THE ACOUSTO-ELECTRIC FIELD ANALYSIS
OF MULTILAYERED
SURFACE ACOUSTIC WAVE DEVICES
Walter Johan Ghijsen
TR diss
1565
^ ft. d--« K ^ '
THE ACOUSTO-ELECTRIC FIELD ANALYSIS
OF MULTILAYERED
SURFACE ACOUSTIC WAVE DEVICES
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan
de Technische Universiteit Delft, op gezag van
de Rector Magnificus, prof. dr. J.M. Dirken,
in het openbaar te verdedigen ten overstaan van
een commissie aangewezen door het College van Dekanen
op donderdag 17 september 1987 te 16.00 uur_
door
Walter Johan Ghijsen
geboren te Geldrop
elektrotechnisch ingenieur
T R d i s s ^
1 5 6 5
Dit proefschrift is goedgekeurd door de promotor
prof. dr. ir. P.M. van den Berg
CONTENTS
CHAPTER 1 . INTRODUCTION 1
1 . 1 . Objectives 2 1.2. The a c o u s t o - e l e c t r i c f i e l d a n a l y s i s . . . 4
1 . 3 . The d i r e c t s o u r c e problem of t h e m u l t i l a y e r e d s t r u c t u r e . . 5
1 . 1 . The unknown s o u r c e problem 7 1 . 5 . Numerical r e s u l t s for a two-dimensional c o n f i g u r a t i o n . . . 8
References 9
CHAPTER 2 . THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 11 2 . 1 . The b a s i c e q u a t i o n s in t h e time domain 12
2 . 2 . Energy r e l a t i o n s 20 2 . 3 . The slowness 21 2 . 4 . The r e l a t i o n s i n t h e time Laplace-transform domain 26
2 . 5 . The time L a p l a c e - t r a n s f o r m domain r e c i p r o c i t y " t h e o r e m . . . 30 2 . 6 . Network f o r m u l a t i o n of t h e a c o u s t o - e l e c t r i c f i e l d problem i n
the time L a p l a c e - t r a n s f o r m domain 34
2 . 7 . The s t e a d y - s t a t e a n a l y s i s 37
References 44
CHAPTER 3 . THE DIRECT SOURCE PROBLEM OF THE MULTILAYERED STRUCTURE 4 7
3 . 1 . The g e n e r a l l a y e r e d s t r u c t u r e 48
3 . 2 . S p e c t r a l a n a l y s i s 53 3 . 3 - The s p e c t r a l f i e l d s o l u t i o n in a s o u r c e - f r e e l a y e r e d
s t r u c t u r e 63 3 . 4 . The d i r e c t source problem in the s p e c t r a l domain 78
3 . 5 . The e l e c t r i c - p o t e n t i a l component of G r e e n ' s s p e c t r a l vector 86
3.A. The system matrix A 91 3.B. Up- and downgoing waves 95 3.C. Propagation coefficients for lossless materials In the
steady-state analysis 104
References 106
CHAPTER 1». THE UNKNOWN SOURCE PROBLEM 109 4.1. The dual boundary value problem 110 4.2. The spectral iterative technique 114 4.3. The error criterion for an approximate solution 116
4.4. Direct minimization of the integrated square error using
a series of expansion functions 118 4.5. An iterative technique minimizing the root-mean-square error 121
4.6. The source truncation technique for the iterative generation
of the variational fields 130
References 133
CHAPTER 5. NUMERICAL RESULTS FOR TWO-DIMENSIONAL CONFIGURATIONS 135
5.1. The analysis of two-dimensional configurations 136 5.2. The two-dimensional direct source problem 146 5.3. The partition of power of lossless configurations in the
steady-state analysis 162 5.4. The two-dimensional unknown source problem 172
5.5. Numerical results for a single substrate configuration . . . 178
5.6. Numerical results for a layered configuration 196
Appendix 219 5.A. The matrix A of the two-dimensional configuration 219
5.B. Ingebrlgtsen's approximation 222
References 225
SAMENVATTING 229
CHAPTER 1
INTRODUCTION
Since their introduction, Surface Acoustic Wave (SAW) devices have been used in a wide range of applications and have evolved into many
different geometries. In this thesis, we pursue an efficient
computational method for the modeling of a large class of configurations which can serve as a tool in the design of SAW devices, including those involving several layers of arbitrary anisotropy. The basis of the method is the aeousto-electric field description by the quasi-static approximation of Maxwell's equations and by the elastodynamic wave equations. The theory takes the electric and elastic properties as well as the piezoelectric properties of all involved media fully into account.
INTRODUCTION
2
1.1. OBJECTIVES
The original Surface Acoustic Wave (SAW) devices introduced by White and Voltmer (1965) consist of two uniform interdigital transducers on a piezoelectric substrate such as lithium niobate or quartz. A uniform interdigital transducer consists of a number of thin, plane and parallel electrodes, having the same length and width and constant distance, and being alternately interconnected. One interdigital transducer is used to "transmit" an acoustic surface wave by connecting it to an electric source, while the other is used to "receive" this wave by connecting it to an electric load. The transduction of electric energy of the source to acousto-electric energy contained by the surface wave and the transduction of the acousto-electric energy into electric energy dissipated by the electric load is strongly frequency-dependent. This phenomenon makes SAW devices suitable for use as electronic filters, which is indeed its major application. The velocity of a surface wave in a quartz substrate is approximately 3.5 km/s, which is nearly a factor
5
10 less than the velocity of light. Therefore, another important application of SAW devices is the delay line as used in color television and radar.
The geometries of SAW devices have become complex. In filter applications, the interdigital transducers are not uniform but consist of electrodes that may have varying lengths and widths in order to attain high stop-band rejection. With the developments in integrated circuit devices, much effort has been put into exploring ways to integrate SAW devices directly on a silicon substrate, close to
electronic circuits. This way then circumvents losses due to long signal paths, which benefits the signal to noise ratio. To construct a SAW device on a substrate of non-piezoelectric silicon, a piezoelectric layer is necessary for the generation of acousto-electric surface waves. For practical applications, zinc oxide is the most important
INTRODUCTION
Fig. 1.1. An example of a layered SAW device,
have been investigated extensively (Venema, 1980). The process of making these integrated SAW devices starts with growing a thin layer of silicon dioxide on a silicon substrate. Subsequently, the interdigital
transducers are placed on the oxide layer and a ZnO layer is sputtered on top of the oxide layer and the metal pattern. The resulting
configuration is depicted schematically in Fig. 1.1-1. In this thesis, we therefore consider configurations consisting of an arbitrary number of perfectly conducting electrodes of vanishing thickness in the
interface of a structure consisting of an arbitrary number of
homogeneous plane layers. The present thesis contributes to the analysis and design of all SAW and bulk-wave devices involving a plane metal pattern in a layered structure.
INTRODUCTION 14
1.2. THE ACOUSTO-ELECTRIC FIELD ANALYSIS
The field equations describing the acousto-electric field are Maxwell's equations of electromagnetics and the elastodynamic wave equations. The properties of a medium enter the field description by the constitutive equations. These basic equations do not hold at the interfaces and are supplemented by boundary conditions. In the present theory, we make two assumptions concerning the basic equations of the acousto-electric field. Neglecting the influence of the magnetic field, we replace Maxwell's equations by their quasi-static approximation. The resulting theory is therefore only suitable for fields in which the
electromagnetic field variations in time are small enough. The basic equations are used in their linearized form. We therefore exclude non linear effects, which are the basis of the operation of SAW convolvers. All media are time-invariant, locally reacting, homogeneous and
reciprocal, and may exhibit relaxation effects. The formulation of the basic equations is chosen such that they obey the IEEE standards
(IEEE/ANSI Std 176-1978) in the case of instantaneously reacting
materials. No assumptions concerning the orientation and symmetry of the occurring media are made. All media can be of any kind of anisotropy. Special attention is given to the class of two-dimensional
configurations where the cross-sectional plane is a symmetry plane. This symmetry occurs in the classical configurations supporting Rayleigh waves, and has received much attention in the literature.
The entire configuration is invariant in time. We take advantage of this property and the linearity of the acousto-electric field by
carrying out the analysis in the time Laplace-transform domain. A quantity in this domain is obtained from its representation in the time domain by the Laplace transformation with respect to the time
coordinate. Causality of the field is enforced by requiring all field quantities to be analytic in the right half of the complex plane of the Laplace variable. In their analysis and design, SAW devices are normally
INTRODUCTION
5
characterized by their steady-state behavior. This is obtained
relatively simply from an analysis in the time Laplace-transform domain in the limiting case of a vanishing real part of the Laplace variable.
1.3. THE DIRECT SOURCE PROBLEM OF THE MULTILAYERED STRUCTURE
The only sources in the configuration are the electric surface-charge sources in the electroded region. For a given configuration, the values of the potential at each electrode are given, and the pertinent surface-charge distribution follows from these. Before considering the
calculation of the surface-charge sources, we first deal with the acousto-electric field problem of the layered structure in which a given surface-charge distribution is present.
In the analysis of this direct source problem, we take advantage of the spatial invariance of the layered structure with respect to the directions parallel to the interfaces. The acousto-electric field of each homogeneous medium is considered in the spectral domain. A quantity in this domain is the two-dimensional Fourier transformation of its representation in the time Laplace-transform domain with respect to the spatial variables parallel to the interfaces. In the spectral domain, the basic equations of each homogeneous medium are reduced to a linear, homogeneous, first-order differential equation for a field vector in the spatial variable normal to the interfaces. The eight elements of the field vector are the quantities that are continuous across the source-free interfaces. The spectral field problem is subsequently formulated as a standard eigenvalue problem of the system matrix. This approach is similar to that described recently by Fryer and Frazer (1981)) for the
elastodynamic case. In many other approaches, the basic equations are written in a system of differential equations for the displacement components and the electric potential only. These Christoffel's
INTRODUCTION
6
equations (Auld, 1973) do not lead to a standard eigenvalue problem. For the analysis of plane waves in homogeneous media of infinite extent, these equations prove to be very useful (Musgrave, 1970).
Two properties of the acousto-electric field receive special attention, the field-reciprocity theorem and the power-reciprocity theorem. Their differential form is formulated in a more general way than is usually encountered. These forms directly lead to the field-reciprocity propagation invariant for homogeneous, source-free and reciprocal media and to the power-reciprocity invariant for homogeneous, source-free and lossless media in the steady-state analysis. These fundamental invariants are used to identify important symmetry
properties of the system matrix and orthogonality of the spectral field solutions. These invariants have been reported in earlier work, e.g. by Ingebrigtsen and Tonning (1969) for surface waves in the elastodynamic case, but were derived by inspection of the system matrix for reciprocal or lossless media.
We have applied two different formalisms to solve the direct source problem of the multilayered structure. Both are based on the continuity of the field vector across a non-electroded interface and on the general solution of a homogeneous domain. In the propagator matrix formalism (Gilbert and Backus, 1966; Kraut, 1969; Fahmy and Adler, 1973) the field vectors at two different locations are expressed in each other, while in the scattering matrix formalism (Kennett, 1975) the ingoing and outgoing waves of an interface are related. The propagator matrix formalism leads to a very straightforward procedure to acquire Green's spectral vector, which relates the field vector to a given source distribution. The scattering matrix formalism appears to be more suitable for numerical applications. Another method encountered in the literature (Campbell and Jones, 1968; Farnell, 1970; Cisternas et al., 1973) is the one in which all solutions of the homogeneous media and all boundary conditions are expressed in one large system of equations. The scattering matrix formalism is an efficient way to solve the large band system of equations formulated by Cisternas et al. (1973). In the scattering
INTRODUCTION 7
m a t r i x and t h e propagator matrix formalisms the number of e q u a t i o n s in t h e g e n e r a l a c o u s t o - e l e c t r i c c a s e i s equal t o e i g h t .
In b o t h formalisms, the e l e c t r i c - p o t e n t i a l component of G r e e n ' s s p e c t r a l v e c t o r i s w r i t t e n as a q u o t i e n t of two determinants of 8-by-8 m a t r i c e s having f i n i t e e l e m e n t s . In two-dimensional s t r u c t u r e s , the r o o t s of t h e s e d e t e r m i n a n t s a r e a s s o c i a t e d with wave s o l u t i o n s t h a t may p r o p a g a t e u n a t t e n u a t e d i n the d i r e c t i o n s along the i n t e r f a c e s , such as s u r f a c e waves. With a g e n e r a l i z e d form of the M i t t a g - L e f f l e r expansion t h e a c o u s t o - e l e c t r i c f i e l d i s shown t o c o n s i s t of p a r t s a s s o c i a t e d with t h e s e r o o t s and a p a r t a s s o c i a t e d with o t h e r s i n g u l a r i t i e s of G r e e n ' s s p e c t r a l v e c t o r .
T.H. THE UNKNOWN SOURCE PROBLEM
I n o r d e r t o compute the e l e c t r i c s u r f a c e - c h a r g e d i s t r i b u t i o n from given v a l u e s of t h e e l e c t r i c p o t e n t i a l a t the e l e c t r o d e s , the a c o u s t o - e l e c t r i c f i e l d problem i s reduced t o a dual boundary value problem. This problem c o n s i s t s of boundary c o n d i t i o n s for t h e e l e c t r i c p o t e n t i a l a t t h e e l e c t r o d e s , t h e e l e c t r i c s u r f a c e - c h a r g e d e n s i t y o u t s i d e the e l e c t r o d e s and a r e l a t i o n between t h e s e q u a n t i t i e s i n t h e s p e c t r a l domain, obtained from the d i r e c t source problem. This problem has been s u b j e c t e d t o many c o n s i d e r a t i o n s of c o n f i g u r a t i o n s r e p r e s e n t i n g SAW d e v i c e s . In many a p p r o a c h e s , t h e e l e c t r i c s o u r c e d i s t r i b u t i o n i s r e p l a c e d by i t s weak-c o u p l i n g approximation (Coquin and T i e r s t e n , 1966; I n g e b r i g t s e n , 1969), which i s t h e d i s t r i b u t i o n p e r t a i n i n g t o t h e c o n f i g u r a t i o n with zero p i e z o e l e c t r i c i t y . The f i e l d problem of s i n g l e s u b s t r a t e c o n f i g u r a t i o n s t a k i n g t h e p i e z o e l e c t r i c p r o p e r t . e s f u l l y i n t o account has been
c o n s i d e r e d by a few a u t h o r s . De Jong (1972) has d e a l t with PZT-4 s u b s t r a t e c o n f i g u r a t i o n s covered by a metal l a y e r . For t h i s s p e c i a l s t r u c t u r e , s p a t i a l G r e e n ' s f u n c t i o n s have been o b t a i n e d a n a l y t i c a l l y .
INTRODUCTION
8
Milsom et al. (197^) have considered general single substrate configurations where a spatial Green's function has been.evaluated numerically. The evaluation of the electric source distribution in these methods and that of Quak (1977) has been performed by using the method of moments. In this thesis, we use a numerical technique based on the minimization of the integrated square error. This principle, first introduced by Van den Berg (1981»), avoids the laborious method of
moments and achieves a desired precision in a natural way. In each step of the iterative scheme, the estimate of the electric surface-charge distribution is the superposition of that of the previous step and a correction function, such that the integrated square error in the satisfaction of the boundary conditions of the pertinent potential is minimal. The generation of the correction function is inspired by the spectral iterative technique used by Hartmann and Secrest (1978) in a study on end effects in SAW devices.
1.5. NUMERICAL RESULTS FOR A TWO-DIMENSIONAL CONFIGURATION
The presentation of numerical results is.restricted to two-dimensional configurations where the cross-sectional plane is a symmetry plane. The theory of the direct-source problem presented and the proposed iterative procedures have been implemented in computer programs. In this
implementation, the configuration is discretized in such a way that the efficient Fast Fourier Transform technique can be used for the numerical evaluation of the Fourier transform and its inverse. The efficiency of the iterative technique is demonstrated for a simple reference
configuration, and its performance is compared to other related
techniques suggested by Van den Berg (198^). The steady-state behavior of a few-PZT1 substrate configurations is computed by using the present methods. The pertinent numerical results are compared to those of the
INTRODUCTION 9
numerical t e c h n i q u e s p r e s e n t e d by Quak ( 1 9 7 6 ) . The behavior of s e v e r a l m u l t i l a y e r e d c o n f i g u r a t i o n s i s i n v e s t i g a t e d . These c o n f i g u r a t i o n s may serve i n the a n a l y s i s of p r a c t i c a l i n t e g r a t e d SAW d e v i c e s . In t h e s e numerical r e s u l t s , t h e e x c i t a t i o n of s u r f a c e waves r e c e i v e s s p e c i a l a t t e n t i o n . The approximation of the c o u p l i n g c o e f f i c i e n t given by I n g e b r i g t s e n (1969) i s compared t o t h e ' e x a c t v a l u e s , and the g e n e r a t i o n of s u r f a c e waves i s r e p r e s e n t e d in s e v e r a l ways. The r e s u l t s show t h a t the method of a n a l y s i s used and the i t e r a t i v e schemes proposed can handle t h e computation of the f i e l d q u a n t i t i e s in m u l t i l a y e r e d SAW d e v i c e s (and o t h e r s i m i l a r c o n f i g u r a t i o n s ) very e f f i c i e n t l y .
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Auld, B.A. ( 1 9 7 3 ) , "Acoustic f i e l d s and waves i n s o l i d s , Vols. 1 and 2 " . New York: John Wiley & Sons, 1-1)23 and 1-111).
Campbell, J . J . , and W.R. Jones ( 1 9 6 8 ) , "A method f o r e s t i m a t i n g optimal c r y s t a l c u t s and p r o p a g a t i o n d i r e c t i o n s for e x c i t a t i o n of p i e z o e l e c t r i c s u r f a c e waves", IEEE T r a n s . Sonics U l t r a s o n . , SU-15, 209-216.
C l s t e r n a s , A., 0. Betancourt and A. Leiva (1973), "Body waves in r e a l e a r t h . Part I " , B u l l . Seism. Soc. Am., v o l . 6 3 , 115-156.
Coquin, G.A., and H.F. T i e r s t e n (1966), "Analysis of the e x c i t a t i o n and d e t e c t i o n of p i e z o e l e c t r i c s u r f a c e waves in Quartz by means of s u r f a c e e l e c t r o d e s " , J o u r n a l Acoust. Soc. Am., v o l . 1 1 , 921-930.
De J o n g , G. ( 1 9 7 1 ) , " G e n e r a t i o n of Bleustein-Gulyaev waves along a semi-i n f semi-i n semi-i t e m e t a l - c o a t e d p semi-i e z o e l e c t r semi-i c medsemi-ium", IEEE T r a n s . Sonsemi-ics
U l t r a s o n . , v o l . SU-21, 187-195.
Fahmy, A.H., and E.L. Adler (1973)> " A c o u s t o e l e c t r i c surface-waves in m u l t i l a y e r s , a matrix approach", Proc. 1973 IEEE U l t r a s o n . Symp., c a t . no. 73 CHO 807-8SU, 271-271.
F a r n e l l , G.W. (1970), "Symmetry c o n s i d e r a t i o n s for e l a s t i c l a y e r modes propagating i n a n i s o t r o p i c p i e z o e l e c t r i c crystal:-,", IKES T r a n s . Sonics
INTRODUCTION 10
F r y e r , G . J . , and L.N. Frazer (1984), "Seismic waves i n s t r a t i f i e d a n i s o t r o p i c media", Geophys. J . R. A s t r . S o c , v o l . 78, 691-710. G i l b e r t , F . , and G.E. Backus (1966), "Propagator m a t r i c e s I n e l a s t i c wave and v i b r a t i o n problems", Geophysics, v o l . 31, 326-332.
Hartmann; C . S . , and B.C. S e c r e s t (1972), "End e f f e c t s in i n t e r d i g i t a l surface wave t r a n s d u c e r s " , P r o c . 1972 IEEE U l t r a s o n . Symp., c a t . no. 72-CHD-8SU, 413-416.
I n g e b r i g t s e n , K.A. (1969), "Surface waves i n p i e z o e l e c t r i c s " , J o u r n a l Appl. Physics, v o l . 40, 2681-2686.
I n g e b r i g t s e n , K.A., and A. Tonning (1969), " E l a s t i c s u r f a c e waves in C r y s t a l s " , Physical Review, v o l . 184, 942-951
Kennett, B.L.N. (1983), "Seismic wave propagation i n s t r a t i f i e d media". Cambridge: Cambridge U n i v e r s i t y P r e s s , 1-342.
Kraut, E.A. (1969), "New mathematical formulation f o r p i e z o e l e c t r i c wave p r o p a g a t i o n " , Physical Review, v o l . 188, 1450-1455.
Mllsom, R.F., N.H.C. R e i l l y and M. Redwood ( 1 9 7 7 ) , " A n a l y s i s of g e n e r a t i o n and d e t e c t i o n of s u r f a c e and bulk a c o u s t i c waves by
i n t e r d i g i t a l t r a n s d u c e r s " , IEEE T r a n s . Sonics and U l t r a s o n - , v o l . SU-24, 147-166.
Muagrave, M.J.P. (1970), " C r y s t a l a c o u s t i c s " . San F r a n s i s c o : Holden-Day, 1-288.
Quak, D. (1977), " I n t e r d i g i t a l e x c i t a t i o n of s u r f a c e waves on the b a s a l plane of a s e m i - i n f i n i t e , hexagonal, p i e z o e l e c t r i c c r y s t a l " , Wave E l e c t r o n i c s , v o l . 3, 107-123.
Van den Berg, P.M. (1984), " I t e r a t i v e computational t e c h n i q u e s i n s c a t t e r i n g based upon the i n t e g r a t e d square e r r o r c r i t e r i o n " , IEEE T r a n s . Ant. P r o p a g a t . , v o l . AP-32, 1063-1071.
Venema, A. (1980), "Transduction and propagation of s u r f a c e a c o u s t i c waves in t h r e e - l a y e r e d media with an e l e c t r i c a l l y c o n d u c t i v e s u b s t r a t e " . PhD. t h e s i s , Delft U n i v e r s i t y of Technology, D e l f t , The N e t h e r l a n d s , 1-221.
White, R.M., and F.W. Voltmer (19f.5), " D i r e c t p i e z o e l e c t r i c coupling t o s u r f a c e e l a s t i c waves", Appl. Phys. L e t t . , v o l . 7, 31'1-316.
ANSI/IEEE Std 176-1978 (1984), "IEEE S t a n d a r d s on p i e z o e l e c t r i c i t y " , IEEE T r a n s a c t i o n s on Son. U l t r a s o n . , v o l . SU-31, 1-55.
CHAPTER 2
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION
The acousto-electric field analysis is governed by the equations of
elastodynamics and Maxwell's equations of electromagnetics, while the
relevant properties of the materials involved are expressed by
c o n s t i t u t i v e r e l a t i o n s . All physical quantities are written in SI u n i t s .
This chapter deals with the formulation of these basic equations in
linearized form; we assume that the amplitudes of the field quantities
involved are so small that the f i r s t - o r d e r terms sufficiently accurately
account for the effects studied (Thurston, 19614). We assume that the
f i e l d variation in time i s slow enough to neglect the influence of the
magnetic f i e l d , thus replacing Maxwell's equations by t h e i r q u a s i - s t a t i c
approximation. Further, we discuss the energy r e l a t i o n s of our wave
motion problem.
In accordance with physical experience, a l l of the quantities
involved depend on time in a causal way. This condition has to be taken
into account in a l l considerations of the acousto-electric f i e l d . Since
we deal with time-invariant media, we can take advantage of the time
Laplace-transformation in which t h i s causality constraint i s replaced by
conditions on the transformed field q u a n t i t i e s . The complex
representation of the quantities involved in the steady-state analysis
i s regarded to be a limiting case of the general time Laplace-transform
case. In the Laplace-transform domain we discuss the f i e l d - r e c i p r o c i t y
r e l a t i o n .
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION
12
2.1. THE BASIC EQUATIONS IN THE TIME DOMAIN
e l
The general configuration
In our general field analysis of acousto-electric transducers, we
consider a bounded domain V containing e l a s t i c media and a number of N
electrodes (Fig. 2.1-1). The media may or may not have piezoelectric
properties, or can be vacuum. The domain V is bounded by the surface 8V.
All physical field quantities involved are functions of space and time.
The Cartesian coordinates (x , x
?, x , ) ( a l l specified in m) with respect
to the right-handed orthogonal Cartesian reference frame ( i . , i
p, i , )
locate a point in space, while t (in s) represents the time of
observation. The vectors and tensors which occur are usually given in
subscript representation. So that no confusion can a r i s e , t h e i r direct
notation, indicated by bold symbols, may be employed to simplify the
pertinent formulas. In p a r t i c u l a r , x = x . i . + x . i + x . i denotes the
position vector. Unless specified otherwise, lower-case Latin subscripts
e l e c t r o d e
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 13
can t a k e t h e values 1, 2 or 3 , and t h e summation convention for r e p e a t e d s u b s c r i p t s i s u n d e r s t o o d . We s h a l l not use t h e s o - c a l l e d compressed m a t r i x n o t a t i o n ( o r V o i g t ' s n o t a t i o n ) , which i s a s i m p l i f i e d n o t a t i o n of second- or h i g h e r - r a n k t e n s o r s with a reduced number of s u b s c r i p t s . P a r t i a l d i f f e r e n t i a t i o n with r e s p e c t t o x i s denoted by 3 . and the symbol 3 i s r e s e r v e d f o r p a r t i a l d i f f e r e n t i a t i o n with r e s p e c t t o t i m e .
The f i e l d e q u a t i o n s
In the e n t i r e s p a t i a l domain, t h e r e a r e no mechanical s o u r c e s ; hence the s o u r c e - f r e e forms of the e q u a t i o n of motion and t h e deformation equation s u f f i c e i n the p r e s e n t c o n s i d e r a t i o n s . The l i n e a r i z e d , s o u r c e - f r e e e q u a t i o n of motion can be w r i t t e n as (Thurston, 1964)
3.T. . - p V v . = 0, (2.1-1) J 1J t i where T . = s t r e s s ( i n P a ) , v. = p a r t i c l e v e l o c i t y ( i n m / s ) , p = volume d e n s i t y of mass ( i n kg/m ) , 3 . = p a r t i a l d e r i v a t i v e with r e s p e c t t o x. ( i n m , J *J _ i 3 = p a r t i a l d e r i v a t i v e with r e s p e c t t o t ( i n s ) .
On account of the absence of body t o r q u e s , t h e s t r e s s T , . i s a symmetric t e n s o r . The l i n e a r i z e d , s o u r c e - f r e e deformation e q u a t i o n of an e l a s t i c s o l i d can be given as
2( 3jvi +3i V. ) - 3t S i J = 0 , ( 2 . 1 - 2 )
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 14
S . . = s t r a i n ( d i m e n s i o n l e s s ) .
This r e l a t i o n makes t h e s t r a i n a l i n e a r , symmetric t e n s o r . The e l e c t r i c c o n t i n u i t y e q u a t i o n i s given as V i = 3tp e > ( 2 . 1 - 3 ) i n which Ji = Ji + 3tDi ' ( 2 . 1 - 4 ) with 2 J. = volume density of the total medium electric current (in A/m ) ,
e 3 p = volume density of the electric source charge (in C/m ) ,
J, = volume density of the electric conduction current (in A/m ) ,
1 2
D. - electric flux density (in C/m ) .
Note that J. is the true electric current density firstly defined by Maxwell (1873). In the quasi-static field analysis, the influence of the magnetic field is neglected. Hence the electric field can be written as
Et - - 8 . * , (2.1-5)
in which
E = electric field (in V / m ) , $ = electric potential (In V ) .
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 15
The c o n s t i t u t i v e r e l a t i o n s
The p r e s e n t a n a l y s i s e x c l u s i v e l y d e a l s with c o n f i g u r a t i o n s c o n s i s t i n g of l i n e a r , t i m e - i n v a r i a n t and l o c a l l y r e a c t i n g media. We assume t h a t these media show time r e l a x a t i o n e f f e c t s i n a l l m a t e r i a l t e n s o r s , and we consider i n s t a n t a n e o u s l y r e a c t i n g m a t e r i a l s as a s p e c i a l c a s e of t h i s . In t h e c o n s t i t u t i v e e q u a t i o n s , c o n v o l u t i o n - t y p e i n t e g r a l s s e r v e t o express t h i s type of behavior m a t h e m a t i c a l l y (Boltzmann, 1876). Of the four s t a n d a r d e x p r e s s i o n s conforming t o the IEEE s t a n d a r d s on
P i e z o e l e c t r i c i t y (IEEE/ANSI Std 176-1978), we s e l e c t the s e t of c o n s t i t u t i v e e q u a t i o n s T ( x , t ) Di( x , t ) J ^ ( x . t ) K. . ( x , t ' ) S ( x , t - t ' ) d t ' i j p q PQ K . . ( x . t1) E ( x , t - t ' ) d t ' , P i j P K ( x , t ' ) S ( x , t - t ' ) d t ' ipq pq K. . ( x , t ' ) E . ( x , t - t ' ) d t ' , < , ( x , t ' ) E . ( x . t - t ' ) d t ' , ( 2 . 1 - 6 ) ( 2 . 1 - 7 ) ( 2 . 1 - 8 )
where the c o n s t i t u t i v e parameters,
<.. - stiffness relaxation tensor (in Pa/s),
èJ p q 2
K. - p i e z o e l e c t r i c relaxation tensor (in C/m s ) , K . . = permittivity r e l a x a t i o n tensor (in F/ms), K . - conductivity relaxation tensor (in S/ms),
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 16
are characteristic for the media involved. Causality of the media has been enforced by restricting the interval of integration to 0 < t < ». The values of the components T. .(x,t), D.(x,t) and J°(x,t) of the stress, the electric conduction current, and the electric flux density at a certain instant t depend on the values of the components E (x,t') and S.,(x,t') of the electric field and the strain at instants t' prior to t only. Mathematically, one can express the same property by
requiring that K °j p q( x , f ) = 0 , -» < f < 0, Kp (x.t*) = 0 , — < t' < 0, K?,(x,t') - 0, — < t» < 0, K°.(x,t') = 0 , -» < t' < 0. (2.1-9) Considering i n s t a n t a n e o u s l y r e a c t i n g m a t e r i a l s as a l i m i t i n g c a s e , t h e i r r e l a x a t i o n f u n c t i o n s approach a d e l t a f u n c t i o n behavior i n t i m e : * i j p q " Ci j p q( x ) 6 ( t )'
K p V W
3 0 6 ( t ) -Ki j = ei /X^ 6 ( t )' K° - o1 (x) 6 ( t ) , (2.1-10) with c. . = s t i f f n e s s ( i n P a ) , ijpq 2 e, = p i e z o e l e c t r i c t e n s o r ( i n C/m ) , ipq e. . = p e r m i t t i v i t y t e n s o r ( i n F/m),THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 17
o. . = c o n d u c t i v i t y t e n s o r (in S/tn).
For t h e s e m a t e r i a l s the c o n s t i t u t i v e r e l a t i o n s reduce t o (cf. IEEE/ANSI Std 176-1978),
V
X , t ) = Ci j p q
U ) Sp q
( X , t )~
ep i j ° ° V
X , t )'
( 2-
1 _ 1 1 ) D ( x . t ) = e (x) S ( x , t ) + e (x) E ( x , t ) , (2.1-12)J ° ( x , t ) - o ^ C x ) E j ( x . t ) . (2.1-13)
The symmetry of t h e s t r e s s t e n s o r and of the s t r a i n t e n s o r l e a d s t o the symmetry r e l a t i o n s
° i WX , t ) ■ CJ i p q( X , t ) = Ci j q p( x > t ) = W X > t ) * ( 2-1"1")
^ p q ^ ' ^ - « i q p ^ ' ^ - ( 2-1"l 5 )
The material tensors generally depend on the spatial coordinates. In a homogeneous subdomain, they are independent of the spatial coordinates x. .
l
The boundary conditions
At a point in space where the elastic, piezoelectric or electric constitutive parameters are discontinuous, as at an interface of two homogeneous media, Eqs. (2.1—1)—(2.1-5) are no longer valid since at least one of the partial derivatives is undefined. In this case, these equations have to be supplemented by boundary conditions. Table 2.1-1 displays the relevant boundary conditions as encountered in the
description of the configurations of present interest. In this table, v denotes the unit vector along the normal to the interface.
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION
18
Table 2.1-1
General boundary conditions
Type of mechanical Traction T..v. ij J boundary
Particle velocity v.
Interface of two media with firm contact
Continuous across interface
Continuous across interface
Traction free Vanishes at boundary Undetermined at
boundary (of vacuum boundary medium) Type of electrical boundary Normal current
density J .v.
J J
Electric potential <j>
Interface of two
media without
surface charge
Continuous across
interface
Continuous across
interface
Perfectly conducting Undetermined
interface
Constant at boundary
The electrodes acting as e l e c t r i c sources
In the general configuration (Fig. 2.1-1), a l l electrodes are of
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 19
electrodes are electrically perfectly conducting, the electric potential at them has a constant value
<(i(x,t) » * ( t ) , at the n-th electrode (2.1-16)
and depends on time only. We take the enclosing surface of the n-th electrode 3V in this situation as depicted in Fig. 2.1-2. It consists of two surfaces S and S_ on either side of the electrode and a surface S. around its edge. The unit normal v. to S is identical of
o> i + / •.
that of S_ at the opposite side of the electrode, pointing from S_ to S . The electric current I fed into the n-th electrode is defined as the integrated value of the time derivative of the electric source charge density over the domain V of the electrode. The electric continuity equation (2.1-3) and the application of Gauss' theorem to the time-invariant domain V , interior to the surface 3V , lead to the result
.(n)
r(n) dV = , , J.v. dA, JJ3 v(n) j j (2.1-17)
~(n) .' 6 >
av(n)=s(tn)tS(n)^s(n)
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 20
i n which v. i s the u n i t v e c t o r along the normal of 3V p o i n t i n g i n t o V. This expression can be reduced t o the form
l(n
WJ
s
(n)
L
W-«'\l\
s
M*f"-
(2J
-
18)
by letting S^ and S_ approach the face S of the n-th electrode, and by taking 6 ■+ 0. In this result,
e 2 3p = surface density of the electric source charge (in C/m ) ,
and [J.v.]_ is the jump in the normal component of the electric current density, being the difference of the value of the electric current density at S and its value at S_ . The electric current density and the surface charge density may exhibit a singular behavior at the edges of the electrodes. However, they are integrable over S since the total electric charge over the electrodes must be finite.
We further assume that the traction T..v. and the particle velocity
ij J +
v are continuous across the electrodes; hence the jump [T .v.]_ in the traction and the jump [v.]_ in the particle velocity across the
electrodes vanish:
[T..v.]_ = 0 at an electrode, (2.1-19)
[v.]* = 0 at an electrode. (2.1-20)
Note that the boundary conditions (2.1-19) and (2.1-20) at the
electrodes fit in the general description of the boundary conditions at an interface (Table 2.1-1).
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 21
2 . 2 . ENERGY RELATIONS
p For a c o u s t o - e l e c t r i c f i e l d s , P o y n t i n g ' s v e c t o r i n g e n e r a l i z e d form S. in the time domain can (analogously t o Holland, 1967) be defined as t h e sum of t h e mechanic and e l e c t r i c energy f l u x e s . In t h e q u a s i - s t a t i c
a p p r o x i m a t i o n , t h i s r e d u c e s t o (Auld, 1973, p . 312)
SP - - T . .v, + J.<)>, ( 2 . 2 - 1 )
J i j i J
in which the first term accounts for the mechanical part and the second term accounts for the electrical part. A power balance in differential form, in which this quantity occurs, can be derived by multiplying the equation of motion (2.1-1) by -v. and the electric continuity equation (2.1-3) by <(>. Combining these results in the expression for
3.[-T. .v + J.i))], we arrive at
V "
Ti j
vi
+V
] = ( 3t p
e )* - 5
3t
(p
r av
i )- T1 J3Jv1 ♦ J j 8 j* . ( 2 . 2 - 2 )
To i n t e r p r e t t h i s r e s u l t , we c o n s i d e r t h i s power balance i n i n t e g r a l form f o r a domain V with e n c l o s i n g s u r f a c e 3V. I t i s o b t a i n e d from t h e power b a l a n c e in d i f f e r e n t i a l form, using Gauss' divergence theorem, as
Pa e = w3 - 3t(Wk + W1) - Wh, ( 2 . 2 - 3 )
where
P = [-T. .v, + J.,ij>]v. dA, outward a c o u s t o - e l e c t r i c power flow through 3V,
'3V
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 22
,k 1
III •■ '•
v dV, k i n e t i c e n e r g y s t o r e d i n V, 3 W1 + W11 {T. . 3 . v . - J.3.<t>} dV,i j J i J J
time rate of internal energy change and power irreversibly dissipated into heat in V,
in which v. is the unit vector along the outward normal to the surface 3V. The material properties determine the precise form of the internal
i 'h energy W and the power dissipated into heat W . For instantaneously
reacting materials, with material tensors c.. (x), e. (x), e..(x) and o..(x), the internal energy consists of two constituent parts as
Wa + We, ( 2 . 2 - 4 ) i n w h i c h ,a 1
f l •«
( 3 . u . ) ( 3 u ) dV, e l a s t i c d e f o r m a t i o n e n e r g yII
reversibly stored in V,e ., (3 . <(>) (3 .<}>) dV, electric field energy reversibly 'ij i1
stored in V, and the power W dissipated into heat is given by
oi ( 3 ^ ) 0 <(>) dV (2.2-5)
The particle displacement u. (in m) is related to the particle velocity v. as v. - 3,u. . Note that the internal energy does not contain a part
l l t l 6 J
accounting for piezoelectric interaction. For the materials of interest, the terms in the power balance in differential form that are associated with the internal energy satisfy (Tiersten, 1969)
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION
23
C
i j
Pq
( 3j
Ui
) ( 8q
Up
) +W ^ j * '
> 0'
(2"
2"
6)for any possible combination of non-zero u and <(>. This result,
originating from thermodynamic considerations, puts restrictions on
the admissible values of the elements of the stiffness tensor and the
permittivity tensor.
To exclude the precise behavior of the electrical sources from the
power balance (2.2-3), they can be isolated from the domain V by
( r\ \
enclosing each n-th electrode by a surface 3V (cf. Fig. 2.1-2). In
this case (2.2-3) holds for the domain V outside the electrodes, in
which the power delivered by the sources is given as
^ = 1
, > [-T. .v. + J.<f.] v. dA. (2.2-7)
3 v
( n ) lj l
y 3
In this expression, v. denotes the unit vector along the normal to
(n)
J3V , pointing into the domain V, as shown schematically in Fig. 2.1-2.
In the limit 6 ■» 0, the contribution of the first term in the integral
(2.2-7) over 3V vanishes on account of the continuity of the traction
T v and the particle velocity v . Therefore, the power delivered by
all electrodes (of vanishing thickness) is found as
*
S
= £ l |(
9W
C„)
J
j V
dA
"
l
C ff
s
(n)
[J
jV- *
dA
£ JLn)
{
V ^
e ) 1
*
dA
•
(2
-
2
"
8)
The l a t t e r r e s u l t i s considered with the expression for W of Eq.
(2.2-3).
The power balance in differential form does not hold at a point in
space where the e l a s t i c , piezoelectric or e l e c t r i c properties are
discontinuous. In view of the boundary conditions of Table 2.1-1, the
normal component of Poynting's vector is continuous in such points, and
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 24
t h e power balance in i n t e g r a l form can cope with t h e s e d i s c o n t i n u i t i e s . T h e r e f o r e , the v a l i d i t y of t h i s form can be extended t o domains
c o n s i s t i n g of more than a s i n g l e medium.
2 . 3 . THE SLOWNESS
In t h i s s e c t i o n , we d i s c u s s t h e plane-wave s o l u t i o n s t h a t s a t i s f y the s o u r c e - f r e e e q u a t i o n s of an i n s t a n t a n e o u s l y r e a c t i n g , l o s s l e s s , homogeneous, p i e z o e l e c t r i c medium. The e q u a t i o n of motion ( 2 , 1 - 1 ) and t h e e l e c t r i c c o n t i n u i t y e q u a t i o n ( 2 . 1 - 3 ) of such a medium i s given by
Vu -
pVi ■ °-
{2-
3"
1)j
3.D. = 0, (2.3-2) I
and the constitutive equations (2.1-11) and (2.1-12) are given by
3..T. . - c, . 3 v + e .,3 3,6, (2.3-3) t ij ijpq q p pij p tv'
3.D. = e. 3 v - e. .3.3,6. (2.3-4) t i ipq q p ij J t*
Note that the material tensors c., , e .. and z, . are constants. The
i j p q ' p i j i j
g e n e r a l form of a uniform plane w a v e - s o l u t i o n can be w r i t t e n as (-T ( x , t ) , - D . ( x , t ) , v . ( x , t ) , 3t< ( > ( x , t ) }
l - T ^ . - D ^ . v ^ . a ^ ^ } m - sqxq) , (2.3-5)
where {-T. ,-D ,v. , 3 4 } denotes the amplitude of the plane wave 1 J 1 1 b
a t t = 0 and x = 0 , where f ( t - s x ) i s the normalized wave shape and Q q q
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 25
where s i s t h e slowness vector of t h e r e l e v a n t wave. S u b s t i t u t i n g Eq. ( 2 . 3 - 5 ) i n ( 2 . 3 - 1 ) - ( 2 . 3 - 4 ) , and e l i m i n a t i n g T: and D. , we a r r i v e at c. . s.s - p 6 i j p q J q ip | e . .s.s q i j J q e. s.s j p q J q -e .s.s qj j q ,(0)
M
(07 (2.3-6)Upon w r i t i n g s = n / c , where n i s the ( r e a l ) u n i t v e c t o r along t h e d i r e c t i o n of p r o p a g a t i o n of the plane wave (n n = 1 ) , and c i s the wave speed of t h e p l a n e wave, we o b t a i n o , , n . n e . . n . n i j p q J q i q i j J q e . n.n j p q J q I -e . n . n qj j q m 2 P c i p
' ° 1
I I l 0 1r
v ( 0 )i
Py
o >
.
0 , (2.3-7)where 8. denotes the Kronecker symbol; <5,
i p ip 1 i f i = p and 6 i p 0 i f i * p . Equation (2.3-7) d e f i n e s , for given n , a g e n e r a l i z e d l i n e a r
e i g e n v a l u e problem for t h e a d m i s s i b l e v a l u e s of p c . After m u l t i p l y i n g Eq. ( 2 . 3 - 7 ) by {v .S,.* } ( t h e a s t e r i s k denotes the complex
(0)* (0) c o n j u g a t e ) , and e l i m i n a t i n g t h e terms v, e , .n n 3 $ and
( 0 ) * (0) q i j J q t 3 , * e . n . n v , we a r r i v e a t tT j p q J q P o , : n . nV;0 )%( 0 ) +e n n ( 3 / ° V o V0 ) ) i j p q j q i p qj j q ty ty m 2 ( 0 ) * (0) p c vi vt (2.3-8) Since o, . n n and e, . a r e p o s i t i v e d e f i n i t e (cf. ( 2 . 2 6 ) ) , t h e l e f t -i j p q j q -i j 2
hand side of (2.3-8) is always positive and hence c must be positive for any {v ,3.<t> }. Therefore, there exist six admissible plane wave solutions with wave speeds c = ± c. , c = ± c_ and c = ± c„.
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 26
2 . 1 . THE RELATIONS IN THE TIME LAPLACE-TRANSFORM DOMAIN
Since we are i n t e r e s t e d i n t h e behavior of the a c o u s t o - e l e c t r i c f i e l d i n a l i n e a r and t i m e - i n v a r i a n t c o n f i g u r a t i o n , one can take advantage of t h i s s i t u a t i o n mathematically by c a r r y i n g out a Laplace t r a n s f o r m a t i o n with r e s p e c t t o time and c o n s i d e r i n g t h e e q u a t i o n s i n t h e corresponding time Laplace-transform domain or s-domain. We assume t h a t the
c o n f i g u r a t i o n and s o u r c e s a r e a t r e s t when t < tn. The s-domain
r e p r e s e n t a t i o n F ( x , s ) of a t i m e - v a r y i n g q u a n t i t y F ( x , t ) i s defined by means of the time L a p l a c e - t r a n s f o r m a t i o n with complex Laplace transform
parameter s as
r
F ( x , s ) = F ( x , t ) e x p ( - s t ) d t . ( 2 . 1 - 1 ) \
For Re(s) > 0 the i n t e g r a l i s convergent for p h y s i c a l (bounded) q u a n t i t i e s . In the s-domain, the time c o o r d i n a t e has been e l i m i n a t e d , l e a v i n g a f i e l d problem i n space i n which t h e Laplace transform parameter s occurs as a parameter, and i n which d i f f e r e n t i a t i o n w i t h r e s p e c t t o time i s r e p l a c e d by m u l t i p l i c a t i o n by s . C a u s a l i t y of t h e f i e l d i s taken i n t o account by t a k i n g Re(s) > 0, and r e q u i r i n g t h a t for a l l x a l l c a u s a l f i e l d q u a n t i t i e s be a n a l y t i c f u n c t i o n s of s in t h e r i g h t h a l f 0 < Re(s) < » of the complex s - p l a n e . The time Laplace-t r a n s f o r m a Laplace-t i o n (2.1-1) Laplace-then h a s , c o n s i d e r e d as an i n Laplace-t e g r a l equaLaplace-tion in F ( x , t ) , a unique s o l u t i o n ( v i z . F ( x , t ) f o r t > t and 0 f o r t < t ) . At the imaginary a x i s i n t h e s - p l a n e (s = joo, io r e a l ) , the r e a l and imaginary p a r t s F ' ( x , u ) and F"(x,w) of the Laplace transform F(X,JOÜ) of a causal function F ( x , t ) ( F ( x , t ) = 0 for t < 0) a r e i n t e r r e l a t e d by the Kramers-Kronig r e l a t i o n s (Landau and L i f s h i t z , 1960).
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 27
The f i e l d e q u a t i o n s i n t h e time L a p l a c e - t r a n s f o r m domain
In t h e time Laplace-transform domain, t h e e q u a t i o n of motion and t h e deformation equation a r e given as
S J T J . - pmsv. = 0, ( 2 . 4 - 2 )
-2
(Yi
+ 3i V - ^ j ■ °-
( 2^ -
3 )while the electric continuity condition and electrostatic field equation are given as
3 ^ . = s pe, (2.4-4)
J. = J° + 3D1 > (2.4-5)
E. = -a.*. (2.4-6)
The c o n s t i t u t i v e r e l a t i o n s i n t h e time L a p l a c e - t r a n s f o r m domain
The c o n s t i t u t i v e r e l a t i o n s i n t h e time L a p l a c e - t r a n s f o r m domain of l i n e a r , t i m e - i n v a r i a n t , and l o c a l l y r e a c t i n g m a t e r i a l s can be w r i t t e n as
T. . = c . . S - e . .E , ( 2 . 4 - 7 ) i j iJPQ pq P i j P
S
i °
;i
wV
;u V
(2-*-
8)'I°-VJ-
(2-'
,-'
9)The t e n s o r s c. . , e. , e. . and o, . a r e t h e time L a p l a c e - t r a n s f o r m s of iJPQ i p q ' i £ i j £
the r e l a x a t i o n t e n s o r s of K, . , K, , <.. and K . , , r e s p e c t i v e l y . In
i j p q i p q i j ï j
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 28
^ i n n ^ ' J " ^ ' ei D a ^x' Jü )^ * e 1 i( * » Ju) a n d a (x.jw), for real to, satisfy
the Kramers-Kronig relations (Landau and Lifshitz, 1960). For instantaneously reacting materials, the material tensors are (real) constants independent of s. In this case we have
W
X'
S ) = Ci J p q
( X )'
;j p q( X > S ) = ej P Q( X )'
Ë j - U . s ) = E j . ( X ) ,
o . . ( x , s ) = 0 i j( x ) . ( 2 . 4 - 1 0 )
Introduction of the generalized flux density
In the s-domain, the acousto-electric field can be completely described by the stress T , the generalized electric flux density B., defined as
D = s "1 J , ( 2 . 4 - 1 1 )
the particle displacement u. (= s v. ) and the electric potential <|>. The relevant basic equations are given as the set of linear differential equations 3 . ^ . - pms2 U i - 0, (2.4-12) 3iBi - pe, (2.4-13) T, . = o, . 8 u + e . .3 d>, (2.4-14) ij iJPQ q P PiJ Py D, = e. 3 u - e. .3.<f>, (2.4-15) i ipq q P ij y '
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 29
i n which t h e g e n e r a l i z e d p e r m i t t i v i t y e . i s defined as
I. . = e. . + s"1 o . . . (2.1-16)
This description (2.4-12)-(2.4-15) of our field problem proves to be
advantageous in the further treatment of the acousto-electric f i e l d , as
outlined in Chapter 3.
The boundary conditions in the time Laplace-transform domain
As in the time domain, the f i e l d equations and constitutive r e l a t i o n s
have to be supplemented by boundary conditions in order to deal with
s p a t i a l discontinuities in material properties. The boundary conditions
of Table 2.1-1 can l i t e r a l l y be transformed into the time
Laplace-transform domain by replacing the time-domain notation by the proper
time Laplace-transform domain notation. The boundary conditions applying
to the generalized e l e c t r i c flux density D. are identical t o those of
the t o t a l medium current density. The normal component D.v. of the
generalized e l e c t r i c flux density i s continuous across an interface
without surface charge, and i t is undetermined at a perfectly conducting
i n t e r f a c e . From Eq. (2.1-18) i t i s seen that the current I fed into
the n-th electrode i s related t o the jump in the normal component of the
e l e c t r i c flux density or the surface charge density by
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 30
2 . 5 . THE TIME LAPLACE-TRANSFORM DOMAIN RECIPROCITY THEOREM
The f i e l d - r e c i p r o c i t y theorem d e s c r i b e s a fundamental p r o p e r t y of the a c o u s t o - e l e c t r i c f i e l d . I t i n t e r r e l a t e s t h e f i e l d q u a n t i t i e s o c c u r r i n g i n two d i f f e r e n t p h y s i c a l s t a t e s A and B of a bounded domain V with e n c l o s i n g s u r f a c e S ( F i g . 2 . 5 - 1 ) . In t h e s e s t a t e s , both t h e f i e l d q u a n t i t i e s and t h e m a t e r i a l p r o p e r t i e s may d i f f e r . We s h a l l g i v e t h e f i e l d - r e c i p r o c i t y theorem i n t h e time L a p l a c e - t r a n s f o r m domain. In our t h e o r y , t h i s theorem may be encountered e i t h e r i n d i f f e r e n t i a l form or in i n t e g r a l form. The p r e s e n t f o r m u l a t i o n of the f i e l d - r e c i p r o c i t y theorem w i l l be used t o d e r i v e p r o p e r t i e s of t h e a c o u s t o - e l e c t r i c f i e l d of media having a d i r e c t i o n of t r a n s l a t i o n i n v a r i a n c e . The d i f f e r e n t i a l form i n t e r r e l a t e s the f i e l d q u a n t i t i e s of s t a t e A a t a p o i n t x , and t h e f i e l d q u a n t i t i e s of s t a t e B a t a point x ' . Point x ' i s r e l a t e d t o x v i a a c o n s t a n t t r a n s l a t i o n v e c t o r a such t h a t x* = x + o . The fundamental
*A ~B =A"B "B "A i n t e r a c t i o n q u a n t i t y i n t h i s form i s 3 . ( - T . . u. - D. é + T. . u.
-B ~A J i j i J i J i + D. cfi ) , i n which t h e q u a n t i t i e s with s u p e r s c r i p t A r e f e r t o q u a n t i t i e s of s t a t e A a t p o i n t x, and t h e ones with s u p e r s c r i p t B denote t h e f i e l d q u a n t i t i e s of s t a t e B a t x ' . Applying t h e d i f f e r e n t i a t i o n r u l e s t o t h e fundamental i n t e r a c t i o n q u a n t i t y r e s u l t s i n 3
. ( - ^ ;
B- D
A?
+? . uf
+B
B;
A)
J i j i 3 i j i 3 T" 3 . UB- D =A3 . ;B + TB. 3 iuA +DB3 . ;A i j J i J J i J J i J J * B " A * R =A " A ~ B ~A =R u 3 JA. - ,)> 3.D + uA 3 . T . + <(> 3-D , x a n d x ' i n V. ( 2 . 5 - 1 )The f i r s t four terms on the right-hand side of t h i s equation can be rewritten using the c o n s t i t u t i v e equations (2.4-14) and (2.4-15) for the two different s t a t e s . The l a s t four terms can be rewritten using the d i f f e r e n t i a l equations (2.4-12) and (2.4-13) for the two different s t a t e s . Collecting the r e s u l t s , we formulate the d i f f e r e n t i a l form of
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 31
State A State B
F i g . 2 . 5 - 1 . The two s t a t e s of the domain V in r e c i p r o c i t y c o n s i d e r a t i o n s .
the f i e l d - r e c i p r o c i t y theorem f o r the f i e l d of s t a t e A in x and the f i e l d of s t a t e B i n x' as Ae , A * B * e , B * A - - p ' 4> + p ' ID 2. m,A m,B, *A *B ,*A AB w. " Aw„ ~B, a (p - p ) ut ut - < *1jP q- < W j) (V p) ( 3Jui)
^ i V ^ p o ^ ^ i ^ ^ V p ) ♦ <v
B><v;>}
+< = f j - * i > < y
A> » A
<3 P x a n d x ' i n V. ( 2 . 5 - 2 )THE ACO US TO-ELECTRIC FIELD DESCRIPTION 32
I n t h i s e q u a t i o n , t h e m a t e r i a l parameters with s u p e r s c r i p t s A a r e those of s t a t e A at point x , and t h e m a t e r i a l parameters with s u p e r s c r i p t s B a r e t h o s e of s t a t e B a t p o i n t x ' .
Apart form the source t e r m s , t h e r i g h t - h a n d s i d e of Eq. ( 2 . 5 - 2 ) vanishes i f the media of the two s t a t e s s a t i s f y p ' = p ' , c , . "B "A "B =A -B i f f » i j p q
c . . , e. = e, and e. . = e . . . I f t h i s i s the c a s e , t h e medium of p q i j * ïpq ipq i j j i s t a t e A i s s a i d t o be a d j o i n t t o t h e medium of s t a t e B, and v i c e v e r s a . A medium i s c a l l e d r e c i p r o c a l or s e l f a d j o i n t if t h e s e r e l a t i o n s hold for x = x ' , t h u s , Ci jPq( X'S ) = Cp q i J( X , 3 )' Ê ( x , s ) « e . . ( x , s ) , f o r r e c i p r o c a l media. ( 2 . 5 - 3 )
Note t h a t the p i e z o e l e c t r i c t e n s o r i s a l r e a d y chosen i n a r e c i p r o c a l way. For d i f f e r e n t p o i n t s x and x! i n a s o u r c e - f r e e , homogeneous and r e c i p r o c a l medium, t h e r i g h t - h a n d s i d e of t h e d i f f e r e n t i a l form of the r e c i p r o c i t y theorem of Eq. ( 2 . 5 - 2 ) f o r d i f f e r e n t s t a t e s A and B a l s o v a n i s h e s . When we t a k e x = x ' , t h e requirement of homogeneity i s s u p e r f l u o u s .
In the s p e c i a l case i n which x = x ' , an i n t e g r a l form of t h e r e c i p r o c i t y theorem i s o b t a i n e d by i n t e g r a t i n g t h e d i f f e r e n t i a l form over domain V. A p p l i c a t i o n of Gauss' theorem l e a d s t o t h e d e s i r e d r e s u l t :
ff (-TA. uB - 5* ? ♦ TB uA + DB ? ) V . dA J J3 V 1J i J i j i 3 3
■ill
, " e , A - B " e , B ^A> ... t - p <p + p <|> ) d vIII.'-
2, m,A m,B. "A "B ,"A "B w~ TAWi ~B\s (p - p ) u .U i - ( cl j p q- cp q i. ) Oqup) 0 . u . )
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 33
+ ( ë * - e ^ ) ( 3 y ^ O ^6 ) ] dV, when o - 0. ( 2 . 5 - 1 )
The m a t e r i a l parameters with s u p e r s c r i p t s A and B a r e t h o s e applying t o t h e s t a t e s A and B, r e s p e c t i v e l y . Note t h a t for r e c i p r o c a l media t h e l a s t i n t e g r a l on the r i g h t - h a n d s i d e of Eq. ( 2 . 5 - 1 ) v a n i s h e s .
The s t i f f n e s s t e n s o r c, . c o n s i s t s of 81 e l e m e n t s . The symmetry of t h e s t r e s s and s t r a i n t e n s o r s i m p l i e s t h a t t h e s t i f f n e s s t e n s o r can at most have 36 elements with d i f f e r e n t v a l u e s . Reciprocal behavior reduces t h i s number t o 2 1 . The p i e z o e l e c t r i c t e n s o r , e. , which i n t h e
i p q '
c o n s t i t u t i v e r e l a t i o n s i s defined in a r e c i p r o c a l way, can c o n t a i n a t most 18 d i f f e r e n t e l e m e n t s , while the c o n d u c t i v i t y and t h e p e r m i t t i v i t y t e n s o r s a. . and e. . of such a m a t e r i a l can c o n s i s t of a t most 6
d i f f e r e n t e l e m e n t s . A d d i t i o n a l c r y s t a l symmetry p r o p e r t i e s of m a t e r i a l s reduce t h e s e numbers of d i f f e r e n t elements of t h e i r t e n s o r s . According t o Neumann's p r i n c i p l e (Nye, 1957), g e o m e t r i c a l c r y s t a l symmetry r e s u l t s in a corresponding symmetry of t h e p h y s i c a l p r o p e r t i e s of a m a t e r i a l . This p r i n c i p l e s e r v e s as the b a s i s f o r the c l a s s i f i c a t i o n of m a t e r i a l s
i n t o d i f f e r e n t c l a s s e s , as given i n t h e IEEE s t a n d a r d s (IEEE/ANSI Std 176-1978). For example, t h e p i e z o e l e c t r i c t e n s o r of a m a t e r i a l
e x h i b i t i n g c e n t e r symmetry v a n i s h e s a l t o g e t h e r . Consequently, i s o t r o p i c m a t e r i a l s have no p i e z o e l e c t r i c p r o p e r t i e s a t a l l , while t h e i r s t i f f n e s s t e n s o r h a s t h r e e d i f f e r e n t e l e m e n t s . I n t h e l i t e r a t u r e , e x p e r i m e n t a l l y determined values of the elements of the m a t e r i a l _tensors a r e defined with r e s p e c t t o C a r t e s i a n r e f e r e n c e frames i n a s t a n d a r i z e d manner. The t e n s o r s w i t h r e s p e c t t o frames d i f f e r e n t from t h i s s t a n d a r d one a r e o b t a i n e d from t r a n s f o r m a t i o n r u l e s of t e n s o r s ( J e f f r e y s , 1981).
S i m i l a r l y t o t h e v a l i d i t y of the power balance i n the time domain, t h e f i e l d - r e c i p r o c i t y theorem i n d i f f e r e n t i a l form does not hold a t a p o i n t i n space where t h e m a t e r i a l p r o p e r t i e s are d i s c o n t i n u o u s . The f i e l d - r e c i p r o c i t y theorem i n i n t e g r a l form, however, a p p l i e s t o domains t h a t c o n t a i n i n t e r f a c e s of d i s c o n t i n u i t y . The f i e l d - r e c i p r o c i t y r e l a t i o n i s often a p p l i e d as a p h y s i c a l and mathematical t o o l . In t h e f o l l o w i n g s e c t i o n i t i s a p p l i e d t o d e r i v e a network f o r m u l a t i o n for c o n f i g u r a t i o n s
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 3^
i n which e l e c t r o d e s a r e p r e s e n t . I n the past (De Jong, 1972), i t has been used t o formulate i n t e g r a l e q u a t i o n s i n t h e r e l e v a n t s t e a d y - s t a t e a n a l y s i s .
2 . 6 . NETWORK FORMULATION OF THE ACOUSTO-ELECTRIC FIELD PROBLEM IN THE TIME LAPLACE-TRANSFORM DOMAIN
The field-reciprocity r e l a t i o n in integral form (2.5-2) can be applied
to characterize the e l e c t r i c response of the general acousto-electric
transducer of Fig 2.1-1. We consider a domain V, bounded by a surface
3V, in which only reciprocal materials occur. The medium exterior to V
el
i s vacuum. In domain V, there are N e l e c t r i c a l l y perfectly conducting
electrodes of vanishing thickness (Fig. 2.1-2). Outside the electrodes,
there are no sources. The n-th electrode i s enclosed by a surface 8V ,
and i t s e l e c t r i c potential i s <(>
The reciprocity theorem in integral form applied to t h i s
configuration yields
■ £ fJ„<n> " J J
:
« - °i
;B
* ^
;
f * n *\ «■ <
2
-
6
-"
To eliminate the surface integral on the lefthand s i d e , the f i e l d
-reciprocity theorem i s applied to the vacuum domain V' exterior to 8V
and interior to a sphere S that completely encloses V, with i t s center
at the origin and radius A (see Fig. 2.6-1). Because a l l sources which
occur are located in V, a l l e l e c t r i c f i e l d quantities at S show for
Re(s) > 0 exponential decay as A ■» », and the integral over the sphere
S vanishes in the l i m i t A + •». In the field-reciprocity theorem applied
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 35
F i g . 2 . 6 - 1 . The vacuum domain V' e x t e r i o r t o 3V and i n t e r i o r t o the s p h e r e S .
t o V ' , b e i n g a vacuum domain i n s i t u a t i o n s A and B and c o n t a i n i n g no s o u r c e s , t h e volume i n t e g r a l s v a n i s h . The s u r f a c e i n t e g r a l over 3V thus e q u a l s t h a t over S in the l i m i t A ■» «°, hence i t v a n i s h e s . In view of the c o n t i n u i t y of (T..U.+ D.d>)v. a c r o s s 3V, the i n t e g r a l on the l e f t
-1 J x J J "A B
hand s i d e of (2.6-1) i s equal t o z e r o . The e l e c t r i c p o t e n t i a l s $ ' of s t a t e s A and B are c o n s t a n t over the e l e c t r i c a l l y p e r f e c t l y conducting
"A B "A B e l e c t r o d e s , and the t r a c t i o n s T.'. v. and p a r t i c l e d i s p l a c e m e n t s u . ' of
i j J i t h e s e s t a t e s are c o n t i n u o u s a c r o s s t h e e l e c t r o d e s of v a n i s h i n g t h i c k n e s s . Hence, u s i n g t h e r e p r e s e n t a t i o n (2.1-17) for t h e c u r r e n t I (n) (n) fed i n t o t h e n-th e l e c t r o d e with p o t e n t i a l w r i t t e n as , Eq. ( 2 . 6 - 1 ) can be
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 36
Y
N
e l-;(n)
A: ( n )
B YN
e l";(n)
B; ( n )
A, .
Taking s t a t e A as the a c t u a l s i t u a t i o n and s t a t e B as some a u x i l i a r y one i n which not a l l p o t e n t i a l s and c u r r e n t s v a n i s h , i t i s observed t h a t , for the a c t u a l s i t u a t i o n , v a n i s h i n g c u r r e n t s imply v a n i s h i n g p o t e n t i a l s , and v i c e v e r s a . Since the b a s i c e q u a t i o n s permit t h e s u p e r p o s i t i o n of a c o u s t o - e l e c t r i c f i e l d s , the r e l a t i o n s h i p between t h e c u r r e n t s and the p o t e n t i a l s i s a l i n e a r one. Hence, t h e p o t e n t i a l s and t h e c u r r e n t s are r e l a t e d i n a unique way, e i t h e r through the impedance m a t r i x Z of t h e e l e c t r o d e d c o n f i g u r a t i o n as
e l
"Mm) v-N *(n) e l ,_ , _*
<f> " Zn = 1 Zm n I , m = 1,2 N , ( 2 . 6 - 3 )
or through the admittance matrix Y of the electroded configuration as mn
" ( m ) VN " ( n ) e l . , h*
" ^n=1 mn * ' = ' ' ( 2 . 6 - 4 )
A p p l i c a t i o n of the f i e l d - r e c i p r o c i t y theorem (2.6-2) t o the impedance f o r m u l a t i o n (2.6-3) l e a d s t o
„ e l „el «, . B », .A „ e l „el „ , .A ~, NB
IN , IN , Z I( m ) I( n ) - IN , f , Z I( m ) I( n ) , ( 2 . 6 - 5 )
Ln=1 '111=1 mn Ln=1 Lm=1 mn '
which holds i n any s i t u a t i o n ; hence, the impedance m a t r i x i s s y m m e t r i c -S i m i l a r l y , the a d m i t t a n c e matrix Y can be proved t o be symmetric.
The e l e c t r o d e s a r e assumed t o be connected t o an e x t e r n a l
e l e c t r i c a l network, such t h a t the t o t a l e l e c t r i c c u r r e n t fed i n t o the e l e c t r o d e s v a n i s h e s , a c c o r d i n g t o K i r c h h o f f ' s law. T h i s i s e q u i v a l e n t t o
THE ACOUSTO-ELECTRIC FIELD DESCRIPTION 37
which means t h a t t h e net e l e c t r i c charge in the c o n f i g u r a t i o n i s equal t o z e r o .
2 . 7 . THE STEADY-STATE ANALYSIS
The b e h a v i o r of the SAW devices of i n t e r e s t i s often c h a r a c t e r i z e d by t h e r e s u l t s of a s t e a d y - s t a t e a n a l y s i s . In such an a n a l y s i s , a l l p h y s i c a l q u a n t i t i e s a r e t a k e n t o depend s i n u s o i d a l l y on time with a common a n g u l a r frequency, say u . Each purely r e a l q u a n t i t y F ( x , t ) can then be a s s o c i a t e d with a complex r e p r e s e n t a t i o n F ( x , j u ) and a common t i m e f a c t o r e x p ( j u t ) . In doing s o , the o r i g i n a l q u a n t i t y i s found from t h i s complex one a s
F ( x , t ) = Re{ F(x,jw) e x p ( j u t ) }. (2.7-1)
The b a s i c e q u a t i o n s i n the s t e a d y - s t a t e a n a l y s i s
S u b s t i t u t i o n of the complex q u a n t i t i e s of the type F(x,jio) expCjut) in t h e b a s i c e q u a t i o n s i n t h e time domain ( 2 . 1 - 1 ) - ( 2 . 1 - 8 ) y i e l d s , except f o r t h e common time f a c t o r e x p ( j w t ) , a s e t of b a s i c e q u a t i o n s i d e n t i c a l t o t h a t of the time Laplace-transform domain ( 2 . 4 - 2 ) - ( 2 . 4 - 1 6 ) . In t h e s e e q u a t i o n s , F stands for t h e complex r e p r e s e n t a t i o n of a q u a n t i t y F, the common t i m e f a c t o r exp(jiot) i s o m i t t e d , and s = jio. We i n t e r p r e t the s t e a d y - s t a t e a n a l y s i s as a l i m i t i n g c a s e of the time Laplace-transform domain i n which s •+ j u v i a Re(s) > 0. In t h i s way we have ensured t h a t t h e c a u s a l i t y c o n d i t i o n s remain s a t i s f i e d .
