Model Hamiltonian for topological insulators

Abstract

In this paper we give the full microscopic derivation of the model Hamiltonian for the three-dimensional topological insulators in the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ family of materials (${\mathrm{Bi}}_2{\mathrm{Se}}_3$, ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ and ${\mathrm{Sb}}_2{\mathrm{Te}}_3$). We first give a physical picture to understand the electronic structure by analyzing atomic orbitals and applying symmetry principles. Subsequently, we give the full microscopic derivation of the model Hamiltonian introduced by Zhang et al. [ Nat. Phys. 5, 438 (2009)] based both on symmetry principles and the $\mathbf{k}\cdot \mathbf{p}$ perturbation theory. Two different types of $k^3$ terms, which break the in-plane full rotation symmetry down to threefold rotation symmetry, are taken into account. An effective Hamiltonian is derived for the topological surface states. Both bulk and surface models are investigated in the presence of an external magnetic field, and the associated Landau level structure is presented. For a more quantitative fitting to the first principle calculations, we also present a model Hamiltonian including eight energy bands.