JOURNAL OF CHEMICAL PHYSICS

VOLUME 119, NUMBER 7

15 AUGUST 2003

Dissipative dynamics of spin-dependent electron–hole capture in conjugated polymers Stoyan Karabunarlieva) and Eric R. Bittnerb) Department of Chemistry and Center for Materials Chemistry, University of Houston, Houston, Texas 77204-5003

共Received 19 February 2003; accepted 21 May 2003兲 Spin-dependent electron–hole (e – h) recombination in poly共p-phenylenevinylene兲 chains is modeled by the dissipative dynamics of the multilevel electronic system coupled to the phonon bath. The underlying Hamiltonian incorporates the Coulomb and exchange interactions of spin-singlet and spin-triplet monoexcitations in Wannier-orbital basis and their coupling to the prominent Franck–Condon active modes. In agreement with experiment, we obtain that the ratio of singlet versus triplet exciton formation rates is strongly conjugation-length dependent and increasing on going from the model dimer to the extended chain. The result is rationalized in terms of a cascade interconversion mechanism across the electronic levels. In parallel to the direct formation of spin-dependent excitons, e – h capture is found to generate long-lived charge-transfer states, whose further phonon-mediated relaxation to the bottom of the density of states is hindered by the near e – h symmetry of conjugated hydrocarbons. Being nearly spin independent, such states most likely form an intersystem crossing pre-equilibrium, from which the singlet e – h binding channel is about ten times faster than the triplet one. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1591719兴

I. INTRODUCTION

scopic measurements that put T 1 at about 0.7 eV below S 1 in PPV13,14 and other conjugated polymers.15–17 Long-lived triplet excitons may still annihilate each other18 or decay via interactions with polarons.19 However, generation of emissive species from such secondary processes is negligible20 and cannot adequately explain the enhanced EL in these systems. Moreover, by combining photoinduced absorption and magnetic resonance 共PA/PADMR兲, Wohlgenannt et al.21 have directly measured the cross section ratio r⫽ ␴ S / ␴ T for a variety of conjugated systems and obtained values ranging from 2 to 5. The observed variation of r has been initially correlated with the effective bond alternation22 of the aggregated conjugated chains. More recent experiments have shown that large r is associated with extended intramolecular ␲ conjugation. Wilson et al.23 have measured r⬇4 in a Ptcontaining conjugated polymer, but only r⬇0.9 in the corresponding monomer. New direct PA/PADMR studies24 in polythiophenes have also revealed a dramatic rise of r with oligomer length. The spin dependency of e – h capture reflects the difference between S 1 and T 1 . In simulating intrachain collisions of opposite polarons, Kobrak and Bittner25 have demonstrated that S 1 is energetically more accessible. Beljonne et al.26 have considered intermolecular recombination and shown that the singlet interchain Davydov exciton is a better match for the separated e – h pair. Quantum chemical studies by Shuai et al.27 and Ye et al.28 suggest that interchain highorder correlations can also favor singlet e – h capture. In a recent work, Tandon et al.29 have indicated that regardless of the primary nature of the process—intramolecular or intermolecular—the direct transition to S 1 should be more facile because of its lower binding energy relative to T 1 .

Among the various applications of conjugated polymers as organic semiconductors, electroluminescence 共EL兲 continues to be the most exploited one.1 In difference to photogenerated singlet species, injected electrons and holes are spinuncoupled and recombine to form both spin-singlet and spintriplet bound excited states or excitons. Since triplet 共T兲 excitations in hydrocarbon polymers are practically nonemissive,2,3 EL is proportional to the fraction of electron–hole (e – h) pairs which recombine as singlet excitons rather than as triplets. e⫹h

再

␴S

k rad

→ S 1 → S 0 ⫹h ␯ 3␴T

k ISC

.

→ T1 → S0

The triplet spin-multiplicity dictates a singlet generation fraction ␹ S ⫽ ␴ S /( ␴ S ⫹3 ␴ T ) that will amount to only 25% if the recombination process is spin independent, ␴ S ⫽ ␴ T . However, EL efficiencies compatible with ␹ S in excess of 50% have been achieved in highly ordered polymers from the poly共p-phenylenevinylene兲 共PPV兲 family.4 – 6 Therefore it has been inferred4 that either the effective e – h binding is very weak as suggested by electroabsorption and photocurrent studies7,8 or the singlet exciton formation rate ␴ S substantially outweighs the triplet one ␴ T . 9 The latter notion seems more adequate since T 1 shows strong e – h binding although the S 1 binding energy is relatively small.10 Earlier estimates in this respect11,12 have been supported by spectroa兲

Author to whom correspondence should be addressed; electronic mail: [email protected] b兲 Electronic mail: [email protected] 0021-9606/2003/119(7)/3988/8/$20.00

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© 2003 American Institute of Physics

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J. Chem. Phys., Vol. 119, No. 7, 15 August 2003

Spin-dependent electron-hole capture

Preference to singlet e – h binding is also implied by the different effective separation of the electron and hole in the two spin configurations. Due to the repulsive nature of spin exchange in the S wave function, S 1 exhibits larger e – h delocalization30,31 and hence provides a better electronic overlap with the separated e – h pair. Nevertheless, the observed increase of r with conjugation length 共CL兲 remains poorly understood from a mechanistic perspective. In a previous work31 we have employed the two-band model of a nondegenerate-ground-state polymer to assess the excited states in PPV and the related electron-vibrational spectra of absorption and fluorescence. The Wannierfunction based configuration interaction 共CI兲 technique used there revealed the difference between S 1 and T 1 binding energies and effective e – h separations. The approach has been extended to include the phonon-mediated internal conversion kinetics in the excited state space. This allows us to model nonradiative electronic relaxation processes, including e – h capture. Preliminary results32 showed that exciton formation times and binding energies are related, so that the ratio of the latter is proportional to r. Herein we refine the model by using an electron–phonon coupling strength consistent with the vibronic progressions in spectra31 and explore the effect of e – h symmetry breaking. II. METHODOLOGY

The underlying model introduced in Ref. 31 is based on the system-plus-bath Hamiltonian Eq. 共1兲, where H el describes the relevant electronic system, H ph corresponds to the phonon bath, and H el–ph is the linear electron–phonon 共el– ph兲 coupling H⫽H el⫹H el–ph⫹H ph ⫽

0 ⫹V mn 兲 兩 m典具 n兩 ⫹ 兺 共 F mn 兺 mn mn

␮

⫹

1 2

冉 冊

⳵ F mn Q ␮ 兩 m典具 n兩 ⳵q␮

兺␮ ␻ 2共 Q ␮2 ⫹␭Q ␮ Q ␮ ⫹1 兲 ⫹ P ␮2 .

The localized orbital basis of H el is constructed from the Wannier functions 共WFs兲36 of the highest ␲ and lowest ␲ * bands of extended one-dimensional PPV. The WFs are not eigenstates of the periodic Fock operator f, and H el in Eq. 共1兲 includes off-diagonal one-body terms which corresponds to the intraband hopping of conduction electrons and valence holes ¯ h 兩 f 兩¯n h 典 F mn ⫽ ␦ m h n h 具 m e 兩 f 兩 n e 典 ⫺ ␦ m e n e 具 m ⫽ ␦ m h n h f m e n e ⫹ ␦ m e n e¯f m h n h .

共1兲

Here H el is the CI33 of the singlet or triplet monoexcitations from the Hartree–Fock ground state S 0 for a generic twoband polymer.34,35 The spin-adapted basis wave functions 兩 m典 ⫽ 兩 m e m ¯ h 典 represent an electron localized in the antibonding orbital 兩 m e 典 of monomer m e and a hole in the bonding orbital 兩 m ¯ h 典 of monomer m h . We further distinguish between the neutral monoexcitations 共or intraunit e – h pairs兲 with m e ⫽m h and the charge-transfer 共CT兲 monoexcitations 共or separated e – h pairs兲 with m e ⫽m h .

共2兲

Since the frontier bands of PPV are nearly cosine shaped, we take into account only on-site electron 共hole兲 energies f mm (¯f mm ) and nearest-neighbor hopping integrals ¯ f mm⫾1 ( f mm⫾1 ). The two-body interaction of the e – h pairs of singlet or triplet spin multiplicity is given by S ⫽⫺ 具 m e¯n h 兩 ␯ 兩 n e m ¯ h 典 ⫹2 具 m e¯n h 兩 ␯ 兩 m ¯ hn e典 , V mn

共3兲

T V mn ⫽⫺ 具 m e¯n h 兩 ␯ 兩 n e m ¯ h典 .

共4兲

Under the premise that the frontier WFs of PPV are well localized, we resolve the integrals in Eq. 共3兲 in zerodifferential overlap approximation for WFs pertaining to different repeat units. The approximation preserves the Coulomb attraction of the localized electron and hole, their short range spin exchange, and the 共transition兲 dipole–dipole coupling of the singlet neutral monoexcitations. The electron–phonon coupling H el–ph is purely empirical and assumes that local distortions Q ␮ modulate the local band gap via the on-site energies and hopping integrals

冉 冊 冉 冊 ⳵ f mn ⳵Q␮

0

3989

0

⫽

⳵¯f mn ⳵Q␮

0

S ⫽ 共 2ប ␻ 3 兲 1/2共 ␦ m ␮ ⫹ ␦ n ␮ 兲 . 2

共5兲

A satisfactory description of the vibronic progressions and Stokes shifts in the optical spectra of PPV37 and its oligomers38 has be obtained31 assuming two Franck–Condon active phonon branches in the frequency range of CvC stretches 共⬇1600 cm⫺1兲 and ring torsions39 共⬇100 cm⫺1兲 with Huang–Rhys parameters S⫽0.6 and S⫽4, respectively. Charge-conjugation or electron–hole symmetry of conjugated hydrocarbons implies identical behavior of ␲ * electrons and ␲ holes, expressed by the relation ¯f mn ⫽ f nm .

共6兲

As long as Eq. 共5兲 is valid, the electronic eigenstates of H can be classified according to their parity with respect to transposition of the electron and hole. The scheme below sketches two e – h symmetry-adapted combinations for a model dimer.

States of even parity are invariant upon transposition of the electron and hole and contain the strongly bound intraunit e – h pairs in their expansions. In comparison, the Downloaded 03 Sep 2003 to 129.7.158.16. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

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S. Karabunarliev and E. R. Bittner

states of odd e – h parity are composed exclusively of separated e – h pairs, for which Coulomb binding is weak and spin exchange is vanishing. After computing the eigenstates of H el on the one hand, and the normal modes of H ph on the other, the system-plusbath Hamiltonian Eq. 共1兲 can be written down in the diabatic representation H⫽

g ab ␰ Q ␰ 兩 a 典具 b 兩 兺a ␧ a兩 a 典具 a 兩 ⫹ 兺 ab ␰

⫹

1 2

兺␰ ␻ 2␰ Q ␰2 ⫹ P 2␰ .

共7兲

Here 兩a典 are the vertical-excited spin states with energies ␧ a , and ␻ ␰ are the phonon frequencies. Because of the presence of a small oscillator–oscillator coupling parameter ␭ in Eq. 共1兲, ␻ ␰ are weakly dispersed in the two bands around 1600 and 100 cm⫺1, respectively. The transformed H el–ph part contains diagonal coupling terms g aa ␰ which have the effect of displacing the adiabatic excited electronic origins from the ground-state minimum at Q ␰ ⫽0. The resulting configuration of the S a potential energy surfaces has allowed us to compute one-photon spectra with vibrational resolution and compare them with experiment.31,40 In what follows we ignore the adiabatic corrections to ␧ a and concentrate on the offdiagonal interstate vibronic couplings g ab ␰ that are responsible for the phonon-mediated interstate transitions. The soadopted diabatic approximation constitutes the most severe limitation in our approach since el–ph coupling is generally too strong to be perturbatively treated. However, it allows us to integrate the motion of the vibrational bath and work with the reduced density matrix of the electronic system. The evolution of an arbitrary electronic superposition state with density matrix ␳ is described in energy representation by the dissipative form of the Liouville–von Neumann equation, also known as the Redfield equation41 ⳵␳ ab ⳵␳ ab ⫽⫺i ␻ ab ␳ ab ⫹ ⳵t ⳵ t diss

冉 冊

⫽⫺i ␻ ab ␳ ab ⫺

R ab,cd ␳ cd 共 t 兲 . 兺 cd

共8兲

Except for the case when the relevant system states are only several, the solution of Eq. 共8兲 is a formidable task.42 Uninterrupted ␲ conjugation in PPV-type polymers extends over several tens of repeat units, and a segment with n units has n 2 singlet and n 2 triplet states within the CI model. Thus even the sheer storage of the n 2 ⫻n 2 density matrix ␳ is problematic for relevant n, not to mention the complete n 4 ⫻n 4 relaxation tensor R or the computational efforts needed for integration of Eq. 共8兲. Therefore we turn to the secular 共or Bloch兲 approximation which separates the evolutions of populations ␳ aa from coherences ␳ ab by leaving only those dissipative terms in R that correspond to population transfer and coherence decay R aa,bb ⫽⫺k ab ⫹ ␦ ab R ab,ab ⫽

1 2

兺c

兺c k ac ,

共 k ac ⫹k bc 兲 .

共9兲

Here we have introduced the rate k ab for the elementary transition from state 兩a典 to state 兩b典. Under such premise, coherences have analytical solutions in the form of decaying oscillations, and the time-dependent populations can be obtained by integrating the n 2 -dimensional master equation

冉 冊

⳵␳ aa ⳵␳ aa ⫽ ⳵t ⳵t

⫽⫺ diss

兺b R aa,bb ␳ bb共 t 兲 .

共10兲

Photoluminescence studies43 have shown that intramolecular vibrational relaxation of the excited species in PPV is fast and falls into the fentosecond time range. Intermolecular thermal equilibration of the excited chromophore with the environment can be less rapid. Hot exciton dissociation data suggest an effective vibrational cooling time of ⬃1 ps for a PPV derivative.44 However, low-temperature transient photoluminescence studies45 are more consistent with a vibrational cooling time of several tens of picoseconds, presumably due to the presence of ring-torsional degrees of freedom in PPV. With this caveat in mind, we assume that electronic processes are still slower than the thermalization of the phonons and derive the rates of the one-phonon interstate transitions in the Markov approximation41 ic k ab ⫽␲

兺␰

2 g ab ␰

ប␻␰

关 1⫹ ␩ 共 ␻ ab 兲兴

⫻ 关 ⌫ 共 ␻ ␰ ⫺ ␻ ab 兲 ⫺⌫ 共 ␻ ␰ ⫹ ␻ ab 兲兴 .

共11兲

Here ␩ is the Bose–Einstein distribution at room temperature T. The rates of Eq. 共11兲 obey the principle of detailed balance, which guarantees that the system also evolves towards the equilibrium with Boltzmann population of the electronic states. In Eq. 共11兲 the discrete phonon spectrum is smoothed by an empirical Lorentzian broadening ⌫ of half width 30 meV. Thus for an elementary one-phonon transition to take place, there must be a mode close to the transition frequency ␻ ab ⫽(␧ a ⫺␧ b )/ប. Because of this, such processes occur between the closely spaced excited levels, but not across the electronic gap which is much wider than the phonon quanta. Spontaneous one-photon radiation processes have rates, determined by the transition dipoles ␮ab and the photon density of vacuum rad ⫽ k ab

2 3 ␮ab 2ប ␻ ab . 关 1⫹ ␩ 共 ␻ ab 兲兴 6␧ 0 ប 2 ␲c3

共12兲

Since ␮ab can be readily computed in WF basis,31 it is in principle possible to incorporate the radiative rates in Eq. 共9兲. However, we currently neglected them as being relatively slow compared with the one-phonon processes from Eq. 共11兲. Even if some ␮ab between excited states could be subrad will be suppressed by the stantial, the corresponding k ab 3 . Thus we are left with small density factor proportional ␻ ab the optical transitions to the ground state for which transition frequencies ␻ a0 are large. However, as summarized by Kasha’s rule, fluorescence occurs essentially from S 1 , regardless of the initial photoexcitation. Our model turns out to be consistent with that rule, since the computed optical coupling between the excited manifold S a and S 0 is almost fully dominated by S 1 共see Fig. 1兲. Because of this we can safely

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J. Chem. Phys., Vol. 119, No. 7, 15 August 2003

Spin-dependent electron-hole capture

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TABLE I. Computed singlet/triplet excitons ␧ XT , charge-transfer states ␧ CT , and their separations ␧ CT–XT as a function of chain length n. Given are also the fitted exciton formation times ␶ 1 and ␶ 2 along the symmetryallowed and symmetry-forbidden conversion pathways.

FIG. 1. Singlet and triplet excited states and densities for PPV32 . Density of states of even e – h parity 共dark gray兲; odd e – h parity 共light gray兲; vertical one-photon absorption intensity 共dotted line兲.

assume that the nonradiative relaxation of higher lying S a is much faster than their emissive lifetime, so that their depopulation from direct radiative decay to S 0 is negligible. This assumption of ours should not be misinterpreted with regard to the dynamic Stokes shifts observed in transient photoluminescence.43,45 Such spectral shifts are relatively small in comparison with electronic energies, and have been attributed either to the incoherent migration of the exciton43 or to the slow thermalization of the libration motion.45 Both processes remain beyond the scope of our treatment.

III. RESULTS AND DISCUSSION A. Density of states

The distribution of the vertical-excited states for a model system with n⫽32 phenylenevinylene units (PPV32) is shown in Fig. 1. Because we deal with over 1000 levels in each spin configuration, the density of states 共DOS兲 is given in a pseudocontinuous representation. Hereafter we label the singlet and triplet excitons by S XT and T XT . Thereby we mean essentially S 1 and T 1 , except when we refer to the S T and ␳ XT in a evolving superposition state. In populations ␳ XT that case we tacitly include all the same-spin states that are S T and ␳ XT are thermally accessible from S 1 or T 1 . Thus ␳ XT asymptotically equal to one when the corresponding state space is in a pseudoequilibrium. In Fig. 1 we also delineate as S CT and T CT the lowest CT excitations, i.e., the lowest eigenstates whose e – h parity is odd. The two-band polymer model of noninteracting electrons would predict a spin-independent DOS with square-root singularities at the edges.7 Figure 1 implies substantial departure from this picture, especially in the low-energy part of the spectrum. We obtain S XT and T XT as intermediate excitons affected by e – h attraction and spin exchange. Coulomb attraction gives preference to the intraunit e – h pairs in the expansion of the exciton wave functions. Exchange interaction, which is repulsive for two antiparallel spins at short distance, renders S XT more weakly bound than T XT , in agreement with their substantial separation in nondegenerate S T and ␧ XT conjugated polymers.13–17 The exciton energies ␧ XT

n

S/T ␧ XT 共eV兲

␧ S/T CT 共eV兲

␧ S/T CT–XT 共eV兲

␶ S/T 共ps兲 1

␶ S/T 共ns兲 2

2 4 8 16 32

3.41/1.92 2.79/1.58 2.59/1.48 2.54/1.46 2.53/1.45

4.24/4.50 3.49/3.36 3.06/3.00 2.97/2.91 2.95/2.90

1.09/2.33 0.70/1.78 0.47/1.51 0.42/1.46 0.42/1.45

96/44 111/98 90/80 101/158 104/127

0.6/7.6 0.5/8.6 1.1/11 1.3/12 1.4/14

for PPV32 共see also Table I兲 already compare well with the PPV spectroscopy,14 –16 although lattice relaxations are herein neglected.46 As a consequence of Coulomb binding the triplet DOS 共T DOS兲 shows a gap that separates the exciton band from the mobility edge marked by T CT . An exciton band can be discerned in the singlet case, as well; however it overlaps with the manifold of unbound states and the S DOS is gapless. Being very weak for separated e – h pairs,31 spin exchange is not essential for the lowest CT states. S CT and T CT differ negligibly, especially in the long chains 共see Table I兲. Apart from this, e – h interaction is predicted to substantially narrow the one-photon absorption spectrum of the S DOS. As shown in Fig. 1, the optical coupling with S 0 is concentrated in S XT which contains the strongly dipole-allowed neutral monoexcitations in a constructive combination. While the odd e – h parity states are generally dipole forbidden from S 0 , higher lying even-parity S a also show weak dipole coupling because the transition dipoles of the configurations involved are either small or tend to cancel. B. Relaxation kinetics

Before turning to the CL dependence of the recombination process, we illustrate it for PPV32 by the evolution of the population density in S and T DOS 共see Fig. 2兲. An initial balanced charge injection is modeled by an electron and hole localized on the outermost units of the chain. The configuration is a superposition state in energy representation. Since the separated electron and hole are not exchange coupled, the starting population distributions in S and T state spaces are practically identical 共see the population distributions at time t⬇0 in Fig. 2兲. Long-range Coulomb interaction favors a mutual approach of the electron and hole, and the resulting relaxation process is predicted to a occur on a ⬃100 ps time scale. Singlet and triplet e – h captures are foreseen to follow the same scenario. Half of the population density relaxes directly to the corresponding exciton, but the other half remains locked in the same-spin CT state. The latter is conversion stable because its vibronic coupling with the lower states of even parity is symmetry forbidden. The exactly balanced XT:CT branching ratio of 1:1 is characteristic for the initially injected e – h pair, which can be represented as a combination of two symmetry-adapted wave functions of opposite parities. Note that the optical S CT→S 0 transition is dipole forbidden, although by point symmetry. Therefore we

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FIG. 2. Evolution of the population density for singlet 共left兲 and triplet 共right兲 e – h capture in PPV32 . In the insets: time-dependent populations ␳ XT and ␳ CT of the exciton and the charge-transfer state.

tentatively identify the CT states herein with the long-lived nonemissive species that are found to bridge the dissociated e – h pairs with the spin-coupled excitons.47 We also point out that the predicted ␧ CT⬇3.0 eV position is consistent with the states generated from relaxing two-photon excitations in derivatized PPV.48 The evolutions of the populations ␳ XT and ␳ CT are shown in the insets of Fig. 2.49 While S CT , S XT , and T CT are practically formed in parallel, the formation of T XT is subS reaches stantially delayed. Whereas the time in which ␳ XT 25% 共or half of its final population under strict e – h symmeS try兲 is ␶ XT ⫽87 ps, the half-formation time for T XT is over twice as long. The formation kinetics of the states clearly deviate from first-order law, especially in early relaxation stages 共see Fig. 3兲. This is consistent with a cascade relaxation process across the intermediate levels that separate the injected e – h pair from the conversion-stable lowest XT and CT states. The difference in the formation rates roughly reflects the difference in dissipated electronic energies. The S ⫽0.45 eV; hence S XT state is slightly lower than S CT , ␧ CT-XT S XT formation is delayed minimally relative to S CT . The T ⫽1.4 eV is much larger and separation of the triplets ␧ CT-XT part of it is even missing intermediate exciton-like levels. As a result T XT is formed over two times slower than T CT . If we take the inverse half-formation time as a measure of recomT S bination efficiency, the obtained ratio ␶ XT / ␶ XT ⫽2.3 is similar to the r values measured in highly ordered polymers.4 – 6,21,24

FIG. 3. Singlet 共left兲 and triplet 共right兲 exciton populations ␳ XT as a function of time for model PPV oligomers with different number of repeat units n.

C. Chain length dependence

Figure 3 shows the family exciton formation curves obtained for model oligomers with n⫽2, 4, 8, 16, and 32. The results clearly show that S XT and T XT formation rates are differently n dependent. Thus T XT formation outpaces S XT in the model dimer PPV2 . Therein intermediate conversion steps are practically missing because of the scarcity of states. The first-order population transfer to S XT or T XT occurs directly from the upper levels of same spin and parity. Initially S T ␳ XT is larger than ␳ XT because S XT is more delocalized and overlaps better with the injected e – h pair. However, T XT population growth quickly outpaces S XT because of the stronger mutual vibronic coupling of the triplets. This theoretical result is consistent with the more efficient triplet exciton generation established from ␹ S and ␹ T in monomers with four unsaturated bonds in conjugation.23 As oligomer length increases, exciton formation changes gradually from first-order rate law to the consecutive kinetics characteristic for the presence of one or more intermediate states. From Fig. 3 we see that whereas the S XT curve changes mainly in shape, the T XT formation is systematically delayed with increasing n. In Fig. 4共a兲 we plot the exciton population ratio ␹ S / ␹ T as a function of time with account of triplet multiplicity. Except for the first ⬃50 ps when the curves largely reflect the initial match between the exciton and the injected e – h pair, ␹ S / ␹ T shows a systematic rise on going from

S T FIG. 4. Singlet vs triplet exciton generation ratio ␹ S / ␹ T ⫽ ␳ XT /3␳ XT as a function of time for PPVn with: 共a兲 strict e – h symmetry and 共b兲 broken e – h symmetry.

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J. Chem. Phys., Vol. 119, No. 7, 15 August 2003

Spin-dependent electron-hole capture

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FIG. 5. Evolution of singlet 共left兲 and triplet 共right兲 population density for e – h capture in PPV32 with weak e – h asymmetry. In the insets: timedependent populations ␳ XT and ␳ CT of the exciton and the charge-transfer state.

PPV2 to PPV32 . At t⬇100 ps for instance, the population ratio for PPV2 is less than 1/3 in favor of the triplet; for PPV32 it is over one. The result is consistent the conjugationlength dependent r values measured in several types of conjugated systems.21,23,24 It has been also shown32 that r as a function of CL is in near linear relationship with the ratio of T S ␧ CT-XT to ␧ CT-XT . From such a perspective the spindependent ␧ CT⫺XT gap in molecules can be viewed as a discrete analog of the exciton binding energy in a polymer. The model rests on the assumption that e – h capture is predominantly intramolecular, whereas in an actual device the process may be both intra- and intermolecular. Since intermolecular recombination models26,29 also give r⬎1 for polymers and r⬇1 for short oligomers, the physics of the process should be viewed in light of simple energy and scaling arguments. Regardless of its detailed nature, recombination in organic semiconductors is a transition from a delocalized excitation to a localized state known to be essentially intramolecular from the spectroscopy. In a small molecule or chromophore the spin configuration of this final state is immaterial since the latter is confined to the dimensions of the molecule or chromophore. In a long polymer S XT is intrinsically more extended than T XT because of spin exchange; consequently S XT provides a wider, more efficient trap for the initial excitation. The inverse relationship between efficiency of e – h capture and binding energy is generally valid because the latter effectively measures the final state localization. D. Breaking of electron–hole symmetry

The e – h symmetry of conjugated polymers is not only approximate, but also destroyed extrinsically when bias is applied to drive the carriers in a device. Since both materialspecific and extrinsic effects are difficult to estimate, we incorporate e – h asymmetry in a rather open way by assuming that ␲ * electrons are slightly more mobile than ␲ holes. This is achieved by modifying the matrix elements of the oneparticle operator f and its derivatives in Eq. 共5兲 according to the equations

⬘ ⫹¯f nm ⬘ ⫽2 f mn , f mn ⬘ ⫺¯f nm ⬘ ⫽2w f mn , f mn

共13兲

where the asymmetry parameter w is taken to be equal to 5%. The excitation spectrum does not change significantly because in an eigenstate with initially equivalent distribu-

tions of the electron and hole, stabilization of the former is compensated by destabilization of the latter, or vice versa. However, the lifting of the symmetry weakly allows the vibronic coupling between states of opposite e – h parity. As exemplified for PPV32 in Fig. 5, the CT state is no longer metastable and exciton formation proceed to a completion with ␳ XT approaching one. In addition to the direct symmetry-allowed XT formation process, we delineate a second slower channel, which is nearly symmetry forbidden and proceeds via the CT transient. The fast channel involves essentially the even parity admixture of the injected e – h pair and completes in 100–200 ps time. The relaxation along the second pathway falls already in the nanosecond time range due to the slow CT→XT conversion. Remarkably, the breaking of e – h symmetry not only opens the CT→XT conversion channel, but also changes in a spin-dependent manner the XT:CT branching ratio of the initial fast symmetryS surallowed relaxation. As shown by the plots in inset, ␳ XT S passes ␳ CT prior and independently of the depopulation of T the latter in the symmetry-forbidden process. In contrast, ␳ CT T grows initially much faster than ␳ XT , and the resulting disproportion is not compensated via the CT→XT population transfer in the time scale of the calculation. The slowness of the T CT→T XT conversion in the extended chains underscores the importance of the predicted gap in the T DOS 共see Fig. 1兲. Our excited-state treatment of PPV is limited to the frontier bands only; hence it cannot be ruled out that triplet excitations involving other bands fall within that gap. From a theoretical perspective such a situation is unlikely because the next-highest and next-lowest bands lie ⬃2 eV apart from the primary band edges in the underlying one-particle HF description.31 More importantly, the predicted separation of 1.45 eV between T XT and T CT in PPV32 happens to match exactly the gap in the T 1 →T a absorption spectrum of the polymer.13,50 In Fig. 6 we plot the time-dependent exciton populations for the model oligomers PPVn with weak e – h asymmetry. As in the symmetric case, the initial S XT formation rates are very close, but those of T XT monotonously decrease with increasing n. The nearly symmetry-forbidden CT→XT conversion is invariably faster for singlets than for triplets, showing no substantial dependency on CL. Also, unlike in the symmetric case, the disproportion between S XT and T XT is more persistent. As shown in Fig. 4共b兲 the ␹ S / ␹ T population ratios exceed the statistical limit for at least 3 ns. Extended conjugation is foreseen to enhance the initial XT:CT

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exciton, T CT is more likely to undergo intersystem crossing to its near degenerate S CT counterpart, which on its turn relaxes much more rapidly into S XT . IV. CONCLUSIONS

FIG. 6. Singlet 共left兲 and triplet 共right兲 exciton populations ␳ XT for PPVn with broken e – h symmetry.

branching ratio in favor of S XT and disfavor of T XT . Except in the very early stage, PPV32 shows the highest ␹ S / ␹ T value with a maximum exceeding one. In comparison, the ␹ S / ␹ T curves for PPV2 and PPV4 lie much lower and converge faster to a statistical limit of one third. However, the dependence of ␹ S / ␹ T on n is no longer monotonic throughout the time range as in the symmetric case. To estimate the exciton formation times ␶ 1 and ␶ 2 along the fast and slow conversion pathways, we further fit the population curves from Fig. 6 by the double-exponential function ˜␳ XT共 t 兲 ⫽1⫺a 1 e ⫺t/ ␶ 1 ⫺a 2 e ⫺t/ ␶ 2 .

共14兲

Least-square-fit optimization of the four parameters is performed under the constraint that ˜␳ XT(0) exactly matches the initial population ␳ XT(0). The trial function Eq. 共14兲 implies pure first-order kinetics, which is certainly inadequate for the slow consecutive CT conversion pathway. However ␶ 2 is much longer than the time in which CT is formed and therefore encompasses almost exclusively its relaxation lifetime. Because of this obtained regression fits are relatively good, with levels of ␹ 2 significance better than 1% in all the cases. The results for ␶ 1 as summarized in Table I confirm the trend found under strict e – h symmetry. Namely, the ␶ T1 / ␶ S1 ratio which reflects the relative efficiency of singlet versus triplet exciton formation in the picosecond range increases almost threefold on going from the dimer to the polymer. According to the ␶ 2 results, both S CT and T CT lifetimes increase about twofold on going from PPV2 to PPV32 , but T CT is invariably about ten times longer lived than its singlet counterpart. We note also that in the long chains S CT and T CT are almost isoenergetic 共see Table I兲 and consequently indistinguishable. These results combined can be interpreted in two ways, but lead to the same overall conclusion. If we consider the CT S T ⬇␧ CT , then they states spin independent in view of ␧ CT should all relax predominantly into S XT , so that the final ␹ S fraction should approach the ␹ T fraction from direct T XT formation from below. If we still treat S CT and T CT as different, then the predicted T CT conversion lifetime of over 10 ns exceeds the time of intersystem crossing processes even for hydrocarbons.51 Thus, rather than conversion into a triplet

Proceeding from a diabatic model of a multilevel electronic system coupled to a phonon bath, we have simulated e – h recombination in short and extended PPV-type chains. Exciton formation from an injected electron and hole is delineated as the 共consecutive兲 interconversion across samespin excited states with energy dissipating into the salient Franck–Condon active modes. The proposed relaxation mechanism is compatible with the established temperature near independence of the process. Moreover, the computed exciton formation times are in good agreement with the observed singlet-to-triplet generation ratios and their pronounced increase with the 共effective兲 conjugation length. The latter trend has been rationalized in terms of a process, which changes from first-order kinetics in a molecule with several excited states to a cascade relaxation in a polymer with a dense excited manifold. Of special importance are the difference in e – h binding of the singlet and triplet excitons and the appearance of a gap in the triplet density of states for the polymer. The approximate e – h symmetry of conjugated systems is found to have important implications for the e – h capture process. It leads to branching of the conversion pathways and the formation of long-lived CT states whose fate is all but clear. The results obtained for weak e – h asymmetry reveal a mechanism of the persistent enhancement of the singlet generation fraction in long conjugated chains. Namely, in view of long interconversion lifetimes and near degeneracy, singlet and triplet CT states are likely to be in an intersystem crossing quasiequilibrium, from which the singlet exciton formation is about ten times more efficient than the triplet one. ACKNOWLEDGMENTS

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