Anisotropic electrical and thermal magnetotransport in the magnetic semimetalGdPtBi

Clemens Schindler,1, ∗ Stanislaw Galeski,1 Walter Schnelle,1 Rafa l Wawrzyńczak,1 Wajdi Abdel-Haq,1 Satya N.

Guin,1 Johannes Kroder,1 Nitesh Kumar,1 Chenguang Fu,1 Horst Borrmann,1 Chandra Shekhar,1

Claudia Felser,1 Tobias Meng,2 Adolfo G. Grushin,3 Yang Zhang,1 Yan Sun,1 and Johannes Gooth1, †

1Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany2Institute of Theoretical Physics, Technische Universität Dresden, 01062 Dresden, Germany

3Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000, Grenoble, France(Dated: March 24, 2020)

The half-Heusler rare-earth intermetallic GdPtBi has recently gained attention due to pecu-liar magnetotransport phenomena that have been associated with the possible existence of Weylfermions, thought to arise from the crossings of spin-split conduction and valence bands. Onthe other hand, similar magnetotransport phenomena observed in other rare-earth intermetallicshave often been attributed to the interaction of itinerant carriers with localized magnetic momentsstemming from the 4f -shell of the rare-earth element. In order to address the origin of the mag-netotransport phenomena in GdPtBi, we performed a comprehensive study of the magnetization,electrical and thermal magnetoresistivity on two single-crystalline GdPtBi samples. In addition, weperformed an analysis of the Fermi surface via Shubnikov-de Haas oscillations in one of the samplesand compared the results to ab initio band structure calculations. Our findings indicate that theelectrical and thermal magnetotransport in GdPtBi cannot be solely explained by Weyl physics andis strongly influenced by the interaction of both itinerant charge carriers and phonons with localizedmagnetic Gd-ions and possibly also paramagnetic impurities.

I. INTRODUCTION

Weyl fermions can be realized as low-energy quasi-particles in certain semimetals with topologically pro-tected crossing points of two inverted electronic bandswith linear dispersion1–3. They occur in pairs of inde-pendent nodes, separated in momentum space with op-posite chirality - a quantum number defining the ‘hand-edness’ of a quasiparticle’s spin relative to its momen-tum. Classically, the particle number of each chiral-ity is separately conserved. However, at the quantumlevel electromagnetic fields can violate the conservationof the particle number at individual nodes due to quan-tum fluctuations. This phenomenon is known as thechiral anomaly4,5, physically interpreted as simultane-ous production of particles of one chirality and anti-particles of the opposite chirality. In the context of Weylsemimetals, the chiral anomaly is expected to induce asteady out-of-equilibrium flow of quasiparticles betweenthe left- and right-handed nodes, leading to a reductionof electrical and thermal magnetoresistivity6–10 in mag-netic fields H aligned with the electric field E or thermalgradient ∇T , respectively. By now, signatures for thechiral anomaly-induced negative magnetoresistance havebeen reported for electrical measurements of a numberof semimetals11–19. One of the challenges for identify-ing the chiral anomaly is to exclude other mechanismsleading to a similar effect, such as inhomogeneous cur-rents paths in high-mobility materials20,21. Concerningsignatures of the chiral anomaly in thermal transport, themain experimental challenge is the extremely low densityof electronic states in Weyl semimetals, which rendersthermal transport heavily dominated by phononic con-

duction. Nonetheless, a negative magnetothermal resis-tivity has recently been observed in a Bi-Sb alloy, wherethe Fermi level has been tuned to be exactly located atthe band crossing points22.

A prominent example for the observation of the chi-ral anomaly in electrical and thermoelectrical magne-totransport is the semimetal GdPtBi. GdPtBi is amember of the half-Heusler RPtBi series, where R de-notes a rare-earth element. Density functional theory(DFT) calculations for these compounds yield an elec-tronic band structure similar to the binary semiconduc-tor HgTe23, whereby touching of conduction and va-lence band occurs near the Fermi level. Dependingon the strength of the spin-splitting, the RPtBi com-pounds can be driven from a topologically trivial gappedstate to a band-inverted state with topologically pro-tected crossing points, potentially hosting Dirac or Weylfermions23. Alongside the predictions concerning thesingle-electron bands, the RPtBi series exhibits a varietyof many-body phenomena ranging from magnetism24–29,superconductivity30,31 and heavy fermion behavior25–27.Among the RPtBi series, GdPtBi probably is themost studied compound and thought to be a zero-gapsemimetal, which upon application of an external mag-netic field exhibits a band inversion32. There have beenreports on a chiral anomaly-induced negative longitudi-nal magnetoresistivity32,33 and anisotropic planar Hallresistivity34 as well as anomalous features in the Hall33,35

and (magneto-)resistivity29,32,33,35. Many studies25,36–38

have also been carried out on the magnetism of GdPtBi,exploring the antiferromagnetic ordering of the local mo-ments stemming from Gd‘s half-filled 4f -shell with L = 0and J = S = 7/2. Furthermore, optical methodshave been employed, confirming the presence of elec-

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tronic bands with nearly linear dispersion39. As stated inref. 23, the many-body correlations of materials with par-tially filled f -shells cannot be treated in the local-densityapproximation used for calculating the band structurein refs. 23 and 32. Therefore it remains an open ques-tion whether interactions of the Gd 4f -electrons with theitinerant carriers play a role in the observed magneto-transport phenomena in GdPtBi, or whether the single-electron picture is a valid assumption, acting as a foun-dation for the Weyl fermion scenario.

In order to address this question, we performed a com-prehensive study of the physical properties of two dis-tinct GdPtBi samples. In Section II, the crystal growthand related issues as well as measurement techniques aredescribed. Section III is about peculiarities in the mag-netization of the two samples, as well as basic character-ization of the electrical and thermal transport. In Sec-tion IV, the Shubnikov-de Haas oscillations occuring inone of the samples are analyzed and results of our DFTresults are shown. The analysis of the electrical and ther-mal magnetotransport are given in Sections V and VI,respectively.

II. METHODS

a b

c

Gd

Bi

Pt

500 µm

a-axis b-axis c-axis

6.68 �

a b

cSample 2

FIG. 1. a) Crystallographic unit cell of GdPtBi. b) Bottom:Photograph of single-crystalline GdPtBi (Sample 2) cut to abar-shaped rod. Top: SEM image. The bright areas in theBSE-mode show Bi-flux inclusions at the edge of the rod, con-firmed by EDXS analysis. c) Single crystal X-ray diffractionimages for the main crystal directions.

GdPtBi crystallizes in a fcc-lattice as shown in Fig. 1awith a lattice constant of a = 6.68 Å. The single-crystalline samples used in our study were grown inBi self-flux40. Two ∼ 3 mm-sized samples could beextracted from two different batches (Sample 1 and

Sample 2), which have been cut along crystalline axeswith a wire-saw prior to orientation using X-ray diffrac-tion. The X-ray diffraction pattern (Fig. 1c) shows thecharacteristic fcc-reflections; the rings most likely stemfrom powder contaminations on the surface due to thepolishing process. The stoichiometry of the sampleshas been confirmed with energy-dispersive X-ray spec-troscopy (EDXS) analysis on multiple points on all sur-faces of the samples, however, patches of < 100µm ofenclosed Bi-flux are sometimes revealed upon polishingthe sample. They appear as shiny silver, distinct fromthe more bronze GdPtBi-phase, and can also be identi-fied as bright patches under the electron microscope inthe back-scattered electron mode (Fig. 1b). We cut allvisible Bi-flux in the investigated GdPtBi-samples awayand broke the samples in half after the measurementsto check for flux at the edges, however, a residual riskof having undetected enclosures always remains. Themagnetization M was measured in a Quantum DesignMagnetic Property Measurement System (MPMS) mag-netometer equipped with a 7 T magnet. The transportmeasurements have been performed in a Quantum DesignPhysical Property Measurement System (PPMS) witha 9 T magnet. Electrical and thermal currents, J andJh, were applied along the same axes for each sample,which is along [100] for Sample 1 and along [211] forSample 2. Electrical resistivity and Hall effect measure-ments were performed in a 4-probe configuration usingStanford SR-830 lock-in amplifiers with a reference fre-quency of 77.77 Hz and an excitation current of 1 mA.Thermal resistivity measurements were performed withthe steady-state method using the commercial ThermalTransport Option (TTO) sample holder from QuantumDesign, whereby a 2 kΩ Ruthenium oxide resistor is at-tached to one end of the sample as a heater, two Cernoxthermometers are attached in the middle part ∼ 1 mmapart, and the lower end of the sample is attached to aCu-block acting as a heat sink. To achieve good thermalcoupling to the sample for thermal resistance measure-ments, we have fabricated four thermal contacts using0.6 × 0.25 mm2 gold-plated Cu-bars for the heater andthe heat sink; and 0.1 mm Pt-wire for the thermometers.In order to avoid thermal smearing, the heater power wasset such that the temperature gradient ∆T does not ex-ceed 3% of the base temperature T . In addition, in bothelectrical and thermal measurements homogeneity of thecurrent flow was ensured by covering the facets connectedto the current leads with silver paint, as well as attach-ing the voltage and thermometer leads across the wholesample width at an expense of a geometrical error dueto the contact size (not more than 10%). The magneto-electrical resistivity (MR) ρxx and magneto-thermal re-sistivity (MTR) wxx have been symmetrized with respectto H = 0, and the Hall resistivity ρxy antisymmetrized,respectively, to eliminate small misalignment effects.

3

III. SAMPLE CHARACTERIZATION

- 0 . 0 30 . 0 00 . 0 30 . 0 60 . 0 90 . 1 20 . 1 5

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 0 . 0 00 . 0 30 . 0 60 . 0 90 . 1 20 . 1 50 . 1 8

- 3- 2- 1012

- 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0- 2- 10123

- 6 0 - 4 0 - 2 0 0 2 0 4 0 6 00 . 0 3 00 . 0 3 50 . 0 4 00 . 0 4 5

0 1 0 2 0 3 0 4 0 5 0 6 06

9

1 2

S a m p l e 1 , ~ 1 0 0 O e S a m p l e 2 , ~ 1 0 0 O e C u r i e - W e i s s S a m p l e 1 C u r i e - W e i s s S a m p l e 2

M/H (

cm3 /m

ol)

a

T ( K )

M/H (

cm3 /m

ol)

M (µ B

/f.u.)

c

S a m p l e 1S a m p l e 2

T = 2 K

T = 2 K

b

H ( k O e )

M (µ B

/f.u.)

dM/dH

(µB/f

.u./kO

e)

H ( k O e )

S a m p l e 1S a m p l e 2

d

(M/H

)-1 (m

ol/cm

3 )

T ( K )

~ 1 0 0 O e7 0 k O e

S a m p l e 1

FIG. 2. a) Molar susceptibility M/H(T ). The dotted linesshow the Curie-Weiss law for orientation. b) Inverse suscep-tibility (M/H)−1(T ) of Sample 1 at 100 Oe and 70 kOe. c)Magnetization M(H). d) Differential susceptibility χ(H) =dM/dH(H). Sample 1 shows a pronounced non-linearity ofM(H) in low fields.

The magnetic susceptibility M/H at 100 Oe as a func-tion of temperature T is shown in Fig. 2a. The antifer-romagnetic phase transition is clearly observed for bothsamples at TN ∼ 8.9 K. Above TN, Sample 2 follows theCurie-Weiss law41

M/H =C

T −ΘCW+ χ0, (1)

where C is the Curie constant, ΘCW the Curie tempera-ture and χ0 is the T -independent part of the susceptibil-ity (e.g. Van-Vleck susceptibility).

In contrast to Sample 2, Sample 1 exhibits a positivedeviation from the Curie-Weiss law41 below 44 K, indi-

cating the presence of a minor ferromagnetic impurityphase. The extra signal is not visible in a higher ap-plied field, most apparent in the inverse susceptibilitycurve (see Fig. 2b), supporting the ferromagnetic char-acter of the impurity. The transition is not very sharp,thus it most likely stems from a Gd-Pt alloy phase, sincethey all order ferromagnetically near 44 K (e.g. GdPt at68 K42, GdPt2−x at 36-46.5 K

43, GdPt5 at 13.9 K44). We

estimate that 30 ppm of ferromagnetic impurity phase issufficient to explain the strength of the observed signal,thus detection via EDXS is below the achievable resolu-tion.

We note that there is a systematic field error of thesetup of up to 20 Oe, causing significant deviation of theabsolute susceptibility values at low fields. We there-fore fitted the M/H(T ) curve recorded at 70 kOe inthe range 30-300 K for extracting of the Curie-Weiss pa-rameters. With C = 1.571 × 10−6g2J(J + 1)41, theeffective paramagnetic moment meff = gµB

√J(J + 1)

can be estimated, yielding (8.00 ± 0.10)µB for Sample1 and (8.08 ± 0.04)µB for Sample 2. These values areslightly larger than the moment of the free Gd3+ ion7.94µB (with g = 2 and J = 7/2). The extractedCurie temperatures are (−38.2± 0.5)K for Sample 1 and(−40.2±0.2)K for Sample 2, yielding a moderate frustra-tion parameter45 f = −ΘCW/TN ∼ 4. The Curie-Weissparameters of our samples are in good agreement withprevious reports25,29,33,35,36,46,

The slight upturn ofM/H(T ) below 2.5 K indicates thepresence of a paramagnetic impurity in Sample 1, addi-tional to the ferromagnetic impurity. It is also manifestedin the M(H) curve (Fig. 2b), where Sample 1 shows afield-symmetric non-linearity around 1 T, better visible inthe derivative χ(H) = dM/dH(H) (Fig. 2c). Again, wecould not clarify the origin of the paramagnetic impurityin Sample 1, as it falls below the resolution of our EDXS.Ultimately, enclosed Bi-flux can be excluded as an origin,since it would just enhance the diamagnetic contribution.We note that similar signatures in the M(H)-curves havebeen reported for GdPtBi samples in ref. 46, suggestingthat paramagnetic impurities are a common issue whengrowing GdPtBi single-crystals.

The electrical resistivity ρ(T ) of both samples showsthe behavior expected for a zero-gap semimetal (Fig. 3).The sharp cusp at TN ≈ 9 K has been reported to oc-cur due to the formation of a small superzone gap of1 meV at the antiferromagnetic ordering of Gd’s 4f -moments25,29,35–37,46. As stated in ref. 32, the robust-ness of TN against different doping levels suggests thatthe conduction carriers are not involved in the couplingmechanism of the Gd-moments. However, the reverse im-plication, that the conduction carriers are not affected bythe localized Gd-moments, is not necessarily the case asthe cusp in ρ(T ) signifies. Sample 2 exhibits a strong in-crease in ρ(T ) below TN, indicating a Fermi level situatedcloser to the band touching point compared to Sample 1.Above TN, ρ(T ) can be fitted with a simplified two-carriermodel, where the Fermi level is positioned at the edge of

4

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0

1 . 0

1 . 5

2 . 0

2 . 5

3 . 0 5 1 0 1 5

1 . 1 61 . 1 8

��(m

Ω cm

)

T ( K )

S a m p l e 1 S a m p l e 2 F i t S a m p l e 1 F i t S a m p l e 2

FIG. 3. Zero-field resistivity ρ(T ). For Sample 1, the re-gion around the antiferromagnetic transition at TN ∼ 9 K isenlarged in the inset. The dashed lines show fits with Eq. 3.

a hole band and electrons are thermally excited acrossEg, with Eg being the energy difference of the electronband minimum and the Fermi level. In that way, Eg canbe understood as a measure for the energy distance ofthe Fermi level from the band touching point. The totalnumber of carriers is modeled as47

n(T ) = n0 +N√kBT ln 2[kBT ln (1 + eEg/kBT )− Eg],

(2)where n0 is the number of temperature-independent car-riers and N the density of states of the individual bands.The resistivity then holds47

ρ(T ) =ρ0n0 +AT

n(T ), (3)

where the term AT accounts for the phonon scatteringcontribution, which is assumed to be linear above ΘD/10,where ΘD is the Debye temperature.

The fits (see Fig. 3) yield Eg = (42±3)meV for Sample1 and (15 ± 5)meV for Sample 2, which gives a roughestimate of how close the Fermi level is positioned tothe band touching point and thus the size of the Fermisea. Consistently, the Hall resistivity ρxy(H) exhibits anincreasingly non-linear slope upon increasing T (see Fig.4a,d). To extract carrier mobilities µ and densities n ofthe individual hole and electron bands, the Hall data wasfitted with a two-carrier Drude model48

ρxy =1

D

[R1σ

21 +R2σ

22 +R1R2σ

21σ

22(R1 +R2)H

2], (4)

whilst fitting the transverse MR using the same parame-ters with

ρxx =1

D

[(σ1 + σ2)

2 + σ1σ2(σ1R21 + σ2R

22)H

2], (5)

with D = (σ1 + σ2)2 + (σ1σ2)

2(R1 + R2)2H2, where

σ1,2 = (nqµ)1,2 are the Drude conductivities and R1,2 =

(1/qn)1,2 the Hall coefficients of the individual elec-tron/hole channels.

The extracted carrier densities are displayed inFigs. 4b,e; the mobilities in Figs. 4c,f. In both samples,hole-type transport is dominant, yielding carrier densitiesin the order of 1018 cm−3 for Sample 1 and 1017 cm−3 forSample 2, consistent with a Fermi level closer to the bandtouching point and thus a smaller Fermi sea in Sample2. This equates to only ∼ 10−4 majority carriers perunit cell. Mobilities and carrier densities of the electronpockets are much less reliably determined, however, ei-ther density or mobility of the electrons are much lowerthan the hole values; or both.

The zero-field temperature dependence of the thermalconductivity κ(T ) (left axes in Fig. 5a,b) is comparable tothe κ(T )-curves reported in refs. 32 and 46. Concerningthe low density of electronic carriers, it is not surpris-ing that the overall T -dependence of the thermal trans-port in GdPtBi can be fully explained by phonon conduc-tion: upon warming from 2 K, κ(T ) increases with T 3 dueto the increasing lattice heat capacity. Near 15 K, κ(T )reaches a maximum of around 95 WK−1m−1 in Sample1 (60 WK−1m−1 in Sample 2) and then starts to de-crease exponentially due to the onset of phonon Umk-lapp scattering. At higher T , the exponential drop is re-placed by a slower 1/T power law-dependence. Estimat-ing the electronic contribution of κ via the Wiedemann-Franz law κel ≈ L0T/ρ, with the Lorenz number L0 =2.44 · 10−8 WΩK−2, it is apparent that thermal trans-port in GdPtBi is dominated by the lattice contribu-tion. The ratio κel/κ is illustrated on the right axis inFig. 5. At 150 K, the estimated electronic contributionstill contributes ∼2-5%, however, it decreases stronglyto less than 0.1% below 30 K. We note that violationsof the Wiedemann-Franz law due to strong correlationsas well as carrier compensation usually result in a re-duced Lorenz number49–51, and we are not aware of anycase (except for quasi-one-dimensional conductors) whereLorenz numbers larger than 25L0 are observed

52. Even ifsuch gross violation of the Wiedemann-Franz law wouldbe the case, we can still assume that the electronic con-tribution at low temperatures is negligible.

IV. BAND STRUCTURE AND SHUBNIKOV-DEHAAS OSCILLATIONS

We performed DFT based first principles calculationsusing the code of Vienna ab-initio simulation package(VASP) with the projected augmented wave method53,54.The exchange and correlation energies were considered inthe generalized gradient approximation (GGA) with thePerdew-Burke-Ernzerhof-based density functional55; the4f -electrons in Gd were considered as the core state. Wehave projected the Bloch wavefunctions into the maxi-mally localized Wannier functions (MLWFs)56 and con-structed the effective tight binding model Hamiltonian.The resulting band structure (Fig. 6a) is in good agree-

5

- 9 - 6 - 3 0 3 6 9- 4

- 2

0

2

4

- 9 - 6 - 3 0 3 6 9- 6- 3036

1 1 0 1 0 0

1 E 1 8

1 1 0 1 0 0

1 E 1 8

1 1 0 1 0 00

1 0 0 0

2 0 0 0

3 0 0 0

1 1 0 1 0 00

1 0 0 0

2 0 0 0

3 0 0 0

� xy�(m

Ω cm

)

� 0 H ( T )

2 K 1 0 K 3 0 K 5 0 K 1 0 0 K 1 5 0 K 2 0 0 K 3 0 0 K

aS a m p l e 1 S a m p l e 1 S a m p l e 1

S a m p l e 2 2 K 1 0 K 3 0 K 5 0 K 1 0 0 K 1 5 0 K 2 0 0 K 3 0 0 K

� xy�(m

Ω cm

)

� 0 H ( T )

dS a m p l e 2 S a m p l e 2

n (cm

-3 )

T ( K )

h o l e s e l e c t r o n s

b

n (cm

-3 )

T ( K )

h o l e s e l e c t r o n s

e

� (cm

2 V-1 s

-1 )

T ( K )

h o l e s e l e c t r o n s

c

� (cm

2 V-1 s

-1 )

T ( K )

h o l e s e l e c t r o n s

f

FIG. 4. (a,d) Hall resistivity ρxy(H) for H ⊥ J for Samples 1 (a) and 2 (d). (b,e) Carrier densities n of holes and electronsextracted by fitting ρxy(H) with Eq. 4. (c,f) Extracted Drude mobilities µ from the fit with Eq. 4.

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 002 04 06 08 0

1 0 0

0 . 00 . 51 . 01 . 52 . 02 . 5

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 001 02 03 04 05 06 07 0

012345

T ( K )

� (W

K-1 m

-1 )

aS a m p l e 1

S a m p l e 2

��L 0T

/��(%

)

T ( K )

� (W

K-1 m

-1 )

b

��L 0T

/��(%

)

FIG. 5. Thermal conductivity κ(T ) of Sample 1 (a) and Sam-ple 2 (b). The left axes show κ(T ), the right axes show theestimated electronic contribution κel from the Wiedemann-Franz law.

ment with previous reports23,35 and conforms to the pic-ture of a zero-gap semimetal with touching conductionand valence bands at the Γ-point. Consistent with pre-

- 0 . 6- 0 . 30 . 00 . 30 . 6

- 0 . 0 50 . 0 00 . 0 5

- 0 . 0 50 . 0 00 . 0 5

Energ

y (eV)

ΓKWΓUXΓL ΣΛ ∆

a

Γ

Γ Λ

G d P t B i

Energ

y (eV)

b

Y P t B i

T P

T P

Energ

y (eV)

c Λ

d

FIG. 6. a) Electronic band structure of GdPtBi calculatedfrom DFT. b) and c) Triple points along the Γ − L-line inGdPtBi and YPtBi. d) Fermi surface of GdPtBi. The inter-calated red and green cube-shaped pockets are hole-like, theblue and yellow tear-shaped pockets are electron-like.

vious calculations57 and recent optical measurements on

6

GdPtBi39, our model yields a triply degenerated pointnear the Γ-point along the Γ-L line (Fig. 6b), situatedaround 10 meV below the Fermi level. To check, whetherthe triple point crossings are exclusive to GdPtBi andcan thus be considered as an origin for the anomalousmagnetotransport features58, we also calculated the bandstructure for the non-magnetic YPtBi (Fig. 6c). A sim-ilar triple point near the Fermi level is found, yet nosignatures of anomalous magnetotransport have been re-ported for YPtBi33, thus indicating the insignificanceof the triple point crossings for electronic transport inGdPtBi .

The calculated Fermi surface (Fig. 6d) shows two in-tercalated cube-shaped hole pockets centered around theΓ-point, and 8 pairs of small, teardrop-shaped electronpockets at the corners of the hole pockets. We note thatthe size and shape of the pockets sensitively depends onthe position of the Fermi level; when it is sufficiently low,the electron pockets fully disappear.

Sample 1 showed clear Shubnikov-de Haas (SdH) os-cillations in the MR at moderate magnetic fields, allow-ing for an experimental reconstruction of the Fermi sur-face. The oscillations in ρxx are periodic in 1/H with afrequency59

F =~

2πeAext, (6)

Aext being the extremal cross-sectional area of the Fermisurface in the plane perpendicular to the applied mag-netic field H.

The oscillatory part ρ̃xx/ρxx was extracted by sub-tracting a smooth polynomial background from the ρxx-curve and plotting the residual versus 1/H. By assign-ing resistivity maxima (and minima) with (half-)integersand fitting a linear function, F can be extracted fromthe slope. The extracted SdH oscillations versus F/Hare displayed for several fields rotated in the plane per-pendicular to J ‖ [100] in Fig. 7a. Figs. 7b,c show thevariation of F upon rotation of H in the plane perpen-dicular to [100] and [010], respectively. The frequencieshave been extracted within a 180◦ range, as the remain-ing range should be symmetric (duplicated values dis-played in blue in Figs. 7b,c). The four-fold rotationalsymmetry of the underlying crystal lattice becomes ap-parent, as F undergoes a maximum of ∼35 T when His aligned along one of the main axes, and a minimumof ∼26 T at 45◦ from the main axes. As expected, therotations in the [100] and [010] planes yield nearly iden-tical frequencies. Assuming a cube-shaped pocket, thevolume enclosed by the Fermi surface can be estimated

via A3/2ext , Aext = (3.3± 0.2) 10−3Å

−2extracted via Eq. 6

for H ‖ [100]. The carrier density can then be estimatedwith n = 2/(2π)3A

3/2ext = (1.53 ± 0.10) 1018 cm−3. This

is in agreement with the hole density extracted from theHall measurement (1.54±0.02) 1018 cm−3 at 2 K. Conclu-sively, the SdH oscillations stem from the hole pocket. Incontrast to the DFT results (see Fig. 6d), where at leasttwo distinct SdH frequencies stemming from the two dif-

ferently sized hole pockets would be expected, only onefrequency is observed. This indicates that the single-electron picture used for calculating the band structuredoes not fully account for the actual shape of the Fermisurface and many-body effects might need to be includedin the theoretical treatment for better agreement betweentheory and experiment23.

For further analysis, the magnetoconductivity σxx =ρxx/(ρ

2xx + ρ

2xy) has been calculated, as the oscillatory

part σ̃xx/σxx is proportional to the oscillations in thedensity of states and can be described by the Lifshitz-Kosevich formalism59.

In the simplest form (neglecting many-body interac-tions) it holds59

σ̃xxσxx∝∞∑p=1

(H

p

)1/2RDRTRs cos

[2πp

(F

H− β

)± π

4

],

(7)where the phase factor β depends on the details of theband structure and the ± accounts for the extremal areabeing a mimimum (+) or a maximum (-).

The Dingle damping factor RD accounts for a finiterelaxation time and holds

RD = exp

[−p π

µcH

], (8)

where µc is the mobility in case of a semiclassical cy-clotron motion.

The damping factor RT accounts for a finite tempera-ture and holds

RT =λ(T )

sinh[λ(T )],with λ(T ) = 2π2p

mceH

kBT

~, (9)

where mc is the cyclotron mass.The damping factor Rs accounts for the effect of spin-

splitting and holds

Rs = cos

(1

2pπg

mcm0

), (10)

where g is the Landé-factor and m0 the free electronmass. The extracted oscillatory part of σxx is shownin Fig. 7d for different temperatures. The characteris-tic damping of the amplitude allows to extract mc byfitting σ̃xx/σxx versus T at fixed 1/H with Eq. 9 (seeFig. 7e). It has been determined to mc = (0.30±0.02)m0for H ‖ [100], and mc = (0.26 ± 0.02)m0 for H ‖ [001].These results are similar to those reported in refs. 32and 46. A fit of the full SdH oscillation at 2 K withEq. 7 is shown in Fig. 7f, only taking the fundamen-tal oscillation (p = 1) into account. The extractedmobility µc = (3200 ± 100) cm2V−1s−1 is comparableto the Drude mobility extracted from the Hall analysisµ = (3000± 200) cm2V−1s−1.

The extraction of the g-factor would be very useful forthe investigation of the magnetic interactions in GdPtBi.However, we were not able to draw any conclusions about

7

3 4 5 6

01 02 03 04 05 0

0 . 1 2 0 . 1 4 0 . 1 6 0 . 1 8 0 . 2 0 0 . 2 2 0 . 2 4

2 4 6 8 1 0

0 . 1 2 0 . 1 4 0 . 1 6 0 . 1 8 0 . 2 0 0 . 2 2 0 . 2 4

� xx/�

xx (a.u

.)

F / � 0 H

a

J || [ 1 0 0 ]

[ 0 0 1 ]H J || [ 1 0 0 ]H J || [ 1 0 0 ]

[ 0 0 1 ]H9 0 ° ( H || [ 0 1 0 ] )

8 5 °

8 0 °

7 5 °

7 0 °

6 5 °6 0 °5 5 °

5 0 °

4 5 °4 0 °

3 5 °3 0 °

2 5 °

2 0 °

1 5 °1 0 °5 °

0 ° ( H || [ 0 0 1 ] )

T = 2 KS a m p l e 1J || [ 1 0 0 ]

0 0 1

0 1 0

0 0 1

0 1 0

0 0 10 0 1

1 0 0

1 0 0

b

F (T)

c

� xx/�

xx (a.u

.)1 / � 0 H ( T - 1 )

2 K 4 K 6 K 8 K 1 0 K

d

Amplit

ude (

a.u.)

T ( K )

H || [ 0 0 1 ] H || [ 1 0 0 ]

e� x

x/�xx (

a.u.)

1 / � 0 H ( T - 1 )

2 K , H || [ 0 0 1 ] L i f s h i t z - K o s e v i c h f i t

f

FIG. 7. Shubnikov-de Haas (SdH) oscillations in Sample 1. a) Extracted oscillatory part ρ̃xx/ρxx versus F/H for appliedmagnetic fields H in the [100]-plane, rotating from [001] to [010]. (b,c) SdH frequency F versus direction of H for rotationin the [100]-plane (b) and the [010]-plane (c). The black dots are extracted from the experimental data, the blue dots areduplicated according to symmetry. d) σ̃xx/σxx versus 1/H for different temperatures T . e) Temperature-damping of σ̃xx/σxxat fixed 1/H. The solid lines represent fits with Eq. 9. f) Fit of the SdH oscillation at 2 K with the Eq. 7.

the spin-splitting59, since the higher harmonics (p > 1)are dampened too strongly in the field range accessible inour setup. High-field MR measurements on GdPtBi havebeen performed by Hirschberger46, showing a frequencychange above a critical field of µ0Hc ∼ 25 T that couldnot be associated with a spin-splitting. At this criticalfield, also the M(H)-curves below TN are reported

35,46 toshow a clear kink and slope change, attributed to the sat-uration of the magnetic moments which potentially leadsto strong alterations of the band structure46. Indications

for field-induced modifications of the band structure havealso been observed in the antiferromagnetic sister com-pound CePtBi28, where the amplitude of the SdH oscil-lations drastically changes above the critical field of 25 T.

In Fig. 7a it becomes apparent that the phase of theSdH oscillations in GdPtBi is strongly varying upon ro-tation of H, which has also been reported in ref. 46.Possible mechanisms for this phase variation can be59:(i) the variation of the extremal orbit from maximum tominimum orbits causing a phase jump of ±π/4 accord-

8

ing to Eq. 7, (ii) the peculiarities of the band structurecausing a deviation of β from the value 1/2 for parabolicbands, (iii) the strength of the spin-splitting and inter-actions with the magnetic ions.

V. ELECTRICAL MAGNETOTRANSPORT

The transverse MR (TMR) (H ⊥ J) for different tem-peratures is displayed in Fig. 8a for Sample 1 and Fig. 8dfor Sample 2. Above 100 K, the TMR can be adequatelydescribed by the two-carrier model (Eq. 5), whereby com-pensation of electron and holes yields a positive TMRwith quadratic H-dependence in low fields. Below 100 K,a pronounced dip evolves around 1-4 T which deepensupon cooling down and ultimately leads to a negativeTMR over a field range of ∼2 T at 2 K. Although the ex-act shape is different for both samples, the main char-acteristics of the dip are similar and have also beenconsistently reported for GdPtBi despite varying dopinglevels29,32,33,35.

The longitudinal MR (LMR) (H ‖ J) for Sample 1and 2 is shown in Figs. 8b,e. Both samples exhibit abell-shaped negative LMR at low temperatures, at 9 Tand 2 K resulting in 20−30% of the zero-field value. Thenegative LMR weakens upon warming up and ultimatelyturns positive (in the accessible field range) around 100-150 K. The low-field behavior is different for both sam-ples: At 2 K, Sample 1 shows a dip around zero-field,which disappears upon warming and leads to a plateau-shape in the low-field LMR curve above 10 K; Sample2 exhibits a dip around zero-field as well, however, uponwarming it becomes more pronounced, resulting in a ‘M’-shaped LMR-curve up to 100 K. The strong sample vari-ation of the LMR-shapes can also be seen by compar-ing the LMR data of previous reports32,33. Additionally,the low-T LMR curves of Sample 1 exhibit a noticeablefeature around 2-3 T, in a similar field range where thepronounced feature in the TMR appears. Such a featurealso appears in the LMR of the GdPtBi sample in ref. 33.The evolution of this feature upon rotation of H suggestsa similar origin as the dip in the TMR (Figs. 8c,f).

The overall negative LMR, which successively dis-appears upon rotating H towards the TMR configu-ration, has previously been associated with the chiralanomaly32,33. Initially32, the Weyl node creation wasthought to be induced by conventional Zeeman splitting,requiring a large spin-orbit coupling with a g-factor ofthe order of ∼40 to explain the persistence of the ef-fects up to 150 K. However, the absence of the negativeLMR and/or anomalous features in the TMR in the non-magnetic sister compounds LuPtBi29,60, LaPtBi28 andYPtBi30,33 despite a similar band structure strongly in-dicated that conventional Zeeman-splitting cannot ac-count for the observed effects and that they are related tothe unfilled f -shell of the rare-earth element. Therefore,magnetic exchange enhancing the spin-splitting has beensuggested to be responsible for a potential Weyl node

creation33. The energy scale of the exchange interactionhas been estimated by gJµBHc with g = 2 and J = 7/2,yielding ∼120 K35. This is congruent with a magneticorigin causing the appearance of anomalous MR featuresbelow this temperature. The transport coefficients in achiral anomaly scenario6 yield a positive quadratic func-tion in H‖ for σxx in the limit of H → 0 and a positive lin-ear function in H‖ for H →∞. The contribution is tiedto the scalar product of E ·H, i.e. only H‖ = H cos(φ)with φ = ^(H,J) contributes to the anomalous cur-rent generation. In agreement with the chiral anomalyscenario, the longitudinal magnetoconductivity σxx,‖ hasroughly a quadratic field dependence over a range of fieldsup to 5 T in both samples (neglecting the peak at zerofield), whereby the range varies with temperature (seeFig. 9). However, the model cannot explain the satu-ration of σxx,‖(H) in Sample 1 at low temperatures aswell as the differences in the low field TMR and LMRbetween the two samples. In contrast to Sample 1, thehigh-field σxx,‖(H) in Sample 2 is linear in the accessiblefield range, congruent with the chiral anomaly model.An argument that has been brought forward in ref. 32is that the magnetic field-induced changes of the bandstructure forming the Weyl nodes might cause the devia-tions in the low-field LMR as well as the dip in the TMR.Following this argument, magnetic interactions might in-fluence the effective carrier density and relaxation time aswell beyond the field range where the dip in the TMR isobserved. Moreover, in case of aligning H perpendicularto J the positive TMR contribution might superimposeon the otherwise negative MR, unveiled when aligning Hand J .

A way of tracing the field-induced changes of theelectronic properties is to fit the ρxx,‖(T )-curves withthe phenomenological two-carrier model (introduced inSec. III, Eq. 3) for constant H. The fits are shown inFigs. 10a,b and agree well with the data above TN. Thefit parameters n0, A, and N are constant within the errortolerance and only Eg and ρ0 show a smooth evolutionwith H. The fitting parameters Eg and ρ0 versus H areshown in Figs. 10c,d. Eg is increasing with H, giving anestimate of the shifting of bands (see Sec. III); whereas ρ0is decreasing, indicating increased relaxation times withincreased fields. Magnetic-field induced changes of theband structure would also explain the reported32 corre-lation between the strength of the negative LMR andthe position of the Fermi level: the closer the Fermi levelis positioned near the band touching point, the biggerthe change in the density of states upon shifting of thebands in magnetic fields will be and hence, the biggerthe effect on the LMR. Our samples are exception to thisobservation, whereas Sample 2 is found to have a LMR∼10% stronger than Sample 1, with the Fermi level being∼25 meV closer to the band touching point. Ultimately,the electron pockets seem to be influenced much more se-vere, given the relatively constant (within our resolution)SdH frequency of the hole pocket observed in Sample1 above 5 T (Fig. 7f). It could also be that the field-

9

0

2 5

5 0

7 5

1 0 0

1 2 5

1 5 0

- 1 0 0- 7 5- 5 0- 2 5

02 55 07 5

1 0 01 2 5

0 . 5

1 . 0

1 . 5

- 9 - 6 - 3 0 3 6 9

0

2 5

5 0

7 5

1 0 0

1 2 5

1 5 0

- 9 - 6 - 3 0 3 6 9

- 1 0 0- 7 5- 5 0- 2 5

02 55 07 5

1 0 01 2 5

- 9 - 6 - 3 0 3 6 9

1

2

3

4

5

- 0 . 4 0 . 0 0 . 4

1 0 01 0 11 0 2

- 0 . 4 0 . 0 0 . 4

1 0 0 . 01 0 0 . 5

��xx/�

xx,0 (%

)

2 K 4 K 6 K 8 K 1 0 K 1 5 K 2 0 K 3 0 K 4 0 K 5 0 K 7 5 K 1 0 0 K 1 2 5 K 1 5 0 K 2 0 0 K 3 0 0 K

a

S a m p l e 1 J || [ 1 0 0 ]

H || [ 0 0 1 ]

��xx/�

xx,0 (%

)

bS a m p l e 1 , T = 2 K

S a m p l e 2 , T = 2 K

J || [ 1 0 0 ]H

S a m p l e 1

S a m p l e 2

� xx�(m

Ω cm

)

0 ° 1 0 ° 2 0 ° 3 0 ° 4 0 ° 5 0 ° 6 0 ° 7 0 ° 8 0 ° 9 0 °

c

J || [ 1 0 0 ]

[ 0 0 1 ] H

��xx/�

xx,0 (%

)

� 0 H ( T )

d

S a m p l e 2

H || [ 1 1 1 ]

J || [ 2 1 1 ]

��xx/�

xx,0 (%

)

� 0 H ( T )

e

J || [ 2 1 1 ]H

� xx�(m

Ω cm

)

� 0 H ( T )

f

J || [ 2 1 1 ]

[ 1 1 1 ]H

FIG. 8. Anisotropic magnetoresistance (MR) ∆ρxx(H)/ρxx(0) for Samples 1 and 2. (a,d) Transverse MR (H ⊥ J) for differenttemperatures. (b,e) Longitudinal MR (LMR) (H ‖ J). The insets enlarge the low-field range of the LMR-curves below 10 K.The curves in (a,b,d,e) are shifted for better visibility. (c,f) MR ρxx(H) upon rotation of H from perpendicular to longitudinalconfiguration at 2 K.

- 9 - 6 - 3 0 3 6 9

1

2

3

- 9 - 6 - 3 0 3 6 9

0 . 5

1 . 0

1 . 5S a m p l e 1H || J || [ 1 0 0 ]

� xx (m

Ω-1 c

m-1)

� 0 H ( T )

2 K 4 K 6 K 8 K 1 0 K 1 5 K 2 0 K 3 0 K 4 0 K 5 0 K 7 5 K 1 0 0 K

� xx (m

Ω-1 c

m-1)

� 0 H ( T )

S a m p l e 2H || J || [ 2 1 1 ]

FIG. 9. Longitudinal magnetoconductivity σxx,‖ = 1/ρxx,‖for different temperatures.

induced changes in Sample 1 happen mostly at fields be-

low 5 T, which would be congruent with the bell-shapedLMR curves almost saturating above 6 T (Fig. 8b).

Another intriguing observation is that both LMR andTMR do not seem to change upon undergoing the anti-ferromagnetic ordering at 9 K, showing that ordering ofthe Gd moments is apparently not crucial for the MR inGdPtBi. This is surprising, as the opening of the super-zone gap leads to a sharp increase of the zero-field ρ(T )(see Fig. 3). Taking these facts together, a simple spin-disorder scattering mechanism is precluded as an originof the negative MR, deduced also from the observationthat the size of the LMR increases with decreasing carrierdensity32.

10

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00 . 3

0 . 6

0 . 9

1 . 2

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00 . 51 . 01 . 52 . 02 . 53 . 0

0 2 4 6 8

2 0

4 0

6 0

0 2 4 6 80 . 00 . 51 . 01 . 52 . 02 . 53 . 0

S a m p l e 2H || J || [ 2 1 1 ]

� xx�(m

Ω cm

)

T ( K )

0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 TaS a m p l e 1H || J || [ 1 0 0 ]

� xx�(m

Ω cm

)

T ( K )

0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 Tb

S a m p l e 1 S a m p l e 2

E g (m

eV)

� 0 H ( T )

c S a m p l e 1 S a m p l e 2

� 0 (m

Ω cm

)

� 0 H ( T )

d

FIG. 10. (a,b) Longitudinal magnetoresistance versus T for different magnetic field strenghts for Sample 1 (a) and 2 (b). Thesolid lines show fits with Eq. 3. (c) Extracted fitting parameter Eg versus H. (d) Fitting parameter ρ0 versus H.

VI. THERMAL MAGNETOTRANSPORT

In contrast to electrical transport, which can onlybe carried by electronic excitations, heat transport inGdPtBi can also be carried by phonons and, below TN,magnons. Depending on the different scattering mecha-nisms, phonons and magnons can in principle also exhibitmagnetic field-dependencies and cause a non-monotonicMTR. The MTR of the two samples is shown in Fig. 11.During the MTR measurements, the mean temperatureTmean = (Thot + Tcold)/2 was found to increase in mag-netic field with respect to the base temperature by up to∼ 0.25% at 9 T, independent of the H-orientation. Thesystematic error of this heating has been accounted for bymultiplying the slope dw(T ′)/dT with (Tmean − T ′)/T ′,yielding the error bars in Fig. 11.

Above 50 K, a positive MTR is observed, quadraticin low fields and appearing only for the transverse H-configuration. At 150 K, it leads to an increase of ∼4-5% of the MTR at 9 T in both samples (Figs. 11a,c).Given the estimated contribution of κel of 2-5% (Fig. 5),the percentage increase is in the order of the positiveelectrical TMR (Fig. 8a,d). The positive transverse MTRtherefore likely results from κel; the thermal two-carriercompensation can be described in a similar way than theelectrical TMR. The electronic thermal conductivity of atwo-carrier system in a transverse magnetic field holds48

κxx =∑

κ1,2 +T

D

[σ1σ2(σ1 + σ2)(∆S1,2)

2 − σ1σ2

×∆N1,2[(σ1 + σ2)∆N1,2 − 2σ1σ2∆S1,2(R1 +R2)]H2],

(11)with

∑κ1,2 = κ1 +κ2 being the sum of the thermal con-

ductivities of the individual channels. ∆S1,2 = S1 − S2and ∆N1,2 = N1 − N2 are the differences of the indi-vidual Seebeck and Nernst coefficients, respectively. As-suming κxy � κxx (where κxy is the thermal Hall con-ductivity), the inverse of Eq. 11 describes the two-carrierMTR, resulting in a behavior similar to the two-carrierMR (that is quadratic in low fields and saturating or lin-ear in high fields in case of perfect compensation, respec-tively). In contrast to Eq. 5, there is a T -proportionality,leading to a decrease of the positive MTR upon cooling.Additionally, κel/κ strongly decreases with decreasing T(see Fig. 5). Both effects contribute to the rapid van-ishing of the positive transverse MTR upon cooling from150 K to 50 K. In Sample 1, a small negative MTR of∼1% is observed for the longitudinal configuration below100 K and for the transverse configuration below 50 K.The shape varies depending on the configuration, froma bell-shaped negative MTR for H ‖ Jh to a ‘W’-shapefor H ⊥ Jh. Sample 2 showed no field-dependence inthe longitudinal configuration above 50 K, and only smallpositive MTR of less than 1% between 30 and 50 K forboth H-orientations.

Below 30 K, κel/κ drops below 0.1%, any resolvablefield-dependencies of κ can therefore no longer originate

11

0

2

4

6

8

0

2

4

6

- 9 - 6 - 3 0 3 6 9

- 1 0

0

1 0

2 0

3 0

4 0

- 9 - 6 - 3 0 3 6 9

- 2 0

- 1 0

0

1 0

2 0

3 0

�wxx/w

xx,0 (%

)a S a m p l e 1

J h || [ 1 0 0 ]

�wxx/w

xx,0 (%

)

b H

HJ h || [ 2 1 1 ]

2 K 3 K 4 K 6 K 8 K 1 0 K 1 5 K 2 0 K 3 0 K 4 0 K 5 0 K 7 5 K 1 0 0 K 1 5 0 K 2 0 0 K

�wxx/w

xx,0 (%

)

� 0 H ( T )

c

S a m p l e 1

S a m p l e 2 J h || [ 2 1 1 ]

H || [ 1 1 1 ]

�wxx/w

xx,0 (%

)

� 0 H ( T )

d

S a m p l e 2

J h || [ 1 0 0 ]

H || [ 0 0 1 ]

FIG. 11. Anisotropic magnetothermal resistivity (MTR) wxx(H)/wxx(0) for Samples 1 and 2. (a,c) Transverse MTR (H ⊥ Jh)for different temperatures. (b,d) Longitudinal MTR (H ‖ Jh). The curves are shifted for better visibility.

from the electronic contribution. As spin-waves onlyoccur in the ordered phase, an abrupt change of κ(T )when undergoing the antiferromagnetic phase transitionwould be expected in case of substantial heat transportby magnons. Since there was no discernible feature atTN (see Fig. 5), we assume that the magnon contribu-tion is negligible. However, even if magnons in GdPtBido not carry much heat themselves, they can still scatterwith phonons and therefore be responsible for the field-dependency of the MTR below TN. The MTR of Sample2 was measured down to T = 2 K, showing similarities inthe general trend for both H-configurations. Below TN,a ‘M’-shaped MTR in both transverse and longitudinalMTR (disregarding the additional anomalous features)evolves, which might be explained by phonon scatteringfrom magnons: If at H = 0 phonon and magnon dis-persion intersect at an energy less than the maximum ofthe phonon distribution around 4kBT , the thermal resis-tivity will first increase with rising H and then decrease

due to the shift of magnon branches in magnetic field61.A κ(H)-curve of GdPtBi with H ‖ Jh at 500 mK hasbeen reported in ref. 46, showing a similar behavior thanthe 2 K-curve of the transverse MTR. In the longitudi-nal configuration, the MTR of Sample 2 shows additionalfeatures which are not present when H ⊥ Jh: (i) A peakaround zero-field evolves upon cooling below 4 K; (ii) adip around 3 T is observed at 2 K; (iii) a deviation fromthe smooth curve (compared to the transverse MTR) ap-pears at 6 T below 6 K. These features are probably notrelated to spin-reorientation processes41, considering thefeatureless M(H)-curve of Sample 2 (Fig. 2). Such dipsmight be attributed to resonant scattering with paramag-netic impurities, where the number of scattered phononsdepend on the Zeeman splitting at a given H, as observedfor example in the MTR of Holmium ethylsulphate61,62.As paramagnetic impurities can generally be present inGdPtBi, especially indicated by in the magnetic suscep-tibility of Sample 1 (see Sec. III), this might also serve as

12

an explanation for the features in the longitudinal MTRof Sample 2. In case of an anisotropic g-factor of theimpurity, this effect might only appear for H along cer-tain directions. It might also be that the dispersion ofdifferent magnon branches along certain directions leadsto field-dependent transitions. The magnon spectrum ofGdPtBi has recently been studied via inelastic neutronscattering38. Two modes with similar energy at theirminima were revealed, but depending on the directionthey disperse quite differently.

The MTR in the intermediate temperature range fromTN to 30 K, where neither magnons nor electrons are ex-pected to have an influence, might originate from phononscattering from the paramagnetic Gd-ions as well as im-purities. Scattering from paramagnetic impurities mightexplain the sample-dependent MTR in that range. Therich MTR features and the anomalies in the magnetiza-tion (Fig. 2) in Sample 1 could therefore be related andmight also explain why the MR of Sample 1 exhibits ad-ditional features compared to Sample 2. Furthermore,frustration might also lead to a field-dependent phonondensity45, however, the moderate frustration of the Gd-ions is found to be relieved already by the next-nearestneighbour-interactions38, GdPtBi thus exhibits only lowfrustration. Magnetostriction might have an effect on theMTR as well. Despite the vanishing crystal field of Gd,magnetoelastic effects in Gd intermetallics can sometimesbe of the same order of magnitude as in other rare-earthcompounds42. Hence, they might as well be significantin GdPtBi.

VII. SUMMARY

We have studied the magnetization, thermal and elec-trical magnetotransport in two single-crystalline GdPtBi

samples and analyzed the Shubnikov-de Haas oscillationsoccurring in one of the samples. Our findings show thatthe complex, anisotropic magnetotransport in GdPtBicannot be solely explained within the single-particle Weylmodel, but requires to take magnetic interactions of bothitinerant charge carriers and phonons with the localizedGd 4f -moments and possibly also with paramagnetic im-purities into account. These results further stimulatethe exploration of the interplay between topological bandstructure features and many-body interactions.

ACKNOWLEDGMENTS

Cl. S. thanks R. Koban for providing technical sup-port and C. Geibel for engaging in clarifying discussions.This work was financially supported by the ERC Ad-vanced Grant No. 291472 Idea Heusler and ERC Ad-vanced Grant No. 742068 TOPMAT. Cl. S. acknowl-edges financial support by the International Max PlanckResearch School for Chemistry and Physics of Quan-tum Materials (IMPRS-CPQM). A. G. G. thanks J.Cano, B. Bradlyn and R. Ilan for engaging in clarify-ing discussions. A. G. G. acknowledges financial sup-port from the Marie Curie programme under EC Grantagreement No. 653846. T. M. acknowledges funding bythe Deutsche Forschungsgemeinschaft through the EmmyNoether Programme ME 4844/1-1 and through SFB1143.

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Anisotropic electrical and thermal magnetotransport in the magnetic semimetal GdPtBiAbstractI IntroductionII MethodsIII Sample CharacterizationIV Band structure and Shubnikov-de Haas oscillationsV Electrical MagnetotransportVI Thermal MagnetotransportVII Summary Acknowledgments References