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Submitted on 1 Jan 1981
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HYBRIDIZATION OF THE TWO-PHONON BOUND
STATE WITH THE LOCAL MODE IN IMPERFECT
CRYSTALS
S. Behera, Sk. Samsur
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, suppZ6ment au n o 12, Tome 42, dicembre 1982 page C6-528
HYBRIDIZATION OF T H E TWO-PHONON B O U N D S T A T E WITH T H E LOCAL MODE IN IMPERFECT CRYSTALS
S.N. Behera and S k . S a m s u r
Institute of Physics, Bhubaneswar-751007, India.
Abstract.- The p o s s i b i l i t y of t h e h y b r i d i z a t i o n of a two- phonon bound s t a t e w i t h an impurity l o c a l mode i n an anharmonic i m p e r f e c t c r y s t a l , is demonstrated. The one-phonon d e n s i t y of s t a t e around t h e l o c a l mode frequency shows t h e two peak s t r u c - t u r e because of t h i s h y b r i d i z a t i o n .
1, I n t r o d u c t i o n
.
-
Recently it h a s been shown t h a t two-phonon bound s t a t e s could e x i s t i n anhannonic i m p e r f e c t c r y s t a l s , l which can be d e t e c t e d i n e i t h e r t h e second o r d e r i n f r a r e d o r Raman s p e c t r a . how eve^t h e s e can a s w e l l be seen i n t h e f i r s t o r d e r spectrum, through t h e i r h y b r i d i z a t i o n t o s u i t a b l e s i n g l e phonons2, Evidence i n support o f t h i s h a s been r e p o r t e d i n t h e l i t e r a t u r e 3 . Light mass s u b s t i t u t i o n a l impu- r i t i e s , g i v e r i s e t o l o c a l modes of v i b r a t i o n w i t h f r e q u e n c i e s h i g h e r t h a n t h e maximum allowed phonon frequency of t h e h o s t l a t t i c e , Hence a two-phonon bound s t a t e caused by t h e anharmonic i n t e r a c t i o n s (which f a l l s s l i g h t l y above t w i c e t h e maximum allowed phonon frequency of t h e h o s t ) can h y b r i d i z e with t h e l o c a l mode provided t h e l a t e r h a s n e a r l y t h e same frequency a s t h e former. T h i s c o n d i t i o n can be achieved by s u i t a b l y choosing t h e mass of t h e s u b s t i t u t i o n a l Impurity atom, In t h e p r e s e n t paper: w e r e p o r t t h e c a l c u l a t i o n of t h e one-phonon d e n s i t y of
s t a t e i n t h e presence of such h y b r i d i z a t i o n .
2. Theory ,
-
The a n h a r m n i c , imperfect c r y s t a l i s c h a r a c t e r i z e d by amodel Hamiltonian with c u b i c and q u a r t i c anhannonic terms with coupl- i n g c o n s t a n t s
1
and6
r e s p e c t i v e l y and t h e s u b s t i t u t i o n a l i m p u r i t i e s which a r e i s o t o p i c i n n a t u r e a r e d e s c r i b d by t h e W s s d e f e c t para- meterh
= (M,-M)/M, l, M and M a r e r e s p e c t i v e l y t h e masses of t h eI
impurity and h o s t atoms. Furthermore, t h e i n p u r i t y c o n c e n t r a t i o n i s
assumed t o be low. For such a system t h e one-phonon Green's f u n c t i o n s a t i s f i e s t h e equation,
where (3)
-.l
ak(h,ol
v=-
C ~ ~ Z ~ [ ~ + T ~ I J ' Z
( 4 )anA
- 2-L
B
Gilo)
= ( & ~ / r )
[
w2-
Wk
1
( 5 )i s t h e f r e e phonon propagator and V ( k , -k,p) is t h e c o e f f i c i e n t of t h e c u b i c anharmonic term. I n w r i t i n g eqn. (1) it i s assumed t h a t t h e q u a r t i c anharmonic term simply renormalizes t h e phonon f r e q u e n c i e s t o
Y
(Jk
.
From eqn. ( l ) t h e phonon s e l f energy can be w r i t t e n asI t i s c l e a r f mm eqns. (l) and (5) t h a t t h e phonon s e l f energy i n v o l v e s b e s i d e s t h e impurity c o n t r i b u t i o n
(A
, a t h e two-phonon p r o p a g a t o r~ : ; - ~ , ~ ; - ~ [ a ? ) which in t u r n c a r r i e s t h e information r e g a r d i n g t h e tuo- phonon bound s t a t e brought about by t h e q u a r t i c anharmonic i n t e r a c - t i o n . Because of o u r i n t e r e s t i n t h e two-phonon bound s t a t e t h e l a t e r need be e v a l u a t e d simply f o r t h e p e r f e c t anharmonic ( q u a r t i c ) c r y s t a l s . T h i s h a s been c a l c u l a t e d i n r e f .l (eqns. ( l 7 )-(19)) and w e s h a l l use t h a t r e s u l t .
I n o r d e r t o s i m p l i f y t h e c a l c u l a t i o n s we adopt t h e E i n s t e i n O s c i l l a t o r model f o r t h e h o s t l a t t i c e with a s i n g l e o p t i c a l phonon of frequency U,
.
With t h i s s i m p l i f y i n g approximation t h e impurity l o c a l mode frequency as c a l c u l a t e d from eqn. ( 3 ) t u r n s o u t t o bev 2
(7 ) and t h e two-phonor. bound s t a t e frequency is given by 1
where
7
=
bwcls
(9)t h e l a t e r being e v a l u a t e d a t z e r o temperature. The dimensionless q u a r t i c anharmonic coupling c o n s t a n t
5
b e i n g a small q u a n t i t y , t h e bound s t a t e appears j u s t above 2.
It is obvious from eqn.(7) t h a t f o rh
= -3 i.e. M== M/4, t h e l o c a l mode frequency becomes 24,
t h u s g i v i n g r i s e t o t h e p o s s i b i l i t y of i t s h y b r i d i s a t i o n with t h e two- phonon bound s t a t e.
C6-530 JOURNAL DE PHYSIQUE
around t h e frequency o r WRS
,
t h e Green ' S f u n c t i o n of eqn. (1) and( 6 ) can be approximated a s
1
G,.
(W)=
(a1-
W:
)(A
h&)/
3T
(3,D(w)
(10) whereLd,4 (11) with
yf
v'+$.
Equating D(&) = 0 one o b t a i n s the two h y b r i d i z e d f r e q u e n c i e s a s(12)
2
I n c a l c u l a t i n g eqn. (12) terms of o r d e r
6
a r e neglected. Making use of t h e r e s u l t (12) t h e one-phonon d e n s i t y of s t a t e s around t h e frequency 2Q can be e a s i l y shown t o bef,
(U)= 8 .
+
E
( m
-
Q.,+
13-
6
(CC-U-)
(13) where[(CIJI
+ 7 ' ) ( \ 0 7 5 2 ~
$1221
(~7:
7 ~ ( ~ \ ~ ~ ) y ~ ]
-L .
B%
=
12
(5
Y
-
7
)'12[8
-+?l
4
- + - ~ l$ l ( l ~ ~ ~ - 2 7 )
C
~A\$~EJ-
14) Thus we s e e t h a t t h e r e w i l l be two peaks of unequal s t r e n g t h . It i s obvious from eqn.
( 12) t h a t i n t h e degenerate c a s e of t h e h y b r i d i z a t i o n f i r s t s h i f t s t h e frequency s l i g h t l y and t h e n t h e mode s p l i t t s synanetrically. Such a h y b r i d i z a t i o n p e r s i s t s even when t h e two-phonon propagator is e v a l u a t e d f o r t h e imperfect c r y s t a l . 43. Discussion.
-
It h a s been shown t h a t i n an anharmonic c r y s t a l con- t a i n i n g a low c o n c e n t r a t i o n of l i g h t m a s s s u b s t i t u t i o n a l i m p u r i t i e s iti s p o s s i b l e f o r t h e l o c a l mode t o h y b r i d i z e with the two-phonon bound s t a t e and a c q u i r e a s t r u c t u r e . Such h y b r i d i z a t i o n can be b e s t observed i n imperfect f e r r o e l e c t r i c c r y s t a l s , because of their s t r o n g a n h a m - n i c n a t u r e , and o f the a v a i l a b i l i t y o f a s o f t mode whose frequency d e c r e a s e s with temperature above t h e t r a n s i t i o n temperature. Hence t h e two-soft mode bound s t a t e can be swept a c r o s s t h e l o c a l mode t o hybri- d i z e with it.
g e f e r e n c e s
.
-
1. Behera S.N. and Samsur Sk. Pramana
