title: Pseudospin-lattice coupling and electric control of the square-lattice iridate Sr2IrO4
authors: Feng Ye Christina Hoffmann Wei Tian Hengdi Zhao G. Cao
text: 
  
  
  
  
  
  
  Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
  
  
  Department of Physics, University of Colorado at Boulder, Boulder, Colorado 80309, USA
  
  
  
      Sr_2IrO_4 is an archetypal spin-orbit-coupled Mott insulator and has
      been extensively studied in part because of a wide range of predicted
      novel states. Limited experimental characterization of these states thus
      far brings to light the extraordinary susceptibility of the physical
      properties to the lattice, particularly,  the Ir-O-Ir bond angle.  Here,
      we report a newly observed microscopic rotation of the IrO_6 octahedra
      below 50 K measured by single crystal neutron diffraction. This sharp
      lattice anomaly provides keys to understanding the anomalous
      low-temperature physics and a direct confirmation of a crucial role that
      the Ir-O-Ir bond angle plays in determining the ground state. Indeed, as
      also demonstrated in this study, applied electric current readily weakens
      the antiferromagnetic order via the straightening of the Ir-O-Ir bond
      angle, highlighting that even slight change in the local structure can
      disproportionately affect the physical properties in the
      spin-orbit-coupled system.
  
  Pseudospin-lattice coupling and electric control of the square-lattice iridate Sr_2IrO_4
      G. Cao
      December 30, 2023
  ========================================================================================
  
  
  
  Strong spin-orbit interactions (SOI), along with appreciable Coulomb
  interactions, crystalline electric field, and large orbital hybridization in
  5d-electron based oxides has produced a wide range of quantum phenomena such
  as spin liquid phases <cit.>, superconductivity
  <cit.>, and Kitaev magnetism
  <cit.>. One prominent example is the observation of
  the so-called effective j_ eff=1/2 Mott insulating state in iridates
  <cit.>, where the magnetism is attributed to an isotropic
  pseudospin <cit.>.  Unlike the situation in the
  3d transition metal oxides which have a distinct energy scale in both
  orbital and spin parts and sequential orbital and magnetic orders, the notion
  of spin-orbital separation in the 5d systems is no longer valid due to the
  strong SOI. Furthermore, the Jahn-Teller effect, which measures the couples
  between the lattice and orbital degrees of freedom and is a common occurrence
  in 3d-electron based oxides, remains largely unexplored in the iridates.
  However, recent experimental and theoretical studies indicate that the
  magnetic properties and low-energy spin dynamics <cit.>, can be
  better explained using a pseudospin-lattice coupling mechanism. The
  introduction of such a term in the spin Hamiltonian not
  only correctly describes the metamagnetic transitions <cit.>, but
  also explains a finite in-plane magnon gap <cit.>.
  It also predicts an orthorhombic structural distortion in the order of
  10^-4, which can be probed using a high resolution Larmor precession
  technique <cit.>.
  
  The extraordinarily strong coupling between the electronic and structural
  properties in the iridates provides unprecedented opportunities to uncover
  novel quantum phases by merely modifying the local structure. It is
  particularly interesting that application of hydrostatic pressure steadily
  reduces the charge gap of approximately 500 meV and diminishes the
  antiferromagnetic (AFM) state existent in the square lattice Sr_2IrO_4
  (Sr-214), yet retains the insulating state <cit.>.  A
  possible quantum paramagnetic state at pressures greater than 20 GPa is
  proposed, attributing to the suppression of the inter-layer exchange coupling
  and enhanced magnetic frustration within the IrO_2 layer <cit.>.
  Alternative approaches to modify the structure using strain engineering are
  also reported in epitaxial grown thin films of Sr-214
  <cit.>, where a compressive or tensile
  strain can drastically enhance or reduce the transition temperature T_N. In some cases, the strengthened in-plane exchange interaction can
  promote a short range magnetic correlation well above the nominal transition
  <cit.>. In this paper, we report a newly observed lattice anomaly at
  50 K. Given the exceptionally strong coupling between the lattice and physical
  properties in Sr-214, this structural anomaly, which has escaped all previous
  studies until now. Our study provides much-needed keys to better understand the
  low-temperature magnetic and transport behavior, which remains a focus of
  current studies on the iridates. Indeed, the susceptibility of the physical
  properties to the lattice is further revealed by the application of in-plane
  dc electric current as a new stimulus that straightens the Ir-O-Ir bond angle,
  thus suppresses the canted AFM order. These results reveal
  crucial insights into the crystal structure and demonstrates that the subtle
  changes in the crystal structure, whether ambient or induced, dictate the physical
  properties in this spin-orbit-coupled Mott insulator.
  
  Single crystals of Sr-214 were grown using the self-flux method <cit.>.  The magnetic susceptibility and specific heat were measured using a
  Quantum Design Magnetic Property Measurement System. The neutron diffraction
  measurements were carried out at the HB1A triple axis spectrometer at the High
  Flux Isotope Reactor and the TOPAZ diffractometer at the Spallation Neutron
  Source, ORNL. For measurements at HB1A, a sample assembly of over 40
  single crystals (mass ∼ 25 mg, mosaicity ∼1.5^∘) was aligned at
  various scattering planes to probe the magnetic and nuclear reflections. A
  closed-cycle refrigerator was used to regulate the sample temperature (T).
  For measurements at TOPAZ, a plate-like single crystal with dimensions of
  1.8 mm×1.8 mm×0.55 mm was used. The sample temperature is
  controlled using the Oxford cryostream cooler Cobra 800. Electric current
  was applied using the power supply from BioLogic science potentiostat with a
  voltage range of ± 12 V and current range ± 200 mA.
  
  Figures 1(a)-1(b) show the main structural features of the Sr-214 with Ir ions
  forming a square-lattice network.  The system exhibits a characteristic
  staggered rotation of IrO_6 octahedra along the c axis at about ≈
  11.8^∘ <cit.>.  Because of the layered structure, the
  corresponding IrO_6 octahedra elongate slightly along the c-axis
  causing a crystal electric field (CEF) level Δ_ CEF∼200 meV
  <cit.>. Neutron scattering studies reveal a long-range
  magnetic order below 240 K [Fig. 1(e)]. Magnetic structure refinement
  indicates that the spins of Ir ions form a canted AFM configuration.
  The in-plane magnetic components closely track the rotation of the IrO_6
  <cit.>, which results directly from the SOI; the canting
  angle is governed by a characteristic ratio, ϕ∼ D/2J, of the
  antisymmetric Dzyaloshinskii-Moriya (DM) interaction D to the isotropic
  Heisenberg interaction J <cit.>.
  
  
  
  It is known that the onset of long range magnetic order at T_N causes no
  anomaly in ρ_ab or ρ_c <cit.>, specific heat (data
  not shown), and the lattice parameters [Figs. 1(c)-1(d)]. The conventionally
  anticipated correlation between the structural and physical properties is
  conspicuously  missing at T_N. However, such a correlation is instead established
  below a newly observed lattice anomaly, T_M=50 K.  As shown in
  Fig. 1(c), both the in-plane and c-axis resistivity exhibit noticeable kinks
  below T_M.  Recently, independent experiments have confirmed the anomalous
  character at low T. Among them, a magnetic study shows that the residual
  magnetization increases below T_N, peaks at T_M, and diminishes on
  further cooling <cit.>.  Such behavior was initially attributed to
  the reorientation of the canted spin moment <cit.>.  A muon spin
  rotation study by Franke et al. shows that the oscillation of single
  precession signal below the magnetic transition splits into high and low
  frequency components below 20 K <cit.>.  The authors of
  Ref. [franke11] suggest that the modification of the magnetic
  structure leads to structurally equivalent muon sites that experience
  increasingly distinct local field at low temperatures.
  
  Nevertheless, the abnormal low-T behavior demands a detailed examination of
  the crystal structure. The neutron diffraction measurements indicate that the
  system remains tetragonal at all temperature and does not show a lower
  symmetry below the magnetic transition <cit.>.
  Since the neutron coherent scattering length of the light oxygen atom (b_ O
  = 5.8 fm) is comparable to the much heavier strontium (b_ Sr=7.02 fm)
  and iridium ( b_ Ir=10.6 fm), the subtle change of the IrO_6
  rotation can be readily characterized using neutron diffraction and used to
  quantify its coupling to the magnetic order.  The structure factor of
  the nuclear scattering at a wavevector transfer q⃗ is
  
      |F(q⃗)|^2     =     |∑_i b_i exp(r⃗_i ·q⃗)|^2,
  
  where b_i is the coherent scattering length for individual atom inside the
  unit cell and r⃗_i is the corresponding atomic coordinate.
  The rotation of the IrO_6 octahedra leads to the nonzero peak
  intensities forbidden for the space group I4/mmm, e.g., (2,1,2n+1),
  and is solely determined by the in-plane oxygen atom O_1 located at
  (1/4-δ,1/2-δ,1/8) and equivalent positions, where
  δ≈0.05 characterizes the deviation from the undistorted IrO_6 scenario.  The intensity of the superlattice peak (2,1,1), I_211, can
  be derived as b_O (sinδ+sin 3δ)^2 and is proportional to
  δ^2 in the small rotation limit.
  
  
  
  The red squares in Fig. 2(c) show the (2,1,1) peak intensity 
  upon cooling. The gradual increase in intensity implies an
  enhancement of the IrO_6 rotation. Surprisingly, the monotonic evolution
  experiences an abrupt change below 50 K.  To further verify the feature,
  θ-2θ scans across the peak are performed (blue circles).  The
  marked decrease at low-T reveals that the suppression of IrO_6 octahedral
  rotation is intrinsic.  Comparing to 50 K, the value at 5 K is reduced by 5%,
  which translates to a suppression of rotation angle by 0.3^∘.  As
  the moment strictly follows the octahedral rotation, one expects a sudden
  reduction of the rotation or straightening of the Ir-O-Ir bond angle. This can
  be probed by examining the magnetic reflection (0,0,3) that is
  exclusively sensitive to the canted component <cit.>. Fig. 2(b) displays
  the T-dependence of the magnetic peak with long counting time.  A
  similar kink is observed below 50 K, reinforcing the structural anomaly in
  Fig. 2(c). This is also consistent with the abrupt reduction of the
  magnetization shown in Figure 2(a) with an external magnetic field of μ_0H = 0.1 T applied along either the [1,0,0] or the [1,1,0]
  directions, which also agrees well with our earlier report <cit.>.  These
  data point out a close association between the crystal structure and magnetic
  order, confirming an essential role of the pseudospin-lattice
  coupling that dictates the low-energy magnetic properties.
  
  
  
  This key characteristic of the lattice degree of freedom provides an effective
  “knob” to control the physical properties of the iridates using small
  external stimuli that readily couple to the lattice.  Electrical current, as a
  novel stimulus, is particularly effective in controlling the lattice, and thus
  the magnetic properties <cit.>. The previous x-ray study has shown a
  prominent lattice expansion for the undoped Sr-214 under electric current,
  accompanied by remarkable reduction in both transition temperature and
  in-plane magnetization. No structural anomaly is observed in Tb-doped Sr-214
  where long range magnetic order is absent <cit.>. To better
  characterize the structural response, we perform single crystal neutron
  diffraction measurements with in situ electric current application.  The
  orientation of the Sr-214 crystal is determined using the x-ray Laue method to
  ensure the currents are applied in the basal plane [inset of Fig. 3].  Since
  the diffractometer TOPAZ has a white incident neutron beam with large
  reciprocal coverage, the sample is mounted such that the characteristic
  (4,0,0) and (0,0,12) peaks can be simultaneously accessed
  [Fig. 3(a)]. In such fixed-orientation configuration, we collected a total
  number of around 2000 nuclear reflections at each temperature that allow a
  reliable structural determination. A electric current of 80 mA (current
  density ∼ 8 A/cm^2) is employed at 300 K and the sample is cooled to
  100 K while the electric current is maintained. The crystal is subsequently
  warmed to 300 K at a rate of 0.5 K/min.  Figs. 3(b)-3(c) display the
  T-dependence for the out-of-plane and in-plane lattice parameters with and
  without electric current, respectively. It is evident that the c-axis
  lattice parameter is insensitive to the current. In contrast, the in-plane
  lattice parameter displays a dramatic response to the current, consistent with
  the fact that the magnetic moments lie within the basal plane. The overall
  T-dependence of lattice parameter a at 80 mA stays above the one at I=0.
  While both data can be well described by a quadratic form, a slope change near
  the AFM transition is clearly visible [see the dashed curve in Fig. 3(c)]. The
  detailed structure refinement reveals that the Ir-O bond distances remain
  unchanged at I=80 mA.  The corresponding IrO_6 octahedron also remains
  slightly elongated, with a ratio of the Ir-O_2 to the Ir-O_1 bond
  distances being 1.04 throughout the temperature range measured.
  
  
  
  Although the IrO_6 octahedra remain rigid irrespective of the applied
  electric current, the in-plane Ir-O-Ir bond angle between the corner-sharing
  IrO_6 octahedra undergoes a considerable modification. Figure 4(b)
  compares the refined bond angle between the two situations. The angle at 0 mA
  stays smaller than 157^∘ and decreases with further reduction of
  temperature. A slope change occurs at the magnetic transition T_N at
  I=0 mA.  In sharp contrast, the bond angle becomes notably straighter
  or relaxed by 0.3^∘ when a current of I=80 mA is applied in the basal
  plane. In addition, these bond angle values show a smooth decrease upon
  cooling; the anomaly observed near T_N at the ambient condition is
  significantly suppressed. Indeed, the corresponding in-plane magnetic
  susceptibility displays the behavior consistent with the structural data
  [Fig. 4(a)].  The χ_a at 80 mA shows reduction in amplitude along with
  the suppression and broadening of the transition temperature.  The reduced
  magnetization is expected because the DM interaction D becomes smaller as
  the Ir-O-Ir bond angle relaxes due to the applied current (recall
  ϕ∼ D/2J).
  
  Tuning the physical properties, in particular magnetism using
  electric current, is highly desirable. It opens a new frontier for studies of
  the correlated and spin-orbit-coupled systems <cit.>.
  For instance, a reorientation of the AFM order is observed in the
  metallic Fe_1/3NbS_2 at a remarkable low current density
  (∼ 10^4 A/cm^2) showing the potential to build 
  AFM spintronics devices <cit.>. Spatially inhomogeneous
  rotation of the skyrmion lattice in the metallic helimagnet MnSi occurs when a
  current flows across the sample. This emphasizes the role of friction near
  the sample edges for skyrmion-based applications <cit.>. Much efforts
  have also been devoted to insulating materials such as ruthenate
  oxide Ca_2RuO_4 <cit.>.  Zhao et al. have shown that a extremely small
  electric current density (∼0.15 A/cm^2) can reduce the orthorhombic
  distortion and AFM order, and further induce a new orbital
  state featuring a simultaneous jump in both magnetization and electrical
  resistivity <cit.>. The authors propose that orbital occupancies
  stabilized by the nonequilibrium current drive the lattice changes and the
  novel phenomena.
  Because of the distinct energy hierarchy in the SOI system, the narrow band
  Mott insulator Sr-214 is more susceptible to such a perturbative approach.  It
  is anticipated the crystallographic details through this structural study
  would feed critical information for the electronic band structure calculations,
  and assist to explain the emerging phenomena including the negative differential
  resistivity, reversible resistance switching in Sr-214 <cit.>, and the nonlinear conductivity in other spin-orbit-coupled
  iridates <cit.>.
  
  In summary, neutron diffraction work of the square lattice Sr-214 reveals
  a pronounced reduction of the staggered IrO_6 distortion below 50 K.
  This important observation will help better understand the low-temperature
  physics of the archetypal spin-orbit-couple Mott insulator as the magnetic and
  transport properties closely tracks the underlying lattice. This point is
  further strengthened in that applying a steady electric current at 8 A/cm^2 readily reduces the in-plane IrO_6 octahedral rotation, which
  aligns well with the diminishing bulk magnetization. Our study demonstrates
  that the application of electric current is an effective and appealing route
  to tune the lattice and probe the rich physics in the SOI system.
  
  We thank Dr. George Jackeli for stimulating discussion. We acknowledge Dr.
  Bing Hu for her help in the magnetization measurement and the technical support for
  the electric current measurement from Junhong He, Gerald Rucker and Harley
  Skorpenske.  This research used resources at the Spallation Neutron Source and
  the High Flux Isotope Reactor, which are DOE Office of Science User Facilities
  operated by the Oak Ridge National Laboratory. Work at Univ. of Colorado was
  supported by the U.S. National Science Foundation via grant DMR-1903888.
  
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