Charge-transfer fluctuation, d-wave superconductivity, and the ${\mathit{B}}_{1\mathit{g}}$ Raman phonon in cuprates

The Raman spectrum of the ${\mathit{B}}_{1\mathit{g}}$ phonon in the superconducting cuprate materials is investigated theoretically in detail in both the normal and superconducting phases, and is contrasted with that of the ${\mathit{A}}_{1\mathit{g}}$ phonon. A mechanism involving the charge-transfer fluctuation between the two oxygen ions in the ${\mathrm{CuO}}_2$ plane coupled to the crystal field perpendicular to the plane is discussed and the resulting electron-phonon coupling is evaluated. Depending on the symmetry of the phonon, the weight of different parts of the Fermi surface in the coupling is different. This provides the opportunity to obtain information on the superconducting gap function at certain parts of the Fermi surface. The line shape of the phonon is then analyzed in detail both in the normal and superconducting states. The Fano line shape is calculated in the normal state and the change of the linewidth with temperature below ${\mathit{T}}_{\mathit{c}}$ is investigated for a ${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$ pairing symmetry. Excellent agreement is obtained for the ${\mathit{B}}_{1\mathit{g}}$ phonon line shape in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_7$.