The Rashba effect in several surface systems, Au(111), Au(110), Ag(111), Sb(111) and Si(111)-Bi, is studied by means of first-principles relativistic density-functional calculations. The importance of the asymmetric behavior around the surface atom is emphasized as a crucial factor to determine the magnitude of Rashba spin splitting in addition to the size of the spin-orbit coupling. The Rashba effect at the Brillouin-zone boundary is generally described with time-reversal symmetry. Distinctive features in the spin splitting and spin direction for a two-dimensional hexagonal system are discussed with the use of symmetry in the double group of k.

A peculiar Rashba effect is found at a point in the Brillouin zone, where the time-reversal symmetry is broken, though this symmetry was believed to be a necessary condition for Rashba splitting. This finding obtained experimentally by photoemission measurements on a Bi/Si(111)-(sqrt(3) x sqrt(3)) surface is fully confirmed by a first-principles theoretical calculation. We found that the peculiar Rashba effect is simply understood by the two-dimensional symmetry of the surface, and that this effect leads to an unconventional nonvortical Rashba spin structure at a point with time-reversal invariance.