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A: Molecular Structure, Quantum Chemistry, and General Theory

Dynamical Formation of BipolaronExciton Complexes in Conducting Polymers Luiz Antonio Ribeiro Junior, Fabio Ferreira Monteiro, Bernhard George Enders, Antonio Luciano de Almeida Fonseca, Geraldo Magela e Silva, and Wiliam Ferreira da Cunha J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b12185 • Publication Date (Web): 02 Apr 2018 Downloaded from http://pubs.acs.org on April 2, 2018

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The Journal of Physical Chemistry

Dynamical Formation of Bipolaron–Exciton Complexes in Conducting Polymers Luiz Antonio Ribeiro Junior,†,‡ Fábio Ferreira Monteiro,‡ Bernhard Georg Enders,¶ Antonio Luciano de Almeida Fonseca,‡ Geraldo Magela e Silva,‡ and Wiliam Ferreira da Cunha∗,‡ †Department of Physics, Chemistry and Biology, Linköping University, SE-58183 Linköping, Sweden. ‡Institute of Physics, University of Bras´ılia, 70.919-970, Bras´ılia, Brazil. ¶University of Bras´ılia, Campus Planaltina, PPG-CIMA, 73345-010, Bras´ılia, Brazil. E-mail: [email protected] Abstract The recombination dynamics of two oppositely charged bipolarons within a single polymer chain is numerically studied in the scope of a one-dimensional tight-binding model that considers electron-electron and electron-phonon (e-ph) interactions. By scanning among values of e-ph coupling and electric field, novel channels for the bipolaron recombination were yielded based on the interplay between these two parameters. The findings point to the formation of a compound species formed from the coupling between a bipolaron and an exciton. Depending on the electric field and e-ph coupling strengths, the recombination mechanism may yield two distinct products: a trapped (and almost neutral) or a moving (and partially charged) bipolaron–exciton. These results might enlighten the understanding of the electroluminescence processes in organic light-emitting devices.

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Introduction Organic semiconductors are recognized as the next generation of materials in developing more efficient green energy solutions, 1,2 particularly when it comes to designing a new class of optoelectronic devices. 3–5 Among them, conjugated polymers are of current interest mostly by presenting important traits for light-emitting purposes, such as, very fast optical responses, high optical damage thresholds, and large nonlinear optical figure of merit. Their architectural flexibility — along with synthesis advantages and mechanical features — is not found in inorganic materials conventionally used by the electronics industry, which makes them of particular interest for both academia and industry. Furthermore, since the discovery of the polymer poly-(p-phenylene vinylene) and its electroluminescence properties, 6 intensive efforts have been accomplished so far to improve, for instance, the electroluminescence efficiency in polymer light-emitting diodes. 7 In conjugated polymers, the understanding of the recombination process promoted by oppositely charged carriers may be the key step behind the electroluminescence enhancement in light-emitting devices. 8,9 Knowing this, some theoretical studies have been performed to elucidate the impact of the polaron recombination mechanism to the electroluminescence phenomena. 10–17 However, considerably less attention has been paid towards the role played by bipolarons in the process of light emission. Therefore, a theoretical insight into the intrachain recombination of bipolarons at an atomic scale is strongly desirable. Recently, some relevant theoretical studies were devoted in investigating the intra and interchain recombination of bipolaron 18,19 pairs in conjugated polymers. Di and colleagues have used a modified version of the Su-Schrieffer-Heeger model to study the recombination mechanism of two oppositely charged bipolarons in a polyacetylene chain. 18 Their results shown that the bipolaron pair can scatters into a bipolaron–exciton specie considering both intra and interchain systems. In an independent yet parallel approach, Sun and Stafström have studied the same system, but in the presence of electron-electron interactions. 19 Their results shown that there are four channels for the bipolaron recombination and, among them, 2


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the bipolaron–exciton is the one with highest yield. Our last researches have also shown that impurities 20 and temperature 21 may favor the formation of such complex via intrachain recombination of oppositely charged bipolarons. Nonetheless, there is still no consensus regarding which is the most dominant effect in promoting (or not) the bipolaron recombiation and the subsequent generation of bipolaron-excitons, such as: electric field, 22 impurities, 20 temperature, 21,23 interactions with other carrier species, 24 electron-lattice, 25 electron-electron, 22 and interchain 19,26 interactions, among others. As a matter of fact, the very nature of the transport mechanism is observed to be dependent on this property, which is a reason why it is crucial to devote further attention to the issue. 27 Of particular importance is the interplay between electric field and e-ph interactions on the recombination mechanism of a bipolaron pair and the subsequent formation of bipolaron–excitons in conjugated polymers. Such is the goal of the present work. In this work, the intrachain recombination of two oppositely charged bipolarons in a model poly-(p-phenylene vinylene) (PPV) lattice is numerically studied by using a modified version of the Su-Schrieffer-Heeger model, named ”Unified Hamiltonian for Conducting Polymers” and developed by Botelho and coworkers. 28 Our computational protocol is based on performing a scan for values of electric field and e-ph interactions to realize critical regimes where, unambiguously, trapped or moving bipolaron–excitons are formed. The physical insight provided by our outcomes may provide guidance for a better understanding about the electroluminescence processes in polymer-based optoelectronic devices.

Methodology The one-dimensional tight-binding Hamiltonian developed here starts from a modified version of the Su-Schrieffer-Heeger Hamiltonaian 29,30 developed by Botelho and colleagues. 28 The total Hamiltonian (H) has the form: H = H1 + H2 , where H1 address the electronic and the lattice parts whereas H2 denotes the Hubbard term introduced to account the con-

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tribution of the electron-electron interactions, 31 which is not originally implemented in the Botelho’s approach. The H1 contribution to the overall Hamiltonian can be written as

H1 = −

X

† (tn,n+1 Cn+1,s Cn,s

n,s

√ 1 X 2 K X 2 4 γα yn − + p . + h.c.) + 2 n πK 2M n n

(1)

In Equation 1, n indexes the lattice sites and tn,n+1 denotes the hopping integral of a πelectron between nearest-neighboring sites and can be placed as

tn,n+1 = e−iηA(t) [γt0 −

√

γαyn ] .

(2)

It is worthwhile to notice that our hopping term differs of the one in the Botelho’s model just by the inclusion of the time-dependent vector potential A(t), which is considered in order to simulate the field included dynamics of quasi-particles and is related to the electric field by ˙ In Equation 2, η ≡ ea/(}c), where e is the absolute electronic charge, a is E = −(1/c)A. the lattice constant, and c is the speed of light. t0 is the hopping parameter for π-electrons in undimerized lattices, α is the e-ph coupling constant, and yn equivun+1 − un describes the relative displacements for lattice sites, where un is the displacement of a n − th site. γ is a dimensionless parameter, which scales t0 and α so that γ 6= 1 represents the C-C bonded pairs in the six-membered rings of a PPV lattice. 28 The first term in Equation 1 accounts † (Cn,s ) creates (annihilates) a π-electron with the electronic contribution and operator Cn,s

spin s at the n − th site. The lattice part, in its turn, is addressed in a harmonic approximation using the second and third terms of Equation 1. In these expressions, K is the harmonic constant that describes a σ bond, M is the site mass, and pn denotes the conjugated momentum to un . The last contribution to the overall Hamiltonian, H2 , accounts the on-site (U ) and nearest-neighbor (V ) Coulomb repulsion strengths, just as described in reference. 31 The initial configuration for the lattice contains two oppositely charged bipolarons in their ground state arrangements. This condition is reached by employing the self-consistent procedure 4


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described in reference. 31 H2 stands for the electron–electron interactions which, in the present work, are described by the extended Hubbard Model as 32–34 X † X 1 1 † H2 = U Ci,↑ Ci,↑ − Ci,↓ Ci,↓ − +V (ni − 1) (ni+1 − 1) , 2 2 i i

(3)

† † here, ni = Ci,↑ Ci,↑ + Ci,↓ Ci,↓ . U stands for the onsite electron–electron Coulombian interac-

tion, and V for the neighboring sites. This form of presenting the e-e interaction is meant to preserve the electron-hole symmetry of the Hamiltonian. Bipolarons arise as self-consistent solutions associated to the electronic configuration. A positively charged bipolaron, named “hole bipolaron”, is formed by removing two electron from the polymer lattice. A negative one (an “electron bipolaron”), by including two electron. In either case, the distortion in the chain caused by the new electronic configuration is manifested by the rising of energy levels within the band gap, as represented in Figure 1. The left energy levels (i.e., the filled levels) are attributed to the electron bipolaron, whereas the right ones (empty levels) are due to the hole bipolaron. A polymer chain with both an electron and a hole bipolaron is represented by the complete picture shown in Figure 1. One can see that the energy levels associated to the hole-bipolaron — the HOMO and LUMO+1, where the former means Highest Occupied Molecular Orbital whereas the latter means Lowest Unoccupied Molecular Orbital — are both empty, thus producing a +2e charge. Conversely, an electron-bipolaron is represented by the doubly occupied levels HOMO-1 and LUMO, thus yielding a -2e charge. It is straightforward to notice that the initial lattice is neutral. For the sake of clarity, Figure 1 presents a schematic representation of the energy levels for a polymer lattice containing a hole-bipolaron and an electron-bipolaron. This diagram might help the understanding of the time evolution of the electronic occupation, that is discussed later. The system dynamics is governed by an Ehrenfest Molecular Dynamics approach, according to reference. 31 The set of parameters settled here for the

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model Hamiltonian are: t0 = 2.5 eV, M = 1349.14 eV×fs2 /˚ A2 , K = 21 eV/˚ A2 , γ = 1.11, U = 1.0 eV, V = 0.3 eV, and a = 1.22 ˚ A. The e-ph coupling α and the electric field E0 vary within the intervals [4.0,6.0] eV/˚ A (with step-size of 0.2) and [0.5,2.5] mV/˚ A (with step-size of 0.5), respectively. It worth to stress that some of these parameters were successfully used in other studies. 24,26,35–42 Conduction Band LUMO LUMO+1 HOMO HOMO-1

Valence Band

Figure 1: Schematic representation of the energy levels for a chain containing two oppositely charged bipoalrons.

Results We begin our discussions by showing the first channel obtained in the simulations performed here, the trapped bipolaron–exciton. The case which yielded such situation considers a polymer lattice with 200 sites with periodic boundary conditions — containing two oppositely charged bipolarons — and subjected to an electric field and e-ph coupling strengths of 1.0 mV/˚ A and 4.8 eV/˚ A, respectively. Although the generation mechanism of charge carriers is of considerable importance, 43 in this work our focus is to investigate the dynamics of two polarons as a given initial condition of the system. The initial state containing these quasiparticles was obtained by the self consistent procedure described in the literature. 31 The positive and negative bipolarons are centered in the site 150 and 50, respectively. The simulations take place until the first picosenconds. In this way, Figure 2 displays the dynamical evolution of the staggered bond-length parameter, which is calculated as y n ≡ (−1)n (2un − un+1 − un−1 )/4. In this figure one can see that the bipolaron pair starts 6


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to move slowly once the field strength is adiabatically turned-on until reach its full value at 200 fs. After this transient time, the bipolarons reach their saturation velocity, for this particular electric field, by compressing the charge and increasing, consequently, the bipolaron’s localization. It is worthwhile noticing that, during the first stage of motion, the moving particles polarize the lattice locally — around the 50 and 150 sites — by creating phonons with amplitudes oscillating in time like ”breathers” (yellow and red contours). This behaviour is clearly noted until the collision moment. At about 600 fs, the bipolarons approach very closely each other and start to recombine. During the collision, the interaction between the bipolarons produces even more energetic phonons that modify the breather profile according displayed in Figure 2(a). Furthermore, just after their encounter, the two lattice deformations associated with the presence of a bipolaron (green regions) rapidly disappear giving rise to only one deeper deformation (light-blue region). This new specie is a composite state that contains a neutral-excited and a charged structure, which is called bipolaron–exciton. It is known that, in organic materials, neutral quasi-particles are more stable than charged ones. 44 Such feature is manifested in the lattice deformation degree, in a such way that more stable structures present deeper lattice deformations. Alongside this evidence, it can be noted from Figure 2(a) that the bipolaron–exciton does not respond systematically to the action of the applied electric field by yielding, therefore, a trapped structure. In this way, the deeper lattice deformation and the unresponsive trend to the electric field lead us to conclude that the new structure formed after the recombination process is an almost neutral bipolaron–exciton. The temporal evolution of the system’s mean charge density, which is presented later, helps to confirm this assumption. Finally, as we will notice from the occupation number analysis, to be discussed shortly, the bipolaron–exciton arising from figure 2(a) is observed to be an excited state structure. In the scope of the employed model, we mimic a given polymer under specific conditions by tuning its parameters. For our purposes, the most parameters are the electron-phonon coupling constant and the electron electron interaction. Our set of parameters are optimized

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2 0 0

2 0 0

(a )

(b )

1 5 0

1 5 0

1 0 0

1 0 0

S ite

S ite

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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5 0

5 0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

T im e [f s ]

T im e [f s ]

Figure 2: Time evolution of the staggered bond-length parameter that depicts the bipolaron– exciton formation through two different channels by using (a) an electric field and e-ph coupling strengths of 1.0 mV/˚ A and 4.8 eV/˚ A, respectively, and (b) an electric field and e-ph coupling strengths of 2.0 mV/˚ A and 4.2 eV/˚ A, respectively. to describe PPV, but it is known that depending on the specific regime under which the system is subjected, some deviations in its performance might arise. By choosing not a specific value but a considerable span of values around those used in the literature, 24,26,35–42 we have the flexibility necessary to present the system under more realistic conditions. In this sense, another interesting channel is obtained by redefining the set of parameters, as depicted in Figure 2(b). Now, by still considering a 200-site lattice endowed of two oppositely charged bipolarons, the electric field strength is 2.0 mV/˚ A and the e-ph coupling constant is settled to be 4.2 eV/˚ A. A markedly difference between the two cases can be noted by observing the colors at both Figure 2(a) and 2(b). Albeit the same color scheme has been used to generate them, the background colours are slightly different. Once the e-ph coupling value adopted for the case in Figure 2(a) is higher than the one used in Figure 2(b), the relative displacements for the lattice sites (yn ) are slightly smaller in the latter. Moreover, another key difference — that impacts directly the carrier dynamics — is the changing in the initial localization for the bipolarons, which is promoted by the different strengths adopted for the e-ph interactions among the cases shown in Figures 2(a) (higher localization) and

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2(b) (smaller localization). Apart of these differences, one can see that the case displayed in Figure 2(b) yields after two collisions just one moving structure that respond to the applied electric field. This new carrier is also a composite quasi-particle that contains a neutralexcited structure mixed with a charged one, i.e., a bipolaron–exciton. Note that the lattice deformation associated with the complex is similar to the one associated with the moving bipolaron. This fact suggests that the bipolaron–exciton formed in this case has more charge coupled to its lattice deformation in comparison to the final structure yielded in the case presented in Figure 2(a). It can also be seen that the complex formed in Figure2(a) deforms more the lattice than the one generated in Figure2(b). It is another clear evidence that the moving bipolaron–exciton is most similar to a charged quasi-particle than a neutral one. Differently from what was discussed about the phonons dynamics in the first case, in Figure 2(b) one can see that no breather-like behavior is obtained. Instead, due to the higher electric field strength considered in the latter, the two consecutive collisions between the bipolarons generates substantially energetic phonons that disturb the lattice ending by suppress the breather-like trend, which is observed in Figure 2(a). In order to better distinguish between these two channels for the bipolaron recombination, in the following we present other ways of understanding the processes displayed in Figures 2(a) and 2(b). These manners of interpreting the results highlight the signatures of the two distinct bipolaron–excitons yielded in our simulations. In this way, Figure 3 shows the time evolution for the mean charge density that is calculated as ρn ≡ (ρn−1 + 2ρn + ρn+1 )/4, where ρ is derived according the reference. 31 The color scheme adopted in this figure states the blue color for the negative charge and the red color for the positive one. Moving on with the discussions about the differences among the yielded bipolaron–excitons, in Figure 3(a) it can be noted that there are, in the beginning, two very clear portions of charge with opposite signs that are sufficiently separated from each other in order to avoid a premature overlap between them. As soon as the field reach its full strength, the charges come closer and the recombination process occurs at about 600 fs. At this moment, the positive bipolaron

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(red contour) vanishes completely and a small part of its charge, which was not canceled, is scattered through the lattice. Concomitantly, a substantial part of the charge associated with the negative bipolaron (blue contour) is also canceled but, however, a very small part of its charge remains coupled to the to the lattice defect of the yielded structure, i.e., the trapped bipolaron–exciton. Once this remaining negative charge is too small, its response to the applied electric field is not strong enough to overcome the potential barrier imposed by the polymer backbone in order to transport adiabatically the composite state of charge and lattice. Therefore, the bipolaron–exciton gets trapped around the 100-site. 2 0 0

2 0 0

(a )

(b )

1 5 0

1 5 0

1 0 0

1 0 0

S ite

S ite

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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5 0

5 0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

2 0 0

T im e [fs ]

4 0 0

6 0 0

8 0 0

1 0 0 0

T im e [fs ]

Figure 3: Time evolution of the mean charge density for (a) the case shown in Figure 2(a) and (b) the case displayed in Figure 2(b). A very distinct behavior can be noted by examining Figure 3(b). After the first collision, some part of both the positive and negative charges are canceled. However, the remaining concentration of charge coupled to the bipolarons is enough to drift the carriers for another collision. In the second collision the positively charged bipolaron vanishes spreading charge through the lattice analogously to the case reported in Figure 3(a). However, due to the interplay between the electric field and the e-ph coupling strengths, instead to cancel the most part of the charges the collision process spreads more positive charge avoiding, therefore, the mutual annihilation between them. Consequently, a larger portion of negative charge remains

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coupled to the product yielded after recombination, i.e, the moving bipolaron–exciton. Now the amount of negative charge associated to this compound state can respond systematically to the applied electric field, thus adiabatically transporting through the lattice the new composite structure formed of both a neutral-excited and a charged state. Electronically saying, the differences between the neutral-excited and charge states, that are mixed in order to form the bipolaron–exciton, are clarified next. As a final note, the issue of the nature of the arising structures represented on Figures 2 and 3 should be discussed. One can conclude that these are, indeed, bipolaron-excitons even before analyzing the occupation number pattern evolution. The idea is to pay close attention to both figures, i.e., to charge as well as lattice distortions. Note that even after the two initial bipolarons recombine, the scattered species presents nonzero net charge. Therefore, it can not be characterized as an exciton for the later is defined as a neutral state. However, the order parameter does shows a distorted signature of an exciton. This complex is, then, named bipolaron-exciton. Now, we turn to the results for the time evolution of the occupation number (occ. number), as depicted in Figure 4. In the present work, the occ. number is derived as explained in reference. 31 Here, we examine and represent only the bipolaron levels within the band gap (i.e., the HOMO/LUMO+1 levels for the hole-bipolaron and the HOMO-1/LUMO levels for the electron-bipolaron) as represented in Figure 1. This is because those are the only levels directly involved in the recombination process. It should be remarked that the LUMO+1 and LUMO levels have exchanged their positions already in the first step of the simulation. The intra-gap configuration for the energy levels obtained by the geometry optimization procedure is the one depicted in Figure 1. However, due to the proximity in energy between the HOMO-1/HOMO and LUMO/LUMO+1 levels, the exchanging process of positions is actually expected. Indeed, such phenomenon was already reported in other theoretical works that investigated the recombination between pairs of polarons 45 and bipolarons. 21 Therefore, the dynamical configuration for the energy levels associated with the

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charge carriers is HOMO-1/LUMO+1 for the electron-bipolaron and HOMO/LUMO for the hole-bipolaron. Bearing this new configuration for the energy levels in mind, we can now move to the discussions on the time evolution of the occupation numbers. To make it clear, Figure 4 is divided into four different panels, each of which presents the time evolution for the occupancy of a particular level separately. In these figures, the black lines are related to the case shown in Figure 2(a) whereas the red lines are associated to the case illustrated in Figure 2(b). By examining the trends of the black lines, one can realize that there is a noticeable migration of electrons between different energy levels, starting from the collision instant (around 600 fs). From this instant on, the electrons that occupy the HOMO-1 level start to move towards the HOMO level. For a very brief moment (about 30 fs) the two electrons are fully transferred from HOMO-1 to HOMO. Once there is a really small mismatching in energy between the bipolarons levels HOMO-1/HOMO and LUMO/LUMO+1, one could say that the electrons were excited due to an energy transfer from lattice to electrons, mostly generated by the collision between the charge carriers. This is a process due to the presence of e-ph coupling term that is responsible to connect the electronic and the lattice degrees of freedom. After 630 fs, the occ. number for both HOMO levels oscillate symmetrically until 710 fs. From then on, each HOMO level remains occupied just by one electron. Conversely, when it comes to LUMO levels, one can note that the LUMO+1 occupancy drops almost to zero whereas, symmetrically, the LUMO occ. number evolves until nearly reaching the value of two. Note that due to energy conservation, and by the virtue of the electron-phonon interaction, electronic energy is transferred to the lattice. The abovementioned process takes place, analogously, for the case depicted in Figure 2(b). However, some important differences between the two cases are observed. It is clearly shown in this figure that the migration of the electrons among the levels (red lines) happens earlier. Also, the transient time for the oscillations in the occ. number is larger than the one described by the black lines. As discussed above, the encounter between the bipolarons occurs faster due to the higher strength of the applied electric field for the case considered in Figure 2(b).

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Furthermore, the multiple collisions between the bipolarons contribute to the production of more oscillations in the occ. numbers in this case. Another clear difference between these two channels is that the LUMO/LUMO+1 transference is not fully achieved in the sense that just 1.5 electrons move from LUMO+1 towards LUMO. This partial transference of electrons is the main responsible by the increase in the net negative charge associated with the yielded bipolaron–excitons, when both cases — Figures 3(a) and 3(b) — are compared. Regarding the occ. number for the HOMO-1/HOMO level, one can see that the final values are similar for both cases, which reinforces the above assumption that the main responsible to yield a moving bipolaron–exciton is the partial transference of electrons between the LUMO levels.

O c c . N u m b e r

2 (a ) H O M O -1

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2 (c ) L U M O + 1

1

0

0 2 O c c . N u m b e r

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(d ) L U M O

1

0

0

T im e [fs ]

Figure 4: Time evolution of the occupation number for the levels represented in Figure 1.

Finally, we turn our attention for the trend presented, in time, by the intra-gap energy levels. This analysis help us to derive the main conclusion about the species present before and after the bipolaron recombination mechanism. As can be seen below, Figure 5 shows the time evolution of the intra-gap energy level for the cases presented in Figure 2. In Figures 5(a) and 5(b) one can be noted that there is a small mismatching between the HOMO-1/HOMO and LUMO/LUMO+1 levels. This degeneracy breaking is imposed by 13



the electron-electron interactions. Moreover, the noticeable oscillations in energy are related, firstly, to the inclusion of an external electric field and, secondly, to collision between the carriers. The moment in which the recombination takes place in both cases can be directly inferred from Figures 5(a) and 5(b), In Figure 5(a) the HOMO-1 and LUMO+1 levels return to the valence and conduction bands, respectively, after 600 fs. This picture states that one quasi-particle is annihilated whereas the another one remains stable until the end of the simulation. Noticeably, the levels HOMO and LUMO shift upward and downward, respectively, within the band gap. This more localized signature for the intra-gap levels revels the formation of a more stable specie, i.e., a quasi-particle that deforms the lattice in a deeper fashion (the trapped bipolaron–exciton). This final configuration for the intragap energy levels is a signature for the presence of, at least, a neutral-excited specie in the polymer lattice. Similarly, in Figure 5(b) the levels HOMO-1 and LUMO+1 move to their respective bands after the recombination of the bipolarons. It occurs sooner (about 400 fs) due to the higher strength of the applied electric field in this case, which drives the oppositely charge particles to their encounter in a faster fashion. However, differently of the case in Figure 5(a), the HOMO and LUMO levels are not substantially shifted towards inside the band-gap. There is, instead, a small shift which denotes that the yielded specie (the moving bipolaron–exciton) resembles more a charged carrier than a neutral excitation.

Conclusions In summary, the intrachain recombination of oppositely charged bipolarons in a model PPV lattice is theoretically investigated by using a one-dimensional tight-binding approach that includes lattice relaxation. The numerical strategy is based in finding critical strengths for the electric field and e-ph interactions that can yield different species after the recombination process, named trapped and moving bipolaron–excitons. Our results clearly show that these two sorts of quasi-particles are obtained depending on the interplay between the electric field

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0 ,8

(a ) 0 ,6

0 ,4

E n e rg y [e V ]

0 ,2

H O M H O M L U M L U M

0 ,0

-0 ,2

O -1 O O O + 1

-0 ,4

-0 ,6

-0 ,8 0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

6 0 0

8 0 0

1 0 0 0

T im e [fs ] 0 ,8

(b ) 0 ,6

0 ,4

0 ,2

E n e rg y [e V ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


H O M H O M L U M L U M

0 ,0

-0 ,2

O -1 O O O + 1

-0 ,4

-0 ,6

-0 ,8 0

2 0 0

4 0 0

T im e [fs ]

Figure 5: Time evolution of the intra-gap energy levels for (a) the case shown in Figure 2(a) and (b) the case displayed in Figure 2(b). and e-ph interactions. Importantly, the e-ph coupling plays the role of transfer the energy from the lattice to electrons being the most important parameter in obtaining the different species formed after the recombination between the bipolarons.

Acknowledgement The authors gratefully acknowledge the financial support from Brazilian Research Councils CNPq, CAPES, FAP-DF, and FINATEC and CENAPAD-SP for provide the computational facilities. B.G.E. gratefully acknowledges the financial support from FAP-DF grants

15



193.001.234/2016 and 193.001.556/2017 and L.A.R.J. of grant 0193.000.942/2015 as well as of the Brazilian Ministry of Planning, Budget and Management (Grant DIPLA 005/2016).

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O c c . N u m b e r

2 (a ) H O M O -1

1

0 0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

O c c . N u m b e r

2 (b ) H O M O

1

0

0 2 O c c . N u m b e r

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(c ) L U M O + 1

1

0

0 2 O c c . N u m b e r

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(d ) L U M O

1

0

0

T im e [fs ]

Figure 6: TOC Graphic

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