We study the electronic structure of heterostructures formed by a graphene nanoribbon (GNR) and a transition metal dichalcogenides (TMD) monolayer using first-principles. We consider both semiconducting TMDs and metallic TMDs, and different stacking configurations. We find that when the TMD is semiconducting the effects on the band structure of the GNRs are small. In particular the spin-splitting induced by proximity on the GNRs bands is only of the order of few meV irrespective of the stacking configuration. When the TMD is metallic, such as NbSe2, we find that the spin-splitting induced in the GNRs can be very large and strongly dependent on the stacking configuration. For optimal stacking configurations the proximity-induced spin-splitting is of the order of 20 meV for armchair graphene nanoribbons, and as high as 40 meV for zigzag graphene nanoribbons. This results are encouraging for the prospects of using GNR-TMD heterostructures to realize quasi one-dimensional topological superconducting states supporting Majorana modes.
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