  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.
