Surface states and topological invariants in three-dimensional topological insulators: Application toBi1−xSbx

We study the electronic surface states of the semiconducting alloy bismuth antimony $\left({\text{Bi}}_{1-x}{\text{Sb}}_x\right)$. 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 ${\mathbb{Z}}_2$ “parity invariants,” which include the four ${\mathbb{Z}}_2$ invariants defined by time-reversal symmetry. The extra invariants determine the “surface fermion parity,” which specifies which surface TRIM are enclosed by an odd number of electron or hole pockets. We provide a simple proof of this result, which provides a direct link between the surface-state structure and the parity eigenvalues characterizing the bulk. Using this result, we make specific predictions for the surface-state structure for several faces of ${\text{Bi}}_{1-x}{\text{Sb}}_x$. We next show that mirror-invariant band structures are characterized by an integer “mirror Chern number” $n_{\mathcal{M}}$, which further constrains the surface states. We show that the sign of $n_{\mathcal{M}}$ in the topological insulator phase of ${\text{Bi}}_{1-x}{\text{Sb}}_x$ is related to a previously unexplored ${\mathbb{Z}}_2$ parameter in the $L$ point $\mathbf{k}\cdot \mathbf{p}$ theory of pure bismuth, which we refer to as the “mirror chirality” $\eta$. The value of $\eta$ predicted by the tight-binding model for bismuth disagrees with the value predicted by a more fundamental pseudopotential calculation. This explains a subtle disagreement between our tight-binding surface-state calculation and previous first-principles calculations of the surface states of bismuth. This suggests that the tight-binding parameters in the Liu-Allen model of bismuth need to be reconsidered. Implications for existing and future angle-resolved photoemission spectroscopy (ARPES) experiments and spin-polarized ARPES experiments will be discussed.