Nontrivial spin texture of the coaxial Dirac cones on the surface of topological crystalline insulator SnTe

We present first-principles calculations of the nontrivial surface states and their spin textures in the topological crystalline insulator SnTe. The surface state dispersion on the [001] surface exhibits four Dirac cones centered along the intersection of the mirror plane and the surface plane. We propose a simple model of two interacting coaxial Dirac cones to describe both the surface state dispersion and the associated spin texture. The out-of-plane spin polarization is found to be zero due to the crystalline and time-reversal symmetries. The in-plane spin texture shows helicity with some distortion due to the interaction of the two coaxial Dirac cones, indicating a nontrivial mirror Chern number of $-2$, distinct from the value of $-1$ in a $Z_2$ topological insulator such as Bi/Sb alloys or Bi${}_2$Se${}_3$. The surface state dispersion and its spin texture would provide an experimentally accessible signature for determining the nontrivial mirror Chern number.

Figure 1

Crystal structure and surface bands of SnTe. (a) Rocksalt crystal structure of SnTe. A [001] surface plane is shown. (b) fcc Brillouin zone (BZ) of SnTe and the [001] surface BZ. The [1 $-1$ 0] mirror plane crosses the 2D surface BZ along $\overline{\Gamma }$$\overline{\mathrm{X}}$. (c) Electronic structure of SnTe [001] surface around $\overline{\mathrm{X}}$. The surface bands are indicated by thick green lines and bulk bands by the shaded purple area. A Dirac point of the surface state appears along the $\overline{\Gamma }$$\overline{\mathrm{X}}$ direction, while the surface band is gapped along $\overline{\mathrm{X}}$$\overline{\mathrm{M}}$.

Figure 2

Two interacting coaxial Dirac cones. Schematic diagram for a noninteracting and interacting coaxial Dirac cone centered at the $\overline{\mathrm{X}}$ point is shown in (a) and (b), respectively. As the interaction between the two Dirac cones is turned on, gaps open up except at the two new Dirac points on the two sides of the $\overline{\mathrm{X}}$ point. The new gapless Dirac cones are protected by the presence of the mirror plane which intersects the surface BZ along $\overline{\Gamma }$$\overline{\mathrm{X}}$ (bold blue line). Maps of the fraction of the partial charges on Sn and Te atoms for the valence surface bands are shown in (c) and (d), respectively, while those for the conduction surface bands are shown in (e) and (f). Dashed lines are the constant energy contours. (g) Surface band dispersion with colors indicating the fraction of partial charges on the Sn and Te atoms.

Figure 3

Nontrivial spin texture of SnTe surface states. Spin textures of the conduction and valence surface bands are shown in (a) and (b), respectively. Energy is represented by the color scale, and constant energy contours are shown. Green line indicates the $\overline{\Gamma }$$\overline{\mathrm{X}}$ direction and green dots denote the Dirac points. Schematic band structures for the conduction and valence surface bands along the $\overline{\Gamma }$$\overline{\mathrm{X}}$ direction are shown in (c) and (d), respectively. Solid lines are Te-like Dirac cones, and the dashed lines are Sn-like. Red (blue) arrows indicate the spin along positive (negative) $y$ direction.

Figure 4

Schematic diagrams for spin textures of the lower Dirac cones in distinct topological phases associated with various values of the mirror Chern number $n_m$: (a) $n_m=-2$; (b) $n_m=2$; (c) $n_m=-1$; and (d) $n_m=1$. Blue lines indicate the axis of mirror symmetry.