Magnetic Raman scattering of insulating cuprates

We study the $B_{1g}$ and $A_{1g}$ Raman profiles of $M_2{\mathrm{CuO}}_4$ (with $M=\mathrm{La},$ Pr, Nd, Sm, Gd), ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_{0.5}{\mathrm{Y}}_{0.5}{\mathrm{Cu}}_2{\mathrm{O}}_{8+y},$ ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.2},$ and ${\mathrm{PrBa}}_2{\mathrm{Cu}}_{2.7}{\mathrm{Al}}_{0.3}{\mathrm{O}}_7$ insulating cuprates within the Loudon-Fleury theory, in the framework of an extended Hubbard model for moderate on-site Coulomb interaction $U.\mathrm{}$ We calculate the nonresonant contribution to these Raman profiles by using exact diagonalization techniques and analyze two types of contributing mechanisms to the line shapes: four-spin cyclic exchange and spin-phonon interactions. Although these interactions contribute to different parts of the spectra, together, they account for the enhanced linewidth and asymmetry of the $B_{1g}$ mode, as well as the non-negligible intensity of the $A_{1g}$ Raman line observed in these materials.