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 chiralmagnetic effect. However, it remains an open question to which extent this effect survives when chirality is not welldefined. Here, we establish the detailed Fermisurface topology of the recently identified WSM TaP via combined angleresolved quantumoscillation spectra and bandstructure calculations. The Fermi surface forms bananashaped electron and hole pockets surrounding pairs of Weyl points. Although this means that chirality is illdefined in TaP, we observe a large negative longitudinal magnetoresistance. We show that the magnetoresistance can be affected by a magnetic fieldinduced inhomogeneous current distribution inside the sample.
Introduction
In a semimetal the conduction and valence bands touch at isolated points in the threedimensional (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)^{1,2,3,4,5,6,7,8,9} or a Dirac semimetal^{10,11,12}, 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 timereversal 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 particle^{13}. In condensedmatter 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 handedness^{14}. In the presence of nonorthogonal magnetic (B) and electric (E) fields (that is, E·B is nonzero), the particle number for a given chirality is not conserved quantum mechanically, inducing a phenomenon known as the Adler–Bell–Jackiw anomaly or chiral anomaly in highenergy physics^{13,15,16}. In WSMs, the chiral anomaly is predicted to lead to a negative magnetoresistance (MR) due to the suppressed backscattering of electrons of opposite chirality^{17,18}. Theoretically, the chiral anomaly only appears, if chirality is welldefined, that is, the Fermi energy is close enough to the Weyl nodes that pairs of separate Fermi surface pockets with opposite chirality exist^{17}. Observing the chiralanomaly induced negative MR requires the applied magnetic and electric field 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 effect^{2,19,20,21}, nonlocal transport properties^{22,23} and sharp discontinuities in angleresolved photoemission spectroscopy (ARPES) signals^{24} in WSMs.
The discovery of various WSM materials has stimulated experimental efforts to confirm the chiral anomaly in condensedmatter physics. Recently, a negative MR has been reported in two types of WSMs: WSMs induced by timereversal symmetry breaking, that is, Dirac semi metals in an applied magnetic field, for example Bi_{1−x}Sb_{x} (x≈3%)^{25}, ZrTe_{5} (ref. 26) and Na_{3}Bi^{27}, and the noninversionsymmetric WSMs TaAs^{28,29}, NbP^{30} and NbAs^{31}. However, a clear verification of whether the Fermi surface topology supports the chiral anomaly or not, is still lacking in most of the above systems.
In the noncentrosymmetric WSMs of the TaAs family two types of Weyl nodes exist at different positions in reciprocal space^{5,6} and energies. Therefore, the Weyl electrons generally coexist with topologigcally trivial normal electrons. In principle, small changes of the Fermi energy (E_{F}), as induced by doping or defects, can change the Fermisurface topology significantly due to the low intrinsic charge carrier density in semimetals. Therefore, a precise knowledge of E_{F} and the resulting Fermisurface topology is required when linking the negative MR to the chiralmagnetic effect. Extensive ARPES studies have shown the existence of Fermiarc surface states and linear band crossings in the bulk band structure of all four materials from the TaAs family^{7,8,9,32,33}. However, because of an insufficient energy resolution (>15 meV (ref. 32)), ARPES is not able to make any claims about the presence or absence of quasiparticles with welldefined chirality at the Fermi level. In contrast, quantumoscillation measurements have the advantage of a millielectronvolt resolution of the Fermienergy level.
In this work, we reconstruct the 3D Fermi surface of TaP by combining sensitive Shubnikovde Haas (SdH) and de Haasvan Alphen (dHvA) oscillations with ab initio bandstructure calculations reaching a good agreement between theory and experiment. We reveal that E_{F} is such that the electron and hole Fermisurface pockets contain pairs of Weyl nodes and the total Berry flux through the Fermi surface Φ_{B}=0. Although this means that chirality is not welldefined, a large negative MR is observed. We discuss possible explanations for this result also considering the magnetic field dependence of the current distribution in our samples^{34}.
Results
Singlecrystal synthesis and characterization
We synthesized highquality single crystals of TaP by using chemical vapour transport reactions and verified TaP as a noncentrosymmetric compound in a tetragonal lattice (space group I4_{1}md, No. 109). The temperaturedependent resistivity exhibits typical semimetallic behaviour. For more details, see Methods and Supplementary Figs 1–4.
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 bismuth^{35}, 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 field orientations. These oscillations are periodic in 1/B. Their frequency (F) is proportional to the corresponding extremal Fermi surface crosssection (A) that is perpendicular to B following the Onsager relation F=(Φ_{0}/2π^{2})A, where Φ_{0}=h/2e is the magnetic flux quantum and h is the Planck constant. Figure 1a shows the resistivity as a function of the magnetic field for different field orientations. When the electric current and magnetic field are perpendicular , the magnetoresistance is very high as typical for other WSM (for example, ref. 36) and normal semimetals^{37,38}. This implies a very high mobility of the charge carriers. Figure 1b depicts the magnetic torque oscillations for the same field orientations. Figure 2a represents the magnetic dHvA oscillations in the magnetization as a function of the inverse magnetic field and their corresponding Fourier transform (Fig. 2b) for Bc. The observable fundamental frequencies are F_{α}=15 T, F_{β}=18 T, F_{γ}=25 T, and F_{δ}=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 Ba the main frequencies are F_{α}=26 T, F_{γ}=34 T, F_{γ′}=105 T F_{δ}=147 T and F_{γ″}=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 fitting the temperature dependence of the oscillation amplitude (inset of Fig. 2b) with the Lifshitz–Kosevich formula (see ref. 39 and Supplementary Figs 5 and 6 as well as Supplementary Note 1). The values of the effective masses are =(0.021±0.003)m_{0}, =(0.05±0.01)m_{0} and =(0.11±0.01)m_{0} for Bc, whereas they are a factor of 4–10 greater for Ba with =(0.13±0.03)m_{0}, =(0.35±0.03)m_{0} and =(0.4±0.1)m_{0}, where m_{0} 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 Cd_{3}As_{2} or graphene^{40,41}. These low masses, together with long scattering times are the reason for the high mobility and the huge transverse magnetoresistances seen in semimetals.
Fermisurface topology
To reconstruct the shape of the Fermi surface, the full angular dependence of the quantumoscillation frequencies was compared with bandstructure calculations (Fig. 1c). The exact position of E_{F} was determined by matching the calculated frequencies and their angular dependence to the experimental ones (see Supplementary Figs 7–10 and Supplementary Note 2). The best fit is obtained when E_{F} lies 5 meV above the ideal electron–hole compensation point, in agreement with the resulting carrier concentrations from Hall measurements (Fig. 2c,d). At this E_{F}, calculations reveal two bananashaped 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 rings^{5,6} on the k_{x}=0 and k_{y}=0 mirrorplanes in the BZ. The rather isotropic frequencies F_{α} and F_{γ} 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 field angles departing from Bc in the (100)plane is explained by the existence of four bananashaped pockets in the BZ, two for each mirror plane for both E and Hpockets. The splitting of the frequency F_{δ} seen in the experiment is not reproduced by the calculation. One possibility for this discrepancy is that a waist may appear in the Epocket at k_{z}=0.
In addition, in the dHvA experiment the mobility of the hole orbits (F_{α} and F_{β}) was extracted via the width of the Fouriertransform peaks, which is given by the exponential decrease of the oscillations with 1/B (the socalled Dingle term; see Supplementary Note 3). The deduced mobility is μ_{h}=3.2 × 10^{4} (±20%) cm^{2} V^{−1} s^{−1} (star in Fig. 2d).
Chargecarrier density and mobility
We extract information on the carrier density and mobility from magnetic field and temperaturedependence 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 twocarrier model (see ref. 42) to fit the Hall conductivity (σ_{xy}) by making use of the longitudinal conductivity at zero field (σ_{xx}) 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=(2±1) × 10^{19} cm^{−3} with an absolute error bar (greyshaded 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 E_{F} lying 5 meV above the chargeneutral 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 μ=(2–5) × 10^{4} cm^{2} 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 reflected in the sign change of the highfield Hall resistivity (see Supplementary Figs 11 and 12) and is similar to the Hall effect observed in TaAs^{28,29} and NbP^{30,36}. Furthermore, the Hall mobilities agree well with the mobility determined by the Dingle analysis (the star in Fig. 2d).
Discussion
The experimental Fermi surface topology and chargecarrier concentrations described above converge to the same statement that the Fermi energy of our samples lies 5 meV above the ideal chargeneutral 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 possible^{43} and can explain the small shift of E_{F}. 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 k_{z}=0 plane (labelled as W1) and eight pairs are located in planes close to k_{z}=±π/c (where c is the lattice constant) (labelled as W2). One can see that the W1 points are far below E_{F} by 41 meV and are included in the Epocket. The W2type Weyl points are 13 meV above E_{F} and are included in the Hpocket. Such pairs of W2 points merge slightly into the head position of the Hpocket, leading to a twohornlike crosssection (see F_{γ} 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 F_{γ} plane in Fig. 3d. There are no independent Fermisurface pockets around the W2 Weyl points and therefore the chiral anomaly is not welldefined. 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 magnetoresistance of three samples from the same crystal batch. The current (I) was applied along the crystallographic c (sample S2) 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 fixed electric current) obtained for the two crystallographic directions. As long as the current flows homogeneously in the sample, MR* is equal to the usual MR=[ρ−ρ(B=0)]/ρ(B=0). At 1.85 K, MR* first 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 θ≤2° and is positive otherwise (see Supplementary Fig. 13 and Supplementary Note 5). This small θ window for the apparent negative MR is similar to that observed in Na_{3}Bi^{27}.
In light of the given Fermienergy position in our crystals, it is not possible to link the negative MR to the chiral anomaly, simply because the former is not a welldefined concept when the Fermi surface connects both Weyl nodes. Although this does not rule out the presence of a nontrivial negative MR^{44,45,46}, we find that the observed negative MR*, is strongly affected by the geometric configuration.
This becomes evident when we examine the apparent longitudinal MR for three different voltage contact configurations on sample S4 as illustrated in Fig. 4c. A clear voltage decrease in magnetic field 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 fields. 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 crosssection of the sample, this leads to a fieldinduced stearing of the current to the direction of the magnetic field, which is along the line connecting the current contacts when current and magnetic field are parallel. This effect is known as current jetting^{34,47,48,49,50,51}. 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 confirmed 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 fieldindependent intrinsic longitudinal MR, the model qualitatively reproduces the three MR* curves without any free parameters (see Fig. 4d). Therefore, the magnetic field dependence of the longitudinal voltage is largely induced by the strong transverse MR, if the current is not homogeneously injected into the whole crosssection of the sample.
In summary, we determined the Fermisurface topology of the inversionasymmetric WSM TaP. The Fermi surface consists of bananashaped spinpolarized electron and hole pockets with very light carrier effective masses. Despite the absence of independent Fermisurface 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 fields.
Methods
Singlecrystal growth
Highquality 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 singlecrystals 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}.
Xray analysis and structure characterisation
The crystal structure and orientation of TaP crystals were determined by Xray diffraction at room temperature. For this, TaP single crystals were mounted on a fourcircle Rigaku AFC7 Xray diffractometer with a builtin Saturn 724+ CCD detector. A suitable sample edge was selected where the transmission of Mo−K_{α} (λ=0.71073 Å) radiation seemed feasible. The intensities of the obtained reflections were corrected for absorption by using a multiscan 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, confirmed the appropriate choice of the unit cell and showed the excellent crystal quality. Supplementary Figs 1 and 2 show Xray diffraction patterns of the S1 and S2 TaP crystals, which were used in our transport measurements. Structure refinement was performed by fullmatrix leastsquares on F within the program package WinCSD, Version 4.14 (ref. 53 and revealed the noncentrosymmetric crystal structure with space group I4_{1}md and lattice parameters a=b=3.30 Å, c=11.33 Å at room temperature.
To confirm 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 Xray diffraction method. Supplementary Fig. 3 shows the corresponding Laue diffraction image indexed with the I4_{1}mdstructure and room temperature lattice parameters. The Laue diffraction image shows sharp reflections, which confirm the excellent quality of the sample. The presence of domains or twinning can be ruled out by indexing all reflections 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 experiments^{39}. The frequency of these oscillations F=A_{ext} × ħ/2πe is proportional to the extremal Fermi surface crosssection A_{ext} perpendicular to the respective magnetic field direction^{39,54,55}. 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. SQUIDVSM in the temperature and magnetic field range of 2–50 K and ±7 T.
Angular dependencies were measured using the Quantum Design Inc. piezoresistive torque magnetometer (TqMag^{56}) 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 confirmed by Xray diffraction. The crystal alignment was verified by photometric methods and showed typical misalignments <2°. Magnetic torque measurements were performed up to 14 T in the same temperature range. Here the sample is mounted on a flexible lever which bends when torque is applied. The sample magnetization M induces a magnetic torque , which bends the lever and can be sensed by piezoresistive elements which are microfabricated 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 Ω 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 lockin amplifiers as readout electronic. This way a resolution of one in 10^{7} was achieved. The magnetic torque was determined for magnetic fields in the (100), (001) and (110)plane in angular steps of 2.5° and 5°. Measurements were taken during magnetic field down sweeps from 14 to 0 T. The magnetic field 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 field, this method can only be applied to samples with strong Fermi surface anisotropy and is insensitive when the magnetic field 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 signals and performing a Fourier transformation of the residual signal over the inverse magnetic field. 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 ACTransport 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 verified by Xray diffraction. The samples were named S1 (I a), S2 (I c), S3 (Ia) and S4 (I a). The physical dimensions of S1, S2, S3 and S4 are (width × thickness × length) 0.42 × 0.16 × 1.1 mm^{3}, 0.48 × 0.27 × 0.8 mm^{3}, 0.5 × 0.2 × 3.0 mm^{3} and 0.79 × 0.57 × 3.2 mm^{3}, respectively. Contacts to the crystals were made by spot welding 25 μm platinum wire (S1 and S2) or gluing 25 μm gold wire to the sample using silver loaded epoxy (Dupont 6,838). The resistance and Hall effect were measured in sixpoint geometry using a current of about 3 mA at temperatures of 1.85–300 K and magnetic fields 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 fields. Almost symmetrical resistivities were obtained for positive and negative magnetic fields when current and magnetic field 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 fields, external Stanford Research SR830 lockin amplifiers were used. Typical excitation currents of a 2–5 mA were applied at frequencies of ∼20 Hz.
Band structure calculations
The ab initio calculations were performed using densityfunctional theory with the Vienna ab initio simulation package^{57}. Projectoraugmentedwave potential represented core electrons. The modified Becke–Johnson exchange potential^{58,59} was employed for accurate band structure calculations. Fermi surfaces were interpolated using maximally localized Wannier functions^{60} in dense kgrids (equivalent to 300 × 300 × 300 in the whole BZ). Then angledependent extremal crosssections of Fermi surfaces are calculated to compare with the oscillation frequencies according to the Onsager relation.
Data availability
The data that support the findings of this study are available from the corresponding authors B.Y. and E.H. upon request.
Additional information
How to cite this article: Arnold, F. et al. Negative magnetoresistance without welldefined chirality in the Weyl semimetal TaP. Nat. Commun. 7:11615 doi: 10.1038/ncomms11615 (2016).
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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 financially supported by the Deutsche Forschungsgemein schaft DFG (Project No. EB 518/11 of DFGSPP 1666 Topological Insulators, and SFB 1143) and by the ERC (Advanced Grant No. 291472 Idea Heusler). R.D.d.R. acknowledges financial support from the Brazilian agency CNPq.
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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 fitting. A.G.G. simulated the current density and potential distribution. M.S. grew the single crystals. D.S. and H.B. measured Laue and Xray 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 scientific planning and discussions.
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Supplementary Figures 118, Supplementary Table 1, Supplementary Notes 16 and Supplementary References (PDF 2195 kb)
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Arnold, F., Shekhar, C., Wu, SC. et al. Negative magnetoresistance without welldefined chirality in the Weyl semimetal TaP. Nat Commun 7, 11615 (2016). https://doi.org/10.1038/ncomms11615
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DOI: https://doi.org/10.1038/ncomms11615
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