density). If two samples happen to have the same maximum mobility or the same mobility at one fixed density, there is no reason to expect them to have the same quality with respect to all experimental properties at arbitrary densities. This obvious aspect of mobility versus quality dichotomy has not much been emphasized in the literature.

It should be clear from the discussion above that in the ‘trivial’ (and experimentally unrealistic in 2D semiconductor structures) situation of the system having just one type of impurities uniquely defining the applicable impurity distribution, both mobility and quality, by definition, would be determined by exactly the same impurity configuration since there is just one set of impurities by construction. Such a situation is, in fact, common in 3D semiconductors where the applicable disorder is almost always described by a random uniform background of uncorrelated quenched charged impurity centers, which can be uniquely characterized by a single 3D impurity density $n_i$. Obviously, in this situation both mobility and quality are uniquely defined by $n_i$, and thus increasing (decreasing) $n_i$ would decrease (increase) both mobility and quality (although not necessarily by the same quantitative factor since mobility and quality are likely to be different functions 