, 46,51, brain19,22,45, muscle22,45,46, skin22,45,46,51, eye22,45,46,51, gill s45, heart45, liver45, \nintestine45, lateral line22,45. The organs of non -migratory fish, in which the presence of BMNs is \nexperimentally shown: ethmoid bone, brain, lateral line24,25. \n \nAll these numerous experimental data on the detection of BMN s in analogous and homologous \norgans of organisms of different phylogenetic groups confirm the existence of the theoretically \npredicted single genetically programmed mechanism of biosynthesis of BMN s in all organisms42,52,53. \nAt the same time, the theoretically predicted presence of BMNs in a number of human organs and \ntissues by the methods of comparative genomics50is confirmed by experimental studies on the presence \nof BMNs in human organs34,37,38, 54-56, as well as analogous and homologous organs and tissues of \nanimals8,14,19, 22,25,33, 44-47,49,50 (Table 1). \n \nTable 1. Human organs, as well as analogous and homologous organs of animals, in which BMNs are \nfound. \n \nHuman organs \nwith theoretically \npredicted BMNs \npresence (+) Experimentall\ny confirmed \nBMNs \npresence in \nhuman organs Experimentally confirmed BMNs presence in relevant \nor analogous fish organs \nBrain50 (+) \n Brain34,54,55 Brain of the Oncorhynchus nerka19, brain of the whale \nMegaptera novaeangliae44, \nhead Pachycondyla marginata14, brain of the silver carp \n(Fig. 9) \nHeart50 (+) Heart37 Heart of the Sus domestica49, fish heart45 \nLiver50 (+) \n Liver37 Liver of the Sus domestica49, fish liver45, liver of the \nhouse mouse Mus musculus (Fig. 2, 3) \nSpleen50 (+) Spleen37 Spleen of the Sus domestica50 \nEthmoid bone50 (+) \n Ethmoid \nbone56 Ethmoid bone of the Salmo salar , ethmoid bone of the \nCyprinus carpio , ethmoid bone of the Esox lucius25, \nethmoid bone of the Oncorhynchus nerka8, ethmoid bone \nof the Delphinus delphis33, ethmoid bone of the \nThunnus albacares46, antennas of the \nPachycondyla marginata14 \nAdrenal glands50(+) Adrenal \nglands38 \nLung50(+) Lung of the Sus domestica49, fish gills45 \nKidney50 (+) Kidney of the Sus domestica50 \nIntestine50 (+) Intestine of the fish45, abdominal cavity of the ants14, \nintestine of the house mouse Mus musculus (Fig. 5) \nMuscle tissue50(+) Muscle of the fish22,46 \nSkin50 (+) Skin of the Salmo salar22, skin of the Oncorhynchus \nnerka47, skin of the Thunnus albacares46 \nEyes Eyes of the fish22,45 \nJoints Joints of the Pachycondyla marginata14 \n \nIn this work, the localization of BMN s was investigated using atomic force (AFM) and magnetic \nforce microscopy (MFM) in a number of organs of various organisms (animals, plants, fungi) in order \nto identify which systems of multicellular organisms include BMNs . \n \nExperiment and discussion \nIn this paper, the following samples were examined using AFM and MFM , namely, animal samples : \nliver, intestine, pancreas of mouse , lungs, kidneys, spleen of pig , brain of carp ; plant samples: leaves \nand root of tobacco , stem and tuber of potato ; and fungi samples: fruit bodies of the common and \nshiitake mushroom s. The cells and other structural components (shown in the images by arrows) in the \nsamples of the listed organs were id entified and the localization of BMNs in the indicated organs was \nshown. \nThe results of studies of samples of the liver of a mouse with the use of AFM and MFM showed \nthat BMNs in the liver of a mouse are located in the wall of sinusoids (capillaries of the liver) (Fig. 2 -3 \nand Table 2). Sinusoids57, sinusoidal endothelial cells of the liver58, nuclei of sinusoidal endothelial \ncells of the liver57,59, hepatocytes, nuclei of hepa tocytes57, fenestra e58-61presented in Fig. 2 -3 in this \npaper have typical morphology and dimensions, characterized in the works57-61. \n \n \n \n \n \n Table 2. Dimensional characteristics of the structura l components of the mouse liver \n \nStructural components \n Calculated dimensions \n(see Fig. 2 -3), µm Dimensions in \nliterature , µm References \nCapillary density 400-600/µm2 400-500/µm2 62 \nSinusoids 5-20 5-15 63-65 \nSinusoidal endothelial cells 15-25 20-25 58 \nHepatocytes 15-25 20-30 58 \nHepatocyte nuclei 5-8 6-8 58 \nFenestrae 200-500 50-500 58-61 \n \n \n \nFig. 2. AFM (left) and MFM (right) images of a sample of the liver of the house mouse Mus musculus : \n1 hepatocyte, 2 nuclei of hepatocyte s and 3 sinusoids. \n \n \n \nFig. 3. AFM (left) and M FM (right) images of a sample of the liver of the house mouse Mus musculus : \n1 sinusoidal endothelial cells of the liver, 2 nuclei of sinusoidal endothelial cells of the liver, 3 \nfenestra e and 4 sinusoids. \n \nSimilar results were obtained on the localization of BMNs in the samples of the pancreas and \nintestines of a mouse. So, BMNs in the pancreas and the intestines of the mouse are located in the wall \nof the capillaries as can be seen from Figures 4 -5 and data on the size and dist ance between the \ncapillaries in Tables 3 -4. The capillaries of the pancreas are shown in Fig. 4 in this paper and have \ntypical morphology and size66-68. \n \n \n \n \n \n \n1 2 1 \n3 \n1 \n2 3 4 Table 3. Dimensional characteristics of the structural components of the mouse pancreas \n \nStructural components \n Calculated dimensions \n(see Fig. 4), µm Dimensions in literature , \nµm \n References \nDistance between the \ncapillaries 10-30 20-50 66 \nCapillaries 4-10 3-7 66-68 \n \n \n \nFig. 4. AFM (left) and MF M (right) image s of a sample of the pancreas of the house mouse Mus \nmusculus : arrows indicate the capillaries. \n \n Capillaries of the intestine are presented in Fig. 5 in this work and have also typical \nmorphology a nd size, described in the work69,70. \n \nTable 4. Dimensional characteristics of the structural components of the mouse intestine \n \nStructural components \n Calculated dimensions \n(see Fig. 5), µm Dimensions in literature , \nµm References \nDistance between the \ncapillaries 5-30 30-40 71,72 \nCapillaries 3-6 3-5 69,70 \n \n \n \nFig. 5. AFM (left) and MFM (right) images of a sample of the intestine of the house mouse Mus \nmusculus : arrows indicate the capillaries. \n \n AFM and MFM examinations of porcine specimens gave similar results, namely, BMNs in the \nlungs (Fig. 5 and Table 6), kidneys (Fig. 6 and Table 7) and spleen (Fig. 7 and Table 8) of the pig are \nlocated in the capillary wall. So, capillaries of lungs, are pre sented in fig. 6 in this paper and have \ntypical morphology and size, described in the works73,74. \n \nTable 5. Dimensional characteristics of the structural components of the pig lung \n \nStructural \ncomponents Calculated dimensions \n(see Fig. 6), µm Dimensions in literature , \nµm References \nDistance between the \ncapillaries 5-25 10-20 73 \nCapillary density 600-900/µm2 700-900/µm2 75-78 \nCapillaries 5-10 5-20 73,75 \n \n \n \nFig. 6. AFM (left) and MFM (right) images of a sample of the lungs of a domestic pig Sus domestica : \narrows indicate the capillaries. \n \n The capillaries of the kidneys are presented in Fig. 7 in this paper and have typical morphology \nand size, described in79. \n \nTable 6. Dimensional characteristics of the structural components of the pig kidney \n \nStructural components \n Calculated dimensions \n(see Fig. 7), µm Dimensions in literature , \nµm \n References \nCapillary density 100-300/µm2 25-220/mm2 80,81 \nCapillaries 7-10 3-20 82,83 \n \n \nFig. 7. AFM (left) and M FM (right) image s of a sample of thekidney of the domestic pig Sus \ndomestica : arrows indicate the capillaries. \n \n The c apillaries of a spleen, are presented in fig. 8 in this work, have typical morphology and \nsize, described in84,85. \n \nTable 7. Dimensional characteristics of the structural component s of the pig spleen \n \nStructural components \n Calculated dimensions \n(see Fig. 8), µm Dimensions in literature , \nµm References \nDistance between the \ncapillaries 5-10 10-30 86 \nCapillaries 4-7 4-10 87-90 \n \n \n \nFig. 8. AFM (left) and MFM (right) images of a sample of the spleen of the domestic pig Sus \ndomestica : arrows indicate the capillaries. \n \nAs noted above, BMNs are most thoroughly investigated in the organs and tissues of fishes. \nHowever, only transmission electron microscopy (TEM) and MFM provide information on the \nlocation of BMNs among all experimental methods that are used today to study BMNs34,35,91 -114. This \npaper presents the results of AFM and MFM studies of samples of the silver carp brain, and it is shown \nthat BMNs are located in the capillary wall (Fig. 9 and Table 8) in the brain of the silver carp, as in the \norgans of mice and pigs . The c apillaries of brain are p resented in fig. 9 in this work and have typical \nmorphology and dimension s, characterized in115. \n \nTable 8. Dimensional characteristics of the structural components of the silver carp brain \n \nStructural components \n Calculated dimensions \n(see Fig. 9), µm Dimensions in literature , \nµm References \nDistance between the \ncapillaries 10-30 15,3 116 \nCapillaries 3-9 3-8 116,117 \n \n \n \nFig. 9. AFM (left) and MFM (right) images of a sample of the silver carp brain of Cyprinus carpio : \narrows indicate the capillaries. \n \nThe BMNs in the human ethmoid bone are located in the sinuses of the bone56, through which \nthe capillaries and nerves pass118-120. The location of BMN s in the ethmoid bone of fish es is similar25. \nThus, a common feature of the location of BMNs in the ethmoid bone and in other organs of various \nanimals (Fig. 2 -9, Table 1) is their localization in the vicinity of the capillaries. \nDue to the fact that for many years, BMN s was investigated mainly in connection with the \nideas about magnetotaxis and magnetoreception, BMN s in plants and fungi has been little studied. For \nexample, the paper11presents experimental data on the detection of magnetically sensitive \nnanoparticles in plants, but the authors of this work consider it more likely that they have detected \nphytofer ritin and its aggregates, and not plant BMNs. As for fungi, BMN s is experimentally detected \nonly in the microfungi Fusarium oxysporum and Verticillium spp .121. \nThe study of BMNs in plants was carried out on the example of tobacco, as the most studied \nmodel organism among plants, as well as potatoes. The BMNs in the leaf and the roots of tobacco (Fig. \n10-11) are located on the membrane of phloem sieve tubes. Phloem is a conducting plant tissue that \nforms a network of sieve tubes through which organic substances , synthesized by leaves during \nphotosynthesis , are transported to all plant organs122-126, unlike xylem, which transports water and \nminerals from the soil127-130. Sieve tubes of tobacco leaf are presented in Fig. 10 in this paper and have \nthe typical morphology and dimensions described in131. \n \n \n \nFig. 10. AFM (left) and M FM (right) image s of a sample of a leaf of the cultivated tobacco Nicotiana \ntabacum . Elements of sieve tubes, including pores, are visible in the AFM image. White arrows in the \nAFM image indicate the membrane of the sieve tubes, red arrows indicate the pores of the sieve tubes. \n \n \n \nFig. 11. AFM (left) and MFM (right) images of a sample of a root of the cultivated tobacco Nicotiana \ntabacum . Elements of sieve tubes, including pores, are visible in the AFM image. White arrows in the \nAFM image indicate the membrane of the sieve tubes, red arrows indicate the pores of the sieve tubes. \n \nSimilar data on the location of BMNs were obtained on the s amples of potatoes. The BMNs in \nthe potato stalk are located in the walls of the conducting tissue (phloem) (as can be seen from Fig. 12 -\n13), wh ich is characterized in132. BMNs in a potato tuber are located at the boundary between starch \ngrains133 and conducting tissue (phloem). \n \n \n \nFig. 12. AFM (left) and MFM (right) images of a sample of the stem of a potato Solanum tuberosum : \nconducting tissue/ sieve tubes (white arrows) phloem, cell wall (red arrows). \n \n \n \nFig. 13. AFM (left) and MFM (right) images of a sample of the tuber of a potato Solanum tuberosum : \nconducting tissue/sieve tubes (white arrows) phloem, cell wall (red arrows). \n \nThe study of BMNs in mushrooms was carried out using the example of higher mushrooms, \nsuch as mushroom s Agaricus bisporus and Lentinula edodes , which are the most common edible \nmushrooms. As can be seen from Fig. 14 -15 BMN s in the mushroom are located in the walls of the \nvascular hyphae, which are characterized in the works134-136. \n \n \n \nFig. 14. AFM (left) and MFM (right) images of a sample of the common mushroom Agaricus \nbisporus : arrows indicate the hyphae. \n \n \n \nFig. 1 5. AFM (left) and MFM (r ight) images of a sample of the shiitake mushroom Lentinula edodes : \narrows indicate the hyphae. \n \nThus, studies of BMNs in samples of organs and tissues of animals, plants and fungi, conducted \nin this work showed that: \n• BMN in the organs of multicellular organisms form chains; \n• BMN in multicellular organisms are part of the transport system. Thus, BMNs in animals are located \nin the walls of capillaries (all examined organs and tissues except the ethmoid bone) or in the vicinity \nof the capillaries (the ethmoid bone). BMNs in plants are located in the wall of the conducting tissue, \nnamely in the wall of sieve tubes of the phloem. BMNs in fungi are located in the wall of the \nconducting tissue, namely in the walls of the vascular hyphae134-136. \n \nConclusions \nThe chains of BMN are components of the cells that form the w alls of the capillaries of animals and \nthe walls of the conducting tissue of plants and fungi. At the same time, the functions of the capillaries \nof animals and the functions of the conducting tissue of the phloem of plants, as well as the vascular \nhyphae of fungi, are similar122,137,138. In particular, capillaries (exchange vessels) exchange nutrients, \ngases, liquids, metabolites, signal substances (hormones), immune cells, etc. between the blood and \nbody tissues137,139 -141. Conducting tissue of the phloem of plants serves to transfer organic substances, \nhormones, etc. throu gh the body122-126. The same functions have vascular hyphae of fungi138,142 -144. The \nlocation of BMNs in various organs and tissues of animals, plants and fungi as p art of systems with \nsimilar functions cannot be accidental, taking on account that the genetically programmed mechanism \nof biosynthesis of BMNs arose at the beginning of evolution42,43,145 -147. Such localization of BMNs \nargues in favour of the idea that BMNs chains are directly involved in metabolic processes25,42,43,148 -150 \nand perform vital functions. In addition, bioinformatics analysis showed that BMNs in multicellular \norganisms, as well as in magnetotactic bacteria151-154, form inside the cell and are associated with the \ncell membrane8,9,15,34,39. This means that the chains of BMNs in multicellular organisms are part of a \nnew type of organelles with the ferrimagne tic properties – the ferrimagnetic organelles of a specific \npurpose. Such ferrimagnetic organelles create scattering magnetic fields of the order of several \nfractions of T and gradients of magnetic field induction of the order of 105-106 T/m in their vicinity, \nwhich can significantly affect the mass transfer processes near the cell membrane of vesicles, \ngranules41,42,150, organelles, structural elements of the membrane and others. \nIn connection with the foregoing, it is clear that an extremely important task is to search for the \ngeneral functions of BMNs in various organs and tissues of multi cellular organisms. To solve this \nproblem, it is necessary to identify the types and functions of cells containing BMNs in various organs \nof multicellular organisms. \n \nMaterials and Methods \nAtomic force microscopy and magnetic force microscopy. Determinatio n of the presence of BMNs \nand the study of their localization in the investigated samples was carried out using the «Solver PRO -\nM» scanning probe microscope by atomic force microscopy (AFM) and magnetic force microscopy \n(MFM). \nThe magnetic probe MFM_LM series with chip size 3.4x1.6x0.3mm, coated by CoCr was used. \nThis probe was used both for AFM and MFM imaging. The non -contact AFM (NC -AFM) mode was \napplied. The MFM scanning was carried out at the constant distance from the samp le surface after \nAFM scanning. The probe “lift” height was 100 nm. The cantilever was calibrated using the test \nsamples. Calibration of the probe was carried out immediately before the measurements. \nThe biological material was prepared before AFM and MFM s canning. Fixation of biological \nmaterial was carried out in a 10% formalin solution. The duration of fixation was 24 hours. 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" }, { "title": "1303.5585v1.Linear_perturbation_renormalization_group_method_for_Ising_like_spin_systems.pdf", "content": "arXiv:1303.5585v1 [cond-mat.stat-mech] 22 Mar 2013CondensedMatterPhysics,2013,Vol.16,No1,13704:1–8\nDOI:10.5488/CMP.16.13704\nhttp://www.icmp.lviv.ua/journal\nLinearperturbationrenormalizationgroupmethod\nforIsing-likespinsystems\nJ.Sznajd\nInstituteforLowTemperatureandStructureResearch,Poli shAcademyofSciences,Wroclaw\nReceivedJuly4,2012,infinalformJanuary3,2013\nThelinearperturbationgrouptransformation(LPRG)isuse dtostudythethermodynamicsoftheaxialnext-\nnearest-neighborIsingmodelwithfourspininteractions( extendedANNNI)inafield.TheLPRGforweaklyin-\nteractingIsingchainsispresented.Themethodisusedtost udyfinitefieldpara-ferrimagneticphasetransitions\nobservedinlayereduraniumcompounds,UAs 1−xSex,UPd 2Si2orUNi 2Si2.Theabove-mentionedsystemsare\nmadeofferromagneticlayersandthespinsfromthenearest- neighborandnext-nearest-neighborlayersare\ncoupledbytheantiferromagneticinteractions J1<0and J2<0,respectively.Eachofthesesystemsexhibitsa\ntriplepointinwhichtwoorderedphases(ferrimagneticand incommensurate)meettheparamagneticone,and\nallundergothehighfieldphasetransitionfrompara-toferr imagnetic( ++−)phase.However,ifinUAs 1−xSex\nthepara-ferriphasetransitionisofthefirstorderasexpec tedfromthesymmetryreason,inUT 2Si2(T=Pd,Ni)\nthistransitionseemstobeacontinuousone,atleastinthev icinityofthemulticriticalpoint.WithintheMFA,the\ncriticalcharacterofthefinitefieldpara-ferrimagnetictr ansitionatleastatoneisolatedpointcanbedescribed\nbytheANNNImodelsupplementedbyanadditional,e.g.,four -spininteraction.However,inLPRGapproxima-\ntionfortheratio κ=J2/J1around 0.5thereisacriticalvalueofthefieldforwhichanisolatedcri ticalpoint\nalsoexistsintheoriginalANNNImodel.Thepositivefour-s pininteractionshiftsthecriticalpointtowardshigher\nfieldsandchangestheshapeofthespecificheatcurve.Inthel attercaseforthefieldssmallenough,thespecific\nheatexhibitstwo-peakstructureintheparamagneticphase .\nKeywords:ANNNImodel,renormalizationgroup,isolatedcriticalpoi nt\nPACS:75.10.Hk,75.40.Cx\n1.Introduction\nTheLinearPerturbationRenormalizationGroup(LPRG)[1]m ethodusesasimpleone-dimensional\ndecimationtostudyuniversal(critical)andnon-universa lsuchasalocationofthecriticaltemperature\nandtemperatureorfielddependenceofthepropertiesofther modynamicquantitiesofseveralclassical\nandquantumhigher-dimensionalmodels.Forthefirsttime,t hiskindofmethodwasproposedbySuzuki\nandTakano(ST)[2].IntheSTapproachtheone-dimensionald ecimationiscombinedwiththeMigdal-\nKadanoff(MK)bondmovingapproximation.Thedisadvantage softhelattermethodespeciallyforthe\nquantumsystemswerediscussedbyBarmaetal.[3]andCastel lanietal.[4].Here,wewishonlytoremind\nthattheMKapproachgivesratherpoorquantitativeresults evenforthetwo-dimensionalIsingmodel\nandthereisnopossibilitytoconstructanysystematicappr oximationprocedurewithinthismethod.\nForexample,theMKproceduregivesfortheIsingmodelonthe squarelatticethevaluesoftheinverse\ncriticaltemperature kc≈0.61whereastheexactvalueis kc≈0.44andforthe s=1\n2XYmodel kc≈\n1.2[2]muchlargerthanthevalue kc≈0.64estimatedfromthehigh-temperatureseriesexpansion[5],\nkc≈0.71(67)foundfromMonteCarlosimulationsbyfittingtotheexponent iallaw[6,7]andpowerlaw\n[8],respectivelyorrotationallyinvariantnon-linear(b lock)transformation kc≈0.62[9,10].TheLPRG\nmethodhasbeenproposedfortheso-calledquasi-one-dimen sionalmagnetsmadeofspinchainswith\ntheintrachaincoupling kandmuchweakerinterchaincoupling k1k1>0.15 k,thedeviationfromthecriticaltemperatureexactvaluesi sless\nthan 2%andfor 0.5k>k1>0.15 kevenlessthan 1%[11].\nInthispaper,theLPRGisusedtostudythethermodynamicsan dtheexistenceofacriticalpointin\ntwo-dimensionalaxialnext-nearest-neighbourIsingmode lwithfourspininteractions(extendedANNNI)\ninafield.\n2.LPRG\nWefirstdescribeindetailtheLPRGapproachinthesimplestp ossiblecase,i.e.,theIsingchainswith\nintrachaininteraction Jcoupledbytheweakinterchaininteractions J1definedbytheHamiltonian\nH=k/summationdisplay\n〈i j〉Si,jSi,j+1+k1/summationdisplay\n〈i j〉Si,jSi+1,j, (2.1)\nwherethelabel ireferstorowsand jreferstocolumns,thefactor −1/kBThasalreadybeenabsorbed\nintheHamiltonian( k≡J/kBT,k1≡J1/kBT),and k1hc(h=0.15infigure3),specificheathasonlyabroadhump.\nThismeansthattheconsideredANNNIinthepresenceoftheex ternalfieldcanexhibitacontinuous\nphasetransitiontotheferrimagneticphase (++− )ath=0and h=hc≈0.11.Thus,withintheLPRG\nunlikeinMFAitisnotnecessarytosupplementtheoriginalA NNNImodelbyanadditionalfour-spin\ninteractiontoreachanisolatedcriticalpointfora finitefieldaround h=0.11[21].\nNow,weproceedtotheextendedANNNImodelwith k=1,k1=−0.6,k2=−0.3and k4/nequal0.Infigure4\nwepresentthetemperaturedependencesofthespeci ficheatinzerofieldforseveralvaluesofnegative\nandpositivevaluesof k4.Asseen,thefour-spininteractionsimplyshiftsthecriti calpointtowardshigher\ntemperaturefor k4>0andtowardslowertemperaturefor k4<0.Figure5showsthetemperaturede-\npendenceofthespeci ficheatforthemodelwith k4=0.2infinitefields.SimilarlytothestandardANNNI\nmodelcase,theapplicationoftheexternal fieldchangesthedivergenceofthespeci ficheatintoamaxi-\nmumfor 0 0)\nExternal easy-axis field (Oe)-200-1000100200M1x / MS01-0.50.5-1−200−1000100200−1−0.500.51Measured\nH+1H\u00001H+2H\u00002(1)\n1H+1H\u00001H+2H\u00002(1)\n1H+1H\u00001H+2H\u00002(1)\n1H+1H\u00001H+2H\u00002(1)\n1ε < 0−200−1000100200−1−0.500.51c\nSymmetric case−200−1000100200−1−0.500.51\nEasy−axis field (Oe)Simulated easy−axis hysteresis Mx/Ms−200−1000100200−1−0.500.51\nEasy−axis field (Oe)Simulated easy−axis hysteresis Mx/Ms−200−1000100200−1−0.500.51\nEasy−axis field (Oe)Simulated easy−axis hysteresis Mx/Ms−200−1000100200−1−0.500.51\nEasy−axis field (Oe)Simulated easy−axis hysteresis Mx/Ms−200−1000100200−1−0.500.51\nEasy−axis field (Oe)Simulated easy−axis hysteresis Mx/Ms−200−1000100200−1−0.500.51\nEasy−axis field (Oe)Simulated easy−axis hysteresis Mx/Ms−200−1000100200−1−0.500.51\nEasy−axis field (Oe)Simulated easy−axis hysteresis Mx/Ms−200−1000100200−1−0.500.51\nEasy−axis field (Oe)Simulated easy−axis hysteresis Mx/Ms00.20.40.60.81x 10−8−180−90090180\n Top free layerBottom free layer\n00.20.40.60.81x 10−8−180−90090180\n00.20.40.60.81x 10−8−180−90090180\n00.20.40.60.81x 10−8−180−90090180\n2468100Time (ns)Simulated magnetic response of layers during switching-90-180090180Angle with EAbBottom free layerTop free layer\nStart excitationFIG. 4: (Color online) a, Switching probability map for the\nsample parameters given in the \fgure, showing a clear separa-\ntion of the two AP states due to a biasing \feld and thickness\nasymmetry. b, Time trace of the two macrospins in the bilayer\nduring a switching event, showing that the switching occurs\nvia an in-phase rotation (acoustical mode), with superposed\noptical oscillations of smaller amplitude. c, Simulated easy-\naxis hysteresis for a sample with a thickness imbalance and\nbiasing \feld asymmetry for \" >0 (main panel), correspond-\ning to our typical experimental con\fguration (left inset), and\nfor\" < 0 (top-right inset). In contrast, an ideal, symmet-\nric spin-\rop bilayer shows a strictly anti-symmetric response\n(bottom-right inset; simulated).\nvious work on the energetics of the SAF system1,28. The\nmagnetization vectors evolve in time driven by torques\nof the respective e\u000bective magnetic \felds, with a suit-\nably small time step of 5 ps. A thickness imbalance and\nfringing \feld are the two asymmetry parameters. The\nsimulations include thermal agitation at T= 300 K.\nIn the case of no thickness imbalance, with the only\nasymmetry due to the unequal biasing \felds on the indi-\nvidual free layers (from the reference layer built-in into\nthe nanopillar), the numerically simulated switching be-\nhavior agrees well with that predicted analytically above.\nThe optical resonance is clearly split, although the mag-\nnitude of the splitting is somewhat smaller.6\nA more pronounced resonance splitting is obtained by\nadding a thickness-imbalance to the magnetic asymme-\ntry of the system. Figure 4a shows a typical simulated\nswitching map. The area where switching occurs only in\none direction is where only one ground state is dynami-\ncally stable. The general form of the stability regions sim-\nulated here for large-signal microwave excitation agrees\nwell with the analytical results.\nFigure 4b shows the real-time trace of the magneti-\nzation of the individual layers during a switching event.\nIt is clear that the microwave-pumped oscillations are of\noptical nature but the switching itself is an \"acoustical\"\nin-phase rotation. The oscillations of the magnetization\nin the initial ground state under a resonant excitation\nof amplitude suitably large to produce switching are, in\nfact, smaller in the switched-to ground state since this\nstate is now o\u000b-resonance with the microwave \feld. This\ne\u000bect of resonant state selection is a direct consequence\nof the magnetic asymmetry of SFi.\nThe full macrospin model further yields important in-\nsights into the quasistatic magneto-resistance properties\nof the junctions. Using Eq. S1 the quasistatic switching\n(H\u0006\n1) and spin-\rop \felds ( H\u0006\n2) can be obtained and used\nfor extracting the asymmetry parameters (individual\nlayers' thicknesses and e\u000bective biasing \felds) from\nthe measured data. The interrelation between the\nthicknesses and fringing \felds is (in notations of Fig.\n4(c)):\nH\u0006\ni\n4\u0019MS=\u0007(\u00001)ij\"j(Ny\u0000Nx\u0000\rx)\u0000hf\n\u0006s\u0012\nNy\u0000Nx+\rx+hu\u0006(\u00001)i\"\nj\"jhd\u00132\n\u0000(1\u0000\"2)\r2y;\nj\"j\u0019h+\n1\u0000h\u0000\n1\u0000h+\n2+h\u0000\n2\n4(Ny\u0000Nx\u0000\rx); hf\u0019\u0000(h+\n1+h\u0000\n1+h+\n2+h\u0000\n2)\n4:\nHereH\u0006\n1= 4\u0019MSh\u0006\n1is the \feld for which only one AP\nground state becomes unstable and H\u0006\n2= 4\u0019MSh\u0006\n2is\nthe spin-\rop \feld, at which both AP ground states be-\ncome unstable. Note that the actual values of H\u0006\n1;2are\nslightly smaller than the analytical values due to thermal\n\ructuations.\nIV. EXPERIMENTS\nA. Samples\nThe samples used in the experiments were spin-\rop\ntype magnetic random access memory (TMRAM) cells\nwith lateral dimensions of 450x375 and 420x350 nm, and\nelliptical in-plane shape. The stack consists of a magnet-\nically soft SFi separated from a magnetically hard SAF\nreference layer by a thin Al-O tunnel barrier. The spin-\n\rop bilayer is composed of two dipole coupled permalloyferromagnets (free layers) with thicknesses t1\u0019t2\u00195\nnm (excluding the magnetically 'dead' layers at the \flm\nsurfaces29) separated by a \u00181 nm thin TaN spacer, for\nwhich there is no interlayer exchange coupling, J= 0.\nHeret1andt2are the e\u000bective thicknesses of the bottom\nand top free layer, respectively. The SFM reference layer\nis nearly ideally \rux-closed and is therefore essentially\ninsensitive to external \felds of strengths used in the ex-\nperiments discussed in this paper. The samples used in\nthis work had a weak fringing \feld from the reference\nlayer acting on the two free layers. The fringing \feld\noriginated predominantly from the top \fxed layer. The\nresistance of the stack was \u00181 k\n, and the magnetoresis-\ntance\u001820%.\nB. Measurement setup\nIn-plane quasi-static \felds were applied using an exter-\nnal toroidal magnet30. High-frequency \felds were gen-\nerated by contacting the on-chip integrated 50 \n bit-\nand word-lines using surface probes with 0-40 GHz band-\nwidth. The bit- and word-lines were oriented at \u000645\u000e\nwith respect to the the easy-axis of the stack in the so-\ncalled \"toggling\" con\fguration. A current through the\nbit- or word-line results in an in-plane magnetic \feld\nperpendicular to the line. In the results presented be-\nlow, the high frequency excitation was applied using the\nword line only, which was electrically decoupled from the\nstack. The applied high-frequency \felds were AC pulses\nof desired frequency ranging from 1 MHz to 10 GHz. The\nAC \felds were generated using an Agilent E8247C PSG\nCW Signal Generator with a bandwidth of 250 kHz - 20\nGHz. The wave-duration was controlled using an RF-\nswitch driven by an Agilent 33250A AWG. The shortest\npossible pulse-length with this setup was 8 ns. The state\nof the sample was determined by measuring the resis-\ntance of the sample and thereby the relative orientation\nof the magnetization of the bottom free and top \fxed\nlayers. To separate the DC-signal from the RF-signal, a\nbias-tee was used.\nC. Experimental results\nThe samples were SFi bilayers of approximately 5 nm\nthick Permalloy, separated by 1 nm thick TaN, inte-\ngrated into approximately 350x400 nm in-plane elliptical\nnanopillars, also containing an Al-O read-out junction\nwith a nominally \rux-closed magnetic reference layer.\nThe microwave excitation was applied using a closely\nspaced \feld-line making a 45\u000eangle with the SFi easy\naxis. The ferromagnetic resonance frequencies were mea-\nsured using transport spectroscopy, where the junction\nresistance versus frequency showed a maximum or a min-\nimum for the low-resistance or high-resistance ground\nstate, respectively. The resistance was measured while\nslowly sweeping the frequency of a continuous-wave ex-7\n01.02.03.04.005101520Time (s)Resistance response of 420x350nm sample during a sequence of100ns excitations with 58G amplitude and frequencies 3.15GHz, 3.65GHz Resistance (1)\n05101520Applied frequency (GHz) DC-response of 420x350 nm sample during 100 ns excitationsHy = 58 Oe, f1 = 3.15 GHz, f2 = 3.65 GHz, Hx = -20 Oe1.81.61.71.41.5Time (s)02015105Resistance (kΩ)Frequency (GHz)c\nMeasured switching probability 300ms excitation,450x375 nm sample, -20 Oe easy axis field applied\nFrequency (GHz)a u (41p2)h / (41p2)Stability criterion underresonant excitation\n \n−60600.1 l = ( 2, 0, 0 ) l = ( −2, 0, 0 )\na u (41p2)h / (41p2)Stability criterion underresonant excitation\n \n−60600.1 l = ( 2, 0, 0 ) l = ( −2, 0, 0 )Low → HighresistanceHigh → LowresistanceAmplitude (Oe)Switch probability\n01b706080\n503.23.64.04.4\n402080600080Time (s)Resistance response of 420x350nm sample during a sequence ofExcitation amplitude Resistance (1)\n020406080Applied frequency (GHz) 080DC-response of 420x350 nm sample during continuous excitation withf = 4.4 GHz, Hy = 32 Oe, applied EA-field alternates: Hx = -35, 0, 35 Oe1.91.71.81.6\nTime (s)-35035Resistance (kΩ)EA-field (Oe)320Amplitude (Oe)dExperimental optical resonance frequencies of 420x350 nm sampleExcitation amplitude of Hy = 25 Oe\nEA-field (Oe)Resonance frequency (GHz)3.84.2\n3.64.0\n-3030-20020-1010a\nLow-resistanceHigh-resistance\nFIG. 5: (Color online) a, Measured resonance frequencies of a 420x350 nm spin-\rop bilayer. The intensity map (red and blue\nfor the two ground states) is normalized to the maximum oscillation amplitude for the given EA-\feld. b, Switching probability\nfor the two ground states subject to 300 ms long excitation pulses with -20 Oe static EA-\feld applied. Black color denotes\nthe frequency-amplitude points, for which switching in both directions occurred. The inset shows the section of the analytical\nswitching map (Fig. 3) covered in the measurement. c, Real time resistance traces of a sample excited by short microwave pulses\nof alternating frequencies in resonance with the split optical mode, showing controlled resonant state selection. d, Resonant\nstate selection using resonant microwave pumping of \fxed frequency and alternating external EA-\feld, tuning the two ground\nstates in and out of the optical spin resonance.\ncitation of \fxed amplitude (typically 25 Oe). Figure 5a\nshows the measured resonance frequencies for the two AP\nground states of a SFi with M0<0,Hm\n1\u00197 Oe,Hm\n2\u00193\nOe andt1\u0000t2\u00190:9 nm (values deduced from quasistatic\nEA-hysteresis as shown in \fg 4c). The splitting of the op-\ntical spin resonance at \u00194 GHz is in excellent agreement\nwith the predicted behavior (Fig. 3).\nFigure 5a shows the measured oscillation amplitude as\na function of the excitation frequency and applied bias\n\feld. The amplitude is normalized with respect to the\nmaximum value for each bias \feld. This diagram clearly\nshows that both \feld-controlled and frequency-controlled\nswitching is possible.\nFrequency-amplitude-switching maps were measured\nfor both AP states (low- to high-resistance and high-\nto low-resistance) for di\u000berent excitation pulse durations\nand static EA-\felds. Figure 5b shows the switching maps\nfor both AP states of the soft bilayer under 300 ms pulses\nof resonant excitation, with Hx=\u000020 Oe. Switching\nat frequencies lower than 3.9 GHz only occurs from the\nlow- to high-resistance AP state ( fL!H), while switch-\ning at above 3.9 GHz only occurs from the high- to low-\nresistance state ( fH!L). Shortening the duration of theexcitation pulse requires a higher amplitude for switch-\ning the junction, at the same time leading to an increase\nin the error-rate likely due to a more non-linear and\nmore non-uniform process for higher-amplitude pumping.\nNevertheless, even for short excitations, the optical res-\nonance is clearly split and amplitude-frequency regions\nexist where the switching is strictly one-directional. The\nweaker higher-order oscillations in the probability maps\nof Fig. 5b are due to secondary spin-wave excitations\nin our nanoparticles that are somewhat bigger than the\ntrue single-domain limit, which we also observe in our\nmicromagnetic simulations as a small superposition on\nthe well-de\fned macrospin behavior.\nFigure 5c shows a realtime trace of the junction resis-\ntance during the application of a sequence of microwave\npulses of 100 ns duration, of frequency alternating be-\ntween the split optical spin-resonance peaks. A pair of\npulses is sequentially applied at each frequency, demon-\nstrating the strictly unidirectional character of resonant\nswitching at a given frequency (for a given AP state of\nthe SFi).\nFigure 5d shows a realtime resistance trace (blue) un-\nder continuous microwave excitation of \fxed frequency8\nnear the centre of the optical resonance, with the EA-\n\feld of amplitude su\u000ecient for tuning the two AP ground\nstates in and out of the resonance. Between the EA-\npulses the excitation is turned o\u000b shortly to con\frm the\nstate the sample is in (microwave amplitude shown in red;\nEA-\feld in green). The EA-\feld pulses were generated\nwith an external electromagnet, so their shortest dura-\ntion was 300 ms. The data show that the \feld-controlled\nresonance state selection is highly reliable.\nBoth measurements were repeated many times with\nthe resulting error-rate smaller than 10\u00003(1000 mea-\nsurements were performed without any error). The ef-\nfect is thus essentially deterministic for the sample and\nfrequency-\feld parameters used, and still is expected to\nimprove (e.g., in terms of speed) on scaling down the\nsample size.\nV. CONCLUSIONS\nWe show how external microwave and static biasing \felds\ncan be used to tune the energetics and dynamics of a\nspin-\rop system between symmetric and asymmetric be-havior. We demonstrate controlled state selection in a\nsynthetic ferrimagnet using a frequency modulated reso-\nnant microwave \feld or a uniform external biasing \feld.\nThis e\u000bect is explained analytically using a small-signal\nanalysis of the SFi spin dynamics, predicting a tunable\nsplitting of the optical spin resonance, which is in good\nagreement with our numerical arbitrary signal-strength\nsimulations. The e\u000bect is fast, robust, and o\u000bers a new\nway of magnetic switching of spin-\rop nanodevices used\nin such large scale applications as magnetic random ac-\ncess memory.\nOur analysis shows that the switching process is fast,\n\u00181=2fa\u0018500 ps, and proceeds through an in-phase\n(acoustical) spin rotation with superposed out-of-phase\n(optical) spin oscillations. Experimentally we demon-\nstrate essentially error-free controlled switching down to\n100 ns in pulse duration of the resonant microwave ex-\ncitation. Smaller, more single-domain-like samples (sub-\n100 nm range), with optimized magnetic asymmetry and\nincreased aspect ratio are expected to show signi\fcantly\nfaster resonant switching due to reduced generation of\nunwanted spin-wave modes and higher acoustical fre-\nquencies.\n\u0003Corresponding author: vk@kth.se\n1M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau,\nF. Petro\u000b, P. Etienne, G. Creuzet, A. Friederich, and\nJ. Chazelas. Phys. Rev. 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Dev., 50, 41, (2006).10\nSupplementary: Controlled resonant switching of a synthetic ferrimagnet\nMacrospin theory\nConsidering the free layers of the synthetic ferrimagnet\n(SFi) as single-domain particles, with the the z-axis out-\nof-plane, the EA and a uniaxial anisotropy \feld along\nthe x-axis, the total magnetic energy ( E) of the two free\nlayers can be written as1:\nE\n2\u0019M2\nS=2X\ni;j=1;i6=jVi\b\nNixm2\nix+ (Niy+hu)m2\niy+Nizm2\niz\n+\rjxm1xm2x+\rjym1ym2y+\rjzm1zm2z\n\u00002(hx+hm\nix)mix\u00002hycos (!t)miyg;\nwhereMSis the saturation magnetisation (assumed\nsame in both layers); Vi=\u0019abti=4 the volume of\nlayeriwitha; b andtithe length, width and thick-\nness of that layer, respectively; Ni\u000bthe demagnetising\nfactors of layer iin the\u000b-dimension withP\n\u000bNi\u000b= 1\nandNifx;yg=nfx;ygti=bwherenfx;ygare the reduced\nfactors2;mithe cartesian unit magnetisation vector of\nthei-th layer;hu=Hu=4\u0019MSthe normalised intrin-\nsic uniaxial anisotropy \feld along the easy-axis; hy=\nHy=4\u0019MSthe normalised amplitude of the applied AC-\n\feld in the y-direction; !the frequency of the AC-\feld;\nhx=Hx=4\u0019MSthe normalised external easy-axis DC-\n\feld;hm\ni=Hm\ni=4\u0019MSthe normalised fringing \feld act-\ning on layer iin thex-direction;\ri\u000b=r\u000bNi\u000bthe inter-\nlayer exchange \feld from layer iacting on the other layer\nin the\u000b-direction3. In our case \ri\u000bis solely of dipolarnature. Since in a thin magnetic \flm the in-plane de-\nmagnetising factors are proportional to their thickness,\nthey can be rewritten as\nNifx;yg=\u0002\n1\u0000(\u00001)i\"\u0003\nNfx;yg; Niz=1\u0000Nix\u0000Niy;\nwhere\nNfx;yg=(t1+t2)nfx;yg\n2b; \"=t1\u0000t2\nt1+t2:\nThe same can be done for the volume and interlayer ex-\nchange \feld of the individual layers:\nVi=\u0002\n1\u0000(\u00001)i\"\u0003\nV=2;\rifx;yg=\u0002\n1\u0000(\u00001)i\"\u0003\n\rfx;yg;\nFor the description of the dynamics of the magnetisa-\ntion of the system, it is useful to convert the magnetisa-\ntion into polar coordinates:\nmi= ( cos'isin\u0012i;sin'isin\u0012i;cos\u0012i);\nSince in the used geometry the thicknesses are much\nsmaller than the lateral dimensions and the sample is\nlonger along the x-dimension than the y-dimension, the\nfollowing holds:\nNix \n3m3c with n1a3c as the main molecular field coefficient , (d) m1a < 3m3c with n1a3c as the main molecular field coefficient , \n(e) m1a > 3m3c with n1a1a as the main molecular field coefficient and (f) m1a > 3m3c and n3c3c as the main molecular \nfield coefficient . The collinear model is used here . In a noncollinear model the chief difference in the shape of the \ncurve is the non -zero slope at low temperatures (see Fig. 2c) . \n \n2. Methods \nThe high purity (> 99.99%) elements of Mn and Z = Cu, Ga, Ge, In, Sn were arc-melted \ntogether five times to prepare homogeneous polycrystalline ingots. Additional Mn (2%) was added \nto compensate the loss due to its high vapor pressure. The ingots were then ground into powder and \nreacted with N 2 (> 99.99%) at 750 – 800 ℃ at a pressure of 50 kPa for 1 day. We found that if the \nN2 pressure is too large (100 kPa ) a Mn 2N impurity phase will form in some samples with small \nvalue s of x. Nitrogen deficiency can lead nitrogen vacancies or formation of γ- or β-Mn type \nimpurit ies. Additional heat treatment (anneal ing at 660 ℃ in vacuum for one day was needed for \nMn-Cu and Mn -Ge ingots before grinding them into powder to transform γ-Mn into β-Mn, owing \nto the ductile mechanical properties of γ-Mn which makes it difficult to grind . \nThe composition of the polycrystalline sample was checked by energy -dispersive X -ray \n4 \n spectroscopy. The crystal structure was characterized by powder X -ray diffraction (XRD) that \nshowed a single -phase cubic structure. Magnetization measurements were conducted using a \nsuperconducting quantum interference device magnetometer (SQUID, Quantum Design). \nAb-initio calculations based on density functional theory were carried out using norm -\nconserving pseudopotentials and pseudo -atomic localized basis function s implemented in the \nOpenMX software package [27]. Calculations were based on a minimal 5 atom basis cell of the \ncubic structure using 13 × 13× 13 k -points to evaluate the total energies. Pre -generated fully \nrelativistic pseudo potentials and the pseudo -atomic orbitals with a typical cut -off radius of 6 atomic \nunits (a.u.) were used with s3p3d3 for the metal and s3p3d2 for the metalloid elements , respectively. \nA energy cut -off of 300 Ry was used for the numerical integratio ns. The convergence criterion for \nthe e nergy minimization procedure was set to 10−8 Hartree. In the case of the non-collinear \ncalculations, we show results without spin -orbit interaction (SOI) , whose influence on the total \nenergy is negligible compared with the exchange interaction \n \n3. Result s \n3.1 Non-collinear ferrimagnetism in Mn 4N \nThe origin of non -collinear ferrimagnetism can be deduced from the magnetic interaction s and \nthe crystal structure , identified by X -ray diffraction in Fig. 2a. The lattice parameter a0 = 3.865 Å is \nalso the nearest -neighbour distance between two Mn1a atoms d1a1a. The nearest distance s between \nMn3c and Mn1a or Mn3c d1a3c and d3c3c are both equal to a0/√2 = 2.733 Å. Generally, Mn atoms \nseparated by 2.5-2.8 Å have delocalized electrons and couple antiferromagnetically while Mn atoms \nwith longer separations (> 2.9 Å) couple ferromagnetically . Therefore, the Mn1a moment s lie parallel \nto each other, wh ereas the small d1a3c distan ce favors antiparallel coupling between the sublattices. \nThe separation of nearest -neighbor Mn3c atoms d3c3c is responsible for the non -collinear triangular \nantiferromagneti sm of the 3c sublattice . Together, these interactions lead to the umbrella -like spin \nstructure, and the overall non-collinear ferrimagnetism. \nMn 4N has a high Curie temperature TC (780 K) and a small saturation moment mtot = 1.1 μB/f.u. \nalong a [111] direction , as shown in Fig. 2c. The measured moment mtot is the difference of the m1a \nand three times the ferr imagnetic component of Mn3c mcFiM, which are 3.8 μB and -0.9 μB per Mn , \nrespectively [13]. It should be noted that the net moment in Fig . 2c remains a constant below 50 K \nand then drops with increasing temperature . By 160 K ( T/TC = 0.2), the moment has fallen by 13% \nof the 4 K value, in agreement with literature [18]. This is quite unusual, because according to the \ncollinear mean -field model , the decrease at T/TC = 0.2 should be smaller than 1% (see Fig. 1) . The \ninability to fit the P-type curve to a collinear mean -field model for Mn 4N [19] is a strong indication \nof the noncollinear nature of the magnetic order . 5 \n \nFig. 2. Noncollinear magnetic structure of Mn 4N. (a) Crystal and magnetic structure showing the Γ4g triangular \nferrimagnetism. Grey, blue and red atoms represent N, Mn1a and Mn3c, respectively. (b) Kagome lattice of Mn3c in a \n(111) plane showing Γ4g-like magnetic structure . The out -of-plane magnetic component is not shown . \n(c) Temperature -dependent magnetization M(T) for Mn 4N. The magnetization remains constant up to 50 K ( green \narrow) but drops significantly above . At 160 K (purple arrow ) the moment has already fallen to 87% of the base \ntemperature value . (d) Energy difference in the calculated magnetic structure as a function of tilt angle θ between \nm3c and (111) plane. (e) Magnetic moment s with varied tilt angle θ. (f) Comparison of c alculated total energies as a \nfunction of lattice constant for the collinear and noncollinear ferrimagnetic structures. \n \n6 \n The noncollinear spin structure is analysed further using a constrained DFT approach, where \nthe direction s of the individual spins are pinned to a selected angle but the magnitude s of the \nmoments are allowed to vary freely in the total energy minimi zation process. The direction of M n1a \nis pinned to the body diagonal [111] and the tilt angle θ of the spins the Mn3c atoms is varied (also \nsee inset of Fig. 5b ). As the angle is rotated from the collinear ferrimagnetic configuration (θ = 90°) \ninto the (111) plane ( θ = 0°, Γ4g spin structure ) and then towards a ferromagnetic configuration at θ \n= -90° the energy and magnetic moments vary as shown in Fig . 2d and 2e One can visualize this \nangle change like a closed umbrella (FiM) that opens out to close to 0 ° in regular usage and well \nbeyond it on a windy day (FM, θ = -90°). The relative total energy cha nge of the constrained angle \napproach is shown Fig. 2d. Following the total energy minimum curve from the right to left, we \nwitness a decrease in total energy Etot with the Mn3c moments canting away from the antiparallel \nspin arrangement into the (111) plane. The minimum of Etot is found at around θ = 20°. The Etot \ndifference between the collinear ferrimagnetic and noncollinear ferrimagnetic ground state is \nsignificant , about 4.5 meV/atom. The calculations also confirm that the FM arrangement ( θ = -90°) \nof the spins on Mn is very unfavourable with energy difference ~32 meV/atom. Our results suggest \nthat the M n3c sublattice moment s make an angle close to 70° with the Mn1a moments, very far from \nthe simplified picture of collinear ferrimagnetism often assumed . For further analysis, Fig . 2e shows \nthe calculated site -specific magnetic moments together with the total magnetic moment per formula \nunit in our range of interest ( θ = 0 to 90°). A strong dependence of magneti zation for both magnetic \nsites as a function of θ is revealed; the M n1a moment remains ~4 μB down to about θ = 45° and then \na significant reduction to ~3.2 μB occurs on closing towards the (111) plane ( θ = 0°). In contrast, the \nMn3c site moment increases from about 1.1 μB up to 2.4 μB when the angle closes from FiM towards \nthe Γ4g-like configuration. T he Etot minimum suggest magnetic values of m1a=3.65 μB, m3c=2.35 μB \nwith mtot=1.24 μB/f.u. Indeed, the collinear ferrimagnetic spin configuration also yields a value of \nmtot that is close to the experimental one , but we have relaxed both spin configuration for the \nequilibrium lattice parameters from DFT and find a value a0 = 3.75 Å for the collinear ferrimagnetic \nstate that is smaller than that for the non -collinear ferrimagnetic state a0 = 3.82 Å , in Fig . 2f, which \nis closer to the experimental value of 3.8 65 Å at 300 K. The earlier calculation [14] found a greater \ntilt angle and a smaller 3c moment, fixing the lattice parameter and exploring a smaller range \nof . The energy difference can be ascribed to the electronic pressure caused by the altered magnetic \nspin configuration . This exchange striction , like that in FeRh [28 ], explains the significantly \nexpanded lattice constant for Mn 4N (3.86 Å ) compared to its ferromagnetic cousins such as Fe4N \n(3.79 Å ), Co 4N (3.75 Å ) and Ni 4N (3.72 Å ) on the one hand and o n the other hand it is also manifest \nthroughout the rotation of the spins that alters exchange -split band energies by Coulomb repulsion. \nThis non -Heisenberg like behaviour relates to the spin split d-bands crossing the Fermi le vel that \ninfluences the band filling and calculated magnetic moments. \n \n3.2 Doping for compensation \nIn order to achieve compensation, namely to chang e the temperature -dependent magnetization \nfrom Q-type to N-type, the main exchange should change from n1a3c to n3c3c, meanwhile the 1 a site \nmoment m1a should be larger than three times the axial component of the 3c site moment s 3m3cFiM. \nThis means that Mn on the 1a site should be substituted at the appropriate level x in Mn 4-xZxN (Z= \nCo, Ni, Cu, Zn, Ga, Ge, As, Rh, Pd, Ag, Cd, In, Sn, Sb, Pt, Au and Hg with x < 1). Fig. 3a shows \nthe X-ray diffraction (XRD) pattern of Mn 3.76Ga0.24N, and the low-angle data are expanded in Fig. 7 \n 3b. The larger intensity of the (110) superlattice peak indicates that Mn 3.76Ga0.24N crystallize s in a \nwell-ordered structure with Ga atoms occupy ing the 1a site. Nonmagnetic Ga weakens the magnetic \nexchange leading to a decreased TC = 610 K. The net moment of 0.17 μB/f.u. at 4 K indicat es that \neach Ga decrease s the moment by of ~3.8 μB, matching both the moment of m1a from neutron \ndiffraction [13] and our calculation . The compensation temperature is then Tcomp = 408 K. In this \ncase, the 1a sublattice dominates the magnetization at low temperature s, while the 3c sublattice is \ndominant above compensation. The M-H curve s shown in Fig. 3 exhibit very little hysteresis, \nindicating weak cubic magnetocrystalline anisotropy. Note the magnetization at 4 K is not saturated \neven in 5 T, further supporting the non-collinear ferrimagnetic structure where the tilt angle θ \nchanges with magnetic field. \nThe doping efficiency of different elements from Cu to Sn is shown in Fig s. 3e-3i. Unlike Ga, \nthat changes the net moment at the rate of ~3.8 μB/atom, the rates for the other elements are \nsignificantly different. For Sn with x = 0.26, the magnetization is reduced to 0.26 μB/f.u., much more \nthan 0.12 μB/f.u. for Ga with the same x. The magnetization curve shows a large hysteresis at 4 K in \nFig. 3e, attributed to the Γ5g antiferromagnetic configuration that co-exist all the way down from the \nNéel temperature of Mn 3SnN [29]. There is a difference in the M-T curves measured after field -\ncooling (FC) and zero -field-cooling (ZFC) shown in Fig. 3f, which is also observed in the In-doped \nsample (x = 0.26). The compensation temperature is around 400 K for Sn substitution for x = 0.26. \nThe magnetization is less sensitive to x for Sn than for Ga, and this trend towards low doping \nefficiency is more significant for Ge than Sn, as shown in Fig. 3g. Compensation was not observed \nbelow 400 K for Ge with x = 0.35 . On the other hand, Mn 4-xCuxN is very sensitive to the \ncomposition al changes . The M-T curve gradually changes from Q-type to N-type and finally to P-\ntype within a narrow range of x, as illustrated in Fig. 3h. The M-T curves for Cu, Ga, Ge, In and Sn \nwith x = 0.26 are all compared in Fig. 3i ). The Cu-doped sample has the highest doping efficiency \nwith a P-type M-T curve without compensation ( Tcomp < 0 K ). Ge, Ga and Sn doped samples exhibit \nTcomp of 70 K, 291 K and ~ 400 K respectively with N-type M-T curves. Ge leads either to a much \nhigher Tcomp or else to a complete disappearance of compensation ( Q-type M-T curve ). 8 \n \nFig. 3. (a) XRD pattern of Mn 3.76Ga0.24N. (b) Expanded low -angle data with simulations showing \nthe superlattice peak for Mn 3.76Ga0.24N. The experimental data confirm that Ga atoms occupy the \n1a site. ( c) Magnetization curve s for Mn 3.76Ga0.24N at different temperatures. (d) Thermomagnetic \nscans Mn 4-xGaxN (x = 0.24 and 0.26) . (e) Magnetization curve for Mn 3.74Sn0.26N at 4 K and 300 K . \n(f) Zero-field-cooled (ZFC) and field -cooled (FC) thermomagnetic scans for Mn 4-xSnxN. (g) \nThermomagnetic scans for Mn 4-xGexN (x = 0.26 and 0.35) . (h) Thermomagnetic scans for Mn 4-\nxCuxN (x = 0.16 to 0.34) . (i) Thermomagnetic scans for Mn 3.74Z0.26N (Z = Cu, Ga, Ge, In and Sn). \n \nWe plot data for different compositions of Mn 4-xZxN at 4 K in Fig. 4a) including both our own \ndata and previous reports [26,30,31,32]. The slope changes significantly from Ni (-6.20 μ B/atom ), \nCu (-5.01 μB/atom ), Zn ( -4.35 μB/atom ), Ga ( -3.70 μB/atom ) to Ge (-2.52 μB/atom ) with increasing \nvalence electron for dopants in the fourth period. A similar trend is also found for dopants in the \nfifth period : Ag (-7.70 μ B/atom ), In (-4.35 μB/atom ) and Sn ( -3.08 μB/atom ). Based on this trend, the \nmagnetic diagram s for different types of M-T curve are visualized in the maps of Figs. 4c and 4d. \nWith a small concentrations of dopan ts, the interaction between 1 a and 3 c sites dominates and there \nis no compensation below the Curie temperature leading to a Q-type M-T curve. With suitab le x, the \nmoment if the 1a sublattice is still larger than that of the 3c sublattice at low temperature, while at \nhigh temperature the 3 c sublattice wins above compensation. Therefore, an N-type M-T curve is \nfound. When heavily -doped , the 3 c sublattice dominat es throughout whole temperature range, and \nthere is a P-type M-T curve with no compensation. Elements from the fifth period have a greater \nability to compensate than those from the fourth period , and the magnetization is very sensitive to \n9 \n x. Therefore, the boundary for different M-T curves are shifted to the left (lower x) and the useful \nN-type region is narrower . \n \nFig. 4. (a) Summary of the net moment as a function of composition x in Mn 4-xZxN. We include our \nown data (solid points) and previous reports (open points). (b) The slope in (a) showing different \nefficienc ies. Magnetic d iagram for Q, N and P type M-T curves, depending both on x and Z for \nelements from the fourth (c) and fifth (d) period s. \n \n4. Discussion \nThe efficiency of the dopants to compensate is analyzed from the view of magnetic moment , \nnon-collinear angle and lattice constant, from both experimental and theoretical points of view. \n4.1 Lattice constant \nCompounds with the same x but different Z from the same group have the same valance \nelectron number , and the main difference in their effects on the magnetic structure is related to the \nlattice parameter . Fig. 5a shows a0 for Mn 4-xZxN (Z = Cu, Ga, Ge, Ag, Sn In). It is clear that Ag [33], \nIn and Sn [26] lead to a greater increase in lattice parameter than Cu, Ga and Ge for the same x, \nbecause of the ir larger atomic radii. The increased lattice parameter translates to a larger atomic \nseparation in the cubic crystal, leading to reduced p-d hybridisation of Mn3c and increased Mn -Mn \nexchange that produce a relative increase of magnetism of the 3c sublattice . The Mn3c moment is \nlarger and more localized , leading to improved doping efficiency for dopants from the 5th period . \n10 \n \nFig. 5. (a) Lattice parameters for Mn 4-xZxN (Z = Cu, Ga, Ge, Ag, Sn In ), compar ing our data (solid \npoints) and previous reports (open points) [26,33]. (b) Calculated tilt angle θ versus x by Eq. 6. (c) \nCalculated magnetic moment for Mn3c in Mn 3ZN ( x = 1). (d) Tilt angle θ for different m1a and m3c \ndeduced from DFT. \n \n4.2 Magnetic moment \nSince the nearest -neighbor for Mn1a is always Mn3c, the magnetic coupling between Mn1a \natoms is weak. As a result, the moment for the remaining Mn1a is not influenced significantly , as \nalso indicated from neutron diffraction [34]. Therefore the effect of doping on the net magnetization \ncomes mainly from Mn3c. We buil t our DFT model to capture trends in the electronic and magnetic \nstructures and used the simplest 5 -atom unit cell model to model our experimental observation with \ndifferent compositions. This model allows us to compare trends for x = 1, when the 3 c site is fully \noccupied by Mn and the symmetry is cubic. \nThe calculated Mn3c moment as a function of valence electron count in Mn 3ZN from Mn to Ge \nin the 4th period and from Tc to Sn in 5th period is plotted in Fig. 5c. The magnetic behaviour shows \nthe same trend; an initial increase saturates around Ni and Pd , and drops monotonically afterward s. \nIn order to separate the electronic effects from the impact of chemical pressure on the lattice \nparameter, we first fixed a0 of all members of the series to that of Mn 4N (3.82 Å). This is shown by \nthe solid red and black lines for the 4th and 5th period s in Fig. 5c. The peak at Ni, which has three \nelectrons more than Mn, resembles a localized moment picture with striking similar ities with the \n11 \n Slater -Pauling rule. These t hree extra electrons are shared by the three nearby Mn3c atoms, and hence \neach of the Mn3c atom s get one more electron becoming like iron , which shows the largest average \nmoment in 3 d alloys. The influence of the lattice constant on the moment expected on M n3c without \nconstrain t is also drawn in green and blue lines for comparison . The same trend is maintained, with \npeak s at Z = Ni and Pd. The main difference is found on the right hand side of the curves , especially \nfor elements from 5th period , as it relates to the expanded lattice parameters with additional valence \nelectrons compared to the Z = Mn reference. The significan t change in the amplitude of m3c is one \nof the main reason s for the different doping efficienc ies. In addition, the orientation of m3c also \ndepends on its amplitude that further impact s the efficiency of compensation as we discuss it in the \nfollowing section. \n \n4.3 Tilt angle \nTwo questions concerning the tilt angle θ between m3c and the (111) plane are: How d oes θ \nchange with x , and with the different dopings ? \nThe 3c moment has two components , one component m3cFi along the ferrimagnetic [111] axis \nand the other m3cAFM in triangular antiferromagnetic (111) plane . The molecular field acting on 3 c \nsite also has two components , parallel and perpendicular to the [111] axis HFi and HAFM, \ncorresponding to the ferrimagnetism and in -plane antiferromagnetism. They satisfy the relati onship s \nHAFM = -2 n3c3c m3c cosθ cos120° (1) \n HFiM = n1a3c m1a(1 - x) - 2n3c3c m3c sinθ (2) \nm3cFi = m3c sinθ (3) \nm3cAFM = m3c cosθ (4) \ntanθ = HFiM / HAFM (5) \nwhere n3c3c and n1a3c are the Weiss coefficients for interactions between 3c-3c Mn and 1 a-3c Mn, as \nshown from the inset of Fig. 5b. In Eq s. 1 and 2 , the in-plane antiferromagnetism considers the \ninteraction from the other two nearest neighbor Mn3c with 120 ° triangular spin structure ; the \nnegative sign of m3c is already considered. Taking Eqs. 1 and 2 into Eq. 5, we get \nsinθ = n1a3c m1a(1 - x)/(3n3c3c m3c) (6) \nPreviously θ was estimated from neutron diffraction to be about 70° (nearly collinear) , with \nm3cFi = 0.9 μ B, m3cAFM = 0.36 μB, m3c= 0.97 μB, m1a= 3.8 μB but with large error bar s [13]. This means \nthe doping efficiency should be weaker than –(3.8+0.36) = -4.16 μB/atom, if we assum e that the \nmagnetic structure becomes collinear ferrimagnetic after doping . But from both our and previous \nexperiments, dopants of Ni, Ag, Cu are much more effective , indicating that m3cAFM was \nunderestimated . This is also found in our DFT calculations, where θ = 19.5 ° for binary Mn 4N.Thus \nif we plot the relationship between x and θ in Eq. 6 (Fig. 5b ), we find that θ decreases with x almost \nlinearly . The umbrella -like triangular spin structure of Mn3c rotates away from the [111] direction \nand becomes in -plane and the refore the net moment changes at a slower rate , as confirmed by \ncomparative neutron study of Mn 3.2Ga0.8N and Mn 4N [34]. Finally, when x = 1 , Mn1a is completely \nreplaced by the nonmagnetic dopant , Mn 3ZN is a triangular topological antiferromagnet in the (111) \nplane if the crystal remains cubic . \nWhen doped with different elements from Cu to Ge, or from Ag to Sn, the decrease of m3c with \nincrease of valance electrons leads to a rise of θ according to Eq. 6. This can weaken the effect of \ndecreasing m3cFi according to Eq. 3. Similarly, considering the increased lattice constant for dopants \nfrom 5th period, the enhanced m3c can also lead to a drop of θ, weakening the influence on m3cFi. We 12 \n further estimate θ by DFT calculation based on differ ent m1a and m3c manganese site moments , as \nshown in Fig. 5d. We use a fixed spin moment (FSM) approach, where the amplitude of the magnetic \nmoments on both sites is fixed . In Fig 5d , we plot the angles for min imum total energy Etot(m1a , \nm3c ). The general trend is that the larger m3c for a given m1a, the smaller θ. The larger m3c moment \ntends to stay in the (111) plane, and only the increasing moment on the 1a site could compensate \nfor this rotation. This is in qualitative agreement with Eq. 6 and the vanishing moment on m1a with \nincreasing x, from experiment. \n \n4.4 Best dopants for compensated ferrimagnet ism \nMn 4-xZxN thin films are already attract ing increasing attention for spintronics [35,36]. Most \nstudies have been done with Ni or Co [22,23,24,25], but they are not ideal for achieving \ncompensation. Beside the demand for compensation, additional requirements must be considered \nwhen choosing the best dopants. Based on our analysis, Ga appears to be a suitable dopant in Mn 4-\nxZxN films for spintronics for the following reasons: First, earlier elements from 4th period like Ni \ncompensate the moment with small values of x. As the total moment is very sensitive to the \ncomposition , it is difficult to control the composition precisely and homogeneously. Second, for \nelements that have many additional valance electrons like Ge, a large value of x is needed due to the \nlow dopi ng efficiency. As a result, the Curie temperature drops substantially, which is not beneficial \nfor room temperature applications. Third, Ga does not significantly increase the lattice constant \ncompared to elements from 5th period so that a series of thin f ilms can be grown with different \ncompositions x and a similar tetragonal distortion is expected on the same substrate. The slight \ntetragonal distortion ( c/a ~ 0.99) due to biaxial strain imposed at the interface of the film and \nsubstrate is the origin of p erpendicular [001] anisotropy. A smaller lattice constant of the film than \ncommon substrates, such as SrTiO 3 with a001 = 3.91 Å, is the key for the in -plane tensile strain and \nperpendicular anisotropy [37,38], which can be easily realized in Ga -doped samples. Finally, the \ndoping efficiency of Ga , -3.70 μ B/atom coincides with m1a. This is a consequence of a combination \nof an increased m3c and a decreased θ rather than simply the nonmagnetic nature of the dopant. \n \n5. Conclusion \nFrom our experimental and theoretical study of the rare-earth -free noncollinear ferrimagnetic \nmetals Mn 4-xZxN, we conclude that the noncollinear ferrimagnetism originates from the structure of \nthe Mn3c (111) kago me planes with a small Mn-Mn interatomic separation that leads to frustration \nof the antiferromagnetic nearest -neighbor interactions. The tilt angle of the moments from the (111) \nplanes , θ = 20° , is smaller than previous ly thought . There is a choice of substitutions to achieve \nmagnetic compensation at room temperature. The efficiency of different elements in this respect \nrises gradually with increasing valance electrons from group 11 (Cu, Ag) to group 14 (Ge, Sn). The \nMn1a moment is not sensitive to the dopants, while the Mn3c moment peaks at Ni and Pd then drops \nwith further valance electron addition . 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" }, { "title": "2204.09776v1.Ferrimagnet_GdFeCo_characterization_for_spin_orbitronics__large_field_like_and_damping_like_torques.pdf", "content": " \n \n1 \n Ferrimagnet GdFeCo characterization for spin-orbitronics: large field -like and \ndamping -like torques \n \nHéloïse Damas1*, Alberto Anadon1, David Céspedes -Berrocal1,2, Junior Alegre -Saenz1,2, Jean -\nLoïs Bello1, Aldo Arriola -Córdova1,2, Sylvie Migot1, Jaafar Ghanbaja1, Olivier Copie1, Michel \nHehn1, Vincent Cros3, Sébastien Petit -Watelot1* and Juan -Carlos Rojas -Sánchez1* \n \n1Université de Lorraine, CNRS, Institute Jean Lamour, F -54000 Nancy, France \n2Universidad Nacional de Ingeniería, Rímac 15333, Peru \n3Unité Mixte de Physique, CNRS, Thales, Université Paris -Saclay, 91767 Palaiseau, France \n \nCorresponding authors: heloise.damas@univ -lorraine.fr, sebastien.petit@univ -lorraine.fr, \nJuan-Carlos.ROJAS -SANCHEZ@univ -lorraine.fr \n \nH. Damas, Dr. A. Anadon, D. Céspedes -Berrocal, A.Y. Arriola -Córdova, J -L Bello, S. Migot, \nJ. Ghanbaja, Dr. O. Copie, Prof. M. Hehn, Dr. S. Petit -Watelot, Dr. J. -C. Rojas -Sánchez. \nUniversité de Lorraine,CNRS, Inst itute Jean Lamour, F -54000 Nancy, France \nE-mail: heloise.damas@univ -lorraine.fr , sebastien.petit@univ -lorraine.fr , Juan-\nCarlos.ROJAS -SANCHEZ@univ -lorraine.fr \n \nD. Céspedes -Berrocal, A.Y. Arriola -Córdova , J. Alegre -Saenz \nUniversidad Nacional de Ingeniería, Rímac 15333, Peru \n \nDr. V. Cros, \nUnité Mixte de Physique, CNRS, Thales, Université Paris -Saclay, 91767 Palaiseau, France \n \nKeywords: \nFerrimagnet GdFe Co; spin -orbit torque; spin -torque ferromagnetic resonance; spin anomalous \nHall effect; spin Hall effect \n \nSpintronics is showing promising results in the search for new materials and effects to reduce \nenergy consumption in information technology. Among these materials, ferrimagnets are of \nspecial interest, since they can produce large spin currents that trigge r the magnetization \ndynamics of adjacent layers or even their own magnetization. Here, we present a study of the \ngeneration of spin current by GdFeCo in a GdFeCo/Cu/NiFe trilayer where the FeCo sublattice \nmagnetization is domina nt at room temperature. Magn etic properties such as the saturation \nmagnetization are deduced from magnetometry measurements while damping constant is \n \n2 \n estimated from spin -torque ferromagnetic resonance (ST -FMR). We show that the overall \ndamping -like (DL) and field -like (FL) effective fields as well as the associated spin Hall angles \ncan be reliably obtained by performing the dependence o f ST-FMR by an added dc current. The \nsum of the spin Hall angles for both the spin Hall effect (SHE) and the spin anomalous Hall \neffect ( SAHE ) symmetries are: 𝜃𝐷𝐿𝑆𝐴𝐻𝐸+ 𝜃𝐷𝐿𝑆𝐻𝐸=−0.15±0.05 and 𝜃𝐹𝐿𝑆𝐴𝐻𝐸+ 𝜃𝐹𝐿𝑆𝐻𝐸=\n0.026±0.005. From the symmetry of ST -FMR signals we find that 𝜃𝐷𝐿𝑆𝐻𝐸 is positive and \ndominated by the negative 𝜃𝐷𝐿𝑆𝐴𝐻𝐸. The present study paves the way f or tuning the different \nsymmetries in spin conversion in highly efficient ferrimagnetic systems. \n \n1. Introduction \n \nIn the last years, ferrimagnets have attracted growing interest for their potential utility in \nspintronic devices [1]. In particular, GdFeCo ferrimagnetic alloy is extensively studied as it \nexhibits a wide diversity of phenomena arising from the specific properties of rare earth -\ntransition metal (RE -TM) ferrimagnets. Furthermore , the two antiferromagnetically coupled \nsublattices have a different response to external stimuli and the spin -orbit couplin g (SOC) of \nthe Gd 5d state allows the interplay between charge, spin , and orbital transport. The different \nrelaxation times of these two coupled sublattices are thought to be responsible for the all -optical \nhelicity -independent switching (AO -HIS) in GdFeCo demonstrated for almost a decade [2,3] . \nAO-HIS has also been recently observed in TbCo [4]. Nowadays, GdFeCo is used to perform \nthe AO -HIS of Co/Pt [5–7] or CoNi/Pt [8] ferromagnetic multilayers. Moreover, it is possible \nto tune the Dzyaloshinskii -Moriya interaction in thin GdFeCo ferrimagnetic alloys [9], a \nrelevant property for skyrmions formation. It has been shown that GdFeCo ferrimagnet also \nhosts large self-induced spin -orbit torque, or self -torque [10,11] , with recent theoretical \nadvances [12,13] . Ferro and ferrimagnetic materials are the source of spin currents with \ndifferent symmetries [10,12] coming from the s pin anomalous Hall effect (SAHE) [14–16] and \nthe spin Hall effect (SHE) [17]. In the SAHE, the spin polarization of the spin current 𝐽sSAHE is \n \n3 \n parallel to the magnetization while in the SHE it is perpendicular to both the injected charge \ncurrent and the produced spin current 𝐽sSHE. A giant overall spin Hall angle for SAHE -like and \nSHE -like symmetries has been reported in a Gd -rich GdFeCo/Cu at r oom temperature [10]. \nSizable i nterconversion efficiencies ha ve also been reported for other magnetic materials such \nas NiFe [18–20] and CoFeB [15,21] . In the present work, we study room temperature FeCo -\nrich GdFeCo in a //Gd 25Fe65.6Co9.4(8 nm)/Cu(4 or 6 nm)/Ni 81Fe19(4 nm) trilayer by structural, \nmagnetic and spintronics characterization. W e use two complementary ST -FMR techniques to \nreveal the signs and ma gnitudes of the contributions coming from the different spin current \nsymmetries in GdFeCo. Namely, the modulation of the damping along with the shift of \nresonance field to extract the overall parameters (sum of the SHE -like and SAHE -like \ncontributions) and the symmetry of the ST -FMR signal which is sensitive only to the SHE -like \nparameters. We found that damping -like (DL) SAHE spin Hall angle, 𝜃𝐷𝐿𝑆𝐴𝐻𝐸, is negative for \nFeCo -rich GdFeCo . In contrast, the DL SHE -like symmetry , is positive. \n \n2. Structur al and chemical characterization \n \nSamples were grown on thermally oxidized Si wafers using dc magnetron sputtering at room \ntemperature with an Ar gas pressure of 3 mTorr and base pressure of 1x10-7 Torr. GdFeCo \n(Gd 25Fe65.6Co9.4) was co -deposited using separate Gd, Co , and Fe targets. All the samples in the \npresent study were capped with 3 nm of naturally oxidized Al. The c omposition was controlled \nby varying the sputter gun power on each target. The d eposition rate was calibrate d by X -ray \nreflectivity and lift -off and profilometer measurements of the thickness. In order to perform \nstructural characterization, a thin lamella was extracted by focused ion beam (FIB) milling \nusing an FEI Helios Nanolab dual -beam 600i. \nTransmission e lectron microscopy (TEM) investigations were carried out using a JEM - ARM \n200F Cold FEG TEM/STEM (Scanning TEM) operating at 200 kV, coupled with a GIF \nQuantum 965 ER and equipped with a spherical aberration (Cs) probe and image correctors \n \n4 \n (point resoluti on 0.12 nm in TEM mode and 0.078 nm in Scanning TEM (STEM) mode) . High -\nResolution TEM (HRTEM) micrographs were performed to study the atomic structure of the \ndeposit layers as shown in Figure 1 a. The Fast Fourier Transformation (FFT) patterns in Figure \n1b,c confirm that the Cu/NiFe layers are [111] textured along the growth direction while \nGdFeCo is amorphous as evidenced by the diffuse rings. Electron Energy Loss Spectroscopy \n(EELS) maps were carried out systematically on the different samples and confirm the nominal \ncomposition and thickness of the different materials (Figure 1 d). We also evidence a slight \nGdFeCo composition variation along the growing direction as usually observed in RE -TM \nferrimagnets [10,11,22] . The EELS maps displayed in Figure 1 d were performed with \n1ev/channel and a step of 0.3 nm. \n \n \nFigure 1. TEM/STEM characterization of Gd 25Fe65.6Co9.4(8)/Cu(6)/Ni 81Fe19(4)/AlOx . (a) \nHRTEM micrograph of the deposit ed layers. The yellow (blue) square shows where the FFT \nanalysis have been performed on Cu/NiFe (GdFeCo). The FFT patterns ( b) and ( c) indicate the \n[111] growth direction of textured Cu/NiFe and that GdFeCo is amorphous. ( d) High Angle \nAnnular Dark Field (H AADF) -STEM micrograph and the corresponding individual EELS \nelemental maps obtained from the green rectangle area in the HAADF micrograph. Co (yell ow), \nFe (gr een), Gd (orange), O (red), Ni (cyan), Cu (pink). \n \n \n5 \n \n \n \n \n3. Magnetic characterization: magnetic anisotropies in GdFeCo \n \n3. 1. SQUID magnetometry \n \nMagnetization loops were performed at room temperature on a //GdFeCo (8)/Cu(6)/NiFe (4) \nstack with the applied field parallel and perpendicular to the field plane. Both 𝑀(𝐻) \nmeasurements are display ed in Figure 2a show ing an open hysteresis loop . This confirm s that \nthe NiFe magnetization direction 𝒎̂NiFe lies in the plane of the sample while that of GdFeCo , \n𝒎̂GdFeCo , is spontaneously perpendicular to the film plane as shown in the inset . We assume \nthat the 6 nm thick Cu layer decouples the two magnetic layers to extract their distinct saturation \nmagnetization and saturation magnetic field . For NiFe, the saturation magnetization 𝑀sNiFe is \n625 kA/m and the saturation field 𝜇0𝐻sat−zNiFe to place 𝒎̂NiFe out of the plane of the film is 0.85 \nT. In the case of GdFeCo, the saturation magnetization 𝑀sGdFeCo is 115 kA/m and the saturation \nfield 𝜇0𝐻sat−xyGdFeCo to align 𝒎̂GdFeCo along the plane is about 0.13 T which are typical values for \nboth NiFe and Gd25Fe65.6Co9.4 at room temperature [23–25]. From the saturation field and \nmagnetizations , we can also estimate the effective saturation magnetization for both magnetic \nmaterials, it results 𝑀effNiFe=676 kA/m and 𝑀effGdFeCo=103 kA/m. The relatively low \nperpendicular magnetic anisotropy of GdFeCo allows its magnetization to be easily placed \nalong t he plane of the film which is useful for ST-FMR measurements. The trilayer used in the \nnext section has a 4 nm Cu spacer and displays a lower saturation field 𝜇0𝐻sat−xyGdFeCo to align \n𝒎̂GdFeCo along the plane, which is about ~0.047 T. \n \n \n \n \n6 \n \nFigure 2. Bulk magnetization data using a SQUID magnetometer. (a) Magnetic \nhysteresis loop of the Gd25Fe65.6Co9.4(8)/Cu(6)/Ni 81Fe19(4) trilayer. Magnetization \nvalues are normalized by the surface sample. To identify the different saturation fields \nand effective saturation magnetization, we consider that the magnetic layers are \ndecoupled by the 6 nm of Cu . The inset shows a s chematic of the sample with the \nspontaneous magnetization alignment of GdFeCo (out -of-plane) and NiFe (in -plane) \naccording to SQUID results. \n \n \n \n3. 2. Spin -torque FMR study \nWe perform ST -FMR measurements [26–31] on a GdFeCo (8)/Cu(4)/ NiFe (4) trilayer to extract \nproperties such as the damping constant 𝛼 and the Landé g -factor of the magnetic layers. From \nMagneto optic Kerr effect measurements we have verified that this GdFeCo (8) is FeCo -rich at \nroom temperature. The experimental setup is described in Figure 3a. A radiofrequency (rf) \ncharge current , 𝑖rf, is applied along the 𝒙̂ direction and generates an oscillating Oersted field \nwhich triggers the magnetization precession at the resonance condition . A sweeping dc \nmagnetic field 𝐻dc is applied in the xy plane of the device, at an angle of 𝜑𝐻 with respect to the \ncurrent line. At the resonance field 𝐻res, a dc voltage 𝑉mix composed of a mix ing of a \nsymmetric and antisymm etric Lorentzians of amplitude Vsym and Vanti respectively can be \nmeasured using a bias tee. The measured mixed voltage display ed in Figure 3b can be fitted \nwith the following general expression: \n𝑉mix=𝑉offset+𝑉symΔ𝐻2\nΔ𝐻2+(𝐻−𝐻res)2+𝑉anti(𝐻−𝐻𝑟𝑒𝑠)Δ𝐻\nΔ𝐻2+(𝐻−𝐻𝑟𝑒𝑠)2, (1) \n \n \n7 \n where we consider an additional offset 𝑉offset and where Δ𝐻 is the linewidth. In Figure 3b, we \nobserve the two resonance lines corresponding to the NiFe resonance (lower resonance field) \nand the GdFeCo resonance (higher resonance field ). For the sake of clarity, it is only show n at \n8, 12 and 14 GHz . Then, from broadband frequency dependence ST-FMR we can extract the \neffective saturation magnetization Meff (it results negative f or system s where perpendicular \nmagne tic anisotropy dominate s over shape anisotropy ), and the Landé g -factor considering the \nfollowing expression : \n𝑓=𝛾\n2π√(𝐻+𝐻uni)(𝑀eff+𝐻+𝐻uni) , (2) \nwhere 𝛾=𝑔μB\nℏ is the gyromagnetic ratio and where Huni stands for a small in -plane uniaxial \nmagnetic anisotrop y. Equation 2 applies for a thin film ferromagnetic layer with a magnetic \nfield applied in the plane . We fix the NiFe Landé g -factor to 2.10 . We determine the effective \nsaturation magnetization of NiFe , 𝑀effNiFe=569±1kA/m . The difference with previous \nSQUID results comes from the difference in Cu thickness which affect s the NiFe anisotropy . \nWe also evaluate a rather small 𝐻uni=−7±1 Oe. We exploit the same Equation 2 for \nGdFeCo resonance condition to determine the GdFeCo Landé g -factor and its effective \nsaturation magnetization 𝑀effGdFeCo. We obtain g=2.87±0.04, and 𝑀effGdFeCo=−37±4 kA/\nm (−46.5 mT). The fitted experimental data is shown in Figure 3c. Finally, from the frequency \ndependence of the linewidth H, we calculate the Gilbert -type magnetic damping constant 𝛼: \n𝛥𝐻=𝛥𝐻0+2π𝑓\n𝛾𝛼 , (3) \nwhere H0 is the f-independent contribution due to inhomogeneity . We have fixed g -Landé \nfactor for both NiFe and GdFeCo. The fits on the measurements are shown in Figure 3d. The \ndamping of in-plane NiFe is estimated as 𝛼𝑁𝑖𝐹𝑒=0.012±0.001 which is about 8 times \nsmaller than the damping of our out -of-plane Gd25Fe65.6Co9.4 𝛼GdFeCo=0.085±0.006 but \ncomparable with in -plane Gd12.5Fe76.1Co11.4 [32]. Recently, it has been point ed out that actual \n \n8 \n or intrinsic damping in ferrimagnet s is lower than th at measured directly due to different spin \ndensity for each magnetic sublattice in GdFeCo and determine d by domain wall mobility [33]. \nIn the next section , we show how we c an estimate the effective field s that drive the spin -orbit \ntorque from Gd25Fe65.6Co9.4(8)/Cu(4) to Ni81Fe19(4). \n \n \n \n \n \nFigure 3. Determination of Meff and damping using broadband ST -FMR in \nGd 25Fe65.6Co9.4(8)/Cu(4)/Ni 81Fe19(4), and g -Landé factor for GdFeCo . (a) Illustration of a \ntypical ST -FMR device along with the dc magnetic field applied at 𝜑𝐻 to the trilayer slab which \nis along 𝑥̂. (b) Typical ST -FMR spectr a at 8, 12 and 14 GHz. At a higher field s, the GdFeCo \nresonance line is observed . The symmetrical (orange) and antisymmetrical (green) voltage \ncontribution s are shown for NiFe at 8 GHz. Broadband frequency dependence of Hres (c) and \nlinewidth H (d) are used to determine Meff and , respectively. Equation 2 is used for NiFe \nand GdFeCo layer in (c). For Gd 25Fe65.6Co9.4 (8 nm), we estimate 𝑔=2.87±0.04 and 𝛼=\n0.085±0.006. Black (green ) experimental data are obtained for the resonance of NiFe \n(GdFeCo) as depicted in (b). \n \n \n \n \n9 \n \n4. Damping -like and field -like efficiencies determination by ST -FMR techniques \n4. 1. Spin torque symmetries and ST -FMR signal \nAs discussed, there are two symmetries for spin current generation in magnetic materials, \nSAHE -like and SHE -like. When these spin currents are absorbed by a nother magnetic layer, \nthey contribute to the total torque on the magnetization : \n𝚪tot=𝚪SHE+𝚪SAHE . (4) \nIn the geometry of our ST -FMR measurements , the GdFeCo and NiFe magnetizations are both \naligned with an angle of 𝜑𝐻 with respect to the 𝑥̂ axis. The sp in polarization corresponding to \nthe SHE -like symmetry, 𝜎̂SHE, lies along the 𝑦̂ axis regardless of the direction of both \nmagnetizations . The spin polarization direction related to the SAHE-like spin current , 𝜎̂SAHE , \nlies along the direction of 𝑚̂GdFeCo which in turn is aligned with the equilibrium direction of \nthe NiFe magnetization , 𝑚̂NiFe . The different contributions to the total torque can further be \ndivided into two contributions, coming from the damping -like (ℎDL) and the field-like (ℎFL) \neffective fields: \n𝚪SHE\n𝛾𝑀SNiFe= ℎDLSHE𝒎̂NiFe×(𝝈̂SHE⏟\n𝒚̂×𝒎̂NiFe)+ℎFLSHE 𝒎̂NiFe×𝝈̂SHE⏟\n𝑦̂, (5𝑎) \n𝚪SAHE\n𝛾𝑀SNiFe= ℎDLSAHE 𝒎̂NiFe ×( 𝝈̂SAHE⏟ \n𝒎̂GdFeCo×𝒎̂NiFe)+ℎFLSAHE 𝒎̂NiFe×𝝈̂SAHE⏟ \n𝒎̂GdFeCo.(5𝑏) \nThe efficiency of the charge -to-spin current conversion is described by the spin Hall angles \n𝜃DL(FL)SAHE and 𝜃DL(FL)SHE. They are related to the SAHE and SHE -like effective fields generated by \nGdFeCo and acting on NiFe layer as follow s [14,15,34,35] : \nℎDL(FL)SHE=ℏ\n2|𝑒|𝑗cGdFeCo\n𝜇0𝑀sNiFe𝑡NiFe 𝜃DL(FL) SHE, (6𝑎) \nℎDL(FL)SAHE=ℏ\n2|𝑒|(𝒎̂GdFeCo×𝑱cGdFeCo).𝒛̂\n𝜇0𝑀sNiFe𝑡NiFe 𝜃DL(FL)SAHE , (6𝑏) \nwith (𝒎̂GdFeCo×𝑱cGdFeCo).𝒛̂=sin(𝜑𝐻) 𝐽cGdFeCo in our geometry , as depicted in Figure 3a . \n \n10 \n \nWe show in the following subs ections that ST -FMR techniques can be useful tool s to further \nstudy the sign and quantification of the different contributions. Indeed, t he analytical expression \nfor the lon gitudinal voltage obtained by ST-FMR measurements reads [15,27,36,37] : \nVdc=−∆𝑅AMRNiFe\n2sin(2𝜑𝐻)𝐼rf(𝜒𝜑𝜃′𝛿ℎ𝜃+𝜒𝜑𝜑′𝛿ℎ𝜑), (7) \nwhere Δ𝑅AMRNiFe is the anisotropic magnetoresistance amplitude , 𝜒𝜑𝜃′ and 𝜒𝜑𝜑′ are respectively \nthe real part of the 𝜑𝜃 and the 𝜑𝜑 components of the susceptibility matrix of NiFe . And, 𝛿ℎ𝜃 \nand 𝛿ℎ𝜑 are respectively the polar and azimuthal component of the exciting fi eld 𝛿ℎ (whose \nexpression is discussed in the next subsection). We can see that only the transverse components \nof the excitation fields contribute to the ST -FMR voltage . We will discuss the different \ncontributions to the total torque: i) first considering the symmetries of Equation 7 , and ii) \nadding a dc current which will modify the susceptibility components. \nWe highlight here that all the equations and signs that our model describe s have been verified \nby considering the results obtained in a //Pt(5)/NiFe (4) reference system. Namely, in this system , \n𝜃𝐷𝐿=𝜃𝐷𝐿𝑆𝐻𝐸>0 and 𝜃𝐹𝐿=𝜃𝐹𝐿𝑆𝐻𝐸>0 (with a negative Oersted field). \n \n4. 2. Symmetry of the ST -FMR signal \nThe NiFe magnetization resonance is triggered by the rf current induced Oersted field, and the \nspin torques described in Equation 5a and Equation 5b . We gather the different contributions \nunder the general term of the exciting field, 𝛿ℎ. Here, the delta means that the excitation is \nweak. The dynamics around the equilibrium position, which takes place in the (𝑒̂𝜃,𝑒̂𝜑) plane \nin spherical coordinates, is only sensitive to the polar and azimuthal components of the exciting \nfield 𝛿ℎ𝜃 and 𝛿ℎ𝜑. Since 𝝈̂SAHE lies along the NiFe magnetizatio n equilibrium position \n𝒎̂𝑁𝑖𝐹𝑒=𝒆̂r, the associated SAHE effective fields do not contribute to the magnetization \ndynamics. On the contrary, the effective fields associated to the SHE -like symmetry contribute \n \n11 \n to the dynamics since 𝝈̂𝑆𝐻𝐸∥ 𝒚̂ and 𝛿ℎ𝜃=ℎDLSHEcos (𝜑𝐻) and 𝛿ℎ𝜑=cos (𝜑𝐻)(ℎOe−\nℎFLSHE) [36,37] . Since at the resonance 𝜒𝜑𝜑′ is an antisymmetric function of the app lied \nmagnetic field and 𝜒𝜑𝜃′ is a symmetric function, we can express th e symmetrical voltage \n𝑉𝑠𝑦𝑚 amplitude and the antisymmetrical amplitude 𝑉𝑎𝑛𝑡𝑖 introduced in Equation 1 by replacing \nthe suitable expressions in Equation 7 : \n𝑉sym= −sin(𝜑𝐻)1\n4𝐼𝑟𝑓 Δ𝑅AMRNiFe\n𝜇0(2𝐻+𝑀effNiFe) 2π𝑓\n𝛾 ℎDLSHE\nΔ𝐻 , (8) \n𝑉anti= −sin(𝜑𝐻)1\n4𝐼rf Δ𝑅AMRNiFe\n𝜇0(2𝐻+𝑀effNiFe) 2π𝑓\n𝛾 [1+𝑀eff\n𝐻res]1\n2\n ℎOe−ℎFLSHE\nΔ𝐻 .(9) \n \n𝑉𝑠𝑦𝑚 (𝑉𝑎𝑛𝑡𝑖) only depends on ℎ𝐷𝐿𝑆𝐻𝐸(ℎ𝑂𝑒−ℎ𝐹𝐿𝑆𝐻𝐸) but the extraction of the effective fields using \nEquation 8 and Equation 9 is not trivial since the rf current has to be evaluate d. Nevertheless, \nwe can discuss the signs of the SHE effective fields. As depicted in Figure 3b , 𝑉𝑠𝑦𝑚 is positive \nwhich me ans that ℎ𝐷𝐿𝑆𝐻𝐸>0. 𝑉𝑎𝑛𝑡𝑖 is negative , and thus ℎ𝐹𝐿𝑆𝐻𝐸>0 assuming that the Oersted \nfield is lower than the FL effective field. \n \n4. 3. Adding a dc bias in ST -FMR: damping modulation and shift of 𝑯𝐫𝐞𝐬 \n \n \nWhen adding a dc bias to the previous ST -FMR measurement, a constant torque is applied on \nthe oscillating magnetization which results in a change in the expression of its dynamical \nsusceptibility matri x. This change induces a modulation of the linewidth and a shift in the \nresonant field, which can be both probed by the ST -FMR technique with an added dc bias. \nBecause the susceptibility is related to the effective field along which the magnetization lies, \nonly the spin polarizations with a projection along this effective field induce a change in the \nsusceptibility. The modulation of damping technique is thus sensitive to both the SHE and \nSAHE -like symmetries and allows to extract overall parameters. \n \n12 \n In the limit of low current densities where we can neglect strong heating contribution that \ndeformed the linear behavior, we can modify the expressions developed for magnetic tunnel \njunctions [38,39] , to apply it in our system [10,15,26] . For the modulation of the NiFe linewidth , \nit reads : \n𝜕𝛥𝐻NiFe\n𝜕𝑖dc=−𝑓\n𝛾𝑁𝑖𝐹𝑒2\n(2𝐻𝑟𝑒𝑠𝑁𝑖𝐹𝑒+𝑀𝑒𝑓𝑓𝑁𝑖𝐹𝑒)𝑆GdFeCo\n𝑊𝑡GdFeCo (𝝈̂SAHE.𝒎̂NiFe ⏟ \n1𝜕ℎ𝐷𝐿𝑆𝐴𝐻𝐸\n𝜕𝐽𝑐𝐺𝑑𝐹𝑒𝐶𝑜+ 𝝈̂𝑆𝐻𝐸.𝒎̂NiFe⏟ \nsin(𝜑𝐻)𝜕ℎ𝐷𝐿𝑆𝐻𝐸\n𝜕𝐽𝑐𝐺𝑑𝐹𝑒𝐶𝑜), (10) \nwhere the left -hand term in the equation is the slope of the modulation of NiFe linewidth, \n𝛾NiFe=𝑔𝑁𝑖𝐹𝑒𝜇𝐵\nℏ . 𝑆GdFeCo accounts for the shunting of the GdFeCo layer by the other conductive \nlayer s, i.e., the current density flowing in GdFeCo layer is 𝐽𝑐GdFeCo=𝑆GdFeCo\n𝑊𝑡GdFeCo𝑖dc with 𝑊 the \nwidth of the slab (10 m). For simplicity, Equation 10 can also be written in terms of the Hall \nangles using Equation 6a,b in the following way : \n𝜕𝛥𝐻NiFe\n𝜕𝑖dc=−𝑓\n𝛾NiFe2\n(2𝐻resNiFe+𝑀effNiFe)𝑆GdFeCo\n𝑊𝑡GdFeCo ℏ\n2|𝑒|sin(𝜑𝐻) 𝜃DLSAHE+ 𝜃DLSHE\n𝜇0𝑀𝑠NiFe𝑡NiFe,(11) \nThe slope s 𝜕𝛥𝐻NiFe\n𝜕𝑖dc that account for the linewidth modulation at 8 GHz are displayed in Figure \n4b for 𝜑𝐻=135° and 𝜑𝐻=−45°. The resistivities were determined independently through \nthe dependence of the GdFeCo and Cu thicknesse s for the different layers obtaining 𝜌𝐶𝑢=15 \ncm, 𝜌𝐺𝑑𝐹𝑒𝐶𝑜=175 cm, and 𝜌𝑁𝑖𝐹𝑒=40 cm. It follows 𝑆GdFeCo=0.11. We note \nthat the slope s obtained when 𝜑𝐻=135° , for all the different frequencies measured, are \nopposite than the one s measured for the //Pt/NiFe reference sample (not shown) . It indicates \nthat the DL overall spin Hall angle, 𝜃DLSAHE+ 𝜃DLSHE, is negative and opposite to the one of Pt \nwhere only the SHE is present . From the average of positive and negative dc fields , or 135° and \n-45°, and for 8, 12 and 14 GHz , we evaluate the overall DL efficiency 𝜃DL𝑆𝐴𝐻𝐸+𝜃𝐷𝐿𝑆𝐻𝐸=\n−0.15±0.05 for the FeCo -rich GdFeCo interfaced with Cu . \n \n13 \n Furthermore, the same experiment also allows us to obtain the corresponding field -like \nvalue s, ℎFL and 𝜃FL. Based on the work of ref . [38,39] , we also obtain the following expression \nthat account s for the linear displacement of the resonance f ield with an added dc current : \n𝜕𝐻resNiFe\n𝜕𝑖dc=𝑆GdFeCo\n𝑊𝑡GdFeCo (𝝈̂SAHE.𝒎̂NiFe ⏟ \n1𝜕ℎFLSAHE\n𝜕𝐽𝑐GdFeCo+ 𝝈̂𝑆𝐻𝐸.𝒎̂NiFe⏟ \nsin(𝜑𝐻)𝜕ℎFLSHE\n𝜕𝐽𝑐GdFeCo )−𝝈̂𝑆𝐻𝐸.𝒎̂NiFe⏟ \nsin(𝜑𝐻)𝜕ℎOe\n𝜕𝑖dc, (12) \nwhere ℎ𝑂𝑒 is the Oersted field which lies along the −𝒚̂ direction in the geometry of our system. \nIts amplitude can be approximated with ℎOe=−1\n2(𝑗cGdFeCo𝑡GdFeCo+𝑗cCu𝑡Cu). Equation 12 \nreads in terms of the FL Hall angles (Equation 6a,b ): \n𝜕𝐻resNiFe\n𝜕𝑖dc=sin(𝜑𝐻)[𝑆GdFeCo\n𝑊𝑡GdFeCo (ℏ\n2|𝑒|𝜃FLSAHE+ 𝜃FLSHE\n𝜇0𝑀𝑠NiFe𝑡NiFe)− 𝜕ℎOe\n𝜕𝑖dc] , (13) \nThe slope obtain ed from the shift of the resonance field vs. 𝑖dc is displayed in Figure 4c for \ndifferent frequencies . We observe that the slope is frequency -independent in agreement with \nEquation 1 3. Moreover, the slope has the same sign as the one in the //Pt/NiFe reference system . \nThat implies that if there is any FL contribution on the GdFeCo/Cu/NiFe system studied here it \nhas the same sign as for the //Pt/NiFe. The slope is evaluate d as 𝜕𝐻resNiFe\n𝜕𝑖dc=0.037 T/A. The \nOersted field is approximated as 𝜕ℎOe\n𝜕𝑖dc=−0.0476 T/A. Finally, considering Equation 13, the \noverall FL efficiency is assessed a s 𝜃FLSAHE+𝜃FLSHE=0.026±0.005. This value has the same \nsign and is comparable to the one measure d in NiFe/Pt [37,40] . We have independently \nmeasured a control //Cu/NiF e sample with out a sizable effect. We can therefore exclude the \nCu/NiFe interface as the origin behind the FL measured in GdFeCo/Cu/NiFe . The sizable \noverall FL value would indicate that even though GdFeCo is not in contact with NiFe, a \nsignificant FL contribution can still be detected. The origin of the FL effect in the trilayer is not \nclear at this sta ge. \n \n \n14 \n \nFigure 4. Damping modulation and Resonance field shift . (a) Schematic of the NiFe \nresonance condition with additional 𝑖dc current injected . (b) 𝑖dc dependence of the NiFe \nlinewidth for a rf frequency of 8 GHz . (c) Resonance field shift vs. 𝑖dc for three frequencies. \nUnlike the damping or linewidth modulation, we can see that the resonance field shift is \nfrequency independent. \n \n5. Discussion and conclusions \nThe overall efficiencies for FeCo -rich GdFeCo/Cu/NiFe are evaluated 𝜃DL𝑆𝐴𝐻𝐸+𝜃𝐷𝐿𝑆𝐻𝐸=\n−0.15±0.05 and 𝜃FLSAHE+𝜃FLSHE=0.026±0.005. For sake of comparison, the SAHE \nefficiency of a ferromagnet such as CoFeB is 𝜃𝑆𝐴𝐻𝐸𝐶𝑜𝐹𝑒𝐵=−0.14 [15], and the SHE efficiency of \nPt heavy metal is 𝜃𝑆𝐻𝐸𝑃𝑡=0.056−0.076 [29,41,42] . Seki et al . show in FePt that DL \n𝜃𝑆𝐴𝐻𝐸+𝑆𝐻𝐸𝐹𝑒𝑃𝑡=0.25 from the linewidth modulation [43]. \nThe damping -like SAHE contribution dominates over the SHE one: |𝜃𝐷𝐿𝑆𝐴𝐻𝐸|>|𝜃𝐷𝐿𝑆𝐻𝐸| with a \nnegative SAHE contribution for FeCo -rich GdFeCo , and a positive SHE contribution . We also \nshow that the field -like SHE contribution is positive. However, we cannot estimate the \nindividual value of each contribution. We perform the same experiments at 15 K where our \nferrimagnet is Gd -rich and its magnetization aligns in -plane with a field above 0.4 T. From the \nsign of the symmetric contribution we confirm that SHE remains positive when crossing the \nmagnetic compensation temperature. This is consistent with the fact that the SHE does not \ndepend on the GdFeCo magnetic propert ies. In contrast, we cannot conclude of any 𝜃𝐷𝐿𝑆𝐴𝐻𝐸 sign \nchange because the modulation of linewidth experiments at 15 K is hidden by others effect that \n \n \n15 \n are out of the scope of this study. However, the large variation in absolute value between these \nresults and the one previously reported, for a Gd-rich GdFeCo at room temperature, |𝜃𝐷𝐿𝑆𝐴𝐻𝐸+\n𝜃𝐷𝐿𝑆𝐻𝐸|=0.80±0.05 [10], suggest that the si gn of 𝜃𝐷𝐿𝑆𝐴𝐻𝐸 changes between FeCo -rich and Gd -\nrich samples . If so, t he opposite DL -SAHE sign for FeCo -rich GdFeCo might indicate that the \nSAHE spin polarization comes always from the same magnetic sublattice . Despite that , further \nstudies could be carried out to confirm that. \nGdFeCo can thus generate efficient spin currents and the different symmetries allow this \nmaterial to be used in a wide variety of devices for spintronics. For instance, the SHE spin \ncurrent can generate self -torque [10] and can be used for the electrical switching of the \nmagnetization, as shown in epitaxial FePt [44] or CoTb [45]. Also, the total spin current \n(SAHE+SHE) can be used to induce a torque on another magnetic layer or for the manipulation \nof skyrmions. \n \nIn summary, w e have studied FeCo -rich GdFeCo/Cu/NiFe heterostructure at room t emperature . \nFirst, structural, and chemical analyses were performed by HRTEM and EELS. T hen, the \nmagnetic properties and the relevant spin -orbitronics parameters were determined by \ncombining magnetometry, spin-torque ferromagnetic resonance and additional dc current \ndependence . The overall damping -like and field -like efficienc ies, which include the SHE -like \nand the SAHE-like symmetries, are 𝜃𝐷𝐿𝑆𝐴𝐻𝐸+ ����𝐷𝐿𝑆𝐻𝐸=−0.15±0.05 and 𝜃𝐹𝐿𝑆𝐴𝐻𝐸+ 𝜃𝐹𝐿𝑆𝐻𝐸=\n0.026±0.005 at room temperature . We show that SAHE dominates over SHE contribution on \nthe DL torque. Furthermore , this study show s that the SHE contribution does not change sign \nwhen crossing the magnetic compensation temperature while SAHE may change sign \ndepending on the dominant sublattice of the ferrimagnet . All this underlines the importance of \nGdFeCo, and RE -TM ferrimagnets in general, as promising materials in spintronics for the \nexploitation of their strong spin-orbit torque . \n \n \n16 \n \nData availability \nThe data that support the fi ndings of this study are available from the corresponding author on \nreasonable request. \n \nAcknowledgements \nWe acknowledge A. Fert for fruitful discussions. This work was supported from Agence \nNationale de la Recherche (France) under contract ANR -19-CE24 -0016 -01 (TOPTRONIC) , \nANR -20-CE24 -0023 (CONTRABASS), and ANR -17-CE24 -0025 (TOPSKY), from the \nFrench PIA project “Lorraine Universit é d’Excellence ”, reference ANR -15IDEX -04-LUE and \nby the « SONOMA» projec t co-funded by FEDER -FSE Lorraine et Massif des Vosges 2014 -\n2020, a European Union Program . DCB and JAS also thanks 2019 and 2021 Master -LUE \nprogram internship. Devices in the present study were patterned at MiNaLor clean -room \nplatform which is partially s upported by FEDER and Grand Est Region through the RaNGE \nproject. \n \n \nReferences \n[1] S. K. Kim, G. S. D. Beach, K. -J. Lee, T. Ono, T. Rasing, and H. Yang, “Ferrimagnetic \nspintronics” Nat. Mater. 21, 24 (2022). \n[2] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, U. Atxitia, O. Chubykalo -\nFesenko, S. El Moussaoui, L. Le Guyader, E. Mengotti, L. J. Heyderman, F. Nolting, \nA. 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Raveau1 \n \n(1)CRISMAT, UMR 6508, CNRS-ENSICAEN \n 6 Bd Marechal Juin, 14050 Caen, France \n(2) Institut Laue Langevin, 6 rue Jules Horowitz , 38042 Grenoble, France \n \n \nAbstract \n CaBaCo\n4O7 represents a new class of ferrimagnets whose structure is built up of CoO 4 \ntetrahedra only, similarly to other members LnBaCo 4O7 of the “114” series, forming an alternate \nstacking of kagomé and triangular layers. Neutr on powder diffraction reveals, that this compound \nexhibits the largest distortion within the “114” series, characterized by a strong buckling of the \nkagomé layers. Differently from all other members it shows charge ordering, with Co2+ sitting on \ntwo sites (Co2, Co3) and “m ixed valent” cobalt “Co3+/Co2+L” sitting on two other sites (Co1, \nCo4). The unique ferrimagnetic structure of this cobaltite at 4 K can be described as the \nassemblage of ferrimagnetic triple chains (Co1 Co2 Co3) running perpendicular to the kagomé \nlayers, ferromagnetically coupled within the la yers, and antiferromagnetic ally coupled with a \nfourth cobalt species Co4. The lifting of the geometrical frustration towards ferrimagnetism, which \nappears in spite of the triangula r topology of the cobalt lattice, is explained by the very large \nstructural distortion, charge ordering phenomena and large cobalt valence compared to other LnBaCo\n4O7 oxides. \n \nKeywords \n \nCobalt oxides, CaBaCo\n4O7, 114, cobaltite, magnetic structure, ferrimagnetism, neutron. \n * Corresponding author: Dr. Vincent Caignaert \ne-mail: vincent.caignaert@ensicaen.fr Fax: +33 2 31 95 16 00 Tel: +33 2 31 45 26 32 \n \nThe competition between geometrical frustration and magnetic ordering in transition metal \noxides leading to exotic magnetic properties and transitions has been the subject of many \ninvestigations these last few decades [1] that are still to date not completely understood. In this respect, oxides whose metallic subl attice exhibits vertex-sharing tr iangular or tetrahedral topology \nwith antiferromagnetic (AFM) interactions are of great interest. This is for example the case in \nA\n2B2O7 pyrochlores [2-3], whose metal fr amework consists of tetrahedral [A 4]∞ or [B 4]∞ \nsublattices. The spinels AB 2O4 may be also geometrically frustrated magnets when the A cation is \ndiamagnetic [4]. A second important example deal s with the “kagomé” oxides, where the magnetic \nfrustration originates from a 2D triangular topology of their ma gnetic network. This class of \ncompounds is nicely illustrated by the Jarosite family [5], which display either a spin-glass \nbehavior or long-range magnetic order with propagation vectors k=0 and k=(1/3, 1/3, 0). \nThe discovery, some years ago of the cobaltites LnBaCo 4O7 [6-9] and very recently of the \nferrites CaBaFe 4O7 [10] and YBaFe 4O7 [11, 12] opened the route to the study of new \ngeometrically frustrated magnets. Structurally, th ese compounds are closely related to the spinels \nas far as they contain similar triangular subl attices of cobalt or iron atoms with the kagomé \ntopology, but they differ fundamentally from the la tter by the fact that their Co-O or Fe-O \nframework consists exclusively of CoO 4 or FeO 4 tetrahedra, forming two sorts of layers, called \ntriangular and kagomé respectivel y. As a consequence, the magnetic properties of these oxides are \nvery different from those of other “kagomé” oxides and show very complex transitions. Such a behavior is exemplifie d by the cobaltite YBaCo\n4O7 for which a spin glass transition was first \nreported around T f ∼ 66 K [7], whereas long range ma gnetic order was evidenced below T N = \n110K [13], and a magnetic tran sition with short range correlations was revealed above T N in a \nsingle crystal of this phase [15]. Fr om the structural view point, this phase also exhibits a structural \ntransition from orthorhombic to he xagonal above room temperature at T S = 313 K [13], the \northorhombic low temperature form corresponding to a distortion of the high temperature \nhexagonal form. Quite remarkably, all th e magnetic transitions appear below T S and the recent \nstudies of YBaCo 4O7 by single crystal neutron scattering by Manuel et al. [15] demonstrate a very \nimportant feature: At 130 K, i.e. above T N, this phase exhibits a quasi 1D magnetic order along cr, \nwhereas in the kagomé layers (a-b plans) it di splays a strong degeneracy, so that it can be \ndescribed as a new class of 2D frustrated oxide. \nRecently, we synthesized the cobaltite CaBaCo 4O7 [16], which differently from all other \ncobaltites of this series, including Y 0.5Ca0.5BaCo 4O7 [17], exhibits ferrimagnetic properties below \nTC = 70 K. Bearing in mind that the orthorhombic dist ortion of this phase is much larger than any \nobserved for other cobaltites, we have explored its nuclear a nd magnetic structure. We report \n2/19 herein on the crucial role of the structural distortion, of the cobalt valence and of charge ordering \nin the appearance of ferrimagnetic ordering in this phase. We show that the AFM out of plane \ncoupling in the “Co1 Co2 Co3” triple chains running along cris a key parameter for weakening the \nin-plane magnetic frustration and consequent ly for inducing the 3 D magnetic ordering. \n \nThe sample was synthesized from a stoichiometric mixture of CaCO 3, BaCO 3 and Co 3O4 \nfirst heated at 900°C in air for 12 h for decarbona tion. The mixture was then heated in air at \n1100°C for 12 h, and quenched down to room temperature. Specific heat measurements were \ncarried out by means of a commercial Physical Properties Measurements System (PPMS, Quantum \nDesign) using a relaxation method with a 2 τ fitting procedure. Neutron powder diffraction (NPD) \nversus temperature was performed on the D2b diffractometer (ILL, Grenoble) at room temperature \n(λ = 1.59Å) and from 4 to 150 K ( λ = 2.428 Å). \n \nThe NPD patterns registered at various temperat ures reflect an excellent crystallization of \nthe compound, as illustrated for the pattern of CaBaCo 4O7 registered at 4 K (Fig.1). An impurity, \nidentified as CoO, was taken into account, due to the presence of one magnetic reflection \n(2θ=28.2°, d=4.92Å) close to those of CaBaCo 4O7. The amount of CoO was estimated to 2% in \nweight and is probably related to Co 3O4 in excess during the synthesis. Whatever the temperature, \nthe Rietveld refinements of CaBaCo 4O7 evidence the orthorhombic Pbn2 1 symmetry, in the whole \ntemperature range. This is in cont rast to most of the other LnBaCo 4O7 cobaltites [8-9] which \ngenerally show a structural transition from or thorhombic (O) to hexagonal (H) symmetry as the \ntemperature increases, corre sponding to the relations: a O ∼ aH ∼ 6.3 Å, b O ∼ aH3 ∼ 11 Å and c O \n∼ cH ∼ 10.2 Å. The evolution of the cell parameters versus temperature (Fig.2a) shows that the ar \nand parameters decrease with T, whereas the crbr\n parameter increases as T decreases, leading to \nan overall decrease of the cell volume (Fig .2b). The orthorhombic symmetry results from a \ndistortion of the hexagonal cell, wh ich can be quantified as D = (b/ 3-a)/a, corresponding to an \nexpansion of the hexagonal cell along one [110] H direction and a contraction along the [1 10]H \nperpendicular direction. The amplit ude of this distortion D in CaBaCo 4O7 is the largest ever \nobserved in the “114” series of cobaltites. It in creases linearly as T decreases from D=1.05% at \nroom temperature to D=1.8% at 50 K, and remain s constant below this temperature down to 4 K \n(Fig.2b). These values can be compared to th e D values previously observed for the LnBaCo 4O7 \ncompounds at low temperature. For the latter, they are always much smaller, reaching a maximum \nvalue of 0.5% at 10 K for YbBaFe 4O7, i.e. still much smaller than the D value observed for \nCaBaCo 4O7 at room temperature. This high value of the distortion is quite remarkable, since it \ninduces a lifting of the geometrical frustration, wh ich is initially present in the cobalt network of \n3/19 the LnBaCo 4O7 series [6-9], due to its triangular topol ogy. It explains the absence of magnetic \nfrustration at low temperature, and the a ppearance of a ferrimagnetic state in CaBaCo 4O7. \n \nThe atomic coordinates of this oxide, determined at 4 K (Table 1) and at room temperature \nby NPD are in agreement with the results obta ined from X-Ray powder diffraction refinements \n[16], but show a higher accuracy of the oxyge n positions. The crystal structure of CaBaCo 4O7 \nconsists of a 1:1 stacking of kagomé (K) and triangular (T ) layers of CoO 4 tetrahedra along cr. The \nprojections of the structure along (Fig.3a) and arbr\n (Fig.3b) show that the triangular layers (T) \nformed by the Co1 tetrahedra are perfectly flat; in contrast, the kagomé layers (K) are strongly \ncorrugated, the apical oxygen of the Co2, Co3 and Co4 tetrahed ra being located in a plane, \nwhereas the oxygen atoms of the basal planes of these tetrahedra form a waving layer. \nThe interatomic distances (Table 2) show that the four symmetry-indepe ndent cobalt tetrahedra \ncan be classified into two groups : the Co2 and the Co3 tetrahedra belonging to the K layers which \nexhibit larger Co-O distances, i.e. average dist ances ranging from 1.94-1.97 Å (RT) to 1.95 Å (4 \nK), and the Co1 and Co4 tetrahed ra belonging to the T and K la yers respectively, whose Co-O \ndistances are significantly smaller i.e. average distances ranging from 1.87-1.88 Å (RT) to 1.85-1.89 Å (4 K). Such a distribution of the Co-O bond lengths between these four independent \ncrystallographic sites suggests that, whatever the temperature, CaBaCo\n4O7 exhibits charge \nordering in agreement with the stoichiometric formula CaBaCo 22+Co23+O7. Nevertheless, though \nthe bond lengths observed for Co2 and Co3 ar e in agreement with those observed for Co2+ in \ntetrahedral coordination, the Co-O bond lengths observed for Co1 and Co4 are significantly larger \nthan those expected for Co3+ in tetrahedral coordination, which were found typically close to 1.79 \nÅ in oxides. The bond valence sum calculations (BVS), performed accord ing to Alternatt and \nBrown [19], support this viewpoint (Table 3). On e indeed observes that Co2 and Co3 exhibit a \ncharge comprised between +1.9 and +2.1, whatev er the temperature and can be considered as \ndivalent, whereas a charge ranging from +2.4 to +2 .6 is observed for Co1 and Co4, suggesting that \nthese cations should be rather mixed valent, with an average value of +2.5. Such a mixed valence, \nsignificantly smaller than +3, is in contradiction with the stoichiometric formula. Indeed, from \nstructure refinements neither oxygen defi ciency leading to the formula CaBaCo 4O6.5, nor presence \nof protons according to the formula CaBaCo 4O6OH were detected. Bear ing in mind that the \ntetrahedral coordination of Co3+ is very rare, due to the crystal field effect which favors the \noctahedral coordination, the possibi lity of partial charge transfer from cobalt to one oxygen may be \nconsidered. This would lead to a ligand hole configuration ( L) similar to that observed for apical \noxygen in layered cuprates [20], corresponding to a hybridizatio n of the bonds of the Co1 and Co4 \ntetrahedra according to the scheme Co3+-O = (Co3d6) Ù Co2+=O-(Co3d7L). The significantly \n4/19 5/19 larger value of the apical Co-O 7 bond length, of the Co1 and C o4 tetrahedra, ranging from 1.91-\n1.95 Å at RT to 1.97 – 1.89 Å at 4 K, suggests that the hole appears on the O7 atom shared \nbetween one Co4 and one Co1 tetrahedron. A study of this compound by X-ray absorption \nspectroscopy would be necessary to elucidate this point. \nThe geometry of the CoO 4 tetrahedra is significantly modi fied by temperature. The size of \nthe Co1 and Co2 tetrahedra increases as T d ecreases, whereas the si ze of the Co3 and Co4 \ntetrahedra decreases with temperature, as show n from the average bond lengths (Table2). \nThe evolution of the distortion is as well very complex. The distortion of the Co1 and Co3 \ntetrahedra increases significantly as T decrease s, whereas the distortion of the Co2 tetrahedra \ndecreases slightly with T, and no significant va riation of the distorti on with temperature is \nobserved for the Co4 tetrahedra. Clearly, the stru ctural evolution vs T does not influence the \ncharge ordering of CaBaCo 4O7: it can be described as 1:1 charge ordering between Co2/Co3 sites \noccupied by Co2+ and Co1/Co4 sites which are mixed or hybridized Co3+ (3d6) / Co2+ (3d7L). \nThe Ca-O distances of the CaO 6 octahedra, ranging from 2.25 to 2.40 Å at 4 K and from \n2.27 to 2.35 Å at RT are in agreement with th ose usually observed. The coordination of barium \nchanges from 5+7 at RT to 6+6 at 4 K. The 5 or 6 nearest oxygen neighbors are located at usual \ndistances from barium, ra nging from 2.80 to 2.90 Å, whereas the other seven or six oxygen atoms \nsit much further, at distances ranging from 3.36 to 3.58 Å. In spite of this reasonable coordination \nof barium, the computed valence of barium calcu lated from BVS is much smaller than expected \ni.e. +1.4 instead of +2, suggesting that Ba2+ is strongly underbonded. A similar behavior has been \nobserved previously for YbBaCo 4O7 [18] and was attributed to a structural instability, in \nagreement with the transition from hexagonal to orthorhombic symmetry at decreasing \ntemperature. Such a hypothesis does not hold here, since CaBaCo 4O7 does not exhibit any \nstructural transition in the whol e temperature range and moreover the so-obtained barium valence \nof 1.40 for CaBaCo 4O7 at 4K, which exhibits th e largest distortion, is st ill very close to that \nobtained for the hexagonal form of YbBaCo 4O7 (1.33). The origin of th is abnormally weak valence \nof barium remains unexplained. \nThe NPD pattern registered at 4 K shows cl early the presence of magnetic Bragg peaks \n(Fig.1) with a propagation vector k=0, whose in tensity decreases as T increases. The latter \ndisappear above T\nC=70 K, in agreement with the specific h eat measurement (inset Fig.1) and with \nthe ferrimagnetic to paramagnetic transition previ ously observed for this oxi de [16]. To generate \nall the spin configurations comp atible with the crystal symmetry we carried ou t a group theory \nanalysis using the program BasiReps. There are f our irreducible representa tions (IR) associated \nwith the Pbn2 1 space group and k=(0, 0, 0). Among the latter, only thre e IRs allow a ferrimagnetic \nalignment of Co sublattices. The ba sis vectors of these representati ons are listed in Table 4. The results of simulated annealing runs, to solve the ma gnetic structure, lead to a unique solution in the \nΓ4 representation. The final refinement shows that one of the 3 coefficients ( Ψ12 in Table 4), \nrepresenting the 3 basis v ectors, refines to zero leading to a magnetic structure where all the spins \nlie in the ( ab) plane. This solution of the magnetic structure at 4 K confirms its ferrimagnetic \nnature, characterized by a simultaneous ordering of Co spins in both triangular and kagomé layers. \nThus, the magnetic structure at 4 K (Fig.4) can be described in the Pbn2 1 space group, with all Co \nspins lying in the (a,b) plane, i.e. parallel to the kagomé and tr iangular layers. The ferrimagnetic \nstructure of this cobaltite is quite unique w ith respect to all other members of the LnBaCo 4O7 \nseries. In the kagomé layers, the Co2 and Co3 spins form zigzag chains running along b (Fig.4a \nand 7b). In those chains, one Co2 alternates wi th one Co3 species, and two successive Co2 (or \nCo3) spins are oriented at ∼60°, while “Co2Co3” ferromagnetic couples are formed. As a result, \nthe transverse magnetic compone nts of Co2 (and Co3) along ar are antiferromagnetically coupled \nand cancel each other, whereas the longitudinal magnetic components of Co2 and Co3 along br\n are \nall oriented in the same direction. Thus, C o2 and Co3 form ferromagnetic zigzag chains along br\n, \nwhich are themselves all ferromagnetically coupled in the entire kagomé layer. Between those \nchains, the Co4 species have thei r spins oriented antiparallel to the FM resultant of “Co2Co3” \nchains. Consequently, a ferrimagnetic order of the Co2, Co3 and Co4 sp ins takes place in the \nkagomé layers, whose facile ma gnetization is directed along br\n. The Co1 spins of the triangular \nlayers (Fig.4a-b) are practically antiparallel to Co2 and Co3 spins (Fig.4a and 7b), two successive \nCo1 spins, being oriented at 60°, similarly to “C o2 Co3” chains. Thus, their transverse components \nare antiparallel and cancel each other and their longitudinal components are parallel, forming \nferromagnetic chains Co1 running along br\n. The latter chains are antiferromagnetically coupled \nwith the “Co2 Co3” chains. In summary, the 3 D ferrimagnetic ordering of cobalt spins in \nCaBaCo 4O7, consists of “Co2 Co3” zigzag FM chains running along br\n, antiferromagnetically \ncoupled with Co4 and Co1 spins along ar and br\n respectively, the resultant magnetic moments of \ncobalt spins being parallel to the facile magnetization axis br\n. Bearing in mind the description of \nthe cationic framework of CaBaCo 4O7, previously proposed by Chapon et al [13], which consists \nof chains of corner-sharing “Co5” bipyramids running along c (Fig.4b) interconnected through \n“Co3” triangles in the (a,b) plane (Fig.4a), anothe r description of this ferrimagnetic structure can \nbe proposed. It consists of the assemb lage of ferrimagnetic chains running along cr, and \ncorresponding to the AFM coupling of Co1, with Co2 and Co3. In th is description, each triple \n“Co1 Co2 Co3” chain is ferromagnetically c oupled to two other id entical chains, and \nantiferromagnetically coupled to three Co4 species in the (a, b) plane. \n6/19 The evolution of the spontaneous magnetization versus temperature (Fig.5) corroborates the \nprevious magnetic measurements [16]. One ind eed observes a resultant magnetic moment of ∼ 1μB \nper f.u. at low temperature, close to the value observed from M(T) data ( ∼ 0.6μB per f.u.), which \ndecreases abruptly at T C down to ∼0.2μB . \nThe evolution of the magnetic moments of the fo ur cobalt sites, versus temperature (Fig.6), \nshows also an abrupt decrease around T C as expected. More importa ntly, the values of these \nmoments at low temperature supp ort strongly the existence of charge ordering. One indeed \nobserves significantly larger va lues of the magnetic moment s of Co1 and Co4 (2.5 - 3.0 μB per Co) \nin agreement with their valences deduced from BVS’s, i.e. Co2+L/Co3+ (theoretical moment ∼ \n4μB). These values of the magnetic moments, sma ller than the theoretical ones, suggest that a \ndegree of frustration remains, inducing a sl ight disordering of the cobalt spins. \n \n \n7/19 This study shows that CaBaCo 4O7 exhibits, compared to all other “114” LnBaCo 4O7 \ncobaltites, a very unique ferrimagnetic structure, in spite of its close structural relationships with \nthese oxides. The origin of this complex magnetic structure is, most likely, reminiscent of the \ncompetition between geometric frustration that appears in the kagomé layers due to their triangular \ntopology and the antiferromagnetic inte ractions that take place along crof corner-shared tetrahedra \n(Co1, Co2 and Co3). The crucial role of these triple AFM cobalt chai ns in the establishment of the \n3 D magnetic ordering in these compounds is clear ly shown by comparing th e magnetic structure \nof YBaCo 4O7 previously observed at 80 K by Chapon et al [15] (Fig.7a) with the structure of \nCaBaCo 4O7 at 4 K (Fig.7b). Both magnetic structures consist of very similar triple AFM chains \n“Co1 Co2 Co3” running along and ferromagnetically coupled along , i.e. forming \nferrimagnetic layers parallel to (0 10). The two structures differ by the coupling of those layers \nalong , which is antiferromagnetic for YBaCo 4O7 (Fig.7a) and ferromagnetic for CaBaCo 4O7 \n(Fig.7b). The spin orientation of Co4 is also di fferent in the two structures, the moment of Co4 \nbeing oriented at 90° from other spins in YBaCo 4O7 and at ∼ 30° in CaBaCo 4O7. The Co4 spins \nare antiferromagnetically coupled in YBaCo 4O7 (Fig.7a) whereas they are ferromagnetically \ncoupled along b in CaBaCo 4O7 (Fig.7b), so that the magnetic structure of the latter phase remains \nferrimagnetic with b as facile magnetization axis. In both structures, the lifting of the magnetic \nfrustration is explained by the fact th at the in-plane exchange interaction J 1, which favors the 120° \ngeometry in the hexagonal structure, is reduc ed by the orthorhombic di stortion as T decreases. \nThis orthorhombic distortion is much larger for CaBaCo 4O7, as shown from the in-plane Co-Co \ndistances between Co2, Co3 and C o4 ranging from 3.07 to 3.33 Å at RT and from 3.00 to 3.31 Å \nat 4 K, so that the Co triangles are no more equilateral. The 3 D ma gnetic ordering is also crar\nbr\nr\nr8/19 supported by the out of plane J 2 interaction, which tends to favor the existence of AFM triple \nchains “Co1 Co2 Co3” in both structures. The fact that the cobalt valence is larger in CaBaCo 4O7, \nbut especially that Co3+/or Co2+L are preferentially located on the Co1 and Co4 sites may change \nsignificantly J 2. Such a charge ordering may be at the origin of the FM coupling of the chains in \nCaBaCo 4O7, in contrast to the AFM coupling observed for YBaCo 4O7. Thus, the combination of \nthese different factors – structural distortion, mixed valence of coba lt and charge ordering – appear \nas key factors for controlling the magnetic ordering in the “114” cobaltite series. \n Finally, it is worth pointing out that the ferrimagnetism observed for CaBaCo 4O7, is \nsignificantly different from what has been observed for CaBaFe 4O7 [10] where T C and magnetic \nmoments are much higher. The hexagonal symmetry of the latter suggests that th e iron spins in this \nphase might be lying out of plane. \n 9/19 References \n1. A. Ramirez, Ann. Rev. Mater. Sci. 24, 453 (1994) \n2. J.E. Greedan, J. Alloys Compounds 408-412 , 444 (2006) \n3. M. J. Harris and M. P. Zinkin, Modern Phys. Lett. B 10, 417 (1996) \n4. S.-H. Lee, C. Broholm, W. Ratcliff, G. Ga sparovic, Q. Huang, T. H. Kim and S.-W. \nCheong, Nature 418, 856 (2002) \n5. A. S. Wills, Can. J. Phys. 79, 1501 (2001) \n6. M. Valldor and M. Andersson, Solid State Sci. 4, 923 (2002) \n7. M. Valldor, J. Phys. Condens. Matter 16, 9209 (2004) \n8. A. Maignan, V. Caignaert, D. Pelloquin, S. Hébert, V. Pralong, J. Hejtmanek, and D. \nKhomskii, Phys. Rev. B 74, 165110 (2006) \n9. V. Caignaert, A. Maignan, V. Pralong, S. Hé bert and D. Pelloquin, Solid State Sciences 8, \n1160-1163 (2006) \n10. B. Raveau, V. Caignaert, V. Pralong, D. Pelloquin and A. Maignan, Chem. Mater. 20, 6295 \n(2008) \n11. V. Caignaert, A.M. Abakumov, D. Pelloquin, V. Pralong, A. Maignan, G. Van Tendeloo \nand B. Raveau, Chem. Mater. 21, 1116 (2009) \n12. V. Pralong, V. Caignaert, A. Maigna n and B. Raveau, J. Mat. Chem., DOI: \n10.1039/B911611G (2009) \n13. L. C. Chapon, P.G. Radaelli, H. Zheng and J.F. Mitchell, Phys. Rev. B 74 172401 (2006) \n14. M. Soda, Y. Yosui, T. Moyoshi, M. Sato, N. Igawa and K. Kakurai, J. Phys. Soc. Japan 75, \n054707 (2006) \n15. P. Manuel, L.C. Chapon, P.G. Radaelli, H. Zheng and J.F. Mitche ll, Phys. Rev. Lett. 103, \n037202 (2009) \n16. V. Caignaert, V. Pralong, A. Maignan and B. Raveau, Solid State Comm. 149, 453 (2009) \n17. M. Valldor, Solid State Sci. 8, 1272 (2006) \n18. A. Hug, J.F. Mitchell, H. Zheng, L.C. Chapon, P.G. Radaelli, K.S. Knight and P.W. \nStephens, J. Solid State Chem. 179, 1125 (2006) \n19. I. D. Brown and D. Alternatt, Acta Cryst. B 41, 244 (1985) \n20. A. Bianconi, J. Budnick, A.M. Flank, A. Fontai ne, P. Lagarde, A. Marcelli, H. Tolentino, \nB. Chamberland, G. Demazeau, C. Michel and B. Raveau, Phys. Lett. A 125, 285 (1988) \n \n 10/19 Table Captions: \n \nTable1: Atomic and magnetic parameters for CaBaCo 4O7 at 4K. All sites are fully occupied. \nSpace group: Pbn2 1. Cell parameters: a= 6.2613(1) Å, b=11.0399(2) Å, c=10.1642(2) Å. \n \nAtom x y z B \nCa 0.0066(21) 0.6729(13) 0.8708(13) 0.06(13) \nBa 0.0043(14) 0.6651(10) 0.5 0.06(13) \nCo1 0.0155(33) -0.0003(24) 0.9332(25) 0.09(13) \nCo2 -0.0067(26) 0.1688(20) 0.6902(28) 0.09(13) \nCo3 0.2527(33) 0.0955(16) 0.1904(21) 0.09(13) \nCo4 0.2673(25) 0.9209(15) 0.6841(21) 0.09(13) \nO1 -0.0155(27) 0.0044(11) 0.2460(16) 0.64(5) \nO2 -0.0045(20) 0.4914(11) 0.2285(16) 0.64(5) \nO3 0.7817(21) 0.2613(10) 0.7815(19) 0.64(5) \nO4 0.7319(18) 0.7418(12) 0.2141(17) 0.64(5) \nO5 -0.0549(12) 0.1522(8) 0.4997(16) 0.64(5) \nO6 0.2035(13) 0.1097(6) 0.0062(19) 0.64(5) \nO7 0.2645(15) 0.9471(6) 0.5002(22) 0.64(5) \n \nAtom M x M y M \nCo1 -1.78(16) -2.20(16) 2.83(16)\nCo2 1.01(14) 1.77(20) 2.04(19)\nCo3 0.72(22) 1.92(11) 2.05(15)\nCo4 -0.05(20) -2.44(12) 2.44(15)\n Table 2: \nInteratomic distances at 4K for CaBaCo 4O7. \nBond Distance (Å) Bond Distance (Å)\n Ca-O2 2.32(4) Co1-O1 1.90(6) \n O3 2.24(4) O5 1.82(5) \n O4 2.32(4) O6 1.85(5) \n O5 2.35(4) O7 1.98(5) \n O6 2.38(4) Co2-O1 2.00(5) \n O7 2.41(4) O3 1.91(5) \n Ba-O2 3.36(3) O4 2.00(4) \n O2 2.89(3) O5 1.97(6) \n O3 3.54(4) Co3-O1 2.04(5) \n O3 2.72(4) O2 1.99(5) \n O4 3.48(3) O3 1.83(5) \n O4 2.79(3) O6 1.90(6) \n O5 2.82(2) Co4-O1 1.89(5) \n O5 3.45(2) O2 1.78(4) \n O6 3.57(3) O4 1.84(4) \n O6 2.81(3) O7 1.89(6) \n O7 3.51(3) \n O7 2.81(3) \n \n Table 3 : Bond valence sum calcu lations (BVS) at 4K. \nAtom BVS \nCa 2.23(5)\nBa 1.48(3)\nCo1 2.39(8)\nCo2 1.89(7)\nCo3 2.08(7)\nCo4 2.63(9)\nO1 1.97(7)\nO2 1.82(6)\nO3 2.01(7)\nO4 1.74(5)\nO5 1.78(7)\nO6 1.79(6)\nO7 1.59(6)\n \nTable 4: Basis vectors for the space group Pbn2 1 with k=(0,0,0). The decomposition of the \nmagnetic representation for the Co site is . The atoms are defined according \nto 1:( x,y,z), 2:( -x,-y,z+1/2 ), 3:( -x+1/2,y+1/2 ) and 4:( x+1/2,-y+1/2,z+1/2 ). 1\n41\n31\n21\n1 3 3 3 3Γ + Γ + Γ + Γ = Γmag\n \n Γ 1 Γ2 Γ3 Γ4 \n Ψ1 Ψ2 Ψ3 Ψ4 Ψ5 Ψ6 Ψ7 Ψ8 Ψ9 Ψ10 Ψ11 Ψ12 \nAtom m //a m //b m //c m //a m //b m //c m //a m //b m //c m //a m //b m //c \nm1 1 1 1 1 1 1 1 1 1 1 1 1 \nm2 -1 -1 1 -1 -1 1 1 1 -1 1 1 -1 \nm3 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 \nm4 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 \n \n \n11/19 Figure Captions \nFig. 1 (Color online) Rietveld refinement of CaBaCo 4O7 at 4 K: The dots and solid line represent \nthe experimental data points and calculated di ffraction pattern, respectiv ely. The difference is \nshown at the bottom. The row of markers sh ows the positions of the nuclear and magnetic \nreflections for CaBaCo 4O7 and CoO. The thick solid line (blu e online) represents the contribution \nfrom magnetic scattering of CaBaCo 4O7 alone. The inset shows the sp ecific heat C versus T. \nFig. 2 (Color online) Evolution of the cell parameters (a) and of the cell volume and structural \ndistortion (D = (b 3-a)/a)(b) versus temperature of CaBaCo 4O7 \nFig. 3 (Color online) Structure of CaBaCo 4O7 at 4 K showing the arrangement of the kagomé (K) \nand triangular (T) laye rs: (a) projection along a, (b) projection along b. \nFig. 4 (Color online) Magnetic structure of CaBaCo 4O7: projection along c (a) and projection \nalong (b). The cobalt network, built up of “Co5” triangular bipyramids and “Co3” triangles is \ndrawn as a support r\nar\nFig. 5 Evolution of the spontaneous magnetization versus temperature for CaBaCo 4O7. The \nresulting moment is given in Bohr magneton per f.u. The line is a guide to the eye. \nFig. 6 (Color online) Evolution of the magnetic moments of the different cobalt ions versus \ntemperature for CaBaCo 4O7. The lines are guide to the eye. \nFig. 7 Comparison of the magnetic structures of YBaCo 4O7 at 80 K (a) and of CaBaCo 4O7 at 4K \n(b) view along . The ellipses show the similar triple ferrimagnetic chains “Co1 Co2 Co3” and \ntheir relative orientati ons in the two compounds. cr\n \n \n12/19 2-θ (degrees)Intensity\n20 40 60 80 100 120 140 0200400\n25 50 75 100 0 50100150\nT (K)C (J/Kmol)\n \n \n \nFigure 1 \n \n13/19 \n \n \n \nFigure 2 \n \n14/19 \n \nFigure 3 \n \n15/19 \n \n \n \nFigure 4 \n \n16/19 \n \nFigure 5 \n \n17/19 \n \n \nFigure 6 \n \n18/19 \n \n \n \nFigure 7 \n \n \n19/19 " }, { "title": "1903.04432v3.Giant_spin_orbit_torque_in_a_single_ferrimagnetic_metal_layer.pdf", "content": "Giant spin-orbit torque in a single ferrimagnetic metal layer\nSimon Lenne,1Yong-Chang Lau,1Ajay Jha,1Gwenal Y. P. Atcheson,1Roberto E.\nTroncoso,2Arne Brataas,2J.M.D. Coey,1Plamen Stamenov,1and Karsten Rode1,\u0003\n1CRANN, AMBER and School of Physics, Trinity College Dublin, Ireland\n2Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\nAntiferromagnets and compensated ferrimagnets o\u000ber opportunities to investigate spin dynamics\nin the `terahertz gap' because their resonance modes lie in the 0 :3 THz to 3 THz range. Despite some\ninherent advantages when compared to ferromagnets, these materials have not been extensively\nstudied due to di\u000eculties in exciting and detecting the high-frequency spin dynamics, especially\nin thin \flms. Here we show that spin-orbit torque in a single layer of the highly spin-polarized\ncompensated ferrimagnet Mn 2RuxGa is remarkably e\u000ecient at generating spin-orbit \felds \u00160He\u000b,\nwhich approach 0 :1\u000210\u000010T m2=A in the low-current density limit { almost a thousand times\nthe Oersted \feld, and one to two orders of magnitude greater than the e\u000bective \felds in heavy\nmetal/ferromagnet bilayers.\nWe depend on fast, reliable exchange of information\nacross long distances through intercontinental optical \f-\nbres, as well as short-distance connections between the\ncentral processing unit of a computer and its memory.\nThe latter is the bottleneck to the powerful computing fa-\ncilities needed in a future where machine learning and al-\ngorithms aid our daily lives. This bottleneck is di\u000ecult to\novercome because electronics lack a practical chip-based\nsolution to produce and detect electromagnetic waves in\nthe spectral range between 0 :3 THz and 30 THz known\nas the terahertz gap.\nSlonczewski1realised that angular momentum could\nbe transferred from one magnetic layer (a polariser) to\nanother (the analyser) by a spin polarised current.2This\nspin-transfer torque has enabled the scaling of devices\nthat depend on the relative magnetic orientation of two\nferromagnetic layers.3\nSpin electronics exploiting the orbital degree of free-\ndom of the electron is a recent development. A ma-\njor advance was the discovery that the angular momen-\ntum could be supplied by a di\u000busive spin current4,5cre-\nated via the spin Hall e\u000bect6,7in a non-magnetic heavy\nmetal layer adjacent to the ferromagnet. Devices based\non these bilayers require a bare minimum of two lay-\ners on a substrate. Earlier, Dresselhaus8and Bychkov\nand Rashba9had shown that in crystalline or patterned\nstructures lacking inversion symmetry, a current-induced\nspin polarisation (CISP) is a direct consequence of the\nsymmetry of the band structure. This idea was devel-\noped by \u0014Zelezn\u0013 y et al.1011to predict the form of the\ntensor relating the charge current to the CISP in crys-\ntals of di\u000berent symmetry. 90\u000eswitching of the metal-\nlic antiferromagnets CuMnAs12and Mn 2Au13was subse-\nquently observed. These ground-breaking results estab-\nlished the existence of a current-induced, \feld-like (or\nreactive) torque, and allow an estimate of the strength\nof the e\u000bective magnetic \feld by comparing it to the in-\nplane magnetic anisotropy of the material.\nHitherto there has been no quantitative measurement\nof the anti-damping (or dissipative) spin-orbit torquein homogeneously magnetised ferrimagnetic or antifer-\nromagnetic single-layer samples. Here we show, via har-\nmonic analysis of the anomalous Hall e\u000bect,14that in a\nsingle layer of the prototype half-metallic compensated\nferrimagnet Mn 2RuxGa withx= 0:7 (MRG),15{21both\nthe \feld- and damping-like components of the torque\nreach record values, almost two orders of magnitude\nstronger than those obtained in the bilayer ferromag-\nnet/heavy metal systems, or in metallic ferromagnets22\nand semimagnetic semiconductors23. The record values\nof the single-layer SOT and the dominance of the dissi-\npative torque, open a path to sustaining magnetic oscil-\nlations in the terahertz gap.\nThin-\flm samples of MRG grown on MgO by DC-\nmagnetron sputtering from stoichiometric targets crys-\ntallise in a Heusler-like structure, space group F\u001643mil-\nlustrated in FIG. 1a, where the conduction bands orig-\ninate predominantly from Mn in 4 csites.20The \flms\nare patterned into the micron-sized Hall bar structures\nshown in FIG. 1b, where the bias current jis parallel to\nthe MRG [010] axis. Further details on sample growth\ncan be found in the supplementary material and [19].\nWe determine the current-induced e\u000bective \felds via the\nanomalous Hall e\u000bect (AHE), assuming it is proportional\nto thezcomponent of the magnetisation of the Mn4csub-\nlattice. Due to the substrate-induced biaxial strain, the\npoint group of Mn in this position is reduced from \u001643m\nto\u001642m. Here we restrict our analysis to the e\u000bect on one\nsublattice, as the other will follow via inter-sublattice ex-\nchange with a phase-lag. We treat all e\u000bective torques as\nequivalent to external applied \felds. For an in-plane ap-\nplied \feldH, the magnetisation is described by the polar\nand azimuthal angles \u00120and\u001em, with the latter taken to\nbe equal to the azimuthal angle \u001eHof the applied \feld\nbecause the four-fold in-plane anisotropy is weak com-\npared with the uniaxial perpendicular anisotropy. The\ncoordinates describing the magnetic state are shown in\nFIG. 1c. In the presence of a unit charge current density\njk[010], the CISP produces a SOT e\u000bective \feld (seearXiv:1903.04432v3 [cond-mat.mes-hall] 29 Apr 20192\n(a)\nMn4a\nGa4b\nMn4c\nRu4d(b)\nI+\nI\u0000V\u0000\nxy V+\nxy\nV\u0000\nyyV+\nyy\n60µm\n(c)\nxyzm\n\u00160Hmxmy\n\u001eH\u00120(d)\nxyzm\nmxmy\u0000xfl\nmz\u0001xdlmx\u0001xdl\n\u001em\u00120\n1\nFIG. 1. (a) MRG crystal structure. The current is carried\nmainly by electrons in bands originating from Mn in the 4 c\nposition, which has point group symmetry \u001643m. Arrows show\nthe direction of the magnetic moment on each site. (b) Micro-\ngraph of a Hall bar with the contacts labelled. (c) Illustration\nof the coordinate system: \u00120is the polar angle of the magneti-\nsation vector in the absence of the SOT \feld, and \u001em=\u001eH\nis the azimuthal angle of the magnetisation and applied \feld\nvectors, respectively. (d) Illustration of the e\u000bective SOT\n\felds acting on the magnetisation with a bias current along\nMRG [010]ky, the \feld-like (reactive) component in blue,\nand the two damping-like (dissipative) components in green.\nFIG. 1d):\n\u00160hSOT=mzxdlex\u0000xfley+mxxdlez (1)\nwhere eiare unit vectors, miare the components of the\nunit magnetisation vector, and xfl,xdlare the coe\u000e-\ncients of the \feld- and damping-like contributions to the\nspin-orbit \feld, respectively. The units of \u00160hSOT,xfl\nandxdlare then T A\u00001m2. Henry is an equivalent unit.\nWhen the bias current has an alternating component\n(j=jdc+jacsin!t) we detect the e\u000bect of the CISP on\nboth the second and the third harmonic response using\nlock-in demodulation. The conversion from the voltages\ndetected at the di\u000berent harmonics to the magnitude of\nthe e\u000bective \felds is detailed in the supplementary ma-\nterial. In TABLE I we indicate the symmetry of the dif-\nferent contributions to Vxyin the \frst, second and third\nharmonic responses. Contributions from the anomalous\nNernst e\u000bect (ANE) are suppressed by measuring V3!\nxyor\nby taking the di\u000berence of V2!\nxymeasured with positive\nand negative DC bias. The contribution from the homo-\ngeneous temperature variation \u0001 Toscillating at twice\nthe applied frequency is determined from data in FIG. 2,\nas explained in the supplementary material.\nFIG. 2 shows the temperature dependence of the lon-TABLE I. Linear contributions to Vxyup to third order in\ncurrent density. - means no contribution, /a contribution odd\ninjdc,veven injdcand oindependent of current direction.\n\u001bxycontributes implicitly to all four e\u000bects.\nContribution/Harmonic !2!3!\nAnomalous Hall E\u000bect: \u001bxy o - -\nAnomalous Nernst E\u000bect: @T=@z - v -\nHomogeneous \u0001 Toscillating at 2 !:@\u001b=@T v / o\nCurrent-induced \felds: hSOT / v / o\n550575600625650(a)\n12141618(b)\n-600-400-2000200\n0 100 200 300(c)\n-20-15-10-50\n0 100 200 300(d)\nσxx(kS m−1)\nσxy(kS m−1)∂σxx\n∂T(S m−1K−1)\nT(K)\n∂σxy\n∂T(S m−1K−1)\nT(K)\nFIG. 2. Temperature dependence of the longitudinal and\ntransverse conductivity of MRG (a and b), and their tem-\nperature derivatives (c and d). The variation of \u001bxyfollows\nthe variation of the magnetisation of the 4 csublattice.20The\ncompensation temperature Tcomp where the net magnetisa-\ntion changes sign is 175 K (vertical dashed line). Since the\ndata were recorded at remnance, the direction of the sublat-\ntice moments does not change at Tcomp.\ngitudinal and transverse conductivity of MRG, recorded\nin the remnant state after saturation in a positive\n\feld at room temperature. The conductivity \u001bxx\n(FIG. 2a) increases with decreasing T, and its satura-\ntion value of 630 kS m\u00001or [159 µ\n cm]\u00001corresponds\nto the minimum metallic conductivity of a bad metal\nwhere the mean free path is comparable to the inter-\natomic spacing.24The Hall conductivity \u001bxy(FIG. 2b)\nclosely follows the Mn4csublattice magnetisation.25The\nlower panels show the temperature-derivatives of \u001b.\nWe now turn to the SOT. FIG. 3 a and b shows the\nexperimentally observed V3!\nxyand its calculated values\nbased on the experimental \u00120. There is excellent agree-\nment between experiment and the model, which is based\nonly on the site symmetry, the data in FIG. 2, and the\n\frst harmonic response (used to determine \u00120and the\nanisotropy constants). All the features in both the \u00120\nand\u001edependencies are well reproduced: two deep min-\nima around \u00160Hx;max, four maxima that align with the\nfour-fold in-plane anisotropy due to the small value of\nthe in-plane anisotropy constant K0\n2, as well as a weaker\ncentral minimum at small \felds. Qualitatively, the \feld-\ndependence of the SOT can be understood by compar-\ning equation (1) with FIG. 3 (a), and noting that the3\n-2-1012-2-1012(a) (b)\n(c) (d)-2-1012-2-1012(a) (b)\n(c) (d)\n02468\n0 2 4 6 8(a) (b)\n(c) (d)\n00.10.20.3\n0 2 4 6 8(a) (b)\n(c) (d)\nµ0Hx(T)µ0Hy(T)-2-101\nV3ω\nxy(µV)\nµ0Hx(T)µ0Hy(T)-2-101\nV3ω\nxy(µV)V2ω\nxy(µV)\nIac(mA)DC diff.\nDC sum\nIdc(mA)\nFIG. 3. (a) Surface plot and its 2D colour map projection\nof the experimentally observed voltage at the third harmonic\nV3!\nxy. (b) Calculated response based on the experimental val-\nues of cos\u00120. (c) and (d) show AC (with Idc= 3 mA) and\nDC (Iac= 1 mA) current-dependence of the second harmonic\nV2!\nxysignal. By making the di\u000berence and sum of records made\nwith positive and negative DC o\u000bset we isolate the SOT (the\ndi\u000berence) from the anomalous Nernst e\u000bect ANE (the sum).\nmagnitude of \u0001 \u0012depends not only on \u00120itself, but ex-\nplicitly on the competition between the SOT \feld, the\nanisotropy \feld and the applied \feld. At low applied\n\felds (\u00120\u00190) the current-driven wobble of mis de-\ntermined by a combination of the anisotropy and SOT\n\felds. \u0001\u0012is small, however, because cos \u00120\u00191, hence\nthe central minimum is shallow. At higher applied \felds,\n\u00120deviates from 0, but the SOT \feld now has to com-\npete with both the (higher) anisotropy \feld and the Zee-\nman torque provided by the applied \feld acting on the\nnet magnetisation. This gives rise to the characteristic\nfour-fold signal. An exceptional feature appears around\n\u00160Hx\u0019\u00062 T where the damping-like \feld in the zdi-\nrection scaling as mxproduces a \feld strong enough to\ndwarf both the anisotropy \feld and the applied \feld. We\nemphasise not only the qualitative agreement, but also\nthat the absolute magnitude of the signal agrees very well\nwith the model when we \ft the coe\u000ecients of the \feld-\nand damping-like \felds; xfl=\u000015\u000210\u000013T A\u00001m2\nandxdl= 50\u000210\u000013T A\u00001m2.\nWe then determine the dependence of the e\u000bective \feld\nmagnitude on bias current, (FIG. 3 c and d). A cur-\nrent of 1 mA in our \flms is equivalent to jof about\n2:5\u0002109A m\u00002. We expect V2!\nxydue to SOT to scale\nwithIdcandI2\nac, while the e\u000bects due to thermal gradi-\nents should be independent of Idcand scale as I2\nac. In-\ndeed, the DC di\u000berence is quadratic in Iac(FIG. 3c) and\nlinear inIdc(FIG. 3d), while the DC sum is quadratic in\nIacand practically independent of Idc.\nIt is instructive to compare the e\u000bective \felds due tointrinsic SOT with those recorded on conventional bilay-\ners of a heavy metal (typically Pt, Ta or W) and a 3 d\nferromagnet (typically Co, Fe, CoFe or CoFeB). For bi-\nlayers, the damping-like e\u000bective \feld per current density\ncan be written: \u00160Hdl=j= (\u0012SH\u0016h)=(2eMst), where\u0012SHis\nthe spin-Hall angle of the heavy metal, \u0016 his the Planck's\nconstant,eis the electron charge, Msthe magnetisa-\ntion of the ferromagnet and tits thickness. For 1 nm\nof CoFeB (Ms\u00191 MA m\u00001), which has a magnetic mo-\nment equivalent to that of \u001930 nm nearly compensated\nMRG, and \u0012SH= 40 % we obtain an e\u000bective, damping-\nlike \feld of 1 :3\u000210\u000013T A\u00001m2(0:13 pH). We would\nneed a \fctitious spin-Hall angle of 400 % to match the\nvalue of the \feld-like term in MRG and 1200 % to match\nthe damping-like term.\nThis comparison highlights the inherent advantage of\nusing ferrimagnets in combination with intrinsic SOT. In\na bilayer, increasing the thickness of the ferromagnet be-\nyond the spin di\u000busion length (typically <10 nm), does\nnot produce any additional torque. If the ferromagnet is\n2 nm rather than 1 nm thick, the e\u000bective \feld may be\nreduced to half, whereas the \feld in single-layer MRG is\nunchanged with thickness. The volume of MRG can be\nscaled up or down without changing the torque, provid-\ning the current density is constant. The nature of the\nintrinsic torque is staggered acting directly on the Mn4c\nsublattice, hence a more correct comparison might be\nbe to normalise the spin Hall angle using the sublattice\nmagnetisation, which is approximately ten times greater\nthan the net magnetisation at room temperature for the\npresent sample. Furthermore, the torque is maintained\neven in the absence of any net magnetisation at the fer-\nrimagnetic compensation temperature, thus permitting\nGMR- and TMR-based device structures to be excited\nby SOT even in the absence of any net moment of the\nfree layer. This enables targeted control of the dynamics,\nand the excitation of both in-and out-of phase resonance\nmodes.\nThe high e\u000bective \felds found above assuming small,\nlinear, current-driven variations in \u00120, imply that the ac-\ntion of the SOT should also be observable in the non-\nlinear transfer characteristics of our Hall bar device. We\ntherefore proceeded as follows:\nWe \frst recorded a full \feld-in-plane hysteresis loop\nfrom\u000014 T to 14 T to determine the relation between\nmzand the applied \feld and invert this relation numeri-\ncally, to be able to deduce, from mz, the value of the total\ne\u000bective \feld at any given applied current. Then, a con-\nstant external \feld \u00160H= 0:4 T is applied in the sample\nplane and rotated around ez, changing the azimuthal an-\ngle\u001eHfrom 0\u000eto 360\u000ewhile recording mz(inferred from\nVxy). This measurement is repeated for range of current\ndensities from 0 :2\u00021010A m\u00002to 2:5\u00021010A m\u00002. As\nthe action of the SOT \feld depends directly on its direc-\ntion relative to the direction of the magnetisation ( \u0012M\nand\u001eM\u0019\u001eH), we can subtract any variation that is \u001e-\nindependent . This\u001e-independent e\u000bective \feld contains\nall variations that are due to heating. The result, after4\n-2-1012-2-1012\njysinφH(1010A m−2)jycosφH(1010A m−2)-250255075\nLeff(pH)\nFIG. 4. High-current-density e\u000bective induction Lin pH =\npT m2A\u00001as a function of the bias current density and\nthe angle\u001eHbetween exand the applied magnetic \feld\n\u00160Happ= 0:4 T. The e\u000bective inductance reaches \u001875 pH,\ncorresponding to an e\u000bective \feld at j= 2:5\u00021010A m\u00002of\n\u00160He\u000b\u00181:9 T. The interpretation of this striking result is\ndiscussed in the main text.\nsubtraction, is shown in FIG. 4, where we give the e\u000bec-\ntive \feld in terms of the e\u000bective, current-induced induc-\ntance in pH = 1 \u00021012T m2A\u00001. A current density of\nj= 2:5\u00021010A m\u00002can produce an e\u000bective inductance\nLe\u000b\u001975 pH, equivalent to an e\u000bective in-plane \feld of\n1:9 T! We note that this \feld is su\u000ecient to magnetically\nswitch\u00192 % of the sample.\nWe make two important comments on this analysis.\nFirst, by removing the \u001e-independent part of the sig-\nnal, we also remove any SOT that behaves the same\nway. If we again assume the SOT \felds can be de-\nscribed by the tensors reported by \u0014Zelezn\u0013 y et al.11and\nTroncoso et al.26, and expand the relation between he\u000b\nand \u0001\u0012to second order in \u0001 \u0012, we \fnd that we have re-\nmoved a damping-like contribution along ezthat varies\nasm2\nx+m2\ny, which may be considerable. Second, as we\nare normalising with respect to the action of the exter-\nnal,in-plane , \feld, the SOT e\u000bective \feld directed along\nez, remaining after the procedure outlined above, con-\ntributes to our signal /1=cos\u0012Mas seen by the upturn at\n\u001eH= 270\u000ein FIG. 4.\nThe strong e\u000bective SOT \felds in MRG are related to\nits high anomalous Hall angle.16The value is unusual in\nthe sense that MRG does not contain any elements heav-\nier than Ru; in any case the AHE angle does not scale\nwith Ru content x. Furthermore the conduction electrons\nin MRG are predominantly d-like, although it has been\nsuggested that Ga in the Mn-containing Heuslers lends\nsomepcharacter to the bands at the Fermi-level through\nhybridisation, increasing the spin-orbit coupling of the\nconduction electrons27. We have already seen above that\nMRG is at the limit of metallic conductivity. From our\nmeasurements of \u001bxxand\u001bxy, we can deduce the spin-\norbit scattering cross-section and \fnd that it correspondsto 60 % of the unit cell surface area. The very large scat-\ntering cross section is consistent with the very short mean\nfree path.\nSo far we have demonstrated high current-induced ef-\nfective \felds as well as a high ratio ( \u00183) of the dissipative\n(anti-damping) to the reactive (\feld-like) torques. This\nwill allow for the realization of more e\u000ecient magnetic\nswitching28, exchange-bias manipulation29as well as low-\ncurrent control of magnetic textures30. The key question\nis, whether sustained self oscillation can be driven by\nSOT. We address this from two di\u000berent angles, \frst by\nconsidering the results established by Troncoso et al.26,\nnoting that the e\u000bective \felds will act distinctly on the\nmagnetisation and the Nel vectors. Using the numeri-\ncal values of the e\u000bective \felds found in the linear, low-\ncurrent regime, self-oscillations will emerge for current\ndensities that provide a reactive torque which is su\u000ecient\nto overcome the in-plane anisotropy \u00180:1 T for MRG,\nwhich corresponds to j>\u00187\u00021010A m\u00002. The second\nnecessary condition is that the dissipative torque must\novercome the Gilbert damping \u000b. Taking\u000b\u00190:01 we\n\fnd the condition j>\u001810\u00021010A m\u00002. An alternative\napproach is to compare directly the e\u000bective inductance\ncreated by the SOT and the self inductance of the os-\ncillating element. In a shorted Hall bar device, a crude\nestimate of the self inductance for a 500 nm thick \flm\nwith an active length of 20 µm is 0:1 pH { the dimensions\nare chosen to enhance impedance matching to free space\nin a real oscillator. We saw in FIG. 4 that the e\u000bective\ninductance reaches values two orders of magnitude higher\nthan this, ensuring that oscillatory behaviour is possible,\neven in the low-current-density region. The natural fre-\nquency of the oscillator will be determined by the larger\nof the two e\u000bective inductances, that is by the SOT and\nthe magnetic resonance frequency of the material, which\nwe previously estimated as 0 :75 THz.31\nIn summary, we \fnd that current-induced spin orbit\ntorque reaches record values in single-layers of the com-\npensated, half-metallic ferrimagnet Mn 2RuxGa, well in\nexcess of those achieved in bilayer structures. With real-\nistic values of damping, this should allow sustained mag-\nnetic oscillations that could be detected by magnetore-\nsistive e\u000bects, or free-space emission using a suitable an-\ntenna. A cheap, compact, and tunable oscillator operat-\ning in the terahertz gap would break new ground in spin\ndynamics, and could potentially unlock a new realm of\ninformation transfer at bandwidths three orders of mag-\nnitude higher than those of the present day.\nACKNOWLEDGMENTS\nThis project has received funding from the Euro-\npean Union's Horizon 2020 research and innovation pro-\ngramme under grant agreement No 737038, and from Sci-\nence Foundation Ireland through contracts 12/RC/2278\nAMBER and 16/IA/4534 ZEMS as well as the Research\nCouncil of Norway through its Centres of Excellence5\nfunding scheme, Project No. 262633 \\QuSpin\". The authors declare no competing \fnancial interests.\n\u0003Corresponding author:rodek@tcd.ie\n1J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989).\n2J. Slonczewski, J. Mag. Magn. Mat. 159, L1 (1996).\n3C. Chappert, A. Fert, and F. N. Van Dau, Nature Mate-\nrials6, 813 (2007), review Article.\n4I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. 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B 98, 220406 (2018)." }, { "title": "2303.15985v2.Exploring_terahertz_scale_exchange_resonances_in_synthetic_ferrimagnets_with_ultrashort_optically_induced_spin_currents.pdf", "content": "Exploring terahertz-scale exchange resonances in synthetic ferrimagnets with\nultrashort optically induced spin currents\nJulian Hintermayr,∗Youri L. W. van Hees, and Bert Koopmans\nDepartment of Applied Physics, Eindhoven University of Technology,\nP.O. Box 13, 5600 MB Eindhoven, the Netherlands\n(Dated: June 21, 2023)\nUsing spin currents generated by fs laser pulses, we demonstrate excitation of GHz ferromag-\nnetic resonance and THz ferrimagnetic exchange resonances in Co/Gd/Co/Gd multilayers by time-\nresolved magneto-optic Kerr effect measurements. Varying the Gd layer thickness allows for a tuning\nof the resonance spectrum by manipulating the total angular momentum and strength of effective\nexchange fields between the antiferromagnetically coupled layers. Close to the compensation point\nof angular momentum, a minimum in the frequency of the exchange-dominated mode and a max-\nimum in the frequency of the ferromagnetic resonance mode is observed. Finally, to gain better\nunderstanding of the excitation mechanism, we analyze the anomalous variation in the measured\nexchange mode amplitude as a function of its frequency. A peak in this amplitude in the vicin-\nity of the compensation point of angular momentum is explained using a macrospin model, taking\nnonlinear effects at finite precession amplitudes into account.\nI. INTRODUCTION\nFerrimagnets combine a number of advantages of ferro-\nand antiferromagnets, making them a promising platform\nfor fast and easily integratable spintronic devices [1–3].\nStrong, alternating exchange fields are inherent to anti-\nferromagnets, giving rise to high-frequency ( ∼THz) ex-\nchange resonance modes (EXMs) [4, 5]. Their lack of\nnet spin polarization however makes experimental ac-\ncess challenging [6]. Metallic ferrimagnets , in contrast,\ncan exhibit finite conduction spin polarization — even\nat the compensation point of angular momentum — and\ncan thus easily be probed by magneto-optic or magneto-\nresistive effects, given that they consist of sublattices\nwith different atomic species [1].\nWhile the high frequency makes this class of materi-\nals a promising candidate for the development of THz\nspintronic devices, techniques of exciting resonances at\nsuch high frequencies coherently are sparse. In the past,\nEXMs have been excited for instance by thermal laser\nexcitation in combination with an applied field [7, 8]\nand circularly polarized laser pulses [9–11]. The former\napproach only made it possible to excite EXMs in the\n<100 GHz range, whereas the latter approach requires\nmaterials with significant inverse Faraday effect. In ad-\ndition, pulses of intense THz radiation have been shown\nto facilitate excitation of fast spin dynamics [12–14].\nIn this work, we will make use of optically induced\nspin currents that are generated by a neighboring fer-\nromagnet upon ultrafast demagnetization [15–18]. The\noptically induced spin-transfer torques (OSTTs) gener-\nated in this manner have been used to excite exchange-\ndominated standing spin waves with frequencies exceed-\ning 1 THz [19–21]. Moreover, its has been shown that\nsuch spin currents can be used to assist in single-shot\n∗Electronic mail: j.hintermayr@tue.nlall-optical switching of ferrimagnets [22, 23] or even in-\nduce full switching in ferromagnets [23–26]. As we will\nshow, a synthetic ferrimagnetic quadlayer consisting of\nthe transition metal (TM) Co and the rare earth (RE)\nGd can host not only GHz ferromagnetic resonance but\nalso THz exchange resonance modes, in spite of the ar-\nguably weaker exchange coupling compared to CoGd al-\nloys. Co/Gd multilayers are highly relevant for a variety\nof spintronic applications as they exhibit energy-efficient\nsingle-shot all-optical magnetization switching [27]. Ad-\nditionally, ultrafast domain wall motion close to the com-\npensation point of angular momentum has been observed\nvery recently [28]. The coexistence of these properties\nmakes this material a promising candidate for hybrid\nspintronic-photonic memory applications [28, 29].\nIn the following, we will show that OSTTs are an ex-\ncellent tool to excite and study GHz ferromagnetic res-\nonance (FMR) and THz EXMs in Co/Gd multilayers.\nFurthermore, we explain our findings with an analytical\nmacrospin model.\nII. SAMPLE STRUCTURE AND\nCHARACTERIZATION\nThe idea behind the sample design is to layer a ferri-\nmagnet with an in-plane (IP) magnetic easy axis onto\na ferromagnet with perpendicular magnetic anisotropy\n(PMA) that efficiently generates spin currents upon\nultrafast demagnetization [17, 30]. We chose a synthetic\nferrimagnetic quadlayer of Co/Gd/Co/Gd as it offers\nthe possibility of reaching magnetic compensation by\nfine-tuning individual layer thicknesses while maintain-\ning IP anisotropy. The spin current generation layer\nis a Pt/[Co/Ni] 4multilayer which provides PMA at\nrelatively large magnetic volumes [19]. Between the two\nlayers, a Cu spacer is placed to magnetically decouple\nthe two systems while allowing for spin currents to pass.\nThe final material stack used in this study is as follows:arXiv:2303.15985v2 [cond-mat.mes-hall] 19 Jun 20232\nFIG. 1. Cartoon of a, sample architecture showing local arrangements of magnetization and the resonance excitation mechanism\nb, stable spin precession configurations and c, local directions of exchange fields in canted magnetization states between Co\nand Gd, allowing for fast resonance frequencies. d, Demagnetization trace of the [Co/Ni] injection layer, based on which the\noptically induced spin current pulse in eis calculated. f, In-plane hysteresis loops of the Co/Gd/Co/Gd absorption layer for\ndifferent Gd thicknesses measured with L-MOKE. g, Coercive field and MOKE step as a function of Gd thickness. The vertical\ngray line indicates the magnetization compensation point Mcomp.\nSi:B/Ta(4)/Pt(4)/[Co(0.2)/Ni(0.6)] 4/Co(0.2)/Cu(5)/\n[Co(0.8)/Gd( tGd)]2/TaN x(5) (thicknesses in nm) with\nGd thicknesses ranging from 0–3 nm (schematically\nshown in Fig. 1 a). Layers are deposited by dc mag-\nnetron sputter deposition at room temperature. The\ndemagnetization is triggered and measured by ∼100 fs\nlaser pulses with a central wavelength of 780 nm and\n80 MHz repetition rate, using a standard pump-probe\nsetup to measure the time-resolved magneto-optic\nKerr effect (TR-MOKE). As the MOKE of Gd at this\nwavelength is small [31], we mostly probe the signal\narising from Co and Co/Ni layers. Pump and probe\npulses are focused on the sample surface with near\nnormal incidence.\nFirstly, we investigate the demagnetization trace and\nderive the optically induced spin current profile to under-\nstand the excitation mechanism. The resulting trace is\nshown in Fig 1 d, revealing rapid demagnetization within\nthe first 300 fs and a slower remagnetization over several\nps. We assume that the spin current originating from this\nprocess follows the spin-pumping model, also referred to\nas “d M/dt” model [17, 18, 32, 33], where the demagneti-\nzation is explained by excitation of magnons that transfer\ntheir angular momentum to mobile conduction electrons.\nThe shape of such a spin current pulse is thus derived\nfrom the time derivative of the demagnetization curve\n(Fig. 1 a) and shown in Fig. 1 e. De- and remagneti-\nzation give rise to a positive and negative peak in spin\ncurrent respectively, resulting in a bipolar pulse on the\nps timescale. It is thus suited to excite THz-scale dy-namics. To characterize the static magnetic properties of\nthe Co/Gd absorption stack, MOKE hysteresis loops in\nlongitudinal geometry are recorded for different Gd thick-\nnesses. Measurements are shown in Fig. 1 f. We find that\nthe easy axis of the absorption stack lies in-plane (IP),\nwith a magnetic compensation point ( Mcomp) at around\n1.9 nm Gd. At this thickness, the magnetic moments of\nthe antiferromagnetically coupled Co and Gd layers ex-\nactly cancel each other out. The switch in sign of the\nmeasured Kerr ellipticity εmaxis accompanied by a max-\nimum in coercive field, as shown in Fig. 1 g, where Mcomp\nis indicated by a vertical line. The angular momentum\ncompensation point Lcomp is expected to lie at slightly\nlower Gd thicknesses, as gCo> gGd(see Appendix). Due\nto the fact that the exact magnetization profile in the\nstack is not known, it is not possible to precisely de-\ntermine Lcomp. We note that, even though the samples\nare deposited as multilayers, significant intermixing be-\ntween Co and Gd is expected at the interface when sput-\nter depositing at room temperature [34–37]. As the Co\nthickness is only 0.8 nm, the intermixing depth in thin\nRE/TM multilayers is expected to be in a similar order\nof magnitude as the individual layer thickness, leading to\nsignificant alloy-like regions [38].\nIII. SPIN RESONANCE MODES\nHaving discussed the static magnetic properties of the\nsynthetic ferrimagnetic multilayer, we will introduce the3\ntypes of spin resonances that can occur in this system.\nGenerally, two types of precessions are possible within\na two-sublattice spin system with antiferromagnetic cou-\npling. For simplicity, we limit ourselves to the alloy-like\ncase for now, assuming one macrospin per material, and\nneglect magnetic anisotropy. Further discussion on the\nvalidity of this approximation is provided later in this\nwork. In the presence of an external magnetic field,\nFMR-like dynamics are allowed, where the magnetiza-\ntion of Co and Gd sublattices, MCoandMGd, are aligned\nfully antiparallel and precess around the direction of the\nexternal field H(schematically shown in Fig. 1 b). The\nfrequency of this oscillation can be approximated as [39]\nfFMR =µ0\n2πMCo−MGd\nMCo\nγCo−MGd\nγGdH=µ0\n2πγeffH. (1)\nµ0denotes the vacuum permeability, γCo,Gdthe gyro-\nmagnetic ratios of Co and Gd, and γeffthe effective gyro-\nmagnetic ratio. Given that γCo̸=γGd, the magnetization\nand angular momentum of the two sublattices compen-\nsate at different atomic fractions. As we approach the\npoint where the total angular momentum is fully com-\npensated ( Lcomp), the torque which is proportional to\nMeff×Hremains finite, resulting in a divergence in γeff,\nas predicted by eq. (1). Please note that this approxima-\ntion neglects demagnetizing fields and becomes invalid\nvery close to Lcomp. A more involved treatment removes\nthe divergence and leads to only a pronounced maximum\nin FMR frequency at Lcomp [40].\nThe exchange-dominated resonance mode with typi-\ncally much higher frequencies requires MCoto be at an\nangle with the exchange field it experiences. This field\nis proportional to the Weiss constant λ, indicating sign\nand strength of the exchange interaction, and can be ex-\npressed as HGd\nex=λMGd. Figure 1 bandcschematically\nshow the canted alignments of spins and exchange fields\nthat are necessary for this mode to exist. Its frequency\nis given by [39]\nfEXM =µ0λγCoγGd\n2π\u0012MCo\nγCo−MGd\nγGd\u0013\n(2)\nin the limit of small oscillation angles. In contrast to the\nFMR mode, the EXM frequency vanishes at the angu-\nlar momentum compensation point. The change in sign\nin frequency predicted by the dispersion relation indi-\ncates an intriguing change in handedness of the oscilla-\ntion, which has been experimentally verified by Brillouin\nlight scattering [41, 42].\nThe effective damping parameter αeffin this system\nhas been calculated to show a maximum at Lcomp for\nboth FMR and EXM modes [40]. Spin resonances are\ntherefore expected to only be very short-lived close to\ncompensation. If the individual sublattice damping co-\nefficients are identical, that is, αCo=αGd,αeffof FMR\nand EXM will likewise be equal. If they differ from one\nanother, the ratio of damping parameters of FMR and\nEXM will depend on αCo/αGd.We note that the symmetry of the quadlayer system\nallows in principle for higher-order exchange resonances\nwhere, for instance, the two Co layers are precessing out\nof phase. However, the frequencies of such modes are ex-\npected to be far above those of the fundamental EXM.\nWe therefore cannot excite them with our technique, as-\nsuming the wedged Gd does not become thick enough\nto only provide a weak link between the Co layers. Yet\nanother type of exchange-driven modes that can exist in\nthin films are quantum-confined standing spin waves that\nhave been excited and measured with techniques similar\nto those used in this work [19–21], but for a 0.8 nm Co\nlayer expected frequencies ( >10 THz) are too high to be\nexcited by our method. Also, more complex modes de-\nlocalized throughout the whole ferrimagnetic stack have\nnot been identified in our experiments, and we find a de-\nscription in terms of an alloy with only two sublattices\nsufficient to explain all our results.\nIV. RESULTS\nA. Time-resolved magnetization dynamics\nTo investigate the dynamics of the sample, we again\nemploy pump-probe spectroscopy. As mentioned previ-\nously, resonances are excited by injection of spin current\npulses generated in the Co/Ni layer into the Co/Gd ab-\nsorption layer. The current from the injection layer shows\nOOP spin polarization and exerts an OSTT on the IP\nabsorption layer, canting the magnetization OOP away\nfrom its ground state, as illustrated in Fig. 1 a. If the\neffective magnetization and the applied field are at an\nangle in the excited state, the FMR mode is triggered. If\nMCoandMGdare at an angle, the following precession is\ndescribed by the EXM. We note that the ultrafast laser\nexcitation of the Co/Gd quadlayer could lead to an injec-\ntion of an IP-polarized spin current into the Co/Ni layer\nas well. Since the dynamics in the Co/Ni layer induced\nby the following OSTT are expected to be fully IP as\nwell, the dynamics are not probed during the experiment\ndue to the polar measurement geometry.\nBefore the measurement, we saturate the injection\nstack in the positive z-direction and apply an IP field\nfor FMR measurements and no IP field for EXM mea-\nsurements. We then repeat the measurement with the\ninjection stack saturated in the opposite direction and\ncalculate the magnetic signal as the difference of mea-\nsurements with positively and negatively saturated injec-\ntion layer. The sum of the measurements includes non-\nmagnetic artefacts such as the coherence peak that occurs\nduring pump-probe overlap [43, 44] and is disregarded.\nHomogeneous, FMR-like precession modes with an ap-\nplied IP field of 92 mT are shown in Fig. 2 afor a range\nof Gd thicknesses. During the first few ps, the demagne-\ntization of the injection layer dominates the signal. Sub-\nsequently, the oscillations of the absorption layer are ob-\nserved. For a more detailed analysis of the measurements,4\nFIG. 2. aTime-resolved MOKE measurements of homoge-\nneous FMR precessions in the in-plane Co/Gd/Co/Gd layer\nat a magnetic field of 92 mT. Offsets are proportional to the\nGd thickness. The inset shows the precession frequency as\na function of applied field at a fixed Gd thickness of 1.5 nm\nincluding a fit according to eq. (4). bOscillation measure-\nments of exchange modes using complex MOKE. Black lines\nrepresent damped sine fits. Note the different timescales in a\nandb.\nwe fit functions of the form\nAsin(2πft−φ)e−t/τ+Be−t/C+D (3)\nto our data (black lines), allowing an extraction of the\nfrequency fand the effective damping parameter αeff=\n1/2πfτ.Adenotes the oscillation amplitude, φthe\nphase, and τthe decay time of the oscillation. The fol-\nlowing terms capture the remagnetization behavior of the\ninjection layer with an amplitude of B, a decay time of\nCand a constant offset D. A steady increase of the\nFMR frequency as a function of tGdis observed, which\nis explained by an increase in Gd angular momentum,\nwhereby the total angular momentum decreases. In ac-\ncordance to eq. (1) γeffincreases, and with it the fre-\nquency and effective damping, as observed in Fig. 3 a\nandb.\nFor a Gd thickness of 1.5 nm, FMR measurements at\ndifferent fields were recorded. The extracted frequenciesas a function of applied field are plotted in the inset of\nFig. 2 a. As a linear fit according to eq. (1) yielded unsat-\nisfactory results due to neglecting demagnetizing fields,\nwe used the following Kittel-like equation, accurately im-\nplementing the effect of finite demagnetizing effects in\nthin films, where the effective magnetization Meffand\nγeffare fitted:\nfKitt=µ0γeff\n2πp\nH(H+Meff). (4)\nThe best fit is found for γeff/γe= 3.0, with the electron\ngyromagnetic ratio γe. This strong boost of γeffis ex-\npected as Lcomp is being approached. Furthermore, we\nextract Meff= 25 kA /m, which corresponds to ∼2%\nof the saturation magnetization of elemental Co. Ad-\nditional possible anisotropy-related effects arising from\nthe multilayered nature of the sample are also captured\ninMeff. They may stem for instance from interfacial\nanisotropy which could give both positive or negative\ncontributions to Meffas well as shape anisotropy. The\nvery low value of Meffimplies, however, that the sample\nis close to Mcomp and that the anisotropy contributions\nare small.\nUpon increasing the Gd thickness to 2.1 nm, a dras-\ntic discontinuous change in the oscillation frequency is\nobserved (see Fig 2 a). We associate this observation\nwith the onset of the EXM and the simultaneous sup-\npression or over-damping of the FMR mode. To in-\ncrease our sensitivity to the EXM at short time delays,\na quarter-wave plate in the probe beam path is used\nto mostly cancel out the contribution of the demagne-\ntization of the injection layer to the signal. This tech-\nnique is known as complex MOKE, for further informa-\ntion the reader is referred to Refs. [45, 46]. Figure 2 b\nshows oscillation measurements of EXM resonances ac-\nquired in this manner. During the first ps, some un-\navoidable leaking signal from the injection layer and re-\nmanent artefacts from the coherence peak close to zero\ndelay are visible in the signal. Thereafter, damped os-\ncillations are observed. Please note the different scale\nof the x-axis compared to Subfigure a. Increasing Gd\nthicknesses results in a strong decrease of precession fre-\nquency and a variation of the precession amplitude. The\ndecrease of frequency is well explained by an approach\nof the angular momentum compensation point as pre-\ndicted by eq. (2). Plotting the resonance frequencies as\na function of tGd(see Fig. 3 b) shows a highly non-linear\nbehaviour. Two factors give rise to such nonlinearity:\nFirstly, we recall that the Curie temperature of Gd is be-\nlow room temperature. Thus, it only shows a finite mag-\nnetization within the partially intermixed Co/Gd and\nGd/Co interfaces as well as proximity-induced magneti-\nzation in the regions close to the interfaces. As one moves\naway, this induced magnetization—and thus the angular\nmomentum—decreases exponentially [38]. Secondly, in-\ncreasing the Gd thickness decreases the average exchange\nparameter λin eq. (2) which scales inversely proportional\nto the thickness in magnetic multilayers. Please note that5\nthis coupling parameter only influences the EXM and not\nthe FMR frequency. Hence, this effect does not influence\nthe thickness dependence of the FMR mode.\nFIG. 3. aFrequency of the ferromagnetic resonance mode\nin a field of 92 mT and that of the exchange mode in the\nabsence of a field as a function of Gd thickness. Please note\nthe different y-scales. bEffective damping parameter of said\nmodes. Lines are guides for the eye.\nAnother peculiarity of the EXM frequency is that it\ndecreases monotonously, implying either that the angu-\nlar momentum compensation point is being approached,\nyet never crossed, or that oscillation amplitudes are too\nlarge for the small angle approximation to hold that pre-\ndicts a vanishing frequency. The magnetostatic charac-\nterization, on the other hand, clearly revealed a mag-\nnetic compensation point around 1.9 nm Gd. Since\nγCo> γ Gd,Lcomp should lie at even smaller tGdthan\nMcomp. However, the magnetization of Gd is strongly\ntemperature-dependent, especially in TM/Gd multilay-\ners. Shortly after the arrival of the laser pulse it is\ntherefore very well possible that the system is Co dom-\ninant even though the room temperature characteriza-\ntion revealed Gd-dominance. Thus, the absence of an\nangular momentum compensation point in time-resolved\nmeasurements at strongly increased temperatures during\nthe first ps after laser irradiation could result in a Co-\ndominated sample across all investigated Gd thicknesses.\nA dependence of the frequency on the pump energy in\nCoGd alloys was found by Mekonnen et al. [7], which was\nexplained by this effect. The steadily increasing behavior\nofαeffshown in Fig. 3 bis well in line with the expectedmonotonous increase towards Lcomp. Furthermore, non-\nlocal damping effects due to spin-pumping either across\nCo/Gd interfaces or through the Co/Cu interface could\ncontribute to the total effective damping.\nIn terms of the phase of the EXM, a slight variation\nas a function of Gd thickness was observed (not shown),\nwhich could be expected, considering the fact that EXMs\nare excited by a bipolar spin current pulse acting on\ntimescales similar to those of the precessions. A simi-\nlar analysis as in Ref. [32] could be carried out, where\nthe phase of THz standing spinwave excited by the same\nps bipolar spin current pulse is investigated. However,\nthis would go beyond the scope of this work.\nB. Exchange-dominated precession amplitudes\nFinally, we seek to explain the observed variation\nin EXM amplitude for different oscillation frequencies,\nwhich is shown in Figure 4 a, to gain both a better un-\nderstanding of the system and insights into the exact ex-\ncitation mechanism. A naive first guess could be the as-\nsumption that the bipolar spin current resonantly drives\nthe precession mode. The EXM amplitude as a function\nof the frequency should then follow the Fourier Trans-\nform (FT) of the injection pulse. A comparison of said\nFT (continuous black line) to the measured data points\n(orange symbols) does show qualitative agreement in the\nsense that there is a maximum at a certain frequency\nand a decrease in amplitude towards zero and high fre-\nquencies. However, the maxima are located at different\nfrequencies and the nature of the decay does not match at\nall. We note that the shape of the Fourier spectrum of the\nspin current pulse crucially depends on our assumption\nof the d M/dtmodel, which is still a subject of ongoing\ndiscussion. Other models, such as the superdiffusive hot\nelectron model [47, 48], assume generation of even faster\noptically induced spin currents, leading to stronger devi-\nations from experimental data due to theoretical peaks\nat higher frequencies. Our assumption can therefore be\nseen as a conservative estimate for the high-frequency\nbehaviour.\nIn the following, we employ a simple macrospin model\nthat captures the ferrimagnetic nature of the absorption\nlayer and model the OSTT as a canting of the macrospins\nwith respect to the horizontal antiparallel state. We\nagain consider a two sublattice, alloy-like case where the\nGd concentration cGdis varied and both sublattices are\ninitially canted by the same angle δwith respect to the\nfilm plane (shown schematically in Fig. 4 d). While, in\npractice, we are dealing with a quadlayer system, we ar-\ngue that the fact that only two types of spin resonances\nare observed and the strong intermixing between thin\nlayers make the alloy model a reasonable approximation.\nNote that the concentration cGdin the model relates to\nthe magnetically active atomic fraction of Gd only, mak-\ning direct comparisons to Gd thicknesses challenging. By\nusing the same canting angle δfor both lattices, equal6\naverage spin transfer efficiencies for the Co and Gd sub-\nlattice are assumed. While a large fraction of spins are\nabsorbed in the bottom Co layer, the strong intermixing\nwith Gd contributes to a significant absorption of spin an-\ngular momentum by Gd. Furthermore, it has been shown\nthat the spin coherence length in layered ferrimagnets is\nlargely enhanced compared to ferromagnets [49] leading\nto considerable OSTTs even deeper into the Gd layers.\nUsing the model defined above, we aim at understand-\ning the anomalous dependence of the EXM amplitude as\na function of frequency. Following initial canting, con-\nservation of angular momentum dictates the subsequent\nresonance be around the total angular momentum vector\nLtot=LCo+LGd, enclosing an angle of Ω with the film\nplane (Fig. 4 d, green arrow). The vertical precession\namplitude of MCo, which is the one that is probed dur-\ning the experiment, then depends on Ω and the angle θCo\nthat is enclosed by LtotandMCo(schematically shown\nin Fig. 4 d) according to\nAz= 2|sinθCocos Ω|. (5)\nWe consider two cases; in the linear approximation, the\nprecession frequency is determined only by the degree of\nstatic angular momentum compensation of the sample\nwith fEXM given by eq. (2). In other words, we assume\nthe small angle approximation. Azas a function of fre-\nquencies is fitted to the data by first converting from\nfEXM tocGdand optimizing the free parameters λandδ.\nValues for saturation magnetization and g-factors of Co\nand Gd are given in the Appendix. The dashed red line\nin Fig. 4 a, obtained for the linear case with δ= 16.0◦,\nshows good agreement. This result implies that, after\nthe spin current pulse is fully absorbed, the magnetiza-\ntion vectors of Co and Gd are canted out of the film\nplane by this angle. In Figure 4 bandc, the under-\nlying frequency and amplitude dependence on the alloy\nconcentration are shown. To understand the evolution of\nthe amplitude, we start by considering the Co-dominated\ncase, corresponding to low values of cGd(ortGdin the\ncase of the experiment). Due to the strong degree of un-\ncompensation, the oscillation frequency is high and Ltot\nis very close to LCo. As a result, θCois very small, leading\nto small Az. Upon increasing cGd,fEXM decreases and\nLtotis tilted away from MCo, inducing larger precession\namplitudes. When approaching Lcomp, an injection of\nOOP spins tilts Ltotfully OOP. Therefore, the projec-\ntion of the precession onto the z-axis vanishes as cos Ω\napproaches zero (see Figure 4 c). This explains the de-\ncrease of precession amplitude in our experiments at low\nEXM frequencies close to Lcomp.\nSince the model treated thus far is based on excited\nstates far away from the antiparallel equilibrium state,\none may argue that the commonly used small angle ap-\nproximation does not hold here. To account for finite pre-\ncession amplitudes, we derive the dispersion relation for\narbitrary precession angles in the Appendix. The main\ndifference to the linear approximation is that the EXM\nfrequency remains finite for nonzero canting angles, even\nFIG. 4. aAmplitude of measured EXMs, fitted analytical\nmodels, and Fourier intensity of the optically induced spin\ncurrent pulse as a function of frequency. The fits are based on\nthe frequency ( b) and amplitude ( c) dependencies of EXMs\non the alloy concentration. Designated angles are explained\nby the cartoon in d, showing deflected angular momenta of\nCo and Gd for different alloy concentrations.\nat the compensation point of angular momentum. The\nlarger the precession angles are, the higher is the pre-\ndicted frequency. We repeat the fitting procedure and\nobtain the continuous blue curve in Fig. 4 a. Again, the\nfit agrees reasonably with experimental data. For the\nnonlinear case, only a canting of 3 .5◦is required to ex-\nplain the observed trend in amplitude while the frequency\nis not required to cross zero (see Fig. 4 b). To obtain a\nrealistic estimate for the upper bound of the canting an-7\ngle, we consider the amount of angular momentum that\nis dissipated during ultrafast demagnetization and, based\non this, calculate by how much the Co sublattice can cant\nupon full absorption of said angular momentum. From\nMOKE measurements of the injection stack, we extract\na maximum demagnetization of ∼12.5%. Further, we\ntake a thicknesses of 3.4 nm and 1.6 nm for injection and\nthe sum of both Co layers, respectively, and use the satu-\nration magnetizations given in the Appendix. We find a\ntheoretical maximum of the canting angle of 7 .0◦. In light\nof this, the results from the nonlinear model seem more\nrealistic and can explain all experimental observations.\nThe hitherto assumed equal spin transfer efficiency on\nCo and Gd requires further attention. The EXM fre-\nquency predicted by our model is actually independent of\nthe precise ratio of δCo/δGdand thus, differences in spin\ntransfer efficiencies. Instead, the sum δCo+δGdalone is\ndecisive, meaning that a variety of combinations of δCo\nandδGdcan yield the same frequency. This is due to\nthe assumed isotropy of the system. Possible anisotropy\nfields are much lower than exchange fields between Gd\nand Co and would only induce small perturbations. The\nz-component of the precession amplitude, on the other\nhand, does depend on δCo/δCo. We did confirm that the\napplicability of our model is not strictly limited to the\ncase that δCo=δGdby fitting the data with various ra-\ntios of δCo/δGdand obtaining good results. However, it\nwas not possible to reliably extract the best ratio from the\nfits. Furthermore, different efficiencies and variations in\nabsorption length of Co and Gd might have a minor influ-\nence on the excitation efficiency. Since dramatic changes\nin EXM dynamics were observed among adding less than\none monolayer of Gd, we conclude that the mechanisms\ndiscussed thus far dominate.\nAnother interesting aspect of the dynamics at Lcomp\nis that half an oscillation around Ltotcorresponds to\nswitching the magnetization by 180◦in the film plane.\nTheoretical studies on antiferromagnets [50, 51] came to\nthe similar conclusion that a perpendicular spin current\npulse can manipulate the order parameter of a magnetic\nsystem. Since our TR-MOKE setup is insensitive to\nchanges in IP magnetization, we refrain from claims as\nto whether such switching is realistically possible in our\ndevice. We further note that distinct IP easy axes would\nbe required to establish defined states between which the\nmagnetization can be switched.\nFinally, we will put our results into perspective with\nprevious studies on EXMs. While higher resonance fre-\nquencies than the ones we find have been observed in\ninsulating ferri- and antiferromagnets, [52–54] those ma-\nterials are extremely challenging to be implemented into\ndevices due to their poor conductivity and the lack of\nconduction spin polarization. Exchange resonance modes\nin a similar frequency range ( ∼0.4 THz) have not been\nreported in metallic systems at room temperature to the\nbest of our knowledge. Comparable Ru-based synthetic\nantiferromagnetic oscillators only reach frequencies in the\n∼20 GHz range at zero field [55] which is owed to theRKKY coupling being much weaker than the RE-TM ex-\nchange. The sample design used in our study offers not\nonly a high conductivity in general but also ferromagnetic\ninterfacial layers with high conduction spin polarization.\nConsequently, our layer structure could easily be inte-\ngrated into electronic applications, as magnetoresistive\neffects can be used to probe the precession state on-chip.\nFurthermore, our results deepen understanding on intri-\ncacies in the fascinating platform of Co/Gd multilayers\nthat are highly relevant for other types of spintronic ap-\nplications such as magnetic racetrack memory devices.\nIt is worth noting that the coherent excitation of THz\ndynamics by optically induced spin currents is not limited\nto the sample design used in this work. Instead, it can\neasily be adapted to study high-frequency modes in other\ntypes of ferrimagnets or even antiferromagnets.\nV. CONCLUSION\nWe have demonstrated excitation of ferromagnetic res-\nonance and exchange resonance modes in IP synthetic\nferrimagnetic Co/Gd/Co/Gd multilayers using OSTTs\ngenerated in a neighboring perpendicular magnetic layer.\nOptically induced spin currents were proven to be an\nexcellent tool to excite and study those modes. FMR\noscillations in the 10 GHz range and EXM modes with\nfrequencies up to 0.4 THz were observed at room temper-\nature. Varying the thickness of the Gd layers was inves-\ntigated, enabling an easy path for manipulating the spin\nresonance spectrum over orders of magnitude by tuning\nthe total angular momentum in the system. The de-\npendence of the frequency of the exchange mode gives\nunique insight into the excitation mechanism and pos-\nsible nonlinear dynamics close to compensation. Our\nfindings open up new pathways for the development of\nferrimagnetic THz spintronic devices and exploring their\nhigh-frequency response.\nACKNOWLEDGMENTS\nThis project has received funding from the Euro-\npean Union’s Horizon 2020 research and innovation pro-\ngramme under the Marie Sk lodowska-Curie grant agree-\nment No 861300.\nAppendix: Nonlinear exchange resonance dynamics\nThe equations of motion predicting the time evolu-\ntion of two antiferromagnetically coupled magnetic mo-\nments MCo,Gdin the absence of external fields, magnetic\nanisotropy, and Gilbert damping are given by the follow-8\ning coupled Landau-Lifshitz (LL) equations:\ndMCo\ndt=−µ0γCoMCo×λMGd,\ndMGd\ndt=−µ0γGdMGd×λMCo.(A.1)\nFIG. 5. Exchange resonance frequency aas a function of θCo\naccording to eq. (A.6) and bas a function of δas given by\neq. (A.7) for different alloy concentrations. A change in sign\nof the frequency implies a reversal of the mode’s handedness.\nUpon canting the magnetic moments by an angle δ\nwith respect to the film plane, as explained in the main\ntext, conservation of angular momentum requires the\nconsequent precessions to revolve around Ltot, enclosing\nan angle of\nΩ = arctansin(δ)h\nMCo\nγCo+MGd\nγGdi\ncos(δ)h\nMCo\nγCo−MGd\nγGdi (A.2)with the film plane. The angles of MCo,Gdwith respect\ntoLtotare given by θCo= Ω−δandθGd= Ω+ δ(graph-\nically shown in Fig. 4 d). As the angular momentum\nperpendicular to the rotation axis has to vanish, the fol-\nlowing relation between the two angles holds:\nMCo\nγCosinθCo=MGd\nγGdsinθGd. (A.3)\nEquation (A.1) may now be solved for the precession fre-\nquency assuming an oscillatory solution and identifying\nangles between Hex\nGd,CoandMCo,Gd. The LL equations\nthen simplify to\n2πfM CosinθCo=µ0γCoλMCoMGdsin(θGd−θCo),\n(A.4)\n2πfM GdsinθGd=−µ0γGdλMGdMCosin(θCo−θGd).\n(A.5)\nThe constraint given by eq. (A.3) makes the two solutions\nequivalent. A simple rearrangement of the solution for\nthe Co sublattice yields\nf=µ0γCoλMGdsin(θGd−θCo)\n2πsinθCo(A.6)\n=µ0γCoλMGdsin(2δ)\n2πsinθCo. (A.7)\nThe former equation as a function of θCoand the lat-\nter one as a function of the canting angle δare plotted\nin Fig. 5 aandb, respectively for a variety of alloy com-\npositions and an arbitrarily chosen exchange constant of\nλ= 2. Both solutions reproduce the change in sign of\nthe frequency across Lcomp, implying a reversal of hand-\nedness. Increasing θCoandδboth leads to an increase of\n|f|with respect to the value obtained at θCo=δ= 0. In\nthe Co-dominated region, dispersion curves in Fig. 5 a\nshow discontinuities whenever eq. (A.3) has no real value\nsolution for θGd. Figure 5 bis only zero when Lis\ncompensated and δ= 0. Therefore, any canting away\nfrom the antiparallel ground state in a ferrimagnet will\nresult in an exchange-driven spin resonance. One can\neasily show that eq. 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O’Handley, Modern Magnetic Materials (Wiley, 1999)." }, { "title": "2112.07233v1.Ground_State__Magnetization_Process_and_Bipartite_Quantum_Entanglement_of_a_Spin_1_2_Ising_Heisenberg_Model_on_Planar_Lattices_of_Interconnected_Trigonal_Bipyramids.pdf", "content": "arXiv:2112.07233v1 [cond-mat.stat-mech] 14 Dec 2021Ground State, Magnetization Process and Bipartite Quantum Entanglement\nof a Spin-1/2 Ising–Heisenberg Model on Planar Lattices of Interconnec ted\nTrigonal Bipyramids\nLucia G´ alisov´ aa,∗, Michał Kaczorb,c\naInstitute of Manufacturing Management, Faculty of Manufac turing Technologies with the Seat in Preˇ sov, Technical Uni versity of Koˇ sice,\nBayerova 1, 080 01 Preˇ sov, Slovakia\nbThe Doctoral School of University of Rzesz´ ow, University o f Rzesz´ ow, Rejtana 16C, 35-935 Rzesz´ ow, Poland\ncInsitute of Physics, College of Natural Sciences, Universi ty of Rzesz´ ow, Rejtana 16A, 35-935 Rzesz´ ow, Poland\nAbstract\nThe ground state, magnetization scenario and the local bipa rtite quantum entanglement of a mixed spin-1 /2 Ising–\nHeisenberg model in a magnetic field on planar lattices forme d by identical corner-sharing bipyramidal plaquettes is\nexamined by combining the exact analytical concept of gener alized decoration-iteration mapping transformations wit h\nMonte Carlo simulations utilizing the Metropolis algorith m. The ground-state phase diagram of the model involves\nsix different phases, namely, the standard ferrimagnetic phase, fu lly saturated phase, two unique quantum ferrimag-\nnetic phases, and two macroscopically degenerate quantum f errimagnetic phases with two chiral degrees of freedom\nof the Heisenberg triangular clusters. The diversity of gro und-state spin arrangement is manifested themselves in\nseven different magnetization scenarios with one, two or three fracti onal plateaus whose values are determined by the\nnumber of corner-sharing plaquettes. The low-temperature values of the concurrence demonstrate that the bipartite\nquantum entanglement of the Heisenberg spins in quantum fer rimagnetic phases is field independent, but twice as\nstrong if the Heisenberg spin arrangement is unique as it is t wo-fold degenerate.\nKeywords: Ising–Heisenberg model, chiral degrees of freedom, magnet ization process, bipartite quantum\nentanglement, rigorous results\n1. Introduction\nQuantum entanglement has been attracting a lot of attention in the last few years mainly due to its crucial role\nin the development of quantum computers, superdense coding , quantum communication, quantum teleportation, as\nwell as quantum information theory [1–3]. The application p otential of this unique phenomenon also exceeds into the\nquantum biology [4, 5] and quantum metrology [6, 7].\nIn quantum theory, quantum entanglement provides a novel pl atform for exploring long-range quantum correla-\ntions, quantum phase transitions as well as exotic properti es of many-body systems [8–11]. The low-dimensional\nHeisenberg spin models, involving quantum fluctuations bet ween spins, play a significant role in this regard because\nthey have been proven to be ideal candidates for a rigorous in vestigation of the entangled states under the influ-\nence of the external stimuli such as magnetic field (homogene ous or inhomogeneous) and /or temperature [12–24].\nMoreover, many analytical and numerical calculations have been performed to examine the tuning of the quantum\nand thermal bipartite entanglement by varying the exchange anisotropy parameter [19–28], the uniaxial single-ion\nanisotropy [16, 17], the Dzyaloshinskii–Moriya interacti on (spin-orbit coupling) [18–20, 26, 27], the next-nearest -\nneighbour interaction [13, 14, 29], as well as by introducin g impurities into the system [28, 30].\nHowever, the rigorous investigation of the bipartite entan glement in the pure Heisenberg models represents a\ncomplex task, which is considerably limited due to a non-com mutability of spin operators in the Hamiltonian. This\n∗Corresponding author\nEmail address: lucia.galisova@tuke.sk (Lucia G´ alisov´ a )\nPreprint submitted to Entropy December 15, 2021computational problem makes the rigorous study of the pheno menon in general inaccessible across whole parame-\nter space of the systems. On the other hand, replacing some of the Heisenberg spins with three spin components\nby the Ising ones with only one ( z-) component at the nodal lattice sites is the alternative wa y to exactly exam-\nine the entanglement in various simpler mixed-spin Ising–H eisenberg models by using the standard transfer-matrix\nmethod [31] and/or the concept of generalized mapping transformations [32– 35]. Taking into account the fact that the\nfinite Heisenberg clusters formed by three-component Heise nberg spins are indirectly coupled with each other through\nthe intermediate one-component Ising spin(s), one finds tha t the eigenstates of two adjacent Heisenberg clusters are\nseparable. Thus, any quantity measuring the local bipartit e entanglement in the considered mixed-spin model can be\nrigorously calculated for each quantum Heisenberg cluster separately.\nTo date, the bipartite entanglement has been rigorously exa mined in several one- (1D) and two-dimensional (2D)\nmixed-spin Ising–Heisenberg models formed by the identica l Heisenberg dimers or triangular clusters which interact\nwith each other via the intermediate nodal Ising spin(s) [36 –45]. The investigations brought a deeper insight into the\nthermal and magnetic-field-driven changes of the phenomeno n [36–43], the impact of the model’s parameters on the\nphenomenon [36, 39–45], as well as the evolution of the pheno menon near and above second-order (continuous) phase\ntransitions [44, 45] without any artefacts arising from app roximations. Despite their simplicity and the general opin ion\nthat the simpler mixed-spin Ising–Heisenberg systems invo lving isolated local quantum correlations are artificial\nmodels, some of the results were in a very good correspondenc e with ones obtained for more complex Heisenberg\ncounterparts [40] and also with experiments [38, 46, 47].\nIn the present paper, we will rigorously solve a spin-1 /2 Ising–Heisenberg model in a longitudinal magnetic field\non 2D lattices formed by identical corner-sharing trigonal bipyramidal plaquettes. Our recent studies [45, 48] of the\nmodel without magnetic field on the particular lattice with f our inter-connected bipyramidal units have shown that this\nquantum mixed-spin model represents a suitable playground for a rigorous study of various unconventional physical\nphenomena such as the macroscopic degeneracy of the spontan eous long-range order caused by chiral spin degrees\nof freedom, the spin frustration, and the bipartite entangl ement. The aforementioned findings motivated us to extend\nthe investigation of the model also to the e ffect of the longitudinal magnetic field. The goals of the prese nt paper are\nto shed a light on the nature of ground states invoked by the ap plied field, to identify the actual fractional plateaus in\nthe zero-temperature magnetization process, to find out a ge neral formula describing how the values of these plateaus\ndepend on the current number of interconnected bipyramidal plaquettes and, finally, to quantify the bipartite quantum\nentanglement between the Heisenberg spins in individual gr ound states.\nIn addition to the academic interest, our investigation of t he spin-1/2 Ising-Heisenberg model on 2D lattices\nformed by interconnected trigonal bipyramids is motivated by the existence of a class of geometrically frustrated\nstructures, namely cobaltates YBaCo 4O7(Y denotes a rare-earth ion) [49] and anion-radical salts (M DABCO+)(C•−\n60)\n(MDABCO+represents N-methyldiazabicyclooctanium cation, C•−\n60is a radical anion) [50], in which one can clearly\nidentify corner-sharing trigonal bipyramidal clusters. A lthough the mentioned compounds do not represent a precise\nexperimental realization of the magnetic structure propos ed in the present paper, we hope that a targeted design of the\nmagnetic material with a magnetic structure of interconnec ted trigonal bipyramids is feasible. The targeted chemical\nsynthesis involving highly anisotropic spin carriers such Dy3+or Co2+magnetic ions and anion-radical salts could\npossibly afford desiring such a quantum mixed-spin system. The findings p resented in this paper could serve as a\nmotivation for chemists to achieve this goal.\nThe outline of the paper is as follows: in Section 2, a magneti c structure of the investigated model is described and\nthe most important steps of its rigorous treatment combinin g the analytical and numerical approaches are clarified. In\nSection 3, we present the most interesting numerical result s for the ground state and the magnetization process of the\nmodel. The section also includes an analysis of the bipartit e quantum entanglement in the individual ground states.\nFinally, the summary of the most important findings are prese nted in Section 4.\n2. Model and Its Rigorous Treatment\nWe consider a mixed spin-1 /2 Ising–Heisenberg model in a longitudinal magnetic field on 2D lattices consisting\nof identical corner-sharing trigonal bipyramidal plaquet tes, as is schematically depicted in Figure 1 for one particu lar\nlattice with four such plaquettes. In this figure, the common vertices of plaquettes (white circles) are occupied by\nthe Ising spinsσ=1/2 that interact with other spins solely through their z-components. The rest ones (red circles),\nforming internal equilateral triangles oriented perpendi cularly to the plaquette axes, are occupied by the Heisenber g\n2spins S=1/2 that are coupled to each other via x-,y-, and z-components. Assuming qbipyramidal plaquettes share\na common vertex, the total Hamiltonian of the mixed spin-1 /2 Ising–Heisenberg model can be written as a sum of\nplaquette (five-spin cluster) Hamiltonians ˆH=/summationtextNq/2\nj=1ˆHj, where Nlabels the total number of the nodal lattice sites\noccupied by the Ising spins (we consider the thermodynamic l imit N→∞ ). Each plaquette Hamiltonian ˆHjcontains\nall exchange interactions realized within the jth Ising–Heisenberg trigonal bipyramid and Zeeman terms th at describe\nthe influence of the applied external magnetic field on magnet ic moments of the individual spins:\nˆHj=−JH3/summationdisplay\nk=1/bracketleftig\n∆(ˆSx\nj,kˆSx\nj,k+1+ˆSy\nj,kˆSy\nj,k+1)+ˆSz\nj,kˆSz\nj,k+1/bracketrightig\n−JI3/summationdisplay\nk=1ˆSz\nj,k( ˆσz\nj+ˆσz\nj+1)\n−HH3/summationdisplay\nk=1ˆSz\nj,k−HI\nq( ˆσz\nj+ˆσz\nj+1).(1)\nIn the above, ˆSα\nj,k(α=x,y,z) and ˆσz\njare spatial components of the spin-1 /2 operator of the Heisenberg spin from the\njth triangle and z-component of the Pauli operator with the eigenvalues ±1/2 at the jth nodal lattice site, respectively,\nwhich satisfy the periodic boundary conditions ˆSα\nj,4≡ˆSα\nj,1and ˆσz\nNq/2+1≡ˆσz\n1. The parameter JHmarks the XXZ\nHeisenberg interaction within the Heisenberg triangles, ∆is the exchange anisotropy parameter in this interaction,\nandJIlabels the Ising-type interaction between the nearest-nei ghbouring Ising and Heisenberg spins. The last two\nterms HHandHIin the second line of Equation (1) are Zeeman terms, which acc ount for the magnetostatic energy of\nthe Heisenberg and Ising spins in an applied longitudinal ma gnetic field, respectively.\nAs we have shown in our recent work on the zero-field case of the model [45], it is convenient for further calcula-\ntions to introduce the composite spin operators:\nˆtj=3/summationdisplay\nk=1ˆSj,k,ˆtα\nj=3/summationdisplay\nk=1ˆSα\nj,k(α=x,y,z), (2)\nwhich determine the total spin of the Heisenberg triangular clusters and its spatial components, respectively. From\nthe definition of the latter operators, one can easily obtain the spin identity ( ˆtα\nj)2=3/4+2/summationtext3\nk=1ˆSα\nj,kˆSα\nj,k+1. This, in\ncombination with the identity for the square of the total com posite spin operator ( ˆtj)2=ˆtj·ˆtj=(ˆtx\nj)2+(ˆty\nj)2+(ˆtz\nj)2,\nallows one to find the following two relations for the Heisenb erg spin operators from the same triangular cluster:\n3/summationdisplay\nk=1/parenleftigˆSx\nj,kˆSx\nj,k+1+ˆSy\nj,kˆSy\nj,k+1/parenrightig\n=1\n2/bracketleftig\n(ˆtj)2−(ˆtz\nj)2/bracketrightig\n−3\n4,3/summationdisplay\nk=1ˆSz\nj,kˆSz\nj,k+1=1\n2(ˆtz\nj)2−3\n8. (3)\nBearing in mind the above relations and the definition of the z-component of the composite spin operator ˆtz\njlisted\nFigure 1: A schematic representation of the j-th trigonal bipyramidal plaquette and the mixed spin-1 /2 Ising–Heisenberg model on the particular\n2D lattice with four ( q=4) corner-sharing bipyramidal plaquettes. White circles l abel lattice sites occupied by the Ising spin σ=1/2 and red\ncircles denote lattice sites occupied by the Heisenberg spi nS=1/2. Black dashed lines illustrate the Ising-type interactio nJIbetween the Ising\nand Heisenberg spins and red solid lines indicate XXZ Heisen berg exchange interaction JH(∆) between the Heisenberg spins in the plaquette.\n3in Equation (2), the plaquette Hamiltonian (1) can be expres sed in the alternative form:\nˆHj=3JH\n8(2∆+1)−JH∆\n2(ˆtj)2+JH\n2(∆−1)(ˆtz\nj)2−JIˆtz\nj( ˆσz\nj+ˆσz\nj+1)\n−HHˆtz\nj−HI\nq( ˆσz\nj+ˆσz\nj+1).(4)\nIt is easy to prove that the operators ( ˆtj)2,ˆtz\njappearing in Equation (4) satisfy the commutation relation s/bracketleftbigˆHj,(ˆtj)2/bracketrightbig=0\nand/bracketleftbigˆHj,ˆtz\nj/bracketrightbig=0, which implies that they are both conserved quantities wit h well defined quantum spin numbers\ntj(tj+1) and tz\nj={−tj,−tj+1,..., tj}fortj={3/2,1/2}, respectively. In this regard, Equation (4) represents a\nfully diagonal form of the plaquette Hamiltonian (1), which implies that the corresponding energy eigenvalues can be\nexpressed in terms of the respective quantum spin numbers:\nEtj,tz\nj=3JH\n8(2∆+1)−JH∆\n2tj(tj+1)+JH\n2(∆−1)(tz\nj)2−JItz\nj(σz\nj+σz\nj+1)\n−HHtz\nj−HI\nq(σz\nj+σz\nj+1).(5)\nAt this calculation stage, the partition function of the con sidered spin-1/2 Ising–Heisenberg model can be partially\nfactorized due to commuting character of di fferent plaquette Hamiltonians and written in terms of the eig envalues (5)\nof the plaquette Hamiltonian:\nZ=Tr exp/parenleftig\n−βˆH/parenrightig\n=/summationdisplay\n{σn}Nq/2/productdisplay\nj=1Trjexp/parenleftig\n−βˆHj/parenrightig\n=/summationdisplay\n{σn}Nq/2/productdisplay\nj=1/summationdisplay\ntj,tz\njgtjexp/parenleftbig−βEtj,tz\nj/parenrightbig. (6)\nHere,β=1/(kBT) (kBis the Boltzmann’s constant and Tis the absolute temperature of the system), and the summatio n\nsymbol/summationtext\n{σn}denotes a summation over all possible spin configurations of the Ising spins, the product symbol/producttextNq/2\nj=1\nruns over all trigonal bipyramids, and the double summation symbol/summationtext\ntj,tz\njruns over all possible values of the quantum\nnumbers tj,tz\njof the composite spins. Finally, gtjis the degeneracy factor, which takes the value 1 for the quan tum\nnumber tj=3/2 and 2 for the quantum number tj=1/2. After performing double summations over tjandtz\nj,\none gains the effective Boltzmann’s weight w(σz\nj,σz\nj+1), which depends only on the Ising spin states σz\nj,σz\nj+1, and,\nthus, it can be replaced by a simpler but equivalent expressi on using the generalized decoration-iteration mapping\ntransformation [32–35]:\nw(σz\nj,σz\nj+1)=/summationdisplay\ntj,tz\njgtjexp/parenleftbig−βEtj,tz\nj/parenrightbig=2 exp/bracketleftiggβHI\nq(σj+σj+1)−βJH\n4/bracketrightigg\n×/braceleftigg/bracketleftigg\nexp(βJH∆)+2 exp/parenleftigg\n−βJH∆\n2/parenrightigg/bracketrightigg\ncosh/bracketleftbiggβJI\n2(σz\nj+σz\nj+1)+βHH\n2/bracketrightbigg\n+exp(βJH)cosh/bracketleftigg3βJI\n2(σz\nj+σz\nj+1)+3βHH\n2/bracketrightigg/bracerightigg\n=Aexp/bracketleftigg\nβJeffσz\njσz\nj+1+βHeff\nq(σz\nj+σz\nj+1)/bracketrightigg\n.(7)\nThe novel effective parameters A,Jeff, and Heffemerging on the right-hand side of Equation (7) are determin ed by\n’self-consistency’ of the algebraic approach used:\nA=4/radicalig\nw+w−w2\n0,Jeff=kBTlnw+w−\nw2\n0,Heff=kBTq\n2ln/parenleftiggw+\nw−/parenrightigg\n. (8)\nHere, w±=w(±1/2,±1/2) and w0=w(±1/2,∓1/2). After substituting Equation (7) into Equation (6), one o btains the\nrigorous equivalence between the partition function Zof the spin-1/2 Ising–Heisenberg model given by the Hamilto-\nnian (1) and the partition function ZIMof the effective spin-1/2 Ising model on the corresponding q-coordinated 2D\n4lattice given by the Hamiltonian HIM=−Jeff/summationtextNq/2\n/an}bracketle{tj,n/an}bracketri}htσz\njσz\nn−Heff/summationtextN\nj=1σz\nj:\nZ(T,JH,JI,∆,HH,HI)=ANq/2ZIM(T,Jeff,Heff). (9)\nThe mapping relation (9) represents the crucial result of th e rigorous solution of the considered 2D spin-1 /2 Ising–\nHeisenberg model in an external magnetic field because of all important physical quantities clarifying a ground-state\narrangement, magnetization process, and quantum bipartit e entanglement between the Heisenberg spins, namely,\nthe local magnetization mI=/an}bracketle{tˆσz\nj/an}bracketri}htandmH=/an}bracketle{t/summationtext3\nk=1ˆSz\nj,k/an}bracketri}htper nodal Ising spin and Heisenberg triangular cluster,\nrespectively, the total magnetization mper bipyramidal plaquette, as well as the pair correlation f unctions Cxx(yy)\nHH=\n/an}bracketle{tˆSx\nj,kˆSx\nj,k+1/an}bracketri}ht=/an}bracketle{tˆSy\nj,kˆSy\nj,k+1/an}bracketri}ht,Czz\nHH=/an}bracketle{tˆSz\nj,kˆSz\nj,k+1/an}bracketri}htandCzz\nIH=/an}bracketle{tˆσz\njˆSz\nj,k/an}bracketri}ht=/an}bracketle{tˆσz\nj+1ˆSz\nj,k/an}bracketri}ht, can be directly derived from the\nformula for the Gibbs free energy G=−kBTlnZby means of the differential calculus:\nmI=−1\n2N∂G\n∂HI,mH=−2\nNq∂G\n∂HH,m=qmH+2mI\nq, (10a)\nCzz\nHH=2\n3NqJ H/parenleftigg\n∆∂G\n∂∆−∂G\n∂JH/parenrightigg\n,Cxx(yy)\nHH=−1\n3NqJ H∂G\n∂∆,Czz\nIH=−1\n3Nq∂G\n∂JI. (10b)\nThe final analytical expressions of all the physical quantit ies listed in\nEquations (10a) and (10b) depend on the on-site magnetizati onmIM=/an}bracketle{tσz\nj/an}bracketri}htIMand the pair correlation function\nCzz\nIM=/an}bracketle{tσz\njσz\nj+1/an}bracketri}htIMof the effective 2D q-coordinated spin-1 /2 Ising lattice with the temperature-dependent nearest-\nneighbour interaction Jeffin the temperature-dependent magnetic field Heff. Because an exact solution for the 2D\nspin-1/2 Ising model in an external magnetic field still belongs to un resolved issues of condensed matter physics, one\nhas to resort to some numerical algorithm applicable to the 2 D Ising lattices to gain the accurate results for mIMand\nCzz\nIM. In the present paper, we will employ the classical Monte Car lo (MC) simulations implementing the standard\nMetropolis algorithm [51, 52] for the e ffective spin-1/2 Ising lattice of a su fficiently large linear size L.\n3. Discussion of the Numerical Results\nIn this section, we will proceed to a discussion of the most in teresting numerical results for the 2D spin-1 /2 Ising–\nHeisenberg model in an external magnetic field with the antif erromagnetic Ising-type interaction JI<0 between\nthe Ising and Heisenberg spins. For simplicity, we will assu me that the local magnetic fields acting in the Ising and\nHeisenberg spins are identical HI=HH=H. The absolute value of the interaction JIwill be used as an energy unit\nfor defining a relative strength of the Heisenberg interacti onJH/|JI|and the magnetic field H/|JI|.\n3.1. Ground-State Phase Diagrams\nFirst, we take a look at possible magnetic ground-state arra ngement of the model, which can be determined\nby a systematic inspection of the eigenvalues (5) of the plaq uette Hamiltonian (1) for all possible combinations of\nquantum spin numbers tj,tz\njentering therein. The typical ground-state phase diagrams are depicted in Figure 2 in the\nJH/|JI|−H/|JI|parameter plane for two representative values of the exchan ge anisotropy∆=0.5 and 2 by assuming\nfour different numbers qof corner-sharing trigonal bipyramidal plaquettes formin g 2D lattices. As one can see from\nFigure 2a, the ground-state phase diagram of the model with t he easy-axis exchange anisotropy ∆=0.5 contains four\ndifferent ground states. Specifically, two ground states are mac roscopically degenerate quantum ferrimagnetic phases\n|−1/2; 1/2/an}bracketri}htR,Land|1/2; 1/2/an}bracketri}htR,L, which differ from each other only by the orientation of Ising spins with respect to\nthe applied magnetic field as indicated by the corresponding eigenvectors and eigenenergies per plaquette:\n|±1/2; 1/2/an}bracketri}htR,L=Nq/2/productdisplay\nj=1|±/an}bracketri}htσz\nj⊗|1/2,RorL/an}bracketri}ht△j, (11a)\nE|±1/2;1/2/an}bracketri}htR,L=JH\n4+JH∆\n2∓JI\n2−(q±2)H\n2q. (11b)\n5The state vector|1/2,RorL/an}bracketri}ht△jin Equation (11a) describes a quantum superposition of thre e different up-up-down\nspin states of the j-th Heisenberg triangular cluster with two opposite ( Right- and Left-hand side) chiral degrees of\nfreedom:\n|1/2,R/an}bracketri}ht△j=1√\n3/parenleftig\n|↑↑↓/an}bracketri}ht+e2πi\n3|↑↓↑/an}bracketri}ht+e4πi\n3|↓↑↑/an}bracketri}ht/parenrightig\n△j,\n|1/2,L/an}bracketri}ht△j=1√\n3/parenleftig\n|↑↑↓/an}bracketri}ht+e4πi\n3|↑↓↑/an}bracketri}ht+e2πi\n3|↓↑↑/an}bracketri}ht/parenrightig\n△j,(12)\nThe two-fold degeneracy of each Heisenberg triangle result s in the field-independent macroscopic degeneracy 2Nq/2\nof the phases|−1/2; 1/2/an}bracketri}htR,Land|1/2; 1/2/an}bracketri}htR,L, which is obviously highly sensitive to the current number qof inter-\nconnected trigonal bipyramidal plaquettes (Heisenberg tr iangular clusters). The direct relation between the number\nof plaquettes sharing a common vertex and the macroscopic de generacy of the phases |−1/2; 1/2/an}bracketri}htR,L,|1/2; 1/2/an}bracketri}htR,Lis\nalso reflected in a current value of the residual entropy per n odal Ising spin observed in both the phases. Specifically,\nit proportionally grows with q[53]:\nSres\nNkB=lim\nN→∞1\nNln 2Nq/2≈0.347q. (13)\nThe other two ground states are the classical ferrimagnetic phase|−1/2; 3/2/an}bracketri}htand the fully saturated phase |1/2; 3/2/an}bracketri}ht.\nThese two phases again di ffer from each other only by the orientation of the Ising spins w ith respect to the applied\nmagnetic field, while the Heisenberg spins are fully polariz ed into the magnetic field direction without any quantum\ncorrelations between their x- and y-components in both phases:\n|±1/2; 3/2/an}bracketri}ht=Nq/2/productdisplay\nj=1|±/an}bracketri}htσz\nj⊗|↑↑↑/an}bracketri}ht△j, (14a)\nE|±1/2;3/2/an}bracketri}ht=−3JH\n4∓3JI\n2−(3q±2)H\n2q. (14b)\nThe uniqueness of the classical spin arrangements in the pha ses|−1/2; 3/2/an}bracketri}htand|1/2; 3/2/an}bracketri}htgiven by Equation (14a) is\nreflected in the zero entropy per Ising spin S/(NkB)=0 in parameter regions corresponding to these phases.\nIt is obvious from Figure 2a that the classical ferrimagneti c phase|−1/2; 3/2/an}bracketri}htcan be detected in the whole pa-\nrameter region with the ferromagnetic Heisenberg coupling JH/|JI|>0 and partially also in the region with the\nantiferromagnetic Heisenberg interaction JH/|JI|<0. By contrast, the macroscopically degenerate quantum pha ses\n|−1/2; 1/2/an}bracketri}htL,Rand|1/2; 1/2/an}bracketri}htL,Rare stable solely for the antiferromagnetic Heisenberg cou plings JH/|JI|<0. Finally,\nthe saturated phase |1/2; 3/2/an}bracketri}htrepresents the actual ground state at high enough magnetic fi elds regardless of whether\nthe ferro- or antiferromagnetic Heisenberg interaction JH/|JI|is considered.\nOn the other hand, the ground-state phase diagram correspon ding to the model with the easy-plane anisotropy\n∆= 2 is a little more complex (see Figure 2b). It contains two mor e ground states in addition to the previous four,\nnamely, the unique quantum ferrimagnetic phases |−1/2; 1/2/an}bracketri}htand|1/2; 1/2/an}bracketri}htwith the Heisenberg triangular clusters in\na symmetric quantum superposition of three possible up-up- down spin states but an opposite orientation of the Ising\nspins:\n|±1/2; 1/2/an}bracketri}ht=Nq/2/productdisplay\nj=1|±/an}bracketri}htσz\nj⊗1√\n3(|↑↑↓/an}bracketri}ht+|↑↓↑/an}bracketri}ht+|↓↑↑/an}bracketri}ht )△j, (15a)\nE|±1/2;1/2/an}bracketri}ht=JH\n4−JH∆∓JI\n2−(q±2)H\n2q. (15b)\nAs shown in Figure 2b, both the quantum ferrimagnetic phases |−1/2; 1/2/an}bracketri}htand|1/2; 1/2/an}bracketri}htemerge in the ground-state\nphase diagram exclusively in the parameter region of the fer romagnetic Heisenberg interaction JH/|JI|>0. Naturally,\nthe zero entropy per Ising spin S/(NkB)=0 solely can be detected in their stability regions due to uni que quantum\nspin arrangement given by Equation (15a).\n6/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s48/s50/s52/s54/s56/s49/s48/s32/s61/s32/s48/s46/s53/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124/s49/s47/s50/s59/s49/s47/s50\n/s62\n/s82/s44/s76/s72 /s32 /s47/s32/s124 /s74\n/s73/s124\n/s74\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s124/s45/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124/s45/s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76\n/s40/s97/s41/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s48/s50/s52/s54/s56/s49/s48/s32/s61/s32/s50/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124/s49/s47/s50/s59/s49/s47/s50\n/s62\n/s82/s44/s76/s72 /s32 /s47/s32/s124 /s74\n/s73/s124\n/s74\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s124/s49/s47/s50/s59/s49/s47/s50/s62\n/s124/s45/s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76/s124/s45/s49/s47/s50/s59/s49/s47/s50 /s62/s124/s45/s49/s47/s50/s59/s51/s47/s50 /s62/s32/s32 /s113 /s32/s61/s32/s51\n/s32/s32 /s113 /s32/s61/s32/s52\n/s32/s32 /s113 /s32/s61/s32/s53\n/s32/s32 /s113 /s32/s61/s32/s54\n/s40/s98/s41\nFigure 2: The ground-state phase diagram of the spin-1 /2 Ising–Heisenberg model on 2D lattices with three ( q=3), four ( q=4), five ( q=5),\nand six ( q=6) corner-sharing trigonal bipyramidal plaquettes in the JH/|JI|−H/|JI|parameter plane for two representative values of the exchan ge\nanisotropy parameter: ( a)∆=0.5 and ( b)∆=2.\nIn addition to their location in the zero-temperature JH/|JI|−H/|JI|parameter plane, it is also possible to un-\nderstand from Figure 2 how the individual phases develop dep ending on the number qof corner-sharing bipyramidal\nplaquettes. Namely, the quantum ferrimagnetic phases |−1/2; 1/2/an}bracketri}htR,L,|−1/2; 1/2/an}bracketri}htand the classical one |−1/2; 3/2/an}bracketri}ht\nare gradually extended to stronger magnetic fields with incr easing number qof the corner-haring plaquettes. More-\nover, the classical phase |−1/2; 3/2/an}bracketri}htsimultaneously spreads to the regions of stronger antiferr omagnetic (ferromag-\nnetic) Heisenberg interactions JH/|JI|<0 (JH/|JI|>0). The remaining three phases |1/2; 1/2/an}bracketri}htR,L,|1/2; 1/2/an}bracketri}htand\n|1/2; 3/2/an}bracketri}htfaithfully follow the evolution of the adjacent ones |−1/2; 1/2/an}bracketri}htR,L,|−1/2; 1/2/an}bracketri}ht,|−1/2; 3/2/an}bracketri}ht: the quantum\nphases|1/2; 1/2/an}bracketri}htR,L,|1/2; 1/2/an}bracketri}htare gradually shifted to stronger antiferromagnetic and fe rromagnetic Heisenberg in-\nteractions JH/|JI|<0 and JH/|JI|>0, respectively, and the saturated one |1/2; 3/2/an}bracketri}htis shifted to stronger magnetic\nfields.\n3.2. Magnetization Process\nThe rich ground-state phase diagrams depicted in Figure 2 su ggest various zero-temperature magnetization scenar-\nios of the studied spin-1 /2 Ising–Heisenberg model either with one, two or three di fferent plateaus at fractional values\nof the saturation magnetization msat=(3q+2)/(2q). In accordance with the definition of the total magnetizati onm\nper plaquette listed in Equation (10a), the values of these p lateaus are given by a current number qof the trigonal\nbipyramids sharing a common vertex:\nm\nmsat=q−2\n3q+2,q+2\n3q+2,3q−2\n3q+2. (16)\nThe first (lowest) magnetization plateau at m/msat=(q−2)/(3q+2) can be identified at low magnetic fields in\nthe stability regions of the macroscopically degenerate qu antum ferrimagnetic phase |−1/2; 1/2/an}bracketri}htR,Land the unique\nquantum ferrimagnetic phase |−1/2; 1/2/an}bracketri}ht. The second one at m/msat=(q+2)/(3q+2) is a result of the spin\narrangements present in the quantum phases |1/2; 1/2/an}bracketri}htR,Land|1/2; 1/2/an}bracketri}ht, and therefore it can be found at moderate\nmagnetic fields. The highest fractional plateau at m/msat=(3q−2)/(3q+2) relates to the classical ferrimagnetic\nphase|−1/2; 3/2/an}bracketri}ht.\nTo illustrate the above statements, two three-dimensional (3D) plots of the isothermal magnetization curves for the\nparticular version of the lattice with four ( q=4) corner-sharing bipyramidal plaquettes are depicted in F igure 3 at the\nsufficiently low temperature kBT/|JI|=1.5×10−3. The plots are the outcomes of MC simulations for 100 ×100 nodal\nlattice sites (Ising spins), which corresponds to 19,800 co rner-sharing trigonal bipyramidal plaquettes. The adequa te\nnumerical accuracy was achieved by 12 ×104MC steps per node. For easy reference, the interaction ratio JH/|JI|is\nfixed to the same ranges and the anisotropy parameter ∆to the same values as were used in Figure 2. It can be under-\nstood from a comparison of these plots with corresponding gr ound-state phase diagrams in Figure 2 that the displayed\nlow-temperature magnetization curves faithfully reflect u p to seven different types of zero-temperature magnetization\nscenarios with the real 1 /7-, 3/7-, and/or 5/7-plateaus satisfying the general formulas listed in Equat ion (16):\n7i.|−1/2; 1/2/an}bracketri}htR,L−|1/2; 1/2/an}bracketri}htR,L−|1/2; 3/2/an}bracketri}ht,\nii.|−1/2; 1/2/an}bracketri}htR,L−|1/2; 1/2/an}bracketri}htR,L−|−1/2; 3/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht,\niii.|−1/2; 1/2/an}bracketri}htR,L−|−1/2; 3/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht,\niv.|−1/2; 3/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht,\nv.|−1/2; 1/2/an}bracketri}ht−|− 1/2; 3/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht,\nvi.|−1/2; 1/2/an}bracketri}ht−|1/2; 1/2/an}bracketri}ht−|−1/2; 3/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht,\nvii.|−1/2; 1/2/an}bracketri}ht−|1/2; 1/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht\n(see the magnetization curves of di fferent colors). Steep continuous rises between di fferent fractional plateaus as well\nas between fractional plateaus and the saturation magnetiz ation indicate a presence of the discontinuous magnetizati on\njumps that exist at the critical fields corresponding to the fi rst-order phase transitions only at zero temperature. In\nagreement with the ground-state analysis performed in Sect ion 3.1, the first three magnetization processes i.–iii., wh ich\ncontain the macroscopically degenerate quantum phases |±1/2; 1/2/an}bracketri}htR,L, can be observed for the easy-axis exchange\nanisotropy∆= 0.5 and also the easy-plane exchange anisotropy ∆= 2, but only in the parameter region of the\nantiferromagnetic Heisenberg interactions JH/|JI|<0. On the other hand, the magnetization scenario iv., reflect ing\nthe single field-induced transition from the classical ferr imagnetic phase|−1/2; 3/2/an}bracketri}htto the saturated one |1/2; 3/2/an}bracketri}ht,\nappears for both the antiferromagnetic ( JH/|JI|<0) and ferromagnetic ( JH/|JI|>0) Heisenberg couplings. The\nlast three magnetization processes v.–vii., which involve unique quantum ferrimagnetic phases |±1/2; 1/2/an}bracketri}ht, emerge\nin the parameter region of the ferromagnetic Heisenberg cou plings JH/|JI|>0 under the condition ∆>1. It should\nbe noted for completeness that the steep staircase dependen ces of all magnetization curves plotted in Figure 3 are\ngenerally gradually smoothing upon increasing of temperat ure due to a thermal activation of excited states, until they\ncompletely disappear.\n/s124 /s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76/s32 /s32/s124/s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76/s32 /s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124 /s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76/s32 /s32/s124/s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76/s32 /s32/s124 /s49/s47/s50/s59/s51/s47/s50 /s62 /s32 /s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124 /s49/s47/s50/s59/s49/s47/s50 /s62\n/s82/s44/s76/s32 /s32/s124 /s49/s47/s50/s59/s51/s47/s50 /s62 /s32 /s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62/s124 /s49/s47/s50/s59/s51/s47/s50 /s62\n/s105/s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124 /s49/s47/s50/s59/s49/s47/s50 /s62\n/s105/s32/s124 /s49/s47/s50/s59/s51/s47/s50 /s62 /s32 /s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124 /s49/s47/s50/s59/s49/s47/s50 /s62\n/s105/s32/s124/s49/s47/s50/s59/s49/s47/s50 /s62 /s32 /s32/s124 /s49/s47/s50/s59/s51/s47/s50 /s62 /s32 /s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62\n/s124 /s49/s47/s50/s59/s49/s47/s50 /s62\n/s105/s32/s124/s49/s47/s50/s59/s49/s47/s50 /s62 /s32 /s32/s124/s49/s47/s50/s59/s51/s47/s50 /s62\nFigure 3: 3D plots of the total magnetization mof the spin-1/2 Ising–Heisenberg model on the regular lattice with four co rner-sharing bipyramidal\nplaquettes reduced to its saturation value msatas a function of the magnetic field H/|JI|and the interaction ratio JH/|JI|for the exchange anisotropy\n(a)∆= 0.5 and ( b)∆= 2 at the temperature kBT/|JI|=1.5×10−3obtained by MC simulations for the lattice with 100 ×100 nodal Ising spins\n(19,800 bipyramidal plaquettes) by using 12 ×104MC steps per node. The curves of distinct colors refer to di fferent magnetization scenarios listed\nin the legend.\n83.3. Quantum Bipartite Entanglement\nThe discussion in the last subsection will be devoted to a bip artite quantum entanglement of the Heisenberg spins\nin the individual ground states. It is obvious from the plaqu ette Hamiltonian (1) that the spins may be quantum-\nmechanically entangled only within the Heisenberg triangu lar clusters in individual plaquettes. Those from di fferent\nplaquettes can never be entangled due to the Ising spin at the ir common vertices.\nIn general, a degree of the bipartite quantum entanglement b etween the Heisenberg spins at k-th and ( k+1)-th\nvertex of the j-th plaquette can be quantified by the quantity referred to as concurrence [54]. For the studied 2D\nspin-1/2 Ising–Heisenberg model, the concurrence can be simply cal culated from the local magnetization mHof the\nHeisenberg triangular cluster and the corresponding pair c orrelation functions Cxx(yy)\nHH,Czz\nHHdefined by Equations (10a)\nand (10b) through the following formula [55, 56]:\nCk,k+1=max0,4|Cxx(yy)\nHH|−2/radicaligg/parenleftigg1\n4+Czz\nHH/parenrightigg2\n−/parenleftbiggmH\n3/parenrightbigg2. (17)\nOf course, the identical XXZ exchange coupling JH(∆) in a given Heisenberg triangle results in the same degree of\nthe bipartite entanglement of the spin pairs. This is reflect ed in the same values of the corresponding concurrences:\nC1,2=C2,3=C3,1=C. (18)\nThe global picture on a degree of the bipartite quantum entan glement between the Heisenberg spin pairs in the\nindividual ground-state phases is illustrated in Figure 4, which shows the low-temperature ( kBT/|J|=1.5×10−3)\ndensity map of the concurrence Cof the spin-1/2 Ising–Heisenberg model on the regular 2D lattice with four corner-\nsharing bipyramidal plaquettes in the JH/|JI|−H/|JI|plane for the fixed value of the exchange anisotropy paramete r\n∆= 2. The plotted data have again been obtained by MC simulation s performed for the lattice of 19,800 corner-\nsharing bipyramidal plaquettes, whereas 2 ×107MC steps per nodal Ising spin were used to achieve the accurac y\nbetter than 10−3. It is clear from a direct comparison of Figure 4 with the corr esponding ground-state phase diagram\ndepicted in Figure 2b that the Heisenberg spins forming tria ngular clusters are quantum-mechanically entangled only\nif the macroscopically degenerate phases |±1/2; 1/2/an}bracketri}htR,Land the unique phases |±1/2; 1/2/an}bracketri}htare ground states. Due to\nthe macroscopic degeneracy caused by two possible chiral de grees of freedom of each triangular cluster, the bipartite\nentanglement of the Heisenberg spins in former two phases is half weaker than that in latter ones. This is also proven\nby the corresponding zero-temperature asymptotic values o f the concurrenceC|±1/2;1/2/an}bracketri}htR,L=1/3 andC|±1/2;1/2/an}bracketri}ht=2/3.\nOn the other hand, the remaining white region with the zero co ncurrenceC|±1/2;3/2/an}bracketri}ht=0 confirms completely non-\nentangled arrangements of the Heisenberg spins in the class ical ferrimagnetic phase |−1/2; 3/2/an}bracketri}htand the fully saturated\nphase|1/2; 3/2/an}bracketri}ht.\nTo get a deeper insight onto a role of pair correlations betwe en the Heisenberg spins in their bipartite quantum\nentanglement, the concurrence Cas function of the magnetic field H/|JI|and the corresponding dependencies of the\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s48/s50/s52/s54/s56/s49/s48\n/s32/s61/s32/s50\n/s67\n/s124/s32/s32/s49/s47/s50/s59/s49/s47/s50\n/s62\n/s82\n/s44\n/s76/s61/s32/s49/s47/s51/s72 /s32/s47/s32/s124 /s74\n/s73/s124\n/s74\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s67/s124/s32/s32/s49/s47/s50/s59/s49/s47/s50/s62/s32/s61/s32/s50/s47/s51/s67\n/s124 /s32/s32/s49/s47/s50/s59/s51/s47/s50 /s62/s32/s61/s32/s48\n/s43/s43\n/s43\nFigure 4: The low-temperature ( kBT/|J|=1.5×10−3) density map of the concurrence Cof the spin-1/2 Ising–Heisenberg model on the regular\nlattice with four corner-sharing trigonal bipyramidal pla quettes in the JH/|JI|−H/|JI|plane for the exchange anisotropy parameter ∆=2 constructed\nfrom MC simulations performed for the lattice of 19,800 bipy ramidal plaquettes by using 2 ×107MC steps per nodal Ising spin.\n9/s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s74\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s32/s61/s32/s45/s52\n/s109\n/s72\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s67\n/s40/s97/s41/s109\n/s73/s32/s44/s32/s32 /s109\n/s72/s32/s44/s32/s32 /s67/s32/s120/s120 /s40/s121/s121 /s41\n/s72/s72/s44/s32/s32 /s67/s32/s122/s122\n/s72/s72\n/s48/s46/s48/s48/s48/s48/s46/s51/s51/s51/s48/s46/s54/s54/s55\n/s109\n/s73/s67/s32/s120/s120 /s40/s121/s121 /s41\n/s72/s72/s67/s32/s122/s122\n/s72/s72/s67\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s74\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s32/s61/s32/s54\n/s109\n/s72\n/s72/s32 /s47/s32/s124 /s74\n/s73/s124/s67\n/s40/s98/s41/s109\n/s73/s32/s44/s32/s32 /s109\n/s72/s32/s44/s32/s32 /s67/s32/s120/s120 /s40/s121/s121 /s41\n/s72/s72/s44/s32/s32 /s67/s32/s122/s122\n/s72/s72/s109\n/s73/s67/s32/s120/s120 /s40/s121/s121 /s41\n/s72/s72\n/s67/s32/s122/s122\n/s72/s72/s67/s48/s46/s51/s51/s51/s48/s46/s54/s54/s55\n/s48/s46/s48/s48/s48\nFigure 5: The low-temperature ( kBT/|JI|=1.5×10−3) dependencies of the concurrence C, the sub-lattice magnetization mI,mH, and the pair\ncorrelation functions Cxx(yy)\nHH,Czz\nHHon the magnetic field H/|JI|of the spin-1/2 Ising–Heisenberg model on the regular lattice with four co rner-\nsharing bipyramidal plaquettes for the exchange anisotrop y∆= 2 and two particular interaction ratios ( a)JH/|JI|=−4 and ( b)JH/|JI|=6. The\ncurves are results of the MC simulations for the lattice of 10 0×100 nodal Ising spins by using 2 ×107MC steps per node.\npair correlation functions Cxx(yy)\nHH,Czz\nHHare plotted in Figure 5 for the anisotropy parameter ∆= 2 and two selected\ninteraction ratios JH/|JI|=−4 and 6 at the temperature kBT/|J|=1.5×10−3. The variations are completed by low-\ntemperature dependences of the local magnetization mIandmHto facilitate identification of the current ground-state\nspin arrangement. We note for completeness that all the curv es are results of the MC simulations for the lattice of\n100×100 nodal Ising spins by using 2 ×107MC steps per node to achieve accuracy better than 10−3.\nFigure 5a captures the sequence of field-induced phase trans itions|−1/2; 1/2/an}bracketri}htR,L−|1/2; 1/2/an}bracketri}htR,L−|1/2; 3/2/an}bracketri}ht. Ev-\nidently, the nonzero concurrence C=1/3, which can be found at the magnetic fields H/|JI|<9 due to stability of\nthe macroscopically degenerate quantum phases |±1/2; 1/2/an}bracketri}htR,L, is a result of the negative pair correlation functions\nCxx(yy)\nHH=Czz\nHH=−1/12 and the reduced local magnetization mH=1/2. Identical values of the transverse and lon-\ngitudinal correlation functions and their minus sign clear ly indicate that the macroscopically degenerate unsaturat ed\nbipartite entanglement of the Heisenberg spins from the sam e XXZ triangle comes from antiferromagnetic xx(yy)\ncorrelations of these spins, which are of the same strength a nd character as those in z-axis direction.\nA different situation can be found in Figure 5b, which illustrates the sequence of field-induced phase transitions\n|−1/2; 1/2/an}bracketri}ht−|1/2; 1/2/an}bracketri}ht−|1/2; 3/2/an}bracketri}ht. Here, the low-temperature concurrence C=2/3, which can be observed at the\nmagnetic fields H/|JI|<7 due to the presence of the unique quantum phases |±1/2; 1/2/an}bracketri}ht, comes from the positive\nvalue of the transverse pair correlation function(s) Cxx(yy)\nHH=1/6, the negative longitudinal correlation function Czz\nHH=\n−1/12, and the reduced local magnetization mH=1/2. It is easy to understand from the values of Cxx(yy)\nHHandCzz\nHHthat\nthe origin of the local quantum bipartite entanglement of th e Heisenberg spins peculiar to the unique quantum phases\n|±1/2; 1/2/an}bracketri}htlies in ferromagnetic xx(yy) correlations and these are twice as strong as the antiferro magnetic ones along\nz-axis.\n4. Conclusions\nIn the present work, we have comprehensively studied the gro und-state properties, possible magnetization sce-\nnarios, and the local bipartite quantum entanglement of the Heisenberg spins in the individual quantum ground states\nof the mixed spin-1 /2 Ising–Heisenberg model in a longitudinal magnetic field on 2D lattices formed by identical\ncorner-sharing bipyramidal plaquettes. The numerical res ults have been obtained by combining the exact analytical\napproach called the decoration-iteration mapping transfo rmation [32–35] with numerical Monte Carlo simulations\nutilizing the Metropolis algorithm [51, 52].\nIt has been demonstrated that the ground-state phase diagra m of the investigated quantum mixed-spin model\nqualitatively does not depend on its lattice topology (the n umber qof corner-sharing plaquettes). In general, it involves\nin total six different ground states, namely the standard ferrimagnetic pha se, fully saturated phase, two unique quantum\nferrimagnetic phases and two macroscopically degenerate q uantum ferrimagnetic phases with two chiral degrees of\nfreedom of the Heisenberg spins forming triangular cluster s in plaquettes. It is also proven that the diversity of the\n10ground-state phase diagram gives rise to seven di fferent magnetization scenarios with one, two or up to three fr actional\nplateaus. The magnitudes of these plateaus are determined b y the current number qof the corner-sharing plaquettes.\nOther interesting findings are concerned with the bipartite quantum entanglement, which has been quantified by\nthe concept of the concurrence. We have verified that the Heis enberg spins of the same XXZ triangular cluster of a\ngiven plaquette can be entangled either due to stability of t he unique quantum ferrimagnetic phases, where they are in a\nsymmetric quantum superposition of three possible up-up-d own states, or due to macroscopically degenerate quantum\nferrimagnetic phases characterized by two chiral degrees o f freedom of each Heisenberg triangle. The strength of the\nentanglement in all the phases does not depend on the applied magnetic field. Moreover, the corresponding values of\nconcurrence clearly indicate that the entanglement of the H eisenberg spins is twice as strong when their arrangement\nis unique that when it is two-fold degenerate. Thus, it can be concluded that the macroscopic degeneracy of the\nHeisenberg triangles proportionally reduces the bipartit e quantum entanglement of their spins.\nFollowing our recent paper [45], dealing with the spin-1 /2 Ising–Heisenberg model on the planar lattice formed\nby trigonal bipyramids without a magnetic field, there is str ong indication that the bipartite entanglement between\nthe Heisenberg spins observed in the unique ferrimagnetic a nd macroscopically degenerated ferrimagnetic ground-\nstate phases in the present paper will also persist at finite t emperatures. 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Tychko 2* \n1Kyiv Taras Shevchenko National University, Radiophy sics Faculty, Glushkov av.2, build.5, \nKyiv, Ukraine, 01022 \n1E-mail: b.m.tanygin@gmail.com \n2E-mail: pasat@univ.kiev.ua \n \nAbstract. Magnetic symmetry of all possible plane domain wal ls in ferro- and ferrimagnets is \nconsidered. Magnetic symmetry classes of non 180 degree (inclu ding 0 degree) domain walls are \nobtained. The domain walls degeneracy is investigated. The symmetry classification is applied for \nresearch of all possible plane domain walls in crystals of the hexoctahedral crystallographic class. \nPACS: 61.50 Ah, 75.60 Ch \nKeywords: domain wall type, symmetry transformation , magnetic symmetry class, degeneracy \n \n1. Introduction \nThe investigation of static and dynamic properties [1,2] of domain walls (DWs) in magnetically \nordered media is of considerable interest for the p hysical understanding of medium behavior and it is also \nimportant for applications. For sequential examinat ion of these properties it is necessary to take int o \naccount the magnetic symmetry [3,4] of the media. D etermination of the DW magnetic symmetry allows \n \n*Corresponding author. O.V. Tychko. Address: 64 Vladimirskaya str., Taras Shevchenko Kyiv Natio nal \nUniversity, Radiophysics Faculty. 01033 Kyiv, Ukrai ne. Tel/fax : +38-044-526-03-49 E-mail : \npasat@univ.kiev.ua , a.tychko@mail.ru 2 \nto characterize qualitatively some elements of the DW structure and their change. The complete \nsymmetry classification of plane 180 degree DWs (18 0 0-DWs) in magnetically ordered crystals [5] and \nsimilar classification of these DWs with Bloch line s in ferromagnets and ferrites [6] were carried out \nearlier. The plane DWs with width δ[1,7] exceeding the characteristic size a of a unit magnetic cell were \nconsidered. Properties of these DWs in ferro- and f errimagnets are described by the density of magneti c \nmoment M [8] . Their symmetry can be characterized by the magnetic symmetry classes (MSCs) [9] of a \ncrystal containing a DW [5]. The building of a tota lity of the MSCs of all possible [1] plane (i.e. DW with \n0r>> δ, where 0r is the curvature radius of the DW [5]) DWs in ferr o- and ferrimagnets is the purpose of \nthis work. \n \n2. Domain wall symmetry in the magnetically ordered media \nLet m be the unit time-odd axial vector [9] along the ma gnetization vector M: M/Mm=, where \nM is the saturation magnetization. Then 1m and 2m are unit time-odd axial vectors along magnetization \nvectors 1M and 2M in neighboring domains: M/1 1Mm= , M/2 2Mm= . The vectors 1m and 2m \ncoincide with different easy magnetization axes (EM A) of the medium. The angle α2 between these \nvectors determines the DW type ( α2-DW): ()2 1 arccos 2 mm =α . A unit polar time-even vector Wn \nindicates the DW plane normal. It is directed from domain with 1m to domain with 2m. In order to define \nthe unified co-ordinate system we introduce the vec tors 1a and 2a as well as the parameters \n[]Σ Σ×=mnWb and []mnΔ ×=Δ Wb . The unit vectors of the co-ordicate system z y x O~~~ are chosen as \n[][ ]W zyx naaeee ,,,,12~~~ −= . Here the unit vector 1a coincides with the direction of the vector \n()mnnm Δ−ΔWW (at 0≠Δb and 0=Σb) or []Wna×2 (at 0=Δb or 0≠Σb). The unit vector 2a coincides \nwith the direction of vector ()Σ Σ−mnnmWW (at 0≠Σb) or []1an×W (at 0≠Δb and 0=Σb) or else with \nan arbitrary direction in the DW plane ( Wna⊥2 at 0==ΔΣbb ). The time-odd axial vectors mΔ and Σm \nare determined by equalities 12mmm−=Δ and 21mmm+=Σ respectively. 3 \n The MSC kG (here k is a MSC number) of a α2-DW is the magnetic symmetry group including \nall symmetry transformations (here and hereinafter all translations are consider ed as unit operations) that \ndo not change the spatial distribution of magnetic moments in the crystal with DW [5]. The above-\nmentioned group is a subgroup of the magnetic (Shub nikov’s) symmetry group of the crystal \nparamagnetic phase [10]. These transformations do n ot change DW boundary conditions and can be \nclassified by two types [5]. The first type transfo rmations ()1g do not change the directions of the vectors \n1m, 2m and Wn: ()\nWgn1=Wn, ()\n111mm=g , ()\n2 21mm=g . The second type transformations ()2g change \nthese directions: ()\nWgn2=Wn−, ()\n2 12mm=g , ()\n1 22mm=g . In conformity with the terminology of [6] the \nMSC BG of DW boundary conditions is the totality of all t ransformations of the magnetic symmetry \ngroup of the crystal paramagnetic phase that satisf y the mentioned six conditions. It is the MSC of th e \nmaximum possible symmetry of a α2 -DW in the given crystal for a particular mutual o rientation of the \nvectors 1m, 2m and Wn. The other possible MSCs kG of a α2-DW with fixed directions of the vectors \n1m, 2m and Wn result by enumeration of the subgroups of BG : P B k GGG ⊂⊆ , where PG is the MSC \nof the crystal paramagnetic phase. The mutual orien tation of the vectors 1m, 2m and Wn is determined by \nthe set of parameters ()Σ Σ=mnWa , ()mnΔ=Δ Wa , ()CW Camn= , Σb and Δb, where time-even axial \nvector Cm is determined by equality []21mmm ×=C . \nThe possible MSCs kG (1 ≤k≤42) of 180 0-DWs were found earlier [5]. All possible MSCs of \nα2-DWs with α2≠180 0 are presented in table 1. \nFor a certain α2-DW the different MSCs are different groups of magn etic point symmetry \ntransformations. Their representations [11,15] are written in the co-ordinate system z y x O~~~. All \nrepresented MSCs are not interrelated by a rotation over an arbitrary angle around Wn. Also the above-\nmentioned MSCs are not reduced with each other by u nit vectors transformation 2 1aa↔. \nThe possible transformations ()1g or ()2g (column “Symmetry elements” of table 1) of α2-DWs \nwith α2≠180 0 are rotations around two-fold symmetry axes n2 , n2′ or 12 , 12′ or else 22 , 22′ that are 4 \ncollinear with the unit vectors Wn or 1a or else 2a, respectively, reflections in planes n2, n2′ or 12′ or else \n22, 22′ that are normal to the above mentioned vectors, re spectively, rotations around three-, four-, six-\nfold symmetry axes n3, n4, n6 that are collinear with the vector Wn, rotations around three-, four-, six-\nfold inversion axes n3 , n4 , n6 that are collinear with the vector Wn, inversion in the symmetry center 1 \nand identity (symmetry element 1). Here an accent a t symmetry elements means a simultaneous use of the \ntime reversal operation R [9]. For MSCs with 24 ≤k≤39 and 52 ≤k≤64 only generative symmetry \nelements [11] are represented in table 1. \nThere is a correspondence between MSCs of 180°-DWs ( i.e. at 1m=2m−[1]), 0°-DWs (i.e. at \n1m=2m[13]) and α2-DWs with non-collinear orientation of vectors 1m and 2m[1] (hereinafter the last \nDWs will be marked as α′2-DWs). The above mentioned determinations of criteri ons for transformations \n()1g and ()2g can be represented in another identical form: ()\nWgn1=Wn, ()\nΣ Σ=mm1g , ()mmΔ =Δ1g and \n()\nWgn2=Wn−, ()\nΣ Σ=mm2g , ()mmΔ − =Δ2g . These criterions restrict an ensemble of MSCs symm etry \ntransformations for an arbitrary α′2-DW. We have 0=Δm and 0=Σm for 0°- and 180°-DWs, \nrespectively. A pair from the above mentioned crite rions does not restrict the MSCs symmetry \ntransformations of 0°- or 180°-DWs. Therefore the ma gnetic symmetry of α′2-DWs does not exceed the \nmagnetic symmetry of 0°- and 180°-DWs generically. The MSCs of 180°-DWs are the MSCs of α′2-\nDWs if their transformations do not break the symme try of the vector Σm of the α′2-DW (i.e. these \nMSCs must be subgroup of the group mm/m ′′∞, where the infinite-fold symmetry axis is collinea r with \nthe vector Σm). \nThere is an analogy between MSCs of 180°- and 0°-DWs : their transformations ()1g are the same \nsince they belong to a subgroup of axial time-odd v ector symmetry group (MSC mm/m ′′∞), where the \ninfinite-fold symmetry axis is collinear with mΔ or Σm for 180°- or 0°-DWs, respectively . Therefore if \nMSCs consist of the transformations ()1g only then these MSCs are common for 180°- and 0°-D Ws. They \nare marked with sign “-” in column “DW center” of t able 1. A conversion of MSC of 180°-DW into MSC 5 \nof 0°-DW is simply a change of the criterion ()mmΔ − =Δ2g by the criterion ()\nΣ Σ=mm2g . The \ntransformations of corresponding MSCs of these α2-DWs are different by the substitution ()()Rgg ⋅→2 2 \nonly. Therefore, if a pair of MSCs of 180°-DWs and a pair of MSCs of 0°-DWs is connected by the \nabove-mentioned substitution, then these MSCs are c ommon for 180°- and 0°-DWs. \nAs a result the lists of MSCs of 0°-, 180°- and α′2-DWs are intersected in general. Total number \nof MSCs of a α2-DW with arbitrary α2 value (including α2=180 0) in ferro- and ferrimagnets is equal \nto 64. General enumeration of MSCs of 180°-DWs cont ains 42 MSCs: 42 1≤≤k [5]. This enumeration \nholds also for MSCs of α2-DW with °≠180 2α(MSC numbers are bold type in column “MSC number \nk” of table 1) . There are 10 MSCs of α′2 -DWs: 13 7≤≤k and 18 16 ≤≤k. The general list of MSCs of \n0°-DWs includes all 42 MSCs of table 1: k=2, 13 6≤≤k, 19 16 ≤≤k, k=22, 24, 26, 30, 32, 37, 39 and \n64 43 ≤≤k. \n \n3. Domain wall structure \nThe α2-DWs with δ>> a in ferro- and ferrimagnets are described by the ma croscopic density of \nmagnetic moment M(z~) [5]. The transformations ()1g and ()2g (()\nkGg∈1;()\nkGg∈2) impose restrictions \non the kind of coordinate dependence of ()z~m components ( ()()()()zzzzz y x~~~~~ ~ ~ mmmm ++= ) in the DW \nvolume and allow to find this dependence [5]. For t he determination of the kind of coordinate dependen ce \nof ()z~m component of 0°- and α′2-DWs for each MSC (column “Coordinate dependences o f ()z~m \ncomponents” in table 1) the next rules are used: a) if an axial time-odd vector along unit vectors re \n(r≡x~,y~ or z~) is not an invariant of the transformation ()1g then there is no component ()zr~m (figure (-) \nin column “Coordinate dependences of ()z~m components” of table 1); b) if the axial time-odd vector \nalong re is inverted by the transformation ()2g then the component ()zr~m is an odd (A) function of \ncoordinate z~; c) if the axial time-odd vector along re is an invariant of the transformation ()2g then \n()zr~m is an even (S) function of coordinate z~; d) if the axial time-odd vector along re is an invariant of \nthe transformation ()1g then transformation ()1g does not restrict the kind of function ()zr~m (A,S). 6 \nIf the MCS of a α2-DW includes transformations that transpose adjacen t magnetic domains then \nthis DW has a center of symmetry [5]. These MSCs en close the symmetry transformations ()2g. They are \nmarked by coordinate 0~=z in column “DW center” of table 1. \nAs in the case of 180 0-DWs [5], the 00- DWs can be pulsating (i.e. DW with collinear dire ctions of \nvectors M and M≠const in its volume [5]) DWs. The MSCs with k=2, 6, 19-45, 49-64 describe \nsymmetry of pulsating DWs only. In contrast with 180°- and 0°-DWs there are no pulsating DWs among \nthe α′2-DWs, since α′2-DWs require the presence of two “nonzero” ()z~m components. The α′2-DWs \nare rotary (i.e. DW with M=const in its volume) or semi-rotary [5] DWs only. Among r otary or semi-\nrotary DWs there are DWs with only Bloch (i.e DWs w ith MWn=const) [1,14] ( k=7, 8 or 46) and only \nNeel (i.e. DWs with m rotation in the plane containing Wn) [1,15] ( k=9, 12, 17 or 47) laws of m rotation \nin their volume. \nCrystal magnetic ordering is accompanied by phase t ransition and change of crystal magnetic \nsymmetry [3]. In a magnetically ordered crystal kq-multiply degenerate α2-DWs with fixed α2 can be \nobtained [6], where ()()k P k GGq /ord ord = . Functions ()PGord and ()kGord give the order [11] of the \nmagnetic point group of the crystal paramagnetic phase [9,10] and of a α2 -DW in this crystal, \nrespectively. These α2 -DWs have the same energy but different structures (magnetization distribution, \nplane orientation, etc.). The minimum value of kq is 2 in accordance with the invariance of energy f or \ntime reversal operation R. \nAt representation of the PG as the totality of kG (with fixed value k and different symmetry \nelements orientations) the lost transformations (me mbers of adjacent classes) lg [6,12] interrelate the \nabove mentioned kq-multiply degenerate α2-DWs (i.e. lg operation converts an one of such α2-DWs \ninto another). \nThe degeneracy kq of a α2-DW can be written in the form k B kqqq′= (kq′≤kq), where \n()()k B k GGq /ord ord =′ is the number of equal-energy α2 -DWs with fixed boundary conditions, 7 \n()()B P B GGq /ord ord = is the number of possible boundary conditions. Her e ()BGord is the order of the \npoint group of the maximum magnetic symmetry of the α2-DW in the given crystal. \nThe α2-DWs of MSC 16 G (MSC 1) have the maximum degeneracy kq. For 180°- and α′2-DWs \nit is equal to 16 (crystallographic class mmm), 48 (crystallographic class 6/mmm) and 96 \n(crystallographic class m3m) in crystals of lower, medium and higher symmetry singonies (in conformity \nwith terminology of [11]), respectively. The 0°-DWs are formed in spatially inhomogeneous media [13]. \nConditions of occurrence and existence of such DWs demand to take into account medium peculiarities. \n \n4. Magnetic symmetry classes of domain walls in hex octahedral crystals \nAs an example let's consider MSCs of all possible D Ws in magnetically ordered crystals of \nhexoctahedral class (crystallographic point symmetr y group m 3 m in the paramagnetic phase [3]) . This \nclass is assumed to exhibit the largest variety of possible DWs. Furthermore it encompasses widely \ninvestigated and used magnetic media (all cubic sym metry metals, specifically iron and nickel [6], \nmagnetic oxides, specifically ferrites with structu res of spinel [4] and garnet [16], perovskite, magn etite \nand others). \nThe magnetic anisotropy (MA) energy Ke is the invariant of the initial paramagnetic phase of \ncrystal. For the m 3 m crystal this energy is given by ()321,,αααKe =1Ks+2Kp+3K2s+4Ksp +..., \nwhere 1K, 2K, 3K and 4K are first, second, third and fourth MA constants, s=2\n22\n1αα+2\n32\n2αα+2\n32\n1αα, \np=2\n32\n22\n1ααα, 1α, 2α and 3α are the direction cosines of m [16]. The absolute minimum of this energy \ncorresponds to EMAs. Signs of MA constants and rela tion between their values determine EMAs \ndirections. In the framework of the (1K, 2K, 3K) approximation the EMAs directions can coincide wi th \nboth high-symmetric and low-symmetric crystallograp hic directions [17]. In the framework of the two-\nconstant ( 1K,2K) approximation the EMA directions can coincide onl y with high-symmetric <111> or \n<110> or else <100> like crystallographic direction s at 1K≤-32K or 0 ≥1K≥-22K or else 1K≥0 \nrespectively [1,18]. At that 71 0-, 109 0- and 180 0-DWs or 60 0-, 90 0-, 120 0- and 180 0-DWs or else 90 0- and 8 \n180 0-DWs are realized in a m 3 m crystal, respectively [1]. The MSCs and degeneracy Bq of a α2-DW \nboundary conditions with °>90 2α and °≤90 2α are presented in tables 2 and 3 respectively. \nThe earlier obtained MSCs of merely 180°-DWs (bold type numbers in table 2) include elements \n[5]: k=1 - ()()1 , 12 , 2 , 2 , 121′×n; k=4 - ()112 , 2 , 1 , 1′ ′; k=5 -()nn2 , 2 , 1 , 1′ ′; k=14 - ()222 , 2 , 1 , 1′′; k=15 - ()1 , 1′; \nk=23 - ( )()1 , 12 , 2 , 2 , 121′×n; k=29 - ()12 , 3′′n; k=34 - ()n n2 , 2 , 41′′. \nOnly generative symmetry elements are presented for k=29 and 34. Other MSCs of tables 2 and 3 \nare presented in table 1. In these tables the DW pl ane orientation is assigned by different Miller ind exes \nh,k,l> 1. A simultaneous change on negative and/or cyclic permutation of all indexes doesn’t change \nMSCs. \nThere are no common MSCs of maximum symmetrical 180 °- and α′2-DWs in the m3m crystal. It \nis connected with the presence of the 1′ transformation ( α′2-DW vector Σm is changed by this \ntransformation) in the MSCs of such 180°-DW. \n \n5. Conclusions \nThe full magnetic symmetry classification of all po ssible domain walls in ferro- and ferrimagnet \ncrystals includes 64 magnetic symmetry classes: 42 classes of 0 0- DWs, 10 classes of α2-DWs with \n00<α2 <180 0 and 42 classes of 180 0-DWs. Lists of magnetic symmetry classes of all abo ve mentioned \ntypes of DWs are intersected in general case. \n00- DWs can be pulsating, rotary or semi-rotary DWs. The α2-DWs with 0 0<α2<180 0 are rotary \nor semi-rotary DWs only. Among rotary or semi-rotar y DWs there are DWs with Bloch or Neel laws of \nmagnetization rotation in their volume. Pulsating, rotary or semi-rotary DWs can have a cen ter of \nsymmetry in their volume. \nAll possible 180 0- and α2 -DWs with 0 0<α2 <180 0 have even degeneracy (its value is between 2 \nand 96 in general case). \nMagnetic symmetry classes of maximum symmetrical 18 0°-DWs do not meet with such classes of \nα2-DWs with 0 0<α2<180 0 in a m3m crystal. 9 \nReferences \n[1] A. Hubert, Theorie der Domanenwande in Geordnet en Medielen (Theory of Domain Walls in Ordered \nMedia), Springer, Berlin, Heidelberg, New York, 1974 \n A Hubert and R. Shafer, Magnetic Domains. Th e Analysis of Magnetic Microstructures, Springer, \nBerlin, 1998 \n[2] V. Bokov and V. Volkov, Physics of the Solid S tate 50 (2008)198 \n[3] L. Shuvalov, Sov. Phys. Crystallogr. 4(1959)399 \n[4] L. Shuvalov, Modern Crystallography IV : Physi cal Properties of Crystals, Springer, Berlin, 1988 \n[5] V. Baryakhtar, V. Lvov and D. Yablonsky, JETP 87(1984)1863 \n[6] V. Baryakhtar, E. Krotenko and D. Yablonsky, JE TP 91(1986)921 \n[7] B. Lilley, Phil.Mag. 41(1950)792 \n[8] A. Andreev and V. Marchenko, JETP 70(1976)1522 \n[9] L. Landau, E. Lifshitz and L. Pitaevskii, Cour se of Theoretical Physics, vol.8. Electrodynamics o f \nContinuous Media, Pergamon Press, London, 1984 \n[10] V. A. Kopcik, Xubnikovskie Gruppy: Spravoqnik po simmetrii i fiziqeskim svostvam. kristalliqeskih \nstruktur [Shubnikov’s groups: Handbook on the symme try and physical properties of crystalline \nstructures, in Russian], Izdatel’stvo Moskovskogo Universiteta, Moscow, 196 6 \n A.V. Shubnikov and N.V. Belov, Colored sym metry, Pergamon Press, London, 1964 \n B. Tavger and V. Zaitzev, JETP 3(1956)430 \n[11] B. Vanshtein, Modern Crystallography 1: Symm etry of Crystals, Methods of Structural \nCrystallography, Springer, Berlin, 1994 \n[12] E. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, \nAcademic Press, New York, 1959 \n[13] L. Heyderman, H. Niedoba, H. Gupta and I. Puch alska, J. Magn. Magn. Mater 96(1991)125. \n R. Vakhitov, A Yumaguzin, J. Magn. Magn. Ma ter. 215-216(2000)52 \n[14] L.Landau and E.Lifshitz, Sov.Phys. 8(1935)153 10 \n[15] L. Neel, Compt.rend. 2419(1955)533 \n[16] A. Paoletti, Physics of Magnetic Garnets, Esevier, Amsterdam, 1978 \n[17] U. Atzmony and M. Dariel, Phys. Rev. B13(1976) 4006 \n[18] K.P. Belov, A.K. Zvezdin, R.Z. Levitin, A.S. M arkosyan, B.V. Mill’, A.A. Mukhin and A.P.Perov, \nJETP 41(1975)590 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 11 \nTable 1. Magnetic symmetry classes of the plane α2-DWs with °≠180 2α. \nMSC \nnumb. \nk Mutual \norientations \nof the vectors \n1m, 2m and Wn \nSymmetry \nelements Coordinate dependences \nof ()z~mcomponents \nDW \ncenter International \nMSC \nsymbol ()zy~~m ()zx~~m ()zz~~m \n2 0===ΣΔΔaba n2 , 2 , 2 , 12 1′′ (-) (A,S) (-) - mm′2′ \n6 0===ΔΣ Caaa \n22 , 1 (-) (A,S) (-) - m \n7 0==ΔΣaa 1, 12′,22,n2′ (A) (S) (-) 0~=z 2 2′2′ \n8 0==ΔΣaa 1, n2′ (A,S) (A,S) (-) - 2′ \n9 0===ΣΔbaaC 1, 12′,22′,n2 (A) (-) (S) 0~=z m m′2′ \n10 0=Δa 1, 12′ (A) (S) (S) 0~=z 2′ \n11 0===ΣΔbaaC 1, n2 (A) (A) (S) 0~=z m \n12 0=Ca 1, 12′ (-) (A,S) (A,S) - m′ \n13 0=Σa 1, 22 (A) (S) (A) 0~=z 2 \n16 Arbitrary 1 (A,S) (A,S) (A,S) - 1 \n17 0===ΔΣbaaC 1, 12′,22,n2′ (-) (S) (A) 0~=z m′m′2 \n18 0===ΔΣbaaC 1, n2′ (S) (S) (A) 0~=z m′ \n19 0==ΣΔbb 1, n2 (-) (-) (A,S) - 2 \n22 0==ΣΔbb 1, 12′,22′,n2 (-) (-) (A,S) - m′m′2 \n24 0==ΣΔbb n3 (-) (-) (A,S) - 3 \n26 0==ΣΔbb 12 , 3′n (-) (-) (A,S) - m 3′ \n30 0==ΣΔbb n4 (-) (-) (A,S) - 4 \n32 0==ΣΔbb 12 , 4′n (-) (-) (A,S) - mm4′′ \n37 0==ΣΔbb n6 (-) (-) (A,S) - 6 \n39 0==ΣΔbb 12 , 6′n (-) (-) (A,S) - mm 6′′ \n43 0===ΣΔΔaba ()()1 , 12 , 2 , 2 , 12 1×′′n (-) (S) (-) 0~=z mmm′′ \n44 0===ΣΔΔaba n2 , 2 , 2 , 12 1′′ (-) (S) (-) 0~=z mm′2′ \n45 0===ΣΔΔaba 1, 1,22,22 (-) (S) (-) 0~=z 2/m \n46 0===ΣΔΔaba 1, 1,n2′,n2′ (S) (S) (-) 0~=z m / 2′′ \n47 0==ΔΔba 1, 1,12′,12′ (-) (S) (S) 0~=z m / 2′′ \n48 0==ΔΔba 1, 1 (S) (S) (S) 0~=z 1 12 \nTable 1. Magnetic symmetry classes of the plane α2-DWs with °≠180 2α (continue). \nMSC \nnumb. \nk Mutual \norientations \nof the vectors \n1m, 2m and Wn \nSymmetry \nelements Coordinate dependences \nof ()z~mcomponents \nDW \ncenter International \nMSC \nsymbol ()zy~~m ()zx~~m ()zz~~m \n49 0===ΣΔΔbba 1, 1,n2,n2 (-) (-) (S) 0~=z 2/m \n50 0===ΣΔΔbba 1, 12′,22′,n2 (-) (-) (S) 0~=z 2 2′2′ \n51 0===ΣΔΔbba ()()1 , 12 , 2 , 2 , 12 1×′ ′n (-) (-) (S) 0~=z mmm′′ \n52 0===ΣΔΔbba n6 (-) (-) (S) 0~=z 6 \n53 0===ΣΔΔbba 12 , 3′n (-) (-) (S) 0~=z 2 3′ \n54 0===ΣΔΔbba 12 , 6′n (-) (-) (S) 0~=z 2m 6′ ′ \n55 0===ΣΔΔbba 12 , 3′n (-) (-) (S) 0~=z m 3′ \n56 0===ΣΔΔbba nn2 , 4 (-) (-) (S) 0~=z 4/m \n57 0===ΣΔΔbba 12 , 4′n (-) (-) (S) 0~=z 224′′ \n58 0===ΣΔΔbba nn2 , 2 , 41′ (-) (-) (S) 0~=z mm/m 4′′ \n59 0===ΣΔΔbba n4 (-) (-) (S) 0~=z 4 \n60 0===ΣΔΔbba 12 , 4′n (-) (-) (S) 0~=z m24′ ′ \n61 0===ΣΔΔbba nn2 , 6 (-) (-) (S) 0~=z 6/m \n62 0===ΣΔΔbba 12 , 6′n (-) (-) (S) 0~=z 2 2 6′′ \n63 0===ΣΔΔbba nn2 , 2 , 61′ (-) (-) (S) 0~=z mm/m 6′′ \n64 0===ΣΔΔbba n3 (-) (-) (S) 0~=z 3 \n \n \n \n \n \n \n \n 13 \nTable.2. Number k (degeneracy Bq) of MSC of boundary conditions of arbitrary orient ed plane α2-DW \n(°>90 2α) in the cubic m 3 mcrystals at selected domain magnetization direction s. \n \nDW \nplane α2-DW boundary conditions \n180°-DW \n[100],[]00 1 180°-DW \n[110], []0 1 1 180°-DW \n[111], []1 1 1 120°-DW \n[110], []1 1 0 109°-DW \n[111], []1 1 1 \n(100) 34 (6) 14 (24) 14 (24) 16 (96) 9 (24) \n(010) 1 (12) 14 (24) 14 (24) 13 (48) 13 (48) \n(001) 1 (12) 1 (12) 14 (24) 16 (96) 13 (48) \n(111) 14 (24) 14 (24) 29 (8) 16 (96) 12 (48) \n()11 1 14 (24) 4 (24) 14 (24) 13 (48) 12 (48) \n()1 1 1 14 (24) 4 (24) 14 (24) 16 (96) 10 (48) \n()111 14 (24) 14 (24) 14 (24) 13 (48) 10 (48) \n(110) 14 (24) 23 (12) 14 (24) 16 (96) 16 (96) \n(101) 14 (24) 15 (48) 14 (24) 11 (48) 16 (96) \n(011) 1 (12) 15 (48) 14 (24) 16 (96) 17 (24) \n()10 1 14 (24) 1 (12) 5 (24) 16 (96) 16 (96) \n()01 1 14 (24) 15 (48) 5 (24) 13 (48) 16 (96) \n()1 1 0 1 (12) 15 (48) 5 (24) 16 (96) 7 (24) \n(hhl ) 15 (48) 14 (24) 14 (24) 16 (96) 16 (96) \n(hkh ) 15 (48) 15 (48) 14 (24) 16 (96) 16 (96) \n(hkk ) 14 (24) 15 (48) 14 (24) 16 (96) 12 (48) \n()hl h 15 (48) 4 (24) 15 (48) 16 (96) 16 (96) \n()kh h 15 (48) 15 (48) 15 (48) 13 (48) 16 (96) \n()k k h 14 (24) 15 (48) 15 (48) 16 (96) 10 (48) \n(hk 0), ()0k h 14 (24) 14 (24) 15 (48) 16 (96) 16 (96) \n(h0l), ()l h0 14 (24) 15 (48) 15 (48) 16 (96) 16 (96) \n(0 kl ), ()l k0 4 (24) 15 (48) 15 (48) 16 (96) 13 (48) \n(hkl ), ()kl h,()l k h,()l hk 15 (48) 15 (48) 15 (48) 16 (96) 16 (96) \n \n \n 14 \nTable.3. Number k (degeneracy Bq) of MSC of boundary conditions of arbitrary orient ed plane α2-DW \n(°≤90 2α) in the cubic m 3 mcrystals at selected domain magnetization direction s. \n \nDW \nplane α2-DW boundary conditions \n90°-DW \n[100], []0 1 0 90°-DW \n[110], []0 1 1 71°-DW \n[111], []11 1 60°-DW \n[110], [011] \n(100) 12 (48) 9 (24) 17 (24) 16 (96) \n(010) 12 (48) 17 (24) 10 (48) 10 (48) \n(001) 7 (24) 7 (24) 10 (48) 16 (96) \n(111) 13 (48) 16 (96) 12 (48) 10 (48) \n()11 1 10 (48) 16 (96) 12 (48) 16 (96) \n()1 1 1 10 (48) 16 (96) 13 (48) 10 (48) \n()111 13 (48) 16 (96) 13 (48) 16 (96) \n(110) 17 (24) 12 (48) 16 (96) 16 (96) \n(101) 16 (96) 10 (48) 16 (96) 10 (48) \n(011) 16 (96) 13 (48) 9 (24) 16 (96) \n()10 1 9 (24) 12 (48) 16 (96) 16 (96) \n()01 1 16 (96) 10 (48) 16 (96) 18 (48) \n()1 1 0 16 (96) 13 (48) 7 (24) 16 (96) \n(hhl ) 13 (48) 16 (96) 16 (96) 16 (96) \n(hkh ) 16 (96) 16 (96) 16 (96) 10 (48) \n(hkk ) 16 (96) 16 (96) 12 (48) 16 (96) \n()hl h 10 (48) 16 (96) 16 (96) 16 (96) \n()kh h 16 (96) 16 (96) 16 (96) 16 (96) \n()k k h 16 (96) 16 (96) 13 (48) 16 (96) \n(hk 0), ()0k h 12 (48) 12 (48) 16 (96) 16 (96) \n(h0l), ()l h0 16 (96) 10 (48) 16 (96) 16 (96) \n(0 kl ), ()l k0 16 (96) 13 (48) 10 (48) 16 (96) \n(hkl ), ()kl h,()l k h,()l hk 16 (96) 16 (96) 16 (96) 16 (96) \n " }, { "title": "1002.4889v1.Ferromagnetic_Resonance_of_Co_Gd_and_Co_Tb_Multilayers.pdf", "content": "1Ferromagnetic Resonance of Co/Gd and Co/Tb Multilayers\nS. Demirtas1, I. Harward1, R. E. Camley1and Z. Celinski1\n1Department of Physics, University of Colorado at Colorado Springs,\nColorado Springs, CO 80918 USA\nM. R. Hossu2and A. R. Koymen2\n2Department of Physics, University of Texas at Arlington, Arlington, TX 76019 USA\nC. Yu3and M. J. Pechan3\n3Department of Physics, Miami University, Oxford, OH 45056 USA\nAbstract\nThe in-plane dynamics of ferrimagnetic Co/Gd multilayers are investigated by \nmeans of ferromagnetic resonance, magneto-optical Kerr effect and SQUID \nmagnetometry. The power absorbed from these multilayers is strongly temperature \ndependent. For example, the resonant peak for a (Co 40 Å /Gd 40 Å) 8multilayer vanishes \napproximately 50 K below room temperature. We have further investigated Gd/Co/Gd\nand Tb/Co/Tb trilayers with different thicknesses of Gd (5-7 Å), Tb (1-7 Å) and Co (30-\n40 Å). At room temperature, these Co-based trilayers show a shift of approximately 600 \nOe at 24 GHz in the uniform ferromagnetic resonance field, compared to pure Co film, \nindicating the exchange coupling between the Co and Gd. The shift in the field for the \nresonance increases as the temperature is decreased. Furthermore the resonance linewidth \nincreases as the temperature is decreased. The experimental results are in good agreement \nwith our theoretical calculations.2Introduction\nMagnetic multilayers have been an area of active research for a considerable \nperiod, partly because their properties, such as giant magnetoresistance, have lead to \napplications in the recording industry. In addition to the common multilayers involving \ntransition metal ferromagnets with nonmagnetic spacers, there has also been a substantial \nstudy of transition metal / rare earth multilayers where both materials are ferromagnetic. \nThese systems exhibit multiple magnetic configurations and phase transitions between \nthese states at different external magnetic fields and temperatures. \nThe Co/Gd and Fe/Gd multilayer systems [1-21] have a number of known \nmagnetic states. There is a low temperature Gd-aligned state where the Gd magnetization \nis aligned with the external field (and the Co is opposite) and there is a high temperature \nCo-aligned state where the Co magnetization is aligned with the external field (and Gd is \nopposite). Finally there is also a \"twisted\" or canted state where the moments in each \natomic layer have a different angle with respect to the applied field. Therefore, Co/Gd \nand Fe/Gd structures are artificial ferrimagnets where the effective contribution of the \ntwo magnetic components is controllable by changing the layering pattern. At a particular \ntemperature, which is called compensation temperature (T comp), the magnetic moment of \nthe Co and Gd layers are equal in magnitude and total moment goes to zero due to the \nantiparallel alignment.\nAlthough there have been many investigations into the static configurations found \nin Fe/Gd and Co/Gd multilayers, there have been relatively few studies on the dynamical \nmodes [22-31] found in these artificial ferrimagnets. In particular, the ferromagnetic \nresonance (FMR) of alternating ferromagnetic films that are antiferromagnetically \ncoupled at the interfaces may be interesting since the magnetizations of the two materials \nrespond very differently to temperature changes. For example, in the Co/Gd multilayer \nthe average Gd magnetic moment changes from about 7 B to zero when the temperature \nis varied from 0 to around 300 K, while the moment of the Co changes only slightly over \nthe same temperature range.\nMost of the earlier work on the dynamic modes [22-30] was done on FeGd and \nCoGd alloys and concentrated on determining the g-value, the exchange stiffness \nconstant, the magnetic damping parameter and the anisotropy constants. An interesting 3property found for these amorphous alloys was the observation of multi-resonance peaks. \nA reasonable explanation [23] was that the non-uniformity of these amorphous alloys \ncreates the multi peak FMR spectra since the structurally different portions of the \nsamples may absorb the microwave energy at different fields due to the different \nanisotropy constants. High temperature annealing can change the multi-peak spectra of \nthese alloys. An alternative explanation [26] was made under the assumption that the \nsurface of these alloys can act differently than the bulk of the thin (<5000 A) film. \nTherefore surface and the bulk of the film can resonate at two different fields. \nOur attention here focuses on the strong antiferromagnetic coupling at the \ninterface of the Co/Gd multilayers and its effects on the FMR absorption. Furthermore we \ninvestigate the temperature dependence of the FMR signal because there are significant \nchanges in magnetic structure with varying temperature. Indeed, our FMR results are \nhighly sensitive to temperature changes. In fact we will see that the FMR signal for some \nmultilayer structures simply vanishes 50 K below the room temperature, while the FMR \nsignal for other structures can be observed down to very low temperatures.\nIn this paper we measure the FMR of Co/Gd multilayers, Gd/Co/Gd and \nTb/Co/Tb trilayers. Our main results demonstrate substantial shifts in the resonance field \ndue to the antiferromagnetic exchange interaction between the Co and Gd. We show that \nthe shift in the resonant field increases as the temperature is reduced and the linewidth of \nthe absorption peak increases as the temperature is decreased. We have also performed \ntheoretical calculations which explain this behavior and which are in reasonable \nquantitative agreement with the experimental data. \nPart of the motivation for this study is to produce device materials which operate \nat higher resonance frequencies. Such materials are of interest, for example, in high \nfrequency signal processing [32-34] where a large operational frequency is desired at a \nlow external magnetic field. The shift in the resonance field discussed above corresponds \napproximately to a 10-15 GHz boost in resonance frequency at low fields, and represents \na substantial increase of the FMR frequency at low fields. 4Experimental Details\nA variety of Gd/Co multilayers, Gd/Co/Gd and Tb/Co/Tb trilayers were prepared \nin a dc magnetron sputtering system at room temperature. The unbaked base pressure of \nthe UHV deposition chamber was 910Torr. Ultra high purity Argon gas pressure was 3\nmTorr during the deposition. The samples were deposited on Corning glass substrates \nand 50 Å Ag layers were used as buffer and protective cap layers in all samples. The \ndeposition thicknesses were monitored in situ by a quartz thickness gage which was\ncalibrated by a stylus profilometer. Magnetization measurements were taken using a \nSQUID magnetometer and a Magneto-Optical Kerr Effect (MOKE) system starting from \n300 K under a constant in-plane external magnetic field. The dynamic response of the \nsamples was measured by a 24 GHz Ferromagnetic Resonance (FMR) spectrometer \nbetween 30 and 300 K. Measurements were made using a cylindrical resonant cavity \nwhich operates in the TE 012mode. External magnetic field is parallel to the sample \nsurface. The peak to peak linewidth was measured from the differential FMR signal.\nTo begin the magnetic characterization of the samples, we measured the \nmagnetization as a function of temperature. A typical curve of the ferrimagnetic (Co 40 \nÅ /Gd 40 Å) 8multilayer is shown in Fig. 1. The results display significant thermal \nhysteresis, and the compensation temperature, T comp, is around 175 K as seen from the \nmidpoint of the \"bow-tie\" in Fig. 1. Details of the tunable thermal hysteresis can be found \nelsewhere [35-38].\nWe confirmed the thermal hysteresis results of Fig. 1 by measuring the MOKE \nsignal as a function of temperature as shown in Fig. 2. Unlike the complicated thermal \nhysteresis results in Fig. 1, this measurement gives a nearly square hysteresis loop which \nmight have applications in magnetic recording. The reason for the differences in Fig. 1 \nand Fig 2 is that the MOKE rotation angle is more sensitive to the 3d Co spins than the \nlocalized 4f Gd spins. This occurs because the MOKE signal originates from spin-orbit \ncoupling. Since there is LS coupling in Co while in Gd there no spin-orbit contribution \n(L=0), the standard MOKE experiment essentially provides element specific results (Co\nin this case).5Experimental Results\nTo understand the FMR spectra [39-40] of the Co/Gd multilayers, it is helpful to \ncompare the results for a pure Co layer with those of the multilayers. In Fig. (3a) we \npresent the FMR spectrum for a single 400 Å Co film at room temperature. There is a \nnarrow peak centered at 3.5 kOe indicating an effective magnetization of 1.25 kOe for the \npure Co film. This magnetization is found using the formula \n)M4 H(H fS res res . (1)\nHere the gyromagnetic ratio is = 2.92 GHz/kOe, H resis the field at which the FMR \nshows maximum absorption and f = 23.93 GHz is the operation frequency. \nThe FMR spectra for a (Co 40 Å /Gd 40 Å) 8multilayer are shown in Fig. (3b-d)\nfor different temperatures. Compared to the pure Co film, the position of the uniform \nFMR peak is lower by about 1 kOe at room temperature. The reason for this shift is the \nexchange coupling between the Co and Gd as we will discuss later on. As the temperature \nis lowered from the room temperature (Fig 3c and 3d), there are some small changes in \nthe resonance field and the linewidth increases substantially. Around 250 K the FMR \nsignal essentially vanishes.\nIt is not immediately clear what changes should occur as the temperature is \ndecreased. At lower temperatures, the effective Gd moment increases in its thermal \naveraged magnitude and this would reduce the magnetization of the entire structure as \nseen in Fig. 1. Using Eq (1), the reduced magnetization would shift the resonance field to \nhigher values. However, this can not be the only effect. As we have seen, the exchange \nfield produced by the Gd acts on the Co and provides an effective field which \ndramatically lowers the resonance field of the Co/Gd multilayer compared to the pure Co \nfilm. At lower temperatures, one would expect that this exchange field would be larger \nand that the resonance position for the multilayer would be further lowered. For some \nCo/Gd multilayer samples we observed multi-peak spectra where the multiple peaks \noccurred above the uniform resonance field however this was not the case for every \nmultilayer combination. These multi-peak spectra may have different sources as indicated 6in the introduction section. Indeed for example the curve in Fig. 3c can be better fitted by \ntwo Lorentzians numerically instead of one. \nThe in-plane angular dependence of the FMR field for the (Co 40 Å /Gd 40 Å) 8\nmultilayer at 24 GHz is shown in Fig. 4. The (Co 40 Å /Gd 40 Å) 8multilayer can be \nconsidered as polycrystalline since it does not show any easy or hard axis anisotropies. \nThis is consistent with the previous X-ray measurement where the (Co 40 Å /Gd 40 Å) 8\nmultilayer does not show any preferred orientations [36]. In most of our samples we did \nnot observe any significant in-plane anisotropy.\nPart of the motivation to proceed further with covering the Co with atomically \nthin Gd is to better focus on the coupling effects on the resonance properties. This is due \nto the fact that increasing Gd moment with decreasing temperature has the effect of \neliminating the FMR signal as shown in Fig. 3b-d in a small (~50 K) temperature interval \nand this 50 K temperature interval below room temperature is not sufficiently large to \nprocess the data. Our theoretical calculations lead us to use different thickness \ncombinations of the Co/Gd layers. Our goal was to reach large, temperature independent,\nshifts of the FMR field. As mentioned earlier, the motivation for the large FMR field \nshifts is to use this structure in microwave device applications in which the external \nmagnetic field requirement would be substantially lower [38-40]. We therefore created \ntrilayers of Co and Gd where the Co film (30-40 Å) is sandwiched between a few\nmonolayers (ML) of Gd (5-7 Å) and Tb (1-7 Å). To increase the signal we repeated the \ntrilayer structure 10 times putting a 50 Å thick nonmagnetic Ag spacer between the \ntrilayers. The experimental results are shown in Figs. 5-7.\nAs shown in Fig. 5, the ferromagnetic resonance field for a 400 Å thick single \nfilm of Co is around 3.5 kOe at room temperature where resonance frequency f = 23.93 \nGHz. When we sandwiched the 40 Å Co film with 5 Å (~1.5 monolayers) of Gd on both \nsides of the Co the FMR peak shifts about 0.6 kOe lower to 2.92 kOe. The FMR peak \nfield for (Gd 7 Å/Co 40 Å/Gd 7 Å/Ag 50 Å) 10is slightly lower still around 2.86 kOe.\nThis is in agreement with theoretical calculations which show that adding additional Gd \nlayers enhances the downward shift. The (Gd 5 Å/Co 30 Å/Gd 5 Å/Ag 50 Å) 10\nmultilayers have a FMR peak position near 2.9 kOe. As will be discussed later, the reason \nfor these field shifts is the strong antiferromagnetic coupling of the Co and Gd layers. 7Antiferromagnetic exchange coupling acts like an extra effective field that goes into the \nEq. (1) and, at a fixed frequency; this reduces the amount of the applied field which is \nrequired for the resonance.\nWhen we repeated the same experiment with atomically thin Tb films on the \noutside of the 40 Å thick Co film, we could only observe an FMR signal when the Tb \nlayer thickness was 1 Å (~0.3 monolayer). Thicker Tb films above 1 Å damped the FMR \nsignal so that no response was visible. The resonance field shift for the (Tb 1 Å/Co 40\nÅ/Tb 1 Å/Ag 50 Å) 10multilayer is on the order of 0.25 kOe from the Co peak.\nThe temperature dependence of the FMR field for the trilayer structures are\nshown in Fig. 6. The FMR field for Co is nearly constant over the entire temperature \nrange. In contrast, the multilayers show a slight decrease in the FMR fields as the \ntemperature is reduced. This may be due to an increased effective exchange field \nproduced by the Gd and Tb as the temperature is reduced. Again this feature is in \nreasonable agreement with the theoretical results.\nThe FMR linewidth as a function of temperature is shown in Fig. 7 for all the \nsamples. The linewidths generally increase monotonically as the temperatures decreases. \nHowever there is an abrupt discontinuity for the sample in Fig. 7 (top panel) below 150 K \nwhere the Gd thickness is 5 Å. This may be due to the non-uniformity of that particular \nsample. Although the resonance field values shown in Fig. 6 for the trilayer with 30 Å\nand 40 Å of Co when the Gd is 5 Å thick are similar at room temperature the gap widens \nat low temperatures and smaller fields for the resonance are observed when the Co is 30 \nÅ thick. This is meaningful owing to the forthcoming theoretical discussion since the \nproportional amount of the Gd is slightly larger when the Co layer is thinner. However a \nreverse effect is shown on the linewidth pattern shown in Fig. 7 as the temperature is \ndecreased. This time trilayer with the thicker Co deflected more as the temperature is \ndecreased. Furthermore temperature dependent linewidth for the (Gd 5 Å/Co 30 Å/Gd 5 \nÅ/Ag 50 Å) 10multilayer has smaller values than that of the bare 400 Å Co film which we \nuse as the reference film. Therefore correct comparison for the linewidth for the trilayer \ncontaining 30 Å thin Co film can be made most probably with a thinner reference Co \nfilm. Nevertheless the general features of trilayer containing 30 Å of Co with 5 Å of Gd \non each side are in good agreement with other films and also with the theoretical 8discussion such as the downward shift of the resonance field compared to bare Co film \nand the monotonic decrease (increase) of the resonance field (linewidth) on the large \ntemperature range. The FMR linewidth depends on a number of factors including the \ndamping mechanisms and the uniformity of the structure. It is well known that Tb can \nsignificantly increase the damping [41], and indeed we find that even 1 Å of Tb on both\nsurfaces of the Co film causes the linewidth to increase by a factor of two at room \ntemperature and a factor of three at low temperatures. However, Gd doping in transition \nmetals [42] does not produce such a dramatic increase. Nonetheless we find that as the \ntemperature is lowered there is a distinct increase in the linewidth even for the Gd/Co/Gd \nsamples. This is in agreement with the theoretical calculations and will be discussed in \nthe next section. \nTheoretical Calculations\nThere are a number of factors which need to be considered in order to understand \nhow the resonance field measured in the Co film is influenced by the thin Gd layers. \nFirst, if the Gd spins were rigidly coupled to the Co spins by exchange coupling, one \nmight expect either no shift in the resonance field or an upward shift in the resonance \nfield. The reason for this is that, in this case, the Gd moments would be strictly \nantiparallel to the Co moments and the exchange field of the Gd would not contribute to \nthe torque on the Co. Then the net effect of the Gd would be to reduce the net \nmagnetization of the structure. Then using the standard formula for resonance in a thin \nfilm, Eq (1), the required field for resonance would have to be increased. If the \ntemperature would be reduced, the net magnetization would be further reduced and the \nresonance field would have to be further increased. This is not what is seen \nexperimentally.\nExperimentally, one finds a downward shift in the resonance field and the shift is \nincreased as the temperature is reduced, therefore the Co and Gd moments can not be \nstrictly antiparallel. There can be several causes for this. First, the magnetization of the \nGd film is very different from that in a Co film. As is well known, the precession in a thin \nferromagnetic film is elliptical due to the demagnetizing fields. For example, in an 9isolated film the ratio of the x and y amplitudes of the magnetization when an external \nfield H is parallel to the surfaces of the film is given by\nHM41MM\nyoxo . (2)\nSince M is not the same in the Gd and Co, one would not get the same ellipticity in two \nmaterials and the moments in the two materials will not always be antiparallel. This \neventually produces an effective field from the Gd onto Co, thus reducing the value of the \nexternal field H necessary to create the ferromagnetic resonance condition. These \nconsiderations agree well with the experimental results.\nNow we want to consider how temperature influences the measured resonance \nfield. It is well known that the effective exchange coupling between the Co and Gd has \nenergy proportional to J I SGdSCo. The interface exchange constant J I is much larger than \nthe exchange constant within Gd. As a result, the Co magnetization can stabilize the Gd \nmoments right at the interface above the usual Curie temperature of the Gd. The \ncalculations show that when there is only one atomic layer of Gd the ratio of the thermal \naveraged Gd moment at 300 K to that at 0 K is about 0.53. In a bulk Gd sample, of \ncourse, this ratio would be zero since the Curie temperature is 293 K. This explains how \nthere can be a substantial shift in H reseven at room temperature. Furthermore, as the \ntemperature is reduced, the thermal averaged Gd moment increases, thereby increasing \nthe effective field and reducing the resonance field. Again, this behavior is in good \nagreement with the experimental results. \nWe can also qualitatively understand what happens as the number of Gd layers is \nincreased. Adding additional Gd layers also helps stabilize the Gd layer right next to the \nCo. For example, when there are two atomic layers of Gd the ratio of the thermal \naveraged Gd moment at 300 K to that at 0 K is about 0.6, compared to the 0.53 seen \nwhen there is only one atomic Gd layer. Thus one expects to see a larger shift in the \nresonance field. This will be evident in the theoretical results, but the experimental results \nare less clear on this point. 10Finally we would like to understand the behavior of the FMR linewidth as a \nfunction of temperature. The linewidth measures many different factors including \nrelaxation mechanisms and sample uniformity. In the calculations we have chosen the \nlinewidth of the two materials to be the same, so we are investigating the question of how \nthe linewidth is influenced by the coupling of the two materials. Without the Gd, all the \nCo moments can precess in a uniform motion; the effective field acting on each Co is the \nsame. When the Gd is added the outer Co spins see a different field than the interior ones \nand this nonuniformity leads to an enhancement of the linewidth. As the temperature is \nreduced the effective exchange field becomes larger, leading to a larger nonuniformity \nand a larger linewidth. \nBecause the interfacial coupling has an energy proportional to J I , one \nexpects the strength of the effective field acting on the Co at the interface is proportional \nto J I. As the temperature is reduced increases and so does the effective \nexchange field. This increase in effective field then reduces the resonance field at low \ntemperatures, in agreement with the experimental results. \nThe calculations are done using the Landau-Lifshitz-Gilbert (LLG) equations. \nThis technique can give information on both the static and dynamic properties of \nmagnetic multilayers. We write a set of LLG equations for each atomic layer in the \nmultilayer\n \n\n\ndtSdSS)HS (dtSdi\ni\nii ii \n(3) \nHere H iis the effective field acting on the magnetization, S i, in layer i. is the \ngyromagnetic ratio and is a dimensionless damping parameter. The total effective field \nis a sum of the external field, exchange fields, the oscillating field, and dipolar fields and \nis given by\n dti\n1i1i , i 1i1i , i o i h xˆ he S J S JzˆH H \n \n (4)11Here H ois the static magnetic field in the z direction, J i,i+1is the exchange coupling \nconstant between layer i and layer i+1. is the thermal averaged magnitude of the \nspin in layer i. The oscillating field h has an angular frequency and is applied in the x\ndirection. We assume that the main contribution to the dipole field h dis the \ndemagnetizing field of a thin layer acting on itself; thus, yˆM4 hy d , where y is the \ndirection perpendicular to the film surface. This approximation works well in the thin \nfilm or long wavelength limit and is appropriate for FMR calculations. \nOne picks an initial spin configuration and integrates the coupled set of equations \nforward in time using a differential equation solver such as the 4thorder Runge-Kutta \nmethod. Thus one obtains S x(t), S y(t) and S z(t). In order to get sensible results appropriate \nto FMR one has to wait for the system to come into dynamic equilibrium with the driving \nfield. The power absorbed in a period T is then given by\nT\n0y\nT1dtdt) t ( dM\n) t ( h P (5)\nand the integral is done numerically. \nThe temperature dependence is included by renormalizing the magnitude of the \nmagnetization using the Brillouin function. For example the magnetic moment for the Co \nis given by = gBSCoBS(gBSCoHeff/kT), and a similar equation holds for the \nmagnetic moment in the Gd. Here is the thermal averaged magnitude of the \nmagnetic moment in Co, and B s(x) is the Brillouin function given by\n.S2xcothS21\nS2x ) 1 S2 (cothS2) 1S2 () x ( Bs \n\n\n\n\n\n\n\n (6)\nThe effective field, H eff, is the same field that acts in Eq (4), but here the exchange field \nthat has the dominant contribution. In fact, the exchange constants within the Gd and Co \nare set by requiring that the bulk Curie temperatures for the two materials are given \ncorrectly. Initially each step of the time iteration also involves a renormalization of the 12magnetic moments for the Co and Gd, but after about 100,000 time steps the magnitudes \nno longer change substantially. \nThe results of the calculation are present in Figs. 8-9. The key parameters for the \ncalculation are given by M Gd(T=0) = 2.06 kG, M Co = 1.46 kG, J Co= 9,680 kG, \nJGd= 198 kG, and J I= -1100 kG. We assume that g = 2 in both materials and S Gd= 3.5 \nand S Co= 0.86. The Fig. 8 shows the calculated FMR absorption curves for a 40 Å Co \nfilm and for the same film with 1 or 2 ML of Gd on each of the outside surfaces of the \nCo. The results are quite close to those found experimentally. The films with the Gd on \nthe outside exhibit a reduced resonance field and are smaller and broader than the FMR \ncurve for the Co alone. The shift between the pure Co resonance field and the system \nwith 1 ML of Gd on the outside is 0.59 kOe, in good agreement with the experiment. This \nshift has been used to obtain the value for the interface coupling constant J I. It is \nreassuring that this value is, in fact, quite close to values based on magnetization results. \nThe theoretical results for how the linewidth and resonance field depend on \ntemperature are shown in Fig. 9. We find that as T is reduced the resonance field \ndecreases slightly, about 0.5 kOe over the entire temperature range. Again, this is in \nreasonable agreement with the experimental results, and indicates the strengthening of the \nexchange field as the temperature is reduced. The linewidth also changes with \ntemperature, increasing substantially as the temperature is reduced. Again the values are \nin reasonable agreement with the experimental results and with the general discussion \nabove. \nSummary and Conclusions\nWe have performed static and dynamic measurements and also self consistent \ntheoretical calculations for Co/Gd and Co/Tb films. Resonance field for the Co/Gd \nmultilayers are substantially reduced compared to the pure Co film. This shift is due to \nthe antiferromagnetic coupling between the Co and Gd. As the temperature is decreased \nfrom room temperature the effective Gd magnetic moment increases and this reduces the \ntotal magnetic moment of the Co/Gd multilayer system. We have observed the \nbroadening of the FMR linewidth for Co/Gd trilayer and multilayers as the temperature \nreduces from room temperature and the Gd becomes magnetic. FMR signal vanishes 13approximately 50 K below room temperature for Co/Gd multilayers when the Gd layer is \nthicker than a few atomic layers. Therefore Gd/Co/Gd trilayers where the thickness of the \nGd is 1-2 monolayers are better choice to study the temperature dependence of the \nantiferromagnetic coupling dynamically. These Gd doped trilayers also showed a \nsubstantial downward shift of the FMR field on the order of 600 Oe at 24 GHz compared \nto that of pure Co film at room temperature. Likewise multilayers of Co/Gd with thicker \nGd layer shows even greater downward shifts on the order of 1 kOe compared to pure Co \npeak at room temperature. Our theoretical calculations relate this substantial downward \nshift of the FMR field to the effective field from the Gd onto the Co due to the \nantiferromagnetic exchange coupling between them. As the temperature decreases \nresonance peak position at room temperature slightly decreases and the associated \nlinewidth increases gradually at the same time. When the same experiments are repeated \nfor the Tb/Co/Tb trilayers the FMR signal is only observable if the Tb layer thickness is \non the order of 1 Å or less. Higher thicknesses over 1 Å of Tb damp the FMR signal. The \nresults of Tb/Co/Tb trilayers are supportive and comparable with the arguments used for \nthe case of Gd/Co/Gd trilayers. The experimental results are in good agreement with the \ntheoretical calculations.\nAcknowledgements\nThe work at UCCS was supported by DOD Grant # W911NF-04-1-0247. The \nwork at UTA is supported by a grant (No. Y-1215) from The Welch Foundation. 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Thermal hysteresis with increasing external magnetic fields for the (Co 40 Å/Gd \n40 Å) 8multilayer.\nFig. 2. An example of MOKE thermal hysteresis for (Co 40 Å /Gd 40 Å) 8multilayer. \nCooling and heating curves are measured under a constant 150 Oe inplane magnetic field \nis applied. Drifts on either side of the hysteresis are artifacts from the electronics of the \nsetup. \nFig. 3. FMR spectrum of a) 400 Å Co film b-d) (Co 40 Å/Gd 40 Å) 8multilayer as a \nfunction of temperature.\nFig. 4. In plane angular dependence of FMR field for the (Co 40 Å /Gd 40 Å) 8multilayer \nat room temperature.\nFig. 5 . Room temperature FMR spectra for Co 400 Å (full squares), (Gd 5 Å/Co 40\nÅ/Gd 5 Å/Ag 50 Å) 10(full circles), (Gd 7 Å/Co 40 Å/Gd 7 Å/Ag 50 Å) 10(empty circles),\n(Gd 5 Å/Co 30 Å/Gd 5 Å/Ag 50 Å) 10(full triangles) and (Tb 1 Å/Co 40 Å/Tb 1 Å/Ag 50\nÅ)10(empty squares) films.\nFig. 6. FMR field as a function of temperature for various films. For the multilayer films \nthere is a gradual decrease in the FMR field as the temperature is reduced. \nFig. 7. Linewidth of the FMR field as a function of temperature for various films. The \nlinewidth of the multilayer films generally increases as the temperature is reduced. \nFig. 8. Theoretical results for FMR absorption for a 40 Å Co film sandwiched between 0, \n1 or 2 monolayers of Gd on each side. \nFig. 9 . Theoretical results showing how the resonance field and the linewidth changes as \na function of temperature for a 40 Å Co film and for a 40 Å Co film with one or two ML \nof Gd on each side. 17-0.00020.00000.00020.00040.0006\n0.00000.00020.00040.0006\n0 50 100 150 200 250 3000.00000.00020.00040.0006Magnetic Moment [emu]100 Oe(Co 40 Å/Gd 40 Å)8\n200 Oe\nTemperature [K]400 Oe\nFig. 1. Thermal hysteresis with increasing external magnetic fields for the (Co 40 Å/Gd \n40 Å) 8multilayer.1850 100 150 200 250 300 3500.000.020.040.060.080.100.120.14 KerrAngle []\nTemperature [K]150 Oe(Co 40 Å/Gd 40 Å)8\nFig. 2. An example of MOKE thermal hysteresis for (Co 40 Å /Gd 40 Å) 8multilayer. \nCooling and heating curves are measured under a constant 150 Oe inplane magnetic field \nis applied. Drifts on either side of the hysteresis are artifacts from the electronics of the \nsetup.190 1 2 3 4 5-0.00050.00000.00050.00100.00150.00200.00250.0030\nMagnetic Field [kOe]Diode Voltage [V]a) Co 400 Å \n295K\n0.00000.00060.0012\n0.00000.00060.0012\n0 1 2 3 4 5-0.00060.00000.00060.0012295K b)Diode Voltage [V]\n270Kc)\nMagnetic Field [kOe]255K(Co 40 Å/Gd 40 Å)8\nd)\nFig. 3. FMR spectrum of a) 400 Å Co film b-d) (Co 40 Å/Gd 40 Å) 8multilayer as a \nfunction of temperature.20Fig. 4. In plane angular dependence of FMR field for the (Co 40 Å /Gd 40 Å) 8multilayer \nat room temperature.-90 -60 -30 0 30 60 902200230024002500260027002800\n(Co 40 Å/Gd 40 Å)8In Plane Angular DependenceFMR Field [Oe]\nAngle [degree]211 2 3 4 5-0.00050.00000.00050.00100.00150.00200.00250.0030\n(Gd 5Å/Co 40Å/Gd 5Å/Ag 50Å)10(Gd 7 Å/Co40 Å/Gd 7 Å/Ag 50 Å)10Diode Voltage [V]\nMagnetic Field [kOe]Co 400 Å\n1 2 3 4 5-0.00020.00000.00020.00040.00060.00080.0010\n(Gd 5Å/Co 30Å/Gd 5Å/Ag 50Å)10(Tb 1Å/Co 40Å/Tb 1Å/Ag 50Å)10\nMagnetic Field [kOe]Diode Voltage [V]\nFig. 5 . Room temperature FMR spectra for Co 400 Å (full squares), (Gd 5 Å/Co 40\nÅ/Gd 5 Å/Ag 50 Å) 10(full circles), (Gd 7 Å/Co 40 Å/Gd 7 Å/Ag 50 Å) 10(empty circles), \n(Gd 5 Å/Co 30 Å/Gd 5 Å/Ag 50 Å) 10(full triangles) and (Tb 1 Å/Co 40 Å/Tb 1 Å/Ag 50\nÅ)10(empty squares) films.220 50 100 150 200 250 30028003000320034003600\nTemperature [K]FMR Field [Oe]\n(Gd 5Å/Co 30Å/Gd 5Å/Ag 50Å)10(Tb 1Å/Co 40Å/Tb 1Å/Ag 50Å)100 50 100 150 200 250 30028003000320034003600FMR Field [Oe]\nTemperature [K](Gd 5Å/Co 40Å/Gd 5Å/Ag 50Å)10\n(Gd 7 Å/Co 40 Å/Gd 7 Å/Ag 50 Å)10Co 400Å\nFig. 6. FMR field as a function of temperature for various films. For the multilayer films \nthere is a gradual decrease in the FMR field as the temperature is reduced. 230 50 100 150 200 250 300200400600800100012001400\nTemperature [K]Linewidth [Oe] (Gd 5Å/Co 40Å/Gd 5Å/Ag 50Å)10\n(Gd 7 Å/Co 40 Å/Gd 7 Å/Ag 50 Å)10\nCo 400Å\n0 50 100 150 200 250 300200400600800100012001400\nTemperature [K]Linewidth [Oe]\n(Gd 5Å/Co 30Å/Gd 5Å/Ag 50Å)10(Tb 1Å/Co 40Å/Tb 1Å/Ag 50Å)10\nFig. 7. Linewidth of the FMR field as a function of temperature for various films. The \nlinewidth of the multilayer films generally increases as the temperature is reduced. 240 1 2 3 4012345678\nGd / Co / Gd\n2 ML Gd1 ML GdNo GdAbsorption [arb units]\nMagnetic Field [kOe]\nFig. 8. Theoretical results for FMR absorption for a 40 Å Co film sandwiched between 0, \n1 or 2 monolayers of Gd on each side. 2550 100 150 200 250 3000.00.40.81.22 ML Gd\n2 ML GdH [kOe]\nTemperature [K]1 ML Gd1.52.02.53.03.5\nNo Gd\n1 ML GdHres [kOe]Gd/Co/Gd\nFig. 9 . Theoretical results showing how the resonance field and the linewidth changes as \na function of temperature for a 40 Å Co film and for a 40 Å Co film with one or two ML \nof Gd on each side. " }, { "title": "2009.14095v2.Spin_transfer_torque_in_Mn__3_Ga_based_ferrimagnetic_tunnel_junctions_from_first_principles.pdf", "content": "Spin transfer torque in Mn 3Ga-based ferrimagnetic tunnel junctions from \frst\nprinciples\nMaria Stamenova,1,\u0003Plamen Stamenov,1Farzad Mahfouzi,2Quilong Sun,2Nicholas Kioussis,2and Stefano Sanvito1\n1School of Physics and CRANN, Trinity College, Dublin 2, Ireland\n2Department of Physics and Astronomy, California State University, Northridge, California 91330, USA\nWe report on \frst-principles calculations of spin-transfer torque (STT) in epitaxial magnetic\ntunnel junctions (MTJs) based on ferrimagnetic tetragonal Mn 3Ga electrodes, both as analyzer in\nan Fe/MgO stack, and also in an analogous stack with a second Mn 3Ga electrode (instead of Fe)\nas polarizer. Solving the ballistic transport problem (NEGF + DFT) for the nonequilibrium spin\ndensity in a scattering region extended to over 7.6 nm into the Mn 3Ga electrode, we \fnd long-range\nspatial oscillations of the STT decaying on a length scale of a few tens of angstroms, both in the\nlinear response regime and for \fnite bias. The oscillatory behavior of the STT in Mn 3Ga is robust\nagainst variations in the stack geometry (e.g., the barrier thickness and the interface spacing) and\nthe applied bias voltage, which may a\u000bect the phase and the amplitude of the spatial oscillation,\nbut the wave number is only responsive to variations in the longitudinal lattice constant of Mn 3Ga\n(for \fxed in-plane geometry) without being commensurate with the lattice. Our interpretation of\nthe long-range STT oscillations is based on the bulk electronic structure of Mn 3Ga, taking also into\naccount the spin-\fltering properties of the MgO barrier. Comparison to a fully Mn 3Ga-based stack\nshows similar STT oscillations, but a signi\fcant enhancement of both the TMR e\u000bect at the Fermi\nlevel and the STT at the interface, due to resonant tunneling for the mirror-symmetric junction\nwith thinner barrier (three monoatomic layers). From the calculated energy dependence of the\nspin-polarized transmissions at 0 V, we anticipate asymmetric or symmetric TMR as a function of\nthe applied bias voltage for the Fe-based and the all-Mn 3Ga stacks, respectively, which also both\nexhibit a sign change below 1 V. In the latter (symmetric) case we expect a TMR peak at zero,\nwhich is larger for the thinner barriers because of a spin-polarized resonant tunneling contribution.\nI. INTRODUCTION\nThe Fe/MgO-based magnetic tunnel junctions (MTJs)\nare the backbone of modern spintronics and the ide-\nalised crystalline Fe(100)/MgO/Fe MTJ is the theoret-\nical proxy system for the locally structurally-coherent\nCoFeB/MgO/CoFeB MTJs. It is the former structure,\nwhere the spin-\fltering tunneling-magnetoresistance\n(TMR) e\u000bect was predicted theoretically[1] some 20 years\nago and soon after demonstrated experimentally[2, 3].\nIn essence, the TMR e\u000bect, which exploits the di\u000ber-\nence in resistivity between parallelly and anti-parallelly\naligned magnetic layers sandwiching an insulator, in\nthese MTJs, is due the special symmetry-driven spin-\n\fltering of the Fe(100)/MgO composite. In a few atomic\nmono-layers (MLs) of MgO, the transmission of the mi-\nnority spin carriers emanating from Fe(100) is almost\ncompletely eliminated and theoretically the TMR e\u000bect,\nin an ideal Fe/MgO/Fe MTJ, can reach several thou-\nsands of percent[4{6]. The TMR e\u000bect combined with\nthe possibility of switching or exciting precession in the\nfree magnetic layer by current, due to the spin-transfer\ntorque (STT), makes the Fe/MgO MTJs suitable func-\ntional components in magnetic memory elements or high\nfrequency generators [7{9]. Those applications for MTJs\nrequire the optimisation of certain magnetic properties.\nThe combination of high spin-polarisation, low Gilbert\ndamping and large anisotropy is highly desirable for\n\u0003Contact email address: stamenom@tcd.iethe scalability of spintronic applications like high-density\nspin-transfer torque memory (STT-MRAM) or spintronic\noscillators and detectors in the THz range. Mn-Ga alloys\nhave been studied for magnetisation dynamics applica-\ntions because of their relatively high anisotropy, for a\nlow-Zmaterial, and indeed found to exhibit low Gilbert\ndamping coe\u000ecients [10] as well. In addition, the tetrago-\nnal Heusler DO 22form of Mn 3Ga exhibits a low-moment,\nferrimagnetic order and a high spin polarization[11]. A\nfurther reason for studying this system, in particular, is\nits similarity with the prototype fully-compensated half-\nmetallic Mn xRu1\u0000xGa compound, a very topical mate-\nrial exhibiting high spin-polarization, low damping and\nstrong perpendicular anisotropy but as of site-disorder\n{ rather di\u000ecult to simulate. For instance, recently\ncurrent-induced switching with interfacial spin{orbit\ntorque has been demonstrated for ultrathin \flms of the\nlatter Heusler compound, interfaced with Pt[12]. Simi-\nlarly, the DO 22structure of Mn 3Ga is ferrimagnetic, also\nfeaturing two antiferromagnetically-coupled Mn sublat-\ntices { one formed of Mn atoms, labeled as Mn Iin the\n2bWycko\u000b positions, e.g. (0 ;0;1=2), forming Mn I-Ga\nplanes; and the other sublattice of Mn IIatoms in the\n4dpositions, like (0 ;1=2;1=4), forming Mn II-Mn IIplanes\n[see Fig. 1(a)].\nSTT-driven thin Mn 3Ga free-layers, as parts of MTJ\nstacks, are interesting on their own for the construction\nof STT-driven oscillators, because of their comparatively\nlarge e\u000bective anisotropy and corresponding ferromag-\nnetic resonance frequencies of even the in-phase modes.\nThe observed resonance frequencies of stand-alone \flmsarXiv:2009.14095v2 [cond-mat.mes-hall] 7 Jan 20222\nvary from about 0.17 THz to above 0.35 THz, for thick-\nnesses in the range 4 - 15 nm, respectively, with the\nemission bandwidth decreasing monotonically as a func-\ntion of increasing thickness from above 40 GHz to below\n25 GHz.[13, 14] While coherent low-THz range emission\nis still to be demonstrated from this type of moderate\nspin polarisation ( P\u001845%) electrode under current ex-\ncitation, within nano-pillar structures, the nature and\nmagnitude of the STT and theoretical maximal e\u000ecien-\ncies, with which the in-phase and out-of-phase resonance\nmodes can be excited, remain open questions.\nWe consider mesoscopic junctions in which the Mn 3Ga\n\flm is grown on top of the Fe(100)/MgO stack in the\nlongitudinal zdirection [see Fig. 1(a)], while there are\nperiodic boundary conditions in the x-yplane. The\nopen-boundary conditions are applied at the two ends of\nthe scattering region (SR) of the stack, depicted in Fig.\n1(a), via the non-equilibrium Green's function (NEGF)\nmethod, as implemented in the Smeagol code.[15] In\npractice, there are two semi-in\fnite crystalline leads of\nbcc Fe and DO 22tetragonal Mn 3Ga attached to the left\nand to the right end of the SR, respectively. Thus con-\nstructed, the stack is laterally commensurate to the lat-\ntice of bcc Fe with its lattice constant aFe= 2:866\u0017A,\nwhich is, rotated by 45\u000ewith respect to the cubic MgO\nlattice. Hence, we consider the tetragonal Mn 3Ga cast\ninto the Fe-dictated in-plane lattice constants a=b=p\n2aFe= 4:053\u0017A in the lateral directions. This leads\nto a +3:7 % tensile bi-axial strain in Mn 3Ga (from the\nexperimental lateral lattice constant of 3.91 \u0017A[11]) and\na\u00003:8 % compressive bi-axial strain in the MgO. Ge-\nometry relaxations, at the level of the local spin-density\napproximation (LSDA) to the exchange-correlation func-\ntional [16] and constrained to the longitudinal direction\nonly, have resulted in a signi\fcant ( \u001815 %) compression\nof the Mn 3Ga slab with respect to the experimental value\ncexp= 7:1\u0017A. [11] This, in turn, leads to unrealistically\nsmall values of the local magnetic moments. These short-\ncomings of the LSDA geometry for Heusler alloys are\nknown and typically the GGA (PBE) are used.[11, 17]\nHere we are, however, limited to the LSDA for our trans-\nport calculations of multi-layered junctions with non-\ncollinear spin alignments. As a compromise, the value of\nchas been chosen such that, within the LSDA, the cal-\nculated atomically-projected local spins (as per Mulliken\npopulation analysis) are within the experimental ranges\nfor the spins of the Mn atoms, obtained by di\u000berent mea-\nsuring techniques (XMCD or neutron di\u000braction),[11]\nnamelysMnI2(3:2;3:7)\u0016BandsMnII2(\u00002:1;\u00002:7)\u0016B.\nNote that Mn Iis in the planes with Ga, while Mn IIforms\nMnII-Mn IIplanes perpendicular to the direction of the\ntransport [see Fig. 1(b)]. Furthermore, our strategy has\nbeen, instead of limiting ourselves to a single albeit ac-\ncurate geometry optimisation beyond LSDA, to explore\na range of structural parameters in the z-direction, i.e.\nvalues ofcandd{ the distance at the Mn 3Ga-MgO in-\nterface (the other side of the junction, the Fe/MgO, is as\nin Ref. 5). For our main representative structure we havechosenc= 6:6\u0017A, which results in magnetic moments for\nthe two types of Mn atoms in the experimental value\nranges [see Fig. 1(c) for the atomically-resolved (Mul-\nliken population analysis) spin values]. We also consider\nthe two di\u000berent possible Mn 3Ga terminations on the\n(001) interface with MgO, but the representative case\n(used as reference throughout the paper, unless stated\notherwise), depicted in Fig. 1(a), features a Mn I-Ga-\nplane termination (shown in the zoomed-in inset). In\nthis termination Mn Iatoms are placed on top of the\noxygen atoms. In our representative case the interfacial\nMn-O spacing d= 2:215\u0017A corresponds approximately to\nhalf lattice constant of bulk \u000b-MnO.[18] Note, that this\nvalue of interlayer distance at the interface agrees well\nwith the value of 2.265 \u0017A that we found using VASP cal-\nculations with the PBE exchange correlation functional\nfor the Mn 3Ga/MgO slab with Mn I-Ga terminated inter-\nface. [19]\nFIG. 1. (Color online) (a) Schematic of the Fe/MgO/Mn 3Ga\njunction with a close-up of the Mn 3Ga/MgO interface (in\nthe green rectangle), depicting the main geometry investi-\ngated with Mn I-Ga termination. (b) The tetragonal unit\ncell of Mn 3Ga showing the directions of the spins of the\nanti-ferromagnetically (AFM) coupled Mn Iand Mn IIsublat-\ntices. (c) The magnitudes of the local spins at Mn Iand Mn II\nsites in the junction starting from the MgO interface, as de-\npicted in the schematic above. (d,e) Corresponding calculated\natomically-resolved in-plane STTk, as described by Eq. (2)\nfor the Mn Iand Mn IIsublattices, respectively, for the case in\nwhich the moments on the Mn are along z, while the moment\nof the Fe lead is along x. The coe\u000ecient \u0011\u0011e=(\u0016BA), where\nAis the transverse area of the junction, is used throughout\nthe paper to convert our computed STT (or STTk) to units\n\n\u00001m\u00002. (f,g) de\fne what we refer as the anti-parallel (AP)\nand parallel (P) spin state of the junction, respectively.\nAs far as the magnetic state is concerned, there are\ntwo possible collinear-spin con\fgurations of the junction.\nWe disregard the spin-orbit interaction, hence there is\nno coupling between the spatial orientations of the spins3\nand the geometry of the junction. For de\fniteness, and\nin view of the expected perpendicular anisotropy in these\njunctions [20], our quantization axis is oriented along the\ndirection of the transport z, see Fig. 1(a,b). We de\fne\ntwo possible collinear states of the junction: a parallel\n(P) state in which the net moment of Mn 3Ga is parallel\nto Fe, and AP when it is anti-parallel to that of Fe. Note\nthat, as the net spin is parallel to Mn IIand anti-parallel\nto Mn Ithis means, for our preferred Mn I-Ga termina-\ntion, that the Mn Ispin at the interface opposes the Fe\nspin in the P state and is parallel to it in the AP state\n[Fig. 1(f,g)]. For our steady-state transport calculations\nof STT, the typical non-collinear state we consider is with\nthe Fe moment rotated along the x-axis so that there is\na 90\u000emisalignment between spins in the two electrodes.\nThe paper is then organised as follows. In the next\nSection II we outline the formalism used for the calcu-\nlation of the linear response ST-torkance (STTk) and\nthen we present some technical insights of the calculated\natomically-resolved STTk for the Mn 3Ga layer in the self-\nconsistently described scattering region. In Section III,\nwe focus on the e\u000bects of the interface and the barrier\ngeometries on the in-plane STTk. In Section IV we ex-\namine the e\u000bect of the bulk lattice parameters (in par-\nticular, the long axis lattice constant c) of Mn 3Ga and\nelucidate the origin of the observed long-range spatial\noscillation of the STTk. In Section V we discuss the\nspin-polarised transmission at equilibrium, decomposed\nover the transverse two-dimensional Brillouin zone (2D\nBZ) or as a function of energy. Based on that, we then\nevaluate the TMR near equilibrium and draw predictions\nfor its asymmetric bias dependence in the range -1 V to\n1 V. In Section VI we look at a modi\fed junction, where\nwe replace the Fe lead with Mn 3Ga, hence constructing\na mirror-symmetric all-ferrimagnetic MTJ (FiMTJ), for\nwhich we compare analogously-calculated STT and TMR\nproperties to the Fe-based junction. Then we conclude\nwith a discussion and comparison to existing experimen-\ntal data on related MnGa-based tunnel junctions. [21]\nII. LINEAR RESPONSE STT\nWe calculate the linear response STT using the method\nfrom Ref. [22], which is described in greater detail in Ref.\n[6] and implemented in the Smeagol code[15]. In this\nregime a small bias voltage, \u000eVb, is applied across the\njunction. Then the transport part of the density matrix\ninduced by \u000eVbis de\fned as\n\u001atr(\u000eVb)\u0019@\u001a(Vb)\n@Vb\f\f\f\f\nVb=0\u000eVb: (1)\nand the so-called spin-transfer torkance (STTk) acting\non an atomic site n,\u001cn, is de\fned as\n\u001cn=\u000eTn\n\u000eVb\u00192X\n\u000b2nN\fX\n\f=1Re\u0014@\u001atr;\f\u000b\n@Vb\u0002H[\u001acond]\u000b\f\u0015\n;(2)where\u000b;\freplace the full set of quantum numbers index-\ning all the orbitals in the local basis set, as implemented\nin the Siesta code [23], H\u000b\f=H0;\u000b\f1+H\u000b\f\u0001\u001bis the\nLSDA Kohn-Sham Hamiltonian of the SR, with f1;\u001bg\n{ the full set of Pauli matrices, including identity. The\ndensity matrix from Eq. (1) is decomposed similarly:\n\u001a\u000b\f=\u001a0;\u000b\f1+\u001a\u000b\f\u0001\u001b. Note that the integration over\nthe transverse 2D BZ is implicit in Eq. (2). The re-\nquired derivative of the density matrix with respect to\nthe bias voltage, within NEGF, is calculated in the lin-\near response regime for each k-point k\u0011k?2\nBZin\nthe 2D BZ surface:\n@\u001atr;k\n@Vb=1\n4\u0019Gk(EF)[\u0000k;L(EF)\u0000\u0000k;R(EF)]Gy\nk(EF)\f\f\f\nVb=0:\n(3)\nIt is assumed that the Green's function of the scatter-\ning region, G(E), and the \u0000 L(R)(E) matrices, which cou-\nple it to the left(right) lead, are slowly-varying functions\naround the Fermi level, EF. It is also assumed that the\nbias drop in the junction is symmetric, i.e. the chemi-\ncal potentials in the left (right) lead shift by \u0006Vb=2 with\nrespect to the equilibrium.\nThe calculated atomically-resolved in-plane STTk, \u001cx\nn,\nin Mn 3Ga for our representative Fe(100)/MgO/Mn 3Ga\njunction with 45 MLs of Mn 3Ga in the SR and Mn I-Ga\nterminated interface, is shown in Fig. 1d,e for the two\nMn sublattices, starting from the MgO interface. The\nnet magnetic moment[24] in Mn 3Ga is oriented along \u0000z\n(Mn Iat the interface points in z-direction), while the\nmoment of Fe is along x. In contrast to the anticipated\nexponential decay of the STT from the insulating barrier\nin conventional MTJs, here \u001cx\nn(z) is showing a slow os-\ncillatory decay for both magnetic sublattices over many\nMLs of Mn 3Ga. The atomically-resolved in-plane STTk\nin Fig. 1 (d,e) is \ftted to a beating sine wave, which\nis one of many possible \ftting functions. A decaying\nsine wave is also a possibility and the size of the data\nset does not allow to discriminate between those \ftting\nfunctions. However, we observe that an exponential de-\ncay is apparently not as good a \ft to the data (the green\ndashed line), in line with the, so called, spatial preces-\nsion behaviour of STT, identi\fed from basic scattering\ntheory principles in a free electron model in Ref. [25].\nWe are restricted by the feasibility of the calculation to\nfurther extend the self-consistent SR for a more mean-\ningful (quantitatively) non-linear \ft. The \ftted carrier\nwavenumber (2 \u0019=\u0015) value is about 0.38 \u0017A\u00001, which cor-\nresponds to a period of oscillation of some 10 MLs of\nMn3Ga (or 5 layers of each sublattice), without it being\nexactly commensurate with the Mn 3Ga lattice spacing.\nThis is completely di\u000berent from the oscillations in L10\nferromagnets, which remain commensurate to the lattice.\n[26] The oscillations we observe here have identical peri-\nods and are approximately in anti-phase (staggered) on\nthe two magnetic sublattices. The in-plane STTk on the\nMnII-sublattice is nearly twice as large as that on the\nMnIsublattice.\nIn Fig. 2 we demonstrate the dependence of the STTk4\noscillations, which we just described, on the dispersion\nof the evaluated at the Fermi-level derivative in Eq. (2),\nand their decomposition over the 4s, 4p and 3d atomic\norbitals. We introduce parameter \u000eEto de\fne an en-\nergy range EF\u0006\u000eE=2 around the Fermi level in which\nthe right-hand side of Eq. (3) is averaged. It can be seen\nthat, allowing more energy channels in the average for the\nspin-density's derivative with respect to bias, leads to a\nfaster decay of the STT into the Mn 3Ga, without a\u000bect-\ning much its amplitude close to the interface. Increasing\n\u000eEalso smoothens the sharp features of the STTk at the\ninterface, which are due to resonant tunneling between\ninterface states and tend to be suppressed with larger\nMgO barrier thickness (see Fig. 4). The e\u000bect is simi-\nlar for both sublattices. A small di\u000berence between the\nsublattices appears in the atomic orbital decomposition\n[Fig. 2(c,d)], where the relative contribution of the 4s and\n4p orbitals is larger for the Mn Isublattice. However, in\nboth cases the 3dz2character dominates the calculated\natomically-resolved STTk.\nAs long as the adjacent Mn atoms from di\u000berent sub-\nlattices have opposite sign spins, as they do, such stag-\ngered relation of the sublattice STTk is expected to result\nin a net STTk on the net ferrimagnetic moment, domi-\nnated by the Mn IIsublattice. As the atomically-resolved\nin-plane STTk becomes close to zero in a few MLs and\nchanges sign, there will also be torques acting against\nthe ferromagnetic exchange interaction in each sublat-\ntice. Torques against the AFM coupling are more subtle\nbecause of the apparently stable long-range anti-phase\nrelation between in-plane STTk in the two sublattices,\nhence the two sublattices' spins are expected to rotate in\nthe same direction. The direction of the net torque and\nits magnitude is thus expected to depend on the number\nof MLs of Mn 3Ga in the stack. We will demonstrate this\nin Section VI (see Fig. 15). Before we focus on the elec-\ntronic structure mechanism giving rise to this long-range\noscillation of the in-plane STTk, we will investigate its\ndependence on the structural parameters of the Mn 3Ga\nanalyser layer and its iterface to MgO.\nIII. EFFECT OF THE Mn 3Ga-MgO INTERFACE\nAND THE MgO BARRIER THICKNESS\nWe \fnd that the long-range in-plane torkance oscil-\nlation is robust and is not limited to the geometry of\nthe representative stack in Fig. 1. Increasing the barrier\nthickness by 2 MgO MLs (from 3 to 5 MLs) results in\na decrease of the in-plane STTk by an order of magni-\ntude (see Fig. 3). However, there is barely any e\u000bect on\nthe phase of the oscillation or its carrier wavenumber in\nthe \frst one or two periods. Arguably, there is a some-\nwhat longer-ranged decay of the STTk oscillation in the\ncase of a thicker barrier but we do not aim to quantify\nthis e\u000bect, likely related to the enhanced directional and\nspin \fltering. The decay of the total STTk, arising from\nthe integration in the 2D BZ of Eq. (3), is a result of\nFIG. 2. (Color online) (a,b) In-plane STTk calculated as per\nEq. (2), but averaging the Fermi-level expression in Eq. (3)\nover di\u000berent \fnite energy ranges EF\u0006\u000eE=2 around the Fermi\nlevel and (c,d) the representative case for \u000eE= 20 meV de-\ncomposed over orbital character, for the two Mn sublattices,\nas indicated in the panels. Note that 4 sand 4pcomponents\nare scaled by a factor of 10.\nthe self-cancellation from the superposition of sinewave-\nlike oscillations from a wide range of di\u000berent kchannels\n[25]. If we only consider the STTk at the \u0000-point, i.e.\n~\u001c\u0011\u001c[k?= (0;0)], we \fnd a perfect sine wave spatial\noscillation [Fig. 3(c,d)]. Furthermore, at the \u0000 point the\noscillation frequency and phase show no dependence on\nthe thickness of the barrier. The \ftted wavenumber in\nboth cases is \u0014= (0:301\u00060:001)\u0017A\u00001. Note that the\nvalues of the STTk at the \u0000 point are much higher than\nthe integral values, which are normalised to the 2D BZ\narea. As we will see later (e.g. in Fig.8) this is because\nthe \u0000 point has the dominant contribution to the STTk,\nas well as to the transmission in the junction.\nIn the case of Mn II-Mn IIinterface termination (realised\nby removing the \frst Mn I-Ga layer and restoring the\nsame interface spacing to MgO), we \fnd a very similar\nlong-range oscillation (see Fig. 3a,b, where this is com-\npared to the reference geometry case). The termination\nappears to a\u000bect signi\fcantly the STTk in the \frst lay-\ners from the MgO { the interface e\u000bects are stronger in\nthe Mn II-Mn IItermination case. Deeper into the Mn 3Ga\nlayer, the di\u000berence amounts mainly to a phase shift of\nthe oscillation. The period appears very similar, but the\namplitude is somewhat reduced compared to that of the\nMnI-Ga termination case and hence the net torque is also\nexpected to be reduced in the Mn IIterminated junction\n(see also Fig. 15). Again, we do not aim for a quanti-\ntative analysis of the decaying total atomically-resolved\nSTTk. We \fnd that the wavenumber of the oscillation of\nthe STTk at the \u0000-point is not a\u000bected by the interface\ncomposition or the thickness of the barrier [Fig. 3(c,d)].\nFor completeness we present also the real-space density\ndistribution of the \u0000-point STTk in Mn 3Ga [Fig. 3(e)]\nin terms of iso-surfaces and note that these show a dz2-5\nFIG. 3. (Color online) Comparison of atomically resolved in-\nplane STTk in Mn 3Ga for the two Mn sublattices and three\ndi\u000berent junction geometries: two interface terminations for\nthe 3 MLs of MgO and the Mn-Ga terminated junction with 5\nMLs of MgO barrier. [(a) and (b)] the total values of the on-\nsite STTk, integrated over the transverse 2D BZ on a 80 \u000280\nk-point mesh (the straight lines are only guide to the eye); [(c)\nand (d)] STTk component (~ \u001c) calculated at the \u0000 point only.\nPairs of values in brackets correspond to the \ftting parame-\nters (wavenumber, phase) for the sine-wave curves shown in\n(c, d). In (e) is a schematic of the Mn 3Ga part of the junction\nwith the ~\u001ciso-surfaces depicted [red (blue) are for a positive\n(negative) iso-value of the STTk density].\norbital-like angular dependence, in agreement with the\nresults in Fig. 2(c,d). The stack geometry is, in general,\nunrelaxed. As mentioned, the LSDA relaxation, under\nthe lateral constraints of bcc Fe, results in a signi\fcant\nuni-axial compressive strain to the tetragonal Mn 3Ga and\nthe interface to MgO splits apart. However, in order to\nobtain some guidance for the atomic interface reconstruc-\ntion from this level of DFT, we have performed a num-\nber of relaxations of a slab of Mn 3Ga/MgO, subject to\nconstant-volume constraints. In this way we have arrived\nat an interface distance between the Mn I-Ga plane and\nthe \frst MgO plane of d= 2:215\u0017A(an average of the\nMn-O and Ga-O bond lengths), minimizing total energy,\nand this is used in our representative junction geome-\ntry throughout the paper. This value has been further\nsupported by observations described in Section I.\nIn order to rule out possible artefacts related to the\nchosen inter-layer distance, we have also calculated the\natomically-resolved in-plane STTk for two smaller values\nofd, reduced by 0.2 \u0017A and 0.4 \u0017A (see Fig. 4). Without\nanalysing quantitatively the total STTk as a function of\nthe distance from the interface, we see that the long-\nrange oscillation is present for smaller dvalues too and,\nalthough there is phase shift and change in amplitude, the\nperiod of the oscillation appears very similar [Fig. 4(a,b)].\nSimilarly to our representative case, the oscillations in\nthe two sublattices also remain staggered. The \u0000-pointanalysis [Fig. 4(c,d)] con\frms a monotonic phase shift as\ndis decreased, and a small, non-monotonic, change in the\namplitude. Both sublattices are a\u000bected similarly by the\nchange ind. The \ftted wavenumber (2 \u0019=\u0015) in all cases,\nhowever, remains the same: \u0014= (0:300\u00060:001) \u0017A\u00001,\nshowing no sensitivity to interface scattering properties\nand hinting to the likelihood of it to be a manifestation\nof only bulk electronic structure properties of Mn 3Ga.\nFIG. 4. (Color online) Comparison of atomically resolved\nSTTk in Mn 3Ga for the two Mn sublattices and three di\u000berent\nvalues of the Mn I-Ga/MgO interfacial distance, d= 2:215\u0017A\n(reference case), 2.015 \u0017A and 1.185 \u0017A, as indicated.\nIV. EFFECT OF THE STRUCTURAL\nPROPERTIES OF THE Mn 3Ga LEAD\nAs the geometry of our multi-layered stack is con-\nstrained laterally to the lattice constant of bcc Fe, a=\nb=p\n2aFe= 4:05\u0017A, and the Fe/MgO side of the junc-\ntion is \fxed to established structures (see Section I), we\nhave taken the approach to probe the few remaining free\nlongitudinal parameters of our geometry and have started\nby investigating the e\u000bect of the inter-layer spacing din\nthe previous Section III. Here we explore two new values\nfor the tetragonal lattice constant cof Mn 3Ga, namely\na smallerc= 6:4\u0017A and a larger c= 6:8\u0017A. We then\ncompare the calculated in-plane STTk with the original\nreference choice of c= 6:6\u0017A (see Fig. 5).\nWe \fnd that the long-range oscillation of the total in-\nplane STTk is also present for the other c-values and\nits amplitude is practically una\u000bected by c. The phase\nof the long-range STTk oscillation and its period, how-\never, depend on cin a monotonic way. The oscillations\nare again staggered between the two sublattices for all\nc-values. It is interesting, once more, to compare the in-\nplane STTk at the \u0000 point (Fig. 5c,d). There we can\nquantify the phase shift, which is approximately linear\nwithcfor both sublattices. The amplitude is a\u000bected\ndi\u000berently by cin the two sublattices, namely there is a6\nsigni\fcant (approximately linear) increase of amplitude\nwithcfor the Mn Isublattice with a factor of 2.4 be-\ntweenc= 6:4\u0017A andc= 6:8\u0017A, while we \fnd a much\nsmaller and non-monotonic variation of about 7% for the\nMnIIsublattice. This disbalance in sensitivity towards\nthe inter-spin distances (and bond angles), in favor of\nMnI(the sub-lattice with the lower symmetry of the local\nenvironment), has been also evidenced experimentally,\nfor example in the sensitivity of the sub-lattice moments\non temperature in the MnGaRu system[27]. There, the\nMn sub-lattice lacking inversion symmetry (4c) has a\nmuch stronger temperature dependence, when compared\nto the inversion symmetric (4a) position. The sensitiv-\nity of the Mn exchange integrals on bond-lengths and\nbond-angles is well-established for metallic and dielectric\nsystems alike. Here we demonstrate that the same sen-\nsitivity is propagated to the scattering properties of the\nelectrons at the Fermi level, in particular, but not lim-\nited to, the non-directionally-averaged torkance at the\n\u0000-point [see Fig. 5(c,d)].\nFIG. 5. (Color online) E\u000bect of the longitudinal lattice con-\nstantcof the tetragonal Mn 3Ga. Compared to our repre-\nsentative lattice constant ( c= 6:6\u0017A 0) are a smaller and a\nlargerc-value, i.e. c= 6:4\u0017A (blue downward pointing trian-\ngles) andc= 6:8\u0017A (red upward triangles). Figure structure\nis analagous to Figs. 3 and 4.\nThe large variation with cof the STTk amplitude in\nthe Mn Isublattice is not preserved in the total in-plane\nSTTk, which implies di\u000berent contributions from the 2D\nBZ { we will investigate that later. The \u0000-point in-\nplane STTk for all cvalues, however, clearly shows a\nsine-wave oscillation with wavenumber \u0014monotonically\nvarying with c[see in Fig. 5(c,d) insets with \ftted pa-\nrameters]. In order to understand the \u0000-point in-plane\nSTTk oscillation in the 7.6 nm-thick Mn 3Ga layer in the\nSR of our junction, we turn to the bulk properties of\ntetragonal Mn 3Ga with unit cell as the one used for the\nopen-boundary electrode. In Fig. 6 we show the band\nstructures near the Fermi level for the two spin species of\nMn3Ga witha=b=p\n2aFe= 4:053\u0017A and the three in-\nvestigated values of c(6.4\u0017A, the central case of 6.6 \u0017Aand6.8\u0017A) as well as that of bcc Fe with aFe= 2:866\u0017A in\nthe direction of the transport in the stack ( z-direction,\ncorresponding to our \u0000-point transport). Highlighted are\nthe \u0001 1and \u0001 5-symmetry bands for spin-up ( \") and spin-\ndown (#) carriers. The latter two bands comprise for the\nleading evanescent states in the MgO barrier while the\n\u00011states decay more slowly than \u0001 5[4] { the TMR ef-\nfect in Fe/MgO/Fe(100) is largely due to the fact that\nthere is no matching \u0001 1symmetry band at the Fermi\nlevel for the minority spins in bcc Fe [see Fig.6(d)], i.e.\nthe well-established spin-\fltering e\u000bect [4].\nFIG. 6. (Color online) Band structure (\u0000 \u0000Z) in the vicinity of\nEFfor three di\u000berent values of the tetragonal lattice constant\nof bulk Mn 3Ga (a)c= 6:4\u0017A, (a)c= 6:6\u0017A and (c)c= 6:8\u0017A,\nwitha=b=p\n2aFe= 4:053\u0017A. The spins on the Mn atoms\nare aligned as in Fig. 1(b) and spin-up/down are de\fned with\nrespect on the z-axis. Panel (d) shows the corresponding band\nstructure of bcc Fe with aFe= 2:866\u0017A. Marked in thicker\nred and blue curves are the \u0001 1and \u0001 5symmetry bands,\nrespectively. For the bcc Fe case spin-up/down correspond to\nmajority/minority spin species.\nIt is evident from Figs. 6 (a), (b), and (c), that the\nvalue ofchas little e\u000bect on the spin-up band struc-\nture around EFin thez-direction, namely for all cases\nconsidered, there is always a single \u0001 1-symmetry band\ncrossing the Fermi level. In all spin-down band struc-\ntures we \fnd both a \u0001 1and a \u0001 5band crossing the EF.\nThis makes Mn 3Ga di\u000berent from bcc Fe, where only\na \u0001 5minority-spin band crosses the Fermi level [Fig.\n6(d)]. The lack of a minority-spin \u0001 1band e\u000bectively\nunderpins the very large theoretical values of TMR in\nFe/MgO/Fe(100) junctions at low bias[1]. As for Mn 3Ga\nwe also \fnd \u0001 1bands for both spin-up and spin-down\nand we expect these to dominate the transport. Indeed,\nevidently also from Fig. 3(e), the spatial distribution of\nthe calculated \u0000-point in-plane STTk in Mn 3Ga is of \u0001 1\nsymmetry.\nThe values of the Fermi wavevector for the \u0001\"\n1, \u0001#\n1and\n\u0001#\n5bands for the di\u000berent lattice constants care listed in\nTable I. Also given in the table are the \ftted wavenum-\nbers from the in-plane STTk spatial precession in Mn 3Ga7\nFrom transport From bulk\ncSz\nMnISz\nMnII\u0014(\u001cx)Sz\nukF(\u0001\"\n1)kF(\u0001#\n1)kF(\u0001#\n5)\n(\u0017A)(\u0016B)(\u0016B)(\u0017A\u00001)(\u0016B)(\u0017A\u00001)(\u0017A\u00001)(\u0017A\u00001)\n6.4 3.33 -2.20 0.376 -1.98 0.291 0.316 0.128\n6.6 3.44 -2.39 0.301 -2.51 0.249 0.403 0.109\n6.8 3.51 -2.56 0.244 -3.12 0.212 0.468 0.111\nTABLE I. Properties of Mn 3Ga extracted from 0 V transport\nand from bulk ground state calculations for three values of\nthe tetragonal lattice constant c. The average local spins\n(Mulliken population) , Sz\nMnI;II, on the two Mn sublattices in\nthe SR show a good agreement with the net spin Sz\nuof the\ntetragonal unit cell obtained from a LSDA calculation of bulk\nMn3Ga, i.e.Sz\nu'2\u0010\nSz\nMnI+ 2Sz\nMnII\u0011\n. The Fermi wavevectors\nkF(\u0001\";#\n1;5) in the \u0000-Z direction of bulk Mn 3Ga, and the \ftted\nspacial precession wavenumber \u0014of the in-plane STTk at \u0000\n[in Fig.5(c,d)], are further compared graphically in Fig. 7.\nin Fig. 5, as well as the average local moment (from Mul-\nliken population analysis) on the two Mn sublattices in\nthe SR. As we can expect, the increase of the lattice con-\nstant corresponds to an increase of the local moments on\nMn. The Mn IIsublattice is more a\u000bected, namely for the\nincrease of 6.3% in cbetween 6.4 and 6.8 \u0017A, we calculate,\nfor the averaged over all atoms in the SR z-components of\nthe Mn spins Sz, an increase of 5.4% for Sz\nMnIand 16.4%\nforSz\nMnII. At the same time there is about 36% decrease\nin\u0014, the \ftted wavenumber of the in-plane STTk at the\n\u0000 point, showing a signi\fcant sensitivity on c. The basic\nscattering theory considerations in Ref. [25] o\u000ber an in-\nsight into the spatial precession of the STT { for a single\nchannel it is expected to exhibit an oscillatory behaviour\nof the form\u0018exp [i(k\"\u0000k#)z]. Their analytical result\nfor the free-electron model k-space integration gives an\noscillatory decay governed by majority-minority Fermi\nwavevector di\u000berence. In Fig. 7 we compare \u0014as a func-\ntion ofcto di\u000berences of Fermi wavevectors between the\nonly available in the z-direction spin-up band and the two\nmost signi\fcant for the MgO tunneling spin-down bands,\nnamely the \u0001 1and the much more attenuated \u0001 5.\nIt is evident from Fig. 7 that using the \u0001#\n5band\nFermi wavevector, i.e. kF(\u0001\"\n1)\u0000kF(\u0001#\n5), results in a\nmuch smaller spatial oscillation frequency than the \ft-\nted\u0014from Fig. 4(c,d). At the same time, taking directly\nkF(\u0001#\n1)\u0000kF(\u0001\"\n1) does not even reproduce the correct sign\nof the slope. We notice that the group velocities at the\nFermi level along zhave opposite signs for the spin-up\nand spin-down (Fig. 6) in this part ( k >0) of the \frst\nBZ for two of the cvalues. In order to make sure we\nconsider carriers travelling in the same direction we take\nthe negative image of the already calculated kF(\u0001#\n1)>0\n(which currently describes right-going states in the junc-\ntion forc= 6:4\u0017A andc= 6:6\u0017A, and is practically\nequal to the BZ boundary wavevector for c= 6:8\u0017A), i.e.\nwe substitute kF(\u0001#\n1)!\u0000kF(\u0001#\n1), and add a shift by\nthe reciprocal lattice vector 2 \u0019=c. Hence we calculate\nFIG. 7. (Color online) Calculated Fermi wavevectors for bulk\nMn3Ga as a function of the clattice constant as per Table I\nand a few possible di\u000berences of spin-up and spin-down Fermi\nwavevectors (lines are guide to the eye). Dashed lines repre-\nsent 2\u0019=c\u0000kF\u0010\n\u0001\"\n1\u0011\n\u0000kF\u0010\n\u0001#\n1\u0011\nfor all three values of c. The\nblack \"x\" symbols correspond to the \ftted wavenumbers \u0014(c)\nof the \u0000-point in-plane STTk oscillation from Fig. 5c,d.\n2\u0019=c\u0000h\nkF(\u0001\"\n1) +kF(\u0001#\n1)i\n, which is plotted for all three\ncvalues in Fig. 7 and shows a remarkable agreement with\nthe \ftted spatial frequency \u0014. It is clear that at \u0000 the\nspatial precession of the STTk is driven by the mismatch\nof the majority and minority Fermi wavevectors, k\"\nF\u0000k#\nF,\nof the \u0001 1-symmetry band (which in this case results from\nhybridisation between sanddz2orbitals) in Mn 3Ga, as\ndescribed by the free electron model in Ref. 25. This re-\nsult corroborates with the observed dz2-orbital-like char-\nacter of the spatial distribution of the STTk at \u0000, as\nshown in Fig. 3e. The matching wavevectors at \u0000 pro-\nvide su\u000ecient evidence for the nature of the spatial os-\ncillation of the integral STTk that we observe in Mn 3Ga\nand which is a robust e\u000bect persisting for a range of lat-\ntice parameters. In the following Section V we will look\nin more detail at the decomposition of the transmission\ncoe\u000ecients and the atomically-averaged STTk over the\ntransverse 2D BZ. This will further elucidate the special\nrole played by the \u0000 point for the spin-transport proper-\nties of the Mn 3Ga/MgO/Fe(001) junctions.\nV. ANALYSIS OF ZERO-BIAS TRANSMISSION\nAND FINITE-BIAS STT\nIn Fig. 8 we return to our representative structure with\nc= 6:6\u0017A and present the decomposition of transport\nproperties at the Fermi level energy over the 2D trans-\nverse BZ. These include: the numbers of open channels\n(bands crossing the Fermi level) of the two semi-in\fnite\nleads, the transmission coe\u000ecients T\u001b\nk(EF) for\u001b=\";#\nand for two barrier thicknesses, and the in-plane STTk,\nall as functions of k?= (kx;ky). Within the NEGF for-\nmalism, the calculated k?-dependent transmission coef-8\n\fcients are de\fned as (see e.g. Ref. [6])\nT\u001b\nk(EF) = Trh\n\u0000\u001b\nk;L(EF)G\u001by\nk(EF)\u0000\u001b\nk;R(EF)G\u001b\nk(EF)i\n;\n(4)\nwhere \\Tr\" denotes the matrix trace operation and k\u0011\nk?(for compactness). A persistent feature in most of\nthe contour plots, shown in Fig. 8, is the dominant con-\ntribution of the \u0000 point. We \fnd, at the \u0000 point, 6 open\nchannels for both majority and minority spin in bcc Fe,\nwhile there is only one majority spin in Mn 3Ga (see also\nthe band structure in Fig. 6) and 4 open channels for mi-\nnority spin (note, \u0001 5band is doubly-degenerate). The\nlarge peak in the transmission at \u0000 for spin-up AP and\nspin-down P is due to the dominant \u0001 1transmission\navailable for majority spin both in Mn 3Ga and in Fe.\nNote, that if the two sub-lattices in Mn 3Ga were equiva-\nlent, the P an AP states of the junction would di\u000ber only\nby the spin orientation in a single interfacial ML (upon\ncomplete spin-reversal in the junction). The AFM limit\nelucidates the similarities in the diagonal pairs from the\n2\u00022 panels of k?-resolved transmissions in Fig. 8(b). In-\ncreasing the MgO barrier thickness to 5 MLs accentuates\nthe similarities between those pairs of con\fgurations, as\nwell as the apparent dominant contribution of the \u0000 point\ninn\nT\"\nk;AP;T#\nk;Po\n, compared to then\nT\"\nk;P;T#\nk;APo\npair.\nIn Fig. 8(c) we look into analogous contour plots in the\n2D BZ for the in-plane STTk in the 7.6 nm thick layer\nof Mn 3Ga in the SR (as per Fig. 1a). We \frst show the\ntotal in-plane STTk for the two Mn sublattices, which\nis de\fned for each k?as the sum of \u001cn(k?) [from the\nk?-decomposed version of Eq. (2)] for all the Mn atoms\nin the SR from the corresponding sublattice (top panels).\nThen we also evaluate another quantity,Pj\u001cnj, for the\ntwo sublattices. The later contrast is not physically mea-\nsurable, but o\u000bers additional insight about the distribu-\ntion of the main contributions (as absolute values) in the\n2D BZ. It is useful in comparison with the panels above,\nwhere the direct summation of atomically-resolved STTk\nportraits are more sensitive to the length of the SR due\nto the oscillatory nature of the in-plane STTk in space.\nThe 'absolute-value contrast' also elucidates the similari-\nties in the k?-portraits of the STTk and the transmission\nabove { arguably it is an amalgamate of the majority and\nminority transmissions. We \fnd, as expected, that the\nmain contribution to the in-plane STTk arises from the\n\u0000 point. This is more evident in the case of the thicker\nbarrier (5 MLs of MgO), where the transport is even fur-\nther suppressed to a small nearly-circular zone around\n\u0000, which contributes to the STTk in Mn 3Ga. In this\narea we see that the net STTk in the 7.6 nm Mn 3Ga\nslab changes sign in concentric rings around the \u0000 point.\nThe \u0000 point contribution has a di\u000berent sign for the two\nsublattices. Apart from the quantitative di\u000berence and\nsome symmetry-driven shape-shift of the main contribut-\ning area around \u0000, similarities between the two sublat-\ntices (even more evident in the 'absolute-value contrast'),\nsuggest a common underlying transport mechanism for\nFIG. 8. (Color online) Contour plots of k?-dependent prop-\nerties in the transverse 2D BZ: (a) the open channels for the\ntwo spin species in the Fe lead and the Mn 3Ga electrode, re-\nspectively; (b) the transmissions for each spin species in both\nP and AP spin alignments; (c) two di\u000berent sums of the atom-\nresolved in-plane STTk in the Mn 3Ga layer; (d) parameters\nof the sine-wave \fts to the atom-resolved in-plane STTk in\nthe Mn Isublattice for each k?channel. Presented are two\nFe/MgO/Mn 3Ga stacks, with 3 and 5 MLs of MgO in the\nleft- and the right-hand side panels, respectively. Note that\nthe hue color shade in (a,c,d) is on linear scale, but in (b) the\ncolor scale is logarithmic. See the text for details.\nthe in-plane STTk in the two sublattices.\nWe further analyse the calculated k?-resolved in-plane\nSTTk in Mn 3Ga by performing sine-wave \ft to the\natomically-resolved STTk for each k?-point in the 2D\nBZ. Because of the observed similarities of the STTk in\nthe two sublattices, we only examine Mn I, and the re-\nsults are shown in Fig. 8(d). The pattern of the \ftted\namplitude matches that of the 'absolute value contrast'\nin the panels above. Note that the white regions are cut\no\u000b because of the very poor \u001f2\ftting parameter value\n(below certain threshold; in fact, at the boundary with\nthe white region, we tend to \fnd very abrupt apparent\nfailure of the single sine-wave \ft). Besides the clear ev-\nidence of directional \fltering between the 3-ML and the9\n5-ML stacks, we \fnd that the \u0000 point contribution to\nthe STTk in both cases arises at an intermediate spa-\ntial frequency and a markedly di\u000berent phase compared\nto its surrounding area. The signi\fcant area of the 2D\nBZ, where wavenumber can be \ftted, is indicative to the\nfaster decay of the STTk oscillation in the 3-ML case [see\nFig. 3 (a, b)]. We would expect this decay to be clearly\nsuppressed for thicker barriers but our calculations also\nshow signi\fcant active 2D BZ area for STT in the 5ML\ncase and a very small change in the decay rate[28] of the\nin-plane STTk into Mn 3Ga.\nFIG. 9. (Color online) Comparison of linear response STTk\nand \fnite bias STT for two bias voltages Vb=-0.1, -0.2 V,\nscaled by their corresponding Vbvalues, as a function of the\natomic position on the two Mn sublattices of the Mn 3Ga in-\nside the SR. In (a,b) is the in-plane ( x) component of the\nSTT, while in (c,d) is the \feld-like ( y) component. Straight\nlines between datapoints are only guide to the eyes. Note\nthat three datapoints are shown outside their panels and con-\nnected with dashed lines. This is done to maximise resolution\nfor the rest of the dataset.\nTo this moment we have only considered the linear re-\nsponse regime and now in Fig. 9 we present the resulting\nSTTs from self-consistent calculations at two di\u000berent \f-\nnite biases (in this case we have chosen Vb<0, i.e. elec-\ntrons \rowing from the Mn 3Ga lead), scaled by their cor-\nresponding Vbvalues, in comparison to the STTk results.\nOur methodology for the \fnite-bias STT is described in\nRef. [6]. Note that the self-consistent \fnite bias calcula-\ntions are much more challenging numerically and there is\na further faster-than-linear scaling of the computational\ntime with the bias voltage. Hence, the Vbwe can ap-\nply is limited by the already signi\fcant size of the SR.\nIn Fig. 9(c,d) we also present the out-of-plane (\feld-\nlike) STT and STTk. These results demonstrate that\nthe long-range oscillation is not an artefact of the linear\nresponse regime. It can be anticipated that opening the\nbias window dampens the spatial precession (in the senseof Ref. [25]) of the STT because of the additional inte-\ngration over energy (together with that over k?) for the\nspin accumulation. Indeed, such enhanced decay is visi-\nble in all panels of Fig. 9 and it tends to increase with\nthe bias voltage. Even at the highest bias considered\n(-0.2 V), there are at least two full periods of STT oscil-\nlation visible, with similar periods to the ones observed\nin the linear response regime. In fact, it is clear that for\nthe bias voltage considered, the linear response regime of-\nfers quite a good approximation for the magnitude of the\nSTT in Mn 3Ga at low bias, especially close to the inter-\nface. The main e\u000bect of opening the bias window is the\nenhancement of the decay and, arguably, a small shift of\nthe oscillation frequency towards larger wavelengths. Al-\nthough the out-of-plane (\feld-like) torque is notoriously\nchallenging to calculate accurately (requiring very high\nk?-point sampling), we can see that despite the noise it\nclearly shows an oscillatory behaviour too. It appears to\nbe o\u000bset by, roughly, a \u0019=2 phase shift from the in-plane\nSTT and again the two sublattices oscillate in antiphase.\nInterestingly, in our \fnite-bias calculations we \fnd a huge\nout-of-plane torque on the \frst Mn Iatom at the inter-\nface. We know this site in the 3-ML junctions is a\u000bected\nby spin-polarised interface states and it appears that at\n\fnite bias it this is manifested in a very large \feld-like\ntorque. This observation, which is not captured in the\nlinear response regime, deserves a further investigation\nwhich goes beyond the scope of this work.\nWe have seen that the k?-resolved transmission at the\nFermi level shows a signi\fcant spin polarisation. In Fig.\n10 (a,b) we compare the ballistic transmission coe\u000ecients\n[from Eq. (4) integrated over the 2D BZ] for the two spin\nspecies in the two, AP or P, magnetic con\fgurations [see\nFig. 1 (f,g)] of the Mn I-Ga-interfaced junction with 5\nMLs of MgO as function of the energy of the carriers. In\nthe AP state Mn Iis aligned \"up\" which corresponds to\nthe band structure in Fig. 7b, and also parallel to the\nmoment of the Fe (also pointing \"up\"). Therefore, the\nconduction is dominated by the \u0001\"\n1band at the Fermi\nlevel. The drop in the spin-up transmission at around\n0.15 eV corresponds to the \u0001\"\n1band edge at \u0000. The other\ndrop in the T\"\nAPat around -1 eV is due to the other band\nedges of the \u0001\"\n1band in both Mn 3Ga and in bcc Fe. In\ncomparison, the transmission of the spin-down carriers\nin the AP state is signi\fcantly lower around the Fermi\nlevel because it is carried by the \u0001#\n5band.T#\nAP, however,\ndominates in the \u0001\"\n1band gap between 0.15-0.4 eV and\nabove 1.5 eV, where \u0001#\n1appears.\nIn the P state the spin-up transmission is similar to\nthat of the spin-down carriers in the AP state as the spin\npolarisation is largely dictated by the ferromagnet (the Fe\nlead). The spin-down transmission (now corresponding\nto majority spin in bcc Fe) dominates in the P case. This\nis due to the availability of \u0001 1bands, which co-exist in\nMn3Ga spin-down and in bcc Fe spin-up in a wide energy\nrange between around -0.75 eV and 1.5 eV. The band-\nedges of the \u0001\"\n1band in Mn 3Ga are clearly what drives10\nFIG. 10. (Color online) Energy dependence of transport prop-\nerties at 0 V. In (a) and (b) are the total and the spin-\ndependent transmissions of the Mn-Ga terminated junction\nwith 5 MLs of MgO in its AP and P state, respectively. In\n(c) and (d) are the total transmission and TMR e\u000bects [Eq.\n(5)] for three di\u000berent junction geometries (as indicated in\nthe legend: the interface termination, the MgO barrier thick-\nness and the spin state of the junctions). The inset (e) is a\nzoom around the Fermi-level. In (d,e) thick lines correspond\nto TMR 1, while dashed lines depict TMR 2as de\fned in Eq.\n(5).\nthe TMR e\u000bect in these junctions.\nBased on these energy-resolved spin-polarised ballistic\ntransmission coe\u000ecients, we consider possible de\fnitions\nfor the theoretical TMR e\u000bect in the Mn 3Ga/MgO/Fe\njunction at equilibrium (0V). Unlike the prototypical Fe-\nMgO-Fe junction[5], here we do not have a clear choice\nfor the less-transmissive reference state. Note that we\nhave de\fned P and AP based on the net-spin alignment\nbetween the Mn 3Ga and the Fe lead, but they correspond\nto the opposite alignment of the interfacial spins in the\njunction [see Fig. 1 (f) and (g)]. Hence, we consider one\nde\fnition, based on the net spin alignment (TMR 1), and\nthe other (TMR 2), based on the alignment of the spins\nat the interface (in our representative case with Mn I-Ga\ntermination) and these simply correspond to swapping\nthe P and the AP state, thus we de\fne\nTMR 1= (TP\u0000TAP)=TAP\nTMR 2= (TAP\u0000TP)=TP; (5)\nwhereTP;AP(E) =T\"\nP;AP(E) +T#\nP;AP(E) are the totaltransmission coe\u000ecients in the two spin-states of the\njunction. These are calculated, at equilibrium (0 V) as a\nfunction of the energy of the incoming carriers, for three\ndi\u000berent junction geometries, i.e. the two possible termi-\nnations of the Mn 3Ga/MgO interface and an additional\nthickness of 5 MgO MLs for the Mn I-Ga termination,\nand their dependence on the electron energy is shown in\nFig. 10. Note, that this is not the typical TMR e\u000bect as\nfunction of the applied bias voltage Vb, calculated from\nthe current-voltage characteristics (e.g. Ref. [5]), but a\nquantity indicating the spin-polarisation of the zero-bias\ntransmission in the vicinity of the Fermi level. At EF, we\nobserve a small TMR e\u000bect, showing only a small vari-\nation with geometry (see the inset) for both de\fnitions\nof the TMR. Both TMR 1(EF) and TMR 2(EF) exhibit a\nsign change between positive and negative values, or the\nother way around, as the geometry changes between the\nMn-Ga and the Mn-Mn termination. The absolute val-\nues of the TMR close to equilibrium for all studied cases\nremain between 10 and 30 %.\nWe, however \fnd a signi\fcant TMR 1e\u000bect in an 'is-\nland' from about 0.15 eV to about 0 :45 eV above the\nFermi level. These correspond to the gap in the spin-\nup transmission between the band edges of the \u0001\"\n1and\n\u0001\"\n5bands in Mn 3Ga [see Fig. 6(b)]. The e\u000bect occurs for\nall geometries but is especially pronounced for the case\nof the thicker barrier. Similarly, in the case of 5ML MgO\nwe also \fnd a region of increased TMR 2e\u000bect between\n-0.7 eV and -1 eV. This is due to the drop in T\"\nAPat\naround -1 V (because of the \u0001\"\n1band edge in Fe) and the\ndrop inT\"\nAPat around -0.75 eV (because of the \u0001#\n1band\nedge in Mn 3Ga). Based on these observations, we can\nanticipate certain features in the bias dependence of the\n\fnite-bias TMR in the Fe/MgO/Mn 3Ga junctions. Since\nthe feature above the EFis determined mostly by the \u0001\"\n1\nband in the Mn 3Ga lead, we expect this to move down\nin energy with applied bias voltage Vbat a rate of Vb=2.\nThus, this would to cause a peak in the total TMR 1(Vb)\nat positiveVbin the range between 0 :2 to 0:6 V. For large\nnegative biases (possibly above 0.5 V) we expect to see a\nchange of sign in the TMR 1(Vb) due to the shift upward\nof the Mn 3Ga \u0001\"\n1band-edge driven end of the second\nisland-like feature notable in TMR 2(E) below the Fermi\nlevel. More accurate SCF \fnite-bias TMR calculations\nare subject of ongoing work.\nVI. AN ENTIRELY FERRIMAGNETIC\nMn 3Ga-BASED STACK\nHere we consider an analogous MTJ in which the Fe\nlead on the right-hand side is completely substituted with\nMn3Ga, namely a Mn 3Ga/MgO/Mn 3Ga stack, again\nwith nearly 8 nm of Mn 3Ga as analyser on the right-hand\nside [as in Fig. 1(a)]. Note that no lateral dimensions\nare changed in this rearrangement and the junctions are\nmade mirror-symmetric with respect to the middle MgO11\nlayer [we consider only two cases again with odd num-\nber of MgO MLs, i.e. 3 and 5 MLs, and the latter is\nvisualised in Fig. 11(a)]. The e\u000bect of the perfect mir-\nror symmetry in the barrier interfaces about the central\nMgO layer can be immediately observed in the transmis-\nsions in Fig. 11d, where there is no di\u000berence between\nthek?-resolved transmissions for the two spin species in\nthe AP state, de\fned by the alignment of the interfacial\nMnIspins [see Fig. 11(b,c)], or the energy-resolved T\"\nAP\nandT#\nAPin Fig. 13(a,b). Furthermore, TAPappears a lot\nlike a scaled product of T\"\nPandT#\nP, indicating a relative\nindependence of the two spin-channels in this system.\nFIG. 11. (Color online) Schematics of: (a) the fully ferrimag-\nnetic junction Mn 3Ga/MgO/Mn 3Ga, (b) and (c) { the local\nand net spin alignments in the P and AP states, respectively.\nIn (d) are the 2D k?-portraits of the Fermi-level transmis-\nsions for the two spin species in the two spin-states of the two\njunctions (with 3 and 5 MLs of MgO), as indicated in the\npanel.\nThe di\u000berence with the case of an Fe polariser is mainly\nin the spin-down transmissions. While in the case of Fe\n\u0001#\n1band is absent until above 1.5 eV, this is no longer\nthe case in a fully Mn 3Ga FiMTJ and the transmission\nat the Fermi level for both spin species is dominated by\nthe \u0000 point [see Fig. 11(d)]. It is interesting to examine\nthe energy dependence of the spin-polarised transmission\nand the TMR e\u000bect, which we now de\fne uniquely as\nthe TMR 1from Eq. (5), because the net and interface\nspins now are always anti-parallel, so P/AP net moments\ncorrespond to P/AP interface spins (Fig. 11b,c), and\nwe only consider Mn-Ga termination for the all Mn 3Ga\nstacks. We \fnd the same dip in the spin-up transmission\nbetween 0.15 eV and 0.45 eV (see Fig.13a,b) as in the Fe-\nbased stack, due to the band gap for spin-up in the \u0000-Z\ndirection of Mn 3Ga (Fig.6). This reduction of the spin-\nFIG. 12. (Color online) Energy-resolved transmission and\nTMR for Mn 3Ga/MgO/Mn 3Ga stacks. In (a,b) are the spin-\nresolved and total transmissions for the 5ML stack in the P\nand the AP state, respectively. In (c) the total transmissions\nfrom above are compared to the case of the 3ML stack. In\n(d) is a comparison of the calculated TMR e\u000bect for the two\nall-ferrimagnetic junctions, compared to the TMR 1of the Fe-\npolariser junctions from Fig.10(d), and the inset (e) is a zoom\naround the Fermi level.\nup \u0001 1transmission can be seen also in the k?-resolved\nportraits in Fig. 13 at 0.2 eV and it leads to a signi\f-\ncant TMR e\u000bect in this energy range [Fig. 12(d)]. The\nTMR e\u000bect is, in fact, substantially higher than in the\ncase of the Fe-based MTJ, because now the dominating\nspin-down transmission in this energy range is less sup-\npressed, compared to the Fe case, where there is no \u0001#\n1\nband present at these energies. This is especially valid,\nand the TMR enhancement is stronger, for the thicker\nbarrier of 5 MLs. Such structure indeed shows a larger\nTMR signal in almost all the energy ranges with signif-\nicant TMR e\u000bect compared to the other Mn 3Ga-based\nMTJs studied here. It is likely that further design, using\na combination of chemical substitution and strain, could\nlead to yet higher TMR values.\nIn contrast, interestingly, the thinner MgO barrier,\ni.e. the Mn 3Ga/3MgO/Mn 3Ga junction, presents an en-\nhancement of the TMR e\u000bect at the Fermi level com-\npared to all the other cases [see Fig. 12(e)]. This is due\nto the enhanced spin-down transmission in the P state,\nbecause of symmetry-driven interface resonant states at\nthe Fermi level (similar to the well-known theoretically12\nFIG. 13. (Color online) 2D BZ portraits of spin- and energy-\nresolved transmission. Columns represent the same energy, as\nindicated at the bottom (not all energy increments between\npanels are the same). The top two rows are for the two spin-\nspecies in the P state, while the bottom row is for both spin-\nspecies in the AP state (which are identical for symmetry\nreasons). The color code for the transmissions is the same as\nin Fig. 11(d).\ninterface resonances in the minority-spin channel near the\nFermi level in Fe/MgO/Fe MTJs[5]). Such feature man-\nifests itself in Fig. 12(c) as a peak in the total P-state\ntransmission just above EF. This leads to a TMR at the\nFermi level of about 123 % for the 3ML-MgO junction,\nwhich signi\fcantly surpasses the TMR observed in all\nother Mn 3Ga-based junctions we have investigated. In\nall other energy ranges where we \fnd signi\fcant TMR\ne\u000bect, the 5ML-MgO junction shows distinctly higher\nTMR e\u000bect in comparison to the thinner barrier because\nof the enhanced directional spin \fltering. Furthermore,\nas far as the equilibrium TMR is concerned, there is lit-\ntle di\u000berence between the Fe-based and the all-Mn 3Ga\nstack with 5ML MgO [Fig. 12(e)]. This is because the\nfeatures in the transmission in the \u00061 eV vicinity of EF\nare determined entirely by the electronic structure of the\nMn3Ga exhibiting a band-gap between the \u0001\"\n1and \u0001\"\n5\nbands (which is also rather stable to cconstant variations\nin Mn 3Ga, as can be seen from Fig. 6). However, away\nfrom equilibrium the anticipated peak in the TMR( Vb),\nproduced when this feature enters the bias window, is\nlikely to be higher for the symmetric all-Mn 3Ga junction.\nThis is due to the fact that in the case of Fe, the e\u000bect\nof the \u0001\"\n1-\u0001\"\n5band gap in Mn 3Ga is suppressed in the\nAP state transmission because of the higher transmis-\nsion between Fe#and Mn 3Ga#bands [Fig.10(a)]. This\nis compared to the lower transmission between Mn 3Ga\"\nand Mn 3Ga#[Fig. 12(b) and Fig.13] in this energy range,\nwhere there is a gap in the Mn 3Ga\"band structure.\nFinally, we look at the STT e\u000bect in the all-Mn 3Ga\nbased stack and compare that to the Fe-polariser case\n(see Fig. 14). We \fnd a somewhat suppressed STTk\nfor both barrier thicknesses, which is due to the reduced\nspin-polarisation of the transmission at the Fermi level,\ncompared to the Fe-based MTJ. The oscillations in the\nin-plane STTk are still present, but we \fnd that they are\nfurther suppressed with respect to the \u0000-point STTk con-\ntribution, due to the presence of a larger number of open\nFIG. 14. (Color online) Atomically resolved in-plane STTk\nfor the Mn Isublattice in Mn 3Ga-only MTJ stacks, compared\nto previously presented results for the stacks with Fe-polariser\nfor the same thicknesses of 3 MLs and 5 MLs of the MgO bar-\nriers (as indicated in the panels). (a) and (b) are for the total\nSTTk, while (c) and (d) are at the \u0000 point. Note that there\nis a broken y-axis in (a) to show the much higher (absolute)\nvalue at the interfacial Mn Isite and there is a scaling factor\nof 0:1 for all the data from the Fe-based stacks.\nchannels away from \u0000 in the case on an entirely Mn 3Ga-\nbased FiMTJ. We also \fnd a signi\fcant enhancement of\nthe STTk at the interface of the 3ML-MgO stack, which\nis due to spin-polarised resonance states at the Mn 3Ga-\nMgO interface of this symmetric-barrier junction. Note\nthat at the \frst interfacial Mn site we \fnd a nearly seven-\nfold increase of the in-plane STTk compared to the anal-\nogous Fe-polariser junction. In this particular case of\nthe Mn 3Ga-only junction we also \fnd a TMR enhance-\nment at the Fermi level, with a theoretical prediction of\n123 % TMR. Based on the 0 V transmissions we can an-\nticipate a change of sign in the TMR( Vb) below\u00061 V.\nHowever, for an accurate analysis of the TMR( Vb) self-\nconsistent NEGF+DFT \fnite bias calculations are re-\nquired and such will be the subject of another publica-\ntion.\nFor these characteristic long-range oscillatory decays\nof the in-plane STTk in both magnetic sublattices, it is\nalso insightful to look at the net torkance as a function\nof the number of Mn 3Ga layers following the MgO in-\nterface. In Fig. 15 we compare the Fe-based and the\nall-Mn 3Ga MTJs, for the same interfaces and barriers;\nwe also show results for the di\u000berent structural parame-\nters in the Fe-based MTJs. We \fnd that in all cases the\nnet in-plane STTk also shows oscillations, but typically\nthere is an o\u000bset and a tendency to decay towards a \fnite\n(most often positive) value, describing the net STTk in\nthe limit of a thick Mn 3Ga layer. For the thinner bar-\nrier, the large interface STTk on the \frst Mn I-Ga layer,13\nFIG. 15. (Color online) Cumulative sums of atomically-\nresolved in-plane STTk as a function of the number of bi-\nlayersn(one ML of Mn Iand one of Mn II) in Mn 3Ga starting\nfrom the MgO interface, representing net in-plane STTk on\n2nMLs of Mn 3Ga. In (a-d) show sub-lattice speci\fc quan-\ntities, calculated using the results in Fig. 14 for the four\ndi\u000berent junctions (as indicated on the panels), but also not\nshown there in-plane STTk for the Mn IIsub-lattice. The\nnet quantity (black circles), representing the torque on the\nnet moment of 2 nMLs of Mn 3Ga, is calculated as \u001cx\nnet(n) =Pn\ni=0[2\u001cx\ni(Mn II)\u0000\u001cx\ni(Mn I)] from the two sub-lattice-speci\fc\nquantities. In bottom panels are \u001cx\nnet(n) for other Fe-based\njunctions, comparing e\u000bect of (e) the c-constant variation in\nMn3Ga (see also Fig. 5) and (f) the variation of the interface\nspacingd(as in Fig. 4) or Mn 3Ga termination on Mn II\u0000MnII\n(Fig. 3).\na feature that we attribute to interface resonance states,\nresults in a net STTk, which is even somewhat higher\nthan that in the Fe-based MTJ. Note that the atomically-\nresolved STTk away from the interface in the all-FiMTJ\nare almost an order of magnitude smaller than that in\nthe Fe-based MTJs (Fig. 14). This is re\rected in the\nnet STTk for the 5 ML junctions [Fig. 15(c,d)], which\nappear qualitatively identical, albeit scaled by about a\nfactor of 10. It is also interesting to see that despite\ndi\u000berences in the phase and frequency of the spatial os-\ncillations as a function of the c-constant, the net in-plane\nSTTk for junctions with the same barriers tend to sat-\nurate at the same level. Hence, we establish that there\nis little sensitivity of net in-plane torkance to the lat-\ntice parameters of Mn 3Ga. What a\u000bects the net STTk\nmore signi\fcantly is the interlayer distance dand we see\nthat the STTk can change sign with compression of d(although it is likely that such small interface spacings\nare nonphysical). The interface termination is also im-\nportant, although our example of Mn II-Mn IItermination\nmight not be representative of a real situation, since we\nhave only removed the top Mn I-Ga layer, but preserved\nthe interface distance and the MgO termination. This\ncase was only included for illustrative purposes to show\nthe e\u000bect of changing just one of the interfaces in the\njunction. Overall, our results suggest that, despite the\nlong-range oscillations, the staggered alignment between\nthe two magnetic sublattices STTk would be giving rise\nto a net switching torque in the limit of su\u000eciently thick\nMn3Ga slab. Furthermore, in this material, torque mod-\ni\fcation is possible, using either the thickness of the layer\nor interface engineering.\nVII. CONCLUSION\nWe report on \frst principles (SDFT+NEGF) calcula-\ntions of atomically-resolved spin-transfer torque in a fer-\nrimagnetic tetragonal Mn 3Ga in Fe/MgO junction and\nin an all-ferrimagnetic junction based on Mn 3Ga. In a\nscattering region extending to over 76 \u0017A of Mn 3Ga, we\n\fnd a long-range oscillatory decay of the STT, both in\nthe linear response (zero-bias) regime and at \fnite bias.\nThis oscillation is investigated against variations of the\nmaterial parameters and stack geometry, and found to\nbe persistent. It is quantitatively understood from the\nbulk electronic structure of Mn 3Ga and the spin-\fltering\nproperties of the Fe/MgO side of the junction. Spin-\ntransport properties are also compared to the case of an\nanalogous fully Mn 3Ga-based FiMTJ stack, which shows\nsimilar spatial oscillations and decay rate of the in-plane\nSTT, but usually at a lower by about a factor of 0.1\namplitude. The later junctions have been constructed\nto be symmetric with odd number of MgO MLs (3 or\n5) and are found to be prone to interface states sup-\nporting resonant tunneling, especially in the 3 ML case.\nThis leads to enhancements of both the net STT and\nthe TMR e\u000bect at the Fermi level, in comparison to the\nasymmetric Fe-based junctions. We also \fnd a signi\f-\ncant enhancement of the out-of-plane (\feld-like) torque\nat the interface, which becomes even more pronounced\nat \fnite bias. The oscillations in the STTk lead to os-\ncillatory behaviour also of the net in-plane torque as a\nfunction of the thickness of Mn 3Ga layer, but the net\nSTT stabilises for su\u000eciently thick layers. The net in-\nplane torques calculated in Fe/MgO/Mn 3Ga junctions\nsaturate at about 750 \u00021010\n\u00001m\u00002, for 3 MLs of MgO,\nand to 100\u00021010\n\u00001m\u00002, for 5 MLs, which is signi\f-\ncantly larger when compared to the net in-plane STT\nof 2\u00021010\n\u00001m\u00002calculated in analogous Fe/MgO/Fe\njunctions with 6 MLs MgO.[6]\nThe bias dependence of the TMR e\u000bect in any of\nthe Mn 3Ga-based MTJs appears to be largely deter-\nmined by the band-edges of the spin-up (in our conven-\ntion) \u0001 1-symmetry band in the z-direction of tetragonal14\nMn3Ga. Although, we have not calculated TMR( Vb),\nwe have identi\fed key features in the energy-dependent\nspin-polarised transmission coe\u000ecients, also de\fning in\npassing TMR( E), which would give rise to peaks and\nsign-changes in TMR( Vb). In the Fe-based structure, we\n\fnd a modest TMR at equilibrium (up to 50 %), but a\ncharacteristic feature in the transmissions due to a band\ngap in the \u0000-Z direction for tetragonal Mn 3Ga in the\nenergy-range 0.15 - 0.4 eV above the Fermi level, which\nwe expect to result in a peak in the TMR( Vb) near 0.5 V.\nWe also anticipate a change in the TMR sign at negative\nbias (under 1V). Such asymmetric TMR e\u000bect, featuring\na peak and a sign-change in the range -1 V to 1 V, is\nin agreement with experimental observations in similar\nMn-based FiMTJs[21].\nIn the all-Mn 3Ga FiMTJs, which we propose, the\ncorresponding island-like feature in the equilibrium\nTMR(E) appears enhanced because of the higher spin-\ndown transmission in the AP state, compared to the Fe-\nbased AP case, where the minority spin is missing a \u0001 1\nband in the Fe electrode, but a lower transmission for\nboth spin species in the AP state of the former junction\nin this energy range. This enhancement in TMR( E) is\nmuch stronger for the thicker barrier. Interestingly, forthe junction with the thinner barrier we \fnd a much more\nsigni\fcant with respect to the Fe case TMR e\u000bect (some\n123 %) close to equilibrium (0 V) due to surface-state res-\nonant tunneling in this mirror-symmetric junction. Fur-\nther self-consistent \fnite bias calculations can elucidate\nmore accurately the anticipated features of the \fnite bias\nTMR e\u000bect in these systems. However, there are clear\nindications that, both the all-FiM (still to be demon-\nstrated experimentally) and the one-sided Mn 3Ga-based\nMTJs, which we propose, hold high promise for applica-\ntions in STT oscillators and memory cells, with their high\ncurrent-induced torques and encouraging TMR e\u000bect.\nAll authors gratefully acknowledge the joint funding\nfrom the Science Foundation Ireland (SFI Grant No.\n16/US-C2C/3287) and the National Science Foundation\n(NSF ERC-TANMS Grant No. 1160504 and NSF-PREM\nGrant No. DMR-1828019). PS, SS and MS acknowledge\nfunding from the EC H2020 FET-Open project TRAN-\nSPIRE (Grant no. DLV-737038). MS acknowledges a\nStarting Investigator Research Grant by SFI (Grant No.\n18/SIRG/5515). We thank the Irish Centre for High-End\nComputing (ICHEC) and the Trinity Research IT Cen-\ntre (TCHPC) for the provision of computational facilities\nand support.\n[1] W. H. Butler, X.-G. Zhang, T. C. Schulthess, and J. M.\nMacLaren, Phys. Rev. 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B 91, 1 (2015).\n[28] It is di\u000ecult to quantify the decay rate for this system\nsize and the 'rectangular wave packet' \ft from Fig. 1\ndoes not show a di\u000berence in the dispersions on the wave\nnumber between the two barrier thicknesses." }, { "title": "2307.13522v1.Lattice_structure_dependence_of_laser_induced_ultrafast_magnetization_switching_in_ferrimagnets.pdf", "content": "Lattice structure dependence of laser-induced ultrafast magnetization\nswitching in ferrimagnets\nJ. A. V´ elez,1, 2R. M. Otxoa,3, 1and U. Atxitia4,a)\n1)Donostia International Physics Center, 20018 San Sebasti´ an, Spain\n2)Polymers and Advanced Materials Department: Physics, Chemistry, and Technology,\nUniversity of the Basque Country, UPV/EHU, 20018 San Sebasti´ an, Spain\n3)Hitachi Cambridge Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom\n4)Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain\n(Dated: July 26, 2023)\nThe experimental discovery of single-pulse ultrafast magnetization switching in ferrimagnetic alloys, such as\nGdFeCo and MnRuGa, opened the door to a promising route toward faster and more energy efficient data stor-\nage. A recent semi-phenomenological theory has proposed that a fast, laser-induced demagnetization below\na threshold value puts the system into a dynamical regime where angular momentum transfer between sub-\nlattices dominates. Notably, this threshold scales inversely proportional to the number of exchange-coupled\nnearest neighbours considered in the model, which in the simplest case is directly linked to the underly-\ning lattice structure. In this work, we study the role of the lattice structure on the laser-induced ultrafast\nmagnetization switching in ferrimagnets by complementing the phenomenological theory with atomistic spin\ndynamics computer simulations. We consider a spin model of the ferrimagnetic GdFeCo alloy with increasing\nnumber of exchange-coupled neighbours. Within this model, we demonstrate that the laser-induced magneti-\nzation dynamics and switching depends on the lattice structure. Further, we determine that the critical laser\nenergy for switching reduces for decreasing number of exchange-coupled neighbours.\nFast, reliable, and inexpensive data manipulation and\nstorage is the cornerstone for innovation and progress\nof our information-technology-based society. Ultrafast\nmagnetism holds promise for fast and low energy data\nmanipulation solutions1–5. The field was initiated by\nthe discovery of femtosecond laser pulse induced sub-\npicosecond demagnetization in Ni6. To this break-\nthrough followed the demonstration of field-free magne-\ntization switching in GdFeCo alloys using a train of cir-\ncularly polarized pulses7. Later on, a combined theo-\nretical/experimental study showed the possibility of sin-\ngle pulse switching using linearly polarized light8,9. This\nfinding uncovered the purely thermal origin of the switch-\ning process, which in turn was used to demonstrate that\nultrafast heating by picosecond electric pulses is suffi-\ncient to achieve switching in GdFeCo10. Single-pulse\nswitching can also be accomplished in CoTb alloys11,\nGd-based ferrimagnetic multilayers12,13, magnetic tunnel\njunctions of Tb/Co14, and the rare-earth-free Heusler al-\nloy Mn 2RuxGa15,16. For future information technologies\nultrafast magnetization manipulation promises high po-\ntential, as such, further understanding of the microscopic\norigin of single-pulse magnetization switching is key for\nultrafast spintronics applications17.\nSingle pulse magnetization switching in ferrimagnets\nhas been described using computer simulations based on\natomistic spin dynamics (ASD)18–21and phenomenolog-\nical models22–25. A recent work has merged these models\ninto an unified macroscopic theory that describes magne-\ntization dynamics and switching of two-sublattice ferri-\nmagnets upon femtosecond laser excitation26,27. Within\na)Electronic mail: u.atxitia@csic.esthis theory, the switching process becomes possible due\nto an enhancement of the exchange relaxation – angu-\nlar momentum exchange between sublattices – when the\nmagnetization of the sublattices is reduced below a cer-\ntain threshold that depends on material parameters. Af-\nter femtosecond laser photo-excitation, the electron sys-\ntem enters a high temperature regime at which angular\nmomentum dissipation into electron or phonon degrees of\nfreedom dominates. This so-called relativistic relaxation\nis related to the spin-orbit coupling connecting spin and\norbital degrees of freedom. Depending on the laser power,\nthe sublattice magnetization can reduce down the thresh-\nold that leads to magnetic switching. Once the laser pulse\nis gone, the electron system starts to cool down, which\nleads to a local recovery of magnetic order due to the\nexchange coupling between spins. In two sublattice mag-\nnets, the so-called exchange relaxation, through local ex-\nchange of angular momentum between sublattices, drives\nmagnetic switching. Previous computational works us-\ning ASD methods have investigated how switching de-\npends on a variety of parameters, such as element-specific\ndamping20,21, rare-earth concentration11, duration of the\nlaser pulse20,28or the role of the initial temperature29.\nThe impact of the number of exchange-coupled neigh-\nbours on the switching behaviours in GdFeCo has re-\nmained however unexplored.\nIn this work, we provide insights about how the\nsingle-pulse magnetic switching of the ferrimagnetic al-\nloy GdFeCo depends on the lattice structure, in terms\nof number of exchange-coupled spins. We use both a\nsemi-phenomenological theory as well as atomistic spin\ndynamics simulations to demonstrate that by reducing\nthe number of exchange-coupled spins, the laser energy\nnecessary to switch the magnetic state of GdFeCo reducesarXiv:2307.13522v1 [cond-mat.mtrl-sci] 25 Jul 20232\nsignificantly.\nThe magnetization dynamics of a ferrimagnet com-\nposed of two magnetic sublattices, such as GdFeCo, can\nbe described in terms of the sublattice angular momen-\ntumSa=µa⟨sa⟩/γ, where ma=⟨sa⟩andµaits atomic\nmagnetic moment27,30\ndSa\ndt=αaµaHa+αex(µaHa−µbHb) (1)\nwhere a= Fe and b= Gd. Here, αastands for the\nmacroscopic relativistic damping parameter26\nαa= 2λaL(ξa)\nξa. (2)\nHere, L(ξ) stands for the Langevin function and λais\nthe element-specific intrinsic damping parameter. The\nso-called thermal field is defined as ξa=βµaHMFA\na (β=\n1/kBT) within the mean-field approximation (MFA),\nwhere\nµaHMFA\na =zaJaama+zabJabmb. (3)\nThe non-equilibrium fields are defined as\nHa=(ma−m0,a)\nµaβL′(ξa), (4)\nwhere, L′(ξ) =dL/dξ andm0,a=L(ξa). We note that\nin the underlying model behind the MFA the Gd spins\nare randomly located in a regular Fe spin lattice with\na concentration q, and zcorresponds to the number of\nexchange-coupled neighbours. Thus, za=zqrepresents\nthe average number of nearest neighbours (n.n.) of spins\nof type a, and similarly zab=z(1−q) represents the\naverage number of n.n. of type b. Since single-pulse\nswitching has been observed mostly for Gd concentra-\ntion of around 25%, in our model we restrict to that\nconcentration of Gd spins. Within this model, one can\ndefine J0,a=zqJaaandJ0,ab=z(1−q)Jab, that is it,\nµaHMFA\na =J0,ama+J0,abmb. Within the MFA the equi-\nlibrium magnetization me=L(ξe) only depends on the\nvalues of J0,aandJ0,abbut it is insensitive to the lattice\nstructure. In comparison to the MFA, ASD simulations\ncan capture the small differences coming from the lattice\nstructure and as a consequence the equilibrium magne-\ntization slightly depends on the lattice structure31. For\nthe MFA calculations we assume a fixed values for J0,a\nandJ0,abfor all values of z. These assumptions give the\nsame temperature-dependent equilibrium magnetization\nfor all cases.\nThe number of exchange-coupled neighbours zshows\nup explicitly in the so-called exchange relaxation param-\neter\nαex=1\n2z\u0012αa\nma+αb\nmb\u0013\n. (5)\nThe number of neighbours zwill therefore become rele-\nvant for the description of the magnetization dynamics.\n00.20.40.60.81\n0 0.2 0.4 0.6 0.8 1αex=αma\nmbz= 2\nz= 3\nz= 4\nz= 6\nz= 8\nz= 20\n00.20.40.60.81\n0 0.2 0.4 0.6 0.8 1αex=αaαex=αb\nαex=αz= 6ma\nmbFigure 1. (a) Lines separate the regions where αex< α(right\nside) and the region where αex> α (left side) for a range\nof nearest neighbours number z(λa=λb). (b) Similar for\nthe case z= 6, for αa=αbcoincident to (a), and two lines\ncorresponding to αa/αb= 5, one for the case αex=αaand\nanother for αex=αb.\nFor example, in the exact MFA limit, where z→ ∞\nwhile J0,aremains constant, αex→0. The vanishing\nof the exchange relaxation parameter for large number\nof exchange-coupled neighbours directly affects the abil-\nity of the system to switch. By contrast, the exchange\nrelaxation parameter increases as the number of neigh-\nbours reduce, which could make the switching process\nmore energy efficient.\nAs a first approximation, we can assume that the mag-\nnetization relaxation pathway, either of relativistic or ex-\nchange nature, is determined by the value of their corre-\nsponding relaxation parameters. Therefore, the crossover\nbetween relativistic- to exchange-dominated regimes can\nbe estimated by finding the conditions at which αex> αa.\nFor simplicity we consider first the case λa=λbin Eq.\n(2), for which αa≈αb=α32. Under this assumption,\none can simplify the condition α=αexto\nz=1\nma+1\nmb(6)\nFigure 1 (a) shows the lines separating the regions αex<\nαandαex> α. As the number of neighbours zreduce,\nthe region ( ma, mb) for which αex> α aincreases. For\ninstance, for the limit case of z= 2 (spin chain), αex=α\nalready for relatively large values of maandmb. For a\nlarger number of neighbours z, corresponding to simple\ncubic ( z= 6) or face-centered cubic ( z= 12) lattices, αex\nis relatively smaller than αfor a large region ( ma, mb).\nExperimental observations combined with ASD simu-\nlations suggest that in rare earth transition metal alloys,\nthe damping values are element specific20,21. Specifically,\nit was found that using λFe= 0.06 and λGd= 0.01 in\nthe ASD simulations, one can qualitatively reproduce the\nultrafast magnetization dynamics and switching of the\nGdFeCo alloys in a range of Gd concentrations20. Sim-\nilarly, in Gd 22−xTbxCo78alloys, it was found element-\nspecific damping values ( λCo=λTb= 0.05 and λGd=\n0.005−0.05) could describe switching as a function of Tb\ncontent21. For the sake of simplicity, we illustrate the im-\npact of element-specific damping on the relation between\nrelaxation parameters for a particular case, z= 6. Figure3\n30060090012001500\n−1−0.8−0.6−0.4−0.200.20.40.60.81\n0 1 2 3 4 5 6 7T(K)electronmz(Fe)\ntime (ps)z= 4\nz= 6\nz= 8\nz= 20\nFigure 2. (Top) Electron temperature dynamics driven by a\nfemtosecond laser pulse. (Bottom) Iron sublattice magnetiza-\ntion dynamics and switching for systems with different num-\nber of exchange-coupled nearest neighbours z. The sublattice\nmagnetization dynamics is calculated using Eq. (1) with the\nelectron temperature profile in the top figure as an input.\n1 (b) shows αex=αin solid lines, which corresponds to\nFig. 1(a), and αa/αb= 5, which agree to experimental\nobservations. For element-specific damping values, each\nelement (sublattice) will enter the exchange dominated\nregime under different conditions, namely, sublattice a\nwhen αex=αaand sublattice bwhen αex=αb. Our\nmodel predicts that under those circumstances the mag-\nnetization relaxation of the Gd sublattice is dominated\nby the exchange relaxation during the whole demagneti-\nzation process, i.e., by transfer of angular momentum to\nthe Fe sublattice, whereas Fe sublattice angular momen-\ntum relaxation remains mostly relativistic – transfer of\nangular momentum to other degrees of freedom.\nIt is worth noting that the magnetization relaxation\ndynamics described by Eq. (1) not only scales with the\nvalue of the relaxation parameters but also with the non-\nequilibrium effective field Ha[Eq. (4)]. In particular,\nthe relaxation of the sublattice magnetization Eq. (1)\ncan be split into two contributions, the relativistic re-\nlaxation rate: Γr\na=αaµaHa, and exchange relaxation\nrate Γex=αex(µaHa−µbHb). After the application of a\nlaser pulse the temperature quickly increases beyond the\ncritical temperature and the dynamics is dominated by\nthe thermal fields, which translates into Γr\na∼λakBTma\nwhereas Γex\na∼λakBT/z. Thus, a prerequisite for switch-\ning is that the laser pulse produces a high temperature\nprofile (see Fig. 2 (Top)) to quickly reduce the magne-\ntization ma, so that the exchange relaxation takes over,\nΓex>Γr\na. One would expect a more efficient switch-\ning process for lower values of z, and a highly non-\nefficient for high values of z. One can rationalize this\nby analysing the equations of motion for some limiting\nsituations. For the same value of the intrinsic damp-\ning parameter λFe=λGd, by assuming that one of thesublattice demagnetizes faster (Fe in GdFe alloys) than\nthe other, soon after the application of a fs laser pulse\none finds that ma≪mb, from the condition Γex>Γr\na,\none gets maΓr\nbone gets\nma≤1/2z27. For example, for a lattice with fcc+bcc\nstructure, z= 20, ma0.7 Oe, indicating\nthe absence of crystal-field effects in the disks. These results\nsuggest the crystal-field contribution arises from anisotropic\nrelaxation in the patterned bars, which corroborates prior work\nwith V [TCNE ]xnanowires where an additional in-plane crys-\ntal field is reported due to anisotropy in the relaxation of the\ntemplated structures.22\nThe more complicated spectra of the disks suggests that\nthe disks are acting as spin-wave cavities with complex in-\nternal mode structure, Fig. 3(e). Numerical simulations and\nanalytical calculations are carried out to better understand\nthis mode structure. To begin characterizing the mode struc-5\nture, the strongest experimental peaks are compared with the\nodd analytic thickness modes predicted for a thin film in the\nOOP geometry.41The vertical blue lines in Fig. 3(e) repre-\nsent the experimental peak values. Fitting to these peak val-\nues using the mode assignments indicated in Fig. 3(e) and\nthe parameters obtained from the FMR measurements yields\nthe red analytic curve and a value of the exchange stiff-\nness, Aex= (2.2±0.5)×10−10erg/cm. The even thickness\nmodes, shown as dashed red lines, agree well with smaller\npeaks within the experimental data. Analytic disk calculations\nshown in black in Fig. 3(e) further describe the identity of\nthe quantum confined modes and agree well when using this\nAex. The exchange stiffness depends on Ms; an approximate\nform, found by several means,42–44isAex∝M2\ns. The exchange\nlength constant λex=2Aex\nµ0M2sis therefore a better metric to use\nto compare samples with different saturation magnetizations.\nThe difference between the exchange length from this study\nofλex=9.7 nm and the previously reported value of 21 nm20\ncould be due to differences in grain structure between the pat-\nterned and unpatterned films44as well as difficulty in mode\nassignment, n, in prior work where fewer modes are visible.\nNumeric modeling is performed using time-domain micro-\nmagnetic simulations with the open-source GPU-based soft-\nware MuMax3 while using the material parameters deter-\nmined from the fits to the experimental data.45The factors\nthat have the most relevant influence on the simulated peak\nstructure are (i) the sloped sidewalls that (a) have a strong ef-\nfect on the shape of the lowest frequency set of peaks which\nare comprised of a set of closely-spaced radially and lowest-\norder thickness quantized modes and (b) apply an overall shift\nto the thickness confined modes, (ii) the pinning conditions\nof the surfaces that have a strong effect on the amplitudes\nof the thickness-confined modes, and (iii) the exchange stiff-\nness, Aex, that controls the spacing between thickness quan-\ntized modes. Sloped sidewalls are used in the simulations to\nreplicate the shape that occurs due to the slower growth rate\nwithin 1 micron of the resist. The simulations show that the\nposition of the most prominent peak relative to the thickness-\nconfined modes is sensitive to the exact shape of the sidewalls\nand the pinning conditions. To account for small differences\nin the slope of simulated and experimental data, the higher-\norder thickness modes are aligned with experiment instead\nof the uniform mode in Fig. 3(e). Simulations with perfect\npinning at the top and bottom surfaces agree better with the\nexperimentally observed thickness and radial confined mode\nstructure as compared to simulations with top, bottom, or no\npinning; however, the close agreement between the calculated\neven-mode resonance fields and several smaller peaks in the\nexperimental spectrum suggests that one of the surfaces likely\nhas slightly weaker pinning than the other. Additional simula-\ntions can be found in the supplement. The resulting simulated\nfrequency response of the simulation is in green in Fig. 3(e)\nalong with several mode maps at peaks indicated by the green\narrows. These maps reveal quantization in the thickness and\nradial directions in the tapered structure. The lower-order\nthickness modes each show distinct radial quantization. The\nn=7 thickness mode, shows a nearly pure thickness quantiza-\ntion and represents the sum of multiple closely-spaced radials\nFIG. 4. Full-width at half-maximum linewidth vs resonant frequency\nfor various V [TCNE ]xpattern sizes from thin films to 5 µm diame-\nter disks. All linewidths are extracted from the OOP geometry. All\ngrowths were 1-hour long, resulting in a 300 nm thick film for the\n5 µm film and 400 nm thickness for the rest. The patterned thin film\nis a 2 mm by 2 mm patterned patch of V [TCNE ]x.\nmodes that are excited simultaneously. The agreement be-\ntween simulated and experimental spectra demonstrates con-\ntrol over the spin-wave mode structure and lays the foundation\nfor the study and application of magnon cavities with adia-\nbatic boundaries and engineered mode structures.\nIn addition to analyzing anisotropy and mode structure,\nFMR can be used to determine the total magnetic loss, or\ndamping of these magnon modes. This damping potentially\ncontains both homogeneous and inhomogeneous sources as\nparameterized via the Gilbert damping factor, α.46The damp-\ning for the patterned V [TCNE ]xfilms is measured via broad-\nband ferromagnetic resonance (BFMR) performed in a cus-\ntom built microstrip-based system wherein the applied mag-\nnetic field is held constant in the OOP geometry and the mi-\ncrowave frequency is swept across the V [TCNE ]xresonance.\nFigure 4 shows the linewidth vs frequency extracted for rep-\nresentative samples of disks and unpatterned films, with ver-\ntical lines indicated the error in the fits. Representative raw\ndata and fits can be found in the supplemental information.\nThe Gilbert damping is fit using Suhl’s expression for the full-\nwidth at half-maximum (FWHM) FMR linewidth,34\n∆H=α\n|dωres/dH|γ\nM/parenleftbigg\nFθθ+1\nsin2(θ)Fϕϕ/parenrightbigg\n, (5)\nin combination with phenomenological inhomogeneous\nbroadening.47This results in\n∆H=4πα\n|γ|f+∆H0, (6)\nwhen one uses θ=0 for the OOP geometry.46In Eq. 6, ∆H\nis the FWHM linewidth of the resonance, αis the Gilbert\ndamping, and ∆H0is the FWHM contribution from inhomo-\ngeneous broadening. The fits yield α=(3.98±0.22)×10−5\nfor unpatterned films, α=(4.60±0.44)×10−5for 25 µm fea-\ntures, and α=(8.34±0.77)×10−5for 5 µm disks. The thin-\nfilm damping result of (3.98±0.22)×10−5places V [TCNE ]x6\nfilms comfortably alongside YIG films as the lowest magnetic\ndamping material currently available, and the retention of\nthat ultra-low damping after patterning is considerably better\nthan the reported values for patterned YIG structures.10–14In\naddition to low-damping, the high-frequency measurements\nof the thin film and 25 µm disks have Quality ( Q) factors,\nf\n∆f, of over 3,700, competitive with Qfactors for YIG thin\nfilms.48Retaining ultra-low damping and high Qin patterned\nV[TCNE ]xfor features as small as 25 µm and as large as mil-\nlimeters, both relevant length scales for many magnonic cavity\napplications,3,14,49–52combined with the flexibility to deposit\non most inorganic substrates, positions V [TCNE ]xto comple-\nment YIG in magnonic and magnetoelectric devices where in-\ntegration of GGG or high-temperature annealing steps is lim-\niting, such as for small form factors and on-chip integration.\nIn summary, this work demonstrates a method for pat-\nterning the ferrimagnetic coordination compound vanadium\ntetracyanoethylene. Standard electron-beam lithography\nof PMMA/P(MMA-MAA) bilayers is used in conjunction\nwith pre-growth aluminum encapsulation and post-growth\ndichloromethane liftoff to pattern V [TCNE ]xthin films with\nno degradation of the microwave magnetic properties. The\nsidewalls of structures patterned in this way are sloped, al-\nlowing for the investigation and quantitative modeling of spin-\nwave confinement in magnetic structures with soft boundary\nconditions. Patterned V [TCNE ]xfilms with features down to\n25 µm exhibit a high Qof over 3,700 and ultra-low damp-\ning of (4.60±0.44)×10−5which are competitive with un-\npatterned YIG and lower than all existing reports of patterned\nYIG microstructures.10–14The versatility of the patterning and\ndeposition conditions of V [TCNE ]x, in combination with its\nultra-low magnetic damping, position V [TCNE ]xas a promis-\ning candidate for incorporation into magnetoelectric devices\nwhere low-loss, highly coherent, magnon excitation are desir-\nable. Such applications range from microwave communica-\ntions to quantum information.\nSUPPLEMENTARY MATERIALS\nSee supplementary material for the a detailed description\nof sample fabrication, measurement techniques, simulations,\nand analytic calculations.\nACKNOWLEDGMENTS\nThis work is supported by Emerging Frontiers in Research\nand Innovation (EFRI) Grant No. EFMA-1741666. 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Porter, “Ferromagnetic resonance\nline width in yttrium iron garnet single crystals,” Physical Review 110,\n1311–1313 (1958).Low-Damping Ferromagnetic Resonance in Electron-Beam Patterned, High- Q\nVanadium Tetracyanoethylene Magnon Cavities\nAndrew Franson,1Na Zhu,2Seth Kurfman,1Michael Chilcote,1Denis R.\nCandido,3, 4Kristen S. Buchanan,5Michael E. Flatté,3, 4Hong X. Tang,2and Ezekiel\nJohnston-Halperin1\n1)Department of Physics, The Ohio State University, Columbus, Ohio 43210,\nUSA\n2)Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511,\nUSA\n3)Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242,\nUSA\n4)Pritzker School of Molecular Engineering, University of Chicago, Chicago,\nIllinois 60637, USA\n5)Department of Physics, Colorado State University, Fort Collins, Colorado 80523,\nUSA\n1arXiv:1910.05325v1 [physics.app-ph] 11 Oct 2019PATTERNING AND MEASUREMENT METHODS\nPatterning and Growth\nV[TCNE ]xbars and disks are patterned on commercially available C-plane polished sapphire\n(Al2O3) via electron-beam lithographic techniques. The substrates are cleaned with a solvent\nchain of acetone, methanol, isopropanol (IPA), and deionized water (DI water) followed by a 20\nminute ultraviolet ozone clean (UVOC) in a UVOCS T10x10/OES to degrease and remove organic\ncontaminants. A 400 nm layer of MMA (8.5) MAA EL 11 (P(MMA-MAA)) is spun on at 2000\nrpm for 45 seconds then soft baked at 180◦C for 300 seconds. A 140 nm layer of 495PMMA\nA6 (PMMA) is then spun on at 2000 rpm for 45 seconds then soft baked at 180◦C for 60 sec-\nonds. A 10 nm thick layer of aluminum is deposited via thermal deposition at 1 ×10−6Torr.\nThe electron-beam patterning of the PMMA/P(MMA-MAA) bilayer is performed on a FEI Helios\nNanolab 600 Dual Beam Focused Ion Beam/Scanning Electron Microscope with the assistance of\nNanometer Pattern Generation System (NPGS) software. Development of the written pattern is\nachieved with Microposit MF-319 for 40 seconds, DI water for 20 seconds, MF-319 for 40 sec-\nonds, DI water for 20 seconds, Microchem MIBK:IPA (1:3) for 60 seconds, IPA for 20 seconds,\nDI water for 20 seconds. That is followed by a 120 second hard bake at 100◦C. A 3 nm thick layer\nof aluminum is then deposited in the same system as the prior 10 nm layer to prevent outgassing\nof the PMMA/P(MMA-MAA) bilayer during V [TCNE ]xdeposition. The sample is oxidized and\ncleaned with a 10-minute UVOC. This oxidizes the surface of the 3 nm aluminum layer and re-\nmoves potential small sources of contamination from the growth surfaces. Growth of V [TCNE ]x\nis described in previous work.1After transfer of the films from the growth glovebox to the liftoff\nglovebox, liftoff is performed in dichloromethane. For the feature sizes and thicknesses used here,\nliftoff occurred in a few minutes with gentle agitation from a Pasteur pipette.\nMicrowave Measurements\nAngle-resolved ferromagnetic resonance (FMR) measurements are done with a Bruker electron\nparamagnetic resonance spectrometer with an X-band bridge and 10 kOe electromagnet. Before\nmeasurement, the V [TCNE ]xsamples are sealed into a quartz tube with a ceramic holder that aligns\nthe normal plane of the sample. Between each scan, the microwave frequency is tuned between\n9 and 10 GHz to match the resonant frequency of the loaded cavity. The frequency is fixed and\n2measurements are then performed by sweeping the static field with 200 µW of applied microwave\npower and a modulation field of 0.1 Oe. The quartz tube has a pointer fixed to it, allowing for\nalignment within 0.5 degrees of the sample with respect to a custom made goniometer. The larger\nerror of±2 degrees seen in Fig. 3(d) comes from the initial aligning the goniometer with the\nin-plane (IP, θ=90) orientation of the V [TCNE ]xfilm. The IP orientation is taken to be the point\nwhere the resonance field is at its minimum.\nFrequency-resolved microwave measurements are done with a broadband ferromagnetic res-\nonance (BFMR) setup with a B4003-8M-50 microstrip test board from Southwest Microwave\nthat is sourced by an Agilent N5222A vector network analyzer (VNA), transduced via a Krytar\n203BK Schottky diode, and measured with an Ametek 7265 Dual Phase DSP lock-in amplifier.\nThe microstrip is positioned inside a 10 kOe electromagnet in the out-of-plane (OOP, θ=0) field\ngeometry. Measurements are performed with an input power of -10 dBm and a modulation field\nof roughly 0.1 Oe oscillating at 577 Hz.\n3NUMERICAL MODELING\nMicromagnetic simulations are done to gain insight into the measured FMR spectra. The\nmicromagnetic simulations are done using MuMax32with the following parameters for the\nV[TCNE ]x: saturation magnetization 4 πMs=76.57 G (6093 A/m), exchange constant Aex=\n2.2×10−10erg/cm (2 .2×10−15J/m), and a damping parameter of α=0.0001, where αis\nlarger than the smallest measured damping value for V [TCNE ]xand was chosen to ensure that the\nsimulations would converge in a reasonable amount of time. Cells of 40 x 40 x 4.6875 nm3are\nused. Selected simulations are repeated with smaller cells with similar results because the 40 nm\ncell sizes are still small compared to the wavelengths of the spin-wave modes. Aexis determined\nby fitting the mode spacing of the thickness confined modes from the OOP spectra of the disks\nblue anisotropy curves in Fig. 3(c). Fitting is done to theory from Kalinikos et al.3and is shown\nin Fig. 3(e). Fitting for fully pinned and fully unpinned geometries yielded the same value for Aex.\nThe simulations are done with the static field, Hbias, applied OOP. The field magnitudes are\nchosen to result in a resonance frequency near 9.83 GHz to be similar to the measurement fre-\nquency used in the experiments. The structures are relaxed in the presence of Hbiasand a small\nadditional field Hdynam of roughly 50 Oe, chosen such that the spins tilt by about 1% from the\nstatic equilibrium position, is applied in the x-direction. The dynamics are monitored as a func-\ntion of time after removing Hdynam and the simulated spectra are obtained by taking the Fourier\ntransforms of the x-component of the magnetization for each run. Mode profiles are calculated for\nselected peaks in the spectra by running driven simulations at the selected resonance frequency\nand extracting the spin distributions as a function of time for a full period after the simulation has\nreached a steady state.\nSince the pinning of the structures is unknown and the exact profile is uncertain, a matrix of\ndifferent combinations of pinning and cross-sectional profile is investigated. Three geometries\nare considered based on the growth characteristics of the bar in Fig. 1(b, c), a cylindrical disk; a\nlens-shaped, circular disk with a spherically-curved top surface and flat bottom surface (spherical\ncap); and a disk with a 1 µm wide ramp from the outer rim to the inner rim, Fig. S1. For each\nof these geometries, simulations with no pinning of surface spins, perfect pinning on the top\nsurface, perfect pinning on the bottom surface, and perfect pinning on both the top and bottom\nsurfaces is considered. To compare the experimental spectra with the simulations, we considered\nthe prevalence of the thickness modes and the shape of the strongest peak. The mode spacing does\n4FIG. S1. Left, experimental data of the 9.83 GHz OOP FMR scan of the array of 5 µm disks. Right, array of\nthe different simulated geometries considered in this study. Blue lines represent the full data and the orange\nlines are the full data multiplied by 10 and truncated to better show low-intensity, low-field behavior. Mode\nmaps of the vertical cross-sections of the mode maps at the disk center for the largest amplitude peak for\neach simulated geometry are shown in the upper left of each plot, where dark blue (dark), green (medium),\nand yellow (light) represent negative, zero, and positive motion, respectively, at an instant of time. These\nare quantized standing modes.\nnot change appreciably with the disk shape (taper vs. lens), and agrees well with the calculations of\nthe mode spacing for an unpatterned thin film. The spherical cap used for the lens-type simulations\nleads to almost complete suppression of the thickness modes for all of the pinning conditions,\nwhich is not consistent with the experimental spectrum. The strongest peak in the data is smoother\nthan the jagged combo-like structure of the main peak of the cylindrical simulations that occurs\ndue to the radial modes. The two simulations that are the closest to the experimental data are\nthe tapered-disk simulations with top/bottom and bottom-only pinning. The even modes show\nup almost as strongly as the odd modes for bottom-only pinning, whereas only the odd modes\nare present for top/bottom pinning. As show in Fig. 3(e), resonance peaks that correspond to\neven thickness modes are present but weak compared to the odd modes, which suggests that both\nsurfaces are pinned but that pinning is imperfect on one of the two surfaces, likely the top.\n5FITTING METHODS\nAll fitting is performed in Python with the emcee package4within the lmfit package.5\nCavity FMR Angular Anisotropy Fitting\nFitting of the red anisotropy curves in Fig. 3(d) are all performed with a modified Eq 2. Squar-\ning both sides and collecting factors of Hyields\n/parenleftbiggω\nγ/parenrightbigg2\n=H2+H×4πMs/parenleftbig\n−NIPcos(2θ)−NIPcos(θ)2−NOPcos(2ϕ)/parenrightbig\n+\n16π2M2\ns/parenleftbig\nN2\nIPcos(θ)2+NIPNOPcos(2θ)cos(2ϕ)−N2\nOPcos(θ)2cos(ϕ)2sin(ϕ)2/parenrightbig\n,(S.1)\nsolving for Hthen yields\nH=−b±√\nb2−4ac\n2a(S.2)\nwhere\na≡1, (S.3a)\nb≡4πMs(−NIPcos(2θ)−NIPcos(θ)2−NOPcos(2ϕ)), (S.3b)\nc≡16π2M2\ns/parenleftbig\nN2\nIPcos(θ)2+NIPNOPcos(2θ)cos(2ϕ)−\nN2\nOPcos(θ)2cos(ϕ)2sin(ϕ)2/parenrightbig\n−/parenleftbiggω\nγ/parenrightbigg2\n.(S.3c)\nTo account for deviations from high-symmetry directions, θandϕare parameterized in terms of\na new parameter, t. This parameterization allows for fitting through a path along any great circle\nof the unit sphere by using the expression\n/vectorv1=sin(θ1)cos(ϕ1)ˆx+sin(θ1)sin(ϕ1)ˆy+cos(θ1)ˆz (S.4a)\n/vectorv2=sin(θ2)cos(ϕ2)ˆx+sin(θ2)sin(ϕ2)ˆy+cos(θ2)ˆz (S.4b)\nη=arccos (/vectorv1·/vectorv2) (S.4c)\n/vectorvm=sin(η(1−t))/vectorv1+sin(ηt)/vectorv2\nsin(η)(S.4d)\nwhere /vectorvmpoints to a location on the great circle that intersects /vectorv1and/vectorv2. The location is determined\nby the parameter twhich steps from ti=0 totf=π\nηin steps of tstep=tf−ti\n18to produce 10◦steps\nfrom 0◦to 180◦.\n6This correction is only necessary for the bar sample scanned IP to OOP with the applied field\nperpendicular to the bar axis (red diamonds in Fig. 3(d)). A path traveling from the IP posi-\ntion ( θ1=90◦,ϕ1=90◦) to a position 10◦away from OOP ( θ2=10◦,ϕ2=90◦) is required\nto accurately describe the data. This corresponds to an initial θoffset in the x-direction while\nsweeping θin the y-direction and explains why the red-diamond curve does not kiss the red-circle\nand red-square curves at exactly one point.\nBroadband FMR Linewidth Fitting\nFIG. S2. Representative single scans of various V [TCNE ]xfeatures ranging from an unpatterned thin film\nto a 5 µm diameter disk array. All scans are done in the OOP ( θ=0) geometry. All growths were 1-hour\nlong, resulting in a 300 nm thick film for the 5 µm film and 400 nm thickness for the rest. The patterned\nthin film is a 2 mm by 2 mm patterned patch of V [TCNE ]x. The linewidth is extracted from the fit to the\nfurthest right lorentzian except for the 25 µm disks where the linewidth is extracted from the largest peak.\n7Figure S2 shows representative linescans from the BFMR measurement setup. The spectra are\nfit well by a combination of several lorentzian derivatives. Each lorentzian derivative function has\nantisymmetric (absorptive) and symmetric (dispersive) components6represented by\nL/prime\nabs(f) =aΓ3(f−f0)\n(Γ2+4(f−f0)2)2, (S.5a)\nL/prime\ndisp(f) =−dΓ2/parenleftbig\nΓ2−4(f−f0)2/parenrightbig\n(Γ2+4(f−f0)2)2, (S.5b)\nso each derivative lorentzian in the fit is represented by\nL/prime\ntotal(f) =aΓ3(f−f0)\n(Γ2+4(f−f0)2)2−dΓ2/parenleftbig\nΓ2−4(f−f0)2/parenrightbig\n(Γ2+4(f−f0)2)2(S.6)\nwhere ais the height of the derivative of the absorptive component, dis the height of the deriva-\ntive of the dispersive component, Γis the full-width at half-maximum of the lorentzian, fis the\nindependent variable, and f0is the peak of the lorentzian.\n8ANALYTIC RESONANT FIELD CALCULATIONS\nHere we find an analytical expression for the spin-wave (or magnetostatic mode) resonant fields\nfor a normally magnetized cylinder with thickness dand radius R. We first solve Maxwell’s equa-\ntions within the magnetostatic regime. Application of the proper boundary conditions at z=±d\n2,\nthe top and bottom surfaces of our cylinder, yields the following transcendental equations7,8\ntan(kid) =2koki\nk2\ni−k2o, (S.7)\niki√1+κ=ko, (S.8)\nwhere koandkiare the in-plane and out-of-plane wave vectors, respectively, and κ=ΩH\nΩ2\nH−Ω2, with\nΩH=H+Msλex(k2\ni+k2\n0)\nMs,λex=2Aex\nµ0M2sandΩ=ω\nγMs. An analytic expression for the resonant spin-wave\nfields is then obtained from the Maxwell’s equation coupled to the Laudau-Lifshitz equation, with\nthe additional assumption that the magnetization is pinned at r=R, which yields7,8\nJm−1(koR) =0, (S.9)\n→kn\no,m−1=βn\nm−1\nR, (S.10)\nwhere βn\nm−1is the nth-zero of the Bessel function of order m−1. Now using Eq. S.8 and Eq. S.10\nwe obtain8the following expression for the spin-wave resonant fields\nBz\nnml≈µoω\nγ+µoMs−2Aex\nMs/bracketleftBigg\nk2\ni,nml+/parenleftbiggβn\nm−1\nR/parenrightbigg2/bracketrightBigg\n−µoMs\n2/parenleftbigg\n1+k2\ni,nmlR2\n(βn\nm−1)2/parenrightbigg−µo(Ms)2\n8ω\nγ/parenleftbigg\n1+k2\ni,nmlR2\n(βn\nm−1)2/parenrightbigg2,\n(S.11)\nassumingω\nγ/greatermuchMs/bracketleftbigg\n2/parenleftbigg\n1+k2\ni,nmlR2\n(βn\nm−1)2/parenrightbigg/bracketrightbigg−1\n. The indices n,mandlrepresent the radial, angular, and\nthickness mode numbers, respectively. The resonant fields in Eq. S.11 are derived using the SI\nelectromagnetic equations and Bz\nnmlhas units of Tesla (T). To obtain the resonant fields Hz\nnmlin\nOersted (Oe) unit, the values obtained from Eq. S.11 should be multiplied by 1 ×104Oe/T.\nEq. S.11 does not account for the demagnetization field. A good match with the experimen-\ntal data is obtained using the approached described in Kakezei9that considers an effective de-\nmagnetization per mode Nnml. To account for the effect of the demagnetization field, we sub-\nstitute Ms→MsNnmlwith Nnml<1. Fig. S3 shows the resonant fields found using Eq. S.11\nforNnml∈[0.865−0.925]8–10for different d. The best match with the experimental data is for\n9d=250 nm. This is smaller than the nominal thickness of the V [TCNE ]xdisks used in the ex-\nperiment (300 nm), which we attribute to the fact that the lens shape leads to a smaller effective\nthickness. Like the simulations, the analytical calculations also predict a much closer spacing for\nthe radial modes as compared to the thickness-quantized modes (not shown).\nAmplitude\nApplied Field, H (Oe)d=200nm d=250nm\nApplied Field, H (Oe) Applied Field, H (Oe)d=300nm\nFIG. S3. Plot of the resonant fields for the first five thickness modes l=1,3,5,7 for the angular and\nradial modes m=1 and n=1 a) d=200nm, b) d=250nm and c) d=300nm. For all three plots,\nA=2.2×10−10erg/cm, Ms=76.57 G, Nnml∈[0.865−0.925],ω=9.83 GHz, γ=2.73×106MHz/Oe,\nandR=2500 nm. The black line shows the experimental spectrum obtained for the V [TCNE ]xcylinder for\nthe OOP field orientation ( θ=0,ϕ=0).\n10REFERENCES\n1H. Yu, M. Harberts, R. Adur, Y . Lu, P. C. Hammel, E. Johnston-Halperin, and A. J. Epstein,\n“Ultra-narrow ferromagnetic resonance in organic-based thin films grown via low temperature\nchemical vapor deposition,” Applied Physics Letters 105, 012407 (2014).\n2A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyen-\nberge, “The design and verification of MuMax3,” AIP Advances 4, 107133 (2014).\n3B. A. Kalinikos and A. N. Slavin, “Theory of dipole-exchange spin wave spectrum for ferro-\nmagnetic films with mixed exchange boundary conditions,” Journal of Physics C: Solid State\nPhysics 19, 7013–7033 (1986).\n4D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, “emcee: The MCMC hammer,”\nPublications of the Astronomical Society of the Pacific 125, 306–312 (2013).\n5M. Newville, T. Stensitzki, D. B. Allen, and A. Ingargiola, “LMFIT: Non-linear least-square\nminimization and curve-fitting for Python,” (2014).\n6S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva,\nand J. P. Nibarger, “Ferromagnetic resonance linewidth in metallic thin films: Comparison of\nmeasurement methods,” Journal of Applied Physics 99, 093909 (2006).\n7M. Sparks, “Magnetostatic modes in an infinite circular disk,” Solid State Communications 8,\n731–733 (1970).\n8D. R. Candido and M. E. Flatté, to be published.\n9G. N. Kakazei, P. E. Wigen, K. Y . Guslienko, V . Novosad, A. N. Slavin, V . O. Golub, N. A.\nLesnik, and Y . Otani, “Spin-wave spectra of perpendicularly magnetized circular submicron dot\narrays,” (2004).\n10S. V . Nedukh, S. I. Tarapov, D. P. Belozorov, A. A. Kharchenko, V . O. Golub, I. V . Kilimchuk,\nO. Y . Salyuk, E. V . Tartakovskaya, S. A. Bunyaev, and G. N. Kakazei, “Standing spin waves in\nperpendicularly magnetized circular dots at millimeter waves,” Journal of Applied Physics 113,\n17B521 (2013).\n11I. S. Maksymov and M. Kostylev, “Broadband stripline ferromagnetic resonance spectroscopy\nof ferromagnetic films, multilayers and nanostructures,” Physica E: Low-dimensional Systems\nand Nanostructures 69, 253–293 (2015).\n11" }, { "title": "1609.07230v2.Electromagnon_resonance_in_a_collinear_spin_state_of_a_polar_antiferromagnet_Fe2Mo3O8.pdf", "content": "1 \n Electromagnon resonance in a collinear spin state \nof a polar antiferromagnet Fe2Mo 3O8 \n \nT. Kurumaji1, Y. Takahashi1, 2, 3, J. Fujioka2, R. Masuda2, H. \nShishikura2, S. Ishiwata2, 3, and Y. Tokura1, 2 \n \n1RIKEN Center for Emergent Matter Science (CEMS), Wako 351 -0198, Japan. \n2Department of Applied Physics and Quantum Phase Electronics Center ( QPEC ), \nUniversity of Tokyo, Tokyo 11 -8656, Japan. \n3PRESTO , Japan Science and Technology Agency, Chiyoda, Tokyo 102 -8666, Japan \n \nAbstract \nMagnetic excitations are investigated for a hexagonal polar \nmagnet Fe 2Mo 3O8 by terahertz spectroscopy. We observed \nmagnon modes including an electric -field active magnon, \nelectromagnon, i n the collinear antiferromagnetic phase with \nspins parallel to the c axis. We unravel the nature of t hese \nexcitations by investigating the correlation between the evolution \nof the mode profile and the magnetic transition from \nantiferromagnetic to ferrimagnetic order indu ced by magnetic \nfield or Zn -doping. We propose that t he observed electromagnon \nmode involves the collective precession of the spins with oscillating \nin-plane electric pol arization through the mechanism of the linear \nmagnetoelectric effect. \n 2 \n Cross correlation between magnetism and electricity, i.e., magnetoelectric \n(ME) effect, is a key in designing electric -field-controllable spin devices [1, 2]. \nAmong various ME materials [3], multiferroics, which exhibit simultaneous \nmagnetic and ferroelec tric order s, ha ve attracted tremendous interest \nbecause of recent discoveries of strong ME response upon magnetic phase \ntransition as in TbMnO 3 [4] as well as o f room -temperature multiferroic s such \nas BiFeO 3 [5] and hexaferrit e [6]. Entanglement between magnetism and \nelectricity can be extended to elementary excitations , which was theoretically \ndiscussed since 1970’s [7]. In fact, the electric -dipole -active magnon , termed \nthe electromagnon, was observed as an infrared absorption in terahertz \nregion [8]. Such excitation s have been identified in various m ultiferroic \nmaterials [9], and promise new terahertz functionalities of multiferroics \nincluding optical control of magnetism and nonreciprocal directional \ndichroism [10, 11]. \n \nAccording to Khomski i [12], multiferroics can be classified into two types ; \nin type-I multiferroics , the ferroelectricity and the magnetism have distinct \norigins , while the magnetic order itself is the driving force of ferroelectricity \nin type-II multiferr oics. The former group includes BiFeO 3 and hexagonal \nYMnO 3 [5, 13], which show relatively large spontaneous polarization s and \nhigh ferroelectric/magnetic transition temperature s, while the magnetism \nonly modestly influences the polarization and/or dielectric constant . The \nlatter group includ ing orthorhombic (perovskite type) RMnO3 (R: rare earth) \n[4, 14], Ni3V2O8 [15], and MnWO 4 [16] hosts strong ME coupling , while 3 \n tending to show relatively lower transition temperature s partly because of \nthe frustration in spin interactions . \n \nEarly works on type -II multiferroics including RMnO 3 [8], hexaferrite s \n[17], and CuO [18] have clarified that incommensurate spiral mag netic order s \ngenerally exhibit the electromagnon resonances . Therein , a part of \nelectromagnons is driven by the exchange striction mechanism described by \nthe inner product of spins, i.e., 𝑺𝑖⋅𝑺𝑗 [19], while the inverse Dzyaloshinskii -\nMoriya ( DM) mechanism expressed by 𝑺𝑖×𝑺𝑗 also contribute s to the \nelectromagnon resonance with both electric - and magnetic -dipole activities \n[11]. For type -I multiferroics, o n the other hand , the electromagnons, which \ntend to show up less conspicuously in the spectra, are often connected to \ncomplex magnetic structures such as cycloidal magnetic order in BiFeO 3 [20, \n21], and noncollinear multi -sublattice ferrimagnetic order in CaBa Co4O7 [22]. \nHere we report one other type of electromagnon in a type -I multiferroic \nFe2Mo 3O8 with a simple collinear magnetic order of magnetic moments of Fe2+ \nions. We identify the mode characters of magnetic excitations including \nelectromagnon by the polarization selection rule and the comparison with \nferrimagnetic phase induced by chemical doping as well as by the magnetic \nfield. We propose a model of magnetic excitations, where the ME coupling is \ntaken into account, that consistently explains magnetic excitation in \nantiferromagnetic and f errimagnetic phases. \n \nFe2Mo 3O8 forms a hexagonal lattice belonging to a polar space group 4 \n P63mc (Fig. 1(a) ). There exist two types of magnetic site for Fe2+ ion, A and \nB, characterized by the tetrahedral and oc tahedral coordination of oxygen , \nrespectively [23]. AO4 and BO6 polyhedra share their corners to form a \nhoneycomb lattice in the ab-plane. The two magnetic layers , which are \nrelated with a nonsymmorphic operation based on the c-glide plane with each \nother, are involved in the unit cell . Below the N éel point (TN = 60 K ) the \nsystem evolves into the collinear antiferromagnetic ( AF) state (see the inset \nof Fig. 1(b)) [24]. Appli cation of magnetic field (Hdc) along the c axis induc es \na collinear ferrimagnetic (FM) order [25, 26] (the inset of Fig. 1(c)) . \nAlternatively, the FM state is stabilized also by substitution of more than \n12.5 % of Fe with Zn [24, 26, 27]. Coexistence of spontaneous polarization \nand magneti c order below the transition temperature allows strong ME \ncoupling and large linear ME coefficients in both the in-plane and out -of-\nplane components , which promises the characteristic spin wave excitation \nrespond ing to ac electric/magnetic field of light. \n \nSingle crystals of Fe 2Mo 3O8 and (Zn 0.125Fe0.875)2Mo 3O8 were grown by \nchemical vapor transport reaction as described in Ref s. [28, 29] from the \nstoichiometric mixture of MoO 2, Fe, Fe 2O3, and ZnO. Samples with ab-plane \nand ac-plane cut, whose dimensions are typically 2 x 2 mm2, were prepared. \nThe time -domain terahertz spectroscopy was employed to measure the \nrefractive indices in a frequ ency range of 0.5 - 2.8 THz and the details about \nthe experimental setup and procedures are described in Ref. [ 30]. Laser \npulse s with 100 -fs duration from a Ti: sapphire laser w ere split into two paths 5 \n to generate and detect the wave form of terahertz pul ses. A ZnTe (110) \ncrystal and a dipole antenna were used for generation and detection of \nterahertz pulses, respectively. The Hdc was applied to the sample with a \nsuperconducting magnet in Voigt geometry , i.e., a light propagation vector k \nperpendicular to Hdc. \n \nFigures 1(d) -(f) show the spectra of extinction coefficient (imaginary part \nof refractive index) for Fe 2Mo 3O8 in zero field at 4.5 K for three possible \ngeometries. As shown in Fig. 1(d), two clear resonance peaks are observed \naround 1.2 THz and 2. 7 THz for the light polarized 𝐸𝜔⊥𝑐 and 𝐻𝜔⊥𝑐, \ndenoted as EM and MM1, respectively. The characters of magnetic \nexcitations can be deduced by the polarization selection rule derived from the \nresults in Figs. 1(e) -(f); EM is concluded as electric -dipole (E1) active , i.e., \nelectromagnon , because it can be excited by the 𝐸𝜔⊥𝑐 (Fig. 1(e)) but not by \n𝐻𝜔⊥𝑐 (Fig. 1(f)) , while MM1 is active for 𝐻𝜔⊥𝑐 (not with 𝐸𝜔⊥𝑐), \nindicating its magnetic -dipole (M1) active nature . To check the correlation \nbetween the mode profile and the magnetic order, we also measured the \nspectra for the collinear ferrimagnetic phase in the doped sample ( y = 0.125 ) \n(Figs. 1(g) -(i)). This composition shows the FM state even at zero field (Fig. \n1(c)). A single resonance peak is observ ed around 2.6 THz (MM2) (Fig. 1(g) \nand 1(i) ), while no discernible resonance structure is seen around 1.2 THz . \nThus, the electromagnon resonance is absent (Fig. 1(h)) in the current energy \nwindow , while the MM2 is active for 𝐻𝜔⊥𝑐 (Fig. 1(i) ) similarly to the MM1 . \n 6 \n Figure s 2(a)-(h) show the spectra of for y = 0 and y =0.125 at selected \ntemperatures. With increasing temperature, the absorption of each mode \ngradually wanes and disappears above the transition temperature (Fig. 2 (d) \nand (h)) . The temperature dependence of t he spectral weights (∝\n−1\n𝑑∫ln(𝑡−𝑡0)d𝜔) for the respective modes are shown in Figs. 2(i) and 2(j). \nHere, d is the thickness of the sample, t the transmittance, and the angula r \nfrequency of light. t0 is assumed to be a background due to flat absorption . We \ndefine the spectral weights of EM, MM1 and MM2 by the integration between \n1.1 ~ 1.4 THz, 2.5 ~ 2.8 THz, and 2.4 ~ 2.6 THz , respectively. The magnitude \nof these resonance s start to rise upon the magnetic ordering as shown in Fig s. \n2(i) and 2(j). This result indicate s that the observed modes are collective \nexcitation s arising from the magnetic order ing and not from a gap excitations \nrelated to the crystal field . Indeed, the latter excitation was observed in \nnoncentrosym metric B a2CoGe 2O7 [31, 32], which has E1 activity but is \nobservable even above the transition temperature unlike the present case . \n \nTo clarify the mode characters in AF and FM states , the Hdc dependence \nof magnetic resonances are measured at selected temperatures as \nsummarized in Fig. 3. Figure 3( a) shows spectra for AF state ( y = 0, see \nthe phase diagram in Fig. 1(b)) at 4.5 K under Hdc//c for two different light \npolarization s. Figure 3(d) shows field evolution of excitation frequency. EM \nshows little magnetic field dependence , while the MM1 splits into two modes , \nimplying the character of t he conventional antiferromagnetic resonance. \nAlthough the EM may be also doubly degenerate , the possible frequency 7 \n splitting appears to be too small to be detect ed for 0Hdc up to 7 T. \n \nWe also performed the comparative measurements for the FM state \nstabilized by the magnetic field at 50 K for y = 0 and by the chemical doping \nat 4.5 K for y = 0.125 . The data set are displayed in Fig s. 3(b), 3(c), 3(e), and \n3(f). At 50 K, Fe 2Mo 3O8 shows the metamagnetic transition at 0Hdc~ 5.2 T \nas shown by M-Hdc curve in Fig. 3( e). Upon the transition , the EM in 𝐸𝜔⊥\n𝑐 geometry suddenly disappears, while the split branches of the MM1 turn \ninto a single mode (termed here MM2 ) with a slightly lower frequency (see \nthe spectra for 0Hdc= 5.1 T and 5. 3 T with 𝐻𝜔⊥𝑐 in Fig. 3(b) and 3(e) ). The \nsimilar MM2 mode in the FM phase are also exemplified by the FM phase \ninduced by the chemical doping ( y = 0.125), in which the monotonous \nsoften ing of the MM2 is observed as the magnetic field is increased from 0 T \nto 7 T (Fig. 3( c) and 3(f) ). \n \nThe emergence of the electro magnon mode in the AF phase indicates that \nthe magnetic excitation possesses in-plane oscillation of electric polarization. \nThe linear ME effect at the DC limit observed in FM phase [26] can be related \nto the electrical activity of magnon excitation in AF phase . Here, we \nconsider two mechanisms for the linear ME effect as identified in Ref. [2 6], \ni.e., the inverse DM effect and the single -site anisotropy effect . Although the \nconventional inverse DM model in Refs. [33, 34 ] predicts P along the c (z) axis \nfor the adjacent spin on A and B site, the loc al site asymmetry in the present \ncompound allows P in general directions, in accord with descriptions in Refs. 8 \n [35, 36 ]. In the case of the nearest -neighboring A and B sites in a honeycomb \nlayer shown in Fig. 4(a), the symmetry with respect to the zy plane allows in -\nplane electric polarization 𝑝𝑦 proportional to dynamical x component of 𝑺𝐴×\n𝑺𝐵, i.e., δ𝑆𝐴𝑦𝑆𝐵𝑧−𝑆𝐴𝑧δ𝑆𝐵𝑦. On the other hand, the single -site anisotropy \neffect [37, 38 ] induces the in -plane polarization 𝑝𝑖𝑦 at each i-th Fe site due to \ncanting of a spin as 𝑝𝑖𝑦∝𝑆𝑖𝑧δ𝑆𝑖𝑦. \n \nHere we speculate possible modes of magn on excitations to explain the \nobserved resonance including EM, MM1 and MM2. We ignore , to the first \napproximation, the interlayer magnetic interactions , because the stacking \nhoneycomb layers are intervened by a Mo layer. In each magnetic layer , \nspins at neighboring A and B sites prefer to o scillate in an antiferromagnetic \nmanner as shown schematically in Fig. 4(a) . The neighboring spins cant \nwith slightly different angle s into opposite direction , as the result of single -\nion anisotropy at each site . In th is circumstance , electric polarization due to \ninverse DM effect ( 𝑝𝑦) is nonzero since the dynamic al x component of 𝑺𝐴×𝑺𝐵 \n(δ𝑆𝐴𝑦𝑆𝐵𝑧−𝑆𝐴𝑧δ𝑆𝐵𝑦) is nonzero, and the difference o f the transverse component \nof th e respective spins ( δ𝑆𝐴𝑦 and δ𝑆𝐵𝑦 in Fig s. 4(a) ) induces a net \nmagnetization 𝑚𝑦. Note that the mutual relation between 𝑚𝑦 and 𝑝𝑦 is \nopposite for the upper and bottom layer s (Fig. 4(a)) in the unit cell , since their \nrelative positions are interchanged . Next, we take into account the \ninterlayer coupling. In that case, doubly degenerate modes for the upper \nand bottom layer s are coupled in in-phase or out -of-phase manner , resulting \nin the mode splitting . Figure s 4(b) and 4(c) show in-phase and out -of-phase 9 \n oscillations, respectively ; 𝑚𝑦 (𝑚1 and 𝑚2) and 𝑝𝑦 (𝑝1 and 𝑝2) are shown \nfor each layer . The oscillation pattern in Fig. 4( b) induces net 𝑝𝑦 while the \n𝑚𝑦 cancels ; this explains why the EM can be excited by the in -plane electric \nfield but not by the in -plane magnetic field of light. As for the out -of-phase \noscillation (Fig. 4( c)), 𝑚𝑦 remains finite while 𝑝𝑦 is cancel led; this \ncorresponds to the MM1 . Therefore , the configurations shown in Fig s. 4(b) \nand 4 (c) qualitatively explain t he selection rule for the electromagnon and \nmagnon modes observed in the AF phase. In Ref. [2 4], the interlayer \ncoupling energ ies were estimated by the molecular field theory , i.e., the \nantiferromagnetic interlayer coupling between A sublattices is stronger (~57 \nK) than that between A and B sublattices (~38 K). This is consistent with \nthe lower excitation energy of EM than that of MM1 ; EM keeps the \nantiferromagnetic nat ure between interlayer A sites during the oscillation as \nshown in Fig. 4( b), while MM1 violates it (Fig. 4( c)). Note that the in-plane \nelectric polarization s due to the single -site anisotropy effect are un cancel led \nand cancel led for the spin configurations in Fig s. 4(b) and 4(c), respectively, \ngiving the same conclusion on the dipole -activities. \n \nThe above scheme is also applicable to the FM phase, which suggests a \nboth E1 and M1 active mode (Fig. 4( d)) as well as a silent mode ( mode X as \nshown in Fig. 4( e)), although the experimentally observed MM2 appears to be \nM1 active but least E1 active . From the symm etry point of view , four \nmagnetic excitation branches exist for a four-sublattice collinear magnetic \nsystem. Thus, we believ e there is another higher energy mode out of the 10 \n range of this experiment w hich would show strong E1 and weak M1 activit ies, \ncomplementary to the nature of the MM2 . \n \nIn conclusion, we observe two distinct collective magnetic excitations \ndriven by electric and magnetic field of terahertz light , respectively , in the \nantiferromagnet ic phase for polar magnet Fe2Mo 3O8. We have also revealed \ndistinct properties of magnetic excitations for the antiferromagnetic and \nferrimagnetic phases . The origin o f the observed electromagnon is \naccount ed for by the oscillation of electric polarization induced by precession \nof spins through the inverse Dzyaloshi nskii-Moriya interaction and/or single -\nsite anisotropy. Possible spin configuration for the excitations are suggested, \nwhich remains electric polarization uncancel ed because of the out -of-phase \ninterlayer coupling . The present observation s show that the simple collinear \nmagnetic order in type -I multiferroics can host electromagnon mode s, \npromising versatile optical magnetoelectric phenomena in t erahertz region as \nwell as those in type -II multiferroics . \n \n 11 \n \nPMy= 0\nFM\nAF(b)Hdc//c\nPM\nFMy= 0.125\n(c)(ZnyFe1-y)2Mo3O8\n0.5\n\n0.5\n0\n3 2 1 0\nfrequency (THz) 3 2 10.5\ny= 0.125 (4.5 K, 0 T) y= 0 (4.5 K, 0 T)\nEM\nｘ3MM1E⊥c\nH⊥c\nE⊥c\nH//c(d)\n(e)\n(f)(g)\n(h)\nｘ3(i)MM2E//c\nH⊥cO(a)\nMo\nFe/Zn\n(A)\nFe (B)\nabc\n80\n40\n0Temperature (K)\n4 0 8 4 0\nMagnetic Field (T) \nFig. 1 \n(a) Crystal structure of (Zn yFe1-y)2Mo 3O8. (b)-(c) Magnetic field ( Hdc) vs. temperature \nphase diagram s under Hdc//c for y = 0 and y = 0.125, respectively , as reproduced from \nRef. [2 6]. Magnetic structure of each phase is also shown. (d)-(i) Spectra of \n(imaginary part of refractive index, i.e., extinction coefficient) for respective light \npolarization s at 4.5 K in zero field for y = 0 ((d)-(f)) and for y = 0.125 ((g)-(i)). 12 \n \n0.05\n0M (/f.u.)\n80 40 0\nTemperature (K)1\n0spectral weight\n(arb. units)\n0.2\n0.2\n\n0.2\n0.03 2 1 0\nfrequency (THz)0.2\n0.2\n0.03 2 1 0\nfrequency (THz)0.3\n0.3\n0.2MM2(j) y= 0.125 (i) y= 0\nEMMM1y= 0, 0 T\n(a) 4.5 K\n(b) 50 K\n(c) 58 KE⊥c\nH//cE//c\nH⊥c\n(d) 65 Ky= 0.125, 0 T\n(e) 4.5 K\n(f) 30 K\n(g) 45 K ｘ2\n(h) 57 K ｘ2ｘ2E//c, H⊥c\nｘ2\nｘ2\n1\n0spectral weight\n(arb. units)\n80 40 0\nTemperature (K)1.5\n0M (/f.u.) \nFig. 2 \n(a)-(d) Temperature dependence of for y = 0 in zero field. Red (blue ) curves are for EM \n(MM1 ) measured with different light polarization s. (e)-(h) Corresponding spectra of y \n= 0.125 for MM2 . Temperature dependence of spectral weight for (i) EM and MM1 , and \n(j) MM2 in zero field. Magnetization measured with 0Hdc = 0.1 T is also shown for \ncomparison . 13 \n \n \n \n \n \n2\n1\n0\n3 2 1 0\nfrequency (THz)(Fe1-yZny)2Mo3O8, Hdc//c\nE//c\nH⊥c\n5.5(a) y= 0, 4.5 K (b) y= 0, 50 K\n01234567T\n023455.15.96.37 Tｘ2\n5.3E⊥c\nH//c\nAFFM\n3\n2\n1\n0frequency (THz)\n8 6 4 2 0\nMagnetic Field (T)\n3\n2\n1\n0frequency (THz)\n8 6 4 2 0\nMagnetic Field (T)3\n0M (/f.u.)\nAF FM(d) y= 0, 4.5 K (e) y= 0, 50 K\nAFMM1\nEMMM2(f)\n2\n1\n0\n3210\nfrequency (THz)(c)y= 0.125\n4. 5 K\n02455.566.57 T\n3\n2\n1\n086420\nMagnetic Field (T)\nFMMM2y= 0.125\n4. 5 K\n2\n1\n03 2 1 0\nfrequency (THz) \nFig. 3 \n(a)-(c) spectra under various field magnitudes. Data are shifted vertically for clarity. \nRed (blue) curves are for the polarization E⊥c and H//c (H⊥c and E//c). Blue \ncurves in (b) are magnified by two. (d)-(f) Evolution of excitation frequency with Hdc//c: \nred, blue , and green circles are for EM, MM1, and MM2 , respectively. 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Lett. 6, 609 (1961) . \n[38] T. Arima, J. Phys. Soc. Jpn. 76, 073702 (2007) . " }, { "title": "2101.06155v2.Tunable_spin_flop_transition_in_artificial_ferrimagnets.pdf", "content": "arXiv:2101.06155v2 [cond-mat.mes-hall] 1 Feb 2021Tunable spin-flop transition in artificial ferrimagnets\nN. O. Antropov,1,2E. A. Kravtsov,1,2M. V. Makarova,1,2V. V. Proglyado,1\nT. Keller,3,4I. A. Subbotin,5E. M. Pashaev,5G. V. Prutskov,5A. L. Vasiliev,5\nYu. M. Chesnokov,5N. G. Bebenin,1V. V. Ustinov,1B. Keimer,3and Yu. N. Khaydukov3,4,6\n1Institute of Metal Physics, 620180 Ekaterinburg, Russia\n2Ural Federal University, 620002 Ekaterinburg, Russia\n3Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstraße 1, D-70569 Stuttgart, Germany\n4Max Planck Society Outstation at the Heinz Maier-Leibnitz Ze ntrum (MLZ), D-85748 Garching, Germany\n5National Research Center ”Kurchatov Institute”, 123182 Mo scow, Russia\n6Skobeltsyn Institute of Nuclear Physics, Moscow State Univ ersity, Moscow 119991, Russia\n(Dated: February 2, 2021)\nSpin-flop transition (SFT) consists in a jump-like reversal of antiferromagnetic magnetic moments\ninto a non-collinear state when the magnetic field increases above the critical value. Potentially the\nSFT can be utilized in many applications of a rapidly develop ing antiferromagnetic spintronics.\nHowever, the difficulty of using them in conventional antifer romagnets lies in (a) too large switching\nmagnetic fields (b) the need for presence of a magnetic anisot ropy, and (c) requirement to apply\nmagnetic field along the correspondent anisotropy axis. In t his work we propose to use artificial\nferrimagnets in which the spin-flop transition occurs witho ut anisotropy and the transition field\ncan be lowered by adjusting exchange coupling in the structu re. This is proved by experiment on\nartificial Fe-Gd ferrimagnets where usage of Pd spacers allo wed us to suppress the transition field\nby two orders of magnitude.\nAntiferromagnetic (AF) spintronic is nowadays a\nrapidly developing area [1–5]. In addition to non-\nvolatility of conventional ferromagnetic spintronics the\nAF devices can offer immunity to external magnetic dis-\nturbances, absence of cross-talks between small-area de-\nvices and much faster dynamics (THz vs MHz). The\nantiferromagnetic systems are featured by spin-flop tran-\nsition (SFT) when there is the transition from antifer-\nromagnetic ordering to noncollinear (NC) state at mag-\nnetic field exceeding certain value HSP. Creation of non-\ncollinearmagneticstateandpossibilitytoswitchbetween\nAF and NC states may have useful applications by utiliz-\ning anomalous Hall or Nernst effects [6–11]. In addition,\nproximity of noncollinear magnetic texture to supercon-\nducting layer generates long-range triplet superconduc-\ntivity which may also find diverse applications in super-\nconducting spintronics [12–16].\nThe utilization of the spin-flop effect in AF systems\nis overly complicated due to at least two reasons. The\nfirst thing is the existence of SFT in AF requires uniaxial\nanisotropy and an external field applied along the corre-\nsponding axis. Secondly, typical transition fields HSP\nin bulk antiferromagnets are tens of Tesla [17–20] thus\nthey are too high for real applications. The need to have\nanisotropy inside the system can be circumvented by re-\nplacing antiferromagnets with ferrimagnets (FEMs). In\nthe FEMs one does not require presence of anisotropy\nand the SFT takes place at HSP=λ|m1−m2|[21],\nwherem1,2are the magnetic moment of first and second\nsublattices and λis the exchange parameter. In bulk sys-\ntems the HSPare still too high for applications and can\nhardly be tuned.\nIn contrast, artificial ferrimagnets based on magneticheterostructures give a possibility to tune the SFT field\nby varying parameters of ferromagnetic layers and by in-\ntroducing non-magnetic spacers. Heterostructures based\non 3d transition metals (TM) and heavy 4f rare-earth\n(RE) metals, like Fe/Gd, are model ferrimagnetic sys-\ntems demonstrating a rich magnetic phase diagram with\ncomplex types of magnetic ordering [22–27]. Coupling\nbetween 4f electrons of Gd and 3d electronsof Fe leads to\nthe antiferromagneticalignment of TM and RE magnetic\nmoments which due to the difference in magnetic mo-\nments of Fe( ∼2µB) and Gd ( ∼7µB) leads to the emer-\ngence of a one-dimensional ferrimagnetic lattice. The\nspin-flop transition was found in Gd/Fe systems at typ-\nical value HSP∼3kOe [28], which is much smaller than\nthat for bulk FEMs but still quite high for applications.\nFurther tuning of HSPcan be gained by suppression of\ninterlayer exchange coupling which can be performed by\nspacing of Fe and Gd with a non-magnetic material like\nCr [29, 30], Pt [31] or Si [32].\nThe SFT can be detected by integral magnetic tech-\nniques as a kink on a magnetic hysteresis loop at HSP.\nIn case of artificial FEMs magnetic signal from thin films\nis heavilypolluted by dia-orparamagneticsignalofthick\nsubstrates.This makes it difficult, if not impossible at\nall, to use integral magnetometric methods to study the\nSFTs. Neutron scattering, being a depth-selective mag-\nnetometric method is a widely used method for studying\nAFs and FEMs [33–35]. Similar to X-ray and light, neu-\ntrons diffract at periodic lattice with period Daccording\nto the well-known Bragg law nλ= 2Dsinθ. Hereλand\nθare the neutron wavelength and incident angle, and n\nis integer number corresponding to order of Bragg peak.\nPresence of spin one-half makes neutron scattering sen-2\nsitive to the magnetic lattice. In case of antiferromag-\nnetic lattice magnetic peak is doubled comparing to the\nstructural one, so that the magnetic Bragg peak appears\non the positions of n/2 of the structural Bragg peaks.\nApplying spin analysis, that is detecting neutron spin-\nstates before and after scattering, allows one to get ad-\nditional information about magnetic configuration. The\nnon-spin-flip (NSF) channels (++) and (- -) are sensi-\ntive to the sum and difference of nuclear potential and\ncollinear to the neutron polarization part of magnetiza-\ntion. Here first and second sign codes neutron polariza-\ntion along the external magnetic field Hbefore and after\nthe scattering process. Presence of non-collinear magne-\ntization causesspin-flip (SF) scattering(+-)and (-+). In\nBorn approximation the amplitude of the SF scattering\nis proportional to the spatial profile of the noncollinear\nmagnetization in reciprocal space. Thus the SF scatter-\ning is very sensitive channel to detect the SFTs.\nIn our prior work [36] we studied superlattice\n[Fe(3.5nm)/Pd(1.2nm)/Gd(5nm)/Pd(1.2nm)] 12. In the\nneutron experiment we measured intensity of SF scatter-\ning at the position of the first Bragg peak RSF\n1as a func-\ntion of external magnetic field at a temperature of 10K.\nAbove magnetic field of HSP=1.5kOe we detected a 20-\nfold increase of SF scattering which is the direct evidence\nfor the presence of SFT in our system. We note that\ntheHSPfield is much smaller than in spacer free Fe/Gd\nsystems. Subsequent structural studies by transmission\nelectron microscopy and synchrotron radiation [37] indi-\ncated presence of mutual diffusion at Gd/Pd interface.\nFor thin ( ∼1nm) Pd spacers this interdiffusion leads to\nalmost complete dissolution of Pd in Gd. As a result\nthe Curie temperature (and hence exchange energy) of\nthe (nominal) Gd layer decreases from 294K for bulk Gd\nto/lessorsimilar100K. Thus ability of Pd and Gd to form an alloy\nwith controllable suppression of exchange energy paves\nthe way for tuning of SFT by varying thickness of Pd\nspacer. To do this we prepared series of samples of nom-\ninal composition [Fe(3.5nm)/Pd(t)/Gd(5.0nm)/Pd(t)] 12\nvaryingtfrom 1.0 to 1.6 nm (details can be found in our\nprior works [36, 37]). Further we will code samples as\nPdYY, where YY is thickness of Pd layer in Angstroms.\nFig. 1a shows the X-ray low-angle diffraction\npatterns (reflectivities) measured at a wavelength of\nλ=1.54˚A from the samples under study. More than 10\norders of Bragg reflection are seen on the reflectivities,\nwhich indicates goodrepeatability ofthe Fe/Gd unit cell.\nFig. 1b shows the energy dispersive X-ray (EDX) mi-\ncroanalysis of scanning transmission electron microscopy\n(STEM) of Pd12 sample. The EDX analysis shows well-\ndefined Fe layers depicted by blue color and yellow layers\nofGdPd alloyinsteadofseparateredGd layersandgreen\nPd spacers. For the sake of simplicity, we will keep nam-\ning Gd layer, remembering however that in reality the\nlayer is a Gd xPd1−xalloy.\nPolarized neutron reflectometry (PNR) experiment/s50 /s52/s49/s48/s48/s49/s48/s50/s49/s48/s52/s49/s48/s54/s49/s48/s56\n/s110/s61/s57/s110/s61/s56/s110/s61/s54/s110/s61/s52/s110/s61/s51/s88/s45/s114/s97/s121/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s40/s176/s41/s32/s80/s100/s49/s48\n/s32/s80/s100/s49/s50\n/s32/s80/s100/s49/s52/s40/s97/s41\n/s110/s61/s49\n/s40/s98/s41\nFIG. 1. (a) X-ray low-angle diffraction (reflectivity) of sam -\nples under study. Vertical arrows show the position of sever al\nBragg peaks for sample Pd10. (b) The energy dispersive X-\nray (EDX) microanalysis of Pd12 sample.\nwas conducted on the monochromatic ( λ=4.3˚A) reflec-\ntometer NREX of the research reactor FRM-2 (Garch-\ning, Germany). Fig.2 shows the PNR data measured\non sample Pd10 at T=10 K in magnetic field H=1kOe\nand additional SF curve at T=10 K in magnetic field\nH=3kOe (solid line). In the neutron experiment 4 Bragg\npeaks were confidently measured. A large splitting of\n(++) and (--) NSF Bragg peaks indicates the presence\nof a collinear magnetic moment in the system. At the\nsame time we observed a much weaker (1-2 orders below\nNSF signal) SF scattering at Bragg peaks. The origin\nof this small, though not negligible SF signal can be as-\nsociated with noncollinear inhomogeneities at the Fe/Gd\ninterfaces. The data at H=1kOe can be quantitatively\ndescribed by a predominantly collinear AF state with\nmagneticmomentsofGd MGd≈5µBandFeMFe≈2µB\naligned parallel and antiparallel to H. By increasing the\nmagnetic field above HSP=2.3kOe (inset in Fig.2) we\nobserved a 20-fold increase of SF scattering at the first\nBragg peak RSF\n1. This SFT is similar to observed pre-\nviously spin-flop in Pd12 sample though taking place at\n1kOe higher magnetic field.\nBy measuring family of RSF\n1(H) scans at different tem-\nperatureswewereabletoconstructthenoncollinearmag-\nnetic phase diagram for the sample Pd10 in H-Tcoordi-\nnates (Fig. 3a). For this sample we observe a collinear\nAFstateinthetemperaturerangeupto30Kinmagnetic\nfields not exceeding 2 kOe. Above this field, the collinear\nAF state is replaced by a NC spin-flop state. Increasing\nthe temperature to 60K leads to a gradual shift of the\nSFT field towards lower values. Finally, above 60K, the\nspin-flip signal disappearsdue to the absenceof magnetic\nordering in Gd layer. Fig.3b and Fig.3c shows similar\nphase diagrams for Pd12 and Pd14 samples. One can see\nthat the transition field HSPdecreases with increase of\nt. For the samples with t=1.6nm (not shown) we did not\nobserve any detectable SF signal evidencing absence of\ncoupling of Fe and Gd layers.\nTo describe magnetic state of our systems we applied\nextended Stoner-Wohlfarth model widely used for de-3\n/s48 /s50 /s52 /s54/s49/s69/s45/s53/s49/s69/s45/s52/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s110/s101/s117/s116/s114/s111/s110/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s32/s40/s176/s41/s32/s40/s45/s32/s45/s41\n/s32/s40/s43/s32/s43/s41\n/s32/s40/s43/s32/s45/s41/s48 /s50 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s82/s83/s70\n/s49\n/s72/s32/s40/s107/s79/s101/s41/s72\n/s83/s80\nFIG. 2. Polarized neutron reflectivities of sample Pd10 mea-\nsured at T= 10 K at magnetic field H= 1 kOe (symbols)\nand SF curve at T=10 K,H=3kOe (solid line) Inset shows\nthe field dependence of intensity of SF scattering at the first\nBragg peak RSF\n1(H). Vertical arrow denotes the magnetic\nfield at which spin-flop transition takes place.\nscription of magnetic multilayers [8, 38]. Density of mag-\nnetic energy of one Fe/Gd unit cell can be written as\nE(αGd,αFe) =−H[mGdcos(αGd)+mFecos(αFe)]+\nJ1cos(αGd−αFe)+J2cos2(αGd−αFe).\n(1)\nIn Eq.1mX=MXdXis a product of magnetization and\nthickness (magnetic moment), αXis the angle between\nmagnetization and Hof a layer X(X=Fe,Gd). The first\nterm in (1) is Zeeman coupling which tends to align mag-\nnetic moments of the layers along the external field. The\nsecond term is bilinear antiferromagnetic exchange cou-\npling of Fe and Gd layers with strength parameter J1.\nThe third term describes biquadratic coupling tending\nto align the magnetic moments non-collinearly. As seen\nfrom (1) in case J2=0 the transition field can be esti-\nmated as HSP≈J1|mGd−mFe|/mGd·mFe.\nFor every magnetic field Hthe magnetic configuration\nof the system as a function of J1,2can be obtained by\nminimizing energy (1) varying angles αGdandαFe. The\nmagnetizationamplitudes MGd,Feandthicknesses dGd,Fe\nwere taken from PNR and SQUID data and fixed during\ncalculations. The angles α′\nGdandα′\nFecorresponding to\nthe minimum of energy for a given set of HandJ1,2is\nused to construct a theoretical SF reflectivity at the first\nBragg peak in Born approximation:\nRSF\n1,th=c[m2\nGd,⊥+m2\nFe,⊥+\n2mGd,⊥mFe,⊥cosdFe\ndFe+dGd]+Rbg,(2)\nwheremGd(Fe),⊥=mGd(Fe)sinα′\nGd(Fe)is the non-/s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s40/s102/s41\n/s74\n/s49/s44/s32/s74\n/s50/s32/s40/s101/s114/s103/s47/s99/s109/s50\n/s41\n/s116/s32/s40/s110/s109/s41/s32/s74\n/s49\n/s32/s74\n/s50/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s40/s101/s41\n/s74\n/s49/s44/s32/s74\n/s50/s32/s40/s101/s114/s103/s47/s99/s109/s50\n/s41\n/s84/s32/s40/s75/s41/s32/s74\n/s49\n/s32/s74\n/s50/s49/s50/s51/s52\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s32/s72/s40/s107/s79/s101/s41/s40/s100/s41/s32/s80/s100/s49/s48/s32/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110\n/s84/s32/s40/s75/s41/s50/s48/s46/s48/s48\n/s55/s54/s46/s56/s48\n/s49/s51/s51/s46/s54\n/s49/s57/s48/s46/s52\n/s50/s52/s55/s46/s50\n/s51/s48/s52/s46/s48\n/s51/s54/s48/s46/s56\n/s52/s49/s55/s46/s54\n/s52/s55/s52/s46/s52\n/s53/s51/s49/s46/s50\n/s53/s56/s56/s46/s48\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s49/s50/s51/s52/s72/s40/s107/s79/s101/s41/s40/s99/s41/s32/s80/s100/s49/s52/s32/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s52/s46/s48/s48/s48\n/s54/s51/s46/s56/s48\n/s49/s50/s51/s46/s54\n/s49/s56/s51/s46/s52\n/s50/s52/s51/s46/s50\n/s51/s48/s51/s46/s48\n/s51/s54/s50/s46/s56\n/s52/s50/s50/s46/s54\n/s52/s56/s50/s46/s52\n/s53/s52/s50/s46/s50\n/s54/s48/s50/s46/s48/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s49/s50/s51/s52/s72/s40/s107/s79/s101/s41/s40/s98/s41/s32/s80/s100/s49/s50/s32/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s48/s46/s48/s48/s48\n/s52/s54/s46/s56/s48\n/s57/s51/s46/s54/s48\n/s49/s52/s48/s46/s52\n/s49/s56/s55/s46/s50\n/s50/s51/s52/s46/s48\n/s50/s56/s48/s46/s56\n/s51/s50/s55/s46/s54\n/s51/s55/s52/s46/s52\n/s52/s50/s49/s46/s50\n/s52/s54/s56/s46/s48/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s49/s50/s51/s52/s72/s40/s107/s79/s101/s41/s83/s70/s80/s40/s97/s41/s32/s80/s100/s49/s48/s32/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s84/s32/s40/s75/s41/s50/s46/s48/s48/s48\n/s54/s49/s46/s54/s48\n/s49/s50/s49/s46/s50\n/s49/s56/s48/s46/s56\n/s50/s52/s48/s46/s52\n/s51/s48/s48/s46/s48\n/s51/s53/s57/s46/s54\n/s52/s49/s57/s46/s50\n/s52/s55/s56/s46/s56\n/s53/s51/s56/s46/s52\n/s53/s57/s56/s46/s48/s65/s70\n/s78/s111/s32/s99/s111/s117/s112/s108/s105/s110/s103\nFIG. 3. (a)-(c) Experimental ( H,T) maps of RSF\n1for samples\nwith different Pd spacer. (d) Simulated map for Pd10 sample\n(e) Fit-resulted J1andJ2terms vs temperature for Pd10\nsample. (f) Thickness dependence of bilinear and biquadrat ic\nenergies J1andJ2obtained for T=10K.\ncollinearcomponentofmagneticmomentofGd(Fe)layer,\ncis scaling constant and Rbgis background intensity.\nThe latter two values were adjusted manually before the\nfit. We fitted then theoretical RSF\n1,thto the experimental\nH-dependencies RSF\n1by varying J1andJ2. The proce-\ndure was repeated for every Tso that for every sample\nwe obtained temperature dependencies of J1,2. Fig.3d\nshows results of such a fit for sample Pd10. It is rather\nnoticeable that despite of the simplicity of the Stoner-\nWohlfarth approach it allows to reproduce experimen-\ntal features quite well. Fig.3e shows the fit-resulted T-\ndependence of the exchange energies J1andJ2for Pd10\nsample. It can be seen that the bilinear term has a pre-\ndominant contribution, which gradually decreases with\ndecreasing temperature. Thus our analysis showed that\nfor a qualitative description of the SFT, a bilinear term\nis sufficient, but quantitatively the data are described\nbetter by including an additional biquadratic term.\nThe data for the other samples were fitted in a similar\nway. Fig.3fshowsthe dependency ofcouplingenergieson\nthickness of Pd spacer. As followsfrom the figure, the bi-\nlinear energy decreases almost linearly from 1.5 erg/cm2\natt=1nm to 0 at t=1.6nm. Biquadratic energy in turn4\nincreases with t. The obtained values are of the same or-\nders asJ1∼0.8 erg/cm2andJ2∼0.2 erg/cm2obtained\nin Ref.[39] for Gd/Pt/Co multilayers at T=10K.\nThe decrease in the bilinear component with the in-\ncrease in tcan obviously be correlated with a decrease in\nthe effective concentration of Gd in the GdPd layer. At\nthe same time, structural studies carried out earlier [37]\nindicate an increase in structural inhomogeneities with\nincreasing of t. It seems prudent to correlatethis growth\nwith an increase in the biquadratic component.\nIn conclusion, using PNR we performed a\nsystematic study of magnetic configuration of\n[Fe(3.5nm)/Pd(t)/Gd(5.0nm)/Pd(t)] 12heterostruc-\ntures with t=1.0-1.6nm. By measuring neutron spin-flip\nscattering we have detected presence of magnetically\nnon-collinear state at temperatures T/lessorsimilar50 K in mag-\nnetic fields of above H >500 Oe for the samples with\n1nm< t <1.4nm. By using of an extended Stoner-\nWohlfarth model we were able to describe the observed\ntransition as a competition of Zeeman energy, bilinear\ninteraction of order of 1 erg/cm2and biquadratic\naddition of order of 0.5 erg/cm2. The coupling energies\ncan be tuned by varying thickness of spacer between\n1nm and 1.4nm leading to the shift of the transition field\nbelow kilo-Oersted range. Our study opens perspectives\nfor a purposeful design of artificial FEMs with adjustable\nfield of spin-flop transition. Thus, the FEMs systems\nwith low Curie temperature components studied in this\nwork can be used in superconducting spintronics for\ngeneration of triplet superconductivitiy. An additional\nadvantage here is the good compatibility of gadolinium\nwith superconducting niobium [40, 41]. For the room\ntemperature applications one can use well-studied\nsynthetic AFs such as Fe/Cr [33–35], Fe/V [42, 43] or\nCo/Cu [44, 45] where subsequent adjustment can be\ncarried out by tuning of the coupling energy and the\nimbalance of the magnetic moments of the sub-lattices.\nWe would like to thank M.A. Milyaev for assistance\nin preparation of the samples, A.B. Drovosekov and\nD.I. Kholin for fruitful discussion of the results. 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B 72, 054437 (2005)." }, { "title": "1908.07286v2.Peculiarities_in_pseudo_transitions_of_a_mixed_spin___1_2_1___Ising_Heisenberg_double_tetrahedral_chain_in_an_external_magnetic_field.pdf", "content": "Peculiarities in pseudo-transitions of a mixed spin- (1=2;1)Ising-Heisenberg\ndouble-tetrahedral chain in an external magnetic field\nOnofre Rojas,1Jozef Strečka,2Oleg Derzhko,3and S. M. de Souza1\n1Departamento de Física, Universidade Federal de Lavras, CP 3037, 37200-000, Lavras-MG, Brazil\n2Department of Theoretical Physics and Astrophysics, Faculty of Science,\nP. J. Šafárik University, Park Angelinum 9, 040 01 Košice, Slovakia\n3Institute for Condensed Matter Physics, National Academy of\nSciences of Ukraine, Svientsitskii Str. 1, 79011 L’viv, Ukraine\nAbstract\nRecently, it has been rigorously verified that several one-dimensional (1D) spin models may exhibit a peculiar pseudo-\ntransition accompanied with anomalous response of thermodynamic quantities in a close vicinity of pseudo-critical temperature.\nIn the present work we will introduce and exactly solve a mixed spin-(1/2,1) Ising-Heisenberg double-tetrahedral chain in an\nexternal magnetic field as another particular example of 1D lattice-statistical model with short-range interactions that displays\na pseudo-transition of this type. The investigated model exhibits at zero temperature three ferrimagnetic phases, three frus-\ntrated phases, and one saturated paramagnetic phase. The ground-state phase diagram involves five unusual interfaces (phase\nboundaries), at which the residual entropy per site equals to a larger entropy of one of two coexisting phases. Four such inter-\nfaces are between a non-degenerate ferrimagnetic phase and a macroscopically degenerate frustrated phase, while one interface\nis between two non-degenerate ferrimagnetic phases. Though thermal excitations typically destroy all fingerprints of zero-\ntemperature phase transitions of 1D lattice-statistical models with short-range forces, the mixed spin-(1/2,1) Ising-Heisenberg\ndouble-tetrahedral chain is quite robust with respect to thermal excitations and it displays peculiar pseudo-transitions close to\nall five aforementioned interfaces.\nPACS numbers: 05.70.Fh, 75.10.-b, 75.10.Jm, 75.10.Pq\nKeywords: Residual entropy; Quasi-phases; Pseudo-transitions; Ising-Heisenberg\nI. INTRODUCTION\nThere are a few paradigmatic examples of one-\ndimensional (1D) lattice-statistical models with short-\nrange couplings, which exhibit a discontinuous (first-\norder) phase transition at finite temperature. Perhaps\nthe most famous example is 1D KDP model of hydrogen-\nbonded ferroelectrics invented by Nagle [1], which dis-\nplays a discontinuous phase transition between the fer-\nroelectric and paraelectric phases due to assignment of\nan infinite energy to all ionized configurations. Another\nparticular example of this type is the Kittel model [2]\ndefined through a finite transfer matrix, which involves a\nconstraint on zipper corresponding to an infinite poten-\ntial being responsible for a non-analyticity of the free en-\nergy. Owing to a singular character of the potential, the\nKittel model also exhibits a first-order phase transition.\nThe next paradigmatic example is the 1D solid-on-solid\nmodel considered by Chui and Weeks [3], which is ex-\nactlysolvableinspiteofaninfinitedimensionofitstrans-\nfer matrix. By imposing suitable pinning potential the\n1D solid-on-solid model may also display a roughening\nphase transition of first order [3]. Furthermore, Daux-\nois and Peyrard [4] have examined another 1D lattice-\nstatistical model with an infinite dimension of the trans-\nfer matrix, which exhibits a phase transition at finite\ntemperature. Last but not least, Sarkanych et al. [5]\nproposed 1D Potts model with so-called invisible states\nand short-range couplings. It could be thus concluded\nthat all five aforementioned 1D lattice-statistical models\nbreak the Perron-Frobenius theorem, because some off-\ndiagonal transfer-matrix elements become null and thefree energy may consequently become non-analytic at a\ncertain critical temperature.\nVan Hove [6] proposed a theorem that proves absence\nof a phase transition in 1D lattice-statistical models with\nshort-range couplings. Later, Cuesta and Sanchez [7]\ngeneralized the non-existence theorem for a phase tran-\nsition at finite temperatures. Surely, this is not yet\nthe most general non-existence theorem, because mixed-\nparticle chains or more general external fields fall beyond\nthe scope of this theorem.\nThe term \"pseudo-transition\" and \"quasi-phase\" was\nintroduced by Timonin [8] in 2011 when studying the\nspin-ice model in a field. These terms refer to a sudden\nchange in first derivatives and vigorous peaks in second\nderivatives of the free energy although these marked sig-\nnatures are not in reality true discontinuities and diver-\ngences, respectively. Note furthermore that the pseudo-\ntransitions do not violate the Perron-Frobenius theorem,\nbecause the free energy is always analytic. A com-\nmon feature of the pseudo-transitions is that some off-\ndiagonal transfer-matrix elements (Boltzmann factors)\nbecome very small (almost zero), since very high albeit\nfinite energy is assigned to the corresponding states.\nObvious fingerprints of pseudo-transitions were re-\ncently found in several 1D spin or spin-electron models.\nFor instance, the pseudo-transitions were detected in the\nspin-1/2 Ising-Heisenberg diamond chain [9, 10], two-leg\nladder [11], as well as triangular tube [12]. Similarly,\nthe emergence of pseudo-transitions was verified in the\nspin-1/2 Ising diamond chain [13] and the coupled spin-\nelectron double-tetrahedral chain [14–16]. In general, the\nfirst derivatives of the free energy such as entropy, inter-\n1arXiv:1908.07286v2 [cond-mat.stat-mech] 27 Sep 2019nal energy or magnetization show a steep change around\npseudo-critical temperature. This feature is similar to\nthe first-order phase transition, but all thermodynamic\nresponse functions are in fact continuous. Contrary to\nthis, second derivatives of the free energy such as spe-\ncific heat and magnetic susceptibility resemble typical\nbehavior of a second-order phase transition at a finite\ntemperature. Therefore, this peculiar pseudo-critical be-\nhavior drew attention to a more comprehensive study of\nthisphenomenonaimedatelucidatingallitsessentialfea-\ntures [17–19]. Recently, a further attention has been paid\nto uncover the mechanism triggering pseudo-transitions\nbased on a rigorous analysis of the correlation function\n[10] and pseudo-critical exponents [20].\nThe goal of the present study is to investigate a mixed\nspin-(1/2,1) Ising-Heisenberg tetrahedral chain in an ex-\nternal magnetic field, which has a pretty rich ground-\nstate phase diagram and exhibits a number of finite-\ntemperature pseudo-transitions close to some inter-phase\nboundaries.There are some 3D compounds in which,\nwhen we consider one columnar stripe, we could ob-\nserve a double tetrahedral chain structure. Such as\ncobalt oxide RBaCo 4O7, where Rdenotes a rare earth\natom, which has a swedenborgite lattice structure[21].\nAnother compound with a similar structure could be\nthe salt with 3D corrugated packing frustrated spin\n[22] of C\u000f\u0000\n60in (MDABCO+)(C\u000f\u0000\n60)[MDABCO+=N-\nmethyldiazabicyclooctanium cation and C\u000f\u0000\n60radical an-\nions], a stripe of this salt can be viewed also as a double-\ntetrahedral chain.\nThis article is organized as follows. In Sec. II we\nconsider and exactly solve the mixed spin-(1/2,1) Ising-\nHeisenberg tetrahedral chain in a magnetic field. Ther-\nmodynamics in a close vicinity of the pseudo-transition\nis examined in Sec. III, where an influence of the residual\nentropy upon basic thermodynamic quantities is investi-\ngated in detail. Finally, several concluding remarks are\npresented in Sec. IV.\nII. MIXED SPIN-( 1=2;1)ISING-HEISENBERG\nDOUBLE-TETRAHEDRAL CHAIN\nThe coupled spin-electron model on a double-\ntetrahedral chain [14–16], which involves localized Ising\nspins at nodal lattice sites and mobile electrons delo-\ncalized over triangular plaquettes, represents a promi-\nnent example of 1D lattice-statistical model mimicking a\ntemperature-driven phase transition [14]. However, ear-\nlier investigations of the analogous spin-1/2 Heisenberg\n[23–25] and Ising-Heisenberg [26, 27] models on a double-\ntetrahedral chain did not verify anomalous thermody-\nnamic response closely related to a pseudo-transition un-\ntil the latter Ising-Heisenberg model was revisited and\nmore thoroughly studied [18].\nJ0\nJJzSa,i \nSb,i Sb,i +1 Sa,i +1 \nSc,i +1 Sc,i \nσi σi+1 \nIsing-Heisenberg interaction Heisenberg interaction Heisenberg spin-1 Ising spin-1/2 Figure 1: A schematic representation of the mixed spin-\n(1/2,1) Ising-Heisenberg double-tetrahedral chain. Small\nballs correspond to the Ising spins \u001biand large balls corre-\nspond to the Heisenberg spins S\r;i(\r=a;b;c ).\nIn the present work we will examine in particular the\nmixed spin-( 1=2;1) Ising-Heisenberg double-tetrahedral\nchain, which is schematically depicted in Fig. 1 and de-\nfined through the following Hamiltonian\nH=NX\ni=1Hi; (1)\nwith\nHi=\u0000[J(Sb;i;Sc;i)z+J(Sc;i;Sa;i)z+J(Sa;i;Sb;i)z]\n\u0000\u0000\nSz\na;i+Sz\nb;i+Sz\nc;i\u0001\n[hz+J0(\u001bi+\u001bi+1)]\u0000h\n2(\u001bi+\u001bi+1):\n(2)\nIn above,S\u000b\n\r;i(\u000b=fx;y;zg,\r=fa;b;cg) denote the\nspin-1 Heisenberg atoms, \u001bi=\u00061\n2denotes the Ising spin,\nandJ(S\r;i;S\u000e;i)z=JSx\n\r;iSx\n\u000e;i+JSy\n\r;iSy\n\u000e;i+JzSz\n\r;iSz\n\u000e;i.\nThe Hamiltonian (2) is written as a sum of cell Hamilto-\nniansHi, which correspond to spin clusters with the geo-\nmetric shape of two face-sharing tetrahedra (i.e., trigonal\nbipyramid).\nThe overall Hilbert space of the mixed spin-( 1=2;1)\nIsing-Heisenberg double-tetrahedral chain splits into sev-\neral disjoint (orthogonal) subspaces, because the Hamil-\ntoniansHifrom different unit cells commute with each\nother. The Hilbert subspace corresponding to the spin-\n1 Heisenberg triangle from the i-th unit cell is given by\nthe Hamiltonian matrix of dimension 27\u000227and it can\nbe further split into several smaller block-diagonal matri-\nces depending on the z-component of the total spin: for\nSz\nt= 0one has one 7\u00027block matrix, for jSz\ntj= 1two\n6\u00026matrices, forjSz\ntj= 2two3\u00023matrices, and for\njSz\ntj= 3two1\u00021matrices. All eigenvalues and eigenvec-\ntors of spin-1 Heisenberg triangle Hamiltonian are listed\nin Table I. The first column stands for the eigenvalues of\ntheSz\ntoperator, while the counter kis used just to dis-\ntinguish the states with same eigenvalues and the respec-\ntive state degeneracy gkin fourth column. With the help\nof eigenvalues and eigenvectors of the spin-1 Heisenberg\ntriangle reported in Table I one can express the full en-\nergy spectrum per Hiunit cell of the mixed spin-( 1=2;1)\nIsing-Heisenberg double-tetrahedral chain as follows\n\"k(\u001bi;\u001bi+1) =\u000fk\u0000\u0012\nJ0Sz\nt+h\n2\u0013\n(\u001bi+\u001bi+1):(3)\n2Here,\u000fkmarks the respective eigenvalue of the spin-1 Heisenberg triangle listed in Table I.\nTable I: Full spectrum of the spin-1 Heisenberg triangle specified according to the respective eigenvalue, state degeneracy, and\neigenvector. The eigenstates are grouped according to the z-component of the total spin Sz\nt=Sz\na+Sz\nb+Sz\nc. The first column\nstands for the eigenvalues of the Sz\ntoperator, and the second column is just to distinguish the eigenvector with the same Sz\nt.\nThe definition of mixing angles: cot (2\u001e1) =Jz\u0000J\n2J,cot (2\u001e2) =Jz+2J\n4J, and cot (2\u001e3) =Jz\u00002J\n2p\n6J.\njSz\ntjkEnergy (\u000fk) gkState\n00J+Jz 2|0,0i=8\n>><\n>>:1\n2\u0012\f\f\f\f1\n0\n\u00001\u001d\n\u0000\f\f\f\f0\n1\n\u00001\u001d\n\u0000\f\f\f0\n\u00001\n1E\n+\f\f\f\u00001\n0\n1E\u0013\np\n3\n6\u0012\n2\f\f\f1\n\u00001\n0E\n\u0000\f\f\f\f1\n0\n\u00001\u001d\n\u0000\f\f\f\f0\n1\n\u00001\u001d\n\u0000\f\f\f0\n\u00001\n1E\n\u0000\f\f\f\u00001\n0\n1E\n+ 2\f\f\f\u00001\n1\n0E\u0013\n1Jp\n6 cot\u001e3 1|0,1i=p\n6\n6cos\u001e3\u0012\f\f\f\f1\n0\n\u00001\u001d\n+\f\f\f1\n\u00001\n0E\n+\f\f\f\f0\n1\n\u00001\u001d\n+\f\f\f0\n\u00001\n1E\n+\f\f\f\u00001\n1\n0E\n+\f\f\f\u00001\n0\n1E\u0013\n-sin\u001e3\f\f\f0\n0\n0E\n2\u0000Jp\n6 tan\u001e3 1j0;2i=p\n6\n6sin\u001e3\u0012\f\f\f\f1\n0\n\u00001\u001d\n+\f\f\f1\n\u00001\n0E\n+\f\f\f\f0\n1\n\u00001\u001d\n+\f\f\f0\n\u00001\n1E\n+\f\f\f\u00001\n1\n0E\n+\f\f\f\u00001\n0\n1E\u0013\n+cos\u001e3\f\f\f0\n0\n0E\n3\u0000J+Jz 2j0;3i=8\n>><\n>>:1\n2\u0012\n\u0000\f\f\f\f1\n0\n\u00001\u001d\n\u0000\f\f\f\f0\n1\n\u00001\u001d\n+\f\f\f0\n\u00001\n1E\n+\f\f\f\u00001\n0\n1E\u0013\np\n3\n6\u0012\n\u00002\f\f\f1\n\u00001\n0E\n\u0000\f\f\f\f1\n0\n\u00001\u001d\n\u0000\f\f\f0\n\u00001\n1E\n+\f\f\f\f0\n1\n\u00001\u001d\n+\f\f\f\u00001\n0\n1E\n+ 2\f\f\f\u00001\n1\n0E\u0013\n4 2J+Jz 1j0;4i=p\n6\n6\u0012\n\u0000\f\f\f\f1\n0\n\u00001\u001d\n+\f\f\f1\n\u00001\n0E\n+\f\f\f\f0\n1\n\u00001\u001d\n\u0000\f\f\f0\n\u00001\n1E\n\u0000\f\f\f\u00001\n1\n0E\n+\f\f\f\u00001\n0\n1E\u0013\n15\n6\u00002J(1\u0000cot\u001e2)\u0006hz1j\u00061;0i=p\n3\n3\u0014\ncos\u001e2\u0012\f\f\f\f\u00061\n\u00061\n\u00071\u001d\n+\f\f\f\f\u00061\n\u00071\n\u00061\u001d\n+\f\f\f\f\u00071\n\u00061\n\u00061\u001d\u0013\n\u0000sin\u001e2\u0012\f\f\f\u00061\n0\n0E\n+\f\f\f0\n\u00061\n0E\n+\f\f\f\f0\n0\n\u00061\u001d\u0013\u0015\n7\n8\u00002J(1 + tan\u001e2)\u0006hz1j\u00061;1i=p\n3\n3\u0014\nsin\u001e2\u0012\f\f\f\f\u00061\n\u00061\n\u00071\u001d\n+\f\f\f\f\u00061\n\u00071\n\u00061\u001d\n+\f\f\f\f\u00071\n\u00061\n\u00061\u001d\u0013\n+ cos\u001e2\u0012\f\f\f\u00061\n0\n0E\n+\f\f\f0\n\u00061\n0E\n+\f\f\f\f0\n0\n\u00061\u001d\u0013\u0015\n9\n10J(1 + cot\u001e1)\u0007hz 2j\u00061;2i=8\n>><\n>>:p\n2\n2\u0014\nsin\u001e1\u0012\f\f\f\f0\n0\n\u00061\u001d\n\u0000\f\f\f0\n\u00061\n0E\u0013\n+ cos\u001e1\u0012\f\f\f\f\u00061\n\u00061\n\u00071\u001d\n\u0000\f\f\f\f\u00061\n\u00071\n\u00061\u001d\u0013\u0015\np\n6\n6\u0014\ncos\u001e1\u0012\n2\f\f\f\f\u00071\n\u00061\n\u00061\u001d\n\u0000\f\f\f\f\u00061\n\u00071\n\u00061\u001d\n\u0000\f\f\f\f\u00061\n\u00061\n\u00071\u001d\u0013\n+ sin\u001e1\u0012\n2\f\f\f\u00061\n0\n0E\n\u0000\f\f\f0\n\u00061\n0E\n\u0000\f\f\f\f0\n0\n\u00061\u001d\u0013\u0015\n11\n12J(1\u0000tan\u001e1)\u0007hz 2j\u00061;3i=8\n>><\n>>:p\n2\n2\u0014\ncos\u001e1\u0010\f\f\f\u00061\n0\n0E\n\u0000\f\f\f0\n\u00061\n0E\u0011\n+ sin\u001e1\u0012\f\f\f\f\u00061\n\u00071\n\u00061\u001d\n\u0000\f\f\f\f\u00071\n\u00061\n\u00061\u001d\u0013\u0015\np\n6\n6\u0014\nsin\u001e1\u0012\n2\f\f\f\f\u00061\n\u00061\n\u00071\u001d\n\u0000\f\f\f\f\u00061\n\u00071\n\u00061\u001d\n\u0000\f\f\f\f\u00071\n\u00061\n\u00061\u001d\u0013\n\u0000cos\u001e1\u0012\n2\f\f\f\f0\n0\n\u00061\u001d\n\u0000\f\f\f0\n\u00061\n0E\n\u0000\f\f\f\u00061\n0\n0E\u0013\u0015\n213\n14J\u0000Jz\u00072hz 2j\u00062;0i=8\n>><\n>>:p\n2\n2\u0012\f\f\f\u00061\n\u00061\n0E\n\u0000\f\f\f\f0\n\u00061\n\u00061\u001d\u0013\np\n6\n6\u0012\f\f\f\u00061\n\u00061\n0E\n\u00002\f\f\f\f\u00061\n0\n\u00061\u001d\n+\f\f\f\f0\n\u00061\n\u00061\u001d\u0013\n15\n16\u00002J\u0000Jz\u00072hz 1j\u00062;1i=p\n3\n3\u0012\f\f\f\u00061\n\u00061\n0E\n+\f\f\f\f\u00061\n0\n\u00061\u001d\n+\f\f\f\f0\n\u00061\n\u00061\u001d\u0013\n317\n18\u00003Jz\u00073hz 1j\u00063;0i=\f\f\f\f\u00061\n\u00061\n\u00061\u001d\nA. Ground-state phase diagram\nThe ground-state phase diagram shown in Fig. 2(a) to-\ntallyinvolvessevenphasesspecifiedbelow. First, thesat-\nuratedparamagneticphase( SA)hasaccordingtoEq.(3)\nthe following energy per unit cell\nESA=\u00003J0\u00003Jz\u00003hz\u00001\n2h; (4)\nwhich corresponds to the eigenstate defined through the\neigenvectorj3;0iispecified in Table I\njSAi=NY\ni=1j3;0iij+ii: (5)\nObviously, both Ising spin magnetization per unit cell\n(mI=1\n2)andHeisenbergspinmagnetizationperunitcell(mH= 3) are fully polarized, and total magnetization\nper unit cell attains the following value mt=mI+mH=\n7\n2.\nThe ground-state phase diagram shown in Fig. 2(a)\nalso displays three different ferrimagnetic ( FI) phases.\nThe ground-state energy of the first ferrimagnetic phase\nFI1reads\nEFI1=2J+Jz\u00001\n2h; (6)\nwhereas its corresponding eigenvector is given by\njFI1i=NY\ni=1j0;4iij+ii (7)\nwith the eigenvector j0;4iidefined in Table I. In the first\nferrimagnetic phase FI1the Ising spin magnetization is\n3Jz h\n0 10 5 20 15 10 \n020 30 40 50 60 70 \nqF I 1qF I 2qF I 3\nqFR 1qSA \nqFR \n2qFR \n3S(a) (b) \nSA \nFR 1FR \n2FR \n3\nF I 1F I 2F I 3\nJz h\n10 \n020 30 40 50 60 70 \n0 10 5 20 15 Figure 2: (a) Ground-state phase diagram in the Jz\u0000hplane\nby assuming the fixed parameters J=\u000010,J0=\u000010, and\nhz=h; (b) Density-plot of entropy in the Jz\u0000hplane for\nthe same set of parameters as in (a) at T= 0:4.\nmI=1\n2, the Heisenberg spin magnetization equals zero\nmH= 0, and the total magnetization thus becomes mt=\n1\n2.\nThe ground-state energy for the second ferrimagnetic\nphaseFI2can be expressed as\nEFI2=\u0000J0\u00002J(1\u0000cot\u001e2)\u0000hz\u00001\n2h;(8)\nwhere cot (2\u001e2) =Jz+2J\n4Jwith\u0000\u0019\n4<\u001e 2<\u0019\n4. The corre-\nsponding eigenvector reads\njFI2i=NY\ni=1j1;1iij+ii (9)\nwith the eigenvector j1;1iidefined in Table I. The Ising\nspinmagnetizationinthesecondferrimagneticphase FI2\nbecomesmI=1\n2, the Heisenberg spin magnetization is\nmH= 1, and the total magnetization is mt=3\n2.\nThe ground-state energy for the third ferrimagnetic\nphaseFI3is given by\nEFI3=3J0\u00003Jz\u00003hz+1\n2h; (10)\nwhereas its corresponding eigenvector reads\njFI3i=NY\ni=1j3;0iij\u0000ii (11)\nwiththeeigenvector j3;0iibeingdefinedinTableI.Anal-\nogouslytothepreviouscase, theIsingspinmagnetization\nis given by mI=\u00001\n2, the Heisenberg spin magnetization\nequalstomH= 3, andthetotalmagnetizationis mt=5\n2.\nIt should be pointed out that the saturated paramagnetic\nphase as well as all three ferrimagnetic phases are non-\ndegenerate, whichmeansthatthereisnoresidualentropy\nS= 0at zero temperature within those ground states.\nHowever, the ground state of the mixed spin-( 1=2;1)\nIsing-Heisenberg double-tetrahedral chain may be one of\nthree frustrated ( FR) phases with a nonzero residual en-\ntropy. The ground-state energy of the first frustrated\nphaseFR1is given by\nEFR1=J0\u0000J(1 + cot\u001e1)\u0000hz+1\n2h;(12)where cot (2\u001e1) =Jz\u0000J\n2Jwith\u0000\u0019\n4<\u001e 1<\u0019\n4. The corre-\nsponding ground-state eigenvector reads as follows\njFR1i=NY\ni=1j1;3iij\u0000ii; (13)\nwhere two-fold degenerate eigenstate j1;3iiis specified\nin Table I. Owing to this fact, the frustrated phase FR1\nis macroscopically degenerate with the residual entropy\nS= ln(2)per unit cell when the entropy is measured in\nunits of the Boltzmann constant kB. Note that the Ising\nspin magnetization is being mI=\u00001\n2, the Heisenberg\nspin magnetization is mH= 1, and the total magnetiza-\ntion becomes mt=1\n2.\nTheground-stateenergyofthesecondfrustratedphase\nFR2can be expressed as follows\nEFR2= 2J0+J\u0000Jz\u00002hz+1\n2h (14)\nand its respective eigenvector is given by\njFR2i=NY\ni=1j2;0iij\u0000ii: (15)\nThe definition of two-fold degenerate eigenstate j2;0iiis\nreported in Table I, which implies that the second frus-\ntrated phase FR2also has residual entropy S= ln(2).\nThe Ising spin magnetization is mI=\u00001\n2, the Heisen-\nberg spin magnetization is mH= 2, and the total mag-\nnetization results in mt=3\n2.\nThe ground-state energy of the third frustrated phase\nFR3follows from the relation\nEFR3=\u00002J0+J\u0000Jz\u00002hz\u00001\n2h; (16)\nwhereas its respective eigenvector reads\njFR3i=NY\ni=1j2;0iij+ii: (17)\nThe two-fold degenerate eigenvector j2;0iiis defined\nin Table I and hence, the third frustrated phase FR2\nis macroscopically degenerate with the residual entropy\nS= ln(2) per unit cell. The corresponding Ising spin\nmagnetization achieves the value mI=1\n2, the Heisen-\nberg spin magnetization equals to mH= 2, and the total\nmagnetization is given by mt=5\n2.\nUsually, plots can be drawn in units of some parame-\nters likeJ, and then the temperature can be measured\nin unitsJ. However, here for convenience, we set the pa-\nrameters to be J=\u000010andJ0=\u000010, just for scale the\ntemperature by a factor 10. From now on, we will con-\nsider this set of parameters to study the pseudo-critical\ntemperature throughout the article.\nAll dashed lines in Fig. 2(a) represent usual ground-\nstate phase boundaries between two phases. The resid-\nual entropy per unit cell at the phase boundary between\nFR1andFR2becomesS= ln(4). Similarly, the residual\n4entropy at the interface between FR2andFI3equals to\nS= ln(3), while the residual entropy at the phase bound-\nary between FI3andSAequals toS= ln(2). Analo-\ngously, the residual entropy attains the value S= ln(3)\nat phase boundaries between SA\u0000FR3andFR3\u0000FI2.\nFinally, the residual entropy becomes S= ln(2)at the\ninterface between FI2andFI3. In all aforementioned\ncases the residual entropy per unit cell is always higher\nthan the entropy of both individual phases, which coexist\ntogether at a relevant ground-state boundary. By con-\ntrast, solid lines represent all unusual phase boundaries\nbetween two phases. The residual entropy per unit cell\nS= ln(2)can be found at interfaces between the phases\nFR1-FI1,FR2-FI1,FR2-FI2, andFR3-FI3, whereas\ntheresidualentropyperunitcellvanishes S= 0atthein-\nterface between two non-degenerate ferrimagnetic phases\nFI2andFI3.\nIII. THERMODYNAMICS\nThe mixed spin-(1/2,1) Ising-Heisenberg double-\ntetrahedral chain can be mapped onto the effective spin-\n1/2 Ising chain given by the Hamiltonian\nH=\u0000NX\ni=1\u0002\nK0+Ksisi+1+1\n2B(si+si+1)\u0003\n;(18)\nwhereK0,K, andBare effective temperature-dependent\nparameters. Bearing this in mind, thermodynamics of\nthe effective spin-1/2 Ising chain can be expressed in\nterms of the transfer matrix V=\"\nw1w0\nw0w\u00001#\naccord-\ning to the procedure previously discussed in Ref. [17].\nEach element of the transfer matrix (Boltzmann factor)\nwnwithn=f\u00001;0;1g, which will be further referred to\nas the sector, can be defined as\nwn=18X\nk=0gn;ke\u0000\f\"n;k; (19)\nwhere\f= 1=(kBT),kBisBoltzmann’sconstant, Tisthe\nabsolute temperature and the eigenvalues \"n;kare given\nby Eq. (3).\nTo be more specific, the Boltzmann factors are explic-\nitly given by\nwn=un(\nq3;nz6+\u0012\nx4+2\nx2\u0013\nz2q2;n+\u0000\n2t+x\u00004\u0001\nz2\n+1\nz\u0014\u00122y1\nx+x2y2\u0013\nq1;n+x2y3\u0015\u001b\n;(20)\nwherex= e\fJ=2,z= e\fJz=2,u= e\fh=2,t= 2 cosh (\fJ),\nwhile the coefficients yrandqr;nwithr=f1;2;3gare\ndefined as follows\nyr=2 cosh [\fJcsc (2\u001er)]; (21)\nqr;n=2 cosh [r\f(nJ0+hz)]: (22)The transfer-matrix eigenvalues are determined by the\nfollowing equation\n\u0015\u0006=1\n2\u0010\nw1+w\u00001\u0006q\n(w1\u0000w\u00001)2+ 4w2\n0\u0011\n:(23)\nConsidering the effective spin-1/2 Ising chain under a pe-\nriodic boundary condition gives the partition function\nZN=\u0015N\n++\u0015N\n\u0000. Consequently, the free energy can be\nobtained in the thermodynamic limit ( N!1) accord-\ning to the formula\nf=\u00001\n\fln\u0014\n1\n2\u0010\nw1+w\u00001+q\n(w1\u0000w\u00001)2+ 4w2\n0\u0011\u0015\n:\n(24)\nSubstituting Boltzmann’s factors wninto Eq. (24), we\ncan exactly calculate the free energy of the mixed spin-\n(1/2,1) Ising-Heisenberg double-tetrahedral chain at fi-\nnite temperature.\nIt has been recently demonstrated [17] that some 1D\nlattice-statistical models satisfy the following condition\njw1\u0000w\u00001j\u001dw0at low enough temperatures. Under\nthis condition, the free energy of the mixed spin-(1/2,1)\nIsing-Heisenberg double-tetrahedral chain reduces to\nf=\u0000Tlnfmax [w1(T);w\u00001(T)]g:(25)\nThe final formula for the free energy per unit cell (24)\ntakes the following simple form at a phase boundary be-\ntween the individual phases with the same energy \"c\nf=\"c\u0000Tln [max (g1;0;g\u00001;0)]: (26)\nConsequently, the residual entropy per unit cell at a rel-\nevant phase boundary reads\nSc= ln [max (g1;0;g\u00001;0)]: (27)\nKnowing this quantity is sufficient for prediction of a\npseudo-transition at finite temperatures [18].\nIn Fig. 2(b) we illustrate the density plot of the en-\ntropy as a function of Jzandhfor the fixed temperature\nT= 0:4by using the same scale as in the ground-state\nphase diagram shown in Fig. 2(a). It is quite evident\nthat the entropy follows the vestige of zero-temperature\nphase diagram at finite temperatures. The notation for\nthe ground state is changed at finite temperatures by\nadding a prefix \" q\" to the name of respective ground\nstates, which will denote the respective quasi-phase [8]\nbecause of a lack of true spontaneous long-range order at\nfinite temperatures. It could be expected that thermal\nexcitations basically influence the phase boundaries. It\nhas been argued previously that all dashed curves dis-\nplayed in Fig. 2(a) describe standard interfaces, which\nare manifested through an increase of the entropy ex-\nceedingtheentropyvalueofbothcoexistingphases. Con-\ntrary to this, the phase boundaries depicted by solid lines\nin Fig. 2(a) behave quite differently, since they show at\nthe respective interface a sharp rise of the entropy to a\ngreater entropy of one of two coexisting phases.\n5(f) (e) (d) (c) (b) (a) qFR 1\nhT\nqF I 1\nqSA qFR 2\nqF I 3\nhT\nhT\nhT\nhT\nhTqF I 1\nqSA qFR 2\nqF I 3\nqF I 1\nqSA qFR 2\nqF I 3qFR 1qF I 1\nqSA qF I 3qF I 2qFR 2\nqF I 1\nqSA qF I 3qF I 2\nqF I 1\nqSA qF I 3qF I 2\nqFR 3\nmIFigure 3: Density plot of Ising spin magnetization in the\nT\u0000hplane for the fixed values of the coupling constants\nJ=\u000010,J0=\u000010, and several values of Jz: (a)Jz=\u000011;\n(b)Jz=\u000013; (c)Jz=\u000015; (d)Jz=\u000015:65; (e)Jz=\u000019;\n(f)Jz=\u000019:85.\nThedensityplotofIsingspinmagnetizationisdepicted\nin Fig. 3 in the T\u0000Jzplane for the following set of\nparameters J=\u000010andJ0=\u000010. In this figure, yellow\nregion corresponds to spin ’up’ ( mI= 1=2), cyan region\ncorresponds to spin ’down’ ( mI=\u00001=2), and red region\ncorresponds to null Ising magnetization ( mI= 0). Surely\nthe temperature in units of T=jJjwould be divided by a\nfactor 10 in Fig. 3 and the following figures.\n(f) (e) (d) (c) (b) (a) \nqF I 1\nqSA qFR 2\nqF I 3qFR 1 qF I 1\nqSA qF I 3qF I 2\nqF I 1\nqSA qF I 3qF I 2\nqF I 1\nqSA qF I 3qF I 2qFR 2\nqFR 3hT\nhT\nqF I 1\nqSA qFR 2\nqF I 3qFR 1\nhT\nqF I 1\nqSA qFR 2\nqF I 3\nhT\nhT\nhTmH\nFigure 4: Density plot of Heisenberg spin magnetization in\ntheT\u0000hplane for the fixed values of the coupling constants\nJ=\u000010,J0=\u000010, and several values of Jz: (a)Jz=\u000011;\n(b)Jz=\u000013; (c)Jz=\u000015; (d)Jz=\u000015:65; (e)Jz=\u000019;\n(f)Jz=\u000019:85.\nThe density plot of the Heisenberg spin magnetization\nis depicted in Fig. 4 in the T\u0000Jzplane for the same\nset of parameters J=\u000010andJ0=\u000010. The color\ncode for the density plot is as follows: yellow region\ncorresponds to the saturated Heisenberg magnetization\nmH= 3, cyan region corresponds to the null Heisenberg\nmagnetization mH= 0, orange region corresponds to the\nmoderate Heisenberg magnetization mH= 2, and dark\nTξξξξ2\nξ\nqF I 1 qFR 1(a) \nqF I 1 qFR 2(b) \n10 3\n10 2qF I 2 qF I 3 (d) \n5\n0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0qF I 3\nqFR 3(e) qFR 2 qF I 2(c) Figure 5: Correlation length against temperature for the\nfixed parameters J=\u000010,J0=\u000010and several values of Jz\nandhz=h: (a)h= 4,Jz=\u000011; (b)h= 11;Jz=\u000011:5;\n(c)h= 26,Jz=\u000015:6; (d)h= 36:76,Jz=\u000017; (e)h= 52;\nJz=\u000019:9.\nred region corresponds to the moderate Heisenberg mag-\nnetizationmH= 1. It can be seen from Figs. 3 and 4\nthat the pseudo-transitions between the quasi-phases is\naccompanied with abrupt change in the magnetization of\nthe Ising spins and/or the magnetization of the Heisen-\nbergspins. ThedensityplotsshowninFig.4(a)-(f)imply\na full alignment of the Heisenberg spins just within the\nquasi-phases qFI 3andqSA.\nNow, let us analyze the correlation length, which can\nbe calculated according to the following simple relation\n\u0018=\u0014\nln\u0012\u0015+\n\u0015\u0000\u0013\u0015\u00001\n: (28)\nThe correlation length is depicted in Fig. 5 as a func-\ntion of temperature for the fixed parameters J=\u000010,\nJ0=\u000010, andhz=h. It is advisable to follow the\nzero-temperature phase diagram to interpret the relevant\ndependences of the correlation length. In Fig. 5(a) we il-\nlustrate the correlation length for h= 4andJz=\u000011,\nwhereas the shark peak delimits the quasi-phases qFI 1\nandqFR 1in agreement with the ground-state phase\nphase diagram shown in Fig. 2(a). Although the cor-\nrelation length seems to diverge at a pseudo-critical tem-\nperature, it is in fact just a sharp finite peak. In Fig. 5(b)\none observes a similar curve for h= 11andJz=\u000011:5,\nbut now the peak indicates a pseudo-transition between\nthe quasi-phases qFI 1andqFR 2. Fig. 5(c) depicts the\ncorrelation length for h= 30andJz=\u000015:65, whereas\nthe sharp peak determines a pseudo-transition between\nthe quasi-phases qFI 2andqFR 2. Similarly, the corre-\nlation length plotted in Fig. 5(d)-(e) demonstrates that\n6a pseudo-transition between the quasi-phases qFI 3-qFI 2\nandqFI 3-qFR 3are accompanied with a sharp robust\npeak of the correlation length. It is worthy to men-\ntion that the quasi-phases melt smoothly upon increas-\ning temperature when the temperature is higher than the\npseudo-critical temperature.It is quite clear from Eq. (25) that the pseudo-critical\ntemperature Tpcan be alternatively obtained by solving\nthe equation\nw1(Tp) =w\u00001(Tp): (29)\nhpTp\nJz =−10 .3Jz =−12 Jz =−14 Jz \n=\n−15 \n.5\n−15 .565 −15 .58 \n−15 .6\n−15 .65 −15 \n.7−16 .5−18 −18 \n−19 .35 \n−19 .6\n−19 .8\n−19 .9−19 \n.35 \nFigure 6: Pseudo-critical temperature as a function of the\nmagnetic field for the fixed values of interaction parameters\nJ=\u000010,J0=\u000010,hz=hand several values of Jz.\nThe numerical solution of Eq. (29) allows us to plot the\npseudo-critical temperature Tpagainst the magnetic field\nhpforseveralvaluesof Jz(seeFig.6). Forsufficientlylow\nmagnetic fields 0Tp. Contrary to this, the Heisenberg spins almost\ndo not contribute to the total magnetization ( mH= 0)\nbelowthepseudo-criticaltemperature T Tp. Last but not least, the specific heat and mag-\nnetic susceptibility displayed in Fig. 7(d)-(e) in a semi-\nlogarithmic scale serve in evidence of a pseudo-transition\nthrough a strong narrow peak observable at the pseudo-\ncritical temperature.\nTemperature dependences of selected thermodynamic\nquantities are depicted in Fig. 8 by assuming the fixed\nvalues of the interaction parameters J=\u000010,J0=\n\u000010, and (h;Jz) ={(11;\u000011:5),(13;\u000012),(15:5;\u000013),\n(16:9;13:6),(18:81;\u000014:5)} outlined by {black solid, or-\n7TC(T) χ(T) mH(T) mI(T) S(T)(a) \n(b) \n(c) \n(d) \n(e) Figure 7: Temperature dependences of some thermodynamic\nquantities by considering the fixed parameters J=\u000010,\nJ0=\u000010,Jz=\u000011, and several values of the magnetic field\nh=f4;6;8;9;10g(black solid, orange solid, red solid, blue\ndashed, and green dot dashed): (a) entropy S; (b) Ising spin\nmagnetization; (c) Heisenberg spin magnetization; (d) spe-\ncific heat (semi-logarithmic plot); (e) magnetic susceptibility\n(semi-logarithmic plot).\nange solid, red solid, blue dashed, and green dot dashed}\ncurves, respectively. The present choice of the interac-\ntion parameters is consistent with the pseudo-transition\nbetween the quasi-phases qFI 1andqFR 2, which varies\nwith the interaction parameter Jzand magnetic field h.\nIt is obvious from Fig. 8(a) that the entropy S(T)ex-\nhibits a steep increase close to a pseudo-critical tempera-\ntureTp, while the magnetization of Ising spins shown in\nFig. 8(b) is pointing upward ( mI= 0:5) forT Tp. Similarly, the mag-\nnetization of Heisenberg spins illustrated in Fig. 8(c)\nis zero (mH= 0) forT < Tp, while there is a sudden\nchange atT=Tpabove which it strongly depends on\nthe magnetic field hand the coupling constant Jz. Fi-\nnally, sharp narrow peaks can be repeatedly detected at\na pseudo-critical temperature in the respective tempera-\nture dependences of the specific heat [Fig. 8(d)] and the\nmagnetic susceptibility [Fig. 8(e)].\nA pseudo-transition between the quasi-phases qFI 2\nandqFR 2is illustrated in Fig. 9 by considering the fixed\nparameters J=\u000010,J0=\u000010,Jz=\u000015:65, and sev-\neral values of the magnetic field of h=f25;27;28;29;30g\noutlined by {black solid, orange solid, red solid, blue\nTC(T) χ(T) mH(T) mI(T) S(T)(a) \n(b) \n(c) \n(d) \n(e) Figure 8: Temperature dependences of some thermodynamic\nquantities by considering the fixed parameters J=\u000010,\nJ0=\u000010, and (h;Jz) ={(11;\u000011:5),(13;\u000012),(15:5;\u000013),\n(16:9;13:6),(18:81;\u000014:5)} (black solid, orange solid, red\nsolid, blue dashed, and green dot dashed): (a) entropy S;\n(b) Ising spin magnetization; (c) Heisenberg spin magnetiza-\ntion; (d) specific heat (semi-logarithmic plot); (e) magnetic\nsusceptibility (semi-logarithmic plot).\ndashed, and green dot dashed} curves, respectively.\nFig. 9(a) shows the entropy S(T)as a function of tem-\nperature: for T < Tpthe entropy increases significantly\nbut is virtually independent of h(for22>h>30), then\na sudden rise occurs at T=Tpfollowed by a successive\nsmooth increase for T > Tp. The Ising magnetization\ndepicted in Fig. 9(b) is nearly constant mI= 0:5for\nT < Tp, but it becomes almost \u00000:5forT&Tpbefore\nshowing a continuous rise approaching null upon further\nincrease of temperature. Analogously, the Heisenberg\nspin magnetization illustrated in Fig. 9(c) tends to zero\nmH!1forT Tp. The magnetization of Ising spins [Fig. 10(b)]\ndisplays an opposite behavior to the previous one: the\nIsing spins are aligned in opposite to the magnetic field\n(mI=\u00000:5) forT < Tpand they are aligned in the\nmagnetic-field direction ( mI= 0:5) forT > Tp. Simi-\nlarly, the magnetization of Heisenberg spins [Fig. 10(c)]\nis close to its maximal value mH= 3forT < Tpand\nit suddenly drops to mH= 1forT > Tp. Finally, one\nobserves a typical narrow peak in thermal variations of\nthe specific heat and magnetic susceptibility displayed in\nFig. 10(d)-(e).\nLast but not least, let us discuss a pseudo-transition\nbetween the quasi-phases qFR 3andqFI 3exemplified in\nFig. 11 for the fixed values of the interaction parameters\nJ=\u000010,J0=\u000010,Jz=\u000019:9, and several mag-\nnetic fields h=f50;52;54;56;56:5gsketched by {black\nsolid, orange solid, red solid, blue dashed, and green dot\ndashed} curves, respectively. It is noteworthy that ther-\nmal variation of the entropy S(T)displayed in Fig. 11(a)\n2\nTC(T) χ(T) mH(T) mI(T) S(T)(a) \n(b) \n(c) \n(d) \n(e) Figure 10: Temperature dependences of some thermody-\nnamic quantities by considering the fixed parameters J=\n\u000010,J0=\u000010, and (h;Jz) ={(36:76;\u000017),(38:7;\u000017:5),\n(40:6;\u000018),(42:55;18:5),(44:45;\u000019)} (black solid, orange\nsolid, red solid, blue dashed, and green dot dashed)): (a) en-\ntropyS; (b) Ising spin magnetization; (c) Heisenberg spin\nmagnetization; (d) specific heat (semi-logarithmic plot); (e)\nmagnetic susceptibility (semi-logarithmic plot).\nis quite reminiscent of the entropy dependence illustrated\nin Fig. 7(a). In addition, the temperature dependences\nof the magnetization of the Ising and Heisenberg spins\nshown in Fig. 11(b) and (c) are quite similar to the pre-\nvious cases shown in Fig. 10(b) and (c), respectively. Al-\nthough the specific heat shows a strong narrow peak at\nthe pseudo-critical temperature, it often becomes negli-\ngible further away from the pseudo-critical temperature\n[see Fig. 11(d)]. The similar situation can be also found\nin the temperature dependences of the magnetic suscep-\ntibility shown in Fig. 11(e).\nIV. CONCLUSIONS\nThepseudo-transitionsofthemixedspin-(1/2,1)Ising-\nHeisenberg double-tetrahedral chain are examined in de-\ntail at non-zero temperature and magnetic field. The\nground-statephasediagramoftheinvestigatedspinchain\ntotally involves seven phases, three of which can be clas-\nsified as the non-degenerate ferrimagnetic phases, three\nas the macroscopically degenerate frustrated phases, and\none as the saturated paramagnetic phase. Interestingly,\nfive different ground-state boundaries of the mixed spin-\n9TC(T) χ(T) mH(T) mI(T) S(T)(a) \n(b) \n(c) \n(d) \n(e) Figure 11: Temperature dependences of some thermody-\nnamic quantities by considering the fixed parameters J=\n\u000010,J0=\u000010,Jz=\u000019:9, and several values of the mag-\nnetic fieldh=f50;52;54;56;56:5g(black solid, orange solid,\nred solid, blue dashed, and green dot dashed): (a) entropy S;\n(b) Ising spin magnetization; (c) Heisenberg spin magnetiza-\ntion; (d) specific heat (semi-logarithmic plot); (e) magnetic\nsusceptibility (semi-logarithmic plot).(1/2,1) Ising-Heisenberg double-tetrahedral chain repre-\nsent peculiar interfaces, at which the residual entropy\nper unit cell is simply given by the larger entropy of\none of two coexisting phases. This condition seems\nto be sufficient criterion whether or not the pseudo-\ntransition does emerge in a close vicinity of the ground-\nstate phase boundary. In fact, the residual entropy per\nunit cell at the usual ground-state phase boundaries is\nstrictly larger than the residual entropy of both coexist-\ning phases. Although thermal fluctuations usually de-\nstroy in 1D lattice-statistical models with short-range\ninteractions all fingerprints of the ground-state phase\nboundaries, the aforementioned five interfaces are quite\nrobust with respect to thermal fluctuations. In conse-\nquence of that, the mixed spin-(1/2,1) Ising-Heisenberg\ndouble-tetrahedral chain may exhibit in a vicinity of five\naforedescribed ground-state phase boundaries a marked\npseudo-transition manifested by vigorous narrow peaks\nof the specific heat and magnetic susceptibility besides a\nsudden change of the entropy and magnetization.\nAcknowledgments\nThisworkwaspartiallysupportedbyBrazilianAgency\nCNPq and FAPEMIG.\n[1] J. F. Nagle, Am. J. Phys. 36, 1114 (1968).\n[2] C. Kittel, Am. J. Phys. 37, 917 (1969).\n[3] S. T. Chui and J. D. Weeks, Phys. Rev. B 23, 2438\n(1981).\n[4] T. Dauxois and M. Peyrard, Phys. Rev. E 51, 4027\n(1995).\n[5] P. Sarkanych, Y. Holovatch, and R. Kenna, Phys. Lett.\nA381, 3589 (2017).\n[6] L. van Hove, Physica 16, 137 (1950).\n[7] J. A. Cuesta and A. Sanchez, J. Stat. Phys. 115, 869\n(2003).\n[8] P. N. Timonin, J. Exp. Theor. Phys. 113, 251 (2011).\n[9] I. M. Carvalho, J. Torrico, S. M. de Souza, M. Rojas, and\nO. Rojas, J. Magn. Magn. Mater. 465, 323 (2018).\n[10] I. M. Carvalho, J. Torrico, S. M. de Souza, O. Rojas, and\nO. Derzhko, Ann. Phys. 402, 45 (2019).\n[11] O. Rojas, J. Strečka, and S. M. de Souza, Solid State\nCommun. 246, 68 (2016).\n[12] J. Strečka, R. C. Alecio, M. Lyra, and O. Rojas, J. Magn.\nMagn. Mater. 409, 124 (2016).\n[13] J. Strečka, arXiv:1904.10704.\n[14] L. Gálisová and J. Strečka, Phys. Rev. E 91, 0222134\n(2015).\n[15] L. Gálisová, Phys. Rev. E 96, 052110 (2017).[16] L. Gálisová and D. Knežo, Phys. Lett. A 382, 2839\n(2018).\n[17] S. M. de Souza and O. Rojas, Solid State Commun. 269,\n131 (2017).\n[18] O. Rojas, arXiv:1810.07817.\n[19] T. Krokhmalskii, T. Hutak, O. Rojas, S. M. de Souza,\nand O. Derzhko, Towards low-temperature peculiarities\nof thermodynamic quantities for decorated spin chains,\narXiv:1908.06419.\n[20] O. Rojas, J. Strečka, M. L. Lyra, and S. M. de Souza,\nPhys. Rev. E 99, 042117 (2019).\n[21] S.BuhrandtandL.Fritz, Phys.Rev.B 90, 094415(2014)\n[22] A. Otsuka, D. V. Konarev, R. N. Lyubovskaya,\nS. S. Khasanov, M. Maesato, Y. Yoshida, and G. Saito,\nCrystals 8, 115 (2018).\n[23] M. Mambrini, J. Trébosc, and F. Mila, Phys. Rev. B 59,\n13806 (1999).\n[24] O. Rojas and F. C. Alcaraz, Phys. Rev. B 67, 174401\n(2003).\n[25] M. Maksymenko, O. Derzhko, and J. Richter, Acta\nPhysica Polonica A 119, 860 (2011); M. Maksymenko,\nO. Derzhko, and J. Richter, Eur. Phys. J. B 84, 397\n(2011).\n[26] V. Ohanyan, Physics of Atomic Nuclei 73, 494 (2010).\n10[27] D. Antonosyan, S. Bellucci, and V. Ohanyan, Phys. Rev.\nB79, 014432 (2009).\n11" }, { "title": "1103.2939v1.Oxygen_hyperstoichiometric_hexagonal_ferrite_CaBaFe4O7_δ_δ__approx_0_14____coexistence_of_ferrimagnetism_and_spin_glass_behavior.pdf", "content": "1/23 \n \n \nOxygen hyperstoichiometric hexagonal ferrite CaBaFe 4O7+ ( 0.14) : \ncoexistence of ferrimagnetism and spin glass behaviour \n \n \nTapati Sarkar *, V. Duffort, V. Pralong, V. Caignaert and B. Raveau \n \nLaboratoire CRISMAT, UMR 6508 CNRS ENSICAEN, \n6 bd Maréchal Juin, 14050 CAEN, France \nAbstract \n \n An oxygen hyperstoichiometric ferrite CaBaFe 4O7+ ( 0.14) has been \nsynthesized using “soft” reduction of CaBaFe 4O8. Like the oxygen stoichiometric \nferrimagnet CaBaFe 4O7, this oxide also keeps the hexagonal symmetry (space group: \nP63mc), and exhibits the same high Curie temperature of 270 K. However, the \nintroduction of extra oxygen into the system weakens the ferrimagnetic interaction \nsignificantly at the cost of increased magnetic frustration at low tempera ture. Moreover, \nthis canonical spin glass (T g ~ 166 K) exhibits an intriguing cross -over from de Almeida -\nThouless type to Gabay -Toulouse type critical line in the field temperature plane above a \ncertain field strength, which can be identified as the anisot ropy field. Domain wall \npinning is also observed below 110 K. These results are interpreted on the basis of \ncationic disordering on the iron sites. \n \n \n \n \n \n \n \nPACS number: 75.47.Lx \n \nKeywords : “114” ferrites, ferrimagnetism and magnetic frustration, spin glass Ising -\nHeisenberg competition, magnetic anisotropy, domain wall pinning. \n \n * Corresponding author: Tapati Sarkar \ne-mail: tapati.sarkar @ensicaen.fr 2/23 Introduction \n \nThe recent discovery of the new series of “114” oxides, (the cobaltites – \n(Ln,Ca) 1BaCo 4O7 [1 – 8] and the ferrites – (Ln,Ca) 1BaFe 4O7 [9 – 11]) have opened up a \nnew field for the investigation of strongly correlated electron systems. These oxides \nconsist of CoO 4 (or FeO 4) tetrahedra sitting in alternating layers of kagomé and triangular \narrays [10]. The structure can also be described as the stacking of close -packed [BaO 3] \nand [O 4] layers whose tetrahedral cavities are occupied by Co2+/Co3+ (or Fe2+/Fe3+) \nspecies, forming triangular and kagomé layers of CoO 4 (or FeO 4) tetrahedra. This \nstructure has been primarily responsible for the wide variety of magnetic states that has \nbeen observed in this group of oxides, ranging from a spin glass for cubic LnBaFe 4O7 [9, \n10] to a ferrimagnet for orthorhombic CaBaCo 4O7 [5] and hexagonal CaBaFe 4O7 [9] \noxides. \nRecent studies of the “114” cobaltites [12 – 17] have revealed the existence of \nclosely related structures with various crystallographic symmetries, and possibility of \noxygen non -stoichiometry in the range “O 7” to “O 8.5” in those systems. This change of \noxygen stoichiometry, which induces the variation of Co2+:Co3+ ratio in the system, is \nexpected to influence the physical properties of these compounds considerably. This is \nthe case of the oxygen rich “114” cobaltites YBaCo 4O8.1 [15] and YbBaCo 4O7.2 [17], \nwhich were shown to be magnetically frustrated rather than magnetically ordered at low \ntemperatures. \nIn contrast to the cobalt oxides, no report of oxygen hyperstoichiometric “114” \nferrites exists till date, probably due to the fact that Fe2+ gets too easily oxidized into Fe3+, \nthereby destabilizing the “114” structure at the benefit of pure “Fe3+” oxides. We have, \nthus, investigated the possibility to stabilize the mixed valence Fe2+/Fe3+ in the “114” \noxygen hyperstoichiometric CaBaFe 4O7+ ferrite by reducin g the fully oxidized \ncompound CaBaFe 4O8 [18] at low temperature in an argon -hydrogen atmosphere. We \nreport herein on the magnetic properties of the “114” oxygen hyperstoichiometric \nCaBaFe 4O7.14 hexagonal ferrite. We show that, like the stoichiometric phas e CaBaFe 4O7, \nthis oxide also exhibits ferrimagnetism with a T C of 270 K, but that the competition \nbetween ferrimagnetism and magnetic frustration is much more pronounced than for the \nstoichiometric phase, as seen from the decrease of the magnetization . Mor e importantly, \nwe observe that CaBaFe 4O7.14 is characterized by a canonical spin glass behaviour with 3/23 Tg 166 K, and an intriguing cross -over from an Ising to a Heisenberg spin glass type \nbehaviour in the external magnetic field at low temperature. Beside s this competition \nbetween ferrimagnetism and spin glass behaviour , one also observes domain wall pinning \nbelow 110 K. This very different magnetic behaviour of CaBaFe 4O7.14 is explained in \nterms of cationic deficiency and disordering on the iron sites, th e “barium -oxygen” \nhexagonal close packing remaining untouched. \n \nExperimental \n \nThe precursor CaBaFe 4O8 [18] was prepared by the sol gel method. Stoichiometric \namounts of calcium carbonate (Prolabo, 99%) and barium carbonate (Alfa Aesar, 99%) \nwere dissolved in a large excess of melted citric acid monohydrate at ~ 200°C. Iron \ncitrate (Alfa Aesar, 20% of Fe) was separately dissolved in hot water leading to a dark \nbrown solution which was poured on the citrate mixture. The water was then evaporated \nfollowed by d ecomposition of the gel. The gel was calcined at 450 °C under air to obtain \nan amorphous precursor, which was then pressed into pellets before firing at 1200 °C to \nobtain CaBaFe 4O8. \nThe oxygen hyperstoichiometric “114” ferrite, CaBaFe 4O7+ was then obtain ed by \nreducing CaBaFe 4O8 under an Ar/H 2 10% mix at 610 °C for 24 hrs. \nThe oxygen content of the sample was determined by redox titration. The sample \nwas dissolved in hot HCl (3M) flushed with argon to remove the dissolved oxygen. After \ncooling down the sol ution, Fe2+ cations were titrated using 2 10-2 M cerium(IV) sulfate \n(Riedel -de Haën) and 1.10 -phenantroline iron(II) sulfate (Alfa Aesar) as an indica tor \nunder constant argon flow . We obtained = 0.14. \nThe X -ray diffraction patterns were registered wit h a Panalytical X’Pert Pro \ndiffractometer under a continuous scanning mode in the 2 range 10° - 120° and step size \n2=0.017°. The d.c. magnetization measurements were performed using a \nsuperconducting quantum interference device (SQUID) magnetometer with variable \ntemperature cryostat (Quantum Design, San Diego, USA). The a.c. susceptibility, ac(T) \nwas measured with a Physical Property Measurement System ( PPMS ) from Quantum \nDesign with the frequency ranging from 10 Hz to 10 kHz (H dc = 0 Oe and H ac = 10 Oe ). \nAll the magnetic properties were registered on dense ceramic bars of dimensions ~ 4 2 \n2 mm3. \n 4/23 \nResults and discussion \n \nStructural Characterization \n \nThe X -ray diffraction pattern (Fig. 1) revealed that CaBaFe 4O7.14 stabilized in the \nsame hexagonal sy mmetry (space group: P63mc) as the “O 7” phase [9]. The Rietveld \nanalysis from the XRD data was done using the FULLPROF refinement program [19]. \nThe fit is also shown in Fig. 1 (red curve). The bottom blue curve corresponds to the \ndifference between the obs erved and calculated diffraction patterns. Satisfactory \nmatching of the experimental data with the calculated profile of the XRD pattern and the \ncorresponding reliability factors RF = 3.88 % and RBragg = 5.01 % confirm that the fit \nobtained is reasonably a ccurate. The extracted lattice parameters ( a = 6.355 Å, c = 10.372 \nÅ) show a very marginal increase over the “O 7” phase – a increases by ~ 0.11 % while c \nremains virtually unchanged. The refinements of the atomic coordinates, thus, lead to \nresults similar to those previously obtained for CaBaFe 4O7 [9]. The low value and the \nlow scattering factor of oxygen do not allow any oxygen excess or cationic deficiency to \nbe detected from X -ray powder diffraction data. A very careful neutron diffraction study \nmight perhaps allow the issue to be sorted out, but will really be at the limit of accuracy, \nand consequently, is not within the scope of this paper. \n \nD. C. magnetization study \n \nIn Fig. 2, we show the Zero Field Cooled (ZFC) and Field Cooled (FC) \nmagnetization o f CaBaFe 4O7+ recorded under a magnetizing field of 0.3 T. The sample \nshows the same increase in magnetization below ~ 270 K as the oxygen stoichiometric \noxide indicating a similar transition to an ordered magnetic state below 270 K. However, \na careful loo k at the magnetization values reached at the lowermost measured \ntemperature (5 K) immediately reveals a striking difference in the magnetic behaviour of \nCaBaFe 4O7.14 vis – à – vis that of CaBaFe 4O7. While the F.C. magnetization of the \noxygen stoichiometric compound reaches a value of more than 2.5 µ B/f.u. at T = 5 K, the \nmaximum magnetization value of our oxygen rich sample is only 0.93 µ B/f.u., which is \nless by more than a factor of ½. \nThis large difference in the magnetization value at low temperature be tween the \ntwo samples ( = 0 and > 0) prompted us to record the hysteresis curve of our oxygen 5/23 rich sample at low temperature (T = 5 K) and compare it with that of the oxygen \nstoichiometric sample. This is shown in Fig. 3. The magnetization value obtaine d at the \nhighest measuring field of 5 T (2.5 µ B/f.u.) is again different from the oxygen \nstoichiometric sample (3.1 µ B/f.u.), as expected. More importantly, a rather striking \ndifference is seen in the shape of the hysteresis loop. The coercive field (H C) and \nremanent magnetization (M r) of our sample at T = 5 K are 0.77 T and 0.63 µ B/f.u. \nrespectively. While the value of the coercive field compares well with that of the oxygen \nstoichiometric sample, the value of the remanent magnetization is much lower than that \nobtained for CaBaFe 4O7 (M r ~ 1.8 µ B/f.u.). This results in the overall shape of the \nhysteresis loop of our sample (Fig. 3) to be very different from that of the oxygen \nstoichiometric sample (inset of Fig. 3). The degree of magnetic saturation in a sam ple can \nbe roughly quantified from the M -H loop by calculating \n5\nrM H T\nM . While the \noxygen stoichiometric sample had = 1.7, our oxygen rich sample yields = 4.0. This \nhigher value of for the oxygen rich sample indicates an increased lack of magnetic \nsaturation in the sample, or in other words, a weakening of long range order. \nAnother important difference with the oxygen stoichiometric sample is in the \nvirgin curve of the M(H) loop. While the virgin curve of the = 0 sample lies entirely \nwithin the main loop, our oxygen rich sample shows an unusual magnetic behaviour \nwhere a major portion of the virgin curve lies outside the hysteresis loop and meets the \nmain loop only at very high fields. \n \nA. C. magnetic susceptibility study \n \nThe tempera ture dependence of the a.c. susceptibility of CaBaFe 4O7.14 in the \ntemperature range 20 K – 280 K and at 4 measuring frequencies ranging from 10 Hz to \n10 kHz is shown in Fig. 4. The sample shows several interesting features which we will \nnow proceed to disc uss separately: \n(a) At T = 272 K, one can see a sharp peak which is frequency independent i.e. the \nposition of the peak maximum does not shift with a change in the measuring \nfrequency. This peak corresponds to the paramagnetic to ferrimagnetic (PM -FM) \ntransiti on occurring in the sample as it is cooled below 272 K. We note here that \nthe oxygen stoichiometric sample also showed a similar peak at around the same 6/23 temperature. However, in the latter sample, the peak corresponding to the PM -FM \ntransition was the stro ngest (maximum amplitude) compared to the other peaks. In \nour sample, this peak is much smaller in magnitude which corresponds to a \nsignificant weakening of the magnetic ordering (or a smaller volume fraction of \nferrimagnetic domains) which we had mentione d earlier in connection with the \nM(H) loop. \n(b) The oxide CaBaFe 4O7.14 shows a broader peak at lower temperature (~ 166 K) \nwhich shows pronounced frequency dependence. The peak temperature shifts from \n166 K (for a measuring frequency of 10 Hz) to 176 K (for a measuring frequency \nof 10 kHz). This corresponds to a peak shift of 0.02 per decade of frequency shift \n(\nlogf\nfT\nTpf\n = 0.02). This value of the parameter p lies within the range for \ncanonical spin glasses, which indicates that this peak is a sig nature of the sample \nundergoing a spin glass transition. We confirm this by analyzing the frequency \ndependence of this peak using the power law form \n0z\nf SG\nSGTT\nT\n\n , where, 0 is \nthe shortest relaxation time available to the system, TSG is the unde rlying spin -\nglass transition temperature determined by the interactions in the system, z is the \ndynamic critical exponent and is the critical exponent of the correlation length. \nThe actual fittings were done using the equivalent form of the power law: \n\n\n\n\nSGSG f\nTT Tzln ln ln0\n. The fit parameters ( 0 = 7.3 10-10 sec, z = 5.01 and \nTSG = 162.2 K) give a good linear fit (as can be seen in the inset of Fig. 4), and \nconfirms that this peak does correspond to a spin glass transition in the sample. \nWhether the magnetic order disappears in the spin glass phase is not clear at the \nmoment. Our data shows that the spin glass transition occurs within the \nferrimagnetically ordered phase. Whether this transition is accompanied by the \ndestruction of ferrimagnetic long r ange order is an open issue as of now. The \npossibility of coexistence of ferrimagnetic and spin glass orders cannot, however, \nbe ruled out. 7/23 (c) At a lower temperature of ~ 110 K a very broad peak is seen (which broadens out \nto almost kind of a shoulder at lowe r measuring frequency). We will come back to \nthe nature of this feature in the following section. \n \nMagnetic field dependence of the spin glass freezing temperature \n \nIn Fig. 5, a.c. susceptibility at 10 kHz driving frequency is plotted as a function of \ntemp erature for different external magnetic fields H dc ranging from 0 to 0.3 T. As can be \nseen from the figure, χ' is suppressed by the magnetic field. First, we focus our discussion \non the evolution of the spin glass freezing temperature (marked by a black ar row in the \nfigure), which occurs at ~ 176 K at the lowest applied field. This peak temperature shows \na continual shift towards lower temperature as the external magnetic field is increased, \nand reaches a value of 155 K at an external magnetic field of 0.3 T. Concomitantly, the \npeak amplitude keeps decreasing as the external magnetic field is increased from 0 to 0.3 \nT. A further increase of the external magnetic field should eventually suppress the spin \nglass transition completely. \n The purpose of exploring how the freezing temperature responds to external \nmagnetic field is to check the stability of the spin glass system. This is done by \nexamining the field versus temperature phase diagram obtained from the a.c. \nsusceptibility measurement as shown in Fig. 6. As was mentioned above, the spin glass \nfreezing temperature is suppressed by increasing the external magnetic field. \n From a theoretical perspective, de Almeida and Thouless [20] studied the Ising \nspin glass system, and predicted that the spin freezing te mperature ( Tg) depends on H. In \nthe low H range, Tg follows the so -called de Almeida -Thouless (AT) line, expressed as \n\n23\n001\n\n\n\n\n\ngg\nTHTH H\n. In addition, Gabay and Toulouse [21] investigated the H \ndependence of the spin freezing temperature for the Heise nberg spin glass system. This \nled to the so -called Gabay -Toulouse (GT) line, expressed as \n\n21\n001\n\n\n\n\n\ngg\nTHTH H . The \nAT line and the GT line are the two critical lines predicted in the presence of field on the \nH-T plane, which mark the phase transition. The first one occurs for an anisotropic Ising \nspin glass while the second is valid for an isotropic Heisenberg spin glass. 8/23 Our sample shows a very interesting behaviour. At low field values (H dc < 0.15 T), \nTg follows the AT line. This can be seen in Fig. 6, where the red line denotes the AT line. \nSince the AT line predicts that \n32H Tg , so we have plotted H2/3 in the H -T phase \ndiagram. However, with an increase in the field (H > 0.15 T), we find deviation from the \nAT line. Remarkably, it is found that at high field, the variation of Tg(H) agrees with the \nGT line. This can be seen in the inset of Fig. 6, where the blue line denotes the GT line. \nSince the GT line predicts that \n2H Tg , so the H -T phase diagram in the inset is p lotted \nwith H2 in the y -axis. \nThese experimental results can be explained using the theoretical calculation by \nKotliar and Sompolinsky [22], who have predicted that in the presence of random \nanisotropy, the critical behaviour for a spin glass in fields lo wer than the anisotropy field \nis close to Ising type following the AT line, and crosses over to Heisenberg behaviour in \nhigh fields. The fact that we see a crossover in critical lines on the H -T plane for our \nsample indicates the existence of magnetic anis otropy in the system. At higher applied \nfields, the system behaves like a Heisenberg spin glass, where the spins can freeze along \nany direction with respect to the applied magnetic field. However, when the applied field \nis lower than the anisotropy field, the spins are forced to be aligned along the local \nanisotropy axis. The preference of the spin alignment adds an Ising character to the \nassociated spin cluster. \n \nDomain wall pinning at lower temperature \n \nIn this section, we discuss the third feature seen i n the χ'(T) curve – the broad peak \nat ~ 110 K (Fig. 4). This peak at 110 K does not shift (i.e. the peak maximum occurs at \nthe same temperature) with a change in the external magnetic field H dc (see the red line in \nFig. 5). Based on this behaviour, we attr ibute the origin of the feature seen at ~ 110 K to \nenhanced domain wall pinning. The signature of this domain wall pinning can also be \nseen in Fig. 7, where we plot the variation of the coercivity (H C) with temperature. As is \nclear from the figure, the coe rcivity is majorly enhanced below 110 K, which occurs due \nto the domain wall pinning. A close look at the high temperature region, which is \nenlarged and shown in the inset of Fig. 7, reveals that the coercivity also shows an \nenhancement below the paramagne tic to ferrimagnetic phase transition temperature 9/23 (shown by a black arrow), and another enhancement below the spin glass freezing \ntemperature (shown by a blue arrow), as expected. \n At this stage, we need to go back to our earlier observation of an unusual initial \nmagnetization curve in the M(H) loop measured at low temperature (Fig. 3). Such \nunusual magnetic hysteresis behaviour, with the virgin curve lying outside the main \nhysteresis loop, was earlier associated with irreversible domain wall motion in spin el \noxides [23]. Thus, this unusual magnetization curve is an additional confirmation of the \ndomain wall pinning at ~ 110 K that we had mentioned earlier. In fact, we find that the \nvirgin curve lies outside the main M(H) loop for temperatures below 110 K, b ut above \n110 K, it lies completely inside the main hysteresis loop. This is shown in Fig. 8, where \nwe plot the M(H) loops at temperatures slightly below (Fig. 8 (a)), and slightly above \n(Fig. 8 (b)) 110 K. In the figures, the virgin curves are shown in red for the sake of \nclarity. \n \nOrigin of the competition between ferrimagnetism and spin glass behaviour \n \n In order to understand the different magnetic behaviour of CaBaFe 4O7.14 with \nrespect to CaBaFe 4O7, we must keep in mind that the oxygen excess in the for mer \ninduces an increase of the Fe3+ content in the structure i.e., the Fe3+:Fe2+ ratio increases \nfrom 1 in the stoichiometric phase to 1.32 in the oxygen hyperstoichiometric phase. As a \nconsequence, the Fe3+-Fe3+ antiferromagnetic interactions increase in the oxygen rich \nphase, and may decrease the ferrimagnetism in the structure. Bearing in mind the model \npreviously proposed by Chapon et. al. [4] to explain the competition between 1D \nferromagnetism and 2D magnetic frustration in the cobaltite YBaCo 4O7 which has the \nsame hexagonal structure, we must consider the iron framework of our compound. The \nlatter consists of corner -sharing [Fe 5] bipyramids running along “ c” interconnected \nthrough “Fe 3” triangles (Fig. 9). In other words, in both oxides, CaBaFe 4O7 and \nCaBaFe 4O7.14, we can expect, similarly to the hexagonal cobalt oxides LnBaCo 4O7, that \nthe system exhibits an unidimensional magnetic order in the bipyramidal rows along “ c”, \nwhereas the triangular geometry of the iron lattice in the (001) plane induces magnetic \nfrustration as soon as the iron species are coupled antiferromagnetically. Such a model \ncan account for the competition between 1D ferrimagnetism and 2D magnetic frustration \nin both oxides, CaBaFe 4O7 and CaBaFe 4O7.14, and explain that the magnetic frustration 10/23 may be larger in the latter owing to the appearance of larger short range \nantiferromagnetic interactions in the (001) plane . \n Nevertheless, the valency effect alone is not sufficient to explain the appearance of \nthe spin glass behaviour. Two hypotheses can be considered to explain this particular \nbehaviour. The first scenario deals with the fact that CaBaFe 4O7.14 contains interstitial \noxygen in spite of the apparent close packed character of the structure, leading to a local \npuckering of the “ O4” and “BaO 3” layers. As a result, the distribution of iron in the \ncationic sites would be locally disordered, leading to a spin glass behaviour. This local \ndistortion would also change the crystal field and would be responsible for the domain \nwall pinnin g. The second scenario deals with the fact that the “barium oxygen” \nframework remains close packed, but that the compound exhibits a cationic deficiency \naccording to the formula Ca 0.98(Ba 0.98O0.02)Fe 3.93O7. Such an effect would be similar to \nthat observed for “oxidized” spinels -Fe2O3 and Co 3-xO4, which do not contain \ninterstitial oxygen, but were found to be iron or cobalt deficient [24, 25]. This second \nscenario would explain the magnetic behaviour of this phase, which is close to that \nobserved for CaBaF e4-xLixO7 [26]. In both the systems, the doping of the Fe sites with \nlithium or vacancies respectively introduces disordering on the Fe sites, which is in turn, \nat the origin of the appearance of spin glass behaviour at lower temperature. Thus, the \ncompeti tion between 1D ferrimagnetism and spin glass behaviour appears normal. \nSubsequently, the competition between anisotropic (Ising) and isotropic (Heisenberg) \nspin glass can be understood from the peculiar geometry of the [Fe 4] lattice. Finally, the \niron va cancies would change the nature of the crystal field in the structure, playing the \nrole of pinning centres. This explains both, the broad peak at 110 K and the enhanced \ncoercivity below this temperature, which are the signatures of domain wall pinning. \n The small deviation from the stoichiometry does not allow to distinguish the \npossibility of interstitial oxygen vis à vis that of cationic deficiency from a structural \nstudy. Attempts are being made to synthesize similar hexagonal ferrites with larger \noxyge n excess in order to answer this question. \n \n \n \n \n \n 11/23 Conclusion s \n \n This study illustrates the extraordinarily rich physics of the “114” CaBaFe 4O7+ \nferrite, in connection with its ability to accommodate oxygen excess, similar to what is \nobserved for the spine l family, Fe 3O4 – -Fe2O3. The remarkable feature of this “114” \noxide deals with the competition between ferrimagnetism and spin glass behaviour that \ncan be induced by varying the oxygen content, without changing the hexagonal symmetry \nof the structure. Su ch a behaviour can be explained, like for the “114” cobaltites, as due \nto the competition between 1D magnetic ordering along “ c” and 2D magnetic frustration \nin the triangular (001) lattice. Nevertheless, CaBaFe 4O7 differs significantly from \nCaBaCo 4O7, the latter’s ferrimagnetism originating mainly from a lifting of its 2D \ngeometrical frustration through a strong orthorhombic distortion of its initial hexagonal \nlattice. We believe that the scenario of cation disordering on iron sites is the key for \nunderstan ding the magnetism of these materials. Further investigations, especially using \nneutron diffraction and X -ray synchrotron have to be performed in order to further \nunderstand this phenomenon. \n \nAcknowledgement s \n \nThe authors acknowledge the CNRS and the Conse il Regional of Basse Normandie \nfor financial support in the frame of Emergence Program. V. P. acknowledges support by \nthe ANR -09-JCJC -0017 -01 (Ref: JC09_442369). \n \n \n \n \n \n 12/23 References \n \n [1] Martin Valldor and Magnus Andersson, Solid State Sciences , 2002, 4, 923 \n [2] Martin Valldor, J. Phys.: Condens. Matter ., 2004 , 16, 9209 \n [3] L. C. Chapon, P. G. Radaelli, H. Zheng and J. F. Mitchell, Phys. Rev. B , 2006 , 74, \n 172401 \n [4] P. Manuel, L. C. Chapon, P. G. Radaell i, H. Zheng a,d J. F. Mitchell, Phys. Rev. \n Lett., 2009 , 103, 037202 \n [5] V. Caignaert, V. Pralong, V. Hardy, C. Ritter and B. Raveau, Phys. Rev. B , 2010 , 81, \n 094417 \n [6] E. A. Juarez -Arellano, A. Friedrich, D. J. Wilson, L. Wiehl, W. Morgenroth, B. \n Winkler, M. Avdeev, R. B. Macquart and C. D. Ling, Phys. Rev. B , 2009 , 79, 064109 \n [7] N. Hollmann, Z. Hu, M. Valldor, A. Maignan, A. Tanaka, H. H. Hsieh, H. -J. Lin, C. \n T. Chen and L. H. Tjeng, Phys. Rev. B , 2009 , 80, 085111 \n [8] D. D. Khalyavin, L. C. Chapon, P. G. Radaelli, H. Zheng and J. F. 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Caignaert and B. Raveau , Phys. Rev. \n B, 2009, 79, 224407 \n \n 14/23 Figure Captions \n \nFig. 1 X-ray diffraction pattern along with the fit for CaBaFe 4O7+. \nFig. 2 MZFC (T) and M FC (T) curves of CaBaFe 4O7+ measured at H = 0.3 T. \nFig. 3 M(H) curve of CaBaFe 4O7+ measured at T = 5 K. The virgin curve is shown in red \n circles, while the rest of the hysteresis loop is shown in black triangles. The inset \n shows the M(H) curve of the oxygen stoichiometric sample (CaBaFe 4O7) measured \n at T = 5 K. \nFig. 4 The real (in -phase) component of a.c. susceptibility for CaBaFe 4O7+ as a function \n of temperature in the frequency range f = 10 Hz – 10 kHz, at zero static magnetic \n field (H dc) and at a dr iving ac field (H ac) of 10 Oe. The inset shows the plot of ln \n vs \n\n\n\n\nSGSG f\nTTTln for the peak at 166 K. \nFig. 5 The real (in -phase) component of a.c. susceptibility for CaBaFe 4O7+ as a function \n of temperature. The driving f requency was fixed at f = 10 kHz and h ac = 10 Oe. \n Each curve was obtained under different applied static magnetic field (H dc) ranging \n from 0 T to 0.3 T. \nFig. 6 Field vs temperature phase diagram of CaBaFe 4O7+. In order to show the AT l ine, \n we have plotted H2/3 vs Tg. The inset shows H2 vs Tg and the GT line. \nFig. 7 Temperature dependence of coercive field for CaBaFe 4O7+. The inset is an \n enlarged version of the high temperature region. \nFig. 8 M(H) loops of CaBa Fe4O7+ at (a) T = 75 K and (b) T = 135 K. \nFig. 9 Schematic representation of the [Fe 4] tetrahedral framework of hexagonal \n CaBaFe 4O7+ showing the Fe 5 bipyramids sharing corners with Fe 3 triangular \n groups (adapted from Ref. 11). \n \n \n \n \n \n \n 15/23 \n20 40 60 80 1000.02.0x1034.0x1036.0x1038.0x1031.0x104P63mc\na=6.3546(2) Å\nc=10.3721(4) Å\n2 = 1.94\nRBragg = 5.01 %\nRF = 3.88 %\n Intensity (arb. units)\n2 (degree) \nFig. 1 . X-ray diffraction pattern along with the fit for CaBaFe 4O7+. \n \n 16/23 \n0 100 200 300 4000.00.20.40.60.81.0\nH = 0.3 T ZFC\n FCMagnetization ( B/f.u.)\nTemperature (K)\n \n \nFig. 2 . MZFC (T) and M FC (T) curves of CaBaFe 4O7+ measured at H = 0.3 T. \n 17/23 \n \n-4 -2 0 2 4-2-1012\n-4 -2 0 2 4-3-2-10123CaBaFe4O7Magnetization ( B/f.u.)\nMagnetic field (T)\n \nT = 5 KMagnetization ( B/f.u.)\nMagnetic field (T)\n \n \nFig. 3. M(H) curve of CaBaFe 4O7+ measured at T = 5 K. The virgin curve is shown in \nred ci rcles, while the rest of the hysteresis loop is shown in black triangles. The inset \nshows the M(H) curve of the oxygen stoichiometric sample (CaBaFe 4O7) measured at T \n= 5 K. \n 18/23 \n50 100 150 200 2504.0x10-36.0x10-38.0x10-31.0x10-2\n110 K166 K\n272 K\n-3.6 -3.3 -3.0 -2.7 -2.4-9-8-7-6-5-4-3-20 = (7.3 ± 0.1) X 10-10 sec\nTSG = 162.2 K\nz = 5.01 ± 0.01ln \nln{(Tf-TSG)/TSG}\n Hac = 10 Oe\nHdc = 0 10 Hz\n 80 Hz\n 1 kHz\n 10 kHz' (emu/gm)\nTemperature (K)\n \n \nFig. 4 . The real (in -phase) component of a.c. susceptibility for CaBaFe 4O7+ as a function \nof temperature in the frequency range f = 10 Hz – 10 kHz, at zero static magnetic \nfield (H dc) and at a driving ac fie ld (H ac) of 10 Oe. The inset shows the plot of ln \nvs \n\n\n\n\nSGSG f\nTTTln for the peak at 166 K. \n \n \n \n \n \n \n \n \n \n 19/23 \n \n50 100 150 200 250 3002.0x10-34.0x10-36.0x10-38.0x10-31.0x10-2\n 0.2 T\n 0.225 T\n 0.25 T\n 0.275 T\n 0.3 T 0 T\n 0.025 T\n 0.05 T\n 0.075 T\n 0.1 T\n 0.125 T\n 0.15 T\n 0.175 T\nTemperature (K)' (emu/gm)\n \n \n \nFig. 5 . The real (in -phase) component of a.c. susceptibility for CaBaFe 4O7+ as a function \nof temperature. The driving frequency was fixed at f = 10 kHz and h ac = 10 Oe. \nEach curve was obtained under different applied static magnetic field (H dc) ranging \nfrom 0 T to 0.3 T. \n \n \n \n \n \n \n \n \n 20/23 \n155 160 165 170 175 1800.0750.1500.2250.3000.3750.450\n155 160 165 170 1750.000.020.040.060.080.10\nGT line H2 (T2)\nT (K)\nAT lineFM\nSpin glass\n H2/3 (T2/3)\nT (K)\n \n \nFig. 6 . Field vs temperature phase diagram of CaBaFe 4O7+. In order to show the AT line, \nwe have plotted H2/3 vs Tg. The inset shows H2 vs Tg and the GT line. \n \n \n \n \n \n \n \n \n 21/23 \n0 50 100 150 200 250 3000.00.10.20.30.40.50.60.70.8\n120 150 180 210 240 270 3000.020.040.060.080.10\n T (K)HC (T)\n113 K\nTemperature (K)\n HC (T) \n \nFig. 7 . Temperature dependence of coercive field for CaB aFe 4O7+. The inset is an \nenlarged version of the high temperature region. \n \n \n \n \n \n \n \n \n \n \n \n \n 22/23 \n-5 -4 -3 -2 -1 0 1 2 3 4 5-2-1012-2-1012\n(b)Magnetization ( B/f.u.)\nMagnetic field (T)T = 135 KT = 75 K(a)\n \n \n \nFig. 8 . M(H) loops of CaBaFe 4O7+ at (a) T = 75 K and (b) T = 135 K. \n \n \n \n \n \n \n \n \n \n \n \n \n \n 23/23 \n \n \n \nFig. 9 . Schematic representation of the [Fe 4] tetrahedral framework of hexagonal \nCaBaFe 4O7+ showing the Fe 5 bipyramids sharing corners with Fe 3 triangular \ngroups (adapted from Ref. 11). \n \n \n \n \n \n \n \n " }, { "title": "2108.11156v3.Quantum_network_with_magnonic_and_mechanical_nodes.pdf", "content": "Quantum network with magnonic and mechanical nodes\nJie Li,1,\u0003Yi-Pu Wang,1,yWei-Jiang Wu,1Shi-Yao Zhu,1and J. Q. You1\n1Interdisciplinary Center of Quantum Information, State Key Laboratory of Modern Optical Instrumentation,\nand Zhejiang Province Key Laboratory of Quantum Technology and Device,\nDepartment of Physics, Zhejiang University, Hangzhou 310027, China\n(Dated: December 3, 2021)\nA quantum network consisting of magnonic and mechanical nodes connected by light is proposed. Recent\nyears have witnessed a significant development in cavity magnonics based on collective spin excitations in\nferrimagnetic crystals, such as yttrium iron garnet (YIG). Magnonic systems are considered to be a promising\nbuilding block for a future quantum network. However, a major limitation of the system is that the coherence\ntime of the magnon excitations is limited by their intrinsic loss (typically in the order of 1 \u0016s for YIG). Here, we\nshow that by coupling the magnonic system to a mechanical system using optical pulses, an arbitrary magnonic\nstate (either classical or quantum) can be transferred to and stored in a distant long-lived mechanical resonator.\nThe fidelity depends on the pulse parameters and the transmission loss. We further show that the magnonic\nand mechanical nodes can be prepared in a macroscopic entangled state. These demonstrate the quantum state\ntransfer and entanglement distribution in such a novel quantum network of magnonic and mechanical nodes.\nOur work shows the possibility to connect two separate fields of optomagnonics and optomechanics, and to\nbuild a long-distance quantum network based on magnonic and mechanical systems.\nI. INTRODUCTION\nHybrid quantum systems, composed of distinct physical\nsystems with complementary functionalities, provide diverse\nnovel platforms and promising opportunities for applications\nin quantum technologies, quantum-information processing\nand quantum sensing [1–3]. It merits our particular attention\nthat, during the past decade a rapid and significant progress\nhas been made in the field of cavity magnonics, based on\ncoherently coupled microwave cavity photons and collec-\ntive spin excitations in the ferrimagnetic material of yttrium\niron garnet (YIG) [4–35]. Cavity magnonics has now be-\ncome a new platform for the study of strong interactions be-\ntween light and matter, in the context of cavity QED with\nmagnons. As one of the main advantages, the magnonic sys-\ntem shows an excellent ability to coherently interact with di-\nverse quantum systems, including microwave [7–9] or optical\nphotons [12–14], phonons [15, 23, 32, 35], and superconduct-\ning qubits [10, 16, 29, 33]. Hybrid cavity magnonic systems\npromise potential applications in quantum-information pro-\ncessing [4], quantum sensing [36–38], and in searching dark-\nmatter axions [39], to name a few.\nIn this paper, we show the potential to build a quantum net-\nwork [40, 41] based on magnonic systems in view of their\naforementioned excellent properties. A future quantum net-\nwork could be constructed based on single atoms in opti-\ncal cavities [42], or atomic ensembles following the Duan-\nLukin-Cirac-Zoller protocol [43], or trapped atomic ions [44],\netc. Compared to these platforms as quantum nodes, where\nthe atomic energy levels are fixed, a major advantage of\nmagnonic systems lies in the fact that their resonance fre-\nquencies can be continuously adjusted by altering the exter-\nnal magnetic field. This o \u000bers a large flexibility to couple to\n\u0003jieli007@zju.edu.cn\nyyipuwang@zju.edu.cndi\u000berent quantum systems, like superconducting qubits, pho-\ntons, and phonons [4]. Therefore, a quantum network based\non magnonic systems shows its unique advantages. However,\na major obstacle for such a magnon-based quantum network is\nthat its coherence time is limited by its intrinsic loss (typically\nwith damping rate \rm=2\u0019\u00181 MHz), and is in the order of 1\n\u0016s for YIG. The coherence time can indeed be significantly\nextended by transferring the magnonic quantum state to the\nmechanical mode (i.e., the vibrational phonon mode) of the\nsame YIG ferrimagnet [45–48], which can act as a long-lived\nquantum memory [49, 50]. However, this local operation via\nmagnomechanics does not allow to build a quantum network\nwith its nodes distributed in a long distance.\nHere, we show that this obstacle can be eliminated by using\nlight, an optimal candidate for transmitting quantum informa-\ntion over a long distance, through which the magnonic system\nis connected to a distant mechanical system. We, for the first\ntime, prove that light can connect two separate fields of op-\ntomagnonics and optomechanics, and be used to accomplish\nsome basic functions of a quantum network, such as quan-\ntum state transfer and entanglement distribution among di \u000ber-\nent nodes of the network [40, 42, 51]. The quantum network\nwith magnonic and mechanical nodes combines the advan-\ntages of both systems, i.e., the great magnonic compatibility\nand tunability as well as the long mechanical coherence time.\nRemarkably, a recent optomechanical experiment has demon-\nstrated a mechanical coherence time longer than 100 ms [52].\nSpecifically, we show that an arbitrary magnonic state, either\nquantum or classical, can be transferred to a distant mechan-\nical resonator by using optical pulses to successively activate\nthe optomagnonic and optomechanical anti-Stokes processes.\nThis allows the transfer of the magnonic state to the anti-\nStokes optical pulse, and then the mapping of the pulse state\nto the mechanical mode. The magnonic state can be stored in\nthe mechanical mode within its coherence time and retrieved\nby sending a weak red-detuned read pulse to the optomechani-\ncal cavity, of which the output field carries the magnonic state.\nWe study the fidelity in this magnon-to-phonon state transferarXiv:2108.11156v3 [quant-ph] 1 Dec 20212\nprocess, and show its dependence on the system parameters,\ne.g., the pulse strengths and durations, and the transmission\nloss.\nWe further show that the magnonic and mechanical nodes\ncan be prepared in a macroscopic entangled state by using\npulses to successively activate the optomagnonic Stokes and\noptomechanical anti-Stokes processes. The former process re-\nalizes a two-mode squeezed vacuum state of the magnons and\nthe pulse, and the latter process maps the pulse state onto the\nmechanical mode, thus establishing a nonlocal entangled state\nbetween the magnonic and mechanical nodes.\nThe paper is organized as follows. In Sec. II, we introduce\nsome basic interactions in cavity optomagnonics and optome-\nchanics, which are necessities for realizing our protocol. In\nSec. III, we show how to operate the system via optical pulses\nsuch that an arbitrary magnonic state can be transferred to a\ndistant long-lived mechanical resonator. We further analyse\nthe fidelity in this state transfer process and show its depen-\ndence on the system parameters, especially on those related\nto optical pulses. We then study the e \u000bects of the transmis-\nsion loss of the pulse on the transferred mechanical state and\nthe fidelity. In Sec. IV, we show how to prepare a nonlocal\nmacroscopic entangled state between the magnonic and me-\nchanical nodes, and provide a strategy to detect it. Finally, we\ndraw the conclusions in Sec. V.\nII. BASIC INTERACTIONS IN OPTOMAGNONICS AND\nOPTOMECHANICS\nWe start with the introduction of the basic interactions in\ncavity optomagnonics and optomechanics that are key build-\ning blocks for realizing our protocol. These include the opto-\nmagnonic (optomechanical) two-mode squeezing and beam-\nsplitter interactions that are used for realizing the entangling\nand state-swap operations, respectively. We explicitly show\nhow to operate the two subsystems to realize these interactions\nand provide their e \u000bective Hamiltonians. With the successful\nimplementation of these local operations, we prove in the next\nsection that a remote quantum network based on magnonic\nand mechanical systems can be built using optical pulses.\nA. Magnon-induced Brillouin light scattering in\noptomagnonics\nWe consider a cavity optomagnonic system of a YIG\nsphere [12–14] that simultaneously supports a magnetostatic\nmode of magnons and whispering gallery modes (WGMs) of\noptical photons, as depicted in Fig. 1(a). The photons in a\nWGM are scattered by the lower-frequency magnons, typi-\ncally in GHz [12–14], yielding sideband photons with their\nfrequency shifted by the magnon frequency. This process\nis known as the magnon-induced Brillouin light scattering\n(BLS). When the scattered photons go into another WGM\n(the so-called triple-resonance condition), the BLS scattering\nprobability is then maximized. This triple resonance can be\nconveniently achieved by tuning the magnon frequency real-\nYIG\nB\n𝜔𝜔2 𝜔𝜔1𝜔𝜔𝑚𝑚\n𝜔𝜔\n𝜔𝜔𝑑𝑑−∆1\n−∆2(a)\n𝜔𝜔2 𝜔𝜔1𝜔𝜔𝑚𝑚\n𝜔𝜔𝜔𝜔𝑑𝑑∆1∆2\n𝑇𝑇𝑇𝑇\n𝑇𝑇𝑇𝑇\n(b)\n(c)\nto TE\nto TM\nFIG. 1: (a) An optomagnonic system of a YIG sphere supporting\ntwo WGMs and a magnon mode. (b) Mode frequencies of the op-\ntomagnonic Stokes BLS. (c) Mode frequencies of the optomagnonic\nanti-Stokes BLS.\nized by altering the strength of the bias magnetic field. Owing\nto the selection rule [53–56] imposed by the conservation of\nthe angular momenta of WGM photons and magnons, the BLS\nshows a pronounced asymmetry in the Stokes and anti-Stokes\nscattering strengths. This asymmetry is the basis for realiz-\ning the proposals for preparing macroscopic quantum states\nof magnons in optomagnonics [57–62].\nThe magnon-induced BLS is intrinsically a three-wave pro-\ncess, which can be described by the Hamiltonian\nH=H0+Hint+Hd; (1)\nwhere H0is the free Hamiltonian of two WGMs and a magnon\nmode\nH0=~=!1ay\n1a1+!2ay\n2a2+!mmym; (2)\nwith ajandm(ay\njandmy,j=1;2) being the annihilation\n(creation) operators of the WGMs and magnon mode, respec-\ntively, and!i(i=1;2;m) being their resonance frequencies,\nwhich satisfy the relation !m\u001c!jandj!1\u0000!2j=!m,\nimposed by the conservation of energy in the BLS. The inter-\naction Hamiltonian Hintof the three modes is given by\nHint=~=G0\u0000ay\n1a2my+a1ay\n2m\u0001; (3)\nwhere G0is the single-photon coupling rate. This coupling is\nweak due to the large frequency di \u000berence between the optical3\nand magnon modes, but it can be significantly enhanced by\nintensely driving one of the WGMs. The driving Hamiltonian\nis\nHd=~=iEj\u0000ay\nje\u0000i!dt\u0000ajei!dt\u0001; (4)\nwhere Ej=q\nPj\u0014e\nj=~!dis the coupling strength between the\njth WGM (with external decay rate \u0014e\nj) and the driving field\n(with frequency !dand power Pj). To maximize the BLS\nscattering probability, we resonantly pump either the WGM\na1ora2[12–14] (i.e., !d=!1or!2) toselectively activate\nthe anti-Stokes or Stokes scattering, which is responsible for\nthe optomagnonic state-swap or two-mode squeezing interac-\ntion. Note that the selection rule also causes di \u000berent optical\npolarizations of the two WGMs. Without loss of generality,\nwe assume a2(a1) mode to be the TM (TE) mode of a certain\nWGM orbit, and !2(TM)>! 1(TE) due to the geometrical bire-\nfringence of the WGM resonator [12, 61]. It is worth noting\nthat this is true for WGMs with the same angular momentum,\nbut when the optical angular momentum changes the domi-\nnant optomagnonic coupling occurs between WGMs satisfy-\ning!TM1, considering also \u0014m=2\u0019is\nhardly below 1 MHz for YIG [7–9]. However, this deficiency\nof the system can be evaded by using fast optical pulses [60–\n62, 64, 65].\nIn the first part, we aim to realize a distant magnon-to-\nphonon (quantum) state transfer. Specifically, as shown in\nFig. 3(a), by using laser pulses to successively activate the\nanti-Stokes scatterings in the optomagnonic and optomechan-\nical systems, an arbitrary magnonic state can be transferred\nto a mechanical resonator that can have a much longer co-\nherence time. We have introduced the interaction Hamilto-\nnian (10) of the optomagnonic anti-Stokes scattering in Sec. II\nA, associated with the process where a TE polarized pulse5\n𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇\n𝜔𝜔𝑇𝑇𝑇𝑇 𝜔𝜔𝑇𝑇𝑀𝑀=𝜔𝜔𝑇𝑇𝑇𝑇+𝜔𝜔𝑚𝑚\n𝜔𝜔𝐿𝐿=𝜔𝜔𝑐𝑐−𝜔𝜔𝑀𝑀𝜔𝜔𝑇𝑇𝑀𝑀anti-Stokes anti-Stokes \n𝜔𝜔𝑐𝑐 = 𝜔𝜔𝑇𝑇𝑀𝑀\n𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇\n𝜔𝜔𝑇𝑇𝑀𝑀 𝜔𝜔𝑇𝑇𝑇𝑇=𝜔𝜔𝑇𝑇𝑀𝑀−𝜔𝜔𝑚𝑚\n𝜔𝜔𝐿𝐿=𝜔𝜔𝑐𝑐−𝜔𝜔𝑀𝑀𝜔𝜔𝑇𝑇𝑇𝑇 𝜔𝜔𝑐𝑐=𝜔𝜔𝑇𝑇𝑇𝑇Stokes anti-StokesYIG\nYI G(a)\n(b)\n𝜔𝜔𝑇𝑇𝑇𝑇 𝜔𝜔𝐿𝐿𝜔𝜔𝑀𝑀\n𝜔𝜔\n𝜔𝜔𝑇𝑇𝑀𝑀 𝜔𝜔𝑐𝑐 𝜔𝜔𝐿𝐿𝜔𝜔𝑚𝑚\n𝜔𝜔\n𝜔𝜔𝑇𝑇𝑇𝑇\n(c) (d)\n𝜔𝜔𝑀𝑀\n𝜔𝜔𝑇𝑇𝑀𝑀\n𝜔𝜔𝑚𝑚\n𝜔𝜔𝑐𝑐\nFIG. 3: Sketch of the distant magnon-to-phonon state transfer protocol (a) and the corresponding mode frequencies (c). The nonlocal magnon-\nphonon entanglement protocol (b) and the associated mode frequencies (d).\ncouples to a WGM and generates TM polarized anti-Stokes\nphotons in another WGM by annihilating magnons. We con-\nsider laser pulses with duration much shorter than the magnon\nlifetime [66]. In this case, the dissipation of the magnon mode\nwithin the pulse duration is negligibly small, and we thus ne-\nglect it for simplicity. This leads to the following QLEs during\nthe pulse interaction:\n˙a2=\u0000\u00142\n2a2\u0000iG2m+p\u00142ain\n2;\n˙m=\u0000iG2a2:(19)\nTo simplify the model, we consider a flattop pulse and thus\na constant coupling G2during the pulse. Given also the fact\nthat a weak coupling G2\u001c\u00142, one can then adiabatically\neliminate the cavity, and obtain a2'2\n\u00142\u0000\u0000iG2m+p\u00142ain\n2\u0001.\nBy using the input-output relation aout\n2=p\u00142a2\u0000ain\n2[67], we\nobtain\naout\n2=\u0000ip\n2G2m+ain\n2;\n˙m=\u0000G 2m\u0000ip\n2G2ain\n2;(20)\nwhereG2\u00112G2\n2=\u00142. Following [68], we define a set of nor-\nmalized temporal modes for the WGM driven by a pulse of\nduration\u001c2\nAin\n2(\u001c2)=r\n2G2\ne2G2\u001c2\u00001Z\u001c2\n0eG2sain\n2(s)ds;\nAout\n2(\u001c2)=r\n2G2\n1\u0000e\u00002G2\u001c2Z\u001c2\n0e\u0000G 2saout\n2(s)ds;(21)which satisfy the canonical commutation relation\n[Aj;Ajy]=1;j=fin;outg. Therefore, by integrating (20) we\nobtain the following solutions (see Appendix):\nAout\n2(\u001c2)=\u0000ip\n1\u0000e\u00002G2\u001c2m(0)+e\u0000G 2\u001c2Ain\n2(\u001c2);\nm(\u001c2)=e\u0000G 2\u001c2m(0)\u0000ip\n1\u0000e\u00002G2\u001c2Ain\n2(\u001c2):(22)\nFrom these solutions, we can extract a propagator L2(\u001c2)\nthat satisfies Aout\n2(\u001c2)=Ly\n2(\u001c2)Ain\n2(\u001c2)L2(\u001c2) and m(\u001c2)=\nLy\n2(\u001c2)m(0)L2(\u001c2), given by [60]\nL2(\u001c2)=e\u0000ip\nS0Ainy\n2meG2\u001c2(Ainy\n2Ain\n2\u0000mym)eip\nS0Ain\n2my; (23)\nwhere S0=S e2G2\u001c2, with S=1\u0000e\u00002G2\u001c2(00 [14]. This e \u000bect is\nprominent when the cavity detuning \u0001a'\u0000!b. By coupling\nthe second magnon mode ( m2) to the cavity, and using their\nstate-swap interaction, the two magnon modes are expected to\nbe entangled. This is confirmed by Fig. 2 (a) and it manifests\nthat the optimal situation is that the magnon mode is resonant\nwith the cavity, \u00012'\u0001a'\u0000!b[see Fig. 1 (c)]. This is also\nthe case for generating squeezed states of magnons by driv-\ning the cavity with a squeezed microwave field [21]. Figure 2\n(b) and (c) denote the cavity-magnon ( m1) entanglement Eam1\nwith g2=0 and the magnon-magnon entanglement Em1m2with\ng2,0, respectively. The similar patterns of Fig. 2 (b) and (c)\n(d)(a)\n(c)(b)\nFIG. 2: Density plot of the entanglement Em1m2between two magnon\nmodes vs (a) \u0001aand\u00012, (c)\u0001aand˜\u00011, and (d) the ratios of g2=g1and\nG=g1(g1is fixed). (b) The entanglement Eam1between cavity and\nmagnon mode m1vs\u0001aand˜\u00011with g2=0. We take ˜\u00011=0:85!bin\n(a) and (d), g2=2\u0019=2:6 MHz in (a) and (c), \u0001a=\u00000:9!bin (d), and\n\u00012= \u0001 ain (c) and (d). See text for details of the other parameters.\nFIG. 3: Magnon entanglement Em1m2vs (a) \u0001aat 10 mK, and (b)\ntemperature for the two cases of \u0014a=2\u0019=\u00141(2)=2\u0019=1 MHz (solid\nlines) and\u0014a=2\u0019=5\u00141(2)=2\u0019=3 MHz (dashed lines). (c) Critical\ntemperature (below which Em1m2>0) vs\u00141(2)=2\u0019, with always \u0014a=\n5\u00141(2). We take an optimal detuning \u0001a=\u00000:9!bin (b) and (c). The\nother parameters are as in Fig. 2 (a) with \u00012= \u0001 a.\nshow more clearly the magnon entanglement Em1m2is trans-\nferred from the cavity-magnon ( m1) entanglement Eam1due to\nthe state-swap interaction between the cavity and the magnon\nmode m2. We take g2=2\u0019=2:6 MHz in Fig. 2 (a) and (c), and\ntherefore we have g2\n1;g2\n2\u001cj˜\u00011\u0001aj;j\u00012\u0001aj'!2\nb, which leads\nto a rather simple expression of the coupling G'p\n2G0\n!b[see\nEq. (6)]. G=2\u0019=4:8 MHz implies the drive magnetic field\nB0'3:9\u000210\u00005T for G0=2\u0019'0:3 Hz, corresponding to the\ndrive power P'8:9 mW [44]. A larger coupling Gwould\nyield a larger entanglement. However, we take a moderate\nvalue to keep the system stable and avoid unwanted nonlinear\ne\u000bect (we analyse this in the next section). There are opti-\nmal couplings of g2andGfor fixed g1, as shown in Fig. 2\n(d). Since all bipartite entanglements of the subsystems orig-\ninate from the magnon-phonon coupling, there is an interplay\namong the three couplings, g1,g2, and G, of the four modes\nof the system [see Fig. 1 (b)], which results in a maximum\nmagnon entanglement. The magnon entanglement is robust\nagainst environmental temperature and survives up to about\n200 mK, as shown in Fig. 3 (solid lines).\nIV . ANALYSIS OF NONLINEAR EFFECT AND\nSTRATEGIES FOR ENTANGLEMENT DETECTION\nIt should be noted that the results of section III are\nvalid only when the magnon excitation numbers hmy\njmji \u001c\n2Ns=5N. For what we used a 250- \u0016m-diam YIG sphere,\nthe number of spins N'3:5\u00021016, and the coupling\nstrength G=2\u0019=4:8 MHz corresponds to jhm1ij'1:1\u0002\n107, and Rabi frequency \n'7:1\u00021014Hz, such that\nhmy\n1m1i'1:3\u00021014\u001c5N=1:7\u00021017, which is well sat-\nisfied.hmy\n2m2i\u001c 2Nsis also well fulfilled due to the fact\nthatjhm2ij\u001cjh m1ijfor the parameters used in Figs. 2 and 3.5\nThe intense magnon drive may bring about unwanted nonlin-\near e\u000bects due to the Kerr nonlinear term Kmymmymin the\nHamiltonian [20, 45]. The Kerr coe \u000ecientKis inversely pro-\nportional to the volume of the sphere. For a 1-mm-diam YIG\nsphere used in Refs. [20, 45], K=2\u0019\u00190:1 nHz, which im-\nplies thatK=2\u0019\u00196:4 nHz for the spheres used in this paper.\nTo keep the Kerr e \u000bect negligible,Kjhm1ij3\u001c\nmust hold.\nThe parameters in Fig. 2 lead to Kjhm1ij3'5:8\u00021013Hz\n\u001c\n'7:1\u00021014Hz, which means that the nonlinear e \u000bects\nare negligible and our model is valid and a good approxima-\ntion.\nFinally, we discuss how to detect and measure the entangle-\nment. The generated magnon entanglement can be quantified\nby measuring the CM of the two magnon modes, following\nthe strategies used in Refs. [26, 35]. In contrast with Ref. [14],\nwhere the detection of a phonon state cannot be avoided, here\nwe only need to measure the states of two magnon modes.\nThe state of each magnon mode can be read out by coupling\nthe magnons to an additional cavity, and by sending a weak\nmicrowave probe field and homodyning the cavity output of\nthe probe field. This requires that the magnon dissipation\nrates\u00141;2should be much smaller than the cavity decay rate\n\u0014a, such that when the magnon drive is switched o \u000band all\ncavity photons decay the magnon states remain almost un-\nchanged, and then two probe fields are sent. The dashed lines\nin Fig. 3 (a) and (b) show the magnon entanglement for the\ncase of\u0014a=5\u00141(2), where the entanglement is still there and\nsurvives up to about 150 mK. In Fig. 3 (a) and (b) we take\n\u00141(2)=2\u0019=0:6 MHz, which is the lowest value demonstrated\nin the experiment [13]. It would be useful to study the entan-\nglement for larger magnon dissipation rates. In Fig. 3 (c) we\nplot the critical temperature versus \u00141(2)=2\u0019starting from 0.6\nMHz to 3 MHz. The critical temperature means that above\nwhich the entanglement becomes zero due to the degradation\nof the thermal noise. We see that the entanglement is quite\nrobust against the magnon dissipation rates and survives up to\nabout 80 mK for \u00141(2)=2\u0019=3 MHz.\nV . CONCLUSIONS\nWe have presented a protocol to entangle two magnon\nmodes in a cavity magnomechanical system, where the twomagnon modes in two YIG spheres couple to a microwave\ncavity mode, and one of them also couples to a vibrational\nmode of the sphere via magnetostrictive force. We have\nshown that with experimentally reachable parameters the two\nmagnon modes can be prepared in a steady-state entangled\nstate. The entanglement is achieved by exploiting the nonlin-\near magnetostrictive interaction and the linear cavity-magnon\ncoupling. The magnon entangled states in massive YIG\nspheres represent genuinely macroscopic quantum states, and\nare thus useful for the study of quantum-to-classical transi-\ntions and tests of decoherence theories [46]. In either the con-\ntinuous spontaneous localization theory [47, 48], or the gravi-\ntationally induced collapse model [49], the strength of the pos-\ntulated collapse noise is proportional to the size of the object\n(either mass or the number of particles it contains). The col-\nlapse e \u000bect becomes prominent when the size of the object is\nin macroscopic scale, which leads to spatial decoherence and\nlocalization of the superposition states, while it reproduces the\nstandard results of quantum mechanics in microscopic scale.\nTherefore, preparing macroscopic quantum states is a key\nstep to test those decoherence theories in macroscopic scale.\nFurthermore, the magnon entangled states can be applied to\nthe quantum information processing based on magnonic sys-\ntems [10], and can also be used for creating entangled states\nof microwave fields, e.g., by coupling each magnon mode to a\nmicrowave cavity and utilizing their beamsplitter (state-swap)\ninteraction.\nNote added : After the completion of this work, independent\nproposals have been put forward, using di \u000berent mechanisms,\nfor entangling two magnon modes, either in ferrimagnetic\nYIG spheres [50–52] or in an antiferromagnetic system [53].\nVI. ACKNOWLEDGMENTS\nWe thank G. S. 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Yung,\narXiv:1903.02484" }, { "title": "1810.10404v1.Long_spin_coherence_length_and_bulk_like_spin_orbit_torque_in_ferrimagnetic_multilayers.pdf", "content": "1 \n Long spin coherence length and b ulk-like spin-orbit torque in ferrimagnet ic \nmultilayers \n \nJiawei Y u1, Do Bang2†, Rahul Mishra1†, Rajagopalan Ramaswamy1, Jung Hyun Oh3, Hyeon-\nJong Park4, Yunboo Jeong5, Pham Van Thach2, Dong -Kyu Lee3, Gyungchoon Go3, Seo-Won \nLee3, Yi Wang1, Shuyuan Shi1, Xuepeng Qiu6, Hiroyuki Awano2, Kyung -Jin Lee3,4,5*, and \nHyunsoo Yang1* \n1 Department of Electrical and Computer Engineering, National University of Singapore, \n117576, Singapore \n2 Toyota Technological Institute, Tempaku, Nagoya 468 -8511, Japan \n3 Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea \n4 KU-KIST Graduate School of Conversing Science and Technology, Korea University, Seoul \n02841, Korea \n5 Department of Semiconductor Systems Engineering, Korea University, Seoul 02841, Korea \n6 Shanghai Key Laboratory of Special Artificial Macro structure Materials and Technology and \nSchool of Physics Science and Engineering, Tongji University, Shanghai 200092, China \n†These authors con tributed equally to this work. \n*e-mail: eleyang@nus.edu.sg, kj_lee@korea.ac.kr \nSpintronics is a multidisciplinary field whose central theme is the active manipula tion of spin \ndegrees of freedom in solid -state system s. Ferromagnetic spintronics has been a main focus \nas it offer s non-volatile memory and logic applications through current -induced spin-\ntransfer torque s1-4. Enabling w ider application s of such magnetic devices requires a lower \nswitching current for a smaller cell while keeping the thermal stability of magnetic cells for \nnon-volatility . As the cell size reduces , however, it becomes extreme ly difficult to meet th is \nrequirement with ferromagnets because spin-transfer torque for ferromagnets is a surface \ntorque due to rapid spin dephasing5,6, leading to the 1/ferromagnet -thickness dependence of \nthe spin -torque efficiency7. Requirement of a larger switching current for a thicker and thus 2 \n more thermally stable ferromagnetic cell is the fundamental obstacle f or high -density non-\nvolatil e application s with ferromagnets . Theories predicted that antiferromagnets have a \nlong spin coherence length due to the staggered spin order on an atomic scale8,9, thereby \nresolv ing the above fundamental limitation . Despite several spin -torque experiments on \nantiferr omagnets10-12 and ferrimagnetic alloys13-16, this prediction has remained un explored . \nHere we report a long spin coherence length and associated bulk -like-torque characteristic \nin an antiferromagnetically coupled ferrimagnetic multilayer . We find that a transverse spin \ncurrent can pass through > 10 nm-thick ferrimagnetic Co/Tb multilayers whereas it is \nentirely absorbed by 1 nm-thick ferromagnetic Co/Ni mult ilayer. We also find that the \nswitching efficiency of Co/Tb multilayers partially reflects a bulk -like-torque characteristic \nas it increases with the ferrimagnet -thickness up to 8 nm and then decreases , in clear contrast \nto 1/thickness -dependence of Co/Ni multilayers . Our results on antiferromagnetically \ncoupled system s will invigorate researches towards energy -efficient spintronic technologies . \nThe spin -transfer torque ( STT) acting on ferromagnet s (FMs) is a surface torque, based on the \naveraging effect of STT5,6. We note that the same averaging effect occurs regardless of the spin -\ncurrent source , and the spin -orbit torque (SOT)17,18, which we use in our experiment, is also a \nsurface torque for FMs (Extended Data Fig. 1 and Methods ). When a transverse spin current with \na spin orientation non-collinear with the magnetization is injected into a FM, the electron spin \nprecesses rapidly in real space because the wave vectors of the majority (↑) and minority (↓) spins \nat the Fermi surface ar e different (i.e., \nFFkk ). The p recession wavelength s are different for \ndifferent incident angle s of electron s (i.e., the direction of wave vector k), leading to rapid spin \ndephasing when summing over all current -carrying k-states . As a result , the k-integrated transverse \nspin c urrent decays to zero within a distance from the FM surface , called the ferromagnetic 3 \n coherence length (spin coherence length, more generally) , \nc F F kk .19 As \nF Fk k \nbecomes larger for larger exchange splitting, λc is only a few angstroms i n strong FMs (e.g. cobalt \nor iron) for which the STT is almost a surface torque. \nTheories predicted that the spin coherence length is very long in antiferromagnets (AFMs) \nbecause of the staggered spin order on an atomic scale8,9. We use the term of “bulk -like-torque” to \ndescribe the characteristic of spin -torque for AFMs, i.e., spin -current absorption on a larger \nthickness, in contrast to the surface -torque of FMs. A semi -classical explanatio n of bulk -like-\ntorque is that for conduction electron spins, the moments with alternating orientation on an atomic \nscale are seen as the exchange interactions with alternating signs. As a result, an ideal AFM has \nzero net effective exchange interaction whe n averaged ove r two sub -lattices and thus has an \ninfinitely long λc, yielding the bulk -like-torque characteristic. Several experiments have \ninvestigated on STT/SOT effect s in systems including AFMs10-12 and more recently on \nferrimagnetic alloys13-16, but not on the long spin coherence length and associated bulk -like-torque \ncharacteristic. \nWe qualitatively illustrate the spin coherence length in FMs and FIMs (or AFMs) based on the \nspin precession around the local exchange field. Neglecting the spin relaxation, dynamics of non -\nequilibrium spin density s is described by \nex t Hs s / , where \n is the gyromagnetic ratio, \nHex is the effective exchange field that is aligned along the local magnetic moment m. Assuming \n 0, cos,sins\n and \nz H ˆHex , this equation of motion transforms to \nH t t / / , \nwhere the sign of spin precession angle \n follows the sign of H and thus the sign of m (Fig. 1a \nand b). In a FM, an electron spin propagating along the x-direction continuously precesses in the \nsame sense because of the homogeneous exchange field (Fig. 1c). On the other hand, in a FIM, an \nelectron spin precesses counter -clockwise on a lattice corresponding to a positive exchange f ield, 4 \n whereas it precesses clockwise on the next lattice corresponding to a negative exchange field. As \na result, the period (or wavelength) of spin precession in FIMs is longer than that in FMs, resulting \nin much less spin dephasing. \nIn order t o verify th e theoretical prediction of long spin coherence length , we perform \nexperiments with a ferrima gnet (FIM) , i.e., Co/Tb multilayers where both Co and Tb layers are \natomically thin and their moments are coupled antiferromagnetically. We choose a FIM, instead \nof an AFM , for following two reasons. One is that Co/Tb multilayer s can show a longer λc than a \nFM because of the antiferromagnetic alignment of Co and Tb moments ( Extended Data Fig. 1 and \nMethods ), thereby exhibiting a feature of the bulk -like-torque characteristic. As explained above, \nthe STT efficiency of a FM is inversely proportional to the FM -thickness whereas that of an ideal \nAFM is independent of the AFM -thickness. As λc of FIM is located between those of FM and \nAFM, it is expected that the STT efficiency of a FIM first increases and then decreases with the \nFIM-thickness. The other reason to choose FIM s is that various measurement methods established \nfor FM s are applicable to FIM s because of nonzero net moment13,20. However, t he choice of FIM \nalso results in a difficulty. FIMs commonly show a thickness -dependent variation of magnetic \nproperties21, as also observed in Co/Tb multilayer s (Extended Data Fig. 4 and Methods ), which \nmakes a quantitative analysis of spin transport difficult. Even with this difficulty, our thickness -\ndependent S OT measurement s combined with spin pumping measurement s support a long spin \ncoherence length and associated bulk -like-torque characteristic in ferrimagne tic Co/Tb multilayer s, \nas we show below . \nWe fabricate perpendicularly magnetized ferromagnetic [Co/Ni ]N and ferrimagnetic [Co/Tb ]N \nmultilayers (Fig. 2a, b; see Methods for details ), where the total thickness varies with changing the \nrepetition number N. Both Co/Ni and Co/Tb multilayers have an additional Pt layer, and an in -5 \n plane current generates SOT s. We use the harmonic Hall voltage measurements to quantify the \nstrength of SOT effective fields22,23. The longitudinal and transverse measurement schematics are \nillustrated in Fig. 2c and d, respectively. Representative results for the longitudinal (blue line) and \ntransverse (red line) second h armonic voltage s (V2f) from Co/Ni ( N = 2) and Co/Tb ( N = 5) devices \nare shown in Fig. 2e and f, respectively . The anomalous Nernst effect is corrected in the V2f data22. \nWe observe a clear V2f in the longitudinal configuration (H//I), which is mostly determined by the \nanti-damping SOT22,23. The opposite V2f signs in the H//I case for Co/Ni and Co/Tb multilayers \nindicate that the Pt layer is the source of spin currents, as it is pl aced on top of the Co/Tb multilayer, \nbut under the Co/Ni multilayer. In order to rule out the contribution from pure bulk Co/Tb to SOT, \nwe conduct a control experiment without and with the spin current source, Pt ( Extended Data Fig. \n5, 6 and Methods ). We find that there is no noticeable current -induced SOT without the Pt layer, \nsuggesting that the Co/Tb bulk itself cannot directly contribute to SOTs. \nWe extract the spin -orbit effective fields, HL and HT, by fitting V2f,23 where HL and HT \ncorrespond to the anti -damping (longitudinal) and field -like (transverse) component s of SOT s, \nrespectively. The planar Hall effect is considered for the fitting ( Extended Data Fig. 8 and \nMethods ). Devices with different N, corresponding to different thickness es, tFM or tFIM, have been \nmeasured. Absolute SOT effective fields normalized by the current density in the Pt layer (HL/T/J) \nare presented in Fig. 3a and b for the Co/Ni and Co/Tb systems, respectively. We find that both \nHL and HT of Co/Ni multilayers decrease as tFM increases , consistent with the surface -torque \ncharacteristic expected for FMs. However, Co/Tb multilayers show an entirely different trend , in \nwhich both HL and HT increase up to tFIM of 7.9 nm and then decrease for thicker samples . \nWe estimate the effective spin Hall angle \n J teM HFIM FMS L eff / 2/ , where e is the electron \ncharge and ħ is the reduced Planck’s constant, with considering thickness -dependent variation of 6 \n the saturation magnetization MS and current density J in the Pt layer . The Co/Tb multilayer shows \na significant MS variation with the minimum at tFIM of 6.6 nm ( inset of Fig. 3d). We find that θeff \nof Co/Ni multilayer is nearly constant with tFM (Fig. 3c). Similar to the tendency of HL/J, θeff of \nCo/Tb multilayer increases up to tFIM = 9.9 nm and then decreases for a thicker sample (Fig. 3d). \nBesides the distinct thickness -dependence of θeff, another interesting observation is that the Co/Tb \nmultilayer shows a larger θeff than Co/Ni multilayer (θeff of Co/Tb multilayer = 2. 1 at tFIM of 9.9 \nnm and the average θeff of Co/Ni multilayer = 0. 2 ± 0.05). We note that a model calculation with \nconsidering a thickness -dependent variation of the sd exchange in FIMs shows qualitatively \nsimilar trends with the experimental ones ( Extended Data Fig. 2 and Methods), even though the \nmodel is too simple to capture all the details of FIMs. Nevertheless, this qualitative agreement \nbetween model and experiment results indicat es that the distinct behavior of θeff of Co/ Tb \nmultilayer would originate from a combined effect of long spin coherence length and thickness -\ndependent property variation. \nAs an independent test, we perform SOT switching experiments with applying an external field \n(Hext) in the current direction ( θ = 0°) for deterministic switching. The insets of Fig. 3e and f show \nthe representative current -induced switching data obtained from Co/Ni ( N = 2) and Co/Tb (N = 5) \nsamples , respectively . As the switching is governed by domain nucleation and propagation in large \nsamples ( i.e., Hall bar width = 10 m), we estimate the STT efficiency \nJ Hp/ , where Hp is \nthe domain wall depi nning field24. Figure 3e and f show η as a function of tFM and tFIM, respectively. \nFor Co/Ni multilayers, η decreases with tFM, whereas for Co/Tb multilayers it increases and then \ndecreases with tFIM, following similar trends to SOT effective fields (Fig. 3a and b). We note that \nrecently reported fast dynamics at the angular momentum compensation condition in FIMs20 \nwould affect the switching data, but not the harmonic Hall data. 7 \n As different approaches for the estimat ion of spin torque efficiency show qualitatively similar \ntrends, it indicate s that S OT for Co/Tb mul tilayer s is not a surface torque. Moreover, the observed \nthickness -dependence of spin torque efficiency is qualitatively consistent with the model \ncalculation ( Extended Data Fig. 2 and Methods) for the bulk -like-torque characteristic in FIMs ; it \nfirst increase s and then decrease s with the FIM-thickness. However, because of the thickness -\ndependent property variation s in Co/Tb multilayers (inset of Fig. 3d and Methods ), this result is \nnot yet conclusive but it is still possible that another unknown mechan ism is responsible for the \ndistinct thickness -dependence observed in Co/Tb multilayers. \nIn order t o resolve this ambiguity, we perform additional spin pumping experiment s to estimate \nthe spin coherence length λc. We measure a spin -pumping -induced inverse spin Hall voltage (VISHE) \nfor substrate/Pt(10) /[FIM or FM] /Cu(2.4)/Co(20) structure s (numbers in nanometers ; FIM = \n[Co(0.32)/Tb(0.34)] N and FM = [Co(0.3)/Ni(0.6)] N) as shown in Fig. 4a. In these structures, the \nCo/Ni and Co/Tb multilayers are perpendicularly magnetized , whereas the top thick Co layer has \nan in -plane magnetization. In the spin pumping setup (Fig. 4b; see Methods for details) , the top \nCo layer generates a spin -pumping -induced spin current with an in -plane spin polarization (thus \ntransverse to Co/Ni or Co/Tb magnetization direction), which passes through the Cu layer and \nenters the Co/Ni or Co/Tb layer. If λc of the Co/Tb multilayer is long, it is expected that a transverse \nspin current passes through the Co/Tb layer without much spin dephasing and reaches the bottom \nspin sink, Pt , and s ubsequently, VISHE is generated by the inverse spin Hall effect of Pt . On the \nother hand, VISHE is expecte d to be negligible for a thick Co/Ni multilayer because a transverse \nspin current is almost absorbed at the [Co/Ni]/Cu interface . Therefore, the measurement of VISHE \nversus FIM - or FM -thickness provides an estimate of λc. 8 \n We find that the experimental results are consistent with th is expectation. In Fig. 4c, black \nsymbols are the data from a reference Pt/Cu/Co sample , which show s the largest VISHE signal at an \nin-plane bias field Hb. For the Co/Ni -based structure, VISHE signal becomes negligible at a Co/Ni \nthickness of 0.9 nm (blue symbols). In contrast, VISHE signal for the Co/Tb -based structure is finite \nat a much thicker Co/Tb (red symbols, Co/Tb thickness = 5.3 nm as an example). VISHE signal \ndisappears when excluding the top Co layer from the Co/Tb -based structure (green symbol), \nproving that the perpendicularly magnetized Co/Tb itself does not generate a VISHE signal. In Fig. \n4d, spin pumping results are summarized for a wide thickness range of Co/Tb multilayer. It shows \nthat VISHE signal is finite even at 13 nm -thick Co/Tb . This result evidenc es a long spin coherence \nlength in ferrimagnetic Co/Tb multilayers . \nThe spin pumping and spin torque are connected through the Onsager reciprocity25. Therefore, \nthe long spin coherence length observed in spin pumping experiment s suggests that the bulk-like-\ntorqu e characteristic must be present in spin torque experiment s at least partially . Given that the \nthickness -dependent change in the spin torque efficiency follows the trend expected for the bulk -\nlike-torque characteristic (Fig. 3), the spin pumping experiment s combined with the spin torque \nexperiment s allow us to conclude that the antiferromagnetically coupled FIMs show a long spin \ncoherence length and associated bulk -like-torque characteristic. We note that this bulk -like-torque \ncharacteristic and equivalentl y long spin coherence length are also observed for FIM alloys \n(Extended Data Fig. 9 and Methods), which would relate to some ordering in FIM alloys26,27. 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T. & Koon, N. C. Structural origins \nof magnetic anisotropy in sputtered amorphous Tb -Fe films. Phys. Rev. Lett. 69, 1939 -\n1942, (1992). \n41 Hufnagel, T. C., Brennan, S., Zschack, P. & Clemens, B. M. Structural anisotropy in \namorphous Fe -Tb thin films. Phys. Rev. B 53, 12024 -12030, (1996). 12 \n \n \n \nFigure 1 | Schematic illustrations of spin precession from the semi -classical viewpoint . a, b, \nLocal spin precession angle \n in a FM with up magnetic moment (\nz m ˆ// ) and down magnetic \nmoment (\nz m ˆ// ), respectively. Blue dots (Fig. 1a) and red crosses (Fig. 1b) indicate the directions \nof magnetic moments. Precession of an electron spin in the FM layer ( c) and in the FIM layer ( d). \nBlue and red curved arrows indicate \n > 0 and \n < 0, respectively. \n13 \n \nFigure 2 | Film stacks and SOT measurements . a, b, Illustrations of Co/Ni ( a) and Co/Tb ( b) \nmultilayers. The magnetization s of Co, Ni and Tb sub -lattices are presented by the yellow, blue \nand green arrow s, respectively. c, d, The measurement schematics for longitudinal ( c) and \ntransverse ( d) SOT effective fie lds. e, f, Second harmonic voltage s (V2f) obtained from Co/Ni ( e) \nand Co/Tb ( f) multilayer devices, with the blue curves representing the longitudinal signals and \nred curves representing the transverse signals. The i nsets correspond to first harmonic voltage s \n(Vf). \n14 \n \nFigure 3 | SOT effective fields and switching efficiencies . a, b, Longitudinal ( HL) and transverse \n(HT) SOT effective fields as a function of Co/Ni ( a) or Co/Tb ( b) thicknesses. c, d, Effective spin \nHall angle (eff) as a function of Co/Ni ( c) or Co/Tb ( d) thicknesses. Insets in c and d are the \nsaturation magnetization (MS) as a function of ferromagnet - or ferrimagnet -thickness. e, f, \nSwitching efficiencies ( η) as a function of Co/Ni ( e) or Co/Tb ( f) thicknesses. Insets in e and f are \ncurrent -induced switching data, showing the Hall resistance ( RH) as a f unction of appli ed pulse \ncurrent. \n0.20.40.6 HL\n HTHL/T/J (10-8 kOe cm2/A)\n1 2 34681012\n (10-10 Oe m2/A)\ntFM (nm)\n0.00.51.0eff\n012eff\n0510 HL\n HTHL/T/J (10-8 kOe cm2/A)\n2 4 6 810 12 140100200300400500\n (10-10 Oe m2/A)\ntFIM (nm)\n2468101214100200MS (emu/cc)\ntFIM (nm)\n1.0 1.5 2.0 2.5 3.06008001000 MS (emu/cc)\ntFM (nm)a b\nc d\ne f\n-4 -2 2 4-0.50.00.5R ()\nJ (1011 A/m2)\n-6-3 36-0.30.00.3R ()\nJ (1011 A/m2)15 \n \n \nFigure 4 | Spin pumping measurements. a, Spin pumping sample structure. b, Schematic of spin \npumping measurements. S and G indicates signal and ground connection for high frequency \nmeasurements. An in -plane field ( Hb) along the waveguide direction is applied. c, Spin pumping \nsignal s in various structures. d, Inverse spin Hall signal as a function of [Co/Tb] -thickness in \nPt/[Co/Tb]/Cu/Co structures at various frequencies . \n \n-2024681012140.000.020.040.060.080.10 7 GHz\n 8 GHz\n 9 GHzVISHE/R (A)\ntCoTb (nm)\nS\nGVHb\nSubstrateNi or Tb\nCo×N\nPtCu…Coa b\n-0.6 -0.4 -0.2 0.00.00.51.0 Pt/Cu/Co Pt/[Co/Tb](5.3)/Cu/Co\n Pt/[Co/Ni](0.9)/Cu/Co\n Pt/[Co/Tb]/CuVISHE (V)\nHb (kOe)\nc d16 \n Methods \nSample preparation \nSubstrate /[Tb (0.3 4 nm)/Co (0.32 nm)] N/Pt (4 nm) and substrate /MgO (2 nm)/Pt (4 nm)/[Co \n(0.3 nm)/Ni (0.3 nm)] N/SiO 2 (3 nm) multilayers are fabricated on thermally oxidized silicon \nsubstrates using rf and dc magnetron sputtering system with a base pressure of ~ 10-9 Torr. N is \nthe repetition number of Tb/Co or Co/Ni bilayer pairs, which is varied from 4 to 20 for Co/Tb \nsystems and from 2 to 5 for Co/Ni systems. For Co/Tb multilayers, a 4 nm -thick Pt layer is \ndeposited on top as a spin current source which also protects the m ultilayer from being oxidized. \nFor Co/Ni multilayers, a bilayer of MgO (2 nm)/Pt (4 nm) is deposited on the bottom as a buffer \nand spin current source, and a SiO 2 (3 nm) layer is deposited as a capping layer to prevent possible \noxidation of the FM layer. S ubsequent photolithography and ion milling processes are performed \nto fabricate the films into Hall bar devices. \nSecond harmonic and spin pumping measurements \nFor the second harmonic measurements, an ac current Iac with a frequency of 13.7 Hz and a \nmagnitude of 5 mA is injected into the channel of the device31. An external magnetic field Hext is \napplied along (orthogonal to) the current direction with a small out -of-plane tilting of θ = 4 from \nthe film plane in the longitudinal (transverse) configuration. The first and second harmonic Hall \nvoltages are recorded simultaneously by using two lock -in amplifiers triggered at the same \nfrequency by the current source. \nIn the spin pumping m easure ments, a microwave at 7 to 9 GHz is applied to the asymmetric \ncoplanar stripline waveguide by a signal generator. An in -plane field ( Hb) along the waveguide \ndirection is swept around the resonance field ( H0) given by the Kittel formula \n 0 0 S 42f H H M\n, where γ is the gyromagnetic ratio and MS is the saturation 17 \n magnetization. The voltage ( V) is recorded by a lock -in amplifier. V includes the asymmetric \ncomponent ( Vasym) from the anomalous Hall effect (AHE) and the anisotropic magnetoresistance, \nas well as the symmetric components ( Vsym) from the spin pumping induced inverse spin Hall \nvoltage ( VISHE). Thus the measured voltage is fitted by a sum of symmetric and asymmetric \nLorentzian function \n\n2\n0\n22 22\n00sym asymHHV V V\nH H H H \n , from which VISHE is extracted \nas Vsym. " }, { "title": "2106.02025v1.Relationship_between_A_site_Cation_and_Magnetic_Structure_in_3d_5d_4f_Double_Perovskite_Iridates_Ln2NiIrO6__Ln_La__Pr__Nd_.pdf", "content": "Relationship between A-site Cation and Magnetic Structure in 3 d-5d-4fDouble Perovskite Iridates\nLn2NiIrO 6(Ln=La, Pr, Nd)\nT. Ferreira,1, 2S. Calder,3,\u0003D. S. Parker,1M. H. Upton,4A. S. Sefat,1and H.-C. zur Loye2,y\n1Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831.\n2Department of Chemistry and Biochemistry, University of South Carolina, Columbia, SC 2920.\n3Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA.\n4Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA.\nWe report a comprehensive investigation of Ln2NiIrO 6(Ln= La, Pr, Nd) using thermodynamic and transport\nproperties, neutron powder diffraction, resonant inelastic x-ray scattering, and density functional theory (DFT)\ncalculations to investigate the role of A-site cations on the magnetic interactions in this family of hybrid 3 d-\n5d-4fcompositions. Magnetic structure determination using neutron diffraction reveals antiferromagnetism for\nLa2NiIrO 6, a collinear ferrimagnetic Ni/Ir state that is driven to long range antiferromagnetism upon the onset of\nNd ordering in Nd 2NiIrO 6, and a non-collinear ferrimagnetic Ni/Ir sublattice interpenetrated by a ferromagnetic\nPr lattice for Pr 2NiIrO 6. For Pr 2NiIrO 6heat capacity results reveal the presence of two independent magnetic\nsublattices and transport resistivity indicates insulating behavior and a conduction pathway that is thermally\nmediated. First principles DFT calculation elucidates the existence of the two independent magnetic sublattices\nwithin Pr 2NiIrO 6and offers insight into the behavior in La 2NiIrO 6and Nd 2NiIrO 6. Resonant inelastic x-ray\nscattering is consistent with spin-orbit coupling splitting the t 2gmanifold of octahedral Ir4+into a J e\u000b=1\n2and\nJe\u000b=3\n2state for all members of the series considered.\nI. INTRODUCTION\nPerovskites are one of the most studied solid-state materi-\nals due to their modular structure allowing for the incorpora-\ntion of a wide range of elements, within the limitations out-\nlined by the Goldschmidt tolerance factor [1–4]. The ability\nto stabilize a wide variety of elements with different, and of-\nten competing, physical properties within the same material\nmakes the perovskite structure a model system to study a rich\ndiversity of magnetic and electronic properties [5–24]. Hybrid\n3d-5d(4d) based materials that adopt the perovskite structure\ntype host an array of physical properties originating from a\ndelicate balance of interactions. For example, unpaired 3 d\nelectrons strongly correlate to 2 poxygen electrons in a per-\novskite lattice, often resulting in technologically useful prop-\nerties such as ferromagnetism [25], ferroelectricity [26], and\nmultiferroicism [5]. By contrast, the greater orbital extent of\nheavier 5delements, weaker electron correlation strength, and\nstronger spin-orbit coupling (SOC) can lead to metal-insulator\ntransitions [27], topological insulators [28], superconductivity\n[29] and a split of the t 2gmanifold into a J e\u000b=1\n2and J e\u000b=3\n2\nstate, as observed in Sr 2IrO4[30], that can lead to new routes\nto Mott and other exotic insulating states [30–41]. Hybrid\nperovskites containing both 3 dand 5delements have been re-\nported to exhibit a wide range of properties characteristic of\nboth 3dand 5dcontaining oxides, in addition to extremely\nhigh magnetic ordering temperatures, (Curie temperature of\nTc= 725 K) such as that observed in Sr 2CrOsO 6, further mo-\ntivating the study of perovskites as a host lattice to investigate\nthe balance of competing interactions.\nCompared to the single perovskite (ABO 3) system with\nonly one B site, the double perovskite (A 2BB”O 6) allows\n\u0003caldersa@ornl.gov\nyzurloye@mailbox.sc.edufor two crystallographically unique sites on which up to three\nmagnetic ions may reside. Most studies of double perovskites\nlimit the number of magnetic cations to one or two, often\non the B and B’ site for ease of study, although exceptions\ndo exist [8, 13]. This allows for the possibility of studying\nthe interaction between superexchange (B-O-B’) and super-\nsuperexchange interactions (B-O-B’-O-B), such as that stud-\nied in Ca 2MOsO 6(M = Co, Ni) [14, 17]. There it was\ndemonstrated that strong antiferromagnetic coupling between\nOs and Co/Ni stabilize the ferrimagnetic ground state, in-\ndicating strong superexchange interactions, and weak super-\nsuperexchange interactions. Interestingly, the chemical sub-\nstitution of nonmagnetic Ca in these materials for Sr re-\nsults in Sr 2CoOsO 6, which has been shown to exhibit strong\nsuper-superexchange interactions (Os-O-Co-O-Os and Co-O-\nOs-O-Os) resulting in two interpenetrating antiferromagnetic\nmagnetic sublattices [15]. These sublattices have indepen-\ndent magnetic ordering temperatures (Os: T N= 108 K; Co:\nTN= 70 K) and distinct magnetic propagation vectors (Os:\nk=(1\n2,1\n2,0); Co: k=(1\n2,0,1\n2) ) [15], in direct contrast to the\nnearly isostructural and isovalent Ca 2CoOsO 6analog. The\nsubtle structural change associated with substitution of Ca\nfor Sr resulted in a drastic change in superexchange strength,\nmagnetic ordering temperature, and the nature of the long\nrange magnetic order (ferrimagnetic Ca 2CoOsO 6and anti-\nferromagnetic Sr 2CoOsO 6), exemplifying how sensitive these\nhybrid perovskites are to chemical changes [42].\nThe Kanamori-Goodenough rules [43] have provided a set\nof semi-empirical guidelines to understand the complex re-\nlationship between superexchange interactions and magnetic\norder in condensed matter systems. These rules provide a\nmethod for determining the sign of superexchange interac-\ntions, predicting antiferromagnetic order for linear M-X-M\ninteractions (where X is a bridging anionic unit such as a\nchalcogenide or halide) and ferromagnetic order for 90\u000eM-\nX-M interactions. Although these rules have been shownarXiv:2106.02025v1 [cond-mat.str-el] 3 Jun 20212\nto successfully predict superexchange interactions for per-\novskites, poor energetic overlap between magnetic cations,\nsuch as those in mixed 3 d-5doxides, can lead to violations\nof these rules. One such example is the hybrid 3 d-5ddouble\nperovskite Sr 2FeOsO 6, [44, 45] in which the bent Os-O-Fe\nsuperexchange interaction in the ab-plane exhibited antiferro-\nmagnetic order, and these bonds exhibited ferromagnetism in\nthec-axis despite the 180\u000eOs-O-Fe bond angle. Exceptions\nsuch as these continue to motivate the detailed study of hy-\nbrid 3d-5dcomplex oxides, and serve as a motivating factor\nfor this work, which extends to the rarely studied 3 d-5d-4f\ncompositions.\nHere we report a comprehensive investigation of\nLn2NiIrO 6(Ln= La, Pr, Nd). We begin with measure-\nments of all compounds with neutron powder diffraction and\nresonant inelastic x-ray scattering (RIXS) to determine the\nmagnetic structure and explore how SOC affects the t 2gman-\nifold of the Ir ion. The remainder of the manuscript focuses\non Pr 2NiIrO 6using thermodynamic, transport property and\ndensity functional theory (DFT) calculations. The results\nallow insights into the role superexchange plays in these\nscarcely studied hybrid 3 d-5d-4fcompositions with variable\nA site cations. These materials were previously reported by\nsome of the authors of this manuscript [8] and this study seeks\nto elucidate the magnetic structure of all three compositions.\nSeveral magnetic ordered phases are observed as the different\nmagnetic ions order. The presence of independent magnetic\nsublattices in Pr 2NiIrO 6is explored in detail. This approach\nallows us to go beyond the Kanamori-Goodenough rules to\ndetermine the varied magnetic interactions and ground states\nin these related materials as the rare earth ion is altered and\nthe temperature is tuned.\nII. EXPERIMENTAL DETAILS\nA. Sample synthesis\nLn2O3(Alfa Aesar 99.99 %) and Pr 6O11(Alfa Aesar\n99.9%) were all heated in air at 1000\u000eC in a tube furnace\novernight to remove any possible hydroxide or carbonate im-\npurities. Pr 6O11(Alfa Aesar, 99.99 %) was reduced to Pr 2O3\nunder 5 %hydrogen at 1000\u000eC in a tube furnace overnight.\nNiO (Sigma Aldrich, 99.999 %) and Ir powder (Engelhard,\n99.9995 %) were used as received. Polycrystalline samples of\nLn2NiIrO 6were prepared by intimately grinding Ln2O3, Ni,\nand Ir metal in stoichiometric amounts and heating the resul-\ntant powder in air in an alumina crucible with a loose fitting\nlid. The samples were heated to 800\u000eC for 72 hours, 900\u000eC\nfor 72 hours, and then 975\u000eC for 168 hours with intermedi-\nate grindings in a programmable furnace. For Pr 2NiIrO 6, an\nadditional heating at 1025\u000eC for 96 hours with intermediate\ngrindings was necessary.B. Physical property measurements\nTemperature dependent heat capacity was measured using\na Quantum Design physical property measurement system\n(PPMS) on polycrystalline powder of Pr 2NiIrO 6that were\npressed into a pellet and sintered at 400\u000eC for 72 hours. The\nelectrical resistance of pressed and sintered pellets cut into a\nrectangular shape was recorded as a function of temperature\nby the four-probe method. Silver paint electrodes using plat-\ninum wires were used as contact points. The temperature was\ncontrolled from 380 K down to 1.8 K using a Quantum Design\nPPMS.\nC. Neutron powder diffraction\nNeutron diffraction measurements were performed on 5\ngram samples of Ln2NiIrO 6at Oak Ridge National Labora-\ntory on the HB-2A Powder diffraction instrument at the High\nFlux Isotope Reactor (HFIR) [46, 47]. Measurements were\nperformed with the samples loaded into 1 mm Al annular cans\nto reduce neutron absorption from the Ir ion. The outer diam-\neter of the sample cans were 15 mm. A wavelength of 2.41 ˚A\nwas selected with a vertically focusing germanium monochro-\nmator on the Ge(113) reflection. Data were collected over a\n2\u0012angular range of 5\u000e- 130\u000ein steps of 0.05\u000e. The detec-\ntor efficiency was normalized with a vanadium measurement.\nThe La 2NiIrO 6and Nd 2NiIrO 6samples were cooled in a top-\nloading closed cycle refrigerator (CCR) to reach 4 K and a\n4He cryostat was used for Pr 2NiIrO 6to get to the lower tem-\nperature of 1.5 K. FullProf was utilized for the Rietveld re-\nfinement and determination of the propagation vectors (k vec-\ntors) [48]. The magnetic space groups were determined using\nthe Bilbao Crystallographic Server [49, 50]. Representational\nanalysis was also used during the magnetic structure determi-\nnation process with SARAh [51]. See Supplemental Material\nat [52] for the mcif files of the determined magnetic structures.\nD. Resonant Inelastic X-ray Scattering\nRIXS was carried out on the MERIX spectrometer, sector-\n27 at the Advanced Photon Source (APS) [53]. The inci-\ndent energy was tuned to the Ir L 3-edge (11.215 keV) reso-\nnant edge to enhance the Ir scattering. The inelastic energy\nwas measured with the use of a Si(844) analyzer. The energy\nresolution was determined to be 35 meV at full width half\nmaximum (FWHM), based on fitting the quasi-elastic line to\na charge peak. The scattering plane and incident photon polar-\nization were both horizontal, i.e. \u0019incident polarization, with\nthe incident beam focused to a size of 40 \u000225\u0016m2(H\u0002V)\nat the sample position. To minimize elastic scattering mea-\nsurements were performed with 2 \u0012at 90\u000ein horizontal ge-\nometry. All measurements were performed on powder sam-\nples mounted onto an Al block sealed with Kapton paper with\nspace for all three samples in a custom mount. The tempera-\nture was controlled with a CCR and measurements taken at 53\nK, 30 K and 150 K to cover the different regions of magnetic\nordering in the materials.\nE. First Principles Calculations\nFirst principles calculations were performed using the all-\nelectron linearized augmented planewave (LAPW) DFT code\nWIEN2K [54], within the generalized gradient approxima-\ntion of Perdew, Burke and Ernzerhof [55]. LAPW sphere\nradii of 1.62, 1.98, 1.98 and 2.35 Bohr were used respectively\nfor Oxygen, Nickel, Iridium and Praseodymium, respectively,\nwith an RK max value of 8.0 employed. Here RK max is the\nproduct of the smallest sphere radius (in this case Oxygen)\nand the largest plane-wave expansion wavevector. All calcu-\nlations used an optimized structure, with the lattice constants\nand space group taken from the experimental measurement\nand all internal coordinates not dictated by symmetry relaxed\nwithin a ferromagnetic Pr-Ni configuration (note that in this\ncase Ir carries a small negative moment). Sufficient numbers\nof k-points (generally between 200 and 600 in the full Bril-\nlouin zone) to describe the magnetic order were used for all\ncalculations. For the detailed magnetic calculations (not the\noptimization), a U value of 5 eV was applied to the Pr 4f or-\nbitals. This value corresponds with that chosen in recent work\non Pr-containing transition metal, perovskite oxides [56]. We\nalso include straight GGA results as this provides insight re-\ngarding the effect of the Hubbard U on the exchange energet-\nics.\nIII. RESULTS AND DISCUSSION\nA. Magnetic Structure Determination\nNeutron powder diffraction measurements were performed\non allLn2NiIrO 6materials to determine the magnetic struc-\nture in different temperature regimes. The crystal structures\nwere previously determined with single crystal x-ray diffrac-\ntion to beP21=n(#14) [8].\n1. La 2NiIrO 6\nx y z Site\nLa0.008(1) 0.546(5) 0.753(2) 4e\nIr 0 0 0 2a\nNi 0 0 0.5 2b\nO10.084(1) 0.019(8) 0.260(2) 4e\nO20.211(4) 0.280(2) -0.047(1) 4e\nO30.207(4) 0.305(2) 0.540(2) 4e\nTABLE I. Crystal structure of La 2NiIrO 6at 100 K from neutron re-\nfinement in the P21=nspace group with a=5.566(2) ˚A,b=5.630(2) ˚A,\nc=7.888(3) ˚A,\f=90.09(2)\u000e.\nWe first begin with the magnetic structure determination of\nLa2NiIrO 6, that is expected to only contain 3 d(Ni2+) and\n(a)\n(d)(c)\n(b)100 K\n4 K\n0.79 Å-1\nac\nacFIG. 1. Magnetic structure of La 2NiIrO 6. (a) Refinement of neu-\ntron powder diffraction data at 100 K to the P21=ncrystal structure\n(upper tick marks). Lower Tick marks correspond to the Al scatter-\ning from the sample holder. (b) Intensity of the reflection at 0.79\n˚A\u00001as a function of temperature. (c) Magnetic structure model fit\nto the intensity obtained by subtracting the 100 K neutron diffraction\ndata from the 4 K measurement. (d) Polyhedral representation of\nthe magnetic and nuclear structure of La 2NiIrO 6and magnetic-atom\nonly representation with La (blue), Ni (green), Ir (grey) and O (red)\natoms shown. The non-magnetic unit cell is outlined with the blue\ndashed line. The magnetic unit cell is doubled along the aandb-axis.\n5d(Ir4+) magnetic ion ordering and a non-magnetic 4fion\n(La3+). La 2NiIrO 6was reported to undergo an antiferromag-\nnetic transition around 75 K, with indications of further mag-\nnetic anomalies within this phase [8]. A powder sample of\nLa2NiIrO 6was measured at four different temperatures: 4 K,\n40 K, 65 K, and 100 K. This allowed the anomalies in the\nreported SQUID measurements [8] to be explored and disen-\ntangle the evolution of any magnetic ordering. The 100 K\nmeasurement is above the highest observed magnetic transi-\ntion and was used to obtain a structural model in the para-\nmagnetic phase based on the previously reported structure of\nP21=n, shown in Fig. 1(a) and Table I. No impurities were\ndetected in the neutron data. Upon cooling below 80 K, the\ntemperature regime in which magnetic order is expected for\nLa2NiIrO 6, the presence of additional intensity was observed.\nThe intensity change at a forbidden nuclear position was fol-\nlowed in Fig. 1(b) to track the onset of magnetic ordering.\nThis indicated a magnetic ordering below T N=80 K, consis-\ntent with the SQUID data [8]. The same magnetic reflections\nwere present at 65 K, 40 K, and 4 K with no change indica-\ntive of further magnetic transitions within the resolution of the\npresent measurements. Further measurements on single crys-4\ntals or with higher resolution will be of interest to probe these\nsubtle changes observed in Ref.[8].\nA propagation vector of k = (1\n2,1\n2, 0) was determined\nfrom the positions of the magnetic reflections. Using the Bil-\nbao Crystallographic Server and non-magnetic space group\nP21=n(non-standard setting) with the determined k-vector\ngives thePS-1 (#2:7) magnetic space group as the only max-\nimally allowed structure with non-zero moments. The direc-\ntion of the moments is unconstrained in this model. Given the\nnumber of variables, powder averaging inherent in the data\nand small contribution from the Ir ion we attempted to limit\nthe spin directions to uncover the dominant component. Con-\nfining the spins to the a-axis produced the most reasonable\nagreement of any of the trial a,b,cdirections to the data with an\nRmagvalue of 22.1. Allowing the spins to have a component\nalong thec-axis further increased the agreement to the data\nwith an R magvalue of 6.25. This model is shown in Fig. 1(d).\nWhen the moments were allowed to freely refine along all di-\nrections the b-axis produced a value with a large error within\nzero, distinct from the aandcaxis. As such we present a mag-\nnetic model for La 2NiIrO 6with onlya-cspin components,\nhowever we cannot rule out a b-axis component. The refined\nmoment values in our model were 1.53(5) \u0016B=Ni2+with com-\nponents ( ma,mb,mc)= (1.0,0,1.1) and 0.17(3) \u0016B=Ir4+with\ncomponents ( ma,mb,mc)= (0.12,0,0.13). We note that the low\nmoment of Ir4+is beyond the typical limit for this measure-\nment and therefore is presented as the best fit model. The\nerrors from the Rietveld refinement are likely an underestima-\ntion and we cannot rule out the Ir4+having a zero moment\nor the Ir and Ni sublattices ordering at different temperatures.\nFurther measurements sensitive to the Ir ion, such as resonant\nx-ray scattering, would be of interest, as would measurements\non crystals to determine the spin direction of all the moments.\n2. Nd 2NiIrO 6\nx y z Site\nNd0.015(2) 0.565(1) 0.565(1) 4e\nIr 0 0 0 2a\nNi 0 0 0.5 2b\nO10.099(2) 0.029(2) 0.267(3) 4e\nO20.184(4) 0.283(5) -0.057(3) 4e\nO30.202(4) 0.312(5) 0.548(4) 4e\nTABLE II. Crystal structure of Nd 2NiIrO 6at 150 K from neutron re-\nfinement in the P21=nspace group with a=5.429(7) ˚A,b=5.682(7) ˚A,\nc=7.753(9) ˚A,\f=90.17(2)\u000e.\nWe now turn to the compositions with magnetic 3 d-5d-4f\nions. Nd 2NiIrO 6was reported to have a ferromagnetic-like\ntransition around 125 K with a further anomaly at 6 K con-\nsistent with antiferromagnetic interactions, based on SQUID\nmeasurements [8]. To follow the magnetic structure we there-\nfore collected neutron diffraction measurements at 4, 40, and\n150 K. The high temperature measurement, shown in Fig. 2(a)\nand Table II, was used to confirm purity and obtain a non-\nmagnetic structural model in the paramagnetic regime. This\n(a)\n(c) (b)\n(d)\n(f) (e)\n(g)\n150 K\n40 K\n4 K1.41 Å-1\n1.15 Å-1\nFIG. 2. Magnetic structure of Nd 2NiIrO 6. (a) Refinement of the 150\nK neutron diffraction pattern to the crystal structure (upper reflec-\ntions). Lower reflections correspond to Al sample holder scattering.\n(b) Magnetic structure model obtained by subtracting the 150 K neu-\ntron pattern from the 40 K data. (c) Intensity of the reflection at\n1.41 ˚A\u00001as a function of temperature. (d) 40 K magnetic struc-\nture model with Nd (blue), Ni (green) and Ir (grey) atoms shown.\n(e) Magnetic structure model obtained by subtracting the 150 K neu-\ntron pattern from the 4 K data. (f) Intensity of the reflection at 1.15\n˚A\u00001as a function of temperature. (g) 4 K magnetic structure for\nNi/Ir ions (left), Nd ions (middle) and all magnetic ions (right). One\nmagnetic unit cell is shown with the dashed lines correspond to the\nnon-magnetic unit cell.\nwas refined with the P21=nspace group. The 40 K mea-\nsurement revealed additional scattering, which is shown in\nFig. 2(b) where the 150 K data has been subtracted from the\n40 K data. The intensity of the scattering at 1.41 ˚A\u00001was\nfollowed as a function of temperature, see Fig. 2(c). The in-\ncrease in intensity is consistent with the predicted magnetic\nordering at 125 K from bulk data [8]. The additional scat-5\ntering could be indexed to a propagation vector of k = (0,\n0, 0). Given this k vector and P21=nsymmetry of the nu-\nclear structure gives four maximally allowed magnetic space\ngroups:P20\n1=c0(#14:79),P21=c0(#14:78),P20\n1=c(#14:77)\nandP21=c(#14:75). Magnetic space groups #14:77and\n#14:78only allow moments on the Nd ion and could be dis-\ncarded. The remaining two magnetic space groups do not\nconstrain the moments to any fixed axis. The best fit to the\ndata was obtained with a ferrimagnetic arrangement of Ni and\nIr in thebaxis in the magnetic space group P21=c(#14:75).\nMagnetic moments of 1.71(2) \u0016B(Ni) and 0.32(7) \u0016B(Ir) were\ndetermined, corresponding to R magof 25.7. The higher agree-\nment index of this fit compared to previous refinements is due\nto the weaker intensity and reduced number of the magnetic\nreflections in this phase. Attempts to improve this R magby\nintroducing components away from the b-axis did not appre-\nciably improve the fit. A ferromagnetic model is additionally\nin agreement with the data, however the ferrimagnetic model\npresented is more consistent with bulk measurements previ-\nously reported [8]. We note again that the low moment of Ir4+\nis beyond the typical limit for this measurement and therefore\nis presented as the best fit model.\nUpon cooling to 4 K, additional magnetic reflections ap-\npeared in the diffraction pattern, shown in Fig. 2(e) for the\ndifference between the 4 K data and the 150 K data. The in-\ntensity at the most intense reflection position was followed as\na function of temperature in Fig. 2(f). This revealed the on-\nset of magnetic ordering below 7 K, consistent with reported\nSQUID results [8]. The magnetic reflections observed at 4\nK were indexed to a k=(1\n2,1\n2, 0) propagation vector within\nthe non-magnetic space group P21=n. Only one maximal\nmagnetic space group allows moments for Ni/Ir, as well as\nNd:PS-1. A magnetic model with the spins still confined\nto theb-axis but in an antiferromagnetic arrangement for all\nthe ions yields the best fit to the data. Magnetic moments\nof 2.20(4)\u0016B=Nd3+, 1.27(4)\u0016B=Ni2+and 0.32(5)\u0016B=Ir4+\nwere determined.\nThe magnetic behavior of Nd 2NiIrO 6upon cooling is there-\nfore characterized as first undergoing ferrimagnetic ordering\nof the Ni/Ir ions with the magnetic order keeping the unit cell\nsize unaltered. Then only at the low temperature of 7 K does\nthe Nd ion order along with a change in the ordering of the\nNi/Ir magnetic order to antiferromagnetic to create a magnetic\nunit cell doubled in size along the aandbaxis.\n3. Pr 2NiIrO 6\nThe composition Pr 2NiIrO 6was measured at 1.5, 20,\n75, and 125 K temperatures to follow anomalies observed\nin previous SQUID measurements [8]. These indicated\nferromagnetic-like ordering at 105 K with a further transi-\ntion at 5 K. The high temperature 125 K neutron diffrac-\ntion measurement shown in Fig. 3(a) was used to confirm\nsample purity and the P21=nstructural model in the para-\nmagnetic regime, see Table III. Upon cooling below 110\nK, additional Bragg reflections appeared. This is shown in\nFig. 3(b) by following the intensity at 1.41 ˚A\u00001. Figure 3(c)x y z Site\nPr0.012(3) 0.560(1) 0.754(5) 4e\nIr 0 0 0 2a\nNi 0 0 0.5 2b\nO10.094(1) 0.026(1) 0.263(2) 4e\nO20.186(3) 0.296(4) -0.057(1) 4e\nO30.198(3) 0.290(4) 0.532(2) 4e\nTABLE III. Crystal structure of Pr 2NiIrO 6at 150 K from neutron re-\nfinement in the P21=nspace group with a=5.473(2) ˚A,b=5.661(3) ˚A,\nc=7.790(3) ˚A,\f=90.03(2)\u000e.\nshows all the observed magnetic reflections by subtracting\nthe 125 K data from the 20 K data. There was no differ-\nence between the 20 K and 75 K measurements apart from\nincreased intensity of the new magnetic reflections at the\nlower temperature measurement. Both temperatures have a\nk = (0, 0, 0) propagation vector. The scattering is simi-\nlar to that observed for the Nd 2NiIrO 620 K measurement,\nhowever here in the Pr 2NiIrO 6case the signal to noise is\nimproved and additional weaker reflections were observed.\nFollowing an identical analysis described above the mag-\nnetic space group of P21=c(#14:75)used for Nd 2NiIrO 6\nwas found to best fit the Pr 2NiIrO 6data at 20 K, shown\nin Fig. 3(c). The magnetic spins are primarily along the b-\naxis, however to model all the magnetic reflections a com-\nponent along the a-axis needed to be added. This gives the\nferrimagnetic structure shown in Fig. 3(f) with the Ni ions\nordered ferromagnetically and the Ir ions ordered ferromag-\nnetically. Magnetic moments of 1.61(4) \u0016B=Ni2+with com-\nponents ( ma,mb,mc)= (0.6,1.5,0) and 0.34(8) \u0016B=Ir4+with\ncomponents ( ma,mb,mc)= (0.1,0.3,0) are found. Again the\nsmall moment size for Ir is presented as a best fit model and\nwe cannot rule out a zero ordered moment.\nCooling further from 20 K to 1.5 K additional magnetic\nscattering is observed as new intensity at certain reflections,\nwhile other positions such as at 1.38 ˚A\u00001and 1.41 ˚A\u00001, re-\nmain unchanged. The intensity change at 1.96 ˚A\u00001is shown\nin Fig. 3(d). Figure 3(e) shows all the observed magnetic re-\nflections at 1.5 K by subtracting the 125 K data from the 1.5\nK data. The propagation vector is also unchanged from the\nhigh temperature phase, k = (0, 0, 0). This behavior is con-\nsistent with the ordering of the Pr ion while the Ni/Ir ions\nmagnetic order remains unchanged. A clear contrast is ob-\nserved with the Nd 2NiIrO 6composition that showed a change\nin the Ni/Ir ordering at the low temperature magnetic phase\ntransition. To model the 1.5 K data for Pr 2NiIrO 6we keep\nthe same magnetic space group of P21=c(#14:75)and in-\nclude a moment on the Pr ion. The data refined to having the\nPr ion in the ab-plane in a ferromagnetic arrangement simi-\nlar to the Ni/Ir ions. The refinement to the 1.5 K data with\nthe 125 K data subtracted is shown in Fig. 3(e) and the cor-\nresponding spin model in Fig. 3(g). The best fit model corre-\nsponds to magnetic moments of 1.63(4) \u0016B=Ni2+with com-\nponents ( ma,mb,mc)= (0.6,1.5,0) and 0.39(7) \u0016B=Ir4+with\ncomponents ( ma,mb,mc)= (0.1,0.3,0) and 1.58(3) \u0016B=Pr3+\nwith components ( ma,mb,mc)= (1.0,1.2,0).\nThe onset of Pr ordering therefore contributes to the over-6\n(a)\n(b) (c)\n(d) (e)\n(f) (g)125 K\n20 K\n1.5 K\n1.96 Å-11.41 Å-1\nc\nbac\nb\na\nFIG. 3. Magnetic structure of Pr 2NiIrO 6. (a) Refinement of the 125\nK neutron diffraction pattern to the crystal structure (upper reflec-\ntions). Lower reflections correspond to Al sample holder scattering.\n(b) Intensity of the reflection at 1.41 ˚A\u00001as a function of tempera-\nture. (c) Magnetic structure model obtained by subtracting the 125 K\nneutron pattern from the 20 K data. Scattering around 1.6 ˚A\u00001due to\nthe strong nuclear contribution. (d) Intensity of the reflection at 1.96\n˚A\u00001as a function of temperature. (e) Magnetic structure model ob-\ntained by subtracting the 125 K neutron pattern from the 1.5 K data.\nScattering around 1.6 ˚A\u00001due to the strong nuclear contribution. (f)\nMagnetic-atom only representation of the magnetic structure at 20\nK of Pr 2NiIrO 6showing, Ni (green) and Ir (black) ions. The unit\ncell is outlined with the blue dashed line. (g) Polyhedral representa-\ntion of the magnetic (1.5 K) and nuclear structure of Pr 2NiIrO 6and\nmagnetic-atom only representation of the 1.5 K magnetic structure\nof Pr 2NiIrO 6showing Pr (blue), Ni (green) and Ir (black) magnetic\nions.\nall ferrimagnetic ordering within Pr 2NiIrO 6, with the Pr\n1.58(3)\u0016Bordering ferromagnetically along the b axis in a\nzig-zag fashion due to a spin angle of 48.1(1)\u000eoff the b axis.\nThe best fit model indicates the Pr and Ni ordering with the\nspins in the same direction along the b-axis and the Ir in the\nopposite direction. Further measurements on single crystals\nand with elemental specific analysis available with resonant x-\nray scattering will be of interest to test this model and contrast\nit against a fully ferromagnetic ordering of all three magnetic\nFIG. 4. Resonant inelastic x-ray scattering measurements of powder\nLn2NiIrO 6(Ln= La, Pr, Nd) at 5 K on the MERIX spectrometer.\nThe incident energy was 11.215 keV corresponding to the Ir L 3-edge.\nThe data have been offset by a constant factor for clarity.\nions.\nB. RIXS measurements of SOC-Induced t 2gmanifold splitting\nTo gain insight into the electronic ground state of the Ir4+\n(5d5) ion inLn2NiIrO 6RIXS measurements were performed.\nThe energy was tuned to 11.215 keV corresponding to the Ir\nL3-edge which allows for an isolation of the Ir scattering. The\n5 K data is shown in Fig. 4. No change was observed in mea-\nsurements collected at higher temperatures. Each compounds\nspectra consisted of two main features around 0.6 eV and 3.5\neV . The spectra are consistent with similar octahedrally co-\nordinated Ir4+ion and provides all the signatures of a SOC\nsplit J e\u000b=1\n2state [57, 58]. The broad, higher energy (3.5 eV)\npeak corresponds to d-dexcitations from transitions between\nthe t 2gand e gorbitals, which are split because of the crys-\ntal field. The sharper, lower energy scattering consists of two\nseparate peaks, within the 35 meV resolution of the instru-\nment. This scattering can be assigned to d-dexcitations from\nintraband t 2gtransitions due to the splitting of the t 2gmanifold\ninto a J e\u000b=1\n2and J e\u000b=3\n2state, characteristic of many com-\nplex iridates [40]. By fitting these reflections to simple Gaus-\nsian peaks we extract peak energies as: La 2NiIrO 6= 0.60(2)\neV and 0.71(2) eV; Pr 2NiIrO 6= 0.58(1) eV and 0.65(1) eV;\nNd2NiIrO 6= 0.59(2) and 0.71(3) eV . The presence of two re-\nsolvable peaks is consistent with a small departure from an\nideal J e\u000b=1\n2state due to the distortions inherent in the crystal\nstructure that is observed in all reported Ir materials in the lit-\nerature [57, 58]. The singular unpaired electron present in the\nJe\u000b=1\n2level for 5d5Ir4+is commonly observed as possess-\ning a significantly reduced magnetic moment, such as that ob-\nserved inLn2NiIrO 6reported above of \u00180:3\u0016B, further sup-\nporting a J e\u000b=1\n2state.7\n(a)\n(b)150 100 50 0050100150145\n140\n135120 130 140\n246810\nT (K)Cp(JMol-1K-1)\n0246CpT-1(JMol-1K-2)\n150 100 50 0\nT (K)1.6\n1.4\n1.2\n1.0\n0.8\n0.6\n0.4\n0.2110 120 1301.01.11.2\n234561.4\n1.0\n0.6\n0.2\nFIG. 5. (a) Bulk heat capacity (C p) for Pr 2NiIrO 6in zero field. The\nonset of Ni/Ir ordering is show more clearly in the upper left inset,\nand the onset of Pr ordering is shown in the bottom right inset. (b)\nBulk heat capacity divided by temperature (T) plotted against tem-\nperature for Pr 2NiIrO 6.\nC. Heat Capacity, electrical resistivity and DFT investigations\nof Pr 2NiIrO 6\n1. Heat Capacity of Pr 2NiIrO 6\nHeat capacity measurements were undertaken on a pressed\nand sintered pellet of Pr 2NiIrO 6, shown in Fig. 5(a), to fur-\nther investigate the long-range ordering and probe for inde-\npendent Pr and Ni/Ir magnetic sublattices. Two clear transi-\ntions are observed at 123 K and 3.7 K, confirming the nature\nof long-range ordering temperatures. The broadness of the\nhigh temperature transition may be due to thermal fluctuation,\na product of measuring at high temperature, or may be due\nto poor sintering of this sample. In addition we cannot rule\nout this as indicating low dimensional correlations for one or\nmore of the ions. This transition was further resolved by plot-\nting heat capacity (C p) divided by temperature (C p/T) against\ntemperature, shown in Fig. 5(b). Although broad features are\nstill present, the 123 K transition is clear. As this transition is\nconsistent with both neutron and susceptibility measurements\n(a)\n(b)80 100 120 140\nT (K)1x105Resistivity ( Ohms cm )\n2x1053x1054x1055x1056x1057x105\n02x1054x1056x105\n50100 150 200 250 300 350\n0.002 0.006 0.01 0.014\nT-1(K-1)46810121416Ln ρ(Ohms)FIG. 6. (a) Temperature dependence of the electrical resistivity for\nPr2NiIrO 6. The inset depicts that resistivity was not measured below\n82.5 K due to instrumental limits of Ohms-cm capability. (b) Inverse\ntemperature dependence of the natural logarithm of resistance for\nPr2NiIrO 6. Linearity for the case of T-n, such that n = 1, indicates\nthermally activated conduction.\nindicating the onset of ferromagnetic-like order, its magnetic\norigin corresponds to the onset of Ni/Ir magnetic ordering. In-\nterestingly, the transition at 3.7 K was found to be sharp and\nlambda-like, and is consistent with the small transition ob-\nserved in zero-field cooled measurements shown in Fig. 5(b)\nand the 1.5 K powder neutron diffraction data, suggesting the\nonset of Pr magnetic ordering. A small change in slope of heat\ncapacity data can be observed below 2.7 K, but the nature of\nthis transition is unclear. Based on the neutron and suscepti-\nbility data measurements, it does not correspond to any long\nrange nuclear or magnetic order, suggesting possible crystal\nfield effects between the three present magnetic ions in this\nstructure.\n2. Electrical Resistivity of Pr 2NiIrO 6\nTemperature dependent electrical resistivity measurements\nfor Pr 2NiIrO 6are shown in Fig. 6(a). The sharp decrease in\nresistance as a function of increasing temperature indicates8\nthat the material is not metallic. Considering low measured\nresistance of 37 Ohms-cm at 380 K, especially for oxide mate-\nrials [59], semiconducting behavior is possible. This was fur-\nther investigated by assessing the conduction mechanism via\nplotting the natural logarithm of resistance against T-n, such\nthat the value of n indicates the dimensionality and type of\ntransport mechanism. For values of n = 1, linearity indicates a\nsimple thermally activated conduction pathway, whereas val-\nues for n greater than 1 indicate a Mott variable range hopping\nmechanism of variable dimensionality. Fig. 6(b) depicts near\nperfect linearity is exhibited for n = 1, indicating the conduc-\ntion pathway is thermally mediated.\n3. First Principles Calculations for Pr 2NiIrO 6\nIn an attempt to better understand the complex magnetic be-\nhavior for Pr 2NiIrO 6, we have performed first principles cal-\nculations of the magnetic order and energetics. Given both\nthe complex monoclinic physical structure as well as the non-\ncollinear canted magnetic structure, with effectively 3 differ-\nent magnetic ions (the dominant Ni, less dominant Pr, and in-\nduced moment Ir), we make certain simplifications in order to\nrender the problem computationally and analytically tractable.\nFirst, we consider only collinear states. While the actual ob-\nserved ground state in Pr 2NiIrO 6is not collinear, a detailed\nexamination of Fig. 3(f) (Ni/Ir ordering) and Fig. 3(g) (all\nions order) shows that deviations of the respective magnetic\nions from collinearity are less than 30\u000eoff thebaxis for both\nNi and Ir, but is a bit more significant for Pr.\nGiven the monoclinic symmetry, there is a large mani-\nfold of potential exchange interactions, with several potential\nnearly-“nearest-neighbor” interactions, with slightly variable\ndistances between Ni and Ni, Ni and Pr, Pr and Pr, and these\natoms with Ir. To simplify matters we consider only Ni-Ni, Ni-\nPr and Pr-Pr effective exchange interactions and consider the\nseveral nearly degenerate distances in each of these categories\ninto one interaction for each category. We note in passing that\nit is not surprising that this compound exhibits a complex non-\ncollinear magnetic structure in view of the complex physical\nstructure and the three effectively magnetic ions, along with\nthe disparate spin-orbit energy scales of Ni ( \u001850 meV), Pr\n(\u00180.5 eV), and Ir ( \u00181 eV). Note that the Ir atom is explicitly\nincluded in the DFT calculations themselves but for simplic-\nity is not included in the extraction of exchange constants as\nthis would substantially complicate the analysis.\nFor the purposes of determining the ground state and asso-\nciated excited state energetics, five distinct magnetic arrange-\nments were considered. We show in Fig. 7 {\u0000a configu-\nration with Ni-Ni and Pr-Pr near-neighbor pairs antialigned\n(NiAFPrAF). We considered four additional arrangements,\nincluding a ferromagnetic state (FM) and three more com-\nplex arrangements. For these purposes spin-orbit coupling\nwas omitted, though for a detailed examination of the FM\nground state below we include it. Here a crystallographic\nunit cell contains 2 formula units. The arrangements con-\nsidered, in addition to the ferromagnetic case, were as fol-\nlows: a state with Ni and Pr antiparallel to each other –a fer-\nFIG. 7. A depiction of the primary magnetic moment-bearing atoms\ndescribed within the first principles calculations. Praseodymium\natoms are depicted as blue spheres, Nickel atoms as green spheres,\nand Iridium atoms as dark grey spheres (no label), as indicated.\nFor clarity, all spheres are shown as the same size, regardless of\natomic size. The “Ni AFPrAF” state is shown above. For the\n“FM” state, all Ni and Pr atoms are ferromagnetically coupled.\nFor “Ni PrFI”, the Ni and Pr atoms are antiferromagnetically cou-\npled. For “Ni AFPrFM”, the Ni1 and Ni2 atoms are antiferro-\nmagnetically coupled while all Pr atoms are ferromagnetically cou-\npled to Ni1. For “Ni FMPrAF”, Ni1 and Ni2 are ferromagnet-\nically coupled while Pr1 is ferromagnetically coupled to Ni1 and\nNi2 while Pr2 is antiferromagnetically coupled to Ni1 and Ni2. For\n“NiAFPrAF”, Ni1 and Ni2 are antiferromagnetically coupled and\nPr1 and Pr2 are also antiferromagnetically coupled. Note that the\napparent fourfold Pr2 falls on the a-face zone boundary (unlike Pr1,\nwhich is within the cell), so that there are only two Pr2 per unit cell.\nSimilarly, the four Ni1 atoms fall on the zone edge, so that there is\nonly one Ni1 per unit cell, and the two Ni2 atoms fall on the c-face\nzone boundary. The vertical moment orientation is for clarity of pre-\nsentation; moment orientation was not studied in these calculations.\nrimagnetic state (Ni PrFI;l total moment 6 \u0016B=u:c), a state\nwith the two unit cell Ni antiparallel, with the Pr themselves\naligned (Ni AFPrFM, total moment 7.82 \u0016B=u:c); a state\nwith the 2 Ni ferromagnetically coupled, but 2 of the 4 Pr\nanti-aligned to the other 2; (Ni FMPrAF, total moment 2\n\u0016B=u:c) and a state with the 2 Ni antiferromagnetically cou-\npled, and 2 of the 4 Pr antiferromagnetically coupled to the\nother two (Ni AFPrAF, no net moment). Note that of these\nlast 4 states, only the last is truly a zero-moment antiferro-\nmagnetic state. In general, individual spin moment magni-\ntudes within these several magnetic states are generally fairly\nrigid, with little variation ( <3%) between states; typical val-\nues, in the absence of spin-orbit coupling and thereby orbital\nmoments, are 1.96 \u0016B=Prand 1.29\u0016B=Ni. Details are given\nin Table IV. We will see below that despite the smaller mo-\nment and fewer atoms per cell, it is the Ni atoms that are ul-\ntimately dictating most of the magnetic character, due to the\ngenerally much larger spatial extent of the Ni 3 dwavefunc-\ntions, relative to the Pr 4 fwavefunctions, which are much9\nEnergy Energy Total\nState relative to FM relative to FM Spin Moment\nGGA GGA+ U (\u0016B/u.c.,\n(per u.c.) GGA+U)\nFM 0.0 0.0 10.0\nNiPrFI 49.56 41.96 6.0\nNiAFPrFM 51.41 114.86 7.82\nNiFMPrAF 103.48 37.68 2.0\nNiAFPrAF 86.38 86.40 0.0\nTABLE IV . Detailed magnetic properties of several magnetic config-\nurations of Pr 2NiIrO 6studied within density functional theory. The\nconfigurations’ relative orientation of the Ni and Pr spin magnetic\nmoments are described in the text.\nmore localized in the Pr core.\nWe see that from Table IV, the state with the 2 Ni atoms\nantiferromagnetically coupled, but the Pr ferromagnetically\ncoupled, is the highest energy state in this manifold, nearly\n115 meV/u.c. above the ferromagnetic ground state, while the\nreverse (Ni FMPrAF) is only 37.7 meV/u.c. above. This\naccords with our intuitive expectation that Ni-Ni exchange\ninteractions should be stronger than Pr-Pr exchange interac-\ntions, but what is surprising is that this is the case even though\nthe Ni-Ni nearest neighbor distances are of the order of 5.7\n˚Awhereas those for Pr are only of order 4.1 ˚A. This is re-\nflective both of the general localization of the Pr 4f electrons\nwithin the core, away from the Fermi level, and also of re-\ncent findings in Cr 1=3NbS 2[60] where exchange interactions\nmediated through electronegative elements can be much more\nlong range than would commonly be expected. Quantitatively,\nmapping the above energetics to a Heisenberg model (appro-\npriate in view of the rigidity of the moments) finds Ni-Ni, Ni-\nPr and Pr-Pr exchange interactions of -4.32, -3.18 and -0.58\nmeV , (all ferromagnetic) respectively, confirming the expec-\ntation for the dominance of the Ni magnetic interaction here.\nIn particular, the Pr-Pr exchange interaction is relatively weak\nand confirms our general expectation that the Pr 4 felectrons\nare localized in the core and do not interact strongly with other\nPr atoms, reducing the Pr ordering temperature.\nAlso evident from Table IV are substantially altered ener-\ngetics in the “straight GGA” calculations, in which no Hub-\nbard U is applied to Pr. The exchange energetics change con-\nsiderably; for example, the Ni FMPrAF state in the straight\nGGA is now 103.48 meV/u.c. above the ferromagnetic state,\nwhereas in the GGA+U it is just 37.68 meV/u.c. above the\nunit cell, and this state now falls considerably higher (62\nmeV/u.c.) in energy above the Ni AFPrFM state where it\nis some 77 meV/u.c. lower in the GGA+U. Most strikingly,\nextracting from these energetics the Ni-Ni, Ni-Pr and Pr-Pr\nexchange interactions, one now finds the Pr-Pr exchange inter-\naction predominant in magnitude at -8.65 meV , with the Ni-Pr\nexchange at -4.05 meV and the Ni-Ni much smaller at just\n-0.06 meV . Thus the straight GGA would here inaccurately\nclaim the Pr-Pr magnetic interaction to be the predominant\none, a logical consequence of the GGA’s placing the Pr 4f or-\nbitals at or near the Fermi level, where they interact strongly\nwith other atoms, instead of being properly localized in thecore of the Pr atom, as a wealth of experience with rare earth\nions dictates. Thus the application of a substantial U value\n(here chosen as 5 eV) is critical to a proper description of the\nmagnetism in this compound.\nAs mentioned previously, we now give a more complete de-\nscription of the FM ground state with spin-orbit coupling in-\ncluded for all atoms with the GGA+U approach. This changes\nboth the total spin moment significantly (it increases to 10.68\n\u0016B/u.c.) and adds significant orbital moment contributions.\nIn particular, Pr exhibits a large negative orbital moment of\n-0.795\u0016B, while Ni acquires a significant orbital moment of\n0.087\u0016B, and Ir also has a significant negative orbital mo-\nment of -0.197 \u0016B. This yields total moments for these three\natoms of 1.17 \u0016B, 1.45\u0016B, and -0.35\u0016Bwith a unit cell total\nof 7.28\u0016B, or 3.64\u0016B/f.u. The above values are comparable\nto the experimental values of 1.58(3) \u0016Bfor Pr, 1.63(4) \u0016B\nfor Ni, and 0.39(7) \u0016Bfor Ir. It is of interest that the largest\norbital moment magnitudes are for the heavy atoms (Pr and\nIr), corresponding to the generally stronger spin-orbit cou-\npling in these atoms, as well as the localized nature of the Pr\n4fstates. The addition of spin-orbit also changes the Iridium\nspin-moment from -0.30 \u0016Bto a value half this, suggesting the\nparticular relevance of spin-orbit coupling here, as evidenced\nexperimentally in the RIXS measurements discussed above.\nIt is of interest both that the Iridium atom couples antiferro-\nmagnetically to the Pr and Ni atoms and that its spin moment\nis significantly reduced by the application of spin-orbit cou-\npling. Sitting at the center of a distorted Oxygen octahedron,\nits antiferromagnetic coupling is notable in view of the ferro-\nmagnetic coupling of Ni itself, sitting near the center of a sim-\nilar Oxygen octahedron. It is also remarkable that its orbital\nmoment of -0.197 \u0016Bis significantly larger in magnitude than\nits spin moment of -0.15 \u0016B, and this may be understood in\nterms of its spin-orbit energy scale of \u00181 eV effectively out-\nstripping its exchange interaction energy scales, which may be\nexpected to be much smaller than the predominant Ni-Ni ex-\nchange interaction. Note that the Ir orbital and spin moments\nare parallel, in accordance with Hund’s rules. It is likely that\nthe Oxygen atoms intervening between the Ni and Ir atoms\ncause a predominant antiferromagnetic superexchange inter-\naction between these atoms. The Ir atom thus likely plays an\nimportant role in the overall Ni-Ni magnetic exchange inter-\naction, despite carrying a comparatively small moment itself.\nThe total calculated Ir moment of 0.35 \u0016Bis consistent both\nwith the neutron-found value of \u00180.3\u0016Band with the J= 1/2\nstate inferred from the RIXS measurements.\nTo further relate these results to the RIXS data we present\nin Figure 8 the calculated density-of-states (DOS), within the\nmodeled ferromagnetic state, with spin-orbit coupling in the\nGGA+U. For simplicity we present only the majority spin (or\nspin-up) DOS and focus on the RIXS-relevant region around\nthe Fermi level. One immediately notes the presence of sub-\nstantial, in fact predominant Ir character(blue dotted line) in\nthe region within half an eV of E F. Furthermore, we note\ntwo Ir-generated DOS peaks falling approximately -0.25 be-\nlow and 0.15 eV above E F. It is quite likely that 0.4 eV transi-\ntions between these regions correspond to the Ir intraband t 2g\ntransitions observed at \u00180.6 eV in the RIXS, with the differ-10\n-2 -1 0 1 2 3 4\nE- EF (eV)01020304050N(E) (eV-u.c.)-1total\nPr\nNi\nIr\n0.4 eVspin-up\nFIG. 8. The calculated spin-up density-of-states of Pr 2NiIrO 6in\nthe modeled ferromagnetic groundstate, within GGA+U with spin-\norbit coupling applied. Note the RIXS-relevant transition indicated\naround the Fermi level, indicative of the intraband Ir transition found\nin RIXS\nence in energies (0.4 vs. 0.6 eV) reasonably ascribed to our\nsimplified treatment of the magnetism in this system.\nIt is also possible to relate the above GGA+U energetics\nto the observed ordering points of the Ni and Pr atoms, which\nsignificantly differ in temperature. In an approximation where\nthe ordering point of a magnetic atom, in a local moment\napproximation, is estimated at 1/3 the energy difference, per\nmagnetic atom, [55, 61, 62], between configurations with that\natom ferromagnetically and antiferromagnetically coupled to\nthe remainder of the system, we use the Ni FMPrAF state\nfor the Pr atom (relative to the FM ground state) and, cor-\nrespondingly, the Ni AFPrFM for the Ni atom, accounting\nfor the different multiplicity of these atoms, and obtain es-\ntimated ordering points for the Pr and Ni atoms as 36 and\n222 K. While these are somewhat higher than the actual val-\nues (due to our wholesale neglect of fluctuations, among other\nfactors), their relative magnitudes are in accordance with the\nexperimental facts. In particular, as seen in experiment, the\nNickel atoms are driving the magnetism, despite their roughly\nequivalent local moment and substantial nearest-neighbor dis-\ntances. This mainly reflects the spatially extended nature of\nthe 3dstates associated with the Ni atom magnetism and the\ngenerally core-localized nature of the Pr 4 felectrons associ-\nated with the Pr magnetism.\nIV . CONCLUSIONS\nA series of double perovskite iridates of the formula\nLn2NiIrO 6were investigated using thermodynamic and trans-\nport properties, neutron powder diffraction, RIXS, and DFT\ncalculations to elucidate the role superexchange plays in hy-\nbrid 3d-5d-4fcompositions with variable A-site cations. The\ncomposition La 2NiIrO 6was determined to be a non-collinear\nantiferromagnet in the ac-plane with the ordering of Ni/Ir\noccurring simultaneously. For Nd 2NiIrO 6two distinct mag-\nnetic structures were determined. The first high temperaturemagnetic phase consists of ferromagnetically ordered Ni/Ir\nsublattices creating a ferrimagnetic structure primarily along\ntheb-axis. Cooling Nd 2NiIrO 6led to a magnetic structure\nchange where the Nd ion orders and the Ni/Ir ordering also\nchanges into a AFM Ni/Ir sublattices. Two independent mag-\nnetic sublattices (Pr and Ni/Ir) were found in the composition\nPr2NiIrO 6, corresponding to ab-plane ferrimagnetic order be-\ntween Ni and Ir, and a zig-zag ferromagnetic order of Pr along\ntheb-axis, resulting in an overall ferrimagnetic order. The\npresence of two independent magnetic sublattices was corrob-\norated by heat capacity measurements, demonstrating transi-\ntions at 123 K (Ni/Ir ordering) and 3.7 K (Pr ordering). Resis-\ntivity measurements indicated semiconducting behavior and\nthermally mediated conduction for Pr 2NiIrO 6. DFT results\nconfirm the independent sublattice ordering and demonstrate\nthe primacy of the Ni atom in determining the magnetic char-\nacter, despite the Ni-Ni nearest-neighbor distances of some\n5.7˚A. All compositions were measured with RIXS, confirm-\ning that spin-orbit coupling splits the t 2gmanifold of octahe-\ndral Ir4+into a J e\u000b=1\n2and J e\u000b=3\n2state. Collectively the\nresults demonstrate the dramatic changes in magnetic order-\ning that can be induced within structurally similar 3 d-5d-4f\ncompounds as the Lnion is varied and as different tempera-\nture regimes are accessed. As shown in the DFT calculations\nand experimental data the presences of distinct magnetic ions\nwith a spectrum of SOC strength and orbital overlaps leads to\nthe inducing of magnetic interactions that otherwise would not\noccur and goes beyond predictions that apply to simpler sys-\ntems less magnetic ions, such as the Kanamori-Goodenough\nrules. This motivates further investigations into hybrid mate-\nrials with multiple magnetic ions.\nACKNOWLEDGMENTS\nThis research used resources at the High Flux Isotope Re-\nactor, a DOE Office of Science User Facility operated by\nthe Oak Ridge National Laboratory. This work was partly\nsupported by the U.S. Department of Energy (DOE), Of-\nfice of Science, Office of Workforce Development for Teach-\ners and Scientists, Office of Science Graduate Student Re-\nsearch (SCGSR) program. The SCGSR program is adminis-\ntered by the Oak Ridge Institute for Science and Education\nfor the DOE under contract number DE-SC0014664. Re-\nsearch at Oak Ridge National Laboratory (ORNL) was sup-\nported by the DOE, Office of Science, Basic Energy Sci-\nences (BES), Materials Science and Engineering Division.\nSample synthesis and structural characterization performed\nat the University of South Carolina were supported by the\nNational Science Foundation under Awards DMR-1301757\nand DMR-1806279. 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Applied 9,\n034002 (2018)." }, { "title": "2403.07169v1.Magnon_bands_and_transverse_transport_in_a_proposed_two_dimensional__Cu_2F_5__ferrimagnet.pdf", "content": "arXiv:2403.07169v1 [cond-mat.mtrl-sci] 11 Mar 2024Magnon bands and transverse transport in a proposed\ntwo-dimensional Cu2F5ferrimagnet\nPedro G. de Oliveiraa∗and Antˆ onio S. T. Piresa†\naDepartment of Physics, Federal University of Minas Gerais, Belo Horizonte, MG, Brazil.\nAbstract\nThe copper fluoride Cu2F5is a proposed stable compound that can be seen as a layered mag netic lattice of\nS= 1 andS= 1/2 sites, corresponding to copper ions. Intending to cast lig ht on the transport properties of\nferrimagnetic magnons, we use the linear spin wave approach to study the magnon band structure of the 2D\nlattice in a ferrimagnetic off-plane order, as well as the tra nsverse transport of magnons in the crystal bulk.\nThat transverse (Hall-like) transport can be induced by a ma gnetic field or temperature gradient, and within\nthe linear response theory is generated by the Berry curvatu re of the eigenstates. As in most of the cases\nfor magnons, the Berry curvature here is related to Dzyalosh inskii-Moriya interactions between next-near-\nneighbors. The band structure of the system is non-degenera te and the transport coefficients are non-null. We\nalso determine the condition for two transport coefficients t o change sign in response to temperature.\n1 Introduction\nWhen a crystal has magnetic order, perturbations in this\norder propagate through the crystal in the form of spin\nwaves. From a quasiparticle point of view, these spin waves\nare called magnons , which are chargeless bosons with def-\ninite spin, momentum and energy. In the past years there\nhas been a great effort to study the creation and manipula-\ntionof magnons [1–3], as well as their interaction with othe r\nparticles or quasiparticles [4–8] including the formation of\nhybrid modes [9–16]. These efforts rely on the exciting\npossibility of using the spin degree of freedom of magnons\nto transport information in a field known as magnonics\n(magnon spintronics) [17–19]. Magnonics has a significant\nadvantage over electron-based spintronics: as magnons are\nnaturally chargeless, they do not present Joule heating and\ncan propagate large distances with low dissipation.\nWithin this context it became essential to study the\ntransport properties of magnons in different lattices and\nmagnetic orders. It is theoretically established that a mag -\nnetic field or temperature gradient can generate longitudi-\nnal and transverse (Hall-like) transport of magnons in the\nbulk of a crystal [20–24]. These transport effects have been\nintensely studied in ferromagnets (FM) [21–41] and antifer -\nromagnets (AFM) [42–68]. The theoretical study of these\nthermomagnetic properties is well developed, andnowadays\nwe can describe magnonic analogs for the quantum Hall\neffectandquantum spin Hall effect of electrons [24,48],\namong other exotic transport effects. Experimental evi-\ndence have demonstrated the existence of transverse trans-\nport in the bulk of 3D and 2D lattices [69–73].\nIt has also been established that magnonic systems canshow protected edge or surface states, which are robust\nagainst non-magnetic impurities. That comes from the\nwell-known bulk-edge correspondence for topological sys-\ntems, which indicates the existence of topological magnon\ninsulators (TMI) [19,27,33,35,74]. That term has been\nused as an analogy toelectronic topological insulators. Ju st\nlike electronic topological insulators, the TMI are charac -\nterized by topological indices like the Chern number or Z2\ninvariant [24,48,75,76]. Despite that, the bosonic nature\nof magnons excludes the existence of a Fermi level, so the\nsystem is not an insulator strictu sensu . Topological and\nHall-like effects of magnons are related to the spin-orbit\ncoupling of the crystal lattice, and both rely on the Berry\ncurvature of the system, which acts like a fictitious mag-\nnetic field and imparts helical movement to the magnons.\nAlthough widely studied in FM and AFM systems, the\nHall transport of magnons has not been well investigated in\nferrimagnetic (FiM) lattices [77,78]. With that in mind, in\nthis work we investigate the magnon Hall transport in the\n2D layers of the proposed copper fluoride complex Cu2F5,\nwhich stability was predicted recently with first-principl e\nmethods [79–81]. In this crystal, copper ions have two dif-\nferent spin states ( S= 1 ands= 1/2), forming a ferri-\nmagnetic system that we investigate using the spin wave\napproach.\nThis paperisorganized asfollows. Wepresentthecrystal\nstructure and the Hamiltonian which models the 2D mag-\nnetic lattice in Section 2. In Section 3 we discuss the Berry\ncurvature of magnon bands and its implications to the\ntransverse transport of magnons. In Section 4 we present\nthe results of the systems’s band structure and transverse\n1transport coefficients, and in Section 5 we make our final\nremarks.\n2 Model\nThe proposed Cu2F5crystal is composed of CuF6distorted\noctahedra and CuF4plaquettes as shown in Figure 1 [79,\n80]. The inequivalent Cuions form a magnetic crystal. We\ncallCu1 theS= 1 ions in the center of the octahedra,\nandCu2 thes= 1/2 ions in the center of the plaquette.\nThe most energetically favorable configuration is a G-type\nferrimagnet, and DFT+U calculations show that the 3D\ncrystal can beseen as a layeredstructure withthe interlaye r\nexchange parameter five times smaller than the intralayer\nones [80]. That inspires us to study the 2D layers from a\nspin wave point of view. We chose a ferrimagnetic off-plane\nspin configuration, with the S= 1 spins pointing in the + ˆ z\ndirection and the s= 1/2 in the −ˆ zdirection. The stacked\nlayers form a ferrimagnetic C-type configuration, and the\nmagnetic lattice has two inequivalent sites. That is not the\nmost stable configuration, but it is much simpler than the\naforementioned G-type FiM, which has four inequivalent\nsites.\nFigure 1: The crystal structure of the proposed Cu2F5lat-\ntice.Cuions inside the blue octahdera ( Cu1) have spin\nS= 1.Cuions inside the magenta plaquettes ( Cu2) have\nspins= 1/2. Reproduced with permission from Ref. [80].\nThe exchange model which stabilizes the spin order of\nthe layer comprises a FM exchange bond between Cu1\nsites and an AFM exchange bond between Cu1−Cu2\nsites. We add a Dzyaloshinskii-Moriya interaction (DMI)\nbetween next-near-neighbors (NNN) sites and a single-ion\nanisotropy (SIA) in the z-direction. The DMI is responsi-\nble for the transverse transport, and the SIA stabilizes the\noff-plane configuration. The Hamiltonian of the model is:H=−J1/summationdisplay\n/angbracketleftij/angbracketrightSi·Sj+J2/summationdisplay\n/angbracketleftij/angbracketrightSi·sj\n+D/summationdisplay\n/angbracketleft/angbracketleftij/angbracketright/angbracketrightνij/parenleftbig\nSx\nisy\nj−Sy\nisx\nj/parenrightbig\n−A/summationdisplay\ni/bracketleftBig\n(Sz\ni)2+(sz\ni)2/bracketrightBig\n(1)\nc (x)b (y)J2J2J1\nFigure 2: 2D model studied in this paper, corresponding to\na layer in the b-c plane of the crystal structure in Figure 1.\nExchange interactions are represented as J1andJ2. There\nis a Dzyaloshinskii-Moriya interaction between NNN with\nνij= +1(−1) along (against) the arrow. Blue sites have\nspinS= 1, and magenta sites, s= 1/2. The gray region\ncorresponds to the unit cell.\nThe upper (lower) case Si(si) operator denotes the spin\noperators for S= 1 (s= 1/2) sites. The first term ( J1>0)\nrepresents the FM exchange between S= 1 (Cu1) sites,\nand the second term ( J2>0) represents the AFM exchange\nbetweenS= 1 ands= 1/2 (Cu1−Cu2) sites. Both in-\nteractions happen between near-neighbors (NN). The third\nterm is the DMI between NNN sites, where νij=±1 fol-\nlowing the arrow convention in Figure 2. The last term is\nthe SIA. We note that the SIA between 1/2 spins is ineffec-\ntive [68], so only the first term inside the square brackets\nneeds to be considered.\nWe use the linearized Holstein-Primakoff representation\nfor up/down spins:\nS+\ni=√\n2Sai, S−\ni=√\n2Sa†\ni, Sz\ni=S−a†\niai\ns+\ni=√\n2sb†\nj, s−\ni=√\n2sbj, sz\ni=−s+b†\njbj(2)\nso the Hamiltonian is written in terms of the magnon\ncreation and annihilation operators in configuration space .\nThis representation can be applied to collinear AFM and\nFiM with off-plane spins. Performing a Fourier transform\n2XX'M\nkx ky\u0001(k)\u0001(k)(a) (b)\nFigure 3: (a) Band structure of the system when J1=J2= 1.0,A= 0.1, andD= 0.2. The blue (red) line corresponds\nto theω↓(↑)(k) band. (b) 3D band structure.\nenables us to write the momentum-space 4x4 Hamiltonian\nin the block matrix form:\nHk=ψ†\nk/parenleftbiggHI\nk0\n0HII\nk/parenrightbigg\nψk (3)\nwithHII\nk=/bracketleftbig\nHI\nk/bracketrightbig∗. The basis is ψ†\nk=/parenleftBig\na†\nk,b−k,a−k,b†\nk/parenrightBig\n.\nThe first block of the Hamiltonian matrix is:\nHI\nk=/parenleftbiggJ1S(1−γk)+J2s+A˜S√\nsS(J2ηk−2iDmk)√\nsS(J2ηk+2iDmk) J2S/parenrightbigg\n≡/parenleftbiggr(k)+∆(k)f∗(k)\nf(k)r(k)−∆(k)/parenrightbigg\n(4)\nwith˜S≡(2S−1)/2 = 1/2 and structure fac-\ntorsγk=cos(kx/2),ηk=cos(ky/2), andmk=\n−sin(kx/2)sin(ky/2).\nThe Hilbert space was doubled in this procedure, so it\ncarries not only the magnon (pseudo-) particle states but\nalso non-physical hole states. To diagonalize Hamiltonian\n(1) we use a Bogoliubov transformation, where the two\nphysical solutions have eigenvalues:\nω↑↓(k) =/radicalBig\nr(k)2−|f(k)|2∓∆(k) (5)\nThat is the band structure of the system. The up/down\nmagnons carry magnetic dipole momentum σgσµB, with\nσ=±1. For AFM systems, the g-factor gσis the same for\nboth magnon species, so it is usually absorbed into another\nconstant. For ferrimagnets, however, usually g↑/negationslash=g↓due\nto the inequivalence of the spins in the two sublattices, and\nit is essential to carry the g-factors in the expressions [78 ].\nWe must remember that magnons are bosons, so\nthere is no Fermi level. The thermal population nλ(k)\nis given by the Bose-Einstein distribution: nλ(k) =/parenleftBig\ne/planckover2pi1ωλ(k)/kBT−1/parenrightBig−1\n. Both bands are always populated.\nEven if a gap exists between the bands, the system is not a\ntrueinsulator (thisterm canbeapplied onlyas an analogy).3 Berry curvature and transverse\ntransport\nThe Berry curvature of the λ-bandΩλ(k) is a property\nof the energy band responsible (among other things) for\ntransversal transport and topological effects. It can be ob-\ntained from the eigenstates uλ(k) of the Hamiltonian [82]:\nΩλ(k) =i/angb∇acketleft∇kuλ(k)|×|∇ kuλ(k)/angb∇acket∇ight (6)\nThe Berry curvature acts like a fictitious magnetic field\non the magnons and is related to the geometrical Berry\nphaseaccumulatedbythegroundstateeigenfunctionswhen\nevolving in the Brillouin zone. In the case of a two-band\nAFM/FiM Hamiltonian, both bands have the same Berry\ncurvature, whose off-plane component can be obtained an-\nalytically from\nΩ↑↓(k) =−1\n2sinhθk/parenleftbigg∂φk\n∂kx∂θk\n∂ky−∂φk\n∂ky∂θk\n∂kx/parenrightbigg\n(7)\nwhereθkandφkare parameters which can be written in\nterms ofr(k), ∆(k) andf(k) [41].\nWhen an external in-plane field gradient is applied to\nsome systems, it is possible to observe magnon transport\nin a direction transverse to the field gradient. This phe-\nnomenon can be generically called the Hall-like transport\nof magnons . When the external perturbation is a magnetic\nfield gradient we observe a transverse spin current given\nby [20,23]:\njS,B\ny=σxy(−∂xB) (8)\nwhereσxyis the spin Hall conductivity, and this phe-\nnomenon is called the spin Hall effect of magnons.\nWhen we apply a temperature gradient, we observe both\nspin and thermal transverse currents, given respectively b y\n[21–23]:\njS,T\ny=αxy(−∂xT) (9)\njQ,T\ny=κxy(−∂xT) (10)\n3These are, respectively, the spin Nernst effect and the\nthermal Hall effect ofmagnons. The coefficient αxyis called\nthe spin Nernst coefficient, and κxyis the thermal Hall con-\nductivity. Within the linear response theory, the transpor t\ncoefficients for each band are given by integrals (in the con-\ntinuum limit) that involve the Berry curvature [48,78]:\n[σxy]↑↓=−(g↑↓µB)2\n/planckover2pi1VBZ/integraldisplay\nBZd2k n↑↓(k)Ω↑↓(k) (11)\n[αxy]↑↓=−(g↑↓µB)kB\n/planckover2pi1VBZ/integraldisplay\nBZd2k c1[n↑↓(k)]Ω↑↓(k) (12)\n[κxy]↑↓=−k2\nBT\n/planckover2pi1VBZ/integraldisplay\nBZd2k c2[n↑↓(k)]Ω↑↓(k) (13)\nHere,n(k) is the thermal population of the band given\nby the Bose-Einstein distribution, and the functions c1and\nc2are defined as\nc1(x) = (1+x)ln(1+x)−xln(x) (14)\nc2(x) = (1+x)/bracketleftbigg\nln/parenleftbigg1+x\nx/parenrightbigg/bracketrightbigg2\n−(lnx)2−2Li2(−x).\n(15)\nwhereLi2(x) is Spence’s dilogarithm function.\nIt is important to remark that in Refs. [48] and [78] the\nauthors write the transport coefficients without the explici t\nintegrals and in terms of the Chern number of the band. In\nthose references it is assumed that the bands are almost flat\nand the functions of n(k) are factored out of the integrals\nin Eqs. (11)-(13), so the coefficients become proportional\nto the Chern number C=/integraltext\nd2kΩ/2π. We do not make\nthe flat band approximation here, so the functions of n(k)\nare not factored out.\n-0.04-0.0200.02\nkxky\n0 2\u0001 -2\u0001-\u0001\u0001\n0\nFigure 4: Berry curvature of both bands in the Brillouin\nzone. Same parameters as figures above.\n\u0001xy\u0002\u0003\u0004\u0005\u0006\u0007\u0002ℏ\u0006\u0004\b\t\n\u000b\n ℏT/J1 D = 3.0\n D = 2.0\n D = 1.0\n\fxy (10-3 ℏ-1kB\bB)\n\u0002ℏT/J1 D = 3.0\n D = 2.0\n D = 1.0\n\rxy (10-3 kB2\n ℏ-1)\n ℏT/J1 D = 3.0\n D = 2.0\nD = 1.0(a)\n(b)\n(c)\nFigure 5: (a) Spin Hall conductivity σxy, (b) thermal Hall\nconductivity and (c) spin Nernst coefficient as functions\nofT/J1. The parameters are J1=J2= 1.0,A= 0.1,\ng↓= 1.0,g↑= 1.2 and three values of D.\nThetransverse currentsofbothmagnons combinetogen-\nerate the total conductivities of the system:\nσxy= [σxy]↓+[σxy]↑ (16)\nαxy= [αxy]↓−[αxy]↑ (17)\nκxy= [κxy]↓+[κxy]↑ (18)\nThe difference in sign between αxyandκxycan be un-\nderstood as follows. If a perturbation drives both magnons\nin the same direction, the thermal current is additive while\n4the spin current is subtractive, and vice-versa if the pertu r-\nbation drives the magnons in opposite directions (which\nis the case of a thermal gradient in the system studied\nhere). Furthermore, in degenerate systems both currents\nhave the same magnitude, and we can observe a pure spin\ncurrent without thermal current ( pure spin Nernst effect\nof magnons ) [45,46,48,68]. These systems are Z2topo-\nlogical magnon insulators, protected by an effective time-\nreversal symmetry. They are analogs of the quantum spin\nHall states of electrons [83].\n4 Results\nThe band structure of the system can be seen in Figure 3.\nFor thetypical values of A J iit is possible to observe a sign change in\ntheσxyandκxy(subtractive) coefficients, while it does not\noccur forαxy(Fig. 6). That can be explained as follows.\nWhenA>J i, the↑-band is always lower than the ↓-band\n(the bands are gapped, Figure 7). We know that in the\n\u0001xy\u0002\u0003\u0004\u0005\u0006\u0007\u0002ℏ\u0006\u0004\b\t\n\u000b\n ℏT/J1\n ℏT/J1\fxy (10-9 kB2\n ℏ-1)\n ℏT/J1\rxy (10-9 ℏ-1kB\bB)(a) \n(b) \n(c) \nFigure 6: (a) Spin Hall conductivity σxy, (b) thermal Hall\nconductivity and (c) spin Nernst coefficient as functions\nofT/J1. The parameters are J1=J2= 1.0,D= 0.3,\nA= 1.1,g↓= 1.0, andg↑= 1.2. Note that the first two\ncoefficients changes for low temperatures, which is a result\nof the competing conductivities of each individual band.\nlow-temperature limit, the lower band dominates, so the\n↑-band (which has negative transport coefficients) is more\npopulated. That results in negative total transport coeffi-\ncients. As the temperature rises, the population in the top\n5band surpasses the lower band’s, and the total transport\ncoefficients become positive. That behavior does not occur\nforA< J i, as↓-band is the lower one, and its population\ndominates in all temperatures. The change of sign of the\ntransport coefficients, which happens also in other magnon\nsystems, opens the exciting possibility of controlling the\ndirection of magnon flow with the temperature.\n\u0001(k)\u0002XX'M\nFigure 7: Band structure for J1=J2= 1.0,D= 0.3, and\nA= 1.1. ForA>J 1=J2, a gap appears.\n5 Conclusion\nWe have studied a 2D magnetic lattice which describes a\nlayer of the Cu2F5crystal [79–81] using the spin wave ap-\nproach. The magnetic order is ferrimagnetic, with spin-up\nsites (S= 1) pointing in the off-plane direction opposite to\nspin-down ( s= 1/2). The magnon bands are not gener-\nate, as expected for ferrimagnets and in contrast to AFM\ncollinear order, where an effective time-reversal symmetry\nmakes the bands degenerate. A non-null Berry curvature\nis generated by the Dzyaloshinskii-Moriya interaction be-\ntween next-near neighbors. That enables the system to\nshow transverse transport effects. 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B 87 (1) (Jan. 2013).\ndoi:10.1103/physrevb.87.014423 .\nURLhttp://dx.doi.org/10.1103/PhysRevB.87.014423\n9" }, { "title": "2208.08148v2.Polarization_selective_magneto_optical_modulation.pdf", "content": "arXiv:2208.08148v2 [physics.optics] 29 Sep 2022Polarization-selective magneto-optical modulation\nBanoj Kumar Nayak and Eyal Buks\nAndrew and Erna Viterbi Department of Electrical Engineeri ng, Technion, Haifa 32000 Israel\n(Dated: September 30, 2022)\nWe study magneto-optical coupling in a ferrimagnetic spher e resonator made of Yttrium iron\ngarnet. We find that the resonator can be operated in the telec om band as a polarization-selective\noptical modulator. Intermodulation gain can be employed in the nonlinear regime for amplification.\nI. INTRODUCTION\nInformation is commonly transmitted by modulating a\nmonochromatic carrier wave. The method of single side-\nband modulation (SSM) allows reducing both transmis-\nsion power and bandwidth, in comparison with simpler\nmethods such as amplitude, frequency and phase mod-\nulation [1]. In the radio frequency band SSM can be\nimplemented using electronic circuits, however, SSM im-\nplementation in the optical band is challenging, since it\nrequires that different out of phase modulation methods\nare simultaneously applied [2, 3].\nMagneto-optical(MO)coupling[4–10] in ferrimagnetic\nsphere resonators (FSR) can be used for optical modu-\nlation of signals in the microwave band. Such a modu-\nlation has been demonstrated before [11–21] by exciting\nindividual whispering gallery FSR optical modes using\neither a tapered optical fiber or a prism. Here we em-\nployed a modified experimental setup, in which light in\nthe telecom band is transmitted through the FSR bulk.\nDriving the FSR near its resonance generates sidebands\nin the transmitted optical spectrum. We find that the\nFSR can be used as a polarization-selective SSM. The\npolarization selectivity is attributed to angular momen-\ntum conservation in photon-magnon scattering [22–27].\nWe demonstrate that intermodulation (IMD) gain can\nbe exploited in the nonlinear regime for amplification.\nII. EXPERIMENTAL SETUP\nThe experimental setup is schematically shown in\nFig.1. Optical components and fibers are red colored,\nwhereasblue coloris usedto label microwave(MW) com-\nponents and coaxial cables. A MW cavity made of a\nloop gap resonator (LGR) allows achieving a relatively\nlarge coupling between magnons and MW photons [29–\n32]. The LGR is fabricated from a hollow concentric\naluminium tube. A sapphire (S) strip of 260 µm thick-\nness is inserted into the gap in order to increase its ca-\npacitance, which in turn reduces the frequency fcof the\nLGR fundamental mode. An FSR made of Yttrium iron\ngarnet (YIG) having radius of Rs= 125µm is held by\ntwo ceramic ferrules (CF) inside the LGR. The two CFs,\nwhich are held by a concentric sleeve, provide transverse\nalignment for both input and output single mode opti-\ncal fibers. Fiber longitudinal alignment is performed by\nmaximizing optical transmission.\nFigure 1: Experimental setup. Optical fibers are installed o n\nboth sides of the FSR for transmission of light through the\nsphere. Optical components [TL (tunable laser), Att (opti-\ncal attenuator), PC (polarization controller) and PBS (pol ar-\nization beam splitter)] and fibers are red colored, and MW\ncomponents [MWA (microwave loop antenna), S (splitter), C\n(circulator), VNA (vector network analyzer), SA (spectrum\nanalyzer) and SG (signal generator)] and coaxial cables are\nblue colored. TL2 together with two PBSs (labelled as PBS1\nand PBS2) and two differential photo detectors (labelled as\nDPD1 and DPD2) operate as a polarization-selective optical\nspectrum analyzer (OSA) [28]. A power amplifier is serially\nconnected to the SG. The MWA is weakly coupled to the\nFSR-LGR system.\nThe angular frequency of the Kittel mode ωmis ap-\nproximately given by ωm=µ0γeHs, where Hsis the\nstatic magnetic field, µ0is the free space permeability,\nandγe/2π= 28GHzT−1is the gyromagnetic ratio [10].\nThe applied static magnetic field Hsis controlled by ad-\njusting the relative position of a magnetized Neodymium\nusing a motorized stage. The static magnetic field is nor-\nmal to the light propagation direction k, and the mag-\nnetic field of MW drive is nearly parallel to k. The\nLGR-FSR coupled system is encapsulated inside a metal-\nlic rectangular shield made of aluminum. The LGR is\nweakly coupled to a microwave loop antenna (MWA).\nThe plot in Fig. 2exhibits a vector network analyzer\n(VNA) reflectivity measurement of the LGR-FSR cou-\npled system. The static applied magnetic field Hsin this\nmeasurement is varied near the value corresponding to\navoided-crossing between the FSR and LGR resonances.2\nFigure 2: VNA reflectivity in dB units as a function of mag-\nnetic field Hsat applied microwave power of −30 dBm.\n1538.85 1538.9 1538.95−80−60−40−20\nλ2 [nm]IT [dBm]\nFigure 3: The transmitted optical spectrum. For this mea-\nsurement the TL1 is set at optical power of 31mW and wave-\nlengthλLof 1538.887nm, and the driving microwave is set at\nfrequency ωp/(2π) of 3.79GHz and power of Ppof 20 dBm.\nIII. OPTICAL SIDE BANDS\nOptical side bands are observed in the transmission\nspectrumwhenthedrivingmicrowavefrequency ωp/(2π)\nistunedclosetotheFSRresonanceat ωm/(2π). Theplot\nshown in Fig. 3exhibits the measured total optical inten-\nsityIT=IDPD1+IDPD2as a function of the wavelength\nλ2of TL2, where IDPD1andIDPD2are the intensities\nmeasured by the two differential photodetectors (labelled\nas DPD1 and DPD2 in Fig. 1). The side band wave-\nlengths are given by λL±λSB, whereλSB≃λ2\nLωp/(2πc),\nandλLis the TL1 wavelength, which is related to the\nTL1 frequency ωL/(2π) byωL= 2πc/λL, wherecis the\nspeed of light in vacuum. The value of λSB= 30.0pm is\nobtained for TL1 wavelength of λL= 1539nm and FSR\ndriving frequency of ωp/(2π) = 3.79GHz.\nBoth motorized polarization controllers (labelled as\nPC1 and PC2 in Fig. 1) have three optomechanical com-\nponents (paddles), which act as either quarter or half\nwave plates. The paddles’ angles of PC1 (PC2) are de-\nnoted by θ1A,θ1Bandθ1C(θ2A,θ2Bandθ2C). The in-cident light state of polarization (SOP) can be manip-\nulated using PC1. We observe that intensity of lower\nwavelength λL−λSBanti-Stokes sideband and higher\nwavelength λL+λSBStokes side band depend on the in-\nput SOP. SSM in the transmission spectrum, with either\nsingle anti-Stokes side band, or with single Stokes side\nband, canbe obtained byadjusting PC1. The plot shown\nin Fig.4(a) exhibits the measured anti-Stokes side band\nintensity as a function of microwave driving frequency\nfp=ωp/(2π) and PC1 angle θ1Cnear the avoided-\ncrossing region. The plot shown in Fig. 4(c) exhibits si-\nmultaneously measured Stokes side band intensity in the\nsame region. We clearly observe appreciable anti-Stokes\nand Stokes intensity in Fig. 4(a) and (c), respectively,\nwhen driving frequency ωp/(2π) becomes close to FSR\nresonance ωm/(2π). However, they are asymmetric. For\na certain range of PC1 position, SSM is obtained, i.e.\nonly one side band, either anti-Stokes or Stokes, is ob-\nserved. Contrary to other experimental setups, in which\nthe FSR is optically coupled by either a tapered optical\nfiber or a prism, for our setup, for which the measured\noptical transmission only weakly depends on the input\nwavelength λL, SSM can be obtained in wide range of\nλL.\nA rotating lambda plate polarimeter is employed to\nmeasure the input SOP. The polarimeter measurements\nrevealthattheinputSOPforthetwoextremecases(SSM\nof either anti-Stokes or Stokes peak) are orthogonal to\neach other (i.e. separated by a diameter on the Poincar´ e\nsphere).\nFigure 4: Side bands in dBm units. (a) anti-Stokes intensity\nas a function of PC1 angle θ1C. (b) anti-Stokes intensity as\na function of magnetic field Hs. (c) Stokes intensity as a\nfunction of θ1C. (d) Stokes intensity as a function of Hs.\nThe magnetic field Hsin (a) and (c) is tuned near avoided-\ncrossing regime. TL1 is set at optical power of 31mW and\nwavelength of λLof 1537.7nm, and the driving microwave\npower is set at Pp= 20 dBm. In (a) and (c), θ1A= 170◦\nandθ1B= 85◦, andθ1Cis varied from 0◦to 170◦, whereas in\n(b) and (d) ( θ1A,θ1B,θ1C) = (170◦,85◦,60◦) (for this setting\nboth Stokes and anti-Stokes peaks are clearly visible near t he\nFSR resonance).3\nFigure 5: Sideband SOP. The measured intensity IDPD1\n(IDPD2) is shown (in dBm units) in the plots labeled by the\nletter ’a’ (’b’). The intensity at wavelengths λL−λSB,λLand\nλL+λSBis shown in the plots labelled by the numbers ’1’,\n’2’ and ’3’, respectively. The TL1 is set at optical power of\n31mW and wavelength of λLof 1538.9nm, the driving mi-\ncrowave is set at power Ppof 20 dBm.\nThe plots shown in Fig. 4(b) and (d) exhibit anti-\nStokes and Stokes intensity, respectively, as a function of\nmicrowave driving frequency fpand static magnetic field\nHs. The FSR resonance changes as we vary the static\nmagnetic field Hs. Accordingly, from Fig. 4(b) and (d),\nwe see that both anti-Stokes and Stokes intensity gets\npronounced when driving frequency ωp/(2π) is within\nthe bandwidth of FSR resonance at ωm/(2π).\nOur experimental setup (see Fig. 1) allows measuring\nthe SOP of both sidebands. While the plots shown in\nFig.4display the total optical intensity IT=IDPD1+\nIDPD2, the intensity IDPD1(IDPD2) is separately dis-\nplayed in the top (bottom) row plots shown in Fig. 5.\nThese two intensities IDPD1andIDPD2represent two or-\nthogonal SOP, which can be set by adjusting PC2 (see\nFig.1). The left (right) column plotsin Fig. 5displaythe\nmeasured intensity of the left anti-Stokes (right Stokes)\nsideband at wavelength λL−λSB(λL+λSB), whereas\nthe intensity at the central wavelength λLis displayed\nby the central column plots in Fig. 5. For the measure-\nments shown in Fig. 5, PC1 is set to a nearly SSM state.\nBy varying the setting of PC2, we find that the cen-\ntral peak at wavelength λLis maximized (minimized)\nin the same region where the sidebands at wavelength\nλL±λSBare minimized (maximized). This observation\nimplies that in the region of SSM, the SOP of the side-\nbands is nearly orthogonal to the SOP of the incident\nlight. This orthogonality can be exploited at the receiver\nend of a data transmission system based on our proposed\nMO modulation, since it allows demodulation by polar-\nization filtering-out of the carrier at wavelength λL.IV. MO COUPLING\nThe MO coupling giving rise to the optical side-\nbands originatesfrom an interactionterm in the system’s\nHamiltonian, which is denoted by VSB. This term VSB\nis commonly derived from the classical energy density\nassociated with the interaction between magnetization\nand optical modes. For the case where only whispering\ngallery FSR optical modes participate in the interaction,\nthe term VSBwas derived in [11–19], whereas for our ex-\nperimental configurationwe consider the case wherelight\npropagates through the FSR bulk.\nConsider an incident I (scattered S) optical field, hav-\ningrightandleft handedcircularpolarizationamplitudes\nEI+andEI−(ES+andES−), respectively. The time-\naveragedenergydensity umassociatedwith MO coupling\nis given by um= (1/4)ReUm, where\nUm=/parenleftbigE∗\nS+E∗\nS−/parenrightbig\nǫm/parenleftbiggEI+\nEI−/parenrightbigg\n, (1)\nand where ǫm=ǫm0+ǫm+m+′+ǫm−m−′is a trans-\nverse permittivity tensor. The static part ǫm0is given\nby Eq. ( A2) of appendix A. The diagonal elements of\nǫm0give rise to the static Faraday effect, whereas the\nstatic Voigt (Cotton-Mouton) effect originates from the\noff-diagonal elements of ǫm0[see Eq. ( A2)]. The terms\nǫm+m+′andǫm−m−′account for the effect of magne-\ntization precession, where ǫm±is given by Eq. ( A3) of\nappendix A, andm±′representamplitudes ofmagnetiza-\ntionprecession. Note thatthe matrix ǫm±isproportional\ntoe±iϕ, whereϕis the azimuthal angle [see Eq. ( A3)].\nThesphericalsymmetryoftheFSR ispartiallybrokenby\nthe two CFs that are employed for holding it (see Fig. 1).\nIn the semiclassical approximation VSBis derived from\num= (1/4)ReUm[seeEq.( 1)]. Considerapairofoptical\nmodeshavingnormalizedscalarspatialwaveforms,which\nin spherical coordinates are expressed as un′(r,θ,ϕ) and\nun′′(r,θ,ϕ), respectively. The contribution of this pair\nto the total interaction term VSB, which is denoted by\nVn′,n′′, is expressed as\nVn′,n′′=a†\nn′an′′/parenleftbig\ngn′,n′′,+b†+gn′,n′′,−b/parenrightbig\n+h.c. ,(2)\nwherean(b) is an annihilation operator for the n’th opti-\ncal mode (magnon mode), and h .c.stands for Hermitian\nconjugate. The coupling coefficients gn′,n′′,±aregivenby\n(recall that in our experiment the static magnetic field is\nnormal to the light propagation direction)\n/planckover2pi1−1gn′,n′′,±≃g0/integraldisplay\ndr′e±iϕun′(r′)u∗\nn′′(r′),(3)\nwhereg0=ωLQs//parenleftBig\n8n2\n0N1/2\ns/parenrightBig\n, andNsis the number of\nFSR spins ( Ns= 3.4×1016for the FSR under study).\nFor YIG in the telecom band (free space wavelength\nλ0≃1550nm), the refractive index is n0= 2.19, and\nthe dimensionless MO coupling coefficient is Qs≃10−44\n[33], and thus g0/(2π) = 2.7Hz. The overlap integral in\nEq. (3) represents a photon-magnon scattering selection\nrule [11, 12].\nThe ratio of side band output optical power to the in-\nput optical power is denoted by ηSB. The largest value\nofηSBis obtained at the triple resonance [11], for which\nthe MW driving is tuned to the FSR resonance ωm, the\nlaser frequency ωLmatches the frequency of one opti-\ncal mode, and the second one has a frequency detuned\nfromωLbyωm. For this case ηSB≃(2n0Rsg0/c)2Nm\n[it is assumed that the overlap integral in Eq. ( 3) is of\norder unity], where Nmis the averaged number of ex-\ncited magnons in steady state. For the case where the\nMWA is nearly critically coupled to the FSR, at reso-\nnanceNm≃Pp/(/planckover2pi1ωmκm), whereκmistheFSRdamping\nrate. The values of Pp= 20 dBm, ωm/(2π) = 3.8GHz\nandκm/(2π) = 1MHz yield ηSB≃10−5. This rough\nestimate agrees with the experimentally observed value\nofηSB[see Fig. 5].\nV. KERR NONLINEARITY\nMagnetic anisotropy gives rise to Kerr nonlinearity in\nthe FSR response [34]. The nonlinearity can be exploited\nfor modulation amplification [35]. Modulation measure-\nments in the nonlinear regime are shown in Fig. 6. The\nresults indicate that the Kerr coefficient is negative (giv-\ningrisetosoftening). Fortheplotsshowninthetop(bot-\ntom) row of Fig. 6, the microwave driving frequency is\nswept upwards (downwards). The dependency on sweep-\ning direction is attributed to nonlinearity-induced bista-\nbility, which, in turn, gives rise to hysteresis.\nVI. SUMMARY\nIn summary, polarization-selective SSM in the telecom\nband is achieved using an FSR strongly coupled to an\nLGR. The modulator can be used in a wide optical band,\nand it is compatible with ultra low temperatures. Future\nstudy will explore potential applications, including quan-\ntum state readout of superconducting circuits.\nThis work was supported by the Israeli science founda-\ntion, the Israeli ministry of science, and by the Technion\nsecurity research foundation.\nAppendix A: Transverse permittivity tensor\nThe evolution of electromagnetic waves propagating\ninside a magnetized medium is governed by a 3 ×3\npermittivity tensor [36–38]. Consider a Cartesian co-\nordinate system ( x,y,z), for which the propagation di-\nrection is parallel to the zdirection. In this system\nthe static magnetic field (magnetization vector) is par-\nallel to a unit vector denoted by ˆh(ˆ m). The angle\nFigure 6: Spectral peaks (in dBm units) in the nonlinear\nregime as a function of MW driving power Pp. The intensity\nof the left (right) sideband at wavelength λL−λSB(λL+λSB)\nis shown in the plots in the left (right) column, whereas the\nplots in the central column show the intensity of the central\noptical peak (at TL1 wavelength λL). For the plots shown\nin the top (bottom) row, the frequency fpis swept upwards\n(downwards).\nbetween ˆh= (hx,hy,hz) = (sin θcosϕ,sinθsinϕ,cosθ)\nandˆ m= (mx,my,mz) is assumed to be small.\nFrom the 3 ×3 permittivity tensor, a 2 ×2 transverse\npermittivity tensor ǫTcan be derived. In a basis of cir-\ncular SOP ǫTis given by ǫT=n2\n0I+ǫm, wheren0is the\nmedium refractive index, Iis the 2×2 identity matrix,\nand the 2 ×2 matrix ǫm(in a basis of circular SOPs) is\ngiven by [39]\nǫm\nn2\n0=/parenleftbigg\nQsmzQ2\nsm2\n−\nQ2\nsm2\n+−Qsmz/parenrightbigg\n, (A1)\nwherem±= (mx±imy)/√\n2. For YIG in the telecom\nband, the refractive index is n0= 2.19, and the dimen-\nsionless MO coupling coefficient is Qs≃10−4[33].\nThe eigenvalues of ǫm/n2\n0(A1) are given by\n±Qs/radicalBig\nm2z+Q2sm2\n−m2\n+. For the Faraday configuration,\nfor which mx=my= 0 and mz= 1, i.e. ˆ mis parallel\nto the propagation direction, the eigenvectors of ǫm/n2\n0\nrepresent circular SOPs, the corresponding eigenvalues\nare±Qs, and MO coupling gives rise to circular bire-\nfringence, whereas for the Voigt (Cotton-Mouton) con-\nfiguration, for which mz= 0 and m2\nx+m2\ny= 1, i.e.\nˆ mis perpendicular to the propagation direction, the\neigenvectors of ǫm/n2\n0represent colinear SOPs, the cor-\nresponding eigenvalues are ±Q2\ns/2 [note that m2\n−m2\n+=/parenleftbig\nm2\nx+m2\ny/parenrightbig2/4], and MO coupling gives rise to colinear5\nbirefringence. Note that for the Faraday configuration,\nthe SOP rotation angle that is accumulated over a trav-\neling distance of a single optical wavelength is 2 πQs.\nTo describe the effect of magnetization precession on\nǫm, it isconvenienttoexpress ˆ m(magnetizationunitvec-\ntor) as a sum of parallel and perpendicular components,\nwithrespectto ˆh(magneticfieldunitvector). InaCarte-\nsian coordinate system ( x′,y′,z′), for which the static\nmagnetic field is parallel to the z′direction, the unit\nvector parallel to the magnetization vector is expressed\nasˆ m′=mx′ˆ x′+my′ˆ y′+mz′ˆ z′=m+′ˆ u′\n++m−′ˆ u′\n−+\nmz′ˆ z′, where ˆ u′\n±= (ˆ x′±iˆ y′)/√\n2, and where m±′=\n(mx′∓imy′)/√\n2. The unit vectors ˆ mandˆ m′are re-\nlatedby ˆ m=R−1\nˆhˆ m′, whereforagivenunit vector ˆ n, the\nrotation matrix Rˆ nis defined by the relation Rˆ nˆ n=ˆ z,\nand thus ˆ m=m+′ˆ v++m−′ˆ v−+mz′R−1\nˆhˆ z′, whereˆ v±=\nR−1\nˆhˆ u′\n±. The matrix elements of the 3 ×3rotationmatrix\nRˆhare given by R11= 1+(cos θ−1)cos2ϕ,R22= 1+\n(cosθ−1)sin2ϕ,R12=R21= (1/2)(cosθ−1)sin(2ϕ),\nR31=−R13= sinθcosϕ,R32=−R23= sinθsinϕ\nandR33= cosθ. The following holds ˆ v±=cos2(θ/2)ˆ u±−e±2iϕsin2(θ/2)ˆ u∓−2−1/2e±iϕ(sinθ)ˆ z,\nhenceˆ m=µ+ˆ u++µ−ˆ u−+µzˆ z+mz′ˆh, where\nµ±=m±′cos2(θ/2)−m∓′e∓2iϕsin2(θ/2) andµz=\n−2−1/2/parenleftbig\nm+′eiϕ+m−′e−iϕ/parenrightbig\nsinθ, and thus m±=µ∓+\n2−1/2mz′e∓iϕsinθ.\nTheassumptionthattheanglebetweenthestaticmag-\nnetic field and the magnetization vector is small implies\nthatmz′≃1 and|m±′| ≪1. To first order in |m±′|,\nǫmcan be expanded as ǫm=ǫm0+ǫm+m+′+ǫm−m−′,\nwhereǫm0, which is given by [compare with Eq. 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Recent observation of helicity -independent all-optical \nmagnetization switching (HI -AOS) in an exchange -coupled ferromagnet ferrimagnet \n(FM-FEM) heterostructures expanded the range and applicability of such ultrafast heat -\ndriven magnetization switching. Here we report the element -resolved HI -AOS \ndynamics of such an exchange -coupled system, using a modified microscopic three -\ntemperature model. We have studied the effect of i) the Curie temperature of the FM, \nii) FEM composition, ii i) the long -range Ruderman –Kittel –Kasuya –Yosida (RKKY) \nexchange -coupling strength, and iv) the absorbed optical energy on the element -\nspecific time -resolved magnetization dynamics. The phase -space of magnetization \nillustrates how the RKKY coupling strength and the absorbed optical energy influence \nthe switching time. Our analysis demonstrates that the threshold switching energy \ndepends on the composition of the FEM and the switching time depends on the Curie \ntemperature of the FM as well as RKKY coupling st rength. This simulation anticipates \nnew insights into developing faster and more energy -efficient spintronics devices. \n \n1. Introduction \nUltrafast helicity -independent all -optical toggle switching (HI -AOS) has been an \nessential topic of research in the spintronics community for its potential applications in \ndigital data storage [1]. It is being explored in ferrimagnet (FEM) alloys and multilayers \n[2, 3, 4, 5, 6, 7, 8, 9, 10, 11] over the last decade. The magnetization switches in FEMs \ndue to the direct e xchange coupling between the transition metal and rare -earth \nelements. However, the switching is limited within a small range of optical energy and \nthe rare -earth element (mostly Gd) concentration and the utilization of synthetic FEMs \nlift these restrictio ns. [4] Theoretically, time -resolved and element -specific \nmagnetization dynamics in such FEMs have been modeled by various means namely; \ni) atomistic Landau -Lifshitz -Gilbert equation [7, 12], ii) phenomenological mean -field \ntheory [6, 13] , and iii) a modif ied version of the microscopic three -temperature model 2 (M3TM) [14, 15, 4, 5]. Beens et al. [4, 5] have used M3TM to incorporate the direct \nexchange coupling between the transition metal and rare earth and described the time -\nresolved magnetization dynamics of FEM alloy as a function of rare earth concentration \nand absorbed optical energy. \nOn the other hand, magnetization switching in ferromagnets (FM) is generally \nachieved using current -induced directional spin -transfer and spin -orbit torques \n(different from HI-AOS) [16, 17, 18, 19]. The Landau -Lifshitz -Gilbert equation in \ncombination with the terms for spin currents was used to analyze such a switching \nmechanism. Only recently, it was demonstrated that a conventional FM can also exhibit \nHI-AOS either due to i) the non -local ultrafast spin current [20, 21] or due to ii) the \nlong-range Ruderman –Kittel –Kasuya –Yosida (RKKY) exchange coupling [22, 23, \n24]. HI -AOS in such structures is promising as the exchange -coupled FM -FEM \nstructure can work as a free (storage) layer in magnetic memory devices [11, 25] for \nefficient electrical reading due to the enhanced tunneling magneto -resistance. RKKY \nexchange -mediated ultrafast switching in FMs has a rich physics that has hitherto \nlargely remained unexplored via theoretical modeling. Gorchon et al. [22] \nexperimentally demonstrated ultrafast switching of an exchange -coupled FM layer \n(Co/Pt) in ∼7 ps. Recently, Chatterjee et al. [24] have obtained a significantly faster \nswitching in ∼3.5 ps using an optimized structure and introduced a modified M3TM \nsimulation to analyze the switching phenomenon. In these structures, RKKY -type \nexchange coupling [26, 27, 28] has been discussed as the primary source of angular \nmomentum transfer channel between the FM and the FEM for obtaining ultr afast \nmagnetization switching [22, 24]. \nIn this paper, we discuss the theoretical modeling of the modified M3TM in greater \ndetail and simulated the experimentally observed ultrafast magnetization switching in \n∼7 ps by Gorchon et al. [22]. We have analyzed the effect of i) the Curie temperature \n(Tc), ii) the composition of the FEM, iii) RKKY coupling strength, and iv) absorbed \noptical energy on the ultrafast magnetization dynamics of the exchange -coupled \nheterostructure. The main addition in this model compa red to the M3TM model, \n(introduced by Beens et al. [4] to study HI -AOS in FEM alloys and multilayers) is the \nincorporation of a long -distance RKKY exchange coupling term, which connects the \nmagnetization dynamics of the FM and FEM layer and enables us to e xamine the \nexchange coupled heterostructure. The FM is magnetically exchange -coupled with the \nFEM while physically separated by a non -magnetic spacer layer (Pt) of high spin -\nscattering strength. The spacer layer significantly blocks any possible transmissi on of \nspin-current which can affect the magnetization dynamics [22, 24]. We observe that \nRKKY exchange strength, which is two orders of magnitude less than the direct \nexchange between the transition metal and rare earth elements of the FEM, is sufficient \nfor switching the FM. Experimentally the Curie temperature of a Co/Pt ML system can \neasily be tuned by changing the thickness and the repetition of the individual layers. \nWe detect a notable slowing of the Co/Pt switching timescale upon reducing the Tc of \nthe FM. The alloy composition of the FEM mainly affects the switching energy \nthreshold and the dependence of the switching timescale on the RKKY exchange \ncoupling. 3 \n \nFigure 1: Schematic diagram of the simulated sample structure and (b) the different exchange interactions \nacting on the sub -lattices of the sample. \n2. Sample Structure and Simulation Method \nWe have simulated the exchange coupled FM -FEM heterostructure \n(Sub/FMFEM/capping layer) experimentally studied by Gorchon et al. [22 ] and \nreplaced their FEM alloy (Fe65Gd 28Co7) with Fe72Gd 28 for the ease of calculation and \nthe Tc of the FM (Co/Pt) is taken to be 470 K as estimated in their experiment. We have \nextended the simulation to explore the effect of different FEM compositions namely; \nFe72Gd 28 and Co72Gd 28, and henceforth, these two FEMs are denoted as FeGd and \nCoGd. We have simulated the effect of different Tc (470 K, 530 K, and 580 K) of the \nCo/Pt on the magnetization dynamics. The configuration of the sample is \nSubstrate/Ta(5 nm)/ Fe72Gd 28 (or Co72Gd 28)(20 nm)/Pt(1.5 nm)/Co(0.6nm)/Pt(0.7 \nnm)/Co(0.6 nm)/Pt(3 nm), where the thicknesses of each layer are given in nanometers \nand schematically shown in Fig. 1a. The thickness of the Pt spacer layer, which \nseparates the FM and FEM sub -lattices, determines the stren gth and sign of RKKY \nexchange coupling. Experimentally it was varied between 1 to 4 nm to obtain both \nferromagnetic and anti -ferromagnetic coupling. The coupling strength varies non -\nlinearly with the thickness of the Pt layer according to the characteristi c oscillatory \nnature of the RKKY exchange interaction [26, 28] and the effect is simulated by \nvarying the RKKY coupling within a predefined range. \n \nThe different types of exchange interactions acting on the heterostructure are \nschematically represented in Fig. 1b. The electron and phonon sub -systems of the FM \nand the two sub -lattices of the FEM have been described by a two-temperature model \nexpressed by the equations 1 and 2. These two sub -systems (electrons and phonons) of \nthe FM and FEM layer get excited by the ultrafast optical pulse depending on the \nrelative optical absorption of the respective layers. First, we solve the two -temperature \nSi Substrate \nTa (5 nm) \nFe \n72 \nGd \n28 \n/Co \n72 \nGd \n28 \n nm \n) \n (20 \nPt (1.5 nm) \nCo (0.6 nm) \nPt (0.7 nm) \nCo (0.6 nm) \nPt (3.0 nm) \na \n) \n( \n ) \n( \nb \nCo/Pt \nFe \n( \nCo \n) \nGd \nElliot \n- \nYafet \nScattering \nElliot \n- \nYafet \nScattering \nElliot \n- \nYafet \nScattering \nUltrafast Laser \nRKKY Exchange \nJ \nCo \n/Pt \n- \nFe(Co) 4 model to m easure the temporal evolution of the electron ( Te) and phonon temperatures \n(Tp) for the FM and FEM using the following equations, \n \n𝐶𝑒,𝑖𝛿𝑇𝑒,𝑖\n𝛿𝑡= −𝑔𝑒𝑝,𝑖(𝑇𝑒,𝑖−𝑇𝑝,𝑖)+ 𝑃𝑖; 𝑃𝑖=𝐴𝑖\n𝜎√2𝜋𝑒−(𝑡\n𝜎𝜋)2\n \n(1) \n𝐶𝑝,𝑖𝛿𝑇𝑝,𝑖\n𝛿𝑡= 𝑔𝑒𝑝,𝑖(𝑇𝑒,𝑖−𝑇𝑝,𝑖)+ 𝐶𝑝,𝑖(𝑇𝑎𝑚𝑏 −𝑇𝑝,𝑖)\n𝜏𝐷,𝑖; (𝑖∈𝐹𝑀,𝐹𝐸𝑀 ) \n(2) \n \nwhere Ce and Cp are electron and phonon specific heat, τD is the heat dissipation \nconstant which represents the thermal diffusion to the substrate, Tamb is the ambient \ntemperature (295 K), gep is the electron -phonon interaction strength and Ai is the \nabsorbed optical energy in each layer. The laser pulse is simulated as a Gaussian pulse \nwith 100 fs full -width -half-maximum which acts as the energy source to the electronic \nsub-system for each layer. The laser pulse gets absorbed differently in each of the layers \ndepending on the thickness and their optical constants, which has been evaluated by \nsolving the transfer matrix [29] (Fig. S1 in the supplemental inf ormation) of the entire \nheterostructure. The complex refractive indices applied in the calculation are given in \nTable S1 in the supplemental information. It is assumed that the electron sub -system \ninstantly reaches thermal equilibrium after optical excitat ion due to a large Coulomb \ninteraction [30], therefore, it can be represented by specific electron temperatures, \nwhich are assumed to be homogeneous inside each layer. The phonon temperatures \nremain in equilibrium due to phonon -phonon interaction. All the sub-systems are \ninitially at room temperature (295 K). The material parameters for the two -temperature \nmodel are listed in Table S2 in the supplemental information. Heat diffusion to the \nsubstrate is added to the phonon subsystem as an energy dissipation t erm with a 50 ps \ntime constant ( τD) for both layers. There are two magnetic elements such as Fe(Co) and \nGd comprising the FeGd (CoGd) alloy, whereas, for the Co/Pt, there is only one \nmagnetic element. The electrons are modeled as a spinless free electron g as and the \nphonons are described by the Debye model according to the basic postulates of the \nM3TM [31]. For simplicity, it is assumed that all the spin subsystems have the same \nspin half quantum number ( si = 1/2) [4, 5] with their respective atomic magneti c \nmoments. We have used the following analytical expressions of exchange splitting for \ncalculating the exchange coupling using Fermi’s golden rule and the details of the \nmagnetization dynamics are given in section S2 in the supplemental information. \n∆𝐹𝑒(𝐶𝑜)= 𝑥𝐹𝑒(𝐶𝑜)𝛾𝐹𝑒(𝐶𝑜)𝑚𝐹𝑒(𝐶𝑜)+(1−𝑥𝐹𝑒(𝐶𝑜))𝛾𝐹𝑒(𝐶𝑜)−𝐺𝑑𝑚𝐺𝑑 \n∆𝐺𝑑= 𝑥𝐹𝑒(𝐶𝑜)𝛾𝐺𝑑−𝐹𝑒(𝐶𝑜)𝑚𝐹𝑒(𝐶𝑜)+(1−𝑥𝐹𝑒(𝐶𝑜))𝐺𝑑𝑚𝐺𝑑 \n∆𝐶𝑜𝑃𝑡= 𝛾𝐶𝑜𝑃𝑡𝑚𝐶𝑜𝑃𝑡 +𝑥𝐹𝑒(𝐶𝑜)𝛾𝐶𝑜𝑃𝑡 −𝐹𝑒(𝐶𝑜)𝑚𝐹𝑒(𝐶𝑜)+(1−𝑥𝐹𝑒(𝐶𝑜))𝛾𝐶𝑜𝑃𝑡\n−𝐺𝑑𝑚𝐺𝑑 \n \n(3) \nwhere γij = zJijµj;(i,j ∈ Fe(Co), Gd, Co/Pt ) can be calculated from the exchange \ninteraction strength Jij and atomic magnetic moment µj. The percentage of Fe (Co) in 5 the FEM alloy is given by xFe (xCo); where xFe(Co) = 0.72. We apply the Weiss mean field \ntheory to determine the exchange strength which has the form 𝐽𝑖=3𝑘𝐵𝑇𝑐,𝑖\n2𝑧𝑆𝑖𝑆𝑖+1,𝑖=\n𝐹𝑒(𝐶𝑜),𝐺𝑑,𝐶𝑜𝑃𝑡 [6]. Here z = 12 is the number of next nearest neighbors and Tc,i is \nthe Curie temperature of individual elements (Tc,Co = 1388 K, T c,Fe = 1043 K, T c,Gd = 292 \nK, T c,Co/Pt = 470 K, 530 K, and 580 K). The short -range exchange interaction between \nFe and Gd sub -lattices is antiferromagnetic and it has the form: JGd−Fe = − 0.348 × JFe \nwhereas , for CoGd, the coupling between Co and Gd is; JGd−Co = − 0.388 × JCo [3, 24]. \nRKKY exchange coupling strength ha s been varied up to ∼5 % of the FM exchange \ni.e., JRKKY = ± 0.05 × JCoPt for both the heterostructures to include the effect of varying \nPt thicknesses which leads to the oscillating RKKY interaction. There are three \nchannels for angular momentum transfer i n the FEM, i) Elliott -Yafet type spin -flip \nscattering in Fe (Co), Gd and Co/Pt [31, 30, 32], ii) a short -range exchange coupling \nbetween Fe (Co) and Gd and iii) a long -range RKKY type exchange coupling between \nCoPt -Fe(Co) and CoPt -Gd layer [22]. As describ ed in equation 3, we have used the \nRKKY exchange between CoPt -Fe(Co) and CoPt -Gd. However, the dominant RKKY \nexchange contribution comes from the interaction of Gd and Co/Pt , and by including \nFe(Co) in the scheme, the dynamics are only minimally affected ( Fig. S3 of \nsupplemental information). This is because Gd has a much larger atomic magnetic \nmoment (µGd = 7.55µB, µFe = 2.20µB, µCo = 1.72µB, and µCo/Pt = 1.30µB) moment \ncompared to Fe(Co) and thereby dominates the angular momentum transfer process. \n \nThe electronic parameters of the materials in the simulation are obtained from the \nliterature [33, 4, 34] and are given in the Table S3 of supplemental information. The \ntime evolution of the magnetization dynamics has been calculated up to 40 ps after the \noptical excitation with 10 fs time resolution. The FM exchange varies linearly with Tc \nand takes the following values; JRKKY,T c,CoPt =470 K = 1.079×1 0-21 J , JRKKY,T c,CoPt =530 K = \n1.217×1 0-21 J and JRKKY,T c,CoPt =580 K = 1.332×10−21 J. Assuming a typical atomic spacing of \n0.4 nm, the RKKY exchange coupling is varied between ± 5% of the direct exchange, \nwhich results in JRKKY,T c,CoPt =470 K = ± 0.337 mJ.m-2, JRKKY,T c,CoPt =530 K = ± 0.394mJ.m-2, and \nJRKKY,T c,CoPt =580 K = ± 0.416 mJ.m-2 for Tc = 470 K, 530 K, and 580 K respectively. S. S. P. \nParkin [27] studied the exchange coupling strength of various 3d, 4d, and 5d transition \nmetals at room temperature and found that the exchange coupling varies from 0.1 to 5 \nmJ.m−2. Thus, the value of the RKKY exchange coupling strength in our simulation is \nwell within the known range. Although the specific thickness of the Pt layer determines \nthe sign and strength of the RKKY exchange coupling, a small thickness difference of \na couple of nanometers doe s not change the relative absorbed energy between the FM \nand FEM layer by more than 4%. Therefore, for simplicity, we assume that the relative \nabsorbed energy between the FM and FEM layer remains the same while changing the \nPt spacer layer thickness (i.e. with the variation of the RKKY exchange coupling). The \ntypical thickness variation of the Pt spacer layer (or any other spacer layer) would result \nin the modification of the RKKY coupling. Therefore, we have simulated the \nmagnetization dynamics within an a cceptable range of this coupling strength which is \ngenerally obtained using standard RKKY coupling layers [23, 24]. 6 3. Results and Discussions \n3.1. Effect of the Curie Temperature of the Co/Pt Layer \nThe phase diagrams of the magnetization as a function of absorbed optical energy \nand RKKY exchange coupling strength are shown in Fig. 2a -c for different values of \nTc (470 K, 530 K, and 580 K) of the Co/Pt layer. The color scale represents the \nmagnetizatio n reversal time of Co/Pt. The final magnetization state has been calculated \nby measuring the magnetization of each element (Fe, Gd, and Co/Pt) at 20 ps after the \nultrafast laser excitation. The initial magnetization of Fe and Co/Pt is positive and it is \nnegative for Gd before the arrival of the pump laser pulse (0 ps). Ideally, switching can \nbe defined when the magnetization of a particular element crosses zero, however, it has \nbeen noticed that the magnetization tends to stay close to zero for some time be fore \nswitching. Consequently, to avoid any ambiguity in the switching time, we have set a \nsmall non -zero magnetization value as the measure of switching [7]. The set conditions \nare the following in order to determine the different cases which may occur as a final \nmagnetization state: i) relaxed if mFe(t = 20 ps) > 0.01, mGd(t = 20 ps) < − 0.01 and \nmCoPt(t = 20 ps) > 0.01, ii) switched if mFe(t = 20 ps) < − 0.01, mGd(t = 20 ps) > 0.01 \nand mCoPt(t = 20 ps) < − 0.01, and iii) completely demagnetized if 0 .01 ≥ mFe(t = 20 ps) \n≥ −0.01, 0 .01 ≤ mGd(t = 20 ps) ≤− 0.01, and 0 .01 ≥ mCoPt(t = 20 ps) ≥ −0.01. \n \nThe phonon temperature increases beyond the Curie temperature at very high \noptical energy, which makes the system unstable i.e. it doesn’t remember its initial \nstate, and the final state randomly switches between up and down, especially at longer \ntimescale s (> 40 ps). Such behavior has also been identified in the simulated HI -AOS \nproperties of [Tb/Co] multilayers [35]. Hence, we have limited the calculation of the \nfinal switched state to 20 ps after the laser excitation. The FEM undergoes partial \ndemagnetiz ation and the FM shows either partial or full demagnetization before \nremagnetization at smaller absorbed optical energy. Consequently, the final state is the \nsame as the initial state for all the elements suggesting the system is in a relaxed state \nirrespe ctive of the RKKY coupling as represented by the pale -yellow region in Fig. 2a -\nc. In this energy range, an example of the magnetization dynamics of Fe, Gd, and Co/Pt \nis plotted in Fig. S4 of the supplemental information, where we find a partial \ndemagnetiza tion of Fe and Gd, and simultaneously Co/Pt gets fully demagnetized even \nthough it absorbs only ∼23% of the total optical energy. However, we can’t switch \nCo/Pt due to the unavailability of the angular momentum transfer channel with the FEM \n(as FEM doesn’t switch at smaller optical energy). With increasing optical energy, a \nswitched region appears as depicted by the color map in combination with bright blue \nand magenta colors, where the magnetization reversal of both the FM and FEM sub -\nlattices is obtained at the same absorbed energy. The green region suggests the \ncomplete switching of the FEM layer, however, the Co/Pt layer doesn’t switch. \nDetailed discussions about these two regions are provided later. On the other hand, with \nintense optical energy (dark b rown region of the phase space in Fig. 2a -c), only a \npermanent demagnetization of all the sub -lattices is observed as the Tp crosses the Tc \nfor all the magnetic components. Experimentally, in this region, the sample gets \ndamaged and its magnetic properties can be permanently lost if we expose the sample \nto such high optical energy for a longer time. 7 \nFigure 2: The phase -space of magnetization for (a) F1FeGd,Tc,CoPt =470 K, (b) F1FeGd,T c,CoPt =530 K, and (c) \nF1FeGd,T c,CoPt =580 K. The pale -yellow region shows a relaxed state, the green region depicts the switching of \nonly FEM, the bright blue and magenta region suggests the switching of both the FM and FEM, and the dark \nbrown region depicts complete demagnetizatio n of the sample. The color bar of the blue and magenta region \nshows the switching time of Co/Pt. The time -resolved element -specific magnetization dynamics of the \ncorresponding elements of (d -f) the FEM and (g -i) the FM sub -lattices in the FM -FEM heterostru cture at the \nabsorbed optical energy of 33 .2 × 108 J.m−3 for a specific RKKY exchange coupling ( JRKKY = + 0.01×JCoPt). \nThe dynamics are measured at the corresponding black dots in Fig. 2a -c. The yellow and cyan -filled region \nin Fig. 2d -i shows the evolutio n of electron and phonon temperatures of the FM and FEM layers. The red, \ncyan, and green dotted lines are the Tcs of the individual elements (Fe, Gd, and Co/Pt). The laser pulse is \nschematically displayed by the black -filled Gaussian pulse at time zero. \nThe time -resolved element -specific magnetization dynamics of the magnetic \nelements of the FM and FEM layers measured at the black dot on the phase space is \nplotted in Fig. 2d -i for absorbed optical energy of 33 .2 × 108 J.m−3 at a specific RKKY \nexchange coupling strength of JRKKY = + 0.01 × JCoPt. Fe, Gd, and Co/Pt sub -lattices \ndemagnetize at an ultrafast timescale upon ultrafast optical excitation where the \nF1 \nFeGd, Tc(CoPt) = 470 K \n F1 \nFeGd, Tc(CoPt) = 530 K \n F1 \nFeGd, Tc(CoPt) = 580 K \n( \na \n) \n ( \nb \n) \n ( \nc \n) \n( \nd \n) \n ) \n( \ng \n F1 \nFeGd, Tc(CoPt) = 470 K \n( \ne \n) \n ( \nh \n) \n F1 \nFeGd, Tc(CoPt) = 530 K \n( \nf \n) \n ( \ni \n) \n F1 \nFeGd, Tc(CoPt) = 580 K 8 demagnetization rate (at a constant fluence) is proportional to 𝑇𝑐2\n𝜇 [31]. The transition \nmetal (Fe) of the FEM layer, demagnetizes much faster than that of the rare earth \nelement (Gd), and the FEM shows a transient ferromagnetic -like state between ∼0.7-\n1.9 ps when the magnetization of both Fe and Gd points along the same direction. As \nthe intrinsic coupling between Fe -Gd is antiferromagnetic, Gd switches later due to \nangular momentum exchange with Fe. The Co/Pt layer exhibits a significantly faster \ndemagnetization in ∼500 fs due to enhanced Elliott -Yafet scattering via sp in-orbit \ntorque as published earlier [33]. \n \nIn the bright blue and magenta colored region of Fig. 2a -c, the electron temperature \nof Co/Pt rises faster and reaches significantly larger than its Tc (can be seen from the \npeak of the yellow -shaded region much larger than Tc in Fig. 2g -i). As a result, Co/Pt \ngets completely demagnetized and then its magnetization stays close to zero for a \nlonger timescale as the electron -lattice temperature equilibrates and slowly cools down. \nThe switched FEM kicks in to deliver enough angular momentum to ultimately switch \nits magnetization. In this region, the Co/Pt and the FEM switch at the same absorbed \nenergy. The magnetization dynamics of Co/Pt show a two -step process and the \nswitching occurs at a much longer timescale as de tected in Fig. 2g -h measured at the \nrespective black dots in Fig. 2a -b. The observation of such a two -step switching \ncharacteristic of the Co/Pt layer (exchange coupled with a FEM) has been explained as \na signature of the RKKY exchange coupling mediated sw itching of the hot/softened \nCo/Pt [22, 24]. \n \nOne of the main differences among the three phase -space figures in Fig. 2ac is the \nexistence of the green region which increases with increasing Tc of the Co/Pt layer, \nsuggesting that the Co/Pt and the FEM layer doesn’t switch at the same absorbed \nenergy. FEM switches at a smaller energy (when Co/Pt remains demagnetized) and \nthen the Co/Pt requires a larger absorbed optical energy to switch. It is relatively easy \nto understand that Co/Pt cannot be switched in a d ecoupled structure, where there is no \nRKKY exchange coupling with the FEM layer. Hence, we notice the green region in \nFig. 2a -c, where the RKKY exchange is zero (i.e. at the middle of the x -axis). However, \nfor larger Tc (530 K, and 580 K) the green region extends towards a finite RKKY \ncoupling strength (− 0.009 × JCoPt ≤ JRKKY ≤ 0.009 × JCoPt for 530 K, and − 0.045 × JCoPt \n≤ JRKKY ≤ 0.045 × JCoPt for 580 K) as shown in Fig. 2b -c. Now, for a smaller Tc, the \nelectron/phonon temperature requires a long time to cool below Tc, and the Co/Pt layer \nstays completely demagnetized for a longer time. The inter -sublattice (between FM and \nFEM) angular momentum exchange depends on the RKKY exchange strength and the \navailable magnetization in the FEM layer, which is dominated by Gd due to its much \nlarger atomic magnetic moment. At a longer timescale, Gd recovers a larger \nmagnetization value as shown in Fig 2d -f, hence the effective angular momentum \nexchange is larger, and Co/Pt can switch at a smaller RKKY exchange strength \nhowever the switching speed gets slower. Now, with increasing Tc, the electron/phonon \ntemperature gets below Tc much faster as plotted in Fig. 2h, and we get Co/Pt switching \nat a faster timescale. T he effective exchange coupling strength increases with \nincreasing Tc. The value of JRKKY is displayed at the top x -axis of the phase space \ndiagrams in Fig. 2a -c. Next, for an even larger Tc, the electron/phonon temperature 9 cools down much more quickly to b elow Tc, as shown in Fig. 2i, when the available \nmagnetization of Gd for the RKKY exchange is tiny. Then Co/Pt can’t switch and starts \nto remagnetize along its initial direction. Hence, Co/Pt either needs a larger RKKY \nexchange coupling or a larger optical energy to switch. In case of larger RKKY \nexchange strength, the Co/Pt switches faster as discussed later. For larger absorbed \noptical energy, the electron/phonon temperature rise is higher and it takes a longer time \nto cool down below Tc. Within that time , Gd gets sufficiently magnetized in the \nopposite direction to provide the necessary angular momentum transfer even at a \nsmaller RKKY exchange coupling strength. Hence, in general, Co/Pt switches at a \nslower speed with increasing energy as observed in Fig. S5 in the supplemental \ninformation. However, this contradicts the experimental observation by Gorchon et al. \n[22], where the switching speed of Co/Pt got faster with increasing optical energy. The \nswitching dynamics also depend on the RKKY exchange coupli ng strength and we do \nnotice a tiny energy range at smaller RKKY interactions, where the switching speed \nincreases with increasing energy as plotted in Fig. S5 of the supplemental information. \nAnother difference in the phase space among these three structu res, is the larger \nswitched region (bright blue and magenta color region), with increasing Tc. The Co/Pt \nlayer can sustain larger absorbed optical energy before getting damaged, as the Tc \nincreases, hence we detect switching up to larger optical energy. \n \nThe variation of the Co/Pt switching time with the RKKY exchange strength is \nplotted in Fig. 3 for different Tc. The switching timescale slows down exponentially \nupon reducing the exchange coupling. As Co/Pt gets demagnetized much more quickly \n(plotted in F ig. 2d -i) than the FEM elements, it waits for the FEM layer (mainly Gd) to \nremagnetize sufficiently and provide the necessary angular momentum transfer to \nswitch. As discussed earlier, at fixed absorbed optical energy, larger exchange coupling \nstrength mea ns the possibil ity of a larger amount of angular momentum transfer even \nif the Gd recovers a small magnetization in the opposite direction (in early timescale), \nand eventually, it translates to a faster switching. The Co/Pt layer switches ∼10 ps for \nJRKKY = + 0.01 × JCoPt, for F1FeGd,T c,CoPt =470 K, and by increasing the exchange amplitude \nto JRKKY = + 0.05 × JCoPt, the switching gets faster at ∼6 ps as shown in Fig. 3a. \n \n \nFigure 3: The variation of the switching time of Co/Pt with the RKKY exchange coupling strength for (a) Tc \n= 470 K, (b) Tc = 530 K, and (b) Tc = 580 K. All the simulations are performed at absorbed optical energy of \n33.2 × 108 J.m−3. \nF1 \nFeGd, Tc(CoPt) = 530 K \n( \na \n) \nF1 \nFeGd, Tc(CoPt) = 580 K \n F1 \nFeGd, Tc(CoPt) = 470 K \n( \nc \n) \n ( \nb \n) 10 On the other hand, as we have discussed earlier, the FM with higher Tc can’t be \nswitched at a smaller exchange as observed in Fig. 3b -c. A significant difference in the \nswitching timescale (at a fixed exchange strength a nd absorbed optical energy) of the \nCo/Pt is detected for the different values of Tc. The switching speed of the Co/Pt layer \ngets faster with increasing Tc. At the largest RKKY exchange strength and for the fixed \nabsorbed optical energy, with increasing Tc, the electron/phonon temperature reduces \nbelow Tc much faster, and if sufficient angular momentum transfer channel is available, \nCo/Pt switches faster. Similarly, it needs a longer time, for its electron temperature to \ncool sufficiently below a smaller Tc when the transferred angular momentum from the \nswitched FEM sub -lattice ultimately switches its magnetization. Therefore, it is evident \nthat the dueling time of Co/Pt in the demagnetized state (zero magnetization state) is \nsmall for larger Tc, which ultima tely fastens the switching time. Previously we \nmeasured and simulated a differently stacked heterostructure and found the switching \ntimescale of the Co/Pt to be ∼3.5 ps [24]. We have examined the effect of Tc in that \nheterostructure as well and found simil ar dependence of the switching timescale of \nCo/Pt, which is plotted in Fig. S6 of supplemental information. Hence, we can also \nconclude that the effect of the Tc on the switching time of the Co/Pt layer is universal \nand only minimally dependent on the samp le geometry. \n \n3.2. Effect of the Composition of the Ferrimagnet Layer \nNext, the effect of the composition of the FEM on the magnetization switching \ndynamics has been explored by changing the FEM to CoGd (from FeGd). The phase \nspace of magnetization is displayed in Fig. 4a considering CoGd as the FEM while \nkeeping the Tc of the Co/Pt fixed at 580 K. We don’t see the extended green region (for \na finite RKKY coupling) in Fig. 4a suggesting both FM and FEM switches \nsimultaneously even for a larger Tc (significantly different from Fig. 2c). \n \nFigure 4: The phase -space of switching for F1CoGd,Tc,CoPt =580 K. The pale -yellow region indicates a relaxed state, \nthe green region shows the switching of only FEM alloy, the bright blue and magenta region exhibits the \nswitching of both the FM and FEM and the dark brown region shows complete demagnetization of the \nsample. The color bar of the blue and magenta region describes the switching time of Co/Pt. (b) Switching \ntime of Co/Pt as a function of RKKY exchange coupling at the absorbed optical energy o f 50.4 × 108 J.m−3. \n( \na \n) \n ( \nb \n) \n50.4 \n \n× \n10 \n8 \nJ.m \n- \n3 \nRelaxed \nSwitching of Co \n72 \nGd \n28 \nSwitching of both CoPt and Co \n72 \nGd \n28 \nComplete Demagnetization \nF1 \nCoGd, Tc(CoPt) = 580 K 11 The Tc of Co in the CoGd sub -lattice is much larger than that of Fe in the FeGd sub -\nlattice, hence larger absorbed optical energy is required to switch the CoGd FEM itself. \nOn the other hand, the Tc of the Co/Pt layer is the same as bef ore, and ∼23% of the \noptical energy gets absorbed by it. Consequently, a larger optical energy is also \ndelivered in the Co/Pt layer, and the electron/phonon temperature needs a long time to \ncool below its Tc. Within this timescale, Gd in the CoGd sub -lattice attains adequate \nmagnetization value (discussed later), which ultimately leads to the switching of the \nFM and FEM layer at a finite RKKY exchange at the same absorbed optical energy. \nThe dependence of the switching time as a function of the RKKY couplin g is shown in \nFig. 4b and we distinguish a similar (to Fig. 3) fastening of the Co/Pt switching time \nwith increasing RKKY coupling strength, however, the much larger optical energy of \n50.4 × 108 J.m−3 is required for switching due to the high Tc of Co in t he FEM layer. \n \nThe time -resolved magnetization dynamics of the FM -FEM heterostructure with \ndifferent FEM sub -lattices are plotted in Fig. 5a -b at their respective threshold \nswitching energies of 50 .4 × 108 J.m−3 and 33 .2 × 108 J.m−3. The red, cyan, and green \nlines of the figures respectively denote the magnetization dynamics of Fe (Co), Gd, and \nCo/Pt. The RKKY exchange strength is JRKKY = + 0.05 × JCoPt for both systems. The \ndemagnetization rate of Co is slightly faster than Fe (and Gd) due to its higher Tc [31]. \n \n \nFigure 5: The time -resolved magnetization dynamics simulated for (a) F1CoGd,Tc,CoPt =580 K and (b) \nF1FeGd,Tc,CoPt =580 K measured at a particular exchange coupling strength (JRKKY = +0.05 × JCoPt) and for respective \nabsorbed threshold optical energy of 50 .4 × 108 J.m−3 and 33 .2 × 108J.m−3. The yellow and cyan -filled region \ndenotes the evolution of electron and phonon temperatures of the respective layers. The red, cyan, and green \ndotted lines are the Curie temperatures of the individual elements. The optical p ulse is schematically shown \nby the black -filled Gaussian pulse at time zero. \n( \na \n) \n( \nb \n) \n50.4 \n \n× \n10 \n8 \nJ.m \n- \n3 \n33.2 \n \n× \n10 \n8 \nJ.m \n- \n3 \nF1 \nCoGd, Tc(CoPt) = 580 K \nF1 \nF \ne \nGd, Tc(CoPt) = 580 K 12 On top of that, the anti -ferromagnetic intra -sublattice exchange (between rare earth \nand transition metal) is larger for CoGd as; JCo−Gd = −0 .388× JCo = −1 .236×10−21 J than \nin FeGd as; JFe−Gd = −0 .348 × JFe = −0 .821 × 10−21 J. Hence, the angular momentum \ntransfer between Co -Gd is faster than Fe -Gd which results in a shorter transient \nferromagnetic -like state of ∼0.6 ps in CoGd than ∼1.2 ps for the FeGd. Co switches at \n∼0.7 ps and Gd switches at ∼1.2 ps for the CoGd alloy (wher eas Fe switches ∼0.7 ps \nand Gd switches ∼1.9 ps for FeGd alloy). This also helps Gd to attain a significant \nmagnetization in the opposite direction within a shorter time. Now, Co/Pt switches in \n∼4.5 ps in Fig. 5a, which is slower compared to Fig. 5b ( ∼3.8 ps) and can be attributed \nto the increased threshold energy required for F1CoGd,T c,CoPt =580 K and faster recovery time \nof Gd. The maximum electron temperature rise in Co/Pt is larger at the higher th reshold \noptical energy, and it takes longer to cool below its Tc, when the RKKY coupling can \nkick in to switch its magnetization. Therefore, the Co/Pt moments stay longer in the \ndemagnetized state before they are flipped in the opposite direction, resultin g in slightly \nlower and two -step switching dynamics of Co/Pt. Hence, we conclude that the effect \nof the composition is important for the occurrence of reversal of FM at lower RKKY \nexchange, and the effect of Tc of plays an important role in controlling the dynamics \ndetermining the switching time of the different types of exchange coupled \nheterostructures. \n4. Conclusion \nWe have simulated the time -resolved and element -specific magnetization dynamics \nin an exchange -coupled FM -FEM system using a modified M3TM. Ult rafast switching \nof the FM (Co/Pt) isn’t possible if it is decoupled from the FEM sub -lattice, as it needs \nadditional angular momentum transfer from the optically switched FEM sub -lattice. \nFor a lower Tc, both the FM and FEM sub -lattice switches at the sam e threshold energy, \nhowever, with increasing Tc, either larger optical energy or larger RKKY coupling \nstrength is required to switch the FM layer. Our analysis depicts the need for i) the \nelectron/lattice temperature of the Co/Pt layer to go below Tc after optical excitation \nand ii) the RKKY coupling strength and available magnetization from the rare -earth \n(Gd) element of the switched FEM alloy for the efficient angular momentum transfer \nto obtain ultrafast magnetization switching of the FM. \n \nWe have reprod uced the experimentally observed ultrafast switching of the Co/Pt \nlayer in at ∼7 ps by Gorchon et al. [22]. We found that the switching can be made much \nfaster ( ∼3.8 ps) by increasing the Tc of the FM to 580 K. We have also explored the \neffect of FEM compo sition on the magnetization dynamics. While CoGd switches at a \nlarger threshold energy due to its high Tc, it displays a narrower transient ferromagnetic -\nlike state. However, for FeGd, the transient ferromagnetic -like state lasts longer due to \nits weak int ra-sublattice exchange interaction. On the other hand, the switching \ndynamics of the FM layer only get minimally affected due to the composition of the \nFEM layer , and that too can be attributed to the difference in the threshold optical \nenergy. \n 13 We can con clude that the Curie temperature of the FM and RKKY exchange \ncoupling strength are the two most important factors in tailoring the switching timescale \nof the FM in such exchange -coupled structures, while the composition of the FEM alloy \ndetermines the requ ired strength of the RKKY interaction for magnetization switching. \nWe believe our results will help in refining the current understanding of the HI -AOS \nphenomenon in exchange coupled structures and lead the spintronics community \ntowards achieving a faster and more energy -efficient magnetization switching. \nAcknowledgements \nThis work was primarily supported by the Director, Office of Science, Office of \nBasic Energy Sciences, Materials Sciences and Engineering Division, of the U.S. \nDepartment of Energy under Contract No. DE -AC02 -05-CH11231 within the \nNonequilibrium Magnetic Materials Program (MSMAG) (theoretical analysis). 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" }, { "title": "1312.5809v1.Effects_of_the_spin_orbital_coupling_on_the_vacancy_induced_magnetism_on_the_honeycomb_lattice.pdf", "content": "arXiv:1312.5809v1 [cond-mat.str-el] 20 Dec 2013Effects of the spin-orbital coupling on the vacancy-induced magnetism on the\nhoneycomb lattice\nWeng-Hang Leong, Shun-Li Yu, and Jian-Xin Li\nNational Laboratory of Solid State Microstructure and Depa rtment of Physics, Nanjing University, Nanjing 210093, Chi na\n(Dated: October 7, 2018)\nThe local magnetism induced by vacancies in the presence of t he spin-orbital interaction is in-\nvestigated based on the half-filled Kane-Mele-Hubbard mode l on the honeycomb lattice. Using the\nself-consistent mean-field theory, we find that the spin-orb ital coupling will enhance the localization\nof the spin moments near a single vacancy. We further study th e magnetic structures along the\nzigzag edges formed by a chain of vacancies. We find that the sp in-orbital coupling tends to sup-\npress the counter-polarized ferrimagnetic order on the upp er and lower edges, because of the open\nof the spin-orbital gap. As a result, in the case of the balanc e number of sublattices, it will suppress\ncompletely this kind of ferrimagnetic order. But, for the im balance case, a ferrimagnetic order along\nboth edges exists because additional zero modes will not be a ffected by the spin-orbital coupling.\nI. INTRODUCTION\nGraphene and related nanostructured materials have\nattractedmuchinterestin solidstatephysicsrecentlydue\nto their bidimensional character and a host of peculiar\nproperties1. Among them, the investigation of the mag-\nnetic properties in graphene is one of the fascinating top-\nics, as no dandfelements are necessaryin the induction\nofmagnetism in comparisonwith the usual magnetic ma-\nterials. Theoretical predictions and experimental investi-\ngations have revealed that a nonmagnetic defect such as\nanimpurity oravacancycan induce the non-triviallocal-\nized magnetism2–6. Similarly, a random arrangement of\na large number of vacancies which are generated by the\nhigh-dose exposure of graphene to strong electron irradi-\nation7can also induce magnetism theoretically8. These\nstudies not only have the fundamental importance, but\nalso open a door for the possibility of application in new\ntechnologies for designing nanoscale magnetic and spin\nelectronic devices.\nOn the other hand, the topological insulating elec-\ntronic phases driven by the spin-orbital (SO) interaction\nhave also attracted much interest recently. The Kane-\nMele model for the topological band insulator is defined\non the honeycomb lattice9,10which is the same lattice\nstructure as graphene. Possible realization of an appre-\nciable SO coupling in the honeycomb lattice includes the\ncold fermionic atoms trapped in an extraordinary optical\nlattice11, the transition-metal oxide Na 2IrO312and the\nternaty compounds such as LiAuSe and KHgSb13. Topo-\nlogical band insulator has a nontrivial topological order\nandexhibitsabulkenergygapwithgapless,helicalstates\nat the edge14–16. These edge states are protected by the\ntime reversal symmetry and are robust with respect to\nthe time-reversal symmetric perturbations, such as non-\nmagnetic impurities. It is shown that a vacancy, acting\nas a minimal circular inner edge, will induce novel time-\nreversal invariant bound states in the band gap of the\ntopological insulator17–19. Theoretically, it is also shown\nthat the SO coupling suppresses the edge magnetism in-\nduced in the zigzag ribbon of the honeycomb lattice inthe presence of electron-electron interactions20. Thus, it\nis expected that the SO coupling would also affect the\nlocal magnetism in the bulk induced by vacancies.\nIn this paper, we study theoretically the effects of the\nSO coupling on the local magnetism induced by a sin-\ngle and a multi-site vacancy on the honeycomb lattice,\nbased on the Kane-Mele-Hubbard model where both the\nSO coupling and the Hubbard interaction between elec-\ntrons are taken into consideration. This model has been\nextensively studied to explore the effect of the strong\ncorrelation on the topological insulators21–27. Making\nuse of the self-consistent mean field approximation, we\ncalculate the local spin moments and their distribution\naround the vacancies. For a single vacancy, we find that\nthe main effect of the SO coupling is to localize the spin\nmoments to be near the vacancy, so that it will enhance\nthe local spin moments. For a large stripe vacancy by\ntaking out a chain of sites from the lattice, we find that\nthe SO coupling tends to suppress the counter-polarized\nferrimagnetic order induced along the zigzag edges, be-\ncause of the open of the SO gap. As a result, in the case\nof the balance number of sublattices (with even number\nof vacancies), the SO coupling will suppress completely\nthe counter-polarized ferrimagnetic order along the up-\nper and lower edges. While, in the case of the imbalance\nnumber of sublattices (with odd number of vacancies), a\nferrimagneticorder along both edges exists because addi-\ntionalzeromodeswill notbe affectedbythe SOcoupling.\nWewillintroducethemodelandthemethodoftheself-\nconsistent mean-field approximation in Sec.II. In Sec.III\nand IV, we present the results for a single vacancy and a\nmulti-site vacancy, respectively. Finally, a briefsummary\nwill be given in Sec.V.\nII. MODELS AND COMPUTATIONAL\nMETHODS\nWe start from the Kane-Mele model9, in which the\nintrinsic SO coupling with a coupling constant λis in-2\ncluded.\nH0=−t/summationdisplay\n/angbracketleftij/angbracketright,σc†\niσcjσ+iλ/summationdisplay\n/angbracketleft/angbracketleftij/angbracketright/angbracketrightσσ′vijσz\nσσ′c†\niσcjσ′,(1)\nwherec†\niσ(cjσ) is the creation(annihilation) operator of\nthe electron with spin σon the lattice site i,/angbracketleftij/angbracketrightrep-\nresents the pairs of the nearest neighbor sites (the hop-\nping ist) and/angbracketleft/angbracketleftij/angbracketright/angbracketrightthose of the next-nearest neighbors.\nvij= +1(−1) if the electron makes a left(right) turn to\nget to the second bond. The size of our system is consid-\nered to be finite with periodic boundary condition. So,\nthe position of each lattice site can be described specif-\nically by i= Γ(m,n), representing that the lattice site\niis in the mth column and the nth row, and Γ = A,B\nthe sublattice labels. The number of the unit cells is\ndenoted by Nc=L2, therefore the total number of the\nlattice sites is Nl= 2L2. To consider the correlation be-\ntween electrons, we will include the Hubbard term in the\nHamiltonian, which is given by HI,\nHI=U/summationdisplay\niˆni↑ˆni↓, (2)\nwhere ˆniσ=c†\niσciσ. When vacancies are introduced, the\nhoppings between the vacancy and the nearest neigh-\nbors and the on-site interaction on that vacancy are sub-\ntracted from the overall Hamiltonian. Hence the corre-\nsponding number of the lattice sites is Nl= 2L2−Nv,\nwhereNvis the number of vacancies. The total number\nof electrons Neis fixed to be at the half-filling ( Ne=Nl).\nThe Hubbard interaction term is treated with the self-\nconsistent mean field approximation, so that we will ob-\ntain an effective single-particle Hamiltonian where the\nelectrons interact with a spin-dependent potential,\nHI≃U/summationdisplay\ni,σ/angbracketleftˆni−σ/angbracketrightˆniσ−U/summationdisplay\ni/angbracketleftˆni↑/angbracketright/angbracketleftˆni↓/angbracketright.(3)\nAnd the overall mean field Hamiltonian Hmfis then\ngiven by,\nHmf=U/summationdisplay\niσ/angbracketleftˆni−σ/angbracketrightˆniσ+H0. (4)\nAfter diagonalizing the Hamiltonian Hmf, we can de-\ntermine the occupation number /angbracketleftˆni−σ/angbracketrightat each site with\ndifferent spins using the eigenvectors of Hmf, and this\nprocess is carried out iteratively until a required accu-\nracy is reached. Then the magnetic moment of each site\nmi=/angbracketleftˆni↑−ˆni↓/angbracketrightcan be calculated. We note that a\ncollinear magnetic texture is assumed in our system, as\nused before for the investigations of Kane-Mele-Hubbard\nmodel20,21. We have checked the results with the non-\ncollinear magnetic texture and found that the collinear\nmagnetic texture is favored.\nIII. MAGNETISM WITH ONE VACANCY\nThe calculation is carried out on the lattice with Nc=\n14×14 unit cells in which a single vacancy is introduced/s49 /s50 /s51 /s52 /s53 /s54/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54\n/s32 /s61/s48/s46/s48/s48\n/s32 /s61/s48/s46/s48/s53\n/s32 /s61/s48/s46/s49/s48/s32/s109\n/s105\n/s32\n/s32/s114/s47/s97/s40/s99/s41/s40/s98/s41\n/s32/s48/s46/s48/s48\n/s48/s46/s48/s52\n/s48/s46/s48/s55\n/s48/s46/s49/s49\n/s48/s46/s49/s52\n/s48/s46/s49/s56/s32/s40/s97/s41\nFIG. 1: (color online). (a) and (b): Distribution of the spin\nmoments mion lattice sites around asingle vacancy at A(7,7)\nwithU= 1.0t, in which (a) corresponds to the SO coupling\nconstant λ= 0.0 and (b) λ= 0.1t. The area and color of the\nhollow circles represent the magnitude of the spin moments.\n(c)mion theBsublattice as a function of the distance r\naway from the vacancy. The unit ais the distance between\nthe nearest sites.\non the site A(7,7). Figure 1 displays the distribution of\nthe magnetic moment when the Hubbard interaction is\ntaken to be U= 1.0t, in which the size and the color\nof the circle on each lattice site denote the magnitude\nof the local spin moment. From Fig.1(a) where the SO\ncouplingisturnedoff, onecanseethatlocalizedmagnetic\nmoments are induced around the vacancy in the presence\nof a finite Hubbard interaction U. This is in agreement\nwith the prediction of the Lieb theorem28regarding the\ntotal spin Sof the exact ground state of the Hubbard\nmodel on bipartite lattices. It states that the total spin\nSis given by the sublattice imbalance 2 S=|NA−NB|,\nwithNAandNBthe number of atoms belonging to each\nsublattice. With the introducing of a single vacancy on\ntheAsublattice, an imbalance NB−NA= 1 appears\nand a magnetic structure near the vacancy with the total\nspinS= 1/2 will form. Similar results have also been\nobtained in recent studies in graphene2–4.\nIn the presence of the SO coupling, the magnitude of\nthe magnetic moments around the vacancy increases, as\nshown in Fig. 1(b) for λ= 0.1t. At the meantime, if\nwe check the distribution of the magnetic moments, as\nshown in Fig.1(c) where the magnitude of the magnetic\nmoments on sublattice Bas a function of the distance r\nawayfromthe vacancyis presented, one will find that the\nmagnetic moments are more localized with the increase\nof the SO coupling. These features demonstrate that the\nSOcoupling will enhancethe magneticmoments nearthe\nvacancy notably.\nIn order to show the emergence of the magnetism in-\nduced by the vacancy in more detail, we calculate the3\nspin resolved local density of state(LDOS) as defined by,\nDσ(ǫ) = Σn,i|un\ni,σ|2δ(ǫ−ǫn), (5)\nwhereiruns over the lattice sites surrounding the va-\ncancy up to the third-nearest neighbors, as those linked\nby the green line in Fig.1(a) and (b). un\ni,σis the single-\nparticle amplitude on the ith site with spin σand the\ncorresponding eigenvalue is ǫn. The Delta function in\nEq.(2)isreplacedbytheLorentzianfunction forplotting.\nThe results for the LDOS are presented in Fig. 2(a)-(h)\nfor different Hubbard interaction Uand SO interaction\nλ. The red and blue lines represent the LDOS for the\nspin up and spin down components respectively, and the\ndash lines show the LDOS away from the vacancy for\na comparison. In the case of U=λ= 0.0 as shown\nin Fig. 2(a), the LDOS shows a V-shape linear behavior\nnear the Fermi level for those lattice sites far away from\nthe vacancy (denoted by the dashed line) which is the\nconsequence of the linear dispersion relation of the elec-\ntrons, the so-called Dirac fermions. For those around the\nvacancy, a peak at the Fermi level emerges as shown by\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s40/s101/s41/s76/s68/s79/s83\n/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50\n/s40/s103/s41/s76/s68/s79/s83\n/s69/s47/s124/s116/s124/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s40/s104/s41\n/s69/s47/s124/s116/s124/s40/s102/s41/s40/s100/s41\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s40/s99/s41/s76/s68/s79/s83/s40/s98/s41\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s32/s76/s68/s79/s83/s40/s97/s41\nFIG. 2: (color online). LDOS for λ= 0.0 [left column, includ-\ning (a),(c),(e),(g)] and for λ= 0.1t[right column, including\n(b),(d),(f),(h)], where the Hubbard interaction U= 0.0 for\n(a) and (b), U= 1.6tfor (c) and (d), U= 2.6tfor (e) and\n(f), and U= 3.6tfor (g) and (h), respectively. LDOS for\ndifferent spins is resolved, those with the spin up are denote d\nby the blue lines and the spin down the red lines. The grey\ndash lines represent the LDOS on the lattice site away from\nthe vacancy./s49 /s50 /s51 /s52/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s40/s98/s41\n/s77\n/s32 /s108/s111/s99\n/s85/s32/s47/s32/s116/s32 /s61/s48/s46/s48/s48\n/s32 /s61/s48/s46/s48/s52\n/s32 /s61/s48/s46/s49/s48\n/s32 /s61/s48/s46/s49/s54\n/s48/s46/s48/s48 /s48/s46/s48/s52 /s48/s46/s48/s56 /s48/s46/s49/s50 /s48/s46/s49/s54/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48\n/s32/s32\n/s32/s85/s61/s49/s46/s48\n/s32/s85/s61/s50/s46/s50\n/s32/s85/s61/s50/s46/s52\n/s32/s85/s61/s50/s46/s56/s77\n/s32 /s108/s111/s99\n/s32/s47/s32/s116/s40/s97/s41\nFIG. 3: (color online). Local moments Mloc(see text) are\nshownasafunctionoftheSOcoupling λfordifferentHubbard\ninteraction U(a) and of Ufor different λ(b).\nthe solid line, which corresponds to the localized states\ninduced by the vacancy29. After turning on the SO cou-\npling, such as that for λ= 0.1t[see Fig.2(b)], we can see\nthat an energy gap opens for those lattice sites far away\nfrom the vacancy9,10, so that now a U-shape LDOS near\nthe Fermi level occurs. In this way, the mid-gap peak is\nenhanced noticeably because the decay rate of the local-\nized states into the continuum is reduced largely due to\nthe open of the energy gap. This will lead to the increase\nin the spectral weight of the localized states around the\nvacancy. However, for both cases, one will find that the\nLDOS for the spin up and spin down components degen-\nerates, so that the system will not show magnetism as a\nwhole without the Hubbard interaction.\nThe effect of a finite Hubbard interaction Uis to split\nthe spin degenerate LDOS, so that two peaks occur cor-\nresponding to different spins, as shown in Fig. 2(c)-(f).\nConsequently, the localized spin up and down moments\nwill not cancel out in this case, and a net magnetism\naround the vacancy is induced.\nThe magnetism may be quantified by the local mo-\nmentMloc=/summationtext\nimi, where the sum runs over the lat-\ntice sites surroundingthe vacancy up to the third-nearest\nneighbors as used above in the calculation for the LDOS.\nThe results are presented in Fig. 3(a) and (b) for dif-\nferentUandλ, respectively. The local moment Mloc\nshows a monotonic increase with the SO coupling λ, so\nit reinforces our observation that the local magnetism\nis enhanced by the SO coupling as shown in Fig.1. On\nthe other hand, Mlocshows a nonmonotonic dependence\non the Hubbard interaction U, namely it increases with\nUfirstly and then decreases with a further increase of\nUafter a critical value Uc. As discussed above, the lo-\ncal magnetism is determined by the spin-split localized\nstates induced by the vacancy, and it is the Hubbard in-\nteraction Uto split the spin-degenerate states. Because\nthe open of the gap due to the SO coupling will decrease\nthe decay rate of the localized states into the continuum,\nso it will enhance the spectral weight of the localized\nstates[see also Fig. 2], consequently the localized mag-\nnetism. The splitting between the two localized states\nwith different spins is proportional to U, so the two split4\nlocalized states will situate in the SO gap for a small\nU[Fig. 2(c)-(f)]. However, when U > U cthe splitting\nwill be larger than the SO gap, and it pushes the local-\nizedstatestomergeintothecontinuum[Fig.2(g)and(h)],\nso the local magnetism will decrease.\nIV. THE CASE OF MULTI-SITE VACANCY\nThe multi-site vacancy can be formed by removing the\nsites continuously. Here, we consider a large stripe va-\ncancy by taking out a chain of sites from the lattice as\nilluminated in Fig. 4. In this way, the stripe vacancy\nconsists of one upper and one lower zigzag edges. As\nclarified by the Lieb theorem28, the sublattice imbalance\nbetween the number of atoms belonging to different sub-\nlattices will have significant effect on the magnetism. For\nthe stripe vacancy considered here, the imbalance is ex-\npressed by the parity of the number of vacancies, where\nthe number is even ( NA=NB) in Fig. 4(a) and (b), and\nodd (NA/negationslash=NB) in Fig. 4(c) and (d), thus the total spin\nof the system is S= 0 and 1 /2 respectively.\nInthecaseofevennumberofvacancies,aferrimagnetic\nspin order emerges on both the upper and lower zigzag\nedges around the stripe vacancies when there is no SO\ncoupling, as shown in Fig. 4(a). The ferrimagnetic ar-\nrangement and the magnitude of the spin moments on\nthese two edges are symmetric, but they are counter-\npolarized, so they cancel out exactly and the whole sys-\ntem will not show magnetism. This is consistent with\nthe Lieb theorem28. The ferrimagnetic order on a suffi-\nciently long zigzag edge around the stripe vacancies here\nissimilartothespin orderformedattheouteredgeofthe\nzigzagribbon30–33and the graphenenanoisland34. In the\ncase of odd number of vacancies, a similar ferrimagnetic\nspin order is also induced with a slightly large magnitude\n[Fig. 4(c)]. Interestingly, this ferrimagnetic order occurs\nonly on the upper zigzag edge, not on the lower edge.\nThis phenomenon is ascribed to the presence of an extra\nspin when a sublattice imbalance NA/negationslash=NBexists, as\n/s40/s97/s41\n/s32\n/s32\n/s32/s32 /s32/s32\n/s40/s99/s41/s40/s98/s41\n/s32\n/s32/s45/s48/s46/s49/s56/s45/s48/s46/s49/s49/s45/s48/s46/s48/s52/s48/s46/s48/s52/s48/s46/s49/s49/s48/s46/s49/s56\n/s40/s100/s41\n/s32\n/s32/s45/s48/s46/s50/s54/s45/s48/s46/s49/s54/s45/s48/s46/s48/s53/s48/s46/s48/s53/s48/s46/s49/s54/s48/s46/s50/s54\n/s66/s65/s66/s65\nFIG. 4: (color online). Distribution of the spin moments mi\non the lattice sites surrounding the vacancies for U= 1.0tand\nL= 14. A cluster of vacancies is formed with the number of\nmissing sites for (a), (b) Nv= 8 and (c), (d) Nv= 7. SO\ncoupling is set to be λ= 0.0 for (a), (c) and λ= 0.1tfor\n(b), (d). The area and color of hollow circles represent the\nmagnitude of the moments./s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s40/s98/s41\n/s32/s78 /s118/s61/s51\n/s32/s78 /s118/s61/s53\n/s32/s78 /s118/s61/s55\n/s32/s78 /s118/s61/s57\n/s32/s78 /s118/s61/s50\n/s32/s78 /s118/s61/s52\n/s32/s78 /s118/s61/s54\n/s32/s78 /s118/s61/s56\n/s32/s78 /s118/s61/s49/s48/s77\n/s101\n/s47/s32/s116/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48\n/s32/s78 /s118/s61/s51\n/s32/s78 /s118/s61/s53\n/s32/s78 /s118/s61/s55\n/s32/s78 /s118/s61/s57\n/s32/s32/s77\n/s108/s111/s99\n/s47/s32/s116/s40/s97/s41\nFIG. 5: (color online). (a)Local moments Mlocare plotted as\na function of λwhileU= 1.0tandL= 14. The function in\ndifferent size of vacancy is distinguished by different color and\nshape of points. The cases of even Nvare not plotted as local\nmoments are always zero obeying Lieb theorem28. (b) The\nfunction of edge moments Meversusλare given in different\nNv.\ndescribed by the Lieb theorem28.\nAfter turning on the SO coupling, such as for λ= 0.1t,\nthe ferrimagnetic spin order on both the upper and lower\nzigzag edges around the stripe vacancies disappears com-\npletely in the case of even number of vacancies[Fig. 4(b)].\nHowever, the effect of the SO coupling on local mag-\nnetism is quite different for the case of an odd number\nof vacancies. Here, a ferrimagnetic spin order similar\nto that on the upper edge emerges on the lower edge,\nthough the magnitude of the individual spin moment is\nreduced[Fig. 4(d)]. To show variation of the total mag-\nnetism, we plot the quantity Mlocas a function of the\nSO coupling λin Fig. 5(a), here Mlocis the sum of the\nspin moments on the sites which are on the zigzag edges\naround the vacancies. Since Mlocis always zero in the\ncase of even Nv, it is not plotted here. With an odd\nNv, the local moment Mlocincreases with the increase\nofλ, which shows a similar behavior as that in the case\nof a single vacancy. This indicates that the total local\nmagnetism shown in Fig. 4(d) is in fact enhanced with\nthe introduction of the SO coupling and approaches the\nsaturation value 1 finally. From Fig. 5(a), one can also\nfind that Mlocincreases with the increase of the number\nof vacancies Nv. This suggests that the SO coupling will\nlocalize the induced spin moments to those lattice sites\nwhich are neighboring the vacancies.\nTo quantify the variation of the spin moments with\nλon the upper zigzag edge, we also present Meas a\nfunction of λin Fig. 5(b), here Meis the sum of the\nspin moments only on the sites on the upper zigzag edge.\nLet us consider firstly the case of even number of Nv,\nfor a small number of even vacancies, Meis always zero.\nUp toNv≥8, a finite Meoccurs and it increases with\nNvby the formation of the zigzag edges. However, Me\ndrops rapidly to zero after turning on the SO coupling.\nThese results quantify the physical picture derived from\nFig. 4(a) and (b). Now let us turn to the case of odd\nnumber of Nv. Without the SO coupling, Mealso shows5\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s40/s98/s41\n/s32/s101/s110/s101/s114/s103/s121/s47/s116\n/s32/s109/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s40/s97/s41/s32/s32/s101/s110/s101/s114/s103/s121/s47/s116\n/s32/s109\nFIG. 6: (color online). The single-particle energy levels l a-\nbeled with m(see text) near the Fermi level of the non-\ninteracting systems for (a) Nv= 8 and (b) Nv= 7. SO cou-\npling is set to be λ= 0.0 for the red circles and λ= 0.1tfor\nthe blue squares.\nan increase with Nv. With the introduction of the SO\ncoupling, Meshows a decrease with λand saturates to\nnear one half of Mloc.\nIn fact, we can make an analogy between the stripe\nvacancy and the graphene ribbon with zigzag edges. A\nremarkable feature of the graphene ribbon with zigzag\nedges is that it has a flat band localized on the zigzag\nedge1. An important effect of this flat band is that a\ncounter-polarized ferromagnetic order along the upper\nand lower edges will be induced when the Hubbard in-\nteraction between electrons is included30–33. In view of\nthis, we plot the single-particle spectra for the systems\nwith the stripe vacancy without the Hubbard interaction\nin Fig.6(a) and (b) for Nν= 8 and Nν= 7, respectively.\nEach energy level is labeled with m=n−Ne−1/2 in\norder to indicate that the energy level with m <0 is\noccupied by electron. For the systems without SO cou-\npling, we find that there are four near-degeneracy local-\nizedstates[redcirclesindicatedbyarrowsinFig.6(a)and\n(b)] which is near the Fermi level for both Nν= 8 and\nNν= 7. These states will have the same effect as the flat\nband in the zigzag ribbon when a suitable Hubbard U\nis turned on. So, a counter-polarized ferrimagnetic order\nas shown in Fig.4(a) will emerge. However, we note that\nthere are two additional zero modes for Nν= 7 relative\ntoNν= 8, due to the imbalance between the sublattices(NA> NB). Thesezeromodeswillinduceextraspinmo-\nments on both edges, which counteract the antiparallel\nmoments on the lower edge. Thus, in the case of Nν= 7,\nonly the ferrimagnetic order on the upper edge appears.\nAfter turning on the SO coupling, those localized states\n[blue squares indicated by arrows in Fig.6(a) and (b)] are\npushed away from the Fermi level due to the open of the\nSO gap. Thus, as shown in Fig.4(b), a small Hubbard U\nis not enough to induce the counter-polarized ferrimag-\nnetic order on the upper and lower edges. However, the\nzero modes originating from the imbalance of sublattices\nare not affected by the SO coupling [Fig.6(b)]. So, the\nadditional ferrimagnetic order on both edges induced by\nthese zero modes will remain for Nν= 7.\nV. CONCLUSION\nIn a summary, we have studied the local magnetism\ninduced by vacancies on the honeycomb lattice based on\nthe Kane-Mele-Hubbard model. It is shown that the SO\ncoupling tends to localize and consequently enhances the\nlocal magnetic moments near a single vacancy. Further-\nmore, along the zigzag edges formed by a chain of va-\ncancies, the SO coupling will suppress completely the\ncounter-polarized ferrimagnetic order along the edges.\nTherefore, the system will not show any local magnetism\nin the case ofeven number ofvacancies. For an odd num-\nber of vacancies, a ferrimagnetic order along both edges\nexists and the total magnetic moments along both edges\nwill increase.\nAcknowledgments\nThis work was supported by the National Natural\nScience Foundation of China (Grant Nos. 91021001,\n11190023 and 11204125) and the Ministry of Science\nand Technology of China (973 Project grant numbers\n2011CB922101 and 2011CB605902).\n1A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. 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Lett.\n99, 177204 (2007)." }, { "title": "1201.2144v1.Direct_observation_of_magnetic_phase_coexistence_and_magnetization_reversal_in_a_Gd___0_67__Ca___0_33__MnO___3___thin_film.pdf", "content": "arXiv:1201.2144v1 [cond-mat.str-el] 10 Jan 2012Direct observation of magnetic phase coexistence and magne tization reversal in a\nGd0.67Ca0.33MnO 3thin film\nJeehoon Kim, Nestor Haberkorn, Leonardo Civale, Paul Dowden,\nAvadh Saxena, J. D. Thompson, and Roman Movshovich\nLos Alamos National Laboratory, Los Alamos, NM 87545∗\nEvgeny Nazaretski\nBrookhaven National Laboratory, Upton, NY 11973\nWe have investigated the ferrimagnetic domain structure in a Gd0.67Ca0.33MnO3thin film using\nmagnetic force microscopy. We observe clear signs of phase s eparation, with magnetic islands\nembedded in a non-magnetic matrix. We also directly visuali ze the reversal of magnetization of\nferrimagnetic domains as a function of temperature and attr ibute it to a change in the balance of\nmagnetization of anti-aligned Mn and Gd sublattices.\nMixed-valent perovskite manganites A 1−xBxMnO3(A\nand B are rare-earth and divalent alkaline elements, re-\nspectively), such as La-based manganites, have been\nstudied extensively in recent years.[1–4] These materials\nexhibit a colossal magnetoresistance (CMR) effect for a\nwide rangeof dopingscentered at x = 1/3where the dou-\nble exchange mechanism is maximized.[5] The resulting\ncombinationoffascinatingphysicalphenomena anda po-\ntential fortechnologicalapplicationshasbeen the driving\nforce in sustaining high interest in these compounds.[1–4]\nElectronic inhomogeneity and phase separation are ubiq-\nuitous in doped manganites, and the resulting CMR ef-\nfect is driven by percolative transport.[6] CMR manifests\nitself by a dramatic drop in resistivity and a discon-\ntinuous decrease in the equilibrium Mn-O bond length\nat a first order phase transition in an applied magnetic\nfield.[7, 8] Their complex electronic structure and a vari-\nety of competing interactions lead to a rich ensemble of\nground states in this family of compounds.\nIn this Letter we report a low temperature\nmagnetic force microscopy (MFM) investigation of\nGd0.67Ca0.33MnO3(GCMO), a compound with an in-\nsulating ferrimagnetic (FIM) ground state. Compared\nto other ferromagnetic (FM) perovskite manganites,\nGCMO exhibits arelativelylowCurie temperature( TC),\nand its small structural tolerance factor t<0.89[9, 10]\nleadstoarobustinsulating groundstate. Magneticprop-\nerties of the system reflect those of the two sublattices of\nMn and Gd ions (see below). The different temperature\ndependence of magnetization of each of the two sublat-\ntices results in a changeofsign ofthe totalmagnetization\nas a function of temperature at a characteristic compen-\nsation temperature Tcomp, where the Mn and Gd sub-\nlattices have magnetic moments of the same magnitude\nand opposite direction.[9–11] A small tolerance factor,\na structural distortion, and the antiferromagnetic inter-\naction between Gd and Mn sublattices yield remarkable\nproperties, such as a giant magnetostrictive effect in a\n∗Electronic mail: jeehoon@lanl.govwide temperature range[12] and inhomogeneousFIM-like\nbehavior with an exchange bias effect close to Tcomp.[13]\nLow values of the saturation magnetization ( MS) sug-\ngest phase coexistence.[12, 13] MFM studies described\nbelow, with the wide range of field and temperature em-\nployed, allow us to visualize the magnetic structure of\nGCMO and provide direct evidence of phase separation.\nThe magnetization reversal at Tcompofeach individual\ndomain provides strong support for the scenario of anti-\naligned Mn and Gd sublattices with the Gd (Mn) mag-\nnetization dominating below (above) Tcomp.\nThe Gd 0.67Ca0.33MnO3thin film was grownby pulsed-\nlaser deposition (PLD) on a SrTiO 3(100) substrate from\nacommercialtargetwith the samechemicalcomposition.\nThe substrate temperature was kept at 790◦C in an oxy-\ngen atmosphere at a pressure of 200 mTorr. After depo-\nsition, the O 2pressure was increased to 200 Torr, and\nthe temperature was decreased to room temperature at\na rate of 30◦C/min. Bulk GCMO is an orthorhombic\nperovskite (Pbnm (no. 62); a= 5.52˚A,b= 5.34˚A,\nc= 7.50˚A).[13, 14] The GCMO film was examined by\nx-ray diffractometry, and was found to be single phase\nwith a (00 l) orientation. The lattice parameters ( a=\n5.55(2)˚A,b= 5.36(2) ˚A, andc= 7.50(1) ˚A) were de-\ntermined using (00 l), (200), and (020) reflections from\na four-circle diffractometer/goniometer. No additional\npeaks due to secondaryphasesor different crystallineori-\nentationswereobserved. Therockingcurvewidtharound\nthe(004)peakofthefilmwas ∼0.27◦. Thefilmthickness\nof 45 nm was determined by a low-anglex-rayreflectivity\nmeasurement with an angular resolution of 0.005◦.\nA Quantum Design SQUID magnetometer was used\nfor measurements of the global magnetization with the\nmagnetic field oriented perpendicular to the film surface.\nAlllocalizedMFM measurementsdescribedinthisLetter\nwere performed in a home-built low-temperature MFM\napparatus.[15] MFM images were taken in a frequency-\nmodulated mode, with the tip-lift height of 100 nm\nabove the sample surface. High resolution SSS-QMFMR\ncantilevers,[16] magnetized along the tip axis in a field\nof 3 T, were used for MFM measurements; the external\nmagnetic field was always applied perpendicular to the2\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s50/s48/s52/s48/s54/s48/s56/s48/s48/s72\n/s99/s32/s91/s84/s93\n/s84 /s32 /s91/s75 /s93/s84\n/s99/s111/s109/s112\n/s77/s110/s32/s71/s100\n/s40/s98/s41/s32/s48/s46/s48/s48/s49/s32/s84\n/s32/s48/s46/s48/s49/s32/s84\n/s32/s48/s46/s49/s32/s84\n/s32/s48/s46/s53/s32/s84\n/s32/s49/s32/s84/s77/s32 /s91 /s101/s109/s117/s47/s99/s109/s51\n/s93 \n/s40/s97/s41\nFIG. 1. (Color online) (a) Field-cooled M(T) curves in differ-\nent magnetic fields ( H). (b) Coercive field ( Hc) as a function\nof temperature obtained from magnetic hysteresis loops.\nfilm surface and parallel to the MFM tip.\nFig. 1(a) shows the field-cooled (FC) magnetization M\nas a function oftemperature at different values ofapplied\nmagnetic field H. The temperature dependence of mag-\nnetization was discussed previously by Snyder et al.[10]\nGCMO undergoes a phase transition from paramagnetic\ninsulating to ferromagnetic insulating states, associated\nwith the ferromagnetic ordering of Mn cations, at TC≈\n80 K. The local field due to FM order in the Mn sub-\nlattice and the negative f-dexchange interaction on the\nGd spins force the moments on the Gd sublattice to be\nanti-aligned to those in the Mn sublattice. The Mn sub-\nlatticedominatesthemagnetizationathightemperature,\nbut the absolute magnitude of magnetization of the Gd\nsublattice grows faster when the temperature is reduced.\nConsequently, the total magnetization Mreaches a max-\nimum value close to 50 K (see Fig. 1), starts to decrease\nwith decreasing temperature, and goes toward zero at\nTcomp≈15 K in low fields ( Tcompdepends on H), where\nmagnetizations of the Mn and Gd sublattices compen-\nsate each other. Below Tcompthe local magnetization of\nGd is larger than that of Mn, |MGd|>|MMn|, and the\nsign of the total magnetization is determined by the di-\nrection of magnetization of the Gd sublattice. When the\napplied magnetic field His below the coercive field Hcof\nthe system at Tcomp, the magnetization of the Gd sublat-\ntice is locked in a direction opposite to the applied field,\nand the total magnetizationis negativebelow Tcomp. For\n10 K (b) \n(d) \n4 K (a) Δf (Hz) \n1\n15 K (c) \n0Correlation (a.u.) 0\n-80 T< Tcomp (e) \nnon-magnetic Matrix : Gd \n: Mn T> Tcomp (f) : Gd \n: Mn \nnon-magnetic Matrix \nFIG. 2. (Color online) (a)-(c) MFM images acquired at dif-\nferent temperatures. Solid and dashed circles represent th e\nsame sample area. (d) Cross-correlation map between images\nshown in panels (a) and (c). The large negative value at the\ncenterof the mapsignifies the anticorrelation between imag es.\n(e) and (f) Schematical illustration of the temperature evo -\nlution of phase-separated magnetic regions above and below\nTcomp≈12 K in 1 mT. The field of view in the images ((a)-\n(d)) is 6 µm×6µm. Features on the left side are broader\nthan those on the right side because the scan plane is not\nperfectly parallel to the sample surface.\nH > H cthe magnetization is reversed immediately be-\nlowTcomp, producing a characteristic sharp kink and a\nV-shape in the data. This sharp reversal of the change\nin magnetization (from decreasing to increasing with de-\ncreasing temperature) is facilitated by a strong decrease\nofHcatTcomp, as shown in Fig. 1(b), which is deter-\nmined on the basis of an analysis of full hysteresis curves\nat different temperatures (data not shown). The positive\noffsetofMatthe kink at0.5Tand 1Tin Fig.1(a) isdue\nto a paramagnetic background. All magnetic transition\ntemperatures observed in the film are in good agreement\nwith the values previously reported for bulk polycrystal\nand single crystal samples.[10, 13, 14] The data at 0.1\nT has a clear kink as it crosses M= 0 and Tcomp, indi-\ncating that some small number of the magnetic domains\nflip their orientationat Tcomp. This is consistentwith the\nbulk measurementsof Hc≈0.1T atTcomp, and points to\ncoercivefieldin thissystembeingalocalproperty, proba-\nbly dependent on the magnetic domain’s size, shape, and\nenvironment.\nThe MFM images depicted in Figs. 2(a)-(c) were taken\nsequentially at 4 K ( TTcomp), respectively, in a magnetic field of 1 mT\n(belowHc) applied above TC(field-cooled data). The\ndashed circles in Figs. 2(a)-(c) show the same sample re-\ngion (thermal drifts are negligible for images taken below\n15 K, see below). Regions of a non-zero magnetic sig-\nnal, either blue or red, change color as the temperature3\n(e) \n(c) 0.5 T \n1 T (d) \n (f) \n0 T (a) \n 0.1 T (b) Δf (Hz) 1\n0Correlation (a.u.) 0220 \nFIG. 3. (Color online) (a)-(d) Field-cooled MFM images\ntaken at 4 K in different magnetic fields. (e) and (f) Cross-\ncorrelation maps between (a)-(b), and (b)-(c), respective ly;\nno correlation is observed. The field of view in the images\n((a)-(f)) is 6 µm×6µm. Dashed circles correspond to the\nsame sample area.\nchanges from 4 K to 15 K, but the green areas remain\ngreen in all images (a)-(c) in Fig. 2. The sample, there-\nfore, is phase separatedinto FIM (blue and red) and non-\nmagnetic (green) regions.[17] At 4 K (Fig. 2(a)) Gd dom-\ninates the magnetization of FIM domains, as depicted\nschematicallyinFig.2(e). Redislandsin Fig.2(a), there-\nfore, represent those parts of the sample where Gd mag-\nnetic moments point “down”, and the blue regions are\nthosewithGdmagneticmomentspointing“up”. At15K\n(Fig. 2(c)) all of the red regions switch to blue, signaling\na reversal in their magnetization, as Mn magnetization is\ndominant above Tcomp≈12 K. This situation is depicted\nschematically in Fig. 2(f). The small magnetic contrast\nacross the sample at 10 K (see Fig. 2(b)) indicates al-\nmost equal magnetic contributionsofthe anti-alignedGd\nand Mn sublattices in FIM regions near Tcomp. In addi-\ntion, Fig. 2(b) demonstrates the domains’ breakup and\na reduction in their size close to Tcomp(at 10 K). This\nphenomenon is consistent with the exchange bias effect\npreviously observed in single crystals.[13] The reduction\nof the size of FIM domains close to Tcompalso leads to a\ndecrease of the coercive field (see Fig. 1(b)).[14, 18]\nFig. 2(d) shows a cross-correlation map between im-\nages (a) and (c) and allows us to investigate qualitatively\nthe temperature evolution of magnetic domains in the\nsample. The large negative value in the center of the\ncross-correlation map demonstrates the anti-correlation\nbetween 4 K and 15 K magnetization data in Fig. 1(a),\nindicating that red and blue islands reverse their magne-\ntization (and colors) upon the temperature change from\n4 K to 15 K. The central location of the cross-correlation\nminimum also demonstrates the small thermal drift in\nour MFM apparatus.Figs. 3(a)-(d) show MFM images obtained at 4 K after\nfield-coolingtheGCMOsamplethrough TCin 0T, 0.1T,\n0.5 T, and 1 T applied fields. In order to understand the\nthermalandfield evolutionofthesample’smagnetization\nwe evaluated cross-correlationmaps for these images. No\ncorrelation was observed between data sets obtained at 0\nT and 0.1 T (panels (a) and (b)), as shown in Fig. 3(e),\nand 0.1 T and 0.5 T (panels (b) and (c)), as shown in\nFig. 3(f). The lack of cross-correlation indicates signifi-\ncant evolution of the spin magnetization due to the re-\nversal process inside FIM clusters in a field up to 0.5 T.\nOn the other hand, magnetic domains imaged in 0.5 T\nand 1 T FC experiments show a similar pattern, suggest-\ning saturation of the magnetization reversal process as\nwell as a clear phase separation between ferrimagnetic\nclusters and the paramagnetic matrix. (Data taken at\n3 T, not shown, are similar to those at 1 T.) The lack\nof correlation between the images (a)-(c) cannot be the\nresult of thermal drift of the tip position over the sam-\nple, as this was observed repeatedly to be under 1 µm for\nour system (e.g., see panels (c) and (d)). The magnetic-\nnonmagnetic phase coexistence could be attributed to lo-\ncalized disorder or a localized strain distribution, similar\nto observations in Y- and Pr-based manganites with a\ncomparably low tolerance factor [Y 2/3Ca1/3MnO3(t∼\n0.884) and Pr 2/3Ca1/3MnO3(t∼0.91)].[19–22] Results\nof x-ray diffraction measurements on our thin-film sam-\nple, however, are close to those on bulk samples and tend\nto rule out a strain mechanism of phase separation.\nIn conclusion, we have performed MFM experiments\nonaferrimagneticGCMOthin filmanddirectlyobserved\nphase separation in the sample, with magnetic (FIM)\nregions of characteristic dimensions between 0.1 to 0.5\nµmembeddedinanon-magneticmatrix. Thebehaviorof\nmagnetic regions is consistent with the presence of anti-\naligned Mn and Gd magnetic sublattices, forming a FIM\nstate. The observed magnetization reversal in the FIM\ndomains as a function of temperature, for small external\nmagnetic field, is consistent with the Mn sublattice being\ndominant at T > T comp≈15 K, but the Gd sublattice\n(with magnetization locked to be antiparallel to a small\napplied field) is dominant for T < T comp. We attribute\nthe phase separation to localized disorder rather than\na strained state of the sample. These results will have\nsignificant bearing on the potential utilization of GCMO\nand otherrelated compoundsin magneticmemorydevice\napplications.\nWork at LANL (sample fabrication, SQUID measure-\nments, MFM, data analysis, and manuscript prepara-\ntion) was supported by the US Department of Energy,\nOffice of Basic Energy Sciences, Division of Materials\nSciences and Engineering. Work at Brookhaven (data\nanalysis and manuscript preparation) was supported by\nthe US Department of Energy under Contract No. DE-\nAC02-98CH10886. N.H. is a member of CONICET (Ar-\ngentina).4\n[1] S. Jin, T. H. Tiefel, M McCormack, R. A. Fastnacht, R.\nRamesh, and L. H. Chen, Science 264, 413 (1994).\n[2] Patrick A. Lee, Naoto Nagaosa, and Xiao-Gang Wen,\nRev. Mod. Phys. 77, 721 (2005).\n[3] Weida Wu, Casey Israel, Namjung Hur, Soonyong Park,\nSang-Wook Cheong, and Alex De Lozanne, Nat. Mater.\n5, 881 (2006).\n[4] J. P. Zhou, J. T. McDevitt, J. S. Zhou, H. Q. Yin, J. B.\nGoodenough, Y. Gim, and Q. X. Jia, Appl. Phys. Lett.\n751146 (1999).\n[5] Lev P. Gor’kov and Vladimir Z. Kresinc, Phys. Rep. 400\n149 (2004).\n[6] E. Dagotto, T. Hotta and A. Moreo, Phys. Rep. 344, 1\n(2001), and references therein.\n[7] Liuwan Zhang, Casey Israel, Amlan Biswas, R. L.\nGreene, and Alex de Lozanne, Science 298, 805 (2002).\n[8] John B. Goodenough, J. Appl. Phys. 81, 5330 (1997).\n[9] H. Y. Hwang, S-W. Cheong, P. G. Radaelli, M. Marezio,\nand B. Batlogg, Phys. Rev. Lett. 75, 914 (1995).\n[10] G. Jeffrey Snyder,C. H. Booth, F. 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Troncoso,1Wolfgang Belzig,2and Arne Brataas1,y\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Department of Physics, University of Konstanz, D-78457 Konstanz, Germany\nAbstract\nWe present a systematic phenomenological description of Gilbert damping in two-sublattice mag-\nnets. Our theory covers the full range of materials from ferro- via ferri- to antiferromagnets. Fol-\nlowing a Rayleigh dissipation functional approach within a Lagrangian classical \feld formulation,\nthe theory captures intra- as well as cross-sublattice terms in the Gilbert damping, parameterized\nby a 2\u00022 matrix. When spin-pumping into an adjacent conductor causes dissipation, we obtain\nthe corresponding Gilbert damping matrix in terms of the interfacial spin-mixing conductances.\nOur model reproduces the experimentally observed enhancement of the ferromagnetic resonance\nlinewidth in a ferrimagnet close to its compensation temperature without requiring an increased\nGilbert parameter. It also predicts new contributions to damping in an antiferromagnet and sug-\ngests the resonance linewidths as a direct probe of the sublattice asymmetry, which may stem from\nboundary or bulk.\n1arXiv:1808.04385v2 [cond-mat.mtrl-sci] 6 Nov 2018I. INTRODUCTION\nThe fundamental connection1between magnetic moment and spin angular momentum\nunderlies the important role for magnets in nearly all spin-based concepts. An applied mag-\nnetic \feld provides the means to manipulate the state of a ferromagnet (FM), and thus the\nassociated spin. Conversely, a spin-polarized current absorbed by the FM a\u000bects its mag-\nnetization2{5. Exploiting a related phenomenon, switching the state of an antiferromagnet\n(AFM) has also been achieved6. Emboldened by this newly gained control, there has been\nan upsurge of interest in AFMs7{10, which o\u000ber several advantages over FMs. These include\nthe absence of stray \felds and a larger anisotropy-induced gap in the magnon spectrum. The\ntwo-sublattice nature of the AFMs further lends itself to phenomena distinct from FMs11.\nConcurrently, ferrimagnets (FiMs) have been manifesting their niche in a wide range of\nphenomena such as ultrafast switching12{14and low-dissipation spin transport15{22. A class of\nFiMs exhibits the so-called compensation temperature23{28, at which the net magnetization\nvanishes, similar to the case of AFMs. Despite a vanishing magnetization in the compensated\nstate, most properties remain distinct from that of AFMs29. Thus, these materials can be\ntuned to mimic FMs and AFMs via the temperature. In conjunction with the possibility of a\nseparate angular-momentum compensation, when the magnetization does not vanish but the\ntotal spin does, FiMs provide a remarkably rich platform for physics and applications. An\nincreased complexity in the theoretical description29,30hence accompanies these structurally\ncomplicated materials, and may be held responsible for comparatively fewer theoretical\nstudies. Nevertheless, a two-sublattice model with distinct parameters for each sublattice\nqualitatively captures all the phenomena mentioned above.\nDissipation strongly in\ruences the response of a magnet to a stimulus and is thus cen-\ntral to the study of magnetic phenomena such as switching, domain wall motion and spin\ntransport. Nevertheless, magnetic damping has conventionally been investigated via the\nferromagnetic resonance (FMR) linewidth. It is accounted for phenomenologically in the\nLandau-Lifshitz description of the magnetization dynamics via the so-called Gilbert damp-\ning term31, which produces a good agreement with experiments for a wide range of systems.\nThe Gilbert damping represents the viscous contribution and may be `derived' within a\nLagrangian formulation of classical \feld theory by including the Rayleigh dissipation func-\ntional31. While the magnetic damping for FMs has been studied in great detail29,31{35,\n2from phenomenological descriptions to microscopic models, a systematic development of an\nanalogous description for ferri- and antiferromagnets has been lacking in literature. Further-\nmore, recent theoretical results on spin pumping in two-sublattice magnets36and damping\nin AFMs37suggest an important role for the previously disregarded29cross-sublattice terms\nin Gilbert damping, and thus set the stage for the present study. Yuan and co-workers have\nrecently presented a step in this direction focussing on spin torques in AFMs38.\nHere, we formulate the magnetization dynamics equations in a general two-sublattice\nmagnet following the classical Lagrangian approach that has previously been employed for\nFMs31. The Gilbert damping is included phenomenologically via a Rayleigh dissipation\nfunctional appropriately generalized to the two-sublattice system, which motivates intra-\nas well as cross-sublattice terms. The Gilbert damping parameter thus becomes a 2 \u00022\nmatrix, in contrast with its scalar form for a single-sublattice FM. Solving the system of\nequations for spatially homogeneous modes in a collinear ground state, we obtain the decay\nrates of the two eigenmodes \fnding direct pathways towards probing the dissipation mech-\nanism and asymmetries in the system. Consistent with recent experiments28,39, we \fnd an\nenhancement in the decay rates39close to the magnetization compensation in a FiM with\nan unaltered damping matrix28. The general description is found to be consistent with the\nspin pumping mediated damping in the magnet34{36, and allows for relating the Gilbert\ndamping matrix with the interfacial spin-mixing conductances. Focusing on AFMs, we ex-\npress the magnetization dynamics in terms of the Neel variable thus clarifying the origin\nof the di\u000berent damping terms in the corresponding dynamical equations38,40. Apart from\nthe usually considered terms, we \fnd additional contributions for the case when sublattice-\nsymmetry is broken in the AFM36,41{45. Thus, FMR linewidth measurements o\u000ber a direct,\nparameter-free means of probing the sublattice asymmetry in AFMs, complementary to the\nspin pumping shot noise36.\nThis paper is organized as follows. We derive the Landau-Lifshitz-Gilbert (LLG) equa-\ntions for the two-sublattice model in Sec. II. The ensuing equations are solved for the\nresonance frequencies and decay rates of the uniform modes in a collinear magnet in Sec.\nIII. Section IV presents the application of the phenomenology to describe a compensated\nferrimagnet and spin pumping mediated Gilbert damping. The case of AFMs is discussed in\nSec. V. We comment on the validity and possible generalizations of the theory in Sec. VI.\nThe paper is concluded with a summary in Sec. VII. The discussion of a generalized Rayleigh\n3dissipation functional and properties of the damping matrix is deferred to the appendix.\nII. MAGNETIZATION DYNAMICS AND GILBERT DAMPING\nWe consider a two-sublattice magnet described by classical magnetization \felds MMMA\u0011\nMMMA(rrr;t) andMMMB\u0011MMMB(rrr;t) corresponding to the sublattices AandB. The system is\ncharacterized by a magnetic free energy F[MMMA;MMMB] with the magnetization \felds assumed\nto be of constant magnitudes MA0andMB0. Here, the notation F[ ] is employed to emphasize\nthat the free energy is a functional over the magnetization \felds, i.e. an integration of the\nfree energy density over space.\nThe undamped magnetization dynamics is described by equating the time derivative of\nthe spin angular momentum associated with the magnetization to the torque experienced\nby it. The resulting Landau-Lifshitz equations for the two \felds may be written as:\nd\ndt\u0012MMMA;B\n\u0000j\rA;Bj\u0013\n=\u0000_MMMA;B\nj\rA;Bj=MMMA;B\u0002\u00160HHHA;B; (1)\nwhere\rA;B(<0) are the gyromagnetic ratios for the two sublattices, and HHHA;Bare the\ne\u000bective magnetic \felds experienced by the respective magnetizations. This expression of\nangular momentum \row may be derived systematically within the Lagrangian classical \feld\ntheory31. The same formalism also allows to account for a restricted form of damping via\nthe so-called dissipation functional R[_MMMA;_MMMB] in the generalized equations of motion:\nd\ndt\u000eL[\u0001]\n\u000e_MMMA;B\u0000\u000eL[\u0001]\n\u000eMMMA;B=\u0000\u000eR[_MMMA;_MMMB]\n\u000e_MMMA;B; (2)\nwhereL[\u0001]\u0011 L [MMMA;MMMB;_MMMA;_MMMB] is the Lagrangian of the magnetic system. Here,\n\u000eL[\u0001]=\u000eMMMArepresents the functional derivative of the Lagrangian with respect to the var-\nious components of MMMA, and so on. The left hand side of Eq. (2) above represents the\nconservative dynamics of the magnet and reproduces Eq. (1) with31\n\u00160HHHA;B=\u0000\u000eF[MMMA;MMMB]\n\u000eMMMA;B; (3)\nwhile the right hand side accounts for the damping.\nThe Gilbert damping is captured by a viscous Rayleigh dissipation functional parame-\nterized by a symmetric matrix \u0011ijwithfi;jg=fA;Bg:\nR[_MMMA;_MMMB] =Z\nVd3r\u0010\u0011AA\n2_MMMA\u0001_MMMA+\u0011BB\n2_MMMB\u0001_MMMB+\u0011AB_MMMA\u0001_MMMB\u0011\n; (4)\n4whereVis the volume of the magnet. The above form of the functional assumes the damping\nto be spatially homogeneous, isotropic, and independent of the equilibrium con\fguration.\nA more general form with a lower symmetry is discussed in appendix A. Including the\ndissipation functional via Eq. (2) leads to the following replacements in the equations of\nmotion (1):\n\u00160HHHA!\u00160HHHA\u0000\u0011AA_MMMA\u0000\u0011AB_MMMB; (5)\n\u00160HHHB!\u00160HHHB\u0000\u0011BB_MMMB\u0000\u0011AB_MMMA: (6)\nHence, the LLG equations for the two-sublattice magnet become:\n_MMMA=\u0000j\rAj(MMMA\u0002\u00160HHHA) +j\rAj\u0011AA\u0010\nMMMA\u0002_MMMA\u0011\n+j\rAj\u0011AB\u0010\nMMMA\u0002_MMMB\u0011\n; (7)\n_MMMB=\u0000j\rBj(MMMB\u0002\u00160HHHB) +j\rBj\u0011AB\u0010\nMMMB\u0002_MMMA\u0011\n+j\rBj\u0011BB\u0010\nMMMB\u0002_MMMB\u0011\n:(8)\nThese can further be expressed in terms of the unit vectors ^mmmA;B=MMMA;B=MA0;B0:\n_^mmmA=\u0000j\rAj(^mmmA\u0002\u00160HHHA) +\u000bAA\u0010\nmmmA\u0002_^mmmA\u0011\n+\u000bAB\u0010\n^mmmA\u0002_^mmmB\u0011\n; (9)\n_^mmmB=\u0000j\rBj(^mmmB\u0002\u00160HHHB) +\u000bBA\u0010\n^mmmB\u0002_^mmmA\u0011\n+\u000bBB\u0010\n^mmmB\u0002_^mmmB\u0011\n; (10)\nthereby introducing the Gilbert damping matrix ~ \u000bfor a two-sublattice system:\n~\u000b=0\n@\u000bAA\u000bAB\n\u000bBA\u000bBB1\nA=0\n@j\rAj\u0011AAMA0j\rAj\u0011ABMB0\nj\rBj\u0011ABMA0j\rBj\u0011BBMB01\nA; (11)\n\u000bAB\n\u000bBA=j\rAjMB0\nj\rBjMA0: (12)\nAs elaborated in appendix B, the positivity of the dissipation functional implies that the\neigenvalues and the determinant of ~ \u000bmust be non-negative, which is equivalent to the\nfollowing conditions:\n\u0011AA;\u0011BB\u00150; \u0011 AA\u0011BB\u0015\u00112\nAB=)\u000bAA;\u000bBB\u00150; \u000b AA\u000bBB\u0015\u000bAB\u000bBA: (13)\nThus, Eqs. (9) and (10) constitute the main result of this section, and introduce the damping\nmatrix [Eq. (11)] along with the constraints imposed on it [Eq. (12) and (13)] by the\nunderlying formalism.\n5III. UNIFORM MODES IN COLLINEAR GROUND STATE\nIn this section, we employ the phenomenology introduced above to evaluate the resonance\nfrequencies and the decay rates of the spatially homogeneous modes that can be probed in a\ntypical FMR experiment. We thus work in the macrospin approximation, i.e. magnetizations\nare assumed to be spatially invariant. Considering an antiferromagnetic coupling J(>0)\nbetween the two sublattices and parameterizing uniaxial easy-axis anisotropies via KA;B(>\n0), the free energy assumes the form:\nF[MMMA;MMMB] =Z\nVd3r\u0002\n\u0000\u00160H0(MAz+MBz)\u0000KAM2\nAz\u0000KBM2\nBz+JMMMA\u0001MMMB\u0003\n;(14)\nwhereH0^zzzis the applied magnetic \feld. The magnet is assumed to be in a collinear ground\nstate:MMMA=MA0^zzzandMMMB=\u0000MB0^zzzwithMA0>M B0. Employing Eq. (3) to evaluate the\ne\u000bective \felds, the magnetization dynamics is expressed via the LLG equations (9) and (10).\nConsidering MMMA=MAx^xxx+MAy^yyy+MA0^zzz,MMMB=MBx^xxx+MBy^yyy\u0000MB0^zzzwithjMAx;Ayj\u001c\nMA0,jMBx;Byj \u001cMB0, we linearize the resulting dynamical equations. Converting to\nFourier space via MAx=MAxexp (i!t) etc. and switching to circular basis via MA\u0006(B\u0006)=\nMAx(Bx)\u0006iMAy(By), we obtain two sets of coupled equations expressed succinctly as:\n0\n@\u0006!\u0000\nA\u0000i!\u000b AA\u0000\u0010\nj\rAjJMA0+i!\u000b ABMA0\nMB0\u0011\n\u0010\nj\rBjJMB0+i!\u000b BAMB0\nMA0\u0011\n\u0006!+ \n B+i!\u000b BB1\nA0\n@MA\u0006\nMB\u00061\nA=0\n@0\n01\nA; (15)\nwhere we de\fne \n A\u0011j\rAj(JMB0+ 2KAMA0+\u00160H0) and \n B\u0011j\rBj(JMA0+ 2KBMB0\u0000\n\u00160H0). Substituting !=!r\u0006+i!i\u0006into the ensuing secular equation, we obtain the\nresonance frequencies !r\u0006to the zeroth order and the corresponding decay rates !i\u0006to the\n\frst order in the damping matrix elements:\n!r\u0006=\u0006(\nA\u0000\nB) +p\n(\nA+ \n B)2\u00004J2j\rAjj\rBjMA0MB0\n2; (16)\n!i\u0006\n!r\u0006=\u0006!r\u0006(\u000bAA\u0000\u000bBB) +\u000bAA\nB+\u000bBB\nA\u00002Jj\rBjMA0\u000bAB\n!r++!r\u0000: (17)\nIn the expression above, Eq. (16) and Eq. (17), we have chosen the positive solutions of\nthe secular equations for the resonance frequencies. The negative solutions are equal in\nmagnitude to the positive ones and physically represent the same two modes. The positive-\npolarized mode in our notation corresponds to the typical ferromagnetic resonance mode,\nwhile the negative-polarized solution is sometimes termed `antiferromagnetic resonance'25.\n6020406080100120\n0 0.2 0.4 0.6 0.8 10.040.050.060.070.080.090.1FIG. 1. Resonance frequencies and normalized decay rates vs. the applied \feld for a quasi-\nferromagnet ( MA0= 5MB0).j\rAj=j\rBj= 1;1:5;0:5 correspond to solid, dashed and dash-dotted\nlines respectively. The curves in blue and red respectively depict the + and \u0000modes. The damping\nparameters employed are \u000bAA= 0:06,\u000bBB= 0:04 and\u000bAB= 0.\nIn order to avoid confusion with the ferromagnetic or antiferromagnetic nature of the un-\nderlying material, we call the two resonances as positive- and negative-polarized. The decay\nrates can further be expressed in the following form:\n!i\u0006\n!r\u0006=\u0016\u000b(\nA+ \n B)\u00002Jj\rBjMA0\u000bAB\n!r++!r\u0000\u0006\u0001\u0016\u000b; (18)\nwith \u0016\u000b\u0011(\u000bAA+\u000bBB)=2 and \u0001\u0016\u000b\u0011(\u000bAA\u0000\u000bBB)=2. Eq. (18) constitutes the main result\nof this section and demonstrates that (i) asymmetric damping in the two sublattices is\nmanifested directly in the normalized decay rates of the two modes (Figs. 1 and 2), and\n(ii) o\u000b-diagonal components of the damping matrix may reduce the decay rates (Fig. 2).\nFurthermore, it is consistent with and reproduces the mode-dependence of the decay rates\nobserved in the numerical studies of some metallic AFMs37.\nTo gain further insight into the results presented in Eqs. (16) and (18), we plot the\n7resonance frequencies and the normalized decay rates vs. the applied magnetic \feld for a\ntypical quasi-ferromagnet, such as yttrium iron garnet, in Fig. 1. The parameters employed\nin the plot arej\rBj= 1:8\u00021011,MB0= 105,KA=KB= 10\u00007, andJ= 10\u00005in SI units,\nand have been chosen to represent the typical order of magnitude without pertaining to a\nspeci\fc material. The plus-polarized mode is lower in energy and is raised with an increasing\napplied magnetic \feld. The reverse is true for the minus-polarized mode whose relatively\nlarge frequency makes it inaccessible to typical ferromagnetic resonance experiments. As\nanticipated from Eq. (18), the normalized decay rates for the two modes di\u000ber when \u000bAA6=\n\u000bBB. Furthermore, the normalized decay rates are independent of the applied \feld for\nsymmetric gyromagnetic ratios for the two sublattices. Alternately, a measurement of the\nnormalized decay rate for the plus-polarized mode is able to probe the sublattice asymmetry\nin the gyromagnetic ratios. Thus it provides essential information about the sublattices\nwithout requiring the measurement of the large frequency minus-polarized mode.\nIV. SPECIFIC APPLICATIONS\nWe now examine two cases of interest: (i) the mode decay rate in a ferrimagnet close to\nits compensation temperature, and (ii) the Gilbert damping matrix due to spin pumping\ninto an adjacent conductor.\nA. Compensated ferrimagnets\nFMR experiments carried out on gadolinium iron garnet23,39\fnd an enhancement in the\nlinewidth, and hence the mode decay rate, as the temperature approaches the compensation\ncondition, i.e. when the two e\u000bective46sublattices have equal saturation magnetizations.\nThese experiments have conventionally been interpreted in terms of an e\u000bective single-\nsublattice model thereby ascribing the enhancement in the decay rate to an increase in the\nscalar Gilbert damping constant allowed within the single-sublattice model24. In contrast,\nexperiments probing the Gilbert parameter in a di\u000berent FiM via domain wall velocity\n\fnd it to be essentially unchanged around compensation28. Here, we analyze FMR in a\ncompensated FiM using the two-sublattice phenomenology developed above and thus address\nthis apparent inconsistency.\n8020406080100120\n1 2 3 4 500.050.10.150.20.250.3FIG. 2. Resonance frequencies and normalized decay rates vs. relative saturation magnetizations\nof the sublattices. The curves which are not labeled as + or \u0000represent the common normalized\ndecay rates for both modes. The parameters employed are the same as for Fig. 1 with \rA=\rB.\nThe compensation behavior of a FiM may be captured within our model by allowing\nMA0to vary while keeping MB0\fxed. The mode frequencies and normalized decay rates\nare examined with respect to the saturation magnetization variation in Fig. 2. We \fnd an\nenhancement in the normalized decay rate, consistent with the FMR experiments23,39, for a\n\fxed Gilbert damping matrix. The single-sublattice interpretation ascribes this change to a\nmodi\fcation of the e\u000bective Gilbert damping parameter24, which is equal to the normalized\ndecay rate within that model. In contrast, the latter is given by Eq. (18) within the\ntwo-sublattice model and evolves with the magnetization without requiring a modi\fcation\nin the Gilbert damping matrix. Speci\fcally, the enhancement in decay rate observed at\nthe compensation point is analogous to the so-called exchange enhancement of damping in\nAFMs47. Close to compensation, the FiM mimics an AFM to some extent.\nWe note that while the spherical samples employed in Ref. 23 are captured well by our\nsimple free energy expression [Eq. (14)], the interfacial and shape anisotropies of the thin\n9\flm sample employed in Ref. 39 may result in additional contributions to decay rates. The\nsimilarity of the observed linewidth trends for the two kinds of samples suggests that these\nadditional anisotropy e\u000bects may not underlie the observed damping enhancement. Quan-\ntitatively accounting for these thin \flm e\u000bects requires a numerical analysis, as discussed\nin Sec VI below, and is beyond the scope of the present work. Furthermore, domain forma-\ntion may result in additional damping contributions not captured within our single-domain\nmodel.\nB. Spin pumping mediated Gilbert damping\nSpin pumping34from a FM into an adjacent conductor has been studied in great detail35\nand has emerged as a key method for injecting pure spin currents into conductors48. The\nangular momentum thus lost into the conductor results in a contribution to the magnetic\ndamping on top of the intrinsic dissipation in the bulk of the magnet. A variant of spin\npumping has also been found to be the dominant cause of dissipation in metallic magnets37.\nThus, we evaluate the Gilbert damping matrix arising due to spin pumping from a two-\nsublattice magnet36into an adjacent conductor acting as an ideal spin sink.\nWithin the macrospin approximation, the total spin contained by the magnet is given by:\nSSS=\u0000MA0V^mmmA\nj\rAj\u0000MB0V^mmmB\nj\rBj: (19)\nThe spin pumping current emitted by the two-sublattice magnet has the following general\nform36:\nIIIs=~\neX\ni;j=fA;BgGij\u0010\n^mmmi\u0002_^mmmj\u0011\n; (20)\nwithGAB=GBA, where the spin-mixing conductances Gijmay be evaluated within di\u000berent\nmicroscopic models36,49{51. Equating the spin pumping current to \u0000_SSSand employing Eqs.\n(9) and (10), the spin pumping contribution to the Gilbert damping matrix becomes:\n\u000b0\nij=~Gijj\rij\neMi0V; (21)\nwhich in turn implies\n\u00110\nij=~Gij\neMi0Mj0V; (22)\n10for the corresponding dissipation functional. The resulting Gilbert damping matrix is found\nto be consistent with its general form and constraints formulated in Sec. II. Thus, employing\nthe phenomenology developed above, we are able to directly relate the magnetic damping in\na two-sublattice magnet to the spin-mixing conductance of its interface with a conductor.\nV. ANTIFERROMAGNETS\nDue to their special place with high symmetry in the two-sublattice model as well as the\nrecent upsurge of interest7{10,52{54, we devote the present section to a focused discussion on\nAFMs in the context of the general results obtained above. It is often convenient to describe\nthe AFM in terms of a di\u000berent set of variables:\nmmm=^mmmA+^mmmB\n2; nnn=^mmmA\u0000^mmmB\n2: (23)\nIn contrast with ^mmmAand ^mmmB,mmmandnnnare not unit vectors in general. The dynamical\nequations for mmmandnnnmay be formulated by developing the entire \feld theory, starting with\nthe free energy functional, in terms of mmmandnnn. Such a formulation, including damping,\nhas been accomplished by Hals and coworkers40. Here, we circumvent such a repetition and\ndirectly express the corresponding dynamical equations by employing Eqs. (9) and (10) into\nEq. (23):\n_mmm=\u0000(mmm\u0002\rm\u00160HHHm)\u0000(nnn\u0002\rn\u00160HHHn) +X\np;q=fm;ng\u000bm\npq(ppp\u0002_qqq); (24)\n_nnn=\u0000(mmm\u0002\rn\u00160HHHn)\u0000(nnn\u0002\rm\u00160HHHm) +X\np;q=fm;ng\u000bn\npq(ppp\u0002_qqq); (25)\nwith\n\rm\u00160HHHm\u0011j\rAj\u00160HHHA+j\rBj\u00160HHHB\n2; (26)\n\rn\u00160HHHn\u0011j\rAj\u00160HHHA\u0000j\rBj\u00160HHHB\n2; (27)\n\u000bm\nmm=\u000bn\nnm=\u000bAA+\u000bBB+\u000bAB+\u000bBA\n2; (28)\n\u000bm\nmn=\u000bn\nnn=\u000bAA\u0000\u000bBB\u0000\u000bAB+\u000bBA\n2; (29)\n\u000bm\nnn=\u000bn\nmn=\u000bAA+\u000bBB\u0000\u000bAB\u0000\u000bBA\n2; (30)\n\u000bm\nnm=\u000bn\nmm=\u000bAA\u0000\u000bBB+\u000bAB\u0000\u000bBA\n2: (31)\n11A general physical signi\fcance, analogous to \rA;B, may not be associated with \rm;nwhich\nmerely serve the purpose of notation here. The equations obtained above manifest new\ndamping terms in addition to the ones that are typically considered in the description\nof AFMs. Accounting for the sublattice symmetry of the antiferromagnetic bulk while\nallowing for the damping to be asymmetric, we may assume \rA=\rBandMA0=MB0, with\n\u0016\u000b\u0011(\u000bAA+\u000bBB)=2, \u0001\u0016\u000b\u0011(\u000bAA\u0000\u000bBB)=2, and\u000bAB=\u000bBA\u0011\u000bod. Thus, the damping\nparameters simplify to\n\u000bm\nmm=\u000bn\nnm=\u0016\u000b+\u000bod; (32)\n\u000bm\nmn=\u000bn\nnn=\u0001\u0016\u000b; (33)\n\u000bm\nnn=\u000bn\nmn=\u0016\u000b\u0000\u000bod; (34)\n\u000bm\nnm=\u000bn\nmm=\u0001\u0016\u000b; (35)\nthereby eliminating the \\new\" terms in the damping when \u000bAA=\u000bBB. However, the sublat-\ntice symmetry may not be applicable to AFMs, such as FeMn, with non-identical sublattices.\nFurthermore, the sublattice symmetry of the AFM may be broken at the interface41{43via,\nfor example, spin mixing conductances36,45,55resulting in \u000bAA6=\u000bBB.\nThe resonance frequencies and normalized decay rates [Eqs. (16) and (18)] take a simpler\nform for AFMs. Substituting KA=KB\u0011K,\rA=\rB\u0011\r, andMA0=MB0\u0011M0:\n!r\u0006=\u0006j\rj\u00160H0+ 2j\rjM0p\n(J+K)K; (36)\n!i\u0006\n!r\u0006=J(\u0016\u000b\u0000\u000bod) + 2K\u0016\u000b\n2p\n(J+K)K\u0006\u0001\u0016\u000b\u0019(\u0016\u000b\u0000\u000bod)\n2r\nJ\nK+ \u0016\u000br\nK\nJ\u0006\u0001\u0016\u000b; (37)\nwhere we have employed J\u001dKin the \fnal simpli\fcation. The term /p\nK=J has typically\nbeen disregarded on the grounds K\u001cJ. However, recent numerical studies of damping in\nseveral AFMs37\fnd \u0016\u000b\u001d\u0016\u000b\u0000\u000bod>0 thus suggesting that this term should be comparable\nto the one proportional top\nJ=K and hence may not be disregarded. The expression above\nalso suggests measurement of the normalized decay rates as a means of detecting the sublat-\ntice asymmetry in damping. For AFMs symmetrical in the bulk, such an asymmetry may\narise due to the corresponding asymmetry in the interfacial spin-mixing conductance36,45,55.\nThus, decay rate measurements o\u000ber a method to detect and quantify such interfacial e\u000bects\ncomplementary to the spin pumping shot noise measurements suggested earlier36.\n12VI. DISCUSSION\nWe have presented a phenomenological description of Gilbert damping in two-sublattice\nmagnets and demonstrated how it can be exploited to describe and characterize the system\ne\u000bectively. We now comment on the limitations and possible generalizations of the formal-\nism presented herein. To begin with, the two-sublattice model is the simplest description of\nferri- and antiferromagnets. It has been successful in capturing a wide range of phenomenon.\nHowever, recent measurements of magnetization dynamics in nickel oxide could only be ex-\nplained using an eight-sublattice model56. The temperature dependence of the spin Seebeck\ne\u000bect in yttrium iron garnet also required accounting for more than two magnon bands57.\nA generalization of our formalism to a N-sublattice model is straightforward and can be\nachieved via a Rayleigh dissipation functional with N2terms, counting \u0011ijand\u0011jias sepa-\nrate terms. The ensuing Gilbert damping matrix will be N \u0002N while obeying the positive\ndeterminant constraint analogous to Eq. (13).\nIn our description of the collinear magnet [Eq. (14)], we have disregarded contributions\nto the free energy which break the uniaxial symmetry of the system about the z-axis. Such\nterms arise due to spin-nonconserving interactions58, such as dipolar \felds and magnetocrys-\ntalline anisotropies, and lead to a mixing between the plus- and minus-polarized modes30.\nIncluding these contributions converts the two uncoupled 2 \u00022 matrix equations [(15)] into\na single 4\u00024 matrix equation rendering the solution analytically intractable. A detailed\nanalysis of these contributions30shows that their e\u000bect is most prominent when the two\nmodes are quasi-degenerate, and may be disregarded in a \frst approximation.\nIn evaluating the resonance frequencies and the decay rates [Eqs. (16) and (18)], we\nhave assumed the elements of the damping matrix to be small. A precise statement of the\nassumption employed is !i\u001c!r, which simply translates to \u000b\u001c1 for a single-sublattice\nferromagnet. In contrast, the constraint imposed on the damping matrix within the two-\nsublattice model by the assumption of small normalized decay rate is more stringent [Eq.\n(18)]. For example, this assumption for an AFM with \u000bAB= \u0001\u0016\u000b= 0 requires \u0016 \u000b\u001c\np\nK=J\u001c1. This stringent constraint may not be satis\fed in most AFMs37, thereby\nbringing the simple Lorentzian shape description of the FMR into question. It can also be\nseen from Fig. 2 that the assumption of a small normalized decay rate is not very good for\nthe chosen parameters.\n13VII. SUMMARY\nWe have developed a systematic phenomenological description of the Gilbert damping\nin a two-sublattice magnet via inclusion of a Rayleigh dissipation functional within the La-\ngrangian formulation of the magnetization dynamics. Employing general expressions based\non symmetry, we \fnd cross-sublattice Gilbert damping terms in the LLG equations in con-\nsistence with other recent \fndings36{38. Exploiting the phenomenology, we explain the en-\nhancement of damping23,39in a compensated ferrimagnet without requiring an increase in\nthe damping parameters28. We also demonstrate approaches to probe the various forms\nof sublattice asymmetries. Our work provides a uni\fed description of ferro- via ferri- to\nantiferromagnets and allows for understanding a broad range of materials and experiments\nthat have emerged into focus in the recent years.\nACKNOWLEDGMENTS\nA. K. thanks Hannes Maier-Flaig and Kathrin Ganzhorn for valuable discussions. We\nacknowledge \fnancial support from the Alexander von Humboldt Foundation, the Research\nCouncil of Norway through its Centers of Excellence funding scheme, project 262633, \\QuS-\npin\", and the DFG through SFB 767 and SPP 1538.\nAppendix A: Generalized Rayleigh dissipation functional\nAs compared to the considerations in Sec. II, a more general approach to parameterizing\nthe dissipation functional is given by:\nR[_MMMA;_MMMB] =1\n2Z\nVZ\nVd3r0d3rX\np;q=fA;BgX\ni;j=fx;y;zg_Mpi(rrr)\u0011ij\npq(rrr;rrr0)_Mqj(rrr0): (A1)\nThis form allows to capture the damping in an environment with a reduced symmetry.\nHowever, the larger number of parameters also makes it di\u000ecult to extract them reliably\nvia typical experiments. The above general form reduces to the case considered in Sec. II\nwhen\u0011ij\npq(rrr;rrr0) =\u0011pq\u000eij\u000e(rrr\u0000rrr0) and\u0011pq=\u0011qp. Furthermore, the coe\u000ecients \u0011ij\npqmay depend\nuponMMMA(rrr) andMMMB(rrr) as has been found in recent numerical studies of Gilbert damping\nin AFMs37.\n14Appendix B: Damping matrix\nThe Rayleigh dissipation functional considered in the main text is given by:\nR[_MMMA;_MMMB] =Z\nVd3r\u0010\u0011AA\n2_MMMA\u0001_MMMA+\u0011BB\n2_MMMB\u0001_MMMB+\u0011AB_MMMA\u0001_MMMB\u0011\n; (B1)\nwhich may be brought into the following concise form with the notation~_MMM\u0011[_MMMA_MMMB]|:\nR[_MMMA;_MMMB] =1\n2Z\nVd3r~_MMM|~\u0011~_MMM; (B2)\nwhere ~\u0011is the appropriate matrix given by:\n~\u0011=0\n@\u0011AA\u0011AB\n\u0011AB\u0011BB1\nA: (B3)\nConsidering an orthogonal transformation~_MMM=~Q~_M, the dissipation functional can be\nbrought to a diagonal form\nR[_MMMA;_MMMB] =1\n2Z\nVd3r~_M|~Q|~\u0011~Q~_M; (B4)\nwhere ~Q|~\u0011~Qis assumed to be diagonal. 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Due to the ferrimagnetic order, the\nlocal magnetization has an incommensurate oscillation with the position. We show that the spon-\ntaneously magnetized TLL is smoothly connected to the existence of a Nambu-Goldstone boson in\nthe canted ferrimagnetic phase of a two-dimensional frustrated antiferromagnet.\nPACS numbers: 75.10.Jm, 75.30.Kz, 75.50.Gg\nI. INTRODUCTION\nSpontaneous symmetry breaking is one of the most\nfundamental concepts in physics. It provides the mech-\nanism to generate mass of elementary particles and al-\nlows macroscopic alignment of magnetic moments in fer-\nromagnets. Spontaneous breaking of global continuous\nsymmetry is accompanied by a massless excitation, the\nNambu-Goldstone boson [1{3]. Since Nambu-Goldstone\nboson governs low-energy physics at long distance and\nthe low-energy physics is a\u000bected by the geometry of sys-\ntem, the dimensionality of the system has strong in\ru-\nences on Nambu-Goldstone boson.\nSuch e\u000bects are most prominent in one dimension (1D)\nbecause of the large suppression of ordering at \fnite tem-\nperatures [4, 5] and even at zero temperature [6] due to\nquantum e\u000bects. As a result the breaking of a continuous\nsymmetry in 1D is deemed impossible. For systems such\nas a 1D super\ruid, indeed no long range order exists,\nand the proper description is the one of a Tomonaga-\nLuttinger liquid (TLL) [7]. Despite the absence of the\ntrue long-range order [8, 9], Goldstone modes exist, and\nhave a dynamical origin [10]. However in some rare cases,\nsuch as a ferromagnet, the ground state can, even in\n1D spontaneously break a continuous symmetry. This\nprompts immediately for the question of why and for\nwhich systems such phenomena can occur.\nIn order to shed light on the possibility of sponta-\nneous symmetry breaking in 1D, dynamical aspects of\nthe system need to be carefully considered. In addition,\nin view of the recent experimental progresses in realiz-\ning 1D quantum liquids in various situations [11{14], it\nis worthwhile to search for novel manifestations of spon-\ntaneous symmetry breaking in 1D.\nFor quantum magnetism in 1D, one expects a system\nwith quasi-long range antiferromagnetic order to have a\nrelativistic dispersion, which is the case of the TLL, while\na ferromagnet would have a quadratic one. This behav-\nior of the Nambu-Goldstone boson has been formulated\nin a quite general context [15{17]. Finding in 1D a sys-\ntem that would spontaneously break the continuous ro-tational symmetry of the spins, while at the same time\nretaining some TLL behavior would thus be interesting\nand an example of the more general character of Nambu-\nGoldstone boson.\nA very good possibility to realize such a spontaneously\nmagnetized TLL (SMTLL) is o\u000bered by ferrimagnetic\nsystems. Several numerical studies [18{24] have fol-\nlowed this route and found incommensurate ferrimag-\nnetic phases which can be candidates for the SMTLL.\nHowever besides the numerical results, there is still no\nmicroscopic theory that explains the nature of the in-\ncommensurate phase and could relate it to a SMTLL.\nIn this paper, we present such a theory of the SMTLL,\nshowing that one can have simultaneously a spontaneous\nbreaking of the spin-rotation symmetry leading to a \f-\nnite magnetization and a TLL behavior. We demonstrate\nthat this SMTLL phase is realized in the ground state of\nanS= 1=2 geometrically frustrated quantum antifer-\nromagnet on a 1D array of the union jack lattice [see\nFig. 1].\nJ1J1Sj,1Sj,2Sj,3J2↵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\nFigure 1. The three-leg union-jack ladder (1).arXiv:1403.1513v2 [cond-mat.str-el] 27 May 20142\nII. INSTABILITY OF THE\nTOMONAGA-LUTTINGER LIQUID\nWe consider the union jack (UJ) spin Hamiltonian:\nH=J1X\nj3X\na=1Sj;a\u0001Sj+1;a+J1X\njSj;2\u0001(Sj;1+Sj;3)\n+J2X\njS2j;2\u0001(S2j\u00001;3+S2j+1;1)\n+\u000bJ2X\njS2j;2\u0001(S2j\u00001;1+S2j+1;3); (1)\nwhereJ1;2>0 and 0<\u000b\u001c1. The parameter \u000bdenotes\nthe imbalance of the diagonal interactions. Throughout\nthe Paper, we \fx \u000band change the ratio J2=J1from\n0 to +1. Note that this model has a priori full spin-\nrotational symmetry.\nA. Classical ground state\nWe \frst consider the classical ground state minimiz-\ning the energy of a unit cell. The classical analysis on\nthe UJ ladder (1) is similar to the 2D UJ antiferromag-\nnet [25{27]. For 0 \u0014J2=J1<1=2, the classical ground\nstate is the N\u0013 eel state. For 1 =2< J 2=J1, spins on the\n\flled sites in Fig. 1 (a) become canted with a polar angle\n#= cos\u00001(J1=2J2) and the classical ground state in the\ncanted phase has an incommensurate magnetization,\nhSz\nj;ai=~S\n2\u0012\n1\u0000J1\n2J2\u0013\n:\nHereafter we use ~= 1 for simplicity. At the classi-\ncal level, a spontaneous magnetization occurs. However,\nsince quantum \ructuation usually destroys long-range or-\nder in 1D systems (1), we have to take them into account\nto conclude on the existence of a spontaneous magneti-\nzation in 1D.\nB. The Tomonaga-Luttinger liquid\nTo do so we derive the low-energy e\u000bective \feld theory\nof the UJ ladder (1). When the diagonal interaction is\nFigure 2. The shaded area depicts the unit cell of the O(3)\nnonlinear\u001bmodel.small enough, J2=J1\u001c1, the low-energy e\u000bective \feld\ntheory is written as a function of two slowly varying \felds\nn=1\n2S3X\na=1(S2j+2\u0000a;a\u0000S2j+3\u0000a;a); (2)\nl=1\n2a03X\na=1(S2j+2\u0000a;a+S2j+3\u0000a;a): (3)\nHerea0is the lattice spacing. We take a diagonal unit\ncell (Fig. 2) to de\fne nandlalong theJ2bond [28]. The\nnandl\felds denote respectively staggered and uniform\nmagnetization densities and satisfy the constraints\nf(n;l)\u0011n2\u00001\u00001\nS\u0000l2\nS2= 0 (4)\nandn\u0001l= 0. The constraint (4) is usually replaceable to\nn2= 1. However, the l2term will play an essential role\nfor our purpose.\nThe Hamiltonian (1) in the low-energy limit is given\nby\nH=Zdx\n2\u00141P\na;bM\u00001\na;bl2+ 2S2X\napa(@xn)2\n+2SP\na;bpaM\u00001\na;bP\na;bM\u00001\na;b(l\u0001@xn+@xn\u0001l)\u0015\n(5)\n=Z\ndx\u0014gv\n2\u0012\nl\u0000\u0002\n4\u0019@xn\u00132\n+v\n2g(@xn)2\u0015\n;(6)\nwheregis a coupling constant, vis the velocity, \u0002 is a\ntopological angle given by\ng=1\nS\u0014\n2X\na;b;cpaM\u00001\nb;c\u0000\u0012\u0002\n4\u0019S\u00132\u0015\u00001=2\n; (7)\nv=Sa0\u00142P\napaP\nb;cM\u00001\nb;c\u0000\u0012\u0002\n4\u0019S1P\na;bM\u00001\na;b\u00132\u00151=2\n;(8)\n\u0002 = 6\u0019S; (9)\nandpa= 3J1=2 + (J1=2)\u000ea;2. Whilegandvdepend on\na 3\u00023 matrix of microscopic parameters,\nM=0\n@5J1\u0000~J2J1\u0000~J2 0\nJ1\u0000~J26J1\u00002~J2J1\u0000~J2\n0J1\u0000~J25J1\u0000~J21\nA (10)\nwith ~J2= (1 +\u000b)J2=2, the topological angle (9) is deter-\nmined only by the number of legs. The derivation of the\ne\u000bective \feld theory (6) is explained in the case of the\nthree-leg spin ladder in Refs. 28 and 29. We obtain the\ne\u000bective \feld theory (6) by replacing the rung coupling of\nthe three-leg ladder to J1\u0000~J2in the matrix (10). We can\nsee from Eq. (5) that information on the structure of the\nUJ lattice (1) is encoded in the matrix (10). Integrating\nlout, we obtain [30]\nH=v\n2gZ\ndx\u00141\nv2(@\u001cn)2+ (@xn)2\u0015\n+i\u0002Q; (11)3\nwhere n2= 1 and\nQ=1\n4\u0019Z1\n\u00001dxn\u0001@\u001cn\u0002@xn\ngives an integer after integrating with an imaginary time\n\u001c, that is,R1\n\u00001d\u001cQ2Z. The Hamiltonian (11) is the\nO(3) nonlinear \u001bmodel (NLSM). The \u0002 term controls\nthe low-energy limit of the NLSM (11) [31]. When \u0002 \u0011\u0019\n(mod 2\u0019), namely when Sis a half integer, the O(3)\nNLSM (11) is identical to the TLL as a conformal \feld\ntheory with a central charge c= 1 [32],\nH=v\n2\u0019Z\ndx\u0014\nK(@x\u0012)2+1\nK(@x\u001e)2\u0015\n: (12)\nWe use here the notation for the TLL of Ref. 7. The\nnature of the \u001eand\u0012\felds will be discussed in detail\nlater. Since \u0002 is independent of J2, for small enough J2,\nthe diagonal interaction is irrelevant and the UJ ladder\n(1) has the TLL ground state (12), and thus in particular\nzero spontaneous magnetization.\nC. Instability at k= 0\nHowever the diagonal interaction has a serious impact\non the ground state, and lead to an instability of the TLL.\nThe diagonal interaction J2partly compensates J1in the\nmatrix (10) and it reduces the velocity down to v= 0\nwhere the linearization of the dispersion relation !=vk\nbecomes invalid. Let us denote the instability point as\nJc1\n2. The instability point is determined from the zeros\nof the matrix (10). The matrix Mis positive de\fnite for\n~J2= 0. As we increase ~J2, the positive de\fniteness \frst\nbreaks down at ~J2= 7J1=3, namely\nJc1\n2=14J1\n3(1 +\u000b): (13)\nWhen 00 and\nL\u0011l\u0000\u0002\n4\u0019@xn: (18)\nWe introduced the quartic interaction ( L2)2instead of\n(l2)2because their di\u000berence (e.g. f(@xn)2g2) is negligi-\nble as we see below.\nWhenJ2>Jc1\n2, the following inequalities are valid:\ngv=1P\na;bM\u00001\na;b<0; (19)\nv\ng= 2S2X\napa\u0000\u0012\u0002\n4\u0019\u001321P\na;bM\u00001\na;b>0: (20)\nThus, the interaction that the L\feld feels takes a form\nof the wine bottle:\ngv\n2L2+\u0015(L2)2=\u0015\u0012\nL2\u00001\n\u0015jgvj\u00132\n+ const: (21)\nThe potential (21) leads to a nonvanishing expectation\nvalue of L. Note that the instability occurs only in\nthe uniform part of the O(3) NLSM (17) because of\nthe inequalities (19) and (20). The diagonal coupling\nonly changes the sign of the coupling constant of the\nL2[Eq. (19)] keeping that of ( @xn)2positive (Eq. (20)).\nNamely, the instability only occurs at the wave number\nk= 0 and the nonzero expectation value of the L\feld\n(18) is attributed to the magnetization density l. There-\nfore the ground state has a spontaneous magnetization\nper site, M\u0011hli=3, with\njMj=1\n3\u00121\n\u0015jgvj\u00131=2\n/\u0012J2\u0000Jc1\n2\nJ1\u00131=2\n: (22)\nThe transition around Jc1\n2is described by a Ginzburg-\nLandau like theory of second order transitions. Con-\ntrarily to the ( L2)2term, the higher-order interaction4\nf(@xn)2g2is negligible because the lower-order term\n(@xn)2is stable. Since \u0015 > 0 the transition cannot\nbe \frst order. We emphasize that our derivation of the\nspontaneous magnetization (22) fully respects the SU(2)\nrotational symmetry and Mcan point in an arbitrary\ndirection.\nD. Nambu-Goldstone bosons\nLet us explain the nature of Nambu-Goldstone boson\ngenerated from this phase transition. First we focus on\nthek= 0 part. We rewrite the Hamiltonian in terms of\n\ructuation\nm\u0011L\u00003M: (23)\nWe can assume M=M(0 0 1)Twithout loss of gen-\nerality. The longitudinal component mzhas a mass\n\u0001 = 12\u0015M2because of the interaction (21), that is,\n\u0015(L2\u00009M2)2'12\u0015M2(mz)2. The transverse com-\nponent m?\u0011(mx;my;0) is the Nambu-Goldstone bo-\nson generated from the spontaneous magnetization and\nit possesses a nonrelativistic dispersion relation [33, 34].\nDispersion relations of the longitudinal and the trans-\nverse modes are respectively\nEk(k) = \u0001 +k2\n2mk; E?(k) =k2\n2m?: (24)\nWe can \fnd another massless excitation near k=\u0019.\nThen the Hamiltonian (17) turns into\n\u0016H=Z\ndx\u0014\n4\u0015M2m2+v\n2g(@xn)2\u0015\n+H0; (25)\nThe repulsion U(n2\u00001)2is transformed into the con-\nstraint n2= 1 again. The last term of Eq. (25) denotes\nthe anisotropy that the spontaneous magnetization in-\nduces,\nH0=VZ\ndx\b\n2(mz)2\u0000(mx)2\u0000(my)2\t\n; (26)\nwithV= 4\u0015M2, which seemingly equals to the coupling\nconstant of m2. However, after including renormaliza-\ntion due to irrelevant operators, the coupling constant\nVof the anisotropic interaction (26) actually deviates\nfrom that of the isotropic part m2. Here we \frst omit\nthe anisotropy (26) and include it later perturbatively\nbecause it does not modify qualitative features of the ef-\nfective Hamiltonian of the Nambu-Goldstone boson. Let\nus rewrite Hamiltonian (25) in terms of nandl,\n\u0016H=Z\ndx\u0014\u0016gu\n2\u0012\nl\u0000\u0002\n4\u0019@xn\u00132\n+u\n2\u0016g(@xn)2\u00003\u0016guM\u0001l\u0015\n:\n(27)\nHere we introduced the coupling constant \u0016 gand the ve-\nlocityuin a parallel manner as Eq. (6). They are givenby\n\u0016g=2M\nS\u0014\n2\u0015X\napa\u0000\u0012\u0002\n4\u0019S\u00132\u0015P\na;bM\u00001\na;b\u0015\u00001=2\n(28)\nu= 2MSa 0\u0014\n2\u0015X\napa\u0000\u0012\u0002\n4\u0019S\u00132\u0015P\na;bM\u00001\na;b\u00151=2\n:(29)\nNote that both \u0016 ganduare positive and proportional to\nthe magnitude of the spontaneous magnetization jMj.\nThe last term of the Hamiltonian (27) can be seen as\nthe Zeeman energy \u0000he\u000b\u0001Sj;l. For further understanding\nof the e\u000bective Hamiltonian (27) of the Nambu-Goldstone\nboson atk=\u0019, we integrate lout:\n\u0016H=u\n2\u0016gZ\ndx\u00141\nu2(@\u001cn+ihe\u000b\u0002n)2+ (@xn)2\u0015\n+ 9\u0016guZ\ndx(M\u0001n)2+i\u0002Q; (30)\nwhere he\u000bis written as\nhe\u000b= 3\u0016guM: (31)\nIf we include the perturbation H0at lowest order, it gives\na correction\nH0'VZ\ndx(M\u0001n)2[3(nz)2\u00001] (32)\nto the Hamiltonian (30). If we use the value V= 4\u0015M,\nH0replaces the term 9\u0016 gu(M\u0001n)2= 9\u0016guM2(nz)2of\nEq. (30) with 27\u0016 guM2(nz)4, which has no impact on\nqualitative aspects of the e\u000bective \feld theory (30).\nTheO(3) NLSM (30) leads to three important conse-\nquences. First, the spontaneous magnetization Mleaves\n\u0002\u0011\u0019(mod 2\u0019) intact. Second, Mgenerates the easy-\nplane anisotropy. Finally, the O(3) NLSM (30) is semi-\nclassical. While the NLSM (11) in the TLL phase has\na coupling g/1=S, the NLSM (30) has \u0016 g/M=S [see\nEq. (28)]. Thus, the NLSM (30) behaves similarly to a\nspin-Se\u000bHeisenberg antiferromagnetic chain with a large\nhalf-integer spin Se\u000b\u0018S=M .\nThe e\u000bective \feld theory at k=\u0019is thus, the TLL\nunder an e\u000bective magnetic \feld he\u000b=he\u000b(0 0 1)T; that\nis,\n\u0016H=u\n2\u0019Z\ndx\u0014\n\u0016K(@x\u0012)2+1\n\u0016K(@x\u001e)2\u0015\n\u0000he\u000b\n\u0019Z\ndx@x\u001e:\n(33)\nwhich is the e\u000bective model for the SMTLL. The TLL\nparameter \u0016Kis determined from the relation [7]\nM=he\u000b\u0016K\n\u0019u: (34)\nThe TLL parameter\n\u0016K=\u0019\n\u0016g/\u0012J2\u0000Jc1\n2\nJ1\u0013\u00001=2\n(35)5\n⇡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⇡(1\u00002M)AAACa3ichVG7SgNBFD1Z3/GR+ChELcQQ0cJwE0TFSrSxEXzFByqyu07ikH2xuwlo8AdsLSzUQkFE/Awbf8DCTxDBRsHGwpvNgqiod5iZM2fuuXNmRnMM6flEDxGlprauvqGxKdrc0toWi7d3rHh20dVFVrcN213TVE8Y0hJZX/qGWHNcoZqaIVa1wkxlf7UkXE/a1rK/54gtU81bMid11WdqfdORQ+mRzNzwdjxBKQqi/ydIhyCBMObt+BU2sQMbOoowIWDBZ2xAhcdtA2kQHOa2UGbOZSSDfYEDRFlb5CzBGSqzBR7zvNoIWYvXlZpeoNb5FIO7y8p+JOmerumF7uiGHun911rloEbFyx7PWlUrnO3YYffS278qk2cfu5+qPz37yGEi8CrZuxMwlVvoVX1p//hlaXIxWR6kC3pi/+f0QLd8A6v0ql8uiMUTRPkD0t+f+yfIZlJjKVoYTUxNhz/RiF4MYIifexxTmMU8snysiSOc4izyrHQpPUpfNVWJhJpOfAkl+QFvkowDAAACa3ichVG7SgNBFD1Z3/GR+ChELcQQ0cJwE0TFSrSxEXzFByqyu07ikH2xuwlo8AdsLSzUQkFE/Awbf8DCTxDBRsHGwpvNgqiod5iZM2fuuXNmRnMM6flEDxGlprauvqGxKdrc0toWi7d3rHh20dVFVrcN213TVE8Y0hJZX/qGWHNcoZqaIVa1wkxlf7UkXE/a1rK/54gtU81bMid11WdqfdORQ+mRzNzwdjxBKQqi/ydIhyCBMObt+BU2sQMbOoowIWDBZ2xAhcdtA2kQHOa2UGbOZSSDfYEDRFlb5CzBGSqzBR7zvNoIWYvXlZpeoNb5FIO7y8p+JOmerumF7uiGHun911rloEbFyx7PWlUrnO3YYffS278qk2cfu5+qPz37yGEi8CrZuxMwlVvoVX1p//hlaXIxWR6kC3pi/+f0QLd8A6v0ql8uiMUTRPkD0t+f+yfIZlJjKVoYTUxNhz/RiF4MYIifexxTmMU8snysiSOc4izyrHQpPUpfNVWJhJpOfAkl+QFvkowD!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!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\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\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(a)(b)\nFigure 3. Dynamical structure factors (a) Sk(k;!) and (b)\nS?(k;!) in the low-energy region. Outside of the shaded\narea, the dynamical structure factor has zero intensity. The\nred lines represent the linear dispersion of the SMTLL and the\nblue dashed curves represent the quadratic dispersions (24) of\nthe nonrelativistic Goldstone mode [ E?(k)] and the massive\nmode [Ek(k)].\ndiverges at the instability point Jc1\n2. The point Jc1\n2brings\nabout a divergence of the susceptibility,\n\u001f/\u0012jJc1\n2\u0000J2j\nJ1\u0013\u0000\r\n; (36)\nwith the critical exponent \ris given by\r= 1=2 forJ2%\nJc1\n2and\r= 1 forJ2&Jc1\n2.\nE. Dynamical structure factors\nWe now use this theory to compute the dynami-\ncal structure factors in the SMTLL phase. We fo-\ncus on longitudinal and transverse dynamical structure\nfactors,Sk(k;!) =R1\n\u00001dtdxei(!t\u0000kx)hSz\nj(t)Sz\n0(0)iand\nS?(k;!) =R1\n\u00001dtdxei(!t\u0000kx)hS+\nj(t)S\u0000\n0(0)i.\nThe longitudinal and transverse dynamical structure\nfactors near k=\u0019are the same as those of the S=\n1=2 Heisenberg antiferromagnetic chain under magnetic\n\feld [7, 35],\nSk(k=\u0019(1\u00002M) +\u000ek;! )\n=\u00192Ck\nu\u00002(\u0016K)\u0012H(!\u0000uj\u000ekj)\u00124u2\n!2\u0000u2(\u000ek)2\u00131\u0000\u0016K\n;(37)\nS?(k=\u0019+\u000ek;! )\n=\u00192C?\nu\u00002(1\n4\u0016K)\u0012H(!\u0000uj\u000ekj)\u00124u2\n!2\u0000u2(\u000ek)2\u00131\u00001=4\u0016K\n:\n(38)\n\u0012H(z) is the Heaviside's step function and CkandC?are\nnonuniversal constants. Equations (37) and (38) hold\nwhenj\u000ekj \u001c 1. The dynamical structure factor near\nk= 0 is given by\nS\u0017(k=\u000ek;! ) =\u0012H\u0000\n!\u0000E\u0017(\u000ek)\u0001C0\n\u0017\n!\u0000E\u0017(\u000ek);(\u0017=k;?);\n(39)whereC0\nkandC0\n?are constants. Figure 3 shows the longi-\ntudinal and transverse dynamical structure factors in the\nlow-energy region. The dynamical structure factor near\nk= 0 (39) clearly shows di\u000berence between the SMTLL\nand either a TLL under magnetic \feld (e.g. the S= 1=2\nantiferromagnetic chain [7]) or a \feld-induced TLL (e.g.\ntheS= 1=2 two-leg spin ladder [36]), for which the sym-\nmetry has been externally broken.\nIII. COMMENSURATE PHASE\nA. Commensurability condition\nLet us now examine the behavior upon increasing J2\nfurther. The spontaneous magnetization saturates at a\ncertain point Jc2\n2(> Jc1\n2). In the case of the UJ lad-\nder, we can \fnd a saturation condition in the spirit of\nthe Oshikawa-Yamanaka-A\u000feck theory [37]. To do so,\nwe need to clarify the physical meanings of the \u001eand\u0012\n\felds of the SMTLL (33). The de\fnitions (2) and (3) of\nnandlindicate that nandl, equivalently \u001eand\u0012, rep-\nresent a \\center-of-mass\" mode. When one consider the\nUJ ladder (1) as a system of three spin chains weakly cou-\npled by the rung and the diagonal interactions [38], each\nspin chain is equivalent to a TLL written in a compact-\ni\fed boson \u001eaand its dual \u0012a(a= 1;2;3). The bosons\n\u001eand\u0012represent the center-of-mass mode because they\nare given by \u001e=\u001e1+\u001e2+\u001e3and\u0012=\u00121+\u00122+\u00123. The\nother \\relative-motion\" modes, \u001e1\u0000\u001e2and\u001e1+\u001e2\u00002\u001e3,\nare massive and negligible in the low-energy e\u000bective \feld\ntheories (12) and (33). The two-site translational symme-\ntryj!j+2 of the UJ ladder (1) requires the invariance\nof the e\u000bective \feld theory under the translation\n\u001e!\u001e+ 6(S\u0000M)\u0019: (40)\nGiven an incommensurate magnetization Msatisfying\n6(S\u0000M)62Z; (41)\nthe incommensurability condition (41) prohibits relevant\ninteractions of \u001e, for instance cos(2 \u001e), to appear in the\ne\u000bective \feld theory (33). Relevant interactions of \u0012are\nnot allowed from another reason, that is, the U(1) sym-\nmetry of the ground state [37].\nEquation (41) shows that the \u001e\feld can be massive\nwhen the incommensurability condition (41) is violated.\nIncreasingMfrom zero, the condition (41) \frst breaks\ndown when\nM\nMs=1\n3: (42)\nHereMs=Sis the saturated value of M. Thus, a com-\nmensurate phase as the 1 =3-plateau (42) should exist.\nThe commensurate phase has only one massless Nambu-\nGoldstone boson near k= 0 because the SMTLL acquires\na mass from a relevant interaction cos(2 \u001e). The condition\n(42) gives the saturation condition of the UJ ladder.6\nB. Trimer-spin chain\nIn order to complete the above derivation of the spon-\ntaneous magnetization, we show that for large J2it can\nbe shown to occur directly from the lattice model (1).\nThe commensurate phase is identi\fed as a ferromag-\nnetic phase of trimers formed on diagonal J2bonds\n(Fig. 4 (a)). Let us consider the case J2=J1\u001d1. When\nJ1= 0, the UJ ladder (1) is composed of an S= 1=2\ndiamond chain [39, 40] and isolated spins. Three spins\nS2j+1;1;S2j;2andS2j\u00001;3form a trimer [a solid rectangle\nin Fig. 1 (b)] on the strongest J2bond. To describe the\nground state and the lowest-energy excitation, we may\nreplace the three spins with an S= 1=2 pseudo spin [40],\nS2j+1;1=S2j\u00001;3=2\n3Tj;S2j;2=\u00001\n3Tj: (43)\nThe eigenstates, j *ijwithTz\nj= 1=2 andj +ijwith\nTz\nj=\u00001=2 are written as\nj*ij=1p\n6(j#ij;1j\"ij;2j\"ij;3\u00002j\"ij;1j#ij;2j\"ij;3\n+j\"ij;1j\"ij;2j#ij;3); (44)\nj+ij=1p\n6(j\"ij;1j#ij;2j#ij;3\u00002j#ij;1j\"ij;2j#ij;3\n+j#ij;1j#ij;2j\"ij;3): (45)\nHerej\"ij;aandj#ij;aare the eigenstate of Sz\n2j+2\u0000a;a\n(a= 1;2;3). Figure 4 (a) depicts that the mapping from\nspins to a trimer metamorphoses an antiferromagnetic in-\nteraction\u000bJ2S2j;2\u0001(S2j\u00001;1+S2j+1;3) to a ferromagnetic\ntrimer-trimer interaction,\n\u000bJ2S2j;2\u0001(S2j\u00001;1+S2j+1;3) =\u0000JFTj\u0001Tj+1 (46)\nwithJF=4\u000bJ2\n9+O(\u000b2J2) [40]. Since the trimer-trimer\ninteraction is ferromagnetic, the ground state at J1= 0\nhas a nonzero magnetization.\n(b)|*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|*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(a)\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\nFigure 4. (a) A con\fguration of trimers (solid rectangles) at\nJ2=J1\u001d1. Trimers are surrounded by spins (circles). In the\nlow-energy limit, one can regard the trimer as the S= 1=2\npseudo spin. The trimer-trimer interaction is ferromagnetic\nand the trimer-spin interaction is antiferromagnetic. (b) The\ne\u000bective model that describes the commensurate phase. The\nsolid and blank circles represent trimers ( S= 1=2 pseudo\nspins) and S= 1=2 spins, respectively. The thick line repre-\nsents the ferromagnetic coupling (46) of trimers and the thin\nlines are the antiferromagnetic coupling of trimers and spins.AtJ1= 0, the residual spins [depicted as blank circles\nin Fig. 4 (a) and (b)] are isolated from the trimers. The\nnonzeroJ1switches on trimer-spin interactions. The low-\nenergy e\u000bective Hamiltonian for J2=J1\u001d1 is given by\nHferri=\u0000JFX\nj2ZTj\u0001Tj+1+X\ni;j2ZJij~Si\u0001Tj; (47)\nwhere spins not participating in forming trimers are rela-\nbeled as ~Sj. Fig. 4 (b) shows interactions of the e\u000bective\nmodel (47). If we are concerned only with the ground\nstate magnetization of the trimer-spin chain (47), we do\nnot even need the details of coupling constants. We use\nonly three facts, JF>0,Jij\u00150 andP\nl2ZJil>0\nfori;j2Z. These conditions enable us to apply the\nMarshall-Lieb-Mattis theorem [41, 42] to the model (47).\nThis theorem imposes that the ground state of the trimer-\nspin chain (47) must have a \fxed magnetization irrespec-\ntive of parameters JFandJij. The ground state magne-\ntization of the commensurate phase is exactly Eq. (42).\nSince the ferromagnetic order of the trimer is exactly\nthe ferrimagnetic order of spins, the commensurate phase\nis exactly the ferrimagnetic phase. Figure 5 shows the\nground state magnetization of the UJ ladder, which re-\nproduces the numerically derived one [24] in the \u000b= 1\ncase. The Marshall-Lieb-Mattis theorem allows us even\nto take a limit \u000b!1. However, compared to the nu-\nmerical result [24], we overestimated Jc1\n2because of the\nimbalance\u000b\u001c1.\nIV. RELATION TO THE GENERAL THEOREM\nNow that we have a description of the SMTLL and its\nNambu-Goldstone boson, we can compare our result with\nthe general theorem [15{17] claiming that the number of\nbroken generators of the symmetry group determines the\nnumber of Nambu-Goldstone boson. The canted phase\ngenerally has a nonrelativistic Nambu-Goldstone boson\nand a relativistic Nambu-Goldstone boson [16, 43], which\nis true in the 2D UJ antiferromagnet [26]. In the 1D case\n(1), theU(1) symmetry is recovered for the ground state\nas a result of quantum \ructuations. In the TLL phase,\neven the full SU(2) symmetry is recovered. Therefore, we\nconclude that the general theorem [16, 17] is applicable\nto 1D systems at the classical level.\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\nFigure 5. A schematic magnetization curve of the UJ lad-\nder (1).Jc1\n2andJc2\n2represent quantum critical points. The\nSMTLL phase exists in the region Jc1\n2 xM (see Supplementary Fig . S4(a) for more details ) because, when \nx < xM (x > xM), the Fe moment is parallel (antiparallel) to the total magnetization and to a strong applied \nperpendicular magnetic field ( Hz). The 3d states of Fe govern the anomalous Hall effect because the 4 f states of \nTb is expected to be located well below the Fermi level and are less involved in transport phenomena of FeTb. \n \nGiant bulk spin -orbit torque \nWe measure the efficiencies of SOTs in the perpendicularly magnetized FeTb samples using the polar -angle \ndependent harmonic Hall voltage response (HHVR) technique[33,34], after carefully taking into account current -\ninduced heating and thermoelectric effect s (Supplementary Sec. 5 and 6 ). This HHVR technique is accurate when \nthe magnetization rotate s coherently at small polar angles (𝜃M). This condition is fulfilled in the FeTb samples , as \nindicated by a well-defined parabolic scaling of the first harmonic Hall signal versus 𝜃M (Supplementary Fig . S8). \nTo determine the dampinglike SOT, we rotate the magnetization by scanning a fixed magnitude of magnetic field \n(𝐻𝑥𝑧) relative to the sample at small values of θM in the x-z plane ( Fig. 2a ), and collect the first and the second \nHHVRs, Vω and V2ω, as a function of θM under the excitation of a low-frequency sinusoidal electric field E in the x \ndirection . As we discuss in detail in the Supplementary Sec. 6, the HHVR signal s are given by \nVω = VAHE cos 𝜃M, (1) \nV2ω ≈ (1\n2𝑉AHE𝐻DL\n𝐻k+𝐻𝑥𝑧 + 𝑉ANE ,𝑧) sin 𝜃M+𝑉ANE ,𝑥, (2) \nwhere 𝑉AHE is the anomalous Hall voltage, 𝑉ANE ,𝑧(𝑥) is the anomalous Nernst voltage induced by an out-of-plane \n(in-plane) temperature gradient, and 𝐻DL is damping -like effective SOT field. As shown in Fig. 2(b), the measured \nV2ω varies linearly with sin𝜃M for each fixed magnitude of Hxz. The value of HDL can be obtained from the fits of \ndata to Eq. ( 2) as shown in Fig. 2(c). In this determination we ignore the so -called “planar Hall correction”[35] \nbecause the planar Hall resistance ( RPHE) samples is negligibly small compared to RAHE (|RPHE/RAHE| ≤0.04, \nSupplementa ry Fig. S9), and even if this were not the case the planar Hall correction is generally found to give \nincorrect values when it is not negligible[36-38]. As shown in Fig. 2(d) , HDL for the FeTb single layers with different \nx increase much more slowly than 1/Ms scaling upon approaching the magnetization compensa tion point (xM ≈ \n47), which is in sharp contrast to the behavior observed for HM/FM bilayers in which HDL is proportional to 1/ Ms. \nHDL reverses sign betwee n xM and the angular momentum compensation point ( xA ≈ 38, see below ). \nUsing the obtained HDL values, we calculate 𝜉DL𝑗 of these FeTb single layers following : \n𝜉DL𝑗 ≡ js /j = (2𝑒/ℏ)𝐻DL𝑀s𝑡/𝑗 (3) \nwhere e is the elementary charge, ћ the reduced Plank’s constant, Ms the saturation magnetization of the spin \ncurrent detector, t the thickness of the spin current detector, js the spin current density absorbed by the spin current \ndetector, and j = E/ρ xx the current density in the spin current generator with electrical resistivity ρxx (Supplementary \nFig. S10). As we justify in the Supplementary Sec. 10, Eq. (3) hold s for FIMs regardless of the sign of effective \ngyromagnetic ratio ( γeff). As plotted in Fig. 2(e) , 𝜉DL𝑗 of the 20 nm FeTb first increases rapidly from +0.11 at x = \n29 to +0.41 at x = 37, then (like 𝐻DL) suddenly becomes negative for 38 < x < 47, and finally becomes positive 4 \n again and starts to decrease from the value + 0.16 at x = 49 upon further increase of x (see Supplementary Sec. 11 \nfor m ore details of the torque determination ). This sign reversal of the dampinglike spin -orbit torque is re affirmed \nby the opposite polarity of the current -induced magnetization switching of the Fe 57Tb43 and the Fe 67Tb33 \n(Supplementary Sec. 16). We find that the sign reversal appears to be correlated to that of the angular momentum. \nIn Fig. 2(f), we show the effective gyromagnetic ratio for the 20 nm FeTb with different composition as calculated \nusing the relation [39-41] 𝛾eff=(𝑚Fe−𝑚Tb)/(𝑚Fe/|𝛾Fe|−𝑚Tb/|𝛾Tb|), the magnetic moments of the two \nsublattices mFe = (1-0.01x)MFe and mTb = 0.01 xM Tb, and the individual gyromagnetic ratios γFe = -2.1μB/ℏ[42] and \nγTb = -1.5μB/ℏ[43]. The composition of the angular momentum compensation point , where the total angular \nmomentum S = 𝑚Fe/|𝛾Fe|−𝑚Tb/|𝛾Tb| is zero, is estimated to be xA ≈ 38.5 for the 20 nm FeTb at the room \ntemperature . We note that 𝛾eff, xA, and xM in Figs. 2(d) -2(f) are only the 20 nm FeTb samples and different from \nthat for the thicker films (e.g. for the films in Fig. 2(g), xA < xM < 43). The FeTb also shows a field-like torque that \nis relatively small compared to the damping -like torque (Supplementary Fig . S12). \n \nBulk characteristics and m icroscopic origin \nTo analyze these data , we first show that the strong damping -like torque we observe within FeTb is a bulk \neffect. Qualitatively similar to previous measurements of CoPt single layers[26], 𝜉DL𝑗 of the FeTb layers increases \nlinearly with layer thickness when the thickness is greater than about 40 nm as shown in Fig. 2( g) for Fe 57Tb43, \nyielding in the bulk limit a SOT efficiency per thickness of 𝜉DL𝑗/t = 0.036 ± 0.008 nm-1. This behavior is not \nconsistent with an interfacial torque , for which 𝜉DL𝑗 should be approximately independent of the magnet ic-layer \nthickness[24,44]. We also find that this torque is insensitive to the details of the sample interfaces because we measure \nessentially the same value of 𝜉DL𝑗 from symmetric MgO/Fe 61Tb39 20 nm/MgO samples and asymmetric \nSiO 2/Fe 61Tb39 20 nm/MgO samples. We thus conclude from these characteristics that the damping -like spin torque \nin FeTb single layers is a bulk effect . This bulk torque is microscopically distinct from the previously reported \n“interface -engineered” self -torque concluded from a study of GdFeCo [22]. \n We suggest that the source of the strong damping -like SOT in the perpendicularly magnetized FeTb is most \nlikely a strong conventional bulk spin Hall effect (SHE). We have considered the possibility of origins associated \nwith the anomalous Hall effect or planar Hall effect , but these can only generate spin polarization collinear with \nthe magnetization[45-47]. Magnetic and antiferromagnetic spin Hall effects[48,49] are also not relevant because they \nare odd under time reversal , while we find the damping -like torque efficiencies generated in FeTb for a given \napplied electric field does not reverse orientation when the magnetization reverses. We have further verif ied the \nexistence of a strong SHE in FeTb by measuring the spin current emi tted by FeTb layers . We perform ed thickness -\ndependent spin -torque ferromagnetic resonance (ST -FMR) experiments[50,51] on a control sample of Fe 50Tb50 (20 \nnm)/Ti (1 nm)/Fe ( 3.8-10.5 nm) and used the in -plane magnetized Fe layer to detect the spin current emitted from \nthe FeTb ( Fig. 2( h)). Here, the 1 nm Ti spacer layer was used to suppress the exchange coupling between the FeTb \nand the Fe layers (Supplementary Fig. S13(b) ). The PMA FeTb produces no measurable FMR excitation under \nthe condition of small in-plane magnetic field, so the ST -FMR signal we measure from the FeTb /Ti/Fe trilayers \ncorresponds only to magnetic dynamics from the Fe layer . If we define the apparent FMR spin -torque efficiency 5 \n (𝜉FMR) from the ratio of the symmetric and anti -symmetric components of the magnetoresistance response of the \nST-FMR (Supplementary Sec. 13), the actual efficiency of the damping -like torque acting on the Fe layer due to \nthe spin current emitted by the Fe50Tb50 (𝜉DL,ext𝑗\n ) can be determined by the method of ref. [ 51] based on the y -axis \nintercept in a linear fit of 1/ ξFMR versus 1/ tFe. As shown in Fig. 2( i), we measure 𝜉DL,ext𝑗\n = 0.16 ± 0.02 for \nFe50Tb50/Ti/Fe, which is 3 times stronger than that of Pt/Fe bilayers (0.051 ± 0.002 , also shown in Fig. 2(i) ) for Pt \nwith resistivity 38 μΩ cm . We have also measured 𝜉DL,ext𝑗\n for x = 43 (where 𝜉DL𝑗 is negative) and for x = 29 (on \nthe other side of the angular momentum compensation point where 𝜉DL𝑗 is positive again), and we find that the sign \nof 𝜉DL,ext𝑗 is unambiguously positive at all three concentrations. The value of 𝜉DL,ext𝑗\n that we quote for Fe 50Tb50 \nonly represents a lower bound for the internal value of spin Hall ratio because the torque applied to the Fe is \nreduced by spin attenuation in the Ti spacer[26,52], interfacial spin backflow[11], and spin memory loss[53]. Spin \nmemory loss, in particular, should be significant at the Ti/Fe interface because it possesses strong interfacial spin-\norbit coupling[11] as indicated by the large interfacial magnetic anisotropy energy density of 1.43 ± 0.05 erg/cm2 \n(Supplementary Sec. 14). \nA non -zero SOT in a single magnetic layer requires that the sample structure is not symmetric relative to a \nmirror parallel to the sample plane[16,26]. The required broken symmetry within the FeTb layers seems unrelated to \nany vertical composition gradient because there is no evidence of a composition gradient in the EELS studies of \nour films. We also find that a deliberately introduced vertical composition gradient does not enhance the damping -\nlike torque in FeTb. A control sample of 20 nm thick Fe 100-xTbx in which x varied from 27 to 41 with thickness \ngave 𝜉DL𝑗 of 0.05± 0.01 (Supplementary Fig. S15), which is similar to the average d value of whole film over the \nthickness using the composition -dependent values in Fig. 2(e) , but significantly smaller in magnitude than -0.46 \nfor 20 nm Fe 61Tb39. In addition, the source of the symmetry breaking is not a vertical thermal gradient because the \nmagnitude of HDL scales in proportion to the applied electric field ( Supplementary Fig. S7) and thus 𝜉DL𝑗 is \nindependent of the applied electric field (symmetry breaking due to Joule heating would give 𝜉DL𝑗∝𝐸2). \n \nStrong composition dependence and sign change \nWe now turn to analyz e the dependence on x of the bulk anti -damping spin torque efficiency 𝜉DL𝑗 for Fe 100-\nxTbx (Fig. 2(e) ). We observe that |𝜉DL𝑗| for the 20 nm Fe 100-xTbx samples shows a broad peak around x = 43, \nsuggesting an enhanced SHE in the intermediate composition range , near and between the two compensation \npoints for magnetization and angular momentum . |𝜉DL𝑗| reaches 0.5 for 20 nm Fe57Tb43 films and 3 for 90 nm \nFe57Tb43 films . Our result differs from the case of GdFeCo, which was reported to have zero self -torque at the \ncompensation point of angular momentum in a previous temperature -dependen ce study [22]. \nThe sign change of 𝜉DL𝑗 that we observe in the 20 nm Fe 100-xTbx samples between the compensation points for \nmagnetization and angular momentum appears to be correlated with the relative orientation of the magnetization \nand angular momentum vector . Outside the region between the two compensation points the magnetization \n(𝑚𝐹𝑒−𝑚𝑇𝑏) and angular momentum (𝑠Fe−𝑠Tb) are antiparallel (𝛾eff<0), but between the compensation points 6 \n the total magnetization becomes parallel to the total angular momentum (𝛾eff>0). A change in the sign of 𝜉DL𝑗 \nindicates a change in the sign of the spin angular momentum being transferred to the magnet. However, our ST -\nFMR measurements on Fe 100-xTbx/Ti/Fe indicate no sign change in the polarization of the spin current emitted from \nFeTb regardless of composition. The microscopic origin of the sign change of 𝜉DL𝑗 remains a puzzle and worth \nstudy in the future . \n \nPractical impact and self -torque -driven magnetization switching \nFrom technological point of view, a strong bulk torque can be advantageous by itself or in combination with \ninterface -applied torques for applications, such as perpendicular magnetic recording and chiral domain \nwall/skyrmion devices, that require relatively large thickness for high thermal stability. The damping -like SOT \nefficiency per unit thickness that we measure in the bulk limit for Fe 57Tb43, 𝜉DL𝑗/t ≈ 0.036 nm-1, is much greater \nthan previous reports for other magnetic single layers, e.g ~ 0.0017 nm-1 for in-plane NiFe[25], ~ -0.008 nm-1 for \nin-plane CoPt[26], ~ 0.005 nm-1 for in -plane FePt[16], and ~ 0.016 nm-1 for perpendicular GdFeCo[22]. Here we do \nnot compare our 𝜉DL𝑗/t result with those out -of-plane HHVR results obtained by applying a large “planar Hall \ncorrection” (e.g. 0.045 nm-1 for L10-FePt single crystals in ref. [19]), because , as we noted above, the planar Hall \ncorrection is generally foun d to give incorrect values when it is not negligible [36-38]. \nThe bulk SOT of FeTb is sufficiently strong to drive SOT switching of layers with very large thicknesses and \nstrong PMA . In Fig. 3(a)-3(d) we compare magnetic -field-driven switching and SOT switching for both a 20 nm \nFe67Tb33 device (x < xA, Fe-dominated, Ms = 250 emu/cm3, Hk=33.5 kOe, Hc=1.59 kOe) and a 20 nm Fe 42Tb58 \ndevice (x > xM, Tb-dominated, Ms= 16 4 emu/cm3, Hk=17.5 kOe, Hc=1.72 kOe) as two representative examples. \nFigs. 3(a) and 3(b) show the Hall resistance ( RH) of the samples as a function of Hz, which indicate sharp full \nswitching for both samples with ΔRH = 2RAHE = +11 Ω (-12.2 Ω) for the Fe- (Tb-) dominated sample . In Figs. 3(c) \nand 3(d), we show RH of the two samples measured following the application of sequences of current pulses of \ndifferent amplitudes (0.2 seconds in duration) under the application of a constant symmetry -breaking in-plane bias \nfield Hx along the current direction ( x direction). We measure a switching current density of only (8.2± 1.2) × 106 \nA/cm2 for the 20 nm Fe 67Tb33 and (5.5± 0.2) × 106 A/cm for the 20 nm Fe39Tb61. The current -driven switching is \nonly partial ( ∼16% of the full value of Δ RH for magnetic -field-driven switching) , likely because the non-uniform \npinning impedes free motion of domain walls in this domain -wall-mediated switching regime. Full current -driven \nreversal is likely still possible in nanodot device s with improved magnetic homogeneity as recently demonstrated \nin CuPt/CoPt bilayers[54]. Here, the switching chirality is opposite for the Fe -dominated Fe67Tb33 and Tb -dominated \nFe42Tb58, i.e., clockwise (anti -clockwise) for the former but anti -clockwise (clockwise) for the latter when Hx > 0 \n(Fig. 3(c) -3(d)). This is because the SOT fields are of the same sign for the two samples (𝜉DL𝑗> 0), but the \nanomalous Hall resistances are of opposite sign s (ΔRH > 0 for Fe 67Tb33, but <0 for Fe 42Tb58). We also note that the \n20 nm Fe 57Tb43 (xA < x < xM, ΔRH > 0, 𝜉DL𝑗< 0) can be also switched at a low current density of (5.5±0.1)× 106 A/cm \n(Supplementary Fig. S16(c) ), but the switching polarity is opposite to that of the Fe 67Tb33 (x < xA, ΔRH > 0, 𝜉DL𝑗> \n0) due to the negative sign of the bulk spin-orbit torque in the Fe 57Tb43. 7 \n \nConclusion \nWe have demonstrated a giant damping -like SOT arising from the SHE in composition -uniform, amorphous, \nsputter -deposited ferrimagnetic Fe100-xTbx single layers with giant PMA . This bulk torque exhibits no apparent \ncorrelation to the interfaces or the absence/presence of a composition gradient. The torque reaches a constant value \nof efficiency per unit layer thickness in the bulk limit , 𝜉DL𝑗/t ≈ 0.036 nm-1. This is more than twice greater any \nprevious report for other magnetic single layers. The torque varies strongly with composition and achieves giant \nefficiencies |𝜉DL𝑗| of 0.5 for 20 nm Fe61Tb39 and 3 for 90 nm Fe57Tb43. Interestingly , the torque become s negative \nin sign in the intermediate composition range where total angular momentum becomes parallel to the \nmagnetization rather than antiparallel . We also show that the bulk SOT can drive switch ing in tens of nm thick \nFeTb layers with strong PMA and high coercivity . For example, the bulk SOT can switch a 20 nm FeTb at very \nlow current densit ies of a few M A/cm2. Our findings of giant bulk SOT efficiency and intriguing torque -\ncompensation correlation will stimulate study of such unique spin -orbit phenomena in a variety of ferrimagnetic \nhosts. Our work suggest s a promising strategy for self -driven -switching perpendicular ferrimagnetic devices with \nlow power , high density, and straightforward integration with CMOS circuit s because there is no requirement for \nepitaxy or composition gradient . \n \nData availability \nThe data that support this study are available from the corresponding author upon reasonable request. \nAcknowledgement s \nThe authors thank Robert A. Buhrman for support. This work was funded in part by the Office of Naval \nResearch (N00014 -19-1-2143 ), in part by the Defense Advanced Research Projects Agency \n(USDI D18AC00009), and in part by the NSF MRSEC program (DMR -1719875) through the Cornell Center for \nMaterials Research . Device fabrication was performed at the Cornell Nanofabrication Facility, in part by the NSF \n(NNCI -2025233 ) as part of the National Nanotechnology Coordinated Infrastructure , and in part by the Strategic \nPriority Research Program of the Chinese Academy of Sciences (XDB44000000). Q. Liu acknowledges the \nfinancial support by the China Scholarship Council (File No. 201906460052). \n \nConflict of Interest \nThe authors declare no conflict of interest. \n \nAdditional Information \nSupplementary Information is available for this paper at xxxxx . \n \nReferences \n[1] C. Kaiser, A. F. Panchula, S. S. P. Parkin, Phys. Rev. Lett. 95, 047202 (2005 ). \n[2] H. Awano, J. Magn. Magn. Mater. 383, 50 (2015) . \n[3] N. Roschewsky, T. Matsumura, S. Cheema, F. Hellman, T. Kato, S. Iwata, S. Salahuddin, Appl. Phys. Lett. 8 \n 109, 112403 (2016) . \n[4] J. Finley, C. Lee, P. Y . Huang, L. Liu, Adv. Mater. 31, 1805361 (2019) . \n[5] K. Kim, S. K. Kim, Y . Hirata, S. Oh, T. Tono, D. Kim, T. 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(d ) Perpendicular magnetic anisotropy field ( Hk), (e) \nSaturation magnetization ( Ms), (f) Coercivity (Hc), and (g) Anomalous Hall resistance ( RAHE) of Fe 100-xTbx films \nwith varying Tb concentration (x). Here, the values of Hk, H c, and RAHE are determined from transport \nmeasurements. \n \n \n \n11 \n \nFig. 2 Spin -orbit torque s. (a) Geometry of the HHVR measurement. (b) Second HHVR V2ω vs sin θM for a 20 nm \nFe71Tb29 sample under for constant magnitudes of applied magnetic field Hzx = 20 and 80 kOe . (c) dV2ω/dsinθM vs \nVAHE/2(Hk+Hzx) for 20 nm Fe 71Tb29. Dependence on the Tb concentration x for (d) the d amping -like effective SOT \nfield HDL, (e) the d amping -like torque efficiencies per current density 𝜉DL𝑗 and |𝜉DL𝑗|, and (f) the calculated value of \n𝛾eff for the 20 nm Fe 100-xTbx. In (d) -(f), the dashed lines indicate the angular momentum compensation point xA \n(blue dashed line) and the magnetization compensation point xM (red dashed line). ( g) 𝜉DL𝑗vs. the thickness (t) of \nFe47Tb43 samples (𝜉DL𝑗> 0 because the composition x = 43 is located in the Tb -dominated regime when the thickness \nis greater than ≈30 nm, see the Supplementary Fig. S4(e) ). (h) Schematic of ST-FMR measurements on Fe50Tb50 \n(20 nm)/Ti (1 nm)/Fe ( tFe) samples. (i) Inverse FMR efficiency (1/ ξFMR) vs. inverse Fe thickness (1/ tFe) for the \nFe50Tb50 (20 nm)/Ti (1 nm )/Fe (tFe nm) sample and a control sample Pt (4 nm)/Fe ( tFe). \n \n12 \n \nFig. 3 . Anomalous Hall resistance hysteresis of (a) a 20 nm thick Fe67Tb33 single layer (xxM, Tb-dominated, Ms= \n164 emu/cm3, Hk=17.5 kOe, Hc=1.72 kOe ). Current induced magneti zation switching of (c) the Fe67Tb33 and (d) \nthe Fe42Tb58, under a constant in-plane bias field Hx = ±3 kOe that overcomes the DMI field within the domain \nwalls . \n \n1 \n Supplementary Information for \n \nGiant bulk spin-orbit torque and efficient electrical switching in single ferrimagnetic FeTb layers \nwith strong perpendicular magnetic anisotropy \n \nQianbiao Liu1,2+, Lijun Zhu1,2+*, Xiyue S. Zhang2, David A. Muller2, Daniel C. Ralph2,3 \n \n1. State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of \nSciences, Beijing 100083, China \n2.Cornell University, Ithaca, New York 14850, USA \n3. Kavli Institute at Cornell, Ithaca, New York 14850, USA \n \n+These authors contributed equally to this work. \n*ljzhu@semi.ac.cn \n \n \nTable of Contents: \nSection 1. Sample fabrication and characterizations \nSection 2. Thickness gradient of FIB -thinned STEM samples \nSection 3. Magnetic properties of Fe 100-xTbx single layer s \nSection 4. Anomalous Hall resistance and coercivity of Fe 100-xTbx single layer \nSection 5. Estimation of current -induced temperature increase \nSection 6. Subtraction of effects of anomalous Nernst vo ltage from harmonic Hall voltage response \nSection 7. Coherent rotation at small polar angles \nSection 8. Validation of Eq. (3) in the main text \nSection 9. Planer Hall resistance \nSection 10. Resistivity of the 20 nm Fe 100-xTbx \nSection 11. Raw data for representative samples exhibiting different torque signs \nSection 1 2. Field -like spin -orbit torque in Fe 100-xTbx single layer \nSection 1 3. Spin -torque ferromagnetic resonance measurements \nSection 1 4. Interfacial perpendicular magnetic anisotropy energy density \nSection 1 5. HHVR measurement of a 20 nm thick composition -gradient Fe100-xTbx (x = 27→41) \nSection 1 6. More examples of magneti zation switching by the bulk spin -orbit torque \n \n \n \n \n \n \n \n \n 2 \n Section 1. Sample fabrication and characterization s \nA series of Fe 100-xTbx (FeTb for short) single layers with different fixed Tb volume percentages ( x = 15-\n78) were deposited on oxidized Si substrate s by co -sputtering at room temperature , and then protected by capping \nwith MgO (1.6 nm)/Ta (1.6 nm) without breaking vacuum . The Tb volume percentages were calibrated from the \nfraction of the Tb deposition rates over the total deposition rate during the co -sputtering deposition. The argon \npressure was 2 mTorr during the sputteri ng process, and the base pressure was ~10-9 Torr. To make devices for \ndetermin ing the efficiencies of the spin-orbit torques by harmonic Hall voltage response (HHVR) measurements \nand spin -torque ferromagnetic resonance (ST-FMR) measurements , the layers were patterned by photolithography \nand ion milling into Hall bars (5×60 μm2, Fig. S1a ) and simple microstrips (10×20 μm2, Fig. S1b ) followed by \ndeposition of 5 nm Ti and 150 nm Pt as electrical contacts . \n \n \nFigure S1 . Optical images of (a) a Hall bar device and (b) a ST-FMR device . \n \nThe magnetic properties of the films were characterized using a superconducting quantum interference \ndevice -vibrating sample magnetometer (SQUID -VSM). The sample for scanning transmission electron \nmicroscopy (STEM) imaging and electron energy loss spectrum (EELS) measurements was thinned using a \nfocused ion beam (FIB) system using a FEI/Thermo Fisher Titan Themis STEM system at 300 kV. Anomalous \nHall resistance and planar Hall resistance were measured using a physical property measurement system (PPMS). \nDuring HHVR measurements, a Signal Recovery DSP Lock -in Amplifier Model 7625 was used to source a \nsinusoidal current onto the Hall bars and to detect the first and second harmonic Hall voltage responses. During \nthe switching measurements, a pulsed write current with a duration of 0.2 ms was source d to the Hall bar devices \nusing a Keithley 2400 . The anomalous Hall voltage was detected either by a Keithley 2182A after each pulse d \nwrite current (with a reading current of 0.1 mA ) or by a Signal Recovery DSP Lock -in Amplifier Model 7625 \n(with a reading excitation of 0.1 V ). For S T-FMR measurements, the rf current was sourced by signal generator \n(E8257D ) and the mixing voltage was detected using a Signal Recovery DSP Lock -in Amplifier Model 7625. All \nthe measurements were performed at room temperature . \n \n \n3 \n Section 2. Thickness gradient of the FIB-thinned STEM sample \n \nOur standard FIB thinning process typically results in a thickness gradient in the cross -sectional view of \nSTEM sample s, with the top side of film thinner than the substrate side. This thickness gradient may cause \nsignificant variation of thickness -sensitive signals (e.g. the absolute intensity EELS), but it usually does not affect \nthe relative strengths of the signals from different elements . Therefore, the quasi -linear variation of the absolute \nEELS intensity of the 66 nm FeTb sample in Fig. 1(b) of the main text is simply due to the thickness gradient \nformed during the FIB thinning process and should not be misinterpreted as a composition gradient. \nFor a detailed understanding of the formation of the thickness gradient, we now briefly describe the standard \nFIB thinning process we employ in this work. We first protect the initial thin -film sample surface by depositing \n20-30 nm carbon and 0.8 -1 µ m Pt and prepare a lamella on the specimen using a Ga ion beam. The lamella is \ntransferred to a TEM grid u sing a needle and then fixed by sputtering the “ paste ” Pt ( Fig. S2(a)), followed by \nrepeatedly cleaning the cross -section by 30 keV Ga ion milling from both sides until the thickness is down to 200 -\n500 nm. The lamella is then thinned further from both side s using 5 keV Ga ions and smaller milling box (see Fig. \nS2(b) until the protective Pt layer on the top is reduced to 0-20 nm ). Because the lamella is tilted 2 -3 degrees \nduring the Ga ion beam thinning process, the end result includes a thickness gradient with the top of the lamella \nthe thinnest. \n \n \nFigure S2 FIB thinning process . (a) Carton of a lamella attached to the TEM grid before thinning. (b) Scanning \nelectron microscopy of a lamella after thinning from both sides until the top protective Pt layer reaches 0 -20 nm. \nThe thinning box is moved gradually toward the top and the inside of the lamella . A thickness gradient with the \ntop side thinnest is finally formed because of the tilted thinning from both sides . \n \n \n4 \n Section 3. Magnetic properties of Fe100-xTb x single layers \n \nWe obtained FeTb single layers with square perpendicular magnetization hysteresis loops over a wide \ncomposition range (the Tb composition x = 29-61, Fig. S3(a)) and over a wide thickness range (even down to 7 \nnm, Fig. S3(b)). The saturation magnetization and the perpendicular magnetic anisotropy field of the FeTb can be \ntuned strongly by the composition ( Figs. S3(c) and S3(d)) or by the layer thickness (Figs. S3(e)). The thickness \ndependence of the magnetic properties was interesting and was attributed to a varying alignment of the Tb \nmagnetic moments with the layer thickness in a previous report [S1]. \n \n \nFigure S3. Magnetic properties of Fe 100-xTbx single layer s as measured by SQUID -VSM . The out -of-plane \nmagnetization hysteresis loops ( M-H curves ) of (a) 20 nm -thick Fe100-xTbx single layer s with different Tb \nconcentration s (x = 31 -61) and (b) a 7 nm thick Fe57Tb43. (c) The s aturation magnetization ( Ms) and (d) the \nperpendicular magnetic field (Hk) of the 20 nm Fe100-xTbx thin films plotted as a function of the Tb concentration . \nThe values of Hk around x = 47 are not measureable by SQUID . (e) Ms and the perpendicular coercivity of the \nFe57Tb43 films plotted as a function of the layer thickness. \n \n5 \n Section 4. Anomalous Hall resistance and coercivity of Fe100-xTb x single layer s \n \nFigure S4. (a) Anomalous Hall resistance ( RH) hysteresis of 20 nm Fe100-xTbx with different composition s. RH is \nnegative for x < xM and positive for x > xM (xM = 47 for the 20 nm films ). (b) The coercivity of the 20 nm Fe100-\nxTbx as measured from the electrical measurement (red points) and magnetic hysteresis measurement (black points) . \nThe coercivity difference around the magnetic compensation point is attributed to enhanced domain pinning during \ndevice fabrication . Anomalous Hall resistance hysteresis of (c) 7 nm , (d) 20 nm and ( e) 66 nm Fe 57Tb43. (f) \nAnomalous Hall resistance and coercivity of the Fe57Tb43 films with different thicknesses . Thickness -induced sign \nreversal of the anomalous Hall resistance indicat es a compensation thickness of ≈30 nm for the Fe57Tb43. The bias \ncurrent is 0.1 mA during the measurement so that Joule heating is negligible. \n \nSection 5. Estimation of current -induced temperature increase \nWe can estimate the temperature increase of our FeTb samples induced by Joule heating during \nmeasurements of spin-orbit torque and current -driven switching by simultaneously measuring the overall change \nin the sample resistance . We first measure the resist ance of the Hall bar devices as a function of temperature using \na small dc bias of 0.1 mA (see Fig. S5(a) for example of 20 nm Fe 59Tb41) so that the result is not affected by \ncurrent -induced heating . Then, the temperature rise during a harmonic Hall voltage response measurement can be \ncalibrated according to the resistance. As summarized in Fig. 5(b), the current -induced temperature increase (∆T) \nrelative to 300 K varies from 11 K to 22 K, depending on the Tb concentration ( thus resistivity) of the FeTb. This \ntemperature increase is sufficient to affect the saturation magnetization of FeTb during harmonic Hall voltage \nresponse measurements ( Fig. S5(c)). Therefore, in the calculation of spin orbit torque efficiency in our samples \nwe have used the magnetization at the calibrated temperature rather than at 300 K . \n6 \n \nFigure S5. Current -induced temperature increase. (a) Temperature dependence of the longitudinal resistance (Rxx) \nof a 20 nm thick Fe59Tb41 sample measured with a small DC current (0.1 mA in a Hall bar that is 5 µm wide and \n60 nm long). (b) The calibrated temperature increase ( ∆𝑇) relative to 300 K of the FeTb with different Tb \nconcentration during the h armonic Hall voltage response measurements . (c) Variation of the magnetization of a \n20 nm thick Fe59Tb41 layer as a function of temperature. \n \nSection 6. Subtraction of effects of anomalous Nernst voltage from harmonic Hall voltage response \n \nFigure S6. Schematic of measurement coordinates. \n \nA charge current flow in a resistive perpendicularly -magnetized Hall bar device can induced thermal \ngradients in both the out-of-plane (∇Tz) and in -plane direction s (∇Tx) [S2]. Sizable anomalous Nernst voltages can \narise from the se thermal gradients and contribute to measurements of the harmonic Hall voltage response s \n(HHVR s). In a general case, the first and the second HHVR s, Vω and V2ω, are given by [S2]: \nVω =VAHE cosθM +VPHEsin2θM sin2φ, (S1) \nV2ω = 𝑉2ω,SOT+ VANE,x cos𝜃M + VANE,z sin 𝜃Mcos 𝜑. (S2) \nwhere θM is the polar angle of the magnetization (with θM = arccos( Vω/VAHE) if the angle of applied magnetic field \nis chosen so that sin2φ = 0), φ is the azimuthal angle of the magnetization and is the same as that of the magnetic \nfield for a uniaxial anisotropy system, 𝑉2ω,SOT is the second HHVR due to the spin-orbit torque s, VAHE is the \nanomalous Hall voltage coefficient , VPHE is the planar Hall voltage coefficient , VANE,x = cANE ∇Tx and VANE,z = cANE \n∇Tz are the anomalous Nernst voltage s associated with the in -plane and perpendicular thermal gradient s, and cANE \nis the anomalous Nernst coe fficient. VAHE can be determined from the dependence of Vω on a swept out -of-plane \nmagnetic field ( Hz) or on a small sweep in -plane field Hx if θM is determined. VANE, z is equal to the value of V2ω \nwhen θM = 90◦ and φ = 0◦ (current direction), while VANE, x is equal to the value of V2ω when θM=θH = 0◦ (film normal \ndirection ). In Eq. (S2) we have ignored possible contributions from the ordinary Nernst effect that tend s to be \nmuch smaller than the anomalous Nernst effect in samples with metallic ferromagnets [S3]. \n7 \n (i) W hen a magnetic field applied in the xz plane ( Hxz) tilts the magnetization by a small θM, Eqs. (S1) and \n(S2) can be simplified as : \nVω =VAHE cosθM， (S3) \nV2ω = (1\n2𝑉AHE\n𝐻𝑘+𝐻xz𝐻DL + VANE,z) sinθM + VANE,x. (S4) \nIn this case, 𝐻DL can be determined f rom the dependence of 𝜕V2ω/𝜕sinθM on the magnitude of Hxz. \n(ii) When a small in -plane field Hx is applied along the current direction ( with Hx/Hk <<1 so that sinθM≈ \nHx/Hk), Eqs. (S1) and (S2) can be simplified as: \nVω = VAHE (1- 𝐻𝑥2\n2𝐻k2), (S5) \nV2ω = 1\n2𝑉AHE\n𝐻k2𝐻DLHx + 𝑉ANE ,z\n𝐻kHx + VANE,x . (S6) \nNow, since both the first and the second terms in Eq. (S6) are proportional to Hx, the anomalous Nernst contribution \nto the second HHVR cannot be separated from the dampinglike torque contribution simply from the dependences \nof V2ω on Hx. Generally, one can define an apparent longitudinal effective field \nHL = -2𝜕𝑉2ω\n𝜕𝐻𝑥 /𝜕2𝑉ω\n𝜕𝐻𝑥2 = 𝐻DL + 2𝐻k𝑉ANE ,z\n𝑉AHE . (S7) \nBoth HDL and the second term of Eq. ( S7) are proportional to the magnitude of the applied ac electric field E. This \nsuggests that the anomalous Nernst effect should be carefully taken into account in the analysis of resistive \nmagnetic systems with a small SOT field and a large value of 𝐻k. This can be done by measuring each of the \nparameters contributing to the second term in Eq. (S7) separately, as described above. We find that the anomalous \nNernst term is significant for some FeTb single layers (x≤51). As an example, we show the results of VANE,z, VAHE, \nHL, and HDL as a function of E for a 20 nm thick Fe71Tb29 in Fig. S7 (a)-(c). However, we find that for the Pt 5 \nnm/FeTb 20 nm , HL is the same as 𝐻DLwithin the experimental uncertainty , indicating minimal influence of \nthermoelectric effect . \n(iii) When a small in -plane field Hy is applied transverse to the current (Hy/Hk <<1 so that s inθM≈ Hy/Hk), Eq. \n(S2) can be simplified as: \nVω = VAHE (1- 𝐻𝑦2\n2𝐻k2). (S8) \nV2ω = 1\n2𝑉AHE\n𝐻k2𝐻FLHy + VANE,x . (S9) \nFrom this equation, the field-like spin -orbit torque field 𝐻FL can be determined f rom the slope of the linear fit of \nV2ω vs Hy. \nHT = -2𝜕𝑉2ω\n𝜕𝐻𝑦 /𝜕2𝑉ω\n𝜕𝐻𝑦2 = 𝐻FL. (S10) 8 \n \nFigure S7. Electric field (E) dependence of (a) VANE,z, (b) VAHE, (c) HL and HDL for a 20 nm Fe71Tb29 layer. \n \nSection 7. Coherent rotation at small polar angles \n \nFigure S8. Polar -angle dependence of the f irst harmonic Hall voltage response ( Vω) of a 20 nm Fe71Tb29 Hall bar \ndevice under an external magnetic field with a constant magnitude Hzx= 80 kOe in the x -z plane . The s olid parabolic \nline represents the best fit to the expression Vω= VAHEcosθM ≈ VAHE(1-𝜃M2/2), indicating a coherent magnetization \nrotation at small polar angles. \n \nSection 8. Plan ar Hall resistance \n \nFigure S9. Planar Hall resistance of a 20 nm thick Fe55Tb 45 layer . (a) Hall resistance ( RH) for a Fe55Tb45 single \nlayer plotted as a function of the in -plane angle ( 𝜑) of an in-plane magnetic field (90 kOe) relative to the current \ndirection . The planar Hall resistance ( RPHE) is determined from the best fit (red line) of the data to the relation \n𝑅H=𝑅PHEsin2 𝜑+𝐶sin𝜑 (the 𝐶sin𝜑 term comes from the small misalignment of the magnetic field with respect \nto the sample plane). (b) RPHE and (c) the ratio of the planar to anomalous Hall resistance s (RPHE/RAHE) plotted as \na function of the Tb concentration. The |RPHE/RAHE| ratios are ≤ 0.04 for Fe 100-xTbx. \n9 \n Section 9. Resistivit ies of 20 nm thick Fe100-xTb x layers \n \nFigure S10. Resistivity of the 20 nm thick Fe100-xTbx layers with different Tb concentration s. \n \nSection 10. Validation of Eq. (3) in the main text \nWe compare the effect of an external magnetic field to the effect a dampinglike spin-orbit torque on a \nferrimagnet composed of two oppositely oriented magnetic sublattices (1 and 2) with different gyromagnetic ratios \n𝛾𝑖 (i=1,2) . We assume the individual gyromagnetic ratios are negative, so that the angular momentum ( 𝑆⃗𝑖) and \nmagnetization ( 𝑀⃗⃗⃗𝑖) vectors are related as 𝑆⃗𝑖=−𝑀⃗⃗⃗𝑖/|𝛾𝑖| with magnitudes 𝑆𝑖=𝑀𝑖/|𝛾𝑖|. All calculations below \nwill assume a unit area of a sample thin film. \nThe LLG equation for an individual sublattice subject only to a magnetic field has the form \n𝑑\n𝑑𝑡𝑀⃗⃗⃗𝑖=−|𝛾𝑖|𝑀𝑖𝑚̂𝑖×𝐻⃗⃗⃗ + 𝛼𝑖𝑀𝑖𝑚̂𝑖×𝑑𝑚̂𝑖\n𝑑𝑡 . (S11) \nwhere 𝛼𝑖 is the Gilbert damping and 𝑀𝑖 is the magnetic moment. It will be more convenient to write the combined \nequation of motion for the two sublattices in terms of the total angular momentum rather than the total \nmagnetization, because the exchange inter action between the two sublattices will conserve total angular \nmomentum, while it does not conserve the total magnetization. \n \nThe LLG equation for sublattice 1 can be re -written \n𝑑\n𝑑𝑡𝑆⃗1=−𝑀1𝑠̂1×𝐻⃗⃗⃗ − 𝛼1\n|𝛾1|𝑀1𝑠̂1×𝑑ŝ1\n𝑑𝑡 . (S12) \nand for sublattice 2 \n𝑑\n𝑑𝑡𝑆⃗2=−𝑀2𝑠̂2×𝐻⃗⃗⃗ − 𝛼2\n|𝛾2|𝑀2𝑠̂2×𝑑ŝ2\n𝑑𝑡=+𝑀2𝑠̂1×𝐵⃗⃗ − 𝛼2\n|𝛾2|𝑀2𝑠̂1×𝑑ŝ1\n𝑑𝑡. (S13) \nAdding these equations, the equation of motion for the total angular momentum of the ferrimagnet subject only to \na magnetic field is \n𝑑\n𝑑𝑡(𝑆⃗total) =−(𝑀1− 𝑀2) 𝑠̂1×𝐻⃗⃗⃗ −(𝛼1\n|𝛾1|𝑀1+𝛼2\n|𝛾2|𝑀2) 𝑠̂1×𝑑ŝ1\n𝑑𝑡. (S14) \nIf the ferrimagnet is also subject to an antidamping spin -orbit torque, with spin angular momentum in the 𝜎̂ \ndirection, we can add this to the equation of motion \n𝑑\n𝑑𝑡(𝑆⃗total) =−(𝑀1− 𝑀2) 𝑠̂1×𝐻⃗⃗⃗ −(𝛼1\n|𝛾1|𝑀1+𝛼2\n|𝛾2|𝑀2) 𝑠̂1×𝑑ŝ1\n𝑑𝑡+ℏ\n2𝑒𝜉DL𝑗𝑗𝑒(𝑠̂1×𝜎̂×𝑠̂1). (S15) \nSince 𝑆⃗total = (𝑆1−𝑆2)𝑠̂1=(𝑀1\n|𝛾1|−𝑀2\n|𝛾2|)𝑠̂1=(𝑀1−𝑀2)𝑠̂1/𝛾eff, where we define 𝛾eff≡ (𝑀1−𝑀2)/(𝑀1\n|𝛾1|−\n𝑀2\n|𝛾2|), in terms of unit vectors Eq. (S15) can be rewritten as \n10 \n 𝑑\n𝑑𝑡(𝑠̂1) =−𝛾eff 𝑠̂1×𝐻⃗⃗⃗ −𝛼eff𝑠̂1×𝑑ŝ1\n𝑑𝑡+𝛾effℏ\n2𝑒(1\n𝑀1−𝑀2)𝜉DL𝑗𝑗𝑒(𝑠̂1×𝜎̂×𝑠̂1). (S16) \nHere 𝛼eff=𝛾eff(𝛼1\n|𝛾1|𝑀1+𝛼2\n|𝛾2|𝑀2)/(𝑀1−𝑀2). \n \nFrom Eq. (S16) , we can read off that the spin -transfer torque is equivalent to an effective magnetic field \n𝐻⃗⃗⃗DL=−ℏ\n2𝑒(1\n𝑀1−𝑀2)𝜉DL𝑗𝑗𝑒(𝜎̂×𝑠̂1)=ℏ\n2𝑒1\n|𝑀1−𝑀2|𝜉DL𝑗𝑗𝑒(𝜎̂×𝑚̂). (S17) \nIf the effective magnetic field is measured relative to the direction 𝜎̂×𝑚̂, we therefore have that \n \n 𝜉DL𝑗=2𝑒\nℏ𝐻𝐷𝐿|𝑀1−𝑀2|\n𝑗𝑒 ， (S18) \nWith the identification that the total magnetization per unit area |𝑀1−𝑀2| is equal to 𝑀s𝑡, the measured \nmagnetization per unit area , one obtain \n 𝜉DL𝑗=2𝑒\nℏ𝐻DL𝑀s𝑡\n𝑗𝑒 . (S19) \nThis is the same as Eq. (3) in the main text , and indicates that 𝜉DL𝑗 has no dependence on the sign of 𝛾eff. \n \nSection 11. Raw data for representative samples exhibiting different torque signs \n \nFigure S1 1. Raw data for three representative samples: 20 nm Fe 71Tb29 (VAHE > 0, 𝜉DL𝑗>0), 20 nm Fe 61Tb39 \n(VAHE > 0, 𝜉DL𝑗< 0), and 20 nm Fe 47Tb53 (VAHE< 0, 𝜉DL𝑗>0). (a) Frist (V1ω) vs sin θM, (b) Second HHVR (V2ω) vs \nsinθM, and (c) dV2ω/dsinθM vs VAHE/2(Hk+Hzx) for under different applied magnetic field Hzx. The definition of \nsymbols are the same as those in the Supplementary Section 6. \n11 \n Section 12. Field-like spin-orbit torque in Fe 100-xTb x single layer s \n \nFigure S12. (a) Field -like effective torque field for 20 nm thick Fe100-xTbx layers with different Tb concentration s. \n(b) Field -like effective torque field for Fe57Tb43 layers with different thickness es. The field -like torque is relatively \nsmall compared to the damping -like torque ( Fig. 2(d) of the main text). \n \nSection 13. Spin -torque ferromagnetic resonance measurements \nTo confirm the spin Hall effect of the FeTb, we performed spin-torque ferromagnetic resonance (ST -FMR) \nto measure emission of spin current from FeTb layers that is absorbed by an Fe detector layer. During the ST -\nFMR measurements, an in -plane magnetic field ( H) was swept at a fixed angle of 45º with respect to the magnetic \nmicrostrip. As shown in Fig. S13(a), the amplitudes of the symmetric and anti -symmetric components of a ST -\nFMR spectrum, S and A, can be determined by fitting the data to [S4]: \n𝑉mix=𝑆∆𝐻2\n∆𝐻2+(𝐻−𝐻r)2+𝐴∆𝐻(𝐻−𝐻r)\n∆𝐻2+(𝐻−𝐻r)2 , (S11) \nwhere ∆H is the FMR linewidth and Hr the resonance field. From the S/A ratio, we can define as an intermediate \nparameter an effective spin-orbit torque efficiency [S5, S6]: \n𝜉FMR =𝑆\n𝐴𝑒𝑀s𝑡𝑑\nℏ√1+4π𝑀eff\n𝐻r, (S12) \nwhere e is the electron charge, ℏ is the reduced Planck constant, 𝜇0 is the vacuum permeability, Ms is the saturation \nmagnetization. 𝑡 is the layer thickness of the magnetic detector, 𝑑 is the layer thickness of the spin -current -\ngenerating layer , and 4𝜋𝑀eff is the effective demagnetization field of the magnetic detector. The true damping -like \nspin-torque efficiency is determined from plots of 1/ 𝜉FMR versus 1/ t, as described in the main text. Here, there is \nonly negligible Oersted field due to the in -plane current flow in the extremely resistive thin Ti layer [S7]. As \nshown in Fig. S13(b), 4𝜋𝑀eff can be determined from the resonance frequency (f) dependence of Hr following the \nKittel’s equation [S6] \n𝑓=𝛾\n2𝜋√(𝐻𝑟+𝐻ex)(𝐻𝑟+𝐻𝑒𝑥+4𝜋𝑀eff) , ( S13) \nwhere 𝐻ex is the exchange field from the other layer . \n \n12 \n \nFigure S13. (a) ST-FMR spectrum at 15 GHz for Fe 50Tb50 (20 nm)/Ti (1 nm)/Fe (tFe), in which a clea r symmetric \ncomponent (solid blue line) is from the damping -like torque, and the asymmetric component (solid pink line) is \nfrom the field -like torque and Oersted field torque . (b) Frequency dependence of f erromagnetic resonance field Hr \nof the Fe layer in the representative samples Fe50Tb50 (20 nm)/Ti (1 nm)/Fe (9.8 nm ) and Fe50Tb50 (20 nm)/Ti (1 \nnm)/Fe (3.8 nm ). The solid red line represent s fit of the data to Kittel’s equation (Eq. (13) ). (c) A and (d) S of the \nrepresentative samples Fe50Tb50 (20 nm)/Ti (1 nm)/Fe (9.8 nm ) (12 GHz) and Fe50Tb50 (20 nm)/Ti (1 nm)/Fe (3.8 \nnm) (14 GHz) plotted as a function of in -plane angle ( φ) of the magnetic field relative to the rf current . The blue \nand red solid curves represent the best fits of the data to a sin2 φ cosφ dependence. \nSection 14. Interfacial perpendicular magnetic anisotropy energy density \n \nFigure S14. Dependence of the effective demagnetization field (4π Meff) on the inverse thickness of the Fe layer \nin the Fe50Tb50 (20 nm)/Ti(1 nm)/Fe( tFe) sample. The linear fit of the data to the relation 4𝜋𝑀eff ≈ 4πM s -2Ks/MstFM \nyields a total interfacial magnetic anisotropy energy density of Ks=2.07 ± 0.05 erg/cm2 and a saturation \nmagnetization Ms = 1500 emu/cm3. After subtraction of the contribution from the top Fe/MgO interface (≈ 0.64 \nerg/cm2) [S8], we obtain the Ks ≈ 1.43 ± 0.05 erg/cm2 for the Ti/Fe interface . \n13 \n Section 15. HHVR measurement of a 20 nm thick composition -gradient Fe100-xTb x (x = 27→41) \n \nFigure S15 𝑑V2ω/𝑑sinθM versus VAHE/2(Hk+Hzx) for the 20 nm thick composition -gradient Fe100-xTbx (x = 27→41) \nsample , where x is varied continuously from 27 at the bottom of the layer to 41 at the top . Using the extracted \nvalue of HDL obtained from a linear fit to Eq. (3) in the main text, we obtain 𝜉DL𝑗 = 0.05 ± 0.01 for this sample . \nSection 16. More examples o f magnetization switching by the bulk spin -orbit torque \n \nFigure S1 6. Current -induced switching of the 20 nm Fe 67Tb33, Fe61Tb39, and Fe57Tb43 films under the same applied \nbias magnetic field of +3 kOe. The arrows indicate the switching polarity . \n \n14 \n In Fig. S16(a) -(c), we show the c urrent -induced switching of the 20 nm-thick Fe67Tb33, Fe61Tb39, and \nFe57Tb43, under the same applied bias magnetic field of 3 kOe. Since all the three films are Fe-dominated (RAHE > \n0), their anomalous Hall voltages have the same sign. As indicated by the arrows, the switching polarity of the \nFe67Tb33 is opposite to that of Fe57Tb43, which is consistent with the opposite sign of the bulk spin -orbit torques in \nthe Fe67Tb33 (𝜉DL𝑗 >0) and the Fe57Tb43 (𝜉DL𝑗 < 0). Interesting, the Fe61Tb39 shows the same switching polarity as \nthe Fe67Tb33, which is because the current -induced heating during the switching has driven the Fe61Tb39 deep into \nthe Fe -dominated regime where the Fe67Tb33 is also located and the bulk spin -orbit torque efficiency is positive \n(𝜉DL𝑗 >0). It has been reported that the ferrimagnetic CoTb near the compensation point can become Co -dominated \nat high er temperatures and Tb dominated at lower temperatures [ S9]. \n \nReferences \n[S1] B. Hebler, A. Hassdenteufel, P. Reinhardt, H. Karl and M. Albrecht , Front. Mater. 3, 8 (2016). \n[S2] A. Ghosh, K. Garello, C. O. Avci, M. Gabureac, and P. Gambardella, Phys. Rev. Appl. 7, 014004 (2017). \n[S3] M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. Wagner, M. Opel, I. -M. Imort, G. Reiss, A. \nThomas, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 108, 106602 (2012). \n[S4] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. R ev. Lett. 106, 036601 (20 11). \n[S5] C.-F. Pai, Y . Ou, L. H . Vilela -Leao, D. C. Ralph, R. A. Buhrman, Phys. Rev. B 92, 064426 (2015). \n[S6] C. Kittel, Phys. Rev. 73, 155 (1948). \n[S7] L. Zhu and R. A. Buhrman, Phys. Rev. Applied 15, L031001(2021). \n[S8] Y. Iida, J. Okabayashi, and S. Mitani, Appl. Phys. Lett. 113, 252401 (2018) . \n[S9] K. Ueda, M. Mann, P. W. P. de Brouwer, D. Bono, and G. S. D. Beach, Phys. Rev. B 96, 064410 (2017). " }, { "title": "2010.01518v2.Magnetic_field_dependent_cycloidal_rotation_in_pristine_and_Ge_doped_CoCr__2_O__4_.pdf", "content": "Magnetic field dependent cycloidal rotation in pristine and Ge doped CoCr 2O4 \nN. Ortiz Hernandez1*, S. Parchenko1*⁑, J. R. L. Mardegan1,4, M. Porer1, E. Schierle2, E. Weschke2, M. \nRamakrishnan1, M. Radovic 1, J. A. Heuver3, B. Noheda3, N. Daffe1, J. Dreiser1, H. Ueda1 and U. Staub1** \n1 Swiss Light Source, Paul Scherrer Institute, Forschungsstrasse 111, 5232 Villigen PSI, Switzerland. \n2 Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Albert-Einstein-Strasse 15, 12489 Berlin, Germany. \n3 Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747AG, The Netherlands. \n4 Deutsches Elektronen-Synchrotron, Notkestrasse 85 , 22607 Hamburg, Germany. \n \n* Authors with equal contribution. \n⁑ Present address: Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland \n**Corresponding author: urs.staub@psi.ch \n \nAbstract \nWe report a soft x-ray resonant magnetic scattering study of the spin configuration in multiferroic thin \nfilms of Co 0.975Ge0.025Cr2O4 (Ge-CCO) and CoCr 2O4 (CCO), under low- and high-magnetic fields, from 0.2 \nT up to 6.5 T. A characterization of Ge-CCO at a low magnetic field is performed and the results are \ncompared to those of pure CCO. The ferrimagnetic phase transition temperature T C ≈ 95 K and the \nmultiferroic transition temperature T S ≈ 27 K in Ge-CCO are comparable to those observed in CCO. In \nGe-CCO, the ordering wave vector (qq0) observed below T S is slightly larger compared to that of CCO, \nand, unlike CCO, the diffraction intensity consists of two contributions that show a dissimilar x-ray \npolarization dependence. In Ge-CCO, the coercive field observed at low temperatures was larger than \nthe one reported for CCO. In both compounds, an unexpected reversal of the spiral helicity and \ntherefore the electric polarization was observed on simply magnetic field cooling. In addition, we find \na change in the helicity as a function of momentum transfer in the magnetic diffraction peak of Ge-\nCCO, indicative of the presence of multiple magnetic spirals. \n 1. Introduction. \nMagnetoelectric (ME) multiferroics are of enormous interest from a technological perspective for \ndesigning new functionalities such as using electric fields to manipulate magnetic order [1]-[2]. Of \nspecial interest are type II multiferroics, where magnetic ordering drives the electric polarization with \nboth order parameters being strongly coupled. This strong coupling enables switching the polarization \nby a magnetic field or the magnetization by an electric field, which is energetically more efficient. \nCoCr2O4 (CCO) is one of the few known ME multiferroics of type II that exhibits a net magnetization \ndue to its ferrimagnetic state [3]. CCO crystallizes in a spinel structure (AB 2O4), having cubic symmetry \n(Fd3തm) with a lattice constant of 8.33 ܣሶ [3] in bulk. The Co2+ ions sit on the tetrahedral coordinated A \nsites and Cr3+ on the octahedral coordinated B sites, subdivided into B1 and B2 sites. This material has \nbeen well characterized in both bulk [4]-[7] and thin film [8]-[12] forms. Three magnetic phases have \nbeen found below room temperatures for bulk. Below TC ≈ 93 K, where CCO becomes ferrimagnetically \nordered [4]. In this phase, uncompensated magnetic sublattices of Co2+ and Cr3+ yield in a remanent \nnet magnetization. Additionally, a spiral short-range order is reported to coexist within the long-range \nferrimagnetic order [13]. In fact, the two magnetic sublattices B1 and B2 couple antiferromagnetically \nto each other with different opening angles for the cone, resulting in a net magnetic moment that is \nantiparallel to the sublattice A. This produces a net magnetization along [001] direction. Below TS ≈ 26 \nK, CCO gest an additional long-range magnetic spiral component, represented by an incommensurate \nmodulation wave vector ( qq0), with q ≈ 2/3 reciprocal lattice units (r.l.u.). This transverse conical \nmagnetic structure induces a ferroelectric polarization along the [ 1ത10] direction [4]. Around TF ≈ 15 K, \nyet another magnetic phase transition to a commensurate spiral phase has been reported [4]-[7], the \noccurrence of which remains controversial. Choi [5] and Chang [6] reported the occurrence of a \ncommensurate wave vector (2/3 2/3 0) and two additional incommensurate satellites in bulk CCO, \nwith the new incommensurability being along the [110] and [1 1ത0]. A more recent work by Windsor [8] \nreported a single incommensurate spiral below TF in an epitaxially grown strained film. However, in this study, the width of the observed magnetic diffraction peak might have been too large to resolve \nadditional long wavelength satellites. \nThe polarization direction in CCO is directly related to the helicity of the spin spiral as given by Katsura \n[14]: ∝ ො ×(×), where , is the spin canting in the neighboring sites i and j, and ො is the \nunit vector connecting the two sites which is parallel to the magnetic ordering wave vector Q = (qq0). \nFor the case of thin films, the Q direction (out of the film surface) being fixed due to a slight tetragonal \ndistortion caused by lattice mismatch with the substrate, the above relation implies that reversing the \nspin spiral results in a reversal of polarization and vice versa. \nIn this paper, we examine the effect of doping a small fraction of non-magnetic Ge on the long-range \nmagnetic order in CCO. We compare the magnetic properties of epitaxial films of Ge doped CCO with \npure CCO in the multiferroic phase. We also explore the behavior of the magnetic spin spiral in these \nsystems under high magnetic fields. For this, we use resonant soft x-ray scattering (RSXS), an excellent \ntechnique to study complex magnetic structures. Recently, RSXS has been employed to investigate \noxide materials, and particularly, multiferroics [15]-[21]. RSXS has the advantage of being element and \norbital specific while probing long-range electronic ordering phenomena. Moreover, RSXS offers a high \nsensitivity in observing magnetic ordering schemes, even for small sample volumes [22]-[24]. \n2. Experiments. \nThin films of CCO and Ge-doped CCO were grown by pulsed laser deposition, monitored in-situ by \nreflection high-energy electron diffraction. The CCO thin films were grown with a thickness of ~80 nm \non [110]-oriented MgO substrates. A more detailed description can be found in [11]. The same growth \nparameters were employed for the Ge doped CCO with 2.5% Ge doping, grown on [110]-oriented MgO \nsubstrates, resulting in Co 0.975Ge0.025Cr2O4 (Ge-CCO) films with a thickness similar to CCO. \n2.1. Resonant magnetic soft x-ray scattering (reflectivity and diffraction) under low \nmagnetic fields. X-ray magnetic circular dichroism (XMCD) in reflectivity mode and magnetic diffraction data have been \ncollected at the RESOXS end station [25], at the X11MA beamline [26] of the Swiss Light Source (SLS). \nIntensities of reflected circularly polarized x-rays were collected at θ = 5° incidence for photon energies \naround the Co L2,3 edges to obtain the XMCD signal in reflection mode (see experimental layout in \nFigure 1). For the magnetic diffraction experiment, q-scans were performed at 780 eV (Co L3 edge). \nBoth circular and linear x-ray polarizations were used. Data were collected with an IRD AXUV100 \nphotodiode, covered by a 400 nm thick Al filter to suppress visible light and secondary electrons. The \nsample was field cooled (FC) in an external field of 0.2 T along the [001] direction from 300 K to 8 K \nprior to data collection. \n \nFigure 1. Experimental geometry used in RESOXS end-station. \n \n2.2. XMCD on Ge-CCO collected by X-ray exited optical luminescence (XEOL). \nXMCD and magnetic hysteresis measurements were carried out at the X-Treme beamline [27] of the \nSwiss Light Source using XEOL [8][28], taking the advantage of the insulating character of the samples \nand the luminescence of the substrates. For luminescent substrates, XEOL effectively measures the \nXAS in transmission mode [29]. Hysteresis loops were collected at the energy with the largest XMCD \naround the Co L3 edge (777.5 eV) for various temperatures. An incident angle of 30° with respect to \n[001] direction was chosen. The sample was field cooled from room temperature to 10 K in a field of \n-0.2 T along [001]. \n \nFigure 2. Layout of XEOL experiment. \n \n2.3. Resonant soft x-ray scattering on Ge-CCO and pure CCO under high-magnetic fields. \nResonant magnetic x-ray diffraction measurements at the Co L3 absorption edge have been carried \nout on the high-field diffractometer at the UE45_PGM1 beamline of BESSY II synchrotron, at Helmholtz \nZentrum Berlin [30]. Incident linear and circular polarized x-rays were used at energies around the Co \nL2,3 edge. A dome was built with 400 nm thick Al foil (same used in section 2.1), and placed above the \nsample to block visible light and secondary electrons. Both CCO and Ge-CCO samples were \nsimultaneously mounted on the same holder to have the same experimental conditions. During field \ncooling, the magnetic field was applied along [001]. Since the magnets are not rotated during the \nscans, the magnetic field direction changes with respect to the surface during a scan. \nFigure 3 displays a sketch of the experimental geometry. The magnetic field is created by four \nsuperconducting magnet coils, which are rotatable with respect to the sample. As the magnetic \ndiffraction ( qq0) peak requires a large scattering angle and the diffraction peak is very broad with a \nwidth of ≈ 45° in total scattering angle 2 , the magnets were rotated by = 32.5° with respect to the \nincident beam to have the best possible scattering geometry. When the magnetic field points towards \nthe detector, secondary electrons are deflected by the field resulting in an artificial increase of the \nsignal that disappears in the absence of the magnetic field. In order to reduce this effect, the sample \nholder was charged for a few seconds by applying an electric field along the [110] direction. To \nsuppress the specular reflectivity background in the magnetic diffraction signal, the angle was \ndisplaced a quarter of a degree from the specular condition. \n \nFigure 3. Schematic representation of the experimental geometry. For (qq0) reflection, the detector was placed \nat 2 = 140° with an incident angle ( ) of 69.75°. The magnetic field during measurements was rotated by an \nangle α = 32.5° from incoming X-ray beam. \n3. Results. \n3.1. Resonant magnetic soft x-ray scattering (reflectivity and diffraction) under low \nmagnetic fields. \nResonant x-ray scattering provides an element-specific measure of the electronic state and magnetic \nconfiguration of ions in a material [31]. In the case of a transition metal ion such as Co2+, electric dipole \nexcitations at L2,3 absorption edges directly probe electron transitions from the 2 p core to the 3 d \nvalance states. Thus, the spectra are sensitive to the electronic configuration of the 3 d states and its \nspin configuration in the presence of a core hole [32]. The resonant magnetic scattering amplitude for \na single ion can be expressed in the electric dipole approximation (E1-E1) as [32]-[33] \n݂ாଵாଵ ∝[(ࢿොᇱ∗∙ࢿො)−݅ෝ∙(ࢿොᇱ∗×ࢿො)ଵ +(ࢿො∙ෝ)(ࢿොᇱ∗∙ෝ)ଶ] (1) \n \nwhere ࢿො and ࢿොᇱ refer to the incoming and outgoing photon polarizations, ෝ is the unit vector of the \nmagnetic moments and the terms F(n) are scattering tensors of rank n, which depend strongly on \nenergy. Scattered intensity is proportional to | ݂+݂ாଵாଵ|2, ݂ being the sum over the non-resonant \namplitudes. For ferromagnetic order (e.g. solely the Co sublattice), only the second term in eq. (1) \ndepends on the circular x-ray polarization and, to first order, is proportional to the magnetic moment. \nHence, the circular dichroism can be approximated to be proportional to the magnetic moment. The \ncircular dichroism is, therefore, large for small scattering angles when the sample is magnetized along \nthe film plane and this is in the scattering plane, as it is in our case. \n \nFigure 4. (a) X-ray reflectivity of Ge-CCO at T = 10 K around the Co L 2,3 edges when field cooled at 0.2 T, for circular \nright (C+) and left (C-) polarizations. (b) Asymmetry as a function of energy. The dashed line indicates the energy \nwith the largest asymmetry (in absolute values). (c) Magnetic asymmetry as a function of temperature measured \nat Co L 3 edge (energy of the dashed line in b), for a sample field cooled at 0.2 T. The pink dashed line estimates \nTC. The small asymmetry offset above the Curie temperature is presumably due to an imperfect normalization \nprocedure. \nFigure 4a shows the energy spectra of the reflected beam at the Co L2,3 edges for incident circular right \n(C+) and circular left ( C-) polarized light for Ge-CCO. A clear contrast is observed between the two \nspectra. Figure 4b shows the magnetic circular dichroism represented by the asymmetry defined as \nܣ=ூశି ூష\nூశା ூష. The dashed vertical line indicates the energy with maximal asymmetry, which is \napproximately 20% at 779.5 eV. Figure 4c presents the asymmetry at 779.5 eV as a function of \ntemperature. The black dashed lines indicate TC ≈ 95 K, which is estimated by a linear fit (indicated by \nthe pink dashed line) and implies that it is a second order phase transition with a critical exponent ߚ \n≈ 1/2. Above TC the circular dichroism disappears, indicating the transition to the paramagnetic phase, \nsimilar to that reported for pure CCO [8]. This suggests that a small amount of Ge doping does not \naffect the magnetic transition temperature of the ferrimagnetic phase significantly. \nFigure 5a displays the magnetic diffraction intensity of the ( qq0) peak in Ge-CCO versus r.l.u. (bottom \naxis) and in total momentum transfer ( Q) (top axis) for various temperatures. The peak maximum is, \nat least, 0.03 r.l.u. higher compared to pure CCO [8]. \n \nFigure 5. (a) Temperature dependence of the magnetic diffraction peak (qq0) of Ge-CCO upon warming, after \nfield cooling in 0.2 T and using C+ polarized x-rays at 780 eV. The data are presented as a function of both r.l.u. \n(bottom axis) and total momentum transfer (Q) (top axis). (b), (c) and (d) represent the temperature dependence \nof integrated intensity, modulation parameter q and the correlation length ߦ extracted from the data shown in \n(a). \nFrom a Gaussian fit in Figure 5a, we extract the temperature dependence of the integrated intensity \n(Figure 5b), the modulation parameter q (Figure 5c) and the correlation length (Figure 5d) calculated \nas ߦ= 2/ܯܪܹܨ .Figure 5b shows that the static antiferromagnetic component appears around 27 K 0.51 0.54 0.57 0.60 0.63 0.66 0.69 0.720.00.51.01.5\n0.00.40.81.2\n0.680.700.72\n10 15 20 25 302.42.83.2Intensity (arb. units)\nq (r.l.u.) T= 9.2 K\n T= 11.1 K\n T= 13.1 K\n T = 15.4 K\n T= 18.1 K\n T = 20.8 K\n T = 23.6 K\n T= 26.4 K\n T = 29.3 K\n T = 32.3 K(qq0)+ FC, IC+0.54 0.60 0.66 0.72 0.78\nd)c)b)Q (1/Ǻ )\na)\nInt. intensity\n(arb. units)q (r.l.u.)x(nm)\nTemperature (K)suggesting that the material is in the multiferroic state below this temperature, akin to pure CCO. \nFigure 5c shows an increase of the modulation parameter q from 0.67 r.l.u. to 0.72 r.l.u. with reduction \nin temperature. We observe an increase of the correlation length with increasing temperature, which \nis in contrast to pure CCO [8]. This increase is prominent around 18 K, close to TF [4]-[7]. \nFor comparison, we show in Figure 6, the magnetic diffraction ( qq0) peak for circular and linear \npolarization with their corresponding circular ( IC+ - IC-) and linear ( Iπ - Iσ) dichroism for pure CCO (Figure \n6 a- b) and Ge-CCO (Figure 6 c-d) at 779 eV collected at UE45_PGM1 beamline (BESSY II). For pure \nCCO, both linear and circular dichroism attain their maximum around 0.68 r.l.u. In contrast, Ge-CCO \nexhibits an extremum for linear dichroism around q ≈ 0.70 r.l.u. (marked by the green dashed line), \nwhile the extremum for circular dichroism is around q ≈ 0.72 r.l.u. (marked by a blue dashed line). This \ndifference in q for the extrema indicates that the diffraction peak is composed of more than a single \nmagnetic contribution. \n \nFigure 6. Magnetic diffraction (qq0) peak for circular and linear polarizations with their respective linear and \ncircular dichroism for CCO (a,b) and Ge-CCO (c,d). Data collected at 779 eV. \n 0.280.320.360.400.44\n0.64 0.66 0.68 0.70 0.72-0.06-0.030.000.030.060.40.60.81.01.2\n0.64 0.66 0.68 0.70 0.72-0.060.000.060.120.180.24 b)Intensity (arb. units) IC+ Ip\n IC- IsPure CCO\nT = 5 K\nFC = 1 T\na)Dichroism (arb. units)\nq (r.l.u.) Ip - Is\n IC+- IC-\nd)c)Ge-CCO\nT = 5 K\nFC = 1 TIntensity (arb. units)\n IC+ Ip\n IC- Is Dichroism (arb. units)\nq (r.l.u.) Ip - Is\n IC+- IC-3.2. XMCD on Ge-CCO collected by X-ray exited optical luminescence (XEOL). \nThe x-ray intensity transmitted through the film is measured by observing the XEOL signal and it can \nbe described as: \n ܫ(ݖ)=ܫ߉(ܧ)݁ିఓ(ா)௭ (2) \nwhere z is the thickness, ܫ is the incident intensity, μ is the energy dependent absorption coefficient \nof the sample, and ߉(ܧ) is the energy dependent efficiency function of XEOL for the substrate used \n(typical value for MgO at Co L2,3 edge is 0.028 [29]). Taking into account the thickness of the substrate \nis ≈ 1 mm, we can assume that the entire signal is absorbed by the substrate, and therefore, the entire \nmeasured signal is XEOL [29] . Using equation 2, we calculate the XAS from the experimental XEOL \ndata for each helicity of the x-ray ( μ±), which are shown in Figure 7a. Figure 7b displays the \ncorresponding XMCD (μ +-μ-) has a maximum at 777.5 eV. \n \nFigure 7. (a) XAS of Co L 2,3 edge on Ge-CCO collected in XEOL mode at 10 K, under 6.8 T and FC -0.2 T. (b) XMCD \ndata obtained from (a). \nFurthermore, hysteresis loops were recorded by means of XMCD at 777.5 eV for various temperatures, \nas shown in Figure 8. For each applied magnetic field, measurements below the absorption edge (770 \neV) have been used for baseline correction . These data show that for T ≤ 40 K, a magnetic field of 6.8 0.00.20.40.6\n770 775 780 785 790 795 800-0.060.000.060.120.180.24L2\nb)XAS (arb.units) m+\n m-a)\nCo L2,3 edge\n-FC\nH = 6.8 T\nT = 10 K\nL3XMCD (arb. units)\nEnergy (eV)T is insufficient to saturate the magnetization, indicative of a very large coercivity and larger saturation \nfield when compared to pure CCO thin film [8]. \n \nFigure 8. Magnetic hysteresis loop of Co sublattice in Ge-CCO for various temperatures measured in XEOL mode. \n3.3. Resonant soft x-ray scattering on Ge-CCO and pure CCO under high-magnetic fields. \nFigure 9 presents the energy dependence of the magnetic diffraction ( qq0) peak (q ≈ 0.69) on pure \nCCO and Ge-CCO for opposite circular polarizations. Both samples have been field cooled at -6 T from \nroom temperature to 5 K and a magnetic field of -6 T was applied during the measurements. -0.4-0.20.00.20.4\n-6 -4 -2 0 2 4 6-0.4-0.20.00.20.4\n-6 -4 -2 0 2 4 6T = 30 K T = 40 KT = 60 K T = 90 KXMCD (arb. units)\nMagnetic field (T) \nFigure 9. Energy dependence of the magnetic diffraction ( qq0) peak for C+ and C- polarizations for (a) pure CCO \nand (b) Ge-CCO at 5 K, under applied of -6 T after FC at -6 T. Energy dependence of the circular dichroism of the \n(qq0) peak for (c) pure CCO and (d) Ge-CCO. Dashed lines indicate the energy corresponding to the extrema of \ncircular dichroism. \nThe energy dependence of the ( qq0) reflection at q = 0.695 r.l.u. has similar features for both \nmaterials, with a maximum at 779 eV and a shoulder around 778.3 eV. However, in the case of pure \nCCO, the spectrum of the diffraction peak is broader and less pronounced than in Ge-CCO. The \nobserved circular dichroism in Ge-CCO and pure CCO have opposite signs at 779 eV (Figures 9c and \n9d). \nTo learn more about the observed circular dichroism, we studied the field dependence of the ( qq0) \npeak at 779 eV. We collected two data sets with opposite applied fields for each sample. In the first \ndata set, both materials were cooled under a field of -1 T from 120 K to 5 K and data were acquired \nwhile increasing the applied field in several steps from 0 T up to -6.5 T. For the second data set, the \nsamples were cooled at 6.5 T from 60 K (which is fully sufficient for reversing the magnetization) to 5 \nK and same procedure was carried out, with the exception that the measurements were done while \ndecreasing the applied field in steps from 6.5 to 0 T. 0.50.60.70.80.91.01.1\n776 777 778 779 780 781 782-0.09-0.06-0.030.000.030.060.40.50.60.70.80.91.0\n776 777 778 779 780 781 782-0.020.000.020.04Ge-CCOIntensity Norm. (arb. untis) IC+\n IC-\n- FC, H = - 6 T q = 0.695\nd) c) (qq0)\nreflectionCirc. dichroism (arb. units)\nEnergy (eV)Pure \nCCO- FC, H = - 6 TIntensity Norm (arb. untis) IC+\n IC- (qq0)\nreflection\nq = 0.695\na) b)b)Circ. dichroism (arb.units)\nEnergy (eV) \nFigure 10. Intensity of magnetic diffraction peak (qq0) of pure CCO (a) and Ge-CCO (b) for various magnetic fields \nat 779 eV. At 5 K, data collected with C + polarized light, in field cooled at -1 T. The origin of the artifact around q \n≈ 0.712 is explained in the text . \nFigure 10 displays ( qq0) peak under various magnetic fields on pure CCO (Figure 10a) and on Ge-CCO \n(Figure 10b) for the case of negative field cooling and incident C+ polarization. The sharp peak \nobserved around q = 0.71 r.l.u. is an artifact caused by secondary electrons when the magnetic field \npoints to the detector, as explained previously. The application of the field results mainly in a \nreduction and/or distortion of the ( qq0) reflection intensity for both materials. \nThe circular dichroism observed in diffraction peak of a spin spiral defines directly the sign of the \ncycloidal rotation of the magnetic moments similar to those found in TbMnO 3 [34] or DyMnO 3 [35]. \nFrom now on, we refer to the circular dichroism of the magnetic diffraction ( qq0) peak as helicity \ncontrast, defined as IC+ ି IC-\n(IC++IC-)ௗ. \nIn Figure 11, we present the helicity contrast of the (qq0) peak for the different sets of measurements: \nfor pure CCO with negative applied magnetic field ( Figure 11 a) and positive applied magnetic field \n(Figure 11 b). The case of Ge-CCO is shown in Figure 11 c and Figure 11 d for negative and positive \napplied magnetic fields, respectively. 0.64 0.66 0.68 0.70 0.72 0.744.56.07.59.010.5\n0.64 0.66 0.68 0.70 0.72 0.743.03.33.63.94.24.5\nC+, (qq0)\n- FCIntensity (arb. units)\nq (r.l.u.) H = 0T\n H= - 3T\n H = - 6.5Tb)Ge-CCO\n- FCC+, (qq0)Intensity (arb. units)\nq (r.l.u.) H = 0T\n H = - 3T\n H = - 6.5T Pure CCO\na) \nFigure 11. Helicity contrast of the (qq0) peak for various magnetic fields at 5 K on (a,b) pure CCO and on (c,d) Ge-\nCCO at 779 eV. (a) and (c) panels are for negative magnetic field cases and (b) and (d) for positive magnetic field \ncases. Insets in (a,b) panels are the sketch of the magnetic modulation vector ( orange arrows) and magnetic \nmoments (green arrows). \nWhile being significantly different in shape, the helicity contrast shows a mirror effect between \nopposite applied field directions for both materials (Figure 11c-d). Therefore, both materials exhibit a \ndirect correlation between the direction of magnetic field cooling and cycloidal rotation. Insets in \nFigure 11a-b show the helicity sense, indicated by the orange arrows. In addition, the helicity contrast \ndecreases for increasing fields for CCO, possibly due to a field dependent elliptical distortion of the \ncycloid. The behavior of Ge-CCO though is more complex. There are two extrema in the helicity \ncontrast as a function of q. At q1 ≈ 0.72 r.l.u., we observe a maximum/minimum for positive/negative \napplied fields, similar to pure CCO. The helicity contrast is reduced when the magnetic field increases, \nirrespective of the magnetic field direction. At a lower q, the extremum around q2 ≈ 0.68 r.l.u. is \nenhanced at larger applied fields. The two extrema q1 and q2 show opposite helicity to each other. The \nsign change in the helicity contrast as a function of q exhibited by Ge-CCO is not observed in pure CCO, \nwhich only possesses a single maximum or minimum depending on the sign of the applied field. The \nobservation of two extrema in Ge-CCO may be interpreted as the appearance of a second cycloidal \ncomponent with slightly smaller q and opposite helicity. In order to extract field induced changes, the \nsignal collected under H = 0 T has been subtracted from the data collected under field, as shown in -1.8-1.2-0.60.00.61.2\n0.62 0.64 0.66 0.68 0.70 0.72 0.74-1.2-0.60.00.61.21.8-0.60.00.61.21.82.43.0\n0.62 0.64 0.66 0.68 0.70 0.72 0.74-3.0-2.5-2.0-1.5-1.0-0.50.00.5- FCGe-CCO\n+ FCHelicity contrast (arb. units)\nq (r.l.u.)Pure CCO\nd)c)\nb) H= 0T\n H= -1T\n H= -2T\n H= -3T\n H= -4T\n H= -5T\n H= -6T\n H= -6.5T- FC\na)\nHelicity contrast (arb. units)+ FC\nq (r.l.u.)H= 6.5T\nH= 6T\nH= 5T\nH= 4T\nH= 3T\nH= 2T\nH= 1T\nH= 0TFigure 12. Pure CCO shows a subtle extremum around 0.66 r.l.u., which reflects a small change in the \npeak shape. In Ge-CCO, however, a single clear maximum around q = 0.69 r.l.u. is observed, indicating \nthat the magnetic contribution at q1, observed in the zero-field contrast is independent of the applied \nmagnetic field. \n \nFigure 12. Helicity contrast of the (qq0) reflection for various magnetic fields after subtracting the zero-field \nsignal. Results of CCO (a,b) and Ge-CCO (c,d) for negative FC and positive FC, respectively . \n \n4. Discussion and conclusions. \nThe Ge doped CCO thin film exhibits magnetic transitions at temperatures similar to those observed \nin pure CCO [8], with TC ≈ 95 K and the multiferroic phase appearing around TS ≈27 K. However, the \nsmall fraction of Ge doping causes an increase of coercive and saturation field in Ge-CCO. Studying the \n(qq0) magnetic diffraction peak of Ge-CCO, we observe that q is temperature dependent and goes \nfrom a commensurate value of 0.67 r.l.u. values at 27 K to incommensurate 0.72 r.l.u. at 9 K (Figure \n5). These values are larger, at least by 0.03 r.l.u., than the ones reported for the CCO film [8]. \nFurthermore, an increase in the correlation length is observed with increasing temperature. This is \npossibly an indication that Ge-CCO undergoes a magnetic phase transition ( TF) around 18 K, similar to \nwhat has been reported in bulk CCO [4]-[7] but not observed in pure CCO films. This view is supported 0.62 0.64 0.66 0.68 0.70 0.72 0.740.00.40.81.21.62.0-2.0-1.6-1.2-0.8-0.40.0\n0.62 0.64 0.66 0.68 0.70 0.72 0.740.00.40.81.21.62.0-2.0-1.6-1.2-0.8-0.40.0Helicity contrast (arb. units)\nq (r.l.u.)d)c)\nb) H= 6.5T\n H= 6T\n H= 5T\n H= 4T\n H= 3T\n H= 2T\n H= 1T+ FC- FC\nGe-CCO- FC\n+ FCPure CCO\nq (r.l.u.)a)Helicity contrast (arb. units) H= -1T\n H= -2T\n H= -3T\n H= -4T\n H= -5T\n H= -6T\n H= -6.5Tby the linear and circular dichroism extrema observed around q ≈ 0.70 and q ≈ 0.72 r.l.u. in Ge-CCO \n(Figure 6).These observations could be an indication of the coexistence of multiple spin cycloidal \ntextures for temperatures below TF as found in bulk CCO [5]-[6], where the ( qq0) magnetic peak splits \ninto three different components below TF. \nThe multiple spin cycloidal scenario in Ge-CCO gets further support from the results observed in high-\nmagnetic field, where distinct pattern is observed in the helicity contrast possessing a maximum and \na minimum at different values of q (Figure 11). As the diffraction contrast between circular \npolarizations relates directly to the sense of the cycloidal rotation, it indicates that the two extrema \nwith opposite signs, at q1 and q2, represent two different modulation vectors close to (qq0) with \nopposite handed spin rotations. Interestingly, in CCO, the helicity contrast features just one maximum, \nwhose intensity is reduced when magnetic field increases. This may simply reflect a reduction and/or \na distortion (ellipticity) of the cone aperture angle (which defines the moment contribution of the \ncycloid) as the moments try to align with the magnetic field. For the case of Ge-CCO, q1 behaves as \nCCO, while q2 gets enhanced with increase in magnetic field. These results show that Ge-CCO has at \nleast two types of spin cycloids oriented differently with respect to each other, or having a different \nlength of the propagation vector with different helicity. Assuming the low temperature phase, below \nTF, is characterized by the appearance of the commensurate (2/3 2/3 0) modulation and some satellite \nreflections, as reported in [5]-[6], the satellite at q1 may be the order parameter of the low \ntemperature phase for Ge-CCO films. The difference in behavior of Ge-CCO and CCO may be due to \nthe fact that the CCO thin film has much lower TF < 5 K because of the strain, which makes such a \ntransition observable in bulk but not in the pure CCO films. The difference in TF between Ge-CCO and \nCCO implies that Ge doping, directly affects the tiny balance between the magnetic exchange \ninteractions, while having negligible effect on the temperature of the ferrimagnetic and multiferroic \nordering that are more sensitive of the overall scale of the magnetic exchange constants. An interesting observation in our study is a direct correlation between the magnetic field cooling and \nthe spin cycloidal rotation, shown in Figure 11. As explained earlier, the circular dichroism of a \nmagnetic diffraction signal directly relates to the helicity of the cycloidal rotation. According to the \nrelationship of the electric polarization to the spiral rotation, ∝ ×(×) [14], for a thin film, \nthe reversal of the spin cycloidal leads to the reversal of the polarization. Here, we report the reversal \nof the spin cycloid and, as a consequence, the reversal of the polarization through only magnetic field \ncooling. Usually, in multiferroics, a combination of electric and magnetic fields is needed to achieve a \nsingle domain state. For our samples, it is unclear why magnetic field cooling alone produces a single \nmultiferroic domain state. This may be due to the fact that the polarization direction in these thin \nfilms is well defined without the need for fixing it through an electric field cooling process. In general, \nwe can propose a few mechanisms, which leads to a well-defined polarization direction. (1) A bias \ncreated by the difference of voltage applied prior to the measurements to charge the sample could \ndefine the polarization direction. This scenario is, however, unlikely since the applied voltage produces \nan electric field perpendicular to the sample surface (the [110] direction) which does not affect the \npolarization that lies along [ 1ത10]. (2) X-rays polarize the sample, as reported by Schierle [35], leading \nto a defined cycloidal rotation domain. This scenario is again not applicable in our case since we \nobserve a reversal of the cycloidal rotation, i.e. reversal in polarization, for opposite magnetic field \ncooling from above T c. (3) Bias produced by inversion symmetry breaking due to strain at the interface \nis another possibility. However, the strain does not break inversion symmetry along the in-plane [ 1ത10] \ndirection. (4) A plausible explanation could be a bias produced along [ 1ത10] by antisymmetric exchange, \ni.e. Dzyaloshinskii-Moriya interaction (DMI). DMI can produce weak ferromagnetism caused by \ncanting of all the collinear AFM moments towards one direction, produced by spatially alternating DM \nvectors. In this case, inverting the external field may cause the inversion of all DM vectors. In any case, \nfurther investigations on the origin this effect is required to confirm such a hypothesis. \n In summary, our x-ray investigation finds that Ge doping in CCO does not alter the main magnetic \nproperties in the ferrimagnetic state nor the onset of the multiferroic phase. Despite the similarity in \nthe temperature of phase transitions, the ground state of the Ge-doped film shows a more complex \nmagnetic behavior below TS compared to the pure CCO films. We find the occurrence of a second \ncycloidal component in the magnetic structure, which is close to commensurate, which might \nrepresent the phase below TF = 15 K occurring in bulk CCO. Only one of the cycloids observed in the \ndoped system is magnetic field dependent, although surprisingly, both reverse their helicity, which \nalso represents an inversion of the electric polarization, for opposite field cooling. \n \nAcknowledgement. \nWe gratefully thank the X11MA and X07MA beamline staff for experimental support. The financial \nsupport of the Swiss National Science Foundation, and its National Center of Competence in Research, \nMolecular Ultrafast Science and Technology (NCCR MUST) No. 51NF40-183615 is acknowledged \nand M.R. and N. O.H. acknowledge financial support of the Swiss National Science Foundation (SNSF) \n(Sinergia project ’Toroidal moments’ No. CRSII2_147606 and No. 200021_169017, respectively). H.U. \nacknowledges financial support from the European Union’s Horizon 2020 research and innovation \nprogramme under the Marie Skłodowska-Curie grant agreement No. 801459 - FP-RESOMUS - and the \nSwiss National Science Foundation through the NCCR MUST. The research leading to this result has \nbeen supported by the project CALIPSOplus under the GRANT Agreement 730872 from the EU \nFramekwork Programme for Research and Innovation Horizon 2020. \n \nReferences. \n[1] N. A. Spaldin et al., The Renaissance of Magnetoelectric Multiferroics, Science, vol. 309, 391-392 \n(2005). \n[2] M. Fiebig, et. al., The evolution of multiferroics , Nature Reviews Materials, vol. 1, p. 16046, 2016. [3] N. Menyuk et al., Ferrimagnetic Spiral Configurations in Cobalt Chromite , J. Phys. (Orsay, Fr.) 25, 528 \n(1964). \n[4] Y. Yamasaki, et. al., Magnetic Reversal of the Ferroelectric Polarization in a multiferroic Spinel Oxide, \nPhys. Rev. 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" }, { "title": "2205.14346v2.Magnetic_collapse_in_Fe__3_Se__4__under_high_pressure.pdf", "content": "Academic Editors: Liwei Geng and\nYongke Yan\nPublisher’s Note: MDPI stays neutral\nwith regard to jurisdictional claims in\npublished maps and institutional affil-\niations.Article\nMagnetic Collapse in Fe 3Se4under High Pressure\nLyudmila V . Begunovich1,2\n, Maxim M. Korshunov1,2,*\nand Sergey G. Ovchinnikov1,2\n1Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Akademgorodok 50/38,\n660036 Krasnoyarsk, Russia; lyuda.illuzia@gmail.com (L.V .B.); sgo@iph.krasn.ru (S.G.O.)\n2Siberian Federal University, Svobodny Prospect 79, 660041 Krasnoyarsk, Russia\n*Correspondence: mkor@iph.krasn.ru\nAbstract: Electronic structure and magnetic properties of Fe 3Se4are calculated using the density func-\ntional approach. Due to the metallic properties, magnetic moments of the iron atoms in two nonequivalent\npositions in the unit cell are different from ionic values for Fe3+and Fe2+and are equal to M1=2.071 mB\nand M2=\u00002.042 mB, making the system ferrimagnetic. The total magnetic moment for the unit cell is\n2.135 mB. Under isotropic compression, the total magnetic moment decreases non-monotonically and\ncorrelates with the non-monotonic dependence of the density of states at the Fermi level N(EF). For 7%\ncompression, the magnetic order changes from the ferrimagnetic to the ferromagnetic. At 14% compres-\nsion, the magnetic order disappears and the total magnetic moment becomes zero, leaving the system in a\nparamagnetic state. This compression corresponds to the pressure of 114 GPa. The magnetic ordering\nchanges faster upon application of an isotropic external pressure due to the sizeable anisotropy of the\nchemical bondings in Fe 3Se4. The ferrimagnetic and paramagnetic states occur under pressures of 5.0 and\n8.0 GPa, respectively. The system remains in the metallic state for all values of compression.\nKeywords: band structure; magnetic moment; DFT; pressure; ferrimagnet; ferromagnet; iron selenide\nPACS: 62.50.-p; 71.15.Mb; 75.50.Gg; 75.50.Bb\n1. Introduction\nMagnetic collapse associated with the disappearance of magnetic moments in 3dions is\nobserved in many insulating transition metal oxides (Mn–O, Fe–O, Co–O and Ni–O systems).\nA review of experimental data for iron oxides [ 1] revealed that the spin crossover between the\nhigh-spin and low-spin states of the cation in most cases takes place with increasing pressure\nand critical pressure is close to 50–70 GPa. The crystals with Fe2+ions have a low-spin state\nwith a zero spin ( S=0) that leads to the appearance of a nonmagnetic phase. In the case\nof crystals with Fe3+ions, the low-spin state exhibit S=1/2, so the magnetic state can be\npreserved, albeit at a lower critical temperature, such as in FeBO 3[2]. For magnetite Fe 3O4\ncontaining both Fe2+and Fe3+ions, a new nonmagnetic phase was experimentally found [ 3,4]\nat pressures above 25 GPa and room temperature.\nThe iron selenides FeSe x(1\u0014x\u00141.33) form phases with iron vacancies that crystallize\ninto structures derived from the hexagonal NiAs-type structure. Among them, two compounds,\nFe7Se8and Fe 3Se4, have superstructures with ordered Fe vacancies [ 5]. Fe 7Se8has a hexagonal\nstructure whereas Fe 3Se4has a monoclinic structure isomorphic to Cr 3X4(X = S, Se, Te). In\nan Fe 3Se4unit cell, the vacancies of Fe appear in every second iron layer. The presence of\nordered Fe vacancies facilitates the appearance of the ferrimagnetic state in Fe 3Se4[6–9]. The\nexperimental values of the total magnetic moment are in the range from 0.69mBto1.17mBper\nformula unit (f.u.) [ 6–8]. The magnetic moments on Fe ions estimated within the framework of\nthe ionic model [ 7] are too large ( 3.25mBfor site 1 and 1.94mBfor site 2) and are not validated by\nexperimental data [ 7,9,10]. Neutron diffraction on Fe 3Se4gives smaller effective spin values for\ntwo Fe positions, S1=1.08mBandS2=0.71mB[10]. In addition, a study by Mössbauer revealedarXiv:2205.14346v2 [cond-mat.mtrl-sci] 4 Jul 20222 of 14\nthe very low average internal magnetic fields for Fe sites [ 9], which do not correspond to regular\nhigh-spin Fe3+or Fe2+states. All of the abovementioned studies indicate the delocalization of\n3delectrons of iron ions and the inapplicability of the model of localized magnetic moments to\nthe system.\nIn the series Fe 3O4–Fe 3S4–Fe 3Se4, on the one hand, the presence of two non-equivalent\npositions of cations and ferrimagnetic properties are preserved, on the other hand, an increase\nin covalence enhances the metallic properties. Despite the structural similarity between greigite\nFe3S4and magnetite Fe 3O4, their magnetic properties [ 11] and magneto-optical spectra [ 12]\ndiffer significantly. The differences from magnetite become even greater for Fe 3Se4, which has\ninteresting magnetic and electrical properties [ 8–10]. Fe 3Se4is a metallic ferrimagnetic material.\nIts electronic structure was calculated within the density functional theory (DFT) [13]. Tewari\net al. [ 6] declare that the Fe 3Se4material possess half-metallic properties. The spin-down band\ngap ( Eg) and half-metallic energy gap ( EHM) calculated using the HSE06 hybrid functional [ 14]\nwere found to be 1.8 eV and 0.17 eV , respectively. However, the analysis of experimental data\nreveled the extremely small EHM(1.3 meV and 34 meV , depending on the Se stoichiometry),\nthus the low energy is required to occupy minority spin states by the majority spin carriers\nduring spin flip processes. The discrepancy between the theoretical and experimental EHM\nvalues can be associated with an increase in the localization of electronic states when a fraction\nof the Hartree–Fock exchange energy is included. This results in the inaccurate description of\nelectronic structure near the Fermi level. It is known that the HSE06 functional can overestimate\nband gap [ 15,16]. Moreover, Gao et al. [ 17] suggest that hybrid functionals, such as HSE and\nPBE0 are not suitable for studying metal systems, and the LDA or GGA give better results in\ndescribing metallic properties.\nHere we study the properties of Fe 3Se4under high pressure within DFT. We calculated the\nchanges of the magnetic moment on each sublattice with the increasing isotropic compression.\nThe values of the magnetic moments decrease non-monotonically and eventually vanish for\nboth inequivalent positions. Since Fe 3Se4possess metallic properties, such a magnetic collapse\ncannot be represented as the energy-level crossing of the high-spin and the low-spin cation\nstates. In this case, the magnetic collapse is caused by the alignment of the numbers of spin-up\nand spin-down electrons on each cation such that one can call this the itinerant analogue of\nthe spin crossover. The spin crossover occurs faster under compression by isotropic pressure\ncompared to when compression by isotropic strain.\n2. Computational Details\nCalculations of atomic and electronic structure and magnetic properties were performed\nin the framework of density functional theory using the Vienna ab initio simulation package\n(VASP) [ 18]. Exchange-correlation effects were described by the Perdew–Burke–Ernzerh (PBE)\nof generalized gradient approximation (GGA) [ 19]. The ion-electron interactions were rep-\nresented by the projector-augmented wave method (PAW) [ 20], and the plane-wave cutoff\nenergy of 600 eV was applied. The criteria for the total energy minimization and interatomic\nforces were set to 10\u00004eV and 10\u00002eV/Å, respectively. The energy convergence criteria were\ndecreased to 10\u00005eV to obtain a more accurate electronic structure. The first Brillouin zone\n(1BZ) was sampled by 24\u000214\u00028grid using the Monkhorst–Pack scheme [ 21]. The isotropic\ncompressive strain was modelled as the change of the structural parameters. The value of strain\nis defined as x= (l0\u0000l)/l0, where l0andlare the equilibrium and strained lattice constant,\nrespectively. The strain was uniform on three lattice directions, i.e., all lattice constants a,b,\nandcwere changed at the same time by the same value of x. The strained lattice constants did\nnot change during the geometry optimization. Compressive strain ranging from 0 to 14% is\nconsidered. The lattice shape remains unchanged during the compression. Compression by an\nisotropic pressure was studied by adding the external pressure to the diagonals of stress tensors.3 of 14\nThe value of the external pressure was changed from zero up to 8 GPa. Lattice constants, cell\nshape, cell volume, and ionic positions were optimized for these structures. The Visualization\nfor Electronic and Structural Analysis (VESTA) [ 22] software was used for representation of\natomic structures. The “vaspkit” software [23] was used for post-processing.\n3. Results and Discussion\nThe crystal structure of the bulk Fe 3Se4is shown in Figure 1a. The unit cell relates to the\nI2/mspace group with the structural parameters a=6.071 Å,b=3.377 Å,c=11.174 Å, and\nb=92.818\u000e, which are well in agreement with previously reported experimental data [ 6,9,10],\nsee Table 1 for comparison. The unit cell contains six Fe atoms and eight Se atoms, with\niron atoms occupying two non-equivalent positions, namely, octahedral (Fe 2) and distorted-\noctahedral (Fe 1) sites. The Se\u0000Fe2\u0000Se angles for distorted-octahedral sites are equal to 104.0\u000e,\n89.8\u000e, 89.9\u000eand 75.5\u000e. The bond lengths between Fe 2and surrounding Se ions are 2.468 Å and\n2.413 Å (bonds 1 and 2 in Figure 1a, respectively) and Fe 1–Se bond lengths are 2.390 Å,\n2.374 Å, 2.545 Å, and 2.581 Å (bond 3, 4, 5, and 6 in Figure 1a, respectively), see Table 2.\nThe distances between Se atoms closest to the iron vacancy are 2.996 Å and 5.277 Å. The\nmagnetic structure was found to be ferrimagnetic. The coupling between spins in Fe 3Se4\nis accomplished by the mechanism of “through-bond spin polarization” that is a specific\nrealization of the superexchange cation–anion–cation interaction, i.e., an iron atom with spin-\nup (spin-down) density induces spin-down (spin-up) density on the p-orbital of the adjacent\nSe atom directly bonded to it. Thus an iron atom is bonded to a selenium atom through the\np-orbital with the direction of the electron spin opposite to its spin density. The distribution\nof the magnetisation density is shown in Figure 1b. The calculated magnetic moments on\niron in sites 1 and 2 are 2.071 mBand\u00002.042 mB, respectively. The magnetic moments on Se\natoms are very small and equal to \u00000.018 mBand 0.015 mB. The total magnetic moment of\nFe3Se4is2.135 mB/f.u., i.e., in the range of the previously DFT-calculated values [ 6,13,24] but\nlarger than the known experimental values [ 6–8], see Table 1. The discrepancy in magnetic\nmoment between experimental and DFT-predicted values relates to the delocalization of 3d\nelectrons of Fe ions. Our calculations showed that the electronic structure of Fe 3Se4is metallic,\nsee Figure 2a,b, as in [ 13,25], and half-metallic state was not found. The density of states\n(DOS) at the Fermi level are predominantly formed by iron states. The bulk modulus of\nFe3Se4is determined by performing six finite distortions of the lattice with \u00060.5,\u00061.0,\u00061.5%\nmagnitude. The calculated P\u0000Vdata are fitted to the Birch–Murnaghan equation of state\nP=3/2B0h\nu\u00007/3\u0000u\u00005/3i\n\u0001h\n1\u00003/4(4\u0000B0)(u\u00002/3\u00001)i\n, where B0,u=V/V0,V0,V, and B0\nare the bulk module, the dimensionless volume, the reference volume (the initial volume of\nFe3Se4), the deformed volume, and the derivative of the bulk modulus with respect to pressure,\nrespectively. Bulk modulus as a function of volume is shown at Figure 3. The obtained value of\nB0is 68 GPa, which is very similar to that of Fe 3S4(62.8 GPa ) [26] and is in the range of values\nfor isomorphic Cr 3Se4(57.7 GPa) and Cr 3S4(72.9 GPa) [ 27] structures. Its pressure derivative\nwas found to be six.4 of 14\nFigure 1. Atomic structure ( a) and the spin-resolved magnetization density ( b) of Fe 3Se4. The unit cell is\nmarked with black lines. Brown and yellow-green colors correspond to Fe and Se atoms, respectively.\nIn panel ( b), yellow (blue) areas indicate the spin-up (spin-down) density. Isosurface value is 0.002\na\u00003\n0,where a0is the Bohr radius.\nTable 1. Structural and magnetic parameters of monoclinic phase of Fe 3Se4.\nStructural Parameters Magnetic Ref.\nLattice Constant (Å) Angle (\u000e) Moment\na b c b (mB/f.u.)\n6.202 3.532 11.331 91.825\u000e- [9]\n6.113 3.486 11.139 91.66\u000e- [10]\n6.16 3.53 11.10 92.0\u000e- [5]\n- - - - 1.17 [6]\n- - - - 0.9 [7]\n- - - - 0.69 [8]\n6.071 3.377 11.174 92.818\u000e2.128 This work\nTable 2. Bond lengths (Å) and distance (Å) between Se atoms closest to the iron vacancy of the Fe 3Se4\nmonoclinic phase. The atomic numbering scheme is shown in Figure 1a.\nBond or distance (Å)\nStrain (%) 1 2 3 4 5 6 7 8\n0 2.468 2.413 2.390 2.374 2.545 2.581 2.996 5.277\n2 2.419 2.364 2.349 2.355 2.503 2.516 2.895 5.151\n3 2.396 2.342 2.325 2.344 2.480 2.481 2.849 5.091\n3.5 2.384 2.333 2.312 2.338 2.467 2.465 2.828 5.066\n4 2.381 2.335 2.291 2.324 2.440 2.446 2.825 5.068\n4.5 2.370 2.319 2.283 2.323 2.426 2.428 2.796 5.044\n5 2.346 2.301 2.281 2.316 2.424 2.429 2.756 4.994\n5.5 2.338 2.280 2.268 2.330 2.416 2.393 2.729 4.967\n6 2.325 2.276 2.264 2.305 2.397 2.400 2.700 4.943\n6.5 2.314 2.262 2.253 2.303 2.387 2.382 2.671 4.916\n7 2.306 2.245 2.239 2.311 2.379 2.352 2.645 4.888\n8 2.282 2.220 2.220 2.298 2.358 2.327 2.588 4.829\n9 2.258 2.196 2.202 2.283 2.337 2.301 2.535 4.767\n10 2.235 2.172 2.183 2.266 2.315 2.274 2.483 4.705\n11 2.212 2.147 2.164 2.250 2.294 2.248 2.432 4.644\n12 2.189 2.123 2.145 2.231 2.272 2.222 2.385 4.582\n13 2.166 2.100 2.127 2.211 2.250 2.195 2.337 4.519\n14 2.143 2.078 2.109 2.189 2.277 2.169 2.292 4.4555 of 14\nFigure 2. Density of states ( a,b) of Fe 3Se4. Total DOS of Fe 3Se4is shown by black curves. In panel ( a),\npartial DOS of Fe and Se atoms are shown by green and red curves, respectively. In panel ( b), partial\nDOS of iron atoms on site 1 and on site 2 are shown by green and red curves, respectively. Positive and\nnegative values of DOS corresponds to spin-up and spin-down channels, respectively. The Fermi level\ncorresponds to zero.\nFigure 3. Pressure as a function of Fe 3Se4volume. Blue dots indicate values for structures with finite\nstrains\u00060.5,\u00061.0,\u00061.5%. The curve through this values fitted to the Birch–Murnaghan equation is\nmarked by red.\nNext we study the effect of strain on the magnetic and electronic properties of Fe 3Se4.\nFirst approach is to model the isotropic compressive strain via the change of the structural\nparameters. The lattice of Fe 3Se4remains monoclinic and the angles between vectors remain\nunchanged during compression. Isotropic compression along the lattice constant up to 4%\nreduces the total magnetic moment of the system to 0.157 mB/f.u., see Figure 4a and Table 3,\nwhich is caused by a decrease in bond lengths, see Table 2. The reduction in bond lengths\nmakes the covalent character greater than the ionic character, which causes a decrease in the\nspin polarization of atoms. Further compression to 5% increases the magnetic moment by\n0.904 mB/f.u. In this compression range, the bond lengths with iron atoms located in the vacant\nlayer (Fe 2\u0000Se) decrease faster (by near 1.5%) than Fe 1\u0000Se bond lengths (by 0.44, 0.34, 0.66,\n0.70%). This leads to the fact that the magnetic moment on the Fe 2atoms drastically decreases\nby 65.4% upon compression from 4 to 5%. At the same time, the magnetic moment on Fe 1\natoms increases from 0.602 to 0.764 mB. A significant decrease in the magnetization on Fe 2,\nopposite in direction to the magnetization of Fe 1, leads to an increase in the total magnetic\nmoment in the system. With further compression, the change in the Fe 2\u0000Se bond lengths\nslows down and the change in Fe 1\u0000Se bond lengths increases, causing the magnetic moment\nto decrease to 0.124 mB/f.u. (6.5% compression).6 of 14\nTable 3. Magnetic moments (in mB) of iron atoms on two sites, Fe 1and Fe 2, and total magnetic moments\nper Fe 3Se4formula unit of Fe 3Se4under compression. The Wigner–Seitz radius for Fe ions is 1.302 Å\nStrain (%) Fe 1Moment Fe 2Moment Total Moment\n0 2.071 \u00002.042 2.135\n2 1.778 \u00001.631 1.969\n3 1.556 \u00001.417 1.759\n3.5 1.387 \u00001.318 1.460\n4 0.602 \u00001.334 0.157\n4.5 0.282 \u00001.214 0.692\n5 0.764 \u00000.462 1.061\n5.5 0.493 \u00000.297 0.673\n6 0.266 \u00000.209 0.368\n6.5 0.053 \u00000.041 0.075\n7 0.463 0.666 1.626\n8 0.352 0.540 1.286\n9 0.282 0.425 1.038\n10 0.225 0.328 0.823\n11 0.162 0.230 0.585\n12 0.068 0.094 0.243\n13 0.004 0.005 0.009\n14 0.000 0.000 0.0007 of 14\nFigure 4. The dependence of the total magnetic moment ( a) and DOS at the Fermi level ( b) on isotropic\ncompression.\nCompression to 7% leads to a sharp jump in the magnetic moment up to 1.626 mBand the\nmagnetic order changes from ferrimagnetic to ferromagnetic. The magnetic moments on Fe 1\nand Fe 2ions are 2.071 mBand\u00002.042 mB, respectively. The magnetic moments on Se atoms are\npractically absent and equal to 0.006 mB. The bond lengths of Fe 3Se4under 7% strain is unevenly\ndecreased by 2.26–8.87%. The distances between selenium atoms located near the iron vacancy\nare reduced by 7.4 and 11.7% compared to the initial distance, and equal to 4.888 Å and 2.645 Å,\nTable 2. The magnetic moment decreases linearly with the further compression from 7% to 13%\nand vanishes under 14% of compression. The pressure corresponding to this strain is equal to\n114 GPa and the volume of the cell is 145.55 Å3. In this structure Fe–Se bond lengths are 7.79–\n15.96% shorter compared to the bond lengths in the original structure. The distances between\nselenium atoms near the vacancy decrease by 15.6 and 23.5% and are equal to 4.455 Å and\n2.292 Å. Octahedrons formed by Fe 2-Se bonds are disodered, so that Se \u0000Fe2\u0000Se angles are\nequal to 85\u000eand 95\u000e. Selenium atoms located at opposite corners of the octahedron are still in\nthe same plane (Se \u0000Fe2\u0000Se angles are 180\u000e). The Se\u0000Fe2\u0000Se angles are equal to 108.3\u000e, 89.5\u000e,\n86.6\u000e, and 74.7\u000e. Values of the iron magnetic moments on sites 1 and 2 are shown in Table 3.\nThe electronic structure of Fe 3Se4remains metallic throughout the studied compression\nrange. The DOS at the Fermi level N(EF)as a function of strain is presented at Figure 4b.\nSlight compression up to 3.5% leads to an increase in the density of states at the Fermi level.\nFurther, the N(EF)non-monotonically depends on the strain and correlates with the non-\nmonotonic dependance of the total magnetic moment. An increase and decrease in the total8 of 14\nmagnetic moment is accompanied by an increase and decrease in the N(EF). The decrease in\nthe magnetic moment during compression from 8 to 14% is accompanied by an increase in\ntheN(EF). Figure 5 shows DOS of Fe 3Se4at critical points. The redistribution of DOSs are\nobserved, compared with the DOS of the original structure (Figure 2b). Compression up to 7%\nresults in N(EF)decreasing, and the vacant states shifting by 0.6eV to higher energies. The\ndensity of states with spin-up and spin-down is equalized on each Fe cation in the structure at a\ndeformation of 14% (Figure 5b), which leads to the disappearance of the magnetic order. It is the\nitinerant analogue of the spin crossover in the metallic system. The metallic system of itinerant\nelectrons lacks for the long-range magnetic order thus ending in the Pauli paramagnetic state.\nIn this case, the DOS at the Fermi level is increased. The occupied states decrease significantly\nat\u00000.25 eV and increase at lower energies. The vacant states decrease at 0.19 eV and increase\nat higher energies.\nFigure 5. Total (TDOS) and partial (PDOS) densities of states for Fe 3Se4under 7% ( a) and 14% ( b)\ncompression. Positive and negative values corresponds to spin-up and spin-down channels, respectively.\nThe Fermi level corresponds to zero.\nThe energy difference DEper formula unit between compressed and original Fe 3Se4was\ncalculated as DE=Ecompressed\u0000Eoriginal . Here Ecompressed andEoriginal are the total energy of the\nFe3Se4cell under isotropic compression and the total energy of the non-compressed Fe 3Se4cell,\nrespectively. The dependence of DEon compressive strain is shown in Figure 6. The increase\nin the energy is described by the cubic polynomial f=0.0032 x3\u00000.0048 x2+0.0759 x\u00000.0241 .\nThe structure at critical points of 7 and 14% is higher in energy by 1.38 and 8.86 eV/f.u.,\nrespectively.9 of 14\nFigure 6. Energetic stability of the compressed Fe 3Se4structure relative to the original structure, DEin eV\nper Fe 3Se4formula unit.\nSecond approach to studying the effect of compression of Fe 3Se4cells is to apply the\nisotropic pressure by adding the external pressure to the diagonals of the stress tensors. The\nresults of calculations show that the cell shape of Fe 3Se4does not change over the entire interval\nof the external pressure. Lattice constants vary nonuniformly because the cell is anisotropic,\nsee Table 4. The lattice constant cdecreases faster than aandbdue to the presence of vacant\nlayers. As a result the Fe 2–Se bonds and the magnetic moments on Fe 2atoms decrease rapidly,\nsee Tables 5 and 6. This difference is not so significant when the cell is compressed by small\npressure. The cconstant changes by more than 7% at an external pressure of more than 4.8 GPa.\nAt the same time, the bconstant is slightly increased compared to the original structure. In this\ncase, the magnetic moments on iron in both sublattices are co-directed and oppositely directed\nto the magnetic moments on the Se ions, which are equal to \u00000.025 mBand\u00000.003 mB. This\nferrimagnetic ordering is accomplished by the through-bond spin polarization. Iron atoms\nwith a spin-up density induce a spin-down density on the adjacent Se atoms. This leads to\nan increase in the magnetic moment by 0.608 mB. An increase in pressure up to 5.0 GPa leads\nto the disappearance of magnetic moments on Se atoms and the magnetic order of Fe 3Se4\nbecomes ferromagnetic. Then, the total magnetic moment decreases with increasing external\npressure that is associated with a further decrease in bond lengths, see Table 5. At a pressure of\n5.0 GPa, the value of the magnetic moment sharply decreased from 2.617 mBto0.146 mBsince the\nFe1–Se bond lengths starts to decrease faster than previously. The magnetic order disappears\nand the magnetic moment becomes zero at a pressure of 8.0 GPa. The Fe–Se bond lengths are\n2.02–7.59% shorter compared to those in the original structure. The distances between selenium\natoms near the vacancy decrease by 8.9 and 6.0%. Se–Fe 1–Se angles are equal to 97.7\u000e,91.2\u000e,\n93.5\u000e, and 76.8\u000e. Se–Fe 2–Se angles are 86.7\u000eand 93.3\u000e. Selenium atoms located at opposite\ncorners of this octahedron remain in the same plane. The volume of the cell is 201.61 Å3. Thus,\nthe spin crossover in the monoclinic phase of Fe 3Se4occurs faster under compression by the\nisotropic pressure compared to by the isotropic strain. The dependence of the total magnetic\nmoment under compression by an external pressure is shown in Figure 7a.\nTable 4. Lattice constants (Å) of Fe 3Se4under isotropic external pressure (GPa).\nExternal Pressure (GPa)Lattice Constant2.0 3.0 4.5 4.8 5.0 7.5 8.0\na(Å) 6.016 5.993 5.962 5.883 5.892 5.857 5.850\nb(Å) 3.341 3.327 3.311 3.423 3.477 3.460 3.455\nc(Å) 11.065 11.014 10.929 10.382 10.048 10.002 9.98710 of 14\nTable 5. Bond lengths (Å) and distance (Å) between Se atoms closest to the iron vacancy of the Fe 3Se4\nunder isotropic external pressure. The atomic numbering scheme is shown in Figure 1a.\nPressure (GPa) Bond or Distance (Å)\n2.0 2.440 2.391 2.371 2.364 2.529 2.543 2.936 5.232\n3.0 2.429 2.382 2.363 2.360 2.521 2.528 2.912 5.214\n4.5 2.414 2.369 2.350 2.353 2.511 2.505 2.878 5.184\n4.8 2.380 2.339 2.310 2.337 2.499 2.432 2.795 5.071\n5.0 2.377 2.345 2.307 2.333 2.457 2.403 2.772 4.993\n7.5 2.366 2.332 2.296 2.450 2.388 2.736 2.968 4.968\n8.0 2.362 2.329 2.293 2.326 2.447 2.385 2.730 4.959\nTable 6. Magnetic moments (in mB) of iron atoms on two sites, Fe 1and Fe 2, and total magnetic moments\nper Fe 3Se4formula unit of Fe 3Se4under external pressure. The Wigner–Seitz radius for Fe ions is 1.302 Å.\nPressure (GPa) Fe 1Moment Fe 2Moment Total Moment\n2.0 1.942 \u00001.829 2.097\n3.0 1.884 \u00001.738 2.068\n4.5 1.786 \u00001.596 2.009\n4.8 1.065 0.544 2.617\n5.0 0.077 0.005 0.146\n7.5 0.013 0.001 0.017\n8.0 0.000 0.000 0.00011 of 14\nFigure 7. The dependence of the total magnetic moment ( a) and DOS at the Fermi level ( b) on\nexternal pressure.\nDOS at the Fermi level N(EF)correlates with the non-monotonic dependance of the total\nmagnetic moment, see Figure 7b. A decrease in the total magnetic moment is accompanied by\nan increase in the N(EF). A change in the magnetic order from ferrimagnetic to ferromagnetic\nleads to a decrease in N(EF)that was also observed in the structure under compression by the\nisotropic strain. The electronic structure of Fe 3Se4remains metallic at all values of external\npressure. DOS with spin-up and spin-down become equal on each Fe cation under 8.0 GPa\n(Figure 8), resulting in a paramagnetic state. DOS in the vicinity of the Fermi level are increased\ncompared to the DOS of the original structure shown in Figure 2b. DOS redistribution in this\ncase differs from that for the paramagnetic Fe 3Se4structure at 14% compressive strain, since\nthe structural parameters change differently.\nThe energy difference DEbetween compressed and uncompressed Fe 3Se4changes faster\nwith the increasing external pressure than with the increasing strain, compare Figures 9 and 6 .\nUnder the external pressure, the increase in energy is described by the linear dependence12 of 14\nf=1.3141 x+0.1717 . Energy of Fe 3Se4at critical points of 5.0 GPa and 8.0 GPa is higher by\n7.10 and 10.55 eV/f.u., respectively. Thus, the ferromagnetic structure obtained under the\nisotropic strain compression is more stable than the one under the isotropic external pressure\nby\u00005.72 eV/f.u. The energy difference of the paramagnetic states in two regimes is not so\nsignificant and equals to \u00001.69 eV/f.u.\nFigure 8. Total and partial DOS for Fe 3Se4under isotropic external pressure of 8.0 GPa. Positive and\nnegative values correspond to spin-up and spin-down channels, respectively. The Fermi level corresponds\nto zero.\nFigure 9. Energetic stability of the Fe 3Se4compressed by the external pressure relative to the uncom-\npressed structure, DEin eV per Fe 3Se4formula unit.13 of 14\n4. Conclusions\nDFT calculations within GGA for Fe 3Se4show that the ground state is metallic and the\nsystem is not in the half-metal state. That agrees with the conclusions of [ 13,25]. The value\nof the bulk modulus was found to be 68 GPa. By studying the compression effect on Fe 3Se4\nwithin DFT, we found the itinerant analogue of the spin crossover in the metallic system. In\nparticular, we calculated the magnetic moment in each of two iron sublattices of Fe 3Se4in two\nregimes: (a) assuming an increasing isotropic compression by the compressive strain and (b)\nthe isotropic external pressure. For crystals with such an anisotropic chemical bonding as in\nFe3Se4, the regime (b) is more relevant to the experiments where the isotropic external pressure\nis applied. If the deformation of the crystal were isotropic as in the regime (a), the values of the\nmoments would vanish for both inequivalent positions at the 14% of strain that corresponds\nto the pressure of 114 GPa. On the other hand, in the regime (b) under the isotropic external\npressure, the magnetic collapse is expected to occur at a much smaller value of pressure, namely,\nat 8 GPa. Under the compression, the system evolves from the uncompressed ferrimagnetic\nstate first to the ferromagnetic state and then to the paramagnetic state; the process is sketched\nin Figure 10.\nFigure 10. Illustration of the Fe 3Se4transformation under the applied isotropic strain xor the isotropic\nexternal pressure Pexbetween the ferrimagnetic and the paramagnetic states through the ferromagnetic\nstate. Brown and yellow-green colors correspond to Fe and Se atoms, respectively. The arrows indicate\nthe direction of magnetic moments on iron atoms.\nAuthor Contributions: Conceptualization, S.G.O.; calculations, L.V .B.; formal analysis, M.M.K.; writing,\nL.V .B., S.G.O., and M.M.K.; funding acquisition, S.G.O. All authors have read and agreed to the published\nversion of the manuscript.\nFunding: L.V .B. and S.G.O. acknowledge the support of the Russian Science Foundation (Project 18-12-\n00022P).\nInstitutional Review Board Statement: Not applicable.\nInformed Consent Statement: Not applicable.\nData Availability Statement: Not applicable.\nAcknowledgments: We acknowledge the useful discussions with M.A. Vysotin. L.V .B. would like to\nthank the Information Technology Center, Novosibirsk State University, for providing access to their\nsupercomputer facilities.\nConflicts of Interest: The authors declare no conflicts of interest.14 of 14\nReferences\n1. Lyubutin, I.S.; Gavriliuk, A.G. High and Ultra-High Pressure Research on Phase Transformations in 3d-Metal Oxides: Current\nProgress. Uspekhi Fiz. Nauk 2009 ,179, 1047. doi:10.3367/UFNe.0179.200910b.1047.\n2. Gavriliuk, A.G.; Trojan, I.A.; Lyubutin, I.S.; Ovchinnikov, S.G.; Sarkissian, V .A. 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Phys. 2013 ,138, 204712. doi:10.1063/1.4807614\n27. Guo, S.; Liu, B. Stable half-metallic ferromagnetism in nonstoichiometric cubic binary chromium chalcogenides. Europhys. Lett. 2009 ,\n88, 67007. doi:10.1209/0295-5075/88/67007" }, { "title": "1810.00763v1.Trimers_of_MnO6_octahedra_and_ferrimagnetism_of_Ba4NbMn3O12.pdf", "content": "1 \n Trimer s of MnO 6 octahedra and ferri magnetism of Ba4NbMn 3O12 \nLoi T. Nguyen , Tai Kong and R.J. Cava \nDepartment of Chemistry, Princeton University, Princeton, New Jersey 08544, USA \n \n \nAbstract \nBa4NbMn 3O12 is reported, synthesized by a solid state method in air . The crystal structure , \ndetermined by performing refinements on room temperature powder X -ray diffraction data by the \nRietveld method, consists of Mn 3O12 trimers in the configuration of three face -sharing MnO 6 \noctahedra , with the trimers arranged in triangular planes . An effective moment of 4.82 μB/f.u is \nobserved and competing antiferromagnetic and ferromagnetic interaction s between Mn ions are \ninferred from the Weiss temperature of -4 K and the ferrimagnetic ordering transition of \napproximately 4 2 K. Ba4NbMn 3O12 is a semiconductor with a transport activation energy of 0.37 \neV. \n \n 2 \n Introduction \nMangan ese-based perovskites have been intensively researched due to the metal -insulator (MI) \ntransition s and colossal magnetoresistance ( CMR )1,2,3 ,4 in this family . The interesting properties \nhave been attributed in part to the competition between a ferromagnetic metallic state and an \nantiferromagnetic insulating state , and the presence of charge -ordering . Manganates are also well \nknown due to other structural, electronic , and magnetic characteristics5,6. In the hexagonal \nperovskite RMnO 3 phases, for example, the electric polarization is reported to be the result of a \nstructural transition7 and the improper nature of ferroelectricity to a network of coupled structural \nand magnetic vortices that induce domain wall magnetoelectricity and magnetization8. \nOf particular interest for CMR are materials with lower structural dimension based on the \nRuddlesden -Popper series9, but quasi -1D materials are also of interest10. Instead of sharing corner s \nin a 3D network as in corner -sharing perovskites, the MnO 6 octahedra in the quasi -1D materials \ncan share faces with each other to form Mn 2O9 dimers or Mn 3O12 trimers. These Mn mini-clusters \nare isolated from other clusters in the structure. This type of MnO 6 configuration is relatively \nuncommon because the Mn-Mn distance s of about 2.5 Å within the cluster s are shorter than the \n3.5 Å typically found in corner -sharing geometries11. Oxygen deficient BaMnO 3-x and \n(Ba,Sr)MnO 3-x phases are good examples of materials where clusters of face -sharing MnO 6 \noctahedra are found12. \nIn th e current work, we report the synthes is and initial characterization of the mixed valent \nMn3+/Mn4+ compound Ba4NbMn 3O12. The crystal structure of this previously unreported phase , \nwhich is much like that of Ba 4REMn 3O12 (RE = Ce, Pr )13, consists of face -sharing MnO 6 octahedra \nform ing Mn 3O12 trimers with their long axes aligned along the hexagonal c axis of the trigonal \nsymmetry crystallographic cell. From magnetic susceptibility measurement s, the effective 3 \n magnetic moment is found to be 4.82 μB/f.u., consistent with a simple spin configuration within \nthe trimers. Resistivity measurement s show that Ba 4NbMn 3O12 is semi conducting with a transport \ngap of 0.37 eV . The current phase is chemically and structurally distinct from recently reported \nBa8MnNb 6O24, which, among other differences, does not appear to contain any face shared \noctahedra14. \nMethod s \nPolycrystalline samples of Ba 4NbMn 3O12 were synthesized by solid -state reaction using BaCO 3, \nMnO2, and Nb2O5 (Alfa Aesar, 99.9 %, 99.999 %, and 99.5 %, respectively ) as starting materials . \nReagents were mixed thoroughly in the appropriate ratio, placed in an alumina crucible, and heated \nin air at 900 ℃ for 24 hours. The resulting powder was re -ground, pressed into a pellet and heated \nin air at 1100 ℃ for 24 hours, and then at 1300 ℃ for 12 hours. The phase purity and crystal \nstructure were determined through powder X -ray diffraction (PXRD) using a Bruker D8 Advance \nEco with Cu Kα radiation and a LynxEye -XE detector. The structural refinements were performed \nwith GSAS15. The crystal structure drawing was created by using the program VESTA16. \nThe magnetic susceptibility of Ba 4NbMn 3O12 powder was measured by a Quantum Design \nPhysical Property Measurement System (PPMS) DynaCool equipped with a VSM option. The \nmagnetic susceptibility of Ba 4NbMn 3O12, defined as M/H, where M is the sample magnetization \nand H is the applied field, was measured at the field of H = 1 kOe from 1.8 K to 300 K; some \nadditional measurements were performed in an applied field of 100 Oe. (Relatively low fields are \nemployed so as to not overly disrupt the magnetic s ystem through the measurements.) The \nresistivity of Ba 4NbMn 3O12 was measured by the DC four -contact method in the temperature range \n200 K to 3 50 K with the PPMS. The sample was pressed, sintered, and cut into pieces with the \napproximate size 1.0 × 2. 0 × 1.0 mm3. Four Pt contact wires were connected to the sample using 4 \n silver paint. The specific heat was measured from 1.8 K to 200 K by a PPMS DynaCool equipped \nwith a heat capacity option . \nResults and discussion \nThe powder x-ray diffraction pattern and structural refinement of Ba 4NbMn 3O12 are shown in \nFigure 1 . The structure of Ba 4NbRu 3O1217 was used as the starting model. Ba4NbMn 3O12 \ncrystallizes in a rhombohedral structure with the space group R-3m (No. 166). Refinements in \nwhich the Mn to Nb ratio was allowed to vary from 3:1 or Nb/Mn site mixing was permitted were \nnot satisfactory. The lattice parameters and structural parameters for Ba4NbMn 3O12 are \nsummarized in Table 1 . The structure consists of three MnO 6 octahedra connected by face -sharing \nto form Mn 3O12 trimers , as shown in Figure 2 . The crystal structure of Ba 4NbMn 3O12 is similar to \nthose of two known rare-earth containing compounds Ba 4REMn 3O12 (RE = Ce, Pr)13 in the same \nfamily. The corner -sharing NbO 6 and Mn 3O12 trimers in Ba 4NbMn 3O12 alternate along c to \ngenerate the 12 -layer hexagonal perovskite structure . Individual Mn 3O12 trimers are co rner-sharing \nwith the non -magnetic NbO 6 octahedra , and not to other trimers, such that the magnetic coupling \nbetween trimers is of the Mn -O-O-Mn super -super exchange type . Within each trimer, the distance \nbetween Mn atoms is 2.4694(1) Å, quite short due to the face-sharing11, favoring strong magnetic \ninteractions between them. The Mn -O distances are 1.903(4) Å and 1.882(4) Å for the outer MnO 6 \noctahedra , reflecting a small distortion, and 1.895(4) Å for the inner MnO 6 octahedron , which has \nan ideal shape . These Mn-Mn and Mn-O bond lengths are in agreement with those found for the \n10-layer hexagonal perovskite Ba5Sb0.64Mn 4.36O15-δ18, for example . \nThe temperature -dependent magnetic susceptibility of Ba 4NbMn 3O12 and its reciprocal are plotted \nin Figure 3. The magnetic data for Ba 4NbMn 3O12 from 100 K to 275 K are well fit to the Curie -5 \n Weiss law 𝜒=𝐶\n𝑇−𝛳𝐶𝑊+𝜒𝑜, where 𝜒𝑜 is the temperature -independent part of the susceptibility, C \nis the Curie constant, and ϴCW is the Curie -Weiss temperature. The least squares fitting yields 𝜒𝑜= \n2.79x10-3 emu Oe-1 mol-f.u.-1, 𝜇𝑒𝑓𝑓= 4.82 μB/f.u, and 𝛳𝐶𝑊= -4 K. The magnetic hysteresis loops \nfor Ba 4NbMn 3O12 from -2 to 2 T at 2 K, 25 K and 150 K are shown in Figure S 1 (See SI) . A \ncoercive force of 0.2 T and the remnant magnetization of 0.04 μ B/f.u. are observed at 2 K and 0.04 \nT and 0.02 μ B/f.u. at 25 K. The overall behavior is like that of a ferrimagnet. At 150 K, the \nmagnetization is linearly proportional to the field and there is no hysteresis loop. \nBecaus e Ba2+ and Nb5+ (which has no electrons in d orbitals) are non -magnetic, the magnetic \nproperties of Ba 4NbMn 3O12 are determined by the intertrimer and intratrimer interactions of the \nMn 3O12 units. Formally, one Mn3+ and two Mn4+ are found in each Mn 3O12 trimer . Thus, i n the \nparamagnetic state, where the moments fluctuate, the total magnetization per formula unit is \nexpected to be the sum of the magnetization s of two Mn4+ and one Mn3+, which, should all the \nmoments be free to respond to an applied field for temperatures below 300 K , yield a total effective \nmoment of 6.16 (low spin Mn3+) or 7.35 (high spin Mn3+) μB/f.u. This magnetic state is not \ncompatible with the observed data. Figure 4 shows , in contrast, a schematic of two simple \nhypothetical magnetic states within each trimer , which, to be consistent with our magnetic data, in \nthese simple scenarios , either two Mn4+ bearing spin -3/2 point in the same direction and opposite \nto the low spin (S=1) Mn3+ between them or two Mn4+ bearing spin -3/2 point in opposite direction s \nand the Mn3+ (high spin) with the spin -2 is between them . With this proposed magnetic coupling \nwithin the trimer , each trimer has spin -2 and a calculated effective moment of 4.90 μB/f.u. This is \nin excellent agreement with the observed effective moment of 4. 82 μB/f.u in the susceptibi lity \nmeasurement s, indicating that either of these trimer -based spin models is likely to successfully \ndescribe the magnetic state of Ba 4NbMn 3O12 above the three -dimensional magnetic ordering 6 \n temperature near 42 K . Compared to Bi3Mn 3O1119, for example, which has mixed -valent Mn, an \neffective magnetic moment of 6.27 μB/f.u and a Curie -Weiss temperature of 222 K, both the \neffective moment and Curie -Weiss temperature of Ba4NbMn 3O12 are smaller. The strong coupling \nof the Mn spins within the trimers to create a magnetic Mn “molecule” made of three Mn’s in the \ncurrent material is consistent with what has previously been found for (Ba,Sr)MnO 3 phases12,20. \nThe fitted χ0 in this scenario is then a representation of the remnant susceptibility of the magnetic \nsystem after the local -only magnetic ordering has set in within the individual trimers at \ntemperatures above 300 K. \nFigure 5 shows the field -cooled (FC) and zero -field-cooled (ZFC) DC susceptibility in an applied \nfield of 100 Oe for Ba 4NbMn 3O12. The increases of the FC and ZFC susceptibility below 42 K \nreconfirm the magnetic transition temperature observed in the magnetic susce ptibility data shown \nin Figure 3, and the χT vs. T data shown in the inset defines the temperature of the magnetic \ntransition at the same temperature through the dramatic change in the Curie constant . \nThe resistivity of Ba 4NbMn 3O12 is plotted as a function of reciprocal temperature in Figure 6. \nResistivity data from 300 to 350 K were fit to the standard model 𝜌=𝜌𝑜𝑒𝐸𝑎\n𝑘𝑏𝑇, and the transport \nactivat ion energy E a was calculated to be 0.37 eV. The inset shows the increase in resistivity when \ncooling. With the activation energy of 0.37 eV, Ba 4NbMn 3O12 is a semiconductor, similar to other \ntrimer -based compounds (Ba 4NbRu 3O12 and Ba 4LnRu 3O12 and Ba 4LnIr 3O12)17,21. \nFigu re 7A shows the specific heat divided by temperature of Ba 4NbMn 3O12 measured from 1.8 K \nto 200 K in its main panel, with the raw C p data shown in the inset . At 200 K, Cp has not yet \nreached the saturation value of 3NR (N is the number of atoms) , but this is often encountered in \nmaterials where d ifferent atomic mass es and strong bonds between atoms lead to very high 7 \n vibrational frequenc ies 22. Three characteristic features are seen. The λ-anomaly corresponds to \nthe magnetic transition around 42 K and is shown in Figure 7B . In this case we employed a \npolynomial function fit to the data above and below the anomaly to estimate the phonon \nbackground ; at this level of analysis we do not attach any significance to the function employed to \nestimate the phonon part . In this way the magnetic entropy change in the higher temperature range \nis estimated to be 0.45 J mol-1 K-1. This relatively weak anomaly implies that there is a significant \nremnant of magnetic entropy in Ba 4NbMn 3O12 below the 42 K transition. The specific heat in the \nlower temperature range is shown in more detail in Figure 7C, which shows another entropy loss \nat 5-6 K. In this case the phonon part of the specific heat is estimated by a Debye -Einstein model, \nagain to which we attribute no physical significance at this level of analysis . The magnetic entropy \nat this lower temperature , which d oes not clearly yield a feature in the magnetic susceptibility, is \nfound to be 2.67 J mol-1 K-1. For these two temperature regions only, o nly one -fourth of the \nmagnetic entropy expected for an S=2 Heisenberg system is recovered. There is a more subtle \nfeature in C p/T at around 30 K that may hold a significant amount of entropy, but we cannot \nreasonably analyze it at this time. Aside from the most general conclusions described here , analysis \nof the specific heat cannot be considered as well -established at present , as we have not been able \nto make a non -magnetic analog to use to best model the phonon contribution to the specific hea t \nof this material . A straightforward analysis (See the SI , Figure S 2 and S 3) shows that the observed \nmagnetic transition near 42 K cannot possibly be due to the presence of an Mn 3O4 impurity. \n 8 \n Conclusion \nBa4NbMn 3O12, a previously unreported material synthesized by a solid state method, crystallizes \nin a 12-layer hexagonal perovskite unit cell with the R-3m space group. Differences in the Mn -O \nbond lengths in the octahedra, although subtle, may reflect the presence of charge ordering , \nalthough we do not believe that the distinction is great enough to warrant assigning specific charge \nstates to specific octahedra at the present time . The material has a high, temperature -dependent \nresistivity, indicating that it is semiconducting , with a transport activation energy of 0 .37 eV \nmeasured on a sintered polycrystalline pellet . From magnetic susceptibility measurements, the \nCurie -Weiss tempe rature is calculated to be -4 K, indicat ing the presence of competing \nferromagnetic and antiferromagnetic interaction s between Mn trimers . The effective moment of \n4.82 μB/f.u agrees with the value of 4.90 μB/f.u that can be deduced from a simple hypothetical \nmagnetic model in which the spins within the trimer are essentially already ordered by 300 K, \nalthough the trimer -trimer ordering does not occur until much lower temperatures . The ordering \nof the trimer spins with respect to those in other trimers is what most likely gi ves rise to the \nmagnetic transition observed around 42 K. Heat capacity measurement s show a weak anomaly at \naround 42 K, supporting the magnetic ordering transition seen in magnetic susceptibility. Only \nsmall amounts of magnetic entropy are observed through the specific heat, consistent, in principle, \nwith the proposal that most of the magnetic entropy is lost at higher temperature than is studied \nhere. A non -magnetic analog for this system would be of interest to account for the phonon \ncontribution i n the specific heat so that a more detailed interpretation of the entropy will be \npossible . Magnetic neutron diffraction will also be of future interest to establish the nature of the \nmagnetism in this material. As for Mn -based conventional perovskites, he xagonal Mn -based \nperovskites appear to d isplay rich magnetic phenomena. 9 \n Conflicts of interest \nThere are no conflicts of interest to declare. \nAcknowledgement \nThis work was supported in its entirety by the Department of Energy Division of Basic Energy \nSciences, through the Institute for Quantum Matter, grant DE -FG02-08ER46544. The authors \nthank Daniel Khomskii for valuable discussion s about manganite magnetism . 10 \n Reference s \n(1) E. O. Wollan and W. C. Koehler, Physical Review 100, 545 (1955). \n(2) R. V. Helmolt, J. Wecker, B. Holzapfel, L. Schultz, and K. Samwer, Physical Review Letters \n71, 2331 (1993). \n(3) S. Jin, T. H. Tiefel, M. Mccormack, R. A. Fastnacht, R. Ramesh, and L. H. Chen, Science \n264, 413 (1994). \n(4) I. Álvarez -Serrano, M. López, C. Pico, an d M. Veiga, Solid State Sciences 8, 37 (2006). \n(5) A. Sundaresan, A. Maignan, and B. Raveau, Physical Review B 55, 5596 (1997). \n(6) R. Laiho, K. G. Lisunov, E. Lähderanta, P. Petrenko, J. Salminen, V. N. Stamov, and V. S. \nZakhvalinskii, Journal of Physics: Condensed Matter 12, 5751 (2000). \n(7) B. B. V. Aken, T. T. Palstra, A. Filippetti, and N. A. Spaldin, Nature Materials 3, 164 (2004). \n(8) H. Das, A. L. Wysocki, Y. Geng, W. Wu, and C. J. Fennie, Nature Communications 5, 2998 \n(2014). \n(9) Y. Morimoto, A. Asamitsu, H. Kawahara, Y. Tokura, Nature 380 141 (1996). \n(10) J. Vente, K.V. Kamenev, D.A. Sokolov, Phys. Rev. B64 214403 (2001) . \n(11) A. Furrer, T. Strässle, J. P. Embs, F. Juranyi, V. Pomjakushin, M. Schneider, and K. W. \nKrämer, Physical Review Letters 107, 115502 (2011). \n(12) J.J. Adkin and M.A. Hayward, Chem. Mat. 19 755 (2007). \n(13) A. F. Fuentes, K. Boulahya, and U. Amador, Journal of Solid State Chemistry 177, 714 \n(2004 ). \n(14) F. Tao, C. Liang, X. Wang, X. Li, F. Porcher, M. Allix, F. Lu, H. Gong, L. Lu and X Kuang \nInorganic Chemistry 57 5732 (2018). \n(15) B. H. Toby, J. Appl. Crystallogr. 34, 210 (2001). \n(16) K. Momma and F. Izumi, J. Appl. Crystallogr. 44, 1272 (2011). \n(17) L. T. Nguyen, T. Halloran, W. Xie, T. Kong, C. L. Broholm, and R. J. Cava, Physical \nReview Materials 2, 054414 (2018). \n(18) C. Yin, G. Li, W. A. Kockelmann, F. Liao, J. P. Attfield, and J. Lin, Chemistry of Materials \n22, 3269 (2010). \n(19) A. A. Belik and E. Takayama -Muromachi, Journal of the American Chemical Society 131, \n9504 (2009). \n(20) K. Kuroda, N. Ishizawa, N. Mizutani, and M. Kato, Journal of Solid State Chemistry 38, \n297 (1981). 11 \n (21) Y. Shimoda, Y. Doi, M. Wakeshima, and Y. Hinatsu, Journal of Solid State Chemistry 183, \n1962 (2010). \n(22) M. Akaogi and E. Ito, Geophysical Research Letters 20, 105 (1993). 12 \n Table 1. Structural parameters for Ba4NbMn 3O12 at 300 K . Space group R-3m (No. 166). \nAtom Wyckoff. Occ. x y z Uiso \nBa1 6c 1 0 0 0.1284 7(4) 0.0344(3) \nBa2 6c 1 0 0 0.2859 2(4) 0.0304(3) \nNb 3a 1 0 0 0 0.0083(3) \nMn1 3b 1 0 0 ½ 0.0195(3) \nMn2 6c 1 0 0 0.4121 8(4) 0.0146(3) \nO1 18h 1 0.4776 (6) 0.522 4(6) 0.12239 (4) 0.0212(3) \nO2 18h 1 0.4908 (6) 0.509 2(6) 0.29327 (4) 0.0127(3) \na = 5.7182 5(3) Å, c = 28.1158 (3) Å \nRwp = 10.30%, R p = 8.02%, R F2 = 10.80 % \n \n \n 13 \n \n \nFigure 1 (Color online) : Rietveld P owder X -ray diffraction refinement for Ba 4NbMn 3O12 in \nspace group R-3m. The observed X -ray pattern is shown in black, calculated in red, \ndifference (I obs-Icalc) in blue, the background in green, and tick marks denote allowed peak \npositions in pink (Ba 4NbMn 3O12) and in cyan (BaNb 0.5Mn 0.5O3). \n \n14 \n \nFigure 2 (Color online) : The crystal structure of Ba 4NbMn 3O12. The NbO 6 octahedra (dark \ngreen) share corners with the Mn 3O12 (purple) trimers made from face -shared MnO6 \noctahedra, to form what is frequently referred to as a “12 -layer” hexagonal perovskite \nstructure. Barium is green, and oxygen is red. \n15 \n \nFigure 3 (Color online) : The temperature dependence of the magnetic susceptibility and the \ninverse of the difference between the magnetic susceptibility and the temperature \nindependent magnetic susceptibility ( 𝝌𝒐 = 2.79 ×10-3 emu Oe-1 mol- f.u.-1) for Ba 4NbMn 3O12. \nThe applied field was 1 kOe. The red solid line is the susceptibility fit to the Curie -Weiss law \nfor T from 100 K-275 K. \n \n16 \n \nFigure 4 (Color online): Two s chematic s of possible simple hypothetical arrangement s for \nMn3+ (either S=1 for low spin (left model) or S=2 for high spin (right model) ) and Mn4+ \n(always S=3/2) in Ba 4NbMn 3O12. Both hypothetical arrangements yield spin-2 for each \nMn 3O12 trimer, μ eff = 4.90 μ B/f.u., which compares very favorably to the observed μ eff = 4.82 \nμB/f.u. \n \n17 \n \nFigure 5 (Color online) : Field Cooled ( FC) and Zero Field Cooled ( ZFC ) DC magnetic \nsusceptibility in an applied field of 100 Oe for Ba 4NbMn 3O12 from 2 -300 K. The inset shows \nthe first derivative of magnetic susceptibility multiplied by temperature as function of \ntemperature (a measure of the Curie constant) . The magnetic transition temperature is \nfound to be 4 2.4 K by extrapolating the derivative curves. \n \n \n \n18 \n \nFigure 6 (Color online) : The resistivity of a sintered polycrystalline pellet of Ba 4NbMn 3O12 \nas a function of temperature ( Inset) and inverse temperature (Main Panel). The data was fit \nto the model 𝝆=𝝆𝒐𝒆𝑬𝒂\n𝒌𝒃𝑻 (red line) with E a = 0.37 eV. \n \n \n19 \n \nFigure 7 (Color online) : (A) Molar heat capacity divided by temperature of Ba 4NbMn 3O12 \nmeasured from 1.8 K to 200 K. Two transitions resulting in entropy losses are marked. A \nthird one near 30 K is unmarked. Inset: the raw C p vs. T data. (B) The lower temperature \ntransition showing a detai l of the experimental data and an estimate of the phonon \nbackground. Inset: the C p/T values in excess of the estimated phonon heat capacity. (C) The \nhigher temperature transition showing a detail of the experimental data and an estimate of \nthe phonon backg round. Inset the C p/T values in excess of the estimated phonon heat \ncapacity. (D) The entropy sums through the higher and lower temperature transitions, the \nformer being 0.45 J mol-1 K-1 and the latter being 2.67 J mol-1 K-1. \n \n" }, { "title": "2012.14911v2.Spin_polarized_imaging_of_strongly_interacting_fermions_in_the_ferrimagnetic_state_of_Weyl_candidate_CeBi.pdf", "content": "Spin-polarized imaging of strongly interacting fermions\nin the ferrimagnetic state of Weyl candidate CeBi\nChristian E. Matt,1Yu Liu,1Harris Pirie,1Nathan C. Drucker,2Na Hyun Jo,3, 4Brinda Kuthanazhi,3, 4\nZhao Huang,5Christopher Lane,5, 6Jian-Xin Zhu,5, 6Paul C. Can\feld,3, 4and Jennifer E. Ho\u000bman1, 2,\u0003\n1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA\n2School of Engineering & Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA\n3Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA\n4Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA\n5Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n6Center for Integrated Nanotechnology, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n(Dated: April 12, 2022)\nCeBi has an intricate magnetic phase diagram whose fully-polarized state has recently been sug-\ngested as a Weyl semimetal, though the role of fstates in promoting strong interactions has remained\nelusive. Here we focus on the less-studied, but also time-reversal symmetry-breaking ferrimagnetic\nphase of CeBi, where our density functional theory (DFT) calculations predict additional Weyl nodes\nnear the Fermi level EF. We use spin-polarized scanning tunneling microscopy and spectroscopy\nto image the surface ferrimagnetic order on the itinerant Bi pstates, indicating their orbital hy-\nbridization with localized Ce fstates. We observe suppression of this spin-polarized signature at\nEF, coincident with a Fano line shape in the conductance spectra, suggesting the Bi pstates par-\ntially Kondo screen the fmagnetic moments, and this p\u0000fhybridization causes strong Fermi-level\nband renormalization. The pband \rattening is supported by our quasiparticle interference (QPI)\nmeasurements, which also show band splitting in agreement with DFT, painting a consistent picture\nof a strongly interacting magnetic Weyl semimetal.\nI. INTRODUCTION\n1 Merging strong electron interactions with topology\nis a new frontier for fundamental research and advanced\ntechnology [1{3]. Kondo lattice systems are a promis-\ning platform for strongly correlated topological phenom-\nena, exempli\fed by the recent observation of strongly-\nrenormalized Dirac surface states in the Kondo insula-\ntor SmB 6[4, 5], and the proposal for a Weyl-Kondo\nsemimetal phase [6]. In general, a Weyl semimetal [7]\narises when a bulk Dirac point is split into two Weyl\nnodes by breaking inversion or time-reversal symmetry\n(TRS). However, a crystal structure that breaks inver-\nsion symmetry is typically not tunable, while an applied\nmagnetic \feld Bthat breaks TRS yields only a small\nZeeman energy of \u00181 Kelvin/Tesla for a typical g-factor\nof 2. Materials with intrinsic magnetic order may have\nlarger energy scales that drive the Weyl nodes farther\napart and protect their well-de\fned chirality [8]. Such\nTRS-breaking Weyl semimetals were recently discovered\nin ferromagnets [9{11] and antiferromagnets [12]. The\nultimate goal is to combine the higher energy scales and\nstrong correlations with the practicality of external tun-\nability [13]. This goal motivates the search for topologi-\ncal phases in Kondo lattice compounds, which often host\nlarge spin-orbit coupling, strongly interacting electrons,\nand proximate \feld-tunable magnetic order [14{16].\n2 Cerium-monopnictides (Ce X,X= As, Sb, Bi) are\ncorrelated low-carrier-density Kondo lattice systems [17]\n\u0003jho\u000bman@physics.harvard.eduwith cascades of magnetic phase transitions, as shown for\nCeBi in Fig. 1 [18{20]. Though bulk magnetic phase di-\nagrams have been measured by neutron scattering, sur-\nface magnetic order has not been studied. Meanwhile,\nnon-trivial band topology was predicted in CeSb [21, 22]\nand CeBi [23, 24], and signatures of Weyl fermions were\nobserved in transport experiments in the fully-polarized\nmagnetic phase of CeSb [22]. However, the Weyl fermion\nbands have not yet been directly resolved in any phase\nof CeX, because the TRS-broken phases exist only un-\nder external magnetic \feld, which precludes the use of\nangle-resolved photoemission spectroscopy (ARPES). In\ncerium monopnictides, it remains crucial to measure the\nsurface magnetic order, its associated band splitting, and\nits orbital contributions, which could in\ruence the Fermi\narcs and their connectivity to Weyl cones [9]. Further-\nmore, characterizing the interplay between magnetic or-\nder and Kondo physics is essential to understand the pos-\nsible emergence of heavy fermions and \rat bands [25].\n3 Here we use spin-polarized (SP) scanning tunneling\nmicroscopy (STM) and spectroscopy (STS) to image the\nenergy-resolved surface magnetic order on CeBi, at low\ntemperature with applied magnetic \feld B. We extend\nprevious density functional theory (DFT) calculations of\nWeyl nodes in the high- Bfully-polarized phase [24], to\npredict additional Weyl nodes near the Fermi level EF\nin the intermediate- Bferrimagnetic phase. We therefore\nfocus our experiments at B= 3 T, where we image the\nexpected (+++\u0000) pattern of the spin orientation of the\nforbitals on the Ce sites, but only above EF. Surpris-\ningly, we observe the same magnetic pattern on the Bi\nsites below EF. The induced magnetic moments on the\nBipstates, co-aligned with the adjacent Ce fstates,arXiv:2012.14911v2 [cond-mat.str-el] 9 Apr 20222\nPM\nTemperature [K]0 10 20 30B [T]\n012345(a)\nFully polarized\nFerrimagnetic\nQ2a\nQa\nQb\n-1\n0\n1\n1\n0\n-1\n-1\n0\n1\n1\n0\n-1\ntip\n(d)(b)\n(c)\nz\nx\ny\nSP-STM\nB\nqz [2π/c]\n-1\n0\n1\n1\n0\n-1\nQlat\nqx [2π/a]\nc\na\nFIG. 1. (a) Magnetic phase diagram of bulk CeBi, with dots\nfrom magneto-transport measurements [19] marking an intri-\ncate cascade of transitions between magnetic orders. (b-d)\nReal-space structure and simulated Fourier transforms of the\nx\u0000zplane of CeBi in the (+ \u0000+\u0000), (++ \u0000\u0000), and (+++ \u0000)\nmagnetic phases. Here, \\+\"and \\ \u0000\" indicate the direction\nof the Cefnet magnetic moments, which are ferromagneti-\ncally aligned in each x\u0000yplane with varying order along the\nzdirection. Qlatindicates the wavevector of the Ce (or Bi)\nsublattice, which would appear in spin-averaged STM images.\nSpin-polarized STM would be sensitive to additional mag-\nnetic Bragg peaks. (b) Antiferromagnetic (+ \u0000+\u0000) phase for\nT.25 K. (c) Antiferromagnetic (++ \u0000\u0000) phase for T.12:5\nK. (d) Ferrimagnetic (+++ \u0000) phase with magnetic \feld B\napplied along z.\nindicatep\u0000forbital hybridization, commonly referred\nto asp\u0000fmixing [26, 27]. For energies closer to EF\nwe observe suppression of the (+++ \u0000) spin polarization,\ncoinciding with a Fano resonance in our measured con-\nductance (dI=dV ), further supporting p\u0000fhybridiza-\ntion. These observations suggest a competition between\nthe mechanism inducing the co-aligned (i.e. ferromag-\nnetically aligned) moments on the Bi pstates and the\nantiferromagnetic Kondo screening. Finally, we present\nquasiparticle interference (QPI) measurements showing\na\u0018100 meV splitting of the Bi p-band, validating our\nDFT calculations that show the crossing of the Bi pand\nCedbands to form Weyl nodes close to EF. Our QPI\nsuggests a \rattening of the mixed-character p\u0000dband\nthat forms the Weyl cones.\nII. METHODS\n4 We calculated the bulk band structure of CeBi using\nthe generalized-gradient approximation (GGA) as imple-\nmented in the all-electron code WIEN2K [28], with the\naugmented-plane-wave + local-orbitals (APW+lo) basis\nkzkx\nkyZXEnergy [eV]0.2\n-0.2\n-0.40.00.4\nZ ZFully polarizedParamagnetic (PM)Energy [eV]0.2\n-0.2\n-0.40.00.4\nZ Z\nW1Source\nSinkBi 6p\nCe 5d\nXW1 W2lower T\nincrease B\nBi p band bottom\n(b)\n(a)\n(c)\n(d)\n(e)FIG. 2. DFT band structure along the \u0000 \u0000Zdirection in the\n(a) paramagnetic; (b) antiferromagnetic (++ \u0000\u0000); (c) fully-\npolarized (++++); and (d) ferrimagnetic (+++ \u0000) phases.\nLight purple and dark green circles indicate locations of Weyl\nnodes. W1 and W2 indicate the Weyl nodes closest to the\nmeasured Fermi energy ( EF) in the ferrimagnetic phase. The\ncalculated bands in each panel have been rigidly shifted by (a)\n+110 meV, (b) \u0000120 meV, (c) \u0000110 meV, and (d) \u000055 meV\nto better match our QPI experiment. (e) Full 3-dimensional\nBrillouin zone (BZ) in the (+++ \u0000) phase.\nset. In the paramagnetic phase we treated the Ce 4 for-\nbitals as core electrons while in the magnetic phases we\nincorporated the Hubbard Coulomb interaction on the Ce\n4felectrons, with U= 7:9 eV andJ= 0:69 eV chosen\nto make the Ce 4 fenergy level qualitatively consistent\nwith ARPES measurements on CeBi [27]. We included\nspin-orbit coupling in all calculations.\n5 Single crystals of CeBi were grown by the self-\rux\nmethod [19]. We cooled the crystals in zero \feld, cleaved\nthem in cryogenic ultra-high vacuum at \u001830 K to expose\na neutral (010) surface before imaging them at T= 4:6\nK. We prepared non-magnetic PtIr STM tips by ex situ\nmechanical sharpening, then in situ \feld emission on Au\nfoil. We obtained spin-polarized tips by gently dunking\nthem into the sample to pick up a few atoms of magnetic\nmaterial [29{31].\nIII. DENSITY FUNCTIONAL THEORY\nCALCULATIONS\n6 Figure 2 shows the folding and splitting of the two\nBipbands and the Ce dband that cross EF, asTis\nlowered and Bis increased [32]. Our calculations predict\nan induced magnetic moment of \u00180:01\u0016Bon the Bip\nstates in both the ferrimagnetic and the fully-polarized\nphase. We also computed the Berry curvature at each\nband crossing in the fully-polarized and ferrimagnetic\nphases, and circled the sinks and sources that constitute3\n(d)\n(f)\n1.0 pm\n0\n2.5 pm\n0\nqz [2π/c]\nQ2a\nQ2a\n(a)\n(c)\n(e)\n(b)\nB = 0 T\nB = 0 T\nBz = 3 T\nqx [2π/a]\n1\n-1\n1\n-1\n0\nQlat\n1\n-1\n1\n-1\n0\n0\n1\n-1\n1\n-1\n0\n0\nx\nz\nx\nz\n2 nm\n2 nm\n2 nm\nx\nz\nBi\nqx [2π/a]\nqz [2π/c]\nqz [2π/c]\nqx [2π/a]\nTopo.\n1.1 pm\n0\nCe\nCe\ntip\ntip\ntip\nFIG. 3. (a) Topography of CeBi measured with spin-\ndegenerate PtIr tip at zero \feld. (Sample bias Vs= 400 mV,\ncurrent setpoint Is= 200 pA.) (b) Fourier transform (FT) of\n(a) shows Ce sublattice peaks at Qlat, but no magnetic peaks.\n(c,d) Topography and FT with spin-polarized tip (obtained\nby dunking to pick up magnetic material) show new struc-\nture and dominant Bragg peaks at Q2a(red circle), consis-\ntent with the expected bulk antiferromagnetic (++ \u0000\u0000) phase\nthat breaks the cubic symmetry even in zero applied \feld.\n(Vs= 400 mV, Is= 150 pA.) (e,f) By applying a horizontal\n\feld ofBz= 3 T orthogonal to the (++ \u0000\u0000) wavevector of\n(c,d), the spin-polarized topography and FT show reoriented\nstructure and Bragg peaks at Q2a(red circle) consistent with\nthe expected bulk ferrimagnetic (+++ \u0000) phase (Vs= 100\nmV,Is= 500 pA, recorded in vicinity of (c) and with the\nsame spin-polarized tip). Insets in panels (a),(c) and (e) show\noverlaid Bi (cyan) and Ce (yellow and red) atoms, with circles\ndenoting expected non-magnetic orbitals and triangles denot-\ning the expected spin orientations of the Ce forbital, from\nneutron scattering [18].\nWeyl nodes (see also Fig. 11) [24]. In the fully-polarized\nphase we \fnd a band splitting of \u0018100 meV that gen-\nerates Weyl nodes near the calculated EF, however we\ncaution that the true EFmay be signi\fcantly shifted in\nlow-carrier-density Kondo materials. The ferrimagnetic\nphase shows comparable band splitting, but is advanta-\ngeous because its band folding generates additional Weyl\nnodes . The predicted ferrimagnetic-phase Weyl nodes are\nwell-spaced over a larger energy range, making the Weyl\nphase and its low energy signatures more robust to band\nbending. This circumvents the problem of ill-de\fned chi-\nrality that arises from scattering between multiple de-\ngenerate Fermi arcs [33] or from Fermi surfaces encom-\npassing multiple Weyl points [8]. We therefore focus our\nattention experimentally on the ferrimagnetic phase for\nthe remainder of this work.\nIV. EXPERIMENTAL RESULTS\n7Figure 3(a) shows a topography acquired with a non-\n(a)\n(b)\n(c)\nx\nz\n2 nm\n2 nm\n2 nm\n-80 mV\nEF\n200 mV\nSpan: 439 pS\nSpan: 76 pS\nSpan: 105 pS\ndI/dV\ndI/dV [a.u.]\nBias [mV]\n-50\n0\n25\n-25\n50\n0\n1\n0\n1\n0\n1\n(e)\nB = 0 T\nBz = 3 T\nBz = 9 T\n(010) surface\n(001) surface\n200\n100\n0\n-300\n-200\n-100\nBias [mV]\ndI/dV [pS]\n0\n40\n120\n80\nQ2a\nQa\nQb\n(d)FIG. 4. Spin-polarized conductance maps at Bz= 3 T of the\n(a) \flled, (b) Fermi level, and (c) empty states. ( Vs= 300\nmV,Is= 4 nA, bias modulation Vrms= 7:1 mV.) The ener-\ngies of the three maps highlight the \flled and Fermi level Bi\nporbitals, and the unoccupied Ce dorbitals, respectively. As\nin Fig. 3(a,c,e), insets show overlaid Bi (cyan) and Ce (yellow\nand red) atoms, but here the triangles denote the observed in-\nduced spin orientations on the Bi p(a,b) and Ce d(c) orbitals.\n(d) Energy-dependent intensity of magnetic Bragg peaks (as\nde\fned in Fig. 1) on the (010) surface of the (+++ \u0000) phase\nof CeBi. The dip in Q2aaround \u0000150 mV corresponds to\nthe bottom of the outer Bi pband, which is marked by a\nred arrow in Fig. 2(d). (e) Background-subtracted, spatially-\naverageddI=dV curves, \ft to Fano line shape (see Figs. 9 and\n10).\nmagnetic PtIr tip. The corresponding Fourier transform\n(FT) in Fig. 3(b) shows four peaks at Qlat= (\u00061;\u00061),\narising from the Ce sublattice (see Fig. 6 ). After gen-\ntly dunking the tip into the sample, the new topogra-\nphy in Fig. 3(c) shows additional structure, suggesting\nthat the tip has picked up magnetic material, leading to\na spin-polarized tunneling current [30, 36]. The mag-\nnetic structure manifests in the FT in Fig. 3(d) as two\ndominant new peaks at Q2a= (0;\u00061\n2), consistent with\nthe expected bulk antiferromagnetic (++ \u0000\u0000) phase that\nbreaks the cubic symmetry of CeBi even in the absence\nof appliedB[see also Fig. 1(c)]. To verify the SP na-\nture of the tip, we applied an in-plane \feld Bz= 3 T,\nperpendicular to the zero-\feld (++ \u0000\u0000) order, to rotate\nthe magnetic ordering vector of the sample by ninety de-\ngrees into the expected (+++ \u0000) phase, as shown in Fig.\n3(e)-3(f) [see also Fig. 1(d)]. Although the tip spin may\nalso realign due to applied B, our observations in Fig.\n3(c) and 3(e) show that the tip has a component that is\nco-aligned with the sample magnetization in both cases.\nThis con\frms two essential requirements for our study:\nOur tip is sensitive to the expected magnetic order of\nCeBi, and we can tune that magnetic order by applying\nmagnetic \feld.\n8 To determine the energy-resolved orbital charac-4\nEnergy [eV]\nΓ\nZ\nΓ\nZ\n0.1\n-0.2\n-0.3\n0.0\n-0.1\n0.2\n0.3\n(b)\nEF\nEF\nBias [V]\n0.1\n-0.2\n-0.3\n0.0\n-0.1\n0.2\n0\n0.5\n-0.5\nqz [2π/c]\n0.3\n(a)\nhigh\nlow\nα\nβ\nγ\n(c)\nCe d2\npd\ndI/dV [a.u.]\n(d)\nCe f\nα\nβ\nγ\np1\np2\npd\np4\nd1\nd2\nd4\nd3\np3\nFIG. 5. (a) Quasiparticle interference (QPI) intensity along\nthe \u0000\u0000Zdirection of the (+++ \u0000) phase atBz= 3 T (see\nFig. 12 for details). (b) DFT band structure from Fig. 2(d)\nin an extended zone scheme. Predicted Weyl nodes are cir-\ncled in light purple and dark green. The dominant features\nin the QPI data, labeled by three white arcs \u000b,\f, and\rin\n(a), match the three intraband scattering processes p1,p2,\nandp3 marked as gray arrows on the DFT in (b). The pre-\ndominance of these three p-orbital band segments in the QPI\nsignal is consistent with a prior observation that porbitals\nextend farther from the surface than dorbitals, and are more\naccessible to the STM tip [34]. Our QPI data also shows\nthat thep3 segment of this folded outer pband is shifted\nup by \u001870 meV (black arrow) with respect to the p1 and\np2 segments, consistent with band \rattening due to strong\nelectron correlations. (c) DFT band structure close to tilted\n(type II) Weyl cone at EF. (d) Schematic showing the renor-\nmalization (\rattening) of the Bi 6 pband upon hybridization\nwith the Ce 4 fstates [35]. Dashed red line denotes the non-\nrenormalized pband. Gray shading denotes the renormalized\nfstate forming our observed Kondo resonance, in agreement\nwith ARPES [25]. Dashed black line indicates experimental\nEF.\nter of the spins in the ferrimagnetic (+++ \u0000) phase,\nwe mapped the spin-polarized di\u000berential conductance,\ndI=dV . Figure 4(a)-4(c) shows SP- dI=dV maps from\nenergies below, at, and above EF. Away from EF, we\n\fnd a high SP-conductance for three neighboring verti-\ncal columns and a low SP-conductance for one column, as\nexpected in the (+++ \u0000) ferrimagnetic phase. Surpris-ingly, comparison of the simultaneously-acquired images\nin Figs. 4(a) and 4(c) reveals that the magnetic contrast\nhas shifted from the Ce lattice sites above EFto the\nBi lattice sites below EF. These induced magnetic mo-\nments on the Bi psites are co-aligned with the underlying\nmagnetic pattern of the localized, unrenormalized Ce f\nstates, which have been observed \u00183 eV below EFby\nARPES [27]. Our evidence of p\u0000forbital hybridiza-\ntion agrees with our DFT prediction of a Bi pmagnetic\nmoment of\u00180:01\u0016B.\n9 We plot the intensity of the magnetic Bragg peaks\nvs. energy in Fig. 4(d). The Q2apeak is dominant at neg-\native energies, but strongly suppressed at EF, and recov-\ners only weakly above EF. This evolution is also appar-\nent in the colorscale spans of the SP- dI=dV maps. While\nthe (+++\u0000) pattern of induced magnetic moments is\ndominant at Vs=\u000080 and +200 meV in Figs. 4(a) and\n4(c), the map at EFin Fig. 4(b) shows a di\u000berent motif\nwith maxima on every second Bi atom. We speculate\nthat this (+\u0000+\u0000) structure may be caused by a residual\nout-of-plane ordering of surface spins (see Fig. 8). The\nsuppression of the Q2aintensity also coincides with a\n180\u000ephase \rip of the real-space pattern (see Fig. 6).\n10 Kondo screening can arise from p\u0000fhybridization,\nas suggested by previous Hall e\u000bect and ARPES measure-\nments [27, 37], so we search for its possible signatures\nindI=dV spectroscopy. Fig. 4(e) shows three spatially-\naverageddI=dV spectra around EFin the (++\u0000\u0000),\n(+++\u0000), and fully-polarized phases. All spectra have\na similar shape with a shoulder around \u000018 meV and a\ndip nearEF, characteristic of the asymmetric Fano line\nshape describing a Kondo resonance,\nF(E)/[q+ (E\u0000E0)=\u0000]2\n1 + [(E\u0000E0)=\u0000]2; (1)\nwhereE0is the energy of the localized many-body reso-\nnance (which we \fnd to be consistent with EF; see Table\nI),qis the Fano factor that describes the tunnelling ratio\nbetween the localized fstate and the itinerant conduc-\ntion electrons, and \u0000 is proportional to the hybridiza-\ntion between the localized and itinerant states [38]. Sim-\nilar line shapes have been observed in other Kondo lat-\ntice systems such as YbRh 2Si2[39], URu 2Si2[40], and\nSmB 6[5]. In all three magnetic phases of CeBi, we \fnd\n\u0000\u001810 meV, consistent with a resistivity upturn at \u0018100\nK, above the N\u0013 eel temperature [19]. However, the large\nresidualdI=dV at the Fano minimum (see Fig. 9) sug-\ngests that only 5 \u000010% of the conduction electrons par-\nticipate in Kondo screening in CeBi, consistent with the\nlocal-moment-like behavior of the Ce sublattice [18, 19].\n11 In CeBi the itinerant electrons closest to EFare\nof both Ce 5 dand Bi 6pcharacter, so we do not know\na priori which itinerant states participate in the partial\nKondo screening of Ce fmoments. However, the disap-\npearance of (+++ \u0000) order from the Bi sites at EFis\nconsistent with the involvement of Bi pstates in Kondo\nsinglet formation. Furthermore, the Kondo resonance is\nfacilitated by the shared symmetry of the Bi 6 porbitals5\nand the Ce f5=2\u00008multiplet [35, 41], while d\u0000fhy-\nbridization is forbidden on-site. We thus conclude that\nthe Bi 6pstates are the primary conduction electrons\nthat couple to the Ce 4 fmoments and participate in the\nformation of Kondo singlets.\n12 Fig. 4 highlights several competing interactions\nin CeBi. First, the Fano lineshape in our dI=dV spec-\ntra suggests a Kondo resonance in which some conduc-\ntion electrons anti-align with and partially screen the\nlocal moments. However, the observed shift of the same\n(+++\u0000) order from the Ce sites in Fig. 4(c) to the Bi\nsites in Fig. 4(a) demonstrates that the induced mag-\nnetic moments on the Bi pstates are co-aligned with\nthe localfmoments. These opposite magnetic inter-\nactions compete for the same pstates. Second, long-\nrange order can arise when localized moments couple\nto each other via the polarized conduction electrons,\nthrough the Ruderman-Kittel-Kasuya-Yosida (RKKY)\ninteraction. The competition between the screening of\nthe local moments (Kondo) and formation of long-range\nmagnetic order (RKKY) is typically determined by both\nthe conduction ( c) electron density and the coupling\nJf\u0000c[14, 16]. However, we observe long-range order in\nFigs. 4(a) and 4(c) coexisting with the Kondo resonance\nin Fig. 4(e). Our data suggests that these competing in-\nteractions in CeBi each dominate at separate energies ,\nin contrast with CeSb where the simultaneous observa-\ntion of Kondo screening and long-range order has been\nexplained by phase separation in momentum space [25].\n13 To determine the e\u000bect of p\u0000fhybridization on\nthe predicted Weyl fermion bands in the CeBi (+++ \u0000)\nphase, Fig. 5(a) shows our quasiparticle interference\n(QPI) measurement of the band dispersion along \u0000 \u0000Z.\nThe dominant QPI features show excellent agreement\nwith the calculated bands of majority Bi porbital char-\nacter shown in Fig. 5(b). From this band assignment,\nwe make several observations. First, the QPI-observed\n\u000band\fbands support the DFT-predicted \u0018100 meV\nsplitting of the hole-like Bi pband around \u0000 into p1\n(\u000b) at 230 meV and p2 (\f) at 120 meV, consistent\nwith the formation of induced magnetic moments on\nthe Bipsites by orbital hybridization. Second, the\nQPI-observed \rband can be attributed to scattering\nacross the BZ boundary between the folded p3 portion\nof the same Bi pband. However, the QPI-observed\n\rband is\u001870 meV higher than the corresponding\nDFT-predicted p3 band, supporting a scenario of p\u0000f\nhybridization that renormalizes (\rattens) the pband\nand the band of mixed pdcharacter that makes up half\nof the Fermi-level Weyl cone. We thus infer that the\nWeyl cones of CeBi are strongly renormalized compared\nto the DFT calculations, consistent with the enhanced\ne\u000bective mass of 4 :3meobserved in quantum oscillation\nexperiments on CeSb [42]. DFT calculations notoriously\nunderestimate band-renormalization e\u000bects in strongly\ninteracting materials, so our QPI experiment serves as a\ncrucial reality check.V. CONCLUSION\n14 Weyl cones are expected to be robust under renor-\nmalization, which can spread their position in momen-\ntum space but not lift the degeneracy of the requisite\ncrossings [43]. Neither p\u0000fnord\u0000fhybridization\nyields an exact realization of the proposed Weyl-Kondo\nsemimetal [6], in which the felectrons are directly in-\nvolved in the formation of the Weyl cones. However,\nour work shows ( i) induced magnetic moments on the Bi\npstates, co-aligned with the local Ce fmoments, from\nspin-polarized dI=dV images of the (+++ \u0000) order on\nthe Bi sites; ( ii)p\u0000fhybridization from dI=dV spec-\ntroscopy of the Fano resonance; ( iii)\u0018100 meV band\nsplitting from QPI that con\frms p\u0000fhybridization and\nTRS breaking; ( iv) and band \rattening from QPI. This\ncomprehensive evidence supports a consistent picture of\nCeBi as a strongly interacting magnetic Weyl semimetal.\nACKNOWLEDGEMENTS\nWe thank Jason Ho\u000bman, Robert Jan-Slager, Daniel\nMazzone for insightful discussions. Experimental and\ntheoretical work was supported by the Center for the\nAdvancement of Topological Semimetals (CATS), an\nEnergy Frontier Research Center funded by the U.S.\nDepartment of Energy (DOE), O\u000ece of Science, Basic\nEnergy Sciences (BES) through the Ames Laboratory\nunder its Contract No. DE-AC02-07CH11358. H.P. was\nfunded by the Gordon and Betty Moore Foundation's\nEPiQS Initiative through Grant GBMF4536. C.E.M.\nwas supported by the Swiss National Science Foundation\nunder fellowships P2EZP2 175155 and P400P2 183890.\nThe theory work was carried out under the auspices of\nthe U.S. DOE National Nuclear Security Administration\nunder Contract No. 89233218CNA000001. C.L. was\nsupported by Los Alamos National Laboratory (LANL)\nLDRD Program. The theory was also supported in\npart by the Center for Integrated Nanotechnologies, a\nDOE BES user facility, in partnership with the LANL\nInstitutional Computing Program for computational\nresources.\nAll data underlying Figs. 1-13 can be accessed in\nRef. [32].\nAppendix A: Surface atom and spin identi\fcation\nTo identify the atomic sublattice imaged in Figs. 3 and\n4, we investigated La-doped CeBi samples, where La is\nexpected to replace Ce. Measurements and DFT calcu-\nlations in Fig. 6 show that at large positive bias volt-\nage we are tunneling predominantly into Ce sites, while\nbelow\u00180:1 V we are tunneling predominantly into Bi\nsites. Fig. 7 shows the full energy dependence of the\nspin-polarized dI=dV maps in Fig. 4. Fig. 8(a) shows6\ntopo (300 mV)\ndI/dV (200 mV)\ndI/dV (-200 mV)\n0.3\n0.2\n0.1\n0.00.1Bias [V]\n(g)\n(h)\n(i)\n(j)\nhigh\nlow\n1 nm\n1 nm\nBi\nCe\n(b)\n(c)\n(e)\n(d)La-doped CeBi\n(a)\n1 nm\n1 nm\n4\n1\n3\n2\n1\n2\n3\n4\nLa\nLa\nvacancy\nLa\nTopo (800 mV)\nTopo \nhigh\nlow\ndI/dV \n-4 -3 -2 -1 0 1 20481216DOS [a.u.]\nEnergy [eV]\nVswitchVswitch\nCe\nBi\n(f)\n0 -0.5 0.5\nFIG. 6. (a-e) Topography of CeBi with 2.5% (nominal) La dopants expected at the Ce sites, recorded with sample bias Vs= 800\nmV and tunneling current setpoint Is= 600 pA. The spacing of the visible atomic lattice (periodic bright spots) in a clean\nregion is 4.6 \u0017A, consistent with x-ray di\u000braction measurements of the lattice constant [20]. (b) Zoom around a typical La\nadatom that is centered between bright lattice spots. (c) Two La dopants in the top surface layer, located on the bright lattice\nsites. (d) La in the subsurface layer, laterally centered on a dark lattice site of the top surface layer. (e) Vacancy of two bright\nlattice sites of the top surface layer. All four observations in (b-e) suggest that at the large positive sample bias of Vs= 800\nmV, we observe the Ce sublattice. (f) DFT-calculated density of states (DOS) of bulk CeBi in the ferrimagnetic (+++ \u0000)\nstate, resolved according to elemental contribution. At negative energies, the DOS of Bi dominates while on the positive side,\nCe dominates. (g-j) Topography and dI=dV maps of nominally pristine (undoped) CeBi, to identify the shift of dominant\nsublattice depending on sample bias voltage. (g) Topography recorded at Vs= 300 mV. (h-j) Simultaneously recorded dI=dV\nmaps at +200 mV and \u0000200 mV, respectively. (j) dI=dV linecut spatially averaged between the vertical dotted lines in (i),\nillustrating the spatial switching of highest-conductance location at Vswitch\u00180:1 V. Average magnitude of horizontal rows in\n(j) has been normalized for visual purposes. All maps in (g-j) have been simultaneously recorded at Bz= 3 T with Vs= 300\nmV,Is= 4 nA, and Vrms= 7:1 mV.\na second spin-polarized dI=dV map at the Fermi level,\nacquired with identical measurement parameters as Fig.\n7(a11) but with di\u000berent tip termination. Together, Figs.\n7(a11) and 8(a) suggest a canting of the surface spins.\nThe Fermi level dI=dV map in Fig. 8(a) shows a pat-\ntern with maxima on every second Bi atom, which is\ndi\u000berent from the (+++ \u0000) pattern of the bulk magnetic\nmoments expected from the phase diagram of Fig. 1(d).\nThis Fermi level dI=dV corresponds to dominant Qaand\nQbmagnetic Bragg peaks, as shown in Fig. 8(b). Fig-\nure 7(c) shows a corresponding suppression of the Q2a\nBragg peak associated with the (+++ \u0000) order for ener-\ngies close to the Fermi level. The suppression of the Q2a\nintensity also coincides with a 180\u000ephase \rip of the real-\nspace pattern in Fig. 7(a). The suppression of Q2aat\nthe Fermi level implies that the (+ \u0000+\u0000) pattern seen at\nthat energy is not connected to the (+++ \u0000) order, but\nof di\u000berent origin. We suggest that this (+ \u0000+\u0000) struc-ture might be caused by a residual out-of-plane ordering\nof surface spins, as sketched in Fig. 8(e), and simulated\nin Figs. 8(c)-(d).\nAppendix B: Kondo lineshape\nFigure 9(a) shows the spatially-averaged raw dI=dV\nspectra around EFin the (++\u0000\u0000), (+++\u0000), and fully-\npolarized phase from which we subtracted the back-\nground (gray dashed lines) to obtain the Fano lineshapes\nshown in Fig. 4(e), and re-displayed here in Fig. 9(b).\nThe qualitative features of the Fano lineshape, with a\nshoulder around \u000018 meV and a dip near EF, are ap-\nparent in the raw spectra. But to quantify the Fano\nlineshape parameters in Table I, we \ft Eqn. 1 added to\na polynomial background. By comparing the residual\nconductance at EFwith the shoulder around \u000020 meV7\n200\n100\n0\n-300\n-200\n-100\nBias [mV]\ndI/dV [pS]\n0\n40\n120\n80\nQ2a\nQa\nQb\n(c)\nPhase [degree]\n360\n180\n0\n-180\n-360\n(d)\nQlat\n-1\n0\n1\n1\n0\n-1\nQ2a\nQa\nQb\nQlat\nqz [2π/c]\nqx [2π/a]\n(b)\n2 nm\ndI/dV\nlow\nhigh\nFIG. 7. (a1-a25) All energy layers of the conductance map presented in Fig. 4, showing the (010) surface of the (+++ \u0000)\nphase atBz= 3 T. Sample bias is indicated in top right corner of each layer. Setup parameters are Vs= 300 mV, Is= 4 nA,\nVrms= 7:1 mV. (b) Simulated Fourier transform of (+++ \u0000) magnetic order on the (010) surface with magnetic ( Qa,Qb,Q2a)\nand structural lattice ( Qlat) Bragg peaks as indicated. (c-d) Energy-dependent (c) intensity and (d) phase of magnetic and\nstructural Bragg peaks, calculated by Fourier transforming each measured conductance map in panels (a1-a25).8\nSpan: 172 pS\ndI/dV\n(a)\ndI/dV at E F\nQb\nQa\nQlat\nQb\nQa\nQlat\n(b)\n(c)\nSimulation\n(d)\n(e)\nCe\nBi\nFIG. 8. (a) Spin-polarized dI=dV map atBz= 3 T, at the\nFermi level, shows alternating magnetic (+ \u0000+\u0000) contrast,\nwhich is di\u000berent from the expected in-plane (+++ \u0000) mag-\nnetic order from the phase diagram of Fig. 1(d). This map\nwas extracted from the same dataset presented in Fig. 6(g-j),\nwithVs= 300 mV, Is= 4 nA,Vrms= 7:1 mV. Although\nthis map and Fig. 7(a11) were recorded with identical pa-\nrameters, the tip termination had a di\u000berent direction of the\nmagnetic moment (spin-DOS), so the real space images do not\nappear identical, but QaandQbare prominent in the Fourier\ntransform in both cases. (b) Corresponding Fourier transform\nshows that the intensity of the Q2amagnetic Bragg peaks is\nsuppressed, indicating suppression of the bulk (+++ \u0000) mag-\nnetic order on the Bi pstates due to the (partial) formation\nof Kondo singlet states. (c-d) Simulation of the (+ \u0000+\u0000)\nmagnetic order. (e) Possible scenario to explain the (+ \u0000+\u0000)\norder atEF: spins on top (010) surface may have a residual\nout-of-plane component, which would produce the intensity\npattern in our simulation (c-d).\nin Fig. 9(a), it can be seen that only 5 \u000010% of the\nconduction electrons participate in the Kondo screening.\nFurthermore, the point spectra in Fig. 10 recorded in the\nantiferromagnetic ground state ( B= 0 T) and in the\nferrimagnetic state ( Bz= 3 T) all show a Fano-like line-\nshape with only slight spatial variation at larger binding\nenergies due to the magnetic order.\nTABLE I. Fano line shape parameters determined by a least-\nsquare \ft. The Kondo temperature is estimated as TK=\n\u0000=kBwherekBis the Boltzman constant.\n\u0000 [meV] E0[meV]qTK[K]\nB= 0 T 10.2\u00064\u00001:9\u00064\u00000:3118.37\nBz= 3 T 8.8\u000610\u00001:6\u000610\u00000:6102.12\nBz= 9 T 7.7\u00062.5 0.8\u00062.5\u00000:498.35\n(010) surface\n100 50 0 50 100\nBias [mV]0.00.51.01.52.02.53.0(a)\n50 25 0 25 50\nBias [mV]010101(b)dI/dV [a.u.]dI/dV [nS]Bz = 3 T\nBz = 9 TB = 0 T\n(001) surfacex0.25\nBz = 0 T\nBz = 3 T, (010) sur face\nBz = 9 T, (001) sur faceFIG. 9. Kondo resonance in CeBi (a) Spatially-averaged\ndI=dV spectra around EFwith varying applied B. Ac-\nquisition parameters are: (blue, B= 0 T)Vs= 50 mV,\nIs= 1 nA,Vrms= 2:82 mV; (green, Bz= 3 T)Vs= 300 mV,\nIs= 2 nA,Vrms= 7:1 mV; (orange, Bz= 9 T),Vs= 200 mV,\nIs= 0:4 nA,Vrms= 1:8 mV. Blue and green curves are spa-\ntially averaged spectra from DOS maps recorded on the (010)\nsurface with a spin-polarized tip, while orange curve is nom-\ninally a point spectrum with a non-spin-polarized tip on the\n(001) surface, it was recorded over an extended time, so it\nis e\u000bectively spatially averaging due to lateral piezo drift in\nnm range. Black lines show the \ft of a Fano lineshape (Eqn.\n1) on top of a polynomial background (parabolic for 0 T and\n9 T, but third order polynomial for 3 T data), depicted by\ngray dashed lines. (b) Background-subtracted dI=dV , over-\nlaid with \fts to Fano lineshape, as shown in Fig. 4(e).9\n(b) (c)\n(d)(e) (f)\nSpan: 390 pS\ndI/dV\nSpan: 240 pS\ndI/dV\n1.21.41.61.82.02.22.42.62.83.0dI/dV [nS]\naverage\nFano+backg. fit\nbackground\n0246810 dI/dV [a.u.]\n2 nm\n10121416182022dI/dV [nS]\n0246810 dI/dV [a.u.]\n2 nm-50 mV(a)\n-50 mVBz = 3 TB = 0 Taverage\nFano+backg. fit\nbackground\n-50 -50 50 50 0 0\nBias [mV] Bias [mV]\n-50 -50 50 50 0 0\nBias [mV] Bias [mV]\nFIG. 10. Point conductance spectra at 0 T and 3 T in\nCeBi (a) Conductance map recorded at -50 mV bias ( B= 0\nT,Vs= 50 mV,Is= 1 nA,Vrms= 2:82 mV) indicating the\nlocation of the point spectra presented in (b). (b) Raw point\nspectra vertically o\u000bset for clarity. Blue spectrum is spatial\naverage within the \feld of view in panel (a), gray dashed\nline is polynomial background \ft, and black line is Fano line-\nshape \ft plus background. (c) Point spectra vertically o\u000bset\nfor clarity after background subtraction, indicating Fano-like\nlineshape at all locations. (d)-(f) Similar image and point\nspectra recorded at Bz= 3 T (Vs= 300 mV, Is= 2 nA,\nVrms= 7:1 mV).\nAppendix C: Comparing DFT to QPI\nFigure 11(a) shows our DFT calculation of the band\nstructure in the ferrimagnetic (+++ \u0000) phase of CeBi,\nover a larger energy range than in Fig. 2(d). To \fnd the\nWeyl nodes, we calculate the Berry curvature at each\ncrossing, and circle the sources (pink) and sinks (green).\nIn Figs. 11(c-e) we plot the Berry curvature of the Weyl\npoints W1 and W2 closest to the Fermi level.\nWe investigate the band structure experimentally by\nimaging quasiparticle interference (QPI), which probes\nelastic momentum transfer, predominantly originating\nfrom intra-band scattering, as shown in Fig. 12(a).\nTherefore, all scattering vectors appear around q= 0\nin a QPI measurement, as shown in Fig. 12(b). Figure\n12(c) and 12(d) show energy vs. qdispersion along the\nqzdirection measured in two di\u000berent energy ranges with\nsimilar setup conditions. The two datasets are combined\nin Figs. 5(a) and 12(e), where we overlay the calculated\nband dispersion ( \u000b,\f,\r) from the Bi porbitals. After a\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00\nZ ZW2 W1Energy [eV]\nkyZX\nXW1 W2\nZ Z\nW2 W1\nkx\nkz\n(c)\nW2 W1Sink Source\n(b)\n(d)\n(e)\n(a)kx\nkz Source\nSinkBi 6p\nCe 5dFIG. 11. Prediction of Weyl points in the (+++ \u0000)\nphase of CeBi (a) Band structure along the \u0000 \u0000Zdirection\nin the ferrimagnetic (+++ \u0000) phase, calculated using density\nfunctional theory (DFT). Open circles indicate the locations\nof Weyl nodes, colored to indicate sources (pink) and sinks\n(green) of Berry curvature. (b) Three-dimensional Brillouin\nzone in the (+++ \u0000) phase of CeBi, showing the Weyl nodes\nW1 and W2 closest to the Fermi level, around 50 meV [gray\ndashed line in (a)]. (c) Berry curvature \feld in kx\u0000kzplane of\nthe gray highlighted band in (a), con\frming the source (W1)\nand sink (W2) of Berry curvature. (d)-(e) Zoom on Berry\ncurvature around W1 and W2.\nslight upshift of the \rband, presumably caused by elec-\ntron correlations, we \fnd a good agreement between our\nDFT calculations and our QPI measurements.10\nα\nβ\nγ\nEner gy [eV]\nΓ\nΓ\nZ\n0.1\n-0.2\n0.0\n-0.1\n0.2\n0.3\n(a)\nα\nβ\n(b)\nγ\n0\n0.5\n0.5\nZ\nq-space (mom entum tran sfer)\nk-space ( absolute momentum )\n0\n0.5\n-0.5\n1\nQPI\nDFT\n-kα\nkα\n-kβ\nkβ\nqγ = kγ - (-kγ) = 2kγ\nqα = 2kα\nqβ = 2kβ\nkγ\nqz [2π/c]\nkz [π/c]\n-kγ\nhigh\nlow\ndI/dV [a.u.]\nBias [V]\n0.1\n-0.2\n0.0\n-0.1\n0.2\n0\n0.5\n-0.5\nqz [2π/c]\n0.3\n(c)\nα\nβ\nγ\n0\n0.5\n-0.5\nβ\nγ\n(d)\n0\n0.5\n-0.5\n(e)\n-0.3\nqz [2π/c]\nqz [2π/c]\nFIG. 12. Quasiparticle interference (QPI) in the\n(+++ \u0000) phase at Bz= 3T.(a-b) Relation between k-\nspace andq-space in QPI measurements. (a) DFT-calculated\nsegments of the Bi outer pband, replicated from Fig. 2(d), but\nwith the\rband shifted up by \u001870 meV to match our mea-\nsured data. Intra-band quasiparticles scatter from \u0000kto +k\nwith a total momentum transfer of q= 2k, shown as gray ar-\nrows. (b) The momentum transfer vectors ( q) originate from\nzero at the tops of all three bands, \u000b,\f, and\r. Therefore,\npockets around Zink-space appear around zero momentum\ntransfer inq-space. (c-e) Experimental QPI data and compar-\nison to DFT. (c) QPI data set #1 along the \u0000 \u0000Zdirection,\nreproduced in Fig. 5(a) for bias voltages above 80 mV. Setup\nparameters: Vs= 300 mV, Is= 2 nA,Vrms= 7:1 mV. (d)\nQPI data set #2 along the \u0000 \u0000Zdirection, reproduced in\nFig. 5(a) for bias voltages below 80 mV. Setup parameters:\nVs= 300 mV, Is= 4 nA,Vrms= 7:1 mV. (e) Direct overlay\nof DFT-calculated band structure on combined QPI measure-\nment. The bands with Bi porbital character, which de\fne the\nintra-band scattering vectors \u000b,\f, and\r, are consistent with\nthe observed QPI intensity. Here the calculated \rbands are\nshifted up by \u001870 meV with respect to the calculated \u000band\n\fbands from Fig. 2(d), indicating the presence of electron\ncorrelations.\nAppendix D: Zero padding\nIn order to improve atomic visibility, the following\nmaps were interpolated by Fourier-transforming, zero-\npadding the FT, then inverting the FT: Fig. 3(a), Fig.\n4(a)-(c), Fig. 6(f)-(j), Fig. 7(a), Fig. 8(a), Fig. 10(a),\nand Fig. 10(d). The comparison between raw and inter-\npolated data is shown in Fig. 13.\n(c)\n(d)\n(e)\nx\nz\n2 nm\n2 nm\n2 nm\n-80 mV\nEF\n200 mV\n(f)\n(g)\n(h)\nx\nz\n2 nm\n2 nm\n2 nm\n-80 mV\nEF\n200 mV\nSpan: 439 pS\nSpan: 76 pS\nSpan: 105 pS\ndI/dV\n(b)\n2 nm\nx\nz\nBi\nCe\nTopo.\n1.1 pm\n0\ntip\n(a)\nB = 0 T\n2 nm\nx\nz\nBi\nCe\ntip\ntip\nBz = 3 T\nFIG. 13. 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Ba-\nlents, Correlated quantum phenomena in the strong\nspin-orbit regime, Annual Review of Condensed Matter\nPhysics 5, 57 (2014)." }, { "title": "1310.8079v2.Geometrical_origin_of_ferrimagnetism_and_superparamagnetism_in_Fe_based_double_perovskite_multiferroics.pdf", "content": "Geometrical origin of ferrimagnetism and superparamagnetism in Fe-based double\nperovskite multiferroics\nR.O. Kuzian,1, 2V.V. Laguta,1, 3and J. Richter4\n1Institute for Problems of Materials Science NASU, Krzhizhanovskogo 3, 03180 Kiev, Ukraine\n2Donostia International Physics Center (DIPC), ES-20018 Donostia-SanSebastian, Spain\n3Institute of Physics, AS CR, Cukrovarnicka 10, 16253 Prague, Czech Republic\n4Institut f ur Theoretische Physik, Otto-von-Guericke-Universit at Magdeburg,\nPF 4120, D - 39016 Magdeburg, Germany\n(Dated: 16.01.14)\nWe show that a superstructure of antiferromagnetically interacting Fe3+(S= 5=2) ions in double\nperovskites AFe 1=2M1=2O3exhibits a ferrimagnetic ordering below Tfe\u00195:6J1(J1=kB\u001850 K),\nwhich is close to room temperature. Small clusters of the same structure exhibit a superparamagnetic\nbehavior at T.Tfe. The possibility of formation of such clusters explains the room-temperature\n(superpara)magnetism in 3 d-metal based oxides.\nPACS numbers: 75.10.-b, 75.20.-g, 75.50.Gg, 75.50.Lk, 75.85.+t\nI. INTRODUCTION\nAn experimental quest to \fnd a room-temperature\nmultiferroic with high magnetoelectric coupling is stim-\nulated by wide prospects they open for applications in\nthe \feld of information and energy-saving technologies.\nThey may form the basis for a fabrication of novel func-\ntional devices: highly sensitive magnetic sensors, ca-\npacitance electromagnets, elements of magnetic memory\nswitched by electric \feld, nonreciprocal microwave \flters,\nand others.1,2Spintronics, an emerging branch of micro-\nand nanoelectronics which manipulates the electron spin\nrather than its charge, has need for a room-temperature\nferromagnetic semiconductor.3\nThe rich family of Fe-based double perovskites\nAFe 1=2M1=2O3=A2FeMO 6(with non-magnetic ions\nA=Pb,Ca,Sr,Ba, and M=Nb,Ta,Sb) is in the focus of\nthe studies as it includes PbFe 1=2Nb1=2O3(PFN) and\nPbFe 1=2Ta1=2O3(PFT) systems, where the multiferroic-\nity was reported more then \ffty years ago.4,5\nIn AFe 1=2M1=2O3compositions, Fe3+and M5+cation\npositions may be ordered or disordered within simple cu-\nbic B-sublattice of perovskite structure ABO 3. The de-\ngree of chemical ordering depends on the strength of elec-\ntrostatic and elastic energies and, in particular, on the\nionic radii of these cations. It is commonly accepted that\nPFN and PFT are chemically disordered compounds due\nto almost equal ionic radii of Fe3+and Nb5+or Ta5+,6\nwhile Sb-contained compounds can be chemically ordered\nup to 90% because Sb5+is much larger than Fe3+.7Mag-\nnetism of the compositions is due to Fe3+,S= 5=2 ions\nthat occupy half of octahedral sites of the perovskite lat-\ntice. The magnetic moments of the Fe3+ions interact\nvia various superexchange paths,\n^H=1\n2X\nR;rJr^SR^SR+r: (1)\nThe disorder prevents an experimental access to the val-\nues of the interactions. In a recent publication, some ofus have argued that the largest superexchange values are\nthe nearest-neighbor (NN) Fe-Fe interaction (Fe ions are\nseparated by the edge of perovskite unit cell and inter-\nact via the shortest Fe-O-Fe path) J1\u001850\u000070 K and\nthe next-nearest-neighbor interaction (Fe ions are sep-\narated by the face diagonal of the cell) J2'0:04J1.8\nThe interaction values J1,J2are similar to the values in\northoferrite RFeO 3(R=Y or a rare earth)9{13and bis-\nmuth ferrite BiFeO 314compounds. Note that both ex-\nchange couplings have antiferromagnetic sign. We thus\nhave two substantially di\u000berent magnetic energy scales:\nS(S+ 1)J1= 8:75J1, which corresponds to temperatures\nof several hundred Kelvins, and S(S+ 1)J2=kB\u001820 K.\nNote that many of Fe-based double perovskites have an\nantiferromagnetic phase transition in the latter temper-\nature range.15{20It means that the probability to \fnd\na pair of Fe ions separated by the face diagonal of the\nperovskite cell is much higher than to \fnd a nearest-\nneighbor Fe pair that is caused by partial chemical or-\ndering of cations. Two multiferroic compounds, PFN\nand PFT, exhibit a magnetic transition at TN\u0018150 K.\nThis means that the probability to \fnd a pair of NN Fe\nions is enhanced in these compounds. But it leads to the\nincrease of the temperature, at which the antiferromag-\nnetic order is established.21{23For instance, in the more\nconcentrated compound PbFe 2=3W1=3O3it increases up\nto 380 K.24\nRecent reports on room-temperature multiferroicity of\nPFT/lead zirconate titanate (PZT)25,26and PFN/PZT27\nand [Pb(Fe 2=3W1=3)O3]/PZT28solid solution systems\nare a real challenge for the solid state theory. One of the\nquestions is the nature of large room-temperature mag-\nnetic response of the systems (non-linear magnetization\ncurves and hysteresis loops) that imply the existence of\nFe spins alignment in a part of the sample with uncom-\npensated magnetic moment. On the qualitative level, it\nwas suggested that the clustering of Fe ions is responsi-\nble for the appearence of the uncompensated magnetic\nmoment.25{29We should mention that the clustering ofarXiv:1310.8079v2 [cond-mat.mtrl-sci] 15 May 20142\n 0 1 2 3 4 5 6 7 8\n 0 0.5 1 1.5 2χ-1(gµB)2\nkBT/J1S(S+1)a\n 0 1 2 3 4 5 6 7 8\n 0 0.5 1 1.5 2χ-1(gµB)2\n kBT/J1S(S+1) b\nFIG. 1. (Color online) a: Inverse subsceptibility \u001f\u00001for a periodic arrangement of PFB2 chemical order with two inequivalent\nS= 5=2 Fe3+ion positions (red solid line - [4,4] Pad\u0013 e approximant of the 8th order HTE series). The susceptibility exeeds\nthe Curie-Weiss (CW) asymptotic (green dashed line) and diverges at Tfe\u00190:640J1S(S+ 1) (shown by the vertical line)\ncorresponding to a transition into a ferrimagnetic phase. The black thin solid line shows the susceptibility of 1:1 ordered\nPFB0 con\fguration, where Fe spins interact with J2= 0:05J1. Inset: Unit cell of the PFB2 chemical ordering, only Fe\n(open circles) and M (\flled circles) cations are shown. Fe1(Fe2) positions are depicted by up(down) arrows, respectively. b:\nInverse subsceptibility \u001f\u00001(red solid line) for a small cluster of a PFB2 con\fguration (as shown in the inset) obtained by full\nexact diagonalization. The susceptibility shows a crossover between CW (dashed green) and superparamagnetic (dashed black)\nbehavior. In both parts, the blue dotted line shows the Curie law for independent spins, \u001f\u00001\np/T.\nFe ions30,31forms locally fragments of AFeO 3structure,\nwhere Fe spins form the simple cubic lattice. Thus, it can\nlead only to G-type antiferromagnetic ordering within the\nfragments, and produces a small or vanishing uncompen-\nsated magnetic moment. It can not convincingly explain\nthe observation of room-temperature hysteresis loops.\nA small canting of predominantly antiferromagnetic Fe\nspins due to the antisymmetric Dzyaloshinskii-Moriya in-\nteraction ^HDM=D\u0001[S1\u0002S2] causes weak ferromag-\nnetism in ortoferrites RFeO 3, R3+being Y or a rare\nearth ion. It was suggested that the canting may cause\nalso the uncompensated magnetic moment in AFeO 3\nstructure that is formed by the Fe ions clustering in\nthe double perovskites.25,32But the moment seems to\nbe too small to explain the e\u000bect.33,34In the ordered\nstate of RFeO 3, the canting angle \u001e\u001810 mrad results\nin the moment \u001b\u00180:05\u0016Bper Fe ion.35,36But such\na moment was never observed in the antiferromagneti-\ncally ordered state of PFN neither in magnetic18,20nor\nin neutron21{23studies. A possible reason is that the\nDzyaloshinskii-Moriya vector for a Fe-O-Fe bond may\nbe written as33,34D=d[r1\u0002r2], wheredis a scalar\nvalue, and riis a unit vector in the direction from oxy-\ngen to spin Si. Thus, its value depends on the Fe-O-Fe\nbond angle D/sin\u0012, which is substantially larger inAFe 1=2M1=2O3(170\u000e<\u0012 < 180\u000e)7,22than in the ortho-\nferrites (140 <\u0012< 157)37.\nIn this paper, we quantitatively consider another\nscenario for the room-temperature magnetism of bulk\nPFT/PZT and PFN/PZT systems25{27and superpara-\nmagnetism often observed in PFN nanoparticles or even\nceramics and thin \flms.29,38,39We explain it by the\nexistence of regions with a special chemical order (a\nsub-nano-size superstructure) that results in a ferrimag-\nnetic ordering of antiferromagnetically interacting Fe3+\nS= 5=2 spins. This explanation was implicitly as-\nsumed in Ref. 29, where the observed slightly asymmet-\nric EPR line shapes above room temperature were simu-\nlated by a model involving the presence of thermally \ruc-\ntuating superparamagneticlike nanoclusters. Note that\nour explanation does not demand the clusterization, as\nthe stoichiometry AFe 1=2M1=2O3is retained within the\n2\u00022\u00022 supercell of the superstructure. Using the\nhigh-temperature expansion (HTE),40,41we show that a\nmacroscopic number of spins orders at about the room\ntemperature, whereas small clusters (studied by exact di-\nagonalization method) exhibit a crossover between para-\nmagnetic and superparamagnetic behavior.3\na)\nb)\ncJc)\n 0 0.5 1 1.5 2\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7χ-1(gµB)2\n kBT/J1S(S+1) d)\na\nb\nc1c2\np\nFIG. 2. (Color online) ( a,b) The fragments of PFB2 con\fgu-\nration. ( c) The model simulating two interacting clusters of\nPFB2 con\fguration. The coupling strength J1(Jc) is denoted\nby black solid (red dashed) lines. Arrows indicate spin-spin\ncorrelations in the ground state of each cluster. ( d) The tem-\nperature dependence of the inverse susceptibility of all clusters\n(full exact diagonalization data for S= 1=2). The vertical\nline showsTfe. AtT.Tfethe susceptibility of the clusters\nexceeds the susceptibility of independent spins (line p) down\nto the lowest temperatures. The lines aandbcorrespond to\nclusters with 7 spins aand 13 spins brespectively. The lines\nc1andc2correspond to 14-spin cluster cwithJc= 0:25J1\nand 0:5J1respectively.\nII. METHODS\nWe use the method and the program packages pre-\nsented earlier in the Refs. 40, and 41 for the eighth-\nand tenth-order high-temperature expansion (HTE) of\nthe magnetic susceptibility \u001ffor a general Heisenberg\nmodel with up to four di\u000berent exchange parameters\nJ1;J2;J3;J4. The input for the HTE package is the de\f-\nnition \fle where all bonds within a cluster or a L\u0002L\u0002L\n(L=16,20) super-cell of a periodic Heisenberg lattice are\nenumerated with the indication of a corresponding valueof the exchange interaction. We use an originally devel-\noped C++ program, for the generation of the de\fnition\n\fles for spin structures studied in this work.\nIn order to simulate the behavior of fragments of PFB2\ncon\fguration in the Fe-based double perovskite mate-\nrial, we have performed full exact diagonalization studies\n(ED) of thermodynamic properties of clusters shown in\nFig. 1b, and in Fig. 2 using J. Schulenburg's spinpack .\nThe susceptibility \u001f(T) is calculated as the ratio of the\ninduced magnetization Mto the \feld H. We use the\n\"vanishing\" magnetic \feld H= 10\u00005J1=g\u0016Bunless oth-\nerwise noted.\nIII. RESULT AND DISCUSSION\nA. Ferrimagnetic superstructure\nThe simplest way to model the (partial) disorder in the\ndistribution of Fe and M ions between the sites of the\nB-sublattice of the perovskite structure is to consider a\nperiodic lattice with a supercell containing several per-\novskite cells and study such periodic systems with di\u000ber-\nent versions of chemical order (ion distributions). Such\nan approach was suggested in Ref. 31 for a 2 \u00022\u00022 su-\npercell, where 6 con\fgurations PFB0. . . PFB5 (see Fig. 3\nof Ref. 31, and Fig. 2 of Ref. 8) of chemical ordering are\npossible in the double perovskites. It was shown that the\ntotal energy is substantially di\u000berent for di\u000berent con\fg-\nurations. Moreover, the hierarchy of the energies depends\non the type of M-ion. In Ref. 8, it was found that the\nPFB2 con\fguration shown in the inset of Fig. 1a has an\nenergy close to the most stable con\fgurations (PFB5 for\nM=Nb,Ta and PFB0 for M=Sb), and has a ferrimag-\nnetic ground state (see Table II of Ref. 8). Below, we\nconsider the ferrimagnetism of PFB2 superstructure in\nmore detail.\nThe PFB2 chemical order has two inequivalent Fe\nsites. Within the B-sublattice of the perovskite struc-\nture, Fe1 has six Fe2 NN ions, whereas three Fe2 sites\nin the supercell has only two Fe1 NN ions (insets in Fig.\n1a,b). In other words, Fe2 sites form a superstructure\nof corner-shared octahedra, Fe1 sites being in the cen-\nter of each octahedron. The interaction value between\nthe two sublattices is J1, and within Fe2 sublattice is\nJ2\u001cJ1. Thus, the spin system satis\fes the require-\nments of the Lieb-Mattis theorem42withg2\nLM=J2=4\n(see Eq.(2) of the Ref. 42). Moreover, it is close to the\nspecial case g2\nLM= 0. According to the theorem (see\nalso the consideration of frustration J26= 0 in the Ref.\n43), the PFB2 ground state corresponds to a ferrimag-\nnetic ordering of Fe spins with a magnetic moment of\n2g\u0016BS\u001910\u0016Bper supercell, or 2 :5\u0016Bper Fe ion. This\nmoment value is much larger than the value provided by\nDzyaloshinskii-Moriya interaction for realistic values of\nlocal lattice distortions.33,34,36\nWe use the [4,4] Pad\u0013 e approximant of the HTE se-\nries to analyse the susceptibilty data.40For a magnetic4\nsuperstructure with the PFB2 spin arrangement the tem-\nperature dependence of the inverse susceptibility \u001f\u00001(T)\nforS= 5=2 is shown in the Fig. 1a. Only NN interaction\nJ16= 0 was taken into account. A reasonable estimate\nof the temperature for the transition into the ferrimag-\nnetically ordered phase Tfeis given by that point where\n\u001f\u00001(Tfe) = 0. The precision of the determination of crit-\nical temperatures by the zero of \u001f\u00001was estimated to be\nabout 10%.41The values of Tfefor di\u000berent spin values\nare given in the Table I.\nTABLE I. The ferrimagnetic transition temperature for\nPFB2 con\fguration, obtained from the [4,4] Pad\u0013 e approxi-\nmant of the 8th order HTE series.\nSpin,S k BTfe=J1S(S+ 1) kBTfe=J1\n1/2 0.61 0.46\n1 0.69 1.4\n3/2 0.64 2.4\n2 0.64 3.8\n5/2 0.64 5.6\n 0 0.5 1 1.5 2 2.5\n 0 0.2 0.4 0.6 0.8 1χ-1=H/M\n kBT/JS(S+1) S=0.5\nS=1.0\nS=1.5\nS=2.0\nS=2.5\nh=10-5J, S=2.5\nFIG. 3. (Color online) The inverse susceptibility \u001f\u00001(T) =\nH=M for the cluster shown in the Fig. 2a and the \feld\nH= 3J1=[(5S+ 1)g\u0016B] for di\u000berent spin values. For com-\nparison, an S= 5=2 curve for a vanishing \feld, i.e. \u001f\u00001\u0019\n(@M=@H )\u00001(H= 0), is given by the black solid line.\nFor Fe-based double perovskites Tfeis of the order\nof the room temperature, as J1=kB\u001850 K. From\nthe graph shown in Fig. 1a we see that in the range\nTfe< T < T\u0003\u00190:92J1S(S+ 1)=kB, the magnetic\nsusceptibility of the PFB2 phase exceeds the value for\nindependent spins, \u001f(T)> \u001f p(T) =S(S+ 1)=(3kBT),\ndespite the antiferromagnetic character of the exchange\ninteraction, which suppress the magnetic response at high\nFIG. 4. (Color online) Examples of ferrimagnetic superstruc-\ntures that may be formed by magnetic impurities (arrows)\nsubstituting for cations (blue circles) in zinc blend (left) and\nwurtzit (right) lattices.\n 0 10 20 30 40 50\n 0 2 4 6 8 10χ-1(gµB)2\nkBT/J2S(S+1)\nFIG. 5. Main panel: The inverse magnetic susceptibility (per\nspin)\u001f\u00001(red solid line - [4,4] Pad\u0013 e approximant of the 8th\norder HTE series) for the ideal 1:1 chemical order (PFB0,\nshown in the insert). It shows a minimum at T\u0018TIidicated\nby the arrow. The Curie-Weiss asymptotic is shown by the\ngreen dotted line. Black dotted line shows the bare HTE se-\nries. Inset: The super-cell for the PFB0, only M5+(closed\ncircles) and Fe3+ions (open circles) are shown. Arrows indi-\ncate the distribution of spins in the I-type ordering.\ntemperatures T\u001dJ1. For comparison, the black thin\nsolid line shows the susceptibility \u001ffcc(T) of 1:1 ordered\nPFB0 con\fguration, where Fe spins form a face centered\ncubic lattice, and interact with J2= 0:05J1. We see that\n\u001ffcc(T)<\u001f p(T) at all temperatures (see Appendix A).\nB. Superparamagnetism\nA sample of a disordered double perovskite compound\nmay contain some regions with PFB2 chemical order. In\nthe ground state, such a region possesses the total spin\nSg= (N2\u0000N1)S, whereN1,N2are the numbers of Fe1,\nand Fe2 sites in that region.42In order to simulate the be-\nhavior of fragments of PFB2 con\fguration in a Fe-based\ndouble perovskite material, we show in Figs. 1b, 2 full5\n 0 5 10 15 20 25\n 0 1 2 3 4 5 6χ-1(gµB)2\nkBT/JS(S+1)S=1.0\nS=1.5\nS=2.0\nS=2.5\nS=0.5\n 0 0.2 0.4 0.6 0.8 1\n 0.6 0.62 0.64 0.66 0.68 0.7χ-1(gµB)2\nkBT/JS(S+1)S=1.0\nS=1.5\nS=2.0\nS=2.5\nS=0.5\nFIG. 6. Temperature dependence of inverse magnetic suscep-\ntibility per spin for PFB2 chemical order for systems with\ndi\u000berent spin values S. [4,4] Pad\u0013 e approximant of the 8th\norder HTE series are shown for S > 0:5, and [4,6] Pad\u0013 e ap-\nproximant of the 10th order HTE series for S= 0:5\nexact-diagonalization data of thermodynamic properties\nof clusters shown in Figs. 1b and 2(a-c). Since we have\nfound that the dependence of the inverse susceptibility as\na function of normalized temperature kBT=J 1S(S+1) on\nthe spin value Sis weak (see Appendix A), the ED data\nfor the simplest S= 1=2 case can be considered as repre-\nsantative for higher values of S. The 7-site cluster shown\nin the Fig. 2a contains one Fe1 site interacting with six\nFe2 sites via J1exchange. This is a particular case of the\nHeisenberg star model.44,45ForT\u001dJ1S(S+ 1)=kBthe\nsusceptibility per spin tends to the Curie-Weiss asymp-\ntotic\u001fCW=\u001fp=[1+4S(S+1)J1=7kBT]. In the opposite\nlimit, the system shows a super-paramagnetic behavior,46\ni.e. it behaves as a single super-spin Sg= 5S, and the\nsusceptibility is \u001fSPM=Sg(Sg+1)\u001fp=[(N2+N1)S(S+1)]\n(see Fig. 1b). At temperatures T\u0018Tfethe system ex-\nhibits a crossover between the two regimes. The suscep-\ntibility exceeds the independent-spin value for T \u001f p(T) atT.Tfeand tends to the superpara-\nmagnetic behavior down to low temperature, where it\nexhibits a maximum (minimum at \u001f\u00001(T) curve). Below\nthe maximum, a singlet ground state of two interact-\ning super-spins is formed. In reality, for large number\nof interacting clusters the disorder in the system favors\na super-spin glass formation18,23,46at temperatures gov-\nerned by the low energy scale T >J\nT<5 ps, the minimum waiting time between pulses\ndepends approximately linearly on \u001cd. In this regime\nheat di\u000busion seems to be the dominant factor, however\na non-trivial dependence on the power of both pulses\nis also found. Comparing the blue squares and green\ncircles, for 40 and 50 \u0002108J m\u00003respectively, it is\nfound that the second switch can happen faster if the\npower of both pulses is higher. Individual time traces\nfor di\u000berent pulse powers (not shown here) indicate\nthat the switch occurs faster for higher pulse powers,\ngiving more time for relaxation towards saturation.\nFor\u001cd<5 ps, the minimum time needed for the\nsecond pulse starts to increase, which is attributed to\na hindrance of the switching mechanism in general by\nthe unrealistically fast dissipation of heat. Here, the\nsystem already cools down enough for the sublattices to\nstart remagnetizing at the timescale at which the switch\nwould normally take place. From these results it is clear\nthat although e\u000ecient heat dissipation is essential for\nachieving high switching repetition rates, the power also\nneeds to be carefully controlled. Combining this with\nthe experimental observation that a slight increase in\nlaser power can already be detrimental, this highlights\nthe narrow range of laser powers for which consistent as\nwell as rapid double switching is possible. Finally, we\nnote that double switching within 10 ps, as observed in\n1b, is not reproduced in modelling even for very e\u000ecient\nheat dissipation. We believe this to be an intrinsic limitof the system modelled here, as this probably represents\na minimum time needed for su\u000ecient recovery of the\nGd magnetization. Past work has shown that including\nintermixing, which is undoubtedly present in the real\nsystem, leads to more e\u000ecient transfer of angular\nmomentum between the two sublattices20. Hence, we\nexpect that in an intermixed system both Co and Gd\ncould remagnetize more rapidly than in a bilayer with\nperfect interfaces, potentially reducing the waiting time\nneeded for the second switch. It should be noted that\nswitching back within a few picoseconds, as was reported\nusing a di\u000berent model14, does not seem to be pos-\nsible using the M3TM for any combination of parameters.\nIn conclusion, we have investigated the timescales\nfor repeated all-optical switching in synthetic ferri-\nmagnetic Co/Gd bilayers, and have demonstrated a\nminimum waiting time of 10 ps between two subsequent\nsuccessful switching events, implying writing speeds\nof up to 100 GHz. We have shown that the layered\nnature of these systems need not be a hindrance to\nachieve similar writing speeds as in alloys, explained\nby the notion that the slower remagnetization of Gd is\ncompensated by a less critical dependence of AOS on the\nGd moment. Furthermore, by changing the substrate\nwe have con\frmed the importance of engineering heat\ndi\u000busion away from the magnetic system. Finally, with\nmodelling e\u000borts using the M3TM we have resolved the\nrole of heat di\u000busion in ultrafast repeated switching, but\nwe also stress that controlling the laser power is critical\nto reliable integration in future optically written data\nstorage devices.\nACKNOWLEDGEMENTS\nWe gratefully acknowledge M. Beens for assistance on im-\nplementation of the M3TM and discussion of simulation\nresults. This work is part of the Gravitation programme\n`Research Centre for Integrated Nanophotonics', which\nis \fnanced by the Netherlands Organisation for Scien-\nti\fc Research (NWO).\nAUTHOR DECLARATIONS\nCon\rict of Interest\nThe authors have no con\ricts to disclose.\nData availability\nThe data that support the \fndings of this study are\navailable from the corresponding author upon reasonable\nrequest.5\n1A. V. Kimel and M. Li, \\Writing magnetic memory with ultra-\nshort light pulses,\" Nature Reviews Materials 4, 189{200 (2019).\n2C. D. Stanciu, F. Hansteen, A. V. 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Man-\ngin, \\Ultrafast magnetization manipulation using single fem-\ntosecond light and hot-electron pulses,\" Advanced Materials 29,\n1703474 (2017)." }, { "title": "1704.07537v2.A_variety_of_elastic_anomalies_in_orbital_active_nearly_itinerant_cobalt_vanadate_spinel.pdf", "content": "arXiv:1704.07537v2 [cond-mat.str-el] 18 Jul 2017Variety of elastic anomalies in an orbital-active nearly it inerant cobalt vanadate spinel\nTadataka Watanabe1,∗Shogo Yamada1, Rui Koborinai2, and Takuro Katsufuji2,3\n1Department of Physics, College of Science and Technology (C ST), Nihon University, Tokyo 101-8308, Japan\n2Department of Physics, Waseda University, Tokyo 169-8555, Japan and\n3Kagami Memorial Research Institute for Materials Science a nd Technology, Waseda University, Tokyo 169-0051, Japan\n(Dated: July 25, 2018)\nWe perform ultrasound velocity measurements on a single cry stal of nearly-metallic spinel\nCo1.21V1.79O4which exhibits a ferrimagnetic phase transition at TC∼165 K. The experiments\nreveal a variety of elastic anomalies in not only the paramag netic phase above TCbut also the\nferrimagnetic phase below TC, which should be driven by the nearly-itinerant character o f the\norbitally-degenerate V 3 delectrons. In the paramagnetic phase above TC, the elastic moduli ex-\nhibit elastic-mode-dependent unusual temperature variat ions, suggesting the existence of a dynamic\nspin-cluster state. Furthermore, above TC, the sensitive magnetic-field response of the elastic mod-\nuli suggests that, with the negative magnetoresistance, th e magnetic-field-enhanced nearly-itinerant\ncharacter of the V 3 delectrons emerges from the spin-cluster state. This should be triggered by\nthe inter-V-site interactions acting on the orbitally-deg enerate 3 delectrons. In the ferrimagnetic\nphase below TC, the elastic moduli exhibit distinct anomalies at T1∼95 K and T2∼50 K, with a\nsign change of the magnetoresistance at T1(positive below T1) and an enhancement of the positive\nmagnetoresistance below T2, respectively. These observations below TCsuggest the successive oc-\ncurrence of an orbital glassy order at T1and a structural phase transition at T2, where the rather\nlocalized character of the V 3 delectrons evolves below T1and is further enhanced below T2.\nPACS numbers: 62.20.de, 72.55.+s, 75.25.Dk, 75.50Gg\nI. INTRODUCTION\nVanadate spinels AV2O4with divalent A2+ions have\nattracted considerable attention owing to the interplay\nbetween the orbital degree of freedom and geometrical\nfrustration [1]. The trivalent magnetic V3+ions are char-\nacterized by double occupancy of the triply-degenerate\nt2gorbitals ( t2\n2g), and form a sublattice of corner-sharing\ntetrahedra (pyrochlore lattice). Upon cooling, AV2O4,\nwith nonmagnetic A= Zn, Mg, and Cd, undergoes a\nstructural phase transition followed by an antiferromag-\nnetic phase transition [2–10]. With magnetic A= Mn\nand Fe, owing to the presence of additional A2+-V3+\nexchange interactions, AV2O4exhibits a more complex\nstructural and magnetic behavior, with successive struc-\ntural and ferrimagnetic phase transitions [11–24].\nForAV2O4(A= Zn, Mg, Cd, Mn, and Fe), the struc-\ntural phase transition is considered to arise from a long-\nrangeorderingofthe V t2gorbitals, wherethe loweringof\nthe lattice symmetry should result in the release of frus-\ntration (magnetic ordering). The orbital order to explain\nthe structural and magnetic orders for AV2O4is still un-\nder debate both theoretically [25–31] and experimentally\n[4,7,12,15]; it is considered to be driven by the competition\namong Jahn-Teller coupling, Kugel-Khomskii exchange\ninteraction [32], and relativistic spin-orbit coupling.\nCoV2O4(Co2+:e4t3\n2, V3+:t2\n2g) is a unique vanadate\nspinel with its highest electrical conductivity among the\ninsulatingorsemiconducting AV2O4, whichisconsidered\nto arise from the nearly-itinerant character of the V 3 d\nelectrons [33–39]. CoV 2O4exhibits a ferrimagnetic order\nbelowTC∼150 K, which is similar to AV2O4with mag-\nneticA= Mn (TC= 57 K) and Fe ( TC= 110 K). How-\never, while AV2O4(A= Zn, Mg, Cd, Mn, and Fe) lowersthe cubic lattice symmetry by the structural phase tran-\nsition, CoV 2O4is unique in that the loweringofthe cubic\nlattice symmetry is absent down to low temperature. Al-\nthoughaveryrecentneutron scatteringstudy ofCoV 2O4\ndiscoveredavery-weakstructuralphasetransitionat ∼90\nK within the ferrimagnetic phase, this structural transi-\ntion was characterized as a short-range distortion of the\nO octahedra, which does not lower the global cubic sym-\nmetry of the crystal [40]. The very small magnitude of\nthe structural distortion in CoV 2O4is considered to be\nrelevant to the nearly-itinerant electron character.\nFor CoV 2O4, the previous reports of the experimen-\ntal studies in the single-crystalline and polycrystalline\nsamples suggested the presence of additional transition\nwithin the ferrimagnetic phase [34,35,38,39]. However, the\nreported transition temperatures, also including the fer-\nrimagnetic transition temperature TC, differ from study\nto study. This variation in the transition temperature\nvalues seems to arise from the off-stoichiometry of the\nsamples. In the crystal growth of the cobalt vanadate\nspinel, the Co : V ratio of the grown crystal is prone to\ndeviate from the stoichiometric ratio of 1 : 2 to 1+ x:\n2-xwith excess Co x. Very recently, the measurements\nof the magnetization, magnetostriction, neutron scatter-\ning, and dielectric constant for the single-crystalline and\npolycrystalline Co 1+xV2−xO4with various xextracted\nthe intrinsic magnetic and structural properties irrespec-\ntiveoftheoff-stoichiometry[39]. Itwassuggestedthat, in\naddition to the ferrimagnetic transition at TC, two other\npossible transitions occur within the ferrimagnetic phase\n[39].\nFig. 1(*) depicts the ferrimagnetic transition tempera-\ntureTCand two other transition temperatures T1andT2\nin Co1+xV2−xO4as functions of off-stoichiometry x(the2\nsolidlines), whichweresuggestedfromthemagnetization\nand magnetostriction measurements [39]. By increasing\nthe amounts of the off-stoichiometry x,TCincreases but\nT1andT2decrease. T1corresponds to a spin canting\ntemperature, where a collinear-to-noncollinear ferrimag-\nnetic ordering occurs upon cooling [39]. In Fig. 1(*), the\nstructural transition temperature of ∼90 K observed in\nthe neutron scattering experiments for the stoichiometric\nCoV2O4is also indicated as a circle [40], indicating the\noccurrence of the structural transition at T2.\nForAV2O4(A= Mn and Fe), the long-range or-\ndering of the V orbitals is considered to be accompa-\nnied by a canting of the V spins, which was observed\nas the appearance of a noncollinear ferrimagnetic order\n[15,21]. For Co 1+xV2−xO4, very recent neutron scatter-\ning experiments revealed the occurrence of a collinear-\nto-noncollinearferrimagneticordering (a spin canting) at\nT1[Fig. 1(*)] upon cooling to within the ferrimagnetic\nphase [39]. However, at the spin canting temperature T1,\nthe magnetostriction of Co 1+xV2−xO4exhibits a change,\nwhichissubtlerandmoregradualthanthatof AV2O4(A\n= Mn and Fe) [39]. Furthermore, the dielectric constant\nof Co1+xV2−xO4exhibits a frequency variation within\nthe noncollinear ferrimagnetic phase, indicating slow re-\nlaxation dynamics [39]. This dielectric behavior is simi-\nlar to that observed in the orbital glass state of FeCr 2S4\n[41]. From the results of the neutron scattering, magne-\ntostriction, anddielectric constantmeasurements, forthe\nnearly-itinerantCo 1+xV2−xO4, itissuggestedthatanor-\nbital glassy state exists in the noncollinear ferrimagnetic\nphase [39].\nHerein, we study the interplay of orbital, spin,\nand lattice degrees of freedom in the nearly-itinerant\nCo1+xV2−xO4by means of ultrasound velocity measure-\nments, which measure elastic moduli of this compound.\nThe elastic modulus of a crystal is a thermodynamic ten-\nsor quantity, and thus the ultrasound velocity measure-\nments in all the symmetrically-independent elastic mod-\nuli in a crystal can provide the symmetry-resolved ther-\nmodynamic information. Furthermore, since the ultra-\nsound velocity can be measured with a high precision of\n∼ppm, the ultrasound velocity measurements can sensi-\ntively probe elastic anomalies driven by phase transition\nand excitations. For the frustrated spinels, the ultra-\nsound velocity measurements have proven to be a useful\ntool for studying not only the ground state but also the\nexcited states [42–47].\nFor the vanadate spinels, the ultrasound velocity mea-\nsurementsinMgV 2O4[44]andMnV 2O4[45]wererecently\nreported. These measurements revealed the presence of\nanomalous elastic behavior in the paramagnetic phase\n(the magnetically disordered phase) and its absence in\nthe magnetically ordered phase. In contrast, the present\nstudy reveals that Co 1+xV2−xO4exhibits a variety of\nelastic anomalies in not only the paramagnetic phase\nbut also the magnetically ordered phase. Furthermore,\nthe present study also reveals that the anomalous elastic\nbehavior in the paramagnetic phase of Co 1+xV2−xO4isuniquely different from that of MgV 2O4and MnV 2O4.\nThe present study strongly suggests that the anoma-\nlous elastic behavior in Co 1+xV2−xO4is relevant to the\nnearly-itinerant character of the orbitally-degenerate V\n3delectrons, which causes the orbital glassiness within\nthe magnetically ordered phase.\nII. EXPERIMENTAL\nSingle crystals of Co 1+xV2−xO4(0≤x≤0.3) were\nprepared by the floating-zone method, where the chem-\nical compositions of the grown single crystals were esti-\nmated by the inductively coupled plasma analyses [39].\nFor the present experiments, we applied a large single\ncrystal of Co 1.21V1.79O4(x= 0.21), where the magne-\ntization, magnetostriction, and neutron scattering mea-\nsurements suggested the occurrence of magnetic and\nstructural transitions at TC∼165 K,T1∼100 K, and\nT2∼50 K [the solid lines in Fig. 1(*)] [39]. The ul-\ntrasound velocity measurements were performed by a\nhome-built apparatus, where the phase-comparison tech-\nnique was used with longitudinal and transverse sound\nwaves at a frequency of 30 MHz. The ultrasound waves\nwere generated and detected by LiNbO 3transducers\nglued on the parallel mirror surfaces of the crystal. We\nmeasured the sound velocities in all the symmetrically-\nindependent elastic moduli in the cubic crystal, specifi-\ncally, compression modulus C11, tetragonal shear modu-\nlusC11−C12\n2≡Ct, and trigonal shear modulus C44. From\nC11andCtdata, we also obtained the bulk modulus\nCB=C11+2C12\n3=C11−4\n3Ct. The respective measure-\nments of C11,Ct, andC44were performed using longi-\ntudinal sound waves with propagation k/bardbl[100] and po-\nlarization u/bardbl[100], transverse sound waves with k/bardbl[110]\nandu/bardbl[1¯10], and transverse sound waves with k/bardbl[110]\nandu/bardbl[001]. The sound velocities of Co 1.21V1.79O4mea-\nsured at room temperature (300K) are 6740m/s for C11,\n2750 m/s for Ct, and 3770 m/s for C44. In the present\nstudy, we also performed the electrical resistivity mea-\nsurements for the single-crystalline Co 1.21V1.79O4using\nthe conventional four-probe method.\nIII. RESULTS\nA. Elastic modulus\nFigures 1(a)-1(c) respectively present the temperature\n(T) dependence of the elastic moduli, CB(T),Ct(T),\nandC44(T), with zero magnetic field ( H= 0) in\nCo1.21V1.79O4. Here,CB(T) =C11(T)−4\n3Ct(T) is ob-\ntained from C11(T) [the inset in Fig. 1(a)] and Ct(T)\n[Fig. 1(b)]. From these experimental results, a variety of\nelastic anomalies are observed.\nFirst, the elastic moduli shown in Figs. 1(a)-1(c) ex-\nhibit anomalies at ∼165 K for all the elastic moduli,\nat∼95 K for Ct(T), and at ∼50 K for C44(T). As3\n37.6\n37.4\n37.2\n37.0\n36.8(b) CtCt (GPa)TC\nT1H = 0180\n179\n178CB (GPa)(a) CB\n177TCTC\n300250200150100500230\n229\n228\n227C11 (GPa)C11 \nH = 0H = 0\n72.0\n71.0\n70.0\n69.0\n300250200150100500\n T (K)(C) C44 H = 0C44 (GPa)TC T2150 \n100 \n50 TC\nT2(*) T (K)\n xT1\n0 0.1 0.2 0.3 \nFIG. 1: (Color online) (*) Transition temperatures TC,T1,\nandT2for Co 1+xV2−xO4as functions of off-stoichiometry x,\nwhich were reported in Ref. [39] (the solid lines). The circle\nin (*) indicates a structural transition temperature for th e\nstoichiometric CoV 2O4reported in Ref. [40]. The vertical\ndashed line in (*) indicates x= 0.21, corresponding to the x\nvalue for the single-crystalline sample applied in the pres ent\nstudy. The crosses in (*) indicate transition temperatures TC,\nT1, andT2, which are obtained from the experimental results\nshown in (a)-(c). (a)-(c) Elastic moduli of Co 1.21V1.79O4as\nfunctions of TwithH= 0. (a) CB(T), (b)Ct(T), and (c)\nC44(T). The insetin(a) depicts C11(T)ofCo 1.21V1.79O4with\nH= 0. The arrows in (a)-(c) indicate TC,T1, andT2.\nindicated in Fig. 1(*), these temperatures [the crosses\nin Fig. 1(*)] respectively agree with TC,T1, andT2re-\nported in Ref. [39] [the solid lines in Fig. 1(*)]. At the\nferrimagnetic transition temperature TC, all the elastic\nmoduli exhibit a discontinuous anomaly, as marked by\nthe arrows in Figs. 1(a)-1(c). Additionally, within the\nferrimagnetic phase ( T < T C),Ct(T) andC44(T) respec-\ntively exhibit a rather gradualanomaly at T1∼95 K and\na discontinuous anomaly at T2∼50 K, as marked by the\narrows in Figs. 1(b) and 1(c). The gradual anomaly at\nT1∼95 K inCt(T) should be driven by the occurrence\nof the orbital glassy order accompanied by the spin cant-\ning, which was suggested in Ref. [39]. The discontinuous\nanomaly at T2inC44(T) indicates the occurrence of a\nphase transition at this temperature. It should be noted\nthat the neutron scattering experiments in the stoichio-\nmetric CoV 2O4observed a structural transition at ∼90\nK [the circle in Fig. 1(*)] [40]. Since this structural tran-\nsition temperature agrees with the temperature of T2forCo1+xV2−xO4withx= 0 [the solid line in Fig. 1(*)] [39],\nthe elastic anomaly at T2inC44(T) for Co 1.21V1.79O4\nshould be driven by the structural transition, which is\nidentical to that observed in the neutron scattering ex-\nperiments for the stoichiometric CoV 2O4[40]. Compar-\ning the elastic anomalies at TC,T1, andT2, the rather\ngradual observation of the anomaly at T1is compatible\nwith the occurrence of the orbital glassy order, which is\nin contrast to the discontinuous anomalies at TCandT2\ndriven by the ferrimagnetic and structural phase transi-\ntions, respectively.\nIn addition to the elastic anomalies at TC,T1, andT2,\nas shown in Figs. 1(a)-1(c), the elastic moduli exhibit\nelastic-mode-dependent unusual Tvariations in both the\nparamagnetic and ferrimagnetic phases. Upon cooling in\nthe paramagnetic phase ( T > T C),CB(T) exhibits ordi-\nnal hardening [48], butCt(T) andC44(T) exhibit anoma-\nlous softening. Here, the magnitudes of the softening in\nCt(T) andC44(T) are ∆Ct/Ct∼1.5 % and ∆ C44/C44∼\n0.5 %, respectively. In the ferrimagnetic phase ( T < T C),\nthe elastic moduli exhibit nonmonotonic Tvariations;\nCB(T) andCt(T) exhibit minima at ∼130 K and ∼\n30 K, and C44(T) exhibits a minimum at ∼130 K.\nIn the present study, we also investigated the magnetic\nfield effect on the elastic properties of Co 1.21V1.79O4. In\nthe ferrimagnetic phase ( T < T C), the ultrasound echo\nsignalsweretoostronglyattenuated bythe applicationof\na magnetic field to perform the ultrasound velocity mea-\nsurement, which should arise from the magnetostriction\n[39]. However, in the paramagnetic phase ( T > T C), we\nwere able to detect the ultrasound echo signals indepen-\ndent of whether the magnetic field was applied or not.\nThus, in the present study, the ultrasound velocity mea-\nsurements under magnetic field were performed only in\nthe paramagnetic phase ( T > T C).\nFigures 2(a)-2(c) respectively depict CB(T),Ct(T),\nandC44(T) with magnetic field H||[001] in the para-\nmagnetic phase ( T > T C) of Co 1.21V1.79O4. Here,\nCB(T) =C11(T)−4\n3Ct(T) is obtained from C11(T) [the\ninset in Fig. 2(a)] and Ct(T) [Fig. 2(b)]. In the zero-\nfield (H= 0) paramagnetic phase, as described above\nin conjunction with Figs. 1(a)-1(c), Ct(T) andC44(T)\nexhibit softening with decreasing T, whileCB(T) ex-\nhibits ordinal hardening [48]. Furthermore, in Figs. 2(a)-\n2(c),CΓ(T) exhibits Hvariation below ∼230 K;CB(T)\nbecomes softer with increasing H, but the softening in\nCt(T) andC44(T) is relaxed with H, as indicated by the\nsolid arrows.\nB. Magnetoresistance\nFigure 3(a) depicts the Tdependence of the magne-\ntoresistanceat 7 T, MR=ρ(7T)−ρ(0)\nρ(0), for Co 1.21V1.79O4\nwith the current I||[100] and the magnetic field H||[001]\n(I⊥H), which is obtained from the Tdependence of\nthe electrical resistivities with H= 0 (ρ(0)) and 7 T\n(ρ(7T)) depicted in Figs. 3(b) and 3(c). The Tdepen-4\n71.2\n71.1\n71.0\n70.9\n300 250 200175 \n T (K) 7 T\n 5 T\n 3 T\n H = 0(c) C44 C44 (GPa)HH || [001] \n Eq. (1) \n C44 = 71.4 GPa \n EJT = 0.2 K \n θ = 192.9 K 0 Eq. (1) \n Ct = 38.1 GPa \n EJT = 1.4 K \n θ = 141.5 K 037.7\n37.6\n37.5\n37.4\n37.3 7 T\n 5 T\n 3 T\n H = 0(b) CtCt (GPa)\nHH || [001] \n300 250 200175 \n T (K)H = 0\n 3 T\n 5 T\n 7 T(a) CB179.5\n179.0\n178.5\n178.0\n177.5\n300 250 200175 \n T (K)H\nH || [001] 211.0\n210.5\n210.0\n209.5\n300 250 200175 H = 0\n 3 T\n 5 T\n 7 THH || [001] C11 C11 (GPa)CB (GPa)\nFIG. 2: (Color online) CΓ(T) of Co 1.21V1.79O4withH||[001] in the paramagnetic phase ( T > T C). (a)CB(T), (b)Ct(T), and\n(c)C44(T). The inset in (a) depicts C11(T) of Co 1.21V1.79O4withH||[001] in the paramagnetic phase ( T > T C). The solid\narrows in (a)-(c) are guides to the eye, indicating the varia tion ofCΓ(T) with increasing H. The dotted curves in (b) and (c)\nare, respectively, fits of the zero-field experimental Ct(T) andC44(T) to Eq. (1) in 250 K < T <300 K. The respective values\nof the fit parameters are also listed in (b) and (c).\ndence of the magnetoresistanceshown in Fig. 3(a) agrees\nwith the previous report on the stoichiometric CoV 2O4\n[37]; upon cooling, Co 1+xV2−xO4exhibits a large neg-\native magnetoresistance at temperatures around ∼TC,\na change to positive magnetoresistance at ∼T1, and a\nfurther enhancement of the positive magnetoresistance\nbelow∼T2.\nIV. DISCUSSION\nA. Paramagnetic phase ( T > T C)\nWe will discuss the origins of the elastic anomalies\nobserved in Co 1.21V1.79O4. First we address the ori-\ngins of the elastic anomalies in the paramagnetic phase\n(T > T C) [Fig. 2].\nIn zero magnetic field ( H= 0),Ct(T) andC44(T)\nsoften upon cooling in the paramagnetic phase, while\nC11(T) exhibits ordinalhardening [48]. One probable ori-\ngin for this softening in Ct(T) andC44(T) is a precursor\nto the Jahn-Teller (JT) structural transition, which has\nbeen observed in MgV 2O4[44] and MnV 2O4[45].\nFor the JT effect, a JT-active ion and its set of lig-\nands are considered to be a structural unit, where the\ncrystal-field striction mechanism leads to the JT distor-\ntion. In the JT magnets, the degenerate ground state\nis considered to couple strongly and selectively to the\nelastic modulus CΓ, which has the same symmetry as\nthe JT distortion. For such a JT-active elastic mode,\nTdependence of the elastic modulus CΓ(T) above the\nJT transition temperature is explained by assuming the\ncoupling of the ultrasound to the JT-active ions through\nthe crystal-field striction mechanism, and the presence of\ninter-JT-active-ion interactions. A mean-field expressionofCΓ(T) in the JT magnets is given as [49–51]\nCΓ(T) =C0,Γ(1−EJT\nT−θ), (1)\nwithC0,Γthe elastic constant without the JT effect, EJT\nthe JT coupling energy, and θthe inter-JT-active-ion in-\nteraction.\nAccording to Eq. (1), CΓ(T) of the JT system exhibits\na Curie-type ( ∼ −1/T-type) softening at temperatures\nabovethe JT transitiontemperature. ForCo 1.21V1.79O4,\nhowever, Ct(T) [Fig. 2(b)] and C44(T) [Fig. 2(c)] exhibit\nthe non-Curie-type softening in the paramagnetic phase\naboveTC. As a comparison, fits of the zero-field exper-\nimentalCt(T) andC44(T) to Eq. (1) in 250 K < T <\n300 K are presented in Figs. 2(b) and 2(c) as dotted\ncurves, respectively. It is evident in these figures that\nthe experimental Ct(T) andC44(T) both deviate from\nthe dotted curves below ∼250 K, which rules out the JT\neffect as a possible origin for the softening in Ct(T) and\nC44(T) in the paramagnetic phase above TC. Indeed,\nfor the stoichiometric CoV 2O4, the X-ray and neutron\ndiffraction experiments reported the absence of a struc-\ntural distortion at TC, indicating the absence of a JT\nstructural transition at TC[33,40].\nTheotherpossibleoriginforthesofteningin Ct(T)and\nC44(T) is the coupling between the correlated paramag-\nneticstateandtheacousticphonons. Itisnotedherethat\nthe inelastic neutron scattering experiments in the stoi-\nchiometric CoV 2O4observed quasielastic magnetic exci-\ntations in the paramagnetic phase [40]. A similar type of\nlocal spin excitations was also observed in AV2O4and\nACr2O4(A= Mg and Zn), which has been character-\nized as spin-cluster excitations on the V/Cr pyrochlore\nlattice [52–54]. Furthermore, for these compounds, the\nnon-Curie-type softening was observed in CΓ(T) in the\nparamagnetic phase, which is considered to be driven by5\nH = 0 \n7 T 0.4 \n0.3 \n0.2 \n0.1 \n30025020015010090 80 70 60 50 40 30 ρ (Ω cm)\n110100 H = 0 \n7 T (b) (c) \nT2\nT1TC\n I ||[100] \nH||[001] I ||[100] \nH||[001] \nT1\n30 300 250 200 150 100 50\nT (K)MR (%)10\n8\n6\n4\n2\n0\n−2\n−4\n−6TCT2\n I ||[100] \nH||[001] \n(a) (d) \neg\nt2g eg\na1g \nFIG. 3: (Color online) (a) Magnetoresistance ( MR) of\nCo1.21V1.79O4at 7 T for I||[100] and H||[001] (I⊥H) as a\nfunction of T. (b), (c) Electrical resistivities of Co 1.21V1.79O4\nwithH= 0 and 7 T ( I||[100] and H||[001]) as functions of T:\n(b) 30 K < T <100 K and (c) 100 K < T <300 K. The ar-\nrows in (a), (b), and (c) indicate the transition temperatur es\nTC,T1, andT2. The horizontal dashed line in (a) is a guide\nto the eye indicating MR= 0. (d) Energy-level scheme of the\nV 3dorbitals in Co 1+xV2−xO4. The arrows in (d) depict the\nHund-rule filling of 2 electrons/V.\nthe coupling of the lattice to the spin-cluster excitations\n[42,44]. Therefore, the non-Curie-type softening in Ct(T)\nandC44(T) in the paramagnetic phase of Co 1.21V1.79O4\nshould stem from the coupling of the lattice to the spin-\ncluster excitations on the V pyrochlore lattice. Taking\ninto account that the quasielastic magnetic excitations\nof CoV 2O4andAV2O4(A= Mg and Zn) were both ob-\nserved centered around a wave vector of Q∼1.3˚A−1\n[40,52], the shape of the spin cluster in these compounds\nmight be identical, which should be confirmed in a future\nstudy.\nThe softening in CΓ(T) driven by the spin-cluster ex-\ncitations is generally explained as the presence of a finite\ngapfortheexcitations, whichissensitivetostrain[42]. In\nthe mean-field approximation, CΓ(T) in the spin-cluster\nsystem is written as [42]\nCΓ(T) =C0,Γ−G2\nΓNχΓ(T)\n[1−KΓχΓ(T)],(2)\nwithC0,Γthe backgroundelastic constant, Nthe density\nofspin clusters, GΓ=|∂∆/∂ǫΓ|the couplingconstant for\nasinglespinclustermeasuringthestrain( ǫΓ)dependence\nofthe excitationgap∆, KΓthe inter-spin-clusterinterac-\ntion, and χΓ(T) the strain susceptibility of a single spincluster. From Eq. (2), when CΓ(T) strongly couples to\nthe excited state at ∆, this elastic mode exhibits soften-\ning upon cooling roughlydown to T∼∆, but recoveryof\nthe elasticity (hardening) roughly below T∼∆;CΓ(T)\nexhibits a minimum roughly at T∼∆. According to\nEq. (2), for Ct(T) andC44(T) of Co 1.21V1.79O4respec-\ntively shown in Figs. 2(b) and 2(c), the softening upon\ncooling down to TCindicates that this softening arises\nfrom the coupling of the lattice to the gapped excitations\nwith ∆< TC, which is compatible with the observation\nof the spin-cluster excitations at energies below ∼4 meV\n(< kBTC) in the paramagnetic phase of the stoichiomet-\nric CoV 2O4[40]. For the experimental results shown in\nFig. 2, the future identification of the shape of the spin\ncluster in Co 1+xV2−xO4by the inelastic neutron scat-\ntering study will enable the quantitative analyses using\nEq. (2).\nAs observed in Fig. 2, CB(T),Ct(T), andC44(T) in\ntheparamagneticphaseallexhibitthemagnetic-field( H)\nvariations. Notably, while the softening in Ct(T) and\nC44(T) is present already at H= 0 and relaxed with in-\ncreasing H, the softening in CB(T) is absent at H= 0\nbut induced by H. As described above, the softening in\nCt(T) andC44(T) withH= 0 is attributed to the cou-\npling of the lattice to the spin-cluster excitations. Thus\nthe relaxation of this softening by Hindicates that the\ncoupling of the lattice to the spin-cluster state is relaxed\nbyH. A similar type of this Heffect was also observed\nin MgV 2O4, which is also considered to be a result of\nthe relaxation of the spin-cluster-lattice coupling by H\n[44]. In contrast, the H-induced softening in CB(T) for\nCo1.21V1.79O4is a unique elastic anomaly, which was not\nobservedin other AV2O4(A=Mg [44] and Mn [45]) com-\npounds. Taking into account that, among the insulating\nor semiconducting AV2O4, Co1+xV2−xO4is closest to\nthe itinerant-electron limit [31,33], theH-induced soften-\ning inCB(T) unique to Co 1+xV2−xO4should be relevant\nto the nearly-itinerant character of the V 3 delectrons.\nAs shown in Fig. 3(a), Co 1.21V1.79O4exhibits the T-\ndependent magnetoresistance, which should arise from\nthe nearly-itinerant V 3 delectrons. Comparing Fig. 2(a)\nwith Fig. 3(a) in the paramagnetic phase ( T > T c), it is\nevident that the H-induced softening in CB(T) develops\nupon cooling below ∼230 K in accordance with the de-\nvelopment of the negative magnetoresistance. We note\nhere that, for the cubic crystal of Co 1+xV2−xO4,CB\nis a symmetry-conserving isotropic elastic mode, while\nCtandC44are symmetry-lowering anisotropic elastic\nmodes, and that the application of Hcauses the devel-\nopment of the softening for CB(T) and the relaxation\nof the softening for Ct(T) andC44(T). This elastic-\nmode-dependent Heffect indicates that the magnetoe-\nlastic coupling becomes rather isotropic with increasing\nH, whichshouldbe drivenbythe H-enhanceddelocaliza-\ntion of the strongly-directional V 3 delectrons. Thus, for\nCo1+xV2−xO4, theH-induced softening in CB(T) and\nthe negative magnetoresistance suggest the presence of\nthe intersite V 3 delectron interactions, which causes the6\nH-enhanced electron delocalization in the paramagnetic\nphase.\nAs is evident from the comparison between Fig. 2 and\nFig. 3(a), in the paramagnetic phase of Co 1.21V1.79O4,\nnot only the H-induced softening in CB(T) but also\ntheH-induced relaxation of the softening in Ct(T) and\nC44(T) coincide with the development of the negative\nmagnetoresistance below ∼230 K. As already described\nin conjunction with Figs. 2(b) and 2(c), the softening in\nCt(T) andC44(T) withH= 0 in the paramagneticphase\nis concluded to be driven by the coupling of the lattice\nto the spin-cluster state. Thus the H-induced relaxation\nof the softening in Ct(T) andC44(T) suggests the oc-\ncurrence of the H-induced ”bond dissolution” in the V\nspin clusters, which should be driven by the H-enhanced\ndelocalization of the V 3 delectrons.\nIn Co1+xV2−xO4, the O octahedra surrounding the V\natoms are trigonally distorted from the regular octahe-\ndral shape even in the cubic crystal structure [33,36,40].\nAdditionally, for the V 3 dorbitals, the small trigonal\ncrystal field splits the triply-degenerate t2gorbitals into\na localized a1gorbital and delocalized doubly-degenerate\ne′\ngorbitals [Fig. 3(d)]. Thus it is expected that the delo-\ncalizede′\ngorbitals are responsible for the nearly-itinerant\ncharacter of the V 3 delectrons [36]. Thee′\ngorbitals are\nalso expected to be responsible for the elastic anomalies\nin the paramagnetic phase [Fig. 2].\nFinally, we note that the softening in CΓ(T) driven by\nthe spin-cluster-lattice coupling is observed in not only\nthe orbital-degenerate Co 1+xV2−xO4and MgV 2O4[44]\nbut also the orbital-nondegenerate ACr2O4(A= Mg\nand Zn) [42] and ZnFe 2O4[43], which indicates that the\nspin-clusterstatecanuniversallyemergeinthefrustrated\nspinels. It is also noted that this type of elastic softening\nis relaxed by Hin Co1+xV2−xO4and MgV 2O4, but in-\nsensitive to HinACr2O4and ZnFe 2O4, indicating that\nthe orbital sector is responsible for the Heffect on the\nspin-cluster state. For Co 1+xV2−xO4and MgV 2O4, as\ndescribed above, it is suggested that the application of\nHresultsin the”bonddissolution”intheVspinclusters,\nwhich should be driven by the enhancement of the delo-\ncalized character for the V 3 delectrons. Here, compar-\ning Co 1+xV2−xO4and MgV 2O4, only MgV 2O4exhibits,\nin addition to the spin-cluster-driven elastic anomalies,\nthe Curie-type softening in CΓ(T), which is the JT ef-\nfect (a precursor to the structural transition) expressed\nby Eq. (1) [44]. Taking into account that the JT ef-\nfect is driven by the coupling of the elastic deformations\nto the localized magnetic moments, the presence of the\nJT effect in MgV 2O4but its absence in Co 1+xV2−xO4\nindicates that the localized character of the V 3 delec-\ntrons is stronger in MgV 2O4than Co 1+xV2−xO4, which\nis compatible with the poorer electrical conductivity of\nMgV2O4than Co 1+xV2−xO4[31,33].B. Ferrimagnetic phase ( T < T C)\nIn this section, we discuss the origins of the elastic\nanomalies observed in the ferrimagnetic phase ( T < T C)\nof Co1.21V1.79O4. As shown in Fig. 1, Co 1.21V1.79O4\nexhibits nonmonotonic Tdependence of the elastic mod-\nuli in the ferrimagnetic phase, which is in contrast to\nthe monotonic Tdependence observed in the magneti-\ncally ordered phase of other AV2O4(A= Mg [44] and\nMn [45]) andACr2O4(A= Mg and Zn [42]). This non-\nmonotonic elastic behavior in Co 1.21V1.79O4is expected\nto be relevant to the nearly-itinerant character of the V\n3delectrons.\nFirst, one of the remarkable elastic anomalies shown in\nFig. 1 is the elastic moduli minima at ∼30 K and ∼130\nK. Since there is no additional phase transition at these\ntemperatures, theelasticmoduli minimashouldoriginate\nfrom the coupling of the ultrasound to the magnetic ex-\ncitations, rather than to the static order. Indeed, in the\nferrimagneticphase,magnonexcitationswereobservedin\nthe inelastic neutron scattering experiments [40]. How-\never, in the ferrimagnetic phase of Co 1+xV2−xO4, there\nremains the possibility for the coexistence of the magnon\nexcitations and the spin-cluster excitations. Thus, at\npresent, we cannot identify which excitation is the ori-\ngin for the respective elastic moduli minima at ∼30 K\nand∼130 K.\nNext, asalreadymentionedinSec. IIIAinconjunction\nwith Fig. 1, the Tdependence of the elastic moduli sug-\ngests the successive occurrence of the orbital glassy order\natT1∼95 K and the structural phase transition at T2∼\n50 K. As shown in Figs. 1(a)-1(c), the elastic anomaly\natT1is observed only in the tetragonal Ct(T) withEg\nsymmetry. We note here again that, for Co 1+xV2−xO4,\nthe trigonal crystal field splitting of the V 3 dorbitals is\npresent even in the cubic crystal structure [Fig. 3(d)]\n[33,36,40]. Thus the elastic anomaly at T1only inCt(T)\nsuggests that the nearly-itinerant doubly-degenerate e′\ng\norbitals [Fig. 3(d)] play a dominant role for the occur-\nrence of the orbital glassy order. Further, as shown in\nFig. 3(a), the magnetoresistance exhibits a negative-to-\npositive sign change at T1, which indicates that, in the\norbital glassy state, the application of Henhances the\nlocalizedcharacterofthe V3 delectrons. While the nega-\ntive magnetoresistance in the paramagnetic phase is sug-\ngested to be driven by the H-induced ”bond dissolution”\nin the V spin cluster, the positive magnetoresistance in\nthe orbitalglassystatemight be drivenbythe H-induced\n”bond reinforcement” in the V spin clusters.\nFor the structural phase transition at T2, from Figs.\n1(a)-1(c), the selective observation of the elastic anomaly\nin the trigonal C44(T) suggests the occurrence of a trigo-\nnallatticedistortionat T2. Thusthestructuraltransition\natT2is expected to further enhance the trigonal crystal\nfieldsplittingoftheV3 dorbitals,whichispresentevenin\nthe cubic crystal structure for Co 1+xV2−xO4[Fig. 3(d)]\n[33,36,40]. Notably, for Co 1+xV2−xO4, Ref. [40] revealed\nthat the structural transition at T2causes a short-range7\n37.5\n37.4\n37.3179.8\n179.6\n179.4\n179.2\n70.6\n70.4\n70.2\n70.0\n69.8\n69.6\n90 80 70 60 50Ct (GPa) CB (GPa) C44 (GPa)H = 0\nH = 0\nH = 0\n T (K)(a) CB\n(b) Ct\n(C) C44 Eq. (1) \n Ct = 37.6 GPa \n EJT = 0.2 K \n θ = 30 K 09080706050229.8\n229.6\n229.4\n229.2C11 (GPa)C11 \nH = 0\nFIG. 4: (Color online) CΓ(T) of Co 1.21V1.79O4withH=\n0 inT2< T < T 1[from Figs. 1(a)-1(c)]. (a) CB(T), (b)\nCt(T), and (c) C44(T). The inset in (a) depicts C11(T) of\nCo1.21V1.79O4withH= 0 inT2< T < T 1[from the inset in\nFig. 1(a)]. The solid curve in (b) is a fit of the experimental\nCt(T) to Eq. (1) in 50 K < T <90 K. The values of the fit\nparameters are also listed in (b).\ndistortion of the O octahedra, but does not lower the\nglobal cubic symmetry of the crystal. Thus the C44(T)\nanomaly at T2should be driven by the short-range trig-\nonal lattice distortion. It should be noted that Refs. [39]\nand [40] suggestedthe existence ofthe orbitalglassystate\nattemperaturesofnotonly T2< T < T 1butalsoT < T 2.\nTaking into account that the localized electron character\nat low temperatures of T < T 2was revealedby the obser-\nvation of the spin gap in the inelastic neutron scattering\nexperiments [40], it is implied that the localized electron\ncharacter in the orbital glassy state is further enhanced\nbelowT2, which seems to give rise to the further en-\nhancement of the positive magnetoresistance below T2\n[Fig. 3(a)].\nFinally, we note that, in the orbital glassy phase in\nthe temperature range of T2< T < T 1,Ct(T) exhibits\nthe Curie-type softening, which is the JT effect expressed\nby Eq. (1). Figs. 4(a)-4(c) respectively depict CB(T),\nCt(T), andC44(T) inT2< T < T 1[from Figs. 1(a)-\n1(c)]. The Curie-type softening selectively observed in\nthetetragonal Ct(T)withEgsymmetryshouldarisefrom\nthe coupling of the ultrasound to the doubly-degenerate\ne′\ngorbitals [Fig. 3(d)]. Taking into account that the JT\neffect is driven by the coupling of the elastic deforma-\ntions to the localized magnetic moments, the presence\nof the Curie-type softening below T1is compatible with\nthe existence of the orbital glassy state below T1, wherethe V 3delectrons should become rather localized. In\nFig. 4(b), a fit of the experimental Ct(T) to Eq. (1)\nis depicted as a solid curve, which reproduces the ex-\nperimental data very well. Comparing the fit values of\nthe onsite JT energy EJTand the intersite orbital in-\nteraction θfor Co 1.21V1.79O4[Fig. 4(b)] with those for\nMgV2O4[44] and MnV 2O4[45], whileθforCo 1.21V1.79O4\n(30 K) is comparable to those for MgV 2O4(15 K) and\nMnV2O4(5 K),EJTfor Co 1.21V1.79O4(0.2 K) is much\nsmaller than those for MgV 2O4(10 K) and MnV 2O4(22\nK). Such a small value of EJTfor Co 1.21V1.79O4seems\nto be compatible with the rather delocalized character of\nthe V 3delectrons compared to MgV 2O4and MnV 2O4.\nBelowT2,CΓ(T) of Co 1.21V1.79O4exhibits the absence\nof the Curie-type softening, and instead exhibits an elas-\ntic moduli minimum at ∼30 K arising from the magnetic\nexcitations [Figs. 1(a)-1(c)]. For Co 1+xV2−xO4, the JT\nfluctuations might be suppressed below T2by the further\nstabilization of the orbital glassy state, which is accom-\npanied by the lattice distortion.\nAlthough the observation of the Curie-type soften-\ning inCt(T) inT2< T < T 1[Fig. 4(b)] indicates\nthe presence of the JT interaction, the e′\ngelectron [Fig.\n3(d)] is also expected to experience the onsite spin-\norbit interaction. The very small fit value of EJTfor\nCo1.21V1.79O4(0.2 K) [Fig. 4(b)] might reflect, in addi-\ntiontotheratherdelocalizedelectroncharacter,thecom-\npetition/coexistence of the onsite JT and spin-orbit in-\nteractions. Additionally, the disappearance of the Curie-\ntype softening in Ct(T) belowT2[Fig. 1(b)] might be\na result of the enhanced contribution of the spin-orbit\ninteraction below T2. For Co 1+xV2−xO4, the intricate\ninterplay of the JT, spin-orbit, and intersite spin-orbital\ninteractions is expected to play an important role for the\nemergence of the nearly-itinerant electron character and\nthe orbital glassy state, which remains to be understood.\nV. SUMMARY\nUltrasound velocity measurements of Co 1.21V1.79O4\nrevealed a variety of the elastic anomalies in both the\nparamagnetic phase ( T > T C) and the ferrimagnetic\nphase (T < T C). In the paramagnetic phase above TC,\nthe present study revealed the elastic-mode-dependent\nunusualtemperaturevariationsoftheelasticmoduli, sug-\ngesting the existence of the dynamic spin-cluster state.\nFurthermore, above TC, the present study revealed the\nsensitive magnetic-field response of the elastic moduli,\nsuggestingthat, withthenegativemagnetoresistance,the\nmagnetic-field-enhanced nearly-itinerant characterof the\nV 3delectrons emerges from the spin-cluster state, which\nshould be triggered by the inter-V-site interactions act-\ning on the orbitally-degenerate 3 delectrons. In the ferri-\nmagnetic phase below TC, the elastic anomalies at T1∼\n95 K and T2∼50 K were found to coincide, respec-\ntively, with the sign change of the magnetoresistance at\nT1(positive below T1) and the enhancement of the posi-8\ntive magnetoresistance below T2. These observations be-\nlowTCsuggest the successive occurrence of the orbital\nglassy order at T1and the structural phase transition\natT2, where the rather localized character of the V 3 d\nelectrons evolves below T1and is further enhanced be-\nlowT2. Further experimental and theoretical studies are\nindispensable if the spin and orbital states of the nearly-\nmetallic Co 1+xV2−xO4in both the magnetically disor-\ndered and ordered phases are to be understood.VI. ACKNOWLEDGMENTS\nThis work was partly supported by a Grant-in-Aid\nfor Scientific Research (C) (Grant No. 17K05520) from\nMEXT of Japan, and by Nihon University College of Sci-\nence and TechnologyGrants-in-Aidfor ProjectResearch.\n∗tadataka@phys.cst.nihon-u.ac.jp\n1P. G. Radaelli, New J. Phys. 7, 53 (2005).\n2Y. Ueda, N. Fujiwara, and H. Yasuoka, J. Phys. Soc. Jpn.\n66, 778 (1997).\n3M. Reehuis, A. Krimmel, N. Buttgen, A. Loidl, and A.\nProkofiev, Eur. Phys. J. B 35, 311 (2003).\n4S.-H. Lee, D. Louca, H. Ueda, S. Park, T. J. 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Tomiyasu, T. Yokobori, Y. Kousaka, R. I. Bewley, T.\nGuidi, T. Watanabe, J. Akimitsu, and K. Yamada, ibid.\n110, 077205 (2013)." }, { "title": "1808.04326v1.Application_of_two_sublattice_bilinearly_coupled_Heisenberg_model_to_the_description_of_certain_ferrimagnetic_materials.pdf", "content": "arXiv:1808.04326v1 [cond-mat.other] 13 Aug 2018Application of two-sublattice bilinearly coupled Heisenb erg model to\nthe description of certain ferrimagnetic materials\nHassan Chamati chamati@issp.bas.bg\nand Diana V. Shopova sho@issp.bas.bg Corresponding author\nAddress:Institute of Solid State Physics, Bulgarian Academy of Scie nces, 1784 Sofia,\nBulgaria\nKeywords; ferrimagnetism, Landau theory, Heisenberg model, ph ase diagram.\nPACS: 75.10.Dg, 71.70.Gm, 75.50.Gg\nAbstract\nWestudyphenomenologically onthebasisoftwobilinearlyc oupledHeisen-\nbergmodelsthephasediagramofsomeferrimagneticsubstan ces. Calculations\nare performed with the help of Landau energy obtained throug h applying the\nHubbard-Stratonovich transformation to the initial micro scopic Heisenberg\nHamiltonian. Thephase transitions within the model are of s econd order with\nthe emergence of a compensation point at lower temperatures for some values\nof parameters of the system. The main phase is a two-sublatti ce collinear\nferrimagnet but also a metastable non-collinear phase is pr esent within the\nexchange approximation presented here. The numerical resu lts give a detailed\ndescription of temperaturedependenceof magnetization on the strength of in-\ntersublattice interaction and the difference between the effec tive exchanges of\ntwo ferromagnetically ordered sublattices.\n1 Introduction\nFerrimagnets are substances made of various components having different magnetic\nproperties. Thedifferences inmagneticmoments leadtoageometric frustrationthat\nmay arise because either different elements occupy the lattice sites or the same ele-\nment occupies nonequivalent crystallographic sites surrounded by adifferent number\nor type of non-magnetic ions, which effectively results in different ma gnetic prop-\nerties. For complex alloys a combination of both may take place (For a n extensive\nreview see Ref. [1] and references therein). Within mean-field appr oach it is gen-\nerally accepted that ferrimagnets can be modeled with the help of se veral interpen-\netrating sublattices each ordered ferromagnetically with effective antiferromagnetic\ncoupling between them. In the pioneering works of N´ eel on ferrima gnetism within\n1the molecular ��eld approach (see e.g. [2]), a two sublattice model is u sed to compute\nthe thermal magnetization behaviour of ferrimagnets, and six pos sible magnetiza-\ntion curves were derived. Special attention there is paid to iron gar nets, where the\nspontaneous magnetization in comparison with experiment can be int erpreted by\napplying a three-sublattice model. In view of experimental study of magnetocaloric\neffect of rare-earth based ferrimagnets [3] which has great pote ntial for technologi-\ncal applications in environmentally-friendly refrigeration, the theo retical mean-field\ndescription of such alloys with three sublattices, is further elabora ted [4]. Such stud-\nies are based on considering microscopic classical Heisenberg models with different\nexchange and spin-orbit interactions depending on the crystal st ructure, chemical\ncomposition of the particular alloy under study.\nThere is another theoretical mean field approach based on conside ring mixed spin\nIsing model fordescription of ferrimagnets, see for example [5, 6, 7], where a very de-\ntailed review of the literature on this approach is presented. In the present paper we\nwill consider ferrimagnets which can be described by different magne tic ions sitting\non two interpenetrating sublattices in a body centred cubic struct ure. The interac-\ntion of ions on each sublattice is supposed ferromagnetic, while ions o n the different\nsublattices are coupled antiferromagnetically. The magnetic prope rties will be inves-\ntigated on the basis of bilinearly coupled Heisenberg classical model in a mean-field\napproximation which is treated using the Hubbard-Stratonovich tr ansformation for\nobtaining the respective Landau free energy and its analysis.\nThe rest of the paper is organized as follows: in Section 2 we describe in detail how\nwe calculate the Landaufreeenergy fromclassical Heisenberg mod el with competing\ninteractions on the basis of previously applied approach [8] for deriv ation of mean\nfield approximation. In Section 3 the solutions of equations of state obtained by\nthe minimization of Landau energy derived in Section 1 are discussed. Section 4\nsummarizes the results both in strong and weak-coupling limit for fer rimagnetic\nsubstances under consideration. Section 5 generalizes the conclu sions and possible\nfurther development of our study.\n2 The model and derivation of Landau free en-\nergy\nThe microscopic Heisenberg Hamiltonian which describes two coupled s ubsystems\nconsisting of classical spins with different magnetic properties which antiferromag-\n2netically between them through a bilinear term can be written in the fo llowing form\n:\nH=−1\n22N/summationdisplay\nij/bracketleftBig\nJ(1)\nijS(1)\ni·S(1)\nj+J(2)\nijS(2)\ni·S(2)\nj+2KijS(1)\ni·S(2)\nj/bracketrightBig\n.(1)\nHereS(1,2)\nj, aren-component classical Heisenberg spins whose magnitude is nor-\nmalized on the unit sphere in spin space through the condition |S(1,2)\nj|= 1. The\nexchange parameters J(1,2)\nij,Kijin the general case are N×Nsymmetric matrices\nwithN- the number of lattice sites considered equal for both subsystem s. This\ncondition simplifies the consideration as makes the system symmetric with respect\nto the interchange of the subsystems. The exchange matrices J(1,2)\nijdenote the inter-\naction between magnetic atoms of the same sort and Kij- between magnetic atoms\nof different sorts.\nThe above Hamiltonian may be applied to the description of magnetic sy stems\nwhich consist of two different magnetic materials and no matter what is the mi-\ncroscopic origin of this difference, it is effectively described by differe nt exchange\ninteractions within the two subsystems. There may be other situat ion when the\nsubstance is made only of one type of magnetic ions but they occupy two different\ncrystallographic positions in the Bravais lattice and are separated b y a number of\nnon-magnetic atoms. Such substance may also be considered as bu ilt of two mag-\nnetic subsystems with different exchange interactions within them.\nIn order to analyse the behaviour of magnetization and the phase t ransitions in sys-\ntems that can be described by the above microscopic Hamiltonian we h ave to find\nthe mean-field energy for the Hamiltonian (1) by calculating the part ition function\nwhich in this case is represented by functional integral in n-dimensio nal spin space,\nwheren- is the number of spin components. To do this we apply the Hubabrd\n-Stratonovich transformation; see for example [10] and the pap ers cited therein. We\nhave used this approach in [8] for ferromagnetic coupling between the two magnetic\nsubsystems where a detailed description of procedure is given. Her e we will just\noutline the important steps in the derivation of Landau free energy , especially in\nrelation of antiferromagnetic coupling between the subsystems.\nWe present the two interacting different magnetic subsystems on a body-centered\ncrystal lattice, for which the corners of elementary cube are occ upied by one sort of\nmagnetic atoms, and at the center of the cube atoms of different m agnetic sort are\nlocated. So the nearest neighbours belong to different magnetic su bsystems and the\nnext-nearest neighbour tosubsystems 1 and2, respectively. Th us thesystem maybe\ndescribed as two interpenetrating sublattices,consisting of differe nt magnetic atoms\nand we assume that the interaction within the sublattices J1,2\nijis ferromagnetic and\n3between them, Kij, it is antiferromagnetic. The Hubbard-Stratonovich transfor-\nmation renders the initial microscopic Hamiltonian in new n-component variables\nΨ(1)\ni,Ψ(2)\nidefined in real space, directly connected with the initial spins (see [8 ]),\nnamely:\nH=1\n22N/summationdisplay\nij/parenleftBig\nJ(1)\nijΨ(1)\ni·Ψ(1)\nj+J(2)\nijΨ(2)\ni·Ψ(2)\nj+2KijΨ(1)\ni·Ψ(2)\nj/parenrightBig\n(2)\n−ln/bracketleftBigg2N/summationdisplay\niIn/2−1(x(1)\ni)(x(1)\ni)\n2)−n/2Γ(n\n2)/bracketrightBigg\n−ln/bracketleftBigg2N/summationdisplay\niIn/2−1(x(2)\ni)(x2\ni\n2)−n/2Γ(n\n2)/bracketrightBigg\n.\nHereIn/2−1(x(1,2)\ni) is the modified Bessel function, and Γ(n\n2) is the Gamma function.\nIn the above expression the exchange parameters J(1,2)\nijandKijare connected to\nthose in the initial Hamiltonian (1) by the relations:\nJ(1,2)\nij=J(1,2)\nij\nT\nKij=Kij\nT(3)\nwithT- the temperature.\nWe denote by x(1)\niandx(2)\niin (2) the following expressions:\nx(1)\ni=/vextendsingle/vextendsingle/vextendsingleJ(1)\nijΨ(1)\nj+KijΨ(2)\nj/vextendsingle/vextendsingle/vextendsingle;x(2)\ni=/vextendsingle/vextendsingle/vextendsingleJ(2)\nijΨ(2)\nj+KijΨ(1)\nj/vextendsingle/vextendsingle/vextendsingle\nThe terms containing Bessel functions in (2) will be further used on ly in the form\nof expansion with respect to x(1,2)\niup to forth order by using the relation:\nΓ(n\n2)(2\nx)(n/2−1)In/2−1(x) = 1+∞/summationdisplay\nk=1(x2/4)k\nk!(k+n/2+1)(k+n/2−2)...n/2\nThe next step is to perform Fourier transformation to k-space , and pass to con-\ntinuum limit in kas the finite size effects will not be considered at this stage. The\nquadratic part of the obtained Hamiltonian again contains a bilinear te rm with re-\nspect toΨ(1,2)(k) andwe have todiagonaliseit. This isdonewiththehelp ofunitary\nmatrixˆS:\nˆS=/parenleftBigg\nS0(k)−S1(k)\nS∗\n1(k)S0(k)/parenrightBigg\n. (4)\nThe eigenvalues of the matrix ˆSread:\nλ1,2(k) =1\n2/bracketleftBig\nJ1(k)+J2(k)±/radicalbig\n(J1(k)−J2(k))2+4K(k)2/bracketrightBig\n, (5)\n4whereJ1,2(k) andK(k) are the Fourier transforms of J(1,2)\nijandKij, respectively. In\norder to compute the integral we use the steepest-descent met hod,i.e.the integra-\ntion contour is taken around the maxima of the eigenfunctions (5). The calculation\nfor bcc structure show that if we take the nearest neigbour inter action between\natoms of the same sort and the nearest neighbour interaction bet ween the atoms of\ndifferent sort, λ1,2(k) has a maximum in the centre of the Brillouin zone that gives\nferromagnetic ordering for the sublattices with antiferromagnet icK <0 interaction\nbetween them which is the focus of our consideration below. We shou ld note that\nλ1,2(k) has a maximum also at the border of the Brillouin zone k=π/a(ais the\nlattice constant) which supposes antiferromagnetically ordered s ublattices. There\nmay be also some local maximum inside the Brillouin zone, which may give so me\nincommensurate ordering within the sublattices, but this case is bey ond the scope\nof the present study.\nAfter performing the reverse Fourier transform to real space w e obtain the dimen-\nsionless Landau energy in the following form:\nF\nT=t1\n2− →ψ2\n1+t2\n2− →ψ2\n2+g\n4/bracketleftBig\n(− →ψ2\n1)2+(− →ψ2\n2)2/bracketrightBig\n+b\n2− →ψ2\n1− →ψ2\n2+b(− →ψ1·− →ψ2)2+w(− →ψ2\n2−− →ψ2\n1)(− →ψ1·− →ψ2).(6)\nThe coefficients of the Landau energy are expressed by the compo nents of the ma-\ntrix (4) and its eigenvalues (5) for k= 0:\nS0=1\nD/parenleftBig\nJ1−J2+/radicalbig\n(J1−J2)2+4K2/parenrightBig\n,\nS1=2K\nD(7)\nwhereDis introduced to satisfy the condition /bardblˆS/bardbl= 1, namely S2\n0+S2\n1= 1:\nD=√\n2/bracketleftbig\n(J1−J2)2+4K2/bracketrightbig1/4/bracketleftBig\nJ1−J2+/radicalbig\n(J1−J2)2+4K2/bracketrightBig1/2\n.(8)\nWe will write here the explicit expressions for the coefficients of landa u energy as we\nwill need them further in solving the mean field equations and discussio n of obtained\nresults;\nt1,2=1\nλ1,2−1\nn(9a)\ng=u\n2(S4\n0+S4\n1) (9b)\nb=uS2\n1S2\n0 (9c)\nw=u\n2S0S1(S2\n0−S2\n1); (9d)\n5hereu=n2(n+2) withn- the number of order parameter components. The real\nvector fields− →ψ1and− →ψ2in the Landau free energy (6) play the role of two coupled\norder parameters, and the averaged sublattice magnetizations a re related to them\nby the equations:\n− →m1=S0\nλ1− →ψ1−S1\nλ2− →ψ2 (10a)\n− →m2=S1\nλ1− →ψ1+S0\nλ2− →ψ2 (10b)\n3 Solving mean-field equations\nThe initial microscopic Hamiltonian is symmetric with respect to the rot ation of\nall spins through the same angle. The application of Hubbard-Strat onovich trans-\nformation for derivation of landau free energy , given in previous se ction, preserves\nthe symmetry of initial hamiltonian also with respect to field variables− →ψ1,2which\nmeans that we can find the magnitude and the mutual orientation be tween order pa-\nrameters− →ψ1,2but not their orientation with respect to crystallographic axes. Th is\nmaybedoneforparticularmagneticsubstancebyincluding intheinitia lmicroscopic\nHamiltonian terms accounting for the magnetic anisotropy. For pur e exchange inter-\nactions we can introduce the following notations [8]:− →ψ1i=|− →ψ1|βiand− →ψ2i=|− →ψ2|δi,\nwhere|− →ψ1|=ψ1,|− →ψ2|=ψ2are the magnitudes of the vector fields, and βi, δiare\nthe respective direction cosines, which fulfil the condition:\n3/summationdisplay\ni=1β2\ni= 1 and3/summationdisplay\ni=1δ2\ni= 1. (11)\nThe equations of state then will be:\n∂f\n∂Xi= 0,whereXi={ψ1,ψ2,βi,δi} (12)\nSolvingtheaboveequationswithrespect todirectioncosines βi, δigives twopossible\norientations between the vector fields− →ψ1,− →ψ2forK <0:\n1. The collinear phase with/summationtext\niβiδi=−1, that is,− →ψ1and− →ψ2are antiparallel, and\n2. Thenon-collinearphasewith/summationtext\niβiδi= 0, thatis,− →ψ1and− →ψ2areperpendicular.\nBelow we will discuss in detail the non-collinear phase 2. The angle betw een the\norder parameters− →ψ1and− →ψ2,i.e.,is;\n3/summationdisplay\niβ1δi=−w(ψ2\n2−ψ2\n1)\n2bψ1ψ2\n6and is defined only when ψ1/ne}ationslash= 0 andψ2/ne}ationslash= 0. ForK <0, the analysis shows\nthat the non-collinear phase exists only when the order parameter s− →ψ1and− →ψ2are\nof equal magnitudes, meaning that the order parameters− →ψ1and− →ψ2are mutually\nperpendicular. The magnitude ψ=ψ1=ψ2for the non-collinear phase in analytical\nform reads:\nψ2=−t1+t2\nu. (13)\nThen the sublattice magnetization magnitudes calculated using the a bove expres-\nsions for the non-collinear phase will be:\n|− →m1|=ψ/radicalBigg\nS2\n0\nλ2\n1+S2\n1\nλ2\n2, (14a)\n|− →m2|=ψ/radicalBigg\nS2\n1\nλ2\n1+S2\n0\nλ2\n2. (14b)\nNote that the sublattice magnetizations are not perpendicular but form an angle\n∠(− →m1,− →m2) =γwith each other, expressed by\ncos(γ) =S0S1(λ2\n1−λ2\n2)/radicalbig\n(S2\n0λ2\n2+S2\n1λ2\n1)(S2\n1λ2\n2+S2\n0λ2\n1).\nThe calculations show that this non-collinear phase for K <0 within the exchange\napproximation has no domain of stability. We should mention here that the free\nenergy (6) is very sensitive to the sign of interaction Kbetween the sublattices.\nWhenK >0,i.e., the interaction between the sublattices is ferromagnetic there is\nsmall domain in which the respective non-collinear phase is stable [9].\nFor antiparallel− →ψ1and− →ψ2it is obvious that the sublattice magnetizations (10) will\nbe also antiparallel. We may write the resulting equations for the magn itudes of the\norder parameters ψ1andψ2of the collinear phase and K <0 in the following form:\nt1ψ1+gψ3\n1+3bψ1ψ2\n2−wψ2(ψ2\n2−3ψ2\n1) = 0, (15a)\nt2ψ2+gψ3\n2+3bψ2\n1ψ2−wψ1(3ψ2\n2−ψ2\n1) = 0, (15b)\nwith the stability conditions given by:\nt1+3gψ2\n1+3bψ2\n2+3wψ1ψ2>0 (16)\n(t1+3gψ2\n1+3bψ2\n2+6wψ1ψ2)(t2+3gψ2\n2+3bψ2\n1−6wψ1ψ2)\n−9[w(ψ2\n1−ψ2\n2)+2bψ1ψ2]2≥0 (17)\nWe will make some remarks on the dependence of solutions of above s ystem on\nthe magnitude of exchange parameters J1, J2andK. WhenJ1<|K|,J2<|K|,\n7the leading interaction is determined by the antiferromagnetic coup ling between the\ntwo sublattices. This may be called a strong coupling limit for which the e igenvalue\nλ2(k= 0)\nλ2=1\n2/bracketleftBig\nJ1+J2−/radicalbig\n(J1−J2)2+4K2/bracketrightBig\n, (18)\nbecomes negative. This isequivalent totheinequality K2−J1J2>0. The coefficient\nt2in front ofψ2\n2becomes positive; see (9), and the field− →ψ2becomes redundant. The\nLandau free energy (6) will be:\n(F\nT)s=fs=t1\n2− →ψ12+g\n4(− →ψ12)2(19)\nThe minimization of above equation gives for− →ψ1the solution:\n(− →ψ1)2=−t1\ng(20)\nwhich exists and is stable for t1<0.\nThe sublattice magnetizations:\n− →m1=S0\nλ1− →ψ1 (21)\n− →m2=S1\nλ1− →ψ1\nwill be antiparallel as S1∼K/DandK <0. The phase described by the above\nequations will be presented by two antiparallel sublattices with differ ent magnitudes\nof sublattice magnetizations.\nIn the weak coupling limit for antiparallel configuration, i.e., whenJ1>|K|,J2>\n|K|, or equivalently J1J2> K2, the system of equations (15), together with the\nstability conditions (18), (19) should be solved. This can be done num erically and\nthe results will be presented in the next section.\n4 Results and discussion\nThe analytical result for sublattice magnetizations in the limiting case of strong\ncoupling (21) gives for the magnitude of total magnetization |− →M|=|− →m1+− →m2|the\nfollowing expression:\n|− →M|=S0|ψ1|\nλ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+S1\nS0/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nwith|ψ1|2, given by (20). The phase transition is obviously of second order an d\nthe total magnetization behaviour with temperature is smooth res embling the one\n8of Weiss ferromagnet with the exception that no saturation is reac hed forT= 0.\nAccording to the Neel’s classification of ferrimagnets, see [2], the c hange of magneti-\nzation with temperature in the strong coupling limit falls within R-type c urve. For\nexample, similar curve is obtained theoretically and compared with the respective\nexperiment for Y 3Fe5O12[4] where two sublattice model with strong antiferromag-\nnetic coupling is considered.\nIn the limiting case when J1=J2=Jthe relation will hold S0=−S1= 1/√\n2 and\nan antiferromagnetic structure with− →m1=−− →m2will appear, only if− →ψ2≡0 and\n|− →ψ1|2=−2t1/u. The transition temperature for antiferromagnetic ordering will b e\ngiven by:ta\nc= (J+|K|)/n.\nFurther we will present the numerical results for the temperatur e dependence of\nsublattice magnetizations and the total magnetization of the syst em in the weak-\ncoupling limit which we define here in the following way: J1>|K|andJ2>|K|.\nSuch a situation is present, for example in some ferrimagnetic compo unds like\nGdCo 12B6[11]. It is experimentally found that the exchange constants within s ub-\nlattices are ferromagnetic and larger than the antiferromagnetic coupling between\nthe sublattices; moreover there the magnetic anisotropy is small.\nExperiments for some R-T compounds where R is a rare earth elemen t and T is\na transition element, show that the exchange in the transition meta l sublattice is\nleading in magnitude, while the exchange in the rare-earth ion sublatt ice can be\nsafely ignored and considered as negligible. The intersublattice inter action is also\nsmall see, for example, DyFe 5Al7[12], ErFe 11TiH [13], RCo 2(R = Tb and Gd and R\n= Er, Ho, and Dy) [14]. In our notations the relation between the exc hange integrals\nin this case will be J1>|K| ≫ J 2, so this does not fall into our assumption of weak\ncoupling and will not be considered here.\nIn order to solve numerically the equations of state (15) for weak c oupling between\nsublattices we introduce the following dimensionless parameters.\nt=T\nJ1+J2, (22a)\nα=J1−J2\nJ1+J2, (22b)\nβ=K\nJ1+J2, (22c)\nwitht– the dimensionless temperature. In the above expression we have supposed\nthatJ1>J2which in view of symmetry in interchanging the sublattices does not\nlimit the consideration; then α>0 andβ <0 asK<0.\n9The weak coupling between the sublattices, namely J1>|K|andJ1>|K|may be\nexpressed by the parameters from (22) by the relation:\nα2+β2<1.\nThe parameter αis a measure for the difference in exchange parameters of the two\nsublattices and by its definition 0 <α<1.\nThe total magnetization of the system is the sum of sublattice magn etizations (10):\n− →M=− →m1+− →m2=S0+S1\nλ1− →ψ1+S0−S1\nλ2− →ψ2. (23)\nHereafter we will use the following notations for magnitudes of subla ttice magneti-\nzations and total magnetization both in the text and in figures:\nm1=|− →m1|;m2=|− →m2|;andM=|− →M|\nThecalculationsshowthatthephasetransitiontoorderedferrima gneticstateoccurs\nat temperature :\ntc=1\n6(1+/radicalbig\nα2+β2)\nwhich grows when either the difference between the exchange inter actions in sublat-\ntices grows, or when the antiferromagnetic coupling is bigger, or bo th. The phase\ntransition from disordered to ordered phase is of second order.\nWe want to note that within the exchange approximation used here f or the regime of\nweak coupling defined above with the decrease of temperature a co mpensation point\nappears no matter how small is the difference between the exchang e interactions of\nsublattices. At the compensation temperature tcomp, the sublattice magnetizations\n− →m1and− →m2areequalinmagnitudeandantiparallel, so M= 0. Therelationbetween\nthe order parameters magnitudes there is defined by:\nψ1=λ1\nλ2(S0−S1)\n(S0+S1)ψ2. (24)\nAs the calculations show ψ2< ψ1for all values of αandβ, butψ1grows with the\ndecrease of temperature in a monotonic way, while ψ2grows more rapidly. The\nquantity\nλ1\nλ2(S0−S1)\n(S0+S1)=(6tc)2\n1−α2−β2/parenleftBigg/radicalbig\nα2+β2−β\nα/parenrightBigg\nis always>1 asβ <0∼Kandα >0 so at some temperature tcomp< tc,\nand values of ψ1, ψ2the condition (24) is fulfilled. In the following figure we\nshow the change of net magnetization magnitude M(t) with the temperature for\n100.000.020.040.06\n0.00 0.05 0.10 0.15 0.20α = 0.1M\ntβ = - 0.08\nβ = - 0.11\nβ = - 0.13\nβ = - 0.20\nFigure 1: The dependence of net magnetization Mon reduced temperature tfor for\nfixedαand the different values of antiferromagnetic coupling β\n.\nα= 0.1,i.e.,J2= 0.83J1and different values of β. It is seen from Fig.1 that\nthe increase of antiferromagnetic coupling between the sublattice s slightly shifts the\ncompensation temperature to higher values, and M(t) grows more rapidly below\nthe compensation temperature and reaches higher values as t−→0. We suppose\nthat within the exchange approximation and in weak coupling limit the ke y factor\nfor the compensation point to appear is the weakness of antiferro magnetic exchange\nbetween the sublattices compared to the ferromagnetic exchang e of sublattices 1\nand 2 , respectively.\nWe will discuss in more detail the influence of difference between the m agnitudes\nof exchange interaction in the sublattices, represented by the pa rameterαonM(t)\nand sublattice magnetizations m1,andm2. Forα= 0.08,i.e.,J2= 0.85J1,M(t),\nm1,andm2are shown in Fig. 2. At tcthe transition is of second order and when\nlowering the temperature a compensation point tcompappears, which is located close\ntotc. Sublattice magnetizations change with temperature in monotonic w ay, and in\nthe temperature interval tcompm2, as expected as in sublattice 1 the exchange interaction J1>J2.\nBelowtcompthe magnetization of weaker sublattice m2becomes bigger than m1.\nFor intermediate values of α= 0.4,orJ2= 0.43J1, see Fig.3, with decrease of\n110.000.050.100.150.200.25\n0.00 0.05 0.10 0.15 0.20α = 0.08, β = -0.3\ntc tcomp.\ntm1m2M\nFigure 2: The dependence of net magnetization Mand sublattice magnetizations\nm1andm2on reduced temperature tfor for small difference between sublattice\nexchange parameters\n.\ntemperature below the compensation point the total magnetizatio n rapidly grows in\nnon-monotonic way as t→0, similar to V-curve according to Neel’s classification.\nThe behaviour of sublattice magnetizations with temperature is quit e different; m1\n, which is the sublattice magnetization with stronger exchange inter actionJ1grows\nin smooth way with decrease of temperature, while m2for weaker sublattice inter-\nactionJ2, below compensation point grows drastically in non-monotonic way an d\nin the temperature interval 0 |K| ≫ J 2, that is, when one of the\nsublattices is very weak.\n140.00.20.40.6\n0.000.050.100.150.200.250.30β = - 0.08M\ntα = 0.1\nα = 0.4\nα = 0.6\nα = 0.7\nFigure 5: The dependence of net magnetization Mfor fixed antiferromagnetic ex-\nchangeβand growing difference αbetween the ferromagnetic exchanges of the two\nsublattices\n.\nAcknowledgments\nThis work was supported by the Bulgarian National Science Fund und er contract\nDN08/18 (14.12.2017).\nReferences\n[1] R. 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Mater.\n439 (2017) 269. doi:10.1016/j.jmmm.2017.05.033 .\n17" }, { "title": "1705.02871v1.Non_local_magnon_transport_in_the_compensated_ferrimagnet_GdIG.pdf", "content": "Non-local magnon transport in the compensated ferrimagnet GdIG\nKathrin Ganzhorn,1, 2,\u0003Tobias Wimmer,1, 2Joseph Barker,3Gerrit E. W.\nBauer,3, 4, 5Zhiyong Qiu,4Eiji Saitoh,3, 4, 6, 7Nynke Vlietstra,1, 2Stephan Gepr ags,1\nRudolf Gross,1, 2, 8Hans Huebl,1, 2, 8and Sebastian T.B. Goennenwein1, 2, 8, 9, 10\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n4WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n5Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The Netherlands\n6CREST, Japan Science and Technology Agency, Tokyo 102-0076, Japan\n7Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\n8Nanosystems Initiative Munich, 80799 Munich, Germany\n9Institut f ur Festk orper- und Materialphysik, Technische Universit at Dresden, 01062 Dresden, Germany\n10Center for Transport and Devices of Emergent Materials,\nTechnische Universit at Dresden, 01062 Dresden, Germany\nWe study the di\u000busive transport of magnons through the compensated ferrimagnetic insulator\nGd3Fe5O12(GdIG). The magnons are injected and detected electrically in a non-local measurement\ncon\fguration via two parallel Pt strips deposited on top of the ferrimagnet. GdIG exhibits a\nrich magnon spectrum, with several thermally populated magnon bands at room temperature. We\nobserve a strong temperature and \feld dependence of the non-local voltage in the detector strip. Just\nbelow the magnetization compensation temperature we \fnd that the increasing magnetic \feld causes\nan unexpected enhancement of the non-local signal. A comparison with GdIG spin wave spectra\nobtained from atomistic modeling indicates that the thermal magnon population is important for\nunderstanding the non-local voltage signal.\nSeveral experimental and theoretical investigations of\ndi\u000busive magnon transport through the ferrimagnetic in-\nsulator yttrium iron garnet (Y 3Fe5O12, YIG) have re-\ncently been conducted, using two electrically isolated Pt\nstrips for electrical injection and detection [1{7]. Upon\nrunning a dc charge current through a Pt injector strip,\nan electron spin accumulation is generated at the YIG jPt\ninterface via the spin Hall e\u000bect (SHE) [8, 9]. This spin\naccumulation induces a non-equilibrium magnon accu-\nmulation in the magnet beneath the injector which dif-\nfuses away from the injector and is detected in a sec-\nond Pt strip via the inverse spin Hall e\u000bect (ISHE). This\nso-called non-local magnon mediated magnetoresistance\n(MMR) has been studied as a function of magnetic \feld\n[10] and temperature in YIG/Pt bilayers [2, 11], giving\ninsight into the length scales associated with magnon\ntransport. However, the microscopic mechanisms respon-\nsible for the MMR are still under investigation. One key\nquestion is which magnons or magnon bands provide the\nmost signi\fcant contributions to di\u000busive spin transport\nand the non-local voltage signal induced in the detector\nstrip. Furthermore, the MMR has only been measured in\ncollinear magnetic systems such as YIG or NiFe 2O4[12]\nso far. The in\ruence of non-collinear magnetic con\fgu-\nrations, such as a canted ferrimagnet, has not yet been\ninvestigated.\nHere, we study the non-local magnon transport in\nthe compensated ferrimagnet gadolinium iron garnet\n(Gd 3Fe5O12, GdIG), where the non-magnetic Y3+in\nYIG is substituted with magnetic Gd3+. GdIG there-\nfore consists of three magnetic sublattices: two antifer-romagnetically coupled Fe3+sublattices (FeA and FeD),\nand a Gd sublattice which is weakly antiferromagneti-\ncally coupled to the FeD moments [13, 14]. GdIG ex-\nhibits a magnetization compensation temperature Tcomp,\nwhere the remanent magnetization exactly vanishes due\nto the di\u000berent temperature dependencies of the antifer-\nromagnetically coupled net Fe and Gd sublattice magne-\ntizations [13]. Far away from the compensation temper-\nature, GdIG is a collinear ferrimagnet and the sublattice\nmagnetizations are all aligned (anti)parallel to the exter-\nnal magnetic \feld. Close to Tcomp, a canted phase can\nbe induced with magnetic \feld magnitudes accessible in\nexperiments, where the magnetic moments on the dif-\nferent sublattices are no longer collinear with the \feld,\nor one another [13{16]. The corresponding magnetic\nphase diagram can be found in Ref. [14]. Recently it was\nshown that the spin Hall magnetoresistance e\u000bect [17{20]\n(SMR) can electrically probe the interface magnetization\ntexture, since the SMR is sensitive to the orientation of\nthe individual sublattice magnetic moments relative to\nthe polarization of the spin accumulation in the adjacent\nPt [14, 21]. Non-local magnon transport or spin di\u000bu-\nsion has not yet been studied in a non-collinear magnetic\nstructure and will be presented in this Article. Aside\nfrom the magnetic compensation point, GdIG also di\u000bers\nfrom YIG in that the spin wave spectrum is richer in the\nlow THz regime. Several additional modes are present\ndue to the magnetic Gd sublattice. Local spin Seebeck\ne\u000bect (SSE) measurements in GdIG showed strong evi-\ndence that two spin wave modes with opposite polariza-\ntion dominate the thermal and dynamical behaviour ofarXiv:1705.02871v1 [cond-mat.mes-hall] 8 May 20172\nthe spin system, resulting in a sign change of the SSE\nvoltage [22, 23]. GdIG is therefore a good material to\ninvestigate the in\ruence of di\u000berent magnon modes and\ntheir polarization on spin di\u000busion. Our experimental\nresults suggest that the complex magnon spectrum of\nGdIG indeed qualitatively impacts the \feld and temper-\nature dependence of the MMR.\nThe investigated sample is a 2 :6µm thick GdIG \flm\ngrown on top of a (111)-oriented Gd 3Ga5O12substrate\nvia liquid phase epitaxy (LPE). After cleaning in Pi-\nranha solution and annealing at 500\u000eC in an oxygen at-\nmosphere of 75 µbar for 40 min, a 10 nm thick Pt \flm\nwas deposited onto the GdIG via electron beam evapo-\nration (see Refs. 2 and 24 for details). Two Pt strips\nwith edge-to-edge separation d= 200 nm and strip width\nw= 500 nm (Fig. 1 (a)) were de\fned by electron beam\nlithography followed by Ar ion etching. The sample was\nmounted in the variable temperature insert of a super-\nconducting magnet cryostat allowing dc transport mea-\nsurements at temperatures between 5 K < T < 300 K\nand magnetic \felds up to 15 T. For the electric trans-\nport measurements a charge current Ic= 100 µA (cor-\nresponding to a current density of 2 \u00021010A m\u00002) was\napplied to the left strip using a Keithley 2400 sourceme-\nter. The local ( Vloc) and non-local voltage drop ( Vnl) in\nthe injector and detector strip were simultaneously mea-\nsured with two Keithley 2182 nanovoltmeters (see Fig. 1\n(a)). To eliminate thermal signals due to Joule heat-\ning in the local strip (e.g. spin Seebeck e\u000bect [25]) and\nto increase the signal-to-noise ratio, a current switching\nmethod was used, as described in Ref. [2]. To obtain\nthe SMR and MMR amplitude, angle dependent mag-\nnetoresistance (ADMR) measurements were carried out\nby rotating the external magnetic \feld of constant mag-\nnitude in the thin \flm plane, while recording Vlocand\nVnl.\nThe local response Vlocmeasured at 300 K as a func-\ntion of the angle \u000bbetween IcandHis shown in\nFig. 1 (b) for \u00160H= 1 T (grey) and 7 T (green). From\nthese ADMR measurements, the SMR amplitude de\fned\nas \u0001Vloc=Vloc;min= (Vloc(0\u000e)\u0000Vloc(90\u000e))=Vloc(90\u000e) was\nextracted and plotted as a function of temperature for\nmagnetic \feld strengths \u00160H= 1;3;5 and 7 T in Fig. 1\n(d). Cooling to lower temperatures, the SMR ampli-\ntude decreases by a factor 2 compared to room tem-\nperature, consistent with measurements in YIG/Pt [26]\nand (In, Y) doped GdIG/Pt bilayers [14]. In a narrow\ntemperature range around the compensation tempera-\nture (Tcomp = 268 K determined via SQUID magnetom-\netry), the SMR decreases to zero. A similar behaviour\nhas been observed in (In, Y) doped GdIG [14] and was\nattributed to the formation of the canting phase in GdIG\nclose to the compensation temperature, where the sublat-\ntice magnetizations are no longer collinear. In Ref. [14] a\nsign change of the SMR was observed and interpreted as a\nperpendicular alignment of the sublattice moments with\n090180270360-1.50E-007-1.00E-007-5.00E-0080.00E+0005.00E-0080901802703600.66440.6646\n0501001502002503000.00E+000-5.00E-008-1.00E-007-1.50E-007-2.00E-007\n0 100 200 300\nTemperature (K) /uni0394Vloc /Vloc, min (x 10-4)\n0246\n1T\n3T\n5T\n7T200\n100150\n050/uni0394Vnl (nV )\n0 100 200 3000.00E+000-5.00E-008-1.00E-007-1.50E-007-2.00E-007\n0 100 200 3000.00E+000-5.00E-008-1.00E-007-1.50E-007-2.00E-007\n0 100 200 3000.00E+000-5.00E-008-1.00E-007-1.50E-007-2.00E-007\n0 100 200 3000.00E+000-5.00E-008-1.00E-007-1.50E-007-2.00E-0079T\n11T\n13T\n15T(a)\n(c)\n50 150 250TcrossTcomp0° 90° 180° 270° 360°664.4664.6Vloc (mV)\n0\n-50\n-100Vnl (nV)\n(d)\n(e)α\nVloc+\n-Ic+\n- Vnl+\n-(b)\nH\n α\n7T7T1T\n1T\nα (deg)026\n260 280\nTemperature (K) /uni0394Vloc /Vloc, min\n(x 10-4)4FIG. 1. (a) Schematic representation of the GdIG(green)/Pt\n(grey) nanostructures consisting of two parallel Pt strips of\nwidthwand edge-to-edge separation d. (b) Local voltage\nVlocas a function of the angle \u000bbetween IcandHat 300 K.\nDue to the weak \feld dependence of the local signal only\nexperimental data taken at \u00160H= 1 T and 7 T are shown.\n(c) Non-local voltage Vnlas a function of \u000bat 300 K for\n\u00160H= 1;3;5 and 7 T. (d) Temperature dependent SMR am-\nplitude (Vloc(0\u000e)\u0000Vloc(90\u000e))=Vloc(90\u000e) extracted from the\nlocal voltage amplitude for di\u000berent magnetic \felds up to\n7 T. The inset is a close-up of the SMR amplitude around\nthe compensation temperature Tcomp = 268 K (red vertical\nline). (e) Temperature dependence of the MMR amplitude\n\u0001Vnl=Vnl(0\u000e)\u0000Vnl(90\u000e) obtained from ADMR measure-\nments for magnetic \felds up to 15 T.\nrespect to the external \feld. The absence of such a sign\nchange in the present data may be attributed to magnetic\ndomain formation for temperatures close to Tcomp, such\nthat on average the SMR modulation vanishes. How-\never the broadening of the dip in the SMR vs. temper-3\nature curves with increasing magnetic \feld (see inset of\nFig. 1 (d)) is in good agreement with the observations in\nRef. [14] and is expected for the magnetic canting phase\nin GdIG [15]. As reported in Ref. [14], in the collinear\nphase the SMR in Fig. 1 (d) hardly depends on the ap-\nplied magnetic \feld. This is consistent with the estab-\nlished SMR model [18, 27], where only the orientation of\nthe sublattice moments is relevant [14] [28].\nThe angular dependence of the non-local response Vnl\nat 300 K is shown in Fig. 1 (c) for di\u000berent magnetic \feld\nstrengths. The non-local voltage signature has the same\ndependence on the external magnetic \feld orientation\nas in previous MMR measurements in YIG/Pt nanos-\ntructures for the same wiring scheme [2]. The MMR ef-\nfect amplitude extracted as \u0001 Vnl=Vnl(0\u000e)\u0000Vnl(90\u000e) =\n\u0000Vnl(90\u000e)>0 is plotted as a function of temperature in\nFig. 1 (e) for \felds up to 7 T. In contrast to the local\nSMR, where no substantial \feld dependence is observed\nin the collinear phase, the non-local MMR signal displays\na more complex \feld and temperature dependence. We\nhave therefore taken additional non-local data in \felds\nup to 15 T for temperatures between 150 K and 300 K,\nas shown in Fig. 1 (e). We \frst focus on the data taken\nat 1 T (grey open squares in Fig. 1 (e)). At low tem-\nperatures \u0001 Vnlvanishes, similar to recent observations\nin YIG/Pt [2] where a T3=2dependence of \u0001 Vnlwas ob-\nserved. This temperature dependence in YIG is in agree-\nment with theoretical expectations based on the magnon\ndensity of states and distribution function [3, 4, 29],\ni.e. general properties of the magnonic system. At low\ntemperatures, we therefore expect GdIG to behave very\nsimilar to YIG.\nWith increasing temperature, the non-local signal mea-\nsured at 1 T increases up to 150 nV just below the com-\npensation temperature. The MMR amplitude then drops\nbelow the experimental resolution of about 5 nV [2] at\nTcomp and recovers a \fnite value above Tcomp, similar\nto what is observed in the local SMR (see Fig. 1 (d)).\nWe attribute the vanishing MMR signal at Tcomp to the\nchange of the magnetic structure of GdIG into the canted\nphase. A vanishing non-local signal in the canted phase\nsuggests two possible scenarios, either (i) the magnon\ninjection/detection becomes ine\u000ecient due to the non-\ncollinear alignment of the magnetic sublattice moments\nand/or (ii) the damping close to the compensation point\nis enhanced [30] leading to shorter magnon lifetimes and\nthus shorter di\u000busion lengths. Additional magnon scat-\ntering e\u000bects may arise due to magnetic domain forma-\ntion in the canting phase as discussed above, which fur-\nther suppress magnon transport.\nWe now turn to the magnetic \feld dependence of the\nMMR e\u000bect. The relative \feld dependence \u000e(\u00160H) of the\nMMR amplitude \u0001 Vnlis de\fned as\n\u000e(\u00160H) =\u0001Vnl(\u00160H)\u0000\u0001Vnl(1 T)\n\u0001Vnl(1 T)(1)\n050100150200250300\n1.00.50.0-0.5\nRelativesuppression\n7T\n15T\n0 100 200 300\nTemperature (K)50 150 2500\n-10050\n-50Tcomp\nTcross7T\n15TRelative suppression \nδ(µ0H) (%)FIG. 2. Relative magnetic \feld dependence \u000e(7 T) (green tri-\nangles) and \u000e(15 T) (purple hexagons) of the MMR e\u000bect de-\n\fned by Eq. (1). A negative (positive) value corresponds to\na suppression (enhancement) with increasing \feld. The grey\nshaded region indicates the temperature range around Tcomp\nin which the MMR drops to values close to or even below the\nnoise level, impeding a reliable quanti\fcation of \u000e.\nand plotted as a function of temperature in Fig. 2 for\n\u00160H= 7 T (green triangles). A negative value corre-\nsponds to a suppression of the non-local voltage, while\na positive value represents an increase of the MMR with\nincreasing magnetic \feld. The \feld dependence is calcu-\nlated relative to the measurements at 1 T, which is the\nlowest magnetic \feld studied here. We distinguish three\ntemperature regimes: (i) for 0 < T < T cross\u0019210 K,\nthe MMR amplitude is suppressed with applied mag-\nnetic \feld by about \u000e(7 T) = 25%, (ii) in the range\nTcross< T < T comp an enhancement with \feld is ob-\nserved, while (iii) for Tcomp\u0014 j;\u0014m, leading to cavity-magnon\npolaritons [15–22]. The Rabi frequency \n =p\n5\n4\r0p\nNB 0[24]\ndenotes the coupling strength between the magnon mode and\nits driving magnetic field with frequency !0and amplitude B0,\nwhere the total number of spins N=\u001aVwith the spin density\nof YIG\u001a=4:22\u00021027m\u00003and the volume of the sphere V.\nFor convenience, we switch to the rotating frame with re-\nspect to the drive frequency !0, and by including input noises\nand dissipations of the system, we obtain the following quan-\ntum Langevin equations (QLEs)\n˙aj=\u0000i\u0001jaj\u0000igjm\u0000\u0014jaj+q\n2\u0014jain\nj;(j=1;2)\n˙m=\u0000i\u0001mm\u0000iX\nj=1;2gjaj\u0000iG0mq+ \n\u0000\u0014mm+p\n2\u0014mmin;\n˙q=!bp;\n˙p=\u0000!bq\u0000G0mym\u0000\rp+\u0018;\n(2)where \u0001j=!j\u0000!0,\u0001m=!m\u0000!0,\ris the mechanical\ndamping rate, and ain\nj,minare input noise operators with\nzero mean value acting on the cavity and magnon modes,\nrespectively, which are characterized by the following corre-\nlation functions [32]: hain\nj(t)ainy\nj(t0)i=\u0002Nj(!j)+1\u0003\u000e(t\u0000t0),\nhainy\nj(t)ain\nj(t0)i=Nj(!j)\u000e(t\u0000t0), andhmin(t)miny(t0)i=\u0002Nm(!m)+1\u0003\u000e(t\u0000t0),hminy(t)min(t0)i=Nm(!m)\u000e(t\u0000t0).\nThe Langevin force operator \u0018, accounting for the Brownian\nmotion of the mechanical oscillator, is autocorrelated as\nh\u0018(t)\u0018(t0)+\u0018(t0)\u0018(t)i=2'\r\u00022Nb(!b)+1\u0003\u000e(t\u0000t0), where\na Markovian approximation has been taken valid for a large\nmechanical quality factor Qm=!b=\r\u001d1 [33]. The equi-\nlibrium mean thermal photon, magnon, and phonon numbers\nareNk(!k)=h\nexp\u0010~!k\nkBT\u0011\n\u00001i\u00001(k=1;2;m;b), with kBthe\nBoltzmann constant and Tthe environmental temperature.\nBecause the magnon mode is strongly driven, it has a large\namplitudejhmij\u001d 1, and further owing to the cavity-magnon\nbeamsplitter interactions the two cavity fields are also of large\namplitudes. This allows us to linearize the system dynamics\naround semiclassical averages by writing any mode operator\nas a c-number plus its fluctuation operator O=hOi+\u000eO,\n(O=aj;m;q;p), and neglecting small second-order fluctua-\ntion terms. Substituting those linearized mode operators into\nEq. (2), the equations are then separated into two sets of equa-\ntions, respectively, for semiclassical averages and for quan-\ntum fluctuations. The solutions of the averages are obtained,\nwhich arehpi=0,hqi=\u0000G0\n!bjhmij2,haji=\u0000igj\ni\u0001j+\u0014jhmi, andhmi\nis given by\nhmi=\n\n(i\u00011+\u00141)(i\u00012+\u00142)\n(i˜\u0001m+\u0014m)(i\u00011+\u00141)(i\u00012+\u00142)+g2\n1(i\u00012+\u00142)+g2\n2(i\u00011+\u00141);(3)\nwith ˜\u0001m= \u0001 m+G0hqithe e\u000bective detuning of the magnon\nmode including the frequency shift caused by the magne-\ntostrictive interaction. It takes a simpler form\nhmi'i\n\u0001 1\u00012\n\u0000˜\u0001m\u00011\u00012+g2\n1\u00012+g2\n2\u00011; (4)\nwhenj\u0001jj;j˜\u0001mj \u001d\u0014j;\u0014m. Let us introduce the quadratures\nof the quantum fluctuations ( \u000eX1;\u000eY1;\u000eX2;\u000eY2;\u000ex;\u000ey;\u000eq;\u000ep),\nwhere\u000eXj=(\u000eaj+\u000eay\nj)=p\n2,\u000eYj=i(\u000eay\nj\u0000\u000eaj)=p\n2,\u000ex=\n(\u000em+\u000emy)=p\n2, and\u000ey=i(\u000emy\u0000\u000em)=p\n2, and the input\nnoise quadratures are defined in the same way. The QLEs of\nthe quadrature fluctuations can be cast in the matrix form\n˙u(t)=Au(t)+n(t); (5)\nwhere u(t)=\u0002\u000eX1(t);\u000eY1(t);\u000eX2(t);\u000eY2(t);\u000ex(t);\u000ey(t);\u000eq(t);\u000ep(t)\u0003T,\nn(t)=\u0002p2\u00141Xin\n1(t);p2\u00141Yin\n1(t);p2\u00142Xin\n2(t);p2\u00142Yin\n2(t);p2\u0014mxin(t);p2\u0014myin(t);0;\u0018(t)\u0003Tis the vector of noises entering the sys-3\ntem, and the drift matrix Ais given by\nA=0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@\u0000\u00141\u00011 0 0 0 g1 0 0\n\u0000\u00011\u0000\u001410 0\u0000g10 0 0\n0 0\u0000\u00142\u00012 0 g2 0 0\n0 0\u0000\u00012\u0000\u00142\u0000g20 0 0\n0 g1 0 g2\u0000\u0014m˜\u0001m\u0000G0\n\u0000g10\u0000g20\u0000˜\u0001m\u0000\u0014m0 0\n0 0 0 0 0 0 0 !b\n0 0 0 0 0 G\u0000!b\u0000\r1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA;(6)\nwhere G=ip\n2G0hmiis the e \u000bective magnomechanical cou-\npling rate. By using the result of Eq. (4), we obtain\nG'p\n2G0\n\u0001 1\u00012\n˜\u0001m\u00011\u00012\u0000g2\n1\u00012\u0000g2\n2\u00011; (7)\nwhich shows that the coupling can be significantly enhanced\nwith a strong magnon drive.\nWe are interested in the quantum correlation of two MW\nfields in the steady state. The steady state of the system is a\nfour-mode Gaussian state due to the linearized dynamics and\nthe Gaussian nature of input noises. Such a state is fully char-\nacterized by an 8\u00028 covariance matrix (CM) Cwith its entries\ndefined asCi j(t)=1\n2hui(t)uj(t0)+uj(t0)ui(t)i(i;j=1;2;:::;8).\nIt can be obtained straightforwardly by solving the Lyapunov\nequation [34]\nAC+CAT=\u0000D; (8)\nwhereD=diag\u0002\u00141(2N1+1);\u00141(2N1+1);\u00142(2N2+1);\u00142(2N2+\n1);\u0014m(2Nm+1);\u0014m(2Nm+1);0;\r(2Nb+1)\u0003is the di \u000busion ma-\ntrix, whose entries are defined through hni(t)nj(t0)+\nnj(t0)ni(t)i=2=Di j\u000e(t\u0000t0).\nWe adopt the logarithmic negativity [35] to quantify the en-\ntanglement between the two MW cavity fields. It is a full\nentanglement monotone under local operations and classical\ncommunication [36] and an upper bound for the distillable\nentanglement [35]. The logarithmic negativity for Gaussian\nstates is defined as [37]\nEN:=max\u00020;\u0000ln 2˜\u0017\u0000\u0003; (9)\nwhere ˜\u0017\u0000=min eigji\n2˜Cmwj(with the symplectic matrix\n\n2=\b2\nj=1i\u001byand the y-Pauli matrix \u001by) is the minimum sym-\nplectic eigenvalue of the CM ˜Cmw=PC mwP, withCmwthe\nCM of the two MW fields, which is obtained by removing in\nCthe rows and columns associated with the magnon and me-\nchanical modes, and P=diag(1;\u00001;1;1) is the matrix that\nperforms partial transposition on CMs [38].\nMW entanglement and its detection . In Fig. 2 we present\nthe main results of the entanglement between two MW cav-\nity fields. The stationary entanglement is guaranteed by the\nnegative eigenvalues (real parts) of the drift matrix A. Fig-\nure 2(a) shows clearly that the maximum entanglement is\nachieved when the two cavity fields are respectively reso-\nnant with the two mechanical sidebands [see Fig. 1(b)], i.e.,\n-2-1012-2-1012Δ\n1/ωbΔ2/ωb0\n0.0350.0700.110.140\n.00.30.60.91.21.50.61.21.82.43.0(b)( a)κ\n2/κ1g2/g10\n0.0450.0900.130.18FIG. 2: (a) Density plot of the entanglement ENbetween two MW\ncavity fields vs (a) \u00011and\u00012, (b)\u00142=\u00141andg2=g1(\u00141,g1are fixed).\nWe take ˜\u0001m=0:9!b,\u00142=\u00141,g2=g1in (a), and \u00011=\u0000\u00012=!bin\n(b). See text for the other parameters.\n\u00011=\u0000\u00012'\u0006!b, where “\u0006” sign is taken due to the sym-\nmetry of the two cavity fields. And the magnon mode reso-\nnant with the blue sideband ˜\u0001m'!bcorresponds to the anti-\nStokes process, which significantly cools the phonon mode,\nthus eliminating the main obstacle for observing entangle-\nment [24]. We have employed experimentally feasible param-\neters [23]:!m=2\u0019=10 GHz,!b=2\u0019=10 MHz,\r=2\u0019=102\nHz,\u0014m=2\u0019=\u00141=2\u0019=1 MHz, g1=2\u0019=3:8 MHz, G=2\u0019=4:5\nMHz, and T=20 mK. We use a strong magnon-phonon cou-\npling G> \u0014 mto create magnomechanical entanglement. This\nmeans that a strong magnon driving field should be used, and\nin order to avoid unwanted magnon Kerr e \u000bect [39, 40] the\nbare coupling rate G0should not be too small [24, 25]. For the\noptimal case \u00011=\u0000\u00012'\u0006!bin Fig. 2(a), a driving power\nof 6.3 mW (0.57 mW) should be used to yield G=2\u0019=4:5\nMHz for G0=2\u0019=0:3 Hz (1 Hz), while keeping the Kerr ef-\nfect negligible. In Fig. 2(b), we analyse the optimal coupling\nrates g1;2and decay rates \u00141;2, and find that in both situations\n\u00011=\u0000\u00012'!band\u0000!b, close coupling rates should be\nused, and the cavity that is resonant with the red (blue) me-\nchanical sideband should have a smaller (larger) decay rate\nthan the other. Such an asymmetric feature is due to the dif-\nferent roles of the two sidebands. The entanglement is robust\n0.000 .050 .100 .150 .200.00.10.2 \n ENT\n (K)\nFIG. 3: MW entanglement ENvs temperature T. The parame-\nters are those with which the maximum entanglement is achieved\nin Fig. 2(b).4\n-2-1012-2-1012Δ\n1/ωbΔ2/ωb1\n.861.891.921.941.972.002\n4680.51.01.52.02.5(b)( a)κ\n1/2π (ΜΗz)κ2/2π (ΜΗz)1\n.801.841.881.921.962.00\nFIG. 4: Density plot of h\u000eX2\n+i+h\u000eY2\n\u0000ivs (a) \u00011and\u00012, (b)\u00141and\u00142.\nThe blank areas denote h\u000eX2\n+i+h\u000eY2\n\u0000i>2. The parameters are the\nsame as in Fig. 2(a), and we take optimal detunings \u00011=0:9!band\n\u00012=\u00001:1!bin (b).\nagainst environmental temperature and survives up to \u0018140\nmK, as shown in Fig. 3, below which the average phonon\nnumber is always smaller than 1, showing that mechanical\ncooling is thus a precondition to observe quantum entangle-\nment in the system [24]. The generated MW entanglement\ncan be detected by measuring the CM of two cavity output\nfields. Such measurement in the MW domain has been real-\nized in the experiments [12, 41].\nAlternatively, one can also verify the entanglement by using\nthe Duan criterion [42], which requires simpler experimental\noperations, i.e., one does not have to measure all the entries\nof the 4\u00024 CM, but measure only two collective quadra-\ntures [43]. Specifically, a su \u000ecient condition for entangle-\nment is that the two collective quadratures satisfy the inequal-\nity\nh\u000eX2\n+i+h\u000eY2\n\u0000i<2; (10)\nwhere X+=X1+X2, and Y\u0000=Y1\u0000Y2. Figure 4(a) shows\nthat in two areas around \u00011=\u0000\u00012'\u0006!bthe inequality is\nfulfilled, indicating that the two cavity fields are entangled.\nExperimental implementations . We now discuss possible\nconfigurations that could realize the proposal. Two MW cav-\nXY\n45°Loop antenna\nFIG. 5: A YIG sphere is placed near the maximum magnetic fields\nof two MW cavity fields in a cross-shape cavity. A loop antenna at\nthe end of a superconducting MW line is used to drive the magnon\nmode [39].ities and each cavity containing a cavity mode are preferred.\nIn this situation, the frequencies of the cavity fields can be ad-\njusted flexibly to match the two mechanical sidebands. The\ntwo cavities could be placed perpendicularly in the horizon-\ntal plane with the YIG sphere located in the intersection (near\nthe maximum magnetic fields) of the cavity fields. This can be\nrealized in a planar cross-shape cavity [44] or coplanar waveg-\nuide [45], see Fig. 5. Taking the “X”-cavity [44] as an exam-\nple, one can set the bias magnetic field along the z(vertical)\ndirection, the magnetic fields of two cavity modes along the\nxandydirection, respectively, and the driving magnetic field\nin the x-yplane and of e.g., 45 degrees with both the xand\nydirection. For directly driving the magnon mode, one may\nadopt a superconducting MW line with a small loop antenna\nat its end [39]. In this case, the loop antenna will also cou-\nple to the two cavity modes leading to increased cavity de-\ncay rates. However, owing to its relatively small dimension\ncompared with the cavity setup the influence is only moder-\nate [46]. Besides, the cross configuration of the cavity may\nalso reduce the Qfactor of the cavities induced by the damage\nto boundary conditions. Taking into account the aforemen-\ntioned e \u000bects, we study the Duan criterion for taking larger\ncavity decay rates in Fig. 4(b). It shows that with much larger\ndecay rates, the two cavity fields are still entangled. Given\nthe flexibility of the cavity resonant frequencies, the mechan-\nical frequency can be freely chosen in a large range (always\nkeeping it much smaller than the magnon frequency). The\nresults presented in this work employed a \u001810 MHz mechan-\nical mode of a 250- \u0016m-diameter YIG sphere [23]. For such\na large sphere, the bare magnomechanical coupling is small,\nbut it can be increased by using a smaller sphere such that the\npump power required is reduced, which can weaken both the\nunwanted nonlinear e \u000bect and the by-e \u000bect of the coupling of\nthe loop antenna to the cavity modes.\nConclusions. We present a new mechanism for creating\nMW entangled states based on magnetostrictive interaction in\na ferrimagnetic YIG sphere. The mechanism makes use of\nthe nonlinearity of such a magnomechanical interaction. The\nentanglement is in the steady state and robust against cavity\ndissipations and environmental temperature. We show strate-\ngies to detect the entanglement and a possible configuration\nthat is promising to realize the proposal. We analyse in de-\ntail various practical imperfections which would help future\nexperimental realizations. This work may find applications in\nquantum information science, quantum metrology, and quan-\ntum tasks that require entangled CV MW fields.\nAcknowledgments . We thank Junjie Liu and Yi-Pu Wang\nfor fruitful discussions on potential experimental realiza-\ntions. This work has been supported by the National Key\nResearch and Development Program of China (Grants No.\n2017YFA0304200 and No. 2017YFA0304202), the Royal\nSociety Newton International Fellowship (NF170876) of UK,\nand the European Research Council project (ERC StG Strong-\nQ, 676842).5\n\u0003j.li-17@tudelft.nl\n[1] C. H. Bennett et al ., Phys. Rev. Lett. 70, 1895 (1993); D.\nBouwmeester et al., Nature 390, 575 (1997); A. Furusawa et\nal., Science 282, 706 (1998).\n[2] V . Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics 5,\n222 (2011).\n[3] B. Hensen et al., Nature 526, 682 (2015); M. Giustina et al.,\nPhys. Rev. Lett. 115, 250401 (2015); L. K. Shalm et al., Phys.\nRev. Lett. 115, 250402 (2015).\n[4] Z. Y . Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys.\nRev. Lett. 68, 3663 (1992); P. G. Kwiat et al., Phys. Rev. Lett.\n75, 4337 (1995).\n[5] J. E. Sharping, M. Fiorentino, and P. Kumar, Opt. Lett. 26, 367\n(2001); X. Li, P. L. V oss, J. E. Sharping, and P. 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Antennas Propag. 57, 2972\n(2009).\n[46] Yi-Pu Wang (private communication)." }, { "title": "2005.09864v1.Frequency_mixing_in_a_ferrimagnetic_sphere_resonator.pdf", "content": "arXiv:2005.09864v1 [quant-ph] 20 May 2020Frequency mixing in a ferrimagnetic sphere resonator\nCijy Mathai,1Sergei Masis,1Oleg Shtempluck,1Shay Hacohen-Gourgy,2and Eyal Buks1\n1Andrew and Erna Viterbi Department of Electrical Engineeri ng, Technion, Haifa 32000 Israel\n2Physics Department, Technion, Haifa 32000 Israel\n(Dated: May 21, 2020)\nFrequency mixing in ferrimagnetic resonators based on yttr ium and calcium vanadium iron gar-\nnets (YIG and CVBIG) is employed for studying their nonlinea r interactions. The ferrimagnetic\nKittel mode is driven by applying a pump tone at a frequency cl ose to resonance. We explore\ntwo nonlinear frequency mixing configurations. In the first o ne, mixing between a transverse pump\ntone and an added longitudinal weak signal is explored, and t he experimental results are compared\nwith the predictions of the Landau-Zener-Stuckelberg mode l. In the second one, intermodulation\nmeasurements are employed by mixing pump and signal tones bo th in the transverse direction for\nstudying a bifurcation between a stable spiral and a stable n ode attractors. Our results are appli-\ncable for developing sensitive signal receivers with high g ain for both the radio frequency and the\nmicrowave bands.\nI. INTRODUCTION\nThe physics of magnons in ferromagnetic resonators\n[1–3] has been extensively studied in the backdrop of\nBose-Einstein condensation [4], optomagnonics [5–7], and\nspintronics [8]. Owing to the high magnon life time of\nthe order of a few microseconds, such ferromagnetic in-\nsulators have become the natural choice of microwave\n(MW) resonators in synthesizers [9], narrow band filters\n[10], and parametric amplifiers [11]. Exploring the non-\nlinearity associated with such systems is gaining atten-\ntion. A variety of magnon nonlinear dynamical effects\nhave been studied in the context of auto-oscillations [12],\noptical cooling [13], frequency mixing [14, 15] and bista-\nbility [16–20]. Applications of nonlinearity for quantum\ndata processing have been explored in [21, 22]. Nonlin-\near interactions between the electromagnetic (EM) MW\nfield coherent photons and these resonators can be sig-\nnificantly enhanced with relatively low power around the\nresonance frequency of the oscillator. Studying such non-\nlinear interactions is important due to the realization of\nhybrid quantum systems for quantum memory and opti-\ncal transducer related applications [23–28].\nHere we study the nonlinear frequency mixing process\nin these ferromagnetic resonators based on two config-\nurations. In the first configuration we study frequency\nmixing of transverse and longitudinal driving tones that\nare simultaneously applied to the magnon resonator. The\nsignal tone is in the radio frequency (RF) band, and it\nis applied in the longitudinal direction, parallel to the\nexternal static magnetic field. This process can be em-\nployed for frequency conversion between the RF and the\nMW bands. Here we find that the measured response can\nbe well described using the Landau-Zener-Stuckelberg\nmodel [29, 30]. In the second configuration, Kerr non-\nlinearity that is induced by magnetic anisotropy is stud-\nied by intermodulation measurements. This is done by\nsimultaneously applying in the transverse direction an\nintense pump and a weak signal tones both having fre-\nquencies close to resonance. The observed intermodula-\ntion frequency conversion reveals a bifurcation between a\nFIG. 1: (a) A schematic of the experimental setup used for\nstudying the ferrimagnetic (FM) sphere. Longitudinal and\ntransverse driving are applied using a radio frequency ante nna\n(RFA) and a microwave antenna (MWA), respectively. (b)\nReal image of the DUT.\nstable spiral and a stable node [31]. These nonlinear ef-\nfects may find applications in signal sensing, parametric\namplification and other related applications.\nThe spherical resonators under test are made of yt-\ntrium iron garnet (YIG) and calcium vanadium bismuth\niron garnet (CVBIG) with a radius of Rs= 125µm. They\nhost magnonic excitations with relatively low damping\nand large spin densities. These spheres are anisotropic\nferrimagnetic crystals with strong Faraday rotation an-\ngles and high refractive index as compared to other iron\ngarnets. A schematic image of our device under test\n(DUT) is shown in Fig. 1. The ferrimagnetic sphere\nis held by vacuum through a ferrule. A fixed magnet is\nemployed for fully magnetizing the sphere. A loop an-\ntenna (coil) is used to apply a transverse (longitudinal)\ndriving in the MW (RF) band. All measurements are\nperformed at room temperature.2\nII. LANDAU-ZENER-STUCKELBERG\nINTERFEROMETRY\nLandau-Zener-Stuckelberg interferometry is based on a\nmixing process between transverse and longitudinal driv-\ning frequencies that are simultaneously applied to a res-\nonator [29, 30]. The polarization vector Pevolves in\ntimetaccording to the Bloch-Landau-Lifshitz equation\ndP/dt=P×Ω+Γ, whereΩ=γeBis the rotation vec-\ntor, with Bbeing the externally applied magnetic induc-\ntion and γe= 28GHzT−1being the gyromagnetic ratio,\nand the vector Γ=−Γ2Pxˆ x−Γ2Pyˆ y−Γ1(Pz−Pz,s)ˆ z\nrepresents the contribution of damping, with Γ1= 1/T1\nandΓ2= 1/T2being the longitudinal and transverse re-\nlaxation rates, respectively, and Pz,sbeing the steady\nstate polarization. Consider the case where Ω(t) =\nω1(cos(ωt)ˆ x+sin(ωt)ˆ y) +ω0ˆ z. Here ω1andωare\nboth real constants, and ω0oscillates in time accord-\ning toω0=ωc+ωbsin(ωmt), where ωc,ωbandωm\nare all real constants. Nonlinearity of the Bloch-Landau-\nLifshitz equation gives rise to frequency mixing between\nthe transverse driving at angular frequency ωand the\nlongitudinal driving at angular frequency ωm. The reso-\nnance condition of the l’th order frequency mixing pro-\ncess reads ω+lωm=ωc, where lis an integer. The\ncomplex amplitude P+(in a rotating frame) of the cor-\nresponding l’th side band is given by (see appendix D of\nRef. [32])\nP+=iω1ζ\nΓ2\n2/parenleftBig\n1+iωd\nΓ2/parenrightBig\nPz,s\n/parenleftBig\n1+ω2\nd\nΓ2\n2/parenrightBig\nΓ1\nΓ2+/parenleftBig\nω1ζ\nΓ2/parenrightBig2, (1)\nwhereζ=Jl(ωb/ωm),Jlis thel′th Bessel function of\nthe first kind, and the detuning angular frequency ωdis\ngiven by ωd=ω+lωm−ωc.\nThe schematic of the experimental setup employed to\nexplore this frequency mixing process is shown in Fig.\n2(a). The device under test (DUT 1) comprises of the\nferrimagnetic resonator coupled to both the MW loop\nantenna and the RF coil. The Kittel mode frequency is\ntuned by the static magnetic field to the value 2.305GHz .\nThe sphere is simultaneously driven by a pump with a\npower of 0 dBm that is applied to the MW loop antenna\nand an RF signal with a frequency of 0.5MHz that is ap-\nplied to the RF coil. Spectrum analyzer measurements of\nthe signal reflected from the MW loop antenna are shown\nin Fig. 2(b) as a function of the spectrum analyzer angu-\nlar frequency ωSAand the driving MW angular frequency\nωMWthat is injected into the loop antenna. The theoret-\nical prediction that is derived using Eq. ( 1) is presented\nby Fig. 2(c). The values of parameters that are used\nfor the calculation are listed in the caption of Fig. 2.\nThe comparison between the measured [see Fig. 2(b)]\nand calculated [see Fig. 2(c)] response yields a good\nagreement.\nFIG. 2: Landau-Zener-Stuckelberg interferometry. Experi -\nmental (b) and theoretical (c) spectral response obtained b y\nthe frequency mixing of transverse and longitudinal drivin g\nsignals applied simultaneously to the magnon resonator. Th e\ntheoretical color-coded plot (c) is derived using Eq. (1), u s-\ning the following parameters’ values ωm/(2π) = 0.5MHz ,\nω1/(2π) = 0.5MHz ,ωb/(2π) = 0.5MHz ,Γ1/(2π) = 1.0MHz\nandΓ2/(2π) = 2.0MHz .\nIII. ANISOTROPY-INDUCED KERR\nNONLINEARITY\nThe experimental setup used for intermodulation mea-\nsurements is shown in Fig. 3(a). Here the device un-\nder test (DUT 2) is the same as that shown in Fig. 1,\nwhere the RF antenna (RFA) is removed from the setup.\nThe nonlinearity gives rise to bistability, which in turn\nyields a hysteretic resonance curve, which is obtained via\nthe forward and backward sweeping directions [see Fig.\n3(b)]. The measured response becomes bistable when\nthe input pump power Ppis of the order of mW. The\nsubsequent idler tones generated due to the nonlinear\nfrequency mixing of pump and signal tones in the ferri-\nmagnetic resonator are shown in Fig. 3(c).\nThe technique of Bosonization can be applied to model\nthe nonlinearity in ferrimagnetic sphere resonators [33].\nIn this approach, the Hamiltonian HMis expressed in the\nform/planckover2pi1−1HM=ωcNM+KMN2\nM+QMN4\nM+···, where\nωc=µ0γeHis the angular frequency of the Kittel mode\n[7, 34],µ0= 4π×10−7NA−2is the permeability of free\nspace,His the externally applied uniform magnetic field\n(which is assumed to be parallel to the ˆ zaxis),NMis a\nnumber operator, KMis the so-called Kerr frequency, and\nQMis the coefficient of quartic nonlinearity. When non-\nlinearity is taken into account to lowest nonvanishing or-\nder only, i.e. when the quartic and all higher order terms\nare disregarded, the response can be described using the\nDuffing-Kerr model. This model predicts that the re-\nsponse of the system to an externally applied monochro-3\nmatic driving can become bistable.\nFIG. 3: Intermodulation. (a) An intense pump and a rela-\ntively weak signal are simultaneously injected into the MW\nloop antenna, and the reflected signal is measured using a\nspectrum analyzer. (b) Experimentally obtained hystereti c\nresonance curve (with no signal) showing bistability corre -\nsponding to the forward and backward microwave frequency\nsweep directions. (c) Spectrum analyzer measurement of the\nreflected signal intensity as a function of the detuning fre-\nquency with respect to the pump frequency. The idler peaks\nare generated as a result of the nonlinear pump-signal mixin g.\nIn general, the number of magnons /angbracketleftNM/angbracketrightin a reso-\nnantly driven sphere having total linear damping rate\nγcwith pump power Ppis given for the case of critical\ncoupling by /angbracketleftNM/angbracketright ≃Pp/(/planckover2pi1ωcγc). On the other hand,\nthe expected number of magnons /angbracketleftNM/angbracketrightat the onset of\nDuffing-Kerr bistability is ≃γc/|KM|[see Eq. (42) of\nRef. [35] and note that, for simplicity, cubic nonlinear\ndamping is disregarded]. Thus, from the measured val-\nues of the linear damping rate γc/(2π)≃1MHz and\nPp≃1mW , at the bistability onset point one obtains\nKM/(2π)≃ −2×10−9Hz(the minus signs indicates\nthat the Kerr nonlinearity gives rise to softening). Note,\nhowever, that the above estimate, which is based on the\nDuffing-Kerr model, is valid provided that the quartic\nand all higher order terms can be disregarded near (and\nbelow) the bistability onset. For the quartic term this\ncondition can be expressed as |QM| ≪ |KM|3/γ2\nc.\nThe values of KMandQMare estimated below for\nthe case where nonlinearity originates from magnetic\nanisotropy. The Stoner–Wohlfarth energy EMis ex-\npressed as a function of the magnetization vector M=\nMˆ uM, and the first-order Kc1and second-order Kc2\nanisotropy constants as [36]\nEM\nVs=−µ0M·H+Kc1sin2φ+Kc2sin4φ , (2)\nwhereVs= 4πR3\ns/3is the volume of the sphere hav-\ning radius Rs, andφis the angle between ˆ uMand theunit vector ˆ uAparallel to the easy axis. It is assumed\nthat the sphere is fully magnetized, i.e. |M| ≃Ms,\nwhereMsis the saturation magnetization. In terms\nof the dimensionless angular momentum vector Σ=\n−2MVs/(/planckover2pi1γe)≡(Σx,Σy,Σz)Eq. ( 2) is rewritten as\nEM−/planckover2pi1ωK1(1+Kc2/Kc1) =HM, where\n/planckover2pi1−1HM=ωcΣz\n2+/parenleftbigg\n1+2Kc2\nKc1/parenrightbiggKM(Σ·ˆ uA)2\n4\n+Kc2\nKc1K2\nM(Σ·ˆ uA)4\n16ωK1,\n(3)\nωK1=/planckover2pi1−1VsKc1andKM=/planckover2pi1γ2\neKc1//parenleftbig\nVsM2\ns/parenrightbig\nis the Kerr\nfrequency [17].\nIn the Holstein-Primakoff transformation [37], the\noperators Σ±= Σx±iΣyandΣzare expressed as\nΣ+=B†(Ns−N′\nM)1/2,Σ−= (Ns−N′\nM)1/2Band\nΣz=−Ns+ 2N′\nM, where Nsis the total number\nof spins, and where N′\nM=B†Bis a number opera-\ntor. If the operator Bsatisfies the Bosonic commu-\ntation relation/bracketleftbig\nB,B†/bracketrightbig\n= 1then the following holds\n[Σz,Σ+] = 2Σ +,[Σz,Σ−] =−2Σ−and[Σ+,Σ−] =\nΣz. The approximation (Ns−N′\nM)1/2≃N1/2\nsleads to\nΣ·ˆ uA=N1/2\ns/parenleftbig\nB†uA++BuA−/parenrightbig\n+2NMuAz, whereuA±=\n[(ˆ uA·ˆ x)∓i(ˆ uA·ˆ y)]/2,uAz=ˆ uA·ˆ z, and the magnon\nnumber operator NMis defined by NM=N′\nM−Ns/2.\nThis approximation is valid near the bistability onset pro-\nvided that γc/(|KM|Ns)≪1. For YIG, the spin density\nisρs= 4.2×1021cm−3, thus for a sphere of radius Rs=\n125µmthe number of spins is Ns=Vsρs= 3.4×1016,\nhence for the current experiment γc/(|KM|Ns)≃10−1.\nThis estimate suggests that inaccuracy originating from\nthis approximation may be significant for the current ex-\nperiment near and above the bistability threshold.\nSecond-order anisotropy gives rise to a quartic non-\nlinear term in the Hamiltonian ( 3) with a coefficient\nQM≃(Kc2/Kc1)/parenleftbig\nK2\nM/ωK1/parenrightbig\n(the exact value depends\non the angle φbetween the magnetization vector and\nthe easy axis). Near or below the bistability onset the\nquartic term can be safely disregarded provided that\n(Kc2/Kc1)/parenleftbig\nγ2\nc/(ωK1|KM|)/parenrightbig\n≪1. When this condition\nis satisfied the Hamiltonian ( 3) for the case where ˆ uAis\nparallel to ˆ z(i.e.uAz= 1anduA+=uA−= 0) approxi-\nmately becomes\n/planckover2pi1−1HM=ωcNM+KMN2\nM. (4)\nThe term proportional to KMrepresents the anisotropy-\ninduced Kerr nonlinearity.\nFor YIG Ms= 140kA /m,Kc1=−610J/m3at297K\n(room temperature), hence for a sphere of radius Rs=\n125µmthe expected value of the Kerr coefficient is given\nbyKM/(2π) =−2.0×10−9Hz. This value well agrees\nwith the above estimation of KM/(2π)based on the mea-\nsured input power at the bistability onset. For YIG\nKc2/Kc1= 4.8×10−2(Kc2/Kc1= 4.3×10−2) at a tem-\nperature of T= 4.2K(T= 294K ) [7]. Based on these4\nvalues one finds that for the sphere resonators used in the\ncurrent experiment (Kc2/Kc1)/parenleftbig\nγ2\nc/(ωK1|KM|)/parenrightbig\n≃10−6,\nhence the second-order anisotropy term (proportional to\nKc2) in Eq. ( 3) can be safely disregarded in the vicinity\nof the bistability onset.\nIV. STABLE SPIRAL AND STABLE NODE\nTo explore the regime of weak nonlinear response, con-\nsider a resonator being driven by a monochromatic pump\ntone having amplitude bcand angular frequency ωp. The\ntime evolution in a frame rotating at the pump driving\nfrequency is assumed to have the form\ndCc\ndt+Θc=Fc, (5)\nwhere the operator Ccis related to the resonator’s anni-\nhilation operator AcbyCc=Aceiωpt, the term Θc=\nΘc/parenleftbig\nCc,C†\nc/parenrightbig\n, which is expressed as a function of both\nCcandC†\nc, is assumed to be time independent, and Fc\nis a noise term having a vanishing expectation value.\nThe complex number Bcrepresents a fixed point, for\nwhichΘc(Bc,B∗\nc) = 0 . By expressing the solution as\nCc=Bc+ccand considering the operator ccas small,\none obtains a linearized equation of motion from Eq. ( 5)\ngiven by\ndcc\ndt+W1cc+W2c†\nc=Fc, (6)\nwhereW1=∂Θc/∂CcandW2=∂Θc/∂C†\nc(both deriva-\ntives are evaluated at the fixed point Cc=Bc).\nThe stability properties of the fixed point depend on\nthe eigenvalues λc1andλc2of the2×2matrixW, whose\nelements are given by W11=W∗\n22=W1andW12=\nW∗\n21=W2[see Eq. ( 6)]. In terms of the trace TW=\nW1+W∗\n1and the determinant DW=|W1|2− |W2|2of\nthe matrix W, the eigenvalues are given by λc1=TW/2+\nυWandλc2=TW/2−υW, where the coefficient υW\nis given by υW=/radicalBig\n(TW/2)2−DW. Note that in the\nlinear regime, i.e. when W2= 0, the eigenvalues become\nλc1=W1andλc2=W∗\n1. For the general case, when both\nλc1andλc2have a positive real part, the fixed point is\nlocally stable. Two types of stable fixed points can be\nidentified. For the so-called stable spiral, the coefficient\nυWis pure imaginary [i.e. (TW/2)2−DW<0], and\nconsequently λc2=λ∗\nc1, whereas both λc1andλc2are\npure real for the so-called stable node, for which υWis\npure real. A bifurcation between a stable spiral and a\nstable node occurs when υWvanishes.\nFurther insight can be gained by geometrically an-\nalyzing the dynamics near an attractor. To that\nend the operators ccandFcare treated as complex\nnumbers. The equation of motion ( 6) for the com-\nplex variable cccan be rewritten as d¯ξ/dt+W′¯ξ=\n¯f, where ¯ξ=/parenleftbig\nReal/parenleftbig\ncceiφ/parenrightbig\n,Imag/parenleftbig\ncceiφ/parenrightbig/parenrightbigTand¯f=/parenleftbig\nReal/parenleftbig\nFceiφ/parenrightbig\n,Imag/parenleftbig\nFceiφ/parenrightbig/parenrightbigTare both two-dimensional\nreal vectors, and where the rotation angle φis real.\nTransformation into the so-called system of principle axes\nis obtained when the angle φis taken to be given by\ne2iφ=W1W∗\n2/|W1W2|. For this case the 2×2real ma-\ntrixW′becomes\nW′=/parenleftbigg\ncosθ1−sinθ1\nsinθ1cosθ1/parenrightbigg/parenleftbigg\nW+0\n0W−/parenrightbigg\n,(7)\nwhereθ1= arg(W1)and where W±=|W1|±|W2|. Thus,\nmultiplication by the matrix W′can be interpreted for\nthis case as a squeezing with coefficients W±followed by\na rotation by the angle θ1.\nThe flow near an attractor is governed by the eigen-\nvectors of the 2×2real matrix W′. For the case where\nυWis pure real the angle αWbetween these eigenvectors\nis found to be given by sinαW=υW/|W2|. Thus at the\nbifurcation between a stable spiral and a stable node, i.e.\nwhenυW= 0, the two eigenvectors of W′become parallel\nto one another. In the opposite limit, when υW=|W2|,\ni.e. when W1becomes real, and consequently the ma-\ntrixWbecomes Hermitian, the two eigenvectors become\northogonal to one another (i.e. αW=π/2).\nThe bifurcation between a stable spiral and a stable\nnode can be observed by measuring the intermodulation\nconversion gain GIof the resonator. This is done by\ninjecting another input tone (in addition to the pump\ntone), which is commonly referred to as the signal, at\nangular frequency ωp+ω. The intermodulation gain is\ndefined by GI(ω) =|gI(ω)|2, wheregI(ω)is the ratio be-\ntween the output tone at angular frequency ωp−ω, which\nis commonly referred to as the idler, and the input signal\nat angular frequency ωp+ω. In terms of the eigenvalues\nλc1andλc2the gain GIis given by [35]\nGI=/vextendsingle/vextendsingle/vextendsingle/vextendsingle2γc1W2\n(λc1−iω)(λc2−iω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (8)\nwhereγc1is the coupling coefficient (in units of\nrate) between the feedline that is used to de-\nliver the input and output signals and the res-\nonator. For the case of a stable spiral, i.e.\nwhenλc2=λ∗\nc1, one has |(λc1−iω)(λc2−iω)|2=/bracketleftBig\nλ′2+(λ′′−ω)2/bracketrightBig/bracketleftBig\nλ′2+(λ′′+ω)2/bracketrightBig\n, where λ′= Reλc1\nandλ′′= Imλc1(i.e.λc1=λ′+iλ′′), whereas for the case\nof a stable node, i.e. when both λc1andλc2are pure real,\none has|(λc1−iω)(λc2−iω)|2=/parenleftbig\nλ2\nc1+ω2/parenrightbig/parenleftbig\nλ2\nc2+ω2/parenrightbig\n.\nFor the case of a resonator having Kerr nonlin-\nearity and cubic nonlinear damping Θcis given by\nΘc= [i∆c+γc+(iKc+γc3)Nc]Cc+i√2γc1eiφc1bc,\nwhere∆c=ωc−ωpis the driving detuning, the total\nrate of linear damping is γc=γc1+γc2, the rate γc1\ncharacterizes the coupling coefficient between the feed-\nline and the resonator, γc2is the rate of internal lin-\near damping, γc3is the rate of internal cubic damping,\nKcis the Kerr coefficient, Nc=A†\ncAcis the resonator\nnumber operator, and φc1is a phase coefficient charac-5\nFIG. 4: Stability map of a driven Duffing-Kerr resonator.\nThe BOP is the point/parenleftbig\n∆c/(∆c)BOP,bc/(bc)BOP/parenrightbig\n= (−1,1).\nIn both 'C'(dark blue) and 'R'(light blue) regions, there is\na single locally stable attractor, whereas there are two in t he\nregions 'CC'(light green), 'CR'(orange) and 'RR'(red). The\nletter 'C'is used to label a stable spiral, whereas the letter\n'R'labels a stable node.\nterizing the coupling between the feedline and the res-\nonator [35]. The rates W1andW2are given by W1=\ni∆c+γc+2(iKc+γc3)|Bc|2andW2= (iKc+γc3)B2\nc.\nThe condition Θc(Bc,B∗\nc) = 0 can be expressed as a cu-\nbic polynomial equation for the number of magnons Ec=\n|Bc|2given by/bracketleftBig\n(∆c+KcEc)2+(γc+γc3Ec)2/bracketrightBig\nEc=\n2γc1|bc|2. The eigenvalues can be expressed in terms\nofEcasλc1,2=TW/2±υW, where TW/2 =γc+\n2γc3EcandυW=/radicalbig\n(∆−−∆c)(∆++∆c), where∆±=/parenleftbigg/radicalBig\n1+(γc3/Kc)2±2/parenrightbigg\nKcEc. The stability map of the\nsystem is shown in Fig. 4. Both driving detuning ∆cand\ndriving amplitude bcare normalized with the correspond-\ning values at the bistability onset point (BOP) (∆c)BOP\nand(bc)BOP[see Eqs. (46) and (47) of Ref. [35]]. Inside\nthe regions 'C'and'R'of mono-stability ( 'CC','CR'\nand'RR'of bistability) the resonator has one (two) lo-\ncally stable attractors. A stable spiral (node), for which\nλc2=λ∗\nc1(bothλc1andλc2are pure real), is labeled by\n'C'('R').\nIn the bistable region, the cubic polynomial equation\nhas 3 real solutions for Ec. The corresponding values\nof the complex amplitude Bcare labeled as C1,C2and\nC3. In the flow map shown in Fig. 5, which is obtained\nby numerically integrating the equation of motion ( 5)\nfor the noiseless case Fc= 0, the point C1is a stable\nnode, the point C2is a saddle point and the point C3\nis a stable spiral. The red and blue lines represent flow\ntoward the stable node attractor at C1and the stable\nspiral attractor at C3, respectively. The green line is the\nseperatrix, namely the boundary between the basins of-8-6-4-202468-10-8-6-4-2\nC3C1\nC2\nRe(C)Im(C)\n(a)\n4.74.754.84.854.94.9555.05-2.7-2.6-2.5-2.4-2.3-2.2-2.1-2\nC1\nC2\nRe(C)Im(C)\n(b)\nFIG. 5: Flow map of a Duffing oscillator in the region of\nbistability. The point C1is a stable node, the point C2is a\nsaddle point, and the point C3is a stable spiral. A closer view\nof the region near C1andC2is shown in (b).\nattraction of the attractors at C1andC3. A closer view\nof the region near C1andC2is shown in Fig. 5(b).\nThe intermodulation conversion gain GIinduced by\nthe Kerr nonlinearity is measured with the ferrimagnetic\nresonator DUT 2[see Fig. 3(a)], and the results are com-\npared with the theoretical prediction given by Eq. ( 8).\nIn these measurements the pump frequency ωpis tuned\nclose to the resonance frequency ωc. The measured gain\nGIis shown in the color-coded plots in Fig. 6(for three\ndifferent values of the pump frequency ωp) as a function\nof the detuning between the signal and pump frequencies\nω/(2π)and the pump power Pp.\nThe overlaid black dotted lines in Fig. 6indicate the\ncalculated values of the imaginary part of the eigenvalues\nλ′′= Imλc1and−λ′′= Imλc2. The calculation is based\non the above-discussed Duffing-Kerr model. At the point6\nFIG. 6: Intermodulation gain GIas a function of detuning\nbetween the signal and pump frequencies ω/(2π)and pump\npowerPp(in dBm units). The pump frequency ωp/(2π)is\n(a)3.8674GHz (b)3.8704GHz and (c)3.8734GHz . The sig-\nnal power is −15dBm. Note that GIis measured with ω >0\nonly, and the plots are generated by mirror reflection of the\ndata around the point ω= 0. Note also that for clarity the\nregion near the pump frequency, i.e. close to ω= 0, has been\nremoved from the plot. The width of this region, in which\nthe intense pump peak is observed, depends on the resolu-\ntion bandwidth setting of the spectrum analyzer. The black\ndotted lines indicate the calculated values of the imaginar y\npart of the eigenvalues λ′′= Imλc1and−λ′′= Imλc2. The\npump amplitude bcand pump detuning ∆cused for the cal-\nculation of the eigenvalues are determined from the measure d\nvalue of Pp= 0.4dBm for the pump power and the value of\n∆c/(2π) = 1.3MHz for the pump detuning at the BOP.\nwhereλ′′vanishes, a bifurcation from stable spiral to sta-\nble node occurs. As can be seen from comparing panels(a), (b) and (c) of Fig. 6, the pump power Ppat which\nthis bifurcation occurs depends on the pump frequency\nωp. This bifurcation represents the transition between\nthe regions 'CC'and'CR'in the stability map shown in\nFig.4. A bifurcation from the bistable to the monos-\ntable regions occurs at a higher value of the pump power\nPp. This bifurcation gives rise to the sudden change in\nthe measured response shown in Fig. 6. In the stability\nmap shown in Fig. 4, this bifurcation corresponds to the\ntransition between the regions 'CR'and'C'.\nV. CONCLUSION\nWe present two nonlinear effects that can be used for\nsignal sensing and amplification. The first one is based\non the so-called Landau-Zener-Stuckelberg process [29]\nof frequency mixing between transverse and longitudi-\nnal driving tones that are simultaneously applied to the\nmagnon resonator. This process can be employed for fre-\nquency conversion between the RF and the MW bands.\nThe second nonlinear effect, which originates from mag-\nnetization anisotropy, can be exploited for developing in-\ntermodulation receivers in the MW band. Measurements\nof the intermodulation response near the onset of the\nDuffing-Kerr bistability reveal a bifurcation between a\nstable spiral attractor and a stable node attractor. Above\nthis bifurcation, i.e. where the attractor becomes a stable\nnode, the technique of noise squeezing can be employed\nin order to enhance the signal to noise ratio [35].\nVI. ACKNOWLEDGMENTS\nWe thank Amir Capua for helpful discussions. 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Yurlov,1, 2K.A. Zvezdin,2, 3,\u0003and A.K. Zvezdin2, 3\n1Moscow Institute of Physics and Technology, Institutskiy per. 9, 141700 Dolgoprudny, Russia\n2New Spintronic Technologies, Russian Quantum Center,\nBolshoy Bulvar 30, bld. 1, 121205 Moscow, Russia\n3Prokhorov General Physics Institute of the Russian Academy of Sciences, Vavilova 38, 119991 Moscow, Russia\n(Dated: December 13, 2022)\nWe report of a theoretical model for calculating the H-T phase diagrams of a rare-earth ferri-\nmagnet, taking into account anisotropies originated by both magnetization sublattices' and by the\nsurface. The possibility of an exchange spring formation due to surface anisotropy is considered.\nThis situation is realized in heterostructures containing a ferrimagnet and a heavy metal. We derive\nthe stability lose lines of the collinear phase from the free energy of the two sublattice ferrimagnet.\nWe numerical calculate the magnetic phase diagrams for the cases when the magnetic \feld applied\nalong and perpendecular to the easy axis. We demonstrate that tricritical point down at the low \feld\nrange due to surface anisotropy e\u000bect. Moreover, the line of the \frst order phase transition between\nangular and collinear phases reduces due to surface anisotropy. In the case when magnetic \feld is\napplied perpendicular to the easy axis we show the possibility of the \frst order phase transition\nbetween two collinear phases in contrast to the phase diagram without surface anisotropy.\nI. INTRODUCTION\nRare-earth-transition metal (RE-TM) compounds is\na class of magnetic materials that attracts particu-\nlar attention in a wide range of di\u000berent areas such\nas spintronics[1, 2], optospintronics[3] and ultrafast\nmagnetism[4]. This rare-earth ferrimagnetic (FiM) thin\n\flms can be applied for technological uses such as ul-\ntrafast memory devices[5] or high density recording[6{\n8]. Depending on the composition, FiM \flms may have\nthe magnetization compensation point TMwhere the an-\ntiferromagneticaly coupled RE and TM magnetizations\ncompensate each other[9]. This point plays an impor-\ntant role for studying the magnetic phase transitions or\nmagnetization dynamics of the FiM \flms[10{12].\nStudying of the magnetic phase diagrams[13] is partic-\nular of a interest for a better understanding the magne-\ntization dynamics in ferrimagnets. Recent experiments\nwith ferrimagnets such as GdFeCo, GdCo and TbFe\ndemonstrate anomalous hysteresis loops near the mag-\nnetization compensation point[14{17]. In particular, in\nthe GdFeCo ferrimagnet, triple hysteresis loops are ob-\nserved above the magnetization compensation tempera-\nture [14]. At the same time, experiments with TbFeCo\nwith Ta capping layer[18] show that triple loops can ap-\npear to the left of the compensation point. To explain\nthis anomalous hysteresis loop, theoretical models[19]\nwere constructed, in which the interplay of the surface\nanisotropy, and anisotropies of both sublattices led to a\nmodi\fcation of the phase diagrams. The thickness and\n\fnite size of a ferrimagnetic \flm can also signi\fcantly\na\u000bect the spin-reorientation transitions and change the\nphase diagram. Theoretical [20, 21] and experimental[22]\n\u0003zvezdin.ka@phystech.edustudies on the e\u000bect of the surface on the spin dynamics\nwere carried out for example for nanowires[23] and for\nvarious ferrimagnetic materials[24, 25]. However, given\nthe new experimental and theoretical results, this area\nrequires further study, and the in\ruence of the surface\ne\u000bects on the FiM phase diagram deserves particular at-\ntention.\nIn this work we study the magnetic phase diagram for\nthe FiM layer taking into account anisotropies of the both\nmagnetization sublattices and the surface anisotropy. We\nderive the stability lose lines of the collinear phase from\nthe free energy of the two sublattice ferrimagnet. In-\n\ruence of the surface exchange anisotropy can be taken\ninto account by introducing the dimensionless parameter\nwhich modi\fes the e\u000bective anisotropy of the ferrimag-\nnet. We numerical calculate the magnetic phase diagram\nfor two di\u000berent directions of the external magnetic \feld\nand show the lines of the second and the \frst order phase\ntransitions. We demonstrate that surface anisotropy can\nshift down at lower \feld range the tricritical point and\nreduce the line of the \frst order phase transition between\nangular and collinear phases. In the case when magnetic\n\feld is applied perpendicular to the easy axis we show\nthat the surface anisotropy enables the \frst order phase\ntransition between two collinear phases.\nII. MODEL AND BASIC EQUATIONS\nTo obtain the magnetic phase diagram for ferrimagnets\nwith the surface anisotropy we use the e\u000bective thermo-\ndynamic potential[13]. We assume that transition metal\n(d-sublattice) is saturated due to large d-d interactions\n(of the order of 106\u0000107Oe) and rare-earth (f-sublattice)\nis considered as a paramagnetic in the e\u000bective mag-\nnetic \feld. The applicability of this model is substanti-\nated by the hierarchy of exchange interactions[26]. Thus,arXiv:2109.12377v2 [physics.app-ph] 12 Dec 20222\nthe e\u000bective thermodynamic potential without surface\nanisotropy can be written as:\n\b =\u0000MdH\u0000ZHeff\n0Mf(x)dx\u0000Ka+ \bex;(1)\nwhere Mdis magnetization of the transition metal sub-\nlattice, Mfis magnetization of the rare-earth ions which\nare saturated in the e\u000bective \feld Heff=H\u0000\u0015Md,\n\u0015is the f-d exchange constant, His external magnetic\n\feld,Kais magnetic anisotropy energy and \b exis non-\nuniform exchange energy. The magnetization of the f-\nsublattice can be described by the Brillouin function\nMf(x) =\u0016BgJBJ\u0010gJ\u0016Bx\nkT\u0011\n, wheregis Lande g-factor,\nJis total angular momentum of the rare-earth ions, \u0016B\nis Bohr magneton. The anisotropy energy of the ferri-\nmagnet is[14, 19]:\nKa=\u0000Kdsin2 \u0000Kf\u0010\u0015Mdsin \nHeff( )\u00112\n; (2)\nwhereKdandKfare unaxial anisotropy constants of\nd- and f- sublattices, respectively, is an angle be-\ntween magnetization of the d-sublattice and the easy\nmagnetization axis. Non-uniform exchange energy can\nbe written as \b ex=A(r )2whereAis the ex-\nchange sti\u000bness constant. Now we should take into ac-\ncount the surface e\u000bect as induced exchange magnetic\nanisotropy. This energy takes the form Fs=kdsin2 s+\nkf(\u0015Mdsin s)2=H2\neff( s), where sis magnitude of the\nangle on the surface of the \flm, kdandkfconstants\nof the d- and f- sublattice surface magnetic anisotropy.\nFurther expressions is written in the assumption that\nmagnetic \flm is homogeneous in the plane of the \flm.\nWe consider the \flm with thickness jzj< d. Exchangesurface anisotropy is equal at the edges of the \flm\nkd(\u0000d) =kd(d) andkf(\u0000d) =kf(d) which means that\nthe symmetrical arrangement of magnetization is most\nbene\fcial. As a result we can say that d =dzj0= 0.\nFinaly, we obtain the free energy of the ferrimagnet by\nintegrating (1) over the \flms volume\nF=Zd\n0\bdz+Fs=Zd\n0n\nA(r )2\u0000MdH\u0000\nZHeff\n0Mf(x)dx+Kdsin2 +Kf\u0010\u0015Mdsin \nHeff( )\u00112o\ndz\n+kdsin2 s+kf\u0010\u0015Mdsin s\nHeff( s)\u00112\n:\n(3)\nEquation (3) will be used below to construct the magnetic\nphase diagram.\nIII. MAGNETIC PHASE DIAGRAM\nWe obtain here the magnetic phase diagram of ferri-\nmagnetic material with surface anisotropy using the free\nenergy (3). We consider two cases with di\u000berent direc-\ntions of the external magnetic \feld H: (a) when the mag-\nnetic \feld is applied along the easy axis and (b) magnetic\n\feld is perpendicular to the easy axis. For the case (a) we\nsuppose that =\u0012and for the case (b) { =\u0019=2\u0000\u0012,\nwhere\u0012is the angle between magnetization of the d-\nsublattice and external magnetic \flm. Without loss of\ngenerality, we consider the case (a) when the magnetic\n\feld is aligned with the easy axis. For the case (b), the\nconclusions given below are also valid.\nWe vary the functional (3) and obtain the Euler-\nLagrange equations and boundary conditions to study\nthe free energy:\n\u0001\u0012=MdH\n2Asin\u0012n\n1\u0000\u0015\u001f(\u0012) +Kf\u0010\u0015Md\nHeff(\u0012)\u001121\nMdH\u0010\n2 cos\u0012\u0000\u0015MdHsin2\u0012\nH2\neff(\u0012)\u0011\n+2Kd\nMdHcos\u0012o\n;\nd\u0012\ndz\f\f\f\f\ns=\u0000kf\n2A\u0010\u0015Md\nHeff(\u0012s)\u00112\nsin\u0012s\u0010\n2 cos\u0012s\u0000\u0015MdHsin2\u0012s\nH2\neff(\u0012s)\u0011\n\u00002kd\n2Asin\u0012scos\u0012s\nd\u0012\ndz\f\f\f\f\n0= 0;(4)\nwhere\u001f=Mf(\u0012)=Heff(\u0012), indexsin the second equa-\ntion de\fnes the \flm boundary. Note that the analytical\nsolution of these equations has some di\u000eculties associ-\nated with the de\fnition of the \frst integral of the di\u000ber-\nential equation. However, the e\u000bect of surface anisotropy\ncan be taken into account by considering the lines of sta-\nbility loss of the collinear phases. Having this is mind\nwe linearize the (4) near the lines of stability loss \u0012= 0\nand\u0012=\u0019. Let us give analytical expressions only forthe case\u0012= 0. We look for a solution of the linearized\nequations in the following form:\n\u0012=\u0012(z) expif{xx+{yyg; (5)\nwhere {{{is the vector lying in the plane of the magnetic\n\flm. We obtain second order di\u000berential equation for\neigenvalues of the Sturm{Liouville type. After some cal-\nculations we obtain a transcendental expression for the3\nH\nMf Md\nTM\nRP\nP'CB'A'\nC' E'A\nBH\nMf Md\nTMR'E\nC'H\nMf MdAB\nB'\nE'EA'\nCRP\nTMH\nMf Md(a)\n(b) (c)\nFIG. 1. (a) H\u0000Tphase diagram of the ferrimagnet in the\nmagnetic \feld applied along the easy axis in the high \feld\nrange; solid lines show the second order phase transition be-\ntween collinear and angular phases, the dotted lines show\nthe second order phase transition between collinear phases;\n\u0012is an angle between external magnetic \feld and magneti-\nzation of the d-sublattice. b) The zoomed-in phase diagram\nnear the tricritical points PandP0; lineTMR0Rshows the\n\frst order phase transition between collinear phases, lines RP\nandR0P0show the \frst order phase transition between an-\ngular and collinear phases. c) The zoomed-out phase dia-\ngram near the magnetization compensation temperature TM.\nLinesACandBEis the stability lose lines when the surface\nanisotropy is zero; lines A0C0andB0E0is the stability in the\npresence of the surface anisotropy. All diagrams constructed\nfor 00,keff<0.\nvector {{{:\np\n{2d2+\u00142d2tanhf{2d2+\u00142d2g=\n\u0000d\nAn\nkf\u0010\u0015Md\nHeff(0)\u00112\n+kdo\n;(6)\nwhere\u00142=MdH\n2Af1\u0000\u0015\u001f(0) +2Kf\nMdH(\u0015Md\nHeff(0))2+2Kd\nMdHg.\nThe stability condition for collinear phases (when mag-\nnetizations of both sublattice are parallel and \u0012= 0 or\n\u0012=\u0019) is that the equation has no real solutions. Car-\nrying out similar reasoning for the \u0012=\u0019, we obtain the\nlines of stability loss of the collinear phases:\n(B0E0) : 1\u0000\u0015\u001f(0) +Keff(0)\nMdH(1\u0000heff) = 0;\n(A0C0) : 1\u0000\u0015\u001f(\u0019)\u0000Keff(\u0019)\nMdH(1\u0000heff) = 0;(7)\nPHMf Md\nTMHMf MdTM\nA\nCA'\nC'B\nEE'B'A\nCE'B' A'\nC'B\nE(a)\n(b)FIG. 2. a) H\u0000Tphase diagram of the ferrimagnet in the\nmagnetic \feld directed perpendecular to the easy axis near\nthe magnetization compensation point; solid lines show the\nsecond order phase transition between collinear and angular\nphases, the dotted lines show the second order phase transi-\ntion between collinear phases; lines ACandBEis the sta-\nbility lose lines when the surface anisotropy is zero; lines\nA0C0andB0E0is the stability in the presence of the sur-\nface anisotropy; this diagram is constructed for heff<\u00001,\nKeff>0,keff<0;\u0012is an angle between external magnetic\n\feld and magnetization of the d-sublattice. b) The magnetic\nphase diagram near the compensation temperature TM; dia-\ngram is constructed for heff>1,Keff>0,keff>0.\nwhereKeff=Kf(\u0015Md\nHeff)2+Kd,heffis the solution of the\nequationj\u000esj= (\u001b\nheff)1=2tanh\u00001(\u001b\nheff)(1=2)andkeff=\nkf(\u0015Md\nHeff)2+kd<0,\u001bis material surface parameter with4\npositive value, \u000es= (kf(\u0015Md\nHeff)2+kd)d=A. Similarly, it is\npossible to obtain lines of stability loss when the magnetic\n\feld is perpendicular to the easy axis of the ferrimagnet.\nFor this case, the stability loss lines will be written in the\nform:\n(B0E0) : 1\u0000\u0015\u001f(0)\u0000Keff(0)\nMdH(1 +heff) = 0;\n(A0C0) : 1\u0000\u0015\u001f(\u0019) +Keff(\u0019)\nMdH(1 +heff) = 0;(8)\nwhereheff is the solution of the equation \u000es=\n(\u001b\nheff)1=2tanh\u00001(\u001b\nheff)(1=2). Here we should note that\nthe present theory applicable in the microscopic range\nford\u001810\u00007\u000410\u00006m.\nRecent researches show that anisotropy of the RE\nsublattice can be larger then the one of the TM\nsublattice[19]. Therefore, let us investigate how lines of\nthe \frst and second phase transitions are changed due\nto the exchange surface anisotropy. By using the equa-\ntions (7), Euler-Lagrange equations (4), free energy (3)\nand methods which are described in [13] we calculate nu-\nmerically the magnetic phase diagram of ferrimagnetic\n\flm with the exchange surface anisotropy (see in Fig. 1\nand Fig. 2). For the calculations we use the GdFeCo\nparameters: Md(0) = 4:5\u0016B=f:u: ,Mf(0) = 7\u0016B=f:u: ,\nKd= 0:1\u0001105erg=cc ,Kf= 0:9\u0001105erg=cc ,Hex=\n\u0015Md\u0018106Oe,TM\u0019263K. Magnetic phase diagrams\nin Fig. 1 and Fig. 2 represent the areas of the collinear\nphase where \u0012= 0 (purple area in Fig. 1 and Fig. 2),\n\u0012=\u0019(green area in Fig. 1 and Fig. 2) and noncollinear\nphase\u0012=\u0012(T;H) (yellow area in Fig. 1 and Fig. 2).\nBlue are in Fig. 1 and Fig. 2 shows the di\u000berent between\ncollinear and noncollinear phases for the ferrimagnet with\nand without e\u000bect of the surface anisotropy.\nIV. RESULTS AND DISCUSSION\nMagnetic phase diagram in Fig. 1 and Fig. 2 demon-\nstrate three di\u000berent phases of the ferrimagnetic \flm.\nThis phases are: the collinear phase at the high tem-\nperature range ( \u0012= 0 purple area), the collinear phase\nat the low temperature range ( \u0012=\u0019green area) and\nthe noncollinear phase \u0012=\u0012(T;H) which is indicated as\nyellow area.\nLet us discuss now the case (a) when external magnetic\n\feld is co-directed with easy axis of ferrimagnet. Fig.\n1(a) show the phase diagram at the high \feld range. Dot-\nted lines show the second order phase transition between\ntwo collinear phases. The zoomed area of the diagram\nin Fig. 1(a) is shown in the Fig. 1(c). Lines ACand\nBEshow the lines of a second-order phase transition be-\ntween collinear and non-collinear phases. Note that the\nsolid lines denote the second-order phase transition be-\ntween the collinear and the angular phases. The dotted\nlines indicate the phase transition between two collinear\nphases. The grey doted line in Fig. 1(a) demonstratethe situation when Heff(T) = 0. These lines ( ACand\nBE) demonstrate the case when the surface anisotropy is\nzero (heff= 0). If the surface anisotropy a\u000bects the mag-\nnetic system, than the e\u000bective anisotropy Keffchanges,\nas follows from the (7) and the ACandBElines turn\ninto theA0C0andB0E0. In Fig. 1(b) and Fig. 1(c), the\nblue color indicates the di\u000berence between two cases de-\nscribed above. Fig. 1(b) and Fig. 1(c) show that surface\nanisotropy plays a signi\fcant role near low values of the\nmagnetic \feld H\u0003\u0018(2Keff\u0015)1=2. With an increase of\nthe magnitude of the magnetic \feld, the lines AC,BE\nandA0C0,B0E0quickly approach to each other. Fig. 1 is\nplotted for the heff\u00180:5. TheTMR0Rline in Fig. 1(b)\nshows a \frst-order phase transition line between collinear\nphases whereF(0) =F(\u0019). TheRPline is the \frst-order\nphase transition line between the angular and collinear\n\u0012= 0 phases. Pis the tricritical point. Note that this\nline is located to the right of the magnetization compen-\nsation point TMdue to the in\ruence of the anisotropy\nof the rare-earth sublattice. Note, that tricritical point\nPmay located to the left from the compensation due to\nmodifying the surface of the ferrimagnetic by the heavy\nmetal \flm such as Ta[18]. The line RPtransforms into\nanR0P0due to the exchange surface anisotropy. The\n\frst-order phase transition between collinear and non-\ncollinear phases is reduced under the in\ruence of surface\nanisotropy. Thus, the regions of phase transitions near\nthe compensation point can change signi\fcantly due to\nsurface anisotropy.\nThe similar situation realises for the case (b)when the\nmagnetic \feld is perpendicular to the easy axis of the\nferrimagnet. Fig. 2(b) shows that the angular phase\nexpands due to surface anisotropy when the heff>0.\nHowever, the most interesting e\u000bect can be seen if the\nheff<0. In this case, the spins are pinned on the surface\nof the ferrimagnet. As a result, the transition between\ncollinear phases in the low-\feld region can occur through\nthe \frst order phase transition. This e\u000bect is demon-\nstrated in the Fig. 2(a). In the case described above,\nthe linesACandBEturns intoA0C0andB0E0. It also\nshould be noted, that if the anisotropy of the d-subluttice\nis higher than the one of the f-sublattice than the \frst\norder phase transition between the collinear phase \u0012=\u0019\nand the angular phase is possible because tricritical point\nin this case is lower than magnetization compensation\ntemperature.\nV. CONCLUSION\nThe magnetic phase diagram are studied for the\nGdFeCo ferrimagnet in presence of the surface magnetic\nanisotropy for the two di\u000berent cases: magnetic \feld ap-\nplied along and perpendecular to the easy axis. The sta-\nbility lose lines are derived from the free energy of the fer-\nrimagnet in the assumption that f-sublattice anisotropy\nis larger than d-sublattice one. In this particular case\nthe tricritical point lies above the compensation temper-5\nature. We numerical calculate the phase diagram and\nshow the lines of the second and \frst order phase tran-\nsition. We show that in the case when magnetic \feld is\nalong the easy axis the stability lose lines and tricriti-\ncal point are falling down in the low \feld range due to\nsurface anisotropy. Moreover, the area of the \frst order\nphase transition between angular and colliniar phase nar-\nrows due to surface e\u000bects. 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