  We have performed electronic-structure and lattice-dynamics calculations on
the AB and AA structures of bilayer graphene. We study the effect of external
electric fields and compare results obtained with different levels of theory to
existing theoretical and experimental results. Application of an external field
to the AB bilayer alters the electronic spectrum, with the bands changing under
bias from a parabolic to a "Mexican hat" double-well structure. This results in
a semi-metal-to-semiconductor phase transition, with the size of the induced
electronic band-gap being tuneable through the field strength. A reduction of
continuous symmetry from a hexagonal to a triangular lattice is also evidenced
through in-plane electronic charge inhomogeneities between the sublattices.
When spin-orbit coupling is turned on for the AB system, we find that the bulk
gap decreases, gradually increasing for larger intensities of the bias. Under
large bias the energy dispersion recovers the Mexican hat structure, since the
energy interaction between the layers balances the coupling interaction. We
find that external bias perturbs the harmonic phonon spectra and leads to
anomalous behaviour of the out-of-plane flexural ZA and layer-breathing ZO'
modes. For the AA system, the electronic and phonon dispersions both remain
stable under bias, but the phonon spectrum exhibits zone-center imaginary modes
due to layer-sliding dynamical instabilities.
