Drude model

# Drude model

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The Drude model of electrical conduction was proposed in 1900 by Paul Drude
Paul Karl Ludwig Drude
Paul Karl Ludwig Drude was a German physicist specializing in optics. He wrote a fundamental textbook integrating optics with Maxwell's theories of electromagnetism.- Education :...

to explain the transport properties of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s in materials (especially metals). The model, which is an application of kinetic theory
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

, assumes that the microscopic behavior of electrons in a solid may be treated classically and looks much like a pinball
Pinball
Pinball is a type of arcade game, usually coin-operated, where a player attempts to score points by manipulating one or more metal balls on a playfield inside a glass-covered case called a pinball machine. The primary objective of the game is to score as many points as possible...

machine, with a sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions.

The two most significant results of the Drude model are an electronic equation of motion,
and a linear relationship between current density
Current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...

and electric field ,
Here is the time and , , and , and are respectively an electron's momentum, 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
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

, one of the most ubiquitous relationships in all of electromagnetism, should be true.

The model was extended in 1905 by Hendrik Antoon Lorentz (and hence is also known as the Drude–Lorentz model) and was supplemented with the results of quantum theory in 1933 by Arnold Sommerfeld
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...

and Hans Bethe
Hans Bethe
Hans Albrecht Bethe was a German-American nuclear physicist, and Nobel laureate in physics for his work on the theory of stellar nucleosynthesis. A versatile theoretical physicist, Bethe also made important contributions to quantum electrodynamics, nuclear physics, solid-state physics and...

.

## Assumptions

The Drude model considers the metal to be formed of a mass of positively-charged ions from which a number of "free electrons" were detached. These may be thought to have become delocalized when the valence levels of the atom came in contact with the potential of the other atoms.

Notwithstanding the origin of such free electrons from the contact of different potentials, the Drude model neglects any long-range interaction between the electron and the ions and assumes that the electrons do not interfere with each other. The only possible interaction is the instantaneous collision between a free electron and an ion, which happens with a fixed probability per unit time.

The Drude model is a purely classical model, and treats both electrons and ions as solid spheres.

### 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 traveling 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
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

mentioned above:

### Time-varying analysis

The dynamics may also be described by introducing an effective drag force. At time the average electron's momentum will be
because, on average, electrons will not have experienced another collision, and the ones that have will contribute to the total momentum to only a negligible order.

With a bit of algebra, this results in the differential equation

where denotes average
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....

momentum, m the effective mass and q the charge of the electrons. This, which is an inhomogeneous differential equation, may be solved to obtain the general solution of
A system in a steady state has numerous properties that are unchanging in time. This implies that for any property p of the system, the partial derivative with respect to time is zero:...

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
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

with a DC
Direct current
Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...

-conductivity :

The Drude model can also predict the current as a response to a time-dependent electric field with an angular frequency , in which case

Here it is assumed that

In other conventions, is replaced by in all equations. The imaginary part indicates that the current lags behind the electrical field, which happens because the electrons need roughly a time to accelerate in response to a change in the electrical field. Here the Drude model is applied to electrons; it can be applied both to electrons and holes; i.e., positive charge carriers in semiconductors. The curves for are shown in the graph.

## Accuracy of the model

This simple classical Drude model provides a very good explanation of DC and AC conductivity in metal
Metal
A metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...

s, the Hall effect
Hall effect
The Hall effect is the production of a voltage difference across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current...

, and thermal conductivity
Thermal conductivity
In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....

(due to electrons) in metals near room temperature. The model also explains the Wiedemann-Franz law of 1853. However, it greatly overestimates the electronic heat capacities of metals. In reality, metals and insulators have roughly the same heat capacity at room temperature. Although the model can be applied to positive (hole) charge carriers, as demonstrated by the Hall effect, it does not predict their existence.

One note of trivia surrounding the theory is that in his original paper Drude made a conceptual error, estimating electrical conductivity to in fact be only half of what it classically should have been.

• Free electron model
Free electron model
In solid-state physics, the free electron model is a simple model for the behaviour of valence electrons in a crystal structure of a metallic solid. It was developed principally by Arnold Sommerfeld who combined the classical Drude model with quantum mechanical Fermi-Dirac statistics and hence it...

• Arnold Sommerfeld
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...

• Classical and quantum conductivity
Classical and quantum conductivity
Classical and quantum mechanical views of conductivity have both described the movements of electrons in a metallic solid. The free electron gas that is present in metallic solids is the reason for an important property of all metals: conductivity...

• Electrical conductivity