Comparative theoretical study of the doping of conjugated...

PHYSICAL REVIEW B VOLUME 26, NUMBER 10 15 NOVEMBER 1982

Comparative theoretical study of the doping of conjugated polymers:Polarons in polyacetylene and polyparaphenylene

J. I.. BredasLaboratoire de Chimie Theorique Appliquee, Facultes UniUersitaires Notre-Dame de la Paix,

rue de Bruxelles, 61 B-5000 Namur, Belgiumand Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

R. R. ChanceCorporate Research Center, Allied Corporation, Morristown, New Jersey 07960

R. SilbeyDepartment of Chemistry and Center for Materials Science and Engineering,

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139(Received 26 March 1982)

Defect-state calculations on all-trans polyacetylene and polyparaphenylene have beenperformed in the framework of the adiabatic Huckel Hamiltonian with o.-bond compressi-bility. In polyacetylene, the study of the energetics of the separation of the radical (neutraldefect) —ion (charged defect) pair induced upon doping indicates that the two defects tendto remain in close proximity, resulting in the formation of a polaron. The binding energyof the polaron is estimated to be about 0.05 eV with this model. Absorption spectra at low

doping levels are shown to be compatible with polaron formation, thus demonstrating thenonuniqueness of the previously proposed soliton model in explaining these absorptiondata. At higher doping levels, interaction between polarons leads to the formation ofcharged solitons carrying no spin. In polyparaphenylene, defects are always correlated inpairs due to the absence of a degenerate ground state. At low doping, polarons with a bind-

ing energy estimated at 0.03 eV are formed on ionization of polyparaphenylene. The relat-ed deformation of the lattice is relatively soft, in agreement with crystallographic data onbiphenyl anions, and extends over about five rings. Increasing the doping level leads to theformation of bipolarons (doubly charged defects) that require a stronger deformation of thelattice and carry no spin. The possibility of a conduction mechanism in polyparaphenyleneinvolving motion of bipolarons is consistent with magnetic data indicative of very low Pau-li susceptibility in the metallic regime of SbF5-doped polyparaphenylene.

I. INTRODUCTION

In recent years, a number of conjugated organicpolymers have been discovered that can be madeelectrically conductive through doping with eitherstrong electron acceptors or donors. Polymers with

doped derivatives which have been reported to haveconductivities larger than l S/cm include polyace-tylene, ' polyparaphenylene, polypyrrole, and vari-ous polyphenylene chalcogenides. "' Interest, bothacademic and industrial, has further increased bydemonstration of rechargeable batteries based ondoped polyacetylene and polyparaphenylene.

It should be stressed that doped organic poly-mers, although displaying phenomena in many

ways similar to conventional semiconductors, are, infact, very different from the usual doped inorganicsemiconductors. ' These polymers possess highlyanisotropic interactions which result in low dirnen-sionality of electronic motion and can lead to col-lective instabilities typical of quasi-one-dimensionalmaterials, such as Pejerls instabilities. In addition,because these are organic compounds, we can antici-pate that electronic excitations or charge-transferprocesses will markedly affect the atomic geometry,just as in typical organic molecules; therefore,rigid-band models will be incorrect. Finally, the po-lycrystalline or amorphous character of the presentpolymers provokes a substantial disorder, which in-creases further during the doping process. As a re-sult, localized states can play a major role. Note

26 5843 1982 The American Physical Society

J. L. BREDAS, R. R. CHANCE, AND R. SILBEY 26

that disorder provides a barrier for determining ac-curately critical aspects of the structures and prop-erties of conducting polymers. Thus most of thetheoretical work on doped organic polymers hasbeen highly idealized and constructed in the frame-work of crude methods, such as Hiickel-typemodels.

Polyacetylene has been the most studied corn-

pound, both experimentally and theoretically. Areason for this is that the degenerate ground statethat exists in all-trans polyacetylene suggests thepresence of topological kinks or so-called soliton de-fects. ' ' On the basis of the soliton model, manyunusual properties have been predicted for all-transpolyacetylene, including a conductivity mechanisminvolving charged solitons that carry no spin. ' 'However, other polymers, such as polypara-phenylene, do not possess degenerate ground statesand are not expected to accommodate solitons;neuertheless, doped polyparaphenylene displays trans

port properties which are very similar to doped po-lyacetylene

In this paper we have studied, within a singletheoretical framework, all-trans polyacetylene, asystem where soliton excitations could appear, and

polyparaphenylene, which has no degenerate groundstate, in order to make a detailed comparison of thechanges in electronic and geometric structures thatappear upon doping in both systems. The theoreti-cal model we have used is based on the approachesof Pople and Walmsley' and of Su, Schrieffer, andHeeger. '

Section II is devoted to polyacetylene (PA). Wedetail the model and apply it to the study of the en-

ergetics of separation of pairs of defects inducedupon doping. We compute the interactions betweenpairs of defects and the band-structure evolution fora lattice of polarons and a lattice of charged soli-tons. Experimental data for optical absorption ofradical anions in finite-chain polyenes are discussed.

In Sec. III, we treat polyparaphenylene (PPP).After pointing out the difference in connectivity be-

tween PPP and PA, we describe the energetics ofseparation of defects and optimization of the bondlengths around the defects. Finally, we emphasizesome unusual characteristics of doped PPP.

It must be borne in mind that the models we useare based on a simple one-electron theory andneglect correlation effects which can be important.Interchain interactions and explicit presence of thedopant ions are also neglected. Results shouldtherefore be viewed qualitatively with emphasis ontrends and internally consistent comparisons.

II. DEFECT STATES IN PGLYACETYLENE

In the framework of one-electron models, all-trans PA is Peierls distorted in its ground state, sin-gle and double carbon —carbon bonds alternatingalong the chain. As a result, two energeticallyequivalent resonance forms (hereafter referred to asphase A and phase 8) can be derived and lead to adouble-well potential energy curve, as illustrated inFig. 1. The potential energy (E) can be expressed asa function of the degree of bond-length alternation(Ar):

E = a(hr) —b (b,r) +c, a, b & 0 .0

The experimental estimate of Ar is 0.10+0.03 A, '

while ab initio double-zeta geometry optimizationdue to Karpfen and Petkov' gives Dr=0. 100 Awith rc=c= 1 346 A and rc-c= 1 446 A TPeierls distortion opens up a gap at the Fermi level;the optical absorption spectrum of all trans PA-

shows an edge at 1.4 eV and peaks at about 1.8y 18

It has been recognized by Su, Schrieffer, andHeeger' (SSH) that because the ground state of thedimerized all-trans PA chain is twofold degenerate,solitons' can be expected, satisfying a wave equa-tion akin to the P field-theory equation. Thiscan be inferred from relation (1). Solitons are exci-tations of a system leading from one minimum ofthe potential to another minimum of same energy.In all-trans PA, they correspond to topologicalkinks that extend over several bonds and graduallylead from a phase-A chain segment to a phase-8chain segment. Displacement from equilibrium ofthe atomic positions within the soliton have beenusually simulated by a hyperbolic tangent functionwhich is solution of the P field-theory equation.

FIG. 1. Sketch of potential energy curve for transpolyacetylene showing two energetically equivalentstructures.

26 COMPARATIVE THEORETICAL STUDY OF THE DOPING OF. . . 5845

Note that solitons in PA have peculiar charge-spinrelations, since a neutral soliton corresponds to aradical and has spin- —,, while charged solitons carryno spin. '

All-trans PA appears then unique in having twopotential energy minima of the same energy, an in-

dispensable requirement for soliton formation. Cispolyacetylene does not possess this characteristic,the cis-transoid backbone being slightly more stablethan the trans-cisoid backbone as suggested by vari-ous experiments ' and by ab initio calculations.

Study of the energetics of formation of a solitonis usually performed in the context of Huckeltheory with 0-bond compressibility. This is true forthe SSH adiabatic Hamiltonian as well as for bond-order —bond-length relationships of the Coulson

type. In a bond-order —bond-length relationship,which we will use in this paper, the P resonance in-

tegral values are expressed as

P(r) =A exp( r /8),— (2a)

where r is the bond length. The energy of the cr

framework is

f(r)=CP(r)(r ro+B), — (2b)

where A, B, and C are parameters to be optimized,and ro usually corresponds to the length of a "puresp —sp single bond" (around 1.50 A) (Ref. 24) butcan also be optimized. We have optimized theparameters in order to reproduce a band gap of 1.4eV, a total bandwidth of 10 eV and a bond-lengthalternation of 0.14 A (rc c——1.47 A; rc c1.33A), as in the SSH work. ' We obtain A=34.56 eV,B=0.531 A, and C=6.78 A

With these parameters, we first compute the ener-

gy of creation of an isolated soliton defect and com-pare with previous calculations. We then addressthe question of the response of the electronic andgeometric structure of the chain to the dopant-induced charge-transfer process. For the numericalcalculations throughout this paper, we considermainly the acceptor doping process. However, thediscussion and results apply equally well to donordoping. Calculations on polyacetylene are per-formed on cyclic polyene molecules containing atleast 110 carbon atoms —more than enough for theresults to be directly compared to the infinite-chaincase.

Pn, n+ i(r}=P 1 4+ ( —1)"2

ntanh—I

+ tanhn+1

(3)

B. Interacting defects

0where hr (=0.14 A) denotes the degree of bond-length alternation far away from the defect, n indi-cates the site location from the defect, and l is amodulation factor to be optimized and roughly cor-responding to the halfwidth of the defect. (We donot use a cutoff procedure such as that used bySSH. ' } The energy of creation of a soliton excita-tion is lowest, 0.45 eV, for I =7, in excellent agree-ment with the SSH results. The presence of the sol-iton introduces a localized electronic state atmidgap, i.e., 0.7 eV above the valence-band edge.As a result, the energy of creation of the solitondoes not depend on whether the soliton is neutral(state singly occupied), negatively charged (statedoubly occupied), or positively charged (state emp-ty). Localization of the electronic state is con-firmed by the fact that the total probability densityof the radical or charge associated with the solitonis 0.855 over the 15 sites around the defect.

Note that the energy to create a positive isolatedsoliton defect is 0.25 eV smaller than the 0.7-eV en-

ergy required by a vertical ionization process. ' Wealso note that ESR experiments on undoped trans-PA indicate the presence of about 1 radical (neutralsoliton} per 3000 carbon atoms2s a ratio muchlarger than what could be expected on the basis ofsimple thermodynamics. This implies, as does thefact that this spin concentration shows little tem-perature dependence, that neutral solitons in un-doped trans-PA are mainly extrinsic defects createdduring cis-trans isomerization (and trapped withrespect to recombination between crosslinks).

A. Isolated soliton defect

Resonance integral P(r) values around the defectare expressed as

We now consider the way the chain can accom-modate extra charges created by charge-transferprocesses toward acceptor dopants. It is importantto recognize that the transfer of an electron from

5846 J. L. BREDAS, R. R. CHANCE, AND R. SILBEY

the valence band during the doping creates a po-laron, or radical cation on the chain, i.e., a pair ofdefects and not an isolated defect. (At very low

doping levels of course, transfer can also take placefrom the extrinsic neutral-soliton states, therebyleading to charged solitons on the chain. ) It is thenrelevant to study the energetics of separation of that

I

pair of defects, that can be considered as a soliton-antisoliton pair.

In the region between the two defects, the pvalues are computed as a function of the product ofhyperbolic tangent functions, in the spirit of whathas been worked out by Takayama et al. ' as fol-lows:

p„„+&(r)=p 1.4+( —1)" tanh —+tanh~ 6T Pl Pl + 1

2 I Itanh +tanh

k —n k —n —1(4)

where k is the number of sites (odd) separating thetwo defects. Outside that region, p(r) values arecomputed as in relation (3). We have, in addition,made calculations using a difference of tanh func-tions as is used in Ref. 26 instead of a product as in(4), and find little difference between these and theresults using (4).

For the sake of completeness, we have treated theseparation of two neutral defects, two positive de-fects, and a neutral and a positive defect. In everycalculation, the value of 1 has been optimized. Re-sults are presented in Fig. 2. When the two defectsare widely separated, we naturally retain the isolat-ed defect results, i.e., the energy of creation of thepair of defects is 2X (0.45 eV) =0.9 eV and 1-7.

1. Separation of two neutral defects

Two neutral defects formed by the breaking of adouble bond tend not to separate but to recombine,leaving no deformation (1—+0) on the chain.

2. Separation of two positiue defects

Two identically charged defects repel each otherand lead to two isolated charged solitons. When thetwo defects are close to each other, the deformationthey provoke is strong, extending over 35 bonds(1-17). When they are widely separated, the corre-sponding energy is 0.5 eV lower than two verticalionization processes. Note that Coulomb repulsionis not included in these calculations.

++1.0—

O.90—

O

e 0.65K4lX

o.s

7

+ ~

//

// ~ ~

//

//g~o

//

/( I I I I I

10 20 30 40 50SEPARATloN

FIG. 2. Energetics of separation of two defects onpolyacetylene chain with "." indicating two radicals{neutral solitons), "++"indicating two cations {chargedsolitons) and "+-" indicating a radical cation (or po-laron for small separations). The quantity I defines theextent of the chain deformation due to the defect. Theunits for the abscissa and I are number of carbon atoms.

3. Separation of a radical cation (polaron)

We find that the energy is smallest, 0.65 eV, when

the two defects are in close proximity, correspondingto the formation of a polaron. Our result that theradical and the ion (i.e., charged and neutral soli-tons) are attractive is consistent with the trendspredicted by Lin-Liu and Maki in the continuumlimit for large defect-pair separations. Withrespect to a vertical ionization process, at 0.7 eV,the polaron binding energy (related to the deforma-tion of the lattice around it) is 0.05 eV. The op-timal I value is again about 7, i.e., the deformationis relatively soft, extending over 15 sites for twonearest-neighbor defects. The charge that the po-laron carries is also very localized, the 15 sites bear-ing 0.757e. The optimal 1 value (-7) remains thesame independent of separation.


It is interesting to note that all these results arefully consistent with the results obtained indepen-dently by Bishop et al. from a relativistic field-theory approach.

The presence of a polaron introduces two defectlevels in the gap, symmetrically placed 0.4 eV aboveand below the Fermi level rather than both at theFermi level as for noninteracting defects. Thisnumber is in agreement with the purely electronicrelaxation energy on electron injection calculated bySu and Schrieffer. ' For acceptor doping the bond-

ing state is singly occupied and the antibondingstate empty. In Fig. 3, we present the difference inband picture going from two isolated defects to twointeracting defects. Our results indicate that at low

doping levels, the extra charges on the chain formpolarons and not charged solitons (except forcharge-transfer processes from the few extrinsicneutral soliton defects).

1.8—

1.6-e

1.4—KUJZUJ

~2

/IIII

IIIII

& low1

1 dOPingI

11

1.0—I

1.0 0.5 0SEP ~ ~/SEP + +

FIG. 4. Interaction between two radical cations {orpolarons) in polyacetylene. The horizontal axis is theratio of the radical separations to the charge separationsso that the right extreme of the figure represents theformation of two charged solitions from two polarons.Doping levels as mole per cent are indicated.

4. Interactions between polarons

We now address the question of the interactionbetween two polarons to understand the evolutionwhen the doping level increases. To study this in-teraction, we have considered two polarons and lo-cated the charged defects at fixed positions, to cor-respond to the pinning of the plus charges at thepositions of the negatively charged dopants. Theneutral defects are gradually pulled away from thecharged defects, approach each other, and finallyrecombine leaving two charged solitons (Fig. 4).

For two polarons widely separated [E-2X(0.65eV)=1.3 eV], that process leads first to the forma-tion of four isolated defects [E-4X (0.45eV) =1.8 eV], before eventually the recombinationof the two radicals leaves only two charged solitons[E-2X(0.45 eV) =0.9 eV]. Thus there is a barrierto recombination as indicated in Fig. 4. At 2.0%

CB

" ////l////////, L///////////

Sep ~ oo Sep ~1FIG. 3. Interaction of charged and neutral soliton in

polyacetylene. States appear at midgap for infiniteseparation but interact and split as indicated when theseparation approaches one bond {polaron formation).

(homogeneous) doping level (charges about 50 sitesapart), the barrier is still of the order 0.1 eV. Thebarrier disappears at around 3% doping level

(charges about 35 sites apart) and, at 4% dopinglevel, neutral defects readily recombine leading to alattice of charged solitons.

Let us consider the evolution of the band struc-ture during the doping process. At low doping lev-

els, we form bonding and antibonding polaron states0.3 eV above the valence band and 0.3 eV below theconduction band. As doping increases, two process-es take place: The polaron states form a band [seeFig. 5(a)] and some polarons close to one anotherwill form charged solitons. If the latter process isneglected, the bonding polaron band would mergewith the valence band at -5% doping (and the an-

tibonding polaron band will merge with the conduc-tion band at -5%). At —10% doping the gapwould disappear. However, the formation ofcharged solitons intervenes and we shift graduallyfrom a polaron lattice [Fig. 5(a)] to a charged-soliton lattice [Fig. 5(b)]. Note that polarons andcharged solitons should coexist until a doping levelof 3—4 /o. The charged soliton states at midgapform a band that may be responsible for the con-duction mechanism without spin occurring justabove the semiconductor-metal (SM) transition(which occurs experimentally at —1% doping' ).In our model,

'

the soliton band merges with thevalence band at —10% doping so that carriers withspin can contribute to the conductivity. This is inagreement with the delayed appearance of signifi-

5848 J. L. BREDAS, R. R. CHANCE, AND R. SILBEY

CB

y Mergingat ~ 5%l Merging

at I0%

~ Merging

CB

Merging

at~ II'%

CB

"'//////////////

2 3

'////I/l//I//

SOlltOh8 Polaroh

FIG. 5. Band-structure scheme at intermediate dop-

ing for a lattice of polarons (a) and a lattice of charged

solitons (b). At higher doping levels, bands in the gapbroaden and this leads in our model for case (a) to the

merging of the bonding (antibonding) band with the

valence (conduction) band at -5%%uo doping level while

the bonding and antibonding bands merge at —10%%uo

doping. For case {b), the soliton band merges with the

valence and conduction bands at —11%%uo doping, in

agreement with Mele and Rice calculations (Ref. 44).Following these authors, inclusion of interchain coupling

could reduce the doping level where merging occurs tobelow 5% where it is observed experimentally.

cant Pauli susceptibility above the SM transition.At very high doping levels, our ab initio Hartree-Fock calculations on Li-doped polyacetylene show

that bond-length alternation is completelydepressed, solitons are no longer present, and con-ductivity is due to the closure of the Peierls gap.

The midgap absorption which appears at about0.7 eV when polyacetylene is doped ' ' has beentaken as evidence for the generation of solitons on

doping. Since our model shows that the formationof polarons is energetically favored over solitons atlow doping levels, the question arises as to whetheror not polarons are responsible for, or contribute to,the observed midgap absorption. A polaron in po-lyacetylene introduces three new transitions as de-

picted in Fig. 6. Table I gives our calculated ener-

gies for these transitions (normalized by the energy

gap) for radical-ion separations of 1, 3, 5, and 7bonds. Note that the average absorption energy ex-

FIG. 6. Schematic representation of doping-inducedoptical transition in polyacetylene for solitons and po-larons.

pected with the polaron model is also about mjdgap.We now consider charged-defect absorption ob-

served in model compounds for polyacetylene.We choose the diphenylpolyenes (DPP),C6H&(HC=CH)„C&H&, since there are rather exten-sive data available for these molecules and theirions with n =1—6. Because we are interested inthe extrapolation of oligomer data to the polymer,we must appropriately "correct" the DPP data totake into account the fact that the phenyl groups(CsH5) effectively extend the conjugation length

beyond the polyene sequence. We accomplish thiscorrection by defining an effective conjugationlength, niff —n +v, such that the absorption energyof DPP with n double bonds is equal to that of apolyene with n +v double bonds. The absorptionenergies are plotted versus 1/n, rf, which accordingto theory and experiment should yield a roughlylinear relationship. Results are shown in Fig. 7.We find v=2.7; in other words, stilbene (n =1) ab-sorbs at about the same energy as octatetrene(n +v=4). The extrapolation to n, ff= oo yields

Eg =1.8 eV, in good agreement with experiment. 'We also show in Fig. 7 the transition energies for

radical anions as obtained by Hoijtink and van derMeij. The data are roughly linear with 1/n, rf andextrapolate to 0.5, 0.8, and 1.0 eV. These data, afternormalizing by the 1.8-eV band gap, are comparedto theory in Table I. These radical anions corre-spond to negatively charged polarons, i.e., to donor

TABLE I. Energies (E~), defect level position (ed), and transition energies for charged de-

fects in polyacetylene. The transition energies (E&, E2, and E3) are normalized by the band-

gap energy for comparison to experiment.

Separation

1

357

Experiment

{eV)

0.6520.6560.6670.680

(eV)

+0.434+0.395+0.357+0.318

El

0.200.230.250.280.30

E

0.620.560.500.450.44

E3

0.810.790.750.730.55


3

C9KNzIll

O 2I-

0thCl

I

0.1 0.2 0.3

FIG. 7. Optical absorption energy vs the reciprocalof the "effective" number of double bonds for diphenylpolyenes (neutral molecules and radical anions). Thesolid line through the neutral molecule data representsdata for polyenes (with extrapolation) taken from theliterature (Ref. 33).

doping of polyacetylene for which a broad (-1-eVwide) transition centered at about 0.7 eV has beenobserved. ' Thus the extrapolated radical anion ad-sorption data are completely consistent with the ex-perimental observations for donor-doped polyace-tylene. We may conclude that polaron absorptionoffers a satisfactory explanation for the midgap ab-sorption in doped polyacetylene and that the solitonexplanation is nonunique.

III. DEFECT STATES INPOLYPARAPHENYLENE

Polyparaphenylene consists of benzene ringslinked in the para position. Crystallographic dataon oligomers indicate that carbon —carbon bond

0

lengths within the rings are about 1.40 A, and thosebetween the rings are about 1.50 A. In the solidstate, two successive benzene rings are tilted withrespect to each other by about 23'. This torsion an-

gle constitutes a compromise between the effect ofconjugation and crystal-packing energy, whichfavor a planar structure, and the steric repulsion be-tween orthohydrogen atoms, which favors a non-planar structure. The band gap in PPP is about 3.5eV or about twice that of PA. A resonance formcan be derived for PPP that corresponds to aquinoid structure; however, contrary to a11-trans

PA, the benzenoid and quinoid forms are not ener-

getically equivalent, the quinoid structure beingsubstantially higher in energy (Fig. 8).

When we discuss defects in PPP, it is importantto recognize that connectivity in PPP makes it verydifferent from trans an-d cis PA-, even at thelowest-order level (Hiickel theory). This is demon-strated when we address the possibility of anymidgap (a =0) state in PPP, by considering the de-fect displayed in Fig. 9. As detailed in the Appen-dix, we find that for physical values of the reso-nance integrals, a midgap e =0 state as in trans- orcis-PA, is not possible in PPP. A very importantresult is that in PPP defects will always be stronglyinteracting and correlated in pairs; thus a soliton-antisoliton pair will be confined. Also contributingto this confinement is (also true by analogy in cisPA and polydiacetylene) that the more two defectsseparate, the more rings with quinoid structure ofhigher energy appear between the two defects.Thus there will be a compromise between thestrength and the extension of the relaxation of thelattice due to the presence of defects.

Calculations on PPP are performed on 21 ringchains (126 carbon atoms) in the framework ofHuckel theory using a bond-order —bond-length re-lationship of the Coulson type. Parameters A, B,and C from relations (2) are optimized in order toreproduce in the absence of any defect: (i) the ex-

perimental geometry of the benzenoid structure, (ii)the band gap of 3.5 eV, and (iii) the bandwidth ofthe highest occupied n. orbital of the order of 3.2eV. ' Note that this procedure parallels that usedfor polyacetylene though the parameters need not bethe same. Under these conditions, parameters A, B,and C are, respectively, 46.0 eV, 0.531 A, and 4.30A '. With this choice of parameters, the total en-

ergy difference per ring between the benzenoidstructure and a "pure" quinoid structure, where

rc c——1.38 A and rc c 1.45 A,——is 7.2 kcal/mol(0.31 eV), in good agreement with the extended-

DEFORM ATION COORD IN ATE

FIG. 8. Sketch of potential energy curve for po-lyparaphenylene showing energetically inequivalentstructures.

5850 J. L. BREDAS, R. R. CHANCE, AND R. SILBEY 26

2 -I -2 -5 -6 -9-4

3

6 5 2 I -I, "2 -5 -6 -9p„„+&

( o ) =p 1.40+a tanh —tanhb n 4E —n

I I

FIG. 9. Sketch of the defect considered when search-

ing for a localized midgap state in polyparaphenylene.

P„'„+,(r) =P 1.50

(Sb)

Huckel-theory estimate of Whangbo et al.The model we have chosen to deal with pairs of

defects derives directly from what we used for po-lyacetylene and is consistent with crystallographicdata on biphenyl and biphenyl anions. These dataindicate that the changes in the geometry in form-ing the ( —1) biphenyl anion are in the directionprovided by an increased admixture of the quinoidalresonance form:

The bonds parallel to the chain direction (bonds aand c) shorten and bonds inclined to that directionelongate. While bonds a and b are nearly equal inthe neutral species (a =b-=1.40 A), these bondsdiffer by about 0.05 A in the anion (a =1.38 A andb—= 1.43 A). Bonds b and c, which in the neutral

0species differ by about the same amount (0.10 A) asdo adjacent bonds in polyacetylene, become nearlyequal in the anion (c —1.44 A).

In our model, we consider that the deformationtoward the quinoid structure is maximum in themiddle of the defect pair and gradually decreasestoward the edge of the defect to lead back to thebenzenoid structure (Fig. 10). Such a behavior canbe simulated by a product of hyperbolic tangentfunctions, as in the case of interacting solitons intrans PA Th-e re.sonance integral values for thethree types of bonds within the defect are expressedas

p„„+~ (r) =p 1.40 —a tanh —tanh

(5a)

Qo-Q=Q=Q=Q-Qo

Extension of Defect

FIG. 10. Deformation model for polyparaphenylenechain based on product of hyperbolic tangent functions.

—2n tanh —tanhn 4E —n

l l(Sc)

In these expressions, X indicates the number ofrings over which the defect extends, n is the site lo-cation from one end of the defect (4E n —is thenthe separation from the other end), I modulates theamplitude of the deformation (note that the largerthe value of I, the smaller the deformation), and adenotes the maximum bond-length difference with

respect to the original bond lengths. In agreementwith the crystallographic data on biphenyl anions,bond c is allowed to vary twice as much as bonds aand b Optim. ization of the value of a has led to thechoice a =0.05 A. The optimization of the energyfor a pair of defects depends on two factors: thenumber of rings X over which the defect extendsand the value of I.

Results for the same three different pairs of de-fects as studied in PA are presented in Fig. 11 anddiscussed below.

A. Two neutral defects

In this case, the minimum of energy is obtainedfor values of I tending to infinity, i.e., there is nodeformation left on the chain, the two radicalsrecombine.

B. Radical cation (polaron)

For a charged defect interacting with a neutraldefect, a polaron is formed whose binding energy isof the order of 0.03 eV (to be compared with 0.05eV in the case of trans PA). The extension o-f thedefect is over about five rings. The deformation ofthe lattice associated with the polaron is relativelysoft (I =13 for X =5); in the middle of the defectwhere deformation is largest, the bonds within the

0

ring change by some 0.02S A, between the rings byO.OS A. These changes are fully consistent with theones observed in the biphenyl anions.

It should be stressed that the relative flatness ofthe energy curves in Fig. 11 comes from the factthat for each value of X, the I value adjusts in such


LE (eV)Oa00 j,

-0.01—POLARON

FIG. 12. Bipolaron on polyparaphenylene chain.

-0.02—

-0.03—

-0.04QKLLIzLll

-0.2—

BIPOLARON

(b)

-0 3—

04-

a way that the maximal deformation in the middleof the defect is always of the same order. If, for agiven N value, the maximal deformation is variedaround its optimal value, the energy rises moresharply.

The presence of the polaron introduces twostates, bonding and antibonding, in the gap. Thebonding state, half-occupied for a radical cation, is0.2 eV above the top of the valence band. Thecharge is localized; its probability density over thefive rings of the defect is 0.791.

C. Two positively charged defects

3 4 5 6NLJMBER OF RINGS

FIG. 11. Energy gain with respect to vertical ioniza-tion processes obtained in polyparaphenylene throughthe formation of a polaron for a singly charged defect(a) and of a bipolaron for a doubly charged defect (b), asa function of the extension of the defect, in number ofrings N. For each N value, the optimal l value is shownin parentheses.

as that of the polaron, but the deformation is muchstronger, about twice as large as for the polaron.The gain in total energy with respect to two verticalionizations is of the order of 0.4 eV, to be comparedwith 0.5 eV in the case of two charged solitons intrans-PA.

Considering explicitly neither Coulomb repulsionnor screening due to the counter ions, the formationof a bipolaron is much preferred over the formationof two polarons (the 0.4-eV energy gain of a bipo-laron with respect to two vertical ionizations is tobe compared with 2X0.03 eV for two polarons).This can be understood by the fact that the strongerdeformation that exists in the bipolaron case pushes

up the bonding state in the gap 0.56 eV above thevalence band, thereby making the ionization processeasier. The charges of the bipolaron are also verylocalized, being located at 90.6% within the fiverings.

Note that as in the case of charged solitons intrans-PA, the bipolaron carries no spin and thebonding state it induces in the gap is empty. As thedoping level increases, interactions between bipo-larons could broaden the bipolaron states in the gapand lead to the formation of a band. Conductivitycould arise from the motion of bipolarons carryingno spin. This picture is consistent with recent mag-netic data, which indicates a very low Pauli sus-

ceptibility in the metallic regime of SbF5-dopedPPP.

The fascinating possibility of formation of bipo-larons in doped PPP, whose interest is further in-

creased by the work on bipolaronic superconductivi-

ty, calls for more attention to be given to thestudy of the electronic properties of PPP upon dop-ing. Unfortunately, detailed experimental analyseshave so far been hindered by the inhomogeneouscharacter of the doping process in PPP and the im-possibility to have precise control of the doping lev-els. Note that in cis-PA, where the same kind ofbehavior as in PPP. could be expected, isomerizationtakes place upon doping, leading to the trans form.

Contrary to what happens in trans-PA where thetwo charged defects separate, we observe in PPP theformation of a bipolaron (or correlated charged-soliton —charged-antisoliton pair), Fig. 12. The ex-tension of the bipolaron is of the same order (N =5)

IV. CONCLUSIONS

By model calculations on the defect states in-duced upon doping in all-trans polyacetylene andpolyparaphenylene, we have given a picture of the


evolution of the electronic properties of both sys-tems upon doping and have shown that, despite thepresence in all tra-ns PA of a degenerate groundstate which is absent in PPP, PA and PPP are muchmore similar than generally thought. At low dop-ing levels, polaron formation is expected for bothPA and PPP with similar polaron binding energiesand defect extensions. At higher doping levels, po-larons interact to produce uncorrelated charged soli-tons in PA and bipolarons or correlated charged-soliton —charged-antisoliton pairs in PPP. As nei-ther charged solitons nor bipolarons carry spin, thisis consistent with the absence of significant Paulisusceptibility observed in both systems at dopinglevels just above the transition to high conductivi-ties (or "semiconductor-to-metal" transition).

We have demonstrated that the response of theatomic geometry to electron excitation processes,i.e., the electron-phonon coupling that we and oth-ers have previously emphasized, ' ' plays a fun-damental role in the conductivity mechanism ofdoped organic polymers. This is valid even in thecase of the macroscopic SM transition suggested tobe due to the percolation threshold of metallicgrains. '

Finally, we believe more attention should begiven to PPP and analogous systems in order to ver-ify the presence of bipolarons and of a conductionmechanism based on them. This is especially signi-ficant since it has been recently pointed out that bi-polarons, formed by strong electron-phonon interac-tion can be considered as localized spatially non-overlapping Cooper pairs and can have, under cer-tain conditions, superconducting properties.

Our results on polyacetylene are in excellentagreement with the field-theoretic results of Bishopand co-workers. Our results on polypara-phenylene, which have not been treated by a contin-uum field-theoretic model, show that, even in thecase of nondegenerate ground states, results qualita-tively similar to degenerate-ground-state systemscan be obtained when polaron and bipolaron statesare considered.

ACKNOWLEDGMENTS

We acknowledge stimulating discussions withProfessor J. M. Andre and Dr. R. H. Baughman,Dr. H. Eckhardt, Dr. R. L. Elsenbaumer, Dr. J. E.Frommer, and Dr. L. W. Shacklette. This work hasbeen partly supported by NSF under Grant No.DMR-7908307 and by the Belgian National ScienceFoundation, Fonds National Beige de la Recherche

Scientific (FNRS). One of us (J.L.B.) acknowledgessupport also by FNRS.

APPENDIX

We are searching for the condition of appearanceof a localized midgap @=0 state in PPP, in theframework of a Hiickel Hamiltonian. We considerthe defect displayed in Fig. 9 with resonance in-

tegral P] corresponding to a "long" bond, P2 to a"short" bond, and ](]] to a "benzene" bond. To theleft of the defect, we obtain the following set ofequations:

ECp =C]P] +C]P]+CpP ]

EC] C2P2——+COP I

EC] =C2Pp +CoP]

EC2 =C]p2+ C3pl

EC~ =C-]$2+C3p]

Ec3=c~p]+C213]+C4p2 ~

(Al)

or (A2)

C „=(2p]/p2)"C

C4. +2 = 2"(P]/P2)'—"+'Co .

In order to have a localized state, we need

2p]/pz &1 or v 2p] &p2 .

0With the short bond taken as 1.35 A, this implies inour parametrization the long bond should be largerthan 1.53 A (i.e., a C—C bond as in alkanes) or, inCoulson-Golobiewski parametrization, larger than

01.48 A. This means a very strong quinoid structureof high energy.

Furthermore, it is relevant to consider that theshort bonds between the rings are actually longer

Imposing @=0 results for a nontrivial solution inall coefficients on odd numbered sites being zero,we get the following relations:

C2 =C2 = p] Co/Pz-

C4 = 2p] C2/p2 =—2P]co/P2 ~

Cs = —2P ]Cp/P2

Cs =4P ]Co/P2

COMPARATIVE THEORETICAL STUDY OF THE DOPING OF. . . 5853

than the short bonds within the rings. With the in-troduction for the former of a resonance integral

p3 (p2 the condition for a localized state becomes

v 2pi (v'P2P3

thus rendering the condition even more difficult tofulfill.

To the right of the defect, the same process leadsto the relations

C i ——C i= PiCo/P

C 3=P i-Co/P,

C 5= —P)Co/2P

C 7 =p i Co /2p

C 9=-P—i Co/4P',

1C (4„+,)————(Pi/P)" 'Co .

pn

This constitutes an exponential decrease of the coef-ficients to the right of the defect.

From the search for a localized state in PPP, weconclude that the @=0 state does not occur forphysical values of the parameters. It only appearsif the rings to the left of the defect in Fig. 9 are ex-tremely quinoid, a situation which is energeticallyhighly unfavorable.

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