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kxeXz. Our diagonalized equation transforms into the following algebraic eigen value equation £33J
I
I —^k \
2
£33
£33
(83)
-z&jk £33
For the eigenvalue A we obtain the two solutions in (84) £33
£33
with the corresponding eigenvectors given in (85).
X {k)
~ " { eP)k\)
*+<*>" ( - * i | * | )
(85)
Superposition%ads to the following result for (ip, D3)*:
(^,)- a " > w (" 1 w) e , - + " m w (-ii*i)* i *'
m
Remark: The reader may say that "layered" boundary value problems (specified by interfaces z = zW = const) represent a small subclass of boundary value problems, and that, in practice, the boundary shapes are much more complicated, and thus the above-described concepts can no longer be applied. Our answer is that in the case of boundary shapes other t h a n planes, the underlying singular integral equations involve, in the majority of cases, infinite-domain Green's functions (and possibly their spatial derivative). Due to the fact that infinite domains can be composed from two semi-spaces, we can apply the above concept. Furthermore, it should be noted that many engineering problems can be modeled by a combination of layered structures, and subdomains with fairly complicated boundary shapes. In these cases, the Green's functions used are field solutions in layered structures sandwiched between two semi-spaces with proper material constitution. These kinds of Green's functions can easily be constructed by the theory developed here. 3.2.
Boundary
conditions
The boundary conditions on an interface z = z^ 9
= const are as follows:
In Ref. 9 these ideas are generalized and it is shown there that this concept applies with the same simplicity to fields in inhomogeneous media (eij(x,z)).
105
972 A. R. Baghai-Wadji (i) the electric potential is a continuous function. (ii) the normal component of D has a jump discontinuity equal to the charge density function p(x) in this plane. Fourier transforming the boundary and interface conditions, and solving for the involved unknown coefficients oq (k), a.^ (k), a\2'(k) and ctjjjik), in regions I, II, and III we obtain closed-form expressions for
U?-$r%^k\y&'-*A
GAX m -J-J. G (a! z| )
'
' * - 2 e M|*|^W + e W e xe
+e
J
j[-(x-a)+-jK{z-c)]k '33
(87)
e
33;
xe
1
1
(88)
("ft" 2 —fo c )l f c l
Gm(*,z|fc) = - r a (nrrje^ e p + £ p II
«ss
e
J'[-(x-a) + (-^ Z —^yc)]fc
Ss
<33
(89)
We find that Gi(x, z\k) and Gu(x, z\k) have the same analytical form, except for the second exponential terms, i.e., exp(—£p / e ^ (z — c)\k\) and exp(ep / e ^ (z — c)|fc|), which seem t o be different. However, the validity range of Gi(x, z\k) is z > c and thus the equality z — c = \z — c\ follows. We can therefore write the following equation for the second exponential term in Gi(x, z\k):
.(») —fo-(z~ c )l fc l e *W
e(«) —fe-l*-<=ll*l
=e
'as'
(90)
Gu(x, z\k) is valid in the range 0 < z < c and thus we have z — c = — \z — c\ in this region. Consequently, we can write the following for the second exponential term in Gu(x, z\k):
106
Theory and Applications of Green's Functions -c)|fc| e 33
973
-c||fc|
= e
(91)
°33
Thus, by introducing \z—c\, we cannot distinguish between the analytical expressions for Gi(x, z\k) and GJI(X, z\k). This is not surprising, because the introduction of the plane z = c and thus the subdivision of the upper semi-space into regions J and 77 was merely conceptual, and was not based on a physical necessity. Regions / and II are both specified by e ^ and together they build a semi-space into which a line charge has been embedded. In the following we will write G^{x,z\k) in order to refer to Gi(x, z\k) as well as to Gu(x, z\k). The superscript (u) denotes that the validity range of G^u\x, z\k) is restricted to the "upper" half-plane (z > 0). Similarly, as the validity range of Giu(x,z\k) is the lower half-plane (z < 0), we write G^(x,z\k) for Giu(x,z\k). By adding to the Green's function a second superscript "u" or "1", we can make clear on which side of the interface plane (z = 0) the line charge is located, "u" also refers to the upper half-space here, while "1" signifies the lower one. According to this convention, in the present case we can write G^u'u\x,z\k) and G^'u\x,z\k). uu G( ' '(x,z\k) is the electric potential in the upper half-plane as the response to a line charge which is located in the upper half-plane. G"'u>(x, z\k) is the electric potential in the lower half-plane as the response to a line charge which is located in the upper half-plane. So far we have used the constants a and c as the co-ordinates of the line charge. The choice of the "constants" a and c makes it possible to consider the position of the line charge and consequently the location of the upper interface plane to be fixed. However, having already constructed the formulae for the Green's functions, we can perform the substitutions a -> 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<^(s,z|^z'|A) = - ? - ^ - ? j T _ <_">+el*
c
(94)
MID,
1'
',M
1
G Hx zlx zlk)
1 ( 4 ° - ^
e
-l1^11'1
> > ^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 <&, we obtain for the charge neutrality condition N
J
J dxdzp(x, z) = J f dxdz V J qi8(x — Xi)8{z — Zi) =
—oo —oo *
— oo—oo
^=1 N
oo oo
= ]L* / /
dxdzS x x s z
( - i) (
-zi) = J2Qi=°-
( 103 )
Eq.(103) can alternatively be called the balance-law, the conservation law or the regularization condition. Interchanging t h e locations of the source and t h e observation points (a;', z' o x,z), G'"'™) and G^*^ remain unaltered, G{u'u\x',z'\x,z) (l
l
G > \x',z'\x,z)
= G(u'u\x,z\x',z')
(104a)
= GV'l)(x,z\x',z').
(104b)
The substitutions x', z' <-» x, z, however, transform G^' u ) into G ( "' Z) and G*"'^ into G('-") G{l'u)(x', M
z'\x, z) = G(u'l\x,
G (x',z'\x,z)
u
z\x', z')
= G^ \x,z\x',z').
110
(105a) (105b)
Theory and Applications of Green's Functions
977
These two properties result from the validity of the reciprocity property in the boundary value problem under consideration. A formal proof of these facts is omitted and is left to the reader. 3.3. Real-space
Green's
3.3.1. Preparatory
functions
calculations
Let us consider the following table integral oo
e-ak
_ e-/3fc
{dk
COSbk =
k
b2 + /32
i
2 l n 6^W
(1 6)
°
Re[/?] > 0; Re[a] > 0. (Formula 3.951/3 of Ref. 27.) In the special case that a is zero, Eq.(106) becomes c» 1 __ e-?k 1 h2 4- B2 Jdk=-~ cosbk = j-ln Zf .
for
Z
K
Q
(107)
0
We now consider integrals of the following type which arise in the expressions G^a'b\x,z\x',z'\k): f e~0k I = 4-dk—-—cosbk J k
f e~^k = lim / dk——- cosbk = lim 7 £ . e-+o+ J k £->o+
0
(108)
£
The subsequent steps are self-explanatory. 00
f Ic =
00
e~l3k dk—-—cosbk=
00
f 1 - e~Pk f 1 — / dk cosbA; 4- / dk—cosbk
e
e
e
00
00
= — J dk-—^
cos&A: + J dk^^~.
0
(109)
£
Using (107) we can write 1, b2 + (32 f „ cosbk h= — ~ m — r ^2 1- / dk1 b ) k OO
2
2
= - i l n ( 6 + /3 ) + l n | 6 | + y ^ ^ .
(110)
e
Now let us focus on the integral on the right-hand side. By employing the variable transformation bk = t, this integral becomes 00
00
„ cosbk
/
dk
f , cost
-k~ = J dt~ 111
.
(U1
)
978
A. R.
Baghai-Wadji
Then based on the relation 1 , cost
,
f , cost - 1
,
/ dt
(112)
/ dt = - 7 - lnz (|arg{z} | < 7r) we come to the result oo
eb
„ cosbk
/
dk——
,
, ,,,
r
= - 7 - lne - ln|6| -
cost — 1
/ dt
£
,
.
(113)
0
Using this formula, we obtain for It (as given by the second equation in (110)) the following: eb
1 , ,.o. . „ • > * . , , , , h= - i l n ( 6 2 + /32) + ln|6| - 7 - lne - ln|6| -
f f
costt
1
0 eb
= -iln(62 + / 3 2 ) - 7 - l n e - | d t ^ L l i .
(114)
0
At the limit e —>• 0
+
the integral on the right-hand side vanishes and we obtain I = -hn(b2
+ j32) - 7 -
lim lne.
2
(115)
e-»0+
Using the definition equation for / , Eq.(108), we obtain the following formula which plays a central role in our analysis: T e-P" -/- dk——cosbk J 0
1 = - 7 - lim lne - - l n | 6 2 + (52\.
K
e—•()
J,
(116)
By considering the fact that the constituent integrals of the various Green's functions G^a,b\x,z\x',z') are of this type, and collecting together the singular terms, it is not difficult to obtain the expression for the Green's function G^u'u\x.z\x' ,z') in real-space: G{u'u\x,z\x',z')= »
_ ,,(0
£p
ln{[(* i - £p r
1 •*7T£:p 1
4™#>
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'G
+C
(123)
(p = u, I.) The functions ip^(x, z) and ^l\x, z) have the following meaning:
OO
OO
OO OO
x[jf f dx'dz'p(u\x',z')+
f
— OO — O O OO OO
+ / /
from (121) and collect-
J
dx'dz'pM(x',z')]
— OO — OO
dx'dz'G^u)(x,z\x',z')pM(x',z')
— OO — O O OO
OO
+ / / dx'dz'G^'l\x,
z\x', z')pV\x',
z') + C
(124)
(p = u, I.) We see that if, and only if, the charge conservation law Eq.(122) is valid, that the terms carrying (—7 — lim lne) vanish and the above equations make sense. We further recognize the relevance of the singular terms appearing in a common form in the pairs Ge (x, z\x', z') and Ge {x, z\x', z') and in the pairs G[l'u\x,z\x',z')
G{IJ)(X,Z\X',Z').
and
Therefore, under the assumption that the charge neutrality condition holds, we can omit the singular terms and operate with G^a,b\x,z\x',z'). 3.4. Periodic
Green's
3.4.1. Construction tions in real space
functions
of periodic Green's functions
using nonperiodic
Green's func-
We assume a sequence of infinite number of line charges which are located on the plane z = z', Fig. (3). The line charges are assumed to be positioned in a parallel fashion, at a distance P from each other. The charge magnitude per unit length is taken t o be unity. The surrounding medium is an isotropic dielectric with a dielectric constant e. We refer t o t h e strip defined by - P / 2 < x < P/2 as t h e fundamental cell.
114
Theory and Applications of Green's Functions
981
a periodic array of line charges
an isotropic infinite-space characterized by
8
Fig. 3. An infinite number of line charges.
The Green's function associated with a line charge with co-ordinates (x',z') the fundamental cell has the form
G £ ( x , z | z ' , z ' ) = - - r - ( - 7 - lim \ne) - ~ln[(x 47re e->o+ 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<*-*'>fc c -l*-*'ll*l.
(128)
Using this equation, we easily obtain the Green's function associated with the n t h line in the wavenumber domain, simply by changing x' to x' — nP, that is G(X,Z\X'
-nP,z'\k)
= Ll-e3[x-(x'-nP))ke-\z-z'\\k\, 2iZ ]rC|
(129)
Summation over n, ranging from —oo to oo, yields the wave number domain representation of the periodic Green's function we are looking for: oo V
Gper(x,z\x',z'\k)=
±±-eJ[x-(*'-nP)]ke-\z-z'\\k\_
(13Q)
^—' 2e \k\ n=—oo
' '
Integration over k yields the Green's function in real space oo
Gpel(x,z\x',z')
Jt
Gpel{x,z\x',z'\k)
= / — _ ™ 27T
f _ 9-rr
J
= 7—
V^ Z ^ Or |£.| n~—oo
}_J_e3[x-{x'-nP)]kp-\z-z'\\k\
— — e3(x-x')k
-\z-z'\\k\
y^
_ J 00 2 7 r2e|fc| 0 0
'
^
'
gjnPfc
Q31N
•
\
i
72 = — O O
By using the relation oo
oo
_L J2 e~^Pk = ± £ n=—oo
n=—oo
where Kn is defined by Kn — n2ir/P we obtain
116
S(k-Kn)
(132)
Theory and Applications of Green's Functions
G(x,z\x',z')=
j d k ~ S ' - ' — °° 1
= p
>ke^-*
Nfcl-i J2
' '
983
S k K
( ~ n)
n=—oo
°°
E
G(x,z\x',z'\Kn).
(133)
n=—oo
Remark: Once the Green's function in wavenumber domain, that is G(x,z \x',z'\k), is available, it is an easy task t o construct its periodic counterpart in spatial domain: we essentially have to sample G(x, z\x', z'\k) at discrete points k = K„ and to add the sampled data. Remember that the derivation of G(x, z\x',z'\k) is an algebraic calculation and basically demands the eigenvalues and eigenvectors associated with the partial differential equations under consideration. The above series representation is a convenient form for further manipulations involved in BEM-applications. This results from the facts that the variables x and the z are separated in the expression of Gper(x,z\x', z'), and t h a t the functions involved are exponential in nature. These properties make differentiation and integration easy tasks. We know t h a t these properties also hold true if we consider the Fourier representation of G(x, z\x', z'), associated with a single line charge, i.e. G(x,z\x',z')=
f —G(x,z\x',z'\k).
(134)
-oo27T
The reader may ask what is peculiar about the series representation. The answer can be found in the last tliree chapters in Ref. 9, where we are concerned with selfaction calculations. It is worth mentioning that the above-mentioned properties get lost in the spatial domain: the Green's functions in spatial domain are not generally "variableseparated" , and their functional structures are fairly complicated in the majority of cases. In one particular case the transformation of non-periodic Green's functions into real space becomes very easy. This is the case when we transform into real space the asymptotic of a non-periodic Green's function in wavenumber for k ^> 1. As mentioned above, in BEM applications the self-action calculation is essential. By definition, self-action determination demands that the observation point coincides with the source point (x —• x' and z —» z'). For reasons of simplicity, let us assume that the source point resides at the origin of the co-ordinate system; thus we can assume that x,z -C 1. Then, in order t o calculate the self-action, we have t o investigate the field behavior in the neighborhood of the origin of the co-ordinate system. For this purpose we recall the property that the Fourier transformation projects the vicinity of the origin of the (x, z)-plane into the far zone in the /c-plane. Thus, self-action calculation requires the asymptotic of Green's functions for k S> 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
(135b)
Here, only one single piezoelectric constant, that is eis, couples the Newton equation of motion for 1*2 and the Laplace equation for tp. In the following we will assume a harmonic time dependence according to e~ J "'', define a transversal bulk velocity ct (c2 = Cu/p), and use the Laplace operator A = d2/dx2 + d2/dz2. In this section we have the problem of demonstrating a way for directly constructing infinite domain Green's functions, and of making the reader familiar with k ;» 1—asymptotics of dynamic Green's functions, and their relations t o static solutions. Whenever we are interested in 2D infinite domain Green's functions, it is preferable to use a 2D Fourier transform. This is the way the calculations are carried out here. The definition equation for the infinite domain Green's functions Gu, G21, G\2 and G22 associated with Eqs.(135) is
We use a 2D Fourier representation for the Green's functions in the form OO
Gmn(x,Z\x',z')=
OO ,1U
AU
J J ^^gmn(ki,kzyk^~*Vfc»<*-*'>
-00-00
™
(137)
^
(m,n = 1,2). Transforming (136) into the Fourier domain we obtain
pu? - c44(k2 + k2) -e15(k2 + k2)
- e15(k2 + k2) \fgu en(k2 + k2) ){g21
012 \ _ ( 1 0 \ G22)-\Q I)'
(138)
Using the notation k\\ = kf + k2 and solving for Qmn, we obtain Gu G12 \ G21 G22 )
=
J_ ( sukl det \ e15k\
e i5*f \ / 1 0 poj2 - CiAk\ / I 0 1
(139)
where det stands for the determinant of the far left matrix in (138), and is thus det = (snk2)(puj2 — C^k2), where we have defined the stiffened elastic constant C44 using the equation: C44 = C44 + e\5/e\\. From this we define the transverse velocity ct thus: c2 = C44/P, as the characteristic bulk velocity in a stiffened elastic medium specified by C44 and the mass density p. Note that the term e^k2 represents the Laplace operator in a dielectric specified by e n , while the term pus2 — C44&M corresponds to the Helmholz operator in a stiffened elastic medium characterized by C44 and p. From (139) we then obtain
119
986
A. R.
Baghai-Wadji
&i(*ll)=
^
2
,2
(140a)
:i5/y,2
g21(ftii)=g12(feii)=
fea(*i)=rJgr+
(nob)
(e 15/ )2 2 y ,2-
<140c)
We notice that the functions Qmn are built up from the following two fundamental types: r
i(fcll)=
2 V
a n d
1.2
^ l l ) = THfl
(141)
Before we proceed with the transformation of these functions into real space, we will discuss the asymptotic properties of 0 n , Q21 Q12 a n d 022 for k\\ ~> 1. These asymptotics are of importance because they describe the behavior of the fields in the immediate vicinity of the line sources, and are thus of major importance in self-action calculations. The appearance of only one characteristic velocity, that is Zt, simplifies the calculation, and we obtain
(142a)
lim 0 i i ~ - = - T 2 - - I r - r - j L 2 fc
||»l
O44 «||
O44 £llK||
lim g21 = lim g12 ~ - ^ ^ = -^r-^rs fc||»l
*||-K»
lim fe - -5-4 *ii»i
3.5.2. Transformation
k
C44
(ei5
e n kf,
-/
\\
£ll)2
C44
C44^H«|
^ = £±*
K
(U2h) (142c)
Cuenki
into real space
We have found out that the functions 7i(ky) and 72(A;||) (Eqs.(141)) are building blocks of Green's functions Qn, g21, 012 and 022- Thus, in order to construct these functions in real space, it suffices to calculate T\(x, z\x',z') and T2(x,z\x',z') as their inverse transforms. In Ref. 9 it is shown that we have:
T^xMx',y')
= -3-r±-H™(±Rq) 4 G44
T2(x,y\x',y')
(143a) c
= -—L-bxRH + ^ - { l n 2 - 7 -
lim lnc}
(143b)
Remembering the charge neutrality condition, the singular term on the righthand side can be omitted, and we are left with T2(x,y\x',y')
= --^—lnR]].
120
(144)
Theory and Applications of Green's Functions Having calculated Tx(x, z\x', z') and T2(x, z\x',z'), Green's function matrix in the following form: / Sn{x,y\x',y')
Q12{x,y\x',y')
\ S2i(x,y\x',y')
G22(x,y\x',y') )
987
we can set up the desired
\
(145)
_(
-^^(fi?,,)
-i^H^ifR^
3.6. Infinite-domain Green's functions tion in purely elastic media 3.6.1. form
Construction
of infinite-domain
Preparatory considerations: integrals of the form
i>(xux3)=
Green's functions
with scalar
wave
equa-
using ID Fourier
trans-
In BEM applications we are dealing with surface
/ dx^n's
+
associated
\
dJ
X^.4)
J dx^^^hiA^
(146)
n\ and 713 denote, respectively, unit normal vectors parallel to the x\- and x3axis, and G(a;i,ar3|a^,a^) is defined by
G(X1,XM,4)
=_/
-iL-j
(147)
with a t (&) = (&2 — (u,/ct)2)1^2Because our interest is the investigation of the results for k 3> 1, we have chosen this branch of the square root for at(k). ct is a characteristic transverse velocity and u> is the angular frequency. We consider the spectral component ejk(xi-x[)e-at(k)\x3-x'3\
-2ut(k) in the integral representation of G(xi,x3\x\, x'3). Apparently the symmetry with respect to (xi — x\) and (£3 — x'3) has been broken. However, adding together all the "^-components" we obtain in real space a function which is invariable with respect to an interchange of (xi — x\) and (2:3 — x'3). The validity of this statement is obvious in the present cases: as we know, the integral in (147) is the Fourier representation of the Hankel function HQ
" * " < ! * . > - / 2TT
121
'(f-R\\)
-2at{k)
988
A. R.
Baghai-Wadji
which is a function of R\\ (Euclidian distance between (a;i,a;3) and (x^x^)), therefore symmetric in (xi — x^) and (2:3 — x'3). Thus we can write
G(x1,x3\x,1,x3)
= H^(^Ri)=
[
-I
e>k^-Xl~x'^e~at^k^Xi~x^ -2at(k)
dk 2K dk
and is
e-
Q
'(fc)lxi-3:ileJ*:(x3-x3)
27r
-2at(k)
'
(148)
This fact is important in the further development of our ideas in the BEM applications. 3.7. Near-field asymptotic limits of Green's function derivative with respect to x\ and x3
G and of its
spatial
There are two alternative ways to calculate the near-field asymptotic limits of G(x\,x3\x\,x3) and of its spatial derivatives. • standard procedure: This is the way we performed our calculations in the proceeding section. The method is based on the Green's function G(xi,x3\x[, x'3) in real space. Having constructed G(xi,x3\x'1,x3), we can find the asymptotics of G, dG/dx[ and dG/dx'3 directly at the limit R\\
behavior
of Green's function
G
Our goal in this section is the investigation of the near-field behavior (i?y -C 1) of G{x-i, x3 \x\, s 3 ) by considering the asymptotic limits of the Green's functions in the wave number domain for k 3> 1. For this purpose consider the integral representation of G{xx,2:3la/j,x'3) as given in (147), and perform successively the above mentioned steps. We find that at(k) behaves like |fc| for k » 1, that is at(k) ~ |fc|. Consequently we can write ejk(x1-x'l)
x x
-at(k)\xs-x'3\
eJk(
-2at(h)
l- i)e-\k\\x3-x
-2|fc|
Let us now construct a function S(xi,x3\x'1,x3) in real space by using the function on the right-hand side of the above O (order)-equation, that is
122
Theory and Applications of Green's Functions 989 x
1 +00
S(x1,x3\x[,x'3)
ejk{
1-x'l)
e -|fc||Z3-
x
3l
= --+dk
.
(149)
The imaginary part of this integrand is an odd function of k and thus its contribution t o the integral vanishes. With Tle{ejk{xi-X'l)}
= cosk{xl
- x\) = cos^llx! - x'x\
we then obtain S(x1,x3\x[,x'3)
= - — ^dk
!
ji
.
(150)
We compare this integral with the integral in (147): the variable k appears in the oscillatory part and in the decaying part in Eq.(150) as it stands, and, unlike the case in (147), not as a function of k. Further notice the polynomial appearance of k in the denominator in (150), which is much simpler than k in at(k) as is the case in (147). Recalling the result +00
e-0k
-fdk——cosbk Q
1
lira lne - - l n | 6 2 + /3 2 |
= -7 -
k
e—fO"*"
(151)
2
we can write
5(a;i,a;3|a;' 1 ,4)= ~ ^ ~ { - 7 -
" m lne - - l n [ ( x ! - x[)2 + (x3 - x'3)2]}
This is the near-field representation for the Hankel function HQ (fR^) the near-field representation of our Green's function G(xj,x3\x'1, x'z). Gix^x^x'^x's)
_hxR,. ~
Z7T
+
and thus
( 7 + i i m in€). 2.TT
(153)
e->-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«
frl^Rl)
'l«1
Then (154) gives dG(x1,x3\x'1,x3)
fin
dx\
< 1
a;i - x[
~
*~RJP'
(155)
This technique requires the Green's function G(xi,x3\x'1,x'3) in real space to find the asymptotics of its spatial derivatives for R •< 1. In the following we show an alternative procedure with a promising applicability range. 3.9.2. Wavenumver
domain
analysis
By using Fourier integral representation of G(xi,x3\x'l,x'3) dG(x1,x3\x'1,x'3)_ dx\
we can write
dkeM^-Oe-0*™**-*'^ -2at(k)
d °? ~ 'dx~{J002ir
jke:>kl-x'i~x'^e'~at<'k^X3~x'^ 2Mk)
°° dk = ! * ;
(156)
Again with at(k) ~ \k\ for k ~^> 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<-x'1)e-\k\\x3-x'3\ 2at(k) 2\k\ Denoting this limit by W(a;i,a;3|a;i,a^|fc) we can write Wix^x^x'^x'^
^
°° dk J
—W(x1,x3\x'1,x'3\k)
= J dk-^-reSM-'ih-Wte-^.
124
(157)
Theory and Applications of Green's Functions 991 In contrast to the previous case the imaginary part of ePk^x^~x^> 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
=
v r ^ 2^ y
(173)
-Ax/2
3.12. Semi-infinite domain dyadic Green's tion in piezoelectric media with semi-space shear horizontal waves
functions for wave inhomogeneity:
propaga-
We again focus our attention on the propagation of shear horizontal polarized waves which are specified by the transversal mechanical displacement component 112 and the electrical potential tp. As excitation sources we consider both line charges and line forces. The following problems will be covered: a piezoelectric half-space excited by a line charge and a line force. The sources are embedded in the piezoelectric medium. Our concern will be the determination of the mechanical displacement distribution U2(x, z) in the substrate and the potential distribution
0 0 _C,._2L_,v..2 44557 - pu)
_1 C4_
ei5_/en C4
C44
C44
-ei5Q^s
\
/
\
(
f T23
0 0
«2
/
\D3 J
«2 f
dz
T23
\D3 J
A staightforward calculation shows that the corresponding eigenpairs are:
128
\
(174)
Theory and Applications of Green's Functions (
0
/
\
0
\
1
1 A=
995
(175)
A = - | & | <=>
\k\& e15|fc|
-eisl^l
V -£ii|*l
/
(
\
1
V en|*|
/
/
1
\
ei5/eii
eis/eii
(176)
- A t <=>
Xt<=>
C44A4
—CnXt
0
0
Having the eigenpairs, we can set up the solution ansatz for the field quantities which will enter the interface and boundary conditions /
u2(x,z\k)
I
\
0
\
23
e>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
\
/
V -£u|*|
/
V £n\k\ J
/
\
t
1
W3)(*0
e>«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|
~
G<£\x,z\xT,Zr\k) 1 1 =-ei5 £o + £ n | f c | C
jfc(a;-x ff )
eo+e n
1
1 15 £ l l £0 + £ l l |fc| Q
ejk(x-
-|fc||z-^„
£0 en
1 At
ffcf
e Jfc(a;-x a ) e -(|fc||z|+At|2,
eo_£^ eo+en e n
pjHx-xT)
-(\k\\z\+Xt\zT\)
2«__£n_£Li **|fc|
(178)
eo+en e n
The electric potential response in the lower semi-space (z < 0):
131
(179)
998
A. R.
Baghai-Wadji
G<$(x,z\xa,za\k) ••
= J-^eJ^-'^e-l*!!*-^! 2 e n |*|
•, g p - g n g . . ^ t , -J £2 £15. -*• e o + e n 4 4 | f c | e p + e n e n 7fe(a-z„)
1
2en |*|
44
P2 C 15_
fc| |fe|
ee oo + + ec nn eeitl
1 x
2 e n 2 C 4 4 At
_Yfc(a-x^1c-A,lz-z„ _£fl £lfi. e 0 + £ l l e i l .,-fc(x-x„1_-A,lz+z.,l
1 ^ T ^fc| r + 2
2 e i i C 4 4 At cI
e
£
15
44
^ _ _^a_£n |fc|
eo+en en
1
°
I
E l l 2 £0 + e n | * | CAA& 44
1*1
+_e|^—eo— 1
+ |fc|z t7 |)
|fc|
M*-x„)e-m\z\+xtu„\)
(18Q)
EO+EII e n
= _ _ £ " _ le*<*-^>e-A'l*-^
G^(x,z\xT,zT\k)
2 e i i G 4 4 At
1 / ^ .Ai_ 4-
eis_
ejfc(x-xa)e-(A,|z|
^— &• eo+eu e n
1 44
eo
1 ^44 |fc| 1- e o + £ l l
2 £ n C 4 4 At £ +
-|fc||z+z„l
c 4 4 n?r - - S o - S u5..
A, 44
e „2
i5
e u c J f c ( 3 ; _ X T ) c _ A t |z+z T |
£0_£LL
|fc|
eo+en en
£i£_i2_J e n eo + £11 1*1 £ , , £ . 4 4
W
I
e J*(*-*i-) e -(|fc||z|+A t |z,-|)
Q
8 1
S
S2—& £o+£ll eil
The elastic displacement response in the lower semi-space (z < 0): G^%(x,z\xa,za\k)
= __!^_lei*(*-x.)e-At|z^| 2 e n C 4 4 At .
2
eis^ 1 W4]fc[ -r 2 £ n C 4 4 At £ A. 44
I
e
l
5
£n
£ o
|fc[
£ Q + e i i £ l l c7 - fc(a! - a;
£a_£L eo+en e n
1
L _ _ _ _ e J ^ - ^ ) e - ( A t | z | + |fc||z„|)
eo + C11 1*1 CAA& 44
G^r(x,z\xT,zT\k)
|fc|
^ —
,182)
^
e o + e i i £11
= - - L - U i*(*-xT)c-*.l*-^ Z U 4 4 At
6
J. 1 44ffi + e 0 +£ii^ e . 7 fc(x-x T ) e -A t |z+z T | 2C 4 4 At cA. 2± - g " £ k °44|fc|
eo+eil£ll
132
(183)
Theory and Applications
of Green's Functions
999
a line tliarye
1 ree sp.it. e
X
a piezo-electric substrate with material parameters £
11 ,
e
! 5 , C-44
Fig. 7. A piezoelectric half-space when the exciting line charge with the cartesian co-ordinates a and c is located above the substrate surface.
3.13.2. Line charges reside outside the substrate In this section our concern is the derivation of Green's functions associated with a piezo-electric half-space when the exciting line charge with the cartesian co-ordinates (xa,za) is located above the substrate surface, Figs. (7) and (8). The electric potential response in the upper semi-space (z > 0):
G^\x,z\xa,za\k)
,„ >(T , . , - , „ „ , - „ , . . , . 1
+•
1
1 1 = J_J_ei*<*-*->e-l*H*- 2 -l 2eo(Jfep
.. e n - e n /-» A, 1 eo+eii 44l*= I
_£fl
£li
eo+cii eiLgj'*:(3:-3:^) e -|fc||^+z
(184)
T/ie electric potential response in the lower semi-space (z < 0):
1
G^(x,z\xa,za\k)
C
1
*4]k\
'°44|fc| 1 £lie0
+ £ii\k\
1 eMx-xa)e-(Xt\z\ Q
At
ejfc(x-xa)e-|*:||2-z„|
eo+eiien
+ W\z,\) £fl_£ii
77&e elastic displacement response in the lower semi-space (z < 0):
133
(185)
1000
A. R.
Baghai-Wadji
region 1 a line charge
tree space
1 region 11
c
'k
tiee spate
J
X
!
region III £
1 1 , e i 5 , *~-44
Fig. 8. Subdivision of the above geometry.
1
G^x,z\xa,za\k) =
ei5
1
ejk(x-xa)e-(\t\z\
_ — — _ - _ £O + £II\K\
-j
CAAH
3.13.3. Dyadic Green's functions for two semi-infinite electric substrates supporting stonely waves
+ \k\\za\)
(186)
£fi_£ia
piezo-
Consider two semi-infinite piezo-electric substrates which occupy the regions z > 0 and z < 0, Figs. (9) and (10). Let the semi-space z > 0, the upper region, be characterized by parameters C 4 4 , e" 5 and e"j, and the semi-space z < 0, the lower ^ . . region, be characterized by the parameters C 4 4 , e 1 5 and e n . Consider first a line charge and a line force within the upper half-plane, and then a combination of a line charge and a line force which are both located in the lower half-plane. (The reader may find interesting details regarding the derivation of the associated dyadic Green's functions in Ref. 9.) Here, we merely give two representative examples for the Green's functions.
134
Theory and Applications
of Green's Functions
a line charge + a line force
a piezo-electric substrate with material parameters o ( u ) p (u) C l l , e 1 5 , <-44
X a piez.o-electric substrate with material parameters
eO)
C
ll ,
~«
C
r(1)
15, ^44
Fig. 9. Two piezoelectric semi-spaces.
region I o
C
(u)
ll ,
a line charge + a line force
p (u) e
1 5 , <-44
1f
region II o p (u) r(u) C l l , e l 5 , C44 (u)
(
Z
1
X region III n0) C
ll ,
e
1 5 , L.44
Fig. 10. Subdivision of the above geometry.
135
1001
1002
A. R.
Baghai-Wadji
G^Hx,z\X
i
+
= _l_±e**(*-*.>e-l*ll*-*.l
£
I gn+en C44T^ + G44]^ -
e
ii£ii(£ii+£n)
'i J "
ii
f
iieiiveii+eii)
xe J'fc(a:-a:a) e -|fc||z+z^|
( e 15)
1
(e?s) 2
1 1^4|fc|-g44|fc|+£j] £g(£j;£y)
ik(x-x„)
-A?|z-zJ
o 4 4 | f c | +o 4 4 | f c |
, e « i e {;' (e v; e r i)
xeJk(x-xa)e-\f\z+za\ e
gnei5-£iieis
i5
£
( ?i)
2
£
£
n+ ii
x
pjk(x~xa)p-(\k\\z\
1*1 7 ^ , 7 ^
+ XY\z^\)
(efreJB-e^efr)' 11
£ ll£ll(£ll+£ll) jfe(»-^) -(AJ'|zH-|fe||z e -fc(«-*e ) -(Ayizi | ||z.i e?5 £ y i e ' 1 5 - £ ' e y 5 i eJ g e + fc < 7 |)) \M \l £ lru ( £ ?i) 2 £+4i 1*17^„ A? ^ ; A] - « ^J. rJl p-u >27J« *£. _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<5(a; — x')S(z — z')e"z and of the associated elastic displacement fields which we denote by G-j,3(x,z\x',z') and Gs3(x, z\x', z'). Defining the longitudinal and transversal bulk velocities Q and ct to be c 2 = C\\/p and c 2 = Cu/p, and the quantities X^t = (k2 — ui2/c2 J 1 / 2 we obtain the following eigenpairs:
/
5k K*>
jk
\
2c 2 A;
V -^tJk
<*
-2c 2 A;
V -2
- ft /
.ih.
At
A+o
\
*>
-2k2^ + if
2k2£ - if \
-2c2tjk
)
\
-2c2jk
J
The eigenvector in our diagonalized form is (Ui(k), U3(k), T13(k), T33(k)Y. Using these eigenpairs and satisfying the interface conditions on z = z' we obtain the Green's functions in (189).
G11 (3:3 -x'3\k)
= -
-ai\xx-x'<,\ 2u; 2 a,
, +
a t -at|x3-x^| 2UJ2
Gl3(x3
jk - x'3\k) = - ^ s i g n ^ s - z'sXe-"'!* 3 "^! _ e - « t | * 3 - 4 l )
G33(x3
- x'3\k) = ^ e ~
a
^ - ^
k2 2w2at
a-at\x3-x'3\
(189a) (189b) (189c)
Notice that in these expressions the multiplicative exponential term e^Xl~x^ has been suppressed for the sake of simplicity. The decaying constants in 2-direction, i.e., the eigenvalues aitt are defined as a^t = (k2 — w 2 / ^ ) 1 / 2 , where ci and ct, respectively, denote the longitudinal and transversal bulk wave velocities.
137
1004
A. R.
Baghai-Wadji
3.15. Asymptotic limits of the Green's functions wavenumber domain k ^> 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/<*z,t and exp(—&ij\x 3 — x3\) in the expressions for the Green's functions in Eqs.(189), we obtain their asymptotic limits for k » 1. We start with G\\. Substituting (198), performing the multiplications, neglecting higher-order terms, and rearranging, we obatin
+ 4 ) 1 * 3 - 33|e-lfcN*»-*sl
S i i ( z 3 - x'3\k) = \(-~
-k$+&w\e~Wi"~*-
(199)
Similarly, we obtain S13(x3 - x'a\k) = - h \ - \)(x3 v
4 cf
ct
- x'3)jSign(k)e-^x^x'^,
(200)
and
S33(x3
- x'3\k) = T ( — 2 + -5)\x3 *
c
t
- 4|e-l fc H*3-*3l
H
- 7 ( 4 + 4)^T eHfc||X3 '" 41 4 c?
( 201 )
k
4 \ \
So far we have determined the asymptotics of the Green's functions in the far field in the wavenumber domain (k >• 1). Our next concern is the transformation of the above results into real space. As is obvious from the above results, t h e functional structure of the asymptotics of the Green's functions is much simpler than that of the Green's functions themselves. This property significantly simplifies the transformation into real-space. The following integrals will play an important role in further discussion: 31
140
Theory and Applications
of Green's Functions
1007
oo
/
dke-pkcosqk
=
0
P
dke-pksinqk
=
0
„
(202a)
„
(202b)
o
I oc
/
dk
e
9
pk
cosqk 1 j - ^ - = --HP2 f
2
2\ + qf)
(202c)
(For the conditions under which the third equality is valid please refer to the exhaustive discussion in the previous section.) Based on these integrals, the transformation of S\\(xz — x'3\k), 513(2:3 — x'3\k) and 533(2:3 — x'3\k) into real space is an elementary task. Note that in the process of this transformation the oscillatory exponential term x x ejk( -L- i) m u s t be accounted for. We omit the details, and summarize our results:
J _ JL_ _ 1 (x1 -x'Jjxa X)
^3(x|x') = ^ ( 3 - 4 ) V 1 1
«
=^-3
3.17. Real-space near-field Green's functions
1
+
l.(x3-x'3)2
T~"
- x's) 1 1
^-"nf^ + T^
asymptotics
(203b) +
of the spatial
1 1
^2lnRi derivatives
(203c)
of
In order to find the spatial-domain near-field asymptotics of the spatial derivatives of Green's functions an important property can be utilized: the asymptotic limit-process and the process of derivation are commutative in real space. Thus, simply by calculating the Xi and £3 derivatives of the asymptotic expressions S'ii(x|x / ), 5i3(x|x') and 533(x|x / ), we find the required asymptotics. For the details the reader is referred to Ref. 9. 3.18. Calculation of i?<^Cl—asymptotic limits of the Green's functions based on k^$>l—asymptotic limits of the underlying eigenvalues and eigenvectors The primary objective of this section is t o show t h a t the near-field behavior (R -C 1—asymptotic limits) of the Green's functions in real-space can be obtained from the far-field behavior (k ~^> 1—asymptotic limits) of the underlying eigenquantities in wavenumber domain in the following way: • find the k 3> 1—asymptotic limits of the eigenvalues and eigenvectors. • Construct the relevant field variables by superposing the limits obtained from the first step. This results in the k 2> 1—limits of the field variables. (A certain
141
1008
A. R.
Baghai-Wadji
number of unknown coefficients enter into our calculations, which have to be determined from the boundary conditions.) • Using the k S> 1—limits of the field variables fulfills the problem's boundary conditions by assuming a delta function excitation. This leads to the values of the unknown coefficients from which the k 3> 1—limits of field variables can be calculated. As will be seen in this section, the resulting field variables are the k » 1—asymptotic limits of the Green's functions which we are seeking. This fact will prove the following commutativity property: the process of satisfying the boundary conditions, and the k 3> 1—process are commutable. • Once we have found the k » 1—asymptotic limits of the Green's functions, simply by inspection a Fourier-transformation yields the R -C 1—asymptotic limits of the Green's functions (in the near field in real space). Remember that they are exactly these limits which play an essential role in the self-interaction analysis in BEM-applications*
3.19. Line force source excitation of an unbounded, medium for sagittal polarzied wave excitation: calculation of i?
elastically-isotropic functions
We assume that a mechanical force (per unit mass density) excites an infinitedomain isotropic elastic medium. Furthermore, we assume that the line force oscillates in time according to exp(ju>t) and that it acts at the point (x',zf) in a :r2-co-ordinate system. There is variation in y-direction (d/dy = 0). We denote the components of this force vector in x- and ^-direction by TJ and T%, respectively. The problem is to find the resulting elastic-displacement functions Ui(x, z) and 1/3(2:, z) (which are already our Green's functions, due to the particular J-function excitation) in the ultimate vicinity of the line force. Thus, our goal is the presentation of an alternative way for the determination of the R -C 1—asymptotic limits of the Green's functions. 3.20. Solution
strategy
The analysis follows the exact same line as for the construction of the infinitedomain Green's functions in the previous sections. The only difference is that here we fulfill the boundary conditions with k 3> 1—asymptotic limits of the field variables, instead of with the field variables themselves. For this purpose we proceed as follows: • Subdivide the (a;, z)-infinite domain into two semi-spaces z > z' and z < z' by the plane z = z'. (Note that z' is the z-coordinate of the line force considered.) fc
It is possible to convert a linear system of governing partial differential equations, together with the constitutive equations, into an eigenvalue equation for a reduced number of field variables. 9 The solution of the resulting eigenvalue equation leads to a certain number of eigenvalues and the associated eigenvectors. In terms of equivalent relationships we can say that: 1. partial differential equations (PDEs), together with the constitutive equations, can be equivalently transformed into an eigenvalue equation. 2. The information contained in an eigenvalue equation is equivalent to the information carried by its eigenvalues and eigenvectors.
Therefore we infer that the information contained in a given system of PDE and constitutive equations is uniquely determined by the system-characterizing eigenquantities. The boundary conditions merely dictate which of the eigenquantities and how strong in competetion with each other they are present in our formulae.
142
Theory and Applications of Green's Functions
1009
In the following, we call the half-space z > z' region I, while the semi-space z < z' will be refered to as region II. • Find the k » 1—asymptotic limits of the eigenvalues and the associated eigenvectors. • Use the k ~S> 1—asymptotic limits of the eigenquantities t o construct the field variables in regions I and / / . (A certain number of unknown coefficients enter into our calculations.) • Impose the boundary conditions so as to obtain the unknown coefficients involved. This leads t o the k 3> 1—asymptotic limits of the Green's functions involved. • Use inverse Fourier transformation to obtain the R
3.21. k 3> 1—Asymptotic
limits
2
2
2
of the eigenvalues
and
eigenvectors
1,/2
Having A;,t = (k — w / c t ) , using asymptotic expansions which are valid for k 3> 1, and restricting ourselves t o two leading terms, we can find the k ~S> 1—limits of the above-mentioned eigenvectors.
/
{
\
Sf-i^M
\
-Sf + i ^ M
A+*>
\
«*
-2c2\k\ +
fc?!*!-^
V
^ -2<&*-£ J
\+4>
«•
i^M
I
jk
\ J1
I
I
27[jk\k\
-2c t |fc| +
-2c2jk
J
1 |fc| _
\
2c? 1*1
^
-2<&* - £
/
Sf + ^ M
^
2^[JT\W
-2c2jk 3.22. k 3> 1—Asymptotic
limits
of the
field
variables
The reader should be reminded t h a t the components of the eigenvectors stand for Ui(k), U3(k), Tiz(k)/p and T33(k)/p. We denote the k > 1-asymptotic limits of these field variables by Sx(k), S3 (A), t\z(k)/p and tzz{k)/p, respectively. Then, in region I (z > z'), we can write the following relationships:
143
1010 A. R. Baghai-Wadji S{(x,z\k)
= A(k)ejkxex>
z
+ B(k)ejkxe^
z
(204a)
2
+B
kx jk ~ 2cf jfc|fcj ^-fk-\^Tk^ ^
t
1
^^=A(k)(-2cm + B(k)(-
+
-i
^ V
_
p^)e^A. z 1*1 <
lu4l 1 2c \k\ + _ ^ _ _ ) e ^ A - * 2
(204c)
2 — " \„ik \elkxXeeXl jk J
2 * 3 3 ( M * ) _ A(k)( „ W -2co-2.jk- t \
+ B(k) ( - 2c? j * ) ePkxex*z
(204d)
Terms associated with A; and \f are omitted here, in order t o satisfy the Sommerfeld's radiation condition at infinity {z —• +oo). Analogously, by using terms associated with Xf and A+, we obtain the limits in region II (z < z'). 3.23. Interface fields
conditions
applied
to the k 3> 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<-x-^e-^z-z'^
=S3i(x-x',z-z'\k)
SS3(x-x',z-z?\k)=
\j{-s
L
*
(210)
- ~s)\z -
c
i
(209)
c
A
t
- 1 ( 4 + L) J_]e^(*-*')e-l*llW|
(2U)
Eqs.(209-211) constitute the matrix £_, which is the k » 1—asymptotic limit of the dyadic Green's function Q_. The elements of £ can be transformed into the real domain simply by inspection, to obtain the real-space near-field asymptotics of the Green's functions involved. This transformation, however, has already been performed in a previous section and will not be repeated here. There the elements of <S were derived directly from the Green's functions and not, as was the case here, from the k 2> 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.
148
Theory and Applications of Green's Functions
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
Melting Point (°C)
— — — 1670
— — —
Reference 16 11 17 3 18 10 18
2.3. Crystal growth The growth of gallium orthophosphate has been described in detail by several researchers.81019"24 Gallium orthophosphate crystals have been grown hydrothermally in silica, pyrex, and teflon,24 and tantalum-lined19 autoclaves using sulfuric, phosphoric, and hydrochloric acid solutions. The material features a retrograde solubility in all acids; nutrient material is dissolved in the cold region and crystal growth occurs in the hot 19,20,25 Starting materials have been prepared by solid state region of the autoclave. reaction of Ga 2 0 3 and NH 4 H 2 P0 4 with thermal treatment in H 3 P0 4 ,' reaction of gallium hydroxide with phosphoric acid, and reaction of gallium metal first with nitric then phosphoric acid.19 Early results reported the use of berlinite seeds;19'20 large size, defectfree gallium orthophosphate seeds have been reported recently.23 Variations of hydrothermal growth reported for gallium orthophosphate include the slow heating (SHT),19'22 horizontal composite gradient (HTG),8'19 and vertical reverse temperature gradient (VTG) methods.19 SHT and HTG growth typically occur in phosphoric acid over the temperature range of 150°C to 180°C with a temperature gradient of 1°C-6°C. VTG growth typically occurs in sulfuric acid at a temperature near 230°C with a temperature gradient also of 1°C-6°C. The growth habit varies markedly
154
New Piezoelectric Substrates for SAW Devices
1021
with the solvent used. Growth in hydrochloric acid exhibits a favorable Vx/Vz ratio, but an overall slow growth rate on the order of less than 0.05 mm/dxface. Growth in sulfuric acid occurs at higher temperatures and hence results in reduced OH content of the grown crystals, but the Vx/Vz ratio is always undesirably high. Growth in phosphoric acid exhibits a favorable Vx/Vz ratio and an overall good growth rate, however with elevated OH content due to the reduced growth temperature as compared to growth in sulfuric acid. All solvents yield growth rates Vy less than 0.01 mm/dxface. At present, growth in phosphoric acid is recommended as most suitable for commercial production.23 An alternative technique using direct solubility in orthophosphoric acid above 335°C has also been reported.26 The perfection of as-grown gallium orthophosphate crystals has been studied by IR spectroscopy819'22 and X-ray topography.1019'2127 A solution of 5% HF and 10% HC1 has been used as an etchant to reveal twins.19 Both Dauphine and Brazil twins have been observed. An absorption band at 3167 cm"1 has been associated with the OH content of the as-grown crystal; the absorption coefficient is then calculated as
a:
log V"cm J
^800
(1)
T
V 3167
where d is the sample thickness in centimeters and T represents the infrared transmission at the specified wavenumber.19 The correlation between a and material Q has yet to be established for gallium orthophosphate. Phosphoric acid has been used as an etchant to reveal etch pits.22
2.4. Material properties The form of the elasto-piezo-dielectric constants matrices (augmented by the entropytemperature relations) for symmetry class 32 are shown in Figure l.4 The material constants of primary interest for piezoelectric device design are the dielectric, piezoelectric, and elastic constants of the material, along with their temperature coefficients. Reported values for these constants are listed in Tables 5 and 6, 7, and 8 respectively. The dielectric constant data are separated into two tables to emphasize the difference between the values sT at constant traction and the values s s at constant strain. The dielectric constants are typically measured using a capacitance bridge at a frequency of 1 kHz; these values correspond to sT. However, the commonly used constitutive relations require the values of es. These values are calculated from the measured sT in conjunction with the elastic and piezoelectric constants as
s'W-ex^xfeJ E
1
'
(2)
where e and c represent respectively the piezoelectric and isagric elastic constants matrices and [e]T represents the transpose of the piezoelectric constants matrix. The reported values of the elastic and piezoelectric constants have been obtained from a variety of methods, including Brilloin scattering, pulse-echo, and resonator methods.
155
1022
J. A.
Kosinski Table 5. GaP0 4 Relative Dielectric Permittivities at Constant Traction.
E E
T
ET
ll 7.8 6.5 5.2±7%
E33
10.3 6.7 5.1 ±7%
Frequency
T„(°C)
Reference
10 kHz 1 kHz 1 kHz
r.t.
24 16 28
— r.t.
Table 6. GaP0 4 Relative Dielectric Permittivities at Constant Strain. Es
•fl
Frequency
ToCC)
Reference
1MHz
r.t.
— —
—
24 29 13
E33
10.3 6.7 5.818
7.8 6.3 5.154
25
Table 7. GaP0 4 Piezoelectric Constants (C/m2). qi 0.20 0.209 0.2238
ei4 0.03 0.107 0.1235
To(°C)
Reference
—
16 30 13
26 25
Table 7(a). First Order Temperature Coefficients of GaP0 4 Piezoelectric Constants (ppm/°C). ei4 -170.0
e
ll 49.52
T„(°C)
Reference
25
13
Table 8. GaP0 4 Elastic Constants (GPa).
«fl
66.7 70.7 66.58 ±0.37 67 66.36
-12.9 6.6 21.81 ±0.70 21
66.35 69.09
21.65 21.97
14.56 19.10
<1 E 4
-3.5 -17.8 3.91 ±0.33 -4 6.56 4.2 4.20 4.66
E C c
c
33 103.8 58.3 102.13 ±0.55 102 101.24
44 62.5 41.9 37.66 ±0.27 37 37.80
101.31 106.96
37.80 39.33
156
T„(°C)
Reference
39.8 32.1 22.38 ±0.32
— —
31 29 30
—
—
22.35
27 27 25 25?
4s
22.35 23.56
26°
16 15 15 13 32
New Piezoelectric
Substrates for SAW Devices
1023
Table 8(a). First Order Temperature Coefficients of G a P 0 4 Elastic Constants (ppm/°C).
•fl
<,E2
-37+1 -51.2
— -277 -242
«& —
—
-27.8
520
,E c 33 -118+10 -123.0
487
-54
<£
c
«fc
44
-— -4.4
-31+10 62.7 47.2
-147
-6.3
29
T„(°C)
Reference
—
29 33 33 15 15 14 14 32
26 26 27
-156
-63 -65 -17
-257.2 -143
-180 459.6
-135 -103.0
-62
69
200.4
825
-14.1
28.1
-44
414
-65
2
41
25 25?
Table 8(b). Second Order Temperature Coefficients of G a P 0 4 Elastic Constants (ppb/°C 2 ).
<E1 -47
«£ -123 -198
<E3 -37
<1E4 -196
CE
-51
°44 -40 9.4 -48
c33
350.4
-62
-34
-151
-88
-105
-1550 2000
-84
5
197
-243
-158
-497
320 -64
T„(°C)
Reference
26 26 25 27
33 33 33 15 15 14 3.2
«& -9.5 27.1 34.7
-12 -207
0
25?
Table 8(c). Third Order Temperature Coefficients of G a P 0 4 Elastic Constants (ppt/°C 3 ).
C,E1
85
4
283
<1E3 -121
«£
E C C33
C44
4e
TofC)
Reference
25
89
-13
-7
25?
32
The variations in the reported constants arise from three distinct sources: 1) limitations imposed by the quality of the measured data, 2) limitations in the data analysis, and 3) real variations in the quality and properties of the material samples. Material quality has been related to variations in the reported dielectric constants and all of the temperature coefficients,34 whereas measurement peculiarities have been related to variations in the reported piezoelectric constants. In particular, it has been observed that the piezoelectric constants appear to be underestimated.23 Measured values of BAW velocities for selected orientations are listed in Table 9. The differences between the various sets of measured velocities illustrate the magnitude of experimental errors in the underlying data. It is unclear which set of room temperature constants is "best". Detaint indicates a preference for the constants determined by Palmier for BAW calculations but provides no supporting data.34 In contrast, Briot demonstrates best agreement between calculations using the constants determined by Wallnofer and measured SAW data for Xaxis propagation on rotated Y-cuts.35 Perhaps less disagreement is found with respect to the "best" set of temperature coefficients. Detaint demonstrates best agreement between BAW calculations and measurements over a broad range of orientations using the temperature coefficients from Wallnofer,34 while other temperature coefficients provide
157
1024
J. A.
Kosinski
good results only near the AT-cut. Briot shows good agreement for SAW calculations using the temperature coefficients from Palmier near the AT-cut, but not for the Y-cut.35 Table 9. GaPQ4 BAW Wave Velocities (mis). Direction
Polarization
X X X Y Y Y Z Z (YXl)+45° (YXl)+45° (YXl)+45° (YXl)-45° (YXl)-45° (YXl)-45°
long fast shear slow shear long fast shear slow shear long shear long fast shear slow shear long fast shear slow shear
Ref. 30 T„ = 26°C 4352±3 3280±5 2440±7 4333±3 3228±9 2550±11 5348+2 3242.5±4.5 5051 3118 2766 5195
Ref. 36
—— — 4344 3220 2527 5344 3220
Ref. 15 r.t. 4337 3241 2417 4358 3228 2546 5327 3254
— 2779
Some caution is in order when reading the literature on gallium orthophosphate material constants. First, there are varying standards and sign conventions used by different authors. The magnitudes of the constants may be compared directly, but the sign of c14 is different between the convention used by Bechmann for quartz,37 and the recent IEEE standard.38 Second, the measurement conditions for the constants are not always specified. For example, many of the elastic constants are listed simply as "c" without clarification as to whether they are cE (required for calculations of device properties) or cD (measured from piezoelectrically stiffened BAW modes). Similarly, the dielectric constants are often not clearly presented as either s s or sT but simply as "e". Lastly, the reported values of temperature coefficients are only valid with respect to the specified reference temperature, and must be translated to a common reference temperature for any proper comparison. 2.5. Device properties Gallium orthophosphate displays good device characteristics in terms of large piezoelectric coupling and zero temperature coefficient of frequency orientations for both BAW and SAW. Calculated values for zero temperature coefficient orientations are listed in Table 10. Measured properties for BAW devices are listed in Table 11. Analagous to quartz, a gallium orthophosphate AT-cut exists in the vicinity of (YX1)-12.5° and a BTcut exists in the vicinity of (YXl)-79°.32'33 These data anticipate favorable STW characteristics near (YX1)77.5° with a zero temperature coefficient of frequency and piezoelectric coupling larger than that of STW on quartz. A number of temperature compensated SAW orientations have been calculated by Briot; these are listed in Table 12.35 Measured properties of SAW devices are listed in Table 13. The metallized-
158
New Piezoelectric
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unmetallized Rayleigh wave velocity difference for the ST-cut of gallium orthophosphate at u= 57° is nearly three times that of the ST-cut of quartz.33 Note that most data on GaP0 4 devices, including the data in Tables 10-13, are reported using IEEE 176-1987,38 wherein the AT-cut of quartz is at 9 = -35.25°. The reported values of the Qxf product are listed in Table 14. These are distinctly the lowest of the three candidate materials and are only 1% of that observed for quartz. Improved material growth with reduced OH content may lead to an increased Qxf product, but it is unclear whether the Qxf product of this material will ultimately meet or exceed that of quartz.
Table 10. GaP0 4 Calculated Zero Temperature Coefficient BAW Orientations. Mode shear shear shear shear
Orientation (YX1)-16° (YXl)-73.3° (YX1)-11.8° (YXl)-85°
Coupling 12.4%
— — "low"
Reference 33 33 32 32
Table 11. Measured Properties of Significant GaP0 4 BAW Orientations. Orientation (YX1)-17.4°
Polarization slow shear
(YX1)-16°
slow shear
(YX1)-15°
slow shear
Properties k=12.4% Turnover = 40.5°C Parabola constant = -18.28 ppb/°C2 k=12.4% Turnover = 25°C Parabola constant = -18 ppb/°C2 v=2548.2 m/s k=15.97%
Reference 16
33
20
T ^ = -2.39 ppm/°C (YX1)-12.5°
slow shear
v=2539.4 m/s k=15.98%
(YX1)-11.8°
slow shear
(YX1)-15.77°
slow shear
(YX1)-13.1°
slow shear
v=2540 m/s k=13% Turnover = 25°C Parabola constant = -10 ppb/°C2 v=2540 m/s k=13% Turnover = 25°C Parabola constant = -10 ppb/°C2 v = 2540 m/s k = 16.7%
19
T£> = 3.37 ppm/°C
fr
fr 3)
T^ =-100ppt/°C 3
159
32
34
2
1026 J. A. Kosinski Table 12. GaP04 Calculated Zero Temperature Coefficient SAW Orientations Featuring Zero Power Flow Angle. Orientation (YXwlt)0°,-25°,0° (YXwlt)10°,-35°,0° (YXwlt)20°,-45°,10° (YXwlt)30°,±50°,20° (YXwlt)40°,50°,10° (YXwlt)50°,40°,0° (YXwlt)-10°,-40°,0° (YXwlt)-50°,40°,0°
Velocity (m/s) 2360 2390 2440 2490 2450 2400 2400 2400
k2 (%) 0.30 0.30 0.24 0.20 0.30 0.30 0.30 0.30
Reference 35 35 35 35 35 35 35 35
Table 13. Measured Properties of Significant GaP04 SAW Orientations. Orientation \i. = 5T
Propagation X
Properties "ST-cut" k2 = 0.29%
Reference 33
T^ 2) = -50 ppb/°C2 Z-cut
^=15
(YX1)-15°<9<5°
X
k' = 0.06% Turnover = 75°C Parabola constant = -25 ppb/°C2 temperature compensated
15
35
Table 14. GaP04 Qxf Product. Type BAW BAW BAW
Direction (YX1)-12.5° (YX1)53° (YX1)-13.1°
Polarization slow shear slow shear slow shear
Frequency fa
f. fr
Qxf(xlO n ) 0.0064 0.0488 0.0375
Reference 19 33 2
3. Calcium Gallo-Germanates - Langasite, Langanite, and Langatate 3.1. General comments The calcium gallo-germanate family of materials has been the focus of intense development during the past decade. Materials belonging to this family have been found to possess moderately high piezoelectric coupling jointly with zero temperature coefficient of frequency orientations,39 and extremely low acoustic loss. Of highest interest for piezoelectric devices are langasite (lanthanum silicon gallate La3Ga5Si014), langanite (lanthanum niobium gallate La3Ga5 5Nbo50i4), and langatate (lanthanum tantalum gallate La3Ga5 5Ta0.5Oi4). These materials are a-quartz isotypes in that they belong to symmetry class 32. However, they are not a-quartz homeotypes and exhibit a completely different internal structure.41 It is particularly important to note that none of these three materials are proper single crystals.41 They are, instead, disordered structures wherein one of the cation sites is occupied either by gallium or respectively silicon,
160
New Piezoelectric
Substrates for SAW Devices
1027
niobium or tantalum. On a macroscopic level, the cation distribution in the shared site is 50:50 between the two species. However, on a microscopic level, the distribution from unit cell to unit cell is entirely random, and hence there are significant issues to be resolved in defining homogeneity and uniformity for these materials. The mythology of the calcium gallo-germanate family provides an interesting footnote to the history of the Cold War. These materials went largely unnoticed in the West until the presentation of two papers on langasite devices at the 1991 Frequency Control Symposium.24,27 To some, this represented the piezoelectric version of a Soviet "secret weapon" that had been deliberately hidden and hence triggered a "materials race" as the West rushed to catch up. In reality, however, the development of the calcium gallogermanates had not been hidden,41 and the article "A New Piezoelectric Material, Langasite (La3Ga5SiO!4), with a Zero Temperature Coefficient of the Elastic Vibration Frequency" by Andreev and Dubovik was readily available in the West. 3.2. Crystallography As quartz isotypes, langasite, langanite, and langatate belong to symmetry class 32, characterized by a single three-fold symmetry axis and three equivalent two-fold symmetry axes as illustrated in Fig. 1. However, unlike quartz, the three-fold axis is not a screw axis, resulting is a single form belonging to space group P321.44 The lack of handedness eliminates the possibility of optical (Brazil) twins, but electrical (Duaphine) twins remain possible. The internal structure of the calcium germano-gallates has been studied in detail.41 It should be noted that langasite differs from langanite and langatate with respect to the location of the disordered site. In the former, Ga and Si randomly occupy the 2d site featuring 4-fold oxygen coordination, while in the latter the Ga and Nb or Ta occupy the la site featuring 6-fold oxygen coordination. Measured values of lattice constants of langasite are listed in Table 15. There is a single formula weight per unit cell. Analysis of the data near room temperature lead to values of a=8.1685±0.0054A and c=5.0920±0.0100A at 25°C corresponding to an x-ray density of 5742.4±18.8 kg/m3 which is consistent with the previously reported values as listed in Table 16. Langasite thermal expansion data are listed in Table 17.
Table 15. LajGasSiOw Lattice Constants.
a (A)
c(A)
M°C)
Reference
8.162 8.162+0.005 8.168+0.003 8.16656 8.16787 8.1662 8.1783 8.1684±0.0001 8.1685+0.0054
5.087 5.087±0.005 5.095+0.001 5.0964 5.0970 5.074 5.1014 5.0975±0.0001 5.0920±0.0100
— — — — — —
44 41 41 45 45 46 47 48
25
this work
0 25
1028
J. A.
Kosinski Table 16. La3Ga5SiOi4 Mass Density. Mass Density (kg/mJ) 5754 5743 5737 5751 5764 5764 5742.4118.8
Method x-ray hydrostatic weighing measured x-ray
T„(°C)
— 17
— — —
— —
20 25
x-ray
Reference 41 49 50 51 52 53 this work
Table 17. La3Ga5SiOi4 Thermal Expansion.
a® (ppm/°C) 5.11 5.84 5.15 5.07 5.08 5.046
a\f (ppb/°C2)
a
3 3 (PPm/°C)
3.61 4.03 3.65 3.60 3.49 3.455
3.2
.... .... ....
—
4 f (ppb/°C2)
To(°C)
Reference
— 2.7
20 27
.... .... — —
20 25 20
54 55 56 57 58 53
....
Published data on the thermal stability of langasite are listed in Table 18. All of these compounds are phase stable up to their melting points at or above 1470°C. In consequence, devices fabricated using these materials may be processed or operated at significantly higher temperatures than comparable quartz devices. Table 18. La3Ga5SiOi4 Critical Temperatures. Reference 44 59
Melting Point (°C) 1470+20 1470
Transition Point (°C) None None
3.3. Crystal growth The calcium gallo-germanates of interest here are congruently melting compounds in the ternary oxide systems of La203-Ga203-Si02, La203-Ga203-Nb205, and La 2 0 3 -Ga 2 0 3 Ta 2 0 5 . The phase relations for langasite and langanite have been determined. ° The phase relations for langatate have not been determined, but are expected to be similar to those of langanite.50 The homogeneity range of these compounds is rather narrow. Congruently melting compounds are appropriate for melt growth techniques. To date, only the ,
• ,
I
^
j
x-
,,
e
1
v „ 39,41,44,47-49,51,59-68
x
v
Czochralski method has been reported for growth of langasite. Ine growth of langanite or langatate is largely similar to that of langasite, with small accomodations for the slightly higher melting points.65
162
New Piezoelectric
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Langasite crystals have been grown using several different types of pulling apparatus and feed materials synthesized by different methods. Langasite feed material is formed by solid phase synthesis (SPS) and melting of a mixture of the constituent oxides with an optional addition of metallic gallium,47'49' 59'64 or by a self-distributing high temperature synthesis (SHS) method.63 The purity of the starting materials has generally been reported qualitatively as "highly pure",49'59 however quantitative results typically describe ordinary 3N or 4N purity constituents.48'62"64 The feed mixtures are often provided with an excess of Ga 2 0 3 or La 2 0 3 in order to compensate for the volatility of these components. Seeds for initial growth have been formed by quenching on Pt wires. Z-axis growth has been reported most often,39'47'49'61,62'65'66 although X- and Y-axis growth are possible.39'49'61'62 The growth rate along the Z-axis is more than twice that of the Yaxis, which is in turn larger than that along the X-axis.49 Langasite has been grown using both Pt and Ir crucibles.39'44'47'63"65 Due to their slightly higher melting points, langanite and langatate are grown only in Ir crucibles.65 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. Langasite crystals typically have been grown both in air39'59 or in an inert gas ambient (N2 or Ar) with a small (1% - 4%) partial pressure of oxygen to suppress dissociation of the volatile oxide components, however growth in a pure N 2 ambient has been described recently.65'66 The choice of crucible material and growth ambient, along with the melt stoichiometry require a compromise. Pt crucibles are less susceptible to oxidation than Ir crucibles and hence allow a higher partial pressure of oxygen, but melt at substantially lower temperature leading to Pt inclusions in the as-grown crystals. Ir crucibles are more resistant to absorption into the melt, but are less resistant to oxidation and hence limit the allowable partial pressure of oxygen. The stoichiometry of the melt can be adjusted to obtain a slight excess of Ga 2 0 3 or La 2 0 3 in order to compensate for their volatility, with the required deviation from stoichiometry dependent upon the partial pressure of oxygen and vice-versa. It has been reported that the deviation from stoichiometry should remain small at less than 0.1% in order to avoid growth defects.47'59 Interestingly, in the single detailed analysis of the stoichiometry of an as-grown crystal, the observed deviation from stoichiometry was much larger than 0.1%: La 2 0 3 was 48.6% as-grown vice 48% ideal, Ga 2 0 3 was 48.9% as-grown vice 46.1% ideal, and Si0 2 was 3.1% as-grown vice 5.9% ideal.69 Reported crystal growth rates range from 1 mm/hr to 6 mm/hr, with general agreement on 1 mm/hr to 2 mm/hr as providing quality growth at an acceptable rate.39'41'48'49'60'63'65 The rotation rate depends upon the pulling rate, with rotation rates as high as 50 rpm being used for the highest growth rates as compared to 10 rpm at the lowest rotation rates. The allowable variation in rotation rate is rather small; a deviation of 1 rpm from ideal causes serious growth defects.48 All of the described pulling apparatus use RF heating, in most cases with an after heater. There is general agreement on the need to limit the temperature gradient above the melt in order to avoid cracking during growth, with specific recommendations to limit the gradient to less than 20°C/cm.48'63 All of the described pulling apparatus employ weight measurement as the independent variable for controlling the diameter of the crystal during growth. The most recent reports indicate improved crystal quality as a result of more sophisticated processing of the weight measurement.66 Observed growth defects in langasite crystals include twins, inclusions, block formation, cracking, and growth striations. Hot pyrophosphoric acid has been used as an
163
1030
J. A. Kosinski
etchant to reveal twins.6 Electrical twins occasionally have been observed in langasite, but more commonly are found in langanite and langatate.66 Inclusions have been associated with two sources, namely metallic Pt or Ir dissolved from the crucible into the melt, and other phases precipitated from the melt.47,61 Block formation and low angle growth defects have been associated with large temperature gradients in the growth process and hence may be controlled through suitable choice of growth parameters. Cracking has typically been associated with block formation due to the inherent thermal expansion mismatch between the differing growth regions. In some cases, bubble inclusions and microcracks have been observed along the central core of the crystal.39'47'61 Periodic growth striations have been reported widely.10'47'61'70 The striations are commonly associated with perturbations in the growth conditions such as temperature fluctuations in the melt. The color of langasite crystals ranges from nearly colorless to light yellow through amber to brown.48,49'59,61'62'65'71 The coloration results from an absorption band at 0.36 um to 0.41 urn. The coloration has been associated with Rh dissociated from either the crucible or the thermocouple used to measure the temperature of the melt.62 The coloration can be modified by sweeping and annealing.71 Charge compensation using Al doping eliminates the coloration and improves the acoustic properties.62 The perfection of as-grown langasite crystals has been studied by Raman spectroscopy,72 IR spectroscopy,62 X-ray diffraction,47'61,73 X-ray topography,10,70,74 and optical activity.73,75 The X-ray results are the most significant to date. X-ray diffraction has determined that the periodic growth striations commonly observed correspond to changes in the lattice constants on the order of Ac/c « 10"4 with a period of 35 um to 70 um.48,67,73 This implies a velocity inhomogeneity on the order of 50 ppm. Measured values for velocity homogeneity are substantially larger, on the order of 3000 ppm.69 Xray topography has demonstrated the significant effects of the growth striations on the vibrations of acoustic wave devices.10'74 3.4. Processing Reported values for the hardness of langasite are listed in Table 19. Langasite is significantly softer than quartz. Cutting has been reported using a slurry saw with SiC abrasive followed by lapping using 3 um and 1 um aluminum oxide abrasive.77 Mechanical polishing of langasite is best accomplished using colloidal silica.78 Chemical polishing yielding a 5 nm surface has been demonstrated using a 1:50:150 solution of HC1:HF:H20 at 70°C.78 Langasite is non-hydroscopic.59 Both acids and alkali may be used as solvents.10
Table 19. La3Ga;SiOi4 Hardness. Mohs 6.5-7 5.5 6.6
Vickers (GPa)
—
10
— ——
Reference 39 51 76 79
New Piezoelectric
Substrates for SAW Devices
1031
3.5. Material properties oflangasite Many of the comments concerning the material properties of gallium orthophosphate apply equally to the calcium germano-gallates. The form of the elasto-piezo-dielectric constants matrices (augmented by the entropy-temperature relations) for symmetry class 32 is shown in Fig. 1. Reported values for the linear dielectric, piezoelectric, and elastic constants and their temperature coefficients for langasite are listed in Tables 20 and 21, 22, and 23 respectively. The dielectric constant data are separated into different tables for eT and e s . The reported values of the elastic and piezoelectric constants have been obtained from pulse-echo and resonator method measurements, inclusive of SAW device measurements. Reported values for various nonlinear material constants are presented in Tables 24-27.
Table 20. La3Ga5SiOi4 Relative Dielectric Permittivities at Constant Traction.
E E
T
ll 18.99 20+1 18.86 18.89 18.89 18.97 19.06 19.05 18.96
ET
Frequency
T„(°C)
Reference
1kHz 1kHz
17
—
80 81 55 69 82 57 58 53 this work
E33
49.32 48±2 49.10 49.0 49.3 52.00 51.60 51.81 50.19
27 20 20 20. 25 20 25
— — — 1 kHz
— — —
Table 20(a). First Order Temperature Coefficients of LajGasSiOn Dielectric Permittivities at Constant Traction (ppm/°C). ET
ET E
ll 447 153 153 137 135.5 134.5
T„(°C)
Reference
27 r.t. 20 20 25 20
55 54 69 57 58 53
E
33 -618 -760 -760 -795 -783.4 -787.0
Table 20(b). Second Order Temperature Coefficients of La3Ga5SiO,4 Dielectric Permittivities at Constant Traction (ppb/°C2).
E E
T
ll -2490 82 118.0 118.0
E
33 -66.8 1076 661.2 658.6
T„(°C)
Reference
27 20 25 20
55 57 58 53
165
1032
J. A.
Kosinski Table 21. La3Ga5SiOi4 Relative Dielectric Permittivities at Constant Strain.
«?1 18.87 18.97 18.92 18.92 18.92
£33
49.32 49.24 50.701 50.7 50.19
To(°C)
Reference
17
80 56 83 84 this work
— — 20 25
Table 21(a). First Order Temperature Coefficients of La3Ga5SiOi Dielectric Permittivities at Constant Strain (ppm/°C).
E8
T
ll
150
ET
E33 -772.5
T„(°C)
Reference
20
84
Table 22. La3Ga5SiOi4 Piezoelectric Constants (C/m2). en -0.45 0.45 -0.44 -0.45 -0.44 -0.41 -0.431 -0.44 -0.438 -0.4371 -0.461
en 0.077 -0.07 0.07 0.077 -0.08 0.08 0.108 -0.08 0.104 0.1039 0.257
TofC)
Reference
17
80 40 55 56 83 69 57 84 58 53 this work
— 27
— — 20 20 20 25 20 25
Table 22(a). First Order Temperature Coefficients of La3Ga5SiOu Piezoelectric Constants (ppm/°C). eil -2400 456 587.5 587 464.5 469.8
ei4 -5000 -628 625 625 -700.3 -713.8
T„(°C)
Reference
27 20 20
55 57 84 85 58 53
— 25 20
Table 22(b). Second Order Temperature Coefficients of La3Ga5SiOi4 Piezoelectric Constants (ppb/°C2). eil 1032 -427.5 -428.5
ej4 1480 1600 1594
T0(°C)
Reference
20 25 20
57 58 53
166
New Piezoelectric
Substrates for SAW Devices
1033
Table 23. La,Ga 5 SiOi 4 Elastic Constants (GPa).
<1E1
&
190.9 191 190.2 ±1.3 190.2 188.9 190.2 188.75 190.1 189.3 188.75 189 188.9 189.75
106.3 106.3 106.3 +2.6 106.3 104.6 106.3 104.75 103.0 105.0
— 104 104.2 104.95
«f3
104.2 104.2 91.9 ±3.7 99.39 96.8 91.9 95.89 95.24 95.28 95.89 102 101.5 96.12
& 15.2 15.6 14.7 +1.4 14.70 14.3 14.7 -14.12 15.7 14.93 -14.12 14.4 14.42 13.84
cCE 33 261.9 261.9 262.1 ±0.6 262.1 262.2 261.4 268.3 262.4 261.4 268 268.3 261.68
<4E4 52.4 52.4 53.82 ±0.40 53.82 53.9 53.4 53.5 53.31 53.84 53.5 53.3 53.30 53.07
To(°C)
Reference
17
80 40 54
«& 43.2 43.2 42.0 ±1.3 41.98 42.2 42.35
42 43.5 42.16
42 42.4 42.37 42.40
.... r.t.
20 27
.... .... 20 20. 20 25 20 25
54 55 56 83 69 57 84 58 53 this work
Table 23(a). First Order Temperature Coefficients of La3Ga5SiOi4 Elastic Constants (ppm/°C).
<&
-58.7 -47 -52.7 -53 -66.3 -64.1 -53 -58.4 -57.54
-218 -100 -107.4 -92
— — -85 -98.7 -98.15
E cC
13 -101 -130 -14.4 -88 -75 -75 -100 -82.3 -81.61
<E4 -154 -370 -176 -205 -334.8 -350.9 -310 -307.0 -307.5
E cC
33 -135 -94 -105 -104 -94.1 -91 -94 -105.0 -104.0
<4E4 -79.7
-30 -57.6
-62
4
139 36 12
ToCC)
Reference
27
55 54 69 57 85 85 86 58 53
r.t.
20 20
-62.8 -58.7
-4.7 -13.46 -13.69
-55
—
.... .... —
-58.7 -57.39
-8.77 -7.576
25 20
Table 23(b). Second Order Temperature Coefficients of La3Ga5SiOi4 Elastic Constants (ppb/°C2).
<E1 -113 -55.4 -4.2 -82.7 -82.65
E C C33
«f2
«?3
<1E4
1085 48.1 25 -51.9 -51.83
-117 156.8 -131 -68.1 -68.10
452
65
44 102
-10.4
-62.8 -109 -89.7 -89.61
-133 -111 -103.6 -130.5
870 95.5 95.34
c
4
-1564 -178 -40.7 -119.4 -119.4
T„(°C)
Reference
27 20 20 25 20
55 69 57 58 53
Table 24. La 3 Ga 5 SiO H Nonlinear Dielectric Permittivities at Constant Strain (10"2" F/V). T„(°C)
Reference
20
84
E
lll -0.5
1034
J. A.
Kosinski \ Table 25. La3Ga5SiOi4 Nonlinear Piezoelectric Constants (C/m2).
e
e
l 11 9.3
e
eil4 1.0
l 13 -3.5
122 0.7
e
ei34 6.9
ei24 -4.8
e
144 -1.7
315 -4
To(°C)
Reference
20
84
Table 26. La3Ga5SiOu Electrostriction Constants (10"' N/V2). Hll -26
Hi 4 -43
Hl3 20
Hl2 65
H31 -24
H41 -170
H33 -40
H44
T„(°C)
Reference
-44
20
84
Table 27. La3Ga5SiO,4 Third Order Elastic Constants (GPa). E C c
l 11 -972
E C c
l 12
7
E C c
l 13 -116
E C c
114 -22
E C c
123
9
CE c
124 -28
C cE 133
-721
To(°C)
Reference
20
84
Table 27. (cont.) La^GasSiOu Third Order Elastic Constants (GPa). E cC
134 -41
E cC
144 -40
CE c
155 -198
E cC
222 -965
CE
C333 -1834
E cC
344 -389
CE C444
202
T„(°C)
Reference
20
84
The reported values of the dielectric constants vary as much as 8%, the reported piezoelectric constants vary as much as 37%, and the reported elastic constants vary between 1% and 12%. Larger variations are observed in the reported temperature coefficients. As with gallium orthophosphate, the variations in the reported material constants arise from three distinct sources: 1) limitations imposed by the quality of the measured data, 2) limitations in the data analysis, and 3) real variations in the quality and properties of the material samples. Material quality has been related to variations in the reported dielectric and piezoelectric constants, and all of the temperature coefficients. Variations in reported piezoelectric constants additionally have been related to intrinsic difficulties of measurement.34 An extensive data set exists for the measured values of BAW velocities and their temperature coefficients for selected orientations as listed in Tables 28 and 29. The differences between the various sets of measured velocities illustrate the magnitude of experimental errors in the underlying data used to derive the elastic constants and in some cases the piezoelectric constants. The values for the material constants listed in the tables are as reported in the literature. The same cautions are in order regarding the material constants of the calcium gallo-germanates as for those of gallium orthophosphate. Unlike the case of gallium orthophosphate, most of the data on the calcium gallo-germanates has been reported using the IRE-1949 convention.37 The magnitudes of the constants obtained using the different standards may be compared directly, but the sign of c14 is different between the convention used by Bechmann for quartz and the recent IEEE standard.37'38 Additional confusion arises with respect to the sign of the piezoelectric constants. Second, the measurement conditions for the constants are not always specified. For example, many of
168
New Piezoelectric
Substrates for SAW Devices
1035
the elastic constants are listed simply as "c" without clarification as to whether they are cE (required for calculations of device properties) or cD (measured from piezoelectrically stiffened BAW modes). Similarly, the dielectric constants are not always clearly presented as either e s or sT. Lastly, the reported values of temperature coefficients are only valid with respect to the specified reference temperature, and must be translated to a common reference temperature for any proper comparison.
Table 28. La3Ga5SiOu BAW Wave Velocities (m/s). Orientation X-cut X-cut X-cut Y-cut Y-cut Y-cut Z-cut Z-cut (YXl)+45° (YXl)+45° (YXl)+45° (YXl)-45° (YX!)-45° (YXl)-45°
Polarization long fast shear slow shear long fast shear slow shear long shear long fastshear slow shear long fast shear slow shear
Ref. 83 5752.8 3311.7 2379.9 5754.9 3010.7 2738.8 6749.4 3053.6 5965.7 3283.3 3232.2 6313 3333.6 2431.4
Ref. 80 5784 3344 2344 5792 2971 2767 6753 3021 5998 3323 3141
Ref. 57 5753 3317 2370 5762 3016 2742 6753 3059 5985 3291 3228
Ref. 69 5746 3305 2394 5756 3003 2750 6826 3043 5949 3281
Ref. 55 5748 2380 5754 3015 2745 6750 3061
Ref. 40 5784 3344 2344 5792 2971 2277 6753 3021
3325 2427
Table 28. (cont.) La3Ga5SiOu BAW Wave Velocities (m/s). Orientation
Polarization
Ref. 89
Ref. 88
Ref. 87
X-cut X-cut X-cut Y-cut Y-cut Y-cut
long fast shear slow shear long fast shear slow shear
5748.7 3311.5 2379.6 5755.3 3009.9 2738.0
578x±10 331x±10 234x±10 579x
578x+10
277x±10
277x±10
Z-cut Z-cut (YXl)+45° (YXl)+45° (YXl)+45° (YXl)-45° (YXl)-45° (YXl)-45°
long shear quasi-long quasi-shear pure shear long fast shear slow shear
6746.7 3052.2
675x+10 301x±10
675x±10
169
579x+10
Ref. 2 Table 6
5769 3005 2775
Ref. 80 Table 7 5769
Ref. 82
2760 2764
276x
578x
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J. A.
Kosinski Table 28. (cont.) La3Ga5Si014 BAW Wave Velocities (mis).
Orientation
Polarization
X-cut X-cut X-cut Y-cut Y-cut Y-cut Z-cut Z-cut (YXl)+45° (YXl)+45° (YXl)+45° (YXl)-45° (YXl)-45° (YXl)-45°
long fast shear slow shear long fast shear slow shear long shear quasi-long quasi-shear pure shear long fast shear slow shear
Ref. 80 N. 5784
Ref. 52
Ref. 2 Table 5
Ref. 74 Table 2
Ref. 71
Ref. 51
2764
2762
2753
2754
2767
2760
Table 29(a). First Order Temperature Coefficients of La3Ga5SiOi4 BAW Wave Velocities (ppm/°C). Type
Direction
Polarization
BAW BAW BAW BAW BAW BAW BAW BAW BAW BAW BAW
X X X Y Y Y Z Z (YXl)+45° (YXl)+45° (YXl)+45°
long fast shear slow shear long fast shear slow shear long shear long quasi-shear pure shear
Ref. 69 T() = 20°C -24.9 -53.2 49.8 -31.0 -18.8 6.6 -50.0 -26.5 -35.1 -52.2
Ref. 57 To = 20°C -17 -50 +50 -32 -17 15 -45 -24 -29 -23 -50
Ref. 89 T„ = ??°C -21.3 -43.7 53.5 -29.9 -14.1 13.6 -40.3 -27.8
Table 29(b). Second Order Temperature Coefficients of La3Ga5SiO,4 BAW Wave Velocities (ppb/°C2). Type
Direction
Polarization
BAW BAW BAW BAW BAW BAW BAW BAW BAW BAW BAW
X X X Y Y Y Z Z +45°YZ +45°YZ +45°YZ
long fast shear slow shear long fast shear slow shear long shear long quasi-shear pure shear
Ref. 69 To = 20°C -24.7 -39.1 -131.0 -42.0 -57.5 -89.0 -29.2 -64.9 -31.2 -87.0
Ref. 57 T„ = 20°C -2.3 -110 -57 -64 -39 -40 -50 -50 -17 -67 -29
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3.6. "Best" material constants for langasite As seen from the time progression of the reported values of the material constants and their temperature coefficients, the reported values appear to be converging toward a "best" set of material constants. Briot demonstrates very good agreement between calculations of room temperature behavior using material constants from Sakharov57 and experimental data for SAW propagation on Z-cut langasite, but poor agreement for the temperature behavior.90 The observed temperature behavior was bracketed by that calculated using two sets of temperature constants from Silvestrova.54,69,90 More recent results for Y+45°, Y+50°, and Y+55° rotated Y-cuts confirm the good room temperature agreement using material constants from Sakharov.57 These results also obtain a reasonable agreement between temperature behavior calculated using the earlier temperature coefficients from Silvestrova54 and the measurements.91 Similar good agreement was shown by Inoue between calculations of room temperature behavior using material constants from Sakharov57 and experimental data for SAW propagation on Xcut, Y-cut, and 50° rotated Y-cut langasite,92 also with poor agreement for the temperature behavior. An even better agreement has been shown by Kadota93 and Naumenko86 between room temperature calculations using material constants from Ilyaev55 and experimental data for LSAW and LPSAW propagation on rotated Y-cut langasite. However, the temperature coefficients of both Sakharov57 and Ilyaev55 provided poor agreement for the temperature behavior observed by Kadota; the experimental results for the orientation with a zero temperature coefficient of frequency at room temperature were bracketed by the calculated values.93 An ad hoc set of temperature coefficients developed by Naumenko was shown to provide good agreement for the temperature behavior of the Y-46.5° rotated Y-cut substrate. 86 Constants and temperature coefficients presented by Bungo53'58 demonstrate exceptional agreement between calculations and measurements of all SAW properties for the 50° rotated Y-cut of langasite. It is clear that the room temperature values of the material constants are close to those of Ilyaev,55 Sakharov,57 Sorokin,84 and Bungo,53,58 with the largest uncertainties in the dielectric and piezoelectric constants. The values of the temperature coefficients are likely close to those reported by Bungo.53,58 The aggregate of the measured values of bulk wave velocities listed in Table 28 provides a "best" set of measurement data for BAW propagation. This data, in conjunction with the aggregated dielectric constant and lattice constant data, may be used to determine a "best" set of material constants more accurate than those of any single experiment. Such values are listed in the tables as "this work". The listed values should predict wave velocities to better than 0.5% for most orientations, with the errors highest at orientations near 0=±45°. 3.7. Material constants for langanite and langatate There have been two measurements of the material constants of langanite,94,95 and one complete measurement of the material constants of langatate.96 These results are consolidated in Appendices A and B respectively. Calculations using these values have been generally in good agreement with measurements of devices using these materials.77,97 Langanite and langatate both feature larger piezoelectric coupling than langasite, with langatate the larger of the two. Both of these materials also feature zero
171
1038
J. A. Kosinski
temperature coefficient of frequency orientations, however device development using these materials has only recently begun.77'97 3.8. Langasite device properties Langasite displays good device characteristics in terms of large piezoelectric coupling and zero temperature coefficient of frequency orientations for both BAW and SAW. The calculated zero temperature coefficient BAW orientations are summarized in Table 30. The orientations of greatest interest to date for BAW are near the Y-cut, which features a zero temperature coefficient near room temperature jointly with nearly maximum piezoelectric coupling for the slow thickness shear mode. Measured properties for BAW devices are listed in Table 31. These data anticipate favorable STW characteristics near the Z-cut with a zero temperature coefficient of frequency and piezoelectric coupling larger than that of STW on quartz.
Table 30. La3Ga5SiOi4 Calculated Zero Temperature Coefficient BAW Orientations. Mode long long long long fast shear fast shear fast shear fast shear fast shear fast shear fast shear fast shear slow shear slow shear slow shear slow shear slow shear slow shear slow shear slow shear slow shear slow shear slow shear slow shear slow shear slow shear slow shear
Ti f(r D _- 0u
T«=0
TP=O ft
42Ko
k (%)
Ref.
(YX1)-61.8° (YX1)9.7° (YXwl)30°,±47.0° (YXwl)30°,±28.4°
0 0 1 4 3 0 0 0 0 0 4 5 7 17 17 3 17 3
55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 76 76 76 76 76 76 51 51 98
(YX1)53.7° (YXl)-85.5° (YXI)-45.6° (YX1)9.4° (YX1)36.1° (YXwl)30°,+85.7° (YXwl)30°,+41.0° (YXwl)30°,±24.3° (YXl)-47.6° (YX1)-12.7° (YX1)1.7° (YXwl)30°,±47.9° (YXl)-0.9° (YXwI)30°,±48.3° (YXl)-48° (YX1)-12.5° (YX1)0° (YX1)45° (YX1)0° (YX1)36°
— — .... — .... — 15 15 14.8
(YXl)-84° (YX1)6° (YX1)3°
172
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Table 31. Measured Properties of Significant La3Ga5SiOi4 BAW Orientations. Orientation (YX1)1.5°
Polarization slow shear
Y-cut
slow shear
Properties v = 2760 m/s k=15% "Optimum for temperature-induced frequency excursion over the range -60°C to +85°C" v = 2762 m/s k = 14.8% Tl f(r l ) _- 00
Reference 51
52
T^r2) = -63 ppb/°C2 T ® = -26 pptTC3 Y-cut
slow shear
Y-cut
slow shear
T„ = 25°C Turnover = 68.7°C Parabola constant = -68.9 ppb/°C2 Turnover = 64.7°C Parabola constant = -66.2 ppb/°C2
77 97 from N=3,5,7,9 data
A variety of temperature compensated SAW orientations have been reported recently for langasite,86 these are listed in Table 32. Reported properties of SAW devices are listed in Table 33. The orientations of greatest interest to date for SAW are near the 50° rotated Y-cut. One of the most interesting properties of the calcium gallo-germanates is an anticipated high intrinsic Q based on extremely low measured values of acoustic attenuation as listed in Tables 36 and 37. The reported values of the Qxf product for langasite, langanite, and langatate are listed in Table 38. The observed device Q of langasite has been somewhat less than anticipated. However, langanite and langatate have readily yielded Q values close to or greater than the maximum possible Q for the Xpolarized shear mode in quartz.77'97 It has been suggested that the disordered nature of these crystals limits the achievable material Q, and the development of fully ordered crystals with a calcium gallo-germanate structure has been recommended.67
Table 32. LasGasSiO^ Calculated Zero Temperature Coefficient SAW Orientations. Orientation (90°,90o,25°) (0°,140o,25°) (ZXtw)95°,26° X-prop. (ZXtw)25°,53° X-prop. (0°,154°,24°) (5°,150o,29°) (10°,150°,34°) (5°,145°,29°) (0°,140°,25.5°) (0°,138.5°,26.3°)
Properties Zero TCD Zero TCF Zero TCD Zero TCD TCF = 7 ppm/°C TCF = 3 ppm/°C TCF = -lppm/°C TCF = -2 ppm/°C TCF = 1 ppmTC TCF = 1 ppm/°C
Velocity (m/s)
k' (%)
2765 2798 2829 2793 2747 2743
0.35 0.18 0.183 0.51 0.50 0.47 0.42 0.37 0.34
173
Reference 99 92 84 84 86 86 86 86 86 86
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J. A.
Kosinski Table 33. Measured Properties of Significant La3Ga5SiOi4 SAW Orientations.
Orientation (90°,0 o ,-6°) (90°,90 o ,0°) (90°,90 o ,25°) X-cut (0°,90°,0°) Y-cut (0°,140°,22.5°) 50° rotated Y-cut (0°,142°,24.5°) (0°,143 o ,24°) (0°,140°,24°)
k 2 (%) 0.477 0.355
Reference 59 59 99
0.35
92
2742.1
0.32
92
2754
0.41
93
2756
0.42
93
Velocity (m/s) 2382.8 2287.82
Properties
zero T C D TCD = -42 ppm/°C T„ = 39°C TCF(2) = -52 ppm/°C 2 TCF=1.83ppm/°C PFA=0 TCF=1.63ppm/°C PFA = 0 To=28.8C TCF(2)=-67ppb/°C2
53
Table 36. La3Ga5SiOi4 BAW Acoustic Attenuation. Direction X
Polarization long
X X Y
fs ss long
Y
shear
Z
long
z
shear
Ref. 82 0.26 dB/cm @ 500 MHz
Ref. 40 0.5 dB/cm @ 900 MHz
Ref. 87 0.52±0.04 dB/(us GHz 2 ) @ 1 GHz
0.75+0.05 dB/(us GHz 2 ) @ 1 GHz 1.05±0.05 dB/(ns GHz 2 ) @lGHz
0.44 dB/cm @ 500 MHz
1.9 dB/cm @ 900 MHz
Ref. 59 0.8 B cm"' GHz 2 '"'88.8 MHz
Ref. 88 0.52+0.05
1.31 0.71 1.30
0.8 B cm - ' GHz 2 r'*88.8 MHz
0.85
0.75
1.30+0.04 dB/(us GHz 2 ) @lGHz
1.05
Table 37. La3Ga5SiOi4 SAW Acoustic Attenuation. Orientation Y' X'
59 0.8 B c m - G H z z @ 8 8 . 8 MHz 0.8 B cm"' G H z ' @ 88.8 MHz
Propagation X Y
Table 38. BAW Qxf Product for Selected Calcium Gallo-germanates. Material
Orientation
Polarization
Frequency
La 3 Ga 5 Si0 1 4 La 3 Ga 5 SiOi4 La 3 Ga 5 SiOi4 La 3 Ga 5 SiOi4 La 3 Ga 5 SiOi4 La 3 Ga5SiOi4 La 3 Ga 5 5 Nbo50i4 La3Ga5.5Tao5Oi4
Y-cut Y-cut Y-cut Y-cut Y-cut Y-cut Y-cut Y-cut
slow slow slow slow slow slow slow slow
fa f, fa fr fr
shear shear shear shear shear shear shear shear
174
fr fr fr
Qxf(xl0") 0.983 1.65 1.81 1.34 1.29 2.1 2.15 2.92
Reference 10 2 74 77 97 100 97 77
New Piezoelectric
Substrates for SAW Devices
1041
4. Lithium Compounds - Diomignite 4.1. General comments Lithium niobate, and to a lesser extent, lithium tantalate, have achieved prominence as piezoelectric materials featuring higher piezoelectric coupling than quartz. Both are limited, however, by a lack of temperature compensation. More recently, diomignite (tetragonal Li2B407) has been developed as an alternative material also featuring piezoelectric coupling significantly higher than quartz. This new material exhibits temperature compensated orientations for both bulk and surface waves, and has demonstrated high intrinsic material Q. Tetragonal Li 2 B 4 0 7 is best known as lithium tetraborate. It has alternatively been designated lithium borate, lithium diborate, and dilithium tetraborate, and has been abbreviated as either LBO or DLTB. The mineral name diomignite was approved by the Commission on New Minerals and Mineral Names, International Mineralogical Association in 1987 following identification of natural diomignite crystals in pegmatitie.101102 The name derives from the Greek dios mignen for "divine mix." 4.2. Crystallography Diomignite belongs to space group I4,cd within tetragonal symmetry class 4mm,103104 characterized by a single four-fold symmetry axis and two equivalent mirror planes as illustrated in Fig. 3.4 As with class 3m, the mirror plane symmetry causes all class 4mm materials to be pyroelectric. Diomignite lacks handedness, eliminating the possibility of optical (Brazil) twins. Electrical (Duaphine) twins remain possible. The internal structure of the diomignite has been studied in detail.103105106 Measured values of lattice constants of diomignite are listed in Table 39. There are eight formula weight per unit cell. Analysis of the data near room temperature lead to values of a=9.4746±0.007lA and c=10.2867±0.0192A at 25°C corresponding to an x-ray density of 2433.0±8.1 kg/m3 which is consistent with the previously reported values as listed in Table 40. Diomignite thermal expansion data are listed in Table 41. Sk,
Ek
AT
: : : : T
y,x2
6 3 2 2 1
• Di XfX]
AS
•-• •-•
V:
« . . . • « . . . •
• • • •
Fig.3. Class 4mm symmetry elements and matrices for equilibrium properties.'1
175
• •
J. 15
1042
J. A.
Kosinski Table 39. Li 2 B 4 0 7 Lattice Constants.
a (A)
c(A)
T„(°C)
9.47 9.479±0.003 9.477±0.005 9.475 9.49 9.47 9.4570±0.00025 9.479±0.003 9.479±0.010 9.474±0.006 9.477 9.477 9.47±0.03 9.4795 9.4757±0.0002 9.4794 9.470±0.004 9.464±10.003 9.4746±0.0071
10.26 10.280±0.004 10.286±0.006 10.283 10.30 10.26 10.28590±0.00020 10.290±0.004 10.340±0.007 10.297±0.006 10.286 10.296 10.26±0.08 10.312 10.2850±0.0002 10.2850 10.279±0.005 10.275±10.004 10.2867±0.0192
.... .... .... .... — 20
— 25 25
— — — r.t. 7
.... .... 25
Reference 103 105 107 108 109 110 111 106 112 112 113 114 115 116 117 118 102 102 this work
Table 40. Li2B407 Mass Density. Mass Density (kg/mJ) 244x 243x 244x 2451 245x 2420 2440 2439 2450 2433±1 2430±10 2436±4 2440 2439 2424 2431 2432 2437 2441 2433.0±8.1
Method x-ray x-ray measured measured x-ray measured measured measured measured x-ray x-ray measured measured x-ray x-ray measured x-ray x-ray x-ray
176
To(°C)
— — — — — — — — — 20 18 18
— r.t. r.t. 25 25
— — 25
Reference 103 107 107 104 108 119 119 109 110 120 112 112 121 122 116 123 123 102 102 this work
New Piezoelectric
Substrates for SAW Devices
1043
Table 41. U2B4O7 Thermal Expansion.
aj 1 / (ppm/°C)
a(f (ppb/°C2)
a^j (ppm/°C)
ag) (ppb/°C2)
13 13.0 11.1 12.5 5.1(1) 19.8(8) 12.7(5)
— —
«4 -1.5 -3.74 -2.5 -2.3(1) -1.62(9) -5.3(5)
— —
5.6
— — — —-
T„(°C)
Reference
—
108 124 125 110 120 112 126
20 25 25 20 25 22
20.8
— — — ....
Published data on the thermal stability of diomignite are listed in Table 42. There is no disagreement with respect to the stability of diomignite in terms of both phase transitions and melting point. Diomignite devices may be processed or operated at significantly higher temperatures than comparable quartz devices.
Table 42. Li2B407 Critical Temperatures. Transition Point (°C) None
Melting Point (°C) 917+2
Reference 127
4.3. Crystal growth Diomignite is a congruently melting compound in the binary oxide system of Li 2 0B 2 0 3 , 127 and appears alternatively as a phase within the ternary oxide system Li 2 0-B 2 0 3 Si0 2 The phase relations for the former system have been determined in detail,127128 the latter only approximately.129 Congruently melting compounds are typically synthesized using melt growth techniques; diomignite has been grown using both Czochralski and Bridgman methods. Interestingly, diomignite also has been grown hydrothermally. Feed material for melt growth may be synthesized by reaction of lithium carbonate and boric acid or boron trioxide,107,113,114'130"133 by reaction of lithium hydroxide and boric acid, or by reaction of lithium nitrate and boron trioxide.131 Alternatively, Li 2 B 4 0 7 powders are available in grades ranging from 99.9% to 99.999% purity. 131 ' 108 ' 1H 119'135'136> 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
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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 (<0.5 mm/hr).108113'114119-131132'134-140144-146 The rotation rate is adjusted to achieve a flat or slightly concave solid-liquid interface, and is typically on the order of only a few rpm due to the high viscosity (1.5 poise) of the melt.108'1,3'114'119>,31-14°'144-146 There is general agreement on the need for a large temperature gradient on the order of 200°C/cm at the melt surface,113114133'135 and the need also to limit the temperature gradient to on the order of 10°C/cm in the cooling zone above the meit.108.U4133-135.137 Heat shields and/or after heaters have been used to establish the required thermal profile.119135137138 Many of the described pulling apparatus employ weight measurement as the independent variable for controlling the diameter of the crystal during growth,119135139 with substantial improvements in crystal quality attributed to the controlled growth regime.119 Nearly flawless crystals have been grown using an alternative control scheme based on measuring the instantaneous electrical current and hence torque developed by the motor driving the pulling rod.137 The inherent advantages of the latter method appear unappreciated. For the weight control technique with 0.0lg sensitivity as reported, at 1 mm/hr growth rate there is a distinguishable control readings every three to four seconds. In contrast, the torque control technique provides distinguishable control readings at the sampling rate of the current sensor. Hence, the time constant of the torque control system can be orders of magnitude faster than that of the weight control system. Diomignite crystals have been reported exclusively as clear and colorless. The main growth defects observed in diomignite crystals grown by the Czochralski method are coring and cracking,108'114'119''31'132'134'135'139'142'147 and occasionally inclusions,119'131'139 and voids.108135139 These defects have been related primarily to growth parameters such as pulling rate and rotation rate, but also to temperature fluctuations in the melt.132'147 The incidence of defects may be reduced by the use of higher purity starting materials.119'134137 Hot water'35 or glycerine and water114 have been used as etchants to evaluate crystal quality. Defect densities of 2000/cm2 to 9000/cm2 have been observed, typically screw or edge dislocations.135148 The viability of Czochralski grown diomignite wafers for mass production has been demonstrated via 200 ppm to 500 ppm wafer-towafer consistency and single wafer uniformity of the SAW velocity. Bridgman growth using both vertical and horizontal adaptations has been reported in detail.149"159 Both platinum149157 and graphite152153 vessels have been used for Bridgman growth. The starting materials reported for Bridgman growth have been limited to 99.9% or 99.99% purity,149151'157 with less precise control of stoichiometry and/or excessive water content.149150'151 Crystals have been grown from X-axis, Z-axis, and 45° rotated X-cut seeds.149150157'158 Growth rates below 3 mm/hr are required for high quality growth.149154 Many of the same defects observed in crystals grown by the Czochralski method have been observed in crystals grown by the Bridgman method: cracks and cores,149 dislocations,156 and microcracks.159 A 25% acetic acid etch has been used to reveal defects in Bridgman grown crystals;150'156 the most significant defects have been electrical twins to an extent not reported in Czochralski grown crystals. The quality of the as-grown crystals has been related directly to the purity of the starting materials, and the stability of the power supply and hence temperature profile during crystal growth.156159 The hydrothermal growth of diomignite crystals has been investigated systematically.115129160 Crystals have been grown in autoclaves with teflon liners, at 250°C and 100 bar. Various combinations of mineralizers and starting materials have been considered, with detailed results obtained for the relationships between crystal
178
New Piezoelectric
Substrates for SA W Devices
1045
growth rate, morphology, and growth conditions. Crystals as large as 12 mm x 12 mm x 10 mm have been obtained. The perfection and properties of as-grown diomignite crystals have been studied by ultrasound,112'161"163 Raman spectroscopy,144164 X-ray diffraction,111112117143 luminescence spectrometry,141 optical extinction,132 reflection spectrometry,165166 dilatometry,126 acoustic emission spectroscopy,167 thermal analysis,168 and synchrotron radiation.148 Absorption bands at 3550 nm, 4100 nm, and 4690 nm have been identified with disruptions in crystal stoichiometry and excess water content.141 Several studies observed anomalies at low temperatures possibly associated with phase transitions stimulated by thermal cycling,111'117'126'143'161'162'168 however these results are not consistent and thus far are not clearly explained. The results of Raman spectroscopy studies are in agreement that high temperature phase transitions cannot be ruled out theoretically, but are unlikely to be observed in practice.144'
4.4. Processing Reported values for the hardness of diomignite are listed in Table 43. Diomignite is significantly softer than quartz. Lapping has been reported using a cast iron plate with a SiC slurry. Mechanical polishing has been reported using a polyurethane foam sheet with a colloidal silica slurry.133 Diomignite is insoluble in organic solvents such as acetone or alcohol,108134 and etches slightly in deionized water or alkaline solutions. It is easily soluble in almost all acids,108134'169 In consequence, standard SAW device fabrication techniques using acid etch to pattern a deposited aluminum film are inappropriate for this material. Several alternative fabrication processes have been reported, including two- and three-layer lift-off techniques,135142170 alkaline etch using a single layer resist,170, and ionbeam milling.142. Most recently, the use of standard processing has been proposed in conjunction with the deposition of a thin Si0 2 passivation layer.171
Table 43. U2B4O7 Hardness. Mohs *6 6-7
Vickers (GPa)
— —
Reference 108 154
4.5. Material properties of diomignite The form of the elasto-piezo-dielectric constants matrices (augmented by the entropytemperature relations) for symmetry class 4mm is shown in Fig. 3. Reported values for the linear dielectric, piezoelectric, and elastic constants and their temperature coefficients for diomignite are listed in Tables 44 and 45, 46, and 47 respectively. The dielectric constant data are separated into different tables for eT and s s . The reported values of the elastic and piezoelectric constants have been obtained from pulse-echo and resonator method measurements, inclusive of SAW and BGW device measurements.
179
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Kosinski Table 44. Li 2 B 4 0 7 Relative Dielectric Permittivities at Constant Traction.
E E
T
ll 9.33 9.13 9.32 8.966 9.33 7.81(2) 7.46(2)
....
ET
Frequency
T„(°C)
Reference
10 kHz
109 162 172 123 154 120 120 110
E
33 9.93 10.34 10.1 9.680 10.1 10.25(2) 9.97(2) 10
1 kHz
25 r.t. 25 25
—
—
10 kHz 100 kHz 100 kHz
20 20 25
.... ....
Table 44(a). First Order Temperature Coefficients of Li 2 B 4 0 7 Dielectric Permittivities at Constant Traction (ppm/°C). ET E
ll -4.31
ET
T„(°C)
Reference
25
134
E
33 163
Table 44(b). Second Order Temperature Coefficients of Li 2 B 4 0 7 Dielectric Permittivities at Constant Traction (ppb/°C2).
E E
T
ll 2600
ET
T„(°C)
Reference
25
134
E
33 350
Table 45. Li2B407 Relative Dielectric Permittivities at Constant Strain.
E
ll 8.97±0.05 8.5 8.90 9.04±0.11 8.88 8.91 8.91 8.662
—
43
8.15+0.06 8.2 8.07 8.81+ 0.11 8.46 8.13 8.07 7.893 8.8
180
T„(°C)
Reference
20
104 108 109 173 162 172 154 123 121
— 25 20 r.t. 25
— 25
....
New Piezoelectric
Substrates for SAW Devices
Table 45(a). First Order Temperature Coefficients of Li 2 B 4 0 7 Dielectric Permittivities at Constant Strain (ppm/°C)
£
•fl
33 -33 120 545
-110
-92 97.1
TofC)
Reference
20 20 25
124 173 109
Table 45(b). Second Order Temperature Coefficients of Li 2 B 4 0 7 Dielectric Permittivities at Constant Strain (ppb/°C2).
«?! 2800
T„(°C)
Reference
25
109
E
33
2900
Table 46. Li2B407 Piezoelectric Constants (C/m2).
ei5
e
e33
T„(°C)
Reference
0.36±0.01 0.472 0.278+ 0.005 0.35 0.36 0.39 0.46 0.3918
0.19±0.01 0.290 0.10+0.04 0.38 0.19 0.24 0.23 0.2544
0.89±0.01 0.928 0.77±0.04 0.88 0.87 0.93 0.91 0.8820
20 25 20
—
0.2
0.9
0.492 0.37
— —
—
104 109 173 162 120 172 154 123 174 175 176
31
r.t.
20 25
— 25 25
— —
—
Table 46(a). First Order Temperature Coefficients of Li 2 B 4 0 7 Piezoelectric Constants (ppm/°C). e
15
-1300 -1320 -349 -1050
e31
e
T„(°C)
Reference
-1300
-1000
—
—
-2300 -573
385
20 25 25 25
124 177 173 109
33
-600
1047
1048
J. A.
Kosinski
Table 46(b). Second Order Temperature Coefficients of Li2B407 Piezoelectric Constants (ppb/°C2). «33
T„(°C)
Reference
—
—
-6910
-6500
25 25
177 109
e3i
ei5 5163 -900
Table 47. Li 2 B 4 0 7 Elastic Constants (GPa). C cE3 3
«Fi
•£
«£
126.7+0.1 135 127.1+0.11 135.2 135.27+0.12 126.8 134.2 127.6
0.5+0.2 3.57 6±2 0.8 0.109+0.06 1.0
30+3 33.5 29.4±1.5 31.9 31.8610.10 23.9
8.9
32
128.4 135.823
4.6 -0.285
29.7 31.982
53.910.3 56.8 53.8+8.3 54.9 54.8+0.10 56.2 57.4 54.7 55.5 54.2 54.152
——
— —
— —
— —
c
44 55.010.2 58.5 53.810.4. 55.9 57.3910.08 58.1 47.3 57.5 57.4 57.8 57.072 56.0 57.3
4s 46.010.2 46.7 57.410.5 47.3 47.38+0.06 45.7
T„(°C)
Ref.
20 25 20 r.t. 20 25
104 109 173 162 120 112 121 172 122 154 123 175 176
— 48.2
25 r.t.
47.5 47.680
25
— —
-— —
—
Table 47(a). First Order Temperature Coefficients of Li 2 B 4 0 7 Elastic Constants (ppm/°C).
£
°Fi
-83901100 14000 -67310 1600 3370 20000+10000
-6515 -125 -70.26 -51 -81.1 -80+20
£
cCE 13
E C C33
-485160 350
....
—
—
390 465 5501100
-21 364 400160
10812 -23 -14.53 -22 -18.1 1316
354
4s -22015 -480 -418.6 -200 -272 -480140
ToCC)
Reference
25 20 25 20 25 20
112 124 177 173 109 120
Table 47(b). Second Order Temperature Coefficients of Li2B407 Elastic Constants (ppb/°C2).
«fl
1E2
E C c
-193.1 -440
114939 -17400
—
13
-2300
E C C33 -1800
£
-398.1 500
182
4s 504.4 -450
T„(°C)
Reference
25 25
177 109
New Piezoelectric
Substrates for SAW Devices
1049
The room temperature values of the material constants of diomignite are known with good accuracy and precision.178 The most reliable determination employed an improved resonator method with a total of 67 samples covering seven different orientations.123 All three BAW modes of each sample were measured using both thickness- and lateral-field excitation, typically on the first through ninth harmonics, for a total of 1000 data points used in deriving the elastic and piezoelectric constants.179 The crystals used in the measurements were of exceptional perfection.137 Independent measurements of BGW and SAW devices have confirmed the superior accuracy of this room temperature data set.176'180181 Limited data have been published on the BAW wave velocities of diomignite; these are listed in Table 48.
Table 48. Li2B407 BAW Wave Velocities (m/s). Orientation X-cut X-cut X-cut Y-cut Y-cut Y-cut Z-cut Z-cut (YX1)28.2°
(YX1)45°
(YX1)56°40'
(YXw)45°
(YXwl)45°,45°
(YXwl)45°,56.1°
Polarization long fast shear slow shear long fast shear slow shear long shear long fast shear slow shear long fast shear slow shear long fast shear slow shear long fast shear slow shear long fast shear slow shear long fast shear slow shear
Ref 124
Ref. 121 7415 4896 4410
4407 51xx 4850
Ref. 112 7224 4720 4337 7224 4720 4337 5297 4889
Ref. 178 7458 4922
— 7458 4922
— 5176 4844 7534 4524 4066 7306 4642 3436
— ——
3400±100 6675 5114 4697
6888 4924 7138 5072 3240 6868 4994 3198
1050
J. A.
Kosinski
The temperature coefficients of the diomignite material constants are known with lesser accuracy.140177 While there is a rough agreement on the first order temperature coefficients, there is disagreement concerning even the sign of some of the second order temperature coefficients. In practice, calculations of temperature behavior have proven to be only approximate, and device developments have required significant experimental refinement to accurately identify favorable temperature compensated orientations.108'182'124'183
4.6. Diomignite device properties Diomignite displays good device characteristics in terms of large piezoelectric coupling and zero temperature coefficient of frequency orientations for both BAW and SAW. In both cases, there is an important dependence upon the electrode metallization arising from the extremely high piezoelectric coupling. The calculated zero temperature coefficient BAW orientations are summarized in Table 49. Measured properties for these BAW devices are listed in Table 50. The measured data are in reasonable agreement with respect to the predicted first order temperature behavior, but are in poor agreement with respect to the predicted second order temperature behavior. The measured properties of the TA-cut are particularly interesting.183 The c-mode features nearly zero first and second order temperature coefficients of frequency, and a third order temperature coefficient only half that of quartz. The b-mode features a Qxf product larger than that of AT-cut quartz.
Table 49. Li2B,t07 Calculated Zero Temperature Coefficient BAW Orientations. Mode
Overtone
T(D_o
_U
long long long long long slow slow slow slow slow slow
fund 3rd fund -ira
fund fund
shear nra shear shear fund shear fund shear i nrtl shear fund
'fr (YX1)±38°49' (YX1)±71°52' (YX1)32° (YX1)68°
'fr " ' f r
k (%)
Ref.
.... — — —
124 124 109 109 184 124 124 185 109 109 184
-U
(YXwl)40°,33° (YXl)+48°48' (YX1)±68°17' (YX1)49°14' (YX1)53° (YX1)68°
19.7
25.9
— — (YXwl)19°,56°
26.6
New Piezoelectric
Substrates for SAW Devices
Table 50. Measured Properties of Significant Li 2 B 4 0 7 BAW Orientations. Orientation (YX1)56°40'
Polarization slow shear
(YX1)49°14'
slow shear
(YX1)30°
long
Properties T&) = -300 ppb/°C
2
T( r 2 ) =-280ppb/°C 2
T £ ) =0.7 ppm/°C
Reference 124 185 109
TJ?)=190ppb/°C2 (YXI)30° (YX1)51°
long (fund) slow shear
T^=2.1ppm/°C T ^ = -17ppm/°C
146 109
T< 2) = 320 ppb/°C2 (YX1)51°
slow shear
T ^ = 0 ppm/°C
186
T^ 2 ) =-260ppb/°C 2 (YX1)54°
slow shear
T ^ = 14 ppm/°C
109
T^ 2) = -220 ppb/°C2 (YX1)54° (YX1)56°40' (YX1)68° (YXI)68° (YXwl)40°,33°
slow shear (fund) slow shear (fund) long (3rd OT) slow shear (3rd OT) long
T^ 1 ) =-16.4ppm/°C T^ 2 ) =-300ppb/°C2 T^=0.1ppm/°C T( r 1) =-9.1ppm/°C T ^ = 14 ppm/°C
146 124 146 146 184
T ^ = -220 ppb/°C2 (YXwl)19°,56°
slow shear
T ^ = 2 ppm/°C T (2) =
185
_277 ppb/°C2
184
1051
1052
J. A. Kosinski
Temperature compensated orientations have been calculated for SAW, leaky SAW, and longitudinal leaky SAW in diomignite; these are listed in Table 51. Due to the inaccuracies in the temperature coefficients of the material properties used in the calculations, the temperature compensated orientations observed experimentally are slightly different from those as calculated. Significant experimental results are listed in Table 52. As noted, the observed temperature coefficient of delay depends strongly upon the thickness of the transducer metallization; all results must be read carefully to determine the range of metallization over which the data are valid. Recently, NSPUDT properties have been identified for the (0°,18°,90°) and (0o,51o,90°) orientations.187 Partial NSPUDT behavior also has been demonstrated for the temperature compensated orientation (0°,78o,90°).187
Table 51. Li2B407 Calculated Zero Temperature Coefficient SAW Orientations. Orientation X-Z (90°,90°,90°) X-Z (90°,90°,90°) X-Z (90°,90°,90°) X-Z (90°,90°,90°) 28 X-Z (90+28°,90°,90°) 17 X-Z (90°+17°,90°,90°) 18 X-Z (90o+18°,90°,90o) (90°,90°+8°,90°) (90°,90°,90°+8°) 45° rotated X-Z 45" rotated X-Z X-Z' 12" rotated prop (0°,73°,75°)
k-1 (%) 0.8±0.1 0.97±0.08 1.1
TCD=10ppm/°C
Velocity (m/s) 3510130 3500±50 3500 3562 3436
1.0
Ref. 188 104 108 189 108
SAW
zero TCD
3465
1.0
108
SAW
zero TCD independent of metallization zero TCD zero TCD metallized zero TCD
1.0
182
1.2 1.0
182 182 190 180 190 191
Mode SAW SAW SAW SAW SAW
Properties parabolic TCD TCD = -llppm/°C TCD = -9 ppm/°C
SAW SAW SAW SAW SAW LSAW
—
— metallized zero TCD zero TCD short and open
3433 3401
—
0.92 1.7
o.oi7 mix (0°,75°,75°) (0°,75°,75°) (0°,75°,42°) (0°,75°,75°) (45°,90°,70°)
LSAW LSAW LSAW LSAW LSAW
(45°,90°,70°) (45°,46o,90°)
LSAW longitudinal LSAW longitudinal LSAW longitudinal LSAW
(0°,45°,90°) (0°,47.3°,90°)
zero TCD zero TCD zero TCD zero TCD zero TCD 0.67 dB/X zero TCD TCD = 10 ppm/°C 0.0001 dB/a. 0.0003 dBIX
4220 4225
1.3 1.8
—
—
4235 3780
1.8 3.6
109 192 192 136 193
3680 6980
1.6
192 194
7074
1.34
195
TCD = -3 ppm/C 0.0005 dB/X
6789.5
1.2
196
186
New Piezoelectric
Substrates for SAW Devices
1053
Table 52. Measured Properties of Significant Li 2 B 4 0 7 Surface Wave Orientations. Orientation X-Z (90o,90°,90°)
Mode SAW
X-Z (90°,90°,90°) X-Z (90°,90°,90°)
SAW SAW
X-Z (90°,90°,90°)
SAW
X-Z (90°,90°,90°) 28° X-Z (90°+28°,90°,90°) 25" X-Z (90°£25°, 90°±10°, 90°+8°) (0°,78°,90°) (0°,90°J8°) (45°,90°,90°) 45° rotated X-Z 45° rotated X-Z (90°+45°,90°,90°) (90°,90°±15°,90°) (0°,75°,75°) (0°,75°,75°) (45°,90°,70°) (0°,47.3°,90°)
SAW SAW
Properties T„=19°C TCD(2) = 310ppb/°C2 TDC = 4 ppm/°C Oppm/C TCD(2) = 230 ppb/°C2 parabolic TCD T„ = 55°C TCD = 6.2 ppm/°C T„ = 20°C TCD(2) = 270 ppb/°C2 TCD < ±5 ppm/°C
SAW SAW SAW SAW SAW SAW SAW SAW LSAW LSAW LSAW longitudinal LSAW
Velocity (m/s) 3510+6
k2 (%) 1.2
Ref. 188
3515 3515
1.06 1%
104 104
3512
0.7%
108
3503 3470
— 0.8
189 108
1
197
zero TCD zero TCD zero TCD 3436 3465
—
TCD(2) = 270 ppb/°C2 TCD<±5ppm/°C TCD < +5 ppm/°C TCD = -1.5ppm/°C TCD = 10 ppm/°C
4120 4224 3670 6676
— TCD=10ppm/°C
1 1.0 1 1 1.6 1.7
— 2.8
109 109 109 180 198 199 199 109 192 192 196
Measured values of acoustic attenuation for diomignite are listed in Table 53. The reported values of the Qxf product are listed in Table 54. The observed device Q of diomignite is close to the maximum possible Q for the X-polarized shear mode in quartz.
Table 53. Li2B407 BAW Acoustic Attenuation. Direction X Z
Polarization long long
Attenuation 15.2 dB/cm@300 MHz 13.2 dB/cm@300 MHz
Reference 121 121
Table 54. Qxf Product for Li2B407. Mode BAW BAW BAW SAW
Orientation (YX1)56°40' (YX1)49°14' (YX1)40°,33° 45° rotated X-Z
Polarization slow shear slow shear fast shear SAW
Frequency fr fr fr fr
Qxf (X 10") 0.172 0.136 2.04 0.465
Reference 124 185 183 190
1054
J. A. Kosinski
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Appendix A - Langanite Material Properties
Table Al. La3Ga5.5Nbo.5O14 Critical Temperatures. Melting Point (°C) 1430±20 1470 1450 1510
Transition Point (°C)
— none
— —
Reference 50 A2 67 66
Table A2. La3Ga5.5Nbo.5O14 Lattice Constants. a (A)
c(A)
T„(°C)
8.218±0.005 8.235+0.005
5.122±0.005 5.129+0.002
20
—
Reference 41 Al
Table A3. La3Ga5.5Nbo.5O14 Mass Density. Mass Density (kg/m') 5934 5934.7+13 5903
Method x-ray x-ray measured
Reference 41 this work 50
ToCQ 20 20
—
Table A4. LasGas sNb0 5Oi4 Thermal Expansion. aj 1 / (ppm/°C)
(X33 (ppm/°C)
4.32
3.09
T„(°C)
Reference
20
95
Table A5. La3Ga55Nbo5Oi4 Relative Dielectric Permittivities at Constant Traction. £8
T
11 20.3 20.2
C33 79.3 79.4
Frequency
T„(°C)
Reference
1 kHz 1 kHz
20 20
94 95
199
1066
J. A.
Kosinski Table A5(a). First Order Temperature Coefficients of I^GassNbosOu Dielectric Permittivities at Constant Traction (ppm/°C).
E
E
33 -1500
ll
30
T0(°C)
Reference
20
95
Table A5(b). Second Order Temperature Coefficients of La3Ga5.5Nbo.5O14 Dielectric Permittivities at Constant Traction (ppb/°C2). s
E
ll 180
33 2360
T„(°C)
Reference
20
95
Table A6. La3Ga5.5Nbo.5O14 Relative Dielectric Permittivities at Constant Strain.
*Sn
E33
19.53 19.57
79.3 79.4
T„(°C)
Reference
20 20
this work using 94 this work using 95
Table A7. La3Ga5.5Nbo.5O14 Piezoelectric Constants (C/m2). e14 0.12 0.05
eil -0.467 -0.44
T0(°C)
Reference
20 20
94 95
Table A7(a). First Order Temperature Coefficients of La3Ga5.5Nbo.5O14 Piezoelectric Constants (ppm/°C). eil 750
en 39500
T„(°C)
Reference
20
95
Table A7(b). Second Order Temperature Coefficients of La3Ga5.5Nbo.5O14 Piezoelectric Constants (ppb/°C2). ei4 -83900
eil -4500
ToCC)
Reference
20
95
Table A8. La3Ga5.5Nbo.5Ou Elastic Constants (GPa).
<& 108.6 109.3
°?3
ClE4
102.3 98.2
10.99 13.5
c
C33
259.0 260.5
200
44 48.57 50.4
& 40.0 40.7
T„(°C)
Reference
20 20
94 95
New Piezoelectric
Substrates for SA W Devices
1067
Table A8(a). First Order Temperature Coefficients of La3Ga5.5Nbo.5O14 Elastic Constants (ppm/°C).
'11 -77
«£
4
E C C33
C44
-112
-84
-502
-108
-50
4
-29
To(0C)
Reference
20
95
Table A8(b). Second Order Temperature Coefficients of La3Ga5.5Nbo.5Ou Elastic Constants (ppb/°C2).
<1E1
41
«f2
cc
E
6.4
CE
«£
13 348
C33 -22
-850
c
44 -942
4
885
T„(°C)
Reference
20
95
Appendix B - Langatate Material Properties Table Bl. La3Ga5.5Tao.5O14 Critical Temperatures. Transition Point (°C)
— — .... —
Melting Point (°C) 1470+20 -1500 1470 1510
Reference 50 96 67 66
Table B2. La3Ga5.5Tao.5O14 Lattice Constants. Reference c(A) T„(°C) 20 41 8.225+0.005 5.123±0.005 8.151 5.105 50 —
a (A)
Table B3. La3Ga55Tao5Oi4 Mass Density. Mass Density (kg/mJ) 6164 6150 6167.0±13.5
Method x-ray x-ray x-ray
Reference 41 Bl this work
T„(°C) 20 20 20
Table B4. La3Ga5.5Tao.5O14 Relative Dielectric Permittivities at Constant Traction. E E
T
ll 18.5
ET
E33 60.9
Frequency
T„(°C)
Reference
1 kHz
20
96
Table B5. La3Ga5.5Tao.5O14 Relative Dielectric Permittivities at Constant Strain.
«?1 17.53
E33 60.9
T„(°C)
Reference
20
this work using 96
201
1068
J. A.
Kosinski Table B6. La3Ga5.5Tao 5 0u Piezoelectric Constants (C/m2). e14 0.07
eil -0.54
T„(°C)
Reference
20
96
Table B7. La3Ga5 sTaosOu Elastic Constants (GPa).
«f.
<&
«£
<£
189.4 191.2
108.4 111.22
132.0 108.5
13.7 13.5
c
33 262.9 267.6
c
44 51.25 51.52
4e 40.52 39.99
T„(°C)
Reference
20 20
96 Bl
International Journal of High Speed Electronics and Systems, Vol. 10, No. 4 (2000) 1069-1109 © World Scientific Publishing Company
PSEUDO AND HIGH VELOCITY PSEUDO SAWs MAURICIO PEREIRA DA CUNHA Dept. of Electronic Systems Eng., Escola Politecnica da Universidade de Sao Paulo, Av. Prof. Luciano Gualberto, Trav.3, n3. 158, 05508-900, Sao Paulo, Brazil
This article discusses the characteristics of pseudo surface waves (PSAWs) and high velocity pseudo surface waves (HVPSAWs). The fundamental properties of these waves, the matrix method formulation, the different solution types due to crystal symmetry, early experiments on HVPSAWs, and practical applications of pseudo surface acoustic waves serve as an introduction. The solutions to the pseudo modes are discussed by analyzing the boundary condition function for several orientations. The relation between the radiating partial modes and the sagittal plane bulk slowness reveals new characteristics of the different symmetry types of HVPSAW, and helps classify and understand the pseudo modes. The acoustoelectric Poynting vector is used along different crystal symmetry orientations to reveal and discuss the pseudo SAW characteristics: penetration depth, declination of the power vector, and an estimation of power flow angle. Experimental data and numerical solutions of HVPSAW and PSAW along selected planes and orientations are discussed. This article concludes with a brief analysis of layered structures on different symmetry type substrates and a discussion of the layered pseudo surface wave properties.
1. Introduction This article discusses pseudo surface acoustic waves or pseudo SAWs, also known as "leaky SAWs" or "leaky waves." Particular attention is given to the high velocity pseudo surface acoustic wave, or HVPSAW. The work is divided into four major sections. Section 1 is an introduction, addressing the basic concepts of the pseudo SAWs, the different solution types, early experiments on HVPSAW, and current pseudo SAW applications. Section 2 discusses pseudo SAW fundamentals. The basic equations used in the matrix method formalism are revisited and the solution to the propagation problem focuses on pseudo SAWs. The classification of the orientation symmetries is highlighted as fundamental in understanding pseudo SAW behavior and properties. Section 3 discusses the pseudo SAW solutions. Starting with an analysis of the boundary condition function for different boundary conditions, the section proceeds with a discussion of the Poynting vector. Important pseudo SAW characteristics for practical applications are described, and the concepts of penetration depth, declination of the Poynting vector, and estimated power flow angle are reviewed. The relationship between the radiating partial modes, RPM, responsible for the power radiation into the substrate, and the bulk slowness is addressed. Pseudo SAW solutions in the plane and along specific orientations of practical interest conclude this section. Solutions in layered structures, their importance and characteristics for the pseudo SAWs, are the focus of Section 4. The relevance of the layer thickness and material in practical device applications is discussed, and properties of the pseudo SAW solutions in layered structures are analyzed. Section 5 concludes this work.
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1.1. Basic Concepts The HVPSAW and the pseudo SAW, PSAW, first discussed by Engan et al and by Lim and Farnell, share the same fundamental propagation characteristics.1'2 Both the HVPSAW and the PSAW are pseudo surface waves in the sense diat they are a form of coupled modes that can exist on anisotropic free surfaces and consist of a linear combination of phase matched decaying and radiating partial modes that together satisfy the mechanically free surface boundary conditions. The decaying partial modes, DPM, are the components that decay with depth, whereas the radiating partial modes, RPM, are the components that radiate power into the substrate, resulting in an attenuation of the field amplitudes as the wave propagates. This attenuation is only due to radiation of power and is not dissipation due to viscosity, an effect that is neglected in this article. When the contribution of the radiating term(s) is(are) sufficiently small, the pseudo modes are detectable in standard surface acoustic wave devices. The early experiments and numerical treatments mention that the type of PSAW analyzed has a higher phase velocity than the transverse bulk wave in the direction of propagation.1'2 To account for this property, the authors of references 1 and 2 suggested and verified that a combination of decaying partial modes and a radiating partial mode is required to satisfy thefreesurface boundary condition. 1.2. Different Solution Types Although the pseudo SAWs are sometimes called leaky SAWs in the literature, the present work does not use the term "leaky" which is also used to refer to liquid-solid boundary conditions where energy leaks from the solid into the liquid. The present work deals with radiation of power into the semi-infinite solid. To avoid confusion with the leakage of energy into liquids, a topic not treated in this work, the term pseudo SAW, introduced in Ref. 2, is used for both the PSAW and the HVPSAW. The PSAW is a coupled mode that contains only one radiating partial mode, all others being decaying partial modes. The RPM radiates inside the solid to the slow shear bulk mode in the sagittal plane. Regarding the particle displacement fields, the mechanically free PSAW usually has the shear horizontal component of the particle displacement field as the dominant component. Another characteristic of the mechanically free PSAW is that the value of the phase velocity is usually close to the fast shear bulk wave velocity in the direction of propagation. The HVPSAW has one or two radiating partial modes, depending on the symmetry of the orientation of propagation discussed in Section 2. If only one RPM is present the HVPSAW radiates inside the solid to the fast shear bulk mode in the sagittal plane. If two RPMs are present the HVPSAW radiates inside the solid to the slow quasishear bulk mode and to the fast quasi-shear bulk mode in the sagittal plane. The particle displacement field for the mechanically free HVPSAW usually has the longitudinal component as the dominant term. The designation high velocity PSAW was originally chosen because the value of the phase velocity for this wave lies close to the longitudinal bulk wave velocity in the direction of propagation.3'4 The term SAW is used here exclusively for the true surface acoustic wave solution, which is strictly guided by the surface.
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1.3. Early Experiments on HVPSA W Like the PSAW, the experimental evidence for the HVPSAW goes back over two decades. Some of the early experiments that detected the HVPSAW classified this wave as: "longitudinal bulk wave," "surface skimming bulk wave," "shallow bulk acoustic wave," or "quasi-bulk acoustic surface wave"; mostly due to the proximity of the phase velocity to the planar longitudinal bulk wave velocity in the direction of propagation.5,6'7,8,9,1° The name "surface skimming bulk wave" has also been used for both PSAW and HVPSAW in early publications.611 Since a bulk acoustic wave, BAW, in a piezoelectric material does not satisfy the mechanically free boundary conditions, the mode experimentally identified could not be a BAW . The existence of an independent high-velocity pseudo SAW mode was identified by researchers from Brillouin scattering experiments on cubic materials.12 Jhunjhunwala also discussed the pseudo SAWs, grouping them as first or second order pseudo SAWs, according to the number of radiating partial modes.13 As mentioned in 1.2, the HVPSAW might have one or two radiating partial modes, depending on the symmetry of the orientation of propagation. Therefore a classification according to the number of radiating partial modes is not appropriate. Reference 12 estimates the HVPSAW to have high loss for the gallium arsenide, GaAs, orientations discussed in that paper. This is not the case for other materials and orientations as discussed in this paper and in recent ~ „ W i ; ™ * ; ~ , . „ 4,14,15,16,17,18,19,20
publications. 1.4. Applications of Pseudo SA Ws One major reason for the recent interest in pseudo SAWs lies in their higher phase velocities. The PSAW has phase velocities around 40% higher than the SAW, and the HVPSAW, 100% higher than the SAW. For the most common SAW materials and orientations used, the values of SAW phase velocity is around 3.0 to 3.5 Km/s, which limits the frequency of SAW devices to a few GHz. For a minimum dimension in the photolithographic process, which defines the wavelength, ZMN, a higher value of phase velocity, vpMAX, extends the operating frequency limit of the SAW devices, JMAX, hi accordance to vpMAX = XMIN ./MAX -3'21 Harmonic operation of SAW devices has been used in the past to achieve higher frequency of operation. This approach presents difficulties in designing high performance devices, such as: less control in the desired device response, higher losses, higher spurious levels, and greater sensitivity to surface defects. These problems are avoided with the proper orientation selection of a PSAW or HVPSAW device. Several recent electronic systems require higher operating frequencies, high performance, and low loss devices. Examples of such . systems are: wireless communication equipment, including cellular phones, mobile phones, local area networks, cordless phones, pagers, and global positioning systems (GPS); security systems; and controlling and remote sensing systems.22 The PSAW and the shear horizontal (SH) modes, with vp about 40% higher than the SAW, have been used in recent high frequency applications, as a means of extending the operating frequency of SAW devices.23'24 The HVPSAW devices and applications appeared recently in GPS and
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communication systems, further extending the operating frequency of SAW devices.14'15'17 An interesting characteristic of the low propagation loss pseudo modes for high frequency devices is that the pseudo SAW orientations have a higher fraction of the power carried deeper inside the material, as compared to the SAW mode. This characteristic, named penetration depth in Ref. 3, is addressed in this work. A larger penetration depth means that the wave power is less confined to the surface. This is suitable for high frequency devices, since the pseudo waves, PSAW and HVPSAW, are less sensitive to surface defects than the SAWs. The smaller sensitivity to surface defects translates into lower losses and higher performance devices, provided the selected pseudo SAW orientation has low propagation loss and that damping is negligible. Still considering high frequency low loss devices, one should mention the potential pseudo SAW applications of metallic layered structure;25'26 the ZnO/diamond/Si, where values of vp above 14 Km/s are found for the HVPSAW;27'28 and the ZnO/SiC, vp HVPSAW^I 1 Km/s.29 The film thickness is a critical parameter to achieve low propagation loss for the pseudo SAWs. Materials like diamond and SiC have high stiffness constants, uius higher phase velocity modes, when compared to commonly used SAW orientations on LiNb03, LiTa03, and quartz. Finally, the pseudo SAWs have application for liquid sensors, due to their particle displacement polarizations. The SAW wave has a large vertical component of particle displacement field, and is generally strongly attenuated by a fluid on the device's surface. The PSAW orientations which are interesting for liquid sensor applications have a large shear horizontal particle displacement, which is not severely attenuated in liquid environments.30 The HVPSAW, with high longitudinal particle displacement, parallel to the surface, is potentially interesting for liquid sensor applications, with low attenuation expected. 2. Fundamentals of Pseudo SAWS This section discusses the fundamental properties of the pseudo SAW modes. Subsection 2.1 introduces the orientation classification that is important in the wave classifications and in finding the acoustic solutions. Sub-section 2.2 reviews the basic equations used in the matrix method. The types of HVPSAW solutions based on the orientation classification are discussed in Sub-section 2.3. The boundary function characteristics, the complex slowness, and the solution approach are discussed in Subsections 2.4, 2.5, and 2.6, respectively. 2.1. Symmetry Classification: Uncoupling ofModes Figure 1 shows the type of structure considered in the present work and the coordinate system adopted. The substrate is considered semi infinite, and may have a mechanically free surface, electrically shorted or open, or yet multiple layers on top of its surface. Although only one layer is shown in Fig. 1, the matrix method analysis adopted in this work can handle multiple layers.31 In order to refer to orientations, the commonly used Euler angles are adopted.32 The material constants used are taken from multiple
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z
hX
|
y /
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propagation ^
layer
substrate
Fig. 1. Structure considered and coordinate system adopted.
An arbitrary orientation of propagation, x axis in Fig. 1, can be classified according to the symmetry type specified by the crystal and the selected orientation.38 This also determines the number of roots, or partial modes, also called partial waves, involved in a surface acoustic wave problem. The classification is fundamental to properly sort out the several types of acoustic waves, thus allowing a better comprehension of the PS AW and the HVPSAW solutions. Table 1 presents the substrate orientation classification, including the associated wave types and the conditions on the elastic stiffness constants (C;J) and the piezoelectric constants (e^) with respect to a propagation direction.3 Recalling that the layer material considered in Fig. 1 may also assume any of the symmetry types enumerated, in the case of a layered problem the type of the solution will be that of the lowest symmetry possible for the problem. Symmetry type 1, or Sym 1, describes the general acoustic wave case of lowest crystal symmetry. In this situation the material is piezoelectric, and the wave particle motion has components in all three directions u x , u y , and u z . A SAW propagating in this orientation is named Generalized SAW (GSAW), and this orientation is called a GSAW direction. Symmetry type 2, or Sym 2, represents non-piezoelectric materials, where the GSAW problem must be solved. Symmetry type 3, or Sym 3, portrays the situation where the sagittal components decouple from the shear horizontal component. The stiffened sagittal SAW, also referred to as Rayleigh wave, is excited for a mechanically free surface. A pure mechanical shear horizontal wave also exists in a layered situation. An example of this propagation symmetry is the well-known YZ-LiNb03 orientation. Symmetry type 4, or Sym 4, refers to another uncoupled situation, in which the stiffened shear horizontal wave and a pure mechanical sagittal SAW wave, also referred to as Rayleigh wave, are present. This is the case for the 36.3°-Y rotated AT quartz (Euler angles: [0° -54.7° 90°]);12'13 the ST quartz (Euler angles: [0° 132.75° 90°]);13 the Bleustein-Gulyaev wave (CdS Euler angles: [0° 90° 0°]).39 Finally, symmetry type 5 refers to the pure mechanical acoustic waves in non-piezoelectric substrates, where the sagittal-plane displacement components uncouple from the transverse component. Before relating the PSAW and the HVPSAW to the symmetry classification of Table 1, a review of the matrix method equations is carried out next.
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M. P. da Cunha Table 1. Substrate crystal symmetry types and wave types relative to Fig. 1: c- are the elastic stiffness constants and e- are the piezoelectric constants for a propagation direction.
SYMMETRY
WAVE TYPES
Type 1 (piezoelectric substrate) Type 2 (non-piezoel. substrate)
lowest symmetry; general case.
Type 3 (piezoelectric substrate)
stiffened sagittal Rayleigh wave and pure mechanical shear horizontal wave stiffened shear horizontal wave and pure mechanical Rayleigh wave
Type 4 (piezoelectric substrate)
CONDITIONS
general elastic anisotropy (but non-piezoel.)
eij
=o
c
14 = c 16 = c 34 = c 36 = = c 45 = c 56 = ° e 14 = e 16 = e 34 = e 36 = ° c
14 = c 16 = c 34 = c 36 = = c 45 = c 56 = ° e i i = e , 3 = e15 = e31 = = e 33 = e 35 = °
Type 5 (non-piezoel. substrate)
pure mechanical acoustic waves
eij c
14
= c
=o
= c
16 34 = c 3 6 = c 45 = c 56 = °
2.2. Basic Equations In this sub-section the major statements in the matrix method formalism applied to SAW propagation problems are summarized.40,41 A comprehensive treatment by Eric L. Adler appears as a companion article. One of the key advantages of the method is that it reduces the acoustoelectric equations to a set of first-order vector-matrix ordinary differential equations of at most size 8, 6 for non-piezoelectrics, in the variables that must be continuous across interfaces. In the layered case, one such matrix exists for each layer, and die interfacial boundary conditions are automatically satisfied by a matrix multiplication that preserves the dimensionality of the problem. The coordinate system is shown in Fig. 1. Although the method can handle any number of layers, for simplicity only one layer is shown in this figure. The z axis is normal to the layer interfaces. The assumed solution form, for sinusoidal travelling waves along x at a radian frequency co radians per second, is f(z) exp U(
kx x)J = f(z) exp [ja>(t-—)]
(1)
with vp and kx=o^vr representing the phase velocity and propagation constant along x respectively. The vector of normal stresses, Tz, and the particle velocity vector, v, are respectively represented (components in their r.m.s. form) by
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(2)
T2 =
(3) These are the six mechanical variables, which must be continuous across interfaces. For piezoelectrics the time derivative of the electric potential (jcofy and the normal electric displacement, Dz, are chosen as the two electric variables which must be continuous. These 6 or 8 variables are a sufficient independent set and all other variables can be expressed in terms of them, using the set of dynamic and constitutive equations of a material. For problems unbounded normal to the z plane with the propagation direction along x in the xy plane, the problem is two-dimensional, there are no variations in the y direction, hence d/dy=0 and d/dx=-ja/vp. The vector r is defined as the six-component vector for nonpiezoelectric materials (4) For piezoelectric materials the corresponding eight-component vector is defined
\Tz ] Dz V
T„
(5)
L^vJ
_yco
with T„=[ Tz Dzf and rv=[v jaxfif, and the superscript "t" indicating transpose; r„ and rv are 4x1 vectors. The differential equation at a> radians/second describing transverse, or z, variations in the sinusoidal steady-state for the material is — = y© AX (6) dz where the system matrix A is a function of the phase velocity vp and of material constants rotated to the coordinate system of Fig. 1. For multilayer problems, (6) applies in each material and r must be continuous at each interface. The matrix A is T'3 X
G-
- r " xr3')
A=
(7) XT
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where the submatrices X and .Tin the case of non-piezoelectric materials are 3x3 containing, respectively, compliance and stiffness constants referred to the coordinate system of Fig. 1; and the submatrix G0=pl3x3, with p the mass density of the material and I3x3 the 3x3 identity matrix. For piezoelectric materials the submatrices X and /"are 4x4 and contain compliance, stiffness, piezoelectric, and permittivity constants; G0 is also 4x4 having zeros in the fourth row and column. The submatrices X, r, and G0 are given by c
m
c
c
21Vc
c
212k
c
2l3k
e
c
31t
c
3l2k
c
3/3t
e
e
e
e
lVc
c
V2k
l2k
V3k
l3k
e
kll k21
(8)
k3l
~elk
X=(T33 rl
(9)
p 0 0 0 0 p 0 0 G = 0 0 p 0 0 0 0 0
(10)
The solution to (6) is 1(z + dh) = exp(j(x>Adh ) X(z) = U exp(\dh )V'x(z)
= ^1X(z)
(11)
with &r=exp(jwAdh) the material state transition matrix that maps x across a distance dh in the z direction within the same material, U the 8x8 eigenvector matrix of ihejaA material matrix, the columns of {/are the eigenvectors ofjcaA, A is the diagonal eigenvalue matrix of jcaA, and r =U'T is the normal mode or uncoupled state vector. For SAW problems, and considering that the half-space occupies z<0, all fields must vanish as z goes to negative infinity. For the substrate, since only four of the eigenvalues decay with depth, it follows that at a semi-infinite substrate surface the state vector is T(0) = USp Tv(Q)
(12)
where USp is the 8x4 submatrix corresponding to decaying modes of the substrate eigenvector matrix, Us; and TJJ(0) is the corresponding 4x1 half of the normal state vector, with the "u" index indicating upper half.
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2.3. The Boundary Condition Function (BCF) This sub-section refers to symmetry Type 1 problems, since this is the most general type. The discussion of the other symmetry types is deferred to Sub-section 2.6, when the pseudo SAW modes are classified. The boundary conditions require that at mechanically free surfaces Tz vanishes, and for piezoelectric materials with short-circuited surface <j)=0 at the top surface "0 0 0 = « U
V°)
= &TOTAL USp 1v(0)
(13a)
v 0 with &roTAL obtained by multiplying the layer transmission matrices. A matrix QT is defined from {0TOTAL UsP) by interchanging the fourth and eighth rows in (13a). Defining QTu by selecting the first four rows of QT 0 0 0 0
= QTuTu(0)
(13b)
TOP
This equation has non-trivial solutions if and only if det[QTu]
=0
(14)
which is the boundary conditions (BC) determinantal equation for the multilayer on halfspace problem. For the mechanically free half-space SAW problem, the boundary determinant equation is directly obtained from (12) using the boundary conditions as in (13), and defining the matrix QTS from USp in (12) by interchanging the fourth and eighth rows of USp. Defining QTSu as the first four rows of QTS, det[(grs„]=0 is obtained. For electrically open-circuited surfaces the electric fields must satisfy Laplace's equation in the free space. The solution for D^free space) is given by s0coftvv and by subtracting Dz(free space) from the fourth element in rmP=[0 0 0DZ vjcofi]' for the free surface, an equation equivalent to (13) is written. The resultant numerical procedure consists in searching for vp such that the boundary conditions, expressed by (14), are satisfied. The four mechanically free boundary conditions statements are taken from (13). For a GSAW on symmetry Type 1 the four partial modes that decay with depth, are selected out of the eight partial modes in the substrate. The resulting boundary condition statement is expressed by (14), and is referred to diroughout this work as the Boundary Condition Function (BCF). There are other ways of expressing the boundary condition statement (14), and the effective permittivity is one of these alternative ways.31
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All the other variables are expressed in terms of the six, non-piezoelectrics, equation (4), or eight, piezoelectrics, equation (5), variables chosen. For the fields in the direction of propagation, x, the vector TP is defined and its relation with x given by
T13 X
r"AT J ' - r-//
•pT
(15)
For the fields in they direction, normal to the sagittal plane, the vector xh is defined and its relation with T given by xy
T yy
Y23 X
T23xr31-r»
ht1
(16)
T
y*
[Dy\ where matrices p and h t , defined in (15) and (16) respectively, are 4x8 for piezoelectric materials and are functions of material constants and phase velocity only. For nonpiezoelectric materials, T P =T X =[ T^ Txy Txz ]' and T i ,=T y =[T yx T yy T yz ] t , and p and h, are 3x6.
2.4. Complex Slowness One might ask what happens to the boundary condition function, BCF, in the vicinity of the expected PSAW and HVPSAW velocities if only decaying partial modes are considered. For the PSAW case, selecting only the decaying modes, a minimum in the BCF is often observed near the expected PSAW velocity.2 Sometimes a minimum is also observed near the expected HVPSAW velocity. In Fig. 2 the BCF for the real slowness is expressed as the effective permittivity, and one can notice the quartz AT GSAW solution, and two minima near die PSAW and HVPSAW velocities, respectively. These minima, however, cannot be decreased to zero by adjusting the values of a real vp parameter. If one of the four partial modes in the substrate is a radiating partial mode, i.e., an eigenvalue of jcoA that grows with depth, a complex slowness must be used. The complex slowness is written as y= l/vp-j a/co, in exp(-ja>pc) = exp{-x (a+jkx)}, with kx=cofvv, and a, the attenuation constant. By considering a complex slowness, the BCF determinant can be decreased to zero, in the sense that further iterations result in smaller values of the determinant, until the computer accuracy is reached. The linear system of equations, (6), now has complex coefficients since the slowness is complex. The matrix A becomes complex when l/vp is replace by y. The fields of the resulting solution attenuate in the x direction as this mode propagates, and the solution is a PSAW in a symmetry Type 1 situation. If two radiating partial modes are used in the substrate for a symmetry Type 1 problem, the solution is a HVPSAW.
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^
8.
10 5
,*
/ '
* t>
1079
0
£ o
3.5
4
4.5 5 Phase velocity [Km/s]
5.5
I 10 I 5 .1 0
0
' •
1 "5 o c
o>-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.
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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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-0.1
Attenuation [Np/Km]
2
Phase Velocity [Km/s] (a)
35 a) 30 •D
125 20
o
en
|
in
Boundary Function
§
-10 -IC
3
4
5
6
7 8 9 Phase Velocity [Km/s]
10
11
(b) Fig. 3. Plot of the HVPSAW BCF magnitude as a function of vp and a along quartz ST-25° (Euler angles: [0° 132.75° 25°]) for the mechanically free shorted surface: (a) spatial view; (b) phase velocity axis view.
216
Pseudo and High Velocity Pseudo
SAWs
•a
1 30 D)
•g 20 c o 10
0-I ^>-10 •g -20 O CO -30:
0.1
0.05
10 -0.05
Attenuation [Np/Km]
-0.1
2
Phase Velocity [Km/s] (a)
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.
217
1084
M. P. da Cunha
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.
218
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
219
1086
M. P. da Cunha
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
Pseudo and High Velocity Pseudo SAWs 700 CM
E
600
8 500 a.
£400
I
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-80
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-80
-60 -40 -20 normalized depth z/wavelength
0
(b) Fig. 7. Plots of the Poynting vector [TW/m2] in the direction of propagation as a function of the normalized depth for the mechanically free shorted surface symmetry Type 1 quartz AT (Euler angles: [0° -54.7° 0°]): (a) PSAW; (b) HVPSAW.
221
1087
1088
M. P. da Cunha 700 CN
E;
600
S500 Q_
8 400
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8 | 200 100
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-80
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20
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£
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-60 -40 -20 normalized depth z/wavelength (b)
Fig. 8. Plots of the Poynting vector [TW/m2] in the direction of propagation as a function of the normalized depth for an aluminum layered surface (h/ ^SAW =1%) symmetry Type 1 quartz AT (Euler angles: [0° -54.7° 0°]): (a) PSAW; (b) HVPSAW.
222
Pseudo and High Velocity Pseudo SAWs
1089
16 14 ri.12 S Q.
° 10 6 jD
"5 8
E 6 % S 4 o a. -
1
0
-
8 6 4 2 normalized depth z/wavelength
0
Fig. 9. Plots of the Poynting vector [TW/m2] in the direction of propagation as a junction of the normalized depth for the mechanically free shorted surface symmetry Type 3 symmetry Type 3 Li2B407 (Euler angles: [45° 45° 90°]).
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
M. P. da Cunha
V) LU III
y
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n Q -20 o
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1090
-80
-60 -40 -20 normalized depth z/wavelength (a)
-100
-80
-60 -40 normalized depth z/wavelength (b)
Fig. 10. Plots of the declination of the Poynting vector as a Junction of the normalized depth for the mechanically free shorted surface symmetry Type 1 quartz AT (Euler angles: [0° -54.7° 0°]): (a) PSAW; (b) HVPSAW.
224
Pseudo and High Velocity Pseudo SAWs
-
1
0
-
1091
8 - 6 - 4 normalized depth z/wavelength
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.
225
1092
M. P. da Cunha
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
<
l
l
or equivalently Vm=vpcos0i
(21)
Equations (19) to (21) allow the calculation of the orientation inside the semi-infinite substrate where a specific radiating partial mode radiates, and the phase velocity of the associated bulk mode. Table 2 gives, in the third column, the radiating partial modes for both PSAW and HVPSAW along some illustrative free surface orientations, namely: the symmetry Type 3 Li 2 B 4 0 7 (Euler angles: [0° 45° 90°]); the symmetry Type 1 Li 2 B 4 0 7 (Euler angles: [0° 75° 75°]); symmetry Type 1 quartz AT (Euler angles: [0° -54.7° 0°]); and symmetry Type 1 quartz ST-X (Euler angles: [0° 132.75° 0°]). The fourth column gives the 0, and vBi calculated from (19) and (21), and associated with the respective radiating partial mode from the third column. The fifth column shows the Euler angles associated to the direction of propagation $ in sagittal plane, and the three BAW phase velocities calculated at 6t.
226
Pseudo and High Velocity Pseudo SAWs
1093
K<=co/vp
Fig. 12. Wave vectors representation in the sagittal plane.
Although both HVPSAW along symmetry Type 3 and PSAW have only one radiating partial mode, a very distinct characteristic of the HVPSAW with respect to the PSAW may be observed from Table 2 regarding the type of bulk mode that radiates inside the solid. Selecting for instance the HVPSAW along symmetry Type 3 Li2B4C>7 (Euler angles: [0° 45° 90°]), note from Table 2, column 4, that the vBi associated with the radiating partial mode that radiate power inside the semi-infinite substrate is 5.014 Km/s. From Table 2, column 5, one sees that this radiating partial mode radiates power to the fast quasi-shear bulk mode along the corresponding orientation in the solid. Figure 13a illustrates the radiation of this radiating partial mode to the fast quasi-shear bulk. The direction of the Poynting vector associated with the radiating partial mode and normal to the fast quasi-shear bulk slowness curve, can also be inferred from Fig. 13a to be around 45°, as calculated in Ref. 4. Observing now PSAW along symmetry Type 1 Li2B407 (Euler angles: [0° 75° 75°]), the respective vBi from Table 2, column 4, is 3.535 Km/s. From Table 2, column 5, one sees that this radiating partial mode radiates power to the slow quasi-shear bulk mode along the corresponding orientation in the solid. The radiation of the PSAW radiating partial mode to the slow quasi-shear bulk mode is depicted in Fig. 13b. Therefore, while the radiating partial mode in a HVPSAW along symmetry Type 3 radiates power to the fast quasi-shear bulk mode, the PSAW radiating partial mode radiates power to the slow quasi-shear bulk mode. Figure 13b also depicts the HVPSAW along symmetry Type 1 Li2B407 (Euler angles: [0° 75° 75°]) and the two associated radiating partial modes presented in Table 2. Note that one of the HVPSAW radiating partial modes radiates power to the slow quasishear bulk mode, whereas the other radiates power to the fast quasi-shear bulk mode, as predicted in Table 2.
227
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M. P. da Cunha
Table 2. PS AW and symmetry Types 1 and 3 HVPSAW radiating partial modes (third column); calculation of 6! and v a using (19) and (21), respectively (fourth column); sagittal plane slowness: verification of (21) and relation to the bulk modes radiation shown double underlined. Material and orientation (Euler angles) symmetry TYPE
Wave Partial modes that grow Type with depth (free surface) [s/Km]
Li2B407 [0° 45° 90°] TYPE 3
HVPSAW
Li2B407 [0° 75° 75°] TYPEl
PSAW
-7.41e-7+j 0.14067
^•n(eql9) ### vBi [Km/s] (eq.21)
Euler angles [°] @ 0t Bulk wave velocities [Km/s] @ 9t
44.86
[90 -90 -0.1416] [4.376, 5.014, 7.440]
### 5.014 -3.29e-5+j0.1556
33.37
###
[86.0 284.5-41.13] [3.535. 4.743, 7.452]
3.535 -2.15e-5+j0.1779
48.08
### HVPSAW
[86.0 284.5 -26.42] [4.183. 4.596, 7.544]
4.183 -1.74e-4+j0.1439
42.01
###
[86.0 284.5 -32.49] [3.901,4^2,7.5341
4.652 quartz AT-X [0° -54.7° 0°] TYPEl
-2.88e-6+j0.117
30.77
###
PSAW
[0 215.3 30.77] [4.381.4.830, 5.349]
4.381 -1.08e-5+j0.137
38.10
### HVPSAW
[0 215.3 38.10] [4.521. 4.730, 5.374]
4.521 -1.72e-5+j0.115
33.45
###
[0 215.3 33.45] [4.470, 4.794, 5.327]
4.794 quartz ST-X [0° 132.75° 0°] TYPEl
-1.53e-4+j0.115
30.3
###
PSAW
[0 215.3 30.3] [4.385. 4.761, 5.447]
4.385 -1.72e-5+j0.115
36.68
### HVPSAW
[0 215.3 36.68] [4.607. 4.624, 5.442]
4.607 -6.6e-6+j0.1284
36.41
### 4.623
228
[0 215.3 36.41] [4.608, 4.623, 5.439]
Pseudo and High Velocity Pseudo SAWs
1095
90 86 DEC
180
270
(a)
180
270
(b) Fig. 13. Slowness curves [s/Km]. a.) Sagittal plane of Li2B407 Euler angles [0° 75° 75°]: Li 2 B 4 0 7 Euler angles [86.0° 284.5° -74.5°], represented at zero degrees, b.) Sagittal plane of Li 2 B 4 0 7 Euler angles [0°45°90°]: Li 2 B 4 0 7 Euler angles [90°-90°-45°], represented at zero degrees. The angles exhibited indicate the orientations of the radiating terms inside the material, calculated from the PSAW and HVPSAW growing partial modes (Table 2).
229
1096
M. P. da Cunha
3.4. PSAW and HVPSAW Solutions The first experimental evidences for the HVPSAW were published during the late seventies.5,6' 7'8 The experimental HVPSAW data available on rotated Y-cuts of quartz has also been reported in subsequent works.9'419,20 Figure 14 shows phase velocity and attenuation for both pseudo modes on the entire quartz ST plane, Euler angles [0° 132.75° v|/]. The phase velocities for the GSAW, PSAW and HVPSAW are shown in Fig. 14a, together with those of the three BAWs. Figure 14b gives the propagation attenuation of the HVPSAW and PSAW modes for the ST plane. Note that the attenuation of both PSAW and HVPSAW vary in the plane, reaching low enough values for practical devices around several orientation intervals. Figures 15 and 16 illustrate the GSAW, PSAW and HVPSAW solutions for the X axis boule cut Euler angles [<> | 8 vj/]= [0° 6 0°] on a promising new SAW material, langatate (La3Ga5 sTao.sOu), that has a temperature behavior comparable to quartz, but with higher piezoelectric coupling.36,37,45 Note from Fig. 15 that both GSAW and PSAW solutions cease to exist on the orientations where these modes approach the slow or the fast BAW velocity value, respectively, whereas the HVPSAW exist for the entire plane, although with relatively high losses, Fig. 16a. The relative fractional phase velocity, K2=2|Avp|/Vp = 2|vshort-Vfree|/vfree, for the pseudo PSAWs is shown in Fig. 16b. The HVPSAW has received considerable attention in the past few years on Li2B407.14,15,10,3 Experimental HVPSAW devices have also been built on LiNb03 and LiTa03, and the mode has also been identified in other materials like GaAs.16,17,1812 Table 3 presents HVPSAW solutions along selected orientations in order to discuss some characteristics of this mode. The first column gives the selected orientations; the respective HVPSAW solution for vp and a are given in the second and third columns for both electrically shorted and free surfaces. The values of the three BAW modes and the value of the SAW solution are included in the first and second column in brackets for reference purposes. Column 4 lists the particle displacement field for both electrically shorted and free surfaces. The orientation symmetry type is also included in this column. The estimated powerflowangles according to the procedure discussed in Subsection 3.2, and the 2|Avp|/vp are displayed in column five. In addition to the HVPSAW characteristics regarding the radiating partial modes and how they radiate inside the semi-infinite substrate, other attributes of the HVPSAW solutions are now discussed. One relevant HVPSAW characteristic is the proximity of the phase velocities to the quasi-longitudinal bulk wave phase velocity, which makes the fabrication of high frequency SAW devices easier. As seen in column 2, the HVPSAW phase velocities reach values 100% higher than those of the SAW. Another property of the HVPSAW is the strong longitudinal particle displacement field obtained for most solutions. From column 4, the magnitude of the longitudinal particle displacement, ux, is usually dominant. An additional characteristic for certain orientations is the fact that for the HVPSAW to behave like a surface wave, it may require a guiding structure, like a shorted surface or a grating, to concentrate the power close to the surface as discussed in Sub-section 3.2.4,16,17,18 Quartz ST-250, LiNb03 (Euler angles: [90° 90° 36°]), and LiTa03 (Euler angles: [90° 90° 31°]), given in Table 3, are such examples. For completeness, Table 4 includes PSAW solutions for selected materials and orientations. Table 4 has the same form as Table 3. An important PSAW characteristic is
230
Pseudo and High Velocity Pseudo SAWs
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70
80
90
(b) Fig. 14. Quartz ST plane Euler angles [0° 132.75° y]: (a) Phase velocity for the GSAW, PSAW and HVPSAW modes, also including the three BAW; (b) attenuation for the PSAW and HVPSAW modes.
231
1097
1098
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140
160
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20
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80
100
120
140
160
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THE [DEGRES]
(b) Fig. 15. X axis boule cut Euler angles [<)i8x))]= [0°8 0°] phase velocities [Ktn/s]: (a) HVPSAW and PSAW, including the longitudinal and fast shear BAW modes; (b) GSAW, including the slow shear BAW.
232
Pseudo and High Velocity Pseudo SAWs
HVPS/ w PSAW
/-"~
/ /-
\
PSAW
\ \
/
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20
40
60
80 100 THE [DEGRES]
120
140
160
180
140
160
180
(a)
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40
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80 100 120 THE [DEGRES]
Fig. 16 X axis boule cut Euler angles [$ 61|/]= [0°8 0°]: (a) PSAW and HVPSAW attenuation [dB/>.]; (b) PSAW and HVPSAW K2 [%].
233
1099
1100
M. P. da Cunha Table 3.
High velocity pseudo-SAW (HVPSAW) for some selected materials and orientations.
VBULK [Km/s]
V
P short/ open
ill
Material and orientation (Euler angles, [°])
< V SAW >
Li2B407
[Km/s] 6.9879 7.0418 (3.0998)
Fields @ z=0 u
*>Vuz>
short/open (magnitude [Km] ) symmetry type
PFA[°] short/ open 2|Av p |/v p [%]
6.1e-3 5.8e-5
1 0 0.03 1 0 0.01 TYPE 3
0 0 1.53
7.0267 7.0740 (3.2658)
l.le-2 2.9e-4
1 0 0.061 1 0 0.037 TYPE 3
0 0 1.34
6.8641 6.9120 (3.2557)
1.3e-2 6.7e-4
1 0 0.09 1 0 0.07 TYPE 3
0 0 1.39
[0 75 75 ] [3.689 4.936 6.439]
6.2405 6.2611 (3.3090)
0.460 0.041
1 0.686 0.364 1 0.675 0.307 TYPEl
-20.3 -27.1 0.66
GaAs [45 -90 25 ] [2.654 2.990 5.376]
5.3987 5.3986 (2.5339)
1.9e-2 1.6e-3
1 0.04 0.234 1 0.04 0.233 TYPEl
2.8 2.9 0.004
quartz AT-X [0 -54.7 0 ] [3.298 5.100 5.744]
5.7447 5.7454 (3.1510)
1.0e-2 2.5e-3
1 0.12 0.077 1 0.12 0.080 TYPEl
-0.2 -0.1 0.023
quartz ST-X [0 132.75 0 ] [3.298 5.100 5.744]
5.7446 5.7449 (3.1576)
6.0e-3 1.2e-3
1 0.066 0.081 1 0.056 0.084 TYPEl
-0.5 -0.7 0.011
quartz ST-25 [0 132.75 25 ] [3.365 4.032 6.604]
6.5262
1.8e-5
1 0.35 0.090
14.1
[45 45 90] [3.238 5.046 7.238] Li2B407 [0 45 90 ] [3.446 4.644 7.409] Li2B407 [0 47.3 90] [3.479 4.632 7.437] Li2B407
LiNb0 3 [90 90 36 ] [4.0214.029 7.315] LiTa0 3 [90 90 31 ] [3.336 3.365 6.316] LiTa0 3 [90 90 42 ] [3.347 3.359 6.347]
TYPE 1
(3.2475) 6.8085 7.3404 (3.6616)
3.8 6.4e-2
1 0.34 0.48 1 0.08 0.24 TYPEl
-12.7 -0.08 14.5
6.2442 6.3179 (3.1420)
0.56 8.0e-3
1 0.19 0.25 1 0.09 0.13 TYPEl
1.3 3.5 2.3
6.2922 6.3512 (3.1420)
0.59 1.0e-2
1 0.14 0.26 1 0.04 0.16 TYPEl
-0.7 -0.5 1.9
the proximity of the phase velocities to the fast quasi-shear bulk mode phase velocity. The PSAW phase velocities reach values around 40% higher than the SAW values, as seen in column 2. In the PSAW case, column 4 shows that the magnitude of the shear horizontal particle displacement, uy , is usually dominant.
234
Pseudo and High Velocity Pseudo SAWs Table 4.
1101
Pseudo-SAW (PSAW) solution for some selected materials and orientations.
Material and orientation (Euler angles, [°]) VBULK [Km/s]
Li2B407 [0 75 75 ] (3.689 4.936 6.439)
V
P short/ open rvFREEi [Km/s] 4.1958 4.2329 (3.3090)
a short/ open [dB/X.]
Fields @ z=0 ux,uy,uz, short/open (magnitude [Km] ) symmetry type
PFA[°] short/ open 2|Av p |/v p [%]
3.0e-2 1.6e-2
1 0.92 0.46 1 1.05 0.44 TYPEl
25.7 27.3 1.8
[0 -54 0 ] (3.3514.227 5.589)
4.1091 4.2267 (3.1252)
2.7e-4 2.1e-4
1 15.2 1.85 1 26.7 0.47 TYPEl
-0.053 -0.006 5.6
quartz AT-X [0 -54.7 0 ] (3.298 5.100 5.744)
5.0994 5.0996 (3.1510)
1.8e-3 1.3e-3
1 6.70 0.33 1 6.66 0.31 TYPEl
-0.034 -0.035 0.008
quartz ST-X [0 132.75 0 ] (3.298 5.100 5.744)
5.0781 5.0789 (3.1576)
7.9e-2 7.8e-2
1 2.40 0.37 1 2.39 0.36 TYPEl
0.040 0.055 0.033
quartz ST-25 [0 132.75 25] (3.365 4.032 6.604) 64 LiNbOj
4.0166 4.0181 (3.2475)
7.8e-2 8.1e-2
1 2.34 0.60 1 2.32 0.59 TYPEl
-28.8 -28.7 0.074
4.4505 4.6919 (3.6799)
3.7e-3 5.2e-2
1 8.58 3.23 1 3.58 0.60 TYPEl
0.6 2.2 10.3
4.3781 4.7515 (3.6412)
1.4e-2 2.4e-4
1 51.2 20.0 1 23.8 0.79 TYPEl
-0.83 0.02 15.7
[0 90 29.2 ] (3.432 3.678 5.989)
3.5123 3.5444 (3.2056)
7.0e-l 8.2e-l
1 3.42 1.45 1 3.49 1.69 TYPEl
2.54 2.54 1.8
quartz [0 15 0 ] (3.298 5.100 5.744)
3.9547 3.9569 (3.2911)
1.7e-4 1.4e-4
1 0.33 0.93 1 0.34 0.93 TYPEl
-0.053 -0.048 0.11
36 YX-LiTa0 3
[0 -26 0 ] (4.0314.752 6.547) 41 LiNbOj [0 -49 0 ] (4.0314.752 6.547) LiTaOj
4. Layered Solutions This section addresses the importance of layer thickness and materials on pseudo SAW devices. It focuses on a discussion of the layer parameters that affect a pseudo SAW device performance and highlight the properties of layered solutions with examples.
4.1. The Importance of the Layer Thickness and Materials in Practical Devices Any surface acoustic wave device is influenced by a material layer, such as: the metallic film of the transducer, a film deposited in the propagation path, or the device might be a layered structure. The layer thickness and material directly affects the pseudo SAW complex slowness solution, vp and a. The layer thickness also modifies the penetration depth, the power flow angle, the particle field Dolarization, and the values of 2|AvD|/vD. 235
1102
M. P. da Cunha
For high performance devices, especially if one is considering portable applications where the systems are battery operated, the importance of low loss devices is evident. In the next section some examples are given, showing that a proper choice of the layer material and thickness leads to significantly reduced values of a. A knowledge of the dispersion characteristic is fundamental in high performance device design and in group delay response prediction. A layer usually has a strong effect on the penetration depth. As shown in Figs. 7 and 8, a layer may concentrate the power associated with the propagating wave closer to the surface, and thus cause the pseudo SAWs to behave more like a surface mode. Other practical consequences that result from the knowledge of the layer effect on a certain mode penetration depth are: the required quality of the surface mechanical polishing; and the substrate thickness in order for the substrate to be appropriately considered semiinfinite. In references 25 and 26 the particle displacement fields are tabulated as a function of thickness for the HVPSAW on an aluminum layered symmetry Type 1 LiNb03 (Euler angles: [90° 90° 164°]), and for the PSAW on an aluminum layered symmetry Type 1 LiTa03 (Euler angles: [0° -26° 0°]). Knowing the particle displacement fields is important in calculating practical devices parameters, such as the electrode reflection coefficient, fundamental in SAW device design. Finally, the power flow angle estimation allows proper placement of the transmitting and receiving transducers in practical devices where the PFA^Q. The PFA as a function of thickness has also been addressed in references 25 and 26, using the HVPSAW LiNb03 and the PSAW LiTa03 examples cited in the preceding paragraph. One should also mention the importance of layered structures such as ZnO/diamond/Si and ZnO/SiC in achieving phase velocities as high as 14 Km/s for the HVPSAW in high frequency applications.46'47> 48 4.2. Properties of Solutions in Layered Structures This subsection discusses properties of the layered pseudo SAWs solutions which have an impact on both practical devices design and in the fundamental understanding of the pseudo modes. Figures 17 and 18 show the phase velocity and attenuation as a function of the thickness times frequency, hf, for two aluminum layered orientations: symmetry Type 3 Li2B407 (Euler angles: [0° 47.3° 90°]), Fig. 17; and symmetry Type 1 LiNb03 (Euler angles: [90° 90° 164°]), Fig. 18. The upper part of Fig. 17 shows velocity plots for the HVPSAW, three lossless pure sagittal particle motion Rayleigh modes, and the phase velocity values of the three BAW modes. The HVPSAW propagation attenuation is plotted in the lower part of Fig. 17. Note that since this orientation refers to a symmetry Type 3 there is no PSAW. The upper part of Fig. 18 shows velocity plots for the HVPSAW, the PSAW and a higher order PSAW, five generalized SAW modes and the phase velocity values of the three BAW modes along that orientation. The lower part of Fig. 18 plots the propagation loss for the HVPSAW, the PSAW and a higher order PSAW mode. Note from Fig. 17 that a minimum in the propagation loss occurs close to hf=0.2 Km/s (vp=6.7413 Km/s), h/^=3%, a reduction of more than two orders of magnitude when compared to the mechanically free electrically shorted surface. This is a relevant
236
Pseudo and High Velocity Pseudo SAWs
1103
7.5 quasi-l ingitud nal 7 6.5 6
v
5.5
/ \ V
s
5
fast shear
4.5 4 3.5 . 1
3
0
1
~2
2 3 4 5 6 7 T H I C K N E S S x F REQU ENCY
3
V ¥
8 9 (hfHKm/sl
10
7 8 9 (hf)[Km/s]
10
101
/ '
§10° z
LU 1 UJ
|
10"1
OQ Z
-2
O 10
/
1 in4
1
2 3 4 5 6 THICKNESSx FREQUENCY
Fig. 17. Layered substrate: phase velocity (upper part) and attenuation (lower part) versus (thickness x frequency). Aluminum layered symmetry Type 3 Li 2 B 4 0 7 (Euler angles: [0° 47.3° 90°]. HVPSAW: dash-dot; lossless Rayleigh modes (pure sagittal particle motion): Solid (first and second), and star (third); larger-dots: BAW phase velocities.
237
1104
M. P. da Cunha 7 quasi longitu dinal
6.5 6 1
5.5
8
test c uasi-sl iear
Qj 4.5
> ui
3
4
\ 0
X
0
slaw iuasi-s tear
x
^ ^ ° 9 * +*•*»
1
Q-
5
4 2
3.5 1 3 1.5 2.5
1 1.5 2 2.5 3 3.5 THICKNESS x FREQUENCY
4 4.5 ( h f ) [Km/s]
5
5.5
5
5.5
10
x iO LU
/
I
> 0
o
°o0
_„0
-i 10
§
0
m -o z
Q , Sc 10
10'
pI \
0.5
1 1.5 2 2.5 3 3.5 THICKNESS x FREQUENCY
4 4.5 ( h f ) [Km/s]
Fig. 18. Layered substrate: phase velocity (upper parts) and attenuation (lower parts) versus (thickness x frequency). Aluminum layered symmetry Type 1 LiNbOj (Euler angles: [90° 90° 164°]). HVPSAW: dash-dot; higher order PSAW: circles; PSAW: dashed; first to fourth GSAW: solid; fifth GSAW: star; largerdots: BAW phase velocities.
238
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
239
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M. P. da Cunha
Numerical and experimental HVPSAW and PSAW orientations and planes are listed and analyzed. Finally, solutions for selected orientations in layered structures and the relevance of layer materials and thickness are discussed based on experimental and numerical results.
Acknowledgments The author is indebted to many suggestions and proofreading of the manuscript by Eric L. Adler, and to fruitful discussions with Prof. Donald C. Malocha. The support of the Brazilian National Council for Scientific and Technological Development (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico - CNPq); the Sao Paulo Research Foundation (Fundagao de Amparo a Pesquisa do Estado de Sao Paulo - FAPESP); and the Advanced Materials Processing and Analysis Center - AMPAC, University of Central Florida, USA, which enable the realization of different parts of the work reported in this article, are gratefully acknowledged.
References 1. H. Engan, K. A. Ingebrigtsen, and A. Tonning, "Elastic surface wave in a-quartz: observation of leaky surface waves," Appl Phys. Lett., vol. 10, pp. 311-313, June 1967. 2. T. C. Lim and G. W. Farnell, "Character of pseudo surface waves on anisotropic crystals," The Journal of the Acoustic Society of America, April 1969, vol. 45, No. 4, pp. 845-851. 3. M. Pereira da Cunha, "Extended investigation on high velocity pseudo surface waves," IEEE Trans. Ultrason. Ferroelec. Freq. Contr., vol. 45, no. 3, May. 1998, pp. 604-613. 4. M. Pereira da Cunha and Eric L. Adler, "High velocity pseudosurface waves (HVPSAW)," IEEE Trans. Ultras., Ferroel, Freq. Contr., vol. 42, No. 5, Sept. 1995, pp. 840-844. 5. T. I. Browning and M. F. Lewis, "New family of bulk-acoustic-wave devices employing interdigital transducers," Electronics Letters, March 1977, Vol. 13, No. 5, pp. 128-130. 6. M. Lewis, "Surface skimming bulk waves - SSBW," in Proc. IEEE Ultras. Symp., 1977, pp. 744-752. 7. K. H. Yen, K. F. Lau, and R. S. Kagiwada, "Recent advances in shallow bulk wave devices," in Proc. IEEE Ultras. Symp., 1979, pp. 776-785. 8. A. Ballato and T. J. Lukaszek, "Shallow bulk acoustic wave progress and prospects," IEEE Trans, on Micr. Theory and Tech.., vol. 27, No. 12, Dec. 1979, pp. 1004-1012. 9. C. A. Flory and R. L. Baer, "Surface transverse wave analysis and coupling to interdigital transducers," in Proc. IEEE Ultras. Symp., 1987, pp. 313-318. 10. N. F. Naumenko, "Quasihorizontal polarized quasibulk acoustic surface waves piezoelectric crystals," Sov. Phys. Crystallogr., Vol. 37, No. 2, March-April 1992, pp. 220-223.
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Pseudo and High Velocity Pseudo SAWs
1107
11. K. Y. Hashimoto, M. Yamaguchi, and H. Kogo, "Experimental verification of SSBW and leaky SAW propagating on rotated Y-cuts of LiNb0 3 and LiTa03," in Proc. IEEE Ultras. Symp., 1983, pp. 345-349. 12. G. Carlotti, D. Fioretto, L. Giovannini, F. Nizzoli, G. Socino, and L. Verdini, "Brillouin scattering by pseudosurface acoustic modes on (l l l) GaAs," Journal of Physics: Condensed Matter, 1992, Vol. 4, pp. 257-262. 13. A. Jhunjhunwala, "Spectrum of Acoustic Waves Excited in Single and Multiple Layered Crystalline Media," Ph.D. Thesis, University of Maine, USA, 1977. 14. M. Adachi, T. Shiosaki, K. Ohtsuka, and A. Kawabata, "Leaky SAW propagation properties on Li 2 B 4 0 7 substrates," in Proc. IEEE Intern. Freq. Contr. Symp., 1994, pp. 296-300. 15. T. Sato and H. Abe, "SAW device applications of longitudinal leaky surface waves on lithium tetraborate," IEEE Trans. Ultras., Ferroel., Freq. Contr., vol. 45, No. 6, Nov. 1998, pp. 1506-1516. 16. S. Tonami, A. Nishikata, and Y. Shimizu, "Characteristics of leaky surface acoustics waves propagating on LiNbCO and LiTa03 substrates," Jpn. J. Appl. Phys., vol. 34, Part 1, No. 5B, May 1995, pp. 2664-2667. 17. Y. Kobayashi, N. Tanaka, K. Matsui, H. Okano, T. Usuki, K. Shibata, and Y. Shimizu, "1.9GHz-Band surface acoustic wave device using second leaky wave on LiTa0 3 and LiNb0 3 ," in Proc. IEEE Freq. Contr. Symp., 1996, pp. 240-247. 18. A. Isobe, M. Hikita, and K. Asai, "Propagation characteristics of longitudinal leaky SAW in Al-grating structure," in Proc. IEEE Ultras. Symp., 1997, pp. 17-21. 19. D. Zhang, X. Tong, and H. Qin, "Study of propagation properties of quasi-longitudinal leaky surface acoustic wave propagating on Y-rotated cut quartz substrates," in Proc. IEEE Ultras. Symp., 1999 (to be published). 20. G. Behme, E. Chilla, and H.-J. Frohlich, "Investigation of longitudinal leaky surface acoustic waves by scanning acoustic force microscopy," in Proc. IEEE Ultras. Symp., 1999 (to be published). 21. R. Veith, "The fabrication technology of Rayleigh wave devices," in Rayleigh-Wave Theory and Application, vol. 2, E. A. Ash and E. G. S. Paige, Ed., Springer series on Wave phenomena, Leopold B. Felsen, Ed., Germany, Springer-Verlag Berlin Heidelberg, 1985, pp. 254-272. 22. C. C. W. Ruppel, R. Dill, A. Fischeraurer, G. Fischeraurer, W. Gawlik, J. Machui, F. Miiller, L. Reindl, W. Ruile, G. Scholl, I. Schropp, and K.C. Wagner, "SAW devices for consumer applications," IEEE Trans. Ultras., Ferroel., Freq. Contr., vol. 40, Sep. 1993, pp. 438-452. 23. K. Yamanouchi and M. Takeuchi, "Applications for piezoelectric leaky surface waves", in Proc. IEEE Ultras. Symp., 1990, pp. 11-18. (see also practically all Proc. IEEE Ultras. Symp. from 1990 to date: several works and different authors). 24. I. D. Avramov, "Gigahertz range resonant devices for oscillator applications using shear horizontal acoustic waves," IEEE Trans, on Ultr. Ferr. and Freq. Control, vol. 40, no. 05, September 1993, pp. 459-468.
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25. M. Pereira da Cunha, "Effects of layer thickness for GSAW, PSAW, and HVPSAW devices," in Proc. IEEE Ultras. Symp., 1997, pp. 239-244. 26. M. Pereira da Cunha, "Effects of layer thickness for GSAW, PSAW, and HVPSAW devices," IEEE Trans, on Ultras. Ferroel. and Freq. Control, (to be published in 2000). 27. Eric L. Adler and Leland Solie, "ZnO on diamond: SAWs and Pseudo-SAWs," in Proc. IEEE Ultras. Symp., 1995, pp. 341-344. 28. D.L. Dreifus, R.J. Higgins, R.B. Henard, R. Almar, LP. Solie, "Experimental Observation of High Velocity Pseudo-SAWs in ZnO/Diamond/Si Multilayers," in Proc. IEEE Ultras. Symp., 1997, pp. 191-194. 29. I. S. Didenko, F. S. Hickernell, and N. F. Naumenko, "Theoretical aspects of GSAW and HVPSAW propagation properties for Zinc Oxide films on silicon carbide and correlation with experimental data," in Proc. IEEE Ultras. Symp., 1999 (to be published). 30. J. C. Andle, M. G. Schweyer, L. A. French, and J. F. Vetelino, "Acoustic plate mode properties of rotated Y-cut quartz," in Proc. IEEE Freq. Contr. Symp., 1996, pp. 532-540. 31. E. L. Adler, "SAW and Pseudo-SAW properties using matrix methods," IEEE Trans. Ultras., Ferroel, Freq. Contr., vol. 41, No. 6, Nov. 1994, pp. 876-882. 32. H. Goldenstein, Classical Mechanics, Addison-Wesley Publishing Company Inc., Cambridge, Massachusetts, USA, 1950, pp. 107-109. 33. A. J. Slobodnik, E. D. Conway, and R. T. Delmomco, Microwave Acoustics Handbook, vol. 1, Surface Wave Velocities, TR-73-0597, AD780172, NITS, Springfield, VA 22151,1972. 34. G. Kovacs, M. Anhorn, H. E. Engan, G. Visintini, and C. C. W. Ruppel, "Improved material constants for LiNb03 and LiTa03 ," in Proc. IEEE Ultras. Symp., 1990, pp. 435-438. 35. M. Adachi, T. Shiosaki, H. Kobayashi, O. Ohnishi, and A. Kawabata, "Temperature compensated piezoelectric lithium tetraborate crystal for high frequency surface acoustic wave device applications," in Proc. IEEE Ultras. Symp., 1985, pp. 228-232. 36. Yu. V. Pisarevsky, P. A. Senushencov, B. V. Mill, N.A. Moiseeva, "Elastic, piezoelectric, dielectric properties of La3Ga5 5Tao 5 0i 4 single crystals", in Proc. IEEE Inter. Freq. Contr. Symp., 1998, pp. 742-747. 37. M. Pereira da Cunha, "Surface and pseudo surface acoustic waves in langatate," in Proc. IEEE Ultras. Symp., 1999 (to be published). 38. G.W. Farnell and E.L. Adler, "Elastic wave propagation in thin layers," Physical Acoustics, vol. 9, W. P. Mason and R. N. Thurston, Ed., New York, Academic Press, 1972, pp. 35-127. 39. J. L. Bleustein, "A new surface wave in piezoelectric materials," Applied Physics Letters, 15 December 1968, pp. 412-413. 40. E. L. Adler, "Matrix Methods Applied to Acoustic Waves in Multilayers," IEEE Trans. Ultrason. Ferroelec. Freq. Contr., Vol. UFFC-37, No 6, November 1990, pp. 485^190. 41. M. Pereira da Cunha, "SAW propagation and device modelling on arbitrarily oriented substrates," PhD Thesis, McGill University, Montreal, Canada, July 1994.
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42. N. E. Glass and A. A. Maradudin, "Leaky surface-elastic waves on both flat and strongly corrugated surfaces for isotropic, nondissipative media," J. Appl. Phys., 1992, Vol. 54, no. 02, pp. 796-805. 43. B.A. Auld, Acoustic Fields and Waves in Solids, vols. I & II, Robert E. Krieger Publishing Company, Malabar, Florida, 1990, 2nd. edition. 44. G. A. Coquin and H. F. Tiersten, "Analysis of the excitation and detection of piezoelectric surface waves in quartz by means of surface electrodes," J. Acous. Soc. Amer., vol. 41, 1967, pp. 921-939. 45. R.C. Smythe, R.C. Helmbold, G.E. Hague, and K.A. Snow, "Langasite, langanite, and langatate resonators: recent results", in Proc. IEEE Inter. Freq. Contr. Symp., 1999, (to be published). 46. Eric L. Adler and Leland Solie, "ZnO on diamond: SAWs and Pseudo-SAWs," in Proc. IEEE Ultras. Symp., 1995, pp. 341-344. 47. D.L. Dreifus, R.J. Higgins, R.B. Henard, R. Almar, L.P. Solie, "Experimental Observation of High Velocity Pseudo-SAWs in ZnO/Diamond/Si Multilayers," in Proc. IEEE Ultras. Symp., 1997, pp. 191-194. 48. I. S. Didenko, F. S. Hickernell, N. F. Naumenko, "Theoretical aspects of GSAW and HVPSAW propagation properties for zinc oxide films on silicon carbide and correlation with numerical experimental data," in Proc. IEEE Ultras. Symp., 1999 (to be published). 49. V. P. Plessky and C. S. Hartmann, "Characteristics of leaky SAWs on 36° YX-LiTa03 in periodic structures of heavy electrodes," in Proc. IEEE Ultras. Symp., 1993, pp. 1239-1242. 50. V. P. Plessky, C. S. Hartmann, and J. Koskela, "Suppression of the leakage effect in 64° YXLiNb03 for thick aluminum electrodes," in Proc. IEEE Ultras. Symp., 1996, pp. 1603-1606. 51. K. Hashimoto, M. Yamaguchi, S. Mineyoshi, O. Kawachi, M. Ueda, G. Endoh, "Optimum leaky-SAW cut of LiTa0 3 for minimised insertion loss devices," in Proc. IEEE Ultras. Symp., 1997, pp. 245-254.
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International Journal of High Speed Electronics and Systems, Vol. 10, No. 4 (2000) 1111-1142 © World Scientific Publishing Company
SAW DEVICE BEYOND 5 GHz HIROYUKI ODAGAWA Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan KAZUHIKO YAMANOUCHI Tohoku Institute of Technology, Sendai 982-8577, Japan Here we describe recent high frequency SAW technology based on ultra-fine fabrication techniques using electron beam (EB) exposure direct writing and low-loss wide-band filter applications beyond 5 GHz. We especially consider the propagation property of SAWs in the 10 GHz range, which is important for low-loss characteristics, the fabrication technology, and some examples of low-lossfiltersusing unidirectional transducers and ladder type SAW filters.
1.
Introduction
Surface Acoustic Wave (SAW) have been applied to electrical communication systems as various electronic devices, such as band bass filters, high frequency oscillators, convolvers, correlators, switching devices for optical applications, and so on. In mobile communication systems, SAW devices have been also applied and contributed the development. Now the frequency range of the mobile communication system is in the 2 GHz-range, and the SAW devices for the next generation mobile system are studied actively. Considering the remarkable increase of subscribers of mobile communication and the increase of the required date transfer rate, we can expect that the frequency range of mobile communication systems will be expanded to 5 to 10 GHz in the near future. Other applications, such as timing pick-up filters for light wave communication are also requiring devices in the 5 to 10 GHz-range. Accompanying the extension of the frequency range of the systems, SAW devices, which are important elements of electrical communication systems, must also expand their frequency range. Here we describe a SAW technology based on ultra-fine fabrication techniques for low-loss wide-band filters beyond 5 GHz. We especially consider the propagation property of SAWs in the 10 GHz range, which is important for low-loss characteristics and the fabrication technology. Moreover some examples of low-loss filter using unidirectional transducers and ladder type SAW filters are discussed.
245
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Yamanouchi
Propagation Loss in the GHz-Range for SAW Devices
It is necessary to estimate the losses in GHz-range SAW devices in order to obtain low-loss devices beyond the 5 GHz range. Especially, the propagation loss of SAWs up to the 10 GHz should be evaluated because the propagation characteristics is an essential physical property in SAW devices, and the loss in the 10 GHz-range is one of the greatest interests for SAW device engineers. In this section we describe the losses of SAW propagation in the GHz-range.
2.1.
Losses in SAW devices
The total insertion losses in SAW devices can be arranged according to different mechanisms ' as shown in Table 1.
Table I.
Losses in SAW Devices
Interaction with thermally excited elastic waves [Temperature dependent] Propagation loss due surface wave attenuation
to
Scattering by crystalline defects, impurities, and pits and scratches in the optical polish [Temperature independent] Energy lost to air adjacent to the surface [Pressure dependent]
Losses due to transducer configuration and geometry
Bi-directional loss using conventional two-IDT Diffraction of surface wave Conduction loss in electrode
Transducer loss
Spurious wave excitation loss Impedance mismatching loss
Among these mechanisms, propagation loss is the most essential to the SAW devices. The propagation loss is caused by interaction with thermally excited elastic waves (temperature dependent), scattering by crystalline defects, impurities, and pits and scratches in the optical polish (temperature independent) and energy lost to air adjacent to the surface (pressure dependent). The air loading loss is evaluated by measurements under low-pressure conditions. The temperature dependent propagation loss is evaluated by the change of the insertion loss from room temperature to lower temperatures.
246
SAW Devices Beyond 5 GHz
2.2.
1113
Measurement method for the propagation loss
In this section, the temperature dependency of the attenuation for a 10 GHz SAW propagating along the X-axis on 128° rotated Y-cut LiNb0 3 2,3 is described. The total insertion loss / of a two-transducer filter is given by ' / = EfAEfB
exp(-orP L),
(1)
where EfA and EfB are the transducer efficiencies of the launching and the receiving transducers (temperature dependent), ap is the surface wave attenuation per unit length and L is the distance between the transducers. If EfA and EfB have no temperature dependency, the attenuation constant aP is evaluated by the two-transducer system. Unfortunately, the variations of the EfA and EfB are large due to the temperature dependency of the resistivity on the transducer electrodes. Therefore, a three-transducer system as shown in Fig. 1 is necessary to evaluate the propagation loss. In this figure, the SAW excited from transducer A is received by transducers B and C. The ratio of the insertion losses is given by ^ ' B
= -^-exp{-ap(Lc-LB)}.
(2)
^ fB
In this equation, if the temperature dependency of Ef!/EfB is sufficiently small, av can be obtained by measuring the IB and / c . Therefore, if the temperature dependency of the insertion loss between transducer A and B (IB) and the loss between transducer A and C (/c) are measured as shown in Fig. 2(a), the difference between lB and lc, as shown in Fig. 2(b) does not contain the transducer characteristics. In Fig. 2(b), a, (-aP - a0) is the temperature dependent propagation loss caused by the interaction with thermally excited elastic waves, and a0 is temperature independent loss caused by crystalline defects, impurities, and pits and scratches in the optical polish.
LB B
Lc A
C
Fig. 1. Schematic diagram of 3-transducer.
247
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H. Odagawa & K.
Yamanouchi |
i
|
1
Loss (dB)
i
c o C oo c
a
1
I
^IB"
oc
a OB i
.
i
i
C1
300 Temperature (K)
(a) Propagation Loss (dB/unit length)
'
i
•
i
i
1
/
-
a£ <* 0
i
.
i
i
1
•
300
(D
Temperature (K)
(b) Fig.2.
2.3.
Measurement method of the propagation loss.
Measurement results of propagation loss
The 3-transducer structure fabricated on 128° Y-X LiNb0 3 is used in the experiment. The wavelength of the transducers is X = 0.4 (im, and the propagation length LB and Lc are 105 X and 300 X, respectively. The pair number is 10, the aperture is 20 X and the film thickness of the Al electrodes is 30 nm. Table 2 shows the insertion loss of the 3-transducer device at room temperature (293 K) and at low temperature (7 K). In order to control the temperature, a helium flow cryostat is used. In this method, the sample space is directly cooled by temperature controlled helium. The measurement at 7 K is carried out after the sample is stabilized at 7 K for 1 hour, and then after the sample is warmed up to 293 K in a period of more than 3 hours, it is measured at 293 K.
248
SAW Devices Beyond 5 GHz 1115 Table 2.
Insertion loss of 3-transducer device at room temperature (293 K) and low temperature (7 K)
Frequency (GHz)
Propagation length (A.)
Insertion loss at 293 K
Insertion loss at7K
9.63
105
31.3
28.3
9.7
300
34.2
30.1
The experimental value at 10 GHz obtained from table 2 is plotted on Fig. 3. In this figure, the results at 1 to 5 GHz obtained in previous work 4 are also plotted. As shown in this figure, the attenuation (dBM,) is almost proportional to the 1.25 power of the frequency, and the propagation loss in the 10 GHz-range is about 1.5 dB / 100 X. This value includes the air loading loss and the temperature independent losses. This result shows that SAW propagation is low-loss even in the 10 GHz-range, and that SAW devices should be applicable in this frequency range. Figure 4 shows the frequency dependency of only the temperature dependent loss due to the interaction with thermal phonons. The new result at 10 GHz is obtained in the present experiment while the other results were obtained in Ref. 4. The loss (dB/cm) is almost proportional to the square of frequency. From these experimental results, we can conclude that the propagation loss and air loading loss are not the main contributors to the insertion loss in 10 GHz-range SAW devices.
0.020
1
I -
T
1
1
1 1 1 |
1.5dB/100A atlOGHz
,/
^ 0.010 -< 0.008 3 c
0.005 0.004
• 2 0.003
0.56dB/100A / at 5 G H z . / /
/0.33dB/10OA at3GHz "
§ 0.002 0.001 ,0.0008 - 0.08dB/100A at 1 GHz 1 2 3 5 7 10 Frequency (GHz) Fig.3. Frequency dependency of the propagation loss on 128° Y-X LiNbO? at room temperature.
249
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H. Odagawa & K.
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200
3 GO
I
|
•
'
'
'
|
y
141dB at 10GHz
a 100
o PQ
I
50
29dB at 3GHz
20
o
10 5 o
7.9dB at 2GHz
-t—>
03
U)
2dB at 1GHz
OH
O
VH
1
1
2
5 10 Frequency (GHz)
Fig.4. Frequency dependency of the loss due to the interaction with thermal phonons on 128° Y-X LiNbOj at room temperature.
3.
Influence of Electrode Resistance in SAW Devices
In the previous section we confirmed that the propagation loss in the 10 GHz-range was small. The value was about 1.5 dB / 100 A,, which included the interaction with thermal phonons, scattering by crystalline defects, impurities, and pits and scratches in the optical polish, and energy lost to air adjacent to the surface. In this section, we describe the electrode resistance which largely affects the insertion loss of SAW device beyond 5 GHz-range.
3.1.
Influence of the resistance ofAl thin film in 8GHz-range SAWfilters
In order to consider the influence of the electrode resistance, filters using a floating electrode type unidirectional transducer (FEUDT) 5'6,7,8 are fabricated and measured at room temperature and low temperature. The structure and features of FEUDTs will be described in Section 5.1. Figure 5 (a) and (b) show the frequency response of a 8 GHz A.,/10 type FEUDT 8 at room temperature (300 K) and low temperature (7 K), respectively. These are result obtained at the second harmonic operation. ' The FEUDT is fabricated by electron beam exposure and lift-off process. The fabrication
250
SAW Devices Beyond 5 GHz
1117
techniques are described in the Section 4. The substrate is 128° Y-X LiNbO^, the wavelength at second harmonic is ^2=0-45 u.m and the propagation length is 46 A.2. The pair number is 20, the aperture is 46 A,2, the film thickness of the Al electrode is 30 nm, and the electrode width is 0.09 u,m. These frequency responses are obtained by using a time gate to reject the feedthrough of electromagnetic waves. The triple transit echo (TTE) is included in these results.
log MOG
CENTER SPAN
9.500000000 OHz
(a) <& 0 . 0 dB A 1 0 . 0 OB-*•
$ / \
MORI<ER E . 3 2 : 7 5 C Hz poir,
/ ;
A \P H 'V 1
^/
V/wV \ A / V \
CENTER SPPN
\
i
V
A, ^
V
/V
/
V
S. 2000650000 Ota 3.000000000 GHz
(b) Fig. 5. Frequency responses of a 8 GHz-range A.i/10-type FEUDT filter, (a): measured at 300 K, (b): measured at 7 K.
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Comparing both minimum insertion losses, we can see that the minimum insertion loss decreases at low temperature. The insertion loss at 7 K is about 8.5 dB, which is 3.4 dB lower than that at 300 K. Since the propagation length is 46 X2 in this case, the propagation loss due to the thermal phonons is only 0.26 dB according to Fig. 4. Therefore, the major reason of the decrease in the insertion loss is not caused by the propagation loss. Instead, we fined that the insertion loss is largely affected by the electrode resistance. Actually, this resistance, which is measured by DC voltage at 7K, decreases to 1/3 in comparison with its value at 300 K. This means that a reduction of the electrode resistance is very important to obtain the low-loss characteristics in the high frequency range. For example, the use of the wide electrodes, and the development of low resistance metal films are necessary to obtain low loss devices, especially beyond 5 GHz.
3.2.
Superconducting IDT using Nb thin film
One of the experiments to deal with the electrode resistance was carried out using a superconducting IDT electrode 2'3 instead of a conventional Al electrode. In this experiment, a Nb thin film is used in a 5 GHz-range Ji,/]0 type FEUDT filter. It is fabricated by electron beam exposure and lift-off technique described later. The substrate is 128° Y-X LiNbOj, the wavelength at second harmonic is ^.2=0.8 (im and the propagation length is 40^2- The pair number is 20, the aperture is 40^.2 and the film thickness of the Nb electrode is 100 nm. Figure 6 (a) and (b) show the frequency responses of the Nb FEUDT filter at 293 K and 5 K, respectively. Comparing both minimum insertion losses, we can see that the minimum insertion loss decreases at low temperature. The insertion loss at 5 K is about 13.3 dB, which is 12.1 dB lower than that at 293 K. The plots in Fig. 7 show the temperature dependency of the 5 GHz-range Nb FEUDT filter. The broken line shows the relative resistivity of the Nb film. As the temperature becomes lower, the insertion loss becomes lower, and finally, it changes markedly by about 2.2 dB at 7 K because the Nb resistance changes from normal to superconducting. The changes in the insertion loss and in the relative resistivity correspond well at the lower temperature. Although these results do not directly show that the superconducting Nb electrode is suitable for low-loss filters, they are important, because they suggest a direction of research on low loss filters beyond 5 GHz.
252
SAW Devices Beyond 5 GHz
CM! 3 g l
lop HAO
10 OB/
M
l Oct 1908 Si: 13: 39 BEF 0 OB A; -23.4Q3 dB 4 o o e . loo oc 4 M H l
Cor
J
/\ / \
G«t Hid
/
IVy U
V
/
\
J
j
f\
/
• ^ - ^
^
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V
START 3 000.000 000 MHl
STOP 3 OOO.OOO OOO MHl
(a) CHI Sg 1
lop MAG
10 OB/
H
1 Oct 199B 17: SO: 04 P.CF O OB Jj -13.310 OB A 0 7 6 . OO o : 2 MHz
pnm
A
Cor
'V
Gat Hid
./V AJ !\\ A/i/ i«n''V
vf!
Vf
w 1
r V V A
~
V
\ v
i START 3 OOO.000 OOO MHz
STOP 9 OOO.OOO OOO I
(b)
Fig. 6. Frequency responses of a 5 GHz-range X/10-type FEUDT filter using Nb electrodes, (a): measured at 293 K, (b): measured at 5 K.
253
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H. Odagawa & K.
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PQ
3 S/3
O
c o C
100 200 Temperature (K)
10 Temperature (K) Fig. 7. Temperature dependency of the insertion loss of a 5 GHz-range X/10-type FEUDT filter using Nb electrodes (a), and relative resistance of the Nb film (b).
4.
Fabrication Process
In the previous section, we considered some mechanisms for the insertion loss beyond 5 GHz. The next important concern for low-loss characteristics is the structure of the device. However, for discussing the device structure, we always have to consider whether a fabrication technique is available to realize the device structure. Therefore, before describing the structure of the filter, we describe in this section the fabrication technology for obtaining ultra-fine IDT electrodes around 0.1 (im.
4.1.
Fine lithography technology
SAW device can be fabricated by applying integrated circuit fabrication techniques using lithographic processes. Many types of fine lithography are proposed. Photolithographic techniques using the Hg-sourced G-line/H-line series to I-systems
254
SAW Devices Beyond 5 GHz 1121
(wavelength of 300 nm to 450 nm) have been developed, and are usually used in modern SAW device fabrication. A more developed method uses shorter wavelength ray sources, such as the excimer-based KrF (248 nm), ArF (193 nm) systems. Wavelengths of the above sources are shown in Fig. 8. In Ultra-Large-ScaleIntegration (ULSI) technology, the 0.18 u.m process has started using the KrF excimer laser. Now the next interest is 0.13 and 0.1 u.m technology. Especially, in the 0.1 u.m process, there are several selections, such as electron beam lithography, X-ray lithography using Synchrotron Orbital Radiation (SOR), and F2 excimer laser (157 nm) lithography. Within a few years, the techniques of the 0.1 u.m process will become the mainstream.
ultra-violet
Fi
ArF
KrF
XeCI
1 1 1
1 l 1
• I •
•
visible
Hg-source high• voltage laop .
.
ligft
beam
.—•-
far-ultra-violet
g
h I
o
Intensity
P
1 100
200
_L 300
*
1 400
500
vavelength of generated l i g h t (nm) Fig. 8. Wavelength of the ray source.
4.2.
Electron beam exposure process for SAW applications
The direct Electron Beam (EB) writing techniques provide the finest line width, and an almost established technology to obtain 0.1 ^m-width electrodes. One of the drawbacks of the EB process is the throughput, because the conventional EB system uses only one focused electron beam to scan the exposure area. However, the size of the SAW device is inversely proportional to the frequency range of the device, and because it is suffices to use the EB lithography only for the interdigital electrode and it is not necessary to use for the bonding-pads and most of the lines for the electrical connection, which occupy the most of the area in the SAW device. The large area can be exposed by conventional photolithography after (or before) exposing or fabricating
255
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the interdigital electrodes by EB lithography. In this case the time to expose the IDT of one device is only a few seconds. Moreover, the writing speed of EB systems is increasing, and resist materials for EB are being developed. Therefore, it is just a matter of time before SAW device processing using EB lithography may be seriously considered for practical applications. Figures 9 and 10 show a flowchart of the etching and the lift-off techniques used for EB lithography. Each method has its own features. The most important feature of the etching process is that there are no limitations on the deposition of metal thin film for the electrode, such as deposition method and deposition temperature. Therefore, we can deposit the low resistivity metal film by the best method and the best condition available, and then we fabricate the electrode by etching. However in the etching process, the process parameters such as the etching rate of the metal film and the resist film, the determination of the etching stop as well as the process parameters in the lithography must always be carefully considered.
(1)
I
(2)
I Expose and Development
(3) Dry or Wet Etching
Fig. 9.
Flowchart of etching process.
256
SAW Devices Beyond 5 GHz 1123 On the other hand, with the lift-off process the electrode can be fabricated by a simpler procedure. In other words, the process parameters in lift-off process are fewer than that in etching process. When the inverse tapered resist pattern is obtained reproductively, the process is almost finished in the lift-off process. And the lift-off process does not cause the mechanical damage on the surface in contrast to the dry etching techniques. However, in the SAW device fabrication using EB lithography, the lift-off process requires additional film deposition to avoid charge-up by the electrons since the substrates of SAW devices are insulator materials. Figure 10(a) shows the case where a Cr film is used under the resist film, and Fig. 10(b) shows the case where an Au film is used on top of the resist film. 10 Both methods use the 0 2 plasma ashing technique.
10
Fine Al electrodes with a width of less than 0.1 u.m and a thickness of
about 30 nm were obtained both with the lift-off and the etching process. Figure 11 shows one of the Al electrodes for a 10 GHz-range ladder type SAW filter (described in Section 6), which is made by the lift-off process shown in Fig. 10(a), and the line and space width is 0.09 H-m.
EB exposure
Resist Development, 0 2 plasma ashing |. | *i^&smmmmmmmvmii
1
1
1 m>.:
^^^^^^t^t^^f^^^^^^^^m^i
Cr etching, 0 2 plasma ashing
Al evaporation
Lift off
Cr etching
Fig. 10(a). Flowchart of lift-off process for dielectric materials.
257
1124 H. Odagawa & K. Yamanouchi
E.BEAM
ill
Hi /
/
/
i ':*:*:*:*: / / /
Au (100 X)
•"•
P^aiySfxgxfj
/
/
/
V - RESIST
/
***
SUBSTRATE
Au ETCHING V -RESIST
•Y.-Y.yjyyS.-.'.'V.-i
SUBSTRATE
DEVELOPING and 0» PLASMA ASHING
1}
77V / 77V / 7 / """ SUBSTRATE
\
/
V /
/
/
/
/
/
Al EVAPORATION
ya /
/
/
-RESIST /
SUBSTRARE
V
LIFT-OFF ^—Al
/ / 7 T / / 7~7~> / SUBSTRATE
Fig. 10(b). Flowchart of lift-off process for dielectric materials.
258
SAW Devices Beyond 5 GHz
015906
20.0kV
X30.0K
1125
aa'enm
Fig. 11. Fabricated Al electrodes for 10 GHz-range SAW filter. (0.09 |im L/S)
4.3. Advanced fabrication process of narrow-gap structures using anodic oxidation techniques The electrode width of a conventional IDT with line and space ratio of 1:1 is A/4 as shown in Fig. 12(a), A being the wavelength at the center frequency f0. In contrast, the electrode width of a comparatively narrow gap IDT (NG-IDT) l0 shown in Fig. 12(b) is about A/2, and the operating frequency is twice that of an IDT using the same width of the electrodes. The operating frequency of NG-IDTs can, therefore, be increased, and the electrode resistivity reduced to about half that of conventional IDTs. Similar to the case of double electrodes, the reflections from the electrodes due to mass loading is reduced. Therefore, the narrow-gap structure is very useful for high frequency SAW
Fig. 12.
Structures of a conventional bidirectional transducer (a) and a
narrow-gap transducer (b).
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devices. Comparative narrow gaps can be obtained by separating the electrodes using an anodic oxidized film in the lift-off anodic oxidation method. Figure 13 shows the fabrication techniques using this method. The electrode separation using an anodic oxidized film is more reliable and has a better insulation between electrodes. It can be applied to the EB lithography process. An example of a fabricated narrow-gap electrode is shown in Fig. 14. " This Al electrode is made on 128° Y-X LiNb0 3 , and the period of the electrode is 0.16 urn and the thickness is 30 nm. This electrode works as a 12 GHz-range bidirectional transducer, and the experimentally measured insertion loss of the filter is about 16 dB, which includes the bidirectional loss and mismatching loss as shown in Fig. 15. In this case, because the transducers are bidirectional, the large ripple caused by the TTE appears in the pass band. However Fig. 15 shows that we can excite and receive efficiently the SAW in the 12 GHz range, and we can expect that low-loss and wide-band characteristics will be obtained if we use a unidirectional transducer. The filters using unidirectional transducers with narrow-gap structures will be described in Section 5. The narrow-gap structure and the fabrication process using anodic oxidation are useful techniques in GHz-range SAW devices.
RESIST
AUO
SUBSTRATE (a) Conventional etching, Anodic oxidation
SUBSTRATE (b)
Al evaporation
SUBSTRATE (c)
Lift off
Fig. 13. Fabrication process of narrow gap IDTs using the anodic oxidation technique.
260
SAW Devices Beyond 5 GHz 1127
n
r%. *% Tci Y > 3 ^ * •
iilllillliif:MllM^^ Fig. 14. Fabricated Al electrodes for a 12 GHz-range transducer using narrow gap structure.
1
S8 to 3 m m
*
1
•
»
"
1
*
Min.l.LsiMMB
1
"T
*
'
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ty ™f-<4
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ui
5 io k 300
1
-
.
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i
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.
i
••
,
i
•
.
ii
12
.
;••• • •'•
J
i S4
FREQUENCY [GHzJ Fig. 15. Frequency response of 12 GHz SAW filter using bidirectional transducers with narrow gap structure.
261
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5. SAW Filter Using Unidirectional Transducer Beyond 5 GHz Conventional IDTs have an inherent minimum insertion loss of 6 dB because of bidirectionality and strong pass band ripple due to TTE. Moreover, in the high frequency range, the reflection of SAWs at the electrodes become large, causing multiple reflections of SAW in the IDT similarly to what happens in SAW resonators because the relative electrode thickness to the wavelength become large. As a result, the radiation efficiency decreases markedly, and wide band characteristics cannot be obtained for low-loss wide-band filter. Therefore use of unidirectional transducers is a practical choice. Important considerations for low-loss and wide band filters using unidirectional transducers beyond 5 GHz-range are a minimum electrode width, directivity, and radiation efficiency. The minimum electrode width is an important parameter from the viewpoint of low-loss characteristics, since a reduction of the electrode resistance is indispensable for low-loss devices. Moreover, from the viewpoint of the fabrication, the electrode width depends directly on the frequency limitation. The directivity and the radiation efficiency should be chosen by considering the total specification of the required characteristics. To study unidirectional transducers is very important, not only for achieving low-loss filters but also for the development of novel SAW devices, since there are certain characteristics that can only be achieved by unidirectional filters. Also such properties are indispensable in the SAW filter with good phase characteristics and functional device such as SAW convolver and switching devices for optical applications. In this section, we describe some low loss filters using unidirectional transducers beyond 5 GHz-range.
5.1.
FEUDT
One of the most practical unidirectional transducers which are applicable beyond the 5 GHz range is the FEUDT (Floating electrode type unidirectional transducer). 5'6'7'8 Generally, the frequency response of the FEUDT depends on the reflection coefficients of SAWs at the electrodes, which are caused by both mechanical and electrical effects. The A712 type FEUDT 5'6 shown in Fig. 16(a) has unidirectional characteristics mainly due to electrical reflection at the open and shorted floating electrodes. This type has good characteristics, but unfortunately, the electrode width is too narrow beyond the 5 GHz range. Also the 3^/10 type FEUDT 7 shown in Fig. 16(b) has good characteristics. This structure can excite SAWs efficiently in second harmonics operation. Therefore, the minimum electrode width is A,2/5 in second harmonic operation. Moreover, this type can be realized in narrow gap structure using anodic oxidation techniques as shown in Fig. 16(c). 8 I ° However, the characteristics of this
262
SAW Devices Beyond 5 GHz
1129
structure largely depend not only on electrical reflection but also on mechanical reflection at the electrodes. Therefore, an increase of the electrode thickness results in a too large directivity because of the increase in the reflection coefficient with increasing film thickness. A large directivity produces a large variation of the admittance, which causes a ripple in the main lobe of the frequency response. Generally, when the mismatch at an electric port is large, the reflection of the SAW at an acoustic port of the receiving transducer increases with increasing directivity. Therefore, a directivity with a value of around 7 dB is preferred in the FEUDT to x,
(a) r
«=•
|S-*|
|6
x,
3 x,
10
10
(c) <>
-*-
2X<
(d)
£=>
x, 10 Fig. 16.
Structure of FEUDT.
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H. Odagawa & K.
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obtain low ripple characteristics. Since thin electrodes make the insertion loss large, thick electrodes relatively to the wavelength are required beyond 5 GHz. It is necessary to use a A.|/l0 type FEUDT which corresponds to a suitable directivity for the thick electrode beyond the 5 GHz range. 8 Figure 16(d) shows the configuration of a X\/\0 type FEUDT. This transducer also works efficiently in second harmonic operation. It has a symmetrical arrangement of electrodes in terms of the mass loading effect, with a directivity that does not depend very much on the electrode thickness. Moreover, this A.,/10 type FEUDT is advantageous in terms of fabrication, especially for electron beam lithography, since all the ratios of line to space are 1.0. The electrode width is X2/5 in second harmonic operation. Figure 17(a) shows a frequency response of the X^/10 type FEUDT filter working in second harmonic operation in the 8 GHz range. This filter is made by the lift-off process. This is a connecter-to-connecter characteristic. Al wire bonding is used to connect the electrical port to the SMC connectors. The length of the Al wire is chosen to cancel the susceptance in the electrical port of the transducer. This characteristic includes the TTE signal while the feedthrough signal is rejected by time gating. Figure 17 (b) shows a Smith chart of this filter. The electrical matching is not satisfactory as shown in this figure. The insertion loss will be reduced when the electrical matching is carried out completely. The substrate is 128° Y-X LiNb0 3 , the wavelength at second harmonic is ^2=0.45 |i,m and the propagation length is 46A,2. The pair number is 20, the aperture is 46X.2) the film thickness of the Al electrode is 30 nm, and the electrode width is 0.09 |a.m.
REF 0.0 dB 1 10.0 CB' V -11.9B9 OB
1 MOW poirv .
395
..
t-
5 Gl-
—
— .
1 /
\j\
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\
\A' iV w ** I Y IA r
I
I CENTER SPON
Fig. 17(a).
n
V 4*\ij
\
\
j
f I
1 1 1 1 1 8. 200000000 GHz 3. 00000O0O0 GHz
Frequency response of XVIO type FEUDT filter.
264
}
SAW Devices Beyond 5 GHz
1131
1/0 Units 300.0 mUnits^
CENTER SPPM
Fig. 17(b).
5.2.
B. 200000000 GHz 3.000000000 GHz
Smith chart of \l 10 type FEUDT filter.
A/4 electrode thickness difference type SPUDT
An electrode configuration similar to that of the conventional 2 fingers/A. IDT is also applicable for high frequency single phase unidirectional transducers (SPUDTs), since the 2/A. IDT has the highest radiation efficiency and wide electrodes. This is important from the points of view of the fine pattern fabrication and for decreasing the electrode resistance. In order to realize such SPUDTs, electrodes of different thickness are useful because they shift only the center of reflection while the radiation efficiency remains high. We have previously reported some practical fabrication processes for electrodes of different thickness. 1213 Figure 18(b) is an electrode thickness difference type SPUDT (ETD-SPUDT) u with an electrode pattern width ofk/4. Figure 18(a) shows a split finger IDT which has unidirectional characteristics due to the difference in the reflection coefficients of the electrodes of different thickness or different material. The mechanism of the unidirectionality for this transducer has been described by Hartmann and Wright. 15 It has high efficiency, and the fabrication for the GHz range is easy if electrochemical fabrication techniques are used. I2 On the other hand, Fig. 18(b) is a narrow gap version of Fig. 18(a) made using anodic oxidation techniques. The mechanism of the unidirectionality is the same as that of the structure in Fig. 18(a), and the radiation characteristics are the same as those of the conventional narrow gap IDT. This IDT is well suited for the high frequencies since the electrode width of the structure in Fig. 18(b) is twice that of the structure in Fig. 18(a), making the electrode resistance approximately half that of the
265
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the conventional 2 finger/A, IDT.
fn\
(a)
/h\
\P)
X
if
'1
r
m
+
m
-
x
l«-
.i
1*
i +
m n
1
+
—
—
Fig. 18. Configuration of ETD-SPUDTs.
(a) Split finger type,
! + \ (b) XIA type.
The fabrication process of the X/4 ETD-SPUDT is shown in Fig. 19. Only the X/4 line and space processes are used, and no accurate alignment exposures are required. One of the fabrication processes is as follows. (a) (1) An Al film with a thickness of h, (Al-1) is evaporated onto the piezoelectric substrate and resist patterns (main electrode patterns) are formed on the Al-1. (2) The Al-1 film is etched by conventional wet or dry etching, and Al electrodes (Electrode 1) are fabricated. (3) Without removing the resist pattern, anodic oxidation is carried out in order to make an A1 2 0 3 film for electrical separation of the + and the - electrodes. (b) (4) A second Al film with a thickness of h2 (A 1 -2) is evaporated from an oblique direction ' 6 shown by the arrow in the figure, without removing the resist film. (5) The areas in the shadow of the resist, which are shown as white areas in Fig. 15(b), are not evaporated by the Al film. (c) After that, a lift-off process is carried out. (d) Electrodes 1 and 2 are connected when a thick Al film for bonding pads is fabricated by a lift-off process and the desired structure is obtained. Experimental results obtained using 128° Y-X LiNb0 3 substrate and Al electrodes are as follows. Figure 20 is a scanning electron microscope (SEM) micrograph of the electrodes of the fabricated 5 GHz range X/4 ETD-SPUDT. The period of the electrodes is 0.185 (im. In this case, the thickness of Al-1 and Al-2 are 60 nm and 45 nm, respectively.
266
SAW Devices Beyond 5 GHz 1133 /K\
(a)
\
n
n
/
Oblique angled evaporation V
W
Resist
W
W
AJ =Z
?:£•*• Al - 1 Substrate
'Electreode 1
LJ
Substrate
(c) Pad layer Electrode 2
Fig. 19. Fabrication process of A/4 ETD-SPUDT.
Fig. 20.
SEM micrograph of the Al electrode of 5 GHz-range A/4 ETD-SPUDT.
Figure 21 shows the directivity of the A/4 ETD-SPUDT measured in the 500 MHz range as a function of the pair number (N) for various film thicknesses. It shows that an IDT with 40 finger pairs has a large directivity (more than 8 dB) within the practical thickness range.
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Figure 22 shows the frequency response of a 5 GHz range X/4 ETD-SPUDT filter. The Al thicknesses are 70 nm (9.5 % of the wavelength) and 50 nm (6.8 % of the wavelength), and the pair number is 30. This characteristic is extracted from the experimental S parameter measured by a microwave probe, assuming that ideal matching circuits are added at the electrical ports. In this case, a minimum insertion loss of 5.6 dB is obtained in the 5 GHz range. 10
~\—i i i | i—i—i
i |
i—m—|—i—i—r
X /4 = 2 li m :00nm/50nm ..-•'"" 150nm/50nm 100nm/50nm
5 2 0 I
i
i
i
i
10
20
30
40
50
Pair Number Fig. 21. Directivity of ETD-STUDT.
Min. I.L. = 5.6 d B
/"
MOR CER
m 20 T3 to
s -1
S
/
O tz
*5 * .93^ 'JA
polrr
\
\rv ^
A
A s 60 V A
40
.r
/
/
•
i
A,, ^N. VI A/ ' V
\
1 100 4.5
5
5.5
Frequency (GHz) Fig. 22. Frequency response of 5 GHz ETD-SPUDT
268
SAW Devices Beyond 5 GHz 1135
6.
Ladder Type Filter
Another type of SAW filter, called ladder type filter, 16 does not rely on SAW propagation, but instead uses SAW resonators as impedance elements. Using ladder type filters, low-loss characteristics have obtained beyond the 5 GHz-range. ' ' In this section, low-loss ladder type filters in the 10 GHz-range are described. In order to study the characteristics of a ladder type filter from equivalent circuit model analysis, we determined the parameters of the equivalent circuit experimentally, as no studies have been carried out on the reflection characteristics of SAWs for thick electrodes. Figure 23(a) shows the admittance characteristic of a 5 GHz-range SAW filter using a A/4 width conventional bidirectional IDT on 128° Y-X LiNb0 3 . The finger pair number of the IDT is 18, the aperture is 30 X, and the thickness of the Al film is 60 nm (0.08A.), where X is 0.75 urn. Large ripples are caused by the reflection in the receiving IDT. We determined the parameters by fitting the calculated results to the «EF 6 . 0 Limits 3.0 ntlnits'' •v
REF B.O U~ut.» A 3.B n O - i l t ^
n.aea wu
MOR poi«
*j
A.ITG
nu
<ER L . 0&•& 7 ; 5 o lz
lA 'I I Jw •A
•
.
vf
^~-
v
\\
-^r
CENTER 3.0650000000 O-fct ?>CN
2.000000000 GHZ
(b) ,6,20 03
< 0.8
1
1.2
Normalized Frequency Fig. 23.
Admittance characteristics of the 5 GHz-range filter using conventional
bidirectional IDT. (a) experimental, (b) calculated results.
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
Port 2 (OUT) o /
• IN
IN
/
I Mil
ntm
A
ntm ntm
Fig. 24.
A
IS ntm
Structure of the ladder type filter.
Table 3. Parameters of the ladder type filter in 10 GHz.
Pair number
Aperture
Period (wavelength)
Resonator A
120.5
11.8 XP
0.38 urn
Resonator B
100.5
lO.OXs
0.37 urn
270
SAW Devices Beyond 5 GHz
1137
0 m
10
o _i
|
30 o>
in
•E 40 50 0.95
Fig. 25.
1.00 1.05 Normalized Frequency
1.10
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.
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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
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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.
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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
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-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
(4)
are determined instead of the propagation time, where f0 in Eq. 4 stands for the operating frequency. In order to eliminate the influence of the radio channel affecting the sensor signal, only time or phase differences are evaluated ("on-chip reference").
I
2
3
4 5 S delay time §ts]
a
9
10
Fig. 13. Time domain response of a temperature sensor at 869 MHz.
A duration between the first and the last reflector of 2.27 |is, a delay temperature coefficient of 85 ppm/°C and a phase resolution of the transceiver unit of ± 5° allows a temperature resolution of 100 mK to be achieved. If the temperature varies by more than 6°C (Fig. 14), the phase shift is greater than 2 % and a phase ambiguity occurs. As a consequence, a minimum of three reflectors is needed if absolute temperature values have
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to be measured. If the sensors are to operate over a broad temperature range, even more than three reflectors would be preferable. The shortest time difference Axmin = AT3-AT2 of the reflected pulses is 30 ns. The measurement range is thus approximately 450°C.
I c o
7^
~
Ax = 2.27 us
3 ^ AT, = 0.84ns
>
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0
: : :
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I I I I I I I I.I I I I I I I
I I. I I
15 20 25 30 35 temperature variation [°C]
I I I I I I I I I.I I I I
40
45
50
Fig. 14. Phase shift vs. temperature change of reflected pulses.
The temperature sensors were successfully tested from -196°C, the boiling point of liquid nitrogen, to several hundred °C. The temperature limits of the various sensor compounds are listed in Tab. 1. Table 1. Temperature limits of sensor compounds. Compound Standard solder material PTFE-cable with clamped connectors FR4-laminate for increased temperatures Silicone adhesive High-temperature solder 82Aul 8In Metal and ceramic package A1203 substrate Al electrodes Pt/Ti electrodes Li2B407 LiNb03, LiTa03 a-Quartz La3Ga5Si014 (LGS)
Temperature limit 170 °C = 200 °C 260 °C, 1 h 350 °C, 1 h 450 °C 500 °C-700 °C 1200-1450 °C 450 °C 1000°C 230 °C 300 °C 573 °C 1470 °C
Standard package technology can be used up to approximately 200 °C. Above this temperature, special adhesives, solder materials and printed circuit boards are required.23 Aluminum electrodes withstand temperatures up to 450 °C. SMD packages resist extreme conditions for at least several hours at about 500 °C - 600 °C. At these temperatures, an increased mobility of metal ions (Au, Ni, Fe, Co) out of the package leads occurs. Thus Ni migrates onto the chip and affects the SAW propagation characteristics. The dielectric isolation of metal package pins and the metal multi-layers in ceramic SMD packages are destroyed at about 700 °C. Noble metals with a high melting point, such as Pt, Rh, or Ir,
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are best suited for electrodes in high-temperature (HT) applications. The adhesion on the wafers can be promoted by a Ti layer. Li2B407 exhibits increased ionic conductivity at elevated temperatures. An enhanced decomposition of LiNb03 can be observed above 300 °C. Nevertheless, between 400 °C and 500 °CS temperatures at which semiconductor devices are irreversibly damaged, LiNb03 substrates remain stable for several days. The well-known phase transition from a-quartz to (i-quartz occurs at 573 °C. LGS seems to be the most promising HT substrate. There is neither a phase transition between room temperature and its melting point of 1470 °C nor any decomposition over an observation period of several weeks at an oven temperature of 850 °C. Figure 15 shows a temperature sensor on LGS sealed with a vitreous silica ampule.
Fig. 15. 433 MHz radio thermometer for high-temperature measurements.
2.4 SA Wpressure sensors A schematic drawing of a SAW pressure sensor is shown in Fig. 16. The SAW propagates on a quartz diaphragm while bending under hydrostatic pressure. A constant reference pressure must be applied to the other side of the diaphragm in order to bend it in a defined manner.24,25 This is implemented by fabricating a hermetically sealed cavity containing the reference pressure: a sand-blasting unit was used to make a blind hole in a quartz cover plate of the same substrate material as the diaphragm. The diaphragm was then attached to the cover plate with an epoxy adhesive.
Fig. 16. Schematic drawing of a SAW pressure sensor.
This all-quartz package sensor is expected to have minimal thermal stress and thus a low temperature sensitivity. To produce pressure sensors with an operating life of several years, use is also made of the glass frit technology already described.26
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Fig. 17. FEM analysis of the membrane deformation just before breaking at a pressure of 600 kPa. The dark frame represents the adhesion area of the membrane and the cover plate.
A Finite Element Method (FEM) program is used to determine the geometry of the sensor, since it is very difficult to compute the mechanics of anisotropic materials analytically. The bending of the diaphragm (Fig. 17) and the cover plate as well as the stress/strain distribution were calculated (Fig. 18). It can be seen from Fig. 18 that both compressed and stretched sections exist. The phase velocity of the SAW is lower in the stretched section and higher in the compressed section.
0
2
4
6
8
10
12
14
16
18
20
location si diaphragm / mm
Fig. 18. Calculated stein of the diaphragm at a pressure of 250 kPa, positions of the mterdigital transducer and the ten reflectors.
The reflective delay line requires at least three reflectors. In an optimal reflector configuration, two of these are ideally positioned at the transitions between the stretched and compressed sections. Because this pressure sensor was used as a test device, the delay path was divided by ten instead of three reflectors to obtain more detail. The measured and calculated phase differences between pulses reflected from adjacent reflectors are compared in Fig. 19. Very good agreement was achieved except at the
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boundary of the membrane where this was prevented by mechanical deformations caused by the joining of the quartz plates.
100 80 60 40 20 0 -20 -40-] -60
1
1 l,rl,rhi,rl,n,rl,l I,
Fig. 19. Comparison of phase differences Aq> 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
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- w, "
' '
, ,
t"
f
''
• 'O
•
^
^
*
+ 160"
ft
+ 140v f-120"
1
•HOO1-
Fig. 22. Lines of constant phase
A new high-Q capacitive differential pressure sensor based on two quartz plates was developed.27 Figure 23 shows the measured phase difference A
2 3 p [100 kPa] Fig. 23. Measured phase difference A
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2Ǥ SAW torque sensors Contactless slip-ring-free torque measurements can be performed with passive RF SAW torque sensors without the need to break up the driving shaft.30"31 SAW torque measurement systems can also operate in industrial environments subject to strong electromagnetic interference thanks to their high operating frequencies of between several hundred MHz and approximately 3 GHz and the inherently broadband nature of the SAW sensors. This has been confirmed by torque measurements in a 750 kW highvoltage motor as well as in an identification system for railroad applications where no interference from the parasitic radiation from high-power inverters could be observed. Like resistive strain gauges, SAW sensors measure the torque indirectly by detecting the strain or stress distribution generated by a torque acting on the shaft (Fig. 24).
M Sf
The fact that the strain components have opposite signs in the ± 45° directions relative to the shaft axis can be used for temperature compensation. In our experiments, sensor elements were made with two and three overlapping SAW paths oriented in different directions on a single chip. Figure 25 shows a torque sensor on quartz.
Fig. 25. SAW torque sensor mounted on a steel plate.
The SAW paths are orthogonal to each other. The acoustic tracks are slightly slanted due to the beam-steering effect. Because it is impossible in a production environment to
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mount the sensors directly on the shaft, we generally mount them on a steel ground plate and attach the packaged sensors to the shaft with two annular clamps.
3. SAW Identification and Sensor Systems The range of ID applications covers the sectors of security/access control, manufacturing/logistics management and transportation: these include personal access, parking-lot access and anti-theft systems, electronic milestones, container and trailer identification, fleet management, identification of paper and cloth rolls or even the identification of trains and railroad car axes. In the same way as SAW identification systems offer new innovative system solutions for ID applications, wireless SAW sensor systems can also play an important role in improved control, monitoring, maintenance and diagnosis systems by reducing waste, labor and energy costs, protecting expensive equipment, extending equipment life, optimizing process efficiency, avoiding unscheduled downtimes and allowing maintenance on demand. 1 1 SAW IB systems for railroad applications One of the first identification systems of this type was installed in 1989 at toll stations on expressways near the city of Oslo, Norway, where vehicles are identified remotely without stopping.32 Our latest ID system was optimized for railroad applications. A chirp radar operating at 2.45 GHz was developed. It uses a pair of dispersive SAW filters for pulse expansion and compression and has a maximum readout distance of 2.5 m. Its peak output power is lOdBm, which is much lower than that emitted from mobile phones. Because a readout procedure requires only a few microseconds, 100,000 interrogations can be performed per second, thus permitting reliable identification of particularly fastmoving objects. The maximum velocity of a train passing by can be 350 km/h. Figures 26 and 27 give an impression of the installation used in Munich's subway system, where the identification system has operated highly successfully for several years.
Fig, 26. Munich subway car with radar antenna
Figure 28 shows the ID tag used in detail. The plastic housing contains a microwave patch antenna with the SAW reflective delay line mounted on it. The antenna was designed for a gain of 6 dBi. The SAW device has a total of 20 reflectors, providing a code space of 16 bits and a 4 bit parity check.
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Fig. 27. SAW ID tag mounted on the car chassis (white).
Another interesting application is the identification and temperature control of freight train axles. Axle maintenance, especially the surveillance of lubricants and operation time, represents a major factor in railroad security. However, the problem arises that a majority of these trains do not have any electrical installations to supply power to the sensor devices. So the advantage of passive operation becomes important. Figure 29 shows a photograph of a patch antenna that fits around train axles. A number of recent experiments have confirmed the feasibility of this form of remote temperature measurement and axle identification.
Fig. 28. SAW ID-Tag in a plastic housing for railway applications. The dimensions of the housing are 158x90x22,5 mm3.
Fig. 29. Axis antenna for identification and temperature sensing at freight train wagons.
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3J SAWID systems far manufacturing and logistics At the end of 1999 a field test was carried out at the automotive manufacturing plant of DaimlerChrysler in Sindelfingen, Germany. Thirty seven SAW ID tags were mounted on skids which passed the reader antenna approximately 125000 times over a period of several weeks. More than 99.9 % of them were read out correctly. The measurement setup is presented in Fig. 30. The reader electronics was assembled in a portable aluminum box together with a laptop computer for processing and visualizing the data (Fig. 31). A low-cost tag was developed for this application. The SAW devices are mounted on thin metal sheets with slot antennas (Fig. 32). The intriguing features of the SAW ID system are its long readout distance and the temperature stability of the SAW devices, allowing highly flexible assembly and the need for only a single ID system in the entire production process.
Fig. 30. Measurement setup in an automotive manufacturing plant.
Fig. 31. Reader and laptop computer in portable box.
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Fig. 32. SAW devices mounted on metal sheets with slot antennas.
3.3 The intelligent tire An "intelligent tire", i.e. a tire which provides information about its current state will considerably improve drivers' safety as well as vehicle stability. Thus it is common knowledge that 80 percent of all flat tires result from a gradual loss of pressure rather than actual punctures caused by nails or stones. If such a creeping decrease in tire pressure can be detected in time, many accidents could be prevented. Figure 33 visualizes the interaction between a tire and road/chassis.
¥#h«et# ^
^
Ufhtcfe
*&*n £&£***
SiirfcK* Suffact — Fig. 33. The interaction of a tire with the vehicle and road.
Several battery-powered systems have hitherto been implemented33,34 for the measurement of tire pressure. However, they suffer from a limited operating life of the pressure sensor unit due to the extreme tire environment. This problem can be circumvented with passive wireless SAW pressure sensors. As described in section 2.4S we developed a hybrid SAW pressure sensor with the ability to satisfy the requirements of the automotive industry.35 The prototype sensor was designed for a pressure range of 100 kPa up to 400 kPa and an excess pressure stability of 600 kPa. It can be seen in Fig. 34 how the prototype module with an antenna is attached to the rim. Measurements of road friction with SAW sensors directly vulcanized into the tread elements of tires supply valuable information for future driver-assistance systems
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and for optimizing slip-control and vehicle-stability systems.36,37 Indeed, this can be accomplished more precisely and conveniently than with the data supplied by antiblocking systems (ABS). Figure 35 shows a schematic drawing of the SAW friction sensor, principally a housed bending beam with a small metal nose at the end. Deformations of the tread elements were investigated on surfaces with different friction coefficients |i (Fig. 36). Curve A represents the behavior under diy conditions (high p), whereas curve B represents a wet surface (low jx). Both curves show similar behavior while the tire is rolling over the tread except in the area around the curve maximum where the phase shift is much greater on the dry surface.
Fig. 34. SAW sensor module for pressure measurement mounted on the rim. Tread Element Deformation Flag
Sensor Housing
SAW Bending Beam
Fig. 35. SAW road friction sensor integrated in tread element.
distance / cm
Fig. 36. Phase variation of sensor signals for a dry (A) and a wet (B) surface.
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3.4 Motor diagnosis and control SAW sensor systems can also be used to advantage for the measurement of vibrations, rotor temperature or torque in electric motors. We successfully tested our SAW sensors in a small 11 kW asynchronous motor as well as in a high-power 750 kW two-pole machine. The temperature sensor that measured the curve shown in Fig. 37 was mounted on the short-circuiting ring of the rotor of the 11 kW motor. The SAW sensor was read out via a wire-loop antenna mounted on the bearing plate. The motor was loaded with an eddy-current brake. During run-up and operation under load, the motor warms up as expected. When the motor is stationary, the temperature curve shows an exponential decay. The crosses mark temperature measurements made with a reference thermometer. Because the thermal contacts of the reference thermometer and the SAW device are not identical, there is a slight difference between the measured temperature values. Figure 38 shows the behavior of the torque after a step excitation of the nominal value of the inverter. Because the test stand was inverter-controlled, the torque increases relatively slowly with time. The solid line shows the measurement results of the SAW system, whereas the dashed curve represents the measured values of a reference-shaft torque sensor connected between the driving and load machines.
^ . - ^ — reference measurement
^
rotation operation under load
-/
time [min] Fig 37. Temperature behavior of the rotor of a 11 kW asynchronous motor during operation.
100 80 60
A ft A
40
/*/ V .
70 0 -20 -40
, t
/•w~ v «-•..* )
0.02
0.04
0.06 0.08 time [s]
0.1
0
Fig. 38. Change in the measured torque after a step excitation of the nominal value of the inverter (— SAW sensor, reference sensor).
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3J High-voltage applications The possibility of reading SAW sensors out over long distances makes them an ideal candidate for high-voltage applications, as isolation problems can be easily avoided. The advantage of SAW sensor systems for power transmission and distribution systems can be seen from the following example.38 At the end of 1997, a long-term field test of a SAW temperature monitoring system for high-voltage surge arresters started in cooperation with Preussen Elektra at a 420 kV substation near the nuclear power plant of Bnmsbuttel, Germany. In the meantime, a second prototype system has been installed at the Celilo converter station in Oregon, USA. Surge arresters are used to protect equipment from excessive overvoltages. In Fig. 39 a surge arrester built up of two stacked units can be seen. Metal oxide (MO) varistors are stacked inside the porcelain insulators (Fig. 40). Some of the varistors were substituted by aluminum housings containing a SAW sensor. The slot in the wall of the housing acts as a sensor antenna. Each porcelain-housed arrester unit was equipped with a SAW temperature sensor located approximately at the center of the MO varistor inserts. A third SAW sensor was mounted outside the insulators at the intermediate flanges of the arrester, as indicated in Fig. 39.
Fig. 39. 420 kV surge arrester in a PneussenEfektaa substation, equipped with a SAW temperature monitoring system.
The antenna of the radar unit and the RF front-end module are attached to an overhang beam at the arrester base. The RF front-end is connected to the signal processing part of the radar unit by a coaxial cable laid in the ground. A wireless radio modem transmits the
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measured temperature data to Berlin, where it is further processed. The main features of this SAW temperature measurement system are that any possible long-term temperature increase (aging), sudden temperature rises (event counter) and energy absorbed by discharge currents (energy monitor) of the arrester during its operating life can be easily detected.
Fig. 40. SAW temperature sensor in a stack of MO varistors.
The energy monitor - a feature which has not hitherto been available for this application - is based on the simple correlation existing between temperature rise and absorbed energy. Figure 41 shows the measured temperatures of both units and the ambient temperature during the first year. Only night temperatures (a single reading taken between midnight and 5 a.m.) were used for this evaluation. The effects of rapid temperature changes, often observed during the day, and in particular the disturbing influence of solar radiation, were thus eliminated.
Fig. 41. Measured temperatures (night values).
Other possible high-voltage applications include temperature measurements of overhead lines, contact surveillance of high-voltage air-insulated disconnectors and temperature monitoring of medium-voltage high-power semiconductor switches.
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4. Event-Driven SAW Sensors 4.1 Overcoming some limitations of radar-type SA W sensors Although there are many applications for the radar-type sensors described above, two main limitations prevent this sensor system from being used on an even wider scale: 1) A maximum reading range of about 10 meters (with directional antennas) 2) A maximum of about five sensors per transceiver The first limitation is simply a result of the high signal attenuation along the RF signal path, as described in this simplified form of the radar equation: P, X2 G2 P '= A D<
(6)
The power of the received signal Pr increases with the square of the wavelength, the transmission signal power P, and a geometrical factor G depending on the antenna configuration. However, it decreases with the SAW element attenuation A (usually about 30 to 50 dB) and the fourth power of the reading distance D, which also explains why large reading ranges cannot be achieved with the limited transmission power of 10 mW stipulated by the radio frequency control restrictions. The second limitation results from the synchronized answer mode of different sensors to a single RF request. Since all sensors within the detection range of a transceiver device respond immediately, just like several mirrors in a room to a light flash, there is no way of distinguishing more than a few echo signals. To do so would require the use of differentiated sensors which respond within different time slots - and would allow the use of up to five different sensors, each with a limited number of reflectors and therefore a limited identity-code capacity. To overcome these two main limitations, 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 changes, movements or torque forces, temperature change etc., then why not simply use this energy to operate the sensor instead of supplying it via radio waves? The new sensor type described below therefore does not reflect radar signals, but generates its own RF signals using energy drawn from the sensed process. We have called it the event-driven sensor. In order to operate, this self-sustaining sensor transmitter needs to perform the following steps: • • • •
It must generate a high voltage by "tapping" the energy of the event to be sensed. It must produce a single, damped high-frequency oscillation. It must encode the high-frequency pulse with the SAW element, thus adding identity code and the absolute value of the reading to the signal. It must transmit the encoded high-frequency pulse via an antenna.
The following sections describe how this can be achieved with a simple device and no active electronic components.
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4.2 Operating principle
reflectors
pyro- or piezoelectric energy converter
piezoelectric
RF Bandpass filter
Fig. 42. Basic Setup of Event Driven SAW Sensors.
The basic circuit of such an event-driven sensor is illustrated in Fig. 42. A high voltage is initially generated by using the piezoelectric or pyroelectric effect - the required energy being obtained from a force or temperature change respectively. This voltage causes a breakdown of a gas-filled spark gap, thus exciting an oscillation in the high-frequency bandpass filter. A single damped high-frequency pulse is then generated. The pulse enters the interdigital transducer (IDT) on the piezoelectric crystal and is then transformed into a mechanical surface acoustic wave. This SAW propagates along the piezoelectric crystal and is partially reflected by reflectors placed in the acoustic path. The reflected parts are reconverted into electromagnetic pulses by the IDT and the resulting signal chain is radiated out by the antenna - just as in case of the radar-type sensor described above. The only difference is the origin of the requesting highfrequency signal, which is no longer generated by the radar device but in the sensor itself. The signal formed by the SAW reflections is again an image of the geometrical arrangement of the reflecting structures on the SAW element surface, and the minute mutual distance between the separate pulses further depends on the velocity of the acoustic wave on the SAW element. This sound speed is influenced by temperature, mechanical tension and other conditions affecting the crystal. The signal thus contains an identity mark (in the pattern of the reflections, up to 32-bit binary code space, i.e. 232 different codes) and information on the sensed property contained in the minute distance between the pulses. Both the code and the sensed property can be determined by processor-based evaluation of the transmitted signal, which takes place in a receiver unit. Let us now take a closer look at the new components added to this sensor.
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4.3 Radio frequency waves without semiconductors: the spark-gap transmitter The spark-gap transmitter was an early device for generating electromagnetic radio waves which operated with no amplifying elements such as vacuum tubes or semiconductors. First demonstrated by David Edward Hughes in 1879, it became widely known thanks to Marconi's famous transatlantic radio transmission experiments in 1901 39 and soon played a key role in the dissemination of radio transmission systems The basic assembly of such an historic transmitter is shown in Fig. 43.
SU.4****
Loading 'tfiffucttnK*
t? Secondary * 17" '•-Power Tr0rtsfmrnf>^f Fig. 43. Basic circuitry of an early spark gap transmitter.
A high-voltage source charges the capacitor until the breakdown voltage of the spark gap is reached. The spark gap rapidly changes its resistance state from high to low and the resulting current through the primary coil stimulates a damped high-frequency oscillation in the secondary, which is radiated out by the antenna. We employ the same principle in our event-driven sensors - with a number of modifications as regards power level, frequency range, high-frequency filters and power source. The spark gap turned out to be the crucial part of the whole concept - because for our application we wanted to excite frequencies in the UHF (ultra high frequency) wave range and not in the long to short wave range. So the switching time of our spark gap had to be in the sub-nanosecond range, which was difficult to achieve. Moreover, the "off state of the spark gap had to offer a very high resistance in the 10 TQ (1013 ohms) range because of the high impedance of the potential voltage sources. These high demands meant that the gap could not be operated in the open air like the historical examples but had to be sealed in a gas-proof housing with optimized gas type and pressure in order to ensure stable conditions and high insulation in the "off state. The properties resulting from this optimization are shown in Fig. 44. The current in the "off" state, known as the dark current of the spark gap, increases exponentially with the applied voltage until the breakthrough point is reached. This means that a small current always flows before igniting as can be seen in the curve; it is caused by free ions in the gas filling.
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In our spark gap, the dark current has a maximum value of about 10"13 amps at 2,500 volts, which is sufficient to prevent interference from the voltage-source impedance as explained in the section on pyroelectric energy generation. After the ignition, the plasma state of the gas between the electrodes remains stable until the current falls below a certain level - in our case when the capacitor is almost completely discharged. Current [A]
1
0
500
1
I
1
1000
1500
2000
L
^~
2500 Voltage [V]
Fig. 44. Characteristic curve of the developed spark gap.
The next optimization steps aimed to produce a spark gap which generated highfrequency oscillations in the UHF range with high efficiency. To do this, theoretical and experimental methods were applied to examine the geometry, spacing and material of the electrodes. The primary task was to produce very fast switching to the "on" state in less than one nanosecond.
1b 10 -
mplitude
,, .-* re
5 0 -5 -10 -15 -
0
Jlkl»„iL,„„
|]j|ll|!|| II, ji||!|l IJlIilii , \
i ;, .1 ih i'iJii'nJMllllilimiJILlUil
IPP^ 11 ll'ln 25
ll™
50
75
100
time [nsec]
Fig. 45 Spark gap generated 433 MHz RF pulse (time domain view).
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Figure 45 shows a time-domain signal of the damped oscillation which is generated by our finally optimized spark gap. The oscillation time is about 50 ns, which means a signal bandwidth of 20 MHz, thus perfectly matching the high bandwidth requirements of the SAW element. A frequency range up to 2.5 GHz was investigated. Figure 46 gives an impression of the peak power which can be generated by discharging a capacitor of about 3 pF and 2500 V into the oscillating circuit.
3 JC C3 3
cpow er[dl
C o [UI$ ca
,KJ
OH
500
1000
1500
2000
2500
3000
frequency [MHz] Fig. 46. Measured peak power of the spark gap generated RF pulses, measured with 10 MHz.
The diagram shows the peak power attained, measured with a spectrum analyzer at a bandwidth of 10 MHz for frequencies between 400 MHz and 2.5 GHz. Taking into account that the signal-encoding process with the SAW element causes a signal power attenuation of about 30 dB, it can be seen that the useful frequency range for these kinds of sensors is from 100 MHz up to about 1 GHz. The power of the sensor reflections is reduced to -5 dBm at 1 GHz, which gives a sufficient detection range of the signal for many applications, as will be shown by the following considerations: Assuming that the propagation range is limited mainly by the thermal noise level, which is -174 dBm for each 1 Hz of bandwidth, the maximum path loss can be calculated by adding: + 70 dB: + 14 dB: + 10 dB: + 0 dBi: + 5 dBm:
for a system bandwidth of 10 MHz signal-to-noise ratio SNR required for a correct evaluation receiver noise figure (2 dB) and external noise sources (8 dB) antenna gain transmission power level of the sensor signal - 5 dBm at 869 MHz
•=> The maximum path loss which can be overcome with the system requirements set to the above values is 75 dB. Figure 47 shows the transmission path loss for free-space RF propagation (calculated) and for propagation in buildings of different types (measured) at 869 MHz. The horizontal line indicates the maximum path loss of 75 dB. The maximum measurement
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range in meters that can be achieved can be seen at the points of intersection. So the maximum transmission range in the case of free-space propagation distance can be estimated to be about 200 meters. The maximum propagation range in the case of reinforced concrete buildings with two brick walls in the propagation path is about 30 meters. attenuation [dB] free space propagation propagation in large, furnished rooms propagation in corridors (2x2 Meter) propagation at single floor level(reinforced concrete, 2 massive brick walls in the propagation path)
-40"-
-50--
-60—
-70—
^sT
i max. tolerable attenuation: 75 dB -80" (-> max. range)
10 50 100 distance in meters 20 Fig. 47. Sensor transmission path loss and maximum detection range in buildings and free space at 869 MHz.
We will now examine how the power of the signal to be sensed is converted into the high voltage required. As already noted, a small capacitor of about 3 pF must be charged at 2,500 volts in order to ignite the spark gap. The energy in this capacitor is as low as: E=2
C V2=-
2
3 10-12
25002Wsec =9.4nWsec
(7)
In order to compare this with dimensions that can be more easily visualized, it may be mentioned that the energy needed to heat a drop of water (about 20 mg) by IK is some 13,000 times greater. This explains why such a tiny amount of energy can be easily drawn from the sensed processes. The reason why this amount of energy, which is hardly sufficient to power a modern semiconductor-based RF transmitter, can power our sensor transmitter lies in the high bandwidth or very short transmission time. Because the energy stored in the capacitor is released in a very short time, considerable transmission powers are produced.
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4.4 Piezoelectric energy conversion and piezoeieciricmlly powered sensors Many processes which we want to sense contain mechanical movements that can be "tapped" by our sensor using the piezoelectric effect: pressure changes in liquids or gases, weight changes, torque changes on axles or hand-operated keys, vibrations on mechanical parts etc. This effect is exhibited by some crystals and polyciystalline materials and consists in the generation of an electric voltage as result of applying mechanical force to the material in a certain direction. The piezoelectric effect was first examined thoroughly by Jacques and Pierre Curie in 1880 and is now well understood and widely used in commercial applications such as ultrasonic transducers and tweeters, piezoelectric motors and transfomiers, quartz crystal time bases, piezoelectric fuel-injection valves and many others.40
high voltage connector
Q
«^ construction with two levers Fig. 48. High voltage spark generator for light switches.
Piezoelectric ceramics are polycrystalline materials which offer a particularly easy way of converting mechanical "events" into the required high voltage. Commercial examples of high-voltage generation are gas-fuelled cigarette lighters with piezoelectric ignition and piezoelectric lighters for gas cookers. To demonstrate the operation of our system, we decided to implement light switches that can be used in private homes, for example. These switches consist of a piezoelectric energy converter that acts when the rocker switch is pressed and activates the spark gap transmitter with SAW element coding described above. We used commercially available piezoceramic cylinders 4 mm in diameter and 15 mm in length with metallization on their end faces. Because the maximum permissiblemechanical deformation of this element is in the region of several pn, it was necessary to develop the force transducer based on a two-lever principle which can be seen in Fig. 48.
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ip*w I T • I * t Fig. 49. Both cirquit board sides of the experimental spark gap transmitter for wireless switch applications.
The properties achieved for this spark generator are: two sparks with 3,000 V at 5 pF load capacitance a mechanical input energy of 5 N and a lever distance of 5 mm The spark-gap transmitter used can be seen in Fig. 49, which is a photograph of both sides of the 869 MHz high-frequency board containing the spark gap, a SAW element and a high-frequency bandpass filter in stripline technology. The spiral antenna combines small size, low cost and acceptable efficiency. This antenna type is also preferred in mobile phone devices. The results obtained with this experimental version of the switch in terms of detection range and code recognition probability are almost as good as expected on the < basis-of the theoretical considerations discussed in Section 5.8. The main potential for improvements can be seen in the efficiency of the piezoelectric energy converter and the code capacity and attenuation of the SAW element. The model built is not yet a prototype suited for production, so one of the next tasks will be to develop a technology that allows production at a competitive cost level. Our laboratory version attained the following parameters:41
-
energy at the rocker on-switch for generating two transmissions: 5N, 5 mm frequency of operation: 869 MHz bandwidth 20 MHz RF pulse power delivered by the spark gap measured at the bandpass filter: 27 dBm RF pulse power of the pulse chain, resulting from the SAW encoding: -7 dBm code capacity of the SAW: 16 bits possibility of extracting sensory information (e.g. temperature or force from the transmitted signal (not used in these experiments, see next section)) duration of one transmission: 5 p e c antenna gain: 0 dBm
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We see many potential applications for the components described here, i.e. piezoelectrically driven switches and sensors that transmit an identifying RF code and a sensor value when a force is applied. The following list is intended to give some ideas for potential applications. As the technique is still in its beginnings, this list is certainly incomplete. However, some applications that seem reasonable at the present time may well turn out to be impractical or unfeasible. It must be emphasized that all measurements accessible to radar-type sensors can also be performed by these event-driven sensors provided that a small source of energy may be tapped by the piezoelectric converters, which must be adapted to the specific environment. Private home market: Light switches, state monitoring of windows, shades, alarm switches, remote controls or electronic keys. All elements operate without wiring or a battery and transmit signals to a single receiver which performs the specified tasks. Production control: Hand-operated switches and remote controls, event-operated switches (by vibration, knock, excess force or pressure), state transmission of balances or scales. Transportation: Force measurements such as tire pressure, torque, weight, hand-operated switches in cars and trains, event-operated switches (by vibration, knock, excess force or pressure). Energy: Wireless state monitoring of high-voltage parts such as circuit breakers. Healthcare: Personal alarm switches, access control. 4.5 Pyroelectric energy conversion andpyroelectrically powered sensors The other mechanism we utilize to power the event-driven sensors is the pyroelectric effect. Pyroelectricity may be found in some crystals and ceramic compounds, which often also show piezoelectric properties. Although not as well known as the piezoelectric effect, it was already mentioned by the ancient Greek philosopher Theophrastus 2,300 years ago. He described the pyroelectric effect of tourmaline, which attracted small particles of ash when heated up and repelled them when cooled down. The term pyroelectricity was coined by the physicist Brewster in 1824, who derived it from the Greek word for fire, namely pyr. The effect consists of a directed separation of electric charges at opposite faces of pyroelectric crystals as a result of a temperature change. The crystal dimensions and a photograph of some of the pyroelectric materials used can be seen in Figs. 50 and 51.
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z- axis
^
4-
c
Fig. 50. Crystal dimensions and dc equivalent circuit.
Let us take a look at some properties of these materials which are important for selecting those which offer optimum performance for our purpose. The strength of the pyroelectric effect is characterized by the material constant p [C m"2 K"1], which means that the separated electrical charge Q depends on the temperature change and the crystal space ab as follows: Q(AT)=p
a b AT
m
Further, if we know the electric capacitance of the crystal plate capacitor, given by
C=eQ-er
a-b
(9)
d
then we can calculate the resulting voltage as a function of the temperature change d
V(AT)=-
AT
(10)
which yields the result that the voltage does not depend on the crystal dimensions a and
••|pK8
Fig. 51. Pyroelectric materials: ceramics (dark) and monocrystals.
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b, but solely on the dimension d along the z-axis. The energy generated by a temperature change can now be determined to be a b d (p AT)2 F,/AT1 E(AT) = ^ — 2 e0 er
(11)
This shows a linear influence of the volume on the transformed energy as well as the importance of the pyroelectric constant p. Another important parameter is the leakage resistance R of the material, given by the electrical resistivity p as R =Q
d
(12)
a b
This means that the crystal will discharge by 36% within a time tdischarge (discharge = ^ 0
%r
G
(13)
This duration must be greater than the time needed for generating the spark-gap breakthrough voltage in the event of slow temperature changes - otherwise the breakthrough will not be reached. Interestingly, this time does not depend on the crystal shape and thus cannot be influenced by means of the architecture. The last property we need to know about in this context is the Curie temperature of the pyroelectric material. If the material is heated up to this point, the pyroelectric effect disappears, so our system must be maintained below this temperature. Some crystal materials are listed in Table 2 below in order to determine the most suitable one for our application: barium titanate BaTi03, lithium niobate LiNb03 and lithium tantalate LiTa03. Table 2. Some material constants of the investigated crystals.
Material
Curie temperature T c 120 to 160 °C* 1210 °C 603 °C 1
BaTi03 LiNb03 LiTa03
Permittivity e 1600 29 43
Resistivity P lE+12Qm 5E+12Qm 5E+12Qm
Pyroelectric constant p 20 nC crn2.rC' 4 nC cm2.K_1 19 nC cm llC 1
The important properties, in particular the voltage and energy generated for these three materials, can be calculated with the aid of the formulas derived above. The basis of the calculation is a temperature change of 1 K, a crystal thickness of 1 mm and a crystal face of 1 cm2. The results are summarized in Table 3.
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Material
Pyroelectric voltage
Pyroelectric Capacita Insulation energy nee
Discharge time
BaTi0 3
14 V
141 nJ
1.41 nF
10 TQ
LiNb0 3
155 V
311 nJ
25 pF
50 TQ
LiTa0 3
499V
4740 nJ
38 pF
50 TQ
14,160 s (=4h) 1,250 s (=21 min) 1,900 s (=33 min)
On this basis, we want to calculate the crystal volume needed for a given energy output of 4.77 uJ K"2 which is generated when the output voltage of the crystals with a face of 1 cm2 changes by 500 V per K temperature change (Table 4). Table 4. Required crystal sizes for a given energy change for different crystals.
Lithium tantalate LiTaCh
Lithium niobate LiNbCK
Barium titanate BaTiCh
a b = 10- 10 mm2 d= 1 mm Volume = 100 mm3 V = 500V.K-1 E=4,77 uJ.K-2 C=40 pF R=50 TQ
a.b = 21 -21 mm2 d = 3,22 mm Volume = 1420 mm3 V = 500V.K-1 E=4,77 uJ.K-2 C=35 pF R=36 TQ •discharge = 21 min
a.b = 9,6 • 9,6 mm2 d = 35 mm Volume = 3175 mm3 V = 500V.K-1 E=4,77 uJ.K-2 C=37 pF R=380 TQ tdischarge = 234 min
tdischarge
=
33
min
m
/
L7
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The "winner" of our comparison is clearly lithium tantalate because of its high energy density which allows the construction of relatively small energy converters. Again, we have to charge the 3-pF capacitor at 2,500 volts. This time the energy E =2
C V2=-
3 10-n
25002Wsec =9.4uWsec
(14)
2
must be found from a temperature change of the pyroelectric material. Let us consider our piece of lithium tantalate from the chart with a volume of 0.1 cm . To obtain the required energy of 6.4 uWsec, we must heat it up by 1.34 K. Since the voltage generation depends on the height of the crystal and the energy on its volume, we have now the basis for the design of a temperature-powered sensor which drives an RF transmission containing identity and temperature data every time that the designated temperature change is reached.
Fig. 52. Pyroelectrically powered temperature sensor, with SAW sensor element, spark gap transmitter and spiral antenna, RF bandpass filter on the back side.
Figure 52 shows a photograph of a temperature sensor designed to transmit RF telegrams. It contains a 16-bit identity code and detects the absolute temperature at every 0.5 K. temperature change. The housings of the pyroelectric energy converter with spark gap and SAW element are opened to allow a view onto the inner assembly. Every temperature change in either direction of 0.5 K generates a voltage of 2,500 volts at the metallized faces of the crystal. This causes the spark gap, consisting of thin platinum wires, to ignite. The stripline bandpass filter (on the reverse side, not visible) is excited and the damped oscillation is conducted to the SAW sensor element where it is encoded and transmitted by the spiral antenna. The sensor parameters are: one transmission per 0.5 K temperature change minimum temperature change rate to reach the breakthrough voltage: 0.1 K/min operating range -20 °C to +150 °C frequency of operation: 869 MHz, bandwidth 20 MHz
313
1180 F. Schmidt & G. Scholl RF pulse power supplied by the spark gap measured at the bandpass filter: 27 dBm RF pulse power of the pulse chain, resulting from the SAW encoding: - 7 dBm code capacity of the SAW: 16 bits measurement accuracy without sensor calibration 0.5 K duration of one transmission: 5 usee antenna gain: 0 dBm This device has potential applications for temperature measurement in many fields, especially where wiring is not feasible or too expensive: Private home market: Temperature monitoring: inside and outside the house, oven, refrigerator and cooking pots, fire alarm, heat-cost allocation. Production control: Temperature monitoring between - 50 °C to about 300 °C in process chambers, on or in containers, smoke monitoring, on moving or rotating parts. Transportation: Brake temperature, axle temperature, overhead line temperature, tire temperature. Energy: Temperature measurements on high-voltage parts such as overvoltage protectors, rotor windings, aerial lines and gas insulated high-voltage coaxial cables. Healthcare: Temperature measurements in rooms, sterilization chambers, X-ray tubes, autoclaves and refrigerators.
4.6 Receiver concept and signal evaluation In order to complete our sensor system, we have developed a receiver with digital signal processing. The block diagram can be seen in Fig. 53. antenna
f V
Y
digitizing
LNA
\
C
/
V_
HfxJ:?) t ' 856 > MHz
buffer data processing memory serial DSP FIFO 56002 interface -
t sequencer
. LO J
iL
i
J
Fig. 53. Block diagram of the developed receiver and evaluation unit.
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The sensor signal is received by the antenna, bandpass filtered and amplified by a lownoise amplifier. It is then down-converted to an intermediate frequency of 13 MHz and digitized at a sampling rate of 40 Ms/sec. Because of the high dynamic range of the signal, we chose an amplitude resolution of 12 bit. After storage, the signal is ready for evaluation by the signal processing unit. Figure 54 shows such a received sensor signal which is encoded by a SAW with ten reflectors.
Fig. 54. Received time domain signal of a temperature sensor with identity code.
This signal in Fig. 54 was obtained from a temperature sensor at a distance of about 10 meters. The first three reflections are used to determine the temperature, whereas the following seven reflections form a 16-bit identity code. The encoding is realized by setting each reflector on one of several possible positions, thus allowing a code space of more than pn to be implemented where n is the reflector number and p the number of positions for each reflector. For the signal evaluation, the following steps have to be carried out by the signal processor: digital bandpass filtering adaptive notch filtering for suppression of small bandwidth interference sources Hilbert transformation and demodulation of the signal interpolation of reflector positions, code evaluation, parity check calculation of phase values and sensed values such as temperature, force etc. transferring the results via a serial interface
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Figure 55 shows the phase constellation of the first four reflections in the signal. Our concept uses a Hilbert transformation of the received signal to obtain the phase information, which makes it possible to manage with only one receiver channel instead of two as in the case of I/Q concepts. The phase values obtained in this manner are used to calculate the corresponding value of the physical property to be sensed, such as the temperature or force. At this point, the questions arise as to where the accuracy limitations of this method lie and how the system must be designed in order to achieve optimum measurement accuracy. These questions will be examined in the next section.
-1
-J
5
0
05
1
i
Fig. 55: Example of a measured phase constellation of the first four reflectors
5. Measurement Accuracy The measurement accuracy of the sensor systems described here will now be discussed on the basis of SAW temperature sensors. However, the results - which generally refer to single measurements - are transferable to other SAW sensor applications such as torque and pressure sensors. Stochastic measuring errors can of course be reduced by averaging. The measurement system will be considered as ideal except for the considered interferences.41 5.1 Basic relations for error qualification The time domain response signal e(t) of a SAW delay line with R reflectors can be described as a chain of carrier pulses weighted with the envelope curve h(t). This pulse chain is the result of the reflection of the single high-frequency pulse by the various reflectors of the SAW element. Assuming that the reflection is ideal (R((u) = const.), noise h,(t) is added to the measurement.
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e(t)HF =J^Ar V ^ - ^ i .A(r-T r )+|(/)
1183
(15)
In the case of Gaussian pulses of the bandwidth Af, the enveloping curve takes the form
h(t) = e~
41n2
.
(16)
The high-frequency measurement signal e(t)HF will usually be mixed down with a fixed carrier frequency: e(t)NF =e(t)HF
• eJ*-* ' = | v
^ " ^ ^
-h(t-xr)
+ ^t) (17^
r=l
(<°NF
=0)
HF
C)mix)
The run times on the SAW chip xr change as a function of the temperature: Tr=Tr0-(l + Cr-(T-r0)),
(18)
where %o is the run time at temperature T = T0 and Cx is a material constant with a value of about 100 ppm/K for lithium niobate. This temperature-dependent run time constitutes our measurement effect. As the following calculation shows, it makes no sense to measure this run time shift Atr directly due to the very small changes of, say, ATr = 0.1 nsec/K for a reflector distance of 1 usee. The attainable accuracy of the time measurement in this case would depend directly on the noise level £ of the signal. The signal-to-noise ratio SNIt^^ is given by the ratio of the maximum noise level £ to the maximum signal amplitude e :
|n=|andSNRmax=201ogl|l.
(19)
If we now want to measure the time shift Atr of the reflections directly with the maximum of the envelope curve, we obtain the following fluctuations caused by Gaussian signal noise :
AT
>
2Vln(2) ln(l/(l-|J v v v . n-Af
(20)
To perform measurements with a typical bandwidth of Af = 10 MHz to an accuracy of 0.1 nsec or 1 K we would need a SNR^ax of 250 dB! This is clearly impossible and indicates that another method of time-shift measurement will have to be found.
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Let us now consider a phase-sensitive evaluation of the carrier signal. Here we measure and evaluate the phase differences at discrete time positions Txr of the signal (preferably x„ = z^). To eliminate constant interference, only phase changes between two reflections will be considered, thus producing a differential measurement. The signal phase (per of the received pulse chain at a position of the reflector r at a time T„ is then: (Per =^S{eNF(t=t„)}
= a)NF -Txr -0)HF -Xr +(p0r.
(21)
Taking into account that t xr is generally not exactly equal to x^, we write the time positions of the phase measurement as t xr as t xr = 1,0 + 8,. Together with (18), this yields:
Xr0 .(1 + CT • (T - TQ )) +
(22)
Assuming a stable mixing frequency, this term can be simplified to (Per =(Pcr
+ ®NF ' &r ^HF
-tr0
• CT • (T-TQ)
.
(23)
The phase difference A$jk between two reflections (with index j and k) is then A
$jk
= (Pej ~ Vek = A 0 A O + ^NF (8 j ~ 8 k) + C0HF • Cj • (T - TQ) • (Tk0 - TJ0 ) . (24)
The terms (pcr and A-djko are constants in this case.
5.2 Phase measurement errors caused by amplitude noise For measurement signals of the form (17) we estimate the phase measurement error to be: _ sin(e) + | n [sin(e)l 8m„ = a tan^ : n [ - a tan-^ — ^ - } . |cos(e) + ^ n J lcos(e)J
,„,. (25)
This term reaches its maximum when sin(e) + %n= -cos(e) - £ n ,
(26)
resulting in a maximum phase measurement error of 5
(27)
The temperature-dependent phase shift at a reflection distance of 1 |xs and at a measurement frequency of 433 MHz is about 16° (phase) per °K (cf. 24). A moderate SNRnax of 15 dB is thus needed to measure the temperature to an accuracy of 1 °K.
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5.3 Phase measurement errors caused by frequency fluctuations Our calculations have hitherto been based on the assumption that the frequency does not fluctuate within the duration of a single measurement. We will now investigate the effect of short-term frequency fluctuations known as phase noise. Equation (24) provides the basis for this procedure. Let us assume the ideal condition where 8j = 8k = 0, or at least 8j = c\. It is then easy to see that the temperature measurement error T-T0 increases in proportion to the fluctuations of the frequency COHF- Changes in the oscillator frequency of 1% also cause changes of 1% in the measured temperature. Given a measurement range of ±100° K, this yields a maximum temperature error of 1° K . Much greater interference occurs in the case where 8j * 8k. If the phase readings Txr are not performed at the exact time positions, the frequency fluctuations result in serious measurement errors. For example, a difference between the 8j and 8k position of 100 nsec causes a ±10 °K temperature error when the frequency deviates by 1%. It is therefore important to choose the time positions xxr for phase measurement with care and to keep the time difference between them strictly constant. As this can be achieved by careful hardware assembly, we only have to consider the interference on 8j and 8k which is unavoidable because of the limits of technological accuracy in fabricating SAW elements. The main error sources of this kind and their influence on the accuracy of the measurements are listed in the following chart. Table 5. Error sources and their influence on the accuracy of temperature measurements.
Error source
Error dimension
Effect on 8j k = | § - 5k]
Resulting temperature error ST, Example for COHF = 433 MHz ± 1 %
|Tko- tjo| = 1 Us fsampie = 5 0 MHz Errors caused by digitization
1 / fsampie (can be improved up to ten times by signal interpolation)
8jk <
Positioning errors of the wafer stepper
ca. 1 urn (speed of sound on the substrate = 3480 m/s) 1 m/s o ca. 300 ppm
Sjk < 30 ps
Fluctuations of sound velocity on the substrate Aging a) oxidation b) degasifying of the adhesive Main effect of sound-speed change due to temperature
100 ppm/year 300 ppm / year
A# ik = ± 32° => 8T « 2 K (or 8T « 0.2°K with interpolation) AAk = ± 0.005° => ST < 0.003 K
8jk < 300 ppm |%T tj0|
A#ik = ± 0.5° =* 8T < 0.03 K
Sjk < 300 ppm !%>- xj0| / year
8T < 0.3 K in 10 years
In proportion to the frequency change
8T = |T-T 0 |' 1 % Measuring range T 0 ± 100K,8T< I K
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As can be seen in Table 5, the accuracy is not seriously affected by the parameters investigated here. When the frequency is determined by the quality of the SAW element as a frequency filter, the calculated errors can be reduced even further. 5.4 Interference by multi-path propagation RF signal reflections from the surroundings are responsible for the multi-path propagation of the radio signals transmitted by the sensor. The different propagation paths vary in length and the various signal components superimpose at the receiver unit. We will now investigate the circumstances under which multi-path propagation can cause measurement errors and what dimensions these may have. The model based on (15) can be extended to include different propagation paths of variable length:
e(t)HF =tta?
-Ar •** [^{'-^M-Kt-Tr
-T,) + S(0,
(28)
p=\ r=l
where P is the number of propagation paths, dtp the attenuation and tp the signal delay of the path p. Assuming that all reflectors R on the SAW element are at a distance that prevents the reflected signal components from overlapping in the time domain and neglecting the amplitude shape of the signal, we can write (28) in the following form: p
e(t) HF
' R
^ap-e-^- £^. e ;[<%H<-M+<
= p=\
(29)
_r=l
We can see here that the signal phase change caused by multi-path propagation affects the whole signal in the same manner. As stated before, the measurement uses the phase difference between two reflections produced by the SAW element. Equation (29) makes it obvious that multi-path propagation has no direct influence on the phase difference and therefore cannot affect the measurement. It must be emphasized that this is only true when the distance between the reflectors on the SAW chip is greater than the run-time differences caused by multi-path propagation. This factor must be taken into account in designing the SAW elements. In the next step we will examine how multi-path propagation influences the signal shape at the receiver. The shape and amplitude of the signal can be considerably affected by the interference effects of multi-path propagation. Because these effects are very difficult to formulate analytically, we have simulated multi-path propagation on the basis of equation 28 and performed a statistical analysis. The following assumptions were made for the simulation: 433 MHz measurement frequency 3 MHz signal bandwidth 100 m detection range 500 reflectors at stochastically evenly distributed distances between 0 and 100 m the amplitude of the reflected echoes decreases linearly with the path length
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Wireless SAW Identification
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The results can be seen in Fig. 56. The graphs show the signals received by our simulated SAW element equipped with three reflectors. The distances between the SAW reflectors are 0.5 usee and 0.6 usee respectively. The variations in shape and amplitude resulting from different propagation-path configurations can be clearly recognized. The amplitude is scaled with respect to the direct propagation-path amplitude. The results indicated that in spite of the extreme conditions underlying the simulations, the signal can be received correctly in more than 99% of all simulated cases. Figure 57 shows the statistical probability distribution of signal amplitudes for a test set of 2,000 measurements based on the same assumptions as before, i.e. a great variety of 500 different reflecting structures surrounding our propagation area. Surprisingly, there are only a few cases where the amplitude resulting from multi-path propagation is lower than that of a single direct transmission. The fluctuations in power amplitude are smaller than 20 dB. However, the results change when there are only a few different propagation paths of similar length - the extreme case would be that of two signals whose path lengths differ by a multiple of the half wavelength (0.7 m in our example). Unlike the total cancellation which would occur in this case with timecontinuous signals, the high bandwidth pulse-shaped signals used here cannot be canceled out by destructive interference. The higher the bandwidth, the more of the signal amplitude remains in such a case. Figure 58 shows the result of a simulation with five different paths whose lengths vary between 0 and 10 meters. The test simulates 10,000 measurements with various reflector sizes and positions. All other assumptions remain the same as before. The amplitude values are considerably smaller than in the configuration with a multitude of propagation paths. However, the error probability for code recognition is still below 1%. The lower amplitude values reduce the signal-to-noise ratio SNR and therefore the phase measurement accuracy. Our last error consideration deals with the relationships between SNR and phase measurement accuracy. Figure 59 shows the effect of multi-path propagation on the phase error probability. The simulation is based on 500 paths varying in length between 0 and 100 m, a frequency of 433 MHz, 3 MHz bandwidth and 0.5 usee time difference between the reflectors on the SAW chip. Two thousand individual measurements with varying surrounding reflectors were performed to obtain the distribution shown. Most of the measurement errors are within a phase error range of ±2°, which results in temperature errors of ±0.25° K. Summarizing, we can state that multi-path propagation does not seriously affect the measurement accuracy provided that the time distances of the reflectors on the chip are not too small. Fluctuations in amplitude and SNR also have little effect on the measured phase differences when the SNR is above 15 to 20 dB, depending on the configuration of reflectors in the propagation area.
321
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5
n
>jtArw*A A*V^A) fr^Wv
0
0.5
1
1.5
2
U A WU/ftlrAiVAJ^V, A X^Afl/VA^f^AJ)(WvA
2.5
3
3.5
4
4.5
5
— ^ t/M-S
r
-
-
-
-
-
-
J\
^
— .~.^*4-^.,J
0
\/
y\
1.5
1
^
V_
V
0.5
^
2.5
3
3.5
4
4.5
5
— * t/Ms
r
3
I 2 1
-
-
-
-
-
-
-
/
0 0
0.5
/)
-
A
w l/% ">A
/\ 1
1.5
;
2
-
2.5
3
3.5
4
—> t/|xs
4.5
5
Fig. 56. Simulation results: signal shape influenced by multipath propagation with different echo distributions in the environment (500 echoes, distributed between 0 and 100 m).
1 if \ **>
fe^V' :'
•:':•:
I.
10 A/A Fig. 57. Influence of multipath propagation (500 paths) on the signal amplitude.
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Wireless SAW Identification
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10 A/A Fig. 58. Influence of multipath propagation (5 paths) on the signal amplitude.
10
A/A Fig. 59. Phase error probability distribition resulting of different multipath arrangements (500 propagation paths, 0,5 usee reflector distance, 2000 simulated transmissions.
Summary SAW systems offer new and innovative solutions for identification and measurement tasks. The principles of wireless sensing based on wireless, passive, rugged and crystal stable SAW devices have been discussed. It has been shown that with event-driven sensors the limitations of radar-type SAW identification or sensor systems can be overcome. SAW ID tags as well as binary sensors and sensors for the non-contact measurement of temperature, pressure and torque were investigated. A broad variety of possible applications for industrial and domestic were presented.
Acknowledgment The authors would like to thank their partners in the business units for the support of their work. There are also many people in our group not standing on the author list which contributed significantly to the success of the experiments and test installations described
323
1190 F. Schmidt & G. Scholl above. Therefore we want to thank the head of our department W.E. Bulst for constant inspiration and encouragement and our colleagues R. Abl and C. Seisenberger (DSPprogramming, prototype systems), A. Albsmeier (DSP unit), F. Dollinger (automotive applications), C. Korden (antenna design and development), M. Mohr and L. Reindl (tag and sensor design), T. Ostertag (RF electronics) , K. Pistor (torque measurement), K. Riek (precision engineering), E. Riha (material investigations), O. Sczesny (railway applications), M. Vossieck (theoretical considerations), U. Wolff (applications in extreme environments) for their support and excellent work as well as our highly engaged PHD students H. Scherr (quartz pressure sensor) and G. Schimetta (SAW hybrid sensors). We also want to thank our university partners for their basic work and their inventiveness in Vienna, Linz, Munich and Clausthal-Zellerfeld. References 1. C.C.W. Ruppel, R. Dill, A.Fischerauer, G. Fischerauer, W. Gawlik, J. Machui, F. Muller, L. Reindl, W. Ruile, G. Scholl, I. Schropp, K.C. Wagner, "SAW Devices for Consumer Applications," IEEE Transactions on UFFC, vol. 40, no. 5, 1993, pp. 438-452. 2. A. J. Ricco, S. J. Martin; "Thin metal film characterization and chemical sensors: monitoring electronic conductivity, mass loading and mechanical properties with surface acoustic wave devices," Thin Solid Films, vol. 206, no. 94,1991, pp. 94-101. 3. G. Fischerauer, "Surface Acoustic Wave Devices" in: W. Gopel, J. Hesse, J. N. Zemel (Series Eds.), H. Meixner, R. Jones (Volume Eds.), Sensors. A Comprehensive Survey, vol. 8. Weinheim: VCH, 1995. 4. H. Hurwitz, US Patent #3,273,146, Sept. 13,1966. 5. M. Epstein, US Patent #4,059,831, Nov. 22,1977. 6. K. Vogel, US Patent #4,623,595, April 21,1981. 7. F. Seifert, W.-E. Bulst, C.C.W. Ruppel, "Mechanical Sensors Based on Surface Acoustic Waves," Sensors and Actuators A44,1994, pp. 231-239. 8. F. Schmidt, O. Sczesny, L. Reindl, V. Magori, "Remote sensing of physical parameters by means of passive surface acoustic wave devices ("ID-TAG")," IEEE Ultrasonics Symposium Proceedings, 1994, pp. 589-592. 9. L. Reindl, G. Scholl, T. Ostertag, C.C.W. Ruppel, W.-E. Bulst, F. Seifert, "SAW Devices as Wireless Passive Sensors," IEEE Ultrasonics Symposium Proceedings, 1996, pp. 363-367 10. M.I. Skolnik: Introduction to Radar Systems, 2. Ed. 1980, McGraw-Hill. 11. R. M. White, "Acoustic Sensors for Physical, Chemical and Biochemical Applicati Frequency Control Symposium, 1998, pp. 587-594. 12. Special Issue on Sensors and Actuators. IEEE Transactions on UFFC, vol. 45, no. 5,1998. 13. T. Yamashita, Kenya Hashimoto, M. Yamaguchi, " Highly Piezoelectric Shear-HorizontalType Boundary Waves," Japanese Journal of Applied Physics, vol. 36,1997, pp. 3057-3059. 14. V. P. Plessky, "Two Parameter Coupling-of-Modes Model for Shear Horizontal Type SAW Propagation in Periodic Gratings," IEEE Ultrasonics Symposium Proceedings, 1998, pp. 195200. 15. M. V. Belubekyan, A. R. Mukhsikhachoyan, "Shear Surface Waves in Weakly Inhomogeneous Elastic Media," Acoustical Physics, vol. 42, no. 2, 1996, pp. 155-158. 16. H. Skeie, P. A. Nysen, US-Patent #4,625,208, Nov. 25,1986. 17. G. Ostermayer, A. Pohl, C. Hausleitner, L. Reindl, F. Seifert „CDMA for wireless SAW sensor applications," Proceedings of the IEEE Symposium on Spread Spectrum Techniques & Applications (ISSSTA), 1996, pp.795-799. 18. P. A. Nyson, M. R. McCoy, US Patent # 4,734,698, March 29, 1988. 19. L. Reindl, W. Ruile, "Programmable Reflectors for SAW-ID-Tags, "IEEE Ultrasonics Symposium Proceedings, 1993, pp. 125-130. 20. E. Ehrmann-Falkenau, K. Langeder, Europ. Patent #0088400, Sept. 15,1983.
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35. 36. 37. 38. 39. 40. 41.
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L. Reindl, T. Ostertag, W. Ruile, C.C.W. Ruppel, A. Lauper, R. Bachtiger, H. Ernst, "Hybrid SAW-Device for a European Train Control System," IEEE Ultrasonics Symposium Proceedings, 1994, pp. 175-179. R. Seindl, A. Pohl, F. Seifert, "Impedance Loaded SAW Sensors Offer a Wide Range of Measurement Opportunities, MTT-S Digest, pp. 1453-1456, Anaheim, June, 1999. J. Hornsteiner, E. Bom, G. Fischerauer, E. Riha, "Surface Acoustic Wave Sensors for HighTemperatrure Applications," IEEE Frequency Control Symposium, 1998, pp. 615-620. R. M. Taziev, E. A. Kolosovsky, A. S. Kozlov, "Deformation-Sensitive Cuts for Surface Acoustic Waves in a-Quartz," IEEE International Frequency Control Symposium, 1993, pp. 660-664. H. Scherr, G. Scholl, F. Seifert, R. Weigel, "Quartz Pressure Sensor Based On SAW Reflective Delay Line," Proceedings of the IEEE Ultrasonics Symposium, 1996, pp. 347-350. G. K. Montress, T. E. Parker, J. Callerame, "A Miniature Hybrid Circuit SAW Oscillator Using An All Quartz Packaged Resonator," Proceedings of the IEEE Ultrasonics Symposium, 1985, pp. 277-282. G. Schimetta, F. Dollinger, R. Weigel, "A Wireless Pressure Measurement System Using A SAW Hybrid System," to be published in the MTT-S Digest, Session TH-2A-1, Boston, June 11-16,2000. G. Tobolka, "Mixed matrix representation of SAW-transducers," IEEE Transactions on UFFC, vol. 26, 1979, pp. 426-428. D. P. Morgan: Surface Wave Devices for Signal Processing. Elsevier, Amsterdam, pp. 92f and 199f, 1985. T. Sachs, R. GroBmann, J. Michel, E. Schriifer, "Remote Sensing Using Quartz Sensors," Proceedings of the SPIE Symposium on Smart Structures and Materials, vol. 2718A-05, 1996. U. Wolff, F. Schmidt, G. Scholl, V. Magori, "Radio Accessible SAW Sensors for NonContact Measurement of Torque and Temperature," Proceedings of the IEEE Ultrasonics Symposium, 1996, pp. 277-282. Technology Focus: Norway Implements RF Vehicle Identification, Microwave and RF Engineering, Feb. 1989, pp. 42-43. B. DeGreeter, O. Nys, M. Chevroulet et. al. "A Wireless Tyre Pressure and Temperature Monitoring System," Sensor Expo Conference Proceedings, 1996, pp. 61-63. R. Grelland, H. Jakobsen, B. Liverod, "A Low Cost Fully Signal Conditioned Pressure Sensor Microsystem with Excellent Media Compatibility," in D. Ricken, W. Gessner (Eds): Advanced Microsystems for Automotive Applications '99, Springer, New York, 1999, pp. 121-131. G. Schimetta, F. Dollinger, R. Weigel, " Tire Pressure Measurement Using A SAW Hybrid Sensor," to be published on the 16th IMEKO World Congress, Sept. 2000, Vienna. W. Hosp, D. Zittlau, M. Prenninger, L. Reindl, A. Pohl, R. Steindl, "Passive SAW Radio Sensors for the Intelligent Tyre, " VDI Berichte, no. 1494,1999, pp. 209-221. A. Pohl, F. Seifert, "Wirelessly Interrogable Surface Acoustic Wave Sensors for Vehicular Applications," IEEE Transactions on Instrumentation and Measurement, no. 4, vol. 46,1997, pp. 1031-1038. V. Hinrichsen, G. Scholl, M. Schubert, T. Ostertag, "Online Monitoring of High-Voltage Metal-Oxide Surge Arresters by Wireless Passive Surface Acoustic Wave (SAW)Temperature Sensors, "11th ISH London 23.-27.08.1999. R.W.Simons, "Guglielmo Marconi and Early Systems of Wireless Communication", GEC Review, Vol. 11, No. 1, pp. 37-55,1996 M. v. Ardenne, "Effekte der Physik und ihre Anwenndungen", VEB Deutscher Verlag der Wissenschaften, pp. 477-478, 1988 F. Schmidt, M. Vossiek, E. Chemisky, "Ergebnisbericht der Studie Heizkostenverteilung mit Oberflachenwellen-Funksensoren", Siemens AG Miinchen, internal report, pp, 21-30,1998
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International Journal of High Speed Electronics and Systems, Vol. 10, No. 4 (2000) 1193-1227 © World Scientific Publishing Company
INTERACTION OF SURFACE ACOUSTIC WAVES, ELECTRONS, AND LIGHT ACHIM WIXFORTH Sektion Physik and Center for NanoScience, Ludwig-Maximilians-Universilat Miinchen, 80539 Miinchen,
Germany
The interaction of surface acoustic waves with free carriers in semiconductor nanostructures has turned out to yield a powerful tool not only for the investigation of the dynamic conductivity of such quantum systems. The latter has been shown in the study of the dynamics of the fractional and integer quantum Hall effect and many other interesting physical phenomena. The interaction is based on a relaxation type and impedance matching effect. However - to make practical use of this strong interaction, the electromechanical coupling coefficients of state-of-the-art semiconductor layered systems are too small. A hybrid technique, merging the strong piezoelectricity of LiNbC>3 or similar substrates with the excellent electronic properties of band gap engineered semiconductor quantum wells tackles this problem. Based on this new hybridization technique, several acoustoelectric high frequency devices have been realized. But also optically generated free electrons and holes in a semiconductor efficiently interact with the piezoelectric fields and potentials accompanying the surface wave. Those are able to field-ionize optically generated excitons leading to an acoustically induced quenching of the photoluminescence of a semiconductor quantum well, and to a system in which photonic signals can be efficiently converted into spatially separated electrons and holes which then can be transported over macroscopic distances along the quantum well. Finally - at a predetermined time and location on the sample - they can be reassembled into photonic signals. But also much simpler photonic devices can be realized using surface acoustic waves on semiconductor samples. For instance, we report on a simple, yet efficient camera type of device for pattern recognition and image processing.
327
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Introduction
Surface acoustic waves (SAW) propagating at the surface of piezoelectric substrates are accompanied by strong piezoelectric fields. If the substrate is a semiconductor, or as in our case a layered semiconductor system providing low-dimensional quantum system,1 these fields can either directly interact with free carriers in the semiconductor or impose a strong potential modulation to its band structure. Both effects have been exploited to open a new field of research in semiconductor physics but also to develop new concepts for high frequency devices. It turns out that the interaction of SAW with free carriers in a piezoelectric semiconductor yields a powerful tool to investigate their dynamic conductivity, being especially interesting for semiconductor systems exhibiting quantum effects. The interaction can be basically described by a change of the electrical boundary conditions of the SAW caused by the finite conductivity of the free carriers. For a specific, critical conductivity of the electron system it leads to a strong absorption of the SAW, induced by screening currents, and to a renormalization of the speed of sound.4 Both quantities can be used to actually map out the interaction and gain profound insight into the properties and nature of the dynamical conductivity of quantum systems.5 Momentum transfer between the SAW and free carriers in a semiconductor leads to the occurrence of acoustoelectric effects, i.e. sound induced currents and potentials in the electron system, which also can be used to probe the quantum nature of the system under consideration. Applying the same concepts as used in basic quantum research, the combination of SAW and semiconductor layered systems can also be used to develop new concepts for high frequency devices. Here, too, the interaction between SAW and free carriers in the semiconductor can be exploited for many different effects leading to a whole class of novel tunable SAW devices. As free carriers in a semiconductor can also be generated by incoming photons, an interaction between SAW on a semiconductor and light in general can be expected. On the one side, again, the direct interaction between the free charges in the substrate and the SAW can be observed. It leads to a intensity and energy dependent attenuation of the SAW and to a renormalization of the speed of sound, both quantities can be strongly modulated via the intensity and/or energy of the incoming light. This type of interaction can be used for optical signal processing and image reconstruction. On the other hand, the strong piezoelectric potentials accompanying the SAW lead to a strong modulation of the band structure of the semiconductor layered system. This modulation leads to a completely new type of interaction between light and SAW and also to a complete change of the optical properties of the semiconductor system. Many different effects arise from this interaction, that can be exploited to, e.g., acoustically induce the storage of light in a solid. The present review article is organized as follows: After a short introduction into the basic theory of the interaction between SAW and free carriers in a semiconductor layered system, we proceed in section 3 with a description of the use of SAW to probe the dynamic conductivity of a low-dimensional electron system under quantizing conditions. Here, we address our early experiments related to the quantum Hall effect. However, not the quantum effects themselves will be on focus, they will only be used to elucidate the interaction between the SAW and a low-dimensional electron system in a semiconductor quantum well. It turns out that the interaction between the SAW and the free charges in a semiconductor system is strongly governed by some material constants of the substrate, which in our case only allow for a quite moderate strength of the
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interaction itself. The discussion of the above points together with the urge to incorporate the described effects into real devices, lead us to the need for the development of hybrid systems between semiconductor thin films and piezoelectric substrates for SAW devices.8 Bringing up the strength of interaction by nearly two orders of magnitude as compared to the monolithic case leads us to the description of many different devices and new acousto-electric effects that might be exploited for future devices. Section 4 is devoted to such new concepts. In section 5, we address the interaction between SAW and light in general again mediated by a semiconductor layered system. We discuss the effect of bipolar photo generated charges in a semiconductor quantum well and compare it to the case of a unipolar doped system. The strong piezoelectric potentials, that accompany a SAW on a piezoelectric substrate lead to a strong and tunable modulation of the band structure of the semiconductor under consideration. We show that this leads to a completely new class of acousto-optic effects like the possibility to quench the photoluminescence of a semiconductor quantum well, acoustically driven storage of light in a solid and to the development of novel acousto-optic devices for pattern recognition. Section 6, finally, addresses some future prospects of the interaction between SAW, electrons and light and reports on some initial experiments and thoughts about even more exotic types of interactions that might be exploited for new and powerful devices and analytic tools based on SAW technology.
2.
Electrical boundary conditions for SAW in two dimensions
It has been known for quite some time that the electrical boundary conditions at the surface of a substrate on which a SAW is propagating strongly influence the propagation parameters of the SAW itself. In the literature, the limiting cases are often referred to as 'open' and 'shorted' boundary condition of the surface, depending on the conductivity of a thin layer on top of the substrate. To give an introduction to the basics of the interaction between a SAW and free charges in a semiconductor, we start with a short description of this interaction in terms of the electrical boundary conditions for the SAW as imposed by an electron system of variable sheet conductivity close to the surface of the substrate. In contrast to the case of an isotropic solid, where all properties of a SAW are independent of the choice of the surface and propagation direction of the SAW, for anisotropic solids like semiconductor crystals, one has to include this anisotropy into the description of the SAW itself. For crystals with the lack of inversion symmetry (like the zincblende lattice as in GaAs), additional effects arise from the piezoelectricity of such materials. Regarding such a piezoelectric crystal and ignoring free charges for a moment, the wave equation for the SAW is usually written in terms of a modified elastic constant c*, taking into account the effect of piezoelectricity.
d2u , d2 — c * ^—5=0 ; dt1 dr1
c* = c V
= c(l + K2)
(1)
c-ej
Here, p, c, and e denote the components of the piezoelectric, the elastic, and the dielectric 2 2 tensor. Usually, these material constants are combined into a single constant K =p /ce, describing the amount of piezoelectricity of the respective substrate. The effect of
329
1196
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Wixforth
piezoelectricity hence slightly stiffens the substrate, leading to a somewhat higher sound 2 velocity v=v„+Av/vo, being connected to the bulk coupling coefficient K via
v0~
2
~
(2)
2
To distinguish between the constant K used in equ. 2, and defining the piezoelectric stiffening in bulk material, the index eff is introduced for the effective electromechanical coupling to surface waves. We now turn to the description of the effect of a thin, massless metallization at the surface of our piezoelectric substrate, on which a SAW is propagating. This has been discussed in great detail by several authors before,2'4-5'9 so we can restrict ourselves here to just give the result. It turns out that a thin, massless metallic layer at the surface of a piezoelectric substrate leads to a conductivity dependent attenuation T and a shift of the SAW velocity Av/v0 given by:
2
l+(o/cO
v0
2 l + (o-/am)2
where k=2rc/X denotes the wave vector of the SAW, o the sheet conductivity of the metallic layer, and o m =v„e0 (l+£ s ) the 'natural' conductivity of the problem. In Fig. 1, we plot the result for the attenuation T and the renormalization of the sound velocity Av/v„ as a function of the sheet conductivity a. We see that for c= am maximum attenuation occurs, whereas Av/v0 exhibits a sharp step like increase around this value. This is exactly the effect of piezoelectric stiffening mentioned above. The physical meaning of the critical conductivity a m is Des t described in terms of a transmission line model describing the energy flow from the SAW into the conducting layer. a m then represents the wave impedance of the SAW delay line, at which impedance matching is obtained. Comparing am to the field impedance Z0 =377 Q. of the vacuum yields: -^L = ^ A W ( l + £r) IQ C
(4)
where Y0=l/Z0 . From this, it becomes clear that c m actually plays the role of a field impedance if transformed from the speed of light to the speed of sound. For GaAs, a m =3.3-10"7Q_1, the largest interaction between the SAW and a conducting layer at the surface is hence expected to occur at very small conductivities.
330
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1197
0.25 0.2
o i l 0.15 0.1 0.05
0.5 04 "fcCB
3*02 0.1
0.01
0.1
1
10
100
Fig. 1. Conductivity dependent attenuation (a) and relative change of sound velocity (b) for a SAW on a piezoelectric substrate, on which a thin conductive layer is deposited. Maximum attenuation occurs at 0 =G,„, where the impedance matching condition is met.5
To illustrate the effect of a conducting layer on top of a piezoelectric substrate even further, we plot in Fig. 2 the magnitude of the piezoelectric potential as a function of depth into the substrate for GaAs. The two limiting cases 'open (o =0)' and 'shorted (0=°°)' are shown. One can see that the perfect screening in the shorted case is related to a vanishing potential directly at the surface. For open conditions, the piezoelectric potential has its maximum value exactly at the surface. However, one can also note that for larger depths below the surface, the piezoelectric potential partially recovers until at about 3/4 of the wavelength no difference between open and shorted is left. We will come back to this important fact when we discuss gated semiconductor systems.
331
1198
A.
Wixforth
GaAs (100) cut [011] propag. in
'c X)
a. < GO
9-
0
0.25
0.5
0.75
1
DEPTH ZASAW Fig. 2: Normalized piezoelectric potential for a SAW on GaAs for the given crystal cut. Both the potential for the 'open' condition (free surface) and the 'shorted' condition (metallized surface) is shown. Note that for a finite depth into the substrate the potential recovers from the screened case directly at the surface.
So far, we have discussed the effect of a thin conducting layer on top of the surface of a piezoelectric substrate. Once this conductivity is tunable, we would expect to also tune the degree of attenuation and phase velocity accordingly. This tuneability of the sound velocity in a crystal by external means is of great importance for the readability of novel high frequency devices. However, as we can see from our simple model, the degree of tuneability strongly depends on the strength of the electromechanical coupling. As we will see below, here, too, some tuneability can be achieved.
3.
SAW probing the dynamic conductivity of low-dimensional electron systems
A perfect system to study the effect of a tunable sheet conductivity onto the propagation parameters of a SAW is the quasi two-dimensional electron system in a semiconductor heterostructure. Moreover, the quasi two-dimensionality of such electron systems leads to the occurrence of quantum effects that have been a major subject of modern semiconductor physics over the last years.' A semiconductor heterostructure consists of different semiconductor materials, which are, preferably lattice matched, deposited in a layer by layer process on a host substrate. Most commonly, this is done during an epitaxial growth, for instance by molecular beam epitaxy. The most common semiconductor heterostructure is the single interface heterojunction, where a material of lower bandgap (e.g. GaAs) is capped by a material of larger bandgap (e.g. AlxGa,.xAs). The resulting conduction band discontinuity results in an approximately triangular quantum well of about lOnm width, in which free electrons can occupy quantized electron levels. These electrons are introduced into the structure by remote doping of the larger bandgap material, resulting in an unusual high electron mobility, especially at low
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temperatures, where phonon scattering is strongly suppressed. The second heterostructure used in our experiments is a so-called quantum well, where a thin layer of low bandgap material is sandwiched between two layers of higher bandgap material serving as potential barriers. This structure strongly resembles the textbook example of a onedimensional potential well with well defined quantized energy levels in both the conduction- and the valence band. In Fig. 3, we schematically depict the band diagrams of such heterostructures. As mentioned above, the quasi two-dimensional electron gas (2DEG) in such structures exhibit very interesting quantum effects, one of which is the quantum Hall effect, which was discovered in 1982 by von Klitzing10 who later was awarded with the Nobel prize in physics for this discovery. Connected with the quantum Hall effect, though conceptually different, is the fractional quantum Hall effect, for which the Nobel prize of 1998 was given to Stormer, Tsui, and Laughlin. Without going into the details of these very exiting physical quantum effects, both provide a nice test system for the discussion of the interaction between a SAW and a 2DEG on a semiconductor heterojunction.
AlxGaj_xAs
GaAs
Ai Ga
x l-xM
Fig. 3. Schematic representation of the semiconductor layered systems used in our experiments. The left figure shows a simple semiconductor quantum well. It consists of a low bandgap material (e.g. GaAs) sandwiched between two layers of a material of higher bandgap (e.g. AlxGai_xAs). The finite thickness of the well leads to quantized energy levels in the conduction (E c ) and valence band (E v ). The right sketch shows the band structure resulting from a semiconductor single heterojunction of two different materials. Once the higher bandgap material is doped, the self-consistent potential leads to the formation of a nearly triangular potential well at the interface between both materials. Here, too, quantized levels in the conduction band exist. The lower part of the figure indicates the layer sequence that can be tailored during the epitaxial growth of the structure.
The occurrence of the quantized Hall effect at low temperatures and in high magnetic fields is connected with deep minima in the magneto conductivity of a 2DEG. These deep minima are related to gaps in the magnetic field induced density of states in the energy spectrum of the electron system. Here, the 2DEG undergoes a form of a metalinsulator transition whose details are very important for the precise understanding of the physics underneath.
333
1200
A.
Wixforth
Fig. 4. Sketch of the sample layout as used in the magnetoconductivity measurements as explained in the text. The active area of the semiconductor heterojunction is shaped into the form of a 'Hall-bar' by a wet etching process. This allows for an independent determination of the magneto conductivity tensor components. To the left and the right of the active area, two interdigital transducers (IDT) to launch and detect a 70MHz SAW.5
In Fig. 4, we show the sample layout for our investigations of the quantum Hall effect using surface acoustic waves. Two interdigital transducers for the generation of surface acoustic waves are prepared to the left and the right of the sample. Between the transducers, the active area of the sample containing the 2DEG is prepared in a wet etching process. Ohmic contacts to the 2DEG allow for the static measurement of the magneto conductivity. Surface acoustic waves can be excited at one transducer and after passing the area with the 2DEG can be analyzed in terms of transmitted intensity and phase. The result of such an experiment is shown in Fig. 5, where we plot the transmitted SAW intensity and the relative change of the sound velocity together with the longitudinal magneto conductivity component axx. As briefly described above, this magneto conductivity exhibits strong quantum oscillations (Shubnikov-de Haas effect) as a function of the external magnetic field B. Deep minima or even zeroes in c xx are a clear indication of the quantum Hall effect in the system. We observe that these quantum oscillations of the conductivity are accompanied by strong quantum oscillations of both the transmitted SAW intensity as well as the sound velocity. Whenever the magneto conductivity shows a minimum, the transmitted SAW intensity also drops and the sound velocity increases. An interesting feature is the splitting of the quantum oscillations in the SAW intensity for very high magnetic fields. As it turns out, this splitting is a sensitive test of the theoretical description of the interaction. At very high magnetic fields the minima in the magneto conductivity are very deep, virtually approaching zero for specific magnetic fields. These zeroes in the magneto conductivity indicate the full quantization of the 2DEG in the quantum Hall effect. To theoretically describe our observations, we use eq. (3) and replace a by the measured magneto conductivity tensor componen c xx . The result is shown as dotted lines in the figure. Clearly, the agreement is quite perfect, given the simplicity of our model. Even the splitting of the high magnetic field oscillations in the transmitted SAW intensity are reproduced. They can be understood in terms of a double crossing o om, if the magneto conductivity approaches zero in this case. The result of this experiment shows that the relaxation type interaction as given by eq. (3) seems to be a very good description of the physics behind the interaction. As it turns out later, this description has to be modified somehow, especially for higher frequencies to account for the fact that the 2DEG is not exactly located at the surface of the semiconductor but rather in same depth below the surface. Then, finite thickness effects lead to a frequency dependence of both Keff2 and Gm.
334
Interaction
T
of Surface Acoustic
Waves, Electrons, and Light
-
1
1
r
2
3
4
5
6
MAGNETIC FIELD
7
8
1201
. -
9
10
B(T)
Fig. 5. Magnetic field dependent SAW attenuation (a), change of SAW velocity (b), and magneto conductivity (c) of a high mobility quasi two dimensional electron system (Q2DES) in a semiconductor heterojunction. The complete quantization of the electron system in high magnetic fields and at low temperatures (here T=4.2K) leads to a strongly oscillating conductivity o"xx. These quantum oscillations are directly reflected ion the SAW transmission parameters. Note that the interaction between SAW and Q2DES is strongest, when a x x approaches zero, i.e. under the conditions of the quantum Hall effect.3
So far, we have used the quantum Hall effect as a mean to modulate the sheet conductivity of a 2DEG. This modulation was reflected in a strong modulation of both the sound velocity and the SAW attenuation. Another way of gradually changing the conductivity of such a 2DEG is the use of a gate electrode on top of the sample.12 Here, similar to a field effect transistor, the carrier density in the quantum well can be changed by application of a gate bias. However, a metallic gate at the surface of the semiconductor would, according to Fig. 2, completely screen out the piezoelectric fields which are essential for the interaction described here. To overcome this problem, the 2DEG can be buried a little bit deeper into the substrate material. As we have seen, at larger depths, the
335
1202
A.
Wixforth
piezoelectric fields and potential recover partially, which in turn leads to a recovery of the interaction with depth. To prove this design criterion, we show in Fig. 6 the transmitted SAW intensity for a gated sample, where the 2DEG was buried somewhat below the physical surface of the sample. Also, the SAW frequency in this case was 800 Mhz, corresponding to a shorter wavelength. By application of a gate bias, the carrier density in the electron channel could be changed from N$=3 10 cm"2 down to zero. The transmitted SAW intensity clearly exhibits the strong peak as expected from eq. (3) and Fig. 2, when the conductivity reaches am. At the same time, the relative change of the sound velocity shows its step like behavior, which is not shown in the figure. It should be noted, however, that the maximum attenuation in this case is smaller than in Fig. 5, although the frequency dependence of T should lead to a larger attenuation. This apparent inconsistency can be explained by the effect of the metal gate on top of the sample. It screens out part of the piezoelectric potential, as indicated in Fig. 2., leading to a reduction in the effective electromechanical coupling coefficient being responsible for the strength of the interaction in general. This is a major drawback for the implementation of the interaction in a exploitation in form of a device. As we will show below, this problem can be overcome by a hybridization technique as explained in the next section.
| M
asa
craO t
I I ,
-2
1
,',,,'
1
|
I
,,-i
1
-1.5
1
1
1
1
1
^ °
a>n
M
|
I
I
1
I
1
L
1
1
1
-1
|
I
I
I
I
1
1
L
>J.
I
-0.5
0
Vga«e M
Fig. 6. Gate bias dependent attenuation of a SAW on a semiconductor heterojunction with variable carrier density as depicted in the inset. Note the strong SAW absorption around V Q = - 1 . 1 Volt, close to the threshold voltage for depletion of the electron system. The maximum attenuation in this case is only about 1 dB/cm, caused by the weak piezoelectric coupling in GaAs.
336
Interaction
4.
of Surface Acoustic
Waves, Electrons, and Light
1203
Hybrid systems
As discussed in the last chapter, the interaction between a SAW and the electron system in a semiconductor heterostructure is governed by the size of the electromechanical coupling coefficient Keff2. For a given material, crystal cut, and type of the surface wave, this material constant is fixed and cannot be altered externally. Following an old idea, where we used a sandwich of a strong piezoelectric (LiNb03) and a GaAs/AlGaAs heterojunction to investigate the quantum Hall effect in such systems, we recently developed a technique, where the coupling coefficient can be enhanced by nearly two orders of magnitude as compared to the monolithic case. The idea is simple: We excite the surface wave on the strong piezoelectric (large Keff2) and then bring the electron system into close vicinity to the surface. The evanescent fields of the SAW then couple into the semiconductor layer and mediate the interaction between the SAW and the electron system. This is achieved by removing the active layers of the semiconductor heterojunction from their natural substrate and transfer them on the piezoelectric SAW delay line. The removing of the active layers is possible by employing a technique introduced by Yablonovitch,8 the so-called epitaxial lift-off (ELO). Here, below the active layers of the heterojunction a sacrificial layer is grown, which can be selectively etched away, hence leaving the substrate and the thin active layers separated. The typical thickness of such an ELO film ranges between 100 nm and 500 nm, pieces as large as 5 5 mm2 have been successfully removed from their substrates this way. Once established, the procedure is reproducible and results in high quality semiconductor heterojunctions which are bonded to the surface of the LiNb0 3 substrate. The process is so well understood that we even could design the growth of the semiconductor layered system taking into account the new free surface and the corresponding surface states, which are not present before the lift-off. For this purpose, a one-dimensional self-consistent Schrodinger - Poisson calculation has to be performed. The thin film needs not to be glued or pressed to the substrate surface as van der Waals forces are strongly attaching the film to the SAW delay line. Both pre-processing as well as post-processing of the semiconductor film is possible, if one needs to alloy contacts or to evaporate a gate on top of the semiconductor. The semiconductor film can also be etched selectively against the LiNb0 3 substrate, allowing for a welldefined geometry and alignment of the film. In Fig. 7, we depict a sketch of the architecture of such a hybrid sample. The semiconductor layers in this example form a voltage tuneable field effect transistor to allow for a defined variation of the carrier density in the heterojunction. In Fig. 8, we depict a measurement of the SAW transmission parameters of such a device.14 The sample layout is given in the inset. Here, we have deposited a semiconductor quantum well sample which has been optimized for the ELO process on a LiNb0 3 SAW delay line with split-four interdigital transducers, allowing for the generation of four different frequencies between 100 MHz and 700 MHz. The semiconductor layer is equipped with a gate electrode and an Ohmic contact to apply a bias voltage between the gate and the electron system in the well. The sample is optimized to operate at room temperature. As the gate bias is swept between VG=0V and some negative bias, the electron channel becomes gradually depleted, leading to a decreasing sheet conductivity. As expected from eq. (3) and Fig. 1, we observe a strong increase of attenuation around VG= -6 V, indicating that the conductivity approaches the critical conductivity am. At the same time, the sound velocity exhibits its typical step
337
1204
A.
Wixforth
contact to 2DES
gate electrodes
gate electrode
conduction / band AIGaAs
2DES
InGaAs
LiNbCX
quantum 'well Fermi-level
Fig. 7: Sketch of a hybrid system consisting of a strongly piezoelectric substrate and a high mobility semiconductor heterojunction. The active semiconductor layers have been selectively removed from their natural substrate. The epitaxial lift-off (ELO) technique was used for this purpose.
like change around this very gate bias or equivalently sheet conductivity. The dashed lines in the figure represent again the result of our simple model calculation. This time, however, eq. (3) had to be modified to account for the effective coupling constant and the effective critical conductivity in the hybrid structure. This modification, based on a perturbative ansatz, is described in detail in.'6, ' The most striking observation is the fact that the effective coupling in this artificial hybrid system is enhanced by nearly two orders of magnitude as compared to the monolithic case. The maximum change in velocity reaches nearly 2%, indicating a very useful system for potential device applications. One of the first real devices fabricated this way, a voltage controlled oscillator, is shown in Fig. 9. The transducer layout in this case is somewhat more complicated as in the previous cases, as it is designed for a high Q oscillating circuit. In the resonator sound path a semiconductor film with gate electrode and Ohmic contact is prepared, allowing for a tunable change of the sound velocity by application of a gate bias. Operating at a center frequency of 434.4 MHz, the semiconductor phase shifter allows for a tuning of the resonance frequency by about 0.8 MHz, only a few Volts of gate bias are necessary for this tuning range. A very nice feature of the device structure is that we can integrate the RF amplifier and the other necessary electronic components directly into the GaAs/AlGaAs MISFET structure of the ELO film.18
338
Interaction
-8
of Surface Acoustic
-G gate voltage
Waves, Electrons, and Light
1205
-4 V [V]
Fig. 8. SAW transmission and velocity change for a semiconductor / piezoelectric hybrid structure as depicted in the inset. Using a gate electrode, the carrier density in the quantum well can be tuned, leading to a tuneable conductivity in the electron channel, as shown in the lower panel. Using this conductivity, we can calculate the expected SAW transmission parameters as shown as dashed lines in the upper panel. The agreement with the experimentally obtained data (full lines) is quite perfect, given the fact that no fitting parameter is used. Note the very large absorption and change of sound velocity in exceed of 1%. Compared to the monolithic case, nearly two orders of magnitude in the strength of interaction are gained.
Another very useful application of these new hybrid systems with large coupling coefficient is the use as a radio-interrogable voltage sensor. As has been shown before, the coupling efficiency of LiNb0 3 SAW delay lines is so high, that the SAW can be remotely excited by simply connecting the input transducer to an appropriate antenna. A radio transmitted RF pulse is picked up by this antenna and efficiently converted into a surface sound wave. Provided a reflector at the end of the sound path, the reflected SAW is then reconverted into an electromagnetic signal which is again emitted by the antenna. Thus, an interrogative RF pulse to the sample is converted into a reflected SAW pulse that can be picked up at a remote location by a radio receiver circuit. Once not only a single reflector is used but a series of separate reflectors, the incoming pulse can be even converted into a train of reflected pulses that might be used to reveal the identity of the SAW delay line ("ID tag"). Such arrangements can serve as identification tags, so to speak the electromagnetic analogue to the bar code that is used on many items nowadays 339
1206
A.
Wixforth
for identification purposes. The most intriguing feature of such devices is that they can be operated without any energy source connected to them. It would be nice, however, if not only the identity of the SAW delay line could be retrieved but also some physical measurement for remote sensing applications.
434_0
I—•
-10
.
•
i—.—i—.
-8
i
. . .
t
. . .
-6 -4 Gate voltage V [V]
i
-2
. . .
I
0
Fig. 9. A voltage controlled oscillator (VCO) based on a semiconductor / piezoelectric hybrid structure. The large effective electromechanical coupling coefficient of this composite material allows for a large tuning range of the resonant frequency of the device. The inset schematically shows the sample layout.
This is, where we can again use our voltage controlled SAW phase shifter. The sensor element in this case is designed to operate as a Mach-Zehner interferometer with two interferometer branches. In one of the branches, a thin semiconductor ELO film is deposited, which again is equipped with a gate electrode and an Ohmic contact to allow for a voltage controlled phase shift or change in sound velocity, respectively. Every sensor element providing an output voltage being proportional to some physical quantity that the sensor is monitoring, can in principle be connected to the remotely interrogable SAW delay line. We have constructed such a remote sensing device and have been able to show that different physical quantities like pressure, temperature or light intensity can be remotely monitored with very high sensitivity using this scheme. One major problem with the devices addressed above, however, has to be clarified at this point: As we have seen form Fig. 1, Both the change in sound velocity as well as the attenuation of the SAW scale with the effective coupling coefficient Ketr • As both quantities are physically connected to each other via a Kramers-Kronig relation, it seems to be unavoidable that an increase in the maximum obtainable change in sound velocity is linked to a very large attenuation of the SAW at the same time. This large attenuation would principally hamper the operation of a voltage controlled oscillator or specially the remote sensor described above. In fact - this appearing problem made it nearly impossible for us to realize real operating devices in the first place. Hence, one has
340
Interaction
of Surface Acoustic
Waves, Electrons, and Light
1207
to think about a way around this obstacle. One way to decrease the overall attenuation of the tunable SAW phase shifter would be to simply shorten the interaction length to a minimum length that is consistent with the maximum change in phase needed for a useful operation. For many purposes, like for instant a phase modulator, a maximum phase change of 90° is sufficient for most applications. In this case, the ELO film may be made very short in the direction of sound propagation, hence minimizing the attenuation that accompanies the phase shift in the sample. However, for many other applications like the VCO or the remote sensor element, much larger voltage controlled phase shifts are necessary. In this case the large attenuation would dampen out the SAW which in turn would not allow the device to reliably operate. A second way around this problem would be a variable amplifier in the feedback loop of the VCO, keeping the oscillator level constant by adjusting the amplitude of the SAW to a constant level, independent of the internal attenuation of the device. This, however would ask for some additional electronic circuitry which is not really desired for such a simple device.
.
.
i
,
i
-11 -10 -9
.
i — . — i — . — i — . — i — i — • — i — i — • — i — • — i — • — i
-8
-7 -6 -5 -4 Gate voltage [V]
-3
-2
-1
0
Fig. 10. Gate voltage dependent phase shift (upper panel) and SAW intensity (lower panel) for a hybrid sample with a split gate electrode. If both gates are connected and held on the same potential, the large phase shift in this case is accompanied by a strong attenuation around V G = - 7 . 5 Volt. Biasing only a single gate (either gate 1 or gate 2) leads to a significant reduction of the attenuation, but also the phase shift. The situation changes, if both gate electrodes are biased at slightly different voltages. Here, the overall attenuation is approximately the same as for a single gate, the phase shift, however, adds up in this case, leading to the desired result. The voltage difference needed to obtain this positive effect is approximately given by the width of the attenuation peak on the voltage scale.
341
1208
A.
Wixforth
We overcame this apparent problem using a very simple trick: Looking at Fig.l, we realize that the attenuation as a function of the sheet conductivity of the 2DEG in the MISFET exhibits a maximum at a=a m . For sheet conductivities larger than a m the attenuation grows with decreasing conductivity, whereas for conductivities a
342
Interaction
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Waves, Electrons, and Light
1209
point out below. The most striking effect of the strong periodic lateral density modulation of the electron system is a significant reduction of the electronic attenuation as compared to the small signal regime. As we could show in a hydrodynamic approach,22 this reduction of the attenuation at high SAW levels can be described in terms of a strong reduction of the screening currents, being responsible for the electronic attenuation. In the extreme case of a completely separated electron system in the form of periodically arranged stripes of charge, no screening currents at all are induced in the electron system. Hence, the electronic attenuation becomes very small in this case.
Fig. 11. Gate bias dependent attenuation and change of phase shift of a SAW on a semiconductor heterojunction with variable carrier density. For low SAW power, the interaction is still linear, the traces for the attenuation and the phase shift ressemble those shown in, e.g., Fig.9. For high acoustic powers, however, the situation completely changes: The attenuation is strongly reduced, whereas the over all phase shift remains the same as for low power. This is a new effect, arising from a strongly non-linear interaction between the SAW and the 2DES in the semiconductor thin film transistor. The density of the 2DES becomes so strongly modulated by the intense SAW that eventually it breaks up into separated stripes.
In Fig. 11, we depict the basic results for such kind of experiments. The sample in this case is a standard ELO film on top of a LiNb0 3 SAW delay line, again equipped with a gate electrode and Ohmic contacts to provide the possibility to tune the electron density
343
1210
A.
Wixforth
in the quantum well. For a fixed carrier density in the well, we observe with increasing SAW power a strong reduction of the electronic attenuation as compared to the small signal case. The relative change in sound velocity, however, remains approximately constant. Also, with increasing SAW power, the voltage at which the interaction is maximum shifts to smaller values. The important parameter for the density modulation of the electron system is the ratio of the height of the potential modulation and the Fermi-level or chemical potential |i s in the 2DEG. This ratio R = USAW/ |^S can be tuned either by tuning the SAW power or by tuning the carrier density in the well. Once R becomes unity, the separation of the formerly connected 2DEG into stripes can take place, at least to first order. As the theoretical description for this new non-linear effect is not exactly straightforward, we refer the reader to ref. 24. The indirect indication that the 2DEG in the semiconductor quantum well can be broken up into an array of independent stripes of charge by an intense surface sound wave can also be proved and observed directly. For this purpose, we prepare a somewhat more complicated sample as depicted in Fig. 12.
Vinjection gate
V
—I detection O v o s c o p e |~gg" "' Ohmic transport gate contact
MJz quantum well with 2DES
Fig. 12: Sketch of a semiconductor / piezoelectric hybrid sample for the investigation of acoustic charge transport. The SAW is excited by interdigital transducers, as usual. Two Schottky gates act as charge injector (left) and charge sensing diode (right). The major part of the sample forms a transport channel, whose potential depth can be adjusted by a bias voltage to the transport gate. The insets show schematically the resulting, laterally modulated band structure, and the layer sequence of the hybrid device.
Here, actually three different gate electrodes are involved. One electrode, the longest in the direction of SAW propagation serves to define a 'transport channel', in which the propagating SAW can convey the stripes of electrons to the other end of the sample. The ratio R = USAW/ (XS in this case can be very well adjusted by a combination of appropriate gate bias to this 'transport gate' and the SAW power on the other hand. Two other, very short gate electrodes to the left and the right of the transport gate serve as charge injector (left) and charge detector (right). Injection of charge into the moving
344
Interaction
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Waves, Electrons, and Light
1211
SAW potential superlattice is achieved by removing the negative gate bias from the injecting gate for a short moment. As a result, the depletion below this gate is lifted for a short moment of time, the resulting pulse of charge can now be picked up by the SAW traveling potential and convey it to the other side of the transport channel. Here, a Schottky diode is used to capacitively detect this arriving charge. The result of a typical experiment with this sample is shown in Fig. 13. Here, we plot the detected charge at the Schottky gate as a function of time after the injection of charge at the other side of the transport channel for different SAW power. The transport gate bias is adjusted such, that the 2DEG below the transport gate is just depleted. The SAW is launched as a continuous wave. For quite small SAW powers, we observe the arrival of a tiny and broad charge pulse at very large times, much longer as the propagation time of the SAW from left to right. With increasing SAW power, the pulse becomes larger and narrower. At the same time, it's arrival to the detection gate happens at shorter times. Eventually, the amplitude and width of the detected pulse of charge saturates, as does the time of arrival. In other words, for the highest SAW powers shown in this figure all the charge is transported with the same velocity to the detection gate. This velocity turns out to be identical to the SAW velocity, as expected. For smaller SAW powers, where the trapping of charge is incomplete, only some average momentum is transferred to the electron system. As this involves a statistical process, the charge pulse is much broader and also the average time of flight is longer as compared to the complete trapping of charge in the moving potential wells of the SAW. There are many more applications of the novel hybrid structures described in this chapter, which will be omitted here for the sake of space. We did not address any effects arising from the fact that even in the small signal limit not only energy is transferred from the SAW to the electron system but also momentum transfer takes place. This phenomenon leads to the occurrence of acousto electric effects, like sound induced currents and potentials in the plane of the electron system. As might be expected, those acousto electric effects as originally described by Weinreich, are of unusual large amplitude in our hybrid system.25 On the other hand, also momentum from a drifting electron system can be transferred to the SAW. This can be achieved by passing a current through the electron layer, either in the direction of sound propagation or anti-parallel to it. The resulting effect is an increased attenuation of the SAW, exceeding the one that has been addressed before for the static electrode system (see. eq. 3). If the direction of the current is reversed, acoustic amplification can occur.26 Based upon this, we proposed a non reciprocal SAW device, where the transmission of RF power strongly depends on the direction of SAW propagation. Again, we have to refer the reader to the respective original publications for further reading. 26
345
1212
A.
Wixforth 1
i ' i • i < i
14dBm—•* increase of RF •power
30.5 dBm •••••••
•
< ~ca c
114 MHz 340 MHz Theory
'
\O <
u -? -4 -6 -8
»
a) a> m *-» o > fl) « O) c o o a) r
•10
- 1 0 1 2 3 4 5 6 7 8 9
Time (|xs) Fig. 13. Acoustic charge transport (ACT) in the semiconductor / piezoelectric hybrid as shown in Fig. 12. The ACT-signal is plotted as a function of the time-of-flight of the detected charge. For low SAW power, a small signal arrives at a very late time at the detector diode. The higher the SAW power, the larger is the signal and the earlier it arrives. Eventually, at the highest SAW power, a strong signal arrives exactly after the SAW delay time, indicating that all charge is conveyed across the transport channel. The inset shows the measured average velocity as a function of the amplitude of the SAW potential together with the theoretical prediction of a hydrodynamic model.
As we have shown in the last paragraph, the combination of high quality band gap engineered semiconductor layered structures with SAW devices based on strong piezoelectric leads to a completely new class of SAW devices with superior performance over many standard applications reported so far. However, the process of the ELO and wafer fusing itself needs to be optimized in the future to also make these novel devices really competitive. It is worth to be noted at this point that the actice layer in the hybrids, containing the free charge not necessarily has to consist of GaAs based material. As we are using a hybridization technique, this layer even does not have to be a piezoelectric. One could imagine that Silicon based layered structures or even semiconducting polymer films could prove the same performance as discussed above.
346
Interaction
5.
of Surface Acoustic
Waves, Electrons, and Light
1213
Interaction between SAW and light
In the previous sections, we have seen how efficiently surface acoustic waves can couple to free charge carriers in a semiconductor layered system. So far, we have discussed doped semiconductor quantum wells and heterojunctions, in which free carriers were present all the time. In this chapter, we would like to discuss undoped quantum wells, where electrons and holes are generated by illumination of the sample. If a photon with an energy hv > EG larger than the band gap energy EG is absorbed by a direct bandgap semiconductor, an electron-hole pair is created. These photo generated charges usually decay after some lifetime while re-emitting a secondary photon. This process is called photoluminescence (PL). Electron-hole pairs may be generated in a quantum well but also in the substrate or barrier material, depending on the energy of the incoming primary photon. To simplify the following discussion, we would like to restrict ourselves to the simplest case of a selective pumping of the quantum well in a semiconductor system similar to the one sketched in Fig. 3. Here, a thin InGaAs - layer is sandwiched between two GaAs barriers, forming a one-dimensional potential well along the growth direction. Once this quantum well is illuminated with light of sufficiently high energy, but not high enough to also create electron-hole pairs in the barrier material, photo generated charges are only present in the well. These bipolar charges, electrons in the conduction band states and holes in the valence band states may now interact with a SAW propagating across the structure. In Fig. 14, we depict one of the first experimental results that have been obtained while studying the influence of a SAW on the optical properties of a semiconductor quantum well. We measure the photoluminescence of such a quantum well and watch its behavior once a SAW is propagating through the optically excited area of the sample. The sample in this case is a single InGaAs quantum well between GaAs barriers, exhibiting a strong and narrow PL line at low temperatures and with no SAW present. The energetic position, the intensity and the width of this line tells us that it is of excitonic origin, as expected for these experimental conditions. An exciton may form from a photo generated electron-hole pair, if the temperature of the experiment is low enough to be exceeded by the exciton binding energy. This exciton binding energy can be estimated from a modified hydrogen model and turns out to be of the order of 10 meV in our case. The temperature of the depicted experiment was 4.2 K, being low enough to allow for efficient exciton formation. The effect of a SAW traversing the illuminated region on the sample, where excitons are formed by laser illumination and from where the PL light originates is clearly seen from the figure:27 With increasing SAW power, the intensity of the PL strongly decreases, whereas the width and also the energetic position of the line remains nearly unaffected. Eventually, at the highest SAW power shown in the figure, the PL becomes completely quenched. How can we explain this surprising findings? First, we have to discuss the effect of an external electric field on an exciton in a semiconductor quantum well. As has been shown before, such an external electric field actually acts on the exciton due to its internal structure. The exciton consists of a negatively charged electron, and a positively charged hole, bound together by their mutual Coulomb attraction. A typical exciton radius is about 10 nm for the materials discussed here. As the wave function overlap between the electron and the hole is very large at these short distances, there is a finite probability for the exciton to decay and to emit a photon, which then is detected as PL light. However, an external electric field polarizes the exciton complex, hence reducing the wave function overlap and reducing the probability of radiative recombination by some amount. Eventually, for strong enough electric fields (which have to exist in the
347
1214
A.
Wixforth
plane of the quantum well in our quasi two-dimensional case) the exciton might even break apart, or field ionize. A simple estimate of the necessary electric field for ionization of an exciton in a semiconductor quantum well yields E=104 V/cm.29 As one can also easily show, such electric in-plane electric fields are obviously achievable with a SAW of moderate power even on weakly piezoelectric GaAs substrates.
• P 1 = -28.5 dBm " ^
IDTT
IDT 2
Fig. 14. Photoluminescence of a semiconductor quantum well under the influence of a surface acoustic wave. With increasing SAW power, the intensity of the PL strongly decreases, whereas its energetic position and line width remain nearly unchanged.
Ionization, however, means that the electron and the hole, the fragments of the exciton, are spatially separated by an electric field. This spatial separation in turn very strongly reduces the wave function overlap between the electron and the hole, and the probability for radiative recombination. Hence, it results in a strongly suppressed PL intensity. This is exactly what we observe in our sample. How can we prove that it is indeed SAW induced exciton ionization that leads to the remarkable suppression of the PL in our quantum well? For this proof, we remember the fact that a SAW can be very efficiently used to probe the dynamical conductivity of a system of free charges. This conductivity, however, should be very different for neutral excitons and for ionized excitons consisting of free electrons and holes. An experiment to prove the ionization process directly, is depicted in Fig. 15: Here, two SAW delay lines with propagation directions that are perpendicular to each
348
Interaction
of Surface Acoustic
Waves, Electrons, and Light
1215
other have been prepared on a GaAs substrate containing an undoped InGaAs quantum well.29 The two delay lines, as indicated in the figure, operate at different frequencies to avoid possible spurious signals arising from cross talk on the sample. One SAW, say the one from left to right, serves as the 'pumping' SAW for exciton ionization as shown in Fig. 14. The second, perpendicularly propagating SAW (from top to bottom) acts as a probe beam for the measurement of the dynamic conductivity in the illumination area at the junction. i i | i i i i—p-1—r-r-f
=3
0.1 dB
• J—I
a 1 <
"—'•'•s,
PLIntensity (a.u.) r
ni>»Mil
-60 -50 -40 -30 -20 -10 0 p
10
l,pump(dBm)
Fig. 15. Direct prove for the exciton ionization in the strong lateral piezoelectric fields accompanying the surface acoustic wave even on GaAs substrate, we plot the transmitted SAW intensity of a probe SAW beam as a function of the amplitude of the pumping SAW being responsible for the exciton ionization and spatial separation of the photogenerated electrons and holes. The inset shows the sample layout and the intensity of the PL in the pumping SAW (cf Fig. 14). At around Pl=-6 dBm, a sharp increase of the probe SAW attenuation indicates exciton ionization by the pump SAW.2'
However, at about Pi= -6 dBm, we observe an abrupt attenuation of the probe SAW. Recalling our results as explained in sections 2 and 3, we interpret this abrupt increase in SAW attenuation as an abrupt increase of the dynamical conductivity in the interaction area. This is exactly compatible with the assumption of exciton ionization as pointed out before. As long as the photo generated excitons are still intact, i.e. at relatively low pumping SAW power, they behave a s more or less neutral particles, being unable to screen the electric fields of the probe SAW. As soon as the excitons become ionized at large enough in-plane electric fields, they break up into free electrons and holes which in turn, are able to screen the piezoelectric fields of the probe SAW. This screening induces currents in the electron-hole system, which have been attributed to the attenuation of a SAW in sections 2 and 3. With further increasing SAW power we even
349
1216
A.
Wixforth
observe the reduction of the attenuation as expected once the average conductivity of the electron-hole system exceeds the critical conductivity ora. Now that the process of SAW field induced exciton ionization seems to be settled, let's further think about the consequences for our experiments: We have seen that the electric fields in the SAW and in the plane of the quantum well can be strong enough to spatially separate the photo generated electrons and holes. In figure 15, we depict the transmitted SAW power of the probe SAW as a function of the power of the 'pumping SAW. Over a wide range of pump power Pi, there is no change in the transmitted probe SAW intensity, indicating no significant change in the local dynamic conductivity at the illuminated cross junction. As the potential modulation of the quantum wells band structure is periodical in space and time, we also expect the electron and hole density distribution to be modulated. In other words, after ionization, the electrons will drift to a local potential minimum in the periodically modulated conduction band states, whereas the holes will drift to a local potential minimum in the respective valence band state. Like in the charge conveyor belt shown in Fig. 12, we may expect to now have spatially separated strips of charges in the moving potential wells of the SAW. In contrast to the experiment shown in Fig. 12 and Fig. 13, however, we now have bipolar charges, stripes of electrons and holes, that are spatially separated by approximately one-half acoustic wavelength. This dynamic structure resembles a two-dimensional lateral n-i-p-i system,30 where n-type carriers are followed by an intrinsic region, which is followed by a p-type system and again by an intrinsic region. In Fig. 16, we try to illustrate this remarkable situation. Once the spatial separation between the electrons and the holes is large enough, the wave function overlap will be vanishingly small, as will be the probability for radiative recombination.
Fig. 16. Sketch of the laterally modulated conduction and valence band in the semiconductor quantum well, leading to a spatial separation of the photogenerated charges.
The first effect of this dramatic increase of the carrier lifetime is the above-seen suppression of the PL of the system. However, there is another, even more intriguing consequence of this spatial separation and the trapping of charges in the moving lateral potential wells of the SAW: We are able to also transport this trapped charge over macroscopic distances across our sample. If we are able to intentionally lift this lateral potential modulation of the SAW at some location on the sample, being remote from the location of optical generation of the electrons and holes, we should be able to re-convert the trapped bipolar charges into light. The experimental setup and the result of this
350
Interaction
of Surface Acoustic
Waves, Electrons,
and Light
1217
experiment is shown in Fig. 17. In the upper part of the figure, we depict a sketch of the experimental setup and the sample layout. To the left of the active layers containing an InGaAs quantum well as described above, an interdigital transducer is deposited to excite an intense SAW pulse at t=0, being strong enough to completely ionize all excitons that are generated by illumination.
gated detector
pulsed lase
H
1
IDT
metal
" ,.y
^/A^-— 1
x
100
1v-—•—
p
^VSAW "
; in
x
A
out at
75 3
ro
'
:
•
01
50
- Intens
CO
£•
_J D.
/
t=t
RF
\
v/
0 l
V t=t 2
Laser i
400
800
time (ns) Fig. 17. A 'photon conveyor belt' realized in a semiconductor layered structure as shown in the upper part of the figure. At t=0, a short SAW pulse is launched at the transducer which arrives at t=t| at the point, where a laser beam excites electrons and holes. Even after the laser has been long shut off, we observe a bright luminescence at the other end of the sample, and after a long delay time. Here, a thin metal layer acts to screen the piezoelectric fields of the SAW, thus lifting the spatial separation of the electrons and holes in the traveling potential wells of the SAW."
Somewhat to the right of this transducer ( at x = x i n ) , we hence focus a laser beam onto the sample, providing the necessary photons to generate electrons, holes and excitons in the quantum well. Some millimeters further to the right, in the direction of sound propagation, we have evaporated a thin, semitransparent NiCr metal layer to serve as a screening electrode for the in-plane SAW fields. Once the SAW pulse and the trapped electrons and holes arrive at this electrode, we expect a complete screening of the
351
1218
A.
Wixforth
confining fields and a flattening of the bands in the quantum well. Hence, the electrons and hole are no longer trapped and, free to move again, drift against each other, driven by their mutual Coulomb attraction. The consequence is a strong increase in wave function overlap and connected to this, a dramatic increase of the probability for radiative recombination. The result of the experiment together with its timing sequence is shown in the lower part of the figure. A photomultiplier tube selectively monitors the PL intensity coming from the remote recombination area on the sample. Several hundred nanoseconds after the generation of electron hole pair by the laser, we observe an intense flash of light coming from the recombination gate, indicating a time delayed PL as expected from our above reasoning. One can regard this experiment as a realization of acoustically induced 'storage of light in a solid', ' or equivalently, the demonstration of a semiconductor based delay line for photonic signals. At one end of a semiconductor sample, a photonic signal is fed into the structure and after a time delay of many hundred nanoseconds, this optical signal leaves the sample at the other end. If we compare the intensities of the PL at the site of generation with the one of the time delayed PL at the site of recombination, we obtain the result depicted in Fig. 18. Here, we plot the integrated intensities of both PL lines as a function of the SAW power. Like in Fig. 14, we observe the threshold behavior of the ionization process in both traces. At the same power level, where the 'direct' PL intensity decreases, the 'delayed' PL intensity increases. i
•
•
•
•
i
1
«• * o o «
to
C
f
N
Us
SAW =
*
8 4 0
M H Z
delayed PL direct PL
1
0 Acoustic Power P-, [dBm]
-30 -25 -20 -15 -10
10
• f
o
Fig. 18. Comparison between the intensity of the 'direct' PL as observed at the location of excitation and the 'delayed' PL intensity as observed at the location of generation (x = x0,„ in Fig. 17). The decrease of the 'direct' PL intensity is directly connected to the increase of the intensity of the 'delayed' PL. This indirectly indicates the transport mechanism as discussed in the text. The data are normalized. In reality, the delayed PL intensity is only about 20% of the direct one.31
This is a clear indication for the validity of our model of charge trapping and conveyance to a remote location. Absolutely, however, only about 20% of the intensity of the 'direct' PL can be recovered at this site. We attribute this 'loss of charge' to some small, but finite probability for non-radiative recombination, mainly caused by traps at the rough edges of the active quantum well region on the sample.
352
Interaction
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Waves, Electrons, and Light
1219
So far, however, the delay time of this 'photon conveyor belt' is fixed and basically given by the length of the SAW path between injection point and recombination region. For practical applications, it would be highly desirable to be able to tune the delay time. This can be done by a different approach to 'flatten out' the lateral potential modulation of the SAW in which the electron hole pairs are trapped: For this experiment, we use a second, counter propagating SAW apart from the one that serves as a conveyor belt. The result of this experiment is shown in Fig. 19. we plot the intensity of the detected PL at the generation site of the sample as a function of the power of the second counter propagating SAW. The first SAW is used to partially ionize the generated excitons and its power is adjusted such that the PL is already quenched to about half its maximum intensity. At low power of the counter propagating SAW (which has the same frequency as the first one), this PL intensity is nearly unaffected. However, at a power level, where both SAWs have approximately the same amplitude, the PL intensity strongly increases, indicating a complete reassembly of the electrons and holes and a subsequent radiative recombination. t
1
1
1
r-
f SAW = 840 MHz P-l = +4.5 dBm
3 CO
CO
c
CD
P1>P2 propagating
/
w°vy
0v
p1=p2 standing wave •
O*X>00G0 *
<°\)eG =z>
-20
-10 0 10 Acoustic Power P2 (dBm)
20
Fig. 19. Direct PL intensity at x = x c as a function of the acoustic power P2 for a constant acoustic power P\ = 4.5 dBm. For the standing wave geometry at P\ = P2 the spatial separation between electrons and holes is lifted, resulting in a recovered PL intensity due to the increased transition probability.
The experiment is easily understood in terms of interference between the two SAWs: Once they both have the same frequency and amplitude, the interference pattern is the one of a standing wave. In such a standing wave, the bands are temporarily flattened and radiative recombination of the formerly trapped charges may occur. Apart from this demonstration of the tuneability of the delay time, this experiment is also a proof for the fact that the exciton ionization is not induced by a possible heating of the sample under
353
1220
A.
Wixforth
the influence of the intense SAW. Although the acoustic power density is increased by some orders of magnitude, the PL comes back at these high powers, which would not be possible for a purely thermal effect. It should be noted, that the standing wave experiment can also be done for equal intensities but slightly different frequencies of the counter propagating SAWs. Here, it turns out that the frequencies have to be equal to one part in 10 to allow for radiative recombination! Then, and only then, a true standing wave pattern ist formed by the interference of the two waves. It is also possible to deliberately change the energy of the time delayed PL in such kind of experiments: as we are able to induce radiative recombination of trapped charges at a location of the sample being remote from the one of their generation, we can deliberately alter the optical properties of the quantum well at this point. This can be achieved by several wellknown means, like the quantum confined Stark effect,32 or deliberate deformation of the semiconductor crystal at this site by the deposition of a stressor layer. Many other possibility useful applications of this remarkable effect of acoustically induced storage of light have been thought of and realized in the recent past. Again, at this point we have to refer the reader to the original publications.31,"'34 That a spatial separation of about one micron is enough to induce much longer storage times than the ones reported her, has been recently shown in a very similar experiment: this time, however, using a static lateral potential modulation. In this publication, we were able to demonstrate storage times in excess of 35 |isec,35 meanwhile we have reached several milliseconds. Combining all our knowledge about the interaction between a SAW and free carriers in a semiconductor, we can realize another new set of devices that can be used for imaging processes. We have seen above, that optically generated charged carriers in a semiconductor can effectively couple to the piezoelectric fields accompanying a SAW at the speed of sound. If we were able to detect this kind of interaction spatially resolved, we should be able to reconstruct the intensity distribution of illumination of the sample, hence to reconstruct an image. Recently, we have developed such a device, which will the subject of the last part of this section. Spatial resolution of the interaction between a SAW and free charges in a piezoelectric semiconductor could be principally achieved if one knew the exact location of a SAW pulse on the sample surface. To gain such knowledge, we have used a different type of interdigital transducer, that has been invented to increase the bandwidth of SAW devices. ' The spacing between neighboring transducer fingers in this case is not constant over the aperture of the transducer. Instead, it changes linearly, forming a 'tapered' transducer structure as sketched in Fig. 20. A SAW is only efficiently excited, when the ratio of the sound velocity and the period of the transducer equals the applied frequency. Hence, for each frequency within the bandpass of such a tapered transducer the SAW is excited at a different location along the aperture of the transducer. The frequency response of such tapered transducers is also shown in Fig. 20, where we superimpose the responses of three transducers with different tapering factors t. As expected, the bandpass for the transducer with the largest tapering is largest, approximately 100 MHz in this case. A simulation of the SAW fields excited by such transducers,38 and the mapping of the SAW fields on a real sample using a X-ray technique39 confirmed the validity of our ideas. Using a tapered transducer, one is able to scan a narrow SAW beam over the width of the transducers aperture across the sample. This way, one-dimensional spatial resolution can be achieved. To date, we can demonstrate a spatial resolution of less then about 5(xm, being competitive with state of the art CCD cameras at this point.
354
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T
Waves, Electrons, and Light
610 frequency [MHz]
1221
T
660
Fig. 20. Schematic description of the operation of a 'tapered' interdigital transducer. In this case, the spacing between adjacent fingers is not constant along the aperture of the IDT but varies linearly. The degree of 'tapering't is defined as the percentage of the variation of the finger spacing from top to bottom. The lower panel shows the resulting frequency response of the tapered transducers. As compared to a standard untapered IDT, the bandpass behavior is strongly broadened. This can be explained by a local excitation of narrow sound paths, as sketched in the upper panel.
If we monitor the transmitted SAW in terms of intensity or phase after the interaction with optically generated charges, we can use this signal to extract some information about the number of charges or the intensity of the illumination at a given location of the sample. To also achieve two-dimensional spatial resolution, we again use two SAWs, that are propagating under a right angle to each other.41 Technically, both sets of tapered transducers have been designed such that their individual bandpass filter characteristics directly follow on the frequency scale. This way, the whole sample can be connected to a single high frequency generator. If the input frequency is swept between the limiting frequencies of this combined bandpass filter, first a narrow SAW beam will be swept over the, say x-direction, and then normal to this in the y-direction. If the active area of the sample, i.e. the area of overlapping SAW beams, is illuminated by a focused light spot, the transmitted SAW intensity exhibits two attenuation peaks at two specific frequencies.
355
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reconstructed position of a laser spot on the sample Fig. 21. Spatially resolved detection of photogenerated electron-hole pairs on a GaAs semiconductor sample with two sets of tapered transducers as explained in the text. The frequency passbands of the two orthogonal SAW delay lines are adjusted such that they provide two passbands just behind each other (see upper panel). The low frequency transmission band corresponds to the tapered delay line probing the x- coordinate, the high frequency passband provides information about the y- coordinate of the photogenerated charge plasma. In the lower part of the upper panel the detected SAW attenuation caused by the interaction with free charges is shown for three different positions of the laser spot on the active sample area. The lower panel shows the reconstructed gray-scale images for six different spot positions.
Those frequencies directly correspond to the x- and y-coordinates of the illumination spot. The result of this experiment is shown in Fig. 21, where we plot the reconstructed images for different locations of a laser spot on our sample. The measured quantity in this case was the phase shift of the SAW caused by the interaction with the photogenerated charges in the illumination region. Given the speed of data collection using our simple camera, one could think about an application of this simple device as a two-dimensional position detector for very rapid changes of a deflected light beam. Even more complex images can be obtained if one uses more sophisticated data processing techniques. Also, the projected image has to be scanned from all directions to obtain maximum information about its shape. To do so, we used an optical scheme that rotates the projected image by 180 degrees while the two-dimensional SAW scans are taken. For the subsequent data processing, we incorporate well known software algorithms that are usually applied for computer tomography (CT) purposes, where a similar data collection technique is used.42 The result of such images, where we have projected quite complex shapes like letters onto the active area is shown in Fig. 21. Clearly, there is more to be done to further increase the resolution and the sharpness of the images, but our first results already might indicate a certain potential of this approach.
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Interaction of Surface Acoustic Waves, Electrons, and Light 1223
Fig. 22. Early attempt to reconstruct the images of the letters *S\ 'A*, and W which have been projected onto the active area of our 'camera'. The size of a projected letters was about 200 u,m. The circular sub structure is an artifact caused by the image processing algorithm used.
6.
Summary and future prospects
As I have tried to point out in the last chapters, the combination of SAW and semiconductor layered systems offers a great wealth of possible applications, both in science and technology. It turned out that this combination introduces new de^ees of freedom to either investigate the properties of electronic systems in the semiconductor structure or even to completely alter some of the physical and material parameters of the semiconductor. A good example might be the optical delay line, where the mere presence of a SAW on a very simple semiconductor quantum well system led to a dramatic change of its optical properties. The hybridization of a piezoelectric substrate and a semiconductor thin film device has also led to a new class of novel devices. Here, especially the strongly enhanced coupling coefficient opened the door to very interesting applications. It is this hybridization, that might also be of some importance for future applications. In this review article, I restricted myself more or less to classical semiconductor systems. This it not necessarily the optimum combination. Especially as far as the technology to fabricate real devices is concerned, a lot more work has to be done to make this novel hybrids competitive to the market. However, one could also imagine to replace the semiconductor ELO thin film by other alternatives, for example the meanwhile well established TFT (thin film transistor) technology43 that is widely used in display applications. The enormous advantage of a hybridization technique lies in the fact that the 'active' layers, containing mobile charges need not to be piezoelectric themselves. As I could show, the evanescent fields of the SAW on the piezoelectric substrate are strong enough to effectively couple to the charges in the thin film. Moreover, in the last few years, there has been an evenly enormous progress in the development of organic thin film devices.44 Transistors, light emitting diodes, even lasers4 have been described on this basis. The major advantage of such 'plastic' devices is their tremendously simplified preparation technology as compared to conventional semiconductor devices. One has to wait and see, how far this development will lead us in the future. Another important aspect that I like to address at this point is the emerging rebirth of purely mechanical systems. Nowadays micro- or even nano-technologies enable us to scale down well known mechanical concepts to the micrometer or even nanometer scale.46,47,48 There are already synonyms around that describe the features of this new and fascinating field. MEMS (micro-electro-mechanical-systems) and NEMS (nano-..) are abbreviations, being very well acknowledged, to date. There has been already a considerable amount of reports on the mechanical properties of such small systems and the first results are very promising. Purely mechanical resonators can be made so small
mr
1224
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that their mechanical eigenmodes range to the multi-hundred Megahertz regime. There, these MEMS or NEMS meet the surface acoustic waves. I can very well imagine that a combination of both might again offer a whole wealth of new applications and novel physical effects for device, sensing and signal processing technologies. This might be what R. Feynman meant, when he stated in 1959: "There's plenty of room at the bottom".
7.
Acknowledgments
It is my great pleasure to acknowledge the fantastic co-operation and those many stimulating and brainstorming discussions with many good friends and colleagues that I had the pleasure to work with over the last couple of years. As one can see from the list of co-authors as cited in the references, they are so many that it would be dangerous to list them all by name - too easily I could simply forget one of the most important! Instead, I try to thank them in general: without you, it would have been impossible to follow all these different directions and surely, it would have been very boring! On the other side, I don't want to omit the generous financial support that we had the pleasure to receive by many different funding agencies: The projects described here have been funded by the Deutsche Forschungsgemeinschaft DFG under various different contracts, and the Sonderforschungsbereich SFB 348, the German - Israeli Foundation GIF, the Volkswagenstiftung, the Bayerische Forschungsstiftung. Special thanks to the Siemens Corporate Research group for their unconventional and stimulating support, both financially as well as on the personal level. Finally, I would like to thank Prof. Art Gossard and co-workers, Prof. H. Kroemer and co-workers and Prof. Pierre Petroff and co-workers, all with the University of California, Santa Barbara for their support, help, and fabulous samples!
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