Spin-orbit coupling in quasiparticle studies of topological insulators

Abstract

We present one-shot $GW$ calculations of the bulk electronic structure of the topological insulators Bi${}_{2}$Se${}_{3}$ and Bi${}_{2}$Te${}_{3}$ within the all-electron full-potential linearized augmented-plane-wave formalism. We compare three different ways of treating the spin-orbit interaction in calculating the quasiparticle energies: (i) The spin-orbit coupling (SOC) is already incorporated in the noninteracting system that serves as starting point for the quasiparticle correction. (ii) The SOC is added in a second-variation approach only after the quasiparticle calculation has been performed in the absence of SOC. We found that the approximate treatment (ii) yields most quasiparticle bands with reasonable accuracy but does fail in the important band-gap region, where the SOC gives rise to a band inversion relevant for the topological properties of these materials. For example, Bi${}_{2}$Se${}_{3}$ is just on the brink of becoming a trivial semiconductor within this approximate approach, while it maintains its topological properties in the case of the consistent treatment (i). Finally, we consider another approach (iii), in which the SOC is included in the Green function $G$ as in (i), but neglected in the calculation of the screened Coulomb potential $W$. This approach gives results in very good agreement with the full treatment (i), but with a smaller numerical effort. We conclude that, in the high-symmetry directions studied, bulk Bi${}_{2}$Se${}_{3}$ is a direct-gap and Bi${}_{2}$Te${}_{3}$ an indirect-gap semiconductor with band gaps of 0.20 and 0.19 eV, respectively.