Proximity spin-orbit and exchange coupling in ABA and ABC trilayer graphene van der Waals heterostructures

We investigate the proximity spin-orbit and exchange couplings in ABA ABC trilayer graphene encapsulated within monolayers of semiconducting transition-metal dichalcogenides ferromagnetic semiconductor Cr$_2$Ge$_2$Te$_6$. Employing first-principles calculations we obtain electronic structures multilayer stacks extract relevant proximity-induced orbital spin interaction parameters by fitting low-energy bands to model Hamiltonians. also demonstrate tunability effects a transverse electric field. Using Hamiltonians study mixed spin-orbit/exchange coupling encapsulation, which allows tailor interactions very efficiently applied summarize physics bare ABA, ABC, ABB trilayers, provide, along with results band structures, density states, splittings, electric-field tunabilities bands, qualitative understanding observed behavior realistic as resource for simulations transport correlation graphene.