  The dependences of the transport scattering time {\tau}t, quantum lifetime
{\tau}q, and their ratio {\tau}t/{\tau}q on the density ne of the electron gas
in modulation-doped single GaAs quantum wells with AlAs/GaAs short-period
super-lattice barriers are investigated. The experimental dependences are
explained in terms of electron scattering by remote ionized donors with an
effective two-dimensional concentration n*R and background impurities with a
three-dimensional concentration nB. An expression for n*R(ne) is obtained
including the contribution of X-valley electrons localized in AlAs layers to
the suppression of scattering by the random potential of remote donors. It is
shown that the experimentally observed abrupt increase in {\tau}t and {\tau}q
with an increase in ne above a certain critical value nec is related to a
decrease in n*R. It is established that the drop in {\tau}t/{\tau}q observed
for electron densities ne > nec occurs because scattering by the random
potential of background impurities in this two-dimensional system with a
decrease in n*R limits an increase in {\tau}t more considerably than an
increase in {\tau}q.
