4.10: The Drude or Semi-Classical Model of Charge Transport

Quantum models of charge conduction are rarely applied outside nanoelectronics. For traditional applications, the semi-classical model of the German physicist Paul Drude is usually sufficient. Drude proposed that conductors contain immobile positive ions embedded in a sea of electrons. Unlike the quantum view, where those electrons occupy various states with different energies, Drude viewed electrons as indistinguishable.

In the quantum model of charge transport, current is carried by only that fraction of electrons close to the Fermi energy. The current carrying electrons move at approximately the Fermi velocity, \(v_{F} = \hbar k_{F}/m\). The remaining electrons are compensated, i.e. equal numbers flow in each direction yielding no net current.

Figure \(\PageIndex{1}\): Application of an electric field shifts the quasi Fermi levels for electrons moving with the field (\(F^{+}\)) and against the field (\(F^{-}\)).

But in the Drude model, current is carried by all electrons, moving at an average velocity known as the drift velocity, \(\textbf{v}_{d}\). Thus, the fundamental classical model for charge conduction is

\[ \textbf{J}= qn\textbf{v}_{d} \nonumber \]

where n is the density of electrons.

In the Drude model, all the electrons travel in the direction of the electric field, gathering energy from the field. Eventually each electron collides with something, a positive ion or another electron, at which point, the electron is stopped. It is then accelerated once more by the electric field, traveling in this stop-start manner through the conductor.

Figure \(\PageIndex{2}\): Electron paths through scattering sites. The average time between collisions is the relaxation time, \(\tau_{m}\).

The conductivity of the material is characterized by \(\tau_{m}\), the relaxation time, the mean time between collisions.

The rate at which electrons gain momentum from the field, \(\epsilon\) must be equal to the rate of losses due to scattering:\(^{†}\)

\[ \left. \frac{d\textbf{p}}{dt} \right|_{\text{scattering}} = \left. \frac{d\textbf{p}}{dt} \right|_{\text{field}} \nonumber \]

\[ \frac{mv_{d}}{\tau_{m}} = q\epsilon , \nonumber \]

Rearranging Equation (4.11.3), we can express the drift velocity and current density in terms of the relaxation time.

\[ \textbf{J} = \frac{nq^{2}\tau_{m}}{m}\epsilon \nonumber \]

Comparing to Ohm's law (expressed in terms of the conductivity, \(\sigma = 1/\rho\))

\[ \textbf{J} = \sigma \epsilon \nonumber \]

where

\[ \sigma = \frac{nq^{2}\tau_{m}}{m} \nonumber \]

\(^{†}\)This derivation follows Ashcroft and Mermin, "Solid State Physics", Saunders College Publishing (1976).