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TEMPERATURE COMPENSATION OF MICROWAVE ACOUSTIC RESONATORS (LEAKY WAVE, SURFACE TRANSVERSE (STW), QUARTZ, LITHIUM NIOBATE, SURFACE GRATING (SAGW)

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TUT-\/f-T
Dissertation
I V l l Information Service
University Microfilms International
A Bell & Howell Information C om pany
300 N. Z e e b R oad, Ann Arbor, M ichigan 48106
8619836
Thompson, Daniel Frank
TEM PE R A T U R E C O M PEN SA TIO N O F MICROWAVE A CO U STIC R E S O N A T O R S
PH.D.
S ta n fo rd U n iv e rs ity
University
Microfilms
International
300 N. Zeeb Road, Ann Arbor, Ml 48106
1986
TEM PE R A T U R E CO M PEN SA TIO N O F
MICROWAVE A CO U STIC RESON ATORS
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF ELECTRICAL ENGINEERING
AND THE COMMITTEE ON GRADUATE STUDIES
OF
STANFORD UNIVERSITY IN
PARTIAL
FULFILLMENT OF THE REQUIREMENTS
FOR
THE
DEGREE
OF
DOCTOR OF PHILOSOPHY
By
Daniel Frank Thompson
June 1086
I certify that I have read this thesis and that in my opinion it is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
^
G. G J s
(Principal Adviser)
(Applied Physics)
I certify that I have read this thesis and that in my opinion it is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
(Electrical Engineering)
I certify that I have read this thesis and that in my opinion it is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
(Electrical Engineering)
Approved for the University Committee on Graduate Studies:
Dean of Graduate Studies & Research
DEDICATION
to
Dr. William Edward Thompson
my father
ACKNOW LEDGM ENTS
I would like to express my deepest gratitude to Professor Bert A. Auld for
his guidance and encouragement in the course of this study and especially for his
contribution to the edification of my professional character. I would also like to
express gratitude to Professor David M. Bloom and Professor Von R. Eshleman for
their active interest in this work and review of this manuscript. Just one week
before the deadline for submission of this dissertation I learned that my original
third reader would be unable to read my dissertation. After over coming the initial
shock of this announcement I began to look for a way out of this uneasy situation.
Fortunately Professor Eshleman came to my rescue by reading my dissertation in
just five days which made the timely submission of this dissertation possible. For
his special last minute effort I wish to express a special thanks to Professor Von R.
Eshleman. Thank you.
I would like to thank the many wonderful people at the Hewlett-Packard
Company. The help of people like Bob Bray, Scott Elliot, Tim Bagwell, Catherine
Johnsen, Joan Henderson, Terri McAuley, Larry Pendergrass, Marek Mierzwinski
and many others at the Microwave Technology Division in Santa Rosa, California
made the fabrication of the STW devices possible. Thanks are also due Waguih
Ishak, Mike Tan, Hylke Wiersma, Elena Luiz, King Tut and others at HewlettPackard Laboratories in Palo Alto, California, without their help the packaging
and testing of the STW grating devices would have never taken place. Thank you
a ll!
I would like to thank the late Lawrence R. Thielen and the people at Avantek for
the generous Avantek fellowship which started my doctoral program in September
1082 .
Special thanks are due Judy Clark for putting up with my constant additions
and revisions while preparing this manuscript, and to Dr. Auld for the personal
attention given to the initial review of this thesis.
iv
I also wish to thank Steve Meeks, Hal Kunkel, George Laguna, Pierre Delval,
Mark Gimple and June Wang for the hours of discussion which have shown that,
to fully understand a subject, I must be able to explain it to others.
Thanks are also due some very special people. Thank you Eric Strong, Jim
Murphy, Ron May, Kent Strong, Vera Hromadko, John Nelson and Tom Lund
without your occasional distractions during the past four years I would have surely
lost my sanity.
And finally I would like to thank my parents, especially my mother whose
encouragement and discipline have seen me though a 21 year scholastic career.
Thanks Mom for putting the light in my life which has made this all possible.
v
A B STR A C T
Temperature stability of time references is essential for dependable operation
of all synchronous electronic systems, especially those operating in the microwave
frequency region, such as space communication and radar. This study deals with
the temperature compensation of microwave acoustic resonators used as stable time
references in these and other synchronous electronic systems.
When designing a high frequency surface acoustic wave resonator, tempera­
ture compensated crystal cuts are usually selected, although they may not be op­
timum with respect to other properties—for example, minimum insertion loss, max­
imum piezoelectric coupling or maximum wavelength at a given frequency. In this
thesis a new means of surface acoustic wave resonator temperature compensation
is presented. This new method achieves temperature compensation through the
addition of a periodic corrugated grating on the surface of the resonator crystal.
Several models are developed for predicting the temperature behavior of acoustic
wave propagation beneath these surface gratings. The grating types considered
include grooves, mass loading strips, conducting strips, and combinations thereof.
Temperature characteristics of acoustic wave propagation beneath these gratings
are determined not only by crystalline material properties, mass and stiffness, but
also by the structure and dimensions of the surface grating. It is this latter property
that increases the number of temperature compensated crystal orientations available
for designing surface acoustic wave resonators of this type.
Horizontally polarized shear surface wave or surface transverse wave (STW)
propagation normal to the X-axis on rotated T-cut quartz, lithium niobate, and
lithium tantalate are considered. Grating dimensions required for temperature com­
pensating these wave orientations are presented. A combination of computer model­
ing and experimentation is used to validate the theoretical results. In addition, a
more general surface acoustic grating wave theory is developed. This theory allows
for the investigation of wave propagation under a surface grating on arbitrary crysvi
tal orientations, including leaky wave orientations on the higher coupling materials,
that have yet to be considered for their temperature properties. A computer al­
gorithm developed from this theory is used to design gratings for trapping leaky
surface waves and also gives the first exact analysis of SAW grating waves, used in
conventional SAW resonators.
TABLE O F C O N TE N TS
A c k n o w le d g m e n ts ...................................................................................... iv
A b s t r a c t .......................................................................................................... vi
T able o f C o n t e n t s ................................................................................. viii
L ist o f F i g u r e s .....................................................................................................x
L ist o f T a b l e s ................................................................................................. xiii
I.
I N T R O D U C T I O N ............................................................................................. 1
H.
G R O O V E D G R A T IN G SU R FA C E T R A N S V E R S E W AVES
A.
. .
4
B ack g ro u n d ....................................................................................................4
B. Theory for Grooved G r a t i n g s ......................................................................9
C. Dispersion Relation
....................................................................................14
D.
Near Stopband Approximation
...............................................................27
E.
Temperature C om pensation..................................................................... 35
F.
Resonator Temperature-frequency C h a ra c te ris tic s ...............................41
m . M ETA L S T R IP G R A T IN G SU R FA C E T R A N S V E R S E WAVES . 46
A.
B ack g ro u n d................................................................................................. 46
B. Theory for Metal Strip G ra tin g s ................................................................. 47
C. Dispersion Relation
................................................................................... 49
D.
Temperature C om pensation..................................................................... 53
E.
Experimental Investigation of Phase-temperature Characteristics . . 54
(1) Overview
...............................................................................................54
(2) Frequency r e s p o n s e ...............................................................................58
(3) Phase versus Temperature M easurem ent........................................... 67
(4) Determination of the Turnover T e m p e ra tu re ...................................74
viii
IV.
STATIC S T R A I N ..................................................................................... 77
A. General C onsiderations................................................................................77
B.
Static Strain T h e o r y ....................................................................................79
C. Nonuniformity of Static Strain
............................................................... 81
D. Effect on Surface Wave Temperature C h a ra c te ristic s ........................... 84
V.
VI.
T R A P P IN G LEA KY SURFACE WAVES
....................................... 90
A.
Introduction
B.
TVapping of Leaky Surface W a v e s ........................................................... 94
C.
Numerical R e s u l t s ......................................................................................100
CO N CLU SIO N
............................................................................... 90
....................................................................................... I l l
R E F E R E N C E S ........................................................................................ 113
ix
LIST OF FIGURES
1.
STW dispersion relation for grooved g r a tin g .................................................... 6
2.
STW resonator s t r u c t u r e ....................................................................................8
3.
Space harmonic dispersion curves
4.
Real part of the complex dispersion r e la tio n .................................................. 17
5.
Imaginary part of the complex dispersion relation
6.
Dispersion relation showing numerical solution c o n v e rg e n c e ......................19
7.
Real part of the complex Oth lateral attenuation c o n s ta n t ..........................21
8.
Imaginary part of the complex Oth lateral attenuation constant
9.
Real part of the complex -1th lateral attenuation c o n sta n t..........................23
10.
Imaginary part of the complex -1th lateral attenuation constant
11.
Near stop-band approximation with h /A = 40*
31
12.
Near stop-band approximation with h /A = 30*
32
13.
Near stop-band approximation with h/A = 20*
33
14.
Near stop-band approximation with h/A = 1 0 *
34
15.
Compensation factor vs. T-cut angle for Q u a r t z ..........................................38
16.
Compensation factor vs. T-cut angle for lithium n i o b a t e ........................... 39
17.
Compensation factor vs. T-cut angle for lithium ta n ta la te ........................... 40
18.
Fractional frequency change near the AT-cut
.............................................. 43
19.
Fractional frequency change near the ET-cut
.............................................. 44
20.
Fractional frequency change near the BT-cut
.............................................. 45
21.
Metal strip grating STW dispersion relation
..............................................50
22.
STW Mask d r a w i n g ......................................................................................... 55
.................................................................. 15
x
......................................18
. . . .
. . . .
22
24
23.
STW device p h o t o ..........................................................................................57
24.
73oA STW frequency resp onse......................................................................59
25.
730A STW time domain response
28*
730A STW response with feedthrough gated o u t .......................................... 65
27.
730A STW response with feedthrough and triple transit o u t ......................66
28.
1400A STW response with feedthrough gated out
29.
1400A STW response near the stopband edge amplitude and phase . . .
30.
1400A STW phase change from 2(f C to 7 0 P C ................................................72
31.
Derivative of phase vs. temperature for the 1400A S T W ..............................75
32.
Derivative of phase vs. temperature for the 3000A S T W ..............................76
33.
Typical stress vs. strain r e l a t io n ..................................................................78
34.
Bowing of the metal strip STW d e v i c e ......................................................82
35.
Fractional frequency change vs. strain due to static s t r a i n ......................... 86
36.
Fractional frequency change vs. temperature due to static strain . . . .
37.
Condition for a skimming wave on X-propagating Z-cut trigonal crys­
.................................................................. 61
...................................... 68
tals
38.
Real height-to-period ratio needed
70
89
93
to trap leaky
wave and
SAW
propagating along the X-axis of T-cut q u a r t z .............................................102
39.
Real height-to-period ratio needed
to trap leaky
wave and
SAW
propagating along the X-axis of T-cut lithium n i o b a t e .............................103
40.
Real height-to-period ratio needed
to trap leaky
wave and
SAW
propagating along the X-axis of -48?rotated T-cut lithium niobate . .
104
41. Real height-to-period ratio needed to trap leaky wave and SAW
propagating along the X-axis of 7 -cu t lithium t a n t a l a t e ........................... 105
42. Height-to-period ratio needed for surface skimming wave propagation on
rotated y*cut q u a r t z ............................................................................................ 107
43.
Height-to-period ratio needed for surface
skimming
wave propagation on
rotated 7-cut lithium n i o b a t e ..................................................................108
44.
Height-to-period ratio needed for surface
skimming
wave propagation on
rotated 7-cut lithium t a n t a l a t e .......................................................... .... .
xii
109
LIST O F TABLES
!•
1400A STW device phase temperature d a t a ..................................................71
3000A STW device phase temperature d a t a ..................................................73
CHAPTER I
IN T R O D U C T IO N
“Does anybody really know what time it is?”1
What time is it? This is an everyday question we have all asked and have all
answered many times. But does anyone really know what time it is? At one point
in man’s history, when time was referenced to the movement of the sun across the
sky and hours were measured with sundials and hour glasses, people felt they could
answer the question “W hat time is it?” But could they? The answer is that they
could, but only to a certain degree of accuracy. As man’s intellect developed so did
his desire and need to measure time with greater and greater accuracy. With the
appearance of the first mechanical clocks in the 14th century, accuracy to fractions
of minutes instead of hours of time was possible. The first mechanical clocks linked
time keeping with the frequency of an oscillator, in this case an oscillating pendulum.
With the introduction of the first piezoelectric oscillator (acoustic resonator) by W.
G. Cady in 1021, a chain of technological development began leading to the present
day digital wrist watch which allowed the common man’s time to have an accuracy
of seconds per year. Although, even with the most sophisticated atomic clocks,
accurate to a few nanoseconds per year, man still does not really know what time
it is; he has only reduced his degree of uncertainty. This brings us to the subject
of this dissertation: reducing the degree of uncertainty in the resonant frequency of
surface acoustic wave resonators.
One of the most popular inexpensive frequency references is the quartz crystal
resonator. The first quartz bulk wave resonators realized were a breakthrough
1
in the design and development of electronic circuits requiring stable frequency
references. These bulk wave resonators provided very high quality factors while
maintaining ease of tuning and inexpensive manufacturing.
One of the major
environmental parameters affecting the stability of these crystal resonators was the
temperature. It was found that the temperature characteristics of these resonators
was dependent on the bulk wave propagation direction in the anisotropic quartz
crystals. Because of the need for maximum frequency reference stability against
changes in ambient temperature, crystal orientations giving good wave propagation
temperature characteristics were sought and selected for resonator applications.
With further advances in electronics came a need for higher and higher fre­
quency references. To achieve these higher frequencies the bulk crystal resonator
dimensions had to be reduced. The resulting very small dimensions of the crystals
made manufacture and tuning of these bulk wave resonators a difficult task. To
circumvent this problem external electronics are used to multiply the low resonant
frequency of the bulk wave resonators up to higher frequencies. Using this tech­
nique the microwave frequency references used in present day communication sys­
tems are obtained. However this multiplication process adds to the phase noise of
the oscillator circuit and in some applications requiring very precise phase control
this added phase noise can be devastating. To obtain these high frequencies, while
still maintaining good phase noise characteristics and ease of production, surface
acoustic waves are used instead of bulk acoustic waves.2 The planar structure of
these surface acoustic wave resonators (or SAW resonators) provide easily manufac­
tured high frequency references with better noise characteristics than the bulk wave
resonators. The upper frequency of such resonators is limited by the photolithog­
raphy and metal deposition technology used to deposit the metal fingers of the
interdigital transducers for excitation. The sub-micron processing technology avail­
able today allows fabrication of surface acoustic wave resonators in the frequency
range of 100 to 2000 MHz.3
Of course, these SAW resonators are also subject to frequency variations due
2
to changes in the ambient temperature. To combat this problem crystal orienta­
tions giving the best temperature characteristics are required. For some crystal
orientations the temperature characteristics are such that at a given temperature
T0 the change in resonator frequency with respect to a change in temperature goes
to zero. Resonators with this characteristic are said to be temperature compensated
at the temperature T0. By selecting a particular temperature compensated crystal
orientation, the designer sometimes forfeits other criteria such as high piezoelectric
coupling (which allows a lower insertion loss) or high acoustic wave velocity (which
allows higher frequency operation with the same interdigital transducer dimensions).
If another method were available for controlling the temperature characteristics the
designer could achieve temperature compensation without compromising require­
ments for low insertion loss and high frequency response.
In this dissertation a new method of controlling the temperature behavior of
surface acoustic waves in the microwave frequency range is proposed and inves­
tigated. This involves a modification to the crystal substrate surface on which the
surface acoustic wave propagates. By introducing a periodic corrugated grating
on the surface the temperature characteristics of the surface acoustic wave can be
altered. It is found that for certain crystal materials and orientations that do not
have temperature compensated behavior, the addition of a grating can alter the
temperature behavior in such a way as to achieve complete temperature compensa­
tion. This ability to control temperature behavior by means other than the crystal
orientation allows the design of a whole new class of acoustic devices with crystal
orientations selected to optimize parameters other than the temperature coefficient
of delay and frequency.
This research, conducted at the W. W. Hansen Laboratories of Physics at
Stanford University, was sponsored by The Rome Air Development Center under
Contract No. F19628-83-K-0011. The principal RADC contract monitor was Dr.
Paul Carr.
3
CHAPTER H
GROOVED GRATING SURFACE TRANSVERSE WAVES
A. Background
The surface acoustic wave considered in this study is a horizontally-polarized
shear surface wave or surface transverse wave (STW). Since its development in 1976
this STW has sparked considerable interest.4-23,87 it has several advantages over
the Rayleigh-type surface acoustic wave (SAW) used in conventional SAW devices.
In some materials the STW propagation velocity is much faster than the SAW.
This higher acoustic wave velocity allows higher frequency devices using the same
interdigital transducer geometries. In conventional SAW devices the acoustic wave
penetration is less than an acoustic wavelength, which gives rise to very high acoustic
energy density near the surface. These high energy densities tend to cause nonlinear
effects and increase the noise levels of the SAW device.24 In STW devices the depth
of acoustic wave penetration is adjustable, allowing the STW devices to be operated
at much higher power levels giving better noise performance. In SAW devices there
exists a source of acoustic interference caused by spurious SAW reflection from
the interdigital transducers. Because the propagation region of the STW can be
made to match the IDT regions this source of acoustic interference, called the
triple transit, can be eliminated. It is these advantages, as well as the ability to
temperature compensate the STW, that make it such an attractive alternative to
the conventional Rayleigh-type SAW.
It has been previously shown that horizontally-polarized shear surface waves
(STW) can exist on a semi-infinite substrate with a periodic corrugation or grating
4
on the surface8,9. This phenomenon differs from Rayleigh wave propagation, on
structures currently used in SAW resonator applications, by the fact th at the
shear surface wave does not exist in the absence of the grating. For a smooth
semi-infinite isotropic substrate a shear surface skimming bulk wave (SSBW) does
satisfy the stress-free boundary conditions.12 This SSBW solution can also exist for
certain orientations of anisotropic substrates. Placing a corrugation or grating on
the surface slows down this SSBW in a manner entirely analogous to that in the
corresponding electromagnetic problem,25,28 and thereby converts it into a shear
surface wave—a Surface Transverse Wave (STW). A SSBW, and thus a STW, can
exist on trigonal crystal plates if propagation is normal to the X-axis. A basic wave
theory for this geometry neglecting piezoelectricity has been developed by Renard
and will be referenced later in this discussion.11
The surface transverse wave (STW) has several inherent advantages over the
Rayleigh-type SAW: (1) Higher acoustic velocity, leading to larger IDT periodicities
at the same frequency; (2) Low propagation loss; and (3) Temperature characteris­
tics that are dependent on the grating structure used to trap the wave energy on
the surface, as well as on the crystal orientation. By properly choosing the grat­
ing height, this third advantage allows one to predict temperature compensation
in quartz for a STW propagating normal to the X-axis for almost all rotated In­
cuts. Compensation can also be achieved for several rotated y-cut angles of lithium
niobate and lithium tantalate crystal plates. However, the piezoelectric characteris­
tics of these higher coupling materials are not the same as for quartz, and the
STW orientation stated above is not piezoelectrically active. Therefore, external
transducers would be required to excite the STW in these cases.
This grating temperature compensation effect can be explained by referring to
the dispersion curve in Fig. 1. In the absence of the grating, the dispersion curve is
the (SSBW) line. When the temperature is changed, the slope of this line changes
because of changes in density and elastic constants. In the presence of the grating
a stop-band appears (£cj), with a width that is a function of the crystal stiffness
5
' S S BW LINE( j j / 0 =VS (T)
UJ A
o>o
LlJ ^
(FIXED)
S T W LINE
t u / f t = V g j ^ (T)
7T /A (T )
FIG U R E 1
G rating t e m p e r a t u r e c o m p en satio n m ech an ism for STW. The effect of temperature
on th e SSBW lin e i s b a la n c e d by changes in the stopband width and position on
the /? axis.
6
constants, the crystal density, and the grating dimensions. The stop band refers
to those frequencies at which the STW will not propagate unattenuated along the
surface. Instead of forward propagation, the STW is reflected back in the reverse
direction due to the periodic nature of the grating structure in a way analogous to
Bragg scattering. As the temperature changes, the width of this stopband changes.
To achieve temperature compensation, the width of the stopband must change with
temperature in a way that compensates for the changing slope of the SSBW line,
so that an operating point on the STW curve (uir ) remains stationary. This can
be achieved in many cases by selecting appropriate grating dimensions for a given
crystal type and orientation.
It has been found that the theoretical temperature compensation achieved for
STW propagation normal to the X-axis on rotated V-cut quartz is comparable
to, and for some orientations surpasses that for, the AT cut surface skimming
bulk waves.5 Temperature compensation of STW propagation normal to the X-axis
on rotated T-cut lithium niobate and lithium tantalate crystal plates can also be
achieved with deep gratings. However, these crystal orientations are unfortunately
not piezoelectrically active in the cases of lithium niobate and lithium tantalate and
thus external transducers would be required to excite the STW.
The STW analysis referred to above considered propagation on corrugated sur­
faces of rotated T-cut trigonal class crystals with grating dimensions corresponding
to the cavity region (Lr ) of Fig. 2, where the STW is in a pass-band. The method
used was to apply Floquet’s Theorem, which gives the general form of the charac­
teristic wave solutions, separately to the semi-infinite substrate and to the grating.
Application of appropriate boundary conditions at the grating-substrate interface
then gives the dispersion relation and the relative amplitudes of the various space
harmonics. In applying the boundary conditions a shallow grating was assumed,
and the stress at the bottom of a tooth was calculated using the Datta-Hunsinger
Perturbation Formula.27 A review of this STW theory follows, along with a detailed
analysis of STW behavior within the stop band, a comparison of the near stop-band
7
YC
-
h-W
V
/
, z Cc
Sl - N
Y
L■m
■m
F IG U R E 2
STW grating resonator structure on a rotated T-cut quartz plate with grating tooth
width W = A/2.
8
approximations made in the Renard theory with the exact numerical calculation,
and an investigation of the temperature characteristics of STW propagation.
B. Theory for Grooved Gratings
Starting from the general acoustic field equations, one can derive the following
differential acoustic wave equation governing horizontal shear wave propagation
normal to the X-axis in trigonal rotated V-cut crystal substrates.
d 2vx , n
d2vx ,
d2vx
d 2vx
+ i C m a ^ z + CM' d ? ' = p 'd i*
where
vx
Cij
is the particle displacement velocity
are rotated stiffness constants
p
x, y, and z
is the density of the trigonal crystal
are the space coordinates
t
(l)
is the time coordinate
A solution based on Floquet theory, consisting of an infinite sum of space harmonics,
is assumed in the substrate (Y > 0 in Fig. 2).
oo
vx =
a„te-a„y e-i0 nz j u t
n— oo
(2)
* - /» + *=
where
An is
Pn is
is
0 is
u
the nth space harmonic amplitude
the space harmonic propagation constant
the lateral attenuation constant into the substrate
the Floquet wave propagation constant
is the steady state angular frequency
A is the grating period
Each space harmonic solution is substituted into Eq. 1, resulting in the following
relation for the nth space harmonic lateral attenuation constant
Caa
V c®6
c®8
(3 )
Because of the double root behavior of the square root term in Eq. 3, there is some
ambiguity as to which sign should be selected. This ambiguity can be resolved by
referring to the Floquet solution for the wave displacement velocity (Eq. 2). Here
it becomes clear that, for the wave displacement velocity contribution from the nth
space harmonic term to remain finite, as Y goes to infinity, the sign must be selected
so that the real part of a n is positive. Space harmonics that satisfy this condition
are referred to as “proper” space harmonics. Those space harmonics that become
infinite as Y goes to infinity are referred to as “improper” space harmonics. In
some theoretical discussions of wave propagation these improper space harmonics
are used to describe wave radiation.28 For a practical radiating device such as a
phased array antenna, the electromagnetic fields extend over a finite region, and
thus the improper space harmonics can only be considered within a finite finite
region, eliminating the energy concerns at infinity.28 Further discussion of proper
and improper space harmonics, together with leaky STW’s, appears in Chapter II
section C. For now, both the plus and the minus sign will be retained.
The space harmonic amplitudes an ’s which make up the acoustic wave (Eq.
2) must be selected so as to match the periodic boundary stresses imposed by
the surface grating over the entire surface.
Because of the periodic nature of
this structure, it is a necessary and sufficient condition for the space harmonic
amplitudes to match the surface boundary condition over just one period. Thus only
the region of a single period need be considered. The tangential stress imposed by
the tooth of the grooved grating is given by the following Datta-Hunsinger interface
stress relation27
(4)
n „ = ;V * » « where
T®y
vx
T'xt
h
is the stress in the crystal at the interface Y = 0
is the particle displacement, which is the
same in the tooth as in the substrate
because of the continuity of displacement
boundary condition at Y = 0
is the stress in the grooved tooth at the interface Y — 0
is the height of the tooth
pf
the density of the tooth
u
is the angular frequency
In the case of the grooved grating STW being considered here the density pf
and the stress Txz are the same as those in the substrate. The stress Txz in the
substrate, and thus the stress in a grating tooth T'xz, can be calculated from the
acoustic field equations and expressed in terms of the displacement velocity as
T> —
>x.
xt
ioj dz
iu dy
1'
To calculate the derivative of the stress in the grating tooth T'xg with respect to
2,
it is noted that the stress has a uniform value multiplied by the wave propagation
term exp(—jfinz) over the width of a single tooth (z = 0 to z = W in Fig. 2), and
abruptly goes to zero at the edges of each grating tooth
(2
= 0 and z = W). These
abrupt changes in the stress terms lead to the excitation of evanescent modes around
the edge of each grating tooth which give rise to second-order (fih)2 stored-energy
effects.27 In the present first-order theory these effects will be neglected and the T'xz
stress term is assumed uniform over the width of a grating tooth, abruptly going to
zero at the tooth edges. Because of these abrupt changes the derivative gives rise
to ^-function stresses at the edges of each tooth. When this result is combined with
Eq. 5, the interface stress relation Eq 4 becomes
(ju p h vx - &[$]
=
{\
0
<
2
< A
( 6)
%< z <A
0
11
w h ere
d 2vx ,
4 ’ = c“
d 2vx ,
^
a ? ' + ,:M a ^ I +
- 5))
T h i s surface stress must equal the acoustic field stress in the substrate at Y = 0
in F ig . 2. The acoustic field stress in the substrate is found from the acoustic field
e q u a tio n s and can be expressed in terms of the displacement velocity as
m
_
C58 9 vx t C88 d v x
*xy — “
q
\u o z
r
q
xu ay
{rrX
[•)
To obtain th e a n ’s in Eq. 2, the boundary stress at the surface of the substrate,
E q . 6 is set equal to the boundary stress under the grating Eq. 7. This is converted
t o a set of linear algebraic equations for the an’s by using the orthogonality of the
d if f e r e n t space harmonics. This is done by multiplying each side of the condition
e q u a t i o n by the complex conjugate of the qth. space harmonic term exp[/?g2 ],and
t h e n in tegrating over a single period from Z — 0 to Z = A. If Eq. 3 is used to
e lim in a te the lateral attenuation constants a n, the following infinite set of linear
e q u a tio n s is obtained
± \ J ce f f P % - p u 2 a q = 53 “n K n q
w h e re
(8)
9 = . . . - 2 , - 1, 0 , 1, 2 ..
ce/f
=
C55C68 ~ c58
caa
T h e p lus/m inus ambiguity of the square root will be carried throughout the theoreti­
c a l analysis. T he correct sign will be selected in the numerical analysis to give the
p r o p e r behavior of the lateral attenuation for each space harmonic, as discussed in
c h a p t e r II section C.
12
Equation 8 is the so-called space harmonic equation for STW propagation in
trigonal crystals. The coupling terms K nq, found by evaluating the integral over a
grating tooth, are
{ AtL x /cA«...6
n= q
K nq —
where
n ^q
(9 )
+ En
* A ^ f e l C 0S * ( n - 9 ) - L °1
p u 2 - ce f f 0 l ± i-^ = = P n yJce f f P l - p u 2
E n — CeffPn ±
CeffP 2n - pu2
To simplify the format, the variables in these equations are normalized to the
grating period and the SSBW velocity Vt ,
uA
/?nA
IT
where
This gives the following normalized space harmonic equation.
± \ J p * - u 2 aq = J 2 anK nq
where
W
^ »
( 10)
t^[cos7r(n —g) —1]
and
The characteristic determinant of this set of equations defines the dispersion relation
(cD = /(/?)) for the STW and will be discussed in more detail in the following
13
section. This set of space harmonic equations will also be used as a starting point
in the derivation of the analytical expression for the STW phase velocity found by
Renard.11 A derivation, with a discussion of the simplifying assumptions for the
near-stopband approximation, will be discussed in a later section.
C. Dispersion Relation
An investigation of the general STW dispersion relation is needed to understand
STW behavior in the region where the propagation constant /? is complex (in, and
above, the stopband). Questions concerning the trapping of the STW arise and need
to be considered for a complete description of STW propagation. Complex wave
propagation in periodic corrugated structures has been previously considered for the
case of electromagnetic waves,28,30,31 but, as yet, there exists no such description
for the acoustic problem. In this section wave propagation in the stopband and
at frequencies inside the fast wave region (above the SSBW line in Fig. 1) will be
considered, to explain the trapping characteristics of a complex STW.
The STW dispersion relation is found from the general normalized STW space
harmonic equation Eq. 10. In this investigation Eq. 10 is solved for the dispersion
relation {Q vs. /?) using numerical techniques. Because Eq. 10 is an infinite set of
equations the number of equations considered must be truncated. As the number
of space harmonics considered is increased the numerical solution approaches the
true solution, Eq. 2, consisting of an infinite sum of space harmonics.
Because of the nature of this problem a multiple root solution is obtained. Each
space harmonic considered introduces another quadratic factor in ]3. Thus a solution
found using 10 space harmonics will have 20 /? roots for each value of cD. This kind
of multiple root behavior is illustrated in Fig. 3, where Fig. 3(a), shows dispersion
curves for positive and negative waves in the absence of the grating (SSBW). When
multiple space harmonics are considered, multiple roots occur, as shown in Fig. 3(b).
The values of w correspond to two roots for each space harmonic. For example, if
6 space harmonics are considered, 12 different roots are found.
14
(a)
-S IN G L E
SPACE HflRMONIC RESPONSE
Bo
(b)
- M U L T IP L E
SPACE HARMONIC RESPONSE WITH
COUPLING BETWEEN ADJACENT HARMONICS
-Lo
0
FIG U R E 3
Space harmonic dispersion curves.
15
B,
A numerical Fortran program was developed to solve Eq. 10 for a finite number
of space harmonics. This program combines several general matrix-manipulation
and root-finding routines from Argonne National Laboratory and the International
Mathematical and Statistical Libraries, along with several routines developed at
Stanford University. At present the program can only consider grooved grating
STW propagation on trigonal crystal substrates, neglecting piezoelectricity; but,
because of the structure of the program, it can be easily adapted to consider
arbitrary crystal anisotropy and piezoelectricity.
Before continuing with the discussion, a numerical example will be given. To
maintain simplicity, the isotropic material silicon dioxide is selected for this example.
Figure 4 and 5 show the real and imaginary part of the 0th space harmonic complex
dispersion relation for STW propagation on a silicon dioxide substrate with a grating
height-to-period ratio of 30%. The real part of the dispersion curve gives the real
part of the propagation constant along the surface of the substrate. Note that
the first stopband appears at
= 1. The imaginary part of the curve shows the
attenuation of the propagation constant as a function of frequency. It can be seen
that below the stopband there is no attenuation of the STW. However, in and
above the stopband, attenuation does exist. This is due to energy coupling between
forward and backward space harmonics.
The curves in Fig. 4 and 5 were calculated using 20 space harmonics. To give
an idea of the accuracy of this approximation, curves showing the lower edge of the
stopband for 2, 4, 6, 10, and 20 space harmonics are given in Fig. 6. The results
quickly converge as the number of space harmonics is increased. Using more than 20
space harmonics does not add appreciably to the accuracy of the solution. For this
reason only 20 space harmonics will be considered in the remainder of the analysis.
In the numerical calculation a choice must be made for the sign in the square
root terms of Eq. 10. This sign selection is very important because it determines the
behavior of the wave function with increasing depth into the substrate. In general
both the plus sign and the minus sign give correct mathematical solutions. However
16
REAL P A R T
0M E 6A
1.4
N0RMALIZED
FREQUENCY
1.2
SSBW
1.0
•
STW
0.8
3 0 7. HEIGHT T0
PERIOD RATIO
0.6
1
1.2
NORMALIZED PROPAGATION C O N ST A N T BETA
0.6
0.8
FIG U RE 4
Plot of the Real part of the complex dispersion relation for Silicon Dioxide, using
20 space harmonics and a grating height-to-period of 30%
17
I MAGI NARY P A R T
<
STW
1.4
CD
LU
z:
is
>u
z:
30X HEIGHT T0
PERI0D RATI0
1.2
LU
Z)
a
LU
cr
a
LU
ru
O'
s
z:
1.0
0.8
0.6
0
-0 .0 5
- 0 .1 5
- 0.1
NORMALIZED PR O PA G A TIO N CONSTANT BETA
•
0.2
FIG U R E 5
Plot of the Imaginary part of the complex dispersion relation for Silicon Dioxide,
using 20 space harmonics and a grating height-to-period ratio of 30%
18
30Z HEIGHT T0 PERI0D RATI0
0 .9 0
2 SPACE HARM0NICS
<
o
z:
is
>u
z
111
3
a
UJ
oc
u.
a
UJ
M
*—t
lli
0 .8 9
0.88
4 SPACE HARM0NICS
0 .8 7
10 SPACE HARM0NICS
_l
<
7Z
OC
SI
z
20 SPACE HARM0NICS
0.86
0 .95
1.05
NORMALIZED PROPAGATION CONSTANT BETA
FIG U R E 6
Real part of the Dispersion relation Eq.10 near the lower edge of the stop-band.
Note the convergence of the solution as the number of space harmonics is increased.
10
only certain mathematical solutions correspond to physical complex wave solutions
satisfying the physical boundary conditions. In the slow wave region (below the
SSBW line in Fig. 1), the STW is a true surface wave, which propagates along the
surface unattenuated and decays into the substrate. This solution requires all space
harmonics to decay with depth (i.e., proper space harmonics). The signs in Eq.
10 are selected to be positive in this region. As in the electromagnetic references
cited, it is found that, in order to maintain continuity in the solution from the slow
wave region to the fast wave region, all space harmonics must remain proper for
]3 < 2. For values of ft > 2, the real part of the lateral attenuation constant o _ i
becomes negative, giving an improper or leaky -1st space harmonic. This improper
space harmonic radiates energy away from the surface, resulting in a leaky STW.
A detailed discussion of this radiation phenomenon for the electromagnetic wave
is given in Reference 28. The same result is found for the STW. For the analysis
presented here, only $ values near the first stopband will be considered. Thus only
proper space harmonics need be included in the numerical analysis.
This proper complex solution near the first stopband can be better understood
by referring to a plot of the lateral attenuation constant for the 0th and -1st space
harmonics. Figures 7,8,9 and 10 show these attenuation constants as a function of (D.
It can be seen in these figures th at the imaginary part of the attenuation constant
remains zero below the stopband, corresponding to the purely real attenuation
constant expected of a surface wave. However, inside the stopband this lateral
attenuation constant becomes complex, suggesting a wave propagation component
in the lateral direction. But the sign of the complex component corresponds to
phase delay toward the surface, not away from the surface, (as would be expected
for a leaky wave). To understand this behavior a solution consisting of just two
space harmonics will be examined.
In the case of just two space harmonics the particle displacement velocity
solution Eq. 2 will have the form
20
REAL P A R T
<
1.4
CD
UJ
H
(SI
>
u
z:
STW
30X HEIGHT T0
PERI0D RATI0
1.2
UJ
=3
a
UJ
cz
u.
a
1.0
UJ
rvi
0.8
a:
s
0.6
0 .4
0.8
1
0.6
0.2
0
OTH N0RMALIZED A T T E N U A T I0 N C 0 N S T A N T ALPHA
F IG U R E 7
Plot of the real part of the 0th space harmonic, .normalized lateral attenuation
constant &o versus the normalized angular frequency Q.
21
0MEGA
I MAGI NARY P A R T
STW
1.4
3 0 1 HEIGHT T0
PERIOD RATIO
N0RMALIZED
FREQUENCY
1.2
1.0
0.8
0.6
•1
-0 .4
0.8
0.6
0.2
0
OTH NORMALIZED ATTENUATION C O N ST A N T ALPHA
-
-
-
F IG U R E 8
Plot of the imaginary part of the Oth space harmonic normalized litera l attenuation
constant fi0 versus the normalized angular frequency u).
22
REAL P A R T
<
1.4
CD
30K HEIGHT T0
PERI0D RATI0
LU
z
s
>u
z.
STW
1.2
id
=)
a
LU
on
1.0
Ll
a
LU
rxj
*—t
0.8
<
z
CL
IS
z
0.6
0
0.2
0 .4
0.6
1
0.8
-1TH NORMALIZED A T T E N U A T IO N CONSTANT ALPHA
F IG U R E 0
Plot of the real part of the -1st space harmonic normalized lateral attenuation
constant &_i versus the normalized angular frequency Q.
23
I MAGI NARY P A R T
<
1.4
CD
UJ
STW
30X HEIGHT T0
PERI0D RATI0
X
SI
>u
z
1.2
UJ
ID
a
UJ
cr
Ll
Q
M
i—i
_l
<
X
UJ
0.8
01
SI
z
0.6
0.2
0 .4
0.6
0
0.8
1
-1TH NORMALIZED ATTENUATION C O N STA N T ALPHA
FIGURE 10
Plot of the imaginary part of the -1st space harmonic normalized /aterai attenuation
constant fi_i versus the normalized angular frequency Q.
24
Below the stopband the lateral attenuation constants are pure real and the propaga­
tion constant /? is pure real. Thus the STW is a trapped wave below the stopband
with a slowing (or trapping factor) related to the real parts of the lateral attenuation
constants. Far below the stopband the amplitude of the Oth space harmonic is much
greater than that of the -1st space harmonic. This is a solution that propagates
energy along the surface of the substrate in the + z direction. However, at the
bottom edge of the stopband the coupling between the forward and reverse going
space harmonics is very strong. At this point the -1st space harmonic amplitude
equals the Oth space harmonic amplitude, and the result is a standing wave that
does not propagate energy along the surface. This solution has the following form
for the displacement velocity
v = ae~Rv cos ^/3(z +
(12)
>
where
a
—
lao| = |a -i|
R = a Q= a _ i =
Pure real
Note the forward Oth space harmonic is propagating in the + z direction, the 1st space harmonic is propagating in the —z direction, and the magnitude of the
displacement velocity decays in the Y direction at the same rate for both the Oth
and -1st space harmonics.
The solution within the stopband exhibits a different behavior. It is still a
standing wave ( |a0| = |a_ i|), but the energy trapping effect of the grating is slowly
25
reduced until, at the top of the stopband, the trapping is no longer present. This
can be seen in Fig. 7 and Fig. 9, where the real part of the Oth and -1st lateral
attenuation constants go to zero at the top of the stopband. The displacement
velocity solution at the top of the stopband corresponds to a tilted standing wave
of the form
v = a sin
—I y j
(13)
with
a = |a0| = |a _ i|
i l = a 0 = —a _ i =
pure imaginary
The wave energy in this standing wave is no longer trapped by the grating,
yet it is not truly radiating as a leaky wave either. For a leaky wave the lateral
attenuation constants would have to be negative. This condition of no trapping
is the same as for the solution in the absence of the grating, the SSBW solution.
The SSBW is not a trapped wave and yet is not a leaky wave either because the
lateral attenuation constant is zero and the acoustic energy is evenly distributed
throughout the substrate. Because of this loss of wave trapping, it is to be expected
that the insertion loss of a STW grating device at the top of the stopband differs
from the loss at the lower edge of the stopband. This is because the STW is closely
confined to the surface at the lower band edge and interacts strongly with the
interdigital transducer. At the upper band edge the STW penetrates deep into the
substrate, like a SSBW, and thus has a weaker interaction with the transducer.
This is what is observed experimentally, and will be discussed later.
An understanding of this trapping behavior of the STW is very important when
designing practical devices. The designer must realize that the trapping effect of the
grating is reduced to zero at the top of the stopband, thus limiting the bandwidth of
26
the device. If this effect is considered during the design of STW devices, satisfactory
performance can be achieved.
D. Near Stopband Approximation
If the surface height h of the grating is small and the analysis is limited to
only those values of w and /? near the first resonant point, then the near-stopband
approximation can be made. This approximation includes two basic assumptions.
The first is th at there are only two nonnegligible space harmonic terms and the
second is that the coupling terms in the space harmonic equations are essentially
constant.
The following is a detailed examination of these assumptions and a
comparison of the approximate solution with exact numerical calculations using
the Fortran program discussed in the preceding section.
When there is no grating (h = 0), the right-hand side of Eq. 10 is zero and the
following dispersion relation is obtained.
(,)■ = .
where
Pq = p + 2q
This is the SSBW dispersion relation, but periodically replicated for each value
of q. This is exactly what is expected, since the SSBW is the wave solution in the
absence of the grating. Figure 3(a) shows a graph of Eq. 14 for this case; and Fig.
3(b) shows the SSBW dispersion curves repeated for each space harmonic. (Note
that all of the curves are plotted against the same abscissa fo-) The points where
pairs of curves cross correspond to resonant points between the two space harmonics
comprising the two crossing curves. For example, the point where the forward-going
Oth space harmonic and the backward-going -1st space harmonic lines intersect (R
in Fig. 3) corresponds to the resonance point ft = 1.0 and Q = 1.0 (or, in the
unnormalized parameters, ft = n /A and ui = Vait/A). At this point the left hand
side of Eq. 10 is equal to zero for q = 0 and q = —1, and non-zero for all other
27
values of q. Thus, the magnitude of the Oth and -1st space harmonics have to be
much much greater than any other space harmonics in order to satisfy the space
harmonic equation.
The amplitudes of the space harmonics, of course, cannot become infinite.
Solving Eq. 10 shows th at coupling between the resonant harmonics results in a
bending of the dispersion curves away from the resonant point noted. This creates
a prohibited frequency band or “stopband” where the STW will not propagate, as
shown in Fig. 3(b).
If the height-to-period ratio is small and Q is restricted to values near the center
of the stopband and /J is limited to values near the stop-band. T hat is,
Y<1
A
<D«1
1
In this case, the following terms in Eq. 10,
and
are much smaller than the terms
and
____
Thus the coupling constants in Eq. 10 can be approximated
n = q
+*^(cos7r(n —g) —1] K
Substituting these values for the coupling constants into the space harmonic
equations, and keeping only the Oth and -1st space harmonics terms, reduces the
28
infinite set of equations Eq. 10 to the following coupled linear equations
(15)
These equations are analogous to those in the well-known coupled wave theory used
to describe SAW propagation on similar structures.32
To calculate the STW dispersion curves from these coupled wave equations, a
small frequency perturbation 6<D, and a small propagation constant perturbation
6ft are assumed relative to the Oth and -1st space harmonic resonance point at R
in Fig. 3. These perturbation parameters
Q = 1 + 6Q
ft0 — 6ft + 1
ft- i = 6 f t - l
are substituted into Eq. 15, and it is noted that, when the second order terms 6<D,
and 6ft are neglected, the resonance factors become
ft2 — <D2 ^ 2(6ft — 6Q)
ft2_ l — Q2 £& —2{6ft + 6(D)
and the following near-stopband space harmonic equation is found.
(16)
± \ / —2(6 ft + 5cD) d —i + i2 —»/ - ^ ^ ( l + 6ft) = 0
v
A V c68
Writing Eq. 16 in matrix form gives
±\/2{6ft-6<D )
i2£
f <*o=
. '■ 2 j y W |1 + W
0
+
The dispersion relation for these equations can be found from the zero deter­
minant condition, giving
. 2
Neglecting the second order term 6/3 on the right hand side and squaring both
sides then leads to the near stopband dispersion relation.
6 f-6 u 2= R 2
VV
c68
,l8 )
Substitution of the unnormalized parameters into this result gives the coupled wave
dispersion relation found by Renard11
6/32 = {6u/V,)2 - K 2
(19)
where the surface skimming bulk wave (SSBW) velocity is
V2 =
(C56C88
“ C2a)/pC6s = Cef f / p
the space harmonic coupling constant is
K = 2(TT)pV2
Aces
and
h is the height of the grating
A is the period of the grating
p
cxx
is the density of the crystal
are the crystal stiffness constants
This near-stopband approximation dispersion relation Eq. 18 is plotted in
Figs. 11,12,13 and 14 for height-to-period ratios of 40%, 30%, 20%, and 10%,
respectively. In each case the exact dispersion relation found by solving Eq. 10 with
20 space harmonics is also plotted in each figure as the dashed line. It can be seen
that the approximation is very good for grating height-to-period ratios of less than
10%. For ratios greater than 30% the approximation has an error greater than 1
part in 10. As the ratio is increased beyond 40% the error increases dramatically.
Consequently, the near-stopband approximation should be considered valid only for
grating height-to-period ratios less than 20%.
30
4 0 2 HEIGHT T0 P E R I 0 D R A T I 0
FREQUENCY
0MEGA
1 .0
0 .9
EXACT S0L U T I0N
0.8
N0RMALIZED
0 .7
0.6
NEAR S T 0 P BAND APPR0XIM ATI0N
0 .5
1.2
1
N0RMALIZED PR 0PA G A TI0N C 0N ST A N T BETA
0.6
0.8
FIG U R E 11
Comparison of the near stop-band approximation with the exact numerical STW
solution for a grating height-to-period ratio of 40%.
31
.
3 0 7 HEIGHT T 0 P E R I 0 D RATIO
0MEGA
1.0
0 .9
FREQUENCY
EXACT SOLUTION
0.8
N0RMALIZED
0 .7
NEAR STOP BAND APPROXIMATION
0.6
0 .5
0.6
0.8
1
1.4
1.2
NORMALIZED PROPAGATION CONSTANT BETA
FIGURE 12
Comparison of the near stop-band approximation with the exact numerical STW
solution for a grating height-to-period ratio of 30%.
32
2 0 X HEIGHT T 0 P E R I 0 D R A T I 0
1 .0
FREQUENCY
0ME6A
EXACT S0L U T I0N
0 .9
0.8
N0RMALIZED
0 .7
NEAR S T 0 P BAND APPROXIMATION
0.6
0 .5
0.6
1.4
1
1.2
NORMALIZED PROPAGATION CONSTANT BETA
0.8
FIGURE 13
Comparison of the near stop-band approximation with the exact numerical STW
solution for a grating height-to-period ratio of 20%.
33
10X HEIGHT T 0 P E R I 0 D R ATI 0
1 .0
FREQUENCY
0MEGA
EXACT S0L U T I0N
0 .9
0.8
N0RMALIZED
0 .7
NEAR S T 0 P BAND APPROXIMATION
0.6
0 .5
0.6
0.8
1
1.4
1.2
NORMALIZED PROPAGATION CONSTANT BETA
F IG U R E 14
Comparison of the near stop-band approximation with the exact numerical STW
solution for a grating height-to-period ratio of 10%.
34
Using the above near-resonance approximation, an analytical expression for the
velocity of pure STW propagation normal to the X-axis on rotated y-cut trigonal
crystals is
V ...
---------------- — --------------r
(2 0 )
where the Bragg, or center stopband, frequency (corresponding to the resonance
point R in Fig. 3 ) is
oj0 = jtV,/A
and the STW operating frequency is u.
In the low-frequency passband of the grating (the heavy line in Fig. 1) the
velocity in Eq. 10 must be pure real. From the equation, this obviously requires
that
(wb - u ) 2 > ( V aK ) 2
(21)
Note that the lower and upper edges of the stopband are defined by an equality
sign in Eq. 21. Since the temperature and rotation angle dependence of Vs, ojo, and
K can be obtained from the temperature dependence of the material constants and
grating dimensions, the temperature dependence of the STW velocity can thus be
calculated. Knowing the temperature characteristics of the STW velocity, a grating
can be selected to temperature compensate the surface wave at room temperature,
for crystal cuts at different rotation angles. This will be presented for both grooved
grating and mass loading grating structures in the following sections.
E. Temperature Compensation
The three quantities normally used as a measure of temperature stability
in SAW devices are the temperature coefficient of delay (TCD), the fractional
time delay change A t/t, and the fractional frequency change AF / F 33. In this
investigation of the temperature characteristics of STW, the TCD is used in the
selection of the grating height, and the fractional frequency change A F / F is used to
compare the frequency stability of the various materials and compensation methods.
35
The TCD is defined as34
(22)
where Vatw is the STW velocity and or is the expansion coefficient in the direction of
STW propagation, and T is the temperature. To better facilitate the prediction of
a grating height that achieves a zero TCD crossing at room temperature, a scheme
for relating the TCD curves to the crystal rotation angle was developed, with the
grating dimensions and operating frequency as parameters. If only small changes
with temperature are considered, and higher order terms are neglected when ever
possible, the TCD at frequency u r in Fig. 1 can be approximated as
TCD =
H [ 2 ( V g t e — A te)
Caetc
2/ttc ~l~ Ptc] — J ^ $ t e
A te]
Fo-Fl
(23)
where
V9te
Ate
Pu
and caste are the surface skimming wave velocity and
66 stiffness temperature coefficients in the rotated coordinates;
and htc are the linear expansion coefficients for the
period and height in the rotated coordinates;
is the temperature coefficient of density of the crystal
This expression can be evaluated for the temperature dependence of the TCD, since
the temperature coefficients for c 8b , h and A are known experimentally.36,38 The
skimming bulk wave velocity tem perature coefficient is calculated (from Eq. 19) to
be
C56C5Stc ~ (2c§fltc —C88tc)C56/ca8
- pu/ 2
This equation for the surface skimming bulk wave velocity temperature coefficient
was checked by substituting the stiffness constants for quartz38 and comparing
with published curves for the SSBW temperature coefficient.12 An exact match was
found.
36
Setting the TCD equation Eq. 23 equal to zero gives a condition on the fre­
quency cor in Fig. 1, the crystal rotation angle 9 in Fig. 2, and the grating dimensions
required to achieve a TCD zero crossing. This condition is
J — HA
(24)
where J and H are defined in Eq. 23 and A (the compensation factor) is
A =
2h t e
C68te
Ate
+ P te —
)]
(25)
The compensation factor is a measure of the grating height required for temperature
compensation. The greater the compensation factor, the greater the height of the
grating required to achieve zero TCD. Since J is positive, it is clear from Eq. 24
that only a rotation angle with a positive compensation factor A can be used to
achieve temperature compensation with a grating.
Plots of the compensation factor A vs. T-cut rotation angle are given for
quartz, lithium niobate and lithium tantalate in Figs. 15,16 and 17. Because A for
quartz is positive for almost all the T-cut angles, the STW can be temperature com­
pensated at any of these angles by properly selecting the grating dimensions. The
compensation factor goes to zero at an angle of -5(F and again at an angle of 36°. At
these angles the grating height for compensation is zero because these orientations,
which correspond to the SSBW AT and BT cuts, are already temperature compen­
sated in the absence of the grating. Using this theory, grating dimensions that
will temperature compensate STW propagating normal to the X-axis on rotated
T-cut quartz at essentially all rotation angles can be determined. In the following
section a few of the better known T-cut quartz orientations have been selected for
comparison of the STW temperature properties with those of the SSBW AT and
BT cuts.
This result has applications in many areas where temperature behavior is of
major concern. For example, in reference 37 the temperature characteristics of
acoustic filters are used to compensate for temperature changes in the timing-loop
37
AT
BT
0
-90
0
90
Y-CUT ROTATION ANGLE
FIG U RE 15
Compensation factor A (Eq. 25) versus Y"-cut rotation angle for quartz. Grating
compensation is possible only for A greater than zero (-90° to -5 (f, -4(f to 35°, 35°
to 9(f).
38
COMPENSATION FACTOR FOR LITHIUM NIOBATE
VS
Y-CUT ROTATION ANGLE
-9 0
90
ANGLE
FIG U R E 16
As In Fig. 15, for lithium niobate (-54° to 41°).
30
a
VS
COMPENSATION FACTOR FOR LITHUM TANTALAT
VS
Y-CUT ROTATION ANGLE
COMPENSATION
FACTOR
<
o -go
90
ANGLE
FIG U R E 17
As in Fig. 15, for lithium tantalate (-90° to 5°,
40
12 °
to 83°).
electronics of undersea fiber transmission systems.
Presently, the temperature
characteristics of these filters are controlled by adjusting the orientation of the
quartz substrate—a process which does not allow for slight modifications of the
temperature behavior once the crystal substrates are delivered. If STW filters
were used, the temperature characteristics of the filters could be modified after
the original design by changing the grating dimensions (a process which would
not require ordering quartz wafers cut at different orientations). In addition, the
ability to control STW temperature behavior with the grating dimensions allows the
designer freedom to select the ET-cut (maximum piezoelectric coupling) for lower
insertion loss.
F. Resonator Temperature-firequency Characteristics
Fractional frequency change (AF / F ) versus temperature curves for a resonator,
are calculated from the resonator resonant frequency condition, and are referenced
to the turnover frequency Fo. The turnover frequency is the frequency at which the
temperature derivative of frequency goes to zero. The resonant frequency condition
for a STW resonator is
F{T) = N V ttw/ 2Lr
(26)
where Vttw is the STW velocity, Lr is the length of the resonator region in Fig.
2, and N is an integer. From the temperature coefficients available for quartz in
the literature 35 the temperature dependence of the Vstw and Lr are calculated and
used to evaluate
AF / F =
Fit) ~ Fo
ro
(27)
for various grating compensation geometries determined by Eq. 24.
Figure 18 shows the effect of grating compensation at a Y- cut rotation angle
near the AT-cut (SSBW temperature compensated at 25°C). The turnover tem­
perature for SSBW on a 37 rotated T-cut quartz crystal is at 85°C. By adding a
grating to create a STW the turnover temperature can be effectively shifted to lower
temperature. Proper selection of the grating height shifts the turnover temperature
41
to room temperature (25°C) and thus temperature compensates STW propagation
normal to the X-axis on 37° rotated T-cut quartz.
Figure 19 is a comparison of STW temperature compensation at several
different rotated T-cut orientations. The ET-cut 10 (maximum piezoelectric cou­
pling) has a A F / F curve that is not as flat as the SSBW AT-cut, while the 32° cut
(large surface wave velocity) has a A F / F curve that is flatter than the AT-cut and
thus has a better temperature characteristic near the turnover temperature.
Figure
20
shows the cubic behavior of the -50.5° rotated T-cut SSBW. This
cubic behavior is lost when the grating is introduced to temperature compensate
the wave. Because of this the A F / F curve is not as flat as the SSBW BT- cut.
(However, it should be noted that the temperature curve is flatter for the STW on
this orientation than for any of the cases considered in Fig. 18 and 19.) This loss
of the cubic behavior is also observed when the SSBW is compensated using metal
overlays.33
42
(ppm)
CHANGE AF/F
FRACTIONAL FREQUENCY
0
\
SSBW
37®-CUT
-1 0 0
SSBW
. AT-CUT
-200
37®-CUT V
-3 0 0
400
-4 0
-2 0
0
20
40
60
80
100
120
1 40
TEMPERATURE (°C)
FIG U R E 18
Examples of A F / F curves for STW in quartz. Turnover temperature shift due to
the STW grating near the AT-cut.
43
(ppm)
AF/F
CHANGE
FREQUENCY
FRACTIONAL
-1 0 0
-200
STW
3 2 * -CUT
-3 0 0
|3II
/ET-CUT
SSBW
\
, AT-CUT \
-4 0 0
-4 0
-2 0
20
40
60
80
100
20
140
TEMPERATURE (°C)
FIG U RE 10
Comparison of the STW temperature compensation for the ET-cut and the 32° cut
with the SSBW AT-cut.
44
(ppm)
CHANGE AF/F
FRACTIONAL FREQUENCY
SSBW
BT-CUT
-1 0 0
- 5 0 . 5®-CUT
STW
,- 5 0 . 5 ° -CUT
-200
-3 0 0
-4 0 0
-4 0
-2 0
20
40
60
80
00
20
TEMPERATURE (°C)
FIGURE 20
The effect of the STW grating on the cubic behavior of the -50.5° cut.
45
140
CHAPTER m
METAL STRIP GRATING SURFACE TRANSVERSE WAVES
A. Background
Surface transverse wave propagation under a metal strip grating differs from
propagation under a grooved grating in several ways. In the absence of the grating
the solution is a SSBW, as it was for the grooved grating, but the slowing effect of
the metal strip grating is two-fold. The first slowing effect is that of the grating and
has characteristics similar to the grooved grating STW described above (see Fig.
1 ).
In addition, a second slowing effect due to the different shear wave velocities
in the substrate and the grating strip is also present. If the velocity in the strip
medium is slower than that of the substrate, the surface wave propagating beneath
the strip has an additional slowing term dependent on the material parameters of
the metal strip. This additional slowing effect is similar to the slowing of Love wave
propagation along the surface of a quartz substrate with a thin metal overlay.38
Both of these slowing effects aid in the trapping of the SSBW and give an additional
slowing term in the metal strip STW dispersion relation.
Another slowing effect is due to the piezoelectricity of the substrate and the
conductivity of the metal strips. This piezoelectric shorting effect can effectively
trap acoustic waves in some materials.39 The shorting effect of the metal strip is
very important in the high coupling materials; but it is small for materials with
weak piezoelectric coupling and can usually be neglected. Since quartz is considered
a weak coupling material, the shorting effects of the metal strips will be neglected
in this analysis.
46
A theoretical understanding of the behavior of STW propagation under grat­
ings made with metal strips is very important because manufacturing a metal strip
STW device is relatively simple. The grooved grating STW device requires several
deposition steps, as well as an etching procedure to cut the grooves. For a metal
strip STW the processing procedure is greatly simplified because of the elimination
of the etching step. The metal strip grating and the interdigital transducers needed
to launch the STW can be manufactured with just one deposition process. This
makes the metal strip STW devices very suitable to a mass production market.
In metal strip STW devices there is a change in the temperature behavior due
to the addition of the metal strips, similar to that caused by a grooved grating. This
modification of the temperature behavior needs to be understood when designing
temperature-stable metal strip STW devices. A metal strip STW theory, similar to
that of the grooved grating STW described earlier, will be developed for metal strip
STW propagation on a quartz substrate. Several experimental devices have been
tested to confirm these changes in temperature behavior of the metal strip STW.
B. Theory for Metal Strip Gratings
General mass loading STW theory begins with the acoustic wave equation (Eq.
1)
and assumes the same form of the space harmonic solution (Eq. 2) as in the
grooved theory. Substitution of each space harmonic into the wave equation gives
the same result for the lateral attenuation constants as was found in the grooved
case (Eq. 3). The difference between the two theories lies in the Datta-Hunsinger
boundary condition (Eq. 4) used to find the space harmonic amplitudes. In the case
of a metal strip grating the stress T'xz and density pf in the strip in Eq. 4 differ
from those of the substrate, and thus the coupling terms K nq are dependent on the
material parameters of both the metal strips and the substrate.
The general Datta-Hunsinger stress relation was stated in Eq. 4 as
Tiy = M h v , - h ^
(28)
Isotropic metal strips are assumed here, giving only one nonzero stress term for
47
pure shear horizontal particle displacement. This stress term is
dvx
iu> dz
C44
nri __ ^
1 xz
•
(29)
Following the same arguments and assumptions as for the grooved grating case, Eq.
28 is reduced to
(j(A>f/hvx - &[$]
< z < $
0
(30)
2
< z < A
where
*
=
-
«(* - a / 2 ))
As before, a space harmonic expansion is taken for vx (Eq. 2). To obtain a
set of equations for the a „ ’s in Eq.
2
the boundary stress at the strip-substrate
interface (Eq. 30) is set equal to the boundary stress in the substrate (Eq. 7). The
resulting equation is converted to a set of linear algebraic equations for the an’s
by using orthogonality of the different space harmonics, as in the grooved grating
STW calculation. This gives
— pw 2 d q — )
"an K n q
(31)
where
e- i (0n- p q)z^ z
=
' [“ v “
+
~ 4( 2 - 1) .
note both the + and - sign roots for the square root will be carried for the reason
described in Chapter II. Evaluating the integrals in the coupling terms, and con­
sidering only the two space harmonics 0 and -1, as in the grooved STW case, leads
to
± \ J C e f f f f i — p w 2 °o = a o K 00 + o - i - K —10
(32)
~ P u 2 a- i =
48
O o K o - i + a _ i i C _- 1 - 1
In the near-stopband approximation described earlier, the coupling terms are found
to be
(33)
where
y2
__ ^44
ff
rr2
c*ff
*
P
In this case note that the self-coupling terms Koo and K - i_ i are not zero, as
they were for the grooved grating. These terms contribute to an additional slowing
effect—the Love wave slowing effect. It will be described more fully in the following.
C . D ispersion R elatio n
The dispersion for the metal strip STW is found by writing the coupled space
harmonic equation (Eq. 32) in a matrix form and solving for the zero determinant
condition. The result is the dispersion relation
Koo ± \Jcef f P l —
A _ i_i ± \JCe/fP-i —
= K o - iK - io
(34)
where
Po = P
A plot of this dispersion relation is shown in Fig.
21
as the STW line. The two
slowing effects due to the metal strip grating can be seen in the figure and will be
described below.
If there were a continuous layer of metal on the surface, then the resulting
structure would support pure Love wave propagation under the metal overlay. For a
continuous overlay there are no delta functions in the Vxy term and the integration
is over a full period. In this case the cross coupling terms /Co-i and /£_ xo are
zero and only the self-coupling terms remain. The resulting uncoupled Oth and - 1st
40
METAL STRIP
-CRYSTAL SUBSTRATE
SSBW LINE
(w //3 )2 S( V | )
STOPJ
BAND)
STW LINE
cur
(O J//3 ) 2
F IG U R E
-
S TW
21
Dispersion curve for metal strip STW with Love wave effect
50
space harmonic equations are then
cef f Po ~ p u 2 °o — a0K oo
Z .__________
(35)
± y f c e f f P - i - p u 2 fl-i =
where
_
* • - * » -
_
-O & 'I -© !
Both the n = 0 and n = —1 equations give the same dispersion relation.
( j
where
) 2
= 0 - * i l
r
This is plotted in Fig.
21
(36)
,
as the Love-wave line as it corresponds to Love wave
propagation .38 For this reason the slowing effect due to the self-coupling terms Kq0
and K - i - i is termed the Love wave slo w in g effect.
Equation (36) defines the so-called “Love wave line” in Fig. 21. When a grating
is etched in the metal layer a stopband appears in the dispersion relation (Fig. 21).
This metal strip stopband is not centered about the frequency w0, as for the grooved
grating (Fig.
1 ).
It is centered about the Love-wave line corresponding to a thin
metal overlay of thickness equal to the grating height times the grating height-toperiod ratio; in other words, a thickness equal to the metal strip grating height
averaged over one period. The Love-wave effect and the grating effect combine to
produce the total slowing observed in a metal strip STW.
Again, as in the grooved grating STW, an analytical expression is needed for
the phase velocity of the STW below the lower edge of the stopband. To find this
analytical expression a form of the dispersion relation similar to the grooved grating
Eq. 19 is written, with a modifying factor M in the coupling constant.
The modifying factor M in Eq. 37 is found from Eq. 34 by making the near-stopband
approximation, described earlier for the grooved grating, and setting S/3 equal to
zero. Solving for the values of 6u at this point gives the value of the product M K
and thus an approximation to the dispersion relation. This gives
M
for M in Eq. 37, with K the coupling constant for a grooved grating
From Eqs. (37), (38), and (39) the following expression for the metal strip STW
phase velocity can be derived.
Vs
t w
Vmoj
—
= ---------.: ...............
- \ / K - w)2 - {Vt K M f
(40)
where
irVa
Note this has the same form as for the grooved STW, but with a modified coupling
constant. As for the grooved grating, the following condition must be satisfied for
STW propagation in the passband
(wb - u f X V . K M ) 2
(41)
The above STW velocity for the metal strip ( Eq.40 ) grating reduces correctly
to the velocity for the grooved grating when the density and stiffness of the strip
are the same as the substrate, as expected. As for grooved STW, this result can be
applied to an investigation of the temperature characteristics of metal strip STW
gratings for propagation normal to the X-axis on rotated T-cut trigonal crystal
plates. This will be considered in the next section.
52
D. Temperature Compensation
To investigate the temperature characteristics of these metal strip STW’s,
a numerical program was developed to calculate the turnover temperature for a
metal strip STW resonator. This program is based on theory presented earlier.
The resonant frequency condition used is given by Eq. 26, where the metal strip
STW velocity Vs t w is defined by Eq. 40. Rotated trigonal crystal half space
substrates with arbitrary orientation and isotropic or cubic metal strip gratings with
arbitrary dimensions are considered. The program calculates the temperature value
where the frequency change with respect to temperature goes to zero (the turnover
temperature), and was used to analyze STW gratings and substrate orientations for
the experimental devices being fabricated.
During the development of this program some concern was felt over the selec­
tion of the thermal expansion coefficient for the period of the grating. The expansion
coefficient for the grating height is well-defined. It must be that of the metal strip
material (in this case, aluminum). However the period of the grating is measured
along the interface of the metal strip and the quartz substrate, where the different
materials have different thermal expansion rates. This question was resolved by
assuming thin metal strips th at readily deform and allow the grating period to
expand according to the temperature coefficient of the substrate. The thermal ex­
pansion coefficient of the substrate material was therefore used to calculate the
expansion of the grating period. However, it is known in practice th at relatively
thin metal layers can cause significant substrate deformation with temperature, due
to the difference in expansion coefficients.'40 This effect, which was neglected in this
chapter’s analysis, will be considered in a later section.
Turnover temperatures were calculated for metal strip STW’s on rotated y-cut
quartz substrate with aluminum strip gratings of varying thickness. The rotated
T-cut angle corresponds to th at of the experimental STW devices (near the ATcut), and the quartz and aluminum material constants and temperature coefficients
were found from the literature .38,41,42 For an SSBW, the turnover tem perature was
53
found to be 131° C. This corresponds to a STW with a grating height of zero. As
the grating height is increased, the temperature compensating effect of the STW
shifts the turnover temperature to a new point, resulting in compensation at the
new turnover temperature. The turnover temperature for a grating height of 1400
angstroms was found to be 118 degrees, and a height of 3000A shifted the turnover
temperature down to 32°C. This result predicts that a grating height of 3000 would
effectively temperature compensate a STW resonator at 32°C (just above room
temperature). These numerical calculations will be compared with experimental
data in the following section.
E. Experimental Investigation of Phase-temperature Characteristics
(I)
Overview
Experimental metal strip STW devices fabricated at the Hewlett-Packard
Microwave Technology Division in Santa Rosa, California, were examined for their
phase-temperature characteristics. The author is grateful to Bob Bray, Scott Elliot,
Tim Bagwell, Catherine Johnson, and the many other people at Hewlett-Packard
Santa Rosa who made the fabrication of these devices possible.
Standard photolithographic and metal deposition methods were used to fabri­
cate these simple STW devices.43 The basic structure consists of aluminum strips
deposited onto rotated y-cut quartz plates.
Standard unapodized interdigital
transducers were placed at the ends of the grating structure to launch and receive
the STW signal. The T-cut rotation angle was very near the standard AT cut
(approximately 35°), and the metal strips were oriented parallel to the X crystal
axis . A drawing of the STW mask used in the fabrication process is shown in Fig.
22. The grating was made up of 623 aluminum strips measuring 2 microns by 200
microns and deposited at 4 micron intervals. At each end of the grating the last
61 metal strips were connected to form the interdigital transducers. Three metal
strip grating heights will be considered here (730, 1400, and 3000A). Referencing
the grating height to the STW grating period (4 microns or 40,000A) gives height54
STW Propagation Grating
Bonding Pad
Interdigital T ransducer
f i g u r e 22
Mask design for the STW grating device
55
to-period ratios of 1.8%, 3.4%, and 7.5%.
A photograph of the STW device after mounting and wire bonding is shown
in Fig. 23. The STW grating can be seen as the light area in the center of the
device. Interdigital transducer fingers are connected to the bounding pads at the
top and bottom of the device. The four connecting wires can be seen wire bonded
to the transducer pads. In addition to the STW transducer described above there is
another interdigital transducer with a finger spacing at twice the spatial frequency
of the grating, and having its bounding pads on the far right and far left of the
device.
This second set of transducers was included in the design to allow an
investigation of the STW grating at higher frequencies. For this study this second
pair of transducers was not wire bonded and thus will not be used in this experiment.
The quartz crystal substrate is a transparent material and the shaded ellipse seen
on the under side of the device is the adhesive th at was used to hold it in place.
To determine the temperature behavior of STW propagation beneath this metal
strip grating a transmission measurement was made from one IDT to the other.
A CW electrical signal was fed into one IDT. This excited an acoustic shearing
motion through the piezoelectric effect of the quartz. The spatial geometry of the
IDT was selected so that this shearing motion launched a STW perpendicular to
the grating fingers. This STW propagates along the grating structure toward the
second IDT, where the acoustic energy is converted back into an electrical signal.
The phase difference of the electrical input and output signals was measured as the
temperature of the device was varied from 20*C to 7QPC. From this phase data the
fractional frequency change turnover temperature can be deduced.
The change in phase with temperature is directly related to the temperature
coefficient of delay (TCD): a measure of the relative change of propagation delay
of the STW from one IDT to the other. This parameter is important because the
temperature coefficient of frequency of a STW resonator is equal to the negative
of the TCD. Thus the frequency turnover temperature discussed earlier occurs at
the same temperature as the delay turnover temperature and is therefore the same
56
FIGURE 23
Photo of the STW grating device
57
as the phase turnover temperature. By measuring the transmission phase change
versus temperature of the STW grating device, the turnover temperature of a STW
resonator can be determined as that tem perature for which the phase change with
respect to temperature goes to zero. T h at is, as the temperature is increased from
2 (f C
the phase will advance. As the temperature is increased further the phase
will advance more slowly until a point where it stops advancing and then starts to
retard. The temperature at the point of change from phase advance to phase retard
corresponds to the turnover temperature.
Phase measurements were made at Hewlett-Packard Laboratories in Palo Alto,
California, using an Hewlett-Packard 8510 network analyzer. The author is grateful
to Waguih Ishak, Bill Shreve, and the many people at Hewlett-Packard Laboratories
who made it possible to package the STW devices and make the phase-temperature
measurements. The temperature of the device was maintained to an accuracy of
better than
10
millidegrees Celsius for each measurement, using a Hewlett-Packard
temperature controller operating over the range
20 PC
to 7(f C. The phase turnover
temperature for the the 1400A and the 3000A devices did not occur within this
temperature range. Thus, to estimate the turnover temperature it was necessary
to extrapolate the temperature data, assuming a quadratic temperature behavior
beyond the measured temperature range.
(2)
Frequency Response
The measured frequency response of the transmission coefficient
730A device is shown in Fig 24.
£21
for the
TYansmission response is weak for the 730A
device because the shallow grating does not trap the acoustic energy close to the
surface, and only a small fraction of the energy launched is delivered to the receiving
transducer. The reference line at the top of the figure is the no loss or 0 db insertion
loss line; the maximum response of the device is at about 632.4 MHz and has an
insertion loss of +30 db. This device has a very high noise floor (-60 db),because of
the electromagnetic feedthrough from the first IDT to the second. In addition, there
are spurious signals caused by acoustic reflections within the device. A discussion
58
S21
(d b )
0 >
-20
6 3 2 4 (MHz)
-4 0
-8 0
5 70
630
F requency (MHz)
FIGURE 24
Measured frequency response for the 730A STW device
690
of these spurious signals follows.
Using the time domain feature of the 8510 network analyzer, the time domain
response was determined, and is shown in Fig 25. It is calculated from the frequency
response, using a fast Fourier transform. This time domain response, corresponding
to the impulse response of the acoustic device, is very helpful in understanding the
operation of the STW device and allows the STW transfer function to be separated
from noise and spurious signals.
The theoretical impulse response expected for this STW device gives zero signal
from time zero to about 50 nanoseconds, corresponding to the STW propagation
delay from the first EDT to the second. The ideal time response then makes a rapid
rise to a maximum value corresponding to the arrival of the STW at the second
IDT. A slow decrease then occurs as multiple reflections of the impulse from the
grating fingers continue to arrive, but with diminishing amplitude. This description
does not fit the observed data Fig. 25, which consists of three major signal peaks
corresponding to the electromagnetic feedthrough, the main STW signal and a
reflection at three times the first STW signal delay (the triple transit signal).
The first major single peak in the time domain response, corresponding to the
electromagnetic feedthrough, occurs at time zero or a very short time delay due to
propagation at the speed of light. This feedthrough has a signal strength of about
-60 db, and is a problem found in all SAW devices. It can be reduced by making the
bonding wires as short as possible and maintaining good shielding from the external
electronics. The value of -60 db is actually a low feedthrough signal but because of
the weak acoustic response of the 730A device (-45 db at the main STW peak) it
still causes considerable distortion of the STW frequency.
The second major single peak in the time response is due to the initial STW
impulse single arriving at the second transducer. This occurs at 500 nanoseconds,
corresponding to the time delay of a STW propagating at a velocity of 4500 meters
per second, and has signal strength of -45 db. A small ripple in the STW time
domain response can be seen shortly after the initial rise of the STW impulse.
60
S21
(d b )
-2 0
se c
495
-4 0
-6 0
-8 0
6.0
3.0
0.0
TIME (|J sec)
F IG U R E 25
Time domain response for the 730A STW device
61
This occurs about
100
nanoseconds after the initial STW impulse and is caused
by a reflection of the initial STW impulse from the end of the grating structure,
which is about 240 microns beyond the center of the second transducer.
This
reflection, referred to as the grating edge reflection, propagates back under the
second transducer, where it is picked up as a second smaller impulse delayed by the
related delay time (approximately
100
nanoseconds). The grating edge reflection
occurs because the STW sees an abrupt change in the wave impedance at the
edge of the grating. This impedance discontinuity could be reduced by extending
the grating beyond the second transducer and somehow slowly varying the wave
impedance to match that of a free surface. One way to do this would be to slowly
reduce the grating height after the second transducer. Tapering the grating height
to zero in
1000
microns would remove the wave impedance discontinuity and thus
greatly reduce the grating edge reflection. Another possible solution would be to
slowly decrease the strip width while maintaining the same grating periodicity and
height. This would also have the same impedance tapering effect and would result
in a smaller grating edge reflection. Tapering the wave impedance from the value
under the grating to the value of a free surface would allow the energy in the STW
to be gradually diffracted into the bulk of the quartz. For the free surface the
STW' becomes a SSBW and diffracts the acoustic energy into the bulk, unlike a
Rayleigh-type SAW which can propagate to the end of the crystal plate and be
reflected.
The third major signal in the time domain response of Fig. 25 occurs at 1500
nanoseconds, three times the initial STW single response, (the triple transit signal).
This is caused by the first STW impulse reflecting from the second transducer
and returning to the first transducer, where it again reflects back to the second
transducer. It is then picked up as a triple transit response. As expected from the
above description, the triple transit response occurs at three times the time delay of
the initial STW impulse response at 1500 nanoseconds and has a signal strength of
-55 db. This triple transient response occurs as a spurious signal in many SAW filter
62
designs and must be considered further in separating out the desired first transit
STW signal in Fig. 25.
To find a solution to the triple transient problem an understanding of how
the reflection process takes place is needed. As in almost all acoustic systems, wave
reflections take place at acoustic impedance discontinuities. For Rayleigh-type SAW
devices, where the propagation medium between the transducers is a free surface,
there is the obvious impedance discontinuity due to the the mass loading of the IDT
strips. However, in these STW devices the strip geometry does not change between
the EDT and the propagation grating, and it might be expected that any contribution
to the triple transient reflection due to the mass loading of the IDT grating in SAWs
will not be present. However there exists another effect that does give rise to the
impedance discontinuity in the case of the STW, namely piezoelectrical shorting by
the metal strips.
Under the IDT the STW acoustic impedance is different than that under the
grating, because the fingers of the IDT are connected to each other through the
external electronic device used to launch and receive the STW. This causes a change
in the electrical conditions of the metal strips and thus a change in the acoustic
impedance, due to the piezoelectricity of the quartz. If the propagation path grating
fingers (between the transducers) were connected to a similar external electrical
load, so that the electrical boundary conditions at the grating strips is identical
to th at of the IDT metal strips, then there would be no change in the acoustic
impedance; and the triple transient problem would be eliminated. Duplicating the
exact electrical conditions of the IDT in the grating strips may not be possible.
However, by shorting the grating strips, the electrical loading of the propagation
fingers would be closer to that of the IDT strips, and should greatly reduced the
triple transient problem. The triple transient response could probably be reduced
to a level far below that of other SAW devices, because of the elimination of the
mass loading impedance discontinuity. Beyond aiding in solving the triple transient
problem, shorting the propagation path fingers would aid in the electromagnetic
63
feedthrough problem by effectively placing a ground plane between the two IDTs.
Many other practical STW device considerations could be discussed here but are
not relevant to the main topic of this dissertation.
To eliminate the noise and spurious signals discussed above, the time-gating
feature of the 8510 network analyzer called, time domain gating, was used. This
allows the time response to be modified by eliminating, or gating-out, unwanted
signals and then calculating the frequency response of the gated signal data by
taking another fast Fourier transform. Figure 26 shows the frequency response
of the 730A STW device, with the electromagnetic feedthrough removed by time
domain gating. As can be seen, the noise floor has been lowered to about 80 db. This
increase in dynamic range greatly improves the frequency response measurement,
and the classic sin x /x frequency response of the transducer can be clearly seen.
However, there is still considerable ripple caused by the beating of the initial STW
signal with the triple transient signal. By further using the gating feature of the
8510 as a time-band gate, (that is, a time-band starting after the feedthrough and
ending just before the triple transit), both the electromagnetic feedthrough and the
triple transient single can be eliminated.
Figure 27 shows the frequency response of the 730A with both the feedthrough
and the triple transient removed. Here the sin x /x character of the transducer
response is clear and the stopband behavior of the grating structure is very clearly
visible. There is still a small ripple in the frequency response due to the beating of
the main STW single with the grating edge reflection signal. Because the grating
edge reflection is very close to the main STW single in the time domain, it can not
be eliminated by time domain gating technique. Thus the measurements must be
analyzed with this small ripple included in the frequency response.
In Fig. 27 the stop-band is seen to have a lower edge at 632.4 MHz and an upper
edge at 635 Mhz. The frequency response curve does not return to the same level on
the upper side of the stop band as on the lower side, thus creating an unsymmetrical
curve about the stopband. In the case of a SAW the frequency response curve
64
S 21
(db)
632
-20
-4 0
-6 0
-8 0 -
670
630
690
Frequency (MHz)
FIGURE 26
Frequency response of the 730A STW device with the electromagnetic feedthrough
removed using time domain gating
65
S 21
(db)
0 >
-2 0
63!U (MHz)
-4 0
-6 0
-8 0
690
630
670
Frequency
FIGURE 27
730 A with feedthrough and triple transit gated out.
66
(MHz)
would be expected to return to the same sin x j x response level occurring below the
stopband, giving a symmetrical response. It is very clear from the figure that this is
not the case for STW. The theoretical reason for this was given in Chapterll section
C.
Theoretical stopband calculations based on the theory of Chapter m , section
C were made for the 730A device. The theoretical Love wave slowing term {VeKi,
in Fig. 21) is 3.81 megaradians per second (or 0.6 MHz), and the grating stopband
term was found to be 8.15 megaradians per second (or 1.3 Mhz). This corresponds
to a stopband width of 2.6 MHz. Note that the top of the stopband is in the fast
wave region of the w—^ diagram (Fig. 21). The theoretical behavior of the STW in
this region is described in Chapter II, section C. In this previous discussion it was
predicted that the trapping effect of the grating would not be present at the top of
the stopband and would be reduced at frequencies above the stopband. This is why
the STW frequency response is not symmetrical about the stopband. The STW
below the stopband is a trapped wave which gives a small insertion loss though
the device, +30 db at 632.4 MHz. At the top of the stopband the STW becomes
a bulk wave, and thus the energy is no longer trapped close to the surface where
it can be received by the the second transducer. This results in a much greater
insertion loss (+40 db at the upper edge of the stopband 635 MHz). Beyond the top
of the stopband the trapping effect of the STW is reduced, and thus the frequency
response above the stopband never returns to the corresponding insertion loss level
below the stopband. These frequency calculations show good agreement between
the experiment and theory but, again, the stopband calculations are not the major
concern of this study.
(3)
Phase versus Temperature Measurement
To investigate the temperature behavior of the STW, a narrow frequency range
just below the lower edge of stop-band will be considered. Figure 28 shows the
frequency response of the 1400A device with the electromagnetic feedthrough gatedout. Note that the insertion loss of the this device, +19.7db, is much smaller than
67
S21
(db)
hp
20
STW 1 4 0 0 A
MARKER
6 2 9 . L MI-
40
60
80
i
570
630
.
690
Frequency (MHz)
FIGURE 28
Frequency response of the 1400A STW device with the feedthrough removed using
time domain gating.
68
th at of the 730A device, (which had an insertion loss of +30 db). This is because
the thicker metal fingers trap the STW energy closer to the crystal surface, where
it is transferred to the second IDT more efficiently. Also, note that the increased
grating height has increased the stopband width considerably. The frequency range
of interest here is just below the lower edge of the stopband (629.1 MHz). Figure
29 gives a detailed description of the frequency and phase response in this region.
Phase was measured at five different frequencies just below the stopband, shown by
the five markers in Fig. 29 at 628.7, 629.0, 629.3, 629.6, and 629.9 MHz. Measured
phase values for the 1400A device at these frequencies are listed as a function of
temperature in Table 1.
The relative phase change from 2(f C to 70PC for the 1400A device is shown in
Fig. 30, where the phase line with the markers corresponds to the 7(P measurement.
Here, we see that the net phase change with temperature is greater at the higher
frequencies. For example, marker one (628.7 MHz) shows a net change over the
5(F temperature range of 76.?, while at marker 4 (629.6 Mhz) , the net change is
considerably different (137.?). This confirms the earlier prediction that the tem­
perature compensating effect of the grating will be greater close to the stopband
edge. Unfortunately, the phase turnover temperature could not actually be.achieved
because the range of the temperature controller was limited to a maximum tem­
perature of 7 ? C . It was expected to occur at 118?C for this device. To deduce an
experimental value for comparison with theory a quadratic extrapolation will be
made.
A similar measurement was made for the 3000A device. The results are given
in Table 2 for the five frequencies 617.0, 617.3, 617.6, 617.9, and 618.2 MHz. In the
case of the 3000A device, the turnover temperature was also found to be outside the
range of the temperature controller. This was surprising, because it is theoretically
calculated to be at 32°C. A discussion of this result, with possible reasons for the
discrepancy is given below.
69
-2 0
S 21
(db)
-4 0
-6 0
-8 0
630
570
690
Frequency (MHz)
phase
180
180
FIGURE 20
Frequency and Phase response of the 1400A STW device just below the Stop-band.
70
1 4 0 0 A PHASE MEASUREMENT*
FREQUENCY (MHz)
TEMPERATURE (°C)
6 2 8 .7
20
6 2 9 .0
6 2 9 .3
-1 15 .6 3
-
194.65
51 .20
6 2 9 .7
9 5 .3 2
-
40
-
8 4 .8 8
-
161.68
8 9 .4 8
60
-
5 4 .1 0
-
127.44
12 6 .0 0
2 4 .6 2
-1 14.90
142.77
4 2 .3 2
109.74
149.70
4 8 .2 5
70
-
3 9 .4 2
75
-
33.31
-
-
39.93
*P h ase measured in degrees
TABLE 1
1400A STW device phase-temperature data
71
PHASE
180
20
70
0
-1 8 0
690
630
570
F re q u e n c y
(MHz)
FIG U R E 30
Relative Phase change from2 CPC to 7<FC for the 1400Adevice.
72
3 0 0 0 A PHASE MEASUREMENT*
TEMPERATURE (°C)
FREQUENCY (MHz)
20
6 ) 7 .0 0
6 1 7 .3 0
104.32
73.15
7 4 .0 0
108.00
4 8 .0 0
-
30
40
-
6 1 7 .6 6
6 1 7 .9 0
134.86
-
17.97
-
95 .1 8
-
21.71
140.10
-
58.12
3 2 .7 9
165.25
-
2 5 .5 0
75.70
-
50
-
2 6 .7 4
60
-
11.39
-
176.49
-
2.45
10 5 .8 4
0.71
-
162.37
13.14
128.40
70
*P h ase measured in degrees
TABLE 2
3000ASTW device Phase-Temperature data
73
(4)
Determination of the Turnover Temperature
To determine the turnover temperatures of these two devices, a quadratic ex­
trapolation of the phase data was made. This was done by plotting the change in
phase with respect to temperature versus temperature and then linearly extrapolat­
ing the data to a value of zero. Note that a quadratic behavior in the phasetemperature characteristics will result in a linear behavior for the first derivative
of phase with respect to temperature. The change in phase with respect to tem­
perature was calculated by taking the difference between two adjacent phase data
points and dividing by the temperature change between the points. This value
was then plotted versus the temperature. These curves appear in Figs. 31 and
32. Extrapolating the curves to zero gives an approximation to the turnover tem­
perature. For the 1400A device a turnover temperature of
for the 3000A device a turnover temperature of
88f* was
110PC
was found and
found. As expected, the
larger grating height had a greater temperature compensating effect and shifted
the turnover temperature closer to room temperature. However both extrapolated
turnover temperature values differed from the theoretical values of 118PC for the
1400A device and 32l°C for the 3000A device. The large discrepancy suggests the
presence of additional forces and temperature effects not included in the present
theory.
One of the temperature effects not considered in the original theory, but which
turns out to have a substantial effect, is caused by the difference in expansion
coefficients of the dissimilar finger and substrate materials. As the temperature
of the device changes these two material try to expand at different rates, thereby
causing a considerable change in the deformation as a function of temperature. A
detailed discussion of this effect, referred to as the static strain effect, will be given
in Chapter IV.
74
PHASE
2. 0
ANGSTROM
DEVICE
OF
U 00
FIRST
DERI V I T I V E
L. 5
I. 0
.5
0
50
70
90
30
120
L10
100
TEMPERATURE ( d e g C)
FIG U R E 31
Extrapolation to the turnover temperature for the 1400ASTW device.
75
PHASE
OF
DERIVITIVE
FIRST
. 0
3000
.
ANGSTROM
DEVICE
0
.0
0
0
0
0
30
40
50
60
70
80
90
L00
TEMPERATURE C d e g C)
F IG U R E 32
Extrapolation to the turnover temperature for the 3000A STW device.
76
C H A P T E R IV
STATIC STRAIN
A. Genera] Considerations
In a metal strip or mass loading grating the grating is made of a different
material than the substrate and thus will, in general, have different acoustic and
thermal characteristics. A difference in thermal expansion coefficients causes the
metal strip to expand or contract at a different rate than that of the substrate
when the temperature is varied. If the metal strip continues to adhere to the
substrate without cracking, the substrate and the metal strip must undergo small
deformations to allow for the different thermal expansion rates.
Because this
deformation or strain field varies slowly with time compared to the strain fields
of the propagating STW, it is referred to as a static strain. This static strain is
a function of the temperature, the thermal and elastic characteristics of the two
dissimilar materials, and the method used to deposit the metal strips on to the
substrate. If the deformation or static strain is small (that is if it remains in the
linear region of the stress vs. strain curve of the material, Fig. 33), it will have little
effect on the STW behavior, other than the change in grating dimensions already
accounted for, and thus be negligible. However, if the material is strained to the
point where the stress strain curve is no longer linear then the stress strain relation
Ti =
cjjS j
(42)
which governs the STW is changed and the STW analysis must be modified to
account for this effect. In the case of aluminum strips on quartz this added strain
77
Fracture
point
Elastic
lim it '
Linear
■>
Stress
Nonlinear
Elastic deform ation
Plastic
deform ation
F IG U R E 33
Typical stress-strain relation for a solid material.
78
effect is not negligible and must be incorporated into the STW temperature theory
to predict accurate STW temperature behavior. This will be done by introducing a
temperature-dependent static strain term in the crystal stiffness constants used in
evaluating the STW velocity. A discussion of this static strain term follows, along
with a calculation of the static strain effects on metal strip STW propagation.
B. Static Strain Theory
The static strain theory developed here is based on the assumption that the
particle displacement is the superposition of a small dynamic displacement and a
larger static displacement. The small dynamic displacement is due to the stress
fields of the acoustic wave and the large static displacement is caused by the forces
created by differential expansion of the metal strips and the crystal substrate. This
theory can be thought of as the mechanical analog of the small signal circuit analysis
in electronic circuit theory, where there is a small AC single superimposed on a large
DC bias. In each case a linear theory is used to describe the small signal behavior,
even though the large signal behavior is nonlinear.
The theory begins with the assumption that the total particle displacement
field near the interface of the two materials is a superposition of two particle
displacement fields. The first is a static displacement caused by crystal particle
displacements relative to the natural or initial crystal particle positions. This static
displacement includes any crystal deformation experienced during processing of the
device, caused by previous thermal expansion history. The second is the dynamic
particle displacement which is caused by the stress fields of the STW. The static
strain is assumed to vary with time at a very slow rate compared to the strain
fields of the STW and is assumed to be constant with time relative to the dynamic
displacement. The spatial and temporal particle displacement can be written
U ( x ,v ,z , t) =
U (x ,v ,z ) + U (x ,v ,z,t)
(43)
where U* is the static stress, which is independent of time, and U d is the dynamic
displacement which is a function of time. To justify the small signal analysis, the
79
magnitude of the dynamic displacement must be much smaller than that of the
static strain.
U*>Ud
In addition it is assumed that the net displacement is still governed by the linear
acoustic field equations
d 2U
V -T = rg p
(4*0
V.U = S
(446)
The constitutive stress strain relation includes the nonlinear stress strain be­
havior and has the form
Ti =
c ij S j
+
c jj k Sk
Sj
(45)
where c/y and c/y/c are the second- and third-order elastic constants which can be
found in the literature .35,30,44-48 Substituting the assumed particle displacement
U into the strain relation Eq. 44b we find
S = Sa+ S d = V 'U a+ V ,U d
which has the sameform as the particle displacement (i.e.,
(46)
a superposition of a
large static strain and a small dynamic strain). If we substitute this strain into the
constitutive relation we find the stress to be of the form
Ti = T1 + T j
where
.
.
Ti =
cijS j
+
cijkS j
Td =
cijS j
+
cukSk
(47)
Sk
S j + ciJK S K
a S dj
If it is assumed that the second-order term in the dynamic stress equation is
negligible, then the dynamic constitutive relation becomes
T id = c u S dj + ciJ K S*K S dj
(48a)
T j = [cu + c u K S K \Sdj
(486)
80
Substitution of these constitutive relations into the divergence of stress relation (Eq.
44a) separates the analysis into two problems, a static problem
vr* = 0
(49)
r = c u S j9 + c u K S K
m S9j
and a dynamic problem
dt2
(50)
T d = ce/j f S dj
where
ci V — c i J + c i j k S * k
The dynamic acoustic equation can be applied to the STW problem, resulting in a
wave equation that includes the static stress eifect
V •c f f
: V.U d
=
P
~
(51)
This is a simplified static strain analysis, giving a wave equation of the form used
in the STW theory described. This allows the static strain effect to be readily
incorporated into the STW temperature theory already prescribed, by using the
modified effective stiffness constants given above. This simplified static strain theory
will be adequate for this study. A more detailed discussion of static strain effects,
including changes in the material density and nonlinear acoustic field equations,
can be found in reference 47.
C. Nonuniformity of Static Strain
If the effective stiffness has a constant uniform value throughout the crystal
substrate, then the problem in Eq. 51 has the same solution as the previous STW
solution, but with modified stiffness values. However the static strain field imposed
by static forces on the crystal surface is not uniform throughout the crystal sub­
strate, and such an assumption needs justification.
Figure 34 shows static strain in the quartz substrate due to the static stress
imposed by the aluminum strips. As the temperature is lowered the aluminum strips
81
(a)
A LU M INU M
S T R IP
S T A T IC
QUARTZ ^
SUBSTRATE
STRESS
IN
A LU M IN U M
BOW C A U S E D BY
SUBSTRATE D EFORMATIO N
(b)
S TA T IC S TR A IN
P A R A L L E L TO
IN Q U A R T Z
S TR IP
S T A T I C S T R A I N IN Q U A R T Z
P E R P E N D I C U L A R TO S T R I P
F IG U R E 34
S t a t ic stress a n d s tra in s in q u a r tz d u e to a lu m in u m s trip s
82
contract faster than the quartz substrate, owing to the larger thermal expansion
coefficient in aluminum. Because of this, static stresses developed in the aluminum
strips are imposed on the quartz substrate (Fig. 34(a)). These static stresses on
the surface of the substrate cause the substrate to deform as shown in Fig. 34(b).
From the laws of static elasticity, the strains are expected to decay exponentially
in the interior of the crystal, to vary periodically about an average value along the
direction of STW propagation, and to be uniform in the direction of the metal strip
fingers.
The substrate static strain perpendicular to the metal strips will be compressive
under a metal strip and extensive between metal strips, as shown in the figure.
The effective small signal stiffness values in the presence of this perpendicular
strain component are expected to increase where the static strain is compressive
(underneath a grating tooth) and decrease where the static strain is expansive
(between grating teeth).
Over a distance of many grating periods this periodic
increase and decrease of the stiffness averages out to the unstrained stiffness, which
is uniform along the direction perpendicular to the grating teeth. In other words,
the effect of the perpendicular static strain component on the effective stiffness
values tends to cancel, so that this part of the static strain has a negligible effect
on the acoustic wave velocity.
The strain parallel to the metal strips (or normal to the STW propagation
direction) is uniform over the entire length of the metal strip (Fig 34(b)). This
produces a substantial spatially-averaged deformation of the substrate. It is welldocumented that quartz plates with aluminum films on the surface will cause a
bowing of the plate .3,40 This same bowing effect will occur in metal strip STW
gratings. Because the width (or aperture) of the grating is much larger than an
acoustic wavelength, the static strain and the effective stiffness values should not
change appreciably within a few wavelengths of the substrate surface. If the grating
height is sufficient to trap the STW energy within a few wavelengths of the surface,
as it is designed to do in a practical STW device, then the effective stiffness values
83
seen by the STW are nearly uniform as a function of depth. Limiting our discussion
to wide aperture devices with many grating fingers and a well-trapped STW, we
can therefore assume that the effective stiffness is uniform in the region of influence
of the STW. In addition, it can be assumed that the only nonnegligible component
of the surface strain will be the static strain component parallel to the metal strips.
This strain component corresponds to Sxx (or Si) component for the rotated Y-cut
geometry considered here (Fig. 2).
D. Effect on surface 'wave Temperature Characteristics
The major strain component caused by the expansion or contraction of the
metal strip fingers will be parallel to the fingers and transverse to the STW propaga­
tion direction. This strain component, acting over the full aperture, has large par­
ticle displacements and gives the major contribution to the nonlinear stress strain
relation affecting the change in the stiffness values. Those strain components per­
pendicular to the fingers tend to be relieved by the gaps between the metal fingers,
and do not have the opportunity to build up the large particle displacements of
the components parallel to the fingers. Thus they will have a smaller effect on the
STW velocity. To maintain simplicity in the analysis, the smaller strain component
oriented along the wave propagation direction (or perpendicular to the fingers) is
neglected, and the static strain caused by the metal strips is assumed to be uniform
and acting parallel to the fingers (or in the X-direction, so that the static strain is
Si).
The effective trigonal crystal stiffness constants, used in the STW calculations
of chapter H, for a uniform Si strain, become
=
c65 + Cssi'S'i
= css + cgai^i
cl^ =
caa + caoi'S'i
From the crystal symmetry
84
(52)
Cg51 =
C155
C581 = [—2cm —C112 + 3 C222] /4
C881 =
and
cijk
(53)
[C ll4 + 3C1241/2
can be found in the literature for quartz44and lithium niobate.48
Stress changes as high as 150 million newtons per square meter per degree
celsius have been observed in thin films of aluminum.40 A stress change of this
magnitude would result in Si strain changes on the order of 1700 ppm in quartz.
Substituting a Si strain of just 100 ppm into Eq. 52 gives rise to a change in the
quartz elastic constant C55 of 300 ppm per degree C. Comparison with the elastic
temperature coefficients of 177 ppm per degree C for C55 in quartz 38 shows that
static strain effects are far from negligible.
Figure 35 shows a plot of the theoretic fractional frequency change of a STW
resonator versus static shear strains. This calculation is based on the effective
stiffness changes described above. It can be seen that frequency changes on the
order of hundreds of parts per million occur for strain changes of the same order.
These frequency changes are comparable to those observed for small temperature
changes. Therefore, to accurately predict the acoustic wave temperature behavior
of a metal strip STW one must include this static strain effect in any analysis of
temperature compensation.
Anomalous temperature behavior in acoustic devices has long been observed
at Hewlett-Packard .24’48 Theoretical temperature coefficients and the measured
parameters were found to be in disagreement, and temperature-dependent static
strain was proposed as the cause of this discrepancy. An example of this unex­
pected behavior was observed in quartz SAW resonators. Identical resonators ex­
hibited varying drifts in resonant frequency with age. It was discovered that if the
resonators were put though an annealing process before the final frequency trimming
step, this frequency drift was reduced. This observed behavior can be explained by
static strains. The drift in frequency is caused by a relaxation with aging of the
very large initial stresses in the aluminum strips. During the annealing processes the
initial stress is reduced so that the change in strain with aging is greatly reduced.
85
500
F RACTI ONAL
FREQUENCY
CHANGE
CPPM)
600
200
100
200
800
400
SHEAR
STRAIN
F IG U R E 35
Fractional frequency change vs. strain.
86
1000
CPPM)
After annealing, all the devices are brought to the same static stress level, so that
the behavior observed in each individual device is consistent with the other identical
devices. These phenomena indicate the importance of considering static stress in
this temperature compensation investigation.
Before the effective stiffness result described above can be substituted into
the earlier STW theory to incorporate the effect of the static strain Si on the
wave velocity, it is necessary to determine the strain Si and its temperature varia­
tion. One method is to consider the published thermal expansion rates and elastic
constants of the substrate and the aluminum, and then to calculate the induced
strain, assuming no strain at an initial temperature. This approach is not adequate
for several reasons. First, the thermal expansion and elastic coefficients of thin
aluminum strips are not the same as the published values for bulk aluminum. Also,
in thin films these coefficients vary greatly with film thickness and composition. In
addition, the initial strain state of the device is nonzero because of the processing
steps and past temperature history. This initial strain state may vary greatly from
device to device and has to be determined to characterize the temperature behavior
of each new device.
For these reasons Si must be determined empirically for each device. When
the strain state of the device has been determined, the effective stiffness can be
substituted into the STW theory described previously and a prediction made of
the STW temperature behavior. Once the experimentally-determined metal strip
grating STW temperature characteristics are obtained they can then be applied to
the STW temperature compensation theory described in previous sections.
To estimate the effects of the static strain on the temperature behavior of
STW’s the empirical stress values found by Castro will be used.40 Castro’s work
considered 1 micron aluminum films on a silicon substrate. These empirical stresstemperature results are expected to be very similar to those found for 1 micron
aluminum strips on quartz, because of the similarity of the expansion coefficients of
quartz along the X-crystal axis (7.97 ppm) and silicon (7.63 ppm). The temperature
87
stress coefficient taken from Castro is 15X107 N/M 2 °C. Using the stiffness values
of quartz the static shear strain temperature coefficient is calculated to be 1700
ppm per degree C. Using these values, the fractional frequency change in a STW
resonators can be calculated and is shown verses temperature in Fig. 36. As can
be seen from the figure, the fractional frequency changes due to the static strain
effect (in the temperature range ( f C to 5(f C) are comparable to the changes due
to the grating dimensions calculated in the earlier STW theory. Thus this static
strain behavior for aluminum strips on quartz substrates must be determined before
a complete STW temperature theory can be developed.
88
CPPM)
CHANGE
FREQUENCY
FRACTI ONAL
600
0
10
20
30
50
40
T E M P E R AT U R E
C d e g C)
F IG U R E 36
Fractional frequency change vs. temperature.
80
CHAPTER V
TRAPPING LEAKY SURFACE WAVES
A. Introduction
In rotated Y-cut trigonal crystals, surface skimming bulk waves (SSBW) exist
only for propagation normal to the X-axis. Addition of a surface grating converts
these surface skimming bulk shear waves into pure shear (or transverse) surface
waves (STW). As noted earlier, temperature-compensation of these waves on quartz,
lithium niobate, and lithium tantalate can be achieved by proper selection of the
crystal orientation and the grating dimensions. For lithium niobate and lithium
tantalate, however, these cuts are not piezoelectrically active, and cannot be excited
with an interdigital transducer.
Piezoelectric coupling is present for propagation along the X-axis of rotated
Y-cut lithium niobate and lithium tantalate crystal plates,49 and these cuts are
found to have the attractive feature of very strong piezoelectric coupling (AY f V =
0.02776 for -48° rotated Y-cut50). For this direction of propagation pure SSBW and
STW exist only for certain specific rotation angles and for most angles only leaky
waves exist. Consequently, to study temperature compensation of surface grating
waves on piezoelectrically active cuts it is necessary to extend the previous theory
(developed for pure shear waves, polarized parallel to the crystal surface) to the
case of general polarizations.
A direct numerical solution to the differential wave equation and boundary
conditions for arbitrarily polarized surface wave propagation under grooved gratings
on anisotropic crystals will be considered. In general, some of these orientations
90
cannot support true surface waves on a smooth surface, because the wave radiates
or leaks power into the bulk of the substrate. This so-called leaky wave theory will
be developed and discussed.
The importance of propagation along the X-axis of rotated T-cut trigonal
crystals was noted earlier. Along this direction of propagation there exists a leaky
surface wave that reduces to a pure SSBW for certain rotation angles.15,51,52 These
waves have very high coupling constants and have aroused interest because of this
fact.14 Another type of leaky wave reported in the literature is the pseudosurface
wave, which reduces to a perfectly trapped Rayleigh wave for certain particular
crystal cuts. These exist for a number of materials and crystal cuts (see, for example,
References 52 and 53).
Although the existence of the leaky surface waves described in the pre­
vious paragraph has been known for some time, especially the pseudo SSBW X propagating waves on rotated Y- cut lithium niobate and lithium tantalate, they
have not been successfully exploited in device applications because of their exces­
sive radiation losses. On quartz, where pure SSBW have found some applications in
delay lines at microwave frequencies despite their diffraction losses, a severe limita­
tion is the weak coupling. It has already been noted th at pure rotated V-cut SSBW
propagation normal to the X —axis cannot be excited piezoelectrically in high cou­
pling lithium niobate and lithium tantalate crystals. Exploration of the strongly
coupled pseudo SSBW on lithium niobate and lithium tantalate is very important
for the development of low insertion loss broad band delay line devices at microwave
frequencies. This requires trapping of the leaky wave on the surface by means of
a grating structure. In addition, the same grating might also be used to tempera­
ture compensate the quasi-STW realized in this way. A theory developed for this
purpose could also extend the range of materials and cuts available for practical
applications, by similarly trapping and compensating pseudosurface waves of SAW
type on arbitrary crystals.52,53
In view of the ultimate importance of the leaky wave problem, for reasons
01
already stated in the preceding paragraphs, it was decided to develop an efficient
leaky grating wave Floquet theory and numerical algorithm. For a certain height
of grating structure the leaky wave becomes a “skimming wave,” with no energy
leakage. Beyond this height, the wave becomes trapped as a quasi-STW. As a first
step in following this leaky wave theory, the height of the grating required to create a
skimming wave is found. Before developing this approach, some attention was given
to the idea of first calculating the leaky wave fields for a smooth surface, and then
trying to build a Floquet theory on this base. It soon became clear that, to do this,
it would be necessary to numerically compute the fields for each space harmonic.
Since the leaky wave calculation is itself complicated and time consuming, this did
not appear to be a viable method. Instead, the relatively simple direct method for
calculating the height of the grating required to convert a pseudo SSBW into a true
SSBW with mixed polarization was used. This method, based on the use of crystal
slowness surfaces, is outlined in the following paragraph.
For simplicity, we will consider only the case of X - propagating T-cut waves.
However the program developed later in this section is completely general and can
calculate grating dimension for waves on a substrate of any crystal symmetry and
orientation. The Floquet theory used resembles that used for the theory for pure
STW propagation normal to the X-axis on rotated T-cut crystals, but it is more
complicated. In the substrate the fields are expanded in space harmonics, and the
coupled space harmonic equations are developed by means of the Datta-Hunsinger
boundary condition for the grating.
For pure STW, each space harmonic has
only an SH polarization. For the case of quasi-STW, space harmonics of all three
polarizations exist. These can be visualized on the slowness curve Fig. 37, by taking
OJ
w
This allows one to visualize which space harmonics are leaky and which are not. The
grating wave becomes nonleaky when Pq/ u first reaches the extremum of the slow
shear wave. At this point, the Oth fast shear and longitudinal space harmonics are
02
k2 l u
( k f u) x Quasishear
<k/ui7 Quasishear
ikiui)} Quesilongitudmal
- - 1 x 10” '* s/m
CJ
FIG U R E 37
Illustrates the condition for a skimming wave on X-propagating Z-cut trigonal
crystals.
93
already nonleaky and all of the higher space harmonics are nonleaky. To calculate
the height of grating required to make this a viable solution, the coupled mode
approximation is made, including in this case Oth and -1st space harmonics of
all three types. The grating height must then be evaluated by applying the zero
determinant condition to the set of coupled equations for these six space harmonic
amplitudes. In this procedure, unlike the programs developed for calculating surface
wave velocities, the values of /3z/cj for the various space harmonics are obtained
from the slowness curves. They do not have to be guessed by successive trials, as in
numerical analysis of Rayleigh waves on a smooth surface, and can be used directly
to find the field values needed in setting up the boundary condition equation for
h. As in the space harmonic problems already treated, the accuracy of the coupled
mode solution can be subsequently tested and improved by including additional
space harmonic amplitudes.
B. Trapping of Leaky Surface Waves
Chapters II and ID dealt with horizontally-polarized shear surface waves (or
surface transverse waves, STW) under surface gratings fabricated on crystal cuts
that support a pure shear skimming wave (SSBW). In this section a numerical
algorithm is developed for calculating grating surface waves on arbitrarily oriented
crystal cuts. The main purpose is to realize a procedure for trapping leaky sur­
face waves by means of a surface grating. (Leaky surface waves have very large
piezoelectric coupling constants in certain cases and would be of great technological
importance if their radiation' losses could be eliminated.) The computation pro­
cedure developed gives all wave solutions, including the STW and SAW grating
waves (or, in general, Surface Acoustic Grating Waves (SAGW) solutions) when
the crystal cut permits these solutions. SAW grating waves are routinely used in
conventional SAW resonators, and this algorithm gives the first numerically exact
solution for these waves. Although the purpose of this work is to study grating
waves on piezoelectric substrates the piezoelectric effect has been neglected in this
94
first version of the grating wave algorithm. This simplification can be removed in
further generations of the program, now that the basic principle has been tested
and verified.
Consider plane surface wave propagation along the Z-axis of a nonpiezoelectric
anisotropic crystal half space with a rotated coordinate system oriented such that
the X — Z plane corresponds to the surface of the half space and the T-axis is
directed into the crystal. As in the STW theory we consider a surface perturbation
of this structure, in the form of a shallow surface grating that is infinite in extent
and oriented normal to the direction of plane surface wave propagation. Floquet
theory will be combined with the method of superposition of partial waves to solve
this single crystal grating problem.
In the STW case we had a simple unperturbed wave solution consisting of a
shear horizontal displacement velocity, the SH mode. Following Floquet theory, we
assumed the perturbed wave solution to consist of a summation of space harmonics,
where each space harmonic is a pure SH wave of a different spatial frequency. For
the more general wave solution now under consideration, we know the unperturbed
wave solution can be expressed as a superposition of three partial waves (quasi­
longitudinal, fast quasi-shear, and slow quasi-shear). Following Floquet theory, we
can express the perturbed solution as a sum of space harmonics, where each space
harmonic is a pure unperturbed wave at a different spatial frequency. The grating
wave, or perturbed wave, then has a displacement velocity solution of the form
F *F •
e—dt " if +* a „
vF
ne
n V
where
95
+ a * v S e - a »''\ ,e- i ( 3 n z e iw t
(55)
a£ is the amplitude of the n-th quasi-longitudinal space harmonic.
aF is the amplitude of the n-th fast quasi-shear space harmonic.
a% is the amplitude of the n-th slow quasi-shear space harmonic.
v% is the unit displacement velocity for the quasi-longitudinal
partial wave.
vF is the unit displacement velocity for the fast quasi-shear,
partial wave.
t)j[ is the unit displacement velocity for the slow quasi-shear
a%, a F and o f
/3n
u
partial wave.
are the complex lateral attenuation constants for
the quasi- longitudinal, fast quasi-shear, and slow
quasi-shear, respectively.
is the n-th space harmonic propagation constant along the Z-axis.
is the grating wave frequency.
The first step in finding a solution to the leaky wave problem is to find the partial
wave lateral attenuation constants and displacement velocity vectors. A partial
wave solution of the form described above is assumed.
an vne - a”y
(56)
This general partial wave solution is substituted into the acoustic wave equation,64
neglecting the body forces F
V-c:V.» = ^
Following an approach similar
to the one used to
(57)
derive the Christoffel equation for
triclinic crystals,56 the following system of linear equations relating the attenuation
constants and displacement vectors to w and /? for the three partial wave solutions
in Eq. 55 was found.
A
D
£T ■v*'
D
B
F vv
£
F
c.
96
Vz.
A = C&&/32
B =
-
CQOa l - i2C 560n <Xn ~ P ^ 2
C44 0 n - c2 2 « n ~ *2c24/?n<*n ~ ( M *
C = c330 l
- c44ttn ~ i2c34^„a„ -
pu2
D =
C4 5 /?^ — C 28«n ~ *(c 48 + <?25)/?n<*n
£
C z& ftn — c 4a a ^
=
E =
C34^
— 1 (0 4 5 +
(58)
C 3 e ) ^ n o rn
- C 24«n - *(c44 + C 2 z ) P n * n
Cjj = the rotated stiffness constants.
Using numerical methods, this set of linear equations can be solved for the
six complex values of a„, given /?„. We then select the three lateral attenuation
constants values that correspond to surface wave solutions (i.e., those solutions
which go to zero when Y is inflnite). These three values correspond to at„ a £ , and
in Eq. 55.
These solutions are an extension of the slowness surface calculation.59 If the
lateral attenuation constants are restricted to be pure imaginary (corresponding to
wave propagation constants that are pure real) the solution to the above system of
equations would result in the crystal slowness surfaces discussed in the literature.
This allows for a check of the numerical program by selecting wave velocities that
are greater than the slowest shear wave and then comparing the solutions with the
corresponding crystal slowness surface discussed in the literature. In general, the
lateral attenuation constants can be complex, corresponding to wave propagation
with attenuation.
Once the three partial wave attenuation constants have been found, then they
can be substituted into Eq. 58 to find the corresponding particle displacement
velocity vectors [vx, vy,v z\. In general, these velocity vectors are complex and the
numerical computation must allow for this possibility, as well as for possible pure
mode displacement velocity vectors.
Using the algorithm described above, three partial wave lateral attenuation
constants and displacement vectors can be found for each space harmonic, once the
crystal orientation and wave velocity have been specified. T hat is to say, if the
numerical values for the wave frequency w, the propagation constant /?<>, crystal
density p and the crystal stiffness values
c jj
97
are known, then the three partial wave
lateral attenuation constants and the three particle displacement velocity vectors
can be determined. This partial wave calculation is then repeated for each space
harmonic propagation constant (3n. These partial wave solutions can now be used
to solve the boundary condition equation at the surface of the substrate.
The partial wave amplitudes of each individual space harmonic must be selected
so that the boundary condition at the surface of the half space is satisfied. In
the general geometry, Datta-Hunsinger boundary condition equations, including
all three polarization displacements, are applied. As in the STW case, this gives
coupled space harmonic equations; but, unlike the STW case, there are now three
scalar equations corresponding to each space harmonic. Using the orthogonality
properties of the space harmonics and the Datta-Hunsinger boundary conditions,
as was done in the STW case, the three following coupled space harmonic equations
were found
a ty q 1 + a q 1>2q F + «£
S +
(59«)
n
a ty ? +
+ «?*}*- J E
S +
+ o f* * ?
(506)
n
aL
,r
,L+<$*’/ + «f
= A
x
E a»K% + “»
n
*
»
£
+
<58c>
where
i)1
/ = |( c /6a^ + icIbPq]vJx + \cn OtJq + iCuPq\vJy + \cIAa Jq + ICj3 /? ,]t)'j
T/’IJ
nq
G ln
_
1 —cos ir(n —q)'
fib ~ @q
=
[V n * i
C= x , y , z
+
for
GnPq -
)VL + (C /2«n + * P n S u ) v { n +
( c ^ + i f i n ^ l z ) v Jtn \
I — 5,4,3
In the previous analyses slow (or surface) wave solutions occurring below the first
stopband of the grating are calculated by assuming all but the 0th and -1st space
98
harmonic amplitudes are of negligible amplitude. The infinite set of coupled space
harmonic equations Eq. 59 is then reduced to six linear equations with the following
characteristic determinant.
X -io
X -io
X -io
Xoo
v ®*X
-io
V®F
X
-io
X-10
Xoo
Xoo
Xoo
Xoo
Xoo
v'IS’
©
Xoo
Xoo
v 4L
V 4S
V4L
X
-io
X
V 2L
Xoo
v 3F
X -io
y3S
v 5S
v 2S
V2F
V4F
V4o5-i X - i - i X
- i - i X - 1- 1
X o-i X
o -i X
v 3F
y4S
V3S
v 3L
V4F
-i-i X -i-i
X o - i X o - i X o -i X - i - i X
n
flo
'
4
«o
<*i
<£i
v 5 L V 6 F v 5 S yOL
VOS'
v eF
- x - i . .« £ i.
-i-i X
. X o-i X
o - i X o -i X - i - i X
where
V I J — ,1,1J _
Xoo — vo
yiJ
X -x-i — v -i
X -io —
A 00
y
“ I-1
A
,,/y _ h g r u
X o - i — —a 0-1
7 = 1, 2, 3 , 4 , 5,6
£ = 4,3,5 for I = 2,4,6, respectively
J = L ,F ,S
Numerical methods were used to solve for the height-to-period ratio which
satisfies Eq. 60 at a specified wave velocity and crystal orientation—that is, for
specified u , 0 , crystal stiffness constants and crystal density. This height-to-period
ratio corresponds to the theoretical grating dimensions needed to trap the leaky
wave, or in the case of a non-leaky wave, the grating dimensions required to slow
the surface wave velocity to the velocity specified.
Equation (60) given above is of sixth order in the height-to-period ratio, and
therefore has six different solutions. If more space harmonics were included, there
would be a corresponding increase in the number of roots to the equation. Roots
99
giving a real positive height-to-period ratio correspond to the physical grating
dimensions that would slow the wave to the velocity specified in the program. If
that velocity is slower than the slowest bulk wave, then the wave is a trapped surface
wave as described above. Complex roots are not physically realizable, and must be
regarded as spurious. In cases where none of the six roots are real and positive, then
the wave velocity specified at the beginning of the numerical calculation cannot be
achieved with a physical grating. However, if there is at least one positive real root
(or, in some cases, more than one) then these correspond to grating heights that
trap leaky waves at the specified velocity and crystal orientation.
C. Numerical Results
A Fortran program based on the above algorithm has been developed and
tested. The program calculates the grating height-to-period ratio needed to slow
a leaky wave to a specified wave velocity. If the prespecified wave velocity is less
than the slowest shear bulk wave velocity found from the slowness surface, then
the grating height-to-period ratio found corresponds to one that will trap a leaky
wave. The program can consider arbitrary crystal cuts and propagation directions
but neglects piezoelectricity. Neglecting piezoelectricity sometimes can present a
problem when comparing the results to other published calculations of leaky waves
on smooth surfaces. These published results included the piezoelectric effect in
the higher coupling materials considered (lithium niobate and lithium tantalate)
because the piezoelectric effect can change the wave velocity on the order of 20%.
To make completely accurate comparisons, the piezoelectrical potential terms will
have to be incorporated into the program in the future.
The program was applied to the problem of wave propagation along the X-axis
of F-cut trigonal crystal half spaces of quartz, lithium niobate and lithium tantalate.
The real positive height-to-period ratio needed to trap the leaky waves or slow
existing surface waves was calculated for various assumed wave velocities. There
are four major wave velocity regions, two of which are of primary interest. Region
100
One has wave velocities greater than the slow bulk shear wave velocity. Region
Two has wave velocities greater than the SAW velocity, but less than the slow bulk
shear wave velocity. Region Three has wave velocities less than the SAW velocity,
but greater than the minimum trapped wave velocity. The minimum trapped wave
velocity is that velocity which is achieved with an infinite grating height and thus
the wave can not be slowed to a velocity less than the minimum trapped wave
velocity. Finally, Region Four has wave velocities less than the minimum trapped
wave velocity.
In Region One the program fails to find any roots because some of the partial
wave lateral attenuation constants needed in the partial wave solution are pureimaginary. The program bases its selection of the three attenuation constants a
on the sign of the real parts of the a ’s. If the real part of any a is zero, then the
program rejects the initially chosen velocity, because lateral attenuation constants
suitable for trapped leaky waves require roots with a positive real part.
In Regions Two and Three, the real parts of the a ’s are non-zero and the
height-to-period ratio is found to have real positive roots. These are the areas of
major interest and they have been plotted for wave propagation along the X-axis of
F-cut crystal half spaces of quartz, lithium niobate, and lithium tantalate in Figs.
38,39,40 and 41. In each figure there exists a grating SAW solution and a trapped
leaky wave solution. The grating SAW has a real positive height-to-period ratio
that descends from infinity, at the minimum trapped wave velocity, to a value of
zero, at the wave velocity corresponding to the SAW velocity on a smooth surface,
neglecting piezoelectricity. The trapped leaky wave has a height-to-period ratio
that descends from infinity, at the minimum trapped leaky wave velocity, to a
minimum height-to-period ratio needed to trap the leaky wave. This minimum
height-to-period ratio needed to trap the leaky wave is the height-to-period ratio
needed to slow the leaky surface wave velocity, which is greater than the slow shear
bulk wave velocity (and thus leaky) to the slow shear bulk wave velocity where it
become a surface skimming wave. When the grating height is increased beyond this
101
. 0
<
cl
Q
O
CL
UJ
CL
0
GRATING
SAW
O
X
LD
. 0
. 0
.0
----------
. 0 I
1800
20 00
r-CUT
2200
QUARTZ
,--------------------------------,—
2400
26 00
28 00
X-PROPAGATING
WAVE
3000
-----!-----32 00
34 00
V eT ocY t
FIGURE 38
Real height-to-period ratio needed to trap leaky wave and SAW propagating along
the X-axis of y-cu t quartz.
102
o
. 5
TRAPPED
LEAKY
WA V E 1
o
o
. 5
SLOW
BULK
. 0
GRATING
SHEAR
WA V E
SA
. 5
.
0—
2200
2400
2600
2800
3000
3200
34 00
3600
(m eter/second)
Y-CUT
LITHIUM
NIOBATE
X - P R O P A G A T I ON WAVE
VELOCITY
F IG U R E 30
Real height-to-period ratio needed to trap leaky wave and SAW propagating along the X-axis of Y'-cut lithium niobate.
103
o
. 5
TRAPPED
a
o
LEAKY
WAVE
. 0
0.
o
o
5
B U L K WAVE
SLOW S H E A R
0
GRATING
SAW
5
0
—
2200
-48
degree
2400
ROTATED
2600
Y - CUT
2800
LITHIUM
3000
NI OBATE
3200
3400
3600
X - P ROP A G aTt m G ^ V E L O C I TY
F IG U R E 40
Real height-to-period ratio needed to trap leaky wave and SAW propagating along
the X-axis of -48° rotated F-cut lithium niobate (maximum coupling A v /v =
0.02775).
104
TRAPPED
LEAKY
WA V E
Q_
O
O
GRATING
1800
Y- CUT
2000
LITHIUM
2200
SLOW
SHEAR
BULK
WAVE-
SAW
2400
TANTALATE
2600
2800
X-PROPAGATI NG
3000
3200
e
3400
nd
(m
m e t e r / sse c o n d ))
WAV
WAVE
VELOC
VELOCITY
FIGURE 41
Real height-to-period ratio needed to trap leaky wave and SAW propagating along
the X-axis of y -cu t lithium tantalate.
105
minimum, the surface skimming wave becomes a trapped leaky wave.
In Region Four, the prespecified wave velocity is so slow that there are no
real positive height-to-period ratios that can slow the leaky wave to the specified
velocity. Because no real roots are present in this region it is not of interest for
trapped leaky wave analysis and will not be discussed further.
These encouraging results show that the grating dimensions needed to trap
leaky waves can be calculated using the algorithm developed. However, the mini­
mum grating height-to-period ratios needed to trap X-propagating leaky waves on
T-cut crystals are relatively large (quartz h/A = 1.3; lithium niobate h fA = 0.38;
lithium tantalate h / \ = 0.29). The calculations for these large grating heights
must be questioned because of the shallow grating assumption made in the DattaHunsinger boundary conditions. Any attem pt to add additional grating height to
temperature compensate these trapped leaky waves will result in grating depths
that contradict the Datta-Hunsinger assumption.
To alleviate the deep grating problem described above, the program was used
to locate crystal orientations where trapped leaky waves can be supported with
shallow gratings. The orientations considered are propagation along the X-axis of
rotated y -cu t quartz, lithium niobate, and lithium tantalate crystal half spaces.
The prespecified wave velocity used in the program is the slow shear bulk wave
velocity, which gives the grating height-to-period ratio needed to just trap the leaky
wave resulting in a wave with lateral attenuation similar to that of the surface
skimming bulk wave described earlier. This corresponds to the minimum grating
height-to-period ratio needed to trap the leaky wave. These height-to-period ratios
are plotted as a function of crystal rotation angle for quartz, lithium niobate, and
lithium tantalate in Figs. 42, 43 and 44.
In lithium tantalate there is only one real positive root for the grating height-toperiod ratio needed to turn the leaky wave into a skimming wave. The grating height
goes to zero over a short range of y-cut angles near 13(f. This angle corresponds to
the surface skimming bulk wave reported in the literature13 (Note: The angle quoted
106
3. 0
TRAPPED
LEAKY
WA V E
cc
LlJ
Q.
2. 0
o
LU
. 5
.
0
.5
0
20
40
QUARTZ
60
80
ROTATED
100
Y-CUT
120
ANGLE
140
160
180
degrees)
FIGURE 42
Minimum height-to-period ratio needed to trap leaky wave propagation parallel to
the X-axis on rotated y-cut quartz.
107
. 0
. 8
. 6
.
TRAPPED
LEAKY
WAVE
4
0
20
40
60
80
1 00
120
140
160
180
(degrees)
LITHIUM
N I O B A T E . ROTATED
r-CUT
A NGL E
F IG U R E 43
Minimum height-to-period ratio needed to trap leaky wave propagation parallel to
the X-axis on rotated y-cut lithium niobate.
108
. 0
.8
. 6
.
TRAPPED
4
LEAKY
WA V E
. 2
0
20
40
LITHIUM
60
60
TANTALATE
100
ROTATED
120
Y- CUT
140
160
180
A N G L E Cda9reBS>
FIGURE 44
Minimum height-to-period ratio needed to trap leaky wave propagation parallel to
the X-axis on rotated y-cut lithium tantalate.
109
in the literature, calculated using the piezoelectric potential, is 125°; the angle found
here, neglecting piezoelectricity, is 13(F.) The grating dimensions near this rotated
T-cut correspond to those of a shallow grating and are accurately predicted with
this leaky wave theory. The grating height can be increased to trap the skimming
wave and thereby create a surface grating wave on lithium tantalate. This type
of surface wave has the high coupling of lithium tantalate and also temperature
characteristics that are a function of the grating dimensions.
In the case of quartz and lithium niobate there are no rotated y-cut angles
found that can trap an X-propagating leaky wave with a grating height-to-period
ratio less than 40%. Because of the deep grating problems described above, the
temperature compensation investigation of trapped leaky waves on these materials
is open to question. However, with the piezoelectric effect included in the program,
there may be an additional solution that may allow for trapping with shallow
gratings. To develop this computation would require a piezoelectric counterpart
of the Datta-Hunsinger equations.
The above analysis demonstrates the needed for further work in this area. It
already shows that gratings can, in principle, be used to trap leaky surface acoustic
waves, but a more refined theory will be required for quantitative grating design
of sufficient accuracy for devices. Such a further development would permit a
systematic evaluation of piezoelectric coupling and temperature compensation in
trapped leaky waves for all doubly-rotated crystal cuts. Since some leaky waves are
already known to have very large piezoelectric coupling, such an evaluation could
well yield new crystal cuts providing optimized combinations of high coupling and
temperature compensation for surface wave deiay lines.
110
CHAPTER VI
CONCLUSION
A new method for temperature compensating microwave acoustic resonators
has been presented. It was found th at the introduction of a periodic corrugated
grating on the surface of the resonator crystal alters the temperature behavior of
the surface wave. In certain cases the addition of the surface grating can achieve
complete temperature compensation at room temperature.
A theory describing the temperature characteristics of horizontally polarized
shear surface wave (or surface transverse wave STW) propagation was presented.
This theory was applied to the design of temperature compensated STW resonators
on rotated Y-cut trigonal crystal plates with wave propagation normal to the X axis. It was found that complete temperature compensation could be achieved for
essentially all rotated Y-cut angles of quartz substrates with grooved gratings. This
result allows the designer to selecte crystal orientations that will optimize resonator
performance in the area of piezoelectric coupling and surface wave velocity, without
forfeiting temperature characteristics. In addition, the low acoustic energy density
of the STW will allow resonators to be driven at much higher power levels resulting
in better noise performance. These results will allow the design of quartz resonators
with higher resonant frequencies, lower insertion loss, and better noise figures while
still maintaining good temperature stability.
Several metal strip STW devices were fabricated. The temperature compen­
sating effect of the surface grating was confirmed by phase-temperature measure­
ments on these devices. In the case of the metal strip grating there was found to be
111
additional tem perature variation due to the material deformation at the interface
between the aluminum strips and the quartz substrate. This deformation or strain
varies slow ly when compared with the strain fields of the surface wave; therefore,
it is termed a static strain. These static strain fields are caused by the different
thermal expansion characteristics of the dissimilar materials. As the temperature
varies, the aluminum expands relative to the quartz causing a static deformation
of the quartz crystal. A simplified theory describing this effect was developed and
applied to STW propagation under aluminum strips. These results showed that
the static strain effect was on the order of the temperature effects observed in the
experiment, thus demonstrating the need to include these static strain effects in the
design of metal strip STW devices.
In order to investigate surface grating effects on acoustic wave propagation in
the highly piezoelectric material lithium niobate and lithium tantalate, a numerical
program describing arbitrarily polarized surface acoustic wave propagation under
corrugated surface gratings was developed. An analysis based on this program
was applied to leaky wave propagation along the X-axis of rotated y-cut lithium
niobate and lithium tantalate crystal plates. It was shown th at the grating structure
could effectively trap the otherwise leaky surface waves that propagate in these
materials. This result will permit a systematic evaluation of piezoelectric coupling
and temperature compensation in trapped leaky waves for all doubly-rotated crystal
cuts. Such an evaluation could well yield new crystal cuts providing optimized
combinations of high coupling and temperature compensation in lithium niobate
and lithium tantalate.
112
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S.Elliott and R.C.Bray, 1984 IEEE Ultrasonics Symposium Proceedings, pp.
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3.
R.C.Bray, L.L. Pendergrass, C.A. Johnsen, T.L. Bagwell, and J.L. Henderson,
1985 IEEE Ultrasonics Symposium Proceedings, pp. 247-252.
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D. F. Thompson and B. A. Auld, 1985 IEEE Ultrasonics Symposium
Proceedings, pp. 203-206.
5.
B. A. Auld and D. F. Thompson, 1984 IEEE Ultrasonics Symposium
Proceedings, pp. 213-217.
6. B. A. Auld and D. F. Thompson, Ginzton Laboratory Report No. 3815, Air
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