Phase diagram of hole-doped high-$T_c$ superconductors: Effects of Cooper-pair phase fluctuations within fluctuation-exchange theory

Using the Hubbard model Hamiltonian we study spin-fluctuation exchange-induced superconductivity of d-wave symmetry. Results are presented for the characteristic temperature $T^{*}$ at which a gap appears in the spectral density, for $T_c^{*}$ at which Cooper-pairs are formed, for $T_c$ at which Cooper pairs become phase coherent, and for the superfluid density $n_s.$ We find that, with increasing doping, for $x>\mathrm{}0.15\mathrm{}$ the phase coherence energy becomes larger than the Cooper-pair condensation energy. Accordingly, one has $T_c\propto n_s$ for $x<\mathrm{}0.15\mathrm{}$ and $T_c\propto \Delta$ for the overdoped cuprates. We use our results to discuss dynamics and recent dynamical conductivity and ultrafast nonequilibrium measurements.