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

## TL;DR

This paper develops a theoretical framework to understand how static electric fields influence the critical temperature of superconducting films, emphasizing the importance of film thickness and electron-phonon interactions.

## Contribution

It introduces a proximity Eliashberg theory model for electrostatic field effects on superconductors, highlighting parameter-free predictions in weak fields and the impact of surface layer thickness.

## Key findings

- Thin superconducting films with increased electron-phonon coupling can have higher critical temperatures.
- The theory provides parameter-free predictions in the weak electrostatic field limit.
- Optimal enhancement of Tc occurs with very thin, strongly charged superconducting layers.

## Abstract

We calculate the effect of a static electric field on the critical temperature of a s-wave one band superconductor in the framework of proximity effect Eliashberg theory. In the weak electrostatic field limit the theory has no free parameters while, in general, the only free parameter is the thickness of the surface layer where the electric field acts. We conclude that the best situation for increasing the critical temperature is to have a very thin film of a superconducting material with a strong increase of electron-phonon (boson) constant upon charging.