GATE 2022 EC – Question 56
A $p$-type semiconductor with zero electric field is under illumination (low level injection) in steady state condition. Excess minority carrier density is zero at $x=\pm2l_n$, where $l_n=10^{-4}$ cm is the diffusion length of electrons. Assume electronic charge, $q=-1.6\times10^{-19}$ C. The profiles of photo-generation rate of carriers and the recombination rate of excess minority carriers ($R$) are shown. Under these conditions, the magnitude of the current density due to the photo-generated electrons at $x=+2l_n$ is ________ mA/cm$^2$ (rounded off to two decimal places).
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Correct answer: 0.58 to 0.60
Explanation
By symmetry the current is zero at $x=0$, and in steady state $\frac{1}{q}\frac{dJ}{dx}=G-R$. The current at $x=2l_n$ equals the net generation-minus-recombination integrated from 0 to $l_n$ (both $G$ and $R$ vanish beyond $l_n$ in the plots): $|J|=|q|\int_0^{l_n}\left(10^{20}-10^{20}e^{-x/l_n}\right)dx=|q|\,10^{20}\,l_n\,e^{-1}$. Numerically $|J|=1.6\times10^{-19}\times10^{20}\times10^{-4}\times0.368\approx5.9\times10^{-4}$ A/cm$^2$ $=0.59$ mA/cm$^2$.