We have investigated the nature of surface states in the Bi${}_2$Te${}_3$ family of three-dimensional topological insulators using first-principles calculations as well as a model Hamiltonian approach. When the surface Dirac cone is warped due to Dresselhaus spin-orbit coupling in rhombohedral structures, the spin acquires a finite out-of-the-plane component. We provide a simple, minimal model to describe the in-plane spin texture of the warped surface Dirac cone observed in experiments where spins are seen to be not aligned perpendicular to the electron momentum. Our $k·p$ model calculation reveals that this in-plane spin texture requires fifth-order Dresselhaus spin-orbit coupling terms.

• Received 24 April 2011

DOI:https://doi.org/10.1103/PhysRevB.84.121401