Free-electron Drude model: scattering time τ, mobility μ, σ(ω), mean free path ℓ = v_Fτ, skin depth δ(ω), and DC Hall σ_xx(B), σ_xy(B).
About this model
The Drude–Sommerfeld model describes electrical transport in metals as free electrons accelerated by electric fields and relaxed by a phenomenological scattering time τ. In the relaxation-time approximation, the DC conductivity is σ₀ = ne²τ/m, the mobility is μ = eτ/m, and the mean free path is ℓ = v_F τ using the Fermi velocity v_F. The complex AC conductivity σ(ω) = σ₀/(1 − iωτ) predicts a Lorentzian roll-off of Re σ at ω ∼ 1/τ and a peak in Im σ — the origin of frequency-dependent AC resistivity and the skin effect, where the skin depth δ = √(2/(ωμ₀ Re σ)) shrinks as conductivity increases at low frequency. In a perpendicular magnetic field, the Drude tensor gives σ_xx(B) = σ₀/(1 + (ω_cτ)²) and σ_xy(B) = σ₀ ω_cτ/(1 + (ω_cτ)²) with cyclotron frequency ω_c = eB/m, connecting to the Hall effect and magnetoresistance. This simulator plots σ(ω), Hall components versus B, δ(ω), and a schematic mean-free-path bar. Pedagogical units use e = m = 1; skin depth maps model σ to a copper-scale S/m value and maps one plotted ω unit to 2π MHz for micrometre readouts comparable to the electricity/skin-effect page.
Who it's for: Advanced undergraduate solid-state or introductory condensed-matter students after free-electron theory and before band-structure transport or the full Boltzmann equation.
Key terms
Drude model
Sommerfeld theory
Scattering time
Mobility
AC conductivity
Mean free path
Skin depth
Hall effect
How it works
Drude–Sommerfeld transport: scattering time τ, mobility μ, AC conductivity σ(ω), mean free path ℓ = v_Fτ, skin depth link, and DC Hall σ_xx(B), σ_xy(B).
Frequently asked questions
What does τ represent?
τ is the average time between electron collisions with impurities, phonons, or defects. Shorter τ means lower mobility and higher resistivity.
Why does Re σ(ω) decrease at high frequency?
Electrons cannot follow the oscillating field fast enough when ωτ ≳ 1, so they do not fully respond and the in-phase (dissipative) conductivity falls — AC resistivity rises.
How does this relate to the skin effect?
High-frequency currents penetrate only a depth δ set by electromagnetic diffusion into a conductor with conductivity Re σ(ω). The Drude roll-off of σ at high ω is part of why δ depends on frequency.
What is the Hall signature in Drude theory?
A transverse σ_xy builds up proportional to B until ω_cτ is large, while σ_xx is suppressed — the standard Hall bar picture before quantum Hall plateaus.