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

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 CuO2 plane with potential V(k)∝exp[-2kda], where da is the distance to the apical plane. The pairing has a maximum at the crystallographic Γ point leading to d-wave channel. The idea is generalized here to include the CuO2 breathing and half-breathing modes in a multilayer cuprate generating the pairing in an adjacent CuO2 layer of the same multilayer. 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 Tcmax 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 on-site Coulomb on site repulsion at optimal doping is to obtain the observed values of Tcmax 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 Tc superconductivity.