Drude Free-Electron Model

Equivalent forms

\rho=1/\sigma=\frac{m}{ne^2\tau}

A 1900 classical billiard-ball picture still gives the right functional form for the conductivity of copper — you only need quantum mechanics to get τ right.

Symbol	Name	SI unit	Dimension	Range
$\sigma$	Conductivityoutput Electrical conductivity of the metal	S/m	M^-1L^-3T^3*I^2	—
$n$	Electron density Free electrons per unit volume	m^-3	L^-3	1e+28 – 1e+29
$\tau$	Relaxation time Mean time between collisions	s	T	1e-15 – 1e-13
$e$	Elementary charge Electron charge magnitude	C	I*T	—
$m$	Electron mass Mass of the electron	kg	M	—

Where it holds

Good for DC/low-frequency conductivity of simple metals. Fails for electronic heat capacity (needs Fermi statistics), strong fields, and the sign of the Hall coefficient in some metals.

Dimensional analysis

\sigma \to [M^-1*L^-3*T^3*I^2] (S/m)

ne^{2}\tau /m \to [L^-3][I\cdot T]^{2}[T]/[M] = [M^-1*L^-3*T^3*I^2]

Both sides match Siemens per metre.

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Discovery

Paul Drude · 1900

Just three years after the electron's discovery, Drude applied kinetic gas theory to the newly found charge carriers in metals. His model explained Ohm's law and the Wiedemann–Franz ratio of thermal to electrical conductivity. Its failures (heat capacity, magnetoresistance) were only fixed by Sommerfeld's 1927 quantum version using Fermi–Dirac statistics.

Research status: stable

Real-world applications

Estimating wire resistance and Joule heating
Modeling the optical/plasma response of metals (Drude permittivity)
Baseline for semiconductor transport before quantum corrections

Common misconceptions

Drift velocity is tiny $(\sim mm/s)$ even though the signal travels near light speed
Electrons do not collide with each other in the model, only with the lattice
$\tau is$ not a classical mean free path over speed — quantum mechanically the relevant speed is the Fermi velocity

Experimental verification

Reproduces measured room-temperature conductivities of metals within an order of magnitude and the Wiedemann–Franz law

(\kappa /\sigma T

≈ constant) remarkably well.

Derivation

Newton's law for an electron with a drag term:

m dv/dt = -eE - mv/\tau

.

In steady state

dv/dt = 0

, so the drift velocity

v_d = -e\tau E/m

.

The current density

J = -nev_d = (ne^{2}\tau /m)E

.

Comparing with

J = \sigma E

gives

\sigma = ne^{2}\tau /m

.

The

1/\tau

term encodes randomizing collisions that reset the electron's velocity.

What if…

What if temperature rises?

More lattice vibrations shorten $\tau$ , lowering $\sigma$ — why metals' resistance climbs roughly linearly with T.

What if you cool toward absolute zero?

$\tau$ grows until limited by impurities, giving a residual resistance; in some metals it collapses entirely (superconductivity), which the model cannot explain.

1

Relaxation time of copper

Given ·

σ:: 5.96e7 S/m
n:: $8.5e28\,\mathrm{m}^{-3}$

Find · collision time

\tau

Steps

$\tau = m\sigma /(ne^{2})$
$= (9.11e-31 \times 5.96e_{7})/(8.5e28 \times (1.602e-19)^{2})$
$\approx 2.5e-14\,\mathrm{s}$

Result ·

\tau = m\sigma /(ne^{2}) \approx 2.5e-14\,\mathrm{s}

Ohm's Law→Band Theory of Solids→hall voltage modernFermi-Dirac Distribution→