of system parameters in general). In such a simplistic situation, mobility and quality are likely to monotonically connected by a unique relationship, and hence enhancing system mobility should always improve the system quality since the same disorder determines both properties. In fact, this simple situation always applies if the mobility/quality are both limited by purely short-range disorder in the system.

By contrast, 2D semiconductor structures almost always have several qualitatively distinct disorder mechanisms affecting transport and other properties arising from completely different physical origins. For example, it is well-known [@andormp] that Si-MOSFETs have at least three different operational disorder mechanisms: random charged impurities in the insulating SiO$_2$ oxide layer, in the bulk Si itself, and random short-range surface roughness at the Si-SiO$_2$ interface. There may still be other distinct scattering mechanisms associated with still different disorder sources in Si-MOSFETs such as neutral defects or impurities, making the whole situation quite complex. In MOSFETs, low-density carrier transport is controlled by long-range charged impurity scattering whereas the high-density carrier transport is controlled by short-range surface roughness scattering, and this dichotomy may very well lead to situations where a measured electronic property (i.e., “quality”) does not necessarily correlate with the high-density maximum mobility 