Drude Model

Definition

In 1900, just three years after the electron was discovered, Paul Drude proposed a startlingly simple picture of a metal. Strip away the chemistry and imagine a gas of free electrons rattling around inside a fixed scaffold of heavy positive ions. The electrons fly in straight lines at high speed, occasionally smacking into an ion, bouncing off in a random new direction, and starting over. The model borrows kinetic theory wholesale — the same machinery that describes air molecules in a box — and applies it to the conduction electrons in a chunk of copper.

Two numbers carry the whole theory. The first is the relaxation time tau, the average time between collisions, around 1×10⁻¹⁴ seconds in a typical metal. The second is the carrier density n, the number of free electrons per cubic meter — for copper, a staggering n = 8.5×10²⁸ m⁻³. From just these, plus the electron charge and mass, the model spits out the conductivity:

σ = n e² τ / m

and its reciprocal, the resistivity rho = m / (n e² tau). Everything else — Ohm's law, the thermal conductivity of metals, the Hall coefficient — follows from this one engine.

How it works

Consider one conduction electron. With no applied field, it moves in a random direction, collides, picks a new random direction, collides again. Average its velocity over many collisions and you get zero — no net current, just thermal jitter. The electrons are fast (their speed is around 1×10⁶ m/s) but they go nowhere on average.

Now switch on an electric field E. Between collisions the electron feels a steady force F = −eE and accelerates. But every collision randomizes its direction, effectively wiping out the velocity it had gained from the field. So the electron is repeatedly nudged in the field direction, then reset. Averaging this stop-start drift gives a small steady velocity:

m (dv/dt) = −eE − m v / τ       (Drude equation of motion)
steady state (dv/dt = 0):
v_d = −e τ E / m              (drift velocity)

The term −m v / tau is the genius move: it models all those collisions as a single, smooth frictional drag with time constant tau. In steady state the field's push balances the drag, and the electron settles into a constant drift velocity v_d superimposed on its fast random motion. The drift is tiny — under a millimeter per second for ordinary currents — yet it is the entire current.

Multiply the drift velocity by the charge density to get current density:

J = −n e v_d = (n e² τ / m) E = σ E

That last equality is Ohm's law. Notice that we did not assume resistance was constant — the linear J–E relationship emerged from averaging electron motion, and the conductivity sigma = n e² tau / m popped out as the proportionality constant. This is the single most cited result in solid-state physics.

Worked example: copper

Let's put real numbers in. Copper has n = 8.5×10²⁸ m⁻³, e = 1.6×10⁻¹⁹ C, and m = 9.1×10⁻³¹ kg. Take the relaxation time as tau = 2.5×10⁻¹⁴ s (the value you extract from copper's measured conductivity). Then:

σ = n e² τ / m
  = (8.5e28)(1.6e-19)²(2.5e-14) / (9.1e-31)
  ≈ 6.0e7 S/m

ρ = 1/σ ≈ 1.7e-8 Ω·m

The measured resistivity of copper at room temperature is 1.68×10⁻⁸ Ω·m. The Drude model nails it — because we tuned tau to fit, but the structure of the formula is what matters: it correctly says copper conducts well because it has a huge n and a relatively long tau.

Now estimate the drift velocity in a household copper wire. A 1 mm² wire carrying 10 A has current density J = 10 / (1×10⁻⁶) = 1×10⁷ A/m². Then:

v_d = J / (n e)
    = 1e7 / (8.5e28 × 1.6e-19)
    ≈ 7.4e-4 m/s  ≈ 0.74 mm/s

Less than a millimeter per second. The electrons crawl. What travels near the speed of light down the wire is the electric field / signal, not the electrons themselves — a counter-intuitive fact that trips up most students. Multiply tau by the Fermi speed (~1.6×10⁶ m/s for copper) and you get a mean free path of about 40 nm — the electron flies past roughly a hundred ion spacings before scattering.

Variants and regimes

The bare Drude model is the start of a family of refinements:

Model / regime	What it adds	Key result	When it matters
Drude DC	Static field only	σ = n e² τ / m	DC resistivity, room-temperature metals
Drude AC	Oscillating field E(ω)	σ(ω) = σ₀ / (1 + iωτ)	Optical / microwave response, plasma frequency
Drude + magnetic field	Lorentz force term	Hall coefficient R_H = −1/(n e)	Measuring carrier density and sign
Drude thermal	Electrons carry heat too	Wiedemann–Franz: κ/σT ≈ const	Why good electrical conductors conduct heat
Sommerfeld (1927)	Fermi–Dirac statistics	Correct electronic specific heat	Low-temperature heat capacity, thermopower
Band theory	Periodic lattice potential	Effective mass m*, holes, insulators	Semiconductors, wrong-sign Hall metals

The AC version is especially elegant. At low frequency the metal looks like a resistor; above the plasma frequency the electrons can no longer keep up with the field, the conductivity becomes imaginary, and the metal turns transparent — which is exactly why metals are shiny in visible light but transparent to ultraviolet.

Resistivity versus temperature

Resistivity is rho = m / (n e² tau). Since n and m barely change with temperature in a metal, the temperature dependence lives entirely in tau. Heat the metal and the ions vibrate harder — more phonons — so a drifting electron scatters more often and tau shrinks. Above the Debye temperature the phonon count grows linearly with T, so tau ∝ 1/T and:

ρ(T) ≈ ρ₀ + a·T      (linear rise above ~Debye temperature)

The constant rho₀ is the residual resistivity from scattering off impurities and defects — it survives even at absolute zero, when phonons freeze out. This is Matthiessen's rule: scattering rates add, so 1/tau_total = 1/tau_phonon + 1/tau_impurity, and the two contributions simply sum in the resistivity. The interactive visualization shows this directly with a resistivity-vs-temperature inset: turn up the temperature and watch the ions jitter, the scattering rate climb, and the rho(T) curve rise.

Common pitfalls and misconceptions

• "Electrons travel near light speed down a wire." No — the drift velocity is under a millimeter per second. The field propagates fast; the electrons barely move. Confusing the two is the single most common error.
• "Tau is the time between hitting ions." Subtle but important: tau is the momentum-relaxation time, the time over which directed velocity is randomized, not literally the time between geometric collisions with ions. In a perfect lattice an electron would not scatter at all — it is phonons and defects that cause scattering, a fact the classical model glosses over.
• "The Drude model is a quantum theory." It is purely classical — Maxwell–Boltzmann statistics, billiard-ball collisions. Its successes are partly luck (errors in electron speed and heat capacity cancel in some formulas).
• Predicting a giant electronic specific heat. Classically every electron contributes (3/2)k_B, which would dominate a metal's heat capacity. Experiment sees roughly 1% of that. This specific-heat catastrophe is the model's most famous failure, fixed only by Fermi–Dirac statistics.
• Assuming the Hall sign is always negative. Drude predicts R_H = −1/(n e), always negative, yet aluminum and beryllium behave as if carriers were positive. Holes and band structure are needed.
• Treating sigma = n e² tau / m as exact. It is an order-of-magnitude estimator. The right m is the effective mass m*, and the right statistics are quantum — but the formula's scaling with n and tau is correct and indispensable.

Applications

• Estimating metallic conductivity. Plug in n and tau to get sigma to within a factor of a few for any simple metal — the back-of-envelope tool every condensed-matter student learns first.
• Hall-effect carrier counting. Measuring R_H = −1/(n e) directly returns the carrier density and sign — the workhorse of semiconductor characterization.
• Optical properties of metals. The AC Drude conductivity and plasma frequency explain why metals reflect visible light, why thin gold films look greenish in transmission, and underpins plasmonics and metamaterials.
• Thermal management. The Wiedemann–Franz law (kappa/sigma·T constant) tells engineers that good electrical conductors are also good heat conductors — central to heat-sink and power-electronics design.
• Skin effect and high-frequency design. The frequency-dependent conductivity sets how deeply AC currents penetrate a conductor, shaping the design of antennas, transmission lines, and inductors.
• Teaching springboard to quantum theory. Every modern treatment of semiconductors, the Sommerfeld model, and band theory starts by fixing the Drude model's failures — making it the most pedagogically valuable wrong theory in physics.

Performance and derivation analysis

Why does such a crude classical model work at all? Two big errors quietly cancel. The Drude model badly overestimates the contribution of electrons to heat capacity (by ~100×) and badly underestimates their typical speed (it uses the thermal speed ~10⁵ m/s instead of the much larger Fermi speed ~10⁶ m/s). In the conductivity formula these two mistakes never both appear, so sigma = n e² tau / m survives the upgrade to quantum mechanics essentially unchanged — only the interpretation of tau and the relevant speed shift.

The Sommerfeld correction (1927) keeps the entire Drude framework but swaps Maxwell–Boltzmann statistics for Fermi–Dirac. The consequence is profound: only electrons within about k_B·T of the Fermi energy can scatter into empty states, so only a thin shell of electrons — not all of them — participates in transport and heat capacity. This single change fixes the specific-heat catastrophe (giving the correct linear-in-T electronic heat capacity) while leaving the conductivity formula standing. The mean free path becomes the Fermi velocity times tau, and tau is reinterpreted as the time between scattering events off phonons and defects rather than off the ions themselves.

// Drude conductivity and drift velocity — order-of-magnitude estimator
const e = 1.602e-19;   // C
const m = 9.109e-31;   // kg (use effective mass m* for real materials)

function drudeConductivity(n, tau) {
  return (n * e * e * tau) / m;   // S/m
}

function driftVelocity(J, n) {
  return J / (n * e);             // m/s
}

const nCu = 8.5e28;    // copper carriers per m^3
const tauCu = 2.5e-14; // relaxation time, s

const sigma = drudeConductivity(nCu, tauCu);
console.log('sigma  =', sigma.toExponential(2), 'S/m');      // ~6.0e7
console.log('rho    =', (1 / sigma).toExponential(2), 'ohm.m'); // ~1.7e-8

// 10 A through a 1 mm^2 wire => J = 1e7 A/m^2
const vDrift = driftVelocity(1e7, nCu);
console.log('v_drift =', (vDrift * 1000).toFixed(3), 'mm/s'); // ~0.74 mm/s

// Recover tau from a measured conductivity
function relaxationTime(sigmaMeasured, n) {
  return (sigmaMeasured * m) / (n * e * e);
}
console.log('tau    =', relaxationTime(5.96e7, nCu).toExponential(2), 's'); // ~2.5e-14

The take-away: the Drude model is computationally trivial, dimensionally honest, and right about scaling even when it is wrong about magnitudes. That is the hallmark of a great physical model — it captures the essential mechanism (free carriers, scattering, drift) in two parameters you can measure, and its failures are signposts pointing precisely at the quantum mechanics that completes the story.