Drude model

The Drude model of electrical conduction in metals was proposed in 1900 by Paul Drude. The Drude model attempts to explain conduction 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 is an application of kinetic theory. It assumes that when electrons in a solid are exposed to the electric field, they behave much like a pinball machine. The sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions produce a net collective motion in the direction opposite to the applied electric field.

In modern terms this is reflected in the valence electron model where the sea of electrons is composed of the valence electrons only, 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. 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.

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

The model gives better predictions for metals, especially in regards to conductivity, 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.

The two most significant results of the Drude model are an electronic equation of motion, $${\displaystyle {\frac {d}{dt}}\langle \mathbf {p} (t)\rangle =q\left(\mathbf {E} +{\frac {\langle \mathbf {p} (t)\rangle }{m}}\times \mathbf {B} \right)-{\frac {\langle \mathbf {p} (t)\rangle }{\tau }},}$$ and a linear relationship between current density J and electric field E, $${\displaystyle \mathbf {J} ={\frac {nq^{2}\tau }{m}}\,\mathbf {E} .}$$

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.