  Unlike single layer graphene, in the case of $AB$-stacked bilayer graphene
(BLG) one can induce a non-zero energy gap by breaking the inversion symmetry
between the two layers using a perpendicular electric field. This is an
essential requirement in field-effect applications, particularly since the
induced gap in BLG systems can be further tuned by the magnitude of the
external electric field. Doping is another way to modify the electronic
properties of graphene based systems. We investigate here BLG systems doped
with boron and nitrogen in the presence of external electric field, in the
framework of density functional theory (DFT) calculations. Highly doped BLG
systems are known to behave as degenerate semiconductors, where the Fermi
energy depends on the doping concentration but, in addition, we show that the
electronic properties drastically depend also on the applied electric field. By
changing the magnitude and the orientation of the electric field, the gap size
and position relative to the Fermi level may be tuned, essentially controlling
the effect of the extrinsic doping. In this context, we discuss in how far the
external electric field may suitably adjust the effective doping and,
implicitly, the conduction properties of doped BLG systems.
