Localization of Electronic States in Polymers and Molecular Solids

6 Localization of Electronic States in Polymers and Molecular Solids Photon, Electron, and Ion Probes of Polymer Structure and Properties Downloaded from pubs.acs.org by UNIV OF ARIZONA on 06/08/17. For personal use only.

C. B. DUKE Xerox Webster Research Center, Xerox Square-114, Rochester, N Y 14644

The n a t u r e o f t h e e l e c t r o n i c s t a t e s a s s o c i a t e d w i t h i n j e c t e d c h a r g e s and o p t i c a l e x c i t a t i o n s i n p o l y m e r i c and m o l e c u l a r s o l i d s has been a s u b j e c t o f renewed i n t e r e s t i n recent years both b e c a u s e o f t h e i n c r e a s i n g use o f such m a t e r i a l s i n e l e c t r o n i c d e v i c e s and b e c a u s e a q u e s t i o n h a s a r i s e n c o n c e r n i n g t h e a p p r o p r i a t e n e s s o f t r a d i t i o n a l o n e - e l e c t r o n e n e r g y band t h e o r y f o r t h e d e s c r i p t i o n o f t h e s e s t a t e s (_1, 2> 3). I n p a r t i c u l a r , i t has been suggested t h a t under many c i r c u m s t a n c e s c h a r g e s i n j e c t e d into p o l y m e r i c and m o l e c u l a r m a t e r i a l s a r e more a n a l o g o u s t o l o c a l i z e d m o l e c u l a r ions i n s o l u t i o n than to the extended " B l o c h " s t a t e s w h i c h a r e t h o u g h t to be a s s o c i a t e d w i t h e l e c t r o n i c m o t i o n i n m e t a l s and common c o v a l e n t s e m i c o n d u c t o r s Q , 2). Consequently, a s u s p i c i o n has been g r o w i n g t h a t s e m i c o n d u c t i n g o r g a n i c s o l i d s s h o u l d n o t be r e g a r d e d as " o r g a n i c s e m i c o n d u c t o r s " i n t h e t r a d i t i o n a l s e n s e , i n s p i t e o f an e x t e n s i v e e a r l y l i t e r a t u r e i n w h i c h e n e r g y - b a n d models were a p p l i e d t o i n t e r p r e t measurements o f t h e i r o p t i c a l and t r a n s p o r t p r o p e r t i e s (4). The p u r p o s e o f t h i s p a p e r i s t h e p r e s e n t a t i o n o f a b r i e f o v e r v i e w o f r e c e n t l i t e r a t u r e i n w h i c h new m o d e l s o f e l e c t r o n i c s t a t e s i n p o l y m e r s and m o l e c u l a r s o l i d s have b e e n p r o p o s e d (1^, 2, 3_ 5-16). S i n c e l o c a l i z e d ( e . g . , m o l e c u l a r - i o n ) s t a t e s seem p r e v a l e n t i n these m a t e r i a l s , I i n d i c a t e i n S e c . I I the p h y s i c a l phenomena w h i c h l e a d t o l o c a l i z a t i o n . S e c . I l l i s d e v o t e d t o t h e d e s c r i p t i o n o f a model w h i c h p e r m i t s t h e q u a n t i t a t i v e a n a l y s i s o f t h e l o c a l i z e d - e x t e n d e d c h a r a c t e r o f e l e c t r o n i c s t a t e s and t o t h e i n d i c a t i o n o f the r e s u l t s o f s p e c t r o s c o p i c d e t e r m i n a t i o n s o f the p a r a m e t e r s i n t h i s m o d e l f o r v a r i o u s c l a s s e s o f p o l y m e r i c and molecular materials. I c o n c l u d e w i t h t h e m e n t i o n i n S e c . I V o f an i m p o r t a n t p r a c t i c a l a p p l i c a t i o n o f t h e s e c o n c e p t s and m o d e l s : The c o n t a c t c h a r g e exchange p r o p e r t i e s o f i n s u l a t i n g p o l y m e r s (_7, 17, 18, 19). y

0097-615 6/ 81 /0162-0065$05.00/0 © 1981 American Chemical Society

66

PHOTON, ELECTRON, AND ION PROBES

Photon, Electron, and Ion Probes of Polymer Structure and Properties Downloaded from pubs.acs.org by UNIV OF ARIZONA on 06/08/17. For personal use only.

R e l a x a t i o n and

Fluctuations

When a c h a r g e ( o r o p t i c a l l y - e x c i t e d e x c i t o n ) i s injected i n t o a m o l e c u l a r s o l i d , i t i n d u c e s changes i n t h e electronic c h a r g e d e n s i t y and a t o m i c p o s i t i o n s b o t h on t h e m o l e c u l a r s i t e w h i c h i t o c c u p i e s ( i n t r a m o l e c u l a r r e l a x a t i o n ) and on n e i g h b o r i n g molecular s i t e s (intermolecular r e l a x a t i o n ) . T h i s phenomenon i s c a l l e d " r e l a x a t i o n " and l e a d s t o a l o w e r i n g o f t h e e n e r g y o f t h e c o m p o s i t e s y s t e m o f added c h a r g e p l u s m o l e c u l a r s o l i d by an amount c a l l e d the " r e l a x a t i o n e n e r g y , " E . The r e l a x a t i o n e n e r g y i s f o r m a l l y d e f i n e d f o r an i n j e c t e d c h a r g e ( e . g . , m o l e c u l a r i o n ) as the d i f f e r e n c e b e t w e e n t h e g r o u n d s t a t e e n e r g y o f t h e i o n and t h e Hartree-Fock molecular o r b i t a l eigenvalue ( i n the canonical b a s i s ) w h i c h c o r r e s p o n d s to the f r e e - i o n s t a t e . B o t h the i n t r a m o l e c u l a r and i n t e r m o l e c u l a r c o n t r i b u t i o n s to t h e relaxation e n e r g y c a n be l a r g e f o r m o l e c u l a r s o l i d s : I.e., E (intra) = E ( i n t e r ) = l - 2 e V ( 1 , 2, 5). T h e r e f o r e the e n e r g i e s o j f m o l e c u l a r a n i o n s ( c a t i o n s ) i n condensed m o l e c u l a r media are about 2-4eV lower ( h i g h e r ) t h a n the c o r r e s p o n d i n g f r e e - m o l e c u l e orbitals. D e t a i l e d models o f the v a r i o u s c o n t r i b u t i o n s to the r e l a x a t i o n e n e r g y a r e g i v e n e l s e w h e r e (2_, .5, 8 ) . An i m p o r t a n t f e a t u r e o f the i n t e r m o l e c u l a r c o n t r i b u t i o n s t o the r e l a x a t i o n e n e r g y i s t h e i r dependence on t h e l o c a l a t o m i c s t r u c t u r e i n the v i c i n i t y o f the i n j e c t e d i o n o r e x c i t o n . For example, a surface i o n e x h i b i t s o n l y a b o u t 60% o f t h e b u l k r e l a x a t i o n e n e r g y b e c a u s e o f the a b s e n c e o f p o l a r i z a b l e n e i g h b o r i n g s p e c i e s o u t s i d e the c o n d e n s e d phase ( 1 , 6). This v a r i a t i o n i n t h e v a l u e o f t h e m o l e c u l a r - i o n s i t e e n e r g y a t the s u r f a c e r e l a t i v e t o the b u l k l e d t o t h e p r e d i c t i o n (1_) and s u b s e q u e n t o b s e r v a t i o n (20) o f l o c a l i z e d s u r f a c e s t a t e s f o r m o l e c u l a r f i l m s ( s p e c i f i c a l l y , anthracene). More g e n e r a l l y , l o c a l v a r i a t i o n s i n the compos i t i o n and s t r u c t u r e o f p o l y m e r s and m o l e c u l a r g l a s s e s c a u s e s p a t i a l f l u c t u a t i o n s i n the s i t e e n e r g i e s o f m o l e c u l a r i o n s and e x c i t o n s i n these m a t e r i a l s . For i n t r i n s i c m o l e c u l a r - i o n s t a t e s , the m a g n i t u d e o f t h e s e f l u c t u a t i o n s may be i n f e r r e d v i a a n a l y s e s o f the w i d t h s o f v a l e n c e - e l e c t r o n p h o t o e m i s s i o n l i n e s (_2, 6> 8) o r o f c o n t a c t c h a r g e exchange s p e c t r a (2^, 6, L 8 , 19)* T y p i c a l l y one finds E - 0.5-1.0eV. The i m p o r t a n c e o f t h e s e f l u c t u a t i o n s i s t h e i r c r e a t i o n o f l o c a l i z e d m o l e c u l a r - i o n s t a t e s i n p o l y m e r s and g l a s s e s (1_, 2, 3). In p a r t i c u l a r , i n t r i n s i c l o c a l i z e d moleculari o n s t a t e s are thought to be i n v o l v e d i n the c o n t a c t charge exchange b e h a v i o r o f p e n d a n t group p o l y m e r s (T_, _T7, 1 8 , 19) and e x t r i n s i c l o c a l i z e d m o l e c u l a r - i o n s t a t e s dominate the t r a n s p o r t p r o p e r t i e s o f m o l e c u l a r l y doped p o l y m e r s (21^, 2 2 , 2 3 ) . r

A n o t h e r s i g n i f i c a n t s o u r c e o f v a r i a t i o n s i n the l o c a l s i t e e n e r g i e s o f m o l e c u l a r i o n s and e x c i t o n s i n c o n d e n s e d m e d i a i s t h e m o d u l a t i o n o f t h e s e e n e r g i e s by the t h e r m a l v i b r a t i o n s e i t h e r o f the medium ( e . g . , a c o u s t i c a l phonons and l i b r o n s ) o r o f the molecular ion (exciton) i t s e l f (intramolecular v i b r a t i o n s ) . A model H a m i l t o n i a n which i n c o r p o r a t e s e l e c t r o n i c i n t e r a c t i o n s w i t h


6.

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67

Localization of Electronic States

the v a r i o u s b r a n c h e s o f the v i b r a t i o n a l s p e c t r u m was p r o p o s e d by R i c e et a l . , (lfJ) and s u b s e q u e n t l y was e x t e n d e d and a p p l i e d t o p r o v i d e an i n t e r p r e t a t i o n o f UV a b s o r p t i o n and p h o t o e m i s s i o n s p e c t r a ( 8 , _12, _24). Three fundamentally d i f f e r e n t phenomena occur: t h e m o d u l a t i o n o f l o c a l m o l e c u l a r - i o n e n e r g i e s by v i b r a t i o n s c h a r a c t e r i s t i c o f the medium i n w h i c h i o n i s embedded (8>, 9^), t h e i r m o d u l a t i o n by modes i n t e r n a l t o the i o n i t s e l f (9^, 1 0 , 2 4 ) , and the m o d u l a t i o n o f the s i t e - t o - s i t e e l e c t r o n transfer i n t e g r a l s by v i b r a t i o n s c h a r a c t e r i s t i c o f the medium C 3 , j ) , 1 0 ) . E a c h o f t h e s e phenomena a f f e c t s p h o t o e m i s s i o n , UV a b s o r p t i o n , and t r a n s p o r t i n i t s own u n i q u e f a s h i o n so t h a t a c o m p l e t e d e s c r i p t i o n o f any o f t h e s e measurements must encompass a l l t h r e e . A c o m p a r i son o f the n a t u r e and c o n s e q u e n c e s o f the t h r e e e f f e c t s h a s b e e n g i v e n by Duke ( 3 , 9). Localization The l o c a l or e x t e n d e d n a t u r e o f m o l e c u l a r - i o n ( o r e x c i t o n ) s t a t e s i n m o l e c u l a r s o l i d s i s d e t e r m i n e d by a c o m p e t i t i o n b e t w e e n f l u c t u a t i o n s i n the l o c a l s i t e e n e r g i e s o f t h e s e s t a t e s ( w h i c h t e n d to l o c a l i z e them) and t h e h o p p i n g i n t e g r a l s f o r inter-site e x c i t a t i o n t r a n s f e r (which tend to d e l o c a l i z e them). I n order to d e f i n e t h i s f l u c t u a t i o n - i n d u c e d l o c a l i z a t i o n c o n c e p t more p r e cisely, consider the model defined by the one-electron Hamiltonian H

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i n which n i s a s i t e index; a d e s i g n a t e s the m o l e c u l a r o r b i t a l ; V , d e s i g n a t e the i n t e r m o l e c u l a r h o p p i n g i n t e g r a l s ; and a is tfie a n n i h i l a t i o n o p e r a t o r f o r an e l e c t r o n o c c u p y i n g the o r b i ? a l a a t the s i t e n . B o t h the m o l e c u l a r i o n e n e r g i e s ( i . e . , s i t e e n e r g i e s , e ) and the h o p p i n g i n t e g r a l s , V , f o r m d i s t r i b u t i o n s b e c a u s e o f tfie l o c a l v a r i a t i o n s i n c o m p o s i t i o n and s t r u c t u r e . While a proper a n a l y s i s o f the r e s u l t i n g model i s c o m p l i c a t e d ( 2 5 , 2 6 ) , the q u a l i t a t i v e f e a t u r e s o f i n t e r e s t t o us c a n be d e f i n e d i n terms o f a mean s i t e e n e r g y , t h e rms d e v i a t i o n from t h i s mean e

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and a mean h o p p i n g i n t e g r a l , V . V a r i a t i o n s i n t h e s i t e e n e r g i e s from the mean ( d e s c r i b e d by A ) a r e r e f e r r e d t o as " d i a g o n a l d i s o r d e r " w h e r e a s a n a l o g o u s v a r i a t i o n s o f the h o p p i n g i n t e g r a l s are c a l l e d " o f f - d i a g o n a l d i s o r d e r . " S i m i l a r i l y , i f these v a r i a t i o n s a r e c a u s e d by l o c a l t i m e - i n d e p e n d e n t f l u c t u a t i o n s i n comp o s i t i o n or s t r u c t u r e one s p e a k s o f " s t a t i c d i s o r d e r " w h i l e i f


68


t h e y a r e g e n e r a t e d by t h e r m a l v i b r a t i o n s one e m p l o y s the t e r m "dynamic d i s o r d e r . " Whereas s t a t i c d i a g o n a l and o f f - d i a g o n a l d i s o r d e r c a n o c c u r i n p o l y m e r s , o n l y dynamic d i s o r d e r s h o u l d be present i n molecular c r y s t a l s . A m o d e l p r o p e r l y embodying dynami c d i s o r d e r i s more c o m p l i c a t e d t h a n E q . ( 1 ) ( 3 , 8, 9). The i m p o r t a n t p o i n t f o r our p r e s e n t p u r p o s e , h o w e v e r , i s t h e r e c o g n i t i o n t h a t a l l s o u r c e s o f d i s o r d e r must be c o n s i d e r e d i n o r d e r t o d e t e r m i n e t h e l o c a l i z e d or e x t e n d e d c h a r a c t e r o f t h e e l e c t r o n i c s t a t e s a s s o c i a t e d w i t h charges (or e x c i t o n s ) i n j e c t e d i n t o p o l y mers and m o l e c u l a r m a t e r i a l s , and t h a t t h e s e s o u r c e s o f d i s o r d e r a r e d e s c r i b e d by c h o o s i n g s u i t a b l e d i s t r i b u t i o n s o f t h e and V , p a r a m e t e r s i n E q . ( 1 ) o r an a p p r o p r i a t e generalization tRereof. I n terms o f the s i m p l i f i c a t i o n o f E q . ( 1 ) d e f i n e d b y c o n s i d e r i n g only nearest-neighbor hopping i n t e g r a l s ( i . e . , n e g l e c t ing long-range hops), i t c a n be shown ( 2 5 , 26) t h a t injected charges are localized, i.e., they form m o l e c u l a r c a t i o n s or a n i o n s w i t h i n the s o l i d , i f A>czV

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i n w h i c h z i s the c o o r d i n a t i o n number o f the ( p r e s u m a b l y i d e n t i c a l ) m o l e c u l a r s i t e s and c i s a d i m e n s i o n l e s s number o f the o r d e r o f u n i t y w h i c h depends b o t h on t h e c o n n e c t i v i t y ( i . e . , d i m e n s i o n a l i t y ) o f t h e m o l e c u l a r s y s t e m and on the e x t e n t o f o f f - d i a g o n a l disorder. T y p i c a l l y c - 2.5 for ( i s o t r o p i c ) three dimensional systems, 1.5 f o r two d i m e n s i o n a l s y s t e m s , and z e r o f o r onedimensional systems. I n e q u a l i t y 03) i s b e l i e v e d t o be s a t i s f i e d i n most p o l y m e r s : I n t h i s case charges i n j e c t e d i n t o the polymer form l o c a l , m o l e c u l a r a n i o n s and c a t i o n s , r e s p e c t i v e l y , r a t h e r than extended m o b i l e s t a t e s l i k e those c h a r a c t e r i s t i c o f c r y s t a l l i n e covalent semiconductors. C o n s e q u e n t l y , under these c i r c u m s t a n c e s t h e e l e c t r o n s and h o l e s a r e more a c c u r a t e l y v i s u a l i z e d as i o n s i n s o l u t i o n w h i c h move, h o w e v e r , by c a r r i e r h o p p i n g r a t h e r t h a n i o n i c d i f f u s i o n . On t h e o t h e r h a n d , s t a t i c d i a g o n a l d i s o r d e r s h o u l d be z e r o i n b u l k o r g a n i c c r y s t a l s , so t h a t i n e q u a l i t y ( 3 ) i s n o t n e c e s s a r i l y s a t i s f i e d and e x t e n d e d b a n d - l i k e s t a t e s a r e n o t ruled out. I n any c a s e , i f A and V a r e known, an a s s e s s m e n t o f t h e l o c a l i z a t i o n o f t h e i n j e c t e d c h a r g e s c a n be made. D i f f e r e n t c l a s s e s o f p o l y m e r s and m o l e c u l a r j s o l i d s e x h i b i t s y s t e m a t i c a l l y d i s t i n c t r a n g e s o f v a l u e s o f A and V . F o r t y p i c a l p e n d a n t - g r o u p and m o l e c u l a r l y doped p o l y m e r s , O . l e V A leV, V 8 ) . A s i m i l a r s i t u a t i o n seems t o p r e v a i l f o r m o l e c u l a r glasses, a l t h o u g h the p a r a m e t e r v a l u e s a r e n o t y e t f i r m l y e s t a b l i s h e d i n this case. For b u l k m o l e c u l a r c r y s t a l s i n the absence o f d e f e c t s

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t h e s t a t i c d i s o r d e r c o n t r i b u t i o n t o A v a n i s h e s so one e x p e c t s A = C'KT i n w h i c h c is a constant. T h i s o b s e r v a t i o n l e d to the prediction (_1) and s u b s e q u e n t o b s e r v a t i o n (_3, 2 7 , 28) o f a t r a n s i t i o n w i t h d e c r e a s i n g t e m p e r a t u r e from h o p p i n g m o t i o n b e tween l o c a l i z e d s t a t e s to band m o t i o n i n e x t e n d e d s t a t e s i n n a p h t h a l e n e and d e u t e r a t e d n a p h t h a l e n e . F i n a l l y , for segregatedstack l i n e a r c h a i n charge t r a n s f e r s a l t s ( e . g . , TTF-TCNQ) and non-saturated-backbone p o l y m e r s ( e . g . , p o l y ( s u l f u r n i t r i d e ) and p o l y a c e t y l e n e ) A < V so t h a t t r a n s p o r t v i a d e l o c a l i z e d s t a t e s i s p o s s i b l e p r o v i d e d the e l e c t r o n i c m o t i o n i s not s t r i c t l y one dimensional. I n some c a s e s , h o w e v e r , the q u a s i - o n e - d i m e n s i o n a l n a t u r e o f the m a t e r i a l s s t r o n g l y s u g g e s t s the o c c u r r e n c e o f Fermi-surface i n s t a b i l i t i e s (10-15, 29). For example, i n p o l y a c e t y l e n e P e i e r l ' s i n s t a b i l i t i e s , which manifest themselves as b o n d - a l t e r n a t i n g backbone s t r u c t u r e s i n undoped m a t e r i a l s , seem t o p r e d o m i n a t e ( L 5 , _16) • I t f u r t h e r has b e e n s u g g e s t e d (_L3, 1 4 , 3 0 ) t h a t t h e s e i n s t a b i l i t i e s g e n e r a t e r a t h e r d i f f e r e n t phenomena upon d o p i n g t h a n t h o s e w h i c h o c c u r i n " t r a d i t i o n a l " o n e - e l e c t r o n semiconduc t o r s . We c o n c l u d e , t h e r e f o r e , t h a t a l t h o u g h t h e o p t i c a l and t r a n s port properties o f o r g a n i c p o l y m e r s and m o l e c u l a r s o l i d s may a p p e a r a n a l o g o u s to t h o s e o f t y p i c a l c o v a l e n t s e m i c o n d u c t o r s , t h e m a t e r i a l s s h o u l d n o t be t h o u g h t o f as " o r g a n i c s e m i c o n d u c t o r s " i n the J i r a d i t i o n a l o n e - e l e c t r o n s e n s e . Most commonly ( i . e . , when A>> V) t h e y a r e more a c c u r a t e l y r e g a r d e d as F e r m i g l a s s e s c h a r a c t e r i z e d by l o c a l i z e d i n t r i n s i c a n d / o r e x t r i n s i c m o l e c u l a r i o n s t a t e s w i t h e n e r g i e s near the Fermi e n e r g y . Sometimes ( e . g . , i n t h e c a s e o f p o l y a c e t y l e n e ) t h e y a r e b e s t c o n s i d e r e d t o be o r g a n i c m e t a l s w h i c h have become s e m i c o n d u c t o r s b e c a u s e o f F e r m i - s u r f a c e instabilities. O c c a s i o n a l l y , as i n t h e c a s e o f u l t r a p u r e n a p h t h a l e n e and d e u t e r o n a p h t h a l e n e a t T £_100K, o r g a n i c c r y s t a l s do seem to e x h i b i t t r a d i t i o n a l o n e - e l e c t r o n e n e r g y - b a n d transport. T h i s s i t u a t i o n i s , however, the e x c e p t i o n r a t h e r than the r u l e .


f

C o n t a c t Charge Exchange An important p r a c t i c a l problem which r e q u i r e s for i t s s o l u t i o n the concepts developed i n the p r e c e e d i n g s e c t i o n i s t h a t o f c o n t a c t c h a r g e exchange b e t w e e n i n s u l a t o r s . Long v i e w e d i n terms o f the t r a d i t i o n a l s e m i c o n d u c t o r m o d e l (31^, 3 2 ) , c h a r g e i n j e c t i o n and m o t i o n i n h i g h p o l y m e r s have s u c c e s s f u l l y e l u d e d q u a n t i t a t i v e i n t e r p r e t a t i o n u n t i l t h e r e c o g n i t i o n i n r e c e n t y e a r s (_5, 6, 18, 19) t h a t the p o l y m e r i c m a t e r i a l s i n v o l v e d a r e b e s t d e s c r i b e d as Fermi glasses. Specifically, the l o c a l i z a t i o n of injected c h a r g e s p r e d i c t e d by the F e r m i - g l a s s m o d e l i s r e q u i r e d b o t h t o u n d e r s t a n d t h e good i n s u l a t i n g p r o p e r t i e s o f p o l y m e r i c m a t e r i a l s ( 3 2 ) and to i n t e r p r e t the n o n - e q u i l i b r i u m f e a t u r e s o f c o n t a c t c h a r g e exchange C 7 , 1 8 , _19, 3 3 , 3 4 ) . The b a s i c c o n s t r u c t o f the F e r m i - g l a s s m o d e l i s a s p a t i a l l y a v e r a g e d d e n s i t y o f d o n o r and a c c e p t o r s t a t e s w h i c h i s a f u n c t i o n

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of energy. T h i s s t a t e d e n s i t y e i t h e r may be c a l c u l a t e d from t h e m i c r o s c o p i c models n o t e d above ( 2 , 6 , _7> .§> 2) ^ ~ mined p h e n o m e n o l o g i c a l l y v i a p h o t o e m i s s i o n ( 8 ) or m e t a l - p o l y m e r c o n t a c t c h a r g e exchange (6, 8 , 1 8 , 19) m e a s u r e m e n t s . Once t h e d e n s i t i e s o f s t a t e a r e known f o r two i n s u l a t o r s , t h e i r steadys t a t e c o n t a c t c h a r g e exchange c a n be p r e d i c t e d (_7). A phenomenol o g i c a l v e r s i o n o f t h i s F e r m i - g l a s s m o d e l has b e e n shown t o p r e d i c t s u c c e s s f u l l y the c o n t a c t c h a r g e exchange o f v a r i o u s c o p o l y m e r s o f p o l y s t y r e n e and p o l y ( m e t h y l m e t h a c r y l a t e ) , and o f c a r b o n - b l a c k p i g m e n t e d p o l y m e r s (J)The m o d e l a l s o has b e e n u s e d to i n t e r p r e t q u a n t i t a t i v e l y p h o t o e m i s s i o n and UV a b s o r p t i o n d a t a for polystyrene and p o l y ( 2 - v i n y l p y r i d i n e ) (2, 8), and to c a l c u l a t e hopping a c t i v a t i o n energies i n p o l y ( 2 - v i n y l p y r i d i n e ) (35). C o n s e q u e n t l y , i t c o n s t i t u t e s the f i r s t s u c c e s s f u l a t t e m p t to p r o v i d e a s i m p l e , u n i f i e d t h e o r y o f o p t i c a l a b s o r p t i o n , c h a r g e i n j e c t i o n , and c h a r g e t r a n s p o r t i n h i g h p o l y m e r s .


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Abstract The extended or localized character of charges injected into solids is determined by a competition between fluctuations in the energies of these charges from one site to another and their probability of extending over more than one site. We illustrate for trinitrofluorenone and triphenylamine the fact that injected charges can exhibit widely different transport behaviors in condensed molecular systems characterized by similar geometrical and electronic structures. A model of charge localization in condensed molecular solids is developed. This model embodies distributions of site energies and of electronic overlap integrals. A procedure for estimating these distributions using spectroscopic data is indicated and applied to a variety of polymers and molecular solids. The model predicts the systematic occurrence of localized states in certain types of materials (e.g., aromatic pendant-group polymers like polystyrene) but not in others (e.g., crystalline, non-saturated-backbone polymers like polyacetylene). Fermi-surface instabilities occur, however, in linear polymer systems which exhibit extended states. A survey is given of the varied behaviors expected for localized and extended-state polymeric materials. As an example of their utility, the results of the analysis are applied to an important practical problem: the description of contact charge exchange between polymeric insulators.

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Duke, C.B., Surface Sci., 1978, 70, 674.

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Duke, C.B. and Schein, L.B., Physics Today, 1980, 33(2), 42.


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Meier, H., "Organic Semiconductors", Weinheim, 1974).

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Duke, C.B. and Fabish, T.J., Phys. Rev. Lett., 1976, 37, 1075.

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Duke, C.B., Fabish, T.J. and Paton, A., Chem. Phys. Lett., 1977, 49, 133.

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Duke, C.B. and Fabish, T.J., J. Appl. Phys., 1978, 49, 315.

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Duke, C.B., Salaneck, W.R., Fabish, T.J., Ritsko, J.J., Thomas, H.R. and Paton, A., Phys. Rev. B., 1978, 18, 5717.

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Duke, C.B., in "Tunneling in Biological Systems", B. Chance et al., eds. (Academic Press, New York, 1979), pp. 31-66.

10. Rice, M.J., Duke, C.B. and Lipari, N.O., Solid State Commun., 1975, 17, 1089. 11. Madhukar, A., Chem. Phys. Lett., 1974, 27, 606. 12. Rice, M.J., Solid State Commun., 1977, 21, 757; 1978, 25, 1083. 13. Rice, M.J., Phys. Lett., 1979, 71A, 152. 14. Su, W.P., Schrieffer, J.R. and Heeger, A.J., Phys. Rev. Lett., 1979, 42, 1678. 15. Kertesz, M., Koller, J. and Azman, A., J. Chem. Phys., 1977, 67, 1180; 1978, 68, 2779. 16. Duke, C.B., Paton, A., Salaneck, W.R., Thomas, H.R., Plummer, E.W., Heeger, A.J. and MacDiarmid, A.G., Chem. Phys. Lett., 1978, 59, 146. 17. Fabish, T.J., Saltsburg, H.M. and Hair, M.L., J. Appl. Phys., 1976, 47, 930; 1976, 47, 940. 18. Fabish, T.J. and Duke, C.B., J. Appl. Phys., 1977, 48, 4256; in "Polymer Surfaces", D.T. Clark and W.J. Feast, eds. (Wiley-Interscience Chichester, 1978), pp. 109-119. 19. Duke, C.B., J. Vac. Sci. Technol., 1978, 15, 157. 20. Salaneck, W.R., Phys. Rev. Lett., 1978, 40, 60. 21. Hoegl, H., J. Phys. Chem., 1965, 69, 755.


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22. Gill, W.D., J. Appl. Phys., 1972, 43, 5033; in "Photoconductivity and Related Phenomena", J. Mort and D. Pai, eds. (Elsevier, Amsterdam, 1976), pp. 303-334. 23. Pfister, G., Phys. Rev. B., 1977, 16, 3676. 24. Duke, C.B., in "Synthesis and Properties of Low-Dimensional Materials", J.E. Miller and A.J. Epstein, eds. (New York Academy of Sciences, New York, 1978), pp. 166-178. Photon, Electron, and Ion Probes of Polymer Structure and Properties Downloaded from pubs.acs.org by UNIV OF ARIZONA on 06/08/17. For personal use only.

25. Anderson, P.W., Rev. Mod. Phys., 1978, 50, 191. 26. Weaire, D. and Srivastave, V., in "Amorphous and Liquid Semiconductors", W.E. Spear, Ed. (G.G. Stevenson, Ltd., Dundee, 1977), pp. 286-290. 27. Schein, L.B., Duke, C.B. and McGhie, A.R., Phys. Rev. Lett., 1978, 40, 197. 28. Schein, L.B. and McGhie, A.R., Phys. Rev. B., 1979, 20, 1631. 29. Chaikin, P.M., in "Synthesis and Properties of Low Dimensional Materials", J.S. Miller and A.J. Epstein, eds. (New York Academy of Sciences, New York, 1978), pp. 128-144. 30. Epstein, A.J. and Miller, J.S., Solid State Commun., 1978, 27, 325. 31. Harper, W.R., "Contact and Frictional Electrification", (Clarendon Press, Oxford, 1967), pp. 185-222; 245-268. 32. Wintle, H.F., Trans. IEEE., 1977, EI-12, 97. 33. Ruckdeschel, F.R. and Hunter, L.P., J. Appl. Phys., 1975, 46, 4416. 34. Lowell, J., J. Phys. D: Appl. Phys., 1976, 9, 1571; 1979, 12, 1541. 35. Duke, C.B. and Meyer, R.J., Bull Am. Phys. Soc., 1979, 24, 309; Phys. Rev., (submitted). RECEIVED December 22,

1980.

Localization of Electronic States in Polymers and Molecular Solids

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