Chapter 50 Band-Structure Calculations on Polymeric Chains
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William J. Welsh Department of Chemistry, University of Missouri—St. Louis, St. Louis, MO 63121
E l e c t r o n i c band structures were calculated for several polymeric chains s t r u c t u r a l l y analogous to polyacetylene (-CH=CH) and carbyne (-C=C). The present c a l c u l a t i o n s use the Extended Hückel molecular orbital theory w i t h i n the t i g h t binding approximation, and values of the calculated band gaps Ε and band widths BW were used to assess the p o t e n t i a l applicability of these materials as e l e c trical semiconductors. Substitution of F or C l atoms for H atoms i n polyacetylene tended to decrease both the Ε and BW values ( r e l a t i v e to that for polyacetylene). Rotation about the backbone bonds in the chains away from the planar conformations l e d to sharp increases i n Ε and decreases in BW. Substitution of -SiH or -Si (CH ) groups for Η in polyacetylene i n v a r i b l y l e d to an increase in Ε and a decrease i n BW, as was generally the case for i n s e r t i o n of Ύ' (where Y = O , S, N H , CH , SiH ) in carbyne to give [ - C = C - Y - C = C ] . The degree of bondlength a l t e r n a t i o n appears to play a major role i n determining the electrical conductivity of a conjugated polymer, with values of Ε decreasing (favoring conductivity) as bond-length a l t e r n a t i o n decreases. g
g
g
3
3 3
g
2
2
g
P o l y a c e t y l e n e , ( - C H - C H ) , the s i m p l e s t o r g a n i c polymer w i t h a f u l l y c o n j u g a t e d backbone, has g e n e r a t e d c o n s i d e r a b l e i n t e r e s t due t o i t s unusual e l e c t r o n i c p r o p e r t i e s . S p e c i f i c a l l y , through s e l e c t i v e d o p i n g o f the polymer i t s e l e c t r i c a l c o n d u c t i v i t y can be made t o v a r y many o r d e r s o f magnitude, from i n s u l a t o r t o semiconductor ( 1 3 ) . The s t r u c t u r e o f the polymer c h a i n appears t o be one o f t h e k e y d e t e r m i n a n t s f o r the e l e c t r o n i c p r o p e r t i e s o f p o l y m e r / d o p a n t systems ( 1 - 3 ) . Recent s t r u c t u r a l e v i d e n c e and t h e o r e t i c a l c a l c u l a t i o n s (1-3) s u g g e s t a p l a n a r backbone s t r u c t u r e f o r c i s and t r a n s forms o f ( - C H - C H ) . However, many a s p e c t s o f the s t r u c t u r e and the c o n f i g u r a t i o n a l c h a r a c t e r i s t i c s o f (-CH-CH) a r e n o t w e l l - d e f i n e d due t o i t s i n t r a c t a b i l i t y , i t s i n s t a b i l i t y t o o x i d a t i o n , and i t s i n s o l u b i l i t y i n most s o l v e n t s . Many o t h e r p o l y m e r i c systems have t h u s been proposed or a c t u a l l y i n v e s t i g a t e d w i t h regard t o t h e i r p o t e n t i a l as c o n d u c t o r s o r s e m i - c o n d u c t o r s ( 1 - 3 ) , among t h e s e the h a l o g e n substituted polyacetylenes (4). F o r example, Z e i g l e r has r e c e n t l y s y n t h e s i z e d the p e r f l u o r o p o l y a c e t y l e n e ( - C F - C F ) ( 4 ) . I t appears t h a t e l e c t r i c a l c o n d u c t i v i t y i s p a r t i c u l a r l y
0097-6156/87/0346-0600S06.00/0 © 1987 A m e r i c a n C h e m i c a l Society
Bowden and Turner; Polymers for High Technology ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
50.
WELSH
Band-Structure
Calculations
on Polymeric
601
Chains
s e n s i t i v e t o the degree o f c o n j u g a t i o n and p l a n a r i t y a l o n g the c h a i n backbone (1-3,5,6). I n the case o f s u b s t i t u t e d p o l y a c e t y l e n e s i n p a r t i c u l a r , i t i s c r u c i a l for e f f e c t i v e conductivity that the s u b s t i t u e n t ' s s t e r i c b u l k n o t cause a p p r e c i a b l e d e v i a t i o n s from p l a n a r i t y i n a n attempt t o reduce s t e r i c c o n f l i c t s . W i t h r e g a r d t o the h a l o g e n - s u b s t i t u t e d p o l y a c e t y l e n e s , t h e f l u o r i n e atom i s j u s t s m a l l enough U » 1.30 Â) t o render t h e F...F i n t e r a c t i o n s a t t r a c t i v e even f o r the p l a n a r c h a i n i n w h i c h case t h e four-bond F...F i n t e r a t o m i c d i s t a n c e i s c l o s e s t (2.60 À). However, w i t h s u b s t i t u t i o n o f c h l o r i n e ( r -w. » 1.80 Â) s t e r i c c o n f l i c t s between pendant c h l o r i n e atoms w i l l render the p l a n a r c o n f o r m a t i o n h i g h l y r e p u l s i v e , and t h i s e f f e c t would become more severe i n the case o f B r and I substitution.
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v d w
I n t h i s s t u d y , quantum mechanical methods were u s e d t o c a l c u l a t e t h e e l e c t r o n i c band gaps Ε and band w i d t h s BW o f t h e (-CF-CF) c h a i n and compared w i t h those s i m i l a r l y c a l c u l a t e d f o r (-CH-CH) i t s e l f . Other h a l o g e n - s u b s t i t u t e d p o l y a c e t y l e n e s c o n s i d e r e d a r e t h e c h a i n s (-CF-CH) and (-CF-CF-CH-CH) and t h e i r c h l o r i n a t e d a n a l o g s . C a l c u l a t i o n s were c a r r i e d o u t a s a f u n c t i o n o f r o t a t i o n about t h e s i n g l e bonds a l o n g the c h a i n backbone i n o r d e r t o a s s e s s the depen dence o f c o n d u c t i v i t y on c h a i n p l a n a r i t y . L i k e w i s e , t h e s e n s i t i v i t y o f t h e c a l c u l a t e d band gaps t o s m a l l changes i n s t r u c t u r e ( i . e . , bond a n g l e s , bond l e n g t h s ) h a s been i n v e s t i g a t e d . Depending on t h e s i z e o f t h e s u b s t i t u e n t s and t h e i r sequence, the e x t e n t o f s t e r i c i n t e r f e r e n c e between the 1- and 3 - s u b s t i t u e n t s ( F i g u r e 1) and between t h e 2- and 4 - s u b s t i t u e n t s i n the p l a n a r t r a n s c o n f o r m a t i o n may v a r y c o n s i d e r a b l y . I n the p l a n a r c i s c o n f o r m a t i o n , t h e c r i t i c a l i n t e r a c t i o n s i n terms o f p o s s i b l e s t e r i c c o n f l i c t s a r e between t h e 2...3 and 1...4 s u b s t i t u e n t s , where t h e l a t t e r i n t e r a c t i o n becomes p a r t i c u l a r l y s i g n i f i c a n t due t o the d e c r e a s e d i n t e r atomic s e p a r a t i o n . F o r a g i v e n halogen-atom s u b s t i t u e n t X, s t e r i c i n t e r f e r e n c e s w i l l g e n e r a l l y i n c r e a s e i n the o r d e r (-CH-CH-CX-CX), (-CH-CX), (-CX-CX). Comparison o f t h e s i z e o f t h e s u b s t i t u e n t atoms H ( r
1
β
2
0A
)
F ( r
l e 3
A
)
0 1 1 ( 1C l ( r
l e 8
A
)
vdW " ' vdw " ° vdW " ° 9 9 t h a t s t e r i c e f f e c t s w i l l be much more s i g n i f i c a n t i n t h e c h l o r i n a t e d derivatives. S
U
E
S
T
S
A n o t h e r s t r u c t u r a l l y analogous group o f s u b s t i t u t e d p o l y a c e t y l e n e s a r e o f t h e type [-CH-CX1-CH-CH-CX2-CH] w i t h X I and X2 chosen from among H, C H , S i H ^ and S i ( C H ) . The c h o i c e o f S i ( C H ) as a s u b s t i t u e n t has been suggested (4) a s a means t o improve t h e s o l u b i l i t y and hence the p r o c e s s i b i l i t y o f the o t h e r w i s e i n t r a c t a b l e polyacetylene. 3
3
3
3
3
O t h e r c o n j u g a t e d p o l y m e r i c systems have been suggested a s p o s s i b l e e l e c t r i c a l c o n d u c t o r s (1-3,6,7), i n c l u d i n g (-C«C-X-C*C-) where X may be a group IV, V o r V I element. Undoped o r g a n o s i l i c o n p o l y y n e s have been shown t o p o s s e s s r e s i s t i v i t i e s t h a t c l a s s i f y them a s o r g a n i c semiconductors. Consequently, band s t r u c t u r e c a l c u l a t i o n s on c h a i n s o f t h e type (-C*C-X-C«C-) a r e i n c l u d e d h e r e f o r t h e c a s e s X - O, S, NH, CH- and S i H - . F o r comparison, c a l c u l a t i o n s have a l s o been c a r r i e d o u t on carbyne (-C»C) t o a s s e s s the e f f e c t o f t h e 'Y' atom o r group on t h e o t h e r w i s e f u l l y c o n j u g a t e d system.
Bowden and Turner; Polymers for High Technology ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
POLYMERS FOR HIGH T E C H N O L O G Y
602 Theory
For any m o l e c u l e , i n c l u d i n g polymers, t h e LCAO a p p r o x i m a t i o n and B l o c h ' s theorem can be used t o d e s c r i b e t h e d e l o c a l i z e d c r y s t a l l i n e o r b i t a l s Ψ (10 as a p e r i o d i c c o m b i n a t i o n o f f u n c t i o n s c e n t e r e d a t t h e a t o m i c n u c l e i . F o r a one-dimensional system i n w h i c h N . - l c e l l s ( r e p e a t u n i t s ) i n t e r a c t w i t h t h e r e f e r e n c e c e l l and f o r a b a s i s s e t o f l e n g t h ω d e s c r i b i n g t h e wave f u n c t i o n w i t h i n a g i v e n c e l l , t h e n t h c r y s t a l o r b i t a l Ψ (10 i s d e f i n e d a s (8-11) η
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η
where {Φ } i s t h e s e t o f B l o c h b a s i s f u n c t i o n s
1
(Ν -1)/2 Ί
^
lr- " -T7Tl Ji—(Ν 1)/2 (k
ω
Σ
1
e x p ( i k . R . ) C ( k ) Y (r-R.) μ-1 3 ημ μ n /
N
Γ
(4)
where C ( k ) i s t h e e x p a n s i o n c o e f f i c i e n t o f t h e l i n e a r c o m b i n a t i o n . The b a s i f i f u n c t i o n s χ a r e e x p o n e n t i a l f u n c t i o n s o f t h e S l a t e r form. The p r e s e n t one-dimensional c a l c u l a t i o n s i n c l u d e d a l l t h e v a l e n c e a t o m i c o r b i t a l s o f t h e H, C, N, and F atoms b u t f o r a l l o t h e r atoms o n l y s and ρ o r b i t a l s c o u l d be c o n s i d e r e d . U s i n g t h e extended Huckel a p p r o x i m a t i o n , we o b t a i n t h e c o r r e s p o n d i n g e i g e n v a l u e s Ε ( k ) and c o e f f i c i e n t s C ( k ) from t h e e i g e n value equation ^
H(k)C (k) - S(k)C (k)E (k) n
n
n
(5)
where H(k) and S ( k ) , a r e r e s p e c t i v e l y , t h e H a m i l t o n i a n and o v e r l a p m a t r i c e s between B l o c h o r b i t a l s d e f i n e d a s
Bowden and Turner; Polymers for High Technology ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
50.
WELSH
Band-Structure
V
k
)
Calculations
k
•
