Critical temperature and specific heat for Cooper pairing on a spherical surface

TL;DR

This paper provides an exact solution for the specific heat and critical temperature of Cooper pairing on a spherical surface, revealing a characteristic temperature for the onset of pair correlations and an odd-even parity effect in large electron systems.

Contribution

It introduces an exact solution for BCS-type pairing on a sphere and identifies a characteristic temperature and parity effects relevant to multielectron bubbles.

Findings

Critical temperature for pair correlations is 10-100 mK.

Specific heat exhibits a 4-6% odd-even parity effect.

Parity effect persists up to 10^6 electrons.

Abstract

Based on an exact solution of the Bardeen-Cooper-Schrieffer type Hamiltonian on a spherical surface, we calculate the specific heat for the electron system with pair correlations on a sphere. We find that it is possible to extract from the specific heat a temperature above which many-body states with broken Cooper pairs get populated. Therefore, we define this temperature as the characteristic temperature signalling the onset of a BCS-type pair-correlated state for electrons on a spherical surface. Such spherical electron systems are realized in multielectron bubbles in liquid helium, for which the above-mentioned characteristic temperature is found to be of the order of 10-100 mK. Both the specific heat and the critical temperature show a pronounced (4-6%) odd-even parity effect that persists even for numbers of electrons as large as 10$^6$.