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

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 {\bf k} except for {\bf k} in high-symmetry directions. The size of the electron spin depends on the direction of {\bf k} due to anisotropic variation of the noncollinearity of the electron state. 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 tological insulators where different surfaces meet.