Proximity-induced Ferromagnetic Order in Graphene on Transition Metal Dichalcogenides

We start with a graphene (GTMD) monolayer on two dimensional transition metal dichalcogenides (such as, MoS 2 / MoSe 2 /WS 2 /WSe 2 ) Hamiltonian (H G ) which is built on the orbital Hamiltonian ( H ) for pristine graphene. The former, apart from H , comprises of the staggered potential term ( H ∆ ) describing the effective orbital energy difference on A and B sub-lattices of graphene, the sub-lattice-resolved intrinsic spin- orbit coupling (H SO )− a next-nearest neighbor hopping much larger than that in the hydrogenated graphene, the pseudo-spin inversion asymmetry related spin-orbit coupling term− a next- nearest neighbor hopping with the spin-flip, and the Rashba-type spin-orbit coupling ( λ' R )which is actually the nearest-neighbor spin-flip hopping. The pen-ultimate term takes care of the spin- splitting away from the Dirac points K and K'. The term H G is basically a low-energy effective Hamiltonian around K and K' in the basis (A ↑ , B ↓ , A ↓ , B ↑ ) in momentum space. We calculate the electronic band dispersion of GTMD exchange coupled to the magnetic impurities, such as Fe atoms deposited to the graphene surface. The presence of the giant spin-orbit coupling fields give rise to large spin-polarization and inverted band structure (see Figure1). The inverted band structure is a precursor of the quantum spin Hall effect. We find that the band inversion feature is robust against the proximity-induced ferromagnetic order.