Superconductivity-induced effects on phononic and electronic Raman scattering in twin-free ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-x}$ single crystals

Raman scattering spectra have been investigated in twin-free ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-x}$ single crystals with various oxygen content. Superconductivity-induced effects are found for all $A_g$ phonons in the overdoped crystals, but they rapidly weaken with reducing oxygen content. The 340 ${\mathrm{cm}}^{\mathrm{-}1}$ phonon line is analyzed in detail by means of the Green function, as well as the standard Fano formula. The results provide much evidence for the nodal structure of a superconducting gap. The polarization dependence, which is significant in the highly doped crystals, illustrates the x-y anisotropy of the electronic response $\chi \left(\omega \right)=-R\left(\omega \right)+i\rho \left(\omega \right)$ and implies a difference of the superconducting gap amplitudes ${\Delta }_x\ne {\Delta }_y$ in the ${\mathbf{k}}_x$ and ${\mathbf{k}}_y$ directions in the two-dimensional Brillouin zone for orthorhombic ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-x}.$ The gap energy difference $\left({\Delta }_y-{\Delta }_x\right)$ decreases as the doping level is deduced.