  We investigate the proximity-induced spin-orbit coupling in heterostructures
of twisted graphene and monolayers of transition-metal dichalcogenides (TMDCs)
MoS$_2$, WS$_2$, MoSe$_2$, and WSe$_2$ from first principles. We identify
strain, which is necessary to define commensurate supercells, 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 conduction (valence) TMDC 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. This
procedure 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-based
heterostructures exhibit a maximum of the coupling at around 20$^{\circ}$. The
maximum for W-based stacks is at 0$^{\circ}$. The Rashba coupling is in general
weaker than the valley-Zeeman coupling, except at angles close to 30$^{\circ}$.
We also identify the Rashba phase angle which measures the deviation of the
in-plane spin texture from tangential, and find that this angle is very
sensitive to the applied transverse electric field. We further discuss the
reliability of the supercell approach with respect to atomic relaxation
(rippling of graphene), relative lateral shifts of the atomic layers, and
transverse electric field.
