BCS theory for finite size superconductors

TL;DR

This paper investigates how finite size and shape influence superconductivity in metallic grains by analyzing the BCS order parameter and excitation spectrum, revealing mesoscopic fluctuations and shell effects.

Contribution

It introduces a combined approach using BCS self-consistency, semiclassical expansion, and corrections to study finite size effects in superconducting grains.

Findings

Mesoscopic fluctuations cause smooth variations in the order parameter.

Shell effects lead to large energy gap changes with small electron number variations.

Finite size and shape significantly impact superconducting properties.

Abstract

We study finite size effects in superconducting metallic grains and determine the BCS order parameter and the low energy excitation spectrum in terms of size, and shape of the grain. Our approach combines the BCS self-consistency condition, a semiclassical expansion for the spectral density and interaction matrix elements, and corrections to the BCS mean-field. In chaotic grains mesoscopic fluctuations of the matrix elements lead to a smooth dependence of the order parameter on the excitation energy. In the integrable case we observe shell effects when e.g. a small change in the electron number leads to large changes in the energy gap.