Phase diagram of underdoped cuprate superconductors: effect of Cooper-pair phase fluctuations

In underdoped cuprates fluctuations of the phase of the superconducting order parameter play a role due to the small superfluid density. We consider the effects of phase fluctuations assuming the exchange of spin fluctuations to be the predominant pairing interaction. Spin fluctuations are treated in the fluctuation-exchange approximation, while phase fluctuations are included by Berezinskii-Kosterlitz-Thouless theory. We calculate the stiffness against phase fluctuations, ns(omega)/m*, as a function of doping, temperature, and frequency, taking its renormalization by phase fluctuations into account. The results are compared with recent measurements of the high-frequency conductivity. Furthermore, we obtain the temperature T*, where the density of states at the Fermi energy starts to be suppressed, the temperature Tc*, where Cooper pairs form, and the superconducting transition temperature Tc, where their phase becomes coherent. We find a crossover from a phase-fluctuation-dominated regime with Tc proportinal to ns for underdoped cuprates to a BCS-like regime for overdoped materials.