Bulk Acoustic Wave Theory and Devices ppt

Bulk
Acoustic
Wave
Theory
and
Devices
The Adech House Acoustics Library
Fennick, John
L.,
Quality Measures and the Design
of
Kleppe, John
A.,
Engineering Applications
of
Acoustics
Rosenbaum, Joel,
Bulk Acoustic
Wave
Theory
and Devices
Rossi, Mario,
Acoustics and Electroacoustics
Telecommunications Systems
Bulk
Acoustic Wave
Theory
and
Devices
by
Joel

F.
Rosenbaum
Artech
House
Boston London
Library
of
Congress
Cataloging-in-Publication
Data
Rosenbaum, Joel, 1945 Dec. 9-
Bulk acoustic wave theory and devices
/
Joel Rosenbaum.
p. cm.
Bibliography:
p.
Includes index.
1.
Acoustic surface wave devices. 2. Acousto-optical devices.
I.
Title.
TK5981.R64 1988
ISBN 0-89006-265-X
621.38’0412
-
dc19 88-6324
Copyright
@
1988

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INC.
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February 16,2000
Joel Rosenbaum
701 1 96‘h Avenue
Seabrook, MD 20706
Dear Mr. Rosenbaum,
Enclosed please find your copy of the
Reversion
of
Copyright
and

Licensing Agreement
for your book “Bulk Acoustic Waves: Theory and Devices.”
Please countersign the Licensing Agreement and return to Artech for our records.
If
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Editor, at

or the number listed above.
Kind regards,
6ssica McBride
Editorial Assistant, Acquisitions
Enc: Lic. Agreement
(2)
Reversion of Copyright (1)
Dr.
John
Vig
UFFC-s
Dear
Dr.
Vig,
I
have enclosed the copyright agreement that
I
received, signed
and
returned to the publisher
(Artech). As per your e-mail

of
21-Jan
I
understand that
I
must give
formal
permission in writing.
So
how
about this:
I
hereby give formal permission to reproduce the
book
"Bulk Acoustics
Waves: Theory
and
Devices" Artech House
1988.
Hope that is sufficient.
Sincerely,
P
oel Rosenbaum
P.S.
sorry for the long delay;
it
took longer than
I
expected
to

get
an
answer from the publisher and then
the paperwork got misplaced
on
my desk.
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FOR
MICHAEL, SARA, AND RIVKA
Contents
PREFACE
CHAPTER
1
1.1
Introduction
1.2
1.3
1.4
1.5
Power Relations
1.6
Strain in Three Dimensions
1.7
The Stress Matrix
1.8
Transformations
1.9
Contracted Notation and the Dynamical Equations in Three
Dimensions
CHAPTER
2
PROPAGATION
OF
ACOUSTIC WAVES

IN
CRYSTALS
2.1
Introduction
2.1.1
Hooke’s Law
in
Three Dimensions
2.2
Symmetry
of
the Stiffness and Compliance Matrices
2.3
The Stiffness and Compliance Matrices in an Isotropic
Medium
2.4
The Christoffel Equation
2.5
Acoustic Propagation in Anisotropic Crystals
2.5.1
Cubic Symmetry
2.5.2
Tetragonal Symmetry
2.5.3
Orthorhombic Symmetry
THE ACOUSTIC EQUATIONS
OF
MOTION
Stress and Strain
in

One Dimension
Mechanical Equation of Motion in One Dimension
1.3.1
Phase Relations
Attenuation
of
an Acoustic Wave
xi
1
1
1
6
9
10
16
20
25
31
39
45
45
45
51
53
57
66
68
76
84
vii

viii
CHAPTER 3 COMPUTER-AIDED ANALYSIS
OF
ACOUSTIC PROPAGATION
Computer Solution of the Christoffel Equation
3.2.1 Isotropic Propagation
3.2.2 Cubic Propagation
3.2.3 Tetragonal Propagation
3.2.4
3.2.5 Orthorhombic Propagation
Properties
of
the Slowness Curve
3.3.1 Power
Flow
Angle
3.3.2 Ray Velocity Curves
Deviation Angle from Pure Mode
3.1 Introduction
3.2
Paratellurite and Anomalously
‘‘Slow’’
Modes
3.3
3.4
CHAPTER 4 PIEZOELECTRICALLY ACTIVE ACOUSTIC
4.1
4.2
4.3
4.4

4.5
4.6
4.7
4.8
4.9
PROPRAGATION
Introduction
Static Piezoelectricity
Symmetry of the Piezoelectric Matrices
Christoffel Equation for Piezoelectric Crystals
Hexagonal and Trigonal Symmetries
Piezoelectric Stiffening and the Electromechanical Coupling
Constant
Examples of Piezoelectric Coupling
Computer- Aided Analysis of Piezoelectrically Stiffened
Modes
Piezoelectric Coupling in Doubly Rotated Cuts
CHAPTER
5
ELECTRICAL CHARACERIZATION
OF
WIDEBAND ACOUSTIC DEVICES
5.1
Introduction
5.2 One-Dimensional Equations for a Nonpiezoelectric Slab
5.3 One-Dimensional Equations for
a
Piezoelectric Slab
5.4 Closed-Form Expression
for

the Input Impedance
5.5 Input Impedance Examples
CHAPTER 6 COMPUTER-AIDED ANALYSIS AND
DESIGN
OF
WIDEBAND ACOUSTIC DEVICES
6.1 Introduction
6.2 The Smith Chart
6.3 Return
Loss
6.4 One-Dimensional Mason Model
6.4.1
6.4.2
Lumped Element Circuit Representation of Piezo-
and Nonpiezolayers
Computer- Aided Analysis of Acoustic Devices
87
87
88
92
94
95
96
100
1
03
105
115
117
125

125
125
131
136
143
145
148
157
163
167
1
67
167
173
184
195
201
201
202
205
213
213
218
ix
6.5 Effect
of
Ground Plane Metalization
6.6 Effect
of
Finite Substrate Length

6.7 External Tuning Elements
6.7.1 Series Inductor
6.7.2
6.8 Series and Parallel Connections
of
Acoustic Radiators
6.9 Use of the Ground Plane as a Matching Element
CHAPTER 7 INTERACTION
OF
ACOUSTIC AND OPTIC
MODES
7.1 Introduction
7.2 Electromagnetic waves in an Anisotropic Medium
7.3 Computer- Aided Solution
of
Optical Modes
7.4 The Index Ellipsoid
7.5 Perturbations to the Index Ellipsoid
7.6 The Piezo-Optic Effect
7.7 Polarization Rotations
7.8 Coupled Mode Theory of Acousto-Optic Interaction
7.9 Wave Vector Diagrams for Acousto-Optic Interactions
7.10 Acousto-Optic Interaction
in
a Birefringent Medium
8.1 Introduction
8.2 Efficiency
8.3 Bandwidth
8.4 Resolution
8.5

Acousto-optic Devices
Parallel Inductors and the Admittance Chart
CHAPTER 8 APPLICATIONS
OF
ACOUSTO-OPTICS
8.5.1 Beam Deflectors
8.5.2 Modulators
8.5.3
Acousto-Optic Spectrum Analysis and Signal
Processing
8.6 Performance Enhancements
8.6.1
Birefringent/Acousto-Optic
Interactions
8.6.2
8.6.3 Birefringent Interaction in Paratellurite
8.6.4 Use of Acoustic Anisotropy
8.6.5 Acoustic Phase Arrays
8.6.6 Multiple Acoustic Beams
8.6.7
8.6.8 Use
of
New Materials
Design of a Birefringent Gallium Phosphide Bragg
Cell
Multiple Acoustic Beams from a Single Transducer
CHAPTER 9 COMPUTER-AIDED ANALYSIS
OF
ACOUSTO-OPTIC INTERACTIONS
9.1 Introduction

223
229
230
230
232
234
237
245
245
246
257
258
261
265
269
274
283
285
293
293
294
296
301
303
303
304
305
307
307
311

313
315
316
318
320
320
325
325
X
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
Acousto-Optic Interactions in Isotropic and Cubic Systems
Tetragonal Interactions
Orthorhombic Interactions
Trigonal Symmetry
Arbitrary Acoustic Directions
Computer-Aided Analysis of Complex Interaction
Geometries
Design Examples
Electro-optic Corrections
to
the Photoelastic Matrix
CHAPTER
10

RESONATORS I: BASIC THEORY
10.1 Introduction
10.2 Thickness Excitation
of
Acoustic Transducers
10.3 Lateral Field Excitation
10.4 The Coupling Constant and the
C
Ratio
10.5 Butterworth-Van Dyke Equivalent Circuit
10.6 Acoustic Attenuation and the Motional Resistance
10.7 Quality Factor
of
a Resonator
10.8
Resonator Figure
of
Merit
10.9 Frequency Pulling
10.10Crystal Quartz
10.11Thermal Stability
CHAPTER
11
PERFORMANCE AND ADVANCED APPLICATIONS
11.1
Introduction
11.2 Mason Model Approach to Resonator Analysis and Design
11.3 Techniques for Improving Resonator
Q
RESONATORS 11: HIGH FREQUENCY

11.3.1 BVA
11.3.2 Lateral Field Excitation
11.4 Calculation of
k2
for Lateral Field Modes
11.5 Lateral Field Excitation
in
Lithium Niobate
11.6 Energy Trapping and Spurious Responses
11.7 Composite Resonators
11.7.1 High Overtone Bulk Acoustic Resonators
11.7.2 Film Bulk Acoustic Resonators
11.7.3 Applications
of
Film Bulk Acoustic Resonators
Appendix
A
Index
326
333
335
336
342
346
356
366
371
37
1
372

376
386
389
392
397
398
399
400
402
411
411
412
418
418
420
42
1
429
430
435
435
437
442
45
1
459
Preface
This book is an outgrowth
of
lecture notes developed for a course in

the applications
of crystal acoustics at Johns Hopkins University. The
course serves the needs
of
engineers and scientists working in the area of
defense electronics, primarily in radar, electro-optics, and electronic war-
fare systems. Students generally are quite knowledgeable
in
system re-
quirements, but lack the theory
of
device physics. The course and this
book attempt to fill this gap.
Approximately the first half of the book is dedicated to a discussion
of
the theory
of
crystal acoustics, with the latter half consisting of ap-
plications to the important areas of acousto-optics and crystal resonators.
Organization in this
form
allows the applications section to reflect the
theory in a set of “symmetric” analogues in which knowledge
of
one half
promotes the learning of the other. Examples
of
these relations are:
I.
Crystal Acoustics

f,
(Chapters 1-6)
(Chapter 6)
(Chapters
43,
and 10)
(Chapter
7)
11.
Wideband Operation
++
111.
Thickness Excitation
f,
IV.
Electro-optic Effect
f,
Crystal Optics
(Chapter
7)
Narrowband Operation
(Chapter
10
and
11)
Lateral Field Excitation
(Chapters
10
and
11)

Photoelastic Effect
(Chapters
7-9)
To
illustrate this principle further, the theory
of
crystal optics can be
developed by using the same formalism as crystal acoustics. Whereas in
acoustics there
are
three possible modes, with three orthogonal polari-
zations, in the optic case there are two polarizations: one is a pure shear
direction called the ordinary mode; the second is a quasishear direction
called the extraordinary
mode.
Understanding the structure
of
the acoustic
xi
xii
modes provides important insights into the optic case. Likewise, knowledge
of wideband low-Q operation leads naturally to narrowband (high-Q res-
onator) operation, thickness excitation to lateral field excitation, and elec-
tro-optic perturbation to the photoelastic effect. The narrowband-
wideband analogue is especially interesting. It is certainly not intuitively
obvious that the
same
crystal can operate in both low-Q (delay line) and
high-Q (resonator) configurations, depending on the boundary conditions.
The same attribute (high piezoelectric coupling) that optimizes perfor-

mance in the low-Q environment also enhances resonator performance in
certain applications.
The theory section is further organized into two basic sections. The
first four chapters deal with the solution
of
the three-dimensional wave
equation without boundary conditions, and Chapters
5
and
6
consider the
solution of the one-dimensional wave equation with boundaries in the
propagation direction. These two basic approaches are continued in the
application section, as illustrated in the following table.
mechanical equations
(Chapter
1)
I
I
three-dimensional propagation
(Chapter
3)**
Christoffel equation
(Chapter
2)
.1
.1
optics and piezoelectricity
+
acousto-optics (Chapter

4)**
(Chapters
7
and
8)
.1
acousto-optic
interaction
(Chapter
9)**
one-dimensional equations
(Chapter
5)
I
electrical characteristics
(Chapter
6)**
.1
resonators
(Chapters
10
and
11)**
thickness and lateral
field coupling constants;
single- and double-rotated
cuts
(Chapters
4
and

11)**
.1
**Indicates computer program listing.
In ths light, the book forms a “closed-loop system,” with the appli-
cation sections reinforcing concepts developed in the theory section. It is
intended for self-study as well as formal classroom training.
A
program
that emphasizes applications to acousto-optics would include Chapters
1
to
4
and
6
to
9,
and a resonator study would include Chapters
1
through
6,
10,
and
11.
xiii
The computer programs reflect the dual approach. They are thus
1.
Solution of the Christoffel equation for arbitrary acoustic propaga-
tion direction (three-dimensional equation without boundaries);
2. Solution of the electrical impedance of acoustic devices (one-dimen-
sional equation with boundaries).

The
first
group
of
programs includes:
1.
Determination of the phase velocity and inverse velocity (slowness)
as functions
of
propagation direction, i.e., the acoustic eigenvalue
(Chapter
3).
2. Determination of properties of acoustic waves as functions of prop-
agation direction; these include
power flow angle (Chapter
3),
energy velocity (Chapter
3),
0
deviation from pure mode direction (Chapter
3).
3.
Determination of the effects
of
piezoelectricity on the propagation
of
acoustic waves (Chapter
4).
4.
Determination of the electromechanical coupling constant as a func-

tion
of crystal orientation (Chapter
4).
5.
Determination of the effective photoelastic constant for arbitrary
direction
of
acoustic and optic beams and acoustic and optic polar-
izations (Chapter
9).
These programs are based on rotation matrices
developed by Dr. Rob Bonney.
6.
Determination of the coupling constant for lateral field excitation
modes for arbitrary direction
of electric and piezoelectric plate ori-
entations (Chapter
11).
The second group of programs includes:
1.
Determination
of
the electrical impedance for an infinite-length sin-
gle-port delay line structure with metal loading, i.e., the Mason
model (Chapter
6).
2.
Determination
of
the electrical impedance for a finite-length single-

port structure showing acoustic standing waves (Chapter
6)
3.
Determination
of
motional elements of the equivalent circuit
of
acoustic resonator with and without metalization (Chapter
11).
4.
Determination
of
electrical characteristics of composite resonator
structure, including film bulk and high overtone acoustic resonators
(Chapter
11).
The programs are written in True BASIC language (version
2.0)
with
listing of the eigenvalue and Mason model programs included in detail in
Chapters
3
and
6.
The other programs are formed by modifying these base
divided into
two
categories:
xiv
programs as explained in the text. These programs are meant to be used!

They provide a valuable learning tool that complements the theory and
adds a level
of
excitement and discovery to a theory that, admittedly, tends
to be at times a bit dry.
All
of the figures in the text that involve physical
parameters were drawn with these programs on a
HP
plotter. The field of
crystal acoustics and its applications
is
still evolving; the best (and probably
only) way to test new device structures is through computer simulation.
This is especially true of the relatively complex configurations
of
coupling
constants of doubly rotated piezoelectric crystals and interactions of acous-
tic and electric fields with optic beams
of
arbitrary direction and polari-
zation. Typically,
20% of class time is spent working with specific
computer-aided designs. All of the programs are available on disk in sev-
eral languages. TrueBasic was chosen because it includes full matrix op-
erations and uses a code that closely follows the text equations, Execution
time per calculation varies with the program and computer, but the average
time is less than
.5
s/pt on an IBM AT running at

6
MHz.
Even a modest project of this sort does not materialize overnight in
a vacuum. It evolves over a number of years with the help and support
of
many talented people working together. Coming from a fabrication back-
ground,
I
am deeply indebted to my former colleagues at Litton Amecom
who patiently taught me the theory of acousto-optics. Special thanks to
Dr. Michael Price, Dr. Rob Bonney, Dr. Otis Zehl, Mr. Zigmund Turski,
Mr. Pradeep Wahi, and
Mr.
Jerry Long.
I
have also benefited greatly from
discussions with my colleagues at Westinghouse, including Dr. Harry
Salvo, Dr. Robert Moore, Dr. David Blackwell, Dr. Dickron Mergerian,
Mr. Michael Driscoll, Mr. Irwin Abramowitz, Mr. Paul Smith, Mr. Dana
Bailey, and Mr. Steven Brown. Many of the illustrations were expertly
drawn by Lisa Carter of Brimrose Corporation.
Unfortunately, a book of this nature cannot be written without a
certain level of mathematical sophistication.
I
have assumed that the reader
is familiar with electromagnetic theory at the undergraduate level.
I
have,
however, attempted to work out all the complex algebra in as much detail
as possible consistent with space limitations;

I
regret any oversights that
make the developments difficult to follow.
To
quote from Professor J.
Gordon: “What we find difficult about mathematics is the formal symbolic
presentation of the subject by pedagogues with a taste for dogma, sadism
and incomprehensible squiggles.”
I
have tried to minimize the sadism, and
I
sincerely hope that the reader will find that most
of
the “squiggles” are
not completely incomprehensible.
Lanham,
MD
May
1988
Chapter
1
The Acoustic Equation
of
Motion
1.1
INTRODUCTION
In this chapter,
we
lay the groundwork for the study of crystal acous-
tics by developing fundamental mechanical equations in one dimension.

We show that the presence
of
an unbalanced system
of
time-varying stresses
results in the propagation of
an
acoustic wave with its propagation velocity
dependent on the material properties of the body. Attenuation, material
quality factor
Q,
and the energy relations are more easily handled in one
dimension and can be extended to three dimensions
if
necessary. The strain
and stress matrices are formulated
in
three dimensions, and rotational
transformations, which will be increasingly important later, are developed.
The three-dimensional equations
of
motion are developed by using the
stress and strain matrices. The coupling between these matrices is not
developed beyond the point
of
stating that they are linearly related. Thus,
the dynamical three-dimensional equations of motion as developed in this
chapter do not include the generalization of Hooke’s law.
1.2
STRESS

AND
STRAIN IN
ONE
DIMENSION
In Newtonian mechanics, a
force
on a
rigid
body results in an
accel-
eration
of the body. Because the body is assumed to be rigid, the external
force is instantaneously transmitted to all
of
the body’s internal parts.
No
consideration is given to the
internal
structure
of
the body, nor to the
bonding forces that hold the body together. These issues are dealt with in
the science of
strength
of
materials
or
mechanics
of
deformable bodies,

which examine the relation between external forces, sometimes called
body
forces,
and the resulting internal effects. The effect of the body forces is
the creation of internal forces, called
stresses,
and deformations, called
strains,
in the atomic structure of the body.
1
In Newtonian mechanics, there is a causal relation between body
forces and acceleration.
If
a body is accelerating, it must be acted on by
a net force, but a body in equilibrium may
be
acted on by many forces
while at rest. In this sense, we may think of force as the independent
variable and acceleration as the dependent variable.
Stress does not cause strain (nor does strain cause stress), but the
two are coupled to each other. Internal deformations for example, can be
“caused” by thermal gradients, dislocations, and defects in the crystal
lattice or by the presence of dopant atoms that are significantly larger or
smaller than the host atoms and thus deform the lattice structure. In such
cases, internal forces are established, and
it
would be proper to refer to
these stresses as being the result
of
the strains. Nonetheless, it is usually

more convenient (as well as precise)
to
refer to the coupling of stress and
strain; the presence
of either necessarily implies that the other is also
present.
Because stress is intimately related to deformation or distortion in
the internal structuri:
of a body, the magnitude of stress is related to internal
forces divided by the area over which the forces act. The nature of the
deformation depends on the orientation of the area (recall that area is a
vector with direction as defined by the surface normal) with respect to the
stress.
A
compressive stress tends to push the internal particles together,
a tensile stress tends to pull them apart, and a shear stress tends to cut.
Compressive and tensile stresses form the class of
longitudinal
messes.
This is illustrated in Figure
1.1.
Note that the orientation of the area (as
defined by its normal vector) determines whether the stress
is
shear or
longitudinal.
A
further distinction between stress and force comes from the fact
that stresses always occur in opposite (but not always equal) pairs. These
stress components are individually referred to as traction forces, and, like

stress, they are denoted by the letter
T.
A
positive traction force points
to the right, and a negative traction force points to the left, in agreement
with conventional notation. The units of traction forces as well as stress
are N/m2. Both compressive and tensile stresses are clearly composed of
two traction forces, one positive and one negative. We define a compressive
stress as negative and a tensile stress as positive. This definition is quite
logical because in a compressive stress the traction forces are both in a
direction opposite to the area (defined as the outward normal). In the
static case, the stresses are equal because there is no net motion
of any
internal volumes. In the dynamic case (e.g., the propagation of an acoustic
wave), the opposite stresses are not generally equal.
3
I I
I
I
I
I
I
I
I
I
I
I
-
-
-

-
- -
-

I
0
0
-
rl
Figure
1.1
Orientation of traction forces relative to the area
of
an internal
volume of an isotropic medium.
Consider Figure 1.2. There are two regions, labeled
1
and
2.
Each
region (which in general contains many particles) consists of a mass element
connected by springs to two nearest neighbors. The equilibrium distance
between them is denoted as
A
L,
which is small enough
so
that the masses
may be approximated by a continuum and
AL

-
dz (Figure 1.2(a)). If a
z-directed external force, which may be either positive (directed toward
the right) or negative (directed toward the left), is applied, internal forces
will be established, moving the particles from their equilibrium positions.
This situation is shown in Figure 1.2(b). The new distance between the
masses is
Al,
and the internal forces are described by stress components
TI
and
T2
(which are not necessarily equal) in Figure 1.2(b). The individual
forces are given by
4
I
I
I
I
I
L
/
0/
,
\
X
dx
I
I
I

H
U1
IC-
k
U2
*
z
Z+U1
z+AL
Z+Ui+AI
(Cl
Figure
1.2 Section
of
internal volume element of isotropic medium: un-
distorted volume element; (a) representation
of
internal cou-
pling as springs;
(b)
particle distortions due to traction forces.
where
dA
is the cross-sectional area with dimensions
dx
and dy.
If
the new
positions are such that
u1

=
u2,
then
AL
=
Al,
and there is no relative
movement of the masses and thus no
distortion
of region
1
relative to
region
2.
This situation results from a translation of the
body
and is not
of
practical interest. If, however,
u1
$.
u2,
and
Au
#
0,
then we can define
the distortion
9
as

5
9
=
(Al)2
-
(AL)2
=
(AL
+
Au)~
-
=
(AL
+
-AL)2
-
au
az
=
+
2
az
9
=
-
(-
+
2)
(AL)2
au

au
az az
The strain
S
is defined as
9
=
2(AL)2
S
(1.2)
If we assume that (U does not change rapidly with position)
aulaz
e 1,
then.
au
az
9
=
2
-
(AL)2
=
2
S
(AL)’
(1.3)
The assumption that the particle displacement
U
changes gradually with
position is an example of linearization and is valid only if the strains are

small, which may
(from
(1.3))
as
not be realistic in practical cases. We define the strain
(1.4)
The presence
of
a strain implies that the particle displacement from equi-
librium changes with position (i.e., there is a distortion in the body). From
(1.3),
we can also write the strain as
1
(Al)2
-
(AL)2
s=-(
2
(W2
6
1.3
MECHANICAL EQUATIONS
OF
MOTION
IN
ONE
DIMENSION
In this section we derive a self-consistent set of equations that de-
scribes the propagation of a mechanical strain in a one-dimensional solid.
The mechanical variables corresponding

to
the electromagnetic variables
E,
D,
H,
and
B
are
stress
=
T
strain
=
S
particle displacement
=
U
particle velocity
=
v
Just as Maxwell’s equations are a set of four relations between the four
electromagnetic variables, we require four equations for completely char-
acterizing the mechanical properties. They are as follows.
1.
Newton’s
law:
Consider the slab
of
Figure
1.2

of cross section
dA
=
dr
dy. If the stresses
TI
and
T2
are not equal, there is a net force on the
slab given by
Newton’s law is written as
m
a
-
dF
-
t
1Ic_
_c_
t
t
aT
a2u
az
at2
-drA
=
pA
dz
-

(F
=
AT)
or
aT
a2u
az
-

where
p
is the density in kg/m3.
2.
Particle velocity is the time derivative of particle displacement:
au
at
v=-
(1.7)