The Derivation of the Fundamental Superconducting Interaction and the Phase Diagram of Cuprates

We show that an electronic phase separation (EPS) transition described by the an appropriate theory yields regions of low free energy forming grains of low and high charge densities. These local differences in the potential energy are studied numerically and used here for the first time as the origin of an attractive interaction that gives rise to local pair formation. The resistivity transition $T_c$ occurs due to Josephson coupling among these superconducting regions or grains. Using this approach within the Bogoliubov-deGennes calculations we derive the Gaussian shape of $T_c$ against the entire doping level $p$. We find that every pairing gap develops locally at a temperature $T_p$, following the relation $2\mathrm{\Delta }\mathrm{/}{k}_B{T}_p\approx 8.0$ in close agreement recent measurements.}