PHYSICS OF UNCONVENTIONAL METALS AND SUPERCONDUCTORS

Fermi surface of Weyl semimetals and negative longitudinal magnetoresistance in semimetals Elena Hassinger#, F. Arnold, M. O. Ajeesh, M. Baenitz, C. Felser, H. Borrmann, N. Kumar, M. Naumann, M. Nicklas, R. D. dos Reis, M. Schmidt, Y. Sun, C. Shekhar, D. Sokolov, S.-C. Wu, B. Yan

In order to find experimental signatures of Weyl in semimetals, one has to identify materials in which the Weyl nodes are close to the Fermi energy. By studying quantum oscillations in measurements of TaAs and TaP, in combination with band structure calculations, we have determined the Fermi surfaces of both compounds and hence quantified the distance of the Weyl nodes to the Fermi energy to be 10-15 meV in both compounds. In TaAs, separate Fermi surface pockets surround each node in a pair of Weyl nodes, whereas in TaP, large pockets surround pairs. This translates to a well-defined chirality in TaAs but not in TaP. Hence, TaAs is the compound of choice for a search for the . However, we have also demonstrated that the longitudinal magnetoresistance is extremely difficult to measure in clean compensated semimetals, due to their strong field-induced conductivity anisotropy. This hinders the experimental detection of the chiral anomaly in the TaAs family of compounds. Traditionally, phase transitions to, for example, chirality or handedness given by the relative superconducting or magnetic states have been des- orientation of the spin with respect to the momentum. cribed and classified by and its spontaneous Realizations of Weyl fermions were first predicted in breaking. Since the discovery of the integer quantum pyrochlore iridates [7] and in alternating layers of Hall state, a different classification scheme has emerged topological and normal insulators [8]. The subject had that is based on the topology of the quantum mechanical its breakthrough in the beginning of 2015 after the wave function [1-3]. prediction of Weyl nodes in the band structure of a Systems with non-trivial topology can appear when family of compounds, comprised of TaAs, TaP, NbAs, band inversion occurs in the presence of certain and NbP [9, 10]. These compounds, for which large symmetries. A hallmark of the non-trivial state is that single crystals can be grown, have a non- it is impossible to tune the system to a topologically centrosymmetric crystal structure that lifts the spin trivial one by changing material parameters smoothly degeneracy and gives rise to separate Weyl nodes of without passing through a quantum phase transition opposite chirality. The predicted Fermi arc surface states [4]. Edge states with a structure appearing were confirmed by angle-resolved photo emission at the boundary to a topologically trivial system, for spectroscopy (ARPES), establishing that these materials example the vacuum, are a characteristic feature of the indeed have Weyl nodes and are topological [11, 12]. non-trivial k-space topology. As opposed to Since then, a quickly increasing number of other topological insulators that are insulating in the bulk and materials have been proposed to be Weyl semimetals have conducting states only on the surface, topological [13]. semimetals additionally possess extra-ordinary bulk Evidence for Weyl fermions in bulk properties can be states [5]. In these systems, the band inversion of the obtained via the chiral anomaly [14, 15]. In particle valence and conduction bands leads to band crossing physics, this is an imbalance of the number of points (nodes) in the Brillouin zone which have a left-handed and right-handed fermions in the presence three-dimensional linear dispersion. In the presence of of parallel electric and magnetic fields. In a Weyl time-reversal symmetry and inversion symmetry, the semimetal, it will induce a bulk negative longitudinal bands are spin degenerate and the material is named a magnetoresistance [16, 17]. Consequently, there have Dirac semimetal. However, when at least one of these been a number of reports of strong negative longitudinal symmetries is broken, the spin degeneracy is lifted due magnetoresistance as proof of the chiral anomaly in all to spin-orbit coupling, the Dirac node splits into two four members of the TaAs family [18-22]. Our work Weyl nodes and the material becomes a Weyl on the TaAs family is a part of this search for semimetal [5]. In that case, the near the experimental bulk signatures of Weyl fermions. There nodes behave effectively like relativistic Weyl are theoretical predictions for Weyl signatures in other fermions, which were predicted by in physical quantities, but the chiral anomaly is the 1929 as fundamental particles when he solved the best-known and most studied one. for the massless case [6]. In addition to the linear dispersion, Weyl fermions have a certain PHYSICS OF UNCONVENTIONAL METALS AND SUPERCONDUCTORS

Chirality and topology of the Fermi surface when the appropriate method is used. Importantly, the pockets surround pairs of Weyl nodes. As First of all, since quantum oscillations are the schematically shown in Fig. 1b, this means the chirality specialization of our group, we determined the Fermi is not well defined. surface of TaP and TaAs on samples grown by Marcus Schmidt in our institute. The oscillations were detected In TaAs, in contrast, a separate Fermi surface pocket in resistivity as well as magnetization and magnetic surrounds each Weyl node of a pair [24] (see Fig. 1 torque for various directions of the magnetic field at d - f). Here, chirality is well defined, a prerequisite for low temperature. In combination with band structure the chiral anomaly and therefore, TaAs and TaP should calculations carried out by the group of Binghai Yan in show different responses in the longitudinal Claudia Felser’s department, this allows for a magnetoresistance, with a sign of the chiral anomaly reconstruction of the Fermi surface of both materials. only in the former. Our group was the first one to point Notably, due to the low carrier density, i.e. the small out these fundamental differences within this class of size of the Fermi surface pockets, the calculations alone materials because ARPES results lack the energy are not able to predict the shape of the Fermi surface as resolution to pinpoint the Fermi surface. the details depend on the method used in the The Fermi surface of NbP is very similar to TaP [25] calculation. Instead, experiments are needed to find the and the Fermi surface of NbAs is not completely right method of calculation so that the Fermi surface is resolved yet. In both materials, very small pockets, best reproduced. which could be chiral, appear. However, in the Nb In a nutshell, our main results are the following. In TaP compounds, experimental results and theoretical the Fermi surface consists of banana-shaped electron calculations do not agree as well as in the Ta com- and hole pockets [23], see Fig. 1a – c. These pockets pounds. Hence, further study is needed to address this are well described by the calculations discrepancy.

Fig.-1: a) and d): Schematic energy dispersion of a Weyl semimetal along the k-line connecting the Weyl nodes. b) and e): Depending on the level of the Fermi energy, the Fermi surface consists of either a large non-chiral Fermi surface pocket enclosing two Weyl nodes (a)-c)) or two separate chiral pockets (d)-f)). Arrows depict the spin structure responsible for chirality. c) and f): Realizations of each case are found in TaP for non-chiral Fermi surfaces or TaAs for chiral ones, respectively (blue: electron pockets, red: hole pockets).

PHYSICS OF UNCONVENTIONAL METALS AND SUPERCONDUCTORS

Longitudinal magnetoresistance of semimetals As mentioned before, all four members of the TaAs family have been reported to display a negative longitudinal magnetoresistance [18-22]. With the absence of chirality in TaP, established by our Fermi surface study, a negative magnetoresistance is not expected and should have an origin other than the chiral anomaly. We investigated the longitudinal magnetoresistance in our samples of TaP and NbP [23, 26] in order to find out more about the origin of the signal. This study was carried out in collaboration with the group of M. Nicklas, on samples prepared by M. Schmidt. When the magnetic field is parallel to the current, we find that the voltage contact position influences strongly the measured magnetoresistance (Fig. 2). This directly proves a field-induced inhomogeneous current distribution in the material. Its origin can be explained by classical physics in compensated semimetals. For a magnetic field perpendicular to the current direction, the conductivity in a compensated semimetal strongly decreases with field due to the orbital effect, whereas the conductivity parallel to the field remains approximately constant. Consequently, when the current enters the sample at a point-like contact, it is strongly steered in the direction of the field, where conductivity remains high. It forms a jet, hence the name “current jetting” [27]. Decisive for the appearance of an inhomogeneous current distribution is a conductivity anisotropy, regardless of its cause. If the chiral anomaly was present in the semimetal, it would increase the conductivity Fig.-2: a): Longitudinal normalized voltage change anisotropy even more. In our paper we give clear [MR* = (V – V0) / V0 , V0 = V (B = 0)] as a function of evidence that current jetting strongly affects magnetic field for three pairs of voltage contacts at longitudinal resistivity measurements in all four different positions on the sample (inset). The current member of the TaAs family. In particular, the angular is injected through I- and I+. The voltage measured at dependence measured at constant field in NbP shows V1, far away from the current injection point, the positive and negative peaks that are a telling decreases in high field as the current concentrates signature of current jetting effects. In addition, the along the line connecting the current contacts conductivity anisotropy is caused predominantly by whereas the voltage at V2 increases due to the the decrease of the transverse resistance rather than the increased current along this line. b): Electric increase of the longitudinal conductivity as we would potential distribution for different conductivity expect from the chiral anomaly. We can support this by anisotropies A. A = 1 corresponds to the situation in reproducing the experimental data with finite element zero magnetic field. The current flows homogenously calculations of the potential distribution without taking through the sample as the potential planes are into account the chiral anomaly. The simulated signal parallel. A = 100 can be reached for typical can reproduce the experimental data rather well. laboratory fields depending on the sample studied. Avoiding the inhomogenous currents by making The potential is strongly distorted, with a current homogeneous contacts over the whole cross section of flowing along the line connecting the current contacts. the samples is extremely difficult and is one of our aims Note that in the experiment, the current was injected for the near future. For now, the intrinsic longitudinal in the top right corner of the face surface whereas in magnetoresistance, including information on the chiral the simulations it is injected in the top center [23, 26]. anomaly, remains undetermined in the TaAs family. PHYSICS OF UNCONVENTIONAL METALS AND SUPERCONDUCTORS

The chiral anomaly was also reported in other [16] The Adler-Bell-Jackiw anomaly and Weyl fermions in a materials. Our work has lead to a realization of most crystal, H. B. Nielsen and M. Ninomiya, Phys. Lett. B 130 (1983) 389. scientists in the field that these measurements have to [17] Chiral anomaly and the classical negative be checked very carefully to rule out false signals for the magnetoresistance of Weyl metals, D. T. Son and longitudinal magnetoresistance induced by current B. Z. Spivak, Phys. Rev. B 88 (2013) 104412. jetting. [18] Observation of the Chiral-Anomaly-Induced Negative Magnetoresistance in 2D Weyl Semimetal TaAs, External Cooperation Partners X. C. Huang, L. X. Zhao, Y. J. Long, P. P. Wang, D. Chen, Z. H. Yang, H. Liang, M. Q. Xue, H. M. Weng, J. H. Bardarson (MPI-PKS, Germany); A. G. Grushin Z. Fang, X. Dai and G.F. Chen, Phys. Rev. X 5 (2015) 031023. (MPI-PKS, Germany) [19] Signatures of the Adler-Bell-Jackiw chiral anomaly in a Weyl semimetal, C.-L. Zhang, S.-Y. Xu, References I. Belopolski, Z. Yuan, Z. Lin, B. Tong, G. Bian, [1] The renormalization group and the ε expansion, N. Alidoust, C.-C. Lee, S.-M. Huang, T. R. Chang, K. Wilson and J. Kogut, Phys. Rep. 12, (1974) 75. G. Chang, C.-H. Hsu, H.-T. Jeng, M. Neupane, [2] L. D. Landau, E. M. Lifshitz and M. Pitaevskii, D. S. Sanchez, H. Zheng, J. Wang, H. Lin., C. Zhang., Butterworth-Heinemann (1999). H.-Z. Lu, S.-Q. Shen, T. Neupert, M. Z. Hasan and S. Jia, [3] Quantized Hall Conductance in a Two-Dimensional Nat. Commun. 7 (2016) 10735. Periodic Potential, D. J. Thouless, M. Kohmoto, [20] Chiral anomaly induced negative magnetoresistance in M. P. Nightingale and M. den Nijs, Phys. Rev. Lett 49 topological Weyl semimetal NbAs, X. Yang, Y. Liu, (1982) 405. Z. Wang, Y. Zheng and Z.-A. Xu, arXiv:1506.03190 [4] Topological Order and the Quantum Spin Hall Effect, (2015). C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95 (2005) [21] Large unsaturated positive and negative 146802. magnetoresistance in Weyl semimetal TaP, J. H. Du, [5] Phase transition between the quantum spin Hall and H. D. Wang, Q. Chen, Q. H. Mao, R. Khan, B. J. Xu, insulator phases in 3D: emergence of a topological Y. X. Zhou, Y. N. Zhang, J. H. Yang, B. Chen, C. M. Feng gapless phase, S. Murakami, New J. Phys. 9 (2007) 356. and M. H. Fang, Sci. China-Phys. Mech. Astron. 59 [6] Elektron und Graviation, H. Weyl, Zeitschrift f. Phys. 56 (2016) 657406. (1929) 330. [22] Helicity protected ultra high mobility Weyl fermions in [7] Topological Semimetal and Fermi-arc surface states in NbP, Z. Wang, Y. Zheng, Z. X. Shen, Y. H. Lu, H. Y. Fang, the electronic structure of pyrochlore iridates, X. Wan, F. Sheng, Y. Zhou, X. J. Yang, Y. P. Li, C. M. Feng and A. M. Turner, A. Vishwanath and S. Y. Savrasov, Phys. Z. A. Xu, Phys. Rev. B 93 (2016) 121112(R). Rev. B 83 (2011) 205101. [23]* Negative magnetoresistance without well-defined [8] Topological nodal semimetals, A. A. Burkov, M. D. Hook chirality in the Weyl semimetal TaP, F. Arnold, and L. Balents, Phys. Rev. B 84 (2011) 235126. C. Shekhar, S.-C. Wu, Y. Sun, R. D. dos Reis, N. Kumar, [9] A Weyl Fermion semimetal with surface Fermi arcs in the M. Naumann, M. O. Ajeesh, M. Schmidt, A. G. Grushin, transition metal monopnictide TaAs class, S.-M. Huang, J. H. Bardarson, M. Baenitz, D. Sokolov, H. Borrmann, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, BK. Wang, M. Nicklas, C. Felser, E. Hassinger and B. Yan, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S. Jia, Nat. Commun. 7 (2016) 11615. A. Bansil, H. Lin and M. Z. Hasan, Nat. Commun. 6 [24]* Chiral Weyl pockets and Fermi surface topology of the (2015) 7373. Weyl semimetal TaAs, F. Arnold, M. Naumann, S.-C. Wu, [10] Weyl Semimetal Phase in Noncentrosymmetric Y. Sun, M. Schmidt, H. Borrmann, C. Felser, B. Yan and Transition-Metal Monophosphides, H. M. Weng, E. Hassinger, Phys. Rev. Lett. 117 (2016) 146401. C. Fang, Z. Fang, B. Andrei Bernevig and X. Dai, Phys. [25] Quantum oscillations and the Fermi surface topology of Rev. X 5 (2015) 011029. the Weyl semimetal NbP, J. Klotz, S. Wu, C. Shekhar, [11] Experimental discovery of Weyl Semimetal TaAs, Y. Sun, M. Schmidt, M. Nicklas, M. Baenitz, M. Uhlarz, B. Q. Lv, H. M. Wang, B. B. Fu, X. P. Wang, H. Miao, J. Wosnitza, C. Felser and B. Yan, Phys. Rev. B 93 (2016) J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, 121105(R). Z. Fang, X. Dai, T. Qian and H. Ding, Phys. Rev. X 5 [26]* On the search for the chiral anomaly in Weyl semimetals: (2015) 031013. The negative longitudinal magnetoresistance, [12] Evolution of the Fermi surface of Weyl semimetals in the R. D. dos Reis, M. O. Ajeesh, N. Kumar, F. Arnold, transition metal pnictide family, Z. K. Liu, L. X. Yang, C. Shekhar, M. Naumann, M. Schmidt, M. Nicklas and Y. Sun, T. Zhang, H. Peng, H. F. Yang, C. Chen, E. Hassinger, New J. of Phys. 18 (2016) 085006. Y. Zhang, Y. F. Guo, D. Prabhakaran, M. Schmidt, [27] Magnetoresistance in Metals, B. Pippard, Cambridge Z. Hussain, S.-K. Mo, C. Felser, B. Yan and Y. L. Chen, University Press (1989).

Nat. Mater. 15 (2016) 27. ______[13] Topological Materials: Weyl Semimetals, B. Yan and # C. Felser, Annu. Rev. Condens. Matter Phys. 8 (2017) [email protected] 337. [14] Axial Vector Vertex in Spinor Electrodynamics, S. L. Adler, Phys. Rev. 177 (1969) 2426. [15] A PCAC puzzle; π0 -> γγ in the σ-model, J. S. Bell and R. Jackiw, Nuovo Cimento A 60 (1969) 47.