Weak topological insulators in PbTe/SnTe superlattices

It is desirable to realize topological phases in artificial structures by engineering electronic band structures. In this paper we investigate ${\left(\mathrm{PbTe}\right)}_m$${\left(\mathrm{SnTe}\right)}_{2n-m}$ superlattices along the [001] direction and find a robust weak topological insulator phase for a large variety of layer numbers $m$ and $2n-m$. We confirm this topologically nontrivial phase by calculating $Z_2$ topological invariants and topological surface states based on the first-principles calculations. We show that the folding of Brillouin zone due to the superlattice structure plays an essential role in inducing topologically nontrivial phases in this system. This mechanism can be generalized to other systems in which band inversion occurs at multiple momenta, and gives us a brand new way to engineer topological materials in artificial structures.

Figure 1

(a) The primitive cell of ${\left(\mathrm{PbTe}\right)}_1$${\left(\mathrm{SnTe}\right)}_1$ [001] superlattice. (b) Brillouin zone (BZ) and TRIM for PbTe and ${\left(\mathrm{PbTe}\right)}_1$${\left(\mathrm{SnTe}\right)}_1$ [001] superlattice: The black lines are the BZ of PbTe primitive cell with the eight TRIM marked by unprimed Greek letters (the red parallelopiped). The vertices of the blue parallelopiped are TRIM marked with primed Greek letters in the BZ of the superlattice. (c) The primitive cell of ${\left(\mathrm{PbTe}\right)}_3$${\left(\mathrm{SnTe}\right)}_1$ [001] superlattice.

Figure 2

(a) Band structure evolution from PbTe to ${\left(\mathrm{PbTe}\right)}_1{\left(\mathrm{SnTe}\right)}_1$ superlattice, by taking the Te 5$s$ band along the $\Gamma X_1$ as an example. (b) Schematic evolution of energy levels around the Fermi surface, which leads to band inversion. In a superlattice structure, the states from $L_1$ and $L_2$ are brought together and the degeneracy is removed by doping Sn, resulting in a band inversion. The $+$ or $-$ sign denotes the parity of state at TRIM.

Figure 3

The band structure of ${\left(\mathrm{PbTe}\right)}_1{\left(\mathrm{SnTe}\right)}_1$ [001] superlattice, with the high symmetry points defined in Fig. 1. The Mexican-hat shape of dispersion around $L_0^{\prime }$ indicates a band inversion. The band gap is at the $L_0^{\prime }\text{−}{R}^{\prime }$ line.

Figure 4

The energy dispersion of a slab configuration for the ${\left(\mathrm{PbTe}\right)}_1{\left(\mathrm{SnTe}\right)}_1$ [001] superlattice. The shadow indicates the regime of bulk dispersion. A surface state with a Dirac cone appears in the bulk band gap at ${\overline{X}}_1^{\prime }$. The inset shows the surface BZ of the slab. ${\overline{\Gamma }}^{\prime }$, ${{\overline{X}}_1}^{\prime }$, ${{\overline{M}}_1}^{\prime }$, and ${{\overline{X}}_2}^{\prime }$ are the projections from ${\Gamma }^{\prime }$, $M_1^{\prime }$, $X_2^{\prime }$, and $M_2^{\prime }$ on the (001) surface BZ. $L_0^{\prime }$ is projected to ${{\overline{X}}_1}^{\prime }$. ${{\overline{X}}_1}^{\prime }$, and ${{\overline{X}}_2}^{\prime }$ are equivalent due to the mirror symmetry with respect to the (100) plane (the plane along the ${\overline{\Gamma }}^{\prime }\text{−}{\overline{M}}_1^{\prime }$ line).