  Bilayer graphene has drawn significant attention due to the opening of a band
gap in its low energy electronic spectrum, which offers a promising route to
electronic applications. The gap can be either tunable through an external
electric field or spontaneously formed through an interaction-induced symmetry
breaking. Our scanning tunneling measurements reveal the microscopic nature of
the bilayer gap to be very different from what is observed in previous
macroscopic measurements or expected from current theoretical models. The
potential difference between the layers, which is proportional to charge
imbalance and determines the gap value, shows strong dependence on the disorder
potential, varying spatially in both magnitude and sign on a microscopic level.
Furthermore, the gap does not vanish at small charge densities. Additional
interaction-induced effects are observed in a magnetic field with the opening
of a subgap when the zero orbital Landau level is placed at the Fermi energy.
