We introduce a topological boundary condition to study the surface states of topological insulators within a long-wavelength four-band model. We find that the Dirac point energy, the band curvature, and the spin texture of surface states are crystal-face dependent. For an arbitrary termination of a bulk crystal, the energy of the symmetry protected Dirac point is determined by the bulk physics that breaks particle-hole symmetry in the surface normal direction and is tunable by surface potentials that preserve time reversal symmetry. For a model appropriate to ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ the constant energy contours are generically elliptical with spin textures that are helical on the cleavage surface, collapsed to one dimension on any side face, and tilted out of plane otherwise. Our findings identify a route to engineering the Dirac point physics on the surfaces of real materials.

• Received 28 March 2012

DOI:https://doi.org/10.1103/PhysRevB.86.081303