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EFFECTS OF OFF-DIAGONAL DISORDER ON SOLITON- AND POLARON-LIKE STATES IN
TRANS-POLYACETYLENE
C. White, M. Elert, J. Mintmire
To cite this version:
C. White, M. Elert, J. Mintmire. EFFECTS OF OFF-DIAGONAL DISORDER ON SOLITON- AND
POLARON-LIKE STATES IN TRANS-POLYACETYLENE. Journal de Physique Colloques, 1983,
44 (C3), pp.C3-481-C3-484. �10.1051/jphyscol:1983396�. �jpa-00222835�
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Colloque C3, supplement au na6, Tome 44, juin 1983 page C3-481
EFFECTS OF OFF-DIAGONAL DISORDER ON S O L I T O N - AND POLARON-LIKE STATES I N TRANS-POLYACETYLENE
C.T. White, M.L. Elert and J.W. Mintmire
Naval Research Laboratory, Washington, DC 20375, U.S.A.
RQsumd
-
Nous montrons que le dssordre non diagonal peut avoir un effet significatif sur les limites de bande de Peierls du polyac6tylSne ainsi que sur les 6tats du gap de type polaron. Nos rssultats montrent aussi qu'un tel d6sordre n'affecte pas la plupart des plus importantes propri6tGs des d6fauts de type soliton.A b s t r a c t
-
We r e p o r t on some exact r e s n l t s we have o b t a i n e d t h a t show o f f - diagonal d i s o r d e r can have a s i g n i f i c a n t e f f e c t on t h e P e i e r l s band edges of p o i y a c e t y l e n e a s w e l l as polaron-like gap s t a t e s i n t h i s system. Our r e s u l t s a l s o prove t h a t such d i s o r d e r l e a v e s many of t h e more important p r o p e r t i e s of s o l i t o n - l i k e d e f e c t s l a r g e l y u n a f f e c t e d .1. I n t r o d u c t i o n . I d e a l i z e d i n f i n i t e c h a i n s of trans-polyacetylene a r e expected t o e x h i b i t a dimerized ground s t a t e due t o a P e i e r l s d i s t o r t i o n
111.
I n t h i si n s t a n c e t h e hopping m a t r i x elements {Vn,n+lf d e f i n i n g t h e n e a r e s t neighbor t i g h t - b i n d i n g model
(S denotes t h e s p i n ) used e x t e n s i v e l y t o d e s c r i b e t h e p h y s i c a l l y i n t e r e s t i n g h a l f - f i l l e d n-band of t h i s m a t e r i a l , adopt one of two p o s s i b l e v a l u e s . These v a l u e s , which we term V, and V,, correspond r e s p e c t i v e l y t o C-C ' s i n g l e ' and
'doable' bonds t h a t a r e of d i f f e r e n t l e n g t h s . These s i n g l e and double bonds a l t e r n a t e down t h e c h a i n l e a d i n g t o a gap of
~ ~ v , - v , I
a t t h e Fermi l e v e l .I n a c t u a l trans-polyacetylene f i l m s it could w e l l be t h a t t h e d i m e r i z a t i o n of any p a r t i c u l a r c h a i n i s not i d e a l , due t o i t s v a r i a b l e environment f o r
example. T h i s can l e a d t o a s t a t e of a f f a i r s where t h e double and s i n g l e bonds, although remaining d i s t i n g u i s h a b l e and a l t e r n a t i n g , a r e of v a r y i n g l e n g t h s
c e n t e r e d around t h e i r r e s p e c t i v e i d e a l i z e d v a l u e s so t h a t t h e corresponding m a t r i x elements can be viewed a s random v a r i a b l e s . Such bond l e n g t h f l u c t u a t i o n s can have important e f f e c t s on t h e e l e c t r o n i c s t r u c t u r e of t h e system i f , on t h e s c a l e of t h e P e i e r l s gap, t h e y s h i f t a p p r e c i a b l y s t a t e s i n t h e neighborhood of t h i s gap w h i l e not changing n e a r l y a s s i g n i f i c a n t l y t h e t o t a l energy of t h e c h a i n p e r u n i t l e n g t h . It i s p l a u s i b l e t h a t t h e s e requirements a r e met i n t h e v i c i n i t y of t h e expected i d e a l i z e d geometry of p o l y a c e t y l e n e . More e x p l i c i t l y , r e c e n t ab i n i t i o X a c a l c u l a t i o n s 121 p r e d i c t t h a t t h e f r e e energy of t h e c h a i n a s a f u n c t i o n of t h e
Su. S c h r i e f f e r , Heeger / I / (SSH) d i m e r i z a t i o n c o o r d i n a t e , a , i s v e r y f l a t f o r u's between -.02
R
and -.03 y e t t h e P e i e r l s gap. El, i n t h i s r e g i o n behaves a s E 23-flu. where j3 l i e s somewhere between 32 and 60EVIR.
Here we d i s c u s s some exact r e s u l t s we have o b t a i n e d r e g a r d i n g t h e e f f e c t s of t h e above type of off-diagonal ( l a t e r a l ) d i s o r d e r on t h e e l e c t r o n i c s t r u c t u r e of t h e rt-band of trans-polyacetylene when modeled by H. These r e s n l t s i n v o l v e not o n l y t h e e f f e c t s on t h e P e i e r l s gap of t r a n s - p o l y a c e t y l e n e b u t a l s o on s t a t e s a s s o c i a t e d w i t h s o l i t o n - and polaron-like d e f e c t s i n t h e system.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983396
C3-482 JOURNAL DE PHYSIQUE
2 . R e s u l t s and D i s c u s s i o n . We have d e r i v e d an e x a c t s e t of coupled f u n c t i o n a l i n t e g r a l e q u a t i o n s , t h e s o l u t i o n of which y i e l d s t h e d e n s i t y of s t a t e s (DOS) of H i n t h e presence of l a t e r a l d i s o r d e r . These e q u a t i o n s were o b t a i n e d by f i r s t assuming t h a t t h e d i s o r d e r e d double and s i n g l e bonds remain d i s t i n g u i s h a b l e and t h e p r o b a b i l i t y d e n s i t i e s of a l l t h e double ( s i n g l e ) bonds a r e a l i k e and t h e n p a r a l l e l i n g Schmidt's a n a l y s i s / 3 / of a r e l a t e d problem. R e s u l t s we have o b t a i n e d from t h e s e e q u a t i o n s when t h e d i s o r d e r e d double and s i n g l e bonds a r e c o n s t r a i n e d t o a l t e r n a t e down t h e c h a i n a2e shown i n F i g u r e 1.
ENERGY (eV)
F i g u r e 1. The DOS p e r s i t e , p ( E ) , i n t h e neighborhood of t h e P e i e r l s gap o b t a i n e d by assuming W=.2 eV, V,=-2.32 eV, and V,=-3.18 eV. The Smooth l i n e i s t h e DOS f o r t h e corresponding i d e a l l y dimerized c a s e . The histogram p l o t r e s u l t s from
numerical d i f f e r e n t i a t i o n of t h e d i r e c t l y c a l o u l a t e d i n t e g r a t e d DOS.
I n a r r i v i n g a t t h e s e r e s u l t s we have assumed f o r s i m p l i c i t y t h a t each double and s i n g l e bond m a t r i x element obeys a r e c t a n g u l a r p r o b a b i l i t y d e n s i t y of half-width W=.2 eV c e n t e r e d around t h e i r r e s p e c t i v e i d e a l i z e d v a l u e s of V,-3.18 eV and V,=-2.32 eV. Within t h e SSH model t h i s choice of W corresponds t o abo t a 25%
s t a n d a r d d e v i a t i o n i n t h e e q u i l i b r i u m d i m e r i z a t i o n c o o r d i n a t e . uo=.03
8.
i f wesup o s e , a s h a s been most r e c e n t l y i n d i c a t e d / 4 / , t h a t a of t h a t model e q u a l s 7.1 eV/fj. We s e e from F i g u r e 1 t h a t t h i s amount of d i s o r d e r i s s u f f i c i e n t t o s i g n i f i c a n t l y smear out t h e square r o o t divergence o c c u r i n g a t t h e band edges i n t h e o r d e r e d system. T h i s r e s u l t confirms e a r l i e r numerical r e s u l t s / 5 / o b t a i n e d by embedding f i n i t e d i s o r d e r e d c h a i n s i n a p e r i o d i c medium and shows t h a t t h e presence of such d i s o r d e r could p a r t i a l l y account f o r t h e e x p e r i m e n t a l l y observed smearing of t h e o p t i c a l a b s o r p t i o n edge a s h a s been suggested i n Reference 5 .
The i n t e g r a l e q u a t i o n s s o l v e d t o o b t a i n F i g u r e 1 can be s t r a i g h t f o r w a r d l y modified t o s t u d y t h e WS i n t h e neighborhood of t h e P e i e r l s gap when s o l i t o n - l i k e kinks a r e p r e s e n t i n t h e system. This i s achieved through i n t r o d u c t i o n of a s i n g l e parameter, P, r e p r e s e n t i n g t h e p r o b a b i l i t y of having a s i n g l e bond-like m a t r i x element g i v e n a double bond-like m a t r i x element and v i c e v e r s a . I n F i g u r e 2 we p r e s e n t r e s u l t s o b t a i n e d i n t h i s way by assuming P=.999 and F . 9 9 and a l l o t h e r parameters t h e same a s t h o s e used t o d e r i v e F i g u r e 1. Note t h a t f o r P c l o s e t o 1, (1-P) g i v e s t h e c o n c e n t r a t i o n of k i n k s .
The most s t r i k i n g f e a t u r e of F i g u r e 2 i s t h e r a t h e r small amount of broadening a s s o c i a t e d w i t h t h e k i n k induced s t a t e s a t midgap. T h i s r e s u l t a l s o 8ug8est8 t h a t t h e primary broadening mechanism f o r t h i s band i s simply kink-kink i n t e r a c t i o n s . I n f a c t , f o r t h i s example we have found t h a t t h i s i s t h e c a s e because we have shown t h a t a s P
->
1, t h e midgap band approaches a d e l t a f u n c t i o n of measure 2(1-P) [Compare F i g u r e 2a t o F i g u r e 2 b l . We have o b t a i n e d s i m i l a r r e s u l t s by i n t r o d u o i n g k i n k s i n d i f f e r e n t ways. These r e s u l t s imply t h a t t h e i n s e n s i t i v i t y of k i n k d e f e c t s t o t h e p r e s e n t type of off-diagonal d i s o r d e r i s a v e r y g e n e r a l p r o p e r t y of t h e model. T h i s i s indeed t h e c a s e a s we have proved,i n o l u d i n g s p i n degeneracy, t h a t i f t h e m a t r i x elements d e f i n i n g
H
a r e a l l bondingFigure 2. DOS per site, p(E), in the neighborhood of the Peierls gap for the disordered system described in the text assuming two different kink
concentrations -(l-P): (a) F0.999, (b) P=0.99 with W=0.2 eV for both cases.
Note, here p(E) = p(-E), the smooth line is the DOS for the corresponding ideally dimrrized oare with no kinks.
(i. e., less than zero) and are either single bond-like ( T ~ < V ~ , ~ + ~ < U , ) or double bond-like (-c,<V n+l<az) and if the double and single bonds are distinguishable
(a,<.c,) and folfbw one another except across a kink center, then
Theorem: Any chain containing Ng kinks exhibits exactly 2NK states between E-=a,-r, and E+=z,-a,,
provided the matrix elements connecting either end site to its nearest neighbor is double bond-like, the kink concentration is low 161, and the matrix elements bordering the kink are as described below. Furthermore, we have shown that the 2N states between E+ and E- will cluster around midgap. The requirement that the chain terminating matrix elements are both double bond-like is certainly expected K o n physical grounds and is consistent with prediction based o n the S S H model 171.
Details of the proof of this theorem are tedious and will be reported elsewhere.
Note, however, that in carrying out the proof w e allow for the relaxation expected in the vicinity of each kink. Explicitly, the sole restriction w e impose o n the kink bordering matrix elements is that if this defect is of the type that when unrelaxed corresponds t o two double bonds in a row and occurs at a site n, then both V
,
V<
(u,-c,). As such our analysis is general enough to include what~&6Ei:ia
ben&e appropriate disordered version of SSH topological solitons.As a n immediate consequence of the above theorem, we have that all even membered nndoped chains containing n o kinks have a gap between occupied and unoccupied states of at least 2(.cl-a,). Furthermore, all odd membered chains containing a single kink have exactly two states between E+ and E-. As in the ideal case one can show that these two states occur precisely at midgap. In addition, by counting the electrons w e see that this isolated kink will exhibit spin 112 when neutral and spin 0 when charged. Thus, isolated kinks continue to exhibit several of their more important properties found in the ideal case. This result could be partially anticipated by first noting that all odd membered polyacetylene chains must be neutral free radicals regardless of the type of disorder present. Furthermore, we see that all odd membered chains in the presence of off-diagonal disorder must continue to exhibit at least one state for each spin at E=O. This follows because the spectrum of H for each spin species continues t o be symmetric about E=O. Finally, by extrapolating results for the ideal case one can attempt to relate these states to the kink. However, this analysis does not show that there is only one such midgap spin state in the polyacetylene chain that is in fact associated with the kink in real space and
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separated from all other similar spin states. Our analysis, however, does establish these properties as well as provide the appropriate generalization when there is more than one kink present in the chain.
In contrast to the effects of off-diagonal disorder on kink defects w e have shown by using an approach similar to that employed in obtaining Figures 1 and 2 that lateral disorder produces significant smearing of polaron-like induced gap states. These results demonstrate that if polaron-like states are observable then they could provide a sensitive probe to the importance of off-diagonal disorder in polyacetylene. Based on our results, we conjecture that kinks should be freer to move along the chain than polarons. This follows because our results show that
the kink induced gap states contribution to the kinks total energy is unchanged regardless of the defect's position in the chain while such is certainly not the case for polarons.
3. Concluding Remarks, The results detailed in the last section have a number of interesting additional consequences. First of all, they show that any kink caused peak observed near midgap should exhibit much less temperature dependence than.
for example, the Peierls band edges and along related lines not couple in the usual way to phonons. Second, these results demonstrate the possibility of using disorder to account for the observed smearing of the optical absorption edges and the momentum independence of the absorption threshold without assuming any significant broadening of the midgap kink band. Such smearing, of course, would make this band difficult to observe. Finally, the generality of the theorem that we have presented indicates that kinks should maintain several of their more
important properties, not only in the presence of static fluctuations but also dynamic fluctuations.
We thank ONR for support of this work through contract N001481WK10010. JWM thanks the National Research Council for support through the NRC-NRL.Cooperative Postdoctoral Associateship program, and MLE thanks the NRL-USNA Cooperative Program for Scientific Exchange for financial support.
4. References.
1.
W.
P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett.42,
(1979) 1698.2. J. W. Mintmire and C. T. White, Phys. Rev. Lett. 50. (1983) 101.
3. H. Schmidt, Phys. Rev.
105.
(1956) 425.4. C. R. Fincher, C.-E. Chen, A. J. Heeger. A. G. MacDiarmid, and J. B. Hastings.
Phys. Rev. Lett.
48
(1982) 100.5.
D.
Vanderbilt and E. J. Mele. Phys. Rev. B22
(1980) 3939.6. Exactly how low the concentration has to be for our analysis to apply depends in a complicated fashion on a, and r, and will be described elsewhere.
7.
