Twist-angle dependent proximity induced spin-orbit coupling in graphene/transition-metal dichalcogenide and graphene/topological insulator heterostructures*

We investigate the proximity-induced spin-orbit coupling in twisted heterostructures of graphene/transition-metal dichalcogenides (MoS₂, WS₂, MoSe₂, and WSe₂) [1] as well as graphene/topological insulators (Bi₂Se₃ and Bi₂Te₃) from first principles. The strain in graphene, which is necessary to define commensurate supercells, is identified as the key factor affecting the band offsets and thus magnitudes of the proximity couplings. We establish that for biaxially strained graphene the band offsets between the Dirac point and the substrate bands vary linearly with strain, regardless of the twist angle. This relation allows to identify the apparent zero-strain band offsets and find a compensating transverse electric field correcting for the strain. The resulting corrected band structure is then fitted around the Dirac point to an established spin-orbit Hamiltonian, yielding the twist angle dependencies of the spin-orbit couplings. <!-- x-tinymce/html -->While for most structures a mix of Rashba and valley-Zeeman spin-orbit coupling is present, we also witness the emergence of Kane-Mele spin-orbit coupling in graphene/topological insulator structures at 30° twist angle.

[1] Naimer et al., Phys. Rev. B 2021, 104, 195156