Electrically tunable exchange splitting in bilayer graphene on monolayer Cr$_2$X$_2$Te$_6$ with X=Ge, Si, and Sn

We investigate the electronic band structure and the proximity exchange effect in bilayer graphene on a family of ferromagnetic multilayers Cr$_2$X$_2$Te$_6$, X=Ge, Si, and Sn, with first principles methods. In each case the intrinsic electric field of the heterostructure induces an orbital gap on the order of 10 meV in the graphene bilayer. The proximity exchange is strongly band dependent. For example, in the case of Cr$_2$Ge$_2$Te$_6$, the low-energy valence band of bilayer graphene has exchange splitting of 8 meV, while the low energy conduction band's splitting is 30 times less (0.3 meV). This striking discrepancy stems from the layer-dependent hybridization with the ferromagnetic substrate. Remarkably, applying a vertical electric field of a few V/nm reverses the exchange, allowing us to effectively turn ON and OFF proximity magnetism in bilayer graphene. Such a field-effect should be generic for van der Waals bilayers on ferromagnetic insulators, opening new possibilities for spin-based devices.