Eur. Phys. J. B (2016) 89: 164
https://doi.org/10.1140/epjb/e2016-70029-3

Regular Article

From granules to bulk superconductors using Richardson-Gaudin equations

¹ Institut des NanoSciences de Paris, Université Pierre et Marie Curie, CNRS, 2 place Jussieu, 75005 Paris, France
² Institut de Recherche Mathématiques de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

^a e-mail: monique.combescot@insp.jussieu.fr

Received: 15 January 2016
Received in final form: 18 April 2016
Published online: 11 July 2016

Abstract

We provide a compact expression of the ground-state energy of N-Cooper pairs valid from small to large sample volumes, as checked by numerically solving Richardson-Gaudin equations which give the exact eigenstates of BCS superconductors. This expression contains a contribution linear in the potential amplitude, dominant for small samples, and an exponential contribution dominant when the number of states available for pairing gets larger than a material-dependent threshold independent from sample size. These “available states” are the states feeling the BCS potential, reduced by the Pauli exclusion principle through a “moth-eaten effect” which comes from the composite boson nature of Cooper pairs. This work also presents an elegant derivation of the N-Cooper pair energy obtained recently, which makes use of the roots of the degree-N Hermite polynomial.