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 $\overline{K}$ and ${\overline{K}}^{\prime }$ points in honeycomb lattices. Our calculation (i) explains the recent experiment of the $\text{Tl}/\text{Si}\left(111\right)\text{−}\left(1\times 1\right)$ surface alloy [K. Sakamoto et al., Phys. Rev. Lett. 102, 096805 (2009)], where abrupt upstanding spin states near $\overline{K}$ are observed and (ii) predicts an electrically reversible out-of-plane surface spin polarization.

Figure 1

(Color online) The spin components (a) ${⟨S_x⟩}_{\mu \nu }$, (b) ${⟨S_y⟩}_{\mu \nu }$, and (c) ${⟨S_z⟩}_{\mu \nu }$ of energy eigenstates of a Rashba spin-orbit-coupled monoatomic honeycomb lattice with its band structure shown in (d). The parameters are set $t_R/\left|U\right|=0.12$. The Brillouin zone is shown in the inset in (d).

Figure 2

(Color online) Spin configuration for the $\nu =+1$ branche. Each arrow is given by $\left({⟨{\sigma }^x⟩}_{\mu +},{⟨{\sigma }^y⟩}_{\mu +}\right)$ and the color shading is determined by ${⟨{\sigma }^z⟩}_{\mu +}$.

Figure 3

(Color online) Energy dispersion and ${⟨{\sigma }^z⟩}_{\mu \nu }$ (indicated by arrows) for (a) monoatomic and (b) diatomic honeycomb lattices. In (a) the black dashed lines are the approximated dispersions given by Eq. (11); $E_M=\sqrt{U^2+4t_R^2}$. In (b) we set ${\Delta }_p=0.1\left|U\right|$; ${\epsilon }_M=\sqrt{{\Delta }_p^2+U^2+4t_R^2}$ and ${\epsilon }_K=\sqrt{{\Delta }_p^2+{\left(3t_R\right)}^2}$. The inset shows a sox-terminal bilayer honeycomb channel with black (gray) dots the upper (lower) sublattice.

Figure 4

(Color online) Surface spin polarization in a 6-terminal bilayer honeycomb channel with (a) ${\overline{K}}^{\prime }$ biasing, (b) and (c) $\overline{K}$ biasing, and (d) $\overline{M}$ biasing. Red/gray (green/light gray) dots represent a $⟨{\sigma }^z⟩>0\left(⟨{\sigma }^z⟩<0\right)$ local spin density with the dot size proportional to $\left|⟨{\sigma }^z⟩\right|$. $\overline{⟨{\sigma }^z⟩}$ and $\overline{⟨{\sigma }^{\parallel }⟩}$ as functions of the bias $e{V}_0$ are shown in (e) for $\overline{K}/{\overline{K}}^{\prime }$ biasing and in (f) for $\overline{M}$ biasing.