Advances in Surface Acoustic Wave Technology, Systems and Applications Volume 2

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ADVANCES IN SURFACE ACOUSTIC WAVE TECHNOLOGY, SYSTEMS AND APPLICATIONS (Voi.2)

SELECTED TOPICS IN ELECTRONICS AND SYSTEMS Editor-in-Chief: M. S. Shur

Published Vol. 1:

Current Trends in Integrated Optoelectronics ed. T. P. Lee

Vol. 2:

Current Trends in Heterojunction Bipolar Transistors ed. M. F. Chang

Vol. 3:

Current Trends in Vertical Cavity Surface Emitting Lasers ed. T. P. Lee

Vol. 4:

Compound Semiconductor Electronics: The Age of Maturity ed. M. Shur

Vol. 5:

High Performance Design Automation for Multichip Modules and Packages ed. J. Cho and co-ed. P. D. Franzon

Vol. 6:

Low Power VLSI Design and Technology eds. G. Yeap and F. Najm

Vol. 7:

Current Trends in Optical Amplifiers and Their Applications ed. T. P. Lee

Vol. 8:

Current Research and Developments in Optical Fiber Communications in China eds. Q.-M. Wang and T. P. Lee

Vol. 9:

Signal Compression: Coding of Speech, Audio, Text, Image and Video ed. N. Jayant

Vol. 10: Emerging Optoelectronic Technologies and Applications ed. Y.-H. Lo Vol. 11: High Speed Semiconductor Lasers ed. S. A. Gurevich Vol. 12: Current Research on Optical Materials, Devices and Systems in Taiwan eds. S. Chi and T. P. Lee Vol. 13: High Speed Circuits for Lightwave Communications ed. K.-C. Wang Vol. 14: Quantum-Based Electronics and Devices eds. M. Dutta and M. A. Stroscio Vol. 15: Silicon and Beyond eds. M. S. Shur and T. A. Fjeldly Vol. 16: Advances in Semiconductor Lasers and Applications to Optoelectronics eds. M. Dutta and M. A. Stroscio Vol. 17: Frontiers in Electronics: From Materials to Systems eds. Y. S. Park, S. Luryi, M. S. Shur, J. M. Xu and A. Zaslavsky Vol. 18: Sensitive Skin eds. V. Lumelsky, Michael S. Shur and S. Wagner Vol. 19: Advances in Surface Acoustic Wave Technology, Systems and Applications (Two volumes), volume 1 eds. C. C. W. Ruppeland T. A. Fjeldly

Selected Topics in Electronics and Systems - Vol. 20

ADVANCES IN SURFACE ACOUSTIC WAVE TECHNOLOGY, SYSTEMS AND APPLICATIONS (Voi.2)

Editors

Clemens C. W. Ruppel Siemens AG, Germany

Tor A. Fjeldly Norwegian University of Science and Technology, Norway

fe World Scientific ll

Singapore • New Jersey London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

ADVANCES IN SURFACE ACOUSTIC WAVE TECHNOLOGY, SYSTEMS AND APPLICATIONS — VOL. 2 Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4538-6

Printed in Singapore.

PREFACE ADVANCES IN SURFACE ACOUSTIC WAVE TECHNOLOGY, SYSTEMS AND APPLICATIONS - 2

Surface acoustic wave (SAW) devices are recognized for their versatility and efficiency in controlling and processing electrical signals. Basically, we may think of a SAW device as consisting of a solid substrate with an input and an output transducer. The input transducer converts the incoming signal by the inverse piezoelectric effect into acoustic waves, which propagate along the planar surface of the solid. At the output transducer, the surface acoustic waves are reconverted to an electrical signal. Hence, a fundamental property of the SAW device is to act as a signal delay line. The relatively slow propagation velocity of the surface acoustic waves of typically 3500 m/s allows delays of several microseconds on a small chip. However, the versatility of the SAW technology lies in the great flexibility in configuring the transducers, the substrate, and the path of the propagating surface acoustic wave. Since the introduction of the first SAW devices in the mid-1960's, this flexibility has given room for a great deal of ingenuity in the design of different types of devices. This has resulted in a multitude of device concepts for a wide range of signal processing functions, such as delay lines, filters, resonators, pulse compressors, convolvers, and many more. As a consequence, the production volume has risen to millions of devices produced every day, as the SAW technology has found its way into mass markets such as TV receivers, pagers, keyless entry systems, and cellular phones. For such high-volume applications, the unit price of packaged SAW band-pass filters is in the range of up to a few US dollars. At the other end of the scale, we find specialized high performance signal processing SAW devices for satellite communication and military applications, such as radar and electronic warfare, that may run into thousands of dollars per unit. In two issues of IJHSES, we present an overview of recent advances in SAW technology, systems and applications by some of the foremost researchers and engineers contributing to this exciting field today. The first issue on Advances in Surface Acoustic Wave Technology, Systems and Applications included the following contributions: "A History of Surface Acoustic Wave Devices" by David P. Morgan, "Thin-Films for SAW Devices" by Fred S. Hickernell, "Bulk and Surface Acoustic Waves in Anisotropic Solids" by Eric L. Adler, "Analysis of SAW Excitation and Propagation under Periodic Metallic Grating Structures" by Ken-Ya Hashimoto, Tatsuya Omori, and Masatsune Yamaguchi, "High-Performance Surface Transverse Wave Resonators in the Lower GHz Frequency Range" by Ivan D. Avramov, "SAW Antenna Duplexers for Mobile Communication" by Mitsutaka Hikita, and "Ladder Type SAW Filter and its Application to High Power SAW Devices" by Yoshio Satoh and Osamu Ikata. Here follows a survey of the seven contributions included in the second issue. The design of modern high-performance SAW devices requires precise and efficient simulation tools. Among the several phenomenological methods proposed for the modeling of SAW structures, the coupling-of-modes (COM) model has been a favorite in practical design work. In Chapter 1, Victor Plessky and Julius Koskela review the COM approach, discuss issues related to parameter extraction, practical design, and unresolved modeling problems. This chapter contains sufficient detail to make it a valuable

vi

Preface

introduction to the COM approach for analyzing SAW devices. Potentially more accurate, although less efficient, are the numerical simulations based on finite element (FEM) and Green's functions techniques. Hashimoto et al. discussed the FEM technique in the first issue, while Ali R. Baghai-Wadji considers the powerful Green's functions approach in Chapter 2 of this issue. In the latter, the author discusses the diagonalization of the three-dimensional governing and constitutive equations in transversally inhomogeneous piezoelectric media, the Green's function theory, and the calculation of self-actions in the boundary elements. The strive for better performance in SAW devices also involves a search for more suitable materials, better technology, and the application of alternative modes of acoustic wave propagation. In Chapter 3, John A. Kosinski reviews the progress in new piezoelectric substrates for SAW devices, where the focus has been on finding "ideal" materials featuring simultaneously properties such as high piezoelectric coupling, zero temperature coefficient of frequency orientations, high intrinsic Q, zero power angle, and minimized diffraction effects, etc. The candidate materials considered in detail are gallium orthophosphate (a quartz homotype), calcium gallo-germanites (quartz isotypes), and diomignate (trigonal symmetry class 4mm). The demand for higher operating frequencies in modern communication systems and other applications has created a renewed interest in so-called pseudo surface acoustic waves (PSAWs), also known as "leaky SAWs". Of particular interest are the high-velocity pseudo surface acoustic waves (HVPSAWs). As explained by Mauricio Pereira da Cunha in Chapter 4, the superior phase velocity of these waves has made it possible to further extend the operating frequency of SAW devices. One of the fastest growing areas for SAW devices today is in mobile communication systems, where SAW filters are used in large quantities. At present, such systems operate into the 2 GHz range (see the contributions by M. Hikata and by Y. Satoh and O. Ikata in the first issue). However, owing to the rapid growth of mobile communication systems, their operating frequencies are expected to expand to the 5 to 10 GHz range in the near future, which poses a great challenge for the SAW technology. In Chapter 5, Hiroyuki Odagawa and Kazuhiko Yamanouchi describe a SAW technology based on ultra-fine fabrication techniques for low-loss filters beyond 5 GHz. Passive SAW sensors can communicate by radio link to a tranceiver unit over distances of 20 meters or more, without the need for wire connection and a battery. Hence, such devices are well suited for use in a wide range of sensor and identification systems. In Chapter 6, Frank Schmidt and Gerd Scholl discuss this interesting and rapidly growing application of SAW technology. A number of passive wireless SAW sensor and identification systems and their applications are presented. The interaction of surface acoustic waves, electrons and light may give rise to a host of new phenomena of interest both to fundamental science and to applications. One example is the use of SAWs to study the dynamic conductivity of quantized, lowdimensional electron systems in semiconductor layered systems. Another example involves exotic acousto-optic effects where photoelectrically generated electron-hole pairs are separated in the strong piezoelectric SAW field and later quenched to obtain a de facto light storage and delay device. These and other related phenomena and devices are discussed by Achim Wixforth in Chapter 7. With the two special issues on Advances in Surface Acoustic Wave Technology, Systems and Application, we have altogether presented 14 reviews on a wide range of topics related to contemporary surface acoustic wave technology. The topics range from

Preface

vii

basic theory, materials and phenomena to very advanced applications in communication systems and several other areas. The editors have been very fortunate to be able to solicit contributions from some of the foremost scientists and engineers in the field, and use the opportunity to thank each one of the contributors. We also would like to thank W. Ruile and U. Rosier for their support in the selection of the different topics and authors, and for reviewing several contributions. We are certain that this collection of up-to-date information on SAW technology will be of great interest, not only to all those working with SAW related problems, but also to many more who stand to benefit from an insight into the rich opportunities that this technology has to offer.

CLEMENS C. W. RUPPEL SIEMENS AG, ZTMS 1, Otto-Hahn-Ring 6 81730 Munich/Germany TOR A. FJELDLY UniK - Center for Technology at Kjeller Norwegian University of Science and Technology N-2027 Kjeller, Norway

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Special Issue EDITORS

t

Clemens C.W. Ruppel was born in Munich, Germany, in 1952. In 1978 he received the Diploma in mathematics from the Ludwig-Maximilians University of Munich, Germany. Afterwards he has participated in research projects, solving mathematical problems related to bio chemistry and power plant safety, at the university. In 1981 he joined the microacoustics research group at Siemens AG as a doctorate student. In 1986 he received his Ph.D. degree for works on the design of surface acoustic b waves (SAW) filters from the Technical University of Vienna, Austria. In ,-..! 1984, he became member of the micro-acoustics group at the Corporate Research and Development of Siemens AG in Munich. In 1990, he became Group Manager. He was responsible for the development of software for the simulation and synthesis of SAW filters. Since 1991, he has been a member of the Technical Program Committee of the IEEE Ultrasonics Symposium, and since 1997 of the IEEE Frequency Control Symposium. In 2000 he has become an elected committee member of the IEEE UFFC AdCom. He has been a voting member of IEEE 802.11. His research interests include all SAW related subjects, especially the design of bandpass filters, dispersive transducers, low-loss filters, and mathematical procedures and algorithms needed for the design and simulation of SAW devices. He is author/co-author of approximately 50 papers (including 7 invited papers) on the design and simulation of SAW filters, and sensors based on SAW devices. In his leisure time he likes to play guitar in a rock band, and enjoys cooking and dining. s. jr

.,

' Tor A. Fjeldly received the M. Sc. degree in physics from the Norwegian Institute of Technology, 1967, and the Ph.D. degree from Brown ' | Jfc University, Providence, RI, in 1972. From 1972 to 1994, he was with Max. W$ Planck-Institute for Solid State Physics in Stuttgart, Germany. From 1974 ** «£ h • t 0 1983, he worked as a Senior Scientist at the SINTEF research organization in Norway. Since 1983, he has been on the faculty of the Norwegian University of Science and Technology (NTNU), where he is a Professor of Electrical Engineering. He is presently with NTNU's Center for Technology at Kjeller, Norway. He was Head of the Department of Physical Electronics at NTNU, and he also served as an Associate Dean of the Faculty of Electrical Engineering and Telecommunication. From 1990 to 1997, he held the position of Visiting Professor at the Department of Electrical Engineering, University of Virginia, Charlottesville, VA, and from 1997 he has been Visiting Professor at the Electrical, Computer and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, NY. His research interests have included fundamental studies of semiconductors and other solids, development of solid-state chemical sensors, electron transport in semiconductors, modeling and simulation of semiconductor devices, and circuit simulation. He has written about 150 scientific papers, several book chapters, and is a co-author of the books Semiconductor Device Modeling for VLSI (Englewood Cliffs, NJ: Prentice Hall, 1993) and Introduction to Device Modeling and Circuit Simulation (New York, NY: Wiley & Sons, 1998). He is also a co-developer of the circuit simulator AIM-Spice. Since 1998, he has been a Co-Editor-in-Chief of the International Journal of High Speed Electronics and Systems, Singapore. Dr. Fjeldly is a Fellow of IEEE and a member of the Norwegian Academy of Technical Sciences, the American Physical Society, the European Physical Society and the Norwegian Society of Chartered Engineers.

IX

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CONTENTS

Preface

v

Coupling-of-modes Analysis of SAW Devices V. Plessky and J. Koskela

1

Theory and Applications of Green's Functions A. R. Baghai-Wadji

83

New Piezoelectric Substrates for SAW Devices J. A. Kosinski

151

Pseudo and High Velocity Pseudo SAWs M. P. da Cunha

203

SAW Devices Beyond 5 GHz H. Odagawa and K. Yamanouchi

245

Wireless SAW Identification and Sensor Systems F. Schmidt and G. Scholl

277

Interaction of Surface Acoustic Waves, Electrons, and Light A. Wixforth

327

International Journal of High Speed Electronics and Systems, Vol. 10, No. 4 (2000) 867-947 © World Scientific Publishing Company

COUPLING-OF-MODES ANALYSIS OF SAW DEVICES

VICTOR PLESSKY Thomson Microsomes, SAW Design Bureau, Fahys 9 2000 Neuchatel, Switzerland and JULIUS KOSKELA Materials Physics Laboratory, Helsinki University of Technology P. O. Box 2200 (Technical Physics), FIN-02015 HUT, Finland

The coupling-of-modes approach for modeling and analyzing surface-acoustic wave devices is reviewed. We discuss the established formalism and survey the modifications introduced to account for phenomena such as resistivity, dispersion and in particular, the effects related to surface transverse wave and leaky surface-acoustic wave devices. The extraction of the COM parameters from experiments and theoretical simulations are considered. The design of various SAW devices such as resonators and resonator filters as well as practical aspects are discussed. Finally, the unresolved modeling problems are addressed.

1

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V. Plessky & J. Koskela

1. I n t r o d u c t i o n During the last two decades, demands set by the expansion of the mobile telecommunication industry have resulted in the introduction of a new generation of surfaceacoustic wave (SAW) filters. These typically feature small losses, very small size, and operating frequencies up to 2 GHz and above. The working principles of modern SAW devices differ essentially from those of the conventional transversal filters. Due to the high operating frequencies, the electrode thickness is no longer negligible in comparison to the acoustic wavelength. Consequently, the mass loading by the electrodes is significant, resulting in a slowing of the wave and, most importantly, in strong reflections. The reflections serve t o t r a p the acoustic energy inside the device. In this sense-in contrast to the transversal filters-the finite reflectivity of the electrodes is fundamental to the device operation. The design of high-performance SAW devices requires precise and efficient simulation tools. Several phenomenological methods have been proposed for modeling and analysis of low-loss structures. These include the coupling-of-modes (COM) model, P-matrix model, equivalent circuit models, and angular spectrum of waves model, see a recent review. l Purely numerical simulators are also being developed by many a u t h o r s . 2 - 1 4 These typically describe the wave motion in the electrodes and the piezoelectric substrate with the finite element method (FEM) and/or Green's function techniques, and they derive device properties directly from the material's constants and the device geometry. They are potentially more accurate than phenomenological models but require intensive computation. Consequently, they are very slow a t the moment and, unfortunately cannot be directly used to optimize the device performance. The comparison of the phenomenological models 1 shows that, if used correctly and with accurate parameters, almost identical results are obtained with all of them. For the case of a classic Rayleigh-type SAW on quartz substrate, with weak interactions and narrow frequency band, all models give excellent results. On the other hand, all models provide-in the best case-only satisfactory descriptions of devices employing surface transverse waves (STWs) and leaky surface-acoustic waves (LSAW) in LiTa03 substrate. The common drawback of the phenomenological models is the underlying heuristic approximations (such as ignoring higher spatial Floquet harmonics, bulk-wave scattering at the ends of an interdigital transducer (IDT), transversal acoustic radiation, etc.). Often these simplifications are reasonable, but in some cases the ignored effects result in deteriorated or even unexpected device behaviour. The choice of the model for design and analysis of SAW devices remains, to some extent, a matter of taste. In our view, the coupling-of-modes model is physically the most adequate for the phenomena being considered: excitation, propagation and scattering of surface-acoustic waves. Really, it deals with wave quantities such as amplitudes and phases in electrode structures and it makes the phenomena easy to interpret—contrary to equivalent circuits, where the waves are represented by currents in artificial networks. Yet, the COM model avoids making too strict hy-

2

Coupling-of-Modes

Analysis

of SAW Devices

869

potheses on the wave behavior on each electrode. The accuracy of the results basically depends on the precision of the parameters used in the modeling. Both experimental data and numerical calculations are widely used to determine the COM parameters. In this review paper we concentrate on the COM formalism and its applications to SAW device design and analysis. The advantages of the COM approach can be summarized as follows: • For industry, the COM model provides an efficient and a highly flexible approach for modeling various kinds of SAW devices. In research, it serves as a conceptual background for the interpretation of the results of more advanced models. • The COM model is physically transparent: initially ignored parasitic and secondary effects are easily incorporated in the theory. • Accurate results are obtained—good agreement is guaranteed for Rayleigh waves in long periodic structures within narrow frequency band and rather weak interaction. However, good results are often obtained even when the model is not supposed to work. • In many important cases, algebraic formulas may be obtained for the quantities of interest. Thus, COM enables extremely fast computer simulations and it may directly be used in optimization algorithms, even if thousands of frequency points are needed in the filter design. For these reasons, the COM model is widely utilized in practical design work in many leading SAW companies. The structure of this paper is as follows. The rest of this Section concentrates on the background of the coupling-of-modes approach: the history of the theory and the phenomenology of waves propagating in a periodically perturbed medium. The second Section reviews the established coupling-of-modes formalism for surfaceacoustic wave devices, whereas Section three discusses the modifications suggested to the theory in order to include various parasitic and secondary effects in the framework. The practically important issue of parameter extraction is considered in Section four, while the actual design and analysis of SAW devices are the topics of Section five. Sections six concludes the paper.

1.1.

History

The coupling-of-modes formalism is a particular branch of the highly developed theory of wave propagation in periodic media, which has an exciting history of more t h a n 100 years. We will not review here this theory in general. An excellent review of many theoretical aspects of the wave propagation in periodic media and applications was written by C. Elachi. 15 The theory covers an amazing variety of wave phenomena, including the diffraction of EM waves on periodic gratings,

3

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V. Plessky & J. Koskela

their propagation in periodic waveguides and antennas, optical and ultrasonic waves in multilayered structures, phonon propagation and X-ray scattering in crystals, quantum theory of electron states in metals, semiconductors and dielectrics. Later applications include distributed Bragg reflectors for lasers, SAW propagation in periodic electrode structures and grooves, magnetostatic waves, arrays of domains, acoustooptics, fiber optics, liquid crystals, etc. For a wave incident on a periodically perturbed region, a particularly strong reflection occurs if the period of the grating, p, is equal or close to half of the wavelength, A: A = 2p. This is the Bragg condition, fundamental for all types of waves. Under this condition, the reflected waves are in phase and interfere constructively. Therefore, even for weak reflectors with a small reflection coefficient r < l , the total reflectivity may be close to unity if the number of reflectors N is sufficiently large, Nr ~ 1. Propagation of waves into the media is prohibited; this is so-called stopband phenomenon. Very early it was recognized that many problems concerning wave propagation in an infinite periodic medium may be formulated mathematically as a Mathieu equation, and it was shown that the solutions may generally be treated in the form of Bloch waves. With the Bloch waves decomposed as a discrete sum of space harmonics, the governing equation can be represented as an infinite set of algebraic equations, coupling the harmonics to each other. The matrix truncation approach may be followed to find a solution: supposing that the high-order harmonics exceeding some threshold are negligible, they may be ignored and the resulting finite system of linear equations may be solved numerically. Moreover, in most cases of interest only the incident wave and one reflected wave have significant amplitudes, while all other harmonics remain small. Then, only the two strongly interacting harmonics may be considered: this is known as the coupling-of-modes approximation. The two equations obtained may be presented algebraically or, if the changes in the amplitudes on one period are small, the equations may be written in the form of differential equations. The coupling-of-modes approach has been extensively used since the 1950's in various problems related to optics and electromagnetism, 1 5 , 1 6 mainly for the description of the wave propagation in periodically perturbed media, or structures with periodic geometry. The case of SAW devices resembles lasers in that the excitation and reflection of waves are distributed and occur in one and the same basic element, which for SAWs is the interdigital transducer 1 7 (IDT). As a rough attempt to sketch the history of COM for SAW we cite a few basic contributions, without pretending to give all important references or establishing exact priorities. The model was introduced to the SAW field by Suzuki 18 and Haus. 1 9 ' 2 0 ' 2 1 Subsequently, transduction and current generation were included, and an analytic solution for uniform structures was found by Hartmann 2 2 ' 2 3 and others. 2 4 Chen and Haus showed that all parameters are determined by the open- and closed-circuit conditions. 25 Wright formulated the theory for spatially varying parameters and included the propagation loss and finite electrode resistivity into the formalism. 26 Abbott estimated

4

Coupling-of-Modes Analysis of SAW Devices 871 the COM parameters based on theoretical grounds. 2 7 Biryukov et al. derived the COM equations in the frame of the perturbation theory using the surface impedance formalism. 28 The progress made in 90s mainly concerns the extraction of the COM parameters 2 9 ' 3 0 ' 3 1 and the extension of the model to cover effects related to shear horizontal surface-acoustic waves, such as surface transverse waves ' ' (STWs) and the commercially extremely important leaky surface-acoustic waves 35 (LSAWs) on rotated Y-cut lithium tantalate (LiTaOa) and lithium niobate (LiNbOa) substrates. 1.2.

Wave propagation

in periodic

structures

1.2.1. Loaded wave equation Although in this paper we concentrate on the coupling-of-modes approximation, we start with the more accurate and general Floquet approach. 1 5 Consider the loaded wave equation (^+kij^) = ~ax)k20ip(x), (i) where ko = OJ/VO, U> = 2irf is the angular frequency, and C(x) is the p-periodic load density along the coordinate x: o denotes the SAW wave velocity in an unloaded substrate and the load density is interpreted to describe the electric and/or mechanical loading due to the presence of the metal electrodes or grooves on the surface. The field (x).

(5)

The generalization of the theorem for vector arguments is the famous Bloch's theorem in solid-state physics. Owing to the p-periodicity of the function (x), it has the Fourier series representation +00

*(*) = J2 3 may be drastically reduced by an optimal selection of the cut angle and the thickness of the aluminum electrodes. For the thickness h/\o « 8%, minimal losses are attained for cut angle of about 42°. However, a problem remains, although the attenuation for LSAW should be diminished (in theory, to less than 2 ' 1 0 - 4 dB/wavelength, or 2.3-10 - 5 Neper/wavelength), electrical measurements of test devices in the 1 GHz frequency range yield attenuation parameters that are almost 2 orders higher (about 10~ 3 Neper/wavelength). The reasons for such high attenuation are not clear for the moment. Besides the other mentioned mechanisms, also waveguide losses are expected to contribute: although the attenuation is minimized for X-propagation, the losses increase radically for propagation directions offset from the crystal X-axis. 2.3.3.

Reflectivity

The magnitude of the normalized reflectivity, KP, is equal to the absolute value of the reflection coefficient per period, and it determines the relative width of the stopband, A / / / O = | K P | / 7 T . For many important substrates, including ST-cut quartz, 36°-42°YX-cut L i T a 0 3 and 64°-YX cut L i N b 0 3 , the sign of KP is negative, but it may be also positive, e.g. for thick aluminium electrodes on 128°YX-cut LiNb03. The phase of K yields the reflectivity center of the period, see below. The reflection coefficient for a thin electrode of width a in a periodic array has been studied by many authors. 2 5 ' 4 2 - 4 3 ' 4 4 - 4 5 ' 5 5 ' 5 6 ' 4 8 ' 5 7 ' 5 8 It can also be represented as a series expansion on the relative electrode height h/\o: r = inp = iKp/2 « iRe + ii? m sin(7ra/p) — . Ao

(77)

The first term describes the reflectivity due to the piezoelectric loading by an ideally reflecting finger. The term depends on the width of the electrode and it is proportional to the piezoelectric coupling coefficient, see the comprehensive discussion in Ref. 46. The second term is due to the mechanical loading and it includes mass

25

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V. Plessky & J. Koskela

loading and stress generation due to film deformation. Even film of the same material as the substrate (that is: an array of grooves) creates additional stresses, and the reflection coefficient is finite in the first order on H/XQ. Example values for Re and Rm are shown in Table 2. However, the formulas should be used with caution, keeping in mind that perturbation theory applies only for small perturbations and weak piezoelectric coupling. Numerous other configurations have been studied with perturbation theory. We will mention one interesting result: for oblique incidence of Rayleigh SAW from a reflectangular strip on an isotropic medium, the reflection coefficient is of the form 59 r = i/2 [ 1 - 4 ( — ) sin 2 6» j s i n / V - c o s t f j ^ - .

(78)

Here, 6 is the angle of incidence, and up. and Vt denote the velocities of Rayleigh wave and transverse bulk-acoustic wave, respectively. One can see that there is a Brewster angle, 6* = arcsin(v R /2w t ), between 25.9°-28.5°, for which the first order reflection coefficient vanishes. For the case of normal incidence (8 = 0), the result reduces to that in Eq. (77). 2.3.4. Transduction

coefficient

The normalized transduction coefficient, ap, measures the excitation of waves due to piezoelectric coupling in a unit cell of length Ao, which in the simplest case is formed by a pair of electrodes. It has the dimension of f 2 - 1 / 2 . For a very short transducer consisting only of one period, from Eq. (66) 13

+ia*,

P 2 3 « -iap.

(79)

Thus, the absolute value \ap\ is equal to the magnitude of waves generated by the period under a drive voltage of unity, while the phase of the coefficient defines the location of the transduction center, see below. As it was shown by Abbott, 2 7 the transduction coefficient ap is proportional to the piezoelectric coupling coefficient K and to the square root of the aperture W. With VR being the Rayleigh wave velocity and £s(°o) denoting the static permittivity of the substrate, 2 7 , 6 0

^0(¥) e s ( o o )

(8o)

with the factor a being of the order of unity. To remove the dependence on aperture, the normalized value an — ap/^W/Xo is defined. It depends on the strength of the piezoelectric coupling and on the metallization ratio. For LSAWs on LiTa03 and LiNb03 substrates, the total mass of the electrodes also has an influence as well. Typical values of an are about 3.3-10 - 5 fi_1/2 in quartz, and 80-110-lO - 5 ft"1/2 for LSAWs in L i T a 0 3 and L i N b 0 3 .

26

Coupling-of-Modes Analysis of SAW Devices 893 2.3.5. Capacitance

parameter

The normalized capacitance parameter, Cp, measures the electrostatic storage of energy in the structure per unit period. Due to the long range of electrostatic forces, the capacitance actually depends on the length and aperture of the structure, but in long transducers the value is practically proportional to the device length. Therefore it is convenient-especially in device design-to introduce normalized capacitance Cn = Cp/W, t h a t is: capacitance per period (electrode pair) and per unit length of aperture. If the aperture is expressed in micrometers, the values for LSAW-cuts in LiTaC>3 and LiNb03 substrates are about 48-64-lO" 5 p F / p m . For materials with weak piezoelectric coupling, such as quartz, the COM capacitance parameter Cp is very close to the static capacitance of the transducer per electrode pair. For this reason, the confusing term 'static capacitance' is frequently used in the literature. However, for strongly piezoelectric materials the situation is more complicated. 61 It is more appropriate to consider Cv as a parameter to be fitted so as to most accurately describe the capacitive contribution to the admittance in Eq. (71) in the frequency range of interest. 2.3.6.

Unidirectionality

In Section 2.2 the components of the P-matrix for a uniform structure were found to depend on the phases of the reflectivity and the transduction coefficient, 8T = LK and 6e = la, respectively. The phases depend on the locations of the reflectivity and transduction centers of the considered period, and they are connected to the directionality of the structure. Consider the period shown in Fig. 8, with reflectivity and transduction coefficients KP and ap, respectively, and with terminals located at Xo ± Ao/2. The reflectivity center xT is defined such t h a t the period is imagined t o be replaced by a symmetric reflector with reflectivity r and located at the reflectivity center. Between the terminals and the reflector the waves are assumed to propagate with some reference wavenumber, say fco = u/v. Since the reflected waves travel the distance between the terminal and the reflection center twice, the reflection coefficients P n and P22 may be expressed as p _ re—iko2(xr—xo+Ao/2) PP 2 2 =re -ifco2(x 0 +A 0 /2-a :r )_ 22 On the other hand, from COM we obtain P n « i/c* and P22 « inp. comparison shows that 0 r = 2k0 (xT - xo) + nir, such that

f P n = ±i|/«P|e-i2fc»(^o),

\ P22 = ±iK|e+ i 2 f c °^°>.

(81) Careful (82)

(S6)

The excitation center is defined in a similar fashion as the reflectivity center: waves are imagined to be excited at the excitation center, from where they propagate to

27

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V. Plessky & J. Koskela

rl=2£ 0 (Xr-*l) ''
gap are the gap parameters discussed above.

56

(128)

Coupling-of-Modes

Analysis

of SAW Devices

923

stopband. As a drawback, the reflectors increase the ripples in the admittance at frequencies below the resonance and close to the high-frequency edge of the stopband. • Finally, the position of the resonance at the low-frequency edge of the stopband makes it more distant from frequencies with pronounced BAW generation. The phase velocity is minimal at the left edge of the stopband, improving the conditions for waveguiding. According to the waveguide theory of Haus 2 1 this difference in velocities is sufficient for waveguiding to occur.

5.2.2. Synchronous

'hiccup'

resonators

The synchronous uniform resonator structure considered above is not very good if maximal Q-factor and small size are required and the resonance-antiresonance distance is of little concern, for example for frequency stabilization in oscillators. In this case the design criteria are cost, frequency tolerance, admittance value at the resonance, and size. 109 Quartz is practically the only material used for this application. The disadvantages of the absolute synchronous resonator are all related to the fact that the resonance occurs at the low-frequency edge of the stopband: • The reflectivity of reflectors at this point is not maximal. This either leads to additional losses or longer gratings. • The device may be too sensitive to variations in the thickness and width of the aluminum electrodes, present because of the limited precision of manufacturing technology. The position of the resonance at the edge of the stopband makes it sensitive to changes in both velocity and reflectivity. To correct these problems, the synchronous 'hiccup' resonator was proposed, see Fig. 20 and Ref. 109. In the center of the transducer there is a gap, which breaks the periodicity and increases the center-to-center distance between the center electrodes by A/4. The center electrodes have the same polarity. Under these conditions the resonance occurs at the center of the stopband, as one can easily demonstrate by considering the phase change of the wave circulating between the two parts of the transducer. Such resonators may without any difficulty be modeled based on COM formulas. One can first connect in parallel two parts of transducer with the gap between them, and then use equation (55) to add the reflectorsf The advantage of having the resonance at the center of the stopband is t h a t the resonance is then insensitive to the finger reflectivity, increasing the yield in the mass production of the device. 109 The price paid is that the electromechanical coupling e

We leave this as an exercise to the reader, as well as the demonstration of the fact that the 'hiccup' resonator always has only one pronounced resonance.

57

924

V. Plessky & J. Koskela

P~l

.-•j p/Kn+j)

Fig. 20. Synchronous one-port hiccup resonator.

0.15

p =4|im

ReY(/)

o.n

a/p= 0.5 „

0.05 \

ImY(/)

v

a

i . - -?%^u .—-j -0.05 V

-0.1 460 480 500 520 540

_J

i—

560 580 600 620 640 660

/ [MHz] Fig. 21. Admittance of a synchronous hiccup resonator on 64°YX-LiNbC>3.

of the resonance to the transducer is reduced: the SAW amplitudes decay exponentially from the center to the ends of the device and the outermost parts of the transducer are not efficiently coupled to the amplitude distribution. Consequently, increasing the number of electrodes increases the magnitude of the admittance at resonance only up to some limit, in contrast to the synchronous uniform resonators where the admittance is roughly proportional to the number of electrodes. Again, for strong piezoelectric materials the situation is not so cloudless. Fig. 21 shows the admittance of a 'hiccup' resonator admittance on a 64°YX-cut LiNbC>3 substrate. In the figure the position of the resonance can be seen to have slightly but visibly shifted from the center of the stopband, indicating that the velocities inside the gap and in the periodic structures are significantly different. Although strongly piezoelectric substrates are rarely employed in resonators, they are used in a few notch filters and voltage controlled oscillators (VCO).

58

Coupling-of-Modes

reflector 1 ',

calibration reflector

WT .

1

Analysis

\

reflector \

\

u

•

**

•

of SAW Devices

925

2

A_

p \

1

'•••

JLg < j^i < L>2

£ s

H

i Li/*.

(140)

Here, A;D is the wavenumber in the gap D. The second half of the gap D, as well as the grating G and the gap S are attached to the transducers T2:

11

-y11

+

p(2) _ p(T2) (. 13 P l 8 +

-

i-pp(G)Je P£2)R(G)

e

{ l-P^R(0))

63

(141)

'

\ e-ikDD/2

'

,U9x (142)

930

V. Plessky & J. Koskela

PJ22)RiG)

p(2) _ p (T2) /

p(2)_p(T2)

\

-ikDD/2

,u„x

. (P£2))W

(1U)

Finally, the resulting two-port elements, described through P-matrices P ' 1 ' and P ' 2 \ are cascaded to obtain the two-port Y-matrix (admittance matrix) of the device: 9P(1)P(1)P(2) 11-38

+

l-(Pff+P1(M?)' y _ 2o(2) . 2PfPlCPiil±Pill) +^TTTTTTrT' P£)PZ *22 - 2P 3 3 + 1-{PS> 7ZjTS

(

}

fl46 .

(146)

(2) p ( l )

U2

2R 13 *31

i-^+pfMV

(147)

(1) p(2)

v

2P:^P

*~i-(*/+kv®-

3

(26)

The projection of D onto the a;-axis is given by (27). D1=n\D

(27)

A relationship between D and the irreducible field variables can be established by using the constitutive equation in (28). D = etS + gE = e%u - gVip

(28)

e and e denote the (6 x 3) piezoelectric- and (3 x 3) dielectric matrix, respectively. In transition from the first to the second equation we have adopted Auld's notation for the strain S, i.e. S = Vu, and the equation E = — V! = n'je'Yw - n\gV W3- This leads to 16 Green's functions ( 4 x 4 dyadic Green's functions). However, not all of these Green's functions are independent. The reciprocity principle dictates certain relationships between the Green's functions. P u t simply, this condition states that by interchanging the positions of the excitation (s: source) and observation (f: field) points we measure the same reaction, field response. The application of this principle leads to the relationships in (80). G«(r/-r,) = G«(r,-r/)

(80a)

Gij(rf

(80b)

- r . ) = Gji(r. - rf)

To understand these equations in the Fourier domain consider the simple case where the source- and field points reside on, say, the z = 0 plane. Denote k|| = (kx,kyY, Eqs.(80) read S«(k||)=S«(-k||)

(81a)

G0-(k||)=^(-k||).

(81b)

Near-field Asymptotic Limits: One of the major advantages of using diagonalized forms is their utilization for determining the asymptotic limits of the Green's functions near the source region in the spatial domain. 9 Green's functions may have singularities. The occurance of these singularities may make the field analysis in terms of surface integral equations a challenge, since generally the near-field asymptotic expansions of the Green's functions are needed. The above eigenform in the wavenumber domain offers an interesting possibility to find these asymptotics. Instead of first constructing the Green's functions and then determining their near-field asymptotics we may alternatively proceed as follows:

101

968

A. R.

Baghai-Wadji

• Calculate the eigenpairs for large values of the wavenumbers (corresponding to the near-field in the spatial domain). • Satisfy the boundary and interface conditions by the resulting eigensolutions.

asymptotic

These steps lead to asymptotic expensions for the Green's functions in the wavenumber domain. Generally these "low frequency" limits for the Green's functions can be transformed into the real space simply by inspection. 9 The resulting functions in the spatial domain represent the asymptotic limits of the Green's functions near-source region. In the next section we investigate these properties in greater detail by studying several examples. (For details see Ref. 9.) 3. A D i s c u s s i o n on Green's Functions In the singular surface integral method Green's functions and/or their spatial derivatives link solutions in the interior of a domain to the values on the domain's bounding surface, and possibly to the sources within the domain as well. The key steps in the application of the singular surface integral method to boundary value problems are the derivation of Green's functions, and the calculation of self-actions. This section is devoted to familiarizing the reader with details regarding these steps. We will focus on four types of problems. • Electrostatic problems in semi-infinite media: 2D electrostatic problems in anisotropic media with a j u m p discontinuity in material parameters along a plane surface. This type of problem is interesting for the following reasons: 1. The underlying algebraic manipulations are tractable and the calculations can be carried out entirely in analytical form; this is useful for gaining physical insight into the problems. 2. The results can be directly implemented to solve a variety of modern engineering problems. 3. Finally, and most importantly, we wish to emphasize the following motivation: dealing with dynamic problems, in the ultimate proximity of the source region, static results represent asymptotic limits of solutions for dynamic solutions; many examples will illustrate this fact. We will employ this property for calculating self-actions arising in BEM. In fact, it turns out that self-actions are static in nature. This property leads to a technique with a promising applicability in practice. Important features of this technique will be shown by comparing the static results with those obtained from asymptotic limits of the scalar wave equation. It should be mentioned that only in connection with vector wave equations do the whole aspect of the underlying relations become clear. • Acousto-electric problems in infinite media: we will continue by considering the acousto-electric dynamic equations in the simplest possible form. It turns out that the calculations here can also be performed entirely in analytical form. This makes possible the investigation of the asymptotic nature of solutions in the ultimate proximity of the source points, in analytical form. The close relation with static solutions will be demonstrated. • Elastic problems in infinite media: the third type of problem involves the dynamic equations of motion in purely elastic isotropic media, Refs. 24 and 25. The isotropy assumption leads to Green's functions in analytical form with the above mentioned advantages.

102

Theory and Applications of Green's Functions

969

• Acousto-electric problems in semi-infinite media: Examples considering piezoelectric half-spaces conclude our discussion. Here also we will start with the corresponding diagonalization equation. Having determined the resulting four eigenvalues and the corresponding eigenvectors, along with our results from the section on electrostatic problems in semi-infinite media, we will be in a position to solve the following semi-space problems: 1. Free space (z > 0)/piezo-electric semi-space (z < 0)-configuration. A line charge and a line force embedded in the piezo-electric substrate will excite the medium. 2. Free space (z > 0)/piezo-electric semi-space (z < 0). A line charge in the free space will act as the source. 3. Piezo-electric semi-space (z > 0)/piezo-electric semi-space (z < 0). A line charge and a line force located in the upper half-space will excite the media. 4. Piezo-electric semi-space (z > 0)/piezo-electric semi-space (z < 0). A line charge and a line force embedded in the lower semi-space will excite the media. Although we will restrict ourselves to 2D problems in this chapter , the solution schemes are also applicable to 3D problems, see chapters two and three in Ref. 9. Furthermore, periodic boundary value problems, and the derivation of the associated periodic Green's functions are briefly discussed in this section. (Regarding the calculation of the self-actions in these problems the reader is referred t o Ref. 9.) 3 . 1 . Green's functions continuity of material

in anisotropic parameters

dielectric

media

with a jump

dis-

Statement of the problem: We consider the following problem which is illustrated in Fig.(l). The region z > 0 is occupied by an anisotropic dielectric which is specified by a symmetric, positive definite matrix ef-u\ Region z < 0 is filled with a dielectric being characterized by e}1'. The superscripts " u " and " 1 " refer to the upper and lower medium relative to the (z = 0)-plane, respectively. In the upper medium we assume that a line charge with co-ordinates x = a and z = c excites the system under consideration. Thus for the source function p(x, z) in our problem we can write p(x, z) = 5(x — a)S(z — c). Our goal is the calculation of the resulting potential function in the entire (x, z)-plane. In mathematical terms, we are interested in the solution of a potential problem subject to boundary conditions at infinity, and to certain interface conditions, with a delta-function "source." The resulting potential distribution will be called the Green's function associated with our boundary value problem. Solution: Technically it is useful to subdivide the (x, 2)-plane into three regions; z > c (region I), 0 < z < c (region II), and z < 0 (region III), Fig.(2). In the following we will construct solutions in each region which satisfy homogeneous differential equations. The unknown coefficients involved will then be determined by imposing the boundary and interface conditions. In this construction, eigenvalues and the corresponding eigenvectors associated with the underlying differential operator will play a central role. 3.1.1.

Diagonalization

For non-piezoelectric, homogeneous, anisotropic, and source free dielectrics diagonalized equation (75b) simplifies to the equation in (82).

103

970

A. R.

Baghai-Wadji

e (u)

a line charge

f C

7 i k

|

X d

(1)

JB

Fig. 1. Geometry of interest.

_e (u)

region I

aline charge

f _e (u)

region II

.

z

C

i

|

X a _e (1)

region III

Fig. 2. Subdivision of the (x, z)-plane into three regions; z > c (region I), 0 < z < c (region II), and z < 0 (region III).

104

Theory and Applications £13 9 £33 9x

££.

2

d

£33 8x'J

L.

\

£33

_£ia^. £33 9x

'

\

/

(

V3 "\ -

U ,

)~dz(

d

(

of Green's Functions

for £116:33 — £i3 2 . The form in (82) is well-structured: it is a diagonalized equation with only those field variables which are relevant to the interface conditions. We substitute for

x' and c —> z' in order to emphasize that the derived formulae are valid for any choice of x' and z' for a line charge with z' > 0. Performing these substitutions, we now extend the list of arguments of Green's functions from (x, z\k) to (x, z\x', z'\k) in order to emphasizing the dependency of Green's functions on the co-ordinates of the line charge:

G

(*.*l f l ! . a ! l*)- 2 e W| f c |^W + e » e xe

,, , , , G^\x,z\x',z'\k)=

'ss

1 l (u)

( !

+ e

J (92)

1 (-&*—fey2')!*! j[-(*-*')+(4fy*--JV')]* 4 ? e 4 7 4 7 (93) ) ^ 1

Discussion: Non-oscillatory exponential terms involved in the expression for £(".«). We first consider the exponential function: exp(—£p /e^ (z + z')\k\). The positive-definiteness of e implies that £33 and Ep are positive. Furthermore, we have z 4- z' > 0, because the conditions z > 0 and z' > 0 hold, and thus the equality z + z' = \z + z'\ is valid. Consequently, this exponential function decays

107

974

A. R.

Baghai-Wadji

for any value of z and z', chosen from the definition range, and thus we can write for it the form: exp(—Sp /e 3 g \z + z'\\k\). Our next concern is the exponential function: e x p ( — S p / e ^ \z — z'\\k\). The appearance of the magnitude sign in \z — z'\ ensures that this exponential function decays for any values of z and z'. Non-oscillatory exponential function appearing in (?('•"); that is: exp (sp /e^\z— £p /e^ z')\k\). By definition we have here the equations z — —\z\ and z' = \z'\. (Remember that the definition range of G^l'u^ is the lower half-space, and t h a t the line charge is located in t h e upper half-space.) Thus this term becomes exp(—(s p /egg \z\ + £p /S33 \z'\)\k\), which is an exponentially decaying function. For a line charge source located in the lower half-plane we denote the resulting potentials in the upper and lower half-planes, respectively, by (?("'') and G^l,l\ The reader can verify that (?(">') and G^l'V> have the forms:

G > ^w\[t^ j[-(x-x')

Xe

+

-$\*-*'\w\

+e

)

-tfr(z-z')]k

*33

(95)

Arranging the above four Green's functions in matrix from, we can define a matrix Green's function G as / G^u\x,z\x',z'\k) a(x,z\x',z'\k)=\

\ G^u\x,z\x',z'\k)

G^l\x.z\x',z'\k) \ .

G^(x,z\x',z'\k)

(96)

J

3.2.1. Properties of G_ The multiplicative factor l/\k\, which is an even function of k, appears in all elements G^a'b^ with a = u,l and b = u,l. Note that ±|fc| are the eigenvalues of the Laplace operator in isotropic media. The non-oscillatory exponential functions involved depend on \k\, and thus are even functions in k. In order t o carry out further properties of G^a'b\ we consider their integration over k to transform them into real-spacei* ''•Strictly speaking, the analysis in the remaining of this subsection and in the following two subsections is not mathematically rigorous. A more careful analysis based on the theory of distributions would be outside the scope of this elementary treatment. The final results are, however, correct and extensively tested.

108

Theory and Applications —e

G^bHx,z\x',z')=\im\

975

oo

f ^G^b\x,z\x',z'\k) e—>0 I J

of Green's Functions

~G^b\x,z\x',z'\k)\

+ [

2f

J

Z7T

J

e

—oo

oo

= -f-^G^b\x,z\x',z'\k)

(97)

—oo

Because of the first two of the above properties, we recognize that only the real parts of the oscillatory exponential functions contribute t o the integrals. Then, by writing

we have

1

G^u\x,z\x',z')=

~(«) _ J O + o o

,.£f.

1

^^rffcfe

-'-E-r\z+z'\k

'»

e («) xcos|(x-a:')--^-(z-z,)l&

£ 33

+

,(") +°° 1 —^iylz—z'tA £ («) ^4-dk-e 'M c o s i ^ - a ; ' ) . 13 ( Z _ / ) | A ; 2weP o -G33 1

Gil'u\x,z\x',z')

=

, .

... 4-dk-e

^i

4F

(0 («) xcosl(s-s')-(%*-%*')!* s

, n

33

,

e

7r(ep

+~

+£p ) 0

J«0 JO xcosKx - x') - (e-f-)Z 33

(100)

33

1

£

(99)

e

33

109

1 -(-&yl 2 l+-frl z 'l)fc K

£

-f)Z')\k

(101)

976

A. R.

Baghai-Wadji

(0 ~{z - z')\k

xcos|(a: - a;')

-33

l

4 -

- ^ | 2 " Z ' fc

1

^

+—'-^-f-dkye 2 ^-F 0 u *

'as'

JO £i ' cosKz-zO-^-fz-*')!* c- ^ 33

(102)

+oo

where the symbol -f- has t o be understood in the sense that o oo

+oo + oo /• -A-dA; • • • = l i m / dk • • •.

o

^-W

From these representations the reader can immediately deduce further facts and properties of the Green's functions: The above integrals do not exist (even in the Cauchy sense ), as we will soon see. This is a consequence of assuming a single (isolated) line source for our problem. In the subsequent discussion it is shown that the above integrals allow a meaningful interpretation if, and only if, we consider a collection of line charges which is charge neutral. Considering a group of N lines with charge magnitudes per length (in y—direction) being denoted by

p-.

'

"•"

7 7 r ( - 7 - lim lne) -(«)

> )

/N

£

- *') " e% ( * - -')]2 + efe(* + z')}2}

1 3 /-,

33

33

g

,/\l2 , f P

/•„

^/M2

W K * - *') - ^ ( z - z')] 2 + lF £ y ( * - ^ O H -

^

4«>

1

(117)

T h i s formula h a s been derived by combining t h e formula (5.2.27), pp.232 w i t h t h e formula (5.2.2), pp.231 from H a n d b o o k of M a t h e m a t i c a l Functions, Abramovics a n d S t e g u n . 2 7 Here 7 , b e i n g equal t o 0.577256649, d e n o t e s t h e Euler c o n s t a n t .

112

Theory and Applications of Green's Functions 979 The expression for G{l'u)(x,z\x',z')

G^\x,z\x>,

can readily be written in the form

l

z>) =

( - 7 - £Hm In.) £_ J "

ir(ep' +£p)

2lT(£p

+ £p

)

£33

£33

(118)

e

£33

33

Remark: We have added the symbol e t o these functions in order t o emphasize the existence of the e-dependent terms in their expressions. As pointed out earlier, by substituting £ ( u ) £ ( ' } , G^'u) (x, z\x', z') transforms into G{l'l)(x,z\x',z')

while G(!'u) (x,z\x',z')

G^Hx,z\x',z')

transforms into G(^'l) (x,z\

* m ( - 7 - Um Inc) £ 7r(eF'+ep) ^-° («) JO _(«) 2 U V ;J + v few MK* *' ) ^ T * %*')] ,._/>) , J 0 / ' >) JO ' >> Z7T^£p

=

+EpJ

1

£33

C33

p(') _ p(«)

^7U^ 1 (I)

*7reP

gp

~

x>)

JO

+ JO ^rl-'l)2} £33

JO

{z

z )f + l

- 7* ~ '

-t-5p

(119)

t33

Fm

ln{[ix

t/iep

x',z').

s33

7i{z + z')]2}

e33

JO JO u v - - x') ~J ~ - -z')f /J + . 1 fer{z ( J )(v z (0» ]n{[(x - % - z')f] e e 33

(120)

33

These forms allow the recognition of further properties of Green's functions. All t h e four Green's functions possess a common constant singular term. Thus we can write

G{a'»\x,z\x',z')

=

} n{eP

+ey)

( - 7 - hm lne) + G^b\x,z\x',z'). £ +0+ -

(121)

This fact is crucial and plays an important role together with t h e charge neutrality condition: We show that for charge-neutral systems (physically realizable systems) the above mentioned singular term vanishes. Consider in t h e upper and lower semi-spaces, respectively, charge distributions p(u\x,z) and p^l\x, z) in such a way that the charge neutrality condition holds 00

00

00

00

/

/ dxdzp(u\x,z)+

J

J dxdzp(l\x,z)

113

= 0.

(122)

980

A. R.

Baghai-Wadji

Based on the linearity of our problem, and on the definition of Green's functions Ge (x,z\x',z') as responses t o a single line charge, we can write the following equations for tp(u\x, z) and tp(l\x,z):

V? (p) (x,z)= /

Jdx'dz'GiP'u^(x,z\x',z')p^(x',z')

—oo—oo oo oo

+ J J dx'dz'Go+ Aire

- x')2 + (z - z')2].

in

(125)

By considering the fact that in physically realizable systems the fundamental cell will contain a charge-neutral system, we omit the singular term to obtain G(x, z\x', z') = ^-— ln[(z - x1)2 + (zAire

z')2].

(126)

We now come back to our line-charge array and superpose the associated potentials to obtain the periodic Green's function GpeT(x,z\x' ,z'), which characterizes the aforementioned line charge array. Using (126), we readily obtain

GpeT(x,z\x',z')=

oo

£ G(x,z\x' n=—oo

I 47T£

-nP,z')

£ ln{[x - (x' - nP)]2 + (zn=—oo

z')2}.

(127)

We see that whenever the Green's function associated with a single line charge (point charge in 3D problems) is available in real space in closed form, it is an easy

115

982

A. R.

Baghai-Wadji

task to construct the corresponding periodic Green's function. Note that in practice the series involved have t o be truncated, by letting the dummy index n run from —N to TV. Numerical calculations indicate that in electrostatic problems, for values of N of the order of 10, we obtain satisfactorily results. 2 8 The construction of GpeT(x,z\x',z') relying on the real-space Green's function G(x, z\x', z') of a single line charge is one of the two alternatives. In the following discussion we will be acquainted with a different way which merely requires the Green's functions in wave number domain. 3.4.2. Construction number domain

of periodic Green's functions

using Green's functions

in wave-

We consider again the line array from the previous section. In order to construct the associated periodic Green's function, we consider first the Green's function in the wave number domain of a line charge located in the fundamental cell. The latter function is G(x,z\x',z'\k)

= -l-I- 1. The last section in this chapter is devoted to this technique. Calculations will be performed entirely in analytical form, and thus the reader will be provided with many useful details in connection with self-action analysis. Phased Periodic Green's Functions: For completeness it might be mentioned that in a variety of applications it is necessary to construct Green's functions associated with periodic phased-line arrays. The latter are periodically arranged, localized sources, that are driven in such a way that the potential of n t h line charge is given by exp(.;'A:oa;n). Here xn is the a;-co-ordinate of the n t h line charge and ko is assumed to vary from —w/P to ir/P. The Green's functions associated with phased arrays are closely connected with the aforementioned "ordinary" periodic Green's functions,

117

984

A. R.

Baghai-Wadji

1ioe spate

X a piezoelectric semi-space supporting SHW

-

*

•

•

a line charge + a line force Fig. 4. A piezoelectric semi-space with a line charge and a coinciding line force located at point (x1, z') beneath the surface plane. Above the surface is free space.

and from a theoretical point of view, provide no significant contribution to BEM theory. For this reason we will not be explicitly concerned with phased periodic Green's functions in this work. For details the interested reader may refer to Refs. 9 and 11.

3.5. Infinite-domain wave problems

Green's functions

associated with

Bleustein-Gulyaev

The treatment of Green's functions in this section provides the reader with further information concerning the construction of elemental solutions of partial differential equations. Statement of the Problem: We consider a piezoelectric semi-space with a line charge and a coinciding line force located at point (x', z') beneath the surface plane. Above the surface is free space, Fig. (4). It is known that under certain circumstances «i and U3 decouple from u^ and if, and it is possible to excite surface waves («2, f) which propagate along the interface z = 0. This type of waves are called Bleustein-Gulyaev Waves, Refs. 29 and 30. 3.5.1. form

Construction

of infinite-domain

Green's functions

using 2D Fourier

trans-

The analysis in this section is devoted to the construction of the infinite domain Green's functions. For this purpose we assume a line charge and a line force be located at the point (a/, z'), and find the resulting elastic displacement component M2 and the electrical potential ip, associated with an unbounded piezoelectric medium. The simplest possible equations for describing this type of motion are the following:

118

Theory and Applications of Green's Functions 985

d2 { p

d2 + Ci4

-W

d^

2

d ( e ^

d2 + C

d2

)U2

+ ( e i 5

"^

2

2

d d +e ^ ) u 2 - ( e n - ^

^

d2 + e i 5

5^ =°

(135a)

2

d + en^)-0+

In the light of the above derivation it is instructive to summarize the relevant steps: • Consider the representation: G(xi, x3 \x\ ,x'3) o G(xi, x3 \x[, x'3 \k). • Find the asymptotic limit of G{x-l,x3\x'1,x'3\k) S(xi,x3\x'1,x'3\k).

for k > 1: G{x1,x3\x'1,x'2i\k)

~

• Extrapolate S(x1,x3\x'1,x'3\k) over the whole definition range of k, that is (-00,00): which means letting S(x1,x3\x'1,x3\k) be valid for any value of k. • Calculate the inverse transform of

S(x1,x3\x'1,x3)

S(xi,x3\x'1,x'3\k)

°° dk = / -

123

m

2-K

—S(x1,xs\x'ux'3\k).

990

A. R.

Baghai-Wadji

• The function S(x\x')

is the near field representation of

G(x1,x3\x'1,x3)

3.9. Near-field

behavior

3.9.1. Real-space

analysis

"

S(x1,x3\x'1,x'3).

of the ^-derivative

Using the relation (d/dx)H^\x)

of Green's

= -H^ix)

flgfri,xalx'^x'z)

~ ~ *~c

functions

we have

w (Xl - x\) 3

^

G(x\x')

fl„

w

(1)

Fl

{ R

(154)

7^

At the limit R« kl-x'i~x'^e'~at 1 we can determine the asymptotic of the above integrand which is AUf,jk(x/,—x/,)

•7fee

e

-at(k)\x3-x'3\

_-r.

JJL.eJH here contributes to the integral and we obtain 1 °° / Wix^x^x'x'z) = - — fdksiakix! - x'^e'^3-^.

(158)

27T 0

We use the table integral °° Jdxe

1 px

sin(qx

j (QCOS^

+ A) = —2

+ psinA)

(159)

(p > 0.) (Formula 3.893/1 of Ref. 27.) which in the special case A = 0 reads oo

a

fdxe-pxsm(qx) o

=

02

0. P +Q2

(160)

Employing this result we obtain (161) which gives the near-field behavior of dG_ dx[

dGjdx'x

i?|| » 1 ~

_ 1 xi - x\ 2?r ijy 2

(162)

This result proves that the processes • spatial domain differentiation with respect to x'x • wave number domain integration over k • spatial domain asymptotic order-calculation (R^ -C 1) and • wave number domain asymptotic order-calculation (k\\ S> 1) commute in the following way: Asymp

. d

°? dk _,.

, , ,,,.,

-Is 7" i^-*"*»i 3.10. Asymptotic

behavior

of dG/dx'3

Consider the following self-explanatory calculations in the wave number domain:

125

992

A. R.

Baghai-Wadji

dG(x1,x3\x'1,x'3)_ dx'3

d °? d f c e J f c ^ i - < ) e - t t ' W N - ^ l "d^J^Tr -2at(k) 1

OO

= - — t a g n ( x 3 - 4 ) / d * c o s * | a ; i - a;' | e -«t(*)|*s-* s l 2lT 0 1 °° ~ - — sign(z 3 - x'3)Jdkcosk\xi - a^le-*1*3-^! 2-K

(164)

0

Using the table integral oo Sdxx^e-f'coBbx

an = { - V T W

¥

a E -

(i65)

(Re/3 > 0; b > 0) in the special case n = 0, that is oo

a

JdxcoBbxe-P" o

= -^SJ b2 + P2

(166)

we obtain 0G(a;i,a:3|a:i,a:'3) 1 12:3-2:3 ~ - ^2-K s i g ns ( av": 3 - 2:i ;3)( - x\f + {x - 4 ) 2 0*3 Xl 3 1 2:3—2:3 ~ _ 2 ^ r iJ,, 2

3.11.

Calculation

of self-action

in

(167)

BEM

In certain BEM calculations we are concerned with integrals of types TI \ I(x1,x3)=n1 J(x1,x3)

t , ,dG{x1,x3\j^1^ dx3 i—J-J-+ = -n3

, aG(a:i,2;3|2:i,4) / dx\

> (168a) (168b)

which for coinciding observation a n d source points deserve special a t t e n t i o n . 3.11.1. Calculation

of I-type

integrals

Substituting x'3 = x3 + u and x\ = x\ + e the following relations are valid: 2:3 — x'3 = — u

xi — x\ = —e

In the ultimate proximity of the observation point (2:1,2:3) we can employ the asymptotic expansion of dGjdx\ from the previous subsection. Thus we can write

126

Theory and Applications of Green's Functions 993 23-A3/2

,dG(x1,x3\x'1,x'z) c

dx[

X3+A3/2 X3-A3/2

1 '27T

f J

, , si - x\ ^ ( x . - x ^ + ixz-xtf

X3+A3/2 -A3/2

= — 2ir

/ J

A3/2

du-x 7; = — / e2 + u2 2TT J

A3/2

du-r. ,. e2 + u2

(169)

-A3/2

By substituting u = ev and thus du = edv (169) becomes A3/2c

2TT

J

A3/2e

e2 + e2?;2

-A3/2e

2n

1 + ^2

J

(170)

-A3/2e

At the limit of e -4 0 we can write

1=

1 °° 1 lim h = — f du~. e-+o+ 27r_i 0 1 + u2

(171) v '

By using the table integral

we obtain

7=i;arctan(,)|-00 = i - ^ - ( - f ) ]

= i,

(172)

which is valid for any order of A3. (This results from an implicit assumption that the surface in the vicinity of the observation point is flat.) 3.11.2. Calculation of J-type

integrals

In a similar fashion we can write the following relations: X\ — x\ = —u The following steps are self-explanatory:

127

X3 — x'3 = —e

994 A. R. Baghai-Wadji zi+Ai/2 Jt=

J dx\,

-Tl3

dG{x1,xz\x'1,x'z)

xi-Ai/2 x1 + A 1 /2

=

_J_

dx> 1

f

27T

J

H-Aa/2

(Xl

J.x

A t /2

-At/2

2^ y

-x3

z

- X[ )2 + (13 - 4 ) 2

d

=

kxe-e 1 5 |fc|

e15|fc|

T (x,z\k) )

\

ejkxe\k\z+a(2)(kj

a^\k)

D3(x,z\k)

0 1

1

«xeMZ

W( +a^>(k)

1 C15A11

\ gjfcig-Ajz

—C^\t

Ca\t

0

0

(177) The unknowns ay\k) 3.13. Semi-infinite

will be determined from the interface conditions.

dyadic

Green's functions

for piezoelectric

half-spaces

We are now prepared to derive Green's functions associated with a semi-space piezoelectric substrate which is capable of supporting shear horizontal waves along the substrate surface. Based on the piezoelectricity assumption, there are two ways of exciting the semi-space: we can excite the substrate either by an electrical line charge or equally

129

996

A. R.

Baghai-Wadji

lice spate

X a piezoelectric substrate with material parameters £

11 ,

e

i 5 , ^44

-*•

a line charge + a line force Fig. 5. A line charge and a line force which coincide at the point {i' = a,z' = c) within the substrate in the lower half plane (z < 0).

well by a mechanical line force. While the location of the line force is limited to the substrate region, the line charge can reside everywhere in the space. In this subsection we focus our attention on sources which are located within the substrate. To cover problems arising in practice, the next subsection will be concerned with the medium excitation by a line charge above the substrate? Results from these two sections will allow the investigation of the properties of the involved Green's functions. However, as the reader will see, it is necessary to consider the excitation of two welded piezoelectric semi-spaces, in order to investigate the underlying reciprocity properties of the Green's functions involved. Two welded piezoelectric semi-spaces will be analyzed in the final part of this section. 3.13.1. Line source excitation: sources are located in the lower half space We consider a line charge a at {xa, za) and a line force r positioned at (xT, zT) within the substrate (in the lower half plane (z < 0)), Figs. (5) and (6). We obtain the following results for the Green's functions. The electric potential response in the upper semi-space (z > 0):

i A piezoelectric substrate enclosed in a metallic package is the basic building block of most microacoustic devices. Under the assumption of ideal electric conductivity the metallic parts in the devices can be regarded as a collection of line charges. In an analogous way mechanical loading of the substrate and the mechanical tensions between the substrate and the package can be modeled by a collection of line forces with a •priori unknown strengths.

130

Theory and Applications

of Green's Functions

997

region I liee\p.u.e

Z ii

region II £

11 ,

e

x

C

i 5 , ^44

a line charge + 1 a line force 1

region III £

11 ,

e

!

•

1

..

1 1 .k '

•

i 5 , ^44

Fig. 6. A plane z = c together with the interface plane z = 0 divides the geometry into three homogeneous regions.

^*tp,a \^i Z\%ai Z01 Za\k)

1 £0 + £11 e

1

^44^

1*1° 4 4 |

fc|

~

G27J« *£. _i_ n ±L _ t £ U e 1 5 ~ £ U e 1 5 ) i(£ii+£ii)

l

;

G & J W l * * >**!*) = 1 £

£

£

i1eyr,-£ne'R 1

^

£

'n

£

n+ n

n 1*1 7T* A,

|*| ^ & +c" £ _ (£'i^s-£ne'5)2 W4| f c | + ^ 4 |fc| ^ ^ ^

T 7 « A,

C 4 4 | 4 + G 4 4 -i^ -

3.14. Infinite-domain tal polarized wave

e**(*-*,) e -(A?W+|k||z,|)

(188)

(£iiei5-£ne1B)2 £ ." £ '" £ j ' l £ '" ii ii( ii+ n)

dyadic Green's functions propagation

for the analysis

of

sagit-

Considering a line force embedded in an isotropic elastic half-space, the derivation of Green's functions for sagittal polarized wave propagation has already been discussed in Ref. 24. We denote the distance of the line force in the interior of the domain from the surface of the semi-space d. By letting d go to infinity, we can then obtain the infinite-domain Green's functions from the half-space Green's functions. This section is devoted to a direct derivation of the aforementioned infinitedomain dyadic Green's function. The derivation is based on eigenvalues and associated eigenvectors of the governing and constitutive equations.

136

Theory and Applications of Green's Functions

1003

Statement of the Problem: We assume the entire space to be filled with an elastic medium characterized by the elastic constants C\\ and C44 and the mass density p. Furthermore, we assume that a line force located at the point (x',z'), and oriented in x-direction (TIS(X — x')S(z — z')e"x), oscillates in time according to exp(ju;i), and excites the medium. The excitation of the medium is uniquely determined by the resulting elastic displacements, i.e., Ui(x,z) and u3(x,z) functions. In order t o signify the localized, delta-function-like nature of the source, we use G\{x,z) and G3(x,z) instead of U\(x,z) and u3(x, z). In addition, in order to indicate the direction of the force, which in the present case is the rc-axis, we write Gu(x, z) and Gzi{x,z). Finally, by writing Gn(x,z\x',z') and G3i(x,z\x',z'), we also provide the co-ordinate values of the above-mentioned line force. Analogously we speak of a line force T3 1 3.16. Self-action

analysis

in vector

field

in the far field in

the

problems

In preceding subsections, which dealt with horizontal scalar waves, we found that the self-interaction calculation can be performed either in the spatial domain or in the wavenumber domain. Furthermore, we have demonstrated that the calculation in the wavenumber domain is significantly simpler. This is an encouraging result and gives rise to the next question: is it possible to extend our ideas to include vector fields? The main objective in this subsection is to show that our solution concept, proposed in the foregoing section for scalar waves, is also valid for vector fields. However, it turns out that, in contrast to scalar waves, more t h a n one term must be retained in the asymptotic series in order to adequately reflect the fine scale structure of the problem. In many engineering problems we will be concerned with the interaction analysis of sagittal polarized waves with surface disturbances, formed as ridges or grooves, on the surface of a piezoelectric semi-space. The analysis employs the BEM and involves infinite domain Green's functions associated with the problem. It should be emphasized that the underlying Green's functions have to be known in real space. Generally, while the analysis of mutual interaction is fairly simple and straightforward, the self-interaction calculation is rather cumbersome. The major steps in calculating the self-interaction using real-space Green's functions are given below. Standard Procedure:

• Transform the underlying system of inhomogeneous partial differential equations from the spatial domain into the wavenumber domain.

• Calculate the Green's functions in the wavenumber domain by inverting the inhomogeneous algebraic system of equations (derived in the first step).

• Transform the Green's functions into real space.

• Calculate the derivatives of the Green's functions with respect to the spatial variables. • Calculate the asymptotic limits of the derivatives of the Green's functions in the near field in real space.

• Integrate the asymptotic limits to obtain the self-interactions.

Assuming an isotropic elastic medium (characterized by the velocities c; and ct), and denoting the asymptotic limits of the infinite domain Green's functions Gij(x\x') in the near field (R\\ < 1) by 5 y ( x | x / ) , we obtain the following result:

138

Theory and Applications

dx[ dx\ 3 an - a^ , 2 ( j ! - a^) 3 ~

„2

T->2

"•" .2

. aGn(x|x')

^ f t i W

.9a;; 1 x3 — x'z

dx\ 2 (0:3 — x'3)3

fl2

c2

1 13 - X3

Ri

c2

c;

#j

. dG 33 (x|x')

47T

da^

C?

~ 47T

i?2

^ dx'3 3 x 3 - x'3

r2

R2

itf

C2

a5 3 3 (x|x0 Sajj

C2

l T 0533(xl^)

~

ylyU>

2 (a?3 — x'3)

-+c 272

JJ2

D#4

(,j.yi;

(192)

=

+

*

^2

+

p4

^4

c ?

^

=

dx'3 2 (13 - a/3)3 , I 1 3 - 4 „2

(193)

2 (a?i - a;;)3

Sii-ii c?

l r r a5 3 3 (xlxQ

l i i - a^ 2 (a^ — a^) r>*' ^+ 32 R2 R4

=

2 (si - a^) 3

_ l i i - i j C2

p4

/

) ^ l 7 r a5 1 3 (x|x') 9a:3 da:3 1 zi — a;i 2 (a;i — x^)3 ,2

.2

3 i 3 - 4 , 2 (a 3 - x'3)3 c2 i? 2 ^h-j j 53* c2

4 , ^ 3 ^ „

l j r 0G 1 3 (x|x

n>2

=

2{x3-x'3)3 c? flJ

itf

c

r2

1005

dSn^x')

_lx3-x(i

.2

2 ( g l - a;;)3

h i - » ; +

p4

^ ^ 47T

47T

of Green's Functions

"+" „2

R2

2 (x 3 - x'3)3 „2

p4

Uy{V

Preparatory Calculations: Based on the Taylor power expansions \/l - 1

a:2

2 p

1

a;

VI-a;2

139

x< 1

(196a)

a; < 1

(196b)

2

1006

A. R.

Baghai-Wadji

and ex « 1 + x

x< 1

(197)

which are valid for vanishingly small values of the argument x, the following manipulations are self-explanatory:

„,

lw2 1

, (198a)

^ " ^ ' - ^ j * ! i _ i u3 ai,t * |fc| +' 2o — c\tT \k\ e-a,lt|*3-^3l

(198b) lw2 1

„ e -|A=ll-3-.T 3 | ( 1 + ±!%--L\x3 2c ; , t I'M

- X'3\).

(198c)

Substituting these asymptotic expressions for aij, l/ 1— asymptotic

limits

of the

The involved coefficients can be determined from the interface conditions on the plane z = z'. Thereby we use the (k 3> 1) asymptotics for the displacement and stress components, rather than these components themselves. A straightforward calculation gives the following result:

S{(x-x',z-z'\k) -_]tLTlP3Hx-x') _

= -X,(z-z') _ I_L_L TlP J*(z-a:')p-A,(2-z')

2u?1 _J^T3e3Hx-x')e-\l(z-z') 2UJ2 _ll.-LTieJHx-x')e-*t(z-z')

Ac2\k\l

4C2|*| JtLTie}k(x-x')e-Xt{z-z')

+

2ui2

+

, J^_n(Jk(x-x')

2^2T3e

-\t(z-z')

/ 2 05)

(M0

>

The appearance of (x — x') and (z — z') in the second equation in (205) suggested the introduction of the form S[{x — x',z — z'\k), instead of S{(x, z\k). As the inequality z — z' > 0 holds true in region / , we can write z — z' = \z — z'\. Using this relation, substituting the k ^> 1—asymptotics of exp(—\i\z — z'\) and exp(—Xt\z — z'\), and reordering, and arranging the terms associated with T\ and T3 into two separate groups, we obtain:

144

Theory and Applications of Green's Functions

1011

S[(x-x',z-z'\k)

= [-!i 'I -\^^)UeM'"%^"'^ + f- l(\ L

4 cl

- \)jsign(k)\z-Z'\]e^x-x\-^z-2\3 ct

(206)

J

Identification of the terms multiplied by T\ and T3 : The term multiplied by Ti represents the k » 1—asymptotic limit of the Green's function Gn(x — x',z — z'\k) in region I (z — z' > 0). Likewise, the term multiplied by T3 is the k 3> 1—asymptotic limit of the Green's function Gi3(x — x',z — z'\k) in the aforementioned region. Denoting these limits by 5f 1 (x — x',z — z'\k) and S{3(x — x',z — z'\k), we can write:

Sl^x

- x \ z - z'\k) = [ - i ( 4 - 4 ) | z - A L

* cl

c

t

- l ( \ + 1) ±_\eM*-x')e-\k\\z-z'\ 4 xcf cf \k\l Si3(x -x',z-

z'\k) = - \ { \ 4

- \)jsign(k)\z

Cl

-

( 207a)

z'\eM—')e-\k\\z-z'\

Ct

(207b)

In order to investigate the behavior of the limits Gu(x — x', z — z'\k) and G\3(x — x',z — z'\k) in region II (z — z' < 0), we consider the term S[! (x,z\k). A similar analysis leads to the results

SU(x-X',z-z'\k)=\--(-s--3)\z-z'\

- i ^ + ^lSi]^'^"1""""'1 SU(x -x',z-

z'\k) = \ ( \ 4

C;

- \)jsign{k)\z

-

(2 8a)

°

z>\eJKx-x')e-\k\\z-z'\_

Ct

(208b)

The expressions in (207) and (208) can be unified by using the fact that \z z'\ = z - z' in region I and that \z - z'\ = -(z - z') in region II. The k > 1-asymptotic limits of the remaining Green's functions, i.e., G3i and G33, can be found analogously. Summarizing our results we can write:

145

1012

A. R.

Baghai-Wadji

Sll(x-x',z-z'\k)=\-\(±-±)\z-z'\

~l(\

S13(x-x',z-z'\k)

=~\(^

+ \)^l}eMx~X

~ ^)jsign{k)(z

Vl*"*-*'!

-

z')ejk 1—asymptotic limits of the eigenquantities. This completes our proof of demonstrating that the process of satisfying the boundary conditions and the k 3> 1—process are commutable. 4. S u m m a r y In this chapter we have shown that the governing equations in anisotropic, and transversally inhomogeneous piezoelectric materials can be diagonalized. Details regarding a newly developed symbolic notation, and a recipe for the construction of diagonalized forms have been discussed in Section 2, following a brief introduction in Section 1. Although the presentation of the diagonalized form in Section 2 is selfcontained, it remains restricted to the piezoelectric media. Further complementary discussions on the diagonalization of Maxwell's equations in anisotropic media can be found in Ref. 9. The reader is also referred to the Refs. 32 and 33 which are devoted to the diagonalization of Maxwell's equations in bi-anisotropic inhomogeneous media and Laplace's equation in the anisotropic dielectric and magnetic media, respectively. The latter forms have been developed to analysis waves and fields in large amplitude corrugated periodic structures. Furthermore, the reader may find additional applications of the diagonalized forms in Ref. 34. There, among others, the propagation of electromagnetic waves in photonic crystals with defects has been addressed. Diagonalized forms in Fourier domain represent standard or generalized algebraic eigenvalue equations, which lead to the eigenvalues and eigenvectors corresponding to the underlying differential operator. An efficient algorithm for the calculation of higher-order derivatives of eigenvalues and eigenvectors is discussed in Ref. 35.

146

Theory and Applications of Green's Functions

1013

Section 3 has been devoted to a brief discussion on Green's functions. Two methods for the construction of Green's functions in infinite and semi-infinite media have been presented. The first method, based on the inversion of the underlying differential operator, is suitable for infinite-domain Green's functions. The second method, utilizing the diagonalized forms from section 2, can be chosen to construct both the infinite doamin and the semi-infinite domain Green's functions. Several boundary value problems have been considered, as useful examples, to demonstrate the details pertaining the construction of Green's functions. Much attention has been devoted to the Green's functions associated with Laplace's operator because of their far-reaching significance. The charge neutrality condition, as a regularizing balance law, has been emphasized. These considerations are followed by two recipes for the construction of periodic Green's functions. Many useful applications of periodic Green's functions can be found in Ref. 28. Section 3 closes with a discussion on the self-action calculation which arises in the boundary element method applications. Although here the discussion on self-action analysis seems exhaustive, it only scratches the surface of this pre-eminent research topic. The material presented is an adaptation of my ideas compiled and discussed in Ref. 9. Space limitation has prevented the inclusion of any of my results have been obtained since 1995. Selected topics concerning the self-action calculations, regularization of singular surface integrals, near-field calculations around the edges, wedges, and corners can be found in my articles in the IEEE Ultrasonics Conference Proceedings published in the years 1995-2000. As a possible future research direction I would like to emphasize the construction of Green's functions-based wavelets. 10 Acknowledgments It is my privilege to thank Professor Martti Salomaa, Director of Materials Physics Laboratory at Helsinki University of Technology, for inviting me and initiating a Visiting Professorship (Oct. 1999 through Dec. 2000), which has been sponsored jointly by TEKES, a National Technology Agency, and, the Nokia Research Foundation. It is also my pleasure to extend my thanks to all the department members for excellent support and for making my stay in this distinguished pedagogical and research environment an invaluable and enriching experience. Furthermore, it is my pleasure to thank the editors Prof. Tor A. Fjeldly and Dr. Clemens C.W. Ruppel for their kind invitation to author this chapter. References

1. R.F. Harrington, Field Computation by Moment Methods, Macmillan, New York, 1968. 2. C.A. Brebbia, J.C.F. Telles, and L.C. Wrobel, Boundary Element Techniques, Springer Verlag, 1984. 3. E. Stein and W.L. Wendland (Editors), Finite Element and Boundary Element Techniques From Mathematical and Engineering Point of View, Springer Verlag, 1988. 4. N.I. Muskhelishvili, Singular Integral Equations, P. Noordhoff N.V. - Groningen Holland, 1953. 5. S.G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965. 6. R.E. Collin, Foundations for Microwave Engineering, McGraw-Hill International Editions, 1966, Electrical & Electronic Engineering Series. 7. G.F. Roach, Green's Functions, 2nd ed., Cambridge University Press, 1967.

147

1014

A. R.

Baghai-Wadji

8. I. Stackgold, Green's Functions and Boundary Value Problems, A Wiley-Interscience Series of Texts, Monographs & Tracts, J o h n Wiley & Sons, 1979. 9. A. R. Baghai-Wadji, A Unified Approach for Construction of Green's Functions. Habilitation manuscript (lecture notes), Vienna University of Technology, Vienna, 1994. 10. A. R. Baghai-Wadji, G. Walter, "Green's Function-Based Wavelets." Accepted for presentation a t t h e IEEE International Ultrasonics Symposium, San Juan, P u e r t o Rico, Oct., 2000. 11. A. R. Baghai-Wadji, Bulk Waves, Massloading, Cross-Talk, and Other Second-Order Effects in SAW-Devices (a short-course manuscript). IEEE International Ultrasonics Symposium, San Antonio, Texas, Nov., 1996. 12. A.H. Fahmy, and E.L. Adler, " P r o p a g a t i o n of Acoustic Surface Waves in Multilayers: A Matrix Description." Appl. Phys. Lett., vol. 22, pp. 495-497, 1973. 13. E.L. Adler, "Analysis of Anisotropic Multilayer Bulk-Acoustic-Wave Transducers." Electron. Lett, vol. 25, pp. 57-58, J a n . 1989. 14. E.L. Adler, " M a t r i x Methods Applied t o Acoustic Waves in Multilayers." IEEE Trans. Ultrson. Ferroelec. Freq. Contr., vol. UFFC-37, no. 6, pp. 485-490, 1990. 15. E. Langer, PhD Dissertation. Vienna University of Technology, Vienna, 1986. 16. R.F. Milsom, N.H.C. Reilly, and M. Redwood, "Analysis of Generation and Detection of Surface and Bulk Acoustic Waves by Interdigital Transducers." IEEE Trans. Sonics Ultrson., vol. SU-24, p p . 147-166, 1990. 17. A.M. Hussein, and V.M. Ristic, " T h e Evaluation of the I n p u t Admittance of SAW Interdigital Transducers." J . Appl. Phys., vol. 50, no. 7, p p . 4794-4801, July 1979. 18. K. Hashimoto, a n d M. Yamaguchi, "Precise Simulation of Surface Transverse Wave Devices by Discrete Green Function Theory." Proc. IEEE Ultrason. Symp., pp. 253258, 1994. 19. P. Ventura, J.M. Hode, and B. Lopes, "Rigorous Analysis of Finite SAW Devices with Arbitrary Electrode Geometries." Proc. IEEE Ultrason. Symp., pp. 257-262, 1995. 20. P. Ventura, J.M. Hode, and M. Solal, " A New Efficient Combined F E M and Periodic Green's Function Formalism for t h e Analysis of Periodic SAW Structures Characterization." Proc. IEEE Ultrason. Symp., p p . 263-268, 1995. 21. R.C. Peach, " A General Green Function Analysis for SAW Devices." Proc. IEEE Ultrason. Symp., pp. 221-225, 1995. 22. V.P. Plessky, and T. Thorvaldsson, "Periodic Green's Function Analysis of SAW a n d Leaky SAW Propagation in a Periodic System of Electrodes on a Piezoelectric Crystal.", IEEE Trans. Ultrson. Ferroelec. Freq. Contr., vol. UFFC-42, p p . 280-293, 1995. 23. B.A. Auld, Acoustic Fields and Waves in Solids, vol. I and II, John Wiley & Sons, 1973. 24. N.E. Glass, R. Loudon, and A. A. Maradudin, " P r o p a g a t i o n of Rayleigh Surface Waves across a Large-Amplitude Grating.", Physical Review B, vol.24, no.12, 1981. 25. A.R. Baghai-Wadji, and A.A. Maradudin, "Shear Horizontal Surface Acoustic Waves on Large Amplitude Gratings", Appl. Phys. Lett, 59 (15), 7 October 1991. 26. R.P. Kanval, Generalized Functions, Series on Mathematics in Science and Engineering, vol. 171, Academic press, 1983. 27. M. Abramowitz and LA. Stegun (editors), Handbook of Mathematical Functions, Dover Publications, Inc., New York. 28. A.R. Baghai-Wadji, H. Reichinger, H. Zidek, a n d Ch. Mecklenbrauker, "Green's Function Applications in SAW Devices." Proc. IEEE Ultrason. Symp., p p . 11-20, 1991. 29. J.L. Bleustein, Appl. Phys. Lett., 13, 412, 1968. 30. Yu.V. Gulyaev, Pisma v Z h T F , 9, 63, 1969. 31. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 1980.

148

Theory and Applications of Green's Functions

1015

32. B. Jakoby, and A.R. Baghai-Wadji, "Analysis of Bianisotropic Layered Structures with Laterally Periodic Inhomogeneities - an Eigenoperator Formulation." IEEE Trans. Antenn. Propag., vol. AP-44, no. 5, pp. 615-621, May 1996. 33. M.T. Manzuri-Shalmani, A.R. Baghai-Wadji, and A.A. Maradudin, "Noise-free Static Field Calculations in Corrugated Periodic Structures." Proc. IEEE-AP, Antenn. Propag. Symp., pp. 1089-1092, 1993. 34. A. R. Baghai-Wadji, Photonic Crystals, lecture notes, Helsinki University of Technology, Helsinki, Fall 2000. 35. S. Ramberger, and A.R. Baghai-Wadji, "Calculation of Higher-order Derivatives of Eigenvalues and Eigenvectors." (In preparation), IEEE Trans. Antenn. Propag.

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International Journal of High Speed Electronics and Systems, Vol. 10, No. 4 (2000) 1017-1068 © World Scientific Publishing Company

NEW PIEZOELECTRIC SUBSTRATES FOR SAW DEVICES JOHN A. KOSINSKI U.S. Army Communications-Electronics Command, AMSEL-RD-IW-S, Fort Monmouth, NJ 07703-5211, USA Recent developments in single crystal piezoelectric materials have focused on the search for "ideal" materials with zero temperature coefficient of frequency orientations featuring jointly high piezoelectric coupling, high intrinsic Q, zero power flow angle, and minimized diffraction effects. In addition, the desired materials should have no low temperature phase transitions, and a physical chemistry conducive to repeatable, low cost growth and wafer scale device production. As difficult as it might seem to find such "ideal" materials, three completely different but strong candidate materials have emerged recently: the quartz homeotype gallium orthophosphate, the quartz isotype calcium gallogermanates (langasite, langanite, langatate, etc.), and diomignite (lithium tetraborate). The current state-of-the-art and prospects for future development of these materials are considered.

1. Introduction To date, single crystal quartz (oc-Si02) and lithium niobate (LiNb03) are the most widely used single crystal piezoelectric substrates for SAW devices. Each of these materials has certain properties that make it attractive for specific applications. In the case of quartz, the most interesting features are the zero temperature coefficient of frequency jointly with zero power flow angle, high Q, and non-zero piezoelectric coupling for the ST-cut; these characteristics lead to widespread use in narrowband filters and precision resonators in the VHF and UHF range. In the case of lithium niobate, the most interesting feature is large piezoelectric coupling leading to widespread use in broadband and low insertion loss filters. However, the lack of a zero temperature coefficient of frequency is a serious limitation for the use of lithium niobate. Recent developments in single crystal piezoelectric materials have focused on the search for and development of "ideal" materials. Traditionally, "ideal" materials were considered to be those featuring jointly high piezoelectric coupling, zero temperature coefficient of frequency orientations, and high intrinsic Q. However, in the context of substrates for SAW devices, additional criteria have been imposed. The zero temperature coefficient of frequency orientations must also feature zero power flow angle jointly with minimized diffraction effects, no low temperature phase transitions, and the physical chemistry of the material must admit repeatable, low cost growth and wafer scale device production. As difficult as it might seem to find such "ideal" materials, three completely different but strong candidate materials have emerged recently: the quartz homeotype gallium orthophosphate, the quartz isotype calcium gallo-germanates (langasite, langanite, langatate, etc.), and diomignite (lithium tetraborate). Gallium orthophosphate and the calcium gallo-germanates both belong to trigonal symmetry class 32, and hence have much in common with ct-quartz. Diomignite belongs to tetragonal symmetry class 4mm as do certain piezoelectric ceramics. The current state-of-the-art and prospects for future development of these materials are considered. 151

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Kosinski

2. Quartz Homeotypes - Gallium Orthophosphate 2.1. General comments Alpha-quartz is unquestionably the most successful single crystal piezoelectric material, with nearly "ideal" characteristics in many respects. It follows naturally that quartz homeotypes should be investigated for similar or potentially superior characteristics, and in fact it has been found that berlinite (a-AlP0 4 ) demonstrates superior piezoelectric coupling as compared to quartz. Unfortunately, extreme difficulty has been experienced in growing high quality berlinite crystals of commercially viable size. Recently, gallium orthophosphate (oc-GaP04) has been proposed as another quartz homeotype of interest. This new material belongs to the same family of M 3+ X 5+ 0 4 crystals as berlinite, constructed by the alternate replacement of half of the silicon atoms by trivalent gallium and the other half by pentavalent phosphorous atoms.1 Preliminary results indicate similar good characteristics for gallium orthophosphate, with the coupling coefficient of the GaP0 4 AT-cut larger than that of berlinite, and approximately twice that of quartz.2 Further, gallium orthophosphate demonstrates a superior thermal stability, transitioning directly to a p-cristobalite form at 933°C as compared to the o>P phase transition near 580°C for quartz and berlinite.3 2.2. Crystallography As a quartz homeotype, gallium orthophosphate belongs to symmetry class 32, characterized by a single three-fold symmetry axis and three equivalent two-fold symmetry axes as illustrated in Fig. 1.4 The three-fold axis is also a screw axis, leading to right- and left-handed enantiomorphs belonging to space groups P3i21 and P3221 respectively,1 hence both electrical (Duaphine) and optical (Brazil) twins are possible. As noted, the crystal structure is similar to that of a-quartz,5 with half of the silicon atoms replaced by trivalent galjium and the other half replaced by pentavalent phosphorous atoms. Consequently, the unit cell extent along the c-axis is twice that of quartz. Measured values of lattice constants are listed in Table 1. There are three formula weight per unit cell. Analysis of the data near room temperature lead to values of a=4.901+0.003A and c=11.046±0.008A at 25°C corresponding to an x-ray density of 3571±7 kg/m3 which is consistent with the previously reported values as listed in Table 2. Thermal expansion data are presented in Table 3. Published data on the thermal stability of the a-phase of gallium orthophosphate are listed in Table 4. There is substantial variation in the reported phase-transition temperature data. The phase relations in gallium orthophosphate are illustrated in Fig. 2.12 The behavior of gallium orthophosphate is distinctly different than that of quartz and berlinite. The material transitions directly from the a-phase to a P-cristobalite form at 933°C, whereas quartz and berlinite undergo an intermediate a-p phase transition near 580°C.3 In consequence, gallium orthophosphate devices may be processed or operated at significantly higher temperatures than comparable quartz or berlinite devices.

152

New Piezoelectric Substrates for SAW Devices

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Fig. 1. Class 32 symmetry elements and matrices for equilibrium properties [Nye, 1960].

Table 1. GaP04 Lattice Constants. a (A) 4.874±0.002 4.899+0.001 4.901±0.001 4.897+0.001 4.934±0.002 4.973+0.002 4.90 4.905 4.901±0.003

c(A) 11.033±0.004 11.034±0.002 11.048+0.001 11.021+0.002 11.075+0.005 11.105+0.006 11.05 11.050 11.046+0.008

To(°C) -100 20 20 100 500 750 r.t.

— 25

Reference 6 7 1 7 1 1 8 9 this work

Table 2. GaP0 4 Mass Density. Mass Density (kg/mJ) 356x 357x 3570 361x 358x 3570 3571+7

Method measured x-ray

To(°C) r.t. r.t.

—

—

x-ray x-ray

-100 20 25 25

— x-ray

Reference 8 8 1 7 7 10 this work

Table 3. GaP0 4 Thermal Expansion.

aj 1 / (ppm/°C)

a f f (ppb/°C2)

a^

17.9 5.3 10.52 9.02 10.15

—

4.6 1.2 1.70 3.38 3.34

35.4 35.4 14.58

(ppm/°C)

a g } (ppb/°C2)'

T„(°C)

Reference

....

....

11 12 13 14 15

2.0 2.0 2.52

153

r.t. 25 27

1020

J. A. Kosinski

P-phase P6222

thermal decomposition >1327°C

(1687 °C) P-cristobalite F43m 933 °C 578 °C 533 °C a-phase P3t21

a-cristobalite C222!

Fig. 2. Phase relations in gallium orthophosphate.12

Table 4. GaP04 Critical Temperatures. Transition Point (°C) 1077 1000 48 4.2. Properties of Solutions in Layered Structures This subsection discusses properties of the layered pseudo SAWs solutions which have an impact on both practical devices design and in the fundamental understanding of the pseudo modes. Figures 17 and 18 show the phase velocity and attenuation as a function of the thickness times frequency, hf, for two aluminum layered orientations: symmetry Type 3 Li2B407 (Euler angles: [0° 47.3° 90°]), Fig. 17; and symmetry Type 1 LiNb03 (Euler angles: [90° 90° 164°]), Fig. 18. The upper part of Fig. 17 shows velocity plots for the HVPSAW, three lossless pure sagittal particle motion Rayleigh modes, and the phase velocity values of the three BAW modes. The HVPSAW propagation attenuation is plotted in the lower part of Fig. 17. Note that since this orientation refers to a symmetry Type 3 there is no PSAW. The upper part of Fig. 18 shows velocity plots for the HVPSAW, the PSAW and a higher order PSAW, five generalized SAW modes and the phase velocity values of the three BAW modes along that orientation. The lower part of Fig. 18 plots the propagation loss for the HVPSAW, the PSAW and a higher order PSAW mode. Note from Fig. 17 that a minimum in the propagation loss occurs close to hf=0.2 Km/s (vp=6.7413 Km/s), h/^=3%, a reduction of more than two orders of magnitude when compared to the mechanically free electrically shorted surface. This is a relevant

236

Pseudo and High Velocity Pseudo SAWs

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Fig. 17. Layered substrate: phase velocity (upper part) and attenuation (lower part) versus (thickness x frequency). Aluminum layered symmetry Type 3 Li 2 B 4 0 7 (Euler angles: [0° 47.3° 90°]. HVPSAW: dash-dot; lossless Rayleigh modes (pure sagittal particle motion): Solid (first and second), and star (third); larger-dots: BAW phase velocities.

237

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Fig. 18. Layered substrate: phase velocity (upper parts) and attenuation (lower parts) versus (thickness x frequency). Aluminum layered symmetry Type 1 LiNbOj (Euler angles: [90° 90° 164°]). HVPSAW: dash-dot; higher order PSAW: circles; PSAW: dashed; first to fourth GSAW: solid; fifth GSAW: star; largerdots: BAW phase velocities.

238

Pseudo and High Velocity Pseudo SAWs

1105

HVPSAW characteristic that can be advantageously used in device design and fabrication. Such behavior has also been observed in PSAW orientations, like quartz ST25° (Euler angles: [0° 132.75° 25°]), 36° YX LiTa0 3 (Euler angles: [0° -54° 0°]), and 64° YX LiNb0 3 (Euler angles: [0° -26° 0°]).4'25' 26 ' 49 ' 50 For 0.2CN

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(b) ,6,20 03

< 0.8

1

1.2

Normalized Frequency Fig. 23.

Admittance characteristics of the 5 GHz-range filter using conventional

bidirectional IDT. (a) experimental, (b) calculated results.

269

1136

H. Odagawa & K.

Yamanouchi

experimental results. First, the susceptance B s , which represents phase shifts caused by the energy storage effect and the piezoelectric shorting, 20 is determined as the center frequency is fitted to experimental results, and then the normalized acoustic impedances of the electrodes (Zm) is determined as the admittance characteristics are fitted to the experimental results. Figure 23(b) shows the calculation results. In this case, Bs is 0.16 and Zm is 1.064. Figure 24 shows the structure of the ladder type filter. It has 6 SAW resonators which have reflectors of 20 strips with widths of X/4 at both sides of the resonators. However, the effects of the number of gratings are small because the reflection coefficients of the electrodes are large. The parameters are shown in Table 3. The ratio of the static capacitance of resonator A to that of resonator B is 1.7. The frequency response calculated using the obtained equivalent circuit parameters is shown in Fig. 25.

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Port 2 (OUT) o /

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A

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A

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Table 3. Parameters of the ladder type filter in 10 GHz.

Pair number

Aperture

Period (wavelength)

Resonator A

120.5

11.8 XP

0.38 urn

Resonator B

100.5

lO.OXs

0.37 urn

270

SAW Devices Beyond 5 GHz

1137

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1.00 1.05 Normalized Frequency

1.10

Calculated frequency response of the ladder type filter.

The micrograph of the electrodes in resonator A observed by SEM was shown in Fig. 11 in Section 4.2. The electrode width is 95 nm, and the thickness of the Al film is 30 nm (0.079k). They are fabricated on 128° Y-X LiNb0 3 using the electron beam exposure system and the lift-off process. The resolution of the EB system is 40 to 50 nm and the minimum step of the beam scan is 10 nm in the setting of this experiments. Because of the limitation of the scan step, we use the structure shown in Fig. 26 (b) as the resonator B for the purpose of adjusting the resonant frequency of resonator B to the antiresonant frequency of resonator A. Figure 27 shows the frequency responses of the 10 GHz-ladder type filters. They are measured by microwave probes. A low loss characteristic with a minimum insertion loss of 3.4 dB is obtained as shown in Fig. 27(a). Figure 27(b) shows the characteristic of another filter. A considerable reduction by 25 dB of the electromagnetic feed-through is obtained in the 17 GHz range. The reduction characteristics in the high frequency range largely depend on the structure of the connection electrodes and the structure of the bonding pads. It is necessary to optimize the structures. These results show the feasibility of SAW devices for communication systems in the 10 GHz-range.

271

1138

H. Odagawa & K.

Yamanouchi

(a)

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