Magnetic, Dielectric and Optical Properties

Optical absorption and transmission

When light crosses a medium, its intensity falls because the medium absorbs photon energy and because some light is scattered out of the forward beam. For a uniform absorbing medium the intensity follows the Beer–Lambert law, I = I_0 e^{-\alpha x}, where \alpha is the absorption coefficient (in \text{cm}^{-1} or \text{m}^{-1}) and x is the path length. The transmitted fraction through a slab of thickness d is therefore e^{-\alpha d}, apart from losses at the two surfaces. At each air–glass boundary the reflectance is R = \left(\dfrac{n-1}{n+1}\right)^2. For n = 1.5, R = 0.04. The two surfaces give a factor (1-R)^2 that does not depend on \alpha, so it cancels when two transmissions are compared as a ratio. The ratio of transmitted fractions for two absorption coefficients is then e^{-(\alpha_1-\alpha_2)d}. Key terms: absorption coefficient — the fractional intensity lost per unit length; transmittance — the transmitted fraction after surface and bulk losses. Example: for \alpha = 2\ \text{cm}^{-1} and d = 0.1 cm, the bulk transmission is e^{-0.2} \approx 0.82.

In a semiconductor, interband optical absorption needs a photon with energy at least equal to the band gap E_g, because each absorbed photon must lift an electron from the valence band to the conduction band. Below the gap the crystal is transparent to that light, and above the gap the absorption rises steeply, forming an absorption edge at E_g. For a semiconductor with E_g \approx 2 eV, the absorption curve therefore stays near zero up to about 2 eV and then increases sharply. A metal, by contrast, absorbs strongly at all photon energies because free electrons are always available.

The threshold condition for absorption is that the photon energy E_p exceeds the band gap, which is written E_g < E_p. Photons with E_p < E_g pass through the fundamental transition unabsorbed. Electron–hole recombination and electron–electron recombination are emission-related or many-body processes; they are not the condition for absorption.

Phenomena that remove intensity from a beam during transmission include electronic absorption, in which photon energy is transferred to bound electrons; photon–phonon interaction, in which energy goes into lattice vibrations; and Rayleigh scattering, in which elastic scattering from particles much smaller than the wavelength redirects light out of the forward direction. Stimulated emission does the opposite: it adds coherent photons to the same mode, so it increases the beam intensity rather than reducing it.

Dielectric polarization mechanisms

A dielectric material becomes polarized when an electric field induces or aligns electric dipoles in it. The polarization P is the dipole moment per unit volume, with unit \text{C m}^{-2}, and it is related to the relative permittivity by P = \varepsilon_0(\varepsilon_r - 1)E. The main mechanisms are the following. Electronic polarization is the displacement of the electron cloud relative to the nucleus. It is very fast, responds up to optical and ultraviolet frequencies, and occurs in all materials. Ionic polarization is the relative displacement of cations and anions in an ionic solid. It is slower and is active up to the infrared range. Orientational (dipolar) polarization is the alignment of permanent dipoles and is slow, so it is active only at lower frequencies. Space-charge (interfacial) polarization comes from charge accumulation at boundaries and grain interfaces and is effective only at very low frequencies. Mechanical stress and temperature are not separate polarization mechanisms; temperature changes how much each mechanism contributes.

At ultraviolet frequencies the field reverses too quickly for ions or permanent dipoles to follow it. Only the electron cloud can respond, so the dielectric constant at these frequencies is governed by electronic polarizability.

The electronic polarization is the dipole moment per unit volume, P = N Z e d, where N is the number density of atoms, Z is the number of electrons per atom, e is the electron charge and d is the displacement of the electron cloud. The number density in a face-centred cubic cell is N = 4/a^3, since the cell contains four atoms. Example: with a = 0.4 nm, N = 4/(0.4\times10^{-9})^3 \approx 6.25\times10^{28}\ \text{m}^{-3}. Units must be checked so that P comes out in \text{C m}^{-2}.