Electronic structure and anisotropic Rashba spin-orbit coupling in monolayer black phosphorus

We investigate the electronic structure of the monolayer black phosphorus (BP) using density-functional methods both with and without an applied electric field. We find that a simple one-band tight-binding Hamiltonian based on the ${p}_{z}$ orbitals and nearest-neighbor hopping is sufficient to describe the band structure in the gap region rather well and justification for this is given from symmetry arguments. The anisotropic nature of the band structure leads in turn to an anisotropic Rashba effect, where the magnitude of the spin splitting caused by an applied electric field is not only momentum dependent, but also depends on the direction of $\stackrel{P\vec}{k}$. The Rashba Hamiltonian is generalized for the anisotropic case, which reads: ${H}_{R}={\ensuremath{\alpha}}_{R}(\stackrel{P\vec}{\ensuremath{\sigma}}\ifmmode\times\else\texttimes\fi{}{\stackrel{P\vec}{k}}^{\ensuremath{'}})\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$, where the scaled momentum ${\stackrel{P\vec}{k}}^{\ensuremath{'}}$ contains the anisotropy effect. The Rashba effect is studied quantitatively for BP from ab initio density-functional calculations in the presence of an applied electric field. A byproduct of this work is the demonstration that the strength of the spin-orbit coupling for the outermost electrons in the atoms, which are relevant for the solids, increases only as the Landau-Lifshitz ${Z}^{2}$ scaling with the atomic number $Z$, rather than the higher power ${Z}^{4}$ scaling, as sometimes thought.