DYNAMICS OF COMPOSITE EXCITON-PHONON STATES IN DOPED MOLECULAR CRYSTALS

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DYNAMICS OF COMPOSITE EXCITON-PHONON STATES IN DOPED MOLECULAR CRYSTALS

J . Singh

To cite this version:

J . Singh. DYNAMICS OF COMPOSITE EXCITON-PHONON STATES IN DOPED MOLECULAR CRYSTALS. Journal de Physique Colloques, 1985, 46 (C7), pp.C7-61-C7-65.

�10.1051/jphyscol:1985712�. �jpa-00224961�

DYNAMICS OF COMPOSITE EXCITON-PHONON STATES I N DOPED MOLECULAR CRYSTALS

J. Singh

Department o f P h y s i c s , NationaZ U n i v e r s i t y o f Singapore, Kent Ridge, S i n g a p o r e 0511

A b s t r a c t - The t r a n s p o r t p r o p e r t i e s o f a composite exciton-phonon (CEP) i s presented by c a l c u l a t i n g t h e r a t e o f i t s t r a p p i n g a t i m p u r i t i e s by e m i t t i n g a phonon. It i s demonstrated f o r t h e f i r s t time t h a t a Debye-Waller type f a c t o r reduces t h e t r a p p i n g p r o b a b i l i t y i n molecular c r y s t a l s i n t h e same way as t h a t does i n the e l e c t r o n i c t r a n s p o r t processes i n atomic c r y s t a l s .

I. INTRODUCTION

The t r a n s p o r t o f e x c i t a t i o n energy and charge c a r r i e r s i n disordered s o l i d s has become an i n t e r e s t i n g t o p i c 11-4/ because o f i t s p o t e n t i a l a p p l i c a t i o n s i n e l l e c t r o n i c processes o f i n d u s t r i a l value. I n c r y s t a l s where d i s o r d e r i s created b y s u b s t i t u t i n g i m p u r i t i e s s t u d i e s about t r a n s p o r t o f e x c i t a t i o n s provide leads t o the understanding of o p t i c a l p r o p e r t i e s l i k e quenching o f fluorescence and phosphorescence. There are two approaches which may be f o l l o w e d t h e o r e t i c a l 1y t o study disordered s o l i d s : ( 1 ) s t a r t w i t h a c r y s t a l l i n e m a t e r i a l and i n t r o d u c e i n i t the e f f e c t o f disorders a t a l a t e r stage, o r ( 2 ) s t a r t w i t h t h e disordered system r i g h t from the beginning.

L a t t e r approach i s u s u a l l y f o l l o w e d f o r numerical c a l c u l a t i o n s /4/. Here however we use t h e f i r s t approach which i s more s u i t a b l e f o r a n a l y t i c a l work. F i r s t we study t h e p r o p e r t i e s o f e x c i t o n s t a t e s i n pure c r y s t a l l i n e s o l i d s and then t h e i n f l u e n c e o f s u b s t i t u t i o n o f i m p u r i t i e s on t h e e x c i t o n s t a t e s . Composite e x c i t o n - phonon (CEP) s t a t e s a r i s e i n pure molecular c r y s t a l s due t o t h e c o u p l i n g between Frenkel e x c i t o n s and phonons 151. We consider a c r y s t a l doped w i t h i s o t o p i c i m p u r i t i e s i n low concentration so t h a t the i m p u r i t y - i m p u r i t y i n t e r a c t i o n can be neglected. As a CEP e x c i t e d i n pure c r y s t a l r e g i o n moves away i t may encounter an i m p u r i t y whose e x c i t e d s t a t e s energy i s lower than t h a t o f host, and due t o e x c i t o n , phonon and i m p u r i t y i n t e r a c t i o n t h e CEP gets trapped. The excess energy o f CEP i s e m i t t e d as phonons. Emission o f a phonon o r a number o f phonons would depend on the t r a p depth and l a t t i c e temperature. The t r a n s i t i o n r a t e from a f r e e CEP t o a trapped e x c i t o n can be c a l c u l a t e d provided one has a s u i t a b l e operator o f e x c i t o n phonon and i m p u r i t y i n t e r a c t i o n which was n o t d e r i v e d f o r molecular e x c i t o n s u n t i l r e c e n t l y / 6 / .

The prupose o f t h i s paper i s t o demonstrate f o r t h e f i r s t time t h a t i n d e r i v i n g t h e exciton-phonon-impurity i n t e r a c t i o n o p e r a t o r f o r molecular c r y s t a l s one gets a f a c t o r which i s s i m i l a r t o t h e Debye-Waller f a c t o r known i n t h e case o f e l e c t r o n - phonon i n t e r a c t i o n i n atomic c r y s t a l s / 7 / . A s i m i l a r f a c t o r i s a l s o shown by P o t t and Davis /8/ t o appear i n r e l a t i o n w i t h the t r a n s i t i o n between Anderson's l o c a l i z e d

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985712

C7-62 JOURNAL DE PHYSIQUE

s t a t e s i n n o n - c r y s t a l l i n e s o l i d s . We have c a l c u l a t e d the r a t e o f t r a p p i n g o f a CEP a t i m p u r i t i e s by e m i t t i n g a phonon a p p l y i n g the zeroth order o p e r a t o r o f e x c i t o n - phonon-impurity i n t e r a c t i o n . The e f f e c t o f t h e above f a c t o r on t h e t r a n s i t i o n p r o b a b i l i t y i s discussed.

11. EXCITON-PHONON-IMPURITY INTERACTION

Consider a molecular c r y s t a l w i t h an i m p u r i t y molecule a t one o f the l a t t i c e s i t e s a t

,,

where A i s t h e t r a p depth d e f i n e d such t h a t n o t o n l y i t accounts f o r t h e d i f f e r e n c e betweenPthe e x c i t a t i o n energy o f i s o l a t e d h o s t and i m p u r i t y molecules b u t a l s o t h a t o f t h e i r i n t e r m o l e c u l a r i n t e r a c t i o n energies. Assuming t h a t t h e i m p u r i t y molecule occupies t h e l a t t i c e s i t e such t h a t no t r a n s l a t i o n a l sy metry i s destroyed (e.g.

i s o t o p i c i m p u r i t y ) we can expand t h e e x c i t o n o p e r a t o r B as

T

-. P B~ = N-3 1 exp(-i?.p)Bk t ,

P k

- - -

where k i s t h e r e c i p r o c a l l a t t i c e vector, and N i s t h e number o f molecules i n t h e

-

c r y s t a l . Here we consider a c r y s t a l w i t h one molecule p e r u n i t c e l l .

I n t r o d u c i n g t h e l a t t i c e v i b r a t i o n s and assuming t h a t t h e displacement vectors from p o i n t s o f l a t t i c e e q u i l i b r i u m a r e small we may assume t h a t t h e t r a n s l a t i o n a l symmetry i s preserved i n a v i b r a t i n g l a t t i c e . Thus one may take t h e l a t t i c e s i t e a t p t o be instantaneously a t p

-

A -P

d e r i v e H i n the f o l l o w i n g form: -

,p

where b i s t h e c r e a t i o n operator o f a phonon w i t h wavevector t g , u n i t p o l a r i z a t i o n

s

vector e ( q ) and frequency ~ ( 9 ) . 1 ~ and v index t h e order o f t h e i n t e r a c t i o n operator e.g. p=v=O i s z e r o t h order, p=0, v=1 and u=1, v=O f i r s t o r d e r e t c . Z(q,p)

- -

In i s t h e mass c o e f f i c i e n t o f t h e i m p u r i t y molecule. It i s t o be noted t h a t t h e skcond exponential f a c t o r i n ( 3 ) does n o t depend on the l o c a t i o n o f t h e i m p u r i t y molecule i n t h e c r y s t a l .

The c a l c u l a t i o n o f a general t r a p p i n g r a t e i n which any number o f phonons can be e m i t t e d i s very complicated t h e r e f o r e i t w i l l be published elsewhere. Here however we w i l l focus our a t t e n t i o n t o c a l c u l a t e the t r a p p i n g r a t e o f a CEP a t an i m p u r i t y by e m i t t i n g a phonon. The t r a p p i n g p r o b a b i l i t y R i s given by:

"0

where HD i s t h e zeroth o r d e r term (u=v=O) o f ( 3 ) and p(w) i s t h e d e n s i t y o f phonon

state^.^ Here we consider emission o f o n l y low energy phonons. I K;n> i s t h e s t a t e v e c t o r o f a CEP w i t h wavevector _K created i n i t i a l l y i n t h e pure c y y s t a l region; n representing t h e i n i t i a l scheme o f phonons occupation ( I n > = In1 , n2, .. ^.n

..

The i n i t i a l s t a t e v e c t o r \K;n> i s w r i t t e n as 151: 9'

where Co and C1 are r e s p e c t i v e l y the probabi 1 i t y amp1 i tude c o e f f i c i e n t s o f c r e a t i n g a pure e x c i t o n s t a t e accompanied by no phonons, and c r e a t i n g an e x c i t o n s t a t e accqmpanied by the c r e a t i o n o r a n n i h i l a t i o n o f a s i n g l e phonon.

F i n a l l y when t h e CEP i s trapped a t t h e i m p u r i t y and a phonon o f wavevector q i s e m i t t e d t h e f i n a l s t a t e v e c t o r can be w r i t t e n as / l o / :

-

where G = (I+!~)-$ i s a n o r m a l i z a t i o n constant and 6 conserves t h e momentum o f

9 ." !!*_K

t h e system.

Using ( 3 ) , (6) and ( 7 ) we g e t t h e t r a n s i t i o n m a t r i x element < p ; n l i O l ~ ; n > as:

-

where ii i s t h e average phonon p o p u l a t i o n i n t h e i n i t i a l s t a t e . The f u n c t i o n f(.k)

9

i s obtained as:

I n (9) f ( k ) has t h e same form as t h e Debye-Waller f a c t o r / 7 / known i n the e l e c t r o n i c t r a n s p o r t processes mediated by phonons i n c r y s t a l l i n e s o l i d s . Also a s i m i l a r f u n c t i o n i s shown by M o t t and Davis /8/ t o appear i n the t r a n s i t i o n m a t r i x element

C7-64 JOURNAL DE PHYSIQUE

between Anderson's l o c a l i z e d s t a t e s i n n o n - c r y s t a l l i n e s o l i d s . F o l l o w i n g /7/ we can d e r i v e f ( k ) f o r low frequency phonons as:

f ( k ) = exp

[

]

'IpkB @D

where k g i s the Boltzmann constant and

mD

3 3

t r a n s p o r t i n atomic s o l i d s /7/ one o b t a i n s a f a c t o r o f i n s t e a d o f i n the exponential o f (10). It i s because t h e r e are s i x v i b r a t i o n a l ( i n t e r m o l e c u l a r ) degrees o f freedom f o r a molecule and o n l y t h r e e f o r atoms i n a c r y s t a l l a t t i c e . Using (8) i n ( 5 ) we o b t a i n R as ( d e t a i l s o f t h e d e r i v a t i o n can be seen i n / l o / and

(Singh, t o be published):

where

E(o) i s the energy o f an e x c i t o n a t k=O and i t i s measured from t h e c e n t r e o f t h e

- - -

e x c i t o n band, C i s t h e i m p u r i t y concentration and B i s h a l f o f t h e unperturbed e x c i t o n band width. P It i s t o be noted t h a t the exponent o f f ( k ) (10) i s n o t zero and t h e r e f o r e R < Ro i n (11).

I V . DISCUSSION

The motion o f a CEP i n mixed o r doped molecular c r y s t a l s w i t h low i m p u r i t y concentra- t i o n i s presented here. I t i s invoked f o r t h e f i r s t time t h a t i n t h e t r a n s p o r t o f CEP a l s o a Debye-Waller type f a c t o r occurs which reduces t h e t r a n s i t i o n p r o b a b i l i t y o f CEP hopping from one s t a t e t o another. It i s d i f f i c u l t t o c a l c u l a t e f ( k )

a c c u r a t e l y . However, as t h e molecular mass c o e f f i c i e n t s I occurs i n t h e denominator which e.g. f o r naphthalene i s ^Q, 20 times g r e a t e r than t h e Patomic mass, f ( k ) i s expected t o be r e l a t i v e l y h i g h e r than the Debye-Waller f a c t o r f o r e l e c t r o n i c

processes i n atomic c r y s t a l s . The exponent o f f ( k ) (10) i s t h e mean square displace- ment o f i m p u r i t y molecules from e q u i l i b r i u m . According t o o u r assumption of small l a t t i c e displacement vectors i t i s expected t h a t a t low temperatures t h e f a c t o r of Ro i n (11) might o n l y be equal t o a f r a c t i o n . I n naphthalene f o r example Ro i s c a l c u l a t e d o f t h e o r d e r o f l o 7 - 10' a t C % / l o / . The e f f e c t o f f ( k ) might o n l y reduce Ro a t the most by an o r d e r o f magnitude a t lower temperature (% 4.Z°K). P A t h i g h e r temperature however t h e e f f e c t would be more prominent. A more accurate c a l c u l a t i o n o f the f a c t o r 1 f ( k ) (11 ) can be done using numerical methods.

k

For mu1 tiphonon processes, the c a l c u l a t i o n o f f ( k ) i s much more complicated. For very deep t r a p s i . e . w i t h t r a p depth equal t o energy o f several phonons one can express t h e f i n a l phonon s t a t e s as coherent s t a t e s . The t r a p p i n g r a t e thus obtained

however lengthy t h e r e f o r e they w i l l be published elsewhere.

We have presented here t h e r a t e of t r a n s i t i o n using only A"ut c a l c u l a t i o n s aredone using 9; (Singh, t o be p u b l i s h e d ) , 9: (Singh and Thilagam, t o be published) and P fi;

( ~ i n ~ h , ~ t o be published) a s well. ~ k i s leads t o t h e p r o b a b i l i t y of a CEP t r a n s i t ; o n from one s t a t e t o another dependent on several v a r i a b l e s p a r t i c u l a r l y i n multiphonon processes. However f o r shallow t r a p s t h e p r o b a b i l i t y of s i n g l e phonon emission i s usually l a r g e r than t h a t of multiphonon emission.

The s u b s t i t u t i o n of i m p u r i t i e s i s known t o c r e a t e l o c a l i z e d s t a t e s i n a c r y s t a l l i n e m a t e r i a l s . The form of t h e t r a n s i t i o n matrix element with t h e exponential f a c t o r suggests t h a t CEP s t a t e s can a l s o be l o c a l i z e d 181.

ACKNOWLEDGEMENT: I have b e n e f i t t e d by a d i s c u s s i o n with Dr. J . Mahanty about t h e Debye-Waller f a c t o r .

REFERENCES

/ I / A . Blumen and R . S i l b e y , J . Chem. Phys. 69 (1978) 3589.

121 C . R . Gochanaur, H.C. Anderson and M.D. Fayer, J . Chem. Phys. 70 (1978) 4254.

/3/ A. Blumen and R. S i l b e y , J. Chem. Phys. 70 (1979) 3707.

/4/ J . K l a f t e r and R. S i l b e y , J . Chem. Phys.

z,

/5/ D . P . Craig and J . Singh, Chem. Phys. L e t t e r s 82 (1981) 405 and J . Singh, Solid S t a t e Physics 38 (1984) 295.

161 J. Singh, Phys. Rev. 830 (1984) 7267.

/7/ See e.g. J.M. Zeeman, " P r i n c i p l e s of t h e Theory of S o l i d s " (University P r e s s , Cambridge, 1972).

/8/ N.F. Mott and E.A. Davis, " E l e c t r o n i c Processes i n Non-Crystalline Materials (Clarendon P r e s s , Oxford, 1979) pp. 77.

/9/ A.S. Davydov, "Theory of Molecular Excitons" (Plenum, N . Y . 1971).

/ l o / J . Singh, Sing. J . Phys. 2 (1985) 33.