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We investigate the electronic band structure and the proximity exchangeeffect in bilayer graphene on a family of ferromagnetic multilayersCr$_2$X$_2$Te$_6$, X=Ge, Si, and Sn, with first principles methods. In eachcase the intrinsic electric field of the heterostructure induces an orbital gapon the order of 10 meV in the graphene bilayer. The proximity exchange isstrongly band dependent. For example, in the case of Cr$_2$Ge$_2$Te$_6$, thelow-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 theferromagnetic substrate. Remarkably, applying a vertical electric field of afew V/nm reverses the exchange, allowing us to effectively turn ON and OFFproximity magnetism in bilayer graphene. Such a field-effect should be genericfor van der Waals bilayers on ferromagnetic insulators, opening newpossibilities for spin-based devices.

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