Photon, Electron, and Ion Probes of Polymer Structure and Properties

carrier bandwidths may well be much larger than phonon bandwidths ... the exciton site shift or charge carrier polarization energy, in ..... crease in...
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4 Exciton States and Exciton Transport in Molecular Crystals R. SILBEY Department of Chemistry, Massachusetts Institute of Technology, Cambridge, M A 02139 Downloaded by CORNELL UNIV on September 6, 2016 | http://pubs.acs.org Publication Date: August 13, 1981 | doi: 10.1021/bk-1981-0162.ch004

R. MUNN Department of Chemistry, UMIST, Manchester, M60 1QD, United Kingdom

E l e c t r o n i c transport i n molecular crystals is a complicated phenomenon f o r two r e a s o n s . The f i r s t i s t h e c o m p l e x i t y o f t h e vibrations. I n m o l e c u l a r c r y s t a l s t h e phonons i n c l u d e m o l e c u l a r modes and t r a n s l a t i o n a l and v i b r a t i o n a l l a t t i c e modes, e a c h w i t h i t s c h a r a c t e r i s t i c f r e q u e n c y , b a n d w i d t h , and mechanism o f e l e c tron-phonon c o u p l i n g , again with d i f f e r e n t strengths. The s e c o n d r e a s o n why t r a n s p o r t i s c o m p l i c a t e d i s t h e a b s e n c e o f any c l e a r o r d e r i n g o f the d i f f e r e n t p a r a m e t e r s . The e l e c t r o n i c b a n d w i d t h s may r a n g e from b e i n g l a r g e r t h a n most phonon f r e q u e n c i e s and bandwidths ( f o r charge c a r r i e r s ) to being s m a l l e r than e i t h e r (for t r i p l e t excitons). The e l e c t r o n - p h o n o n c o u p l i n g e n e r g y may b e l a r g e o r s m a l l compared w i t h e l e c t r o n i c and v i b r a t i o n a l e n e r g i e s , and i f l a r g e may c a u s e t h e e l e c t r o n i c b a n d w i d t h t o n a r r o w r a p i d l y with i n c r e a s i n g temperature, so c h a n g i n g the p a r a m e t e r ordering. E a r l y t r a n s p o r t t h e o r i e s were r e s t r i c t e d i n s c o p e and d i d not r e f l e c t these c o m p l i c a t i o n s . As w e l l as t r e a t i n g o n l y a s i n g l e phonon b a n d , f o r s i m p l i c i t y , t h e t h e o r i e s w o u l d assume a p a r t i c u l a r p a r a m e t e r o r d e r i n g and a t r a n s p o r t mechanism ( 1 - 1 1 ) . T r a n s p o r t has been d e s c r i b e d by the m e a n - s q u a r e p a r t i c l e d i s p l a c e m e n t as a f u n c t i o n o f t i m e , so p e r m i t t i n g s t u d y o f the c l o t h i n g o f the p a r t i c l e by phonons and the d e v e l o p m e n t o f diffusive motion. These a p p r o a c h e s a l s o r e v e a l t h e change from h o p p i n g to band m o t i o n as t h e t e m p e r a t u r e i s l o w e r e d i n s y s t e m s with strong electron-phonon coupling. H o w e v e r , t h e a v a i l a b l e t h e o r i e s have s t i l l b e e n r e s t r i c t e d to s e l e c t e d parameter o r d e r i n g s . I n p a r t i c u l a r , i t has been assumed i n t h e o r i e s o f e x c i t o n t r a n s p o r t t h a t t h e e x c i t o n b a n d w i d t h i s n a r r o w e r t h a n t h e phonon b a n d w i d t h , and t h i s a s s u m p t i o n has been c a r r i e d over to t h e o r i e s o f c a r r i e r t r a n s p o r t . In fact, c a r r i e r b a n d w i d t h s may w e l l be much l a r g e r t h a n phonon b a n d w i d t h s a t low t e m p e r a t u r e s , b e c o m i n g s m a l l e r t h a n phonon b a n d w i d t h s as the temperature is raised, o w i n g to p o l a r o n band n a r r o w i n g

0097-6156/81/0162-0045$05.00/0 © 1981 American Chemical Society Dwight et al.; Photon, Electron, and Ion Probes of Polymer Structure and Properties ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

PHOTON, ELECTRON, AND ION PROBES

46

effects. T h e r e i s t h e r e f o r e a need f o r a t h e o r y w h i c h c a n t r e a t b o t h w i d e and n a r r o w e l e c t r o n i c bands i n m o l e c u l a r c r y s t a l s w i t h e i t h e r s t r o n g o r weak e l e c t r o n - p h o n o n c o u p l i n g . We have a l r e a d y d i s c u s s e d how s u c h a t h e o r y may be d e v e l o p e d , (_12) and h e r e we d e s c r i b e the t h e o r y i n d e t a i l . Principles

take

We s t a r t h=l:

from

the

f o l l o w i n g m o d e l H a m i l t o n i a n , i n w h i c h we

+

+

H =£ e a a + Z J a a n n n n,mnmn n + £ GO ( b b q q q q

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+

+ 1/2)

+ N"

1 / 2

n

+

+

£ g oo ( b + b ) a a nq q q q q n n 6

(2.1)

+

Here the o p e r a t o r s a and a c r e a t e and d e s t r o y an e l e c t r o n i c e x c i t a t i o n ( e x c i t o n o r c h a r g e c a r r i e r ) o f e n e r g y u) a t s i t e n , w h i l e the o p e r a t o r s b and b c r e a t e and d e s t r o y a phonon o f f r e q u e n c y w and w a v e v e c t o r j | . The q u a n t i t y J =J is the t r a n s f e r i n t e g r a l b e t w e e n s i t e s n and m. The l a s t t e r m i s t h e e l e c t r o n - p h o n o n c o u p l i n g t e r m , ^ o f ^ m a g n i t u d e d e t e r m i n e d by the d i m e n s i o n a l e s p a r a m e t e r s g = g e ^* n , where £ i s the p o s i t i o n v e c t o r o f s i t e n . The c o u p r i n f i s l o c a l ( d i a g o n a l ) i n e x c i t a t i o n site. Such a c o u p l i n g c a n a r i s e from m o l e c u l a r d i s t o r t i o n i n t h e e x c i t e d o r i o n i z e d s t a t e , i n w h i c h c a s e g and u) a r e e x p e c t e d t o be a l m o s t i n d e p e n d e n t o f q , o r from v i b r a t i o n a l ? f l u c t u a t i o n s i n the e x c i t o n s i t e s h i f t or charge c a r r i e r p o l a r i z a t i o n e n e r g y , i n w h i c h c a s e g and u) may v a r y more m a r k e d l y w i t h q . I f the eflectror? phonon c o u p l i n g i s weak (| g |, changing e v e n t u a l l y to T when k _ T » w . /t

A c o u s t i c Phonons. We assume t h a t a c o u s t i c phonons c a n be a d e q u a t e l y d e s c r i b e d by a Debye s p e c t r u m c u t - o f f f r e q u e n c y and the e l e c t r o n - p h o n o n c o u p l i n g i s g i v e n by t h e deformation p o t e n t i a l approximation g

q

=

The s c a t t e r i n g

(3.8) rate

is

then

r £ = 2 7 r A N ~ y > { ( n +1)6(E. -E,+w ) + kk q q k-q k q a

n

q

1

6(E, -E,-co )} k-q k q

(3.9)

I n some l i m i t s , r e s u l t s c a n be o b t a i n e d more d i r e c t l y . In the conventional semiconductor limit B>>k T»w^ we have n % k T / ( A ) >>1, and | E _ - E ^ | »/k

T(m*) ]V

(3.11)

5

15

5/2

49

3/2

so t h a t D'vB /A^B i f we assiime A i s p r o p o r t i o n a l t o B. Eq. (3.11) g i v e s the standard T temperature dependence f o r the mobility (13). In molecular crystals one may a l s o require the limit k T>>u) >>B. Then i n E q . ( 3 . 9 ) n o n z e r o c o n t r i b u t i o n s a r i s e o n l y f o r f r e q u e n c i e s u) ^ ( ^ B a ) , we o b t a i n Z

k

N p

(o))=3w t

l

3 /u) .

Taking

D/a =a)^/127rAk T, 2

(3.13)

B

3

similarly

D

3

so t h a t D ^ a ^ / A ^ o ^ / B ^ w i t h t h e p r e v i o u s a s s u m p t i o n A B) and t h e m o b i l i t y v a r i e s as T . The n a r r o w e x c i t a t i o n b a n d i n t h i s l i m i t g r e a t l y r e d u c e s t h e number o f a l l o w e d o n e - p h o n o n s c a t t e r i n g p r o c e s s e s , so t h a t t h e d i f f u s i o n c o e f f i c i e n t may b e l a r g e , a s t h e r a t i o ^ /B i n d i c a t e s . Finally, i n t h e l i m i t k T>>B>>u) , t h e s c a t t e r i n g r a t e s a r e g i v e n by E q . ( 3 . 1 0 ) b u t t h e t h e r m a l a v e r a g e s i n D hay^e t o b e ^ t a k e n over a narrow excitation band. We take v, ^(%Ba) and N ( E M / B , obtaining 0C

D

B

e

x

D

k

D/a =B /(167rAk 2

3

15

T)

(3.14) -2

so t h a t t h e m o b i l i t y v a r i e s as T . T h i s r e s u l t accords with the n a r r o w e x c i t a t i o n band t r e a t m e n t s o f G l a r u m ( 1 4 ) and F r i e d m a n (15). _

Transformed Coupling Transformation. The t r a n s f o r m a t i o n o f t h e H a m i l t o n i a n ( 2 . 1 ) w h i c h y i e l d s a weak r e s i d u a l e x c i t a t i o n - p h o n o n c o u p l i n g e v e n when t h e g a r e l a r g e h a s b e e n d i s c u s s e d s e v e r a l t i m e s ( 4 , T_ 16^> 1 7 ) . I t p r 8 d u c e s a u n i f o r m s h i f t i n t h e e x c i t a t i o n e n e r g y l e v e l s and a d i s p l a c e m e n t i n t h e e q u i l i b r i u m p o s i t i o n o f t h e phonons c o r r e s p o n d i n g to t h e f o r m a t i o n o f a p o l a r o n . Since the t r a n s f e r interactions J compete w i t h t h i s t e n d e n c y t o f o r m a l o c a l i z e d s t a t e , t h e optimum t r a n s f o r m a t i o n s h o u l d be d e t e r m i n e d v a r i a t i o n 9

Dwight et al.; Photon, Electron, and Ion Probes of Polymer Structure and Properties ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

50

PHOTON, ELECTRON, AND ION PROBES

a l l y ^6, J 7 ) . H o w e v e r , f o r p r e s e n t p u r p o s e s we use t h e full c l o t h i n g t r a n s f o r m a t i o n w h i c h i s e x a c t f o r J=0 and y i e l d s t h e c o r r e c t u n t r a n s f o r m e d r e s u l t s f o r l a r g e J and weak c o u p l i n g . The r e s u l t s a r e q u a l i t a t i v e l y s i m i l a r t o t h o s e w h i c h w o u l d be o b t a i n ed w i t h the f u l l v a r i a t i o n a l t r a n s f o r m a t i o n , b u t a r e s i m p l i f i e d by the a b s e n c e o f the t e m p e r a t u r e - d e p e n d e n t v a r i a t i o n a l p a r a meters . After the transformation, the excitation part of the Hamiltonian (2.2) is 2

H

- * ( - r t c * |g | z \e*'%n) k q n i s a l a t t i c e v e c t o r and

e x

e

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where R ^n

+

aja

(4.1)

k


h n + h , n n+h n

(4.2)

j l U

J

= exp [ N

1 / 2

n

+

E(g )*(b -b )] q q -q

Y

n

(4.3)

q


B. The s e c o n d t e r m i n E q . ( 4 . 6 ) r e p r e s e n t s t h e h o p p i n g c o n t r i b u t i o n , which i n c r e a s e s w i t h i n c r e a s i n g temperature through I g ( y ) and t h e e x p o n e n t i a l f a c t o r . H o w e v e r , once B becomes much l e s s t h a n f,^ t h e e x p o n e n t i a l f a c t o r becomes c o n s t a n t , and t h e d e c r e a s e i n B o u t w e i g h s t h e i n c r e a s e i n I ( y ) , so t h a t e v e n t u a l l y the second term a l s o d e c r e a s e s w i t h i n c r e a s i n g t e m p e r a t u r e . The o c c u r r e n c e o f t h i s d e c r e a s e i s d i s c u s s e d i n more d e t a i l b e l o w . I n the l i m i t f>>B assumed i n most p r e v i o u s w o r k , we f i n d , D/a

2 z

=

f j—i (2^) [l (y)-l]

^[I (y)-1]B 2 4f

2

0

+

J s

0

(

4

-

?

J

T h i s i s e s s e n t i a l l y o f t h e form d e r i v e d e a r l i e r ( 4 ) . We n o t e t h a t i n E q . ( 4 . 7 ) f, o r e q u i v a l e n t l y A, c a n n o t t e n d t o z e r o ( w h i c h w o u l d make D i n f i n i t e ) , b e c a u s e t h e e q u a t i o n i s v a l i d o n l y f o r f>>B. On t h e o t h e r h a n d , J( c a n t e n d t o z e r g , i n w h i c h c a s e o n l y t h e band t e r m r e m a i n s . A s t h e v e l o c i t i e s v i n E q . ( 2 . 3 ) tend to z e r o , so do t h e s c a t t e r i n g r a t e s £ • b o t h f a c t o r s v a r y as % , so t h a t t h e i r r a t i o t e n d s to a c o n s t a n t . Le >e written

as 2

D/a

2

= (

B

Q 8Ag 2 2

(4.8)

2

sinh | ^ exp(-2g tanh

2

I n the

opposite

D/a

2

limit

B

- L _ ^[l (y)-l] x

Q

> >

T

>

we o b t a i n t h e

+ 4

new

[T(y)-1] °

result

Be-

B 2

«

2 / 8

H e r e f does n o t o c c u r and so c a n t e n d t o z e r o , b e i n g a l r e a d y assumed much s m a l l e r t h a n B . I f t h e n B t e n d s to z e r o , the d i f f u s i o n c o e f f i c i e n t v a n i s h e s , as e x p e c t e d . V i b r a t i o n a l d i s p e r s i o n ( n o n z e r o A ) i s t h e r e f o r e not e s s e n t i a l to o b t a i n d i f f u s i v e m o t i o n and f i n i t e t r a n s p o r t c o e f f i c i e n t s i n t h i s l i m i t , b u t to show t h i s has r e q u i r e d a s u f f i c i e n t l y g e n e r a l t h e o r y . Discussion I n t h i s p a p e r we have shown how e l e c t r o n i c t r a n s p o r t in molecular crystals can be t r e a t e d much more g e n e r a l l y than hitherto. The p r e s e n t t r e a t m e n t a v o i d s ( i ) t h e a s s u m p t i o n t h a t v i b r a t i o n a l r e l a x a t i o n i s f a s t compared w i t h e x c i t a t i o n t r a n s f e r

Dwight et al.; Photon, Electron, and Ion Probes of Polymer Structure and Properties ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

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52

PHOTON, ELECTRON, AND ION PROBES

(f>>B, as u s u a l l y assumed i n p o l a r o n and e x c i t o n transport t h e o r i e s ) , and ( i i ) the a s s u m p t i o n t h a t t h e e x c i t a t i o n - p h o n o n c o u p l i n g i s weak (gtjon