# The interaction of radio-frequency fields with dielectric materials at macroscopic to mesoscopic scales.

The goal of this paper is to overview radio-frequency (RF) electromagnetic interactions with solid and liquid materials from the macroscale to the nanoscale. The overview is geared toward the general researcher. Because this area of research is vast, this paper concentrates on currently active research areas in the megahertz (MHz) through gigahertz (GHz) frequencies, and concentrates on dielectric response. The paper studies interaction mechanisms both from phenomenological and fundamental viewpoints. Relaxation, resonance, interface phenomena, plasmons, the concepts of permittivity and permeability, and relaxation times are summarized. Topics of current research interest, such as negative-index behavior, noise, plasmonic behavior, RF heating, nanoscale materials, wave cloaking, polaritonic surface waves, biomaterials, and other topics are overviewed.Relaxation, resonance, and related relaxation times are overviewed. The wavelength and material length scales required to define permittivity in materials is discussed.

Key words: dielectric; electromagnetic fields; loss factor; metamaterials; microwave; millimeter wave; nanoscale; permeability; permittivity; plasmon; polariton.

Accepted: August 25, 2011

Published: February 2, 2012

http://dx.doi.org/10.6028/jres.117.001

1. Introduction

1.1 Background

In this paper we will overview electromagnetic inter-actions with solid and liquid dielectric and magnetic materials from the macroscale down to the nanoscale. We will concentrate our effort on radio-frequency (RF) waves that include microwaves (MW) and millimeter-waves (MMW), as shown in Table 1. Radio frequency waves encompass frequencies from 3 kHz to 300 GHz. Microwaves encompass frequencies from 300 MHz to 30 GHz. Extremely high-frequency waves (EHF) and millimeter waves range from 30 GHz to 300 GHz.

Table 1. Radio-Frequency Bands (1) frequency wavelength band 3 - 30 kHz 100 - 10 km VLF 30 - 300 kHz 10 - 1 km LF 0.3 - 3 MHz 1 - 0.1 km MF 3 - 30MHz 100 - 10 m HF 30 - 300 MHz 10 - 1 m VHF 300 - 3000 MHz 100 - 10 cm UHF 3 - 30GHz 10 - 1 cm SHF 30 - 300 GHz 10 - 1 mm EHF 300 - 3000 GHz 1 - 0.1 mm THz

Many devices operate through the interaction of RF electromagnetic waves with materials. The characterization of the interface and interaction between fields and materials is a critical task in any electromagnetic (EM) device or measurement instrument development, from nanoscale to larger scales. Electromagnetic waves in the radio-frequency range have unique properties. These attributes include the ability to travel in guidedwave structures, the ability of antennas to launch waves that carry information over long distances, possess measurable phase and magnitude, the capability for imaging and memory storage, dielectric heating, and the ability to penetrate materials.

Some of the applications we will study are related to areas in microelectronics, bioelectromagnetics, home-land security, nanoscale and macroscale probing, magnetic memories, dielectric nondestructive sensing, radiometry, dielectric heating, and microwave-assisted chemistry. For nanoscale devices the RF wavelengths are much larger than the device. In many other applications the feature size may be comparable or larger than the wavelength of the applied field.

We will begin with an introduction of the interaction of fields with materials and then overview the basic notations and definitions of EM quantities, then progress into dielectric and magnetic response, definitions of permittivity and permeability, fields, relaxation times, surfaces waves, artificial materials, dielectric and magnetic heating, nanoscale interactions, and field fluctuations. The paper ends with an overview of biomaterials in EM fields and metrologic issues. Because this area is very broad, we limit our analysis to emphasize solid and liquid dielectrics over magnetic materials, higher frequencies over low frequencies, and classical over quantum-mechanical descriptions. Limited space will be used to overview electrostatic fields, radiative fields, and terahertz interactions. There is minimal discussion of EM interactions with nonlinear materials and gases.

1.2 Electromagnetic Interactions From the Microscale to Macroscale

In this section we want to briefly discuss electromagnetic interaction with materials on the microscale to the macroscale.

Matter is modeled as being composed of many uncharged and charged particles including for example, protons, electrons, and ions. On the other hand, the electromagnetic field is composed of photons. The internal electric field in a material is related to the sum of the fields from all of the charged particles plus any applied field. When particles such as biological molecules, cells, or inorganic materials are subjected to external electric fields, the molecules can respond in a number of ways. For example, a single charged particle will experience a force in an applied electric field. Also, in response to electric fields, the charges in a neutral many-body particle may separate to form induced dipole moments, which tend to align in the field; however this alignment is in competition with thermal effects. Particles that have permanent dipole moments will interact with applied dc or high-frequency fields. In an electric field, particles with permanent dipolemoments will tend to align due to the electrical torque, but in competition to thermal randomizing effects. When EM fields are applied to elongated particles with mobile charges, they tend to align in the field. If the field is nonuniform, the particle may experience dielectrophoresis forces due to field gradients.

On the microscopic level we know that the electro-magnetic field is modeled as a collection of photons (2). In theory, the electromagnetic field interactions with matter may be modeled on a microscopic scale by solving Schrodinger's equation, but generally other approximate approaches are used. At larger scales the interaction with materials is modeled by macroscopic Maxwell's equations together with constitutive relations and boundary conditions. At a courser level of description, phenomenological and circuit models are commonly used. Typical scales of various objects are shown in Fig. 1. The mesoscopic scale is where classical analysis begins to be modified by quantum mechanics and is a particularly difficult area to model.

The interaction of the radiation field with atoms is described by quantum electrodynamics. From a quantum-mechanical viewpoint the radiation field is quantized, with the energy of a photon of angular frequency [omega] being E = h[omega]. Photons exhibit waveduality and quantization. This quantization also occurs in mechanical behavior where lattice vibrational motion is quantized into phonons. Commonly, an atom is modeled as a harmonic oscillator that absorbs or emits photons. The field is also quantized, and each field mode is represented as a harmonic oscillator and the photon is the quantum particle.

The radiation field is usually assumed to contain a distribution of various photon frequencies. When the radiation field interacts with atoms at the appropriate frequency, there can be absorption or emission of photons. When an atom emits a photon, the energy of the atom decreases, but then the field energy increases. Rigorous studies of the interaction of the molecular field with the radiation field involve quantization of the radiation field by expressing the potential energy V(r) and vector potential A(r, t) in terms of creation and annihilation operators and using these fields in the Hamiltonian, which is then used in the Schrodinger equation to obtain the wavefunction (see, for example, (3)). The static electromagnetic field is sometimes modeled by virtual photons that can exist for the short periods allowed by the uncertainty principle. Photons can interact by depositing all their energy in photoelectric electron interactions, by Compton scattering processes, where they deposit only a portion of the energy together with a scattered photon, or by pair production. When a photon collides with an electron it deposits its kinetic energy into the surrounding matter as it moves through the material. Light scattering is a result of changes in the media caused by the incoming electromagnetic waves (4). In Rayleigh elastic light scattering, the photons of the scattered incident light are used for imaging material features. Brillouin scattering is an inelastic collision that may form or annihilate quasiparticles such as phonons, plasmons, and magnons. Plasmons relate to plasma oscillations, often in metals, that mimic a particle and magnons are the quanta in spin waves. Brillouin scattering occurs when the frequency of the scattered light shifts in relation to the incident field. This energy shift relates to the energy of the interacting quasiparticles. Brillouin scattering can be used to probe mesoscopic properties such as elasticity. Raman scattering is an inelastic process similar to Brillouin scattering, but where the scattering is due to molecular or atomic-level transitions. Raman scattering can be used to probe chemical and molecular structure. Surface-enhanced Raman scattering (SERS) is due to enhancement of the EM field by surface-wave excitation (5).

Optically transparent materials such as glass have atoms with bound electrons whose absorption frequencies are not in the visible spectrum and, therefore, incident light is transmitted through the material. Metallic materials contain free electrons that have a distribution of resonant frequencies that either absorb incoming light or reflect it. Materials that are absorbing in one frequency band may be transparent in another band.

Polarization in atoms and molecules can be due to permanent electric moments or induced moments caused by the applied field, and spins or spin moments. The response of induced polarization is usually weaker than that of permanent polarization, because the typical radii of atoms are on the order of 0.1 nm. On application of a strong external electric field, the electron cloud will displace the bound electrons only about [10.sup.-16] m. This is a consequence of the fact that the atomic electric fields in the atom are very intense, approximately 10 (11) V/m. The splitting of spectral lines due to the interaction of electric fields with atoms and molecules is called the Stark effect. The Stark effect occurs when interaction of the electric-dipole moment of molecules interacts with an applied electric field that changes the potential energy and promotes rotation and atomic transitions. Because the rotation of the molecules depends on the frequency of the applied field, the Stark effect depends on both the frequency and field strength. The interaction of magnetic fields with molecular dipole moments is called the Zeeman effect. Both the Stark and Zeeman effects have fine-structure modifications that depend on the molecule's angular momentum and spin. On a mesoscopic scale, the interactions are summarized in the Hamiltonian that contains the internal energy of the lattice, electric and magnetic dipole moments, and the applied fields.

In modeling EM interactions at macroscopic scales, a homogenization process is usually applied and the classical Maxwell field is treated as an average of the photon field. There also is a homogenization process that is used in deriving the macroscopic Maxwell equations from the microscopic Maxwell equations. The macroscopic Maxwell's equations in materials are formed by averaging the microscopic equations over a unit cell. In this averaging procedure, the macroscopic charge and current densities, the magnetic field H, the magnetization M, the displacement field D, and the electric polarization field P are formed. At these scales, the molecule dipole moments are averaged over a unit cell to form continuous dielectric and magnetic polarizations P and M. The constitutive relations for the polarization and magnetization are used to define the permittivity and permeability. At macroscopic to mesoscopic scales the permittivity, permeability, refractive index, and impedance are used to model the response of materials to applied fields. We will discuss this in detail in Sec. 4.5. Quantities such as permittivity, permeability, refractive index, and wave impedance are not microscopic quantities, but are defined through an averaging procedure. This averaging works well when the wavelength is much larger than the size of the molecules or atoms and when there are a large number of molecules. In theoretical formulations for small scales and wavelengths near molecular dimensions, the dipole moment and polarizability tensor of atoms and molecules can be used rather than the permittivity or permeability. In some materials, such as magnetoelectric and chiral materials, there is a coupling between the electric and magnetic responses. In such cases the time-harmonic constitutive relations are [~.B]([omega]) = [micro][~.H]([omega]) + [[eta].sub.1][~.E]([omega]) and [~.D]([omega]) = [epsilon][~.E]([omega]) + [[eta].sub.2][~.H]([omega]). In most materials the constitutive relations [~.B]([omega]) = [[micro].sub.0]([~.M]([omega]) + [~.H]([omega])) and [~.D]([omega]) = [[epsilon].sub.0] [~.E]([omega]) + [~.P]([omega]) are used.

In any complex lossy system, energy is converted from one form to another, such as the transformation of EM energy to lattice kinetic energy and thermal energy through photon-phonon interactions. Some of the energy in the applied fields that interact with materials is transfered into thermal energy as infrared phonons. In a waveguide, there is a constant exchange of energy between the charge in the guiding conductors and the fields (6).

When the electromagnetic field interacts with material degrees of freedom, a collective response may be generated. The term polariton relates to bosonic quasiparticles resulting from the coupling of EM photons or waves with an electric or magnetic dipole-carrying excitation (4), (5). The resonant and nonresonant coupling of EM fields in phonon scattering is mediated through the phonon-polariton transverse-wave quasi-particle. Phonon polaritons are formed from photons interacting with terahertz to optical phonons. Ensembles of electrons in metals form plasmas and high-frequency fields applied to these electron gases produce resonant quasiparticles, commonly called plasmons. Plasmons are a collective excitation of a group of electrons or ions that simultaneously oscillate in the field. An example of a plasmon is the resonant oscillation of free electrons in metals and semiconductors in response to an applied high-frequency field. Plasmons may also form at the interface of a dielectric and a metal and travel as a surface wave with most of the EM energy confined to the low-loss dielectric. A surface plasmon polariton is the coupling of a photon with surface plasmons. Whereas transverse plasmons can couple to an EM field directly, longitudinal plasmons couple to the EM field by secondary particle collisions. In the microwave and millimeter wave bands artificial structures can be machined in metallic surfaces to produce plasmons-like excitations due to geometry. Magnetic coupling is mediated through magnons and spin waves. A magnon is a quantum of a spin wave that travels through a spin lattice. A polaron is an excitation caused by a polarized electron traveling through a material together with the resultant polarization of adjacent dipoles and lattice distortion (4). All of these effects are manifest at the mesoscale through macroscale in the constitutive relations and the resultant permittivity and permeability.

1.3 Responses to Applied RF Fields

If we immerse a specimen in an applied field and the response is recorded by a measurement device, the data obtained are usually in terms of a digital readout or a needle deflection indicating the phase and magnitude of a voltage or current, a difference in voltage and current, power, force, temperature, or an interference fringe. For example, we deduce electric and magnetic field strengths and phase through Ampere's and Faraday's laws by means of voltage and current measurements. The scattering parameters measured on a network analyzer relates to the phase and magnitude of a voltage wave. The detection of a photon's energy is sensed by an electron cascade current. Cavities and microwave evanescent probes sense material characteristics through shifts in resonance frequency from the influence of the specimen under test. The shift in resonance frequency is again determined by voltage and power measurements on a network analyzer. Magnetic interactions are also determined through measurements of current and voltage or forces (4), (7-9). These measurement results are usually used with theoretical models, such as Maxwell's equations, circuit parameters, or the Drude model, to obtain material properties.

High-frequency electrical responses include the measurement of the phase and magnitude of guided waves in transmission lines, fields from antennas, resonant frequencies and quality factors (Q) of cavities or dielectric resonators, voltage waves, movement of charge or spin, temperature changes, or forces on charge or spins. These responses are then combined through theoretical models to obtain approximations to important fundamental quantities such as: power, impedance, capacitance, inductance, conductance, resistance, conductivity, resistivity, dipole and spin moments, permittivity, and permeability, resonance frequency, Q, antenna gain, and nearfield response (10-16).

The homogenization procedure used to obtain the macroscopic Maxwell equations from the microscopic Maxwell equations is accomplished by averaging the molecular dipole moments within a unit cell and constructing an averaged continuous charge density function. Then a Taylor series expansion of the averaged charge density is performed, and, as a consequence, it is possible to define the averaged polarization vector. The spatial requirement for the validity of this averaging is that the wavelength must be much larger than the unit cell dimensions (see Sec. 4.6). According to this analysis, the permittivity of an ensemble of molecules is valid for applied field wavelengths that are much larger than the dimensions of an ensemble of molecules or lattice, assuming one can isolate the effects of the molecules from the measurement apparatus. This metrology is not always easy because a measurement contains effects of electrodes, probes, and other environmental factors. The concepts of atomic polarizability and dipole molecular moment are valid on a smaller scale than are permittivity and permeability.

In the absence of an applied field, small random voltages with a zero mean are produced by equilibrium thermal fluctuations of random charge motion (17). Fluctuations of these random voltages create electrical noise power in circuits. Analogously, spin noise is due to spin fluctuations. Quasi-monochromatic surface waves can also be excited by random thermal fluctuations. These surface waves are different from blackbody radiation (18). Various interesting effects are achieved by random fields interacting with surfaces. For example, surface waves on two closely spaced surfaces can cause an enhanced radiative transfer. Noise in nonequilibrium systems is becoming more important in nanoscale measurements and in systems where the temperatures vary in time. The information obtained from radiometry at a large scale, or microscopic probing of thermal fluctuations of various material quantities, can produce an abundance of information on the systems under test.

1.4 RF Measurements at Various Scales

At RF frequencies the wavelengths are much larger than molecular dimensions. There are various approaches to obtaining material response with long wavelength fields to study small-scale particles or systems. These methods may use very sensitive detectors, such as single-charge or spin detectors or amplifiers, or average the response over an ensemble of particles to obtain a collective response. To make progress in the area of mesoscale measurement, detector sensitivity may need to exceed the three or four significant digits obtained from network analyzer scattering parameter measure ments, or one must use large ensembles of cells for a bulk response and infer the small-scale response. Increased sensitivity may be obtained by using resonant methods or evanescent fields.

Material properties such as collective polarization and loss (19) are commonly obtained by immersing materials in the fields of EM cavities, dielectric resonators, free-space methods, or transmission lines. Some responses relate to intrinsic resonances in a material, such as polariton or plasmon response, ferromagnetic and anti-ferromagnetic resonances, and terahertz molecular resonances.

Broadband response is usually obtained by use of transmission lines or antenna-based systems (12-14), (19), (20). Thin films are commonly measured with coplanar waveguides or microstrips (14). Common methods used to measure material properties at small scales include near-field probes, micro-transmission lines, atomic-force microscopes, and lenses.

In strong fields, biological cells may rotate, deform, or be destroyed (21). In addition, when there is more than one particle in the applied field, the fields between the particles can be modified by the presence of nearby particles. In a study by Friend et al. (22), the response of an amoeba to an applied field was studied in a capacitor at various voltages, power, and frequencies. They found that at 1 kHz and at 10 V/cm the amoeba oriented perpendicular to the field. At around 10 kHz and above 15 V/cm the amoeba's internal membrane started to fail. Above 100 kHz and a field strength of above 50 V/cm, thermal effects started to damage the cells.

1.5 Electromagnetic Measurement Problems Unique to Microscale and Nanoscale Systems

Usually, the electrical skin depth for field penetration is much larger than the dimensions of nanoparticles. Because nanoscale systems are only 10 to 1000 times larger than the scale of atoms and small molecules, quantum mechanics plays a role in the transport properties. Below about 10 nm, many of the continuous quantities in classical electromagnetics take on a quantized aspect. These include charge transport, capacitance, inductance, and conductance. Fluctuations in voltage and current also become more important than in macroscopic systems. Electrical conduction at the 10 nanoscale involves movement of a small number of charge carriers through thin structures and may attain ballistic transport. For example, if a 1 [micro]A charge travels through a nanowire of radial dimensions 30 nm, then the current density is on the order of 3 x [10.sup.9] A/[m.sup.2]. Because of these large current densities, electrical transport in nanoscale systems is usually a nonequilibrium process, and there is a large influence of electron-electron and electron-ion interactions.

In nanoscale systems, boundary layers and interfaces strongly influence the electrical properties, and the local permittivity may vary with position (23). Measurements on these scales must model the contact resistance between the nanoparticle and the probe or transmission line and deal with noise.

2. Fundamental Electromagnetic Parameters and Concepts Used in Material Characterization

2.1 Electrical Parameters for High-Frequency Characterization

In this section, the basic concepts and tools needed to study and interpret dielectric and magnetic response over RF frequencies are reviewed (24).

In the time domain, material properties can be obtained by analyzing the response to a pulse or impulse; however most material measurements are performed by subjecting the material to time-harmonic fields.

The most general causal linear time-domain relationships between the displacement and electric fields and induction and magnetic fields are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [[[left right arrow].f].sub.p] (t) is a polarization impulse-response dyadic,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [[[left right arrow].f].sub.m] (t) is a magnetic impulse-response dyadic.

The permittivity [[[left right arrow].[epsilon]]([omega]) dyadic is the complex parameter in the time-harmonic field relation [~.D] ([omega]) = [[[left right arrow].[epsilon]]([omega]) * [~.E]([omega]) and, is defined in terms of the Fourier transform of the impulse-response function. For isotropic linear media, the scalar complex relative permittivity [[epsilon].sub.r] is defined in terms of the absolute permittivity [epsilon] and the permittivity of vacuum [[epsilon].sub.0] (F/m), as follows [epsilon] ([omega]) = [[epsilon].sub.0][[epsilon].sub.r] ([omega]), where [[epsilon].sub.r] ([omega]) = [[epsilon] [.sub.r[infinity]] + [X.sub.r] ([omega]) = [[epsilon]'.sub.r] ([omega]) - i[[epsilon]".sub.r] ([omega]), and [[epsilon] [.sub.r[infinity]] is the optical-limit of the relative permittivity. The value of the permittivity of free space is [[epsilon].sub.0] [equivalent to] [1/[[micro].sub.0][c.sub.v.sup.2] [approximately equal to] 8.854 x [10.sup.-12] (F/m), where the speed of light in vacuum is [c.sub.v] [equivalent to] 299792458 (m/s) and the exact value of the permeability of free space is [[micro].sub.0] = 4[pi] x [10.sup.-7] (H/m). Also, tan [[delta].sub.d] = [[epsilon] ".sub.r]/[[epsilon]'.sub.r] is the loss tangent in the material (25).

Note that in the SI system of units the speed of light, permittivity of vacuum, and permeability of vacuum are defined constants. All measurements are related to a frequency standard. Note that the minus sign before the imaginary part of the permittivity and permeability is due to the [e.sup.i[omega]t] time dependence. A subscript eff on the permittivity or permeability releases the quantity from some of the strict details of electrodynamic analysis. The permeability in no applied field is: [micro]([omega]) = [[micro].sub.0]([[micro]'.sub.r]([omega]) - i[[micro]".sub.r]([omega]) and the magnetic loss tangent is tan [[delta].sub.m] = [[micro]".sub.r]([omega])/[[micro]'.sub.r]([omega])

For anisotropic and gyrotropic media with an applied magnetic field, the permittivity and permeability tensors are hermitian and can be vexpressed in the general form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

For a definition of gyrotropic media see (4). The offdiagonal elements are due to gyrotropic behavior in an applied field.

Electric and magnetic fields are attenuated as they travel through lossy materials. Using time-harmonic signals the loss can be studied at specific frequencies, where the time dependence is [e.sup.i[omega]t]. The change in loss with frequency is related to dispersion.

The propagation coefficient of a plane wave is [gamma] = [alpha] + i [beta] = ik = i[omega][square root of ([epsilon][micro])]. The plane-wave attenuation coefficient in an infinitely thick half space, where the guided wavelength of the applied field is much longer than the size of the molecules or inclusions, is denoted by the quantity [alpha] and the phase is denoted by [beta]. Due to losses of a plane wave, the wave amplitude decays as [absolute value of E] [varies] exp(-[alpha]z). The power in a plane wave of the form E(z, t) = [E.sub.0] exp(-[alpha]z) exp (i[omega]t-i[beta]z), attenuates as P [varies] exp (-2[alpha]z). For waves in a guided structure: [gamma] = i [square root of (t[k.sup.2] - [k.sub.c.sup.2])], where [k.sub.c] [[omega].sub. c] / c = 2[pi] / [[lambda].sub.c] is the cutoff wavenumber, and speed of light c. Below cutoff, the propagation coefficient becomes [gamma][square root of ([k.sub.c.sup.2] - [k.sup.2])]. [alpha] of a plane wave is given by

[alpha] = [omega] /c[square root of (2)] [square root of ([[epsilon]'.sub.r] [[micro]'.sub.r] (([(tan[[delta].sub.d] tan [[delta].sub.m] -1).sup.2] +[[(tan [[delta].sub.d] +tan [[delta].sub.m]).sup.2]).sup.1/2] + [(tan [[delta].sub.d] tan[[delta].sub.m] -1)).sup.1/2], (4)

and has units of Np/m. [alpha] is approximated for dielectric materials as

[alpha] = [[omega]]/c[absolute value of 2] [absolute value of [[epsilon]'.sub.r]] [[micro]'.sub.r]] [absolute value of [absolute value of 1+[tan.sup.2] [[delta].sub.d]]-1]]. (5)

In dielectric media with low loss, tan [[delta].sub.d] [much less than]1, and [alpha] reduces in this limit to [alpha] [right arrow] [omega][absolute value of [[epsilon]'.sub.r]] [[micro]'.sub.r]]/2c. The skin depth is the distance a plane wave travels until it decays to 1/e of its initial amplitude, and is related to the attenuation coefficient by [[delta].sub.s] = 1/[alpha]. The concept of skin depth is useful in modeling lossy dielectrics and metals. Energy conservation constrains a to be positive. The skin depth is defined for lossy dielectric materials as

[[delta].sub.s] = [c[absolute value of 2]]/[omega]][1/[absolute value of [[epsilon]'.sub.r]] [[micro]'.sub.r]] [absolute value of [absolute value of 1+[tan.sup.2] [[delta].sub.d]]-1]]] (6)

In Eq. (6), [[delta].sub.s] reduces in the low-conductivity limit to to[[delta].sub. s] [right arrow] 2c/([omega][absolute value of [[epsilon]'.sub.r]] [[micro]'.sub.r]] tan[[delta].sub.d]). The depth of penetration [D.sub.p] = [[delta].sub.s]/2 is the depth where the plane-wave energy drops to 1/e of its value on the surface. In metals, where the conductivity is large, the skin depth reduces to

[[delta].sub.s] = 1/[square root of ([pi] f [[micro].sub.0][[micro]'.sub.r] [[delta].sub.dc])] (7)

where [[sigma].sub.dc] is the dc conductivity and f is the frequency. We see that the frequency, conductivity, and permeability of the material determine the skin depth in metals.

The phase coefficient [beta] for a plane wave is given by

[beta] = [+ or -][[omega]/c[absolute value of 2]] [absolute value of [[epsilon]'.sub.r]] [[micro]'.sub.r]] (([(tan [[delta].sub.d] [tan[delta].m] -1)).sup.2] + [(tan [[delta].sub.d] + [tan[delta].m]).sup.2]).sup.1/2] - [(tan [[delta].sub.d] [tan[delta].m] -1)).sup.1/2]. (8)

In dielectric media, [beta] reduces to

[beta] = [+ or -][[epsilon]/c[absolute value of 2]] [absolute value of [[epsilon]'.sub.r]] [[micro]'.sub.r]] [absolute value of [absolute value of 1+[tan.sup.2] [[delta].sub.d] +1]. (9)

The imaginary part of the propagation coefficient defines the phase of an EM wave and is related to the refractive index by n = [+ or -] [absolute value of [[epsilon]'.sub.r]] [[micro]'.sub.r]] In normaldielectrics the positive square root is taken in Eq. (8). Veselago (26) developed a theory of negative-index materials (NIM) where he used negative intrinsic [[epsilon]'.sub.r]] and [[micro]'.sub.r]], and the negative square root in Eq. (8) is used. There is controversy over the interpretation of metamaterial NIM electrical behavior since the permeability and permittivity are commonly effective values. We will use the term NIM to describe materials that achieve negative effective permittivity and permeability over a band of frequencies.

The wave impedance for a transverse electric and magnetic mode (TEM) is [absolute value of [micro]/[epsilon]]; for a transverse electric mode (TE) is i[omega][micro]/[gamma], and for a transverse magnetic mode (TM) is [gamma]/i[omega][epsilon]. The propagating plane wave wavelength in a material is decreased by a permittivity greater than that of vacuum; for example, for a TEM mode, [lambda] [approximately equal to] [c.sub.vac]/([absolute value of [[epsilon]'.sub.r] [[micro]'.sub.r]] f). In waveguides theguided wavelength [[lambda].sub.g] dependson the cutoff wavelength [[lambda].sub.c] and is given by [[lambda].sub.g] = 1/[absolute value of [epsilon]'[micro]'[f.sup.2]/[c.sup.2] -1/[[lambda].sub.c.sup.2] = [absolute value of 1 -[([lambda]/[[lambda].sub.c]).sup.2].

The surface impedance in ohms/square of a conducting material is [Z.sub.m] = (1 + i)[sigma][[delta].sub.s]. The surface resistance for highly lossy materials is

[R.sub.s] = 1/[[R.sub.s][[delta].sub.dc]] = [[absolute value of [pi]f[[micro].sub.0][[micro]'.sub.r]/[[delta].sub.dc]]. (10)

When the conductors on a substrate are very thin, the fields can penetrate through the conductors into the substrate. This increases the resistance of a propagating field because it is in both the metal and the dielectric. As a consequence of the skin depth, the internal inductance in a highly-conducting material decreases with increasing frequency, whereas the surface resistance [R.sub.s] increases with frequency in proportion to [absolute value of f].

Any transmission line will have propagation delay that relates to the propagation speed in the line. This is related to the dielectric permittivity and the geometry of the transmission line. Propagation loss is due to conductor and material loss.

Some materials exhibit ionic conductivity, so that when a static electric field is applied, a current is induced. This behavior is modeled by the dc conductivity [[sigma].sub.dc], which produces a low-frequency loss ([varies] 1/[omega]) in addition to polarization loss ([[epsilon]".sub.r]). In some materials, such as semiconductors and disordered solids, the conductivity is complex and depends on frequency. This is because the free charge is partially bound and moves by tunneling through potential wells or hops from well to well.

The total permittivity for linear, isotropic materials that includes both dielectric loss and dc conductivity is defined from the Fourier transform of Maxwell's equation: i[omega] [~.D] ([omega]) + [~.J] ([omega]) [equivalent to] i[omega][epsilon] [~.E] ([omega]) + [[sigma].sub.dc] [~.E] ([omega]) [equivalent to] [[i[omega][epsilon].sub.tot] [~.E] ([omega]), so that

[[epsilon].sub.tot] = [[[epsilon]'.sub.r][[epsilon].sub.0] - i[[epsilon]".sub.r][[epsilon].sub.0] - i([[epsilon]".sub.r][[epsilon].sub.0] + [[[sigma].sub.dc]]/[omega]]). (11)

In plots of RF measurements, the decibel scale is often used to report power or voltage measurements. The decibel (dB) is a relative unit and for power is calculated by 10 [log.sub.10] ([P.sub.out]/[P.sub.in]). Voltages in decibels are defined as 20 [log.sub.10] ([V.sub.out]/[V.sub.in]). [alpha] has units of Np/m. The attenuation can be converted from 1 Np/m = 8.686 dB/m. dBm is similar to dB, but relative to power in milliwatts 10 log(P/mW).

2.2 Electromagnetic Power

In the time domain the internal field energy U satisfies: [partial derivative]U/[partial derivative]t = [partial derivative]D/[partial derivative]t * E + [partial derivative]B/[partial derivative]t * H. Using Maxwell's equations with a current density J, then produces Poynting's Theorem: [partial derivative]U/[partial derivative]t + [nabla] * (E X H) = - J * E, where the time-domain Poynting vector is S(r, t) = E(r, t) X H(r, t). The complex power flux [(W/m.sup.2]) is summarized by the complex Poynting vector [S.sub.c]([omega]) = (1/2)([~.E] ([omega]) X [~.H] ([omega])). The real part of [S.sub.c] represents dissipation and is the time average over a complete cycle. The imaginary part of [S.sub.c] relates to the reactive stored energy.

2.3 Quality Factor

The band width of a resonance is usually modeled by the quality factor (Q) in terms of the decay of the internal energy. The combined internal energy in a mechanical system is the kinetic plus the potential energy; in an electromagnetic system it is the field stored energy plus the potential energy. In the time domain the quality factor is related to the decay of the internal energy for an unforced resonator as as (27)

[dU(t)]/[dt] = -[[[omega].sub.0]/[Q.sub.0]]U(t). (12)

The EM field is modeled by a damped harmonic oscillator at frequencies around the lossless resonant frequency [[omega].sub.0] and frequency pulling factor (the resonant frequency decreases from [[omega].sub.0] due to material losses), [DELTA][omega] as (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Taking a Fourier transform of Eq. (13), the absolute value squared becomes

[[absolute value of E([omega])].sup.2] = A/[[([omega] - [[omega].sub.0] - [DELTA][omega]).sup.2] + [([omega]/2[Q.sub.0]).sup.2]], (14)

and therefore [[absolute value of E([omega])].sup.2], which is proportional to the power, is a Lorentzian. This linear model is not exact for dispersive materials, because [Q.sub.0] may be dependent on frequency. The quality factor is calculated from the frequency at resonance [f.sub.0] as [Q.sub.0] = [f.sub.0] / 2([absolute value of [f.sub.0] - [f.sub.3dB]]), or from a fit of a circle when plotting [S.sub.11]([omega]) on the Smith chart. The quality factor is calculated from [Q.sub.0] = [f.sub.0]/f, where f is the frequency difference between 3 dB points on the [S.sub.21] curve (28). For resonant cavity measurements, the permittivity or permeability is determined from measurements of the resonance frequency and quality factor, as shown in Fig. 2. For timeharmonic fields the Q is related to the stored field energies [W.sub.e], [W.sub.h], the angular frequency at resonance [[omega].sub.r], and the power dissipated [P.sub.d] at the resonant frequency:

Q = [[omega].sub.r] [[W.sub.e] + [W.sub.h]]/[P.sub.d]. (15)

Resonant frequencies can be measured with high precision in high-Q systems; however the parasitic coupling of the fields to fixtures or materials needs to be modeled in order to make the result meaningful. Material measurements using resonances have much higher precision than using nonresonant transmission lines.

The term antiresonance is used when the reactive part of the impedance of a EM system is very high. This is in contrast to resonance, where the reactance goes to zero. In a circuit consisting of a capacitor and inductance in parallel, antiresonance occurs when the voltage and current are in phase.

3. Maxwell's Equations in Materials

3.1 Maxwell's Equations From Microscopic to Macroscopic Scales

Maxwell's microscopic equations in a media with charged particles are written in terms of the microscopic fields b, e and sources j, and [[rho].sub.m] as

[nabla] x b = [[epsilon].sub.0][[micro].sub.0] [[[partial derivative]e]/[[partial derivative]t]] + [[micro].sub.0] j, (16)

[nabla] x e = - [[[partial derivative]b]/[[partial derivative]t]], (17)

[[epsilon].sub.0] [nabla] * e = [[rho].sub.m], (18)

[nabla] * b = 0. (19)

Note, that at this level of description the macroscopic magnetic field H and the macroscopic displacement field D are not defined, but can be formed by averaging dielectric and magnetic moments and expanding the microscopic charge density in a Taylor series. In performing the averaging process, the material length scales allow the dipole moments in the media to be approximated by continuously varying functions P and M. Once the averaging is completed, the macroscopic Maxwell's equations are (see Sec. 4.6) to obtain (27), (29), (30)

[nabla] x H = [[[partial derivative]D]/[[partial derivative]t]] + J, (20)

[nabla] x E = -[[[partial derivative]B]/[[partial derivative]t]], (21)

[nabla] * H = [rho], (22)

[nabla] * B = 0. (23)

J denotes the current density due to free charge and source currents. Because there are more unknowns than equations, constitutive relations for H and D are needed. Even though B and E are the most fundamental fields, D usually is expressed in terms of E, and B is usually expressed in terms of H.

3.2 Constitutive Relations

3.2.1 Linear Constitutive Relations

Since there are more unknowns than macroscopic Maxwell's equations, we must specify the constitutive relationships between the polarization, magnetization, and current density as functions of the macroscopic electric and magnetic fields (31), (32). In order to satisfy the requirements of linear superposition, any linear polarization relation must be time invariant, further, this must also be a causal relationship as given in Eqs. (1) and (2).

The fields and material-related quantities in Maxwell's equations must satisfy underlying symmetries. For example, the dielectric polarization and electric fields are odd under parity transformations and even under time-reversal transformations. The magnetization and induction fields are even under parity transformation and odd under time reversal. These symmetry relationships place constraints on the nature of the allowed constitutive relationships and requires the constitutive relations to manifest related symmetries (29), (33-39). The evolution equations for the constitutive relationships need to be causal, and in linear approximations must satisfy time-invariance properties. For example, the linear-superposition requirement is not satisfied if the relaxation time in Eq. (4) depends on time. This can be remedied by using an integrodifferential equation with restoring and driving terms (40), (41).

The macroscopic displacement and induction fields D and B are related to the macroscopic electric field E and magnetic fields H, as well as M and P, by

D = [[epsilon].sub.0]E + [P.sub.d] - [nabla] * [[left right arrow].Q] + ... [equivalent to] [[epsilon].sub.0]E + P, (24)

and

B = [[micro].sub.0]H + [[micro].sub.0]M. (25)

In addition,

J = J (E, H), (26)

where J is a function of the electric and magnetic fields, and [[left right arrow].Q] is the macroscopic quadrupole moment density. [P.sub.d] is the dipolemoment density, whereas P is the effective macroscopic polarization that also includes the effects of the macroscopic quadrupolemoment density (27), (29), (30), (32), (42). The polarization and magnetization for time-domain linear response are expressed as convolutions in terms of the macroscopic fields. For chiral and magneto-electric materials, Eqs. (24) and (25) must be modified to accommodate crosscoupling behavior between magnetic and dielectric response. General, linear relations defining polarization in non-magnetoelectric and non-chiral dielectric and magnetic materials in terms of the impulse-response dyadics are given by Eqs. (1) and (2). Using the Laplace transform L, gives

L[absolute value of [[approximately equal to].N] ([omega]) = [[[left right arrow].X].sub.er] ([omega]) [~.E] ([omega]), (27)

Where

[[[left right arrow].X].sub.er] ([omega]) = [[integral].sub.0.sup.[infinity]] [[[left right arrow].f].sub.p] ([tau])[e.sup.-i[omega]t]dt = [[[left right arrow].X'].sub.er]([omega]) - i[[[left right arrow].X"].sub.er]([omega]). (28)

So the real part is the even function of frequency given by

[[[left right arrow].X'].sub.er]([omega]) = [[integral].sub.0.sup.[infinity]] [[[left right arrow].f].sub.p](t) cos([omega]t) dt, (29)

and the imaginary part is an odd function of frequency

[[[left right arrow].X"].sub.er]([omega]) = [[integral].sub.0.sup.[infinity]] [[[left right arrow].f].sub.p](t) sin ([omega]t) dt, (30)

and therefore

[[[left right arrow].X"].sub.er]([omega]) = [[[left right arrow].I] + [[[left right arrow].X'].sub.er]([omega]) - i[[[left right arrow].X"].sub.er]([omega]); (31)

Also

[[[left right arrow].X'].sub.er] (0) = [[integral].sub.0.sup.[infinity]] [[[left right arrow].f].sub.p](t) dt, (32)

[[[left right arrow].X"].sub.er] (0) = 0. (33)

The time-evolution constitutive relations for dielectric materials are generally summarized by generalized harmonic oscillator equations or Debye-like equations as overviewed in Sec. 5.2.

3.2.2 Generalized Constitutive Relations

Through the methods of nonequilibrium quantumbased statistical-mechanics it is possible to show that the constitutive relation for the magnetization in ferro-magnetic materials is an evolution equation given by

[[partial derivative]M(r, t)]/[[partial derivative]t] = -[absolute value of [[gamma].sub.g]] M (r, t) x [H.sub.eff] (r, t) - [integral][d.sup.3] r' [[integral].sub.0.sup.t] [[left arrow].K].sub.m] (r, t, r', [tau]) * [X.sub.0][H.sub.eff] (r', [tau]) d[tau] (34)

where [[[left right arrow].K].sub.m] is a kernel that contains of the micro-structural interactions given in (43), [[gamma].sub.g] is the gyro-magnetic ratio, [X.sub.0] is the static susceptibility, and [H.sub.eff] is the effective magnetic field. Special cases of Eq. (34) reduce to constitutive relations such as the Landau-Lifshitz, Gilbert, and Bloch equations. The Landau-Lifshitz equation of motion is useful for ferromagnetic and ferrite solid materials:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

where [alpha] is a [[partial derivative]M (r, t)]/[[partial derivative]t] [approximately equal to] - [absolute value of [Y.sub.g] M (r, t) x [[H.sub.sub.eff]] (r, t) nother special case of Eq. (34) reduces to the Gilbert equation

[[partial derivative]M (r, t)]/[[partial derivative]t] [approximately equal to] -[absolute value of [[gamma].sub.g]]M(r, t) x [H.sub.eff](r, t) + [alpha]/[M.sub.s]M(r, t) x [[partial derivative]M(r, t)]/[[partial derivative]t]. (36)

In electron-spin resonance (EPR) and nuclear magnetic resonance (NMR) measurements, the Bloch equations with characteristic relaxation times [T.sub.1] and [T.sub.2] are used to model relaxation. [T.sub.1] relates to spin-lattice relaxation as the paramagnetic material interacts with the lattice. [T.sub.2] relates to spin-spin interactions:

[[partial derivative]M(r, t)]/[[partial derivative]t] [approximately equal to] -[absolute value of [[gamma].sub.g]]M(r, t) x H(r, t) - [[left right arrow].X].sub.b] * M(r, t) + [[M.sub.s]/[T.sub.1]], (37)

where [[left right arrow].X].sub.b] has only the diagonal elements [X.sub.b(11)] = 1/[T.sub.2], [X.sub.b(22)] = 1/[T.sub.2], [X.sub.b(33)] = 1/[T.sub.1], and [M.sub.s] = [M.sub.s] [[vector].z]. An equation analogous to (34) can be written for the electrical polarization (46) as (43)

[[partial derivative]P(r, t)]/[[partial derivative]t] = -[integral][d.sup.3]r'[[integral].sub.0.sup.t][[[left right arrow].K].sub.e](r, t, r', [tau]) x (P(r', [tau]) - [[[left right arrow].X].sub.0] * E (r', t, d[tau]). (38)

The Debye relaxation recovered from Eq. (38) when [[[left right arrow].K].sub.e] (r, t, r', [tau]) = [[[left right arrow].I][delta](t - [tau])[delta](r, r')]/[[tau].sub.e].

4. Electromagnetic Fields in Materials

4.1 The Time-Harmonic Field Approximation

Time-harmonic fields are very useful for solving the linear Maxwell's equations when transients are not important. In the time harmonic field approximation, the field is assumed to be present without beginning or end. Periodic signals over -[infinity] < t < [infinity] are nonphysical since all fields have a beginning where transients are generated, but are very useful in probing material response.

Solutions of Maxwell's equations that include transients are most easily obtained with the Laplace transform. Note that the Laplace or Fourier transformed fields do not have the same units as the time-harmonic fields due to integration over time. In Eq. (1), causality is incorporated into the convolution relation for linear response. D(t) depends only on E(t) at earlier times and not future times.

4.2 Material Response to Applied Fields

When a field is suddenly applied to a material, the charges, spins, currents, and dipoles in a medium respond to the local fields to form an average field. If an EM field is suddenly applied to a semi-infinite material, the total field will include the effects of both the applied field, transients, and the particle backreaction fields from charge, spin, and current rearrangement that causes depolarization fields. This will cause the system to be in nonequilibrium for a period of time. For example, as shown in Fig. 3, when an applied EM field interacts with a dielectric material, the dipoles reorient and charge moves, so that the macroscopic and local fields in the material are modified by surface charge dipole depolarization fields that oppose the applied field. The macroscopic field is approximately the applied field minus the depolarization field. Depolarization, demagnetization, thermal expansion, exchange, nonequilibrium, and anisotropy interactions can influence the dipole orientations and therefore the fields and the internal energy. In modeling the constitutive relations in Maxwell's equations, we must express the material properties in terms of the macroscopic field, not the applied or local fields, and therefore we need to make clear distinctions between the interaction processes (40).

Materials can be studied by the response of frequency-domain or time-domain fields. When considering time-domain pulses rather than time-harmonic fields, this interaction is more complex. The use of timedomain pulses have the advantage of sampling a reflected pulse as a function of time, which allows a determination of the spatial location of the various reflections.

Time-harmonic fields are often used to study material properties. These have a specific frequency from time minus infinity to plus infinity, without transients; that is, fields with a [e .sup.i [omega]t] time dependence. As a consequence, in the frequency domain, materials can be studied through the reaction to periodic signals. The measured response relates to how the dipoles and charge respond to the time-harmonic signal at each frequency. If the frequency information is broad enough, a Fourier transform can be used to study the corresponding time-domain signal.

The relationships between the applied, macroscopic, local, and the microscopic fields are important for constitutive modeling (Fig. 3). The applied field originates from external charges, whereas the macroscopic fields are averaged quantities in the medium. The displacement and inductive (or magnetic) macroscopic fields in Maxwell's equations are implicitly defined through the constitutive relationships and boundary conditions. The local field is the averaged EM field at a particle site due to both the applied field and the fields from all of the other sources, such as dipoles, currents, charge, and spin (47). The microscopic field represents the atomic-level EM field, where particles interact with the field from discrete charges. Particles interact with the local EM field that is formed from the applied field and the microscopic field. At the next level of homogenization, groups of particles interact with the macroscopic field. The spatial and temporal resolution contained in the macroscopic variables are directly related to the spatial and temporal detail incorporated in the constitutive material parameters. Constitutive relations can be exact as in (40) and Eqs. (34) and (38), but usually, to be useful, are approximate.

Plane waves are a useful approximation in many applications. Timeharmonic EM plane waves in materials can be treated either as traveling without attenuation, propagating with attenuation, or evanescent. Plane waves may propagate in the form of a propagating wave [e.sup.i([omega]t - [beta]z)], or a damped propagating wave [e.sup.i([omega]t - [beta]z) - [alpha]z], or an evanescent wave [e.sup.i[omega]t - [alpha]z]. Evanescent fields are exponentially damped waves. In a waveguide, this occurs for frequencies below any transverse resonance frequencies (24), (48), when [k.sup.2] - [k.sub.c.sup.2] < 0, where [k.sub.c] is the cutoff wave number calculated from the transverse geometry and k = [omega][absolute value of [epsilon][micro]] = ([omega] / c) [absolute value of [[epsilon]'.sub.r][[micro]'.sub.r]]. Evanescent and near field EM fields occur at apertures and in the vicinity of antennas. Evanescent fields can be detected when they are perturbed and converted into propagating waves or transformed by dielectric loss. Electromagnetic waves may convert from near field to propagating. For example, in coupling to dielectric resonators the near field at the coupling loops produce propagating or standing waves in a cavity or dielectric resonator. Evanescent and near fields in dielectric measurements are very important. These fields do not propagate and are used in near-field microwave probes to measure or image materials at dimensions much less than [lambda]/2 (49), (50) (see Fig. 17). The term near field usually refers to the waves close to an waveguide, antenna, or probe and is not necessarily an exponentially damped plane wave. In near-field problems the goal is to model the reactive region. Near fields in the reactive region, (L < [lambda]/2[pi]), contain stored energy and there is no net energy transport over a cycle unless there are losses in the medium. By analogy, the far field relates to radiation. These remove energy from the transmitter whether they are immediately absorbed or not. There is a transition region called the radiative near field.

Because electrical measurements can now be performed at very small spatial resolutions, and the elements of electrical circuits are approaching the molecular level, we require good models of the macroscopic and local fields. This is particularly important, because we know that the Lorentz theory of the local field is not always adequate for predicting polarizabilities (51), (52). Also, when solving Maxwell's equations at the molecular level, definitions of the macroscopic field and constituative relationships are important. A theoretical analysis of the local EM field is important in dielectric modeling of single-molecule measurements and thin films. The effective EM fields at this level are local, but not atomic-scale, fields.

The formation of the local field is a very complex process whereby the applied electric field polarizes dipoles in a molecules or lattices and the applied magnetic field causes current and precession of spins. Then, the molecule's dipole field modifies the dipole orientations of other molecules in close proximity, which then reacts back to produce a correction to the molecule's field in the given region. This process gets more complicated for behavior that depends on time. We define the local EM field as the effective, averaged field at a specific point in a material, not including the field of the particle itself. This field is a function of both the applied and multipole fields in the media. The local field is related to the average macroscopic and microscopic EM fields in that it is a sum of the macroscopic field and the effects of the near-field. In ferroelectric materials, the local electric field can become very large and hence there is a need for comprehensive local field models. In the literature on dielectric materials, a number of specific fields have been introduced to analyze polarization phenomena. The electric field acting on a nonpolar dielectric is commonly called the internal field, whereas the field acting on a permanent dipole moment is called the directing field. The difference between the internal field and directing fields is the average reaction field. The reaction field is the result of a dipole polarizing its environment (53).

Nearly exact classical theories have been developed for the static local field. Mandel and Mazur developed a static theory for the local field in terms of the polarization response of a many-body system by use of the T-matrix formalism (54). Gubernatis extended the T-matrix formalism (55). However, the T-matrix contributions are difficult to calculate. Keller's review article (56) on the local field uses an EM propagator approach. Kubo's linear-response theory and other theories have also been used for EM correlation studies (40), (53), (57).

If the applied field has a wavelength that is not much longer than the typical particle size in a material, an effective permittivity and permeability is commonly assigned. The terms effective permittivity and permeability are commonly used in the literature for studies of composite media. The assumption is that the properties are "effective" if in some sense they do not adhere to the definitions of the intrinsic material properties. An effective permittivity is obtained by taking a ratio of some averaged displacement field to an averaged electric field. The effective permeability is obtained by taking a ratio of some averaged induction field to an averaged magnetic field. This approach is commonly used in modeling negative-index material properties when scatterers are designed in such a manner such that the scatterers themselves resonate. In these situations the wavelength may approach the dimensions of the inclusions.

4.3 Macroscopic and Local Electromagnetic Fields in Materials

The mesoscopic description of the EM fields in a material is complicated. As a field is applied to a material, charges reorient to form new fields that oppose the applied field. In addition, a dipole tends to polarize its immediate environment, which modifies the field the dipole experiences. The field that polarizes a molecule is the local field [E.sub.l] and the induced dipole moment is p = [[[left right arrow].[alpha]].sub.b] * [E.sub.1], where [[left right arrow].[alpha]] is the polarizability. In order to use this expression in Maxwell's equations, the local field needs to be expressed in terms of the macroscopic field. Calculation of this relationship is not always simple.

To first approximation, the macroscopic field is related to the external or applied field ([E.sub.a]), and the depolarization field by

E = [E.sub.a] - [1/3[[epsilon].sub.0]]P. (39)

The local field is composed of the macroscopic field and a material-related field. In the literature, the effective local field is commonly modeled by the Lorentz field, which is defined as the field in a small cavity that is carved out of a material around a specific site, but excludes the field of the observation dipole. A wellknown example of the relationship between the applied, macroscopic, and local fields is given by an analysis of the Lorentz spherical cavity in a static electric field. For a Lorentz sphere the local field is the sum of applied, depolarization, Lorentz, and atomic fields (4), (56), (58):

[E.sub.t] = [E.sub.a] + [E.sub.depol] + [E.sub.Lorentz] + [E.sub.atom]. (40)

For cubic lattices in a spherical cavity, the Lorentz local field is related to the macroscopic field and polarization by

[E.sub.l] = E+ [1/3[[epsilon].sub.0]] P. (41)

In the case of a sphere, the local field in Eq. (39) equals the applied field.

For induced dipoles,

P = N[alpha][E.sub.l], (42)

where N is the density of dipoles, and Eq. (41) yields [E.sub.l] = E/(1 - N[alpha]/3[[epsilon].sub.0]) = P/N[alpha].

Onsager (53) generalized the Lorentz theory by distinguishing between the internal field that acts on induced dipoles and the directing field that acts on permanent dipoles. If we use P = [[epsilon].sub.0] ([[epsilon].sub.r] - 1)E in Eq. (41), we find [E.sub.l] = (([[epsilon].sub.r] + 2)/3)E. Therefore, for normal materials the Lorentz field exceeds the macro-scopic field. For a material where the permittivity is negative we can have [E.sub.l] [less than or equal to] E. In principle, we can null out the Lorentz field when [[epsilon].sub.r] = - 2. Some of the essential problems encountered in microscopic constitutive theory center around the local field. Note that for some materials, recent research indicates that the Lorentz local field does not always lead to the correct polarizabilities (51). We expect the Lorentz local field expression to break down near interfaces. For nanoparticles, a more complicated theory needs to be used for the local field.

A rigorous expression for the static local field created by a group of induced dipoles can be obtained by an iterative procedure (53), (59) using [p.sub.i] = [[alpha].sub.i][E.sub.l] ([r.sub.i]) and

[E.sub.i]([r.sub.j]) = [E.sub.a] + [N.summation over i = l, i[not equal to]j][E.sub.ij]([r.sub.j]), (43)

where

[E.sub.ij]([r.sub.j]) [approximately equal to] [1/4[pi][[epsilon].sub.0]], [[[3([r.sub.j] - [r.sub.i])p([r.sub.i])]/[[absolute value of [r.sub.j] - [r.sub.i]].sup.5]] - [p([r.sub.i])]/[[absolute value of [r.sub.j] - [r.sub.i]].sup.3]]. (44)

If there are also permanent dipoles, they need to be included as p([r.sub.i]) = [p.sub.perm]([r.sub.i]) + [[alpha].sub.i][E.sub.l]([r.sub.i]).

4.4 Overview of Linear-Response Theory

Models of relaxation that are based on statistical mechanics can be developed from linear-response theory. Linear-response theory uses an approximate solution of Liouville's equation and a Hamiltonian that contains a time-dependent relationship of the field parameters based on a perturbation expansion. This approach shows how the response functions and relaxation are related to time dependent polarization correlation functions. The polarization P(t) is related to the response dyadic [[left right arrow].[phi]] (t) and the driving field E(t) by (53), (60)

P(t) = [[integral].sub.-[infinity].sup.t][[left right arrow].[phi]](t - [tau]) * E ([tau])d[tau], (45)

where [[left right arrow].[phi]] (t - [tau]) = 0 for t - [tau] < 0. The susceptibility is defined as

[[left right arrow].X]([omega]) = [[integral].sub.0.sup.[infinity]] [[left right arrow].[phi]]([atu])[e.sup.-i[omega][tau]]d[tau] = [[left right arrow].X']([omega]) - i[[left right arrow].X"]([omega]), (46)

where the response in volume V is related to the correlation function for stationary processes in terms of the microscopic polarization

[[left right arrow].[phi]]([tau]) = -V[d/d[tau]] [< P(0) P[tau] > 0]/[[k.sub.B]T], (47)

and therefore for microscopic polarizations

[[left right arrow].X]([omega]) = [[integral].sub.0.sup.[infinity]][e.sup.-i[omega][tau]][[left right arrow].[phi]]([tau])d[tau] = -V[[integral].sub.0.sup.[infinity]][e.sup.-i[omega][tau]][d/d[tau]] [[< P(0)P[tau] > 0]/[[k.sub.B]T]]d[tau] = [[V[omega]]/[[k.sub.B]T]][[[integral].sub.0.sup.[infinity]][< P(0)P([tau]) >.sub.0]sin([omega][tau])d[tau] - i[[integral].sub.0.sup.[infinity]][< P(0)P([tau]) >.sub.0]cos([omega][tau])d[tau]]. (48)

Once the correlation functions are determined then the susceptibility can be found. An approach that models relaxation beyond linear response is given in (40), (43), (44), (61). The method of linear response has exceeded expectations and has been a cornerstone of statistical mechanics.

4.5 Averaging to Obtain Macroscopic Field

If we consider modeling of EM wave propagation from macroscopic through molecular and sub-molecular to atomic scales, the effective response at each level is related to different degrees of homogenization. At wavelengths short relative to particle size the EM propagation is dominated by scattering, whereas at long wavelengths it is dominated by traveling waves. In microelectrodynamics, there have been many types of ensemble and volumetric averaging methods used to define the macroscopic fields obtained from the microscopic fields (27), (29), (30), (40), (54). For example, in the most commonly used theory of microelectromagnetics, materials are averaged at a molecular level to produce effective molecular dipole moments. The microscopic EM theories developed by Jackson, Mazur, and Robinson (27), (29), (30) average multipoles at a molecular level and replace the molecular multipoles, with averaged point multipoles usually located at the center-of-mass position. This approach works well down to near molecular level, but breaks down below the molecular to submolecular level.

In the various approaches, the homogenization of the fields are formed in different ways. The averaging is always volumetric rather than a time average. Jackson uses a truncated averaging test function to proceed from microscale to the macroscale fields (27). Robinson and Mazur use ensemble averaging (29), (30) and statistical mechanics. Ensemble averaging assumes there is a distribution of states. In the volumetric averaging approach, the averaging function is not explicitly determined, but the function is assumed to be such that the averaged quantities vary in a manner smooth enough to allow a Taylor-series expansion to be performed. In the approach of Mazur, Robinson, and Jackson (27), (29), (30) the charge density is expanded in a Taylor series and the multipole moments are identified as in Eq. (49). The microscopic charge density can be related to the macroscopic charge density, polarization, and quadrupole density by a Taylor-series expansion (27)

< [[rho].sub.micro] (r, t) > [approximately equal to] [[rho].sub.macro] (r, t) - [nabla] - P(r, t) - [nabla] * ([nabla] * [[left right arrow].Q])(r, t), (49)

where [[left right arrow].Q] (r, t) is the quadrupole tensor. In this interpretation, the concepts of P and [[rho].sub.macro] are valid at length scales where a Taylor-series expansion is valid. These moments are calculated about each molecular center of mass and are treated as point multipoles. However, this type of molecular averaging limits the scales of the theory to larger than the molecular level and limits the modeling of induced-dipole molecular moments (40). Usually, the averaging approach uses a test function [f.sub.a] and microscopic field e given by

E = [integral] dr'e(r - r')[f.sub.a](r'). (50)

However, the distribution function is seldom explicitly needed or determined in the analysis. The macroscopic magnetic polarization is found through an analogous expansion of the microscopic current density.

In NIM materials, effective properties are obtained by use of electric and magnetic resonances of embedded structures that produce negative effective [[epsilon]'.sub.eff] (62). In Sec. 4.6 the issue of whether this response can be summarized in terms of material parameters is discussed. Defining permittivity and permeability on these scales of periodic media can be confusing. The field averaging used in NIM analysis is based on a unit cell consisting of split-ring resonators, wires, and ferrite or dielectric spheres (62), (63).

In order to obtain a negative effective permeability in NIM applications, researchers have used circuits that are resonant, which can be achieved by the introduction of a capacitance into an inductive system. Pendry et al. (63-65) obtained the required capacitance through gaps in split-ring resonators. The details of the calculation of effective permeability are discussed in Reference (63). Many passive and/or active microwave resonant devices can be used as sources of effective permeability in the periodic structure designed for NIM applications (66). We should note that the composite materials used in NIM are usually anisotropic. Also, the use of resonances in NIM applications produce effective material parameters that are spatially varying and frequency dispersive.

4.6 Averaging to Obtain Permittivity and Permeability in Materials

The goal of this section is to study the electrical permittivity and permeability in materials starting from microscopic concepts and then progressing to macroscopic concepts. We will study the limitations of the concept of permittivity in describing material behavior when wavelengths of the applied field approach the dimensions of the spaces between inclusions or inclusion sizes. When high-frequency fields are used in the measurement of composite and artificial structures, these length-scale constraints are important. We will also examine alternative quantities, such as dipole moment and polarizability, that characterize dielectric and magnetic interactions of molecules, atoms, and that are still valid even when the concepts of permittivity and permeability are fuzzy.

The concepts of polarizability and dipole moment p in p = [alpha][E.sub.l] are valid down to the atomic and molecular levels. Permittivity and permeability are frequencydomain concepts that result from the microscopic timeharmonic form of Maxwell's equations averaged over a unit cell. They are also related to the Fourier transform of the impulse-response function. The most common way to define [[left right arrow].[epsilon]] is through the impulse-response function [[[left right arrow].f].sub.p](t).

Statistical mechanics yields an expression for the impulse-response function in terms of correlation functions of the microscopic polarizations p. For linear response (53)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (51)

where V is the volume, [L.sub.0] is Liouville's operator, p denotes i[L.sub.0]p, and < > 0 denotes averaging over phase. From this equation, we can identify the impulseresponse dyadic [[[left right arrow].f].sub.p] from P(t) = V[[integral].sub.0.sup.[infinity]] [< p (t)p([tau]) >.sub.0] * E([tau])d[tau] / [k.sub.B]T, and for a stationary system, [[[left right arrow].f].sub.p](t) = V[< p(0)p(-t) >.sub.0] / [k.sub.B]T (53).

Ensemble and volumetric averaging methods are used to obtain the macroscopic fields from the microscopic fields (see Jackson (27) and the references therein). For example, in the most commonly used theory, materials are averaged at a molecular level to produce effective molecular dipole moments. When deriving the macroscopic Maxwell's equations from the microscopic equations, the electric and magnetic multipoles within a molecule are replaced with averaged point multipoles usually located at the molecular center-of-mass positions. Then these effective moments are assumed to form a continuum, which then forms the basis of the macroscopic polarizations. The procedure assumes that the wavelength in the material is much larger than the individual particle sizes. As Jackson (27) notes, the macroscopic Maxwell's equations can model refraction and reflection of visible light, but are not as useful for modeling x-ray diffraction. He states that the length scale [L.sub.0] of 10 nanometers is effectively the lower limit for the validity of the macroscopic equations. Of course, this limit can be decreased with improved constitutive relationships.

For macroscopic heterogeneous materials the wave-lengths of the applied fields must be much longer than individual particle or molecule dimensions that constitute the material. When this criterion does not hold, then the spatial derivative in the macroscopic Maxwell's equations, for example, ([nabla] x H), and the displacement field loses its meaning. Associated with this homogenization process at a given frequency is the number of molecules or inclusions that are required to define a displacement field and thereby the related permittivity.

When the ratio of the dipole length scale to wavelength is not very small, the Taylor's series expansion is not valid and the homogenization procedure breaks down. When this criteria is not satisfied for metafilms, some researchers use generalized sheet transition conditions (GSTC's) (67-70) at the material boundaries; however, the concept of permittivity for these structures, at these frequencies, is still in question and is commonly assigned an effective value. Drude and others (67), (68) compensated for this by introducing boundary layers. In such cases, it is not clear whether mapping complicated field behavior onto effective permittivity and permeability is useful, since at these scales, the results can just as well be thought of as scattering behavior.

When modeling the permittivity or permeability in a macroscopic medium in a cavity or transmission line, the artifacts of the measurement fixture must be separated from the material properties by solving a relevant macroscopic boundary-value problem. At microwave and millimeter frequencies a low-loss macroscopic material can be made to resonate as a dielectric resonator. In such cases, if the appropriate boundary-value problem is solved, the intrinsic permittivity and permeability of the material can be extracted because the wavelengths are larger than the constituent molecule sizes, and as a result, the polarization vector is well defined. However, many modern applications are based on artificial structures that produce an EM response where the wavelength in the material is only slightly larger than the feature or inclusion size. In such cases, mapping the EM response onto a permittivity and permeability must be scrutinized. In general, the permittivity is well defined in materials where wave propagation through the material is not dominated by multiple scattering events.

5. Overview of the Dielectric Response to Applied Fields

5.1 Modeling Dielectric Response Upon Application of an External Field

Dielectric parameters play a critical role in many technological areas. These areas include electronics, microelectronics, remote sensing, radiometry, dielectric heating, and EM-assisted chemistry (20). At RF frequencies dielectrics exhibit behavior that metals cannot achieve because dielectrics allow field penetration and can have low-to-medium loss characteristics.

Using dielectric spectroscopy as functions of both frequency and temperature we can obtain some, but not all of the information on a material's molecular or lattice structure. For example, measurements of the polarization and conductivity indicate the polarizability and free charge of a material and polymer mobility of side chains can be studied with dielectric spectroscopy. Also, when a polymer approaches a glass transition temperature the relaxation times change abruptly. This is observable with dielectric spectroscopy. In addition, the loss peaks of many liquids change with temperature.

When an EM field is applied to a material, the atoms, molecules, free charge, and defects adjust positions. If the applied field is static, then the system will eventually reach an equilibrium state. However, if the applied field is time dependent then the material will continuously relax in the applied field, but with a time lag. The time lag is due to screening, coupling, friction, and inertia. An abundance of processes are occurring during relaxation, such as heat conversion processes, latticephonon, and photon phonon coupling. Dielectric relaxation can be a result of dipolar and induced polarization, lattice-phonon interactions, defect diffusion, higher multipole interactions, or the motion of free charges. Time-dependent fields produce nonequilibrium behavior in the materials due both to the heat generated in the process and the constant response to the applied field. However, for linear materials and time-harmonic fields, when the response is averaged over a cycle, if heating is appreciable, nonequilibrium effects such as entropy production relate more to temperature effects than the driving field stimulus. The dynamic readjustment of the molecules in response to the field is called relaxation and is distinct from resonance. For example, if a dc electric field is applied to a polarizable dielectric and then the field is suddenly turned off, then the dipoles will relax over a characteristic relaxation time into a more random state.

The response of materials depends strongly on material composition and lattice structure. In many solids, such as solid polyethylene, the molecules are not able to appreciably rotate or polarize in response to applied fields, indicating a low permittivity and small dispersion. The degree of crystallinity, existence of permanent dipoles, dipole-constraining forces, mobility of free charge, and defects all contribute to dielectric response. Typical responses for high-loss and low-loss dielectrics are shown in Figs. 4, 5, and 6.

A material does not respond instantaneously to an applied field. As shown in Fig. 4, the real part of the permittivity is a monotonically decreasing function of frequency in the relaxation part of the spectrum, far away from intrinsic resonances. At low frequencies, the dipoles generally follow the field, but thermal agitation also tends to randomize the dipoles. As the frequency increases to the MMW band, the response to the driving field generally becomes more incoherent. At higher frequencies, in the terahertz or infrared spectrum, the dipoles may resonate, and therefore the permittivity rises until it becomes out of phase with the field and then drops. At RF frequencies, materials with low loss respond differently from materials with high loss (compare Fig. 4 for a high-loss material versus a low-loss material in Figs. 5 and 6). For some materials, at frequencies at the low to middle part of the THz band, [[epsilon]'.sub.r] may start to contain some of the effects of resonances that occur at higher frequencies, and may start to slowly increase with frequency, until resonance, and then decreases again.

The local and applied fields in a dielectric are usually not the same. As the applied field interacts with a material it is modified by the fields of the molecules in the substance. Due to screening, the local electric field differs from the applied field and therefore theories of relaxation must model the local field (see Sec. 4.3).

Over the years, many models of polar and nonpolarmaterials have been developed that use different approximations to the local field. The Clausius-Mossotti equation was developed for noninteracting, nonpolar molecules governed by the Lorentz equation for the internal field. This equation works well for nonpolar gases and liquids. Debye introduced a generalization of the Clausius-Mossotti equation for the case of polar molecules. Onsager developed an extension of Debye's theory by including the reaction field and a more comprehensive local field expression (53). For a dielectric composed of permanent dipoles, the polarization is written in terms of the local field as Eq. (42)

There are electronic, ionic, and permanent dipole polarizability contributions, so that [[[vector].[micro]].sub.d] = ([[alpha].sub.el] + [[alpha].sub.ion] + [[alpha].sub.perm]) [E.sub.l], [[alpha].sub.el] = 4[pi][[epsilon].sub.0][R.sup.3] / 3, [[alpha].sub.ion] = [e.sup.2] / Y[d.sub.0]. Here, Y is Young's modulus, R is the radius of the ions, [d.sub.0] is the equilibrium separation of the ions, and [[alpha].sub.perm] = [[absolute value of [[[[vector].[micro]].sub.e].sup.2] / 3[k.sub.B]T, where [[[vector].[micro]].sub.e] is the permanent dipole moment. There may also be a contribution to the polarizability due to excess charge at microscopic interfaces. Using the Lorentz expression for the local field, the polarization can be written as

P = N[alpha](E + [P/[3[[epsilon].sub.0]]], (52)

or

P = [[N[alpha]E]/[1- N[alpha]/3[[epsilon].sub.0]]] = ([[epsilon].sub.r] -1) E]. (53)

This is the Clausius-Mossotti relation that is commonly used to estimate the permittivity of nonpolar materials from atomic polarizabilities:

[[[epsilon].sub.r] - 1]/[[[epsilon].sub.r] + 2] = [N[alpha]]/[[epsilon].sub.0], (54)

or

[[epsilon].sub.r] = [3[[epsilon].sub.0] + 2N[alpha]]/[3[[epsilon].sub.0] - N[alpha]]. (55)

The Clausius-Mossotti relation relates the permittivity to the polarizability. The polarizability is related to the vector dipole moment [[[vector].[micro]].sub.d] of a molecule or atom and the local field [E.sub.l], [[[vector].[micro]].sub.d] = [alpha] [E.sub.l]. In principle, once the polarizability is determined for a group of molecules, then the permittivity of the ensemble can be calculated with the implicit assumption that there are many molecules located over the distance of a wavelength. Typical polarizabilities of atoms are between 0.1 and 100 F[m.sup.2] (72). Polarizabilities of molecules can be higher than for atoms. The local field for a sphere is related to the polarization by Eq. (41).

A generalization of the Clausius-Mossotti equation to include a permanent moment [[[vector].[micro]].sub.e] is summarized in what is called the Debye equation that is valid for gases and dilute solutions:

[[[epsilon].sub.r] - 1]/[[[epsilon].sub.r] + 2] = [N/3[[epsilon].sub.0]] ([alpha] + [[[absolute value of [[[[vector].[micro]].sub.e]].sup.2]/[3[k.sub.B]T]]). (56)

The Debye equation could be used to estimate the permittivity of a gas if both the polarizability and the dipole moment were known from experiment.

For a specific dipole immersed in an environment of surrounding dipoles, the dipole will tend to polarize the surrounding dipoles and thereby create a reaction field. Onsager included the effects of the reaction field into the local field and obtained the following relationship for the static field that, and unlike the Debye equation, can be used to model the dipole moment of some pure liquids:

[[absolute value of [[[vector].[micro]].sub.e]].sup.2] = [[9[k.sub.B]T[[epsilon].sub.0]]/N][[([[epsilon].sub.s] - [[epsilon].sub.[infinity]])(2[[epsilon].sub.s] - [[epsilon].sub.[infinity]])]/[[epsilon].sub.s][([[epsilon].sub.[infinity]] + 2[[epsilon].sub.0]).sup.2]]], (57)

where [[epsilon].sub.[infinity]] is the optical limit of the permittivity. The Onsager equation is often used to calculate dipole moments of gases. Both atoms and molecules can polarize when immersed in a field. Note that Eq. (57) uses the permittivity of the liquid, which is a macroscopic quantity to estimate the microscopic dipole moment.

5.2 Dielectric Relaxation and Resonance

5.2.1 Simple Differential Equations for Relaxation and Resonance

A very general, but simplistic equation, for modeling polarization response that depends on time is given by a harmonic-oscillator relation:

[1/[[omega].sub.0.sup.2]][[[d.aup.2]P]/[d[t.sup.2]]] + [tau][[dP]/[dt]] + P = [X.sub.0] E, (58)

where P is polarization, [tau] is the relaxation time, [[omega].sub.0] is the natural frequency ([[omega].sub.0] = [square root of (k / m]))], and [X.sub.0] = [[epsilon].sub.s] - [[epsilon].sub.[infinity]]. Various special cases of Eq. (58) serve as simple, naive models of relaxation, resonance, and plasmonic response. The first term relates to the effects of inertia, the second to dissipation, the third to restoring forces, and the RHS represents the driving forces. A weakness of Eq.(58) is that the simple harmonic oscillator model assumes only a single relaxation time, and resonance frequency. This equation can be generalized to include interactions, (see Eq. (117)). In most materials, the molecules are coupled and have a broad range of relax-ation frequencies that widens the dielectric response. For time-harmonic fields Eq. (58) is

[~.P]([omega]) = [[[X.sub.0]/[1 - [[omega].sup.2]/[[omega].sub.0.sup.2] + i[omega][tau]]][~.E]([omega]). (59)

A resonance example is shown in Fig. 7. Intrinsic material resonances in ionic solids can occur at high frequencies due to driving at phonon normal-mode frequencies and relate to the mass inertial aspect in [[omega].sub.0] = [square root of (k / m)] of the positive and negative charges of sublattices.

If we eliminate the inertial interaction when [[omega].sub.0.sup.2] [much greater than] [[omega].sup.2], we have the time-domain Debye differential equation for pure relaxation:

[tau][[dP]/[dt]] + P = [X.sub.0] E. (60)

For time-harmonic fields, the Debye response is

[~.P]([omega]) = [[[X.sub.0]/[1 + i[omega][atu]]][~.E]([omega]). (61)

Except for liquids like water, dielectrics rarely exhibit the response of Eq. (61) since there is no single relaxation time over RF frequencies.

We generally assume that dipoles reorient in an applied field in discrete jumps as the molecule makes transitions from one potential well minimum to another with the accompanied movement of a polaron or defect in the lattice. The Debye model of relaxation assumes that dipoles relax individually with no interaction between dipoles and with no inertia, but includes frictional forces. The real part of the permittivity for dipolar systems generally does not exhibit single-pole Debye response, but rather a power-law dependence. The origin of this difference can be attributed to manybody effects that tend to smear the response over a frequency band.

If we eliminate the restoring force term in Eq. (59), we have an equation of motion for charged plasmas,

[1/[[omega].sub.0.sup.2]][[[d.sup.2]P]/[d[t.sup.2]]] + [tau][[dP]/[dt]] = [X.sub.0] E (62)

For time-harmonic fields, this becomes

[~.P]([omega]) = [[[X.sub.0]/[[-[[omega].sup.2]/[[omega].sub.0.sup.2]] + i[omega][tau]]][~.E]([omega]). (63)

5.2.2 Modeling Relaxation in Dielectrics

The polarization of a material in an applied field depends on the permanent and induced dipole moments, the local field, and their ability to rotate with the field. Dielectric loss in polar materials is due primarily to the friction caused by rotation, free charge movement, and out-of-phase dipole coupling. Losses in nonpolar materials originate mainly from the interaction with neighboring permanent and induced dipoles, intrinsic photon-phonon interactions with the EM field, and extrinsic loss mechanisms caused by defects, dislocations, and grain structure. Loss in many high-purity crystals is primarily intrinsic in that a crystal will vibrate nearly harmonically; however, anharmonic coupling to the electric field and the presence of defects modifies this behavior. The anharmonic interaction allows photon-phonon interaction and thereby introduces loss (73). High-purity centrosymmetric dielectric crystals, that is, crystals with reflection symmetry, such as crystalline sapphire, strontium titanate, or quartz, have generally been found to have lower loss than crystals with noncentrosymmetry (74).

A transient current may be induced if an electric field is applied, removed, or heated. This can be related to the dielectric response. The depolarization current for many lossy disordered solids is nonexponential and, at time scales short relative to the relaxation time of the media, can satisfy a power law of the form (75), (76)

I(t) [varies] [t.sup.- n], (64)

and satisfy a power law at long times of the form

I(t) [varies] [t.sup.(1+m)], (65)

where 0 < n, m < 1. In this model, a short time scale corresponds to frequencies in the microwave region ([tau] [varies] 1/f < 1 X [10.sup.-9] s) and long relaxation times refer to frequencies less than 10 kHz ([tau] [varies] 1/f < 1 X [10.sup.-4] s). In order to satisfy theoretical constraints at very short periods the current must depart from Eq. (64). There are exceptions to the behavior given in Eqs. (64) and (65) in dipolar glasses, polycrystalline materials, and other materials (77). The susceptibility of many lossy disordered solids typically behave at high frequencies as a power law

X'([omega]) [varies] X"([omega]) [varies] [[omega].sup.(n-1)] (66)

This implies X" / X' is independent of frequency. On the other hand, measurements of many ceramics, glasses, and polymers exhibit a loss tangent that increases approximately linearly with frequency as shown in Fig. 6.

Dissado and Hill conclude that nonexponential relaxation is related to cluster response (75). In their model, molecules within a correlated region react to the applied field with a time delay. The crux of this approach is that in most condensed-matter systems the relaxation is due not to independently relaxing dipoles, but rather that the relaxation of a single dipole depends on the state of other dipoles in a cluster. Therefore their model includes dipole-dipole coupling. This theory of disordered solids is based on charge hopping and dipolar transitions within regions surrounding a defect and between clusters (75). The effect is to spread out the response over time and therefore to produce nonexponential behavior. Dissado and Hill developed a representation of a correlation function that includes cluster interaction. According to this theory, the time-domain response for short time scales is Gaussian [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

At longer periods there are intra-cluster transitions that follow a power law of the form [t.sup.-n]. At still longer periods there are inter-cluster transitions with a Debyetype response [e.sup.[-t/[tau]]], and finally at very long periods there is response of the form [t.sup.[-m-1]] (75).

Jonscher, Dissado, and Hill have developed theories of relaxation based on fractal self-similarity (78), (79). Jonscher's approach is based on a screened-hopping model where response is modified due to many-body charge screening (80). In the limit of weak screening, the Debye model is recovered.

Nonexponential response has been obtained with many models. In any materials where the dipoles do not rotate independently, the relaxation is nonexponential.

Nonexponential response has also been reproduced in computer simulations for chains of dipoles by means of a correlation-function approach with coupled rate Eqs. (81-83).

Note that nonexponential time-domain response is actually required for over some bands in order to have a causal-function response over all frequencies. This is a consequence of the Paley-Wiener theorem (84). According to this theorem, the correlation or decay function cannot be a purely damped exponential function for large times. If C(t) is the decay function then

[[integral].sub.-[infinity].sup.[infinity]]d[tau] [[absolute value of log]C([tau])[parallel]]/[1 + [[tau].sup.2]], (67)

must be finite. This requires the decay function to vanish less fast than a pure exponential at large times, C(t) [approximately equal to] exp (- c[t.sup.q]) where q < 1 and c is a constant. We can show that at short times, decay occurs faster than exponential (85).

Nigmatullian et al. (86), (87) used the Mori-Zwanzig formalism to express the permittivity in a very general form:

[epsilon](i[omega]) = [[epsilon].sub.[infinity]] + [[[[epsilon].sub.s] - [[epsilon].sub.[infinity]]]/[1 + [R.sub.[+ or -]](i[omega])]], (68)

and concluded that for most disordered materials, the response is similar to that of a distributed circuit with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [V.sub.i] are constants determined by numerical fits. In the formulation of Baker-Jarvis et al. (88), [R.sub.[+ or -]] corresponds to the complex relaxation times [tau]([omega]) as [R.sub.+](i[omega]) = i[omega][tau]([omega]) (see Sec. 11). A [(i[omega][tau]).sup.(n-1)] frequency dependence of the complex relaxation periods corresponds to a impulse-response function of the form [t.sup.-n].

In addition, in analyzing dielectric data the electric modulus approach is sometimes used where M([omega]) = M'([omega]) + iM"([omega]) = 1/[[epsilon].sub.r] = [[epsilon]'.sub.r]/([[epsilon]'.sub.r.sup.2] + [[epsilon]".sub.r.sup.2]) + i[epsilon]"/([[epsilon]'.sub.r.sup.2] + [[epsilon]".sub.r.sup.2])

Dielectric relaxation has also been described by Kubo's linear-response theory that is based on correlation functions. This is an example of a relaxation theory derived from Liouville's equation. The main difficulty with these approaches is that the correlation functions are difficult to approximate to highlight the essential physics, and gross approximations are usually made in numerical calculations. The linear expansion of the probability-density function in Kubo's theory also limits its usefulness for highly nonequilibrium problems. Baker-Jarvis et al. have recently used a statistical-mechanical projection-operator method developed by Zwanzig and Robertson (89) to model dielectric and magnetic relaxation response and the associated entropy production (19), (40), (41), (43), (44).

6. The Distribution of Relaxation Times (DRT) Model for Homogeneous Materials

There are many models used to fit measured frequency-dependent dielectric relaxation data for homogeneous materials. These models are usually general enough to fit many types of response. When dealing with heterogeneous materials, mixture equations are commonly used (Sec. 22). The DRT model is restricted to relaxation, and it assumes there is a probability distribution y (t) that underpins the relaxation response with a relaxation time [tau]. In this model, the permittivity can be written as

[epsilon]([omega]) = [[epsilon].sub.[infinity]] + ([[epsilon].sub.s] - [[epsilon].sub.[infinity]]) [[integral].sub.0.sup.[infinity]] [[y([tau])]/[1 + (i[omega][tau])]]d[tau], (69)

where

[[integral].sub.0.sup.[infinity]]y([tau])d[tau] = 1. (70)

Note that DRT is a single-pole model and cannot be used for resonances. We see that in the DRT, Debye relaxations are weighted by a probability-density function. Equation (69) can be inverted by the Laplace transform as shown in the Appendix of Bottcher (53).

The DRT approach is sufficiently general that most causal, relaxation dielectric-response phenomena can be described by the model for Debye and power-law response. In the DRT the slope of [[epsilon]'.sub.r] ([omega]) is always negative (90). This is consistent with causality. It also indicates that the model is only valid for relaxation and not resonance. Around resonance [[epsilon]'.sub.r] ([omega]) can increase with frequency and become negative as indicated in Fig. 7.

Equation (69) can fit the relaxation response of many dielectrics because the Debye equation originates from a rate equation based on thermodynamics containing the essential physics, and Eq. (69) is a distribution of Debye relaxations. The DRT then extends this into a multi-relaxation period rate equation. We consider various special cases of Eq. (69) below. For other special cases please see Bottcher (53). In any complex dielectric material, we would expect there would be a broadening of relaxation times due to heterogeneity of the molecular response, and in this context the DRT model makes sense. This approach is often criticized, because it is not always possible obtain a physical interpretation of the distribution function (75).

6.1 Debye Model

The simplest case of the distribution function in Eq. (69) is an uncorrelated approximation where

y(t) = [delta](t - [tau]), (71)

which yields the Debye response

[epsilon] = [[epsilon].sub.[infinity]] + [([[epsilon].sub.s] - [[epsilon].sub.[infinity]])]/[1 + i[omega][tau]]. (72)

In this case, the pulse response function is

f p(t) = exp(- t / [tau]) / [tau]. (73)

In terms of components,

[[epsilon]'.sub.r]([omega]) = [[epsilon].sub.[infinity]] + [([[epsilon].sub.s] - [[epsilon].sub.[infinity]])]/[1 + [[omega].sup.2][[tau].sup.2]]. (74)

and

[[epsilon]'.sub.r]([omega]) = [[epsilon].sub.[infinity]] + [([[epsilon].sub.s] - [[epsilon].sub.[infinity]])[omega][tau]]/[1 + [[omega].sup.2][[tau].sup.2]]. (75)

If [omega][tau] is eliminated in the Debye model, and the equations for [[epsilon]'.sub.r] ([omega]) and [[epsilon]".sub.r] ([omega]) are plotted against each other, we obtain the equation for a circle:

[[[epsilon]'.sub.r]([omega]) - [[[[[epsilon].sub.s] - [[epsilon].sub.[infinity]]]/2].sup.2] + [[epsilon]".sup.2] ([omega]) = [[[epsilon].sub.s] - [[epsilon].sub.[infinity]]]/2].sup.2]. (76)

The center of the circle is on the horizontal axis.

The reasons why the Debye equation is a paradigm in dielectric relaxation theory is because it is simple and contains the essential physics and thermodynamics in relaxation. That is, it models idealized relaxation, and it yields predictions on the temperature dependence of the relaxation time [tau] = A exp ([E.sub.a] / RT), where [E.sub.a] is the activation energy.

6.2 Cole-Cole Model

The Cole-Cole model has been found useful for modeling many liquids, semisolids, and other materials (53). In this case,

y([tau]) = [1/2[pi]] [[sin[pi][alpha]]/cosh[(1 - [alpha])ln[tau]/[[tau].sub.0] - cos[pi][alpha]]] (77)

and

[epsilon]([omega]) = [[epsilon].sub.[infinity]] + [[[epsilon].sub.s] - [[epsilon].sub.[infinity]]]/[1 + (i[omega][[tau].sub.0])].sup.1 - [alpha]], (78)

where [alpha] < 1. The pulse response function is [f.sub.p] (t) = (1/[[tau].sub.0])[[SIGMA].sub.m=0.sup.[infinity]] [(-1).sup.m]/[GAMMA]((m + 1)[(1 - [alpha]))(t/[[tau].sub.0]).sup.m(1 - [alpha]) - [alpha]]. The real and imaginary parts of the permittivity can be separated into

[[epsilon]'.sub.r]([omega]) = [[epsilon].sub.[infinity]] + ([[epsilon].sub.s] - [[epsilon].sub.[infinity]]) [1 + ([omega][[tau].sub.0]).sup.1-[alpha]] sin ([pi][alpha]/2)]/1 + ([omega][[tau].sub.0]).sup.2(1-[alpha])] + 2 ([omega][[tau].sub.0]).sup.1-[alpha]] sin([pi][alpha]/2)'] (79)

[[epsilon]".sub.r]([omega]) = ([[epsilon].sub.s] - [[epsilon].sub.[infinity]])[([omega][[tau].sub.0]).sup.1-[alpha]] sin ([pi] [alpha]/2)]/[1 + ([omega][[tau].sub.0]).sup.2(1-[alpha])] + 2 ([omega][[tau].sub.0]).sup.1-[alpha]] sin([pi] [alpha]/2)]. (80)

A plot of [[epsilon]'.sub.r] ([omega]) versus [[epsilon]".sub.r] ([omega]) yields a circle, where the center is below the vertical axis.

6.3 Cole-Davidson Model

The Cole-Davidson model has also been found useful for modeling many liquids, semisolids, and other materials (53). If we consider the case [tau] [less than or equal to] [[tau].sub.0]:

y([tau]) = [1/[pi]][[[tau]/[[[tau].sub.0] - [tau]].sup.[beta]]sin[pi][beta] (81)

and zero otherwise. The permittivity is

[[epsilon].sub.r]([omega]) = [[epsilon].sub.[infinity]] + [([[epsilon].sub.s] - [[epsilon].sub.[infinity]])]/[(1 + i[omega][[tau].sub.0]).sup.[beta]], (82)

where [beta] < 1. The pulse response function is

[f.sub.p](t) = ([1/[[tau].sub.0]][GAMMA]([beta]))[(t/[[tau].sub.0]).sup.[beta]-1] exp(-t/[[tau].sub.0]). (83)

The real and imaginary parts of the permittivity can be separated into

[[epsilon]'.sub.r]([omega]) = [[epsilon].sub.[infinity]] ++ ([epsilon].sub.s] - [[epsilon].sub.[infinity]])[(1 + [[omega].sup.2] [[tau].sub.0.sup.2]).sup.-[beta]/2] cos([beta]Arg[1 + i[omega][[tau].sub.0]]), (84)

[[epsilon]".sub.r]([omega]) = ([epsilon].sub.s] - [[epsilon].sub.[infinity]]) [(1 + [[omega].sup.2] [[tau].sub.0.sup.2].sup.-[beta]/2) sin ([beta]Arg[1 + i[omega][[tau].sub.0]]), (85)

The plot of [[epsilon]'.sub.r]([omega]) versus [[epsilon]".sub.r]([omega]) maps out a skewed arc rather than a circle.

6.4 Havrilak-Negami Model

The Havrilak-Negami distribution has two parameters to fit data and is very general. It can be used to fit the response of many liquids and non-Debye solid materials (53). In the special case [alpha] = 0 it reverts to the Cole-Davidson model. The distribution function is

y([tau]) = [1/[pi]][[[([tau]/[[tau].sub.0]).sup.[beta](1-[alpha])]sin[beta][theta]]/[[[([tau]/[[tau].sub.0]).sup.2(1-[alpha])] + 2[([tau]/[[tau].sub.0]).sup.(1-[alpha])]cos[([pi](1 - [alpha])) + 1].sup.[beta]/2]]], (86)

and [theta] = [tan.sup.-1]{sin [pi] (1 - [alpha])/([tau] / [[tau].sub.0] + cos [pi] (1 - [alpha]))}

[[epsilon].sub.r]([omega]) = [[epsilon].sub.[infinity]] + [([[epsilon].sub.s] - [[epsilon].sub.[infinity]])]/[(1 + [(i[omega][[tau].sub.0]).sup.1-[alpha]]).sup.[beta]], (87)

where 0 < [alpha] [less than or equal to] 1 and 0 < [beta] [less than or equal to] 1, and

[[epsilon]'.sub.r]([omega]) = [[epsilon].sub.[infinity]] + ([[epsilon].sub.s] - [[epsilon].sub.[infinity]]) x [[cos[beta][theta]]/[1 + 2[([omega][[tau].sub.0]).sup.(1-[alpha])]sin([1/2][pi][alpha]) + ([omega][[tau].sub.0]).sup.2(1-[alpha]))]].sup.[beta]/2]], (88)

[[epsilon]".sub.r]([omega]) = [[epsilon].sub.[infinity]] + ([[epsilon].sub.s] - [[epsilon].sub.[infinity]]) x [[sin[beta][theta]]/[1 + 2([omega][[tau].sub.0]).sup.(1-[alpha])]sin([1/2][pi][alpha]) + ([omega][[tau].sub.0]).sup.2(1-[alpha]))].sup.[beta]/2]]. (89)

7. Loss and Conductivity

Loss originates from the conversion of EM field energy into heat and radiation through photon-phonon interactions. In dielectrics the heating is caused by the transformation of electromagnetic energy into lattice kinetic energy, which is seen as frictional forces on dipoles and the motion and resulting friction of free charges in materials. Major mechanisms of conduction in dielectrics in the RF band are ionic or electrolytic migration of free ions, impurities or vacancies, electrophoretic migration of charged molecules, and electronic conduction of semi-free electrons that originate from jump processes of polarons. At low frequencies, dipoles can respond to the changes in the applied field, so dielectric losses usually are low and the stored energy is high, but as the frequency increases, the dipole response tends to fall behind the applied field and, therefore, the loss usually increases and the stored energy decreases. This is related to the phasing between the current and voltage waves, in analogy to the heating an electric motor encounters when the phase between the voltage and current changes.

Ionic conduction in insulating dielectrics is due to the migration of charged ions. The migration takes place through tunneling or jumps induced by the applied field, or by slow migration under the applied field. In solid polymers it may proceed by jumps from one vacancy to another or by electronic conduction. In oxide glasses it is the movement of positively charged alkali ions in the applied field. In many materials, the dielectric losses originate in vacancy-vacancy and vacancy-impurity relaxations.

At high frequencies, lossy semiconductors, superconductors, and metals have a complex free-charge ac conductivity that is explained by the Drude model. This can cause the effective permittivity to become negative (27). To understand this, consider Maxwell's equation,

[[partial derivative]D]/[[partial derivative]t] + J = [nabla] x H. (90)

We can define an effective charge current as

[J.sub.eff](t) = [[integral].sub.-[infinity].sup.t][sigma](t - [tau])E ([tau])d[tau] + [[partial derivative]D]/[[partial derivative]t], (91)

or for time-harmonic fields

[[~.J].sub.eff]([omega]) = [sigma]([omega])[~.E]([omega]) + i[omega][~.D]([omega]). (92)

Combining ac J with the displacement field produces an effective real part of the permittivity that can be negative over a region of frequencies. For example in plasmas and superconductors, the effective conductivity satisfies i[omega][~.D] ([omega]) + [~.J]([omega]) = [i[omega]([epsilon]'([omega]) - i[epsilon]"([omega])) + [sigma]'([omega]) - i[sigma]"([omega])][~.E]([omega]), yielding

[[epsilon].sub.eff](r)([omega]) = [[epsilon]'.sub.r] ([omega]) - [[[sigma]"([omega])]/[[epsilon].sub.0] ([omega])], (93)

where [sigma]' [approximately equal to] [[sigma].sub.dc] and [sigma]" relates to the reactive part of the surface impedance. A large [sigma]" can produce a negative real part of the total permittivity such as what occurs in superconductors (91).

A total conductivity has been used in the literature to model either the ac effects of the free charge and partially bound free charge in hopping and tunneling conduction, or as another way of re-expressing the complex permittivity. Because some charge is only partially bound, the distinction between conductivity and permittivity can, at times, get blurred. This blurring points out the mesoscopic property of the permittivity. Most models of ac conductivity are based on charged particles in potential wells where energy fluctuations determine whether the particle can surmount a potential barrier and thereby contribute to the conductivity. In conducting liquids, human tissue, and water-based semisolids the conductivity is generally flat with increasing frequency until megahertz frequencies, and then it increases, often in a nearly linear fashion.

There are a number of distinct models for [[sigma].sub.tot]. The Drude model of the complex conductivity of electrons or ions in a metal is approximately modeled as

[[sigma].sub.tot] = [Ne.sup.2]/(m([[gamma].sub.0] + i[omega])) = [sigma]' - i[sigma]" , (94)

where [[gamma].sub.0] is the collision frequency, N is the electron density, m is the ion mass, and e is the electronic charge (27). Note that the dc conductivity is [[sigma].sub.dc] = [Ne.sup.2] / m[[gamma].sub.0]. The net dielectric response is a sum of the dipolar contribution and that due to the ions, where [[epsilon]'.sub.eff] = [[epsilon]'.sub.d] - [Ne.sup.2] / [m([[gamma].sub.0.sup.2] + [[omega].sup.2])] and = [epsilon]" ([omega]) = [[Ne.sup.2][[gamma].sub.0]] / [m[omega]([[gamma].sub.0.sup.2] + [[omega].sup.2]) + [[epsilon]".sub.d]([omega])]. Therefore, for metals, the real part of the permittivity is negative for frequencies near the plasma frequency, [[omega].sub.p] = [square root of [Ne.sup.2]/[[epsilon].sub.0] m]. The plasma frequency in metals is usually well above 100 GHz. The conductivity is thermally active and can be modeled for some ionic materials as (92)

[[sigma].sub.dc] = [[[n.sub.c][e.sup.2][b.sup.2][v.sub.0]]/[[k.sub.B]T]] exp(-[[DELTA]G]/[[k.sub.B]T]), (95)

where [n.sub.c] is the ion vacancy, b is the ion jump distance, [v.sub.0] is a characteristic ion frequency, and [DELTA]G is the Gibb's free energy.

For plasmas at high frequencies

[epsilon]'([omega]) [right arrow] [[epsilon].sub.0][1 - [[omega].sub.p.sup.2]/[[Y.sub.0.sup.2] + [[omega].sup.2]]]]. (96)

For disordered solids, where hopping and tunneling conduction takes place with a relaxation time [[tau].sub.e], the ac conductivity can be expressed as (93), (94)

[[sigma].sub.tot]([omega]) = [[[sigma].sub.0]i[omega][[tau].sub.e]] / [ln(1 + [i[omega][[tau].sub.e])] = [[sigma].sub.0][[[[omega][[tau].sub.e] arctan([omega][[tau].sub.e])]/[[1/4][ln.sup.2](1 + [[omega].sup.2][[tau].sub.e.sup.2]) + arctan [.sup.2]([omega][[tau].sub.e])]] - i[[omega][[tau].sub.e]ln(1 + ([omega][[tau].sub.e]).sup.2]0)]/[[1/2][ln.sup.2](1 + [[omega].sup.2][[tau].sub.e.sup.2]) + arctan [.sup.2] ([omega][[tau].sub.e])]]. (97)

8. Double Layers and Conducting Materials Near Metal Interfaces

Conducting and semiconducting dielectric materials at interfaces or metallic contacts can be influenced by the effects of double layers. Measurements on conducting liquids are complicated by the effects of electrode polarization, which are the direct result of the double layers (95). Double layers and electrode polarization are due to the build up of anions and cations at the interface of electrodes and conducting materials, as shown in Fig. 8. Modeling ionic solutions near electrodes is complicated, because the charge is mobile and depends on the potential.

Two conducting dissimilar materials can have different electronic affinities. When these dissimilar materials are in contact, a potential gradient frequently develops between the materials. As a result an electrical double layer forms at a material interface. This interface could be between liquid and metal electrodes or the layer between a biomolecule and a liquid. The potential difference will attract ions of opposite charge to the surface and repel like charges. For a double layer, the charge density depends nonlinearly on the applied potential and is modeled at low frequencies by the Poisson-Boltzmann equation for the potential (96), (97) ([[nabla].sup.2][psi] = - [rho]([psi])/[epsilon]). The potential decreases roughly exponentially from the surface as [psi](x) = [[psi].sub.0] exp (- x / [[lambda].sub.D]), where [[lambda].sub.D] is the Debye screening length or skin depth. The region near the electrode consists of the Stern layer and a diffuse region beyond the Stern layer where the potential decays less rapidly. It is known that the Poisson-Boltzmann equation is of limited use for calculating the potential around many biomolecules due to molecular interactions and the effects of excluded volume (97).

At the interface of conductive materials and electrodes, electrode polarization produces a capacitive double-layer region in series with the specimen under test. The presence of electrode polarization results in [[epsilon]'.sub.eff] being much greater than the value for the liquid by itself. Because the electrode capacitance is not a property of the material under test, but rather the interface, it can be treated as a systematic uncertainty and methods to remove it from the measurement can be applied. Double layers also form at the metal interface with semiconducting materials where the conductivity is a function of applied voltage.

The effects of electrode polarization can strongly affect dielectric measurements up to around 1 MHz, but the effects can be measurable up into the low gigahertz frequencies. Any electrode influencing the calculated permittivity should be treated as a systematic source of uncertainty. Alternatively, the permittivity with the electrode effects could be called the effective permittivity.

The effects of electrode polarization capacitance as commonly analyzed with the following model (98)

C = [C.sub.s] +[1/[[[omega].sup.2][R.sup.2][C.sub.p]]], (98)

R = [R.sub.s](1 + [[omega].sup.2][R.sup.2][C.sup.2]) + [R.sub.p], (99)

where C and R are the measured capacitance and resistance, [C.sub.p] and [R.sub.p] are the electrode double-layer capacitance and resistance, and [C.sub.s] and [R.sub.s] are the specimen capacitance and resistance. A way to partially eliminate electrode polarization is to measure the capacitances [C.sub.1] and [C.sub.2] and resistances [R.sub.1] and [R.sub.2] at two separations [d.sub.1] and [d.sub.2]. Because [C.sub.p] is the same for each measurement and [C.sub.s] can be scaled as [C.sub.s2] = [(d.sub.1] / [d.sub.2])[C.sub.s1], we can obtain the specimen capacitance. Another way of minimizing the effects of electrode polarization is to coat the capacitor plates with platinum black (99). This lessens the influence of electrode polarization by decreasing the second term on the right hand side of Eq. (98). However, both the coating and two-distance methods schemes do not completely solve this problem. For biological liquids, often the buffer solution is first measured by itself and then again with the added biological material and the difference between the measurements is reported.

For dielectric measurements, probably the best approach is to bypass much of the electrode-polarization problem altogether and use a four-probe capacitor system as shown in Fig. 9. The four-probe capacitance technique overcomes electrode problems by measuring the voltage drop away from the plates and thereby avoiding the double layer (100).

9. Relationships of the Permittivity Components: Causality and Kramers-Kronig Equations

Kramers-Kronig relations relate the real and imaginary parts of the permittivity. These equations are a result of causality and analytic functions. There are many forms of the Kramers-Kronig conditions (101), below are standard relationships

[[epsilon]'.sub.r]([[omega].sub.0]) - [[epsilon].sub.[infinity]] = [2/[pi]][integral].sub.0.sup.[infinity]][[[[epsilon]'.sub.r]([omega])[omega] - [[epsilon]".sub.r]([[omega].sub.0])[[omega].sub.0]] / [[[omega].sup.2] -[[omega].sub.0.sup.2]]]d[omega], (100)

[[epsilon]".sub.r]([[omega].sub.0]) = [[2 [[omega].sub.0]]/[pi]][[integral].sub.0.sup.[infinity]][[[[epsilon]'.sub.r]([omega]) - [epsilon]'([[omega].sub.0])] / [[[omega].sup.2] - [[omega].sub.0.sup.2]]]d[omega], (101)

For example, if we neglect any dc conductivity, the dc permittivity must [satisfy

[epsilon].sub.s] -[[epsilon].sub.[infinity]] = [2/[pi]] [[[integral].sub.0.sup.[infinity]][[epsilon]"(x)/x]dx. (102)

We should note that [[sigma].sub.dc] is not causally related to the permittivity and, therefore, before Kramers-Kronig analysis is performed, the contribution of conductivity to the loss should be subtracted.

As a consequence of causality, the permittivity satisfies the condition [epsilon]*([omega]) = [epsilon](-[omega]). Causality and second law of thermodynamics requires that when the response is averaged over a cycle, for a passive system [epsilon]"([omega]) > 0 and [micro]"([omega]) > 0. However, [epsilon]'([omega]) or [micro]' ([omega]) can be greater or less than zero. Also, the real part of the characteristic impedance must be greater than zero.

10. Static and Dipolar Polarization

10.1 Static Polarization

The total kinetic energy (K) plus potential energy of a dipole in a static applied field is approximately

U = K - [[[vector].[micro]].sub.d] * E. (103)

The probability that a dipole is aligned at angle [theta] to the directing electric field is

p([theta]) = Aexp([[[[vector].[micro]].sup.e] * E] / [3[k.sub.B]T]). (104)

The average moment for N dipoles is therefore

P = N[absolute value of [[[vector].[micro]].sub.e]] [[infinity].sub.0.sup.[pi]]p([theta])sin[theta]cos[theta]d[theta] / [[infinity].sub.0.sup.[pi]] p([theta])sin[theta] d[theta]. (105)

or

P = N[absolute value of [[[vector].[micro]].sub.e]][coth[absolute value of [[[vector].[micro]].sub.e]] [absolute value of E] / [k.sub.B]T - [k.sub.B]T / [absolute value of [[[vector].[micro]].sub.e]] [absolute value of E] = N[absolute value of [[[vector].[micro]].sub.e]]L([absolute value of [[[vector].[micro]].sub.e]] [absolute value of E] / [k.sub.B]T)

where L(x) = coth(x) - 1/x N[- or +] x/3 - [x.sup.3]/45 + [x.sup.5]/945 ... is the Langevin function. At high temperatures or weak fields, the Langevin function is approximated as

P = N[absolute value of [[[vector].[micro]].sub.e]]L([absolute value of [[[vector].[micro]].sub.e]] [absolute value of E] / [k.sub.B]T) [approximately equal to] N[absolute value of [[micro].sub.e.sup.2]]E / 3[k.sub.B]T, (107)

and in the approximation we assume [absolute value of [[[vector].[micro]].sub.e]] [absolute value of [E]/k.sub.B]T < 0.1. Note that the model shows that the polarizing effect of the applied field affects < cos[theta] >, and there is a lesser effect on the direction of the individual dipole moments. At room temperature this corresponds to an electric field of about 3 x [10.sup.7] (V/m), which is a very strong field. In intense fields or low temperatures, higher-order terms in the Langevin function must be included (53).

Using a similar analysis, the magnetic moment for noninteracting paramagnetic materials has the same form as Eq. (107)

> Mh < = [absolute value of [[micro].sub.h.sup.2] [[micro].sub.0] H] / 3[k.sub.B]T

10.2 Deriving Relaxation Equations by Analyzing Dipolar Orientation in an Applied Field

Upon application of an electric field, dipole moments, impurities, and vacancies can change positions in the lattice potential wells. This is the origin of rotation, conduction, and jump reorientation (53), (102).

Consider the density of N[- or +] molecules where there are [N.sub.[- or +]] dipole moments that are aligned either parallel (+) or antiparallel (-) to the applied field. The time evolution of the numbers of dipoles is described by the number of dipoles flipping one direction minus the number flipping the other direction characterized by the transition rates [v.sub.[- or +]], where [v.sub.+] denotes the rate of going from a + state to a - state

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (109)

In equilibrium and in the absence of an electric field, the number of transitions in either direction is the same so that [v.sub.+] [N.sub.+] = [v.sub.-] [N.sub.-], where [N.sub.+] + [N.sub.-] = N. In an electric field, the transition rates are given by

[v.sub.[+ or -]] = [v.sub.[infinity]]exp(-([U.sub.0]/[k.sub.B]T [+ or -] N[[micro].sub.e] E / 3[k.sub.B]T)), (110)

where [v.sub.[infinity]] is the maximum transition rate and the factor 3 is related to isotropic polarization pE [right arrow] [absolute value of [[[vector].[micro]].sub.e] * E] /3. At high temperatures and [v.sub.0] = [v.sub.[infinity]] exp (- [U.sub.0] / [k.sub.B]T)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (111)

Therefore, for molecules that each have a permanent electric dipole moment [[[vector].[micro]].sub.e], the net polarization is P(t) = [absolute value of [[[vector].[micro]].sub.e]](N.sub.+] - [N.sub.-]) = [absolute value of [[[vector].[micro]].sub.e]](2[N.sub.+] - N), and

[[dP]/[dt]] + 2[v.sub.0] P = [2[v.sub.0][absolute value of [[[vector].[micro]].sub.e]].sup.2]NE]/[3[k.sub.B]T]. (112)

The relaxation time is [tau] = 1/2[v.sub.0] = (1/2[v.sub.[infinity]]) exp ([U.sub.0] / [k.sub.B]T). In this model the susceptibility is

[X.sub.s] = [N[[micro].sub.E.sup.2]]/[3[k.sub.B]T]. (113)

Therefore Eq. (112) reduces to the Debye equation

[[dP]/[dt]] + P/[tau] = [X.sub.s]E/[tau]. (114)

Note that such a simple model can describe to a remarkable degree the polarization and yields a relaxation time with a reasonable dependence on temperature. This indicates the basic physics is correct.

11. Relaxation Times

11.1 Background

When a field is applied to a material, the material responds by re-arranging charge, causing spin precession, and currents. The characteristic time it takes for the response is called a relaxation time. Relaxation times are parameters used to characterize both dielectric and magnetic materials. Dielectric relaxation times are correlated with mechanical relaxation times (103). Magnetic relaxation in NMR and ESR is modeled by spin-spin ([T.sub.2]) and spin-lattice ([T.sub.1]) relaxation times.

In the literature, dielectric relaxation times have been identified for molecules and bulk materials. The first is a single molecule relaxation time [[tau].sub.s] and the other is a Debye mesoscopic relaxation time [[tau].sub.D]. For magnetic nanoparticles in a fluid, where the magnetic moment is locked in place in the lattice, the Brownian time constant is defined as [[tau].sub.B] = 3v[V.sub.H] / [k.sub.B]T, where v is the fluid viscosity and [V.sub.H] is the hydrodynamic volume of the particle (104). The Neel relaxation time is for crystals where the magnetic moment is free to rotate in the field. Dielectric relaxation times are related to how the dipole moments and charge are constrained by the surrounding material. The characteristic relaxation time for a polarized material that was in an applied field at t = 0 to decay to a steady state is related to the coupling between dipoles and details of the lattice. At high frequencies, the electric response of a material lags behind the applied field when the field changes faster than the relaxation response of the molecules. This lag is due to long and short-range forces and inertia. The characteristic Debye relaxation time [[tau].sub.D] can be obtained from the maximum of the loss peak in Eq. (61). Relaxation times are usually defined through the decay of the impulse-response function that is approximated by a Debye response exp (- t / [tau]). Debye used Onsager's cavity model to show that [[tau].sub.D] / [[tau].sub.s] = ([[epsilon].sub.s] + 2) / ([[epsilon].sub.[infinity]] + 2) (105), (106). Arkhipov and Agmon (105) showed that [[tau].sub.D] / [[tau].sub.s] = (3[k.sub.B]T / [[micro].sub.d.sup.2] [[rho].sub.c])([[epsilon].sub.s] - [[epsilon].sub.[infinity]])(2[[epsilon].sub.s] + [[epsilon].sub.[infinity]])/[[epsilon].sub.s], where [[rho].sub.c] is the density of molecules, and [[micro].sub.d] is the dipole moment. In their review, Arkhipov and Agmon also discuss the relationship between macroscopic and microscopic relaxation times from various perspectives (105). This theory predicts that the macroscopic and microscopic relaxation times are related by [[tau].sub.D] / [[tau].sub.s] [approximately equal to] (2[[epsilon].sub.s] + [[epsilon].sub.[infinity]]) / 3[[epsilon].sub.s]. Debye showed that the microscopic relaxation time for molecules of radius a is related to the viscosity [eta] and the friction constant [zeta] by [[tau].sub.s] = 4[pi][a.sup.3][eta] / [k.sub.B]T = [zeta]/2[k.sub.B]T. The Arrenhius relaxation rate is modeled as [tau] = [[tau].sub.0] exp (U [k.sub.B]T)) The Vogel-Fulcher relaxation time is used to model relaxation near polymer glass transition temperatures as [tau] = [[tau].sub.0] exp U [k.sub.B](T - [T.sub.0])). The relaxation time can also be related to changes in the activation entropy S, Helmholtz energy of activation [DELTA]H, and free energy [DELTA]F as [tau] = (h/[k.sub.B]T) exp ([DELTA]F / RT), and the entropy of activation is related by [DELTA]S = ([DELTA]H -[DELTA]F) / T. Therefore, we have [tau] = (h / [k.sub.B]T) exp ([DELTA]H / RT - [DELTA]S / R). So, by fitting the relaxation times obtained by dielectric measurements as a function of temperature we can extract changes in the entropy [DELTA]S and the Helmholtz free energy [DELTA]H for an activation process.

The typical relaxation time [T.sub.1] in NMR experiments is longer than in EPR (107). In EPR experiments, relaxation times are generally less than milliseconds. In dielectrics, the relaxation times of liquids can be picoseconds, as indicated in Table 2, but in some glasses they can be seconds and longer. The characteristic relaxation times have been found to change with the frequency of the applied field (88). This is due to the restoring and frictional forces acting differently under different field conditions. In the past researchers have realized this and resorted to using phenomenological DRT models as in Eq. (69).

Table 2. Relaxation Times of Common Liquids (105) Material [[epsilon].sub.s] [[epsilon].sub. [[tau].sub.D] [infinity]] (ps) water (22 78.4 6.2 8.3 [degrees]C) methanol (22 32.6 5.9 51.5 [degrees]C) ethanol(22 24.3 4.5 163.0 [degrees]C) 1-propanol (22 20.4 3.7 329.0 [degrees]C) 2-propanol (22 20.4 2.4 59.0 [degrees]C) Material [[tau].sub.s](ps) water (22 1.0 [degrees]C) methanol (22 7.1 [degrees]C) ethanol(22 9.0 [degrees]C) 1-propanol (22 15.0 [degrees]C) 2-propanol (22 -- [degrees]C)

11.2 Relaxation Time Based Model in Fields of Varying Frequency

A very general approach to modeling the susceptibility can be obtained by the Laplace transform of the time-invariant approximation to Eq. (38). This yields a permittivity in terms of complex relaxation times [tau]([omega]) = [tau]'([omega]) - i[tau]"([omega]) (46):

[[epsilon]'.sub.r]([omega]) = [[epsilon].sub.r[infinity]] + ([epsilon].sub.rs] - [[epsilon].sub.r[infinity]]), [[1 - [omega][tau]"([omega])]/([omega][tau]'([omega])).sup.2] + (1 - [omega][tau]"([omega])).sup.2]], (115)

[[epsilon]".sub.r]([omega]) = [([epsilon].sub.rs] - [[epsilon].sub.r[infinity]]) x [omega] [tau]'([omega]) / [([omega] [tau]'([omega]).sup.2] + [(1 - [omega] [tau]"([omega])).sup.2] + [[sigma].sub.s] / [[epsilon].sub.0][omega] (116)

The assumption of this model is that at RF frequencies the relaxation has a dependence on the frequency of the driving field. This frequency dependence originates from the applied field acting on the molecules in the material that keeps the molecules in a nonequilibrium electromagnetic state. Equations (115) and (116) have the same form as the Laplace transform of a linear harmonic oscillator equation of motion. However, this model contains additional information through the frequency dependence of the relaxation times. For a real, frequency-independent relaxation time ([tau]' constant and [tau]" = 0), Eq. (38) is the Debye equation. In the special case where [tau]' is constant, the ensemble response function is of the form exp (-t/[tau]') and we have classical Debye relaxation. This can be traced to the fact that the Debye model assumes there is no inertia, and therefore, a purely damped motion of dipoles. Performing the inverse Laplace transform of the time-invariant approximation to Eq. (38) we obtain another form for the polarization equation,

[[integral].sub.0.sup.t] [bar.[tau]] (t - [theta])[[dP([theta])]/[d [theta]]]d[theta] + P(t) = [X.sub.s] E (t). (117)

Equation (117) highlights the physics of the interaction with materials and is useful in determining the underlying differential equation related to phenomenological models. For this equation the Debye model is obtained if [tau] (t) = [[tau].sub.0][delta](t). Relaxation phenomenological models such as Cole-Davidson can be related [tau]([omega]). Therefore the underlying differential equations can be cast into the form of Eq. (117). Because they are complex pairs, it is not possible to extract the time-domain functions of [tau]'([omega]) and [tau]"([omega]) independently.

It is important to study the origin of the frequency-domain components. Whereas [tau]'([omega]) models the out-of-phase behavior and loss, [tau]"([omega]) models the effects of the local field on the restoring forces. If [tau]"([omega]) is positive it is related to inertial effects. If [tau]"([omega]) is negative, it is related to the local field interaction that tends to decrease the polarization through depolarization. The relaxation times are

[tau]'([omega])[omega] = ([[epsilon]".sub.r] ([omega]) - [[[sigma].sub.s]]/[[epsilon]'.sub.0] ([omega])) [[([[epsilon].sub.s] - [[epsilon].sub.r[infinity]])]/([[([[epsilon]'.sub.r] [omega]- [[epsilon].sub.r[infinity]]).sup.2] + ([[epsilon]".sub.r] ([omega]) - [[[[sigma].sub.s]]/[[epsilon].sub.0] [omega]]).sup.2], (118)

[tau]"([omega])[omega] = - ([[epsilon]'.sub.r]([omega]) - [[epsilon].sub.r[varies])([[epsilon].sub.s] - [[epsilon]'.sub.r]([omega])) -[[epsilon]".sub.r]([omega]) - [[sigma].sub.s]/[[[epsilon].sub.0][omega]]).sup.2]/[[[epsilon]".sub.r]([omega]) -[[epsilon].sub.r[varies]).sup.2] + [[epsilon]".sub.r]([omega]) - [[sigma].sub.s]/[[[epsilon].sub.0][omega]]).sup.2]. (119)

In Fig. 10 we plot the relaxation times extracted from dielectric measurements as well as measurements given in Reference (108). We see that the measured [tau]" ([omega]) values are all negative. We see that for ethanediol, [tau]" is very small and [tau]' is nearly frequency-independent. Therefore ethanediol, is well modeled by the Debye equation. The physical significance of [tau]'([omega]) relates to the effective time for the material to respond to an applied electric field. [tau]" ([omega]) > 0 at resonance corresponds to an effective ensemble period of oscillation and [tau]" ([omega]) < 0 corresponds to a characteristic time scale for charge depolarization and screening effects. An interpretation is that in relaxation the effects of the local field on the short-range restoring forces and screening may have a frequency dependence. This frequency dependence can manifest itself as the commonly observed frequency shift in the loss peak relative to the Debye model. We also see that [tau]" < 0 can be interpreted as the effects of the local field on the short-range electric restoring forces, which tend to reduce the permittivity and modify the position of the maximum in the loss curve relative to the Debye maximum condition ([omega] [tau]' = 1). The behavior for [tau]"([omega]) < 0 is analogous to what is seen in longitudinal optical-phonon behavior that yields a local field that tends to reduce polarization. Over frequencies where mass-related inertial interactions are important, [tau]"([omega]) > 0. This occurs in polaritonic resonances at terahertz to infrared frequencies and in negative-index materials. In this case the local field tends to enhance the polarization through the effects of inertia that counteract restoring forces (5). When [tau]"[omega] = 1, the real part of the susceptibility goes to zero, indicating the system is going through resonance. In general, just as in the Debye and other phenomenological models, the relaxation times can depend on temperature (A exp ([U.sub.0]/[k.sub.B]T)).

11.3 Surface Waves

Electromagnetic surface waves occur in many applications. Surface waves can be supported at the interface between dielectrics and conductors. These waves travel on the interface, but decay approximately exponentially away from the surface. There are many types of surface waves, including ground waves and surface plasmons polaritons (SPP) that travel at the interface between a dielectric and conductor, surface plasmons on metals, and Sommerfeld and Goubau waves that travel on coated or uncoated wires. SPP's require the real part of the permittivity of the metal to be negative (109). A Goubau line guides a surface wave and consists of a single conductor coated with dielectric material (110). A Sommerfeld surface wave propagates as a TM mode around a finitely conductive single bare conductor. Plasmonic-like surface waves can form from incident microwave electromagnetic energy on subwavelength holes in metal plates. We will examine plasmonic surface waves in Sec. 14.2.

11.4 Electromagnetic Radiation

Classical electrodynamics predicts that accelerated charged particles generate EM waves. This occurs in antennas where charged particles oscillate to produce radiation. Linearly or elliptically polarized radiation waves are determined by the type of acceleration the source charged particles undergo. If the charge particle undergoes oscillation from a nonlinear restoring force, the emitted radiation may not be monochromatic.

11.5 Thermal Noise and Blackbody Fields

Due to the continual Brownian motion of microscopic charges, thermal Johnson noise fields are produced over a broad distribution of frequencies (111), (112). There are also many other sources of noise such as phase noise and shot noise. Thermal movement and blackbody radiation are a source of electrical noise and was described theoretically by Nyquist (112). This theory was expanded by Callen (113). A blackbody has an emissivity of near unity and is an excellent absorber and emitter of radiation. The spectral distribution of blackbody radiation follows the Planck distribution for the energy density u (T, f) = (8[pi]h[f.sup.3]/[c.sup.3])/(exp (hf/[k.sub.B]T - 1)). Examples of blackbody radiation include radiation from intergalactic space, as well as black cavities with an aperture. Typical blackbody materials have some free electrons and a distribution of molecular resonant frequencies and, as a result, are useful in converting optical energy into heat energy. They are also good radiators of infrared thermal energy. Most materials only partially reflect any incident energy. Therefore, they do not radiate as much power as a blackbody at the same temperature. The ratio of the energy radiated by a material relative to that of an ideal black body is the emissivity. In a frequency band [DELTA]f, the emissivity is defined as e = P/([k.sub.B]T[DELTA]f). The emissivity satisfies 0 [less than or equal to] e [less than or equal to] 1. The brightness temperature is [T.sub.B] = eT, where T is the physical temperature. Nyquist/Johnson noise in the RF band has only a weak frequency dependence. It is modeled for voltage fluctuations in a transmission line terminated by resistors R over a frequency band [DELTA]f by <[v.sup.2]/R = 4[k.sub.B]T[DELTA]f (112).

Radiometers in the RF band are usually receiving antennas that collect noise power from the direction they are pointed and infer the brightness temperature. The goal of radiometry is to infer information about the remote source of noise from the brightness temperature (111).

Quantum-field theory models the vacuum as filled by quantum fluctuations that contain a spectrum of frequencies having energy (1/2)h[omega]. In this model, fluctuations give rise to virtual photons and spontaneous emission of short-lived particles. Virtual photons and short-lived particles are allowed by the uncertainty principle between energy and time: [DELTA]E[DELTA]t [approximately equal to] h.

Vacuum fluctuations can produce attractive forces between nanometer-spaced parallel electrodes. This Casimir effect is commonly explained classically by the cutoff of EM modes between the plates so that the external radiation pressure exceeds the pressure between the plates (114). A more complete and satisfactory description can be derived with quantum mechanics. The force is extremely short range. It has also been shown that the force can be made repulsive by changing one of the plates from a metal to a dielectric such as silica (115). In addition, there has been speculative research where NIM materials are used for the microscopic plates to produce levitation of nanoparticles (116). Casimir effects may play a role in future modeling of microelectronics because the electrode separations are close to where these effects become important.

12. Magnetic Response

12.1 Overview of Magnetism

In this section, we will very briefly overview the basic elements of magnetic phenomena needed in our applications to RF interactions. Magnetism has a quantum-mechanical origin intimately related to the spin and angular momentum and currents of electrons, nuclei, and other particles. Stern and Gerlach (4) proved the existence of discrete magnetic moments by observing the quantized deflection of silver atoms passing through a spatially varying magnetic field. Electrons orbiting a nucleus form a magnetic moment as well as the intrinsic spin of the electron. Magnetic moments are caused either by intrinsic quantum-mechanical spin or by currents flowing in closed loops m [infinity] (current)(area).

Spins react to a magnetic field by precessing around the applied field with damping (117). For spins of the nucleus, this precession forms the study of nuclear magnetic resonance (NMR); for paramagnetic materials it is called electron-spin or ESR or electron-paramagnetic resonance or EPR; and for ferromagnetic materials it is called ferromagnetic resonance or FMR. The dynamics in spin systems are tied phenomena such as spin precession, relaxation, eddy currents, spin waves, and voltages induced by domain-wall movements (7-9), (118).

Paramagnetism originates from spin alignment in an applied magnetic field and relates to the competition between thermal versus magnetic energy (m * B/[k.sub.B]T) (see A in Fig. 11). Paramagnets do not retain significant magnetization in the absence of an applied magnetic field, since thermal motion tend to randomize the spin orientations.

The origin of diamagnetism in materials is the orbital angular momentum of the electrons in applied fields. Diamagnetic materials usually do not have a strong magnetic response, although there are exceptions. In ferromagnetic materials, exchange coupling allows regions of aligned spins to be formed (119). Ferro-magnetic and ferrimagnetic materials may have spin resonances in microwave to millimeter wave frequencies (120). Ferrimagnetic materials consist of two overlapping lattices whose spins are oppositely directed, but with a larger magnetic moment in one lattice than the other. Antiferromagnetism is a property of many transition elements and some metals. In these materials the atoms form an array with alternating spin moments, so the average spin and magnetic moment are zero. Antiferromagnetic materials are composed of two interpenetrating lattices. Each lattice has all spins more or less aligned, but the lattices, as a whole, are inverse. Resonances in antiferromagnetic materials may occur at millimeter wave frequencies and above. Antiferromagnetic materials are paramagnetic above the Neel temperature.

James Baker-Jarvis and Sung Kim

Electromagnetics Division, National Institute of Standards and Technology, Boulder, Colorado 80305

jjarvis@boulder.nist.gov

sung.kim@boulder.nist.gov

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Title Annotation: | p. 1-31 |
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Author: | Baker-Jarvis, James; Kim, Sung |

Publication: | Journal of Research of the National Institute of Standards and Technology |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jan 1, 2012 |

Words: | 21277 |

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