Magnetic Field tuning of low energy spin dynamics in the single-atomic magnet Li$_2$(Li$_{1-x}$Fe$_x$)N
S. A. Braeuninger A. Jesche S. Kamusella F. Seewald M. Fix R. Sarkar A. A. Zvyagin H. -H. Klauss

APS/123-QED


Institute of Solid State and Materials Physics, TU Dresden, D-01069 Dresden, Germany

Institute of Physics, University Augsburg, D-86135 Augsburg, Germany

Institute of Solid State and Materials Physics, TU Dresden, D-01069 Dresden, Germany

Institute of Solid State and Materials Physics, TU Dresden, D-01069 Dresden, Germany

Institute of Physics, University Augsburg, D-86135 Augsburg, Germany



Institute of Solid State and Materials Physics, TU Dresden, D-01069 Dresden, Germany

Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer
Str. 38, D-01187 Dresden, Germany
B.I. Verkin Institute for Low Temperature Physics and
Engineering of the National Academy of Sciences of Ukraine,
Nauky Ave. 47, 61103 Kharkiv, Ukraine
V.N. Karazin Kharkiv National University, Svobody sq. 4, 61022 Kharkiv, Ukraine

[]henning.klauss@tu-dresden.de
Institute of Solid State and Materials Physics, TU Dresden, D-01069 Dresden, Germany


We present a systematic  ^57Fe-Mössbauer study on highly diluted Fe centers in single-crystals as a function of temperature and magnetic field applied transverse and longitudinal with respect to the single-ion anisotropy axis.
Below 30 K the Fe centers exhibit a giant magnetic hyperfine field of B̅_A=70.25(2) T parallel to the axis of strongest electric field gradient V̅_zz=-154.0(1) V/Å^2. Fluctuations of the magnetic hyperfine field are observed between 50 K and 300 K and described by the Blume two-level relaxation model. From the temperature dependence of the fluctuation rate an Orbach spin-lattice relaxation process is deduced.
An Arrhenius analysis yields a single thermal activation barrier of E̅_A=570(6) K and an attempt frequency ν̅_0=309(10) GHz. Mössbauer spectroscopy studies with applied transverse magnetic fields up to 5 T reveal a large increase of the fluctuation rate by more than one order of magnitude.
In longitudinal magnetic fields a splitting of the fluctuation rate into two branches is observed consistent with a Zeeman induced modification of the energy levels. The experimental observations are qualitatively reproduced by a single-ion effective spin Hamiltonian analysis  assuming a Fe^1+ d^7 charge state with unquenched orbital moment and a J=7/2 ground state. It is demonstrated that  a weak axial single-ion anisotropy D of the order of a few Kelvin can cause a two orders of magnitude larger energy barrier for longitudinal spin fluctuations. 




74.70.Xa, 74.25.nj, 76.60.Gv, 76.60.Es
                             

                              
Magnetic field tuning of low energy spin dynamics in the single-atomic magnet 
    H.-H. Klauss
    December 30, 2023
==============================================================================





§ INTRODUCTION

Single-molecule magnets (SMMs) and single-atomic magnets (SAMs) are model systems to study fundamental aspects of magnetic relaxation including quantum tunneling of magnetization and quantum coherence in nanoscale systems<cit.>. Moreover, SMMs and SAMs are proposed for novel data storage devices in spintronics or possible applications for quantum computing <cit.>.

Isolated magnetic moments with strong axial anisotropy exhibit a bistability of magnetization with an energy barrier to magnetization reversal resulting in slow magnetic relaxation. Fluctuation of magnetization is driven by  spin-lattice relaxation or hyperfine coupling processes. For thermally activated fluctuations the spin relaxation time τ at a temperature T is described by 

    τ =τ_0exp(E_A/k_BT),

controlled by an effective energy barrier E_A, an 'attempt time' τ_0, and with the Boltzmann constant k_B. 
The experimentally observed energy barrier E_A can be very different from the magnetic anisotropy energy E_m = D S^2. Magnetic quantum tunneling processes via degenerate S_z states above the ± S ground state can considerably reduce E_A. External magnetic fields applied longitudinal to the magnetic anisotropy axis can be used to tune the magnetic system through the level crossing condition for specific S_z states via Zeeman interaction. However, quantum tunneling can only occur if off-diagonal elements in the spin Hamiltonian cause a mixing of these S_z eigenstates and an avoided level-crossing with a finite minimum energy gap Δ_m,m' is generated. 
The Landau-Zener model <cit.> is applied to describe the magnetic quantum tunneling observed via magnetization steps in low temperature magnetization experiments, e.g. in nanomagnets such as Mn_12-ac <cit.> or [Fe_8O_2(OH)_12(tacn)_6]^8+, briefly Fe_8 <cit.>. In an external magnetic field applied transverse to the anisotropy axis, the quantum tunneling can be modulated by quantum interference <cit.>. 
This topological quantum phase interference, described by a Berry phase, is experimentally observed <cit.>. 

crystallizes in a hexagonal symmetry (space group P6/mmm) and alternating planes of (Li_2N) and (Li_1-xFe_x) are stacked along the crystallographic c-axis <cit.>. Figure <ref> shows the enhanced unit cell emphasizing the hexagonal symmetry of the Fe site and the corresponding linear N-Fe-N geometry. In Li_3N each N^3- ion is surrounded by eight Li^+ ions. Six Li^+ are located in-plane in a hexagonal geometry (Li-2c sites). Two Li^+ (Li-1b sites) are located between the planes leading to a hexagonal-bipyramidal geometry. The Fe ions occupy only the Li-1b site in between the Li_2N planes.
Studies on polycrystalline samples of concentrated with x≈0.16 and x≈0.21 by Mössbauer spectroscopy revealed a static hyperfine field below T_C≈ 65 K and long-range ferromagnetic ordering was proposed on the basis of magnetization studies <cit.>.
More recently, magnetization studies on large single crystals of with x = 0.001-0.3 were reported <cit.>.
  Large magnetic moments exceeding the spin-only value with a strong axial anisotropy parallel to the c-axis are found. 
  These magnetic moments can be associated with isolated Fe ions linearly coordinated with two Nitrogen ions in covalent N-Fe-N bonds <cit.>. 
From low temperature magnetization experiments on single crystals a magnetic anisotropy field of μ_0H≈88 T (x≈0.0032) was estimated together with a large effective magnetic moment μ_eff=6.5(4)μ_B per Fe atom parallel to the c-axis, largely independent of the Fe concentration <cit.>. 
For x≈0.28 an even larger magnetic anisotropy field of μ_0H≈220 T was reported <cit.>.
The deduced value of μ_eff is in agreement with the fully spin-orbit coupled Hunds rule value of an Fe^1+ configuration <cit.>. 
The observation of steps in magnetic hysteresis loops and relaxation phenomena with an energy barrier E_B≈430 K indicate a SMM-like behavior. The relaxation time is only weakly temperature-depended below 10 K indicating the importance of quantum tunneling in this temperature range <cit.>. However, the microscopic process of the thermally excited relaxation is not known. At low Fe doping concentrations data suggest  that the spontaneous magnetization and hysteresis is not caused by a collective magnetic ordering but rather due to the strong axial magnetic anisotropy in the linear N-Fe-N moiety <cit.>. 
A recent study reports a slow paramagnetic relaxation stressing the proposed ferromagnetic nature of nondiluted (x≈0.30) <cit.>.



Xu et al. <cit.> performed electronic structure calculations for which reveal large magnetic anisotropy energies of 305 K for an Fe^2+ d^6 with J=4 configuration and 360 K for Fe^1+ d^7 with J=7/2 configuration. Moreover, the authors propose that an Fe^2+ state could dominate at low x≪ 1 whereas the Fe^1+ state should play the major role at larger x. However, it is not clear how such strong axial anisotropy energies around 300 K can be reconciled with the observation of electronic level crossings in the magnetic hysteresis experiments at very low magnetic fields of μ_0H = 0.15, 0.55 and 3 T <cit.>, i.e. energy scales μ B/k_B of several Kelvin only.

To address these questions, in this manuscript we report a detailed ^57Fe-Mössbauer investigation on single crystals of highly diluted Fe in with x = 0.0275, 0.0109, 0.0099, and 0.0013.
The measurements were performed at temperatures 2 K ≤ T ≤ 300 K in magnetic fields 0 T ≤ B≤ 5 T applied transverse and longitudinal to the crystallographic c-axis (magnetically easy-axis). Below T = 30 K the Fe centers exhibit a giant magnetic hyperfine field of B̅_A=70.25(2) T parallel to the axis of strongest electric field gradient V̅_zz=-154.0(1) V/Å^2.
We demonstrate that the diluted Fe ions in  indeed form isolated single-ion paramagnets consistent with an Fe^1+ d^7 charge state and an unquenched orbital moment, i.e. total angular momentum J = 7/2. 
A continuous slowing down of the spin fluctuations is observed by Mössbauer spectroscopy below T = 300 K, which can be described by a thermally activated  Orbach process with an activation barrier of E̅_A = 570(6) K. 
The fluctuation rate is very sensitive to magnetic fields of the order of a few Tesla even at elevated temperatures of T ∼ 70 K. A quasistatic magnetic hyperfine field is observed below 50 K. A clustering of nearest neighbor Fe ions is ruled out by studies on samples with four different x≤ 0.028 proving the single atomic magnet behavior. The experimental observations are qualitatively reproduced by a single-ion spin Hamiltonian analysis. 
It is demonstrated that, for dominant magnetic quantum tunneling relaxation processes, a weak axial single-ion anisotropy D of the order of a few Kelvin can cause a two orders of magnitude larger energy barrier E_B for longitudinal spin fluctuations.



§ EXPERIMENTAL

Four single crystals (SCs) were investigated by ^57Fe-Mössbauer spectroscopy in this work.
The crystals were grown out of lithium rich flux <cit.>. 
The starting materials Li_3N powder (Alfa Aesar, 99.4 %), Li granules (Alfa Aesar, 99 %) and Fe granules (Alfa Aesar 99.98 %) were mixed in a molar ratio of Li:Fe:Li_3N = 6-x:x:1 with x = 1.5,0.5,0.5 and 0.1 for samples SC 1, SC 2, SC 3, and SC 4, respectively.
The mixtures with a total mass of roughly 1.5 g were packed into a three-cap Nb crucible  <cit.> inside an argon-filled glovebox. 
The crucibles were sealed in ∼ 0.6 bar Ar via arc welding and finally sealed in a silica ampule in ∼ 0.2 bar Ar. 
The mixtures were heated to T = 900 within 5 h, cooled to T = 750 over 1.5 h, slowly cooled to T = 500 over 60 h and finally decanted to separate the crystals from the excess flux. The composition was determined by inductively-coupled-plasma optical-emission-spectroscopy (ICP-OES) using a Vista-MPX. To this end the samples were dissolved in a mixture of hydrochloric acid and distilled water. Obtained Fe concentrations based on the measured Li:Fe ratio are given in Table <ref>. Magnetization measurements were performed using a 7 T Magnetic Property Measurement System (MPMS), manufactured by Quantum Design.




Mössbauer measurements were carried out in CryoVac and Oxford instruments helium flow cryostats in under-pressure mode or normal mode, respectively. We used a WissEl Mössbauer spectrometer. The detector was a proportional counter tube or Si-PIN-detector from KETEK and the source a Rh/Co source with an initial activity of 1.4 GB. The superconducting coil was powered by an Oxford instruments IPS 120-10 power supply with an applied magnetic field parallel or perpendicular to the γ-beam with an angle error of <5^∘. The absorber SC 1 exceeded the thin absorber limit requiring a transmission integral fit. The analysis was done using the Moessfit analysis software <cit.>. 
All measurements were performed with the γ-beam parallel parallel to the crystallographic c axis. The single crystals were protected by paraffin wax to avoid oxidation.


§ RESULTS




 §.§ Macroscopic Magnetization

Figure <ref> shows the isothermal magnetization of SC 1 measured at different temperatures for magnetic field applied parallel to the crystallographic c-axis, B_L ∥ c. The effective sweep rate for the full loops was 2.9 mT/s with 10 mT/s between the measurements. Hysteresis emerges for temperatures T < 50 K. At T = 2 K, steps appear at μ_0 B_L ≈ 0 and ± 3.3 T as well as for μ_0 B_L = 0.55 T, with the latter being recognizable only in the derivative dM/dB. At lower x, additional steps appear at μ_0 B_L = 0.15 T and the anomalies become sharper <cit.>. 
The M-H measurement shown in Fig. <ref> was performed after the Mössbauer experiment and is in good agreement with results published earlier <cit.>.




 §.§ Low temperature ^57Fe-Mössbauer spectroscopy at base temperature T_B


Mössbauer spectroscopy was performed at base temperature T_B≤4 K in zero-field (ZF) on the crystals SC 1-4. 
At this temperature the lifetime of the electronic states exceeds that of the nuclear states. Therefore, the hyperfine interactions are effectively stationary.

Fig. <ref> shows the ^57Fe-Mössbauer measurement at T_B in ZF of SC 1-4. For SC 1, two Fe sites A (green) and B (red) are observed. 
The black line is given by the total transmission integral fitting function


    T(v)=∫^∞_-∞L(E,v)e^-σ (E)t_a dE


with the normalized Breit-Wigner resonance cross section σ(E) depending on the energy E and an effective thickness t_a=2.39(10) reflecting a non-thin absorption limit. 
Therefore, the black line represents the transmission integral fit whereas the lines for the sites A and B show the natural line L(E,v).
T(v) is proportional to the line intensity of the recoil-free γ-ray, a function of the absorber thickness, and v is the Doppler velocity, for details see appendix. 
A thin absorber approximation is only valid for t_a<1 and then is the line shape described by a Lorentzian L(E,v) <cit.>. The fit is for SC 2-4 closer to the full thin absorption limit since the black line is replaced and shown by the green single Fe site A. 
The model used in Fig. <ref> is the static crystal Hamiltonian ℋ̂=ℋ̂_M+ℋ̂_Q+ℋ̂_Z
assuming the same electric monopole ℋ̂_M and  quadrupole interaction ℋ̂_Q for the two Fe sites A and B and independent Zeeman terms ℋ̂_Z. 
We deduced an isomer shift of δ=0.100(21) mm/s with respect to α-Fe at room temperature assuming a negligible second-order Doppler shift δ_SOD≈0 of the absorber at this temperature.



The electric monopole interaction between the nuclear charge Ze=26e of ^57Fe with the charge number Z and the s-electron charge density ρ_e(0)=-e|ψ(0)|^2_A/S at the nucleus is shifted by the energy E_A of the absorber material relative to the energy shift E_S of the source

and leads to an effective energy shift 


    δ=2π/5Ze^2S(Z){|ψ(0)|^2_A-|ψ(0)|^2_S}(R_(e)^2-R_(g)^2).


Here, R_(g)^2= ⟨r^2_(g)⟩ and R_(e)^2=⟨r^2_(e)⟩ are the mean square values of the radii of the ^57Fe nucleus of the excited state (e) with nuclear spin 3/2 and ground state (g) with nuclear spin 1/2, respectively. S(Z) is the dimensionless relativity factor which takes the spin-orbit coupling into account, e.g. for ^57Fe around S(26)≈1.32 or for neptunium S(93)≈19.4. 
These values vary slightly depending on the oxidation state.


The monopole interaction ℋ̂_M is given by a scalar ℋ̂_M≡δ+δ_SOD(T) as a function of the temperature. δ_SOD is the second-order Doppler shift and a direct consequence of the time dilation according to the relativity theory of the lattice dynamics. The γ-photon frequency ω_0 is shifted according to the transverse Doppler effect in the laboratory frame to

    ω=ω_0√(1-v^2/c^2)/1-vcosα/c≈ω_0(1+v/ccosα-v^2/2c^2)

where v is the velocity of the nucleus, α the angle between the movement of the nucleus and γ-photon absorption and c the speed of light. The last term assumes v≪ c. This yields in the Debye approximation the expression

    δ_SOD=-C[Θ_D+8T(T/Θ_D)^3∫_0^Θ_D/Tx^3dx/e^x-1]

with

    C=9k_BE_γ/16M_effc^2

where M_eff is the effective mass. Using this expression to analyze the temperature dependence of the central shift in SC 1 yielded a Debye-temperature of Θ_D=315(8) K. For details see appendix. 
To describe the electric quadrupole interaction ℋ̂_Q, e.g. SC 1 has a principle axis of the largest component of the EFG (electric field gradient) of V_zz=-154.1(2) V/Å^2,
denoted as usual,

    |V_zz|≥|V_xx|≥|V_yy|

and the introduced asymmetry parameter

    η=|V_xx|-|V_yy|/|V_zz|≤ 1.


This leads to the reduced quadrupole Hamiltonian

    ℋ̂_Q=eQV_zz/4I(2I-1)[3Î^2_z-Î^2+1/2η(Î^2_++Î^2_-)]

with the quadrupole moment Q and the raising and lowering spin operators Î_±=Î_x ± iÎ_y.
 
The pure quadrupole energy eigenvalues are given by

    E_Q(m)=eQV_zz/4I(2I-1)[3m^2-I(I+1)]√(1+η^2/3)

with I=3/2.
The negative sign of V_zz<0 corresponds to an elongation of the EFG charge distribution and an excess of negative charges in c-axis, the elongated case of the EFG ellipsoid <cit.>. 
The asymmetry parameter η=0  assuming axial symmetry was used due to the hexagonal structure. 

The magnetic hyperfine or Zeeman term ℋ̂_Z of the Hamiltonian ℋ̂_hf is given by ℋ̂_Z=-g_Nμ_NÎ·𝐁̂
with nuclear Landé factor g_N, the nuclear magneton μ_N=eħ/2m_pc, the proton mass m_p and the magnetic field 𝐁.

Taking the scalar |𝐁|=B and expresses ℋ̂_Z by the polar angle Θ and the azimutal angle Φ of 𝐁 relative to direction of V_zz yields

    ℋ̂_Z=-g_Nμ_NB(Î_+e^-iΦ+Î_-e^+iΦ/2sinΘ
    
    +Î_zcosΘ).

The values of the magnetic hyperfine fields for the two Fe subspecies converged to B_A=70.21(1) T and B_B=65.0(2) T. Site A is the dominant Fe site. Site B  is only observed in SC 1 containing the highest Fe concentration with an  intensity fraction of 5.9(3)%.

The two transitions

    m=±1/2→ m=±1/2

with Δ m=0 are not observed in the spectra of Fig. <ref>.

The relative line intensities W(β) depend on the polar texture angle β describing the direction of the incident γ-ray with respect to the magnetic hyperfine field direction, W(β)∝sin^2β. For the analysis Θ=0 was taken.  
The angle β between the principle axis (largest component) V_zz of the EFG tensor and the γ-beam was assumed to be an identical fit parameter for the monomer site A and the dimer site B. The resulting value β = 0 proves that the magnetic hyperfine field and V_zz are aligned parallel to the c-axis.
The result is the observed ratio of the spectral line intensities of 3:0:1:1:0:3.

Fig. <ref> shows the measurements of SC 2-4 at T_B. 
No indications for Fe site B are observed in SC 2 as well as in SC 3 and SC 4. 
The green line is the fit of the model of the static crystal Hamiltonian ℋ̂_hf with an isomer shift δ=0.119(20) mm/s and a principle axis of the EFG of V_zz=-154.2(4) V/Å^2. The asymmetry parameter is assumed to be η=0. 
The fit yielded a magnetic hyperfine field B_A=70.24(1) T parallel to V_zz of the EFG tensor and parallel to the γ-beam as well. Table <ref> shows the obtained hyperfine parameters of SC 1-4 and the calculated mean values of V̅_zz, B̅_A, lnν̅_0 and E̅_A. 
The hyperfine parameters are nearly concentration-independent. 
The  absolute values of the magnetic hyperfine fields B_A and B_B are above typical spin-only values in solid state systems and can be understood in terms of a strong unquenched orbital contribution.   
The analysis to obtain the fluctuation rate parameters lnν̅_0 and E̅_A of the Arrhenius temperature dependence are described in appendix <ref>.





 §.§ Zero Field ^57Fe-Mössbauer Spectroscopy for T_B< T <300 K




Above 50 K the ^57Fe nucleus interacts with a fluctuating magnetic hyperfine field. Fig. <ref> shows representative spectra between 60 K and 200 K of SC 1 with x=2.75%. In the following we will only consider Fe site A, site B is neglected in this analysis. The fit represents a Blume dynamic line shape model in the presence of quadrupole hyperfine interactions for two states, described by absorption cross section

    σ=-σ_aΓ_0/2Im∑_αSp(V̂_α⟨W|Â^-1(ω,Θ)|1⟩V̂_α^+).

Here, V̂_α is the operator of hyperfine interactions of the γ-beam with polarization α and the nucleus, σ_a the effective absorber thickness and ⟨W| and |1⟩ as described by Chuev and therein <cit.>. The superoperator

    Â(ω,Θ)=ω̃+iΓ_0/2-𝐋̂(Θ)+i𝐏̂.

is defined by the Liouville operator of hyperfine interactions 𝐋̂(Θ), the resonance transition energy E_0 is given by the corresponding frequency ω̃=ω-E_0/ħ, Γ_0 the width of the excited nuclear level and 𝐏̂ the matrix of hyperfine transitions <cit.>.

The initial conditions for the analysis are identical to the static case at 2 K. A two level relaxation model was used taking into account an electronic spin reversal process. The magnetic hyperfine field fluctuates with the fluctuation frequency ν between the two values +B_A and -B_A. 
Above 60 K the spectral lines begin to broaden due to the fluctuations, see Fig. <ref>. With increasing ν at 70 K the two internal lines collapse first  yielding a singlet at 100 K. 
At 150 K and above, the left resonance line of the quadrupole doublet, which is expected to appear in the fast relaxation limit ν→∞, results from the collapse of the external lines <cit.>. The Arrhenius parameter E_A and lnν_0 are obtained by an Arrhenius analysis

    lnν=lnν_0-E_A/k_B(1/T)

of the extracted fluctuation frequencies lnν of SC 1-4.
In this analysis, the values of ν for T<50 K are not considered since these value reflect the lower bound of the fluctuation rate which the Mössbauer spectra analysis can resolve. 
This yielded a thermal activation barrier of E_A=552(26) K and lnν_0=12.36(32) GHz for SC 1. 
The fluctuation frequency ν of Fe site A is essentially concentration-independent in SC 1-4. 
Table <ref> shows the Arrhenius plot fit parameter of SC 1-4, for details see appendix.



 §.§ ^57Fe-Mössbauer Spectroscopy in transverse magnetic Fields B_T
 

In the following we present the results of systematic Mössbauer spectroscopy experiments under applied transverse magnet field B_T. These experiments were performed on sample SC 1. For eight temperatures between 30 K and 247 K a magnetic field up to 5 T was applied perpendicular to the normal vector of the sample plate and therefore to the crystallographic c-axis and perpendicular to the γ-beam, see discussion. Therefore, the field was applied perpendicular to the quantization axis of the Fe spins which is identical to the low temperature orientation axis of the magnetic hyperfine field at the Fe nucleus. 
In this geometry, an increasing field B_T leads to an increasing mixture of the S_z-eigenstates of the electronic spins and an increasing fluctuation rate of the magnetic hyperfine field is expected supported by a theoretical treatment based on the minimal spin Hamiltonian of a single-ion in the next subsection.

Figure <ref> shows four typical Mössbauer spectra in different transverse magnetic fields (TF) up to 5 T. The experimental data clearly reveal an increase of the fluctuation frequency ν with increasing field strength. The temperature and field range for these experiments was chosen such that the slowly fluctuating magnetic hyperfine field of ≈70 T can be regarded as the dominant hyperfine interaction with the ^57Fe nuclei and the Blume model of axial fluctuations of the magnetic hyperfine field described in the former subsection can be used for the quantitative analysis (solid lines in Fig. <ref>). For higher fields the vector sum of the external field and the internal magnetic hyperfine field must be considered.

At T = 70 K, ν has increased in 1 T by a factor 2 and in 5 T by a factor 8. This documents a strong transverse field sensitivity.


The dependence of the fluctuation frequency ν on the transverse magnetic field B_T and temperature T is investigated in detail for SC 1. Fig. <ref> shows the logarithmic frequency log_10 as a function of the inverse temperature 1/T for different transverse external fields and Fig. <ref> shows the logarithm of ν as a function of the transverse magnetic field for different temperatures.
The used fit function is discussed in the spin-Hamiltonian part below and in the appendix. In Fig. <ref>, at low temperatures 1/T > 0.01 K^-1, a pronounced field-induced non-linear deviation from the zero-field Arrhenius line is observed. 
For high temperatures 1/T<0.01 K^-1 the data converge to the zero-field Arrhenius line, i.e. the temperature-induced fluctuations are dominant.
This is also seen in Fig. <ref>: the change of ν with increasing B_T is enhanced by lowering the temperature. Note that for the lowest temperatures (30 K and 40 K) the determined fluctuations rates are close to the lower bound of the frequency window of the Mössbauer method due to the effective time window.




To describe the change of the spin fluctuation rates induced by the applied transverse magnetic field  we considered a simplified perturbation proportional to B_T   J_x ≡ B_T   (J_++J_-)/2 . Such a term can mix the states with different J_z, however with one difference from the O_6^6 processes discussed in section <ref> below, namely Δ J_z=1. We have described our experimentally observed fluctuation rate data by a function linear in B_T. The data can be described with the phenomenological model function

    ν(T,B_T)=ν_0e^-Δ_1/T+ξ B_Te^-Δ_2/T.

The first term describes the field-independent temperature-activated Arrhenius-contribution observed in the ZF experiments using ν̅_0 and Δ_1=E̅_A/k_B 
(see Fig. <ref> and black line in Fig. <ref>). The second term describes the increase of ν due to the transverse field scaling linear 
with B_T. In a global fit ν̅_0, ξ and Δ_i are constant parameters. The result is ξ=2332(995) MHz/T and Δ_2=251(20) K. We associate this relaxation process with a second Orbach process which is observed by ac susceptibility below 30K (see Fig. 13). The applied transverse field increases the attempt frequency so that it becomes detectable  within the Mössbauer frequency window. 




 §.§ ^57Fe-Mössbauer Spectroscopy in longitudinal magnetic Fields B_L

^57Fe-Mössbauer spectroscopy measurements were performed with applied longitudinal magnetic fields (LF) at 100 K up to 3 T with the γ-beam parallel to the applied field parallel to the c-axis of the crystal.




Fig. <ref> shows the ^57Fe-Mössbauer measurements at 100 K up to 2 T longitudinal magnetic field (LF). The measurements at 0.5 T and 1 T show an increase of the linewidth of the central absorption line compared to the ZF spectrum. The spectra at 2 T clearly reveals a splitting into two lines corresponding to two different fluctuation rates.


The analysis model to describe the LF spectra is the Blume two-state spin reversal fluctuation model between the states with hyperfine fields B_h= B_A+B_L and B_h=- B_A+B_L. Since the Zeeman interaction will lift the degeneracy between the "spin up" and "spin down" transitions two different fluctuation frequencies ν_up describing the frequency to flip the spin into longitudinal magnetic field direction and ν_down to flip it against the applied field direction are considered. The population of the two states are assumed to be the same as shown by the equal central line intensities at 2 T in Fig. <ref>. Note that a small static external field at the ^57Co-source  caused by the Helmholtz magnet leads to a slight increase of the linewidth (0.24(2) mm/s at 2 T). 


Fig. <ref> shows the deduced  frequencies log_10ν_up and log_10ν_down as a function of the longitudinal magnetic field B_L. The observed change of the fluctuation rate is one order of magnitude smaller than in the case of applied transverse fields. The data show a linear dependence of log_10ν_up and log_10ν_down as a function of B_L up to 3 T. We clearly observe an asymmetry of the observed positive and negative frequency changes, i.e. a stronger increase of ν_up(B_L)
than decrease of ν_down(B_L). This  cannot be explained by the Zeeman-induced decrease of the energy differences for the transition J_z = -7/2→ J_z=5/2 and increase of the energy difference for J_z=7/2→ J_z = -5/2 since these changes are of equal absolute value. The experimental slopes are given by 0.084(5) log_10[MHz]/T for ν_up and -0.015(3) log_10[MHz]/T for ν_up, respectively. The theory curves (orange lines) shown in Fig. <ref> will be discussed in the  section <ref>.



 §.§ Effective single-ion J_eff=7/2-Hamiltonian calculation of spin dynamics
 

A striking result of the temperature and transverse magnetic field dependent Mössbauer spectroscopy is that  the activation energy scale for thermal fluctuations of the individual electronic Fe spins E_A ≈ 570 K is two orders of magnitude larger than the Zeeman energy g_tμ_B B_x S_x∼5 K which is needed to induce similar changes of the fluctuation rate.  
Moreover it is important to note, that the low temperature longitudinal magnetic field magnetization data on this system presented in Fig. <ref> also reveal an energy scale for longitudinal magnetic field induced system changes of the order of 1 to 5 K from the appearance of level crossing induced magnetization steps at μ_0H_z = 0.15, 0.55, and 3 T.

For a qualitative understanding of the transverse magnetic field and temperature dependence of the spin fluctuation frequency ν we present a calculation of the spin dynamics using a single-ion spin Hamiltonian model. 
We demonstrate that an axial anisotropy of energy scale D≪ 570 K, consistent with the Zeeman response of the system, can indeed give rise to an effectively two orders of magnitude larger energy barrier for thermal fluctuations. Moreover, qualitatively, the obtained results can be extended to a broad class of SAM and SMM by the introduced effective mixing term.

We consider the single-ion properties of Fe ions in Li_2(Li_1-xFe_x)N. 
Considering spin-orbit interaction and the hexagonal point symmetry of the Fe site (1b Wyckoff site, point symmetry 6/mmm <cit.>), the crystalline electric field yields the single-ion magnetic anisotropy  H_an = B_2^0O_2^0 +B_4^0O_4^0 +B_6^0O_6^0 +B_6^6O_6^6, where O_n^m are Stevens' operators, and B_n^m are the parameters of the magnetic anisotropy <cit.>. 
The Fe ions in Li_3N can be either Fe^+ (which, according to Hund's rules have the lowest multiplet with S=3/2, L=3 and J=9/2), or Fe^2+ (with S=2, L=2, and J=4). 
For diluted Fe in α-Li_3N, however, we can apply the arguments used in Refs. Klatyk2002, PhysRevB.66.184434, where a ground state of |M_L=-2,M_S=-3/2⟩ was proposed. The oxidation state Fe^1+ d^7 is consistent with our obtained hyperfine parameters, see discussion.
The arguments, namely, the strong uniaxial anisotropy of the hexagonal lattice (due to O_2^0, O_4^0 and O_6^0 operators, which distinguish only z∥ c axis) lifts the degeneracy, and only L=2 states are coupled with S=3/2 for Fe^+ in α-Li_3N. 
It yields the effective total moment J_eff = 7/2.
The splitting of the Mössbauer lines (see Fig. <ref>) also confirms that assumption: The splitting is too large for pure spin states S=3/2 or S=2. In what follows we call J a pseudo-spin and drop the index “eff” for simplicity. However, qualitatively, the calculated results below are independent of the detailed value of J.

The most important part of the Hamiltonian H_an  can be written as  H_an≈ -DJ_z^2, where D is the parameter of the magnetic anisotropy.  We can conclude from the magnetization experiments and the spin-reversal Blume model with axis c ||B_A||V_zz, that we deal with “easy-axis” magnetic anisotropy, D > 0. Consider the Hamiltonian of the Fe ion in the external magnetic field B_z=B_L, directed along the “easy” axis, namely H_0 = -DJ_z^2 -g_lμ_BB_zJ_z, where μ_B is the Bohr magneton, and g_l is the (longitudinal) z-component of the effective g-tensor. The levels of that Hamiltonian cross each other at several values of B_z, depending on the  value. The only Stevens' operator from H_an, which does not commute with J_z, and, hence, which can mix states with different values of J_z and lift the degeneracies at the crossover points, is O_6^6 ≡ (J_+^6 +J_-^6), where J_± =J_x± iJ_y. Such a mixing is the crucial point for the quantum tunneling <cit.>. Notice that according to standard quantum mechanics in the basis with diagonal action of the operator J_z the eigenstates of O_6^6 for J=3/2 and J=2 are zero. The operator O_6^6 corresponds to the processes with Δ J_z=6, hence connecting the states J_z = -7/2 with J_z=5/2, and J_z = -5/2 with J_z=7/2. 

Unfortunately, the explicit results for the relaxation rate due to quantum tunneling cannot be realized for O_6^6 because of the numerical effort. 

Therefore, to mimic the action of the operator O_6^6 we consider a more simplified perturbation related to the transverse magnetic field, for example, J_x ≡ (J_++J_-)/2. Such a term also can mix the states with different J_z, however with one difference from the O_6^6 processes, namely  Δ J_z=1. This substitution of O_6^6 by J_x, while giving the opportunity to obtain a qualitative agreement with the results of our experiments, still cannot give a full quantitative description of Li_2(Li_1-xFe_x)N. 

Summarizing, we consider, an effective Hamiltonian, which permits quantum tunneling in Li_2(Li_1-xFe_x)N. It has the form

    H = -DJ_z^2 -g_ttμ_BB_xJ_x - g_lμ_BB_zJ_z,

where g_t is the value of the effective g-tensor in the plane, transverse to the easy axis. 
Note that B_x can include not only the effective field, introduced to mimic the action of the O_6^6, but also internal (dipole) or external magnetic fields applied transverse to the z axis, i.e. perpendicular to the crystallographic c axis. 
According to <cit.> the lowest (2J+1) eigenvalues and eigenfunctions of that Hamiltonian coincide with those of the discrete spectrum of a quantum particle in the effective potential

    U= DB^2/4[sinh (x) -C/B]^2-DB/2 (2J+1)cosh (x),

where B=g_tμ_BB_x/D and C=g_lμ_B B_z/D. 
The spin quantum tunneling in that approach is totally equivalent to the tunneling of that quantum particle between the minima of the potential U. The tunneling rate can be calculated using the Euclidean version of dynamical equations, using dynamics of instantons of the Eulcidean action, i.e., solitons, connecting two minima of the potential U with each other <cit.>. Consider the range of the field values, limited by the region [B^2/3 +C^2/3]^3/2 < 2J+1, in which the potential U has two minima (the lowest minimum is related to the stable state, and the highest one to the metastable state). The energy barrier between the minima is finite, hence there exists a probability for the metastable state to decay due to the quantum tunneling. It is possible to calculate the values of the relaxation rate due to the quantum tunneling <cit.>, expanding the expression for U near the position of the metastable minimum. The decay rate is determined by the analytic continuation of the energy value to the complex plane. 
Analyzing the results obtained this way, we conclude that two regimes, T_0 ≤ T ≤ T_1, and T ≥ T_1, where T_0 =  D√(a)/π k_B, and T_1 = 2D√(a)/π k_B can be related to the conditions of our experiments with Li_2(Li_1-xFe_x)N. 
Here and below we use the notations

    a= 3^1/2/2^3/2 BC^1/3(2J+1)^2/3δ^1/2 , 
          b=1/12 B^4/3 C^1/3(2J+1)^1/3 , 
          δ = 1 - ( B^2/3 +C^2/3)^3/2/2J+1 .

For T_0 ≤ T ≤ T_1, i.e., at low temperatures for our experiment, the relaxation rate can be approximated as, according to <cit.>,

    ħγ_1 = D √(a)sinh(D√(a)/k_B T)/πsin (D√(a)/k_B T)×
          ×exp[ -4Da^3/27b^2k_B T] .

On the other hand, for higher temperatures T ≥ T_1 the relaxation rate is

    ħγ_2 = D √(a)/πexp[ -4Da^3/27b^2k_BT].

This higher temperature behavior of the relaxation rate caused by the quantum spin tunneling is similar to the Orbach relaxation <cit.>, i.e., it has the Arrhenius form. 
Notice that the “true” quantum spin tunneling-induced relaxation rate exists only at T = 0 <cit.>.

Given that the pre-expontial factor, ν_0, obtained from Mössbauer spectroscopy is temperature-independent (as the pre-exponential factor in equation <ref>), the most essential regime for our experiments with Li_2(Li_1-xFe_x)N is the region with T ≥ T_1. 
We see that the relaxation rate γ_2 follows an Arrhenius law in the temperature dependence, γ_2 = ν_0 exp(-E_A/k_BT) with the prefactor ν_0 and the activation energy E_A determined as

    ν_0 = D√(a)/πħ∼ B_x^1/2 B_z^1/6δ^1/4 , 
          E_A = 4Da^3/27b^2∼ (B_xB_z)^1/3δ^3/2 .

In detail, 

    E_A≈ 4.89B^1/3C^1/3(2J+1)^4/3δ^3/2

where the factor (2J+1)^4/3 is the dominant scaling factor for J>3/2. 

Note that we cannot use the limits B_x → 0, and B_z → 0 in the expression for the relaxation rate caused by the spin tunneling, because the latter is absent there: Without B_z there is no crossover, and without B_x there is no lifting of the degeneracy of crossover points). However, we can evaluate the field dependence of the activation energy, not taking into account the limiting cases B_x → 0 and B_z → 0.
A weak effective tilted magnetic field can  originate, e.g., from the long-range magnetic dipole-dipole interaction in the mean field approximation. 
We also suppose that the region of applicability of the results can be expanded to all δ < 1, which implies the  difference between the potential U and its expansion near the position of the metastable state being small (this difference produces higher-order quantum corrections). 
The result is shown in Fig. <ref> for J=7/2.

We see that for very small but finite values of the components of the external magnetic field the activation energy is much larger than the value of the magnetic anisotropy D. 
It explains the observation of the giant activation energy for the relaxation rate in our Mössbauer studies of Li_2(Li_1-xFe_x)N. 
Furthermore, we see that the application of the external field of the order of (2J+1)D reduces drastically the value of the activation energy. 

Now we can compare the transverse field dependence of the relaxation rate, extracted from Mössbauer experiments in Li_2(Li_1-xFe_x)N with the calculated one. 
In Fig. <ref> a) the logarithm of the relaxation rate γ_2 is plotted as a function of the applied transverse field B_x at g_lμ_BB_z=0.001 for J=7/2 and several values of the temperature. 
To have better agreement with experiment we have to add the constant g=3.5 to log_10(γ_2), which implies additional sources of relaxation that are temperature- and magnetic field-independent.










 Fig. <ref> b) shows the logarithm of the relaxation rate γ_2  as a function of the inverse temperature, 1/T, for several values of the transverse external magnetic field B_x.









We see that the general tendency is well described by our simplified theory, while there is no quantitative agreement. 


We conclude that this single-ion theory, based on the spin properties of Fe impurities, which at low energies produce quantum spin tunneling, well reproduces the most dramatic feature of dynamical experiments in Li_2(Li_1-xFe_x)N: the giant value of the activation energy in the Arrhenius law for the temperature dependence of the relaxation rate, and much smaller values of the external magnetic field, which drastically change that relaxation rate. 



 §.§ Zeeman Analysis of Spin Dynamics in longitudinal Fields B_L
 

The splitting of the  resonance line in longitudinal fields (Fig. <ref>) can be understood as a consequence of the Zeeman term in the effective spin Hamiltonian. 
For B_z=0 the relevant relaxation processes with Δ J_z=± 6 introduced in the last section are equivalent. However, a finite longitudinal field B_z=B_L removes the degeneracy of the ± J_z energy levels via Zeeman interaction. 
The values of the energy differences between the states J_z = -7/2 and J_z=5/2, and between the states J_z = -5/2 and J_z=7/2 become non-equal (see Fig.  <ref>).

To calculate these energy differences we need to specify the longitudinal g-factor 
g_l. Here, we model the complex many electron state of the Fe ions including strong spin-orbit interaction  by a simplified  J = 7/2 system with a fixed Landé-g-factor value of g_l = 1.55 (based on L=2 and S=3/2, see appendix) for both, the ground state (assumed to be J_z = ± 7/2) and the excited state (assumed to be J_z = ± 5/2). This choice of g_l leads to an effective paramagnetic moment μ_eff=g_l μ_B√(J(J+1))≈ 6.15  μ_B which is consistent with the experimental value.


 





According to the Arrhenius law the change 
of the spin fluctuation rate with respect to its ZF value can be calculated according to 

    ν (B_L) / ν (B_L = 0) = exp(E_Z/k T).


Here, E_Z=± g_lμ_B Δ J_z B_L is the Zeeman term with the transition rule ±Δ J_z. The ±-sign corresponds to the positive and negative branches in Fig. <ref>. To calculate  the Zeeman induced change of the spin fluctuation rate ν (B_L), we use Δ J_z = 6 according to the hexagonal symmetry. The resulting slope of ν vs. B_L is

    Δlog_10ν/Δ B_L≈±0.027/T.

The results of this calculation are included in Fig. <ref> as orange lines. The calculated values for ν(B_L) are below the experimental results for the branch ν_up(B_L) as well as the branch ν_down(B_L). This is expected since a second contribution stemming from the B_L dependence of the B^6_6O^6_6 term in the crystal field spin Hamiltonian is not included in this model. Such a term  is always positive, linear in B_L and identical for both branches.
Therefore, the  Zeeman contributions to ν_up and 
 ν_down must be located below the experimental values, ideally shifted by  identical values with respect to the experimental values.
 The latter is not fulfilled (crf. Fig. <ref>), however increasing the absolute value of both slopes (equation <ref>) by ≈ 80% would lead
 to such a situation. Note that a correction of this size is feasible since from all experiment performed on so far we cannot  determine the ground state and excited state values of J_z exactly. Moreover, also the longitudinal g-factors g_l for both states in this effective spin Hamiltonian approach can be modified strongly due to the subtle interplay of the crystal electric field with the spin-orbit coupling in this 3d^7 state. 
 A more realistic many-body electronic structure calculation is needed to calculate the effective crystal field energies as well as the longitudinal and transverse g-factors of the ground and excited states seperately.



§ DISCUSSION




 §.§ Mössbauer Sites and Sample Homogeneity


Two Fe sites A and B are observed in the low temperature Mössbauer spectroscopy on sample SC 1. The main site A is associated with monomer Fe sites without relevant magnetic exchange with other Fe ions since it is observed also in samples SC 2-4 which contain an up to one order of magnitude lower Fe concentration x.
Site B is not observed in SC 2-4. We associate site B with a nearest neighbor in-plane or out-of-plane Fe-dimer site. The magnetic hyperfine field  for the two Fe subspecies is determined to B_A=70.21(1) T and B_B=65.0(2) T at 2 K. These values are in agreement with Refs. <cit.>, in which Klatyk et al., Ksenofontov et al. have performed a powder study of x > 0.15 proposing ferromagnetic ordering for T<65 K.


The temperature-dependence of the Mössbauer spectrum shown in Fig. <ref> is consistent with the expected behavior of SAM. The observed spin fluctuations are consistently described by a thermal activation crossover rather than by a cooperative long-range ordering transition. However, this does not exclude by itself that Fe site A arises from small cluster-like SMM units like Fe_i clusters in the Li_3N matrix with ferromagnetic interaction between the Fe ions with various size numbers i of Fe depending on x. The deduced hyperfine parameters are within error bars identical for samples SC 1-4. The spin dynamics described by the fluctuation frequency ν (T), ν_0 and E_A are concentration-independent for Fe site A of SC 1-4. The invariant parameters as a function of x proves well isolated Fe sites like in a SAM. 




 
 
  
  

A combinatorial expression to calculate the probability for n Li ions among six neighbors in the [001] plane for the Fe concentration x yields

    W_n=6![n!(6-n)!]^-1(1-x)^nx^6-n≈ 14%

for x = 0.027 and n= 5, i.e. an in-plane Fe-dimer <cit.>. This value is twice as large as the observed value. The area contribution of site B is overestimated in this statistical treatment in which every kind of Coulomb repulsion is neglected. Either due to Coulomb repulsion a more homogeneous mononuclear SAM is preferred or an out-of-plane Fe-N-Fe dimer configuration is the observed site B.

Interestingly, the total contribution of the Fe-N-Fe in a binomial distribution is supposed to be ≈ 5.3% which is closer to the experimentally determined value of 5.9(3)% of Fe site B. 
A systematic Mössbauer study on a series of  with larger x on single-crystals is needed to identify nearest neighbor Fe-cluster configurations in this system. 
However, this is beyond the scope of this work.




 §.§ Oxidation and Spin State of Fe Ions in 




The observed isomer shift value around 0.10 mm/s is unconventional for a Fe oxidation states of Fe^1+ or Fe^2+. 
It can possibly arise from the linear N-Fe-N low-coordinated electronic structure of Fe in . Because of the paramagnetic behavior a Fe^2+ low spin state S=0 can be excluded.





A ^57Fe-Mössbauer study was performed on the linear complexes [K(crypt-222)][Fe(C(SiMe_3)_3)_2] and [Fe(C(SiMe_3)_3)_2] with a similar Fe linear coordination by carbon <cit.>. 
[Fe(C(SiMe_3)_3)_2]^1- in [K(crypt-222)][Fe(C(SiMe_3)_3)_2] is proposed to contain Fe^+1, whereas Fe^2+ is present in [Fe(C(SiMe_3)_3)_2]. 
The assumed asymmetry parameter η=0 is according to the axial symmetric EFG tensor discussed by Lewis et al. <cit.> (and references therein).

Table <ref> shows the values of the isomer shift δ, the quadrupole splitting Δ E_Q and the magnetic hyperfine field B. 
The smaller δ-value of Fe site A can be explained by the increase of s-electron density at the nucleus and the 4s-mixing. 
The EFG value V_zz, here given by Δ E_Q of Fe-diluted Li_2(Li_1-xFe_x)N (site A) and the Fe^1+-SMM, [Fe(C(SiMe_3)_3)_2]^1-, are very close to each other whereas the Fe^2+-SMM shows only half of this value. 
Moreover, also the magnetic hyperfine fields of [Fe(C(SiMe_3)_3)_2]^1- and Fe-diluted Li_2(Li_1-xFe_x)N are comparable. 
Therefore, we conclude a strong similarity of the Fe electronic systems in these two systems with oxidation state Fe^+1 for . An oxidation state of Fe^1+ is also consistent with calculated electronic band structure <cit.>.

















 §.§ Energy Barrier and Spin Dynamics


Fig. <ref> shows the temperature-dependence of the spin fluctuation rate of SC 1, determined from Mössbauer spectroscopy, ac-susceptibility  and direct magnetization relaxation measurements, respectively. 
At three Mössbauer data points at 1/T ≫ 0.05 K^-1 essentially static Mössbauer spectra are measured, i.e. the fluctuation rate becomes smaller than the lower bound of the frequency window of the method and these data points are not shown in Fig. <ref>.








The relaxation rates, ν, obtained by Mössbauer and ac susceptibility measurements can be well described by a single effective energy barrier of E̅_A=515(20) K.

Note that there is another, larger peak observable in the temperature-dependent ac susceptibility that corresponds to a faster relaxation with an effective energy barrier of E̅_A = 270(12) K (grey stars in Fig. <ref>).
The pre-exponential factor of the Arrhenius behavior amounts to ν_0 = 150(10) GHz.
We conclude that this relaxation process is enhanced  by the application of transverse magnetic fields and becomes visible in the Mössbauer frequency window (see section <ref> above). The deduced activation energies are consistent within error bars.
For T < 10 K the relaxation rates were determined by fitting the time-dependent magnetization to stretched exponential M(t) = M_0 exp[-(t ν)^β].

At zero external field the two allowed phonon-assisted relaxation processes J_z = -7/2 →  J_z=5/2 and J_z=7/2 → J_z = -5/2 have equal energy differences. They become non-equal under applied longitudinal magnetic field as presented above.
The direct quantum tunneling regime  is reached below 10 K (see Fig. <ref>).

In table <ref> we compare the thermal activation energy barriers for several SAM systems with large energy barriers E_A compared to Fe-diluted Li_2(Li_1-xFe_x)N. The thermal activation barrier E_A is often associated with a two-phonon Orbach process <cit.>. Above 50 K the dominant character of this process is plausible: the direct spin transition process in the Debye model accompanied by the creation or annihilation of a single phonon is dominant only for lower temperature T<50 K with τ_S^-1∝ T.


In the literature the energy barrier is often identified with the zero-field splitting value D rather than considered as an effective experimental quantity, which depends on different microscopic parameters as discussed above. However, as demonstrated by our spin Hamiltonian approach, the energy barrier is a function of the (internal or applied) transverse magnetic field
 and a general scaling proportional to (2J+1)^4/3 (crf. eqn. (23)). The effective spin Hamiltonian calculation presented in this work can qualitatively account for the temperature and transverse field dependencies of the experimentally observed spin fluctuation rates. 

The spin dynamics in applied longitudinal magnetic fields can be understood considerung the Zeeman shift of the J_z states which induces a splitting of the spin fluctuation rate into two branches. The observed experimental asymmetry is expected theoretically and is caused by higher-order Stevens' operator terms produced by the hexagonal symmetry of the lattice.

























§ CONCLUSIONS


In this work, we present ^57Fe-Mössbauer studies on

diluted Fe centers in a linear N-Fe-N configuration along the crystallographic c-axis in single crystalline specimen of hexagonal . 

The homogeneity of the nanoscale distributed isolated Fe centers is shown and the single-atomic magnet nature confirmed. 
Below 30 K the magnetically isolated single-ion Fe centers exhibit a large quasistatic magnetic hyperfine field of B̅_A=70.25(2) T parallel to the c-axis which is the strongest principle axis of the electric field gradient V̅_zz=-154.0(1) V/Å^2.

Fluctuations of the magnetic hyperfine field clearly observed in the Mössbauer spectra between 50 K and 300 K are described by a Blume two-level relaxation model. The spin dynamics in is concentration-independent for x ≤ 0.028.
 From the temperature dependence an Orbach process is deduced as the dominant spin-lattice relaxation process. An Arrhenius analysis ν=ν_0e^-E_A/k_BT yields a thermal activation barrier of E̅_A=515(20) K and an attempt frequency ν̅_0=150(10) GHz. Mössbauer spectroscopy studies with applied transverse magnetic fields up to 5 T reveal a huge increase of the fluctuation rate by two orders of magnitude. In applied longitudinal magnetic fields a characteristic splitting of the spin fluctuation frequency is observed. These experimental observations are qualitatively reproduced by a single-ion spin Hamiltonian analysis. 
 It demonstrates that for dominant magnetic quantum tunneling a weak axial single-ion anisotropy D of the order of a few Kelvin can cause a two orders of magnitude larger energy barrier for temperature-induced longitudinal spin fluctuations.
We think that this is one of the most spectacular manifestations of the macroscopic quantum spin tunneling observed in the solid-state based single-atomic magnet Li_2(Li_1-xFe_x)N.
The experiments suggest  as a candidate for novel functional magnetic materials, e.g. for quantum computing or spintronic devices.



§ ACKNOWLEDGMENTS

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through SFB 1143 and JE748/1. Special thanks to J. Schnack, M. Baker and E. Bill for helpful and constructive remarks.



§ APPENDIX



 §.§ Mass absorption coefficients

The Fe concentration of sample SC 4 is with below 0.2% rather small, even for a ^57Fe-Mössbauer experiment of a non-^57Fe-enriched sample. Fortunately,   contains only light elements with small absorption coefficients of the 14.41 keV radiation energy, see table <ref>. The mass absorption coefficient μ_a describes the exponential Intensity reduction of the initial γ-ray intensity I_0,

    I(d)=I_0(1-f_s)e^μ_ad,

where d is the absorber thickness and f_s the recoil-free fraction of transitions. μ_a describes the non-resonance atomic absorption, mainly by the photoelectric effect. For a comparison, the value of Osmium represents a heavy element in table <ref> showing the rather small mass absorption coefficient μ_a of Li and N since Fe is highly-diluted. We have used for this reason large crystals of a thickness of a few millimeter and the effective thickness t_a=2.39(10) of SC 1 reflects still absorption far away from the saturation limit. The Fe concentration of SC 4 is even below the concentration of Ho in LiY_0.998Ho_0.002F_4 or at least in the same order which is a prominent example for a SAM in a solid crystal <cit.>.



 §.§ Magnetic Hyperfine field

The results of the calculations are discussed assuming the Fe^+ oxidation state <cit.>.
In general, the total magnetic hyperfine field B is the sum of different contributions

    B_h=B_c+B_orb+B_dip+B_lat.

The sign of the Fermi contact contribution B_c is negative and arises from the spin-polarization of the s-electrons by unpaired valence electrons. B_orb is the orbital contribution scaling with the orbital quantum number L which is expected to be important because of the exceeded spin only value of the magnetic moment. B_dip is the dipolar contribution arising from nonsperical electron spin density contribution which is approximately proportional to V_zz. B_lat is the lattice contribution, i.e., the magnetic field generated by neighbor electronic moments in the lattice. This contribution can be neglected in the diluted system.   The detailed values vary strongly on the used computational method and estimations <cit.>, however, a tendency is given by

    B_orb>B_dip≈ |B_c|>B_lat≈ 0

or even B_orb≫ B_dip,|B_c| which is based the Fe^1+ (S=3/2, L=2) assumption <cit.>.















 §.§ Magnetization hysteresis loops

The presented hysteresis loops of magnetization were measured at different temperatures for magnetic fields applied parallel to the crystallographic c-axis, B ∥ c. 

The obtained data were corrected for the diamagnetic sample holder (sample sandwiched between two torlon discs and fixed inside a straw) for which the magnetization was determined separately using a similar setup. 

The diamagnetic contribution of the α-Li_3N host was subsequently subtracted from the sample holder corrected data using ρ(Li^1+)= -8.8· 10^-12 m^3mol^-1  ref. <cit.> and ρ(N^3-)=1.63· 10^-10 m^3mol^-1   ref. <cit.>.



 §.§ Breit-Wigner formula

The cross-section

    σ(E)=σ_a(E)/σ_0

of the transmission integral is given by the Breit-Wigner formula

    σ_0(E)=σ_0Γ_a^2/4/(E-E_0)^2+Γ_a^2/4

where

    σ_0=(λ^2/2π)1+2I_(e)/1+2I_(g)1/1+α

is the maximum cross section, e.g see Chen <cit.>. Here, α is the internal conversion coefficient, I_(e),I_(g) are the nuclear spin numbers of the ground state and excited state, respectively, and λ the energy of the γ-ray. σ(E) and σ_0(E) are given as a function of the photon energy E, E_0 is the energy of the γ-ray corresponding to the Mössbauer transition. The excited state is not strict monochromatic and has a natural distribution given by a Lorentzian line

    L(E)dE=Γ_s/2π1/(E-E_0)^2+Γ_s^2/4dE

with

    ∫ L(E)dE=1.

Γ_s is the natural linewidth of the Mössbauer nuclei and Γ_a is the natural linewidth of the absorber. Here,

    L(E,v)=L(E-v/cE_0)

is the relation to our notation with the speed of light c.


 §.§ Arrhenius plot
 
Fig. <ref> shows the Arrhenius plot (reciprocal T-scaling)

    lnν=lnν_0-E_A/k_B(1/T)

of the extracted fluctuation frequencies lnν of SC 1-4 in MHz. 





The fluctuation frequency is concentration independent as reflected by the parameter E_B and lnν_0 of table <ref>.




 §.§ Landé factor

To estimate the Zeeman splitting, it is important to recall the large effective magnetic moment μ_eff=6.5(4)μ_B per Fe atom parallel to the c-axis  <cit.> which is close to the full Hunds' rule value of Fe^+. This indicates the validity of the Hunds' rules in this system. Using Russel-Saunders coupling and the proposed spin quantum number L=2 and S=3/2, we get the Landé factor for J=7/2,

    g_J=3/2+S(S+1)-L(L+1)/2J(J+1)≈1.55.

Here, g_L=1 and g_S=2 are used. 








 §.§ Comparison with ferrous halides

Axtmann et al. have found a linear relationship between the Pauling electronegativity and the isomer shift δ in ferrous halides is discussed <cit.>. The difference of the ligand electronegativity is related to the isomer shift. This is treated as direct evidence for the participation of 4s electrons in the formation of the chemical bonds <cit.>. Fig. <ref> shows the presented ferrous halides by Axtmann et al. and the isomer shift of SC 1. The observed isomer shift deviates strongly. In the ferrous halides the electronic configuration is 3d4s^x where x measures the ionicity. The ionicity increased with x <cit.>. The electric monopole hyperfine interaction in is far away from the values of the Fe^2+ ferrous halides. The quadrupole splitting in the ferrous halides behaves linear as a function of the isomer shift as well <cit.>. The values are between 1.4 mm/s (FeI_2) and 2.6 mm/s (FeF_2). For conversion <cit.> one can use

    Δ v_QS=ceQ/2E_γV_zz√(1+η^2/3)

with

    ceQ/2E_γ≈ 0.0167mm/s/V/Å^2.

The value of Fe-diluted is -2.572(2) mm/s which shows a comparable electric quadrupole hyperfine interaction with respect to the amount of V_zz.






 §.§ Determination of the Debye-temperature Θ_D


Fig. <ref> shows the center shift as a function of temperature obtained in ZF of SC 1. The center shift is here δ+δ_SOD without α-Iron correction and therefore relative to the ^57Co-source. The temperature dependence of SC 1 yielded a Debye-temperature of Θ_D=315(8) K which is a measure of the collective motion of the surrounding atoms of the Mössbauer nucleus. One should keep in mind the special geometry with the γ-beam parallel to the crystallographic c-axis and therefore the phonic excitations in c-direction are considered according to the Debye-Waller factor. In table <ref> we compare this value with the aforementioned linear C-Fe-C compounds. The values for [Fe(C(SiMe_3)_3)_2]^1- and are similar. This fact further supports the conclusion of a similar electronic configuration of the Fe ion drawn from the values of the quadrupole splitting Δ E_Q and the magnetic hyperfine field B_A in compared to those of [Fe(C(SiMe_3)_3)_2]^1-. 





















 §.§ ^57Fe-Mössbauer measurement at 80 K in 2 T LF of 

Fig. <ref> shows a comparison of the ^57Fe-Mössbauer measurements at 80 K in ZF and in an applied longitudinal field of 2 T of SC 1. The analysis is done in the same way as discussed in the main text. The intermediate relaxation line splits  at 80 K in a magnetic field of 2 T LF. The doublet is weakly adumbrated because of the scattering of the data (lower magnitude of absorption) and not so well pronounced like in the presented 100 K measurement at 2 T. However, a splitting is confirmed. The grey fit is the result of free convergence of the mentioned two-frequency spin reversal model. The relaxation frequencies are log_10ν_up=2.63(8)log_10[MHz] and log_10ν_down=2.36(6)log_10[MHz], therefore ν_up≈427 MHz and ν_down≈229 MHz.




