---
title: "Diffusion coefficient of electrons at 30°C"
subject: "Electrical Engineering Material"
chapter: 6
key: "semester_3/electrical_engineering_material:diffusion-coefficient-30c"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/diffusion-coefficient-30c
site: BCT Engineering
---

# Diffusion coefficient of electrons at 30°C

Electrical Engineering Material · Chapter 6: Semiconductors

- Asked in: 2072 Kartik, 4 marks
- Video: [Einstein Relation in Semiconductor (Basics, Statement, Meaning & Proof) Explained](https://www.youtube.com/watch?v=t2sRYkVDZaQ) — Engineering Funda
- 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-30c
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-6

## Question

Calculate the diffusion coefficient of electrons at 30°C in silicon doped with 10¹⁵ Arsenic atoms cm⁻³. Given that the drift mobility of electron with 10¹⁵ cm⁻³ dopants is 1300 cm²V⁻¹s⁻¹.

## Short answer

Using the Einstein relation D = μ kT/e. At 30°C kT/e = 0.02612 V, so D_e = 1300 × 0.02612 = 34.0 cm²/s.

## Answer

The diffusion coefficient is found from the **Einstein relation**, which links diffusion and drift of the same carrier:

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

**Given:** $\mu_e = 1300\ \text{cm}^2\text{V}^{-1}\text{s}^{-1}$ (for $10^{15}$ cm⁻³ As), $T = 30^\circ\text{C} = 303.15$ K, $k = 8.617\times10^{-5}$ eV K⁻¹.

#### Thermal voltage

$$
\frac{kT}{e} = 8.617\times10^{-5}\times303.15 = 0.02612\ \text{V}
$$

#### Diffusion coefficient

$$
\begin{aligned}
D_e &= \mu_e\frac{kT}{e} = 1300\times0.02612\\
&= 33.96\ \text{cm}^2\text{s}^{-1}
\end{aligned}
$$

**Answer: $D_e \approx 34.0\ \text{cm}^2\text{s}^{-1} = 3.40\times10^{-3}\ \text{m}^2\text{s}^{-1}$.**

Arsenic doping of $10^{15}$ cm⁻³ is light, so the material is non-degenerate and the Einstein relation in this simple form is valid. The doping matters only through the mobility value given.

## 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 = mu x kT x e
   D. D = mu/(kT)
   Answer: A. The Einstein relation is D_e/mu_e = kT/e. (Key point: Einstein relation)

2. What is D_e for mu = 1300 cm^2/Vs at 30 C (kT/e = 0.02612 V)?
   A. About 0.34 cm^2/s
   B. About 3.4 x 10^4 cm^2/s
   C. About 34 cm^2/s
   D. About 1300 cm^2/s
   Answer: C. D = 1300 x 0.02612 = 33.96, about 34 cm^2/s. (Key point: Result)

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