First-principles determination of spin-orbit coupling parameters in two-dimensional materials

Spin-orbit coupling (SOC) is fundamental to many phenomena in solid-state physics. Two-dimensional materials and van der Waals heterostructures provide researchers with exquisite control over this interaction; the ability to fine-tune SOC has impacts on spin transport and relaxation, topological states, optoelectronics, magnetization dynamics and even superconductivity and other correlated states. This Technical Review covers both the theoretical methodology and experimentally relevant phenomenology of SOC in 2D materials, by providing essential insights into the process of extracting the spin interactions from the underlying electronic structure obtained from first-principles density functional theory calculations. This Technical Review begins with graphene. Its SOC has a surprisingly complicated origin yet graphene remains the benchmark for other elemental centrosymmetric 2D materials in which SOC leads to a mixing of spin-up and spin-down components of the Bloch states. We then discuss spin-orbit materials, such as transition-metal dichalcogenides, in which strong SOC and the lack of space-inversion symmetry yield large spin splittings of the valence and conduction bands. This enables highly efficient optical spin orientation or robust valley Hall effect in transition-metal dichalcogenides. Next, we give guidelines for extracting the spin-orbit characteristics of van der Waals heterostructures, such as graphene/WSe2, which serve as a platform for SOC engineering. For these representative systems, we highlight the essentials of first-principles-based methodology, including supercell formation, strain artefacts, twisting, gating and lattice relaxation. Finally, we briefly discuss the effects of proximity exchange coupling, which is another relevant spin interaction for spintronics.