$6\mathrm{.}13$ This problem is about testing the quality of the quasi$-$equilibrium approximation across the SCR of an ideal PN junction under bias. This is to be done by comparing the magnitude of the minority carrier current, which is equal to the imbalance between drift and diffusion currents inside the SCR$,$ with either one of these two current components.

Consider an $N^{+}$P junction with "long" QNRs$.$ Focus on electron current. Assume negligible SCR generation and recombination.

$($a$)$ Derive a first$-$order expression for the electron drift current at the metallurgical junction,