PHONON CONDUCTIVITY DUE TO NONDIAGONAL ENERGY-FLUX OPERATOR

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PHONON CONDUCTIVITY DUE TO

NONDIAGONAL ENERGY-FLUX OPERATOR

G. Srivastava

To cite this version:

G. Srivastava. PHONON CONDUCTIVITY DUE TO NONDIAGONAL ENERGY-FLUX OPERA- TOR. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-253-C6-255. �10.1051/jphyscol:1981673�.

�jpa-00221609�

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JOURNAL DE PHYSIQUE

ColZoque C6, supplgment au n012, Tome 42, &eembre 1981 page C6-253

PHONON CONDUCTIVITY DUE TO NONDIAGONAL ENERGY-FLUX OPERATOR

G.P. Srivastava

Physics Department, The NelJ University of Ulster, Coleraine, N. Ireland BT52 ISA, U.K.

Abstract. Using the Zwanzig-Mori projection operator method we present calculations of renormalised phonons and their contribution to the thermal conductivity of Ge from the nondiagonal part of the heat-flux operator given by Hardy.

Recently we1 have used the Zwanzig-Mori projection operator method to obtain expressions for the lattice thermal conductivity of an anhar- monic crystal from the diagonal and nondiagonal parts of the heat-flux operator given by ~ardy,. In the van Hove limit the diagonal contribution is the well known single-mode relaxation time result3, in which only the true phonon frequency appears. But the nondiagonal contribution includes renormalised phonons, with shifted frequencies. Here we present results of our calculations of renormalised phonons and their contribution to the thermal conductivity of Ge from the nondiagonal part of the heat-flux operator.

In the notation of Ref. 1 the expressions for the diagonal and nondiagonal contributions to the lattice thermal conductivity are as follows

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981673

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C6-254 JOURNAL DE PHYSIQUE

-+ -+

Here i 5 ( q , s ) , j ( q , s ' ) , u i i j = W ~ ? W ~ , A i j I m ( 6 i )

-

^{I m ( 6}^{. )}⁼^{A i}

-

^{A j}

3

-+ -+

-

¹

and c i j

C;ss

'

is a g e n e r a l i s e d g r o u p v e l o c i t y . Ai and r i a r e t h e ZrequenGy s h i f t and t h e i n y e r s e r e l a x a t i o n time o f a phonon i n mode i and a r e , r e s p e c t i v e l y , c a l c u l a t e d from t h e imaginary and r e a l p a r t s of Bi where

w i t h Ti a s a measure of c u b i c a n h a r m o n i c i t y : H = Hharm + ; T . A . A s

I ~i

i n o u r p r e v i o u s p a p e r i n t h e s e p r o c e e d i n g s we e x p r e s s Ti i n terms o f t h e Griineisen c o n s t a n t o f t h e m a t e r i a l

where

q ( q , s ) . With t h i s t h e n -+

With e q u a t i o n s (4-6) and a l i t t l e b i t o f a l g e b r a , we c a n e x p r e s s 6 _i f o r t h r e e phonon p r o c e s s e s a s

From t h i s A and can be e v a l u a t e d u s i n g t h e r e l a t i o n

q q

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where P d e n o t e s t h e p r i n c i p a l p a r t . I t is i n t e r e s t i n g t o n o t e t h a t

-

¹

T c a l c u l a t e d i n t h i s way i s t h e same a s t h e r e s u l t 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 p e r t u r b a t i o n method3 q

'

^4.

I n F i g . 1 we have p l o t t e d A and T - I f o r t h e l o n g i t u d i n a l mode a t

q q

300 K . Although r - ' i s an i n c r e a s i n g f u n c t i o n o f f r e q u e n c y , A shows

q _q

a pronounoed s t r u c t u r e a t w ?r 2 5 x

ld

^s-', Vc f i n d khat i n g e n e r a l A > r-' and ( w +A ) > > r

-',

implying t h a t a pseudoharmonic model5 i s

q '3 4 q q

a good d e s c r i p t i o n f o r phonons i n Ge. F u r t h e r m o r e , o u r c a l c u l a t i o n s show t h a t K~~ i s n e g l i g i b l y s m a l l i n comparison t o K i n t h e tempera-

d

t u r e r a n g e 10-900 K. A s i m i l a r c o n c l u s i o n was r e a c h e d by Hardy2, and Semwal and sharma6 u s i n g q u a l i t a t i v e arguments.

F i g . I: The i n v e r s e r e l a x a t i o n t i m e ( r - I ) and f r e q u e n c y s h i f t ( A ) f o r t h e l o n g i t u d i n a l a c o u s t i c phonon mode i n Ge a t 300 K a s a f u n c t i o n of f r e q u e n c y ( w )

.

R e f e r e n c e s

' ~ r i v a s t a v a , G.P. and P r a s a d , M., Phys. Rev. B 2 3 , 4723 (1981).

'Hardy, R., Phys. Rev.

132,

168 ( 1 9 6 3 ) .

$ s r i v a s t a v a , G . P . P h i l o s , Mag.

34,

795 ( 1 9 7 6 ) .

+ ~ r i v a s t a v a , G.P. J. Phys. Chem-Solids,

41,

357 (1980).

' ~ e i s s l a n d , J.A. The P h y s i c s o f Phonons ( W i l e y , New York, 1 9 7 3 ) . ' ~ e m w a l , B.S. and Sharma, P.K., Phys. Rev. Bz, 3909 ( 1 9 7 2 ) .