LETTERSPUBLISHED ONLINE: 27 JUNE 2016 | DOI: 10.1038/NMAT4684

The chiral anomaly and thermopower of Weylfermions in the half-Heusler GdPtBiMax Hirschberger1*, Satya Kushwaha2, ZhijunWang1, Quinn Gibson2, Sihang Liang1,Carina A. Belvin1†, B. A. Bernevig1, R. J. Cava2 and N. P. Ong1*The Dirac and Weyl semimetals are unusual materials inwhich the nodes of the bulk states are protected against gapformation by crystalline symmetry1–4. The chiral anomaly5,6,predicted to occur in both systems7–10, was recently observedas a negative longitudinal magnetoresistance (LMR) in Na3Bi(ref. 11) and in TaAs (ref. 12). An important issue is whetherWeyl physics appears in a broader class ofmaterials.We reportevidence for the chiral anomaly in the half-Heusler GdPtBi. Inzero field, GdPtBi is a zero-gap semiconductor with quadraticbands13,14. In a magnetic field, the Zeeman energy leads toWeyl nodes15. We have observed a large negative LMR withthe field-steering properties specific to the chiral anomaly.The chiral anomaly also induces strong suppression of thethermopower.We report a detailed study of the thermoelectricresponse functionαxx ofWeyl fermions. The schemeof creatingWeyl nodes from quadratic bands suggests that the chiralanomaly may be observable in a broad class of semimetals.

The Dirac semimetal Na3Bi exhibits two bulk Dirac conesin zero field4,16. Each cone resolves into two Weyl nodes withdistinct chiralities χ =±1. In a magnetic field B, the Weyl nodesseparate in k (momentum) space to act as monopole source andsink of the Berry curvature Ω(k) (which acts as an effectivemagnetic field in k space17). As observed in Na3Bi (ref. 11), theapplication of an electric fieldE ‖B produces a negative longitudinalmagnetoresistance (LMR) produced by the chiral anomaly. Therecently discovered Weyl semimetals18,19 are similar, except thatthe Weyl nodes are already separated at B= 0 because inversionsymmetry is broken12. Here we demonstrate a third route towardsWeyl nodes, starting with a material with Td symmetry anddisplaying quadratic bands that touch15. In finite B, the Zeemanenergy leads to band crossings and the formation of Weyl nodes.In the half-Heusler GdPtBi, this scheme results in the appearanceof the chiral anomaly (in samples with the Fermi energy EF muchcloser to the Weyl node than in previous experiments13,20,21).

The unit cell of GdPtBi is comprised of Pt-Gd tetrahedra arrayedin the zincblende structure (Supplementary Section 1). The low-lying states involve only the four Bi 6p bands, |j,mj〉=|3/2,±1/2〉and |3/2,±3/2〉, which are four-fold degenerate at the 0 point (thelattice has Td symmetry at 0). Combining ab initio calculations(Methods) with the k ·pmodel in finite B, we find that the Zeemanenergy results in crossings (Fig. 1a). The number of low-lying nodesdepends on whether B is aligned ‖ [110] or [111] (SupplementaryFig. 4). Details of the k · p calculations are given in SupplementarySection 2 and in ref. 22.

Crystals of GdBiPt were cut with the axis x of the longest edgeeither ‖ [110] or ‖ [111] (see Methods). Altogether, we measured

15 samples (A, B, . . ., Q), with the current density J or heat currentdensity JQ applied ‖ x in all samples (see Supplementary Table 1).Samples C, E, F and G (cut from the same boule) have very similarcarrier densities. Figure 1b shows curves of the zero-B resistivity ρversus temperature T in zero B in three samples (K, C and M) withFermi energy EF<0 (p-type) and one sample (L) that is n-type. Thenon-metallic profile and the sharp decrease of the Hall density nHare consistent with a zero-gap material (Fig. 1b, inset). At 4K, allsamples display a prominent negative LMRmeasuredwithB‖E ‖x .Figure 1c plots the longitudinal resistivity ρxx versus T at selectedvalues of B‖[110] in Sample G. The field suppression of ρxx onsetsat T∼150K and increases strongly as T→ 2K. The profile of ρxxversus B is a bell-shaped curve with a half-width δB∼ 2.5 T below10K (Fig. 1d). Raising T rapidly increases δB, but the negative LMRremains observable up to 150K. The negative LMR, observable to150K, is unrelated to the antiferromagnetic state that appears belowthe Néel temperature TN= 8.8 K, which is insensitive to nH (seeSupplementary Fig. 9 and Methods).

First we show that the negative LMR goes away when B istilted away from E . As shown in Fig. 2a, increasing the tilt angleθ rapidly broadens the bell-shaped LMR profile. As θ→ 90, themagnetoresistance (MR) becomes positive apart from a low-fieldoscillatory feature (see below). The angular variation of the plume inthe conductivity σxx at fixed B (Fig. 2b) is consistent with the chiralanomaly (the plume here is slightly broader than that observed inNa3Bi; ref. 11). In weak B, the LMR profile differs between B‖[111]and [111], consistent with band calculations (see SupplementarySection 2 and Supplementary Figs 2 and 13).

A striking property of the anomaly is that when the direction ofE ‖ J is rotated to a new crystal axis, the plume direction movesto the new axis. In ref. 11, E was changed by selecting a differentpair of current contacts on the same crystal. Here, we compare ρxxmeasured in the two crystals measured with J aligned with [110]versus [111]. Figure 2c plots ρxx versus φ (the angle between Band J in both cases) for B fixed at 2 and 9 T. In both crystals,ρxx attains a minimum only when B‖ J , consistent with the chiralanomaly. (We have also confirmed (Supplementary Fig. 11) thefield-steering property using just one crystal and alternating thecurrent contacts as in ref. 11.) The results together confirm the field-steering property.

As a third check, we verify that the LMR is suppressed whenEF is far from the node. When |EF| exceeds the Weyl energyscale, the chiral anomaly becomes unresolvable. Using the zero-Bthermopower S, nH and the Shubnikov–de Haas (SdH) period, wehave determined EF in six crystals. The magnitude of the LMR,measured by ρ(0T )/ρ(9T ), is largest when |EF| is closest to the

1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA. 2Department of Chemistry, Princeton University, Princeton,New Jersey 08544, USA. †Present address: Wellesley College, 106 Central Street, Wellesley, Massachusetts 02481, USA.*e-mail: hirschberger.max@gmail.com; npo@princeton.edu

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LETTERS NATUREMATERIALS DOI: 10.1038/NMAT4684

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Figure 1 | The field-induced band crossing in GdBiPt and the chiral anomaly in its longitudinal magnetoresistance (LMR). a, Left, a sketch showing thefour-fold degeneracy at 0 in zero B of the bands |3/2,±3/2〉 (blue curve) and |3/2,±1/2〉 (red curve). Au: Right, in finite B (‖ to the axis as shown), thelarger Zeeman shift of |3/2,±3/2〉 leads to Weyl nodes with opposite χ (red and grey cones). b, Non-metallic resistivity profiles (at B=0) in samples K, C,M and L. The Hall density nH in Sample C falls by a factor of 50 between 300 K and 2 K (inset). c, In a longitudinal field B‖ J (see inset), ρxx decreases withincreasing B below 170 K. d, ρxx(B) shows a negative LMR at 6 K with a bell-shaped profile, which remains resolvable to>150 K. The evidence stronglysupports identification of the chiral anomaly as the origin of the LMR.

node (Sample K with LMR= 11). This variation is plotted inSupplementary Fig. 16 (also Supplementary Figs 14 and 15). Thesetests confirm that the LMR is associated with the chiral anomaly(they also establish that the LMR is distinct from the isotropic MRseen in gapped half-Heuslers23).

Lastly, we show that the LMR is an intrinsic effect rather than aspurious consequence of inhomogeneous J (x) caused by disorder(Methods). In Sample G, we retained the large current contacts(A and B) and added small voltage contacts (1,. . .,10) (Fig. 3d, insetand Supplementary Fig. 18). As plotted in Fig. 3d, the potentialdrops Vi,j (where i, j run over the eight nearest-neighbour pairs)all show closely similar LMR profiles. Furthermore, from extensivesimulations (Supplementary Section 7 and Supplementary Fig. 21),we find that the mobility µ estimated from the Hall angle θH is fartoo small for ‘current jetting’ to be the origin of the LMR (fromSupplementary Fig. 12a, µ=1,500 cm2 V−1 s−1 at 6 K).

GdBiPt provides a platform to explore the thermoelectricresponse of Weyl fermions. Figure 3a shows the T dependence ofthe Seebeck coefficient Sxx measured in sample G with B ‖ x (inall samples, both the thermal gradient −∇T and JQ are ‖ x). Inlongitudinal B, Sxx is strongly suppressed starting at 150K, highabove TN. The suppression of Sxx is highly directional, rapidlydiminishing as B is tilted away from x (Fig. 3b).

We first focus on the zero-B curve in Fig. 3a. As T decreasesfrom 300K, the (hole-like) thermopower S rises monotonically,accelerating below 40K to attain a large peak value of 215 µVK−1at 18K. Below 15K, S decreases linearly with T , consistent witha metal with an unusually small EF. Using the Mott relationS= (π2/3)(k2B/e)(βT/EF), we infer EF = 3.1meV (where kB isBoltzmann’s constant, e the electron charge and β∼ 1.5 an expo-nent). Combining this with the weak-B Hall coefficient RH, we cal-culate an effectivemassm∗/m0'1.8. In large field (>6 T), the onsetof SdH oscillations provides an independent determination of thecyclotron mass mcyc from the damping of the SdH amplitudes withT (Fig. 3c,d and Supplementary Fig. 12b). Interestingly, we find thatmcyc/m0=0.23±0.03 (a factor of eight smaller thanm∗). This largediscrepancy—unexpected in a conventional metal—is strong evi-dence that themoderately heavymass in zero B changes to the smallmass of Dirac states, as predicted in the inducedWeyl node picture.

As noted in Fig. 3b, the field suppression of Sxx is strongest whenB is aligned with−∇T (θ→0). The half-widths of the bell-shapedprofiles δB broaden rapidly as T is raised above 80K. Also, the sup-pression of Sxx goes away in a transverse field. These features implythat the strong suppression of Sxx arises from the chiral anomaly.

At low B, however, the process of node formation (which leads toa sharp peak in the density of states N (E) near EF; Supplementary

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NATUREMATERIALS DOI: 10.1038/NMAT4684 LETTERS

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Figure 2 | Dependence of the LMR in GdBiPt on field-tilt angles ( , ). a, Curves of ρxx versus B in Sample G at 6.46 K, for selected values of θ (anglebetween B and J in the polar plane x–z; see inset). As θ deviates from 0, the bell-shaped profile broadens. For θ∼90, the MR is positive. The low-fieldoscillations reflect changes caused by Zeeman shift of the bands. b, Angular width of the conductivity σxx versus azimuthal angle φ for selected B inSample F at 2.5 K (φ is defined in inset). The ‘axial plume’ is similar to that in Na3Bi, but broader in angular width. c, Plot of ρxx versus azimuthal angle φillustrating field steering by comparing the angular plots in samples C (with x‖[111]; red) and F (‖[110]; blue) at two values of B (2 and 9 T). In both cases,the LMR is seen only when B aligns with J‖x. d, Plot of ρxx versus azimuthal angle φ, showing that the magnitude of the negative LMR is large in Sample K(EF closest to the node), but steadily decreases as |EF|moves away from the node (samples F, M and L), consistent with the chiral anomaly.

Fig. 5) differs between the two cases B ‖[110] versus [111]. Thedifferences may underlie the strong anisotropy observed in Sxx . InFig. 4a,b we compare the field profiles of Sxx versus B in Sample G(B ‖ [110]) and Sample E (B ‖[111]). In Sample G, Sxx decreasesmonotonically with B at all T (apart from a tiny dimple in weak B).By contrast, in Sample E, Sxx initially increases in weak B, attainingpeaks at±3.5 T before decreasing rapidly in large B.

This striking anisotropy may reflect the differences in thethermoelectric response function αxx(B) which directly relates thegradient to J via Ji = αij(−∂jT ) (αxx = Sxx(B)/ρxx(B)). For Weylfermions, recent Boltzmann equation calculations yield for αxx (atone node)24–26

αxx= eτ∫ d3k(2π)3

∂f0∂Ek

(Ek−EF)

T[vk+ eBvk·Ω]2

(1+ eB·Ω)(1)

where f0 is the Fermi–Dirac distribution, Ek the energy, and vk is theband velocity in state k with τ the relaxation time.

In Fig. 4c,d, we compare the profiles of αxx versus B inSample G measured with B ‖ −∇T ‖[110] against Sample E(B‖−∇T ‖ [111]). Below 30K, the profiles in both samples arequalitatively similar: αxx increases as B2 to attain a broad maximumat the peak field Bp, which separates two distinct field regimes.

In low fields (B<Bp), the changes toN (E) lead to an increasingαxx in both geometries. The difference in Bp (6 versus 8 T) sufficesto produce the V-shaped profile in Sxx(B) of Sample E, but not inSample G. The increase in αxx in the range 0<B<Bp is a signatureof the node creation process that is not understood.

In the regime above Bp the Weyl nodes are fully formed withwell-resolved Landau levels (LLs), as indicated by the quantumoscillations (curves below 11K in Fig. 4a). The dominant featureis the steady decrease in αxx with increasing B, which drives Sxxtowards zero at large B (in Sample G, we enter the n= 0 LLabove 25 T). A characteristic of Weyl states in the quantum limit isthe one-dimensional dispersion along the B axis, which implies adensity of states N ∼ eB/v that is independent of E. Consequently,αxx ∼ ∂N /∂E vanishes (by contrast Sxx increases linearly with Bin the lowest LL for a massive Dirac system27). We interpret thedownward trend in both αxx and Sxx above Bp as consistent withcharge pumping between the Weyl nodes associated with the chiralanomaly. The curves of αxx can provide sharp tests of equation (1)in the intermediate field regime.

We have observed in GdBiPt a large, negative longitudinal MRwhen EF is near zero. By varying both the directions of B and E , weconfirm that the enhanced conductance is confined to a plume cen-tred at B‖E . Moreover, it is steerable by the direction of B. Finally,

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LETTERS NATUREMATERIALS DOI: 10.1038/NMAT4684

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Figure 3 | Variation of the thermopower with B and quantum oscillations in GdBiPt. a, Seebeck coecient Sxx(B) versus T in Sample G at selected B withB‖x‖ JQ (the heat current density). The field suppression of Sxx, largest at 6 K, is resolvable up to 150 K. In zero B, the T-linear dependence of S≡Sxx(0)below 15 K yields EF=3.1 meV. b, Change in the field profile of Sxx at 6.45 K when B is tilted by angle θ with respect to x in the x–z plane (see inset). In thecurves at θ=−2 and−3, quantum oscillations appear above 6 T. c, Oscillatory part of ρxx (ρosc; expressed as a percentage with ρbg a smoothbackground curve) in Sample C, showing the damping of the SdH oscillations with increasing T (ref. 28). d, Resistances R(B) inferred from the potentialdrops Vi,j across eight pairs of nearest-neighbour pairs (inset). The close similarity of the LMR profiles is evidence that the LMR is intrinsic, rather thancaused by distortions of current paths in a disordered crystal.

we show that the LMR is most prominent when |EF| is closest to thenode. The three tests together with the measurements ofm∗ presenta strong case for the chiral anomaly in Weyl nodes. The observedthermoelectric response function αxx shows a decrease in large B,consistent with N of a chiral n= 0 LL. The anisotropy of the ther-moelectric response is consistent with the anisotropic nature of thenode creation, although a full accounting of the results awaits fur-ther analysis.More broadly, we have shown that field-induced cross-ing of degenerate bands can result inWeyl nodes and the associatedchiral anomaly. The results imply that zero-gap semiconductorswith large spin–orbit interaction (for example, half-Heuslers, greytin and HgCdTe) are prime candidates for exploring Weyl physics.

MethodsMethods and any associated references are available in the onlineversion of the paper.

Received 4 December 2015; accepted 27 May 2016;published online 27 June 2016

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Figure 4 | Anisotropy of the thermoelectric response in GdBiPt in a longitudinal field. a, Curves of Sxx versus B in Sample G (with B‖ JQ ‖[110]). Thecurves decrease monotonically with B at all T (apart from a tiny dimple in zero B below 20 K). Quantum oscillations are resolvable in the curve at 6.74 K.b, Curves of Sxx versus B in Sample E (with B‖[111]) in contrast are non-monotonic (we plot Sxx/T to minimize overlap). Below 25 K, the low-field region inSample E is dominated by a V-shaped profile. c,d, Curves of the thermoelectric response function αxx=Sxx/ρxx in samples G and E, which unlike Sxx(B) arebroadly similar below 30 K between the two geometries. In Sample G, αxx rises to a broad maximum at Bp'6 T (arrow) before falling steeply, withquantum oscillations appearing for B>Bp (c). In Sample E, αxx shows a similar profile except that Bp'8 T (d).

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AcknowledgementsWe are indebted to J. Cano, B. Bradlyn and J. Xiong for discussions, and S. Koohpayeh,J. Krizan and W. Xie for technical assistance. The research is supported by a MURI awardfor topological insulators (AROW911NF-12-1-0461) and by the Army Research Office(AROW911NF-11-1-0379). The growth and characterization of crystals were performedby S.K. and R.J.C., with support from the National Science Foundation (NSF MRSECgrant DMR 1420541). C.A.B. was an REU participant funded by the NSF-MRSEC grantDMR 1420541. N.P.O. acknowledges the support of the Gordon and Betty MooreFoundations EPiQS Initiative through Grant GBMF4539. B.A.B. acknowledges supportby NSF CAREER DMR-095242, ONR-N00014-11-1-0635, NSF grant DMR 1420541,Packard Foundation and a Keck grant.

Author contributionsM.H. performed most of the measurements with early assistance from C.A.B. Thecrystals were grown and characterized by S.K. and R.J.C. Analyses of the results weredone by M.H., Z.W., Q.G., B.A.B. and N.P.O. Simulations of current distributions wereperformed by S.L. The manuscript was written by M.H. and N.P.O., with contributionsfrom all authors.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints.Correspondence and requests for materials should be addressed to M.H. or N.P.O.

Competing financial interestsThe authors declare no competing financial interests.

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LETTERS NATUREMATERIALS DOI: 10.1038/NMAT4684

MethodsElectronic structure calculations were performed in the framework of densityfunctional theory (DFT) using the WIEN2k code29. (The program packageWIEN2k allows users to perform electronic structure calculations of solids usingdensity functional theory.) We used a full-potential linearized augmented planewave and local orbitals basis with the PBE parametrization of the generalizedgradient approximation30. The plane-wave cutoff parameter RMTKmax was set to 7.The modified Becke Johnson (mBJ) functional31 was used. Spin orbit coupling(SOC) was included in the calculation as a perturbative step. A large exchangeparameter Ueff=7eV was applied to the Gd 4f states (with open core treatment ofthe 4f electrons), in effect removing these states far away from the Fermi energy EF.

Single crystals of pure and Au-doped GdPtBi were grown using Bi-self flux.Stoichiometric mixtures of the elements were placed in an alumina crucible andsealed inside a quartz tube under vacuum. The ampoules were heated to 1,130 Cand kept for 12 h at constant temperature to get a homogeneous solution. Thesamples were cooled at a rate of 1.5 Ch−1, to 940 C and then centrifuged toremove Bi-flux. Single crystals of 2mm in size were successfully obtained.

We have used the Hall effect to characterize our crystals, and to obtainestimates of the carrier density, assuming conduction from a single band at lowtemperature (∼4K). The Hall resistivity at 4 K was fitted in the low-field limit to aline to extract the Hall density nH=1/(eRH), where RH is the Hall coefficient. Athigh temperatures, thermally activated carriers suppress the amplitude of RH

(enhance nH), but the saturated value of nH at low T may be used to estimate theintrinsic carrier concentration (results are shown in Supplementary Information).The onset of the low-temperature regime where nH saturates is expected to scalewith the Fermi energy EF in a zero-gap material.

Altogether, we investigated single crystals from three batches of pristine GdPtBi(batches 1–3), all of which exhibited intrinsic p-type behaviour in the Hall effectand in the thermopower. The highest-quality crystals (estimated by the amplitudeof quantum oscillations, and the enhancement of the thermal conductivity κxx atlow temperature) have the lowest carrier concentrations. We also investigated threebatches of crystals doped slightly with Au (0.5, 5, 10% for batches 4, 5, 6respectively) to substitute for Pt. We found that only a small fraction of the Au instarting materials is incorporated in the final crystals, as determined bymeasurements of the Hall density. For example, the low-temperature Hall densitiesmeasured for the samples in batch 5 are (1.4–3.5)×1018 1 cm−3. But if 5% of the Pthad been replaced by Au, we would expect densities of around−6.7×1020 cm−3,using the experimental lattice parameter a=6.68Å(ref. 32). Batch 4 containedcrystals which had very low Hall densities (close to compensation).

The so-called ‘18 electron rule’ for half-Heusler materials33 implies that heavydoping with atoms of a different valence count is prohibited. In practice, thisimplies a solubility limit for Au doping in our case. We have found that crystalsfrom batch 5 (5% Au for Pt in the starting boule) have Hall densities similar tobatch 6 (10% Au), that is,−nH=1.5×1018 to 4×1018 cm−3.

The relatively large sizes of the single crystals used for thermopowerexperiments (∼2mm on a side) allowed us to cut and orient samples accuratelyusing the Laue X-ray set-up at Johns Hopkins University, Baltimore, Maryland(with the assistance of S. M. Koohpayeh and J. W. Krizan). In particular, samples C,D, E, F and G were oriented in this way.

For all other crystals, we used the edges of hexagonal facets of the single crystals([111] direction of the cubic structure) to determine the orientation; the angularerror of cut planes may be larger for such samples, and we estimate it at δφ∼10.These samples may be as small as∼0.5mm. After cutting and polishing, electricalcontacts to the platelets (thickness t∼0.1mm) were made by silver paint and thingold wires. In some cases, the contacts cover a considerable fraction of the surfacearea. This implies limitations of determining the correct sample geometry (usingan optical microscope) for calculating the Hall resistivity ρyx , and by extension theHall density nH. An error of up to δnH/nH∼30% may be incurred for thesmallest crystals.

To test for inhomogeneous current distribution, we remounted Sample G,replacing the previous voltage contacts with ten new, small voltage contact pads.The current contacts A and B are sufficiently large to cover the shorter edges of thecrystal. At T=2K, we measured simultaneously the potential difference Vi−j

between the eight pairs of nearest-neighbour contacts (V1–2, V2–3, · · · ,V9–10) as B(applied ‖ J ‖x) is varied.

The eight curves for the relative change in Vi−j are nearly identical below 3T,displaying only slight, non-systematic deviations above 5 T (Fig. 3d andSupplementary Information). To us, the striking agreement across the eightcontacts provides strong evidence for the uniformity of J (r) throughout the crystal.From the eight curves, we infer that the current density is uniform throughout thecrystal in the LMR experiment. This implies that the observed negative LMR is anintrinsic electronic effect rather than arising from strong distortions of thecurrent paths.

A final concern is ‘current jetting’, which can lead to field-induced changes inthe longitudinal MR in very high mobility semimetals and metals. To see if currentjetting can be a plausible origin of the negative LMR observed in GdPtBi, we haveperformed extensive simulations for our sample geometry and mobility valuesµ=1,500–2,000 cm2 V−1 s−1. From the equations∇ · J=0 and Ji=σijEj, whereEj=−∂jψ , the potential function ψ(x ,y) satisfies the anisotropic 2D Laplaceequation (see Supplementary Information). The simulations show that, below 10T,current jetting has a negligible effect (a few %) in a sample withµ=2,000 cm2 V−1 s−1. This is far too small to account for the large LMR inGdPtBi. Note that from Fig. 3d, ρxx has decreased by a factor of two at the low fieldof 2 T. We would need B>50 T to achieve the same suppression if current jettingwere the origin (see simulations in Supplementary Information).

References29. Blaha, P., Schwarz, K., Madsen, G. K. H., Kvasnicka, D. & Luitz, J. WIEN2k

package; http://www.wien2k.at30. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation

made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).31. Becke, A. D. & Johnson, E. R. A simple effective potential for exchange.

J. Chem. Phys. 124, 221101 (2006).32. Dwight, E. in Proc. 11th Rare Earth Research Conf. Vol. 2 (eds Haschke, J. M. &

Eick, H. A.) 642 (US Atomic Energy Commission, 1974).33. Graf, T., Felser, C. & Parkin, S. S. P. Simple rules for the understanding of

Heusler compounds. Prog. Solid State Chem. 39, 1–50 (2011).

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