Magneto-electric properties and low-energy excitations of multiferroic FeCr2S4
A. Strinic, S. Reschke, K. V. Vasin, M. Schmidt, A. Loidl, V. Tsurkan, M. V. Eremin, J. Deisenhofer
MMagneto-electric properties and low-energy excitations of multiferroic FeCr S A. Strinic, S. Reschke, K. V. Vasin, M. Schmidt, A. Loidl, V. Tsurkan,
1, 3
M. V. Eremin, ∗ and J. Deisenhofer † Experimentalphysik V, Center for Electronic Correlations and Magnetism,Institute for Physics, Augsburg University, D-86135 Augsburg, Germany Institute for Physics, Kazan (Volga region) Federal University, 420008 Kazan, Russia Institute of Applied Physics, MD-2028 Chi¸sin˘au, Republic of Moldova (Dated: September 22, 2020)We report on the low-frequency optical excitations in the multiferroic ground state of polycrys-talline FeCr S in the frequency range 0.3-3 THz and their changes upon applying external magneticfields up to 7 T. In the ground state below the orbital-ordering temperature T OO = 9 K we observethe appearance of several new modes. By applying the external magnetic field parallel and perpen-dicular to the propagation direction of the THz radiation, we can identify the strongest absorptionsto be of predominantly electric-dipole origin. We discuss these modes as the low-energy electronicexcitations of the Fe ions (3 d , S = 2) in an tetrahedral S − environment. The eigenfrequenciesand relative intensities of these absorption lines are satisfactorily reproduced by our calculation as-suming an effective exchange field of 12 . − at the Fe -ions sites. The direction of the exchangefield is found to be slightly tilted out of the ab -plane. With our approach we can also describe previ-ously reported results from Mssbauer studies and the order of magnitude of the electric polarisationinduced by orbital and non-collinear spin ordering. I. INTRODUCTION
In condensed matter physics there are some materi-als, which were revisited again and again throughoutseveral decades of research and always revealed excit-ing new properties. One of these materials is the spinelFeCr S , which first came into the focus of researchin the 1960s as a ferrimagnetic semiconductor with a T C = 170 K . Subsequently, the competing spin-orbitand electron-phonon interactions of the Jahn-Teller ac-tive Fe -ions in tetrahedral environment were studiedin the 1970-80s and around the change of the millen-nium colossal magneto-resistance effects were reported and the magnetic structure was shown to be more com-plex than that of a simple collinear ferrimagnetic arrange-ment of the Cr ions (3 d , S = 3 /
2) on the octahedralsites and the Fe ions (3 d , S = 2) on the tetrahe-dral ones (see Fig. 1 for the cubic crystal structure withspace group F d ¯3 m ) . In particular, a non-collinearmagnetic structure is realized below T M = 60 K . Inaddition, a giant magneto-optical Kerr rotation was re-ported for the Fe d − d transitions in the mid-infraredfrequency range . More recently, it was recognised thatFeCr S belongs to the class of materials with a multifer-roic ground state, because the emergence of a finite po-larization was reported below the orbital ordering tran-sition at T OO =9 K . The electric polarization wasreported to consist of two different contributions P and P , with the latter arising directly at the transition tem-perature T OO , while the former appears at lower temper-atures of around 4 K : In contrast to P the contribution P depends strongly on an external magnetic field and,therefore, was assigned to originate from a non-collinearspin configuration in the ground state. The component P might then be a direct consequence of the structuralJahn-Teller distortion related to the orbital ordering.While over the decades no direct evidence for clear deviations from cubic symmetry throughout the knownphase transitions could be obtained , some of us re-ported recently, that the transition temperatures T M and T OO are accompanied by a splitting of infrared-activephonons and the emergence of new modes, confirmingthe expected lack of inversion symmetry in the multi-ferroic ground state . In Fig. 2 we show an updated H − T -phase diagram adapted from Ref. .In this work we used THz-time domain spectroscopy toinvestigate possible optical magneto-electric effects in themultiferroic ground state and shed light on the mecha-nism involved in the formation of this state. We observed FIG. 1. Structural unit cell of cubic FeCr S with spacegroup F d ¯3 m . a r X i v : . [ c ond - m a t . s t r- e l ] S e p c u b i c ?c o l l i n e a rF i M c u b i cP MO O / i s o t r o p i c F i M / m u l t i f e r r o i c n o n - c u b i c / O L / n o n - c o l l i n e a r F i M F e C r S m
0H (T)
T e m p e r a t u r e ( K ) n o n - c e n t r o s y m m e t r i cO O / n o n - c o l l i n e a r F i M / m u l t i f e r r o i c
FIG. 2. H − T phase diagram of FeCr S updated with re-spect to Ref. by taking into account phonon-anomalies inzero-field reported in Ref. .(FiM: ferrimagnet, PM: param-agnet, OO: orbital order, OL: orbital liquid) the emergence of several excitations in the multiferroicorbitally ordered state below T OO and studied their de-pendence on external magnetic fields. In addition, wediscuss a theoretical approach to model some of the ob-served excitations in terms of the low-energy electronicexcitations of the Fe ions (3 d , S = 2) in an tetrahe-dral S − environment. II. EXPERIMENTAL DETAILS AND SAMPLEPROPERTIES
We used polycrystalline samples with a high densityof 3.85 g/cm obtained by spark-plasma sintering (SPS)technique. . The density of the SPS sample is very closeto the density of single crystals and, therefore, the mea-sured absorption coefficients should not depend on thepolycrystalline nature of the sample. Transmission mea-surements in the frequency range from 10-105 cm − wereperformed using THz-time-domain-spectroscopy with aToptica Tera-flash spectrometer and an Oxford Instru-ments cryomagnet in external magnetic fields up to 7 T.The polycrystalline sample was polished to platelets witha thickness of 100 µ m.Due to the lack of single crystals which are largeenough for long-wavelength optical measurements and atthe same time exhibit the orbital ordering transition at9 K, the measurement and light polarization configura-tion of the thin polycrystalline sample were restricted tothe Faraday configuration with the wave vector k of theincoming linearly polarized THz pulse parallel to the ex-ternal magnetic field H and the Voigt configuration with k ⊥ H . In the latter case the light polarization was var-ied to be either parallel E ω (cid:107) H or perpendicular E ω ⊥ H to the applied magnetic field.Before discussing our THz results in magnetic fields H ^ EH I I E D (cid:1) m H ( T ) D (cid:1) I I D (cid:1) ^ M (µB/f.u.) FIG. 3. Magnetic-field dependence of ∆ ε (cid:48) defined by Eq. 1with the probing microwave electric field parallel (∆ ε (cid:48)(cid:107) , redsolid line) or perpendicular (∆ ε (cid:48)⊥ , blue solid line) to the exter-nal magnetic field H and the magnetization M (black dashedline) taken from Ref. 19. The solid symbols represent corre-sponding estimates obtained by Eq. 3 using the THz spectraas described in the text. we want to recall the magnetic-field dependence of themagnetization and the dielectric constant of the poly-crystalline sample. The magnetic-field dependence ofthe real-part of the dielectric constant at frequencies inthe kHz-range was reported in Refs. 19 and 20 and usedas an estimate of the static magnetic-field dependentvalue ε (cid:48) H ( ω ≈ ε (cid:48) H ( ω ≈
0) in an applied magnetic field at 5 K byshowing the difference∆ ε (cid:48) ( H ) = ε (cid:48) H ( ω ≈ − ε (cid:48) H =0 ( ω ≈
0) (1)for the two cases that the external magnetic field wasapplied parallel and perpendicular to the electric field,denoted as ∆ ε (cid:48)(cid:107) and ∆ ε (cid:48)⊥ , respectively. The value of ε (cid:48) H =0 ( ω ≈
0) used here was determined as 21.4 in Ref. in good agreement with 22.5 reported in Ref. .It was shown in Ref.19 that ∆ ε (cid:48)(cid:107) increases with respectto H = 0 and ∆ ε (cid:48)⊥ decreases with increasing magneticfield. In addition, it was reported that the magnetic-fielddependence of the magnetization can be scaled on top of∆ ε (cid:48)(cid:107) suggesting that the magnetic-field induced spin andmagnetic domain reorientation is also responsible for thefield-dependence of ∆ ε ( H ).Looking at the M − H -dependence of the poly-crystalline sample, it is important to note that insingle-crystal FeCr S there is a clear magnetocrystallineanisotropy with the easy axis coinciding with the (cid:104) (cid:105) direction, while the (cid:104) (cid:105) and (cid:104) (cid:105) directions are con-sidered to be hard axes . The data of the polycrystalrepresent a statistical average, where nearly all domainsbecome coaligned in fields below 1 T and the increasetowards higher fields is due to the successive alignment E E E M M F e C r S a (cm-1) w a v e n u m b e r ( c m - 1 )
2 K 2 5 K 1 0 0 K 1 8 5 K ( a )( c ) ( d )( b ) * T = 2 K E E E E M M E w I I HH w ^ H a (cm-1) w a v e n u m b e r ( c m - 1 ) M ' * T = 2 KF a r a d a y E w ^ HH w ^ H M M E E E a (cm-1) w a v e n u m b e r ( c m - 1 )
0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T E E * E w ^ HH w I I H E E E E M a (cm-1) w a v e n u m b e r ( c m - 1 ) M T = 2 K * FIG. 4. Absorption spectra for (a) different temperatures upon zero-field cooling, (b) at 2 K for increasing magnetic fields inVoigt configuration with E ω (cid:107) H, H ω ⊥ H and (c) for E ω ⊥ H, H ω (cid:107) H . (d) Absorption spectra at 2 K for increasing magneticfields in Faraday configuration with E ω ⊥ H, H ω ⊥ H . of the magnetization in the domains where the externalmagnetic field has to overcome the magneto-crystallineanisotropy. It is anticipated that the reported increaseof the magnetization in the range of 4 . < H < . S . III. EXPERIMENTAL RESULTS ANDDISCUSSIONA. Temperature dependence
In Figure 4(a) we show the absorption coefficient ofFeCr S as a function of temperature. The tempera-ture dependence upon cooling the sample in Figure 4(a)reflects the sequence of the different magnetic phasesshown in the phase diagram in Fig. 2(b). The mono-toneous increase of the absorption coefficient with fre-quency at 185 K and 100 K representing the param-agnetic and collinear ferrimagnetic phases, respectively,is attributed to the lowest-lying infrared-active phonon at about 125 cm − , which narrows and undergoes ablueshift with decreasing temperatures . As a re-sult the accessible frequency range for transmission mea-surements within our experimental sensitivity increaseswith decreasing temperatures and in the non-collinearmagnetic phase for 9 K < T <
60 K spectral weight isshifted to the THz-range indicated by a broad absorptioncontinuum, which appears below 85cm − in the spec-trum at 25 K. This absorption continuum increases inintensity on approaching the orbital ordering transitionat T OO = 9 K and develops into a strong absorption bandwithin the multiferroic ground state, where in the spec-trum at 2 K a minimal set of four distinct modes E − E may be distinguished by the shoulders and maxima of theabsorption band as indicated in in Fig. 4(a). In addition,two more modes, M and M , appear in the polar groundstate at 90 cm − and 100 cm − , respectively.Note that for T > T OO there is a clear periodic modu-lation of the spectra, which can be explained as a result ofinternal reflections in the platelet-shaped sample. For ex-ample, the maxima in the spectra at 100 K are separatedby 11 cm − in agreement with a thickness of 100 µ m anda refractive index of n = 4 .
5. We interpret the featurebetween E and E (marked by an asterisk) at about27 cm − as a remainder of this Fabry-Perot interferencepattern. Consequently, it should mainly be affected bythe magnetic field due to a change of the refractive indexand the changing contributions of E − E . The fielddependence shown in Fig. 5 supports this assignment. B. Field dependence
In Figs. 4(b) and (c) we show the evolution of thespectra measured at 2 K in increasing magnetic fieldsin Voigt configuration with E ω (cid:107) H, H ω ⊥ H and E ω ⊥ H, H ω (cid:107) H , respectively, and in Faraday config-uration with E ω ⊥ H, H ω ⊥ H , in Fig. 4(d):In the configuration E ω (cid:107) H, H ω ⊥ H in Fig. 4(b),there is a huge increase of the intensity of the E absorp-tion band in the frequency range 20-80 cm − , while themaxima and shoulders of E − E do not exhibit a dis-cernible shift with increasing fields. For E ω ⊥ H, H ω (cid:107) H in Fig. 4(c), again, the most significant changes takeplace in the frequency range 20-80 cm − and between 0Tand 2T, where the excitations E − E become stronglysuppressed with increasing magnetic fields. Note thatthe absorption spectrum at 7 T in this frequency rangeclearly resembles the zero-field cooled spectrum at 25 Kshown in Fig. 4(a). In Faraday configuration with E ω ⊥ H, H ω ⊥ H the changes of E − E with increasingfields are similar to measurements in Voigt configuration E ω ⊥ H, H ω (cid:107) H . This indicates that the orientation ofthe THz-electric field E ω with respect to the static mag-netic field H is the decisive factor for the huge changesin intensity of the E − E absorptions.In contrast, the behavior of E is more complex andshown in detail in Fig. 5 for T = 2 K: For E ω (cid:107) H, H ω ⊥ H (Fig. 5(a)) we first identify two absorption peaks E a and E b at 13.5 and 16.0 cm − (dashed lines) in the zero-field spectrum, respectively. With increasing field in therange 0-2 T the intensity of E b decreases and remainsat a lower level for higher fields. For E ω ⊥ H, H ω (cid:107) H (Fig. 5(b)) both absorption peaks are also visible andan increase in intensity for E b in the range 0-2 T canbe anticipated. In Faraday configuration the spectrumexhibits a single maximum at the eigenfrequency of E b and does not change significantly with increasing mag-netic field. As mode E a does not discern itself by aclear peak, it is impossible to judge about its presence inthis case. Although changes in this frequency range areevident, we want to point out that the strong changes ofthe absorptions E − E clearly influence the changes inthe region of E .Interestingly, a similar behavior can be observed for M , and M which is highlighted in Fig. 6 for all threeconfigurations: While M shows an increase in absorp-tion with increasing magnetic field E ω (cid:107) H, H ω ⊥ H (Fig. 6(a)), mode M is not visible anymore at 2 T,but a somewhat weaker absorption feature emerges againfor fields larger than 4 T and shifts to lower frequen- w I I HV o i g t
7 T6 T5 T4 T2 T0 T a (cm-1) w a v e n u m b e r ( c m - 1 )* E E V o i g tE w ^ H
7 T6 T5 T4 T0 T w a v e n u m b e r ( c m - 1 )
2 T E E * T = 2 K F a r a d a y ( a ) E w ^ H
1 T3 T7 T6 T5 T4 T2 T0 T w a v e n u m b e r ( c m - 1 )( b ) ( c ) E E * FIG. 5. Magnetic field dependence of mode E at 2 K for(a) with E ω (cid:107) H, H ω ⊥ H , (b) for E ω ⊥ H, H ω (cid:107) H , and (c)for E ω ⊥ H , H ω ⊥ H . The spectra were shifted by a verticaloffset with respect to the zero-field curve for clarity. Dashedlines indicate absorption peaks in zero-field as described inthe text. cies with further increasing fields. For the configurations E ω ⊥ H, H ω (cid:107) H shown in Fig. 6(b), there is first an in-crease in intensity of M between 0 T and 2 T, and thena decrease in intensity for field larger than 2 T. Thereis a shift to lower frequencies with increasing fields. In-tensity changes of M in this configuration can not betracked reliably due to the lower transmission in this fre-quency range. In Faraday configuration (see Fig. 6(c))the lineshapes of M and M are better resolved and theintensity seems to be slightly reduced in comparison toFig. 6(b). Again, M gains intensity in the range 0-2 Tand shifts to lower frequencies with increasing magneticfield. Significant intensity changes of M are, however,still difficult to track.Before we discuss the possible origin and optical ac-tivity of the observed modes, we want to point out thatthe most significant changes in magnetic field, the in-crease in intensity of the E -band and the disappearanceof M happen between 0T and 2T, coinciding with thefield regime, where the majority of the magnetic and fer-roelectric domains are getting aligned as discussed abovefor the magnetization M and ∆ ε (cid:48)(cid:107) in Fig. 3. The reap-pearance of M for H >
2T is tentatively assigned to thechanges in the field regime 4 . < H < . . C. Discussion
The observed increase or decrease of the intensity ofthe E-band for the respective configurations E ω (cid:107) H and E ω ⊥ H and the agreement with the field variation of thestatic real part of the dielectric function ∆ ε ( H ) shown inFig. 3 strongly suggests that the excitations are predom-inantly of electric-dipole origin. As a consequence we w I I HV o i g t
7 T6 T5 T4 T2 T0 T a (cm-1) w a v e n u m b e r ( c m - 1 ) M M E w ^ HV o i g t
7 T6 T5 T4 T0 T a (cm-1) w a v e n u m b e r ( c m - 1 )
2 T M M ( a ) T = 2 KE w ^ HF a r a d a y
1 T3 T7 T6 T5 T4 T2 T0 T a (cm-1) w a v e n u m b e r ( c m - 1 )( b ) ( c ) M M FIG. 6. Magnetic field dependence of modes M and M at2 K for (a) with E ω (cid:107) H, H ω ⊥ H , (b) for E ω ⊥ H, H ω (cid:107) H ,and (c) for E ω ⊥ H, H ω ⊥ H . The spectra were shifted by avertical offset with respect to the zero-field curve for clarity.Solid lines are fitting curves of the baseline as described inthe text. assume that the E -band excitations can be assigned tothe imaginary part of the dielectric constant ε (cid:48)(cid:48) ( ω ), whichis related via the Kramers-Kronig transformation to thestatic real part of the dielectric constant ε (cid:48) ( ω ): ε (cid:48) H (0) − π (cid:90) ∞ ε (cid:48)(cid:48) H ( ω ) ω dω (2)To check the validity of this assumption, we evalu-ate the experimental data by converting the time-domainspectra in terms of the dielectric function and integratefrom ω = 21 cm − to ω = 76 cm − , thus concentratingon the excitations E - E , which show the most promi-nent changes. Hence, we compare the obtained dynami-cal quantity δε (cid:48) ( H ) = 2 π (cid:90) ω ω (cid:18) ε (cid:48)(cid:48) H ( ω ) ω − ε (cid:48)(cid:48) H =0 ( ω ) ω (cid:19) dω + C (3)with ∆ ε (cid:48) ( H ) in Fig. 3. Thereby, we assume that thedependence of ε (cid:48)(cid:48) H ( ω ) on the magnetic field outside thisfrequency range can be approximated by a constant con-tribution C , which can be different for the two configura-tions E ω (cid:107) H and E ω ⊥ H . Using the values C (cid:107) = − . C ⊥ = +1 . H > E ω ⊥ H . In the case E ω (cid:107) H the maxima of the strong absorption bands could not beresolved anymore, resulting in a larger uncertainty dueto a possible underestimation of the dielectric strengthof the E -band. This should, in principle, produce lowervalues in comparison to the microwave data, which is notthe case. Indeed, we even had to used the negative value C (cid:107) = − . mode ω Faraday at 7 T Voigt at 7 T activity [cm − ] E ω ⊥ H E ω ⊥ H E ω (cid:107) HH ω ⊥ H H ω (cid:107) H H ω ⊥ HE × × (cid:88) E ω (cid:107) HE × × (cid:88) E ω (cid:107) HE × × (cid:88) E ω (cid:107) H mode ω Faraday at 2 T Voigt at 2 T activity [cm − ] E ω ⊥ H E ω ⊥ H E ω (cid:107) HH ω ⊥ H H ω (cid:107) H H ω ⊥ HE a (cid:88) (cid:88) E ω ⊥ H , E ω (cid:107) HE b (cid:88) (cid:88) × E ω ⊥ HM (cid:88) (cid:88) × E ω ⊥ HM (cid:88) (cid:88) (cid:88) E ω ⊥ H , E ω (cid:107) H TABLE I. Eigenfrequencies in zero field ω and observed op-tical activity for Voigt and Faraday configurations. Upperpart: The notation (cid:88) and × indicates that the mode’s inten-sity is strongly increased or strongly suppressed in a magneticfield of 7 T, respectively. Lower part: The notation (cid:88) and × indicates that the mode is present or strongly suppressed in amagnetic field of 2 T, respectively. The notation n.d. signifiesthat the mode was not clearly descernible. follow a similar trend. As it will be discussed below insection IV, the underlying electronic levels of the E -bandexcitations allow also for magnetic-dipole contributionsand interference effects, which strongly affect the inten-sity of the strongest E -band excitations and shows thelimits of this comparison. However, we think that thisis clear evidence for a predominant electric-dipole ori-gin of the excitations E - E . Their eigenfrequencies inzero field are denoted by ω in the upper part of Tab. Itogether with their occurrence or suppression in fieldsof 7 T. Assuming that the primary effect of the externalmagnetic field is to align magnetic domains and overcomethe magnetic anisotropy, the increase of the absorptionof the E -band should be a result of aligning the elec-tric dipoles responding to the electric-field component ofthe THz radiation. This is in agreement with the in-terpretation of the reported polarization component P originating from the non-collinear spin configuration inthe orbitally ordered state , with the corresponding fer-roelectric domains strongly linked to the magnetic ones.Similarly, the changes of M and M with increasingfield in the three configurations can be understood interms of the magnetic field dependence of the magneti-zation. In order to compare the relative changes of theintensity of M and M , we used polynomial fits (shownas black lines in Fig. 6) outside of the frequency regionof M and M to subtract the underlying absorptionbackground due to the lower-lying E -band and higher-lying phonon contributions and then integrated over theabsorption peaks. The results for M are displayed inFig. 7(a), where the values for 2 K and for 5 K (spectranot shown) are shown.Clearly, the magnetic field dependence reveals the sup-pression of M in fields of 2 T for E ω (cid:107) H, H ω ⊥ H and an increase for E ω ⊥ H, H ω (cid:107) H . For further in-creasing fields the integrated intensity again decreasesfor E ω ⊥ H, H ω (cid:107) H and increases for E ω (cid:107) H, H ω ⊥ H .Notably, plotting the sum of the intensities of M for thetwo configurations can, however, be considered constantwithin the experimental uncertainty. This implies thatdomain reorientation in the applied magnetic field caninduce a configuration in a field of about 2 T, which de-termines a selection rule for the optical activity of M ,namely E ω (cid:107) H, H ω ⊥ H . For further increasing fieldsthis metastable configuration is again lost, presumablydue to the interplay of the magnetocrystalline anisotropyand the external magnetic field .The integrated intensity of M in Faraday configura-tion E ω ⊥ H, H ω ⊥ H follows a similar but somewhatless pronounced trend as for E ω ⊥ H, H ω (cid:107) H . Hence,we conclude that in the field range 0 < H < M is mainly active for E ω ⊥ H as denoted in the lower partof Tab. I. In the case of mode M the integrated intensityshown in Fig. 7(b) has to considered with care, because inboth Voigt configurations the maxima of the absorptionpeaks could not be resolved as discussed above. How-ever, it seems that the intensities for E ω ⊥ H, H ω (cid:107) H and E ω ⊥ H, H ω ⊥ H are in agreement with each otherand do not depend strongly on the applied magnetic field.For E ω (cid:107) H, H ω ⊥ H the intensity of mode M showsan increase with increasing magnetic field and an overallhigher intensity than for the other configurations, sug-gesting the presence of an additional excitation mecha-nism for M . Unfortunately, the exact nature of M and M cannot be established based on the present data andmeasurements on single crystals will be necessary to de-termine, if they correspond to collective magneto-electricmagnon modes.A similar ambiguity remains for the optical activityof excitation E a , while for E b below 2 T the opticalactivity is assigned as E ω ⊥ H in Tab. I. Finally, themagnetic-field dependence of the eigenfrequencies of M and M are shown in Fig. 7(c), showing that M exhibitsa similar shift to lower frequency with increasing fieldfor all configurations, while the eigenfrequency of M re-mains approximately constant. IV. THEORETICAL MODEL FOR THE E -BANDEXCITATIONS In the following we will discuss a model, which de-scribes the excitations between the lowest-lying elec-tronic d − d - levels of the Fe -ions in tetrahedralenvironment . We will discuss the effective single-ion Hamiltonian and compare the resulting absorptionscheme with the experimentally observed excitationsforming the E -band.
05 0 01 0 0 01 5 0 02 0 0 005 0 01 0 0 01 5 0 02 0 0 02 5 0 0 M integrated intensity (cm-2) ( b )( a ) E w I I H , H w ^ H E w ^ H , H w I I H E w ^ H , H w ^ H M integrated intensity (cm-2) M M eigenfrequency (cm-1) m a g n e t i c f i e l d ( T )( c ) FIG. 7. Magnetic field dependence for the three configura-tions E ω (cid:107) H, H ω ⊥ H (blue), E ω ⊥ H, H ω (cid:107) H (red), and E ω ⊥ H, H ω ⊥ H (green) of (a) the integrated absorption co-efficient of mode M at 2 K (spheres) and 5 K (stars) and thesum for the two Voigt configurations (black) and (b) the inte-grated absorption coefficient of mode M at 2 K (diamonds)and 5 K (triangles). In (c) the eigenfrequency of the modesfor the different polarization configurations at 2 K and 5 Kare shown. Lines are to guide the eyes. A. Effective Hamiltonian for the low-lying Fe states in the presence of electric and magnetic fields The Fe -ions occupy the tetrahedral sites with elec-tronic configuration t e in the ground state, while theCr -ions at the centre of S − octahedra are in t g con-figuration, which has no orbital degrees of freedom. Inthe FeCr S unit cell there are two tetrahedral fragmentsFe(1)S and Fe(2)S which are rotated relative to eachother by 90 ◦ around the c -axis.It is well known that the level scheme of the low-lying E-states depends on the competition of spin-orbit cou-pling, the Jahn-Teller-effect, and, in the case of magneti-cally ordered systems, on the internal magnetic exchangefields at the Fe -sites .For the calculation of the Fe -ions energy levels weused the effective Hamiltonian H eff = − ξ (cid:110) [3 S z − S ( S + 1)] U θ + √
32 ( S + S − ) U ε (cid:111) + ζ (cid:16) S x S z S y + S y S z S x + S z S x S y + S z S y S x ++ S x S y S z + S y S x S z (cid:17) U α + V ρ (cid:16) U θ cos( φ ) + U ε sin( φ ) (cid:17) + (cid:88) J F e,j S (cid:104) S j (cid:105) + (cid:16) g s − λ ∆ (cid:17) µ B BS − λµ B ∆ ×× (cid:110) (3 S z B z − BS ) U θ + √ B x S x − B y S y ) U ε (cid:111) . (4)The first two terms account for the spin-spin and spin-orbit interactions with ξ = ρ S + λ / ∆ + 2 λ / ∆ and ζ = √ λ / ∆ . The parameter ∆ denotes the crystal-fieldsplitting between the exited T and ground E states,and λ is the spin-orbit coupling constant. The Pauli-like matrices U θ = | ε (cid:105)(cid:104) ε | − | θ (cid:105)(cid:104) θ | and U ε = | ε (cid:105)(cid:104) θ | + | θ (cid:105)(cid:104) ε | , U α = i ( | θ (cid:105)(cid:104) ε |−| ε (cid:105)(cid:104) θ | ) describe the E-orbital ground statewithin the orbital doublet states | θ (cid:105) and | ε (cid:105) . The thirdterm takes into account a possible distortion of the FeS − tetrahedra at low temperatures due to a linear Jahn-Teller coupling of the E-orbital states. The last threeterms in Eq. (4) describe the exchange interaction be-tween Fe ions and surrounding Cr and Fe ions and theinteraction with an external magnetic field, respectively.The effective operator of the interaction of the Fe ionwith an electric field, which takes into account the mixingof states with opposite parity, i.e. 3 d and 3 d p as wellas 3 d S − and 3 d S − , where S refers to the electronicshell of the sulfur ions, is written as follows : H E = (cid:88) p,tk =2 , (cid:110) E (1) U ( k ) (cid:111) ( p ) t × (5) × (cid:88) j d (1 k ) p ( R j ) ( − t C ( p ) − t ( ϑ j , ϕ j )The curly brackets denote the Kronecker product ofthe spherical tensor of the electric field ( E (1)0 = E z , E (1) ± = ∓ ( E x ± iE y ) / √
2) with the unit irreducible ten-sor operator acting on the 3 d electronic states. Thespherical coordinates R j , ϑ j , ϕ j denote the positionsof the lattice ions (as in crystal field theory), and C ( p ) t ( ϑ j , ϕ j ) = (cid:112) π/ (2 p + 1) Y pt ( ϑ j , ϕ j ) are the compo-nents of the spherical tensors. The quantities d (1 k ) p ( R j )are calculated in the local coordinate system with the c -axis along the 3 d -ion ligand direction. For iron ions in anundistorted tetrahedral environment p = 3, and, there-fore, there are only two intrinsic parameters : d (12)3 and d (14)3 , which we will try to extract based on relative in-tensities of transitions in the absorption spectra (Notethat d (1 k ) p includes a Lorentz local-field correction fac-tor). The effective coupling operator of Fe with an E -state in an applied electric field is then written as H = − λ (cid:114) (cid:110) d (12)3 − (cid:114) d (14)3 (cid:111) ES U α (6)Considering the results of Mssbauer studies in the or-bitally ordered ground state for T < T OO the orbitaldegeneracy of the lowest-lying Fe states is lifted . Themeasured electric field gradient V zz has a negative signand the associated asymmetry parameter is about 0 . E θ .Within the states of the E θ multiplet, the effectiveoperator (6) is supplemented by the following expression H = − λ (cid:114) (cid:110) d (12)3 − (cid:114) d (14)3 (cid:111) | θ (cid:105)(cid:104) θ | × (7) × E x (cid:104) S F e × ( S (1) ξ (cid:48) + S (2) ξ (cid:48) + S (3) η (cid:48) + S (4) η (cid:48) ) (cid:105) x − E y (cid:104) S F e × ( S (1) ξ (cid:48) + S (2) ξ (cid:48) + S (3) η (cid:48) + S (4) η (cid:48) ) (cid:105) y − E x (cid:104) S F e × ( S (1) ξ (cid:48) + S (2) ξ (cid:48) − S (3) η (cid:48) − S (4) η (cid:48) ) (cid:105) y + E y (cid:104) S F e × ( S (1) ξ (cid:48) + S (2) ξ (cid:48) − S (3) η (cid:48) − S (4) η (cid:48) ) (cid:105) x Here we have used the notations S (1) ξ (cid:48) = (cid:80) Cr ( J (1) ξ (cid:48) ,Cr / ∆) S Cr , S (3) η (cid:48) = (cid:80) Cr ( J (3) η (cid:48) ,Cr / ∆) S Cr etc.The operator H describing the coupling of Cr and Fespins with an electric field (7) was obtained by combiningEq. (5) with the operator of exchange and spin-orbital in-teractions in the third order of perturbation theory, sim-ilar to the approach described in Ref. 38. The upperindex at the parameters of the exchange interaction cor-responds to the numbering of the sulfur ions in Fig. 8,through which the superexchange interaction of the ex-cited state of the Fe ions with the nearest three Cr ions istaking place. Variables with an upper prime refer to thelocal coordinate system with the x (cid:48) -axis directed parallelto the edge of the tetrahedron.The effective operator H contains a vector product ofFe and Cr spin operators and, therefore, turns to zero inthe case of a collinear order of spins. Experimentally, thepresence of non-collinearity between spin orientations inFeCr S was found in Ref. 17. The exact spin configu-ration and, thus, the canting angle between Fe and Crspins angles is not yet known.Therefore, we determine the magnitude and directionof the exchange field of the Cr ions based on the ob-served relative intensities and positions of the absorp-tions E − E in the spectrum. The direction of the Fespin is calculated on the basis of the ground state wavefunction of the Hamiltonian (4), while the direction ofthe Cr spins is assumed to be coinciding with the direc-tion of the effective exchange field. The angle betweenCr and Fe spins obtained by this procedure is 170 ◦ .Finally, there is another effective on-site interaction ofthe Fe spin with the electric field, which in our case is Fe Cr S (cid:31)(cid:30)(cid:29) (cid:28) (cid:27) (cid:26)(cid:25) (cid:24) FIG. 8. Fragment of the FeCr S crystal structure illus-trating the dominant superexchange coupling (black arrow)of the Fe(1) excited state ξ (cid:48) to a neighbouring Cr ions via the p -state of the sulphur ligand with number 2. described by the following operator H = E z λ ∆ (cid:114) d (12)3 ( S x S y + S y S x ) | θ (cid:105)(cid:104) θ | . (8)This operator H is also derived in the third order of per-turbation theory, when we take into account the virtualexcitation processes caused by the spin-orbital interac-tion and the action of an induced electric field within theexcited T -states of the Fe ions. B. Extracting d (12)3 /d (14)3 ratio from opticalconductivity spectrum To gain more information on the parameters of theHamiltonian H E given by Eq. (5), we start with thediscussion of the optical d − d transitions between the E and T multiplets, which were observed in Ref. 40at T = 300 K . At this temperature the Fe tetrahe-dron is not distorted and there is no internal (molecular)exchange field due to the absence of orbital and mag-netic ordering at room temperature. The splitting of theground E multiplet due to the second order of spin-orbitcoupling and spin-spin interaction is much smaller thanthe energy interval between E and T states and thesplitting of the T -multiplet is caused by the first orderin spin-orbit coupling. According to the optical con-ductivity spectrum can be approximated as follows σ = (cid:88) j S j ω ω j γ j ( ω j − ω + γ j ) + 4 ω γ j , (9)where ω j and γ j correspond to the eigenfrequenciesand damping constants for the excitations between the -1 ) σ ( Ω - c m - ) - Exp. - Sim.
FIG. 9. Optical conductivity spectrum at T = 300 K : blackcurve experimental data taken from , red cruve - our simula-tion using Eq. (9), blue curve - calculated superfine structurefor γ j ∼ ground multiplet and the T -states split by the spin-orbitinteraction with ζ (cid:39)
596 cm − . We expanded the sug-gested model by taking into account a superfine struc-ture caused by the mixing of E and T states due tospin-orbit coupling. This mixing leads to a violation ofthe Lande interval rule for exited states and actually in-creases the number of possible transition frequencies ω j .We calculated the transitions with wavefunctions in a ba-sis of states | D, M l , M s (cid:105) by numerical diagonalization ofthe 25 ×
25 matrix. Using the pure operator of interac-tion with an electric field (5) we expressed the oscillatorstrength S j using the two parameters d (12)3 and d (14)3 S j ∝ Z ( e − EikT − e − EjkT ) ω ij |(cid:104) j | H E | i (cid:105)| , (10)where Z is the partition function summing over all 25states. Our results are presented in Fig. 9. We assumedthe same γ j (cid:39) cm − for all oscillators. The obtainedsuperfine structure is shown as blue spikes in Fig. 9 de-noting underlying transitions.By varying the positions of the excitations we ob-tained the parameter ∆ (cid:39) cm − , which is some-what smaller than the value reported in Ref. 40. Therelative intensities of the broadened lines allowed us tofix the ratio d (12)3 /d (14)3 ∼ C. Simulation of absorption spectra and estimationof the resulting polarization
Having discussed the relevant coupling terms, we sim-ulated the absorption spectra corresponding to the pre-sented Hamiltonians.To reduce the number of unknown parameters, we con-sidered λ (cid:39) − cm − according to the estimates in Ref.30 for the related system FeSc S and J (1) ξ (cid:48) ,Cr (cid:39) cm − (see discussion in Ref. 39). The energy intervals betweenthe electronic levels are mainly determined by the value ofthe exchange molecular field. The wave functions of theenergy levels are labeled by the spin quantum numbers( S, M S ). The spin-orbit interaction destroys the equidis-tant level spacing and leads to the mixing of states withdifferent quantum numbers M S . In order to describe thetransition from the ground state ( M S = −
2) to the ex-cited state ( M S = 0), which corresponds to the most in-tense spectral line E , the presence of a sufficiently largecomponent of the exchange field at the site of the Fe ions in the ab -plane must be assumed.The observed absorptions of the E -band where sim-ulated as excitations from the ground state | (cid:105) to theexcited states | m (cid:105) using α ( ω ) ∝ (cid:88) m |(cid:104) | H + H + H + H M | m (cid:105)| ×× ω g ( ω m − ω ) , (11)where H M = ng s | µ b | S (cid:2) k ω k × E ω (cid:3) describes the interac-tion of the effective spin S = 2 with the magnetic field H ω of the incident THz radiation, giving rise to magnetic-dipole transitions. The refractive index n sets the rela-tion between E ω and H ω in Gaussian units. The sum( H + H + H ) represents the effective interaction be-tween the spin and the electric field component E ω ofthe radiation. Note that Boltzmann occupation factorswere omitted in Eq. (11), since the energies of the exitedstates of the effective Hamiltonian (4) are all above 10cm − . To simulate the experimental spectrum the shapefunction g (∆ ω ) (11) was treated as a Gaussian lineshapewith the same width for all transitions. The calculatedcontributions to the absorbtion spectrum of the E -bandin zero magnetic field is shown in Fig. 10. A very stronginterference between the different contributions to the to-tal intensity of the absorption was observed. For exam-ple, considering only H and neglecting the contributionsof H and H M in Eq. (11) results in a strong increasein intensity in the high-frequency absorptions as demon-strated by the magenta curve in Fig. 10. The intensity oftransitions caused by H is negligible compared to oth-ers, so it is not shown separately. However, the operator(7) has a significant contribution to the p x , p y compo-nents of the electric polarization which will be discussedbelow.The background contribution to the absorption spec-trum due to infrared active phonons and possible magnonmodes was approximated by a straight cyan line.The obtained set of parameters is V ρ (cid:39)
150 cm − , φ (cid:39) ◦ , J F e,j (cid:104) S j (cid:105) (cid:39) . − , d (12)3 (cid:39) − . a.u .The set of angles determining the equivalent directionsof the exchange field are given in Table II as spherical an-gles ϑ , ϕ . The contribution due to electric-dipole transi-tions determined by Eq. (8) dominates the spectrum.Now we turn to the discussion of the change in the ab-sorption spectrum in the external magnetic field. Conse-quently, the intensities of electric dipole transitions de-fined by expressions (6) and (8) are roughly speakingproportional to the square of the dielectric permittivity. α ( c m - ) TotalEl. - dip H El. - dip H Mag. - dip.Exp.wave number (cm -1 ) Subtracted background FIG. 10. Absorption THz spectrum at T = 2 K . Black curveexperimental data, blue - calculation using Eq. (6), cyan -subtracted background, magenta - Eq. (8), green - contribu-tion of magnetic-dipole transitions, red - simultaneously takeninto account magnetic and electro-dipole transitions. ϑ ϕ (cid:104) S a (cid:105) (cid:104) S b (cid:105) (cid:104) S c (cid:105) (cid:104) p a (cid:105) (cid:104) p b (cid:105) (cid:104) p c (cid:105) . ◦ ◦ -1.85 -0.17 0.21 0.20 2.60 -3.77106 . ◦ ◦ . ◦ ◦ . ◦ ◦ -1.85 0.17 0.21 -0.20 2.60 3.7773 . ◦ ◦ -1.85 -0.17 -0.21 -0.20 -2.60 -3.7773 . ◦ ◦ . ◦ ◦ . ◦ ◦ -1.85 0.17 -0.21 0.20 -2.60 3.77 TABLE II. Calculated expectation values for the spin compo-nents of the Fe ions and for the electric polarization vector p (in 10 − a.u. ) per one Fe site. The angles refer to the exchangefield acting on Fe(1). The projections of spin and polarizationonto the c -axis are the same for the Fe(2) position, while the a and b components change sign. According to Fig. 3, when the magnetic field increasesfor E ω (cid:107) H , the dielectric constant ε (cid:48) ∼ n increases anddecreases for E ω ⊥ H . Qualitatively, this corresponds tothe trend of changes in the absorption intensity in Figs.4(b)-(d) and agrees with the conclusion that the absorp-tion lines E , E and E are mainly due to the electriccomponent of the electromagnetic wave.Passing on to the description of the microscopic theoryof changes in the absorption spectrum when an externalmagnetic field is applied, we note the following. From theexperimental spectra we can assume that the absorptionspectrum of the E -band almost does not change its shapein an applied magnetic field. Moreover, we determinedwith our simulation that the line positions and relativeintensities are mainly determined by the internal molec-ular field.We believe that the external magnetic field makes pref-erence to domains where the c -axis is parallel to the ap-plied field. The calculated spectra for such domains are0 α ( c m - ) E ω ┴ HE ω || H wave number (cm -1 ) FIG. 11. Calculated THz absorption spectrum at 2 K fordomains, aligned along the c -axis. Blue curve Faraday ge-ometry: H ω (cid:107) c , E ω ⊥ c , red Voigt configuration: E ω (cid:107) c , H ω ⊥ c . shown in Fig. 11. The tendency of the change in ab-sorption in applied external fields is in agreement withexperiment.It is important to emphasize that our model providesa consistent description of reported Mssbauer data re-garding the asymmetry parameter ( η = 0 .
23) and thesign of the electric field gradient at the Fe nucleus ( V zz < V and φ . It is alsointeresting to note the following. The minimum energyof the operator (4) depends on the direction of the ex-change (molecular) field acting on iron spin. By adjust-ing the magnitude and direction of the exchange fieldaccording to the observed absorption spectrum, we wereable to calculate the magnitude and direction of the ironspins as a result of the diagonalization of the Hamiltonian(4). Performing such kind of calculations, we found thatthe angle between the directions of the molecular field,which is presumably determined by the total direction ofchromium spins, and the Fe spins is about ∼ ◦ . Thisis an interesting mechanism for the formation of a non-collinear spin arrangement of chromium and iron spins,which to the best of our knowledge, has not been reportedbefore.As one can see from Table II, in the absence of anexternal magnetic (electric) field, there are eight ener-getically equivalent spin configurations differing in rela-tive orientation of iron, chromium spins and spontaneouselectric polarization. The absolute values of the spin-induced electric polarization components along the c -axisare equal, but they differ in sign. Therefore, we can spec-ulate about two different types of electrically polarizeddomains with opposite electric polarization in FeCr S .When an external electric (magnetic) field is switchedon, the equivalence of these domains gets broken. In thisregard, we can understand why the evaluated electric po-larization along the c -axis for the monodomain case (after averaging over Fe(1)S and Fe(2)S fragments) is about P = 255 µC/m , i.e. it is larger by the factor ∼ . T < J F e,j (cid:104) S j (cid:105) (cid:39) . cm − is rathersmall. Comparing this value with the exchange fieldof chromium spins 12 J (1) ξ (cid:48) ,Cr (cid:104) S Cr (cid:105) one can conclude thatthe average projection of Cr spins (cid:104) S Cr (cid:105) along the direc-tion of the Fe spin, perhaps, is reduced due to the non-collinearity between Cr spins. These issues obviouslyrequire further investigation. V. SUMMARY
We identified six low-frequency modes in the multi-ferroic ground state of FeCr S by THz-spectroscopy andstudied their behavior in magnetic fields up to 7 T. Theintensity dependence of the three most intense modes E − E on the relative orientation of the light polar-ization and the external magnetic field allowed to con-clude that they are predominantly electric-dipole active.Modes M and E b are active for E ω ⊥ H , while for M and E a no clear selection rules could be determined. Inaddition, a theoretical model is introduced to describethe excitations E − E in terms of the low-energy elec-tronic excitations of the Fe -ions (3 d , S = 2) in antetrahedral S − environment. Reproducing the eigenfre-quencies and relative intensities of these absorption linesgives a good agreement for the strongly field-dependentmodes E − E , but overestimates the intensity of mode E . The obtained parameters and effective Hamiltoni-ans also allow to reproduce experimental parameters ofprevious Mssbauer studies and the order of magnitude ofthe electric polarisation induced by orbital ordering andnon-collinear spin ordering. The additionally observedmodes M and M are not described within our theo-retical approach. Hence, further theoretical and experi-mental work on single crystals will be needed to decide,whether they correspond to collective magneto-electricmagnon modes of the ground state. ACKNOWLEDGMENTS
We acknowledge support by the Deutsche Forschungs-gemeinschaft via TRR 80 (project no. 107745057). Thework of M.V.E. and K.V.V. was supported by the Rus-sian Science Foundation (Project No. 19-12-00244).1 ∗ [email protected] † [email protected] G. Shirane, D. E. Cox, and S. J. Pickart, J. Appl. Phys. , 954 (1964). F. K. Lotgering, A. M. van Diepen, and J. F. Olijhoek,Solid State Commun. , 1149 (1975). R. Englman and B. Halperin, Phys. Rev. B , 75 (1970). M. R. Spender and A. H. Morrish, Solid State Commun. , 1417 (1972). L. Brossard, J. Dormann, L. Goldstein, P. Gibart, andP. Renaudin, Phys. Rev. B , 2933 (1979). L. F. Feiner, J. Phys. C: Solid State Phys. , 1495 (1982). M. Eibschutz, S. Shtrikman, and Y. Tenenbaum, Phys.Lett. A , 563 (1967). G. Hoy and K. Singh, Phys. Rev. , 514 (1968). A. P. Ramirez, R. J. Cava, and J. Krajewski, Nature ,156 (1997). V. Tsurkan, M. Lohmann, H.-A. Krug von Nidda, A. Loidl,S. Horn, and R. Tidecks, Phys. Rev. B , 125209 (2001). V. Tsurkan, M. Baran, R. Szymczak, H. Szymczak, andR. Tidecks, Physica B , 301 (2001). V. Tsurkan, J. Hemberger, M. Klemm, S. Klimm, A. Loidl,S. Horn, and R. Tidecks, J. Appl. Phys. , 4639 (2001). D. Maurer, V. Tsurkan, S. Horn, and R. Tidecks, J. Appl.Phys. , 9173 (2003). M. Mertinat, V. Tsurkan, D. Samusi, R. Tidecks, andF. Haider, Phys. Rev. B , 100408 (2005). C. Shen, Z. Yang, R. Tong, G. Li, B. Wang, Y. Sun, andY. Zhang, J. Magn. Magn. Mater. , 3090 (2009). V. Tsurkan, O. Zaharko, F. Schrettle, C. Kant, J. Deisen-hofer, H.-A. Krug von Nidda, V. Felea, P. Lemmens, J. R.Groza, D. V. Quach, F. Gozzo, and A. Loidl, Phys. Rev.B , 184426 (2010). G. M. Kalvius, A. Krimmel, O. Hartmann, R. W¨appling,F. E. Wagner, F. J. Litterst, V. Tsurkan, and A. Loidl, J.Phys.: Condens. Matter , 052205 (2010). K. Ohgushi, T. Ogasawara, Y. Okimoto, S. Miyasaka, andY. Tokura, Phys. Rev. B , 155114 (2005). J. Bertinshaw, C. Ulrich, A. Guenther, F. Schrettle,M. Wohlauer, S. Krohns, M. Reehuis, A. J. Studer,M. Avdeev, D. V. Quach, J. R. Groza, V. Tsurkan,A. Loidl, and J. Deisenhofer, Scientific Reports , 6079(2014). L. Lin, H. X. Zhu, X. M. Jiang, K. F. Wang, S. Dong,Z. B. Yan, Z. R. Yang, J. G. Wan, and J. M. Liu, Scientific Reports , 6530 (2014). J. Deisenhofer, F. Mayr, M. Schmidt, A. Loidl, andV. Tsurkan, Phys. Rev. B , 144428 (2019). V. Fritsch, J. Deisenhofer, R. Fichtl, J. Hemberger, H.-A. Krug von Nidda, M. M¨ucksch, M. Nicklas, D. Samusi,J. Thompson, R. Tidecks, V. Tsurkan, and A. Loidl, Phys.Rev. B , 144419 (2003). V. Felea, S. Yasin, A. Guenther, J. Deisenhofer, H.-A. K.von Nidda, E.-W. Scheidt, D. V. Quach, J. R. Groza,S. Zherlitsyn, V. Tsurkan, P. Lemmens, J. Wosnitza, andA. Loidl, J. Phys.: Condens. Matter , 486001 (2014). K. Wakamura, Sol. St. Comm. , 1033 (1989). T. Rudolf, K. Pucher, F. Mayr, D. Samusi, V. Tsurkan,R. Tidecks, J. Deisenhofer, and A. Loidl, Phys. Rev. B ,014450 (2005). G. Slack, F. Ham, and R. Chrenko, Phys. Rev. , 376(1966). G. Slack, S. Roberts, and J. Vallin, , 511 (1969). S. Wittekoek, V. Staepele.RP, and A. Wijma, Phys. Rev.B , 1667 (1973). L. Mittelstaedt, M. Schmidt, Z. Wang, F. Mayr,V. Tsurkan, P. Lunkenheimer, D. Ish, L. Balents,J. Deisenhofer, and A. Loidl, Phys. Rev. B , 125112(2015). N. J. Laurita, J. Deisenhofer, L. Pan, C. M. Morris,M. Schmidt, M. Johnsson, V. Tsurkan, A. Loidl, andN. P. Armitage, Phys. Rev. Lett. , 10.1103/Phys-RevLett.114.207201 (2015). T. T. Mai, C. Svoboda, M. T. Warren, T. H. Jang,J. Brangham, Y. H. Jeong, S. W. Cheong, and R. V.Aguilar, Phys. Rev. B , 224416 (2016). J. Vallin, Phys. Rev. B , 2390 (1970). F. Varret, H. Kerner-Czeskleba, F. Hartmann-Boutron,and P. Imbert, J. Phys , 549 (1972). P. Bonville, C. Garcin, A. Gerard, P. Imbert, and G. Je-hanno, Phys. Rev. B , 4310 (1981). L. F. Feiner, J. Phys. C: Solid State Phys. , 1515 (1982). K. Ono, S. Koide, H. Sekiyama, and H. Abe, Physical Re-view , 38 (1954). T. Fujiwara and Y. Tanabe, Journal of the Physical Societyof Japan , 1512 (1974). M. V. Eremin, Physical Review B , 140404 (2019). M. V. Eremin, Jetp Lett. , 249 (2019). K. Ohgushi, Y. Okimoto, T. Ogasawara, S. Miyasaka, andY. Tokura, J. Phys. Soc. Jpn.77