Dielectric Properties of Solids

Abstract

When an external electromagnetic disturbance is introduced into a solid, it will produce induced charge density and induced current density. These induced densities produce induced electric and magnetic fields. We begin with a brief review of some elementary electricity and magnetism. In this chapter we will neglect the magnetization produced by induced current density and concentrate on the electric polarization field produced by the induced charge density.

Appendices

Problems

8.1

Suppose an electric field $\mathbf {E}=E\hat{z}$ is applied to a hydrogen atom in its ground state $\psi _{100}(r,\theta ,\phi )=\frac{1}{\sqrt{2}}a_0^{-3/2}\mathrm{e}^{-r/a_0}$, where $a_0$ is the Bohr radius. In the presence of an external electric field, the electron cloud of the hydrogen atom is displaced in the opposite direction of the field to an induce dipole moment. Evaluate the atomic polarizability $\alpha$ of the hydrogen atom assuming semiclassically that the atom remains in its ground state.

8.2

In the presence of an external electric field $\mathbf {E}=E\hat{z}$, the ground state in Problem 1 is no longer $\psi _{100}(r,\theta ,\phi )$, but is perturbed to be $\tilde{\psi }_0$ due to an additional term $-qEz$ in the Hamiltonian. Evaluate the atomic polarizability of the hydrogen atom by calculating $\langle \tilde{\psi }_0|qz|\tilde{\psi }_0\rangle$ to first order in E. Note the selection rule of $\varDelta n=\mathrm any~ value$, $\varDelta \ell =\pm 1$, and $\varDelta m=0$.

8.3

A degenerate polar semiconductor contains $n_0$ free electrons per unit volume in the conduction band. Its dielectric function $\epsilon (\omega )$ is given by

$$ \epsilon (\omega ) = \epsilon _{\infty } \frac{\omega ^2-\omega _\mathrm{L}^2}{\omega ^2-\omega _\mathrm{T}^2} -\frac{\omega _\mathrm{p}^2}{\omega ^2} $$

where $\omega _\mathrm{L}$ and $\omega _\mathrm{T}$ are the LO and TO phonon frequencies, and $\omega _\mathrm{p} = \sqrt{\frac{4\pi n_0 e^2}{m}}$.

• (a) Show that $\epsilon (\omega )$ can be written as $\epsilon (\omega ) = \epsilon _{\infty } \frac{(\omega ^2-\omega _-^2)(\omega ^2-\omega _+^2)}{\omega ^2(\omega ^2-\omega _\mathrm{T}^2)},$ and determine $\omega _-^2$ and $\omega _+^2$.

• (b) Make a sketch of $\epsilon (\omega )$ versus $\omega$; be sure to indicate the locations of $\omega _\mathrm{T}$, $\omega _\mathrm{L}$, $\omega _-$, $\omega _+$, $\epsilon _0$, and $\epsilon _{\infty }$.

• (c) Determine the dispersion relation of the longitudinal and transverse modes, i.e. $\omega$ as a function of q. In which regions of frequency are the transverse waves unable to propagate?

8.4

Evaluate the reflectivity for an S-polarized and a P-polarized electromagnetic wave incident at an angle $\theta$ from vacuum on a material of dielectric function $\epsilon (\omega )$ as illustrated in the figure above. One can take $\mathbf E=(E_x, 0, 0)\mathrm{e}^{i\omega t-i\mathbf q\cdot \mathbf r}$ and $\mathbf E=(0, E_y, E_z)\mathrm{e}^{i\omega t-i\mathbf q\cdot \mathbf r}$ as the S- and P-polarized electric fields, respectively. Remember that $\mathbf q \cdot \mathbf E =0$ and $\mathbf q=(0, q_y, q_z)$.

8.5

   (a) Consider a vacuum–degenerate polar semiconductor interface. Use the results obtained in the text to determine the dispersion relations of the surface modes.

• (b) Make a sketch of $\omega$ versus $q_y$ ($q_y$ is parallel to the interface) for these surface modes and for the bulk modes which have $q_z=0$.

Summary

In this chapter we studied dielectric properties of solids in the presence of an external electromagnetic disturbance. We first reviewed elementary electricity and magnetism, and introduced concept of local field inside a solid. Then dispersion relations of self-sustaining collective modes and reflectivity of a solid are studied for various situations. Finally the collective modes localized near the surface of a solid are also described and dispersion relations of surface plasmon-polariton and surface phonon-polariton modes are discussed explicitly.

When an external electromagnetic disturbance is introduced into a solid, it will produce induced charge density and induced current density. These induced densities produce induced electric and magnetic fields. The local field $\mathbf E_{\mathrm {LF}}(\mathbf r)$ at the position of an atom in a solid is given by

$$ \mathbf E_{\mathrm {LF}} = \mathbf E_0 + \mathbf E_1 + \mathbf E_2 + \mathbf E_3, $$

where $E_0$, $E_1$, $E_2$, $E_3$ are, respectively, the external field, depolarization field $(= -\lambda \mathbf P)$, Lorentz field $({=}\frac{4\pi \mathbf P}{3})$, and the field due to the dipoles within the Lorentz sphere $(= \sum _{i \in \mathrm {L.S.}} \frac{3\left( \mathbf p_i\cdot \mathbf r_i\right) \mathbf r_i -r_i^2\mathbf p_i}{r_i^5})$. The local field at the center of a sphere of cubic crystal is simply given by

$$E_{\mathrm {LF}}^\mathrm{sphere} = E_0 - \frac{4\pi }{3}P +\frac{4\pi }{3}P +0=E_0. $$

The induced dipole moment of an atom is given by $\mathbf p = \alpha \mathbf E_{\mathrm {LF}}$. The polarization $\mathbf P$ is given, for a cubic crystal, by $\mathbf P = \frac{N\alpha }{1-\frac{4\pi N\alpha }{3}}\mathbf E \equiv \chi \mathbf E,$ where N is the number of atoms per unit volume and $\chi$ is the electrical susceptibility. The electrical susceptibility and the dielectric function ($\epsilon =1+4\pi \chi$) of the solid are

$$ \chi = \frac{N\alpha }{1-\frac{4\pi N\alpha }{3}} ; \text { } \epsilon = 1+\frac{4\pi N\alpha }{1-\frac{4\pi N\alpha }{3}}. $$

The relation between the macroscopic dielectric function $\epsilon$ and the atomic polarizability $\alpha$ is called the Clausius–Mossotti relation:

$$ \frac{\epsilon -1}{\epsilon +2} = \frac{4\pi N \alpha }{3} $$

The total polarizability of the atoms or ions within a unit cell can usually be separated into three parts: (i) electronic polarizability $\alpha _\mathrm{e}$: the displacement of the electrons relative to the nucleus; (ii) ionic polarizability $\alpha _\mathrm{i}$: the displacement of an ion itself with respect to its equilibrium position; (iii) dipolar polarizability $\alpha _\mathrm{dipole}$: the orientation of any permanent dipoles by the electric field in the presence of thermal disorder.

In the presence of a field $\mathbf E$, the average dipole moment per unit volume is given by $\bar{p}_z = p\, \mathcal {L}\left( \frac{pE}{k_\mathrm{B}T}\right) ,$ where $\mathcal {L}(\xi )$ is the Langevin function. The dipolar polarizability $\alpha _\mathrm{dipole}$ shows strong temperature dependence. The electronic polarizability $\alpha _\mathrm{e}$ and the ionic polarizability $\alpha _\mathrm{ion}$ are almost independent of temperature.

In a metal, the conduction electrons are free and the dielectric function becomes

$$ \epsilon (\omega ) = 1 - \frac{4\pi Ne^2/m}{\omega ^2-i\omega /\tau }=1-\frac{\omega _p^2}{\omega ^2-i\omega /\tau }. $$

In an ionic crystal, we have

$$ \epsilon (\omega ) =\epsilon _\infty \left[ \frac{\omega ^2 -\omega _\mathrm{L}^2}{\omega ^2 -\omega _\mathrm{T}^2}\right] . $$

Here, $\omega _\mathrm{L}$ and $\omega _\mathrm{T}$ are the TO and LO phonon frequencies, respectively. We note that $\omega _\mathrm{L} > \omega _\mathrm{T}$ since $\epsilon _0 > \epsilon _\infty$ in general.

For the propagation of light in a material characterized by $\epsilon (\omega )$, the external sources $\mathbf j_0$ and $\rho _0$ vanishes. Therefore, we have

$$ \mathbf \nabla \times \mathbf E = -\frac{i\omega }{c}\mathbf B; \text { } \mathbf \nabla \times \mathbf B = \frac{i\omega \epsilon (\omega )}{c}\mathbf E. $$

The two Maxwell equations for $\mathbf \nabla \times \mathbf E$ and $\mathbf \nabla \times \mathbf B$ can be combined to give a wave equation:

$$ \left( \frac{\omega ^2}{c^2}\epsilon (\omega ) - q^2\right) \mathbf E + \mathbf q \left( \mathbf q \cdot \mathbf E\right) = 0. $$

For an infinite homogeneous medium of dielectric function $\epsilon (\omega )$, a general dispersion relation of the self-sustaining waves is written as

$$ \epsilon (\omega ) \left[ \frac{\omega ^2}{c^2}\epsilon (\omega ) -q^2\right] ^2 =0. $$

The two transverse modes and one longitudinal mode are characterized, respectively, by

$$ \omega ^2 = \frac{c^2q^2}{\epsilon (\omega )} ; \text { } \epsilon (\omega )=0. $$

For the interface ($z=0$) of two different media of dielectric functions $\epsilon _\mathrm{I}$ and $\epsilon _\mathrm{II}$, the boundary conditions give us the general dispersion of the surface wave:

$$ \frac{\epsilon _\mathrm{I}}{\alpha _\mathrm{I}} + \frac{\epsilon _\mathrm{II}}{\alpha _\mathrm{II}} =0 \text {or} \frac{\epsilon _{o}}{\alpha _{o}} + \frac{\epsilon (\omega )}{\alpha } =0. $$

where

$$ \alpha _\mathrm{o} = \sqrt{q_y^2 -\frac{\omega ^2}{c^2}\epsilon _\mathrm{o}} \text {and} \alpha = \sqrt{q_y^2 -\frac{\omega ^2}{c^2}\epsilon (\omega )}. $$