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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 rac{d}{d t}\langle ec{v} angle = q ec{E} - \gamma \langle ec{v} angle

where \langle ec{v} angle denotes Average velocity, m the effective mass and q the charge of the charge carriers.

The Steady State solution ( rac{d}{d t}\langle ec{v} angle = 0) of this differential equation is:

:\langle ec{v} angle = rac{q au}{m} ec{E} = \mu ec{E}

where:

\, au = rac{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:

: ec{J} = nq\langle ec{v} angle

The material can now be shown to satisfy Ohm's Law with a DC -conductivity \, \sigma_0.

: ec{J} = rac{n q^2 au}{m} ec{E} = \sigma_0 ec{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) = rac{\sigma_0}{1 + i\omega au}

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 \, au 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.


INADEQUACIES OF MODEL


This simple classical model provides a very good explanation of DC and AC conductivity in Metal s, 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.


SEE ALSO