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 onl y changes the electronic states, but it also generates a channel of the ele ctron-phonon interaction due to electron scattering from vibrating impuriti es. Impurity-induced electron-phonon coupling results in significant modifi cations of the phonon self-energy. For a d-wave superconductor the phonon a ttenuation coefficient (the imaginary part of the phonon self-energy) has b een calculated in the Born approximation and in the unitary limit for elect ron-impurity scattering. In the case of weak electron-impurity potential, t he attenuation decreases if temperature reduces below T-c, while an increas e of the attenuation in the superconducting state has been found in the uni tary limit. The theory shows a good agreement with measured disorder-depend ent coupling of the B-1g phonon mode to superconducting electrons in high-T -c samples: the Born approximation well describes temperature dependence of the attenuation in YBaCuO single crystals, the unitary limit corresponds t o polycrystals and twinned crystals.