Dynamical Study of Polaron–Bipolaron Scattering in Conjugated Polymers

Organic Electronics 11 (2010) 279–284

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Organic Electronics

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Dynamical study of polaron–bipolaron scattering in conjugated polymers

Z. Sun a,Y.Lia, K. Gao a, D.S. Liu a,b,*,Z.Anc,*, S.J. Xie a a School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China b Department of Physics, Jining University, Qufu 273155, China c College of Physics, Hebei Normal University, Shijiazhuang 050016, China article info abstract

Article history: We simulate the scattering processes between a negative polaron and a positive bipolaron Received 12 July 2009 in a conjugated polymer chain using mixed quantum classical molecular dynamics. The Received in revised form 1 October 2009 simulations are performed based on the Su–Schrieffer–Heeger (SSH) model modified to Accepted 3 November 2009 include electron–electron interactions, a Brazovskii–Kirova symmetry-breaking term, and Available online 10 November 2009 an external electric field. It is found that there exist a critical electric field, below which the polaron and bipolaron can scatter into an excited polaron. If the external electric field Keywords: is higher than the critical electric field, the polaron and bipolaron will pass through each Polymers other and continue moving as isolated ones. Because the excited polaron can decay to Polarons Bipolarons the polaron state through emitting a photon, our results indicate that the interactions between polarons and bipolarons can open a channel for electroluminescence. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction which the singlet exciton is radiative. However, to our knowledge, less attention has been paid to the recombina- Since the discovery that a conjugated organic polymer, tion and products of polarons and bipolarons. poly(paraphenylene vinylene) (PPV), can be used as the ac- Bipolarons were hypothesized to exist in conjugated tive component in light-emitting diodes (LEDs) [1], much polymers almost 30 years ago [2] and studied extensively effort has been devoted to the study of the properties of or- over the years. It was well accepted that bipolarons can ganic conjugated materials. Conjugated polymers, as qua- be formed in doped polymers [4–7], while their formation si-one-dimensional materials, have property that their in pristine photoexcited polymers was still an open ques- structure can be easily distorted due to the strong elec- tion [8]. In addition, in studies on bipolarons it appeared tron-lattice interaction [2]. As a result, charges added to that the stability of a bipolaron is questionable in the ab- the polymer chain by doping, charge injection, or photoex- sence of oppositely charged ions [9]. In this paper, we are citation will induce self-localized electron states called interested in the recombination processes between a po- polarons or bipolarons. A polaron has a spin 1=2 and a laron and a bipolaron with opposite charges. Total energy charge ±e, whereas a bipolaron is spinless with charges estimates indicate that two extra electrons may go either ±2e [3]. It has been generally accepted that the polarons into two independent polarons or into a bipolaron [10]. can recombine to form singlet and triplet excitons, in There is a subtle balance between the two situations. The optical data of some polymers showed clearly signals involving polarons and bipolarons [3,11], which testiﬁes the coexistence of polarons and bipolarons in conjugated * Corresponding authors. Address: School of Physics, Shandong Uni- polymers. Therefore, there is a possibility that polarons versity, Jinan 250100, China. Tel.: +86 531 8837035x8369 (D.S. Liu), tel.: collide with bipolarons, and investigating the interactions +86 311 86268257 (Z. An). E-mail addresses: [email protected] (D.S. Liu), [email protected] between polarons and bipolarons seems to be necessary (Z. An). and signiﬁcant.

Recently, Kadashchuk and coworkers reported an where K is the elastic constant of a r bond and M the mass experimental study of trions in a PPV derivative [12]. They of a CH group. pointed out that the capture of a positive bipolaron by a The second term of Eq. (1) expresses the electron–elec- deeply trapped negative polaron is one of the mechanism tron interactions: of trion formation. Onodera [13] investigated the interac- X tions among polaron and bipolaron excitations within the y 1 y 1 Hee ¼ U c ci;s c ci;s framwork of the Brazovskii and Kirova continuum model i;s 2 i;s 2 i;s of cis-polyacetylene. The results showed that polarons X y 1 y 1 and bipolarons can freely pass through each other without þ V c ci;s c 0 ciþ1;s0 ; ð3Þ i;s 2 iþ1;s 2 attractive or repulsive interactions. Swanson et al. [14] i;s;s0 pointed out that a positive (negative) polaron and a nega- where U and V are the on-site and nearest-neighbor Cou- tive (positive) bipolaron produce a polaron and phonons. lomb repulsion strengths, respectively. In this paper, these However, the results are simply based on a conjecture, extended-Hubbard-type interactions are treated within not be convinced yet. Very recently, Ge et al. [15] theoret- the unrestricted Hartree–Fock approximation. ically investigated the scattering process between a nega- The electric field is included in the Hamiltonian as a tive polaron and a positive bipolaron in a single polymer scalar potential, which gives the following contribution chain. They employed an adiabatic molecular dynamics to the Hamiltonian: method (in which the electrons are in their eigenstates at any instant), and thought the reaction between a polaron X 1 H ¼ jje E ðÞia þ u cy c : ð4Þ and a bipolaron produces a positive polaron and emits a E i i;s i;s 2 photon. From these works, we can see that all the results i;s for the polaron–bipolaron interaction are not consistent, The model parameters we use in this work are those so that further investigations are needed. generally chosen for polyacetylene, [2] t0 ¼ 2:5 eV,a ¼ In this paper, we present results from a numerical study 2 4:1eV=Å, te ¼ 0:05 eV, K ¼ 99:0eV=Å , M ¼ 1349:14 of the dynamics of scattering between a negative polaron 2 eVfs2=Å and a ¼ 1:22 Å. As for the electron–electron inter- and a positive bipolaron in a polymer chain under the action, U ¼ 0:5t , V ¼ U=3. influence of an external electric field. The dynamics is per- 0 Within the unrestricted Hartree–Fock approximation, formed by using a nonadiabatic evolution method where the time-dependent Schrödinger equations for one-particle electron transitions between instantaneous eigenstates wave functions are expressed in the following form: are allowed in contrast to the so-called adiabatic dynamics with fixed level occupation. We mainly give our attention o ih w ðÞ¼i; t Hs ðtÞw ðÞi þ 1; t to the products and their yields for the polaron–bipolaron ot k;s i;iþ1 k;s reaction. s s þ Hi;i1ðtÞwk;sðÞþi 1; t Hi;iðtÞwk;sðÞi; t ; ð5Þ The paper is organized as follows. Section 2 describes the model and parameters used. The results are presented with in Section 3. The summary and conclusions are given in s Section 4. Hi;iþ1ðtÞ¼ti;iþ1 Vssði; tÞ; ð6Þ Hs ðtÞ¼t Vsði 1; tÞ; ð7Þ i;i1 i1;i s 2. Model and method s 1 Hi;iðtÞ¼U qsði; tÞ X 2 We use the following model Hamiltonian: þ V ½qs0 ðÞþi þ 1; t qs0 ðÞi 1; t 1 ; ð8Þ s0 H ¼ HSSH þ Hee þ HE: ð1Þ where the density q ði; tÞ and bond density s ði; tÞ are ex- The first term of Eq. (1) expresses the Su–Schrieffer– s s pressed as Heeger (SSH) Hamiltonian [16]: X 0 X qsði; tÞ¼ wk;sði; tÞw ði; tÞ; ð9Þ y y k;s HSSH ¼ ti;iþ1 ci;sciþ1;s þ ciþ1;sci;s Xk i;s 0 ssði; tÞ¼ wk;sði; tÞwk;sði þ 1; tÞ: ð10Þ 1 X 1 X þ K ðÞu u 2 þ M u_ 2; ð2Þ k 2 iþ1 i 2 i i i Here the sum over the index k is over those occupied sin-

i gle-particle states. The equation of motion for the lattice is where ti;iþ1 ¼ t0 aðuiþ1 uiÞþð1Þ te, t0 is the transfer integral of electrons in a regular lattice, is the elec- Mu€ Ku t u t 2u t p a i ¼ ½Xiþ1ð Þ i1ð Þ ið Þ tron-lattice coupling constant, and ui is the lattice displace- þ a fg½ssði; tÞssði 1; tÞ þH:c: ment of the i-th site from its equidistant position. t is e s introduced to lift the ground-state degeneracy for non- X 1 y degenerate polymers. The operator c c creates (annihi- þjejE qsði; tÞ : ð11Þ i;sð i;sÞ 2 lates) a p electron with spin s at the i-th site. The last two s terms on the right hand side of Eq. (2) represent the lattice The above set of equations are numerically solved by harmonic potential energy and the lattice kinetic energy, discretizing the time with an interval Dt which is chosen Z. Sun et al. / Organic Electronics 11 (2010) 279–284 281 to be sufficiently small so that the change in the electronic els for a polymer chain containing a negative polaron and a Hamiltonian during that interval may be negligible small. positive bipolaron. We can see that there are four levels in u d For all the results presented below, we choose a time step the gap. The left two levels ep and ep come from the polar- of Dt ¼ 0:005 fs. on, in which the upper one is occupied by one electron and u By introducing electron eigenstates at each moment, the lower one occupied by two. The right two levels ebp and d the solutions of the time-dependent Schrödinger equations ebp come from the bipolaron, and they are empty. can be put in the form: In our simulations, a 200-site chain is considered, which "# is long enough to contain two independent excitations at X X the beginning. Before the external electric field turns on, wk;s n; tjþ1 ¼ /l;sðÞm wk;s m; tj l m a negative polaron is located at the 60th site while a posi- iðÞ Dt=h tive bipolaron at the 140th site. They are well enough sep- e l / ðÞn ; ð12Þ l;s arated to ensure that they are non-interacting at the where f/l;sg and fl;sg are the eigenfunctions and the eigen- beginning. Starting from the initial conditions, the scatter- values of the electronic part in the Hamiltonian H at a gi- ing processes between them driven by the external electric ven time tj. The lattice equations are written as field are investigated. In the following, we will focus on the evolution of the FiðtjÞ 2 u ðt Þ¼u ðt Þþu_ ðt ÞDt þ ðDtÞ ; ð13Þ staggered lattice configuration rnðtÞ as i jþ1 i j i j 2M Fiðtjþ1ÞþFiðtjÞ n un1ðtÞþunþ1ðtÞ2unðtÞ u_ ðt Þ¼u_ ðt Þþ Dt: ð14Þ rnðtÞ¼ð1Þ : ð17Þ i jþ1 i j 2M 4 Hence, the electronic wave functions and the lattice dis- 3. Results and discussion placement at the ðj þ 1Þth time step are obtained from the jth time step. Simulations are carried out for a number of different

At time tj, the wave functions fwk;sði; tjÞg can be ex- field strengths. In these simulations, a critical electric field 5 pressed as a series expansion of the eigenfunctions f/l;sg strength, EC ¼ 0:8 10 V=cm , is found corresponding to at that moment: markedly different behavior of the polaron and bipolaron dynamics. If the external electric field is lower than the XN s critical electric field, the polaron and bipolaron scatter into wk;sði; tjÞ¼ Cl;kðtjÞ/l;sðiÞ; ð15Þ l¼1 a new state. If the external electric field is higher than the critical electric field, the polaron and bipolaron pass s where Cl;k are the expansion coefficients. The occupied through each other, and continue moving in more or less number for eigenstate /l;s is the same way as before. X 0 s 2 Fig. 2 shows the lattice evolution for the polaron–bipo- nl;sðtjÞ¼ jCl;kðtjÞj : ð16Þ k laron scattering process in the external electric fields of 0:7 105 V=cm and 0:8 105 V=cm. As shown in Fig. 2a, nl;sðtjÞ contains information concerning the redistribution driven by the external electric field, the polaron and bipo- of electrons among the energy levels. laron are accelerated and quickly approach their saturation It is well known that removing or adding electrons to velocity. At about 160 fs, they begin to contact. Due to the conjugated polymers creates lattice distortion in the form high kinetic energies of them, the polaron first passes of polarons or bipolarons [17,2]. The formation of a polaron through the bipolaron. However, after that, the polaron or bipolaron state not only induces lattice distortion but turns around and moves back following with the bipola- also lets the highest occupied molecular orbit (HOMO) ron. This explicitly shows that there is a strong attraction and the lowest unoccupied molecular orbit (LUMO) enter between the polaron and bipolaron. A short time later at the original energy gap and become localized deep levels about 300 fs, the polaron and bipolaron recombine into a [2].InFig. 1, we show the schematic diagram of energy lev- new state. We can see that such a state corresponds to only one lattice deformation. What are the characteristics of this new state? We will analyze it later. In a slightly higher electric field, as shown in Fig. 2b, the polaron and bipolaron simply pass through each other when they come into collide. After that, they recollect and continue moving as a polaron and a bipolaron along the chain. We also find that the critical electric field is sensitive to the electron–electron interaction parameters U and V. The critical electric field strength increases as the electron– electron interaction parameters increase. This is easy to be understood. As U and V increase, the attraction between electron and hole gets strong, i.e., the attraction between the polaron and bipolaron becomes strong. Thus, passing Fig. 1. The schematic diagram of energy levels for a polymer chain through each other becomes difficult for the polaron and containing a negative polaron and a positive bipolaron. bipolaron. 282 Z. Sun et al. / Organic Electronics 11 (2010) 279–284

Fig. 3. Time evolution of the occupied number of the intragap energy levels, E ¼ 0:7 105 V=cm.

sion. As indicated by the results displayed in Fig. 3,we found that there are mainly three electronic states mixed together after collision, which are shown in Fig. 4. State (a) denotes the initial electron distribution. So this state still contains a polaron and a bipolaron. State (b) denotes u u the electron in level ep transferred to level ebp. This state actually contains an excited polaron. As for state (c), it de- u notes the electron in level ep being excited to the conduction band. So this state contains a free carrier and a bipolaron. We do not draw the other possible states as

Fig. 2. Time dependence of rn for the polaron–bipolaron scattering in we have found that the yields of other states are very small different external electric fields, (a) E ¼ 0:7 105 V=cm and (b) compared with these three states. 5 E ¼ 0:8 10 V=cm. In the following, we will calculate the yields for these three states using a projection method [18]. The evolved wavefunction of the whole system jWðtÞi can be con- In addition, if the electron–electron interaction is ne- structed by the single electron evolutional wavefunction glected (U ¼ V ¼ 0), we find that the polaron and bipolaron fwkðn; tÞg as a Slater determinant. After each evolution can easily pass through each other even in very weak elec- step, the evolved state jWðtÞi is projected onto the space tric fields. These results are quite consistent with the re- of eigenstates of the system. The relative yield IK ðtÞ for a gi- sults obtained by Onodera [13]. ven eigenstate jUK i is then obtained from: To recognize the new state formed after the polaron– 2 IK ðtÞ¼jjhiUK jWðtÞ : ð18Þ bipolaron collision, we first display the time evolution of u d u d The eigenstate jUK i is constructed by f/ ðnÞg which are occupied number of the intragap levels ep, ep, ebp and ebp l;s in Fig. 3. Before 300 fs, the occupied numbers of the four the eigenfunctions of the electronic part of the Hamilto- levels are almost invariable, which means that the polaron nian H. jUK i can be any state of interest, e.g., the states and bipolaron all keep their isolation. After 300 fs, how- (a), (b) and (c). u ever, the occupied number of level ep rapidly drops to u about zero, while the occupied number of level ebp quickly increases to about 0.3. At the same time, the occupied d d number of the other two levels ep and ebp still remain invariable. These results indicate that the reduced elec- u u trons in level ep may transfer to level ebp or level in conduction band with some probability after collision. Explicitly, u electron transfer from polaron level ep to bipolaron level u ebp and levels in conduction band will induce forming excited polarons and free carriers. As shown above, the polaron–bipolaron interaction induces electron redistribution among levels. It also can be said that there are other electronic states mixed into the Fig. 4. The initial states and other mixed states after the polaron– initial electronic state after the polaron–bipolaron colli- bipolaron collision. Z. Sun et al. / Organic Electronics 11 (2010) 279–284 283

Fig. 5. Time dependence of the yields for states (a)–(c) which are shown Fig. 7. Average yields of state (b) as a function of electric ﬁeld strength for 5 in Fig. 4, E ¼ 0:7 10 V=cm. different electron–electron interactions.

Fig. 5 shows the time dependence of the yields for states However, just above the critical electric field strength, the (a), (b) and (c) during the polaron–bipolaron interaction. It yield of state (a) sharply increases to about 90% , while can be seen that the yield of state (a) keeps 100% before the yields of states (b) and (c) decrease quickly all to about 200 fs. This is because the polaron does not interact with 4%. As we have known from Fig. 4(a), state (a) contains a po- the bipolaron before this time. From 200 to 300 fs, the yield laron and a bipolaron. It shows that the external field dom- of state (a) drops about 10%, which means that some other inates the motion of the polaron and bipolaron in this case. states, e.g., states (b) and (c), must mix into state (a). After In Fig. 7, we display electric field dependence of the 300 fs, the yield of state (a) sharply drops. At the same average yields of state (b) for different electron–electron time, the yields of states (b) and (c) increase to about interactions. It can be seen that the variation of the elec- 30% and 40%, respectively. This explicitly shows that states tron–electron interaction parameters U and V does not lead (b) and (c) appear after the collision. As we have known, to significant change in the qualitative behavior of the state (b) contains an excited polaron, and state (c) contains yield of state (b). In addition, the results also show that a free carrier. Thus, the results indicate that excited polar- the critical electric field increases as the electron–electron 5 on and free carrier can be produced by the polaron–bipola- interaction parameters increase: EC ¼ 1:0 10 V=cm for 5 ron interaction. U ¼ 0:6t0 and V ¼ U=3, EC ¼ 1:3 10 V=cm for U ¼ 0:7t0 We also calculate the yields of states (a), (b) and (c) for and V ¼ U=3. different electric fields. We average the yields for these From the results above, an important channel for the three states over the last 200 fs since their yields distinctly polaron–bipolaron reaction can be written as increased. The results are displayed in Fig. 6. Below the crit- 2þ þ ical electric field strength, the yields of states (a), (b) and (c) P þ BP ! P ; ð19Þ oscillate in a large scope. It shows that the attraction be- where P denotes a negative polaron, BP2þ a positive bipo- tween the polaron and bipolaron dominates their motions. laron, and Pþ an excited positive polaron. It is worth not- ing that the excited polaron can decay to the polaron state through emitting a photon. Therefore, it is expected that the polaron–bipolaron reaction can contribute to electroluminescence in conjugated polymers. From the above results, it can be seen that the electronic state of the system also have large probability to keep the initial state especially under weak electric fields, as shown in Fig. 6. In this condition, the polaron and bipolaron coa- lesce into a bound state due to the electron–phonon interaction and the Coulomb interaction between them. The bound state formed by a polaron and a bipolaron with opposite charges is called a trion [12] or a charged exciton [19]. Obviously, the polaron–bipolaron bound state has lower energy than separate ones so that it is more stable.

4. Conclusion

Fig. 6. Average yields for states (a)–(c) as a function of electric field We have investigated the scattering processes between strength. a negative polaron and a positive bipolaron in a conjugated 284 Z. Sun et al. / Organic Electronics 11 (2010) 279–284 polymer chain. This study was carried out through numer- References ical calculations using an improved version of the SSH model modified to include electron–electron interactions [1] J.H. Burroughes, D.D.C. Bradley, A.R. Brown, R.N. Marks, K. Mackay, R.H. Friend, P.L. Burn, A.B. Holmes, Nature 347 (1990) 539. via an extended Hubbard Hamiltonian, a symmetry-break- [2] A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.P. Su, Rev. Mod. Phys. 60 ing term and an external electric field. The time-dependent (1988) 781. unrestricted Hartree–Fock equation and the equation of [3] P.A. Lane, X. Wei, Z.V. Vardeny, in: N.S. Sariciftci (Ed.), Spin- motion for the time-dependent lattice displacements form dependent Recombination Processes in p-Conjugated Polymers, World Scientific, Singapore, 1997, p. 292. a coupled set that was numerically integrated over the [4] F. Genoud, M. Guglielmi, M. Nechtschein, E. Genies, M. Salmon, Phys. time in a self-consistent way. Rev. Lett. 55 (1985) 118. Initially, the negative polaron and the positive bipola- [5] Yukio Furukawa, J. Phys. Chem. 100 (1996) 15644. [6] Yukihiro Shimoi, Shuji Abe, Phys. Rev. B 50 (1994) 14781. ron are accelerated by an external electric field. They [7] C.H. Lee, G. Yu, A.J. Heeger, Phys. Rev. B 47 (1993) 15543. quickly approach saturation velocity before collision. A [8] E.M. Conwell, H.A. Mizes, ***Synth. Meter. 78 (1996) 201. critical electric field is found corresponding to markedly [9] E.M. Conwell, in: N.S. Sariciftci (Ed.), Are Bipolarons Photogenerated in PPV?, World Scientific, Singapore, 1997, p 489. different behavior of polaron and bipolaron dynamics. Be- [10] J. Cornil, D. Beljonne, J.L. Brédas, J. Chem. Phys. 103 (1995) 842. low the critical electric field, the polaron and bipolaron can [11] P.A. Lane, X. Wei, Z.V. Vardeny, Phys. Rev. Lett. 77 (1996) 1544. scatter into an excited polaron with high yield. The excited [12] Andrey Kadashchuk, Vladimir I. Arkhipov, Chang-Hwan Kim, Joseph Shinar, D.-W. Lee, Y.-R. Hong, Jung-II Jin, Paul Heremans, Heinz polaron can decay to the polaron state through emitting a Bässler, Phys. Rev. B 76 (2007) 235205. photon, so that the polaron–bipolaron reaction shows a [13] Y. Onodera, Phys. Rev. B 30 (1984) 775. contribution to the electroluminescence in conjugated [14] L.S. Swanson, J. Shinar, A.R. Brown, D.D.C. Bradley, R.H. Friend, P.L. Burn, A. Kraft, A.B. Holmes, Phys. Rev. B 46 (1992) 15072. polymers beside the polaron recombination. Above the [15] Li Ge, Sheng Li, Thomas F. George, Xin Sun, Phys. Lett. A 372 (2008) critical electric field, the polaron and bipolaron will pass 3375. through each other and continue moving as isolated ones. [16] W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. Lett. 42 (1979) 1698; W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. B 22 (1980) 2099. [17] S.A. Brazovskii, N. Kirova, Sov. Phys. JETP 33 (1981) 4; Acknowledgments D.K. Campbell, A.R. Bishop, K. Fesser, Phys. Rev. B 26 (1982) 6862. [18] Z. Sun, Y. Li, S.J. Xie, Z. An, D.S. Liu, Phys. Rev. B 79 (2009) 201310(R). This work was supported by the National Basic [19] Li Ge, Thomas F. George, Xin Sun, Phys. Status Solidi B 246 (2009) 1642. Research Program of China (Grant No. 2009CB929204), the National Natural Science Foundation of China (No. 10874100) and the Hebei Province Outstanding Youth Science Fund (Grant No. A2009001512).