Surface states and topological invariants in three-dimensional topological insulators: Application to Bi 1 − x Sb x

We study the electronic surface states of the semiconducting alloy bismuth antimony $({\text{Bi}}_{1\ensuremath{-}x}{\text{Sb}}_{x})$. Using a phenomenological tight-binding model, we show that the Fermi surface for the 111 surface states encloses an odd number of time-reversal-invariant momenta (TRIM) in the surface Brillouin zone. This confirms that the alloy is a strong topological insulator in the (1;111) ${\mathbb{Z}}_{2}$ topological class. We go on to develop general arguments which show that spatial symmetries lead to additional topological structure of the bulk energy bands, and impose further constraints on the surface band structure. Inversion-symmetric band structures are characterized by eight…