  Motivated by recent experiments reporting the suppression of the critical
current in superconducting Dayem bridges by the application of strong
electrostatic fields, in this work we study the impact on the superconducting
gap of charge redistribution in response to an applied electric field in thin
crystalline metals. By numerically solving the BCS gap equation and the Poisson
equation in a fully self-consistent way, we find that by reducing the pairing
strength we observe an increased sensitivity of the gap on the applied field,
showing sudden rises and falls that are compatible with surface modifications
of the local density of states. The effect is washed out by increasing the
pairing strength towards the weak-to-moderate coupling limit or by introduction
of a weak smearing in the density of states, showing the evolution from a clean
crystal to a weakly disordered metal.
