Topological electronic state and anisotropic Fermi surface in half-Heusler GdPtBi

Half-Heusler alloys possess unique and desirable physical properties due to their thermoelectricity, magnetism, superconductivity, and weak antilocalization effects. These properties have become of particular interest since the recent discovery of topological Weyl semimetal state for which the electronic bands are dispersed linearly around one pair of Weyl nodes, with opposite chirality (i.e., chiral anomaly). Here, we report the transport signatures of topological electronic state in a half-Heusler GdPtBi single crystal. We show that the non-trivial π Berry phase, negative magnetoresistance and giant planner Hall effect arise from the chiral anomaly and that the Shubnikov–de Haas oscillation frequency in GdPtBi is angle-dependent with an anisotropic Fermi surface (FS). All transport signatures not only demonstrate the topological electronic state in half-Heusler GdPtBi crystals, but also describe the shape of the anisotropy FS.


Introduction
Ternary half-Heusler compounds crystallize in a cubic MgAgAs type structure, which belongs to the space group F43m and exhibits tunable electron properties as varying the components in the chemical formula XYZ (where X and Y are transition or rare earth metals and Z is the main-group element) [1][2][3]. Then, many fantastic properties of the compounds, including the thermoelectricity [4], unusual magnetism [5], superconductivity [6], weak anti-localization effect [7] are 3 Authors to whom correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. being widely explored due to the variable electronic states. This becomes more attractive since the discovery of topological Weyl semimetal states in half-Heusler compounds [8], in which the electronic band can be dispersed linearly around one pair of Weyl nodes with opposite chirality (i.e., chiral anomaly). The signi cance of Weyl/Dirac semimetal as a representative quantum material has been extensively studied for their potential applications and fascinating physics [9,10]. For example, in Dirac semi-metals, each Dirac cone is the result of the superposition of two Weyl nodes with opposite chirality. The Weyl nodes could be separated in the momentum space once the time-reversal or inversion symmetry, or both, are broken, making it impossible to generate a fully gapped insulating state [11]. In theory, the massless Weyl fermions can drive exotic topological phenomena and chiral anomaly related quantum transport properties, if they are located close enough to the Fermi level [12,13]. Thus, studying the chiral anomaly and recognizing the Fermi surface (FS) become ever more important not only for Weyl semimetals but also for half-Heusler compounds.
The topological Weyl semimetal state has two transport signatures, in which the relation of negative magnetoresistance (NMR) to the suppressed backscattering electrons with opposite chirality [14,15], as well as the origination of the giant planar Hall effect (PHE) from the chiral magnetic effect [16,17] are being theoretically studied and con rmed. However, the exact origin of these transport phenomena and the shape of the FS remain unclear, especially in the Weyl semimetal GdPtBi. Herein, we perform magneto-transport experiments on a half-Heusler GdPtBi crystal, which is an essential member of half-Heusler compounds featuring a frustrated antiferromagnetic lattice and an inverted band structure [18]. We study the non-trivial π Berry phase, NMR and PHE and nd that the amplitude of PHE can be divided into two eld-regions, in agreement with theoretical predictions. Our temperaturedependent transport study allows us to conclude that the PHE in GdPtBi should originate from the chiral anomaly. In addition, the shape of Fermi pocket in GdPtBi sketched by the angle-dependent magnetoresistance (MR) measurements is shown to be anisotropic. Thus, our results not only demonstrate the topological electronic state of the half-Heusler GdPtBi, but also describes the shape of the FS.

Experimental detail and methods
Single GdPtBi crystals were synthesized by a Bi-ux method. First, we mixed high-purity raw materials of Gd (ingot, 99.99%), Pt (ingot, 99.99%) and Bi (ingot, 99.99%) at a molar ratio of 1:1:20 using an alumina crucible. Prior to mixing, the oxide layer on the Gd ingot surface was removed by grinding. Note that the Bi act as not only the reactant but also solvent (i.e., ux) during the GdPtBi crystals growth, that is why the excess Bi should be ensured in the precursors. The whole procedures were carried out in an argon-lled glove box, in which the content of O 2 and H 2 O was strictly controlled below 0.5 ppm. The alumina crucible and raw materials were sealed inside a tantalum tube under a proper Ar pressure; the tantalum tube was then further sealed into an evacuated quartz tube. The crystal growth was carried out in a furnace by heating the tube from room temperature to 1150 • C, for 10 h, and kept at 1150 • C for 48 h, before being slowly cooled to 650 • C, at a rate of 2 • C h −1 . Lastly, the excess of Bi ux was removed by centrifuging the tube at 650 • C.
The crystal structure, orientation, crystalline quality and chemical composition of the GdPtBi single crystal was analyzed using an x-ray diffraction (XRD) instrument and transmission electron microscope (TEM, FEI Titan Cs probe). High-angle annular dark-eld scanning transmission electron microscopy (HAADF-STEM) was carried out with an inner semi-angle of 80 mrad and a 250 mrad outer semi-angle. The TEM specimens were prepared using an FEI dual-beam focused ion beam-scanning electron microscope (FIB/SEM, FEI Helios G4 UX). STEM HAADF image simulations were performed with the Dr. Probe software package [19]. Models of the atomic structure were built based on a cubic unit cell (F43m space group) with a lattice constant a = 6.682 Å. The high voltage, C s coef cient, C c coef cient, thickness and defocus were 300 kV, 50 µm, 0 mm, 40 nm and 0 nm, respectively. The supercells used in the calculations extended over 4 × 4 unit cells projected on the (100) plane.
The GdPtBi single crystal was polished to a thickness of 0.1 mm for the transverse MR measurement (as shown in gure S1) (https://stacks.iop.org/JPCM/32/355707/mmedia). The four-probe method was applied for measurement, and the electrical leads were attached to the sample by platinum wires, using silver paste. To minimize the misalignment of the device for the PHE measurement, the FIB/SEM technique was used to prepare the Hall bar device (in (111) plane) that was lifted out from a bulk GdPtBi single crystal. The width, length and thickness of the device were 5 µm, 10 µm, and 1.5 µm, respectively. The MR and PHE were measured using a quantum design physical properties measurement system (PPMS) equipped with a temperature range of 2-300 K.

Results and discussion
The polyhedral characteristic of as-prepared GdPtBi crystals with a size of 3.0 × 2.3 × 0.5 mm 3 is presented in gure 1(a). The molar ratio of Gd, Pt, and Bi was con rmed to be 32.3:33.3:34.1 by the energy-dispersive x-ray (EDX), demonstrating the accuracy of the chemical component and formula in GdPtBi ( gure 1(c)). Speci cally, we found that the GdPtBi was the single crystal that crystallized in a cubic MgAgAs type structure along the [111] direction, as con rmed by the (lll) re ection in the XRD pattern ( gure 1(d)). The crystal structure of the single-crystal was further studied by the HAADF-STEM image and selected area electron diffraction (SAED) pattern. These results demonstrate that there are no bismuth impurities, stacking faults, and/or amorphous regions in the GdPtBi crystal ( gures 1(c) and (d)). The atomic columns contrast between the ideal cubic GdPtBi (the inset of gure 1(e)) and the experimental STEM image were also found to be consistent, fully demonstrating the good crystal quality of GdPtBi. All characterizations con rmed the single-crystal features of our GdPtBi in a cubic MgAgAs-type structure of high quality.
The FS topology (i.e., chiral anomaly) of GdPtBi was investigated rst by the means of MR, with the current applied parallel to the surface of the single crystal (i.e., (111) plane). The angle-dependent longitudinal resistivity (ρ xx ) measured at 2 K is presented in gure 2(a), in which the θ is de ned as the angle between current (I) and the applied magnetic eld (B) (inset of gure 2(a)). We nd that the MR increases linearly up to 285% at 14 T, when the tilted angle θ = 85 • ; on the other hand, the MR decreases as the tilted angle θ decreases from 85 • to 25 • . Moreover, the NMR appeared in the low-eld region where θ is lower than 25 • (i.e., θ < 25 • ). This phenomenon becomes the most prominent when θ decreases to 0 • (magni ed view in gure 2(b)). Deeper physical insights can be gained through the transport properties of the device in the presence of collinear external electric and magnetic elds (θ = 0 • ). Figure 2(c) shows  the temperature dependence ρ xx at the selected eld, which is consistent with the previous results [8]. Besides, the antiferromagnetic ordering in GdPtBi at about 9.7 K (i.e., in ection points in ρ xx curves) was also observed, which should originate from the frustrated fcc lattice caused by Gd atoms [18]. The temperature of the phase transition is consistent with the hysteresis loops ( gure S2). We also nd that the NMR can be suppressed as the temperature increases (i.e., a prominent NMR at the low temperature), and can ultimately disappear at 200 K, as con rmed by the LMR for a temperature range of 2-300 K ( gure 2(d)). These behaviors are similar to those observed in Dirac/Weyl semi-metals (e.g., Cd 3 As 2 , WTe 2 , TaAs) [12,20,21], demonstrating the existence of chiral anomaly associated fermions in GdPtBi.
In addition, the topological electronic state in GdPtBi was further con rmed by the origination of the NMR. Generally, the chiral anomaly [21], current jetting effect [21], or magnetism [22] can give rise to the NMR. We rstly exclude the possibilities originating from the current jetting effect (relate to the geometry or size effects of samples). This is because there are no characteristic dips, hump or negative voltage of the current jetting effect that can be observed in the angular dependence ρ xx ( gure S3). Besides, the Néel temperature of the pronounced antiferromagnetic ordering (∼9.7 K) is much lower than 100 K for the appearance of the NMR, which also excludes the origination from the antiferromagnetism of GdPtBi. Thus, we conclude that the NMR arises from the suppressed backscattering electrons with opposite chirality (i.e., Weyl fermions) in GdPtBi.
The pronounced Shubnikov-de Hass (SdH) oscillations can be also observed in the LMR curves ( gures 2(b) and (d)) at low temperatures, by which the inherent quantum mechanical nature of GdPtBi can be further analyzed. We calculated d 2 ρ/dB 2 to reveal these SdH oscillations and explore the physics of the quantum transport underlying the experimental data. We found that the SdH oscillations were more obvious in the curves of d 2 ρ/dB 2 vs B −1 at 2 K ( gure 3(a)). Then, the Berry phase and the shape of the FS can be extracted and reconstructed, respectively. In more detail, the Landau fan diagram was plotted according to the Lifshitz-Onsager quantization rule [23], F/B = n − γ + δ, wherein F and n are the oscillation frequency and Landau index, respectively ( gure 3(b)). The Onsager phase γ = 1/2 − φ B /2π and the additional phase shift factor δ changed from 0 to ±1/8, depending on the curvature of the FS in the third direction. Figure 3(d) presents a detailed Landau fan at θ = 0 • , which gives an intercept of about 0.06 and corresponds to a 0.88 π Berry phase. These values demonstrate the nontrivial Berry phase feature when B//I. Note that the SdH oscillation frequency of this GdPtBi crystal is about 24.71 T ( gure 3(b)). Then, we studied the angulardependent of SdH oscillations at 2 K to gain more insight into the topology of the FS in GdPtBi ( gure 3(c)). A Periodic oscillation can be observed in all directions, but with slightly different periods, indicating the anisotropic nature of the FS of GdPtBi, at high eld. Besides, we nd that the cross-sectional areal of the FS reaches a maximal value at θ = 20 • , and then decreases to a minimum value at θ = 90 • ( gure 2(e)). This angle-dependent behavior of the FS is similar to that of half-Heusler LuPtBi [24]. Thus, the shape of the FS of Weyl semimetal GdPtBi can be sketched, based on the angular-dependent oscillation frequency ( gure 3(f )).
Another phenomenon related to the topological electronic state is the PHE, where the current, Hall voltage and magnetic eld are coplanar. The angular dependence of PHE in Weyl semimetals can be formulated as [16]: where ϕ is the angle between the magnetic eld and the current, ∆ρ chiral = ρ ⊥ − ρ // is the resistivity anisotropy induced by the chiral anomaly, ρ ⊥ and ρ // are the resistivities for the magnetic eld in perpendicular and parallel directions to the current, respectively. Herein, a typical GdPtBi device and measurement con guration of PHE are represented in gure 4(a), in which the current I is applied along the longitudinal direction of the device. The angle-dependent planar Hall resistivity ρ xy are recorded at different elds and temperatures when the sample is rotated in the elds-plane from 0 • to 360 • . The angular dependence of ρ xy (B) curves under magnetic elds of 9 T and −9 T at 2 K are presented in gure 4(b). We nd that the shape of ρ xy (B) curves is asymmetric and angulardependent, which can be attributed to the normal Hall effect arising from the perpendicular component of the magnetic eld. Several steps need to be taken, as shown below, to get the intrinsic PHE.
The rst, the normal Hall resistivity can be removed by symmetrizing the ρ xy (B) curves via ρ xy sym = 0.5 × [ρ xy (B) + ρ xy (−B)] when the perpendicular component is introduced by the misalignment between the device plane and the magnetic eld (i.e., the green and red curves in gure 4(b)). We nd that the ρ xy sym becomes more symmetric (the blue curve in gure 4(b), but the data still cannot be described by equation (1) due to an obvious resistivity shift away from zero. This shift should originate from the misalignment of the Hall bar, which can generate a planar LMR component coupled to the ρ xy planar . Thus, we describe the ρ xy sym by the modi ed equation (2), after considering this misalignment [17]: The three terms in equation (2) correspond to the intrinsic PHE caused by the chiral anomaly and the in-plane anisotropic MR and LMR offset originating from the geometric misalignment of the Hall device, respectively. The symmetrized ρ xy sym can be tted well using equation (2), as demonstrated by the blue curve in gure 4(c). Hence, the intrinsic PHE can be obtained after subtracting the normal Hall resistivity and planar LMR component ( gures 4(d) and (e)). We nd that the ρ xy planar can be increased monotonically when increasing the eld, and that the ρ xy planar can be decreased as increasing the temperature under various magnetic elds at 2 K ( gures 4(d) and (e)). It should be noted that the ρ xy planar decreases monotonically with increasing temperature ( gure 4(e)). This is attributed to the fact that the Weyl points of GdPtBi can move farther away from the Fermi level when the temperature increases ( gure 4(e)). Thus, these phenomena are consistent with those observed for other Weyl semi-metals previously reported [25].
More physical insights of the PHE in GdPtBi can be obtained from the extracted amplitudes of PHE (ρ xy amp ), which are directly related to the anisotropic resistivity originating from the chiral anomaly. The ρ xy amp versus B extracted from the ρ xy sym at 2 K ( gure 4(d)) and 50 K ( gure S4) are shown in gure 4(f). We nd that the results cannot follow the simple quadratic magnetic eld dependence, which is different for traditional magnetic materials [26]. This is because the theoretical prediction of the magnitude of PHE in a Weyl/Dirac semi-metal can be divided into two eld-dependent regions: the weak magnetic-eld regime and the sample size related to the stronger-eld regime. In the weak magnetic eld region, l a ≫ l c , the PHE is given by [16], For the more complicated, stronger eld-region, where l a ≪ l c and l x > l a , the PHE has the following form: where l a = D ΓB , l c = √ Dτ c and l x are the chiral anomaly related magnetic length, chiral charge diffusion length and Hall bar length, respectively. The parameters of D, B, Γ and τ c are the diffusion coef cient, magnetic eld, transport coef cient and chiral charge relaxation time, respectively. We nd that the ρ amp xy are t well with a B 2 and − 1 B 2 − 1 Bl x function in the low eld regime (B 5 T) and high eld regime (B 6 T), respectively. The consistency between the experimental observations and the theory further demonstrates the existence of chiral anomaly in GdPtBi.

Conclusion
In summary, we have observed the transport signatures of a non-trivial π Berry phase, NMR and the giant PHE in a half-Heusler GdPtBi single-crystal, in which the NMR and PHE were con rmed to be induced both by the chiral anomaly. We nd that the SdH oscillation frequency is angle-dependent, and it show that the Fermi-surface is anisotropic in GdPtBi. We also nd our experimental results to be consistent and in good agreement with theoretical predictions. Therefore, our study demonstrates the topological electronic state of half-Heusler GdPtBi crystals and also describes the shape of the anisotropic FS experimentally.