  Extensive muon spin relaxation measurements have been performed to determine
the magnetic field penetration depth in high-Tc cuprate superconductors with
simple hole doping, Zn-doping, overdoping, and formation of static SDW nano
islands. System dependence of $n_{s}/m^{*}$ (superconducting carrier density /
effective mass) reveals universal correlations between Tc and $n_{s}/m^{*}$ in
all these cases with / without perturbation. Evidence for spontaneous and
microscopic phase separation was obtained in the cases with strong
perturbation, i.e., Zn-doping. overdoping and SDW nano-islands. The length
scale of this heterogeneity is shown to be comparable to the in-plance
coherence length. We discuss implications of these results on condensation
mechanisms of HTSC systems, resorting to an analogy with He films, on regular
and porous media, reminding essential features of Bose-Einstein, BCS and
Kosterlitz-Thouless condensation/transition in 2-d and 3-d systems, and
comparing models of BE-BCS crossover and phase fluctuations. We propose a new
phase diagram for HTSC systems based on distinction between pair formation and
superconducting phase fluctuations in the pseudogap region and spontaneous
phase separation in the overdoped region. We also remind anomaly in BEDT and
A3C60 systems similar to that in overdoped cuprates, seen in the evolution from
superconducting to metallic ground state.
