From colossal to zero: Controlling the Anomalous Hall Effect in Magnetic Heusler Compounds via Berry Curvature Design

Since the discovery of the anomalous Hall effect (AHE), the anomalous Hall conductivity (AHC) has been thought to be zero when there is no net magnetization. However, the recently found relation between the intrinsic AHE and the Berry curvature predicts other possibilities, such as a large AHC in non-colinear antiferromagnets with no net magnetization but net Berry curvature. Vice versa, the AHE in principle could be tuned to zero, irrespective of a finite magnetization. Here, we experimentally investigate this possibility and demonstrate that, the symmetry elements of Heusler magnets can be changed such that the Berry curvature and all the associated properties are switched while leaving the magnetization unaffected. This enables us to tune the AHC from 0 {\Omega}-1cm-1 up to 1600 {\Omega}-1cm-1 with an exceptionally high anomalous Hall angle up to 12 %, while keeping the magnetization same. Our study shows that the AHC can be controlled by selectively changing the Berry curvature distribution, independent of the magnetization.

In conventional metals or semiconductors, a transverse voltage is generated due to the Lorentz force when a magnetic field is applied perpendicular to an applied electric current. This effect leads to an anti-symmetric contribution to the off-diagonal electrical resistivity, which is generally known as the Hall effect. In ferro-or ferrimagnets it is believed that the spontaneous magnetization generates an additional transverse voltage due to the material's spin-orbit coupling (SOC), which causes carriers to be deflected by the magnetic moments of the host solid. This effect is known as the anomalous Hall effect (AHE) [1][2][3]. The AHE has two different contributions, an extrinsic contribution from scattering and an intrinsic contribution from the band structure [1,4].
For a long time it was believed that the anomalous Hall conductivity (AHC) scales with the sample's magnetization. According to this notion, any ferromagnetic material exhibits an AHE, but it is zero for an antiferromagnet, owing to the compensation of the magnetic sublattices (M = 0). Therefore, the AHE has been considered as the key signature of finite magnetization in ferromagnets or ferrimagnets. However, only recently, it was realized that the intrinsic contribution to the AHE is not directly related to the sample magnetization of a material but derives more generally from its net Berry curvature.
The Berry curvature distribution in materials is a property of the band structure that determines the topological aspects of a material [2,[5][6][7]. A necessary condition for a finite net Berry curvature and thus a nonzero AHE is the absence of symmetries that reverse the sign of the local Berry curvature in the Brillouin zone (BZ) when reversing the sign of the momentum vector, e.g. time-reversal symmetry and mirror operations. In contrast to previous believes it is thus possible to control the Berry curvature and intrinsic AHE by suitable manipulations of symmetries and band structures, independent of the finite value of the magnetization [8]. As a result of these considerations, a strong AHC was predicted in the non-collinear antiferromagnetic systems like Mn3Ir [9], Mn3Ge and Mn3Sn [10] etc. and then experimentally observed in Mn3Ge [11] as well as Mn3Sn [12,13]. Moreover, very recently, a large intrinsic AHE has been found in magnetic Weyl semimetals with broken time-reversal symmetry that depends on the separation of the Weyl nodes in momentum space [1,[14][15][16][17]. In Weyl semimetals, the Weyl point acts as the monopole of Berry curvature, where a topological invariant, the Chern number, can be assigned to each Weyl node [18,19]. Therefore, one can tune the AHC via the symmetry and topological band structure without considering net magnetic moments [20].
In ferromagnetic materials, the distinction whether the magnetization or the Berry curvature is the origin of the AHE is more subtle: While most ferromagnets show a finite AHE that is proportional to the net magnetization of the sample, the AHE in principle could be tuned to zero, irrespective of the magnetization value. For a set of bands, the Berry curvature depends on how it is connected throughout the BZ [21]. In a magnetic semiconductor without band inversion, the AHC vanishes, as the occupied valence states can be adiabatically connected to the topologically trivial vacuum (Fig. 1a). In a doped magnetic semiconductor or topologically trivial magnetic metal, the AHC can take small to large values depending on the details of the band structure; it usually follows the density of states (DOS) (Fig. 1b). The energy dependence of the AHC is constrained if there is a trivial band gap close in energy, where the AHC has to vanish. In topological metals, such as Weyl semimetals or nodal line semimetals, the connectivity of the occupied bands is nontrivial owing to the band crossings. While the Berry curvature exhibits a strong peak close to the crossing points, the DOS vanishes, and the AHC peaks sharply in energy, with giant values (Fig. 1c). The inverse relation between the DOS and AHC leads to a large anomalous Hall angle (AHA), unlike the small AHA in regular magnetic metals.
At a symmetry-protected crossing, e.g., in nodal line semimetals, the removal of the protecting symmetry leads to a gapping of the crossings, and the band structure becomes topologically equivalent to that of a semiconductor (Fig. 1d). We define a band structure as semiconducting in a topological sense, if it can be adiabatically deformed into the band structure of a gapped trivial semiconductor, even if it would be metallic from a transport perspective.
In this paper, we demonstrate these principles theoretically and experimentally and show that the AHC can be selectively tuned from 0 to 1600 Ω −1 cm −1 in magnetic Heusler compounds via suitable manipulations of the symmetries and band structures of the materials. In contrast to previous studies, where a finite AHE has been observed without a net magnetization, we here show that the AHE can be tuned to zero despite a finite magnetization. Magnetic Heusler compounds are excellent materials for this purpose owing to the diverse possibilities for tuning the electronic and magnetic structure by varying the composition [22,23]. The large AHC and high Curie temperature (TC) allow us to obtain giant AHAs of up to 12% at room temperature.
Ternary Heusler compounds with the formula X2YZ (where X and Y are transition metals, and Z is a main group element) crystallize in a face-centred cubic lattice with either space group (SG) 3 ̅ (225) (the regular Heusler structure) or SG 4 ̅ 3 (216) (the inverse Heusler structure). In the regular Heusler structure, the X atoms occupy the Wyckoff position 8c ( ) and 4a (0,0,0) with Cu2MnAl (L21) as the prototype [24]. However, in the inverse Heusler structure, atoms in the 4b position replace half of the 8c atoms, and the revised atomic arrangement becomes X [4d ( )] Z [4a (0,0,0)], with Li2AgSb as the prototype. The magnetic ground state (ferromagnetic, ferrimagnetic, or antiferromagnetic) of Heusler compounds is controlled by the interatomic distances of the corresponding X and Y atoms and follows the Slater-Pauling rules [25,26]. Further, TC generally scales with the sum of the local moments and can reach 1200 K [27].
We compare the structures of regular and inverse Heusler compounds in Fig. 2a (Fig. 1d). Being topologically nontrivial, Co2VGa and Co2MnGa should display an AHE with a peak close to the Fermi level (Fig. 1c). In contrast, the AHE of Mn2CoGa should vanish at an energy close to the Fermi level, as it is topologically equivalent to a regular semiconductor (Fig. 1a). Having established their similar saturation magnetization, we now discuss the AHE of both compounds. The total Hall resistivity, is expressed as: where R0 is the linear Hall coefficient and is the total anomalous Hall contribution. In absence of any non-collinear spin textures, is commonly believed to scale with the spontaneous magnetization of materials [1,4,28]. However, this doesn't hold true for the topological magnetic materials. We calculate the Hall conductivity from the diagonal and off-diagonal components of the resistivity tensor as where is the longitudinal resistivity. All known materials with ferromagnetic or ferrimagnetic ordering that exhibit spontaneous magnetization show an AHE [1]. Co2VGa obeys this rule very well, as seen in the summary of magneto-transport measurement in Fig In contrast, Mn2CoGa shows very different anomalous Hall behaviour. Fig. 3d illustrates the Hall conductivity at various temperatures. Interestingly,  increases linearly with the field, similar to a normal Hall effect. This is an exceptional observation in a metallic magnetic material with a large magnetic moment of 2 and differs markedly with other similar compounds reported in the literature [1,29]. The calculated R0 is 0.035 cm 3 /C at 2 K for Mn2CoGa and decreases sharply (by a factor of 10) to 0.0035 cm 3 /C at 300 K. Consequently,  ( ) decreases remarkably to a negligible value at 300 K compared to that at 2 K (Fig. 3d). The charge carriers are of the hole type.
For all the above measurements, the magnetic field B and current were applied along the [001] and [100] directions, respectively. However, B along [011] or [1 ̅ 11] show similar properties, indicating small anisotropy in the sample.
We now want to understand the unusual behavior of Mn2CoGa theoretically. As discussed earlier, the inverse Heusler Mn2CoGa can be considered as the symmetry-reduced counterpart of regular Heusler Co2VGa, which has the same NV. Co2VGa belongs to the space group of 3 ̅ , and the topological properties in Co2VGa are due to the three mirror planes , and .
Without considering spin orbit coupling, the full Hamiltonian can be decomposed into the direct product of a spin-up Hamiltonian and spin-down Hamiltonian, i.e. the band degeneracy of each spin channel is decided by the corresponding Hamiltonian. Owing to the mirror symmetry, the band inversion between the bands with opposite mirror eigenvalue forms three gapless nodal lines in the = 0, = 0 and = 0 mirror planes respectively. Compared to Co2VGa, the crucial difference in Mn2CoGa is the absent of these three mirror planes. Though they have the same NV, the mirror plane protected gapless nodal lines don't exist in Mn2CoGa due to the lack of the three mirror planes.
After taking SOC into consideration, the symmetry of the system is reduced depending on the direction of magnetization, which leads to band anti-crossings of the nodal lines and generates Weyl nodes near EF for regular Heusler (like Co2VGa, Co2MnGa etc.) [21,30]. This is due to the fact that the mirror planes perpendicular to the magnetization do not preserve the direction of the spins, while the ones parallel do. For example, if we consider a magnetization along the ̂ direction (as used in the experiment), the mirrors , are no longer allowed symmetry operations, whereas remains a symmetry operation. Thus, an anti-crossing band gap will only appear from the two nodal lines in the = 0, = 0 planes. However, some linear crossing points are still allowed due to the combined time-reversal and rotational symmetries, which are just the Weyl points [21,30,31]. In comparison with regular Heusler compounds such as Co2VGa, the absence of nodal lines in inverse Heusler compounds (like Mn2CoGa) without spin-orbit interaction generally leads to the absence of Weyl points. Mn2CoAl is a spin-gapless semiconductor, in which one spin channel is gapped, whereas the other spin channel is semimetallic with a vanishing DOS at the Fermi energy [32].
We now discuss these band structure effects on the AHC of both compounds (for details of the calculation, see the Methods section). In the limit of weak SOC, the AHC is given as the sum of the conductivities each spin species, A ( ) = ↑A ( ) + ↓A ( ), where is the chemical potential [33]. For the half-metallic ferromagnet Co2VGa, the states around the Fermi level arise only from the majority states, whereas the minority states exhibit a band gap of about 0.2 eV. Thus, the contribution to the AHC of the minority carriers remains constant throughout the band gap energy window (Fig. 4c). Because most of the slightly gapped nodal lines from SOC that generate a large Berry curvature, lie far from the Fermi level (Fig. 2d), the absolute value of the integrated Berry curvature of the spin-up channel is not very large. We calculate an intrinsic AHC of A ( ) = 140 Ω −1 cm −1 for Co2VGa, which is consistent with the experiment.
In Mn2CoGa, however, both spin species possess a finite AHC around the Fermi level (Fig.   4d) The obtained results show that the interplay of the crystal symmetry with the topological and geometrical properties of the Berry curvature provides a powerful framework to control the AHC, independent of the magnetization [17,33]. We condense our findings in Fig. 5, which presents a strategy for tuning the electronic and topological properties of Heusler compounds. In regular Heusler compounds, we find a half-metallic nodal line protected by mirror symmetry at different valence electron counts. By suitable chemical substitution, one may destroy the mirror symmetry of the compound (Fig. 5a). The removal of certain mirror symmetries changes the electronic structure of a half-metallic nodal line semimetal into a trivial band structure very close to that of a spin-gapless semiconductor by removing the nodal lines. These changes induce corresponding changes in the Berry curvature and thus lead to a small or vanishing AHC. Because of the excellent tunability, one can easily manipulate the chemical potential in Heusler compounds by changing NV (Fig. 5b). For example, A changes from ~140 Ω −1 cm −1 for Co2VGa (NV = 26) to ~1600 Ω −1 cm −1 for Co2MnGa (NV = 28), which also possesses nodal lines around the Fermi level, although the compounds have different NV.
Again, keeping the same NV, when the crystal symmetry is altered from ℎ in Co2VGa to in Mn2CoGa, A decreases to zero (Fig. 5c). The maximum AHA,  = ∆ A / , reaches a giant value of ~12% for Co2MnGa at room temperature, at which the nodal line is closest to EF (Fig. 5d). We conjecture that the nodal line dispersion and charge carrier concentration ( Fig. 2d and 2e) control the value of  , which is why  for Co2VGa is only about 2% (Fig. 5d).
Owing to the extensive tunability of Heusler compounds and the topological nature of our Heusler compound will generally be large, whereas that of the inverse Heusler compound will be close to zero, as we illustrate for CoFeMnSi and Co2MnGa, which have NV = 28 ( Fig. 5g and 5h, respectively).
Vice versa to previous studies, where a finite AHE has been observed without a net magnetization, we have showed that the AHE can be tuned to zero despite a finite magnetization.
Thus, our results complement the recent investigations on the Berry curvature origin of the AHE.
Our findings are general and can be extended to other classes of materials with finite magnetization, as one can control the topology by changing the magnetic space group. Thus, our work is relevant to the recent interest in topological classification of magnetic materials, as the topology often constrains the Berry curvature distribution [6]. By symmetry engineering, a metallic magnet can be converted to a topologically trivial semiconductor with zero AHC. The possibility to tune the AHE from zero to a colossal value, independent of the magnetization of the material, may be interesting for next-generation topo-spintronics applications. Additionally, topological semimetals with a high Curie temperature and large AHA, such as Heusler compounds, are excellent candidate materials for a confinement-induced quantum AHE in thin films.

Methods
Single crystals of Mn2CoGa, Co2VGa, and Co2MnGa were grown using the Bridgman-Stockbarger crystal growth technique. First, stoichiometric amounts of high-purity metals were premelted in an alumina crucible using induction melting. Then the crushed powder was packed in a custom-designed sharp-edged alumina tube, which was sealed in a tantalum tube. Before crystal growth, the compound's melting point was determined using differential scanning calorimetry The electronic band structure was calculated using density functional theory (DFT) with the localized atomic orbital basis and the full potential as implemented in the full-potential localorbital (FPLO) code [35]. The exchange and correlation energies were considered in the generalized gradient approximation (GGA), following the Perdew-Burke-Ernzerhof parametrization scheme [36]. To calculate the AHC, we projected the Bloch wave functions into high-symmetry atomic-like Wannier functions, and constructed the tight-binding model Hamiltonian. On the basis of the tight-binding model Hamiltonian, we calculated the AHC using the Kubo formula and clean limit [14]:        The quality of the grown Mn2CoGa, Co2VGa and Co2MnGa single crystals was analyzed at room temperature by a white beam backscattering Laue X-ray diffractometer and a Bruker D8 VENTURE x-ray diffractometer using Mo-K radiation. The crystals were cut in a plate-like shape and fixed with glue on a glass slide for the measurement. All samples show sharp and well-defined Laue spots that can be indexed with a single pattern, suggesting excellent quality of the grown crystals. Fig. S3 presents the measured Laue diffraction patterns of the crystals superposed with simulated ones.

Magnetization:
The magnetization measurements were performed in a commercial Quantum Design, vibrating sample magnetometer. Fig. S4 (a), (b)

Electrical resistivity:
The electrical resistivity measurements were performed in a commercial Quantum Design, physical property measurement system equipped with the AC transport option. The crystals were cut in a rectangular bar shape in desired crystallographic directions. For resistivity measurements four-probe contacts geometry and for Hall resistivity, five-probe contacts geometry were made by 25 m Pt wire and spot-welded. Fig. S5 (a), (b)  Field dependence of the longitudinal resistivity of all the samples were measured in the temperature window from 2 to 300 K with magnetic field upto 9 T. In Fig. S6   In order to estimate the intrinsic anomalous Hall conductivity ( int ) contribution in the regular-Heusler compounds Co2VGa and Co2MnGa, which is free from the extrinsic scattering effects, we fit the experimental anomalous hall resistivity ( A ) data with the longitudinal Hall resistivity ( ) using the equation: Here ( 0 ) is a function of the residual resistivity 0 , which incorporates the contributions due to the skew scattering as-well-as the side jump effects, and int accounts for the anomalous Hall signal purely due to the Berry curvature effect [3,4]. Here A at each temperature is estimated by extrapolating the high field data of 0 ( ) to the B → 0 limit. Now TC of Co2VGa is ~ 327 K and from its ( ) curve in fig. S5(d) we see that the A increases up to ~ 250 K as normally observed in other compounds. However, above ~ 250 K we believe that the magnetic transition comes into play and A decreases at higher temperature. So for a good comparison, the scaling plot between A vs 2 in Fig. S7 is restricted in the temperature window 2 to 250 K. Evidently the experimental data fits very well in this temperature window with the linear relation of equation (1). From the slope we estimated int as 1164 Ω -1 cm -1 for Co2MnGa and 95 Ω -1 cm -1 for Co2VGa single crystals.

More compounds:
Here we list a series of suggested spin-gapless semiconducting compounds in Heusler family in Table-