Overview

• Concise summary of charge and heat transport in metals using the Drude and Sommerfeld pictures.
• Key relations: conductivity, relaxation time, mean free path, Wiedemann–Franz law.

Drude model — electrical conductivity

• Drude formula for DC electrical conductivity:

\(\(\sigma = \frac{n e^2 \tau}{m}\)\)

where n = carrier density, e = electron charge, \tau = relaxation time, m = electron mass. - Interpretation: conductivity grows with more carriers, longer scattering time, and lower effective mass.

Relaxation time (\tau)

• Solve Drude formula for \tau:

\(\(\tau = \frac{m\,\sigma}{n e^2}\)\) - Physical meaning: average time between momentum-relaxing scattering events. - Typical scattering mechanisms: phonons (dominant at high T), impurities, defects, grain boundaries.

Mean free path (\ell)

• Definition in transport theory:

\(\(\ell = v\tau\)\)

where v is the characteristic electron speed used for transport.

Which velocity to use

• Classical Drude (thermal classical gas): use root-mean-square speed

\(\(v_{\mathrm{rms}} = \sqrt{\frac{3 k_B T}{m}}\)\)

• Sommerfeld (quantum, low T) model: use the Fermi velocity

\(\(v = v_F\)\)

where \(v_F\) is set by the Fermi energy; thermal excitations occur near the Fermi surface.

Why a large mean free path is surprising

• In many metals measured \ell spans several lattice constants, implying electrons travel many unit cells before scattering.
• A perfect periodic lattice would not scatter Bloch states; observed resistance requires deviations from periodicity.

Scattering mechanisms that set \tau

• Phonon scattering: increases with temperature; dominant at higher T.
• Impurities and defects: temperature-independent (residual scattering at low T).
• Grain boundaries and microstructure: can shorten \ell independent of intrinsic scattering.

Wiedemann–Franz law (electrical ↔ thermal transport)

• Relates electronic thermal conductivity \(\kappa\) to electrical conductivity \(\sigma\):

\(\(\kappa = L\,T\,\sigma\)\)

where \(L\) is the Lorenz number. - Experimental Lorenz number (metals, low T limit):

\(\(L \approx 2.44\times 10^{-8}\ \mathrm{W\,\Omega\,K^{-2}}\)\)

When the law fails

• At intermediate temperatures scattering becomes energy-dependent, so heat and charge currents weigh electron energies differently.
• Electron–phonon scattering and inelastic processes break the simple proportionality between \(\kappa\) and \(\sigma\).

Temperature dependence of transport (qualitative)

• As temperature increases:
• \tau decreases (more phonon scattering).
• \sigma decreases (lower electrical conductivity).
• \ell decreases (shorter mean free path).
• Resistivity (\rho) generally increases.

Perfect lattice and Bloch waves

• In a perfectly periodic potential electrons form Bloch waves which do not decay; an ideal lattice gives no resistance.
• Only deviations from periodicity (phonons, impurities, defects) produce momentum-relaxing scattering and finite resistivity.

Quick practical notes

• To estimate \ell from measurements: measure \(\sigma\), compute \(\tau\) from \(\tau=\tfrac{m\sigma}{ne^2}\), then use appropriate \(v\) to get \(\ell=v\tau\).
• Use \(v_{\mathrm{rms}}\) at high T or when treating electrons classically; use \(v_F\) for low-temperature, quantum-degenerate electrons.

Common tags / topics to review

• Drude model, Sommerfeld model, relaxation time, mean free path, Wiedemann–Franz law, Lorenz number, scattering mechanisms