quality (i.e., $\tau_q$) and the other type to control the mobility (i.e., $\tau_t$). Of course, whether such a distinction actually applies to a given situation or not depends entirely on the details of the sample parameter (i.e., the specific values of $n_R$, $d_R$, $n_B$) as well as the carrier density $n$, but the possibility certainly exists for $d_R$ to be large enough so that the remote impurity scattering is almost entirely small-angle scattering (thus only adversely affecting $\tau_q$ in an appreciable way) whereas the background impurity scattering determines $\tau_t$. If this happens, then mobility ($\tau_t$) and quality ($\tau_q$) could very well be very different in 2D samples, and may have little to do with each other.

To bring out the above physical picture explicitly, we now provide some analytical calculations for the integrals in Eq. (\[eq12\]) defining mobility ($\tau_t$) and quality ($\tau_q$). We rewrite Eq. (\[eq12\]) as $$\tau^{-1}_{t,q} = I^{(R)}_{t,q} + I^{(B)}_{t,q},
\label{eq16}$$ where $$\begin{aligned}
I^{(R)}_{t,q}  = n_RV_0  \frac{2\pi}{\hbar} & & 
\int\frac{d^2k'}{(2\pi)^2} \frac{\delta(E_{\bf k}-E_{\bf k'})}{(q_{TF} + |{\bf k}-{\bf k'}| )^2}
 \nonumber  \\ 
& &\times   f_{t,q}(\theta)  e^{-2|{\bf k}-{\bf k'}| d_R},
\label{eq17}\end{aligned}$$ and $$I^{(B)}_{t,q}  = n_B V_0  \frac{2\pi}{\hbar}
\int\frac{d^2k'}{(2\pi)^2} \frac{\delta(E_{\bf k}-E_{\bf k'})}{(q_{TF} + |{\bf k}-{\bf k'}| )^2}
 f_{t,q}(\theta),
\label{eq18}$$ where $V_0=(2\pi e^2/\kappa)^2$. 