Weyl Node and Spin Texture in Trigonal Tellurium and Selenium

We study Weyl nodes in materials with broken inversion symmetry. We find based on first-principles calculations that trigonal Te and Se have multiple Weyl nodes near the Fermi level. The conduction bands have a spin splitting similar to the Rashba splitting around the $H$ points, but unlike the Rashba splitting the spin directions are radial, forming a hedgehog spin texture around the $H$ points, with a nonzero Pontryagin index for each spin-split conduction band. The Weyl semimetal phase, which has never been observed in real materials without inversion symmetry, is realized under pressure. The evolution of the spin texture by varying the pressure can be explained by the evolution of the Weyl nodes in $\mathit{k}$ space.

Figure 1

(a) Crystal structure of trigonal Te (Se) in the $P{3}_{1}21$ structure. (b)–(d) Isosurface of the maximally localized Wannier functions for the Te-$5p$ orbitals.

Figure 2

Electronic structures of (a) Te and (b) Se at ambient pressure obtained by the $GW+\mathrm{SO}$. (c) and (d) are magnified figures for Te and Se, respectively. (e) and (g) are electronic structures of Te at 2.18 and 3.82 GPa, respectively, and (h) is that of Se at 14.0 GPa. (f) is the Brillouin zone. $P_1$ ($P_3$) on the $H-K$ line and ${\mathrm{\Delta }}_1$ (${\mathrm{\Delta }}_3$) on the A-$\mathrm{\Gamma }$ line have the same $k_z$. $S_1$ ($P_2$) is on the $H-A$ ($H-K$) line. Weyl points are indicated by the black dashed (white solid) circles having a positive (negative) monopole charge (calculated from lower eigenstates around the degenerate point). The energy is measured from the Fermi level.

Figure 3

Motion of Weyl nodes by increasing pressure. The filled and empty circles represent monopoles and antimonopoles for the HOS, respectively. (a) When pressure is increased, a monopole-antimonopole pair is created at each $P$ point [$P_2$ and $P_3$ in Fig. 2] at 2.17 GPa, and the system becomes a Weyl semimetal. They run along the $H-K$ lines, and (b) two monopoles meet at each of the $H$ points at 2.19 GPa. (c) Then they evolve into three monopoles along the $H-A$ lines [$S_1$ in Fig. 2], and one antimonopole residing at the $H$ point.

Figure 4

Fermi surface and spin structure of Te. (a) is the Brillouin zone around the $H$ point. The $\mathit{k}$ point on the red line is unchanged under the $S_3$ or $C_2$ operations. The Fermi level is shifted by 0.31 eV in (b) and (c). (d) is the Fermi surface at 3.82 GPa. Arrows surrounded by circles represent the spin projected onto each plane.