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FOCUSING OF DISPERSIVE PHONONS IN Ge
G. Northrop, W. Dietsche, A. Zdetsis, J. Wolfe
To cite this version:
G. Northrop, W. Dietsche, A. Zdetsis, J. Wolfe. FOCUSING OF DISPERSIVE PHONONS IN Ge.
Journal de Physique Colloques, 1981, 42 (C6), pp.C6-209-C6-211. �10.1051/jphyscol:1981660�. �jpa-
00221596�
JOURNAL DE PHYSIQUE
CoZZoque C6, supple'ment au n012, Tome 42, de'cembre 1981 page C6-209
FOCUSING OF D I S P E R S I V E PHONONS I N Ge
G.A. Northrop, W. D i e t s c h e , A.D. Z d e t s i s and J . P . Wolfe
Physics Deparhent and MateriaZs Research Laboratory, University of IZZinois, Urbana, I L . 61801, U.S.A.
A b s t r a c t . - S i n g u l a r i t i e s i n t h e phonon f l u x a r e observed and c a l c u l a t e d f o r d i s p e r s i v e phonons i n Ge.
The i n t e n s i t y of b a l l i s t i c h e a t p u l s e p r o p a g a t i o n i s h i g h l y a n i s o t r o p i c f o r most c r y s t a l s , a n e f f e c t known a s phonon-focusing. Phonon f o c u s i n g h a s been s t u d i e d i n many m a t e r i a l s , b o t h w i t h f i x e d s o u r c e and d e t e c t o r , and more r e c e n t l y w i t h l a s e r s a s movable h e a t - p u l s e A q u a n t i t a t i v e e x p l a n a t i o n
o r i g i n a l l y o f f e r e d by Maris was based on c a l c u l a t i n g a n enhancement f a c t o r , 3
+ +
A = Ida,/d~,l, t h e r a t i o of k-space t o V-space s o l i d a n g l e s . T h i s f a c t o r d e s c r i b e s how t h e phonon f l u x i s i n c r e a s e d o r d e c r e a s e d o v e r t h e i s o t r o p i c c a s e . The l o c u s o f p o i n t s dQv = 0 forms l i n e s t h a t mark d i r e c t i o n s of peak phonon f l u x .
These s i n g u l a r i t y
lines
have been observed by s c a n n i n g a p u l s e d l a s e r a s a h e a t s o u r c e o v e r o n e s u r f a c e of a c r y s t a l and r e c o r d i n g , a s a c o n t i n u o u s f u n c t i o n of p r o p a g a t i o n d i r e c t i o n , t h e h e a t p u l s e i n t e n s i t y r e c e i v e d by a f i x e d d e t e c t o r on t h e o p p o s i t e f a c e . L T h i s t e c h n i q u e , known a s b a l l i s t i c phonon imaging, was u s e d t o produce t h e p i c t u r e i n F i g . l a f o r Ge. Here, X-Y p o s i t i o n o n t h e sample f a c e c o r r e s p o n d s t o a r a n g e of p r o p a g a t i o n d i r e c t i o n s , and b r i g h t n e s s i s p r o p o r t i o n a l t o phonon i n t e n s i t y . The b r i g h t l i n e s bounding l i g h t a r e a s a r e s i n g u l a r i t y l i n e s , i n good agreement w i t h t h e t h e o r e t i c a l l y p r e d i c t e d s i n g u l a r i t y l i n e s i n F i g . 2.Fig.1: A1 d e t e c t o r (low-v) phonon image. Pb-0-Pb d e t e c t o r (high-v) phonon image.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981660
C6-210 JOURNAL DE PHYSIQUE
I n t h i s paper we extend t h e s e experimental and t h e o r e t i c a l r e s u l t s t o i n c l u d e t h e e f f e c t s of v e l o c i t y d i s p e r s i o n which occur a t h i g h e r phonon
f r e q u e n c i e s . To observe t h e e f f e c t s of d i s p e r s i o n on t h e s i n g u l a r i t y d i r e c t i o n s i n Ge, two m o d i f i c a t i o n s have been made t o our phonon imaging technique:* 1 ) A t h i n sample (0.5 mm) and g r e a t e r time r e s o l u t i o n (60 n s ) a r e used t o o f f s e t t h e s m a l l e r mean-free-path of l a r g e r - k phonons. -+ 2) A t u n n e l - j u n c t i o n (Pb-0-Pb) d e t e c t o r i s used
t o d e t e c t phonons w i t h v > 700 GHz. An A 1 bolometer i s used f o r d e t e c t i o n of low-v phonons
.
Comparison of t h e two images f o r t h e s e two d e t e c t o r s , F i g . l a and F i g . l b , r e v e a l s t h r e e major d i f f e r e n c e s : 1 ) The wedge shaped r i d g e s (PTA mode) extending h o r i z o n t a l l y and v e r t i c a l l y from t h e two diamond shapes a r e wider i n t h e high frequency image than i n t h e low frequency image. 2) The two diamond s t r u c t u r e s (STA mode) about t h e
<loo>
d i r e c t i o n s i n t h e low frequency image a r e n o t i c e a b l y rounded a t h i g h e r f r e q u e n c i e s . 3 ) The s h a r p , t h r e e - f o l d symmetric p a t t e r n (STA mode) t o t h e l e f t and r i g h t of c e n t e r i n F i g . l a h a s disappeared i n Fig. l b .Fig. 2 : Zero frequency s i n g u l a r i t i e s . Fig. 3 : FTA mode frequency dependence.
I n t h e non-dispersive phonon-focusing t h e o r y an a r e a ( s o l i d a n g l e ) i n
+ +
k-space transformed o r mapped i n t o an a r e a ( s o l i d a n g l e ) i n V-space. This was
-f +
independent on Ik(, and
l$l
was f i x e d f o r each d i r e c t i o n of k . I n t h e d i s p e r s i v e c a s e t h e s e r e s t r i c t i o n s a r e removed and a more g e n e r a l f o r m u l a t i o n i s r e q u i r e d . Thus we d e f i n e t h e d i s p e r s i v e enhancement f a c t o r A =ld3$/d331. T h i s i s t h e r a t i o-+ 3 -f
of an i n f i n i t e s i m a l volume i n k-space t o i t s corresponding volume i n V-space. By analogy t o t h e non-dispersive c a s e s i n g u l a r i t i e s w i l l occur when d V 3+ = 0, and, due
t o t h e i n c r e a s e d number of dimensions, t h e l o c u s of s i n g u l a r i t i e s w i l l be s u r f a c e s
+ +
i n k-space and V-space. I n our experiment, we study t h e s e s u r f a c e s by s e c t i o n i n g them a t a f i x e d frequency
.
The computation of A3 i s based upon a model4 of t h e f u l l B r i l l o i n - z o n e phonon band s t r u c t u r e f o r G e . T h i s extended Born-Von Karmen scheme p a r a m e t e r i z e s t h e l a t t i c e dynamics w i t h 20 f o r c e c o n s t a n t s r e p r e s e n t i n g up t o f i f t h n e a r e s t neighbor i n t e r a c t i o n s . With t h e s e c o n s t a n t s a d j u s t e d t o f i t i n e l a s t i c n e u t r o n
+ +
s p e c t r a t h e model p r o v i d e s w(k) f o r a l l modes and any v a l u e of k.
Fig.4: STA mode a t 800 GHz. Expected STA p a t t e r n a t 1500 GEz.
+ -+ +
S i n c e V(k) = Vkw(k), and A3 i s t h e J a c o b i a n of t h i s f u n c t i o n , A i n c l u d e s
-+ 3
second d e r i v a t i v e s o f w(k), and is h e r e d e t e r m i n e d by f i n i t e d i f f e r e n c e methods. A two d i m e n s i o n a l r o o t - f i n d i n g r o u t i n e t h e n f i n d s l i n e s f o r which w = c o n s t a n t and A;' = 0. These l i n e s a r e p r o j e c t e d i n V-space -+ i n t o t h e e x p e r i m e n t a l geometry.
These c a l c u l a t i o n s have been done f o r b o t h t r a n s v e r s e modes f o r f r e q u e n c i e s between 700 and 1600 GHz. The r e s u l t s f o r two f r e q u e n c i e s a r e shown i n F i g s . 3 and 4.
The b r o a d e n i n g of t h e FTA r i d g e s t n g u l a r i t i e s w i t h i n c r e a s i n g f r e q u e n c y c a n be s e e n i n F i g . 3. The e x p e r i m e n t a l Pb-0-Pb d a t a a g r e e s w e l l w i t h t h e p r e d i c t i o n f o r v .:800 GHz. The change of t h e STA s i n g u l a r i t y p a t t e r n i s more complex, a s shown i n F i g . 4a. Note t h e rounding of t h e c o r n e r s o f t h e diamond s t r u c t u r e a s o b s e r v e d i n F i g . l b . The t h r e e - f o l d p a t t e r n a b o u t t h e < I l l > d i r e c t i o n i n Fig. 4 a h a s changed l i t t l e , t h u s i t s a b s e n c e from F i g . l b must b e d u e t o r e a s o n s o t h e r t h a n phonon-focusing. F i g . 4b i s t h e V-space (STA) p a t t e r n e x p e c t e d a t 1500 GHz. +
I n summary, we h a v e o b s e r v e d s h i f t s d u e t o d i s p e r s i o n i n t h e phonon- f o c u s i n g s i n g u l a r i t y p a t t e r n s i n Ge. These s h i f t s a g r e e w i t h p r e d i c t i o n s o b t a i n e d from a model o f t h e f u l l l a t t i c e dynamics i n G e .
T h i s work was s u p p o r t e d by t h e N a t i o n a l S c i e n c e Foundation u n d e r Grant DMR-80-20250.
REFERENCES
1. J. C. H e n s e l and R. C. Dynes, Phys. Rev. L e t t .
42,
1033 (1979).2. G. A. Northrop and J. P. Wolfe, Phys. Rev. L e t t .
2,
1424 (1979);Phys. Rev. B g , 6169 (1980).
3. H. J. Maris, J. Acoust. Soc. Am.
2,
812 (1971).4. A. D. Z d e t s i s , P r o c . I n t . Conf. o n L a t t i c e Dynamics, p. 45, P a r i s (1978).
