Calculations of optical conductivity in a two-band superconductor: Pb

We demonstrate the effect of band structure on the superconducting properties of Pb by calculating the strong-coupling features in the optical conductivity sigma(omega) due to the electron-phonon interaction. The importance of momentum dependence in the calculation of the properties of superconductors has previously been raised for MgB2 [H. J. Choi , Phys. Rev. B 66, 020513(R) (2002); Nature (London) 418, 758 (2002)]. Pb resembles MgB2 in that it is a two-band superconductor in which the bands' contributions to the Fermi surface have very different topologies. We calculate sigma(omega) by calculating a memory function [W. Gotze and P. Wolfle, Phys. Rev. B 6, 1226 (1972)], which has been recently used to analyze sigma(omega) of Bi2Sr2CaCu2O8+delta. [J. Hwang , Nature (London) 427, 714 (2004)]. In our calculations, the two components of the Fermi surface are described by parametrizations of de Haas-van Alphen data. We use a phonon spectrum that is a fit to neutron scattering data. By including the momentum dependence of the Fermi surface, a good agreement is found with the experimentally determined strong-coupling features, which can be described by a broad peak at around 4.5 meV and a narrower higher peak at around 8 meV of equal height. The calculated features are found to be dominated by scattering between states within the third band. By contrast, scattering between states in the second band leads to strong-coupling features in which the height of the high energy peak is reduced by similar to 50% compared to that of the low energy peak. This result is similar to that in the conventional isotropic (momentum independent) treatment of superconductivity. Our results show that it is important to use realistic models of the band structure and phonons and to avoid using momentum averaged quantities in calculations in order to get quantitatively accurate results.