  We study the electron spin relaxation in intrinsic and $p$-type (001) GaAs
quantum wells by constructing and numerically solving the kinetic spin Bloch
equations. All the relevant scatterings are explicitly included, especially the
spin-flip electron-heavy hole exchange scattering which leads to the
Bir-Aronov-Pikus spin relaxation. We show that, due to the neglection of the
nonlinear terms in the electron-heavy hole exchange scattering in the
Fermi-golden-rule approach, the spin relaxation due to the Bir-Aronov-Pikus
mechanism is greatly exaggerated at moderately high electron density and low
temperature in the literature. We compare the spin relaxation time due to the
Bir-Aronov-Pikus mechanism with that due to the D'yakonov-Perel' mechanism
which is also calculated from the kinetic spin Bloch equations with all the
scatterings, especially the spin-conserving electron-electron and
electron-heavy hole scatterings, included. We find that, in intrinsic quantum
wells, the effect from the Bir-Aronov-Pikus mechanism is much smaller than that
from the D'yakonov-Perel' mechanism at low temperature, and it is smaller by no
more than one order of magnitude at high temperature. In $p$-type quantum
wells, the spin relaxation due to the Bir-Aronov-Pikus mechanism is also much
smaller than the one due to the D'yakonov-Perel' mechanism at low temperature
and becomes comparable to each other at higher temperature when the hole
density and the width of the quantum well are large enough. We claim that
unlike in the bulk samples, the Bir-Aronov-Pikus mechanism hardly dominates the
spin relaxation in two-dimensional samples.
