CONTRIBUTION OF LOW-FREQUENCY LONGITUDINAL PHONONS TO THE LATTICE THERMAL CONDUCTIVITY OF DIELECTRIC SOLIDS

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CONTRIBUTION OF LOW-FREQUENCY LONGITUDINAL PHONONS TO THE LATTICE

THERMAL CONDUCTIVITY OF DIELECTRIC SOLIDS

R. Brito-Orta, P. Klemens

To cite this version:

R. Brito-Orta, P. Klemens. CONTRIBUTION OF LOW-FREQUENCY LONGITUDINAL

PHONONS TO THE LATTICE THERMAL CONDUCTIVITY OF DIELECTRIC SOLIDS. Journal

de Physique Colloques, 1981, 42 (C6), pp.C6-256-C6-258. �10.1051/jphyscol:1981674�. �jpa-00221610�

JOURNAL DE PHYSIQUE

CoZZoque C6, suppl6ment a u n012, Tome 42, ddcembre 1982 page C6-256

CONTRIBUTION OF LOW-FREQUENCY LONGITUDINAL PHONONS TO THE L A T T I C E THERMAL CONDUCTIVITY OF D I E L E C T R I C SOLIDS

R. Brito-Orta and P.G. Klemens

Dept. o f P h y s i c s and I n s t . of M a t e r i a l S c i e n c e , U n i ~ . o f C o n n e c t i c u t , S t o r r s , CT 06268, U.S.A.

A b s t r a c t . - C o n t r i b u t i o n s o f l o w - f r e q u e n c y l o n g i t u d i n a l phonons t o 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 a r e e s t i m a t e d u s i n g t h e Po- meranchuk mean f r e e p a t h a n d a c u t - o f f g i v e n by t h e A k h i e s e r t h e o r y . The t e m p e r a t u r e d e p e n d e n c e s a r e compared w i t h d a t a f o r r u b i d i u m h a l i d e s , germanium a n d s i l i c o n .

1. I n t r o d u c t i o n . - R e s t r i c t i o n s o f f r e q u e n c y and w a v e - v e c t o r c o n s e r v a - t i o n i n t h r e e - p h o n o n anharmonic p r o c e s s e s f o r b i d l o w - f r e q u e n c y l o n g i - t u d i n a l phonons t o i n t e r a c t d i r e c t l y w i t h t h e r m a l phonons. T h i s l e n g t h e n s t h e i r mean f r e e p a t h and c a u s e s them t o make a l a r g e con- t r i b u t i o n t o 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 . ' However, a t v e r y low f r e q u e n c i e s , i . e . a t w < w , = l / ~ ~ ~ , where T~~ i s t h e r e l a x a t i o n t i m e o f t h e r m a l phonons, t h e c o n s e r v a t i o n c o n d i t i o n s a r e r e l a x e d , and l o n g i - t u d i n a l and t r a n s v e r s e phonons h a v e e s s e n t i a l l y t h e same mean f r e e p a t h . The e x c e s s c o n t r i b u t i o n o f l o n g i t u d i n a l phonons i n c r e a s e s t h e t e m p e r a t u r e dependence o f 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 .

The p r e s e n t t r e a t m e n t i s c o n f i n e d ' t o t e m p e r a t u r e s a b o v e t h e De- bye t e m p e r a t u r e .

2. Phonon Mean F r e e P a t h . - For f r e q u e n c i e s above w c l o n g i t u d i n a l pho- nons a r e s c a t t e r e d i n a t w o - s t e p p r o c e s s . F i r s t t h e y p a r t i c i p a t e i n a t h r e e - p h o n o n normal p r o c e s s w i t h t r a n s v e r s e phonons o f a b o u t t h e i r own f r e q u e n c y . The t r a n s v e r s e phonons t h e n i n t e r a c t w i t h phonons o f t h e r m a l f r e q u e n c y . The mean f r e e p a t h o f a l o n g i t u d i n a l phonon o f f r e q u e n c y w and v e l o c i t y v I , a s l i m i t e d by t h e f i r s t p r o c e s s , i s e s - t i m a t e d '

,

where y i s t h e ~ r G n e i s e n c o n s t a n t , wD t h e Debye f r e q u e n c y and To=Mv 2 /k, where M i s t h e a v e r a g e a t o m i c mass and k t h e Boltzmann c o n s t a n t . The mean f r e e p a t h o f a t r a n s v e r s e phonon o f f r e q u e n c y w and v e l o c i t y v I I , a s l i m i t e d by t h e second p r o c e s s , i s ⁴

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981674

where B i s a c o e f f i c i e n t o f o r d e r u n i t y which depends on t h e s t r u c - t u r e of t h e c r y s t a l and on t h e form o f t h e d i s p e r s i o n c u r v e s . I t w i l l b e t r e a t e d a s a n e m p i r i c a l p a r a m e t e r . Thus, t h e t o t a l mean f r e e p a t h o f t h e l o n g i t u d i n a l phonons i s RI=R1+$II. T h a t o f t h e t r a n s - v e r s e phonons i s s i m p l y R

11'

Below wc t h e p i c t u r e o f i n t e r a c t i o n s between i n d i v i d u a l phonons i s no l o n g e r v a l i d . A phonon of v e r y low f r e q u e n c y i n t e r a c t s w i t h a l l o t h e r phonons, a s i n t h e A k h i e s e r t h e o r y o f t h e deformed g a s o f t h e r m a l phonons, and t h e mean f r e e p a t h s of l o n g i t u d i n a l a n d t r a n s - v e r s e phonons a r e o f comparable m a g n i t u d e , i n d e p e n d e n t o f tempera- t u r e and a p p r o x i m a t e l y g i v e n b y 2 y 3

where v i s a n a v e r a g e v e l o c i t y .

3. The Thermal C o n d u c t i v i t y . - Given t h e p r e v i o u s c o n s i d e r a t i o n s , and u s i n g a Debye model f o r t h e s p e c t r a l s p e c i f i c h e a t C(w), w i t h

vI=vII=v f o r s i m p l i c i t y , t h e k i n e t i c t h e o r y e x p r e s s i o n f o r 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 may be w r i t t e n a s

( 4 ) where CI(w)=C(w)/3. Taking wc=v/RII(wD)=(1/2By )(T/To)wD and s u b s t i - 2 t u t i n g t h e e x p r e s s i o n s f o r LA, RI and R I I i n t o Eq. ( 4 ) we f i n d

where

i s , e x c e p t f o r t h e c o n s t a n t B , t h e t h e r m a l c o n d u c t i v i t y d e r i v e d by L e i b f r i e d and ~ c h l b m a n n , ~ and a 3 i s t h e volume p e r atom.

The second t e r m i n Eq. ( 5 ) g i v e s t h e a d d i t i o n a l c o n t r i b u t i o n o f t h e l o n g i t u d i n a l phonons due t o R1. The l a s t t e r m E , r o u g h l y p r o p o r - t i o n a l t o T, i s u s u a l l y s m a l l b u t may be a p p r e c i a b l e a t v e r y h i g h t e m p e r a t u r e s f o r s o l i d s o f a p p r o p r i a t e l y h i g h m e l t i n g p o i n t . It i s a r e s u l t o f LA b e i n g t h e mean f r e e p a t h o f a l l phonons o f f r e q u e n c y below wc.

4 . Comparison w i t h E x p e r i m e n t s . - The d e v i a t i o n o f A from t h e 1 / T be- h a v i o r a t h i g h t e m p e r a t u r e s i s i n p a r t due t o t h e r m a l e x p a n s i o n . The r e m a i n i n g p a r t h a s o f t e n been a s c r i b e d t o four-phonon p r o c e s s e s . However, four-phonon p r o c e s s e s a r e t o o weak, and we a t t r i b u t e t h e r e m a i n i n g d e v i a t i o n t o t h e second t e r m i n E q . ( 5 ) . Using t h a t equa-

C6-258 JOURNAL DE PHYSIQUE

tion, with .A corrected for thermal expansion, we have calculated the thermal conductivities of RbBr, RbI, and RbCR, and we have com- pared them to recent m e a s u ~ e m e n t s ; ~ those of Ge and Si were similar- ly compared. Selected ratios with respect to a reference tempera- ture are shown in the table below. The theory seems to give agree- ment with observed temperature variations except for Ge (and for Si at temperatures higher than 1100 K.). However, no correction has been attempted for a possible electronic component.

One of us ( R . B . 0 ) is on leave of absence from the Universidad Autonoma de Puebla, Mexico, and acknowledges support by CONACYT

(Mexico 1.

RbBr RbI RbCR G e S i

TR(K) 360 360 3 6 0 600 600

T(K) 280 160 280 160 280 160 800 1100 800 1100 Exp. 1.47 3.09 1.38 2.86 1.43 3.00 0.72 0.63 0.67 0.44 Calc. 1.40 2.98 1.39 2.91 1.39 2.95 0.68 0.46 0.67 0.43 Table 1: Experimental and calculated values of A(T)/A(TR).

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5 ,

A. Akhieser, J. Phys. (USSR)

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P.G. Klemens, in Physical Acoustics, Ed. W.P. Mason, vol. 3B, Academic Press, New York (19651, pp 201-234.

G. Leibfried and E. SchlEmann, Nachr. Akad. Wiss. ~gttingen, 2a(4), 71 (1954).

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J.P. Moore, R.K. Williams, and R.S. Graves, Phys. Rev. B

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3107 (1975).

C.J. Glassbrenner and G.A. Slack, Phys. Rev.

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