Structural stability and topological surface states of the SnTe (111) surface

We perform first-principles calculations to study the stability and electronic structure of the (111) surface of SnTe, a representative topological crystalline insulator (TCI). We find three stable surface phases, which support two qualitatively different types of topological surface states: type I with four Dirac points at Γ̄ and three M̄ points and type II with two Dirac points nearby but not at Γ̄. Their appearance can be controlled by varying growth conditions. Under an Sn-poor condition, the Te-terminated surface without reconstruction is stable, resulting in the type-I surface states. While under an Sn-rich condition, the (2×1)-reconstructed Sn-terminated surface becomes more stable. The reconstruction folds the surface Brillouin zone and effectively induces interactions between the Dirac points at the Γ̄ and M̄ points. Surface states thus change from type I to type II accompanied by a Lifshitz transition. Under intermediate growth conditions, the (3×3)-reconstructed Sn-terminated surface gets stabilized, which recovers the type-I surface states. Our work suggests a promising alternative way to control the topological surface states of TCIs besides selecting different surface orientations.