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Mean free time of electrons in Si

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.

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.

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

μe=eτme∗⇒τ=μeme∗e\mu_e = \frac{e\tau}{m_e^*} \quad\Rightarrow\quad \tau = \frac{\mu_e m_e^*}{e}

Given: μe=1400 cm2V−1s−1=0.14 m2V−1s−1\mu_e = 1400\ \text{cm}^2\text{V}^{-1}\text{s}^{-1} = 0.14\ \text{m}^2\text{V}^{-1}\text{s}^{-1}, me∗=0.33me=0.33×9.109×10−31=3.006×10−31 kgm_e^* = 0.33 m_e = 0.33 \times 9.109 \times 10^{-31} = 3.006 \times 10^{-31}\ \text{kg}.

Mean free time

τ=0.14×3.006×10−311.602×10−19=2.63×10−13 s\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 (∼2.5×10−14\sim 2.5 \times 10^{-14} s). The conduction electrons in Si move at thermal speeds (about 10510^5 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 τ\tau and partly from the small effective mass.

Answer: τ≈2.63×10−13 s\tau \approx 2.63 \times 10^{-13}\ \text{s} (0.263 ps).

✓ Answer checked by Claude Opus 5.5 (high effort)

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