Figure 1

Atomic structure of TlBiSe${}_2$ in rhombohedral and hexagonal bases (left), and charge-density distribution in (11$\overline{2}$0) plane of the hexagonal cell (right). The Brillouin zone for the rhombohedral cell is shown in the lower left corner.Reuse & Permissions

Figure 2

Bulk band structure containing all TRIMs in the Brillouin zone calculated by the VASP (lines) and the FLEUR code (circles). A sketch of the Brillouin zone can be found in Fig. 1. All calculations were performed including spin-orbit coupling.Reuse & Permissions

Figure 3

(a) Band structure of a symmetric 39-layer TlSbTe${}_2$ slab with a Te-terminated (0001) surface (black lines) and the projected bulk band structure (gray) in the background (FLEUR result). The area of the bulk-projected band gap around the $\mathrm{\Gamma ̄}$ point is indicated with the yellow background. The size of the symbols reflects the weight of the states in the vacuum and the color indicates the spin direction, i.e., parallel or antiparallel to $\mathbf{n}\times \mathbf{k}$, if $\mathbf{n}$ is the surface normal and $\mathbf{k}$ is the wave vector of the state. Inset: Slab with H termination on one of the sides; dotted lines show the state that connects the valence and conduction bands, full lines show a normal, Rashba split surface state. (b) Surface band structure in the vicinity of $\mathrm{\Gamma ̄}$ for a 39-layer TlSbTe${}_2$(0001) (VASP calculation). Gray areas mark the region of projected bulk states. (c) $\mathit{xy}$-integrated charge-density distribution for Dirac state (green line) and for trivial surface state (violet line); horizontal lines indicate the atomic-layer positions. (d) Surface band spectrum for a 63-layer TlSbTe${}_2$(0001) slab. (e) Results for an unrelaxed 39-layer film.Reuse & Permissions

Figure 4

Band structure of a symmetric 39-layer TlSbSe${}_2$ film with an Se-terminated (0001) surface. The color coding follows the one in Fig. 3. The right panel shows the spatial distribution of the $\mathrm{\Gamma ̄}$ Dirac-state charge density in the (11$\overline{2}$0) plane in the upper half of the film and integrated over the $(x,y)$ plane (right).Reuse & Permissions

Figure 5

(a) FLEUR-calculated band structure of a symmetric 39-layer TlBiTe${}_2$ film with the Te-terminated (0001) surface. The color coding follows the one in Fig. 3. (b) The right panel shows the spatial distribution of the $\mathrm{\Gamma ̄}$ Dirac-state charge density in the (11$\overline{2}$0) plane and integrated over the $(x,y)$ plane. (c) VASP-calculated surface electronic structure of TlBiTe${}_2$ at (around) the $\mathrm{\Gamma ̄}$ point with indicated cut energies shown in Fig. 5. (d) Constant energy cuts of the Dirac cone with energies indicated in (c) and given with respect to the Dirac point (DP).Reuse & Permissions

Figure 6

Band structure of symmetric 39-layer TlBiSe${}_2$ (0001) films with different surface terminations: (a) $\dots$-Tl-Se-Bi-Se, (b) $\dots$-Se-Tl-Se-Bi, (c) $\dots$-Se-Bi-Se-Tl, and (d) $\dots$-Se-Bi-Tl-Se. Red lines indicate the band structure of the films and gray areas are the projected bulk bands.Reuse & Permissions

Figure 7

Spin polarization of the Dirac-cone states of a TlBiSe${}_2$ (0001) surface with Se termination. Shown are constant energy cuts at $90,$ $140$, and $190$ meV above the Dirac point, where the spin precesses clockwise around the $\mathrm{\Gamma ̄}$ point.Reuse & Permissions

Figure 8

Bulk band structure of InBiTe${}_2$ and InSbTe${}_2$ calculated by the VASP (lines) and the FLEUR codes (circles) with spin-orbit coupling included.Reuse & Permissions

Figure 9

Band structure of a symmetric 39-layer InBiTe${}_2$ film with Te-terminated (0001) surface. The color coding follows that in Fig. 6. The right panel shows the spatial distribution of the surface-state charge density at the $\mathrm{\Gamma ̄}$ point (circle) in the (11$\overline{2}$0) plane and integrated over the $(x,y)$ plane.Reuse & Permissions