Effective Hamiltonian for Extrinsic Spin-Orbit Coupling in 2D Materials

Inversion symmetry of a two-dimensional (2D) electron system can be broken by a perpendicular external electric field which lifts the spin degeneracy while it preserves the time-reversal symmetry. The resulting spin splitting, known as the Bychkov-Rashba (BR) spin-orbit coupling, can be used for spin manipulation solely using electric fields without an external magnetic field. However, the main limitation of the phenomenological BR Hamiltonian is that it is inapplicable to two-dimensional (2D) monolayer materials in which the crystal structure is different than that of conventional zinc-blende and wurtzite semiconductors. In this work, we use the theory of invariants to derive the effective Hamiltonian for 2D materials, such as phosphorene and transition metal dichalcogenides, without resorting to any phenomenological prefactor as that in the BR model. Owing to their vertical and lateral scalability and high electron mobility, 2D materials provide an excellent platform to realize semiconducting spintronics devices. We determine the bands that contribute to the extrinsic spin-orbit coupling of conduction electrons and calculate the prefactors in terms of k.p parameters.