Title: Theory of superconductivity in thin films under an external electric field

Abstract: The supercurrent field effect is experimentally realized in various nano-scale devices, based on the superconductivity suppression by external electric fields being effective for confined systems. In spite of intense research, a microscopic theory and explanation of this effect is missing. Here, a microscopic theory of phonon-mediated superconductivity in thin films is presented, which accounts for the effect of quantum confinement on the electronic density of states, on the Fermi energy, and on the topology of allowed states in momentum space. By further accounting for the interplay between quantum confinement, the external static electric field, the Thomas-Fermi screening in the electron-phonon matrix element, and the effect of confinement on the Coulomb repulsion parameter, the theory predicts the critical value of the external electric field as a function of the film thickness, above which superconductivity is suppressed. In particular, this critical value of the electric field is the lower the thinner the film, in agreement with recent experimental observations. Crucially, this effect is predicted by the theory when both Thomas-Fermi screening and the Coulomb pseudopotential are taken into account, along with the respective dependence on thin film thickness. This microscopic theory of the supercurrent field-effect opens up new possibilities for electric-field gated quantum materials.