Electronic Structure and Rashba Spin-Orbit Coupling in Black Phosphorus

We investigate the electronic structure of black phosphorus using both the first-principles density-functional methods as well as a tight-binding model. The electronic structure in the gap region is described by a tight-binding Hamiltonian keeping the nearest-neighbor hopping and the $p$ orbitals. The calculated bond-centered Wannier functions lead to the bonding picture in terms of the occupation of the $p_\sigma$ bond orbitals along the phosphorous-phosphorous bonds. We find that a symmetry-breaking external electric field introduces a Rashba spin-orbit coupling; however, its magnitude is small, phosphorous being a small-Z atom. The magnitude is enhanced significantly if the phosphorous is replaced by the larger-Z bismuth.