Figure 4: Low-energy band structure of graphene on WS₂ near the K/K′ point.

(a) Comparison of the result of the ab initio calculations with the continuum model Hamiltonian discussed in the main text. The black dots represent the low-energy dispersion relation for graphene on WS₂ as obtained from our density functional theory calculations, which can be fit with excellent precision with the dispersion relation obtained from equation (2) (red lines). (b–d) Evolution of the low-energy dispersion relation of graphene as a function of SOI. (b) The usual Dirac cone for spin-degenerate charge carriers in isolated graphene close to the K/K′ point. (c,d) At an interface with a WS₂ substrate, the dispersion relation is modified by the effect of the induced SOI. Ab initio calculations show that the low-energy Hamiltonian in equation (2) accurately describes the modifications to the band structure of graphene. Two SOI terms, with coupling constant λ and λ_R, are induced by interfacial interactions. Our calculations indicate that λ∼5 meV and λ_R∼1 meV. With these values the dispersion relation of electrons becomes the one shown in c that, at charge neutrality, corresponds to the band structure of an insulator (with non-trivial topological properties). The size of the gap is determined by the value of λ_R (as long as λ>>λ_R). Since the gap is likely small in our devices as compared with electrostatic potential fluctuations, λ_R can be neglected in a first approximation, in which case the dispersion relation becomes the one shown in d.