f p

Yand D

Electronlattice interaction

eegergend tef imtral

ers hich thand pst. Espemisst in codevelopurpog dev

the charge carrier, the polaron transports under the inuence of an

negatively charged one, they will collide and recombine to form a

beenle inerfor-ntro-rpose

charge layer [18,19]. Although a considerable amount of experi-

Contents lists available at SciVerse ScienceDirect

.el

Physic

Physica B 421 (2013) 1317focus on the dynamical process of the scattering of the polaronE-mail address: zhaoh@fudan.edu.cn (H. Zhao).external electric eld. When a positively charged polaron meets a mental work has been reported concerning exciton formation,systematically theoretical investigation of the internal mechan-isms, and, especially, effects of impurity on the recombinationprocesses have not yet been well understood. In this paper, we

0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physb.2013.04.007

n Corresponding author. Tel.: +86 21 65984890.accepted that injected charge (electron and/or hole) from themetal electrodes deforms the polymer chain to form a polaron, as

enables the electrochemical doping of the polymer at the interfaceto occur [16,17] or facilitates the formation of an ionic spacetions or photoexcitations. Based on the fundamental importance inorganic optoelectronic devices, there have been considerableamounts of research works devoted to the study of non-linearelementary excitations in conjugated polymers [7]. For example,in polymer-based light-emitting diodes, it has been generally

limit imposed by the formation of triplet excitons [8,9]. It hasproved that the ionic impurities do play a very important roboth the properties of the conjugated polymer and the pmance of the polymer optoelectronic devices [1015]. The iduction of ionic species into the polymer active layer on pucated with a luminescent polymer sandwiched between metallicelectrodes. Because of the strong electronlattice interactions andinstability, low-dimensional conjugated polymers can be easilydistorted so that self-trapped elementary excitations [2], such assolitons [35] and polarons [6], will be induced by charge injec-

because the electroluminescent efciency of OLEDs dependsmainly on the number of excitons. On the other hand, controlledand stable doping is a prerequisite for the realization and theefciency of many organic-based devices. So far, phosphorescentdyes have been used in organic LED's to overcome the efciency1. Introduction

In recent years, conjugated polymattention in global society, for whof both metal and semiconductorsof large-scale applications at low coelectroluminescence (EL), that is theted by the ow of an electric currenhas provided a new impetus for thedevices (LEDs) for display and other

These organic-based light-emittinave attracted sustainedey possess propertieserformance advantagesecially, the discovery ofion of light when exci-njugated polymers [1],pment of light-emittingses.ices are normally fabri-

neutral exciton state, in which the electron and the hole arebounded in a self-trapped lattice deformation, and have betterstability than two polarons. Then, the exciton state can decay tothe ground state by emitting a photon. Therefore, the scatteringprocess of oppositely charged polarons and the quantum yield ofneutral excitons are believed to be of vital importance for polymerbased light-emitting diodes.

Clearly, understanding the mechanisms underlying excitonformation from injected charge in organic semiconductors is anoutstanding challenge in the eld of organic electroluminescence,Dynamics study of the recombination owith impurities

Qing Lu a, Hui Zhao a,n, Yuguang Chen a, Yonghonga Key Laboratory for Advanced Microstructure Materials of the Ministry of Education ab Department of Physics, Shaoxing University, Shaoxing 312000, China

a r t i c l e i n f o

Article history:Received 30 November 2012Received in revised form15 March 2013Accepted 8 April 2013Available online 12 April 2013

Keywords:PolaronExcitonPolymer

a b s t r a c t

Within a SuSchriefferHprocess of oppositely chaevolution method. It is fouweak electric eld, and theBut under the assistance ostrengths. The yield of neucent efciency of OLEDs.

journal homepage: wwwolaron pairs in polymer chain

n b

epartment of Physics, Tongji University, 1239 Siping Road, Shanghai 200092, China

r model modied to include an external electric eld, the dynamicald polarons in a polymer chain is simulated by using a non-adiabatichat the polaron pair can combine with neutral exciton in the presence of ald requirement for achieving exciton is very strict in a pure polymer chain.purities, the exciton can be produced under a wide range of applied eldexciton can reach as high as 96%, which is signicant for electrolumines-

& 2013 Elsevier B.V. All rights reserved.

sevier.com/locate/physb

a B

the static structure by the minimization of the system energy. The

ing Newtonian equations of motion:

Q. Lu et al. / Physica B 421 (2013) 131714total energy is obtained by the expectation value of the Hamilto-nian equation (1) at the ground state jg,

Et gjHel Himpjg K2iui1ui2: 6

The electronic states are determined by the electronic, and thepair under the inuence of electric elds and impurities, andsimulate the generation mechanism of the exciton by using thelattice dynamics method [2028]. Our results show that theexciton can be produced under a wide range of applied eldstrengths with the assistance of impurities.

2. Model and method

For the calculations, we use an extended SuSchriefferHeeger(SSH) model plus electric eld, the Hamiltonian of this modelconsists of four sections,

HHel Hlatt Himp HE: 1The rst part is to describe the electron energy, which containsboth the electronlattice coupling, modelled by a SSH-type Hamil-tonian

Hel i;sti;i1ci;sci1;s h:c:; 2

where tit0ui1ui 1ite is the hopping integral betweensites i and i 1; describes the electronlattice coupling betweenneighboring sites due to the lattice bond stretch or compression;ui is the monomer displacement of site i from its undimerizedequilibrium position, and te is the BrazovskiiKirova symmetry-breaking term introduced for non-degenerate polymers [6].ci;sci;s is the creation (annihilation) operator of an electron withspin s at the site i. As usual, the electronelectron interaction isthought to be not negligible. Actually, it is really a challenge toinclude the electronic correlation in a dynamical study while noqualitative difference be found if the interaction is treated onlywithin a mean-eld approximation, that is the reason why we donot include the interaction in this work.

The second part in Eq. (1) is to describe lattice elastic potentialenergy and kinetic energy, respectively,

Hlatt K2iui1ui2

M2i

_u2i ; 3

where K denotes the force constant originating from the s-bondbetween carbon atoms and M the mass of an unit-cell. Theimpurity term is described as [25]

Himp iV ici;ci; c

i;ci;; 4

where Vi is the strength of impurity potential at the ith site. Theelectric eld E(t) is included in the Hamiltonian as a scalarpotential. This gives the following contribution to the Hamilto-nian:

HE Etjeji;sia ui ci;sci;s

12

; 5

where e is the electron charge and a the lattice constant. Themodel parameters we use in this work are those generally chosenfor trans-polyacetylene [2]: t02.5 eV, te0.05 eV, K21.0 eV/2, 4:1 eV=, a1.22 , M1349.14 eV fs2/2, the frequency of thebare optical phonon is Q

4K=M

p 0:25 fs1. Moreover, a

typical damping coefcient 0:04Q is introduced to avoid thelattice vibration [29].

In the absence of an external electric eld, we can determinelattice conguration of the polymer fuig is determined by theM uit K2uitui1tui1t 2ssi;i1tsi;i1t

jejEts

si;it12

; 9

in which si:i is the element of the density matrix dened as

si;it sni;tf ssi;t; 10

where f s is the time-independent distribution function as deter-mined by the initial electron occupation. The coupled differentialEqs. (8) and (9) can be solved numerically using the RungeKuttamethod of order 8 with step-size control.

We can then obtain the charge density distribution (the chargedensity of the ground state in the same electric eld has beensubtracted) at every time step

it ;sf sjsi;tj21: 11

The yield of the singlet exciton can be attained by using aprojection method which has been proved to be an efcient way[3032]. The evolutional wave function jt is a Slater determi-nant consisting of the occupied single-electron evolved wavefunction. The relative yield IK(t) for a given eigenstate jk is thenobtained from

IK t jtjkj2: 12Generally, the state jk can be any state of interest.

3. Results and discussions

We specialize to the case of the collision process of oppositelycharged polaron pairs under the inuence of electric eld andimpurities. In this work, we consider the impurity affects the itinerant electrons in the range of several sites, i.e., Vi Vcexpjnncj=nw [25]. Here nw serves as the impurity width, Vcis the impurity potential strength and nc its location. In our simu-lations, a polymer chain with total sites N300 is considered, andthe impurity is set to locate at the 125th site with nw5 withoutloss of generality. Before the applied electric eld turns on, apositively charged polaron is located at the left side while anegatively charged polaron is located at the right side of thepolymer chain. They are well enough separated to ensure that theyare almost non-interacting at the beginning. Starting from theinitial conditions, the scattering processes between them drivenby the external electric eld are investigated.

The external electric eld is then applied, which will driveminimization of the total energy in the above expression

ui1ui

Kssi;i1 si1;i ; 7

where is a Lagrangian multiplier to guarantee the polymerchain length unchanged, i.e., iui1ui 0. si;i is the element ofdensity matrix, which will be given below. The initial congura-tion of a polaron-pair in the following dynamical evolution willbe chosen from the solution of the above self-consistent equa-tions (6) and (7).

Now, we briey describe the non-adiabatic dynamical methodthat has been widely used for the dynamics of soliton and polaronin an electronlattice interacting system. The time-dependentSchrdinger equations for one-particle wave functions areexpressed in the following form:

i _ si;t ihsi;itsi;t: 8

The lattice displacements are determined classically by the follow-these two polarons come close to each other and then collide.

In order to reduce the lattice vibration in the accelerated processof the polaron, the electric eld is turned on smoothly, that is, theeld strength changes as Et E0 expttc2=t2w for 0ototc ,Et E0 for tcot with tc being a smooth turn-on period, tw thewidth. Here, we take tw 25 fs, tc 75 fs. Before their collision,the motion for both the electron polaron and the hole polaron isthe same as that reported in Ref. [20], that is, the polaron pair isaccelerated at rst and then moves as one entity consisting of bothcharge and lattice defect.

We initially consider systems with no impurities, i.e., Vc0.0.We nd that the external electric eld plays an important role inthe scattering between polarons, and the eld requirement forachieving singlet exciton is very strict in a pure polymer chain. Atexternal electric elds weaker than a critical one (about E00.02 mV/), the oppositely charged polaron pair can move into adistance where they cannot move any more. This case is generatedby weak kinetic energy and a less overlap between two wave-packets. At a slightly larger electric eld, such as E00.05 mV/,these two oppositely charged polarons can recombine to form aneutral exciton. As can be seen in Fig. 1(a), displaying the time

overlap between electronhole pairs, and makes charge transfermore easy. In contrast to the weak eld case, not only does theamount of charge localized in the lattice deformations decrease,but also their signs have been changed after collision. These twoseparated particles can go far and far away in the external electriceld. Finally, the oppositely charged polarons will scatter into amixed state including the polaronexciton state.

The situation is quite different when the effect of impurity istaken into account. We nd that the neutral exciton can beproduced under a wide range of applied eld strengths with theassistance of impurities. Fig. 1(c) shows the dynamical evolution ofthe charge density for the case Vc0.28 eV with E01.0 mV/.One should note that the electronhole symmetry is broken byintroducing the impurity. Moreover, it is clear that the impurity isrepulsive to the negative charge while it is attractive to thepositive charge when the strength Vi being positive. As a result,the holepolaron is trapped by the attractive impurity rst atabout 200 fs (see the dark regions in Fig. 1(c)). Subsequently, thevelocity of electronpolaron slows down due to the repulsiveinteraction, and then enters this region that binds the hole

Q. Lu et al. / Physica B 421 (2013) 1317 15dependence of the smoothed excessive electron charge densitynt 14 n1 2n n11, the velocity of these two polaronsdecreases when they are close to each other, because theirpotential energy increases due to the lattice interactions. There-fore, the two polarons cannot move much closer, and their wavefunctions have a smaller overlap. In turn, the charge transferbetween them depends on the overlap and the time they stayclosely. Only the amount of charge localized in the polarons isslightly decreased, their signs cannot be changed after rebounding.Thus, the two deformation cannot separate far away due to thepresence of external electric eld. Then they meet again andrecombine to form a neutral exciton deformation, and the latticedeformation of the exciton, in which an electron and a hole arebound together, is more localized than that of the negativepolaron, which contains an electron.

Upon further increasing the strength of electric eld, we ndthat the formation of the exciton depends sensitively on theapplied electric eld, and a larger electric eld is disadvantageousfor the formation of exciton because of the larger pre-collisionvelocity of the polarons. Once the eld exceeds a threshold valueof about E00.2 mV/, there will be no neutral exciton survive. InFig. 1(b), we show the time evolution of nt for E01.0 mV/. Dueto the stronger electric eld, the negative polaron and positivepolaron can move more close than that in the weak eld case forhaving more kinetic energy, which increases the wave function

Fig. 1. Dynamical evolution of the charge density nt with time for:

(a) E00.05 mV/, Vc0; (b) E01.0 mV/, Vc0; (c) E01.0 mV/, Vc0.28 eV.polaron (see the white regions in Fig. 1(c)). If the attractiveinteraction between these two polarons and the magnitudes ofthe electric elds are suitable for forming a stable state, they willcombine with each other to form an exciton. Furthermore, theimpurity interaction induces a polarization of the charge distribu-tion of the exciton.

The evolution of energy levels and occupation numbers mayprovide us with the concrete formation process of exciton. As anexample, the evolution of the energy levels with time correspond-ing to Fig. 1(c) is plotted in Fig. 2. Fig. 2 shows that there are fourlocalized electronic energy levels in the energy gap, which arelocalized electronic states corresponding to the two lattice defectsbefore the collision. We use L1R1 and L2R2 to denote thelower level and the upper level corresponding to the polaron atthe left (right) side, respectively. Initially, L1 and R1 (L2 and R2)are degenerate in the absence of the electric eld, because thesetwo polarons are well enough separated. As the time increases, thedegeneracy is broken both by the electric eld due to the Starkeffect and by the coupling of the two polarons when they moveclose to each other. Since the lattice distortions involved in apolaron and in a exciton are quite different, and the energies oftwo isolated polarons are greater than that of a exciton. Latticedisplacements accompanying the formation of a exciton-likedistortion will cause excitations of lattice vibrational modes(phonons) which should primarily dissipate through damping,

Fig. 2. Localized electronic energy level evolution with time corresponding to thescattering described in Fig. 1(c). The inset shows the schematic diagram of the four

localized levels at the initial.

with the assistance of impurities. The yield of the exciton could

Q. Lu et al. / Physica B 421 (2013) 131716thus carrying away the excess energy. As a result, these energylevels exhibit a damped oscillation with time, corresponding to theamplitude oscillation of the lattice deformations. Simultaneously,two localized levels L2 and R1 gradually merge into conductionband and valence band, respectively. The remainder L1 and R2 godeeper into the gap to form a self-trapped exciton state corre-sponding to a single large lattice deformation.

In Fig. 3, we display the time evolution of occupation numbersof the localized energy levels. In the initial state, L1 and R2 areboth singly occupied with opposite spin, while R1 is doubleoccupied and L2 is empty. This indicates that a hole polaron islocated at the left side of the chain and an electron polaron at theright side. We can nd that the occupation numbers for two lowerlocalized energy levels L1 and R1 are almost unchanged withtime, which remains 1 and 2, respectively. Moreover, L2 (whichevolves into the conduction band) only increases slightly when thecollision occurs, and then decays to about 0.01. When the excitonlattice distortion is formed after collision, the electron populationin R2 changes to 0:93. From the above analysis, the mainproducts after the scattering of the oppositely charged polaronsare the singlet exciton state plus a few free charges excited into theconduction band levels during the combination process.

Now we discuss the effect of the strength of the impurity onthe yield of exciton by systematically varying the value of Vc in thepresence of an electric eld, and the yields can be calculated byusing a projection method mentioned above in Eq. (12). Fig. 4shows the dependence of the exciton yield on the impurity

0

Fig. 3. Occupation numbers of localized energy level evolution with time corre-sponding to the scattering described in Fig. 1(c).potential for several eld strengths. For E 1 mV/, one can ndthat the yield of the neutral exciton rst increases with increasingVc, until it reaches its maximum value at about Vc0.28 eV, thendecreases as the strength of impurity becomes larger. For a largevalue of Vc, the repulsive interaction of impurity will hindereffective wave function overlap of the polarons, thus resulting ina lower efciency.

4. Summary and discussion

In summary, we have studied the collision of oppositelycharged polarons in conjugated polymers with impurities bysolving the time-dependent Schrdinger equation combining thelattice equation of motion simultaneously and non-adiabaticallywith the SSH Hamiltonian and an additional contribution for anexternal electric eld. As in An et al. work [33], we nd that theproducts formed by the collision and the combination of oppo-sitely charged polarons depend sensitively on the applied electriceld, and the eld requirement for achieving neutral exciton isreach as high as 96%. The theoretical results are expected toprovide useful predictions concerning which polymers with prop-erly impurity-assisted interactions are likely to be more suitablefor use in OLEDs.

Finally, it should be stated that the explicit electronelectroninteractions are ignored in the Hamiltonian, thus singlet andtriplet excited states are degenerate and not distinguishable. Dueto the electron correlations between delocalized states on poly-mers, the recombination could become spin dependent [34]. Forexample, Tandon et al. suggest that irrespective of the recombina-tion process, interchain or intrachain, the direct transition to formsinglets should always be easier than triplets due to its smallerbinding energy relative to the triplet [35]. Moreover, recent workshave shown that the interchain charge transfer effect [36], theinteractions between various elementary excitations (such as abipolaronexciton pair [31] and/or a polaronbipolaron pair [32])may open channels that enhance the quantum efciency ofelectroluminescence. The relative yields of light emissive singletexcitons versus non-emissive triplet excitons in electrolumines-cence are still an open issue.

Acknowledgmentsvery strict in a pure polymer chain. However, the neutral excitoncan be produced under a wide range of applied eld strengths

Fig. 4. Dependence of the yield of exciton on the strengths of impurity.This work was supported by National Natural Science Founda-tion of China. Y.Y. was supported by Qianjiang Talent Project(Grant no. 2012R10074).

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Q. Lu et al. / Physica B 421 (2013) 1317 17

Dynamics study of the recombination of polaron pairs in polymer chain with impuritiesIntroductionModel and methodResults and discussionsSummary and discussionAcknowledgmentsReferences