Topological electronic structure of twin boundaries and twinning superlattices in the SnTe material class

The topological electronic structure of a single twin boundary and coherent twinning superlattices (TSLs) based on the SnTe class of material is calculated and discussed within a supercell implementation. The superlattices consist of two twin planes (TPs) in the supercell arranged in such a way that each of the boundaries forms a mirror plane for the entire structure. Two types of TP boundary, cationic and anionic, can exist, and so three types of supercells can be constructed. We study the topological phases of each twinning configuration using the tight-binding approximation and calculating the topological invariants. We show that they differ by topological properties. We find that all-cationic TSLs are topologically distinct from the anionic case due to the opposite sign of the Berry curvature around the $ \mathrm{ \Gamma} $ point of the TSLs Brillouin Zone. Our findings are consistent with a complementary analysis of (111)-oriented slabs with a single twin boundary in the presence of the Zeeman field. They are also consistent with the number of spin-polarized Dirac-like edge states of both superlattices and slabs. We conclude that each type of TP forms the 2D mirror-plane-protected topological crystalline insulator.