Anomalous Hall effect and planar Hall effect in half-Heusler GdPdBi

Abstract

Half-Heusler intermetallic compounds have emerged as a fertile platform for investigating correlated quantum phenomena, encompassing superconductivity, quantum criticality, heavy fermion physics, and topological states. Here, we conduct a systematic comparative study between the trivial semimetal GdPdBi and its topological counterpart GdPtBi − a prototype Weyl semimetal, motivated by their isostructural nature and analogous magnetic configurations. While both compounds exhibit similar characteristics in electrical resistivity, magnetic susceptibility, and heat capacity measurements, striking differences emerge in their magnetotransport properties. The angular dependence of magnetoresistance, anomalous Hall angle evolution, and planar Hall resistivity demonstrate pronounced material-specific variations. Through this comparative analysis, we elucidate the critical interplay between topological band structure and magnetic ordering in modulating emergent transport phenomena. Specifically, we can distinguish the effect of topologically non-trivial bands on MR, anomalous Hall effect, and planar Hall effect from that of usual magnetic scattering. These findings provide fundamental insights into tailoring quantum transport properties through crystallographic and electronic structure engineering.

Graphical abstract

Keywords

half-Heusler compounds / magnetic topological materials / anomalous Hall effect / chiral anomaly

1 Introduction

Topological materials exhibit a range of unique properties stemming from their non-trivial band structures, which have garnered significant interest for potential applications [1–5]. A key class of topological materials is Weyl semimetals, where the conduction and valence bands show linear dispersion at their intersection, known as the Weyl point [6]. Weyl semimetals typically manifest distinct behaviors in electrical transport measurements, such as negative magnetoresistance (MR) when the electric field (E) is parallel to the magnetic field (B) [7, 8], as well as the planar Hall effect (PHE) [9, 10] and the anomalous Hall effect (AHE) [11]. These macroscopic transport phenomena provide crucial insights for identifying the presence of Weyl points in the material.

The realization of non-trivial topological phases typically requires band inversion, a phenomenon driven by strong spin-orbit coupling (SOC) that is intimately linked to the atomic weight of constituent elements [12–14]. This mechanism is best exemplified by the prototypical CdTe/HgTe system, where the heavier Hg atoms induce strong SOC to realize a topological state, in contrast to the lighter, trivial CdTe [14]. Guided by this principle, the search for topological materials has largely focused on systems containing heavy elements, leading to significant discoveries in Bi-based compounds such as Bi{}_2Se{}_3, Na{}_3Bi, YbMnBi{}_2 and GdPtBi [15–18]. Within the half-Heusler family, this SOC-driven topological tunability is demonstrated in the isostructural pair YPtBi and YPdBi: substituting the heavy Pt element with the lighter Pd reduces the SOC strength, switching the ground state from topological to trivial [19]. Following this logic, in GdPtBi, the strong SOC from heavy Pt atoms induces band inversion, and thus makes it topologically nontrivial, creating the prerequisite for magnetic-field-induced Weyl nodes [18]. While transport measurements on GdPtBi, including negative MR [20, 21], a large PHE [22], and a significant AHE [21, 23], have been presented as evidence for Weyl points, disentangling these signatures from inherent magnetic contributions remains challenging [24–26].

To isolate the effect of magnetic scattering on the transport properties in GdPtBi, we investigate GdPdBi as an ideal reference compound. Its suitability arises from the fact that GdPtBi and GdPdBi are isostructural (space group F\overline{4}3m, No. 216), share nearly identical lattice parameters (6.68 Å) [27, 28], and similar magnetism from the same Gd{}^{3+} ions [27, 29]. The results of single-crystal neutron diffraction indicate they share the same antiferromagnetic (AFM) structure [30]. The AFM transition temperatures (T_N) are 9 K for GdPtBi and 13 K for GdPdBi [29–31]. However, in GdPdBi, the substitution of Pt with the lighter element Pd significantly reduces the SOC strength. As systematically classified in the RPdBi series [31] and consistent with the behavior of the Y(Pt,Pd)Bi system [19], this reduction restores the normal band order, rendering GdPdBi topologically trivial [31]. The nearly identical magnetism with GdPtBi but trivial band structure in GdPdBi provides an ideal reference system for investigating the interplay between topology and magnetism in GdPtBi. Therefore, we systematically investigated MR, AHE, and PHE in GdPdBi and compared the results with those reported for GdPtBi, aiming to isolate the potential contributions originating from Weyl physics in the latter. This direct comparison provides a new perspective to uncover the complex transport results in magnetic topological materials.

2 Experimental methods

Single crystals of GdPdBi were grown by self-flux method. Gadolinium, palladium and bismuth powders were mixed in an alumina crucible, and then sealed in an evacuated quartz ampoule. The quartz ampoule was heated to 1373 K, held for 12 hours, and then slowly cooled at a rate of 2 K/h down to 873 K. Excess Bi flux was removed using a centrifuge at 873 K, yielding large single crystals. The resulting single crystals exhibited pronounced cubic morphology, with typical dimensions of approximately 5 mm \times 5 mm \times 5 mm. A sample with dimensions of approximately 3 mm \times 0.6 mm \times 0.2 mm was cut using a wire saw, preserving two natural facets. These natural facets facilitated sample orientation for subsequent measurements. The crystal facets were confirmed using single-crystal X-ray diffraction (XRD) on a PANalytical Empyrean diffractometer with Cu-K{}_{\alpha } radiation. The elemental composition was analyzed using energy-dispersive X-ray spectroscopy (EDS). The standard four-probe method was applied to measure the resistivity and Hall effect. All the transport and heat capacity measurements were performed using a Quantum Design Physical Property Measurement System (PPMS-DynaCool). The magnetic susceptibility was measured using the Magnetic Properties Measurement System(MPMS-5).

3 Results and discussion

Figure 1(a) shows XRD patterns from two different directions. The peak positions obtained from XRD patterns on two different facets were identical. This is consistent with the face-centered cubic lattice structure of GdPdBi. The refined lattice parameter is 6.68 Å, which is also consistent with the value previously reported [28]. A photograph of a representative GdPdBi single crystal is shown in Fig. 1(b). The crystal orientation, determined by XRD, is schematically illustrated in Fig. 1(c). This schematic also defines two key surfaces, labeled Surface 1 and Surface 2, which correspond to preserved natural facets on such crystals. In addition, in our subsequent measurements, the electrical current J was applied along the [100] direction, and the external magnetic field B was rotated in two configurations: In the first configuration, B was rotated within the plane containing [100] and [010] (Surface 2), where \theta is the angle between B and J ([100]). In the second configuration, B was rotated within the plane containing [100] and [001] (Surface 1), where \psi is the angle between B and J ([100]).

We now turn to the magnetic properties of GdPdBi. The AFM transition is marked by a cusp at T_N = 13 K in Fig. 2(a), consistent with previous literature [29, 31]. Fitting the inverse susceptibility 1/(\chi -{\chi }_0) in the high-temperature paramagnetic region using the modified Curie−Weiss law:

(1)\chi ={\chi }_0+{\displaystyle \frac{C}{T-{\mathrm{\Theta }}_{CW}}},

where {\chi }_0 is a temperature-independent term, C is the Curie constant, and {\mathrm{\Theta }}_{CW} is the Weiss temperature. We find that it fits well at temperatures above T_N. According to the optimal fitting, C = 7.43 emu\cdotK/(mol\cdotOe), {\mathrm{\Theta }}_{CW} = -45 K, and the obtained effective moment {\mu }_{eff}=7.7{\mu }_B/Gd, which is close to the theoretical value for a free Gd{}^{3+} ion. Figure 2(b) presents isothermal magnetization curves M(H) at different temperatures of GdPdBi. The magnetization increases almost linearly with the applied field up to 5 T, similar to the behavior observed in GdPtBi [21, 25]. Previous studies on GdPtBi showed that its magnetization remains unsaturated up to 25 T at 1.4 K [21]. Given the strong similarity in magnetism between the two compounds, it is reasonable to infer that GdPdBi exhibits a similar robust response to high magnetic fields, and achieving its magnetic saturation would also require fields beyond our experimental limits. The field derivative of magnetization, \mathrm{d}M/\mathrm{d}H, is plotted in Fig. 2(c). A broad hump-like feature is observed below 0.5 T at temperatures well within the AFM state. This feature diminishes as the temperature approaches T_N and completely vanishes in the paramagnetic state. The confinement of this anomaly to the AFM phase suggests it could be a weak, field-induced metamagnetic transition. Such transitions, where a small applied field reconfigures the AFM spin structure, are not uncommon in rare-earth intermetallic compounds such as CePd{}_2Al{}_2 [32].

The resistivity {\rho }_{xx}(T) increases upon cooling from room temperature [Fig. 3(a)], characteristic of semimetallic or semiconducting behavior. This behavior is similar to that reported for GdPtBi, and the magnitude of resistivity is also comparable [25]. Significant negative MR is observed in a wide temperature range, as shown in Fig. 3(b). Here, MR is defined as \text{MR}=[\rho (B)-\rho (0)]/\rho (0)\times100%, where \rho (B) and \rho (0) are the resistivities in an applied magnetic field B and in zero field, respectively. Notably, a large negative MR, reaching approximately −84% under a field of 9 T at 2 K, is observed. The large negative MR could stem from either the chiral anomaly associated with Weyl points or the suppression of spin-disorder scattering. To distinguish these mechanisms, we performed angular-dependent MR measurements. The negative MR contribution from the chiral anomaly is expected to be maximal for B//J and vanish for B\perp J. However, our data in Figs. 3(c) and (d) reveal a nearly isotropic response as the magnetic field rotates within both Surface 1 and Surface 2. Furthermore, negative MR is a universal feature in the magnetic R-Pd-Bi series, appearing in both topologically trivial and non-trivial members [30]. This isotropy and universality strongly suggest that the negative MR in GdPdBi originates primarily from the suppression of spin-disorder scattering rather than the chiral anomaly.

The specific heat (C_{total}) measurement reveals a clear \lambda-like anomaly peaking near 12 K (Fig. 4), consistent with the AFM transition temperature T_N determined from the susceptibility. The magnetic entropy S_m(T), obtained by integrating (C_m/T)\mathrm{d}T after subtracting the estimated phonon (C_{ph}) and electronic (C_{el}) contributions from the total specific heat (C_{total}), shows a sharp rise around T_N and approaches saturation above approximately 30 K. The saturation value of S_m approaches R\ln (2J+1)=R\ln 8 (where J=7/2 for Gd{}^{3+}, and the right axis is scaled by this value), confirming the ordering of localized Gd{}^{3+} moments. The entropy release profile suggests dominant contribution from long-range ordering below T_N with possibly some short-range correlations extending above T_N, similar to the observations in GdPtBi [21].

Figure 5(a) displays the Hall resistivity, {\rho }_{yx}, measured at various temperatures. At low temperatures (e.g., below 20 K), {\rho }_{yx}(B) exhibits significant non-linearity, particularly a change in slope around 2 T, which is diminished at higher temperatures (e.g., 75 K). The low-field behavior at 2, 20, and 75 K is highlighted in Fig. 5(b), where the contrast between the non-linear curves at low temperatures and the relatively linear curve at 75 K is evident.

This temperature-dependent non-linearity strongly points to the presence of an AHE. The total Hall resistivity in a magnetic material is generally described by {\rho }_{yx}=R_{0}B+{\rho }_{yx}^{\mathrm{A}}, where R_0 is the ordinary Hall coefficient and {\rho }_{yx}^{\mathrm{A}} represents the anomalous contribution. The latter term, often scaling with magnetization M(B), is responsible for the non-linear behavior observed at temperatures below or near the magnetic ordering temperature. At 75 K, well into the paramagnetic state, the AHE contribution diminishes, and the transport is dominated by the ordinary Hall effect, resulting in a quasi-linear {\rho }_{yx}(B) response. However, deviations from strict linearity persist at high fields, likely due to the multi-band nature of the material. Figure 5(a) shows that the field range over which {\rho }_{yx}(B) appears approximately linear increases with temperature. This trend is consistent with a general decrease in carrier mobility at higher temperatures due to enhanced phonon scattering, which extends the magnetic field range satisfying the low-field condition (\mu B\ll 1) where the ordinary Hall response is typically linear.

We now focus on the non-linear behavior of {\rho }_{yx} at low temperatures, indicative of an AHE. To better visualize the AHE contribution, we plot the derivative \mathrm{d}{\rho }_{yx}/\mathrm{d}B in Fig. 5(c). A distinct hump is observed around 2 T at low temperatures. This feature broadens and decreases in amplitude as temperature increases. The hump disappears around 75 K and evolves into field-independent behavior. This hump represents the main signature of the AHE in GdPdBi. Additionally, a closer look at the low-field region at low temperatures [Fig. 5(d)] reveals a sharper, weaker feature below 0.5 T. The field scale of this feature corresponds roughly to the subtle anomaly observed in \mathrm{d}M/\mathrm{d}B [Fig. 2(c)]. Therefore, this low-field structure in \mathrm{d}{\rho }_{yx}/\mathrm{d}B might be related to initial magnetization processes, perhaps involving impurities or spin reorientations. However, the prominent transport anomaly around 2 T in \mathrm{d}{\rho }_{yx}/\mathrm{d}B appears distinct from the bulk magnetization behavior, as M(B) exhibits a monotonic increase without a sharp feature at this specific field [Figs. 2(b) and (c)]. This deviation from simple M-scaling suggests that the AHE in GdPdBi is not governed solely by the net magnetization, distinguishing it from conventional ferromagnets. One compelling possibility in AFM systems is that the applied magnetic field induces a non-collinear spin texture. Such a non-collinear arrangement can generate a substantial Berry curvature in momentum space, leading to a large intrinsic AHE that is not directly proportional to the net magnetization [33]. This scenario implies that Hall transport acts as a sensitive probe for field-induced magnetic structures that are subtle in bulk magnetization. Confirmation of such complex spin textures would require further investigation, for instance, using neutron diffraction in a magnetic field.

The Hall angle, defined as \tan {\mathrm{\Theta }}_H={\rho }_{yx}/{\rho }_{xx}, relates the transverse Hall response to the longitudinal resistivity [34]. Figures 6(a) and (b) plot \tan {\mathrm{\Theta }}_H versus B. Compared to the relatively linear behavior at 75 K, the curves at 2 K and 20 K show a pronounced non-monotonic feature around 2−3 T. This feature directly reflects the AHE contribution seen in {\rho }_{yx}. To estimate the anomalous Hall angle (AHA), we subtracted the ordinary Hall effect contribution, which was approximated using the scaled Hall angle data from 75 K where the AHE is assumed to be negligible. The resulting AHA as a function of magnetic field for different temperatures is presented in Fig. 6(c). The estimated AHA reaches a maximum magnitude exceeding 0.1 at 2 K. Compared with GdPtBi, the field-dependent AHA in GdPdBi evolves more gently, lacking the broad peak structure reported for its counterpart [21]. This peak in GdPtBi is a key signature of its Weyl physics, attributed to the intense Berry curvature from field-induced Weyl points. The absence of this feature in the topologically trivial GdPdBi is consistent with its conventional band structure, thus providing a clear distinction between their magnetotransport responses.

PHE stems from the difference in resistivity between magnetic field perpendicular and parallel to the current, \mathrm{\Delta }\rho ={\rho }_{\perp }-{\rho }_{//}. The angular dependence of planar Hall resistivity {\rho }_{yx}^{\mathrm{P}\mathrm{H}\mathrm{E}} can be represented as [26]

(2){\rho }_{yx}^{\mathrm{P}\mathrm{H}\mathrm{E}}=-\mathrm{\Delta }\rho \sin \theta \cos \theta ,

in which \theta is angular between current and applied magnetic field. On the other hand, the angular dependence of longitudinal resistivity ({\rho }_{xx}^{\mathrm{P}\mathrm{H}\mathrm{E}}) can be described as

(3){\rho }_{xx}={\rho }_{\mathrm{\perp }}-\mathrm{\Delta }\rho {\cos }^2\theta .

The small conventional Hall signals can be eliminated by taking the average of the signals under positive and negative magnetic fields. The interference of {\rho }_{xx} could be eliminated by exploiting the symmetry inherent in the PHE, using the relation: {\rho }_{yx}^{\mathrm{P}\mathrm{H}\mathrm{E}}=[{\rho }_{yx}^{\mathrm{P}\mathrm{H}\mathrm{E}}(\theta )-{\rho }_{yx}^{\mathrm{P}\mathrm{H}\mathrm{E}}(\pi -\theta )]/2. After these processes, we could get more reasonable {\rho }_{yx}^{\mathrm{P}\mathrm{H}\mathrm{E}}, as presented in Figs. 7(a), (c) and (e). The slight phase shift observed in the 1 T curve [Fig. 7(c)] may be attributed to magnetic hysteresis at low temperature. We use Eq. (2) to fit {\rho }_{yx}^{\mathrm{P}\mathrm{H}\mathrm{E}}, and the fitting parameter \mathrm{\Delta }{\rho }_{yx} is presented in Figs. 7(b), (d) and (f). We can observe that the parameter \mathrm{\Delta }\rho experiences a process of first rising and then dropping as the temperature increases under 8 T in Fig. 7(b). The value of \mathrm{\Delta }\rho reaches its maximum at around 30−40 K. On the one hand, \mathrm{\Delta }\rho does not change monotonically with temperature, which is probably related to the change in magnetic ordering. On the other hand, it is easy to distinguish the large PHE caused by magnetism or chiral anomaly. The \mathrm{\Delta }\rho caused by chiral anomaly often decreases with increasing temperature. In addition, the PHE caused by the chiral anomaly will satisfy the relation: \mathrm{\Delta }\rho \propto B^2 [10]. However, \mathrm{\Delta }\rho does not satisfy the relation at 2 K or 40 K, as presented Figs. 7(d) and (f). \mathrm{\Delta }\rho is not monotonic as the magnetic field increasing at 2 K. Compared with PHE in GdPtBi, the most prominent distinction is the magnitude of \mathrm{\Delta }\rho. The magnitude is \mathrm{m}\mathrm{\Omega }\cdot \mathrm{c}\mathrm{m} in GdPtBi, but only μ\mathrm{\Omega }\cdot \mathrm{c}\mathrm{m} in GdPdBi [22]. This orders-of-magnitude difference, observed between two materials with nearly identical magnetic properties, suggests that conventional anisotropic MR from magnetism alone is insufficient to explain the giant PHE in GdPtBi. Instead, this stark contrast highlights the crucial role of its non-trivial band topology in generating the anomalously large planar Hall response in GdPtBi.

4 Conclusions

We performed a comprehensive study on the magnetotransport properties of GdPdBi and compared them with the isostructural Weyl semimetal GdPtBi. Both compounds share similar lattice parameters, magnetic properties, and resistivity profiles. However, the lighter atomic mass of Pd in GdPdBi results in weaker spin-orbit coupling. This prevents band inversion and establishes GdPdBi as an ideal topologically trivial reference system. By contrasting their transport behaviors, we successfully disentangled topological signatures from the magnetic background. First, the negative magnetoresistance in GdPdBi is isotropic. This confirms its origin from spin-disorder scattering rather than the chiral anomaly. Second, the anomalous Hall angle in GdPdBi exhibits a gentle field evolution, lacking the broad peak structure seen in GdPtBi. Finally, the planar Hall resistivity in GdPdBi is orders of magnitude smaller than the giant signal in its topological counterpart. These distinct differences demonstrate that magnetism alone cannot explain the exotic transport anomalies in GdPtBi. Instead, our results confirm that these features originate from the non-trivial band topology and the presence of Weyl nodes.

References

[1] M. Z. Hasan and C. L. Kane , Colloquium: Topological insulators, Rev. Mod. Phys. 82(4), 3045 (2010)

[2] X. L. Qi and S. C. Zhang , Topological insulators and superconductors, Rev. Mod. Phys. 83(4), 1057 (2011)

[3] Z. Rao , H. Li , T. Zhang , S. Tian , C. Li , B. Fu , C. Tang , L. Wang , Z. Li , W. Fan , J. Li , Y. Huang , Z. Liu , Y. Long , C. Fang , H. Weng , Y. Shi , H. Lei , Y. Sun , T. Qian , and H. Ding , Observation of unconventional chiral fermions with long Fermi arcs in CoSi, Nature 567(7749), 496 (2019)

[4] A. S. Fuhr , P. Ganesh , R. K. Vasudevan , K. M. Roccapriore , and B. G. Sumpter , Digital twins and deep learning segmentation of defects in monolayer MX₂ phases, Appl. Phys. Lett. 124(3), 031901 (2024)

[5] Q. Wang , Y. Xu , R. Lou , Z. Liu , M. Li , Y. Huang , D. Shen , H. Weng , S. Wang , and H. Lei , Large intrinsic anomalous Hall effect in half-metallic ferromagnet Co₃Sn₂S₂ with magnetic Weyl fermions, Nat. Commun. 9(1), 3681 (2018)

[6] X. Wan , A. M. Turner , A. Vishwanath , and S. Y. Savrasov , Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates, Phys. Rev. B 83(20), 205101 (2011)

[7] H. Kim , K. Kim , J. F. Wang , M. Sasaki , N. Satoh , A. Ohnishi , M. Kitaura , M. Yang , and L. Li , Dirac versus Weyl fermions in topological insulators: Adler−Bell−Jackiw anomaly in transport phenomena, Phys. Rev. Lett. 111(24), 246603 (2013)

[8] A. A. Burkov , Chiral anomaly and diffusive magnetotransport in Weyl metals, Phys. Rev. Lett. 113(24), 247203 (2014)

[9] A. A. Burkov , Giant planar Hall effect in topological metals, Phys. Rev. B 96(4), 041110 (2017)

[10] S. Nandy , G. Sharma , A. Taraphder , and S. Tewari , Chiral anomaly as the origin of the planar Hall effect in Weyl semimetals, Phys. Rev. Lett. 119(17), 176804 (2017)

[11] A. A. Burkov , Anomalous Hall effect in Weyl metals, Phys. Rev. Lett. 113(18), 187202 (2014)

[12] S. Chadov , X. Qi , J. Kübler , G. H. Fecher , C. Felser , and S. C. Zhang , Tunable multifunctional topological insulators in ternary Heusler compounds, Nat. Mater. 9(7), 541 (2010)

[13] H. Lin , L. A. Wray , Y. Xia , S. Xu , S. Jia , R. J. Cava , A. Bansil , and M. Z. Hasan , Half-Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena, Nat. Mater. 9(7), 546 (2010)

[14] B. A. Bernevig , T. L. Hughes , and S. C. Zhang , Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314(5806), 1757 (2006)

[15] H. Zhang , C. X. Liu , X. L. Qi , X. Dai , Z. Fang , and S. C. Zhang , Topological insulators in Bi₂Se₃, Bi₂Te₃ and Sb₂Te₃ with a single Dirac cone on the surface, Nat. Phys. 5(6), 438 (2009)

[16] Z. K. Liu , B. Zhou , Y. Zhang , Z. J. Wang , H. M. Weng , D. Prabhakaran , S. K. Mo , Z. X. Shen , Z. Fang , X. Dai , Z. Hussain , Y. L. Chen , and Discovery of a three-dimensional topological Dirac semimetal , Na₃Bi, Science 343(6173), 864 (2014)

[17] S. Borisenko , D. Evtushinsky , Q. Gibson , A. Yaresko , K. Koepernik , T. Kim , M. Ali , J. van den Brink , M. Hoesch , A. Fedorov , E. Haubold , T. Kushnirenko , I. Soldatov , R. Schäfer , and R. J. Cava , Time-reversal symmetry breaking type-II Weyl state in YbMnBi₂, Nat. Commun. 10(1), 3424 (2019)

[18] C. Felser and B. Yan , Magnetically induced, Nat. Mater. 15(11), 1149 (2016)

[19] J. C. Souza,M. V. Ale Crivillero,H. Dawczak-Debicki,A. Ptok,P. G. Pagliuso,S. Wirth, Tuning the topological character of half-Heusler systems: A comparative study on YTBi (T = Pd, Pt), Phys. Rev. B 108(16), 165154 (2023)

[20] M. Hirschberger , S. Kushwaha , Z. Wang , Q. Gibson , S. Liang , C. A. Belvin , B. A. Bernevig , R. J. Cava , and N. P. Ong , The chiral anomaly and thermopower of Weyl fermions in the half-Heusler GdPtBi, Nat. Mater. 15(11), 1161 (2016)

[21] C. Shekhar , N. Kumar , V. Grinenko , S. Singh , R. Sarkar , H. Luetkens , S. C. Wu , Y. Zhang , A. C. Komarek , E. Kampert , Y. Skourski , J. Wosnitza , W. Schnelle , A. McCollam , U. Zeitler , J. Kübler , B. Yan , H. H. Klauss , S. S. P. Parkin , and C. Felser , Anomalous Hall effect in Weyl semimetal half-Heusler compounds RPtBi (R = Gd and Nd), Proc. Natl. Acad. Sci. USA 115(37), 9140 (2018)

[22] N. Kumar , S. N. Guin , C. Felser , and C. Shekhar , Planar Hall effect in the Weyl semimetal GdPtBi, Phys. Rev. B 98(4), 041103 (2018)

[23] F. H. Murdaka , Y. Wicaksono , E. Suprayoga , A. M. Pradipto , B. Prijamboedi , I. Watanabe , and A. A. Nugroho , Tuning anomalous Hall conductivity via antiferromagnetic configurations in GdPtBi, Phys. Chem. Chem. Phys. 27(4), 2123 (2025)

[24] C. Schindler , S. Galeski , W. Schnelle , R. Wawrzyńczak , W. AbdelHaq , S. N. Guin , J. Kroder , N. Kumar , C. Fu , H. Borrmann , C. Shekhar , C. Felser , T. Meng , A. G. Grushin , Y. Zhang , Y. Sun , and J. Gooth , Anisotropic electrical and thermal magnetotransport in the magnetic semimetal GdPtBi, Phys. Rev. B 101(12), 125119 (2020)

[25] T. Suzuki , R. Chisnell , A. Devarakonda , Y. T. Liu , W. Feng , D. Xiao , J. W. Lynn , and J. G. Checkelsky , Large anomalous Hall effect in a half-Heusler antiferromagnet, Nat. Phys. 12(12), 1119 (2016)

[26] Q. Liu , F. Fei , B. Chen , X. Bo , B. Wei , S. Zhang , M. Zhang , F. Xie , M. Naveed , X. Wan , F. Song , and B. Wang , Nontopological origin of the planar Hall effect in the type-II Dirac semimetal NiTe₂, Phys. Rev. B 99(15), 155119 (2019)

[27] P. C. Canfield , J. D. Thompson , W. P. Beyermann , A. Lacerda , M. F. Hundley , E. Peterson , Z. Fisk , and H. R. Ott , Magnetism and heavy fermion-like behavior in the RBiPt series, J. Appl. Phys. 70(10), 5800 (1991)

[28] T. Sekimoto , K. Kurosaki , H. Muta , and S. Yamanaka , Thermoelectric properties of half-Heusler type LaPdBi and GdPdBi, Mater. Trans. 48(8), 2079 (2007)

[29] K. Gofryk , D. Kaczorowski , T. Plackowski , A. Leithe-Jasper , and Y. Grin , Magnetic and transport properties of rare-earth-based half-Heusler phases RPdBi: Prospective systems for topological quantum phenomena, Phys. Rev. B 84(3), 035208 (2011)

[30] O. Pavlosiuk , X. Fabreges , A. Gukasov , M. Meven , D. Kaczorowski , P. Wiśniewski , Magnetic structures of REPdBi half-Heusler bismuthides (RE = Gd , Tb , Dy , and Ho , Er), Physica B 536, 56 (2018)

[31] Y. Nakajima , R. Hu , K. Kirshenbaum , A. Hughes , P. Syers , X. Wang , K. Wang , R. Wang , S. R. Saha , D. Pratt , J. W. Lynn , and J. Paglione , Topological R PdBi half-Heusler semimetals: A new family of noncentrosymmetric magnetic superconductors, Sci. Adv. 1(5), e1500242 (2015)

[32] M. Klicpera , P. Doležal , J. Prokleška , J. Prchal , and P. Javorsky , Magnetic and transport properties of CePd₂Al₂ single crystal, J. Alloys Compd. 639, 51 (2015)

[33] H. Chen , Q. Niu , and A. H. MacDonald , Anomalous Hall effect arising from noncollinear antiferromagnetism, Phys. Rev. Lett. 112(1), 017205 (2014)

[34] J. Yang , Y. Shang , X. Liu , Y. Wang , X. Dong , Q. Zeng , M. Lyu , S. Zhang , Y. Liu , B. Wang , H. Wei , Y. Wu , S. Parkin , G. Liu , C. Felser , E. Liu , and B. Shen , Modulation of the anomalous Hall angle in a magnetic topological semimetal, Nat. Electron. 8(5), 386 (2025)

RIGHTS & PERMISSIONS

Higher Education Press