The emergence of interface states in graphene/transition metal dichalcogenides heterostructure with lateral interface

The relative strength of different proximity spin-orbit couplings in graphene on transition metal dichalcogenides (TMDC) can be tuned via the metal composition in the TMDC layer. While Gr/MoSe$_2$, has a normal gap, proximity to WSe$_2$ instead leads to valley-Zeeman-driven inverted bands. Although the $\mathbb{Z}_2$ index vanishes, these systems enable a concentration-dependent topological crossover with band gap closure when graphene is stacked on a composite or alloyed TMDC layer. This is due to a nonzero Berry curvature at the individual valleys and a change of the valley Chern index at a critical composition ratio. Therefore, inherently, we also expect that stacked heterostructures of graphene on composite TMDC layers should host localised boundary modes due to the presence of Gr/WSe$_2$- and Gr/MoSe$_2$-like domains with opposite valley Chern indices. In this study, we show that a Gr/(Mo-W)Se$_2$ heterostructure with a lateral interface in the TMDC layer can indeed host topologically protected in-gap propagating modes, similar to those at the border of commensurate AB and BA domains in biased minimally-twisted bilayer graphene. However, the stability of these modes depends crucially on the system size. We demonstrate that the electronic behaviour of Gr/(Mo-W)Se$_2$ heterostructures evolves from a homogeneous effective medium to a superposition of domain-localised bands and zero-energy branch crossings as the domain size in the alloyed TMDC layer is increased.