PHONON SCATTERING AT FREQUENCIES NEAR THE EXCITON FREQUENCY OF TWO-LEVEL DEFECTS

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PHONON SCATTERING AT FREQUENCIES NEAR

THE EXCITON FREQUENCY OF TWO-LEVEL

DEFECTS

J .W. Tucker

To cite this version:

JOURNAL DE PHYSIQUE

CoZZoque C6, suppZ6ment

au

n o 2 2 , Tome

42,

d6cembre 1981 page C6-478

PHONON SCATTERING AT FREQUENCIES NEAR THE EXCITON FREQUENCY OF

TWO-LEVEL DEFECTS

J .W. Tucker

Physics Department, SheffieZd University, SheffieZd 5 3 7RH, EngZand.

Abstract.- I n t h e diagrammatic temperature-dependent p e r t u r b a t i o n t h e o r y of coupled spin-phonon systems an e s s e n t i a l q u a n t i t y determining t h e p o l e s of t h e phonon Green f u n c t i o n i s t h e p o l a r i s a t i o n m a t r i x , M, of t h e g e n e r a l i s e d s p i n s u s c e p t i b i l i t y . Near t h e s p i n e x c i t o n f r e q u e n c y t h e resonance c o n t r i - b u t i o n from t h e p o l a r i s a t i o n m a t r i x t o a l l o r d e r s i s r e q u i r e d . H i t h e r t o only t h e chain-like diagrams t h a t occur i n t h e t h e o r y have been r e t a i n e d . F o r M

we show t h a t

an

important c o n t r i b u t i o n a l s o a r i s e s from o t h e r diagrams

4

p r e v i o u s l y n e g l e c t e d , whose i n c l u s i o n i s e s s e n t i a l if M i s t o remain f i n i t e i n t h e h i g h temperature l i m i t . 4

1. I n t r o d u c t i o n . - I n r e c e n t y e a r s t h e r e h a s been c o n s i d e r a b l e i n t e r e s t i n t h e e x c i t a t i o n spectrum of l o c a l i s e d d e f e c t s ( f o r example, paramagnetic i o n s ) i n t e r a c t i n g with t h e phonons of t h e h o s t l a t t i c e i n which t h e y a r e embedded. I n a d d i t i o n t o knowledge of t h e e x c i t a t i o n f r e q u e n c i e s t h e l i f e t i m e s of t h e coupled e x c i t a t i o n s a r e

of i n t e r e s t because t h e y e n t e r i n t o

a

c a l c u l a t i o n of t h e thermal t r a n s p o r t p r o p e r t i e s of t h e system / I / a s w e l l a s c o n t r i b u t i n g t o t h e l i n e w i d t h i n s p e c t r o s c o p i c measure- ments, f o r example, i n phonon spectroscopy /2/ o r i n n e u t r o n s c a t t e r i n g experiments

/ 3 / .

One system t h a t h a s been widely s t u d i e d i s t h a t of paramagnetic i o n s having a n e f f e c t i v e s p i n

Q

coupled t o t h e phonons of t h e h o s t c r y s t a l through an i n t e r a c t i o n l i n e a r i n t h e s t r a i n . Among t h e t h e o r e t i c a l approaches t o t h e problem of d e t e r m i n i n g t h e l i f e t i m e of t h e coupled mode e x c i t a t i o n s a r e t h o s e based on t h e a p p l i c a t i o n of diagrammatic temperature-dependent p e r t u r b a t i o n t h e o r y /2,4,5/. The work embodied i n /2/ and /4/ was p r i m a r i l y concerned w i t h f i n d i n g t h e l i f e t i m e of t h e coupled mode spectrum i n t h e phonon regime s u f f i c i e n t l y removed from t h e s p i n resonance f r e q u e n c y f o r a low-order p e r t u r b a t i o n t h e o r y t o b e a p p l i c a b l e . The work of Care and Tucker

/5/

was an attempt t o extend t h e t h e o r y t o f r e q u e n c i e s c l o s e t o t h e e x c i t o n f r e - quency of t h e two-level d e f e c t s b y i n c o r p o r a t i n g h i g h e r o r d e r terms i n t h e s e l f energy of t h e phonon Green f u n c t i o n . An e x p r e s s i o n was o b t a i n e d f o r t h e l i f e t i m e of t h e e x c i t a t i o n s i n t h a t regime, b u t r e c e n t work by Memos and Loudon

/ 6 /

p o i n t s out

an u n s a t i s f a c t o r y f e a t u r e of t h e r e s u l t as far as t h e wave-vector dependence of t h e r e l a x a t i o n r a t e s

i s

concerned. While t h e r e s u l t of Memos and Loudon f o r t h e l i f e t i m e of t h e e x c i t a t i o n s over t h e whole frequency range i s i n t u i t i v e l y more a t t r a c t i v e it

h a s n e v e r t h e l e s s n o t been deduced r i g o r o u s l y i n a p r o p e r coupled mode t h e o r y but i s

r a t h e r an e x t r a p o l a t i o n of r e s u l t s o b t a i n e d i n two l i m i t i n g regimes. Although an

improved t h e o r y i s s t i l l awaited, t h e work of Memos and Loudon h a s a t l e a s t caused

u s t o re-examine our e a r l i e r work i n an attempt t o r e s o l v e some of t h e c o n f l i c t i n g t h e o r e t i c a l p r e d i c t i o n s .

2. The P o l a r i s a t i o n Matrix.- I n

/5/

t h e p o l e s of t h e phonon Green f u n c t i o n were expressed i n terms of t h e p o l a r i s a t i o n m a t r i x a s s o c i a t e d with t h e g e n e r a l i s e d s p i n s u s c e p t i b i l i t y . Only t h e z e r o t h and second-order c o n t r i b u t i o n s t o t h e p o l a r i s a t i o n ma.trix M were d e r i v e d completely, which i s a l l t h a t i s needed i n t h e low-order t h e o r y a p p l i c a b l e t o t h e phonon regime w e l l away from t h e s p i n resonance frequency. I n t h e s p i n e x c i t o n frequency range h i g h e r o r d e r terms i n t h e p o l a r i s a t i o n m a t r i x have t o be included. I n t h e work of

/5/

t h e important c o n t r i b u t i o n s ( r e f e r r e d t o below a s resonance t e r m s ) were i s o l a t e d from t h e h i g h e r o r d e r terms i n t h e p o l a r i s a t i o n m a t r i x and a summation of t h e s e was achieved i n t h e low temperature l i m i t f o r t h e f u l l y c o n c e n t r a t e d l a t t i c e . More r e c e n t l y , t h e complete e x p r e s s i o n f o r t h e f o u r t h - o r d e r c o n t r i b u t i o n t o M h a s been r e p o r t e d /7/. However, when s i n g l i n g o u t t h e resonance-like terms i n

/5/

it was thought t h a t t h e only diagrams t h a t c o u l d c o n t r i - b u t e were t h o s e having a simple chain-like s t r u c t u r e . However we now r e a l i s e a f t e r a more c a r e f u l examination of t h e complete e x p r e s s i o n f o r

M

t h a t t h i s i s n o t t h e

4

c a s e , a p o i n t a l s o not a p p r e c i a t e d i n

h/.

I n t h e p r e s e n t work we have t h e r e f o r e e v a l u a t e d t h e resonance c o n t r i b u t i o n t o t h e p o l a r i s a t i o n m a t r i x a r i s i n g from t h e diagrams of M n o t having a chain-like c h a r a c t e r . By t h e resonance terms of M we

4

3

4

mean t h o s e c o n t a i n i n g a denominator (w~-o)

.

3.

The Resonance c o n t r i b u t i o n t o M4.- The resonance c o n t r i b u t i o n s t h a t were missed f r o *

T(h)

a r i s e i n f a c t from t h e t e r n s

i n t h e n o t a t i o n of e q u a t i o n (12) of

/ 7 / .

On performing t h e a n a l y t i c c o n t i n u a t i o n i h

-,

w + i 6 t h e e v a l u a t i o n of t h e wave-vect o r summations over p and q l e a d s t o both r e a l and imaginary p a r t s . The wave-vector summations a r e e v a l u a t e d i n t h e Debye

l i m i t and only t h e l e a d i n g t e r m s i n an i n v e r s e expansion w i t h r e s p e c t t o t h e Debye frequency a r e r e t a i n e d . From e q u a t i o n (1

)

we f i n d t h a t t h e r e a l p a r t of

hiCX

h a s a

4

c o n t r i b u t i o n

C6-480 JOURNAL DE PHYSIQUE

w i t h

K(W) =

(~/~)~*(N/V)~Y~(W/W~)~/~B

( 0 ~ 4 ) ~

1

A s found i n

/7/,

t h e chain-like diagrams ~ ( 3 ) , ~ ( l ) , ~ ( 3 ) and ~ ( 2 ) g i v e a resonance c o n t r i b u t i o n through t h e terms

of

~ ~ ( h ) .

From e q u a t i o n ( 4 ) t h e r e a l and imaginary p a r t s of $X

₄

of i n t e r e s t a r e

and

r e s p e c t i v e l y . It

i s

n o t i c e d t h a t t h e c o n t r i b u t i o n s , e q u a t i o n s ( 2 ) and ( 3 ) from t h e non chain-like diagrams e x a c t l y c a n c e l t h e t e r m s i n v o l v i n g cosech(@W ) i n b o t h e q u a t i o n s

(5)

and (6). T h i s h a s t h e s a t i s f y i n g r e s u l t t h a t t h e t o t a l resonance term from remains f i n i t e i n t h e i n f i n i t e temperature l i m i t . I n t h e d i l u t e s i t u a t i o n when th: b(")are r e p l a c e d by bC(") t h e same c a n c e l l a t i o n occurs. That i s , c o n t r i - b u t i o n s from t h e non chain-like diagrams e x a c t l y c a n c e l t h e cosech(@W ) terms i n

b o t h e q u a t i o n s (18) and (19) of

h/.

References

/I/ E l l i o t t R.J. and Parkinson J.B., Proc. Phys. Soc.,

2 (1967) 1024-39

/2/ Tucker J.W., J. Phys. C.,

2

(1972) 2064-76

/3/ C h a l l i s L.J., F l e t c h e r J.R., J e f f r i e s D. J., Sheard F.W., Toombs G.A., De Go&

A.M., and Hutchings M.T., Phys. Rev., B (1979) 296-9

/4/

Toombs G.A. and S h e a d F.W., J. Phys. C., (1973) 1467-88

/5/

Care C.M. and Tucker J.W., J. Phys. C.,

4

(1976) 2681-93

/6/ Memos T. and Loudon R., J. Phys. C.,

3

(1980) 1657-68