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There is growing evidence that the unconventional spatial inhomogeneities inthe doped high-Tc superconductors are accompanied by the pairing of electrons,subsequent quantum phase transitions (QPTs), and condensation in coherentstates. We show that these superconducting states can be obtained from phaseseparation instabilities near the quantum critical points. We examine electroncoherent and incoherent pairing instabilities using our results on exactdiagonalization in pyramidal and octahedron Hubbard-like clusters undervariation of chemical potential (or doping), interaction strength, temperatureand magnetic field. We also evaluate the behavior of the energy gap in thevicinity of its sign change as a function of out-of-plane position of theapical oxygen atom, due to vibration of apical atom and variation of inter-sitecoupling. These results provide a simple microscopic explanation of(correlation induced) supermodulation of the coherent pairing gap observedrecently in the scanning tunneling microscopy experiments at atomic scale in$Bi_2Sr_2CaCu_2O_{8+\delta}$. The existence of possible modulation of localcharge density distribution in these materials is also discussed.

Phase separation instabilities and pairing modulations in $Bi_2Sr_2CaCu_2O_{8+\delta}$