  Proximity orbital and spin-orbit effects of bilayer graphene on monolayer
WSe$_2$ are investigated from first-principles. We find that the built-in
electric field induces an orbital band gap of about 10 meV in bilayer graphene.
Remarkably, the proximity spin-orbit splitting for holes is two orders of
magnitude---the spin-orbit splitting of the valence band at K is about 2
meV---more than for electrons. Effectively, holes experience spin-valley
locking 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 this
effect and allows, instead for holes, electrons to be spin-valley locked with 2
meV spin-orbit splitting. Such a behavior constitutes a highly efficient
field-effect spin-orbit valve, making bilayer graphene on WSe$_2$ a potential
platform for a field-effect spin transistor.
