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

Abstract

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 ${\ensuremath{\varepsilon}}_{\ensuremath{\infty}}$ and by holes within the ${\mathrm{CuO}}_{2}$ planes. We get a superconducting state with ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ 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.