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POLARONS IN QUASI-ONE-DIMENSIONAL CONDUCTING POLYMERS
Adrian Bishop, D. Campbell, K. Fesser
To cite this version:
Adrian Bishop, D. Campbell, K. Fesser. POLARONS IN QUASI-ONE-DIMENSIONAL CON- DUCTING POLYMERS. Journal de Physique Colloques, 1983, 44 (C3), pp.C3-423-C3-428.
�10.1051/jphyscol:1983384�. �jpa-00222822�
JOURNAL DE PHYSIQUE
Colloque C3, supplement au n06, T o m e 44, juin 1983 page C3-423
POLARONS IN QUASI-ONE-DIMENSIONAL CONDUCTING POLYMERS
A.R. Bishop, D.K. Campbell and K. Fesser
T h e o r e t i c a l D i v i s i o n and Center f o r Nonlinear S t u d i e s , Loo Alamos National L a b o ~ a t o r y , Los Alamos, P!eu Mexico 8 7 5 4 5 . U.S.A.
R6sum6
-
Nous discutons ici certains aspects des excitations du polaron dans les polymbres conducteurs quasi-unidimensionnels.Nous dicrivons la structure des polarons pour les chaines
trans-
et *-(CH) , ainsi que la correspondance existant entre les polarons co$ventionnels mono-bande et ceux issus des modbles du type SSH. Finalement, nous 6tablissons l'existence de polarons dans les polym&res diatomiques conjugu6s du type fA=Bjx, ainsi que leurs propriit6s.
Abstract
-
We discuss several aspects of polaron excitations in quasi-one-dimensional conduting polymers. We review the structure of polarons intrans-
and &-(CH),
illustrate the relation of polarons in SSH-type models to cznventional one-band polarons, and establish the existence and properties of polarons in conjugated diatomic polymers of the fA=Bjx type.I. INTRODUCTION
In current theoretical models [l-61 of quasi one-dimensional polymers, localized, nonlinear excitations play a critical role. Indeed, many contributions to these proceedings are devoted to various aspects of these excitations, in particular to the central question of whether experiments support the theoretical expecta- tions. In the past, most of the effort in this area has focused on the "kink"
solitons predicted by models of trans-(CH) [I-61; the exotic spinlcharge relation of these kinks (S = 112, Q = 0; S = 0, = ?e) heightens the interest in con- firming experimentally their existence in, and relevance to, the real materials.
More recently, it has been shown [7-111 that the same theoretical models--specifi- cally, the SSH model [3] and its continuum limit 14-51
--
that predict kink solitons in trans-(CH) also predict "polaron" solitons. Although more conven- tional than the kinks 3f trans-(CH) , the polarons are no less important. First, because they are they lowest energ$ excitations available to a single electron, polarons are expected to play crucial roles in experimental properties at low doping levels, including optical absorption [12,13] and electrochemical dop- ing [13]. Second, polarons are expected to be "generic" in quasi-one-dimensional conducting polymers, since (unlike kink solitons) they do not require the (a- typical) degeneracy of the ground state found in =-(CH).
In particular, polaron solutions have been found in theoretical models for &-(CH) [9-111 and, as we shall see, for diatomic polymers of the form fA=Bj [9,1$]. Further, polarons can be anticipated to be relevant for a wider clags of polymers, in- cluding polydiacetylenes and polyparaphenylene. In this brief report we can touch on only selected aspects of polarons in quasi-one-dimensional conducting polymers. We have chosen to focus on two of the most recent results: (1) the relation [14] of polarons in SSH-like models to the familiar polarons of the (one-band) molecular crystal model and (2) the nature and properties [15] of the polaron in an fA=Bj polymer chain. To clarify these two results, we must start with a short review %f polarons intrans-
and c&-(CH) [ 9 - 1 1 1 .Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983384
11. POLARONS IN zS_AND -CLS-(CH)y
As first shown by Brazovskii and Kirova [9], a natural and simple generalization of the SSH [2-41 model provides a framework for discussing nonlinear excitations in both
trans-
and ~is-(cH)~. In the continuum limit, the adiabatic mean field Hamiltonian is (9,111-where o. is the i~ Pauli qatrix* w2/g2 is the net effective electron-phonon th couplini constant, Jc (x) = (u (x),v (xy) represents the two components (right and left moving) of the electron field, and vF is the Fermi velocity. In (1) A.
represents the "intrinsic" gap which is driven by the Peierls instability due tz the (n-) electron-phonon coupling, whereas A is an "extrinsic" gap which is not influenced by the n-electron dynamics [9,f 11. For A = 0, the Hamiltonian possesses the symmetry A(x) -+ -A(x), (u,v) + (u,-v), and %he ground state has the two-fold degenerate structure appropriate to trans-(CH)
.
For A # 0, this symmetry is broken, and the ground state is unique, as i? *-(CH)~~ and related polymers.Varying H leads to the equation for the single particle electron wave functions
and the self-consistent equation for the the intrinsic gap
The polaron solution to these equations is [9-111
hp(x)
=
Ai(x) + A =i? -
k v [tanh ko(x+xo)-tanh ko(x-xo)]e o O F (3)
where tanh 2k x = kovF/io. Here
xo
is the value of the gap in the ground state0 0
and satisfies A Qn
2
/Ao = he/zo with A. = W exp(-A-I), where W is the full band width and h = g /nv 2 F wqQ'
When Ae = 0, as in =-(CH)*,5
= Ao.Recalling that the gap parameter for kink solution for trans-(CH) has the form A (x) = A tanh A x/v 12-51, we see thatshe polaron has a structuze similar to a kmk/rntigkink p A r . k Indeed, for k v /A + 1, %(x) approaches pr-ciaely an infinitely separated !is configuratio2.F
12
the opposite limit, k v /A << 1--
which will be relevant to our later discussion of the reduction ?oF th% conven- tional polaron form
--
$(x) becomes a small localized "dimple" on the ground state.The electronic states in the presence of the "distortion" &(x) consist of the
2 2 - 2 % 52 :2a
valence (&(k) = -(k v +A ) ) and conduction (~(k) = +(k2v2+~ ) ) bands and two
F o F o
intra-gap states at &+
- =
two, with wo = .'):v:k-:?i( As we have discussed exten- sively elsewhere [11,16], these two localized states lead to interesting structure in optical absorptionbelow
the inter-band edge. There is some evidence that parts of this structure are seen in optical experiments at low doping levels112,171-
It is vital to stress that a "polaron" excitation in this model consists of the distortion $(x), the filled valence band, the appropriate electronic occu- pation of in ra-gap states, all self-consistently solving the equations (2a) and (2b). Indeed, as we remarked earlier. the value of kovF is determined by the gap
equation (2h). Let n , (I?.) be the numbe_r of electrons in_ the localized state
c + ( E - > . 1 f i : r ~ d i i c ~
e
J U L ~ :hat kovF = A sin0 aiii: w = CI ius6. T i i r ~ i 8, ar~dhence kovF, is determined by [9-11)
For trans-[CHj
,
y = 0 and the two polaron solutions have 0 = 7114 and n+ = 1, n- = 2--
the "elexctron polaron" state--
and n+ = 0, n- = 1--
the "holejolaron".Note that when n+ = n-
--
for n+ = 0 , 1, or 2--
8 = n/2, so k v = A,
wo = 0and x + m. Hence these would-be ffbipolbronff states in
=-&fix
!?e sslmplyinfinftely separated SS pairs! For +-(CHI
,
y # 0 and for combmat-ion of n+ and n- in (5) one canJind a solution fo? k vF satisfying 05
k v <4
In particular, since kovF < A,
there could never Be an infinitely sepa~afed ~g'pair in cis-(CH).
This correc6?ly reflects the result that kink-like solutions do not exist in &-(cH)~ and that only polaron-type nonlinear excitations (single polarons, bipolarons, etc.) are present.111. RELATION OF SSH POLARONS TO CONVENTIONAL POLARONS
To those readers acquainted with conventional one-band polarons in quasi-one- dimensional systems, the rather complicated structure of the polarons in the SSH- type models may at first seem both unfamiliar and formidable. In fact, the SSH polaron is a natural generalization to the two-band case
--
that is, the case in which the valence band electron effects are explicitly included--
of the conven- tional polaron in quasi-one-dimensional systems and, as_we shall show, reduces to the conventional polaron in the weakly bound (k v /A << 1) limit. Here we simply sketch this relation; further details are av2ifabfe in the literature [14].In the conventional one-band polaron
--
as discussed in the context of the mole- cular crystal model [18], for example--
one starts from the qualitative picture that a single electron added to a deformable crystal creates a molecular displace- ment which self-consistently localizes the single electron wave function. Thus one concentrates on a single localized electronic level just below the conduction band: an "electron polaron". The resulting equations for the single electron wave function, xo(x), and molecular displacement, yp(x), can be written in scaled form aswhere Y~(X),~=
I X
(x)lL is the self-consistent potential. Equation (5) is the well-known nonltnear Schrsdinger equation".To see that the full equations (2) reduce to those in (5) for kov /Zo << 1, it is 72) =
convenient to introduce the new combinations
(il)
= (un+vn)/&,a
-i(un-vn)/n, for which Eq. (2.a) becomes
a(:')
(2)and E):$n = -vF
-
(~l~(x)+A~)(~For a weakly bound polar_on, we expect
hp
A.(,x)+Ae to differ only slightly from its ground stat: value A.
Defining 6p(x) 2 A-
%(x), we can study Eqs. ( 7 ) in powers of l / A o For eqectronic states near t%e bo tom of the conduction band,-
E = .A + 0(k2). Thus, since cn +
hp
E 2Z0, (7.b) implies"(1)
amn
/ax)/2ao so that I$(~) is of order l/aO relative to (I(') and fnrther, thatC3-426 JOURNAL DE PHYSIQUE
this leading term in 8 be calculated directly from $I('). Hence to leading order i:: !,'Ao, ~ i zhc y Lq. 7 . c i s . Fociisirjj on a weak:y L-..-'
Yn U U U I L "
state just below the conduction band with E =
0 F substituting for ac(2)/ax
-
by differentiating ((lP a second tine--
valid toleading arder in k v
- -
we obtein the equaticn for t h e aingle independent component of the elgc wave functionwhich is precisely the same Schrb'dinger-like form found i n z q . (6.9). Similar manipulations can be used to show that, in the limit k v /A << 1, the self- consistent gap equation reduces to the relation O F 0
so that the SSH polaron
does
reduce, in the limit k v /ao << 1, to the convention- al one-band result. Since the explicit proof of0(6) IS too lengthy to include here, let us give (a perhaps more valuable) intuitive argument for this result.The obvious question arising from comparison of (9) and (2.b) is, "What happened to all the states in the valence band and Lo the negative energy localized state
(at E - = -wo)?" Clearly, for small k v /A these states are separated by a
"large" energy (- 2a0) from the state a?
%+
P 7 + w.
Further, for the case of the electron polaron we are considering, these states are fully occupied in the polaron configuration just as they are in the ground state. This motivates the"frozen valence band" approximation, in which one argues that, since 4(x) differs only slightly from the ground-state value, bp(x) is small and thus the shifts of the states in near the valence band are small, so that one can approximate the sum over all states with energies less than zero by its value in the ground state. Hence these terms all cancel in the equation for 6p(x), which then takes the form shown in (9). That this "frozen valence band" approximation is correct to leading order in kovF/ao is proved elsewhere 1141.
Finally, note that for --like polymers in which y is very large, this reduction is of more than academic interest, because as (4) shows, in this case k v /A can
F 0
indeed be very small, and the conventional one-band polaron picture can ge quanti- tatively accurate.
IV. POLARONS IN THE fA=B+- POLYMER
Recently Rice and Mele [19] have developed a coupled electron-phonon model [9] of a diatomic conjugated polymer, fA=Bj
.
They find two different types of kink solitons which have irrational chargesx, the values of which depend on the differ- ence (2a = E -E ) of the energy levels of the atomic p orbitals of the two con- stituent ato$s,B~ and B. In this section we show that the fA=B+ polymer also admits a polaron solution 19,151. For small a, the polaron is as u&al the lowest energy state-available to a single electron added to the ground state. However, for a > A ~ / J Z , where 2A is band gap that would exist due to the electron-phonon coupling ~f a were 0, 'the polaron breaks up into a kinklantikink pair. The striking consequence of this is thct, if a physical realization of the model system is found with a > A 142, the addition of a single electron to the system would lead to the creatio: of two independent solitons with irrational charge.In the continuum limit the coupled electron-phonon equations become, following the notation of Rice and Mele [19],
and
The r e l a t i o n between t h e e l e c t r o n wave f u n c t i o n components (A ,B ) i n t h i s c o n v e n t i o n - and t h o s e i n o u r p r e v i o u s e q u a t i o n s i s A = ( u +?v
q/\l?,
B =(u - i v ) / 4 2 . When a = 0 , Eqs. (10) r e d u c e t o Eqs. ( 2 ) , w i t h A =
'b.
"I'he c o n s f a n t c % s $ r o p o r t i o n a l t o t h e s q u a r e o f t h e e l e c t r o n - p h o n o n coupl?ng i n t h e o r i g i n a l l a t t i c e model.The p o l a r o n s o l u t i o n t o t h e s e e q u a t i o n s t a k e s t h e (by now) f a m i l i a r form
$(x) = A
-
kovF[tanh ko(x+xo>-
t a n h ko(x-xo>] ( 1 l . a )2 2 + ,
w i t h t a n h 2k0x0 = kovF/A and kovF = ( ~ ~ + a ~ - ( ) ' = (Ao-%) ( 1 l . b ) s i n c e
n2
= A'-
a 2.
The s e l f c o n s i s t e n t gap Eqs. (1O.c) y i e l d s kovF = A ~ / & = w . A s u s u a f , t h e p o l a r o n h a s two l o c a l i z e d s t a t e s s y m m e t r i c a l l y p l a c e d I n t h e g$p a t E+ = +w.
F o r l a t e r comparison t o t h e k i n k s t r u c t u r e s , we, n o t e t h a t t h e wave f u n c t i o nPA
,B ) f p r t h e s t a t e a t +w i s A = i ( k (w + a ) / & )' s e c h k (x-x ) and B = (k (w -&)/%w ) * s e c h k (x+x ) . f u r t h e ? , t h e 8 a v 8 f u n c t f o n ( A - o , ~ O o ) f 8 r t h e s p a t e a 2 -u? s a t i s % i e s A-o = O B ~ (go + - X ) and B-o = Ao(xo + - x o ) .The e l e c t r o n p o l a r o n h a s n+ = 1 , n- = 2 , whereas t h e h o l e p o l a r o n h a s n+ = 0 , n- = 1. I n e i t h e r c a s e t h e p o l a r o n s have s t a n d a r d change and s p i n - a s s i g n m e n t s : Q = r e and S = 4 . F u r t k e r , t h e t o t a l p o l a r o n e n e r g y i s Ep = (242/n)Ao, i n d e - p e n d e n t of a ( f o r a < A0/J2)!
These r e s u l t s f o r p o l a r o n s i n t h e fA=Bj polymer a r e n e a r l y i d e n t i c a l t o t h o s e f o r
trans-
and cis-(CH) w i t h t h e veryX i m p o r t a n t d i f f e r e n c e t h a t whereas t h e l o c a l i z e d p o l a r o n gap l e c e l s a r e f i x e d a t w = A142,
t h e s i z e and s t r u c t u r e o f t h e p o l a r o n ( s e e ( 1 1 ) ) a r e r e l a t e d t o b o t hSn
a n 8 A ( o r a ) . C o n s i d e r , i n p a r t i - c u l a r Eq. ( l b R e c a l l i n g t h a t A = ( ~ t - a ~ j ' and kovF = A,/@ f o r t h e p o l a r o n s o l u t i o n , we s e e t h a t u n l e s s a <~ ~ / \ l ? ,
t h i s e q u a t i o n h a s no s o l u t i o n ( f o r r e a l x,
t h e " s e p a r a t i o n " o f t h e k i n k and a n t i k i n k y h i f h + " m a k e up" t h e p o l a r o n ) . Thus a?though t h e r e e x i s t s a d i m e r i z a t i o n [A = (A -a )_ ] i n t h e polymer f o r any a <Ao, t h e p o l a r o n s o l u t i o n e x i s t s o n l y f o r a < So/J2. C o n s i d e r i n g a a s a v a r i a b l e parameter_ (by v a r y i n g t h e chemical c o m p o s i t i o n of t h e polymer) we n o t e t h a t a s a + A /J2 from below, x + and t h e p o s i t i v e and n e g a t i v e e n e r g y e l e c t r o n wave- f u n c t f o n s o f t h e p o l a r o E go smoothly t o t h o s e of t h e a p p r o p r i a t e A and B k i n k s o l u t i o n s of R i c e and Meie [ 1 9 ] . A t t h i s poi.nt, t h e t o t a l p o l a r o n e n e r g y , which i s always E f o r a < A /J2, becomes p r e c i s e l y e q u a l t o t h e e n e r g y o f t h e i n f i n i t e - l y s e p a r a t e d P (AB) sol?ton p a i r . F o r a > A 1 4 2 , t h e l o w e s t e n e r g y s t a t e a v a i l a b l e t o a s i n g l e e l e c t r o n i s a p a i r of k i n k s w!th i r r a t i o n a l c h a r g e d e t e r m i n e d by a . T h i s " f r a g m e n t a t i o n o f e l e c t r o n i c charge" is a n o v e l and p o t e n t i a l l y v e r y e x c i t i n g f e a t u r e o f t h e fA=B3 system.
ACKNOWLEDGMENTS
We a r e v e r y g r a t e f u l t o S e r g e i B r a z o v s k i i , David Emin, Ted H o l s t e i n , Baruch H o r o v i t z , Gene Mele, Michael R i c e , and Leonid T u r k e v i c h f o r v a l u a b l e comments.
REFERENCES
1. K o t a n i , A . , J. Phys. Soc. J a p a n 42 (1977) 408 and 416.
C3-428 JOURNAL DE PHYSIQUE
brazovskii, S. A . , JETP Letters
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(1978) 606 (trans. of Pisma Zh. Eksp.'Lecz. Fiz. 'ib ( 1 9 7 8 ) 656); ar.2 ScS:iet Pk;s. E T P 5 1 "3O0) \ A u 342 ( t r a i i s . of Zh. Eksp. Teor. Fiz. 78 (1980) 677).
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(1980) 2099.Takayama, H., Lin-Liu, Y. R. and Maki, K., Phys. Rev. B (1980) 2388;
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(1981) 742.Rice, M. J., Phys. Lett. 71A (1979) 152; Rice, M. J., and Timonen, J., Phys.
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(1981) 4 (trans of Pisma Zh. Eksp. Teor. Fiz. 3 (1981) 6); Brazovskii, S., these proceedings. The elegant "combined Peierls systems", framework developed in these references contains also the results for the fA=Bj chain, although this explicit physical application is not_ mentioned. ~urrher, the interesting instability of the polaron for a > A /J2 is not indicated.Campbell, D. K., and ~ i % h o ~ , A. R., Phys. Rev. B z (1981) 4859; Nuc. Phys.
B200 [FS4] (1982) 297.
Bishop, A. R., and Campbell, D. K., in Nonlinear Problems: Present and Future, ed. A . R. Bishop, D. K. Campbell, and B. Nicolaenko (North Holland, Amsterdam, 1982) 195; Bishop, A. R., Campbell, D. K., and Fesser, K., Mol.
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(1981) 253.Etemad, S., these proceedings; Etemad, S., Feldblum, A., Heeger, A. J., Chung, T.-C., HacDiarmid, A . G., Bishop, A. R., Campbell, D. K., and Fesser, K., Phys. Rev. B (to be published).
Heeger, A. J., these proceedings.
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(1982) 6862.Campbell, D. K., Phys. Rev. Lett. (to be published).
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For a different possible interpretation, see Orenstein, J., these proceedings.
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