Acoustoelectric effect in anisotropic piezoelectric semiconductors

Volume 28A. n u m b e r 6

PHYSICS

LETTERS

ACOUSTOELECTRIC ANISOTROPIC PIEZOELECTRIC R. KLEIN

30 December 1968

EFFECT IN SEMICONDUCTORS *

RCA L a b o r a t o r i e s . Princeton. New J e r s e y Received 5 November 1968

The phenomenological theory of the a c o u s t o e l e c t r i c effect has been generalized to take into account the anisotropic dielectric and piezoelectric p r o p e r t i e s of semiconductors. Results a r e given for CdS and the relevance of off-axis s h e a r waves in r e c e n t e x p e r i m e n t s is pointed out.

T h e i m p o r t a n c e of o f f - a x i s s h e a r w a v e s f o r t h e a c o u s t o e l e c t r i c e f f e c t h a s b e e n m e n t i o n e d in t h e p a s t b y s e v e r a l a u t h o r s [1]. R e c e n t B r i l l o u i n s c a t t e r i n g e x p e r i m e n t s [2] a n d o t h e r o p t i c a l p r o b e m e a s u r e m e n t s w h i c h c o u p l e to t h e s t r e s s f i e l d [3] s h o w c l e a r l y t h a t f o r t h e c a s e of s t a t i c e l e c t r i c f i e l d E (°) a p p l i e d p a r a l l e l to t h e c a x i s of h e x a g o n a l c r y s t a l s l i k e C d S a n d Z n O s h e a r w a v e s a r e a m p l i f i e d w h i c h p r o p a g a t e in a n o f f a x i s d i r e c t i o n . T h e t h e o r y of t h e a c o u s t o e l e c t r i c e f f e c t b y W h i t e [4] i s a o n e - d i m e n s i o n a l l i n e a r t h e o r y w h i c h we w a n t to g e n e r a l i z e to a n a n i s o tropic piezoelectric semiconductor. For elastic w a v e s p r o p a g a t i n g in a g e n e r a l d i r e c t i o n o n e no longer has a clear distinction between longitudinal and transverse waves. Consider a plane cont a i n i n g t h e c a x i s of a h e x a g o n a l c r y s t a l ( s y m m e t r y C6v). F o r e v e r y w a v e v e c t o r i n t h i s p l a n e , there is one purely transverse (T1), one pred o m i n a n t l y t r a n s v e r s e ( T 2), a n d o n e p r e d o m i n a n t ly l o n g i t u d i n a l m o d e (L). T h i s c h a n g e in t h e c h a r a c t e r of t h e w a v e s f r o m p u r e w a v e s in h i g h s y m m e t r y d i r e c t i o n s to m i x e d w a v e s in g e n e r a l d i r e c t i o n s r e p r e s e n t s t h e m a i n p r o b l e m in u n d e r s t a n d i n g t h e a c o u s t o e l e c t r i c i n t e r a c t i o n in t h e a n i s o t r o p i c c a s e . F o r t h e p r o b l e m of n o i s e a m p l i f i c a t i o n o n e h a s to c o n s i d e r a l l F o u r i e r c o m p o n e n t s of t h e n o i s e , a n d o n l y t h o s e f o r w h i c h t h e gain is highest will be predominantly amplified a n d g i v e r i s e to a d o m a i n . T h e a i m of o u r t h e o r y i s , a m o n g o t h e r t h i n g s , t h e c a l c u l a t i o n of t h e g a i n of m i x e d w a v e s p r o p a g a t i n g a t a n a n g l e ¢ off t h e c a x i s a s a f u n c t i o n of t h e r a t i o oF d r i f t v e l o c i t y to s o u n d v e l o c i t y (both taken along the c axis). The basic equations * P e r m a n e n t a d d r e s s : L a b o r a t o r i e s RCA. Zurich. Switzerland. 428

3.0 2.5 8fD(

2.0 1.5

i.I

I.o

0.5

0

50

60

70

80

90

-o.5

-I.o Fig. 1. The attenuation (or gain) of predominantly t r a n s v e r s e waves (T 2 mode) in CdS. propagating at an angle ¢ with r e s p e c t to the c axis, for s e v e r a l values of rd/t'(0}. The angle of maximum gain i n c r e a s e s with i n c r e a s i n g e l e c t r i c field. D and E are isotropic values for the diffusion constant and the dielectric constant. respectively. f o r t h i s p r o b l e m a r e t h e g e n e r a l i z a t i o n s to t h e a n i s o t r o p i c c a s e of t h e e q u a t i o n s u n d e r l y i n g W h i t e ' s t h e o r y . T h e e q u a t i o n of m o t i o n f o r a n anisotropic elastic continuum together with the p i e z o e l e c t r i c e q u a t i o n s of s t a t e c o m b i n e s t h e elastic displacement field with an electric field E k = E(°}Sk3 + E k (1}, w h i c h i s t h e s u m of t h e a p p l i e d c o n s t a n t f i e l d E ~0} a l o n g t h e c a x i s a n d a

Volume 28A, number 6

P H Y SI C S LETTERS

f i e l d E k (1) w h i c h i s s e t up by the s o u n d w a v e . T h e c o o r d i n a t e s y s t e m i s s u c h t h a t x 1 is an a x i s in t h e b a s a l p l a n e and x 3 i s p a r a l l e l to the c a x i s of t h e h e x a g o n a l s t r u c t u r e . T h e f i e l d Ek(1) c a n b e e x p r e s s e d in t e r m s of t h e d i s p l a c e m e n t f i e l d Uk, so t h a t t h e e q u a t i o n of m o t i o n b e c o m e s an e q u a tion for uk alone. Using plane wave representat i o n s f o r t h e d i s p l a c e m e n t f i e l d and t h e a c e l e c t r i c f i e l d , we get a s y s t e m of h o m o g e n e o u s e q u a tions. The corresponding secular determinant g i v e s the f r e q u e n c y - w a v e v e c t o r r e l a t i o n s f o r t h e w a v e s p r o p a g a t i o n in a f i x e d d i r e c t i o n . T h e a t t e n u a t i o n (or gain) f o r t h e T 2 m o d e c a n t h e n b e c a l c u l a t e d f r o m t h e d i s p e r s i o n r e l a t i o n w = aJ (k). T h e s o l u t i o n of t h e d e t e r m i n a n t a l e q u a t i o n f o r ¢o2/k 2 g i v e s f o r t h e T 2 m o d e

(1)

~c =

a (~;w)

2v(o! ~ 2 + (~2c/~O2) (I + o:2/~e~0D) 2

c(,p).

30 December 1968

shear electromechanical coupling factor evaluated in c o n n e c t i o n w i t h c e r t a i n CdS t r a n s d u c e r p r o b l e m s [7]. T h e r e s u l t s f o r a(O;COma x) a r e shown in t h e f i g u r e . It s h o u l d be p o i n t e d out t h a t t h e a n g l e of m a x i m u m g a i n i n c r e a s e s with i n c r e a s i n g v a l u e s of the e l e c t r i c field. T h i s r e s u l t should s h e d s o m e l i g h t on the e x p e r i m e n t s m e n t i o n e d e a r l i e r [2]. F o r e x a m p l e , a c h a n g e of Vd/V(0) s h o u l d s h i f t t h e i n t e n s i t y d i s t r i b u t i o n I(co) of t h e B r i l l o u i n s c a t t e r e d light. M o r e r e c e n t e x p e r i m e n t s by M o o r e [8] show t h a t t h e a n g l e b e t w e e n t h e a c o u s t i c d o m a i n and t h e a p p l i e d f i e l d v a r i e s q u a l i t a t i v e l y a s p r e d i c t e d by o u r c a l c u l a t i o n s . A d d i t i o n a l r e s u l t s on t h e a c o u s t o e l e c t r i c e f f e e t w i l l b e p u b l i s h e d e l s e w h e r e . I w i s h to t h a n k R. W. S m i t h , A. R. M o o r e and L. R. F r i e d m a n f o r discussions.

References T h i s r e s u l t h a s t h e f o r m of t h e g a i n e x p r e s s i o n o r i g i n a l l y d e r i v e d by White f o r t h e o n e - d i m e n s i o n a l i s o t r o p i c c a s e *. H e r e , ~ = 1 - ( V d / V ( ~ ) ) × × c o s C w i t h v d = ~ 3 3 E(°} f o r the d r i f t v e l o c i t y , and v(~P) i s t h e p h a s e v e l o c i t y of t h e T 2 m o d e . ~ c and ~ D a r e g e n e r a l i z a t i o n s to t h e a n i s o t r o p i c c a s e of t h e c o r r e s p o n d i n g q u a n t i t i e s in W h i t e ' s paper. C(o) replaces the electromechanical c o u p l i n g c o n s t a n t a n d i s g i v e n in t e r m s of t h e v a r i o u s c o m p o n e n t s of the p i e z o e l e c t r i c , d i e l e c t r i c , and e l a s t i c t e n s o r s t o g e t h e r with t r i g o n o m etric functions. U s i n g the m a t e r i a l c o n s t a n t s [6] f o r CdS we h a v e e v a l u a t e d a ( ¢ ;~0) a c c o r d i n g to eq. (1) a s a f u n c t i o n of ~ f o r s e v e r a l v a l u e s of Vd/V(O= O) and f o r t h e f r e q u e n c y of m a x i m u m g a i n w z = oJ2ax,,, = ~ e ( O ) ~ D ( ~ ) . N o t e t h a t t h i s f r e q u e n c y i s a n i s o t r o p i c . T h e f a c t o r C ( o ) in eq. (1) f o r a ( o ; o : ) t u r n s out to a g r e e with t h e s q u a r e of the * An expression equivalent to eq. ~1) was reported by Kikuchi et al. [5] as was pointed out by the referee.

1. J . H . M c F e e , J.Appl. Phys.34 (1963} 1548: N . I . M e y e r and M.H.Jorgensen. Phys. Letters 20 0966) 450: J. H. McFee, P . K . Tien and H. L. Hodges. J. Appl. Phys.38 (1967} 1721: J. Zucker and S. Zemon. Appl. Phys. Letters 9 (1966) 398:

R. W. Smith, Phys. Letters. to be published. 2. W. Wettling and M. Bruun, Phys. Letters 27A (1968} 123;

3. 4. 5. 6. 7.

8.

R. W. Smith, Conf. on Light scattering, New York. September 1968. A.R. Moore, Appl. Phys. Letters. August 15 (1968). D. L. "White, J. Appl. Phys. 33 (1962} 2547. Y. Kikuchi, N. Chubachi and H. Sasaki at the 1968 Sendal Symposium on Acoustoelectronics. D. Berlincourt. H. Jaffe and L. R. Shiozawa. Phys. Rev. 129 (1963} 1009. R.W. Gibson. Electronics Letters 2 (1966) 213; T. R. Sliker and D. A. Roberts. J. Appl. Phys. 38 (1967) 2350: N. F. Foster, G.A. Coquin. G.A. Rozgonyi and F.A. Vannatta. IEEE Transact. Sonics and Ultrasonics SU-15 (1968} 28. A.R. Moore. private communication.

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