Finite-temperature real-energy-axis solutions of the isotropic Eliashberg integral equations

A fast, efficient algorithm has been developed for calculating the finite-temperature real-energy-axis solutions of the Eliashberg integral equations for an arbitrary form of the electron-boson coupling function and Coulomb repulsion. Using this algorithm, the complex superconducting gap function Δ(ω, T), and the complex renormalization function Z(ω, T), have been obtained for a variety of forms of the electron-boson coupling spectrum. In addition, by calculating Δ(ω, T) at finite temperatures, the superconducting critical temperature Tc has been obtained for a variety of model systems. These results compare well with the approximate analytic expression derived by Allen and Dynes for values of λ less than 0.75. The solution of the Eliashberg equations has also been obtained for a model in which there are two well separated peaks in the electron-phonon coupling spectrum. This form of coupling spectrum is found to be particularly effective in raising the Tc of the model system. Further, this model has been extended and the solution of the Eliashberg equations has been obtained with an electron-boson coupling spectrum consisting of both an electron-phonon component and a high-energy electronic electron-boson component. This form of the electron-boson coupling function may have special significance in the field of high-temperature superconductivity.