Theoretical analysis of Cooper-pair phase fluctuations in underdoped cuprates: a spin-fluctuation exchange study

We study Cooper-pair phase fluctuations in cuprate superconductors for a spin fluctuation pairing interaction. Using an electronic theory we calculate in particular for the underdoped cuprate superconductors the superfluid density $n_s(T)$, the superconducting transition temperature $T_c(x) \propto n_s$ below which phase coherent Cooper-pairs occur, and $T_c^{*}(x)$ where the phase incoherent Cooper-pairs disappear. Also we present results for the penetration depth $\lambda(x,T)$ and for the weak pseudogap temperature $T^{*}(x)$ at which a gap structure occurs in the spectral density. A Mei{\ss}ner effect is obtained only for $T < T_c$. We find that phase fluctuations become increasingly important in the underdoped regime and lead to a reduction of $T_c$ in good agreement with the experimental situation.