Phonon renormalization and symmetry of the superconducting order parameter

We present a method for the calculation of the renormalization of the phonon energy in the superconducting state. The expression is derived in the weak-coupling case and takes into account the k dependency of the order parameter ${\mathrm{\Delta }}_{\mathbf{k}}$. The phonon shift as well as the linewidth are calculated. The influence of the anisotropy of the gap in the ab plane of the high-${\mathit{T}}_{\mathit{c}}$ superconductors on the renormalization of the q→0 phonon energies is studied. The anisotropies considered have either s-, d-wave, or mixed character. It is shown that an anisotropic gap with s-wave symmetry leads to a renormalization of the phonon energy below ${\mathit{T}}_{\mathit{c}}$ that is in agreement with the Raman measurements. The renormalization resulting from the d-wave case instead is incompatible with these experiments. Finally, the strength of the electron-phonon coupling is estimated for the Raman observed phonons.