Impurity effects on polaron-exciton formation in conjugated polymersLuiz Antonio Ribeiro Jr., Wiliam Ferreira da Cunha, Pedro Henrique de Oliveira Neto, Ricardo Gargano, and

Geraldo Magela e Silva

Citation: The Journal of Chemical Physics 139, 174903 (2013); doi: 10.1063/1.4828726 View online: http://dx.doi.org/10.1063/1.4828726 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Scattering process between polaron and exciton in conjugated polymers J. Chem. Phys. 134, 044906 (2011); 10.1063/1.3548667 Recombination of polaron and exciton in conjugated polymers J. Chem. Phys. 131, 244502 (2009); 10.1063/1.3274680 Interchain coupling effects on dynamics of photoexcitations in conjugated polymers J. Chem. Phys. 128, 184903 (2008); 10.1063/1.2912190 Screening and polaronic effects induced by a metallic gate and a surrounding oxide on donor and acceptorimpurities in silicon nanowires J. Appl. Phys. 103, 073703 (2008); 10.1063/1.2901182 Polaron–excitons and electron–vibrational band shapes in conjugated polymers J. Chem. Phys. 118, 4291 (2003); 10.1063/1.1543938

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THE JOURNAL OF CHEMICAL PHYSICS 139, 174903 (2013)

Impurity effects on polaron-exciton formation in conjugated polymersLuiz Antonio Ribeiro, Jr.,a) Wiliam Ferreira da Cunha, Pedro Henrique de Oliveira Neto,Ricardo Gargano, and Geraldo Magela e SilvaInstitute of Physics, University of Brasilia, 70.919-970 Brasilia, Brazil

(Received 18 July 2013; accepted 19 October 2013; published online 5 November 2013)

Combining the one-dimensional tight-binding Su-Schrieffer-Heeger model and the extended Hub-bard model, the collision of two oppositely charged polarons is investigated under the influence ofimpurity effects using a non-adiabatic evolution method. Results show that electron-electron inter-actions have direct influence on the charge distribution coupled to the polaron-exciton lattice defect.Additionally, the presence of an impurity in the collisional process reduces the critical electric fieldfor the polaron-exciton formation. In the small electric field regime, the impurity effects open threechannels and are of fundamental importance to favor the polaron-exciton creation. The results in-dicate that the scattering between polarons in the presence of impurities can throw a new light onthe description of electroluminescence in conjugated polymer systems. © 2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4828726]

I. INTRODUCTION

Conjugated polymers have attracted considerable inter-est from the scientific community since the discovery of elec-troluminescence properties on phenyl-based organic semicon-ductors. Their optoelectronic features combined with easeof synthesis, low temperature processing, tunability via syn-thetic chemistry, and low cost, make them attractive materi-als for the electronics industry, particularly concerning thedevelopment of a new display technology. Among the sev-eral possible application are Organic Photovoltaics devices(OPVs)1–3 and Polymer Light Emitting Diodes (PLEDs).4–6

In these devices, the generation of excited states is one of themost fundamental physical process.7–11 Lei et al.12 studiedthe dependence of exciton formation rate on the spin orien-tation of polarons by simulating the collision process of twooppositely charged polarons, using a modified version of Su-Schrieffer-Heeger (SSH) model. The role played by the spinconfiguration in the collision dynamics of these charge carri-ers is a question of major importance for electroluminescenceprocess in PLEDs. Their results show that the yield of sin-glet excitons from parallel spin or antiparallel spin polaronsconfiguration is not influenced by the e-e interactions. On theother hand, in the manufacture of spin PLEDs to improve theluminescent efficiency, the e-e interactions should be criticalfactors, whose role must be clarified. Particularly, understand-ing how internal effects such as e-e interactions and impurityeffects can favor or unfavor the polaron-exciton formation viascattering of oppositely charged polarons is an issue that isbelieved to be crucial for the design of more efficient deviceswith respect to the electroluminescence. Thus, this point re-quires a better phenomenological description.

Some relevant theoretical studies carried out by Anand co-workers13 have shown that external electric fieldhas a significant influence on polaron-exciton formation via

a)Electronic mail: ribeirojr@fis.unb.br

polaron-pair scattering in conjugated polymers. The goal wasto identify the generation mechanism of the self-trappedpolaron-exciton. Their results show that, for the scatteringprocesses of the two polarons initially presented on a samepolymer chain, three regimes of the applied electric fieldstrength were identified: (1) at field strength smaller than0.2 mV/Å, the polaron-pair scatter into an exciton stateafter 1300 fs; (2) for an electric field strength between0.2 mV/Å and 1.2 mV/Å, the polaron-pair scatters into a pairof independent particles and each of them is a mix of po-larons and excitons (in this case, the yield of the neutral ex-citon depends sensitively on the electric field strength); and(3) at electric field strength greater than 1.2 mV/Å, the twopolarons break into irregular lattice vibrations after their col-lision. When two polymer chains are taken into account, thetwo polarons will combine together to form a self-trapped ex-citon in one of the two coupled polymer chains when theylie initially on the different polymer chains. The results indi-cate that the interchain interaction favors the formation of thepolaron-exciton. It is important to remark that these studieshave been focused on specific cases with idealized conditions.A theory that widely holds true for real materials needs fur-ther verifying by addressing some realistic effects, for exam-ple, the order degree or molecules, temperature and impurityeffects. Also, from these works, we can see that all the resultsfor the polaron-pair interaction are not fully described, so thatfurther investigations that take into account some of these ef-fects are needed.

In this paper, a systematic numerical investigationof polaron-exciton formation was performed in a cis-polyacetylene chain in terms of a non-adiabatic evolutionmethod. The collision of oppositely charged polarons was in-vestigated on a conjugated polymer chain subjected to differ-ent field strengths, one-site, and nearest-neighbor Coulombinteractions. We carried out the molecular dynamics by us-ing an one-dimensional tight-binding model including lat-tice relaxation. Combined with the extended Hubbard model

0021-9606/2013/139(17)/174903/6/$30.00 © 2013 AIP Publishing LLC139, 174903-1

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174903-2 Ribeiro, Jr. et al. J. Chem. Phys. 139, 174903 (2013)

(EHM), an extended version of the SSH model was used toinclude external electric fields and Brazoviskii-Kirova sym-metry breaking term. The aim of this paper is to give aphysical picture of the polaron-exciton formation in conju-gated polymers, when the e-e interactions and impurity effectsare taken into account, and contribute to the understandingof these important processes that may provide guidance forimproving the electroluminescence efficiency in PLEDs.

II. METHODOLOGY

A polyacetylene chain in cis configuration14 was usedto study collision between oppositely charged polarons un-der the influence of impurity effects in conjugated polymers.The overall Hamiltonian is given by

Htotal = HSSH + Hee + Himp. (1)

The first term in Eq. (1) is the SSH-type Hamiltonian mod-ified to include an external electric field and the Brazovskii-Kirova symmetry-breaking, which has the following form:

HSSH = −∑n,s

(tn,n+1C

†n+1,sCn,s + h.c.

)

+∑

n

K

2y2

n +∑

n

p2n

2M, (2)

where n indexes the sites of the chain. The operator C†n,s (Cn, s)

creates (annihilates) a π -electron state at the nth site withspin s; K is the harmonic constant that describes a σ bond andM is the mass of a CH group. The parameter yn is defined asyn ≡ un + 1 − un where un is the lattice displacement of anatom at the nth site. pn is the momentum conjugated to un andtn, n + 1 is the hopping integral, given by

tn,n+1 = e−iγ At[(

1 + (−1)nδ0)t0 − αyn

], (3)

where t0 is the hopping integral of a π -electron betweennearest-neighbor sites in the undimerized chain, α is theelectron-phonon coupling, and δ0 is the Brazovskii-Kirovasymmetry-breaking term, which is used to take the cis sym-metry of the polymer into account. γ ≡ ea/(¯c), with e be-ing the absolute value of the electronic charge, a is the latticeconstant, and c is the speed of light. The relation between thetime-dependent vector potential A and the uniform electricfield E is given by E= −(1/c)A.

The second contribution in Eq. (1) denotes e-e interac-tions and can be written as

Hee = U∑

i

(C

†i,↑Ci,↑ − 1

2

)(C

†i,↓Ci,↓ − 1

2

)

+V∑

i

(ni − 1)(ni+1 − 1), (4)

where U and V are the on-site and nearest-neighborCoulomb repulsion strengths, respectively, andni = C

†i,↑Ci,↑ + C

†i,↓Ci,↓. It should be noted that the in-

clusion of the additional constant factors (related to theconventional description of the Hubbard model) is necessaryin order to maintain the electron hole symmetry of theHamiltonian.

The last contribution in Eq. (1) represents the on-site im-purity effects and can be written in the form

Himp = ZjC†j,sCj,s . (5)

Zj is the strength of an impurity, which is locatedin jth site. The parameters used here are t0 = 2.5 eV,M = 1349.14 eV × fs2/Å2, K = 21 eV Å−2, δ0 = 0.05,α = 4.1 eV Å−1, a = 1.22 Å, and a bare optical phonon en-ergy ¯ωQ = ¯√4K/M = 0.16 eV. These values have beenused in previous simulations and are expected to be valid forconjugated polymers in general.15–19

In order to solve these equations numerically, first a sta-tionary state that is self-consistent with all degrees of freedomof the system (the lattice plus electrons) is obtained. We beginby constructing the Hamiltonian from an arbitrary {yn} set ofpositions. By solving the time dependent Schrödinger equa-tion, a new set of coordinates {y’n} is obtained. Iterative rep-etitions of this procedure yields a self-consistent initial statewhen {y’n} is close enough to {yn}.

The time evolution of the system is described by theequations of motion. The electronic wave function is the so-lution of the time-dependent Schrödinger equation

i¯ψk,s(i, t) = − [t∗i,i+1 + V τs(i, t)

]ψk,s(i + 1, t)

− [ti−1, i + V τ ∗

s (i − 1, t)]ψk,s(i − 1, t)

+{U

[ρ−s(i, t) − 1

2

]+ Zj

+ V∑s ′

[ρs ′ (i + 1, t) + ρs ′ (i − 1, t) − 1]

}

×ψk,s(i, t), (6)

where k is the quantum number that specifies an electronicstate,

ρs(i, t) =∑

k

′ψ∗k,s (i, t) ψk,s (i, t) , (7)

and

τs(i, t) =∑

k

′ψ∗k,s (i + 1, t) ψk,s (i, t) . (8)

The prime symbol specifies that the sum runs over occupiedsingle-particle states. It is important to note that, consideringthe hydrocarbon nature of our system, an unrestricted Har-teee Fock scheme is sufficient to correctly address the desiredcharge transport properties.

The equation of motion that describes the site displace-ment and provides the temporal evolution of the lattice isobtained by a classical approach.14 The nuclear dynamics iscarried out by considering the Euler-Lagrange equations

d

dt

(∂〈L〉∂un

)− ∂〈L〉

∂un

= 0, (9)

where

〈L〉 = 〈T 〉 − 〈V 〉. (10)

Equation (9) leads to

Mun = Fn(t), (11)

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174903-3 Ribeiro, Jr. et al. J. Chem. Phys. 139, 174903 (2013)

where

Fn (t) = −K[2un (t) − un+1 (t) − un−1 (t)

]+α

[Bn,n+1 − Bn−1,n + Bn+1,n − Bn,n−1

]. (12)

Fn(t) represents the force on the nth site. Here,

Bn,n′ =∑k,s

′ψ∗k,s (n, t) ψk,s

(n′, t

)(13)

is the term that couples the electronic and lattice solutions.The total time dependent wave function is constructed by

means of a combination of instantaneous eigenstates of theelectronic Hamiltonian. The solutions of the time-dependentSchrödinger equation can be put in the form20

ψk,s

(n, tj+1

) =∑

l

[∑m

φ∗l,s

(m, tj

)ψk,s

(m, tj

)]

× e(−iεl�t/¯)φl,s

(n, tj

). (14)

{φl(n)} and {εl} are the eigenfunctions and the eigenvaluesof the electronic part for the hamiltonian at a given time tj.Equation (12), which governs the evolution of the lattice, maybe numerically integrated using the following method:21

un(tj+1) = un(tj ) + un(tj )�t, (15)

un(tj+1) = un(tj ) + Fn(tj )

M�t. (16)

Hence, the electronic wave functions and the lattice displace-ments at the (j + 1)th time step are obtained from the jth timestep. At time tj, the wave functions {ψk, s(i, tj)} can be ex-pressed as a series expansion of the eigenfunctions {φl, s} atthat moment

ψk,s(i, tj ) =N∑

l=1

Csl,kφl,s(i), (17)

where Csl,k are the expansion coefficients. The occupation

number for eigenstate φl, s is

ηl,s(tj ) =∑

k

′|Csl,k(tj )|2. (18)

ηl, s(tj) contains information concerning the redistribution ofelectrons among the energy levels.22, 23

III. RESULTS AND DISCUSSION

In this work, we performed a series of simulationsconcerning collisional processes between oppositely chargedpolarons in a 200-site cis-polyacetylene chain. In order tomitigate end effects of the chain end, periodic boundaryconditions were imposed to our systems. This way, wesimulate a chain of infinite length, thus reducing effects ofnumerical nature, due to the symmetry breaking imposed bythe edge. The aim was to investigate the effects of impurities,electric field, and Coulomb interactions over the consequentformation of polaron-exciton after the interaction of thepolaron-pair. For the electric field, the values used in thesimulations varied from 0.1 to 1.0 mV/Åwith a increment

of 0.1 mV/Å, whereas the on-site e-e interactions valuesconsidered are 0.1, 0.2, 0.3, 0.4, and 0.5 eV. The nearest-neighbor Coulomb repulsion strength was defined using therelation V = U/2. In the present work, we considered theimpurity in the 100th site and created oppositely chargedpolarons initially at sites 50 and 150. It is important toremark that this procedure is carried out without loss ofgenerality, because the impurities act like a long rangecoulombic potential that affects all its neighborhood. Ouranalysis is carried out by considering a mean charge den-sity defined by ρ(t) = 1 − [

ρn−1(t) + 2ρn(t) + ρn+1(t)]/4

and a mean order parameter of the lattice distortiony(t) = (−1)n

[yn−1(t) − 2yn(t) + yn+1(t)

]/4. The goal is to

provide a better visualization of the simulations and conse-quently to perform a more accurate analysis of the results. Weshould emphasize that, although spins effects are known tobe of fundamental importance for the recombination processof polarons, as described in the literature,23 our goal is toinvestigate effects that are not spin dependent.12

In this context, Figure 1(a) shows the schematic diagramof energy-levels for a polymer chain containing an oppo-sitely charged polaron-pair with antiparallel spin configura-tion. These states are constructed by occupying the differ-ent energy levels, which correspond to consider the sum inEq. (13) over different states. There are four levels in the gap.The two left levels, ε1

L and ε2L, represent the electronic con-

figuration for an electron-polaron, in which the upper one isoccupied by one electron and the lower one occupied by two.The right levels, ε1

R and ε2R , represent the electronic configu-

ration for a hole-polaron, where only the ε1R level is occupied

by one electron. Figure 1(b) represents an alternative routeto creating a oppositely charged polaron-pair, with parallelspin configuration. It is well known that the exciton forma-tion rate depends on the spin orientation of polarons. Also,it is accepted that the yield of singlet excitons from paral-lel spin or antiparallel spin configuration of a polaron-pairis not influenced by the e-e interactions.12 A first importantresult that we can remark is that, indeed, no qualitative dif-ference is found on the dynamical behavior of a polaron-paircreated through mechanism of Figure 1(a) or 1(b), when im-purity effects are taken into account. We thus present resultssolely concerning the band structure configuration shown inFigure 1(a) throughout this paper.

We begin by discussing the dynamic features of thecollisional process between oppositely charged polaronsconsidering the contribution of impurity effects to thismechanism. Figure 2 presents a set of simulations for thechain endowed with a 0.2 eV impurity in the 100th site. We

FIG. 1. The schematic diagrams of energy levels for a polymer chain con-taining a electron-polaron and a hole-polaron with (a) antiparallel spin and(b) parallel spin.

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174903-4 Ribeiro, Jr. et al. J. Chem. Phys. 139, 174903 (2013)

FIG. 2. Collision between oppositely charged polarons in a polymer chainin the presence of an impurity: (a1) and (a2) E = 0.2 mV/Å and U = 0.1 eV;(b1) and (b2) E = 0.2 mV/Åand U = 0.2 eV; (c1) and (c2) E = 0.4 mV/Åand U = 0.2 eV.

choose this value because it is the typical value observed forthe potential barrier of PLEDs and for means of comparison,since it has recently been used in other theoretical work.24 Inthis case, one can observe a difference on the lattice distortionbetween different Hubbard terms, a feature that is representedon the upper part of Figure 2 (Figures 2(a1), 2(b1), and 2(c1)).Figures 2(a1)–2(b2) represent the simulation of an electricfield of 0.2 mV/Åand Hubbard terms of 0.1 and 0.2 eV,respectively, whereas Figures 2(c1) and 2(c2) show the resultof the application of a 0.4 mV/Åfor the electric field and aHubbard term of 0.2 eV. These values were chosen due to thefact that they guarantee the stability of the polaron-exciton

throughout the whole simulation. After a small transient timefor the electric field response, the polarons begin to movetowards one another due to the opposite charges. The polarondynamics before the collision processes occurs as describedby Stafström and Johansson.25, 26 At approximately 200 fs,when the collision takes place, one can see that the positivepolaron passes through the negative polaron, a process thatyields the formation of a mixed state of polaron and excitonafter the collision—thus the name “polaron-exciton.” Wenote that the polaron-exciton emerging from the collision iscreated at approximately the same time. As a matter of fact,we noted an even more general pattern: for a pristine chainand electric fields smaller than 0.2 mV/Å, no polaron-excitoncreation is seen to take place before 1500 fs,13 whereas whenthe impurity is considered, the creation time falls to at most400 fs. This is the first evidence that, in these systems, im-purity acts as an exciton creating enabler. Another importanteffect that impurity has over the systems dynamics is readilynoted by observing Fig. 2. When impurity effects are takeninto account, three formation channels are observed: (1) Afterthe collision one of the polarons becomes trapped by thepotential of the impurity. This feature can be observed bothin the lattice distortion Figure 2(a1) and in the charge densityFigure 2(a2). An interesting observation not presented bythe figure is that this pattern is observed to take place forU = 0.1 eV regardless the electric field considered. (2)Figure 2(b) is the situation in which we only increased theHubbard interaction to 0.2 eV. One can see that after the

FIG. 3. Time evolution of the occupation number for the cases shown in Figure 2.

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174903-5 Ribeiro, Jr. et al. J. Chem. Phys. 139, 174903 (2013)

collision one of the polarons ceases to be present whereas thecharge tends to be a little more delocalized over the chain.(3) Finally, Figure 2(c) represents the previous case with anincrease on the electric field, as can be inferred by the greaterslope on the order parameter of Figure 2(c1). In this case, wecan see that no polaron becomes trapped by the impurity noris destroyed in the collision process.

The previous discussion on the creation of polaron-excitons is confirmed by an analysis of the occupation num-bers presented in Figure 3. Note that this figure is not tobe confused with the schematic representation of Figure 1.Whereas Figure 1 presents the static energy levels distribu-tions for the chains containing different charge carriers in theinitial state, Figure 3 represents the time evolution of the oc-cupation numbers of the levels related to Figure 1(a). Thisfigure represents the whole system: ε1

L and ε2L, stand for the

states of the negatively charged polaron while ε1R and ε2

R , rep-resent the states of the positive one. After the transient periodin which the occupation number also oscillates due to the col-lision, we note a considerable degree of symmetry between ε1

L

and ε1R and also between ε2

L and ε2R , which is an indication of

an electron exchange between these levels. The crossed andhorizontal electron exchange between the levels representsthe formation of a polaron-exciton structure. By analyzing theoccupation number itself, one can see that the final states arenot covered by integer numbers, which is an evidence of par-tial transfer. Even so, our simulations yield excitations with

a better rate than those previously reported in the work byAn.13 We believe that the degeneracy breaking provided bythe inclusion of Hubbard terms, absent in the work of An,13 isresponsible for this better yield of excitation. In the absenceof an impurity in the lattice, studies reported that there existsonly one channel for the polaron-exciton formation: the scat-tering of the polaron-pair after the interaction.12, 13 However,when impurity effects are taken into account, the polaron-exciton formation occurs even in the collision of the polaron-pair. This fact can be noted observing Figures 3(a) and 3(b),where the time evolution of the occupation numbers indicatesan electron exchange between the levels that represent theformation of a polaron-exciton structure.

While the occupation number analysis is the most suit-able tool in studying the process of polaron-exciton creation,the stability of the quasi-particle is better described by meansof the energy levels time evolution. We finish our discussionby presenting in Figure 4 the energy levels time evolutionprofile of the simulations correspondent to an interactionbetween polarons shown in Figure 2. The collisional processis noted to take place at approximately 200 fs, when theresulting phonons are represented by the oscillation of theenergy levels. The signature of the loss of stability of oneof the polarons is related to the red states returning to theconducting and valence bands. As in Figures 4(b) and 4(c),we see that the other polaron remains stable through therest of the simulation, since the blue polarons levels remain

FIG. 4. Time evolution of the energy levels for the cases shown in Figure 2.

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174903-6 Ribeiro, Jr. et al. J. Chem. Phys. 139, 174903 (2013)

consistently inside the bandgap. In Figures 2(a1) and 2(a2),both polaron-excitons keep the integrity until the end of simu-lation. This can be seen in Figure 4(a), where the energy levelsthat correspond to the polarons remain inside the bandgap.

IV. CONCLUSIONS

In summary, it was presented a modified version ofthe SSH model to include an external electric field, theBrazovskii-Kirova symmetry breaking term, one-site andnearest-neighbor Coulomb interactions, and impurities in or-der to investigate the effects of these properties over the scat-tering process of oppositely charged polarons in cis-symmetryconducting polymers chains. Using a non-adiabatic evolutionmethod, within an one-dimensional tight-binding model, thecritical electric field regimes of polaron-exciton formation isdescribed. The influence of e-e interactions on the polaron-exciton formation is also discussed and it is found to play animportant role on the yield of excitations after the polaron-pair scattering process. The charge coupled to the polaron-exciton lattice deformation is sensitive to on-site and nearest-neighbor Coulomb interaction. The higher the Coulombinteraction values the less the charge remains coupled to thepolaron-exciton lattice deformation and the excitation yield isimproved. When the impurity effects are taken into account,the critical electric field to polaron-exciton formation is lowerthan in the absence of these effects. Also, the results indicatethat the presence of impurities in a conjugated polymerchain, for all electric fields regimes, improves the excitationyield and facilitates the polaron-exciton formation in electricfield regimes smaller than 0.7 mV/Å. These properties mayprovide guidance for improving the electroluminescence inpolymer light-emitting diodes by defining a path consideringimpurities and materials with greater electronic correlation.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial supportfrom the Brazilian Research Councils CNPq, CAPES, and FI-NATEC.

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