Semiconductors — Question 8

Question 8

A concentration gradient requires an electric field for zero electron current

In a one-dimensional isothermal region, electron current obeys the supplied drift-diffusion law $J_n=en\mu_nE+eD_n\,dn/dx$. Use $D_n/\mu_n=k_BT/e$. Let $n(x)=n_0e^{x/L}$ and impose $J_n=0$.

1. Find the electric field and potential difference V(L)-V(0).

2. Find the electron drift and diffusion current densities separately, and verify cancellation.

Question 8 - Solution

A concentration gradient requires an electric field for zero electron current

(a) Method and calculation

$$E=-\frac{D_n}{\mu_n}\frac1n\frac{dn}{dx}=-\frac{k_BT}{eL}.$$$$V(L)-V(0)=-\int_0^L E\,dx=k_BT/e.$$

(b) Interpretation and further calculation

$$J_{\mathrm{drift}}=en\mu_nE=-n\mu_nk_BT/L.$$$$J_{\mathrm{diffusion}}=eD_n(n/L)=+n\mu_nk_BT/L,\quad J_n=0.$$

Physical check

Zero total electron current does not force each transport mechanism to vanish separately.