LATTICE PHONON LIFETIME CALCULATIONS

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HAL Id: jpa-00221580

https://hal.archives-ouvertes.fr/jpa-00221580

Submitted on 1 Jan 1981

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LATTICE PHONON LIFETIME CALCULATIONS

J. Henkel

To cite this version:

J. Henkel. LATTICE PHONON LIFETIME CALCULATIONS. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-146-C6-148. �10.1051/jphyscol:1981644�. �jpa-00221580�

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JOURNAL DE PHYSIQUE

CoZZoque C6, stcp"p2e'ment a u n012, Tome 42, de'cembre 1981 page C6-146

L A T T I C E PHONON L I F E T I M E C A L C U L A T I O N S

J.H. Henkel

Department o f P h y s i c s a n d Astronomy, U n i v e r s i t y o f Georgia, A t h e n s , G e o r g i a 30602, U. S.A.

A b s t r a c t . - I t i s shown t h a t phonon l i f e t i m e s can b e c a l c u l a t e d u s i n g f i n i t e p e r i o d i c i t i e s and d i s c r e t e f r e q u e n c i e s w i t h o u t going c o m p l e t e l y t o t h e l i m i t o f i n f i n i t e p e r i o d i c i t i e s o r q u a s i - c o n t i n u o u s f r e q u e n c y d i s t r i b u t i o n s . I n a p p l y i n g t h e Golden Rule e q u a t i o n i n time-'dependent p e r t u r b a t i o n t h e o r y t h e r e i s a t i m e i n t e r v a l o v e r which t h e t r a n s i t i o n r a t e i s very n e a r l y indepen- d e n t o f t i m e and e n e r g y l e v e l d i f f e r e n c e s ( o r p e r i o d i c i t y ) . A s t h e p e r i o d i c i t y o f t h e l a t t i c e i n c r e a s e s t h e t i m e i n t e r v a l o v e r which t h e t r a n s i t i o n r a t e i s i n d e p e n d e n t o f t i m e i n c r e a s e s and approaches i n f i n i t y i n t h e l i m i t o f i n f i n i t e p e r i o d i c i t y . C a l c u l a t i o n s a r e p r e s e n t e d .

The Golden Rule o f time-dependent p e r t u r b a t i o n t h e o r y r e l a t i n g t r a n s i - t i o n r a t e s from an i n i t i a l u n p e r t u r b e d e n e r g y e i g e n s t a t e E i O ) t o one o f t h e o t h e r e n e r g y e i g e n s t a t e s ):E i n d u c e d by t h e p e r t u r b i n g Hamil- t o n i a n H' w i t h m a t r i x e l e m e n t s II& i s g i v e n by 1

where

E ( 0 ) ( 0 )

W = m - En

mn b ⁽²

and where t h e prime on t h e sum means t h a t t h e t e r m f o r m = n i s ex- c l u d e d from t h e sum. The Golden Rule can a l s o b e w r i t t e n a s

d 2 s i n ( w mn t )

w =

,

_m

^I

^lam(.)

^{l 2}

⁼

_;n2

^-_m

¹

_Wm

F o r q u a s i - c o n t i n u o u s e n e r g y d i s t r i b u t i o n s t h e Golden Rule t a k e s t h e form

w = - 1

_P

ZIT H~

I 2 ~ ( ~ i 0 ) ) ^,

^{( 4 )}

where p i s t h e d e n s i t y o f s t a t e s e x p r e s s e d a s a f u n c t i o n o f E'O) and where t h e r i g h t hand s i d e i s i n d e p e n d e n t o f t i m e t .

To s e e how phonon l i f e t i m e s can b e c a l c u l a t e d u s i n g t h e Golden Rule e q u a t i o n w i t h o u t f i r s t t a k i n g t h e l i m i t o f q u a s i - c o n t i n u o u s e n e r g y

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981644

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l e v e l d i s t r i b u t i o n s c o n s i d e r t h e f u n c t i o n s F ( a , t , L) and F ( a , t , L)

1 2

g i v e n by

L s i n - ( a - n ) t 2T F ( a , t , L ) ₁ = - n = l Z ( *

1

^h- n )

L (5

and

L 1 - c o s ~ ( n 2T - n ) t

F 2 ( a , t , L ) = - - _{t L}

1

^{2 T} ₂ ( 6 )

IL(a

-

^{n )}I

I n t h e l i m i t L + m t h e s e f u n c t i o n s a r e u n i t y i n d e p e n d e n t o f t i m e t pro- v i d e d 0 < a < L b u t n o t one o f t h e i n t e g e r s 1, 2 , .

. .

^L. F o r f i n i t e L t h e f u n c t i o n s a r e t o w i t h i n a b o u t 1 0 % e q u a l t o u n i t y i n d e p e n d e n t o f a and t f o r t h e i n t e r v a l 1 < t < L and f o r 0 < a < L w i t h a # 1, 2 , .

. .

^L. ^A

p l o t o f a t y p i c a l F ( a , t , L ) v e r s u s t i s shown i n F i g . 1 w h i l e T a b l e I 1

l i s t s c a l c u l a t e d v a l u e s o f Fl ( a , t , 1 2 )

.

The f u n c t i o n F ( a , t , L ) i s z e r o 1

dF1(a,O,L) a t t = O , w i t h a s l o p e

d t ⁼2,and l e v e l s o f f t o a p p r o x i m a t e l y 1 a t t = l . I t remains u n i t y w i t h i n s m a l l v a r i a t i o n s up t o t = L independent o f a p r o v i d e d 0 < a < L and a # 1 , 2 , .

. .

^L.

Two i m p o r t a n t c o n c l u s i o n s r e s u l t i n examining t h e p r o p e r t i e s o f

F1 (a , t , L )

.

^One ^{i s}t h a t t h e v a l u e o f F ( a , t , L ) f o r ₁ 1 < t < L _{i s}a p p r o x i - m a t e l y t h e same a s t h e l i m i t i n g v a l u e o f F l ( a , t , m ) . The o t h e r i s t h a t t h e convergence i n L i s v e r y r a p i d . The main r e s u l t i n i n c r e a s i n g L i s t h e i n c r e a s e i n t i m e i n t e r v a l o v e r which F1 i s c o n s t a n t .

The p o t e n t i a l e n e r g y e x p r e s s i o n o f a c r y s t a l can b e expanded i n s e r i e s form i n v o l v i n g t h e a t o m i c d i s p l a c e m e n t s . That p a r t o f t h e p o t e n t i a l i n c l u d i n g t h e t h i r d d e g r e e t e r m s i n t h e d i s p l a c e m e n t s can be used a s t h e p e r t u r b i n g p o t e n t i a l u t i l i z e d i n t h e Golden Rule e q u a t i o n t o c a l - c u l a t e t i m e d e r i v a t i v e s o f phonon o c c u p a t i o n d e n s i t i e s . When t h i s i s done t h e d e r i v a t i v e t a k e s t h e f o l l o w i n g form 2

[ ( n o + l ) n o , n u , ,

-

n o ( n u , + 1) (nu,, + 1) I , ^{( 7 )} where

A w = w -w,, u - w 0'' ⁽⁸⁾

T h i s d e r i v a t i v e can b e e v a l u a t e d f o r d i f f e r e n t t i m e s by d i r e c t sum- mation o f f i n i t e sums o b t a i n e d u s i n g f i n i t e p e r i o d i c boundary

c o n d i t i o n s . The above a n a l s i s o f t h e p r o p e r t i e s o f F l ( a , t , L ) i n d i - c a t e s t h a t t h i s d e r i v a t i v e

51,

2 u s i n g f i n i t e p e r i o d i c i t y s h o u l d b e

d t

roughly c o n s t a n t o v e r a t i m e i n t e r v a l E < t <

,

where

1

^Aw

1,

^{i s a}

minimum v a l u e o f

1

Aw

1

=

I

mu - wU

-

wU,,l. The a n a l y s i s a l s o i n d i c a t e s t h a t t h e convergence i n L s h o u l d b e v e r y r a p i d and t h a t t h e c o n s t a n t

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C6-148 JOURNAL DE PHYSIQUE

v a l u e o b t a i n e d s h o u l d b e w i t h i n a b o u t 1 0 % o f t h e v a l u e o b t a i n e d i n t h e l i m i t o f i n f i n i t e p e r i o d i c i t y .

An advantage o f t h e u s e o f f i n i t e p e r i o d i c boundary c o n d i t i o n s w i t h t h e r e s u l t i n g f i n i t e sums i s t h a t t h e complex phase r e l a t i o n s i n v o l v - i n g c o n s e r v a t i o n o f momentum and e n e r g y i s a u t o m a t i c a l l y i n c l u d e d i n t h e sums. A

F1

/

I 3 t

F i g . 1: P l o t o f t y p i c a l v a l u e o f F ( a , t , L ) v e r s u s t. Independent o f a f o r

o

< a < L , a # 1 , 2 , f . . ~ .

TABLE I . C a l c u l a t e d v a l u e s o f F l ( a , t , 1 2 )

References:

1. E. Merzbacher, Quantum Mechanics (John Wiley and Sons, I n c . , New York, London, 1 9 6 1 ) , p. 470.

2. P. G. Klemens, J . Appl. Phys.

2,

4573 ( 1 9 6 7 ) .

LATTICE PHONON LIFETIME CALCULATIONS

HAL Id: jpa-00221580

https://hal.archives-ouvertes.fr/jpa-00221580

LATTICE PHONON LIFETIME CALCULATIONS

J. Henkel

To cite this version:

,

I

l 2

;n2

1

P

I 2 ~ ( ~ i 0 ) ) ,

1

1

-

. .

. .

.

. .

.

-

51,

,

1

1,

1

1

I

-

/

o

2,

^I

^{l 2}

_;n2

¹

_P

I 2 ~ ( ~ i 0 ) ) ^,