Drude model
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.
[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.
[Ashcroft & Mermin 2] 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.
[Ashcroft & Mermin 5] The two most significant results of the Drude model are an electronic equation of motion,
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: Drude used Maxwell–Boltzmann statistics for the gas of electrons and for deriving the model, which was the only one available at that time.
[6] In his original paper, Drude made an error, estimating the Lorenz number of Wiedemann–Franz law to be twice what it classically should have been, thus making it seem in agreement with the experimental value of the specific heat.
Later it was supplemented with the results of quantum theory in 1933 by Arnold Sommerfeld and Hans Bethe, leading to the Drude–Sommerfeld model.
[Ashcroft & Mermin 11] This is a generic method in solid state physics, where it is typical to incrementally increase the complexity of the models to give more and more accurate predictions.
It is less common to use a full-blown quantum field theory from first principles, given the complexities due to the huge numbers of particles and interactions and the little added value of the extra mathematics involved (considering the incremental gain in numerical precision of the predictions).
[8] Drude used the kinetic theory of gases applied to the gas of electrons moving on a fixed background of "ions"; this is in contrast with the usual way of applying the theory of gases as a neutral diluted gas with no background.
[Ashcroft & Mermin 12] The core assumptions made in the Drude model are the following: Removing or improving upon each of these assumptions gives more refined models, that can more accurately describe different solids: The simplest analysis of the Drude model assumes that electric field E is both uniform and constant, and that the thermal velocity of electrons is sufficiently high such that they accumulate only an infinitesimal amount of momentum dp between collisions, which occur on average every τ seconds.
The Drude model can also predict the current as a response to a time-dependent electric field with an angular frequency ω.
In engineering, i is generally replaced by −i (or −j) in all equations, which reflects the phase difference with respect to origin, rather than delay at the observation point traveling in time.
This happens because the electrons need roughly a time τ to accelerate in response to a change in the electrical field.
included above The following are Maxwell's equations without sources (which are treated separately in the scope of plasma oscillations), in Gaussian units:
The plasma frequency can be employed as a direct measure of the square root of the density of valence electrons in a solid.
Observed values are in reasonable agreement with this theoretical prediction for a large number of materials.
Above the plasma frequency the light waves can penetrate the sample, a typical example are alkaline metals that becomes transparent in the range of ultraviolet radiation.
[Ashcroft & Mermin 17] One great success of the Drude model is the explanation of the Wiedemann-Franz law.
[Ashcroft & Mermin 18] Solids can conduct heat through the motion of electrons, atoms, and ions.
(This derivation ignores the temperature-dependence, and hence the position-dependence, of the speed v. This will not introduce a significant error unless the temperature changes rapidly over a distance comparable to the mean free path.)
In addition to these two estimates, Drude also made a statistical error and overestimated the mean time between collisions by a factor of 2.
The correct value of the Lorenz number as estimated from the Drude model is[Ashcroft & Mermin 19]
Drude formula is derived in a limited way, namely by assuming that the charge carriers form a classical ideal gas.
The conductivity predicted is the same as in the Drude model because it does not depend on the form of the electronic speed distribution.
Also, the Drude model does not explain the scattered trend of electrical conductivity versus frequency above roughly 2 THz.
In a conventional, simple, real metal (e.g. sodium, silver, or gold at room temperature) such behavior is not found experimentally, because the characteristic frequency τ−1 is in the infrared frequency range, where other features that are not considered in the Drude model (such as band structure) play an important role.
[12] But for certain other materials with metallic properties, frequency-dependent conductivity was found that closely follows the simple Drude prediction for σ(ω).
[12] This is the case for certain doped semiconductor single crystals,[14] high-mobility two-dimensional electron gases,[15] and heavy-fermion metals.
