Prediction of Weyl semimetal in orthorhombic ${\mathrm{MoTe}}_2$

We investigate the orthorhombic phase $\left(T_d\right)$ of the layered transition-metal dichalcogenide ${\mathrm{MoTe}}_2$ as a Weyl semimetal candidate. ${\mathrm{MoTe}}_2$ exhibits four pairs of Weyl points lying slightly above $\left(\sim 6\mathrm{meV}\right)$ the Fermi energy in the bulk band structure. Different from its cousin ${\mathrm{WTe}}_2$, which was recently predicted to be a type-II Weyl semimetal, the spacing between each pair of Weyl points is found to be as large as 4% of the reciprocal lattice in ${\mathrm{MoTe}}_2$ (six times larger than that of ${\mathrm{WTe}}_2$). When projected onto the surface, the Weyl points are connected by Fermi arcs, which can be easily accessed by angle-resolved photoemission spectroscopy due to the large Weyl point separation. In addition, we show that the correlation effect or strain can drive ${\mathrm{MoTe}}_2$ from a type-II to a type-I Weyl semimetal.

Figure 1

(a) Orthorhombic crystal lattice structure of $T_d\text{−}{\mathrm{MoTe}}_2$ in the space group of ${Pnm2}_1$. (b) Brillouin zone (BZ) in the $k_z=0$ plane. WPs with positive and negative chiralities are marked as green and gray dots. The evolution of Wannier charge centers between the $\mathrm{\Gamma }$ and $S$ points is calculated along the red curve $C$. (c) 3D bulk BZ and the projected surface BZ to the (001) plane.

Figure 2

Bulk band structure around the $\mathrm{\Gamma }$ point in the $Y\text{−}\mathrm{\Gamma }\text{−}X$ direction (a) without and (b) with the inclusion of SOC. (c) Berry curvature in the $k_z=0$ plane around two types of WPs. The sizes and directions of the arrows represent the magnitudes and orientations of the Berry curvature, respectively. WPs with positive and negative chirality are denoted by the green and gray points. (c) The evolution of Wannier charge centers between two time-reversal-invariant points of $\mathrm{\Gamma }$ and $S$ along the $C$ curve [see Fig. 1]. The Wannier charge centers cross the reference horizontal line once, indicating the nontrivial $Z_2$ invariant in the $C\text{−}{k}_z$ plane. (e) Band structure crossing two types of WPs. The two points of $K$ and $K^{\prime }$ are shown in (c). Black and orange bands in (a), (b), and (e) are the lowest $N\mathrm{th}$ bands and other higher bands, respectively.

Figure 3

3D plot of WPs in (a) $k_z=0$, (b) $k_x=0.1024\frac{2\pi }{a}$, and (c) $k_x=0.1001\frac{2\pi }{a}$ planes. (d)–(g) The evolution of FS around two WPs in the $k_z=0$ plane. WP-related Fermi surface linear crossings can be at $E_F+6$ meV and $E_F+59$ meV, respectively. The orange and blue FSs come from the electron and hole pockets.

Figure 4

Fermi surfaces and surface band structures. (a)–(c) Surface FS and (d) the illustration of Fermi arcs at $E_F+6$ meV. (f), (g) Surface FSs and (h) corresponding schematics at $E_F+59$ meV. (i) Surface energy dispersion along the high symmetry lines of $\overline{Y}\text{−}\overline{\mathrm{\Gamma }}\text{−}\overline{X}$. (j) Surface energy dispersion along the $\mathrm{\Sigma }\text{−}{\mathrm{\Sigma }}^{\prime }$ crossing WPs. Brighter colors represent the higher LDOS. The exact projections of W1 and W2 are denoted as solid gray and green dots. The end points of Fermi arcs are marked as open circles. The Fermi arcs are highlighted by dashed lines in (b) and (f).

Figure 5

Electronic structure calculated from the hybrid functional. (a) Bulk band structure along two WPs in the $k_y$ directions, and (b) corresponding surface energy dispersion. (c) Schematic of WP evolution with correlation and compression strain. (d) Surface FS at the energy level of $E=E_F+45$ meV and (e) corresponding schematics for the surface Fermi arcs. The two points of $L$ and $L^{\prime }$ are shown in (d). The red and green circles represent WPs with opposite chirality.