Abstract
The quantum mechanical (Berry) phase of the electronic wavefunction plays a critical role in the anomalous^{1,2} and spin Hall effects^{3,4}, including their quantized limits^{5,6,7}. While progress has been made in understanding these effects in ferromagnets^{8}, less is known in antiferromagnetic systems. Here we present a study of antiferromagnet GdPtBi, whose electronic structure is similar to that of the topologically nontrivial HgTe (refs 9,10,11), and where the Gd ions offer the possibility to tune the Berry phase via control of the spin texture. We show that this system supports an anomalous Hall angle Θ_{AH} > 0.1, comparable to the largest observed in bulk ferromagnets^{12} and significantly larger than in other antiferromagnets^{13}. Neutron scattering measurements and electronic structure calculations suggest that this effect originates from avoided crossing or Weyl points that develop near the Fermi level due to a breaking of combined timereversal and lattice symmetries. Berry phase effects associated with such symmetry breaking have recently been explored in kagome networks^{14,15,16,17}; our results extend this to halfHeusler systems with nontrivial band topology. The magnetic textures indicated here may also provide pathways towards realizing the topological insulating and semimetallic states^{9,10,11,18,19} predicted in this material class.
Main
The ordinary Hall effect is due to the Lorentz force bending of charge carriers perpendicular to a magnetic field. In systems where timereversal symmetry (TRS) is spontaneously broken, it typically can be overwhelmed by a different class of mechanisms for transverse velocity. In such systems, there are contributions to transverse velocity from both extrinsic effects due to spindependent scattering^{13} and intrinsic effects related to real space^{20,21} and momentum space^{2} Berry phase mechanisms. The former is relevant in systems with noncoplanar spin textures with finite scalar spin chirality χ_{ijk} = S_{i} ⋅ (S_{j} × S_{k}), where S_{n} are spins, while the latter generically occurs in TRSbroken systems originating from the spin–orbitinteractioninduced Berry curvature of the filled bands. The anomalous Hall effect (AHE) due to magnetic texture is most often associated with finite χ_{ijk} and tends to exhibit relatively small anomalous Hall angles Θ_{AH} ≲ 0.01 (such as SrFeO_{3} (ref. 22) or Pr_{2}Ir_{2}O_{7} (ref. 23)), while intrinsic bandstructurebased effects are common in ferromagnetic systems and can be significantly larger^{13}. Recent theoretical work has suggested that effects that rely on both magnetic texture and strong spin–orbit coupling may exist in noncollinear antiferromagnets that lead to significant Hall responses^{14}. Singlecrystal studies of Mn_{3}Sn and Mn_{3}Ge have been shown to support Θ_{AH} ≲ 0.02 and 0.05, respectively, originating from its inverse triangular spin structure and electronic structure^{16,17}.
Here we study single crystals of GdPtBi, a member of the family RPtBi (R is a rare earth element) known to exhibit antiferromagnetic ordering^{24}. As shown in Fig. 1a, this system has the halfHeusler structure consisting of interpenetrating face centred cubic (fcc) lattices of each constituent element (space group ) with lattice constant a = 6.68 Å for R = Gd. In general, the antiferromagnetic fcc lattice is frustrated, with the Gd ions forming a triangular lattice when viewed along the 〈111〉 direction. From the hightemperature magnetic susceptibility we find a Curie–Weiss temperature θ_{CW} = −38 K, while the transition to longrange magnetic order occurs at T_{N} = 9.2 K, as shown in Fig. 1b, consistent with previous reports^{24}. This yields a moderate frustration parameter f ≡ − θ_{CW}/T_{N} ∼ 4, similar to insulating fcc systems FeO and MnSe^{25}. As shown in Fig. 1c, highfield torque magnetometry reveals a distinct kink at a magnetic field B_{C} ≍ 25 T (seen more clearly in d^{2}τ/dB^{2}, which is proportional to the magnetic susceptibility), which is independent of orientation, as shown in the inset of Fig. 1c. This is suggestive of fieldinduced alignment of the spins and an energy scale gμ_{B}SB_{C} = 120 K, where g = 2 is the gfactor, μ_{B} is the Bohr magneton, and S = 7/2 is the Gd spin (see Supplementary Information). This is approximately three times larger than θ_{CW}, indicating cancelling exchange interactions, as has been theoretically discussed for this material^{26}. The structure of the magnetic ordering in B = 0 has recently been probed by Xray and neutron scattering, which suggests an antiferromagnetic ordering (typeII) in which the Gd moments order ferromagnetically in sheets in the (111) planes, which in turn are stacked antiferromagnetically^{27}. Theoretical work has suggested that other orderings are of comparable energy^{26}; the structure at finite B has not been previously reported.
The focus of the present study is the interplay of the magnetic structure and strong spin–orbit coupling in GdPtBi. The heavyelement PtBi sublattice is expected to provide the system with a similar electronic structure to that of HgTe (refs 9,10,11), sensitive to the magnetic ordering of the Gd ions through the spin–orbit interaction. The bulk electronic structure of GdPtBi is known to be semimetallic^{28}. As shown in Fig. 1b, the observed resistivity as a function of temperature ρ(T) has a moderate increase with decreasing T, while at low temperature it shows a clear kink concomitant with the magnetic transition at T_{N}. A richer perspective of the coupling of the conduction electrons to the magnetic order is provided by magnetotransport. The longitudinal resistivity as a function of magnetic field ρ_{xx}(B), with B along the [001] direction, is shown in Fig. 2a. At temperatures T > T_{N} we observe an approximately quadratic dependence on B, indicating an average mobility μ_{avg} ≍ 1,000 cm^{2} V^{−1} s^{−1}. As T is lowered, we observe a dip in ρ_{xx} that develops near B = 4 T, progressively deepening to T = 2.5 K. Measurements to B = 31 T (see Supplementary Information) show that this is not related to Landau quantization, as no continued pattern in B^{−1} connected to this feature is observed with the appearance of Shubnikov–de Haas (SdH) oscillations for B > 10 T.
The field dependence of the transverse resistivity ρ_{yx}(B) is shown in Fig. 2b. At T = 50 K we see a Hall effect consistent with multiband conduction, as would be expected from the electronic structure^{28}. Upon lowering T, we see a prominent feature develop in the vicinity of B = 4 T, which corresponds to an enhancement in ρ_{yx} of 55% at T = 2.5 K. This feature draws a distinct contrast with the magnetization M. As shown in Fig. 2c, for T both above and below T_{N}, we see an approximately linear and featureless M(B). In magnetic systems, it is conventional to describe three separate contributions to ρ_{yx} = R_{0}B + R_{S}M + ρ_{yx}^{T}, where R_{0} is the ordinary Hall coefficient, R_{S} is the anomalous Hall coefficient coupled to M, and ρ_{yx}^{T} is the Hall effect arising from spin textures. As described above, the final among these include noncoplanar textures that generate finite χ_{ijk} (refs 29,30,31) as well as noncollinear textures that allow finite momentumspace Berry curvature^{14,15,16,17}. Structure in ρ_{yx}(B) from Berry phase effects normally mirrors that in M unless there are contributions related to spin texture. Mathematical separation of these components is often performed in the literature to isolate ρ_{yx}^{T} (ref. 32). Performing such analysis we estimate ρ_{yx}^{T} = 0.18 mΩ cm (see Supplementary Information), significantly larger than has been previously reported in any antiferromagnetic system, both in terms of absolute value and size relative to ρ_{xx} (see Supplementary Information). We note that the empirical relationship used for additivity of the contributions to ρ_{yx} is valid only in the limit of small Hall angle, whereas at all values of the Hall angle it is possible to discuss additivity in the transverse conductivity σ_{xy}. Here we quantify the AHE using the anomalous Hall angle Θ_{AH} = σ_{xy}^{A}/σ_{xx} = (σ_{xy} − σ_{xy}^{N})/σ_{xx}, where A(N) denotes the anomalous (normal) part of the Hall conductivity. We estimate σ_{xy}^{N} by using σ_{xy}(T = 50 K), which is far above T_{N} and reflects the ordinary contribution to the Hall effect, and refer to the resulting anomalous Hall contribution as Δσ_{xy}, as shown in Fig. 2e (similar behaviour is obtained by fitting the σ_{xy}^{N} at large B). The value of Θ_{AH} peaks at more than 0.15 at B = 4 T, which is larger than has been previously observed in antiferromagnetic systems. A view of this transport response in the space of B and T is shown in Fig. 3a. The region with the largest Δσ_{xy}, corresponding to the peak feature we observe, occurs near B = 4 T and below T_{N} = 9.2 K. However, the anomalous Hall response persists to significantly higher temperatures, with a reduced magnitude, finally disappearing between 25 and 30 K.
The most common contribution of magnetic texture to the AHE is that of finite χ_{ijk} appearing in systems that have skyrmionic spin textures stabilized by the Dzyaloshinskii–Moriya interaction^{30,31}. It has also been recognized that frustrated magnets may play host to similar textureinduced Hall effects^{23}. More recently, theoretical work has suggested that, in antiferromagnets, a bandstructureinduced intrinsic AHE may arise in the presence of noncollinear spin structures that break the combined timereversal and lattice symmetries of the simple antiferromagnetic state^{14}. Such a situation is realized in GdPtBi as the B = 0 typeII antiferromagnetic spin structure cants in finite B. Neutron scattering measurements confirm both the typeII ordering in fields up to 10 T (see Supplementary Information) and a ferromagnetic canting, as evidenced by an increase in intensity at the nuclear Bragg positions with increasing B, as shown for (111) in Fig. 2d along with its scaling with M^{2}. Additionally, an applied field produces an enhancement in the magnetic Bragg peaks, as shown for (0.5 0.5 2.5) in Fig. 2d, which saturates above 6 T, similar to the field scale for disappearance of the anomalous Hall response. For antiferromagnetic systems with such broken timereversal and lattice symmetries and large spin–orbit coupling, the anomalous Hall response is predicted to be significant—calculated to be 218 Ω^{−1} cm^{−1} for Mn_{3}Ir (ref. 14). The variation of the Hall response across five different GdPtBi crystals is shown in Fig. 3b; the observed Δσ_{xy} = 30–200 Ω^{−1} cm^{−1} are comparable to that predicted for the intrinsic AHE case. We note that the transport observations also parallel predictions for the distorted fcc lattice with finite orbital magnetization^{33}. However, this requires the presence of a typeI magnetic ordering and finite χ_{ijk}, which is inconsistent with our neutron scattering measurements (see Supplementary Information).
In terms of scaling of the AHE, we find a temperaturedependent scaling factor α (defined as Δσ_{xy} ∼ σ_{xx}^{α}), as shown in Fig. 3c. At low T, we find α = 1.2, intermediate between the value of 1.6 expected for the intrinsic AHE in multiband disordered metals^{34,35} and the σ_{xx}invariant intrinsic AHE in moderately clean systems. This crossover is predicted to occur in ferromagnets in the vicinity of σ_{xx} ∼ 10^{3} Ω^{−1} cm^{−1}, similar to the scale observed here. The existence of frustrated magnetic interactions in GdPtBi, as shown by the moderate value of f, suggests that shortrange correlations persist above T_{N} (ref. 26); such correlations in the absence of longrange order have been shown to affect transport in PdCrO_{2} (ref. 36) and Pr_{2}Ir_{2}O_{7} (ref. 23). We hypothesize that this extends to the anomalous Hall effect observed here, as such fluctuations will break the same symmetry as the ordered phase, which is distinct from that expected for Mn_{3}Ge and Mn_{3}Sn, where fluctuations due to symmetric exchange cancel (see Supplementary Information).
To investigate the origin of the large AHE allowed in GdPtBi in finite B, we consider the effect of the spin texture on the electronic structure. Bandtouching points in the bulk electronic structure of GdPtBi have been previously reported in calculations with an uncanted magnetic structure^{37} which, if driven to anticrossings near the Fermi level, could be a source of significant AHE as in ferromagnetic systems^{35}. We have performed electronic structure calculations for magnetically ordered GdPtBi using the rhombohedral magnetic unit cell shown in Fig. 4a. The approximate location of E_{F} is determined from the ordinary Hall effect and k_{F} from SdH oscillations (see Supplementary Information), and the B dependence of the spin structure is calculated using the results of the magnetic and neutron scattering experiments. For B = 0 we find that the typeII ordered system exhibits band touching at Γ, as shown in Fig. 4b. The structures with spin canting for B = 4 T and 8 T along the [001] direction are shown in Fig. 4c and d, respectively. The bands near the Fermi level originate from Pt and Bi (see Supplementary Information). The combination of the large spin–orbit coupling and critical band alignment of the system amplifies the physical consequence of the Gd spin texture on these bands; inspection of the band structure shows the development of avoided crossings near E_{F} which carry significant Berry curvature. This suggests that the evolution of the electronic structure due to spin texture will generate significant Berry phase contributions to the AHE. The requirement that the combined symmetry of the antiferromagnetic state be broken to observe the AHE necessitates the application of finite B (ref. 14)—the calculations here indicate that the particularly large response relies on the detailed band structure of GdPtBi itself. We note that Weyl points have been predicted to appear in the electronic structure of the HgTe class compounds^{18}, and can be a source of significant anomalous Hall conductivity^{15,16,38}. While such points do not appear along the highsymmetry directions in the calculated structure, here they may be expected to appear at other points in momentum space^{39}. If sufficiently isolated Weyl points exist near E_{F}, they may also contribute to the AHE in a manner similar to avoided crossings (magnetotransport possibly indicative of Weyl points is observed in these crystals as shown in the Supplementary Information). Further theoretical work is needed to determine if such conditions are met in GdPtBi. More broadly, the symmetryallowed momentumspace Berry phase mechanism provides a consistent description for the transport, scattering and calculation results presented herein.
It has been predicted that spin–orbitdriven band inversion in the bulk of RPtBi may give rise to a topological insulating state^{9,10,11}, although thus far experiments using photoemission to probe the band structure have been unable to resolve the topological nature of the surface bands^{28}. The symmetrybroken ground states known to occur in RPtBi are then of heightened interest, as they would be projected on to a potentially topologically nontrivial ground state. The evolution of the bulk band structure is relevant to the AHE presented here; the band topology of RPtBi provides a band structure acutely sensitive to gap formation via the magnetic texture and strong spin–orbit coupling of the system. Additionally, the antiferromagnetic ordering itself has been predicted to give rise to a new type of topological insulating state where a combined timereversal and translational symmetry is the source of protection for the surface modes^{19}, a scenario further enriched by the observations here. This highlights GdPtBi as a candidate platform not only for hosting topological edge modes in the antiferromagnetic state^{19}, but also as a novel TRSbroken topological insulator with noncollinear spin texture in finite B (ref. 6).
The combination of spin–orbit and correlation effects is expected to reveal new behaviour beyond that possible for singleelectron physics. Here, the combination of strong spin–orbit coupling and symmetry breaking probed by neutron scattering gives rise to an extremely large anomalous Hall response that connects to recent theories for emerging effects in antiferromagnetic systems^{14} and the calculated electron structure. This effect and its scaling with conductivity^{12} provide an antiferromagnetic example to generalize the notion of phasespace Berry phase effects in TRSbroken systems^{40}. Furthermore, it provides an example of a robust AHE driven by the combined role of spin–orbit coupling and spin texture that demonstrates the ubiquity of the Berry phase mechanism in symmetrybroken systems.
Methods
Single crystals of GdPtBi were grown using a flux method following ref. 24. A mixture of 99.99% Pt wire, 99.9% pure Gd, and 99.999% Bi needles in a molar ratio of 1:1:20 was put in an alumina crucible and sealed in an evacuated quartz tube with 0.016 MPa Argon gas. The sample was heated at 1,100 °C for 2 h and then cooled slowly between 900 °C and 600 °C at a rate of 2 °C h^{−1}, followed by centrifuging to dissociate the GdPtBi crystals from the Bi flux. For neutron scattering experiments, 97.7% isotopeenriched ^{160}Gd (with a purity of 99.9%) was used. All the obtained crystals were confirmed to be of single phase by a powder Xray diffraction method. The crystals were aligned by Xray Laue backreflection and/or singlecrystal Xray diffraction techniques. Transport measurements were performed with a conventional fourprobe method. To correct for contact misalignment, the measured longitudinal and transverse voltages were field symmetrized and antisymmetrized, respectively. Magnetization measurements were performed using a commercial superconducting quantum interference device (SQUID) magnetometer. Torque magnetometry was performed using a BeCu cantilever, with the deflection detected by a capacitance bridge. Neutron scattering measurements were carried out on the tripleaxis spectrometer BT7 at the NIST Center for Neutron Research^{41}. Electronic structure calculations were performed using the projector augmented wave method and exchange–correlation potential with the generalized gradient approximation of Perdew–Burke–Ernzerhof parameters, implemented in the Vienna ab initio simulation package^{42}.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We are grateful to Y. Lee, L. Fu, L. Savary and H. Sakai for fruitful discussions. This research is funded in part by the Gordon and Betty Moore Foundation EPiQS Initiative, Grant GBMF3848 to J.G.C., material development by NSF grant DMR1554891, and instrumentation development with ARO grant W911NF1610034. W.F. acknowledges support from the NSF of China (No. 11374033). D.X. acknowledges support from DOE Basic Energy Sciences Grant No. DESC0012509. A.D. acknowledges support by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR1231319. J.G.C. acknowledges support from the Bose Fellows Program at MIT. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR1157490, the State of Florida, and the US Department of Energy. We acknowledge the National Institute of Standards and Technology, US Department of Commerce, in providing the BT7 neutron facility used in this work.
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T.S. and Y.T.L. synthesized the crystals. T.S., A.D. and Y.T.L. performed the electrical measurements. T.S., R.C. and J.W.L. performed the neutron scattering experiments. W.F. and D.X. performed the electronic structure calculations. All authors participated in discussion of the results and writing the paper. J.G.C. coordinated the project.
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Suzuki, T., Chisnell, R., Devarakonda, A. et al. Large anomalous Hall effect in a halfHeusler antiferromagnet. Nature Phys 12, 1119–1123 (2016). https://doi.org/10.1038/nphys3831
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