Question

Aluminium has three valence electrons per atom, an atomic weight of 0.02698 kg mol−1,
a density of 2700kgm-3, and a conductivity of 3.54× 107 W−1m−1. Calculate the relaxation
time in Aluminium.

Answer 1

Answer on Question #51560, Physics, Solid State Physics

Aluminium has three valence electrons per atom, an atomic weight of 0.02698 kg·mol⁻¹, a density of 2700 kg·m⁻³, and a conductivity of 3.54 × 10⁷ W⁻¹ m⁻¹. Calculate the relaxation time in Aluminium.

Solution

Boltzmann transport equation in the relaxation time approximation leads to the conduction electron gas to the Drude formula:

$$\sigma = \frac{n \cdot e^2 \cdot \tau}{2 m_e}$$

$\sigma$ - Electrical conductivity; $n = 3 \cdot \rho N_A / \mu = 3 \cdot (2700 \cdot 6.02 \cdot 10^{23}) / 0.02698 = 1.81 \cdot 10^{29} \, \text{m}^{-3}$ - electron density; $e = 1.6 \cdot 10^{-19} \, \text{C} = 1.6 \cdot 10^{-19} \, \text{A} \cdot \text{s}$ - elementary charge; $\tau$ - the momentum relaxation time (the time at which the electron "forgets" about which way to go); $m_e = 9.1 \cdot 10^{-31} \, \text{kg}$ - the effective mass of the electron.

Then

$$\tau = \frac{2 m_e \sigma}{n \cdot e^2} = \frac{2 \cdot 9.1 \cdot 10^{-31} \, \text{kg} \cdot 3.54 \cdot 10^7 \, \text{W}^{-1} \, \text{m}^{-1}}{1.81 \cdot 10^{29} \, \text{m}^{-3} \cdot (1.6 \cdot 10^{-19} \, \text{C})^2} = 1.4 \cdot 10^{-14} \, \text{s}$$

Answer: $\tau = \frac{2 m_e \sigma}{n \cdot e^2} = 1.4 \cdot 10^{-14} \, \text{s}$

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