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

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.