Dielectrics and Electrooptics

Abstract

The present chapter describes the physical properties of dielectrics and includes the following data:

• 1. Low-frequency properties , i. e., density and Mohs hardness, thermal conductivity, static dielectric constant, dissipation factor (loss tangent), elastic stiffness and elastic compliance, and piezoelectric strain

• 2. High-frequency (optical) properties , i. e., elastooptic and electrooptic coefficients, optical transparency range, two-photon absorption coefficient, refractive indices and their temperature variation, dispersion relations (Sellmeier equations), and second and/or third-order nonlinear dielectric susceptibilities.

A large amount of the information in this chapter is taken from the compilations of low- and high-frequency properties of dielectric crystals in Landolt–Börnstein, Group III, Vols. 29 and 30, especially Vol. 30b. Since 1992–1993, the date of publication of the first of these volumes, a large amount of new data on the physical properties of dielectrics has appeared in the literature. In particular, various linear and nonlinear optical properties of new crystals in the borate family have been (re)measured. Among these are: beta barium borate (GlossaryTerm

BBO

), lithium triborate (GlossaryTerm

LBO

), cesium borate (GlossaryTerm

CBO

), cesium lithium borate (GlossaryTerm

CLBO

) and new organic crystals such as deuterated l-arginine phosphate (GlossaryTerm

DLAP

) and 2-methyl-4-nitro-N-methylaniline (MNMA). This situation has encouraged us to refresh the knowledge about these crystals by adding new data from publications from the last decade. We have also included the most commonly used isotropic materials. The criteria for the selection of 124 dielectrics out of several hundred were their wide range of application and the availability of most of the above-mentioned data.

One of our aims was to produce a reader-friendly compilation. For this purpose, all data on dielectrics are presented in a unified form in tables that are similar for both isotropic materials and crystals of various symmetry classes. Moreover, all data for each particular material are collected together in one place. Sections 23.1–23.2 serve as a brief introduction to the various physical phenomena and to the definitions, symbols, and abbreviations used. All numerical data are presented in Sect. 23.4. The tables in this chapter are arranged according to piezoelectric classes in order of decreasing symmetry. Guidelines for searching for (and finding!) the required parameter in these tables can be found in Sect. 23.3.

The following Table 23.1 presents a list of the 124 different substances which have been selected to be described in Sect. 23.4.

The physical quantities used to describe the properties of the dielectric substances are drawn up in Table 23.2.

Table 23.1 Alphabetical list of described crystals and isotropic dielectrics

Table 23.2 Used physical quantities, their symbols and their units

1 Dielectric Materials: Low-Frequency Properties

1.1 General Dielectric Properties

1.1.1 Density

The density of a substance is defined as the mass per unit volume of the substance

$$\varrho=\frac{m}{V}\;,$$

(23.1)

where V is the volume occupied by a mass m. The density is thus a measure of the volume concentration of mass.

1.1.2 Mohs Hardness Scale

In 1832, Mohs introduced a hardness scale ranging from 1 to 10, based on ten minerals:

• 1. Talc, Mg₃H₂SiO₁₂

• 2. Gypsum, CaSO₄ ⋅ 2H₂O

• 3. Iceland spar, CaCO₃

• 4. Fluorite, CaF₂

• 5. Apatite, Ca₅F(PO₄)₃

• 6. Orthoclase, KAlSi₃O₈

• 7. Quartz, SiO₂

• 8. Topaz, Al₂F₂SiO₄

• 9. Corundum, Al₂O₃

• 10. Diamond, C.

1.1.3 Thermal Conductivity

A temperature gradient between different parts of a solid causes a flow of heat. In an isotropic medium, the heat flux of thermal energy h (i. e., the heat transfer rate per unit area normal to the direction of heat flow) is given by

$$h=-\kappa\text{ grad }T\;,$$

(23.2)

where κ is the thermal conductivity. In a crystal, this expression is replaced by

$$h_{i}=-\kappa_{ij}\frac{\partial T}{\partial x_{j}}\;.$$

(23.3)

Here κ_ij is the thermal-conductivity tensor. Note that other notations are also used in the literature, e. g., κ ≡ λ or κ ≡ k.

1.2 Static Dielectric Constant (Low-Frequency)

In isotropic and cubic dielectric materials, the electric displacement field D, the electric field E, and the polarization P are connected by the relation

$$D=\varepsilon_{0}E+P=\varepsilon_{0}(1+\chi)E\;,$$

(23.4)

where $\varepsilon_{0}={\mathrm{8.854\times 10^{-12}}}\,{\mathrm{C/(V{\,}m)}}$ is the dielectric constant (or permittivity) of free space (vacuum), and χ is the dielectric susceptibility. The relative dielectric constant of the material is defined as

$$\varepsilon=1+\chi\;,$$

(23.5)

and therefore (23.4) becomes

$$D=\varepsilon_{0}\varepsilon E\;.$$

(23.6)

In anisotropic crystals, these equations should be written in tensor form

$$D_{i}=\varepsilon_{0}\varepsilon_{ij}E_{j}\;,\quad\varepsilon_{ij}=1+\chi_{ij}\;.$$

(23.7)

The following relations are valid

$$\varepsilon_{ij}=\varepsilon_{ji}\;,\quad\chi_{ij}=\chi_{ji}\;.$$

(23.8)

Note that other notations are also used in the literature, e. g., ε ≡ ε_r, or ε ≡ κ and ε₀ ≡ κ₀.

1.3 Dissipation Factor

The capacitance C of a capacitor filled with a dielectric is

$$C=\frac{\varepsilon_{0}\varepsilon A}{d}\;,$$

(23.9)

where A is the area of the two parallel plates and d is the spacing between them. For a lossy dielectric the relative dielectric constant ε can be represented in a complex form

$$\varepsilon=\varepsilon^{\prime}-\mathrm{i}\varepsilon^{\prime\prime}\;.$$

(23.10)

The imaginary part is the frequency-dependent conductivity

$$\sigma(\omega)=\omega\varepsilon_{0}\varepsilon^{\prime\prime}\;,$$

(23.11)

where ω is the frequency. The dissipation factor (or loss tangent) is defined as

$$\tan\delta=\frac{\varepsilon^{\prime\prime}}{\varepsilon^{\prime}}\;.$$

(23.12)

and in anisotropic crystals

$$\begin{aligned}\displaystyle\tan\delta_{1}&\displaystyle=\frac{\varepsilon^{\prime\prime}_{11}}{\varepsilon^{\prime}_{11}}\;,\quad\tan\delta_{2}=\frac{\varepsilon^{\prime\prime}_{22}}{\varepsilon^{\prime}_{22}}\;,\\ \displaystyle\tan\delta_{3}&\displaystyle=\frac{\varepsilon^{\prime\prime}_{33}}{\varepsilon^{\prime}_{3}}\;.\end{aligned}$$

(23.13)

The quality factor Q of the dielectric is the reciprocal of the dissipation factor

$$Q=\frac{1}{\tan\delta}\;.$$

(23.14)

1.4 Elasticity

Hooke's law states that for sufficiently small deformations the strain is directly proportional to the stress. Thus the strain tensor S and the stress tensor T obey the relation

$$S_{ij}=s_{ijkl}T_{kl}\;,$$

(23.15)

where s_ijkl is called the elastic compliance constant (or compliance, or elastic constant). The elastic stiffness constant (or stiffness, or Young's modulus) is the reciprocal tensor

$$c_{ijkl}=s_{ijkl}^{-1}\;,$$

(23.16)

and for the stress tensor we have

$$T_{ij}=c_{ijkl}S_{kl}\;.$$

(23.17)

In the matrix notation for the elastic compliance and stiffness, we have

$$S_{m}=s_{mn}T_{n}\quad\text{ and }\quad T_{m}=c_{mn}S_{n}\;,$$

(23.18)

where

$$\begin{aligned}\displaystyle s_{ijkl}&\displaystyle=s_{mn}\quad\text{ when both }m\text{ and }n\text{ are }1,2,\text{ or }3\;,\\ \displaystyle 2s_{ijkl}&\displaystyle=s_{mn}\quad\text{ when either }m\text{ or }n\text{ is }4,5,\text{ or }6\;,\\ \displaystyle 4s_{ijkl}&\displaystyle=s_{mn}\quad\text{ when both }m\text{ and }n\text{ are }4,5,\text{ or }6\;,\\ \displaystyle c_{ijkl}&\displaystyle=c_{mn}\quad\text{ for all }m\text{ and }n\;,\end{aligned}$$

(23.19)

and

$$\begin{aligned}\displaystyle S_{ij}&\displaystyle=S_{m}\quad\text{ when }m\text{ is }1,2,\text{ or }3\;,\\ \displaystyle S_{ij}&\displaystyle=\frac{1}{2}S_{m}\quad\text{when }m\text{ is }4,5,\text{ or }6\;,\\ \displaystyle T_{ij}&\displaystyle=T_{m}\quad\text{ for all }m\;.\end{aligned}$$

(23.20)

For relations between tensor and matrix notation, see Table 23.3. The sound velocity v ^s _{m n} in the direction mn in a crystal is given by

$$v_{mn}^{\text{s}}=\sqrt{\frac{c_{mn}}{\varrho}}\;.$$

(23.21)

Table 23.3 The relations between ij (tensor notation) and m (matrix notation), jk and n, and kl and n

1.5 Piezoelectricity

The phenomenon of the development of an electric moment P_i if a stress T_jk is applied to a crystal is called the direct piezoelectric effect

$$P_{i}=d_{ijk}T_{jk}\;,$$

(23.22)

where d_ijk is the piezoelectric strain tensor (or the piezoelectric moduli). The relation d_ijk = d_ikj reduces the number of independent tensor components to 18. The matrix notation is introduced for the piezoelectric strain as follows

$$\begin{aligned}\displaystyle d_{ijk}&\displaystyle=d_{in}\quad\text{ when }n=\text{ 1, 2, or 3}\;,\\ \displaystyle 2d_{ijk}&\displaystyle=d_{in}\quad\text{ when }n=\text{ 4, 5, or 6}\;,\end{aligned}$$

(23.23)

and thus

$$P_{i}=d_{in}T_{n}\;.$$

(23.24)

The relations between jk and n are presented in Table 23.3.

The converse piezoelectric effect is described by

$$S_{jk}=d_{ijk}E_{i}$$

(23.25)

and, correspondingly,

$$S_{n}=d_{in}E_{i}\;.$$

(23.26)

2 Optical Materials: High-Frequency Properties

2.1 Crystal Optics: General

The dielectric properties of a medium at optical frequencies are given by

$$D=\varepsilon_{0}\varepsilon E\;,$$

(23.27)

where ε₀ is the dielectric constant of free space and ε is the relative dielectric constant of the material. From Maxwell's equations, the velocity of propagation of electromagnetic waves through the medium is given by

$$v=\frac{c}{\sqrt{\varepsilon}}\;,$$

(23.28)

where c is the velocity in vacuum (the relative magnetic permeability is taken as 1). The refractive index $n=c/v$ is therefore $n=\sqrt{\varepsilon}$.

In an anisotropic medium

$$D_{i}=\varepsilon_{0}\varepsilon_{ij}E_{j}\;.$$

(23.29)

In this general case, two waves of different velocity may propagate through the crystal. The relative dielectric impermeabilities are defined as the reciprocals of the principal dielectric constants

$$B_{i}=\frac{1}{\varepsilon_{i}}=\frac{1}{n_{i}^{2}}\;.$$

(23.30)

2.2 Photoelastic Effect

The photoelastic effect is the effect in which a change of the refractive index is caused by stress. The changes in the relative dielectric impermeabilities are

$$\Updelta B_{ij}=q_{ijkl}T_{kl}\;,$$

(23.31)

where the q_ijkl are the piezooptic coefficients. The photoelastic effect can also be expressed in terms of the stress

$$\Updelta B_{ij}=p_{ijrs}S_{rs}\;,$$

(23.32)

where the p_ijrs = q_ijklc_klrs are the (dimensionless) elastooptic coefficients. In matrix notation,

$$\Updelta B_{m}=q_{mn}T_{n}\quad\text{ and }\quad\Updelta B_{m}=p_{mn}S_{n}\;.$$

(23.33)

Note that q_mn = q_ijkl when n = 1,2, or 3 and q_mn = 2q_ijkl when n = 4,5, or 6; p_mn = p_ijrs (Table 23.3).

2.3 Electrooptic Effect

The electrooptic effect is the effect in which a change in the refractive index of a crystal is produced by an electric field

$$n=n_{0}+aE_{0}+bE_{0}^{2}+\ldots\;,$$

(23.34)

where a and b are constants and n₀ is the refractive index at E₀ = 0. The linear electrooptic effect (Pockels effect) is due to the first-order term aE₀. In isotropic dielectrics and in crystals with a center of symmetry, a = 0, and only the second-order term bE ²₀ and higher even-order terms exist (Kerr effect).

The changes in the relative dielectric impermeabilities are

$$\Updelta B_{ij}=r_{ijk}E_{k}\;,$$

(23.35)

where the r_ijk are the electrooptic coefficients. Since r_ijk = r_jik, the number of independent tensor components is 18, and the above formula can be written in matrix notation (Table 23.3)

$$\Updelta B_{m}=r_{mk}E_{k}\quad(m=1,2,\ldots,6,\ k=1,2,3)\;.$$

(23.36)

2.4 Nonlinear Optical Effects

The dielectric polarization P is related to the electromagnetic field E at optical frequencies by the material equation of the medium

$$\boldsymbol{P}(\boldsymbol{E})=\varepsilon_{0}(\chi^{(1)}\boldsymbol{E}+\chi^{(2)}\boldsymbol{E}^{2}+\chi^{(3)}\boldsymbol{E}^{3}+\ldots)\;,$$

(23.37)

where $\chi^{(1)}=n^{2}-1$ is the linear dielectric susceptibility, and χ⁽²⁾ , χ⁽³⁾, etc. are the nonlinear dielectric susceptibilities.

The Miller delta formulation is

$$\varepsilon_{0}E_{i}(\omega_{3})=\delta_{ijk}P_{j}(\omega_{1})P_{k}(\omega_{2})\;,$$

(23.38)

where the Miller coefficient

$$\delta_{ijk}=\frac{1}{2\varepsilon_{0}}\frac{\chi_{ijk}^{(2)}(\omega_{3})}{\chi_{ii}^{(1)}(\omega_{1})\chi_{jj}^{(1)}(\omega_{2})\chi_{kk}^{(1)}(\omega_{3})}\;,$$

(23.39)

has a small dispersion and is almost constant for a wide range of crystals.

For anisotropic media, the coefficients χ⁽¹⁾ and χ⁽²⁾ are, in general, second- and third-rank tensors, respectively. In practice, the tensor

$$d_{ijk}=\left(\frac{1}{2}\right)\chi_{ijk}$$

(23.40)

is used instead of χ_ijk. Usually the plane representation of d_ijk in the form d_il is used; the relations between jk and l are presented in Table 23.3.

The Kleinman symmetry conditions

$$\begin{aligned}\displaystyle d_{21}&\displaystyle=d_{16},\quad d_{23}=d_{34},\quad d_{14}=d_{25}=d_{36}\;,\\ \displaystyle d_{26}&\displaystyle=d_{12},\quad d_{31}=d_{15},\quad d_{32}=d_{24},\quad d_{35}=d_{13}\end{aligned}$$

(23.41)

are valid in the case of no dispersion of the electronic nonlinear polarizability.

The following three-wave interactions in crystals with a square nonlinearity (χ⁽²⁾ ≠ 0) are possible:

• Second-harmonic generation (GlossaryTerm

  SHG

  ), $\omega+\omega=2\omega$

• Sum frequency generation (GlossaryTerm

  SFG

  ) or up-conversion, $\omega_{1}+\omega_{2}=\omega_{3}$

• Difference frequency generation (GlossaryTerm

  DFG

  ) or down-conversion, $\omega_{3}-\omega_{2}=\omega_{1}$

• Optical parametric oscillation (GlossaryTerm

  OPO

  ), $\omega_{3}=\omega_{2}+\omega_{1}$.

For efficient frequency conversion, the phase-matching condition $\boldsymbol{k}_{1}+\boldsymbol{k}_{2}=\boldsymbol{k}_{3}$, where the k_i are the wave vectors for ω₁, ω₂, and ω₃, respectively, must be satisfied. Two types of phase matching can be defined

$$\text{type I:}\,\text{o}+\text{o}\to\text{e}\text{ or }\text{e}+\text{e}\to\text{o}\;;$$

and

$$\text{type II:}\,\text{o}+\text{e}\to\text{e}\text{ or }\text{o}+\text{e}\to\text{o}\;.$$

These can be represented with a shortened notation as follows

$$\begin{aligned}\displaystyle&\displaystyle\text{ooe}{:}\,\text{o}+\text{o}\to\text{e}\quad\text{ or }\quad\text{e}\to\text{o}+\text{o}\;;\\ \displaystyle&\displaystyle\text{eeo}{:}\,\text{e}+\text{e}\to\text{o}\quad\text{ or }\quad\text{o}\to\text{e}+\text{e}\;;\\ \displaystyle&\displaystyle\text{eoe}{:}\,\text{e}+\text{o}\to\text{e}\quad\text{ or }\quad\text{e}\to\text{e}+\text{o}\;;\\ \displaystyle&\displaystyle\text{oeo}{:}\,\text{o}+\text{e}\to\text{o}\quad\text{ or }\quad\text{o}\to\text{e}+\text{o}\;.\end{aligned}$$

In the shortened notation ($\text{ooe},\text{eoe},{\ldots}$), the frequencies satisfy the condition $\omega_{1}<\omega_{2}<\omega_{3}$, i. e., the first symbol refers to the longest-wavelength radiation, and the last symbol refers to the shortest-wavelength radiation. Here the ordinary beam, or o-beam is the beam with its polarization normal to the principal plane of the crystal, i. e., the plane containing the wave vector k and the crystallophysical axis Z (or the optical axis, for uniaxial crystals). The extraordinary beam, or e-beam is the beam with its polarization in the principal plane. The third-order term χ⁽³⁾ is responsible for the optical Kerr effect.

2.4.1 Uniaxial Crystals

For uniaxial crystals, the difference between the refractive indices of the ordinary and extraordinary beams, the birefringence Δn, is zero along the optical axis (the crystallophysical axis Z) and maximum in a direction normal to this axis. The refractive index for the ordinary beam does not depend on the direction of propagation. However, the refractive index for the extraordinary beam n^e(θ), is a function of the polar angle θ between the Z axis and the vector k

$$n^{\text{e}}(\theta)=n_{\text{o}}\left(\frac{1+\tan^{2}\theta}{1+\left(n_{\text{o}}/n_{\text{e}}\right)^{2}\tan^{2}\theta}\right)^{1/2},$$

(23.42)

where n_o and n_e are the refractive indices of the ordinary and extraordinary beams, respectively in the plane normal to the Z axis, and are termed the principal values. If n_o > n_e the crystal is called negative, and if n_o < n_e it is called positive. For the o-beam, the indicatrix of the refractive indices is a sphere with radius n_o, and for the e-beam it is an ellipsoid of rotation with semiaxes n_o and n_e. In the crystal, in general, the beam is divided into two beams with orthogonal polarizations; the angle between these beams ρ is the birefringence (or walk-off) angle.

Equations for calculating phase-matching angles in uniaxial crystals are given in [23.1, 23.2, 23.3, 23.4].

2.4.2 Biaxial Crystals

For biaxial crystals, the optical indicatrix is a bilayer surface with four points of interlayer contact, which correspond to the directions of the two optical axes. In the simple case of light propagation in the principal planes XY, YZ, and XZ, the dependences of the refractive indices on the direction of light propagation are represented by a combination of an ellipse and a circle. Thus, in the principal planes, a biaxial crystal can be considered as a uniaxial crystal; for example, a biaxial crystal with $n_{Z}> n_{Y}> n_{X}$ in the XY plane is similar to a negative uniaxial crystal with n_o = n_Z

$$n^{\text{e}}(\varphi)=n_{Y}\left({\frac{1+\tan^{2}\varphi}{1+(n_{Y}/n_{X})^{2}\tan^{2}\varphi}}\right)^{1/2}\;,$$

(23.43)

where φ is the azimutal angle. Equations for calculating phase-matching angles for propagation in the principal planes of biaxial crystals are given in [23.3, 23.4, 23.5, 23.6].

3 Guidelines for Use of Tables

Tables 23.5–23.23 are arranged according to piezoelectric classes in order of decreasing symmetry (Table 23.4 ), and alphabetically within each class. They contain a number of columns placed on two pages, even and odd. The following properties are presented for each dielectric material: density ϱ, Mohs hardness, thermal conductivity κ, static dielectric constant ε_ij, dissipation factor tan⁡δ at various temperatures and frequencies, elastic stiffness c_mn, elastic compliance s_mn (for isotropic and cubic materials only), piezoelectric strain tensor d_in, elastooptic tensor p_mn, electrooptic coefficients r_mk (the latter two at 633 nm unless otherwise stated), optical transparency range, temperature variation of the refractive indices dn ∕ dT, refractive indices n (the latter two at 1.064 μm unless otherwise stated), dispersion relations (Sellmeier equations), second-order nonlinear dielectric susceptibility d_ij, and third-order nonlinear dielectric susceptibility χ ⁽³⁾ _{i j k} (for isotropic and cubic materials only) (the latter two at 1.064 μm unless otherwise stated). For isotropic materials, the two-photon absorption coefficient β is also included.

Table 23.4 Number of independent components of the various property tensors

The numerical values of the elastic and elastooptic constants are often averages of three or more measurements, as presented in [23.10, 23.11, 23.7, 23.8, 23.9]. In such cases, the corresponding Landolt–Börnstein volume is cited together with the most reliable (latest) reference. The standard deviation of the averaged value is given in parentheses. Vertical bars ∥ mean the modulus of the corresponding quantity. The absolute scale for the second-order nonlinear susceptibilities of crystals is based on [23.12, 23.13, 23.14]. The second-order susceptibilities for all crystals measured relative to a standard crystal have been recalculated accordingly. In particular, all previous measurements relative to KDP and quartz have been normalized to $d_{36}\,(\text{KDP})={\mathrm{0.39}}\,{\mathrm{pm/V}}$ and $d_{11}\,(\mathrm{SiO_{2}})={\mathrm{0.30}}\,{\mathrm{pm/V}}$. These data lead to an accurate, self-consistent set of absolute second-order nonlinear coefficients [23.14]. All numerical data are for room temperature (300 K) and in SI units.

4 Tables of Numerical Data for Dielectrics and Electrooptics

Table 23.5 Isotropic materials

Table 23.6 Cubic, point group m3m (O_h) materials

Table 23.7 Cubic, point group $\bar{4}3m$ (T_d) materials

Table 23.8 Cubic, point group 23 (T) materials

Table 23.9 Hexagonal, point group $\bar{6}m2$ (D_3h) materials

Table 23.10 Hexagonal, point group 6mm (C_6v) materials

Table 23.11 Hexagonal, point group 6 (C₆) materials

Table 23.12 Trigonal, point group $\bar{3}m$ (D_3d) materials

Table 23.13 Trigonal, point group 32 (D₃) materials

Table 23.14 Trigonal, point group 3m (C_3v) materials

Table 23.15 Tetragonal, point group 4 ∕ mmm (D_4h) materials

Table 23.16 Tetragonal, point group 4 ∕ m (C_4h) materials

Table 23.17 Tetragonal, point group 422 (D₄) materials

Table 23.18 Tetragonal, point group $\bar{4}2m$ (D_2d) materials

Table 23.19 Tetragonal, point group 4mm (C_4v) materials

Table 23.20 Orthorhombic, point group mmm (D_2h) materials

Table 23.21 Orthorhombic, point group 222 (D₂) materials

Table 23.22 Orthorhombic, point group mm2 (C_2v) materials

Table 23.23 Monoclinic, point group 2 (C₂) materials