completely defining the underlying disorder. We note that writing the second term in Eq. (\[eq11\]) as $n_B \delta(z-d_B)$, and thus introducing an additional length parameter into the model, is completely unnecessary since, as we will see below, the physics of the mobility versus quality duality in high-quality 2D structures is entirely dominated by scattering from near impurities (controlling mobility) and far impurities (controlling quality), which allows us to put $d_B=0$ in the minimal model (thus making the near impurities very near indeed). Putting Eqs. (\[eq3\]) – (\[eq11\]) in Eqs. (\[eq1\]) and (\[eq2\]), we can combine them into a single 2D integral, obtaining $$\begin{aligned}
\frac{1}{\tau_{t,q}} = \frac{2\pi}{\hbar} \left (\frac{2\pi
  e^2}{\kappa} \right )^2 & &  
\int\frac{d^2k'}{(2\pi)^2} \frac{\delta(E_{\bf k}-E_{\bf k'})}{(q_{TF}
  + |{\bf k}-{\bf k'}| )^2} 
 f_{t,q}(\theta) \nonumber  \\ 
& &\times   \left \{ n_R e^{-2|{\bf k}-{\bf k'}| d_R} + n_B \right \},
\label{eq12}\end{aligned}$$ where $f_t(\theta) = 1- \cos \theta_{kk'}$ and $f_q(\theta) =1$. For completeness, we mention that $E_{\bf k} = \hbar^2 k^2/2m$.

Before proceeding further, we emphasize that the “only" difference between mobility (i.e., $\tau_t$) and quality (i.e., $\tau_q$) is the appearance (or not) of the angular factor $(1-\cos \theta)$ inside the double integral in Eq. (\[eq12\]) for $\tau_t$ ($\tau_q$). This arises from the 