Diffusion coefficient and diffusion current density

The diffusion coefficient follows from the Einstein relation

$$\frac{D_e}{\mu_e} = \frac{kT}{e}$$

Data used: electron mobility in Si $\mu_e = 1350\ \text{cm}^2\text{V}^{-1}\text{s}^{-1}$ (not given, standard value), $T = 300$ K, $kT/e = 0.02585$ V.

Diffusion coefficient

$$\begin{aligned} D_e &= \mu_e\frac{kT}{e} = 1350\times0.02585\\ &= 34.9\ \text{cm}^2\text{s}^{-1} \end{aligned}$$

Diffusion current density

The gradient "$10^3$ electrons per centimetre" is read as $dn/dx = 10^{3}\ \text{cm}^{-3}$ per cm $= 10^3\ \text{cm}^{-4}$.

$$\begin{aligned} J_{diff} &= eD_e\frac{dn}{dx}\\ &= 1.6\times10^{-19}\times34.9\times10^{3}\\ &= 5.58\times10^{-15}\ \text{A cm}^{-2} \end{aligned}$$

Answer: $D_e \approx 34.9\ \text{cm}^2\text{s}^{-1}$ ($3.49\times10^{-3}\ \text{m}^2\text{s}^{-1}$); $J \approx 5.6\times10^{-15}\ \text{A cm}^{-2}$.

The current is tiny because the given gradient is extremely small; $J$ is directly proportional to $dn/dx$, so a realistic gradient such as $10^{20}\ \text{cm}^{-4}$ would give $J \approx 560\ \text{A cm}^{-2}$. The current flows in the direction of increasing $n$ (electrons diffuse the other way, and their charge is negative).