  Considering scattering by unintentional background charged impurities and by
charged dopants in the modulation doping layer as well as by GaAs acoustic
phonons, we theoretically consider the practical intrinsic (phonons) and
extrinsic (background and dopants) limits to carrier mobility in modulation
doped AlGaAs-GaAs 2D semiconductor structures. We find that reducing background
impurity density to $10^{12}$ cm$^{-3}$ along with a modulation doping
separation of 1000 \AA or above will achieve a mobility of $100 \times 10^6$
cm$^2$/Vs at a carrier density of $3\times 10^{11}$ cm$^{-2}$ for T=1K. At T=4
(10)K, however, the hard limit to the 2D mobility would be set by acoustic
phonon scattering with the maximum intrinsic mobility being no higher than 22
$(5) \times 10^6$ cm$^2$/Vs. Detailed numerical results are presented as a
function of carrier density, modulation doping distance, and temperature to
provide a quantitative guide to experimental efforts for achieving ultra-high
2D mobilities.
