  A nonequilibrium Green's function approach is employed to investigate the
spin-Hall effect in diffusive two-dimensional electron systems with Rashba
spin-orbit interaction. Considering a long-range electron-impurity scattering
potential in the self-consistent Born approximation, we find that the spin-Hall
effect arises from two distinct interband polarizations in helicity basis: a
disorder-unrelated polarization directly induced by the electric field and a
polarization mediated by electron-impurity scattering. The disorder-unrelated
polarization is associated with all electron states below the Fermi surface and
produces the original intrinsic spin-Hall current, while the disorder-mediated
polarization emerges with contribution from the electron states near the Fermi
surface and gives rise to an additional contribution to the spin-Hall current.
Within the diffusive regime, the total spin-Hall conductivity vanishes in {\it
infinitely large} samples, independently of temperature, of the spin-orbit
coupling constant, of the impurity density, and of the specific form of the
electron-impurity scattering potential. However, in a {\it finite-size} Rashba
two-dimensional semiconductor, the spin-Hall conductivity no longer always
vanishes. Depending on the sample size in the micrometer range, it can be
positive, zero or negative with a maximum absolute value reaching as large as
$e/8\pi$ order of magnitude at low temperatures. As the sample size increases,
the total spin-Hall conductivity oscillates with a decreasing amplitude. We
also discuss the temperature dependence of the spin-Hall conductivity for
different sample sizes.
