THE EFFECT OF STRUCTURAL DISORDER ON PHONON LIFETIMES

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THE EFFECT OF STRUCTURAL DISORDER ON PHONON LIFETIMES

D. Walton, J. Vanderwal, H. Teh

To cite this version:

D. Walton, J. Vanderwal, H. Teh. THE EFFECT OF STRUCTURAL DISORDER ON PHONON LIFETIMES. Journal de Physique Colloques, 1982, 43 (C9), pp.C9-545-C9-548.

�10.1051/jphyscol:19829109�. �jpa-00222413�

Colloque C9, supplément au n°12, Tome 42, décembre 1982 page C9-545

THE EFFECT OF STRUCTURAL DISORDER ON PHONON LIFETIMES

D. Walton, J.J. Vanderwal and H.C. Teh*

Physios Department, MoUastev University,Hamilton, Ontario, U.S.A.

Résumé•- Nous avons mesuré le temps de vie des phonons thermiques dans la silice vitreuse par spectroscopie Brillouin. Les résultats montrent que, en addition à celle due aux systèmes à deux niveaux, il existe une contribution substantielle due à la diffusion qui est indépendante de la température et probablement d'origi- ne structurale.

abstract.- The lifetimes of thermal phonons in fused silica have been measured using Brillouin spectroscopy. The results show that in addition to that due to two level systems there is a substantial contribution to the scattering which is temperature independent and is probably structural in origin.

Introduction.- A wave travelling through a structurally disordered material must be attenuated by the disorder. It is difficult in general to calculate the prop- erties of phonons in disordered systems, but it is possible to do so in the long wavelength limit using a Debye approximation and treating the phonons as plane waves. Such a calculation is detailed in reference (1). We will begin by out- lining the work reported in (1). It reveals that amorphous materials are unique in that, in addition to the usual scattering which increases in strength as the fourth power of the phenon frequency, the so-called "Rayleigh scattering", there is a contribution that increases linearly with frequency. This latter contribu- tion is unique to strongly scattering systems.

The hamiltonian for any solid may be written

I I _ £ J-i- + • > . 7~- B ,' I ^Ut UJ + * ><<»<t<- order terms (1) Where i is a site index, OC a component of displacement W ; , ^^,1 a 25 order force constant, and p. and M^ the momentum and mass of the atom at site i.

If this series is truncated after the 2r order term it may be readily diagonalized if the atoms are on a regular lattice. If the solid is disordered, though, this is not possible because we don't know how to choose an appropriate set of wave vectors. On the other hand, if long wave lengths alone are of interest, the Debye approximation may be employed. In this case one obtains in addition to the hamiltonian for the ordered lattice, additional terms arising from summing over a disordered lattice. Expressing the result in terms of annihilation and creation operators a. and aj£ corresponding to wave vector k one obtains.

H "- H

+H ' H * r f. **

U^l + 'A. ) (2 >

Where 4>^ i s the frequency of the wave, and € it's polariza- tion vector.

The first two terms in equation 3 contain the effect of density fluctuations.

The third arises from the fluctuations in force constant produced by the changing interatomic spacing. The effect of density fluctuations is negligible in glasses, but the fluctuations in force constant are not and lead to a strong attenuation of the phonon. The attenuation and energy shift can be obtained from the imaginary and real parts of the self energy We seek it's matrix elements:

*Permanent address : Physics Dept. National University of Singapore.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19829109

C9-546 JOURNAL DE PHYSIQUE

where G is t h e propagator corresponding t o H.

L e t t i n g t h e propagator correspondin t o t h e unperturbed h a m i l t o n i a n b e Go i t i s p l a u s i b l e t o approximate G by

( 8 )

Where

Atv)

T h i s approximation simply e x p r e s s e s phenomenologically t h e f a c t t h a t a phonon of frequency C*) i s a t t e n u a t e d i n t r a v e l l i n g from i t o j. The u s e of t h i s e x p r e s s i o n f o r

6

A

Where \/c&) i s t h e phonon group v e l o c i t y I

J?;~

(q-5

I n common with a l l s c a t t e r i n g c a l c u l a t i o n s t h e r e a r e two c o ? t r i b u t i o n s t o from e q u a t i o n s ( 5 ) one &he u s u a l i n c o h e r e n t f a c t o r ( f o r i= J and m =

1

A

.

b

4

, and makes no n e t c o n t r i b u t i o n t o t h e s c a t t e r i n g . However i f t h e s c a t t e r i n g i s s t r o n g

A

Experimental evidence f o r t h i s c o n t r i b u t i o n t o t h e phonon l i f e t i m e w i l l now be considered. We w i l l b e seeking a temperature independent term which i s i n addi- t i o n t o t h e e f f e c t of two l e v e l systems and r e l a x a t i o n p r o c e s s e s .

Experimental.- We have measured t h e phonon l i f e t i m e i n v i t r e o u s s i l i c a u s i n g B r i l l o u i n s c a t t e r i n g . T h i s t e c h n i q u e h a s t h e advantage t h a t i t can r e v e a l t h e presence of a temperature independent background, which i s d i f f i c u l t f o r a c o u s t i c methods. S i m i l a r measurements were a l s o performed by Vacher e t a l . ( 3 ) u s i n g two F a b r y - P e r o t ' s i n tandem.

We have employed a method of s t u d y i n g narrow l i n e s s u g g e s t e d by Bradberry and vaughant4' : i t depends on t h e f a c t t h a t t h e r e s o l u t i o n f u n c t i o n of a multipassed i n s t r u m e n t i s t h e s i n g l e p a s s f u n c t i o n taken t o a power e q u a l t o t h e number of p a s s e s . T h e r e f o r e t h e r e s o l u t i o n f u n c t i o n d r o p s o f f more q u i c k l y i n t h e wings t h a n t h e e x c i t a t i o n l i n e being s t u d i e d and, provided s u f f i c i e n t s i g n a l i s a v a i l - a b l e , t h e l i n e w i d t h of t h e s c a t t e r i n g l i n e i s o b t a i n e d by e s s e n t i a l l y measuring i n t h e wings of t h e s c a t t e r e d l i n e . Using a f i v e - p a s s Fabry-Perot i t h a s been shown t h a t it i s p o s s i b l e t o o b t a i n widths of l i n e s only one s i x t h a s wide a s t h e i n s t r u m e n t a l f u n c t i o n (5) provided s u f f i c i e n t i n t e n s i t y i s a v a i l a b l e . The specimen was a p i e c e of " S u p r a s i l " f u s e d q u a r t z 1 cm i n diameter by 2 cm long mounted i n a c o n v e n t i o n a l o p t i c a l c r y o s t a t . A 4 w a t t argon-ion l a s e r o p e r a t - i n g a t 5145

2

The s c a t t e r i n g a n g l e was 1 7 8 f 0 . 0 5 ~ . The c o l l e c t i o n a p e r t u r e was r e c t a n g u l a r , and provided a c o l l e c t e d beam 0 . 4 cm i n width by 0.8 cm i n h e i g h t . A minimum of 1000 counts a t t h e peak was n e c e s s a r y t o p r o v i d e adequate accuracy i n deconvolution.

The r e s u l t i n g s t a t i s t i c a l u n c e r t a i n t y i n t h e phonon l i n e w i d t h was about 5%.

was t h a t o b t a i n e d f o r t h a t spectrum. Because of changes i n f i n e s s e from r u n t o run it i s impossible t o achieve h i g h accuracy u n l e s s t h i s i s done.

Results.- The deconvoluted l i n e w i d t h s a r e p l o t t e d a g a i n s t temperature i n F i g . 1 These were c o r r e c t e d f o r t h e s m a l l broadening due t o f i n i t e a p e r t u r e u s i n g t h e procedure d e s c r i b e d by D a n i e l m e ~ e r ( ~ ) .

TEMPERATURE I K ) F i g u r e 1.

Phonon l i n e width a s a f u n c t i o n of temperature. The s o l i d l i n e i s a f i t t o t h e d a t a i n c l u d i n g a temperature independent term of magnitude 6.5 HZ.

The dashed l i n e i s a f i t w i t h o u t t h i s term.

The s o l i d l i n e through t h e d a t a i s t h e r e s u l t of a l e a s t s q u a r e s f i t t o t h e s e p o i n t s of t h e f u n c t i o n

A t c x L ( T P / ~ ) 4

i 3

c r 3

where T 2 kU/2k, 0 i s t h e phonon frequency, T i s t h e temperature.

The f i r s t term P i s t h e c o n t r i b u t i o n of two l e v e l systems, t h e t h i r d t h a t of r e l a x a - t i o n p r o c e s s e s ( 2 r 6 ' 7 ) and t h e second i s t h e temperature independent s t r u c t u r a l c o n t r i b u t i o n . The dashed l i n e i s t h e b e s t f i t which i s o b t a i n e d i f t h e co- e f f i c i e n t of t h e 2d term i s s e t 0 .

-

A = J 3.5 & # H z )3

=

C =

The s t a n d a r d d e v i a t i o n i s 0.8 M H z which i s about 7% and a g r e e s w i t h t h e s t a t i s t i c a l u n c e r t a i n t y of 5% when e r r o r s due t o v a r i a t i o n s i n s c a t t e r i n g geometry a r e added.

The l a t t e r a r e d i f f i c u l t t o e s t i m a t e , b u t should a l s o amount t o about 5%. We have assumed t h a t B i s temperature independent. On t h e o t h e r hand, i f i t i s n o t , t h e temperature dependence of t h e thermal c o n d u c t i v i t y w i l l b e a f f e c t e d .

The c o e f f i c i e n t of t h e f i r s t t e r m i n e q u a t i o n ( 6 ) y i e l d s N

9:

C9-548 JOURNAL DE PHYSIQUE

systems c o n t r i b u t i n g t o t h e s c a t t e r i n g and

Y

Phonons of f r e q u e n c i e s 30 GHz c o n t r i b u t e t o t h e thermal c o n d u c t i v i t y a t low tem- p e r a t u r e . T h i s i s t h e only independent measurement a v a i l a b l e which y i e l d s t h e l i f e t i m e of phonons of comparable energy. The l i f e t i m e of l o n g i t u d i n a l phononshas been e s t i m a t e d from t h e thermal c o n d u c t i v i t y . To do t h i s it was necessary t o a l s o e s t i m a t e t h e r e l a t i v e c o n t r i b u t i o n of t h e t r a n s v e r s e modes. T h i s was done u s i n g t h e r a t i o of t h e t r a n s v e r s e t o l o n g i t u d i n a l phonon l i f e t i m e o b t a i n e d from u l t r a - s o n i c d a t a (').

The l i f e t i m e o b t a i n e d i n t h i s way corresponds t o a width of 19 MGz i n good agree- ment w i t h t h e z e r o temperature l i m i t of our f i t t e d curve.

To summarize, our r e s u l t s from B r i l l o u i n s c a t t e r i n g r e v e a l t h e p r e s e n t of a n addi- t i o n a l c o n t r i b u t i o n t o t h e phonon l i f e t i m e of roughly e q u a l magnitude t o t h a t due t o t h e two l e v e l systems.

Discussion.- I t is of c o u r s e i n c o n c e i v a b l e t h a t phonons can propagate through a s t r u c t u r a l l y d i s o r d e r e d l a t t i c e w i t h o u t being s c a t t e r e d . When t h e c o n t r i b u t i o n of t h i s p r o c e s s t o t h e i n v e r s e m.f.p. i s c a l c u l a t e d it i s found t o b e temperature independent and p r o p o r t i o n a l t o t h e phonon frequency (I). Indeed t h e proportion- a l i t y of t h e s t r u c t u r a l c o n t r i b u t i o n t o t h e phonon frequency i s necessary t o p r e s e r v e t h e T~ temperature dependence of t h e thermal c o n d u c t i v i t y .

On t h e b a s i s of t h e B r i l l o u i n s c a t t e r i n g r e s u l t s a l o n e it i s of c o u r s e impossible t o exclude a dependence of t h i s a d d i t i o n a l mechanism on o t h e r f u n c t i o n s of and T which w i l l r e s u l t i n a T~ dependence of t h e thermal c o n d u c t i v i t y . However, s i n c e t h e o r y p r e d i c t s a s t r u c t u r a l s c a t t e r i n g c r o s s - s e c t i o n which i s independent of T and p r o p o r t i o n a l t o &J , i t seems Likely t h a t t h i s i s r e s p o n s i b l e f o r t h e a d d i t i o n a l s c a t t e r i n g we have obtained.

I f a c o n t r i b u t i o n t o t h e s c a t t e r i n g i s temperature independent i t i s d i f f i c u l t t o observe d i r e c t l y , u s i n g u l t r a s o n i c t e c h n i q u e s , and h a s been d i s c a r d e d o r ignored i n p r e v i o u s work. I n f a c t , t h e a d d i t i o n of a temperature independent c o n t r i b u t i o n of comparable magnitude would remove t h e d i s c r e p a n c y noted by Golding e t a l . between t h e experimental thermal c o n d u c t i v i t y and t h a t c a l c u l a t e d from t h e i r d a t a . They a l s o remark t h a t two c o n t r i b u t i o n s were d i s c a r d e d : one a background of roughly 8% a s c r i b e d t o d i f f r a c t i o n e f f e c t s , t h e o t h e r t o r e l a x a t i o n a l l o s s e s which become important below 0.2O~, and should t h e r e f o r e a l s o b e of o r d e r 10%.

Acknowledgement.- The a u t h o r s would l i k e t o thank M r . Andy Duft and M r . P a u l Waterhouse f o r t e c h n i c a l a s s i s t a n c e , and t h e N a t u r a l S c i e n c e s and Engineering Research Council f o r i t s support.

References.-

(1) Walton, D . , Phys. Rev. B s , (1977) 3723.

( 2 ) B u t l e r , W.H. and Kohn, W. I. Res. N a t l . Bun. Stand.

A 7 4

( 3 ) Vacher, R . , S u s s n e r , H. and Hunklinger, S . , Phys, Rev. B g , (1980) 5850.

(4) Bradberry, G.W. and Vaughan, J . M . , O p t i c s Comm. 20, ⁽¹⁹⁷⁷⁾ 307

(5) Vanderwal, J . , Mudare, S.M. and Walton, D . , O p t i c s Comm.

37,

(6) Jackson, H.E. Rand, S. and Walton, D . , S o l i d S t a t e Comm.

21,

(7) P e l o u s , J . and Vacher, R . , S o l i d S t a t e Comm.

19,

(8) Danielmeyer,H.G., J. Acous. Soc. Am.

47,

(9) Golding, B . , Graebner, J . E . and S c h u l t z , R.J. Phys. Rev. B C , (1976) 1660.