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Proximity orbital and spin-orbit effects of bilayer graphene on monolayerWSe$_2$ are investigated from first-principles. We find that the built-inelectric field induces an orbital band gap of about 10 meV in bilayer graphene.Remarkably, the proximity spin-orbit splitting for holes is two orders ofmagnitude---the spin-orbit splitting of the valence band at K is about 2meV---more than for electrons. Effectively, holes experience spin-valleylocking due to the strong proximity of the lower graphene layer to WSe$_2$.However, applying an external transverse electric field of some 1 V/nm,countering the built-in field of the heterostructure, completely reverses thiseffect and allows, instead for holes, electrons to be spin-valley locked with 2meV spin-orbit splitting. Such a behavior constitutes a highly efficientfield-effect spin-orbit valve, making bilayer graphene on WSe$_2$ a potentialplatform for a field-effect spin transistor.

Proximity effects in bilayer graphene on monolayer WSe$_2$: Field-effect spin-valley locking, spin-orbit valve, and spin transistor