Bulk Acoustic Wave Theory and Devices pptx

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, Acoustic

Bulk Acoustic Wave

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

Boston London

Rosenbaum, Joel, 1945 Dec 9-

Bulk acoustic wave theory and devices / Joel Rosenbaum

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FOR MICHAEL, SARA, AND RIVKA

1.6 Strain in Three Dimensions

1.7 The Stress Matrix

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

Properties of the Slowness Curve

3.3.1 Power Flow Angle

3.3.2 Ray Velocity Curves

Deviation Angle from Pure Mode

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

6.5 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

Parallel Inductors and the Admittance Chart

CHAPTER 8 APPLICATIONS OF ACOUSTO-OPTICS

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

Arbitrary Acoustic Directions

Computer-Aided Analysis of Complex Interaction

Geometries

Design Examples

Electro-optic Corrections to the Photoelastic Matrix

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

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

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:

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

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

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, M D May 1988

The Acoustic Equation of Motion

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

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

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 A u # 0, then we can define the distortion 9 as

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

1 Newton’s law: Consider the slab of Figure 1.2 of cross section dA =

dr dy If the stresses T I and T2 are not equal, there is a net force on the slab given by

Newton’s law is written as

where p is the density in kg/m3

2 Particle velocity is the time derivative of particle displacement:

au

at

3 Equation (1.4), which defines S as the gradient (spatial rate of change)

of particle displacement with respect to position:

4 Relation between the stress T and the strain S: We assume there is a

linear relation between the internal stresses and the deformation, and we write

where C is called the stiffness constant and has units of stress (since strain

is dimensionless) Equation (1.8) defines the properties of the connecting springs For given stress components, a stiff spring results in a relatively small strain, whereas a compliant spring results in a large strain

Equations (1.4), (1.6), (1.7), and (1.8) allow us to solve for the four

variables T , S, v , and U As.for Maxwell’s equations, there are two fun-

damental physical laws, (1.4) and (1.6), and two constitutive equations,

(1.7) and (1.8) In the static case, the gradient of the stress is zero, just

as in electrostatics where V x E = 0 From (1.6) and (1.7):

Differentiating S = au/az with respect to t and using (1,8), we obtain

Both the acoustic and electromagnetic systems are composed of two

“fundamental” physical laws and two “constitutive” equations In the electromagnetic system, the solutions of Maxwell’s equations in Cartesian coordinates are plane waves with either one (for an optically isotropic

medium) or two (for an optically anisotropic medium) polarizations In the one-dimensional acoustic system, we have seen that the solution is also

a plane wave with acoustic polarization (defined as the direction of either particle displacement or velocity) in the direction of wave propagation (a longitudinal wave) In the three-dimensional system, we will see that there are, in general, three possible acoustic polarizations

Because the phase velocity of an electromagnetic wave is from four

to six orders of magnitude greater than that of an acoustic wave, the presence of boundaries plays a much more significant role On the other hand, acoustic propagation is complicated not only by the presence of three acoustic modes, but also because the phase velocities are complex functions of the propagation direction This directional dependence is due,

in part, to the relatively complex nature of Hooke's law (1.8) in three dimensions as compared to the corresponding constitutive relations for electromagnetics

1.3.1 Phase Relations

For an electromagnetic wave propagating in an isotropic, lossless medium, the displacement D, the electric field E, the magnetic field H, and the induction B are all in phase In the acoustic system, the particle displacement is not in phase with the particle velocity (1.7) If we let

U = U,, ei(~~-Sz)

other phase relations are

To find the phase relation between S and v , we use (1.9):

10

Because T and S are related by a constant (C), they are in phase In summary, T and S are in phase, and all are 90" out of phase with U and 180" out of phase with v The ratio of T and v is called the acoustic impedance of the medium (2) and is

(the minus sign is included so that the impedance will be positive, because

T and v are 180" out of phrase), but juu = v , so

From ( l l l ) , V, = va, so

(1.12)

The units of impedance are kg/s m2 Like the phase velocity, the acoustic impedance is a property of the medium Corresponding to (1.11) and (1.12), the electromagnetic relations €or phase velocity and impedance are given by the well-known formulas:

z, = J' E

phase velocity

impedance

1.4 ABSORPTION OF AN ACOUSTIC WAVE

If a solid medium obeyed Hooke's law (T = CS) precisely, there would be no acoustic absorption In a real medium, there are viscous damping forces and nonlinearities, which cause energy to be extracted

from the wave in the form of heat We have already encountered a non-

linearity in the definition of strain We can include these forces in the wave equation by modifying Hooke's law:

We thus expect that the attenuation will be proportional to frequency and inversely proportional to velocity We can formally demonstrate this by recalling the one-dimensional equations of motion (1.6):

For all practical cases, q and thus a are very small compared with p

and o, so the last two terms can be neglected Equation (1.20) reduces to

= 6 = v, (the phase velocity)

Equation (1.22) can be written in the form:

where we define the quality factor Q as

an order of magnitude We will study the properties of this important crystal in detail later In general, materials with high velocities tend to have low absorptions and high Q-factors

The degree of compliance of acoustic attenuation with frequency is

an excellent measure of material quality A poorly grown crystal, e.g., will generally be highly stressed and will contain a high density of grain boundaries, air pockets, and impurities These defects will usually have dimensions comparable with an acoustic wavelength and will scatter the wave, thus reducing the frequency variation to a linear dependence In practice, then, the frequency dependence is o n , where 1.2 < n < 1.4 for

a poor-quality material and n > 1.8 for a high quality material If n < 1.7,

it is usually safe to assume that improved crystal growth conditions will result in a significant reduction in absorption

Absorption is determined experimentally by performing a pulse echo measurement in which a crystal sample is excited by an impulse and the reduction in the resulting pulse train is observed In some situations in which low absorption is critical, the sample is cooled to a very low tem- perature, which dramatically reduces a For longitudinal modes at room temperature, the de endence, as derived by Woodruff and Ehrenreich, is more fundamental physical property of the crystal than a is, and k is the thermal conductivity (which is inversely proportional to T so there is n o net temperature dependence) [ 5 ] The dependence on k follows from the fact that high conductivity facilitates the transfer of energy from regions

of compression to regions of extension This mechanism is not operative for shear modes (because there is no compression or extension) Hence,

it may be inferred that the absorption of shear modes is less than that of longitudinal modes; the lower acoustic velocities of shear modes usually more than compensate for the lower absorptions

proportional to y20 P kT, where y is called the Gruneisen constant and is a

I6

The dependence of a on thermal conductivity has led to a search for techniques to reduce k by, for example, doping the crystal; the results have been only partially successful At very low temperatures, the ab- sorption is extremely low and its dependence is of the form o T 4 Between

the high temperature and low temperature regions, the absorption follows

a dependence characteristic of relaxation behavior as shown in Figure 1.4 Unfortunately, for most materials TI falls between 30 and 100 K

In electromagnetic theory, the divergence of the Poynting vector

P = (E x H) determines the power in the electromagnetic wave:

and BH are unit vectors in the directions of E and H, respectively,

and the Poynting vector is (because E and H are orthogonal)

Multiplying (1.6) by v and (1.9) by T and adding gives

kinetic energy density

Equation (1.28) is the acoustic analogue of (1.25) Comparing (1.28) with (1.25), we define the acoustic Poynting vector as

For a lossless medium using (1.28) and (1.29):

w(stored energy) dissipated energy per cycle

Q

Because the maximum strain and kinetic energies are equal, substituting from (1.30) yields

where uo is the particle displacement amplitude (maximum displacement)

Because the particle velocity is the derivative of particle displacement , we write

20

(1.35)

Example 1.1 Consider a longitudinal acoustic wave with a power of 20

dBm (0.1 W) propagating in ( x ) lithium niobate (v, = 6.6 - I d d s and

p = 4.6 - I d kg/m3) Find the particle displacement at 1 GHz, 100 M E , and 10 MHz

Because (1.32) and (1.35) require the acoustic intensity, we must make reasonable assumptions about the cross-sectional area of the acoustic waves At 1 GHz the area will be between 1 lo-' m2 (5 mils x 5 mils) The acoustic impedance is

s ! For less acoustic power or larger radiating area, the displacements are even smaller The particle velocity is given by (1.34) and typically ranges from 10 to 100 c d s ; because the displacements are so small, the particle accelerations are on the order of lo7 d s 2 !

In the previous section we dealt with strains of the form auldz De- pending on the coordinates of the one-dimensional mass-spring system, there are three such terms: &,lax, du,lax, and au,laz Now consider Figure

1.5 The definitions are identical to the one-dimensional case We allow the displacement (U) to be a function of y and z as well as x This is illustrated in Figure 1.6 We write

9 = (A/)' - ( A L ) 2

(in the limit of infinitesimal displacements) From Figure 1.5, we have immediately,

dLy)i + 2dL, (” dL, + - au, dLy

From (1.40) and the definition of the deformation, we have

9 = ( 2 ) ’ d L : + (z) au, dL$ + 2 au, au, dL,dL, + 2-dL: au,

a ~ , aL, JL,

+ 2-dL,dLy au, +

a LY

au au

+ 2 dL,dLy + 2 au, dL$ + 2 au, dL,dL, (1.43)