One of the most persuasive arguments in support of this explanation was the absence of the splitting at r (the center of the Brillouin zone), which seemingly indicated that the splitting of the surface bands for [k.sub.[parallel]] [not equal to] 0 was allowed due to an absence of inversion symmetry at the surface and thus the lifting of Kramers' degeneracy.
Second, any real crystal or film always has an opposite surface, which restores the inversion symmetry in a general case.
It is just this splitting of the surface bands that was interpreted as a Rashba-type spin-splitting, thus apparently supporting the concept of the brake of the inversion symmetry at the surface and hence the lifting of the spin degeneracy of the bands.
The Rashba splitting could be switched on by breaking the inversion symmetry of the slab with a help of adsorbed layer, so we have calculated semirelativistic and relativistic band structures for Au(111) film with one surface covered by H (Figure 4) (similar method was adopted in , for Bi(111) films, to eliminate, as it was suggested, the interaction between surface states).
In contrast to the films with clean surfaces, the slab with one surface covered by adsorbate (hydrogen in the present case) has no inversion symmetry. Consequently, the SOC coupling lifts the spin degeneracy and causes the spin-splitting of the bands (Figure 4(c)).
Among various material candidates for valleytronics, spatial inversion symmetry broken two-dimensional (2D) honeycomb lattice systems such as graphene and monolayer Mo[S.sub.2] are predicted to be the most useful.
In finite doping concentration the existence of dopant introduced exchange field breaks the time inversion symmetry and decouples the energetically degenerated valley into nondegenerate and elucidating the occurrence of valley polarization.
Niu, "Valley-dependent optoelectronics from inversion symmetry breaking," Physical Review B: Condensed Matter and Materials Physics, vol.