strict 1-impurity model of underlying sample disorder, mobility ($\tau_t$) and quality ($\tau_q$) are both affected by the same impurity density, and hence they must behave monotonically (but not necessarily identically) with changing disorder, i.e., if the impurity density is decreased (increased) at a fixed carrier density, both mobility and quality must increase (decrease) as well.

As emphasized already, however, the 2-impurity model (near and far impurities or background and remote impurities) introduces a new element of physics by allowing for the possibility that mobility and quality could possibly be affected more strongly by different types of disorder, for example, mobility (quality) could be dominated by near (far) impurities in the 2-impurity disorder model, thus allowing, in principle, the possibility of mobility and quality being completely independent physical properties of realistic 2D semiconductor samples at least in some situations. We find this situation to be quite prevalent in very high-mobility 2D semiconductor structures where the mobility (quality) seems to be predominately determined by near (far) impurities. In low mobility samples, on the other hand, the situation is simpler and both mobility and quality are typically determined by the same set of impurities (usually the charged impurities close to the 2D layer 