Transition of polaron to bipolaron structure in conjugated polymers

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Transition of polaron to bipolaron structure in conjugated polymers

Matheus Paes Lima, Geraldo Magela e Silva *

Institute of Physics, University of Brasilia, 70.917-970, Brasilia, Distrito Federal, Brazil

Received 15 September 2006; accepted 19 December 2007Available online 8 January 2008

Abstract

We describe the effects of removing one more electron from conjugated polymers with polarons. We use a Su-Schrieffer-Heeger modelmodified to include an external electric field and electron–electron interactions via a Parr-Pariser-Pople mean field Hamiltonian. Withinthe Unrestricted Hartree–Fock approximation, we study the dynamics performing numerical simulations. We have found that removingone electron from a conducting polymer bearing a single polaron leads to the creation of a structure that contains a bipolaron coupled toa breather. The bipolaron energy levels in the gap oscillate with higher amplitude due to the bipolaron–breather coupling. The bipola-ron–breather pair can be decoupled through the action of an external electric field. Lattice oscillation modes are created due to the bipo-laron motion and the energy levels associated to the bipolaron still oscillate, nevertheless, with smaller amplitude. The effects of theexternal electric field on the bipolaron–breather dynamics are studied by changing the electric field intensity.� 2008 Elsevier B.V. All rights reserved.

1. Introduction

In recent years, conjugated polymers have been shownto have properties that allow the fabrication of organiclight-emitting diodes, transistors and lasers [1,2]. Theirsemiconductor properties are related to the nonlinear elec-tronic response of the coupled electron–lattice system [3,4].These non-degenerate ground state p-electron materials areable to form, by the electron–lattice interaction, selflocalized electron states called polaron and bipolaron.The non-linear electronic excitations of the p-electronsystem distinguish the semiconductor physics of thesematerials from that of three-dimensional inorganic semi-conductors. Bipolarons and polarons are though to playthe leading role in determining the charge injection, opticaland transport properties of conducting polymers [5,6].

Bipolarons and polarons are self-localized particle-likedefects associated with characteristic distortions of thepolymer backbone and with quantum states deep in theenergy gap due to strong electron–lattice coupling. A

polaron has a spin ±1/2 and an electric charge ±e, whereasa bipolaron is spinless with a charge ±2e.

A consistent description of the dynamics of the mecha-nism of creation, stability and transition of polarons andbipolarons constitutes a critical problem in the understand-ing of these materials.

Recently, UV–Vis–NIR spectroscopy studies on poly(p-phenylene vinylene) combined to the follow-up of thekinetics of doping with iodine vapor were reported andinterpreted as direct observations of the formation of pola-ronic charge carriers [1]. However, by following differentdoping levels with I2 doping, bipolaron formation is iden-tified as well showing that polarons and bipolarons coexistin the oxidized polymer. These results corroborate the find-ings of Steinmller et al. [7] where the evolution of the gapstates of bithiophene as a model system for polythiophenefor different n-doping levels was followed by ultravioletphoto-emission spectroscopy (UPS) and electron-energy-loss spectroscopy (EELS). In both spectroscopies thebehavior of the states formed in the gap indicated two dop-ing regimes. The EELS results suggested a polaron to bipo-laron transformation. The UPS results were consistent withthis interpretation, showing first in the low-doping regimethe appearance of two states in the band gap which move

0166-1280/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.theochem.2007.12.045

* Corresponding author.E-mail address: [email protected] (G.M. e Silva).

www.elsevier.com/locate/theochem

Journal of Molecular Structure: THEOCHEM 852 (2008) 15–21


close together in the higher-doping regime. This constituteddirect observation of the a transition from a polaron tobipolaron states. Optical spectroscopy experiments [3] offield-induced charge (charge injection in semiconductordevice structures as the means for introducing chargedexcitations into conjugated polymers other that doping orphoto-excitation) in poly(3-hexyl thienylene) also indicatedoptical-absorption bands assigned to polarons with addi-tional bands assigned to bipolarons.

The polaron–bipolaron transition problem was explicitlyaddressed by Cik et al. in poly(3-dodecyl thiophene) in con-nection with temperature changes [8]. They found that whenthe sample was heated and subsequently cooled, there wasan amplification of the diamagnetic inter- and intra-chainbipolarons. Kaufman et al. study of polypirrole [9] by opti-cal-absorption spectroscopy and ESR also pointed that themetastable states possess spin, while the stable states do not.Their data revealed a slow transition, consistent with thediffusion rate limited by the mobility of the dopant.

Many efforts have been devoted to describe the polaron–bipolaron conundrum theoretically. Electronic structurecalculations, [10] extensions of the Su-Schrieffer-Heegermodel, [11,12] the Pariser-Parr-Pople model, [13] as wellas combinations of them [14] have been used to determinethe relative prevalence of each excited state in variousregimes. The dependence of the transition between pola-rons and bipolarons on the electron–electron Coulombinteractions (on-site Hubbard U and inter-sites V) has pro-vided contradictory results. While strong Coulomb interac-tions destabilize the bipolaron in Refs. [11,12], it favorsbipolaron in comparison to polaron in Ref. [13]. On theother hand, there exist a remarkable agreement concerningthe role of impurities in the stabilization of bipolarons.Several different approaches [11,14–16] point to bipolaronsystem being more stable than the polaron system whendopants are taken into account.

Two mechanisms have been put forward to explain thetransition from polaron to bipolaron states: Polaronsrecombination into bipolaron, [8,9,17] where the bipolaronis generated when polarons with the same electric chargemeet each other; and single-polaron to bipolaron transi-tion, [1,15,18] where the polaron structure is transformedby the addition of one extra charge. The realm of eachone of these two mechanisms is still subject of controver-sive discussion in the literature.

In this work, we report the results of dynamical calcula-tions on polaron–bipolaron transition. We use theSu-Schrieffer-Heeger model [19] modified to include theCoulomb interaction via extended Hubbard model, Braz-ovskii–Kirova (BK) symmetry breaking terms and theaction of an external electric field [14]. The time-dependentequations of motion for the lattice sites and the p-electronsare numerically integrated within the Time-DependentHartree–Fock approximation.

In agreement with UV–Vis–NIR spectroscopy [1], UPSand EELS measurements [7], our theoretical studies of thetransition indicate that the single-polaron to bipolaron

transition is the preferred mechanism of polaron–bipolarontransition in conjugated polymers. Besides that, once thetransition is achieved the bipolaron shows no signs of disso-ciation into two polarons in our simulations, where no addi-tional counterion or photo-excitation beyond the gap wereconsidered. We find that a breather mode of oscillation iscreated at the lattice in connection with the transitionaround the bipolaron. The breather amplitude is associatedwith how fast the extra charge is added to the system. Thepresence of this breather coupled to the single-polaron–bipolaron transition could possibly be detected in the IRspectrum giving additional evidence of the underlying tran-sition mechanism. Moreover, the created bipolaron istrapped by the breather. We show that the bipolaron isdepinned from the breather for electric fields stronger thana given value. The same transition occurred for different U

and V values (from U = 0 to U = 4.5 eV and V = U/2)pointing at the generality of this transition in conjugatedpolymers under this regime. It should also be noted thatbipolarons are very stable and that there was no need ofinterchain interactions to stabilize it.

2. Model

A SSH type Hamiltonian modified to include an exter-nal electric field is used. The Coulomb interactions areintroduced by the extended Hubbard model. The Hamilto-nian is given by:

HðtÞ ¼ �X

i;s

ðti;iþ1Cyiþ1;sCi;s þ H :c:Þ þ U

�X

i

ni" �1

2

� �ni# �

1

2

� �þ V

Xi

ðni � 1Þ

� ðniþ1 � 1Þ þX

i

K2

y2i þ

Xi

M2

_u2i ; ð1Þ

where Cyi;s (Ci,s) is the creation (annihilation) operator of ap electron with spin s at the ith lattice site, ni;s � Cyi;sCi;s isthe number operator, and ni ¼

Psni;s � yn � unþ1 � un,

where un is the displacement of nth CH-group from equilib-rium position in the undimerized phase. tn,n+1 = exp(�icA)[(1 + (�1)nd0)t0 � ayn], t0 is the transfer integral be-tween the nearest neighbor sites in the undimerized chains,a is the electron–phonon coupling, d0 is the BK symmetry-breaking parameter. M is the mass of a CH group, K is thespring constant of a r-bond, U and V the on-site and near-est-neighbor Coulomb repulsion strengths, respectively.c � ea/(�hc), e is the absolute value of the electronic charge,a the lattice constant, and c the light velocity. The relationbetween the time-dependent vector potential A and the uni-form electric field E is given by E ¼ � 1

c � A. We use asparameters the commonly accepted values for conjugatedpolymers: t0 = 2.5 eV, K = 21 eVA�2, a = 4.1eVA�1,U = 0–1.8t0, V = U/2, a = 1.22A, d0 = 0.05t0, and a bareoptical phonon energy �hxQ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi4K=M

p¼ 0:16 eV.

Within the unrestricted time dependent Hartree–Fockapproximation, the single electron equations are given by:

16 M.P. Lima, G.M. e Silva / Journal of Molecular Structure: THEOCHEM 852 (2008) 15–21


i�h _wk;sði; tÞ ¼ � ½t�i;iþ1 þ V ssði; tÞ�wk;sðiþ 1; tÞ� ½ti�1;i þ V s�s ði� 1; tÞ�wk;sði� 1; tÞ

þ U q�sði; tÞ �1

2

� �þX

p

V pdi;p

(

þX

s0½qs0 ðiþ 1; tÞ þ qs0 ði� 1; tÞ � 1�

)wk;sði; tÞ

ð2Þ

with

qsði; tÞ ¼X0

k

w�k;sði; tÞwk;sði; tÞ

ssði; tÞ ¼X0

k

w�k;sðiþ 1; tÞwk;sði; tÞð3Þ

The dynamics of the nuclear part is made with the Euler–Lagrange equations

d

dtohLio _un

� �� ohLi

oun¼ 0; ð4Þ

where

hLi ¼ hT i � hV i: ð5ÞThe Eq. (4) leads to:

M€un ¼ F nðtÞ; ð6Þwhere

F nðtÞ ¼ �K½2unðtÞ � unþ1ðtÞ � un�1ðtÞ� þ a½eicAðtÞðBn;nþ1

� Bn�1;nÞ þ e�icAðtÞðBnþ1;n � Bn;n�1Þ�: ð7Þ

Here Bn;n0 �P0

k;sw�k;sðn; tÞwk;sðn0; tÞ. The prime on the sum-

mation means that the sum is taken over the occupied sin-gle-particle states in the initial stationary state.

In order to perform the dynamics, an initial self-consis-tent state is prepared solving the Eqs. (2) and (6) simulta-neously. Periodic boundary conditions are considered.The initial state is taken in equilibrium (E = 0). Therefore,we have _un ¼ 0 for all n in the initial state.

The equations of motion (2) and (6) are solved by dis-cretizing the time variable with a step Dt. The time stepDt is chosen so that the change of ui(t) and A(t) during thisinterval is always very small in the electronic scale.

The Eq. (2) is integrated by introducing single-electroneigenstates at each moment. Therefore, the solutions ofthe time dependent Hartree–Fock equations can then beput in the form,

wk;sðn; tjþ1Þ ¼X

l

Xm

/�l;sðm; tjÞwk;sðm; tjÞ" #

� e�ielDt

�h /l;sðn; tjÞ; ð8Þ

where {/l} and {el} are the eigenfunctions and the eigen-values of the single-electron equations at a given time tj.

The lattice equations are written as:

uiðtjþ1Þ ¼ uiðtjÞ þ _uiðtjÞDt ð9Þ

_uiðtj þ 1Þ ¼ _uiðtjÞ þF iðtjÞ

MDt: ð10Þ

Hence, the electronic wave functions and the displacementcoordinates at the j + 1-th time step are obtained from thej-th time-step.

3. Simulation results

We removed one more electron from the polymer chainbearing already a positively charged polaron. Sincecharged excitations defects can be created by quite differentmeans: photoexcitations, chemical doping or direct chargeinjection via electronic device, we performed simulationswhere the second electron is taken from the system duringdifferent time intervals (DT). We varied DT from 0 to100 fs. The shorter time intervals simulate photoexcitationsand the direct charge injection. The longer time intervalsaccount for the different impurity addition proceduresassociated with chemical doping. The electron is takenfrom the highest occupied level using the followingexpression

OCðtÞ ¼ 1

21þ cos

pðt � tiÞDT

� �� ð11Þ

for t between ti and ti + DT. Here, ti is the time when thehole injection begins and OC(t) is the occupation numberof the Fermi level.

We also systematically varied the external electric field E

and the Coulomb interaction parameters U and V. In allconsidered regimes, the removal of one extra electron leadsto single-polaron to bipolaron transition. Here, no simula-tion yielded a pair of polarons.

We have considered polymeric chains with N = 104 sitescontaining initially one single positively charged polaron inall simulations. We use a mean charge density �qiðtÞ, derivedfrom the charge density qi(t), and the order parameter �yiðtÞto analyze the simulations

qiðtÞ ¼X

s

qsði; tÞ;

�qiðtÞ ¼ 1� qi�1ðtÞ þ 2qiðtÞ þ qiþ1ðtÞ4

;

�yiðtÞ ¼ ð�1Þi yi�1ðtÞ � 2yiðtÞ þ yiþ1ðtÞ4

;

ð12Þ

where qs(i, t) is given in Eq. (3). These order parameters areconvenient to vizualize the charge distribution associatedwith polarons and bipolarons; and the bond length alter-ation also associated with them.

The calculated energies in each simulation are the elec-tronic energy, the lattice potential energy, and the latticekinetic energy. The lattice total energy is the sum of theseenergies. The dynamics of the system is followed during100,000 time steps spanning 400 fs.

We find that the adiabatic removal (DT > 80 fs) of thesecond electron leads to a smooth transition of the single-polaron to bipolaron in the chain. Fig. 1 presents the orderparameter �yiðtÞ and the mean charge density �qiðtÞ forDT = 80 fs and no external electric field. The surfaces pres-ent the time evolution of �yiðtÞ and �qiðtÞ. It should be noted

M.P. Lima, G.M. e Silva / Journal of Molecular Structure: THEOCHEM 852 (2008) 15–21 17


that there is no lattice oscillation associated with this adia-batic transition.

In Fig. 2 is presented the behavior of the system energylevels around the gap and inside it. It can be clearly seen thetransition in time from the characteristic polaron levels tobipolaron levels. The tiny oscillation in the bipolaron levelsis associated to the small oscillation in the order parameterafter the bipolaron appearance (Fig. 1(a)).

Fig. 3 shows the spin parameter during a non-adiabatictransition (DT = 0). The half spin distribution of thepolaron vanishes and the spinless distribution of a bipola-ron takes its place. The small oscillation in the spin distri-bution is related to a breather mode.

The fast removal of the second electron (DT < 80 fs)leads to the appearance of a breather oscillation mode inthe lattice. This breather appears at the bipolaron position.

a

b

Fig. 1. Time evolution of the order parameter (a) and mean charge density (b). The extra electron is adiabatically taken from the chain.



There exists an effective interaction between the bipolaronand the breather. As a matter of fact, the bipolaron istrapped by the breather. We varied the external electricfield, enhancing its intensity, until the bipolaron gotuntrapped from it. We find a critical valueEc = 6.5 � 105 V/m for the untrappment.

Fig. 4 shows the time evolution of the transition with abipolaron being bonded to a breather even in the presenceof an electric field E1 = 1.3 � 105 V/m (E1 < Ec). In Fig. 5astronger electric field E2 = 7.8 � 105 V/m (E2 > Ec) candepin the bipolaron from the breather and the bipolaroncan roam freely in the chain. It should be noted that thebipolaron leaves a track of small oscillations in the chain

as it is accelerated through it. The use of periodic boundaryconditions makes the bipolaron reappears at the right sideof the chain once it leaves the chain at the left side.

The variation of the Coulomb interaction parameters U

and V, from small values until around the reasonable val-ues of U = 1.8t0 and V = U/2, does not lead to any signif-icant change in the reported qualitative behavior of thetransition and the trapping of the polaron–bipolaron sys-tem. The extension of U values beyond this limit leads toa regime where bipolarons are not stable in this model withUHF calculations [20]. It should also be noted that, for theparameters adopted, a bipolaron has a lower energy thantwo polarons in the static case.

4. Conclusions

We have studied the transition of polarons to bipolaronson a single conjugated polymeric chain. This study was car-ried out through numerical calculations. A modified SSHtype Hamiltonian considering an external electric field,Coulomb interactions and symmetry breaking terms wasused. The initial configuration is fully self-consistent andthe time evolution of the system is carried out using theSchrdinger and the Euler–Lagrange equations. This modelresults are considered to describe the polaron–bipolarontransition behavior in conjugated polymers in general cap-turing the essential physics of the process.

We present theoretical results pointing to direct single-polaron to bipolaron transition as the favored mechanismof bipolaron formation. This result is in accordance withprevious results on polaron and bipolaron dynamics calcu-lations [14] where a pair of polarons do not expontaneously

0 200 400

time [fs]

-1.25

0

1.25

Ene

rgy

Lev

els

[eV

]

Fig. 2. Evolution of the energy levels inside and around the gap in anadiabatic transition. The spin up levels are shown. The system changesfrom polaron levels (t < 40 fs) to bipolaron levels configuration(t > 120 fs).

Fig. 3. Spin density behavior of non-adiabatic polaron–bipolaron transition.



merged to produce a bipolaron. The other transition mech-anism, namely the merging of polarons, was indisputablyobserved in experiments only under exceptional conditions,[8] where the sample was heated above 450 K and subse-quently cooled, what is not the usual condition of bipola-ron formation.

In the simulations, bipolarons once formed showed nosign of dissociation into two polarons, remaining the

non-linear excitation that transport the excess of chargein the chain.

Since we considered different characteristic time inter-vals for the hole insertion in the chain, in order to simulatedifferent ways of bipolaron generation (photoproduction,chemical doping or direct charge injection), we obtaineddifferent responses from the lattice. It is found that thenon-adiabatic electron removal led to the formation of an

Fig. 4. Time evolution of the order parameter for a bipolaron trapped on a breather subjected to an external electric field E < Ec.

-

Fig. 5. Order parameter evolution of bipolaron depinning from breather with the application of E > Ec.



associated breather oscillation mode in the chain. More-over, the breather interacts with the newly formed bipola-ron trapping it around its position.

The depinning of the bipolaron from the breather isobtained by external electric fields whose strength is abovea calculated threshold Ec = 6.5 � 105 V/m. The trappingand depinning of bipolarons from breathers have directinfluence on the mobility of that charge carrier in the chain.

It should be noted that within our calculations, bipola-rons are more stable than polarons and the changing ofthe Coulomb interaction parameters, from small valuesup to the commonly accepted values for conjugated poly-mers (U = 1.8t0 and V = U/2), did not lead to any alter-ation in the relative prevalence of bipolarons and polarons.

Since we considered only one single chain, the interchaininteraction seems not necessary to the stability of bipola-rons nor to the transition mechanism. Therefore, these phe-nomena appear to be essentially one-dimensional bound.

Based in our numerical simulations, we suggested twoexperimental approaches to better understand thepolaron–bipolaron transition mechanism. First, the verifi-cation of the presence and quantity of breathers associatedwith different transition regimes. Second, the change inbipolarons mobility due to the trapping effect of breathers.

Acknowledgement

We thank CAPES and CNPQ for financial support.

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Transition of polaron to bipolaron structure in conjugated polymers

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