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Submitted on 1 Jan 1984
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ACOUSTIC SURFACE WAVES IN PIEZOELECTRIC CRYSTALS WITH THIN OVERLAYERS
V. Velasco, F. Garcia-Moliner
To cite this version:
V. Velasco, F. Garcia-Moliner. ACOUSTIC SURFACE WAVES IN PIEZOELECTRIC CRYS- TALS WITH THIN OVERLAYERS. Journal de Physique Colloques, 1984, 45 (C5), pp.C5-65-C5-68.
�10.1051/jphyscol:1984508�. �jpa-00224126�
ACOUSTIC SURFACE WAVES I N P I E Z O E L E C T R I C CRYSTALS W I T H T H I N OVERLAYERS
V.R. Velasco and F. Garcia-Moliner
I a s t i t u t o d e ~ i s i c a deZ Estado ~ 6 Z i d o (CSICI, Serrano 123, Madrid-6, Spain
Resume - Nous S t u d i o n s d e s c r i s t a u x p i 6 z o G l e c t r i q u e s ( s y m g t r i e 6 mm) a v e c d e s couches minces e n s u r f a c e d e mBme symGtrie, e n u t i l i s a n t une n o u v e l l e v e r s i o n d e l a m6thode " S u r f a c e Green F u n c t i o n Matching" (SGFM). Nous c o n s i d 6 r o n s les combinaisons p o s s i b l e s d e s s u r f a c e s e t i n t e r f a c e s m 6 t a l l i s e e s / n o n m g t a l l i - s e e s .
A b s t r a c t - P i e z o e l e c t r i c s y s t e m s o f 6 mm symmetry w i t h t h i n o v e r l a y e r s o f t h e same symmetry a r e c o o s i d e r e d by u s i n g a r e -
c e n t e x t e n s i o n o f t h e s u r f a c e Green f u n c t i o n matching (SGFM) method. The p o s s i b l e c o m b i n a t i o n s o f m e t a l l i z e d / n o n m e t a l l i z e d s u r f a c e s and i n t e r f a c e s a r e c o n s i d e r e d .
1 . I n t r o d u c t i o n . The o n e - i n t e r f a c e problem.
S u r f a c e waves a t t h e s u r f a c e s o r i n t e r f a c e s o f p i e z o e l e c t r i c mate- r i a l s c a n be s t u d i e d / I / by means o f t h e S u r f a c e Green F u n c t i o n Mat- c h i n g (SGFM) method / 2 / . Here i s a b r i e f summary i n a c o n c i s e n o t a t i o n a d a p t e d f o r t h e e x t e n s i o n t o t h e two i n t e r f a c e problem. C o n s i d e r a me- dium w i t h e l a s t i c s t i f f n e s s c o e f f i c i e n t s C i j k l , p i e z o e l e c t r i c c o e f f i - c i e n t s e i j k . and d i e l e c t r i c c o e f f i c i e n t s c i j . We d e f i n e a t e t r a v e c t o r V = (VM , V E ) = (u ,$ ) i n a manifold spanned by g r e e k symbols a , 6 , . . . = I , 2 , 3 , 4 , i n which u i s t h e mechanical ( M ) p a r t , i . e . t h e e l a s t i c wave ampli- t u d e w i t h components u i n t h e m a n i f o l d M spanned by l a t i n symbols
i
i , j.. . =I , 2 , 3 . The f o u r t h component o f t h e complete m a n i f o l d i s t h e e l e c t r o s t a t i c p o t e n t i a l @. We d e f i n e a l s o t h e t e t r a v e c t o r W = ( W M r W E ) =
- - -
= ( f ,q), where f = f o r c e / v o l u m e and q=charge/volume. The e l a s t i c wave e q u a t i o n and P o i s s o n ' s e q u a t i o n c a n be compacted i n t o one f o r m a l equa- t i o n o f motion:
I n F o u r i e r t r a n s f o r m (summation o v e r r e p e a t e d i n d i c e s )
whence t h e Green f u n c t i o n G (k,w) , i . e . t h e ~ e c i p r o c a l o f L ( k , w ) . We p u t a s u r f a c e a t x 3 = 0 a n d , from t h e c o n s t i t u t i v e r e l a t i o n s , d e f i n e
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984508
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which i n v o l v e s normal d e r i v a t i v e s . Let A ( 2 ) i n d i c a t e t h e s u r f a c e pro- j e c t i o n o f A w i t h t h e s i g n s p e c i f i c a t i o n c o r r e s p o n d i n g t o t h e s i d e from which normal d e r i v a t i v e s a r e t a k e n . Then
The o b j e c t s e n t e r i n g t h i s formula a r e now 4 x 4 s u p e r m a t r i c e s i n t h e ( M , E ) m a n i f o l d , a s i n d i c a t e d . Knowing G one e x t r a c t s from it t h e de- s i r e d p h y s i c a l i n f o r m a t i o n i n t h e s t a n d s r d way / 2 / . I n p a r t i c u l a r , t h e s u r f ace mode d i s p e r s i o n r e l a t i o n (SMDR) i s ( r = p r o j e c t i o n o f k p a r a l l e l t o t h e s u r f a c e )
We s h a l l c o n s i d e r t h e c a s e o f 6 m m symmetry ( e . g . CdS, ZnO) w i t h t h e c - a x i s i n t h e Ox2 d i r e c t i o n , c o n t a i n e d i n t h e s u r f a c e x 3- -0 and normal t o t h e p r o p a g a t i o n d i r e c t i o n Oxl. For t h i s geometry t h e o n l y c o e f f i - c i e n t s needed a r e p , c 4 4 = c , E ~ I = E and e l s = e . For a f r e e s u r f a c e , me- dium 1 i s t h e vacuum, w i t h p l = c l = e l = O and ( i n r a t i o n a l i s e d MKS u n i t s )
E 1 = E o .
Whether f o r a f r e e s u r f a c e o r f o r an i n t e r f a c e , w i t h t h i s geometry t h e s u p e r m a t r i c e s a r e d i a g o c a l i z e d i n 2 x 2 b l o c k s / I / c o r r e s p o n d i n g t o a d e c o u p l i n g o f t h e Rayleigh s a g i t t a l mode ( ~ 1 ~ and t h e B l e u s t e i n - ~ 3 )
Gulyaev / 3 / p i e z o e l e c t r i c mode (un , $ ) . The a n a l y s i s of / I / w i l l now be extended t o t h e c a s e o f two i n t e r f a c e s a f i n i t e d i s t a n c e a p p a r t . 2 . The t w o - i n t e s a c e problem.
T h i s i s o f p r a c t i c a l i n t e r e s t , a s it i n c l u d e s t h e c a s e o f a l a y e r of f i n i t e t h i c k n e s s h d e p o s i t e d on a s u b s t r a t e o f a d i f f e r e n t mate- r i a l . Formally we c o n s i d e r medium 1 i n x < 0 , 2 i n 0<x3<h and 3 i n
J: > h . For t h e f i l m problem medium 1 i s tl?e vacuum. The formula f o r t a e g e n e r a l c a s e ( t h r e e d i f f e r e n t media) can be w r i t t e n down / 4 / i n t e r m s o f p r o j e c t o r s I z (2 f o r Z e f t , s u r f a c e a t x 3 = 0 ) and IF ( r f o r r i g h t , s u r f a c e a t z 3 = h ) w i t h c r o s s t e r m s d e s c r i b i n g t h e c o u p l i n g b e t - ween t h e two s u r f a c e s . The formula can be r e a d i l y u s e d , a s given i n / 4 / , f o r c o m p u t a t i o n a l p u r p o s e s . When t h e problem can be s o l v e d a n a l y - t i c a l l y , a s i s t h e c a s e f o r t h e Bleustein-Gulyaev mode w i t h t h e 6 rnm symmetry, t h e n it may be more p r a c t i c a l t o u s e an a l t e r n a t i v e procedu- r e . F i r s t c o n s i d e r 1 and 2 matched a t x 0. T h i s y i e l d s ( 4 ) from which, when b o t h r and r ' a r e on t h e sid; x3>0 we have /2/
The i d e a i s t o match t h e medium 1 / 2 , d e s c r i b e d by ( 6 ) , w i t h medium 3 and t o e f f e c t t h i s matching a t x 3 = h .
L e t G , e t c i n d i c a t e p r o j e c t i o n s a t x3=0 and g , e t c i n d i c a t e p r o j e c - t i o n s a t x3=h. S i n c e G Z and G3 d e s c r i b e homogeneous media, t h e s e Green f u n c t i o n s and t h e i r normal d e r i v a t i v e s have t h e same p r o j e c t i o n s at
x3=0 and a t x -h. Now, from ( 4 ) and ( 6 ) we can o b t a i n g s l hence g s l , 3 -
and a s + . ( A l l t h e i n f o r m a t i o n needed a b o u t G 2 and i t s d e r i v a t i v e i s g i v e n i n / I / ) . Let G i n d i c a t e t h e Green f u n c t i o n o f t h e one-sur- f a c e system which r e s u l s z when ( 4 ) is matched w i t h G 3 a t x3=h. The same a n a l y s i s y i e l d s
s u r f a c e s . I n o u r n o t a t i o n t h e b l o c k c o r r e s p o n d i n g t o t h i s mode i s l a - b e l l e d w i t h a , . ,.=2,4. For media 2 ( f i l m ) and 3 ( s u b s t r a t e ) we d e f i n e 6 = e 2 / & , d=c+e', d = d / ~ and
The r e s u l t f o r ( 7 ) i s o f t h e form
where f i s a s c a l a r f a c t o r g i v e n by
w i t h E = E ~ + E O , E ' = E Z - E ~ , E l = e x ~ ( - R ~ h ) , E ~ = e x p ( - ~ h ) , ~ 3 = ~ 1 ' ,
E 9 = E 2 ' , E5=E1E2, E 6 = E S 2 , and t h e m a t r i x e l e m e n t s o f m a r e
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w h e r e
U s i n g ( 1 1 ) a n d ( 1 0 ) i n ( 9 ) w e o b t a i n t h e s e c u l a r e q u a t i o n - 1
d e t gss = 0 ( 1 3 )
w h i c h y i e l d s t h e n o r m a l m o d e s o f t h e f i l m s y s t e m . T h e case o f s h o r t c i r c u i t e d s u r f a c e s o r i n t e r f a c e s / 5 / c a n be l i k e w i s e s t u d i e d b y m o d i - f y i n g t h e e l e c t r i c a l b o u n d a r y c o n d i t i o n s a n d d o i n g t h e same m a t c h i n g a n a l y s i s . Work on these p r o b l e m s i s i n progress.
R e f e r e n c e s
1 . - VELASCO V . R . , S u r f a c e S c i . , 128 ( 1 9 8 3 ) 1 1 7 . 2.- GARCIA-MOLINER F., A n n . P h y s i q u e 2 ( 1 9 7 7 ) 1 7 9 .
3.- BLEUSTELN J . L . , A p p l . P h y s . L e t t e r s 13 ( 1 9 6 8 ) 4 1 2 ; GULYAEV YU.V., S o v i e t P h y s . J E T P L e t t e r s 2 ( 1 9 6 9 ) 3 7 .
4.- VEI.ASC0 V.R. a n d GARCIA-MOLINER F . , P h y s i c a S c r i p t a 20 ( 1 9 7 9 ) 1 1 1 . 5.- FARNEL G.W. a n d ADDLER E . L . , i n P h y s i c a l A c o u s t i c s , v o l . I X , E d .
M a s o n W.P. and T h u r s t o n R . N . , A c a d e m i c P r e s s , New Y o r k ( 1 9 7 2 ) .
