Influence of electron-phonon interaction on spin-fluctuation-induced superconductivity

We investigate the interplay of the electron-phonon and the spin-fluctuation interaction in the superconducting state of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7.$ The spin fluctuations are described within the nearly antiferromagnetic Fermi-liquid theory and the phonons are treated using a shell-model calculation of all phonon branches. The electron-phonon coupling is calculated using rigidly displaced ionic potentials screened by a background dielectric constant ${\varepsilon }_{\infty }$ and by holes within the ${\mathrm{CuO}}_2$ planes. We get a superconducting state with $d_{x^2-y^2}$ symmetry whose origin is antiferromagnetic spin fluctuations. The phononic contribution of all phonon modes of the system to the d-wave pairing interaction is attractive. This gives a positive isotope effect. The size of the isotope exponent depends strongly on the relative strength of the electron-phonon and spin-fluctuation coupling. Due to the strong electronic correlations no phononic induced superconducting state of s-wave character, is possible.