1.Microwave ElectronicsMeasurement and Materials CharacterizationL. F. Chen, C. K. Ong and C. P. Neo National University of SingaporeV. V. Varadan and V. K. Varadan Pennsylvania State University, USA

2. Microwave Electronics 3. Microwave ElectronicsMeasurement and Materials CharacterizationL. F. Chen, C. K. Ong and C. P. Neo National University of SingaporeV. V. Varadan and V. K. Varadan Pennsylvania State University, USA 4. Copyright 2004 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777Email (for orders and customer service enquiries): cs-books@wiley.co.ukVisit our Home Page on www.wileyeurope.com or www.wiley.comAll Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form orby any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright,Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 TottenhamCourt Road, London W1T 4LP, UK, without the permission in writing of the Publisher. 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Some content that appearsin print may not be available in electronic books.British Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryISBN 0-470-84492-2Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, IndiaPrinted and bound in Great Britain by Antony Rowe Ltd, Chippenham, WiltshireThis book is printed on acid-free paper responsibly manufactured from sustainable forestryin which at least two trees are planted for each one used for paper production. 5. ContentsPreface xi1 Electromagnetic Properties of Materials 11.1 Materials Research and Engineering at Microwave Frequencies 11.2 Physics for Electromagnetic Materials 21.2.1Microscopic scale21.2.2Macroscopic scale61.3 General Properties of Electromagnetic Materials111.3.1Dielectric materials111.3.2Semiconductors161.3.3Conductors171.3.4Magnetic materials191.3.5Metamaterials 241.3.6Other descriptions of electromagnetic materials 281.4 Intrinsic Properties and Extrinsic Performances of Materials 321.4.1Intrinsic properties321.4.2Extrinsic performances32References 342 Microwave Theory and Techniques for Materials Characterization372.1Overview of the Microwave Methods for the Characterization of Electromagnetic Materials37 2.1.1Nonresonant methods 38 2.1.2Resonant methods402.2Microwave Propagation42 2.2.1Transmission-line theory42 2.2.2Transmission Smith charts 51 2.2.3Guided transmission lines 56 2.2.4Surface-wave transmission lines 73 2.2.5Free space832.3Microwave Resonance87 2.3.1Introduction87 2.3.2Coaxial resonators93 2.3.3Planar-circuit resonators 95 2.3.4Waveguide resonators97 2.3.5Dielectric resonators103 2.3.6Open resonators1152.4Microwave Network 119 2.4.1Concept of microwave network 119 2.4.2Impedance matrix and admittance matrix 119 6. vi Contents2.4.3 Scattering parameters1202.4.4 Conversions between different network parameters 1212.4.5 Basics of network analyzer 1212.4.6 Measurement of reection and transmission properties 1262.4.7 Measurement of resonant properties 134References 1393Reection Methods 142 3.1 Introduction142 3.1.1Open-circuited reection 142 3.1.2Short-circuited reection143 3.2 Coaxial-line Reection Method 144 3.2.1Open-ended apertures 145 3.2.2Coaxial probes terminated into layered materials 151 3.2.3Coaxial-line-excited monopole probes 154 3.2.4Coaxial lines open into circular waveguides157 3.2.5Shielded coaxial lines 158 3.2.6Dielectric-lled cavity adapted to the end of a coaxial line 160 3.3 Free-space Reection Method 161 3.3.1Requirements for free-space measurements 161 3.3.2Short-circuited reection method 162 3.3.3Movable metal-backing method 162 3.3.4Bistatic reection method164 3.4 Measurement of Both Permittivity and Permeability Using Reection Methods 164 3.4.1Two-thickness method 164 3.4.2Different-position method165 3.4.3Combination method 166 3.4.4Different backing method 167 3.4.5Frequency-variation method 167 3.4.6Time-domain method 168 3.5 Surface Impedance Measurement 168 3.6 Near-eld Scanning Probe170 References1724Transmission/Reection Methods175 4.1Theory for Transmission/reection Methods1754.1.1 Working principle for transmission/reection methods 1754.1.2 NicolsonRossWeir (NRW) algorithm 1774.1.3 Precision model for permittivity determination 1784.1.4 Effective parameter method 1794.1.5 Nonlinear least-squares solution 180 4.2Coaxial Air-line Method1824.2.1 Coaxial air lines with different diameters 1824.2.2 Measurement uncertainties1834.2.3 Enlarged coaxial line185 4.3Hollow Metallic Waveguide Method 1874.3.1 Waveguides with different working bands1874.3.2 Uncertainty analysis 1874.3.3 Cylindrical rod in rectangular waveguide 189 4.4Surface Waveguide Method 190 7. Contents vii4.4.1 Circular dielectric waveguide 1904.4.2 Rectangular dielectric waveguide1924.5 Free-space Method 1954.5.1 Calculation algorithm 1954.5.2 Free-space TRL calibration1974.5.3 Uncertainty analysis1984.5.4 High-temperature measurement1994.6 Modications on Transmission/reection Methods2004.6.1 Coaxial discontinuity 2004.6.2 Cylindrical cavity between transmission lines 2004.6.3 Dual-probe method 2014.6.4 Dual-line probe method2014.6.5 Antenna probe method2024.7 Transmission/reection Methods for Complex Conductivity Measurement 203References2055 Resonator Methods 2085.1 Introduction2085.2 Dielectric Resonator Methods2085.2.1Courtney resonators2095.2.2Cohn resonators2145.2.3Circular-radial resonators 2165.2.4Sheet resonators 2195.2.5Dielectric resonators in closed metal shields2225.3 Coaxial Surface-wave Resonator Methods2275.3.1Coaxial surface-wave resonators2285.3.2Open coaxial surface-wave resonator2285.3.3Closed coaxial surface-wave resonator2295.4 Split-resonator Method2315.4.1Split-cylinder-cavity method 2315.4.2Split-coaxial-resonator method 2335.4.3Split-dielectric-resonator method2365.4.4Open resonator method2385.5 Dielectric Resonator Methods for Surface-impedance Measurement2425.5.1Measurement of surface resistance2425.5.2Measurement of surface impedance 243References2476 Resonant-perturbation Methods 2506.1Resonant Perturbation250 6.1.1Basic theory250 6.1.2Cavity-shape perturbation 252 6.1.3Material perturbation 253 6.1.4Wall-impedance perturbation 2556.2Cavity-perturbation Method 256 6.2.1Measurement of permittivity and permeability256 6.2.2Resonant properties of sample-loaded cavities 258 6.2.3Modication of cavity-perturbation method 261 6.2.4Extracavity-perturbation method 2656.3Dielectric Resonator Perturbation Method 2676.4Measurement of Surface Impedance 268 8. viii Contents 6.4.1 Surface resistance and surface reactance268 6.4.2 Measurement of surface resistance 269 6.4.3 Measurement of surface reactance275 6.5 Near-eld Microwave Microscope278 6.5.1 Basic working principle 278 6.5.2 Tip-coaxial resonator 279 6.5.3 Open-ended coaxial resonator280 6.5.4 Metallic waveguide cavity 284 6.5.5 Dielectric resonator284 References2867Planar-circuit Methods288 7.1Introduction 2887.1.1Nonresonant methods 2887.1.2Resonant methods290 7.2Stripline Methods2917.2.1Nonresonant methods 2917.2.2Resonant methods292 7.3Microstrip Methods 2977.3.1Nonresonant methods 2987.3.2Resonant methods300 7.4Coplanar-line Methods3097.4.1Nonresonant methods 3097.4.2Resonant methods311 7.5Permeance Meters for Magnetic Thin Films 3117.5.1Working principle 3127.5.2Two-coil method 3127.5.3Single-coil method3147.5.4Electrical impedance method 315 7.6Planar Near-eld Microwave Microscopes 3177.6.1Working principle 3177.6.2Electric and magnetic dipole probes 3187.6.3Probes made from different types of planar transmission lines 319References 3208Measurement of Permittivity and Permeability Tensors323 8.1Introduction 3238.1.1Anisotropic dielectric materials3238.1.2Anisotropic magnetic materials325 8.2Measurement of Permittivity Tensors3268.2.1Nonresonant methods 3278.2.2Resonator methods 3338.2.3Resonant-perturbation method336 8.3Measurement of Permeability Tensors3408.3.1Nonresonant methods 3408.3.2Faraday rotation methods3458.3.3Resonator methods 3518.3.4Resonant-perturbation methods 355 8.4Measurement of Ferromagnetic Resonance 3708.4.1Origin of ferromagnetic resonance 3708.4.2Measurement principle 371 9. Contents ix 8.4.3 Cavity methods 373 8.4.4 Waveguide methods374 8.4.5 Planar-circuit methods 376 References 3799Measurement of Ferroelectric Materials 382 9.1Introduction3829.1.1Perovskite structure 3839.1.2Hysteresis curve 3839.1.3Temperature dependence 3839.1.4Electric eld dependence 385 9.2Nonresonant Methods 3859.2.1Reection methods3859.2.2Transmission/reection method386 9.3Resonant Methods3869.3.1Dielectric resonator method3869.3.2Cavity-perturbation method 3899.3.3Near-eld microwave microscope method390 9.4Planar-circuit Methods3909.4.1Coplanar waveguide method3909.4.2Coplanar resonator method3949.4.3Capacitor method 3949.4.4Inuence of biasing schemes404 9.5Responding Time of Ferroelectric Thin Films 405 9.6Nonlinear Behavior and Power-Handling Capability of Ferroelectric Films 4079.6.1Pulsed signal method 4079.6.2Intermodulation method 409References41210 Microwave Measurement of Chiral Materials414 10.1 Introduction414 10.2 Free-space Method 41510.2.1 Sample preparation 41610.2.2 Experimental procedure 41610.2.3 Calibration41710.2.4 Time-domain measurement43010.2.5 Computation of , , and of the chiral composite samples 43410.2.6 Experimental results for chiral composites 440 10.3 Waveguide Method45210.3.1 Sample preparation 45210.3.2 Experimental procedure 45210.3.3 Computation of , , and of the chiral composite samples 45310.3.4 Experimental results for chiral composites 454 10.4 Concluding Remarks458References45811 Measurement of Microwave Electrical Transport Properties 460 11.1 Hall Effect and Electrical Transport Properties of Materials46011.1.1Direct current Hall effect46111.1.2Alternate current Hall effect 46111.1.3Microwave Hall effect 461 10. xContents11.2Nonresonant Methods for the Measurement of Microwave Hall Effect 46411.2.1Faraday rotation 46411.2.2Transmission method46511.2.3Reection method 46911.2.4Turnstile-junction method47311.3Resonant Methods for the Measurement of the Microwave Hall Effect47511.3.1Coupling between two orthogonal resonant modes 47511.3.2Hall effect of materials in MHE cavity 47611.3.3Hall effect of endplate of MHE cavity48211.3.4Dielectric MHE resonator 48411.3.5Planar MHE resonator 48611.4Microwave Electrical Transport Properties of Magnetic Materials48611.4.1Ordinary and extraordinary Hall effect 48611.4.2Bimodal cavity method48711.4.3Bimodal dielectric probe method489References 48912Measurement of Dielectric Properties of Materials at High Temperatures 49212.1 Introduction492 12.1.1 Dielectric properties of materials at high temperatures492 12.1.2 Problems in measurements at high temperatures494 12.1.3 Overviews of the methods for measurements at high temperatures 49612.2 Coaxial-line Methods497 12.2.1 Measurement of permittivity using open-ended coaxial probe 498 12.2.2 Problems related to high-temperature measurements498 12.2.3 Correction of phase shift500 12.2.4 Spring-loaded coaxial probe502 12.2.5 Metallized ceramic coaxial probe 50212.3 Waveguide Methods 503 12.3.1 Open-ended waveguide method503 12.3.2 Dual-waveguide method50412.4 Free-space Methods506 12.4.1 Computation of r50712.5 Cavity-Perturbation Methods 510 12.5.1 Cavity-perturbation methods for high-temperature measurements510 12.5.2 TE10n mode rectangular cavity512 12.5.3 TM mode cylindrical cavity 51412.6 Dielectric-loaded Cavity Method 520 12.6.1 Coaxial reentrant cavity 520 12.6.2 Open-resonator method523 12.6.3 Oscillation method 524 References528Index531 11. PrefaceMicrowave materials have been widely used in a variety of applications ranging from communicationdevices to military satellite services, and the study of materials properties at microwave frequenciesand the development of functional microwave materials have always been among the most active areasin solid-state physics, materials science, and electrical and electronic engineering. In recent years, theincreasing requirements for the development of high-speed, high-frequency circuits and systems requirecomplete understanding of the properties of materials functioning at microwave frequencies. All theseaspects make the characterization of materials properties an important eld in microwave electronics. Characterization of materials properties at microwave frequencies has a long history, dating from theearly 1950s. In past decades, dramatic advances have been made in this eld, and a great deal of newmeasurement methods and techniques have been developed and applied. There is a clear need to have apractical reference text to assist practicing professionals in research and industry. However, we realizethe lack of good reference books dealing with this eld. Though some chapters, reviews, and bookshave been published in the past, these materials usually deal with only one or several topics in thiseld, and a book containing a comprehensive coverage of up-to-date measurement methodologies is notavailable. Therefore, most of the research and development activities in this eld are based primarilyon the information scattered throughout numerous reports and journals, and it always takes a great dealof time and effort to collect the information related to on-going projects from the voluminous literature.Furthermore, because of the paucity of comprehensive textbooks, the training in this eld is usually notsystematic, and this is undesirable for further progress and development in this eld. This book deals with the microwave methods applied to materials property characterization, and itprovides an in-depth coverage of both established and emerging techniques in materials characterization.It also represents the most comprehensive treatment of microwave methods for materials propertycharacterization that has appeared in book form to date. Although this book is expected to be mostuseful to those engineers actively engaged in designing materials propertycharacterization methods, itshould also be of considerable value to engineers in other disciplines, such as industrial engineers,bioengineers, and materials scientists, who wish to understand the capabilities and limitations ofmicrowave measurement methods that they use. Meanwhile, this book also satises the requirement forup-to-date texts at graduate and senior undergraduate levels on the subjects in materials characterization. Among this books most outstanding features is its comprehensive coverage. This book discussesalmost all aspects of the microwave theory and techniques for the characterization of the electromagneticproperties of materials at microwave frequencies. In this book, the materials under characterizationmay be dielectrics, semiconductors, conductors, magnetic materials, and articial materials; theelectromagnetic properties to be characterized mainly include permittivity, permeability, chirality,mobility, and surface impedance. The two introductory chapters, Chapter 1 and Chapter 2, are intended to acquaint the readers with thebasis for the research and engineering of electromagnetic materials from the materials and microwavefundamentals respectively. As general knowledge of electromagnetic properties of materials is helpfulfor understanding measurement results and correcting possible errors, Chapter 1 introduces the general 12. xii Prefaceproperties of various electromagnetic materials and their underlying physics. After making a briefreview on the methods for materials properties characterization, Chapter 2 provides a summary ofthe basic microwave theory and techniques, based on which the methods for materials characterizationare developed. This summary is mainly intended for reference rather than for tutorial purposes, althoughsome of the important aspects of microwave theory are treated at a greater length. References are citedto permit readers to further study the topics they are interested in. Chapters 3 to 8 deal with the measurements of the permittivity and permeability of low-conductivitymaterials and the surface impedance of high-conductivity materials. Two types of nonresonant methods,reection method and transmission/reection method, are discussed in Chapters 3 and 4 respectively;two types of resonant methods, resonator method and resonant-perturbation method, are discussed inChapters 5 and 6 respectively. In the methods discussed in Chapters 3 to 6, the transmission lines usedare mainly coaxial-line, waveguide, and free-space, while Chapter 7 is concerned with the measurementmethods developed from planar transmission lines, including stripline, microstrip-, and coplanar line.The methods discussed in Chapters 3 to 7 are suitable for isotropic materials, which have scalar orcomplex permittivity and permeability. The permittivity of anisotropic dielectric materials is a tensorparameter, and magnetic materials usually have tensor permeability under an external dc magnetic eld.Chapter 8 deals with the measurement of permittivity and permeability tensors. Ferroelectric materials are a special category of dielectric materials often used in microwave electron-ics for developing electrically tunable devices. Chapter 9 discusses the characterization of ferroelectricmaterials, and the topics covered include the techniques for studying the temperature dependence andelectric eld dependence of dielectric properties. In recent years, the research on articial materials has been active. Chapter 10 deals with a specialtype of articial materials: chiral materials. After introducing the concept and basic characteristics ofchiral materials, the methods for chirality measurements and the possible applications of chiral materialsare discussed. The electrical transport properties at microwave frequencies are important for the development of high-speed electronic circuits. Chapter 11 discusses the microwave Hall effect techniques for the measurementof the electrical transport properties of low-conductivity, high-conductivity, and magnetic materials. The measurement of materials properties at high temperatures is often required in industry, scienticresearch, and biological and medical applications. In principle, most of the methods discussed in thisbook can be extended to high-temperature measurements. Chapter 12 concentrates on the measurementof the dielectric properties of materials at high temperatures, and the techniques for solving the problemsin high-temperature measurements can also be applied for the measurement of other materials propertyparameters at high temperatures. In this book, each chapter is written as a self-contained unit, so that readers can quickly getcomprehensive information related to their research interests or on-going projects. To provide a broadtreatment of various topics, we condensed mountains of literature into readable accounts within a text ofreasonable size. Many references have been included for the benet of the readers who wish to pursuea given topic in greater depth or refer to the original papers. It is clear that the principle of a method for materials characterization is more important thanthe techniques required for implementing this method. If we understand the fundamental principleunderlying a measurement method, we can always nd a suitable way to realize this method. Althoughthe advances in technology may signicantly change the techniques for implementing a measurementmethod, they cannot greatly inuence the measurement principle. In writing this book, we tried topresent the fundamental principles behind various designs so that readers can understand the process ofapplying fundamental concepts to arrive at actual designs using different techniques and approaches. Webelieve that an engineer with a sound knowledge of the basic concepts and fundamental principles formaterials property characterization and the ability apply to his knowledge toward design objectives, is 13. Preface xiiithe engineer who is most likely to make full use of the existing methods, and develop original methodsto fulll ever-rising measurement requirements. We would like to indicate that this text is a compilation of the work of many people. We cannot be heldresponsible for the designs described that are still under patent. It is also difcult to always give propercredits to those who are the originators of new concepts and the inventors of new methods. The names wegive to some measurement methods may not t the intentions of the inventors or may not accurately reectthe most characteristic features of these methods. We hope that there are not too many such errors and willappreciate it if the readers could bring the errors they discover to our attention. There are many people to whom we owe many thanks for helping us prepare this book. However,space dictates that only a few of them can receive formal acknowledgements. But this should not be takenas a disparagement of those whose contributions remain anonymous. Our foremost appreciation goes toMr. Quek Gim Pew, Deputy Chief Executive (Technology), Singapore Defence Science & TechnologyAgency, Mr. Quek Tong Boon, Chief Executive Ofcer, Singapore DSO National Laboratories, andProfessor Lim Hock, Director, Temasek Laboratories, National University of Singapore, for theirencouragement and support along the way. We are grateful to Pennsylvania State University and HVSTechnologies for giving us permission to include the HVS Free Space Unit and the data in this book.We really appreciate the valuable help and cooperation from Dr. Li Zheng-Wen, Dr. Rao Xuesong, andMr. Tan Chin Yaw. We are very grateful to the staff of John Wiley & Sons for their helpful efforts andcheerful professionalism during this project.L. F. ChenC. K. OngC. P. NeoV. V. VaradanV. K. Varadan 14. 1 Electromagnetic Properties of MaterialsThis chapter starts with the introduction of thethe applications of techniques for materials prop-materials research and engineering at microwave erty characterization in various elds of sciencesfrequencies, with emphasis laid on the signi-and engineering. The importance of the researchcance and applications of the study of the elec-on the electromagnetic properties of materials attromagnetic properties of materials. The fun- microwave frequencies can be understood in thedamental physics that governs the interactionsaspects that follow.between materials and electromagnetic elds is Firstly, though it is an old eld in physics,then discussed at both microscopic and macro- the study of electromagnetic properties of mate-scopic scales. Subsequently, we analyze the gen-rials at microwave frequencies is full of academiceral properties of typical electromagnetic materi-importance (Solymar and Walsh 1998; Kittel 1997;als, including dielectric materials, semiconductors,Von Hippel 1995a,b; Jiles 1994; Robert 1988),conductors, magnetic materials, and articial mate- especially for magnetic materials (Jiles 1998; Smitrials. Afterward, we discuss the intrinsic proper-1971) and superconductors (Tinkham 1996) andties and extrinsic performances of electromagneticferroelectrics (Lines and Glass 1977). The knowl-materials.edge gained from microwave measurements con-tributes to our information about both the macro-1.1 MATERIALS RESEARCH ANDscopic and the microscopic properties of materi-ENGINEERING AT MICROWAVEals, so microwave techniques have been importantFREQUENCIES for materials property research. Though magneticmaterials are widely used in various elds, theWhile technology decides how electromagneticresearch of magnetic materials lags far behind theirmaterials can be utilized, science attempts toapplications, and this, to some extent, hinders usdecipher why materials behave as they do. The from making full application of magnetic mate-responses of materials to electromagnetic elds rials. Until now, the electromagnetic propertiesare closely determined by the displacement of of magnetic properties at microwave frequenciestheir free and bounded electrons by electric eldshave not been fully investigated yet, and this isand the orientation of their atomic moments byone of the main obstacles for the development ofmagnetic elds. The deep understanding and full microwave magnetoelectrics. Besides, one of theutilization of electromagnetic materials have comemost promising applications of superconductors isfrom decoding the interactions between materialsmicrowave electronics. A lot of effort has beenand electromagnetic elds by using both theoretical put in the study of the microwave propertiesand experimental strategies.of superconductors, while many areas are yet to This book mainly deals with the methodologybe explored. Meanwhile, as ferroelectric materi-for the characterization of electromagnetic materi- als have great application potential in developingals for microwave electronics, and also discusses smart electromagnetic materials, structures, andMicrowave Electronics: Measurement and Materials Characterization L. F. Chen, C. K. Ong, C. P. Neo, V. V. Varadan and V. K. Varadan 2004 John Wiley & Sons, Ltd ISBN: 0-470-84492-2 15. 2 Microwave Electronics: Measurement and Materials Characterizationdevices in recent years, microwave ferroelectricityFinally, as the electromagnetic properties ofis under intensive investigation. materials are related to other macroscopic or Secondly, microwave communications are play- microscopic properties of the materials, we caning more and more important roles in military,obtain information about the microscopic orindustrial, and civilian life, and microwave engi-macroscopic properties we are interested in fromneering requires precise knowledge of the elec- the electromagnetic properties of the materials.tromagnetic properties of materials at microwaveIn materials research and engineering, microwavefrequencies (Ramo et al. 1994). Since World War techniques for the characterization of materialsII, a lot of resources have been put into electromag- properties are widely used in monitoring the fab-netic signature control, and microwave absorbersrication procedure and nondestructive testing ofare widely used in reducing the radar cross sectionssamples and products (Zoughi 2000; Nyfors and(RCSs) of vehicles. The study of electromagneticVainikainen 1989).properties of materials and the ability of tailoring This chapter aims to provide basic knowledgethe electromagnetic properties of composite mate- for understanding the results from microwave mea-rials are very important for the design and devel-surements. We will give a general introductionopment of radar absorbing materials and other on electromagnetic materials at microscopic andfunctional electromagnetic materials and struc- macroscopic scales and will discuss the parameterstures (Knott et al. 1993).describing the electromagnetic properties of mate- Thirdly, as the clock speeds of electronic rials, the classication of electromagnetic mate-devices are approaching microwave frequencies,rials, and general properties of typical electro-it becomes indispensable to study the microwave magnetic materials. Further discussions on variouselectronic properties of materials used in elec-topics can be found in later chapters or the refer-tronic components, circuits, and packaging. The ences cited.development of electronic components workingat microwave frequencies needs the electrical 1.2 PHYSICS FOR ELECTROMAGNETICtransport properties at microwave frequencies,MATERIALSsuch as Hall mobility and carrier density; andIn physics and materials sciences, electromagneticthe development of electronic circuits work-materials are studied at both the microscopicing at microwave frequencies requires accu- and the macroscopic scale (Von Hippel 1995a,b).rate constitutive properties of materials, such At the microscopic scale, the energy bands foras permittivity and permeability. Meanwhile, theelectrons and magnetic moments of the atomselectromagnetic interference (EMI) should beand molecules in materials are investigated, whiletaken into serious consideration in the design of at the macroscopic level, we study the overallcircuit and packaging, and special materials areresponses of macroscopic materials to externalneeded to ensure electromagnetic compatibilityelectromagnetic elds.(EMC) (Montrose 1999). Fourthly, the study of electromagnetic properties1.2.1 Microscopic scaleof materials is important for various elds of sci-ence and technology. The principle of microwave In the microscopic scale, the electrical properties ofremote sensing is based on the reection anda material are mainly determined by the electronscattering of different objects to microwave sig- energy bands of the material. According to thenals, and the reection and scattering proper-energy gap between the valence band and theties of an object are mainly determined by theconduction band, materials can be classied intoelectromagnetic properties of the object. Besides,insulators, semiconductors, and conductors. Owingthe conclusions of the research of electromag-to its electron spin and electron orbits around thenetic materials are helpful for agriculture, food nucleus, an atom has a magnetic moment. Accordingengineering, medical treatments, and bioengineer- to the responses of magnetic moments to magneticing (Thuery and Grant 1992).eld, materials can be generally classied into 16. Electromagnetic Properties of Materials3diamagnetic materials, paramagnetic materials, andband. While for some elements, for example carbon,ordered magnetic materials. the merged broadband may further split into separatebands at closer atomic separation. The highest energy band containing occupied1.2.1.1 Electron energy bandsenergy levels at 0 K in a solid is called the valenceAccording to Bohrs model, an atom is characterized band. The valence band may be completely lledby its discrete energy levels. When atoms are or only partially lled with electrons. The electronsbrought together to constitute a solid, the discretein the valence band are bonded to their nuclei.levels combine to form energy bands and the The conduction band is the energy band above theoccupancy of electrons in a band is dictatedvalence energy band, and contains vacant energyby Fermi-dirac statistics. Figure 1.1 shows the levels at 0 K. The electrons in the conduction bandrelationship between energy bands and atomicare called free electrons, which are free to move.separation. When the atoms get closer, the energy Usually, there is a forbidden gap between thebands broaden, and usually the outer band broadensvalence band and the conduction band, and themore than the inner one. For some elements, for availability of free electrons in the conduction bandexample lithium, when the atomic separation ismainly depends on the forbidden gap energy. If thereduced, the bands may broaden sufciently forforbidden gap is large, it is possible that no freeneighboring bands to merge, forming a broader electrons are available, and such a material is calledan insulator. For a material with a small forbiddenenergy gap, the availability of free electron in theconduction band permits some electron conduction,and such a material is a semiconductor. In a [Image not available in this electronic edition.]conductor, the conduction and valence bands mayoverlap, permitting abundant free electrons to beavailable at any ambient temperature, thus givinghigh electrical conductivity. The energy bands forinsulator, semiconductor, and good conductor areshown schematically in Figure 1.2.Figure 1.1 The relationships between energy bandsand atomic separation. (a) Energy bands of lithiumInsulatorsand (b) energy bands of carbon. (Bolton 1992) Source:Bolton, W. (1992), Electrical and Magnetic Properties For most of the insulators, the forbidden gapof Materials, Longman Scientic & Technical, Harlow between their valence and conduction energy bands [Image not available in this electronic edition.]Figure 1.2 Energy bands for different types of materials. (a) Insulator, (b) semiconductor, and (c) goodconductor. (Bolton 1992). Modied from Bolton, W. (1992), Electrical and Magnetic Properties of Materials,Longman Scientic & Technical, Harlow 17. 4 Microwave Electronics: Measurement and Materials Characterizationis larger than 5 eV. Usually, we assume that an conductor, the density of free electrons is on theinsulate is nonmagnetic, and under this assump- order of 1028 m3 . Lithium is a typical example oftion, insulators are called dielectrics. Diamond, a a conductor. It has two electrons in the 1s shellform of carbon, is a typical example of a dielectric. and one in the 2s shell. The energy bands of suchCarbon has two electrons in the 1s shell, two in theelements are of the form shown in Figure 1.1(a).2s shell, and two in the 2p shell. In a diamond,The 2s and 2p bands merge, forming a largethe bonding between carbon atoms is achievedband that is only partially occupied, and under anby covalent bonds with electrons shared between electric eld, electrons can easily move into vacantneighboring atoms, and each atom has a share in energy levels.eight 2p electrons (Bolton 1992). So all the elec- In the category of conductors, superconduc-trons are tightly held between the atoms by thistors have attracted much research interest. In acovalent bonding. As shown in Figure 1.1(b), thenormal conductor, individual electrons are scat-consequence of this bonding is that diamond has tered by impurities and phonons. However, fora full valence band with a substantial forbiddensuperconductors, the electrons are paired withgap between the valence band and the conduc-those of opposite spins and opposite wave vec-tion band. But it should be noted that, graphite, tors, forming Cooper pairs, which are bondedanother form of carbon, is not a dielectric, but atogether by exchanging phonons. In the Bardeenconductor. This is because all the electrons in the CooperSchrieffer (BCS) theory, these Coopergraphite structure are not locked up in covalentpairs are not scattered by the normal mechanisms.bonds and some of them are available for conduc-A superconducting gap is found in superconduc-tion. So the energy bands are related to not only the tors and the size of the gap is in the microwaveatom structures but also the ways in which atomsfrequency range, so study of superconductors atare combined. microwave frequencies is important for the under-standing of superconductivity and application ofsuperconductors.SemiconductorsThe energy gap between the valence and conduction 1.2.1.2 Magnetic momentsbands of a semiconductor is about 1 eV. Germaniumand silicon are typical examples of semiconductors. An electron orbiting a nucleus is equivalent to aEach germanium or silicon atom has four valence current in a single-turn coil, so an atom has aelectrons, and the atoms are held together by magnetic dipole moment. Meanwhile, an electroncovalent bonds. Each atom shares electrons with also spins. By considering the electron to be aeach of four neighbors, so all the electrons aresmall charged sphere, the rotation of the chargelocked up in bonds. So there is a gap between a fullon the surface of the sphere is also like a single-valence band and the conduction band. However,turn current loop and also produces a magneticunlike insulators, the gap is relatively small. Atmoment (Bolton 1992). The magnetic properties ofroom temperature, some of the valence electrons a material are mainly determined by its magneticcan break free from the bonds and have sufcientmoments that result from the orbiting and spinningenergy to jump over the forbidden gap, arriving of electrons. According to the responses of theat the conduction band. The density of the free magnetic moments of the atoms in a material to anelectrons for most of the semiconductors is in theexternal magnetic eld, materials can be generallyrange of 1016 to 1019 per m3 .classied into diamagnetic, paramagnetic, andordered magnetic materials.ConductorsDiamagnetic materialsFor a conductor, there is no energy gap between The electrons in a diamagnetic material are all pairedthe valence gap and conduction band. For a good up with spins antiparallel, so there is no net magnetic 18. Electromagnetic Properties of Materials5moment on their atoms. When an external magneticeld is applied, the orbits of the electrons change,resulting in a net magnetic moment in the directionopposite to the applied magnetic eld. It should benoted that all materials have diamagnetism since allmaterials have orbiting electrons. However, for dia-magnetic materials, the spin of the electrons does [Image not available in this electronic edition.]not contribute to the magnetism; while for param-agnetic and ferromagnetic materials, the effects ofthe magnetic dipole moments that result from thespinning of electrons are much greater than the dia-magnetic effect.Paramagnetic materials Figure 1.3 Arrangements of magnetic moments inThe atoms in a paramagnetic material have netvarious magnetic materials. (a) Paramagnetic, (b) ferro-magnetic moments due to the unpaired electronmagnetic, (c) antiferromagnetic, and (d) ferrimagneticspinning in the atoms. When there is no exter- materials. Modied from Bolton, W. (1992). Electricalnal magnetic eld, these individual moments areand Magnetic Properties of Materials, Longman Scien-randomly aligned, so the material does not showtic & Technical, Harlowmacroscopic magnetism. When an external mag-netic eld is applied, the magnetic moments areordered arrangement of magnetic dipoles shown inslightly aligned along the direction of the exter- Figure 1.3(b), is quite different from the couplingnal magnetic eld. If the applied magnetic eldbetween atoms of paramagnetic materials, whichis removed, the alignment vanishes immediately.results in the random arrangement of magneticSo a paramagnetic material is weakly magneticdipoles shown in Figure 1.3(a). Iron, cobalt, andonly in the presence of an external magnetic nickel are typical ferromagnetic materials.eld. The arrangement of magnetic moments in aAs shown in Figure 1.3(c), in an antiferromag-paramagnetic material is shown in Figure 1.3(a). netic material, half of the magnetic dipoles alignAluminum and platinum are typical paramag- themselves in one direction and the other half ofnetic materials. the magnetic moments align themselves in exactly the opposite direction if the dipoles are of theOrdered magnetic materials same size and cancel each other out. Manganese, manganese oxide, and chromium are typical anti-In ordered magnetic materials, the magnetic mo-ferromagnetic materials. However, as shown inments are arranged in certain orders. According to Figure 1.3(d), for a ferrimagnetic material, alsothe ways in which magnetic moments are arranged, called ferrite, the magnetic dipoles have differentordered magnetic materials fall into several subcat- sizes and they do not cancel each other. Magnetiteegories, mainly including ferromagnetic, antiferro-(Fe3 O4 ), nickel ferrite (NiFe2 O4 ), and barium fer-magnetic, and ferrimagnetic (Bolton 1992; Wohl-rite (BaFe12 O19 ) are typical ferrites.farth 1980). Figure 1.3 shows the arrangementsGenerally speaking, the dipoles in a ferromag-of magnetic moments in paramagnetic, ferromag- netic or ferrimagnetic material may not all benetic, antiferromagnetic, and ferrimagnetic materi-arranged in the same direction. Within a domain, allals, respectively. the dipoles are arranged in its easy-magnetization As shown in Figure 1.3(b), the atoms in a direction, but different domains may have differ-ferromagnetic material are bonded together in such ent directions of arrangement. Owing to the randoma way that the dipoles in neighboring atoms are allorientations of the domains, the material does notin the same direction. The coupling between atomshave macroscopic magnetism without an externalof ferromagnetic materials, which results in the magnetic eld. 19. 6 Microwave Electronics: Measurement and Materials Characterization The crystalline imperfections in a magneticwalls are pinned by crystalline imperfections. Asmaterial have signicant effects on the magneti-shown in Figure 1.4(b), when an external magneticzation of the material (Robert 1988). For an idealeld H is applied, the domains whose orienta-magnetic material, for example monocrystallinetions are near the direction of the external mag-iron without any imperfections, when a magnetic netic eld grow in size, while the sizes of theeld H is applied, due to the condition of minimumneighboring domains wrongly directed decrease.energy, the sizes of the domains in H direction When the magnetic eld is very weak, the domainincrease, while the sizes of other domains decrease.walls behave like elastic membranes, and theAlong with the increase of the magnetic eld, the changes of the domains are reversible. Whenstructures of the domains change successively, andthe magnetic eld increases, the pressure on thenally a single domain in H direction is obtained.domain walls causes the pinning points to giveIn this ideal case, the displacement of domain wallsway, and the domain walls move by a series ofis free. When the magnetic eld H is removed, the jumps. Once a jump of domain wall happens,material returns to its initial state; so the magneti-the magnetization process becomes irreversible. Aszation process is reversible. shown in Figure 1.4(c), when the magnetic eld H Owing to the inevitable crystalline imperfec-reaches a certain level, all the magnetic momentstions, the magnetization process becomes com- are arranged parallel to the easy magnetizationplicated. Figure 1.4(a) shows the arrangement ofdirection nearest to the direction of the externaldomains in a ferromagnetic material when no magnetic eld H. If the external magnetic eldexternal magnetic eld is applied. The domain H increases further, the magnetic moments arealigned along H direction, deviating from the easymagnetization direction, as shown in Figure 1.4(d).In this state, the material shows its greatest magne-tization, and the material is magnetically saturated. In a polycrystalline magnetic material, the mag- Hnetization process in each grain is similar to thatin a monocrystalline material as discussed above.However, due to the magnetostatic and magne-tostrictions occurring between neighboring grains, (a) (b)the overall magnetization of the material becomesquite complicated. The grain structures are impor-tant to the overall magnetization of a polycrys-talline magnetic material. The magnetization pro-cess of magnetic materials is further discussed inHHSection 1.3.4.1. It is important to note that for an ordered magneticmaterial, there is a special temperature called Curietemperature (Tc ). If the temperature is below theCurie temperature, the material is in a magnetically (c)(d)ordered phase. If the temperature is higher thanFigure 1.4 Domains in a ferromagnetic material. the Curie temperature, the material will be in a(a) Arrangement of domains when no external mag-paramagnetic phase. The Curie temperature for ironnetic eld is applied, (b) arrangement of domains whenis 770 C, for nickel 358 C, and for cobalt 1115 C.a weak magnetic eld is applied, (c) arrangement ofdomains when a medium magnetic eld is applied,and (d) arrangement of domains when a strong mag- 1.2.2 Macroscopic scalenetic eld is applied. Modied from Robert, P. (1988).Electrical and Magnetic Properties of Materials, Artech The interactions between a macroscopic mate-House, Norwoodrial and electromagnetic elds can be generally 20. Electromagnetic Properties of Materials 7described by Maxwells equations: I = jC0wUID = (1.1)B =0 (1.2) C0 U = U0exp(jwt) 90 H = D/t + J(1.3) E = B /t (1.4)with the following constitutive relations: U(a) (b)D = E = ( j )E(1.5) Figure 1.5 The current in a circuit with a capacitor.B = H = ( j )H(1.6) (a) Circuit layout and (b) complex plane showing cur- rent and voltage J = E(1.7)where H is the magnetic eld strength vector; E, properties of low-conductivity materials. As thethe electric eld strength vector; B, the magnetic value of conductivity is small, we concentrateux density vector; D, the electric displacement on permittivity and permeability. In a general case,vector; J, the current density vector; , the charge both permittivity and permeability are complexdensity; = j , the complex permittivity ofnumbers, and the imaginary part of permittivitythe material; = j , the complex perme-is related to the conductivity of the material. Inability of the material; and , the conductivity of the following discussion, we analogize microwavethe material. Equations (1.1) to (1.7) indicate that signals to ac signals, and distributed capacitor andthe responses of an electromagnetic material toinductor to lumped capacitor and inductor (Vonelectromagnetic elds are determined essentially Hippel 1995b).by three constitutive parameters, namely permit-Consider the circuit shown in Figure 1.5(a). Thetivity , permeability , and conductivity . These vacuum capacitor with capacitance C0 is connectedparameters also determine the spatial extent toto an ac voltage source U = U0 exp(jt). Thewhich the electromagnetic eld can penetrate intocharge storage in the capacitor is Q = C0 U , andthe material at a given frequency. the current I owing in the circuit is In the following, we discuss the parameters dQddescribing two general categories of materials: I== (C0 U0 ejt ) = jC0 U(1.8)low-conductivity materials and high-conductivity dtdtmaterials. So, in the complex plane shown in Figure 1.5(b), the current I leads the voltage U by a phase angle1.2.2.1 Parameters describing low-conductivity of 90 .materials Now, we insert a dielectric material into the capacitor and the equivalent circuit is shown inElectromagnetic waves can propagate in a low-Figure 1.6(a). The total current consists of two parts,conductivity material, so both the surface and inner the charging current (Ic ) and loss current (Il ):parts of the material respond to the electromag-netic wave. There are two types of parameters I = Ic + Il = jCU + GU = (jC + G)Udescribing the electromagnetic properties of low-(1.9)conductivity materials: constitutive parameters andwhere C is the capacitance of the capacitorpropagation parameters.loaded with the dielectric material and G is the conductance of the dielectric material. The lossConstitutive parameterscurrent is in phase with the source voltage U . In the complex plane shown in Figure 1.6(b), theThe constitutive parameters dened in Eqs. (1.5) tocharging current Ic leads the loss current Il by a(1.7) are often used to describe the electromagnetic phase angle of 90 , and the total current I leads 21. 8 Microwave Electronics: Measurement and Materials CharacterizationIc = jwCUI Jc = jweEJ IIc IlU C Gddqq Il = GU (a) (b)Jl = weEFigure 1.6 The relationships between charging currentand loss current. (a) Equivalent circuit and (b) complexFigure 1.7 Complex plane showing the charging cur-plane showing charging current and loss current rent density and loss current densitythe source voltage U with an angle less than 90 .and the dielectric power factor is given byThe phase angle between Ic and I is often calledcos e = / ( )2 + ( )2 (1.14)loss angle . We may alternatively use complex permittiv-Equations (1.13) and (1.14) show that for aity = j to describe the effect of dielec-small loss angle e , cos tan e .tric material. After a dielectric material is insertedIn microwave electronics, we often use relativeinto the capacitor, the capacitance C of the capaci-permittivity, which is a dimensionless quantity,tor becomes dened byC0 C0 jC= = ( j )(1.10) r = == r jr = r (1 j tan e ) 0 0 00And the charging current is (1.15)where is complex permittivity, C0C0r is relative complex permittivity,I = j( j ) U = (j + ) U 000 = 8.854 1012 F/m is the(1.11) permittivity of free space,Therefore, as shown in Figure 1.6, the current r is the real part of relative complexdensity J transverse to the capacitor under the permittivity,applied eld strength E becomes r is the imaginary part of relativedE complex permittivity, J = (j + )E = (1.12)dt tan e is dielectric loss tangent, andThe product of angular frequency and loss factor e is dielectric loss angle.is equivalent to a dielectric conductivity: = .Now, let us consider the magnetic responseThis dielectric conductivity sums over all the dis- of low-conductivity material. According to thesipative effects of the material. It may representFaradays inductance lawan actual conductivity caused by migrating chargecarriers and it may also refer to an energy loss asso- dI U =L ,(1.16)ciated with the dispersion of , for example, the dtfriction accompanying the orientation of dipoles. we can get the magnetization current Im :The latter part of dielectric conductivity will bediscussed in detail in Section 1.3.1. U Im = j (1.17) According to Figure 1.7, we dene two parame- L0ters describing the energy dissipation of a dielectricwhere U is the magnetization voltage, L0 ismaterial. The dielectric loss tangent is given bythe inductance of an empty inductor, and tan e = / , (1.13) is the angular frequency. If we introduce an 22. Electromagnetic Properties of Materials 9Ilgiven by Uq j r == 90d 0 0= r jr = r (1 j tan m ) (1.22) Im = j Uwhere is complex complex permeability, w L0 mrIm Ir is relative complex permeability, (a)(b) 0 = 4 107 H/m is the permeabilityFigure 1.8 The magnetization current in a complex of free space,plane. (a) Relationship between magnetization current r is the real part of relative complexand voltage and (b) relationship between magnetizationpermeability,current and loss currentr is the imaginary part of the relativecomplex permeability,ideal, lossless magnetic material with relative tan m is the magnetic loss tangent, andpermeability r , the magnetization eld becomesm is the magnetic loss angle.UIn summary, the macroscopic electric and mag- Im = j (1.18) netic behavior of a low-conductivity material is L0 rmainly determined by the two complex parame- In the complex plane shown in Figure 1.8(a), ters: permittivity () and permeability (). Per-the magnetization current Im lags the voltage mittivity describes the interaction of a materialU by 90 for no loss of magnetic materials. with the electric eld applied on it, while per-As shown in Figure 1.8(b), an actual magnetic meability describes the interaction of a materialmaterial has magnetic loss, and the magnetic loss with magnetic eld applied on it. Both the elec-current Il caused by energy dissipation duringtric and magnetic elds interact with materials inthe magnetization cycle is in phase with U. Bytwo ways: energy storage and energy dissipation.introducing a complex permeability = j Energy storage describes the lossless portion ofand a complex relative permeability r = r jr the exchange of energy between the eld and thein complete analogy to the dielectric case, wematerial, and energy dissipation occurs when elec-obtain the total magnetization currenttromagnetic energy is absorbed by the material. Soboth permittivity and permeability are expressed asU jU ( + j )complex numbers to describe the storage (real part)I = Im + Il = =jL0 r(L0 /0 )( 2 + 2 ) and dissipation (imaginary part) effects of each. (1.19)Besides the permittivity and permeability, anotherparameter, quality factor, is often used to describe Similar to the dielectric case, according to an electromagnetic material:Figure 1.8, we can also dene two parameters r 1describing magnetic materials: the magnetic lossQe == (1.23)tangent given by r tan e r 1 tan m = / , (1.20) Qm == (1.24) r tan mand the power factor given by On the basis of the dielectric quality factor Qeand magnetic quality factor Qm , we can get the cos m = / ( )2 + ( )2 .(1.21)total quality factor Q of the material: In microwave electronics, relative permeability1 1 1is often used, which is a dimensionless quantity=+(1.25)Q Qe Qm 23. 10 Microwave Electronics: Measurement and Materials CharacterizationPropagation parameters For a high-conductivity material, we assume , which means that the conducting current isThe propagation of electromagnetic waves in amuch larger than the displacement current. So,medium is determined by the characteristic waveEq. (1.29) can be approximated by ignoring theimpedance of the medium and the wave veloc-displacement current term:ity v in the medium. The characteristic waveimpedance is also called the intrinsic impedance = + j = j = (1 + j)of the medium. When a single wave propagates j 2with velocity v in the Z-positive direction, the(1.30)characteristic impedance is dened as the ratioWe dene the skin depth:of total electric eld to total magnetic eld at a1 2Z-plane. The wave impedance and velocity can bes = =(1.31)calculated from the permittivity and permeabilityof the medium: The physics meaning of skin depth is that, in a high-conductivity material, the elds decay by an= (1.26)amount e1 in a distance of a skin depth s . At microwave frequencies, the skin depth s is a very 1 small distance. For example, the skin depth of av=(1.27) metal at microwave frequencies is usually on the order of 107 m. From Eqs. (1.26) and (1.27), we can calculate theBecause of the skin effect, the utility and behav-wave impedance of free space, 0 = (0 /0 )1/2 ior of high-conductivity materials at microwave= 376.7 , and the wave velocity in free space, c = frequencies are mainly determined by their surface(0 0 )1/2 = 2.998 108 m/s. Expressing permit- impedance Zs :tivity and permeability as complex quantities leadsto a complex number for the wave velocity (v), Et Zs = Rs + jXs == (1 + j) (1.32)where the imaginary portion is a mathematical con- Ht 2venience for expressing loss.where Ht is the tangential magnetic eld, Et Sometimes, it is more convenient to use the is the tangential electric eld, Rs is the surfacecomplex propagation coefcient to describe resistance, and Xs is the surface reactance. Forthe propagation of electromagnetic waves innormal conductors, is a real number. Accordinga medium:to Eq. (1.32), the surface resistance Rs and the surface reactance Xs are equal and they are = + j = j = j r r = j ncc proportional to 1/2 for normal metals:(1.28)where n is the complex index of refraction, where Rs = Xs =(1.33) is the angular frequency, is the attenua- 2tion coefcient, = 2/ is the phase changecoefcient, and is the operating wavelength in 1.2.2.3 Classication of electromagnetic materialsthe medium.Materials can be classied according to their macroscopic parameters. According to conductiv-1.2.2.2 Parameters describing high-conductivityity, materials can be classied as insulators, semi-materialsconductors, and conductors. Meanwhile, materials can also be classied according to their perme-For a high-conductivity material, for example a ability values. General properties of typical typesmetal, Eq. (1.28) for the complex propagation of materials are discussed in Section 1.3.constant should be modied asWhen classifying materials according to their macroscopic parameters, it should be noted that we = + j = j 1 j (1.29)use the terms insulator, semiconductor, conductor, 24. Electromagnetic Properties of Materials 11and magnetic material to indicate the dominantmaterials. According to their permeability values,responses of different types of materials. Allmaterials generally fall into three categories: dia-materials have some response to magnetic eldsmagnetic ( < 0 ), paramagnetic ( 0 ), andbut, except for ferromagnetic and ferrimagnetic highly magnetic materials mainly including ferro-types, their responses are usually very small,magnetic and ferrimagnetic materials. The perme-and their permeability values differ from 0 by ability values of highly magnetic materials, espe-a negligible fraction. Most of the ferromagneticcially ferromagnetic materials, are much largermaterials are highly conductive, but we call them than 0 .magnetic materials, as their magnetic propertiesare the most signicant in their applications. Forsuperconductors, the Meissner effect shows that 1.3 GENERAL PROPERTIESthey are a kind of very special magnetic materials, OF ELECTROMAGNETIC MATERIALSbut in microwave electronics, people are moreHere, we discuss the general properties of typi-interested in their surface impedance.cal electromagnetic materials, including dielectricmaterials, semiconductors, conductors, magneticInsulatorsmaterials, and articial materials. The knowledgeInsulators have very low conductivity, usually in the of general properties of electromagnetic materi-range of 1012 to 1020 ( m)1 . Often, we assume als is helpful for understanding the measurementinsulators are nonmagnetic, so they are actuallyresults and correcting the possible errors one maydielectrics. In theoretical analysis of dielectricmeet in materials characterization. In the nal partmaterials, an ideal model, perfect dielectric, is often of this section, we will discuss other descriptionsused, representing a material whose imaginary partof electromagnetic materials, which are importantof permittivity is assumed to be zero: = 0. for the design and applications of electromag-netic materials.SemiconductorsThe conductivity of a semiconductor is higher 1.3.1 Dielectric materialsthan that of a dielectric but lower than thatof a conductor. Usually, the conductivities ofFigure 1.9 qualitatively shows a typical behaviorsemiconductors at room temperature are in the of permittivity ( and ) as a function of fre-range of 107 to 104 ( m)1 . quency. The permittivity of a material is related toa variety of physical phenomena. Ionic conduction,Conductorsdipolar relaxation, atomic polarization, and elec-tronic polarization are the main mechanisms thatConductors have very high conductivity, usually contribute to the permittivity of a dielectric mate-in the range of 104 to 108 ( m)1 . Metals arerial. In the low frequency range, is dominatedtypical conductors. There are two types of specialby the inuence of ion conductivity. The variationconductors: perfect conductors and superconduc- of permittivity in the microwave range is mainlytors. A perfect conductor is a theoretical modelcaused by dipolar relaxation, and the absorptionthat has innite conductivity at any frequencies. peaks in the infrared region and above is mainlySuperconductors have very special electromagnetic due to atomic and electronic polarizations.properties. For dc electric elds, their conductivityis virtually innite; but for high-frequency electro-magnetic elds, they have complex conductivities. 1.3.1.1 Electronic and atomic polarizationsElectronic polarization occurs in neutral atomsMagnetic materialswhen an electric eld displaces the nucleus withAll materials respond to external magnetic elds, respect to the surrounding electrons. Atomic polar-so in a broad sense, all materials are magnetic ization occurs when adjacent positive and negative 25. 12 Microwave Electronics: Measurement and Materials Characterization eDipolar and related relaxation phenomenaAtomicElectronice0 103 106 1091012 1015 Microwaves Millimeter Infrared Visible Ultraviolet wavesFrequency (Hz)Figure 1.9 Frequency dependence of permittivity for a hypothetical dielectric (Ramo et al. 1994). Source:Ramo, S. Whinnery, J. R and Van Duzer, T. (1994). Fields and Waves in Communication Electronics, 3rd edition,John Wiley & Sons, Inc., New York er Bare present, the materials are almost lossless at A+2a microwave frequencies. er2Be In the following discussion, we focus on elec-A+w B0 atronic polarization, and the conclusions for elec- A tronic polarization can be extended to atomic polarization. When an external electric eld is erapplied to neutral atoms, the electron cloud of the atoms will be distorted, resulting in the electronicABpolarization. In a classical model, it is similar to a2aw0 a w0 + a spring-mass resonant system. Owing to the small mass of the electron cloud, the resonant frequency 0 w0 w of electronic polarization is at the infrared regionFigure 1.10 The behavior of permittivity due to elec-or the visible light region. Usually, there are sev-tronic or atomic polarization. Reprinted with permis-eral different resonant frequencies correspondingsion from Industrial Microwave Sensors, by Nyfors, E.to different electron orbits and other quantum-and Vainikainen, P., Artech House Inc., Norwood, MA, mechanical effects. For a material with s differentUSA, www.artechhouse.com oscillators, its permittivity is given by (Nyfors and Vainikainen 1989)ions stretch under an applied electric eld. Actu-ally, electronic and atomic polarizations are of sim- (ns e2 )/(0 ms ) r = 1 + (1.34)ilar nature. Figure 1.10 shows the behavior of per- s s 2 + j2 s2mittivity in the vicinity of the resonant frequency0 . In the gure, A is the contribution of higher where ns is the number of electrons per volume withresonance to r at the present frequency range, andresonant frequency s , e is the charge of electron,2B/0 is the contribution of the present resonance ms is the mass of electron, is the operating angularto lower frequencies. For many dry solids, these frequency, and s is the damping factor.are the dominant polarization mechanisms deter- As microwave frequencies are far below themining the permittivity at microwave frequencies,lowest resonant frequency of electronic polariza-although the actual resonance occurs at a much tion, the permittivity due to electronic polariza-higher frequency. If only these two polarizationstion is almost independent of the frequency and 26. Electromagnetic Properties of Materials13temperature (Nyfors and Vainikainen 1989):tan d erNs e 2 r = 1 +2(1.35)tan ds 0 ms ser0Eq. (1.35) indicates that the permittivity r is a erreal number. However, in actual materials, smallerand constant losses are often associated with thistype of polarization in the microwave range. wmax wFigure 1.11 The frequency dependence of the com-1.3.1.2 Dipolar polarizationplex permittivity according to the Debye relation (Robert1988). Reprinted with permission from Electrical andIn spite of their different origins, various typesMagnetic Properties of Materials by Robert, P., Artechof polarizations at microwave and millimeter-wave House Inc., Norwood, MA, USA, www.artechhouse.comranges can be described in a similar qualitative way.In most cases, the Debye equations can be applied,is inversely proportional to temperature as all thealthough they were rstly derived for the special movements become faster at higher temperatures.case of dipolar relaxation. According to Debye From Eq. (1.36), we can get the real and imag-theory, the complex permittivity of a dielectric caninary parts of the permittivity and the dielectricbe expressed as (Robert 1988) loss tangent:r0 rr0 r r = r +(1.40) r = r + (1.36)1 + 2 1 + j r0 r r = (1.41)with 1 + 2r = lim r (1.37) r0 rtan e = (1.42) r0 = lim0 r(1.38) r0 + r 2r0 + 2 Figure 1.11 shows the variation of complex per-=(1.39)mittivity as a function of frequency. At the frequency r + 21 r0 r + 2where is the relaxation time and is the oper-max = , (1.43)ating angular frequency. Equation (1.36) indicates r r0 + 2that the dielectric permittivity due to Debye relax-the dielectric loss tangent reaches its maximumation is mainly determined by three parameters, r0 , value (Robert 1988)r , and . At sufciently high frequencies, as the 1 r0 rperiod of electric eld E is much smaller than thetan max = (1.44)2 r0 rrelaxation time of the permanent dipoles, the orien-tations of the dipoles are not inuenced by electric The permittivity as a function of frequency iseld E and remain random, so the permittivity atoften presented as a two-dimensional diagram,innite frequency r is a real number. As isColeCole diagram. We rewrite Eq. (1.36) asmainly due to electronic and atomic polarization, r0 rit is independent of the temperature. As at suf- r r jr = (1.45) 1 + jciently low frequencies there is no phase differenceAs the moduli of both sides of Eq. (1.45) shouldbetween the polarization P and electric eld E, r0be equal, we haveis a real number. But the static permittivity r0decreases with increasing temperature because of(r0 r )2(r r )2 + (r )2 = (1.46)the increasing disorder, and the relaxation time 1 + 2 27. 14Microwave Electronics: Measurement and Materials CharacterizationAfter eliminating the term 2 using Eq. (1.40),water, the material exhibits a distribution of relax-we get (Robert 1988)ation frequencies. Often an empirical constant, a,is introduced and Eq. (1.36) is modied into the (r r )2 + (r )2 = (r r )(r0 r )following form (Robert 1988): (1.47)Eq. (1.47) represents a circle with its center onr0 rr = r + (1.50)the r axis. Only the points at the top half 1 + (ja )1aof this circle have physical meaning as all thewhere a is related to the distribution of values,materials have nonnegative value of imaginary partand a denotes the most possible value. Theof permittivity. The top half of the circle is calledconstant a is in the range 0 a < 1. When a = 0,ColeCole diagram, as shown in Figure 1.12.Eq. (1.50) becomes Eq. (1.36), and in this case,The relaxation time can be determined fromthere is only single relaxation time. When the valuethe ColeCole diagram. According to Eqs. (1.40)of a increases, the relaxation time is distributedand (1.41), we can getover a broader range. r = (r r )(1.48) If we separate the real and imaginary parts ofEq. (1.50) and then eliminate a , we can nd that r = (1/)(r r0 ) (1.49)the r (r ) curve is also a circle passing throughthe points r0 and r , as shown in Figure 1.13. As shown in Figure 1.12, for a given operatingThe center of the circle is below the r axis with afrequency, the value can be obtained fromdistance d given bythe slope of a line pass through the pointcorresponding to the operating frequency and the r0 rpoint corresponding to r0 or r . After obtaining d= tan (1.51) 2the value, the relaxation time can be calculatedfrom according to Eq. (1.39). where is the angle between the r axis and the In some cases, the relaxation phenomenon line connecting the circle center and the point r :may be caused by different sources, and the dielectric material has a relaxation-time spectrum. =a(1.52)2For example, a moist material contains watermolecules bound with different strength. Depend-Similar to Figure 1.12, only the points above the ring on the moisture and the strength of binding axis have physical meaning. Equations (1.51) and(1.52) indicate that the empirical constant a can becalculated from the value of d or . erb=1 Slope 1/b er Slope bb= erer0b=0 q erd erer + er0 er0 er 2Figure 1.12 The ColeCole presentation for a single Figure 1.13 ColeCole diagram for a relaxation-timerelaxation time (Robert 1988). Reprinted with permis- spectrum. Reprinted with permission from Electricalsion from Electrical and Magnetic Properties of Mate- and Magnetic Properties of Materials by Robert, P.,rials by Robert, P., Artech House Inc., Norwood, MA,Artech House Inc., Norwood, MA, USA,USA, www.artechhouse.comwww.artechhouse.com 28. Electromagnetic Properties of Materials151.3.1.3 Ionic conductivity response of polarization versus electric eld is non- linear. As shown in Figure 1.14(b), ferroelectricUsually, ionic conductivity only introduces losses materials display a hysteresis effect of polarizationinto a material. As discussed earlier, the dielectricwith an applied eld. The hysteresis loop is causedloss of a material can be expressed as a functionby the existence of permanent electric dipoles inof both dielectric loss (rd ) and conductivity ( ):the material. When the external electric eld is ini- tially increased from the point 0, the polarization r = rd +(1.53)increases as more of the dipoles are lined up. When0 the eld is strong enough, all dipoles are lined upThe overall conductivity of a material may con-with the eld, so the material is in a saturationsist of many components due to different conduc- state. If the applied electric eld decreases from thetion mechanisms, and ionic conductivity is usually saturation point, the polarization also decreases.the most common one in moist materials. At low However, when the external electric eld reachesfrequencies, r is dominated by the inuence ofzero, the polarization does not reach zero. Theelectrolytic conduction caused by free ions in the polarization at zero eld is called the remanentpresence of a solvent, for example water. As indi- polarization. When the direction of the electriccated by Eq. (1.53), the effect of ionic conductivityeld is reversed, the polarization decreases. Whenis inversely proportional to operating frequency.the reversed eld reaches a certain value, called the coercive eld, the polarization becomes zero.1.3.1.4 Ferroelectricity By further increasing the eld in this reverse direc- tion, the reverse saturation can be reached. WhenMost of the dielectric materials are paraelectric. the eld is decreased from the saturation point, theAs shown in Figure 1.14(a), the polarization ofsequence just reverses itself.a paraelectric material is linear. Besides, the For a ferroelectric material, there exists a par-ions in paraelectric materials return to their ticular temperature called the Curie temperature.original positions once the external electric eld isFerroelectricity can be maintained only below theremoved; so the ionic displacements in paraelectricCurie temperature. When the temperature is highermaterials are reversible.than the Curie temperature, a ferroelectric material Ferroelectric materials are a subgroup of pyro- is in its paraelectric state.electric materials that are a subgroup of piezo-Ferroelectric materials are very interesting sci-electric materials. For ferroelectric materials, the entically. There are rich physics phenomena near[Image not available in this electronic edition.]Figure 1.14 Polarization of dielectric properties. (a) Polarization of linear dielectric and (b) typical hysteresisloop for ferroelectric materials. Modied from Bolton, W. (1992). Electrical and Magnetic Properties of Materials,Longman Scientic & Technical, Harlow 29. 16Microwave Electronics: Measurement and Materials CharacterizationSilicon and germanium are typical intrinsic semi- Permittivityconductors. An extrinsic semiconductor is obtained byadding a very small amount of impurities to FerroelectricParaelectrican intrinsic semiconductor, and this procedure is statestatecalled doping. If the impurities have a highernumber of valence electrons than that of thee host, the resulting extrinsic semiconductor is calledetype n, indicating that the majority of the mobilecharges are negative (electrons). Usually the host is Tp Temperaturesilicon or germanium with four valence electrons,and phosphorus, arsenic, and antimony with veFigure 1.15 Schematic view of the temperature de- valence electrons are often used as dopants inpendence of a ferroelectric material near its Curie type n semiconductors. Another type of extrinsictemperature semiconductor is obtained by doping an intrinsicsemiconductor using impurities with a number ofthe Curie temperature. As shown in Figure 1.15, valence electrons less than that of the host. Boron,the permittivity of a ferroelectric material changesaluminum, gallium, and indium with three valencegreatly with temperature near the Curie tempera-electrons are often used for this purpose. Theture. Dielectric constant increases sharply to a high resulted extrinsic semiconductor is called type p,value just below the Curie point and then steeply indicating that the majority of the charge carriersdrops just above the Curie point. For example, bar- are positive (holes).ium titanate has a relative permittivity on the orderBoth the free charge carriers and boundedof 2000 at about room temperature, with a sharp electrons in ions in the crystalline lattice haveincrease to about 7000 at the Curie temperature ofcontributions to the dielectric permittivity = 120 C. The dielectric loss decreases quickly when j (Ramo et al. 1994):the material changes from ferroelectric state tone e 2paraelectric state. Furthermore, for a ferroelectric = 1 (1.54) m(v 2 + 2 )material near its Curie temperature, its dielectricconstant is sensitive to the external electric eld.ne e 2 v = (1.55) Ferroelectric materials have application poten-m(v 2 + 2 )tials in various elds, including miniature capaci-where 1 is related to the effects of the boundtors, electrically tunable capacitors and electricallyelectrons to the positive background, ne is thetunable phase-shifters. Further discussions on fer-density of the charge carriers, v is the collisionroelectric materials can be found in Chapter 9.frequency, is the circular frequency, m is themass of the electron, and (ne e2 /mv) equals the1.3.2 Semiconductorslow frequency conductivity . At microwave frequency (2v 2 ), for semi-There are two general categories of semiconduc- conductors with low to moderate doping, whosetors: intrinsic and extrinsic semiconductors. Anconductivity is usually not higher than 1 S/m, theintrinsic semiconductor is also called a pure semi- second term of Eq. (1.54) is negligible. So the per-conductor or an undoped semiconductor. The band mittivity can be approximated asstructure shown in Figure 1.2(b) is that of an intrin-sic semiconductor. In an intrinsic semiconductor, = 1 j(1.56)there are the same numbers of electrons as holes. Intrinsic semiconductors usually have high resis-Besides the permittivity discussed above, thetivity, and they are often used as the starting electrical transport properties of semiconductors,materials for fabricating extrinsic semiconductors. including Hall mobility, carrier density, and con- 30. Electromagnetic Properties of Materials17ductivity are important parameters in the devel-eld; while for a perfect conductor, Eq. (1.57) onlyopment of electronic components. Discussions on applies for time-varying magnetic elds.electrical transport properties can be found inFor a superconductor, there exists a critical tem-Chapter 11. perature Tc . When the temperature is lower thanTc , the material is in superconducting state, and at1.3.3 ConductorsTc , the material undergoes a transition from nor-mal state into superconducting state. A materialConductors have high conductivity. If the con-with low Tc is called a low-temperature super-ductivity is not very high, the concept of per- conducting (LTS) material, while a material withmittivity is still applicable, and the value of per-high Tc is called a high-temperature superconduct-mittivity can be approximately calculated froming (HTS) material. LTS materials are metallicEqs. (1.54) and (1.55). For good conductors withelements, compounds, or alloys, and their criticalvery high conductivity, we usually use penetra- temperatures are usually below about 24 K. HTStion depth and surface resistance to describe the materials are complex oxides and their critical tem-properties of conductors. As the general properties perature may be higher than 100 K. HTS materialsof normal conductors have been discussed earlier, are of immediate interest for microwave applica-here we focus on two special types of conductors: tions because of their very low surface resistance atperfect conductors and superconductors. It should microwave frequency at temperatures that can bebe noted that perfect conductor is only a theoretical readily achieved by immersion in liquid nitrogenmodel, and no perfect conductor physically exists.or with cryocoolers. In contrast to metallic super- A perfect conductor refers to a material withinconductors, HTS materials are usually anisotropic,which there is no electric eld at any frequency. exhibiting strongest superconductive behavior inMaxwell equations ensure that there is also nopreferred planes. When these materials are usedtime-varying magnetic eld in a perfect conduc- in planar microwave structures, for example, thin-tor. However, a strictly static magnetic eld shouldlm transmission lines or resonators, these pre-be unaffected by the conductivity of any value,ferred planes are formed parallel to the surfaceincluding innite conductivity. Similar to an idealto facilitate current ow in the required direc-perfect conductor, a superconductor excludes time-tion (Lancaster 1997; Ramo et al. 1994).varying electromagnetic elds. Furthermore, the The generally accepted mechanism for super-Meissners effect shows that constant magneticconductivity of most LTS materials is phonon-elds, including strictly static magnetic elds, aremediated coupling of electrons with opposite spin.also excluded from the interior of a superconduc-The paired electrons, called Cooper pairs, traveltor. From the London theory and the Maxwellsthrough the superconductor without being scat-equations, we havetered. The BCS theory describes the electron pair- B = B0 ez/L(1.57)ing process, and it explains the general behav-ior of LTS materials very well. However, despitewith the London penetration depth given bythe enormous efforts so far, there is no theory 1that can explain all aspects of high-temperaturem2 L = (1.58)superconductivity. Fortunately, an understanding ne e2 of the microscopic theory of superconductivity inwhere B is the magnetic eld in the depth z, B0 HTS materials is not required for the design ofis the magnetic eld at the surface z = 0, m is the microwave devices (Lancaster 1997; Shen 1994).mass of an electron, is permeability, ne is the In the following, we discuss some phenomeno-density of the electron, and e is the electric charge logical theories based on the London equationsof an electron. So an important difference betweenand the two-uid model. We will introduce somea superconductor and a perfect conductor is that, commonly accepted theories for explaining thefor a superconductor, Eq. (1.57) applies for both responses of superconductors to electromagnetictime-varying magnetic eld and static magneticelds, and our discussion will be focused on the 31. 18 Microwave Electronics: Measurement and Materials Characterizationpenetration depth, surface impedance, and complexequal, and they are proportional to the square rootconductivity of superconductors. of the operating frequency 1/2 .If we want to calculate the impedance of a super-1.3.3.1 Penetration depthconductor using Eq. (1.32), the concept of complex conductivity should be introduced. According toThe two-uid model is often used in analyzingthe two-uid model, there are two types of cur-superconductors, and it is based on the assumption rents: a superconducting current with volume den-that there are two kinds of uids in a superconduc-sity Js and a normal current with volume densitytor: a superconductive current with a carrier den- Jn . Correspondingly, the conductivity also con-sity ns and a normal current with a carrier densitysists of two components: superconducting conduc-nn , yielding a total carrier density n = ns + nn . At tivity s and normal conductivity n , respectively.temperatures below the transition temperature Tc , The total conductivity of a superconductor is giventhe equilibrium fractions of the normal and theby = s + n .superconducting electrons vary with the absoluteThe superconducting conductivity s is purelytemperature T :imaginary and does not contribute to the loss:4nnT 1 = (1.59) s =(1.63)n Tcj2L 4nsTWhile the normal conductivity n contains both =1 (1.60)real and imaginary components and the real partn Tc contributes to the loss: From Eqs. (1.59) and (1.60), we can get the2relationship between the penetration depth L and nn qn n = n1 jn2 =temperature T :mn 1 + j1 4 2 2 nn qn 1 jT=(1.64) L (T ) = L (0) 1 (1.61) mn1 + ( )2Tc where qn is the electrical charge for the normalwith carriers, is the relaxation time for electronms scattering, and mn is the effective mass of the L (0) =2 . (1.62) nqsnormal carriers. Therefore, the total conductivitywhere ms and qs are the effective mass and of a superconductor is then obtained:electrical charge of the superconductive carriers. 2 nn qn 1Eq. (1.62) indicates that the penetration depth has = n + s =mn 1 + ( )2a minimum value of penetration depth L (0) at 2T = 0 K. nn qn 1j j(1.65)mn1 + ( )22 L1.3.3.2 Surface impedance and complex At microwave frequencies (1), Eq. (1.65)conductivity can be simplied asThe surface impedance is dened as the charac- 2 nn qn 1teristic impedance seen by a plane wave incident = 1 j2 = j (1.66)perpendicularly upon a at surface of a conductor.mn2 LAccording to Eqs. (1.32) and (1.33), the surface where 1 and 2 are the real and imaginaryimpedance of normal conductors, such as silver,components of the complex conductivity. Thecopper, or gold, can be calculated from their con- real part of complex conductivity represents theductivity . For a normal conductor, the value of loss due to the normal carriers, whereas itsits conductivity is a real number, and the sur-imaginary part represents the kinetic energy of theface resistance Rs and the surface reactance Xs aresuperconductive carriers. 32. Electromagnetic Properties of Materials 19From Eqs. (1.32), (1.33) and Eq. (1.66), we canm, mmcalculate the surface impedance of a superconductor:Zs = Rs + jXsm (d)(e) 1 (a) (b) j 12==j1+j(1.67) (c)1 j22 2 104 106 108 1010 f (Hz)As usually 1 2 , Eq. (1.67) can be simpli-ed as Figure 1.16 Frequency dependence of permeability Zs = Rs + jXs for a hypothetical ferromagnetic material 1 =+j At different frequency ranges, different physics222phenomena dominate. In the low frequency range 2 2 3 nn qn 2(f < 104 Hz), and almost do not change =L + jL(1.68) with frequency. In the intermediate frequency2mn range (104 < f < 106 Hz), and change 1nn a little, and for some materials, may haveRs = 2 2 3 NL (1.69) 2na maximum value. In the high-frequency rangeXs = L (1.70) (106 < f < 108 Hz), decreases greatly, while increase quickly. In the ultrahigh frequencywhere N is the conductivity of the superconductor range (108 < f < 1010 Hz), ferromagnetic reso-in its normal state: nance usually occurs. In the extremely high fre- 2 nqn quency range (f > 1010 Hz), the magnetic proper- N = (1.71) ties have not been fully investigated yet.mn 2 4 nn qn nn Tn = = N= N (1.72) 1.3.4.1 Magnetization and hysteresis loopmnnTcAccording to Eqs. (1.67) to (1.72), the two- Figure 1.17 shows the typical relationship betweenuid model leads to the prediction that thethe magnetic ux density B in a magnetic materialsurface resistance Rs is proportional to 2 forand the magnetic eld strength H . As discussed insuperconductors, which is quite different from the1/2 frequency dependence for normal conductors.B Saturation1.3.4 Magnetic materials BmAs the penetration depth of metals at microwaveBrfrequencies is on the order of a few microns,the interior of a metallic magnetic material doesnot respond to a microwave magnetic eld.So, metallic magnetic materials are seldom used H Hc0 Hcas magnetic materials at microwave frequencies.Here, we concentrate on magnetic materials withlow conductivity. Br The frequency dependence of magnetic materi-als is quite complicated (Smit 1971; Fuller 1987),Saturationand some of the underlying mechanisms have notbeen fully understood. Figure 1.16 shows the typ-Figure 1.17 The hysteresis loop for a magneticical magnetic spectrum of a magnetic material. material 33. 20 Microwave Electronics: Measurement and Materials CharacterizationSection 1.2.1.2, at the starting point 0, the domainsc-axisare randomly orientated, so the net magnetic uxdensity is zero. The magnetic ux density Bincreases with the increase of the magnetic eldstrength H , as the domains close to the directionof the magnetic eld grow. This continues untilall the domains are in the same direction with themagnetic eld H and the material is thus saturated. q 60 jAt the saturation state, the ux density reachesits maximum value Bm . When the magnetic eldstrength is reduced to zero, the domains in theFigure 1.19 Preferential directions for a ferroxplanamaterial turn to their easy-magnetization directions material (Smit 1971). Source: Smit, J. (editor), (1971),close to the direction of the magnetic eld H, and Magnetic Properties of Materials, McGraw-Hill,the material retains a remanence ux density Br .New YorkIf we reverse the direction of the magnetic eld, be easily achieved, while in the hard-magnetizationthe domains grow in the reverse direction. When direction, high magnetic eld is required forthe numbers of the domains in the H direction saturation. The magnetic eld Ha correspondingand opposite the H direction are equal, that is, the to the cross point of the two magnetization curvesux density becomes zero, the value of the applied is called anisotropic eld.magnetic eld is called coercive eld Hc . FurtherThere are two typical types of anisotropiesincrease in the strength of the magnetic eld in the of magnetic materials: axis anisotropy and planereverse direction results in further growth of the anisotropy for a hexagonal structure. Figure 1.19domains in the reverse direction until saturation in shows the potential directions for a ferrox-the reverse direction is achieved. When this eldplana material. If the easy-magnetization direc-is reduced to zero, and then reversed back to thetion is along the c-axis, the material has uniaxialinitial direction, we can get a closed hysteresis loop anisotropy, usually described by the anisotropicof the magnetic material.eld Ha . If the easy-magnetization direction is in In most cases, magnetic materials are anisotropic the c-plane, the material has planar anisotropy.for magnetization. For a hexagonal ferrite, therePlanar anisotropy is usually described by theexists an easy-magnetization direction and a hard- anisotropic elds H and H , where H is themagnetization direction. As shown in Figure 1.18,magnetic eld required for turning a domain inin the easy-magnetization direction, saturation canone preferential magnetization direction in the c- plane to another preferential magnetization direc- M tion in the c-plane through the hard-magnetization c-axis, and H is the magnetic eld required for Msturning a domain in one preferential magnetiza-1 tion direction in the c-plane to another preferential magnetization direction in the c-plane within the2easy-magnetization plane.The coercive eld Hc is an important param- eter in describing the properties of a magnetic material. The value of coercive eld Hc is mainly governed by two magnetization phenomena: rota-0HaH tion of domain and movement of domain wall. ItFigure 1.18 Magnetization curves for an anisotropicis related to intrinsic magnetic properties, such asmagnetic material. Curve 1 is the magnetization in anisotropic eld and domain-wall energy, and it isthe easy-magnetization direction and Curve 2 is thealso related to the microstructures of the material,magnetization in the hard-magnetization directionsuch as grain size and domain-wall thickness. 34. Electromagnetic Properties of Materials21Besides, the amount and distribution of impuritiesdBdHin the material also affects the value of the coerciveeld Hc .m0 mrm1.3.4.2 Denitions of scalar permeabilitym0 mriAs the relationship between the magnetic uxdensity B and the magnetic eld strength H isnonlinear, the permeability is not a constant butvaries with the magnetic eld strength. Usually, it0 Hm His not necessary to have a complete knowledge ofthe magnetic eld dependence of permeability. InFigure 1.21 The dependence of permeability on mag-the mathematical treatment of general applications, netic eldthe relative permeability is simply a numberdenoted by the symbol r , but for differentcases, permeability has different physical meaning. theoretical value corresponding to a zero eld,On the basis of the hysteresis loop shown inand in a strict meaning, it cannot be directlyFigure 1.20, we can distinguish four denitions measured. Usually, the initial relative permeabilityof scalar permeability often used in materialsis determined by extrapolation. In practice, ri isresearch (Robert 1988). often given as the relative permeability measured The initial relative permeability is dened as in a weak eld lying between 100 and 200 A/m. Figure 1.21 shows the relationship between 1B (dB/dH ) and H corresponding to the dashed lineri = limH 0(1.73) 0 H in Figure 1.20. The (dB/dH ) value point at H = 0It is applicable to a specimen that has never equals the initial permeability discussed above. Atbeen subject to irreversible polarization. It is athe point Hm , which satisesd2 B B = 0,(1.74)dH 2Slope: m0 mrmthe value of (dB/dH ) reaches its maximum value,which is dened as maximum permeability (0 rm ),as shown in Figures 1.20 and 1.21. The value of rm (Hm, Bm) can be taken as a good approximation