Bulk and interface quantum states of electrons in multi-layer heterostructures with topological materials

In this article we describe the bulk and interface quantum states of electrons in multi-layer heterostructures in one dimension, consisting of topological insulators (TIs) and topologically trivial materials. We use and extend an effective four-band continuum Hamiltonian by introducing position dependence to the eight material parameters of the Hamiltonian. We are able to demonstrate complete conduction-valence band mixing in the interface states. We find evidence for topological features of bulk states of multi-layer TI heterostructures, as well as demonstrating both complete and incomplete conduction-valence band inversion at different bulk state energies. We show that the linear k _z terms in the low-energy Hamiltonian, arising from overlap of p _z orbitals between different atomic layers in the case of chalcogenides, control the amount of tunneling from TIs to trivial insulators. Finally, we show that the same linear k _z terms in the low-energy Hamiltonian affect the material's ability to form the localised interface state, and we demonstrate that due to this effect the spin and probability density localisation in a thin film of Sb₂Te₃ is incomplete. We show that changing the parameter that controls the magnitude of the overlap of p _z orbitals affects the transport characteristics of the topologically conducting states, with incomplete topological state localisation resulting in increased backscattering.