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We calculate the effect of a static electric field on the criticaltemperature of a s-wave one band superconductor in the framework of proximityeffect Eliashberg theory. In the weak electrostatic field limit the theory hasno free parameters while, in general, the only free parameter is the thicknessof the surface layer where the electric field acts. We conclude that the bestsituation for increasing the critical temperature is to have a very thin filmof a superconducting material with a strong increase of electron-phonon (boson)constant upon charging.

Proximity Eliashberg theory of electrostatic field-effect-doping in superconducting films