Maximal critical temperature dependence on number of layers due to phonon d-wave pairing in hole doped cuprates

Abstract: Recently an apical oxygen atoms vibrations exchange mechanism of d-wave pairing in cuprates was proposed. The phonon mode in an insulating layer generates attraction of holes in metallic cuper oxygen planes. The pairing has a maximum at the crystallographic gamma point leading to d-wave channel. The idea is generalized here to include the in-plane breathers and half - breather modes in a multi-layer cuprate generating the pairing in an adjacent cuper oxygen layer of the same multi-layer. It is demonstrated that the phonon exchange and the spin fluctuation pairing constructively enhance each other since the paramagnon pairing peaks at crystallographic M point. The phonon contribution explains the maximal critical temperature dependence on the number of layers N. It rises equidistantly by 15K from N=1 to N=3 and then saturates. The strength of the onsite Coulomb on site repulsion at optimal doping is to obtain the observed values of maximal critical temperature in the intermediate range of the effective on-site repulsion U=(1.5-2) eV, smaller than commonly used in purely in-plane (spin fluctuation) theory of high temperature superconductivity.