Detailed Analysis of the Superconducting Gap with Dynes Pair-Breaking Scattering

TL;DR

This paper analytically investigates the energy gap in Dynes superconductors, accounting for pair-breaking scattering, providing simplified formulas and insights into the gap's behavior near critical temperature and scattering limits.

Contribution

It introduces analytical solutions for the superconducting energy gap considering pair-breaking scattering, extending the BCS theory with practical tools for further research.

Findings

Derived analytical form of gap-to-Tc ratio at critical pair-breaking

Established temperature dependence of the energy gap near Tc

Provided approximate formulas for the gap under general scattering conditions

Abstract

We study the energy gap within the Dynes superconductor theory. This model generalizes the Bardeen-Cooper-Schrieffer (BCS) approach by including the pair-breaking scattering, introducing the tunneling in-gap states up to a Fermi level. We analytically solve the energy gap equation in various limit cases. The solution provides simple tools for further studies, compared to more complex numerics, and highlights the basic characteristics of the theory. First, in the critical limit of pair-breaking scattering, we derive an analytical form of zero-temperature gap to transition temperature ratio. Next, we derive the dependence of the energy gap close to critical temperature and look at its behavior for general and critical pair-breaking scattering rate. Furthermore, we compare our result with the numerical solution of the gap equation assuming general temperature. We show the range of temperatures, for which the analytical approximation is valid. In the end, we provide the approximative formula of the gap, assuming general pair-breaking scattering, emphasizing its exact behavior close to transition temperature.