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Theoretical explanation of electric field-induced superconductive critical temperature shifts in Indium thin films

We calculate the effect of a static electric field on the superconductive critical temperature of Indium thin films in the framework of proximity effect Eliashberg theory, in order to explain 60 years old experimental data. Since in the theoretical model we employ all quantities of interest can be computed ab-initio (i.e. electronic densities of states, Fermi energy shifts and Eliashberg spectral functions), the only free parameter is in general the thickness of the surface layer where the electric field acts. However, in the weak electrostatic field limit Thomas-Fermi approximation is still valid and therefore no free parameters are left, as this perturbed layer is known to have a thickness of the order of the Thomas-Fermi screening length. We show that the theoretical model can reproduce experimental data, even when the magnitude of the induced charge densities are so small to be usually neglected.