Surface Acoustic Wave Devices and Their Signal Processing Applications

By Colin Campbell

Surface Acoustic Wave Devices and Their Signal Processing Applications is a textbook that combines experiment and theory in assessing the signal processing applications of surface acoustic wave (SAW) devices. The operating principles of SAW devices are described from a circuit design viewpoint. This book is comprised of 18 chapters and begins with a historical background on surface acoustic waves and a discussion on the merits of SAW devices as well as their applications. The next chapter introduces the reader to the basics of acoustic waves and piezoelectricity, together with the effect of acoustic bulk waves on the performance of SAW filters. The principles of linear phase SAW filter design and equivalent circuit models for a SAW filter are then described. The remaining chapters focus on trade-offs in linear phase SAW filter design; compensation for second-order effects; harmonic SAW delay lines for gigahertz frequencies; and coding techniques using linear SAW transducers. The final chapter highlights Some other significant alternative design techniques and applications for SAW devices. This monograph will be suitable for engineering or physics students as well as engineers, scientists, and technical staff in industry who seek further information on SAW-based circuits, systems, and applications.

• Language: English
• Publisher: Academic Press
• Release date: Dec 2, 2012
• ISBN: 9780323148665

Book preview

Ian

Preface

The suggestion for writing this book came from my long-time friend, Professor Colin diCenzo, FIEEE, to whom I owe many thanks for his encouragement. The outline of the book was started many years ago, as a somewhat sparse series of lecture notes that I used in support of an electrical engineering graduate course at McMaster University. The volume of material expanded rapidly after I started to give lectures on this subject to diverse audiences, through the Continuing Engineering Education division of The George Washington University, in Washington, DC. Mr Cliff Hopkins and Mr Chip Blouin of GWU gave me much support in preparing and presenting these lectures.

I have tried to give this textbook a blend of experiment and theory—in keeping with my own traditional engineering training in Scotland, begun in 1942. Where possible, I have tried to simplify the mathematical coverage while retaining a physical understanding of processes involved. In this way, it is hoped that the material will be suitable for students, as well as engineers, scientists and technical staff in industry, who wish to embark on this subject or gain a greater depth of understanding of SAW devices and applications. In aid of this approach, I have dispensed with the normal procedure of including sets of problems with the subject material. Instead, worked examples are given in support of the design concepts and parameters involved. I also appreciate the illustrative material given with reproduction permission by Andersen Laboratories, Bloomfield, Connecticut; Crystal Technology Inc, Palo Alto, California; Institute of Electrical and Electronic Engineers, New York; SAWTEK Inc, Orlando, Florida; and Hewlett Packard, Toronto, Canada, as well as by Dr. J. H. McClellan, Dr. Peter Smith and Ms Suet Yuen.

In an attempt to keep the book within reasonable size while introducing subject material not covered in other texts, some topics have been omitted, while others have been simplified. For example, a detailed study of piezoelectricity and crystallography is not given here. As well, coverage of thin-film SAW devices has not been included. It is hoped, however, that the depth of coverage in this text will enable the reader to embark readily on further studies of these subjects as required.

I would like to express my deep appreciation to all of the former electrical engineering graduate students of McMaster University who shared this SAW experience with me. These are Mr. Robert Amorosi, Mr. Craig Bailey, Mr. Tony Chu, Mr. Peter Edmonson, Dr. Waguih Ishak, Dr. Pala Nanayakkara, Mr. Patrick Naraine, Dr. Jim Reilly, Dr. Dieter Seiler, Mr. Nick Slater, Dr. Peter Smith, Dr. Mark Suthers, Dr. John Choo Saw, Mr. Joe Sferrazza Papa and Ms Suet Yuen. It was a fun time!

I am indebted to my former mentor at the Massachusetts Institute of Technology, Professor Francis Reintjes, FIEEE, for highlighting my student days there by introducing me to electronic correlation processors in 1951.1 am also indebted to Professor Jack Allen, FRS, of St. Andrews University, Scotland, for guiding me as a graduate student along the many paths of experimental physics.

Most of all, I could not have survived the many years of preparing this book without the understanding and support of my wife Vivian. Thank you, Vivian. Now we can relax.

Colin Campbell,     Ancaster, Ontario, Canada

January 1989

1

Introduction

Publisher Summary

This chapter outlines the operating principles of surface acoustic wave (SAW) devices from the viewpoint of a circuit design. Devices and systems based on SAW technology have several excellent features when compared to competing technologies. SAW devices can generally be designed to provide quite complex signal processing functions within a single package. SAW devices can be mass produced using semiconductor micro-fabrication techniques. As these can be implemented in small, rugged, light, and power-efficient modules, these devices are finding ever-increasing application in mobile and space-borne communications systems. Though SAW devices are analog devices, they can be employed in many digital communications systems. Moreover, SAW filters can be made to operate very efficiently at high-harmonic modes. The signal processing and frequency response characteristics of a SAW device on a piezoelectric substrate are primarily governed by the geometry of the metal-film interdigital transducers (IDTs) deposited on the substrate. The transducers and device structures can be grouped into four general categories. The first three categories relate to the SAW devices on piezoelectric substrates where the signal levels are small enough for the acoustoelectric interactions to be considered linear. The fourth category relates to the SAW devices that utilize the weak nonlinear response of the piezoelectric under high signal level excitation.

1.1 Historical Background

Surface acoustic waves can be generated at the free surface of an elastic solid. This phenomenon has been exploited for electronic analog signal processing over the past 20 years, with the development of a host of devices and systems for consumer, commercial and military applications running at a multimillion dollar annual rate. While this is a comparatively new electronic technology, it has its roots in scientific findings that date back over the past 100 years. Indeed, a mathematical discussion on the propagation of surface acoustic waves at the free surface of a homogeneous isotropic elastic solid was first reported by Lord Rayleigh in an address to the London Mathematical Society on 12 November 1855 [1]. His paper begins with the statement … It is proposed to investigate the behaviour of waves upon the plane free surface of an infinite homogeneous isotropic elastic solid, their character being such that the disturbance is confined to a superficial region, of thickness comparable with the wavelength… It concludes with the observation … it is not improbable that the surface waves investigated play an important part in earthquakes, and in the collision of elastic solids. Diverging in two dimensions only, they must acquire at a great distance from the source a continually increasing preponderance … The presence of these surface acoustic waves (also known as Rayleigh waves) in earthquake shocks was later verified using seismograph recording techniques.

It was not until 1965, however, that the phenomenon of surface acoustic wave (SAW) propagation was first exploited for its applications to electronic devices. This was with the invention of the interdigital transducer (IDT) by White and Voltmer at the University of California [2] as a most efficient technique for the generation and detection of SAW waves on a piezoelectric surface. Before this major discovery, applications of acoustic wave processes to solid-state electronic device technology had been principally restricted to piezoelectric transducers employing acoustic bulk wave phenomena (such as conventional quartz crystals for oscillator circuit design). The advent of the IDT opened up the gateway to a new and most versatile approach to the design of analog electrical filters operating at selected frequencies in the range from about 10 MHz to 1 GHz or above. This immediately generated two major electrical engineering product design challenges, with quite divergent thrusts. At one end of the scale, in the high-volume, low-cost TV component market, the challenge related to whether or not mass-produced SAW filters could be competitive in price and performance with inductance-capacitance (LC) filters currently employed in the intermediate-frequency (IF) circuit stages. At the opposite extreme, relating to low-volume, high-cost components for radar signal-processing, maximum emphasis was given to the efficient implementation of SAW pulse compression filters with very large compression gains. Between these extremes, a wide range of other SAW device configurations and applications started to receive intensive research scrutiny.

Initially, principal research activities in North America in the late 1960s relating to SAW device applications were limited to a few centers such as at Stanford University, Texas Instruments Inc., Hughes Aircraft Company and Lincoln Laboratory of the Massachusetts Institute of Technology. At the same time in England, the Royal Radar Establishment was actively involved in radar systems implementation of this technology. Only a few more years passed, however, before a host of other research laboratories in industry, government and universities around the world became involved in SAW device research.

The extent of this rapid expansion of SAW device research activity can perhaps be highlighted by a listing of some of the participating laboratories publishing research papers on SAW devices within a 10-year period following White and Voltmer’s introduction of the interdigital transducer. By that time published research papers from industrial and government laboratories in North America included (in alphabetical order) those from Air Force Cambridge Research Laboratories, Anderson Laboratories Inc., Bell Laboratories, Hazeltine Corporation, Hughes Aircraft Company, Lincoln Laboratory at the Massachusetts Institute of Technology, Magnavox Company, Motorola Inc., Naval Undersea Center, Raytheon Research

Division, Rockwell International, Sperry Research Center, Tektronix Inc., Texas Instrument Inc., Westinghouse Electric Corporation and Zenith Radio Corporation. Universities in North America also publishing SAW research papers within this period included those of California, Carnegie-Mellon, McGill, Northwestern, Illinois, Pennsylvania, Purdue, Rensselaer Polytechnic, Stanford and Toronto.

In Britain, leading laboratories involved in SAW research at an early date included The General Electric Company, Mullard Research Laboratories, The Plessey Company, The Royal Radar Establishment (now the Royal Signals and Radar Establishment) and Standard Telecommunications Laboratories Limited, together with the University of Edinburgh, Queen Mary College and University College, London. In Norway, pioneering contributions to SAW research were emanating from the Norwegian Institute of Technology. In France, SAW research was well under way at centers such as Thomson-CSF and Centre National d’Etudes des Telecommunications, while in West Germany research activities were reported by the Institut fur Angewandte Festkorper-Physik. In Italy, SAW signal processing studies were conducted at the Istituto di Ricerca sulle Onde Elettromagnetiche CNR in Florence. Japanese laboratories with early publications of SAW research activities included Matsushita Research Institute, Tokyo Institute of Technology, Nippon Electric Company and Tohoku University.

This intensive research activity had rapid results in terms of applications to the consumer, commercial and military markets. In 1977, Wiliamson [3] listed 45 different types of SAW devices that had received development effort by that date, with varying degrees of success, including 10 major devices with exceptional performance that had already received widespread application. Developmental successes continued so that by 1985 (only 20 years after the introduction of the IDT), Hartmann [4] listed nine major consumer applications, nine major commercial applications and 18 major military applications of the technology.

1.2 Merits of Saw Devices

As noted by Hartmann in a review paper on the systems impact of modern SAW device technology [4], many SAW-based devices and systems have several excellent features when compared to competing technologies. These include the following:

1. SAW devices can generally be designed to provide quite complex signal processing functions within a single package containing but a single piezoelectric substrate with superimposed thin-film input and output interdigital transducers. Thus, for example, SAW bandpass filters with outstanding response characteristics can now be routinely designed to achieve responses that would require several hundred inductors and capacitors in conventional LC-filter designs.

2. SAW devices can be mass produced using semiconductor microfabrication techniques. As a result, they can be made to be cost competitive in mass-volume applications, with some products selling for less than $1.00.

3. SAW devices can have outstanding reproducibility in performance, from device to device. This is especially desirable for the design and implementation of channelized receivers for spectral analyses of signals in electronic support measures (ESM). Indeed, the practical implementation of such channelized receivers has only been made possible since the arrival of SAW devices.

4. Since they can often be implemented in small, rugged, light and power-efficient modules, they are finding ever-increasing application in mobile and space-borne communications systems.

5. Although SAW devices are analog devices, they can be employed in many digital communications systems. One example of this is in the use of SAW Nyquist filters in quadrature-amplitude-modulation (QAM) digital radio modems.

6. SAW filters can be made to operate very efficiently at high-harmonic modes [5]. As a result, gigahertz frequency devices can be fabricated using relatively inexpensive photolithographic techniques, rather than the significantly more expensive processes involving electron-beam (E-beam) lithography.

1.3 Outline of SAW Device Applications

The signal processing and frequency response characteristics of a SAW device on a piezoelectric substrate are primarily governed by the geometry of the metal-film interdigital transducers (IDTs) deposited on the substrate. In the period since 1965, when White and Voltmer demonstrated the basic SAW IDT structure of Fig. 1.1, a highly varied (and often bewildering looking) succession of IDT geometries has evolved to cater for a multitude of signal processing functions. Despite this variety, however, the transducers and device structures can be grouped under four general categories, as listed in Table 1.1.

TABLE 1.1

Categories Used for Initial Grouping of Saw Devices

Figure 1.1 Basic SAW delay line fabricated on a piezoelectric substrate. Employs metal thin-film input and output interdigital transducers. Finger period at center frequency f0 corresponds to acoustic wavelength λ0 = v/f0, where v = SAW velocity.

The first three categories relate to SAW devices on piezoelectric substrates, where signal levels are small enough for the acoustoelectric interactions to be considered linear. The fourth category relates to SAW devices that utilize the weak nonlinear response of the piezoelectric under high signal level excitation.

Two design parameters employed in establishing the groupings of Table 1.1 relate to filter insertion loss (IL) and filter fractional bandwidth (BW%). The insertion loss (IL) of a SAW device in decibels (dB) is defined here as

(1.1)

(A more standard definition of insertion loss will be employed later.) The insertion loss values quoted in Table 1.1 are given as typical of representative designs. Likewise, filter fractional bandwidth BW% is defined here as

(1.2)

In eqn (1.2), f0 is the center frequency of the transducer response, while Δf represents the bandwidth between defined amplitude levels about f0. Here, Δf may be designated as measured at the 1 dB, 3 dB or 4 dB amplitude response levels relative to midband.

Table 1.2 subsequently lists a variety of SAW device applications under each of the categories specified for Table 1.1.

TABLE 1.2

Some Signal Processing Applications of Saw Devices

1.4 Aims of This Text

This text was written with a number of objectives in mind. First and foremost, it is intended to provide a treatment of the operating principles of SAW devices from a circuit design viewpoint. In this way, it is hoped that the coverage will be suitable for engineering or physics students who wish to gain both a theoretical and experimental understanding of SAW circuit design principles, as well as engineeers, scientists and support staff in industry who seek further information on SAW-based circuits, systems and applications. To this end, an attempt has been made to provide a balance between the theoretical and experimental coverage. Where appropriate, systems-oriented worked examples are given as an aid to familiarizing the reader with practical SAW device design parameters and constraints.

1.5 References

1. Rayleigh, Lord. On waves propagating along the plane surface of an elastic solid,. Proc. London Math. Soc. November 1885; 7:7–11.

2. White, R. M., Voltmer, F. W. Direct piezoelectric coupling to surface elastic waves,. Appl. Phys. Lett. 1965; 17:17–316.

3. R. C. Williamson, Case studies of successful surface-acoustic-wave devices, Proc. 1977 IEEE Ultrasonics Symposium, Oct. 26–28, 1977, Phoenix, AZ, IEEE Cat. No. 77CH1264–1SU, pp. 460–468, 1977.

4. Hartmann, C. S. System impact of modern Rayleigh wave technology,. In: Ash E.A., Paige E.G.S., eds. Rayleigh-Wave Theory and Application. New York: Springer-Verlag; 1985:238–253.

5. Smith, W. R. Basics of the SAW interdigital transducer,. In: Collins J.H., Masotti L., eds. Computer-Aided Design of Surface Acoustic Wave Devices. New York: Elsevier; 1976:25–63.

2

Basics of Acoustic Waves and Piezoelectricity

Publisher Summary

This chapter highlights the basic properties of surface acoustic waves and their generation by an interdigital transducer (IDT) located on the free surface of a piezoelectric substrate. The excitation of an IDT on a piezoelectric substrate can lead to the generation of bulk acoustic waves as well as surface waves. Bulk waves incident on the receiving IDT induce voltages. These interfere with the voltage signals due to received surface waves. The resultant voltage due to both sources can degrade the in-band performance specifications on amplitude, phase, or group delay response. Surface acoustic waves can be generated at the free surface of an elastic solid. In the SAW devices, the generation of such waves is achieved by the application of a voltage to a metal-film interdigital transducer (IDT) placed on the surface of a piezoelectric solid. Bulk acoustic waves can be characterized in terms of three modal types of mechanical excitation. One of these is a compressional wave, termed the longitudinal bulk wave, that is polarized in the direction of the acoustic wave propagation vector. The other two wave components, termed transverse or shear waves, have their vibrational modes. The chapter also outlines the acoustic bulk wave excitation by an IDT as a highly undesirable effect in SAW filter design that can cause severe degradation of filter performance and can also limit the fractional bandwidth obtainable with normal IDT structures.

2.1 Introduction

This chapter highlights the basic properties of surface acoustic waves and their generation (or detection) by an interdigital transducer (IDT) located on the free surface of a piezoelectric substrate. It omits a detailed mathematical treatment of acoustic wave propogation in piezoelectrics, as this may already be found in a number of texts [1]–[7]. Here, the emphasis is given on conveying an understanding of principles to readers interested in or involved with communications circuits and systems. A mathematical outline of stress and strain relations in piezolectric solids is given in order to relate to the materials aspects of SAW filter design. Sections of Chapter 2 dealing with this subject (as marked with an asterisk *) can be bypassed by the reader without prejudice to the remaining text coverage.

Unfortunately, the excitation of an IDT on a piezoelectric can lead to the generation of bulk acoustic waves as well as surface waves. The presence of the former in SAW filter structures is an undesirable one which, unless controlled, can often seriously degrade the desired filter response. Bulk waves incident on the receiving IDT will also induce voltages. These interfere with the voltage signals due to received surface waves. The resultant voltage due to both sources can degrade the in-band performance specifications on amplitude, phase or group delay response. In addition, the out-of-band rejection can be undesirably reduced.

Bulk acoustic wave interference in SAW filters may be regarded as a second-order effect giving rise to response degradation. While such interference represents but one of a number of second-order effects that can occur in these devices, it is instructive to consider this separately in this chapter.

2.2 Surface Acoustic Waves

2.2.1 Excitation Requirements

Surface acoustic waves can be generated at the free surface of an elastic solid. In the SAW devices considered in this text, the generation of such waves is achieved by application of a voltage to a metal film interdigital transducer (IDT) placed on the surface of a piezoelectric solid. Two IDTs are then required in the basic SAW device configuration sketched in Fig. 2.1. One of these acts as the device input and converts signal voltage variations into mechanical surface acoustic waves. The other IDT is employed as an output receiver to convert mechanical SAW vibrations back into output voltages. Such energy conversions require the IDTs to be used in conjunction with elastic surfaces that are also piezoelectric ones. Note, however, that the surface outside the IDT regions need only be elastic, without being piezoelectric.

Figure 2.1 Outline of a basic SAW filter on a piezoelectric substrate. Note the wax absorbers often used in such designs to suppress spurious SAW transmissions due to the bidirectionality of the IDTs illustrated here.

Input and output IDTs may be likened to electromagnetic transmitting and receiving antennas in that the principle of reciprocity applies to both systems. As a result, signal voltages can be applied to either IDT with the same end result. The aim of all this, of course, is to effect some advantageous signal-processing function through the interaction of acoustic waves, rather than through electromagnetic ones, while enjoying the compact device dimensions attainable with such processing.

Detailed mathematical treatments of SAW propagation on the surface of an unbounded piezoelectric elastic surface are to be found in a number of textbooks [1]–[6], [14]. Only the essentials of these will be considered here in order to highlight two most important practical properties relating to surface wave propagation on a piezoelectric substrate. These are the SAW velocity v and the electromechanical coupling coefficient K² of the piezoelectric.

2.2.2 Mechanical Motion of Surface Acoustic Waves

It may first be noted that the propagation of a SAW wave on an unbounded elastic surface is associated mechanically with a time-dependent elliptical displacement of the surface structure. One component of this physical displacement is parallel to the SAW propagation axis, while the other is normal to the surface. This is depicted in Fig. 2.2, where x relates to the SAW propagation axis, while y is a surface normal axis in a Cartesian coordinate system. (Coordinate notation x, y, z used here should not be confused with piezoelectric crystal axis notation X, Y, Z). Surface particle motion is then predominantly in the y-z-plane in Fig. 2.2. (This is only strictly true for certain crystal cuts, such as Y-cut Z-propagating lithium niobate). The two wave motions are 90° out of phase with one another in the time domain so that when one displacement component is maximum at a given instant the other will be zero. This has ramifications in SAW resonator design, considered later in the text. In addition, the amplitude of the surface displacement along the y-axis is larger than that along the SAW propagation axis x. This can be appreciated intuitively, since it is easier for the crystal structure to vibrate in the unbounded direction than in the bounded one. The amplitudes of both of these SAW displacement components become negligible for penetration depths y greater than a few acoustic wavelengths λ (= v/f) into the body of the solid. (This phenomenon may be considered to be somewhat analogous to that of skin depth, relating electromagnetic wave penetration into a conductor.)

Figure 2.2 Pictorial representation (not to scale) of surface acoustic wave motion on the surface of an elastic solid. Although the illustration relates to a piezoelectric solid, this is not a requirement for SAW propagation.

Example 2.1

A surface acoustic wave is generated on the surface of a piezoelectric YZ-lithium niobate substrate by means of an ac voltage applied to an IDT at a synchronous frequency of 1 GHz. (a) Given that the velocity of propagation of the SAW on this material is v = 3488 m/s, determine the acoustic wavelength λ. (b) Compare the value of this acoustic wavelength with that associated with an electromagnetic (e-m) wave propagating in free space at the same signal frequency. (c) Determine the ratio between the SAW wavelength and the e-m wavelength in this case.

Solution: (a) The SAW wavelength λ is given by λ = v/f = 3488/(1 × 10⁹) = 3.488 × 10−6 m = 3.488 μm, where 1 μm = 1 micron = 10−6 m. (b) The electromagnetic wavelength λe in this case is λe = c/f where c 1.1 × 10−5.

*2.2.3 Stress and Strain in a Nonpiezoelectric Elastic Solid

First of all consider the relations between mechanical stress T and strain S for small static deformations of a nonpiezoelectric elastic solid. Stress is just the force F applied per unit area A of the solid. Moreover, stress, force and area can all be represented as vector quantities (using bold face letter symbols) so that T = F/A. The units of T are N/m² when force F is expressed in newtons (N). In addition, the strain parameter S, which represents the fractional deformation due to force F, can be defined as S = Δ/L (dimensionless), where Δ is the fractional deformation of the solid of length L.

Stresses and strains exerted within an elastic solid can exist in compressional or shear form. With compressional stresses, for example, the applied force F is normal to the surface area A in Fig. 2.3. In either case they can be related proportionally by Hooke’s Law for elastic deformations. For simple compressional stress and strain along the same axis, this can be written as

Figure 2.3 Notation used here for parameters relating to deformation of an elastic solid. Force and area parameters can be vectors.

(2.1)

where c = elastic stiffness coefficient, also known as Young’s modulus (N/m²). In general, however, eqn (2.1) must be formulated to accommodate all possible components of stress and deformation so that

(2.2)

expressed as a tensor equation (with tensor terms identified by symbols{ }).

Thus, for example, an expansion of tensor equation eqn (2.2) to get the value of Txx along the x-axis would yield terms

(2.3)

Since force and area vectors need not be aligned, the stress parameter T can be written as Tjk = Fj/Ak, where the first subscript j denotes the direction of the force F while second subscript k is the direction of the vector representing area A in Fig. 2.3. Likewise, tensor strain terms can be given as Slm = Δ1/Lm for deformation directions defined by the second two suffices. Here, {c} is referred to as a fourth-rank tensor, since its components have four suffices Cjklm. Similarly {T} and {S} are classed as second-rank tensors. Where j = k = l = m is specified along one axis (say the x-axis for example), Txx = cxxxxSxx relates compressional stress and strain along that axis.

We will not dwell long on the concept of tensors except to identify with the crystal classifications of various SAW piezoelectrics. Tensors are used to relate parameters that are dependent on more than one coordinate axis set (e.g., Txy) to those measured in another (e.g., Syz). Tensor equation (2.2) can be reduced to a matrix equation [T] = [c][S] (denoted by symbols [ ]), by redimensioning {T} and {S} so that they each have only one suffix instead of two as in eqn (2.3). To this end, tensor components of T and S can be reduced to matrix components given by

(2.4)

and

(2.5)

In this way, the elastic stiffness constant is reduced to a 6 × 6 matrix [c] with 36 possible independent values relating the six (reduced) components of stress to the six (reduced) components of strain. Moreover, from energy and symmetry considerations, these 36 possible independent terms can be reduced to a maximum of 21 for the most general crystal symmetry examples. A further reduction is made possible by an appropriate choice of reference coordinate axes in relation to crystal axes. For example, cubic crystals with coordinate reference axes x, y, z chosen parallel to crystal axes X, Y and Z have their number of independent elastic constant coefficient terms in [c] reduced from 21 to just three, as shown in eqn (2.6)

(2.6)

Silicon, which is not piezoelectric falls into this class of cubically symmetric crystals, as does bismuth germanium oxide (Bi12GeO20).

On the other hand, the number of independent coefficients in the elastic constant matrix [c] reduces to five when the z reference coordinate is chosen along the Z-axis of a hexagonal crystal. Piezoelectric zinc oxide (ZnO) falls into this hexagonal crystal class. It is used in thin-film SAW circuits in conjunction with silicon technology. Substrate materials such as lithium niobate and quartz are in the trigonal class of crystal structures, with a greater number of independent constants in the elastic coefficient matrix [c].

The elastic constant matrix for an isotropic or polycrystalline solid is the same as for a cubic crystal, except that any choice of reference axes can be employed. For isotropic crystals, the number of independent elastic coefficients reduces from three to two. The piezoelectric ceramic material lead-zirconium-titanate (PZT) is polycrystalline and, therefore, is isotropic elastically. Lithium niobate and quartz piezolectrics come under the trigonal crystal classification with six independent elastic constants.

*2.2.4 Piezoelectric Interactions

The stress-strain relations considered above have been tacitly applied to a nonpiezoelectric dielectric elastic solid. Application of an electrical field to such a solid would have no effect on its mechanical stress-strain characteristics. To review the effect of applying an electric field of intensity E (V/m) to a simple nonpiezoelectric dielectric, consider the electrical relationships for the simple plane-parallel capacitor model of Fig. 2.4 containing a solid insulator. Here, the electric field E established by applied voltage V will cause a distortion of the otherwise neutral molecular charge distributions in the insulator. In turn, this will result in an accumulation of surface charge on the capacitor plates. The surface charge of density D (C/m²) will be proportionally related to E by

Figure 2.4 Displacement density D and electric field intensity E vectors in a simple plane-parallel capacitor containing a solid nonpiezoelectric dielectric.

(2.7)

where εr = relative dielectric permittivity, or dielectric constant (dimensionless), and ε0 = permittivity of free space = 8.856 × 10−12 F/m.

The simple relation of eqn (2.7) no longer holds for piezoelectric dielectrics. Because of coupling between electric and mechanical parameters, the application of an electric field stimulus will give rise to mechanical deformation and vice versa. Mathematically, this interaction can be expressed in terms of a piezoelectric constant matrix [e] (with units of C/m²) such that the electrical displacement density D is given by a matrix equation

(2.8)

where S = strain and E = electric field intensity as before. Here, permittivity ε is as measured at zero or constant strain.

Equation (2.8) is a matrix equation employing the reduced coordinates for strain S in eqn (2.5). The displacement density term is then a three-dimensional one in x, y and z coordinates. Since the S term has six components from eqn (2.5), the piezoelectric constant terms in [e] will form a 3 × 6 matrix with 18 elements. The parameter [ε] relating dielectric permittivity is a 3 × 3 matrix with nine elements.

In addition, for piezoelectric materials the mechanical stress relationships are extended to

(2.9)

where [et] is now a 3 × 6 matrix and is the transpose of the piezoelectric constant [e] in eqn (2.8) (i.e., matrix rows and columns are interchanged). Equations (2.8) and (2.9) are often referred to as constitutive equations.

The element values of [e] will depend on the symmetry properties of the piezoelectric crystal. For lithium niobate and lithium tantalate piezoelectrics with trigonal crystal classification, these are

(2.10)

Also in the trigonal crystal class is quartz, with coefficients of [e] given by

(2.11)

Gallium arsenide, which is both a cubic-compound semiconductor and a piezoelectric, has piezoelectric constant coefficients

(2.12)

For SAW wave propagation in piezoelectrics, it may be shown that the electromechanical coupling coefficient K² can be defined in terms of the piezoelectric coefficient e, elastic constant c and dielectric permittivity ε considered above, where

(2.13)

and tensor subscripts have been dropped in eqn (2.13). Appropriate constants depend on both the crystal cut and the propagation direction of the surface acoustic wave. Additionally, the parameter K² in eqn (2.13) can be derived experimentally, as we shall later consider (see Example 2.3).

2.2.5 SAW Wave Equation

Ordinarily, a description of the mechanism of elastic wave propagation in a piezoelectric medium would require an evaluation of both the mechanical equations of motion as well as the electromagnetic ones governed by Maxwell’s equations. Since the mechanical propagation of SAW waves is at velocities in the order of 10⁵ less than the velocity of light, however, it may be determined that the mechanical solution will dominate such wave transport processes. The wave solutions for the mechanical wave propagation of the surface wave are indeed complex. It transpires, however, that the electrical potential Φ that they induce at the surface of the piezoelectric may be modelled to a high degree of accuracy as a travelling wave of potential Φ (in Volts) such