Kondo $s$-$d$ exchange in the CuO$_2$ plane as the long sought interaction determining $T_c$ in cuprates

The well-known Pavarini et al. [Phys. Rev. Lett. 87, 047003 (2001)] correlation between the critical temperature $T_{c,\,\mathrm{max}}$ and the shape of the Fermi contour of the optimally hole-doped cuprates is explained within the framework of the BCS theory with Kondo exchange interaction incorporated as a pairing mechanism. The strong influence of the relative position of the Cu4$s$ level with respect to the Cu3$d_{x^2-y^2}$ level on the critical temperature $T_c$ reveals the importance of the $s$-$d$ hybridization of the conduction band. This hybridization is proportional to the $s$-$d$ exchange scattering amplitude between the conduction electrons -- the mechanism of $d$-wave pairing in the CuO$_2$ plane. In other words the Kondo interaction considered as a pairing mechanism in the CuO$_2$ plane gives a natural explanation of the correlation between the critical temperature and the shape of the Fermi contour. This result suggests that the long-sought pairing mechanism in high-$T_c$ superconducting cuprates has possibly been found and that the Kondo exchange interaction as a property of strongly correlated quantum matter deserves further attention in the physics of layered cuprates. To test the developed theoretical scheme, we explored in detail the recent results of the scanned Josephson modulation microscopy experiment of the modulation of the order parameter due to apex distance super-modulation. Our analysis shows a satisfactory agreement between our theory and the named experiment.