---
title: "Diffusion coefficient and diffusion current density"
subject: "Electrical Engineering Material"
chapter: 6
key: "semester_3/electrical_engineering_material:diffusion-coefficient-gradient"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/diffusion-coefficient-gradient
site: BCT Engineering
---

# Diffusion coefficient and diffusion current density

Electrical Engineering Material · Chapter 6: Semiconductors

- Asked in: 2074 Chaitra, 4 marks
- Video: [Diffusion Current Example](https://www.youtube.com/watch?v=B7OLkpH6ypM) — Jordan Edmunds Chetty
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/diffusion-coefficient-gradient
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6

## Question

Calculate the diffusion coefficient of electrons at 300K in n-type silicon semiconductor. Also find current density if electron concentration gradient is 10³ electrons per centimeter.

## Short answer

Using Einstein relation, D_e = μ_e kT/e = 1350×0.02585 = 34.9 cm^2/s. Diffusion current density J = e D_e (dn/dx) = 1.6×10^-19×34.9×10^3 = 5.6×10^-15 A/cm^2 for a gradient of 10^3 cm^-4.

## Answer

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).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which relation gives the electron diffusion coefficient from mobility?
   A. D = mu kT / e
   B. D = mu e / kT
   C. D = kT / (mu e)
   D. D = mu^2 kT
   Answer: A. The Einstein relation is D/mu = kT/e. (Key point: Einstein relation)

2. With mu_e = 1350 cm^2/Vs and kT/e = 0.02585 V, what is D_e?
   A. 3.49 cm^2/s
   B. 52 000 cm^2/s
   C. 34.9 cm^2/s
   D. 0.349 cm^2/s
   Answer: C. D_e = 1350 x 0.02585 = 34.9 cm^2/s. (Key point: Value of D_e)

3. Which formula gives the electron diffusion current density?
   A. e n mu E
   B. e D_e dn/dx
   C. e D_e n
   D. kT dn/dx
   Answer: B. J = e D_e dn/dx, proportional to the concentration gradient. (Key point: Diffusion current formula)

4. Why is the computed current density (about 5.6 x 10^-15 A/cm^2) so tiny?
   A. Because D_e is very small
   B. Because silicon is an insulator
   C. Because the electron charge is negative
   D. Because the given gradient of 10^3 cm^-4 is extremely small
   Answer: D. J is directly proportional to dn/dx, and the given gradient is very small. (Key point: Size of J)

## Related in Electrical Engineering Material

- [Diffusion coefficient of electrons at 300 K](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Adiffusion-coefficient-300k) — Electrical Engineering Material, chapter 6
- [Diffusion coefficient of electrons at 30°C](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Adiffusion-coefficient-30c) — Electrical Engineering Material, chapter 6
- [Drift and diffusion current density](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Adrift-diffusion-current-density) — Electrical Engineering Material, chapter 6
- [Electron concentration in an extrinsic semiconductor](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aextrinsic-electron-concentration) — Electrical Engineering Material, chapter 6
- [Electron concentration with equal P and B doping](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aequal-doping-electron-concentration) — Electrical Engineering Material, chapter 6, asked 4×
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