Theory of the singlet exciton yield in lightemitting polymers
Abstract
The internal electroluminescent quantum efficiency of organic light emitting diodes is largely determined by the yield of singlet excitons formed by the recombination of the injected electrons and holes. Many recent experiments indicate that in conjugated polymer devices this yield exceeds the statistical limit of 25% expected when the recombination is spinindependent. This paper presents a possible explanation for these results. We propose a theory of electronhole recombination via intermolecular interconversion from intermolecular weakly bound polaron pairs (or chargetransfer excitons) to intramolecular excitons. This theory is applicable to parallel polymer chains. A crucial aspect of the theory is that both the intramolecular and intermolecular excitons are effectiveparticles, which are described by both a relativeparticle wavefunction and a centerofmass wavefunction. This implies two electronic selection rules. (1) The parity of the relativeparticle wavefunction implies that interconversion occurs from the even parity intermolecular chargetransfer excitons to the strongly bound intramolecular excitons. (2) The orthonormality of the centerofmass wavefunctions ensures that interconversion occurs from the chargetransfer excitons to the lowest branch of the strongly bound exciton families, and not to higher lying members of these families. The interconversion is then predominately a multiphonon process, determined by the FranckCondon factors. These factors are exponentially smaller for the triplet manifold than the singlet manifold because of the large exchange energy. As a consequence, the interconversion rate in the triplet manifold is significantly smaller than that of the singlet manifold, and indeed it is comparable to the intersystem crossing rate. Thus, it is possible for the singlet exciton yield in conjugated polymers to be considerably enhanced over the spinindependent recombination value of
pacs:
78.67.n, 77.22.dI Introduction
The internal electroluminescent quantum efficiency of organic light emitting diodes is largely determined by the yield of singlet excitons formed by the recombination of the injected electrons and holes. Singlet exciton yields in light emitting polymers exceeding the spinindependent recombination value of 25% have now been reported by a large number of groups 1 7 , although its value remains controversial. Lin et al. 8 claim that the singlet yield only exceeds the statistical limit in large electric fields,foot1 while Segal et al. 9 report a singlet exciton yield of only 20%. A photoluminescence detected magnetic resonance investigation17a suggests that interchain (or bimolecular) recombination is spindependent.
Many theoretical attempts have been made to explain the enhanced singlet exciton yield. Bittner et al. 10 12 assume that intrachain electronhole recombination occurs via vibrational relaxation through the band of exciton states between the particlehole continuum and lowest bound excitons. Since vibrational relaxation is faster in the singlet channel than the triplet channel, because the lowest singlet exciton lies higher in energy than the lowest triplet exciton, a faster formation rate for the singlet than the triplet exciton is predicted. Hong and Meng 13 argue that a multiphonon process in the triplet channel also leads to faster intramolecular singlet exciton formation.
The different rates for singlet and triplet exciton formation predicted in the literature for interchain recombination 14 16 arise largely from the assumption that an interchain densitydependent electron transfer term is an important factor in the recombination mechanism. This term couples states of the same ionicity. Since the interchain charge transfer states are predominately ionic, while the intrachain triplet exciton has more covalent character than the intrachain singlet exciton, the rate for the singlet exciton formation is correspondingly greater.
In this paper we develop a model of interchain electronhole recombination between pairs of parallel polymers that involves intermediate, loosely bound (‘chargetransfer’) states that lie energetically between the electronhole continuum and the final, strongly bound exciton states. We argue that as a consequence of electronic selection rules, intermolecular interconversion occurs from the charge transfer to the lowest energy exciton states. This process is then limited by multiphonon emission, which decreases approximately exponentially with the energy gap between the pair of states. Since the lowest singlet and triplet exciton energies are split by a large exchange energy (of ca. 0.7 eVkohler ), while the chargetransfer states are quasidegenerate, the triplet exciton formation rate is considerably smaller than the singlet exciton rate.
Multiphonon emission has already been discussed as a possible factor in determining the overall singlet exciton yield in intramolecular processes19 ; 13 . However, our model differs from these works by emphasizing the important role of the intermediate interchain chargetransfer states.
In the next section we introduce the relevant rate equations and derive an expression for the singlet exciton yield as a function of the characteristic relaxation times. In the following section the microscopic model of intermolecular interconversion is described and the interconversion rates are calculated.
Ii Basic model and the rate equations
Fig. 1 shows the energy level diagram for this model. The electrons and holes are injected into the polymer device with random spin orientations. Under the influence of the electric field the electrons and holes migrate through the device, rapidly being captured (in less than s) to form the weakly bound chargetransfer singlet and triplet excitons, and , respectively. We assume that no spin mixing occurs during this process, and thus the ratio of to is 1:3.foot3 If the intersystemcrossing (ISC) between and (with a rate ) competes with the interconversion from to the triplet exciton, , (with a rate ) and is smaller than the interconversion rate () from to the singlet exciton, , then the singlet yield is enhanced.
The chargetransfer states might be either intramolecular loosely bound excitons or weakly bound positive and negative polarons on neighboring chains. Intramolecular chargetransfer states are MottWannier excitons whose relative electronhole wavefunctions are odd under electronhole exchange20 . A crucial characteristic of these states is that, because of their odd electronhole parity, the probability of finding the electron and hole on the same molecular repeatunit is zero. Thus, they experience very small exchange interactions and therefore the singlet and triplet states are quasidegenerate20 ,foot2 . Similarly, the intermolecular weakly bound positive and negative polarons  although now possessing even electronhole parity  also experience weak exchange interactions as necessarily the electron and hole are on different repeat units, and thus singlet and triplet states are also quasidegenerate. Furthermore, since the chargetransfer excitons are also weakly bound with relatively large electronhole separations, there exist efficient spinflipping mechanisms, such as spinorbit coupling, or exciton dissociation via the electric field or by scattering from free carriers and defects. In this paper we focuss on interconversion to the intramolecular excitons from the interchain chargetransfer excitons.
The strongly bound excitons, and , are intramolecular states. The interconversion process from to and from to depends on the nature of and . The mechanism for bimolecular interconversion is described more fully in the next section. In this section we describe the kinetics by classical rate equations. The use of classical rate equations is justified if rapid interconversion follows the ISC between and , as then there will be no coherence or recurrence between and 24 . We also note that since interconversion is followed by rapid vibrationalrelaxation (in a time of s) these processes are irreversible.
We first consider the case where ISC occurs directly via the spinorbit coupling operator. This operator converts the triplets into the singlet13 , and vice versa. Let , , and denote the number of , , and the and excitons, respectively. is the number electronhole pairs created per second. Then the rate equations are:
(1) 
(2) 
(3) 
and
(4) 
Notice that the component of the exciton is converted directly to the component of the exciton, and cannot contribute to the singlet exciton yield.
When these equations are solved under the steady state conditions that
(5) 
we obtain the singlet exciton yield, , defined by,
(6) 
as
(7) 
where and .
Alternatively, we might consider ISC via a spinrandomization process, whereby the chargetransfer excitons are scattered into chargetransfer triplets with a probability of and chargetransfer singlets with a probability of . Then the rate equations are those of the Appendix of ref17 and the singlet exciton yield becomes17 ,
(8) 
In practice, as we shall show, so . We note that is a function only of the relative lifetimes of and , and the ISC rate. The singlet yield is plotted in Fig. 2 as a function of . We now describe the calculation of the relative rates.
Iii Derivation of the intermolecular interconversion rate
The BornOppenheimer (BO) Hamiltonian for a pair of coupled polymer chains is,
(9) 
where is the intrachain BO Hamiltonian for the th chain and is the interchain BO Hamiltonian. We split the interchain Hamiltonian into two components: the interchain oneelectron Hamiltonian, , and the interchain twoelectron Hamiltonian, . predominately describes the Coulomb interactions between the electrons on neighboring chains. describes electron transfer between chains. For parallel chains with nearest neighbor electron transfer this is,
(10) 
where () creates (destroys) a electron on site of chain and is the interchain hybridization integral. If the chains are weakly coupled we may regard as a perturbation on the approximate Hamiltonian,
(11) 
Within the BornOppenheimer approximation the electronic and nuclear degrees of freedom are described by the BornOppenheimer states. A BornOppenheimer state, , is a direct product of an electronic state, , and a nuclear state associated with that electronic state, :
(12) 
The label indicates that the electronic state is parametrized by the nuclear coordinates.
The stationary electronic states are the eigenstates of the approximate Hamiltonian, . Thus, the perturbation, mixes these electronic states. In particular, it causes an interconversion from the interchain excitons (or weakly bound polaron pairs) to the intrachain excitons by transferring charge from one chain to another.
We take the initial electronic state to be a positive polaron on chain and a negative polaron on chain ,
(13) 
The interchain Coulomb interaction between the chains creates a weakly bound chargetransfer exciton, to be described below. The labels and indicate the independent normal coordinates of chains 1 and 2, respectively.
We consider the situation where the negative polaron is transferred from chain 2 to chain 1 by . Thus, the final state is an intramolecular exciton on chain 1 (denoted by ), leaving chain 2 in its ground electronic state,
(14) 
Before proceeding it will be useful to review the theory of excitons in conjugated polymers. In the weakcoupling limit (namely, the limit that the Coulomb interactions are less than or equal to the band width) the intramolecular excited states of semiconducting conjugated polymers are MottWannier excitons described by20 ,
(15) 
is an electronhole basis state constructed by promoting an electron from the filled valence band Wannier orbital at to the empty conduction band Wannier orbital at ,
(16) 
and are the valence and conduction Wannier orbital operators, respectively, approximately defined by,
(17) 
and
(18) 
where is the unit cell index. The symbol in Eq. (16) refers to singlet () or triplet () excitons.
is the centerofmass coordinate and is the relative coordinate of the effective particle. is a hydrogenlike electronhole wavefunction labelled by the principle quantum number, , which describes the effectiveparticle. This has the property that under electronhole reflection (namely, ) for odd and for even .
(19) 
is the centerofmass wavefunction, which describes the motion of the effectiveparticle on a linear chain. is the number of unit cells. For each principle quantum number, , there is a band of excitons with different pseudomomentum, , where satisfies and is the unit cell distance. Thus, every exciton state label, , corresponds to two independent quantum numbers: and . As described in20 , corresponds to the and families of intrachain excitons, while corresponds to the and families of intrachain excitons. The lowest energy member of each family has the smallest pseudomomentum, namely, foot10 .
It is also convenient to describe the intermolecular weakly bound polaron pairs as chargetransfer excitons described by,
(20) 
where represents the interchain effectiveparticle wavefunction. (i.e. even electronhole parity) for the lowest energy interchain excitons. is an electronhole basis state constructed by promoting an electron from the filled valence band Wannier orbital at on chain 1 to the empty conduction band Wannier orbital at on chain 2,
(21) 
With this background to the theory of excitons we now proceed to derive the transfer rate. The isoenergetic interconversion rate from the initial to the final states is determined by the Fermi Golden Rule expression,
(22) 
where the initial and final BO states are,
(23) 
and
(24) 
respectively.
iii.1 Electronic matrix elements
The corresponding electronic matrix element is,
(25) 
Using Eq. (15) and Eq. (20) this is,
(26)  
This matrix element is evaluated by expressing in terms of the valence and conduction Wannier orbital operators. Retaining terms that keep within the exciton subspace we have,
(27) 
Then,
(28) 
By exploiting the orthonormality of the basis functions,
(29) 
as well as the functions,
(30) 
we have the final result for the electronic matrix element,
(31) 
Eq. (30), Eq. (19) and Eq. (31) demonstrate the very significant result that interconverion via is subject to two electronic selection rules.

Interconversion occurs between excitons with the same centreofmass pseudomomentum, .

Interconversion occurs between excitons with the same electronhole parity. Thus, .
Since the lowest energy interchain excitons have even electronhole parity this implies that connects them to and , and not to the intramolecular and .foot6 Moreover, since the interchain exciton will have relaxed to its lowest momentum state, converts it to the intrachain exciton in its lowest momentum state, and not to higher lying momentum states.
iii.2 Vibrational overlaps
We now discuss the contribution of the vibrational wavefunctions to the total matrix element. Intermolecular interconversion is an isoenergetic process which occurs from the lowest pseudomomentum state of the chargetransfer manifold and the lowest vibrational levels of this state to the lowest pseudomomentum state of the intramolecular excitons at the same energy as the initial level. Thus, the vibrational levels in Eq. (23) are and . However, the vibrational levels in Eq. (24) are determined by the conservation of energy.
Using Eq. (22) the rate is thus,
(32)  
where
(33) 
and
(34) 
are the FranckCondon factors associated with the vibrational wavefunction overlaps of chains 1 and 2, respectively. Likewise,
(35) 
and
(36) 
are the changes in energy of chains 1 and 2, respectively. These changes in energy are illustrated in Fig. 3.
Using the identity,
(37) 
we can rewrite Eq. (32) as,
(38) 
Defining the spectral functions for the donor (chain 2) and acceptor (chain 1) as,
(39) 
and
(40) 
respectively, we have the familiar rate expression for bimolecular electron transfer,
(41) 
A useful simplification to this expression arises by noting that the geometric distortions of the polarons and exciton polarons (namely the or states) from the ground state structure are very similar.32 Thus, the HuangRhys parameter (proportional to , as defined in Fig. 3) for the and states relative to the positive polaron is negligible. Therefore,
(42) 
and thus the change of energy of chain 1 is,
(43) 
where is the energy difference on chain 1 between the final exciton state and the positive polaron. This is illustrated in Fig. 3. By the conservation of energy we therefore have,
(44) 
The vibrational level, , of the final state of chain 2 to which interconversion from the negative polaron initially occurs is given by,
(45)  
where
(46) 
and
(47) 
are the transition energies of the chargetransfer exciton and the state , respectively.
The condition expressed in Eq. (43) implies that the energy integral in Eq. (38) is restricted to the value of , and thus the rate becomes,
(48)  
is the final density of states on chain 2, defined by,
(49) 
which is usually taken to be the inverse of the vibrational energy spacing. Inserting the expression for the Franck Condon factor,
(50) 
we have the final result that,
(51) 
This equation, along with the definition of in Eq. (31), is our final expression for the interconversion rate. is the HuangRhys factor for the polaron relative to the ground state, defined as , where is the reorganization (or relaxation) energy of the polaron relative to the ground state. After the isoenergetic transition there is vibrational relaxation to the lowest vibrational level of the state of chain 2 via the sequential emission of phonons. The number of phonons emitted corresponds to the difference in energies between the initial chargetransfer and final exciton states, given by Eq. (45). This is a multiphonon process. In the next section we estimate these rates.
Iv Estimate of the interconversion rates
Since interconversion from the intermolecular to the intramolecular chargetransfer excitons is forbidden by symmetry, we now only discuss interconversion to the lowest excitons, or . (As remarked in footnotefoot6 , interconversion to higherlying exciton states is allowed, but if this happens recombination is an intramolecular process via the intramolecular chargetransfer excitons.) Thus, the state label is now either or , and the number of phonons emitted, , is either or , as determined by Eq. (45).
Within the MottWannier basis the exciton wavefunction overlaps are easy to calculate. Using eV16 , the interchain distance as and standard semiempirical Coulomb interactions gives
(52) 
and
(53) 
The polaron HuangRhys parameter, , is not accurately known for light emitting polymers. However, we expect it to be similar to the exciton HuangRhys parameter. The relaxation energy of the exciton has been experimentally determined as eV in PPVliess , with a similar value calculated for ‘ladder’ PPP in refmoore . From the figures in refhertel , we estimate the relaxation energy to be eV in ladder PPP (where the phenyl rings are planar) and eV in PPP (where the phenyl rings are free to rotate). Thus, taking the relaxation energy as eV and eV implies that .
Now, using eV (), and assuming that the energy difference between the strongly bound singlet exciton () and the intramolecular chargetransfer excitons of eV is approximately the energy difference between the singlet exciton and the intermolecular chargetransfer excitons, we can estimate the interconversion rate for the singlet exciton. This is s (or ps). Similarly, using an exchange gap of eV gives s (or ns). Thus, the triplet interconversion rate is much slower than the singlet interconversion rate.
The ISC rate is also not accurately known, with quoted values ranging from ns19 , nsfrolov and nsbarford2004 . Nevertheless, despite this uncertainty, we see that the estimated triplet interconversion rate is comparable to or slower than the ISC rate, which from Eq. (7) implies a large singlet exciton yield.
Generally, the ratio of the rates is,
(54) 
Thus,
(55) 
This ratio increases as decreases, because then multiphonon emission becomes more difficult. The ratio also increases as the exchange energy, , increases for any or if .
V Discussion and Conclusions
We propose a theory of electronhole recombination via intermolecular interconversion from intermolecular weakly bound polaron pairs (or chargetransfer excitons) to intramolecular excitons. This theory is applicable to parallel polymer chains. A crucial aspect of the theory is that both the intramolecular and intermolecular excitons are effectiveparticles, which are described by both a relativeparticle wavefunction and a centerofmass wavefunction. This implies two electronic selection rules.

The parity of the relativeparticle wavefunction implies that interconversion occurs from the even parity intermolecular chargetransfer excitons to the strongly bound intramolecular excitons and not to the intramolecular chargetransfer excitons (namely, the first odd parity exciton). (However, if the interchain charge transfer excitons lie higher in energy than the second family of even parity intrachain exciton, recombination will be an intramolecular process.)

The orthonormality of the centerofmass wavefunctions ensures that interconversion occurs from the chargetransfer excitons to the lowest branch of the strongly bound exciton families, and not to higher lying members of these families.
These selection rules imply that interconversion is then predominately a multiphonon process, determined by the FranckCondon factors. These factors are exponentially smaller for the triplet manifold than the singlet manifold because of the large exchange energy.
There is also a contribution to the rates from the overlap of the relativeparticle wavefunctions, which again are smaller in the triplet manifold, because the triplet exciton has a smaller particlehole separation and has more covalent character than its singlet counterpart16 . As a consequence, the interconversion rate in the triplet manifold is significantly smaller than that of the singlet manifold, and indeed it is comparable to the ISC rates. Thus, it is possible for the singlet exciton yield is expected to be considerably enhanced over the spinindependent value of in conjugated polymers.
Any successful theory must explain the observation that the singlet exciton yield is close to for molecules and increases with conjugation length3 ; 5 . This theory qualitatively predicts this trend for two reasons. First, the effectiveparticle description of the exciton states is only formally exact for long chains. This description breaks down when the chain length (or more correctly, conjugation length) is comparable to the particlehole separation. In this case separation of the centerofmass motion and the relativeparticle motion is no longer valid. Then the quantum numbers and (which describe the relativeparticle wavefunction and centerofmass wavefunction, respectively) are no longer independent quantum numbers. Interconversion is then expected to take place between all the states lying between the chargetransfer state and the lowest exciton state. However, as the chain length increases interconversion to higher lying states is suppressed in favor of the lowest lying exciton. This prediction is confirmed by a recent quantum mechanical calcuations by Beljonne et al. 33 . The second reason that the singlet exciton yield is enhanced in polymers over molecules is that the HuangRhys parameters decrease as the conjugation length increases, and thus the relative rate (given by Eq. (54)) increases.
We note that the effectiveparticle description is still valid when there is selftrapping. In this case the centerofmass wavefuctions are not the particleinthebox wavefunctions appropriate for a linear chains (Eq. (19)), but they are the orthonormalized functions appropriate for the particular potential well trapping the effective particle. The key point is that because these are orthonormalized functions interconversion occurs between a pair of states with the same centerofmass quantum numbers, as described in this paper.
This theory has been formulated for an idealized case of sufficiently long, parallel polymer chains. The applicability of this theory for polymer light emitting displays needs verifying by performing calculations on oligomers of arbitrary length and arbitrary relative conformations.
Finally, we remark that this theory presents strategies for enhancing the singlet exciton yield. Ideally, the polymer chains should be well conjugated, closely packed and parallel. The last two conditions ensure that the interchain chargetransfer excitons lie energetically below highlying evenparity families of intramolecular excitons, and thus recombination is an interchain interconversion process and not an intramolecular process via the intramolecular chargetransfer excitons. Intramolecular recombination is not desirable because although interconversion from the intrachain chargetransfer excitons is slower in the triplet manifold than the singlet manifold, both rates are expected to be faster than the ISC rate. The relative intermolecular interconversion rates are also increased when the electronlattice coupling is reduced. This suggests that the singlet exciton yield is enhanced in rigid, wellconjugated polymers.
Acknowledgements.
I thank N. Greenham, A. Köhler, and C. Silva (Cambridge) for useful discussions. I also gratefully acknowledge the financial support of the Leverhulme Trust, and thank the Cavendish Laboratory and Clare Hall, Cambridge for their hospitality.References
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