Theory of Raman scattering with final-state interaction in high-T_c BCS superconductors: Collective modes

A theory of Raman scattering in isotropic superconductors is presented, where the light breaks a Cooper pair. The role of the final-state interaction including the long-range Coulomb interaction is discussed in great detail. It is shown that the Raman spectrum consists of two parts: bound states with different symmetries below the threshold (ω<2Δ) and a continuum above the threshold, where the square-root singularity is removed by the final-state interaction. The small binding energies of the bound states cannot be resolved because of the experimental broadening and the possible lifetime (recombination time) of the excited pair. The spectrum shows very weak dependence on the momentum transfer providing that the transfer q is not too large thus q≪Δ/v_F. The spectrum is sensitive, however, on the polarizations of the incident and scattered light, because the relative weights of the excited pairs with different symmetries are also sensitive. It is also demonstrated that it is not necessary to invoke a large gap anisotropy to explain a broadening of the pair breaking edge at 2Δ. The results previously derived by Klein and Dierker and by Abrikosov and Falkovsky are also discussed. The present results agree with the former ones in the zero momentum transfer limit, while the later ones are reproduced for small momentum transfer in certain energy regions.