Metamagnetic multiband Hall effect in Ising antiferromagnet ErGa₂

We first summarize the model for the MM Hall response. In the presence of more than one type of conduction electrons, the coefficient R₀ in Eq. 1 in principle should reflect a multiband response; here, we consider a representative two-band scenario with hole and electron pockets (their respective carrier densities as n_h and n_e and mobilities as µ_h and µ_e). In the low-field regime (µ_e,hB < 1), this is approximately (19)

However, for magnetic systems, the mobility is further modified by band-dependent spin-electron coupling. When the magnetization rapidly evolves at a metamagnetic transition this can give rise to the MM Hall effect. Importantly, in typical magnetically textured system (in particular those based on rare earth), metamagnetic transitions can take place at a relatively low magnetic field scale and can often be lower than the threshold for typical orbitally driven multiband (OHE) nonmonotonic Hall responses.

For concreteness, we consider a metallic compound with an electron band with 5d/6s orbital characters from magnetic cations and a hole band mainly composed of p-orbital at nonmagnetic (anionic) sites (Fig. 1A). For the case of rare earth intermetallics, the local magnetic moment has the form of 4f levels that strongly couples with the cation conduction band through the intra-atomic 5d-4f interaction, JS · sδ (r − R) (20), where J is the exchange coupling constant, S and s are spins localized at R and for conduction electrons with spatial coordinate r, respectively. The intersite interaction with the anion p-valence band, however, is expected to be relatively weak due to their reduced hybridization (21–23) and energy cost for 4f-p electron hopping (we consider the case where the 4f level is well-localized below the Fermi energy, which is typically realized in rare earth intermetallics apart from Ce, Pr, and Yb; see SI Appendix, section S1 for the opposite extreme where interatomic f-p hybridization becomes relevant). In such cases, the electron-type carriers are more readily scattered from magnetic disorder (Fig. 1B), giving rise to a field modulation in Eq. 2 via relative changes in µ_h and µ_e.

This can be quantitatively illustrated with a simplified two-band model, in which we assume that µ_e is sensitive to while µ_h is immune from magnetic disorder scattering. Within Matthiessen’s rule, the mobility of electron-type carriers is

The inverse of the carrier lifetime is additive: τ⁻¹ = τ₀⁻¹ + τ_mag⁻¹, where τ₀ is for the nonmagnetic (field-independent) scattering due to impurities/defects and phonons, and τ_mag stems from electron-spin scattering. In the magnetization process from the paramagnetic state (PM) to the field-induced ferromagnetic state (Fi-FM), the field-dependence of 1/τ^e_mag can be expressed as

where the spin-disorder scattering is taken to be proportional to the magnitude of disorder upon uniformly aligned spins, viz (1 – M²/M₀²) (24–26); this provides a good description for Ising magnets wherein spin-flip and spin-wave scattering processes are minimal (27–29).

Fig. 1 C–E show the evolution of µ_e, two-band resistivity (ρ_xx), and Hall resistivity (ρ_yx) for a linear evolution of M as a general illustration for the case of constant µ_h and n_e,h. The negative magnetoresistance at low fields is consistent with field-suppression of magnetic disorder (spin MR), while the Hall resistivity exhibits nonlinearity. We note that this model does not include a contribution from M-linear AHE and THE and carrier density change. The modulation of the Hall effect arises from the field-dependence of OHE across the magnetization process. This effect appears to be often overlooked in the conventional treatment of the AHE caused by side jump, skew scattering, and Berry phase mechanism, where the Hall anomaly originates from the second term in Eq. 1 and R₀ is taken as a constant.

The extension of this model to a metamagnetic transitions among antiferromagnetic (AFM), metamagnetic (M), and Fi-FM states is shown in Fig. 1 F–H. At zero temperature, spin-disorder scattering or spin-wave scattering does not contribute to the spin MR. Instead, the order parameter and magnetic unit cell change across the metamagnetic transition result in a reconstruction of the magnetic Brillouin zone (BZ) and give rise to a stepwise change of the renormalized effective-mass of carriers. The magnetoresistivity and Hall resistivity (Fig. 1 G and H) both reflect the characteristic field-dependence of µ_e–the MM Hall effect. We note such affects ascribed to changes in carrier lifetime have been previously reported at metamagnetic transitions for other Ising-like rare earth systems (30, 31). The crucial point of this effect is that it should be distinguished from the THE to avoid the overestimation of the emergent magnetic field (and in turn interpretation in terms of a noncoplanar nature of the underlying magnetic structure).

Here, we present a pronounced MM Hall effect in ErGa₂ as a remarkable example of the above scenario. Rare-earth intermetallics of the type RGa₂ (R: rare earth) are frustrated magnets with an alternative c axis stacking of R triangular-lattice Ga honeycomb layers (Fig. 2A) (32, 33). ErGa₂ exhibits an archetypical two-step metamagnetic transition for H ||c (34), desirable for the current study. This is due to the frustration (18) among the Er moments with strong easy-axis type anisotropy (18, 35). Below the AFM transition at T_N = 7 K (36), commensurate single-q order for the up–down configuration of Er-moments is stabilized in zero field (AFM in Fig. 2A) (37). Application of magnetic field along the easy-axis induces two stepwise transitions (18, 36, 37). At magnetic field of 0.8 T a 3up-1down-type commensurate phase is induced. The network of upward magnetic sites forms a kagome network (Fig. 2A) (18). This phase retains the original magnetic modulation vector (q_AFM = ½a*) as the principal modulation but forms a triple-q state with a new reciprocal lattice unit: a*_kagome = ½a*, b*_kagome = ½b*, and c*_kagome = c*, where a*, b*, and c* are the reciprocal lattice unit of the original lattice. At low temperatures, a half-magnetization plateau remains until the field-induced ferromagnetic state for q_FM = (0, 0, 0) is induced at 2 T (Fig. 2A). Coupling between conduction electrons and Er moments is observed as a sudden drop of resistivity at T_N (38) as well as in magnetoresistance anomalies across the metamagnetic transitions (39).

Fig. 2 B–D show the magnetization and magnetotransport properties of ErGa₂, measured with single crystals for H||c. The half-magnetization plateau in the intermediate kagome spin state (Fig. 2E) and characteristic magnetic magnetoresistance are similar to those previously reported (39). Across the metamagnetic transition, we further observe a nonmonotonic Hall resistivity, which shows a positive enhancement in the intermediate kagome state. As a THE is precluded by the Ising-type magnetism, this arises from some combination of OHE and AHE. The persistence of the nonlinearity at T = 10 K in the paramagnetic phase (gray curve in Fig. 2D) is suggestive of two-carrier behavior with high-mobile hole-type (low density) carriers and electron-type low-mobile (high density) carriers, which sharpens at lower temperatures along with magnetization and ρ_xx (Fig. 2C), i.e., an OHE.

Comparison of the temperature dependence of ρ_yx/B of ErGa₂ and that for isoelectronic and isostructural LaGa₂ (40) (Fig. 2F) shows that the two are of a similar value at all the temperature range (ratio < 1.4) including lack of significant discrepancy below T_N for ErGa₂, suggesting a weakness or absence of AHE in ErGa₂. This is consistent with the absence of skew scattering given the longitudinal conductivity σ_xx ~ 10⁵ S/cm (41), where the intrinsic mechanism rather than the extrinsic scattering mechanism becomes more relevant. Moreover, as shown in Fig. 2F, there is no significant anomaly in ρ_yx/B at T_N in contrast to the factor ρ_xx²M/B (blue circle in Fig. 2F) where the intrinsic Berry curvature would manifest if present (42). This is distinct from the reported features for other rare-earth materials (14, 15, 43–45) and RGa₂ (R = Ce, Sm, Gd) (40, 46), where anomalous Hall contribution was suggested. We note that demagnetization correction may also affect the estimation of R₀ as discussed in SI Appendix, section S2; therein, the Hall coefficient is enhanced at low temperatures and shows a peak at T_N [not explicitly argued in the previously (40, 46)]. The absence of AHE in ErGa₂ may further arise due to small de Gennes factors (∝ (g_J – 1)²J(J+1)), the square root of which roughly determines the scale of the exchange field on conduction electrons from the localized moment of rare earth ions (47, 48).

To illustrate the consistency of the observed effect with the MM Hall effect, we plot M, σ_xx (=ρ_xx/(ρ_xx²+ρ_yx²)), and σ_xy (=ρ_yx/(ρ_xx²+ρ_yx²)) at T = 10 K in Fig. 3 A–C as a function of B (=µ₀H_int + M). We analyze the transport properties within the model introduced above. For simplicity, we introduce the mobility modulation due to spin alignment to the electron-type carriers (dominant in ErGa₂, which we return to below). Fig. 3 A–C show this analysis applied for the PM-to-Fi-FM process at T = 10 K. The field-dependence of σ_xx and σ_xy can be simulated within a conventional two-band model:

where µ_h, n_e, and n_h are fitting parameters. From the field-dependence of M(B), µ_e(B) is captured by fixing $m_e/{\tau }_0^e$ and $m_{\mathrm{e}}/{\tau }_{\mathrm{mag}0}^{\mathrm{e}}$ in Eqs. 3 and 4 (the negligible contributions from the spin-fluctuations and spin-wave-scattering are further discussed in SI Appendix, section S3). This shows reasonable agreement with the observed response, reproducing the spin MR σ_xx and nonmonotonic behavior in σ_xy simultaneously.

We next extend this model to T = 1.8 K within the AFM, kagome, and Fi-FM phases as shown in Fig. 3 D–F, where we assume the proportionality of the field-dependence of ρ_xx(B)/ρ_xx0 to 1/µ_e(B) (Fig. 3D) (31). We note that at low temperatures there are interphase states (IP1 and IP2) across the metamagnetic transitions which give rise to singular peaks in σ_xx (Fig. 2E); as these are proposed to arise from specular domain wall scattering (39), we exclude these in the following analysis (gray-hatched area in Fig. 3 D–F and see SI Appendix, section S4 for a discussion on the IP states). We also note that multistep metamagnetism is observed (SI Appendix, section S2) suggestive of more complex intermediate spin configurations in the IP1 and IP2 regions (this is beyond the scope of our simplified model).

Fig. 3 G and H summarize the transport parameters obtained by performing the analysis using the ansatz ρ_xx(B)/ρ_xx0 ∝ 1/µ_e(B) at various temperatures. For T = 10 and 8 K (>T_N), the parameter set is consistent with those for Eq. 4. We also confirmed the absence of significant anomaly in µ_e(9 T), n_e, n_h, and µ_h at T_N, while µ_e(0 T) is enhanced below T_N due to the suppression of the spin-disorder scattering. n_h shows a slight temperature dependence (Fig. 3H), which might not be an intrinsic effect as the fitting quality does not change by fixing n_e and n_h constants. Fig. 3I shows the zero field transport properties reproduced from the fitting parameters; the two-band Hall coefficient (R₀ in Eq. 2) and resistivity ρ_{xx,0 T} (=σ_xx(0 T)⁻¹ in Eq. 5) are reasonably consistent with the direct observation. The numbers of carrier per unit volume are 1.2 and 3 × 10⁻⁴, for electrons and holes, respectively. The large electron concentration is consistent with the negative slope of ρ_yx at high fields, corresponding to 1.5 × 10²² cm⁻³ (Fig. 2D). Two orders of magnitude higher mobility of the hole pocket is necessary to reproduce the positive Hall coefficient at zero field (Figs. 2F and 3I). We note that these values reflect an oversimplification of the electronic structure of this system as a two-band model and are not directly connected to the volume of Fermi surfaces for the low-field Hall effect (49–51), which is affected by the local curvatures (see below).

To evaluate the relevance of this model in the present intermetallic setting, we calculate the band structure for the nonmagnetic analog LaGa₂ (Fig. 4A). Ellipsoidal hole pockets (FS1 and FS2, see Fig. 4B) at the A point on the BZ edge are ascribed to the Ga-4p band, while the large Fermi surface (FS3, see Fig. 4C) is composed mainly of La-5d orbitals, consistent with the previous studies (52, 53). These two bands correspond to those in the schematic model in Fig. 1A. The Hall coefficient for each Fermi surface is determined by the sign of curvature γ (∝-σ_xy) (49, 50). We map γ onto each Fermi surface as shown in Fig. 4 D and E. For the hole bands (FS1 and 2), it is evident that their contributions to the Hall effect are ${\mathrm{\sigma }}_{xy}^{\mathrm{h}}$ > 0. The FS3 has both characters showing a sign change for γ from negative near k_z = π/c (A point) to positive across around k_z = π/2c to 0 (Fig. 4E), where c is the lattice constant. The representative electron-type carrier orbits at k_z ~ π/8c and k_z ~ 0 are depicted in Fig. 4 F and G, respectively, which confirms the presence of an electron-type orbit deflecting counter-clockwise (SI Appendix, section S5) (51). We note that a three-band treatment (a hole pocket for p-band, an electron and a hole pocket with comparable concentration and mobility for d-band) would be more realistic on the basis of the band calculations. This can be effectively treated by a two-band model by representing the d-band pockets as a large electron pocket and leaving a hole pocket for the high-mobile p-band (SI Appendix, section S6). These results support the application of the two-band model with the asymmetric spin–charge coupling as an ansatz to analyze the magnetotransport properties of RGa₂. We also note that this argument can be applied to the magnetic ErGa₂ on the basis of the rigid band approximation from the nonmagnetic analog LaGa₂, where the 4f electrons are expected to be well-localized (Fig. 1A) and only weakly hybridize with the electrons near the Fermi energy. This is consistent with previous quantum oscillation experiments in various RGa₂ (R = Ce, Pr, Sm, Gd) (46, 54–57). In these studies, the quantum oscillation branches associated with the FS1, 2, and 3 are identified almost unchanged from the nonmagnetic LaGa₂. Further, in CeGa₂, the exchange splitting on the FS1 and FS2 has been observed to be smaller than that of FS3 (54) validating the above simplification for the mobility modulation only on the electron-type carriers in ErGa₂.

We note that the modulation of the OHE has also been considered in dilute magnetic alloys, therein known as the “spin effect” (58). In particular, it has been established that a variation of R₀ arises due to the difference of the field-induced change of scattering rate of conduction electrons with up and down spins. The distinction from the current model is that the former always leads to the enhancement of R₀, but the latter can induce both enhancement and reduction depending on the details of the electronic and magnetic subsystems. In the current analysis, we consider the mobility is field-dependent in the magnetization process. It is known that the mobility can be field-dependent even without the magnetic moments as observed in diluted (semi)metals (59, 60) when the disorder potential smoothly varies compared to the cyclotron radius (61). This effect is expected to be masked in the present metallic system owing to the short Thomas-Fermi screening length (~Å). Magnetic semiconductors/semimetals with reduced carriers are a class of materials where this effect potentially interferes with the MM Hall effect. Beyond the mobility and effective-mass changes discussed above, we also note that there are possible contributions to the field-dependence of carrier density across metamagnetic transitions. This effect could be significant in semimetals with low carrier concentration as discussed previously in Eu compounds (62). ErGa₂, on the other hand, is a metal with large carrier number and not a heavy-fermion system and is anticipated to be insensitive to changes of magnetic q-vector as it is commensurate to the lattice. The valley-dependent carrier emptying (63) as well as the magnetic breakdown (64) is irrelevant in ErGa₂ (and is instead important for the quantum limit in semimetals).

In conclusion, we have observed a MM Hall effect in the Ising-antiferromagnet ErGa₂ and demonstrated its consistency with a realistic two-band model. The orbital character in the band structure is further consistent with the two-carrier feature with different scattering cross section to magnetic moments. The band-induced Hall effect discussed in our work in principle can also be present in intermetallic metamagnetic systems with topological spin textures. For the Hall anomaly in such systems, this shows rather than a straightforward assignment to a THE via the ansatz Eq. 1, that instead it can be captured by a multiband response of the charge carriers. This finding provides a clear example where the ansatz Eq. 1 fails to capture the origin of the Hall response and a simple framework to understand the magnetotransport properties of magnetic intermetallics, which would be useful for more precise evaluation of the emergent field therein.

Single crystals of ErGa₂ were grown via the Pb self-flux method following ref. 65. The starting materials Er, Ga, and Pb were mixed in the molar ratio Er:Ga:Pb = 1:2:10. They were loaded into a 2-mL alumina crucible and sealed in an evacuated quartz tube. The growth ampoule was heated to 1,000 °C and slowly cooled to 400 °C at a rate of 1 °C/min. To increase the size of crystals, the furnace temperature was increased to 850 °C in 2 h and cooled down to 400 °C at 1 °C/min; this process was subsequently repeated between 700 °C to 400 °C. The single crystals were separated by decanting the flux in a centrifuge. Finally, crystals were annealed in an evacuated quartz tube at 600 °C for 6 d. The typical size of the crystals is 1 × 1 × 0.5 mm³ (for a×b×c). The quality of the crystal was checked by the single-crystal x-ray diffraction (XRD) using the synchrotron light source at SPring-8, Japan (SI Appendix, section S7).

Electrical transport measurements were performed by a conventional five-probe method with an AC excitation current of 1 mA at a typical frequency near 15 Hz. The transport response in low temperature and a magnetic field was measured using commercial superconducting magnets and cryostat. The obtained longitudinal and transverse signals were field-symmetrized and antisymmetrized to correct for a contact misalignment, respectively. Magnetization measurements were performed using a commercial superconducting quantum interference device magnetometer.

The electronic band structures of LaGa₂ were calculated by the density functional theory (DFT) code using the Vienna ab initio simulation package (SI Appendix, section S5). The electronic band structures and projections were further computed on a finer 100 × 100 × 90 k-mesh grid to create interpolated Fermi surfaces near the Fermi level for evaluating band characteristics and curvatures.

We thank Y. Nakamura and H. Sawa for supporting synchrotron XRD experiments. The synchrotron radiation experiments were performed at SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (Proposal No. 2023A1882). This research is funded in part by the Gordon and Betty Moore Foundation Emergent Phenomena in Quantum Systems (EPiQS) through Grants GBMF9070 to J.G.C. (material synthesis, DFT calculations), NSF grant DMR-1554891 (material design), ONR Grant N00014-21-1-2591 (instrumentation development), and AFOSR grant FA9550-22-1-0432 (advanced characterization). T.K. acknowledges the support by the Yamada Science Foundation Fellowship for Research Abroad and Japan Society for the Promotion of Science (JSPS) Overseas Research Fellowships. S.F. acknowledges support from a Rutgers Center for Material Theory Distinguished Postdoctoral Fellowship. L.Y. acknowledges the support by the Science and Technology Center (STC) for Integrated Quantum Materials, NSF grant number DMR-1231319, the Heising-Simons Physics Research Fellow Program, and the Tsinghua Education Foundation.

Recently, we became aware of a related work by Y. Ōnuki et al. (67) on transport properties of RGa₂ (R = rare earth) and structurally related systems.