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

код для вставкиСкачатьINFORMATION TO USERS This reproduction was made from a copy of a m anuscript sent to u s for publication and microfilming. While the m ost advanced technology h as been used to pho tograph and reproduce this manuscript, the quality of the reproduction is heavily dependent upon the qualiiy of the material submitted. Pages in any m anuscript may have indistinct print. In all cases the best available copy has been filmed. The following explanation of techniques is provided to help clarify notations which may appear on this reproduction. 1. M anuscripts may not always be complete. When it is not possible to obtain missing pages, a note appears to Indicate this. 2. When copyrighted materials are removed from the m anuscript, a note ap pears to indicate this. 3. Oversize materials (maps, drawings, and charts) are photographed by sec tioning the original, beginning at the upper left hand com er and continu ing from left to right in equal sections with small overlaps. 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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 REFERENCES 1. Chicago, CBS Columbia records, record number 38590. 2. 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