Chemical Physics 426 (2013) 9–15

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Chemical Physics

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Interchain coupling effects on large acoustic polaron in two parallelmolecular chains

0301-0104/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.chemphys.2013.09.005

⇑ Corresponding author. Tel.: +381 116442611.E-mail address: zivic@vinca.rs (Z. Ivic).

Dalibor Cevizovic a, Zoran Ivic a,⇑, Zeljko Przulj a, Jasmina Tekic a, Darko Kapor b

a University of Belgrade, ‘‘Vinca’’ Institute of Nuclear Sciences, Laboratory for Theoretical and Condensed Matter Physics-020, P.O. BOX 522, 11001 Belgrade, Serbiab Department of Physics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovica 4, 21000 Novi Sad, Serbia

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 May 2013In final form 13 September 2013Available online 23 September 2013

Keywords:PolaronSolitonInterchain couplingMolecular chain

We examine the effects of the interchain coupling on the properties of the large adiabatic polaron in thetwo chain model. In dependence of the strength of the interchain coupling (c) two different types ofpolaron solutions were found. In the weak coupling regime (c < 1=2), polaron is confined to a singlechain. As coupling increases continual transition towards the delocalization takes place – polaron ampli-tudes on both chains gradually become equalized, in the same time, its binding energy vanishes. Finally,above the critical coupling strength, polaron is fully delocalized – equally distributed over the both chainswhile its energy lies within the band of free states.

Our the most specific prediction is the substantial impact of the polaron motion on its character. Thisimplies the emergence of the two different regimes in charge transfer processes which may be experi-mentally verified.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

The last few years have witnessed the widespread applicationsof the various quasi-one-dimensional materials (Q1D) in the fabri-cation of the components (i.e. nanowires [1–3]) of the devices inmolecular electronics. Examples are numerous and, among other,include conjugated polymer based electronic and optoelectronicdevices including light emitting diodes [1–6], field effect transis-tors [7,8], photovoltaic cells [5,9], biosensors [2,3], rechargeablebatteries [2], etc. In addition, potential technological applicationsof the biological macromolecules such as DNA and a-helix asmolecular wires [10–12] has been also suggested.

Performances of the molecular devices are determined by theefficiency of the charge transfer processes. This prompted the re-newed interest in the investigations of the properties of nonlinearexcitations such as kink-solitons, polarons and bipolarons in Q1Dmedia. They arise due to the self-trapping of the excess charges,photo generated or created upon doping, and function as chargecarriers [13,14]. Transport properties of these localized excitationsprofoundly differ from those of bare electron. In that respect pos-sible formation of large polarons is of particular importance dueto their robustness and ability to propagate over the large dis-tances in soliton form with minimal loss [12,15–20]. In this way

they may provide an efficient long range charge and energy trans-fer in these substances.

High anisotropy of electronic spectra has been the main argu-ment in benefit of the soliton mechanisms of the charge transportin these materials. Nevertheless, the soliton-like features of thelarge polarons maintain in ideal 1D systems only [21–23] and, asindicated in [24,25], represent just an artefact of the theoreticalconsiderations carried out within the pure 1D models. For that rea-son idealized 1D models cannot be simply applied to actual Q1Dmaterials which, although highly anisotropic in their electronicproperties, are truly 3d substances mostly composed of the certainnumber of coupled molecular chains. In real media interchain cou-pling is inevitable and may have considerable impact on polaron(soliton) features-especially its stability. In particular, on the basisof the theoretical examinations of Emin [25], the existence of sta-ble large polaron, confined to a single chain within the 3D solidbuilt up from a collection of parallel molecular chains, requires thatintrachain transfer integral should exceed the transverse one atleast hundred times. This rises some doubts concerning the rele-vance of the large polaron concept for transport processes in actualQ1D media since the calculations of the electronic structure of con-jugated polymers yield that the interchain transfer integrals mayvary between 25 meV [26] up to 0.6 eV [27], while the average va-lue of the intrachain transfer integral for gross of the conductingpolymers is 2.5 eV [13] so that the conditions for the large polaronstability established in [25] can be hardly satisfied. Nevertheless,the experiments including the infra-red absorption [28], charge

10 D. Cevizovic et al. / Chemical Physics 426 (2013) 9–15

carrier mobility measurements [29] and resonant Raman spectra[30] of conjugated polymers indicate the existence of the largepolarons in various Q1D materials even when conditions proposedby Emin are not satisfied.

The possible origin of polaron stabilization could be the conse-quence of the appearance of chain endings, conjugation breaks, andvarious defects. Moreover, the electron–phonon interaction maycause the additional anisotropy of electron bands due to the expo-nential reduction of the electronic transfer integrals [31,32] whichin final instance may cause polaron confinement to a single chain.Note also that the validity of the Emin’s theoretical arguments fullyrely on the continuum approximation and does not hold when po-laron radius in some particular direction (transfer for example) be-comes of the order of lattice constant and effects of discretenessshould be accounted for. Thus, in contrast to the expectations ofcontinuum theories, interchain coupling may not fully destroy po-laron stability but, more likely, may lead to qualitatively new fea-tures of 1D large polarons whose investigation needs to be carriedout within the more realistic models. Such investigations have at-tracted a considerable interest over the years and include studiesof interchain coupling effects on conjugated polymers [13,33–40],a-helix proteins [41–47] and DNA [48].

In the most cases the system composed of the limited number,two or three, of coupled molecular chains were considered. Practi-cal calculations were carried out within the slightly improved 1Dmodels which were modified to account for the effects of the elec-tron tunneling between the neighboring chains. Such an approachstill imposes certain oversimplification in a theoretical treatmentof the gross of the actual Q1D materials. However, there are somecases when these models could be quite accurate. This concernsthe substances such as aforementioned biological macromoleculescomposed of two and three backbone strands [10–12,15] andpolyacene which may be considered as two polyacetylene chainswith alternate interchain coupling [39,49].

Owing to these efforts there has been certain progress in theunderstanding of the transport processes in chain like materialsincluding biological macromolecules, DNA [10,11] and a-helix[46,47], and particularly Peierls dielectrics which were especiallyextensively studied due to their potentially wide practical impor-tance [13,33–40]. Besides just a few exceptions [25,36,41–45],the majority of these studies have been carried out numericallyemploying different numerical schemes to solve the self-consis-tent system of evolution equations for the polaron wave func-tions and lattice displacement field [49–54]. Each of thesestudies have been carried out for a given (different) set of sys-tem parameters and, therefore, concerns the specific materialand cannot give the comprehensive picture of the polaronbehavior in general case. Yet, summarizing the result of the var-ious specific studies it may be either concluded that polaronmay be confined to a single chain or may be sited between themdepending on the magnitude of the interchain transfer integral.Moreover, it may be concluded that, for each specific choice ofparameters characterizing the pure 1D polaron, there is a certainthreshold value of the transfer integral below which polaron isconfined to a single chain, while, above it polaron is delocal-ized-equally shared among the chains. This implies that theremust exist the precise relationship between system parameterswhich determines that threshold value. Such general relationwas not found in the aforementioned numerical investigationsand we address that issue here and establish the criterion forthe existence of the effectively 1D large acoustic polaron in thesystem consisting of two molecular chains. For that purposewe employ the methods developed for the examination of thesolitons in coupled nonlinear wave equations arising in differentcontexts [55–61]. Our results are in very good qualitative andquantitative agreement with previous related studies [37].

2. The model

We are considering an extra electron (excitation in general) in asystem consisting of the two coupled identical molecular chainseach composed of N � 1 molecules (molecular groups). The essen-tials of the physics of charge transfer in such system can be de-scribed by the common tight binding model modified here inorder to incorporate the interchain coupling of electronic excita-tions on different chains. Only the coupling with acoustic modeswill be considered, and, as for the phonons, they will be accountedin the harmonic approximation. Accordingly, we employ the fol-lowing model Hamiltonian:

H ¼ DX

n;j¼1;2

Byn;jBn;j � JXn;l¼�1j¼1;2

Byn;jBnþl;j þ LX

n;j¼1;2

Byn;jBn;3�j

þ 1ffiffiffiffiNp

Xn;j;q

FqeiqnR0 Byn;jBn;jðaq;j þ ay�q;jÞ þX

q;j

�hxq;jayq;jaq;j: ð1Þ

Here n labels the lattice sites and takes values from �N=2 to N=2,index j enumerates molecular chains; operators Byn;j; ðBn;jÞ corre-spond to the electron creation (annihilation) on nth site. The energyparameters of electronic subsystem are respectively: D – the energyof electronic excitation of nth molecule; J-resonant energy of theelectron (excitation) transfer between the neighboring moleculeswithin the particular chain; finally, L stands for the interchain res-onant integral of the excitation transfer between the nearest mole-cules at different chains. Its sign, depending on the particularmaterial and the type of interaction between the molecules on dif-ferent chains, may be either positive or negative. In the case of theintramolecular energy transfer [12,15,41–45] dipole–dipole interac-tion is responsible for both intra- and interchain coupling and thesigns of corresponding exchange integrals are determined by the di-pole orientation. Within the single chain dipoles are all aligned andJ > 0. On the other side, dipoles at different chains are anti-paralleland L > 0. In the case of charge transfer in Peierls dielectrics it iscommonly taken that L < 0. Nevertheless, in some substances itmay be of the opposite sign or may even alternate along the chainand the extended model with L! Lþ ð�1ÞnL0 [33] has been pro-posed for the examination of the electronic properties of such mate-rials. This possibility will be set aside in our study.

Only the local e–p interaction will be considered hereafter. Forthat reason, we chose the explicit form of the coupling parameter

as Fq ¼ 2viffiffiffiffiffiffiffiffiffi

�h2Mxq

qsin qR0, where v stands for the strength of e–p

interaction while R0 denotes the lattice constant. The effects ofthe nonlocal e–p coupling, i.e. the one coming from the phonon in-duced variation of the resonant intra-chain transfer usually usedwithin the famous Su–Schrieffer–Heeger (SSH) model, were notconsidered since, as far as the large polaron is considered, contin-uum approximation applies and this type of coupling has preciselythe same role as the local one and, if necessary, may be accountedby the renormalization of the strength of the e–p interaction.Lattice subsystem is described in terms of phonon creation (ayq;j)

and annihilation operators (aq;j) and frequency xq ¼ x0 sin qR02 ;

x0 ¼ 2ffiffiffiffiffiffiffiffiffiffiffij=M

pwith j representing the stiffness of the chain, while

M stands for the mass of the molecules (molecular groups) inlattice sites.

3. Variational ansatz and validity of adiabatic approximation

Let us now discuss the conditions for the occurrence of the sol-iton-like large polaron states of the above model. We first recallthat the whole large polaron concept relies on the adiabaticapproximation which assumes the formal treatment of the phononoperators as classical variables. Having in mind the limitations and

D. Cevizovic et al. / Chemical Physics 426 (2013) 9–15 11

certain controversies raised concerning the validity of such anapproach, we outline first the conditions under which it applies.In brief, large polaron formation is possible provided that the sys-tem parameters satisfy adiabatic and strong coupling conditions.This concisely may be expressed through the inequality:

2J � EB � �hx0 ; x0 ¼ 2ffiffiffiffiffijM

r: ð2Þ

Here EB ¼P

qjFq j2�hxq� 4v2

Mx20

denotes the small polaron binding energy

which characterizes the strength of e–p coupling.Adiabatic condition (2J � �hx0) provides that the fluctuations in

electron and lattice subsystems are uncorrelated and may be ne-glected [22]. Consequently, the expectation values of the productsof the electron and lattice operators may be factorized and theoret-ical treatment may be carried out within the semiclassical approx-imation [21,22,62,63]. Strong coupling condition (EB � �hx0)provides the polaron stability; that is, the lattice distortion whichcaptures the electron is deep enough to prevent residual phononfluctuations from destroying so formed bound state. Note thatthe coupling should not be very strong in order to allow the polar-on to span large number of lattice sites. This is achieved in the con-tinuum approximation which applies under the condition 2J � EB

[22,63].On the basis of existing data it looks like that the above criteria

could be satisfied in great deal of Q1D media. Thus, for example, inmost conjugated polymers typical values of the intrasite transfer inte-gral and electron–phonon coupling parameter are J � 2:5 eV andv � 4:1 eV/Å [13], while the phonon energies range between0.12 eV in polyacetylene [13] and 0.2 eV in the double strand polya-cene molecule. Similar situation emerges in the a-helix proteinswhere electron bandwidth is estimated to be of the order of few eVwhile the maximal acoustic phonon energy is about 17 meV [63,64].

Under the above circumstances system dynamics may be de-scribed within the simple time dependent extension of the Pekar’svariational theory allowing the simple factorization ansatz for trialstate:

jWi ¼X

n;j

Wn;jðtÞByn;jj0ie � jaðtÞi; ð3Þ

Here Wn;jðtÞP

n;jjWn;jj2 ¼ 1� �

represents the electron wave function

on jth chain, while the phonon part is chosen in the form of the mul-

timode coherent state jaðtÞi ¼Q

q;jjaq;jðtÞi aq;jjaðtÞi ¼ aq;jðtÞjaðtÞi� �

.Electron wave functions Wn;j and phonon coherent amplitudesaq;jðtÞ are treated as dynamical variables for which we derive theset of evolution equations employing the time dependent varia-tional principle d

R t2t1

dtL ¼ 0. Here

L ¼ i�h2hWðtÞ @

$

@t

������������WðtÞi � hWðtÞjHjWðtÞi; ð4Þ

denotes system Lagrangian. In such a way we derive the set of Hamil-ton’s equations for our dynamical variables i�h _Wn;j ¼ @H

@W�n;j; i�h _aq;j ¼ @H

@a�q;j

,where H ¼ hWjHjWi stands for the Hamiltonian function.

In the explicit form evolution equations read

i�h _Wn;j ¼ DWn;j � JðWnþ1;j þWn�1;jÞ þ LW3�j

þ 1ffiffiffiffiNp

Xq

Fq;jeiqnR0Wn;j aq;j þ a��q;j

� �;

i�h _aq;j ¼ �hxqaq;j þ1ffiffiffiffiNp

Xn

F�q;je�iqnR0 jWn;jj2: ð5Þ

At this stage we pass to continuum limit and assume that electronprobability densities depend on time only through the coordinate in

moving frame: jWjðx; tÞj2 ¼ jWjðx� vtÞj2 (v-polaron velocity). Then,

by mens of the substitution y ¼ x� vt, we transform evolutionequations for phonon coherent amplitudes into the set of ordinarydifferential equations of the first order which may be easily inte-grated to give aq;jðtÞ ¼ acoh

q;j ðtÞ þ ahq;jð0Þe�ixqt [65,66]. Here we shall

consider stationary polaron, therefore, only the particular solution

acohq;j ¼ �

1ffiffiffiffiNp

F�q;j�hðxq � qvÞ

ZdxR0

e�iqxjWjðx; tÞj; ð6Þ

which only participates in polaron formation, will be taken into theconsideration. The homogeneous solution, corresponding to phononfluctuations will be disregarded hereafter in consistency with theadiabatic treatment. Note, however, that its implications on polarondynamics [65,66] may be significant and cannot be fully disre-garded especially at finite temperatures. The examination of thefinite temperature effects is beyond scope of the present paperand will be done separately.

Substituting (6) in the evolution equations for polaron ampli-tudes yields the system of the two coupled nonlinear Schrödingerequations for polaron wave functions

i�h _Wj þ JR20@2Wj

@x2 þ GðvÞjWjj2Wj þ LW3�j ¼ 0; j ¼ 1;2

GðvÞ ¼ 4EB

1� v2

c20

; c0 ¼ R0x0: ð7Þ

Physically irrelevant term D� 2J has been absorbed in timedependent phase factor Wj ! e�

ı�hðD�2JÞtWj.

4. Spectra of linear excitations

Before going into the detailed study of the soliton-like excita-tions of the system it is useful first to discuss the linear modes,i.e. the solutions of the system (5) in the absence of the e–pcoupling. Thus, setting Fq ¼ 0 and taking the polaron amplitudesin the form Wn;jðtÞ ¼ e�ði=�hÞeðkÞtþiknR0 Aj we obtain the homogeneoussystem of linear equations for amplitudes Aj.

ðeðkÞ � Dþ 2J cos kR0ÞAj � LA3�j ¼ 0: ð8Þ

Without loss of generality one may assume that Aj are real. In par-ticular, choosing Aj ¼ ei/j jAjj and separating the real from imaginarypart of the above equation we found that phase difference satisfies/1 � /2 ¼ sp; ðs ¼ 0;1; . . .Þ. This yields the same linear system forjAjj as above but with L! �L. This simple change is irrelevant sinceit does not affect our solutions in any way. In particular, by virtue ofthe above system and normalization condition, we found theeigenvalues and eigenstates of the linear excitations:

elðkÞ ¼ D� 2J cos kR0 þ lL; l ¼ �1;

jA1j2 ¼ jA2j2 � 1=2;

jWli ¼ �1ffiffiffiffiffiffiffi2Np

Xn

ðByn;1 þ lByn;2ÞeiknR0�1�heðkÞt j0i: ð9Þ

Thus, the spectrum of the linear excitations of system splits in twobands, symmetric (l ¼ 1) and antisymmetric (l ¼ �1) ones, sepa-rated by the gap which equals 2L. The corresponding eigenstatesrepresent the linear superposition of the electron states in differentchains and correspond to hybrid, fully delocalized state, in whichelectron probability density is equally distributed on both chainsjA1j2 ¼ jA2j2.

12 D. Cevizovic et al. / Chemical Physics 426 (2013) 9–15

5. Average profile approximation and effective dimer model

We now examine soliton solutions of the coupled system of twononlinear Schrödinger Eq. (7). Analogous model equations appearin various contexts [55–61] and have been intensively studiedespecially within the nonlinear optics [56–61]. Apart the two par-ticular cases a. trivial W1 ¼ 0;W2 – 0; W2 ¼ 0;W1 – 0 and b. sym-metric W1 ¼ W2 and antisymmetric W1 ¼ �W2, the explicitsolutions of this system are yet unknown in general case. Never-theless, properties of their solutions may be pretty accuratelyexamined by means of different approximate approaches. In thatrespect the average profile approximation turns out to be very con-venient[59–61]. In particular, it is simple enough to allow analyticcalculations whose accuracy has been demonstrated by compari-son with the numerical calculations [56,57,59–61]. The main ideabehind such an approach is the assumption that the solutions ofthe coupled nonlinear Schröedinger equations do not substantiallydiffer from those of the uncoupled ones. Thus, they may be chosenin the form Wj ¼ AjðtÞeiðkðx�x0ðtÞÞ�xtÞW. The real function W is chosen

in the form W ¼ffiffiffil2

qsech l

R0ðx� vt � x0Þ;l ¼ G

4J, in order to recover

pure 1d results in the limit L! 0, while, the explicit form for polar-

on quasi-momentum k ¼ m�v�h m� ¼ �h2

2JR20

� �follows immediately after

the substitution of the assumed form of the solutions into (5) andseparating imaginary from the real part.

To determine polaron amplitudes we now derive an effectivesystem Lagrangian expressed in terms of amplitudes AjðtÞ. For thatpurpose it is necessary to eliminate phonon variables from thesystem Lagrangian (4) and rewrite it in terms of polaron wavefunctions (Wjðx; tÞ) as follows:

L ¼ i�h2

Xj

ZdxR0

_WjW�j � _W�j Wj

� �

�X

j

ZdxR0

LW�j W3�j þ JR20@Wj

@x

��������2

� GðvÞ2

Wj

�� ��4 !: ð10Þ

We then substitute the above assumed form of solutions Wj into thelast expression and perform integration over x. This yields

L ¼ i�h2

Xj

_AjA�j � cc:

� ��Heff

Heff ¼ aX

j

jAjj2 �g2

XJ

jAjj4 þ LX

j

A�j A3�j: ð11Þ

a ¼ mv2

2þ JR2

0

ZdxR0

@W@x

� 2

� mv2

2þ G2ðvÞ

48

g ¼ GðvÞZ

dxR0

W4 � G2ðvÞ12J

ð12Þ

Note thatHeff in the above expressions should not be confused withthe expectation value of the model Hamiltonian in the trial state (3)which, after the elimination of phonon variables, reads:

H¼D�2Jþm�v2

2þX

j

JR20

ZdxR0

@Wj

@x

��������2

�GðvÞ2

1� 3v2

c20

1� v2

c20

ZdxR0

Wj

�� ��4þLZ

dxR0

W�j W3�j

24

35

�D�2Jþmv2

2þG2ðvÞ

48J�X

j

G2ðvÞ24J

1� 3v2

c20

1� v2

c20

jAj j4þLA�j A3�j

0@

1A: ð13Þ

6. Results and discussion

Demanding the stationarity of the effective Lagrangian weobtain the following system of coupled differential equations:

i�h _A1 ¼ aA1 þ LA2 � gjA1j2A1;

i�h _A2 ¼ aA2 þ LA1 � gjA2j2A2: ð14Þ

We now search for the stationary solutions of the above system inthe form: Aj ¼ e�ixtBj which, supplemented with the normalizationcondition, yields the following system equations for amplitudes Bj

and x:

ð�hx� aÞB1 ¼ LB2 � gB31; ð�hx� aÞB2 ¼ LB1 � gB3

2;B21 þ B2

2 ¼ 1:

As discussed in the previous paragraph amplitudes Bi may taken tobe real functions.

To find frequency x we multiply the first two equations respec-tively with B1 and B2 and then add them. This, by virtue of normal-ization condition, yields:

�hx ¼ aþ 2LB1B2 � gðB41 þ B4

2Þ: ð16Þ

To evaluate polaron amplitudes we now multiply first of theseequations with B2 and the second one with B1 and then subtractthem from each other. This yields

LðB22 � B2

1Þ ¼ gB1B2ðB21 � B2

2Þ; ð17Þ

implying the existence of two substantially different types of solu-tions in accordance with conditions L ¼ gB1B2; B

21 – B2

2 andB2

1 ¼ B22; L – gB1B2.

The first of these conditions, combined with the normalizationconstraint, yields the biquadratic equation for polaron amplitudes

B4j � B2

j þ c2 ¼ 0; ð18Þ

whose solutions

Bj ¼ �1ffiffiffi2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ð�1Þ3�j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4c2

qr; with �hx ¼ a� g; ð19Þ

define so called asymmetric solutions B21 – B2

2 with polaron pre-dominantly localized on one chain. The parameter c ¼ jLjg has beenintroduced here as a measure of the strength of the interchaincoupling.

The choice of the sign of the each particular solution in theabove expression depends on the sign of the interchain transferintegral. Thus, in order Eq. (17) to be satisfied, solutions (19) haveto be of the opposite sign for L > 0, and of the same sign for L < 0.

Substitution of the above solutions into (13) yields the energy ofthe asymmetric polaron

E ¼ D� 2J þm�v2

2þ G2ðvÞ

48J� 2jLjc� G2ðvÞ

24J

1� 3v2

c20

1� v2

c20

ð1� 2c2Þ: ð20Þ

In order to evaluate ground state energy and effective mass wenow pass to the ’’non-relativistic’’ limit (v2=c2

0 � 1) and expand po-laron energy in powers of v2=c2

0, which, in the lowest order in v2

c20,

yields:

E � D� 2J � G20

48J ð1þ 4c20Þ þ

msolv2

2 ;

msol ¼ m� 1þ G20R2

0

6�h2c20

� :

ð21Þ

The third term in the first expression above denotes the polaronicshift of the ground state energy and its absolute value correspondsto the large polaron binding energy. Finally, the last term representspolaron kinetic energy. The effective mass of asymmetric polaronattains precisely the same form as in the case of an ordinary 1Dpolaron.

Fig. 1. The magnitudes of the probability density (square of polaron amplitudes) ofthe occupation of particular molecular chain versus strength of the interchaincoupling.

Fig. 2. Comparison of the polaron amplitudes on different chains for three differentvalues of the interchain coupling strength.

Fig. 3. Variation of the system ground state energy (measured in units E0GS ¼

G20

48J-ground state energy in the absence of interchain coupling) versus the strength ofinterchain coupling.

D. Cevizovic et al. / Chemical Physics 426 (2013) 9–15 13

The asymmetric solutions exist only in the weak interchain cou-pling regime – 0 < c < 1=2 and correspond to the polaron predom-inantly distributed over the single chain i.e. B2

j > B23�j. With the

increase of the interchain coupling the difference between theseamplitudes gradually vanishes and asymmetric solutions continu-ously transit towards the new type of polaron solutions, the hybridones (Fig. 1 and Fig. 2). They are determined by B2

1 ¼ B22 � 1=2 and

correspond to polaron equally distributed among the chains. Theirenergy spectrum consists of the two branches: symmetric, withboth amplitudes in phase, i.e. both ’’positive’’ (þþ) or both ‘‘nega-tive’’ (��), with �hx ¼ aþ L� g

2, and the antisymmetric ones whenamplitudes are out of phase (þ�; �þ) while �hx ¼ a� L� g

2. Theenergy the hybrid soliton excitations is specified as:

E� ¼ D� 2J þmsolv2

2� L: ð22Þ

Signs þ and � should be associated with symmetric and antisym-metric solutions, respectively.

The effective mass of both asymmetric and the hybrid solitonattains precisely the same form as for pure 1D systems and in thatsense resembles the ordinary 1D polaron. Nevertheless, only theformer one, being confined to a single chain represents the truecounterpart of the ordinary 1D polaron. Its ground state energy liesbelow the bottom of the conduction band which provides its stabil-

ity with respect the free (band) states. On the contrary, the hybridpolaron is delocalized, while its energy is always above the bottomof the conduction band so it represents the excitation within theband of linear modes Fig. 3.

7. Polaron stability: the linear mode analysis

So far, in referring to polaron stability we have addressed the is-sue of its energetic stability, i.e. we were looking for the criteria un-der which particular type solutions correspond to the minima ofthe effective adiabatic functional Heff . Nevertheless, this is onlythe particular aspect of the problem, while the comprehensiveunderstanding demands the analysis of the influence of the linearmodes on polaron stability. This is of particular importance in theview of the assumed role of large polarons in long range transferprocesses. The most correct approach in answering that question(also important from the point of view of accounting for the quan-tum-mechanical corrections to polaron energy and for exploring itsmobility [20]) would be to find the phonon spectrum in the pres-ence of polaron. Formally, such procedure consists in perturbationtreatment of linearized equations for lattice vibrations and elec-tron wave functions. Nevertheless, in the view of the present treat-ment involving the approximation of the average profile, we adopta bit different approach and restrict our attention to the stability ofthe above obtained solutions. Thus, we apply standard linear per-turbation theory [67] and introduce small deviations in the polaronamplitudes as follows:

AjðtÞ ¼ ðBj þ dAjðtÞÞe�ixt ; ð23Þ

with x specified by (15). Deviations dAjðtÞ are both complex andsmall relative to the stationary solutions. In order evaluate thesedeviations we substitute AjðtÞ as proposed above in the system(14). Then, separating imaginary from the real part, and keepingonly the first order terms, we derive the following system of evolu-tion equations for these variables:

ddt

Re dAðtÞIm dAðtÞ

� ¼

0 B

A 0

� Re dAðtÞIm dAðtÞ

� :

Here A; B represents the following 2x2 matrices

B ¼�ð�hx� aþ gB2

1Þ L

L �ð�hx� aþ gB22Þ

!

A ¼�hx� aþ 3gB2

1 �L

�L �hx� aþ 3gB22

!;

14 D. Cevizovic et al. / Chemical Physics 426 (2013) 9–15

while

Re dAðtÞ ¼Re dA1ðtÞRe dA2ðtÞ

� ; Im dAðtÞ ¼

Im dA1ðtÞIm dA2ðtÞ

� :

We now pose dAjðtÞ ¼ ek�htdAjð0Þ and then, for each type of the above

solutions, evaluate instability growth rate (k) from the conditionthat the above system has the non-trivial solutions.

This yields

k2 k2 � 8L2 � 5g2

2

!¼ 0; ð27Þ

for asymmetric solutions and

k2 k2 þ 4L2 � 2gL� �

¼ 0; ð28Þ

for hybrid ones. The upper and lower sign in the above expressioncorrespond to the symmetric and antisymmetric solutions,respectively.

Stable solutions arise when instability growth rate is pure imag-inary. This condition is satisfied always for all types of solutionswhich means that the both types of polaron are linearly stableand may propagate coherently parallel to chains maintaining theirform for a long time.

8. Concluding remarks

We have introduced a simple analytical method which providesan easy yet rather accurate tool for the study of the effects of theinterchain coupling on polaron features in Q1D media. It reliesupon the average profile approximation which allows the reduc-tion of the system of coupled nonlinear Schrödinger equationsfor polaron envelope functions to a system of the coupled nonlin-ear equations for polaron amplitudes. We have considered the sys-tem consisting of just two molecular chains and particularcalculations were carried out within the nonlinear dimer model[67].

In spite of its simplicity such an approach is flexible enough tocover the gross of the previously published particular results andenables their understanding on unified ground. In particular, ourstudy is consistent the previous ones [49–54,37] which predictthe occurrence of two substantially different types of polaron solu-tions. Finally, the simple criterion for their existence each of themis established: in the weak coupling regime (c < 1=2) asymmetricpolaron, predominantly distributed along the single chain and withthe energy below the bottom of the conduction band, appears.Above this critical value we observe new type of solutions corre-sponding to polaron equally distributed among the chains. In thiscase the shift of the ground state energy is zero so that polaron en-ergy lies fully within the band of free states. Such polaron no morecorresponds to energetically the most favorable state. This meansthat the interchain coupling leads to a large polaron destabilizationwhich may be understood as a transition between asymmetric andhybrid polarons. This transition takes place as continuous equaliza-tion of the polaron amplitudes on different chains followed by thegradual disappearance of the polaron ground state energy Figs. 2and 3.

The desired criterion for the existence of stable, effectively 1Dpolaron (i.e. c < 1=2), may be expressed as follows.

E2B

1� v2

c20

� �2 >32

LJ: ð29Þ

For quantitative estimations of the threshold value of the ratio ofthe intra- over the inter-site transfer integrals it is appropriate to

rephrase the above condition, in the static limit v ¼ 0, in terms ofthe polaron width Rpol=R0 1=l ¼ J=EB

JL>

32

Rpol

R0

� 2

: ð30Þ

The analogous criterion for the large polaron existence within thehighly anisotropic 3D media established in [25] reads

JL> 12

Rpol

R0

� 2

: ð31Þ

The difference in the numerical prefactors on the right hand sides ofthese relations arises as a consequence of the fact that our studyconcerns the system composed of the two chains only, while (31)addresses the highly anisotropic but essentially 3D lattice. Appar-ently, the required threshold value for the polaron formation inthe two chain system is substantially lower than that in bulk media.

For the comparison with the previous studies we must pointthat our analysis strictly concerns so called conventional polaron,i.e. single excess electron (hole) in the otherwise empty conductionband, which is fully equivalent to a polaron in Peierls models onlywithin the ‘‘frozen-valence-band’’ approximation [19]. In spite ofthat, the analogous results, qualitatively fully equivalent with ours,were reported previously in a variety of the studies of polarons inconducting polymers [37,49–54]. As for the agreement with quan-titative estimates we refer to the reference [37] where the influ-ence of the interchain coupling on large polaron stability inconducting polymers have been numerically studied within the ex-tended SSH model. Polaron properties in the two parallel chainsand in the cluster of parallel chains were explored. In the two-chain case almost identical results with ours have been reported.In brief, for the small interchain coupling, i.e. up to L=J 0:004, po-laron is practically confined to a single chain. With the increase ofthe interchain coupling the depth of potential well for electron cre-ated by the lattice distortion gradually decreases, in parallel, themagnitude of polaron amplitude on one chain also decreases, whilein the second one increases. This continues until interchain cou-pling approaches L ¼ 0:15 eV. Beyond that, the two amplitudesare symmetric so that L ¼ 0:15 eV corresponds to a critical cou-pling. To strengthen the pointed equivalence we estimate the crit-ical interchain coupling strength. For that purpose we first have toestablish relationship between the values of the system parame-ters exploited in [37] and in the present study. At this stage we re-call that the present model and SSH are equivalent in the strictcontinuum limit, i.e. in approaching the continuum limit one keepsonly the most dominant terms (those R2

0). Under such circum-stances the system of nonlinear Schrödinger equation for polaronwave functions derived within the framework of SSH model isidentical to ours (7). This enables us to identify correspondingparameters – J � tk ¼ 2:5 eV, a ¼ v � 4:1 eV/Å and finally for thestiffness we took j � K ¼ 21 eV/Å

2. In such a way we estimate crit-

ical coupling as c 0:485 which deviates from ours for about11.5%. We consider this as satisfactory agreement, while, the abovediscrepancy arises as a consequence of the fact that the presentmodel (1) and the SSH are equivalent only in the strict continuumlimit so that the established relationship between system parame-ters is only approximate. For that reason this discrepancy rather re-flects the inconsistency of the continuum approximation than itmay be associated with the unreliability of the average profileapproximation.

In connection with the experimental verification of our predic-tions we point to a possible significant impact of the polaronmotion on its character. This is the consequence of the pseudo-relativistic dependence of the coefficient in the nonlinear term ofEq. (7) on polaron velocity whose magnitude, for that reason, in-creases with the velocity and enhances the depth of the potential

D. Cevizovic et al. / Chemical Physics 426 (2013) 9–15 15

well that captures the electron. Thus, polaron motion has the ten-dency opposite to the interchain coupling and, in final instance,may cause its confinement to a single chain. This is the most strik-ing experimental implication of our results. It may be revealedthrough the possible emergence of the two different modes in po-laron motion. In particular, when polaron velocity is small in com-parison with the speed of sound (v=c0 � 1), polaron translationalmotion along the two parallel chains should be accompanied withthe periodic inter-strand transfer, while, when it is close to thespeed of sound, motion should be confined to a single strand. Pre-cisely the same effect has been predicted recently in the numericalsimulations of the charge transfer in two-strand model of the DNAmolecule [48].

According to our predictions it would be possible to achieve thetransition of the localized (asymmetric) polaron into the delocal-ized (symmetric or antisymmetric) one and vice versa by meansof varying its velocity with the help of the applied electric field,for example. We expect that in low fields, directed along thechains, and for coupling strength less than the critical one, transferwould have the quasi oscillatory character – translational motionassociated with the interchain hopping. With the increase of theelectric field only the translational motion along the chains wouldpersist. Described features could be tested by the longitudinal andtransverse polaron mobility measurements at low temperatures inDNA or polyacene. At this stage it is difficult to give definitedescription of the polaron dynamics in the particular experimentalsetup, nevertheless, we expect that sufficiently strong longitudinalelectric fields should suppress the interchain transfer – i.e. l? � 0when electric field exceeds some specific value EC

k .The full validation of our prediction and precise estimates of the

critical values of electric field demands the comprehensive analysisof polaron dynamics in the presence of defects and taking into ac-count the polaron–phonon interaction [17]. These issues are far be-yond the scope of this study and we left them for further analysis.

Acknowledgments

This work was supported by the Serbian Ministry of Educationand Science under Grants Nos. III-45010 and OI-171009.

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