Toward the theory of the higher harmonic in superconducting gap dispersion of hole- and electron-doped cuprates

Numerical solutions of the Bardeen-Cooper-Schrieffer equation are obtained, taking into account superexchange, three site correlation term, spin-fluctuation, plasmon, and phonon mediated mechanisms. Expression for charge and spin susceptibilities derived beyond RPA approximation. For both e- and h- doped cuprates at carrier concentrations close to optimal the energy gap dispersions where approximated by expression ${\Delta }_k$ = ${\Delta }_0$ (B cos(2$\varphi$) + (1-B) cos (6$\varphi$)), where the angle is measured from the antinodal direction in the Brillouin zone. In agreement with ARPES data we have got (1-B)>0 for h- doped cuprates and (1-B)<0 for e-doped one and right value of ${\Delta }_0$. In Bi${}_2$Sr${}_2$CaCu${}_2$O${}_8$ we found that the amplitude before cos(6$\varphi$) is determined by spin-fluctuation and phonon mediated mechanisms. In case Pr${}_{0.89}$LaCe${}_{0.11}$CuO${}_4$ the higher-harmonic is originated from spin-fluctuation and plasmon (Coulomb) pairing mechanisms. Numerically and analytically shown that role of three site correlations is weakening superexchange interaction.