Discovery of a Weyl fermion semimetal and topological Fermi arcs

Tantalum arsenide is a semimetallic material that crystallizes in a body-centered tetragonal lattice system (Fig. 1A) (23). The lattice constants are a = 3.437 Å and c = 11.656 Å, and the space group is $I{4}_{1}md$ ($\#109$, $C_{4v}$), as consistently reported in previous structural studies (23–25). The crystal consists of interpenetrating Ta and As sublattices, where the two sublattices are shifted by $(\frac{a}{2},\frac{a}{2},\delta )$, $\delta \approx \frac{c}{12}$. Our diffraction data match well with the lattice parameters and the space group $I{4}_{1}md$ (26). The scanning tunneling microscopic (STM) topography (Fig. 1B) clearly resolves the (001) square lattice without any obvious defect. From the topography, we obtain a lattice constant a = 3.45 Å. Electrical transport measurements on TaAs confirmed its semimetallic transport properties and reported negative magnetoresistance, suggesting the anomalies due to Weyl fermions (23).

We discuss the essential aspects of the theoretically calculated bulk band structure (9, 10) that predicts TaAs as a Weyl semimetal candidate. Without spin-orbit coupling, calculations (9, 10) show that the conduction and valence bands interpenetrate (dip into) each other to form four 1D line nodes (closed loops) located on the $k_x$ and $k_y$ planes (shaded blue in Fig. 1, C and E). Upon the inclusion of spin-orbit coupling, each line node loop is gapped out and shrinks into six Weyl nodes that are away from the $k_x=0$ and $k_y=0$ mirror planes (Fig. 1E, small filled circles). In our calculation, in total there are 24 bulk Weyl cones (9, 10), all of which are linearly dispersive and are associated with a single chiral charge of $\pm 1$ (Fig. 1E). We denote the 8 Weyl nodes that are located on the brown plane ($k_z=\frac{2\pi }{c}$) as W1 and the other 16 nodes that are away from this plane as W2. At the (001) surface BZ (Fig. 1F), the eight W1 Weyl nodes are projected in the vicinity of the surface BZ edges, $\overline{X}$ and $\overline{Y}$. More interestingly, pairs of W2 Weyl nodes with the same chiral charge are projected onto the same point on the surface BZ. Therefore, in total there are eight projected W2 Weyl nodes with a projected chiral charge of $\pm 2$, which are located near the midpoints of the $\overline{\Gamma }-\overline{X}$ and the $\overline{\Gamma }-\overline{Y}$ lines. Because the $\pm 2$ chiral charge is a projected value, the Weyl cone is still linear (9). The number of Fermi arcs terminating on a projected Weyl node must equal its projected chiral charge. Therefore, in TaAs, two Fermi arc surface states must terminate on each projected W2 Weyl node.

We carried out low–photon-energy ARPES measurements to explore the surface electronic structure of TaAs. Figure 1H presents an overview of the (001) Fermi surface map. We observe three types of dominant features, namely a crescent-shaped feature in the vicinity of the midpoint of each $\overline{\Gamma }-\overline{X}$ or $\overline{\Gamma }-\overline{Y}$ line, a bowtie-like feature centered at the $\overline{X}$ point, and an extended feature centered at the $\overline{Y}$ point. We find that the Fermi surface and the constant-energy contours at shallow binding energies (Fig. 2A) violate the $C_4$ symmetry, considering the features at $\overline{X}$ and $\overline{Y}$ points. In the crystal structure of TaAs, where the rotational symmetry is implemented as a screw axis that sends the crystal back into itself after a $C_4$ rotation and a translation by $\frac{c}{2}$ along the rotation axis, such an asymmetry is expected in calculation. The crystallinity of the (001) surface in fact breaks the rotational symmetry. We now focus on the crescent-shaped features. Their peculiar shape suggests the existence of two arcs, and their termination points in k-space seem to coincide with the surface projection of the W2 Weyl nodes. Because the crescent feature consists of two nonclosed curves, it can either arise from two Fermi arcs or a closed contour; however, the decisive property that clearly distinguishes one case from the other is the way in which the constant-energy contour evolves as a function of energy. As shown in Fig. 2F, in order for the crescent feature to be Fermi arcs, the two nonclosed curves have to move (disperse) in the same direction as one varies the energy (26). We now provide ARPES data to show that the crescent features in TaAs indeed exhibit this “copropagating” property. To do so, we single out a crescent feature, as shown in Fig. 2, B and E, and show the band dispersions at representative momentum space cuts, cut I and cut II, as defined in Fig. 2E. The corresponding $E-k$ dispersions are shown in Fig. 2, C and D. The evolution (dispersive “movement”) of the bands as a function of binding energy can be clearly read from the slope of the bands in the dispersion maps and is indicated in Fig. 2E by the white arrows. It can be seen that the evolution of the two nonclosed curves is consistent with the copropagating property. To further visualize the evolution of the constant-energy contour throughout $k_x,k_y$ space, we use surface state constant-energy contours at two slightly different binding energies, namely $E_{\mathrm{B}}=0=E_{\mathrm{F}}$ and $E_{\mathrm{B}}=20\mathrm{ meV}$. Figure 2G shows the difference between these two constant-energy contours, namely $\Delta I(k_x,k_y)=I(E_{\mathrm{B}}=20\mathrm{ meV},k_x,k_y)-I(E_{\mathrm{B}}=0\mathrm{ meV},k_x,k_y)$, where $I$ is the ARPES intensity. The k-space regions in Fig. 2G that have negative spectral weight (red) correspond to the constant-energy contour at $E_{\mathrm{B}}=0\mathrm{ meV}$, whereas those regions with positive spectral weight (blue) correspond to the contour at $E_{\mathrm{B}}=20\mathrm{ meV}$. Thus, one can visualize the two contours in a single $k_x,k_y$ map. The alternating “red-blue-red-blue” sequence for each crescent feature in Fig. 2G shows the copropagating property, consistent with Fig. 2F. Furthermore, we note that there are two crescent features, one located near the $k_x=0$ axis and the other near the $k_y=0$ axis, in Fig. 2G. The fact that we observe the copropagating property for two independent crescent features that are ${90}^{\circ }$ rotated with respect to each other further shows that this observation is not due to artifacts, such as a k misalignment while performing the subtraction. The above systematic data reveal the existence of Fermi arcs on the (001) surface of TaAs. Just as one can identify a crystal as a topological insulator by observing an odd number of Dirac cone surface states, we emphasize that our data here are sufficient to identify TaAs as a Weyl semimetal because of bulk-boundary correspondence in topology.

Theoretically, the copropagating property of the Fermi arcs is unique to Weyl semimetals because it arises from the nonzero chiral charge of the projected bulk Weyl nodes (26), which in this case is $\pm 2$. Therefore, this property distinguishes the crescent Fermi arcs not only from any closed contour but also from the double Fermi arcs in Dirac semimetals (27, 28), because the bulk Dirac nodes do not carry any net chiral charges (26). After observing the surface electronic structure containing Fermi arcs in our ARPES data, we are able to slightly tune the free parameters of our surface calculation and obtain a calculated surface Fermi surface that reproduces and explains our ARPES data (Fig. 1G). This serves as an important cross-check that our data and interpretation are self-consistent. Specifically, our surface calculation indeed also reveals the crescent Fermi arcs that connect the projected W2 Weyl nodes near the midpoints of each $\overline{\Gamma }-\overline{X}$ or $\overline{\Gamma }-\overline{Y}$ line (Fig. 1G). In addition, our calculation shows the bowtie surface states centered at the $\overline{X}$ point, also consistent with our ARPES data. According to our calculation, these bowtie surface states are in fact Fermi arcs (26) associated with the W1 Weyl nodes near the BZ boundaries. However, our ARPES data cannot resolve the arc character because the W1 Weyl nodes are too close to each other in momentum space compared to the experimental resolution. Additionally, we note that the agreement between the ARPES data and the surface calculation upon the contour at the $\overline{Y}$ point can be further improved by fine-optimizing the surface parameters. To establish the topology, it is not necessary for the data to have a perfect correspondence with the details of calculation because some changes in the choice of the surface potential allowed by the free parameters do not change the topology of the materials, as is the case in topological insulators (21, 22). In principle, Fermi arcs can coexist with additional closed contours in a Weyl semimetal (6, 9), just as Dirac cones can coexist with additional trivial surface states in a topological insulator (21, 22). Particularly, establishing one set of Weyl Fermi arcs is sufficient to prove a Weyl semimetal (6). This is achieved by observing the crescent Fermi arcs as we show here by our ARPES data in Fig. 2, which is further consistent with our surface calculations.

We now present bulk-sensitive SX-ARPES (29) data, which reveal the existence of bulk Weyl cones and Weyl nodes. This serves as an independent proof of the Weyl semimetal state in TaAs. Figure 3B shows the SX-ARPES measured $k_x-k_z$ Fermi surface at $k_y=0$ (note that none of the Weyl nodes are located on the $k_y=0$ plane). We emphasize that the clear dispersion along the $k_z$ direction (Fig. 3B) firmly shows that our SX-ARPES predominantly images the bulk bands. SX-ARPES boosts the bulk-surface contrast in favor of the bulk band structure, which can be further tested by measuring the band dispersion along the $k_z$ axis in the SX-ARPES setting. This is confirmed by the agreement between the ARPES data (Fig. 3B) and the corresponding bulk band calculation (Fig. 3A). We now choose an incident photon energy (i.e., a $k_z$ value) that corresponds to the k-space location of W2 Weyl nodes and map the corresponding $k_x-k_y$ Fermi surface. As shown in Fig. 3C, the Fermi points that are located away from the $k_x$ or $k_y$ axes are the W2 Weyl nodes. In Fig. 3D, we clearly observe two linearly dispersive cones that correspond to the two nearby W2 Weyl nodes along cut 1. The k-space separation between the two W2 Weyl nodes is measured to be 0.08 Å^–1, which is consistent with both the bulk calculation and the separation of the two terminations of the crescent Fermi arcs measured in Fig. 2. The linear dispersion along the out-of-plane direction for the W2 Weyl nodes is shown by our data in Fig. 3E. Additionally, we also observe the W1 Weyl cones in Fig. 3, G to I. Notably, our data show that the energy of the bulk W1 Weyl nodes is lower than that of the bulk W2 Weyl nodes, which agrees well with our calculation shown in Fig. 3J and an independent modeling of the bulk transport data on TaAs (23).

In general, in a spin-orbit–coupled bulk crystal, point-like linear band crossings can either be Weyl cones or Dirac cones. Because the observed bulk cones in Fig. 3, C and D, are located neither at Kramers’ points nor on a rotational axis, they cannot be identified as bulk Dirac cones and have to be Weyl cones according to topological theories (6, 28). Therefore, our SX-ARPES data alone prove the existence of bulk Weyl nodes. The agreement between the SX-ARPES data and our bulk calculation, which only requires the crystal structure and the lattice constants as inputs, provides a further cross-check.

Finally, we show that the k-space locations of the surface Fermi arc terminations match with the projection of the bulk Weyl nodes on the surface BZ. We superimpose the SX-ARPES measured bulk Fermi surface containing W2 Weyl nodes (Fig. 3C) onto the low–photon-energy ARPES Fermi surface containing the surface Fermi arcs (Fig. 2A) to scale. From Fig. 4A we see that all the arc terminations and projected Weyl nodes match with each other within the k-space region that is covered in our measurements. To establish this point quantitatively, in Fig. 4C, we show the zoomed-in map near the crescent Fermi arc terminations, from which we obtain the k-space location of the terminations to be at ${\overrightarrow{k}}_{\mathrm{arc}}=(0.04\pm 0.01{\stackrel{\circ }{\mathrm{A}}}^{-1},\mathrm{ }0.51\pm 0.01{\stackrel{\circ }{\mathrm{A}}}^{-1})$. Figure 4D shows the zoomed-in map of two nearby W2 Weyl nodes, from which we obtain the k-space location of the W2 Weyl nodes to be at ${\overrightarrow{k}}_{\mathrm{W2}}=(0.04\pm 0.015{\stackrel{\circ }{\mathrm{A}}}^{-1},\mathrm{ }0.53\pm 0.015{\stackrel{\circ }{\mathrm{A}}}^{-1})$. In our bulk calculation, the k-space location of the W2 Weyl nodes is found to be at $(0.035{\stackrel{\circ }{\mathrm{A}}}^{-1},\mathrm{ }0.518{\stackrel{\circ }{\mathrm{A}}}^{-1})$. Because the SX-ARPES bulk data and the low–photon-energy ARPES surface data are completely independent measurements obtained with two different beamlines, the fact that they match well provides another piece of evidence of the topological nature (the surface-bulk correspondence) of the Weyl semimetal state in TaAs. In figs. S5 to S7, we further show that the bulk Weyl cones can also be observed in our low–photon-energy ARPES data, although their spectral weight is much lower than the surface state intensities that dominate the data. Our demonstration of the Weyl fermion semimetal state in and Fermi arc surface metals paves the way (30) for the realization of many fascinating topological quantum phenomena.

Work at Princeton University and Princeton-led synchrotron-based ARPES measurements were supported by the Gordon and Betty Moore Foundations EPiQS Initiative through grant GBMF4547 (Hasan). First-principles band structure calculations at National University of Singapore were supported by the National Research Foundation, Prime Minister's Office, Singapore, under its NRF fellowship (NRF Award no. NRF-NRFF2013-03). Single-crystal growth was supported by National Basic Research Program of China (grant nos. 2013CB921901 and 2014CB239302) and characterization by U.S. Department of Energy DE-FG-02-05ER46200. F.C. acknowledges the support provided by MOST-Taiwan under project no. 102-2119-M-002-004. We gratefully acknowledge J. D. Denlinger, S. K. Mo, A. V. Fedorov, M. Hashimoto, M. Hoesch, T. Kim, and V. N. Strocov for their beamline assistance at the Advanced Light Source, the Stanford Synchrotron Radiation Lightsource, the Diamond Light Source, and the Swiss Light Source under their external user programs. Part of the work was carried out at the Swiss Light Source through the external/international facility user program. We thank T.-R. Chang for help on theoretical band structure calculations. We also thank L. Balents, D. Huse, I. Klebanov, T. Neupert, A. Polyakov, P. Steinhardt, H. Verlinde, and A. Vishwanath for discussions. R.S. and H.L. acknowledge visiting scientist support from Princeton University. M.Z.H. acknowledges hospitality of the Lawrence Berkeley National Laboratory and Aspen Center for Physics as a visiting scientist. A patent application is being prepared on behalf of the authors on the discovery of a Weyl semimetal.