Derivation of the Disorder Induced Interaction and the Phase Diagram of Cuprate Superconductors

We show that an electronic phase transition described by the Cahn-Hilliard equation has important applications to cuprate superconductors. The simulations of the local charge density and free energy reveal two main features: i) The segregation process creates tiny isolated regions with potential wells where the holes can be bound in single-particle levels. ii) The clustering process also gives rise to an effective two-body pairing interactions and superconducting amplitudes $\Delta_{sc}(\vec r)$ at low temperatures. The resulting system resembles a granular superconductor with the resistivity transition driven by Josephson coupling among these nanoscale grains. This approach reproduces the well known critical temperature transition $T_c(p)$ as function of the doping level $p$. The derived $p \times T$ phase diagram reproduces the main features measured by several experiments. Furthermore, the local density of states with spatial dependent gaps $\Delta(\vec r)$ is due to the intragrain single-particle bound states that remain above $T_c$, which characterizes the pseudogap phase and reproduces many measurements.