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IMPURITY-INDUCED DEFECT STATES IN POLYACETYLENE
D. Baeriswyl
To cite this version:
D. Baeriswyl. IMPURITY-INDUCED DEFECT STATES IN POLYACETYLENE. Journal de
Physique Colloques, 1983, 44 (C3), pp.C3-381-C3-386. �10.1051/jphyscol:1983377�. �jpa-00223039�
JOURNAL DE PHYSIQUE
Colloque C3, supplément au n°6, Tome 44, juin 1983 page C3-381
IMPURITY-INDUCED DEFECT STATES IN POLYACETYLENE
D. Baeriswyl
Seminar fiir theoretisehe Physik, ETH -Honggerberg, CH-809'i Zttrioh, Switzerland
Résumé - Les e f f e t s d ' impuretés sur l a s t r u c t u r e é l e c t r o n i q u e et l e s d i s t o r s i o n s du réseau sont étudiés en u t i l i s a n t 1 ' Hamiltonien SSH et un p o t e n t i e l d'impureté agissant sur un seul p o i n t du réseau. Dans une chaine dimérisée les impuretés produisent des é t a t s é l e c t r o n i q u e s l o c a l i s é s dont l e s niveaux d ' é n e r g i e tendent vers l e centre du gap pour des valeurs de p o t e n t i e l tendant vers l ' i n f i n i . Simultanément, l ' e x t e n s i o n de ces é t a t s s'approche de c e l l e du s o l i t o n . Dans l a l i m i t e d é c o u p l a g e f a i b l e , l e r é s u l t a t obtenu pour l e modèle d i s c r e t est i d e n t i q u e "a c e l u i d é r i v é pour l e modèle c o n t i n u . L ' a b s o r p t i o n optique engendrée par une impureté se montre plus f a i b l e d'un f a c t e u r 4 que c e l l e amenée par un s o l i t o n . Les m o d i f i c a t i o n s de l a s t r u c t u r e é l e c t r o n i q u e produisent d ' i m p o r t a n t s e f f e t s de r e l a x a t i o n dans l e réseau c r i s t a l l i n d'où on c o n c l u t que les impuretés a f f e c t e n t et l e s é t a t s électroniques e t l e modèle des déplacements du réseau.
A b s t r a c t - The e f f e c t o f i m p u r i t i e s on both the e l e c t r o n i c s t r u c t u r e and the l a t t i c e d i s t o r t i o n s i n polyacetylene i s studied using the SSH
Hamiltonian and an i m p u r i t y p o t e n t i a l a c t i n g on a s i n g l e s i t e . Assuming an i d e a l l y dimerized chain i t i s shown t h a t the i m p u r i t i e s lead to l o c a l i z e d e l e c t r o n i c states w i t h energy l e v e l s tending to mid-gap and extents
approaching the c h a r a c t e r i s t i c s o l i t o n l e n g t h w i t h i n c r e a s i n g s t r e n g t h o f the i m p u r i t y p o t e n t i a l . Both the d i s c r e t e and the continuum model y i e l d the same r e s u l t i n the weak-coupling l i m i t . The i m p u r i t y - i n d u c e d o p t i c a l absorption i s c a l c u l a t e d and found to be weaker by a f a c t o r o f about 4 than the s o l i t o n - i n d u c e d a b s o r p t i o n . F i n a l l y the m o d i f i c a t i o n s i n the e l e c t r o n i c s t r u c t u r e are shown t o lead t o strong l a t t i c e r e l a x a t i o n , and i t i s concluded t h a t the i m p u r i t i e s a f f e c t both the e l e c t r o n i c states and the displacement p a t t e r n o f the l a t t i c e .
1 . I n t r o d u c t i o n
Much o f the excitement about the unusual e l e c t r i c a l , magnetic and o p t i c a l p r o p e r t i e s o f polyacetylene arose from the suggestion t h a t a simple one-dimensional model would o f f e r a framework f o r a s e m i - q u a n t i t a t i v e explanation o f experiments. In p a r t i c u l a r the proposed s o l i t o n i c s t a t e w i t h i t s p e c u l i a r spin-charge r e l a t i o n s h i p s has
s t i m u l a t e d both theory and experiment, as i s well documented by a number o f c o n t r i b u t i o n s w i t h i n these proceedings. There a r e , however, a number o f e f f e c t s which w i l l appreciably modify i f not destroy the simple s o l i t o n concept such as quantum f l u c t u a t i o n s which y i e l d an important c o r r e c t i o n to the s o l i t o n energy / l / , Coulomb i n t e r a c t i o n s which tend to d e s t a b i l i z e the charged s o l i t o n / 2 , 3 / and i n t e r c h a i n coupling leading to confinement o f s o l i t o n - a n t i s o l i t o n pairs / 4 / . Here we w i l l concentrate on the e f f e c t of i m p u r i t i e s which are assumed to be represented by p o t e n t i a l s a c t i n g on s i n g l e s i t e s o n l y . I t is f i r s t shown t h a t , n e g l e c t i n g l a t t i c e r e l a x a t i o n , i m p u r i t i e s produce l o c a l i z e d states i n the gap (Sections 2 and 3) leading to a d d i t i o n a l o p t i c a l absorption below the threshold f o r
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983377
C3-382 JOURNAL DE PHYSIQUE
i n t e r b a n d t r a n s i t i o n s ( S e c t i o n 4 ) . F i n a l l y i t i s argued t h a t t h e l o c a l changes i n t h e e l e c t r o n i c s t r u c t u r e w i l l i n d u c e s t r o n g l a t t i c e d i s t o r t i o n s i n t h e neighbourhood o f i m p u r i t i e s , and i t i s concluded t h a t i m p u r i t i e s a c t as n u c l e a t i o n c e n t e r s f o r l a t t i c e d e f e c t s ( S e c t i o n 5).
2. I m p u r i t y s t a t e s i n t h e d i s c r e t e case
The H a m i l t o n i a n i s chosen as H=Ho+V where Ho i s t h e SSH H a m i l t o n i a n / 5 /
+
=
f i + I
( ~ ; + l s 'ns + 'ns ' n t l s ) +hKP:
0 (1
and V t h e i m p u r i t y p o t e n t i a l
c d e n o t i n g t h e a n n i h i l a t i o n o p e r a t o r f o r a 7 - e l e c t r o n i n t h e Wannier s t a t e In>
aR8 w i t h s p i n p r o j e c t i o n s. For t h e p a r t i c u l a r case where V d e s c r i b e s t h e
h y b r i d i z a t i o n o f i m p u r i t y and carbon o r b i t a l s a t l a t t i c e s i t e n=o t h e p o t e n f i a l a c t s o n l y on a s i n g i e s i t e , Vn= V 6
o n,o' I n t h i s case t h e p r o p a g a t o r G(z)=(z-H) i s found t o be
where G ( z ) = ( z - H ) - I . L a b e l l i n g t h e e i g e n s t a t e s o f H by X, t h e l o c a l d e n s i t y o f s t a t e s ?s g i v e n By
t h u s y i e l d i n g t h e m o d u l i j<n
l 2
and e n e r g i e s E o f t h e e i g e n s t a t e s ofnH. A p p l y i n g t h i s scheme t o t h e p e r f e c t l y d i m e r i z e d c h a i n whebe t = t + 2a(-1) u one f i n d s , besides t h e extended s t a t e s o f v a l e n c e and c o f i d i z t i o n Band, two l o ? a l i z e d e i g e n s t a t e s a t e n e r g i e s E, =+
s i g n (Vo) 1 W +1
whereand A =4au i s t h e usual gap parameter. The c o r r e s p o n d i n g m o d u l i a r e found t o decre3se aP
I ( L
- a ) / ( c + + a ) 1 2 where a i s t h e l a t t i c e c o n s t a n t and 5, =a [ ( 4 t ~ - ~ ~ ) l ( n ~ - ~ ~ ) ] d e s c r i b e s t h e e x t e n t o f t h e w a v e f u n c t i o n around t h e i m p u r i t y s i t e . Far w e i k Coupling, ao<<2to, t h e e q u a t i o n s f o r t h e bound s t a t e i n t h e gap a r e s i m p l i f i e d as f o l l o w s ,
where 5 = 2 t a / n i s t h e s o l i t o n e x t e n t o f t h e continuum t h e o r y . I n t h e l i m i t o f an i n f i a i t e i f l p u r ? t y s t r e n g t h b o t h t h e energy and t h e e x t e n t o f t h e bound s t a t e t e n d t o t h e c o r r e s p o n d i n g v a l u e s o f t h e s o l i t o n i c s t a t e .
3. Continuum l i m i t
The continuum v e r s i o n o f H has been g i v e n b y v a r i o u s a u t h o r s / 6 , 7 / . Proceeding i n t h e same way f o r V we obta?n
where $ ( x ) i s a s p i n o r i n v o l v i n g t h e two f e r m i o n f i e l d s a s s o c i a t e d w i t h e l e c t r o n s around SkF ( w i t h w a v e f u n c t i o n s u X ) and - k F ( w i t h w a v e f u n c t i o n s v X ) , r e s p e c t i v e l y .
I n t r o d u c i n g t h e f u n c t i o n s f + = u X + i v and f =v + i u X t h e Bogoliubov
-
De Gennes e q u a t i o n s a r e o f t h e f o l l o w ~ n g f o r h ( h = l ) - Xwhere v F = 2 t a. D i f f e r e n t i a t i n g once one f i n d s t h e f o l l o w i n g second o r d e r e q u a t i o n 0
F o r a c o n s t a n t o r d e r parameter A=A, t h i s e q u a t i o n e x h i b i t s a bound s t a t e
w i t h e x t e n t 5. and energy c . as g i v e n b y Eqs.(6), d e m o n s t r a t i n g t h a t t h e i m p u r i t y s t a t e s o b t a i n & d from t h e coAtinuum model a r e i n complete agreement w i t h t h o s e o f t h e d i s c r e t e H a m i l t o n i a n i n t h e weak-coupling l i m i t .
The same s o l u t i o n ( 1 0 ) would be o b t a i n e d from Eq.(9) f o r a s o l i t o n - s h a p e A ( X ) = A ta n h ( x / < ) s i n c e i t s a t i s f i e s t h e r e l a t i o n a 2 + v F d ~ / d x = a 2 . T h i s i n d i c a t e s t h a t both0 ener3y and e x t e n t o f t h e i m p u r i t y s t a t e a r e n o t i n f s u e n c e d by t h e presence o f a k i n k . Conversely, as i t i s e a s i l y seen from Eqs.(8) t h e r e i s a bound s t a t e a t E=O i f ~ ( x ) has a k i n k - l i k e f o r m s u g g e s t i n g t h a t t h e i m p u r i t y b a r e l y m o d i f i e s t h e bound s t a t e a s s o c i a t e d w i t h a s o l i t o n . S i m i l a r l y t h e i n t e r f e r e n c e between i m p u r i t y p o t e n t i a l and s o l i t o n i s v e r y weak i n t h e c o n t i n u o s spectrum, b o t h l e a d i n g t o phase s h i f t s w h i c h a r e s i m p l y added. A t t h i s p o i n t i t i s w o r t h w h i l e t o a n t i c i p a t e t h a t t h i s w i l l no l o n g e r be t r u e if one a l l o w s f o r l a t t i c e r e l a x a t i o n around t h e i m p u r i t y which w i l l l o c a l l y d i s t o r t t h e s o l i t o n - s h a p e and screen t h e i m p u r i t y p o t e n t i a l . The c a l c u l a t i o n has been extended t o deal w i t h a Poisson d i s t r i b u t i o n o f i m p u r i t i e s , u s i n g t h e method o f F r i s c h and L l o y d / 8 / . I t i s found t h a t , w i t h i n c r e a s i n g
c o n c e n t r a t i o n , an i m p u r i t y band i s formed i n t h e gap and, c o r r e s p o n d i n g l y , t h e d e n s i t y o f s t a t e s a t t h e band edges i s decreased.
4. O p t i c a l a b s o r p t i o n
The s o l i t o n model has become q u i t e p o p u l a r i n i n t e r p r e t i n g o p t i c a l a b s o r p t i o n experiments on weakly doped p o l y a c e t y l e n e , i n p a r t i c u l a r w i t h r e s p e c t t o t h e dopant- induced "mid-gap" a b s o r p t i o n a t a b o u t 0.6eV. Here we focus on t h e a l t e r n a t i v e p o s s i b i l i t y o f o p t i c a l t r a n s i t i o n s i n v o l v i n g i m p u r i t y s t a t e s . I n c o n t r a s t t o t h e e l e c t r o n i c s t a t e a s s o c i a t e d w i t h a charged s o l i t o n an i m p u r i t y s t a t e would be s i n g l y occupied and t h u s e x h i b i t t r a n s i t i o n s b o t h f r o m t h e v a l e n c e band t o t h e i m p u r i t y s t a t e and from t h e i m p u r i t y s t a t e t o t h e c o n d u c t i o n band. We s t a r t from t h e g e n e r a l f o r m u l a f o r t h e c o n d u c t i v i t y t e n s o r f o r an a r r a y o f p a r a l l e l c h a i n s / 9 /
where p i s t h e s i n g l e - p a r t i c l e d e n s i t y o p e r a t o r , N i s t h e number o f e l e c t r o n s per c h a i n and d i s t h e i n t e r c h a i n d i s t a n c e . The v e l o c i t y o p e r a t o r i s chosen as
Va = Jdx v a ( x ) $ ( x ) s Z ~ ( x )
where v =v i s t h e component p a r a l l e l t o t h e c h a i n d i r e c t i o n and v =-v ~ / ( 2 n t ) t h e cornponefit t e r p e n d i c u l a r t o t h e c h a i n ( b u t w i t h i n t h e p l a n e definedYby t h e zig-gag s t r u c t u r e ) / l O/.
For t h e p e r f e c t l y d i m e r i z e d c h a i n w i t h a s i n g l e i m p u r i t y one f i n d s f o r w:2Ao
JOURNAL DE PHYSIQUE
where g (w) = [ ( W + E . ) " A ~ ] ' [(w'c?) +~E~]~(wIE.-A ) / ( w ' ~ E . ) 3 , t h e upper and l o w e r s i g n r e i e r i n g t o tri!nsit?ons f r o d t h e ~ m p u r i t ; s f a t e t o t h e c o n d u c t i o n band and from t h e v a l e n c e band t o t h e i m p u r i t y s t a t e , r e s p e c t i v e l y . The a n i s o t r o p y , determined by t h e r a t i o v 2 / v 2 = 1 2 t : / ~ i , i s independent o f W. The s o l i t o n - i n d u c e d a b s o r p t i o n f o r A o <w<2Ao i s x g i # e n by
w i t h c o e f f i c i e n t s u =3u =n2/2, v =-v =1/2. The a n i s o t r o p y , g i v e n b y t h e r a t i o 1 2 t 2 / ( w 2 - a 2 ) depend$ onYw i n t h i g caze. T h i s i s due t o t h e f a c t t h a t t h e d i m 8 r i z a t i 8 n a m p l i t u d e ~ ( x ) and t h u s t h e t r a n s v e r s e c o u p l i n g v ( x ) i s l o c a l l y suppressed and, c o r r e s p o n d i n g l y , o i s s t r o n g l y reduced a t t h f e s h o l d . Such a frequency-dependent a n i s o t r o p y o f Y#e mid-gap a b s o r p t i o n may be o b s e r v a b l e f o r very-we1 1 o r i e n t e d samples.
Fig.1
-
I m a g i n a r y p a r t o f t h e d i e l e c t r i c c o n s t a n t E" = 4nReo /a. The f u l l l i n e corresponds t o an i d e a l l y d i m e r i s e d c h a i n w i t h a s i # 4 l e i m p u r f t y (V =10eV), t h e dashed l i n e t o a s i n g l e s o l i t o n (and no i m p u r i t y ) . 0The two r e s u l t s a r e compared i n Fig. 1 and one n o t i c e s t h a t t h e i m p u r i t y - i n d u c e d a b s o r p t i o n i s d o u b l y peaked. T a k i n g s h o r t c h a i n e f f e c t s , quantum f l u c t u a t i o n s and i n t e r c h a i n c o u p l i n g i n t o account t h i s s t r u c t u r e would be smeared o u t ( i n a s i m i l a r way as t h e s q u a r e - r o o t s i n g u l a r i t y o f i n t e r b a n d t r a n s i t i o n s ) and p o s s i b l y r e s u l t i n a broad mid-gap a b s o r p t i o n . However, t h e s t r e n g t h o f t h e i m p u r i t y p o t e n t i a l o f lOeV appears t o be somewhat u n r e a l i s t i c . I n a d d i t i o n we f i n d t h a t , i n t e g r a t i n g up t o m=2AO and t a k i n g t h e l i m i t / V /-, t h e i m p u r i t y - i n d u c e d a b s o r p t i o n i s by a f a c t o r o f 2 . 1 ~ / 6 2 4 s m a l l e r t h a n t h e s o l i t o n - i n d u c e d a b s o r p t i o n . T h i s i s o f p a r t i c u l a r i n t e r e s t s i n c e t h e s o l i t o n model appears t o account n i c e l y f o r t h e observed o s c i l l a t o r s t r e n g t h o f t h e mid-gap a b s o r p t i o n / l l / . It seems t o i n d i c a t e t h a t t h e s i m p l e i m p u r i t y model which n e g l e c t s l a t t i c e r e l a x a t i o n cannot reproduce t h e e x p e r i m e n t a l data.
5. I m p u r i t y - i n d u c e d l a t t i c e r e l a x a t i o n
The q u e s t i o n o f t h e competing e f f e c t s o f i m p u r i t y p o t e n t i a l and l a t t i c e r e l a x a t i o n has been addressed by B r y a n t and G l i c k / l Z / assuming a screened Coulomb p o t e n t i a l and e i t h e r p o l a r o n i c o r k i n k - l i k e d i s t o r t i o n s . On t h e b a s i s o f a numerical d i a g o n a l i z - a t i o n o f t h e d i s c r e t e model t h e y conclude t h a t , i n t h e presence o f t h e i m p u r i t y , t h e p o l a r o n i c s t a t e has l o w e r energy t h a n t h e s o l i t o n i c s t a t e . They appear n o t t o have a l l o w e d f o r m o d i f i c a t i o n s o f t h e shape o f t h e l a t t i c e d e f e c t s ( e x c e p t f o r d i f f e r e n t e x t e n t s ) , and t h e s e may i n d e e d be s m a l l f o r long-range p o t e n t i a l s . The s h o r t - r a n g e p o t e n t i a l o f Eq.(7), however, i s expected t o induce s t r o n g l o c a l d i s t o r t i o n s a f f e c t i n g b o t h o p t i c a l and a c o u s t i c phonon modes. R e p r e s e n t i n g t h e displacements as u n = ( - l ) " +zn and assuming t h a t b o t h t h e o p t i c a l modes y and t h e a c o u s t i c modes z a r e slowTy v a r y i n g i n space one f i n d s , i n t h e continuumnliimit, a t xJO, t h e s e l f - c o R s i s t e n c y e q u a t i o n
where t h e summation i s performed o v e r t h e occupied s t a t e s . For an i m p u r i t y p o t e n t i a l o f l e v t h e f i r s t t e r m i s o f t h e o r d e r o f 0.05A whereas t h e second t e r m corresponds e s s e n t i a l l y t o t h e number o f e l e c t r o n s p e r s i t e w h i c h i s c l o s e t o 1 f o r t h e d i m e r i z e d system and e x a c t l y 1 f o r t h e u n d i m e r i z e d case. From t h i s one has t o conclude t h a t t h e l o c a l d i s t o r t i o n p a t t e r n around t h e i m p u r i t y i s q u i t e independent o f t h e d i m e r i z a t i o n and t h u s w i l i p e r s i s t i n t h e m e t a l l i c phase. T h i s s h o u l d be o f r e l e v a n c e f o r t h e i n t e r p r e t a t i o n o f t h e dopant induced peaks i n t h e I R o p t i c a l a b s o r p t i o n s p e c t r a s i n c e t h e y p e r s i s t u p t o t h e h i g h e s t dopant l e v e l s where t h e i n t e r b a n d t r a n s i t i o n and t h u s presumably a l s o t h e d i m e r i z a t i o n has vanished 1131.
The p r e s e n t approach has i t s l i m i t a t i o n s i n c e t h e s t e p s i n t h e displacement f i e l d s p r e d i c t e d by Eq.(15) a r e c l e a r l y i n c o n s i s t e n t w i t h t h e p r e v i o u s assumption t h a t y and z a r e smooth f u n c t i o n s . N e v e r t h e l e s s , t h e c o n c l u s i o n t h a t 1 o c a l i z e d i m p u r i t i e s i n d u c e s t r o n g l o c a l l a t t i c e d i s t o r t i o n s should be g e n e r a l l y v a l i d . T h i s i s
c o n f i r m e d by a p e r t u r b a t i v e c a l c u l a t i o n o f t h e ground s t a t e energy u s i n g t h e d i s c r e t e model o f S e c t i o n 2. I o n s which a r e a d j a c e n t t o t h e i m p u r i t y a r e d i s p l a c e d by l a r g e amounts and o u t s i d e t h e p e r t u r b a t i o n r e l a x e s on a l e n g t h s c a l e
cs.
6. Conclusion
The p r e s e n t a n a l y s i s shows t h a t l o c a l i z e d i m p u r i t i e s have s t r o n g e f f e c t s on b o t h t h e e l e c t r o n i c s t r u c t u r e and t h e l a t t i c e d i s t o r t i o n s of t h e ground s t a t e o f t h e SSH H a m i l t o n i a n . C o n s i d e r i n g o n l y t h e i r i n f l u e n c e on t h e e l e c t r o n i c s t a t e s one o b t a i n s donor o r a c c e p t o r l e v e l s i n t h e gap w h i c h l e a d t o o p t i c a l a b s o r p t i o n below t h e i n t e r b a n d t h r e s h o l d . I n c o n t r a s t t o p r e d i c t i o n s o f t h e s o l i t o n model t h e i n t e n s i t y o f t h e t r a n s i t i o n s i n v o l v i n g i m p u r i t y s t a t e s appears t o be t o o l o w t o a c c o u n t f o r t h e o s c i l l a t o r s t r e n g t h observed on doped p o l y a c e t y l e n e . Thus t h e i m p u r i t y - i n d u c e d d i s t o r t i o n o f t h e l a t t i c e should be r e l e v a n t f o r i n t e r p r e t i n g t h e o p t i c a l data. The combined s c r e e n i n g e f f e c t s on t h e i m p u r i t y p o t e n t i a l due t o l a t t i c e r e l a x a t i o n and on t h e electron-phonon i n t e r a c t i o n due t o l o c a l changes i n t h e bond o r d e r r e p r e s e n t an i n t e r e s t i n g b u t n o n - t r i v i a l problem which r e q u i r e s f u r t h e r study.
Acknowledgments
I would l i k e t o thank P. Horsch and K. Maki f o r h e l p f u l d i s c u s s i o n s . I n t h e course o f t h i s work I have b e n e f i t t e d f r o m t h e h o s p i t a l i t y o f t h e Max P l a n c k I n s t i t u t e i n S t u a g a r t and t h e U n i v e r s i t y o f Southern C a l i f o r n i a .
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