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

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