Nonequilibrium spin transport on Au(111) surfaces

The well-known experimentally observed $sp$-derived Au(111) Shockley surface states with Rashba spin splitting are perfectly fit by an effective tight-binding model, considering a two-dimensional hexagonal lattice with $p_z$ orbital and nearest-neighbor hopping only. The extracted realistic band parameters are then imported to perform the Landauer-Keldysh formalism to calculate nonequilibrium spin transport in a two-terminal setup sandwiching a Au(111) surface channel. Obtained results show strong spin density on the Au(111) surface and demonstrate (i) intrinsic spin-Hall effect, (ii) current-induced spin polarization, and (iii) Rashba spin precession, all of which have been experimentally observed in semiconductor heterostructures but not in metallic surface states. We therefore urge experiments in the latter for these spin phenomena.

Figure 1

(Color online) (a) Tight-binding energy dispersion $E_{\pm }^{\text{TBM}}$ and the experimentally measured binding energy $E_{\pm }^{\text{exp}}$ of Ref. 11 along the $\overline{\Gamma }\overline{M}$ direction. The surface Brillouin zone is sketched in the inset. (b) Total density of states and $E_{\pm }^{\text{TBM}}$ along $\overline{\Gamma }\overline{K}$ and $\overline{K}\overline{M}$ directions.

Figure 2

(Color online) Local spin density of (a) the out-of-plane component $⟨S_z⟩$ and (b) the in-plane component $\left(⟨S_x⟩,⟨S_y⟩\right)$ in a $398.9\times 63.3$ (total number of sites $N=931$) conducting sample made of Au(111) surface. The size of each local marker depicts the magnitude. In (a), red/dark (green/light) dots denote $⟨S_z⟩>0\left(⟨S_z⟩<0\right)$. The maximum of $⟨S_z⟩$ is $1.2\times {10}^{-3}\left(\hslash /2\right)$ while the mean of $⟨S_y⟩$ is $3.16\times {10}^{-4}\left(\hslash /2\right)$.

Figure 3

(Color online) Local spin density of (a) the out-of-plane component $⟨S_z⟩$ and (b) the in-plane component $\left(⟨S_x⟩,⟨S_y⟩\right)$ in a conducting sample made of Au(111) surface, subject to a ferromagnetic source lead with $+x$ magnetization. (c) $⟨S_x⟩$, $⟨S_y⟩$, and $⟨S_z⟩$ as a function of $x$ at $y=0$, i.e., along the dashed line sketched in (a). Computed values are compared with the previously obtained spin vector formula based on quantum mechanics.