Drude Model

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Drude Model

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 :

:

mrac{\partial}{\partial 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 magnitude.

The Steady State solution (

rac{\partial}{\partial 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 = m/\gamma

is the ''mean free time'' of a charge carrier. μ Is called the ''mobility''. Now, introducing charge carrier density ''n'' (particles per volume unit), we can relate average velocity to current density:

:

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

This leads to the DC -conductivity

\sigma_0

of the material:

:

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 ''ω'', 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})

The imaginary part indicates that the current lags behind the electrical field, which happen 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.

INADEQUACIES OF MODEL

This simple classical model does a surprisingly good job of explaining 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.