Spin-orbit coupling effects in single-layer phosphorene

The electronic band structure of monolayer phosphorene is thoroughly studied by considering the presence of spin-orbit interaction. We employ a multiorbital Slater-Koster tight-binding approach to derive effective k · p -type Hamiltonians that describes the dominant spin-orbit coupling (SOC) effects of the Rashba and intrinsic origin at the high Γ and S high symmetry points in phosphorene. In the absence of SOC effects a minimal admixture of p z and p y atomic orbitals suffices to reproduce the well-known anisotropy of highest valence and the lowest conduction bands at the Γ point, consistent with density functional theory (DFT) and k · p methods. In contrast, the inclusion of the p x and s atomic orbitals are rather crucial for an adequate description of the SOC effects in phosphorene at low energies, particularly at the S point. We introduce useful analytical expressions for the Rashba and intrinsic SOC parameters in terms of the relevant Slater-Koster integrals. In addition, we report simple formulas for the interband dipole strengths, revealing the nature of the strong anisotropic behavior of its lower bands. Our findings can be useful for further studies of electronic and spin transport properties in monolayer phosphorene and its nanoribbons.