Spin texture of topological surface states of side-surfaces in Bi$_2$Se$_3$ from first principles

Topological insulators have recently drawn a lot of attention because of topologically protected surface states with Dirac dispersion and spin-momentum locking induced by time-reversal symmetry and strong spin-orbit coupling, respectively. Recent experiments report interesting transport properties of Bi$_2$Se$_3$ nanowires and nanoribbons with growth directions normal to the (111) surface. However, most of the studies of Bi$_2$Se$_3$ are focused on the (111) surface. Additionally, surfaces other than the (111) surface would facilitate hole doping with adatoms since both Bi and Se atoms are present at these other surfaces. We investigate the spin texture and electronic structure of topologically protected surface states of two representative side-surfaces of Bi$_2$Se$_3$ by using density-functional theory (DFT). In particular, we consider two surfaces normal to the (111) surface, such as (1$\bar{1}$0) and (11$\bar{2}$), where the former has twofold symmetry and the latter has mirror symmetry. We present our calculated spin textures of the surface states of these side surfaces that qualitatively differ from typical Rashba-like features and from the prediction based on the bulk model Hamiltonian by keeping up to quadratic terms in momentum.