  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 layer, flanked by narrow AlAs
layers which capture excess electrons from donors but leave each of them
localized in a compact dipole atom with a donor. Still excess electrons can hop
between host donors to minimize their Coulomb energy. As a result they screen
the random potential of donors dramatically. We numerically model the
pseudoground state of excess electrons at a fraction $f$ of filled donors and
find both the mobility and the quantum mobility limited by scattering on remote
donors as universal functions of $f$. We repeat our simulations for devices
with additional disorder such as interface roughness of the doping layers, and
find the quantum mobility is consistent with measured values. Thus, in order to
increase the quantum mobility this additional disorder should be minimized.
