  In modern GaAs/Al$_x$Ga$_{1-x}$As heterostructures with record high
mobilities, a two-dimensional electron gas (2DEG) in a quantum well is provided
by two remote donor $\delta$-layers placed on both sides of the well. Each
$\delta$-layer is located within a narrow GaAs well, flanked by narrow AlAs
layers which capture excess electrons from donors. We show that each excess
electron is localized in a compact dipole atom with the nearest donor.
Nevertheless, excess electrons screen both the remote donors and background
impurities. When the fraction of remote donors filled by excess electrons $f$
is small, the remote donor limited quantum mobility grows as $f^{3}$ and
becomes larger than the background impurity limited one at a characteristic
value $f_c$. We also calculate both the mobility and the quantum mobility
limited by the screened background impurities with concentrations $N_1$ in
Al$_x$Ga$_{1-x}$As and $N_2$ in GaAs, which allows one to estimate $N_1$ and
$N_2$ from the measured mobilities. Taken together, our findings should help to
identify avenues for further improvement of modern heterostructures.
