  The electronic properties of doped bilayer graphene in presence of bottom and
top gates have been studied and characterized by means of Density Functional
Theory calculations. Varying independently the bottom and top gates it is
possible to control separately the total doping charge on the sample, and the
average external electric field acting on the bilayer. We show that, at fixed
doping level, the band gap at the K point in the Brillouin zone depends
linearly on the average electric field, whereas the corresponding
proportionality coefficient has a non-monotonic dependence on doping. We find
that the DFT-calculated band gap at K, for small doping levels, is roughly half
of the band gap obtained with standard Tight Binding approach. We show that
this discrepancy arises from an underestimate, in the TB model, of the
screening of the system to the external electric field. In particular, on the
basis of our DFT results we observe that, when the bilayer graphene is in
presence of an external electric field, both an interlayer and an intralayer
screening occur. Only the interlayer screening is included in TB calculations,
while both screenings are fundamental for the description of the band gap
opening. We finally provide a general scheme to obtain the full band structure
of the gated bilayer graphene, for an arbitrary value of the external electric
field and of the doping.
