  Proximity effects resulting from depositing a graphene layer on a TMD
substrate layer change the dynamics of the electronic states in graphene,
inducing spin orbit coupling (SOC) and staggered potential effects. An
effective Hamiltonian that describes different symmetry breaking terms in
graphene, while preserving time reversal invariance, shows that an inverted
mass band gap regime is possible. The competition of different perturbation
terms causes a transition from an inverted mass phase to a staggered gap in the
bilayer heterostructure, as seen in its phase diagram. A tight-binding
calculation of the bilayer validates the effective model parameters. A relative
gate voltage between the layers may produce such phase transition in
experimentally accessible systems. The phases are characterized in terms of
Berry curvature and valley Chern numbers, demonstrating that the system may
exhibit quantum spin Hall and valley Hall effects.
