Metals — Question 1

Question 1

Momentum relaxation sets the metallic dc conductivity

A one-band metal has electron density n, effective mass m, and relaxation time $\tau$. Its mean velocity obeys $m\dot v=-eE-mv/\tau$, where $e>0$.

1. Derive the steady conductivity and mobility.

2. For $n=5.00\times10^{28}\,\mathrm{m^{-3}}$, $m=9.11\times10^{-31}\,\mathrm{kg}$, $\tau=2.00\times10^{-14}\,\mathrm s$, and $e=1.60\times10^{-19}\,\mathrm C$, evaluate both.

Question 1 - Solution

Momentum relaxation sets the metallic dc conductivity

(a) Method and calculation

$$\dot v=0\Rightarrow v_d=-e\tau E/m,\qquad J=-nev_d.$$$$\boxed{\sigma=ne^2\tau/m,\quad\mu=e\tau/m}.$$

(b) Interpretation and further calculation

$$\sigma=\frac{(5.00\times10^{28})(1.60\times10^{-19})^2(2.00\times10^{-14})}{9.11\times10^{-31}}.$$$$\boxed{\sigma=2.81\times10^7\,\mathrm{S/m},\quad\mu=3.51\times10^{-3}\,\mathrm{m^2/(V\,s)}}.$$

Physical check

Electron velocity opposes the field but conventional current follows it. The relaxation term transfers electronic momentum to the lattice and defects.