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A soliton in polyacetylene: Optical and dynamical activities
D447
Synthetic Metals, 28 (1989) D447 D456
A SOLITON IN POLYACETYLENE: OPTICAL AND DYNAMICAL ACTIVITIES
Y. WADA, A. TERAI, AND K. IWANO Department of Physics, U n i v e r s i t y of Tokyo, Bunkyo-ku, Tokyo 113 (Japan) Y. Ono Department of Physics, Toho University, Funabashi, Chiba 274 (Japan) Y. O h f u t i Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560 (Japan)
ABSTRACT Doping of a polyacetylene chain is numerically simulated. Schrieffer, and Heeger model is used for the chain. mechanisms are examined.
Two types of doping
One is the doping by " s i t e - t y p e " impurities,
randomly modify the e l e c t r o n i c energy levels at each site. type" impurities.
The Su, They
Other is by "bond-
The e l e c t r o n i c transfers between two adjacent sites are
randomly modified.
For each random configuration of i m p u r i t i e s , we determine
metastable bond configurations, electron concentrations, dynamical conductivity, and infrared active phonon modes.
I t is found that the doping
gives rise to s o l i t o n s which are pinned by the impurities.
Dynamical
c o n d u c t i v i t y shows that the charged solitons, produced by the doping, are playing important roles in the infrared absorption,
INTRODUCTION As well known, the doping of polyacetylene by i m p u r i t i e s gives a dramatic e f f e c t for transport properties.
At low doping level, the e l e c t r i c a l
conduction was observed by the c a r r i e r s without spin [ I ] . fact that the c a r r i e r s are charged solitons.
I t substantiated the
At high doping level, the system
becomes m e t a l l i c and i t is important to understand the r e l a x a t i o n mechanism which leads to the T-2 dependence of the c o n d u c t i v i t y [2]. various t h e o r e t i c a l works.
s o l i t o n s has been proposed as an important mechanism [3]. hopping between dopant levels has been another [4]. of charged solitons, 0379-6779/89/$3.50
There have been
Electronic hopping between the charged and neutral A variable range
Thermally excited motions
trapped by dopants, have also been studied [5]. © Elsevier Sequoia/Printed in The Netherlands
D448 In order to see how these mechanisms would work, i t is important to know s t a t i c states of a single chain of the doped polyacetylene and e x c i t a t i o n s around them.
Recently, one of the authors has studied the SSH model with the
two types of impurities, assuming that the perfect dimerization pattern is being kept [6].
He has found that the magnitude of the pattern reduces, as the
disorder increases, and f i n a l l y i t vanishes.
Presumably, i t would be too
simple to assume the perfect dimerization, since we know experimentally the doping produces solitons which strongly modulate the dimerization pattern, I t would be very i n t e r e s t i n g to perform numerical works which simulate the doping processes on computers.
A single chain of polyacetylene is represented
by the Su, Schrieffer, and Heeger model. d i s t r i b u t e d randomly along the chain.
Suppose the i m p u r i t i e s are
They would be charged, i f they would
release or accept electrons to become donors or acceptors, respectively.
The
Coulomb p o t e n t i a l s would modulate the e l e c t r o n i c energy levels at each s i t e randomly.
This is the " s i t e - t y p e " impurity effect.
Meanwhile, mutual
interactions among the dopants would disrupt the alignment of the atoms in the chain, giving rise to the random modulation of the electron t r a n s f e r between adjacent atoms.
This is the "bond-type" impurity effect.
I f there are cis
segments in the chain, t h e i r effects would also be taken into account as the bond-type impurities. The SSH model is generalized to include these effects.
A random
configuration of i m p u r i t i e s is a r b i t r a r i l y chosen. Static solutions are obtained by i t e r a t i o n procedures. depend on the i n i t i a l
We find many metastable solutions.
They
configurations from which the i t e r a t i o n has started.
It
is c h a r a c t e r i s t i c to the disordered systems. We calculate metastable bond configurations to show that the doping a c t u a l l y gives rise to solitons. solitons are charged. the frequency.
Local concentration of electrons helps find how the Dynamical conductivity is calculated as a function of
Infrared active phonon modes are also obtained.
are manifestly found to be pinned by impurities. modes become infrared active due to the doping.
The solitons
All the localized phonon We can thus conclude that many
features of the doped polyacetylene, observed experimentally, are reproducible by the numerical simulations. MODELS AND METHODS The Hamiltonian of the model of the doped polyacetylene is H = HSSH + Himp, where the f i r s t
(I) term is the SSH Hamiltonian,
D449 HSSH =
+ + _ ~ [ t o - m(Un+I - Un)](Cn+isCns + CnsCn+is) ns + (K/2)Z(Un+ I - Un)2 + (M/2)~U2n . n n
(2)
Here, C+ns and Cns are the electron creation and a n n i h i l a t i o n operators, respectively, at the s i t e n with the spin s, un the displacement of the CH unit at the s i t e n, K the force constant between the adjacent CH units, M the mass of the unit.
Electronic overlapping integral t O is modified by the l a t t i c e
deformation which gives r i s e to an electron-phonon i n t e r a c t i o n with the strength m , the dot above un means the time derivative. The s i t e - t y p e i m p u r i t i e s give
Him p
= nZsVnC+sCns ,
(3)
where Vn is a random quantity,
Him p
The bond type i m p u r i t i e s give
+ + = ~sVb(n)(Cn+IsCns + CnsCn+is),
(4)
where Vb(n ) is a random quantity. We w i l l
discuss the r e s u l t s of three s i m u l a t i o n works.
Two of them are w i t h
the s i t e - t y p e i m p u r i t i e s and the other w i t h the bond-type.
In the f i r s t
work,
the i m p u r i t y p o t e n t i a l Vn is so truncated t h a t i t can take only two values ~V. They represent the Coulomb p o t e n t i a l s by an acceptor and a donor, We choose t o t a l
respectively.
numbers of the s i t e s N, of the electrons Ne, of the i m p u r i t i e s
Nim p which is the number of the s i t e s whose Vn is e i t h e r V or -V.
Periodic
boundary conditions are used. The numerical works are c a r r i e d out as f o l l o w s . ( I ) Select Nim p i m p u r i t y s i t e s randomly to f i x a sample. (2) A s t a r t i n g c o n f i g u r a t i o n {un(O) } is selected randomly. (3) E l e c t r o n i c states are determined by the eigenvalue equation
(c i
Vn)~i(n)
=
-[t O - ~y~kl]~i(n-i ) - [t O -
~'(k)]~i(n+l) -
#n
where yn (k) = Un+l (k) - un(k) and k indicates i t value.
is the k-th i t e r a t e d
With the help of the e l e c t r o n i c wave f u n c t i o n
determined by
(5)
'
~i'
Yn(k+l) is
D450 y~k+l) :
-(2m/K) Z ' @ i ( n ) # i ( n + l ) + (2m/KN) Z Z'@i(m)@i(m+l), i,s m i,s
(6)
where the prime i n d i c a t e s the sum over the occupied states and the second term is necessary to s a t i s f y the p e r i o d i c c o n d i t i o n procedure is i t e r a t e d u n t i l a s u f f i c i e n t s t a t i c s o l u t i o n depends on {un(O)}.
ZYn = O.
convergence is obtained.
I f you s e l e c t a d i f f e r e n t
un(O), you may get a d i f f e r e n t s t a t i c s o l u t i o n .
The
The
set of
I t would be necessary to
examine several s t a r t i n g c o n f i g u r a t i o n s , at least, to gain i n f o r m a t i o n s directly
r e l a t e d to observations.
(4) With the help of the s t a t i c s o l u t i o n , phonon modes are determined in the same way as f o r the i m p u r i t y - f r e e system [7]. (5) Dynamical
c o n d u c t i v i t y is c a l c u l a t e d f o r the i n f r a - r e d frequencies to
f i n d the o p t i c a l a c t i v i t i e s
of phonons, using the method given in [8].
STATIC SOLUTIONS The r e s u l t s are shown in the f i g u r e s from 1 to 9. 2~2/~Kto
The parameters are
= 0.19, V = t o / 2 , N = 199, Ne = 200, and Nim p = 16.
l o c a t i o n s are shown by the c i r c l e s in Figs. 1 and 2. are i m p u r i t i e s w i t h V and the e i g h t closed w i t h -V. are almost overlapped each other.
The i m p u r i t y
The e i g h t open c i r c l e s Two of the closed c i r c l e s
This corresponds to a system w i t h
compensation between donors and acceptors.
Since Ne is l a r g e r than N by u n i t y
and N is odd, there would be a s o l i t o n w i t h a negative charge, i f there were no impurities.
Fig, 1 shows the bond c o n f i g u r a t i o n s ~n:(-l)n(yn-Yn_l ) as a
f u n c t i o n of the s i t e l o c a t i o n n.
There are three s o l i t o n s .
One is close to
the acceptor p o t e n t i a l s w i t h open c i r c l e s and another to the donor p o t e n t i a l s w i t h close c i r c l e s .
The t h i r d one is in an i m p u r i t y - f r e e region.
does produce s o l i t o n s in the numerical simulations, too. also c l o s e l y associated w i t h i m p u r i t i e s . electrons defined by
The doping
Small structures are
Fig. 2 is the l o c a l number d e n s i t y of
~ n : ( Pn_l+2 P n+ Pn+l)/4 to remove the rapid v a r i a t i o n
due to the d i m e r i z a t i o n ,
The h o r i z o n t a l l i n e gives the average value.
The
s o l i t o n at the acceptor p o t e n t i a l s has a p o s i t i v e charge, w h i l e the one at the donor p o t e n t i a l s has a negative charge. PHONONS AND INFRARED ABSORPTION Dynamical c o n d u c t i v i t y gives the i n f r a r e d absorption i n t e n s i t y shown in Fig. 3 as a f u n c t i o n of frequency in an a r b i t r a r y scale. : 0 t o 0.79 ~Q withWQ = ~
The o p t i c a l
Five large peakes are found at ~ = my w i t h
The abscissa extends from
phonon edge i s ~0 = O.63WQ.
v = a, b, c, d, and e.
Each is
due to an i n f r a r e d a c t i v e phonon mode. Fig. 4 is the phonon normal mode f u n c t i o n
D451
/
Fig.
I.
/
....
The bond c o n f i g u r a t i o n
~n=(-l)n(yn-Yn_ I ) as a f u n c t i o n of n.
Fig. 2. Local number d e n s i t y of e l e c t r o n s
Pn=(Pn_l+2Pn+Pn+l)/4.
ga(n) of small v i b r a t i o n s of un around the s t a t i c at ~ = wa in Fig. 3, as a f u n c t i o n of s i t e . o p t i c a l and a c o u s t i c components. ga (n)
:
The f u n c t i o n is an a d m i x t u r e of
frequency ~a = 0.036~Q i n d i c a t e s . It
,
(7)
We thus f i n d the a-peak is due to the Goldstone mode of
the s o l i t o n in the i m p u r i t y f r e e region. Fig. 6.
It
is almost f r e e as i t s low
The o p t i c a l component of gb(n) i s shown in
is l o c a l i z e d around the same s o l i t o n .
mode w i t h w b = 0.60 w ~ and e, r e s p e c t i v e l y . donor region.
which gives the peak
The o p t i c a l component is obtained by
(-l)n(-ga(n-l)+2ga(n)-ga(n+l))/4
and shown in Fig. 5.
solution,
Evidently,
it
i s the t h i r d
Figs. 7-9 show the o p t i c a l components f o r ~ = c, d.
The peaks c and d are associated w i t h the s o l i t o n in the
The former is due to the pinned Goldstone mode w i t h w c
= O.48mQ,
D452
..............
II ....
TT tt ce
clb
Fig. 3. I n f r a r e d absorption i n t e n s i t y ( a r b i t r a r y scale) as a f u n c t i o n of frequency ~ , The abscissa is f o r O
al
Fig. 4. Phonon mode f u n c t i o n ga(n) as a f u n c t i o n of n,
Optical component
a2
2'\. Fig, 5, Optical component g a ( n ) : ( - l ) n ( - g a ( n - l ) + 2 g a ( n ) - g a ( n + l ) ) / 4 "
D453
Opt Ical component
b2
Fig. 6. Optical component gb(n),
Opt I c = l component
c2
Fig. 7. Optical component gc(n).
opt I c a l
commonent
d2
Fig, 8, Optical component gd(n),
while the l a t t e r presumably due to the shape mode w i t h ~ d = 0.57~Q which is activated by the doping,
Fig, 9 shows that the e-peak is due to the strongly
D454
opt ical
component
e2
J~Cv',a,~V'~,~,'~'m~.~.A ), ~,t i,) ~v~r~v~I~q~,,pA '~ih/i)~u'..~h
Fig
9. Optical component g e ( n )
F
~p.~°,°.°.°.°@,o "o
" ""
o
~,~
J'
~ O o o O o o ~"
-]
J
50
100
Fig. I0. The bond c o n f i g u r a t i o n ( - l ) n y n . The open c i r c l e s
f o r odd n, and the
closed f o r even n
300
200
00
0
./I
Fig. I I . Averaged dynamical c o n d u c t i v i t y as a f u n c t i o n of frequency ~ .
Units
of the ordinate is 2(e~ /to)2/M and the abscissa ~ Q
pinned Goldstone mode, w i t h ~ e acceptor r e g i o n .
: 049WQ, associated w i t h the s o l i t o n in the
D455
SITE-TYPE IMPURITIES WITH A LONG-RANGE POTENTIAL We are p e r f o r m i n g two o t h e r independent works. doping by s i t e - t y p e
impurities
q u a n t i t y Vn is w r i t t e n Vn
One of them s t u d i e s the
w i t h a long range p o t e n t i a l .
The random
as
X(:~e2/~Irn-Rjl), J
(8)
where r n is the p o s i t i o n of the n - t h s i t e and Rj is the l o c a t i o n o f e i t h e r the dopant or the c a r r i e r
on o t h e r chains whose charge d e t e r m i n e s the sign ~.
q u a n t i t y Rj is a t h r e e d i m e n s i o n a l random v a r i a b l e . N = lO0.
Suppose t h e r e are f i v e donors, c l o s e to the chain,
with five electrons. distribution. antisoliton
I f the system were i m p u r i t y f r e e , p a i r s w i t h a polaron.
shown in Fig.
lO.
structures
middle.
which p r o v i d e i t
E f f e c t s of o t h e r dopants are a p p r o x i m a t e d by a random
Two s o l i t o n i c positively
The open c i r c l e s
charged.
bond c o n f i g u r a t i o n
structure
They t u r n out t o be
is being d e s t r o y e d in the
d e v i a t e from the closed c i r c l e s
s t r o n g l y each o t h e r .
at n = O.
Fig. I I
u n i t of the o r d i n a t e is 2(e ~ / t o ) 2 / M
I t shows
and polarons
is the dynamical c o n d u c t i v i t y ,
f u n c t i o n of frequency, which is o b t a i n e d as an average o f s i x t y
of the Goldstone modes is r e a d i l y
and
( - l ) n y n is
are f o r odd n and the closed f o r even n.
are found at n ~ 4 and 70.
The d i m e r i z a t i o n
The open c i r c l e s
we would f i n d two s o l i t o n
Typical static
the a c o u s t i c component is enhanced at the r e g i o n where s o l i t o n s interact
The
We t a k e ~; = lO and
and t h a t o f the abscissa
as a
samples. i~ Q.
The
The p i n n i n g
seen.
BOND-TYPE IMPURITIES The t h i r d
work s t u d i e s the bond-type i m p u r i t i e s .
bonds where Vb(n) is not v a n i s h i n g . between -V and V.
The q u a n t i t y
We randomly s e l e c t Nim p Vb(n) takes a random v a l u e
We f i n d t h a t t h i s t y p e of doping can a l s o produce s o l i t o n s .
The Goldstone modes are pinned and the shape modes become i n f r a r e d a c t i v e . gives qualitatively
similar
effect
t o the s i t e - t y p e
It
impurities.
DISCUSSION It
is i n t e r e s t i n g
t o f i n d t h a t some aspects of doping can be reproduced so
w e l l by s i m p l e s i m u l a t i o n s . impurities.
The s o l i t o n s
Even the system, which i s p e r f e c t l y
i n a c t i v e w i t h o u t the i m p u r i t i e s , solitons.
are c r e a t e d and pinned by the
The system, w i t h one charged s o l i t o n
a l s o i n c r e a s e the a c t i v i t y
d i m e r i z e d and i n f r a r e d
becomes s t r o n g l y a c t i v e due t o the c r e a t e d w i t h o u t the i m p u r i t i e s ,
can
due t o the doping. The small peaks on the r i g h t h a n d
s i d e of the a-peak in Fig. 3 are due t o a c o u s t i c phonons which become a c t i v e , m i x i n g w i t h the Goldstone mode [ 9 ] . by the doping.
The shape mode is found t o become a c t i v e
In the work w i t h the s i t e - t y p e
impurities
w i t h a l o n g - range
D456 p o t e n t i a l , the r e s u l t for the smaller i m p u r i t y concentration does not show any change in the a c t i v a t i o n energy of the pinned modes. This paper has discussed the phonon structures and infrared a c t i v i t i e s .
Electronic structures and
optical properties w i l l be reported elsewhere,
This work was p a r t i a l l y
supported by a Grant-in-Aid for S c i e n t i f i c Research from the M i n i s t r y of Education, Science, and Culture.
I t was also p a r t l y sponsored by Research
Association for Basic Polymer Technology. REFERENCES I. S. Ikehata, J. Kaufer, T. Woerner, A. Pro~, M,A. Druy, A. Sivak, A.J. Heeger, and A,G. MacDiarmid,
Phys. Rev. Letters, 45 (1980) 1123.
2. K. Kume, S. Masubuchi, Y, Mochizuki, Y. Hirayama, K, Mizoguchi, and K. Mizuno, Solid State Commun., 58 (1986) 473. 3. S. Kivelson, Phys. Rev. B25 (1982) 3798. T. Yamabe, K. Tanaka, S, Yamanaka, T. Koike, and K, Fukui, J. Chem. Phys., 82 (1985) 5753. 4. K. Ehinger and S. Roth, P h i l . Mag., B53 (1986) 301. 5. H. Fukutome and A. Takahashi, Prog. Theor, Phys., 77 (1987) 1376, 6. A. Terai, Ph.D. Thesis, Tokyo, Japan, 1988. 7. A. Terai and Y. Ono, J. Phys. Soc., 55 (1986) 213. 8. H. I t o and Y. Ono, J. Phys. Soc., 54 (1985) 1194. 9. J.C. Hicks and J.T. Gammel, H,-Y. Choi, and E.J. Mele, Synth, Met., 17 (1987) 57.
Synthetic Metals, 28 (1989) D447 D456
A SOLITON IN POLYACETYLENE: OPTICAL AND DYNAMICAL ACTIVITIES
Y. WADA, A. TERAI, AND K. IWANO Department of Physics, U n i v e r s i t y of Tokyo, Bunkyo-ku, Tokyo 113 (Japan) Y. Ono Department of Physics, Toho University, Funabashi, Chiba 274 (Japan) Y. O h f u t i Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560 (Japan)
ABSTRACT Doping of a polyacetylene chain is numerically simulated. Schrieffer, and Heeger model is used for the chain. mechanisms are examined.
Two types of doping
One is the doping by " s i t e - t y p e " impurities,
randomly modify the e l e c t r o n i c energy levels at each site. type" impurities.
The Su, They
Other is by "bond-
The e l e c t r o n i c transfers between two adjacent sites are
randomly modified.
For each random configuration of i m p u r i t i e s , we determine
metastable bond configurations, electron concentrations, dynamical conductivity, and infrared active phonon modes.
I t is found that the doping
gives rise to s o l i t o n s which are pinned by the impurities.
Dynamical
c o n d u c t i v i t y shows that the charged solitons, produced by the doping, are playing important roles in the infrared absorption,
INTRODUCTION As well known, the doping of polyacetylene by i m p u r i t i e s gives a dramatic e f f e c t for transport properties.
At low doping level, the e l e c t r i c a l
conduction was observed by the c a r r i e r s without spin [ I ] . fact that the c a r r i e r s are charged solitons.
I t substantiated the
At high doping level, the system
becomes m e t a l l i c and i t is important to understand the r e l a x a t i o n mechanism which leads to the T-2 dependence of the c o n d u c t i v i t y [2]. various t h e o r e t i c a l works.
s o l i t o n s has been proposed as an important mechanism [3]. hopping between dopant levels has been another [4]. of charged solitons, 0379-6779/89/$3.50
There have been
Electronic hopping between the charged and neutral A variable range
Thermally excited motions
trapped by dopants, have also been studied [5]. © Elsevier Sequoia/Printed in The Netherlands
D448 In order to see how these mechanisms would work, i t is important to know s t a t i c states of a single chain of the doped polyacetylene and e x c i t a t i o n s around them.
Recently, one of the authors has studied the SSH model with the
two types of impurities, assuming that the perfect dimerization pattern is being kept [6].
He has found that the magnitude of the pattern reduces, as the
disorder increases, and f i n a l l y i t vanishes.
Presumably, i t would be too
simple to assume the perfect dimerization, since we know experimentally the doping produces solitons which strongly modulate the dimerization pattern, I t would be very i n t e r e s t i n g to perform numerical works which simulate the doping processes on computers.
A single chain of polyacetylene is represented
by the Su, Schrieffer, and Heeger model. d i s t r i b u t e d randomly along the chain.
Suppose the i m p u r i t i e s are
They would be charged, i f they would
release or accept electrons to become donors or acceptors, respectively.
The
Coulomb p o t e n t i a l s would modulate the e l e c t r o n i c energy levels at each s i t e randomly.
This is the " s i t e - t y p e " impurity effect.
Meanwhile, mutual
interactions among the dopants would disrupt the alignment of the atoms in the chain, giving rise to the random modulation of the electron t r a n s f e r between adjacent atoms.
This is the "bond-type" impurity effect.
I f there are cis
segments in the chain, t h e i r effects would also be taken into account as the bond-type impurities. The SSH model is generalized to include these effects.
A random
configuration of i m p u r i t i e s is a r b i t r a r i l y chosen. Static solutions are obtained by i t e r a t i o n procedures. depend on the i n i t i a l
We find many metastable solutions.
They
configurations from which the i t e r a t i o n has started.
It
is c h a r a c t e r i s t i c to the disordered systems. We calculate metastable bond configurations to show that the doping a c t u a l l y gives rise to solitons. solitons are charged. the frequency.
Local concentration of electrons helps find how the Dynamical conductivity is calculated as a function of
Infrared active phonon modes are also obtained.
are manifestly found to be pinned by impurities. modes become infrared active due to the doping.
The solitons
All the localized phonon We can thus conclude that many
features of the doped polyacetylene, observed experimentally, are reproducible by the numerical simulations. MODELS AND METHODS The Hamiltonian of the model of the doped polyacetylene is H = HSSH + Himp, where the f i r s t
(I) term is the SSH Hamiltonian,
D449 HSSH =
+ + _ ~ [ t o - m(Un+I - Un)](Cn+isCns + CnsCn+is) ns + (K/2)Z(Un+ I - Un)2 + (M/2)~U2n . n n
(2)
Here, C+ns and Cns are the electron creation and a n n i h i l a t i o n operators, respectively, at the s i t e n with the spin s, un the displacement of the CH unit at the s i t e n, K the force constant between the adjacent CH units, M the mass of the unit.
Electronic overlapping integral t O is modified by the l a t t i c e
deformation which gives r i s e to an electron-phonon i n t e r a c t i o n with the strength m , the dot above un means the time derivative. The s i t e - t y p e i m p u r i t i e s give
Him p
= nZsVnC+sCns ,
(3)
where Vn is a random quantity,
Him p
The bond type i m p u r i t i e s give
+ + = ~sVb(n)(Cn+IsCns + CnsCn+is),
(4)
where Vb(n ) is a random quantity. We w i l l
discuss the r e s u l t s of three s i m u l a t i o n works.
Two of them are w i t h
the s i t e - t y p e i m p u r i t i e s and the other w i t h the bond-type.
In the f i r s t
work,
the i m p u r i t y p o t e n t i a l Vn is so truncated t h a t i t can take only two values ~V. They represent the Coulomb p o t e n t i a l s by an acceptor and a donor, We choose t o t a l
respectively.
numbers of the s i t e s N, of the electrons Ne, of the i m p u r i t i e s
Nim p which is the number of the s i t e s whose Vn is e i t h e r V or -V.
Periodic
boundary conditions are used. The numerical works are c a r r i e d out as f o l l o w s . ( I ) Select Nim p i m p u r i t y s i t e s randomly to f i x a sample. (2) A s t a r t i n g c o n f i g u r a t i o n {un(O) } is selected randomly. (3) E l e c t r o n i c states are determined by the eigenvalue equation
(c i
Vn)~i(n)
=
-[t O - ~y~kl]~i(n-i ) - [t O -
~'(k)]~i(n+l) -
#n
where yn (k) = Un+l (k) - un(k) and k indicates i t value.
is the k-th i t e r a t e d
With the help of the e l e c t r o n i c wave f u n c t i o n
determined by
(5)
'
~i'
Yn(k+l) is
D450 y~k+l) :
-(2m/K) Z ' @ i ( n ) # i ( n + l ) + (2m/KN) Z Z'@i(m)@i(m+l), i,s m i,s
(6)
where the prime i n d i c a t e s the sum over the occupied states and the second term is necessary to s a t i s f y the p e r i o d i c c o n d i t i o n procedure is i t e r a t e d u n t i l a s u f f i c i e n t s t a t i c s o l u t i o n depends on {un(O)}.
ZYn = O.
convergence is obtained.
I f you s e l e c t a d i f f e r e n t
un(O), you may get a d i f f e r e n t s t a t i c s o l u t i o n .
The
The
set of
I t would be necessary to
examine several s t a r t i n g c o n f i g u r a t i o n s , at least, to gain i n f o r m a t i o n s directly
r e l a t e d to observations.
(4) With the help of the s t a t i c s o l u t i o n , phonon modes are determined in the same way as f o r the i m p u r i t y - f r e e system [7]. (5) Dynamical
c o n d u c t i v i t y is c a l c u l a t e d f o r the i n f r a - r e d frequencies to
f i n d the o p t i c a l a c t i v i t i e s
of phonons, using the method given in [8].
STATIC SOLUTIONS The r e s u l t s are shown in the f i g u r e s from 1 to 9. 2~2/~Kto
The parameters are
= 0.19, V = t o / 2 , N = 199, Ne = 200, and Nim p = 16.
l o c a t i o n s are shown by the c i r c l e s in Figs. 1 and 2. are i m p u r i t i e s w i t h V and the e i g h t closed w i t h -V. are almost overlapped each other.
The i m p u r i t y
The e i g h t open c i r c l e s Two of the closed c i r c l e s
This corresponds to a system w i t h
compensation between donors and acceptors.
Since Ne is l a r g e r than N by u n i t y
and N is odd, there would be a s o l i t o n w i t h a negative charge, i f there were no impurities.
Fig, 1 shows the bond c o n f i g u r a t i o n s ~n:(-l)n(yn-Yn_l ) as a
f u n c t i o n of the s i t e l o c a t i o n n.
There are three s o l i t o n s .
One is close to
the acceptor p o t e n t i a l s w i t h open c i r c l e s and another to the donor p o t e n t i a l s w i t h close c i r c l e s .
The t h i r d one is in an i m p u r i t y - f r e e region.
does produce s o l i t o n s in the numerical simulations, too. also c l o s e l y associated w i t h i m p u r i t i e s . electrons defined by
The doping
Small structures are
Fig. 2 is the l o c a l number d e n s i t y of
~ n : ( Pn_l+2 P n+ Pn+l)/4 to remove the rapid v a r i a t i o n
due to the d i m e r i z a t i o n ,
The h o r i z o n t a l l i n e gives the average value.
The
s o l i t o n at the acceptor p o t e n t i a l s has a p o s i t i v e charge, w h i l e the one at the donor p o t e n t i a l s has a negative charge. PHONONS AND INFRARED ABSORPTION Dynamical c o n d u c t i v i t y gives the i n f r a r e d absorption i n t e n s i t y shown in Fig. 3 as a f u n c t i o n of frequency in an a r b i t r a r y scale. : 0 t o 0.79 ~Q withWQ = ~
The o p t i c a l
Five large peakes are found at ~ = my w i t h
The abscissa extends from
phonon edge i s ~0 = O.63WQ.
v = a, b, c, d, and e.
Each is
due to an i n f r a r e d a c t i v e phonon mode. Fig. 4 is the phonon normal mode f u n c t i o n
D451
/
Fig.
I.
/
....
The bond c o n f i g u r a t i o n
~n=(-l)n(yn-Yn_ I ) as a f u n c t i o n of n.
Fig. 2. Local number d e n s i t y of e l e c t r o n s
Pn=(Pn_l+2Pn+Pn+l)/4.
ga(n) of small v i b r a t i o n s of un around the s t a t i c at ~ = wa in Fig. 3, as a f u n c t i o n of s i t e . o p t i c a l and a c o u s t i c components. ga (n)
:
The f u n c t i o n is an a d m i x t u r e of
frequency ~a = 0.036~Q i n d i c a t e s . It
,
(7)
We thus f i n d the a-peak is due to the Goldstone mode of
the s o l i t o n in the i m p u r i t y f r e e region. Fig. 6.
It
is almost f r e e as i t s low
The o p t i c a l component of gb(n) i s shown in
is l o c a l i z e d around the same s o l i t o n .
mode w i t h w b = 0.60 w ~ and e, r e s p e c t i v e l y . donor region.
which gives the peak
The o p t i c a l component is obtained by
(-l)n(-ga(n-l)+2ga(n)-ga(n+l))/4
and shown in Fig. 5.
solution,
Evidently,
it
i s the t h i r d
Figs. 7-9 show the o p t i c a l components f o r ~ = c, d.
The peaks c and d are associated w i t h the s o l i t o n in the
The former is due to the pinned Goldstone mode w i t h w c
= O.48mQ,
D452
..............
II ....
TT tt ce
clb
Fig. 3. I n f r a r e d absorption i n t e n s i t y ( a r b i t r a r y scale) as a f u n c t i o n of frequency ~ , The abscissa is f o r O
al
Fig. 4. Phonon mode f u n c t i o n ga(n) as a f u n c t i o n of n,
Optical component
a2
2'\. Fig, 5, Optical component g a ( n ) : ( - l ) n ( - g a ( n - l ) + 2 g a ( n ) - g a ( n + l ) ) / 4 "
D453
Opt Ical component
b2
Fig. 6. Optical component gb(n),
Opt I c = l component
c2
Fig. 7. Optical component gc(n).
opt I c a l
commonent
d2
Fig, 8, Optical component gd(n),
while the l a t t e r presumably due to the shape mode w i t h ~ d = 0.57~Q which is activated by the doping,
Fig, 9 shows that the e-peak is due to the strongly
D454
opt ical
component
e2
J~Cv',a,~V'~,~,'~'m~.~.A ), ~,t i,) ~v~r~v~I~q~,,pA '~ih/i)~u'..~h
Fig
9. Optical component g e ( n )
F
~p.~°,°.°.°.°@,o "o
" ""
o
~,~
J'
~ O o o O o o ~"
-]
J
50
100
Fig. I0. The bond c o n f i g u r a t i o n ( - l ) n y n . The open c i r c l e s
f o r odd n, and the
closed f o r even n
300
200
00
0
./I
Fig. I I . Averaged dynamical c o n d u c t i v i t y as a f u n c t i o n of frequency ~ .
Units
of the ordinate is 2(e~ /to)2/M and the abscissa ~ Q
pinned Goldstone mode, w i t h ~ e acceptor r e g i o n .
: 049WQ, associated w i t h the s o l i t o n in the
D455
SITE-TYPE IMPURITIES WITH A LONG-RANGE POTENTIAL We are p e r f o r m i n g two o t h e r independent works. doping by s i t e - t y p e
impurities
q u a n t i t y Vn is w r i t t e n Vn
One of them s t u d i e s the
w i t h a long range p o t e n t i a l .
The random
as
X(:~e2/~Irn-Rjl), J
(8)
where r n is the p o s i t i o n of the n - t h s i t e and Rj is the l o c a t i o n o f e i t h e r the dopant or the c a r r i e r
on o t h e r chains whose charge d e t e r m i n e s the sign ~.
q u a n t i t y Rj is a t h r e e d i m e n s i o n a l random v a r i a b l e . N = lO0.
Suppose t h e r e are f i v e donors, c l o s e to the chain,
with five electrons. distribution. antisoliton
I f the system were i m p u r i t y f r e e , p a i r s w i t h a polaron.
shown in Fig.
lO.
structures
middle.
which p r o v i d e i t
E f f e c t s of o t h e r dopants are a p p r o x i m a t e d by a random
Two s o l i t o n i c positively
The open c i r c l e s
charged.
bond c o n f i g u r a t i o n
structure
They t u r n out t o be
is being d e s t r o y e d in the
d e v i a t e from the closed c i r c l e s
s t r o n g l y each o t h e r .
at n = O.
Fig. I I
u n i t of the o r d i n a t e is 2(e ~ / t o ) 2 / M
I t shows
and polarons
is the dynamical c o n d u c t i v i t y ,
f u n c t i o n of frequency, which is o b t a i n e d as an average o f s i x t y
of the Goldstone modes is r e a d i l y
and
( - l ) n y n is
are f o r odd n and the closed f o r even n.
are found at n ~ 4 and 70.
The d i m e r i z a t i o n
The open c i r c l e s
we would f i n d two s o l i t o n
Typical static
the a c o u s t i c component is enhanced at the r e g i o n where s o l i t o n s interact
The
We t a k e ~; = lO and
and t h a t o f the abscissa
as a
samples. i~ Q.
The
The p i n n i n g
seen.
BOND-TYPE IMPURITIES The t h i r d
work s t u d i e s the bond-type i m p u r i t i e s .
bonds where Vb(n) is not v a n i s h i n g . between -V and V.
The q u a n t i t y
We randomly s e l e c t Nim p Vb(n) takes a random v a l u e
We f i n d t h a t t h i s t y p e of doping can a l s o produce s o l i t o n s .
The Goldstone modes are pinned and the shape modes become i n f r a r e d a c t i v e . gives qualitatively
similar
effect
t o the s i t e - t y p e
It
impurities.
DISCUSSION It
is i n t e r e s t i n g
t o f i n d t h a t some aspects of doping can be reproduced so
w e l l by s i m p l e s i m u l a t i o n s . impurities.
The s o l i t o n s
Even the system, which i s p e r f e c t l y
i n a c t i v e w i t h o u t the i m p u r i t i e s , solitons.
are c r e a t e d and pinned by the
The system, w i t h one charged s o l i t o n
a l s o i n c r e a s e the a c t i v i t y
d i m e r i z e d and i n f r a r e d
becomes s t r o n g l y a c t i v e due t o the c r e a t e d w i t h o u t the i m p u r i t i e s ,
can
due t o the doping. The small peaks on the r i g h t h a n d
s i d e of the a-peak in Fig. 3 are due t o a c o u s t i c phonons which become a c t i v e , m i x i n g w i t h the Goldstone mode [ 9 ] . by the doping.
The shape mode is found t o become a c t i v e
In the work w i t h the s i t e - t y p e
impurities
w i t h a l o n g - range
D456 p o t e n t i a l , the r e s u l t for the smaller i m p u r i t y concentration does not show any change in the a c t i v a t i o n energy of the pinned modes. This paper has discussed the phonon structures and infrared a c t i v i t i e s .
Electronic structures and
optical properties w i l l be reported elsewhere,
This work was p a r t i a l l y
supported by a Grant-in-Aid for S c i e n t i f i c Research from the M i n i s t r y of Education, Science, and Culture.
I t was also p a r t l y sponsored by Research
Association for Basic Polymer Technology. REFERENCES I. S. Ikehata, J. Kaufer, T. Woerner, A. Pro~, M,A. Druy, A. Sivak, A.J. Heeger, and A,G. MacDiarmid,
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