from the 2D electron system in the $z$-direction with $d=d_R$ (0) for the remote (background) impurities. We assume the random quenched charged impurities to all have unit strength (i.e., having an elementary charge of $\pm e$ each) with no loss of generality.

Our model thus has four independent parameters with dimensions of length: $n^{-1/2}$, $n_R^{-1/2}$, $n_B^{-1/2}$, $d_R$. In addition to these (experimentally variable) parameters, we also have $m$ and $\kappa$ defining the material system which is fixed in all samples for a given material. In principle, two additional materials parameters should be added to describe the most general situation, namely, the spin ($g_s$) and the valley ($g_v$) degeneracy, but we assume $g_s=2$ and $g_v=1$ throughout the current work (and for all our numerical results) since our interest here is entirely focussed on high-mobility n- and p-GaAs 2D systems where the mobility/quality dichotomy has mostly been discussed. Additional experimentally relevant (but, theoretically non-essential) parameters, such as a finite width of the 2D electron system (instead of the strict 2D limit) and/or a 3D distribution of the random impurities (instead of the 2D distribution assumed above), are straightforward to include in the model and are not discussed further in details.

The mobility 