Rashba spin splitting of Shockley surface states on semi-infinite crystals

Abstract

We study the Rashba spin splitting of the $L$-gap surface state on the (111) surfaces of Au, Ag, and Cu, and that of the topologically protected surface state on Sb(111) by a first-principles calculation based on the embedded Green's function technique. By taking advantage of semi-infinite surface geometry, we address several issues that are difficult to handle with the finite slab calculation. For Au(111), we investigate the nonlinear dependence on wave vector $\mathbf{k}$ of the spin-splitting energy, and also at what wave number each dispersion curve of the two spin-split bands is merged into a surface-projected bulk band. Then, we discuss why Cu(111) exhibits a larger Rashba effect than Ag(111) despite the fact that the atomic orbitals of Ag have larger spin-orbit matrix elements than those of Cu. We reveal that, while the wave function of the $L$-gap surface state has strong ${p}_{z}$ character (the $z$ axis is surface normal), the Rashba spin splitting occurs through the spin-orbit matrix element between the ${d}_{{z}^{2}}$ and ${d}_{uz}$ orbital components hybridized with ${p}_{z}$, where ${d}_{uz}$ denotes a linear combination of ${d}_{xz}$ and ${d}_{yz}$ with its orbital lobes pointing to the $\mathbf{k}$ direction (defined as the $u$ axis). Cu(111) exhibits a larger spin splitting than Ag(111), since the amount of both the ${d}_{{z}^{2}}$ and ${d}_{uz}$ components mixed in its surface-state wave function is much larger than that of Ag(111). In contrast to the (111) surfaces of noble metals, the spin splitting of the topologically protected surface state on Sb(111) arises from the spin-orbit matrix element between the ${p}_{z}$ and ${p}_{u}$ orbital components. We also discuss under what condition the energy dispersion curve of a surface state intersects the boundary line between a projected bulk band and a projected bulk band gap at a large angle, instead of gradually approaching the boundary line with increasing $|\mathbf{k}|$. Finally, we comment on a recent theoretical work of Krasovskii [Phys. Rev. B 90, 115434 (2014)] on the microscopic origin of the Rashba spin splitting.