Single-Dirac-Cone Topological Surface States in the ${\mathrm{TlBiSe}}_2$ Class of Topological Semiconductors

We investigate several strong spin-orbit coupling ternary chalcogenides related to the (Pb,Sn)Te series of compounds. Our first-principles calculations predict the low-temperature rhombohedral ordered phase in ${\mathrm{TlBiTe}}_2$, ${\mathrm{TlBiSe}}_2$, and $\mathrm{TlSb}{X}_2$ ($X=\mathrm{Te}$, Se, S) to be topologically nontrivial. We identify the specific surface termination that realizes the single Dirac cone through first-principles surface state computations. This termination minimizes effects of dangling bonds, making it favorable for photoemission experiments. In addition, our analysis predicts that thin films of these materials could harbor novel 2D quantum spin Hall states, and support odd-parity topological superconductivity.

Figure 1

(A) Crystal structure of idealized fcc ${\mathrm{TlBiTe}}_2$. The Tl surface is emphasized by black lines, the Bi surface by dotted lines, and the Te surface by gray lines. (B) The fcc Brillouin zone. The thick red arrow marks the 111 rhombohedral distortion direction.

Figure 2

(A)–(D) Band structures of PbTe, PbTe in a supercell rhombohedral structure with $4\mathrm{atoms}/\mathrm{unit}$ cell, ${\mathrm{PbSnTe}}_2$, and ${\mathrm{TlSbTe}}_2$ with fcc atomic positions. The fcc symbols for points in the Brillouin zone are used as in Fig. 1b for ease of comparison. The size of the red dots denotes the probability of $s$-orbital occupation on the $M$ atoms. (E) Band inversion at L for SnTe relative to PbTe. Panels (F) and (G) highlight band inversion at the $\Gamma$ and $X$ points for ${\mathrm{PbSnTe}}_2$ (C) and ${\mathrm{TlSbTe}}_2$ (D), respectively.

Figure 3

Panels (A)–(C) show the bulk band structure of ${\mathrm{TlSbTe}}_2$ for $z_{\mathrm{Te}}=0$, 0.21, and 0.33 Å, respectively. The fcc notation for the high-symmetry points is used as in Fig. 1b. Panels (D) and (E) show the direct gaps at $\Gamma$ (blue solid line) and $X$ (red dashed line) as a function of $V_a$ and $z_{\mathrm{Te}}$, respectively. The gray area highlights features related to topological nontriviality. Panel (F) shows the surface state dispersions at an optimized value of $z_{\mathrm{Te}}$ [red (gray) lines]. Projected bulk bands are shown as blue areas.

Figure 4

Band structures based on thin slab calculations for ${\mathrm{TlSbTe}}_2$ (A), ${\mathrm{TlSbSe}}_2$ (B), ${\mathrm{TlSbS}}_2$ (C), ${\mathrm{TlBiTe}}_2$ (D), ${\mathrm{TlBiSe}}_2$ (E), and ${\mathrm{TlBiS}}_2$ (F) with $z_X=0$, but without band-gap corrections. The blue (gray) area marks projected bulk bands. Red dots are 2D bands for the thin slab. Parity analysis of states at $\overline{\Gamma }$ and $\overline{M}$ indicates that thin films in panels (A)–(C) would harbor topologically nontrivial 2D quantum spin Hall states.