# Drude theory

Schematic representation of
the movement of electrons (blue)
in a crystal lattice  (red)
according to the Drude theory,
with (explanations in the text):
v d : drift speed of the electrons
E: direction of the electric field
I: direction of the electric current

The Drude theory (also Drude model , after Paul Drude , published around 1900) is a classic description of the charge transport through an external electric field in metals or, more generalized, through free electrons in solids . When considering alternating electric fields (including light ), the term Drude-Zener theory or model (after Clarence Melvin Zener ) is also used.

With the Drude model, Ohm's law could be explained for the first time , even if the resistance value calculated with this model is about six times greater than the true (measured) resistance value of the respective material. The reason for this is that more electrons are actually available due to quantum statistical processes, since the Fermi energy is reached.

The Drude theory was expanded in 1905 by Hendrik Antoon Lorentz and in 1933 by Arnold Sommerfeld and Hans Bethe the results of quantum mechanics were added.

## description

In the Drude model, an electrical conductor is viewed as an ion crystal in which the electrons can move freely, form an electron gas and are thus responsible for conducting electricity. The term electron gas comes from the similarity between this theory and the kinetic gas theory : if there is no electric field inside the conductor , the electrons behave like gas particles in a container.

By an external electric field , the free electrons in the conductor experience a force effective and accelerated , but not continuously. If this were so, the resistance and the current strength should not be constant and Ohm's law would therefore not apply. After a short time, however, an equilibrium is established in which the mean speed of the electron and thus the electric current is proportional to the field strength . ${\ displaystyle {\ vec {E}}}$${\ displaystyle F _ {\ mathrm {el}} = q \ cdot E}$

This is explained by the Drude model by the fact that the electron collides with a lattice ion and is decelerated. This process is phenomenologically described by an average rush time${\ displaystyle \ tau}$ between two collisions . As the temperature rises , the mean peak time and thus also the electrical conductivity of the metals decrease .

The equation of motion for this is:

${\ displaystyle m {\ dot {v}} + {\ frac {m} {\ tau}} v _ {\ mathrm {D}} = - eE}$

With

• ${\ displaystyle m}$ the electron mass
• ${\ displaystyle v}$ the electron speed
• ${\ displaystyle v _ {\ mathrm {D}}}$the drift speed (e-speed minus the thermal speed) and
• ${\ displaystyle \ tau}$ the rush hour
• ${\ displaystyle e}$the elementary charge .

For the steady state ( ) the following applies: ${\ displaystyle {\ dot {v}} = 0}$

${\ displaystyle \ Rightarrow v _ {\ mathrm {D}} = - {\ frac {e \ cdot \ tau} {m}} E}$

With the charge carrier density , the current density results in : ${\ displaystyle n}$ ${\ displaystyle j}$

${\ displaystyle j = -e \ cdot n \ cdot v _ {\ mathrm {D}} = {\ frac {e ^ {2} \ cdot \ tau \ cdot n} {m}} E}$

The conductivity is therefore: ${\ displaystyle \ sigma}$

${\ displaystyle \ sigma = {\ frac {j} {E}} = {\ frac {e ^ {2} \ cdot \ tau \ cdot n} {m}}}$

This equation is also known as the Drude formula or Drude conductivity .

## Limits

The Drude model, with its assumption that all electrons contribute to the current, contradicts the statements of the Pauli principle , and also seen classically this assumption creates a contradiction: from statistical thermodynamics it follows that all degrees of freedom of a system (here: Solids ) contribute to its internal energy on average . So every electron should deliver. Measurements have shown, however, that the electronic contribution to the total energy is about a thousand times smaller. So not all electrons can be part of the electron gas, and more: the movement of the electron gas is less free than the kinetic gas theory describes. ${\ displaystyle {\ tfrac {1} {2}} k _ {\ text {B}} T}$${\ displaystyle 3 \ cdot {\ tfrac {1} {2}} k _ {\ text {B}} T}$

Apart from the wrongly predicted size of the conductivity or the resistance, the Drude model has other significant weaknesses:

It predicts a proportionality of resistance and electron speed to the square root of the temperature, which in reality is not given.

Furthermore, no statement can be made as to whether a material is a conductor , semiconductor or an insulator . The latter can be seen as an advantage in that the theory can also be applied to the free electrons in the conduction band of a semiconductor. The remedy is the quantum mechanical description using the Sommerfeld model or, further, the band model , in which the band gaps are correctly predicted.

A generalization of the Drude model is the Lorentz oscillator model (also Drude-Lorentz model). Additional absorption maxima are described, which are caused, for example, by band transitions . With the Lorentz oscillator model it is possible to describe the dielectric function of a large number of materials (including semiconductors and insulators).

## Individual evidence

1. Paul Drude: To the electron theory of metals . In: Annals of Physics . tape 306 , no. 3 , 1900, p. 566–613 , doi : 10.1002 / andp.19003060312 ( full text in Internet Archive BookReader - from S 566- .. ).
2. Paul Drude: On the ion theory of metals. In: Physikalische Zeitschrift. Jg. 1, No. 14, 1900, ZDB -ID 200089-1 , pp. 161-165 .
3. Absorption processes in semiconductors ( Memento of the original from March 28, 2017 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (Section 2.1.3)
4. ^ A b Arnold Sommerfeld, Hans Bethe: Electron theory of metals. In: Handbook of Physics. Volume 24, Part 2: Structure of the coherent matter. 2nd Edition. Springer, Berlin 1933, pp. 333-622.
5. Harland G. Tomkins, Eugene A. Irene (Eds.): Handbook of Ellipsometry . Springer et al., Heidelberg et al. 2005, ISBN 3-540-22293-6 .