  Binary bismuth chalcogenides Bi$_2$Se$_3$, Bi$_2$Te$_3$, and related
materials are currently being extensively investigated as the reference
topological insulators (TIs) due to their simple surface-state band dispersion
(single Dirac cone) and relatively large bulk band gaps. Nanostructures of TIs
are of particular interest as an increased surface-to-volume ratio enhances the
contribution of surfaces states, meaning they are promising candidates for
potential device applications. So far, the vast majority of research efforts
have focused on the low-energy (0001) surfaces, which correspond to natural
cleavage planes in these layered materials. However, the surfaces of
low-dimensional nanostructures (nanoplatelets, nanowires, nanoribbons)
inevitably involve higher-index facets. We perform a systematic ab initio
investigation of the surfaces of bismuth chalcogenide TI nanostructures
characterized by different crystallographic orientations, atomic structures and
stoichiometric compositions. We find several stable terminations of high-index
surfaces, which can be realized at different values of the chemical potential
of one of constituent elements. For the uniquely defined stoichiometric
termination, the topological Dirac fermion states are shown to be strongly
anisotropic with a clear dependence of Fermi velocities and spin polarization
on the surface orientation. Self-doping effects and the presence of
topologically trivial mid-gap states are found to characterize the
non-stoichiometric surfaces. The results of our study pave the way towards
experimental control of topologically protected surface states in bismuth
chalcogenide nanostructures.
