Electronic Shell Structure of Nanoscale Superconductors

TL;DR

This paper investigates the electronic shell structure in nanoscale superconductors, revealing shell effects and odd-electron negative gaps through a canonical BCS approach, aligning with recent experimental findings.

Contribution

It introduces a canonical BCS framework to study shell effects in nanoscale superconductors, capturing phenomena missed by grand canonical methods.

Findings

Negative gap for odd-electron particles observed.

Shell structure in energy gap for even-electron particles.

Enhanced gaps at 'magic numbers' of electrons or atoms.

Abstract

Motivated by recent experiments on Al nanoparticles, we have studied the effects of fixed electron number and small size in nanoscale superconductors, by applying the canonical BCS theory for the attractive Hubbard model in two and three dimensions. A negative ``gap'' in particles with an odd number of electrons as observed in the experiments is obtained in our canonical scheme. For particles with an even number of electrons, the energy gap exhibits shell structure as a function of electron density or system size in the weak-coupling regime: the gap is particularly large for ``magic numbers'' of electrons for a given system size or of atoms for a fixed electron density. The grand canonical BCS method essentially misses this feature. Possible experimental methods for observing such shell effects are discussed.