Spatial aspects of spin polarization of structurally split surface states in thin films with magnetic exchange and spin-orbit interaction

A theoretical study is presented of the effect of an in-plane magnetic exchange field on the band structure of centrosymmetric films of noble metals and topological insulators. Based on an ab initio relativistic $\mathbf{k}\cdot\mathbf{p}$ theory, a minimal effective model is developed that describes two coupled copies of a Rashba or Dirac electronic system residing at the opposite surfaces of the film. The coupling leads to a structural gap at $\bar{\Gamma}$ and causes an exotic redistribution of the spin density in the film when the exchange field is introduced. We apply the model to a nineteen-layer Au(111) film and to a five-quintuple-layer Sb$_2$Te$_3$ film. We demonstrate that at each film surface the exchange field induces spectrum distortions similar to those known for Rashba or Dirac surface states with an important difference due to the coupling: At some energies, one branch of the state loses its counterpart with the oppositely directed group velocity. This suggests that a large-angle electron scattering between the film surfaces through the interior of the film is dominant or even the only possible for such energies. The spin-density redistribution accompanying the loss of the counterpart favors this scattering channel.