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PARAMETRIC AMPLIFICATION OF MICROWAVE
PHONONS IN SEMICONDUCTORS
H. Spector
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, suppze'ment au n o 12, Tome 42, de'cembre 1981
PARAMETRIC AMPLIFICATION OF MICROWAVE PHONONS
I N SEMICONDUCTORS
K.N. spector*
IZZinois Institute of TechnoZogy, Chicago, IL, U.S.A.
Abstract.-
Two d i f f e r e n t nonlinear processes involving the i n t e r a c t i o n s of phonons with electrons i n a semiconductor i n t h e presence of a d.c. e l e c t r i c f i e l d have been t h e o r e t i c a l l y investigated, The f i r s t process involves the
mixing
of u l t r a s o n i c iKaves.to generate'n&-ulfrasohid waveis a t 'the sum-or 'dfffefence PrequencXes of t h e o r i g i n a l waves. The mechanisms whfch couples t h e waves together involves the i n t e r a c t i o n of t h e c a r r i e r s bunched by one of t h e waves with t h e f i e l d s induced by the other wave. The second process involves t h e i n t e r a c t i o n between a forward t r a v e l i n g u l t r a s o n i c wave and a microwave r a d i a t i o n f i e l d t o generate a backward t r a v e l i n g u l t r a s o n i c wave (phonon echo) and an amplification of t h e forward trav- e l i n g wave. I n p i e z o e l e c t r i c semiconductors i n the presence of a d.c. e l e c t r i c f i e l d , t h e need f o r a threshold microwave f i e l d f o r the generation of the echo wave and the amplification of t h e forward t r a v e l i n g wave can be eliminated underc e r t a i n circumstances.
Nonlinear processes involving the i n t e r a c t i o n of phonons with each other o r with photons provide a mechanism f o r generating high frequency phonons i n the microwave frequency range. These nonlinear processes can be d r a s t i c a l l y effected by the presence of d r i f t i n g c a r r i e r s i n a semiconductor. The n o n l i n e a r i t i e s which give r i s e t o the parametric amplification of microwave phonons i n semiconductors o r i g i n a t e e i t h e r from the nonlinear p r o p e r t i e s of t h e media (such a s the existence of t h i r d order e l a s t i c c o e f f i c i e n t s , electro-optical c o e f f i c i e n t s , photoelastic c o e f f i c i e n t s , etc.) o r from the nonlinear currents generated when charge c a r r i e r s bunched by one wave i n t e r a c t with the f i e l d s induced by t h e other wave.
The b a s i c equations which ggvern t h e behavior of the system of phonons i n t e r - acting with each other, a s w e l l a s with electrons and photons, a r e t h e equation of motion of the l a t t i c e i n the e l a s t i c continuium approximation
where ui is t h e amplitude of t h e u l t r a s o n i c wave and Tij i s t h e s t r e s s tensor, Maxwell's equations f o r the e l e c t r i c and magnetic f i e l d s , and the c o n s t i t u t i v e re-
+
+
l a t i o n s between t h e s t r e s s tensor, p o l a r i z a t i o n P, magnetization M and current
-+ -+
-+
density J and t h e s t r a i n tensor S e l e c t r i c f i e l d E and magnetic f i e l d H. I n a
i j
'
"Supported by a grant from the NSF Material Research Laboratory Program, Grant No.
76-24466.
n o n f e r r o e l e c t r i c m a t e r i a l , t h e f r e e energy
F
can b e expandedi n
powers of t h e1
s t r a i n and t h e e l e c t r i c f i e l d w h i l e i n a f e r r o e l e c t r i c m a t e r i a l , i t can b e ex-
2
panded i n powers of t h e s t r a i n and t h e p o l a r i z a t i o n
.
The stress t e n s o r can be 1 o b t a i n e d from t h e f r e e energy u s i n g t h e r e l a t i o na
F T..=as
(2) '3 i j 1 2w h i l e t h e e l e c t r i c displacement v e c t o r and e l e c t r i c f i e l d v e c t o r can b e o b t a i n e d from t h e f r e e energy u s i n g t h e r e l a t i o n s
and
The c u r r e n t d e n s i t y induced a t
a
frequency w 3 can a l s o h e expanded in powers oft h e e l e c t r i c f i e l d
where w l h 2 = w 3 The n o n l i n e a r term i n t h e c u r r e n t d e n s i t y a r i s e s from t h e mixing of waves of d i f f e r e n t f r e q u e n c i e s t o g e n e r a t e waves a t t h e sum o r d i f f e r e n t f r e - quency. By expanding t h e f r e e energy up t o terms of t h i r d o r d e r i n t h e product of t h e s t r a i n and e l e c t r i c o r p o l a r i z a t i o n f i e l d s , we can o b t a i n t h e terms i n t h e i n t e r a c t i o n which couple t h e phonons, photons and e l e c t r o n s t o g e t h e r i n t h e lowest o r d e r t o g i v e r i s e t o p a r a m e t r i c a m p l i f i c a t i o n of phonons i n semiconductors.
We now c o n s i d e r two d i f f e r e n t p r o c e s s e s i n which waves can b e coupled t o g e t h e r by t h e n o n l i n e a r terms t o g e n e r a t e new waves a t t h e sum and d i f f e r e n c e f r e q u e n c i e s of t h e o r i g i n a l waves. I n t h e f i r s t p r o c e s s , two u l t r a s o n i c waves a r e coupled
3
t o g e t h e r t o g e n e r a t e e i t h e r harmonics o r subharmonics of t h e o r i g i n a l waves
.
The e l e c t r i c f i e l d s which appear i n t h e second term of t h e c u r r e n t d e n s i t y , Eq.(5), a r e t h e n t h e p i e z o e l e c t r i c f i e l d s induced by each of t h e u l t r a s o n i c waves. I n t h e second p r o c e s s , t h e r e i s an i n t e r a c t i o n between an u l t r a s o n i c wave and a microwave e l e c t r o m a g n e t i c wave t o g e n e r a t e an echo u l t r a s o n i c wavelp2 o r between twoo p p o s i t e l y d i r e c t e d u l t r a s o n i c waves t o g e n e r a t e a microwave r a d i a t i o n f i e l d
.
The e l e c t r i c f i e l d s i n t h e second term of t h e c u r r e n t d e n s i t y a r e t h e r a d i a t i o n f i e l d and t h e p i e z o e l e c t r i c f i e l d accompanying t h e u l t r a s o n i c waves. I f t h e non- l i n e a r coupling terms a r e s m a l l , we can w r i t e t h e displacement due t o t h e u l t r a - s o n i c wave and t h e e l e c t r i c f i e l d s due t o t h e e l e c t r o m a g n e t i c wave a s p l a n e waves modulated by a slowly v a r y i n g amplitude A,E ( z , t )
,
u ( z , t ) a A(z)expi(qz-wt) ( 6 )C6-678 JOURNAL DE PHYSIQUE
Here Ai, A and Ak a r e t h e slowly v a r y i n g amplitudes of t h e waves which a r e coupled
j
t o g e t h e r , a i is t h e l i n e a r l o s s c o e f f i c i e n t f o r t h e i t h wave and q Ls t h e non-
i j k
l i n e a r coupling c o e f f i c i e n t which couples t h e t h r e e waves t o g e t h e r . These coupled e q u a t i o n s have been solved t o o b t a i n t h e g e n e r a t i o n of u l t r a s o n i c waves f o r t h e
1-4
d i f f e r e n t p r o c e s s e s mentioned above
.
The r e s u l t s o b t a i n e d depend upon t h e magnitude, frequency and e l e c t r i c f i e l d dependence of t h e n o n l i n e a r coupling co-5 e f f i c i e n t
nijk
.
For t h e f i r s t p r o c e s s , which i n v o l v e s t h e mixing of u l t r a s o n i c waves t o gene- r a t e u l t r a s o n i c waves a t t h e sum o r d i f f e r e n c e frequency, t h e n o n l i n e a r coupling c o e f f i c i e n t g i v e s r i s e t o a n o n l i n e a r g a i n which depends on t h e s t r a i n f i e l d accompanying t h e i n i t i a l wave
a~~ =
-
a+ M
2q l s c o ) IThe n o n l i n e a r coupling i s enhanced i n p a r a e l e c t r i c and f e r r o e l e c t r i c m a t e r i a l s b e c a u s e of t h e l a r g e r n o n l i n e a r m a t e r i a l parameters i n t h e s e substances. Eq,(8)
g i v e s t h e n o n l i n e a r f o r t h e c a s e of subharmonic g e n e r a t i o n . For harmonic genera- t i o n , t h e i n t e n s i t y of t h e generated wave i s d i r e c t l y p r o p o r t i o n a l t o
n.
For t h e second p r o c e s s , which i n v o l v e s t h e mixing of u l t r a s o n i c waves w i t h a microwave r a d i a t i o n f i e l d t o g e n e r a t e an echo wave and amplify t h e o r i g i n a l u l t r a - s o n i c wave, we have found t h a t f o r p i e z o e l e c t r i c semiconductors i n t h e presence of a n a p p l i e d d.c. f i e l d t h e need f o r a t h r e s h o l d microwave f i e l d f o r g e n e r a t i o n of t h e echo and a m p l i f i c a t i o n of t h e i n i t i a l wave can b e e l i m i n a t e d under c e r t a i n
1
circumstances
.
The c o n d i t i o n f o r t h e a m p l i f i c a t i o n of t h e i n i t i a l wave and t h e g e n e r a t i o n of t h e echo isReN > a.+a
1 e (9)
where
and
%
i s t h e amplitude of t h e microwave f i e l d . I n t h e presence of d r i f t i n g c a r r i e r s , a >a where t h e s u b s c r i p t s i and e denote q u a n t i t i e s a s s o c i a t e d withi e'
t h e i n i t i a l and echo waves, and t h e c o n d i t i o n given i n E q . ( 9 ) can b e s a t i s f i e d w i t h n e g l i g i b l y s m a l l t h r e s h o l d microwave e l e c t r i c f i e l d s .
References
1. J.E. Economou and H.N. S p e c t o r , Phys. Rev. B
18,
5578 (1978).2. 3.E. Economou and H.N. S p e c t o r , Phys. Rev. B
