Apical oxygen vibrations dominant role in cuprate superconductivity and its interplay with spin fluctuations.
Microscopic theory of a high temperature cuprate Bi2212 based on main pairing channel of electrons in CuO planes due to lateral vibrations of the apical oxygen atoms in adjacent the SrO ionic insulator layer is proposed. Similar ionic substrate phonon model was used recently to explain very high critical temperature in novel one unit cell FeSe on perovskite STO. A microscopic vibration theory identifies the 40 meV phonon mode coupled to conducting CuO planes with electron-electron strength 0.5. It naturally explain the kink in dispersion relation observed by ARPES and the and effect of the isotope substitution. To describe the pseudogap physics by a single band fourfold symmetric t-t' Hubbard model, the hopping parameters t'=-0.184 and the on side repulsion energy U=1.9t are chosen. The electronic system is still strongly correlated, but U is weak enough to be effectively described by the mean field model and its perturbative extensions. In particular the fragmentation of the Fermi surface in underdoped samples and the non-circularity of the Fermi Surface are described well within the "symmetrized Hartree - Fock" approximation. The T star transition line dividing the pseudogap (locally antiferromagnetic) and paramagnetic phases and susceptibility (describing spin fluctuations coupling to 2DEG) are also obtained within this approximation. The superconducting gap was calculated in the framework of the weak coupling approximation for both the phonon and the spin fluctuations channels. The dominant "glue" responsible for the d - wave pairing is the phonon mode rather than spin fluctuations, although the later enhances superconductivity by 10-15%. The dependence of the superconducting gap and certain normal state properties, like the kink in dispertion relation,on doping, temperature and effect of the oxygen isotope substitution are obtained.