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We investigate the proximity-induced spin-orbit coupling in heterostructuresof twisted graphene and monolayers of transition-metal dichalcogenides (TMDCs)MoS$_2$, WS$_2$, MoSe$_2$, and WSe$_2$ from first principles. We identifystrain, which is necessary to define commensurate supercells, as the key factoraffecting the band offsets and thus magnitudes of the proximity couplings. Weestablish that for biaxially strained graphene the band offsets between theDirac point and conduction (valence) TMDC bands vary linearly with strain,regardless of the twist angle. This relation allows to identify the apparentzero-strain band offsets and find a compensating transverse electric fieldcorrecting for the strain. The resulting corrected band structure is thenfitted around the Dirac point to an established spin-orbit Hamiltonian. Thisprocedure yields the dominant, valley-Zeeman and Rashba spin-orbit couplings.The magnitudes of these couplings do not vary much with the twist angle,although the valley-Zeeman coupling vanishes for 30$^{\circ}$ and Mo-basedheterostructures exhibit a maximum of the coupling at around 20$^{\circ}$. Themaximum for W-based stacks is at 0$^{\circ}$. The Rashba coupling is in generalweaker than the valley-Zeeman coupling, except at angles close to 30$^{\circ}$.We also identify the Rashba phase angle which measures the deviation of thein-plane spin texture from tangential, and find that this angle is verysensitive to the applied transverse electric field. We further discuss thereliability of the supercell approach with respect to atomic relaxation(rippling of graphene), relative lateral shifts of the atomic layers, andtransverse electric field.

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