Interchain coupling in polyacetylene

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Interchain coupling in polyacetylene

P.L. Danielsen, R.C. Ball

To cite this version:

P.L. Danielsen, R.C. Ball. Interchain coupling in polyacetylene. Journal de Physique, 1985, 46 (2),

pp.131-137. �10.1051/jphys:01985004602013100�. �jpa-00209951�

131

LE JOURNAL DE PHYSIQUE

Interchain coupling ⁱⁿ polyacetylene

P. L. Danielsen and R. C. Ball

Cavendish Laboratory, Madingley Road, Cambridge ^{CB3 OHE, U.K.}

(Reçu ^le 27 juin ^1984, accepté le 17 octobre 1984)

Résumé.

Nous considérons le couplage ^entre deux chaînes de polyacetylène, (i) pour les chaînes ^sans solitons et

(ii) pour les chaînes chacune ^{avec un} soliton neutre. Dans le premier ^{cas nous trouvons une} correction nouvelle pour la grandeur ^{et pour la} position de la bande interdite. Dans le cas avec solitons, ^{nous trouvons une} interaction

entre leurs niveaux au centre de la bande interdite. Les différences entre le cas de dimérisation alignée à ± ^{~ et} le

cas de dimérisation décalée sont démontrées. La séparation ^entre les niveaux est 0394E = t~ l/03BE ^sinh l/03BE, ^{dans le cas} aligné, ^{et ~} 0(e-kf03BE) ^{dans le cas décalé} (où ^{l est la} séparation ^{entre les} solitons, kf ^{le vecteur d’onde de} Fermi, 03BE ^la

taille du soliton, ^et [1] kf 03BE ~ 10). Cette interaction contribue à la localisation tridimensionnelle des solitons,

ce qui ^renforce l’argument ^{contre le modèle} solitonique ^de conduction dans le polyacétylène. ^{Ainsi nous} prévoyons

aussi l’existence d’une densité critique de solitons 03C1c ~ 0,05; où l’ordre local favorisé entre chaînes devient aligné

au lieu de décalé.

Abstract.

The interchain coupling ^{between two} polyacetylene ^{chains is} considered, both in the absence of

solitons, ^{and with one} neutral soliton on each chain. A new correction to the size and position of the band gap is found in the former _case, and when solitons are present it is shown that their mid-gap levels interact. In this case

the differences between the matched (same dimerization pattern ^at ± ~) and the mismatched cases (opposite pattern ^{at ±} ~) ^are clearly illustrated. The mid-gap ^level splitting ^{is found to be 0394E}

t~ l/03BE ^sinh l/03BE, ^{in the}

matched _case, and ~ 0(e-kf03BE) in the mismatched case (where l is the soliton separation, kf ^{the Fermi} wavevector, 03BE the soliton width, ^and [1] kf 03BE ~ 10). This interaction increases the strength of the 3 D pinning ^{of solitons, there-}

fore strengthening ^{the case} against the solitonic model for conduction in polyacetylene. ^{It also} predicts the existence of a critical density 03C1c ~ 0.05 of solitons, ^{at which the} preferred ^local ordering ^{between chains} switches from mismatched to matched dimerization.

J. Physique ⁴⁶ ( 1985) ^{131-137 FTVRIER} 1985,

Classification

Physics ^Abstracts

72 . 80L

1. Introductioa

There has been considerable theoretical interest in

polyacetylene, (CH)x, ^{in recent} years, it being ^the simplest ^{of a now} growing ^{class of} conducting conju- gated polymers. The discrete soliton model of Su, Schrieffer, ^and Heeger [1] (SSH) ^{and of Rice} [2] (and

the extension to the continuum limit by Takayama,

Lin Liu, ^{and Maki} [3] (TLM)) ^is attractive, ^{both for}

its simplicity and for its apparent ^{success in} explaining

much of the magnetic, electric, ^and optical ^{data for} trans-(CH)x, though ^{there are} many discrepancies

with experiment [4]. ^{It is} possible that these could be

explained by including quantum fluctuations, ^electron correlations, ^lattice deformation terms, and three dimensional ordering. The relative size of the intra- and inter-chain hopping terms, to ^and t.1’ ^estimated

to be - 2.5 eV and - 0.1 eV respectively [1, 5],

suggest that conduction in three dimensions should be relatively easy. This, together ^with X-ray scattering

results which suggest ^{that the} dimerisation pattern shows 3 D order [6], ^has drawn much of the recent interest to 3 D effects.

There have been several publications ^{on the} subject [7, 8] ^{which treat} the model of two parallel (CH)x

chains where coupling ^{is due to} interchain hopping

of the n-electrons. They show that this mechanism favours mismatched over matched ordering ^{of the}

chains. Also, the confinement _energy of two solitons

on mismatched (opposite dimerisation pattern ^at + oo ) chains is found to be

where a is the lattice spacing, ^{I the soliton} separation, and ç ^the soliton width. However, ^several important

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004602013100

132

points ^were overlooked, including the interaction of midgap ^states between matched chains, ^{and the} subsequent marked differences between the matched and mismatched cases.

The object ^{of this} paper is to both clarify ^{and to}

extend some of these previous results. Section 2 treats the case of two pristine (CH)x chains and summarizes the derivation of the preference ^of mismatched order-

ing. ^Some interesting ^{new effects on the} dependence

of both the size and position of the band _gap on the

ordering of the chains are then presented ^{Section 3}

treats the confinement _energy of two neutral solitons

on different chains finding ^{some new and} important

terms. We illustrate the differences between the two

possible orderings, and calculate the splitting ^{of the} midgap levels in the matched case. The confinement energy of the two solitons is then found-showing ^that

the matched and mismatched cases differ considerably, leading ^{to a} critical concentration Pc of solitons, ^above which matched ordering ^is preferred. Consequences

of these results are then discussed

2. Interchain coupling ^{between 2} pristine (CH)x ^chains.

The model considered [7] is that of two parallel (CH)x

chains (j

1, 2) ^described individually by ^{the SSH}

Hamiltonian [1] ]

(where ^the spin suffixes have been suppressed ^for simplicity), ^and coupled by ^an interchain hopping

term

We then obtain the simple ^ladder hopping problems

shown in figure ^{1. In} the notation of SSH, _to ^{is the} hopping integral for the undimerized chain, ^and tl

^{2 au where a is the} electron-phonon coupling

constant and u is the displacement of the carbon atoms

Fig. ^1.

Ladder models for : a) ^matched ordering, ^and b)

mismatched ordering.

from their undimerized positions. ^It is assumed

throughout that tl tl.

In the case of a single ^chain (1) ^we had, ^{in the basis}

(01, OC),

In the case of the ladder problems, ^{if we use the basis} (03C8v1, 03C8c1, 03C8v2, 03C8c2), the Hamiltonian takes the form :

where 6 = + /

^{1 for} matched/mismatched ordering.

Clearly, the matched case will simply ^{lead to} bonding

and antibonding states, ^with energies

(where ^Ek

^± (82 ⁺ j2), /2 ^{is the} single ^chain energy)

as shown in figure ^2a. Thus, ^{as both the} bonding

and antibonding ^{states are} fully filled, ^{the net} change

in energy, AE matched

^0.

Fig. ^2.

^{Band structure for :} a) ^matched case, and b) mismatched case.

The splitting ^{of the} energy levels in the mismatched

case is easily ^obtained by diagonalizing (5) ^with

6

1, giving :

which behave as shown in figure ^{2b. In this} (mismat- ched) case, the _presence of two interacting ^chains

will affect the size of tl, ^{but we assume} that this is

negligible compared ^{to the} energy directly ^associated

with the interchain hopping.

So, neglecting the shift in tj, ^the difference in energy from the two individual chains is given by :

giving, ^after integration ^over the first Brillouin _zone,

(in ^agreement with the results obtained by Baeriswyl

and Maki [7] using the substitution

and diagonalizing using ^a Bogoliubov transforma-

tion). Thus mismatched ordering ^is preferred ^to

matched ordering by ^{an amount} W ~ 1.3 x l0-3 eV

or 15 K per ^atom. This is in agreement with results found for coupling ^{due to} coulombic forces [9, 10]

where the mismatched ordering ^was also found to be

preferred. ^Even allowing ^{for the} uncertainty ^{in the}

value of t.1’ which has been reported ^{to be as low as}

25 meV [11] (giving ^{W - 1} K), coupling ^{due to inter-}

chain hopping will be the stronger ^effect.

The effect of the interchain coupling ^{on the band}

structure is shown in figure ^2. This illustrates the two main differences between the two types ^of ordering : Firstly, the size of the band _gap depends ^{on the} ordering :

and as t1 ~ ^{0.35 eV} [1], Eg - ^{1.2 eV and 1.4 eV} respectively, ^{the band} gap being larger ^{in the} preferred

mismatched case.

Secondly, ^the position of the band _gap changes,

from being ^{at the Fermi} wavevector kf ^{to a} position at kf ^± q, where

3. Confinement energy of ^two solitons on different chains.

3.1 WAVEFUNCTIONS FOR THE MATCHED AND MIS- MATCHED CASES.

The problem considered is that of two parallel (CH)x ^{chains, both} containing ^a

soliton defect. There are two (non equivalent) ^cases

of interest, those of the matched and mismatched

cases shown in figure 3, ^where 0 ^{denotes a} soliton,

and A and B are the two different (though degenerate) phases of (CH)x.

’

The wavefunctions of a single chain, ^can be found

using the continuum approximation [3]. ^{For a chain}

of length L, ^{with a} soliton centred at the origin they

are given by [7] (in spinor representation ^where

for the bound midgap state, ^and

for the conduction band (upper sign) ^{and the valence}

band (lower sign) electrons. Here

and

and the wavevectors for the conduction and valence

Fig. ^3.

Soliton-soliton confinement for : a) ^matched

case, and b) mismatched case.

134

bands satisfy ^the boundary ^conditions

respectively, ^with Ok

^tan-’ kvfldo, ^{where n, m are} integers.

What happens ^{when a soliton moves or the dime-} rization pattern changes ?

When a soliton moves the underlying dimerization pattern ^{does not} change; ^{and if one moves} the dimeri- zation origin by any factor of 2a, there should be no

change ^{in the} physical ^results. Thus, ^{there are two}

distinct fixed positions ^{in the} problem, ^{one which}

locates the centre of the soliton, and the other which fixes the dimerization pattern ^{at ± oo.}

Hence, the wavefunctions should read exactly ^as

in equations (13) except the x’s should be replaced by (x - I) ^and equation (12) ^{should now read}

where I is the distance of the soliton from the origin,

and where d defines the dimerization pattern ^at

± oo ; if d ⁼ 0 or a multiple ^{of 2a} the dimerization pattern ^is unchanged, and if d is an odd multiple ^{of a,}

then the dimerization pattern is reversed.

Therefore, ^{for a} change in dimerization pattern (d = a), ^we have for the reverse chain (see Fig. 3)

This corresponds ^{to the} substitutions made by Baeriswyl ^{and Maki} [7], ^where d Z ^was replaced by

-

d 2 ^and t/J 2 by Q3 ql2l leaving the individual Hamil- tonians invariant. The physical significance ^{of this} symmetry ^is easy ^{to see :} changing ^the sign ^of d 2 ^has

the effect of reversing the dimerization pattern, ^which has, ^{as shown} above, the effect of changing ^{the wave-}

function to i 63 t/J 2.

3.2 SOLITON CONFINEMENT ENERGY. - It is argued

here that the interaction of the two (CH)x chains will be dominated by ^{two effects :}

Firstly, when the solitons are close, the interaction of the well localized midgap ^states (this ^{will also} depend, ⁱⁿ general, ^on the soliton occupation numbers, though they ^{are assumed to be} neutral) ; and, secondly,

when the solitons are far appart, the effect of either

creating ^or destroying lengths of the favoured AB

phase. ^{In the matched case} the solitons will be ^encou-

raged ^to separate by the formation of the energetically preferred ^AB region, whereas in the mismatched case

they ^{will be confined} by this effect.

Using degenerate perturbation theory ^the splitting

of the midgap ^states is determined by

i.e.

In the matched _case, for a soliton at x

0 on chain 1, and at x

I on chain 2,

so that

giving

However, in the mismatched ^case the wavefunctions

are now (for ^{the same soliton} positions ^as before)

so that

where kf ç ~ ^{10. Thus, the} midgap splitting ^{is of the}

order of ti- ^{for small soliton} separations in the matched

case (see Fig. 4a), ^{and is a} negligible effect in the

Fig. ^4.

^{a. Different} competing effects in AEconf f in the matched case. b. AEconf for both matched and mismatched

cases (zero of energy in each ^case is arbitrary).

mismatched case, in agreement ^{with the exact results} obtained by Baeriswyl and Maki. As [5] t.1 ~ ^0.05 to the effect in the matched case is expected ^{to be} signi-

ficant.

The details of the treatment of the mismatched case are given ^{in the} appendix, the results being (see ^also Fig. 4b)

which is, ^as expected, asymptotic ^to

at large ( ± ) distances, ^as predicted by simply ^consider- ing the effect of the annihilation of AB phase.

In the matched _case, exact analytic solution of the

problem proves ^to be more involved, though ^some

progress has been made (see ^the appendix). ^Never- theless, the form of AEconf ^is easily ^deduced.

Again, ^at large (ip) distances, ð.Econf ^{will be asymp-}

totic to

whereas the interaction of the midgap ^{states will}

dominate at small separations (see Fig. 4a). AE,:.nf

for the matched case is shown in figure ^4b.

Thus in both _cases, the solitons experience ^{a confi-}

nement energy tending ^to pin ^them together. ^The midgap interaction produces ^a relatively strong ^but local confinement energy in the matched ^case (of depth t.1 ~ ^{0.1 eV}

^{1 150 K, and extent - ± 2.5 ç =}

± 35 a), while the confinement energy in the mis- matched case is given by W

t2/7rto _ ^{1.3 x}

10- 3 eV = 15 K _per site. Even if they ^{are not} pinned

to impurities ^the midgap interaction serves to streng- then the hypothesis that solitons within crystalline regions will be well localized. The above results imply that, ^{at room} temperature, they will be localized within - 100 lattice constants in the mismatched case, and within - 10 lattice constants for matched chains.

Although the interaction of the midgap ^states

strengthens ^the hypothesis that solitons within (CH)x

are effectively pinned by ^{3 D} effects, ^{it does} produce

some other interesting consequences. One such effect

can be illustrated by ^a simple model : consider 2 (CH)x

chains as in figure 5, ^one containing ^a large ^{number of}

solitons in fixed positions, and the other one mobile defect. If the concentration of solitons on chain 1 is greater ^{than some} critical concentration _pc (as ^is illustrated) ^{the most stable} positions for the mobile soliton will be at a locally ^matched position (i.e. ^at positions 1, 3, ^or 5). However, if the concentration is less than pc, the minima ^at positions 2, 4, ^{and 6 will}

have the lowest _energy, and local mismatched ordering

will be preferred. Clearly, ^this argument ^{can be}

generalized ^{to two} arrays of solitons, ^{and to more} complicated systems. Thus, ^even though ^mismatched ordering ^is preferred ^for pristine chains, ^matched ordering ^{will be} equally relevant in the _presence of solitons.

The consequences of the preference of mismatched

ordering ^{have been} pursued by ^{Bohr and} Brazovskij [12] through ^a mapping ^{to an} Ising model. Their treatment neglects ^the competing potential ^{wells at}

matched order, which would give ^a strong ^four spin

term in their model Hamiltonian.

A rough estimate for the critical concentration can

easily ^{be found} Assuming ^{that the solitons are} equally spaced, ^{a distance I} appart, ^{then from} figure ⁵

i.e.

which gives

i.e. roughly ^one soliton defect _{per 200} carbon atoms.

Even though ⁱⁿ reality the situation will be far

more complicated ^than this, this does illustrate that the local soliton density will effect the local ordering (i.e. either matched or mismatched), and, therefore,

the size and position ^{of the} bandgap.

Fig. ^5.

^Soliton density ^{effects on} local chain ordering.

136

4. Discussion and conclusion.

The model considered [7] ^{is the} simplest possible ^form

of interchain hopping, ^{and we have} further restricted the discussion to a pair of chains. Not only ^{the total}

energy but also the size and position of the band _gap

depends ^{on the} matching ^{of the} dimerization, ^as

summarized in table I.

Thus, interchain hopping ^{results in mismatched} ordering being preferred by - 15 K per ^atom. The

bandgap (and ^its position) ^varies by 2 t I - ^0.2 eV, depending ^on the local environment of the (CH)x

chains.

The confinement _energy of two solitons on parallel (CH)x ^{chains can} be studied in the continuum model

of TLM [3]. ^{The exact} wavefunctions available in this model depend explicitly ^on the dimerization pattern

resulting ⁱⁿ different results in the matched (same

dimerization pattern ^{at +} 00) and the mismatched

cases (opposite dimerisation pattern ^at + oo). ^In

both cases, at large separations, Econf is dominated

by the effect of either creating ^or destroying lengths

of the preferred mismatched AB phase. However,

as the solitons approach ^{each other,} the interaction of their localized midgap ^{states becomes} important.

The splitting ^{of the} midgap ^{states is} given by

resulting ^{in a} locally large soliton confinement _energy in the matched case (see Fig. 5b). ^This strengthens

the hypothesis that solitons in crystalline (CH)x

will be effectively pinned by ^{3 D effects} (even ^if they

are not pinned ^to impurities).

Such splitting ^{of the} midgap ^levels implies ^{that the} spin concentration should be partially quenched ^due

to the pairing of electrons in the lower binding ^state

of interacting matched solitons (which ^will always

be present ^{to some extent in} any real material). ^The

characteristic temperature, Tq, ^however, is of the order

well outside the range of experiments ^on (CH)x.

At room temperature ^a soliton will be pinned ^to

-

100 lattice units (mismatched) ^{and to within - 10}

lattice units (matched ^case). This distinction in confi- nement length will be mirrored by the soliton mobi-

lities (the ^solitons appearing ^more mobile in the _very shallow wells of the mismatched case), which should be detectable by ^motional narrowing ^{in ESR measure-}

ments (13). The form of Econf ^{also implies} the existence of a critical concentration of solitons, pc (~ ^{1 soliton}

per 200 carbon atoms) above which local matched

ordering ^{becomes more} favourable than mismatched

ordering. ^This implies that there should be an upper limit on spin density ^{in a} system of neutral solitons,

of approximately ^0.05 spins per carbon atom (as

above pc ^the chains match, ^the midgap ^levels split,

and the electrons at midgap ^become paired). Thus, above _pc one would expect ^a marked decrease in spin density, ^{while the} optical absorption should become

« polaron-like » ^{due to the} splitting ^{of the} midgap

levels.

Unfortunately, however, ^{it is not} possible ^{to control}

the concentration of neutral solitons in polyacetylene ;

and as the concentration in the pristine ^Shirakawa

material is approximately ^{1 in 3 000} [13], well bellow Pc’ testing ^{of the above} theory ^{does not seem} possible

with present ^materials.

In conclusion, ^the splitting ^{of the} midgap ^levels,

in that it predicts ^even stronger ^3D pinning ^of solitons,

serves to strengthen ^{the case} against ^{the solitonic}

model for the electronic properties ^of polyacetylene.

However, ^{it does} produce ^{some other} interesting

effects on the dependence ^{of local} ordering ^{on soliton} density, suggesting that matched ordering ^is just ^as

relevant as mismatched ordering ^{in the} theory ^of

interchain coupling ⁱⁿ polyacetylene.

Acknowledgments.

The present ^{work was} supported by ^a Science and

Engineering Research Council Case Award (Reference

Number 83505170) ⁱⁿ conjunction with I.C.I. New Science Group, ^Runcorn.

Appendix.

ANALYTIC SOLUTION OF SOLITON CONFINEMENT.

As mentioned in section 3.3 the mismatched case has

already been solved analytically by Baeriswyl ^and

Maki [7].

The Hamiltonian for the two interacting ^chains

is given by

Table I.

Band structure for ^two pristine (CH)x ^chains.

where H ff ^is given by

And _so, in the basis 0 = (lkll t/J 2)’ Heff ^{for the two}

chain system ^is given by

so that the perturbation, ^{H’ is} given by

As H’ does not connect individual chains to them- selves, first order perturbative effects will be zero.

Thus,

where the factor of 4 comes from the spin ^summation

over both chains, and the first term comes from intraband transitions, and the second from valence band to midgap transitions.

i.e.

a) Mismatched case.

In this case it is straightforward ^{to show that} [7]

which is independent ^of l, and therefore does not contribute to the confinement _energy.

Also, ^{after some} manipulation [14] using ^the boundary conditions given ⁱⁿ equations (13),

which, ^after transforming ^{sums to} integrals, ^{and some}

further simplifications [7], gives

b) ^{Matched case.}

The analysis ^{in this case becomes more} complicated.

It is straightforward ^{to evaluate} Mk ^and Mkk, using

the appropriate wavefunctions, giving

and, ^where

However, ^further analytic evaluation of 4E from equation (A. 6) appears ^to be difficult.

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