Methods of solving the system of equations for the energy gap in the revisited BCS theory of superconductivity

Dragoş-Victor Anghel; Amanda Teodora Preda

doi:10.1016/j.mex.2021.101388

Methods of solving the system of equations for the energy gap in the revisited BCS theory of superconductivity

MethodsX. 2021 May 19:8:101388. doi: 10.1016/j.mex.2021.101388. eCollection 2021.

Authors

Dragoş-Victor Anghel¹, Amanda Teodora Preda^{1

2}

Affiliations

¹ Horia Hulubei National Institute for Physics and Nuclear Engineering, P.O. Box MG-6, 077126 Măgurele, Ilfov, Romania.
² University of Bucharest, Faculty of Physics, Romania.

Abstract

The Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity has been revisited in a series of papers [1], [2], [3] and in this context the equation for the energy gap was generalized to a system of integral equations. The physical consequences of this change are major, leading not only to the change of the critical temperature and of energy gap, but even to a change of the order of the phase transition and to multiple solutions for the energy gap. Nevertheless, finding the solutions of the proposed system of equations is much more complicated than solving the typical BCS gap equation and requires a careful analysis. This analysis is done here and consists of the following steps:•writing the system of equations at finite temperature ( $k_{B} T$ comparable with the energy gap $Δ$ ) and in the low temperature limit ( $k_{B} T ≪ Δ$ );•separate analysis of the equations and of their solutions in the two temperature ranges, (first) $k_{B} T ≪ Δ$ and (second) $k_{B} T$ comparable with $Δ$ ;•presenting the methods to consistenlty searching the solutions.

Keywords: BCS gap equation; BCS theory; Phase transitions; Quantum statistics; Superconductivity.