  We report on a capacitance study of dual gated bilayer graphene. The measured
capacitance allows us to probe the electronic compressibility as a function of
carrier density, temperature, and applied perpendicular electrical displacement
D. As a band gap is induced with increasing D, the compressibility minimum at
charge neutrality becomes deeper but remains finite, suggesting the presence of
localized states within the energy gap. Temperature dependent capacitance
measurements show that compressibility is sensitive to the intrinsic band gap.
For large displacements, an additional peak appears in the compressibility as a
function of density, corresponding to the presence of a 1-dimensional van Hove
singularity (vHs) at the band edge arising from the quartic bilayer graphene
band structure. For D > 0, the additional peak is observed only for electrons,
while D < 0 the peak appears only for holes. This asymmetry that can be
understood in terms of the finite interlayer separation and may be useful as a
direct probe of the layer polarization.
