  Depositing monolayer graphene on a transition metal dichalcogenide (TMD)
semiconductor substrate has been shown to change the dynamics of the electronic
states in graphene, inducing spin orbit coupling (SOC) and staggered potential
effects. Theoretical studies on commensurate supercells have demonstrated the
appearance of interesting phases, as different materials and relative gate
voltages are applied. Here we address the effects of the real
incommensurability between lattices by implementing a continuum model approach
that does not require small-period supercells. The approach allows us to study
the role of possible relative twists of the layers, and verify that the SOC
transfer is robust to twists, in agreement with observations. We characterize
the nature of the different phases by studying an effective Hamiltonian that
fully describes the graphene-TMD heterostructure. We find the system supports
topologically non-trivial phases over a wide range of parameter ranges, which
require the dominance of the intrinsic SOC over the staggered and Rashba
potentials. This tantalizing result suggests the possible experimental
realization of a tunable quantum spin Hall phase under suitable conditions. We
estimate that most TMDs used to date likely result in weak intrinsic SOC that
would not drive the heterostructure into topologically non-trivial phases.
Additional means to induce a larger intrinsic SOC, such as strain fields or
heavy metal intercalation may be required.
