Negative magnetoresistance without well-defined chirality in the Weyl semimetal TaP

Abstract

Weyl semimetals (WSMs) are topological quantum states wherein the electronic bands disperse linearly around pairs of nodes with fixed chirality, the Weyl points. In WSMs, nonorthogonal electric and magnetic fields induce an exotic phenomenon known as the chiral anomaly, resulting in an unconventional negative longitudinal magnetoresistance, the chiral-magnetic effect. However, it remains an open question to which extent this effect survives when chirality is not well-defined. Here, we establish the detailed Fermi-surface topology of the recently identified WSM TaP via combined angle-resolved quantum-oscillation spectra and band-structure calculations. The Fermi surface forms banana-shaped electron and hole pockets surrounding pairs of Weyl points. Although this means that chirality is ill-defined in TaP, we observe a large negative longitudinal magnetoresistance. We show that the magnetoresistance can be affected by a magnetic field-induced inhomogeneous current distribution inside the sample.

ARTICLE

Received 29 Jan 2016 | Accepted 14 Apr 2016 | Published 17 May 2016

Frank Arnold1, , Chandra Shekhar1, , Shu-Chun Wu1, Yan Sun1, Ricardo Donizeth dos Reis1, Nitesh Kumar1, Marcel Naumann1, Mukkattu O. Ajeesh1, Marcus Schmidt1, Adolfo G. Grushin2, Jens H. Bardarson2,

Michael Baenitz1, Dmitry Sokolov1, Horst Borrmann1, Michael Nicklas1, Claudia Felser1, Elena Hassinger1 & Binghai Yan1,2

Weyl semimetals (WSMs) are topological quantum states wherein the electronic bands disperse linearly around pairs of nodes with xed chirality, the Weyl points. In WSMs, nonorthogonal electric and magnetic elds induce an exotic phenomenon known as the chiral anomaly, resulting in an unconventional negative longitudinal magnetoresistance, the chiral-magnetic effect. However, it remains an open question to which extent this effect survives when chirality is not well-dened. Here, we establish the detailed Fermi-surface topology of the recently identied WSM TaP via combined angle-resolved quantum-oscillation spectra and band-structure calculations. The Fermi surface forms banana-shaped electron and hole pockets surrounding pairs of Weyl points. Although this means that chirality is ill-dened in TaP, we observe a large negative longitudinal magnetoresistance. We show that the magnetoresistance can be affected by a magnetic eld-induced inhomogeneous current distribution inside the sample.

DOI: 10.1038/ncomms11615 OPEN

Negative magnetoresistance without well-dened chirality in the Weyl semimetal TaP

1 Max Planck Institute for Chemical Physics of Solids, Dresden 01187, Germany. 2 Max Planck Institute for the Physics of Complex Systems, Dresden 01187, Germany. These authors contributed equally to this work. Correspondence and requests for materials should be addressed to E.H.

(email: mailto:[email protected]

Web End [email protected] ) or to B.Y. (email: mailto:[email protected]

Web End [email protected] ).

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In a semimetal the conduction and valence bands touch at isolated points in the three-dimensional (3D) momentum (k) space at which the bands disperse linearly. Depending on

whether the bands are nondegenerate or doubly degenerate, such a 3D semimetal is called a Weyl semimetal (WSM)19 or a Dirac semimetal1012, respectively. Correspondingly, the band touching point is referred to as a Weyl or a Dirac point. The Dirac point can split into one or two pairs of Weyl points by breaking either time-reversal symmetry or crystal inversion symmetry. At energies close to the Weyl points, electrons behave effectively as Weyl fermions, a fundamental kind of massless fermions that has never been observed as an elementary particle13. In condensed-matter physics, each Weyl point acts as a singularity of the Berry curvature in the Brillouin zone (BZ), equivalent to magnetic monopoles in k space. Thus Weyl points always occur in pairs with opposite chirality or handedness14. In the presence of nonorthogonal magnetic (B) and electric (E) elds (that is, E . B is nonzero), the particle number for a given chirality is not conserved quantum mechanically, inducing a phenomenon known as the AdlerBellJackiw anomaly or chiral anomaly in high-energy physics13,15,16. In WSMs, the chiral anomaly is predicted to lead to a negative magnetoresistance (MR) due to the suppressed backscattering of electrons of opposite chirality17,18. Theoretically, the chiral anomaly only appears, if chirality is well-dened, that is, the Fermi energy is close enough to the Weyl nodes that pairs of separate Fermi surface pockets with opposite chirality exist17. Observing the chiral-anomaly induced negative MR requires the applied magnetic and electric eld to be as parallel as possible. Otherwise the negative MR will easily be overwhelmed by the positive contribution arising due to the Lorentz force. In addition to the negative MR, the chiral anomaly is also predicted to induce an anomalous Hall effect2,1921, nonlocal transport properties22,23 and sharp discontinuities in angle-resolved photo-emission spectroscopy (ARPES) signals24 in WSMs.

The discovery of various WSM materials has stimulated experimental efforts to conrm the chiral anomaly in condensed-matter physics. Recently, a negative MR has been reported in two types of WSMs: WSMs induced by time-reversal symmetry breaking, that is, Dirac semi metals in an applied magnetic eld, for example Bi1 xSbx (xE3%)25, ZrTe5 (ref. 26)

and Na3Bi27, and the non-inversion-symmetric WSMs TaAs28,29, NbP30 and NbAs31. However, a clear verication of whether the Fermi surface topology supports the chiral anomaly or not, is still lacking in most of the above systems.

In the non-centrosymmetric WSMs of the TaAs family two types of Weyl nodes exist at different positions in reciprocal space5,6 and energies. Therefore, the Weyl electrons generally coexist with topologigcally trivial normal electrons. In principle, small changes of the Fermi energy (EF), as induced by doping or defects, can change the Fermi-surface topology signicantly due to the low intrinsic charge carrier density in semimetals. Therefore, a precise knowledge of EF and the resulting

Fermi-surface topology is required when linking the negative MR to the chiral-magnetic effect. Extensive ARPES studies have shown the existence of Fermi-arc surface states and linear band crossings in the bulk band structure of all four materials from the TaAs family79,32,33. However, because of an insufcient energy resolution (415 meV (ref. 32)), ARPES is not able to make any claims about the presence or absence of quasiparticles with well-dened chirality at the Fermi level. In contrast, quantum-oscillation measurements have the advantage of a millielectronvolt resolution of the Fermi-energy level.

In this work, we reconstruct the 3D Fermi surface of TaP by combining sensitive Shubnikov-de Haas (SdH) and de Haas-van Alphen (dHvA) oscillations with ab initio band-structure

calculations reaching a good agreement between theory and experiment. We reveal that EF is such that the electron and hole Fermi-surface pockets contain pairs of Weyl nodes and the total Berry ux through the Fermi surface FB 0. Although

this means that chirality is not well-dened, a large negative MR is observed. We discuss possible explanations for this result also considering the magnetic eld dependence of the current distribution in our samples34.

ResultsSingle-crystal synthesis and characterization. We synthesized high-quality single crystals of TaP by using chemical vapour transport reactions and veried TaP as a non-centrosymmetric compound in a tetragonal lattice (space group I41md, No. 109).

The temperature-dependent resistivity exhibits typical semimetallic behaviour. For more details, see Methods and Supplementary Figs 14.

Quantum oscillations. The Fermi surface topology of TaP was investigated by means of quantum oscillations. Typically, for a semimetal with light carriers and high mobility, such as bismuth35, prominent oscillations appear in all properties sensitive to the density of states at the Fermi energy. Here, we measured SdH oscillations in transport (Fig. 1a) and dHvA oscillations in magnetic torque (Fig. 1b) and magnetization (Fig. 2a) for different magnetic eld orientations. These oscillations are periodic in 1/B. Their frequency (F) is proportional to the corresponding extremal Fermi surface cross-section (A) that is perpendicular to B following the Onsager relation F (F0/2p2)A,

where F0 h/2e is the magnetic ux quantum and h is the Planck

constant. Figure 1a shows the resistivity as a function of the magnetic eld for different eld orientations. When the electric current and magnetic eld are perpendicular I a; B

k kc

, the

magnetoresistance is very high as typical for other WSM (for example, ref. 36) and normal semimetals37,38. This implies a very high mobility of the charge carriers. Figure 1b depicts the magnetic torque oscillations for the same eld orientations. Figure 2a represents the magnetic dHvA oscillations in the magnetization as a function of the inverse magnetic eld and their corresponding Fourier transform (Fig. 2b) for B||c. The observable fundamental frequencies are Fa 15 T, Fb 18 T, Fg 25 T, and

Fd 45 T. These frequencies are consistent between all three

measurement techniques and different sample batches within the error bars (Supplementary Table 1). This indicates that all samples have a similar chemical potential to within 1 meV. In addition, we can conclude that the resistivity is sensitive to the bulk Fermi surface. For B||a the main frequencies are Fa 26 T, Fg 34 T,

Fg0 105 T Fd 147 T and Fg

00

320 T clearly indicating anisotropic 3D Fermi surface pockets. For most of the detected oscillation frequencies, we derive their cyclotron effective masses (m ) of the carriers by tting the temperature dependence of the oscillation amplitude (inset of Fig. 2b) with the LifshitzKosevich formula (see ref. 39 and Supplementary Figs 5 and 6 as well as Supplementary Note 1). The values of the effective masses are m a (0.0210.003)m0, m b (0.050.01)m0 and

m d (0.110.01)m0 for B||c, whereas they are a factor of 410

greater for B||a with m g (0.130.03)m0, m g0 (0.350.03)m0

and m d (0.40.1)m0, where m0 is the mass of a free electron

(see Supplementary Table 1). These values are small and comparable to the effective masses in other slightly doped Dirac materials such as Cd3As2 or graphene40,41. These low masses, together with long scattering times are the reason for the high mobility and the huge transverse magnetoresistances seen in semimetals.

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a c

[110] [100] [110]

[001]

12

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[afii9829]

TaP

[afii9828]

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E (Theory) H (Theory)

[afii9829]

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b

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0 12

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4 8 10

6 14

10 90 60 30 0 30 60 90 120

B (T)

[afii9835] (deg)

Figure 1 | Quantum oscillations and angular dependence of oscillation frequencies in TaP. (a) SdH oscillations in resistivity for different angles in steps of 10. (b) dHvA oscillations from magnetic torque measurements for the same angles. Curves are shifted for clarity. (c) Full angular dependence of the measured and theoretical quantum oscillation frequencies. Open and closed symbols refer to SdH and dHvA data of ve different samples from two different batches. Lines show the extremal orbits calculated from the banana-shaped 3D Fermi surface topology (solid lines for the pockets lying in the tilting plane of the magnetic eld, dashed lines for the pockets lying perpendicular to it).

a c

b

TaP B || c

F[afii9826]

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6 S m1 )

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[afii9839]E

103

0 0.5 1.0 1.5

0 20 40 60

80

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T (K)

T (K)

Figure 2 | Charge carrier properties from quantum oscillation and Hall data. (a) dHvA oscillations as a function of the inverse eld at T 1.85 K.

(b) Fourier transform of a showing the characteristic quantum-oscillation frequencies. The inset shows the temperature dependence of the quantum oscillation amplitude and LifshitzKosevich temperature reduction term ts. (c) Hall conductivity of sample S1 for different temperatures and two-band model ts (dashed lines). (d) Hole (H) and electron (E) carrier concentrations and mobilities as obtained by tting the Hall conductivities of samples S1 (triangles) and S3 (diamonds), respectively. The grey-shaded areas give the condence intervals of the densities and mobilities. The blue and red dashed lines mark the theoretical electron and hole densities based on the tted Fermi-surface topology. The star marks the hole mobility determined from the Dingle analysis.

Fermi-surface topology. To reconstruct the shape of the Fermi surface, the full angular dependence of the quantum-oscillation frequencies was compared with band-structure calculations (Fig. 1c). The exact position of EF was determined by matching the calculated frequencies and their angular dependence to the experimental ones (see Supplementary Figs 710 and Supplementary Note 2). The best t is obtained when EF lies 5 meV above the ideal electronhole compensation point, in agreement with the resulting carrier concentrations from Hall measurements (Fig. 2c,d). At this EF, calculations reveal two banana-shaped Fermi surface pockets, a hole pocket (H) and a slightly larger electron pocket (E). These two pockets reproduce the angular dependence of the measured dHvA frequencies with great accuracy (see lines in Fig. 1c and the Fermi surface in

Fig. 3). E and H are almost semicircular and are distributed along rings5,6 on the kx 0 and ky 0 mirror-planes in the BZ.

The rather isotropic frequencies Fa and Fg result from a neck and extra humps (head with horns) at the end of the hole pocket (see Fig. 3b). The splitting of all frequencies with eld angles departing from B||c in the (100)-plane is explained by the existence of four banana-shaped pockets in the BZ, two for each mirror plane for both E and H-pockets. The splitting of the frequency Fd seen in the experiment is not reproduced by the calculation. One possibility for this discrepancy is that a waist may appear in the E-pocket at kz 0.

In addition, in the dHvA experiment the mobility of the hole orbits (Fa and Fb) was extracted via the width of the

Fourier-transform peaks, which is given by the exponential

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decrease of the oscillations with 1/B (the so-called Dingle term; see Supplementary Note 3). The deduced mobility is mh 3.2

104 (20%) cm2 V 1 s 1 (star in Fig. 2d).

Charge-carrier density and mobility. We extract information on the carrier density and mobility from magnetic eld- and temperature-dependence of the Hall effect in sample S1 (full temperature range, Fig. 2c) and only at low temperature in sample S3 from a different batch. We employ a two-carrier model (see ref. 42) to t the Hall conductivity (sxy) by making use of the longitudinal conductivity at zero eld (sxx) as an additional

condition (see Supplementary Note 4). As shown in the left panel of Fig. 2d, the carrier concentration for both electrons and holes at low temperature is around n (21) 1019 cm 3 with an

absolute error bar (grey-shaded area) approximately given by the difference between sample S1 and S3. Although TaP is an almost compensated metal the electron density is slightly larger than the hole density, which is in agreement with EF lying 5 meV above the charge-neutral point as determined from the Fermi surface topology. The theoretical values of the carrier densities, given as dashed lines in Fig. 2d, are in good agreement with the experimental data. Note that above 150 K the hole density becomes larger than the electron density. At low temperature, the mobility of both carriers is on the order of m (25) 104 cm2 V 1 s 1, with the

hole mobility higher than the electron one. These high mobilities indicate the very high quality of the single crystals with only few defects and impurities. The slight electron and hole mobility difference is also reected in the sign change of the high-eld Hall resistivity (see Supplementary Figs 11 and 12) and is similar to the Hall effect observed in TaAs28,29 and NbP30,36. Furthermore, the Hall mobilities agree well with the mobility determined by the Dingle analysis (the star in Fig. 2d).

DiscussionThe experimental Fermi surface topology and charge-carrier concentrations described above converge to the same statement that the Fermi energy of our samples lies 5 meV above the ideal charge-neutral point in the calculated band structure. This slight carrier doping is not expected for a completely stoichiometric sample. However, the appearance of some defects/vacancies in this type of material is possible43 and can explain the small shift of EF. The consistency between experiment and theory strongly suggests that we found the true Fermi surface of our TaP samples. We plot the corresponding 3D Fermi surfaces from the ab initio calculations in Fig. 3. As can be seen, E and H are the only two pockets at the Fermi energy.

We shall further investigate the Weyl points in the band structure. In the 3D BZ, there are 12 pairs of Weyl points with opposite chirality: four pairs lie in the kz 0 plane (labelled as

W1) and eight pairs are located in planes close to kz p/c

(where c is the lattice constant) (labelled as W2). One can see that the W1 points are far below EF by 41 meV and are included in the

E-pocket. The W2-type Weyl points are 13 meV above EF and are included in the H-pocket. Such pairs of W2 points merge slightly into the head position of the H-pocket, leading to a two-horn-like cross-section (see Fg in Fig. 3b). Energetically, they are separated by a 16 meV barrier along the line connecting the Weyl points of a pair. We plot the energy dispersion of Weyl bands over the Fg plane in Fig. 3d. There are no independent Fermi-surface pockets around the W2 Weyl points and therefore the chiral anomaly is not well-dened. The Weyl cone is strongly anisotropic in the lower cone region below the Weyl point.

Finally, we discuss the longitudinal magnetotransport properties of TaP. We measured the longitudinal magneto-resistance of three samples from the same crystal batch. The current (I) was applied along the crystallographic c (sample S2)

a c

H

E

W2

EF

13 meV

41 meV

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W2

W1

[afii9829]

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E

[afii9828]

ky kx

W2

Figure 3 | 3D Fermi surface pockets and Weyl points. (a) Fermi pockets in the rst BZ at the Fermi energy (EF) detected in the experiment. The electron(E) and hole (H) pockets are represented by blue and red colours, respectively. (b) Enlargement of the banana-shaped E and H-pockets. The pink and green points indicate the Weyl points with opposite chirality. W1- and W2-type Weyl points can be found inside E and close to H-pockets, respectively. Green loops represent some extremal E and H cross-sections, corresponding to the oscillation frequencies measured, Fa,b,g,d for B||c. (c) Energy dispersion along the connecting line between a pair of Weyl points with opposite chirality for W1 (left) and W2 (right). The deduced experimental EF (thick dashed horizontal line) is 13 meV below the W2 Weyl points and 41 meV above the W1 Weyl points. (d) Strongly anisotropic Weyl cones originating from a pair of

W2-type Weyl points on the plane of Fg. Green and red Weyl cones represent opposite chirality.

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a b c d

3 TaP S2 : I || B || c

S1 : I || B || a

S4 : I || B || a

S4 : I || B || a Theory

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V3

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4 12

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Figure 4 | Negative longitudinal magnetoresistance in TaP. (a) Longitudinal MR [V V(B 0)]/V(B 0) for B||I||c for different temperatures.

(b) Same for B||I||a. The temperatures are the same as in a. (c) Same for B||I||a and three pairs of contacts. The difference in the curves can be explained by an inhomogeneous current distribution induced by the magnetic eld (see text). The contact geometry is shown in the inserts: (b) for S1 and S2, and (c) for S4. (d) Theoretical curves for S4 as in c.

and a axes (samples S1 and S4), respectively. Figure 4a,b represent the apparent longitudinal magnetoresistance (MR [V V(B 0)]/V(B 0), where V is the voltage drop at

a xed electric current) obtained for the two crystallographic directions. As long as the current ows homogeneously in the sample, MR is equal to the usual MR [r r(B 0)]/r(B 0).

At 1.85 K, MR rst increases slightly and soon drops steeply from positive to negative values. The apparent negative MR is very robust against increasing temperature, and MR is negative in a very narrow angular window of yr2 and is positive otherwise (see Supplementary Fig. 13 and Supplementary Note 5). This small y window for the apparent negative MR is similar to that observed in Na3Bi27.

In light of the given Fermi-energy position in our crystals, it is not possible to link the negative MR to the chiral anomaly, simply because the former is not a well-dened concept when the Fermi surface connects both Weyl nodes. Although this does not rule out the presence of a non-trivial negative MR4446, we nd that the observed negative MR , is strongly affected by the geometric conguration.

This becomes evident when we examine the apparent long-itudinal MR for three different voltage contact congurations on sample S4 as illustrated in Fig. 4c. A clear voltage decrease in magnetic eld is observed for pair V1, similar to the low temperature curves in Fig. 4a,b, whereas the two other pairs, denoted V2 and V3, show a higher MR . This points to an underlying inhomogeneous current distribution in the sample becoming important in high magnetic elds. As typical for high mobility semimetals, TaP has a large transverse MR arising from the orbital effect, whereas the longitudinal MR most likely stays of the same order of magnitude. For current contacts smaller than the cross-section of the sample, this leads to a eld-induced stearing of the current to the direction of the magnetic eld, which is along the line connecting the current contacts when current and magnetic eld are parallel. This effect is known as current jetting34,4751. As a consequence, a voltage pair close to this line (V3) detects a higher MR than the intrinsic longitudinal MR whereas a voltage pair far away from it (V1) detects a smaller MR than the intrinsic one. This effect is conrmed by calculations of the voltage distribution for sample S4, taking into account the current jetting by following ref. 51 (see Supplementary Figs 14 and 15 and Supplementary Note 6). Using the experimental transverse MR and assuming a eld-independent intrinsic longitudinal MR, the model qualitatively reproduces the three MR curves without any free parameters (see Fig. 4d). Therefore, the magnetic eld

dependence of the longitudinal voltage is largely induced by the strong transverse MR, if the current is not homogeneously injected into the whole cross-section of the sample.

In summary, we determined the Fermi-surface topology of the inversion-asymmetric WSM TaP. The Fermi surface consists of banana-shaped spin-polarized electron and hole pockets with very light carrier effective masses. Despite the absence of independent Fermi-surface pockets around the Weyl points, an apparent negative longitudinal MR is detected. We show that in such studies, special care is needed to avoid a decoupling of the voltage contacts from the current jet in longitudinal magnetic elds.

Methods

Single-crystal growth. High-quality single crystals of TaP were grown via a chemical vapour transport reaction using iodine as a transport agent. Initially, polycrystalline powder of TaP was synthesized by a direct reaction of tantalum (Chempur 99.9%) and red phosphorus (Heraeus 99.999%) kept in an evacuated fused silica tube for 48 h at 800 C. Starting from this microcrystalline powder, the single-crystals of TaP were synthesized by chemical vapour transport in a temperature gradient starting from 850 C (source) to 950 C (sink) and a transport agent with a concentration of 13.5 mg cm 3 iodine (Alfa Aesar 99,998%)52.

X-ray analysis and structure characterisation. The crystal structure and orientation of TaP crystals were determined by X-ray diffraction at room temperature. For this, TaP single crystals were mounted on a four-circle Rigaku AFC7 X-ray diffractometer with a built-in Saturn 724 CCD detector. A suitable sample edge

was selected where the transmission of Mo Ka (l 0.71073 ) radiation seemed

feasible. The intensities of the obtained reections were corrected for absorption by using a multi-scan technique. The unit cell was assigned by using a 30 images standard indexing procedure. Here oscillatory images about the crystallographic axes allowed the assignment of the crystal orientation, conrmed the appropriate choice of the unit cell and showed the excellent crystal quality. Supplementary Figs 1 and 2 show X-ray diffraction patterns of the S1 and S2 TaP crystals, which were used in our transport measurements. Structure renement was performed by full-matrix least-squares on F within the program package WinCSD, Version 4.14 (ref. 53 and revealed the non-centrosymmetric crystal structure with space group I41md and lattice parameters a b 3.30 , c 11.33 at room temperature.

To conrm the quality of our TaP single crystals additional Laue images from an unperturbed as grown (001)-facet of a TaP crystal were taken. The single crystal was oriented using a white beam backscattering Laue X-ray diffraction method. Supplementary Fig. 3 shows the corresponding Laue diffraction image indexed with the I41md-structure and room temperature lattice parameters. The Laue diffraction image shows sharp reections, which conrm the excellent quality of the sample. The presence of domains or twinning can be ruled out by indexing all reections of the image by a single pattern.

Quantum oscillations. The electronic structure of TaP has been characterized by means of quantum oscillations. Here, the SdH and dHvA effect were measured by electrical resistivity, magnetization and torque magnetometry experiments39.

The frequency of these oscillations F Aext :/2pe is proportional to the extremal

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Fermi surface cross-section Aext perpendicular to the respective magnetic eld direction39,54,55. Quantum oscillation spectra were obtained from magnetization and torque data by discrete Fourier transformation of the background subtracted oscillatory part of the respective signal.

Magnetization and dHvA measurements. The magnetism and magnetic quantum oscillations of TaP along the main crystallographic axes were investigated in a Quantum Design Inc. SQUID-VSM in the temperature and magnetic eld range of 250 K and 7 T.

Angular dependencies were measured using the Quantum Design Inc. piezo-resistive torque magnetometer (Tq-Mag56) in a physical property measurement system (PPMS) with installed rotator option. Magnetization and torque experiments were performed on two large 4.4 and 21.7 mg TaP single crystals. The samples were mounted on the sample holder and torque lever by GE varnish or Apiezon N grease and aligned along their visible crystal facets, which were conrmed by X-ray diffraction. The crystal alignment was veried by photometric methods and showed typical misalignments o2. Magnetic torque measurements were performed up to14 T in the same temperature range. Here the sample is mounted on a exible lever which bends when torque is applied. The sample magnetization M induces a magnetic torque tM B, which bends the lever and can be sensed by piezo-

resistive elements which are micro-fabricated onto the torque lever. These elements change their resistance under strain and are thus capable of sensing the bending of the torque lever. The unstrained resistance of these piezo resistors is typically 500 O and can change by up to a few per cent when large magnetic torques are applied. Probing quantum oscillations, however, requires a higher torque resolution. Thus the standard torque option was altered and extended by an external balancing circuit and SR830 lock-in ampliers as read-out electronic. This way a resolution of one in 107 was achieved. The magnetic torque was determined for magnetic elds in the (100), (001) and (110)-plane in angular steps of 2.5 and 5. Measurements were taken during magnetic eld down sweeps from 14 to 0 T. The magnetic eld sweep rate was adjusted such that the sweep rate in 1/B was constant. The resultant magnetic torque signals are a superposition of the sample diamagnetism, dHvA oscillation and uncompensated magnetoresistance of the piezo resistors. Because of the vector product of the magnetization and magnetic eld, this method can only be applied to samples with strong Fermi surface anisotropy and is insensitive when the magnetic eld and magnetization are aligned parallel, for example, along the crystallographic c direction in TaP.

The quantum oscillation frequencies are extracted from the obtained resistivities and magnetizations by subtracting all background contributions to those signalsand performing a Fourier transformation of the residual signal over the inverse magnetic eld. The resulting spectra show the dHvA frequencies and their amplitude.

Magnetoresistance and SdH measurements. Resistivity studies were performed in a PPMS using the DC mode of the AC-Transport option. Samples with two different crystalline orientations, that is, bars with their long direction parallel to the crystallographic a and c axes, were cut from large TaP single crystals using a wire saw. The orientation of these crystals was veried by X-ray diffraction. The samples were named S1 (I| a), S2 (I| c), S3 (I||a) and S4 (I| a). The physical dimensions of S1, S2, S3 and S4 are (width thickness length) 0.42

0.16 1.1 mm3, 0.48 0.27 0.8 mm3, 0.5 0.2 3.0 mm3 and 0.79 0.57 3.2

mm3, respectively. Contacts to the crystals were made by spot welding 25 mm platinum wire (S1 and S2) or gluing 25 mm gold wire to the sample using silver loaded epoxy (Dupont 6,838). The resistance and Hall effect were measured in six-point geometry using a current of about 3 mA at temperatures of 1.85300 K and magnetic elds up to 14 T. Crystals were mounted on a PPMS rotator option. Special attention was paid to the mounting of the samples on the rotator puck to ensure a good parallel alignment of the current and magnetic direction. The Hall contributions to the resistance and vice versa were accounted for by calculating the mean and differential resistance of positive and negative magnetic elds. Almost symmetrical resistivities were obtained for positive and negative magnetic elds when current and magnetic eld were parallel showing the excellent crystal and contact alignment of our samples. Otherwise, the negative MR was overwhelmed by the transverse resistivity. To increase the sensitivity of the angular dependent SdH measurements at low magnetic elds, external Stanford Research SR830 lock-in ampliers were used. Typical excitation currents of a 25 mA were applied at frequencies of B20 Hz.

Band structure calculations. The ab initio calculations were performed using density-functional theory with the Vienna ab initio simulation package57. Projector-augmented-wave potential represented core electrons. The modied BeckeJohnson exchange potential58,59 was employed for accurate band structure calculations. Fermi surfaces were interpolated using maximally localized Wannier functions60 in dense k-grids (equivalent to 300 300 300 in the whole BZ). Then

angle-dependent extremal cross-sections of Fermi surfaces are calculated to compare with the oscillation frequencies according to the Onsager relation.

Data availability. The data that support the ndings of this study are available from the corresponding authors B.Y. and E.H. upon request.

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Acknowledgements

We are grateful for K. Behnia, Y.-L. Chen, L.-K. Lim, Z.-K. Liu, E. G. Mele,J. Moore, S.-Q. Shen and D. Varjas for helpful discussions. This work was nancially supported by the Deutsche Forschungsgemein- schaft DFG (Project No. EB 518/1-1 of DFG-SPP 1666 Topological Insulators, and SFB 1143) and by the ERC (Advanced Grant No. 291472 Idea Heusler). R.D.d.R. acknowledges nancial support from the Brazilian agency CNPq.

Author contributions

B.Y. conceived the project. E.H., B.Y., M.Ni. and C.F. supervised the project. F.A.,M.Na. and M.B. carried out magnetization and magnetic torque measurements. C.S., R.D.d.R., N.K., M.O.A., F.A. and M.Na. performed magnetoresistance experiments. S.C.W., Y.S. and B.Y. calculated the ab initio band structure. F.A. carried out the quantum oscillation analysis and topology tting. A.G.G. simulated the current density and potential distribution. M.S. grew the single crystals. D.S. and H.B. measuredLaue and X-ray diffraction. F.A., C.S., A.G.G., J.H.B., M.Ni., E.H. and B.Y. wrote the manuscript. All authors contributed to the scientic planning and discussions.

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How to cite this article: Arnold, F. et al. Negative magnetoresistance without well-dened chirality in the Weyl semimetal TaP. Nat. Commun. 7:11615 doi: 10.1038/ncomms11615 (2016).

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