5 Percolation of Molecular Excitons
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RAOUL KOPELMAN Department of Chemistry, University of Michigan, Ann Arbor, MI 48109
Band T h e o r y , L o c a l i z a t i o n and P e r c o l a t i o n B a s i c aspects o f s o l i d s t a t e theory face t h e i r c r i t i c a l t e s t s i n disordered organic s o l i d s . I n t h e F e b r u a r y 1980 i s s u e o f P h y s i c s Today t h e t i t l e o f t h e a r t i c l e by Duke and S c h e i n i s : " O r g a n i c Solids: i s e n e r g y band t h e o r y e n o u g h ? . They show t h e d i f f i c u l t i e s f a c i n g not o n l y the t r a d i t i o n a l e l e c t r o n - b a n d t h e o r y b u t a l s o t h e t r a d i t i o n a l e l e c t r o n h o p p i n g t h e o r y when compared w i t h experiments. Energy ( e x c i t o n , e x c i t a t i o n ) t r a n s p o r t i n o r g a n i c s o l i d s may be e q u i v a l e n t to e l e c t r o n transport both i n i t s i m p o r t a n c e and i n i t s t h e o r e t i c a l s i g n i f i c a n c e . A g a i n t h e t r a d i t i o n a l band and h o p p i n g models a r e s u s p e c t . Two i m p o r t a n t c o n c e p t s i n modern t h e o r i e s o f s e m i c o n d u c t o r s ( i n c l u d i n g amorphous o n e s ) a r e A n d e r s o n l o c a l i z a t i o n (1_) and p e r c o l a t i o n (2> 3) ( b o t h a p p r o a c h e s seem t o have o r i g i n a t e d a t B e l l L a b s a t a b o u t t h e same t i m e , i . e . 1 9 5 7 ) . These two a p p r o a c h e s h a v e o f t e n b e e n r e p r e s e n t e d as i r r e c o n c i l a b l e , even though t h e r e i s no b a s i c c o n t r a d i c t i o n between t h e t w o . Anderson d e r e a l i z a t i o n r e f e r s to the f o r m a t i o n , a t z e r o t e m p e r a t u r e , o f a band ( e l e c t r o n or e x c i t o n ) , r e s u l t i n g i n m e t a l l i c conduction. The h a l l m a r k o f m e t a l l i c ( b a n d ) c o n d u c t i o n i s a negative temperature c o e f f i c i e n t . Our e x p e r i m e n t a l r e s u l t s show t h a t t h i s i s not the case here (see b e l o w ) . A t h e r m a l l y a s s i s t e d h o p p i n g p r o c e s s i s b a s e d on l o c a l i z e d s t a t e s . The l o c a l i z a t i o n c o u l d be due t o A n d e r s o n l o c a l i z a t i o n o r s e l f - t r a p p i n g ( e . g . s m a l l p o l a r o n s ) o r some o t h e r f a c t o r . This process* g i v e s a t y p i c a l " n o n - m e t a l l i c conduction (or migration), e x h i b i t i n g a p o s i t i v e temperature c o e f f i c i e n t . The d e t a i l s o f s u c h a h o p p i n g p r o c e s s may be d e s c r i b e d by a p e r c o l a t i o n m o d e l . The l a t t e r i s b a s e d on t h e c l u s t e r s w i t h i n w h i c h t h e r e i s e f f i c i e n t hopping f o r a given t i m e - s c a l e . The p o i n t on t h e o r d e r parameter a x i s at which the average c l u s t e r s i z e r i s e s s h a r p l y i s the c r i t i c a l p e r c o l a t i o n p o i n t and may d e f i n e a p e r c o l a t i o n t r a n s i t i o n (2^). We g i v e b e l o w a b r i e f o v e r v i e w o f p e r c o l a t i o n c o n c e p t s ( I I ) 11
1 1
0097-6156/81/0162-005 7$05.00/0 © 1981 American Chemical Society In Photon, Electron, and Ion Probes of Polymer Structure and Properties; Dwight, David W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
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and o f the f o r m a l i s m o f e x c i t a t i o n p e r c o l a t i o n ( i l l ) , w h i c h l e a d s t o s c a l i n g and c r i t i c a l b e h a v i o r . E x p e r i m e n t a l r e s u l t s are g i v e n ( I V ) and shown t o f u l l y support the t h e o r e t i c a l predictions (III). A p h y s i c a l p i c t u r e emerges ( V ) , i n v o l v i n g p a r t i a l d e l o c a l i z a t i o n ( " i s l a n d s of coherence") over microdomains, w h i l e e f f i c i e n t m i g r a t i o n t a k e s p l a c e o v e r l a r g e r domains ( w h i c h d o m i n a t e the p e r c o l a t i o n t r a n s i t i o n b e h a v i o r ) . An e p i l o g u e ( V I ) and " E x p e r i m e n t a l " s e c t i o n ( V I I ) conclude t h i s paper.
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What i s E x c i t o n
Percolation?
An example o f m a c r o s c o p i c p e r c o l a t i o n i s t h e f l o w o f m a t t e r t h r o u g h t h e random b u t f i x e d c h a n n e l s i n a p o r o u s g l a s s f i l t e r . An example o f m i c r o s c o p i c p e r c o l a t i o n i s the m i g r a t i o n o f a gas through a s o l i d ( 4 ) . The n o m i n a l d i s t i n c t i o n b e t w e e n d i f f u s i o n and p e r c o l a t i o n i s t h a t the l a t t e r i m p l i e s m i g r a t i o n t h r o u g h a m i c r o s c o p i c a l l y h e t e r o g e n e o u s medium. We a r e i n t e r e s t e d h e r e i n the m i g r a t i o n o f e x c i t o n s , e.g. localized Frenkel excitons ( t i g h t l y bound e l e c t r o n - h o l e p a i r s ) . P e r c o l a t i o n has a l s o been v i e w e d as an a g g l o m e r a t i o n o f small c l u s t e r s i n t o large aggregates ( " i n f i n i t e c l u s t e r s " ) (.2). I n t e r e s t i n g l y enough t h i s i d e a has been a p p l i e d to s p i n c l u s t e r s , i . e . m a g n e t i c phase t r a n s i t i o n s , l o n g b e f o r e i t has b e e n a p p l i e d t o p o l y m e r s (2)* G e l a t i o n and v u l c a n i z a t i o n have been r e c e n t l y t r e a t e d w i t h a s i t e - b o n d p e r c o l a t i o n a p p r o a c h (_5, 6), and a r e l a t e d m o d e l has b e e n a p p l i e d to s u p e r - c o o l e d w a t e r (_7). These are " s t a t i c " p e r c o l a t i o n problems t h a t are r e l a t e d to static c r i t i c a l phenomena (8) r a t h e r t h a n to dynamic ( t r a n s p o r t ) c r i t i c a l phenomena. The l a t t e r d e a l w i t h e l e c t r i c a l c o n d u c t i o n and w i t h energy t r a n s p o r t - the t o p i c o f i n t e r e s t h e r e . The m i g r a t i o n o f F r e n k e l e x c i t o n s i s dynamic i n n a t u r e b u t c a n s t i l l be r e l a t e d t o t h e p r e s e n c e o f l a r g e c l u s t e r s o r a g g r e g a t e s o f the a p p r o p r i a t e m o l e c u l e s ( o f t e n c a l l e d g u e s t , donor o r carrier molecules). The l a r g e r the a v e r a g e c l u s t e r , t h e h i g h e r the e x c i t o n m i g r a t i o n e f f i c i e n c y . S i m u l t a n e o u s l y i t has t o be r e a l i z e d t h a t a g i v e n t i m e s c a l e d e f i n e s an a v e r a g e m i g r a t i o n r a n g e w h i c h c o n c e p t u a l l y i s c l o s e l y r e l a t e d t o an a v e r a g e c l u s t e r size. The r e s u l t i n g m i g r a t i o n c l u s t e r s ( " d y n a m i c c l u s t e r s " ) c a n b e t r e a t e d m a t h e m a t i c a l l y l i k e the c o n v e n t i o n a l ( " s t a t i c " ) c l u s t e r s , w h i c h do n o t depend on t i m e . The e x c i t o n m i g r a t i o n p r o b e s a r e " i n t e r n a l " t o the dynamic c l u s t e r and t h u s t h e r e i s no r e q u i r e m e n t t h a t the e x c i t o n s r e a c h the s u r f a c e o f t h e c r y s t a l o r c r y s t a l l i t e . T h i s i s somewhat a n a l o g o u s t o the AC c o n d u c t i v i t y o f composite m a t e r i a l s ( e . g . s i l v e r loaded t e f l o n ) . The e m p h a s i s i s t h u s on a 3 - d i m e n s i o n a l c l u s t e r ( a " p e r c o l a t i o n v o l u m e " r a t h e r than a p e r c o l a t i o n p a t h ) . Excitation Here
Percolation we
limit
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excitation percolation
in
very
In Photon, Electron, and Ion Probes of Polymer Structure and Properties; Dwight, David W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
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5.
KOPELMAN
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Percolation of Molecular Excitons
simple disordered molecular materials. These b e a r some r e semblance t o n a t u r a l p h o t o s y n t h e t i c a n t e n n a s y s t e m s and may h a v e some b e a r i n g on f u t u r e p h o t o v o l t a i c m a t e r i a l s (9). A t y p i c a l "energy semiconductor" c o n t a i n s three k i n d s o f m o l e c u l e s or m o l e c u l a r u n i t s : E n e r g y d o n o r s , e n e r g y t r a p s and "inert" host. The h o s t ' s e l e c t r o n i c e n e r g y l e v e l s a r e t o o h i g h t o be e x c i t e d . The e n e r g y l e v e l s o f the donor a r e s i g n i f i c a n t l y l o w e r and t h u s a r e e x c i t a b l e . The e n e r g y t r a p s have even l o w e r energy l e v e l s , thus e n a b l i n g e f f i c i e n t quenching o f the donor excitations. The systems o f i n t e r e s t e x h i b i t m u l t i p l e d o n o r donor e n e r g y t r a n s f e r e v e n t u a l l y r e s u l t i n g i n t h e t r a p p i n g o f t h e excitation. The e n e r g y t r a p p i n g may i n i t i a t e some " p h o t o c h e m i s t r y " ( a c t u a l l y , " e x c i t o c h e m i s t r y " ) , as i n p h o t o s y n t h e s i s , o r , on t h e c o n t r a r y , may i n h i b i t p h o t o c h e m i s t r y by o f f e r i n g a more effective route f o r r a d i a t i v e or r a d i a t i o n l e s s decay o f the excitation. One o f the most i n t e r e s t i n g a s p e c t s o f e n e r g y t r a n s p o r t i s t h e e x c i t a t i o n p e r c o l a t i o n t r a n s i t i o n (2^, 9) and i t s s i m i l a r i t y ( 1 0 ) t o m a g n e t i c phase t r a n s i t i o n s and o t h e r c r i t i c a l phenomena (2^, 8 ) . I n i t s s i m p l e s t form t h e p r o b l e m i s one o f c o n n e c t i v i t y . I n a b i n a r y s y s t e m , made o n l y o f h o s t s and d o n o r s , t h e q u e s t i o n is: c a n t h e e x c i t a t i o n t r a v e l from one s i d e o f t h e m a t e r i a l t o the other? The i m p l i c i t a s s u m p t i o n i s t h a t t h e r e a r e e x c i t a t i o n t r a n s f e r - b o n d s o n l y b e t w e e n two d o n o r s t h a t a r e " c l o s e e n o u g h " , where " c l o s e enough" has a p r a c t i c a l a s p e c t ( e . g . d e f i n e d by t h e e x c i t a t i o n t r a n s f e r p r o b a b i l i t y or t i m e ) . O b v i o u s l y , i f t h e r e i s a s u c c e s s i o n o f e x c i t a t i o n - b o n d s from one edge o f t h e m a t e r i a l t o t h e o t h e r , one has " p e r c o l a t i o n " , i . e . a c o n n e c t e d c h a i n o f d o n o r s f o r m i n g an e x c i t a t i o n c o n d u i t . We n o t e t h a t the e x c i t a t i o n - b o n d s s e l d o m c o r r e s p o n d t o r e a l c h e m i c a l b o n d s ; r a t h e r more o f t e n t h e y c o r r e s p o n d to v a n - d e r - W a l l s t y p e bonds and most o f t e n t h e y c o r respond to a d i p o l e - d i p o l e or e q u i v a l e n t quantum-mechanical interaction. F o r a s y s t e m where the d o n o r s a r e d i s t r i b u t e d a t random i t c a n be shown m a t h e m a t i c a l l y t h a t t h e r e e x i s t s a c r i t i c a l c o n c e n t r a t i o n C ( a l s o c a l l e d " p e r c o l a t i o n c o n c e n t r a t i o n " ) below which the p e r c o l a t i o n ( e d g e - t o - e d g e c o n n e c t i v i t y ) has a p r o b a b i l i t y o f z e r o and above w h i c h the p e r c o l a t i o n p r o b a b i l i t y (P ) r i s e s s h a r p l y w i t h donor c o n c e n t r a t i o n ( C ) . A mathematical r e l a t i o n (2^), f o r a s u b s t i t u t i o n a l ^ d i s o r d e r e d b i n a r y l a t t i c e , i s : P
* oo
|C-C | C
3
C>C
(1)
C
1
where the e x p o n e n t 3 i s a b o u t 0 . 1 3 f o r a 2 - d i m e n s i o n a l t o p o l o g y and 0 . 3 5 f o r a 3 - d i m e n s i o n a l one ( 2 , 1 0 ) . We n o t e t h a t 0 " = 0 (for C=C ), 0 . 0 0 1 = 0.41, 0 . 0 r = 0 . 5 5 and 0 . 1 = 0.74. F o r t e r n a r y systems ( h o s t , d o n o r , t r a p ) , w h i c h a r e u s u a l l y o f i n t e r e s t , t h e q u e s t i o n i s no l o n g e r w h e t h e r t h e e x c i t a t i o n w i l l make i t from one edge o f t h e m a t e r i a l to the o t h e r , b u t w h e t h e r t h e e x c i t a t i o n w i l l make i t t o a n e a r - b y ( o r n o t so n e a r - b y ) t r a p . U , i J
,
H
0
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1
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c
In Photon, Electron, and Ion Probes of Polymer Structure and Properties; Dwight, David W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
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A r e l e v a n t c o n c e p t i s t h a t o f a d o n o r c l u s t e r , w h i c h i s d e f i n e d as a s e t o f d o n o r s c o n n e c t e d by b o n d s . The a v e r a g e c l u s t e r s i z e ( I ^ y ) w i l l be i n f i n i t e above the c r i t i c a l c o n c e n t r a t i o n but f i n i t e below i t . T h i s average c l u s t e r s i z e i s m a t h e m a t i c a l l y r e l a t e d to a c r i t i c a l exponent y ( w h i c h i s about 2 . 2 - 2.4 i n 2 dimensions (10)) : I
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Y
|c-C | ~ c
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