Transport, Wiedemann–Franz, and the Hall Effect

The Sommerfeld model gave the equilibrium electron gas: a filled Fermi sphere and a heat capacity set by the thermal shell near $E_{F}$ . Transport asks what happens when the gas is pushed out of equilibrium by an electric field, a temperature gradient, or a magnetic field. The same handful of electrons near the Fermi surface carry every current, and one phenomenological parameter — the relaxation time $τ$ — ties the responses together. The payoff is a chain of results, culminating in the Wiedemann–Franz law and the Hall effect, that made the free-electron picture credible long before its microscopic justification.

The relaxation-time approximation

Between collisions an electron of crystal momentum $ℏ k$ obeys Newton's law under the Lorentz force,

ℏ \frac{d k}{d t} = - e (E + v \times B) .

Collisions — with lattice vibrations, impurities, other electrons — randomize $k$ on a timescale $τ$ . The relaxation-time approximation models this by assuming the distribution $f (k)$ relaxes back toward equilibrium $f_{0} (k)$ at a rate $1/ τ$ :

\frac{\partial f}{\partial t} = - \frac{f - f _{0}}{τ} + (\frac{\partial f}{\partial t})_{drift} .

This is the Boltzmann transport equation with its collision integral replaced by a single relaxation rate. In steady state the drift and relaxation terms balance. The physical content is simplest in $k$ -space: the field displaces the entire Fermi sphere by a small amount $δ k = - e E τ /ℏ$ , and collisions try to push it back. The net drift velocity is $v_{d} = ℏ δ k / m = - e E τ / m$ .

An electric field along minus x displaces the whole Fermi sphere by a small delta-k opposite to the field. Electrons on the leading face have no occupied state ahead of them; scattering returns them to the trailing face. The net shift, exaggerated here, is what carries the current.

Electrical conductivity

With $n$ electrons per unit volume the current density is $j = - n e v_{d}$ , and substituting $v_{d} = - e E τ / m$ gives Ohm's law $j = σ E$ with

σ = \frac{n e ^{2} τ}{m}, ρ = \frac{1}{σ} = \frac{m}{n e ^{2} τ} .

The form is identical to Drude's, but the interpretation is Sommerfeld's: the electrons that shift are those at the Fermi surface, moving at $v_{F}$ , so the mean free path is $ℓ = v_{F} τ$ — set by the fast Fermi velocity, not the slow drift or the classical thermal speed. Because only a perfect lattice fails to scatter, $τ$ is limited by deviations from periodicity. At high temperature phonon scattering gives $τ^{- 1} \propto T$ and $ρ \propto T$ ; as $T \to 0$ the residual impurity scattering leaves a temperature-independent floor, so $ρ = ρ_{phonon} (T) + ρ_{0}$ — Matthiessen's rule, with the two rates adding because the scattering probabilities add.

Thermal conductivity and Wiedemann–Franz

The electron gas conducts heat as well as charge. Kinetic theory gives the thermal conductivity of a gas of carriers as

κ = \frac{1}{3} c_{v} v_{F}^{2} τ = \frac{1}{3} c_{v} v_{F} ℓ,

where $c_{v}$ is the electronic heat capacity per unit volume. Using the Sommerfeld result $c_{v} = \frac{π ^{2}}{2} n k_{B} (T / T_{F})$ together with $\frac{1}{2} m v_{F}^{2} = E_{F} = \frac{3}{2} k_{B} T_{F}$ , the Fermi velocity and temperature cancel in a remarkable way. Forming the ratio of thermal to electrical conductivity,

\frac{κ}{σ} = \frac{\frac{1}{3} c _{v} v _{F}^{2} τ}{n e ^{2} τ / m} = \frac{π ^{2}}{3} (\frac{k _{B}}{e})^{2} T,

so that $κ / σ T$ is a universal constant independent of the metal, the carrier density, and the relaxation time:

L \equiv \frac{κ}{σ T} = \frac{π ^{2}}{3} (\frac{k _{B}}{e})^{2} = 2.44 \times 1 0^{- 8} W Ω K^{- 2} .

The law holds impressively well for good metals near and above room temperature, where scattering is elastic. It measures the Lorenz number to within a few percent for copper, silver, gold, and the alkalis, and its main deviations — at intermediate temperatures where phonon scattering of the heat current is inelastic — are themselves diagnostic.

The Lorenz number kappa over sigma T measured for several metals near room temperature, scattered narrowly about the free-electron value L0 (dashed). The near-constancy across metals with very different densities and relaxation times is the content of the Wiedemann–Franz law.

Cyclotron motion and the Hall effect

Add a magnetic field $B = B \overset{z}{^}$ and no electric field. The Lorentz force $- e v \times B$ bends the electron into a circle in the plane perpendicular to $B$ , orbiting at the cyclotron frequency

ω_{c} = \frac{e B}{m},

independent of the electron's speed or orbit radius. For $B = 1 T$ this is $1.76 \times 1 0^{11} rad/s$ , or $28 GHz$ . Only if the electron can complete an orbit between collisions — $ω_{c} τ ≫ 1$ , a clean sample in a strong field — do the orbits matter for transport; otherwise scattering interrupts them.

In a magnetic field out of the page, the Lorentz force curves an electron into a cyclotron orbit of angular frequency omega-c equal to eB over m. The field points out of the page; the velocity and force are perpendicular.

Now the classic Hall geometry: current $I$ flows along $\overset{x}{^}$ through a flat bar, with $B$ along $\overset{z}{^}$ . The magnetic force $- e v \times B$ pushes the drifting electrons toward one edge ( $- \overset{y}{^}$ ), charge piles up there, and a transverse Hall field $E_{y}$ grows until it cancels the magnetic force in steady state:

- e E_{y} - e (v_{x} B) = 0 \Rightarrow E_{y} = - v_{x} B = \frac{j _{x} B}{n e} .

The Hall coefficient is defined by $E_{y} = R_{H} j_{x} B$ , giving

R_{H} = \frac{E _{y}}{j _{x} B} = - \frac{1}{n e} .

Hall-bar geometry. Current I runs along x, field B along z. The magnetic force deflects carriers to one edge until the transverse Hall field Ey balances it. The sign of the measured Hall voltage fixes the sign of the charge carriers.

The Hall coefficient measures the carrier density directly, and — its sharpest use — the carrier sign. For free electrons $R_{H} < 0$ , and for the alkali metals the measured value matches $- 1/ n e$ within a few percent. But a number of metals — beryllium, zinc, cadmium, aluminium — show a positive Hall coefficient, as if the current were carried by positive charges. Free-electron theory has no explanation; the resolution is that in these metals the Fermi surface reaches the Brillouin-zone boundary and the carriers behave as holes — a signal that the periodic lattice cannot be ignored, taken up in band theory.

Magnetoresistance and the limits of the model

Solving the steady-state equations of motion with both $E$ and $B$ gives the full conductivity tensor. For a single species of carrier the free-electron model predicts no magnetoresistance: the diagonal resistivity $ρ_{xx}$ is independent of $B$ , because the Hall field exactly compensates the transverse deflection. Real metals show a definite magnetoresistance, growing with $B$ and often failing to saturate — another discrepancy that band structure, with multiple carrier types and non-spherical Fermi surfaces, resolves.

The tally so far is favourable: the free-electron gas gives Ohm's law, the Wiedemann–Franz law with the right Lorenz number, the cyclotron frequency, and the Hall coefficient of the simple metals. Its failures — the positive Hall coefficients, the magnetoresistance, the material-dependent $γ$ — all point at the same missing ingredient, the periodic potential. Before that, one more purely electronic effect deserves attention: how the gas screens a foreign charge and oscillates collectively, the subject of the next lesson.