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THERMALIZATION OF HIGH-FREQUENCY
PHONONS IN SILICON SINGLE CRYSTALS
R. Horstman, J. Wolter
To cite this version:
JOURNAL DE PHYSIQUE
CoZloqz~e C6, supple'ment au n o
12,Tome
4 2 ,ddcembre
1981page
C6-813THERMALIZATION OF HIGH-FREQUENCY PHONONS IN SILICON SINGLE CRYSTALS
R.E. Horstman and J . W o l t e r
Philips Research Laboratories, Eindhoven, The NetherZmds
A b s t r a c t . - H e a t p u l s e s g e n e r a t e d by means o f s m a l l c o n s t a n t a n h e a t e r s w e r e t r a n s m i t t e d t h r o u g h l o n g s i l i c o n s i n g l e c r y s t a l s . D e s p i t e i s o t o p e s c a t t e r i n g w e f i n d t h e t r a n s m i t t e d e n e r g y t o depend l i n e a r l y on t h e a p p l i e d h e a t e r power f o r h e a t e r t e m p e r a t u r e s b e t w e e n 5 K and 3 0 K. We o b t a i n e v i d e n c e t h a t r a p i d t h e r m a l i z a t i o n of t h e phonon p u l s e t a k e s p l a c e i n t h e v i c i n i t y of t h e h e a t e r . We r e p o r t on e x p e r i m e n t s p e r f o r m e d t o s t u d y t h e i n f l u e n c e of i s o t o p e s c a t t e r i n g o n t h e t r a n s m i s s i o n of h i g h - f r e q u e n c y phonons t h r o u g h l o n g s i l i c o n c r y s t a l s . The o b s e r v e d p u l s e s h a p e s c a n n o t be d e s c r i b e d w i t h s i m p l e models f o r b a l l i s t i c o r d i f f u s i v e phonon p r o p a g a t i o n . The e x p e r i m e n t s i n d i c a t e t h a t f r e q u e n c y down-conversion of h i g h - f r e q u e n c y phonons t a k e s p l a c e w i t h i n s e v e r a l m i c r o s e c o n d s i n t h e n e i g h b o u r h o o d of t h e phonon g e n e r a t o r . The e x p e r i m e n t s w e r e c a r r i e d o u t on two d i s l o c a t i o n - f r e e s i l i c o n s i n g l e c r y s t a l s w i t h c y l i n d r i c a l s h a p e ( 5 cm d i a m e t e r ) . One c r y s t a l ( 1 0 cm l o n g ) was grown a l o n g t h e [ I l l ] d i r e c t i o n , t h e o t h e r ( 6 c m l o n g ) a l o n g t h e [ l o o ] d i r e c t i o n . C o n s t a n t a n h e a t e r s ( i m p e d a n c e 50 ohms) a a d a l u m i n i u m b o l o m e t e r s (0.5
*
0.2 mm2) w e r e e v a p o r a t e d o n t o o p p o s i t e c r y s t a l e n d s . The c r y s t a l s were immersed i n l i q u i d h e l i u m . Only t h e h e a t e r and t h e b o l o m e t e r w e r e mounted i n vacuum c a n s .V o l t a g e p u l s e s ( 5 0 n s d u r a t i o n ) were a p p l i e d t o t h e h e a t e r . The d e t e c t e d s i g n a l s w e r e r e c o r d e d by means of a B i o m a t i o n 8 1 0 0 t r a n s i e n t r e c o r d e r a n d s t o r e d i n t h e memory of a N i c o l e t 1 1 7 0 d i g i t a l s i g n a l a v e r a g e r c o n n e c t e d t o a c o m p u t e r s y s t e m f o r f u r t h e r d a t a h a n d l i n g . T y p i c a l s i g n a l s f o r s e v e r a l h e a t e r t e m p e r a t u r e s a r e d i s p l a y e d i n F i g . 1 f o r t h e [ l l l ] c r y s t a l . The a m p l i t u d e s of t h e p u l s e s h a v e b e e n s c a l e d t o t h e same h e i g h t t o a l l o w a b e t t e r c o m p a r i s o n of t h e p u l s e s h a p e s . The a r r o w s i n d i c a t e t h e e x p e c t e d a r r i v a l t i m e s f o r b a l l i s t i c phonon p r o p a g a t i o n of t h e t r a n s v e r s e phonon modes [ I ] . T h e s e t i m e s a r e i n good a g r e e m e n t w i t h t h e o b s e r v e d o n s e t s o f t h e h e a t p u l s e s . However, t h e o b s e r v e d s i g n a l s a r e s u b s t a n t i a l l y b r o a d e r t h a n t h e d u r a t i o n (50 n s ) of t h e o r i g i n a l p u l s e a p p l i e d t o t h e h e a t e r . A n e g l i g i b l e p a r t of t h i s b r o a d e n i n g is d u e t o t h e t i m e c o n s t a n t of t h e b o l o m e t e r ( 1 0 0 n s ) and t h e f i n i t e d i m e n s i o n s of t h e h e a t e r and t h e b o l o m e t e r . From t h e d a t a [ 2 ] f o r t h e i s o t o p e s c a t t e r i n g t i m e we c a l c u l a t e t h e a r r i v a l t i m e s f o r t h e maximum a m p l i t u d e i n t h e phonon f l u x a t t h e d o m i n a n t phonon f r e q u e n c y i n t h e P l a n c k s p e c t r u m f o r d i f f e r e n t h e a t e r t e m p e r a t u r e s
C6-8 1 4 JOURNAL DE PHYSIQUE
( T a b l e I). These t i m e s a r e much l o n g e r t h a n t h e t i m e s observed f o r t h e a r r i v a l of t h e maximum phonon f l u x . Thus s i m p l e d i f f u s i v e phonon p r o p a g a t i o n caused by i s o t o p e s c a t t e r i n g does n o t e x p l a i n t h e o b s e r v e d p u l s e s h a p e s .
A
,r
t
I I20
2 1
22
TIME
OFFLIGHT
IDS1
F i g u r e 1: Bolometer s i g n a l f o r t r a n s v e r s e phonons a s a f u n c t i o n of time f o r d i f f e r e n t h e a t e r t e m p e r a t u r e s .
...
T ~ / K fd/THzh / m m
t/ps
-- ---
-,---a---- 3.8 0.2 100 20--
T a b l e I : C a l c u l a t e d mean f r e e p a t hh
and 6.7 0.4 loo e x p e c t e d a r r i v a l time t f l u x f o r t h e dominant peonon f r e q u e n c y f d of t h e maximum phonon i n 12.0 0.7 1 1000 t h e P l a n c k s p e c t r u m of a h e a t e r w i t ht e m p e r a t u r e T 21.0 1.3 0.1 10000
H
We a l s o determined t h e t o t a l e n e r g y a r r i v i n g a t t h e d e t e c t o r by i n t e g r a t i n g t h e phonon s i g n a l f o r d i f f e r e n t h e a t e r t e m p e r a t u r e s . F o r t h e L and T phonons i n t h e [ I l l ] c r y s t a l Fig.2 shows t h e e n e r g y i n t h e phonon s i g n a l a s a f u n c t i o n of t h e e n e r g y a p p l i e d t o t h e h e a t e r . The a b s o l u t e e n e r g y s c a l e was o b t a i n e d from a b s o l u t e measurements i n t h e manner d e s c r i b e d i n r e f . 3 . We f i n d t h a t , i n d e p e n d e n t of h e a t e r a r e a and c r y s t a l l o g r a p h i c d i r e c t i o n , t h e d e t e c t e d e n e r g y is d i r e c t l y p r o p o r t i o n a l t o t h e a p p l i e d h e a t e r energy. From t h e s t r o n g f r e q u e n c y dependence of t h e i s o t o p e s c a t t e r i n g , however, we e x p e c t t h a t , w i t h i n c r e a s i n g h e a t e r t e m p e r a t u r e , a d e c r e a s i n g f r a c t i o n of t h e phonons p r o p a g a t e s b a l l i s t i c a l l y from t h e h e a t e r t o t h e b o l o m e t e r (dashed l i n e i n F i g . 2 ) .
HEATER TEMPERATURE
[ K ]
2
3 4 5 7
10
1L
20 30
I I I 1 1100
-
-
CJ10-
\ 7 0.-
F i g u r e 2: T r a n s m i t t e d e n e r g y f o r>-
--
a
1 -
l o n g i t u d i n a l and t r a n s v e r s er t
phonons a s a f u n c t i o n of a p p l i e d W'
h e a t e r energy. P u l s e l e n g t h 5 0 n s . Z Upper a x i s shows c a l c u l a t e d h e a t e r W t e m p e r a t u r e s . ( 0 ) A b s o l u t e C3 measurements f o r 1*1 mm used f o r W I-0.1
-
c a l i b r a t i o n . R e l a t i v e measurement U1
w i t h h e a t e r s yf dimensions 2*2 mm2 W ( x ) ; 1*1 m m ( A ) ; 0.5*0.5 mrnt-
(+>
.
C a l c u l a t e d t r a n s m i t t e d W 0 e n e r g y : ( f u l l y drawn l i n e ) a l l t r a n s v e r s e phonons p r o p a g a t e0.01
-
b a l l i s t i c a l l y ; (dashed l i n e ) t a k i n g i n t o a c c o u n t i s o t o p e L s c a t t e r i n g0.1
1
10
100
1000
10000
A P P L I E D
HEATER ENERGY
( n ~ l r n d l
To e x p l a i n o u r r e s u l t s we s u g g e s t t h e f o l l o w i n g model. Due t o i s o t o p e s c a t t e r i n g a h o t phonon c l o u d 17.81 i s formed i n f r o n t of t h e h e a t e r . I n s i d e t h e c l o u d s t r o n g phonon s c a t t e r i n g w i t h e f f i c i e n t f r e q u e n c y c o n v e r s i o n t a k e s p l a c e . T h i s i s l i k e l y , because i n many c r y s t a l s anharmonic decay of L phonons a t 1 THz t a k e s p l a c e w i t h i n one microsecond [ 9 ] . D i r e c t decay of T phonons i s much l e s s l i k e l y , b u t due t o i s o t o p e s c a t t e r i n g T phonons c a n be r a p i d l y c o n v e r t e d i n t o L phonons, which s u b s e q u e n t l y decay. The dimensions of t h e c l o u d a r e d e f i n e d by t h e i s o t o p e s c a t t e r i n g . From t h i s c l o ~ d low-frequency phonons e s c a p e and p r o p a g a t e ' b a l l i s t i c a l l y t h r o u g h t h e c r y s t a l . The o b s e r v e d broadening of t h e phonon s i g n a l a t t h e o p p o s i t e c r y s t a l s u r f a c e i s d e t e r m i n e d by t h e l i f e t i m e of t h e high-frequency phonons i n s i d e t h e c l o u d .R e f e r e n c e s : [ I ] R.E.Horstman, Phys.Lett.79A (1980) 229 [ 2 ] M.G.Holland, Phys.Rev.132 (1963) 2461
[ 3 ] J . W o l t e r and R.E.Horstman, Phys.Lett.61A (1977) 238 (41 R.E.Horstman and J . W o l t e r , Phys.Lett.62A (1977) 279
