Question #43460

In the base-emitter P-N junction of an N-P-N transistor at temperature $T$, the depletion region is formed by uniform doping. This results in a position-dependent electric field that peaks at the metallurgical junction and drops linearly to zero at the edges of the depletion region. Under zero bias, the maximum electric field is $E_0$ and the total depletion width is $W_0$. When a forward bias $V_{BE}$ is applied, the maximum electric field reduces to $E_1$ and the depletion width becomes $W_1$.

The emitter current $I_E$ is governed by the probability of electrons overcoming the potential barrier, yielding $I_E \approx I_0 \exp(e V_{BE} / k_B T)$ for $V_{BE} \gg k_B T/e$, where $I_0$ is the reverse saturation current, $e$ is the elementary charge, and $k_B$ is the Boltzmann constant.

Using the work-energy theorem to evaluate the potential barrier from the electric field profile, which of the following expressions correctly represents the emitter current $I_E$?