Influence of Electron-Phonon Interaction on Spin Fluctuation Induced Superconductivity

We investigate the interplay of the electron-phonon and the spin fluctuation interaction for the superconducting state of YBa$_2$Cu$_3$O$_{7}$. The spin fluctuations are described within the nearly antiferromagnetic Fermi liquid theory, whereas 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 $\epsilon_\infty$ and by holes within the CuO$_2$ planes. Taking into account both interactions we get a superconducting state with $d_{x^2-y^2}$-symmetry, whose origin are antiferromagnetic spin fluctuations. The investigation of all phonon modes of the system shows that the phononic contribution to the d-wave pairing interaction is attractive. This is a necessary prerequisite for 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, which is always of s-wave character, is possible.