Fermi arc electronic structure and Chern numbers in the type-II Weyl semimetal candidate ${\mathrm{Mo}}_x{\mathrm{W}}_{1-x}{\mathrm{Te}}_2$

It has recently been proposed that electronic band structures in crystals can give rise to a previously overlooked type of Weyl fermion, which violates Lorentz invariance and, consequently, is forbidden in particle physics. It was further predicted that ${\mathrm{Mo}}_x{\mathrm{W}}_{1-x}{\mathrm{Te}}_2$ may realize such a type-II Weyl fermion. Here, we first show theoretically that it is crucial to access the band structure above the Fermi level ${\varepsilon }_F$ to show a Weyl semimetal in ${\mathrm{Mo}}_x{\mathrm{W}}_{1-x}{\mathrm{Te}}_2$. Then, we study ${\mathrm{Mo}}_x{\mathrm{W}}_{1-x}{\mathrm{Te}}_2$ by pump-probe ARPES and we directly access the band structure $>0.2$ eV above ${\varepsilon }_F$ in experiment. By comparing our results with ab initio calculations, we conclude that we directly observe the surface state containing the topological Fermi arc. We propose that a future study of ${\mathrm{Mo}}_x{\mathrm{W}}_{1-x}{\mathrm{Te}}_2$ by pump-probe ARPES may directly pinpoint the Fermi arc. Our work sets the stage for the experimental discovery of the first type-II Weyl semimetal in ${\mathrm{Mo}}_x{\mathrm{W}}_{1-x}{\mathrm{Te}}_2$.

Figure 1

Chern numbers in type-II Weyl semimetals. A loop (dashed blue line) to show a nonzero Chern number in (a) a type-I Weyl cone and (b) a type-II Weyl cone. Suppose the Weyl point lies above ${\varepsilon }_F$. In the simplest case, it is clear that for a type-I Weyl cone we can show a nonzero Chern number by counting crossings of surface states on a closed loop, which lies entirely below the Fermi level. This is not true for a type-II Weyl semimetal. (c)–(e) Constant-energy cuts of (b). We can attempt to draw a loop below ${\varepsilon }_F$, as in (c), but we find that the loop runs into the bulk hole pocket. We can close the loop by going to $\varepsilon >{\varepsilon }_W$, as in (e). However, then the loop must extend above ${\varepsilon }_F$.

Figure 2

${\mathrm{Mo}}_{0.45}{\mathrm{W}}_{0.55}{\mathrm{Te}}_2$ below the Fermi level. (a)–(g) Conventional ARPES spectra of the constant-energy contour. We observe a palmier-shaped hole pocket and an almond-shaped electron pocket. The two pockets approach each other and we directly observe a beautiful avoided crossing near ${\varepsilon }_F$ where they hybridize, in (a). This hybridization is expected to give rise to Weyl points above ${\varepsilon }_F$. (h)–(k) ARPES measured $E_{\mathrm{B}}\text{−}{k}_y$ dispersion maps along the cuts shown in (f). We expect Weyl points or Fermi arcs above the Fermi level at certain $k_y$ where the pockets approach. (l)–(n) Constant-energy contours for ${\mathrm{Mo}}_{0.4}{\mathrm{W}}_{0.6}{\mathrm{Te}}_2$ from ab initio calculations. The black and white dots indicate the Weyl points, above ${\varepsilon }_F$. Note the excellent overall agreement with the ARPES spectra. The offset on the $k_y$ scale on the ARPES spectra is set by comparison with calculation. (o) Cartoon of the palmier and almond at the Fermi level. Based on calculation, we expect Weyl points above ${\varepsilon }_F$ where the pockets intersect.

Figure 3

${\mathrm{Mo}}_{0.45}{\mathrm{W}}_{0.55}{\mathrm{Te}}_2$ above the Fermi level. (a),(b) $E_{\mathrm{B}}\text{−}{k}_y$ dispersion maps of ${\mathrm{Mo}}_{0.45}{\mathrm{W}}_{0.55}{\mathrm{Te}}_2$ along the $\overline{\mathrm{\Gamma }}\text{−}\overline{Y}$ direction at $k_x=0{\AA }^{-1}$ with and without the pump laser. The sample responds beautifully to the pump laser, allowing us access to the band structure $>0.2\mathrm{eV}$ above ${\varepsilon }_F$, well above the predicted energies of the Weyl points. (c),(d) Dispersion maps of ${\mathrm{Mo}}_{0.45}{\mathrm{W}}_{0.55}{\mathrm{Te}}_2$ with and without pump on a cut parallel to $\overline{\mathrm{\Gamma }}\text{−}\overline{X}$ at (c) $k_y\sim 0.29{\AA }^{-1}$ and (d) $k_y\sim 0.26{\AA }^{-1}$. (e)–(g) The evolution of the almond pocket in energy. We observe that the almond pocket evolves into two nested contours, seen most clearly at $E_B=-0.1\mathrm{eV}$. (h)–(j) Calculation of the Fermi surface for ${\mathrm{Mo}}_{0.4}{\mathrm{W}}_{0.6}{\mathrm{Te}}_2$ above ${\varepsilon }_F$. We see two nested electron pockets, consistent with the measured Fermi surface at $E_B=-0.1\mathrm{eV}$.

Figure 4

Signatures of Fermi arcs in ${\mathrm{Mo}}_x{\mathrm{W}}_{1-x}{\mathrm{Te}}_2$. (a) Pump-probe ARPES spectrum at $k_y=0.225\AA {}^{-1}$. (b) Ab initio calculation at $k_y=k_{\mathrm{W}}1=0.215\AA {}^{-1}$, showing excellent overall agreement with the data. (c),(d) Same as (a),(b) but with key features in the data marked. (e),(f) There are two possible scenarios for the connectivity of the arcs. The scenario in (e) is favored by our ab initio calculations. The surface state electron pocket (3) contains both the topological Fermi arcs and adjacent trivial surface states. While the Fermi arc is formally disconnected from the trivial surface states, in practice the trivial surface state merges into the bulk very close to the Weyl points, so that we do not observe a disjoint arc. Nonetheless, future measurements by pump-probe ARPES may allow us to observe a kink or “ripple” where the topological Fermi arc and trivial surface states meet, demonstrating that ${\mathrm{Mo}}_x{\mathrm{W}}_{1-x}{\mathrm{Te}}_2$ is a type-II Weyl semimetal.