possible, in principle, for samples of different mobility to have the same quality or vice versa (i.e., samples of different quality to have the same 2D mobility).

There is still the vague qualitative expectation, however, that if the sample mobility is enhanced by improving the sample purity (i.e., suppressing disorder), then this should automatically also improve the sample quality (perhaps not necessarily by the same quantitative factor) since reduced disorder should enhance quality. We will show below that this may not always be the case since mobility and quality (for a specific property) may be sensitive to completely different aspects of disorder and therefore enhancing mobility by itself may do nothing to improve quality in some situations. On the other hand, when mobility and quality are both determined by exactly the same microscopic disorder in the sample, increasing (decreasing) one would necessarily improve (suppress) the other although we will see later in this work \[see Fig. \[fig01\](c)\], for example) this is not necessarily true if mobility and quality are both determined by remote dopant scattering arising from long-range Coulomb disorder.

It is important to emphasize a subtle aspect of the mobility/quality dichotomy with respect to experimental samples. Theoretically, we can 