8.1Volume of Electric Dipoles

Let us briefly revise the concept of a dipole. An elementary electric dipole is constructed by a negative and a positive charges of equal value separated by a small distance d (Figure 8.1).

The dipole moment p of this system is given by

The potential fields Φ_p due to such a dipole, when r ≫ d, is given by

and the electric field is given by

For a more general distribution of volume charges in a localized volume, where the total charge is zero, the dipole moment (Figure 8.2) is given by

The potential due to a volume of dipoles in V′ of dipole moment P per unit volume, also called polarization vector (C/m²) (see Figure 8.3), is given by

Equation 8.5 may be transformed into

By comparing Equation 8.6 with similar expressions for potential due to monopole distributions, the following equivalence may be drawn: a volume of dipoles of dipole moment density P is equivalent to a monopole volume charge of density ρ_Vb and a surface charge density ρ_Sb:

(see Figure 8.4).

Gauss’s law in free space with volume of dipoles can be written as

where ρ_V is the volume charge density due to free charges and ρ_Vb is an equivalent volume charge density due to the volume of dipoles.

By substituting Equation 8.7 into Equation 8.9, we obtain

Let us denote

For linear materials, P is proportional to E and let us write this linear dependence in terms of electric susceptibility χ_e:

Substituting Equation 8.12 into Equation 8.11, we obtain

where

and is called relative permittivity.

8.2Frequency-Dependent Dielectric Constant

Under the influence of an electric field, the positive and negative charges inside each atom are displaced from their equilibrium position. Since the mass of the nucleus M is much larger than that of an electron m, M ≫ m, the positive charge is assumed to be stationary. The electron moves and exhibits a friction coefficient ν. The mechanical equivalence of the system is shown [1] in Figure 8.5.

When a time-harmonic electric field of angular frequency ω, E = E₀ e^jωt, is applied to the atom, the force equation is given by

where x is the position of the electron relative to the atom. To find the particular solution, let x = x₀ e^jωt, then

where

and q is the absolute value of the charge and m is the mass of the electron. The displacement of the charge centers gives rise to a dipole moment p = −qx. If there are N dipoles created per unit volume, then the polarization P, which is the dipole moment per unit volume, is given by

The electric susceptibility χ_e defined by P = ε₀χ_eE can be computed as

The complex dielectric constant ε_r = ε_p = 1 + χ_e = n². Because of the presence of ν, ε_r and n are complex. The parameter ν is called the collision frequency and has the unit of radians/second. Denoting the numerator of the right-most part of Equation 8.20 by ${\mathrm{\omega }}_{\text{p}}^2$, the dielectric (constant) function is

where

In simple metals, there are free electrons that are not bound (k = 0) but are free to move, that is, ω₀ = 0, and

On the other hand, for some dielectrics, there are multiple resonances and for such materials, the most general form of the dielectric function is given by

In the above, ε_∞ is the dielectric constant at ω = ∞, ε_s the relative permittivity at DC, ω_R a resonant frequency, and δ_R a damping constant. A sketch of the real part of a dielectric constant of a hypothetical material is given in Figure 8.10. A rapid change in the real part of the dielectric constant around a resonant frequency is indicated in the figure. Around this resonant frequency, the imaginary part of the dielectric function would have a significant value and the electromagnetic wave is strongly absorbed.

8.3Modeling of Metals

Equation 8.23 gives the dielectric function of simple metals such as sodium. Since the electron density N is of the order of 10²³/cm³, the plasma frequency ω_p ≈ 2 × 10¹⁶ rad/s. The collision frequency ν is of the order of 10¹³. Since ν/ω_p ≈ 10⁻³, the metal is modeled as a low-loss plasma. Modeling of such metals may be further subdivided into three frequency regions.

8.3.3Case 3: ω > ω_p (High-Frequency Region)

In this region, the plasma becomes a low-loss dielectric. In the limit ${\mathrm{\nu }}^2\ll \left({\mathrm{\omega }}^2-{\mathrm{\omega }}_{\text{p}}^2\right)$ and ${\mathrm{\nu }}^2\ll {\mathrm{\omega }}^2\left({\mathrm{\omega }}^2-{\mathrm{\omega }}_{\text{p}}^2\right)\text{}/\text{}{\mathrm{\omega }}_{\text{p}}^4$, we can obtain the following approximations:

(8.34)$$n_{\text{R}}={\left(1-\frac{{\mathrm{\omega }}_{\text{p}}^2}{{\mathrm{\omega }}^2}\right)}^{1/2},$$

(8.35)$$n_{\text{I}}=\frac{\mathrm{\nu }{\mathrm{\omega }}_{\text{p}}^2}{2{\mathrm{\omega }}^3}{\left[1-\frac{{\mathrm{\omega }}_{\text{p}}^2}{{\mathrm{\omega }}^2}\right]}^{-1/2},$$

(8.36)$$\mathrm{\delta }=\frac{c}{{\mathrm{\omega }}_{\text{p}}}\left(\frac{2{\mathrm{\omega }}^2}{\mathrm{\nu }{\mathrm{\omega }}_{\text{p}}}\right){\left(1-\frac{{\mathrm{\omega }}_{\text{p}}^2}{{\mathrm{\omega }}^2}\right)}^{1/2}.$$

Figure 8.6 shows the variation of n_R, n_I, α, and β for a laboratory plasma of electron density N = 10¹³/cm³ corresponding to a plasma frequency ω_p ≈ 3 × 10¹⁰. In metals, the electron density is much higher but the features of conducting, cutoff, and dielectric regions are all present.

8.4Plasma Medium

In Equation 8.22, we defined a new quantity ω_p, called the angular plasma frequency, which is proportional to the square root of the electron density. In this section, we will explore the characteristics of the medium, called plasma medium. Plasma state is a fourth state of matter, the other three being solid, liquid, and gaseous states. On heating a solid material, we can successfully obtain liquid and gaseous states. If we heat the material further, we can obtain an ionized gas. The electrons are stripped away from the neutral atoms and in this state of matter we have positive ions and free electrons. Though the electrons are no longer bound in the atom, they are still subject to some restoring forces due to Coulomb forces between positive charges and negative charges. The effect of these long-range Coulomb forces, which vary as 1/r², may be understood by considering the following one-dimensional problem. Consider a plasma medium that has positive ions and electrons with overall charge neutrality to begin with. As shown in Figure 8.7, let us assume that we disturbed the equilibrium situation by displacing a layer of electrons by a small distance z. At one end, there will be an excess of electric charge, −Nqz per unit area and at the other end, there will be an excess of positive ion charge +Nqz per unit area. The electric field created by this imbalance from charge neutrality at the ends may be calculated from the boundary condition D_n = ρ_s = Nqz. Hence,

The force experienced by an electron in the presence of this electric field is given by

Equating this to the inertial force, we can write the force balance equation as follows:

Equation 8.40 is an equation of oscillatory motion with the solution

where ${\mathrm{\omega }}_{\text{p}}^2$ is given by Nq²/mε₀. Thus, ω_p gives the natural frequency of oscillation of the electronic charge layer.

8.5Polarizability of Dielectrics

If there are N molecules for unit volume, then the polarization may be written as

where α_T is the molecular polarizability and g the ratio between local field E_loc acting on the molecule and the applied field E. These two fields differ because of the presence of surrounding molecules. Since ε_r = 1 + χ_e, we obtain from Equation 8.42,

It can be shown that

if the surrounding molecules act in a spherically symmetric fashion. Figure 8.8 shows the applied and local fields for the spherically symmetrical case.

The polarization $P=\widehat{z}P$ creates equivalent charges on the walls. The equivalent charge dq on the walls is given by P · ds, where ds is the vector differential surface element on the sphere given by $\widehat{r}{r}_0^{2}sin\mathrm{\theta }\text{d}\mathrm{\theta }\text{d}\mathrm{\varphi }$. Therefore, the electric field dE_p at the center of the sphere is given by

Since

and

From Equations 8.43 and 8.44, we obtain

This is called the Clausius–Mossotti relation. We have noted from the above equations that the local field is different from the applied electric field. However, in writing the force Equation 8.16, the RHS is electric force on the charge due to the applied electric force. If we use, instead, the force due to local electric field, then Equation 8.20 has to be modified: in Equation 8.20, replace ω₀ by ω₁, where

Figure 8.9 shows the general shape of ε_p versus ω₁, where ε_p is given by Equation 8.21 with ω₁ replacing ω₀.

The complex permittivity

has analytical properties given by the Kramer–Kronig relations:

The polarization α_T has contributions from several effects and may be written as

where the subscripts e, i, and d stand for electronic, ionic, and permanent dipole contributions to the polarizability, respectively. The electronic and ionic components may be generalized in the form

where F_j is the measure of the strength of the jth resonance. The permanent dipole contribution is different in the sense that the force opposing the complete alignment of the dipoles in the direction of the applied electric field is of thermal nature:

where T is the temperature, k_B the Boltzmann constant, p the permanent dipole moment, and τ the relaxation time. Figure 8.10 [4] is a sketch of frequency response of a hypothetical dielectric.

In addition to the three polarization mechanisms discussed above, interfacial polarization or space–charge polarization [3] gives rise to a complex dielectric constant:

This polarization is due to the large-scale field distortions caused by the piling of space charges in the volume or of the surface charges at the interfaces between different small portions of materials with different characteristics. Geophysical media are examples [4] of such materials.

8.6Mixing Formula

Mixing formula gives the effective dielectric constant of dielectric inclusions of a material of dielectric constant ε_r2 embedded in a host dielectric material of dielectric constant ε_r1. Let the inclusions be spheres of radius a. Let the radius a be small when compared to the wavelength. This assumption permits us to ignore the effects of scattering. Let the spheres be sparsely distributed in the host material so that the fractional volume f, which is a fraction of the volume occupied by the spheres, is lower than a few percent, f ≪ 1. This assumption permits us to ignore the consideration of a correlation between spheres. The situation described here is similar to the situation described in Section 8.5, where the dielectric constant of the material consisting of many dipoles is given by

where N is the number of dipoles per unit volume and α the polarizability of the dipole. In our case, the effective dielectric constant ε_e is given by

where N is the number of spheres per unit volume. The polarizability α of the sphere is given by [3]

Equation 8.64 is obtained after making several simplifying assumptions, some of which are stated above.

Since

where V is the volume of the sphere, we obtain the effective dielectric constant as

In the above equation, we have

This is called the Maxwell–Garnet mixing formula. If the inclusion has a nonspherical shape, then the expression for α for that shape should be used. For more information on mixing formulas, see [3,5].

8.7Good Conductors and Semiconductors

The transport of charge gives rise to the flow of current, and the constitutive relation of a simple conductor is given by J = σE. Based on the magnitude of σ, we classify conductors into

• semiconductors (σ ~10⁻² S/m) and

• conductors (metals and alloys) (σ ~10⁻² − 10⁸ S/m).

In general, the charge carriers in a conductor are free electrons, whereas in a semiconductor the charge carriers are electrons as well as holes. We have seen that σ_DC = Nq²/mν. We can use this expression for the conductivity σ provided ω < ν and ${\mathrm{\nu }}^2\ll {\mathrm{\omega }}_{\text{p}}^2$. These conditions are satisfied in metals for frequencies right up to optical frequencies. The conductivity of metals may also be written in terms of its electron mobility μ_e, where

The power dissipation per unit volume is

Let us discuss the collision frequency ν a little more in the context of conductors. The mean free path Λ of an electron can be related to the average drift velocity of the electron:

Here τ is the average collision time and is related to the collision frequency ν by

In the context of conductors, τ is also called relaxation time. Since the conductivity depends on the average collision time, the electrical resistivity ρ = 1/σ increases with temperature. It may be expressed in terms of its value ρ_RT at room temperature T_R as

For pure metals, α_R, the temperature coefficient of resistivity, is about 0.004/°C. For a table of values for α_R, see [5]. As the absolute temperature T approaches zero, the pure metal has zero resistivity and becomes typically a superconductor. The temperature dependence of conductivity σ in pure metals as dictated by the mobility of electrons may be written as

Intrinsic semiconductors belong to group IV elements that include silicon and germanium. The conductivity of intrinsic semiconductors may be expressed as

where N_n and N_p are the intrinsic electron and hole densities, respectively. Theoretical and experimental studies show that σ can be expressed in terms of forbidden gap energy E_g [5]:

Germanium (Ge) has a gap energy of 0.67 eV, while silicon (Si) has an E_g of 1.12 eV. Ge has a higher mobility than Si, hence Ge devices can be operated at higher frequencies. However, Ge devices are more sensitive to temperature changes. Extrinsic semiconductors are made by adding impurities (dopants) to the intrinsic semiconductors. The dopants are from group III or group V elements. The electrical properties of extrinsic semiconductors are strongly influenced by the dopants. N-type semiconductors have dopants, from group V (e.g., phosphorous) and have a set of easily activated electrons in the conduction band. Thus, donor impurities add to the free-electron population in the conduction band, facilitating an increased conductivity of the material. The P-type semiconductors are constituted by the addition of impurity atoms from group III elements (e.g., aluminum) and the impurity atoms are called acceptors. The conductivity of N-type is given by

where N_D is the donor concentration. The conductivity of P-type is given by

where N_A is the acceptor concentration.

Effective masses of electrons and holes in semiconductors are different from free electron mass due to various interactive force fields experienced by the charge carriers. As an example, for silicon, $m_{\text{e}}^{*}\text{}/\text{}{m}_0\approx 0.190-0.260$ and $m_{\text{h}}^{*}\text{}/\text{}{m}_0=0.5$, where $m_{\text{e}}^{*}$ is the effective mass of the electron, $m_{\text{h}}^{*}$ the effective mass of the hole, and m₀ the mass of the free electron. For more details, see [5]. When a semiconductor is subjected to crossed electric (E) and magnetic (H) fields, a small potential field (E_H) orthogonal to H and E is induced. This phenomenon is called the Hall effect and is discussed in Chapter 11.

8.8Perfect Conductors and Superconductors

The resistivity of many metals and alloys drops abruptly when they are cooled to a sufficiently low-temperature T_c (Figure 8.11a). The superconductivity state [5,67] is destroyed and normal conductivity state is restored if the current density J_c or the magnetic field H_c exceeds a threshold value. An empirical law relating H_c with the temperature T is given by [5]

where H_c0 is the critical magnetic field at T = 0. The boundary between superconductivity state and normal conducting state is shown in Figure 8.11b.

The empirical relation between B_max and T_max is given by

Provided that the current density, the magnetic field, and the temperature are less than the threshold values, the resistivity drops to zero and the conductor may be treated as a perfect conductor. As we noted in the previous section, a DC magnified field H_DC can exist in a conductor even though E_DC = 0. However, in a perfect superconductor, it will be shown that H_DC = 0. Thus, a perfect superconductor is not only a perfect conductor, but also a perfect diamagnet. The magnetic field in a specimen is expelled as the temperature is brought below the critical temperature T_c. This phenomenon is called the Meissner effect. We investigate the electromagnetic phenomena in a superconductor based on a macroscopic theory given by London.

Instead of the usual Ohm’s law for normal conductors, that is,

London postulated that in a superconductor, the current density J_s is given by

where C is a constant characteristic of a given material and A the usual vector potential. Taking the curl of Equation 8.81, we have

From Ampere’s law of direct current, we have

Since ∇ · B = 0,

The solution of Equation 8.86 for a semi-infinite superconductor occupying the space z > 0 is given by

where

and B is in x−y-plane. We will show later that

where n_s is the density of superconducting electrons, and q and m are the charge and mass of an electron, respectively. Thus,

which is a real quantity. Equation 8.87 shows that λ_L is the length characterizing the decay of the flux density from its value at the surface of the superconductor. It is called the London penetration depth. For lead, λ_L = 146 Å. In a good normal conductor, the time-harmonic magnetic field penetrates to a depth δ:

where f is the frequency. For direct current, δ = ∞, signifying uniform distribution of DC magnetic field. In a superconductor, even for DC, λ_L is very small; this explains the Meissner effect. Let us next give a phenomenological basis for Equation 8.81. The current density J_s in the superconductor is given by

where $n_{\text{s}}^{*}$ is the density of carriers of charge q_s and v_s is their drift velocity. In the presence of a magnetic field B_s = ∇ × A_s, the total momentum P_s is

where m_s is the mass of the charge carriers. The last term is the additional momentum due to the presence of the magnetic field. From Equations 8.92 and 8.93, we have

For super electrons, the total momentum is zero, provided

Equation 8.95 is obtained by making use of Equation 8.81. To calculate λ_L, we need to find the values to be used for m_s, $n_{\text{s}}^{*}$, and q_s. In a superconducting state, the electrons are loosely associated as pairs [6]. The electrons in a given pair have momentum ℏk and −ℏk, so that the net pair momentum is zero. The electron pair behaves like a single particle, giving rise to

where m, q, and n_s are the mass, absolute value of the charge, and the number density of the electrons in a superconducting state, respectively. For a superconductor, the charge carriers are paired electrons. Substituting Equation 8.96, Equation 8.97 and Equation 8.98 into Equation 8.95, we see that Equation 8.95 is the same as Equation 8.89. From Equation 8.90, we see that λ_L should always be finite. Perfect superconductor, which is a perfect diamagnetic (Meissner effect of flux exclusion), is an idealization of a practical situation where λ_L ≪ l, where l is the relevant dimension of the sample.

A two-fluid model of superconductors at high frequencies is developed by considering that a fraction of the electrons are in superconductive state and they form pairs. In normal conductors, free electrons are assumed to move under the influence of the electric field and experience collisions as described earlier. The superconducting electron pairs, however, are immune from collisions. In the two-fluid model, we write separate momentum equations for each of the fluids, the normal fluid denoted by the subscript n and the superconducting fluid by the subscript s:

For a time-harmonic field, we have from Equations 8.99, 8.100, 8.102, and 8.103 that

where τ = 1/ν is the momentum relaxation time. An equivalent circuit for the above may easily be constructed as shown in Figure 8.12, where the analogy is

In the above circuit, V = jωL_sI_s and from the analogy

From the analogy of the second branch,

and

By comparison,

Note that for ω = 0, ωL_n = 0, and R_n is short-circuited by ωL_s. Thus, for T < T_c, although there are normal as well as superconducting electrons, the resistivity is identically zero. If ω is finite, then R_n is no longer shorted.

Maxwell’s equations for the superconductor, based on the two-fluid theory, are written as

The wave equation for H may be written as

Note that λ_L = ∞, if n_s = 0. The last term on the LHS is negligible for a good conductor. This term arises because of the displacement current in the conductor. Let us obtain a one-dimensional time-harmonic solution, assuming

The magnetic field satisfies the equation

where $\mathrm{\delta }=1\text{/}\sqrt{\mathrm{\pi }f{\mathrm{\mu }}_0{\mathrm{\sigma }}_{\text{n}}}$ is the skin depth.

Assuming further

where

For the direct current case, δ = ∞, θ =1, and γ = 1/λ_L. λ_L is the penetration depth of H_DC in a superconductor. This parameter is the quantitative measure of the Meissner effect. If it is a normal conductor and ω ≠ 0, then δ is finite, λ_L is large, and

thus

as expected.

The surface resistance may be calculated, following the steps in Section 2.5. Alternatively, for a unit width and unit length along y and x axes, respectively,

where

From Equations 8.135 and 8.136,

Since, E₀ is the electric field on the surface at z = 0,

and from Maxwell’s equation 8.115,

Thus, we obtain the relation between E₀ and H₀:

The surface impedance is

8.9Magnetic Materials

Magnetic materials [4,5,6] get magnetized. The quantitative measure of magnetization is given by the magnetization vector M (A/m³), which is the net magnetic dipole moment p_m per unit volume. It is analogous to the polarization P (C/m²), which is the net electric dipole moment per unit volume.

One can think of an analogous situation with reference to magnetic charges and define a magnetic dipole moment

where the magnetic charges +q_m and −q_m are separated by a small distance d. It is worth noting at this time that we cannot isolate magnetic charges, they can appear only as a pair, and the monopole magnetic charge density is zero. When we view currents as the sources of magnetic fields, we can draw the following analogy of a magnetic dipole and a loop of current (see Figure 8.13). At a greater distance, they have the same vector potential:

where

For a more general loop of current,

If the loop is planar and the origin is in the plane (Figure 8.14),

and

If the source is a volume current J, then

The vector magnetic potential due to a volume of magnetic dipoles (Figure 8.15) of dipole density M (A/m) is given by

Equation 8.149 may be manipulated to yield

By comparing Equation 8.150 with the expressions for A due to volume current of density J_b and surface current of density K_b, the following equivalence may be given (Figure 8.16):

A topic of interest connected to a magnetic dipole is the torque T experienced by the dipole when it is brought into an external B field. It is given by

A current loop experiences a torque in the presence of an external B field. The loop will rotate and when the normal to the plane of the loop is parallel to B, an equilibrium is achieved. An electron in an orbit constitutes current. If an electron of charge −q executes an orbital motion of angular velocity ω, the corresponding current is given by

where ω depends on the environment to which the electron is subjected. Consider the simplest case of an electron circulating around a proton of charge +q in the presence of a magnetic field H₀ perpendicular to the circular orbit (Figure 8.17).

From Lorentz force equation,

Assuming that the massive proton is stationary, we have

where we have used the relation v = ωr. The term mω²r on the RHS of the above equation is the counterbalancing centrifugal force due to the circular motion of the electron. Solving for ω,

where

In the above, ω₀ is the frequency of circulation in the absence of H₀. If ω_L ≫ ω₀, then corresponding to the case of large H₀ or large r,

where ω_L is called the Larmor frequency and ω_b is called the cyclotron frequency. On the other hand, if H₀ is small or r is small (ω_L ≪ ω₀), then the case for electrons orbiting in an atom is

Diamagnetic materials have μ_r slightly less than 1. It occurs in materials where the spin magnetization, explained later, is zero. The magnetization is only due to orbital motion. The magnetization due to the electrons spinning on their axis is zero due to even number of spins. The orbital magnetic dipole moment is given by

The net dipole moment per unit volume M that arises from the orbital motion of the electrons is opposite to the original H field and when related to the field through magnetic susceptibility χ_m,

where χ_m will be a negative value. This results in a relative permeability μ_r:

which is slightly less than 1. The above explanation is oversimplified but serves the purpose of showing that magnetization due to the orbital motion only leads to diamagnetization.

A simple explanation for paramagnetism in which μ_r is slightly larger than 1 is in terms of magnetization due to electron spin. The electrons of an atom with an even number of electrons usually exist in pairs, with the members of pairs having opposite spin directions, thereby canceling each other’s spin magnetic moment. If the number of electrons is odd, individual atoms have net magnetic moment. If an external magnetic field is applied, then the individual atoms will align with it subject to the randomizing effect of thermal motion. The magnetic moments partially line up with the external fields giving rise to positive susceptibility. However, in paramagnetic material, the coupling between the magnetic moments of various atoms is relatively weak.

Ferrimagnetic materials have magnetized domains. A magnetized domain of a material (of 10⁻¹⁰ m³ size) has atoms that are aligned parallel to each other (typically of the order of 10¹⁰ atoms in a domain). The domains are randomly oriented but in the presence of external fields they align. Examples of ferromagnetic materials are iron, nickel, and cobalt. The relative permeability is of the order of 3000. They exhibit nonlinearity, residual magnetism, and hysteresis. Soft ferromagnetic materials have narrow hysteresis loops and hence they can be more easily magnetized and demagnetized. Ferromagnetic materials are, in general, metallic in nature and their low resistivity gives rise to high eddy current losses at microwave frequencies. These losses are proportional to the square of the frequency and also proportional to the cube of the sample thickness. Laminating usually reduces these losses. At microwave frequencies, the losses are still large for vacuum-deposited thin ferromagnetic films of ~100–10,000 Å.

Ferromagnetic materials, known as ferrites and garnets, have high resistivity of up to 10⁷ Ω m when compared to 10⁻⁷ Ω m for iron. This material is a mixture of metallic oxides of high resistivity. They have a lower magnetization and are not used in applications requiring high flux density as in power transformers. They are very useful at microwave frequencies. The high-resistivity property arises because, unlike ferromagnetic materials, the constituents of ferrite materials, both metallic and oxygen ions, have no free 3s or 4s electrons available for conduction. Ferrites in the presence of a static magnetic field behave as an anisotropic magnetic material. The wave propagation in such a material is discussed in Appendix 11B.

8.10Chiral Medium [8,9]

Figure 8.18 shows an approximation to a single turn left-handed and right-handed helix. The approximation consists of a circular conducting loop in a plane (x–y-plane) of radius a and a conductor of length 2l along the normal to the plane (z-axis). The distance between the turns (pitch) of the helix is 2l.

Assume a time-harmonic incident electric excitation field E₀ along the z-axis. The voltage induced is given by

The current is

where Z_H is the impedance of the helix. The electric dipole moment due to electric excitation is