Fixed Number and Quantum Size Effects in Nanoscale Superconductors

TL;DR

This paper investigates how fixed electron number and finite size influence superconductivity in nanoscale materials, revealing phenomena like negative gaps and magic number effects that differ from traditional grand canonical models.

Contribution

It applies canonical BCS theory to nanoscale superconductors, demonstrating the importance of fixed electron number and finite size effects, and compares results with exact solutions.

Findings

Negative gap appears naturally in the canonical scheme for odd electron numbers.

Magic electron numbers lead to larger energy gaps in even electron systems.

Canonical results align with exact solutions, unlike grand canonical predictions.

Abstract

In recent experiments on nanoscale Al particles, whose electron number was fixed by charging effects, a ``negative gap'' was observed in particles with an odd number of electrons. This observation has called into question the use of a grand canonical ensemble in describing superconductivity in such ultrasmall particles. We have studied the effects of fixed electron number and finite size in nanoscale superconductors, by applying the canonical BCS theory for the attractive Hubbard model. The ground state energy and the energy gap are compared with the conventional and parity-projected grand canonical BCS results, and in one dimension also with the exact solutions by the Bethe ansatz. The crossover from the bulk to quantum limit is studied for various regimes of electron density and coupling strength. The effects of boundary conditions and different lattice structures are also examined. A ``negative gap'' for odd electron number emerges most naturally in the canonical scheme. For even electron number, the gap is particularly large for ``magic numbers'' of electrons for a given system size or of atoms for a fixed electron density. These features are in accordance with the exact solutions, but are essentially missed in the grand canonical results.