Analytic solutions of Eliashberg gap equations at superconducting critical temperature

Abstract

We perform simple mathematical analysis of the Eliashberg gap equations, and derive the analytic expression at the superconducting critical temperature. In order to perform exact summation of LTc=∑λm12m−1, we use the Einstein and Debye models for the Eliashberg spectral function. This quantity is in good agreement with data derived from experimental and DFT data. To the leading-term approximation, the analytic expression of the superconducting critical temperature is of the form Tc=2eγ−1πωxexp−1+μ*ln4eγmcλ, where ωx=ω2 or ωln. The frequencies ω2 and ωln come out naturally from the models. These equations contain mc as an adjustable parameter. There are no other free parameters and no empirical formula involved. We compare the results of these equations with the celebrated Allen-Dynes modified McMillan equation together with the data from experiments and DFT calculations. The data show strong correlation between our analytic expression and the celebrated Allen-Dynes equation.