IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-28, NO. 6, NOVEMBER 1981 449

Shear Horizontal Surface Waves on Rotated Y-Cut Quartz

Absrracr-A theoretical analysis of shear horizontal surface waves on ated Y-cut quartz has been performed. For the substrate layered ~th S i 0 2 , 2Av/u has been calculated as a function of the thick-

ness/wavelength ratio for the lowest mode surface wave. For the sub strate without the surface layer, leaky Bleustein-Gulayev (BG) waves are shown to exist for the crystal orientations from 14.5O-138'. The BG waves are described in more detail.

S I . INTRODUCTION

HALLOW BULK acoustic wave (SBAW) devices possess many advantages over the usual surface acoustic wave

(SAW) devices including, among other things, higher frequency operation and smaller temperature effects [ l ] , [2]. The dis- advantage is the acoustic beam spreading loss, which occurs because SBAW radiates energy into the crystal bulk. The beam spreading loss can be minimized through coupling the SBAW with the surface modes: SBAW energy can be trapped close to the substrate surface through corrugating [3] the sur- face region between the transducers, or through layering the surface with materials whose acoustic velocity is lower than the acoustic velocity in the substrate material. This paper analyzes shear horizontal surface o r Love waves in rotated Y-cut quartz layered with SiO,. The dispersion relationship 2Av/v and the decay coefficient into the substrate are calcu- lated for the lowest mode surface wave as a function of the SiO, thickness. Here Au is the difference between phase ve- locities for electrically open and electrically shorted substrate surfaces. A large 2Au/u is expected to result in a decreased beam spreading loss.

to zero, surface waves of the Bleustein-Gulayev (BG) type exist on rotated Y-cut quartz for the crystal orientations 0"-14.5" and 138"-180" [4], [5]. There has also been some speculation that pseudo-surface waves may exist in the com- plementary crystal orientations region; however, none were found, at least not in the velocity region greater than the bulk wave cutoff velocity [ 51. This paper shows that in the orienta- tion region 14.5"-138", leaky BG waves exist with the acous- tic velocity lower than the bulk wave cutoff velocity. The waves consist of a linear combination of the quasi-acoustic and quasi-electrostatic modes: the electrostatic mode is nearly evanescent, with its decay length on the order of the acoustic wavelength; the quasi-acoustic mode grows slowly into the

It is well known that as the thickness of the dielectric goes

Fig. 1. (a) Geometry of thin dielectric layer on top of rotated Y-cut quartz. (b) Velocity slowness curve for bulk waves in rotated Y-cut quartz. kc is the cutoff wave vector, cutoff velocity uc w/k,.

substrate. The attenuation and growth coefficients and 2Av/u are evaluated for the surface and leaky BC waves, respectively, in the entire crystal orientation region. The existence of the leaky wave solutions is important for a rigor. ous SBAW analysis [6].

11. THEORY The geometry is shown in Fig. 1. The field equations are

where the usual tensor notation has been used: U and @ are particle velocity and electrostatic potential, and C g L , eij , and eiL are elastic stiffness and dielectric and piezoelectric cou- pling constants, respectively.

Equations (1) and (2) are solved by Fourier analysis:

paper was supported in part by ARO Contract DAAG26-78-C-0043. Manuscript received January 7, 1981; revised March 26, 1981. This we consider propagation perpendicular to the x axis of y- The authors are with TRW Defense and Space Systems Group, rotated quartz, and the film a material with a cubic crystalline

Redondo Beach, CA 90278. ..

symmetry.

0018-9537/81/1100-0449$00.75 0 1981 IEEE

450 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-28, h

Air Equation (2) reduces to Laplace's equation. The solution

that decays in the -z direction is

p ) ( k , , z ) = c$(a)erz (4)

where the 7 = ' k , for Re(k,) 2 0, respectively.

Dielectric Film Equation ( 2 ) reduces to Laplace's equation, yielding - z ) = d(+)ekXz + (5)

For the cubic symmetry, (1) couples particle displacements in the x and z directions, and the y-polarized displacement is de- coupled. Because the x- and z-polarized displacements are de- coupled from the electric potential, for the propagation per- pendicular to the x axis of the rotated Y-cut quartz, we

where n = ac, el for the quasi-acoustic and qui modes, respectively. The total field is the sum

n

The parameter 2Au/u is often of considerable practical ir, terest. We consider the propagation for the substrate/film interface electrically open and electrically shorted.

A. SubstratelFile Interface Electrically Open The boundary conditions are electrostatic potential, partic

displacement, normal components of electrical displacement, and mechanical stress are continuous across the substrate/film interface; @ and D, are continuous at the air/film interface, and the frlm surface (z = - t ) is stress-free. Using (4)-(9) and the usual constitutional relations, the conditions are sum- marized according to

consider only the y-polarized displacement:

and the acoustic wave velocity

' ( F ) = m. Quartz

For the y-polarized displacement, (1) and (2) yield

where

D11 = [ ~ 4 4 k : + 2C46k,kz + c66k: - P O 2 ] ,

D l , = i u [ ( e 1 4 + e 3 6 ) k x k z +e l&: +e&] ,

D,, = W2(Ellk: + 2~1,k ,k , + ~ ~ ~ k j ) .

The diagonal terms in (7) describe pure horizontally polar- ized shear acoustic and electrostatic modes, respectively, in nonpiezoelectric material. In piezoelectric material, the two modes are piezoelectrically coupled.

A nontrivial solution to (7) is obtained by putting its de- terminant equal to zero. This yields

k i - kz @x) ") - (n )

(8)

where and are dielectric constants in the air and film, respectively, and

F, = (e14kx + e3,k?)) + i(c46 k, + ~ , ~ k p ) ) E , . (1 1)

A nontrivial solution to (10) is obtained by putting the de- terminant equal to zero. This yields the dispersion relation- ship k , = k, ( W , t ) .

B . SubstratelFilm Interface Electrically Shorted The boundary conditions are that the mechanical quantities

are continuous and the electrostatic potential is equal to zero at the substrate/film interface, ($ and D, are continuous at the air/film interface, and the film surface is stress-free. Using (4)-(9) one finds that the electrostatic potentials above the metallized interface, @('l, &), and Q(-), are decoupled from those in the substrate, and also from the field in the film.' Mathematically, the (6 X 6) determinant is reduced to two (3 X 3) determinants. A nontrivial solution is obtained by

'This decoupling corresponds physically to shielding off the electro- static potentials above the metallization. Hence, the resulting determi- nant (12) also applies to the case of a metallic layer (of thickness t ) on top of a piezoelectric substrate.

WILCOX AND YEN: SHEAR HORIZONTAL SURFACE WAVES 451

putting a determinant of the following matrix equation equal to zero:

FREE SURFACE VELOCITY IS SLIGHTLY HIGHER THAN METALIZED SURFACE VELOCITY

I l k - 0.01

----SURFACE WAVE VELOCITY /

0

[it, cos (k:F)t)

C. Thickness t = 0 Because of its importance, this case is treated separately.

ro t = 0, boundary conditions reduce to three equations for "open" surface:

l

0 F,, Fe[ d e ' )

and to two equations for the metallized surface

111. RESULTS

A. Surface Waves for the Thickness t f 0 Results obtained for the lowest mode surface wave for SiO,

on rotated Y-cut quartz are shown in Figs. 2 and 3. For t # 0, surface wave solutions are obtained for those crystal cut orientations for which the cutoff bulk wave velocity (U,) is higher than the acoustic velocity in SO,, U, > uSio, . Here U, is defined as the phase velocity of the bulk wave propagat- ing parallel to the substrate surface, U , = w / k , , where kc is the projected (on the surface) component of the cutoff wave vector kc (Fig. l(b)) defined through

d k y ) = 0 at kc .

dkx The phase velocity of these surface waves, or Love waves, in piezoelectric crystals is usio, < U < U,.

Fig. 2 shows the surface wave phase velocity (U E w / k , ) and the parameter 2Au/u(Au Z uOpen - Ushort) as a function of the crystal orientation for the thickness t /h = 0.01, and in the insert, 2Au/u as a function of the ratio t /h for AT-cut quartz. Notice that the "open" surface phase velocity is slightly higher than the "shorted" surface velocity, and that 2Au/u has a maximum at about t /h = 0.1 and asymptotically ap- proaches zero for very large t . This is because for large t? the phase velocities are unaffected by the substrate, eventually approaching usio,. For small t , velocities approach values very near U, and 2Au/u drops to a very small value. The limiting behavior of 2Au/u at t = 0 is discussed below in greater detail.

The decay coefficient 6 E -Im(k!ac)/k,) is plotted in Fig. 3 as a function of the crystal orientation for t/h = 0.01 and in the insert as a function of t /h for AT-cut. The effect of in- creasing the thickness of the dielectric layer is to decrease the wave penetration depth into the substrate; for the same thick- ness, the penetration depth is smaller for the metallized sur- face than for the open surface.

Fig. 2. Lowest mode surface wave in SO2 on rotated Y-cut quartz: phase velocity and 2Av/v.

p -1MAG lkz/kcl P I S NORMALIZEDWITH RESPECTTO kx SUBSTRATE

METALLIZED

LL IDEGI

Fig. 3. Lowest mode surface wave in SiO, on rotated Y-cut quartz: decay constants for the quasi-acoustic mode.

B. Leaky Bleustein-Gulayev Waves for t = 0

For the open surface boundary conditions, Koerber et al. [4] and Jhunjhunwala et al. [ S ] showed that (13) has solu- tions corresponding to surface waves of the BG type for crystal orientations in the region 0"-14.5" and 138"-180". The wave vector k, is real and larger than the cut-off wave vector k c ; i.e., the waves are unattenuated and slower than bulk waves parallel to the substrate surface. The imaginary part of k, is negative; i.e., the waves decay in the direction into the substrate. In the crystal orientation region 15"-138", (13) was searched for pseudo-surface wave solutions in the velocity region U > u c ; the search revealed no such solutions [S]. For the metallized surface boundary conditions, surface wave solutions were found to exist [4], [S] for all crystal orientations from 0" to 180".

[ 6 ] , we have extended our analysis (for t = 0) to complex values of k , (and k , > k c ) ; the procedure employed in the analysis allowed us to search for leaky wave solutions with (in principle) arbitrary values of the wave vector k, .

In general, leaky [7] (or pseudo-surface) waves are charac- terized by a small yet finite negative imaginary part of the propagation wave vector k, , and a positive imaginary part of

Because of their importance for the bulk wave calculations

452 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-28, NO. 6 , NOVEMBER 1981

k, : the waves are attenuated along the surface and "leak" into the bulk of the crystal. For these crystal symmetries, leaky waves provide an additional solution (in addition to normal surface waves) for the boundary-condition determi- nant; the waves usually occur for those propagation directions for which the normal surface wave solution degenerates or almost degenerates into a bulk wave. Mathematically, the presence of a leaky wave is indicated by a second deep mini- mum appearing in a plot of the magnitude of the boundary- condition determinant as a function of assumed values of phase velocity. For this type of leaky wave, the normal sur- face wave velocity is lower than the phase velocities of two transverse (or in general, quasi-transverse) bulk waves propa- gating in the same direction and lower than the leaky wave velocity.

This is different from the case considered here. For the propagation perpendicular to the x axis of rotated Y-cut quartz, the only surface wave is the BG wave. This wave consists of a linear combination of decaying (i.e., modes corresponding to two lower-half complex plane roots k,(")) quasi-electrostatic and quasi-acoustic modes. In our calcula- tions, this linear combination, when substituted into the boundary-condition determinant for the crystal orientations Oo-14.S0 and 138"-180°, drove the determinant through zero; i.e., both the real and imaginary parts of the determinant changed signs at the same value of k,. In this analysis, the size of the stepping interval Ak, was IO-* kc and the surface wave k, was k, >kc, thus recovering the previous [4], [5] results. In the complementary crystal orientation region, taking this linear combination for k, >kc and those ky) which correspond to wave propagation into the substrate for k, < k c , yielded no zero and no sharp minimum on the real k , axis. Even when modified to allow for small imaginary parts of k,, the search yielded no zero and no sharp minimum.

static and a growing acoustic mode did, however, drive the boundary-condition determinant through zero. The growing acoustic mode corresponds to the upper-half-complex plane root of the determinant of equations of motion for k, >kc; for k, < k c , that real root was taken which corresponds to a (bulk) wave tilted into the crystal interior. The determinant went through zero for most crystal orientation, except for those immediately adjacent to the critical orientations 14.5" and 138". What typically happened for these orientations was that one part, say the real part, of the determinant went through zero, while the imaginary part of the determinant nearly became equal to zero yet did not change its sign at the same value of k,. At this value of k,, which, though very close, was always greater than kc, the boundary-condition determinant exhibited a sharp minimum, its value being by at least six orders of magnitude lower than the value of the de- terminant evaluated at k , at Ak, = lo-' kc away from minimum.

The computations were generalized for complex values of k , at this stage. An iterative root solving technique was used that sought to minimize the absolute value of the boundary condition determinant. A root was accepted when two suc-

Taking a linear combination consisting of a decaying electro-

5 x

1 0 4 I I SURFACE BG WAVES 1 L LEAKY WAVEREGION

l

0

J P (DEG)

80

SURFACE BG WAVES

1 80

Fig. 4. Velocity for leaky BG waves in rotated Y-cut quartz for open surface boundary conditions.

O w W IDEG)

Fig. 5. Velocity for surface BG waves in rotated Y-cut quartz with metallized surface.

cessive iterants differed by no more than lo-". Though com- pletely general in principle, the values of k, used in the iterations should not excessively depart from k c , because for complex k,, k, is no longer complex conjugate, and when selecting k y ) it is always convenient to be able to categorize the modes as decaying or growing electrostatic or acoustic waves. This computation essentially recovered the results obtained in the first stage of our computation: complex roots k, whose real parts were equal to the minimum k,, were found for all crystal orientations 15"-135". In all cases, the computed imaginary parts of k, were negative, and Ik: l were on the order of, or less than, IO-' kc. This is qualitatively consistent with leaky BC waves: the waves are only very slightly attenuated in the direction parallel to the surface and because their velocity is very close to U,, only slightly growing into the crystal interior. (Note that for a slab geometry, there would be six boundary conditions and all four roots k:) would have to be considered.) No particular attention was given to the magnitude k i because IO-' kc is less than the accuracy of our computation of kc. The results of the com- putations are shown in Figs. 4-7.

Fig. 4 shows the phase velocity [U G w/Re(k,)] for the (leaky) BG waves for the open surface boundary conditions as a function of the crystal orientation: U < U, for all orienta- tions and U = U, at orientations near 15" and 138O, i.e., at orientations at which the surface BG wave changes into a leaky BC wave. This is to be compared to BG wave phase velocities for the metallized surface boundary conditions [ S ] shown in Fig. 5 . The wave is a surface wave, and umeta1 <

WILCOX AND YEN: SHEAR HORIZONTAL SURFACE WAVES 453

VELOCITY FOLLOW VERY

0 90 Y IDEG)

180

Fig. 6. Velocity and 2Au/u for the (leaky) surface BG waves in rotated Y-cut quartz.

2 x 10-2

10-2

n p! y1

U l-

2 o

SURFACE BG WAVES

LEAKY WAVE REGION - +S x loJ

0

YI Y E

Q

0 90 180

U IOEG)

Fig. 7 . Decay (growth) coefficients in direction normal to surface for (leaky) surface waves in rotated Y-cut quartz.

uopen <U, for all crystal orientations. The parameter 2Au/u is shown in Fig. 6. Fig. 7 shows the values of the imaginary part of the acoustic wave vector k,, E -Im(kpc))/kc; 0 posi- tive and negative corresponds to surface and leaky BC waves respectively. Notice the different scales for the open and metallized surfaces. As expected, the penetration depth into the solid for the metallized surface is less than that for the open surface.

The direction of the energy transport is parallel to the acoustic Poynting vector p'. Using the ratio p z /p,, 6, = tan-' (pz/p,) was also evaluated. The values of 6,. thus ob- tained, though positive (indicating velocity vector tilted into the crystal), were on the order of lo-' or less. Because this is smaller than the realistic accuracy of the computation, the computed values of 6 , were given no further consideration. The program also evaluated the normalized values of the BC amplitudes @@) and U@): (z = 0) was found to be on the same order of magnitude as @((Ic) (z = 0), and d e l )

(z = 0) lower than U(',) (z = 0) by a factor on the order 2Au/u = e$/qiCjj . Because the quasi-electrostatic mode de- cays very rapidly, its penetration depth into the substrate

being on the order of k;' , this mode contribution to the energy transport can be neglected.

Finally, we should note that the results presented in Section 111-A for the finite thickness dielectric referred to those crystal orientations for which the surface wave solutions existed for any finite dielectric thickness, and would exist even with the piezoelectric effect neglected (Love waves). It was also seen that as the consequence of taking the piezoelectric effect into account, surface BC waves occurred for some of the orienta- tions for which the normal Love waves do not exist. It would then be logical if for these orientations, surface wave solutions would occur for small yet finite dielectric layer thicknesses. Thn in fact has indeed been observed; our numerical computa- tions for the crystal orientation of 0' found surface wave solutions with the phase velocity U , uB-G < U < U, for the dielectric layer thickness t less than X, where h is the acoustic wavelength. For the thickness t/h = l 0-4, the sur- face wave velocity reached the value of the cutoff bulk wave velocity, at which point the character of the wave would be expected to change; that is, the surface wave would be ex- pected to change into the leaky wave. Because of the restric- tion of this part of our program to real values of k,, leaky waves for finite t were not calculated.

IV. CONCLUSION An analysis of shear horizontal surface waves on rotated Y-

cut quartz has been performed. For the case of quartz covered with a dielectric layer, the dispersion relationship 2Au/u and the wave penetration depth into the substrate have been calcu- lated for those substrate orientations for which the bulk cutoff velocity is greater than the acoustic velocity in the dielectric; i.e., for those orientations for which surface Love waves exist. When evaluated as a function of the dielectric layer thickness, 2Au/u was shown to exhibit a maximum at about t /h = 0.1, and the wave penetration depth into the quartz was decreas- ing, with the dielectric thickness increasing. This property could be used to decrease the insertion loss of SBAW inter- digital transducers through the coupling between the SBAW and surface waves.

For quartz without the surface dielectric layer, a more de- tailed analysis showed that the boundary-condition determi- nant has leaky BC wave solutions for the crystal orientations 15"-138" and surface wave solutions for the complementary crystal orientations [S]. This is somewhat different from the usual situation for which leaky surface waves occur. The usual leaky wave consists of coupled acoustic modes, some of which decay beneath the surface and some of which represent a bulk wave radiating into the solid. The wave occurs in addition to the normal surface wave at the given propagation direction. Leaky waves in rotated Y-cut quartz consist of coupled quasi- electrostatic and growing shear horizontal quasi-acoustic modes, and occur for those crystal orientations at which the surface BC waves do not exist. The leaky wave velocities are very close to the bulk wave cutoff velocity, with the result that both the wave attenuation along the surface and growth into the solid are very slow; this makes it possible to talk about pseudo-surface waves. The waves are nearly degenerate with the bulk waves; however, whereas (for a line excitation

4 54 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-28, NO. 6 , NOVEMBER 1981

source) the bulk wave amplitude decays with distance as the leaky surface wave amplitude is (practically) unattenuated at the surface.

REFERENCES

[ l ] M. Lewis, “Surface skimming bulk waves,” in Roc. 1977 IEEE Ultrason. Symp., pp. 744-752.

[2] K. F. Lau, K. H. Yen, R. S. Kagiwada, and K. L. Wang, “Further investigations of shallow bulk acoustic waves generated by using interdigital transducers,” in Roc. 1977 IEEE Ultrason. Symp., pp. 996-1001.

[ 3) A. Auld, J . J. Gagnepain, and M. Tan, “Horizontal shear surface

waves on corrugated surfaces,” Electron. Lett . , vol. 12, pp. 650-652, Nov. 1976.

[4] G. G. Koerber and R. F. Vogel, “SH-mode piezoelectric surface waves on rotated cuts,” IEEE Dam. Son. Ultrason., vol. SU-20, pp. 9-12, 1973.

“Theoretical examination of surface skimming waves,” in R o c . 1978IEEE Ultrason. Symp., pp. 670-674.

[6] J. Z. Wilcox, K. H. Yen, and K. F. Lau, “Excitation of shallow bulk acoustic waves by interdigital transducers in rotated Y-cut quartz,” to be published.

[7] T. C. Lim and G. W. Farnel, “Character of pseudosurface waves on anisotropic crystals,” J. Acoust. Soc. Am., vol. 45, pp. 845-85 1, 1969.

[ 5 ] A. Jhunjhunwala, J. F. Vetelino, D. Harmon, and W. Soluch,

Distortion Estimation of SAW Time Inversion System Based on Delta Function Approximation

Absfruct-Time inversion can be obtained using surface acoustic wave (SAW) devices with no more than two chirp filter elements. This time inversion method is based on the fact that the Fourier transform of a linear frequency modulated (FM) signal, whose envelope is modulated by a given time function, has (approximately) the time-inverted func- tion as its amplitude. The different distortions inherent in the method are discussed. Procedures are suggested to eliminate some of these dis- tortions, while others, inherent due to the approximation used, are estimated. It is shown that the average of the output distortion is upper-bounded by w2/16p, where W is the signal bandwidth and p is the chirp filter’s dispersive slope. Simulation results supplement the analysis.

T INTRODUCTION

HE time-inverted replica of a given signal is very impor- tant in many signal processing applications. Using sur-

face acoustic wave (SAW) components, Nudd et al. [ l ] applied two identical inverse chirp-transform systems to a real time sig- nal and obtained an output that is the time-inverted replica of the input. However, implementing the system using chirp de- vices requires at least four chirp filters. Arsenault, e t al. [ 2 ] used the Fresnel transform to obtain time inversion with three chirp filters. Recently, a new SAW time inversion system was

Manuscript received May 15, 1981. This work was supported in part by the Air Force Office of Scientific Research under Contract No.

H. Messer is with the School of Engineering, Tel Aviv University, Ramat-Aviv 69978, Israel. Y. Bar-Ness is with the Valley Forge Research Center, Moore School

of Electrical Engineering, University of Pennsylvania, Philadelphia, PA 19104, on leave from the Department of Electronic Systems, School of Engineering, Tel Aviv University, Ramat-Aviv 69978, Israel.

AFOSR-78-3688.

proposed [3]. It is based on the fact that the Fourier trans- form of a linear frequency modulated (FM) signal, which is envelope-modulated by a given time function, is (approxi- mately) the time-inverted replica of the function. The idea is based on a certain approximated limit expression for the delta function [4]. A detailed comparison between the various time inversion methods was recently published in [ 5 ] .

Here, a general analysis is presented for time inversion sys- tems based on delta function approximation [3]. Two types of distortions are shown to result-image distortion and delta function approximation distortion-and expressions describing them are derived. Conditions are established for the elimina- tion of image distortion using time gating. To estimate the second type of distortion, upper bounds are established on its time average. These bounds depend on the signal energy, the squared signal bandwidth, and the filter’s slope. Using com- puter simulations, the time average of the delta function ap- proximation distortion is computed for different values of sig- nal bandwidth and filter dispersive slopes, and a comparison is made with the calculated upper bounds.

SYSTEM ANALYSIS Consider the system shown in Fig. 1 and let

f ( t ) = f ( t ) , to l < t o + T

= 0, elsewhere

m (t) = exp [it(wl ? 2 p ( t - 2 t 1 - T , ) ) ]

+CC, t1 < t < t l +TI

= 0, elsewhere

0018-9537/81/1100-0454$00.75 0 1981 IEEE