Impact of superconducting gap on the phonon mode mixing itinerant and localized electron states

The influence of superconducting ordering on the optical lattice vibration modified through the mixing of localized states with Hubbard correlation and itinerant band states is analyzed. We consider the situation where the broad electron band is located between lower and upper Hubbard levels. The hardening or softening of phonon dynamics caused by the opening of superconducting gap in the band of itinerant carriers appears in dependence on the disposition of the electron spectrum and chemical potential. The effect varies substantially if the chemical potential approaches the edges of the band of itinerant carriers or the van Hove singularity.