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Consider a long rectangular bar of direct bandgap $p-$type semiconductor. The equilibrium hole density is $10^{17} \; \text{cm}^{-3}$ and the intrinsic carrier concentration is $10^{10} \; \text{cm}^{-3}.$ Electron and hole diffusion lengths are $2 \; \mu\text{m}$ and $1 \; \mu\text{m},$ respectively.

The left side of the bar $(x = 0)$ is uniformly illuminated with a laser having photon energy greater than the bandgap of the semiconductor. Excess electron-hole pairs are generated $\text{ONLY}$ at $x = 0$ because of the laser. The steady state electron density at $x = 0$ is $10^{14} \; \text{cm}^{-3}$ due to laser illumination. Under these conditions and ignoring electric field, the closest approximation (among the given options) of the steady state electron density at $x = 2 \; \mu\text{m},$ is ______________.

- $0.37 \times 10^{14} \; \text{cm}^{-3}$
- $0.63 \times 10^{13} \; \text{cm}^{-3}$
- $3.7 \times 10^{14} \; \text{cm}^{-3}$
- $10^{3} \; \text{cm}^{-3}$