Upstanding Rashba spin in honeycomb lattices: Electrically reversible surface spin polarization

The spin-split states subject to Rashba spin-orbit coupling in two-dimensional systems have long been accepted as pointing inplane and perpendicular to the corresponding wave vectors. This is in general true for free electron model, but exceptions do exist elsewhere. Within the tight-binding model, we unveil the unusual upstanding behavior of those Rashba spins around $\bar{K}$ and ${\bar{K}}^{\mathrm{\prime }}$ points in honeycomb lattices. Our calculation (i) explains the recent experiment of the Tl/Si(111)-$(1\times 1)$ surface alloy [Phys. Rev. Lett. $\mathbf{\text{102}}$, 096805 (2009)], where abrupt upstanding spin states near $\bar{K}$ are observed, and (ii) predicts an electrically reversible out-of-plane surface spin polarization.