Experimental Tests of the Chiral Anomaly Magnetoresistance in the Dirac-Weyl Semimetals Na3bi and Gdptbi

PHYSICAL REVIEW X 8, 031002 (2018)

Experimental Tests of the Chiral Anomaly Magnetoresistance in the Dirac-Weyl Semimetals Na3Bi and GdPtBi

Sihang Liang,1 Jingjing Lin,1 Satya Kushwaha,2 Jie Xing,3 Ni Ni,3 R. J. Cava,2 and N. P. Ong1,* 1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA 2Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA 3Department of Physics and Astronomy and California NanoSystems Institute, University of California, Los Angeles, California 90095, USA

(Received 27 November 2017; revised manuscript received 7 June 2018; published 3 July 2018)

In the Dirac-Weyl semimetal, the chiral anomaly appears as an “axial” current arising from charge pumping between the lowest (chiral) Landau levels of the Weyl nodes, when an electric field is applied parallel to a magnetic field B. Evidence for the chiral anomaly was obtained from the longitudinal magnetoresistance (LMR) in Na3Bi and GdPtBi. However, current-jetting effects (focusing of the current density J) have raised general concerns about LMR experiments. Here, we implement a litmus test that allows the intrinsic LMR in Na3Bi and GdPtBi to be sharply distinguished from pure current-jetting effects (in pure Bi). Current jetting enhances J along the mid-line (spine) of the sample while decreasing it at the edge. We measure the distortion by comparing the local voltage drop at the spine (expressed as the R R R B R resistance spine) with that at the edge ( edge). In Bi, spine sharply increases with , but edge decreases R R (jetting effects are dominant). However, in Na3Bi and GdPtBi, both spine and edge decrease (jetting effects are subdominant). A numerical simulation allows the jetting distortions to be removed entirely. We find that the intrinsic longitudinal resistivity ρxxðBÞ in Na3Bi decreases by a factor of 10.9 between B ¼ 0 and 10 T. A second litmus test is obtained from the parametric plot of the planar angular magnetoresistance. These results considerably strengthen the evidence for the intrinsic nature of the chiral-anomaly-induced LMR. We briefly discuss how the squeeze test may be extended to test ZrTe5.

DOI: 10.1103/PhysRevX.8.031002 Subject Areas: Condensed Matter Physics, Topological Insulators

I. INTRODUCTION feature protected 3D Dirac and Weyl states in the bulk [2,3]. In the past two decades, research on the Dirac states in A crucial step in the search for 3D Dirac states was the graphene and topological insulators has uncovered many realization that inclusion of point group symmetry [with novel properties arising from their linear Dirac dispersion. time-reversal (TR) symmetry and inversion symmetry] In these materials, the Dirac states are confined to the two- allows Dirac nodes to be protected anywhere along dimensional (2D) plane. Interest in three-dimensional (3D) symmetry axes, instead of being pinned to TR-invariant Dirac states may be traced to the even earlier prediction of Nielsen and Ninomiya (1983) [1] that the chiral anomaly momenta on the Brillouin zone surface [4,5]. Relaxation may be observable in crystals (the space-time dimension of this constraint led to the discovery of Na3Bi [6] and C 3 þ 1D needs to be even). The anomaly, which appears as Cd3As2, in which the two Dirac nodes are protected by 3 C B a current in a longitudinal magnetic field B, arises from the and 4 symmetry, respectively. In the absence of , each 4 4 breaking of a fundamental, classical symmetry of massless Dirac node is described by a × Hamiltonian that can be 2 2 fermions—the chiral symmetry. Recent progress in topo- block diagonalized into two × Weyl blocks with χ ¼1 logical quantum matter has led to several systems that opposite chiralities ( ). The absence of mixing between the two Weyl fermions expresses the existence of chiral symmetry. In a strong B, the Weyl states are *Corresponding author. quantized into Landau levels. As shown in Fig. 1(a),a [email protected] distinguishing feature is that the lowest Landau level (LLL) in each Weyl node is chiral, with a velocity v strictly kB (or Published by the American Physical Society under the terms of −B as dictated by χ) [3]. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to As a result, electrons occupying the LLL segregate into the author(s) and the published article’s title, journal citation, two massless groups—left and right movers, with popula- and DOI. tions NL and NR, respectively. Independent conservation

2160-3308=18=8(3)=031002(13) 031002-1 Published by the American Physical Society SIHANG LIANG et al. PHYS. REV. X 8, 031002 (2018)

(a) (b) (50–100 times weaker than in Na3Bi) and their fragility with respect to the placement of contacts, together with the high mobilities of the Weyl semimetals (150 000 to 200 000 cm2=Vs), have raised concerns about current- jetting artifacts [20,21]. As a consequence, there is consid- (c) erable confusion and uncertainties about LMR experiments, in general, and the LMR reported in the Weyl semimetals, in particular. The concerns seem to have spread to Na3Bi and GdPtBi as well, notwithstanding their much larger LMR signal. FIG. 1. (a) The Landau spectrum of Weyl fermions. In a field Bkxˆ, the lowest Landau levels are chiral with velocity v either kB There is good reason for the uncertainties. Among the or −B. Application of EkB transfers charge between them, which resistivity matrix elements, measurements of the longi- ρ Bkxˆ increases the left-moving population NL at the expense of the tudinal resistivity xx (for ) are the most vulnerable to right-moving population NR (the bold blue and red curves inhomogeneous flow caused by current jetting. Even when indicate occupations of the LLs). Panel (b) shows a pair of the LMR signal in a sample is mostly intrinsic, the chiral voltage contacts (blue dots) placed on the line joining the current anomaly produces an intrinsic conductivity anisotropy u, contacts (white circles). A second pair (yellow) is placed along an which unavoidably produces inhomogeneous current dis- J edge. Panel (c) is a schematic drawing of the intensity map of x tributions that distort the observed LMR profile. Given the (with dark regions being the most intense) when current-jetting prominent role of LMR in chiral-anomaly investigations, it J y x effects are pronounced. The profile of x vs (with at the is highly desirable to understand these effects at a quanti- dashed line) is sketched on the right. tative level and to develop a procedure that removes the distortions. 5 of NL and NR implies that the chiral charge density ρ ¼ A major difference between the large LMR systems ðNL − NRÞ=V is conserved, just like the total charge Na3Bi and GdPtBi, on the one hand, and the Weyl semi- ρ ¼ðN þ N Þ=V V density tot L R ( is the sample volume). metals, on the other, is their carrier densities. Because the 17 −3 However, application of an electric field EkB breaks the density is low in both Na3Bi (1 × 10 cm ) and in 17 −3 chiral symmetry by inducing mixing between the left- and GdPtBi (1.5 × 10 cm ), the field BQ required to force right-moving branches [Fig. 1(a)] (for a pedagogical dis- the chemical potential ζ into the LLL is only 5–6T.By cussion, see Ref. [7]). A consequence is that conservation contrast, BQ is 7–40 T in the Weyl semimetals. As shown in of ρ5 is violated by a quantity A called the anomaly term, Fig. 1(a), the physics underlying the anomaly involves the 5 5 5 viz. ∇ · J þ ∂tρ ¼ eA, where J is the axial current occupation of chiral, massless states. Occupation of the density. [From the density of states in the LLL and the higher LLs (when B

031002-2 EXPERIMENTAL TESTS OF THE CHIRAL ANOMALY … PHYS. REV. X 8, 031002 (2018) classical effects is described in Sec. V. To look beyond we found it expedient to remove a thin layer of oxide by Na3Bi and GdPtBi, we discuss how the tests can be lightly sanding with fine emery paper (within the glove- extended using focused ion beam techniques to test box). The sample was then placed inside a capsule made of ZrTe5, which grows as a narrow ribbon. The Weyl G10 epoxy. After sealing the lid with STYCAST epoxy, the semimetals, e.g., TaAs, require availability of ultrathin capsule was transferred to the cryostat. films. The planar angular magnetoresistance (Sec. VI) We contrast two cases. In case 1, the anisotropy u ¼ provides a second litmus test—one that is visually direct σxx=σyy increases in a longitudinal field B because the when displayed as a parametric plot (Sec. VII). In Sec. VIII, transverse conductivity σyy decreases steeply (as a result of we summarize our results. cyclotronic motion), while σxx is unchanged in B. With Bkxˆ, the two-band model gives the resistivity matrix II. THE SQUEEZE TEST ½σ þ σ −1 0 Current jetting refers to the focusing of the current ρ˜ðBÞ¼ e h ð Þ 0 ½σe þ σh −1 1 density JðrÞ into a narrow beam kB arising from the field- Δe Δh induced anisotropy u of the conductivity (the drift of carriers transverse to B is suppressed relative to the (we suppress the z component for simplicity). The zero-B longitudinal drift). To maximize the gradient of J along conductivities of the electron and hole pockets are given the y axis, we select platelike samples with L,w ≫ t, where by σe ¼ neμe and σh ¼ peμh, with n and p the electron w, L, and t are the width, length, and thickness, respectively and hole densities, respectively, and e the elemental charge. [Fig. 1(b)]. The x and y axes are aligned with the edges and Note that μe and μh are the mobilities in the electron Δ ¼ð1 þ μ2B2Þ Bkxˆ. As sketched in Fig. 1(c), the profile of Jx vs y is and hole pockets, respectively, and e e and 2 2 strongly peaked at the center of the sample and suppressed Δh ¼ð1 þ μhB Þ. With Bkxˆ, the off-diagonal elements towards the edges. In the squeeze test, we measure the vanish. In case 1, we assume that σe and σh remain voltage difference across a pair of contacts (blue dots) constant. Hence, the observed resistivity ρxx is unchanged along the line joining the current contacts (which we call in B. However, the transverse conductivity σyy decreases the spine), as well as that across a pair on the edge (yellow (as 1=B2 in high B). The anisotropy arises solely from the dots). To accentuate current-jetting effects, we keep the suppression of the conduction transverse to B by the current contact diameters dc small (dc ≪ w) and place them cyclotron motion of both species of carriers. on the top face of the sample wherever possible. (The Case 2 is the chiral anomaly regime in the Dirac squeeze test cannot be applied to needlelike crystals.) semimetal. Charge pumping between Landau levels (LLs) of opposite chirality leads to an axial current that A. Sample preparation causes σxx to increase with B. Simultaneously, the 1D nature of the LL dispersion suppresses the transverse We provide details on the preparation of Na3Bi, which σ u is by far the most difficult of the three materials to work conductivity yy. Hence, the increase in derives equally σ σ with. Na3Bi crystallizes to form hexagonal platelets with from the opposite trends in xx and yy. the broad face normal to (001). The crystals investigated Denoting field-induced changes by Δ, we have here were grown under the same conditions as the samples used in Xiong et al. [16]; they have carrier densities Case 1∶ Δu>0 ⇔ Δσxx ∼ 0; Δσyy < 0; ð2Þ 17 −3 1 × 10 cm and BQ in the range 5–6 T. These crystals should be distinguished from an earlier batch [22] that have Case 2∶ Δu>0 ⇔ Δσxx > 0; Δσyy < 0: ð3Þ much higher carrier densities (3–6 × 1019 cm−3) for which V V we estimate BQ ∼ 100 T. No evidence for negative LMR In the test, the voltage drops spine and edge are given by was obtained in the highly doped crystals [22]. Z l Because of the high Na content, crystals exposed to V ðBÞ¼−ρ ðBÞ J ðx; w=2; BÞdx ≡ ρ L ; ð Þ edge xx x xx e 4 ambient air undergo complete oxidation in about 5 s. 0 The stainless growth tubes containing the crystals were Z l opened in an argon glovebox equipped with a stereoscopic V ðBÞ¼−ρ ðBÞ J ðx; 0; BÞdx ≡ ρ L ; ð Þ spine xx x xx s 5 microscope, and all sample preparation and mounting were 0 performed within the glovebox. The crystals have the ductility of a soft metal. Using a sharp razor, we cleaved where LeðBÞ and LsðBÞ are the line integrals of Jx along the bulk crystal into platelets 1 × 1 mm2 on a side with the edge and spine, respectively (l is the spacing between thickness 100 μm. Current and voltage contacts were voltage contacts). The intrinsic B dependence [expressed in painted on using silver paint (Dupont 4922N). A major ρxxðBÞ] has been cleanly separated from the extrinsic B difficulty was achieving low-resistance contacts on the top dependence of LeðBÞ and LaðBÞ, which arises from current R face (for measuring spine). After much experimentation, focusing effects.

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The area under the curve of Jx vs y is conserved, i.e., III. RESULTS OF SQUEEZE TEST Z w=2 The results of applying the squeeze test on the three Jxðx; y; BÞdy ¼ I; ð6Þ systems are summarized in Fig. 2. In panels (a) and (b), we w=2 V V show the voltage drops edge and spine measured in pure bismuth (sample B1). The signals are expressed as the with I the applied current. At B ¼ 0, we may take J to be effective resistances uniform with the magnitude J0 ¼ I=ðwtÞ. The line integral L ¼ J l B reduces to 0 0 . In finite , focusing of the current R ¼ ρ ðBÞ:L ðBÞ=I; ð Þ beam implies that the current density is maximum along edge xx e 9 the spine and minimum at the edge, i.e., Jxðx; 0; BÞ > R ¼ ρ ðBÞ:L ðBÞ=I: ð Þ Jxðx; w=2; BÞ. Moreover, Eq. (6) implies that Jxðx; 0; BÞ > spine xx s 10 J0 >Jxðx; w=2; BÞ. Hence, the line integrals satisfy the R inequalities The steep decrease in edge [panel (a)] illustrates how an apparent but spurious LMR can easily appear when the LsðBÞ > L0 > LeðBÞ: ð Þ 6 2 7 mobility is very high (in Bi, μe exceeds 10 cm =Vs at R R 4 K). Comparison of edge and spine measured simulta- If both σe and σh are B independent, as in case 1, we have neously shows that they have opposite trends vs B.Asσe from Eqs. (4), (5), and (8), and σh are obviously B independent in Bi, ρxx is also B V ðBÞ >V >V ðBÞ; ð Þ independent. By Eqs. (9) and (10), the changes arise solely spine 0 edge 8 from Le and Ls. Hence, Figs. 2(a) and 2(b) verify experimentally that V and V display the predicted where V0 is the voltage drop across both pairs at B ¼ 0. edge spine V B V large variations of opposite signs when current jetting is the Clearly, spine increases monotonically with , while edge decreases. Physically, focusing the current density along sole mechanism present (see simulations in Sec. IV). R the spine increases the local E field there. Current con- Next, we consider Na3Bi. In this sample (N1), edge below 20 K decreases by a factor of 50 between B ¼ 0 and servation then requires Jx to be proportionately suppressed V 10 T [Fig. 2(c)]. This is an order of magnitude larger than along the edges. Measuring edge alone yields a negative LMR that is spurious. observed in Ref. [16]. The increase arises from the enhanced current focusing effect in the present contact In case 2, however, ρxx decreases intrinsically with B placement utilizing small current contacts attached to the because of the chiral anomaly, while Ls increases. broad face of the crystal, as well as a larger u. In spite of Competition between the two trends is explicitly seen in R V B the enhanced jetting, spine shows a pronounced decrease in the profile of spine vs . As shown below, in Na3Bi and contrast to the case for Bi. The intrinsic decrease in ρxx GdPtBi, the intrinsic decrease in ρxx dominates, so both V V B dominates the increase in Ls throughout [see Eq. (10)]. spine and edge decrease with . We remark that, from V V Hence, we conclude that there exists a large intrinsic, Eq. (8), spine always lies above edge. Moreover, when the R negative LMR that forces spine to decrease, despite the rate of increase in Ls begins to exceed (in the absolute focusing of JðrÞ along the ridge. Further evidence for the sense) the rate of decrease in ρxx at sufficiently large B, the V ðBÞ competing scenario comes from the weak minimum at 10 T curve of spine can display a broad minimum above V in the curves below 40 K in Fig. 2(d). As anticipated above, which spine increases. B ρ V V in large , xx approaches a constant because of the Hence, if both spine and edge are observed to decrease L B saturation chiral-anomaly term. However, s continues with increasing , the squeeze test provides positive to increase because the transverse conductance worsens. confirmation that the observed LMR is intrinsic. Their R ρ Consequently, spine goes through a minimum before field profiles bracket the intrinsic behavior of xx. R R increasing. This is seen in spine but absent in edge. Conversely, if intrinsic LMR is absent (i.e., σxx is A feature that we currently do not understand is the large unchanged), V and V display opposite trends (the spine edge V-shaped cusp in weak B. At 100 K, the cusp is prominent marginal case when the intrinsic LMR is weak is discussed V V in spine but absent in edge. in Sec. IV). R In Figs. 2(e) and 2(f), we show the field profiles of edge We remark that the current-jetting effects cannot be R eliminated by using very small samples (e.g., using nano- and spine measured in GdPtBi (sample G1). Again, as in ρ lithography). As long as we remain in the classical transport Na3Bi, the anomaly-induced decrease in xx dominates the L R regime, the equations determining the functional form of increase in s, and spine is observed to decrease in B R Jðx; yÞ in strong B are scale invariant. Because intrinsic increasing . The relative decrease in edge is larger than R R length scales (e.g., the magnetic length lB or the skin depth that in spine. Further, spine below 10 K shows the onset of R δs) are absent in classical dc transport, the same flow a broad minimum above 10 T [Fig. 2(f)], whereas edge pattern is obtained on either mm or micron-length scales. continues to fall.

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14 100 Bi (a) 120 Na Bi 2, 4, 10, 20 K (c) GdPtBi 80 (e) 12 3 Edge Edge 40 80 100 K 100 Edge 100 10 60 ) ) 8 80 60 ) Ω Ω

60 Ω (m (m 100 K 60 6 40 60 (m edge edge 40 edge

R 40 R 4 20 R 40 20 2 2K 20 10 20 2K 0 0 0 -8 -6 -4 -2 0 2 4 6 8 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 B (T) B (T) B (T)

160 100 Bi (b) Na Bi (d) 100 (f) 200 140 3 80 Spine Spine 20 80 2K 120 60 150 ) 100 K 40

100 ) 40 Ω 60 Ω (m Ω) 20 60 80 (m (m 100 spine 60 10 spine

R 40 2K spine 60 40 R

R 100 40 50 2, 4, 10, 20 K 20 GdPtBi 20 Spine 0 0 0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 B (T) B (T) B (T)

FIG. 2. Comparison of squeeze-test results in pure bismuth (case 1) and in Na3Bi and GdPtBi (case 2). Panels (a) and (b) display the V V R R R voltage drops edge and spine (expressed as edge and spine, respectively) measured in Bi (sample B1). In panel (a), edge displays a B T R steep decrease with increasing that steepens as decreases from 100 K to 2 K. By contrast, spine increases steeply. The opposite trends in (a) and (b) imply that both arise from pure current jetting, strictly reflecting changes in Le and Ls, respectively [Eqs. (9) and R B ρ ðBÞ (10)]. By contrast, in Na3Bi (sample N1), both edge decrease with [panels (c) and (d), respectively]. This implies that xx decreases uniformly throughout the sample. Nonetheless, current focusing effects (expressed by Le and Ls) are visible. Below 20 K, Le R L R R exaggerates the decrease in edge, while s counters some of the intrinsic decrease in spine. A telling feature is the weak upturn in spine above 10 T. When the intrinsic LMR is saturated, Ls produces a weak increase in Rspine. In GdPtBi [sample G1, panels (e) and (f)], both Redge and Rspine also decrease with increasing B. Below 10 K, the decrease in Redge is much larger than that in Rspine. The latter also attains a broad minimum above 10 T (but not the former). These features confirm that Le amplifies the growth in the intrinsic LMR, while Ls partially counters it.

These features, in accordance with the discussion above, range of u. For simplicity, the simulation is performed for a are amenable to a quantitative analysis that yields the sample in the 2D limit by solving the anisotropic Laplace intrinsic field profiles of both ρxx and u (Sec. IV). equation

2 2 IV. THE INTRINSIC LMR PROFILE ½σxx∂x þ σyy∂yψðx; yÞ¼0 ð11Þ The factorization expressed in Eqs. (9) and (10) allows ψðx; yÞ u us to obtain the intrinsic field profile of ρxxðBÞ in the face of for the potential at selected values of . We used strong current-density inhomogeneity induced by jetting. the relaxation method on a triangular mesh with Dirichlet To start, we note that, once the boundaries are fixed, the boundary conditions at the current contacts [inset in functional form of the inhomogeneous current density JðrÞ Fig. 3(a)]. R0 depends only on the conductivity anisotropy u regardless Figure 3(a) displays the calculated curves of edge and R0 of its microscopic origin. Assuming a constant ρxx ¼ ρ0 spine in a 2D sample with the aspect ratio matched to (i.e., case 1), we first calculate, by numerical simulation, that in the experiment on Na3Bi. As expected, with ρxx set R0 ðuÞ R0 ðuÞ the effective resistances edge and spine over a broad to a constant, the two curves diverge. This reflects the

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1 20 (a) (b) .)

ne F(u) 15 0.1 spi (a.u R 0 0 spine R / spine 0 edge R

10 0 , R

edge 0.01 0 R 5 0 R edge

0 0.001 01020304050 0 1020304050 = σ σ = σ σ u xx / yy u xx / yy

FIG. 3. Procedure to obtain the intrinsic LMR curve Rintr vs B and the intrinsic anisotropy u ¼ ρyy=ρxx from measurements of Redge R R0 R0 u ρ ¼ ρ and spine. Panel (a) plots the calculated resistances edge and spine vs with xx 0 (case 1) [obtained by solving the anisotropic Laplace equation (11)] in a 2D sample (matched to x and y dimensions of the sample). The divergent trends reflect the enhancement of JðrÞ along the spine and its decrease at the edge. The inset shows the triangulation network generated during the simulation. The stubs on the left and right edges are current contacts. Panel (b) plots the calculated template function FðuÞ in the semilog scale [Eq. (12)]. GðBÞ FðuÞ R u Equating the measured ratio to , and using Eq. (13), we derive the intrinsic field profiles of intr and (see Fig. 4).

120 Na Bi (N1) (a) (b) 3 8 Na3Bi (N1) T=2K T=2K ) xx

Ω ρ / yy

(m 6 80 ρ intr = R u , Rspine 4 spine

R Rintr , 40 Redge

edge 2 Anisotropy R

0 0 02468101214 02468101214 B (T) B (T) 3.5 (c) (d) 80 GdPtBi (G1)

) 3.0 T=2K xx Ω ρ / (m

60 yy ρ 2.5 R

spine = edge u R , 2.0

intr 40 R

R intr , 1.5 spine

20 Anisotropy R GdPtBi (G1) Redge T=2K 1.0 0 02468101214 02468101214 B (T) B (T)

FIG. 4. The intrinsic field profiles of ρxxðBÞ and anisotropy uðBÞ derived by applying Eqs. (12)–(14). Panel (a) shows the curves of Rspine (black) and Redge (red) measured in Na3Bi at 2 K. The inferred intrinsic profile Rintr (blue curve) is sandwiched between the measured curves. In this sample (N1), Rintr decreases by a factor of 10.9 between B ¼ 0 and 10 T. We note that Rspine displays a weak minimum near 10 T, which results from the competing trends in ρxxðBÞ and LsðBÞ. Panel (b) displays the intrinsic uðBÞ in Na3Bi derived from Eq. (13). The corresponding curves for GdPtBi (sample G1 at 2 K) are shown in panels (c) and (d). Again, RintrðBÞ in panel (c) is sandwiched between the curves of RedgeðBÞ and RspineðBÞ. Panel (d) plots the intrinsic anisotropy uðBÞ. In GdPtBi, the Weyl nodes only appear when B is strong enough to force band crossing. In G1, this occurs at 3.4 T (u remains close to 1 below this field).

031002-6 EXPERIMENTAL TESTS OF THE CHIRAL ANOMALY … PHYS. REV. X 8, 031002 (2018) simultaneous enhancement of the E field along the spine to create the Weyl nodes. This occurs at about 3.4 T in G1. and its steep decrease at the edge caused by pure current Below this field, the system is isotropic (u close to 1). jetting. From the calculated resistances, we form the ratio V. QUANTUM VS CLASSICAL EFFECTS From the experimental viewpoint, it is helpful to view the FðuÞ¼R0 =R0 ¼ L =L : ð Þ edge spine e s 12 LMR experiment as a competition between the intrinsic anomaly-induced decrease in ρxx (a quantum effect) and The template curve FðuÞ, which depends only on u,is the distortions engendered by current jetting (classical plotted in semilog scale in Fig. 3(b). effect). To observe a large, negative LMR induced by R ðBÞ R ðBÞ Turning to the values of edge and spine measured the chiral anomaly, it is imperative to have the chemical in Na3Bi at the field B, we form the ratio GðBÞ¼ potential ζ enter the LLL. The field at which this occurs, R =R ¼ L =L G edge spine e s. Although is implicitly a function which we call BQ, sets the onset field for this quantum of u, it is experimentally determined as a function of B effect. By contrast, the distortions to J caused by current u B B (how varies with is not yet known). We remark that jetting onset at the field cyc, which is set by the inverse GðBÞ FðuÞ 1=μ B ¼ A=μ and represent the same physical quantity mobility . We write cyc , where the dimension- expressed as functions of different variables. To find u, less parameter A ¼ 5–10, based on the numerical simu- we equate GðBÞ to FðuÞ in the template curve. This process lations [Eqs. (12) and (13)]. B B ( Q cyc), the LLL is entered long after current-jetting effects squeezing factor is always subdominant. With the pro- become dominant. The classical effect effectively screens the cedure described, this subdominant distortion can be quantum behavior. Bulk crystals of Na3Bi and GdPtBi fall safely removed entirely. The corresponding profiles of ρxxðBÞ in the upper half, whereas TaAs and NbAs fall in the lower half. and uðBÞ in GdPtBi are shown in Figs. 4(c) and 4(d), One way to avoid current jetting in the latter is to lower BQ by respectively. Unlike the case in Na3Bi, a finite B is required gating ultrathin-film samples.

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3 30 As BQ approaches B [the diagonal boundary in Fig. 5(a)], 6 cyc (a) classical current jetting becomes increasingly problematical. 5 5 TaAs Rspine In Fig. 5(b), the schematic curve illustrates the trend of how 4 4

T=4K n 3 the quantum behavior can be swamped by the onset of 3 2 2 20 2 current jetting. Integer ) ) B ≫ B Ω Finally, if Q cyc, classical distortion effects onset 1 Ω

R (m long before the LLL is accessed. In this case, spine and 0 (m R 0.0 0.2 0.4 0.6 R B Redge -1 R edge display divergent trends vs . Even if an intrinsic 4 1/B (T ) LMR exists, we are unable to observe it in the face of the 1 10 R dominant (artifactual) change in Vxx caused by classical 3 spine current jetting. In the Weyl semimetals TaAs and NbAs, R 2 B ≃ 7 B ∼ 0 4 edge Q and 40 T, respectively, whereas cyc . T 0 0 (because of their high mobilities). This makes LMR an -4 -2 0 2 4 6 unreliable tool for establishing the chiral anomaly in the B (T) R Weyl semimetals. We illustrate the difficulties with edge R 30 and spine measured in TaAs [Fig. 6(a)] and in NbAs 0.5 (b) [Fig. 6(b)]. – In the initial reports, a weak LMR feature (5% 10% 0.4

overall decrease) was observed in TaAs and identified )

20 ) Ω with the chiral anomaly. Subsequently, several groups Ω (m 0.3

R (m found that both the magnitude and sign of the LMR feature edge edge spine are highly sensitive to voltage contact placement. We can, R in fact, amplify the negative LMR to nearly 100%. To apply 0.2 R the squeeze test, we have polished a crystal of TaAs to the 10 form of a thin square plate and mounted contacts in the NbAs 0.1 R configuration sketched in Fig. 1(b), with small current T=4K spine 80 μ R contacts (about m). As shown in Fig. 6(a), edge 0.0 0 at 4 K (thick blue curve) displays a steep decrease, falling -15 -10 -5 0 5 10 15 to a value approaching our limit of resolution at 7 T. B (T) R Simultaneously, however, spine (red) increases rapidly. The two profiles are categorically distinct from those in Na3Bi FIG. 6. The squeeze test applied to the Weyl semimetals TaAs R R and GdPtBi [Figs. 2(c)–2(f)] but very similar to the curves and NbAs. Panel (a) shows field profiles of edge and spine in B for Bi [Figs. 2(a) and 2(b)]. Moreover, the weak SdH TaAs (measured at 4 K) in an in-plane, longitudinal along the “spine” of the sample. The crystal is a thin plate of dimension oscillations fix the field BQ needed to access the LLL at 2.0 × 1.5 × 0.2 mm3 with spacing l between voltage contacts of 7.04 T [inset in (a)]. Since 1=μ ∼ 0.06 T, we infer that 1.5 mm, and current-contact diameters dc ∼ 80 μm. Between the classical current-jetting effect onsets long before the R 0.5 T and 6 T, edge (thick blue curve, left axis) is observed to quantum limit is accessed. Hence, TaAs is deep in the right- decrease by a factor of about 50 to values below our resolution. R bottom corner of the phase diagram in Fig. 5(a). The However, the spine resistance spine (red curve) increases by a R R current-jetting effects appear well before TaAs attains the factor of 7.8 (the right axis replots spine and edge reduced by a B R quantum limit at Q, and it completely precludes the chiral factor of 10). This implies that the pronounced decrease in edge anomaly from being observed by LMR. is an artifact caused by current jetting, just as observed in pure Bi. Applying the squeeze test to NbAs next, we display the The Landau level indices n of the weak SdH oscillations are curves of R (black) and R (red) at 4 K in Fig. 6(b). indicated by vertical arrows. The index plot (inset) shows that edge spine n ¼ 1 B ¼ 7 04 1=μ ∼ 0 06 Here, both R and R increase with B,butR is reached at Q . T, whereas . T. Panel edge spine spine R R 0

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NbAs are Weyl semimetals. Rather, they demonstrate that y axes) remain fixed to the sides of the sample. The in-plane the negative LMR results reported to date in the Weyl B determines the sample’s orthogonal frame a, b, and c semimetals fall deep in the regime where classical current- (akB is tilted at an angle θ relative to xˆ with ckzˆ). The tilt jetting effects dominate. produces potential drops Vxx and Vyx given by Figure 5(a) suggests a way to avoid the screening effect of current jetting for the Weyl semimetals. By growing 2 Vxx=I ¼ ρbb þ Δρ cos θ; ð15Þ ultrathin films, one may use gating to lower ζ towards zero in the Weyl nodes. This allows the LLL to be accessed at a V =I ¼ Δρ θ θ; ð Þ much lower BQ. Simultaneously, the increased surface yx sin cos 16 scattering of the carriers will reduce μ (hence increase B cyc). By allowing the quantum effect to onset before the where ρaa and ρbb are the resistivities measured along axes classical effect becomes dominant, both trends shift the a and b, respectively, and Δρ ¼ ρaa − ρbb. “ ” B

031002-9 SIHANG LIANG et al. PHYS. REV. X 8, 031002 (2018)

VII. PARAMETRIC PLOTS Although the curves for ρyx are similar to those in Bi, a ρ As in Sec. III, we compare the planar angular MR qualitatively different behavior in xx becomes apparent. At θ ¼ 0, ρxx is suppressed by a factor of about 7 (when the results in the three materials, pure Bi, Na3Bi, and GdPtBi. Figure 7(a) displays the angular profiles ρxx vs θ at selected chiral anomaly appears). In the transverse direction θ ¼ 90 B field magnitudes B measured in Bi at 200 K with Jkxˆ.AsB ( °), the poor conductance transverse to in the ρ is tilted away from alignment with J (θ ¼ 0), ρxx increases LLL raises xx by a factor of about 2.5. In terms of absolute very rapidly at a rate that varies nominally as B2. The magnitudes, the changes to ρxx are comparable along the overall behavior in ρxx is a very large increase with B as two orthogonal directions, in sharp contrast with the case soon as jθj exceeds 10°. However, at θ ¼ 0, a decrease in in Bi. This “balanced” growth leaves a clear imprint in the ρxx of roughly 50% can be resolved. This is the spurious parametric plots. The off-diagonal signal ρyx displays the LMR induced by pure current jetting. The off-diagonal same sin θ: cos θ variation as in Bi. signal ρyx shows the sin θ: cos θ variation described in The plots of ρxx and ρyx for GdPtBi in Figs. 7(e) and 7(f) Eq. (16) (ρyx is strictly even in B). also show a concentric pattern. A complication in GdPtBi is The corresponding traces of ρxx and ρyx measured in that the nature of the Weyl node creation in the field (by the Na3Bi at 2 K are shown in Figs. 7(c) and 7(d), respectively. Zeeman shift of parabolic touching bands) is anisotropic

5 3.0 3.0 11 13.5 Bi 13.5T 9 10 13.5 T 10 13 (a) Na Bi (c) 8 (e) 12 3 10 13 GdPtBi 12 200 K 2.5 7 2.5 12 4 9 11 9 6 11 5 8 8 2.0 4 2.0 7

3 m) 6 cm) c cm)

Ω 7 Ω Ω

m 1.5 1.5 5 m ( ( (m

x 6 x x x x x 4 ρ 2 ρ ρ 5 1.0 0.5 1.0 1 4 2 1 3 3 0.5 0.5 2 1 0 0.0 0.0 -90 -60 -30 0 30 60 90 -90 -60 -30 0 30 60 90 -90 -60 -30 0 30 60 90 θ (deg) θ (deg) θ (deg)

1.0 3 13 11 13.5 3 13.5 Bi (b) 13.5 T 11 (d) 10 (f) 12 Na3Bi GdPtBi 11 10 9 12 7 200 K 12 2 2 10 6 0.5 8 5 9 7 1 1 4 8 6

3 m) cm) c 3 2 1 0.5 T cm) 5 Ω Ω 0 0.0 Ω 0 (m (m

1T (m 4T 2 4 yx yx yx ρ ρ -1 5 ρ Vxx -1 B a θ -0.5 6 7 -2 Vyx 8 -2 -3 9 -1.0 -3 -90 -60 -30 0 30 60 90 -90 -60 -30 0 30 60 90 -90 -60 -30 0 30 60 90 θ (deg) θ (deg) θ (deg)

FIG. 7. Planar AMR measurements in Bi, Na3Bi, and GdPtBi. Panels (a) and (b) show the θ dependence of the diagonal and off- diagonal AMR signals, expressed as effective resistivities ρxx and ρyx, respectively, in Bi (sample B2) at 200 K. In panel (a), ρxx decreases with increasing B when θ ¼ 0 (spurious LMR produced by current jetting). When jθj exceeds 20°, ρxx increases steeply with both jθj and B. In absolute terms, the increase in ρxx at θ ¼ 90° far exceeds its decrease at θ ¼ 0. The off-diagonal MR ρyx follows the standard sin θ: cos θ variation [panel (b)]. Panel (c) shows curves of ρxx for Na3Bi (sample N2) at 4 K. The decrease in ρxx at θ ¼ 0 is roughly comparable with the increase at 90°. A similar balanced pattern is observed in GdPtBi (sample G1) measured at 2 K [panel (e)]. In both case 2 systems, the comparable changes in ρxx measured at 0° and 90° contrast with the lopsided changes seen in Bi (case 1). The off-diagonal ρyx in both Na3Bi and GdPtBi [panels (d) and (f)] show the standard sin θ: cos θ profile. The profiles in panel (f) are distorted by anisotropic features associated with the creation of the Weyl pockets. The inset in panel (b) shows the sample geometry with akB, both at the tilt angle θ to xˆ.

031002-10 EXPERIMENTAL TESTS OF THE CHIRAL ANOMALY … PHYS. REV. X 8, 031002 (2018)

Bi Bi 13.5 T (a) 50 (b) 2.5 200 K 100 K 13

cm) 12 cm) Ω Ω 0.0 567 0 5 6 7 10 (m 11 (m 89 8 9 10 yx yx 11 ρ ρ 12 1 1 2 2 3 4 3 4T -2.5 13.5 T -50

036 0 50 100 ρ Ω ρ Ω xx (m cm) xx (m cm) 1.0 GdPtBi Na3Bi (c) (d) 2 4K 13.5 4K 0.5 1 13.5

cm) 13 cm) Ω Ω

(m 0.0 0 (m

yx 12

yx 12 ρ ρ 11 -1 11 -0.5 0.5 4 1 10 5 10 2 3 9 -2 6 9 4 5 6 7 8 7 8 -1.0 01230123 ρ Ω ρ Ω xx (m cm) xx (m cm)

FIG. 8. Parametric plots of the planar AMR signals. Panel (a) shows the orbits in pure Bi at 200 K (sample B2) obtained by plotting ρxx vs ρyx with θ as the parameter and with B kept fixed. The orbits start at the left (small ρxx) and end at the right (large ρxx). As B increases, the orbits expand without saturation to the right. Panel (b) shows the same plots measured at 100 K. The increased mobility strongly amplifies the lopsided expansion pattern, which resembles a shock wave. The orbits in the parametric plots measured in Na3Bi (sample N2, at 4 K) and GdPtBi (G1) at 2 K [panels (c) and (d), respectively] are nominally concentric around the point at B ¼ 0. The differences between case 1 and case 2 reflect the very different behaviors of ρxx at the extreme angles θ ¼ 0 and 90°, as discussed in Fig. 7.

(dependent on the direction of B). The existence of low-B of an artifactual decrease Vxx in the LMR geometry is a oscillations adds a modulation to the off-diagonal curves, common experience in high-mobility semimetals. To assist which distorts the variation from the sin θ: cos θ form. in the task of disentangling the rare intrinsic cases from Nonetheless, a balanced growth in ρxx is also observed] extrinsic cases (mostly caused by current jetting), we Fig. 7(e)]. described a test that determines the current-jetting distor- Figure 8 compares the parametric plots in Bi at T ¼ tions and a procedure for removing them. The squeeze test 200 B R K [panel (a)] and 100 K [panel (b)]. In each orbit ( set consists of comparing the effective resistance spine mea- at the indicated value), θ starts at 0° on the left limb and sured along the spine (line joining current contacts) with R R ends at 90° on the right. In Fig. 8(a), the steep increase on that along an edge edge. In pure Bi, spine increases the right limb causes the orbits to expand strongly to the B R dramatically with , while edge decreases. However, in right. At 100 K [panel (b)], the higher mobility creates R R both Na3Bi and GdPtBi, both spine and edge decrease. exaggerated skewing of the rightward expansion, leading to Hence, an inspection of the trend in R allows case 1 the emergence of a “shock-wave” pattern. By contrast, the spine (here Bi) to be distinguished from case 2, the two chiral- parametric plots in both Na3Bi [Fig. 8(c) and GdPtBi in anomaly semimetals. From the factorization implicit in panel (d)] show concentric orbits that expand in a balanced Eqs. (9) and (10), we relate the experimental ratio G to the pattern as anticipated above. The contrast between case 1 curve F obtained by numerical simulation. This enables [panels (a) and (b)] and case 2 directly reflects the distinct the intrinsic profiles of both ρxxðBÞ and u to be obtained nature of mechanisms that increase the anisotropy u [see from measurements of R and R . After removal of Eqs. (2) and (3)]. edge spine the distortions, ρxx is seen to decrease by a large factor (10.9) between B ¼ 0 and 10 T, while u increases by 8. VIII. CONCLUDING REMARKS (For simplicity, the numerical simulation was done in the As mentioned in Sec. I, the existence of a negative LMR 2D limit, which we judge is adequate for flakelike samples. that is intrinsic is relatively rare. However, the observation Obviously, this can be improved by adopting a fully 3D

031002-11 SIHANG LIANG et al. PHYS. REV. X 8, 031002 (2018) simulation.) The yes or no nature of the test based on States in the Electronic Structure of Pyrochlore Iridates, R inspection of spine, bolstered by the quantitative analysis Phys. Rev. B 83, 205101 (2011). that removes the subdominant corrections, adds consider- [4] C. Fang, M. J. Gilbert, X. Dai, and B. A. Bernevig, Multi- able confidence that the chiral-anomaly LMR profiles in Weyl Topological Semimetals Stabilized by Point Group Symmetry, Phys. Rev. Lett. 108, 266802 (2012). Na3Bi and GdPtBi are intrinsic. Moreover, the subdomi- [5] B.-J. Yang and N. Nagaosa, Classification of Stable Three- nant distortion factors arising from current jetting can be Dimensional Dirac Semimetals with Nontrivial Topology, effectively removed. Nat. Commun. 5, 4898 (2014). In Sec. V, we described LMR experiments as a com- [6] Z. J. Wang, Y. Sun, X. Q. Chen, C. Franchini, G. Xu, petition between the intrinsic quantum effect arising from H. M. Weng, X. Dai, and Z. Fang, Dirac Semimetal and the chiral anomaly and the classical effects of current Topological Phase Transitions in A3Bi (A ¼ Na, K, Rb), jetting. The former describes a phenomenon intrinsic to Phys. Rev. B 85, 195320 (2012). massless chiral fermions. To see it in full force, the applied [7] Classical Theory of Gauge Fields,editedbyV.Rubakov B should exceed BQ, the field needed to move ζ into the (Princeton University Press, Princeton, NJ, 2002), Chaps. 15 LLL. This point seems worth emphasizing because in and 17. many reports the claimed anomaly seems to appear in [8] S. L. Adler, Axial-Vector Vertex in Spinor Electrodynamics, B ≪ B Phys. Rev. 177, 2426 (1969). weak fields, Q. The experimental concern is that [9] J. S. Bell and R. Jackiw, A PCAC Puzzle: πo → γγ in the B once current jetting appears (at the field scale cyc), it σ-Model, Nuovo Cimento 60A, 47 (1969). inevitably engenders a dominant, negative LMR profile that [10] Introduction to Quantum Field Theory, edited by M. E. R is extrinsic in origin. The divergent field profiles of spine Peskin and D. V. Schroeder (Westview Press, Boulder, R and edge provide a strong warning that the LMR profile is Colorado, 1995), Chap. 19. then highly unlikely to be intrinsic. [11] Anomalies in Quantum Field Theory, edited by Reinhold A. Looking ahead, we discussed in Sec. V how the classical Bertlmann (Clarendon Press, Oxford, 2011). [12] K.-Y. Yang, Y.-M. Lu, and Y. Ran, Quantum Hall Effects in screening effect from current jetting may be avoided by a Weyl Semimetal: Possible Application in Pyrochlore using ultrathin, gateable films of the Weyl semimetals, Iridates, Phys. Rev. B 84, 075129 (2011). which may become available in the near future. For the [13] D. T. Son and B. Z. Spivak, Chiral Anomaly and Classical fourth class of chiral anomaly semimetal ZrTe5 [23,24],we Negative Magnetoresistance of Weyl Metals, Phys. Rev. B propose using a focused ion beam to sculpt platelike 88, 104412 (2013). samples that are 10 μm on a side, and applying micro- [14] S. A. Parameswaran, T. Grover, D. A. Abanin, D. A. Pesin, lithography to attach contacts for the squeeze test. and A. Vishwanath, Probing the Chiral Anomaly with Nonlocal Transport in Three-Dimensional Topological Semimetals, Phys. Rev. X 4, 031035 (2014). ACKNOWLEDGMENTS [15] A. A. Burkov, Negative Longitudinal Magnetoresistance The research was supported by the U.S. Army Research in Dirac and Weyl Metals, Phys. Rev. B 91, 245157 (2015). Office (Grant No. W911NF-16-1-0116) and a MURI award [16] J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan, M. for topological insulators (No. ARO W911NF-12-1-0461). Hirschberger, W. Wang, R. J. Cava, and N. P. Ong, Evidence for the Chiral Anomaly in the Dirac Semimetal Na3Bi, N. P. O. acknowledges support from the Gordon and Betty ’ Science 350, 413 (2015). Moore Foundation s Emergent Phenomena in Quantum [17] M. Hirschberger, S. Kushwaha, Z. Wang, Q. Gibson, S. Systems Initiative through Grant No. GBMF4539. The Liang, C. A. Belvin, B. A. Bernevig, R. J. Cava, and N. P. growth and characterization of crystals were performed by Ong, The Chiral Anomaly and Thermopower of Weyl S. K. and R. J. C., with support from the National Science Fermions in the Half-Heusler GdPtBi, Nat. Mater. 15, Foundation (NSF MRSEC Grant No. DMR 1420541). 1161 (2016). Work at UCLA was supported by the U.S. Department of [18] X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Energy (DOE), Office of Basic Energy Sciences, Award Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, No. DE-SC0011978. Observation of the Chiral-Anomaly-Induced Negative Mag- netoresistance in 3D Weyl Semimetal TaAs, Phys. Rev. X 5, 031023 (2015). [19] C.-L. Zhang et al., Signatures of the Adler-Bell-Jackiw Chiral Anomaly in a Weyl Fermion Semimetal, Nat. [1] H. B. Nielsen and M. Ninomiya, The Adler-Bell-Jackiw Commun. 7, 10735 (2016). Anomaly and Weyl Fermions in a Crystal, Phys. Lett. B 130, [20] R. D. dos Reis, M. O. Ajeesh, N. Kumar, F. Arnold, C. 389 (1983). Shekhar, M. Naumann, M. Schmidt, M. Nicklas, and E. [2] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, Hassinger, On the Search for the Chiral Anomaly in Weyl and A. M. Rappe, Dirac Semimetal in Three Dimensions, Semimetals: The Negative Longitudinal Magnetoresistance, Phys. Rev. Lett. 108, 140405 (2012). New J. Phys. 18, 085006 (2016). [3] X. G. Wan, A. M. Turner, A. Vishwanath, and S. Y. [21] Y. Li, Z. Wang, P. Li, X. Yang, Z. Shen, F. Sheng, X. Li, Y. Savrasov, Topological Semimetal and Fermi-Arc Surface Lu, Y. Zheng, and Z.-A. Xu, Negative Magnetoresistance in

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Weyl Semimetals NbAs and NbP: Intrinsic Chiral Anomaly and T. Valla, Chiral Magnetic Effect in ZrTe5, Nat. Phys. 12, and Extrinsic Effects, Front. Phys. 12, 127205 (2017). 550 (2016). [22] J. Xiong, S. Kushwaha, J. Krizan, T. Liang, R. J. Cava, and [24] T. Liang et al., Anomalous Hall Effect in ZrTe5, Nat. Phys. N. P. Ong, Anomalous Conductivity Tensor in the Dirac 14, 451 (2018).. Semimetal Na3Bi, Europhys. Lett. 114, 27002 (2016). [25] A. A. Burkov, Giant Planar Hall Effect in Topological [23] Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosić, Metals, Phys. Rev. B 96, 041110 (2017). A. V. Fedorov, R. D. Zhong, J. A. Schneeloch, G. D. Gu,

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