$0\mathrm{.}4(20pts)$

Consider the p$-$n junction below with a negligibly small depletion layer width. Intrinsic cartie concentration, acceptor$/$donor dopings and hole'electron diffusion coefficients are given as $n_3,N_{n}T{h}_5,D_{}$ and $D_n,$ respectively. Electron diffusion length $(L_n)$ in the p $-$side is much larger than p $-$side widh $(W_n)$ and hole diffusion length $(L_T)$ in the $n-$ side is much smaller than $n-$side width $(W_n).$ Thermal voltage $(\frac{kT}{q})$ is given as $V_{\frac{T}{E}}$ Excess carrier concentrations at the contacts are zero.

i$)(4$ pts$)$ Write the expressions showing the excess carrier concentration distributions on both sides using only the parameters given in the question. No derivation is required. Plot the distributions on the figure below.

ii$)(9$ pts$)$ Starting from the excess carrier distributions found in $($i$),$ derive the total current passing through the device as a function of applied bias voltage using only the parameters given in the question, Show complete work with necessary explanations. Answer:

iii$)(7$ pts$)$ Assuming p$-$type doping $(N_A)$ is much larger than the n$-$type doping $(N_D),$ derive the expression for the diffusion capacitance of the diode in terms of the diode current $($I$)$ and other parameters given in the question. Show complete work. Answer: