Generalized Kohn-Sham approach for the electronic band structure of spin-orbit coupled materials

Spin-current density functional theory (SCDFT) is a formally exact framework designed to handle the treatment of interacting many-electron systems including spin-orbit coupling (SOC) at the level of the Pauli equation. In practice, robust and accurate calculations of the electronic structure of these systems call for functional approximations that depend not only on the densities and currents but also on spinors explicitly. Here we extend the generalized Kohn-Sham (GKS) approach of [Seidl et al., Generalized Kohn-Sham schemes and the band-gap problem, Phys. Rev. B 53, 3764 (1996)] to SCDFT. This framework entails the prominent cases of hybrid forms and meta-generalized-gradient-approximations. We clarify that the exchange-correlation potentials conjugate to the currents need to be computed within the GKS approach only when the spin currents are included in the functional form explicitly. We analyze the consequence of this fact for various approximations and numerical procedures for the evaluation of SOC effects. The practical power of the extended approach is demonstrated by calculating the spin-orbit induced/enhanced band splittings of inversion-asymmetric single-layer MoSe2 and inversion-symmetric bulk alpha - MoTe2. Key to these results is the capacity to account for SOC self-consistently while employing energy functionals and effective potentials that depend (implicitly or explicitly) on spin currents.