Phonon self-energy in superconductors: Effect of vibrating impurities

The phonon self-energy due to the interaction with electrons is studied for superconductors with s- and d-wave pairing. Addition of impurities not only changes the electronic states, but it also generates a channel of the electron-phonon interaction due to electron scattering from vibrating impurities. Impurity-induced electron-phonon coupling results in significant modifications of the phonon self-energy. For a d-wave superconductor the phonon attenuation coefficient (the imaginary part of the phonon self-energy) has been calculated in the Born approximation and in the unitary limit for electron-impurity scattering. In the case of weak electron-impurity potential, the attenuation decreases if temperature reduces below Tc, while an increase of the attenuation in the superconducting state has been found in the unitary limit. The theory shows a good agreement with measured disorder-dependent coupling of the B1g phonon mode to superconducting electrons in high-Tc samples: the Born approximation well describes temperature dependence of the attenuation in YBaCuO single crystals, the unitary limit corresponds to polycrystals and twinned crystals.