We investigate the spin and spin-orbital textures and electronic structures of topologically protected surface states at side surfaces of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ by using slab models within density-functional theory. This is motivated by recent experiments on nanowires, nanoribbons, and nanoplates of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ with side surfaces. In particular, two representative surfaces normal to the (111) surface, such as $\left(1\overline{1}0\right)$ and $\left(11\overline{2}\right)$ surfaces, are examined, in the presence of time-reversal symmetry and inversion symmetry. The $\left(1\overline{1}0\right)$ surface lying in the mirror plane has twofold $\left({\mathrm{C}}_2\right)$ rotational symmetry, whereas the $\left(11\overline{2}\right)$ surface has only mirror symmetry. For the $\left(1\overline{1}0\right)$ surface, we find that a Dirac cone with strongly anisotropic Fermi velocity is formed at $\mathrm{\Gamma }$ with the Dirac point at the Fermi level, and that the spin texture reveals features of Rashba-type combined with Dresselhaus-type spin-orbit coupling. For the $\left(11\overline{2}\right)$ surface, a Dirac cone is found at either $\mathrm{\Gamma }$ or the $Y$ point (along the mirror symmetry axis) below the Fermi level. In this case, the spin texture of the surface states strikingly differs from that of the (111) and $\left(1\overline{1}0\right)$ surfaces: (i) the in-plane spin polarization dominantly aligned perpendicular to the [111] direction or the mirror symmetry axis, (ii) the Dresselhaus-type spin texture, and (iii) significant out-of-plane spin polarization away from the mirror symmetry axis. Our findings distinctively differ from the previous works based on the effective bulk model Hamiltonian. Our calculated spin and spin-orbital textures and band structures can be observed by spin-resolved angle-resolved photoemission spectroscopy.

• Received 10 December 2015

DOI:https://doi.org/10.1103/PhysRevB.93.085122