Symmetry dependence of phonon line shapes in superconductors with anisotropic gaps

The temperature dependence below ${\mathit{T}}_{\mathit{c}}$ of the line shape of optical phonons of different symmetry as seen in Raman scattering is investigated for superconductors with anisotropic energy gaps. It is shown that the symmetry of the electron-phonon vertex produces nontrivial couplings to an anisotropic energy gap which leads to unique changes in the phonon line shape for phonons of different symmetry. The phonon line shape is calculated in detail for ${\mathit{B}}_{1\mathit{g}}$ and ${\mathit{A}}_{1\mathit{g}}$ phonons in a superconductor with ${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$ pairing symmetry. The role of satellites peaks generated by the electron-phonon coupling are also addressed. The theory accounts for the substantial phonon narrowing of the ${\mathit{B}}_{1\mathit{g}}$ phonon, while narrowing of the ${\mathit{A}}_{1\mathit{g}}$ phonon which is indistinguishable from the normal state is shown, in agreement with recent measurements on ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{CaCu}}_2$${\mathrm{O}}_8$ and ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_7$.