Question #43845

A semiconductor junction is excited by a time-varying optical pulse. The instantaneous drift current responds directly to the pulse as $I_{\text{drift}}(t) = I_0 e^{-t/\tau_1}$ for $t \ge 0$. The diffusion current $I_{\text{diff}}(t)$ is driven by the excess carriers generated by the same pulse and satisfies the continuity equation $\frac{d I_{\text{diff}}}{dt} = k I_{\text{drift}}(t) - \frac{I_{\text{diff}}(t)}{\tau_2}$, where $\tau_2$ is the effective carrier lifetime and $k$ is a coupling constant. Given $I_0 = 2 \text{ mA}$, $\tau_1 = 1 \text{ \mu s}$, $\tau_2 = 0.5 \text{ \mu s}$, $k = 4 \text{ \mu s}^{-1}$, and $I_{\text{diff}}(0) = 0$. At what time $t$ does the total current $I(t) = I_{\text{drift}}(t) + I_{\text{diff}}(t)$ reach its maximum value?