 ## Optical properties of insulators and semiconductors

In an insulator, all charges are bound. By applying an electric field, the electrons and ions can be pulled out of their equilibrium positions. When this electric field is turned off, the charges oscillate as they return to their equilibrium positions. A simple model for an insulator can be constructed by describing the motion of the charge as a damped mass-spring system. The differential equation that describes the motion of a charge is,

### $m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx = qE.$

Rewriting above equation using $\omega_0 = \sqrt{\frac{k}{m}}$ and the damping constant $\gamma = \frac{b}{m}$ yields,

### $\frac{d^2x}{dt^2}+\gamma\frac{dx}{dt}+\omega_0^2 x = \frac{qE}{m}.$

If the electric field is pulsed on, the response of the charges is described by the impulse response function $g(t)$. The impulse response function satisfies the equation,

### $\frac{d^2g}{dt^2}+\gamma\frac{dg}{dt}+\omega_0^2g = \frac{q}{m}\delta(t).$

The solution to this equation is zero before the electric field is pulsed on and at the time of the pulse the charges suddenly start oscillating with the frequency $\omega_1 = \sqrt{\omega_0^2-\frac{\gamma^2}{4}}$. The amplitude of the oscillation decays exponentially to zero in a characteristic time $\frac{2}{\gamma}$.

### $g(t)=\frac{q}{m\omega_1}\exp(-\frac{\gamma}{2} t)\sin(\omega_1 t).$

 $\frac{m \omega_1}{q}\,g(t)$ t [ps]

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 charge is $-q\vec{x}$. The polarization is the electric dipole caused by one charge times the charge density, $\vec{P}=nq\vec{x}$.

### $\chi_E\left(\omega \right)=\frac{P\left(\omega\right)}{\epsilon_0 E\left(\omega\right)}=\frac{nqx\left(\omega\right)}{\epsilon_0 E\left(\omega\right)}$

The susceptibility for the simple model described above can be calculated by assuming a harmonic form for $E(t)$ and $x(t)$, $E\left(\omega\right)e^{i\omega t}$ and $x\left(\omega\right)e^{i\omega t}$. Substituting this into the differential equation yields,

### $\chi_E\left(\omega \right)= \frac{n_{\omega_0}q^2}{\epsilon_0m}\,\frac{1}{\omega_0^2-\omega^2+i\gamma\omega}.$

Here the charge density is written as $n_{\omega_0}$ to indicate that we only include the charges that oscillate with resonance frequency $\omega_0$. Real materials often need to be described by more than one resonance. For instance, ionic crystals typically have a resonance in the infrared corresponding to the motion of the ions (ionic polarizability) and another resonance in the ultraviolet due to electrons oscillating around their equilibrium positions in the atoms (electronic polarizability). When modeling the resonance in the infrared, the existence of a resonance at much higher frequency can be taken into account by adding a constant term to the susceptibility. This constant is typically called $\chi_E(\infty)$ although it corresponds to the relatively constant susceptibility in the frequencies above the resonance in the infrared but below the resonance in the ultraviolet. The expression for the susceptibility can then be written,

### $\chi_E\left(\omega \right)= \chi_E(\infty)+\frac{\omega_0^2(\chi_E(0)-\chi_E(\infty))}{\omega_0^2-\omega^2+i\gamma\omega},$

with $\chi_E\left(0 \right)= \chi_E(\infty)+\frac{n_{\omega_0}q^2}{\epsilon_0m\omega_0^2}$.

 $\chi_E(\omega)$ $\omega/ \omega_0$

The electric susceptibility is proportional to the Fourier transform of the impulse response function. The real and imaginary parts of the susceptibility are related by the Kramers-Kronig relations.

Complex conductivity
The frequency dependent conductivity $\sigma\left(\omega\right)$ is the ratio of the current density to the electric field.

### $\sigma\left(\omega \right)=\frac{j\left(\omega\right)}{E\left(\omega\right)}=nq\frac{v\left(\omega\right)}{E\left(\omega\right)}=\frac{i\omega n q x(w)}{E(\omega)}=i\omega \epsilon_0 \chi_E(\omega)$

 σ(ω)[103 Ω-1 m-1] $\omega/ \omega_0$

The conductivity of an insulator is zero at zero frequency.

Dielectric function
The relative dielectric constant describes the relationship between the electric displacement $\vec{D}$ and the electric field $\vec{E}$, $\vec{D}=\epsilon_r \epsilon_0 \vec{E}= \vec{P}+\epsilon_0 \vec{E}$.

### $\epsilon_r\left(\omega \right)=1+\chi_E=\epsilon(\infty)+\frac{\omega_0^2(\epsilon(0)-\epsilon(\infty))}{\omega_0^2-\omega^2+i\gamma\omega}$

 $\epsilon_r\left(\omega \right)$ $\omega/ \omega_0$

The form below can be used to generate new plots for the impulse response function, the susceptibility, the complex conductivity, and the relative dielectric constant for the specified values of the frequency $\omega_0$, the damping constant $\gamma$, $\epsilon\left(0\right)$ and $\epsilon\left(\infty\right)$. Plots for the the index of refraction, the extinction coefficient, the absorption coefficient, and the reflectance (shown below) are also modified by this form.

 $\omega_0$ =  [rad/s]  $\gamma$ =  [1/s]  $\epsilon(0)$ =   $\epsilon(\infty)$ =    The following buttons can be used to load the values for $\omega_0$, $\epsilon(0)$, and $\epsilon(\infty)$ for some dielectric crystals at 300 K into the form above. In all cases the damping constant is assumed to be $\gamma = \omega_0/10$. The frequency at which the real part of $\epsilon_r$ becomes positive again above the resonance is called $\omega_L = \omega_0\sqrt{\left(\epsilon\left(0\right)/\epsilon\left(\infty\right)\right)}$.

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 $\exp\left(-\alpha x\right)$ where $\alpha=\omega K/c$.

 $\sqrt{\epsilon_r}$ $\omega/ \omega_0$

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,

### $\alpha =\frac{2\omega K}{c}$

 α[106 m-1] $\omega/ \omega_0$

Reflectance
The reflectance of light striking the crystal normal to the surface from vacuum ($\epsilon_r=1$) is,

### $R=\frac{\left(n-1\right)^2+K^2}{\left(n+1\right)^2+K^2}$

 R $\omega/ \omega_0$

See the $n,k$ database (www.ioffe.ru/SVA/NSM/nk/index.html) for data on the refractive index and the absorption constant.