Giant anomalous Hall conductivity in the itinerant ferromagnet LaCrSb3 and the effect of f-electrons

Itinerant ferromagnets constitute an important class of materials wherein spin-polarization can affect the electric transport properties in nontrivial ways. One such phenomenon is anomalous Hall effect which depends on the details of the band structure such as the amount of band crossings in the valence band of the ferromagnet. Here, we have found extraordinary anomalous Hall effect in an itinerant ferromagnetic metal LaCrSb3. The rather two-dimensional nature of the magnetic subunit imparts large anisotropic anomalous Hall conductivity of 1250 S/cm at 2K. Our investigations suggest that a strong Berry curvature by abundant momentum-space crossings and narrow energy-gap openings are the primary sources of the anomalous Hall conductivity. An important observation is the existence of quasi-dispersionless bands in LaCrSb3 which is now known to increase the anomalous Hall conductivity. After introducing f-electrons, anomalous Hall conductivity experiences more than two-fold increase and reaches 2900 S/cm in NdCrSb3.


Introduction
Nontrivial band topology features a unique electronic structure that describes the origin of the quantum Hall effect, which exists with many variants. In the Hall effect, a mutually perpendicular magnetic field and electric current applied in materials causes a voltage perpendicular to them, i.e., the Hall voltage. Similarly, a comparatively large spontaneous Hall effect, termed the anomalous Hall effect (AHE), is known to exist in magnetic materials in which the Bloch wave function of electrons is asymmetric in momentum space. In this scenario, electrons acquire an additional group velocity in the presence of a driving perturbation, such as an external electric field. This anomalous velocity is perpendicular to the applied electric field, giving rise to an additional value to the Hall effect, i.e., AHE. [1] In addition, the group velocity is drastically enhanced by virtue of the Berry phase of nontrivial bands, which provides a strong fictitious field.
A nontrivial band topology arises when band inversion occurs, i.e., the conduction band is beneath the valance band with respect to their natural order in the vicinity of the Fermi level EF (Figure 1a, left). Such inversion can be of several possible combinations among the s-, p-, d-, and f-bands. Figure 1a (right) shows a schematic of various types of band mixing; a resulting band gap arises after considering spin orbit coupling (SOC) regardless of their type. In such cases, the wave function of each band twists in momentum space inducing a non-zero Berry phase. A linear response of conductivity [2,3] from the Berry phase [4] for a 3D system is expressed in the Kubo formula: % is the Berry curvature, which crucially depends on the entanglement of bands. A small convergence of bands is caused by a large contribution from mixed occupied states, whereas its counterpart, i.e., unoccupied states, contribute negligibly. Therefore, AHE is rather large when the SOC-induced gap is small in the vicinity of EF. The material selection and desired band gap depend strongly upon the hybridization strength (by the lattice constant) and the magnitude of SOC (by the atomic charge). Interestingly, the dispersion of bands is a crucial factor and further depends on orbital hybridizations. For example, Figure 1b shows the calculated electronic band structure of LaCrSb3, wherein several bands are relatively dispersionless, quite close to EF along Y-G. Such bands are highly sensitive to perturbation and account for various intriguing phenomena such as nontrivial topology [5][6][7] , hightemperature fractional Hall effect [6,[8][9][10][11] , unconventional superconductivity [12,13] , and unconventional magnetism. [7,14,15] In our present selection of compound i.e. LaCrSb3 the quasi-dispersionless bands facilitate larger mixing of occupied and unoccupied bands close to EF, that induces a large volume of Berry curvatures (BCs). Such BC associated with nontrivial bands as a source of AHE has recently been recognized in various compounds, for example, chiral antiferromagnets Mn3Sn and Mn3Ge [16,17] , ferromagnetic massive Dirac metal Fe3Sn2 [18] , and ferromagnetic nodal line compounds Co2MnGa [19,20] , Co3Sn2S2 [21] and Fe3GeTe2. [22]

Results
LaCrSb3 possesses an orthorhombic centrosymmetric crystal structure and belongs to the Pbcm space group (no. 57). The lattice parameters a, b, and c for LaCrSb3 are 13.18, 6.16, and 6.07 Å, respectively and they decrease smoothly on replacing rare earth metals from La to Lu except for Yb. [23] In the crystal structure as shown in Figure S1a, the atoms are arranged in special manners: The Sb square nets form perpendicular to the [100]; the edge-and face-sharing of CrSb6 octahedra arrange along the b-and c-axes, respectively. CrSb2-magnetic layers in the b-c plane gives rise to a 2D character to the crystal. The anion layers (Sb-square net and CrSb2-layers) are separated by a cation La layer. These atomic arrangements are responsible for anisotropic electrical and magnetic properties. [24] From Figure 1c (left panel), electronic charge density is localized on Cr and La atoms. The 3d 3 states of Cr 3+ in each CrSb6 octahedron experience the highest crystal field energy, and they favorably split into eg and t2g energy levels, which are the main source of magnetism in this material (Figure 1c right panel). These levels are split further among themselves. The orbitals dx 2 -y 2 and dz 2 split about 1.0 eV above and below the EF, respectively and give rise to 1 µB of magnetism. However, the t2g states split and are localized in the order of meV around the EF. This adds a small magnetic moment to the Cr atoms, due to the EF lying in the middle of the band. Therefore, LaCrSb3 is known to exhibit itinerant ferromagnetism. [25] The transition temperature TC is 125 K, and their spins are aligned in the b-c plane. [26] This means that the b-and c-axes are the easy axes, whereas the a-axis is the hard axis. Below 95 K, spins point 18° away from the b-axis in the b-c plane, reminiscent of an antiferromagnetic (AFM) component along the c-axis. Figure S2 shows the temperature-dependent resistivity behavior along different crystallographic axes; the resistivity rapidly decreases after TC. Magnetization measurements are consistent with the b-axis being the easy axis; TC also corresponds to that observed in the resistivity measurement ( Figure S9).
Electrical resistivity is directly related to the density of states of materials at EF, whereas AHE is controlled by the BC concerning all the occupied states below EF. We measured the Hall resistivity ) and longitudinal resistivity as a function of field B along all three crystallographic axes of LaCrSb3 at varying temperatures, as shown in Figure 2a and Figure S3, respectively. ) shows anomalous behavior up to TC (~125 K) along the b-and c-axes (Fig. S6); their respective anomalous values at 2 K are 1.2 and 0.32 µWcm, whereas the a-axis data do not show any anomalous behavior (Figure 2a). The Hall conductivity is calculated according to the relation: Figure 2b, the corresponding anomalous Hall conductivity (AHC) are 1250 W −1 cm −1 for the b-axis and 1150 W −1 cm −1 for the c-axis at 2 K; which gradually decrease to zero as the temperature approaches Tc (Figure 2c). It should be noted that the Hall conductivity remains zero up to 0.17 T along c-axis after which it suddenly rises to attain the saturation. This behavior is consistent with the small AFM interaction along c-axis owing to the fact that the anomalous velocity over all the occupied states in an AFM is zero. It is clear that the measured AHE is strongly anisotropic and appears only when B applies in the b-c plane. Magnetic fielddependent magnetization measurements for LaCrSb3 at 2 K along different crystallographic axes are shown in Figure 2d. The magnetic moments are easily aligned along the b-and c-axes, whereas a-axis is the hard axis, evidencing an anisotropic magnetic behavior. As compared to b-axis, magnetization along c-axis starts to increase slowly at small field, then suddenly jumps to saturation, accounting for the 18° spin canting towards the c-axis in the b-c plane. The saturation magnetization reaches 1.6 µB/f.u. for LaCrSb3, which is the same as previously reported. [25,26] To gain an insight into the giant observed AHC, we used constrained-moment ab-initio calculations with the local spin density approximation (LSDA) exchangecorrelation potential to simulate the band structure of LaCrSb3 with the experimental lattice parameters and magnetic moments with spins pointing along b-axis. We calculated the momentum space BC of the electronic structure, revealing a large volume, which is centered on the Γ point (Figure 3a). This originates further due to the inversion between Cr-d and Sb-p orbitals, forming rather dispersionless bands around this region. We found two interesting features: i) Nontrivial bands in the plane ky-kz that produce large non-zero BC. [10,11] This large volume of BC is different from normal magnetic metals, which show delta-like "hot" spots in the Brillion zone (BZ), for example, bcc Fe. [2] Such a unique and large volume of BC distribution provides a giant AHC. ii) Trivial bands along kx due to the weak coupling between Cr-Sb layers and La layers, producing the large longitudinal resistivity and negligible anomalous Hall effect as we have observed. Figures 3b-d show the electronic band structures for LaCrSb3 along the high symmetric points (shown by dashed green, black and pink lines), depicting many nontrivial nearly dispersionless bands in the vicinity of EF. From their atomic orbital contributions as given in Figure S14, these bands are mainly Cr-d dominated, which play a crucial role for the electric and magnetic properties. These nontrivial bands are localized around the center of the Brillouin zone along kx. A recent study [27] on the effect of reducing band width in the enhancement of AHC is consistent with our observation of large AHC in quasi-dispersionless bands in LaCrSb3 (see Figure S15 for a tight binding method based toy model).

Effect of f-electron:
After measuring the remarkable values of AHC in the parent compound LaCrSb3, it is highly desirable to obtain understanding about the AHE behavior by measuring other compounds from RCrSb3 series, where R is a rare earth element. Therefore, we extended our study to CeCrSb3 and NdCrSb3 possessing f-electrons and measured the field-dependent ) and values ( Figure S4-S7) at various temperatures. For CeCrSb3, the measured anomalous values of ) along band c-axes at 2 K are 4.3 µW cm and 5.2 µW cm, respectively (see Figure S6). The resulting values of ) are plotted in Figure 4 at various temperatures, which shows that the anomalous behavior of ) is quite similar to that of LaCrSb3. The AHC of CeCrSb3 at 2 K is 1550 W −1 cm −1 along both the b-and c-axes (Figure 4a), and the AHC of NdCrSb3 at the same temperature is 2900 W −1 cm −1 for the b-axis ( Figure 4b) and 900 W −1 cm −1 for the c-axis. As expected, these values decrease as temperature reaches at TC (Figure 4c). Even though felectrons introduce finite spontaneous magnetization along a-axis in CeCrSb3 and NdCrSb3, AHE is negligible for both compounds over the entire temperature range, as observed with LaCrSb3. This demonstrates that Cr-d electrons are largely responsible for the AHE in this series of compounds. Moreover, there is no one-to-one correspondence between observed AHC and magnetic moment along various axes. This can be seen from the column plot in Figure 4d, signifying the role of electronic structure for AHC. For example, even though the magnetization of NdCrSb3 is the highest along a-axis at 2K but produces negligible AHC. The similar effect is also observed for CeCrSb3. Among the series of compounds, NdCrSb3 shows giant AHC, which is the highest ever measured in any material to the best of our knowledge. NdCrSb3 has a larger moment as compared to LaCrSb3 and CeCrSb3, and that moment is weakly coupled to the Cr d-states. This is evident from a metamagnetic transition in magnetization and AHE data of NdCrSb3 along b-axis. However, above the ordering temperature of Nd spins (12 K), the effect is lost and the behavior and value of AHC closely follow that of LaCrSb3. Larger magnetic moment in NdCrSb3 compared to LaCrSb3 and CeCrSb3 at low temperature could be one of the reasons for the existence of giant AHC. The observation that the AHC in NdCrSb3 decreases sharply and closely follows that of LaCrSb3 after Nd-spin ordering temperature, indicates that Cr-d electrons dominated in the larger part of the temperature range studied. Owing to the correlation effect for added f-electrons in CeCrSb3 and NdCrSb3, it is not straightforward to estimate AHC from first principles calculations. It should be noted that the sign of anomalous Hall resistivity differs for various axes in all three compounds despite maintaining the same measurement geometry. Hence, this sign is dictated by the sign of Berry curvature for a particular direction of applied magnetic field. For a broad comparison, the observed values of AHC of some notable compounds are shown in Figure S13.
The anomalous behaviors in RCrSb3 not only appear in electrical transport but are also found in thermal transport, i.e., the Nernst thermopower S. Figure  S8 shows the field-dependent measured Sxz of CeCrSb3 at different temperatures, affirming an anomalous behavior. The anomalous value of Sxz is found to be 2.5 µVK −1 at 21 K, which associates these compounds with the non-trivial materials, exhibiting high anomalous Nernst effect. [20,28] Another important parameter, anomalous Hall angle (AHA), defines as how much longitudinal current coverts into the transverse direction. The estimated AHA is 4-10% for the present series of compounds (see Table  1). It is notable that despite a large carrier concentration ~ 10 22 cm -3 , the value of AHA compares to materials like Weyl semimetal Co3Sn2S2 where the carrier concentration is at least two orders of magnitude smaller. [21,29]

Discussion
RCrSb3 is a promising series of quasi-2D compounds that exhibit high anisotropic AHC. The measured AHC is the sum of all the contributions from the entire BZ and that can have both intrinsic and extrinsic origins. From the framework of unified models that are valid for the varieties of compounds having conductivity beyond the range of 10 4 < < 10 6 W -1 cm -1 , extrinsic origins dominate. [30,31] The conductivity of the RCrSb3 series of compounds ranges from 0.7 ´ 10 4 to 5.9 ´ 10 5 W -1 cm -1 , which lie within the moderate range of conductivity. The temperature-dependent data of ' ( vs is neither constant nor linear, excluding the single contribution from the Berry phase or skew scattering ( Figure S12). However, from the power law behavior ' ( ∝ % , n is found to be 1.7 for LaCrSb3 and CeCeSb3 ( Figure S12). For cases where a mixed contribution of Berry phase and side-jump dominates, n is predicted to be 1.6. [32] Surprisingly, our estimation of n = 3 for NdCrSb3 goes beyond this power law and calls for more accurate scaling law.  [30] We found that the values of ℏ 3 and 67 are 0.7 and 0.8, respectively, for the RCrSb3 series of compounds, and they are best matched in the criteria of the resonantly enhanced AHE. These scenarios indicate that the measured AHC of RCrSb3 can arise from a mixture of intrinsic and extrinsic origins that is hard to separate out. The low temperature longitudinal conductivity of all the three compounds is between 10 4 to 10 5 W -1 cm -1 which is still in the moderate conductivity limit, but is close to the boundary of dirty limit. Hence, the most probable cause of the extrinsic contribution can be the side-jump effect. It originates from the change in the momentum of the Gaussian wave packet when it interacts to a sufficiently smooth impurity potential in the presence of spin-orbit interaction. [3] Like the intrinsic effect, it is also independent of the scattering time and hence very difficult to differentiate.

Conclusion
We observed large values of anisotropic AHC in RCrSb3 (R = La, Ce and Nd) series of compounds. Effect of the introducing f-electrons as in CeCrSb3 and NdCrSb3 shows enhancement in AHC. The large magnetic moment in NdCrSb3 can be one of the reasons for the existence of giant AHC. We demonstrate that power law scaling for anomalous Hall conductivity follows ' ( ∝ 1.9 which is valid in the intrinsic and side-jump regime for LaCrSb3 and CeCrSb3 while it goes beyond this scaling for NdCrSb3. The positive aspects of the existence of rather dispersionless bands for observing large value of anomalous Hall conductivity have also been discussed for the first time which will provide motivation for exploring anomalous transport in flatband magnetic systems.         In the above criteria, ** is considered to be sample-dependent (residual value). However, for a fixed sample, ** will vary when temperature dependent *+ ( is measured. The energy broadening of bands which is related to scattering time, increases with increasing temperature and when it is larger than the energy splitting of bands due to spin-orbit interaction, n reaches to 2 i.e. *+ ( ∝ ** # as revealed by Berry phase induced intrinsic behavior (3). Interestingly, the value of n is found to be n = 1.7 for LaCrSb3, n = 1.7 for CeCrSb3 and n = 3 for NdCrSb3, which are below and above the Berry phase induced AHC. The contributions from Berry phase AHC can be deduced from the intercept of straight line of *+ ( vs ** # plot. However, due to a nonlinear behavior in the whole temperature range of measurement, this intercept method is not justified in the present case. While doing so, the intercept from the low temperature data reveals the value of AHC 220 W -1 cm -1 for LaCrSb3 Fig. S12.  Figure S13. Column plot of ! " for some selected compounds and elements that exhibit anomalous Hall conductivity; the column data were taken from Table S1. Among them, NdCrSb3 shows the largest value of ! " .

Band structure of LaCrSb3
Bands around Fermi level, EF, are dominated by the Cr-d orbitals, with dyz crossing EF, and a slight hybridization with the Sb-py orbital. The La-f states are highly localized around 1.5 to 2.0 eV above the Fermi level. Although the Laf states are not occupied, they play a crucial role with that of the Sb-p to sandwich the Cr-d states to be more localized around the EF. Since the f-states above EF is almost forbidden for the Cr-d electrons, the energy window for Cr-d states is compressed and results in the Cr-d dominated flat bands around EF. Furthermore, because of the orthorhombic crystal structure, the crystal field causes the eg, and t2g states to split. These eg, and t2g states are further split among themselves, where the dx 2 -y 2 and dz 2 are split on the order of 1 eV above and below the Fermi energy, respectively, to give 1 µB of magnetism. The t2g states are localized around the EF and split on the order of meV, with hybridization. This adds a small moment to the Cr atoms, due to the EF lying in the middle of the band.

Effect of band flattening on anomalous Hall conductivity
The motivation to study the AHC in materials with quasi-dispersionless bands is based on the effective volume of band anti-crossings at the Fermi level. In most cases, the band anticrossings are not only some isolated points but can also form some finite area in the reciprocal space. The overlap between band anti-crossing and Fermi level gives effective contributions to the intrinsic AHC. We would like to have such kind of overlaps as much as possible. Along with the dispersion of band structure, such kind of anti-crossing areas can also have dispersion in k-space. The overlap between band anticrossings and Fermi level cannot be completely 100%. However, it is possible to increase the percentage of the overlap by reducing the band width with suitable chemical potential. This can be understood from an effective two band tight binding model , where , .
We have set the parameters to guarantee the existence of band anti-crossing between these two bands. Supposing only one electron in this system and keeping the existence of band anticrossing, we decrease the band width of 1 st band, see Figure S15 (a-f). For each case after tuning the band width, we have calculated the DOS and integrated it to get the chemical potential, such that always one electron is filled in the system. When the band-width of 1 st band is large enough, the relationship between bandwidth and AHC is not so clear, such as from 2.7eV to 2.0 eV. However, as the band width becomes much smaller, the trend of AHC becomes much more obvious. From Figure S15 (g), one can find that the AHC increases dramatically with decreasing band width in the range of 2.0 eV to 0.2 eV. The effective change of AHC is 800 S/cm (from ~380 S/cm to ~1180 S/cm) by reducing the band width in the tight binding model. This accounts for an effective enhancement of more than 200 %. Indeed, this change is not small and is large enough to illustrate the important connection between the flatband and large intrinsic AHC.