Drude model
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The Drude model of electrical conduction was developed in the 1900s by Paul Drude to explain the transport properties of electrons in materials (especially metals). The Drude model is the application of kinetic theory to electrons in a solid. It assumes that the material contains immobile positive ions and an "electron gas" of classical, non-interacting electrons of density n, each of whose motion is damped by a frictional force, due to collisions of the electrons with the ions, characterized by a relaxation time τ.
[edit] Explanation
The Drude model assumes that an average charge carrier experiences a `drag-coefficient' γ. Under an applied electric field E this leads to the following differential equation:
where 
denotes average velocity, m the effective mass and q the charge magnitude.
The steady state solution (
) of this differential equation is:
where τ = m / γ is the mean free time of a charge carrier. μ Is called the mobility. Now, introducing charge carrier density n (particles per unit volume), we can relate average velocity to current density:
The material can now be shown to satisfy Ohm's Law with a DC-conductivity σ0.
The Drude model can also predict the current as a response to a time-dependent electric field with an angular frequency ω, in which case
Here it is assumed that
The imaginary part indicates that the current lags behind the electrical field, which happens because the electrons need roughly a time τ 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.
[edit] Inadequacies of model
This simple classical model does a surprisingly good job of explaining DC and AC conductivity in metals, the Hall effect, and thermal conductivity (due to electrons) in metals, although it fails to explain the disparity between the expected heat capacities of metals compared to insulators. In an insulator, one would expect the heat capacity to be zero since there are no free electrons. In reality, metals and insulators have roughly the same heat capacity at room temperature. Also, the Drude model fails to explain the existence of apparently positive charge carriers as demonstrated by the Hall effect.
