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Radiative Decay of Singlet Excitons and Carrier-Fusion-Induced Electroluminescence Enhancement of Polymer Light-Emitting Diodes Sheng Li,*,†,‡ Guo-Ping Tong,† and Thomas F. George*,‡ Department of Physics, Zhejiang Normal UniVersity, Jinhua, Zhejiang 321004, China, and Office of the Chancellor and Center for Nanoscience, Departments of Chemistry & Biochemistry and Physics & Astronomy, UniVersity of MissourisSt. Louis, St. Louis, Missouri 63121 ReceiVed: August 14, 2009; ReVised Manuscript ReceiVed: October 8, 2009
During the radiative decay of singlet excitons in polymer light-emitting diodes (PLEDs), the newly developed molecule dynamics not only describes the evolution of the lattice distortion of the polymer chain but also illustrates the time-dependent electron transitions and electron populations in detail. Furthermore, based on this method, it is revealed that the nonemissive triplet excitons in PLEDs become emissive through the fusion of injected positive polarons. The obtained maximum quantum efficiency is consistent with recent experimental results. I. Introduction Since the pioneering discovery of polymer/organic lightemitting diodes (PLEDs/OLEDs) in 1990,1,2 much progress has been made in improving the efficiency of PLEDs. The light emission of PLEDs/OLEDs is due to the radiative decay of singlet excitons. Yet according to spin statistics, excitons have four possible spin microstates, only one of which is a singlet exciton, which means that the maximum quantum efficiency of the PLEDs/OLEDs would be limited to 25%. There is experimental confirmation that the singlet fraction in OLED is about 22%.3 Thus, how to improve the efficiency of PLEDs has become a topic of great interest. During the past decade, theoretical research showed that the singlet exciton is formed with higher probability than the triplet once the binding energy of the triplet is tuned to become weak.4-6 Fortunately, only by improving the electron transport in OLEDs/PLEDs can the binding energy of an exciton be reduced to a lower level, which provides an opportunity to break the limitation of 25%. Accordingly, experiments have shown the ratio of electroluminescence to photoluminescence to reach 50% after improving the electron transport in paraphenylenevinylene (PPV) films.7 Further, the ratio of singlet to triplet also reaches as high as 45% through tuning the interface at the molecular level in paraphenylenevinylene (PPV).8 Actually, the electroluminescence enhancement of PLEDs caused by electron transport entails a dynamical process. Yet, previous research has focused mostly on the static properties of the electroluminescence of PLED/OLED, which makes it difficult to discover the underlying dynamical mechanism. Thus, the purpose of this paper is to explore the dynamical evolution of excitons in PLEDs. A recent experiment has shown the internal quantum efficiency to be improved to 60% when a hole injection layer is inserted into a PLED,9 and the ratio of singlet to triplet is greatly increased when an electric field is applied.10 This not only indicates the injected charged carriers as the key means for the enhancement of electroluminescence of PLEDs, but also raises * Authors to whom correspondence should be addressed. E-mail:
[email protected] (S.L.),
[email protected] (T.F.G.). † Zhejiang Normal University. ‡ University of MissourisSt. Louis.
the question as to how injected carriers transfer in the presence of an external electric field. Furthermore, on the basis of extensive research on PLEDs/OLEDs, Wohlegenannt had concluded that the ratio of the formation cross sections of singlet to triplet excitons is strongly material-dependent,11 which is opposite to the independence of phosphorescent materials with respect to the thickness of the film, temperature, and electric field.12 To clarify all of these puzzling problems, it is necessary to know how the carrier motion contributes to the electroluminescence of PLEDs, including an answer to the following questions: What is the evolution of the decay of a singlet exciton? How does the injected charged carrier interact with the exciton? What is the result of the interaction between these carriers? The original molecule dynamics fixes the electron occupation, making it difficult to describe electron transitions during the evolution of the exciton. In this paper, we develop molecule dynamics that not only involves time-dependent electron transitions but also describes the evolution of the lattice configuration and charge transport. On the basis of this, the evolution of radiative decay of a singlet exciton is exhibited in detail. Furthermore, we dynamically show the mechanism of electroluminescence enhancement of PLEDs caused by charged carrier injection, illustrating the difference between light emission of singlet exciton and triplet exciton induced by carrier fusion. II. Model and Radiative Decay of Singlet Excitons Due to the one-dimensional nature of a conjugated polymer, the lattice configuration sensitively depends on the electronic state. Once this state is changed, the bond structure of the polymer undergoes distortion. Subsequently, some new electronic bound states are reformed by the lattice distortion, called self-trapping, which can be quantitatively described by the Su-Schreiffer-Heeger Hamiltonian.13 The stability of composite particles, such as polarons and excitons, in the conjugated polymer with a nondegenerate ground state is ensured by the confinement effect, which can be accounted for the BrazovskiiKirova symmetry-breaking term.14,15 To consider the electronelectron interaction in a polymeric molecule, the extended Hubbard Hamiltonian is introduced. Including also the effect
10.1021/jp907886n CCC: $40.75 2009 American Chemical Society Published on Web 10/28/2009
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Figure 1. (a) Electronic spectrum of the singlet exciton. (b) Lattice configuration of a singlet exciton in a polymer light-emitting diode. The vertical axis is the lattice configuration in units of angstroms.
of an electric field applied along with the polymer chain, we then have the following total Hamiltonian H:
H)-
+ cl,s + ∑ {t0 + R(ul+1 - ul) + (-1)lte} × [cl+1,s l,s
K 2
H.c.] + H' ) U
(1)
l
∑ nl,vnl,V + V ∑ nl,snl+1,s'
(2)
∑ Ee(l - N +2 1 )anl,s
(3)
l,s
HE )
∑ (ul+1 - ul)2 + H' + HE l,s,s'
l,s
Here, t0 is a hopping constant; R is an electron-lattice coupling constant; cl,+s (cl, s) denotes the electron creation (annihilation) operator at site l with spin s and corresponding + cl,s; a is a lattice constant; ul is the occupation number nl,s ) cl,s displacement of atom l with mass M, whereby the lattice configuration of the polymer chain is defined as (-1)lul; K is an elastic constant; te is the Brazovskii-Kirova term; and U and V are the on-site and nearest-neighbor Coulomb repulsion strengths, respectively. HE is the interaction of the electrons with the electric field b E directed along the polymer chain, where E is the electric field strength, and N is the number of lattice sites. In order to describe the electron’s behavior, we have to know the its energy spectrum εµ and wave function Φµ, which are functionals of ul and also determined by the eigenequation
HΦµ ) εµΦµ
(4)
Since atoms are much heavier than electrons, based on the Feynman-Hellmann theorem, the atomic movement of the lattice can be described by classical dynamics as 2
M
d ul 2
dt
occ
)-
∂ε
∑ ∂uµl + K(2ul - ul+1 - ul-1)
(5)
polymer is not a strongly correlated system, the electron-electron interaction term H′ can be treated by the Hartree-Fock approximation.16-18 In general, a PLED is structurally composed of three layers, where an electroluminescent polymeric material is inserted between two electrodes. Once the bias voltage between two electrodes is applied, electrons and holes are injected into the polymer from negative and positive electrodes. Then, an electron from the highest-occupied molecular orbital (HOMO) attracts a hole of the lowest-unoccupied molecular orbital (LUMO). Combining the prominent self-trapping effect of the polymer, the HOMO and LUMO states move to the center of the gap, becoming two localized statessΓu and Γdsand leading to a new carrier called an exciton. If the formation of the exciton is a spin-independent process, the resultant excitons are divided into two groups: one is a triplet with S ) 1 (Sz ) -1, 0, 1), and the other is a singlet with S ) 0 (Sz ) 0). The ratio of the formation of singlet to triplet excitons is 1:3. The singlet exciton, as an emissive carrier, is made of two localized states, Γu and Γd, depicted in Figure 1a, whose lattice configuration is distorted, as shown in Figure 1b. During the relaxation process of the decay of the singlet exciton, the electron occupations in the two localized levels are changed due to the transition. However, in the original molecule dynamics description, the electron occupation is fixed, which makes it difficult to describe the electronic transition when the exciton emits light. In order to resolve this, we introduce the electron population rate equations and merge them with the original molecule dynamics. If |u〉 and |d〉 are the wave functions of the localized states Γu and Γd whose energies are Eu and Ed, the transition rate γud between these states is
γud )
4e2(Eu - Ed)3 3p4c3
|〈u|r|d〉| 2
(6)
where r is the dipole operator. If electron populations of Γu and Γd are designated as Pu and Pd, their evolutions due to the electron transition can be expressed as
µ
Assuming an electronic wave function Φµ ) {Zn,s µ}, the s 2 | - n 0, charge distribution can be represented as Fn ) Σ |Zn,µ µ,s where n0 is the density of the positively charged background, and occ stands for the occupation or population of electrons. Since the
dPu ) -γudPu dt
(7)
Pd ) n - Pu
(8)
where n is the total electron number.
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Figure 2. Time-dependent electron populations in the states Γu and Γd during the decay of a singlet exciton.
Figure 3. Time-dependent lattice configuration during the radiative decay of a singlet exciton in a polymer light-emitting diode. The vertical axis is the lattice configuration in units of angstroms.
The transition rates also change during the dynamical process. Further, after substituting ul of eq 5 into the Hamiltonian eq 1 and incorporating the time-dependent electron populations Pu, and Pd into the changed electron occupation (eq 5), the Hamiltonian also becomes a time-dependent variable. The coupled eqs 4-8 can quantitatively describe the dynamics of a conjugated polymer chain. The whole process of the decay of singlet exciton is depicted in Figures 2 and 3. At the beginning of the decay, i.e., at 0 ps in Figure 2, each localized state is occupied by one electron with opposite spin. At this instant, as illustrated in Figure 3, the lattice distortion of 0.013 Å at 0 ps is the most severe as compared with the evolution of the decay of the singlet exciton. When the time reaches 400 ps, due to the electron transition, the electron population of Γu is 0.55, while that of Γd becomes 1.45, as depicted in Figure 2. The associated lattice distortion at 400 ps becomes more gentle than seen before in Figure 3. Up to 1 ns, the electron population of Γu becomes 0.08, which is the end of the
radiative decay of the singlet exciton, indicating the life of the exciton to be about 1 ns. Accompanying with the electron transition, the polymer has been emitting light. III. Carrier Fusion in PLEDs When a conjugated polymeric molecule is in its ground state, the homogeneous dimerization of the lattice configuration produces a 1.0 eV gap between the valence and conduction bands. If an electron in HOMO is removed, the original homogeneous dimerization of the lattice configuration is no longer stable. The self-trapping effect finally produces two localized states inside the gap, Φu and Φd, where Φd is occupied by only one electron while the other is empty, as shown in Figure 4a. The resulting bond structure is also illustrated as Figure 4b. We call this charged carrier a positive polaron. After electrons and holes are injected into the polymer from negative and positive electrodes, the ratio of the formation
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Figure 4. (a) Electronic spectrum of the positive polaron. (b) Lattice configuration of a positive polaron in a polymer light-emitting diode. The vertical axis is the lattice configuration in units of angstroms.
Figure 5. (a) Electronic spectrum of the triplet exciton. (b) Lattice configuration of a positive triplet exciton in a polymer light-emitting diode. The vertical axis is the lattice configuration in units of angstroms.
of singlet to triplet excitons is 1:3. For the triplet, the spin of the electron in the localized state Ψu is as the same as that of the electron in Ψd, as shown in Figure 5a. Moreover, due to self-trapping, the formation of the triplet exciton also distorts the original homogeneous dimerization of the lattice configuration, as shown in Figure 5b. The Pauli exclusion principle forbids an electronic transition between these two localized states, which directly leads to the triplet exciton being a nonemissive carrier. As in a previous experiment,9 a hole-transport layer is inserted into the PLED, which enables positive polarons to be easily injected into the polymer electroluminescent material. When an external electric field E ) 4.0 × 104 V/cm is applied along with the polymer chain, the positive polaron moves along the polymer chain to approach the triplet exciton. Without the restriction of the Pauli exclusion principle for this transition, it now becomes highly possible for the electrons in Ψu and Ψd of the triplet exciton to transit to Φd once the positive polaron moves close to the triplet exciton. With the time-dependent lattice configuration and charge distribution, the whole dynamical process is revealed in detail in Figures 6 and 7, respectively. In Figure 6, up to 1400 fs, the polaron approaches. Two separated distorted lattice
structures, where one is associated with the polaron and the other with the triplet exciton, fuse together to form a new localized lattice distortion, which is the process of carrier fusion. What is the new resultant carrier after the carrier fusion? To answer this, we look at the charge property of the dynamical process depicted in Figure 7. After the fusion, the resultant charge + |e| is still localized. It seems that the resultant carrier is a positive-polaron-like carrier. Further, the overall dynamical process changes the original electron population. As mentioned previously, it is possible for the localized electrons of the triplet exciton to transit to Φd of polaron. The rate equations for the new developed molecule dynamics provide an efficient tool to analyze this process, which is described in Figure 8. As illustrated in Figures 4 and 5, the electron population of Φd in the positive polaron is 1, and the electron populations of Ψu and Ψd in the triplet exciton are each 1. After the positive polaron is injected into the polymer, it moves toward a collision with the triplet exciton. After 5 × 104 fs, as shown in Figure 8, the electron population of Ψu changes to 0.05, close to 0, and the population of Φd of polaron becomes close to 2 (1.95). However, the population of Ψd remains at 1, which means that,
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Figure 6. Time-dependent lattice configuration of the carrier fusion between the positive polaron and triplet exciton under an external field E ) 4.0 × 104 V/cm. The vertical axis measures time in units of femtoseconds. The orientation of the electric field is along the polymer chain.
Figure 7. Time-dependent charge distribution of the carrier fusion between the positive polaron and triplet exciton under an external field E ) 4.0 × 104 V/cm. The vertical axis measures time in units of femtoseconds. The orientation of the electric field is along the polymer chain.
during the carrier fusion, Ψd is like a bridge that helps the electron in Ψu of the triplet exciton transfer to Φd of the polaron,
emitting a photon. Not only from the charge distribution but also from the electron population of the energy spectrum, the
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Figure 8. Time-dependent electron populations in the state Φd of the polaron and in Ψu and Ψd of the triplet exciton during the fusion of these two carriers.
resultant carrier after collision is a positive polaron. It can be described by the formula
TrE + P+ ) P+ + hγ
(9)
where TrE stands for the triplet exciton, P+ is the positive polaron, and hγ is the photon emitted from the PLED. The carrier fusion causes the triplet exciton to be an emissive carrier. Here, it must be emphasized that the fusion-induced emission happens only when the spin of injected polaron is opposite of that of triplet exciton. Yet, because the fusion is a spin-independent process, it means only half of the triplet exciton is involved in the fusion and emits light. Thus, the maximum quantum efficiency of the PLED becomes 25% + 75% × 50% ) 62.5%, which is in excellent agreement with the 60% internal quantum efficiency after the hole injection layer is inserted into the PLED.9 In conclusion, the new developed molecule dynamics describes the radiative decay of singlet exciton in PLEDs, not only exhibiting the evolution of the lattice distortion of polymer chain but illustrating the time-dependent electron transition and electron population in detail. Moreover, it reveals that the nonemissive triplet exciton in PLEDs can become an emissive one through the fusion between triplet exciton and injected positive polaron. The obtained maximum quantum efficiency is well consistent with recent experimental result. Acknowledgment. We thank Shi-Yang Liu and Xin Sun for helpful discussions. Sheng Li also thanks R. B. Tao for a visit to the Center for Quantum Manipulation at Fudan University. This work was supported by the National Science Foundation of China under Grant 20804039 and the Zhejiang Provincial Natural Science Foundation of China under Grant Y4080300.
References and Notes (1) Burroughes, J. H.; Bradley, D. D. C.; Brown, A. R.; Marks, R. N.; Mackay, K.; Friend, R. H.; Burns, R. L.; Holmes, A. B. Nature 1990, 347, 539. (2) Friend, R. H.; Gymer, R. W.; Holmes, A. B.; Burroughes, J. H.; Marks, R. N.; Taliani, C.; Bradley, D. D. C.; Dos Santos, D. A.; Bredas, J. L.; Logdlund, M.; Salaneck, W. R. Nature 1999, 397, 121. (3) Baldo, M. A.; O’Brien, D. F.; Thompson, M. E.; Forrest, S. R. Phys. ReV. B 1999, 60, 14422. (4) Pakbaz, K.; Lee, C. H.; Heeger, A. J. Synth. Met. 1994, 64, 295. (5) Shuai, Z.; Beljonne, D.; Silbey, R. J.; Bredas, J. L. Phys. ReV. Lett. 2000, 84, 131. (6) Ye, A.; Shuai, Z.; Bredas, J. L. Phys. ReV. B 2002, 65, 045208. (7) Cao, Y.; Parker, I. D.; Yu, G.; Zhang, C.; Heeger, A. J. Nature 1999, 397, 414. (8) Ho, P. K. H.; Kim, J. S.; Burroughes, J. H.; Becker, H.; Li, S. F. Y.; Brown, T. M.; Cacialli, F.; Friend, R. H. Nature 2000, 404, 481. (9) Meulenkamp, E. A.; van Aar, R.; van den Bastiaansen, A. M.; van den Biggelaar, A. M.; Borner, H.; Brunner, K.; Buchel, M.; van Dijken, A.; Kiggen, N. M. M.; Kilitziraki, M.; de Kok, M. M.; Langeveld, B. M. W.; Ligter, M. P. H.; Vulto, S. I. E.; van de Weijer, P.; Winter, S. H. P. M. In Organic Optoelectronics and Photonics; Heremans, P. L., Muccini, M., Hofstraat, H., Eds.; SPIE Proc.; 2004; Vol. 5464, p 90. (10) Lin, L. C.; Meng, H. F.; Shy, J. T.; Horng, S. F.; Yu, L. S.; Chen, C. H.; Liaw, H. H.; Huang, C. C.; Peng, K. Y.; Chen, S. A. Phys. ReV. Lett. 2003, 90, 036601. (11) Wohlgenannt, M.; Tandon, K.; Mazumdar, S.; Ramasesha, S.; Vardeny, Z. V. Nature 2001, 409, 494. (12) Wilson, J. S.; Dhoot, A. S.; Seeley, A. J. A. B.; Khan, M. S.; Kohler, A.; Friend, R. H. Nature 2001, 413, 828. (13) Heeger, A. J.; Kivelson, S.; Schrieffer, J. R.; Su, W. P. ReV. Mod. Phys. 1988, 60, 781. (14) Brazovskii, S. A.; Kirova, N. N. JETP Lett. 1981, 33, 6. (15) Sun, X.; Fu, R. L.; Yonemitsu, K.; Nasu, K. Phys. ReV. Lett. 2000, 84, 2830. (16) Li, S.; George, T. F.; Sun, X.; Chen, L. S. J. Phys. Chem. B 2007, 111, 6097. (17) Li, S.; Chen, L. S.; George, T. F.; Sun, X. Phys. ReV. B 2004, 70, 075201. (18) Li, S.; George, T. F.; He, X. L.; Xie, B. P.; Sun, X. J. Phys. Chem. B 2009, 113, 400.
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