# Superconducting size effect in thin films under electric field: mean-field self-consistent model

## TL;DR

This paper presents a self-consistent mean-field model analyzing how an external electric field influences superconductivity in thin films, revealing energy scaling behaviors near subband edges and discussing nonlinear effects beyond simple gating.

## Contribution

It introduces a mean-field self-consistent model for superconducting thin films under electric fields, highlighting energy scaling near subband edges and nonlinear effects, contrasting with experimental observations.

## Key findings

- Electrostatic effects scale as μ^{-1} near subband edges
- Energy scaling follows μ^{-2} away from subband edges
- Discussion of nonlinearities beyond gate effects

## Abstract

We consider effects of an externally applied electrostatic field on superconductivity, self-consistently within a BCS mean field model, for a clean 3D metal thin film. The electrostatic change in superconducting condensation energy scales as $\mu^{-1}$ close to subband edges as a function of the Fermi energy $\mu$, and follows 3D scaling $\mu^{-2}$ away from them. We discuss nonlinearities beyond gate effect, and contrast results to recent experiments.