Drude-Sommerfeld Model

Electronic conduction with scattering, degenerate fermions, and Sommerfeld corrections.

Conceptual context

The Drude-Sommerfeld model joins two layers of thought. Drude treats conduction electrons as carriers accelerated by an electric $E$ and slowed by scattering. Sommerfeld replaces the classical gas by a degenerate Fermi gas: electrons fill quantum states up to the $E_F$, and transport is governed mainly by states near that boundary. The material is not a decorative label. It enters through $n$, $m*$, scattering, band structure, and the way the electronic distribution can be slightly shifted by an external field.

🟢 Essential level

The central question is why a metal conducts strongly, weakly, or almost not at all when an electric $E$ is applied.

The model answers by separating three ingredients. First, $n$ counts how many carriers are available. Second, $\tau$ measures how long electronic momentum survives before scattering with phonons, impurities, or defects erases it. Third, $m*$ translates band inertia inside the crystal. A large $\sigma$ does not mean each electron rushes through the wire. It means that many electrons acquire a tiny collective $v_d$. Sommerfeld adds the quantum lock: the electron gas is degenerate, so $E_F$ and $v_F$, not ordinary thermal agitation, set the internal scale. The Fermi surface becomes the narrow gate through which transport changes and responds measurably.

Paul Drude proposed the classical model in 1900, barely three years after the electron was discovered, treating carriers as a classical gas of particles colliding with the lattice. Arnold Sommerfeld corrected that picture in 1927 by applying Fermi-Dirac statistics, resolving in one stroke several discrepancies of the original model, such as the electronic contribution to heat capacity.

🔵 Formal level

Basic formulas

The central Drude-Sommerfeld relation for electrical transport uses $n$, $e$, $\tau$, and $m*$ to determine $\sigma$. It packs the microscopic reading into one coefficient: more carriers and longer momentum memory increase response, while larger band inertia reduces it. The relation is scalar for an isotropic simple metal.

Drude conductivity

In an isotropic metal, that $\sigma$ connects $j$ and $E$ through the microscopic Ohm law. This is a local linear law, not a statement about the whole wire geometry.

Microscopic Ohm law

The equivalent opposition is $\rho$, useful when the problem is framed through losses or four-probe measurements. Low $\rho$ and high $\sigma$ express the same material response from opposite sides.

Resistivity as inverse conductivity

Real student questions

Why does $e$ appear squared if electrons are negatively charged? Because $\sigma$ uses the positive magnitude of charge; the sign appears in $v_d$ and $R_H$.

Does larger $\tau$ always improve transport? In the linear model yes, but real materials may also change band structure, $T$, or disorder.

Is $E_F$ used directly to compute current? Not always. It justifies degeneracy and helps estimate $v_F$ and $ℓ$.

Why is $v_d$ so small? Because current is a tiny asymmetry built on top of large quantum velocities pointing in many directions.

Final synthesis

Drude-Sommerfeld explains conduction as a competition between electrical acceleration, carrier density, band inertia, and loss of memory by scattering. Its power is turning microscopic material parameters into measurable $\sigma$; its limit is everything that refuses to fit into one relaxing band.