Drude model

The Drude model of electrical conduction was proposed in 1900^[1][2] by Paul Drude to explain the transport properties of electrons in materials (especially metals). Basically, Ohm's law was well established and stated that the current J and voltage V driving the current are related to the resistance R of the material. The inverse of the resistance is known as the conductance. When we consider a metal of unit length and unit cross sectional area, the conductance is known as the conductivity, which is the inverse of resistivity. The Drude model attempts to explain the resistivity of a conductor in terms of the scattering of electrons (the carriers of electricity) by the relatively immobile ions in the metal that act like obstructions to the flow of electrons.

The model, which is an application of kinetic theory, assumes that the microscopic behaviour of electrons in a solid may be treated classically and behaves much like a pinball machine, with a sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions.

In modern terms this is reflected in the valence electron model where the sea of electrons is composed of the valence electrons only,^[3] and not the full set of electrons available in the solid, and the scattering centers are the inner shells of tightly bound electrons to the nucleus. The scattering centers had a positive charge equivalent to the valence number of the atoms.^{[Ashcroft & Mermin 1]} This similarity added to some computation errors in the Drude paper, ended up providing a reasonable qualitative theory of solids capable of making good predictions in certain cases and giving completely wrong results in others. Whenever people tried to give more substance and detail to the nature of the scattering centers, and the mechanics of scattering, and the meaning of the length of scattering, all these attempts ended in failures.^{[Ashcroft & Mermin 2]}

The scattering lengths computed in the Drude model, are of the order of 10 to 100 inter-atomic distances, and also these could not be given proper microscopic explanations.

Drude scattering is not electron-electron scattering which is only a secondary phenomenon in the modern theory, neither nuclear scattering given electrons can be at most be absorbed by nuclei. The model remains a bit mute on the microscopic mechanisms, in modern terms this is what is now called the "primary scattering mechanism" where the underlying phenomenon can be different case per case.^{[Ashcroft & Mermin 3]}

The model gives better predictions for metals, especially in regards to conductivity,^{[Ashcroft & Mermin 4]} and sometimes is called Drude theory of metals. This is because metals have essentially a better approximation to the free electron model, i.e. metals do not have complex band structures, electrons behave essentially as free particles and where, in the case of metals, the effective number of de-localized electrons is essentially the same as the valence number.^{[Ashcroft & Mermin 5]}

The two most significant results of the Drude model are an electronic equation of motion,

$${\displaystyle {\frac {d}{dt}}\langle \mathbf {p} (t)\rangle =q\left(\mathbf {E} +{\frac {\langle \mathbf {p} (t)\rangle }{m}}\times \mathbf {B} \right)-{\frac {\langle \mathbf {p} (t)\rangle }{\tau }},}$$

$${\displaystyle \mathbf {J} ={\frac {nq^{2}\tau }{m}}\,\mathbf {E} .}$$

Here t is the time, ⟨p⟩ is the average momentum per electron and q, n, m, and τ are respectively the electron charge, number density, mass, and mean free time between ionic collisions. The latter expression is particularly important because it explains in semi-quantitative terms why Ohm's law, one of the most ubiquitous relationships in all of electromagnetism, should hold.^{[Ashcroft & Mermin 6][4][5]}

Steps towards a more modern theory of solids were given by the following:

• The Einstein solid model and the Debye model, suggesting that the quantum behaviour of exchanging energy in integral units or quanta was an essential component in the full theory especially with regard to specific heats, where the Drude theory failed.
• In some cases, namely in the Hall effect, the theory was making correct predictions if instead of using a negative charge for the electrons a positive one was used. This is now interpreted as holes (i.e. quasi-particles that behave as positive charge carriers) but at the time of Drude it was rather obscure why this was the case.^{[Ashcroft & Mermin 7]}

Drude used Maxwell–Boltzmann statistics for the gas of electrons and for deriving the model, which was the only one available at that time. By replacing the statistics with the correct Fermi Dirac statistics, Sommerfeld significantly improved the predictions of the model, although still having a semi-classical theory that could not predict all results of the modern quantum theory of solids.^{[Ashcroft & Mermin 8]}

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