Maximal critical temperature dependence on the number of layers in hole-doped cuprates due to phonon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-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 ${\mathrm{CuO}}_{2}$ plane with potential $V(k)\ensuremath{\propto}exp[\ensuremath{-}2k{d}_{a}]$, where ${d}_{a}$ is the distance to the apical plane. The pairing has a maximum at the crystallographic $\mathrm{\ensuremath{\Gamma}}$ point leading to $d$-wave channel. The idea is generalized here to include the ${\text{CuO}}_{2}$ breathing and half-breathing modes in a multilayer cuprate generating the pairing in an adjacent ${\text{CuO}}_{2}$ 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 ${T}_{c}^{max}$ dependence on the number of layers $N$. It rises equidistantly by $15\phantom{\rule{0.222222em}{0ex}}\text{K}$ 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 ${T}_{c}^{max}$ in the intermediate range of the effective on-site repulsion $U=(1.5\ensuremath{-}2)\phantom{\rule{4pt}{0ex}}\text{eV}$, smaller than commonly used in purely in-plane (spin fluctuation) theory of high ${T}_{c}$ superconductivity.