Editing Drude model (section)

=== AC field ===
[[File:DrudeModelComplexConductivity.png|thumb|300 px|right|Complex conductivity for different frequencies assuming that {{math|''τ'' {{=}} 10<sup>−5</sup>}} and that {{math|''σ''<sub>0</sub> {{=}} 1}}.]]
The Drude model can also predict the current as a response to a time-dependent electric field with an angular frequency {{mvar|ω}}. The complex conductivity is
<math display="block">\sigma(\omega) = \frac{\sigma_0}{1 - i\omega\tau}= \frac{\sigma_0}{1 + \omega^2\tau^2}+ i\omega\tau\frac{\sigma_0}{1 + \omega^2\tau^2}.</math>

Here it is assumed that:
<math display="block">\begin{align}
E(t) &= \Re{\left(E_0 e^{-i\omega t}\right)}; \\
J(t) &= \Re\left(\sigma(\omega) E_0 e^{-i\omega t}\right).
\end{align}</math>
In engineering, {{mvar|i}} is generally replaced by {{math|−''i''}} (or {{math|−''j''}}) in all equations, which reflects the phase difference with respect to origin, rather than delay at the observation point traveling in time.

{{math proof|title=Proof using the equation of motion<ref group="Ashcroft & Mermin" name=":16">{{harvnb|Ashcroft|Mermin|1976|pp=16}}</ref>|proof=

Given
<math display="block">\begin{align}
\mathbf{p}(t) &= \Re{\left(\mathbf{p}(\omega) e^{-i\omega t}\right)} \\
\mathbf{E}(t) &= \Re{\left(\mathbf{E}(\omega) e^{-i\omega t}\right)}
\end{align}</math>
And the equation of motion above
<math display="block">\frac{d}{dt}\mathbf{p}(t) = -e\mathbf{E} - \frac{\mathbf{p}(t)}{\tau}</math>
substituting
<math display="block">-i\omega\mathbf{p}(\omega) = -e\mathbf{E}(\omega) - \frac{\mathbf{p}(\omega)}{\tau}</math>
Given
<math display="block">\begin{align}
\mathbf{j} &= - n e \frac{\mathbf{p}}{m} \\
\mathbf{j}(t) &= \Re{\left(\mathbf{j}(\omega) e^{-i\omega t}\right)} \\
\mathbf{j}(\omega) &= - n e \frac{\mathbf{p}(\omega)}{m}=\frac{(n e^2/m)\mathbf{E}(\omega)}{1/\tau -i \omega}
\end{align}</math>
defining the complex conductivity from:
<math display="block">\mathbf{j}(\omega) = \sigma(\omega)\mathbf{E}(\omega)</math>
We have:
<math display="block">\sigma(\omega) = \frac{\sigma_0}{1-i\omega\tau};\sigma_0=\frac{ne^2\tau}{m}</math>
}}

The imaginary part indicates that the current lags behind the electrical field. This happens because the electrons need roughly a time {{mvar|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. The curves for {{math|''σ''(''ω'')}} are shown in the graph.

If a sinusoidally varying electric field with frequency <math>\omega</math> is applied to the solid, the negatively charged electrons behave as a plasma that tends to move a distance {{math|''x''}} apart from the positively charged background. As a result, the sample is polarized and there will be an excess charge at the opposite surfaces of the sample.

The [[dielectric constant]] of the sample  is expressed as
<math display="block">\varepsilon_r = \frac {D}{\varepsilon_0 E} = 1 + \frac {P}{\varepsilon_0 E} </math>
where <math>D</math> is the [[Electric displacement field|electric displacement]] and <math>P</math> is the [[polarization density]].

The polarization density is written as
<math display="block">P(t) = \Re{\left(P_0e^{i\omega t}\right)} </math>
and the polarization density with {{math|''n''}} electron density is
<math display="block">P = - n e x</math>
After a little algebra the relation between polarization density and electric field can be expressed as
<math display="block">P = - \frac{ne^2}{m\omega^2} E</math>
The frequency dependent dielectric function of the solid is
<math display="block">\varepsilon_r(\omega) = 1 - \frac {n e^2}{\varepsilon_0m \omega^2}</math>

{{math proof|title=Proof using Maxwell's equations<ref group="Ashcroft & Mermin" name=":17">{{harvnb|Ashcroft|Mermin|1976|pp=17}}</ref>|proof=

Given the approximations for the <math>\sigma(\omega)</math> included above
* we assumed no electromagnetic field: this is always smaller by a factor v/c given the additional Lorentz term <math> - \frac {e \mathbf{p}}{mc} \times \mathbf{B} </math> in the equation of motion
* we assumed spatially uniform field: this is true if the field does not oscillate considerably across a few mean free paths of electrons. This is typically not the case: the mean free path is of the order of Angstroms corresponding to wavelengths typical of X rays.
The following are Maxwell's equations without sources (which are treated separately in the scope of [[plasma oscillation]]s), in [[Gaussian units]]:
<math display="block">\begin{align}
\nabla \cdot \mathbf{E} &= 0; & \nabla \cdot \mathbf{B} &= 0; \\
\nabla \times \mathbf{E} &= - \frac{1}{c}\frac{\partial \mathbf{B}}{\partial t}; & \nabla \times \mathbf{B} &= \frac{4\pi}{c}\mathbf{j} + \frac{1}{c}\frac{\partial \mathbf{E}}{\partial t}.
\end{align}</math>
Then
<math display="block">\nabla \times \nabla \times \mathbf{E} = - \nabla^2 \mathbf{E} = \frac{i \omega}{c} \nabla \times \mathbf{B} = \frac{i \omega}{c} \left( \frac{4\pi \sigma}{c} \mathbf{E} - \frac{i \omega}{c} \mathbf{E} \right)</math>
or
<math display="block"> -\nabla^2 \mathbf{E} = \frac{\omega^2}{c^2}  \left( 1 + \frac {4\pi i \sigma}{\omega}\right) \mathbf{E}</math>
which is an electromagnetic wave equation for a continuous homogeneous medium with dielectric constant <math>\varepsilon(\omega)</math> in the Helmholtz form 
<math display="block"> - \nabla^2 \mathbf{E} = \frac{\omega^2}{c^2} \varepsilon(\omega) \mathbf{E}</math>
where the refractive index is <math display="inline">n(\omega) = \sqrt{\varepsilon(\omega)}</math> and the phase velocity is <math> v_\text{p} = \frac{c}{n(\omega)}</math>  
therefore the complex dielectric constant is 
<math display="block">\varepsilon(\omega) = \left( 1 + \frac {4\pi i \sigma}{\omega}\right)</math>
which in the case <math>\omega\tau \gg 1</math> can be approximated to:
<math display="block">\varepsilon(\omega) = \left( 1 - \frac{\omega_{\rm p}^2}{\omega^2} \right); \omega_{\rm p}^2 = \frac {4\pi n e^2}{m}  \text{(Gaussian units)}.</math> In [[International_System_of_Units | SI units]] the <math>4 \pi</math> in the numerator is replaced by <math>\varepsilon_0</math> in the denominator and the dielectric constant is written as <math>\varepsilon_r</math>.
}}
At a resonance frequency <math>\omega_{\rm p}</math>, called the '''plasma frequency''', the dielectric function changes sign from negative to positive and real part of the dielectric function drops to zero.
<math display="block">\omega_{\rm p} = \sqrt{\frac{n e^2}{\varepsilon_0 m}} </math>
The plasma frequency represents a [[plasma oscillation]] resonance or [[plasmon]]. The plasma frequency can be employed as a direct measure of the square root of the density of valence electrons in a solid. Observed values are in reasonable agreement with this theoretical prediction for a large number of materials.<ref name="Kittel2">{{cite book|title=[[Introduction to Solid State Physics]]|author=C. Kittel|publisher=Wiley & Sons|year=1953–1976|isbn=978-0-471-49024-1}}</ref> Below the plasma frequency, the dielectric function is negative and the field cannot penetrate the sample. Light with angular frequency below the plasma frequency will be totally reflected. Above the plasma frequency the light waves can penetrate the sample, a typical example are alkaline metals that becomes transparent in the range of [[ultraviolet]] radiation.<ref group="Ashcroft & Mermin" name=":18">{{harvnb|Ashcroft|Mermin|1976|pp=18 table 1.5}}</ref>