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# Drude model

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.

## Explanation

The Drude model assumes that an average charge carrier experiences a `drag-coefficient' $\, \gamma$. Under an applied electric field E this leads to the following differential equation: $m\frac{d}{d t}\langle\vec{v}\rangle = q\vec{E} - \gamma \langle\vec{v}\rangle,$

where $\langle\vec{v}\rangle$ denotes average velocity, m the effective mass and q the charge of the charge carriers.

The steady state solution ( $\frac{d}{d t}\langle\vec{v}\rangle = 0$) of this differential equation is $\langle\vec{v}\rangle = \frac{q \tau}{m}\vec{E} = \mu\vec{E},$

where: $\, \tau = \frac{m}{\gamma}$ is the mean free time of a charge carrier, and $\,\mu$ is the mobility. Now, introducing charge carrier density n (particles per unit volume), we can relate average velocity to current density: $\vec{J} = nq\langle\vec{v}\rangle.$

The material can now be shown to satisfy Ohm's Law with a DC-conductivity $\, \sigma_0$. $\vec{J} = \frac{n q^2 \tau}{m} \vec{E} = \sigma_0\vec{E}$

The Drude model can also predict the current as a response to a time-dependent electric field with an angular frequency $\, \omega$, in which case $\sigma(\omega) = \frac{\sigma_0}{1 + i\omega\tau}.$

Here it is assumed that $E(t) = \Re(E_0 e^{i\omega t});$ $J(t) = \Re(\sigma(\omega) E_0 e^{i\omega t}).$

In other conventions, $\, i$ is replaced by $\, -i$ in all equations. The imaginary part indicates that the current lags behind the electrical field, which happens because the electrons need roughly a time $\, \tau$ 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.