A closer look at how symmetry constraints and the spin-orbit coupling shape the electronic structure of Bi(111).

Fully relativistic density-functional-theory calculations of Bi(111) thin films are analyzed to
revisit their two metallic surface-states branches. We first contrast these metallic branches with surface states arising at gaps in the valence band opened by the spin-orbit coupling (SOC).
We find that the two metallic branches along ΓM do not overlap with the bulk band at the zone
boundary, M. We show that the spin texture observed in such states cannot be traced to the lifting
of Kramers' degeneracy. Instead, we track them to the mj = ±1/2-mj = ±3/2 SOC splitting, the
potential anisotropy for in-plane and out-of-plane states, and the coupling between the opposite
surfaces of a slab occurring near M, which is driven by a spatial redistribution of the four metallic
states composing the two metallic branches. Each of these branches appears to be non-degenerate
at the tested surface, yet each is degenerate with another state of opposite spin at the other surface.
Nevertheless, the four metallic states bear some contribution on both surfaces of the film because of their
spatial redistribution near M. The overlapping among these states near M, afforded by their spatial
redistribution on both surfaces, causes a hybridization that perpetuates the splitting between the
two branches, makes the film's electronic structure thickness dependent near M, extinguishes the
magnetic moment of the metallic states avoiding the magnetic-moment discontinuity at M, and
denies the need or expectancy of the metallic branches becoming degenerate at M. We propose that
the opposite spin polarization observed for the two metallic branches occurs because the surface
atoms retain their covalent bonds and thus cannot afford magnetic polarization. We show that
the Rashba-splitting of the metallic states for inversion-asymmetric films does not have a fixed
magnitude but can be tuned by changing the perturbation breaking inversion symmetry.