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Observation of Weyl nodes in TaAs

Abstract

In 1929, H. Weyl proposed that the massless solution of the Dirac equation represents a pair of a new type of particles, the so-called Weyl fermions1. However, their existence in particle physics remains elusive after more than eight decades. Recently, significant advances in both topological insulators and topological semimetals have provided an alternative way to realize Weyl fermions in condensed matter, as an emergent phenomenon: when two non-degenerate bands in the three-dimensional momentum space cross in the vicinity of the Fermi energy (called Weyl nodes), the low-energy excitations behave exactly as Weyl fermions. Here we report the direct observation in TaAs of the long-sought-after Weyl nodes by performing bulk-sensitive soft X-ray angle-resolved photoemission spectroscopy measurements. The projected locations at the nodes on the (001) surface match well to the Fermi arcs, providing undisputable experimental evidence for the existence of Weyl fermionic quasiparticles in TaAs.

Main

The massless Dirac equation in the three-dimensional (3D) momentum space can be regarded as the overlap of two Weyl fermions with opposite chirality1,2. The Dirac fermionic quasiparticle is stable under the protection of some crystal symmetry in topological Dirac semimetals such as Na3Bi (ref. 3) and Cd3As2 (ref. 4). However, a separated single Weyl node is much more robust and requires no protection of crystal symmetry. An isolated Weyl node is a sink or source of gauge field of Berry curvature, like a monopole in momentum space, and the chirality corresponds to its topological charge5,6,7. Weyl nodes appear in pairs of opposite chirality in a real material due to the ‘No-go theorem’8,9. To obtain isolated Weyl nodes, the spin degeneracy of the electronic bands has to be removed by breaking either inversion symmetry or time-reversal symmetry. Although non-degenerate band crossing is not rare, finding a material with only Weyl nodes near the Fermi energy (EF) is a big challenge. Recently, the noncentrosymmetric and non-magnetic transition-metal monoarsenide TaAs has been predicted to be a Weyl semimetal (WSM), and twelve pairs of Weyl nodes are expected in its 3D Brillouin zone (BZ; refs 10, 11). Compared with other proposals6,7,12,13,14,15,16,17,18,19,20,21,22 for realizing a Weyl state, the TaAs family features easy sample fabrication, a non-magnetic state and no fine-tuning of the electronic states is necessary, making experimental studies of Weyl semimetals possible. Many exotic properties induced by the Weyl nodes have been predicted and observed recently, such as surface states with Fermi arcs23,24 and a negative magneto-resistivity25,26 due to the chiral anomaly27,28,29. However, crucial evidence for Weyl nodes in the bulk states has not been observed. In this paper, by using soft X-ray angle-resolved photoemission spectroscopy (ARPES), which is sensitive to the bulk states, we report the first experimental observation of Weyl nodes in TaAs.

TaAs crystallizes in a body-centred-tetragonal structure with the nonsymmorphic space group I41md. The lattice parameters are a = b = 3.4348 Å and c = 11.641 Å at 300 K. The crystal structure consists of alternating Ta and As layers (Fig. 1a). The adjacent TaAs layers are rotated by 90° and shifted by a/2, leading to a lack of inversion symmetry. First-principles band calculations predict that TaAs is a time-reversal invariant 3D WSM with twelve pairs of Weyl nodes in the 3D BZ (Fig. 1c)10,11. These Weyl nodes with opposite chirality are linked to the Fermi arcs of the surface state in the projected surface BZ.

Figure 1: Crystal structure and electronic structure of TaAs.
figure1

a, Crystal structure of TaAs. b, Bulk BZ of TaAs with the high-symmetry points indicated. The shaded planes represent the mirror-invariant (100) and (010) planes. c, Fermi surfaces from first-principles calculations. The purple banana-like hole pockets are topologically trivial Fermi surfaces that do not enclose a Weyl node. The blue date-like electron pockets enclose the Weyl nodes W1, which are 21 meV below EF. The Weyl nodes W2 in the kz = 0 plane are only 2 meV above EF (the Fermi surfaces enclosing them are tiny and marked by the red dots). d, Schematic band dispersions of one pair of Weyl nodes in a 2D plane of the momentum space. The arrows indicate that the Weyl node is a monopole of Berry flux in the momentum space.

Fermi-arc-like surface states have been identified in recent vacuum ultraviolet (VUV) ARPES experiments23,24, which are remarkably well reproduced by the band calculations for the As termination23. However, due to the short escape depth of the photoelectrons excited by VUV light, the bulk electronic structure was nearly invisible in the VUV ARPES experiments. To probe the bulk electronic states of TaAs, we use soft X-rays to promote valence photoelectrons to a higher kinetic energy with a much longer escape depth, thus representing the bulk band structure. This bulk nature is reflected in the shallow core-level spectra shown in Fig. 2a, where the two As 3d surface peaks, which are strong at photon energy = 70 eV, almost vanish at = 440 eV. In addition, due to the Heisenberg uncertainty principle, increasing the photoelectron escape depth sharpens the intrinsic experimental resolution in the surface-perpendicular momentum (kz; ref. 30), allowing accurate navigation in the 3D momentum space.

Figure 2: Core-level spectra and electronic structure in the kx = 0 plane.
figure2

a, Core-level spectra taken at = 70 and 440 eV, respectively. The inset shows the magnification of the As 3d core-level peaks, indicating that the surface electronic states are drastically suppressed with the incident soft X-rays. The peaks marked as ‘Surface’ are contributed by photoelectrons from the topmost As layer. b, Photoemission intensity plot at EF in the kykz plane at kx = 0. The green lines represent the BZ structure in the kykz plane. c,d, Photoemission intensity plot (c) and energy distribution curves (d) along Γ–Z (C1) indicated in b. e,f, Photoemission intensity plot (e) and energy distribution curves (f) along Γ–Σ(S)–Z (C2) indicated in b. For comparison, the calculated bands along Γ–Z and Γ–Σ(S)–Z are plotted on top of the experimental data in c and e, respectively.

The bulk nature of the electronic states is further confirmed by the strong kz dispersion on varying the incident photon energy (Fig. 2). The Fermi surfaces and band dispersions exhibit a modulation along the kz direction with a period of 2π/c′, where c′ is one half of the c-axis lattice constant of TaAs. The experimental bands along the Γ–Z and Γ–Σ(S)–Z high-symmetry lines are in remarkably good agreement with the bulk bands from first-principles calculations with spin–orbit coupling (SOC) included, as demonstrated in Fig. 2. Just below EF, there are two bands clearly observed along Σ(S)–Γ in Fig. 2e, which are mostly composed of Ta 5d orbitals and well reproduced by calculations. The large band splitting of these two bands is due to the Rashba-like SOC originating from broken inversion symmetry and intrinsic SOC in Ta 5d electrons. The calculations without SOC show that the bands crossing near EF within the mirror-invariant planes are protected by the mirror symmetry, leading to gapless nodal rings in the mirror plane10. As the SOC is turned on, the nodal rings are fully gapped, generating non-zero mirror Chern numbers for the occupied states on the mirror planes and Weyl nodes slightly off the mirror planes10.

To identify the Weyl nodes and their locations in the momentum space, we mapped out the 3D electronic structure with a number of incident photon energies. The Fermi surface map at = 440 eV in Fig. 3b clearly shows two date-like pockets slightly off the mirror plane. The pockets originate from an ‘M’-shaped band perpendicular to the (010) mirror plane (Fig. 3d, e), whose tops touch EF at kx = ±0.06π/a. On moving along the ky direction, the two tops gradually merge together and sink below EF, with this band eventually evolving into a ‘Λ’ shape, as illustrated in Fig. 3h. The band parallel to the mirror plane through one of the pockets exhibits a ‘Λ’ shape with its top at ky = −3.53π/a (Fig. 3f, g). These observed band dispersions in the kxky plane are highly consistent with the calculated Weyl-cone band structure of W1.

Figure 3: Band dispersions near the Weyl nodes W1.
figure3

a, Photoemission intensity plot at EF in the kykz plane at kx = 0. The four red curves from top to bottom indicate the momentum locations taken at = 440, 435, 430 and 420 eV, respectively. b, Photoemission intensity plot at EF taken at = 440 eV. The red arrows indicate the date-like pockets that enclose the Weyl nodes W1. c, Schematic band dispersion of one pair of Weyl nodes in the kxky plane. The vertical planes C1–C6 schematize the ARPES measurements that slice through the Weyl cone at different ky positions. d,e, Photoemission (d) and curvature intensity (e) plots along the kx direction through the two Weyl nodes W1, illustrating the ‘M’-shape dispersion of a pair of Weyl nodes. f,g, Photoemission (f) and curvature intensity (g) plots along the ky direction through one of the Weyl nodes W1. For comparison, the calculated bands through the Weyl nodes W1 along the kx and ky directions are plotted (red curves) on top of the experimental data in d and f, respectively. Note that the calculated bands in d and f are shifted by 0.11π/a along the ky direction to be consistent with the experimental bands. h, Band dispersions along the kx direction at different ky positions. i, Band dispersions along the kx direction at different kz positions. The dotted lines in h and i, which are guides to the eye indicating the band dispersions, show the evolution of the band shape. The dashed lines in h and i track the band tops.

To verify the 3D nature of the cone, we also investigate the band dispersion along the kz direction. In this case, as shown in Fig. 3i, the ‘M’-shaped band also evolves into a ‘Λ’ shape. These results demonstrate that these pockets enclose the 3D Weyl nodes W1 that are located at (±0.06, ±0.47, ±0.58) in units of (π/a, π/a, π/c′), which are pretty close to the calculated locations (±0.059, ±0.556, ±0.592) (ref. 10).

The predicted Weyl nodes near the BZ boundary (labelled as W2) are also identified in Fig. 4. As compared with the W1 nodes, the nodes of the W2 pair are very close to each other in the calculations10. This is consistent with our experimental results, in which we observe a nearly ‘Λ’-shaped band rather than an ‘M’-shaped band along the cut perpendicular to the mirror plane (Fig. 4c, d). The band also exhibits a ‘Λ’ shape along both the ky and kz directions, with a much smaller slope along kz, which is consistent with the band calculations. The excellent agreement between the experimental and calculated band dispersions along all three directions supports the existence of the Weyl nodes W2. From the measured dispersions, the Weyl nodes W2 are determined to be at (0, ±1.05, 0), which are very close to the calculated locations (±0.01, ±1.03, 0) (ref. 10).

Figure 4: Band dispersions near the Weyl nodes W2.
figure4

a, Photoemission intensity plot at EF in the kykz plane at kx = 0. The black curve indicates the momentum location for the data recorded at = 408 eV. b, Photoemission intensity plot at EF taken at = 408 eV. c,d, Photoemission (c) and curvature intensity (d) plots along Cx indicated in b. e,f, Same as c and d, but recorded along Cy indicated in a. g,h, Same as c and d but recorded along Cz indicated in a. For comparison, the calculated bands along the kx, ky and kz directions through the Weyl nodes W2 are plotted (red curves) on top of the experimental data in c,e and g, respectively.

Finally, it is crucial to verify experimentally the relationship between the bulk Weyl nodes and the surface Fermi arcs. When a pair of bulk Weyl cones with opposite chirality are projected onto the surface, there must be a surface Fermi arc connecting them (Fig. 5a)6. As shown in Fig. 5b, the projections of the Weyl nodes W1 onto the (001) surface BZ are exactly located on the loci of the Fermi arcs observed by our VUV ARPES (ref. 23). Therefore, our results have provided convincing experimental evidence that Weyl nodes in certain projected surfaces are the ‘source’ (or ‘drain’) for the surface Fermi arcs, which is a hallmark of a WSM.

Figure 5: Relationship between bulk Weyl nodes and surface Fermi arcs.
figure5

a, Surface projection of a pair of bulk Weyl cones with opposite chirality connected by a surface Fermi arc. Blue, cyan and magenta curves are Fermi arcs at different binding energies. b, Locations of experimentally determined Weyl nodes W1 in the 3D Brillouin zone and their projections onto the measured (001) surface Fermi arcs. The projections of the Weyl nodes are connected to the Fermi arcs, thus proving the existence of Weyl fermions in TaAs. The Weyl nodes are located at (±0.06, 0.47, 0.58) in units of (π/a, π/a, π/c′). Red and blue colours represent Weyl nodes with opposite chirality. The arrows indicate that the Weyl node is a monopole of Berry flux in the momentum space. The Fermi arcs are obtained at = 54 eV using circularly polarized light.

Note added in proof: After acceptance of our paper, we became aware of a related manuscript (ref. 31) showing similar results to this work and those of refs 23, 24.

Methods

Sample synthesis.

Single crystals of TaAs were grown by chemical vapour transport. A previously reacted polycrystalline TaAs sample was filled in a quartz ampoule using 2 mg cm−3 of iodine as transporting agent. After evacuating and sealing, the ampoule was kept at the growth temperature for three weeks. Large polyhedral crystals with dimensions up to 1.5 mm were obtained in a temperature field of ΔT = 1,150 °C–1,000 °C. The as-grown crystals were characterized by X-ray diffraction (XRD) using a PANalytical diffractometer with Cu Kα radiation at room temperature. Single-crystal XRD was used to determine the crystal growth orientation. The average stoichiometry was determined by energy-dispersive X-ray (EDX) spectroscopy. No I2 doping was detected.

Angle-resolved photoemission spectroscopy.

Soft X-ray ARPES measurements were performed at the Advanced Resonant Spectroscopies (ADRESS) beamline at the Swiss Light Source (SLS) with a SPECS analyser, and data were collected using circularly polarized light with an overall energy resolution of the order of 50–80 meV at T = 10 K. VUV ARPES measurements were performed at the Surface and Interface (SIS) beamline at the SLS with a Scienta R4000 analyser. Fresh surfaces for ARPES measurements were obtained by cleaving TaAs samples in situ in a vacuum better than 5 × 10−11 torr.

Band structure calculations.

First-principles calculations were performed using the OpenMX (ref. 32) software package. We have checked that the choices of pseudo-atomic orbital basis set with Ta9.0-s2p2d2f1 and As9.0-s2p2d1, the pseudo-potentials for Ta and As and the sampling of the BZ (10 × 10 × 10 k-grid) describe the electronic structure accurately. The exchange–correlation functional within the generalized gradient approximation parameterized by Perdew, Burke and Ernzerhof has been used33. The optimized lattice constants a = b = 3.4824 Å, c = 11.8038 Å and atomic sites are in agreement with the experimental values.

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Acknowledgements

We acknowledge help in plotting figures from H. Miao and W. L. Zhang. This work was supported by the Ministry of Science and Technology of China (Nos 2013CB921700, 2015CB921300, 2011CBA00108 and 2011CBA001000), the National Natural Science Foundation of China (Nos 11474340, 11422428, 11274362 and 11234014), the Chinese Academy of Sciences (No. XDB07000000), the Sino-Swiss Science and Technology Cooperation (No. IZLCZ2138954), and the Swiss National Science Foundation (No. 200021-137783).

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Contributions

H.D., T.Q. and M.S. conceived the experiments. B.Q.L., N.X. and J.Z.M. performed ARPES measurements with the assistance of C.E.M., F.B., V.N.S. and J.M. H.M.W., Z.F. and X.D. performed ab initio calculations. N.X., B.Q.L., J.Z.M., T.Q. and H.D. analysed the experimental data. N.X., B.Q.L. and H.M.W. plotted the figures. T.Q., H.D., H.M.W., M.S. and P.R. wrote the manuscript. X.C.H., L.X.Z. and G.F.C. synthesized the single crystals.

Corresponding authors

Correspondence to T. Qian, M. Shi or H. Ding.

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Lv, B., Xu, N., Weng, H. et al. Observation of Weyl nodes in TaAs. Nature Phys 11, 724–727 (2015). https://doi.org/10.1038/nphys3426

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