  Doping dependence of the spin fluctuations and the electron correlations in
the effective five-band Hubbard model for iron pnictides is investigated using
the fluctuation-exchange approximation. For a moderate hole doping, we find a
dominant low-energy spin excitation at Q=(\pi,0), which becomes critical at low
temperature. The low-energy spin excitations in the heavily hole-doped region
are characterized by weak Q dependence. The electron doping leads to an
appearance of a pseudogap in spin excitation spectrum. Correspondingly, the
NMR-1/T1 relaxation rate is strongly enhanced on the hole-doped side and
suppressed on the electron-doped side of the phase diagram. This behavior can
be to large extent understood by systematic changes of the Fermi-surface
topology.
