  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.
