BCS superconductivity in metallic nanograins: Finite-size corrections, low energy excitations, and robustness of shell effects

TL;DR

This paper analytically explores how the superconducting gap in metallic nanograins depends on size, shape, and electron number, highlighting shell effects, mesoscopic fluctuations, and robustness of superconductivity in finite systems.

Contribution

It introduces an analytical framework combining BCS theory with semiclassical methods to study size and shape effects on superconductivity in nanograins.

Findings

Shell effects cause large gap variations at specific electron numbers.

Mesoscopic fluctuations smooth out the energy dependence of the order parameter.

Superconducting shell effects are robust against small geometrical deformations.

Abstract

We combine the BCS self-consistency condition, a semiclassical expansion for the spectral density and interaction matrix elements to describe analytically how the superconducting gap depends on the size and shape of a 2d and 3d superconducting grain. 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 find shell effects i. e. for certain values of the electron number $N$ a small change in N leads to large changes in the energy gap. With regard to possible experimental tests we provide a detailed analysis of the dependence of the gap on the coherence length and the robustness of shell effects under small geometrical deformations.