Spin polarization by first-principles relativistic k·p theory: Application to the surface alloys PbAg2 and BiAg2

An effective $\mathbf{k}\phantom{\rule{0.16em}{0ex}}\ifmmode\cdot\else\textperiodcentered\fi{}\phantom{\rule{0.16em}{0ex}}\mathbf{p}$ Hamiltonian is applied to spin-orbit-split states in the surface alloys ${\mathrm{PbAg}}₂$ and ${\mathrm{BiAg}}₂$ on Ag(111) with the aim to infer the microscopic mechanism behind the momentum dependence of the spin polarization beyond the standard Rashba model. The Hamiltonian is derived from the ab initio wave functions at $\overline{\mathrm{\ensuremath{\Gamma}}}$ without introducing any adjustable parameters so as to accurately reproduce the spin of the surface-alloy states, in particular the sign reversal of the spin polarization within a nondegenerate band. We establish the origin of the sign reversal and describe the spin-orbit splitting around $\overline{\mathrm{\ensuremath{\Gamma}}}$ as a combination of the linear and cubic Rashba effects. Furthermore, we predict the behavior of the ${\mathrm{PbAg}}₂$ and ${\mathrm{BiAg}}₂$ states under the influence of an in-plane and out-of-plane magnetization for a surface alloy in contact with a magnetic layer. Apart from the surface-alloy states, we identify a substrate-related surface state originating from the same branch of the complex band structure as the surface state on a clean Ag(111).