---
title: "Mean free time of electrons in Si"
subject: "Electrical Engineering Material"
chapter: 2
key: "semester_3/electrical_engineering_material:silicon-mean-free-time"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/silicon-mean-free-time
site: BCT Engineering
---

# Mean free time of electrons in Si

Electrical Engineering Material · Chapter 2: Free Electron Theory of Conduction in Metal

- Asked in: 2069 Chaitra, 4 marks
- Video: [Important terms in Free electron theory (Drift velocity, Mean free path, Relaxation time) PHYSICS](https://www.youtube.com/watch?v=CI1R3r4sfDY) — Gautam Varde
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/silicon-mean-free-time
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-2

## Question

Electron mobility in Si is 1400 cm² V⁻¹s⁻¹. Calculate the mean free time in scattering of electrons. Effective mass is mₑ*/mₑ = 0.33.

## Short answer

Mean free time tau = (mu_e m_e*)/e. Using mu_e = 0.14 m^2/(V·s) and effective mass m_e* = 0.33 m_e, tau = 2.63×10^-13 s.

## Answer

In a semiconductor the electron moves with an effective mass $m_e^*$, so the drift mobility is

$$
\mu_e = \frac{e\tau}{m_e^*} \quad\Rightarrow\quad \tau = \frac{\mu_e m_e^*}{e}
$$

**Given:** $\mu_e = 1400\ \text{cm}^2\text{V}^{-1}\text{s}^{-1} = 0.14\ \text{m}^2\text{V}^{-1}\text{s}^{-1}$, $m_e^* = 0.33 m_e = 0.33 \times 9.109 \times 10^{-31} = 3.006 \times 10^{-31}\ \text{kg}$.

#### Mean free time

$$
\tau = \frac{0.14 \times 3.006 \times 10^{-31}}{1.602 \times 10^{-19}} = 2.63 \times 10^{-13}\ \text{s}
$$

This is about ten times longer than in copper ($\sim 2.5 \times 10^{-14}$ s). The conduction electrons in Si move at thermal speeds (about $10^5$ m/s), much slower than the Fermi speed in a metal, so they take longer between collisions. The much higher mobility of Si (1400 against 43 cm² V⁻¹ s⁻¹ for Cu) comes partly from this longer $\tau$ and partly from the small effective mass.

**Answer:** $\tau \approx 2.63 \times 10^{-13}\ \text{s}$ (0.263 ps).

## Check yourself

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

1. In Si, mobility uses
   A. free electron mass m_e
   B. effective mass m*: μ = eτ/m*
   C. hole mass
   D. proton mass
   Answer: B. Electrons in a crystal respond with effective mass. (Key point: Effective mass)

2. Mean free time is
   A. τ = μ m*/e
   B. τ = μ e/m*
   C. τ = e/(μ m*)
   D. τ = μ m_e/ e^2
   Answer: A. Rearranging μ = eτ/m*. (Key point: Formula)

3. 1400 cm^2/V/s equals
   A. 14 m^2/V/s
   B. 1.4e7 m^2/V/s
   C. 1400 m^2/V/s
   D. 0.14 m^2/V/s
   Answer: D. 1 cm^2 = 1e-4 m^2. (Key point: Unit conversion)

4. τ is about
   A. 2.63e-14 s
   B. 8e-13 s with m_e instead
   C. 2.63e-13 s
   D. 2.63e-10 s
   Answer: C. 0.14 x 3.006e-31 / 1.602e-19 = 2.63e-13 s. (Key point: Result)

## Related in Electrical Engineering Material

- [Copper drift mobility and scattering time](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acopper-mobility-scattering-time) — Electrical Engineering Material, chapter 2
- [Effective mass of a free electron](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Afree-electron-effective-mass) — Electrical Engineering Material, chapter 1, asked 3×
- [Copper drift mobility from scattering cross-section](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acopper-mobility-from-cross-section) — Electrical Engineering Material, chapter 2
- [Mean free path for 53 cm²/V·s mobility](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Amean-free-path-53-mobility) — Electrical Engineering Material, chapter 2
- [Effective mass of electron in gold](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Agold-effective-mass-justify) — Electrical Engineering Material, chapter 1
- [Si polarizability with 6×10²² electrons/cm³](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Asi-polarizability-6e22) — Electrical Engineering Material, chapter 3

## Related in other subjects

- [Free electron density from Hall emf](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Asilver-strip-hall-free-electron-density) — Engineering Physics, chapter 8
- [Phi-index from mass curve, 3.5 cm runoff](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aphi-index-3-5cm-runoff) — Engineering Hydrology, chapter 3
- [Round bar fixed-free (effective length 2 m): buckling load](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Around-bar-clamped-free-buckling-2m) — Strength of Materials, chapter 8, asked 2×
- [Free body diagram and X1/F, X2/F of two-mass system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afbd-m1-m2-k1k2-b1b2) — Control Systems, chapter 2