Optical sum rule in metals with strong electron-phonon interaction

The properties of a superconductor featuring strong electron-phonon and electron-impurity interactions have been studied in a single-band approximation. It is shown that the elastic scattering of electrons on static impurities decreases the electron-phonon interaction constant in proportion to the ratio of the electron-impurity relaxation rate and the seeding band width. The optical spectral weight (integral of the real part of the optical conductivity) in various energy intervals of a metal in the normal and superconducting states has been calculated as a function of the temperature and the superconducting gap. In the region of large (compared to the phonon) energies, the dependence of the spectral weight on the superconducting gap is weak, while the dependence on the temperature is completely determined by the corresponding dependence of the relaxation rate due to the elastic electron-phonon scattering far from the Fermi surface. It is shown that a difference in behavior of the spectral weight between the normal and superconducting states at lower energies is determined by so-called Holstein’s shift of the feature in the optical conductivity spectrum (rather than by the gap width, as it is commonly believed) and sharply decreases upon the introduction of impurities.