2k_F$ \[see Eq. (\[eq8\])\] because of the constant energy-independent 2D electronic density of states. Since the $\delta$-functions (necessary for energy conservation during the impurity-induced elastic scattering of an electron from the momentum state $|{\bf k}\rangle$ to the momentum state $|{\bf k}' \rangle$ with a net wave vector transfer of ${\bf q} = {\bf k} - {\bf k}'$) in Eqs. (\[eq1\]) and (\[eq2\]) restrict the wave vector transfer $0
\leq q \leq 2k_F$ range (this is simply because we are at $T=0$ so that the maximum possible scattering wave vector is $2k_F$ corresponding to the pure back-scattering of an electron from $+k_F$ to $-k_F$ while obeying energy conservation), the relevant screening wave vector for our problem is purely the Thomas-Fermi wave vector $q_s = q_{TF}$ as follows from Eq. (\[eq8\]) for $q\le 2k_F$.

We can, therefore, rewrite Eq. (\[eq3\]) as $$u_q(z) = \frac{2\pi e^2}{\kappa (q+ q_{TF})} e^{-qz},
\label{eq10}$$ giving the effective screened Coulomb interaction (in the 2D momentum space) between an electron in the 2D layer and a random quenched charged impurity located a distance ‘$z$’ away.

Finally, our 2-impurity disorder model is given by $$N_i(z) = n_R \delta(z-d_R) + n_B \delta(z),
\label{eq11}$$ with three independent parameters $n_R$, $d_R$, and $n_B$ 