ROLE OF THERMAL EXPANSION IN THE PHONON CONDUCTIVITY OF SOLIDS

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ROLE OF THERMAL EXPANSION IN THE PHONON CONDUCTIVITY OF SOLIDS

G. Srivastava

To cite this version:

G. Srivastava. ROLE OF THERMAL EXPANSION IN THE PHONON CONDUC- TIVITY OF SOLIDS. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-149-C6-151.

�10.1051/jphyscol:1981645�. �jpa-00221581�

JOURNAL DE PHYSIQUE

CoZZoque C6, suppldmnt au n O 1 2 , Tome 42, ddcembre 1981 page C6-149

ROLE OF THERMAL EXPANSION I N THE PHONON CONDUCTIVITY OF SOLIDS

G.P. Srivastava

Physics Department, The new University of Ulster, Coleraine, N. IreZund B152 ZSA, United Kingdom

Abstract. We have studied the role of thermal expansion in the thermal conductivity of solids. By treating the Grfineisen constant as a temperature dependent semiadjustable parameter we find that the high temperature lattice thermal conductivity of Ge and Si can be explained. The fitted values of the Grfieisen constants show reasonable agreement with recent experimental and theoretical results.

The high-temperature lattice thermal conductivity of most insulators and semiconductors decreases faster then T-l. Recently welt2 have concluded that consideration of three-phonon processes (both acoustic- acoustic and acoustic-optical interactions), together with mass-defect and boundary scattering, cannot explain this Sehaviour. Inclusion of four-phonon processes does not improve the situation3. It was

suggested by ~csedy' that thermal expansion might be helpful in this respect. In this paper we study the role of thermal expansion on the lattice thermal conductivity of Ge and Si.

The effect of thermal expansion is to make the ~rcneisen constant of a material temperature dependent5r6. In our earlier work1 (here- after referred to as I) we have derived an expression for the Grfineisen constant in terms of the second- and third-order elastic constants of the material. Because of thermal expansion the elastic constants become temperature dependent. This would rec+uire calculating the Grzneisen constant at various temperatures using experimental data for elastic constants provided these were available. In view of the unavailability of elastic constants at different temperatures in our range of interest, we treat the ~ r h e i s e n constant y as a temperature dependent semiadjustable parameter for the thermal conductivity

calculation.

The model conductivity expression in the notation of I is

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981645

JOURNAL DE PUYSIQUE

*- ¹

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Here 6 = l/kBT, < f > = q q q q Cf W (fi +1), and ';T = T~ + ^{T ; ~} + ^{T R '} ,

w i t h q a s t h e e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n f o r phonons i n mode q. Also i n e q u a t i o n (1) T ~ *r e p r e s e n t s t h e single-mode r e l a x a t i o n t i m e of a phonon i n mode q d u e t o three-phonon N p r o c e s s e s

,

t h e r e l a x a t i o n t i m e due t o mass d e f e c t and boundary s c a t t e r i n g .

E x p r e s s i o n s f o r t h e r e l a x a t i o n t i m e s can be o b t a i n e d u s i n g t h e f i r s t - o r d e r t i m e dependent p e r t u r b a t i o n method. A s i n Refs 1 and 2 r e s u l t s f o r t h e s e a r e d e r i v e d w i t h t h e c r y s t a l Hamiltonian e x p r e s s e d w i t h i n an i s o t r o p i c continuum approximation

Here y = y ( T ) i s t h e t e m p e r a t u r e dependent anharmonic (Grfineisen)

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c o n s t a n t , c i s an a v e r a g e phonon s p e e d , A = (a- +a ) e t c . a r e t h e

4 Y q

phonon f i e l d o p e r a t o r s , and o t h e r symbols a r e t h e same a s i n I . For a c o u s t i c - o p t i c a l phonon s c a t t e r i n g we make a c o r r e c t i o n t o t h e model p r e s e n t e d i n I: t h e r e d u c t i o n f a c t o r i n t h e c u b i c anharmonic Hamilton- i a n i s i n c l u d e d o n l y when d e a l i n g w i t h i n t e r a c t i o n s o f t h e t y p e

a c + a c f op.

The p h y s i c a l p a r a m e t e r s and boundary l e n g t h s used i n o u r c a l c u - l a t i o n s a r e t a k e n from I. I n I a c o n s t a n t y was a d j u s t e d t o f i t t h e low t e m p e r a t u r e l a t t i c e t h e r m a l c o n d u c t i v i t y o f m a t e r i a l s from g r o u p s I V , 1 1 1 - V , 1 1 - V I and I - V I I . I t was n o t i c e d t h a t w i t h a c o n s t a n t y t h e i s o t o p e p a r a m e t e r A had t o be a d j u s t e d t o f i t t h e maximum i n t h e e x p e r i m e n t a l c o n d u c t i v i t y c u r v e . I n t h i s work we t a k e t h e experimen- t a l v a l u e f o r A ( 2 . 4 4 x s e c 3 f o r t h e G e b a l l e and H u l l ' s sample o f Ge, and 0.132 x lo-" s e c 3 f o r S i ) and o n l y a d j u s t y ( T ) t o f i t t h e e x p e r i m e n t a l c o n d u c t i v i t y c u r v e b o t h i n low and h i g h t e m p e r a t u r e r e g i o n s . The v a l u e s o f y ( T ) needed t o e x p l a i n t h e c o n d u c t i v i t y r e s u l t s of s i 7 and ~ e ' a r e l i s t e d i n T a b l e 1. I t i s c l e a r t h a t a r e l a t i o n y ( T ) = y o ( l + p T ) w i t h a p p r o p r i a t e yo and e x p l a i n s t h e h i g h t e m p e r a t u r e v a r i a t i o n of t h e t h e r m a l c o n d u c t i v i t y o f t h e samples

s t u d i e d h e r e . The f i t t e d v a l u e s o f y ( T ) a r e i n r e a s o n a b l e a g r e e m e n t w i t h r e c e n t e x p e r i m e n t a l g and t h e o r e t i c a l 6 r e s u l t s .

I n c o n c l u s i o n , w i t h i n a n i s o t r o p i c continum model f o r germanium and s i l i c o n we h a v e shown t h a t c o n s i d e r a t i o n o f t h e r m a l e x p a n s i o n e x p l a i n s b o t h t h e low and h i g h t e m p e r a t u r e l a t t i c e t h e r m a l

c o n d u c t i y i t y a f t h e s e m a t e r i a l s ,

T a b l e 1: T e m p e r a t u r e d e p e n d e n t Griineisen c o n s t a n t s f i t t e d t o e x p l a i n t h e l a t t i c e t h e r m a l c o n d u c t i v i t y o f S i and G e .

' S r i v a s t a v a , G.P., J . Phys. Chem. S o l i d s 41, 357 (1980) 2 ~ r i v a s t a v a , G.P., P h i l o s , Mag.

2,

3 ~ c s e d y , D . J . and Klemens, P.G., Phys. Rev. G I 5957 (1977) ' ~ c s e d ~ , D . J . , i n Thermal c o n d u c t i v i t y 1 4 (Ed. P.G. Klemens, and

T.K. C h u ) , Plenum P r e s s , N . Y . , 1976 p.195

' ~ a t e s , B., Thermal Expansion (Plenum P r e s s , N . Y . , 1 9 7 2 ) 6 ~ o m a , T . , J . Phys. Soc. J p n 42, 1 4 9 1 (1977)

7 ~ o l l a n d , M.G. and N e u r i n g e r , L. J . , P r o c . I n t . Cong. o n t h e Phys. o f S e m i c o n d u c t o r s , p.475 E x e t e r 1962 (The I n s t . o f Phys. and Phys. S o c . , London 1 9 6 2 ) .

' ~ e b a l l e , T.H., and H u l l , G.W., Phys. Rev. 110, 773 ( 1 9 5 8 ) . 9 ~ l a c k , G . A . , and B a r t r a m , S.F., J. Appl. Phys. 46, 89 ( 1 9 7 5 ) .

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0 . 5 0.65

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0.28 0 . 2 7

0.29 0.25

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0 . 4 3 0.33

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0 . 5 5 0.42

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