Low-velocity anisotropic Dirac fermions on the side surface of topological insulators

Abstract

We report anisotropic Dirac-cone surface bands on a side-surface geometry of the topological insulator Bi${}_{2}$Se${}_{3}$ revealed by first-principles density-functional calculations. We find that the electron velocity in the side-surface Dirac cone is anisotropically reduced from that in the (111)-surface Dirac cone, and the velocity is not in parallel with the wave vector k except for k in high-symmetry directions. The size of the electron spin depends on the direction of k due to anisotropic variation of the noncollinearity of the electron state. The anisotropy of electronic structures follows the corresponding anisotropy of the surface atomic structure. Low-energy effective Hamiltonian is proposed for side-surface Dirac fermions, and its implications are presented including refractive transport phenomena occurring at the edges of topological insulators where different surfaces meet.