Classical Free Electron Theory and Conductivity in Metals

By the end you should be able to: Derive the expression for electrical conductivity from the classical free electron model, use it to compute conductivity, mobility and drift velocity for a metal, and state the specific experimental facts the model fails to explain.

A metal conducts. The question is what is actually moving, how fast, and why the answer depends on temperature the way it does.

The model

Drude's picture, from 1900, is deliberately crude:

The valence electrons of the metal atoms detach and move freely through the solid, like the molecules of a gas. Hence free electron gas.
They collide with the fixed positive ion cores. Between collisions they are free; collisions are instantaneous and randomise the velocity completely.
With no applied field, the average velocity is zero. Electrons move fast but in all directions equally, so no net charge flows.

The key parameter is the relaxation time $\tau$ — the average time between collisions.

Drift velocity

Apply a field $E$ . Each electron feels a force $-eE$ , so it accelerates:

a = \frac{eE}{m}

It accelerates only until the next collision, on average a time $\tau$ . So on top of its large random velocity it picks up a small drift velocity in the direction opposite to the field:

v_d = a\tau = \frac{eE\tau}{m}

That is the whole mechanism. The random motion carries no current; the small systematic drift superposed on it carries all of it.

Current density and conductivity

If there are $n$ electrons per unit volume, the current density is

J = n e v_d = \frac{n e^2 \tau}{m} E

Compare this with Ohm's law in its point form, $J = \sigma E$ :

\boxed{\sigma = \frac{n e^{2} \tau}{m}} \qquad \rho = \frac{1}{\sigma} = \frac{m}{n e^{2} \tau}

Two related quantities follow. Mobility is drift velocity per unit field:

\mu = \frac{v_d}{E} = \frac{e\tau}{m}, \qquad \sigma = n e \mu

What the model gets right

Ohm's law itself. Nothing in the derivation assumed a linear relation between $J$ and $E$ — it came out, because $\tau$ does not depend on the applied field for ordinary field strengths. That is a genuine success.

Feed in copper's numbers and the magnitude is right too, as the worked example below shows.

What the model gets wrong

This matters more than the successes, because it is why the module does not stop here.

Specific heat. Treating electrons as a classical gas predicts they contribute $\tfrac{3}{2}R$ to the molar specific heat. The measured electronic contribution is smaller by roughly a factor of a hundred at room temperature. The model does not miss by a little.

Temperature dependence of resistivity. Classically $\tau$ is set by the thermal speed, $v \propto \sqrt{T}$ , giving $\rho \propto \sqrt{T}$ . Real metals obey $\rho \propto T$ over a wide range.

Mean free path. The classical estimate comes out at a few nanometres — a handful of atomic spacings, which sounds reasonable until you cool a pure metal and find the mean free path growing to microns. Electrons are somehow not colliding with a perfectly regular lattice.

Why some solids are insulators. The model has no answer at all. Every solid has electrons; the model gives no reason for any of them to refuse to conduct.

The common root of all four failures is the same assumption: that electrons obey classical statistics, so that every electron can take any energy and all of them respond to heating. They do not. Electrons are fermions obeying the Pauli principle, and only the small fraction near the top of the filled energy states can do anything at all. That is the subject of the next topic.