Abstract
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 nontrivial 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 nontrivial phases. Additional means to induce a larger intrinsic SOC, such as strain fields or heavy metal intercalation, may be required.