Electronic Raman scattering in superconductors as a probe of anisotropic electron pairing

A gauge-invariant theory for electronic Raman scattering for superconductors with anisotropic pairing symmetry is analyzed in detail. It is shown that Raman scattering in anisotropic superconductors provides a wealth of polarization-dependent information that probes the detailed angular dependence of the superconducting ground-state order parameter. The Raman spectra shows a unique polarization dependence for various anisotropic pair-state symmetries which affects the peak position of the spectra and generates symmetry-dependent low-frequency and temperature power laws that can be used to identify the magnitude and predominant symmetry of the energy gap. In particular, we calculate the collective modes and the subsequent symmetry-dependent Raman spectra for a ${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$ superconductor and compare our results to the relevant data on the cuprate systems as well as theoretical predictions for s-wave, anisotropic s-wave, and mixed-state energy gaps. Favorable agreement is shown with the predictions for ${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$ pairing and the experimental data on ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$, ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_8$, and ${\mathrm{Tl}}_2$ ${\mathrm{Ba}}_2$ ${\mathrm{CuO}}_6$.