Energy-Gap Function in the Theory of Superconductivity

Abstract

The relation between the form of the energy-gap function $\ensuremath{\Delta}(\ensuremath{\epsilon})$ and experimental results for the superconductors with stronger coupling, Pb and Hg, is discussed. It is shown that the critical field curve is affected rather strongly by the form of $\ensuremath{\Delta}(\ensuremath{\epsilon})$ near the Fermi surface, but that the form of $\ensuremath{\Delta}(\ensuremath{\epsilon})$ at distances of a Debye energy or more has little effect. The positive deviation of the critical field from a parabolic indicates that $\ensuremath{\Delta}$ is flat or slightly increasing on moving away from the Fermi surface. Solutions of the BCS integral equation are given for the Bogoliubov, Bardeen-Pines, and Eliashberg interactions. When a screened Coulomb interaction is included, all of these solutions have the necessary form at the Debye energy to explain the anomalous tunneling behavior of Pb. The critical field data eliminate both the Bogoliubov and Bardeen-Pines interactions, and favors the Eliashberg interaction. However, the Pb and Hg critical fields can not be reproduced with these solutions since the ratio $\frac{\ensuremath{\Delta}(0)}{k{T}_{c}}$ is much too small.