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 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,
and a linear relationship between current density and electric field,
Here is the time, ⟨p⟩ is the average momentum per electron and, and 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.

History

German physicist Paul Drude proposed his model in 1900. He was inspired by the discovery of electrons in 1897 by J.J. Thomson. He assumed a simplistic model of the solid: positively charged scattering centers and a gas of electrons, giving an electrically neutral solid. The model was extended in 1905 by Hendrik Antoon Lorentz to give the relation between the thermal conductivity and the electric conductivity of metals, and is a classical model.
Drude also presented his work in the 1900 International Congress of Physics during the Exposition Universelle in Paris. During the congress he defended the existence of electrons which was still under debate.
While Drude's model for conductivity is still useful, his calculation of the specific heat capacity of a metal, the amount of energy needed to increase the temperature by one degree is not. In his calculation, 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. This number is about 100 times smaller than the classical prediction but this factor cancels out with the mean electronic speed that is about 100 times bigger than Drude's calculation. It is now known that the electrons in a metal make no contribution to specific heat for temperatures around room temperature.
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 created the free electron model, significantly improving the predictions while still having a semi-classical theory that could not predict all results of the modern quantum theory of solids.
Nowadays the Drude and Sommerfeld models are still significant to understanding the qualitative behaviour of solids and to get a first qualitative understanding of a specific experimental setup. 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.

Assumptions

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. The number density of the electron gas was assumed to be
where Z is the effective number of de-localized electrons per ion, for which Drude used the valence number, A is the atomic mass per mole, is the mass density of the "ions", and N is the Avogadro constant.
Considering the average volume available per electron as a sphere:
The quantity is a parameter that describes the electron density and is often of the order of 2 or 3 times the Bohr radius, for alkali metals it ranges from 3 to 6 and some metal compounds it can go up to 10.
The densities are of the order of 1000 times of a typical classical gas.
The core assumptions made in the Drude model are the following:
  • Drude applied the kinetic theory of a dilute gas, despite the high densities, therefore ignoring electron–electron and electron–ion interactions aside from collisions.
  • The Drude model considers the metal to be formed of a collection of positively charged ions from which a number of "free electrons" were detached. These may be thought to be the valence electrons of the atoms that have become delocalized due to the electric field of the other atoms.
  • The Drude model neglects long-range interaction between the electron and the ions or between the electrons; this is called the independent electron approximation.
  • The electrons move in straight lines between one collision and another; this is called free electron approximation.
  • The only interaction of a free electron with its environment was treated as being collisions with the impenetrable ions core.
  • The average time between subsequent collisions of such an electron is, with a memoryless Poisson distribution. The nature of the collision partner of the electron does not matter for the calculations and conclusions of the Drude model.
  • After a collision event, the distribution of the velocity and direction of an electron is determined by only the local temperature and is independent of the velocity of the electron before the collision event. The electron is considered to be immediately at equilibrium with the local temperature after a collision.
Removing or improving upon each of these assumptions gives more refined models, that can more accurately describe different solids:
  • Improving the hypothesis of the Maxwell–Boltzmann statistics with the Fermi–Dirac statistics leads to the Drude–Sommerfeld model.
  • Improving the hypothesis of the Maxwell–Boltzmann statistics with the Bose–Einstein statistics leads to considerations about the specific heat of integer spin atoms and to the Bose–Einstein condensate.
  • A valence band electron in a semiconductor is still essentially a free electron in a delimited energy range ; the independent electron approximation is essentially still valid, where instead the hypothesis about the localization of the scattering events is dropped.

    Mathematical treatment

DC field

The simplest analysis of the Drude model assumes that electric field 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 between collisions, which occur on average every seconds.
Then an electron isolated at time will on average have been travelling for time since its last collision, and consequently will have accumulated momentum
During its last collision, this electron will have been just as likely to have bounced forward as backward, so all prior contributions to the electron's momentum may be ignored, resulting in the expression
Substituting the relations
results in the formulation of Ohm's law mentioned above:

Time-varying analysis

The dynamics may also be described by introducing an effective drag force. At time the electron's momentum will be:
where can be interpreted as generic force on the carrier or more specifically on the electron. is the momentum of the carrier with random direction after the collision and with absolute kinetic energy
On average, a fraction of of the electrons will not have experienced another collision, the other fraction that had the collision on average will come out in a random direction and will contribute to the total momentum to only a factor which is of second order.
With a bit of algebra and dropping terms of order, this results in the generic differential equation
The second term is actually an extra drag force or damping term due to the Drude effects.

Constant electric field

At time the average electron's momentum will be
and then
where denotes average momentum and the charge of the electrons. This, which is an inhomogeneous differential equation, may be solved to obtain the general solution of
for. The steady state solution,, is then
As above, average momentum may be related to average velocity and this in turn may be related to current density,
and the material can be shown to satisfy Ohm's law with a DC-conductivity :