Enhancement of the anomalous Hall effect by distorting the Kagome lattice in an antiferromagnetic material

Fig. 3B shows the temperature variation of the field-cooled (FC) magnetic susceptibility χ(T) measured at µ₀H = 5 mT for the H || a (χ_a) and H || c (χ_c) axes. The χ_a spontaneously drops below T_N = 11.4 K, indicating a transition from a paramagnetic to an AFM-ordered state, which is consistent with the heat capacity (HC) measurement (SI Appendix, Fig. S3) (35, 42, 43). In contrast, the value of χ_c increases with decreasing temperature.

To gain more insight into the magnetic phase diagram of HoAgGe, the isothermal magnetizations for the field H || a and H || c axes were measured (Fig. 3 C and D). At T = 2.0 K, the magnetization curve for H || a exhibits five plateaus with a small hysteresis. The previously reported metamagnetic transitions can be identified at µ₀H ≈ 0.9, 1.1, 2.2, 3.1, and 3.4 T (35, 42–44). According to the existing neutron diffraction measurements on HoAgGe single crystals in magnetic fields, these phase transitions are associated with canted AFM states (35), which may have a strong impact on the electrical transport properties, as previously observed for canted AFMs Mn₃Sn and Mn₃Ge (11–13). In contrast, the magnetization curve for the H || c axis does not show any sudden jumps in the magnetic field. Therefore, the magnetization measurements indicate that the Ho spins in HoAgGe are more confined to the ab-plane. Our study now focuses on transport measurements for the H || a axis to understand the interplay between magnetism and topology in the system.

At zero magnetic field, the longitudinal resistivity ρ_xx for current along the c-axis of HoAgGe decreases completely linearly with decreasing temperature down to 20 K. This implies a crucial metallic nature of the compound with Planckian-type dissipation involved, which attracts attention for further study (45). The resistivity drops suddenly after exceeding the Néel temperature (T_N ~ 11.0(2) K) due to suppression of spin scattering. Typically, the value of ρ_xx is 0.06 mΩ cm at 300 K and decreases to 0.015 mΩ cm at 2 K, resulting in a residual resistivity ratio [RRR = ρ_xx(300 K)/ρ_xx(2 K)] of 4, indicating a good crystal quality. The field-dependent ρ_xx and the Hall resistivity, ρ_yx, are significantly affected by the change of the magnetic property as shown in Figs. 4 and 6, and see SI Appendix, Figs. S4–S7. These two different behaviors can be classified into two regimes: 1) below T_N and 2) above T_N.

The field dependence of the longitudinal and Hall resistivities of HoAgGe crystals (T = 2 to 12 K) is shown in Fig. 4. The field-dependent resistivity ρ_xx(H) shows a complicated behavior at low temperatures (Fig. 4B). The ρ_xx initially shows a small decrease in the low-field regime (μ₀H ≤ 0.50 T) Second, in an intermediate field range (0.50 T ≤ μ₀H ≤ 3.5 T), several peaks in ρ_xx are observed due to increased spin-canting. Beyond the field range μ₀H > 3.5 T, the spins are fully polarized as at higher temperatures. At 5 K, the ρ_xx increases slowly and then drops sharply at 1.0 T. Similar behavior is observed near ~2, ~3, and ~3.3 T, which cannot be described by standard MR models. However, these abrupt variations in ρ_xx manifest at the critical fields of magnetic structure changes and correspond to the plateaus observed in Fig. 3C. These features appear only below 11 K due to spin–flip transitions and are more clearly visible in MR (MR = [ρ_xx(H) − ρ_xx(0)]/ρ_xx(0)) (Fig. 4C). With increasing temperature, these peaks move to lower fields and eventually disappear above T_N. Based on the magnetization data and a previous report (35, 44), these properties can be explained by in-plane spin flipping. The maximum negative value of the MR was observed to be ~20% at 12 K, close to the Néel temperature. The overall negative MR results from suppressed scattering due to spin disorder and the large positive MR is observed above 5 T due to the dominance of the Lorentz force (44). However, the negative MR is also a smoking gun experiment to prove transport in a Weyl system, but it is hard to confirm a solo contribution in such a strong spin scattering system.

In addition, magnetic ordering in metallic systems strongly influences the electronic band structure and its band topology. For example, the field-induced canted AFM structures of GdPtBi, Mn₃Sn, and YbMnBi₂ produce Weyl nodes, resulting in large AHE due to enhanced BC (11, 12, 46, 47) However, these AHE are typically lower compared to FM systems, such as Co₃Sn₂S₂, Co₂MnGa, and MnAlGe (4, 41, 48). We measured the field-dependent Hall isotherm ρ_yx(H), which shows quite different behavior below and above T_N (Figs. 4–6). In general, the total Hall resistivity ρ_yx of a magnetic system has the following three contributions: (49) ordinary, anomalous, and topological Hall resistivities, i.e., ρ_yx = R₀H + R_SM + ${\rho }_{\mathit{yx}}^T$. In our measured data below T_N, an additional “hump-feature” is observed at fields μ₀H < 3.4 T (Fig. 4D) after subtracting the first two terms, and such a hump is associated with either noncolinear or noncoplanar spin structures. Remarkably, the previous neutron scattering experiments in single crystals of HoAgGe reveal the evidence of noncoplanar structures in the applied magnetic fields (35). Due to the net spin chirality, the conduction electrons moving through noncoplanar structures accumulate a Berry phase, resulting in a topological Hall effect (THE). The maximum topological Hall resistivity of ~1.6 µΩ-cm was obtained at 2.0 K in a field of ~3 T (Fig. 4D). THE is unlikely to be large in AFM states with noncollinear spin structures. It appears that the electronic structure of HoAgGe and its magnetism are strongly related, allowing the observation of multiple topological states when a magnetic field is applied. There are several examples where the noncollinear spin structures induced by BC in the momentum space strongly couple the electronic and magnetic properties of MnBi₂Te₄ and MnBi₄Te₇ in the canted AFM state (49, 50).

The ${\rho }_{\mathit{yx}}^A$ increases stepwise with the field, followed by saturation and a further increase. These steps are similar to the magnetization curve (Fig. 3C). The typical values of ${\rho }_{\mathit{yx}}^A$ at 2.0 K and 12 K are ~0.3 μΩ cm and ~0.8 μΩ cm, respectively, and electrons are the majority charge carriers. Based on the anomalous Hall resistivity (Fig. 4E) and longitudinal resistivity (Fig. 4B), the total AHC was determined using the following equation: ${\sigma }_{\mathit{xy}}^A$ ≈ ${\rho }_{\mathit{yx}}^A$/${\rho }_{\mathit{xx}}^2$, when ${\rho }_{\mathit{yx}}\ll {\rho }_{\mathit{xx}}$. Fig. 4F shows the calculated ${\sigma }_{\mathit{xy}}^A$ and the AHC values at 2.0 K and 12 K of 1,110 Ω⁻¹ cm⁻¹ and 2,250 Ω⁻¹ cm⁻¹, respectively. At 12 K, the AHA (${\sigma }_{\mathit{xy}}^A$/σ_xx) was estimated to be about 4 %.

AHC typically measures the total contributions from various intrinsic and extrinsic mechanisms (10) The extrinsic contributions result from side jumps and skew scattering, while the intrinsic contributions are theoretically determined from electronic band structures (10). However, they can also be separated by using the empirical relationship: ${\rho }_{\mathit{yx}}^A\left(T\right)={\alpha \rho }_{xx0}+{\beta \rho }_{xx0}^2+{\gamma \rho }_{\mathit{xx}}^2(T)$, where ρ_xx0 denotes the residual longitudinal resistivity (10). The first and second terms refer to the extrinsic contribution due to skew scattering and the side-jump mechanism, respectively, while the third term refers to the intrinsic contribution due to BC resulting in nontrivial bands. The intrinsic contribution to the AHC can be determined by plotting ${\rho }_{\mathit{yx}}^A$ versus ${\rho }_{\mathit{xx}}^2$ plot, where an intrinsic AHC of ~ 700 Ω⁻¹ cm^–1 is observed at temperatures below T_N (Fig. 5 A, Upper panel). Furthermore, the zero-field electrical conductivity of HoAgGe is of the order of ~10⁴ Ω⁻¹ cm^–1, which belongs to a good regime for intrinsic anomalous Hall transport. In general, FM systems exhibit a higher AHC than AFM systems, but a few magnetic WSM candidates have also shown large AHC, where BC contributions are enhanced. To date, the maximum observed AHC for AFM systems is 500 Ω⁻¹ cm⁻¹, about half the value for FM systems. However, HoAgGe exhibits an intrinsic AHC of ~700 Ω⁻¹ cm⁻¹ below T_N, surpassing all known AFM systems and comparable to FM systems (Fig. 5B) (4, 12, 13, 30, 51–54). In contrast to Mn₃Sn (12) and Co₃Sn₂S₂ (4), HoAgGe does not exhibit remanent magnetization and behave as soft magnets without hysteresis. At a fixed temperature, there is a similar magnetic field dependence between the anomalous Hall contribution and the magnetization, indicating that the anomalous Hall contribution (proportional to M) dominates. In this case, it is possible to extrapolate the data on the y-axis in order to obtain the zero-field value, usually referred to as the field induced AHE. Similar results have been observed earlier in other magnetic Kagome systems such as LiMn₆Sn₆ (30), Fe₃Sn₂ (51), and YMn₆Sn₆ (54). To gain insight into the origin of the large AHC in AFM HoAgGe, detailed density functional theory (DFT) calculations were performed. Two magnetic states were considered in the electronic structure calculations, namely the FM (Fig. 5C and see SI Appendix, Fig. S8) and AFM (2 Ho-spin in – 1 Ho spin out and 1 Ho-spin in – 2 Ho spin out, spin-ice like configuration possible within the unit cell, cf. See SI Appendix, Figs. S9–S12) states. The FM configuration, with its Ho 4f spin aligned along the a-axis, was calculated including Hubbard U and SOC, along the high-symmetry line Γ-M-K-Γ-A-L-H-A. With GGA+U+SOC, we found a localized total magnetic moment of 9.68 μ_B (spin moment 3.64 μ_B, orbital moment 6.04 μ_B) at the Ho 4f site, which is consistent with the Ho³⁺ (4f¹⁰) charge state and the experimental measured value. Since the 4f-shell is more than half filled in Ho, the orbital magnetic moment is not quenched and is aligned parallel to the spin magnetic moment. Due to the large orbital magnetic moment at the Ho site, the effect of SOC in the electronic structure is found to be substantial.

In the electronic band structure, we observed several band crossings along the high-symmetry line Γ-M-K-Γ-A-L-H-A, within ~200 meV above and below the E_F. It is noteworthy that a nonvanishing net BC is observed when SOC is combined with magnetism in HoAgGe to form AHC. To estimate the intrinsic contribution of the AHC near E_F, a tight-binding Hamiltonian model based on Wannier functions which are maximally localized for Ho-s, d, f, Ag-s, p, and Ge-p was used. Considering the BC, the energy-dependent AHC of HoAgGe are calculated theoretically (Fig. 5C). The estimated value of the AHC is found to be in qualitative agreement with the experimentally observed value, indicating that the intrinsic BC causes the AHC in HoAgGe.

In addition to a large AHC below the T_N, HoAgGe also exhibits a surprisingly large AHC above the T_N, which is quite rare (Fig. 6). From Fig. 6C, the AHC initially increases and reaches maxima at 45 K, which is about 4 T_N. It then decreases and maintains a finite value up to 200 K. The maximum ${\rho }_{\mathit{yx}}^A$ value is approximately 2 μΩ cm at 45 K, which corresponds to a ${\sigma }_{\mathit{xy}}^A$ value of around 2,800 Ω⁻¹ cm⁻¹ with an AHA of 7%. Unlike below T_N, a relationship between the field-dependent Hall resistivities and magnetization is not expected, so the contribution of BC to AHC is not expected above T_N. It seems that the noncoplanar fluctuations can potentially exist above the magnetic ordering temperature (55), which strongly enhances the spin-cluster skew scattering (56). In the present case, the spin-cluster AHC is maximized at a finite temperature (~45 K) in the presence of a constant magnetic field, as shown in Fig. 6B.

For such a large AHC, the skew scattering plays a crucial role, especially in the noncoplanar spin lattices, e.g., spin cluster (57), spin chiral (56), and triangular spin (37). In the spin structure of HoAgGe, when an external field causes magnetic fluctuations to act as scattering centers, the distortion of the local order within a triangular spin cluster or a tiled cluster in a Kagome lattice produces an enhanced skew scattering potential above the T_N (Fig. 6 C, Rightpanel). More precisely, spin cluster skew scattering contributes to the AHC by averaging the spin chirality 〈S_i ⋅ (S_j × S_k)〉 rather than the average chirality 〈S_i〉 ⋅ (〈S_j〉 × 〈S_k〉), similar to the intrinsic mechanism (57). This scenario is also supported by the large orbital magnetic moment of Ho atoms due to SOC, which provides the main centers of skew scattering. The large AHC in HoAgGe over T_N is similar to that in KV₃Sb₅ and EuAs. The former one is a Kagome metal without magnetic order (18) and the latter has a large spin fluctuation in the triangular lattice with large AHC (38). Furthermore, it is evident that the local spin chirality in HoAgGe is gradually reduced by thermal fluctuations above T_N, which may lead to larger AHC in frustrated Kagome systems with finite spin chirality. In view of these results, future theoretical studies may unravel the details of the scaling relations.

In summary, the magnetoelectric transport of AFM HoAgGe containing a distorted Kagome lattice of Ho atoms has been studied. HoAgGe exhibits a large intrinsic AHC of ~700 Ω⁻¹ cm⁻¹ at 11 K close to the Néel temperature, and the AHC value is higher than that of state-of-the-art AFM systems. This large value can be attributed to the enhanced BC and the density of states corresponding to Weyl points close to the Fermi level due to TRS breaking rather than inversion symmetry breaking. Moreover, the large extrinsic AHC of ~2,800 Ω⁻¹ cm⁻¹ and a corresponding AHA of 7% were observed at 45 K due to a spin-chirality-induced skew scattering mechanism, and the temperature is much larger than the AFM ordering temperature. In addition, the large topological Hall resistivity of ~1.6 µΩ-cm was observed at 2 K, arising from the noncollinear magnetic structure of the material. From these results, we can relay that the AFMs with Kagome spin ice state have the potential for even larger AHC, which warrants further investigation to other family members of Kagome magnets in RAgGe compounds in terms of magnetic skyrmions, as well as magnetoelectrical and thermal transport studies. Our present results are relatively simple but realistic distortion of the Kagome lattice is capable of causing a diverse set of interesting and unexpected magnetic and quantum phenomena. ZrNiAl-type intermetallic compounds with space group P-62m are an intriguing class of distorted Kagome lattices, ranging from nonmagnetic to magnetic materials, which may host additional exotic topological physics interplaying between topology, electron correlation, and magnetism.

During the review process of the manuscript, a similar work with AHE in HoAgGe was found (58).

S.R. thanks the Alexander von Humboldt Foundation for a fellowship. This work was financially supported by Deutsche Forschungsgemeinschaft (DFG) under SFB1143 (Project No. 247310070); the European Research Council (ERC) Advanced Grant No. 742068 (“TOPMAT”), and Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter—ct.qmat (EXC 2147, Project No. 390858490). M.G.V. thanks the support to Programa Red Guipuzcoana de Ciencia Tecnologia e Innovacion 2021 No. 2021 CIEN-000070-01 Gipuzkoa Next and the DFG (German Research Foundation) GA 3314/1-1 – FOR 5249 (QUAST). M.G.V. also acknowledges partial support from ERC Grant Agreement No. 101020833. Y.Z. was supported by the start-up fund at University of Tennessee Knoxville. Open access funding provided by the Max Planck Society.