Anisotropic Superconductors and Elastic Impurity Scattering

Abstract

<p>The thermodynamic properties of a number of simple metals in the superconducting state are calculated from the numerical solution of the isotropic Eliashberg equations on the imaginary axis and comparison is made with experiment. The functional derivatives of these properties with respect to change in the electron-phonon interaction are calculated and used in a detailed analysis of the interaction in superconducting Nb. Anisotropy in the electron-phonon interaction is also investigated numerically for some metals using a simple model for the anisotropy. The functional derivative in an anisotropic superconductor is found to be qualitatively different at low frequencies from that of an isotropic system. The difference is shown to be caused by the washing out of anisotropy by thermal phonons. Some comparison with experiment and with previous approximate solutions of the same model is made. For some quantities the previous approximate estimates of the effect of gap anisotropy are shown to be inadequate. A useful expression relating the amount of anisotropy with the impurity dependence of the critical temperature is developed independent of any such model or approximation.</p> <p>Another calculation is done with a realistic calculated interaction for lead that includes anisotropy in the phonons and Fermi surface. An orthonormal basis set for the expansion of this interaction is enumerated and tested, but is found unnecessary for lead because the interaction is very close to separable. It is shown that the anisotropy in the interaction can be approximated by a very simple separable model without significantly affecting the thermodynamic properties. Also, the gap anisotropy on the imaginary axis is found to be identical to that of the gap edge at T=0 calculated on the real axis from the same interaction.</p>