Mobility versus quality in 2D semiconductor structures

We consider theoretically effects of random charged impurity disorder on the {\it quality} of high-mobility two dimensional (2D) semiconductor structures, explicitly demonstrating that the sample mobility is not necessarily a reliable or universal indicator of the sample quality in high-mobility modulation-doped 2D GaAs structures because, depending on the specific system property of interest, mobility and quality may be controlled by different aspects of the underlying disorder distribution, particularly since these systems are dominated by long-range Coulomb disorder from both near and far random quenched charged impurities. We show that in the presence of both channel and remote charged impurity scattering, which is a generic situation in modulation-doped high-mobility 2D carrier systems, it is quite possible for higher (lower) mobility structures to have lower (higher) quality as measured by the disorder-induced single-particle level broadening. In particular, we establish that there is no reason to expect a unique relationship between mobility and quality in 2D semiconductor structures as both are independent functionals of the disorder distribution, and are therefore, in principle, independent of each other. Using a simple, but reasonably realistic, ``2-impurity'' minimal model of the disorder distribution, we provide concrete examples of situations where higher (lower) mobilities correspond to lower (higher) sample qualities. We discuss experimental implications of our theoretical results and comment on possible strategies for future improvement of 2D sample quality.