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for £116:33 — £i3 2 . The form in (82) is well-structured: it is a diagonalized equation with only those field variables which are relevant to the interface conditions. We substitute for
x' and c —> z' in order to emphasize that the derived formulae are valid for any choice of x' and z' for a line charge with z' > 0. Performing these substitutions, we now extend the list of arguments of Green's functions from (x, z\k) to (x, z\x', z'\k) in order to emphasizing the dependency of Green's functions on the co-ordinates of the line charge:
G
(*.*l f l ! . a ! l*)- 2 e W| f c |^W + e » e xe
,, , , , G^\x,z\x',z'\k)=
'ss
1 l (u)
( !
+ e
J (92)
1 (-&*—fey2')!*! j[-(*-*')+(4fy*--JV')]* 4 ? e 4 7 4 7 (93) ) ^ 1
Discussion: Non-oscillatory exponential terms involved in the expression for £(".«). We first consider the exponential function: exp(—£p /e^ (z + z')\k\). The positive-definiteness of e implies that £33 and Ep are positive. Furthermore, we have z 4- z' > 0, because the conditions z > 0 and z' > 0 hold, and thus the equality z + z' = \z + z'\ is valid. Consequently, this exponential function decays
107
974
A. R.
Baghai-Wadji
for any value of z and z', chosen from the definition range, and thus we can write for it the form: exp(—Sp /e 3 g \z + z'\\k\). Our next concern is the exponential function: e x p ( — S p / e ^ \z — z'\\k\). The appearance of the magnitude sign in \z — z'\ ensures that this exponential function decays for any values of z and z'. Non-oscillatory exponential function appearing in (?('•"); that is: exp (sp /e^\z— £p /e^ z')\k\). By definition we have here the equations z — —\z\ and z' = \z'\. (Remember that the definition range of G^l'u^ is the lower half-space, and t h a t the line charge is located in t h e upper half-space.) Thus this term becomes exp(—(s p /egg \z\ + £p /S33 \z'\)\k\), which is an exponentially decaying function. For a line charge source located in the lower half-plane we denote the resulting potentials in the upper and lower half-planes, respectively, by (?("'') and G^l,l\ The reader can verify that (?(">') and G^l'V> have the forms:
G > ^w\[t^ j[-(x-x')
Xe
+
-$\*-*'\w\
+e
)
-tfr(z-z')]k
*33
(95)
Arranging the above four Green's functions in matrix from, we can define a matrix Green's function G as / G^u\x,z\x',z'\k) a(x,z\x',z'\k)=\
\ G^u\x,z\x',z'\k)
G^l\x.z\x',z'\k) \ .
G^(x,z\x',z'\k)
(96)
J
3.2.1. Properties of G_ The multiplicative factor l/\k\, which is an even function of k, appears in all elements G^a'b^ with a = u,l and b = u,l. Note that ±|fc| are the eigenvalues of the Laplace operator in isotropic media. The non-oscillatory exponential functions involved depend on \k\, and thus are even functions in k. In order t o carry out further properties of G^a'b\ we consider their integration over k to transform them into real-spacei* ''•Strictly speaking, the analysis in the remaining of this subsection and in the following two subsections is not mathematically rigorous. A more careful analysis based on the theory of distributions would be outside the scope of this elementary treatment. The final results are, however, correct and extensively tested.
108
Theory and Applications —e
G^bHx,z\x',z')=\im\
975
oo
f ^G^b\x,z\x',z'\k) e—>0 I J
of Green's Functions
~G^b\x,z\x',z'\k)\
+ [
2f
J
Z7T
J
e
—oo
oo
= -f-^G^b\x,z\x',z'\k)
(97)
—oo
Because of the first two of the above properties, we recognize that only the real parts of the oscillatory exponential functions contribute t o the integrals. Then, by writing
we have
1
G^u\x,z\x',z')=
~(«) _ J O + o o
,.£f.
1
^^rffcfe
-'-E-r\z+z'\k
'»
e («) xcos|(x-a:')--^-(z-z,)l&
£ 33
+
,(") +°° 1 —^iylz—z'tA £ («) ^4-dk-e 'M c o s i ^ - a ; ' ) . 13 ( Z _ / ) | A ; 2weP o -G33 1
Gil'u\x,z\x',z')
=
, .
... 4-dk-e
^i
4F
(0 («) xcosl(s-s')-(%*-%*')!* s
, n
33
,
e
7r(ep
+~
+£p ) 0
J«0 JO xcosKx - x') - (e-f-)Z 33
(100)
33
1
£
(99)
e
33
109
1 -(-&yl 2 l+-frl z 'l)fc K
£
-f)Z')\k
(101)
976
A. R.
Baghai-Wadji
(0 ~{z - z')\k
xcos|(a: - a;')
-33
l
4 -
- ^ | 2 " Z ' fc
1
^
+—'-^-f-dkye 2 ^-F 0 u *
'as'
JO £i ' cosKz-zO-^-fz-*')!* c- ^ 33
(102)
+oo
where the symbol -f- has t o be understood in the sense that o oo
+oo + oo /• -A-dA; • • • = l i m / dk • • •.
o
^-W
From these representations the reader can immediately deduce further facts and properties of the Green's functions: The above integrals do not exist (even in the Cauchy sense ), as we will soon see. This is a consequence of assuming a single (isolated) line source for our problem. In the subsequent discussion it is shown that the above integrals allow a meaningful interpretation if, and only if, we consider a collection of line charges which is charge neutral. Considering a group of N lines with charge magnitudes per length (in y—direction) being denoted by
p-.
'
"•"
7 7 r ( - 7 - lim lne) -(«)
> )
/N
£
- *') " e% ( * - -')]2 + efe(* + z')}2}
1 3 /-,
33
33
g
,/\l2 , f P
/•„
^/M2
W K * - *') - ^ ( z - z')] 2 + lF £ y ( * - ^ O H -
^
4«>
1
(117)
T h i s formula h a s been derived by combining t h e formula (5.2.27), pp.232 w i t h t h e formula (5.2.2), pp.231 from H a n d b o o k of M a t h e m a t i c a l Functions, Abramovics a n d S t e g u n . 2 7 Here 7 , b e i n g equal t o 0.577256649, d e n o t e s t h e Euler c o n s t a n t .
112
Theory and Applications of Green's Functions 979 The expression for G{l'u)(x,z\x',z')
G^\x,z\x>,
can readily be written in the form
l
z>) =
( - 7 - £Hm In.) £_ J "
ir(ep' +£p)
2lT(£p
+ £p
)
£33
£33
(118)
e
£33
33
Remark: We have added the symbol e t o these functions in order t o emphasize the existence of the e-dependent terms in their expressions. As pointed out earlier, by substituting £ ( u ) £ ( ' } , G^'u) (x, z\x', z') transforms into G{l'l)(x,z\x',z')
while G(!'u) (x,z\x',z')
G^Hx,z\x',z')
transforms into G(^'l) (x,z\
* m ( - 7 - Um Inc) £ 7r(eF'+ep) ^-° («) JO _(«) 2 U V ;J + v few MK* *' ) ^ T * %*')] ,._/>) , J 0 / ' >) JO ' >> Z7T^£p
=
+EpJ
1
£33
C33
p(') _ p(«)
^7U^ 1 (I)
*7reP
gp
~
x>)
JO
+ JO ^rl-'l)2} £33
JO
{z
z )f + l
- 7* ~ '
-t-5p
(119)
t33
Fm
ln{[ix
t/iep
x',z').
s33
7i{z + z')]2}
e33
JO JO u v - - x') ~J ~ - -z')f /J + . 1 fer{z ( J )(v z (0» ]n{[(x - % - z')f] e e 33
(120)
33
These forms allow the recognition of further properties of Green's functions. All t h e four Green's functions possess a common constant singular term. Thus we can write
G{a'»\x,z\x',z')
=
} n{eP
+ey)
( - 7 - hm lne) + G^b\x,z\x',z'). £ +0+ -
(121)
This fact is crucial and plays an important role together with t h e charge neutrality condition: We show that for charge-neutral systems (physically realizable systems) the above mentioned singular term vanishes. Consider in t h e upper and lower semi-spaces, respectively, charge distributions p(u\x,z) and p^l\x, z) in such a way that the charge neutrality condition holds 00
00
00
00
/
/ dxdzp(u\x,z)+
J
J dxdzp(l\x,z)
113
= 0.
(122)
980
A. R.
Baghai-Wadji
Based on the linearity of our problem, and on the definition of Green's functions Ge (x,z\x',z') as responses t o a single line charge, we can write the following equations for tp(u\x, z) and tp(l\x,z):
V? (p) (x,z)= /
Jdx'dz'GiP'u^(x,z\x',z')p^(x',z')
—oo—oo oo oo
+ J J dx'dz'Go+ Aire
- x')2 + (z - z')2].
in
(125)
By considering the fact that in physically realizable systems the fundamental cell will contain a charge-neutral system, we omit the singular term to obtain G(x, z\x', z') = ^-— ln[(z - x1)2 + (zAire
z')2].
(126)
We now come back to our line-charge array and superpose the associated potentials to obtain the periodic Green's function GpeT(x,z\x' ,z'), which characterizes the aforementioned line charge array. Using (126), we readily obtain
GpeT(x,z\x',z')=
oo
£ G(x,z\x' n=—oo
I 47T£
-nP,z')
£ ln{[x - (x' - nP)]2 + (zn=—oo
z')2}.
(127)
We see that whenever the Green's function associated with a single line charge (point charge in 3D problems) is available in real space in closed form, it is an easy
115
982
A. R.
Baghai-Wadji
task to construct the corresponding periodic Green's function. Note that in practice the series involved have t o be truncated, by letting the dummy index n run from —N to TV. Numerical calculations indicate that in electrostatic problems, for values of N of the order of 10, we obtain satisfactorily results. 2 8 The construction of GpeT(x,z\x',z') relying on the real-space Green's function G(x, z\x', z') of a single line charge is one of the two alternatives. In the following discussion we will be acquainted with a different way which merely requires the Green's functions in wave number domain. 3.4.2. Construction number domain
of periodic Green's functions
using Green's functions
in wave-
We consider again the line array from the previous section. In order to construct the associated periodic Green's function, we consider first the Green's function in the wave number domain of a line charge located in the fundamental cell. The latter function is G(x,z\x',z'\k)
= -l-I-<J 1. The last section in this chapter is devoted to this technique. Calculations will be performed entirely in analytical form, and thus the reader will be provided with many useful details in connection with self-action analysis. Phased Periodic Green's Functions: For completeness it might be mentioned that in a variety of applications it is necessary to construct Green's functions associated with periodic phased-line arrays. The latter are periodically arranged, localized sources, that are driven in such a way that the potential of n t h line charge is given by exp(.;'A:oa;n). Here xn is the a;-co-ordinate of the n t h line charge and ko is assumed to vary from —w/P to ir/P. The Green's functions associated with phased arrays are closely connected with the aforementioned "ordinary" periodic Green's functions,
117
984
A. R.
Baghai-Wadji
1ioe spate
X a piezoelectric semi-space supporting SHW
-
*
•
•
a line charge + a line force Fig. 4. A piezoelectric semi-space with a line charge and a coinciding line force located at point (x1, z') beneath the surface plane. Above the surface is free space.
and from a theoretical point of view, provide no significant contribution to BEM theory. For this reason we will not be explicitly concerned with phased periodic Green's functions in this work. For details the interested reader may refer to Refs. 9 and 11.
3.5. Infinite-domain wave problems
Green's functions
associated with
Bleustein-Gulyaev
The treatment of Green's functions in this section provides the reader with further information concerning the construction of elemental solutions of partial differential equations. Statement of the Problem: We consider a piezoelectric semi-space with a line charge and a coinciding line force located at point (x', z') beneath the surface plane. Above the surface is free space, Fig. (4). It is known that under certain circumstances «i and U3 decouple from u^ and if, and it is possible to excite surface waves («2, f) which propagate along the interface z = 0. This type of waves are called Bleustein-Gulyaev Waves, Refs. 29 and 30. 3.5.1. form
Construction
of infinite-domain
Green's functions
using 2D Fourier
trans-
The analysis in this section is devoted to the construction of the infinite domain Green's functions. For this purpose we assume a line charge and a line force be located at the point (a/, z'), and find the resulting elastic displacement component M2 and the electrical potential ip, associated with an unbounded piezoelectric medium. The simplest possible equations for describing this type of motion are the following:
118
Theory and Applications of Green's Functions 985
d2 { p
d2 + Ci4
-W
d^
2
d ( e ^
d2 + C
d2
)U2
+ ( e i 5
"^
2
2
d d +e ^ ) u 2 - ( e n - ^
^
d2 + e i 5
5^ =°
(135a)
2
d + en^)-0+
In the light of the above derivation it is instructive to summarize the relevant steps: • Consider the representation: G(xi, x3 \x\ ,x'3) o G(xi, x3 \x[, x'3 \k). • Find the asymptotic limit of G{x-l,x3\x'1,x'3\k) S(xi,x3\x'1,x'3\k).
for k > 1: G{x1,x3\x'1,x'2i\k)
~
• Extrapolate S(x1,x3\x'1,x'3\k) over the whole definition range of k, that is (-00,00): which means letting S(x1,x3\x'1,x3\k) be valid for any value of k. • Calculate the inverse transform of
S(x1,x3\x'1,x3)
S(xi,x3\x'1,x'3\k)
°° dk = / -
123
m
2-K
—S(x1,xs\x'ux'3\k).
990
A. R.
Baghai-Wadji
• The function S(x\x')
is the near field representation of
G(x1,x3\x'1,x3)
3.9. Near-field
behavior
3.9.1. Real-space
analysis
"
S(x1,x3\x'1,x'3).
of the ^-derivative
Using the relation (d/dx)H^\x)
of Green's
= -H^ix)
flgfri,xalx'^x'z)
~ ~ *~c
functions
we have
w (Xl - x\) 3
^
G(x\x')
fl„
w
(1)
Fl
{ R
(154)
7^
At the limit R« kl-x'i~x'^e'~at 1 we can determine the asymptotic of the above integrand which is AUf,jk(x/,—x/,)
•7fee
e
-at(k)\x3-x'3\
_-r.
JJL.eJH here contributes to the integral and we obtain 1 °° / Wix^x^x'x'z) = - — fdksiakix! - x'^e'^3-^.
(158)
27T 0
We use the table integral °° Jdxe
1 px
sin(qx
j (QCOS^
+ A) = —2
+ psinA)
(159)
(p > 0.) (Formula 3.893/1 of Ref. 27.) which in the special case A = 0 reads oo
a
fdxe-pxsm(qx) o
=
02
0. P +Q2
(160)
Employing this result we obtain (161) which gives the near-field behavior of dG_ dx[
dGjdx'x
i?|| » 1 ~
_ 1 xi - x\ 2?r ijy 2
(162)
This result proves that the processes • spatial domain differentiation with respect to x'x • wave number domain integration over k • spatial domain asymptotic order-calculation (R^ -C 1) and • wave number domain asymptotic order-calculation (k\\ S> 1) commute in the following way: Asymp
. d
°? dk _,.
, , ,,,.,
-Is 7" i^-*"*»i 3.10. Asymptotic
behavior
of dG/dx'3
Consider the following self-explanatory calculations in the wave number domain:
125
992
A. R.
Baghai-Wadji
dG(x1,x3\x'1,x'3)_ dx'3
d °? d f c e J f c ^ i - < ) e - t t ' W N - ^ l "d^J^Tr -2at(k) 1
OO
= - — t a g n ( x 3 - 4 ) / d * c o s * | a ; i - a;' | e -«t(*)|*s-* s l 2lT 0 1 °° ~ - — sign(z 3 - x'3)Jdkcosk\xi - a^le-*1*3-^! 2-K
(164)
0
Using the table integral oo Sdxx^e-f'coBbx
an = { - V T W
¥
a E -
(i65)
(Re/3 > 0; b > 0) in the special case n = 0, that is oo
a
JdxcoBbxe-P" o
= -^SJ b2 + P2
(166)
we obtain 0G(a;i,a:3|a:i,a:'3) 1 12:3-2:3 ~ - ^2-K s i g ns ( av": 3 - 2:i ;3)( - x\f + {x - 4 ) 2 0*3 Xl 3 1 2:3—2:3 ~ _ 2 ^ r iJ,, 2
3.11.
Calculation
of self-action
in
(167)
BEM
In certain BEM calculations we are concerned with integrals of types TI \ I(x1,x3)=n1 J(x1,x3)
t , ,dG{x1,x3\j^1^ dx3 i—J-J-+ = -n3
, aG(a:i,2;3|2:i,4) / dx\
> (168a) (168b)
which for coinciding observation a n d source points deserve special a t t e n t i o n . 3.11.1. Calculation
of I-type
integrals
Substituting x'3 = x3 + u and x\ = x\ + e the following relations are valid: 2:3 — x'3 = — u
xi — x\ = —e
In the ultimate proximity of the observation point (2:1,2:3) we can employ the asymptotic expansion of dGjdx\ from the previous subsection. Thus we can write
126
Theory and Applications of Green's Functions 993 23-A3/2
,dG(x1,x3\x'1,x'z) c
dx[
X3+A3/2 X3-A3/2
1 '27T
f J
, , si - x\ ^ ( x . - x ^ + ixz-xtf
X3+A3/2 -A3/2
= — 2ir
/ J
A3/2
du-x 7; = — / e2 + u2 2TT J
A3/2
du-r. ,. e2 + u2
(169)
-A3/2
By substituting u = ev and thus du = edv (169) becomes A3/2c
2TT
J
A3/2e
e2 + e2?;2
-A3/2e
2n
1 + ^2
J
(170)
-A3/2e
At the limit of e -4 0 we can write
1=
1 °° 1 lim h = — f du~. e-+o+ 27r_i 0 1 + u2
(171) v '
By using the table integral
we obtain
7=i;arctan(,)|-00 = i - ^ - ( - f ) ]
= i,
(172)
which is valid for any order of A3. (This results from an implicit assumption that the surface in the vicinity of the observation point is flat.) 3.11.2. Calculation of J-type
integrals
In a similar fashion we can write the following relations: X\ — x\ = —u The following steps are self-explanatory:
127
X3 — x'3 = —e
994 A. R. Baghai-Wadji zi+Ai/2 Jt=
J dx\,
-Tl3
dG{x1,xz\x'1,x'z)
xi-Ai/2 x1 + A 1 /2
=
_J_
dx> 1
f
27T
J
H-Aa/2
(Xl
J.x
A t /2
-At/2
2^ y
-x3
z
- X[ )2 + (13 - 4 ) 2
d
=
kxe-e 1 5 |fc|
e15|fc|
T (x,z\k) )
\
ejkxe\k\z+a(2)(kj
a^\k)
D3(x,z\k)
0 1
1
«xeMZ
W( +a^>(k)
1 C15A11
\ gjfcig-Ajz
—C^\t
Ca\t
0
0
(177) The unknowns ay\k) 3.13. Semi-infinite
will be determined from the interface conditions.
dyadic
Green's functions
for piezoelectric
half-spaces
We are now prepared to derive Green's functions associated with a semi-space piezoelectric substrate which is capable of supporting shear horizontal waves along the substrate surface. Based on the piezoelectricity assumption, there are two ways of exciting the semi-space: we can excite the substrate either by an electrical line charge or equally
129
996
A. R.
Baghai-Wadji
lice spate
X a piezoelectric substrate with material parameters £
11 ,
e
i 5 , ^44
-*•
a line charge + a line force Fig. 5. A line charge and a line force which coincide at the point {i' = a,z' = c) within the substrate in the lower half plane (z < 0).
well by a mechanical line force. While the location of the line force is limited to the substrate region, the line charge can reside everywhere in the space. In this subsection we focus our attention on sources which are located within the substrate. To cover problems arising in practice, the next subsection will be concerned with the medium excitation by a line charge above the substrate? Results from these two sections will allow the investigation of the properties of the involved Green's functions. However, as the reader will see, it is necessary to consider the excitation of two welded piezoelectric semi-spaces, in order to investigate the underlying reciprocity properties of the Green's functions involved. Two welded piezoelectric semi-spaces will be analyzed in the final part of this section. 3.13.1. Line source excitation: sources are located in the lower half space We consider a line charge a at {xa, za) and a line force r positioned at (xT, zT) within the substrate (in the lower half plane (z < 0)), Figs. (5) and (6). We obtain the following results for the Green's functions. The electric potential response in the upper semi-space (z > 0):
i A piezoelectric substrate enclosed in a metallic package is the basic building block of most microacoustic devices. Under the assumption of ideal electric conductivity the metallic parts in the devices can be regarded as a collection of line charges. In an analogous way mechanical loading of the substrate and the mechanical tensions between the substrate and the package can be modeled by a collection of line forces with a •priori unknown strengths.
130
Theory and Applications
of Green's Functions
997
region I liee\p.u.e
Z ii
region II £
11 ,
e
x
C
i 5 , ^44
a line charge + 1 a line force 1
region III £
11 ,
e
!
•
1
..
1 1 .k '
•
i 5 , ^44
Fig. 6. A plane z = c together with the interface plane z = 0 divides the geometry into three homogeneous regions.
^*tp,a \^i Z\%ai Z01 Za\k)
1 £0 + £11 e
1
^44^
1*1° 4 4 |
fc|
~
G27J« *£. _i_ n ±L _ t £ U e 1 5 ~ £ U e 1 5 ) i(£ii+£ii)
l
;
G & J W l * * >**!*) = 1 £
£
£
i1eyr,-£ne'R 1
^
£
'n
£
n+ n
n 1*1 7T* A,
|*| ^ & +c" £ _ (£'i^s-£ne'5)2 W4| f c | + ^ 4 |fc| ^ ^ ^
T 7 « A,
C 4 4 | 4 + G 4 4 -i^ -
3.14. Infinite-domain tal polarized wave
e**(*-*,) e -(A?W+|k||z,|)
(188)
(£iiei5-£ne1B)2 £ ." £ '" £ j ' l £ '" ii ii( ii+ n)
dyadic Green's functions propagation
for the analysis
of
sagit-
Considering a line force embedded in an isotropic elastic half-space, the derivation of Green's functions for sagittal polarized wave propagation has already been discussed in Ref. 24. We denote the distance of the line force in the interior of the domain from the surface of the semi-space d. By letting d go to infinity, we can then obtain the infinite-domain Green's functions from the half-space Green's functions. This section is devoted to a direct derivation of the aforementioned infinitedomain dyadic Green's function. The derivation is based on eigenvalues and associated eigenvectors of the governing and constitutive equations.
136
Theory and Applications of Green's Functions
1003
Statement of the Problem: We assume the entire space to be filled with an elastic medium characterized by the elastic constants C\\ and C44 and the mass density p. Furthermore, we assume that a line force located at the point (x',z'), and oriented in x-direction (TIS(X — x')S(z — z')e"x), oscillates in time according to exp(ju;i), and excites the medium. The excitation of the medium is uniquely determined by the resulting elastic displacements, i.e., Ui(x,z) and u3(x,z) functions. In order t o signify the localized, delta-function-like nature of the source, we use G\{x,z) and G3(x,z) instead of U\(x,z) and u3(x, z). In addition, in order to indicate the direction of the force, which in the present case is the rc-axis, we write Gu(x, z) and Gzi{x,z). Finally, by writing Gn(x,z\x',z') and G3i(x,z\x',z'), we also provide the co-ordinate values of the above-mentioned line force. Analogously we speak of a line force T3 1 3.16. Self-action
analysis
in vector
field
in the far field in
the
problems
In preceding subsections, which dealt with horizontal scalar waves, we found that the self-interaction calculation can be performed either in the spatial domain or in the wavenumber domain. Furthermore, we have demonstrated that the calculation in the wavenumber domain is significantly simpler. This is an encouraging result and gives rise to the next question: is it possible to extend our ideas to include vector fields? The main objective in this subsection is to show that our solution concept, proposed in the foregoing section for scalar waves, is also valid for vector fields. However, it turns out that, in contrast to scalar waves, more t h a n one term must be retained in the asymptotic series in order to adequately reflect the fine scale structure of the problem. In many engineering problems we will be concerned with the interaction analysis of sagittal polarized waves with surface disturbances, formed as ridges or grooves, on the surface of a piezoelectric semi-space. The analysis employs the BEM and involves infinite domain Green's functions associated with the problem. It should be emphasized that the underlying Green's functions have to be known in real space. Generally, while the analysis of mutual interaction is fairly simple and straightforward, the self-interaction calculation is rather cumbersome. The major steps in calculating the self-interaction using real-space Green's functions are given below. Standard Procedure:
• Transform the underlying system of inhomogeneous partial differential equations from the spatial domain into the wavenumber domain.
• Calculate the Green's functions in the wavenumber domain by inverting the inhomogeneous algebraic system of equations (derived in the first step).
• Transform the Green's functions into real space.
• Calculate the derivatives of the Green's functions with respect to the spatial variables. • Calculate the asymptotic limits of the derivatives of the Green's functions in the near field in real space.
• Integrate the asymptotic limits to obtain the self-interactions.
Assuming an isotropic elastic medium (characterized by the velocities c; and ct), and denoting the asymptotic limits of the infinite domain Green's functions Gij(x\x') in the near field (R\\ < 1) by 5 y ( x | x / ) , we obtain the following result:
138
Theory and Applications
dx[ dx\ 3 an - a^ , 2 ( j ! - a^) 3 ~
„2
T->2
"•" .2
. aGn(x|x')
^ f t i W
.9a;; 1 x3 — x'z
dx\ 2 (0:3 — x'3)3
fl2
c2
1 13 - X3
Ri
c2
c;
#j
. dG 33 (x|x')
47T
da^
C?
~ 47T
i?2
^ dx'3 3 x 3 - x'3
r2
R2
itf
C2
a5 3 3 (x|x0 Sajj
C2
l T 0533(xl^)
~
ylyU>
2 (a?3 — x'3)
-+c 272
JJ2
D#4
(,j.yi;
(192)
=
+
*
^2
+
p4
^4
c ?
^
=
dx'3 2 (13 - a/3)3 , I 1 3 - 4 „2
(193)
2 (a?i - a;;)3
Sii-ii c?
l r r a5 3 3 (xlxQ
l i i - a^ 2 (a^ — a^) r>*' ^+ 32 R2 R4
=
2 (si - a^) 3
_ l i i - i j C2
p4
/
) ^ l 7 r a5 1 3 (x|x') 9a:3 da:3 1 zi — a;i 2 (a;i — x^)3 ,2
.2
3 i 3 - 4 , 2 (a 3 - x'3)3 c2 i? 2 ^h-j j 53* c2
4 , ^ 3 ^ „
l j r 0G 1 3 (x|x
n>2
=
2{x3-x'3)3 c? flJ
itf
c
r2
1005
dSn^x')
_lx3-x(i
.2
2 ( g l - a;;)3
h i - » ; +
p4
^ ^ 47T
47T
of Green's Functions
"+" „2
R2
2 (x 3 - x'3)3 „2
p4
Uy{V
Preparatory Calculations: Based on the Taylor power expansions \/l - 1
a:2
2 p
1
a;
VI-a;2
139
x< 1
(196a)
a; < 1
(196b)
2
1006
A. R.
Baghai-Wadji
and ex « 1 + x
x< 1
(197)
which are valid for vanishingly small values of the argument x, the following manipulations are self-explanatory:
„,
lw2 1
, (198a)
^ " ^ ' - ^ j * ! i _ i u3 ai,t * |fc| +' 2o — c\tT \k\ e-a,lt|*3-^3l
(198b) lw2 1
„ e -|A=ll-3-.T 3 | ( 1 + ±!%--L\x3 2c ; , t I'M
- X'3\).
(198c)
Substituting these asymptotic expressions for aij, l/ 1— asymptotic
limits
of the
The involved coefficients can be determined from the interface conditions on the plane z = z'. Thereby we use the (k 3> 1) asymptotics for the displacement and stress components, rather than these components themselves. A straightforward calculation gives the following result:
S{(x-x',z-z'\k) -_]tLTlP3Hx-x') _
= -X,(z-z') _ I_L_L TlP J*(z-a:')p-A,(2-z')
2u?1 _J^T3e3Hx-x')e-\l(z-z') 2UJ2 _ll.-LTieJHx-x')e-*t(z-z')
Ac2\k\l
4C2|*| JtLTie}k(x-x')e-Xt{z-z')
+
2ui2
+
, J^_n(Jk(x-x')
2^2T3e
-\t(z-z')
/ 2 05)
(M0
>
The appearance of (x — x') and (z — z') in the second equation in (205) suggested the introduction of the form S[{x — x',z — z'\k), instead of S{(x, z\k). As the inequality z — z' > 0 holds true in region / , we can write z — z' = \z — z'\. Using this relation, substituting the k ^> 1—asymptotics of exp(—\i\z — z'\) and exp(—Xt\z — z'\), and reordering, and arranging the terms associated with T\ and T3 into two separate groups, we obtain:
144
Theory and Applications of Green's Functions
1011
S[(x-x',z-z'\k)
= [-!i 'I -\^^)UeM'"%^"'^ + f- l(\ L
4 cl
- \)jsign(k)\z-Z'\]e^x-x\-^z-2\3 ct
(206)
J
Identification of the terms multiplied by T\ and T3 : The term multiplied by Ti represents the k » 1—asymptotic limit of the Green's function Gn(x — x',z — z'\k) in region I (z — z' > 0). Likewise, the term multiplied by T3 is the k 3> 1—asymptotic limit of the Green's function Gi3(x — x',z — z'\k) in the aforementioned region. Denoting these limits by 5f 1 (x — x',z — z'\k) and S{3(x — x',z — z'\k), we can write:
Sl^x
- x \ z - z'\k) = [ - i ( 4 - 4 ) | z - A L
* cl
c
t
- l ( \ + 1) ±_\eM*-x')e-\k\\z-z'\ 4 xcf cf \k\l Si3(x -x',z-
z'\k) = - \ { \ 4
- \)jsign(k)\z
Cl
-
( 207a)
z'\eM—')e-\k\\z-z'\
Ct
(207b)
In order to investigate the behavior of the limits Gu(x — x', z — z'\k) and G\3(x — x',z — z'\k) in region II (z — z' < 0), we consider the term S[! (x,z\k). A similar analysis leads to the results
SU(x-X',z-z'\k)=\--(-s--3)\z-z'\
- i ^ + ^lSi]^'^"1""""'1 SU(x -x',z-
z'\k) = \ ( \ 4
C;
- \)jsign{k)\z
-
(2 8a)
°
z>\eJKx-x')e-\k\\z-z'\_
Ct
(208b)
The expressions in (207) and (208) can be unified by using the fact that \z z'\ = z - z' in region I and that \z - z'\ = -(z - z') in region II. The k > 1-asymptotic limits of the remaining Green's functions, i.e., G3i and G33, can be found analogously. Summarizing our results we can write:
145
1012
A. R.
Baghai-Wadji
Sll(x-x',z-z'\k)=\-\(±-±)\z-z'\
~l(\
S13(x-x',z-z'\k)
=~\(^
+ \)^l}eMx~X
~ ^)jsign{k)(z
Vl*"*-*'!
-
z')ejk 1—asymptotic limits of the eigenquantities. This completes our proof of demonstrating that the process of satisfying the boundary conditions and the k 3> 1—process are commutable. 4. S u m m a r y In this chapter we have shown that the governing equations in anisotropic, and transversally inhomogeneous piezoelectric materials can be diagonalized. Details regarding a newly developed symbolic notation, and a recipe for the construction of diagonalized forms have been discussed in Section 2, following a brief introduction in Section 1. Although the presentation of the diagonalized form in Section 2 is selfcontained, it remains restricted to the piezoelectric media. Further complementary discussions on the diagonalization of Maxwell's equations in anisotropic media can be found in Ref. 9. The reader is also referred to the Refs. 32 and 33 which are devoted to the diagonalization of Maxwell's equations in bi-anisotropic inhomogeneous media and Laplace's equation in the anisotropic dielectric and magnetic media, respectively. The latter forms have been developed to analysis waves and fields in large amplitude corrugated periodic structures. Furthermore, the reader may find additional applications of the diagonalized forms in Ref. 34. There, among others, the propagation of electromagnetic waves in photonic crystals with defects has been addressed. Diagonalized forms in Fourier domain represent standard or generalized algebraic eigenvalue equations, which lead to the eigenvalues and eigenvectors corresponding to the underlying differential operator. An efficient algorithm for the calculation of higher-order derivatives of eigenvalues and eigenvectors is discussed in Ref. 35.
146
Theory and Applications of Green's Functions
1013
Section 3 has been devoted to a brief discussion on Green's functions. Two methods for the construction of Green's functions in infinite and semi-infinite media have been presented. The first method, based on the inversion of the underlying differential operator, is suitable for infinite-domain Green's functions. The second method, utilizing the diagonalized forms from section 2, can be chosen to construct both the infinite doamin and the semi-infinite domain Green's functions. Several boundary value problems have been considered, as useful examples, to demonstrate the details pertaining the construction of Green's functions. Much attention has been devoted to the Green's functions associated with Laplace's operator because of their far-reaching significance. The charge neutrality condition, as a regularizing balance law, has been emphasized. These considerations are followed by two recipes for the construction of periodic Green's functions. Many useful applications of periodic Green's functions can be found in Ref. 28. Section 3 closes with a discussion on the self-action calculation which arises in the boundary element method applications. Although here the discussion on self-action analysis seems exhaustive, it only scratches the surface of this pre-eminent research topic. The material presented is an adaptation of my ideas compiled and discussed in Ref. 9. Space limitation has prevented the inclusion of any of my results have been obtained since 1995. Selected topics concerning the self-action calculations, regularization of singular surface integrals, near-field calculations around the edges, wedges, and corners can be found in my articles in the IEEE Ultrasonics Conference Proceedings published in the years 1995-2000. As a possible future research direction I would like to emphasize the construction of Green's functions-based wavelets. 10 Acknowledgments It is my privilege to thank Professor Martti Salomaa, Director of Materials Physics Laboratory at Helsinki University of Technology, for inviting me and initiating a Visiting Professorship (Oct. 1999 through Dec. 2000), which has been sponsored jointly by TEKES, a National Technology Agency, and, the Nokia Research Foundation. It is also my pleasure to extend my thanks to all the department members for excellent support and for making my stay in this distinguished pedagogical and research environment an invaluable and enriching experience. Furthermore, it is my pleasure to thank the editors Prof. Tor A. Fjeldly and Dr. Clemens C.W. Ruppel for their kind invitation to author this chapter. References
1. R.F. Harrington, Field Computation by Moment Methods, Macmillan, New York, 1968. 2. C.A. Brebbia, J.C.F. Telles, and L.C. Wrobel, Boundary Element Techniques, Springer Verlag, 1984. 3. E. Stein and W.L. Wendland (Editors), Finite Element and Boundary Element Techniques From Mathematical and Engineering Point of View, Springer Verlag, 1988. 4. N.I. Muskhelishvili, Singular Integral Equations, P. Noordhoff N.V. - Groningen Holland, 1953. 5. S.G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965. 6. R.E. Collin, Foundations for Microwave Engineering, McGraw-Hill International Editions, 1966, Electrical & Electronic Engineering Series. 7. G.F. Roach, Green's Functions, 2nd ed., Cambridge University Press, 1967.
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Baghai-Wadji
8. I. Stackgold, Green's Functions and Boundary Value Problems, A Wiley-Interscience Series of Texts, Monographs & Tracts, J o h n Wiley & Sons, 1979. 9. A. R. Baghai-Wadji, A Unified Approach for Construction of Green's Functions. Habilitation manuscript (lecture notes), Vienna University of Technology, Vienna, 1994. 10. A. R. Baghai-Wadji, G. Walter, "Green's Function-Based Wavelets." Accepted for presentation a t t h e IEEE International Ultrasonics Symposium, San Juan, P u e r t o Rico, Oct., 2000. 11. A. R. Baghai-Wadji, Bulk Waves, Massloading, Cross-Talk, and Other Second-Order Effects in SAW-Devices (a short-course manuscript). IEEE International Ultrasonics Symposium, San Antonio, Texas, Nov., 1996. 12. A.H. Fahmy, and E.L. Adler, " P r o p a g a t i o n of Acoustic Surface Waves in Multilayers: A Matrix Description." Appl. Phys. Lett., vol. 22, pp. 495-497, 1973. 13. E.L. Adler, "Analysis of Anisotropic Multilayer Bulk-Acoustic-Wave Transducers." Electron. Lett, vol. 25, pp. 57-58, J a n . 1989. 14. E.L. Adler, " M a t r i x Methods Applied t o Acoustic Waves in Multilayers." IEEE Trans. Ultrson. Ferroelec. Freq. Contr., vol. UFFC-37, no. 6, pp. 485-490, 1990. 15. E. Langer, PhD Dissertation. Vienna University of Technology, Vienna, 1986. 16. R.F. Milsom, N.H.C. Reilly, and M. Redwood, "Analysis of Generation and Detection of Surface and Bulk Acoustic Waves by Interdigital Transducers." IEEE Trans. Sonics Ultrson., vol. SU-24, p p . 147-166, 1990. 17. A.M. Hussein, and V.M. Ristic, " T h e Evaluation of the I n p u t Admittance of SAW Interdigital Transducers." J . Appl. Phys., vol. 50, no. 7, p p . 4794-4801, July 1979. 18. K. Hashimoto, a n d M. Yamaguchi, "Precise Simulation of Surface Transverse Wave Devices by Discrete Green Function Theory." Proc. IEEE Ultrason. Symp., pp. 253258, 1994. 19. P. Ventura, J.M. Hode, and B. Lopes, "Rigorous Analysis of Finite SAW Devices with Arbitrary Electrode Geometries." Proc. IEEE Ultrason. Symp., pp. 257-262, 1995. 20. P. Ventura, J.M. Hode, and M. Solal, " A New Efficient Combined F E M and Periodic Green's Function Formalism for t h e Analysis of Periodic SAW Structures Characterization." Proc. IEEE Ultrason. Symp., p p . 263-268, 1995. 21. R.C. Peach, " A General Green Function Analysis for SAW Devices." Proc. IEEE Ultrason. Symp., pp. 221-225, 1995. 22. V.P. Plessky, and T. Thorvaldsson, "Periodic Green's Function Analysis of SAW a n d Leaky SAW Propagation in a Periodic System of Electrodes on a Piezoelectric Crystal.", IEEE Trans. Ultrson. Ferroelec. Freq. Contr., vol. UFFC-42, p p . 280-293, 1995. 23. B.A. Auld, Acoustic Fields and Waves in Solids, vol. I and II, John Wiley & Sons, 1973. 24. N.E. Glass, R. Loudon, and A. A. Maradudin, " P r o p a g a t i o n of Rayleigh Surface Waves across a Large-Amplitude Grating.", Physical Review B, vol.24, no.12, 1981. 25. A.R. Baghai-Wadji, and A.A. Maradudin, "Shear Horizontal Surface Acoustic Waves on Large Amplitude Gratings", Appl. Phys. Lett, 59 (15), 7 October 1991. 26. R.P. Kanval, Generalized Functions, Series on Mathematics in Science and Engineering, vol. 171, Academic press, 1983. 27. M. Abramowitz and LA. Stegun (editors), Handbook of Mathematical Functions, Dover Publications, Inc., New York. 28. A.R. Baghai-Wadji, H. Reichinger, H. Zidek, a n d Ch. Mecklenbrauker, "Green's Function Applications in SAW Devices." Proc. IEEE Ultrason. Symp., p p . 11-20, 1991. 29. J.L. Bleustein, Appl. Phys. Lett., 13, 412, 1968. 30. Yu.V. Gulyaev, Pisma v Z h T F , 9, 63, 1969. 31. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 1980.
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1015
32. B. Jakoby, and A.R. Baghai-Wadji, "Analysis of Bianisotropic Layered Structures with Laterally Periodic Inhomogeneities - an Eigenoperator Formulation." IEEE Trans. Antenn. Propag., vol. AP-44, no. 5, pp. 615-621, May 1996. 33. M.T. Manzuri-Shalmani, A.R. Baghai-Wadji, and A.A. Maradudin, "Noise-free Static Field Calculations in Corrugated Periodic Structures." Proc. IEEE-AP, Antenn. Propag. Symp., pp. 1089-1092, 1993. 34. A. R. Baghai-Wadji, Photonic Crystals, lecture notes, Helsinki University of Technology, Helsinki, Fall 2000. 35. S. Ramberger, and A.R. Baghai-Wadji, "Calculation of Higher-order Derivatives of Eigenvalues and Eigenvectors." (In preparation), IEEE Trans. Antenn. Propag.
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International Journal of High Speed Electronics and Systems, Vol. 10, No. 4 (2000) 1017-1068 © World Scientific Publishing Company
NEW PIEZOELECTRIC SUBSTRATES FOR SAW DEVICES JOHN A. KOSINSKI U.S. Army Communications-Electronics Command, AMSEL-RD-IW-S, Fort Monmouth, NJ 07703-5211, USA Recent developments in single crystal piezoelectric materials have focused on the search for "ideal" materials with zero temperature coefficient of frequency orientations featuring jointly high piezoelectric coupling, high intrinsic Q, zero power flow angle, and minimized diffraction effects. In addition, the desired materials should have no low temperature phase transitions, and a physical chemistry conducive to repeatable, low cost growth and wafer scale device production. As difficult as it might seem to find such "ideal" materials, three completely different but strong candidate materials have emerged recently: the quartz homeotype gallium orthophosphate, the quartz isotype calcium gallogermanates (langasite, langanite, langatate, etc.), and diomignite (lithium tetraborate). The current state-of-the-art and prospects for future development of these materials are considered.
1. Introduction To date, single crystal quartz (oc-Si02) and lithium niobate (LiNb03) are the most widely used single crystal piezoelectric substrates for SAW devices. Each of these materials has certain properties that make it attractive for specific applications. In the case of quartz, the most interesting features are the zero temperature coefficient of frequency jointly with zero power flow angle, high Q, and non-zero piezoelectric coupling for the ST-cut; these characteristics lead to widespread use in narrowband filters and precision resonators in the VHF and UHF range. In the case of lithium niobate, the most interesting feature is large piezoelectric coupling leading to widespread use in broadband and low insertion loss filters. However, the lack of a zero temperature coefficient of frequency is a serious limitation for the use of lithium niobate. Recent developments in single crystal piezoelectric materials have focused on the search for and development of "ideal" materials. Traditionally, "ideal" materials were considered to be those featuring jointly high piezoelectric coupling, zero temperature coefficient of frequency orientations, and high intrinsic Q. However, in the context of substrates for SAW devices, additional criteria have been imposed. The zero temperature coefficient of frequency orientations must also feature zero power flow angle jointly with minimized diffraction effects, no low temperature phase transitions, and the physical chemistry of the material must admit repeatable, low cost growth and wafer scale device production. As difficult as it might seem to find such "ideal" materials, three completely different but strong candidate materials have emerged recently: the quartz homeotype gallium orthophosphate, the quartz isotype calcium gallo-germanates (langasite, langanite, langatate, etc.), and diomignite (lithium tetraborate). Gallium orthophosphate and the calcium gallo-germanates both belong to trigonal symmetry class 32, and hence have much in common with ct-quartz. Diomignite belongs to tetragonal symmetry class 4mm as do certain piezoelectric ceramics. The current state-of-the-art and prospects for future development of these materials are considered. 151
1018
J. A.
Kosinski
2. Quartz Homeotypes - Gallium Orthophosphate 2.1. General comments Alpha-quartz is unquestionably the most successful single crystal piezoelectric material, with nearly "ideal" characteristics in many respects. It follows naturally that quartz homeotypes should be investigated for similar or potentially superior characteristics, and in fact it has been found that berlinite (a-AlP0 4 ) demonstrates superior piezoelectric coupling as compared to quartz. Unfortunately, extreme difficulty has been experienced in growing high quality berlinite crystals of commercially viable size. Recently, gallium orthophosphate (oc-GaP04) has been proposed as another quartz homeotype of interest. This new material belongs to the same family of M 3+ X 5+ 0 4 crystals as berlinite, constructed by the alternate replacement of half of the silicon atoms by trivalent gallium and the other half by pentavalent phosphorous atoms.1 Preliminary results indicate similar good characteristics for gallium orthophosphate, with the coupling coefficient of the GaP0 4 AT-cut larger than that of berlinite, and approximately twice that of quartz.2 Further, gallium orthophosphate demonstrates a superior thermal stability, transitioning directly to a p-cristobalite form at 933°C as compared to the o>P phase transition near 580°C for quartz and berlinite.3 2.2. Crystallography As a quartz homeotype, gallium orthophosphate belongs to symmetry class 32, characterized by a single three-fold symmetry axis and three equivalent two-fold symmetry axes as illustrated in Fig. 1.4 The three-fold axis is also a screw axis, leading to right- and left-handed enantiomorphs belonging to space groups P3i21 and P3221 respectively,1 hence both electrical (Duaphine) and optical (Brazil) twins are possible. As noted, the crystal structure is similar to that of a-quartz,5 with half of the silicon atoms replaced by trivalent galjium and the other half replaced by pentavalent phosphorous atoms. Consequently, the unit cell extent along the c-axis is twice that of quartz. Measured values of lattice constants are listed in Table 1. There are three formula weight per unit cell. Analysis of the data near room temperature lead to values of a=4.901+0.003A and c=11.046±0.008A at 25°C corresponding to an x-ray density of 3571±7 kg/m3 which is consistent with the previously reported values as listed in Table 2. Thermal expansion data are presented in Table 3. Published data on the thermal stability of the a-phase of gallium orthophosphate are listed in Table 4. There is substantial variation in the reported phase-transition temperature data. The phase relations in gallium orthophosphate are illustrated in Fig. 2.12 The behavior of gallium orthophosphate is distinctly different than that of quartz and berlinite. The material transitions directly from the a-phase to a P-cristobalite form at 933°C, whereas quartz and berlinite undergo an intermediate a-p phase transition near 580°C.3 In consequence, gallium orthophosphate devices may be processed or operated at significantly higher temperatures than comparable quartz or berlinite devices.
152
New Piezoelectric Substrates for SAW Devices
Ski
Ek
£!!:: •
•
•
AT
\ : : \
• ©
. . . >® x
•
•
V:
• ^5"^®
X,X i
1019
AS
6 2 2 2 0 _! 13
Fig. 1. Class 32 symmetry elements and matrices for equilibrium properties [Nye, 1960].
Table 1. GaP04 Lattice Constants. a (A) 4.874±0.002 4.899+0.001 4.901±0.001 4.897+0.001 4.934±0.002 4.973+0.002 4.90 4.905 4.901±0.003
c(A) 11.033±0.004 11.034±0.002 11.048+0.001 11.021+0.002 11.075+0.005 11.105+0.006 11.05 11.050 11.046+0.008
To(°C) -100 20 20 100 500 750 r.t.
— 25
Reference 6 7 1 7 1 1 8 9 this work
Table 2. GaP0 4 Mass Density. Mass Density (kg/mJ) 356x 357x 3570 361x 358x 3570 3571+7
Method measured x-ray
To(°C) r.t. r.t.
—
—
x-ray x-ray
-100 20 25 25
— x-ray
Reference 8 8 1 7 7 10 this work
Table 3. GaP0 4 Thermal Expansion.
aj 1 / (ppm/°C)
a f f (ppb/°C2)
a^
17.9 5.3 10.52 9.02 10.15
—
4.6 1.2 1.70 3.38 3.34
35.4 35.4 14.58
(ppm/°C)
a g } (ppb/°C2)'
T„(°C)
Reference
....
....
11 12 13 14 15
2.0 2.0 2.52
153
r.t. 25 27
1020
J. A. Kosinski
P-phase P6222
thermal decomposition >1327°C
(1687 °C) P-cristobalite F43m 933 °C 578 °C 533 °C a-phase P3t21
a-cristobalite C222!
Fig. 2. Phase relations in gallium orthophosphate.12
Table 4. GaP04 Critical Temperatures. Transition Point (°C) 1077 1000 137 Starting materials for hydrothermal growth are boric acid plus either lithium borate or lithium hydroxide, with HC0 2 H as the mineralizer.115 Diomignite crystals have been grown by the Czochralski method using either ^119,133-136,138,139 o r r e s i s t a n c e heating.113.114.137.140 Platinum crucibles are used nearly universally, with all recent growth in an air ambient.114,121132,137140 Crucibles are typically as tall as they are wide, and the width is chosen to be about one and one half times the diameter of the desired crystal. The feed mixtures are typically stoichiometric,121'135137139" 141 although a slight excess of boron sometimes is provided.114,119131 Seeds for initial growth have been formed by quenching on Pt wires. Diomignite crystals have been grown commonly along both X- and Z-axes.108,113,114,119'131-132-137,140-144 Growth also has been demonstrated along the [110] direction;135'139144 such boules simplify the manufacture of desirable 45° rotated X-cut devices.
177
1044
J. A.
Kosinski
Reported Czochralski growth rates range from 0.13 mm/hr to 5 mm/hr, with general agreement on higher quality growth at slower growth rates (
1079
0
£ -10 ro E
3.5
4
4.5 5 Phase velocity [Km/s]
5.5
Fig. 2. Plot of the normalized effective permittivity, Stff/Eo, as a function of real slowness along quartz AT (Euler angles:[0° -54.7° 0°]).
Finding the solutions of either a PSAW or a HVPSAW requires finding the zero of the complex BCF of two real variables, vp, and a, the attenuation constant. Once the PSAW or the HVPSAW solution is found, the respective complex slowness, y, is known. If the solution for the pseudo problem has a high attenuation then the minima in the BCF observed in Fig. 2 might not be detected in the vicinity of the pseudo SAW phase velocity solutions. PSAW and HVPSAW practical orientations must have relatively low attenuation so that it makes sense to talk about propagation of such waves. Typical plots of the BCF for complex slowness are discussed in Section 3. 2.5. How the Problem is Solved As is clear from (13) and (14) the BCF has an implicit dependence on the real or complex slowness, therefore solving SAW and pseudo SAW problems requires numerical minimization. For a pseudo SAW problem, an initial guess is assigned for the complex slowness, y= l/vp -j a/co, (6) is solved and the BCF determinant, (14), is calculated and the equality to zero checked. The process is repeated for successive values of vp and a with the aid of a multivariable minimization program until (14) is satisfied to within some predetermined accuracy.
213
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M. P. da Cunha
2.6. Types ofHVPSA WSolutions Based on Symmetry Classification This section classifies the pseudo SAWs in terms of the orientation symmetries given in Sub-section 2.1, and in light of the surface wave problem solution outlined in Sub-section 2.2. Characteristics of the pseudo SAW solutions in terms of the particle displacement polarization, the phase velocities, and the type of BAW that radiates power into the substrate are given in Section 3. For the lowest symmetry Type 1, a surface wave solution is built from a linear combination of four partial modes in the semi-infinite substrate to satisfy (14). For the GSAW four decaying partial modes are selected. If one of the four partial modes chosen is a radiating partial mode, the solution is a PSAW. If two radiating partial modes are selected, the solution is an HVPSAW. Due to the radiating partial modes, the fields attenuate as they propagate, but if the contribution of the radiating terms are sufficiently small, these pseudo modes are observed in standard surface acoustic wave devices. For symmetry Type 2, (14) is 3x3, the material is non-piezoelectric, and a surface wave solution is built from a linear combination of three partial modes in the semi-infinite substrate. For the GSAW, three decaying partial modes are selected. For the PSAW and HVPSAW, one and two of the partial modes, respectively, are selected to be radiating partial modes. For symmetry Type 3, the sagittal components decouple from the shear horizontal ones, and (14) is 3x3 for the piezoelectric active solutions. A stiffened surface wave solution is built from a linear combination of three partial modes in the semiinfinite substrate. For the sagittal SAW, Rayleigh wave, three decaying partial modes are selected. If one of the three partial modes is chosen to be a radiating partial mode, the solution is a HVPSAW. This is not a PSAW for reasons that are outlined in Sub-section 1.2 and exemplified in Section 3. In a symmetry Type 4 orientation, the material is piezoelectric, the stiffened shear horizontal solution decouples from the pure mechanical sagittal solution, and (14) is 2x2 for both solutions. Two decaying partial modes are used to build up the stiffened shear horizontal wave and the other two are used for the pure mechanical Rayleigh wave solution. For symmetry Type 5, the material is non-piezoelectric, and the sagittal components decouple from the shear horizontal one. A surface wave solution is built from a linear combination of two partial modes in the semi-infinite substrate. For the sagittal SAW wave (Rayleigh wave) the two decaying partial modes are selected. If one of the two partial modes is a radiating partial mode the mode is a HVPSAW.4212
3. Characteristics of PSAW and HVPSAW Solutions This section discusses the properties and characteristics of the pseudo SAW solution. Sub-section 3.1 shows typical boundary condition functions and considers their behavior for different materials, boundary conditions, and symmetry types. The Poynting vector declination, the power flow angle calculation, and the penetration depth for the pseudo SAWs are described in Sub-section 3.2. The relationship between the radiating partial modes and bulk slowness is addressed in Sub-section 3.3. Examples of PSAW and HVPSAW solutions are given in 3.4.
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3.1. The Boundary Condition Function (BCF) Recalling that the pseudo SAW BCFs depend on vp and a, the BCFs are plotted as a function of those two real variables. Figure 3 a shows the HVPSAW BCF magnitude for the mechanically free shorted surface of symmetry Type 1 quartz ST-25° (Euler angles: [0° 132.75° 25°]), and Fig. 3b shows the two dimensional view of the same plot with respect to the phase velocity axis. As can be noted from these two plots, the BCF has several local minima, which work as traps for a multi-variable optimization routine. Although the multiple local minima behavior can also be observed in the single variable SAW problem, it is more pronounced in the pseudo SAW multi-variable case. Figure 3b also clearly identifies the actual HVPSAW solution at vp=6.5262 Km/s. Note that the BCF decays gradually around several local minima, whereas around the HVPSAW solution the BCF has a comparatively steeper behavior. This behavior usually poses an additional degree of difficulty in finding and tracking the HVPSAW solutions when varying the direction of propagation. An appropriate starting point is very important in finding the HVPSAW solution. Figure 4 compares the BCF between the free, Fig. 4a, and the shorted, Fig. 4b, surface for the symmetry Type 1 mechanically free Li 2 B 4 0 7 (Euler angles: [0° 75° 75°]). Normally the solution for the shorted surface is easier to find than the solution for the electrically free surface, since under shorted boundary condition the energy of the wave is usually trapped closer to the surface, and the pseudo SAWs behave more like a surface mode. The same behavior is observed in pseudo SAW layered problems, Section 4, or when grating structures are used on top of a semi-infinite substrate.25'17'18 The BCF in the electrically free surface, Fig. 4b, has a larger overall gradient with vp and a when compared to the electrically shorted surface, Fig. 4a. In such cases, the use of bounded minimization routines in the complex slowness plane and the careful selection of the initial guess are very important in finding the solution. Still referring to the symmetry Type 1 mechanically free Li 2 B 4 0 7 (Euler angles: [0° 75° 75°]), Fig. 5 shows the BCFs of the electrically shorted HVPSAW (Fig. 5a) and PSAW (Fig. 5b) around me respective solution regions. One notes that the HVPSAW BCF has a steeper behavior around the solution when compared to the PSAW BCF, a typical behavior for symmetries Type 1, also detailed for quartz ST-25° (Euler angles: [0° 132.75° 25°]) in Ref. 3. Note, however, that along symmetries Type 3, where only one radiating partial mode leads to a HVPSAW solution (Sub-section 2.6), the HVPSAW BCF behaves like a PSAW BCF on a symmetry Type 1 orientation. Figure 6 is an example of a HVPSAW BCF along a symmetry Type 3, the mechanically free electrically shorted Li2B407 (Euler angles: [0° 45° 90°]). Normally when there is only one radiating partial mode instead of two, the energy of the wave is trapped closer to the surface, and the pseudo SAW behaves more like a surface mode. When there are two radiating partial modes, the BCF is usually steeper close to the solution when compared to the one radiating partial mode problem. The steeper behavior of the HVPSAW BCF on symmetry Type 1 with respect to the PSAW BCF along symmetry Type 1 and the HVPSAW BCF on symmetry Type 3, usually implies that it is more difficult to find the solution to the HVPSAW BCF on symmetry Type 1 when compared to the other problems.
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216
Pseudo and High Velocity Pseudo
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Attenuation [Np/Km]
Phase Velocity [Km/s] (b)
Fig. 4. Plot of the HVPSAW BCF magnitude as a function of vp and a along Li2B407 (Euler angl [0° 75° 75°]) for the mechanically free: (a) shorted surface; (b) free surface.
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Attenuation [Np/Km]
Phase Velocity [Km/s]
S-60 HD.01
Attenuation [Np/Km]
Phase Velocity [Km/s] (b)
Fig. 5. Plot of the BCF magnitude as a function of vp and a along Li2B407 (Euler angles: [0° 75° 75°]) for the mechanically free shorted surface: (a) HVPSAW; (b) PSAW.
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Pseudo and High Velocity Pseudo SAWs
Attenuation [Np/Km]
0
6.5
1085
Phase Velocity [Km/s]
Fig. 6. Plot of the HVPSAW BCF magnitude as a function of vp and a along the symmetry Type 3 Li 2 B 4 0 7 (Euler angles:[0° 45° 90°]) for the mechanically free shorted surface.
3.2. The Poynting Vector and the Power Flow Angle This subsection analyzes the characteristics of the acoustoelectric Poynting vector for the pseudo SAWs. The specific topics discussed in this subsection are: (i.) the Poynting vector penetration depth; (ii.) the declination of the Poynting vector, its tilt down into the solid, as a function of depth; (iii.) the power flow angle, PFA. The Poynting vector for piezoelectrics is defined by43 P = Re {- v *. T + 0 (o D )'}
(17)
or equivalently P = Re { - v * . r - 0 H > / Z ) } = Re {-T'V . Tn } (18) with " ' " the transpose conjugate operator; rv and x„ are defined in (5). For the BAW problem, P points in the direction of power flow and is invariant from point to point in the infinite solid. For the SAW problem, there is no power flow normal to the mechanically free surface, since all partial modes are decaying partial modes, allfieldsvanish as z goes to minus infinity, and the wave is strictly guided by the surface.44 The magnitude and the direction of the power are obtained by integrating P.dA, over the area of a unit width strip and infinite depth. The eigenvalues of (j^A) in (11), which correspond to the partial modes of the pseudo SAW problem, can be written as (tfi+jka), in e?q){z(ai+jkzl)}, with k^co/Vzi. For the decaying partial modes, DPM, o, are positive in accordance to the coordinate system adopted in Fig. 1. For the radiating partial modes, RPMs, a ; are negative. Note that the power
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crossing any area in the solid is taken to be I P.dA over the area in question. As pointed out in Ref. 2 for the case of the PSAW, carrying out this integration over any closed surface within the solid, the resultant value must be zero to be consistent with the interpretation of the pseudo surface waves and the non-dissipative assumption regarding the medium. For the pseudo SAW orientations of practical interest, meaning those orientations where the pseudo modes can be generated and detected by standard surface acoustic wave devices, one observes that: i.) The amplitudes of the decaying partial modes are much larger than those of the radiating partial modes; ii.) The magnitude of decay rates of the decaying partial modes are larger than the magnitude of growth rate of the radiating partial modes. When i.) and ii.) apply, the pseudo mode behaves like a surface mode: P is large near the surface; P is almost parallel to the surface close to the surface; and the attenuation a is low. A high propagation loss pseudo SAW solution transfers power to one or two of the radiating partial modes within a short distance. 3.2.1. Penetration depth Figure 7 shows the plots of the Poynting vector [TW/m2] in the direction of propagation as a function of the normalized depth for the PSAW, Figs. 7a, and for the HVPSAW, Figs. 7b, along the mechanically free shorted surface symmetry Type 1 quartz AT (Euler angles: [0° -54.7° 0°]). Figure 8 is equivalent to Fig. 7 for an aluminum layered surface (h/^FSAw=l%)- From these figures, which represent practical PSAW and HVPSAW orientations, one may notice that P is large near the surface, decaying after a few wavelengths inside the semi-infinite substrate, where the only remaining partial modes are the radiating partial modes, only visible in these figures for the HVPSAW. One notices from the comparison between Figs. 7a and 7b that the PSAW power decays to relatively negligible values after approximately 20 wavelengths, whereas a relatively significant fraction of power is observed in Fig. 7b for the HVPSAW after 20 wavelengths. This behavior is expected along this symmetry Type 1 substrate, since the PSAW has only one radiating partial mode, whereas the HVPSAW has two radiating partial modes. Characterizing the distribution of power inside the semi-infinite substrate by penetration depth, as done in Ref. 3, a comparison between Figs. 7a and 8a shows that the aluminum layer reduces the penetration depth from around 20 wavelengths to 10 wavelengths for the PSAW. The HVPSAW, Figs. 7b and 8b, is less affected by the same aluminum layer thickness, reflecting the effect of the two radiating partial modes that typically distributes a relatively larger fraction of power inside the material in this symmetry Type 1 orientation. Figure 9 shows the plot of the Poynting vector for the HVPSAW along the mechanically free shorted surface symmetry Type 3 Li2B407 (Euler angles: [45° 45° 90°]). Since only one radiating partial mode exists, one sees that for this symmetry type 3 HVPSAW the power decays to a relatively negligible value after about 1.5 wavelengths, denoting the strong surface guiding behavior this mode has for this orientation.
220
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222
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The pseudo SAWs have a larger penetration depth when compared to the SAW, since about 90% of SAW wave power is contained in one wavelength. The pseudo SAW modes are thus less sensitive to surface defects than SAWs at high frequencies, a practical advantage of the pseudo SAW modes. At lower frequencies the finite thickness of the substrates may need to be considered in device design in light of the higher penetration depth of the pseudo SAW modes. 3.2.2. Declination of the Poynting vector As observed from Figs. 7 to 9, below some depth, the radiating terms are the only ones left in the solution and the direction of P is mainly dictated by the associated radiating partial modes. Figure 10 shows the declination of the Poynting vector, its tilt down into the solid, for both PSAW, Fig. 10a, and HVPSAW, Fig. 10b, as a function of normalized depth for the mechanically free shorted surface symmetry Type 1 quartz AT (Euler angles: [0°-54.7° 0°]). Figure 11 plots the declination of the Poynting vector for the HVPSAW on the mechanically free shorted surface symmetry Type 3 Li2B407 (Euler angles: [45° 45° 90°]). In Figs. 10 and 11, the declination of the Poynting vector is zero close to the surface, as expected for a PSAW or a HVPSAW orientation of practical interest. For both the PSAW represented in Fig. 10a and the HVPSAW represented in Fig. 11, there is only
223
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Pseudo and High Velocity Pseudo SAWs
-
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Fig. 11. Plots of the HVPSAW declination of the Poynting vector as a function of the normalized depth for the mechanically free shorted surface symmetry Type 3 symmetry Type 3 Li2B407 (Euler angles: [45° 45° 90°]).
one radiating partial mode. In these cases the declination of the Poynting vector is unambiguously dictated by the radiating partial mode after a depth has been reached where the decaying partial modes contributions have become negligible. As can be observed from Fig. 10b, the declination of the Poynting vector for die symmetry Type 1 HVPSAW has the influence of both radiating partial modes, after the DPMs have become negligible. A similar behavior is reported for quartz ST-25° (Euler angles: [0° 132.75° 25°]).4 3.2.3. Powerflowangle Provided the pseudo SAW mode behaves like a surface mode in the sense that that P is large near the surface, which means that the power is concentrated near the surface, an approximate power calculation is performed. As suggested in Ref. 2 for the PSAW and in Ref. 4 for the HVPSAW, the magnitude and the direction of the power flow can be evaluated approximately by performing the integration of P over a strip of unit width and some finite depth. In the case of HVPSAW along symmetry Type 3 and die PSAW a suitable depth is around a couple of wavelengths. For die HVPSAW along symmetry Type 1, the integration usually needs to be done for a larger depth, due to die existence of two radiating partial modes, and the fact that die decaying fields are still significant at a few wavelengths away from the surface. The choice of depth to use in integration varies from orientation to orientation according to die P behavior with depth, and therefore it is helpful to plot die deptii behavior of P before choosing an integration depth.
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Using the procedure mentioned in the previous paragraph, the power components px and py necessary to calculate the azimuth angle between P and the direction of propagation can be approximately evaluated, and the power flow angle, PEA, estimated as PFA=tgA(py /px). Good agreement between the PEA predictions and experimental results are reported in reference 4 when a depth of around 14 wavelengths was considered along the quartz ST-25° orientation. 3.3. The Relationship between Radiating Partial Modes and Bulk Slowness This subsection discusses the relationship between the PSAW and HVPSAW radiating partial modes and the bulk waves that are radiated inside the semi-infinite substrate, according to the type of bulk wave and the direction of propagation. The partial modes that enter the pseudo SAW solution have been genericalfy written as (<Ji+jk^, in exp{z(<Ji+jk„)}, vnthkzi=a>/vz,. In accordance to the coordinate system adopted in Fig. 1, Oj are necessarily negative and kd positive for the RPMs, since these partial modes radiate power towards the interior of the semi-infinite substrate. Recalling from Subsection 2.2 that the wave vector in the direction of propagation is given by k^a/Vp, Fig. 12 defines the tilted radiating partial mode wave vector, kBi, and its angle to the surface, #,, that is calculated using3
e,=#-x' kO
(19) V xJ The wave velocity associated with the radiating partial mode, v B ;, is calculated from Fig. 12 by 1 1 1 — = — + ~T (2°) v v
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Pseudo and High Velocity Pseudo SAWs
1105
HVPSAW characteristic that can be advantageously used in device design and fabrication. Such behavior has also been observed in PSAW orientations, like quartz ST25° (Euler angles: [0° 132.75° 25°]), 36° YX LiTa0 3 (Euler angles: [0° -54° 0°]), and 64° YX LiNb0 3 (Euler angles: [0° -26° 0°]).4'25' 26 ' 49 ' 50 For 0.2.:CN
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269
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experimental results. First, the susceptance B s , which represents phase shifts caused by the energy storage effect and the piezoelectric shorting, 20 is determined as the center frequency is fitted to experimental results, and then the normalized acoustic impedances of the electrodes (Zm) is determined as the admittance characteristics are fitted to the experimental results. Figure 23(b) shows the calculation results. In this case, Bs is 0.16 and Zm is 1.064. Figure 24 shows the structure of the ladder type filter. It has 6 SAW resonators which have reflectors of 20 strips with widths of X/4 at both sides of the resonators. However, the effects of the number of gratings are small because the reflection coefficients of the electrodes are large. The parameters are shown in Table 3. The ratio of the static capacitance of resonator A to that of resonator B is 1.7. The frequency response calculated using the obtained equivalent circuit parameters is shown in Fig. 25.
Port 1 (IN) o
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Table 3. Parameters of the ladder type filter in 10 GHz.
Pair number
Aperture
Period (wavelength)
Resonator A
120.5
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0.38 urn
Resonator B
100.5
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270
SAW Devices Beyond 5 GHz
1137
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Calculated frequency response of the ladder type filter.
The micrograph of the electrodes in resonator A observed by SEM was shown in Fig. 11 in Section 4.2. The electrode width is 95 nm, and the thickness of the Al film is 30 nm (0.079k). They are fabricated on 128° Y-X LiNb0 3 using the electron beam exposure system and the lift-off process. The resolution of the EB system is 40 to 50 nm and the minimum step of the beam scan is 10 nm in the setting of this experiments. Because of the limitation of the scan step, we use the structure shown in Fig. 26 (b) as the resonator B for the purpose of adjusting the resonant frequency of resonator B to the antiresonant frequency of resonator A. Figure 27 shows the frequency responses of the 10 GHz-ladder type filters. They are measured by microwave probes. A low loss characteristic with a minimum insertion loss of 3.4 dB is obtained as shown in Fig. 27(a). Figure 27(b) shows the characteristic of another filter. A considerable reduction by 25 dB of the electromagnetic feed-through is obtained in the 17 GHz range. The reduction characteristics in the high frequency range largely depend on the structure of the connection electrodes and the structure of the bonding pads. It is necessary to optimize the structures. These results show the feasibility of SAW devices for communication systems in the 10 GHz-range.
271
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(a)
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Structure of resonators, (a) Resonator A,
(b) Resonator B.
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272
SAW Devices Beyond 5 GHz
7.
1139
Prospective
In this paper, we described the recent progress in high frequency SAW technology based on the ultra-fine fabrication using EB direct writing techniques. On the other hand, other approaches using high-speed substrates have also been studied and have shown promise for achieving SAW device beyond 5 GHz-range by means of optical lithography. For example, diamond thin film on Si substrate, 2I'22 longitudinal leaky surface wave on Li 2 B 4 0 7 23 and other materials have the possibility to beyond 5 GHz devices without using EB lithography. In order to realize the 10 GHz-range SAW devices commercially, issues such as electromagnetic feedthough rejection, the durability of the electrodes, the electrode resistance, and fine electrode fabrication techniques should be studied. These are important issues not only for the SAW devices, but generally for high frequency miniaturized devices. However, especially in SAW devices, the substrate is a dielectric material, which sometimes has a high dielectric constant. The design technology that will be required in 10 GHz range does not lend itself to a separate evaluation of the IDT characteristics and the packaging, but must instead be based on a 3 dimensional analysis of the electromagnetic field of the total device, which includes both the substrate, the IDTs and the packaging. By optimizing the total device structure, we will be able to achieve a sufficient electromagnetic feedthough rejection. In fact, we experimentally obtained the 50 dB suppression of the feedthough signal for propagation distance of 100 X at the 10 GHz. Furthermore, advanced trimming techniques will also be required in 10 GHz-range, especially when relatively narrow band characteristics are required.
8.
Conclusions
We described SAW device techniques for the 5 to 10 GHz range based on ultra-fine fabrication techniques. The recent advances show that the SAW device technology has reached a level at which practical applications should be considered up to the 10 GHz range. Several approaches may be applied to achieve high frequency SAW devices. Here is shown how SAW devices fabricated by EB lithography may be approaching sufficient maturity to be considered for practical applications.
273
1140 H. Odagawa & K. Yamanouchi References 1. A. J. Slobodnik, P. H. Carr and A. J. Budreau, "Microwave Frequency Acoustic Surface-Wave Loss Mechanism on LiNb0 3 ", J.Appl.Phys., Vol.41, No.l 1 (1970) 4380—4387. 2. K. Yamanouchi, "Generation, Propagation, and Attenuation of 10 GHz-Range SAW in LiNbO,", 1998 IEEE Ultrason. Symp. Proa, Vol.1 (1998) 57—62. 3. K. Yamanouchi, H. Nakagawa, J. A. Qureshi and H. Odagawa, "10 GHz-Range Surface Acoustic Wave Low Loss Filter Measurement at Low Temperature", Jpn.J.Appl.Phys., Vol.38, No.5B (1999) 3270—3274. 4. K. Yamanouchi, "GHz-Range SAW Device Using Nano-Meter Electrode Fabrication Technology", 1994 IEEE Ultrason. Symp. Proc, Vol.1 (1994) 421—428. 5. K. Yamanouchi and H.Furuyashiki, "New Low-Loss Filter Using Internal Floating Electrode Reflection type of Single-Phase Unidirectional Transuducer", Electron. Lett. Vol.20, No.24 (1984)989—990. 6. M. Takeuchi and K. Yamanouchi, "Coupled Mode Analysis of SAW Floating Electrode Type Unidirectional Transducers", IEEE Trans. Ultrason. Ferroelec. Freq. Contr., Vol.40, No.6 (1993) 648—658. 7. K. Yamanouchi, M. Takeuchi, T. Meguro, K. Doi and K. Murata, "Wide Bandwidth Low Loss Filter Using Piezoelectric Leaky SAW Unidirectional Transducers with Floating Electrodes", Jpn. J. Appl. Phys., Vol.30, Supplement 30-1 (1991) 173—175. 8. K. Yamanouchi, C. S. Lee, K. Yamamoto, T. Meguro and H. Odagawa, "GHz-range Low-Loss Wide Band Filter Using New Floating Electrode Type Unidirectional Transducers", 1992 IEEE Ultrason. Symp. Proc, Vol.1 (1992) 139—142. 9. K. Yamanouchi, Y. Cho and T. Meguro, "SHF-Range Surface Acoustic Wave Interdigital Transducers Using Electron Beam Expusre", 1988 IEEE Ultrason. Symp. Proc, Vol.1 (1988) 115—118. 10. K. Yamanouchi, T. Meguro and K. Matsumoto, "Surface-Acoustic-Wave Unidirectional Transducers Using Anodic Oxidation Technology and Low-Loss Filter", Electron. Lett. Vol.25, No. 15 (1989) 958—960. 11. H. Odagawa, N. Tanaka, T Meguro and K. Yamanouchi, "Submicron Fabrication Techniques
Using Electro-Chemical
Effects
274
and
Application
to Unidirectional
SAW
SAW Devices Beyond 5 GHz 1141 Transducers", 1994 IEEE Ultrason. Symp. Proc, Vol.1 (1994) 437—440. 12. H. Odagawa, T. Kojima, T. Meguro, Y. Wagatsuma and K. Yamanouchi, "GHz-Range Conventional )J4 Unidirectional Surface Acoustic Wave Transducers and Their Application to Low-Loss and Zero-Temperature Coefficient Filters", Jpn. J. Appl. Phys., Vol.36, No.5B (1997) 3087—3090. 13. H. Odagawa, T. Meguro and K.Yamanouchi, "5 GHz Range Low-Loss Wide Band Surface Acoustic Wave Filter Using Electrode Thickness Difference Type Unidirectional Transducers", Jpn. J. Appl. Phys., Vol.35, No.5B (1996) 3028-3031. 14. C. S. Hartmann, P. V. Wright, R. 1. Kansy and E. M. Garber, "An Analysis of SAW Interdigital Transducers with Internal Reflections and Application to the Design of Single-Phase Unidirectional Transducers", 1982 IEEE Ultrason. Symp. Proc. (1982) 40—45. 15. K. Yamanouchi, Z. Chen and T. Meguro, "New Low-Los Surface Acoustic Wave Transducers in the UHF Range", IEEE Trans. Ultrason. Ferroelec. Freq. Contr., Vol.34, No.5 (1987)531—539. 16. T. Nishihara, H. Uchishiba, O. Ikata and Y. Satoh, "Improved Power Durability of Surface Acoustic Wave Filters for an Antenna Duplexers", Jpn. J. Appl. Phys., Vol.34, No.5B (1995) 2688—2692. 17. H. Odagawa and K. Yamanouchi, "10 GHz-range Extremely Low-Loss Surface Acoustic Wave Filter", Electron. Lett. Vol.34, No.9 (1998) 865—866. 18. H. Odagawa and K. Yamanouchi, "10 GHz-Range Extremely Low-Loss Ladder Type Surface Acoustic Wave Filter", 1998 IEEE Ultrason. Symp. Proc. (1998) 103—106. 19. S. Lehtonen, V. P. Plessky, M. T. Honkanen, VOvchinnikov, J. Turunen and M. M. Salomaa, "SAW Impedance Element Filters for 5 GHz and beyond", 1999 IEEE Ultrason. Symp. Proc. (1999)395—399. 20. K. Kojima and K. Shibayama, "An Analysis of Reflection Characteristics of the Surface-Acoustic-Wave Reflector by and Equivalent Circuit Model", Proc. 7lh Symp. Ultrasonics, Kyoto 1986, Jpn. J. Appl. Phys., Suppl.26-1 (1987) 117—119. 21. K. Yamanouchi, N. Sakurai and T. Satoh, "SAW Propagation Characteristics and Fabrication Technology of Piezoelectric Thin Film / Diamond Structure", 1989 IEEE Ultrason. Symp. Proc. (1989)351—354.
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22. A. Hachigo, H. Nakahata, K. Itakura, S. Fujii and S. Shikata, "10 GHz Narrow Band SAW Filters using Diamond", 1999 IEEE Ultrason. Symp. Proc. (1999) 325—328. 23. T. Sato and H. Abe, "Propagation Properties of Longitudinal Leaky Surface Waves on Lithium Tetraborate", IEEE Trans. Ultrason. Ferroelec. Freq. Contr., Vol.45, No.l (1998) 136—151.
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International Journal of High Speed Electronics and Systems, Vol. 10, No. 4 (2000) 1143-1191 © World Scientific Publishing Company
WIRELESS SAW IDENTIFICATION AND SENSOR SYSTEMS F. SCHMIDT and G. SCHOLL Siemens AG, Corporate Technology Surface Acoustic Wave Technology and Wireless Systems Otto-Hahn-Ring 6, 81730 Munich, Germany
Identification and sensor systems based on surface acoustic waves exhibit intriguing properties which have hitherto remained unexploited by semiconductor-based systems. They offer a long readout distance of up to more than 20 meters with purely passive surface acoustic wave (SAW) devices. SAW devices operate with no battery or wiring, withstand extreme temperatures and work reliably and maintenance-free over many decades even in harsh industrial environments. Because they operate at frequencies in the GHz range, SAW identification and sensor systems are well protected from the electromagnetic interference that often occurs in the vicinity of industrial equipment such as motors and high-voltage lines. The fundamentals and design rules of numerous passive wireless SAW sensor and identification systems for industrial and domestic applications as well as relevant practical work will be presented.
1. Introduction Surface acoustic wave (SAW) filters play a key role in consumer and communication systems thanks to their high performance, small size and high reproducibility.1 These properties also make them attractive for identification and sensor applications.2'3 Generally, any problem will have several technological approaches and various solutions. It is therefore vital to focus on the specific advantages offered by SAW devices. Their most outstanding property is clearly that they can operate with no wire connection or battery over long distances, as they are connected only by a radio frequency link to a transceiver or reader unit.4"9 This is due to the fact that SAW devices can operate even with very low signal levels at their input. In contrast, semiconductor-based devices require a fixed minimum voltage to operate, thus drastically limiting the range of passive coupled ID tags to about 0.5 meters. In this article, wireless SAW sensor and identification systems will be discussed and various interesting applications will be presented. A schematic drawing of a wireless sensor/identification system is shown in Fig. 1. A high-frequency electromagnetic wave is emitted from an RF (radio frequency) transceiver which is basically a low-cost radar unit.10 This radio wave is received by the antenna of the SAW sensor/ID-tag. The comblike interdigital transducer (IDT) shown in Fig. 2 is connected to the antenna and transforms the received signal into a SAW which propagates along the piezoelectric crystal and is partially reflected by reflectors placed in the acoustic path. The reflected waves are reconverted into an electromagnetic pulse train by the IDT and are then retransmitted to the radar unit. The received signal is amplified and down-converted to the baseband frequency in the RF module. The sensor signals are then analyzed by a digital signal processor. Finally, the measurement results are transferred to a PC or another device for post processing, data storage or to perform specific tasks.
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SAW ID tag
Antenna
Transceiver unit Fig. 1. Schematic drawing of a SAW identification/sensor system. The velocity of a SAW is lower than that of electromagnetic signals propagated in free space by a factor of approximately 100,000. Signals can consequently be efficiently delayed by a small SAW chip. A delay of 1 |_is requires a chip length of only between 1.5 and 2 mm (the path length between the IDT and the reflector is used twice), depending on the substrate material, whereas radio signals in free space propagate 300 m in the same time. The pulse responses of SAW sensors can thus be delayed by up to several microseconds. This enables them to be separated easily from environmental echoes, which typically fade away in less than 1-2 (xs. If the reflectors are arranged in a predefined bit pattern such as a barcode, an RF identification system can be implemented with a readout distance of several meters. By analyzing the fine structure of the response signals, SAW radio ID and sensor systems can also be applied to the measurement of chemical or physical quantities. In the first part of this article, radar-type sensor and identification systems such as those shown in Figs. 1 and 2 will be discussed. Antenna SAW Chip
Transmitted pulse
A») (JUUVLA.
(«
Reflector
SensorfTag response
Fig. 2. Schematic drawing of a SAW reflective delay line. However, such a radar-based system has two drawbacks. Firstly, not many ID tags or SAW sensors can be located within the detection range of the transceiver at the same time. If more than a few sensors are present, their response signals superimpose at the receiver and can no longer be separated. And secondly, the detection range of about 10 meters, although sufficient for many applications, is rather limited.
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Receiver unit Fig. 3. Schematic drawing of an event driven sensor/identification system.
To overcome these two main restrictions, we have developed a different kind of SAW sensor. The basic idea is that, as sensed processes often involve remarkable amounts of energy in the form of mechanical pressure change, movements or torque forces, temperature change etc., why not simply use this energy to operate the sensor instead of supplying it via radio waves? Consequently, the sensor type described from Section 4 onward does not reflect radar signals but generates its own RF signals with energy drawn from the sensed process as shown in the schematic drawing in Fig. 3. The sensor still makes use of SAW elements. The only difference compared with the radartype sensor is the origin of the requesting RF signals, which in this case are generated in the sensor itself. We have called it the event-driven sensor.
2. Passive Wireless SAW ID Tags and Sensors Although the first physical and chemical sensors based on SAW devices were reported in the early 1970s, considerable active research is still under way. ' The basic design of reflective delay lines for wireless identification and sensor systems generally follows the same principles as those established for classical SAW filters. An additional difficulty is that many sensors cannot be sealed hermetically like conventional filters but are exposed to the influence of external perturbations. In the case of physical sensors, an applicationspecific packaging and mounting technology must often be developed. Fortunately, these difficulties do not exist for SAW ID tags. Another advantage is that except for the coding process, the same fabrication techniques as those used for RF SAW filter manufacture can be adopted. Thanks to this compatibility with standard fabrication processes and to the potential size of the high-volume markets for automotive manufacturing, trailer and container tracking, as well as car and personnel access, we believe that SAW identification systems have a very good chance of market success and will pave the way for other SAW-based wireless sensor systems.
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2.1 SAW ID tags Like all wireless communications systems, SAW identification systems must operate according to the radio frequency regulations. The ISM (industrial, scientific and medical) band at 2.45 GHz with a bandwidth of 100 MHz is an internationally available frequency band compatible with SAW technology. Although another ISM band at 433.92 MHz is available and is often used for remote entry systems in the automotive industry, its bandwidth is limited to 1.7 MHz. It is consequently not well suited for SAW ID systems, especially when a large code space is required. Modern RF filters for cellular phones typically employ leaky waves and surfaceskimming bulk waves13"15 on high coupling substrates for the three following reasons. The high electromechanical coupling coefficient allows efficient interdigital transducers and reflectors with only a small number of electrodes and therefore a broad bandwidth to be implemented. Because of the high velocity of leaky waves, the line widths that have to be implemented become greater. The power durability is also much higher than for Rayleigh waves. Although the first two arguments are also relevant for ID tag designs, the most important parameter of an ID tag is its insertion loss, especially at 2.45 GHz. As a result, the well-known classical Rayleigh wave substrates LiNb03-YZ and LiNbOy rotl28° are generally used. However, even on these substrates the propagation loss is 6 dB/us at 2.45 GHz. The propagation loss at 433 MHz is a negligible 0.25 dB/us, whereas that at 900 MHz is approximately 1 dB/us. Which substrate is finally chosen will depend on the specific design and on the specifications which have to be satisfied, so that tradeoffs will have to be made. Basically, there are two ways of arranging the reflectors. Either all the reflectors can be positioned in different tracks (Fig. 4), or several reflectors can be lined up in only a few tracks (Fig. 2).16,17 In the first case, all the reflectors can have 100% reflectivity and thus totally reflect the incoming wave. In the second case, the reflectivity has to be decreased in order to make it homogeneous for all the reflecting elements. The problem of multiple reflections must also be taken into account, thus limiting the dynamic range of on/off-coded ID tags. The optimum number for a 2.45 GHz tag was calculated in Ref.19 as eight reflectors per track. Although the reflectivity of the reflector positioned next to the IDT is typically reduced to -20 dB, we found that a design with multiple reflectors in one track can nevertheless be attractive, especially when a large code space is desired. If the reflectors are placed in different tracks, problems with diffraction effects and track loss arise. Large apertures are desirable in order to minimize diffraction effects, but these lead to problems with the impedance level. On the other hand, if the transducer aperture satisfies the requirements of the termination impedance, the reflector apertures can be smaller than an acoustic wavelength. Millions of RF filters are manufactured every day for cellular phones. More than a thousand fit onto a 3-inch wafer and one dozen are exposed at a time. SAW delay lines are based on the efficient delaying of RF signals. The size of SAW ID tags is thus limited by the required code space, which results in a reduced number of chips/wafer. If on/off coding is selected as the modulation scheme and a code space of 32 bits is required, then 34 reflectors are needed. Two of them, one start and one stop reflector, are generally used for temperature compensation. If each reflector of the tag is also structured with a wafer stepper, the number of exposures required to structure the same number of chips is three orders of magnitude greater than that for an RF filter. It is therefore vital to reduce the number of reflectors and to develop an economical coding process. Phase coding represents one way of doing this.18
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Fig. 4. Reflective delay line with only a single reflector per track.
Figure 5 shows the measured time-domain response of a 433 MHz ID tag with quadrature shift keying. Although phase-coded tags can be implemented easily at 433 MHz, material and fabrication tolerances prevent the use of this technique at 2.45 GHz. The solution is to use pulse position modulation, in which the information is given by the relative position of the reflectors instead of the phase. The time-domain response at 2.45 GHz of a SAW ID tag with 18 reflectors and a code space greater than 232 is presented in Fig. 6. The tag was measured with a network analyzer and then ideally matched to 50 Q. It can be seen that the insertion loss is approximately 53 dB. Excellent amplitude uniformity was achieved. The time-domain response of a 20 bit ID tag at 2.45 GHz with on/off-keying is shown in Fig. 7 for comparison.
100 CD
0
2,
CD CO co
sz a.
I-
2
4
6
-100
8
time [us] Fig. 5. Time-domain response of a phase-coded ID tag at 433 MHz.
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-40
3 4 time [us] Fig. 6. Time-domain response of a pulse-position coded ID tag at 2.45 GHz.
-*0
time [us] Fig. 7. Time domain response of an on/off-coded ID tag at 2.45 GHz.
2.2 Binary SAW sensors and programmable reflectors The first ID tags with on/off-keying were designed with open or short-circuited reflectors, the coding being implemented by either structuring a reflector or not. This involved the inherent problem that the uniformity was affected and code-dependence occurred because the attenuation and bulk wave conversion differ depending on whether a reflector is present or not. A solution was to replace an "off reflector with a nonreflecting structure having the same transmission and bulk-wave conversion loss as the "on" reflector.19 In order to allow the reflector to be dynamically changed from "on" to splitfmger transducer can be used for this component (Fig. 8). If the reflector is open circuited, the transducer reflects an incoming wave. If the reflector is shorted, hardly any reflection occurs. The first and last reflectors in Fig. 8 are shortened to reduce end-effects which generate unwanted reflections when the reflector is "off.
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Fig. 8. Splitfinger reflector with shortened end electrodes.
In Ref. 21 a prototype of a train-control system was implemented in which a reflective SAW delay line with nine splitfinger reflectors was embedded in a commutator network in hybrid technology (Fig. 9). The last reflector was kept permanently "on" to maintain function control. After a trigger burst from the reader unit, an 8-bit data word was transferred from the memory to the commutator network. According to the logical state of each bit, the corresponding reflector was switched "on" or "off and an 8-bit data telegram was retransmitted to the reader unit. Thanks to the coherent storage of the energy in the SAW device, no synchronization was necessary between reader and transponder so that a very high data rate could be achieved.
R
c
L
Fig. 9. Switching network for SAW reflector.
In Fig. 10 a Reed element is connected to a reflective element of a binary SAW sensor. If the sensor is accelerated beyond a specific value, the magnet moves and causes the Reed contact to change from the open-circuited state to the short-circuited state or vice versa.
Reed Element Fig. 10. Binary SAW sensor with switchable reflectors.
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In the left diagram of Fig. 11, all three reflectors have almost the same reflectivity. In the right diagram, the third reflector is shorted by the Reed element. A change of approximately 30 dB in reflectivity can be observed. Sensors of this type can be used as sectoralignment indicators, for checking switch positions, or generally as radio-access switches. In the same way, classical sensors with varying impedance can also be read out in wireless mode when combined with a SAW transponder as in Ref. 22, as will be shown in Section 2.4.
-40
open
-60
-80
2"'
i ' • '' 1
li k
time [lis]
6
7
L
-80 2
8
3
4
5
6
7
8
time [jis]
Fig 11. Time-domain response of a binary SAW sensor with switchable reflectors.
2.3 SAW temperature sensors A change in the environmental temperature AQ results in a variation of the path length 31 and the SAW velocity 3v. Accordingly, the propagation time T changes to
T ~ 1/30
v90
T3 = T C D , A T 3 ,
(1)
where TCDi represents the first-order temperature coefficient of the delay. This equation can be generalized for other physical or chemical sensors to AT
(2)
where the sensitivity S of y to a change z is defined by
^ = liml-^ = i . ^ AZ-»O y
\z
y
(3)
az
The term 31/1 represents the mechanical strain, which can also be caused by factors such as the direct application of a mechanical force or electrostriction. The second term in Eq. 2 is produced by a change in the material parameters. Either effect can be dominant, depending on the geometry and the substrate material. For instance, when YX-LiNbC>3 is subjected to a biasing electric field, the strain contribution exceeds the velocity by far, whereas on YZ-LiNb03 with a delay temperature coefficient of 85 ppm/K the thermal expansion coefficient is an order of magnitude smaller than the temperature coefficient of the velocity. A SAW temperature sensor with four reflectors and a center frequency of
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433 MHz mounted in a standard SMD package is shown in Fig. 12. The time domain response of an equivalent temperature sensor at 869 MHz is depicted in Fig. 13.
Fig. 12. SAWtemperaturesensor with four reflectors mounted in an SMD package.
The readout distance is approximately 0.5 m. The time delay is 2.5 p . The sensor pulses are separated by 0.84 |is, 0.7 ps and 0.73 |is respectively. The reflectors were designed for high reflectivity in order to achieve a low return loss of 28 dB. Consequently, the pulses following the four main response peaks, which are due to multiple reflections between the reflectors themselves and between these and the IDT, are suppressed by only 15 dB. To evaluate the temperature value, the phase variations of reflected pulses A
between reflected pulses at a pressure of 250 kPa (measured results black , calculated results white ).
Although monolithic SAW pressure sensors are extremely stable and offer high pressure sensitivity, several applications require smaller and cheaper sensor modules than can be implemented with standard quartz substrates. Reflective delay lines with electrically loaded splitfinger reflectors offer an attractive alternative in this case.27 An external pressure sensor is connected to the second IDT of the SAW transponder as shown in Fig. 20, so that pressure variations generate a variation in the termination impedance of the IDT which results in amplitude and phase changes in the reflected SAW pulses.
\>x.1 \%4 1
A*
SAW chip
matching circuit
pressure sensor
Fig. 20. Hybrid SAW sensor with external matching circuit and pressure sensor.
The reflectivity of an impedance-loaded reflector as a function of the matched sensor impedance Zsensor can be described by the complex scattering parameter c (7 ^ U ^
2P
) -
'n
P 33 +
(5) 1
sensor)
Z
+7 sensor
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where the P-matrix elements in Eq. 5 characterize the electro-acoustic behavior of the reflector. Zmatd, represents the impedance of the matching circuit. As the sensor impedance modulates the reflected pulse in both amplitude and phase, there are two ways of calculating the sensor information in the time domain. One strategy is to match the sensor impedance to the electrical impedance of the reflector for maximum amplitude modulation. This is outlined in the Smith chart in Fig. 21, where the amplitude of the reflected pulse is shown as a contour plot as a function of the load impedance. The simulation is based on the scattering parameters of an implemented reflective SAW delay line measured by a network analyzer.
. "
-26dB -3QdB -22dB
, '
-35dB ' ' '
'. -40dB
-25dB " -8MB
-45dB «SMB,
'.
•\ -3SdB _ • ' . . ,
'' -45d8 • • . . -SkHp: 50 *; \ *' • ',
I
•
Fig. 21. Impedance dependent acoustical reflection |Su|. The lines represent constant reflection amplitudes.
As can be seen in Fig. 21, a very high impedance variability is needed to modulate the magnitude of S n between -22 dB and -60 dB. Reference should also be made to Section 2.2 "Binary SAW sensors", where the load impedances were switched between short and open-circuited modes. In the case of capacitive sensor elements, a high Q-factor of the sensor impedance is also desirable. With the aid of a matching circuit, the sensor impedance can be transformed to the upper right region of the Smith chart where high reflectivity can be observed. Although a reflectivity change of more than 20 dB could be achieved with correctly dimensioned sensors22, there is a drawback in deriving the sensor information from the amplitude of the reflected pulses. A high dynamic range of the amplitudes results in a low signal-to-noise ratio, thus limiting the readout distance. To overcome this limitation, the sensor impedance can be matched to the acoustic reflector such that the phase modulation is maximized. Figure 22 shows the phase
