Current Inside the Depletion Region Forward Biased PN Junction

Question

For a forward biased PN junction, we assume the hole and electron currents are constant inside the depletion region when there is no generation or recombination in the depletion region (shown from the figure off of Neaman's book). If we only consider the holes, their concentration profile inside the depletion region and the P side can be determined determined from the quasi-fermi level for holes $E_{Fp}$: $$p(x)= n_{i}e^\frac{(E_{i}-E_{F_{p}})}{KT}$$

However, using $J_{p$} = e\mu_{p}p(x)E_{bi}(x)-eD_{p}\frac{dp(x)}{dx}$ where $E_{bi}(x)$ is the built in electric field which is known, I dont get a constant hole current inside the depletion region as the book claims? why is this the case? Is there something I am missing that is taking place inside the depletion region ?

Answer 1

You have probably assumed constant quasi-Fermi levels. But the quasi-Fermi levels of holes and electrons in the depletion zone cannot be exactly constant.This is only an approximation. Exact constancy would imply zero hole and electron current densities. You can most easily see this by rewriting the drift-diffusion equations in terms of the quasi-Fermi level. It suffices to show this for the hole current density.

You insert the hole density
$$p(x)= n_{i}e^\frac{(E_{i}-E_{F_{p}})}{KT}$$ into the diffusion equation $$J_{p} = e\mu_{p}p(x)F(x)-eD_{p}\frac{dp(x)}{dx}$$ with the electric field $F(x)$. Then, using that the slope of the intrinsic energy gives the electric field$$\frac{dE_{i}(x)}{dx}= eF(x)$$ and using the Einstein relation between mobility and the diffusion coefficient $D_{p}= \mu_{p}kT/e$ one obtains for the current equation $$J_{p} = \mu_{p} p(x)\frac{E_{F_{p}}(x)}{dx}$$ Thus the drift and diffusion current is driven by the gradient of the quasi-Fermi level. In particular, when the hole quasi-Fermi level is constant and thus its gradient zero, there is zero current. The assumption of a constant hole quasi-Fermi level in the depletion zone is thus not exactly correct, it is only an approximation.

An analogous consideration holds for the electron current density and the electron quasi-Fermi level.