Motivated by recent experiments reporting the suppression of the critical current in superconducting Dayem bridges by the application of strong electrostatic fields, in this paper 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 the gap becomes sensitive to the applied electric field when the size of the gap becomes an order of magnitude larger than the average level spacing of the spectrum in the normal state. In this case, the gap shows 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 toward the weak-to-moderate coupling limit or by introduction of a weak smearing in the density of states that effectively mimics a thicker sample and a weakly disordered system.
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