$\left(111\right)$ surface states of SnTe

The characterization and applications of topological insulators depend critically on their protected surface states, which, however, can be obscured by the presence of trivial dangling bond states. Our first-principle calculations show that this is the case for the pristine $\left(111\right)$ surface of SnTe. Yet, the predicted surface states unfold when the dangling bond states are passivated in proper chemisorption. We further extract the anisotropic Fermi velocities, penetration lengths, and anisotropic spin textures of the unfolded $\overline{\Gamma }$- and $\overline{M}$-surface states, which are consistent with the theory in Zhang et al. [Phys. Rev. B 86, 081303 (2012)]. More importantly, this chemisorption scheme provides an external control of the relative energies of different Dirac nodes, which is particularly desirable in multivalley transport.

Figure 1

(a) The crystal structure of bulk SnTe. The light-blue plane is a $\left(111\right)$ lattice plane with only Te atoms. (b) The bulk BZ and the $\left(111\right)$ surface BZ. (c) Band structure of SnTe semi-infinite $\left(111\right)$ slab calculated in the iterative Green's function method. The gray value represents the momentum-resolved density of states. The surface BZ and its high symmetry points are shown in the lower panel, where the dotted lines represent the isoenergy contours around Dirac cones.

Figure 2

DFT band structures of SnTe $\left(111\right)$ slabs. (a) Pristine Sn-terminated surface. (b) Pristine Te-terminated surface. (c) Adsorption geometry for iodine on the Sn-terminated surface and that for sodium on the Te-terminated surface. Electronic structures of (d) I-Sn surface, and (e) Na-Te surface. In all panels for band structures, the vertical width of a pink fat band shows the extent of its localization near the surface. The fat band width in (d) and (e) has been magnified by three times, compared with that in (a) and (b).

Figure 3

DOS projected onto the $s$, $p_x+p_y$, $p_z$ orbits of surface atoms. The DOS of $p_x$ and $p_y$ are the same. (a) Projected DOS on the Sn atoms at the bottom surface of the pristine SnTe $\left(111\right)$ slab with 79 atomic layers. (b) The same as (a) but the surface is decorated with iodine. (c) Projected DOS on the bottom iodine adatoms of the decorated slab.

Figure 4

Surface state energy gaps at $\overline{\Gamma }$ and $\overline{M}$, induced by the hybridization between the top and bottom surfaces, as a function of the slab thickness. The solid and dashed lines are the exponential fitting with $E_g^0{e}^{-D/2l_0}$.

Figure 5

The in-plane projection of the surface state spin textures. (a) Conduction-band textures at $\overline{\Gamma }$; (b) valence-band textures at $\overline{\Gamma }$; (c) conduction-band textures at $\overline{M}$; (d) valence-band textures at $\overline{M}$. The numbers denote the energies of constant energy contours in units of meV with the reference being the Fermi energy.

Figure 6

(a) Schematic diagram of the $\overline{\Gamma }$ valley filter, where the top surface of the SnTe film is the (111) surface. (b) Schematic plot of the $\overline{\Gamma }$ valley selecting mechanism. The blue open (red filled) circles denote the electrons in the $\overline{\Gamma }$ ($\overline{M}$) valley.