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 $\Delta (\omega , T)$, and the complex renormalization function $Z(\omega , T)$, have been obtained for a variety of forms of the electron-boson coupling spectrum. In addition, by calculating $\Delta (\omega , T)$ at finite temperatures, the superconducting critical temperature $T_c$ 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 $\lambda$ 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 $T_c$ 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.

• Received 15 June 1995

DOI:https://doi.org/10.1103/PhysRevB.54.6648