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Optical properties of a diffusive metal
It is assumed that electrons in a diffusive metal scatter so often that we can average over the scatering events. The differential equation that
describes the motion of the electrons is,
\( m\frac{d\vec{v}}{dt}+\frac{e\vec{v}}{\mu} = -e\vec{E}. \)
Here $m$ is the mass of an electron, $\vec{v}$ is the velocity of the electron, $-e$ is the charge of an electron, and $\vec{E}$ is the electric field.
When a constant electric field is applied, the solution is,
\( \vec{v}= -\mu \vec{E}. \)
Thus the (negatively charged) electrons move in the opposite direction as the electric field.
If the electric field is pulsed on, the reponse of the electrons is described by the impulse response function $g(t)$. The impulse
response function satisfies the equation,
\( m\frac{dg}{dt}+\frac{eg}{\mu} = -e\delta \left(t\right). \)
When the electric field is pulsed on, the electrons suddenly start moving and then their velocity decays exponentially to zero in a time
$\tau = m\mu /e$.
\( g\left(t\right)=-\frac{e}{m}\exp \left(-t/\tau\right)H(t). \)
Where $H(t)$ is the Heaviside step function.
The scattering time $\tau$ and the electron density $n$ are the only two parameters that are needed to describe many of the optical properties
of a diffusive metal. The form below can be used to input $\tau$ and $n$ and then a script calculates and plots the impulse response function,
the Fourier transform of the impulse response function, the mobility, the dc conductivity, the frequency dependent complex conductivity, the
electric susceptibility, the dielectric function, the plasma frequency, the
index of refraction, the extinction coefficient, the absorption coefficient, and the reflectance.
Impulse response function
The impulse response function describes the velocity of the electrons after the electric field has be pulsed on briefly.
\( g\left(t\right)=-\frac{e}{m}\exp \left(-t/\tau\right)H(t). \)
The impulse response function can be decomposed into its even and odd components.
\( g\left(t\right)=E\left(t\right)+O\left(t\right). \)
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| Generalized susceptibility
The generalized susceptibility $\chi$ is the Fourier transform of the impulse response function. It can be constructed by assuming a harmonic form for the electric field and the velocity, $E\left(\omega\right)e^{i\omega t}$ and $v\left(\omega\right)e^{i\omega t}$. Substituting this into the differential equation yields,
\( \chi\left(\omega \right)=\frac{v\left(\omega\right)}{E\left(\omega\right)}=-\frac{\mu \left(1-i\omega\tau\right)}{1+\omega^2\tau^2} \)
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| Complex conductivity
The current density $\vec{j}$ is proportional to the average velocity $\vec{v}$, $\vec{j}=-ne\vec{v}$. The frequency dependent conductivity $\sigma(\omega)$ is the ratio of the current density to the electric field.
\( \sigma\left(\omega \right)=\frac{j\left(\omega\right)}{E\left(\omega\right)}=-ne\frac{v\left(\omega\right)}{E\left(\omega\right)}=ne\frac{\mu \left(1-i\omega\tau\right)}{1+\omega^2\tau^2} \)
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σ(ω)
[108 Ω-1 m-1] | |
| ω [THz] |
| Electric susceptibility
The electric susceptibility $\chi_E$ describes the relationship between the polarization $\vec{P}$ and the electric field $\vec{E}$, $\vec{P} = \epsilon_0\chi_E\vec{E}$. The electric dipole caused by one electron is $-e\vec{r}$ where $\vec{r}$ is the displacement of the electron from its equilibrium position. The polarization is the electric dipole caused by one electron times the electron density, $\vec{P}=-ne\vec{r}$. Assuming a harmonic form for $\vec{P}$, $\vec{r}$,and $\vec{E}$; the frequency dependent electric susceptibility is,
\( \chi_E\left(\omega \right)=\frac{P\left(\omega\right)}{\epsilon_0 E\left(\omega\right)}=\frac{-ner\left(\omega\right)}{\epsilon_0 E\left(\omega\right)}=\frac{-nev\left(\omega\right)}{i\omega \epsilon_0 E\left(\omega\right)}=-\frac{ne\mu}{\omega \epsilon_0}\left(\frac{ \omega\tau+i}{1+\omega^2\tau^2} \right) .\)
Comparing the electrical conductivity to the electric susceptibility we find,
\( \chi_E\left(\omega \right)=\frac{\sigma\left(\omega\right)}{i\omega \epsilon_0}. \)
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| Dielectric function
The relative dielectric constant describes the relationship between the electric displacement $\vec{D}$ and the electric field $\vec{E}$, $\vec{D}=\epsilon_r \epsilon_0 \vec{E}= \vec{P}+\epsilon_0 \vec{E}$.
\( \epsilon_r\left(\omega \right)=1+\chi_E=1-\frac{ne\mu}{\omega \epsilon_0}\left(\frac{ \omega\tau+i}{1+\omega^2\tau^2} \right)=1-\omega_p^2\left(\frac{ \omega\tau^2+i\tau}{\omega+\omega^3\tau^2} \right) \)
$\omega_p=\sqrt{\frac{ne^2}{m\epsilon_0}}=$
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| $\epsilon_r\left(\omega \right)$ | |
| ω/ωp |
| The index of refraction n and the extinction coefficient K
The real and imaginary parts of the square root of the dielectric constant are the index of refraction and the extinction coefficient.
\( \sqrt{\epsilon_r}= n+iK \)
When waves travel from vacuum into some material, the frequency remains constant. A plane wave moving to the right in vaccuum has the form $\exp\left(i\left(\omega x/c -\omega t\right)\right)$ where $c$ is the speed of light in vacuum. When this wave enters some material, $c \rightarrow c/ \left(n+iK\right)$. The speed of the electromagnetic waves is smaller than the speed of light in vacuum by a factor of $n$. The extinction coeffcient describes the exponential decay of the amplitude of the electromagnetic waves. For waves propagating in the $x$-direction, the amplitude decays like $\exp\left(-ax\right)$ where $a=\omega K/c$.
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| Absorption coefficient $\alpha$
The absorption coefficient describes how the intensity of the light decays. Since the intensity is proportional to the amplitude of the waves squared, the exponential decay of the intensity is $I= I_0\exp\left(-\alpha x\right)$ where,
\( \alpha =\frac{2\omega K}{c} \)
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| Reflectance
The reflectance of light striking the metal normal to the surface from vacuum ($\epsilon_r=1$) is,
\( R=\frac{\left(n-1\right)^2+K^2}{\left(n+1\right)^2+K^2} \)
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