Acousto-electric effect and amplification of sound waves in semiconductors at large sound intensities

Volume 30A, n u m b e r 4

PHYSICS

ACOUSTO-ELECTRIC EFFECT WAVES IN SEMICONDUCTORS

SOUND

LETTERS

20 October 1969

AND AMPLIFICATION AT LARGE SOUND

OF INTENSITIES

Yu. V. G U L A Y E V The Institute o f Radio Engineering and E l e c t r o n i c s . A c . Sci. USSR, M o s c o w , USSR Received 18 September 1969

A simple n o n - l i n e a r theory of stationary amplification of sound waves of a r b i t r a r y amplitude in a finite c r y s t a l is developed, the distributions of sound intensity and e l e c t r i c field along the s p e c i m e n a r e found and its c u r r e n t - v o l t a g e curve is calculated.

W e s h a l l c o n s i d e r t h e a m p l i f i c a t i o n of a n o r i g i n a l l y m o n o c h r o m a t i c s o u n d w a v e of w a v e n u m b e r q in a p i e z o e l e c t r i c s e m i c o n d u c t o r i n t h e classical case. The equation determining the c h a n g e of t h e i n t e n s i t y W of t h e w a v e m a y b e directly obtained from the momentum conservat i o n l a w [e.g. 1 , 2 ] a n d h a s t h e f o r m :

L~

V - 175V ~ " Tmm /

3

(1)

= eNvs( E- Vd/~ ) ,

where ~,N and vd are the mobility, local concent r a t i o n a n d l o c a l d r i f t v e l o c i t y of t h e e l e c t r o n s , v s the sound velocity and E the local electric field. N o w u s i n g i n t h e c a s e of s m a l l s o u n d i n t e n s i t i e s and large electric fields for the electron drift v e l o c i t y t h e e x p r e s s i o n of t h e l i n e a r t h e o r y [3,2], r e p r e s e n t i n g in t h e c a s e of l a r g e s o u n d a m p l i t u d e s t h e s o u n d w a v e b y a n a r r a y of h i g h p o t e n t i a l barriers * travelling with the sound velocity and c a l c u l a t i n g t h e c u r r e n t in t h i s s y s t e m f o r Vd, we obtain the following interpolation formula valid f o r a r b i t r a r y s o u n d i n t e n s i t i e s [2]:

.,"'"

/

v d - vs

[

#E-V

/ 1+

s-

L_

q4r~ W/W c 2 4 4 2 2 2 + P q r D ( l + q r D)

~s i n h

"exp

(2)

Wcc

/ k

I I I I / !

0

I

I I I I I I I I

-t

-2 -3

2

10

",,,

8

I

-4 t

k

3

4

5

-v-O v~

Fig. 1. The output sound intensity and maximum e l e c t r i c field h e a r the anode as functions of the applied voltage and the c u r r e n t voltage curve of the specimen for the c a s e of amplification of the t r a n s v e r s a l sound wave with the frequency f = 500 MHz and initial intensity Wo = = 10 -7 W / c m 2 in CdS c r y s t a l with p = 750 ~ cm, =200 c m 2 / V sec, 7 7 = 3 . 7 × 10 -2 , T = 0 . 0 2 6 e V .

2q2r~l]

Here r D is the Debye length, p = (e/Tq)(g- Vs/~) , T t h e e l e c t r o n i c t e m p e r a t u r e in e n e r g y u n i t s , W c = q 2 r 2 ( T N V s / T 1 ) (~1 i s t h e e l e c t r o m e c h a n i c coupling ~'onstant) and F(W/Wc) is some slowly v a r y i n g f u n c t i o n , w e a k l y d e p e n d e n t on t h e s h a p e of t h e w a v e . • At large sound amplitudes the sound wave is no more sinusoidal [2], which we take into account. 260

!

12

/

/ I

0

// & .....

/" /

/

~) -~.8~Vlcm

E%875V/cm ,,"

dW/dx

/ / / /

...... /--;:? ...... [ = ! t7A/cm

2

//

T h e s y s t e m of e q s . (1) a n d (2) i s n o w s e l f contained and together with the boundary condit i o n s W(0) = Wo a n d L

f E(x)dx = V, O

(L a n d F a r e t h e l e n g t h of t h e s p e c i m e n a n d t h e applied voltage), fully describes the problem. This system may be easily solved for small

PHYSICS

Volume 30A, number 4

s p e c i m e n u n d e r sound a m p l i f i c a t i o n c o n d i t i o n s h a s an S - s h a p e d f o r m with a l l s u b s e q u e n t c o n s e q u e n c e s ( h y s t e r e s i s of W ( L ) , E ( L ) and c u r r e n t I a s f u n c t i o n of t h e a p p l i e d v o l t a g e , p i n c h i n g of t h e c u r r e n t and the sound flux and so on).

(W << Wc) and l a r g e ( W > > W c) sound i n t e n s i t i e s , t h e t o t a l s o l u t i o n b e i n g o b t a i n e d by " m a t c h i n g " t h e two s o l u t i o n s at W ~ Wc. S o m e of the t y p i c a l r e s u l t s of t h e c a l c u l a t i o n a r e g r a p h i c a l l y p r e s e n t e d in fig. 1. L e t u s m a k e the f o l l o w i n g two r e m a r k s ( f o r a d e t a i l e d d i s c u s s i o n of the r e s u l t s s e e r e f . 2): 1) T h e n o n - l i n e a r e f f e c t s a c c o m p a n y i n g t h e sound a m p l i f i c a t i o n in a f i n i t e c r y s t a l a p p e a r o n l y in a r e s t r i c t e d r e g i o n of v o l t a g e s on t h e s p e c i m e n . T h u s , one m a y a l w a y s o b t a i n the l i n e a r a m p l i f i c a t i o n e i t h e r by d i m i n i s h i n g t h e l e n g t h of the c r y s t a l o r by i n c r e a s i n g t h e v o l t a g e on t h e s p e c i men. 2) T h e c u r r e n t v o l t a g e c u r v e of the f i n i t e

INELASTIC

NEUTRON IN

20 October 1969

LETTERS

T h e a u t h o r i s i n d e b t e d to P r o f . V. L. B o n c h B r u e v i c h and h i s c o l l e a g u e s f o r h e l p f u l d i s c u s sions.

References 1. P . K . Tien, Phys. Rev. 171 (1968) 970. 2. Yu. V. Gulayev, Fiz. Tver. Tela, to be published. 3. D. L. White, J. Appl. Phys. 33 (1962) 2547.

SCATTERING ITS GROUND

FROM STATE

LIQUID

HELIUM

J . F. F E R N A N D E Z

Physics Department, University of South Carolina, Columbia, S.C., USA and H. A. G E R S C H *

School of Physics, Georgia Institute of Technology, Atlanta, Georgia, USA Received 18 September 1969

Observed periodic fluctuations in the energy width as a function of momentum transfer of neutrons scattered off ground state helium liquid are explained as due to short range h a r d - c o r e interactions between helium atoms.

C o w l e y and Woods [1] h a v e s c a t t e r e d n e u t r o n s i n e l a s t i c a l l y off of l i q u i d h e l i u m and i m p a r t e d l a r g e m o m e n t u m t r a n s f e r s ( f r o m 2 to 9 A - I ) to t h e h e l i u m a t o m s . T h e s o l i d c u r v e of fig. 1 s h o w s t h e m e a s u r e d e n e r g y width of the n e u t r o n g r o u p s a s a f u n c t i o n of m o m e n t u m t r a n s f e r with the h e l i u m at 1.1°K. T h e d o t t e d s t r a i g h t l i n e i s t h e theory for free helium atoms, assuming ground s t a t e s i n g l e p a r t i c l e m o m e n t a a s c a l c u l a t e d by M c M i l l a n [2], and n e g l e c t i n g any c o n d e n s a t e . This letter offers the explanation that these w i d t h s a r e p e r i o d i c f l u c t u a t i o n s about a l i n e a r d e p e n d e n c e , due to t h e h a r d c o r e i n t e r a c t i o n s b e t w e e n h e l i u m a t o m s . A p p l i c a t i o n of the u n c e r t a i n t y p r i n c i p l e (we t a k e / ~ = 1 e v e r y w h e r e ) to t h e s c a t t e r i n g p r o c e s s s h o w s that the s c a t t e r i n g t i m e 2talk 2 i s not n e g l i g i b l y s m a l l c o m p a r e d to t h e t i m e b e t w e e n h e l i u m c o l l i s i o n s , din~k, d ~ 1 A,

so that the f r e e p a r t i c l e m o d e l i s d e f i c i e n t . C o l l i s i o n s c a n be a c c o u n t e d f o r a p p r o x i m a t e l y by an e f f e c t i v e p o t e n t i a l ~(k), so that the s i n g l e p a r t i c le e n e r g y b e c o m e s Fk = k 2 / 2 m + Z(k). We d e t e r m i n e t h e e f f e c t i v e p o t e n t i a l ~ ( k ) f r o m the s t r u c t u r e f a c t o r S(k), u t i l i z i n g t h e f - s u m r u l e . O u r b a s i c a s s u m p t i o n i s that a H a r t r e e - F o c k d e s c r i p t i o n s u f f i c e s to d e t e r m i n e S(k, w), the d y n a m i c s t r u c t u r e f a c t o r m e a s u r e d in the e x p e r i m e n t :

S ( k , ~o) = S(k) ~kl n k l 5(w-Ekl+k + F k l )

(1)

w h e r e nk i s t h e n u m b e r of h e l i u m a t o m s h a v i n g w a v e v e c t o r k in the g r o u n d s t a t e . T h e f a c t o r S(k) in eq. (1) i s n e e d e d to p r o d u c e t h e e x p e r i m e n t a l r e s u l t upon i n t e g r a t i o n o v e r w, NS(k) = * Supported by National Science Foundation Grant GP 14058. 261