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Dispersion of low frequency surface acoustic waves of different polarizations in multilayered systems
Solid State Communications. Vol. 93, No. 8, pp. 701-705, 1995 Elsevier Science Ltd Printed in Great Britain 003%1098/95 $9.50 + .OO
Mm-1o98ooo687-3
DISPERSION
OF LOW FREQUENCY SURFACE ACOUSTIC WAVES OF DIFFERENT POLARIZATIONS IN MULTILAYERED SYSTEMS V.V. Kosachev and A.V. Shchegrov
Department
of Theoretical Physics, Moscow Engineering Physics Institute, 3 1 Kashirskoye shosse, Mos~cow 115409, Russia (Received 20 June 1994; in revised form 16 September 1994 by S.G. Louie) Dispersion of surface acoustic waves (SAW) of sagittal and shear horizontal (SH) polarizations in a multilayered system of n isotropic layers on an isotropic substrate is investigated by the technique of effective boundary conditions in the framework of perturbation theory. The ratio of the total layer thickness to the wavelength of SAW is chosen to be a small parameter. Under such assumptions the dispersion relations for the :SAW of both sagittal and SH-polarizations are derived. The results for sagittally polarized SAW derived by means of perturbation theory are compared with numerical solution for a bilayered structure. Possibl[e applications of the results obtained are discussed. Keywords: A. surfaces and interfaces, D. acoustic properties, D. elasticity, E. ultrasonics.
1. INTRODUCTION AN INTEREST in the surface acoustic waves (SAW) not abating since the discovery of Rayleigh w.aves is mainly explained by their sensitivity to the perturbations of a surface as well as the ability to control the velocity of SAW by disturbing the surface. One of the most intensively explored problems has been the study of the dispersion relation of SAW propagating in the structure of a semi-infinite substrate and a system of elastic layers. Seismologists were the first to show interest in such structures [l, 21. F’urther investigations of propagating SAW in such structures were prompted by the development of electronic systems of signal processing based on SAW [3,4] and the study of elastic properties of surface layers [5]. The dispersion relation of SAW in the structure of a substrate and an arbitrary number of elastic layers can be derived by means of a matrix technique, originally designed in [ 1,2] and later developed in the investigations of seismic waves [6]. Further it was also extended to study the propagation of SAW in anisotropic media [7]. However, even in the simplest case of a single layer the exact dispersion relation for the SAW of sagittal polarization is a fairly com-
plicated transcendental equation subjected only to numerical analysis. Recently an approach has been developed to the calculation of the dispersion of SAW in layered structures where with the help of expanding the displacement fields along the complete orthogonal system of functions the problem is reduced to the solution of an infinite set of linear algebraic equations [8, 93. However, on the basis of such an approach as well as the matrix technique, one cannot obtain any calculational formulae describing the SAW dispersion in multilayered structures. It should be noted that the situation when the number of layers on the substrate can be considered as a perturbation is frequently realized. The case of a single thin layer perturbing the substrate was theoretically treated in detail [lo-131. The small parameter in it is the relation of the layer thickness to the wavelength of SAW. If only the first mode is taken into account and higher modes are neglected, then the dispersion relation of the sagittally polarized SAW can be written as w(k) = w,,(k) + Au(k), where tic,(k) = cf)k
701
(1) and
cf’
is the velocity
of
DISPERSION
702
OF LOW FREQUENCY
Rayleigh waves in the substrate. The explicit form of the dispersion relation for Rayleigh waves is given in [3, 11, 121.A similar problem takes place for the SAW of shear horizontal (SH) polarization, well-known as Love waves in the case of a single layer. In [13] to derive the dispersion relation of Love wave in the long wavelength region a technique different from the conventional was developed. It allows reduction of the problem to the treatment of the equations of motion in a substrate with effective boundary conditions. The approximate methods mentioned above have not been developed to cover multi-layered structures. However, such results could be useful both as calculational formulae [5] and for the problem of choosing parameters of a multilayered structure, which arises frequently in designing SAW devices [4]. In this work we try to make such a generalization. 2. EFFECTIVE BOUNDARY CONDITIONS AND DISPERSION RELATIONS FOR SAW The geometry of the problem is shown in Fig. 1. On the substrate taken as an isotropic uniform elastic half-space z 5 0 there are n flat parallel uniform isotropic layers of different thicknesses dj (j = 1, . . . , n). The contact on the interface boundary is rigid whereas the surface z = h, is stress-free. The total thickness of all layers h is taken so small that the condition h/X < 1 is satisfied where X is the wavelength of SAW. Let a plane monochromatic SAW of sagittal polarization propagate along the x axis. The displacement vector u(x; t) in such a wave has only two components a,, u; depending on x, z, while uv = 0. The boundary conditions on the stress-free surface z = h, and on the interface z = hi of any two media will correspondingly be (T;#rn-) = 0;
s&z-)
a,(hi-) ox,;(hi-)
= ux(hi+); =
4rz(4+);
= 0;
SURFACE ACOUSTIC WAVES
Vol. 93, No. 8
di
I
/
-x
///I//
SW
Suhstrate
//////
9%
/
Fig. 1. Geometry of the problem: n is the total number of layers on the substrate; dj is the thickness of the jth layer; h = h, = Cj”= 1dj 1s the total layer thickness. field u,,, can be broken at the interface. To eliminate these values from equations (5) and (6) the equations of motion of the nth medium will further be used:
where the point means the time derivative; pn is the mass density of the n th layer. Then we pass to the interface of the nth and (n - 1)th layers z = h,_ 1, where we substitute equations (5) and (6) into the right-hand sides of the boundary conditions (4) z = h,_,: &‘)(h,_,-) x-
= - d,c$‘,z I (h,, _ I+) ,
&“(h,_,-) __
= - d,,CT(“) u,z (h,, _ , +) .
(2)
Then we use equation (8) in equation (10):
U=(hi-) = u;(hi+),
(3)
&‘)(h,_
022(hi-)
(4)
=
Oiz(hi+)j
where oXxz,crZ,are the components of the stress tensor c&3(% B = x, Y,z). Using a small magnitude of d,, we can expand a,@ into the Taylor series in the upper vicinity of the plane 2 = h,_t:
1-)
= -d,,{p,ii;(h,-,
+) - &@n-,
+)I.
(11)
It follows from equation (5) that the value of {d,p~J,x} in equation (1 1) is of the second order in small d, while the accuracy we hold is of the first order. Therefore this term can be neglected in equation (11) and the resulting equation has the form
a(“)@ ,,2(h,-1+)+...=0; xz n_ , +) + d,p(“)
(5)
a(!-‘)(h,_, 2.
&)(h,_
(6)
The situation is more complicated for equation (9). Applying equation (7) alone
,+) + d,p?j2..Z(h,_, +) + . . . = 0,
where the upper index n denotes the number of the medium and notations ga81y = &,,/a~, (xt = x, x2 = y, x3 = z) are used for brevity. It should be pointed out that the derivatives of the displacement
&“(h,_,
-) = -d,,pniiZ(h,_I +).
-)
(14
DISPERSION
Vol. 93, No. 8
OF LOW FREQUENCY
is not enough here because o.y.n; ix includes broken values. It would be useful first to write a,, in the form
using explicit expressions for O.rX,a,, in an isotropic medium. Taking equation (14) into account, we substitute a,,r, .‘( for nth layer into equation (13) keeping the first-order terms in the right-hand side: &‘)(h,_,
-)
= - p,,d, G(h,- I +) - 4cj”“u,,&-
I +)
Cl
w(k) g cIps’k{1 + A(M)} + 0[(kh)2],
*
(20)
where the following notations are used:
Thus the boundary conditions on the plane z ==h,_ I are equations (12) and (15). As the right-hand terms of equations (12) and (15) include only continuous values at the interface z = h,_ ,, it is possible to transfer from z = h, _ , + to z = h, _ I - in them. The boundary conditions on the plane z = h,_ I will finally be &‘)(h,_, X.
703
isotropic medium S. Equations of motion in the substrate are the same, the effective boundary conditions (18) and (19) include the influence of the layers. If all di in equations (18) and (19) are set equal to zero, the expressions transform into the boundary conditions c(s) I_ = aif) = 0 which correspond to a stress-free surface and the waves propagating along the stress-free surface of an elastic half-space are in fact the well-known Rayleigh waves. Then substituting in equations (18) and (19) the solution of the equations of motion in the substrate in form of a surface wave, we obtain the set of linear algebraic equations. The solvability condition of the system gives the dispersion relation of the sagittally polarized SAW to the first order of perturbation theory:
(15)
1
$)4
+4-(;;jr~XIX.Y@-1+)
SURFACE ACOUSTIC WAVES
A =p2/p,;
p, =4
-)
= - p,d,
ii,@,_, -) - ~c?)~u,,,@,-
1-j
(16) 17 Cl &‘)(h”_, -)=-~ndnk(h,-1-). (17) $14
+
4~~XIX.#r~-I
-)
Proceeding further to the interface z = /z~_~, and then z = hn_3, and so on, one can achieve the interrace of the substrate and the layers z = 0. Finally, the effective boundary conditions are given by 0::‘(0-)
= - 2
Pjdj ii, cjj)4
+
4~~+X(O-) Cl
&o-)
= -
2
- 4cv)2u,iX.X(o-)
[
j=l
pjqiz(O-).
1 7
(18) (19)
c$, are the velocities of Rayleigh and bulk longitudinal and transverse waves, respectively; k is a wave number. If there is only one isotropic layer on the isotropic substrate (n = 1) equation (20) agrees with the result of [lo]. It should be noticed that we distinguished the perturbation theory parameter (kh) in equation (20) intentionally. We did so only for convenience since small enough (k/r) always ensures the correctness of all the expansions we carried out when deriving equation (20). We also note that each layer takes part in the first-order correction in equation (20) additively, therefore the arrangement of layers does not play any role to the first order. This effect shows in the second-order correction, whose derivation is similar to the above procedure. Finally, the dispersion relation and phase velocity of the sagittally polarized SAWS to the second order are:
j=l
Thus the problem of sagittally polarized SAW propagating in the multilayered medium shown in Fig. 1 can be reduced to a simpler problem ol? SAW propagating along the surface of semi-infinite elastic
w(k) = $‘k{ 1 + A&h) + B(kh)2},
(21)
W(V) z cf’{ 1 + A(27rvh@)
+ (B - A2)(27r+~‘)2},
(22)
DISPERSION
704
OF LOW FREQUENCY
SURFACE ACOUSTIC WAVES
Vol. 93, No. 8
(2mhlCR’S’) Il.
3450”
I
3150 I)
I IO
0.3 I
0.1 I
I
I 30
I 20
Frequency
I
I
I
40
50
60
70
(MHz)
Frequency
Fig. 2. Dispersion of the phase velocity for the sagittally polarized SAW. The substrate is fused quartz; the 1st layer is Al, di = 2250 nm; the 2nd layer is Ag, d2 = 480 nm. Solid line is for numerical solution; dashed line is for perturbation theory. where v = w/27r is a frequency, and the constant B is defined in Appendix A. Similarly to the case of sagittally polarized SAW we derived the dispersion relation for shear horizontal SAW. The first nonvanishing correction here is of the second order in (k/z): w(k) = cj%[l
- g(kh)2],
(23)
where
For the case of a single layer equation (23) coincides with that of [13]. 3. RESULTS AND DISCUSSION Let us discuss the results obtained in the case of sagittally polarized SAWS for the simplest multilayered structure with n = 2. Figure 2 presents the theoretical and numerical results for the dispersion of SAW in the structure with the substrate of fused quartz; the 1st layer is Al, dl = 2250 nm, the 2nd layer is Ag, d2 = 480 nm. The numerical curve was calculated directly from the transcendental dispersion equation for SAW in the b&layered structure under study. The dispersion curves depicted in Fig. 2 confirm the correctness of the results derived in the framework of the perturbation theory for a small (kh). A noticeable difference of the perturbation curve from the numerical one manifests for the structure beginning from v N 40 MHz [(27r&r/c(Rs’)N 0.201. The study of other bi-layered structures brought that the validity region of the perturbation theory
(MHz)
Fig. 3. Dispersion of the phase velocity for the sagittally polarized SAW, numerical solution. Solid line for the structure: the substrate is fused quartz; the 1st layer is Al, dl = 590nm; the 2nd layer is Cr, d2 = 140nm. Dashed line is for the structure: the substrate is fused quartz; the only layer is Al, dl = 590nm. can increase or decrease appreciably. The formula (22) is always correct in the limit of small frequencies. However, it is rather difficult to define the condition (27r~h/c~‘) < 1 (which formally determines the validity range of the perturbation theory) more accurately because of a great number of parameters in the problem. In addition to the use of equation (22) as a calculational formula in the low-frequency limit, its application for designing layered structures with a minimum dispersion could be no less important. In SAW devices it is often desirable to diminish the perturbation introduced by the layers [4]. Hence Al, whose Rayleigh velocity is close to those of such widespread materials as quartz and lithium niobate, is usually used. It is obvious that even in case of two layers we can minimize the dispersion of the lowfrequency SAW by appropriate choice of not only materials but also their thicknesses. For instance, let us take for the substrate of fused quartz Al as the 1st layer (“slower” than the substrate, cj’) < cl’)), and Cr as the end layer (“faster” than the substrate, cj2) > cj’)). The ratio d,/d2 is taken from the minimum condition for the magnitude of A in equation (22). The following system can serve an example of such a structure: 1st layer is Al, dl = 590nm; 2nd layer is Cr, d2 = 140nm. It is clear that the higher orders of the perturbation theory become essential, therefore we use a numerical solution as an independent check. It follows from Fig. 3 that up to u,,, N 200MHz [(27r~h/c~‘) ~0.271 dispersion in the structure under study is considerably
Vol. 93, No. 8
DISPERSION
OF LOW FREQUENCY
less as compared with that in the structure with only the layer of Al, 590nm. This frequency range can be expanded by diminishing the total thickness of the layers with a fixed ratio d,/d,. Therefore, the advantage of equation (22) here is obvious since it is very difficult to find a multilayered structure with minimum dispersion with only SAW transcendental dispersion equation. Acknowledgements - We are grateful to Prof. P. Hess
and Prof. D. Morgan discussions.
for stimulating
and useful
A.A. Maradudin, in Nonequilibrium Phonon Dynamics (Edited by W.E. Bron), p. 448. Plenum, New York (1985).
13.
APPENDIX B=
2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12.
N.A. Haskell, Bull. Seismol. Sot. Amer. ~43, 17 (1953).
W.T. Thomson, J. Appl. Phys. 21, 89 (1950). A.A. Oliner (editor), Acoustic Surface Waves. Springer, Berlin (1978). D. Morgan, Surface-wave Devices for Signal Processing. Elsevier, Amsterdam (1985). A. Neubrand & P. Hess, J. Appl. Phys. 71, 227 (1992). K. Aki & P. Richards, Quantitative Seismology. Theory and Methods, Vol. 1, Ch. 7. Freeman,
A
l E/p+!54+ps -2
PI
PI
Pl’
$1 $1
2 Qr +r,,-2%
$1 4);’
G, $1
2rf-2r,p+r,-+rrl--+rf-
p3 =4
REFERENCES 1.
; 1
n
p4 = -2r,
c 1 .j=
fj4)
1
W/r,
+ h(r,
-- 1 2+, ‘tci Ps = (1 - P2) $
%#iCi 2
+ Ej)
1
cjdjrt
;
-
w
j=l
San Francisco (1980). S. Crampin, Geophys. J. Royal Astron. SIX. 21,
387 (1970). S. Datta & B.J. Hunsinger, J. Appl. PhjIs. 49, 475 (1978). Y. Kim & W.D. Hunt, J. Appl. Phys. 68, 4993 (1990). H.F. Tiersten, J. Appf. Phys. 40, 770 (1968).
S.V. Bogdanov, M.D. Levin & LB. Jakovkin, Akust. Zh. 15, 12 (1969) [Sov. Phys.-Acoustics 15, 10 (1969)]. B.A. Auld, Acoustic Fields and Waves in Solids,
Vol. 1. Wiley, New York (1973).
105
SURFACE ACOUSTIC WAVES
x
;
-
ir,r, WQ
Mm-1o98ooo687-3
DISPERSION
OF LOW FREQUENCY SURFACE ACOUSTIC WAVES OF DIFFERENT POLARIZATIONS IN MULTILAYERED SYSTEMS V.V. Kosachev and A.V. Shchegrov
Department
of Theoretical Physics, Moscow Engineering Physics Institute, 3 1 Kashirskoye shosse, Mos~cow 115409, Russia (Received 20 June 1994; in revised form 16 September 1994 by S.G. Louie) Dispersion of surface acoustic waves (SAW) of sagittal and shear horizontal (SH) polarizations in a multilayered system of n isotropic layers on an isotropic substrate is investigated by the technique of effective boundary conditions in the framework of perturbation theory. The ratio of the total layer thickness to the wavelength of SAW is chosen to be a small parameter. Under such assumptions the dispersion relations for the :SAW of both sagittal and SH-polarizations are derived. The results for sagittally polarized SAW derived by means of perturbation theory are compared with numerical solution for a bilayered structure. Possibl[e applications of the results obtained are discussed. Keywords: A. surfaces and interfaces, D. acoustic properties, D. elasticity, E. ultrasonics.
1. INTRODUCTION AN INTEREST in the surface acoustic waves (SAW) not abating since the discovery of Rayleigh w.aves is mainly explained by their sensitivity to the perturbations of a surface as well as the ability to control the velocity of SAW by disturbing the surface. One of the most intensively explored problems has been the study of the dispersion relation of SAW propagating in the structure of a semi-infinite substrate and a system of elastic layers. Seismologists were the first to show interest in such structures [l, 21. F’urther investigations of propagating SAW in such structures were prompted by the development of electronic systems of signal processing based on SAW [3,4] and the study of elastic properties of surface layers [5]. The dispersion relation of SAW in the structure of a substrate and an arbitrary number of elastic layers can be derived by means of a matrix technique, originally designed in [ 1,2] and later developed in the investigations of seismic waves [6]. Further it was also extended to study the propagation of SAW in anisotropic media [7]. However, even in the simplest case of a single layer the exact dispersion relation for the SAW of sagittal polarization is a fairly com-
plicated transcendental equation subjected only to numerical analysis. Recently an approach has been developed to the calculation of the dispersion of SAW in layered structures where with the help of expanding the displacement fields along the complete orthogonal system of functions the problem is reduced to the solution of an infinite set of linear algebraic equations [8, 93. However, on the basis of such an approach as well as the matrix technique, one cannot obtain any calculational formulae describing the SAW dispersion in multilayered structures. It should be noted that the situation when the number of layers on the substrate can be considered as a perturbation is frequently realized. The case of a single thin layer perturbing the substrate was theoretically treated in detail [lo-131. The small parameter in it is the relation of the layer thickness to the wavelength of SAW. If only the first mode is taken into account and higher modes are neglected, then the dispersion relation of the sagittally polarized SAW can be written as w(k) = w,,(k) + Au(k), where tic,(k) = cf)k
701
(1) and
cf’
is the velocity
of
DISPERSION
702
OF LOW FREQUENCY
Rayleigh waves in the substrate. The explicit form of the dispersion relation for Rayleigh waves is given in [3, 11, 121.A similar problem takes place for the SAW of shear horizontal (SH) polarization, well-known as Love waves in the case of a single layer. In [13] to derive the dispersion relation of Love wave in the long wavelength region a technique different from the conventional was developed. It allows reduction of the problem to the treatment of the equations of motion in a substrate with effective boundary conditions. The approximate methods mentioned above have not been developed to cover multi-layered structures. However, such results could be useful both as calculational formulae [5] and for the problem of choosing parameters of a multilayered structure, which arises frequently in designing SAW devices [4]. In this work we try to make such a generalization. 2. EFFECTIVE BOUNDARY CONDITIONS AND DISPERSION RELATIONS FOR SAW The geometry of the problem is shown in Fig. 1. On the substrate taken as an isotropic uniform elastic half-space z 5 0 there are n flat parallel uniform isotropic layers of different thicknesses dj (j = 1, . . . , n). The contact on the interface boundary is rigid whereas the surface z = h, is stress-free. The total thickness of all layers h is taken so small that the condition h/X < 1 is satisfied where X is the wavelength of SAW. Let a plane monochromatic SAW of sagittal polarization propagate along the x axis. The displacement vector u(x; t) in such a wave has only two components a,, u; depending on x, z, while uv = 0. The boundary conditions on the stress-free surface z = h, and on the interface z = hi of any two media will correspondingly be (T;#rn-) = 0;
s&z-)
a,(hi-) ox,;(hi-)
= ux(hi+); =
4rz(4+);
= 0;
SURFACE ACOUSTIC WAVES
Vol. 93, No. 8
di
I
/
-x
///I//
SW
Suhstrate
//////
9%
/
Fig. 1. Geometry of the problem: n is the total number of layers on the substrate; dj is the thickness of the jth layer; h = h, = Cj”= 1dj 1s the total layer thickness. field u,,, can be broken at the interface. To eliminate these values from equations (5) and (6) the equations of motion of the nth medium will further be used:
where the point means the time derivative; pn is the mass density of the n th layer. Then we pass to the interface of the nth and (n - 1)th layers z = h,_ 1, where we substitute equations (5) and (6) into the right-hand sides of the boundary conditions (4) z = h,_,: &‘)(h,_,-) x-
= - d,c$‘,z I (h,, _ I+) ,
&“(h,_,-) __
= - d,,CT(“) u,z (h,, _ , +) .
(2)
Then we use equation (8) in equation (10):
U=(hi-) = u;(hi+),
(3)
&‘)(h,_
022(hi-)
(4)
=
Oiz(hi+)j
where oXxz,crZ,are the components of the stress tensor c&3(% B = x, Y,z). Using a small magnitude of d,, we can expand a,@ into the Taylor series in the upper vicinity of the plane 2 = h,_t:
1-)
= -d,,{p,ii;(h,-,
+) - &@n-,
+)I.
(11)
It follows from equation (5) that the value of {d,p~J,x} in equation (1 1) is of the second order in small d, while the accuracy we hold is of the first order. Therefore this term can be neglected in equation (11) and the resulting equation has the form
a(“)@ ,,2(h,-1+)+...=0; xz n_ , +) + d,p(“)
(5)
a(!-‘)(h,_, 2.
&)(h,_
(6)
The situation is more complicated for equation (9). Applying equation (7) alone
,+) + d,p?j2..Z(h,_, +) + . . . = 0,
where the upper index n denotes the number of the medium and notations ga81y = &,,/a~, (xt = x, x2 = y, x3 = z) are used for brevity. It should be pointed out that the derivatives of the displacement
&“(h,_,
-) = -d,,pniiZ(h,_I +).
-)
(14
DISPERSION
Vol. 93, No. 8
OF LOW FREQUENCY
is not enough here because o.y.n; ix includes broken values. It would be useful first to write a,, in the form
using explicit expressions for O.rX,a,, in an isotropic medium. Taking equation (14) into account, we substitute a,,r, .‘( for nth layer into equation (13) keeping the first-order terms in the right-hand side: &‘)(h,_,
-)
= - p,,d, G(h,- I +) - 4cj”“u,,&-
I +)
Cl
w(k) g cIps’k{1 + A(M)} + 0[(kh)2],
*
(20)
where the following notations are used:
Thus the boundary conditions on the plane z ==h,_ I are equations (12) and (15). As the right-hand terms of equations (12) and (15) include only continuous values at the interface z = h,_ ,, it is possible to transfer from z = h, _ , + to z = h, _ I - in them. The boundary conditions on the plane z = h,_ I will finally be &‘)(h,_, X.
703
isotropic medium S. Equations of motion in the substrate are the same, the effective boundary conditions (18) and (19) include the influence of the layers. If all di in equations (18) and (19) are set equal to zero, the expressions transform into the boundary conditions c(s) I_ = aif) = 0 which correspond to a stress-free surface and the waves propagating along the stress-free surface of an elastic half-space are in fact the well-known Rayleigh waves. Then substituting in equations (18) and (19) the solution of the equations of motion in the substrate in form of a surface wave, we obtain the set of linear algebraic equations. The solvability condition of the system gives the dispersion relation of the sagittally polarized SAW to the first order of perturbation theory:
(15)
1
$)4
+4-(;;jr~XIX.Y@-1+)
SURFACE ACOUSTIC WAVES
A =p2/p,;
p, =4
-)
= - p,d,
ii,@,_, -) - ~c?)~u,,,@,-
1-j
(16) 17 Cl &‘)(h”_, -)=-~ndnk(h,-1-). (17) $14
+
4~~XIX.#r~-I
-)
Proceeding further to the interface z = /z~_~, and then z = hn_3, and so on, one can achieve the interrace of the substrate and the layers z = 0. Finally, the effective boundary conditions are given by 0::‘(0-)
= - 2
Pjdj ii, cjj)4
+
4~~+X(O-) Cl
&o-)
= -
2
- 4cv)2u,iX.X(o-)
[
j=l
pjqiz(O-).
1 7
(18) (19)
c$, are the velocities of Rayleigh and bulk longitudinal and transverse waves, respectively; k is a wave number. If there is only one isotropic layer on the isotropic substrate (n = 1) equation (20) agrees with the result of [lo]. It should be noticed that we distinguished the perturbation theory parameter (kh) in equation (20) intentionally. We did so only for convenience since small enough (k/r) always ensures the correctness of all the expansions we carried out when deriving equation (20). We also note that each layer takes part in the first-order correction in equation (20) additively, therefore the arrangement of layers does not play any role to the first order. This effect shows in the second-order correction, whose derivation is similar to the above procedure. Finally, the dispersion relation and phase velocity of the sagittally polarized SAWS to the second order are:
j=l
Thus the problem of sagittally polarized SAW propagating in the multilayered medium shown in Fig. 1 can be reduced to a simpler problem ol? SAW propagating along the surface of semi-infinite elastic
w(k) = $‘k{ 1 + A&h) + B(kh)2},
(21)
W(V) z cf’{ 1 + A(27rvh@)
+ (B - A2)(27r+~‘)2},
(22)
DISPERSION
704
OF LOW FREQUENCY
SURFACE ACOUSTIC WAVES
Vol. 93, No. 8
(2mhlCR’S’) Il.
3450”
I
3150 I)
I IO
0.3 I
0.1 I
I
I 30
I 20
Frequency
I
I
I
40
50
60
70
(MHz)
Frequency
Fig. 2. Dispersion of the phase velocity for the sagittally polarized SAW. The substrate is fused quartz; the 1st layer is Al, di = 2250 nm; the 2nd layer is Ag, d2 = 480 nm. Solid line is for numerical solution; dashed line is for perturbation theory. where v = w/27r is a frequency, and the constant B is defined in Appendix A. Similarly to the case of sagittally polarized SAW we derived the dispersion relation for shear horizontal SAW. The first nonvanishing correction here is of the second order in (k/z): w(k) = cj%[l
- g(kh)2],
(23)
where
For the case of a single layer equation (23) coincides with that of [13]. 3. RESULTS AND DISCUSSION Let us discuss the results obtained in the case of sagittally polarized SAWS for the simplest multilayered structure with n = 2. Figure 2 presents the theoretical and numerical results for the dispersion of SAW in the structure with the substrate of fused quartz; the 1st layer is Al, dl = 2250 nm, the 2nd layer is Ag, d2 = 480 nm. The numerical curve was calculated directly from the transcendental dispersion equation for SAW in the b&layered structure under study. The dispersion curves depicted in Fig. 2 confirm the correctness of the results derived in the framework of the perturbation theory for a small (kh). A noticeable difference of the perturbation curve from the numerical one manifests for the structure beginning from v N 40 MHz [(27r&r/c(Rs’)N 0.201. The study of other bi-layered structures brought that the validity region of the perturbation theory
(MHz)
Fig. 3. Dispersion of the phase velocity for the sagittally polarized SAW, numerical solution. Solid line for the structure: the substrate is fused quartz; the 1st layer is Al, dl = 590nm; the 2nd layer is Cr, d2 = 140nm. Dashed line is for the structure: the substrate is fused quartz; the only layer is Al, dl = 590nm. can increase or decrease appreciably. The formula (22) is always correct in the limit of small frequencies. However, it is rather difficult to define the condition (27r~h/c~‘) < 1 (which formally determines the validity range of the perturbation theory) more accurately because of a great number of parameters in the problem. In addition to the use of equation (22) as a calculational formula in the low-frequency limit, its application for designing layered structures with a minimum dispersion could be no less important. In SAW devices it is often desirable to diminish the perturbation introduced by the layers [4]. Hence Al, whose Rayleigh velocity is close to those of such widespread materials as quartz and lithium niobate, is usually used. It is obvious that even in case of two layers we can minimize the dispersion of the lowfrequency SAW by appropriate choice of not only materials but also their thicknesses. For instance, let us take for the substrate of fused quartz Al as the 1st layer (“slower” than the substrate, cj’) < cl’)), and Cr as the end layer (“faster” than the substrate, cj2) > cj’)). The ratio d,/d2 is taken from the minimum condition for the magnitude of A in equation (22). The following system can serve an example of such a structure: 1st layer is Al, dl = 590nm; 2nd layer is Cr, d2 = 140nm. It is clear that the higher orders of the perturbation theory become essential, therefore we use a numerical solution as an independent check. It follows from Fig. 3 that up to u,,, N 200MHz [(27r~h/c~‘) ~0.271 dispersion in the structure under study is considerably
Vol. 93, No. 8
DISPERSION
OF LOW FREQUENCY
less as compared with that in the structure with only the layer of Al, 590nm. This frequency range can be expanded by diminishing the total thickness of the layers with a fixed ratio d,/d,. Therefore, the advantage of equation (22) here is obvious since it is very difficult to find a multilayered structure with minimum dispersion with only SAW transcendental dispersion equation. Acknowledgements - We are grateful to Prof. P. Hess
and Prof. D. Morgan discussions.
for stimulating
and useful
A.A. Maradudin, in Nonequilibrium Phonon Dynamics (Edited by W.E. Bron), p. 448. Plenum, New York (1985).
13.
APPENDIX B=
2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12.
N.A. Haskell, Bull. Seismol. Sot. Amer. ~43, 17 (1953).
W.T. Thomson, J. Appl. Phys. 21, 89 (1950). A.A. Oliner (editor), Acoustic Surface Waves. Springer, Berlin (1978). D. Morgan, Surface-wave Devices for Signal Processing. Elsevier, Amsterdam (1985). A. Neubrand & P. Hess, J. Appl. Phys. 71, 227 (1992). K. Aki & P. Richards, Quantitative Seismology. Theory and Methods, Vol. 1, Ch. 7. Freeman,
A
l E/p+!54+ps -2
PI
PI
Pl’
$1 $1
2 Qr +r,,-2%
$1 4);’
G, $1
2rf-2r,p+r,-+rrl--+rf-
p3 =4
REFERENCES 1.
; 1
n
p4 = -2r,
c 1 .j=
fj4)
1
W/r,
+ h(r,
-- 1 2+, ‘tci Ps = (1 - P2) $
%#iCi 2
+ Ej)
1
cjdjrt
;
-
w
j=l
San Francisco (1980). S. Crampin, Geophys. J. Royal Astron. SIX. 21,
387 (1970). S. Datta & B.J. Hunsinger, J. Appl. PhjIs. 49, 475 (1978). Y. Kim & W.D. Hunt, J. Appl. Phys. 68, 4993 (1990). H.F. Tiersten, J. Appf. Phys. 40, 770 (1968).
S.V. Bogdanov, M.D. Levin & LB. Jakovkin, Akust. Zh. 15, 12 (1969) [Sov. Phys.-Acoustics 15, 10 (1969)]. B.A. Auld, Acoustic Fields and Waves in Solids,
Vol. 1. Wiley, New York (1973).
105
SURFACE ACOUSTIC WAVES
x
;
-
ir,r, WQ