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Metals: the Drude model of electrical conduction

Due to the quantum mechanical nature of electrons, a full simulation of electron movement in a solid (i.e. conduction) would require consideration of not only all the positive ion cores interacting with each electron, but also each electron with every other electron. Even with advanced models, this rapidly becomes far too complicated to model adequately for a material of macroscopic scale.

The Drude model simplifies things considerably by using classical mechanics and treats the solid as a fixed array of nuclei in a ‘sea’ of unbound electrons. Additionally, the electrons move in straight lines, do not interact with each other, and are scattered randomly by nuclei.

Rather than model the whole lattice, two statistically derived numbers are used:
τ, the average time between collisions (the scattering time), and
l, the average distance traveled between collisions  (the mean free path)

Under the application of a field, E, electrons experience a force –e E, and thus an acceleration from F = m a

For an electron emerging from a collision with velocity v0, the velocity after time t is given by:

\[v =v_{0} - \frac{eEt}{m} \]

Of course, if the electrons are scattered randomly by each collision, v0 will be zero. If we also consider the time t = τ, an equation for the drift velocity is given:

\[v =\frac{-eE\tau}{m} \]

For n free electrons per unit volume, the current density J is: J = -n e v

Substituting v for the drift velocity:

\[J = \frac {ne^{2}\tau E}{m} \]

The conductivity σ = n e μ, where μ is the mobility, which is defined as

\[ \mu = \frac{|v|}{E} = \frac{eE\tau}{mE} = \frac{e\tau}{m} \]

The net result of all this maths is a reasonable approximation of the conductivity of a number of monovalent metals. At room temperature, by using the kinetic theory of gases to estimate the drift velocity, the Drude model gives σ  ~ 106 Ω-1 m-1. This is about the right order of magnitude for many monovalent metals, such as sodium (σ  ~ 2.13 × 105 Ω-1 m-1).

The Drude model can be visualised using the following simulation. With no applied field, it can be seen that the electrons move around randomly. Use the slider to apply a field, to see its effect on the movement of the electrons.

Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.

However, it is important to note that for non-metals, multivalent metals, and semiconductors, the Drude model fails miserably. To be able to predict the conductivity of these materials more accurately, quantum mechanical models such as the Nearly Free Electron Model are required. These are beyond the scope of this TLP

Superconductors are also not explained by such simple models, though more information can be found at the Superconductivity TLP.

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