Electromagnetic Composites Handbook: Models, Measurement, and Characterization [2 ed.] 9781259585043, 1259585042, 9781259585050, 1259585050

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ABOUT THE AUTHOR Rick Moore received his PhD in Physics in 1978 and has performed and coordinated research, development, and engineering at the Georgia Institute of Technology Research Institute. His work is documented in approximately 150 articles, presentations, patents, and reports. He has contributed to patents in photonic structures and fiber material treatments and has authored or coauthored hundreds of peer reviewed publications and technical reports. He retired after 35 years at the Georgia Tech Research Center where he was a Principal Research Scientist, GTRI Research Fellow, and Co-Director of the Georgia Tech Center of Excellence in Ultra-Wideband Technologies. He continues work part-time in the fields of electromagnetic measurements and metamaterial design.

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CONTENTS

Preface Acknowledgments Introduction Outline References Chapter 1. Introduction to Wave Equations and Electromagnetic Constitutive Parameters 1.1 Maxwell’s Equations and Field Sources 1.2 Permittivity and Charge 1.3 Permeability and Current 1.4 Wave Equations for Homogeneous and Inhomogeneous Materials 1.5 Homogeneous Propagation in Linear Media 1.6 Conclusion References Chapter 2. Sources and Dispersion for Polarization and Permittivity ε(f) 2.1 Sources of Permittivity, Resistivity, and Conductivity in Materials 2.2 Modeling Material Complex Permittivity and Its Frequency Dependence 2.3 Small Damping, τ ⇒ 0 2.4 DC, Zero Frequency, and DC Scaled Permittivity 2.5 Combined Models for Permittivity 2.6 Coupling Permittivity and Permeability 2.7 Additional Bound Charge Contributions to Permittivity and Frequency Dispersion 2.8 Permittivity Variation with Temperature References Chapter 3. Sources and Dispersion of Magnetization and Permeability μ(f)

3.1 Sources of Permeability 3.2 Frequency Dispersion in Magnetic Materials 3.3 Susceptibility Models for Data Analysis 3.4 An Overview of Micromagnetic Models 3.5 Kramers–Kronig (KK) Relationships 3.6 Temperature-Dependent Permeability References Chapter 4. Fundamental Observables for Material Measurement 4.1 Introduction 4.2 Scattering of Plane Waves from Homogeneous Planar Boundaries and Material Slabs 4.3 Single Planar Slab of Material 4.4 Scattering: Cascade Matrix Method for Multi-Boundary Material Analysis 4.5 Scattering from a Shunt Planar Impedance Sheet 4.6 Transmission and Reflection from Anisotropic Laminates 4.7 A Numerical Anisotropic Material Example 4.8 Conclusion References Chapter 5. Composites and Effective Medium Theories 5.1 Introduction 5.2 EMT Development Timeline 5.3 Limitations and Derivation of EMTs 5.4 Scattering Functions for Spheres 5.5 Scattering and EMT of Large–Aspect Ratio Particle Geometries 5.6 Layered Inclusions 5.7 Model Choices: Importance of Conduction and Particulate Interaction References Chapter 6. Conducting-Dielectric and Magneto-Dielectric Composites 6.1 Introduction 6.2 Percolation, Dimensionality, Depolarization, and Frequency Dispersion in Semiconducting, Conducting-Dielectric Composites 6.3 Magnetic Effective Media References

Chapter 7. Numerical Models of Composites 7.1 Method of Moment Modeling and Laminated Composites 7.2 Finite Difference Time Domain Simulations 7.3 Comments for Chapters 5 to 7 References Chapter 8. Electromagnetic Measurement Systems Summary for RF–Millimeter Wavelengths 8.1 An Introduction to Wideband Material Metrology 8.2 Error Correction, Calibration, and Causality 8.3 Historical: Von Hippel and the Slotted Line 8.4 Summary of Measurement Techniques 8.5 Nonresonant Techniques: General Transmission Line Measurement Guidelines and Procedures 8.6 Cylindrical Waveguide 8.7 Coaxial Lines 8.8 Stripline Measurements 8.9 Focused Beam Free Space System 8.10 Focused Beam Technical Description 8.11 Calibration, Measurements, and Discussion References Chapter 9. Resonant Techniques for Material Characterization 9.1 Resonant Cavities 9.2 Overview of the TE10p Measurement Technique 9.3 Parallel Plate Stripline (TEM) Cavity 9.4 Closed Reflection Cavity 9.5 Open Cavity: Fabry–Perot Resonator References Chapter 10. Transmission Line, Free Space Focused Beam and TE10N Measurement Details 10.1 Constitutive Parameter Solutions in Coaxial Transmission Line, Rectangular Waveguide, and Free Space 10.2 Extreme Elevated Temperature Reflection Measurements 10.3 Free Space Focused Beam Characterization of Materials

10.4 TE10N Transmission Cavity References Chapter 11. Micrometer and Nanoscale Composites 11.1 Applications and Impetus for Nano Magnetic Composites 11.2 Case Study 1: NiZn and MnZn Ferrites 11.3 Case Study 2: Nano Magnetic Composites 11.4 Case Study 3: Multiscale EMT (Nano to Macro) for Artificial Dielectrics 11.5 Conclusions References Chapter 12. Measured Data of Materials and Composites 12.1 Solid Ceramic versus Frequency 12.2 Solid Ceramic versus Temperature 12.3 Ceramic Fiber versus Temperature 12.4 Two-Phase Ferrite-Polymer Composites and Three-Phase Ferrite-Fe-Polymer Composites 12.5 Composites Demonstrating Percolation 12.6 Solid Semiconductors versus Frequency 12.7 Honeycomb and Foams versus Frequency 12.8 Polymers versus Frequency 12.9 R-Cards versus Frequency 12.10 Micrometer and Nanometer Magnetite Magnetic Composites versus Frequency 12.11 Iron–Polymer Composites versus Frequency 12.12 Ceramic Polymer Fiber versus Frequency 12.13 Dense Ferrites versus Frequency 12.14 Fiber–Polymer Composites versus Frequency Index

PREFACE

Arthur Von Hippel’s book, Dielectric Materials and Application, was published in 1954. At the time, the development of composites for electrical and electromagnetic technologies was just beginning. Thus, dielectric and magnetic theory, models, measurement techniques, and measured data that were presented by Von Hippel emphasized homogeneous isotropic materials composed of a single molecular species or compound. The vast majority of those materials were electrically insulating and nonmagnetic. Semiconductor production was in developmental phase, but samples for waveguide measurements (as used by Von Hippel) were not available and the importance of semiconductors for everyday technology was not yet recognized. Shockley’s patent on the transistor (#2569347) was just 6 years old. Ferrites were known; however, their application in radio and microwave technology for phase shifters, filters, and isolators were just being realized. They are now applied for suppression of radio frequency interference on computer mother boards, integrated circuits, communication networks, and in electrically small antennas. The use of fiber and laminate-based composites in electromagnetic technologies did not begin until the 1970s. The Electromagnetic Composites Handbook is designed as an engineering and scientific handbook that extends the Von Hippel text to include data on additional nonconducting dielectrics, semiconducting, conducting, and magnetic materials and composites composed of two or more molecularly distinct compounds that are distributed in size scales from nanometers to centimeter dimensions. The development of models that attempt to predict composite constitutive parameters, using constitutive parameters of their constituents, is a parallel effort. The models support predictions of and comparison to measured permittivity and permeability. Permittivity, permeability, impedance, and conductivity data for solids and composites are presented for frequencies from about 1 MHz to 1000 GHz. Chapters of this book are devoted to the descriptions of electromagnetic constitutive parameter sources, procedures and equipment to measure the parameters, propagation models in composites, prediction of composite properties, and measured constitutive parameter data for the electromagnetic spectrum of wavelengths larger than a few micrometers but mostly in the meter to millimeter wavelengths. Each chapter concludes with a list of references for that chapter. These are indicated in each chapter ’s text in brackets. MK units are primarily used throughout this book; however, English or CG units may occasionally enter into discussion. The analysis crosses scientific and technological boundaries and thus the scientific complex operator, i, sometimes appears rather than the engineering j for the complex numbers. Note that in the data tables a positive sign, +, is adopted for dielectric and magnetic loss. Modeling and theory chapters discuss various composite models and then apply the most successful

analytical and numerical methodologies to typical electromagnetic design problems that often use electromagnetic composites in their solution, again for wavelengths larger than a few micrometers. Reflection and transmission line measurements, such as those of Von Hippel, are the framework from which composite material measurements began and those measurement techniques are reviewed. The review is followed by a discussion of advances in the measurement technology since 1980. For example, the microwave and millimeter wave application of lens-based open cavities and free space measurements, common for infrared and optical spectra, is one advance. The techniques include Fabry–Perot and etalon derivatives. The adoption of the infrared and optical techniques for millimeter, centimeter, and even meter wavelengths and the use of various multi-mode resonant cavity configurations, was facilitated by the second major technology addition, i.e., the development of the automatic network analyzer (ANA) and digital receivers–transmitters that had modest power (hundreds of milliwatts), broad bandwidth frequency, synthesized sources, and matched adapters. A third advance was microwave and millimeter antennas with bandwidths larger than 20:1. Advances in electromagnetic tools, instrumentation, and “borrowing” of lens-based measurements now allow accurate measurement of isotropic or anisotropic constitutive properties for single samples from a few hundred megahertz to above 100 GHz. Some composites may contain constituents that are distributed in size scales of nanometer to centimeter dimensions. The larger scales make the composite electrically inhomogeneous at higher frequencies since inhomogeneity is determined by the ratio of the physical size of the composite phases and the electromagnetic wavelength. Characterizing the large-scale composites by effective permittivity and/or permeability is not sufficient. In cases where physical scales of the composite components are small but their electrical scale approach unity, diffuse and/or bistatic electromagnetic scatter modeling and measurements may be used to expand understanding of electromagnetic observables (reflection, transmission, and absorption) and calculated, effective magnetic permeability and electrical permittivity of composites. Measurement techniques that apply to some electrically inhomogeneous composites can also be used for isotropic, homogeneous materials. Numerical models will be discussed that give insight into electromagnetic properties of inhomogeneous electromagnetic composites and the problems that may be encountered in their utilization. The advances discussed in this handbook are significant to both electromagnetic engineers and theoreticians. ANA advances now allow continuous measurement and thus material parameter data over 1000:1 or greater bandwidths. With such a dense database, experimentalists and engineers can confidently design broadband meter, microwave, and millimeter wave devices and material constructs. A physicist, chemist, or material scientist benefits from the high data density in verification of electromagnetic composite material theories over bandwidths that encompass multiple physical and electrical scales, material dimensionalities, and material physics. Examples are multiphase magnetics, periodic dielectrics exhibiting photonic bandgaps, and material constructs with negative index behavior. The book concludes by presenting dielectric and magnetic parametric fits to measured data for almost 300 composites and/or composite components. Many gigabytes of data contributed to the preparation of this book and a comprehensive presentation of complex permittivity and

permeability in tabular form were not possible due to space limitations; however, a digital database is planned for the future. For now, the parametric fits of Chap. 12 supply frequency and temperature dispersive data that are presented as analytic equations whose forms are based upon solid-state physics. The frequency and/or temperature range used for each fit are annotated with the equation parameters. Measurements range from 1 MHz to a few hundred gigahertz. Data density was typically at 1 MHz intervals below 100 MHz, 10 MHz spacing from 100 MHz to 1 GHz, and 100 MHz spacing above 1 GHz. The complex magnetic permeability and permittivity are fit to a range of relaxation models. Measurement frequencies are above characteristic solid-state Debye relaxation frequencies and below terahertz to infrared molecular relaxations. Power laws in frequency coupled with a single resonant model produce excellent parameterizations for permittivity data, especially those of composites containing semiconducting components. Overall, the parametric fits aid in spanning measurement frequency gaps and in interpretation of material physics. Selected composite data are presented for measurements made before and during exposure to environmental extremes of temperature. For example, ceramic and ceramic composites are often used in high-temperature environments; thus data are shown from ambient to temperatures in excess of 2200 K. Exponential functions (typical of semiconductors) are used for temperature dependence of ceramics and ceramic fibers. Select materials were chosen to overlap data of Von Hippel and other publications for comparison. Some data are repeated for identical material compositions, but from different suppliers, and thus illustrate unsurprising variability. Data on composites may be for “identical” compositions but are included to illustrate variability in manufacturing and source.

ACKNOWLEDGMENTS

This book has grown over the past 35 years and matured to its final form in the last 5 years. Many colleagues at the Georgia Tech Research Institute (GTRI) contributed to the development of the many GTRI measurement systems that are described or pictured in this text. An attempt to list those who gave special contributions is below. The list is in rough historical order: Drs. Patrick Montgomery and Thomas B. Wells for transmission line analyses, inversion algorithms, and design of Fabry-Perot systems; Mr. Thomas Taylor, Paul Friedrich, and Mrs. Anita Pavadore for design, assembly, and making work, high temperature cavities, transmission lines, and focused beam systems; Drs. Lisa Lust, Paul Kemper, Alexa Harter, Greg Mohler, and Silvia Liong for efforts in material characterization, percolation theory (dielectric and magnetic), effective media modeling (both numerical and analytical), and nanomagnetics; Dr. James Maloney and Mr. Brian Shirley for FDTD advancements; Dr. John Schultz, Mr. Stephen Blalock, and Mr. Edward Hopkins for optimizing focused beam measurement procedures and system design; Dr. John Meadors and Mr. Norm Ellingson for support; My son, Jason Mathew Moore, who assembled and processed measured data into useable spreadsheets and produced many photographs of the Georgia Tech Research InstituteAdvanced Concepts Laboratory owned measurement fixtures; Ms. Kathryn Gilbreath for her graphic arts contribution and Drs. Lon Pringle and Erik Shipton for hours of proofing the document; and Dr. Eric Kuster for 30 years of effort in developing and refining numerical simulations to predict material properties. Finally, I must remember Mr. James Gallagher, the Georgia University System’s first Regents’

Researcher. Jim redirected me down a path of measurement in 1981.

INTRODUCTION

Over the last 50 years and since the Von Hippel’s text [1], multifunctional and frequency dispersive electromagnetic composites have become intertwined in aerospace, computer, sensor, communication, and structural engineering. Aerospace applications are exemplified in the Space Shuttles, Northrop B2, and Boeing 787 aircraft [2–4]. Space shuttle tiles are multifunctional examples that meet structure, impact, temperature stability, and thermal conductivity specifications. However, the same tiles also covered and protected communication and radar antennas and thereby operated as environmentally resistant, temperature stable, radio frequency radomes. Composite laminates are routinely found as circuit substrates. Energy-efficient windows for skyscrapers must be optically transparent and are often infrared reflective. However, if cell phone and wireless access are required, the same window must have transparency from hundreds of megahertz to tens of gigahertz. Alternatively, window specifications may call for radio frequency isolation to limit wireless access within the structure. In either case, the window can incorporate semiconducting film and components to achieve the required frequency dispersions. Magnetic micrometer and nanometer particulates are found in electromagnetic interference materials and are applied as MRI enhancers and for the treatment for cancer. Composites are also used to construct lenses, waveguide, photonic bandgap, and/or structures that produce an effective negative index of the structure. These examples often incorporate semiconducting, artificial dielectric, conducting, and magnetic components. The electromagnetic models, measurement techniques, and measured data of this book are chosen to aid development of multifunctional material designs. Models and measurement apply to composites made with components of sizes from tens of nanometers to centimeters. Goals of this text are to contribute to current and visionary electromagnetic composite applications and supply an extension of the Von Hippel database for composites.

OUTLINE Chapter 1 introduces the following chapters and establishes definitions, terminology, Maxwell’s equations, electromagnetic propagation, and concepts of electromagnetic material constitutive parameters. Physics and physical sources of electromagnetic permittivity, permeability, and conductivity are discussed in Chaps. 2 and 3, respectively, for the electromagnetic spectrum between approximately 1 MHz and a few terahertz. Concepts that are discussed include the field polarization, interfacial contact between composite

constituents, electron spin, magnetic domains, sizes and shapes, periodicities of composition, and free charge carrier density. All play parts in determining frequency dispersive constitutive parameters in the above electromagnetic frequency range and therefore limit the solid-state functional forms that are used in parametrically fitting measured data of Chap. 12. Controlled constitutive parameter temperature dependence can be a significant advantage in composites and therefore various sources and methods for reduction of temperature variation are also discussed in Chaps. 2 and 3. Kramer–Kronig (K–K) relations are introduced in Chap. 3 as a physics-based analysis to evaluate measurement accuracy and causality of measured data. K–K analyses are made easier by the advances in measurement technology arising from wideband, phase-locked electromagnetic sources and digital amplitude and phase analyzers and receivers. However, K–K still has strongest application for characterization at the infrared and optical spectral region. In those wave bands power reflection and transmission amplitude are easily measured over many decades of frequency but continuous frequency phase measurements are not easily obtained. With the advent of network analyzer systems, digital receivers/sources, and free space characterization techniques, 100:1 and even 1000:1 bandwidths of data can be obtained on single material samples in the meter through millimeter wavelength regime. The multi-decade bandwidths support K–K analysis at lower wave bands. Chapter 4 is devoted to descriptions of propagation models that are used to calculate electromagnetic observables (plane wave reflection and transmission coefficients) from constitutive data and also used for the inverse function, i.e., calculating constitutive parameters from measured reflection and transmission. The measurement techniques described in this book acquire measured plane wave reflection and transmission voltage amplitude and phase as their observables. Analyses of Chapter 4 relate that reflection and transmission to frequency dispersive isotropic and/or anisotropic permittivity, permeability, and conductivity of material layers that are contained within laminar structures. Layers may have arbitrary thickness and may themselves be composites or mixtures. Electromagnetic boundary value problems are initially used to develop propagation and reflection equations under the assumption that the materials have electromagnetic components with characteristic electromagnetic scale, much smaller than the interrogating electromagnetic wavelength. Impacts of large-scale discrete components are addressed in later chapters under discussions of numerical modeling. Chapter 4 also presents propagation models in the context of scattering-cascade matrices [5] and shows how to apply these to multilayer structures so long as internal material laminates are electrically homogeneous. Both isotropic and anisotropic propagation models are presented. Example calculations of reflection and transmission coefficients are shown for planar electromagnetic waves, for waves incident from various angles, and for different laminate constructions. The analysis is also described that demonstrates that reflection and transmission measurements, at various angles and sample orientations, can be combined with the geometric construction of the laminate to apply scattering matrices and infer the effective constitutive parameters of a laminate or laminate layers. The chapter includes the discussion of the impact of small variations in constitutive parameters on reflection and transmission. The chapter concludes by presenting discussions, models, and calculated examples of propagation in diagonal and fully anisotropic materials. Calculations are presented to

establish a baseline with which other models and future measurements can be compared. Examples of composite applications and use of scattering matrices are found throughout the book. Chapters 5 to 7 address effective media theory (EMT) methodologies to predict composite properties from those of their constituents. Models presented in these chapters can assist the experimenter in analyses that bound expectations for measured electromagnetic properties. Models also place limits on electromagnetic size scale for material components that are used in composites and yet allow the composite to be characterized by using measured effective homogeneous anisotropic permittivity and permeability that are used in scattering matrix analysis. The chapters are ordered in hierarchical EMT complexity, electrical scale of the composites, parameters to be modeled, and composite chemistry/composition. The coherent potential approximation is used to derive and apply various electromagnetic scattering-based EMT. This application of the coherent potential approximation has been used by many authors but here it will be discussed in the context of two books: Wave Propagation and Scattering in Random Media by Akira Ishimaru [6] and Introduction to Wave Scattering, Localization and Mesoscopic Phenomena by Ping Sheng [7]. Chapter 5 supplies an introduction to effective media concepts and derives common forms of EMTs by application of the coherent potential approximation. General limits are established on dielectric composite composition, component size, and constitutive parameter so that simple EMT can be applied. The emphasis is placed on spherical inclusions since they can easily supply physical insight into particulate coating effects, higher order scattering, and constitutive parameter complexity. The EMTs are applied to simple dielectric composites composed of a matrix and single electrically small particulate which is at a low fractional volume. Particulate shape and aspect ratio are incorporated to illustrate anisotropies that may be encountered in arrayed or physically thin composites. Chapter 6 begins with extension of effective media equations to semiconducting and conducting particulates and high volume fraction compositions. The addition of semiconductor or conducting particulates facilitates the synthesis of artificial dielectrics; however, effective media models must be extended to include electrical percolation within the composite. Correct description of the percolation requires details on particulate and composite geometry. The importance of multiscale modeling (nanometer to centimeter) is illustrated in Chap. 11 by considering micrometer size dielectric spheres that are coated with nanoscale conducting films. Percolation introduces constitutive parameter dependence on composite geometry and dimensionality. In general the permittivities of these composites show a power law frequency dispersion, and have anisotropy in thin layers and their constitutive parameters scale with composite sample size. Additional physical observables may include anomalous power loss that actually arises from diffuse scatter; local electric field strength concentration and “plasmon” resonance. Examples of phenomenon will be discussed to establish foundations for measurement system requirements and equipment for characterizing these composite types. Chapter 6 further extends analysis of artificial dielectrics to the study of artificialdielectric-magnetic materials. Though most EMTS appear to be easily mapped to permeability, they are accurate only when low-volume fractions of multi-domain magnetic

particulates are dispersed in the composite. Examples are given of EMT applied for dense composites containing magnetic components. Fundamentally, EMTs fail to account for magnetic coupling between magnetic particles within a composite. Accurate models that account for complex magnetic particulate coupling and combinations with artificial dielectrics would appear to require numerical approaches rather than simple computations using effective media equations. Numerical methods in composite analysis are topics for Chap. 7. The method of moment (MoM) and finite-difference time domain (FDTD) numerical techniques are reviewed. Properties of artificial dielectrics near the percolation threshold and artificial dielectrics in magnetic media are also discussed. Interactive models of magnetic and conducting components can support negative index material concepts [8–10], photonic bandgap materials [11], and other metamaterials [11–13] which are active areas of research in materials engineering. Examples are given of numerical model applications to predict the electromagnetic observables in dense composites and those that have large electrical scales. The implications of the model’s predictions are that electromagnetic characterization of many artificial and/or metamaterials may require equipment beyond those applied to measure simple isotropic dielectrics or magnetics. Chapter 8 begins discussion of measurement techniques and equipment configurations and provides a summary of a 30-year evolution of equipment and measurement procedures for electromagnetic material characterization. The ability to perform accurate 10:1, 100:1, and even 1000:1 bandwidth electromagnetic measurements was significantly advanced by development of various network analyzers by Hewlett-Packard (now Agilent) and Wiltron; automated multi-port receiver and transmitter systems (e.g., Scientific Atlanta, now MI Technologies of Norcross Georgia), and rapidly scanned frequency-synthesized sources (e.g., Agilent and Wiltron). The computer-controlled network analyzer ’s compact combination of ultra low noise receiver and frequency-power stable source allowed the electrical engineer and/or material scientist to rapidly adapt waveguides or other transmission line, resonant cavity, or antenna systems to make characterizations of the isotropic or anisotropic materials. Chapter 8 discussions focus on the measurement techniques using transmission line and plane wave scattering analysis. Discussions include system designs and configurations, discussion of error correction procedures, sample preparation, problems encountered with high-dielectric or high-permeability materials and inversion algorithms that calculate electromagnetic constitutive parameters from the measurement. Descriptions of reflection and transmission measurements (waveguide, coaxial line, and free space) are presented that allow characterization of homogeneous but inherently anisotropic materials (i.e., at the molecular lattice scale) such as magnetic ferrites. Focused beam free space systems are summarized. The focused beam systems were historically used in gas spectroscopy, plasma, and charged particle beam characterization. Chapter 9 continues the measurement system design but emphasizes resonant measurement techniques and repeats many of the same system studies. Error corrections, perturbative and exact cavity measurements, transmission line, and cavity combinations are discussed in the context of network analyzer utilization. Chapter 10 extends discussions of transmission line, free space, and cavity techniques and applies them for material measurements in low- and high-temperature environments and for

anisotropic magnetic materials. Commercial environmental systems are discussed that allow characterization over modest temperature ranges (e.g., 170 to 500 K) and lower frequencies (1000:1 in the RF, microwave, and millimeter spectra. Figure 2.2 shows an example fit to measured data. Rapid “ripples” are typical of measurement errors due to background scattering, imperfect miss matches, coaxial line gaps, and imperfect calibrations. Parametric fits smooth the data to reveal true and causal dispersion.

FIGURE 2.2 Measured and harmonic oscillator model parameter fit for a frequency dispersive complex permittivity example.

2.2 MODELING MATERIAL COMPLEX PERMITTIVITY AND ITS FREQUENCY DEPENDENCE Models of permittivity and frequency dispersion are introduced in the classical, force equals mass times acceleration, non-quantum mechanical context. The reader is referred to additional texts for the correct quantized models and mathematics. More complete explanations are found in Electronic Properties of Materials by R. E. Hummel, Introduction to Solid State Physics and Quantum Theory of Solids by C. Kittel, and Optical Coherence and Quantum Optics by L. Mandela and E. Wolf [1, 3–5]. By its definition, electromagnetic polarization is a volumetric parameter. Thus, it is proportional to microscopic dipole moment(s), magnitude(s), and the sum over number densities of each dipole moment source type that is to be found in a material. The sketch model of induced polarization in materials (Fig. 2.1) has the external electric field distorting the electron cloud around a nucleus or molecule to produce geometric asymmetry of local charge density. The equilibrium distance between the effective charge concentrations will depend on (1) the force that is exerted by the external electric field, (2) restoring force(s) that attempt to equilibrate the separated opposite charge volumes to produce zero net charge, (3) any dissipative forces that contribute resistance to charge redistribution (i.e., lattice, defects,

or even viscosity), and (4) charges in motion—if charges are in motion, their inertia must be overcome. The overall motion may be oscillatory, continuum, or damped over time and the action will depend on surroundings and oscillator parameters. With this picture in mind, a simple one-dimensional classical physics model is described that still captures the qualitative physics of permittivity sources. To paraphrase Richard Feynman [2], “a strange thing occurs again and again: equations which appear in different fields of physics, and even in other sciences, are often almost exactly the same. The harmonic oscillator is one of these.” In modeling polarization we apply the second-order harmonic oscillator differential equation that describes the sum of forces that act on the charges while remembering that a physically correct model would incorporate the quantum mechanics of charges, spin, angular momentum, nuclei, etc., and electromagnetic field quantization. Newton’s second law states that the rate of change of momentum is equal to the sum of forces acting on a mass. If the electromagnetic damping and resorting forces are summed, one finds

The first term is the acceleration experienced by the charged particles of mass m. The mass may represent light-weight charges, e.g., a single electron bound in an atom, or may have large molecular magnitude mass such as a molecule, e.g., H2O. The molecule may rotate/oscillate in some medium and the surroundings produce friction to damp the rotation. The second term is the damping term that represents energy dissipation due to resistance experienced by the charged mass moving in a medium. We assume relatively slow motion so that energy dissipation is proportional to the first power of velocity with proportionality constant τ. The third term represents all restoring forces. Only forces that are proportional to distance are assumed. These may include coupling between two molecules or the restoring force that a nucleus exerts on surrounding electrons. The external force here is represented by the mass’s charge multiplied by external electric field qE(t). Solutions to Eq. (2.1) are found in many physics and applied mathematics texts for a harmonic driving force qE(t) = qE0eiωt with ω the radial driving frequency. Readers are referred to Applied Mathematics for Engineers and Physicists by L. Pipes and L. Harvill [6] and classical dynamics texts such as Classical Dynamics of Particles and Systems by J. Marion and S. Thorton [7]. The differential equation’s solution consists of a complementary and particular solution x(t) = xc(t) + xp(t). The complementary solution is of the form

and the periodic particular solution

Note that the complementary solution for particle motion is damped to zero over long times and only the steady state, particular solution remains. Recall that polarization, , is proportional to the product of charge and distance over which the charge moves. It is a volumetric quantity and the strength is proportional to the number of dipoles, N (or atoms), per unit volume that have an oscillating charge q. Relative permittivity is defined by ε = 1 + P/(ε0E). Equation (2.3), for position as a function of time, is combined with the definition of polarization and permittivity to obtain the atomic/molecular oscillator forms for real and imaginary relative permittivity, εr ,real, εr ,imag, i.e.,

Equation (2.1),

, assumes that all charges on the N atoms are

the same. However, the electromagnetic field may excite many electrons on each atom, each with its own relaxation, τj, and resonance frequency, . In addition each excitation is characterized by its own “strength” or in quantum terms “transition probability.” Such a multi-excitation model gives a permittivity expressed as a summation over each contributor:

The following sections investigate the single electron case for a number of limiting cases of the oscillator model.

2.3 SMALL DAMPING, τ ⇒ 0 In this limit it is assumed that τ 1 and/or approaches zero. For small τ a strong resonance will be observed for f → f0. In this case εr → ±∞ and εimag takes on a Lorentzian shape centered at f0.

When discussing semiconducting or conducting material this functionality is most often observed in the infrared, optical, and UV portions of the electromagnetic spectrum. However, free charge plasmas in vacuum (e.g., Van Allen belts [8] or in particle accelerators [9]) may also display a strong resonance. As described previously, a macroscopic equivalent of this permittivity functionality can be achieved by mixing a small concentration of thin electrically conducting fibers in a dielectric matrix. The resonant behavior is expected at low concentration in semiconducting or conducting-dielectric composites for the RF, microwave, and millimeter spectra.

2.4 DC, ZERO FREQUENCY, AND DC SCALED PERMITTIVITY When extended to low frequencies, approaching DC, the permittivity is purely real and equal to

This equation fits physical intuition as it suggests that permittivity increases with decreasing restoring constant. Decreased constant would lead to a larger maximum swing d and thus an increased product qd. Real and imaginary parts are often expressed in terms of the DC and infinite frequency permittivity limits (ε(∞) = 1.0). The following is an example:

2.4.1 Free Charge Contributions to Permittivity and Dispersion Bound charges with negligible damping and DC scaled models have been discussed. The case for identically zero restoring force, i.e., k ⇒ 0, leads to a negative real permittivity. This limit describes the “free charge” cases for electrons in semiconductors, plasmas, and conductors and for some charged atoms or molecules found in liquids and solids. The differential equation is revisited for free charges of mass m and the restoring force constant set identically to κ = 0 and equations are rewritten for different parameter scaling in terms of electrical conductivity σ. Through qualitative physics discussion it is noted that after some time period τ charges in conductors achieve an equilibrium drift velocity. That velocity is Vd = eE/τ = σE/eNf [1], where σ and Nf are conductivity and free charge number density and E is electric field strength. The relationship between conductivity and dissipation constant is τ = Nf e2/σ. Dissipation reflects resistance which is inversely proportional to electrical conductivity, σ, and directly proportional to the number of free charges, Nf. These new parameters are applied to the k = 0 limit of the harmonic oscillator differential equation

where ω = 2π f; f is the frequency of the electromagnetic field, and E0 is the magnitude of that field. The transient, complementary solution of the equations is ignored (damping is very rapid) and a particular periodic solution is chosen to have the form x(t) = x0eiωt. This assumed solution is substituted directly into the differential equation and the solution is found in terms of field and material parameters:

Polarization is calculated by the product of charge, e, and distance, x. The relative permittivity is then derived from polarization using the definition:

The plasma frequency is defined as . The relaxation frequency is . The equation for relative permittivity is a function of these two materialdependent frequencies:

Equation (2.13) has interesting implications for the real and imaginary parts of permittivity and is consistent with the previous section’s small restoring force equation. The real and imaginary relative permittivity components are

The real part of this model is zero or negative for and therefore negative/zero for . Figure 2.3 shows measured and modeled data for a semiconducting material just above . Real permittivity in that data exceeds 1 since the free electrons are found within a graphite-loaded foam/core; however, it decreases and approaches zero as frequency is reduced. The imaginary part shows a typical f–a functionality.

FIGURE 2.3 Measured and free electron model of a graphite-loaded core material. Note that the real part decreases to zero as frequency is reduced. The imaginary part increases as 1/f. Data points and fitted functions are near overlay and difficult to separate.

A second measured and modeled data set for a polymer fiber coated with a 0.5-μm nickel coating is shown in Fig. 2.4. The data values are difficult to fit with the pure free electron model but plasma frequency trends are apparent with the real part large and negative, while the imaginary permittivity varies as an inverse power of frequency.

FIGURE 2.4 Measured and free electron model of the frequency dispersive permittivity for a 0.5-μm nickel-coated polymer fiber.

A medium with zero or negative real permittivity will strongly reflect electromagnetic energy and strongly attenuate any field propagating in the media. The effect is well known to HAM radio operators and AM radio stations. Operators and stations reflect their lowfrequency radio waves from the charged particles that make up the Earth’s ionosphere. The reflection allows them to bounce their signals over the horizon and have them received at very long ranges. The phenomena of total reflection below some frequency band also sets limits on those radio frequencies that can be used for Earth to space communications (e.g., SATCOM). Recall that when most spacecraft return to Earth, there is a period of time for radio “blackout.” A plasma forms around the returning spacecraft due to air friction. Since the charge particle density in this plasma is high, radio communications at even S band (e.g., 3 GHz) may be attenuated. Similar frequency dependence is observed in conducting materials such as graphene that has an electron concentration of approximately 1014 cm–3. Its electrons have near ballistic mobility and the graphene plasma frequency lies in terahertz frequency bands. Below that frequency range electromagnetic waves are reflected. However, above the terahertz range electromagnetic energy can propagate through graphene and thus graphene is almost transparent at the shorter, near-infrared, and visible wavebands.

2.5 COMBINED MODELS FOR PERMITTIVITY Many materials contain some free charges but may also have bound electrons. Thus, a model accounting for both free and bound charges is required. A simple model would just sum the two components, Eqs. (2.16) and (2.17).

However, materials are often combinations of differing types of bound and free charges. Charge carriers that scatter not just from lattice points but from material imperfections of all kinds may be involved. In general, this introduces frequency scales of many forms and new parameters. Thus, an expanded model is used to calculate permittivity in composites over many decades of frequency. A proposal for an expanded combined model is provided in the next equation. It identifies nine material parameters rather than seven (fp, fr, q2 N/2πε0, α, τj, f0, 2πm), which are presented in Eqs. (2.16) and (2.17). The numerical values of the nine-parameter model are derived by fitting complex permittivity (not real and imaginary separately) to measured data over 100:1 to 10,000:1 bandwidths. Therefore, parameters are themselves complex numbers.

Some boundary values are expected on the parameters. For example the DC term, A(1), is expected to be purely real. Because fitted data are bounded in frequency, the DC term in Chap. 12 tables sometimes may have a small imaginary part. When using the Chap. 12 parameters, that small imaginary term is often seen to cancel other small imaginary parts of other terms to yield a net purely real DC value. Note that the nine-parameter model does not contain plasma frequency–dependent models explicitly but contains real and imaginary parameters varying with frequency to some power. This functionality was chosen to better fit composites that show “percolation” (see Chap. 6). In Chaps. 5 to 7, analytical models of conducting-dielectric composites are developed that indicate that the power law form of Eq. (2.18) is often a better functional form for composites that contain semiconducting components. These include semiconducting and conducting thin

films measured for this book. Figures 2.5 and 2.6 illustrate typical material dispersions and show measurements fit to the nine-parameter model data.

FIGURE 2.5 Measured and free charge model fit to the real and imaginary complex permittivity of a semiconducting material (0.1 cm thick).

FIGURE 2.6 Measured and free charge model fit to the real and imaginary impedance of a graphite-loaded conducting film of thickness l.

2.6 COUPLING PERMITTIVITY AND PERMEABILITY Materials with free charge carriers are sources of not only permittivity but also magnetic permeability. Simple physical models show that any current source or moving charges produce a magnetic field and thus are a source of magnetization and permeability different from free space. The combination of charge and drift velocity in a conductor produces a current. The current induces a magnetic field with magnitude proportional to current and in a direction that opposes the external electromagnetic H field. Since the induced field opposes the external field, it supplies a negative contribution to permeability. When added to the unit relative permeability found in many conductors, the total is less than unity. The reduction, or “diamagnetism,” may be small but appears in all materials supporting moving charges and is present when an external field source is evident. Chapter 6 discusses the diamagnetic effect in composites that contain conducting particulates. Even in composites containing electrically isolated ferromagnetic metal particulates such as iron, nickel, or cobalt, diamagnetism can dominate permeability at frequencies above a few gigahertz.

2.7 ADDITIONAL BOUND CHARGE CONTRIBUTIONS TO PERMITTIVITY AND FREQUENCY DISPERSION The electronic and atomic polarizability permittivity sources discussed in previous sections often require significant electromagnetic excitation energies (either high field strengths or frequencies in the infrared-visual-UV band). Molecular relaxation also has been discussed and may be observed in long-chain organic polymers at RF frequencies or in a low-density fluid medium where rotational motion of a molecule is not quickly damped. This book places emphasis on the RF, microwave, and millimeter portion of the electromagnetic spectrum. In most nature-generated materials, frequency dispersive permittivity is often not observed or is minimal at these frequencies. Man-made composites and metamaterials [10–15] take prominence when frequency dispersive permittivity is needed in RF through millimeter wave bands. The control of microscopic and mesoscopic interfacial polarizability and the finite distance over which charge may flow are the physical mechanisms that can produce resonance, relaxation, and unusual frequency dispersive behavior in composites at RF, microwave, and millimeter wave. The observed dispersions are of the same form as nature’s dispersions at infrared through ultraviolet frequencies; however, microscopic energy excitations are not involved. Before continuing to discussions of temperature impacts on permittivity the author wishes to spend a bit more effort on permittivity. This short section is also a “teaser” for information to be presented in Chaps. 5 to 7. Composite media, composed of conductor and dielectric mixtures, may demonstrate

strong frequency dispersion near a critical volume fraction, pc, that is called the “percolation threshold.” That concentration reflects a phase transition that is apparent in composites containing either semiconductor or conducting material randomly distributed in a nonconducting (or partially conducting) dielectric composite. When the volume fraction, p, of a conductor or semiconductor approaches pc, conducting paths of statistically distributed lengths are formed in the composites. As previously noted, conducting paths of finite length generate large dipole moments and thus below the critical volume fraction these composites have large permittivity magnitudes. The statistical distribution of conducting lengths produces multiple dipole-like resonances at different frequencies. These are qualitatively similar to the bound electron contributions to permittivity in natural materials. Above the threshold at least one DC conduction path (the conducting “backbone”) is established and a “free electron” like permittivity component appears. The combination of finite length and DC continuity produces a metamaterial whose dispersion can be controlled at RF–millimeter wave frequencies to produce dispersion such as those observed in natural materials at infrared and shorter wavelengths. Material manufacturers of composites have long used low-volume fractions of semiconducting constituents as a dielectric tuning mechanism. At high-volume fractions the same mixtures are used as resistive-conductive paints, dyes, and films. By controlling geometry, intercalation of the semiconducting and dielectric components and concentration, the full range of physically achievable dispersions should be achievable. This statement is a description of modern metamaterial research and Chaps. 5 and 7 address the composites in detail.

2.8 PERMITTIVITY VARIATION WITH TEMPERATURE Permittivity can be affected by temperature in a number of different ways. Material polarization is a volumetric quantity and therefore a temperature-induced expansion or densification (sometimes appearing as a change of state such as crystallization) can change permittivity, even when chemistry of material is unaffected. The conductivity and number of free electrons in a material vary with temperature and change the free charge contribution to permittivity.

2.8.1 Semiconductors and Conductors The permittivity in semiconductors and conductors has been related to carrier density, charge, mass, resistivity, and conductivity. In semiconductors the carrier density is an exponential function of temperature, i.e., Boltzmann factor [1, 3]. A rise in temperature will increase carrier density, increasing plasma frequency, and thus electromagnetic blackout frequency. In general there would be an increase in the imaginary part of permittivity. In addition a temperature rise may change the crystal structure of the material and thereby change the conductivity and resistivity. Real and imaginary parts of permittivity have similar functionality.

The material-specific A and B parameters of Eq. (2.19) are used to fit measured data of complex permittivity versus temperature at fixed frequencies and these parametric fits (A and B) are tabulated in Chap. 12. In some cases, permittivities were measured for frequencies in C, Xn, X, Ku, and Ka bands and for temperatures of 273 to above 2300 K. That data are used to form frequency and temperature functionalities. A(0), B(0) are permittivity values at room temperature. The ideal value for temperature scales (A(3), B(3)) should be 3/2; however, most of the materials discussed in Chap. 12 show different values of temperature scale. The parameters A(4), B(4) are related to the energy bandgap of the material, i.e., 2Δenergy where Δenergy is the energy bandgap(s) of semiconducting constituents of the materials that were tested [3]. The bandgap is that energy which is required to excite (move) a bound electron from an atom and insert it into the conduction band of the material. The conduction band is composed of the electron holes that can contribute to electrical conductivity. Examples of measured and parametric fits to temperature dependence are shown in Fig. 2.7.

FIGURE 2.7 Measured and modeled semiconductor permittivity variation vs. temperature.

There are very few cases of electromagnetic measurements of conductors versus temperature in the Chap. 12 database. One expects that the conductor conductivity will decrease with temperature because electronic atomic-lattice scattering tends to increase with temperature. At high temperature, the conductor versus semiconductor physics behavior is easily distinguished on a logarithmic plot of the data. When log(εimag(T)) or log(σ(T)) is plotted versus log(T), the resulting plot will be linear with negative slope for metals. However for semiconductors, a linear relationship is achieved when the logarithm is plotted versus 1/T. Significant temperature decrease can of course produce superconducting materials. A few examples are discussed in Chap. 10 and data for high Tc ceramic superconductors are shown. The ceramic conductivity undergoes a phase transition with dependence approximately 1/(T – Tc)α [1, 3]. This phase transition is reflected in a “discontinuous” change in transmission coefficient and phase step in measured data. In short the permittivity temperature functionalities of composites discussed in this text are rich.

2.8.2 Dielectric and Ceramic Temperature Functionality Until a phase change in the material occurs, the permittivity of a dispersionless dielectric material such as a polymer is a weak function of temperature. The phase change may be at

temperature TP of solidification, melt, or ignition. An example of a slow change is shown below for an aramid fiber ([–CO–C6H4–CO–NH–C6H4–NH–]) measured for temperatures below its stability point, ∼673 K (Fig. 2.8).

FIGURE 2.8 Microwave permittivity of an aramid fiber measured vs. temperature. Measurement is in the air.

When dealing with ceramics permittivity, temperature functionality may be more complex. Bosman and Havinga [16] derived a model of the temperature dependence for mainly single crystalline ceramics dielectric constants. Ho [17, 18] found that the model also applied for polycrystalline materials and he extended the model to study of the imaginary parts of permittivity. The model proposed by Bosman and Havinga began with the classical Clausius–Mossotti formulae for material dielectric [3]:

where Njαj are number density and polarizability of atoms making up the material. The differentiation with respect to temperature at constant pressure addresses the volume expansion change and inherent polarizability change with temperature. The rate of change in permittivity with temperature, T, at constant pressure, p, is expressed in terms of the molar

polarizability, αm, and molar volume, V.

The first term on the right-hand side of this viral expansion is indicative of the reduced number of polarizable particles per unit volume due to volume expansion. If the material becomes denser, that term is positive. This may occur in centering processes. The second term is proportional to the change of polarizability with volume and is positive if the volume is expanding. Imagine the swing of harmonic oscillator model for polarizability is increasing in length. This second term generally dominates temperature behavior. The third term is for fixed volume and is expected to be small in materials which are not ferroelectric or do not undergo a rapid state change. Note that since the first term has opposite sign from second and third, there is a possibility of net zero change with temperature. This type of material would be valuable in applications related to radomes on systems that move at high velocity, e.g., reentering capsules and space shuttles. A few of the ceramics discussed in Chap. 12 show both positive and negative slope of temperature versus permittivity. Chapter 12 data suggest that composites of magnesium-calcium-titanate and other ceramics could show negligible permittivity change with temperature. Thereby, they might achieve fixed electrical thickness at elevated temperatures in radome structures.

2.8.3 Mixed Phases and Expectation for Composite Permittivity: Temperature Functionality Ceramics are chosen as examples. Their permittivity may demonstrate a mixed-phase temperature dependence if they contain more than one constituent in a solid mixture, have a component that is not fully oxidized (i.e., partially composed of a semiconductor component), or the temperature is very high and melting occurs. A state change may separate metallic/semiconductor phases from the oxide/nitride/boride state and form a suspension that could include semiconductor components that demonstrate a Boltzmann exponential increase in the imaginary part of the permittivity as a function of temperature. Separately, the resonance or critical temperature behavior can be exhibited for ceramics with ferroelectric compositions, e.g., titanates. The mixed-phase temperature dependence leads again to anticipation of “effective medium theories” that are discussed in Chaps. 5 to 7. An overview of effects due to different composition phases for ceramics at high temperature can be given by applying general principles of those theories. If one is dealing with cryogenic temperatures and mixed-phase high Tc ceramics, the behaviors will be similar except phase changes will be determined by the rapid superconducting phase change and not an exponential behavior. At extremely high temperatures one expects that small volumes within a ceramic may “melt” and become semiconducting (electrons excited above the bandgap) or conducting if metal oxides are refined to a pure form. If the total semiconducting/conducting volume

percentage approaches the critical volume threshold, pc, the ceramic will display a dielectricsemiconducting phase transition accompanied by an increase in imaginary permittivity. A dramatic decrease in εr,real may also be observed if the final electron concentration is sufficient to bring plasma frequency into play. Three regions of conducting volume fraction determine the expected frequency and temperature dependence of the mixed-phase material: p pc, p is approximately pc, and p ≥ pc. For p pc the mixed-phase permittivity is dominated by that of the original ceramic composition, εcer. For ceramics with εcer,imag 1, the permittivity trends described in Eq. (2.21) are expected to apply and permittivity will show a slow increase with temperature and small frequency dispersion. As the total volume of “melt” approaches pc, the effective medium theories predict that permittivity of the mixed phase, εmp, will be a product function of the ceramic and melt phase, εmlt, to half the power, i.e., εmp (f, T) approximately {εcer (f, T) εmlt (f, T)}1/2. If the melt phase is semiconducting or conducting, it will dominate frequency and temperature dependence. Above pc the ceramic makes minimal contribution and εmp (f, T) is approximately {εmlt (f, T)S where the power S is about 0.8 and permittivity is essentially that of the melt region. If the melt region is semiconducting, a Boltzmann behavior is expected. In order to account for both the permittivity increases due to expansion and possible mixed phase and/or semiconducting variations in the imaginary permittivity, different temperature functional forms for real and imaginary permittivity are used in the Chap. 12 analysis. The relatively slow variations of the real permittivity with temperature-induced expansion are modeled as a power law. However, best fits to the imaginary part of the permittivity apply combinations of power law and exponential form. Note that if the argument of the exponential is small, the exponential can be expanded in series and a power law form for the imaginary part results. The equations that are fitted to measured data are shown below.

Figures 2.9 to 2.12 are presented to illustrate the wide variation in ceramic temperature functionality. In Fig. 2.9, a power-law dependence is found to be sufficient. The imaginary permittivity only changes by a factor of 10 over a 6:1 rise in temperature. In Fig. 2.10, the ceramic shows a negative slope with rising temperature. Permittivity decreases until a material-dependent temperature is reached, after which the more typical positive temperature derivative is observed. Figures 2.11 and 2.12 illustrate what is believed to be melt and refining behavior. As the measurement of alumina approaches its melting temperature (∼2273 K), the real permittivity suddenly decreases below unity while imaginary permittivity increases more than two orders of magnitude, both a characteristic of the appearance of a conducting phase.

FIGURE 2.9 Typical measured and modeled ceramic permittivity vs. temperature.

FIGURE 2.10 Decreasing real permittivity with temperature as found in some ceramics that have ferroelectric components.

FIGURE 2.11 Microwave real permittivity temperature variation of Al 2 O3 . Decrease occurs near the melting temperature.

FIGURE 2.12 Microwave imaginary permittivity temperature variation of Al 2 O3 . Sudden increase occurs near the melting temperature.

Chapter 3 will continue discussion of temperature impacts but as related to magnetic properties and permeability. Magnetic sources, their frequency dependence, and their temperature dependence will be reviewed. The structure of Chap. 3 is intentionally made similar to Chap. 2.

REFERENCES 1. R. E. Hummel, Electronic Properties of Materials, 3rd ed., Springer Science + Business Media, New York, ISBN 0-38795144-X, 3rd edition (2005). 2. R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. 1, Chap. 21 and Vol. 2, Chap. 11, ISBN 0-8053-9046-4, Addison-Wesley, Menlo Park, California. 3. C. Kittel, Introduction to Solid State Physics, ISBN 0-471-11181-3, Wiley (2005). 4. C. Kittel, Quantum Theory of Solids, ISBN 0-471-62412-8, Wiley. 5. L. Mandela and E. Wolf, Optical Coherence and Quantum Optics, ISBN 0-521-41711-2, Cambridge University Press. 6. L. Pipes and L. Harvill, Applied Mathematics for Engineers and Physicist, 3rd ed. McGraw-Hill, New York. 7. J. Marion and S. Thorton, Classical Dynamics of Particles and Systems, 5th ed., ISBN 10: 0534408966 and ISBN-13: 9780534408961, Academic Press (2004). 8. The Earth’s Trapped Radiation Belts United States. National Aeronautics and Space Administration. Washington DC, QC809 .V3 U58X (1975). 9. Peter Strehl, Beam Instrumentation and Diagnostics, Springer, Berlin, Heidelberg (2006).

10. C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications: The Engineering Approach, ISBN 10-0-471-66985-7, John Wiley & Sons, Hoboken, NJ (2006). 11. K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, Rev. Mod. Phys., 80:1201 (Oct.–Dec. 2008). 12. V. G. Veselago, Sov. Phys. USP. 10:509 (1968). 13. J. B. Pendry, Phys. Rev. Lett., 85:3966 (2000). 14. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett., 84:4184(2000). 15. A. Adibi, A. Scherer, and Shawn-Yu Lin, eds., Proc. SPIE. Int. Soc. Opt. Eng., San Jose, Ca. Vol. 4655 (23–25 Jan. 2002). 16. A. J. Bosman and E. E. Havinga, Phys. Rev., 129:1593 (1963). 17. W. Ho and P. E. D. Morgan, J. Am. Ceram. Soc., 70(9):C-209 (1987). 18. W. Ho, “High-Temperature Millimeter-Wave Dielectric Measurements by Free-Space Techniques,” Conference Digest— International Conference on Infrared and Millimeter Waves, Lake Buena Visa, Florida, USA, pp. 16–17(1987).

CHAPTER 3

SOURCES AND DISPERSION OF MAGNETIZATION AND PERMEABILITY μ(f)

A brief introduction to permeability and its sources has been supplied in Chap. 1. In this chapter, the physical sources of μ are revisited and equations that relate sources of permeability and material frequency dispersion are derived. The derived equations serve as models that are combined with measured data in Chap. 12. The resulting parametric fits identify values of fundamental parameters that determine the macroscopic permeability. The chapter includes discussions of numerical micromagnetic modeling, Kramers–Kronig (KK) relations, and temperature dependence in magnetic materials. Because there is sometimes confusion in the conversion of units for magnetic quantities (i.e., field, magnetization, etc.), a conversion table is provided at the end of Chap. 3 for the reader ’s reference.

3.1 SOURCES OF PERMEABILITY As compared to permittivity, permeability enters Maxwell’s equations in a very different way. The divergence of · is proportional to total charge density ρt. This is indicative of point charge sources of the electric field. However, the divergence of is identically zero, · = 0, indicating the absence of magnetic charge. Magnetic field source is dipolar–multipolar and the creation of a magnetic field requires that the charge has motion that creates current. “Currents” giving rise to magnetism in materials have basis in quantum mechanical sources and thus macroscopic physical model analogues may or may not be easily constructed. The following discussion often makes reference to “electronic energy states,” Planck’s constant, and other terms based in quantum physics. Therefore if the reader is not acquainted with these terms, a brief review (Chap. 2 or Chap. 3 of [1] or Chap. 12 of [2]) may be in order. In Chap. 2, permittivity was found to have different sources but it was always presented as contributors to “permittivity.” Permeability is discussed in five different “flavors”: paramagnetism, diamagnetism, ferromagnetism, ferrimagnetism, and antiferromagnetism. Of

the five flavors, materials presented in Chap. 12 manifest the first four: paramagnetism, diamagnetism, ferromagnetism, and ferrimagnetism. Ferrimagnetic and ferromagnetic materials are emphasized in the text with diamagnetism entering in the discussion of composites. Paramagnetism and antiferromagnetism are addressed as short summaries.

3.1.1 Paramagnetic and Diamagnetic Sources Microscopic sources of magnetic moment, , in an isolated atom, are electron spin, electron orbital angular momentum, and any modification of these two that is induced by an external electromagnetic field. In a classical picture, the quantized electron spin can be conceptualized as one electron charge distributed in a spinning sphere. This model of the spin source is illustrated in Fig. 3.1. The rotation of the charged sphere about an axis produces current which then produces a magnetic moment perpendicular to the local current direction. The magnetic moment and induced field direction are determined by the right-hand rule and the charge’s sign. The fundamental quantum of magnetic moment is the Bohr magneton, an inherently quantum mechanical quantity. It is the ratio of charge, electron mass, and Planck’s constant, A·m2.

FIGURE 3.1 Classical picture of the electron spin source of quantum magnetic moment.

A charge moving in an orbit around a nucleus also gives rise to a current and magnetic moment. That moment is proportional to the angular momentum of the charged particle. As with spin, the direction of moment is determined by the same right-hand rule and is in the direction of angular momentum (Fig. 3.2). The angular momentum of the electron orbit is a multiple of Planck’s constant, [3, 4] with the proportionality constant, the Bohr magneton,

where

.

FIGURE 3.2 The orbit charge contribution to magnetic moment and its proportionality with angular momentum, L.

In the case of spin, the Pauli exclusion principle [2] controls magnetic response. Any of the Refs. [1] to [4] is suggested as a background reading on this topic. This foundation of quantum mechanics, as applied to magnetics, states that two electrons in the energy state of atom cannot have the same spin state. Therefore, atoms that have completely filled electron energy shells do not contribute to magnetic moment because all spins sum to zero since there cannot be net alignment. However, atoms with energy levels that are incompletely filled can have net aligned electron spin since energy levels are available for alike spin electrons to occupy. Alignment has a large impact when associated with incompletely filled outer shells. Outer shell electrons are not shielded by other electrons from external electromagnetic fields. Therefore, they can more easily respond to external electromagnetic sources. These “free spins” can be oriented by an external magnetic field and contribute to the total paramagnetic moment. Without an external field to orient spin or angular momentum-based moments, the directions of the moments in paramagnetic atoms are randomized by temperature and a zero moment is normally observed. When the field is present, thermal excitation continues to drive randomization of magnetic vectors; reduces the net magnetic moment; and gives rise to a distinctive magnetic moment—temperature functionality. In general, the spin contribution of isolated atoms is weakest in free atoms (e.g., plasmas and gas) while angular momentum contributions are smallest in solids. In solids, the outer electron orbitals of atoms on different lattice sites of a material can be strongly coupled since their separation can be on the order of 1 Å. The coupling between lattice sites acts as a stabilizing force and holds or “freezes” the angular momentum vector directions. This reduces angular momentum contribution to magnetic moment when an external field is

present and thus any paramagnetic effect. Thus, spin supplies the largest contribution to magnetic moment in the solid. An external field may couple to either spin or angular momentum and modify their directions over time. That modification becomes an induced source of time-varying magnetic field. The modification may be diamagnetic, i.e., a negative moment. Diamagnetism appears when the external field accelerates the orbiting or spinning electron. Lenz’s law [3] predicts that the acceleration results in a magnetic field that is generated with its direction opposed to that of the external field. The diamagnetic effect is also observed in materials with free electrons for largely the same reason. Two other types of magnetism are observed in materials described in the book. These are ferromagnetism and ferrimagnetism. A third flavor, antiferromagnetism, was not evident in the composites and ferrites discussed in Chap. 12.

3.1.2 Ferromagnetic, Antiferromagnetic, and Ferrimagnetic Sources At the most fundamental level these material magnetic properties arise from the quantum exchange integral or exchange energy [3]. This integral function is a measure of overlap between the outer shell electron wave-functions of different atoms. The integral is conceptualized by considering two atoms, a, b, that are brought close together, near 1 Å, in atomic systems. If they have a stable configuration over time, the configuration exists at a minimum of the total free energy of the atomic system. When two atoms are close together the free energy of the two atom systems has many components. However, there are four major contributors to the electromagnetic energy. One term is the electrostatic interaction between nuclei (a – b). A second is the electrostatic interaction between electrons attached to a and those attached to b. The nucleus a acts on electrons belonging to b and the nucleus b acts on electrons of a. The two atom system reaches an equilibrium configuration of electrons and nuclei when the overall energy is at a minimum. This occurs when the interaction integral is positive. Of course, for a minimization to occur there must be physics that allows energy trades (some terms increase when others decrease). That energy trade occurs because the quantum Pauli exclusion principle is applied to the system. The principle trades energy associated with the electrostatic interactions described in the previous paragraph. Consider the outer d shell electrons of an atom. The alignment of d shell electron moments on two atoms would normally increase the overall energy. However, the alignment of the two electrons can be favored if the nuclei are separated far enough that the energy cost of the Pauli exclusion is weakened and overall electrostatic energy is reduced. This lowers the total energy of a collection of electrons of the same spin (magnetic moment), but they must be at different spatial locations. Figure 3.3 is a qualitative picture of that interaction integral as associated with different materials. The integral magnitude is plotted versus the ratio of nuclei separation and radius of the outer d shell electrons for various atoms. As illustrated in Fig. 3.3, iron, cobalt, and nickel have the largest positive integral and the greatest number of allowed, aligned electron spins. Large numbers of aligned spins produce large permeabilities. The underlying physics does

have a practical trade (e.g., mass) for application of these materials. The necessity for large numbers of electrons in tightly bound d shells implies a large atomic mass and high specific gravity. Thus, weight is always an issue in using solid materials with large magnetic permeability.

FIGURE 3.3 A qualitative drawing of the interaction integral as a function of nuclei separation ratio to d electron shell radius for various atoms. Positive values indicate materials showing ferromagnetism.

The figure indicates some ratios where the exchange integral is less than zero. This occurs for Mn and Cr. In these cases, spins on adjacent atoms are aligned in opposite directions to produce zero net moment. These are examples of antiferromagnetic materials. For ferromagnetic energy minimization to occur, adjacent atoms should have electron spins that spontaneously align. The local alignments produce small regions (called domains) in a ferromagnetic material where the electron magnetic moments are all oriented in the same direction. The domain sizes range from 10 to 1000 nm near room temperature and can be further increased in size by exposing the material to external magnetic fields and/or changing the shape of a material sample. The spontaneous alignment is predicted in numerical simulations that calculate interaction energies for finite size and shape magnetic materials. Figure 3.4 shows example simulations from one such solver, the National Institute of Standards (NIST) OOMMF computer code [5]. The illustrations show a time evolution of the formation of magnetic domains as predicted in micromagnetic simulation. Directions of the local magnetic moment are illustrated by the small arrow graphic. The growth of domains is predicted for two different ferromagnetic particulate shapes as a function of time. The parameters used in the simulations are typical of 80:20 Fe–Ni, Permalloy.

FIGURE 3.4 Micromagnetic simulations using the NIST OOMMF code [5] of magnetic domain growth in two different ferromagnetic metal shapes. Domains are formed as moments interact and relax to equilibrium situations.

The domains are evidenced in measurement using a number of techniques. Figure 3.5 shows transmission electron microscopy (TEM) on the right and magnetic force microscopic (MFM) measurements on the left. The MFM is of a high density iron composite made of 1- to 10-μm particulates. The TEM shows a domain boundary (reversal of spin directions) in a nickel-zinc-ferrite. In the iron composite, striped domain structures are about 1 × 5 μm in size and in the ferrite the boundary transition is 5 to 10 nm in width.

FIGURE 3.5 Magnetic force microscope image of and iron particulate (L) and TEM image of a ferrite grain boundary (R).

The simulation of Fig. 3.4 shows spontaneous alignment, without an external magnetic field. The spins in adjacent domains are misaligned and this forms grain boundaries such as those shown in Fig. 3.5. When materials are exposed to an external field of sufficient magnitude, the magnetic spins of different domains begin to align and any differentiation of small domains disappears. The number of aligned spins grows to make larger domains until they fill a particulate volume. A large domain structure, as predicted by OOMMF, is shown in Fig. 3.6. The alignment of the maximum number of spins defines an upper limit on magnet moment and establishes the saturation magnetization, Ms. Ms has units of magnetic moment per unit volume. Many ferromagnetic metals have coercivity and hold the dipole alignment of magnetic moments after saturation is achieved. The coercivity of a ferromagnetic is that field which is required to bring the magnetization back to zero after the magnetization of the sample has been driven to saturation. When the external field is removed, the aligned spins that remain produce a “permanent” magnet.

FIGURE 3.6 Equilibrium magnetic domain growth when exposed to approximately a 5000-G field for a rectangular 80:20 Fe–Ni particulate.

The number of domains and the size and shape of the domains are determined by a number of geometric factors and magnetic history (i.e., exposure to external fields). The geometric factors include particle aspect ratio, magnetic particle sizes, and shape. The magnetic history includes field exposure and field anisotropy. The material anisotropy originates in the crystal structure of each magnetic material and the effective anisotropy can be strongly impacted by shape. Figure 3.7 shows computed domain structures for a range of particle shapes and sizes. Figure 3.8 is a similar calculation but for different anisotropy field and aspect ratio. A second image shows the computed domains for zero fields and one exposed to a large external field. In general the magnetic domain complexity increases for larger particulate sizes and more complex shapes, if zero field is the initial condition.

FIGURE 3.7 Example simulations of geometrical impact on magnetic domain structure for a soft magnetic material.

FIGURE 3.8 Example simulations of field impact on magnetic domain structure for a soft magnetic material.

The last magnetic material to be discussed is the ferrimagnetic material. As with ferromagnetic materials, ferrimagnetic materials contain small domains which have spins that spontaneously align below some temperature. However, a major difference is that

ferrimagnetic materials often have complex crystal structures and are oxides. Being oxides they may have very small electrical conductivity (as contrasted with ferromagnetic metals). Negligible conductivity facilitates their use in ceramic magnetic applications (e.g., RF phase shifters, isolators, and switches). It was proposed by Neel that these materials are composed of two sub-lattices with each lattice containing spins that are aligned in that lattice. However, the number of magnetic ions in each lattice is different. Thus, even if the spins are aligned antiparallel in the two lattices, the total spin will have a finite value and produce a net magnetic moment.

3.2 FREQUENCY DISPERSION IN MAGNETIC MATERIALS A model is applied to calculate frequency dispersion in magnetic materials, which is qualitatively similar to the harmonic oscillator used for permittivity in Chap. 2. However, in the magnetic case there is no single charge that moves but the direction of magnetic moment vector is modulated by an external electromagnetic field. In Chap. 2, Newton’s laws of motion were applied to formulate a second-order differential equation. Decaying and time harmonic solutions to the equation were found with an external harmonic force. The solutions described time dependence of the dipole moment (thus polarization and permittivity). Various limit solutions were derived for ranges in system parameters. Magnetic moment, magnetization, and permeability also have harmonic and time damped solutions but the force laws from which these are derived can be very complex and coupled. However, if limiting conditions are applied, e.g., small external field magnitudes, approximate solutions to the dynamic equations can be obtained. Though the sources of magnetism are inherently quantum mechanical, a first-order differential equation of a classical dynamic model of magnetic moment gives an excellent working model for frequency dependence observed for permeability. The magnetic model is very similar to solutions for a dynamic spinning top which is placed in a gravitational field. Creating magnetic moment requires charge to be placed in motion (electrons orbiting nuclei or electrons spinning on their axes). If an external magnetic induction field is present it will produce a force on a moving charge (an electron is assumed to have velocity ) given by

. The force is in a direction that is perpendicular to both velocity

and magnetic field. The equation is rewritten for Fig. 3.9. The figure illustrates an electron orbiting at a fixed radius, R, and it has angular momentum . The relationship between magnetic moment, , and angular momentum is derived below for Fig. 3.9, i.e.,

FIGURE 3.9 A charge (assumed an electron) orbiting at fixed radius about a center point and immersed in magnetic induction field

.

Equation (3.1) is derived for a single moment, . In order to obtain a relationship with macroscopic magnetization per unit volume, , all Nm moments within a material must be summed. This is represented by

. If the magnetic induction, , is replaced by field

, the equation is rewritten as

This first-order equation has a simple undamped oscillatory solution. Additional terms must be added to introduce energy loss and better represent the actual physics. When a loss term is added, the equation is identified as the Landau–Lifshitz (LL) and Landau–Lifshitz– Gilbert (LLG) equations [6, 7]. The next section uses the LLG equation to obtain dispersive permeability for ferromagnetic (antiferromagnetic), ferrimagnetic, and paramagnetic systems.

3.2.1 The Landau–Lifshitz Equation and Frequency Dispersion in Magnetic Materials

The dynamics of ferromagnetic, ferrimagnetic, antiferromagnetic, or paramagnetic particulates are described by the LLG equation [3, 6, 7]:

Ms is the material saturation magnetization. Heff is the sum of magnetic fields that are present in the system. It includes DC field components due to magnetization, Hm; fields that arise from particle or crystal anisotropy and geometry, Hk; and external and internal DC fields, H0, HDM. The external field may be from an external permanent magnetic or DC current source. The internal contribution is due to demagnetizing fields, DM. A demagnetizing field will be present for any finite size particle. It originates from the magnetic poles that are formed at surfaces of a ferrimagnetic or ferromagnetic particle. Internal demagnetizing fields can be caused by material imperfections (introducing boundaries) or occur in large multidomain particles or particles with unusual shape (e.g., Fig. 3.7). Lastly, there is the electromagnetic field h(ω) which produces a dynamic periodic precession and modulation- nutation of the magnetic moment about the DC field direction. It is the source of dynamic permeability and susceptibility, μ(ω) = 1 + χ(ω). The parameters in Eq. (3.3) are α, the phenomenological LLG damping parameter, and γ = eμ0/me = 2.8 MHz/oersted, which is the gyromagnetic ratio and has absolute units of m/C. The first term on the right-hand side of Eq. (3.3) describes the torque that a magnetic field exerts on the magnetization and is the same term derived in Eq. (3.2). The direction of that torque vector is perpendicular to both magnetization and effective field. The torque forces the magnetization to precess around the effective field direction. The second term on the right describes a vector that induces wobble of the magnetization during its precession. It is included to account for the intrinsic loss of the system. Loss forces the direction of the magnetization to relax and spiral in toward the field (see Fig. 3.10). The maximum magnitude of is equal to the saturation magnetization Ms.

FIGURE 3.10 Illustration of precession of the magnetization vector about the direction of the summed static magnetic fields. North and South poles indicate surface sources of demagnetization. The anisotropy field is shown internal to the particle. If there is damping (energy loss) the magnetization spirals in and finally aligns with the H field.

A short discussion of numerical solutions to Eq. (3.3) follows in the next section. However, it is preceded by analysis leading to an approximate solution for the magnetic resonant frequency and permeability of ferrimagnetic or ferromagnetic materials. The analytical solution is a function of magnetic parameters of the material; magnetization and anisotropy field, Ms, Hk; and assumptions about the shape and size of the magnetic particulate. The knowledge of shape allows demagnetization fields to be calculated in terms of demagnetization vector factor

As in Chap. 2 analysis of permittivity, an analytical solution becomes the function to which measured frequency dispersive permeability data can be fit for derivation of intrinsic parameters. The derived parameters for magnetic materials measured for this book are tabulated in Chap. 12. The derived parameters identify intrinsic parameters which can be independently verified by other measurements such as DC saturation magnetization measurements [8–10]. Caution should be observed when the derived parameters are over a narrow bandwidth and may not fully encompass the fundamental resonance. In those cases, extrapolation of the parameters outside the original measured band can lead to error. The analytical solution for the dynamic susceptibility can be derived from the LLG equation under the assumption that the DC magnetic fields have added to them a very weak microwave field of characteristic frequency ω, Heff = H0 + Hk + Hm + h0rf ejωt = Hi + h0rf ejωt. The magnetization is also assigned a time independent and time-dependent part,

. For now demagnetization terms are chosen to be negligible (the terms will be added back) and the microwave magnetic field is applied in a direction that is perpendicular to the anisotropy axis of a large, single domain particle of our magnetic material (Fig. 3.11). The h0rf field is assumed much smaller in magnitude than other magnetic fields and also |m| / |MDC| 1. These assumptions are placed in the Landau equation and only terms linear in h0rf and m are retained. Surface terms are ignored, and the LLG equation yields the magnetic dynamic susceptibility χ(ω) = μ(ω)–1 = m / h0rf, [10, Chap. 4]:

FIGURE 3.11 Geometry for the measurement of the susceptibility term.

The frequency ω0 is the radial frequency at which the magnetization precesses around the field direction. For small loss α a sharp resonance is predicted by Eq. (3.5) at At ω = 0.0 and negligible external field, the equation predicts a resonant frequency of ωr = γHk. In addition a relationship can be obtained between the DC bulk susceptibility and ratio of magnetization and anisotropy field. These are called Snoek’s law(s) [11] [Eq. (3.6)]. An integral form, or functional, generalization of the law for isotropic magnetic materials and one for magnetic composites was published by Acher et al. [12, 13] and will be revisited in Chaps. 6 and 7.

Equation (3.6) shows a simple form of Snoek’s law and supplies physical insight about intrinsic properties. It can be extrapolated to suggest limitations on the RF, microwave, and millimeter response of magnetic materials and/or their composites. Equation (3.6) suggests that for a given magnetization, the magnetic resonant frequency, ωr, of a material must decrease for any increase in the DC permeability magnitude, i.e., one cannot have a material with high DC permeability and arbitrarily high resonant frequency. Alternatively one can infer that for equivalent magnetization, a decrease in anisotropy field of material will produce increased DC permeability. Thus, materials with large permeability (like Ni–Fe) are soft magnets with anisotropy fields that approach zero. Either statement has immediate impact on a

magnetic material’s application. If the systems engineer is seeking a material for a low-loss microwave wave phase shifter, e.g., with relaxation in the millimeter bands, a large, low-frequency permeability is not expected. Similarly, if a very high permeability material is needed for low-frequency electromagnetic interference applications (e.g., coaxial chokes and/or surface shielding) the material’s permeability will be near unity at microwave frequencies. In Chaps. 6 and 7 and again in Chap. 11 discussions will address how magnetic composites obey an integral form of this law, i.e., Acher ’s law [13]. These have similar ramifications in the choice of application. The general inverse relationship of resonant frequency and DC susceptibility is illustrated in the measured data of Figs. 3.12 and 3.13. Figure 3.12 shows a modest susceptibility (χDC ≈ 16) Ni–Co–Mn–Ferrite with resonance frequency near 400 MHz. Figure 3.12 is for a Ni–Mn– Zn–Ferrite with χDC ≈ 800, but its resonance has been reduced to approximately 4.5 MHz. An increase of approximately 100:1 in resonant frequency is accompanied by approximately a 50:1 decrease in DC susceptibility—not the perfect scale predicted by Snoek, but in qualitative agreement.

FIGURE 3.12 Measured and model data fit for a Ni.99 Co .01 Mn.02 Fe1.9 O4 ferrite; modest DC susceptibility and about 400 MHz resonance.

FIGURE 3.13 Measured and model data fit for a Ni.4 Zn.5 Mn.02 Fe1.9 O4 ferrite; modest DC susceptibility and about 4 MHz resonance.

3.2.2 Demagnetization, Resonance, and Snoek’s Laws Resonant frequency, line-width, and the form of Snoek’s law are changed if one is dealing with finite size particles of a magnetic material and/or finite size particulates in composites. Particulates have demagnetization fields that must be added to DC and AC fields. Magnetic poles form on the particle’s surfaces and in the volume so as to oppose external and magnetization fields. A DC demagnetization field of the form is created along with a frequency-dependent . When substituted into Eq. (3.3) and if frequency-dependent field components are still small, a modified form for the susceptibility can be derived [10, Chap. 4, Appendix 4-1]. If the anisotropy field and external fields are oriented in the z-direction, and the weak external AC magnetic field is applied in the y-direction, the susceptibility in the x-direction is given by

where ωm = γ4πMs; ω0 = γH0 – Nzωm; ωr2 = {(γH0 + (Nx – Nz)ωm)(γH0 + (Ny – Nz)ωm)}. In Eq. 3.7, H0 includes any external DC field and also the internal crystalline anisotropy

field, Hk. In the following the external DC field is assumed zero. The DC limit is a function of the particulate shape and the demagnetization fields which are represented by demagnetization factors. A different scaling functionality between resonant frequency and magnetization is predicted for particles of different shapes. Spherical and cubic particulates have equal demagnetization factors (Nx = Nz = Ny = 1/3) and the DC limit of susceptibility is χDC = γ4πMs/(γHk) = ωm/ωr which is identical to Eq. (3.6). Flat rectangular particles, with DC fields directed in the plane of the particle, have preferred symmetry and unequal demagnetization factors (Nx = Nz = 0, Ny = 1). Snoek’s law scales as

where fr2 = γHk (γHk + ωm) for a planar particle. The equation shows that for planar particles, DC susceptibility is limited but scales with inverse resonant frequency squared. Another particle shape that is often encountered in composites is a needle shape (e.g., Fe or Ni fibers). Demagnetization factors for a needle are Nx = Ny = 1/2, Nz = 0 and Snoek’s law is

Here the resonant frequency can be quite high; however, demagnetization has a large impact and the DC susceptibility will uniformly approach 2 for soft magnetic materials where Ms Hk. Figures 3.14 and 3.15 show scatter plots of measured DC susceptibility versus resonant frequency for ferrites, ferrite composites, Fe composites, and Fe2O3 composites that are discussed in Chap. 12. In general, the inverse relationship of susceptibility and resonant frequency are followed in the figures.

FIGURE 3.14 Scatter plots of measured resonant frequency and DC susceptibility for ferrites (top) and a single-ferrite composite (bottom). Composites have multidomain particles with volume fractions ranging between 7 and 60 percent.

FIGURE 3.15 Scatter plots of measured resonant frequency and DC susceptibility for Fe composites (top) and nanoscale (5 nm to 60 μm) Fe3 O4 composites (bottom). Composites have multidomain and near single domain particulates with volume fractions ranging between 2 to 50 percent.

The Fe2O3 composites deviate from the Snoek scaling but they have very broad range of sizes (single domain to multidomain); other factors are reflected in the data. Chapter 11 discusses evidence of size-dependent magnetization and anisotropy field. Figures 3.16 and 3.17 show visualizations of the various Snoek’s law forms. Equation (3.7) has been applied to illuminate demagnetization impact on resonant frequency for a soft magnetic material (Ni–Fe) made into various shapes.

FIGURE 3.16 Resonant frequency for various shapes and corresponding demagnetization factors.

FIGURE 3.17 Resonant frequency for various shapes and a continuum of demagnetization factors (4πM s = 10,600 Oe, Hk = 3.5 Oe).

In general as shape is changed to achieve higher resonance the DC susceptibility decreases. Particles that show symmetry in x and z directions have the same DC susceptibility as larger particles. Asymmetric particles sometimes have a reduced susceptibility due to demagnetization effects. These calculations assume that the particles have approximately ellipsoidal shapes. If non- ellipsoidal shapes are present, demagnetization factors may not be uniform and the equations are not accurate. Micromagnetic calculations indicate that for single domain particles the approximations are good. Formation of internal domains begins as the magnetic particle increases beyond a material-dependent dimension. Soft magnetic materials will show closure domains that disrupt the demagnetization effects parallel to the anisotropy field. This effectively renders the demagnetization factor Nz = 0 and the initial susceptibility is dominated by demagnetization effects. The above analysis gives an approximation for the response of a particle to a small excitation field. In reality the resonant spectra is complicated. Micromagnetics modeling has shown that in some particles the presence of multiple domains introduces domain–domain interactions and other excitations that yield a susceptibility frequency spectrum having many “resonances” with differing magnitudes. These complicated spectra are not produced in analytical models but become apparent via micromagnetic simulations.

3.3 SUSCEPTIBILITY MODELS FOR DATA ANALYSIS The magnetic data in Chap. 12 are composed of two material types. Most common are composites that are composed of small particulates (10 nm to 80 μm) that have spherical or ellipsoidal shapes. The second set of materials is solid ferrites and/or ferromagnetic materials. All materials had rectangular solid or toroid shapes when they were measured. Measured data are fit to a scaled form of

where it is rewritten in terms of the DC susceptibility for a spherical particle, i.e., χDC = ωm/(γHk). Factoring ωmγHk from the numerator and (γHk)2 from the denominator produces

In most cases the composites contain almost spherical particles and when combined with a mixture theory we find that a simple three-parameter fitting function is sufficient to model the composite and determine the intrinsic magnetic parameters of the particle used to make that composite. The fitting function that is used will be

When is determined from a data fit, it can be used to calculate relaxation α from α = BC1/2. The magnetization is calculated from Ms = AC1/2. The magnetization, resonant frequency, and anisotropy field were often available for the ferrite samples tabulated in Chap. 12. Therefore, measured data-derived quantities can be compared with those from the manufacturer for the same sample chemistry.

3.4 AN OVERVIEW OF MICROMAGNETIC MODELS Analytical models are not sufficient when large, irregular shaped magnetic particles are under study or when intense electromagnetic fields are involved [14]. In those cases numerical solutions to the complete LLG equation are required. This micromagnetic modeling is based on volumetric finite difference solutions [15] of the equation. Numerical approaches become a necessity to predict magnetic spectra and effective permeability of a material, particulate, and composites using the particulate. A model calculation is performed in steps. First, the zero field, or a resting configuration, of moments and domains is calculated. The resting properties establish the initial conditions from which time-dependent properties evolve. After a stable configuration is obtained, a timedependent electromagnetic field is introduced and the equation is solved in a time-stepping fashion. The magnetic moment and domain structure in external AC and DC magnetic fields are thus calculated as a function of time. Frequency dispersive moments, domains, and permeability can then be calculated from the Fourier transform of the time domain responses. More details of the procedure follow. Micromagnetic solutions to the LL and LLG differential equations begin by calculating the magnetic domain structure. The magnetic sample dimensions are set by the simulation. An initial domain configuration is determined by minimizing the magnetic free energy of the system with respect to the macroscopic magnetic parameter, i.e., magnetization . In general magnetization is a function of position within the material, (x, y, z). The free energy calculation assumes that individual spins have summed moments not larger than Ms, the bulk saturation magnetization of the material chemistry. Local values of (x, y, z) are then calculated by minimizing the free energy under the constraint L = (x, y, z)/Ms = 1.0 [5]. The magnetic free energy function is assumed to have four contributors:

The term EX is the previously mentioned exchange energy and the source of moment alignments in ferromagnets, ferrimagnets, or antiferromagnets. As discussed previously, in ferromagnets moments align with one another since the Pauli exclusion principle encourages aligned electrons with wider separations and decreased electrostatic repulsion. However, opposing moments will decrease magnetic energy but the opposite spin electrons can occupy the same state. They can be closer; however, closeness competes with spin-aligned states since closeness increases electrostatic energy. The equation for the exchange term is

S is a parameter that depends on atomic composition and is in general a function of temperature. Values of S range from 10–12 to 10–11 J/m. The second term in the energy sum is the anisotropy energy EANI. The magnetization is

often preferentially aligned with some geometry, e.g., with select axes of the material’s crystalline lattice. The anisotropy term accounts for the energy loss or increase in misalignment. In materials with a single preferred axis, EANI takes the form

where θ is the angle between the direction of magnetization and preferred crystal axis. The parameters K1 and K2 vary with the atomic composition. Their magnitudes can vary by many factors of 10. In permanent (hard) magnets their values can exceed 107 J/m3. In soft magnetic materials, like Ni–Fe, the magnitude approaches zero. In magnetic materials that have cubic lattices the anisotropy energy term at room temperature is

where m1, m2, m3 are magnetizations projected along each lattice axis. The values of K have smaller variations in cubic lattices. A relative magnetic measure of anisotropy is the anisotropy field k. The direction of the anisotropy field is along the easy axis of the material, i.e., the most energetically favored direction for spontaneous magnetization. Its magnitude is expressed in terms of a multiple of the saturation magnetization, |H|k = 2K1/Ms, and it represents the field required to saturate along the hard axis. The anisotropy field can act as an effective saturating magnetic field and plays a prominent role in determining ferromagnetic and ferrimagnetic resonance. The third energy term is the potential energy contribution for coupling of ferrimagnetic or ferromagnetic magnetization and the external field. This is given by EZ = – · External. The fourth energy contribution is due to demagnetizing fields, DM, that arises from magnetic poles that exist within or on surfaces of a ferrimagnetic or ferromagnetic material sample. This energy term is the magnetostatic energy, EDM, and is calculated from integration over surface and volume where surface integrations increase the energy while volume integrals have the opposite effect. The integrals are

and are surface normal. Note there are multiple integrations involved in each term and for an irregular shaped particle these can be difficult to perform. However, one may address

many conical problems in composites of magnetic materials by choosing a fundamental shape which can be distorted to fit or encompass the shape of many magnetic particulates. Because of the manufacturing processes (grinding, in situ formation, sputtering, chemical etching) the particulates are often ellipsoidal or, in sputtering systems, planar. For infinite planes of magnetic material the demagnetization field is where the unit vector is normal to the plane. The energy for this case is and is the maximum energy that can be associated with demagnetization fields. In case of ellipsoidal particles, where N is a diagonal matrix and the sum of the diagonal elements is equal to unity, Nx + Ny + Nz = 1. The values of Nj are dependent on the ratios of the axes of an ellipsoid (a, b, c) and these are tabulated or can be directly calculated [3, 10]. As shown in previous sections, an accurate description of the particle shape, and therefore Nj is particularly important since the anisotropy terms determine the resonance frequency and DC value of the particle permeability and susceptibility. These factors are repeated in Table 3.1 for several shapes. TABLE 3.1 Demagnetization Factors and Limiting Ellipsoidal Shapes

The equilibrium configuration of a particle’s magnetization is reached when the magnetic energy has been minimized. Recall that the minimization is achieved via the numerical solution of the LL equation under zero external field and the minimization constraints. When using a public domain magnetics solver such as the NIST OOMMF code [5], the system is allowed to relax until the torque on the magnetization effectively disappears, e.g., . OOMMF is referenced in over 1200 publications where many magnetic chemistries, shapes, and sizes have been modeled. A reference list is maintained at the NIST OOMMF website. Selected applications, referenced by the author and collaborators, are found in Refs. [16] to [20]. My collaborators, Drs Harter and Mohler, can be thanked for much of this chapter ’s analysis. The predicted equilibrium domain structure of a soft magnetic rectangle is shown in Fig. 3.18. In this calculation the easy axis was parallel to the width, and the damping coefficient has been arbitrarily set to α = 0.015. No external magnetic fields have been applied. The directions of the local magnetic moment are indicated by the small arrows. At these dimensions the particle has multiple domains that are configured to complete the magnetic loop (circuit). The calculation illustrates “closure domains.”

FIGURE 3.18 Calculated equilibrium domain structure of a rectangular particle (N–Fe, 10 × 20 × 2 μm).

Micromagnetic solvers such as OOMMF perform the relaxation calculation based on a geometry and also the represented magnetic parameters (Ms, α, Hk) that are appropriate to the composition. Figure 3.19 shows a calculation for an irregular shape Ni–Fe. As with Fig. 3.18, the size and parameters lead to closure domains but they are coupled in convolved combinations.

FIGURE 3.19 Equilibrium (zero field) domain configuration for an irregular shaped 10 × 10 × 2 μm Ni–Fe particle.

Frequency dispersive susceptibility is defined as the ratio of dispersive magnetization to

the oscillatory magnetic field component that drives the magnetization. Thus, after the equilibrium field has been reached, a time-dependent magnetic field is added to the LLG equation and computations are run over an extended time period. The time evolution of the magnetization, time-dependent field, and time-dependent susceptibility are all related to their frequency dependence by a Fourier transform. The susceptibility is calculated by dividing the Fourier transform of the magnetization along the direction of the time-dependent field by the Fourier transform of a small-magnitude excitation magnetic field. A gaussian time dependence (pulse) is often chosen for the time-dependent excitation magnetic field. However, the gaussian has a “DC to daylight” Fourier spectrum. “All” electromagnetic frequencies contribute to the excitation. The excitation does not reflect the real world where RF or microwave excitation is often narrow banded. In addition, when using a gaussian pulse, any computation must be run for an extreme number of computational cycles to achieve convergence. Extended computational effort can be mediated and a better representation of RF/microwave excitations can be obtained by applying a differentiated gaussian pulse. Its time evolution is given by

The Fourier transform of hex(t) is

This excitation spectrum is more representative of real-world conditions. The excitation has a maximum at the frequency fωmx = 1/τ. The 10 percent bandwidth extends from approximately .06ωmx to 2.8ωmx. Thus, the frequency-dependent magnetization and ultimately susceptibility can be predicted in a narrower (as compared to gaussian pulse) and userspecified frequency range. As an example, if τ = 10–9 sec, the excitation frequency maximum would occur at approximately 160 MHz. The bandwidth of the excitation would be from approximately 10 to 440 MHz. After the excitation is turned on, the system is allowed to evolve until it relaxes. Changes over each time step approach zero. After near zero change is observed, the frequencydependent permeability/susceptibility can be calculated from the Fourier transform of the magnetization and excitation electromagnetic field. By definition the susceptibility in the direction of the applied field is calculated from the ratioed fast Fourier transforms (FFTs):

This functional combination has been applied in many of the publications referenced at the NIST website for OOMMF. An example of the computation for the rectangular Ni–Fe particle

of Fig. 3.18 is shown in Figs. 3.20 through 3.22. A differentiated gaussian with time constant of 0.022 has been used for hex (t).

FIGURE 3.20 Time domain external field and resulting dispersive magnetization of a Ni–Fe rectangular particle (Fig. 3.18).

FIGURE 3.21 FFTs of external field and dispersive responding magnetization of the Ni–Fe rectangular particle (Fig. 3.18).

FIGURE 3.22 Magnetic susceptibility calculated from the ratio of external field and excited magnetization FFTs for the rectangular Ni–Fe particle of Fig. 3.18.

All micro volumetric numerical computations are size limited. However, parallelization, application of graphical processors (GPU), and field averaging have allowed numerical solvers [21] to solve geometries with tens of millions of finite grid points in a few hours on desktop computers. Thus, arrays and other geometrical distributions of magnetic particles can be modeled to predict effective susceptibility of particle assemblies. Figures 3.23 and 3.24 are included to illustrate capability.

FIGURE 3.23 Finite difference micromagnetic calculation for domain structure of a Ni–Fe rectangle assembly as separation is increased.

FIGURE 3.24 Finite difference micromagnetic calculation for coupling field of a Ni–Fe rectangle assembly as separation is increased.

Figure 3.23 shows an assembly of rectangular ferromagnetic particulates and illustrates the change in their domain structure as the particles move from a closely packed to separated configuration. The size of the computational box is 30 × 30 μm. Figure 3.24 shows the same assembly, but now the color scale is for magnetic field magnitude with red being any field higher than H > 3.5Oe. When the particles reach approximately a 5-μm separation, coupling fields drop well below the 3.5-Oe cutoff. The change in coupling will result in a change in frequency dispersion and susceptibility for the assembly.

3.5 KRAMERS–KRONIG (KK) RELATIONSHIPS Dispersion relations or KK relationships exist between the real and imaginary parts of any

casual function [22–25]. Causality relates a source (external AC electric or magnetic fields) to its impact (i.e., either electrical polarization and/or magnetic magnetization) in a time-ordered manner. Polarization and magnetization time dependence follow the excitation field in time. Electric polarization and magnetization are imaged in the magnitude and frequency dispersion of permittivity-electric susceptibility and/or permeability-magnetic susceptibility ε(ω) – 1 = χe(ω) and μ(ω) – 1 = χm(ω). One common form of KK relationships is shown below. The parameter β(ω) is β(ω) = βr(ω) + jβim(ω) and can be either dielectric or magnetic susceptibility.

Consider the example of low-frequency permittivity dispersion due to polar relaxation in liquids and some solids. In this case, the real and imaginary parts of the permittivitysusceptibility (in the frequency region of the relaxation) are given by

One can check the KK relationship directly for this case. The real part is calculated from the imaginary part by direct substitution in Eq. (3.21) and evaluating the integral. Using the imaginary part to calculate the real part, one finds

P indicates the Cauchy principle value, which is determine by the sum of the residues at the poles of the integrand. The simple poles in the upper half complex plane are found at ν = ω and jτ–1; thus

where, Integ The real part of this calculation is evaluated at the pole at ν = jτ –1 and this substitution gives

Thus the integral directly recovers the real part of Eq. (3.22). The same procedure can be applied to measured data of either real or imaginary parts of a causal function; however, application may not be straightforward. The integral relationships have lower and upper frequency limits of 0 and ∞. Therefore, a calculation of one quantity from the other (either real from imaginary or vice versa) requires measured data of one quantity over extreme bandwidths. Further, the real or imaginary value, at a single frequency, requires integration over all frequencies above and below the one sought. A very dense and extended frequency data set is required. The KK analysis is time tested and has often been applied at far-infrared to ultraviolet wavelengths (about 100 to 0.3 μm, 100 to 300:1 bandwidths) to perform measurements of dielectric parameters. Until recent developments in metamaterials, permeability was assumed to be unity at these wave bands and thus one measurement would suffice. In the infrared to visible bands, materials are thousands of wavelengths thick and thus the material can be considered semi-infinite. Reflection amplitude data are the observable of choice. Measurements can be frequency dense and very broadband. The need for bandwidth is not a problem in the infrared, visible, and ultraviolet spectra. Sources and power detectors have been available for well over 50 years in this spectral region. Infrared and/or optical electromagnetic phase is difficult to measure over wide bandwidths and transmitted power may be negligible especially for semiconducting or conducting materials. Thus, KK dispersion relations are applied to the real valued power reflection coefficient to calculate phase from the dispersion integral. With the knowledge of reflection amplitude and phase, the real and imaginary constitutive parameter components can be calculated. In the far-infrared to ultraviolet spectra, the KK relationship between reflection amplitude and relative reflection phase, δ(f), is described by the form [1]

When both ρ and δ are known, the optical index from

and μ ≈ 1 is calculated

With the development of the network analyzer and digital receivers in the 1980s, measurements of material properties in 7-mm coaxial test fixtures easily exceeded 1000:1 (.01 to 20 GHz) bandwidths and in some test cases 0.001 to 20 GHz became practical. When

combined with other low-frequency analyzer data, the coverage from 1 KHz to 20 GHz (≥106:1) could be achieved. However, the application of KK analysis to infer one constitutive component from another is really a mote question for the HF, RF, microwave, and millimeter wave bands. Two measurements supply complex reflection and transmission coefficients and together these give the experimenter a direct means of calculating permittivity and permeability. Thus, the KK analysis is best applied to validate causality, improve data continuity, smoothing, or supplementing analyzer measurements [26]. Figure 3.25 shows an example of the KK analysis applied directly to measurement. In Figure 3.25 measured, real permeability data from Chap. 12 for Ni.99Co .01Mn.02 Ferrite has been transformed using the second equation of Eq. (3.21). It is plotted against a fit to the actual measured imaginary permeability data. The KK analysis is slightly shifted in frequency and magnitude. However, the integration shows that the real and imaginary parts are consistent with causality requirements.

FIGURE 3.25 Measured imaginary permeability (blue) for Ni.99 Co .01 Mn.02 Ferrite plotted with KK transform of the measured real permeability.

3.6 TEMPERATURE-DEPENDENT PERMEABILITY Permeability versus temperature is not included in data sets to be found in the tabulations of Chap. 12. Magnetic permeability–temperature relationships use historically verified physical models that allow prediction of temperature functionality, if some basic parameters of a magnetic material are known. Those basic relations can be obtained from DC and AC measurements. The expected temperature dependence can be taken directly from theory and the resonance equation that will be used in fitting permeability data of Chap. 12, i.e.,

In this equation, both real and imaginary parts of the susceptibility are proportional to χDC for any frequency ω. It is also recognized that the DC susceptibility is proportional to the magnetization, Ms. Though the impact of temperature on anisotropy field is material dependent, the variation of magnetization with temperature is founded in quantum theory and can be applied to the problem. In ferromagnetic and ferrimagnetic materials magnetization as a function of temperature, M(T), is proportional to the Brillouin function Bs(x) [10]:

where g is the Lande g-factor; μB, kB are Bohr magneton and Boltzmann constant; s is the total spin; T is the temperature (K); and λ is the constant describing spin coupling

Note that the solution of this equation requires a nonlinear calculation since M is contained in the argument of Bs (x), x = gμBM/λkBT. Total spin varies with each individual material and thus solutions are material dependent. However, the general shape of M versus T is very similar for all materials. That shape is shown in Fig. 3.26, as presented by Soohoo [27].

FIGURE 3.26 Brillouin function prediction of normalized magnetization versus normalized temperature for Fe (circles) and for Co, Ni (squares) [27].

In Fig. 3.26 the temperature is normalized to the Curie temperature, Tc, which is that temperature where the ferromagnetic state disappears. Tc varies with the material. For Co the Curie temperature is approximately 1388°C, for Ni 358°C, and for Fe 770°C. The shape of Fig. 3.26 suggests a smooth reduction in magnetization. Therefore, the magnetic susceptibility varies in a similar manner as temperature approaches the Curie temperature. One direct measurement of permeability versus temperature was taken by the author for this text. That data is shown for illustration in Fig. 3.27. A TE10N resonant cavity operating near 11 GHz (to be discussed in Chaps. 9, 10, and 12) was used to measure permeability of a Fe particulate placed in a glass cylinder. The 0.05-cm diameter cylinder was placed in nitrogen to avoid oxidation and the permeability was measured as a function of the temperature. The data shown in Fig. 3.27 show the general dependence of the Brillouin function in Fig. 3.26. Both real and imaginary parts of the permeability are observed to approach unity and zero, respectively, near the stated Curie temperature of Fe.

FIGURE 3.27 Measured permeability versus temperature of a ferromagnetic powder-filled glass cylinder using a cavity resonance technique near 11 GHz.

Helpful Magnetic Terms and Gaussian to SI Conversions (Multiply C by Gaussian to Get SI and Rationalized Units)

REFERENCES 1. R. E. Hummel, Electronic Properties of Materials, 3rd ed., ISBN 0-387-95144-X, Springer, New York (2005). 2. C. Kittel, Introduction to Solid State Physics, ISBN 0-471-11181-3, Wiley, New Jersey (2005). 3. R. F. Soohoo, Microwave Magnetics, ISBN 0-06-046367-8, Harper&Row, New York (1985). 4. C. Kittel, Quantum Theory of Solids, ISBN 0-471-62412-8, Wiley, New York (1987). 5. M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, Interagency Report NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD (Sept. 1999). 6. L. D. Landau and E. Lifshitz, “On the Theory of the Dispersion of Magnetic Permeability in Ferromagnetic Bodies.” Phys. Z. Sowjetunion, 8:153–169 (1935). 7. A. Hubert and R. Shafer, Magnetic Domains: The Analysis of Magnetic Microstructure, Springer, Berlin (2000). 8. J. A. Osborn, Phys. Rev., 67:351 (1945). 9. E. Schlomann, J. Appl. Phys., 33:2825 (1962). 10. B. Lax and K. J. Button, Microwave Ferrites and Ferrimagnetics, McGraw-Hill (1962). 11. J. L. Snoek, Physica (Amsterdam), 14:204 (1948) and generalization. 12. O. Acher and A. L. Adenot, Phys. Rev. B, 62:11324 (2000). 13. O. Acher and S. Dubourg, Phys. Rev. B, 77:104440 (2008). 14. G. Mohler, A. W. Harter, and R. L. Moore, J. Appl. Phys., 93:7456–7458 (May 2003). 15. A. Taflove, Computational Electrodynamics: The Finite-difference Time-domain Method, 2nd ed., Artech House, Boston, MA (2000). 16. A. W. Harter, G. Mohler, R. L. Moore, and J. Schultz, ICCN 2002, International Conference on Computational Nanoscience, San Juan, Puerto Rico (Apr. 2002). 17. M. Zubris, A. W. Harter, G. Mohler, and R. Tannenbaum, Appl. Phys. Lett. (Dec. 2003). 18. R. L. Moore, J. Meadors, and R. Rice, RadarCon2008 Proceedings, Rome, Italy (May 2007).

19. R. L. Moore, J. Meadors, and R. Rice, Paper #1161, EuMC/EURAD01-Proceedings, Amsterdam (Oct. 2008). 20. Silvia Liong and Ricky L. Moore, “DC and AC Measurements of Magnetite Nanoparticulates and Implications for Nonlinear Response,” 2008 Fall Proceedings Section FF of the Materials Research Society, Boston, MA (Dec. 2008). 21. R. Chang, et al., J. Appl. Phys., 109:07D358 (2011); R. Chang, et al., J. Appl. Phys., 111:07D129 (2012). 22. W. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed., Sec. 22-10, ISBN 0201057026, AddisonWesley, Reading, MA (1962). 23. L. Mandela and E. Wolf, Optical Coherence and Quantum Optics, ISBN 0-521-41711-2, Cambridge University Press. 24. L. Pipes and L. Harvill, Applied Mathematics for Engineers and Physicists, 3rd ed., McGraw-Hill, New York (1970). 25. J. Chamberlain and G. W. Chantry, eds., High Frequency Dielectric Measurement Proceedings of a Tutorial Conference on Measurement of High Frequency Dielectric Properties of Materials, National Physical Laboratory, UK (Mar. 27–29, 1972). 26. J. Baker-Jarvis, et al. IEEE Trans. Instr. Meas., 45(5) (Oct. 1992); J. Baker-Jarvis, et al. NIST Technical Note 1355-R (Dec. 1993). 27. F. Tyler Soohoo, Philos. Mag., 11:596 (1931).

CHAPTER 4

FUNDAMENTAL OBSERVABLES FOR MATERIAL MEASUREMENT

The goal of this chapter is to establish the fundamental propagation models that are most often applied in material measurements. These models lead to a hierarchy of observables that must be measured to infer the electromagnetic constitutive material parameters. In general, the constitutive parameters of materials that are isotropic and homogeneous on the physical scale of the electromagnetic wavelength can be measured at individual frequencies by associating one independent complex data value for each constitutive parameter to be determined. Thus, the real and imaginary parts of permittivity for a material slab of unit permeability for a given thickness and at individual frequencies can be determined using a single complex measurement. If the material has magnetic permeability different from free space, two independent complex data values are required. These may be the relative amplitude and phase of the voltage reflection coefficient for two sample thicknesses. An alternative measurement combination might be two measurements of reflection and transmission amplitude for two orientation angles of the sample or two reflection amplitude/phase measurements as functions of angle. In some cases, when material geometry and/or physical parameters are considered, it may prove easier to measure two complex reflection coefficients with a single material sample placed on two different microwave terminations. An example is a sample measured placed on a short circuit followed by a second measurement terminated by a matched load. Modern microwave vector network analyzers easily produce the required independent data sets from measurement of complex voltage reflection and transmission coefficients of a material. It is the experimentalist’s responsibility to choose the correct material geometry and terminations. The various trades among measurement combinations and sample geometry are discussed in this chapter. The analysis of this chapter is applicable across the electromagnetic spectrum but discussions are directed to the radio, microwave, and millimeter wave frequency spectra. In these bands network analyzer methodology is significant to both electromagnetic engineers and theoreticians. Near continuous frequency coverage can be achieved over bandwidths of 104 or 105. That large bandwidth allows the experimentalist and engineer to confidently use measured data in the design of broadband meter, microwave, and millimeter wave devices. The theoretician can benefit by applying the data in verification of electromagnetic composite material theories over bandwidths that encompass multiple scales, material dimensionalities,

and material physics, e.g., multiphase magnetics or periodic dielectrics exhibiting photonic bandgap and negative index behavior.

4.1 INTRODUCTION The chapter begins with the scattering and numerical models that use isotropic homogeneous electromagnetic constitutive parameters to calculate electromagnetic scattering and observables, reflection and transmission coefficient amplitude, and phase. Most measurement techniques described in this book assume that measured reflection and transmission voltage amplitude and phase derive from isotropic parameters. Thus, the boundary conditions and scattering (cascade) matrix [1, 2] analyses of this chapter can be used to invert measured reflection and transmission and solve for frequency dispersive permittivity, permeability, and conductivity of material layers that are contained within laminar structures. It is recognized that material layers may themselves be composites or mixtures. Thus, the isotropic material analyses are extended with presumption that each material layer has electromagnetic components with characteristic electromagnetic scale much smaller than the electromagnetic wavelength. As shown by Sheng’s application of the coherent potential approximation [3, Chap. 4], a small electromagnetic scale allows description of the layers by homogeneous but possibly anisotropic permittivity and permeability. Thus, the scattering matrix formulism can be extended to anisotropic layered laminates. In principle, multiple independent reflection and/or transmission measurements could be applied to infer tensor constitutive parameters from many reflection and/or transmission measurements. The experimental design problem is daunting, as the elements of an inversion matrix are strongly coupled, and thus experimental design remains an area of engineering development. However, the isotropic and anisotropic material observable can be compared (using assumed tensor constitutive parameters) to calculate the impact on reflection and transmission for variations in constitutive parameters on electromagnetic observables. These calculations will be applied as a baseline to compare with actual measured data on anisotropic materials found in Chap. 12. In short determining anisotropic material properties is not easily done. That also implies that measurement efforts may be moot in establishing an application for the material, e.g., performance as a radome or an electromagnetic interference (EMI) layer.

4.2 SCATTERING OF PLANE WAVES FROM HOMOGENEOUS PLANAR BOUNDARIES AND MATERIAL SLABS The measurement of the intrinsic material properties is not performed directly. In this text, the effective constitutive parameters are derived from a measurement of relative amplitude and phase of the field after an electromagnetic wave passes through or reflects from a material.

Propagation and scattering equations are derived by enforcing electromagnetic boundary conditions at each material interface of a planar material sample and combining them with a propagation model that applies between the material interfaces. The inversion of the equations assumes specific propagation models that relate reflection and transmission to assumed tensor forms for permittivity, permeability, and conductivity. When performing an inversion, two assumptions are made. These are a propagation model and material tensor properties. If there is no prior knowledge about material composition and microstructure, either assumption could be incorrect. In Chaps. 5 to 7, discussions of effective medium models will show how knowledge of composition and microstructure allows one to make a good estimate on propagation and material properties and help with experimental design. The simplest geometry for analyzing plane wave scattering is a planar geometry and isotropic material properties. These analyses also allow investigation of Snell’s laws (reflection and transmission at interfaces) for both positive and negative index material as functions of electromagnetic field polarization. Analysis is extended to finite thickness slabs by applying electromagnetic boundary conditions. The boundary condition analysis is repeated later using a scattering matrix formulation. Scattering matrices allow the user to predict reflection and transmission from an arbitrary number of parallel slabs, but here it is applied to derive expressions for a single slab reflection and transmission, e.g., from free space to slab to free space as functions of angle, polarization, and frequency. The special cases for reflections from metallic-backed slabs and reflection and transmission from slabs of negligible electrical thickness are derived. Analyses are then repeated in later sections for anisotropic materials. The simpler case of a single slab characterized by diagonal permittivity and permeability tensors is analyzed first and numerical simulations are presented that illustrate how reflection and transmission of simple anisotropic materials can deviate from the isotropic case as functions of polarization, angle, and frequency. The final step in planar slab analysis assumes a fully anisotropic permittivity and permeability tensor. This problem is addressed by combining information from Refs. [4] to [8]. These papers analyze the propagation, reflection, and transmission by seeking numerical solutions of linear differential equations. In the final section, equations are derived and analysis is described; however, numerical solutions are found elsewhere in Ref. [7]. Note that the author of this text has never found it necessary to apply the fully anisotropic equations in measurement.

4.2.1 Boundary Conditions and Scattering at Planar Surfaces Electromagnetic boundary conditions, as reflected in Maxwell’s equations, are enforced when plane electromagnetic waves propagate from one electromagnetic medium to another (i.e., from free space through a sample and back to free space). Electromagnetic boundary conditions require continuity of the tangential electric, , and magnetic, , fields across an interface between materials of differing intrinsic constitutive parameters. Since harmonic time dependence is assumed in the following analysis, one can apply phasor electromagnetic boundary conditions. As the field passes across the planar boundary between two regions the

following relations hold:

Equation (4.1) reflects the continuity of the normal component of electric field and Eq. (4.2) shows the current discontinuity of normal component of magnetic field. Referring to Fig. 4.1, the normal vector points outward from region 2 into region 1. Equation (4.3) is indicative of the charge discontinuity of the tangential electric field and Eq. (4.4) shows the tangential H field continuity. Equation (4.5) reflects the continuity of charge and current.

FIGURE 4.1 Scattering at a planar dielectric boundary (TE polarization). Region 1 is on the left and region 2 is on the right. A negative index refracted field direction is indicated by the dashed vector.

The specification of the material type promotes some equation simplification. Most composite materials that we will address in this book are described by s = ρs = 0.0 and the equations express simple continuity of the fields across the boundary. If a material is a good conductor, the current is confined in a thin layer (i.e., skin depth, Chap. 1, Sec. 1.3) and can be considered a two-dimensional current sheet rather than a volumetric current. The surface current density has units of Js (A/m) and is the current per unit width with surface vector normal to the direction of current flow. In such planar boundary conditions, an amount of the charge, Δq, is identified with a surface element, Δs, and a surface charge density has units of

Again referring to Fig. 4.1, if region 2 material is a perfect conductor, a superconductor or just a good conductor like metals, the electric and magnetic fields are excluded and the boundary conditions simplify.

Note that some superconductors (e.g., high-temperature YBCO or Bi) are composites that are formed by regions/layers of low conductivity combined with atomic thickness films of zero DC resistivity. This mixed composite morphology can produce a complex frequencydependent conductivity in high-temperature ceramic superconductors. Further, these are often more sensitive to magnetic field strength. In sufficiently high magnetic fields, the superconducting property is lost [9]. Measurements of ceramic superconductors are discussed in Chap. 10. Recently new microscopic techniques (operating at micron and submicron scales) have been developed that leverage magnetic and electric forces combined with modified atomic force microscopy (AFM) to measure intrinsic properties [9]. However in most cases, the electromagnetic measurement of intrinsic material properties is not performed at the microscale. Measuring RF transmitted or reflected fields from a material, as a function of frequency, requires macroscale sizes. The simple measurement may become more complex when negative index, metamaterials, or anisotropic materials are measured. Data versus frequency may be augmented to include variation with angle of electromagnetic wave incidence and even variation in sample geometry (i.e., high aspect cylinders or thin disks may be used). However, certainly the simplest geometry is a planar geometry which is the focus of the rest of this section.

4.2.2 TE Polarization The first calculation is illustrated in Fig. 4.1. A plane electromagnetic wave propagates in the , plane and is incident at an oblique, arbitrary angle of incidence θ1 on the planar interface between two regions [region 1 (left) and region 2 (right)] with different electromagnetic constitutive parameters. The incident plane wave produces a reflected wave that is directed back into the originating region 1, and a transmitted plane wave that propagates into region 2 but at an angle θ2, different from θ1.

The direction of incident electric field vector is normal to the , plane. This linear electric field vector propagates in a direction perpendicular to the plane formed by the surface normal, , and the direction of propagation, . Therefore, the electric field is parallel to × . An incident electric field with this orientation is designated a transverse electric (TE) polarization field. The transverse magnetic (TM) polarization field (where E and H directions are interchanged) is discussed in the following section. Propagation models for the two orthogonal polarizations allow the user to model arbitrary polarization fields since any field can be expanded in the TE, TM orthogonal bases. The magnitude and phase of the fields and direction of propagation in the second volume are governed by the electromagnetic boundary conditions. Those are determined by the differences in ε,μ,σ of the regions. The three propagating fields overlap at the boundary for TE polarization in Fig. 4.1. Positive (+) and negative (–) signs refer to propagation direction, i.e., in or opposite the direction. Each plane wave has electric field vector , a magnetic field vector , and a wave vector . The assumption of homogeneous isotropic materials requires that the fields be independent of . The incident electric and magnetic field vectors are

and

where E1 is the amplitude of the incident electric field and is a position vector from the origin. is the intrinsic impedance of the medium. The reflected electric and magnetic fields are

and

where is the reflection coefficient that is to be determined by meeting tangential boundary conditions. The reflection coefficient superscript designates the boundary (e.g., from region 1 to region 2) and the subscript designates polarization. The transmitted electric and magnetic fields into region 2 are

and

where is the transmission coefficient to be calculated. If s = 0 at the boundary, E, H meet two conditions. First, the choice of the position along is arbitrary and thus field tangential independence is implied. Second, for tangential electric and magnetic fields to be identical in the z = 0 plane, the phases of the tangential electric or magnetic field must be equal. Equating the arguments of the exponential yields Snell’s laws [10, 11] of reflection and refraction, then

therefore,

Note that so long as index of medium 1 is less than medium 2, the transmitted wave is bent toward the axis and θ2 < θ1. If both ε2 and μ2 are negative the diffraction law still holds. In the double negative case field, the continuity requires reflection about the axis and medium 2 is in the – ,+ direction. The refraction in the second medium is on the same side of the axis. This is indicated by the dashed vector in Fig. 4.1. In region 1 there are incident + reflected fields and in region 2 a transmitted field. Snell’s laws are applied to both, and total electric and magnetic field vector components are set equal at z = 0 to give

These linear equations simplify to

which are solved for

and

:

4.2.3 TM Polarization Figure 4.2 illustrates the TM polarization boundary problem. As in the TE polarization case, the negative index refraction is indicated by the dashed vector.

FIGURE 4.2 Scattering at a planar dielectric boundary (TM polarization). Region 1 is on the left and region 2 is on the right. A negative index refracted field direction is indicated by the dashed vector.

The same procedure is followed to solve for reflection and transmission. The incident fields, reflected and transmitted fields, are

The total field in region 1 is incident and reflected and total field in region 2 is the transmitted field. Their tangential vector components are set equal at the boundary and Snell’s laws are applied. The following equation set for , is the product of the analysis.

Equations (4.22), (4.23), (4.30), and (4.31) comprise Fresnel’s equations [10, 11]. The angularand material-dependent impedance and/or index dependence of Fresnel’s equations are illustrated in the following graphs. Figures 4.3 and 4.4 show calculated relative (i.e., to unity) power reflection and transmission coefficients for TE and TM polarizations at the interface of free space and a frequency independent ideal dielectric material. In Fig. 4.3, data are shown versus relative impedance, , and in Fig. 4.4 data are presented versus index squared or dielectric constant. The range of parameters is identical in each figure.

FIGURE 4.3 Relative power reflection and transmission (from free space to a laminate) for TE and TM polarizations. A dispersionless dielectric material is assumed. The relative intrinsic impedance, , is the material parameter.

FIGURE 4.4 Relative power reflection and transmission (from free space to a laminate) for TE and TM polarizations. A dispersionless dielectric material is assumed. The index magnitude is the material parameter.

Notable in these calculations are two characteristics. At normal incidence (angle = 0.0) TE and TM reflections and transmissions are equal. As angle deviates from normal, TE and TM reflections and transmissions differentiate. Between about 40 and 80 degrees, TM polarization shows a distinct reflection minimum. This reflection null identifies the Brewster ’s angle (Sir David Brewster, 1781–1868) and corresponds to a zero in the TM reflection coefficient numerator and this occurs when θ1 + θ2 = π/2. The differential reflection is often applied in polarization filter applications. Since the reflection zero only occurs for TM polarization, the null can be leveraged to filter the reflected TE from TM polarization component for an incident field of arbitrary polarization. The Brewster ’s angle is clearly shown in Fig. 4.5 as the null in calculated ratio of TE to TM reflected field. However, the calculations are performed for a single interface and are realistic for a physically and electrically thick second material. In the RF–millimeter wave bands, most laminates are penetrated and electromagnetic scattering also occurs from the second (back) interface. That scattering is accounted for in the next section. Thus, the polarizer application (using a single interface) is most often applied in the infrared through UV spectrum where materials are electrically thick.

FIGURE 4.5 Calculated TM/TE magnitude ratio illustrating Brewster’s angle for a low-loss dielectric of the indicated magnitude.

Figure 4.6 repeats the calculations; however, the second medium is strongly dispersive. It has a large magnetic and dielectric loss and is typical of materials encountered in electromagnetic suppression applications. Note that the position of the Brewster ’s angle region is somewhat expanded. Figure 4.6 calculations also extend to indices less than unity such as those encountered with plasmas and negative index materials. Increased reflection and decreased transmission are observed.

FIGURE 4.6 Relative power reflection and transmission (from free space to a laminate) for TE and TM polarization material with frequency dispersive (large loss) magnetic and dielectric properties. Relative index squared, εμ, is the material parameter.

4.3 SINGLE PLANAR SLAB OF MATERIAL As indicated previously, it is rare to encounter only a single material interface in an electromagnetic application or problem in the RF through millimeter wave spectrum. Instead the engineer and/or designer deals with a planar material slab of finite thickness. Scattering from a material slab is illustrated in Fig. 4.7 for thickness ℓ, positive index, and TE and TM polarizations. This geometry is often used in the measurement of electromagnetic constitutive parameters.

FIGURE 4.7 A TE (left) and TM (right) polarized plane wave incident on finite thickness slabs. Reflection, propagating, and transmitted fields are shown.

The calculation of the reflected and transmitted fields in-from-through the layer can be performed in a number of ways. First a direct extension of the boundary value solutions for single interfaces is presented where simultaneous frequency domain solutions are found at the front and back interface of a layer. Later an alternative scattering matrix analysis of slabs is described. That analysis provides a straightforward methodology for calculating transmission and reflection from the multilayers, or for inverting measured reflection and transmission to calculate ε, μ.

The problem of a plane wave propagating from region 1 through region 2 and into a different region 3 is illustrated in Fig. 4.7. The boundary conditions at the left and right most interface have an additional field added due to the reflection from the region 2–3 interface, i.e., E2–, H2–. At the left interface and for TE polarization

Simultaneously at the right interface and TE polarization

Note that at the right interface the field has been propagated a distance ℓ along the z-axis and that the propagation sequence is region 1 to region 2 to region 3. If the right-most region differs from the left-most region, the propagation constants and permeability must be changed appropriately since cosθj+1 and where λ0 is the free space wavelength. The boundary equations for TM polarization and regions 1–2 and 2–3 interfaces are

The four equations for each polarization allow a direct algebraic solution to calculate the reflection at interface 1–2, , and transmission coefficient from region 1 through the slab and exiting at the 2–3 interface, . Equations (4.36) and (4.37) are for TE and TM polarizations, respectively. The scattering problem will be revisited in the scattering matrix discussion and different algebraic (but equivalent) equations will be derived.

and

The impact of thickness is illustrated by Figs. 4.8 and 4.9. The figures show calculations of reflection and transmission power coefficients as functions of index magnitude squared and angle. All calculations are for a frequency of 10 GHz and thickness of 0.015 m and air at the first and second interfaces. Figure 4.8 uses the same low-loss dielectrics as used for Fig. 4.3. The Brewster ’s angle is still apparent at TM polarization. However, new angular-dependent reflection nulls (thus transmission peaks) appear for squared indices near 1.5, 4, and 9 for both polarizations. These correspond to slab electrical thicknesses of 1/2, 1, and 3/2.

FIGURE 4.8 Calculated 10-GHz reflection and transmission versus angle through a 0.015-m thick planar slab of the indicated squared index. Color bar is in decibel (dB).

FIGURE 4.9 Calculated 10-GHz reflection and transmission versus angle through a 0.005-m thick planar slab of the indicated squared index and moderate dielectric and magnetic loss (εi,μ i ≈ 0.05 to 1). Color bar is in decibel (dB).

This resonant transmission has direct application for electromagnetic windows and radomes. In those devices, thickness and constitutive parameters are chosen with these ratios to maximize transmitted power. The reflection nulls change with angle and also become much narrower for index magnitude. The angular dependence can be reduced by using anisotropic materials and this will be illustrated in later discussions. Power reflection and transmission for a laminate with moderate dielectric and magnetic loss (imaginary parts 0.05 to 1) material and 0.005 m thickness and 10 GHz are shown in Fig. 4.9. Since loss has now been added to the material, a low reflection magnitude does not always correspond to transmission near unity. There is significant power absorbed as the wave passes through the material. The loss is leveraged for EMI materials and thus a second calculation, where region 2–3 is a material– metal interface will follow. Figure 4.10 shows the reflection (metal backing means zero transmission). Note that as index and loss increase there are regions of index and angle where reflection is reduced by 10 dB or more and thus greater than 90 percent of the power is absorbed in the material. This illustrates a second application of the finite thickness material slab, as an absorber of electromagnetic energy. Lossy magneto-dielectric materials are often used to reduce EMI and as antenna-backing materials. Modern cell phones are physically thin and they often apply magneto-dielectric films to reduce the impact of ground planes and maintain gain for the cell phone antenna or to surround the antenna to increase electrical size. Additional applications of the magneto-dielectrics are to reduce electromagnetic scattering from surroundings or structures (e.g., bridges and test chambers).

FIGURE 4.10 Calculated 10-GHz reflection versus angle through a 0.005-m thick planar slab backed by a metal surface. Squared index is for a moderate dielectric and magnetic loss (εi, μ i ≈ 0.05 to 1) material. Color bar is in decibel (dB).

4.4 SCATTERING: CASCADE MATRIX METHOD FOR MULTI-BOUNDARY MATERIAL ANALYSIS The planar slab analysis of the previous section derived linear equations that arose from continuity conditions of incident reflected and transmitted electromagnetic fields at boundaries. The modeled materials have linear relationships between constitutive parameters and electromagnetic fields. The materials are infinite cross-section slabs and thus are translationally invariant in a plane perpendicular to the wave’s direction of propagation. In the direction of propagation, e.g., the direction, Maxwell’s equations are first order with the field derivative proportional to the electromagnetic field, e.g., (dE/dz) ∝ E. This leads to a solution where the field experiences a change in phase and amplitude that are exponential in distance, time, frequency, and wavelength. As seen in the previous section, E(ℓ) = E(0)e(±jKℓ) with the exponential argument sign determined by the propagation direction. These observations suggest that propagation through one material slab, or structures composed of many slabs, can be modeled by a set of sequential calculations. One calculation satisfies the boundary conditions at each interface and the second represents propagation through each layer. These sequential calculations can be represented by two, 2 × 2 “scattering or cascade matrices” which have specific forms at each interface and also for propagation through each material. When the overall scattering propagations are calculated, the

transmission, Sij, Sji, i ≠ j, and reflection, Sjj, for a laminate structure are calculated by ratios and products of elements of the overall matrix product. The plane wave transmission and reflection through a two-dimensional slab in free space is analogous to propagation in a transverse electric and magnetic transmission line [11, 12]. In the transmission line, the wave description is replaced by the excitation of equivalent voltages and currents. Lengths of transmission line are described by their characteristic impedance and transmission line or network analysis. The measurement locations are defined as the “ports” of a network. Figure 4.11 illustrates a matrix representation of a transmission line network which has incident, reflected, and transmitted voltages a–, b–, a+, b+ for a general two-port network C. The network can be composed of multilayers of material or lumped resistive, capacitive, or inductive elements. Arrows directed toward [C] represent incident waves and arrows exiting [C] represent reflected or transmitted waves. Letters represent waves at a distinct port and the direction is represented by the “+” or “–” designation.

FIGURE 4.11 Two-port network voltages.

Since the model applied here is for a linear system, the two incoming voltages can be related to the outgoing voltages by two equations. The voltage equation coefficients define a 2 × 2 matrix as illustrated in Fig. 4.11. The arguments of the matrix are the coupling coefficients between all voltages. This general two-port network has a number of elemental matrix representations. The ones emphasized in this text are the boundary, propagation, and shunt element.

4.4.1 The scattering Matrix The scattering and cascade matrix differ. The S matrix (“scattering matrix”) relates the wave voltages that are incident on the ports to those reflected and transmitted from the ports. The S parameters are parameters measured directly with a vector network analyzer such as those produced by Agilent [13]. The S matrix is defined in relation to the incident and reflected voltage [Eq. (4.38)] and illustrated in Fig. 4.12.

FIGURE 4.12 Scattering matrix and definition.

In the two-port representation of a homogeneous isotropic material S11 = S22 and S12 = S21. Therefore, for the two-port S matrix, the elements (S11, S22) are reflection coefficients from the left- and right-hand side, respectively, while (S12, S21) are the transmission coefficients from right- and left-hand side, respectively.

4.4.2 The Cascade Matrix of N Laminates The computer analysis of complex networks motivates a rearrangement of the scattering matrix to a “cascade” form. The cascade matrix Cij relates voltage waves on the left (input) of a transmission line, though all laminates and impedances, and connects to voltages on the right (output) side of structure or transmission line. The transmission line problem for propagation and reflection from a series of laminates, scatters, or impedances can be expressed as a multiplication of cascade matrices for each interface, e.g., T1, T2, and region making up the line, Cj.

The input ports of a transmission line are related to the output by

The Cj represents the separate cascade matrix for each interface and propagation through some length. The multiplicative property is observed since the outputs from one cascade are the inputs to the next. This multiplicative property holds for the cascade matrix but not a scattering matrix. Transmission and reflection properties of a multilayer structure are typically determined by calculating separate cascade matrices and multiplying them in the correct order. The resulting, final ordered product cascade matrix is then transformed to the corresponding S matrix followed by the calculation of reflection and transmission coefficients using S matrix elements. Note that this transformation also works in reverse. Assume that S-matrix elements of a transmission line (i.e., reflection and transmission coefficients) that contain some device or construct are measured using a network analyzer or similar equipment. The corresponding cascade matrix can be calculated for that transmission-line device. By knowing the device (or material) cascade matrix, the performance of that device, in combination with other linear elements or materials, can be calculated and ultimately an overall new scattering matrix for the many combinations can be predicted. Cascade elements are determined from measured scattering matrix components by using Eq. (4.39).

Alternatively if the constitutive parameters, thicknesses, and line lengths are applied to calculate an overall cascade matrix of a laminate, the elements of the scattering matrix are calculated from the elements of the total cascade matrix using Eq. (4.40). Recall that S11 and S22 are reflections from the left and right, respectively, while S12 and S21 are the transmission going left to right and right to left, respectively.

In summary, the approach applied in predicting (or inverting) reflection and transmission data is to calculate the cascade matrix for each element of a planar structure; perform an ordered multiplication of the matrices; and transform the result using Eq. (4.40) to calculate overall reflections and transmissions. The following sections show the C- and S-matrix elements for various material interfaces, regions, and impedance types.

4.4.3 Matrix Representation of Plane Wave Scattering of a Planar Region Boundary The previous sections described means to derive reflection and transmission equations through a material interface for oblique incident plane waves and any general polarization. For linear materials, a field of arbitrary polarization can be separated into TE and TM components. The reflection and transmission of each polarization can be calculated and then summed to calculate the arbitrary polarization case. The matrix representations of plane wave reflection are straightforward and are found in subsections 4.4 and 4.5. Fresnel’s equations for incidence at an interface are applied to S to C transforms and the cascade matrix elements for TE polarization are calculated, i.e.,

Note that a notation change has been made from intrinsic impedance to characteristic impedance where . C21 and C11 are equal to their symmetric elements and therefore

Thus the cascade matrix for the TE polarization of a material interface is

The same procedure can be applied for TM polarization yielding

Note that TE and TM polarizations are equal for the case of θ1 = θ2 = 0.0, i.e.,

4.4.4 Matrix Representation for Propagation The C matrix representation for plane wave propagation is the next matrix. This matrix describes movement of the reference plane of a single plane wave mode between geometric planes. This matrix describes the change in phase and dissipated power of a plane wave with the propagation constant . The field propagates through a thickness ℓ. The angle of propagation (compared to interface normal) is θ. Therefore, accounting for thickness and angle the total electrical path length is kl cosθ and the propagation matrix is

The propagation cascade matrices for TE and TM polarizations are equivalent since each describes the movement of a planar wave front.

4.4.5 Matrix Representation of a Shunt Impedance Laminate designs that include electrically thin ( /λ0 1) materials may describe the materials as shunt elements of some impedance. Typical materials are impedance sheets (Rcards), capacitive surfaces (C-sheets), frequency selective surfaces [14], or metamaterial surfaces [15, 16]. These surfaces are often encapsulated by bounding dielectric-magnetic material slabs. The cascade matrices for shunt impedances are derived in a similar manner as those for a dielectric laminate where we allow the electrical thickness to approach zero. Such a limiting calculation will follow in later sections. The shunt impedance representations are

The S-matrix representation of a shunt impedance boundary are

where TE and TM are identical at normal incidence:

When a thin material sheet is in free space, Z0 is the impedance of free space and the value of Zs is approximately Zs = (σℓ)–1 with σ, ℓ the electrical conductivity and material thickness, respectively. Zs has units of ohms per unit area and is referred to by the term ohms/square.

4.4.6 Cascade Matrix Example: Material Slab Reflection and Transmission The elemental interface and propagation cascade matrices form building blocks to predict the electromagnetic scattering for laminates composed of many material layers. There are many layers through which propagation occurs and there are multiple material interfaces. The most basic laminate is a single material slab suspended in free space and this is the geometry often used for measuring intrinsic parameters of a material. This problem has two free space material interfaces and a single translation through the material’s thickness. In this short example the cascade matrices for interfaces and propagation will be applied to derive expressions for reflection and transmission coefficients. The equations produce identical results to Eqs. (4.36) and (4.37) but have different forms. TE and TM slab scattering geometry and polarization are shown in Figs. 4.13 and 4.14. The interface and propagation cascade matrices, overall products and scattering matrix elements follow in subsections 4.4.6.1 and 4.4.6.2. The first (left most) cascade matrix derives from the slab’s first interface. The second matrix describes propagation through the slab. The last (right most) matrix is from the last interface. A shorthand for cosθ1 = Cθ1 and cosθ2 = Cθ2 has been used and the materials obey Snell’s laws, α1 sinθ1 = α2 sinθ2 and

FIGURE 4.13 TE scattering for a material slab.

FIGURE 4.14 TM scattering geometry for a material slab.

4.4.6.1 TE Cascade Matrix

Multiplication of the matrices and application of eja = cos a + j sin a yield

After applying the cascade-scattering transformation, the 11 and 12 elements for reflection, S11, and transmission, S12, are given by

4.4.6.2 TM Cascade Matrix

After applying the cascade-scattering transformation, the 11 and 12 elements for reflection, S11, and transmission, S12, are given by

Equations (4.57), (4.58), (4.62), and (4.63) are the fundamental equations that are leveraged to calculate permittivity and permeability. They will be repeated in various forms throughout discussions of measurement procedures that are described in Chaps. 8 to 10. The next section describes a second very important laminate configuration, a material slab placed on a conducting surface. That “shorted” measurement supplies a third equation which is used to resolve permittivity from permeability measurements and also is the fundamental design equation in EMI suppression and absorber performance.

4.4.7 Material Slab Backed by a Perfect Conductor: An Example Figures 4.15 and 4.16 show TE, TM geometries of a slab backed by a perfect electrical conductor (PEC). In these cases there are no transmissions through the structure and transmission, S12TE,TM, S21TE,TM = 0. Equations simplify since Z1 = 0.0 in the right-hand matrix of Eqs. (4.54) and (4.59). Since is determined by ratios of cascade matrix elements, the apparent division by zero does not contribute. Equations (4.64) and (4.65) predict reflection for TE and TM polarizations, respectively.

FIGURE 4.15 TE and TM polarization diagram for a PEC-backed material slab.

4.5 SCATTERING FROM A SHUNT PLANAR IMPEDANCE SHEET An “impedance sheet” is an approximation often used when describing an electrically thin

material that is semiconducting or conducting. In most examples, the physical thickness, ℓ, is very small (tens of nanometers to tens of microns). Example materials would be transparent conducting coatings, graphene, indium tin oxide (ITO), or carbon nanotube composite films. Others are conducting polymer films such as polyacetylene, polyaniline (PANI), polydioctylbithiophene (PDOT), or thin metals (Ag, Au, Ni, typically tens of nanometer) [17]. These materials typically have ε μ. The shunt element cascade and scattering matrices apply to these materials. However, the expression for the shunt impedance Zs = (σℓ)–1 is only an approximation and better approximations for Zs can be derived by investigating limiting forms of the exact expressions for transmission through a planar slab. In these shunt impedance sheets, the material properties are much greater than unity (εμ 1) but for the shunt approximation to hold, the electrical thickness must simultaneously be small (kℓ 1). The TE, TM transmission coefficients [Eqs. (4.58) and (4.63)] apply to arbitrary material slabs and are the beginning point of the calculation. Since kℓ 1, the simplification begins by expanding each equation to fist order in kℓ.

From Eqs. (4.51) and (4.52) the TE, TM transmissions for shut elements are

Equations (4.66) and (4.67) are brought into a similar form by extracting {0.5 jkℓCθ2(Z1Cθ2/Z2Cθ1) [1 + (Z2Cθ1/Z1Cθ2)2]} from the denominator of Eq. (4.66) and {0.5 jkℓCθ2(Z1Cθ1/Z2Cθ2) [1 + (Z2Cθ2/Z1Cθ2)2]} from the denominator of Eq. (4.67). Note that [1 + (Z2Cθ1/Z1Cθ2)2] ≈ 1 and also [1 + (Z2Cθ2/Z1Cθ1)2] ≈ 1. Therefore,

The third term in the denominators is much smaller than the leading positive terms. Further

After making these substitutions the equations are brought into same form as Eq. (4.68) if the equivalent shunt impedance is . This expression does reduce to (σℓ)–1 at low frequencies or when the real part of the permittivity is negligible with respect to conductivity. However, in general the expression for the impedance indicates that there is expectation for a reactive impedance component. This reactive component is often ignored by suppliers of commercial R-cards. Many only perform a DC measurement of the conductivity or use a four-point probe [18] to measure the DC surface resistance. Therefore, the engineer who uses manufacturing data on R-cards may find errors in high frequency electromagnetic applications. It is at high frequencies where reactive impedance components contribute to tuning of circuit elements. The capacitance is evident in measured data on R-cards in Chap. 12. In those data, additional capacitance is contributed from the thin polymer on which conductive coatings are applied.

4.6 TRANSMISSION AND REFLECTION FROM ANISOTROPIC LAMINATES Any analysis of modern electromagnetic materials and composites must deal with anisotropy. Many composite materials have some degree of preferred orientation for the components which comprise the composite. One example is a fiber-polymer composite made from fabric with different fibers in the warp and weft directions. Another example is composites that contain micro- and/or nanotubes that are grown along a preferred geometric axis of the composite. A common anisotropic electromagnetic and structural material is honeycomb. The honeycomb is made from polymer sheets with edges oriented along one axis. Though parameter differentials are not large, orientation produces anisotropic dielectric and/or magnetic properties in the honeycomb. Though many composites may be characterized by simpler tensor forms, general permittivity and permeability exist as 3 × 3 tensor quantities. Propagation through materials with off-diagonal elements equal to zero is modeled first.

4.6.1 Diagonal Tensor ε, μ The analysis of electromagnetic propagation in general anisotropic materials will be addressed in the last few sections of this chapter. However, the impact of anisotropy can be illustrated by considering a simple single laminate. The geometry to be analyzed is nearly identical to the isotropic homogeneous laminate in previous analysis. Figure 4.16 establishes the electromagnetic field and material interaction. The z-axis is aligned normal to the laminate surface. The permittivity and permeability are diagonal

tensors (εx, εy, εz) (μx, μy, μz) for both TE and TM polarizations. The element values are arbitrary; however, there is translational invariance of properties in the and directions. The incident field propagates at an angle θ1 from the surface normal (z direction). Mediums 1 and 3 are on either side of the laminate and they are assumed to have isotropic and homogeneous permittivities and permeabilities.

FIGURE 4.16 TE (left) and TM (right) propagations through an anisotropic material with diagonal permittivity and permeability tensors.

In dealing with tensor properties, analysis is better presented if the incident fields are shown in their linear matrix forms. TE and TM polarized incident fields are, respectively,

As with the isotropic slab analysis, solutions are sought that satisfy Maxwell’s equations and , and the double bar over permittivity and permeability indicates the 3 × 3 tensors. The tangential fields maintain continuity along the laminate’s front and rear interface. The application of Maxwell’s equations to Eq. (4.71) supplies the z-directed propagation constants in each of the m = 1,2,3 media.

Tangential E and H fields are set equal at the front and rear interfaces. The equalization leads to four independent equations (respectively for both TE and TM polarizations). The four equations of the TE case, at interfaces for regions 1–2 and 2–3, are shown below. The +,– subscripts indicate fields propagating respectively in positive and negative z directions. R12 and R23 are reflection coefficients at the 1–2 and 2–3 interfaces.

The four TE and four TM equations (not shown) are solved separately. Solutions lead to expressions for reflection, , and transmission coefficients, TTE, TM and these are shown below. The TE set is presented first [Eq. (4.74)] and is followed by TM which is in a slightly different form [Eq. (4.75)].

Just as with the isotropic laminate, the equations simplify if the third region is assumed to be a perfect electrical conductor. There is no transmission coefficient and the TM, TE reflections are

It is useful to expand these last equations and explicitly show the dependence on permittivity and permeability. Equation (4.76) is rewritten as

An advantage of anisotropy is best illustrated by considering transmission from free space,

through a laminate and back to free space. This is a simple model of electromagnetic propagation through a radome. Many radomes are geometrically curved such that radiating antennas placed inside the radome view a transmitting surface which presents many incident angles with respect to a local surface normal. In some ogive-shaped radomes (Fig. 4.17), very large incident angles can be involved and this can lead to radar tracking errors (as described in the figure’s origin document as referenced). The errors arise from the differences in TE, TM transmission coefficients at high incident angles. For example, the frequency at which TE transmission shows its highest value shifts to frequencies higher than that observed near a zero angle of incidence. The application of the simple planar transmission equations [Eqs. (4.74) and (4.75)] suggests that this frequency–angle dispersion might be reduced with a locally anisotropic radome material.

FIGURE 4.17 Ogive radome geometry from F. Arpin and T. Ollevier, “Crosspolarization tracking errors of a radome covered monopulse radar,” Microw. Opt. Technol. Lett., 49 (10). pp. 2354–2360 (October 2007). D is the diameter and L is the length.

A simple prediction of reduction in frequency-angle dispersion is illustrated in a calculation of transmission through a panel as function of frequency and angle. The panel is assumed to be nonmagnetic and permittivity and thickness are chosen so that the panel is approximately λ/2 in thickness at 14.8 GHz. This choice produces a transmission peak for TE and TM polarizations at 14.8 GHz. The impact of having unequal normal and planar constitutive parameters is illustrated in

Figs. 4.18 and 4.19. In “anisotropic” prediction figures the constitutive parameters of the panel have been chosen to be approximately three times the planar values of parameters. Thus, for εx,y ∼ 4, μx,y ∼ 1, εz ∼ 12, and μz ∼ 3. In the isotropic case, ε ∼ 4 and μ ∼ 1. Transmission coefficient magnitudes are in decibel and are indicated by the color bar on the right of each plot, and plots are restricted to higher angles of incidence to emphasize the effects.

FIGURE 4.18 Isotropic (left) and anisotropic (right) laminate TM transmission.

FIGURE 4.19 Anisotropic and isotropic laminate TE transmission. Note reduced frequency dispersion at 14.8 GHz for anisotropic materials. Color bar is in decibel (dB).

In the TM isotropic and anisotropic cases, transmission has small dependence on angle, even at 80 degrees, but even at this polarization the maximum transmission shifts to a slightly higher frequency. It is the TE polarization that shows significantly less frequency–angle dispersion for the anisotropic panel. In the isotropic case, peak transmission is shifted above 15 GHz and transmission at 14.8 GHz is reduced by almost –4 dB, while in the anisotropic case the peak transmission is maintained at 14.8 GHz. In the TM case, the isotropic peak is shifted to about 14.9 GHz and transmission at 14.8 GHz is reduced by approximately 0.5 dB, while the anisotropic case again shows a smaller change. The calculation is simple but illustrates how frequency–angle variations of the transmission can be reduced when propagating through an anisotropic material. It is noteworthy that transmission advantages offered with anisotropy are applied in the infrared and visible spectra where nanometer-scale cylindrical-shaped carbon nanotubes, ZnO, and others are being grown on cover glass for solar cells and solar absorbers to increase transmission or absorption [17]. The angular dependence of reflection coefficient for the shorted anisotropic material case is shown in Eq. (4.77). Ratios of planar to component constitutive parameters occur in leading terms and in the arguments of the tangent. The TE, TM angular dependence carries an inverse dependence on the component of permeability or permittivity, respectively. Therefore, an intentional change in the components of the permittivity and permeability tensors can increase or decrease the angular variation of propagation constants. However, TE,

TM properties are also functions of the inverse or proportional dependence found in the cosine function. Any advantage of simple anisotropy for reflection must be treated on individual cases, but in general large components of the permittivity and permeability tensor reduce propagation constant dependence on angle. A carefully chosen case is shown in Fig. 4.20. A reduction in frequency–angle dispersion is shown for one anisotropic absorbing material.

FIGURE 4.20 Shorted TM reflection for isotropic (right) and anisotropic (left) materials. Note reduced frequency dispersion at both reflection nulls for anisotropy. Color bar is in decibel (dB).

4.6.2 Arbitrary Tensor, ε, μ This section addresses transmission and reflection from a slab with arbitrary 3 × 3 tensor ε, μ. The section will not address chirality since that is addressed in many other papers and books [19]. The background analysis will be presented herein. Example applications of the analysis are found in Refs. [4] to [6]. The geometry that is being used is a single slab bounded by regions with isotropic, homogeneous constitutive parameters. A sketch of that geometry is shown in Fig. 4.21.

FIGURE 4.21 Fields and geometry for general anisotropy analysis.

The analysis follows that of the previous section. Solutions to Maxwell’s equations of the following forms are assumed. A large • indicates a vector-matrix product and = over a parameter indicates a 3 × 3 tensor. Subscripts are added to E, H, i.e., Ey,x indicate partial derivatives. The second index identifies the differential with respect to that variable Ey,x = (δEy/δx). The first-order Maxwell’s equations become

Rather than seeking direct solutions of E and H in each medium, the above first-order equations are applied to the vectors, order differential equations for Ez, Hz where

The operation produces first,

.

The explicit differentiation and collection of the like differentials with respect to z (i.e.,

)

yield four linear equations.

where

The second indices on E and H indicate the partial differential with respect to that variable and terms of the form μ1rHr or ɛ2rEr imply summation over the second variable, i.e., μ1rHr = μ11H x + μ12 Hy + μ13 Hz. By substituting these forms into Eq. (4.79), they can be solved simultaneously to yield first-order differential equations for Ex,z, Hy,z, Hx,z, Hy,z. Some algebra is required; however, the terms can be collected into the form

where

m is a 4 × 4 matrix. Its matrix elements are

The first-order equation can be formally integrated with the solution

The exponential with tensor argument relates the vector fields in one plane z1 to the vector fields on a different plane z2. The operation would proceed by expanding the exponential in a power series; raise the tensor argument to the appropriate power, resume the series, and operate on the vector field. This power series can become unstable if the step size z2 – z1 is too large or the series summation is prematurely truncated. A simplification for the expansion can be achieved if the m matrix is multiplied by a second matrix to yield a diagonal product. The matrix U is calculated by solving an eigenvalue equation, i.e.,

As discussed in Refs. [4] to [6], at this point the analysis is formally completed since the fields can be calculated numerically at any plane parallel to the surface of the material via the matrix multiplication(s) that would occur in expansion of the exponential in Eq. (4.82). Reflection and transmission coefficient matrices can then be calculated using ratios of the numerical solutions at the entrance and exit plane of the material. These reflection and transmission scattering matrices can be placed in terms of the (z) matrix elements and this was completed by Morgan et al. [6]. Scattered and incident (at the z = 0 plane) fields and transmitted tangential fields (at the z = ℓ plane) are, respectively, . Morgan defines the L(z), scattering, , and transmission,

, matrices as shown in Eq. (4.84).

After numerically obtaining the 4 × 4 matrix L, its sub 2 × 2 interior matrices (Qj) are identified by the association

Morgan’s paper also uses an impedance matrix for the isotropic regions 1 and 3 given by

Details of the derivation can be found in Morgan’s paper; however, the final equations for the reflection (scattering) and transmission matrices are found to be

There is zero transmission in the case of a slab terminated by a PEC surface and the equations simplify to a scattering (reflection) matrix of

4.7 A NUMERICAL ANISOTROPIC MATERIAL EXAMPLE In 35 years of material measurements, the author cannot think of a single instance where a full anisotropic permittivity and permeability has been required. However, there have been many times when numerical examples would have been useful in verifying analysis. This short section is made possible by Dr Eric Kuster (see Acknowledgements) who kindly volunteered to generate data comparing scattering from a nondispersive, isotropic material prediction to a fully anisotropic nondispersive case. The calculation of scattering matrices is performed by a direct evaluation of the exponential translation function. The test material slab is 1 cm in thickness and calculations are performed at 5 and 10 GHz. The isotropic material parameters are ε = 5 – j.3,μ = 1.6– j.9. The layer is bounded by free space on front and back. The incidence angles from z-axis and x-axis are both 45 degrees. The theoretical anisotropic material tensors are

Since nondispersive properties are assumed, the first calculation required is the 4 × 4 m tensor [Eq. (4.81)]. The numerical values, rounded to two or three decimal places, of the elements are for 10 GHz: T11 = –1.98 – j0.14; T12 = –0.51 –j0.04; T13 = 1843.65 – j88.04; T14 = 415.96 – j194.10; T21 = –0.69 – j0.05; T22 = –1.81 – j0.12; T23 = –1174.38 + j82.74; T24 = – 1843.65 + j88.04; T31 = 0.016 + j0.004; T32 = 0.03 + j0.005; T33 = –1.81 – j0.12; T34 = 0.51 + j0.04; T41 = –0.025 – j0.003; T42 = –0.016 – j0.003; T43 = 0.69 + j0.053; T44 = –1.98 – 0.014. Five scattering matrices are provided in the table below. The first shows the general form and correlates matrix element (S11 and S22 reflections; S12 and S21 transmissions) with scattered fields. The second and third are for the 10-GHz isotropic and anisotropic cases, respectively. The fourth and fifth are the same order but for 5 GHz. Note the upper left 2 × 2 sub-matrix is for TM polarization and lower right 2 × 2 is for TE polarization. The upper right and lower left are the cross polarization values and these are identically zero for the isotropic calculation. The general form of the scattering matrix is shown first. General configuration for scattering matrices

Isotropic material, 10 GHz

Anisotropic material, 10 GHz

Note that even anisotropic reflections of the same polarization are symmetric about the plane of the material; however, this is not true for transmission. The same behavior is found in the 5-GHz calculation that is shown below. Isotropic material, 5 GHz

Anisotropic material, 5 GHz

The purpose of this calculation is to supply a comparison for others who may be evaluating their own computational codes. The calculation also illustrates the TE and TM polarization coupling that is to be expected.

4.8 CONCLUSION This chapter has been presented to establish those electromagnetic propagation models and equations that are often applied in material measurements. The equations for reflection and transmission coefficients are applied in a hierarchy or measurement procedures that are discussed in Chaps. 8 to 11. The equations may be modified for specific measurement configurations. For example, the propagation constants may require inclusion of “cutoff” wavelengths that are appropriate to a specific waveguide test fixture. There are only a few cases where anisotropic propagation models will be used and those are mostly in data presentation on magnetic materials. However, a constant question for the experimentalist (especially with composite materials) is whether an isotropic or anisotropic model should be applied in data analysis. The next three chapters are presented to aid the experimentalist in making that choice. These chapters discuss and present models which, in principle, allow scientists and engineers to calculate constitutive parameters (isotropic or anisotropic) of composites if the fundamentals of the composite components are known. The required data go beyond the bulk electromagnetic constitutive parameters of the components and also will include constituent microscopic shape, size, volumetric distribution, and connectivity. This of course requires more detailed and parametric information, but data are often known when the original composite is formulated. The goal of the composite model discussions will be construction of path(s) that allow the experimentalist to choose the measurement technique and data processing which should be appropriate to the material that is to be evaluated.

REFERENCES 1. R. E. Collin, Foundations of Microwave Engineering, 2nd, Wiley, New Jersey, Arpin ed., ISBN-10 0780360311 (2001). 2. C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications: The Engineering Approach, Wiley-IEEE Press, New Jersey ISBN 10-0-471-66985-7 (2005). 3. P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena, 2nd ed., Springer, BerlinHeidelburg, ISBN 10 3-540-29155-5 (2005). 4. S. Teitler and B. W. Hen, “Refraction in Stratified, Anisotropic Media,” J. Opt. Soc. Am., 60(6) (Jun. 1970). 5. D. W. Berreman, J. Opt. Soc. Am., 62(4) (Apr. 1972). 6. M. Morgan, et al., “Electromagnetic Scattering by Stratified, Inhomogeneous, Anisotropic Media,” IEEE Trans. Ant. Prop. VAP-35(2) (Feb. 1987). 7. D. Yang, IEEE Trans. Ant. Prop., 45(3), (Mar. 1997). 8. C. Kittel, Introduction to Solid State Physics, 8th ed., Chapter 10, Wiley, New Jersey (2005). 9. S. Kalinin and A. Gruverman, eds., Scanning Probe Microscopy, Springer, New York, NY (2007). 10. P. Lorrain and D. Corson, Electromagnetic Fields and Waves, W. H. Freeman, San Francisco, CA. 11. M. Born and E. Wolf, Principles of Optics, Pergamon, NY. 12. S. Ramo and J. R. Whinnery, Fields and Waves in Communication Electronics, Wiley, New Jersey (1994). 13. Solutions for Measuring Permittivity and Permeability with LCR Meters and Impedance Analyzers, Application Note 1369-1, Agilent Technologies, United States. 14. Ben A. Munk, Frequency Selective Surfaces: Theory and Design, Wiley-Interscience, ISBN 0-47 1-37047-9 (2000), New York, NY. 15. C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications: The Engineering Approach, ISBN-10-0-471-66985-7, Wiley, New Jersey (2006). 16. K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, Rev. Mod. Phys., 80:1201 (Oct.–Dec. 2008).

17. Materials Research Society Bulletin, October 2011 (special edition). Numerous articles on R-cards. 18. Dieter K. Schroder, Semiconductor Material and Device Characterization, 2nd ed., Wiley, New York, YK (1998). 19. A. H. Sihvola, A. J. Viitanen, I. V. Lindell, and S. A. Tretyakov, Electromagnetic Waves in Chiral and Bi-isotropic Media, ISBN 0-89006-684-1, Artech House, Boston, MA (1994).

CHAPTER 5

COMPOSITES AND EFFECTIVE MEDIUM THEORIES

5.1 INTRODUCTION An experimentalist may not have any clues about probable constitutive parameter values when measuring composites, especially those composed of nanostructures or complex shaped particulates. The knowledge of theories that can supply a first guess on composite properties can be valuable for interpretation of measured data. Therefore, Chaps. 5 to 7 are designed to supply operational descriptions for a user to apply common theoretical frameworks. This chapter supplies an introduction to effective media concepts and derives common forms of effective medium theories (EMTs). EMTs are introduced in the context of media/material in which electromagnetic propagation can be described by a single plane wave (single mode). Equations that predict the electromagnetic properties of such a material are derived by combining the forward scattering theorem or coherent potential approximation (CPA) with analytical equations that predict electromagnetic scattering from single, isolated particles. The particles may be isotropic or layered and may have a range in geometry (sphere, needle, plate, etc.). Example composites are shown in Fig. 5.1.

FIGURE 5.1 Micro composite geometries that will be addressed in models.

The geometries that will be discussed in model developments are limited to those that

allow closed form solutions for scattering. Numerical solvers allow scattering from even very complex geometry constituents to be modeled and therefore propagation and effective constitutive parameters of composites containing complex shapes can be predicted. A short introduction to relevant numerical solvers is in the references listed in this chapter. Chapters 6 and 7 highlight selected solvers and shows examples of electromagnetic observable predictions (transmission and reflection) for complex material geometries that are not easily described by simple effective media theories. This chapter addresses mixtures containing two constituents. General limits are established on dielectric composite composition, component size, and constitutive parameter magnitude such that simple EMT can be applied. Though scattering from many shapes are tabulated, example calculations emphasize spherical inclusions since they supply physical insight into particulate coating effects, import of higher order scattering, and constitutive parameter complexity. EMTs are applied to simple dielectric composites composed of a matrix and single electrically small particulate of some specific volume fraction. Particulate shape and aspect ratio are incorporated in parametric studies to illustrate anisotropies that may be encountered in arrayed or physically thin composites. In most composites, particles used in a composite have a range in size. Thus, subsection 5.6.2 will address model extensions to include spherical particulate size distributions and terms beyond the lowest order dipole scattering (proportional to r03) to the next order of contribution is r05.

5.2 EMT DEVELOPMENT TIMELINE The derivation of improved EMT models of electromagnetic and mechanical material properties has been pursued for the last approximately 110 years. One classic model is due to Maxwell–Garnett [1] who derived a simple linear analytical solution (MGT equation) for the very low frequency effective dielectric properties of a two component mixture where particulates are much smaller than the electromagnetic wavelength. The MGT mixture dielectric is a product of the matrix permittivity and a term which depends upon volume fraction of the additive and dielectric properties of both additive and matrix. The equation is a good model for real dielectric constant in many two-component composites made from random arrangements or periodic arrays. However, when applied to dielectric matrices with conducting pigments, the pigment volume fraction should be limited to fractions less than or equal to about 30 percent. Particles should be spherical in shape and spacing between particulates should be smaller than the wavelength. The upper limit of volume fraction depends upon the type of matrix, particle, and geometrical arrangement. Mixtures that contain small volume fractions are illustrated in the left-most and right-most of Fig. 5.2 and SEM (scanning electron microscope) photo of the composite in Fig. 5.3. A periodic composite is also shown in Fig. 5.3. Low concentrations and periodic structures represent differentiated structures (i.e., particles and matrix are easily distinguished). Additive particles (or voids) are separated by the matrix which isolates particles or voids. These asymmetric microstructures are called “dispersive microstructures” [2]. The concentrations to which random dispersive microstructures are maintained (and successfully modeled by

MGT) are typically less than 50 percent by volume of one material blended into another. However, periodic composites allow better control of distributions and the microstructure can allow concentrations above 50 percent.

FIGURE 5.2 Left-most and right-most mixtures are appropriate to Maxwell–Garnett theory (MGT). However, in the center either material 1 or 2 could be picked as a matrix.

FIGURE 5.3 SEM of composite of polymer and spheres (top) (20–50 μm, 20 percent by volume) and near periodic array of voids (bottom, left) (30 percent by volume). Voids are separated by polymer as indicated in AFM (atomic force microscope) image (bottom, right). All are appropriate for MGT modeling.

The concentrations of both ε2 and ε1 are large and spheres have random locations in the

center drawing of Fig. 5.2 and composite photos of Fig. 5.4. Some regions of ε1 are surrounded by ε2 and vice versa. One might interchange materials 1 and 2 without seeing an obvious difference in the resulting image. What is the matrix and what is the additive would be difficult to define. This interchangeable mix is associated with the term “symmetric microstructure” [2].

FIGURE 5.4 Metallic particle polymer composite (left) (particles 40 percent by volume) and void-filled conducting foam (right) (voids 70 percent by volume). Each are appropriate for Bruggeman modeling.

A knowledge of any mixture’s microstructure is required for successful modeling, since the type of microstructure determines the EMT that should be applied. In the volume fraction range of 30 to 70 percent, the symmetric phase behavior of the middle figure of Fig. 5.2 often applies and especially when conducting particulates are used. The Bruggeman model [3] is commonly applied at these symmetric phase volume fractions. This book references it as the BEMT model. The BEMT is “self-consistent” since it treats each component equally as a particulate or matrix. The BEMT leads to a quadratic equation solution for effective parameters which has products and powers of each constituent property and volume fraction. It is most often used for a two-component mixture. The microstructure is not the only driver in choosing the appropriate EMT. For example, if accurate electromagnetic loss is required, the MGT is often in error. Recall that it is for low frequency. It fails to predict the dielectric-conducting phase transition that is observed in the measurement of conducting–nonconducting mixtures. The volume fraction, at which a phase change occurs, is termed the percolation threshold and appears in the BEMT. The threshold value is a strong function of conducting particulate shape. Its value may be less than 1 percent for needle shapes or approaches 50 percent for thin square patches. Other factors contributing

to percolation threshold are composite thickness (relative to the particle size) and electromagnetic wavelength. The interest in and advancement of EMT models, post Bruggeman, were guided by the need to design composites to absorb electromagnetic energy and make electromagnetic lenses. During and after World War II, the development of radar led to the development of microwave-absorbing materials. Dielectric and magnetic RAM, i.e. radar absorbing material, were produced using iron and carbon particles. The correct manufacture required models to predict electromagnetic properties and aid in formulation. This material research and engineering were performed in Germany, the United Kingdom, and the United States [at Massachusetts Institute of Technology (MIT) and the Naval Research Laboratory] [4–7]. Two histories of these material developments are found in Ref. [8]. Also of note is the short article by L. Lewin of the Admiralty Signals Establishment [9]. It is often overlooked in the history of EMTs. However, Lewin appears to be the first to publish an EMT which showed coupling of frequency dispersive permittivity and permeability derived from an electromagnetic scattering model of the particulates. The 1950s and 1960s saw mixtures and/or structures of dissimilar dielectrics appearing as fiber reinforced polymers, i.e., fiberglass, and microwave lenses [10]. In the 1960s and 1970s new developments in EMT were significant. Advances in solid-state physics encouraged researchers to evolve EMT beyond the perspective of electrons moving through a random media [11–13]. In many cases media were modeled as “hard sphere” impediments to electron movement. The research had direct extension to modeling photons and electromagnetic fields moving through similar random media composed of spherical dielectric discontinuities. Models were also developed to predict properties of multiphase semiconductors and propagation in micrometer and submicrometer, conductor solid and liquid dielectric mixtures, and coatings [14–17]. Many studies addressed infrared and optical spectral anomalies and visible color changes. Numerical solutions for effective media parameters and descriptions of electron motion in random media became possible in the mid- to late 1970s and earlier EMT analysis was verified in direct numerical solutions [17, 18]. Computer models and simulation evolved with computer improvements in the 1980s and these were applied for numerical EMT of mixtures containing nonspherical geometries and material system of different dimensions [19–21]. Two reviews and two text books that address research through the 1980s are listed in Refs. [22] to [24]. The 1990s epoch in effective media research drew the 1980s numerical models and combined them with advancements in computational electromagnetics (CEM). A fundamental assumption in EMT analytical models is that only a single propagating mode exists in the composite; however, a composite is never perfectly isotropic and homogeneous at all physical and electrical scales. Numerical models facilitated investigations at inhomogeneous scales. They could predict electromagnetic observables, propagation, and scattering in composites while incorporating all modes of propagation and could account for inhomogeneity by leveraging detailed information on each composite’s fine-scale geometry and local electromagnetic properties [25–30]. The utilization of fine-scale (e.g., micro and submicrostructure) information and multimodal information can prove an onerous bookkeeping task. Thus, many initial figure of merit investigations for engineering applications of composites often use the approximate

analytical “working” models as a first approximation. However, numerical models are needed for accurate calculations. Many composites, such as metamaterials, are composed of random and periodic assemblies of components and numerical approaches can leverage the periodicity to achieve high-fidelity calculations. In present day, numerical models are commonly used to perform scattering and effective constitutive parameter EMT predictions and several will illustrated in Chap. 7. However, as an example of their power consider the following appropriate task. Numerical models can be designed around periodic representations of a material which may incorporate random composite characteristics in the representation of a single unit cell. A model’s predictions can then be refined by repeated Monte Carlo calculations which allow variations of the unit cell in separate computer runs. Overall, the advancements in numerical approach, accompanied by many orders of magnitude improvement in computer speed, storage, etc., allow many consecutive simulations that better represented the property variations that are inherent in real composites. The numerical approach combined with periodicity and hierarchical, multiscale predictions facilitated the most recent step in application and new phase in composite formulation and manufacture, i.e., metamaterials [31–34]. This EMT development timeline is summarized in Fig. 5.5.

FIGURE 5.5 Effective media theory timeline.

In the next sections, qualitative limits on inhomogeneity are derived to better understand when an EMT might be applied with confidence. Following that analysis various forms of effective media equations are derived to conclude the chapter.

5.3 LIMITATIONS AND DERIVATION OF EMTS 5.3.1 Inhomogeneous Wave Equation

Chapter 1 proposed wave equations for and propagating through inhomogeneous materials and then reduced them to the more familiar homogeneous forms. In fact, permittivity, and/or permeability are always position dependent in composite medium; however, in certain dimensional limits the composite can be treated as isotropic and homogeneous and described by effective complex constitutive parameters. A description using isotropic homogenous properties implies that the electromagnetic fields will propagate through the material with a single wave vector and that wave vector should be independent (or a very weak function) of position. In such media the fundamental question is “what are the particulate and composite microstructures that satisfy this assumption?” The analysis of the general inhomogeneous wave equations supplies some insight. The inhomogeneous wave equations of Chap. 1 are repeated below:

The pre-factors, α,β, are added as scale parameters that multiply the additions to the common homogeneous forms. Alternatively, one might assume that constitutive parameters have small, position-dependent parts, i.e.,

For now it is assumed that permeability is that of free space (μ = 1.0) and that the media has position-independent conductivity. The assumptions allow one to concentrate on Eq. (5.1) which under these conditions reduces to

The assumption of a linearly polarized electromagnetic field with an isotropic wave vector suggests the following electric field form propagating in the media, i.e.,

Substitution in the differential equation gives a constitutive equation for kz:

Equation (5.5) is composed of two parts. The normal equation for propagation constant is –kz2 + εf ω2 + jσ f ω = 0.0. The first additional term is composed of derivatives and permittivity ratios but all are proportional to Ez, i.e.,

This addition is a simple multiple of Ez and is in general a function of position. However, an EMT treatment implies spatial averaging. An average over position simply combines with the normal form for kz as a perturbation to the calculation of wave number. The average is indicated by the brackets , i.e.,

The second contributing term has a different implication. A modification of the propagating field is required to satisfy that term.

Note that the term is not proportional to Ez and the form suggests a modification in the field polarization. An added polarization suggests creation of a new mode whose vector would be in the direction of propagation; not possible for plane wave propagation. The field may exist only if it is localized in space and is rapidly attenuated. However, the EMT singlemode model is saved by the kz scale factor. The second contribution can be ignored (and thus EMT applies) if the term always makes negligible contribution to the total field. The negligible contribution is assured if the product by kz drives the contribution to zero. The propagation constant kz in the effective media would be proportional to λ–1. Therefore, if the characteristic length over which permittivity varies perpendicular to the propagation direction (i.e., indicated by the differential) is insignificant with respect to the wavelength, the implied change in modal content and additional loss can be ignored and only terms that are proportional to Ez need to be incorporated. This requirement and spatial averaging are the heart of EMTs. The analysis implies that the validity of the EMT assumption, at least for artificial dielectrics, is that the electrical size of inclusions must be less than one wavelength, as measured in the media. In many recent treatments of metamaterials a less stringent requirement has been applied. In those treatments, inhomogeneities are often required to be smaller than the electrical wavelength in free space and the free space wavelength is often larger than that in the media.

5.3.2 Forward Scattering and Coherent Potential Approximation The previous section illustrates physics underlying the description of composites and inhomogeneous materials. The use of effective homogeneous constitutive parameters depends heavily on assumptions of small electromagnetic scale (i.e., inhomogeneity size ÷ wavelength B = C and eccentricity ex = (A2 – B2)1/2/A, PA is parallel to A, and Eq. (5.30) applies.

This equation would apply for electrically short and thin fibers, rods, and capped cylinders. When A = C > B, the particle is an oblate, flattened spheroid and is appropriate to model thin platelets. In this case PA along the B axis is

When the equivalent sphere radius is applied in CPA to determine the EMT forms, the BEMT and MGT are modified. Equations (5.32) and (5.33) show modifications where P is to

be applied along the appropriate particulate axis.

5.6 LAYERED INCLUSIONS Some composites use particulates made from multiple atomic compositions. Particulates may have coatings placed on the surface or have decorated surfaces (see Fig. 5.8) or be multilayered where each layer is a distinct atomic composition. As with previous analysis, so long as the particles and coatings are electrically small and/or thin, analytical equations can be derived that are then used in EMTs. As with different shapes the approach again seeks the correct form for polarizability.

FIGURE 5.8 Decorated spheres for this text. Sphere surface features are 10 to 60 nm.

5.6.1 Layered Spheres The equation for layered spheres was previously applied in deriving the MGT equation. The coated sphere equation from Kerker [Eq. (5.25)] is repeated below:

where

, and z is either permittivity or permeability.

In the electrically small and thin coating limit the equation simplifies to

where z can be permittivity or permeability, subscripts c is coating and in indicates interior, and T, r0 are coating thickness and interior material radius.

5.6.2 Impact of Sphere Size and Distributions in EMTs As demonstrated in Sec. 5.3.4, combinations of the CPA and the lowest order dipole scattering term leads directly to the BEMT model. It was noted that higher order contribution could be added. In this section, the BEMT is modified to include terms proportional to the quadrapole moment or fifth power of spherical particulate radius r05. A distribution in size is also included and it is assumed that permeability is unity. The CPA for the forward scattering, S(0), is rewritten below from Secs. 5.3 and 5.4 and expanded to fifth power:

where j is summed over the number of materials with permittivity εj and sphere radius size distribution nj(r), which is summed over all partial waves and an,bn are the spherical scattering amplitudes from the Mie series. When simplified for dielectric-only particulates these are

When these are incorporated in the forward scattering equation and the relation (4π/3) ∫ nj (r)r3 dr = Vj is applied where Vj is the volume occupied by the jth particle type, a modified form for BEMT is obtained with ƒ1 particulate volume fraction and εe effective permittivity:

The first two terms are the normal BEMT form. The other terms can be evaluated to determine relative magnitude if a choice of size distribution function is made. In many cases the range of particulate radii are characterized by a mode, a small size trailing going to zero with a large size range also tapering to some small percentage. The gamma distribution (Fig. 5.9) furnishes that behavior, a is some constant, α controls the function spread, and rm is the most probable radius.

FIGURE 5.9 Example gamma distributions: mode and spread.

With this choice the integrals over r can be evaluated, i.e.,

where the gamma distribution is used. When integral values are substituted, the following effective media equation, which included gamma functions, results:

The largest impact of the additional terms is expected for conducting particulates. In order to assure a uniformly distributed particulate microstructure, studies have found that particulate sizes should be a few tens of micrometers in diameter. The largest diameters observed in the study case composites to be shown in Chap. 11 have radius modes of about 15 μm with largest values of about 50 μm. These values were used as guidelines in evaluating the impact of the radius-dependent term in Eq. (5.40). At and below about 20 GHz the term changes the effective permittivity answers by about 1 percent for conductive particle and by 30 percent for volume fractions. The small impact is a result of the four order of magnitude difference in the dipole (r/λ)3 and quadrapole (r/λ)5 multipliers.

5.6.3 Layered Dielectric Cylinders Reference [36] provides useful limiting forms when dealing with high aspect spheroids or cylindrical particle geometries such as layered and decorated nanotubes [38]. The model described here is that of Ref. [39] and allows for three cylinder layers, i.e., a core and two layers with the last being an outer coating. The geometry is shown in Fig. 5.10 and presents an extreme aspect—long cylinder that is assumed to approximate a truncated cylinder with two endcaps. Recall that the high aspect of the particle suggests that composites may have anisotropic properties. The magnitude of anisotropy will depend upon the geometric distribution of particulate and polarization of the incident electromagnetic field.

FIGURE 5.10 A geometry for the multilayer cylinder with endcaps.

All layers of the cylinder will be in an electrically parallel circuit with each other for an electromagnetic field with electric field parallel to the layered cylinder axis. It also is assumed that the layers are thin with respect to electromagnetic wavelength. In this limit Kerker ’s description of the layered cylinder is equivalent to an areal average of the permittivities in each layer:

In this equation, τi is the area fraction of the ith material, ri2 is the radius of that layer, and εi,|| is the permittivity of the ith layer parallel to the tube axis. As stated above the parallel model assumes long cylinders and allows for two-layer flat endcaps on the cylinders. Thus, the model parallel to the length can be combined in series with the model for the effective permittivity due to the layered endcaps. This allows calculations that account for the contact between tubes at the ends as well as tubes contacting along the length. There may be differences in contact resistance if the layer is an anisotropic semiconductor, with radial conductivity different than conductivity parallel to the surface. An example might be a layer of graphene. Second, if the high aspect particulate has a dielectric coating it would inhibit conduction of electrons between particulates and decrease the effective conductivity of each particulate. The series model approximates effects by placing effective permittivity due to the tube length in a series circuit with a model for the endcap effective permittivity. The overall parallel permittivity is

where ℓ0 is the length of the cylinder, ℓi and εi,⊥ are the thickness and radial permittivity of the

ith layer on the endcaps as shown in Fig. 5.10. As previously stated, the high aspect leads to anisotropic properties. The effective permittivity for an electric field perpendicular to the layered cylinder long axis, ε⊥,effective, must be calculated separately using boundary conditions very similar to that used for the layered sphere. That calculation leads to

In modeling the composites the anisotropic properties are applied as the input permittivity or permeability in an EMT. However, there remains the choice of EMT, i.e., BEMT, MGT, or other approximations [3]. That choice is largely determined by the outer-layer conductivity, particle coupling, and the particle’s constitutive parameter frequency dispersions.

5.7 MODEL CHOICES: IMPORTANCE OF CONDUCTION AND PARTICULATE INTERACTION The basics of EMT and the most common formulations have been developed at this point. This last section of this chapter attempts to clarify some of the questions that remain concerning the choice of the appropriate model(s), the magnitude of differences between models, and what are expectations for differences between model predictions and measured data. The author believes that initial insight is best developed by comparing measured data for permittivity and permeability (equivalently magnetic susceptibility, χm = μ – 1) and predictions. Permittivity will be addressed first. Figure 5.11 shows measured, low-frequency real permittivity of a mixture of conductive Fe spheres that are distributed in an epoxy matrix. A second set of measured data are shown for a nonconductive ferrite mixed into the same epoxy matrix. Lower volume fraction samples were prepared using high shear mixers. Higher fractions began with a premixed lowdensity sample and added small volumes of particulate while being mixed using high-torque slow rotation. The SEM and visible microscopic imagery of the cured composites showed that particulates had a random uniform volumetric distribution. A bulk sample of the ferrite was measured and found to have a permittivity of about 14. Ferrite particles were 40 to 80 μm in diameter and therefore magnetically multidomain. The Fe particulates were 0.2 to 1.2 μm in diameter and also multidomain. Unlike the ferrite, the permittivity of the Fe could not be measured directly; but estimates of permittivity were made based on DC electrical conductivity measurements of a compacted sample of the Fe particles. The conductivity varied

from 1/100 to 1/10 of the bulk Fe value. That conductivity produces a complex permittivity with very large negative real part and large imaginary part (see Chap. 2). When applied in an MGT model, the large magnitude particulate permittivity produces MGT predictions along a limit line [e.g., εr = εepoxy(1 + 3 ƒ/(1 – ƒ))]. BEMT predictions were performed with a fixed value, εIRON = – 1000 + i1000 ; however, just about any values as large or larger yields very similar predictions.

FIGURE 5.11 Measured low frequency εr of epoxy–Fe (represented by diamonds) and epoxy–ferrite composites (circles); MGT Fe composite εr predictions (squares); MGT ferrite εr predictions (triangles), and BEMT predictions of Fe (stars). The xaxis is the volume fraction of particulates.

In Fig. 5.11, the MGT predictions follow its limit line and are consistently below the measured Fe data. A similar underprediction was found in the vast majority of random mixtures that used conductive particulates and are found in the Chap. 12 database. That finding includes other particle shapes. If the conducting particles are kept electrically isolated (either a dielectric coating or via a periodic configuration) MGT model and predictions are much closer to measured data. In Fig. 5.11, ferrite–epoxy mixture measured data and MGT predictions are close, and often overlap. This is also consistent with the Chap. 12 database; MGT proves accurate in dielectric mixture systems which do not contain conductive particulates and particulates have modest permittivity, i.e., ≤ 50. The BEMT predictions are near measured permittivity for the Fe–epoxy mixture, even when using only the estimate for the iron permittivity. The slope of the BEMT prediction is

overly steep as the volume fraction approaches 33 percent. However, the predicted trend is qualitatively like the measured data. The 33 percent value equates to the 3D percolation threshold for spherical particles and is the inverse of the composite dimensionality, i.e., 3. In these mixtures the precise percolation numerical value is sensitive to composite thickness, particle shape and size, and the particle surface morphology. As percolation is approached from lower volume fractions, the Fe particle system becomes more interconnected and DC electrical conductivity rapidly increases. The measured data and example calculations guide choices for preliminary conclusions. (1) The BEMT appears to be a better model for strongly interacting particles (in this case electrically contracting). However, Ref. [17] notes that if the ratio of matrix and particulate constituent parameters does not lie in the range 0.05 to 20, predictions are still expected to be in error. (2) The knowledge of the percolation threshold is critical since it determines the slope of the permittivity–volume fraction function in the BEMT. (3) The MGT appears most accurate in modeling mixtures containing nonconducting particles. (4) The MGT does have a limit line which cannot be exceeded even with arbitrary value constituent parameters and thus may underpredict composite data for mixtures containing particulates such as TiO2. The analysis now turns to a similar study of several magnetic composites. The BEMT and MGT are symmetric for permittivity and permeability for spherical particulates and results similar to the dielectric are expected. Measured and model magnetic susceptibility data are shown in Fig. 5.12.

FIGURE 5.12 Measured (triangles) and predicted (circles, diamonds, and inverted triangles) low-frequency magnetic susceptibility for ferrite–epoxy and Fe–epoxy mixtures.

The plot on the left of Fig. 5.12 shows measured magnetic susceptibility and BEMT/MGT predictions of low-frequency magnetic susceptibility for a ferrite. This NiZnCu ferrite’s fully dense DC permeability was measured and found to be μ about 872. The permeability of fully dense Fe was taken from the Handbook of Chemistry and Physics, 82 edition, and is approximately 100. Figure 5.12 measurement-model comparison stresses both models by using very large permeability constituents. BEMT and MGT models are symmetric in their predictions of composite permittivity and permeability. The test also has implications on the question on what roll dimensionality, and thus percolation threshold in the BEMT, plays in permeability/susceptibility predictions. Unlike permittivity in conducting-dielectric mixtures where percolation corresponds to current flow, there is no magnetic charge to flow between magnetic particulates. The magnetic coupling is weaker and will occur via magnetic moment to moment. The physics of magnetic particle coupling composites will be deferred until Chap. 6. There is no clear modeling winner when comparing measured and predicted data. BEMT and MGT calculations do not overlap measurements but are observed to “bracket” the measured data (triangles for Fe and ferrite) if magnetic particulate volume fractions are above about 30 percent. MGT predictions are below while BEMT predictions are above

measurement. At volume fractions below 30 percent measurement is greater than either BEMT or MGT predictions. However, if the Fe susceptibility value is relaxed to about 60, the BEMT is more in line with measured data. Further, its erroneous prediction of susceptibility slope versus volume fraction is mediated. This reduction in permeability is probably justified since some volume of the Fe is expected to oxidize. The chemical reaction specifics (and surface of volume reaction) could result in permeability decrease. A relaxation in bulk susceptibility is not justified for the ferrite data. Both BEMT and MGT predictions are in error with BEMT deviating from measurement by the larger value. The error is due to the prediction of percolation and rapid slope increase near the volume fraction of 33 percent (percolation). Any magnetic “percolation” is not evident using multidomain particles and BEMT will always overpredict. The single magnetic domain percolation will be revisited in Chap. 6. The MGT underpredicts magnetic susceptibility throughout the NiZnCu ferrite volume fraction range. However, note that measured data parallel the MGT limit line and MGT appears to correctly predict the sudden slope change at about 70 percent volume fraction. MGT accuracy at high-volume fractions is shown in Fig. 5.13. However in that figure, a lower value DC permeability multidomain NiMn ferrite is tested and compared with the predicted value. MGT predictions nearly overlay measured data.

FIGURE 5.13 MGT predictions and measured magnetic susceptibility data for a NiMn ferrite of measured μ DC = 27. The MGT proves very accurate at these volume fractions.

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CHAPTER 6

CONDUCTING-DIELECTRIC AND MAGNETO-DIELECTRIC COMPOSITES

6.1 INTRODUCTION Having established the basis for effective medium theory (EMT) in Chap. 5, this chapter selects a set of common composites and investigates the space of achievable composite parameters ε, σ and μ. In particular, frequency dispersions of the properties are investigated. Chapter discussions are divided approximately 60:40 between permittivity-conductivity and permeability. The first part of the chapter emphasizes EMT models applied to composites composed of a nonconducting matrix containing semiconducting or conducting particulates. The second part of the discussion addresses magnetic composites. The magnetic composite dispersions prove to be Lorentzian like. As shown at the end of Chap. 5, effective media studies point to MGT-like models to be used for modest-permeability-magnitude nonconducting ferrite-dielectric mixtures. However, neither Maxwell–Garnett theory (MGT) nor the Bruggeman effective medium theory (BEMT) works well with large permeability magnitude multidomain particulates. Chap. 6 expands discussions and utilizes input from numerical calculations. Most dielectrics do not exhibit anything like magnetic dispersion in their radio frequency (RF), microwave, or millimeter wave bands. However, the frequency dispersion possibilities of artificial dielectrics are expanded by using combinations of semiconductor or conducting particulates within a dielectric matrix. In order to emulate measured data, EMT must be extended to include electrical percolation within the composite. The correct description of percolation (both permittivity and permeability) requires microstructure details on particulate and composite geometry. Percolation introduces a high probability for geometric-dependent power law scaled, frequency dispersive anisotropic properties and results in constitutive parameters that are functions of composite size, shape, and thickness. Composites that contain components with large constitutive parameter contrasts (i.e., semiconductor-conductor or conductor-dielectric composites) are important for they can display a wide range of constitutive parameters. However, they also display inherent, anomalous diffuse scattering properties and/or localized electromagnetic modes. The diffuse scattering from composites is often called “material noise.” The localization and material noise are functions of the composite constituent’s electromagnetic size scale and therefore

their electromagnetic constitutive parameters, the constitutive parameters of the matrix surrounding constituents, and physical dimensionality of the system (1D rods, 2D thin films, 3D thick material, or intermediate fractional dimensions). These electrical and dimensional properties must be accounted for in choice of effective medium model, design of experiments to test a model, and choosing equipment and data interpretation. Examples of EMT application often revolve about semiconducting films and composites. Many examples can be found in the literature. One of these, transparent conducting films (TCF) or equivalently transparent conducting coatings (TCC), is highlighted in the October 2011 and December 2012 Materials Research Society Bulletin.

6.2 PERCOLATION, DIMENSIONALITY, DEPOLARIZATION, AND FREQUENCY DISPERSION IN SEMICONDUCTING, CONDUCTING-DIELECTRIC COMPOSITES Examples of effective media models and their predictive accuracy have been shown in Chap. 5. Focus now shifts to the assumptions and physics that underpin and, in some cases, limit application of BEMT and MGT to both conductor-dielectric and magnetic-dielectric composites. The discussion extends to Chap. 7 where numerical effective media models are emphasized. The Lewin [1] effective particle parameters are investigated to delve into coupling between effective media constitutive parameters due to their finite electrical size, appearance, diamagnetism, and particulate shape. The additional terms can change the natural frequency dispersions in composites and is a major reason for their investigation. A quick review of “natural dispersions” (taken from Chaps. 2 and 3) is presented in preparation for investigation of composite dispersion sources.

6.2.1 Review of Frequency Dispersion Sources Permeability frequency dispersion appeared in five different “flavors” of magnetism: paramagnetism, diamagnetism, ferromagnetism, ferrimagnetism, and antiferromagnetism. Measured data of Chap. 12 trace to four of the five sources: paramagnetism, diamagnetism, ferromagnetism, and ferrimagnetism. Microscopically permeability sources are magnetic moment(s), , which is/are produced by charge flow, either motion in a closed path (spinning charge, orbiting charge) or linear motion. Of course, blends of both dynamics occur. Magnetic moments couple to external electromagnetic fields via a frequency-dependent vector product and generate frequency dispersive permeability which numerically represents magnetic moment per unit volume. The magnetic frequency dispersion appears in multiple frequency bands. The magnetic domain motion controls dispersion in approximately the one MHz to one hundred MHz bands. Electronic spin and diamagnetic response often appear at hundreds of megahertz to tens of gigahertz. The dispersion can be shifted by DC internal anisotropy fields or externally imposed DC magnetic fields. Anisotropy can extend dispersion

to the millimeter wave spectrum. The dispersive responses are Lorentzian like with multiple Lorentzian shapes across the entire spectrum. Diamagnetism contributes a weak dispersion and derives from currents excited in conducting regions. Conductivity, with the current flowing in conducting paths, in a composite also impacts its permittivity. Free electronic contributions in conductors and semiconductors have dispersion impacts throughout the electromagnetic spectrum. Free electron frequency dispersion functional forms are Lorentzian like, thus similar to the magnetic dispersions in the megahertz and gigahertz spectrum. However, free electron densities and/or electronic excitation energies dictate that dispersions appear at very short wavelengths in conductors and many semiconductors. However, graphene and CNT may have dispersions appearing at a few terahertz. For example, electronic and atomic polarizability excitations from bound to conduction bands normally require significant electronic excitation energies Eex. These are tenths to tens of electron volts (> 1.6 × 10–20 J). Wavelengths or frequencies associated with excitation phenomena are dictated by Eex = hc/λ or Eex = hƒ where h,c are Planck’s constant (6.63 × 10–34 J-sec) and light speed (∼ 2.9979 × 108 m/sec), respectively and are shorter than thermal infrared, about 12 μm. Other physical mechanisms changing permittivity and dispersion include the following: Electronic polarizability due to electronic excitation in a single atomic species. Atomic polarizability is the effective separation of opposite charge concentrations on different atoms. Molecular relaxation occurs when effective positive and negative charges are separated along molecular chains. Molecular relaxation may be achieved at centimeter and millimeter wave frequencies but is most likely observed in a fluid or low-density medium (e.g. air) where rotational motion is not restricted. Water, CO2 , NH4 and other gas absorption lines are observed in air. These produce dispersion at a few GHz and above but only in liquids and gases. Again the frequency dispersion is Lorentzian like.

This book emphasizes measured data and models for composites at wavelengths longer than 1 mm (about 299 GHz). In this region, the natural homogeneous dielectric dispersions are not evident; thus, there is not means to match the dispersions observed in magnetic materials. Dielectric dispersions are at much lower frequencies (1 to 100 KHz) or at terahertz wavelengths, and above. One concludes that dielectric and magnetic dispersions in natural magnetic materials rarely, if ever, overlap. This poses the question of where/if dispersion overlaps can occur in composites. Since magnetic properties are based on quantum sources, and thus not easily modified by a macro mixture, matching dispersion appears best be achieved by shifting the frequency dispersion of permittivity which can be modified at micro or macro sizes. A large macroscopic contributor to composite dispersion is interfacial polarizability arising from spatial control of conduction in charge-carrying particles, i.e., separated semiconducting or conducting particulates in a dielectric matrix. The analysis of BEMT and MGT indicates control is achieved via the numerical value of the dimensional parameter (d – 1) and/or Kerker ’s depolarization factor P of Chap. 5. These dimensional parameters are incorporated in Clerc’s [2] equivalence relating percolation threshold and dimensionality, i.e., d – 1 = (1 – pc)/pc, and Eq. (5.28) suggests that dimensionality, percolation, and polarizability are all interrelated, d – 1 = (1 – pc)/pc = (1 – P)/P. Percolation threshold and depolarization

factor P are effectively the same and therefore percolation is largely determined by the particulate eccentricity or aspect ratio. These parameters can be controlled macroscopically. Periodic assemblies of conductors in dielectrics can produce Lorentzian frequency response. The periodic engineering of materials leads to photonic bandgap and negative index materials [3]. However, they display dispersion over very narrow bandwidths. Above the resonance the structures are electrically inhomogeneous and behave as Bragg scatters and generate significant bistatic scattering. Composites that blend dielectric and conducting or semiconducting materials also introduce the Lorentzian-like dispersion displayed by magnetic materials. They can produce broader bandwidth responses but the blend needs to be precisely modeled and made. The blend volume concentrations, and particulate size and shape control interfacial polarizability and the finite distance over which charge may flow in the composite. These are the physical sources of the composite dielectric frequency dispersion. A precise model requires detailed knowledge of the components and their interactions are simulated by macroscopic percolation. Overall it appears that combinations of periodic and percolating structures might supply the materials scientist with tools that may be used to design materials that fill the frequency gap between natural dielectric resonant responses and those of magnetic materials.

6.2.2 Coupled Constitutive Parameters, Diamagnetism, and Impact on Dispersion The effective permittivity and permeability of a spherical particulate in a composite have been introduced in Chap. 5 and it has been noted that the form was first published by Lewin and is repeated below. When used in an EMT, the Lewin F function acts to couple with both effective parameters of the composite and permittivity, ε1, permeability, μ1, and radius, r0, of a particulate. In all cases, the radius of the particle is much smaller than a wavelength within the composite.

Note that the equation incorporates diamagnetism encountered in a conducting particulate. F(ϕ) describes the exclusion of field from a conducting particulate whose radius is greater than skin depth of the material. Figure 6.1 shows three calculations of F for a magnetic Fe particulate of 0.2, 2.0, and 20 μm in radius. At the 20 μm radius, real and imaginary parts of F are less than unity. Therefore, when multiplied by the particle permeability, μ1, the effective permeability of the particle is reduce, i.e., diamagnetism has its effect. At 2 μm the function is unity at low frequencies with diamagnetism becoming apparent above 1 GHz. At 0.2 μm the particle is smaller than a skin depth and the function is unity well above 10 GHz.

FIGURE 6.1 Lewin F function for a magnetic Fe particulate at radii from 0.2 to 20 μm.

The permittivity is dominated by the very large Fe conductivity (106 mho) and function multiplier of order 0.1 to 0.5 (like the function value for the 20-μm case) has negligible effect in prediction of composite permittivity. However, the large Fe conductivity combined with radius controls the function dispersion. Therefore, the Fe electrical property is strongly coupled to the effective permeability of the Fe particulate and the composite magnetic permeability. This is illustrated in Fig. 6.2. Three MGT predictions of an epoxy-Fe particulate composite (28 percent by volume) are shown for three particulate radii. As particulate size decreases the magnetic relaxation frequency is predicted to increase. Thus, size scaling should be a parameter for changing the composite magnetic resonance at a fixed volume fraction. This prediction is consistent with measured data of Chap. 12.

FIGURE 6.2 MGT-predicted effective permeability of an epoxy-Fe composite at 28 percent volume fraction as a function of Fe particulate radius.

6.2.3 Percolation and Dielectric Dispersion The next discussion on constitutive parameter frequency dispersion addresses the topic of percolation. In mixtures of semiconductors and/or conductors and dielectrics, the percolation refers to establishment of characteristic lengths and density of paths that allow charge transport through networks of touching conducting constituents. Percolating networks within the composite modify the frequency dispersive characteristics of composite permittivity. Thus, it may offer a means to match magnetic dispersions. Though this text only addresses electromagnetic parameters, the mathematical models that form the bases for predicting material parameters have been applied to diverse problems. These may be models that require description of material(s), fluids, or particles that move or flow in channels. Such diverse problems as predicting the path that a forest fire will take is described by percolation. The expected path depends strongly on the local density of fuel. Oil pools or flows through fractured rocks as controlled by fractures and voids. Of course, there is also the way water flows through the porous particle mix in the filter of your drip coffee maker [4]. The concepts of percolation were introduced in the late 1950s [5] to describe electronic propagation in random media, hopping conduction in periodic lattices, and electronic

scattering from material defects. Seminal work was published by Kirckpatrick [6, 7]. The papers served as sustenance for many of the publications in the late 1970s and throughout the 1980s and 1990s. Numerical simulations of percolation dominated in the 1990s and post1990s. Connected path(s) coalesce at some surface or volume fraction of material to form conduits that allow transport of fluid, particle, fire, etc. The fraction at which continuity is established is a critical point or percolation threshold. When electrical conduction is considered percolation produces an electrical path that extends throughout a material. The DC conductivity is observed. The transition from nonconductor to conductor is a phase transition that is second order. This material phase change is familiar to the reader who has observed condensation of gases/vapors/liquids that demonstrate “critical opalescence.” The optical transparency becomes negligible for tiny changes in temperature, T, pressure, P, or concentration, ƒ, of a particulate. The change in transparency due to index or permittivity varies as (T – Tc)–α, (P – Pc)–β, or (ƒ – ƒc)–τ. Both conducting and dielectric composite phases are observed at the critical fraction, and frequency dispersive permittivity [or AC conductivity σ(ƒ) ] exhibits a mixed phase behavior. The effective conductivity/permittivity of the composite varies with volume fraction, p, just as opalescence near the critical point of the dielectric. The dielectric phase transition varies as ε ∝ εd (p – pc)– s below pc and ε ∝ εc (p – pc)t above pc, where εc, εd are conductor and dielectric permittivity, respectively. The electromagnetic properties of the mixture change continuously but with diverging derivative as relative fraction of the two constituents are increased or decreased about the critical point. The exponents t,s prove to be a strong function of the dimensionality of the composite and weaker function of the composite micro geometry. The critical exponents are reflected in the composite permittivity frequency dispersion. Therefore, physically thin (two dimensional) composites and physically thick (three dimensional) composites will show different dispersions. This fact implies that controlling thickness (for a fixed particulate size) gives access to another source of composite permittivity frequency dispersions, one that is not observed in homogenous materials. Composite dimensionality is controlled by the ratio of average conducting particle size to composite thickness, a0/T. The particulate size also determines the electrical conduction correlation length and below the critical point the characteristic distance over which current may flow varies as ζ ∼ a0|p – pc|–v. In practice, a0 is often “representative” of a size distribution. The physics of composite frequency dispersive properties can also be discussed in terms of the correlation length. When ζ ∼ a0, volume fractions are not near pc; conductive regions are isolated; MGT should model composite permittivity. The composite’s dispersive properties are dominated by the matrix material. For ζ >> a0, but volume fraction just below the critical point, the BEMT should be the best EMT and dispersion is mixed phase. ζ extends over distances comparable to the electrical wavelength near the critical point and most EMTs become inaccurate. The composite may display resonant or Lorentzian-like dispersion. ζ extends throughout the composite above the critical value and dispersion is largely that of the conducting component. Since the phase transition is fundamentally statistical in nature, a

numerically precise prediction is not possible. However, semi-quantitative predictions can be obtained by taking advantage of functional scaling that is often applied to material systems near a critical point. The physics of second-order phase transitions is described by Landau [8] and successfully applied to conductor-dielectric composites by Clerc [2, 9], Zabel-Stroud [10], Koss, [11], Brouers [12], and Neimark [13]. When percolation is treated in light of its phase transition behavior, semi-quantitative models of the composite frequency dispersive permittivity and/or conductivity can be developed as functions of frequency; properties of the constituents; constituent particle size; composite cross section; and composite thickness or dimensionality (thin two-dimensional or thick three-dimensional materials). Models based on phase transition phenomenology must have physically continuous properties across the critical point and should demonstrate agreement with BEMT above and below the critical point. The BEMT is used to identify symmetries and general properties which must be obeyed by scaling laws. Recall that the basic BEMT for a binary composite effective permittivity, εe, with matrix ε2 and inclusion εp is

As discussed previously, the equation is symmetric on interchange of volume fraction and permittivity, εe(ƒ1,εp,1 – ƒ1,ε2) ≡ εe(1 – ƒ1,ε2,ƒ1,εp), and the solutions to the equation can be reduced to dependence on three dimensionless variables ƒ1, ze2 = εe/ε2,z12 = εp/ε2. A scaling function treatment of a composite must show like symmetries and meet the same physical conditions. Like many publications on percolation, e.g., [2, 9, 14], the text initially shifts variables to frequency dispersive conductivity, σe(ƒ),σp(ƒ), and σ2(ƒ). This can be transformed back to permittivity at any time. The reader may exchange permittivity for conductivity at any time with the transform without the loss of generality. σdc is DC conductivity. The inclusion conductivity σp is associated with frequency ω1. The matrix conductivity, σ2, is identified with a frequency ω0. The following scaling discussion parallels the arguments of Ref. [2]. The composite is assumed to have cross-sectional dimensions much larger than a free space wavelength (i.e., infinite and planar). In this case the general scaling form near the percolation threshold can be written as

Φ is a scaling function and may have different functional forms above (+) and below (–)

the critical point but is continuous at pc. The parameters s,t are functions of the composite dimension or thickness [2, 9, 13]. The numerical values have been calculated and/or determined from measurement as near s = t = 1.299 ± 0.002 in two dimensions and s = 0.73 ± 0.02, t = 1.9 ± 0.01 in three dimensions. The scaling function is unique at a given frequency and is symmetric for variations of p about pc· Explicit forms of the functional are known for extremes of volume fraction. Well below percolation the functional matches to the BEMT. Below, but approaching pc, σ ∝ σ2(p – pc)–s. Above percolation σ ∝ σp(p – pc)t and the functional is continuous for all values of p. An alternative form of the scaling function, meeting the same physical conditions on volume fraction, is obtained by making a variable change in the argument of the functional. –1 –1 Note that the argument is a product and h(ω)|p – pc|(s+t) = h(ω)|Δp(s+t) can be changed to –1

h(ω)(s+t) Δp. The same complex contour for variations in h or Δp is followed. By applying this scale change the alternate conductivity and frequency-dependent form is obtained, i.e.

where Y represents an alternative form of the scaling function. The scaling forms can be used to illustrate frequency expectations for the permittivity of the two-component composite. The permittivity is reintroduced for the dielectric matrix. Also assume that the particulate is well described by a DC conductivity. This should be good through the microwave region. In this case h may be rewritten in terms of frequencies ω0, ω1. Substitution in Eq. (6.3) facilitates investigation of frequency dependence via Eq. (6.4).

First note that Eq. (6.4) implies there will be different frequency dependences in two and three dimensions since the power to which frequency is raised varies with the parameters s,t. Irrespective of the dimension, the equation divides the frequency axis into roughly three frequency ranges. At very low frequencies, ω < ω0 < ω1, σe is mostly frequency independent and has a fractional magnitude of the conductor conductivity. As ω approaches and exceeds ω0 dispersion reappears and is mixed in ω0, ω1, i.e., and the functional argument

. The

third region of functionality appears for ω >> ω1 > ω0 where the lead coefficient varies as (ω/ ω1)t/(s+t) and the argument of Y varies as (ω/ω1)1/(s+t). It has been repeatedly said that dispersion depends on dimensionality. That dimensionality is a function of “relative thickness scale” of the composite where the thickness is scaled to the size of a conducting element. The

next discussion includes dependence on thickness scale. A conductive coating that is composed of small conductor particulates supplies an example of how thickness might change the composite model. A two-dimensional surface would be one whose thickness is one conducting particulate diameter thick; e.g., for spherical particulates lying on a flat surface, the scaling dimension would be the diameter of a particulate and percolation would occur when the surface area covered by the particulates is above approximately 50 percent of the available surface. In a composite that is many spherical particulate diameters thick, the volume concentration would be about 33 percent (i.e., 1/d) to achieve percolation. Thus, the scaling function is expected to be different for surfaces and volumes since the difference |p – pc| appears in all the equations. Sketches that illustrate the two- and three-dimensional cases are illustrated in Fig. 6.3. Figure 6.4 shows an actual nanostructured film with surface fraction above the threshold. However, it must be said that high surface fractions do not assure electrical continuity for some pathological cases. Figure 6.5 shows an atomic force microscope (AFM) image of an aluminum film (about 99 percent fraction) on polyester. During manufacture the film was mechanically stressed, fracturing the Al film when the underlying polyester film was elongated. Though the film has Al of about 100 percent on the surface, DC conductivity is negligible.

FIGURE 6.3 The left figure illustrates how percolation might occur on a two-dimensional flat surface having conducting spheres on the surface. On the right percolation is occurring within the material volume, i.e., 3D.

FIGURE 6.4 SEM of a 2D conducting film of silver deposited on silica. Approximately 80 percent of the surface is covered.

FIGURE 6.5 AFM image of a 99 percent Al surface coat but low conductivity. Fracture lines (about 10-μm separations) increase surface impedance from approximately 0.2 to 800 ohms/square.

Equation (6.4) describes scaling at the two- or three-dimensional limits. The correct values of parameters s,t and pc are known for these extremes. However, bridging the gap with corrected parameters for all scaled thicknesses between two and three dimensions is a challenge. The determination of s,t as a function of scaled thickness must be performed numerically on a case by case basis. However, some working models can be produced that allow the investigation of frequency dispersion for composites of different scaled thicknesses, if s,t are assumed to be some average fixed value. Numerical calculations have been performed that indicate the working models can be good approximations to fit a continuous variation in the critical fraction for different composite thicknesses [13, 15–17]. The following discussions reflect these assumptions on parameters

and apply approximate analytical forms that predict changes in pc with scaled thickness. The model will be inserted in Eq. (5.4) but s,t will remain parameters to be determined. The insertion will suggest new frequency dispersive possibilities for the composite. Clerc and Neimark [2, 13] proposed models for changes in percolation threshold from two to three dimensions. In this text, the Neimark model is applied. In essence, Neimark envisioned a finite composite of thickness T as made from cubes of dimension ap. Conducting cubes have volume fraction p. As ap ⇒ T the percolation threshold must change from its three- to two-dimensional value. The working numbers used for particulates will be pc3 = 0.33 and pc2 = 0.59. The percolation scales with the thickness as

where the factor d is the dimensionality of the material, i.e., 2 or 3, and the product dt/(s + t) = v3–1 is the exponent describing the growth of the conducting correlation length in three dimensions. The absolute value of the volume fraction and threshold, |p — pc|, as a function of scale thickness, Ts, is

and δ23 = 0.26 for our choice of thresholds. The scaling function for frequency dispersive conductivity was Eq. (6.4), and when the percolation scaling is included it becomes

by changing

to

and extracting

equation is reformulated to show a third critical frequency

from the absolute sign the

i.e.

This is a complete thickness-dependent model. The thickness dependence extends the critical region about the critical point and identifies a third critical frequency, (ωTs), which is found in the argument of the scale function Y. That frequency is a function of thickness, conductivity, and matrix dielectric constant, (ap/T)tdσp/(2πε0εr). Even for thin materials, this frequency may be at a very short wavelength (infrared or visible). However, by choosing the scale factor and using a reduced conductivity inclusion such as graphene ω(Ts) can be adjusted to appear in the millimeter wave or microwave band. Figure 6.6 shows a measurement of a composite containing semiconducting spheres, diameter near 50 μm, and conductivity near carbon. Though not conclusive the figure shows measured data that does show a frequency inflection near that expected for a prediction of ω(Ts).

FIGURE 6.6 Measured data for a composite with a p /T ~ 20. A change in slope appears near the calculated value for ω(Ts).

6.2.4 Particulate Shape Impacts In the previous section, flat and cubic particulates have been assumed in estimating the percolation thresholds in two and three dimensions. However, the percolation threshold is a function of particulate shape and therefore parameters will require adjustment. In Chap. 5, the impact of particulate shape anisotropy (i.e., long thin fibers) has been included in the effective medium model. The polarizability, P, has been derived from particle shape and P has been included in the BEMT rather than dimensionality, d. As aspect increases, P decreases. This has the effect or reducing effective percolation threshold. The impact of aspect ratio has been studied by many authors. Two papers coauthored by I. Balberg and N. Binenbaum fixed the numerical relationships between particulate aspect and percolation [16, 17] and the second paper of Ref. [17] by McLachlan et al. demonstrated the strong aspect dependence in measurement of nanotube composites. Figure 6.7 is a replica of Fig. 1 of Ref. [16]. The figure shows the critical concentration in three dimensions (i.e., percolation threshold) for capped cylinders of length L that are capped at the ends with a hemisphere of radius r. If L is near r the particles are nearly spherical. If L r the particles are high aspect, i.e., fibers. In the figure, radius varies from much less than 1 (left side, high aspect) to greater than 1 (spherical) for a fixed value of L = 0.15. There are three functional variations in radius observed in the plot: cubic on the right, quadratic in the central region, and linear at the far left. Balberg and Binenbaum concluded that excluded area or volume, rather than real area or volume, should be applied to scale from the disk or sphere percolation threshold and thereby account for particulate aspect ratio. Excluded volume versus real volume is illustrated in Fig. 6.8.

FIGURE 6.7 The dependence of the critical concentration Nc of capped cylinders on their radius in an isotropic system of sticks.

FIGURE 6.8 Illustration of actual and approximate excluded volumes for very high aspect particulates.

Balberg’s, Binenbaum’s, and McLachlan’s publications become especially relevant when one considers technology advances (i.e., in 2012) such as emergence of flat and touch screen

display technology. The transparent conducting films used in these screens are beginning to leverage carbon nanotubes, very high aspect particulates, in their production. Active touch screens are used in tablets, smart phones and laptop computers. By using the B&B analysis, the engineer can design films with high DC conductivity using nanotube concentrations in the 0.1 to 1 percent volume fraction range while simultaneously minimizing optical blockage due to the tubes. Films with 80 percent or more optical transparency and surface impedances of a few hundred ohms per square are produced. The technology may someday replace use of indium-tin-oxide (ITO) for many active displays and flexible conducting materials. An excellent review of film theory and manufacture is found in the Materials Research Society Bulletin, October 2011 and December 2012. As the film technology advances flake-like graphene particulates may replace nanotubes in 5 to 10 years. In summary, the analysis has shown percolating conductor/dielectric composite can introduce frequency dispersion as a design parameter in composites for the effective conductivity, σe, or the effective permittivity, εe(ω, p, pc) = σe(ω, p, pc)/jω. The dispersion is a function of three characteristic critical frequencies, ω0,ωl , and ω(Ts). In general ω1 ω(Ts) > ω0. For good conducting particulates, ω1 lies in the terahertz spectrum. However, for poor conductors (conductivities near graphite and below) all three frequencies should appear in the RF spectra at millimeter wave and below. The analyses suggest that by changing the dielectric matrix and/or the conducting particulate (geometry or conductivity) frequency dependence of permittivity may be controlled in composites. By changing particulate size and composite thickness, ω(Ts), the critical frequency can be positioned in the microwave or millimeter wave electromagnetic spectrum.

6.2.5 Implications of Percolation to Measurement A major goal of the book is to explore electromagnetic material measurement techniques and measured constitutive parameters of electromagnetic composites. Therefore, a short investigation of percolation’s implication for electromagnetic measurements is summarized prior to discussions on modeling of magnetic composites. Additional detail and example data will be supplied in discussion of measurement techniques and data in Chaps. 8 to 12. The physics of percolation places restrictions on measurement technique and measurement procedures. Therefore, insights on measurement impact, derived from analysis, have major benefits to the experimentalist. When comparisons of RF, microwave, and/or millimeter data from a single composite type are made, but for different measurement equipment, data discrepancies may appear. Data from free space transmission or reflection measurements might be compared to resonant cavity measurements of frequency and loss, or transmission line measurements taken using coax and/or waveguide. Discrepancies in composite parameters may be especially evident for materials when a conducting or semiconducting inclusion concentration is near the critical concentration. A second, often unexpected impact, is observation of high levels of bistatic and/or diffuse scattering from planar samples of composites when they are measured with free space techniques. In contrast bistatic scattering from homogeneous, isotropic materials (such as Rexolite, Lexan, or metallic surfaces) shows extremely low levels. Their low scattering level is a reflection of the single mode, ray

propagation nature of reflection from and transmission through a homogeneous material. Constitutive parameter measurements of percolating composites exhibit behaviors which were implied in scalability discussions. A fundamental error in experimental design is the assumption that effective electromagnetic properties will be the same irrespective of measurement procedure. Fixed values for effective parameters apply only when the electromagnetic wavelength within the material, λM, is much greater than any characteristic dimension of an inclusion. When such inhomogeneities are present, measured constitutive values will depend on the dimensionality of the test fixture, sample (sample thickness and/or sample-test fixture cross-sectional dimensions), electromagnetic wavelength, and test equipment and procedures. The inhomogeneity, electrical correlation length, will increase as the concentration of inclusions increases, i.e., ap is the inclusion size; p, pc are the inclusion volumetric and critical concentrations, and under the square root are media constitutive parameters surrounding ap. All are divided by the free space wavelength. The parameter v is a characteristic critical exponent (about 4/3 in 2D and 0.88 in 3D). As discussed previously, the critical threshold scales with thickness and for finite cross-sectional materials, i.e., |pc – ps|∝L–1/v. The critical threshold for a finite sample, ps, is used in calculation of correlation length. The lengths and scale require that ξ = ap |p – pc|–v L. In AC measurements this relation must be replaced by a threefold condition on correlation length, wavelength, and cross-sectional dimension L λ ξ. The first inequality is required to allow a reflection and transmission coefficient of the material to be defined and the second to meet criteria for effective electromagnetic constitutive parameters. Transmission line measurements have sought to meet the first criteria while using small samples (typically less than λ0/2 in size) within a metallic transmission line of circular or rectangular cross section. The metallic fixture geometry is chosen to meet electrical boundary conditions for electrical imaging of the sample within the transmission line walls and thus emulate plane wave propagation through and reflection from the material. However, this is only true for materials which perfectly fit the transmission line and have near perfect electrical contact at the sample transmission line interface. Analytical procedures are available to correct for imperfect fit of nonconducting materials and conducting fillers have been used to successfully fill small gaps at transmission line walls and nonconductive material sides. However, neither of these approaches can correctly reproduce the electrical continuity which is within a composite that contains conducting inclusions. Comparative measurements of transmission coefficient for a conductor-dielectric composite, composed of about 50-μm particulates, at and just above the critical concentration are shown in Fig. 6.9. Data are for a 7-mm sample in coaxial transmission line and a 30 cm × 30 cm sample measured using a free space focused beam that radiates a three to five λ0 size spot on the sample. The measured data discrepancies are typical. Note the transmission line measurement largely misses the conduction which is evidenced by small free space transmission coefficients. This is due to imperfect contact with the transmission line walls.

FIGURE 6.9 Transmission measurement of a sample at and 5 percent above percolation using 7-mm transmission line and free space measurement.

Figure 6.10 illustrates the importance of scaled thickness when measuring composites. The data are DC and AC measurements of relative conductivity for a series of conductordielectric mixtures. Again, about 50-μm conducting particulates are used. Various concentrations of conductor and different thicknesses were measured. The DC data on the left graph shows the characteristic state change as particle concentrations exceed percolation. The DC measurement is for a thick sample. The right figure shows measured surface impedance near 10 GHz for fixed concentration but various thicknesses scaled to the 50-μm particulate. The concentration is fixed below but near the critical point for thick materials.

FIGURE 6.10 The graph on the left shows measured dielectric-conductor phase transition for a thick sample; the graph on the right illustrates thickness-dependent percolation at fixed volume fraction.

Composites that contain high concentrations of inclusions may show significant levels of diffuse scatter. The level of scatter is also correlated with electrical percolation when conducting inclusions are involved. Again the correlation length will be a function of concentration, critical volume fraction, and therefore shape. A free space focused beam measurement is reviewed in Chap. 9. Reference [18] (composed by a colleague of this author) addresses the measurement system in detail. Reference [18] demonstrates the technique as applied to measure diffuse scattering issues in composites. The technique supplies radiation to an area on the sample that is near circular in cross section with diameter of about 3 free space wavelengths or greater at any frequency in the measured spectrum. The measurement satisfies correlation length limitations if the sample size is L ξ (f, p, pc, ap) and 3λ0 ξ(f, p, pc, ap). Figures 6.11 and 6.12 are comparisons of measured diffuse scattering examples from a homogeneous dielectric coated conducting surface and two-dimensional surfaces but composed of conducting fiber and patch surfaces, below, at, and above their percolation threshold. Diffuse scatter is identified with scattering angles greater than about 20 degrees. Below 20 degrees scattering is largely direct backscatter.

FIGURE 6.11 Measured diffuse scatter from a homogeneous polymer on a metallic surface (left) and a conducting fiber mat with fiber concentration above the percolation threshold (right). The color scale indicates scattering magnitude in decibel (dB).

FIGURE 6.12 Relative bistatic scattering amplitude for a 2D surface of small conducting patches (size > 1. As previously observed, all forms of MGT approach a limit line in this case.

Since MGT assumes that the particulates are separated and weakly interact, MGT cannot describe any magnetic phase transition; however, MGT does predict a frequency shift in composites. Measurement shows that this form of MGT performs well in describing certain magnetic composites, particularly multidomain, nonconducting magnetic particles. Many conducting magnetic materials (such as Fe or Ni) have large permeability and MGT may fail since μ1 = 1 + χ1 >> 1. In addition conductivity requires that χ1 be replaced by χ1 = (μ1 – 1)F(ϕ). The F(ϕ) is defined in Chap. 5. This replacement would impose severe restraints on the inversion Eq. (6.8). The magnetic composite will display frequency dispersions similar to any other magnetic material. As described in Chap. 3, the frequency dispersion of a material should be described by a Landau-Lifshitz-Gilbert (LLG) like equation:

where Ms, Hk, Nx, Ny, γ, and α are the particle magnetization, internal anisotropy field, x shape factor, y shape factor, gyromagnetic ratio, and loss factor. Note H0 is the sum of any external DC field and also the internal crystalline anisotropy field, Hk. If the external DC field is zero, one may factor ωmγHk from the numerator and (γ Hk)2 from the denominator and produce a composite susceptibility that is in terms of its DC value, χDC = ωm/(γ Hk)2:

This equation simplifies greatly when the bulk form is considered and it can be written in terms of parameters of the magnetic particulate within the composite. The bulk form for frequency dependence of a material is obtained for equal shape factors, i.e.,

This is further simplified by assuming that one is near resonance and the imaginary loss term in the numerator is small. The composite should have a similar frequency dependence

and should assume a Lorentzian like form with ωrc, ωdc resonant and relaxation frequencies of the composite:

If the MGT solution for χc(0) is expressed in terms of a particle, χ1(0) new expressions for the composite resonant and relaxation frequency are derived,

Here, χ1(ω) also has a Lorentzian frequency dependence and the equations are specialized to spherical particulates. A direct substitution of the upper equation into the lower of Eq. (6.13) yields a frequency-dependent equation for composite susceptibility and allows for a calculation of the composite ωrc, ωdc.

This equation takes the Lorentzian form for the composite [Eq. (6.12)] if the following identifications are made [30, 31]:

For any volume fraction below unity, the resonant frequency of the composite will be higher than that of the particulate and the relaxation frequency increases almost linearly with

volume fraction. Overall, this suggests composites with higher and broader resonances than the particulate used to make them. This is observed in measurements of Chap. 12. Note that a spherical particle shape is assumed above. Shape can be partially added by applying the shape corrected form Eq. (6.9) in the shape-dependent MGT equation

where L is demagnetization. Another indicator of the composite model limitations parallels Snoek’s law for bulk ferrites or spherical ferrite particulates, i.e., (μDC – 1) ƒr = γ 4πMs, in Chap. 3. The law changes for ferrite particles that are not spherical, i.e., for thin magnetic films, . Snoek’s law serves as an upper limit for the product of magnetic resonant frequency and DC magnetic susceptibility. Archer, Rozanov, and Lagarkov [24, 30–32] have derived corresponding equivalents for magnetic composites. These require an integral over the frequency spectrum of the imaginary permeability, i.e., . In this relation p, Ms, μi, ƒ, and k are magnetic particle volume fraction, magnetization of particulate bulk, imaginary part of the composite permeability, frequency, and scaling constant. The integral form acts as the composite limiter. Chapter 3 introduced micromagnetic simulations as a means to predict magnetic properties. Those simulations are near truth and as might be expected may not produce results that are predicted by either Maxwell-Garnett (MGT) or the Bruggeman or BEMT effective media equations. Numerical simulations found that the best fits to numerical simulations required that the MGT equation to be modified to allow different demagnetization factors for different fill fraction regimes. Those micromagnetic simulations were in a two-dimensional system. In the simulations, at low fill fractions, the magnetic system acted like a twodimensional plane of dipoles, and the demagnetization factors scaled as

At high fill fraction, the particles had strong interactions which changed the effective demagnetization factors:

Note that this scaling law is an ansatz that agrees with the simulation data but a physical derivation was not found. 6.3.2.2 BEMT for Magnetic Composites. The BEMT is a robust and common model that is

based on the Chap. 5 fundamental scattering descriptions for obtaining composite effective constitutive parameters. Particulate and matrix carry equal weight in the theory formulation and therefore the model can describe a change of state, e.g., electrical percolation or other systems where constituents strongly interact. Equation (6.18) is a magnetic BEMT where P is associated with demagnetization of an ellipsoidal magnetic particulate and thus the particle shape and size. Unlike treatments of semiconductor or conductor dielectric matrices, the parameter is not associated with composite dimensionality and any percolation threshold in the medium. In this case, P is physically associated with the alignment of the particulate major and minor magnetic axes with the polarization of the incident magnetic field and the resulting coupling strength.

Any critical thresholds and demagnetization factors are descriptive of particulate geometry in the composite. However, any threshold due to particulate coupling may be quite different from the particle demagnetization factor and in some cases the particle geometry may prove so complex that the demagnetization factor cannot be easily specified. Therefore, except in the case of spherical particles (where demagnetization is 1/3) P is treated as a free parameter that is fit to systems, i.e., multidomain particles or single-domain particulates. The need for the “fit” will become clear when measured data are presented in the following sections. The solution of the BEMT involves a quadratic term and therefore the MGT simple relationships between particulate and composite resonant and relaxation frequencies are not reproduced with BEMT. A combined approach was elucidated by Mattei and Le Floc’h [25]. The BEMT does not suffer from the loss of information observed with MGT and its limit line when using particulates with very large permeability (or permittivity). Thus, one might find the effective DC permeability with the BEMT. This value is then used as the guide to modify the inclusion demagnetization factors. The demagnetization fields of an inclusion are heavily dependent on the nature of the magnetic material surrounding the inclusion; if the inclusion is surrounded by an infinite amount of identical magnetic material, then there is no true boundary between the inclusion and the composite, and the demagnetization fields are zero. Reciprocity describes the weakening of the demagnetization fields as the effective permeability of the composite approaches the inclusion value:

where Ni0 is the geometrical demagnetization factors of the particulate along the ith axis, is the effective demagnetization factor, and R = μ1 χe/μe χ1 is the reciprocity. The effective demagnetization factors are then used as the demagnetization factors in an LLG equation, which is then applied to calculate the effective dynamic susceptibility.

6.3.3 Magnetic Properties of Multiphase Materials with Three and Four Phases The MGT and BEMT models were developed to describe binary systems. However, each of them can be qualitatively extended to describe multiphase composites by applying Chap. 5 CPA as summed over multiple scattering types. For clarity, the multiphase EMTs presented here describe static systems. The generalized MGT is

and the BEMT can be generalized to

where

indicates summation over fill fraction of inclusions.

However, many assumptions in deriving EMTs are counter to multiphase systems. The multiphase MGT model as presented is physically incorrect except in cases where different particulates exist in a host matrix and are maintained far from each other. Second, the inclusions have identical geometries to satisfy the requirement that all demagnetization factors be equal. The BEMT also has physicality problems, e.g., what does the critical fraction mean in multiphase composites. For example, if there are two inclusion types, with similar properties, does one define a phase change when one reaches a critical value, or does the sum of volume fractions exceed a critical value? One geometry which has been studied in experiment uses small ferrite particles that cluster around larger metallic magnetic particulates [33, 34] and also the inverse, particulates round larger ferrite particles [35]. The last case is reviewed at the end of this chapter. One must ask which particle naturally clusters around another? The BEMT does not contain physics that describes how one particulate modifies the magnetic property of the another. There are also mathematical assumptions since a solution of more than two phases involves a solution of polynomials of order R where R is the number of distinct particle phases. Two-, three-, and four-component composites have analytical solutions. A multiphase BEMT may not numerically converge on one physically meaningful root especially when there are unknown physical complexities in multiphases. As described below, a working multiphase EMT might be practically treated by sequentially choosing matrix and particulate. For example, a dual magnetic component composite plus a neutral host matrix is modeled by sequentially adding the particulate types where volume fractions are scaled appropriately.

6.3.4 Magnetic Case Study: Examples and Lessons Learned in Measured

Magnetic Composites as Compared to their Magnetic EMT Models At this point the text has described the major EMT models used in composite analysis. The goal of this section is to apply those models and test them in a real-world example. Thereby, a set of guidelines and lessons learned will be established. In order to maximize the lesson learned content, the measured data will address a range of magnetic composite types, multidomain and single-domain magnetic particulates, and particle size scales from tens of nanometers to tens of micrometers. Particulates are treated as spherical in these studies. Conducting magnetic Fe particulate composites may evidence impact of electrical conductivity via skin depth with a resulting coupling of dispersive permittivity, conductivity, and apparent magnetic susceptibility. Calculations that include skin depth found that at diameters ≤2 μm, skin depth impacts could largely be ignored. Therefore, a reader might use measured data from Fe composites with particulates below 2 μm to establish baseline data for Fe particulate magnetic parameters. However, the forthcoming measurements demonstrate that a DC relative permeability of about 50 to 70 is a better fit to data than the historical value found in a CRC Handbook of Chemistry and Physics (CRC Press, NYNY) table, i.e., about 200. Therefore, excluded volume may still be having impact in the measurements. Predictions of EMT for ferrite composites are based on measured permeability and permittivity data from fully dense Ni32Zn48Cu12Fe2O4 ferrite. Predictions for the MGT and BEMT are compared to measured data in these nonconducting composites with the goal to test MGT and BEMT for nonconductors with large DC permeability (about 850). Measurements or a second ferrite NiMn.02Fe1.9O4 were applied for low to modest DC permeability materials (μDC ∼ 30). However, that data is for partially dense ferrite samples at and above 75 percent of full density. Particulate size is also a parameter that can modify composite permeability. This parameter is illustrated by using magnetite, Fe3O4. Particulates were acquired with diameters as small as about 10 nm and as large as 5 μm. The magnetite size scale encompassed near single to multidomain characteristics. The magnetite permeability and effective magnetization were found to decrease with size while its effective anisotropy increased. Thus, large, 5 μm, size particulates had DC permeability near bulk, about 100, while nanoscale particles had DC permeability near 10. Both MGT and BEMT were applied by taking measured composite permeability and attempting to invert the measurements to calculate particulate permeability. Additional data on nanoparticles are presented in Chap. 12. Three phase (epoxy–Fe–Ni32Zn48Cu12Fe2O4) composites were also prepared and measured. Those composites challenge the application of EMT models and may serve as a guide to improve models. The three-phase composites were found to have large resonant bandwidths and three-phase results are qualitatively similar to recent publications [33, 34]. Selected epoxy–Fe–Ni32Zn48Cu12Fe2O4 data were published in Ref. [35]. Rather than showing the many measurements of frequency dispersive permeability, the data from Chap. 12 are used in the analysis. Chapter 12 tabulates electrical and magnetic parameters for the composites. The permittivity was fit to ε(ƒ) = ε DC + ε1(ƒ)x + εh(1 – A(ƒ –

ƒ1)2 + 2iBƒ)–1 with εDC, ε1, x, εh, A, f1, B as fitting parameters. All composite magnetic susceptibility was fit to

The parameters are γ = 2.8 MHz/Oe and γ Hk = ƒr, 2α / γ Hk = ƒd, resonant and relaxation frequencies, respectively. The parameters χDC, α, β, γ Hk are free fitting parameters and f is frequency. Ideally, β and ƒd should be related by β/ƒd = 0.5. In analysis, this ideal ratio was used as a guide for “goodness of fit” with a variance of 20 percent being the maximum allowed before data were subject to further analysis. In most of the composites discussed in this section, permittivity was not frequency dispersive, thus a single complex number is quoted. Figure 6.13 shows example fits for a composite and fully dense ferrite. Overall, the analysis of the composites shows that volume fraction scaling of composite resonant and relaxation frequencies, for composites containing particulates with very large DC permeability, can significantly vary from models [30]. It is hoped that the large measured data constellation that is presented in Chap. 12 will facilitate development of improved models.

FIGURE 6.13 Measured and Lorentzian fits for one composite and one dense ferrites are shown.

Fe particulates were obtained commercially for preparation of the Fe–epoxy composites. Surface oxides were observed and thus the Fe particulate conductivity was reduced. A DC measurement of the compacted powder was consistent with that assumption. Oxide coatings appeared to be on the order of 10 nm. Since particle diameters were about 1 μm, the oxide coating would have a negligible volumetric contribution to magnetic susceptibility. However, the conductivity reduction (approximately a factor of 100) slowed the onset of long-range electrical conductivity and allowed large volume fractions for the composites. Figure 6.14 shows a selected set of measured frequency dispersions and a composite micrograph.

FIGURE 6.14 Measured real permeability versus frequency for Fe composites and typical particle micrograph.

The Ni32Zn48Cu12Fe2O4 ferrite was obtained in fully dense, thin-film platelets form (1 × 1 × 0.2 cm). Platelets were mechanically ground to a 30 to H0 includes any external DC field and also the internal crystalline anisotropy field, Hk 80 μm powder form and composites were made from the powder and epoxy. Fully dense samples of Ni32Zn48Cu12Fe2O4 were also measured. Figure 6.15 shows typical measured data and micrograph. The two-phase Fe or ferrite epoxies and three-phase Fe–ferrite epoxy particulate composites were blended using a high-shear mixer for low volume fractions and by hand at higher (>20 percent) fractions. Three-phase Fe–ferrite–epoxy composites had Fe and ferrite in equal fractions. The data and micrograph are shown in Fig. 6.16.

FIGURE 6.15 Measured real permeability versus frequency for NiZnCu ferrite composites and typical particle micrograph.

FIGURE 6.16 Measured real permeability versus frequency for Fe–NiZnCu ferrite composites and typical particle micrograph.

Sub 100 nm magnetite particulates were synthesized using chemical coprecipitation and blended with epoxy polymer using a high-shear blade at 760 rpm for 10 min. Sonification eliminated air bubbles in the uncured nanocomposite blends. Five-micrometer magnetite particulate was commercially acquired and similarly blended. Published data from Ref. [36] on NiMn ferrite samples containing voids (about 0 to 25 percent voids) were subject to analysis as were composite data for 150-nm magnetite from Ref. [37]. Chapter 12 and Refs. [38] and [39] discuss DC magnetic measurements which were used as a separate check on magnetization and DC permeability calculations. Table 6.2 summarizes the measurements that were performed. Measurement techniques are expanded on in Chaps. 8 through 12. TABLE 6.2 Samples Tested and Measurement Technique

Both BEMT and MGT were used to predict effective permittivity and/or permeability and these are compared to measurement. Recall that in the BEMT formulation for spherical particles in a thick composite, dimensionality of the system determined the system critical threshold and change of state. The threshold is easily observed in measured data by a sharp slope change in plots of volume fraction versus composite permeability. The magnetic phase change from isolated magnetic particles to correlated magnetic moments does not require the physical contact but coupling will require small separation and depends upon the magnetic moment and depolarization of the particle. Therefore, the association of a single unique volume fraction with phase change in all magnetic systems is not expected to be correct. Micromagnetic calculations for single and multidomain ferromagnetic particles indicate a change occurs closer to d – 1 ≅ 0.5 [27–29] and this is also apparent in the following measured data. The previously described scaling relations between particle and composite susceptibility, particle and composite resonant, and relaxation frequencies are good indicators of an appropriate choice of magnetic EMT. If MGT applies, the composite dispersion is Lorentzian in shape; however, if the BEMT is appropriate a Lorentzian shape is not expected, nor should simple scaled fits be good. The difficulty in determining a good EMT for magnetics is apparent in attempts to calculate values for χp from measured data of the composite. Those calculations can vary by orders of magnitude. For MGT and spherical particulates with large χp, the composite susceptibility converges and never exceeds χc = 3ƒV {(1 – ƒV)}–1 but measurements show the limit to be incorrect. Therefore, the application of MGT for high permeabilities is suspect. This also suggests that MGT cannot be used as an inverse solver to calculate particulate susceptibility from measured composite data. Therefore, inference of particulate DC susceptibility from composite data can be a test of the BEMT. The BEMT inversion formulae is χp = (3χc2 + 3χc)/(3ƒ + 3 ƒχc – χc). It was applied in the case of Fe and magnetite composites with an assumption of spherical particles. The composite DC χc was taken from the zero frequency value in the Lorentzian fit to measured composite susceptibility. Table 6.3 shows the inversion calculated values for particulate DC

susceptibilities, resonance frequency, and relaxation frequency. The resonant and relaxation frequencies are derived from MGT scales that relate resonant and relaxation frequencies, ƒr, ƒd, to those of the composite, i.e., TABLE 6.3 Fitted Particulate Parameters for 1.46-μm Fe and Various Diameters of Magnetite

The subscript c indicates the composite and the subscript p indicates the particle. Note from this table that the calculated magnetite susceptibility for Fe and magnetite are smaller than bulk values which are approximately 200 and 90, respectively. The magnetite reductions are consistent with reduced magnetization and increased anisotropy in small magnetite spheres. This is especially evident for the nanoscale magnetite spheres which can have a magnetic dead layer on their surface which comprises a significant percentage of their total volume [38]. The lower calculated value for iron permeability may reflect the reduction in volume due to skin depth. The constitutive parameters ƒr and ƒd for the NiZnCu and NiMn ferrite and DC permeability were measured directly from the fully dense ferrite. These were applied in a forward calculation using the BEMT and MGT to predict composite constitutive parameters and frequency scaling in the composites. These were then compared to composite measurements. MGT and BEMT DC permeability predictions in two-phase composites are shown in Figs. 6.17 to 6.20. The BEMT under predicts at low volume fractions. To fit measured data the assumed DC particulate susceptibility must be reduced or the critical threshold of the system must be changed.

FIGURE 6.17 MGT and BEMT predictions of NiZnCu ferrite DC susceptibility compared to the measurement.

FIGURE 6.18 MGT and BEMT predictions of Fe composites DC susceptibility compared to the measurement.

FIGURE 6.19 MGT and BEMT predictions of NiMn ferrite composites DC susceptibility compared to the measurement.

FIGURE 6.20 MGT predictions of magnetite composites DC susceptibility compared to the measurement. Large-size particulate predictions agree well with the measurement. However, the particulate susceptibility must be reduced to fit nanometer scale composite data. This is consistent with separate DC measurements of reduced magnetization and increased anisotropy fields [38, 39].

The MGT under predicts measured composite susceptibility throughout the volume fraction spectrum. NiZnCu ferrite composite measured values are well above the MGT limit line, but measurements do approximate the MGT curve shape at higher volume fractions. Overall, the measured data are always larger than the MGT limit line for high-permeability particulates and volume fractions below about 70 percent. In the high volume fraction limit, MGT predications are accurate for NiMn ferrite composites and work well for magnetite composites. Both NiMn ferrite and magnetite have mid-range susceptibilities. Figure 6.21 shows data for three-phase composites of Fe and NiZnCu ferrite. In modeling of these, a cascade of MGT and BEMT models were used. The BEMT was first used to predict a two-phase susceptibility by using the NiZnCu ferrite as particulate placed in the nonmagnetic epoxy matrix. That achieved an intermediate permeability. The BEMT was then used a second time, using Fe as the particulate that was assumed mixed in a matrix composed of the ferrite–epoxy mix. This two-step calculation yielded predictions that approached

measurement. A similar MGT calculation failed. The three-phase permeability is often larger than either of the two-phase magnetic mixtures for the same total volume fraction. That observation is qualitatively similar to recent publications [33, 34].

FIGURE 6.21 MGT and BEMT predictions of NiZnCuFerrite-Fe composites DC susceptibility as compared to the measurement.

Figures 6.22 and 6.23 show measured ƒr and ƒd for the ferrite composites. Figure 6.24 reproduces the data but also shows the MGT-predicted scaling for the same ferrites. In all cases, the MGT appears to underestimate the measured frequency scales. Even the modest permeability NiMn ferrite shows that measured ƒd follows the general slope of the ideal scaling at modest volume fractions. However, the sharp increases in ƒr and ƒd at highest volume fractions is not reproduced. Predictions for NiZnCu ferrite are also in error and do not parallel the scaling for either characteristic frequency. For example, in NiZnCu ferrite composites, ƒr and ƒd variations with volume fraction are almost identical. This probably arises from the limiting function that comes into play for these large DC permeability particulates.

FIGURE 6.22 Measured resonant and relaxation frequency of the NiZnCu ferrite composites versus volume fraction.

FIGURE 6.23 Measured resonant and relaxation frequency of the NiMn ferrite composites versus volume fraction.

FIGURE 6.24 Measured composite resonant and relaxation frequencies versus volume fraction. The “ideal” scaling for ferrite composites is shown.

The measured frequency scaling data for the Fe and Fe–ferrite composites are shown in Figs. 6.25 and 6.26. In the Fe composites, scaling of resonance appears to follow prediction until the highest volume fraction. In the Fe–ferrite composite, the frequency scaling is a very strong function of overall volume fraction. Figure 6.27 repeats the data but with an ideal scaling curve added for Fe. Note the scaling curve uses the Fe DC permeability that was derived from the BEMT inversion averaged over all Fe composites.

FIGURE 6.25 Measured resonant and relaxation frequency of the Fe composites versus volume fraction.

FIGURE 6.26 Measured resonant and relaxation frequency of the Fe-NiZnCu ferrite composites versus total magnetic volume fraction.

FIGURE 6.27 Measured composite resonant and relaxation frequencies versus composite volume fraction. The “ideal” scaling for Fe composites is shown. Correct scaling for the three component mixtures is unknown.

Figure 6.28 shows a similar plot for magnetite composites that includes very low volume fractions. Even though the 150-nm and 5-μm magnetite particles should have modest permeability, their characteristic frequency versus volume fraction change does not follow the MGT curvature. However, the predicted trend of frequency increase with decreased volume fraction is reproduced. The 12- and 25-nm measurements are shown for completeness; however, at their small volume fractions and for the limited data set, a measurement versus theory comparison is not conclusive.

FIGURE 6.28 Scaling of ƒ r and ƒ d versus total volume fraction for magnetite particulates of the size identified.

Permittivity of the composites did not have significant frequency dispersion and, thus, selected single, averaged values are quoted in Table 6.4. TABLE 6.4 Measured Permittivities (Nondispersive) for Selected Composites. Particulate and Total Volume Fraction, ƒ, Are Identified

The last real-world lesson will attempt a test of the integral forms of Snoek’s law for composites as presented by Archer–Rozanov–Lagarkov [24, 30–32]. Recall their limit laws applied the integral over the frequency spectrum of the imaginary permeability, i.e., . In this relation p, Ms, μi, ƒ, and k are magnetic particle volume fraction, magnetization of particulate bulk, imaginary part of the composite permeability, frequency, and scaling constant. The integral form acts as the composite limiter. The frequency dispersive measurements of each composite are used to calculate the integral. Magnetization, 4πMs, of the particulates can be supplied from handbooks or ferrite literature. The calculated values of the integral are 21,450, 9089, 17,014, and 4770 for Fe, magnetite, NiMn ferrite, and NiZnCu ferrite, respectively. Using these values the constant k was calculated for a range of composites made with these materials. The calculated ranges are shown in Table 6.5. Some variation in k is observed as volume fraction, ƒv, increases and may indicate the onset of significant magnetic coupling between particulates. TABLE 6.5 Calculated Values of k for Limiting Composite Integral

This last data set concludes the case studies for magnetic composites. As the reader has probably noted, a good analytical approach to predict the magnetic permeability of composites is not identified. Model-measurement deviations are especially large when a large particulate constitutive parameter is involved. The problem is apparent over a wide range of particulate volume fractions. Both BEMT and MGT produce the observed volume fraction trends in the measurement but a reproduction of permeability magnitude is not achieved. This difficulty is also seen in high-volume fraction dielectric composites. However, there the coupling mechanism between particulates is better defined. For these reasons the next chapter reviews alternative numerical approaches to predict constitutive parameters of composites. In Chap. 11, discussions of nano magnetic material are expanded with a return to MGT and BEMT but only to demonstrate methodology requirements needed when nanometer size inclusions are used.

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CHAPTER 7

NUMERICAL MODELS OF COMPOSITES

Discussions of the effective medium theories (EMTs) have emphasized prediction of frequency dispersive permeability of nonconducting mixtures and permittivity of conducting inclusions in dielectrics. Since there is dielectric-magnetic symmetry in the EMT derived from the coherent potential approximation, essentially the same methodology applies to both constitutive parameters. Chapter 6 concluded with a case study comparing composite permeability measurements to Bruggeman (BEMT) and Maxwell–Garnett (MGT) effective media theory predictions. Both formalisms demonstrated qualitative but not quantitative agreement with measurement. The study points to a need for approaches that lead to more precise descriptions of scattered fields in a composite. Numerical approaches are a natural next step. A review of the references listed in Chaps. 6 and 7 will show that the detailed study of numerical approaches for material modeling is in the progressive phase and each approach deserves a complete textbook of its own. Chapter 7 presents brief reviews of selected numerical methodologies for prediction of constitutive parameters. The first numerical approach is a periodic method of moments (MoMs) approach. Similar approaches were applied by Sarychev [1] to study inhomogeneous conducting and semiconducting films near the percolation threshold. In this text the technique is summarized and applied to investigate (1) magnetic and dielectric coupling in systems near percolation, (2) current excitation and field localization in artificial dielectrics, and (3) increased diffuse scatter in composite media. The MoM in this text applies a surface boundary condition with simple triangular basis functions. The approach has received significant attention and has been advanced in the last 10 years. Reference [2] is a summary of that research. The finite difference time domain (FDTD) is the second approach for discussion. It is summarized and then applied to a range of composite problems and material concepts. The FDTD case studies are (1) 2D–3D transitions in conductor-dielectric mixtures, (2) corrections to the BEMT and MGT near percolation, and (3) electromagnetic field localization and resonance in composite structures. FDTD solutions to Maxwell’s equations were championed by Allen Taflov [3] and the use of those solutions tracks advances in computational capability. In the late 1980s it was being applied to a wide range of electromagnetic design, radiation, and scattering problems. The FDTD was used in modeling composites in the mid-1990s [4] and has become a preferred modeling tool in the study of metamaterials and negative index structures [5, 6] and more recently in plasmonics and solar cell structures [7, 8].

7.1 METHOD OF MOMENT MODELING AND LAMINATED COMPOSITES EMTs largely suggest that permittivity and permeability of a composite can be predicted separately using only volumetric loading and the appropriate constitutive parameters of the composite matrix and inclusions. In their classic forms, the models cannot account for effects of enhanced wavelength-dependent electromagnetic coupling among electrically correlated groups of inclusions or complex particulate geometries such as 3D spiral elements studied by V. K. Varadan, V. V. Varadan, and A. Lakhtakia [9, 10]. EMTs do not account for the many types of scattering that appear when particulate correlation lengths approach an electromagnetic wavelength in the material or when geometrical or electromagnetic symmetry is broken. The BEMT and MGT are based on simple and fixed coupling between particulates; therefore, they can never be expected to achieve predictive accuracy in complex composites especially when correlation length is a significant fraction of the electromagnetic wavelength within the composite. That electromagnetic correlation length ξ(λ) is on the order of . This simple approximation does not reflect the strong interdependence that can appear between constitutive parameters. The composites εc, μc are themselves functions of correlation length, thus the relation is actually a nonlinear one. In composites where electrically long path(s) are formed by overlapping or touching particulates, the long-range order is established in the resulting “backbone” structure. It may dominate measured effective constitutive parameters and further violate the EMT founding principle, i.e., “forward scattering only.” Also since wavelength within the composite is scaled by the relative index, and the index is a function of the medium permeability and permittivity, it is expected that an apparent interplay of effective permittivity and matrix permeability will be measured in composites. The MoM (and for that matter FDTD) approach is not limited by the correlation length, conductivity of inclusions, or the matrix which surrounds the inclusions. Presentations on MoM can be found in many texts, e.g., Refs. [11] and [12]. It is applied to solve linear partial differential equations where boundary conditions are known. When applied, they yield an integral equation such as the Green’s function discussed in Chap. 5. Since MoM uses the boundary conditions at a surface, it is a good approach when there is large surface to volume ratio (such as in thin films of composites). The MoM is specific to the surface geometry represented by a numerical mesh surface. That surface is a map of the small current elements whose geometry often forms edge-interconnected triangles. When boundary conditions, e.g., continuity of Etan, Htan, are applied at each triangle forming the interface of two media (such as free space and thin conductor) a large matrix equation is generated that relates the local surface impedance and currents on each triangle to the voltage induced by the incident electromagnetic field. Large surfaces require meshing of many triangles; often ≥20 triangular elements per electromagnetic wavelength. Thus, a large mesh produces large matrices that must be solved. Matrix elements must be individually calculated and stored in memory and the matrix must be inverted to calculate currents. The overall impact is that computers with very large memories and fast processors are required for matrix fill and inversion.

This section will discuss the background and presents a series of test cases where the MoM has been applied to study selected problems in composites. The problems are chosen to investigate composites which cannot be easily modeled by the BEMT and/or MGT. Thus, they all address properties of conductor-dielectric thin films where the conductor concentration is within a few percent of the percolation threshold. This presents a challenging problem and illustrates properties of composites that may not be evidenced by BEMT or MGT simulations. Single-mode transmission and reflection coefficients are calculated through composites that are formed from laminates of near percolating structures and/or near percolating films. At times the films are surrounded by a magnetic matrix. The predicted reflection and transmission are then used to calculate effective constitutive parameters that would be observed in measurement. Measured electromagnetic complex transmission data are presented to demonstrate the experimental validity of the MoM model for film and dielectric laminate geometries. The code is also applied to illustrate unique characteristics of composites, i.e., current/field localization and diffuse scattering from composites. Both calculations evidence phenomena which suggest modification of classical approaches to material measurement. The diffuse scattering is most easily measured using a focused beam approach (Chaps. 9, 10, and Ref. [13]). The measurement of localized fields requires use of a near field probe or “microwave microscopes” [14, 15].

7.1.1 Periodic MoM and Applications to Composites MoM codes have been applied to calculate electromagnetic observables of many inhomogeneous composites. The problem of predicting the transmission and reflection through a composite is approached by dividing it into a “laminate” stack. Each laminate layer is a thin dielectric on which a pattern of thin particulate patches can be attached. Even with this geometrical simplification, calculating properties of an arbitrary inhomogeneous structure is made tractable in the MoM codes only through additional simplifying assumptions. For example, a laminate layer is modeled as a structure which can be assumed periodic in two dimensions. Considerable simplification ensues because any particulate geometry and surface/volume fraction is defined and developed in only one unit cell of the structure. Of course, to have any predictive accuracy the unit cell must be a reasonable facsimile of the actual structure to be modeled. Repeated random replications of the geometry, with fractional contents fixed, allow a full development of average properties and variance of the properties. An example simulation of a deposited metallic microstructure is shown in Fig. 7.1.

FIGURE 7.1 Actual (left) and MoM geometry simulation (right) of a percolating nano metallic granular surface. The small squares with lined triangular patches are conducting elements. Left–right and/or top–bottom square edges are connected if square edges are solid lines.

The Unit cell electrical size, on each surface making up the laminate layer, is important. Ideally good simulation of electromagnetic properties, i.e., emulating an “infinite size” composite panel, would require that a unit cell represent details of the microstructure out to sizes of about some fraction of the wavelength. This could lead to very large matrices to invert. If each element is ≤ 0.05λ in size at the wavelength of interest a square unit cell may be composed of tens of thousands of square elements where each element is divided into two triangular facets. In modeling effective media, a unit cell is often smaller than one wavelength and the periodicity of the unit cell does not strongly impact the calculated electromagnetic properties. At low frequencies it is important to replicate paths or areas of connected particulates or regions of matrix that might appear in a composite. This will be especially true near a critical fraction. Even if one layer of composite does not have intrinsic periodicity, periodic boundary conditions may be applied to connect elements at the unit cell edges. This emulates a large finite panel and effects of connectivity on electromagnetic properties. The alternative approach to periodic boundary conditions would be an iterative solution to the numerical effective field problem. MoM calculations are frequency domain simulations that find solutions to the wave equation by calculating surface currents over all surface(s) that are constructed to simulate a composite. The solution to Maxwell’s equations is obtained in integral equation form. The formulation that follows assumes that all the fields have an exponential +iωt time dependence where the angular frequency is that of the incident plane wave. For now consider a single surface that is made from conducting and dielectric patches. This is illustrated in Fig. 7.2. The

patches are in the , plane and propagation is in the direction.

FIGURE 7.2 A single layer of a composite (vertical patches) has both incoming and outgoing fields indicated by a + , b + , a – , b – . The region to the left of the layer is region 1 and right of the layer is region 2.

The tangential components of the electric and magnetic fields in either left or right region of Fig. 7.2 are written as a modal expansion. By construction, all structures have fundamental periodicity and the fields are expanded in a set of basis functions known as Floquet modes. The Floquet analysis has historically been applied in crystalline solid-state research (see Kittel [16]) but has become a standard approach to solving Maxwell’s equations where periodic boundary conditions are relevant. Examples include phased arrays [17] and many metamaterials [18]. Floquet modes represent a special set of plane waves which are in one-toone correspondence with the points on a reciprocal lattice. They are characterized by two integers which describe the number of times the mode oscillates in crossing a unit cell in each of the two transverse directions. The 00 Floquet mode is just the incident plane wave. For the simple case of Fig. 7.2 the tangential fields can be written as

and

Longitudinal components of the fields are

The longitudinal components of electric field must be used to obtain the magnetic field from the curl of the electric field. The Floquet modes are defined by

and modal admittances are

The tangential field boundary conditions require continuity and are proportional to the surface current density, , and surface impedance, Zsurf, on each element. Magnetic fields have discontinuities across any partially conducting elements. When stated in equation format

The incident field induces surface currents on conducting elements of the structure. The conductivity, position, size, and shape of the elements are used in the method of moments to obtain the scattered fields, and a core portion of the MoM code is designed to produce an approximate solution for induced surface currents at a particular frequency. This situation is to be contrasted with numerical methods, such as FDTD, which solve the partial differential equations for the scattered fields themselves. Boundary conditions are imposed in Eqs. (7.1) to (7.4) and the system Greens function and surface current Fourier transform are defined as below in terms of current amplitudes, In, and triangular current basis functions,

with FFt of , and

The final form of the moment method approach takes the form of a familiar Ohm’s law

expressed in a linear algebra problem Vn = Zns Is and allows for a solution of the current amplitudes, Is,

where impedance matrix, Zns, and voltages, Vn, are identified with

Zns are the matrix coupling individual basis functions vector unknown components Is to the source vector components Vn that are from the electromagnetic field incident on the panel. Current amplitudes are found by inverting the impedance matrix and multiplying by the source voltage vector. The analysis can be extended to any volumetric composite whose geometry allows it to be carved into a cascade of thin laminate panels. This is illustrated in Fig. 7.3. The solution at each laminate interface is satisfied and leads to a cascade or scattering matrix solution (see Chap. 4). However, in the present case the scattering or cascade matrix will also couple all of the excited modes in the Flouquet expansion and can lead to effective parameters which are functions of surface separation and the number of surfaces. Figure 7.4 illustrates how the inherent constitutive parameter variation with thickness might appear.

FIGURE 7.3 The figure shows a complete cascade of composite layers that are surrounded by various matrices. Model coefficients are indicated at each layer. The solution requires a multimode match and coupling at all layers.

FIGURE 7.4 MoM-predicted relative permittivity for a fixed 30 percent surface fraction (well below percolation) on 1 to 5 layers separated by about 0.001 to 0.1 wavelengths.

7.1.2 MoM Application 1: Demonstration of Magnetic-Dielectric Coupling in Composites The basic geometry for all composites that are presented in this section is that of laminate stacks of finite dimension films with unit cells formed from a lattice of elements with complex frequency-dependent impedance or constitutive parameters. These are surrounded by a matrix with a fixed or frequency-dependent permittivity and permeability. MoM simulations of random composites utilize predetermined geometries of electrically small, square platelet inclusions (Fig. 7.5), which are placed randomly on a square lattice to represent a network of a specified inclusion fraction. Three-dimensional composites are modeled as laminates of inclusion surfaces where inclusions on different surfaces may or may not be electrically connected. In the following calculations there is no direct connection between layers but the closely spaced films are coupled via local modes. In the first series of calculations the matrix material, which surrounds the inclusions, possesses frequency dispersive permeability, and inclusions are conducting.

FIGURE 7.5 A typical element configuration within one unit cell of the calculations. The small squares are each formed of two triangular current elements. Blanks indicate vacant area which is occupied by matrix. Squares are DC connected along their broad sides; corner connections are not allowed in model nor test panels.

MoM calculations use the computed 00 mode transmission and reflection coefficients for the slab of composite and apply them to calculate effective dispersive constitutive parameters. Figure 7.6 shows a typical comparison of measured and predicted transmission of a single realization of a multilayer laminate. The data do not overlay in this single calculation. Multiple runs on a variety of realizations of a given volume fraction are often required to find an average system response and typical fluctuations.

FIGURE 7.6 Comparison of predicted (solid circles) and measured (solid lines) normal incidence transmission through a five-laminate composite in a dielectric matrix. Each laminate has a surface conductor fraction of approximately 35 percent.

As an example consider calculations near the critical fraction for conducting inclusions. By carrying out millions of simulations it has been demonstrated that near the critical fraction a composite system can be approximated if the periodic cell size is on the order of the correlation length. Thus, a reasonable choice for the cell size for a computation satisfies L > ξ. However, at the critical threshold the correlation length diverges, so there is little chance of doing a direct computation exactly at this state of a random composite. However, the modeling approach proves to be particularly attractive when composites can be formed by layering of sputtered, photo etched, transfer etched, and even directed growth films. These techniques allow the experimenter to manufacture an exact representation of a modeled laminate. MoM simulations were performed to investigate coupling of magnetic and dielectric parameters in composites containing conducting particulates. The conducting inclusion fractions were in the range 0.001 ≤ |p – pc| ≤0.4. Figures 7.7 to 7.9 are selected to qualitatively illustrate constitutive parameter coupling.

FIGURE 7.7 Calculated effective real permittivity ratios for conducting inclusions (of two conductivities, σa and 0.2σa) placed in a nondispersive magnetic matrices with indicated relative permeabilities. Matrix permittivity is εr = 10;εi = 0.1.

FIGURE 7.8 Calculated effective imaginary permittivity ratios for conducting inclusions (of two conductivities, σa and 0.2σa) placed in a nondispersive magnetic matrix with indicated relative permeabilities. Matrix permittivity is εr = 10;εi = 0.1.

FIGURE 7.9 The figure shows the MoM-predicted effective permittivity ratios for nearly percolating conducting inclusions placed within a dispersive magnetic material with permeability as indicated. The matrix permittivity is fixed and set equal to εr = 10;εi = 0.1. Predictions are shown for three conducting concentrations.

Effective ε, μ were determined by inversion of the MoM-predicted complex transmission and reflection data using the indicated values or postulated dispersion in permeability. In the simulations composites contained eight layers of patterned conducting films and for these calculations the conducting patches on differing films were not DC connected but were separated by a thin (0.1 mm) nondispersive dielectric layer with a permittivity typical of ferrites (i.e., ε ≈ 10) and with the indicted permeability. The frequency dependence and magnitude of the separating layer ’s permeability was one variable. Figures 7.7 and 7.8 illustrate typical behavior when conducting inclusions (here fractional content 8 percent below the critical point) are surrounded by the magnetic matrix. Side-byside figures are for different element conductivities and lines do not represent a numerical fit but are aids to the eye. The figures show the ratio of real and imaginary parts of the calculated effective permittivity, as placed in a dielectric matrix (εr = 10;εi = 0.1), to that of the same conducting patch panels placed in nondispersive magnetic matrix with identical permittivity (εr = 10;εi = 0.1) but with a permeability of the indicated value. These particular calculations are chosen to demonstrate a behavior that cannot be reproduced by the MGT or BEMT. Here the presence of a surrounding magnetic material will impact the effective permittivity of the composite since the electrical correlation length is modified. However BEMT, MGT, and most forms of effective media predict effective constitutive parameters in isolated calculations. Any coupling of constitutive parameters would be represented by skin-depth effects. The MoM predictions indicated that the addition of the magnetic surrounding (though

unrealistically nondispersive) tends to reduce magnitude of the calculated effective permittivity. The largest changes are observed when the highest conductivity patches are used. This trend is believed to arise from the larger current densities that would be excited on these patches. Currents respond to the presence of a magnetic material. The imaginary part of the effective permittivity also sees the largest decrease when the higher conductivity inclusions are used. This again is consistent with stronger currents which would appear on these elements. Figure 7.9 is the last that illustrates behavior which could not be emulated with analytical EMTs. Again effective permittivity ratios (without magnetic and with magnetic) are calculated but now with a dispersive permeability that is similar to what might be observed at frequencies well above a magnetic resonance. Conducting elements have a higher conductivity and the plots are for various conducting patch fractions, all near the percolation threshold. The trend observed in the initial calculations, 8 percent below the critical value, is largely maintained. Permeability decreases with increasing frequency and the permittivity ratios (both real and imaginary) approach unity. The exception is an apparent dielectric resonance which appears just below the critical point and at scaled frequencies near 3. All represent phenomena that are not expected with analytical EMTs.

7.1.3 MoM Simulations: Current Excitation and Field Localization in Composites The MoM is also a powerful tool when investigating the field and current distributions on or within layered composites where composite layers can be represented as random or periodic depositions of small patches on a dielectric substrate. When patches are conducting, structures demonstrate localization where fields on the surface can be many times larger than the average field on an isotropic film. Again this physics requires strongly coupled particulates and is not easily discerned with analytical EMTs. Sarychev et al. [19] demonstrated that plasmonic localization should occur on percolating films with localized enhancements near factors of 104. This phenomena can be leveraged to enhance Raman scattering and nonlinear optical response of molecules and active dyes which are placed on a surface [20–22]. Thus, fractured films allow “custom-designed” surfaces for Raman and surface-plasmon-based ultrasensitive optical sensors. The behavior is evident to a lesser extent in the microwave and millimeter wave spectrum and has been observed using “microwave microscopes” or local probes that measure fields just off the surface of a conductor-dielectric patch surface. Figure 7.10 shows random unit cells of conductor-dielectric film with a near field probe superimposed. The incident field is produced by an offset radiating antenna. The right side of the same figure shows the field and the surface. Notice the large enhancement, which is near 20 dB.

FIGURE 7.10 Physical geometry (left) and measured near field magnitude (right) of a good periodic structure with complex unit cell geometry. A probe is shown in the upper right side of the left-hand image.

Figure 7.11 shows a single unit cell geometry, similar to the one illustrated in Fig. 7.10. It could represent an interconnected group of conductive elements that make up a network. Though it is only one representation of the many structures that might be generated for a given surface fraction, it can be applied to illustrate changes in composite physics and field localization as surface fractions are modified.

FIGURE 7.11 Unit cell geometry; darker scale indicates conductor and lighter or white dielectric.

The following sequence of figures show grey scale (darker indicates high magnitude and white low magnitudes) visualizations of current (H field) magnitudes for a series of fragmented patch surfaces composed of random structures below and up to the 2D critical surface fraction. The current/field visualizations of Figs. 7.12 to 7.14 are qualitatively similar to the measured near fields of Fig. 7.11 and have been restricted to views of a single unit cell. As concentration increases, the excited current pathways and the number of intense localizations (indicated by a darker appearance). Calculations are for wavelengths 20 and 2.9 times the cell size.

FIGURE 7.12 Surface currents (i.e., near fields) of 20 and 30 percent surface fraction cells for two frequencies, fs (20 and 2.9 times the cell dimension). White and black differ by 20 dB.

FIGURE 7.13 Surface currents (i.e., near fields) of 40 and 55 percent surface fraction cells for two frequencies, fs (20 and 2.9 times the cell dimension). White and black differ by 20 dB.

FIGURE 7.14 Surface currents (i.e., near fields) of 58 percent surface fraction (the percolation threshold) cells for two frequencies, fs (20 and 2.9 times the cell dimension). White and black differ by 20 dB.

At the lower surface fractions, highest fields appear at common lattice sites for both frequencies that are shown. Similarity in excitation would indicate little dispersion in the surface’s electromagnetic properties and approximately the same constitutive parameters. As surface fraction increases, excitations differentiate at the two frequencies, with large differences apparent at 55 percent. The differentiation would be parallel to an observation of frequency dispersion in an effective constitutive parameter calculation. In Fig. 7.14, at the critical fraction 58 percent, elements are electrically connected across the cell, between cells, and thus throughout the entire structure. High currents, at the two frequencies, are different in position and magnitude. Note the calculations represent a single realization at the indicated surface fraction. A realistic composite might have many realizations (at fixed conductor fraction) in different cells and calculations must be expanded to determine an expectation of internal current statistics.

7.1.4 MoM Simulations: Diffuse Scattering in Composites A plane wave incident on a uniform isotropic homogeneous material’s planar surface excites translational invariant conducting, polarization, and/or magnetic currents along that surface. Reradiated fields from those current sources add in phase to produce a specular reflected

electromagnetic field from the surface and transmitted field propagating at angles determined by Snell’s law. Composites are inherently inhomogeneous and most are assembled from two or more electrically dissimilar materials. Even if the particulates forming the composites are electrically small, their relative concentrations may vary over small volumes or areas of the composite. The volumetric changes produce a random variation in local reflection and transmission. Variations in electrical parameters also excite localized surface or volumetric currents in the material and at the surface. If currents have amplitude and phase coherency that vary with position (they would be coherent in a homogeneous material), translational invariance is broken. A break in symmetry results in energy radiated in directions other than those predicted by Snell’s law and leads to “diffuse reflection” and transmission. The magnitude of this diffuse scattering is a function of the difference in electrical properties of the composite, the physical length scale L over which variation may occur, the electromagnetic length λ, and the electromagnetic length-scale, L/λ. As the electrical lengthscale of the components approaches unity, diffuse scattering increases. In addition, the larger the ratio of component constitutive parameters, the higher the level of diffuse scattering. For example, one might be conducting and the second dielectric. The physics is illustrated in Fig. 7.15.

FIGURE 7.15 Physics of diffuse scatter.

By their nature historical material measurements were designed to measure a single electromagnetic mode. In free space measurements this is the 00 mode of reflection and transmission. In the case of inhomogeneous composites or mixtures, this singular, standard measurement may lead to a misleading evaluation of material constitutive properties. In free space measurements there may be energy lost in diffuse scattering [13] that is not sampled by a 00 mode measurement. This suggests that composites, which should have very low electrical loss tangents, may appear to have higher loss. An expectation for diffuse scatter can be illustrated using MoM simulations. Accurate calculations of scattering from random composites would best be performed using a specific geometry that is appropriate to a manufactured composite material. However, in the following discussions of this section an approximate simulation is performed. Multiple composite populations using a single unit cell size (much smaller than one wavelength) are generated. Each has identical surface fractions. These are then placed in stacked laminate composites. The scattered field of each is calculated by a Fast Fourier Transform of the surface currents for three surface fractions and a low frequency where the cell is much smaller than a wavelength. The scattered field from each representation is calculated and saved. The fields are then added and averaged over the population of calculations. The average scattered field is a function of incidence and scattering angles. This simulates “diffuse scattering” that might be seen in artificial dielectric materials that contain inclusions. Diffuse reflection is minimized when 100 percent of the scattering area is continuous and conductive. Since excited currents are uniform and coherent, scattered field magnitude is dominated by the Fresnel specular term. Magnitude of a calculated uniform scattered field is shown on the left of Fig. 7.16. A calculation for the same sized plate is shown on the right of Fig. 7.16 but for an average 54 percent of its surface occupied by patches of the same conductivity. The horizontal line indicates the average specular value for the uniform case. The shape of the 54 percent scattered pattern is changed; however, the increase in scatter from 45 to 90 degrees is small. The vertical scale of the fragmented surface scatter is changed to show that the specular scattered magnitude is less than that of the uniform surface and is attributed to the higher surface impedance that is associated with the fragmented conducting patch surface.

FIGURE 7.16 Calculated far zone scattered fields (0-degree incidence) of a uniform conducting surface (left) and a fragmented surface about 4 percent below critical fraction (right).

Figure 7.17 compares scattered field magnitude for the 54 percent case and one very near the percolation fill fraction, i.e., 58 percent. The diffuse scattering increases by 20 dB. In addition, the set of surface realizations near 58 percent has modified the average field phase taper across the unit cell so as to redirect the scattering peak away from the 0 degree incidence with magnitude slightly lower than the uniform surface peak value.

FIGURE 7.17 Calculated far zone scattered fields (0-degree incidence) of a 54 percent conductive patch surface (left) and 58 percent (right).

7.2 FINITE DIFFERENCE TIME DOMAIN SIMULATIONS The FDTD is the second numerical technique that is summarized. It has demonstrated great success by extending materials modeling beyond EMT to model complex composite materials, metamaterials, complex radiating elements, and photonic structures [4–8]. Where MoM or finite element solutions use a boundary mesh and boundary conditions to calculate currents, the FDTD seeks a direct numerical solution to the first-order time-dependent formulation of Maxwell’s equations for and . In this case boundary values of electric and magnetic field are fixed at time t = 0.0 at surfaces of the modeled volume. Field values at internal points of the volume are determined by iteratively solving the partial differential equation on a grid of previously chosen positions. The procedure is described in detail in either Taflov’s [3] or Smith’s book [23]. Smith will be followed for this short introduction to

the FDTD.

7.2.1 1D FDTD Introduction Since our discussions are a review, description of the technique will be restricted to calculation of the time evolution of fields propagating along the single direction within a boundary defined by a maximum time, tmax, and a corresponding maximum distance, zmax = ctmax. Figure 7.18 shows a geometric grid of points (time and one spatial dimension) on which Maxwell’s equations are to be solved.

FIGURE 7.18 Grid geometry of the single-dimension FDTD simulation. Solid circles represent points at which E is calculated and solid triangles are points of a B calculation. Boundary conditions are indicated. For expansion, see Smith [23].

In one spatial dimension Maxwell’s first-order equations are

The differential equations are approximated by their finite difference equivalents and solved simultaneously [23]. When the algebra is carried through, Maxwell’s differential equations for the fields reduce to two difference equations that supply solutions at stepwise alternating grid positions. Using the formalism of Smith but expressing the equations in terms of the H field and constitutive parameters

Both equations express the fields at later times in terms of field values at an earlier time. Boundary conditions set the field values at time zero. The equations are then applied in a “marching in time” process to calculate field values at alternating grid points at all times. Constitutive parameters enter the analysis via the lead factors of each second term in both equations, i.e., {Δt/ε0εrΔz} and {Δt/μ0μrΔz}. As in any numerical iterative solution, these prefactors must remain less than unity (typically much less than unity) to assure solution convergence. That requirement can be met if Δt/Δz is kept small and modified for various problems. This requires the computational electromagneticist to judiciously pick the time increment and spatial increment. FDTD can be directly extended to three dimensions; however, fields are often transitioned from a planar grid to a cubic lattice, i.e., this Yee lattice illustrated in Fig. 7.19. Formulations of the field solutions are found in Refs. [3] and [23] and will not be reproduced here. Computation time for the FDTD method is approximately proportional to N4, where N is the number of Yee cells, and therefore limits on the size of computational volume also must be considered.

FIGURE 7.19 Three-dimensional Yee cell (left) and Yee cell reduced in one dimension (right). Locations for field calculations are indicated by solid circles, squares, and triangles.

Even though it is computationally intense, the application of FDTD in composites modeling is an excellent choice. FDTD simulation uses small volumes and in most cases those spatial regions can be chosen to represent volumes and constitutive parameters that identify composite matrix and inclusions. The inclusions can have user-specified constitutive parameters and therefore multiphase composites can be modeled. An example of simulation for an actual material is shown in Figs. 7.20 to 7.22.

FIGURE 7.20 An FDTD simulation volume of a composite composed of two different materials (black and white). Total thickness is d. This volume becomes the unit cell of a periodic structure when reflection and transmission calculations are needed.

FIGURE 7.21 Two stereolithographically constructed structures applied in verification of FDTD composite predictions. The structure on the left is resistive to allow for percolation and that on the right uses only dielectric cubes.

FIGURE 7.22 FDTD grid of one section of the stereolithography structure such as that on the right of Fig. 7.21.

Applications of FDTD will be illustrated for three cases. The first is a demonstration of the constitutive parameter scaling in finite thickness structures where the thickness is a few to ten times a composite particulate size. This physics has been discussed in Chap. 6. The second application will compare MGT and BEMT with FDTD in composites that contain particles with nondispersive conductivity (often a good approximation below terahertz wavelengths) and those that contain particulates with dispersive conductivity. Dispersion would be appropriate in modeling at far infrared through visible wavelengths. The analysis leads to suggestions for improvement of the BEMT that allow its use at and above the critical threshold. The third and last application of the FDTD will be a simple visualization of volumetric field intensity as a function of time in a finite thickness composite containing inclusions. The visualization is for conducting particulate-dielectric composites that were modeled in the second example and shows evidence of field localization.

7.2.2 FDTD and Finite Thickness Composites FDTD simulations of a material mixture demonstrates the scaling of permittivity with thickness which is predicted in scaling theory [24–28] and discussed in Chap. 6. Composite geometries were realized by randomly placing on a lattice (square/cubic/hexagonal, etc.) a characteristic shape of the fundamental conducting and dielectric elements of size a0. Each element was assigned a characteristic frequency-dependent permittivity and permeability. An electrically large composite panel is modeled similar to the MoM method. The composite geometry is chosen to emulate a composite concentration profile and that geometry becomes the unit cell of a periodic structure. In the first example, the simulation geometry uses a 16 by 16 by 1-10 layer thickness (scaled to a0) unit cell. The particulate is shaped like a three-dimensional plus sign. In these simulations the elements are randomly assigned to be either conducting or dielectric and for

the example calculations, the conductivity of the 3D plus signs is 50 mhos/m and dielectric elements have relative permittivity near 2. Conductive elements occupy either 41 or 32 percent of the FDTD grid sites. As discussed in Chap. 6, the critical threshold in composites varies with the scaled size L/a0. Here L can be either a cross-sectional dimension or thickness. As this ratio becomes large (about >20) critical thresholds approach their theoretical 2D and 3D values (about 58 percent and approximately 31–33 percent). For this small 16 × 16 × 1 unit cell size, 41 percent is just below the 2D critical threshold and above the 3D critical fraction. A choice of 32 percent was made since it is just below both 2D and 3D thresholds. In the calculations of Figs. 7.23 and 7.24, the FDTD was applied to predict the composite transmission coefficient versus scaled frequency. Effective permittivity was then calculated from transmission and the sample thickness.

FIGURE 7.23 FDTD predicted permittivity for a conductor-dielectric composite unit cell of 16 × 16 × t/a 0 with a conductor site fraction of 41 percent.

FIGURE 7.24 FDTD predicted permittivity for a conductor-dielectric composite unit cell of 16 × 16 × t/a 0 with a conductor site fraction of 32 percent.

Thickness-scale dependence of the critical threshold is very evident in the frequency dependence of the imaginary part of the permittivity (left of Fig. 7.23). The single element thickness εimag approaches zero at low frequencies and begins to rise at the highest frequency. The real part shows a relaxation at the highest frequency. However, for the composites of 4, 7, and 10 element thicknesses, the εimag demonstrate an approximate f–1 dispersion of a continuously conducting material. Figure 7.24 shows the 32 percent results. No thickness exhibits the inverse frequency dependence. This is expected since concentration is below both 2D and 3D thresholds. A homogenous material would show a permittivity that is independent of thickness. However, the predicted constitutive parameters and dispersions of these composites are a strong function of scaled thickness. The predictions of Fig. 7.24 have a Debye-like relaxation. The relaxation frequency shifts to lower frequency with an increase in the scaled thickness. This is accompanied by an increase in the DC value of εr. The decrease in relaxation frequency is consistent with the scaling model of Chap. 6. In principle this and similar calculations suggest an alternative approach to making matched filters or impedance structures. Rather than using different thicknesses of different materials, a materials engineer might use a stack of materials, made from the same constituents but each with different scaled thickness and isolated from each other by a thin dielectric layer. Such a stack could make an impedance matching structure.

7.2.3 FDTD and EMT Comparisons The FDTD can serve as a numerical check on EMT which have been discussed in Chaps. 5 and 6. In the simulations of this section the parameters for conductive particulates are

appropriate to Palladium and particulate sizes from 40 to 60 nm. Calculations are appropriate to wavelengths in the near- and far-infrared from about 1 to 100 μm. The choice of metal, size, and wavelength allows testing of the FDTD, MGT, and BEMT for particulates with Drude [29] dispersions where permittivity varies as

and a + iωt is used. A cross section of the unit cell in the composite is shown in Fig. 7.25. This unit cell is replicated in the cross-section dimensions, and directions, and is >10 particle diameters thick. The critical volume fraction is near a 3D threshold, 31 to 33 percent. The figure shows a z, x cross section and many of the spherical particles (represented by many assembled small cubes) are sliced to reveal various cuts.

FIGURE 7.25 and cut of the composite unit cell used in the calculations of this section.

The first set of comparisons uses the low-frequency (long-wavelength) limit of the Drude conductivity model and thus permittivity is iσ/(ωε0). It has been established in Chap. 5 that the MGT is not correct for conducting particulate composites at higher volume fractions and therefore FDTD comparisons are restricted to the BEMT in Fig. 7.26.

FIGURE 7.26 BEMT and FDTD comparisons assuming particulate described by the DC conductivity. Real permittivity (top) and imaginary permittivity (bottom). Volume fractions are 5 to 35 percent.

Figure 7.26 indicates that the BEMT should be a good choice when particulate permittivity can be described by DC conductivity alone. The agreement is largely achieved even beyond percolation which is indicated by the appearance of an inverse wavelength variation in imaginary permittivity at long wavelength (low frequency) for the 35 percent fraction case. Now the Drude dispersion is added to the calculation and concentration is maintained above the critical threshold. BEMT simulations of composite properties, where particulates have resonant dielectric behavior, have been studied by Cohen et al. [30–32] and others. If the FDTD simulations are taken as the standard, Fig. 7.27 shows similar results to Cohen’s with the BEMT missing any resonance resulting from the Drude particulate. Recall that the BEMT was also found to miss prediction of the Lorentzian resonance in magnetic composites. The long-wavelength conductivity dominates any resonant behavior; MGT calculations are added for comparison. As in Cohen’s results, MGT shows a resonance, but at an incorrect wavelength and totally fails in the long-wavelength (low- frequency) bands where DC conductivity does dominate.

FIGURE 7.27 Comparison for MGT, BEMT, and FDTD simulations of permittivity where Drude particulates are included. Drude parameters appropriate to Palladium of 50-nm diameter. Conducting volume fraction is 7 percent above p c.

This calculation is investigated in more detail below. Figure 7.28 shows a grey-scale

graphic of multiple unit cells of the FDTD composite geometry. The dielectric background is white. The conducting particulate fraction is 37 percent. The darkest areas show particulates that are touching and establish long conducting paths, “conducting backbones” through the composite. In this and similar cases, the morphology suggests an effective medium approach that is similar to that which was attempted for multiphase magnetics.

FIGURE 7.28 Multiple unit cell graphic of the FDTD geometry file. Conducting backbone contributors are shown as the darkest elements. Grey indicates conducting elements not on the backbones. White indicates dielectric background.

The Drude artificial dielectric composite is treated as a three-component mixture, i.e., dielectric + particles not on the backbone + the backbone itself. First a new dielectric “matrix” is modeled by using BEMT with the volume fractions of dielectric combined with the volume fraction of non-backbone Drude particulates. Then in a second BEMT calculation, the new matrix has the backbone added as a separate volume element. As shown in Fig. 7.29, the approach, Meta-EMT, does recover the resonance in stepped BEMT calculations. The BEMT

and MGT alone are also shown. Graphic indicators (solid circles, triangles, and diamonds) are shown to aid the eye.

FIGURE 7.29 Comparison of MGT, BEMT, Meta-EMT, and FDTD predictions of a composite’s effective permittivity for Drude particulates (fraction above p c).

7.2.3.1 FDTD Field Visualization. The last example is a straightforward use of FDTD data products. A FDTD simulation of propagation contains the time evolution of fields throughout a computation. If this fact is leveraged in composite modeling, the user can produce graphic representations of electromagnetic fields and amplitudes during propagation and scattering. When this is visualized, the evidence of field localization, resonance, or field enhancement can be observed within the composite. Example visualizations are found in Figs. 7.30 and 7.31. Visualizations supply insight that is not easily obtained by analyzing constitutive parameter plots.

FIGURE 7.30 FDTD-predicted field intensities within a composite of Drude particulates and dielectric. Time evolves from

upper left (external field impacts front surface) to lower right where transmission is nearly complete. Field enhancement is shown by dark elements in this grey scale picture. Low magnitude fields are grey.

FIGURE 7.31 FDTD-predicted field intensities, at one time, viewed along the x,y axes and on one surface. Composite and particulate parameters are similar to those of the previous figure. Field enhancements is indicated by the darkest shades in this grey scale image.

7.3 COMMENTS FOR CHAPTERS 5 TO 7 This book strongly recommends that EMT models should be applied by engineers or scientists in their experimental design, in choice of the measurement technique and equipment for testing a specific material, and/or in application of different composites for engineering problems. The models reveal material physics that dictate requirements for a composite sample’s size and shape, material morphology, relevant and useful ranges of electromagnetic wavelength, and the types of equipment that are required for measurement. For example, composites that contain small volume fractions of a single constituent may be appropriately characterized by a simple reflection or transmission measurement in a transmission line. However, composites containing high volume fractions of a material (especially when that material is conducting) can be electrically inhomogeneous. For these composites a different measurement plan of attack may be required for data that establish a useful figure of merit for application in devices or structures. As an example, consider characterization of structural

foams or core. If foam voids or core cells approach λ/10 in scale, the engineer may require diffuse reflection measurements. These augment simple plane wave reflection and transmission measurements and can explain measurement abnormalities, such as dispersion or excessive absorption in supposedly low electrical loss foam. Chapter 6 applied implications of the composite models as a guide to illustrate how simple transmission line techniques may fail in capturing material physics in a wide range of composite materials. The examples suggest that for certain classes of materials, characterization systems should be modified to utilize configurations that measure multiwavelength scale samples rather than samples with sub-wavelength (0.01 to 0.4) sizes that are typical of coaxial or waveguide. In order to accommodate larger samples the discussions of focused free space reflection and transmission systems are revisited in later chapters. These techniques have been found to be best for characterizing electromagnetic observables of composite materials that have been fabricated into samples greater than or equal to one electromagnetic wavelength in cross section. The focused beam systems can be reconfigured to radiate discrete areas of large material samples and establish expectations of cross sectional homogeneity. By changing lenses, composites with cross section between about 4λ2 to 10λ2 can be measured. Large samples can be rotated to study diffuse scattering and parametrics at different angles of incidence and do so without radiation of sample edges. This capability proves essential in the measurement of composites of semiconductors and dielectrics where electrical continuity may extend to many wavelengths. The space of composite measurements has the capability to measure tensor electrical and magnetic properties by combining broad frequency, angular and measurements for different sample dimensions. In addition, focused beams allow the measurement of diffuse scattering and bistatic scattering. These can be applied to produce a map of surface scattering and in effect image inhomogeneous materials. Thus, the composite morphology can be inferred for those samples formed from macro- or microscale components.

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11. R. F. Harrington, Field Computation by Moment Methods, ISBN 0-7803-1014-4, Wiley-IEEE Press Hoboken, NJ (1993). 12. W. C. Gibson, The Method of Moments in Electromagnetics, ISBN 978-1-4200-6145-1, Chapman & Hall/CRC, NY NY, (2008). 13. J. W. Schultz, Focused Beam Methods: Measuring Microwave Materials in Free Space, Amazon.com ebooks, ISBN: 1460092851 (2012). 14. E. Kuster, R. L. Moore, S. Blalock, B. Cieszynski, J. Swarner, and M. Habib, “Microwave Volumetric Probe Measurements of Field Localization, Zero and Negative Index Within Photonic Bandgap Metamaterial Structures,” Mat. Res. Soc. Symp. Proc., 1343:10–15 (2011), Recent Progress in Metamaterials and Plasmonics. 15. S. Kalinin and A. Gruverman, eds., Scanning Probe Microscopy, Springer, New York (2007). 16. C. Kittel, Introduction to Solid State Physics, Wiley, Hoboken NJ, ISBN 0-471-11181-3 (2005). 17. A. Bhattacharyya, Phased Array Antennas: Floquet Analysis, Synthesis, BFNs, and Active Array Systems, ISBN-13: 978-0-471-72757-6, Wiley Hoboken NJ (2006). 18. S. Sajuyigbe, M. Ross, P. Geren, S. A. Cummer, M. H. Tanielian, and D. R. Smith, IET Microw. Antennas Propag., 4(8):1063–1072 (2010). 19. A. Sarychev, V. Shubin, and V. Shalaev, Phys. Rev. B, 60(24) (Dec. 15, 1999). 20. J. B. Jackson, et al., Appl. Phys. Lett., 82:257 (2003). 21. S. Shultz, et al., Proc. Natl. Acad. Sci. U.S.A., 97:996 (2000). 22. C. A. Rohde, K. Hasegawa, and M. Deutsch, PRL, 96:045503 (2006). 23. G. S. Smith, An Introduction to Classical Electromagnetic Radiation, ISBN 0 521 58698 4, Cambridge Press, NY NY (1997). 24. J. P. Clerc and G. Giraud, Phys. Rev. B, 22(5):2489 (Sept. 1, 1980). 25. I. H. H. Zabel and D. Stroud, Phys. Rev. B, 44 (1992). 26. R. Koss and D. Stroud, Phys. Rev. B, 35:9005 (1987). 27. F. Brouers, Phys. Rev. B, 44:5299 (1991). 28. A. V. Neimark, Sov. Phys. JETP, 71:341 (1990). 29. R. E. Hummel, Electronic Properties of Materials, 3rd ed., ISBN 0-387-95144-X, Springer, NY NY (2005). 30. M. Cohen and J. Jortner, Phys. Rev. Lett., 30(15):649 (April 9, 1973). 31. R. W. Cohen, G. Cody, M. Coutts, and B. Abeles, Phys. Rev. B., 8(8):3689 (Oct. 15, 1973). 32. M. Cohen, I. Webman, and J. Jortner, J. Chem. Phys., 64(5):2013 (Mar. 1, 1975).

CHAPTER 8

ELECTROMAGNETIC MEASUREMENT SYSTEMS SUMMARY FOR RF– MILLIMETER WAVELENGTHS

Chapters 8 to 10 describe measurement procedures, equipment, and data processing that may be used in the RF, microwave, and millimeter frequency spectrum to measure electromagnetic constitutive parameters of materials. This chapter begins with an overview of material and electromagnetic measurement techniques and systems and breaks them into two groups, transmission line and resonant techniques. The second half of this chapter is a review of transmission line techniques that measure material constitutive parameters via the reflection and/or transmission coefficient. The sections describe the technique, the physics that is fundamental to the technique, and then show examples of measured data based on the technique. Comparative data between different measurement techniques are supplied when available. It is noteworthy that during the writing of this book one of my colleagues John Schultz assumed the task of compiling and publishing a detailed monograph on the free space focused beam technique that is summarized at the end of this chapter. The publication details are as follows: Focused Beam Methods: Measuring Microwave Materials in Free Space, ISBN: 1480092851 (2012). Although the discussions in this book contain a summary of the focused beam techniques, John’s publication includes details on system design, error expectations, and the use of the system for measurements other than constitutive parameters. Thus the two books are complementary. A historical review of reflection and transmission line measurements of Von Hippel begins this chapter ’s technical context and is followed by a discussion of measurement technology evolution post-1980. Most of the advances and measurement adaptations were facilitated by the development of digital receivers and refinement of network analyzers. These led to modifications of transmission line, free space, and cavity measurement procedures to optimize interface with the network analyzers and/or receivers. The various measurement fixture and procedural modifications address the characterization of materials with homogeneous or anisotropic constitutive parameters such as magnetic ferrites or composites. However, even the best of the electromagnetic measurements do not eliminate some measurement impediments. As configurations are presented the lessons learned about measurement error sources for various configurations will be annotated as will the

applicability of the measurement techniques to modern composite materials, which often times have characteristic internal geometry and unique dimensional requirements. Constitutive parameter measurements have not been evolutionary but recently were forced to be revolutionary. The material science research community constantly creates new material technologies that are adapted for electromagnetic applications. Frequency selective surfaces (FSS), photonic bandgap (PBG) or photonic crystal (PC) materials, metamaterials, negative index structures, nanotube and grapheme-based materials, ceramic fibers, and artificial magnetic conductors (AMC) are a few examples [1–6]. These materials incorporate controlled anisotropy, inhomogeneities, dimensionality (thin 1D to thick 3D), near-resonant conducting structures, and adaptive/reactive materials to modify electromagnetic reflection and transmission, electromagnetic impedance, and absorption. The material geometries often incorporate unique component dimensions with multiple electromagnetic correlation lengths. Any measurement of the material that measures effective parameters should use material samples which are much larger than the characteristic lengths physically intrinsic to the material (e.g., FSS periodicity, conducting fiber length, and loop dimension). A natural extension of this observation is that measured data, taken from a single measurement technique, may be insufficient to explain electromagnetic observables and/or expose the underlying physics in advanced composite materials. For example, photonic crystals are best understood by combining reflection and transmission measurements with electric field probes within the material [7]. In addition, measurements must be accompanied by better theoretical understanding and numerical models that account for material composition, volume, and surface geometry. Metamaterials and negative index composites are relatively recent material concepts. Historical measurement approaches require improved data processing and computation combined with propagation models to elucidate the underlying physics of these materials. Models and measurement must incorporate numerical simulation of the underpinning material composition and internal geometry to correctly predict observables. Even then modernized measurements cannot rely solely on the historical plane wave scattering models of Chap. 4. The derivation of effective constitutive parameters for a composite may require numerical solvers of Maxwell’s equations. These include the finite difference time domain (FDTD) and method of moments (MoM) models as discussed in Chap. 7 on composite modeling. This or similar combinations of formidable analytical infrastructure and measurement expertise are needed to choose the correct measurement hardware and analyze the measured data. This combination allows the characterization of materials and their composites which facilitate prediction of propagation in geometrical constructs made with those materials. The numerical modeling/measurement combination is the means for advancing the state of the art in materials development and characterization, particularly for the advanced negative index composites that may be electrically inhomogeneous, anisotropic, have unique frequency dispersions, or are responsive to their environment. Having annotated these caveats Chap. 8 proceeds with discussions of historical and modern electromagnetic characterization.

8.1 AN INTRODUCTION TO WIDEBAND MATERIAL METROLOGY In the 1980s, Hewlett-Packard (now Agilent) and Wiltron began production of multiport automated network analyzers (ANA). At approximately the same time, digital receivers and sources (e.g., Scientific Atlanta) appeared on the measurement market. Network analyzers used rapidly stepped, frequency locked sources, and/or frequency synthesized sources with stabilized power levels that approached 1 W. Analyzer-source availability necessitated a revisit of measurement approaches that previously applied slotted waveguide and/or coaxial transmission line with single-frequency phase-locked sources (klystrons). Klystrons and slotted lines had been used by Von Hippel and were the standards described in ASTM technical guidelines [8–10]. The analysis used with slotted lines was adapted to use direct measurement of amplitude and phase rather than amplitude and phase derived from VSWR (voltage standing wave ratio) plane wave interference as observed in the reflection “nulls” measured in a slotted line using a calibrated movable monopole probe in series with crystal detector and voltmeter. The network analyzer and/or digital receiver facilitated accurate direct phase and amplitude measurements over 1000 to 1 bandwidths (e.g., 0.01 to 18 GHz in 7-mm coaxial line). Reproducible phase and amplitude, wideband sources, combined with early digital computers, facilitated error correction and data processing improvements [11]. A single measurement system also had the practical advantage of reduced laboratory space and equipment requirements.

8.1.1 A Wideband Laboratory and Equipment As indicated above, electromagnetic characterization and physical insight of modern composites (and their single-chemistry homogeneous subset) may require measurements over many decades of frequency. Therefore to understand the advantage of the network analyzer, it is worthwhile to visualize the complexity of making reflection and transmission amplitude measurements without the network analyzer over two decades of frequency (1–100 GHz). Figure 8.1 shows a typical configuration which was assembled for the W millimeter wave band, 76 to 110 GHz. A 12-in. ruler is shown in the foreground for scale. The power supply (rectangular box) and swept frequency unlocked IMPATT diode source (black tripod) are on the extreme left. The next is a dial attenuator (black dial). After the attenuator, one observes a –10-dB waveguide coupler. The left side samples the reflection and right hand waveguide coupler arm measures a reference signal. To the right of the calibration is a sample holder made from a short section of straight waveguide. It is followed by another coupler whose left side is terminated by a waveguide load and right waveguide arm is for measurement of the transmitted signal. Note that the waveguide receivers are separated from the waveguide couplers by ferrite isolators (short black cylinders). Isolators are needed to reduce multipath interference due to reflections at imperfect loads or waveguide interfaces. Approximately 1.5 m × 1 m of space is required to implement this W-band reflectometer. Note that the photo includes a single Xn-band (5.85 to 8.4 GHz) waveguide coupler in the background.

FIGURE 8.1 Pre-network analyzer waveguide reflectometer configuration.

The Xn-band coupler is approximately 1 m in length. A simple wavelength dimensional scale of the W-band equipment suggests that at least a 3 m × 1 m area would be required to implement an Xn-band reflectometer, three couplers, sample holder, waveguide detectors, and attenuators. That does not include space required for sources and receivers. A complete configuration with similar wavelength scaled areas would be required for each (at least 12) waveguide bands that might be used to address 1 to 100 GHz. In comparison, a single network analyzer, source, and reflection test set combination replaces the 11 configurations and extends the band width by a factor of 1000 (0.001 to 100 GHz). The ANA replaces source, couplers, attenuators, and phase shifters. If waveguide measurements are to be performed, the geometrical convoluted waveguide-referencemeasurement coupler can be replaced by two coax to waveguide couplers. In summary, the equipment requirements are reduced to one simple test fixture and coaxial-waveguide coupler at each band. The use of network analyzer reduces the required laboratory bench surface requirements to less than 1.5 square meters for S through W waveguide bands. Below S-band coaxial reflectometers would be used. Overall, by using an ANA, the experimentalist obtains a compact electromagnetic material measurement laboratory.

8.1.2 Algorithms and Material Electromagnetic and Physical Scale Assumptions

Calculating effective electromagnetic constitutive parameters of a material from measurement is a derivative computation. The calculation requires some assumptions about electromagnetic propagation and assumptions about the sample under test. Constitutive parameters are derived from meter, micrometer, or millimeter wave reflection and/or transmission coefficient amplitude and/or phase measurements. Constitutive parameter calculations require an electromagnetic propagation model (see Chap. 4). Most propagation models assume that a material’s electromagnetic response is governed by parameters that are independent of the sample size and sample geometry. The assumption is good so long as any sample inhomogeneity size scale, a0, is many times smaller than the electromagnetic wavelength in the material, a0/λ > L, ws >> t. The section concludes with example measurements and configurations. Figure 9.19 is a sketch of one cavity configuration. Figures 9.20 and 9.21 are photographs of a cavity in operation with its first resonance near 150 MHz. Photographs of full band scans and highresolution frequency scans are shown for clarity. Examples of measured data are shown in Figs. 9.22 to 9.25. The first graph shows predicted and measured resonant frequencies from a 78.7-cm-long, 30.48-cm-wide TEM stripline cavity. The center strip is 8 cm wide and 0.8 cm thick. The cavity was designed to produce modes 1 to 5. For center positioned samples odd modes will be sensitive to dielectric properties and even modes to magnetic properties. As Fig. 9.22 shows, predicted and measured resonant frequencies are near overlays with Qs between 200 and 600. Figure 9.23 shows the shift in resonance magnitude and Q when a small quartz fabric sample is inserted at the center position. As proposed, even modes see little change while odd modes shift to lower frequency and smaller quality factor. Figures 9.24 and 9.25 are the measurement end result with calculated values of the effective real and imaginary parts of the quartz fabric permittivity.

FIGURE 9.19 Full perspective of an open-sided cavity. The 60-in.-long, 14-in.-wide cavity would have a fundamental resonance near 100 MHz. Rohacell is suggested as a low dielectric structural foam to keep metallic strips fixed in position.

FIGURE 9.20 A fully enclosed and operating stripline cavity with first resonance near 150 MHz.

FIGURE 9.21 On the left is a photo at a 200 to 1000 MHz span for the enclosed strip line cavity. Five longitudinal modes are shown. On the right is a 745 to 755 MHz scan showing one resonance.

FIGURE 9.22 Measured and predicted resonant frequencies for a 78.7-cm-long, 30.48-cm-wide TEM stripline cavity. Quality factors are measured.

FIGURE 9.23 Resonant frequency and Q shifts for the TEM stripline cavity versus mode number for insertion of a 0.0254cm.-thick quartz fabric material.

FIGURE 9.24 Measured real permittivity for a 0.0254-cm-thick quartz fabric sample in a TEM stripline cavity. Odd mode numbers apply.

FIGURE 9.25 Measured imaginary permittivity for a 0.0254-cm-thick quartz fabric sample in a TEM stripline cavity. Odd mode numbers apply.

9.4 CLOSED REFLECTION CAVITY A typical microwave test configuration for a reflection cavity measurement is shown in the photograph in Fig. 9.26. This configuration uses a network analyzer with a waveguide reflectometer network to measure the reflection coefficient, or input impedance, of a cavity as a function of frequency. A coaxial transmission line is connected from the network analyzer to rectangular waveguide coupler. The coupler is interfaced with a rectangular to circular waveguide transition. The transition ends at the rectangular iris which is formed from a conducting film with a thin (0.15 cm at X band) rectangular slit which opens into the cavity. The slit is aligned parallel to the incident TE field of the waveguide and preferentially excites a TE11 mode electric field in the circular section. The following discussion is restricted to the TE11 mode, but a similar discussion can be found for excitation and use of the TM11 in the third section “Microwave Cavity Methods” of Chamberlain and Chantry’s text [6].

FIGURE 9.26 Reflection cavity.

The network analyzer display shows the characteristic reflection coefficient amplitude versus frequency scan from a reflection cavity. Reflection has a minimum at the resonant frequency of the cavity. Below and above the resonance, the reflection amplitude approaches unity. A sketch of a typical TE11 cylindrical reflection cavity is shown in Fig. 9.27. In the reflection cavity the sample is placed at the shorted end of a section of the cylindrical waveguide of length ℓ. Iris width and cavity length scale with wavelength.

FIGURE 9.27 Sketch of cavity (inside diameter 2a, length ℓ, and test sample of thickness t).

9.4.1 Reflection Cavity Geometry and Fields If material sample dimensions can be prepared with high accuracy and sharp corners, the rectangular to cylindrical transition can be eliminated and a shorted rectangular reflection cavity can be used. Rectangular cavity electric and magnetic fields are the TE10 p modes previously discussed. However, many ceramic samples are hard and fracture easily (e.g., fused quartz). The slow diamond grinding of these materials allows a precision-sized cylinder sample to be made for use in a cylindrical cavity while minimizing any air gaps in the radial direction. Electric and magnetic fields in the cylindrical cavity are periodic in length but have curvature about the center line of the cavity. The fields can be generated from the generating function ψ by the following differential equations [Eq. (9.14)]. The subscript i in the equation has values 1,2 which correspond to air and sample, respectively, and α11 is the first zero of the Bessel function J1:

Resonant frequencies of the TE11n mode in the empty cylindrical cavity are given by Eq. (9.15):

The parameter c0 is the vacuum light speed, ƒc is the cutoff frequency in circular guide, and n is an integer (1, 2, …). The integers correspond to the longitudinal modes. If a rectangular cavity is used, the resonant frequencies are TE10 p. Table 9.1 shows resonant frequencies and dimensions for the nth mode of X through Ka band cylindrical cavities. TABLE 9.1 Cylindrical Cavity Resonant Frequencies (GHz) and Dimensions for X, Ku, K, and Ka bands

The second important cavity parameter is the quality factor and Eq. (9.16) can be used to calculate Q of an air-filled cylindrical cavity. As the diameter-to-length ratio decreases, the Q of the cavity increases but the separation between resonant frequencies can decrease. Thus, there is a trade in cavity design. The length must be short enough that the resonances do not overlap but long enough to produce a high-quality factor for measuring low-loss materials.

The resonant cavity measurements easily adapt to the constant power and frequency stable sources found in modern network analyzers and leverage the capability to measure reflection amplitude and phase. If the cavity resonances are well separated, a parallel circuit model can be referenced to the plane of a detuned short—approximately the plane of the coupling iris. Foster ’s reactance theorem can be applied to describe the impedance and thus parameters of an RLC circuit model of the cavity. The network analyzer can directly convert reflection to impedance which can be fit to RLC circuit. The fitted circuit parameters describe coupling between the cavity and the network analyzer measurement system. A transmission-line model

is used to describe wave propagation inside the cavity. The circuit model describes the cavity behavior from an external viewpoint and the transmission-line model describes the internal behavior. In the circuit model, resonance frequencies are measured frequencies at which the impedance is purely real. The half power frequencies are those at which real and complex impedance values are equal. The half-power frequencies are applied to calculate cavity quality factors. Since the empty cavity has finite loss and the inserted material has complex permittivity and/or permeability, the measured resonant frequencies of empty and samplefilled cavities, ƒ0 and ƒs, will be complex and each is characterized by similar equations, i.e.,

where are the measured, real resonant frequency of empty, subscript 0, and sample inserted, subscript s, cavity. Q0, s are the measured quality factors of empty, subscript 0, and sample inserted, subscript s, cavity. The measured quality factor of the empty cavity is defined by and Δƒ0 is the frequency width at the half power points of the empty TE11 cavity. Calculation of the Q for the sample is somewhat more complex. When the sample under test has a small loss factor, the Q measured with sample inserted, , must be corrected to subtract the loss factor of the empty cavity. Thus, the Q associated with the sample is given by

The cavity experiments can of course be accomplished by using amplitude-only data and fitting that data to a resonant model. However, the admittance approach makes optimum use of a network analyzer. The results are repeatable and noise can be suppressed by data averaging of (e.g., 1028 samples) readings for each frequency.

9.4.2 Reflection Cavity Measurements After the complex frequencies are measured, an electromagnetic propagation model that is appropriate inside the cavity can be used to determine dielectric properties from the complex resonant frequencies. The propagation frequency at resonance is calculated from Eq. (9.17). The sample-loaded waveguide cavity is modeled by a cascaded, short-circuited transmission line. In the empty section of the waveguide that feeds the cavity, the field propagation constant and characteristic impedances are

where

is the free space wave number at the resonant frequency of the

cavity measurement. In the cavity section, containing the sample, the propagation constant and impedance are, respectively,

and

where the subscript s indicates the dielectric property of the sample.

In the transmission line model, the sample-filled transmission line is terminated by a shorting plate with impedance of 0, and on the opposite end from the iris. The cavity loss is characterized as totally inside the cavity. Therefore, the iris is again assumed to have negligible impact on the measurement and is assigned a zero impedance. For our single-mode propagation model, the boundary conditions are satisfied by requiring equal and opposite impedances, looking out from the junction of the input and cavity section of transmission line. That boundary condition produces the modal match equation

and td is the length of the dielectric and where ℓ is the effective length of the empty cavity that is derived from the real empty cavity resonance frequency and Eq. (9.15). This calculation of an effective length is performed since the small reactance that is associated with the iris is represented by a small change in the length of the line. Example data from two reflection cavities are presented in Figs. 9.28 and 9.29.

FIGURE 9.28 Measured resonance frequency and Q of cylindrical reflection cavity with an empty and Rohacell 300 filled cavity.

FIGURE 9.29 Measured resonant frequency and Q shifts of Rohacell 300 sample.

This section concludes by summarizing some measurement errors with the reflection cavity technique. One is accidental or random line-length changes that may be encountered with this particular cavity test configuration and these will impact measured constitutive parameters. Unlike other cavities discussed in this book, the closed reflection cavity requires dismantling (i.e., removal of the short termination) to insert a test sample. The short circuit termination must then be reattached. These procedures imply an inherent time delay between an empty cavity and sample-filled cavity measurement. In addition, the act of replacement may perturb the cavity geometry or “perfection” of the short termination. Any or all of these disturbances can result in a random error in an assumption of the empty cavity resonant frequency, line length, and quality factors. With these cautions in mind, numerical simulations are presented below where all errors are lumped as random variations in resonant frequency and quality factor. This approach includes errors of the previous paragraphs and may include errors due to changes in temperature between measurements. Temperature variation causes random cavity expansion or contraction. Resonant frequencies and Q variations were randomized using a gaussian probability

distribution. As the gaussian distribution width decreases, the percentage variation in parameters decrease with the baseline configuration reached for a zero width. The physical dimensions and measured quantities used for this simulation are shown in Table 9.2. Different levels of variation were assumed in the model. Relative variations in resonant frequency were as follows: TABLE 9.2 Cavity Parameters for Baseline Error Study

0.99 ≤ ƒ/ƒ0 ≤ 1.01, 0.995 ≤ ƒ/ƒ0 ≤ 1.005, 0.999 ≤ ƒ/ƒ0 ≤ 1.001 Quality factor variations were 0.99 ≤ Q/Q0 ≤ 1.01. In addition random errors in cavity length were introduced by assuming a range of cavity temperatures and then applying the thermal expansion coefficient for the cavity metal—here gold. Conclusions are annotated in the following paragraph. Constitutive parameters can be measured to less than 1 percent if temperature can be maintained within ±1°C, resonant frequency measured to 0.1 percent, and Q measured to less than 1 percent. Wall losses that derive from temperature measurement errors (e.g., expansion and change in conductivity) will cause errors in calculated imaginary but not real permittivity. The Q errors will become more significant when a waveguide measurement system with higher resistivity is used and this has significance in higher temperature measurement systems. Measurement errors in resonant frequency will induce errors both in real and imaginary permittivity and those errors are more sensitive to resonant frequency than to errors in temperature measurement (i.e., wall losses). Network analyzer frequency measurements are less than 0.01 percent and Q reproducibility has been measured to less than 1 percent. The error study summary conclusion found that the reflection cavity technique should measure real permittivity to ±0.2 percent and imaginary permittivity to ±5 percent for materials with loss tangents 0.001 and larger.

9.5 OPEN CAVITY: FABRY–PEROT RESONATOR The text has introduced a number of microwave wave reflection, transmission, and cavity measurement techniques. Each can measure permittivity and/or permeability of a material within a set of limitations determined by sample geometry and detector capability. Most techniques can be extended to the electromagnetic millimeter wave or higher frequency bands. However, in almost every technique that has been discussed, the physical dimensions of the samples and components used to construct the test fixtures decrease as the frequency of

measurement increases. At some point (near the millimeter wave bands) the parts become difficult and expensive to fabricate. Maintaining fixtures in pristine condition become onerous. The very small parts (e.g., connectors, transmission line, and waveguide) are difficult to manipulate and align. Further, sample size decreases as frequency increases and the experimenter may have problems with composite samples that have inherent characteristic correlation lengths. The millimeter wave sample can be smaller than a correlation length. Recall that the specimen to be measured must be accurately machined to fit closely into a resonator or waveguide. It has been repeated many times that small air gaps between a sample and transmission line wall can result in large errors in measured data. At millimeter wave and higher frequencies, typical gaps of 0.002 cm become ever larger percentages of a wavelength and such errors become more significant. In closed cavities the quality factor of a cavity resonator may decreases since cavity imperfections play more important roles with increased frequency. Therefore, the measurement of small imaginary parts of permittivity or permeability becomes difficult. A free space transmission technique, such as that discussed in Chap. 8, does not have quality factor issues and could be used without cutting or special preparation of samples. However, the free space transmission technique is not suited for measurements of materials with small imaginary parts of permittivity or permeability. An open cavity resonator (the Fabry–Perot [6, 16, 17]) differs from metallic enclosure resonant techniques. Fabry–Perot reflecting surfaces and component parts are large compared to a wavelength, and currents excited in the reflecting mirror surfaces of the cavity are a much smaller percentage of the available electromagnetic volume. Thus, higher Q values can be obtained and therefore more accurate measurements of low-loss materials can be achieved. Like the free space technique, the open resonator has minimal sample preparation requirements. The semiconfocal Fabry–Perot requires a flat sample which covers (or is larger than) the surface of its flat mirror. Many times free space transmission and Fabry– Perot can use the same sample in testing. Finally, since the open cavity radiates a multiwavelength size area on a sample, it has similar advantages to the free space transmission system for measuring effective constitutive parameters of composites. Figures 9.30 to 9.34 depict basic confocal and semiconfocal Fabry–Perot open resonator configurations. All could be used in material measurements. The first configuration uses half reflecting planar mirrors that are fed by antennas (Fig. 9.30). Figures 9.31 and 9.32 show configurations using two concave mirrors.

FIGURE 9.30 A Fabry–Perot made by using half reflecting planar mirrors that are fed by two antennas.

FIGURE 9.31 Fully confocal Fabry–Perot resonator and geometry.

FIGURE 9.32 Fully confocal Fabry–Perot resonator with centered sample illustrating the beam waist.

FIGURE 9.33 Semiconfocal (left) and fully confocal (right) Fabry–Perot geometry, with a single dielectric placed at the measurement position.

FIGURE 9.34 The semiconfocal Fabry–Perot geometry and the measurement configuration.

The configuration in which one plane mirror is placed at approximately half the radius of the concave mirror is known as a semiconfocal Fabry–Perot. It is shown in Figs. 9.33 and 9.34. In the semiconfocal arrangement the plane mirror is often moved to tune the resonant frequency. The flat mirror is positioned near the center of curvature of the concave mirror and in the plane of symmetry along the z-axis of the confocal design. It has a controllable beam radius and experimenter-specified beam waist (i.e., radiated area) at the plane mirror. The beam waist can be contracted in the design to approach λ/2 in size, and thus the resonator can measure small samples. The energy is typically coupled into and out of the resonator by small coupling holes or irises symmetrically placed about the center of the concave mirror. The semiconfocal Fabry–Perots of Figs. 9.33 and 9.34 are the measurement configurations which will be analyzed in this section. They are also the configurations applied to measure polymer and fabric–polymer millimeter permittivity. Data are presented in Chap. 12. The cavity has a high-quality factor and is sensitive to even very thin (film like) samples placed in the cavity. The test configuration for a thin material is shown in Fig. 9.34. If the material is not self-supporting or cannot be placed on the flat mirror, the material may be placed on a previously characterized supporting dielectric and the combination is placed on the flat conducting mirror.

9.5.1 Open Cavity Geometry and Field Analysis The analysis of the cavity is taken from Cullen and Yu [17]. The fields, resonant frequencies, and quality factors can be derived from the knowledge of the dominant propagating cavity modes and these were derived by Cullen and Yu in a paraxial approximation to the wave

equation.

The equation has solutions of the form

where and ϕ = (m + n + 1)arctan(z/z0). The parameter and R, w0, ƒ, and c0 are the radius of curvature, beam waist, frequency, and light speed, respectively. The variables r and z are radius at location z along the beam axis. These two geometric parameters are shown in Fig. 9.34. The fundamental propagating mode has m = n = 0.0 and the field is

One observes that the field is TEM, i.e., a cutoff wave number is not apparent. There is a gaussian amplitude radial distribution. Near the axis (i.e., r < R) the phase is that of a plane wave propagating in the z direction. The TEM resonance condition is determined by the boundary condition for tangential E to be zero at the surface of the flat and curved mirrors. That condition requires that

D is defined in Fig. 9.34. Notice that the field is very similar to that of the focused beam free space system that has been discussed in Chap. 8. However, the reflecting mirrors generate a cavity in which the field reflects many times. The TEM resonance occurs first at frequencies where cavity length is a multiple of half wavelengths. These are of course restricted to the frequency span of the waveguide that feeds the cavity. In principle the cavity can remain at fixed length and broadband measurements can be performed by replacing waveguide feeds at different bands. If effects of the irises are ignored, many TEMqnm modes (q, m, n run from 0 to ∞) can propagate in the resonator. However, a symmetric positioning of the coupling irises (one transmit and second receive) supports excitation of the TEMq0n modes. Other nodes are inhibited since they have a node of electric or magnetic field at the center of the mirror. The

TEMq0n modes have an antinode of magnetic field at this point. For large values of n, the energy is distributed over larger distances from the axis of the resonator and may be lost by falling outside the reflector. This property can be advantageous since the number of modes which can appear is limited. The resonances having mode numbers q, 0, 0 have axial lengths of roughly (q + 1)D/2 and these are the modes which are used for measurements of dielectric properties. In the semiconfocal cavity that is applied for measurements of Chap. 12 waveguide is used to feed both transmit and receiver irises. That configuration is shown in the photographs of Figs. 9.35 and 9.36. It is desirable to operate the cavity over full waveguide bands since this allows spectrally dense measurement and facilitates detection of even small sources of frequency dispersion. Since this open resonator was designed for high losses in essentially all non-fundamental modes, measurements require only fundamental modes.

FIGURE 9.35 Free-standing semiconfocal Fabry–Perot. Ka band waveguide feeds are shown.

FIGURE 9.36 Curved mirrors showing positioning of center positions of the two irises—best seen on the right-hand mirror photograph.

9.5.2 Open Cavity Design Parameters The Fabry–Perot cavity of Fig. 9.35 was used in acquiring data which appear in the database of Chap. 12 and presented at the conclusion of this section. The resonator has a spherical mirror with the two feed irises placed symmetrically off center. The second mirror is flat. It is located at half the radius of curvature of its spherical mirror and has diameter of approximately 12 cm. This size is much larger than the 2 to 4 wavelength spot size which is generated at the mirror surface. The cavity is 50 to 51 cm in length. The structural components (i.e., tri-legged support) remain fixed but separate spherical mirrors, spherical mirror mounts, and waveguide assemblies are swapped into the structure to cover Ka, V, U, W, D, and F bands, essentially the complete 26.5- to 160-GHz frequency range. The 50- to 51-cm length is used from Ka to U bands and is reduced to about 25 cm using a separate mirror mount for measurements above U band. In order to achieve broadband performance, two coupling irises are used since a single iris reflection cavity performs best when the iris is critically matched to one or few frequencies. Though it has a reduced Q, the dual iris configuration can be used over a full waveguide band operation. The direct coupling between irises has been minimized in design of the spherical mirror and coupling magnitude is not altered when the sample is placed on

the flat mirror. In principle (and often seen in history of the technique), the flat mirror can be moved to tune the cavity resonance frequency. However, the procedure that best leverages the modern network analyzer uses a fixed position flat mirror and measures a frequency scanned S12 (iris to iris). The frequency breadth, scan density, and power stability of the analyzer make the scanned procedure most accurate in locating resonance frequency shifts due to electrically thin samples. Wells et al. [18] examined the interaction of throughput and loss mechanisms as a function of their effects on the variance of the resonant frequency and loss factor. This interaction was modeled by a Monte Carlo analysis of the propagation including statistical errors. The analysis determined that a center-to-center iris separation of roughly four wavelengths at the design frequency represented a compromise between direct iris-to-iris coupling and excitation of the fundamental mode. In the determination of the radii of the reflectors, the radii was to be large enough to ensure a sufficient taper (about 30 dB) of the field distribution at the reflector edge and small enough to attenuate higher order modes. An important determiner for attenuation of higher order modes is the Fresnel number N. It is of , where rm is the radius of the spherical mirror and rf is the radius of the flat mirror. By a proper choice of N, one can discriminate against higher-order modes by increasing their losses relative to that of the fundamental mode. The Fresnel number is typically chosen to be about 1 [17]. Lastly, the relationship between the radius of the curvature of the curved mirror R0 and the resonator length L is chosen to yield a modal spot size large compared to a wavelength. This is appropriate for measurements of composites and allows for finite correlation lengths of the composite to be included within the radiated spot. A correlation length example is found in measurements of honeycomb structures, radome, and fabric-polymer laminates of Chap. 12.

9.5.3 Open Cavity Material Measurements In measurements discussed in Chap. 12, planar samples, either single layer or thin films on support layers, were placed on the plane mirror as indicated in Fig. 9.34. The sample should be planar to avoid scattering of energy outside the cavity. The radius should be large enough to extend beyond the beam waist. The procedure for determining the permittivity at a particular frequency is as follows. With a network analyzer, the resonance frequencies and quality factors for each empty cavity resonance are obtained by measuring S12 from input to output iris throughout the waveguide band. Resonance and Q of the resonator are calculated by fitting amplitude to a Lorentzian function using about 50 to 100 frequencies centered on the apparent resonance. The frequency at the predicted Lorentzian peak and half power frequencies are then used as the resonance, and to determine Q. The sample is then placed in full contact with the flat mirror and the analyzer is again scanned to determine the new sample-loaded resonant frequencies and quality factors. As with other cavities, the empty Fabry–Perot still has finite loss and the inserted material has complex permittivity. Therefore, measured resonant frequencies of the empty sample and

the cavity with the planar sample in positions ƒ0 and ƒs will be complex and each is determined by a similar equation.

As with the closed TE11 cavity of the previous section, are the measured, real resonant frequency of empty cavity, subscript 0, and the cavity with sample, subscript s. Q0,s are the measured quality factors of empty cavity, subscript 0, and the sample inserted cavity, subscript s. The measured quality factor of the empty cavity is defined by and Δƒ0 is the frequency width at the half power points of the empty cavity. The calculation of Q for the sample is somewhat more complex. The Fabry–Perot is used for samples which have small electrical loss. Therefore, Q measured with sample inserted, , must be corrected to subtract the loss factor of the empty Fabry–Perot. Thus, the Q associated with the sample is given by

Since it poses a more general measurement configuration, the cavity configuration in which a thin dielectric is placed on a support dielectric will be considered for analysis. Solutions to permittivity for a single layer are degenerated to dual-layer solution and can be obtained by setting one layer ’s thickness to zero. A cavity resonance corresponds to a pole (matched admittance or impedance) of the cavity input admittance in the complex frequency plane. The test cavity has length L. It contains a flat sample with two layers of dielectric of thicknesses t1, t2. It is assumed that sample 2 is the support layer and lies on the flat mirror. The resonance frequencies correspond to a zero of the following transmission line cascade equation:

The propagation constants βi in each medium are given by . The thickness t0 = L – t1 – t2 is the length of free space between curved mirror and the upper surface of the sample. For maximum accuracy, the cavity length, L, should be determined from an empty cavity resonant frequency which is near the frequency for which the calculation is being made. The ϕi are included for generality and describe phase corrections that can arise from the finite curvature of the spherical mirror. The phase corrections can be

calculated by matching boundary conditions for modal wave functions at each interface. As R0 approaches infinity the corrections approach zero and the equation has the form of the resonant condition for a closed cavity but with TEM modes. In the analysis which calculated material properties in Chap. 12, it was assumed that the sample under test was a dielectric and had the permeability of free space. If magnetic materials (e.g., iron, ferrite composites) are to be measured in the cavity, the procedure must be modified to include a second independent data set using different test sample geometry. This may be different sample thickness or a different supporting dielectric. Two equations like Eq. (9.28) must then be iteratively solved for permittivity and permeability. More detail on measuring permeability can be found in publications by Afsar [19, 20]. Figures 9.37 to 9.39 show examples of data from the Fabry–Perot of Fig. 9.35. Figure 9.37 shows the characteristic linear increase in measured TEM resonant frequency for unit increases in mode number. Ka, V, and U band measurements are shown. Resonance separations are approximately 0.295 GHz as would be predicted for a 50- to 51-cm cavity.

FIGURE 9.37 This figure shows measured resonant frequencies (solid circles) for a 51-cm length cavity (Ka band) and 50.8-cm cavity (V–U band).

FIGURE 9.38 Data show a Lorentzian fit to one Ka-band resonance. The Q is about 100,000 and is characteristic of Ka to U band values (e.g., 50,000 to 100,000).

FIGURE 9.39 Data show a Lorentzian fit to one W-band resonance. The Q is about 20,000 and is characteristic of W–D band values. Data has more noise since a multiplier is used to advance from Ka–U bands to W–D bands with about 20-dB loss.

Examples of frequency scanned S12 are shown in Figs. 9.38 and 9.39. In the lower mmW bands (below U and away from atmospheric absorption bands), Q values were measured to be 50,000 to 100,000. Those bands are accessible with higher power network analyzer sources. For W band and above, frequency measurements used base band sources (i.e., K, Ka, and U) which were used to drive waveguide doublers and tripplers to produce phase-locked sources. However, there was a significant loss in output power and signal-to-noise ratio. Apparent quality factors were reduced to 20,000 to 30,000 (Fig. 9.39). Examples of permittivity measurements carried out with the Fabry–Perot of Fig. 9.35 are shown in the following four figures (Figs. 9.40 to 9.43). Figures 9.40 and 9.41 plot, respectively, real and imaginary parts of the permittivity of a 6.3-mm-thick Teflon glass composite panel, a circuit substrate. Though composed of a glass fiber, PTFE mixture, this material is largely homogenous and isotropic at mmW and lower frequencies. Note the apparent discontinuity in permittivity just above 40 GHz is only .01 a 0.4 percent change.

FIGURE 9.40 Semiconfocal Fabry–Perot measurement of the real permittivity for a 0.25-in. Teflon glass composite panel at selected resonant frequencies.

FIGURE 9.41 Semiconfocal Fabry–Perot measurement of the imaginary permittivity for a 0.25-in. Teflon glass composite panel at selected resonant frequencies.

FIGURE 9.42 Real permittivity measurements of a 3/16–in. cell honeycomb sample at a 0- and 90-degree rotation about the surface normal.

FIGURE 9.43 Imaginary part of the permittivity measurements of a 3/16-in. cell honeycomb sample at a 0- and 90-degree rotation about the surface normal.

Due to their geometry, many dielectric honeycomb materials have anisotropic permittivity but even at mmW wavelengths their unit cell dimension allows them to be approximated by effective permittivity. The anisotropy is small and may not be detected unless a sensitive measurement technique is used. Honeycomb composites are constructed from polymer film (Kevlar, polycarbonate, or Nomex) and are often used as low dielectric constant, low weight structures for aircraft, automobiles, and naval craft. Figures 9.42 and 9.43 demonstrate the high resolution of the Fabry–Perot and its ability to resolve even small anisotropy in materials. Measured data are for a honeycomb with 3/16-in. cell size. The honeycomb sample was prepared by cutting the section perpendicular to its cell axis. Two measurements are shown. In the 0-degree case the incident electric field has been oriented parallel to a cell wall. In the second 90-degree case, the test sample has been rotated about its cell axis by 90 degrees. Anisotropy is evident in both the real and imaginary parts of the permittivity. Steps present data round off. In conclusion, a number of procedures and constraints are repeated about the Fabry–Perot system used to measure data of Chap. 12. 1. A test sample’s constitutive parameters are determined by measuring the change in resonant frequency and the quality factor. However, the two measured parameters are

only sufficient to calculate two (real, imaginary) constitutive parameters. The determination of a complex permittivity and permeability would require two independent measurements of the quality factor, Q1Q2, and two resonant frequencies, . The measurements might be obtained by using two different thickness of the same material. However, the materials tested with the Fabry–Perot (especially at millimeter wavelengths) and listed in the database were nonmagnetic and a single thickness was sufficient. 2. The Fabry–Perot quality factor and resonant frequencies were not taken directly from the network analyzer but S12 were measured at a number (about 100) of frequencies about the apparent resonant frequency. The resonant frequency Q and peak amplitude applied in permittivity calculations were then derived from a Lorentzian function fit to the measured amplitude data by a constrained three-parameter minimization of the sum of the squared deviation. 3. In order to calculate accurate permittivity data from the measurements, the empty cavity resonant frequency was applied to calculate an effective length of the cavity. The inverse Q of the empty cavity is subtracted from that of the loaded cavity to determine the quality factor applied in complex resonance frequency with the sample present. 4. Finally, in calculation the cavity length and empty cavity Q are assumed constant or interpolated to the loaded cavity resonant frequencies. Equation (9.28) is solved iteratively by starting with an initial guess for the sample’s real and imaginary parts of the permittivity.

REFERENCES 1. James R. Birch, et al., IEEE Transac. Microw. Theory Tech., 42(6):956–965 (1994). 2. A. MacDonald, R. L. Moore, and P. Friederich, Antennas and Propagation Society Symposium Digest (Cat. No. 91CH3036-1), 3:1668–1671 (1991). 3. P. Friederich, R. L. Moore, and J. Larsen, Antennas and Propagation Society Symposium 1991 Digest (Cat. No. 91CH3036-1), 3:1672–1675 (1991). 4. R. L. Moore, A. MacDonald, H. Ross Moroz, Microwave Journal, 34(2):67–68, 71, 73, 75–82 (Feb. 1991). 5. R. L. Moore, M. C. Thompson, and T. S. Robbins, Review of Scientific Instruments, 61(3):1136–1142 (Mar. 1990). 6. J. Chamberlain and G. W. Chantry, eds., High Frequency Dielectric Measurements, IPC Science and Technology Press, Surrey UK (1 August 1973). 7. J. D. Jackson, Classical Electrodynamics, 2nd ed., Sec. 7.10, ISBN 0-471-43132-X, Wiley, New York (1975). 8. R. F. Harrington, Time-Harmonic Electromagnetic Fields, ISBN-07-026745-6, McGraw-Hill New York, l (1961). 9. S. Ramo, J. Whinnery, and T. Van Duzer, Fields and waves in Communication Electronics, 3rd ed., ISBN No. 047-1585513, Wiley Hoboken NJ (1994). 10. Jyh Sheen, “Measurements of Microwave Dielectric Properties by an Amended Cavity Perturbation Technique,” Measurement, 42:57–61 (2009). 11. L. F. Chen, C. K. Ong, and B. T. G. Tan, “Amendment of Cavity Perturbation Method for Permittivity Measurement of Extremely Low-Loss Dielectrics,” IEEE Trans. Instru. Meas., 48:1031–1037 (1999). 12. R. A. Waldron, “Theory of the Strip-Line Cavity Resonator for Measurement of Dielectric Constants and Gyromagnetic Resonance Line-Widths,” IEEE-MTT, 123–131 (Jan. 1964); R. A. Waldron, “Theory of the Strip-line Cavity Resonator,” Marconi Rev., 27:30–42 (1964); R. A. Waldron and S. P. Maxwell, “Note on the Measurement of Material Properties by the Stripline Cavity,” IEEE Trans. Microw. Theory Tech., MTT-13:711 (Sept. 1965). 13. C. A. Jones, L. Muth, J. Baker-Jarvis, Y. Kantor, J. DeFord, and P. Wallen, “Stripline Resonator Analysis Using Finite Element Codes,” Abstracts Nat. Radio Sci. Meeting, CO (Jan. 3–7, 1995).

14. C. Wiel, Chriss A. Jones, Yehuda Kantor, and John H. Grosvenor, Jr., IEEE Trans. MTT, 48(2) (Feb. 2000). 15. H. M. Musal, Jr., “Demagnetization Effect in Strip-Line Cavity Measurements,” IEEE Trans. on Magnetics, 28:3129–3131 (1992). 16. W. Culshaw and M. V. Anderson, Proc. IEE, 109:820 (162). 17. A. L. Cullen and P. K. Yu, Proc. R. Soc., A325:493 (1971). 18. T. B. Wells, R. L. Moore, and D. C. Hicks, Millimeter Wave Materials Measurement by Fabry–Perot, Proceedings SPIE International Society for Optical Engineering, 544:185–188 (1985). 19. Shu Chen and M. N. Afsar, “Fabry–Perot Open Resonator Technique for Dielectric Permittivity and Loss Tangent Measurements of Yttrium Iron Garnet,” IEEE Trans. Mag., 43(6):2734–2736 (Jun. 2007). 20. K. A. Korolev, Shu Chen, and M. N. Afsar, “Dielectric and Magnetic Measurements on Ferrite Ceramics at Millimeter Waves,” 2007 IEEE Instrumentation and Measurement Technology Conference Proceedings, pp. 1225–1229 (2007).

CHAPTER 10

TRANSMISSION LINE, FREE SPACE FOCUSED BEAM AND TE10N MEASUREMENT DETAILS

This chapter describes procedures and methods for determining constitutive parameters from the transmission line, or free space, and TE10N cavities. In particular the cavity is investigated in greater detail than in Chap. 9. An extension of analysis is made to the measurement of anisotropic materials and also the use of the cavity for elevated temperature measurements. Elevated temperature transmission line and free space methods are also reviewed. However, accuracies of these two techniques suffer from difficulties with calibration at temperature. It is difficult to make calibration measurements (at temperature) and sample measurement (at temperature) simultaneously (near in time). In some configurations the measurements are significantly disjointed in time, i.e., days. The separation in time leads to variation in electrical line lengths and loss in high-temperature fixtures such as sintering of oven endcaps or oxidation of metal transmission lines. Overall, additional measurement errors appear and measured parameter reproducibility suffers. Mediation approaches for high-temperature transmission line and free space require parallel calibrations separate from the heating zone. An example will be discussed. In contrast, the cavity can be configured to allow calibration and material measurement, at temperature, within tens of seconds to a few minutes. The confluence of measurement and calibration does have costs in sample preparation, equipment alignment, and ability to achieve a dense frequency data set. The cavity is preferred for small samples up to temperatures near 1200°C and its measurement procedures will be discussed in detail. Examples of measurements for all techniques will be presented. Analyses for inverting transmission line or free space reflection and transmission are extensions of plane wave scattering presented in Chap. 4. Chapter 9 added a summarized analysis of common error as applied to transmission lines, i.e., air gaps or imperfect sample fits. National Institute of Standards and Technology (NIST) references [1–3] present very detailed error analysis for transmission line measurements. Therefore, the reader is referred to those for error analysis and correction details. The recent publication by Schultz [4] on free space techniques addresses similar error analysis for focused beam systems and thus the reader is also referred there with one exception. That is the reader might reference Chap. 6’s discussion of errors that can be incurred with composites that contain large inhomogeneous

regions. The determination of permittivity and permeability in waveguide cavities was summarized in Chap. 9. They required accurate cavity volume, sample volume, and measurement of frequency shifts. Difficulties in achieving stable resonance frequency and high-quality factors (necessary for low-loss material measurements) in high-temperature designs are investigated in this chapter. Many high-temperature measurement problems are related to the limited choice of metals that are available to construct transmission lines and cavity. As discussed in Chap. 9, preferred sample shape for a TE10N cavity is a high-aspect cylinder or rectangular solid. These shapes allow anisotropy in parameters to be discerned. Though high-aspect samples are preferred, cavity configurations and analysis can be modified to measure very small volumes with other shapes. These include spheres and thin films. This chapter expands analysis to anisotropic materials and error corrections for materials with large constitutive parameters (e.g., εR or μR > 10). The analysis of Chap. 10 also discusses error correction for the finite dimension of the samples, variation in cavity volume, and undesirable coupling of electromagnetic modes within the cavity. As one may judge from the prior paragraphs, many error contributors remain in transmission line or cavity techniques. However, after 35 years of experience the author still suggests that when equipment availability is limited (e.g., due to cost or space) a combination of free space focused beam reflection and transmission cavity configuration is a best paring for characterizing composite materials. The cavity can address many composites, temperatures, and frequencies so long as the test sample is small and dense frequency data are not required. Focused beam systems measure composite samples with cross section (volume) as small as λ/2 and the technique easily adapts to large, planar samples, with samples often 10 – 100λ2 in cross section. In the case of very large samples, the focused configuration allows the experimenter to easily measure many different regions of large materials in production, at different electrical scales with two orthogonally polarized plane waves. Thus, the experimenter or quality control engineer can acquire data on homogeneity and material anisotropy. When combined with broad frequency band diffuse scattering measurements, composite morphology can be inferred for composites formed from macro or micro scale components [4].

10.1 CONSTITUTIVE PARAMETER SOLUTIONS IN COAXIAL TRANSMISSION LINE, RECTANGULAR WAVEGUIDE, AND FREE SPACE In transmission line configurations, electromagnetic observables are the basis for characterization of composites. This requires accurate, reproducible measurements of voltage reflection and transmission coefficients, R(f) and T(f). Absolute plane wave reflection and transmission measurements require calibration to known standards. Normally reflection is calibrated to materials (conductors) which have a reflection magnitude of unity and a 180degree phase shift relative to the incident field. The reflection calibration is ideally located in

space where the rear surface of the sample under test will be placed. Positioning must be reproducible and the metal should be impervious to surroundings so that reflection magnitude does not vary over time. Transmission measurements require calibration using a transmission line or geometry with known, nonfrequency dispersive line length. The transmission calibration line should not contain any nonlinear elements and the line length (which is translated to phase) should be stable over time. If waveguide or transmission lines are the designated test fixtures, the material samples will be small (3 GHz). The network analyzer records complex reflection data at approximately 1024 frequencies within the specified bandwidth. Reflection data are corrected for waveguide dispersion and Fourier transformed to produce complex data which is a map of characteristic

impulse responses within the transmission line. Impulse is mapped to position.

FIGURE 10.13 Concept for data processing extreme temperature reflectometer measurements.

All mismatches in the transmission produce reflections that are located with respect to the position of the source. The resolution depends on the bandwidth of the measured data and by using more than 3 GHz of bandwidth, responses that are separated by less than 6.5 cm can be resolved in the time-domain data. The frequency bandwidth and time-domain resolution are inversely related; the wider the measured data bandwidth, the better time-domain resolution, so wider band frequency measurements provide better impulse simulation. This system was implemented at X and Ku bands. Since bandwidth was greater in Ku band (typically 4 to 5 GHz) corrections for mismatch was generally easier to achieve. The background (error terms) return is extracted by establishing a reference using the external calibration short Re. The use of Re successfully reduces the number of calibration measurements and significantly improved dielectric measurement accuracy. In practice, elevated temperature measurements of only one empty heated sample holder were required to obtain stable calibrations for multiple sample measurements. By combining the impulse response and external calibration, the reflectometer

measurement procedure is established. Immediately preceding each sample measurement, at temperature Ti, the external calibration short is inserted into the waveguide and a reflection measurement was recorded, Re(Ti, ƒ). That data acquisition was followed by the single measurement of the empty sample holder, Rh(Ti, ƒ). If those data were already acquired, a sample measurement would be taken, Rsa(Ti, ƒ). All files were modified to account for the waveguide dispersion and were then Fourier transformed, FFT, to produce corresponding impulse responses.

In Fig. 10.15, the sample location is identified by the impulse response at time t5 and the external calibration is at time t3. Having identified the return from these two positions, the next step in data processing is the application of numerical filters to isolate the responses.

FIGURE 10.15 Configuration of a focused beam measurement. Photograph 1991–1992.

An inverse Fourier transformation (FFT-1) is then applied to the sample data and external reflection data. The external calibration data are phase-offset by the separation distance between t3 and t5. After the offset, the two reflections are ratioed to establish the reflection coefficient of the sample, , backed by the electrically conducting graphite termination, i.e.,

The sample permittivity at temperature Ti and frequency f is then calculated using an iterative solution of Eq. (10.2). The procedure has been successfully applied to characterize ceramics of Chap. 12 in X, Ku, and Ka waveguide bands. Temperatures up to 2100°C have been achieved. Figure 10.14 is an example of one measurement for alumina. The measurement has been discussed in Chap. 2. The rapid increase in imaginary and rapid decrease in real permittivity above 1500°C result from the alumina degradation and formation of aluminum particulates which then deposit on the walls of the graphite sample holder.

FIGURE 10.14 Alumina εR and εimag (right axis and squares). Data were taken in Ku band using approximately a 5-GHz bandwidth.

10.3 FREE SPACE FOCUSED BEAM CHARACTERIZATION OF MATERIALS Chapter 8 has introduced the focused beam measurement of materials technique and Ref. [4] supplies detail on system design along with material, scattering, and radar cross section measurements. In this section, emphasis is placed on the use of the system to perform elevated temperature measurements of materials over broad frequency bandwidths (15 to 20 GHz). The broad bandwidth is feasible since TEM propagation is characteristic of the system, the extreme data acquisition bandwidths available using modern network analyzers, and the ever increasing bandwidths of antenna feeds. The focused beam free space system can characterize samples that are as small as 1λ in cross-sectional dimension but over narrower bandwidths. Wideband measurements are best performed with larger samples and these are often multiple free space wavelengths in size at the lowest frequency of measurement, i.e., areas greater than approximately 25 λ2. The larger size samples mediate problems encountered with embedded constituent correlation lengths that are characteristic of many composites and metamaterials. However, large samples for elevated temperature measurements suggest large ovens in which the sample can be mounted and also maintain a uniform temperature throughout the material. A focused beam system, i.e., Fig. 10.15, leverages network analyzers or other digital receivers to measure reflection and/or transmission coefficient of the material. It establishes a reflection amplitude and phase reference by placing a metallic plate in the center between its two lenses and measuring at 1028 to 2056 frequencies in the operational bandwidth. The transmission reference, , is established by removing the plate and measuring the straight through amplitude and phase at the same frequencies over the same bandwidth. As with the previously discussed high-temperature reflectometer, the focused beam system leverages the Fourier transform of reflection and transmission data (plate reflection, through transmission, and the same with samples in place) to digitally filter scattering from transmission line transitions, antenna mismatches, and other artifacts such as imperfect transmission from endcaps that are needed for oven operation. Unlike the reflectometer, the propagation is TEM, waveguide dispersion corrections are not necessary, and the position of the sample can be located in the system impulse response to accuracies of 1 cm. After filtering sample and calibration data and inverse transforming them, data ratios can be calculated to measure reflection and transmission. The appropriate combination of Eqs. (10.1) to (10.4) can be used to calculate permittivity and permeability. If a nonmagnetic material is tested, any single equation can be applied. Example room temperature data are shown in Figs. 10.16 and 10.17.

FIGURE 10.16 Free space focused beam measurement of permittivity for a commercial Fe–polymer composite at room temperature.

FIGURE 10.17 Free space focused beam measurement of permeability for the commercial Fe–polymer composite of Figure 10.16 at room temperature.

10.3.1 High-Temperature Focused Beam Measurements One benefit of the focused beam measurement is the ability to make free space measurements with sample size (25λ2 at the lowest frequency) and with a large measurement bandwidth. A second benefit arises from the focusing aspect. At the sample the measurement is performed with negligible illumination at the sample edge. Thus, the sample can be placed in a robust sample holder with minimal impact on the measurement accuracy. As one moves away from the sample position, the beam has a cone-like spread. However, it is still confined within a calculable radius. Therefore, if another, low loss and low dielectric material, with radius larger than the beam, is added some distance before and after the sample, the experimenter only adds additional phase. The combination of beam shaping while maintaining a good plane wave at the sample position allows the sample to be totally surrounded by a cylindrical oven. The oven endcaps should have low dielectric constant, have low electrical loss tangent, and be very good thermal insulators. Figure 10.18 shows a typical configuration.

FIGURE 10.18 Configuration for a high-temperature focused beam measurement.

The choice of the oven is made to maximize diameter and minimize length. The experience has shown that Series 3420 Silicon Carbide split tube furnace which is constructed by Applied Test Systems (ATS) is an acceptable choice. The furnace has a stainless steel shell with nickel-plated end flanges and stainless steel mesh terminal covers. The furnace insulation is a low “K” factor vacuum cast ceramic fiber to provide minimum heat loss, hightemperature capability, rigid structure, and low dielectric mismatch to the transmitted beam. The furnace can be heated to a maximum 1500°C. Figure 10.19 shows a close-up of the furnace as configured with the focused beam measurement system.

FIGURE 10.19 Close-up of the furnace used with the focused beam system.

A temperature control system regulates the power applied to the resistive heating elements to reach and maintain the desired temperature as measured by a control thermocouple. The critical functionality of a furnace design is enabling electromagnetic fields to propagate to and through the sample while surrounded by the furnace. The design of Fig. 10.19 has about 46-cm openings at the furnace front and rear and the diameter is sufficient to fully encompass the focused beam. The ends are capped with a foamed alumina composite ceramic. The material was chosen for its low electromagnetic loss and stability of permittivity with frequency. Though endcaps are largely transparent to electromagnetic energy, properties vary somewhat with temperature and over time. That ultimately reduces measurement accuracy, especially for low electrical loss ceramics. Though calibrations cannot be performed in real time, the amplitude and phase variations or drifts are observed just as with the reflectometer measurement. Sample holders are fabricated from a ceramic fiber board rated to 1600°C. The sample holder is in two pieces to support the sample between them and seat the sample position and orientation. Figures 10.20 and 10.21 show measured data. Typical error bars for real permittivity and material loss tangent measurement are ±5 to 10 percent. The actual error depends upon absolute permittivity magnitude and stability of the calibration for that day.

FIGURE 10.20 Ka-band real permittivity of an alumina foam as a function of frequency.

FIGURE 10.21 Ka-band imaginary permittivity of an alumina foam as a function of frequency.

Of course, the measurement technique is not limited to furnaces operating at elevated temperatures. The furnace configuration may be replaced by a hot-cold environmental chamber such as the one in Fig. 10.22. Again, the chamber which operates from about –100 to 200°C is equipped with large low dielectric constant endcap apertures. The beam is transmitted through the chamber. The sample is mounted as in the furnace. Since the endcaps are not exposed to extreme temperatures in this measurement, calibrations have better stability. Since temperature extremes are limited, endcap properties do not vary significantly with temperature or with time. A typical frequency-dependent permittivity measurement at room temperature is shown in Fig. 10.23. Data were acquired in over two bands; 800 MHz to 6 GHz and 4 to 18 GHz. Note there is excellent data overlap. Room temperature data is followed by measured data for X-band at various temperatures in Fig. 10.24.

FIGURE 10.22 Focused beam measurement configured for environmental chamber material tests.

FIGURE 10.23 Measured frequency dependence of a polymer–fabric composite.

FIGURE 10.24 Measured permittivity of the composite of Fig. 10.23, averaged over X-band, at temperatures from –100 to 200°C.

The same configuration has been extended to the measurement of materials at cryogenic temperatures. By reducing the chamber size to approximately 1000 cc and adding liquid nitrogen cooling with a cold finger attachment, measurements of ceramic materials and superconductors were demonstrated. The significant reduction in chamber size and therefore chamber apertures (i.e., endcaps) does not easily map for measurements at microwave frequencies. Recall that the sample needs to be approximately 5λ in cross section to make a broadband measurement and this would be bigger than the chamber at 10 GHz. However, measurements can be made at millimeter wave frequencies. At W band (75 to 110 GHz) the sample only needs to be about 2 cm in cross section and the aperture 4 to 5 cm. The following figures illustrate some interesting results from such a measurement. Figure 10.25 begins the discussion with the characterization of cordierite, a very low loss tangent dielectric that is used in resonators. Focused beam transmission measurements were performed using a 10-GHz bandwidth (88 to 98 GHz) and the time domain processing was used to extract transmission data for the sample. Calibration transmission measurements were performed a few hours before the sample was measured. Only real permittivity data are shown (93 and 97 GHz) since the calibration varied by tenths of a decibel and thus were not accurate enough to extract loss tangent information.

FIGURE 10.25 Permittivity of cordierite (97 and 93 GHz) from 100 to 300 K.

So long as a free-standing single dielectric is used permittivity measurements are straight forward. However, thin films like high Tc superconductors are often of interest at cryogenic temperatures. Those thin films may be grown on a substrate that acts to match crystal lattice and facilitates the growth of the superconducting crystal. Therefore, appropriate characterization of the thin film necessitates measurement of the substrate. After measuring, the S-matrix analysis of Chap. 4 can be applied to extract the data for the thin film. Yttrium-Barium-Copper-Oxide (Y-Ba-Cu-O) and Tantalum-Calcium-Barium-CopperOxide (Tl-Ca-Ba-Cu-O) films were grown on the matching substrate YSZ (Yttria-stabilized Zirconia). These were then placed in a cold chamber and W-band transmission data were acquired from the laminate. Figures 10.26 and 10.27 show typical transmission measurements.

FIGURE 10.26 Measured W-band transmission coefficient versus the temperature of the thin film YBa2 Cu3 Oxide mounted on YSZ.

FIGURE 10.27 Measured W-band transmission coefficient versus the temperature of the thin film TlBaCaCuOxide mounted on YSZ.

At the superconducting transition temperature, a significant drop in transmission amplitude and phase step was expected. Data do show a drop in the amplitude and phase step near the expected temperatures (i.e., 100 K); however, changes are a few decibel at best. The data reflect lack of knowledge concerning YSZ. YSZ substrates were measured over 90 and 100 GHz and the explanation of small transmission loss was forthcoming. YSZ data are shown in Fig. 10.28. Unexpectedly, YSZ substrates showed a large dielectric resonance precisely in the middle of the measure frequency ranges. Further, the YSZ substrates used for deposition were many millimeters thick. The combination of high-loss, large real permittivity and thickness significantly attenuated the propagating field and left little for the high Tc ceramics. Further investigations found that the superconducting volume fraction within each thin film was 20 to 40 percent. Therefore, the low attenuation was also indicative of the artificial dielectric (not conducting) sample being measured. The lesson learned from the exercise is that measure substrates before any film depositions are made for test.

FIGURE 10.28 Measured YSZ permittivity near ambient and 117 K at W band.

10.3.2 Alternate Free Space Elevated Temperature Configuration In the 1980s and into the 1990s network analyzer systems, especially those operating at

millimeter wave frequencies, became available. However, calibration problems still existed when performing an elevated or reduced temperature measurement of ceramic dielectric properties. Willie Ho and his colleagues at the Rockwell Science Center [8–11] (now Teledyne Technologies of Thousand Oaks, California) implemented another free space configuration that mediated some problems. They configured a simultaneous reflection and transmission free space configuration for measuring dielectric properties at temperature. The configuration could operate in amplitude-only mode or amplitude and phase could be acquired. A calibration propagation path was added external to the measurement and served to monitor system stability. The configuration is shown in Fig. 10.29. The position for a modern network analyzer is inserted by the author.

FIGURE 10.29 Configuration for a mmW high-temperature simultaneous reflection and transmission free space measurement including an external calibration path.

As with the previous discussions, the system measures the transmission and reflection coefficients and/or phase shifts of free space propagating microwave beam incident on a planar sample. When operating in the amplitude measurement mode, the dielectric properties were obtained from fits to reflection and transmission coefficient amplitudes |R|2, |T|2 and angle of incidence ϕ (also the orientation angle of the rotatable sample) as a function of frequency. Spot-focusing lens antennas produced near-diffraction-limited illumination at the focal plane where the sample was centered. The lenses were commercially made and had unity fnumbers. Consistent and long-time calibration of the equipment was facilitated by the use of radiating heat sources and a surrounding furnace. The oven enclosure was made with removable sides and radiant heating, using infrared lamps and/or lasers, was added to achieve extreme temperatures. The configuration has similar advantages to the system of the previous sections. The sample under test is isolated from measurement apparatus and this allows either high or low temperature measurements to be carried out. Both focus beam configurations are suitable for quality control and online monitoring of materials. In either system, focusing elements can be removed to produce a plano-convex lens that collimates the field and can be used to spatially average over the sample surface. Alternately, the collimating beam can be used for homogeneity tests. The simple planar sample geometry is the same for both systems as is the sample rotation and/or dual polarization applied for easy testing of the sample anisotropy. When data analysis applied fits of amplitude as a function of frequency, additional requirements on the sample were determined. As illustrated in Fig. 10.30, data analysis depends upon measurement of reflection minima and transmission maxima as functions of frequency and temperature. The sample needs to be at least λ/2 in thickness for these to appear in measurement. Thick samples are also leveraged to accurately measure low-loss materials that might be used in resonators or transmission apertures.

FIGURE 10.30 Typical reflection (upper) and transmission (lower) amplitudes for a 0.58-in. thick silicon nitride sample. Null and peak amplitudes shift lower in frequency with temperature and indicate increased permittivity.

The calibration was stabilized by combining thick samples, simultaneous reflection and transmission measurement, external calibration line, and radiation heaters. Measurement accuracies of 1 percent were obtained when measuring εR. Variations in εR with temperature of less than 0.1 percent could be observed. Loss tangent was derived from the transmission loss which had measurement sensitivity of 0.02 dB in the transmission coefficient. For materials with εR ≈ 7 and electrical thickness of 2λ0, loss tangents of δ = 2 × 10–4 can be resolved. Figures 10.31 and 10.32 show examples of measured data for several samples as a function of temperature at frequencies in Ka band.

FIGURE 10.31 Relative change in real permittivity versus temperature in Ka band for ZnS, ZnSe, Spinel, Alumina, and ALON.

FIGURE 10.32 Relative change in loss tangent versus temperature in Ka band for SiN and fused quartz, both very low-loss materials.

10.3.3 Free Space Measurements of Anisotropic Materials Either of the free space configurations is sensitive to the planar material sample anisotropic parameters. Components of the permittivity and permeability tensor, through the thickness of a sample, can be obtained by measuring at oblique incidence angles so that there is some component of the electric or magnetic field in that direction. If the permittivity and/or permeability in the plane of the sample are known, the analysis of Chap. 4 can be applied and select tensor components may be extracted. The reader is referred to Ref. [12]. Chapter 4 demonstrated that reflection and transmission are given by

where

where k0 is the wave number in the free space, t is the thickness of the slab, θ is the incidence angle, and μij, εij are the appropriate components of the permeability and permittivity diagonal 3 × 3 tensors. Similar expressions derived for the TE polarization are provided in Chap. 4. The application of the free space systems to measure arbitrary anisotropy continues to be studied; however, to date a definitive inversion procedure has not been elucidated. However, the free space systems can supply some insight to the problem. A measurement might begin with a sample aligned transverse and at 90 degree with respect to the propagation direction. Separate polarization measurements or a 90 degree rotation of the sample about its normal (two reflections and two transmissions) would be used to measure effective complex permittivity and permeability in two orthogonal directions in the plane of the material (i.e., a ε11, ε22, μ11, μ22). These are not defined with respect to a sample axis but to an external reference. Two reflections and two transmissions would also be made at oblique incidence angles (e.g., 45 degrees incidence). Each oblique reflection–transmission set would be applied using ε11, ε22, μ11, μ22 as inputs. These would extract a permittivity and permeability along the normal, hopefully the same for both polarizations. The proposed procedure is expected to work best for low index materials such as foams and core. Angles larger than 45 degrees may be required for high index samples since refraction (Snell’s law) demands that the direction of travel through the sample be close to normal for refractive indices much greater than about 2. For higher permittivity materials with large anisotropies, this methodology may be applicable at incidence angles closer to grazing. Again it should be repeated that the proposed procedure is qualitative and supplies the experimenter with an estimate of anisotropy magnitude. As Chap. 4 indicated, a fully anisotropic propagation requires knowledge of a 4 × 4 complex tensor. At a minimum, a full

inversion would require 16 + 2 independent voltage measurements to determine the two 3 × 3 constitutive tensors. As indicated, that procedure has not been developed.

10.4 TE10N TRANSMISSION CAVITY The last measurement discussion of this chapter revisits the TE10N cavity of Chap. 9 to suggest improvements in accuracy for high index materials; demonstrate the cavity in measurement of anisotropic ferrites; and describe modifications and measurements when the cavity becomes part of an elevated temperature system. The TE10N waveguide cavity proves especially useful when small sample volumes that have a high aspect ratio geometry (long and thin) are to be measured. Many times these samples are developmental materials and come directly from lab samples of a few cubic millimeters. In general, if a thin solid rod shape is used, the aspect should be greater than 10:1.

10.4.1 Cavity Theory Modifications for High Index Materials The first discussion in the section will address suggested modifications for materials with high indices. The need for modification to the measurement procedure has origins in efforts to measure permeability and permittivity of hexagonal ferrite samples. An apparent coupling of permittivity and magnetic field was observed (Fig. 10.33) when the cavity samples were exposed to high external magnetic fields. The external field shifted permeability resonance (with an accompanying large permeability magnitude) into the measured waveguide band. The large magnitude was even observable in the measurement of permittivity (Fig. 10.33). The apparent coupling appeared because the sample has a finite electrical size and in fact modifies the field geometry within the cavity. The apparent coupling reflects violations of the perturbation assumptions that dictate measurement of electrically small samples. Those assumptions are fundamental in the experimental design. Summary results are presented here and details of the analysis can be found in Ref. [13]. Additional measurement technique modifications are suggested in Ref. [14].

FIGURE 10.33 Measured, εR, odd mode numbers versus external magnetic field strength at two X-band frequencies indicative of nonzero RF field strength.

Equation (10.20), also presented in Chap. 9, expresses the calculation of energy density within the cavity. That calculation relates shift in resonant frequency on insertion of a test sample. In the limit for the sample volume approaching zero, the equation is exact. The sample permittivity or permeability is related to the shift in resonant frequency and quality factor of a cavity.

where dVs is the differential volume element to be integrated over the sample volume, dV0 is the differential volume element to be integrated over the empty cavity volume, i, i, e, e are the exact cavity fields internal and external to the sample in the cavity with sample in place, 0, 0 are the fields in the original empty cavity, ε, μ are relative permittivity and permeability of the sample, ƒ0 is the resonant frequency of the empty cavity, and ƒs is the measured resonant frequency of the cavity with the sample present. The lowest order perturbation solution of the equations results from the quasi-static assumption that i = e = 0 and/or

i =

e =

0 . This leads to algebraic solutions for the sample permittivity (odd

mode numbers) or permeability (even mode numbers). Subscripts c and s indicate cavity and

sample volumes, respectively.

For square-shaped samples of the cross-sectional dimension d and the cavity of width w and length l (as assumed later in this section) the dielectric equation becomes

If the sample index is large it can modify the internal field of the cavity. Recall that a permittivity measurement assumes that the sample is a longitudinal mode zero of the RF magnetic field. Therefore, if the sample cross section is too large (e.g., proves difficult to machine) it may overlap the magnetic field null and the sample permeability can interfere with the permittivity measurement and modify the resonant frequency. The later physics will be especially apparent when an external field is applied which forces the permeability through resonance (Fig. 10.33). An improvement in high index measurements or larger crosssectional samples requires an improved estimate of the sample internal fields. One method to improve the field calculation is via a perturbation solution to the modified field equations. Since permittivity is expected to be the largest constitutive parameter (i.e., at high frequencies and away from magnetic resonance), the improved solution is developed for permittivity alone. The empty cavity fields satisfy

The perturbed fields (with the sample present) are to satisfy the equation

and zero elsewhere. The perturbation is structured to be a square shape to maintain symmetry with the cavity. When the same analysis is applied for cylindrical-shaped samples, an equivalent volume square shape is used to calculate impacts. In Eq. (10.24), δ0 is the perturbation strength and b, d are waveguide narrow wall and sample cross-sectional

dimension. The strength is set to be the lowest order frequency shift, i.e.,

and w, l are cavity width and length. To second order the improved solutions are

When these are substituted into Eq. (10.20), one obtains the frequency shift of odd cavity modes in the form

C must be numerically determined. Tables 10.1 and 10.2 show calculated values for one odd and one even mode in an X and Ka band cavities. As one would expect, the numerical correction for C and odd modes increases with sample dimension and permittivity. In the even mode case the correction reflects ΔμR = μR, measured – μR, corrected. For the X-band cavity Δμ is quite small; however, the smaller volume of the Ka-band cavity results in permeability corrections that would approach 10 percent for many ferrites since their permittivity often exceeds a magnitude of 10. The corrections are applied in Fig. 10.34 for a NiZn ferrite with a known εR 14.5. Two samples, 0.38 mm and 1.02 mm in dimension, were measured in an X-band and Ku-band TE10 N cavity. The smaller size is taken as the standard to which the thicker sample data are compared. The figure graphics illustrate the measurement dispersion that is predicted and can be corrected for in the analysis. As the cavity size is decreased (X to Ku band), the apparent lowest order measured permittivity shows an upward trend with frequency. However, when the perturbed field corrections are applied, the large-size sample measurement is shifted to lower values and approaches the smaller standard. The correction is not perfect, but the trend is correct.

FIGURE 10.34 Apparent measured εR and corrected for two size samples in X and Ku band cavities. Diamonds are original, triangles corrected and circles small calibration sample. TABLE 10.1 Calculated C for an 11.938 cm long X band TE10N Cavity vs. εR, N, d

TABLE 10.2 Calculated C for a 4.064-cm-Long Ka-Band TE10 N Cavity Versus εR, N, d

A definitive requirement for the correction is yet to be determined. However, a review of cavity measurements and correction simulations establish a scaling “guide.” The compilation suggests that a correction calculation will be required when the product of mode number – real permittivity and sample size (cm), divided by the cavity width (εRdN / w) ≥ 4.0. Though the obvious answer to the correction problem is to make the sample smaller, preparing a ceramic rectangular solid with length near 2.5 cm and cross section less than 0.5 mm may be a challenge. Therefore, it is suggested that the experimenter should be prepared for a correction especially when measuring ferrites and titanate ceramics. These often have permittivities in the range of 20–100 and may have X to Ka band magnetic resonances. In the next section, hexagonal ferrite measurements are presented and these required correction in the calculation of constitutive parameters.

10.4.2 Anisotropic Permeability Measurements The measurement of diagonal and off-diagonal permeability tensor elements is the topic of the second discussion. Off-diagonal elements are needed since these parameters determine engineering designs of microwave and millimeter wave components that exhibit asymmetric propagation. Circulators, isolators, power limiters, and switches are components that use magnetically biased ferrites. The measured properties are also used in phase shifter and filter designs [15]. The previous section illustrated how magnetic resonance might impact dielectric measurement; however, in some cases (e.g., Ka-band cavities), the high dielectric constants of ferrites might also impact measured magnetic properties. At the birth of ferrite microwave component technology, the analysis used in the measurement of off-diagonal elements was developed by researchers such as Van Trier [16], Waldron [17], and Berk [18]. Van Trier performed detailed analysis of transmission line, rectangular and cylindrical cavities. They describe measurement techniques to determine tensor elements of homogeneous dielectric/anisotropic magnetic spheres. Van Trier ’s “split resonance” technique was successfully applied for small spheres near 20 GHz. In this section, the rectangular waveguide and rod samples are discussed. The “simple” TE10N cavity measurement becomes involved since three complex parameters must be measured. Three experimental configurations are needed; three complex

equations must be solved; a source of external DC magnetic field is needed; and computation of the correction factor C for both odd and even mode numbers must be performed. Figure 10.35 illustrates three measurement configurations that are required to calculate permittivity (assumed isotropic), diagonal tensor elements (μR + jμimag), and off-diagonal tensor elements (KR + jKimag). An external magnetic field is required to pole the magnetization of the ferrite sample. For each configuration, a resonant frequency and cavity quality factor are measured. Thus, there are a total of six data inputs at each frequency that determine the real and imaginary parts of the three complex constituent parameters of the ferrite samples.

FIGURE 10.35 Three configurations required in measuring elements of a ferrite permeability tensor.

The tangential components of electric and magnetic fields within the sample must be known and are calculated from frequency shift and constitutive parameters via Eq. (10.20). The calculation uses the free space boundary value problem of perpendicular and parallel scattering from an isotropic dielectric and anisotropic permeability cylinder. That problem is solved in either of Refs. [19], [20], or [21]. In cases a., b., and c. of Fig. 10.35, the internal electric fields are, respectively,

The tangential magnetic field components are calculated from the tensor product where are respectively the tensor form of permeability and the external magnetic field in the cavity. The forms of permeability tensor for configurations a. to c. are, respectively,

As an example in configuration a. (even mode numbers), the internal magnetic field is

However in case c. (even mode numbers), the internal field is like that of an isotropic material, i.e., . In each case, the expressions for internal field are substituted in Eq. (10.20) and the integrals performed to give three equations that direct the calculation procedure for all parameters. Case b. (odd mode numbers) and case c. (even mode numbers) reproduce the previous section analysis for measuring isotropic permittivity and permeability equations, i.e.,

The o and e subscripts on C indicate perturbed field-calculated corrections for odd or even mode number, respectively. These determine two of the three constitutive parameters. K is calculated by combining the results of Eq. (10.31) with measured resonant frequency and quality factor shifts for configuration a., i.e.,

and w, l N are cavity width, length, and mode number. Vsam, Vc are sample and cavity volume, respectively. The equation remains in complex form since an iterative solution to complex K is used. The equation does not easily separate into real and complex parts. These relations are applied in analyzing measurements of two hexagonal ferrites (BaO6Fe2O3 and BaCo xTixFe12–2xO19) in Figs. 10.36 through 10.38. Both materials have permittivities greater than 20. The specific gravity, anisotropy field, and inferred bulk resonant frequency for BaO6Fe2O3 were 5.18 g/cc, Hk = 17 kiloOe, and 47.6 GHz, respectively. The same parameters for BaCo xTixFe12–2xO19 are 5.21 g/cc, Hk = 9 kiloOe, and 25.2 GHz. Measurements are in C, X, and Ku bands and are below resonance. Small dispersions in permeability is expected.

FIGURE 10.36 Measured, field-corrected permittivities (real and imaginary) for the indicated ferrites.

Both samples were very hard. The smallest cross-sectional dimension that could be ground was 0.5 mm while keeping the sample uniform in cross section over its entire 2.5 cm length. Therefore, perturbed field corrections were required for odd and even modes. However, as the upward frequency trend in εR of Fig. 10.36 illustrates, the calculation likely did not fully correct for field perturbation. Measurements of μ(ƒ), K(ƒ) are shown in Figs. 10.37 and 10.38. Though ferrite resonances lie in Ka band, evidence of resonant onset is indicated by the increase permeability and anisotropy in Ku band.

FIGURE 10.37 Measured, field-corrected permeability (real and imaginary) for the indicated ferrites.

FIGURE 10.38 Measured, field-corrected off-diagonal permeability tensor elements (real and imaginary) for the indicated ferrites.

10.4.3 High-Temperature TE10N Cavity Measurement Procedures This section reviews technical bases for electromagnetic characterization of composites and ceramics and in the frequency range 5 to 40 GHz and for constitutive parameters with high or low loss. The TE10N rectangular cavity continues as the measurement technique of choice. The basic cavity configuration remains constant; however, the choice of cavity construction and system configuration requires change when used at elevated temperatures. At room temperature and below, the priority for cavity construction is given to metals with the highest conductivities. Thus, Ag, Cu, and Au are often the choice for cavity interior. When consideration is given to the conductor robustness in modern atmosphere containing oxygen, sulfur dioxide, etc., Au often is chosen for inertness and conductivity. The room temperature measurement configuration of Chap. 9 is repeated Fig. 10.39. It highlights the placement of a test sample inside a stable, low dielectric, long tube (i.e., quartz). The tube is long enough that an empty section of the tube can be measured, thus serving as calibration to a sample measurement when it is inserted/pulled back into the cavity volume. Quartz was chosen as the carrier tube for its stability at high temperature and nondispersive frequency permittivity.

FIGURE 10.39 The general cavity configuration with sample aligned for a permittivity measurement. The sample can also be inserted parallel to the broad cavity wall for the permeability measurement.

Elevated temperature cavities and waveguide, which operate at 1000°C, have been constructed from Pt and Rh; however, these were at significant expense. After reviewing conductivity, robustness to temperature, thermal expansion, and expense, the alloy nickel 200

was chosen for use in extensive and long-term (multiyear) measurement efforts. The procedure is a modification of what was ANSI/ASTM Standard Test Procedure D-2520-80. The analysis of measurement is a direct application of the previous two sections. The photographs of Figs. 10.40 and 10.41 show a nickel cavity and waveguide configuration in a cylindrical furnace that was used over a 4-year period. It did undergo some significant exterior thermal damage and oxidation; however, the cavity and waveguide interior did not degrade over that time. The interior was filled and constantly purged with flowing Argon while the cavity temperature was elevated.

FIGURE 10.40 Photo of waveguide and TE10N cavity mounted in a cylindrical furnace. Highest elevated temperature near 1200°C.

FIGURE 10.41 Close-up photograph of the nickel-200 cavity exterior. Significant oxidation is observed on the outside. The interior of the cavity is filled and constantly purged with flowing Argon during measurement.

A general configuration for elevated temperature cavity measurements is shown in Fig. 10.42. As with room temperature measurements, a network analyzer and reflection/transmission test set with synthesizer are used as receiver and source to measure the transmission coefficient of a cavity as a function of frequency at a temperature. This configuration differs from others in the full encapsulation of cavity and sample by the furnace. The configuration also has real-time calibration capability. Note that a nickel tube extends beyond the furnace wall. The quartz-sample combination of Fig. 10.39 can be inserted into the heated cavity volume via the tube. The partial insertion of an empty section of quartz, or a separate empty quartz tube, serves as a calibration at temperature and resonant frequency. The calibration normally requires about 30 s. After the calibration is achieved, the sample is immediately inserted. This approach significantly improves on calibration problems encountered in both waveguide and free space elevated temperature measurements.

FIGURE 10.42 Configuration for an elevated temperature TE10N measurement. Sample may be inserted parallel or perpendicular to the internal electric field.

In operation, the frequency is scanned about each resonance at each measurement temperature. Data are then fit to a Lorentzian from which the center frequency and quality factor, Q, of the cavity are derived at a temperature. The measurement is performed for each resonant mode (and temperature) with and without a test sample placed in the cavity. The highaspect samples are inserted into a fused silica tube (10" in length with interior diameter of nominally 1 mm and exterior diameter of 1.2 mm). The silica tube constrains and orients the material tangentially with the electric or magnetic field. After resonant frequencies and quality factors are obtained the cavity equations [e.g., Eq. (10.31)] can be applied. Though this technique requires more sample preparation (requiring plug samples to be precisely ground to uniform rectangular or cylindrical bars), it can be sensitive to loss tangent (i.e., 0.001) and has been used to temperatures near 1200°C. The technique does have problems especially in the measurement of low-loss dielectrics (such as quartz). This is largely due to achievable limits in the cavity quality factor. Since nickel-200 conductivity decreases with temperature, the quality factor also decreases as temperature increases. The temperature also expands the cavity volume which decreases resonant frequency for a given mode. These effects are illustrated in Figs. 10.43 to 10.45.

FIGURE 10.43 Scans of frequency about an X and Ku band resonance as temperature increases. Temperature is in degrees Celsius and frequency in megahertz.

FIGURE 10.44 Comparison of predicted and measured nickel cavity X-band resonance frequency and Q changes versus temperature.

FIGURE 10.45 Measured Ku-band resonant frequency and quality factor changes.

Limits on loss tangent measurement are illustrated in Fig. 10.46. Two glass-like materials are shown, Pyrex and GE 124 quartz. The latter has an extremely low loss tangent (0.0001 or less). The quality factor variations lead to a consistent negative loss tangent which indicates

that loss cannot be resolved. However, the Pyrex loss (0.01) is easily detected.

FIGURE 10.46 X-band cavity measurements of low-loss quartz and higher loss Pyrex as a function of temperature.

Figures 10.47 to 10.49 are a sequence of material measurements for ever-increasing loss tangent. Figure 10.47 shows a very low loss crystalline alumina and a 99 percent dense fired sample of alumina. Both material loss factors are at the limit of detection. Figure 10.48 shows

a Magnesium–Calcium-Titanate with an imaginary permittivity of about 0.05 and data are easily resolved. This ceramic is one with a negative (δε / δT) at temperatures below 1000°C. Figure 10.49 illustrates the cavity technique for three samples of the high-loss ceramic SiC.

FIGURE 10.47 X-band cavity measurements of crystalline and high-density alumina bars. Data show difficulty in measuring low-loss/high-dielectric constant materials.

FIGURE 10.48 X-band cavity measurement of Magnesium-Calcium-Titanate versus temperature.

FIGURE 10.49 X-band cavity measurement of three commercial silicon carbide fibers

10.4.4 Largest TE10N Measurement Error Source A short discussion on error sources and their magnitude concludes this chapter. In the TE10N cavity, determinations of permittivity and/or permeability have identical sources of error. The excessive sample dimension with resulting field perturbation and the loss of cavity quality factor due to increased resistance of the cavity metal have been discussed. Historically, frequency variations in the source led to resolution limits in determining the resonant frequency and quality factor. However, the use of digitally tracked analyzer sources, with the stability of a few hertz, has eliminated frequency tracking as a problem. TE10N measurements are performed when small volumes of material are under test, typically a tenth of a cubic millimeter or less. Samples include machined dielectrics and ferrites, natural and artificial fibers, fiber roving, spun fiber, hair (human and animal), micrometer and submicrometer cross-section semiconducting and conducting wires, nanotube composites, and spun nanofibers and composites of all of these materials. The measurement of powder samples is also common. However, when testing powder it must be placed in a tube to hold it in place. The 1-mm diameter low-loss quartz tube serves well. In the case of fiber, an EMT is applied to scale the measurement and allow for powder packing density. In all of the above, the major remaining error source is variations that are observed in the cross section of the high-aspect samples. For high-aspect samples, the error appears as the sample is inserted into the cavity. A 2.54-cm-long sample placed in a Ku-band cavity (height 0.79 cm) has three distinct positions along its length that can be measured as the sample is inserted further into the cavity. The frequency shift, and thus the measured εR, responds to the total volume in the cavity. If variations in cross section are present, it would appear as measured permittivity changes. However, if appropriately monitored this sensitivity can be leveraged to study material inhomogeneities and especially composites that use blends of fibers. Fiber blends are also a common sample used in these cavities. The blends are twisted together in long tows. The tows may then be spun into a blended thread or be woven into fabric. Modern structural composites are assemblies of fabric and polymer. When tows are made, they may have variations in fiber density parallel to their length. Variations in fiber diameter/shape or fiber count (number of fibers per unit length) are the most common type. Fiber count changes are reflected in the measured permittivity as tows are pulled in the direction of their length. Thus, a TE10N cavity which is placed in a textile production line can serve as a quality control device. The next chapter will address measurements and characterization of even smaller material volumes, i.e., nanostructures and nanoparticles. Typically the nanoparticles are encapsulated in polymer or ceramic matrices. Techniques such as the TE10N cavity are applied to measure a composite after which the EMT of Chaps. 5 to 7 can be applied to calculate the properties of the composites particulates.

REFERENCES 1. J. Baker-Jarvis, et al., IEEE Trans. Instr. Meas., 45(5) (Oct. 1992). J. Baker-Jarvis, et al., Transmission/Reflection and Short-Circuit Line Methods for Measuring Permittivity’ and Permeability NIST Technical Note 1355-R (Dec. 1993). 2. J. Jarvis, M. D. Janezic, J. H. Grosvenor, Jr., and R. G. Geyer, “Transmission/Reflection and Short-Circuit Line Methods for Measuring Permittivity and Permeability,” NIST Technical Note 1355 (May 1992). 3. John W. Schultz, Focused Beam Methods: Measuring Microwave Materials in Free Space, ISBN: 1480092851 (2012). 4. Agilent PNA Family Microwave Network Analyzers Configuration Guide, © Agilent Technologies, Inc. 2012. Published in United States California, Oct. 18, 2012. 5990-7745EN. 5. J. Jarvis, “Transmission/Reflection and Short-Circuit Line Permittivity Measurements,” NIST Technical Note 1341 (Jul. 1990). 6. J. H. Grosvenor, “NISTIR 5006, NIST Measurement Service for Electromagnetic Characterization of Materials” (Aug. 1993). 7. W. W. Ho, Proceedings of the SPIE: The International Society for Optical Engineering, 362:190–195 (1982). 8. W. W. Ho, “Millimeter-Wave Dielectric Properties of Infrared Window Materials,” Proceedings of the SPIE: The International Society for Optical Engineering, 750:161–165 (1987). 9. W. Ho, “High-Temperature Millimeter-Wave Dielectric Measurements by Free-Space Techniques,” Conference Digest: Twelfth International Conference on Infrared and Millimeter Waves (Cat. No.87CH2490-1), 16–17 (1987). 10. D. Rogovin, et al., IEEE Trans. Micro. Theo. Tech., 40(9) (Sept. 1992). 11. D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4X4-Matrix Formulation,” J. Opt. Soc. Am., 62(4), 502–510 (1972). 12. R. Moore, M. C. Thompson, and T. S. Robbins, Rev. Sci. Instrum., 61(3) (Mar. 1990). 13. Jyh Sheen, “Measurements of Microwave Dielectric Properties by an Amended Cavity Perturbation Technique,” Measurement, 42:57–61 (2009). 14. Vincent G. Harris, IEEE Trans. Magn., 48(3):1075–1104 (Mar. 2012). 15. M. Van Trier, Appl. Sci. Res., 3:305 (1953). 16. R. A. Waldron, Convention on Ferrites, p. 307 Institution of Electrical Engineers. Proceedings. Part B. Radio and electronic engineering. Supplement; London (Nov. 1956). 17. A. D. Berk and B. A. Lengyel, IRE, 43:1587 (1955). 18. J. A. Stratton, Electromagnetic Theory, ISBN 07-062150-0, McGraw-Hill New York (original 1941). 19. P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena, ISBN 0933-033X, 2nd ed., Springer-Verlag Berlin (2005). 20. M. Kerker, The Scattering of Light, Academic Press Waltham, Mass. (1969).

CHAPTER 11

MICROMETER AND NANOSCALE COMPOSITES

This chapter addresses micrometer and nanometer scale composite properties, and modeling and measurement of materials and their composites. Conventional materials science of composites often describes the behavior of bulk materials and material-integrated systems that depend on homogeneity at macroscopic scales. The macroscopic properties of bulk materials (such as permittivity, permeability, and conductivity) derive from the stochastic nature of a large assembly of atoms (hundreds to thousands). Single atom properties are averaged and thus statistical mechanics and models (such as EMTs) are often sufficient to predict macroscopic properties. On the other hand, surface effects come into play when predicting magnetic coupling and/or electrical conductivity of particulates. A correct physical description of surface effects may require precise quantum numerical models that solve for electron density matrices. These may dominate physics of nanometer-sized particulates (less than a few nanometer in size). The size range of one to tens of nanometers is a transition range. In this region, effective medium theories (EMTs) may be useful; however, electromagnetic parameters of magnetization, anisotropy field, conductivity, permittivity, and permeability must be adjusted to account for the transition. In order to determine properties of nanoscale particulates they must be synthesized and particle production techniques for the particulates are mature. Techniques to distribute the particles in appropriate matrices are also refined. References [1] to [3] are a few of a multitude of publications that discuss procedures and applications. Future advances in selfassembly may someday allow for nanoparticles to be periodically dispersed; however, at this time nanoparticle composites are most often mixtures. It is difficult to isolate and measure the electromagnetic properties of single nano particulates. However, these and other developmental nano material particulates easily precipitate from solution or can be deposited and trapped within polymer or ceramic matrices and substrates. The electromagnetic sensing of these materials pose a small volume problem for the measurement; however, if measurements are combined with simple effective medium theories, RF characterization can supply fundamental properties of the nano particulates that are contained within the composite. The detection and characterization of nano metallic particulates can leverage optical absorption and characteristic color change. Magnetic nanoparticles may be nonconductors or have greatly reduced conductivity and they often are in composites at low concentrations. Microwave magnetic measurement, using small

volumes, requires the experimenter to return to resonant cavity and stripline measurements. These measurements will be used in one case study of the chapter. The chapter begins with a short review of applications for magnetic particulates and their composites. The second section focuses on ferrites and their analysis in a case study to illustrate the importance of nano- and microscale contributions to macro electromagnetic properties. The ferrites in this study were chosen because their permittivity and permeability data are included in the database of Chap. 12. Micrometer and nanoscale magnetic discussions continue in the third section with focus placed on nano magnetite particles and several composites. The section highlights the importance of size scale in understanding measured magnetic composite properties. This case study may have application where magnetite is heavily investigated for health, energy, electromagnetic interference, and communication applications [1–18]. The last section will focus on nano and micro electrical conductivity and its impact on mesoscale composite permittivity. Here the size-dependent conductivity for particles less than about 50 nm in dimension impacts macro permittivity. That section is the last case study to illustrate need for multiscale (nano throughout the meso) models to fully explain macro electromagnetic observables.

11.1 APPLICATIONS AND IMPETUS FOR NANOMAGNETIC COMPOSITES The formulation and manufacture of fine-scale particulates and geometric configurations of those particulates are empowering technologies. Applications of composites based on microand nanoparticles continue their expansion as components of electromagnetic devices and systems. An excellent illustration is the 60-year investment for increased magnetic storage and transistor size reduction. Increased data density was the impetus to develop submicrometer magnetic structures that have since evolved to nanoscale-patterned magnetic thin films and particulates. Particulates and films have been leveraged to make composites with improved electromagnetic compatibility in electronic circuitry, inductors capacitors, microwave components [7–9], electric generators, transformers, turbines, and ferrofluids [5, 10, 11, 18]. All benefit from composites of reduced-density resistive magnetic particulates. More recently, nanomagnetic particulates have found application in contrast enhancers for magnetic resonance imaging (MRI), for biological identifier magnetic tags for pathogens, and as facilitators in biotherapy [3, 12, 14–18]. Ferromagnetic nano particulates, and presumably any composites made from dispersions of their chemically passivated forms, have clear electromagnetic advantages over micrometer size dispersions and/or ferromagnetic oxides. The nanoscale size increases electrical resistivity in dispersions of the ferromagnetic particulate. The resistivity is desirable in power production, and electromagnetic and digital memory. The increased electrical resistance derives from increased electron collision frequency within the particle, while the composite resistance increases due to the number of high impedance surface contacts between particulates. Theoretical and experimental results [4, 19–22] have demonstrated that

magnetization can be controlled and even enhanced by size-scale changes in some nanoscale ferrites and Fe for sizes from about two to tens of nanometers. The controlability extends above room temperature, even for these small sizes. Reduced magnetic moment, as compared to bulk, is often observed at sizes below about 20 nm. However, ferromagnetic metal oxides still receive significant study due to the difficulty in passivation of metallic ferromagnetic forms, especially for applications requiring oxidation resistance and environmental stability. Magnetite, Fe3O4, is one of these and is the subject of the second case study.

11.2 CASE STUDY 1: NiZn AND MnZn FERRITES The first case study illustrating nano and microscopic impacts on macro electromagnetic properties deals with seven materials that are in the database of Chap. 12. In this study, various NiZn and MnZn ferrites received detailed microstructure study. The study was performed in parallel with EM permittivity and permeability measurements of the same materials. The microanalysis is included in this chapter to aid the reader in relating microstructure to bulk permittivity and permeability and also to illustrate the microstructure details that may be required in laboratory study to obtain full understanding of material properties. This section is a greatly expanded as compared to the introduction to permittivity and permeability sources in Chaps. 2 and 3. This example and similar microstructure studies can feed material development and manufacture so as to enhance material electromagnetic properties. Six NiZn ferrites and two MnZn ferrites were studied. Frequency dispersive permeability and permittivity were obtained from reflection and transmission measurements of samples that were cut from blocks approximately 10 cm3 in volume. Samples were diamond ground to fit (nominal 0.001 gap) in a 7-mm coaxial transmission line. Example data are shown in Figs. 11.1 and 11.2. Expanded dispersion measurements and parametric fit to dispersion relations are found in Chap. 12 tables. Ferrites 6 and 7 were samples of the same material but were acquired from two distinct manufacturing batches. Microscopic constituents showed very small differences and the mesoscopic electromagnetic data were identical within the measurement error. The result gives a high degree of confidence that manufacturing processes generate material that is homogeneous throughout a single batch and is suggestive that there exists only minimal variance between batches.

FIGURE 11.1 Plots comparing the permittivity of individual ferrites. Note that ferrites 6-7 are a single composition.

FIGURE 11.2 Plot comparing the permeability of individual ferrites. Note that ferrites 6-7 are a single composition.

The highlights and correlation of material parameters with Ferrite ID# are shown in Table

11.1. Ferrites #1, #2, #3, and #5 demonstrate a nondispersive real part of the permittivity of εr ≈ 10 – 15 over the frequency range investigated. Ferrites #6–7 and #4 display a finite conductivity that produced a f–α -like dependence in the imaginary part and relaxation in the real part. An investigation of the cause of conductivity was performed to illustrate microstructure impact on permittivity. Ferrites #8 and #9 show much larger permittivity magnitude and have Debye relaxation behavior at high frequencies. TABLE 11.1 Identification of Ferrite Samples with Relevant Parameters

The magnitude of the permeability tends to increase in the ferrite sequence 1–6 and dispersions become narrower banded. Ferrites #8 and #9 had small measured values of μr and have a f–a dispersion in μimag. That combination indicates that the frequency band in the measured data was far above resonance. Subsequent measurements (shown in Chap. 12) proved that observation to be correct. All measured constitutive properties are analyzed in the context of the microstructural and compositional nature of each sample in the following sections. Measured characteristics of ferrites are shown in Table 11.1. Values for polaron and electron hopping activation energies are supplied. These prove important in explanations of permittivity.

11.2.1 Chemical and Physical Characterization Chapter 3 has summarized permeability and permittivity sources and noted that these arise from microstructural and compositional sources. These range from effects of constituent doping on the atomic scale (impacting electron spin) to differences in grain matrix

characteristics at the mesoscopic scale (impacting conductivity and polarization). The NiZn ferrites possess an inverse spinel crystal structure as illustrated in Fig. 11.3:

FIGURE 11.3 Cubic spinel structure characteristic of ferrites 1 to 7.

In the inverse spinel structure the tetrahedral sites are occupied by trivalent metal ions, and the octahedral sites are occupied by half trivalent metal ions, and half divalent metal ions. Since all spinels possess negative exchange integrals, JAA, JBB, JAB [23], the spins favor antiparallel alignment. As the AB interaction is the strongest, A and B sites orient antiparallel in the ordered ground state. In magnetite, Fe3O4, (Fe3+[Fe2+ Fe3+]O4) the octahedral sites are filled by Fe2+ and Fe3+ cations. The ferric (Fe3+) spins in the A and B sites cancel leaving the ferrous (Fe2+) ions to each contribute a spin of 2 or 4 μB (Bohr magnetons [23]) to the saturation moment. The material may be substitutionally doped with Ni2+ and Cu2+, predominately into the octahedral sites, or Mn2+ and Zn2+, predominately into the tetrahedral sites. The electronic conduction in the lattice occurs primarily through electron transfer (Fe3+ + e ⇔ Fe2+) along the 110 crystalline planes (B sites). Energy dispersive x-ray spectroscopy (EDX) spectra were measured on all NiZn ferrites and in each case the point of chemical analysis was centered on a grain matrix region. The probable chemical composition of each ferrite studied in this investigation was taken from EDX data and is shown in Table 11.1.

11.2.2 Microstructure, Permittivity, and Electrical Conductivity

Several conduction mechanisms were explored in attempts to explain the permittivity dispersion of the samples. The dispersive characteristics of ferrites 4 and 6–7 were originally thought to arise from delocalized conduction through copper precipitates that might deposit at ferrite grain boundaries. The ideas were based on a melt-solid phase diagram that was obtained and indicated that Cu might freeze out of the melt during the cooling process of the manufacturing sintering process. Deposits would occur near 1200°C. The solidification process suggested that binary phases could give rise to a structure composed of insulating ferrite grains where grains could be surrounded by surfaces of percolating conducting pathways. This would explain permittivity dispersion. It would be an example of the “dispersion microstructure” discussed in Chap. 5 and in Ref. [24]. This candidate source of conduction received further study with the ferrite samples being mechanically thinned to 30 μm. This mechanical process was followed by ion milling to further reduce the sample thickness. Specimens were cooled with liquid nitrogen during the milling to inhibit artifact formation. The chemical composition and microstructure were examined with an energy dispersive x-ray spectrometer. Clear evidence of Cu precipitates was found in both ferrites 6 and 7 samples. Energy Dispersive Spectroscopy (EDS) data were taken, first centered over a precipitate region and followed up with a repositioning of the EDS spot to be centered on a nearby grain matrix. Data clearly showed that Cu precipitated out of the melt. However, subsequent analysis demonstrated that the precipitates only appeared in a small fraction of the pores that were studied. Precipitates were localized and often separated by 10 to 100 μm. The separation suggests that the Cu precipitates could not contribute significantly to the DC conductivity and thus the observed f–α permittivity dispersion. Grain boundary regions and triple junction regions were also examined and Cu continued to be of minimal fraction as compared to the matrix material. All grain boundaries and most triple junction regions were free of any such precipitates or Cu defects. A grain boundary region image is shown in Fig. 11.4. One side of the junction was measured with the electron beam aligned along the crystal axis. The alignment would reveal even nanometer scale Cu defects. None were observed.

FIGURE 11.4 Grain bound region of ferrite 6-7; no Cu precipitates are evident.

Another possibility is that changes in the stoichiometry of the grain might have occurred near the boundaries. Most likely these would involve some level of oxidation and oxidation state could alter resistivity at the grain boundaries. The EDS was performed over all the samples and data proved homogeneous with respect to the metal constituents (see Table 11.2). However, any measured variations in the oxygen content are more likely to arise from surface oxide layers which give rise to the rich oxygen content found in the samples. TABLE 11.2 Atomic Composition Across Grain Boundary

The impact of oxide layers forming at the grain boundaries has been investigated and discussed in literature. The conclusion is that current data are consistent with prior art. The permittivity and permeability of NiZn ferrites (Ni1-xZnxFe3O4, x = 0.8,0.5,0.2) were compared for two sintering conditions, in air and under nitrogen. It was found that samples

sintered under nitrogen displayed minimal changes in the complex permeability characteristics. However, large differences in the permittivity characteristics were evident. Samples sintered in air possessed dielectric constants of εr ≈ 12, similar to the permittivity for the NiZn ferrites. The materials sintered under nitrogen doubled the dielectric constant magnitude. Their real permittivities were in the range ε r ≈ 20 – 30 in the frequency band 0.1 to 1 GHz. The source of large permittivity values is believed assigned to formation of reduced oxide layers at the ferrite grain boundaries. Reduced oxides would alter the impedance of the grain boundary region and thus polarizability of the grain. Note that ferrites 8 and 9 fit these observations as they were sintered under nitrogen. However, a second permittivity contributor can be their high Mn content and their more favorably balanced ferrous and ferric ion content. Nitrogen sintering and Mn content contribute to permittivity variations. Samples of 8–9 show similar conductivities and they have overwhelming grain matrix region volume to that of grain boundaries. Therefore, both the grain matrix and grain boundaries most likely contribute significantly to conduction. The conduction will be considered in view of two distinct theories that govern the conductivity of ferrites: the small polaron model and the hopping conductivity model of electrons. Each of these has a characteristic quantum energy barrier of the process. Known barrier data are presented in Table 11.1. Magnetite is used as an example. In the case of electron hopping, the conductivity is controlled by valence exchange of charge carriers between Fe3+ and Fe2+ on octahedral localized sites rather than via delocalized band conduction. The conductivity may be expressed as σ = σ0e–Eμ/kT where Eμ is the electron hopping energy threshold. A small polaron model would be appropriate if the charge carrier and its distortion are hopping. The polaron has conductivity, σ = (Nc(1 – c)e2v0)e–Eμ/kT / (akT). In this expression, N is the total density of conducting ions, c is the fraction of conducting ions, v0 is the lattice vibration frequency, and a is the lattice parameter. The conductivity is the highest when c = 1/2. Hopping energies are shown in Table 11.1. In the case of NiZn ferrites, Zn2+ is predominately substituted for Fe3+ in the tetrahedral sites and Ni2+ is predominately substituted for the Fe2+ in the octahedral sites. Hopping mechanisms between ions of different metals possesses higher activation energy. Therefore, substitutionally doping with these different metals should result in higher resistivity or lower conductivity. In the current materials, activation energy was computed for each ferrite by measuring the conductivity over the temperature range 200 K < T < 340 K. Data were then applied to the two potential model parameterizations. The measured data could not distinguish between the two models and for this temperature range. A summary of results for permittivity begins with the simplest trends that were found for the ferrites. Ferrites 1, 2, and 3 increased Zn content with accompanying increased activation energy for DC conductivity. Investigations of the effects of the Zn composition in NiZn ferrites show that the lattice constant of ferrites, with similar composition, increases with an increase in Zn content. That evidence provides a plausible explanation for the measured reduction in hopping activation energy in those samples.

In polaron hopping, the charge carrier plus its associated lattice polarization must migrate from position to position. The activation energy is a function of the energy necessary to free the charge carrier from a site plus the potential barrier formed by polarization, Ehop = EBar + Epol and Epol ∼ a–1. Overall this model yields increased hopping energy and decreased conductivity. The prediction is consistent with data of Table 11.1. An explanation for increased conductivity in ferrites 6–7 remains and alternative explanations can be identified. Cu introduction into the mix has an impact opposite to Zn for the ferrites. Introduction of Cu in the matrix decreases the lattice parameter and therefore decreases the activation energy. This establishes a trend that increases conductivity. Second, Cu, unlike Zn, also modifies the band structure of a ferrite. Cu and Mn distort the crystal lattice. The source of distortion is the Jahn Teller effect [25]. In the presence of lattice deformation, small perturbations on the degenerated state of the system forces the energy level to be split. The energy will decrease linearly with the distortion. The overall energy is stabilized since the elastic energy will increase as the square of the distortion. Since the ion seeks equilibrium with overall energy at a minimum, there is a tendency to remove degeneracy by distortion in the lattice. This particular distortion has been shown to reduce the activation energy between different valence states of d orbital electrons. The reduction in activation increases conductivity and thus one observes the large DC conductivity of ferrites 6 and 7.

11.2.3 Microstructure and Permeability Aside from the atomic constituents, several microstructural characteristics can dominate the permeability of ferrites. These include interfacial stress, grain boundary size, and porosity of the material. In order to illustrate possible impacts, each of these considerations, along with the compositional characteristics, were addressed by using ferrites 1, 2, 3, 6, and 7 as examples and the DC permeability as given by

where Ms is the saturation magnetization, K is the crystal magnetic anisotropy, Λ is the magnetostriction constant, and σstress is the inner stress. Lattice distortion, grain boundary defects due to interstitial precipitates, and exaggerated grain growth are factors which can induce strain in the matrix grain. As the equation indicates, their growth leads to the degradation of the DC permeability. The TEM (transmission electron microscopy) was employed to investigate the potential effects of stress in the ferrites and to analyze grain boundary regions. Under low magnification, grain boundaries for all of the ferrites appeared remarkably clean and strained grain boundary regions were rare. One example is shown in Fig. 11.5. It is a TEM image of a triple grain junction of ferrite 2. One strain boundary region is apparent and is in contrast against the normally smooth regions. High-resolution TEM was used to examine interference fringes to ascertain any potential lattice distortion which would be undetected at low magnification. Again, all ferrites exhibited clean boundary regions. Clean boundaries are illustrated in Fig. 11.6, where the orientation of both grains was visible. The lack of internal

stress suggests that the cooling process after sintering was done slowly and quasi-equilibrium conditions in the ferrites were maintained. Overall, manufacturing conditions did not contribute grain stress with resulting degradation in the permeability.

FIGURE 11.5 Ferrite 2 showing a triple grain boundary with stress indications.

FIGURE 11.6 HRTEM of grain boundary region in ferrite 1.

The magnetic susceptibility of a polycrystalline material is a result of contributions arising from domain wall displacement and rotational effects. As discussed in Chap. 3, domain wall displacement dominates the permeability in ferrites that have grains large enough to support domain walls (D > 1 μm). Their contribution is at low RF frequencies (e.g.,