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Modulating the Exciton Dissociation Rate by up to More than Two Orders of Magnitude by Controlling the Alignment of LUMO + 1 in Organic Photovoltaics Haibo Ma*,† and Alessandro Troisi*,‡ †

Key Laboratory of Mesoscopic Chemistry of MOE, School of Chemistry and Chemical Engineering, Institute of Theoretical and Computational Chemistry, Nanjing University, Nanjing 210093, China ‡ Department of Chemistry and Centre of Scientific Computing, University of Warwick, Gibbet Hill, Coventry CV4 7AL, United Kingdom S Supporting Information *

ABSTRACT: Efficient organic solar cells require a high yield of exciton dissociation. Herein we investigate the possibility of having more than one charge-transfer (CT) state below the first optically bright Frenkel exciton state (FE) for common molecular donor (D)/acceptor (A) pairs and the role of the second-lowest CT state (CT2) in the exciton dissociation process. This situation, previously explored only for fullerene acceptors, is shown to be rather common for other D/A pairs. By considering a phenomenological model of a large aggregate, we reveal that the position of CT2 can remarkably modulate the exciton dissociation rate by up to more than two orders of magnitude. Thus, controlling the alignment of CT2 is suggested as a promising rule for designing new D/A heterojunctions.

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INTRODUCTION The nature of electronic excited states in organic semiconductors governs the dynamics of excitation energy transfer and/or exciton dissociation and is an important ingredient in understanding the fundamental mechanism of organic photovoltaics (OPVs).1−8 With the rapidly growing interest in OPVs in both the scientific and industrial communities, theoretical modeling of the electronic structure of excitons and their accompanying dynamics has been widely implemented and now plays an important role in either interpreting elementary processes or designing new materials.5−17 Because of the large size of the system, the theoretical investigations are mainly based on simplified models with the common assumption that only the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of each molecule participate in exciton dynamics within the donor (D)/acceptor (A) heterojunction, i.e., one initially photoexcited Frenkel exciton (FE) can convert to an excited charge-transfer (CT) state, where the hole occupies the HOMO of the donor (or a linear combination of HOMOs in the case of many donors) and the electron occupies the LUMO of the acceptor (or a linear combination of LUMOs in the case of many acceptors). These CT states evolve into free electrons and holes via a mechanism that is still controversial.18−32 The idea that only the lowest CT state (CT1), which is dominated by the HOMO (D) to LUMO (A) transition, is accessible from the FE state via the internal conversion process is very useful for giving qualitative descriptions of exciton dynamics in many © 2014 American Chemical Society

cases, but it may sometimes be too simplified for quantitative predictions, especially when one considers the relatively dense energy levels of frontier orbitals in medium- to large-sized πconjugated molecules and the structural versatility of chemical substances. For example, it is well known that the fullerene molecule (C60) has three degenerate LUMOs because of its high symmetry; accordingly, the dimer complex of any donor molecule and one fullerene molecule will have three nearly degenerate lowest CT states. Thus, the charge separation efficiency can be expected to be much enhanced because it will be the sum of the contributions from the three CT states. Recent work by Liu and Troisi33 verified that slightly lifted LUMO/LUMO + 1 degeneracy in fullerene derivatives can further increase the charge separation rate by more than one order of magnitude, instead of only three, because the slightly lifted LUMO/LUMO + 1 degeneracy can decrease the Gibbs free energy of the charge separation. This idea was further explored by Savoie et al.16 who suggested that the resonance coupling of photogenerated singlet excitons to a high-energy manifold of fullerene electronic states enables efficient charge generation. However, whether such a second CT state (CT2) or other higher CT states located below the dipole-allowed FE state can exist in other general D/A pairs with organic nonfullerene small electron acceptor molecules is still unknown. If the answer is affirmative, then it may provide further Received: September 28, 2014 Published: October 28, 2014 27272

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indicated that our scheme gives a reasonably good evaluation of the relative positions of frontier molecular orbital (MO) levels and the excitation energies for such small to medium conjugated molecules. (Supporting Information, Table SI). All the quantum chemical calculations in this work were performed using Gaussian 09, Revision D.01.38 Quantum chemistry calculations were used primarily to assess the likelihood that CT2 can be located below the FE state in donor−acceptor pairs. To assess the effect of these states on the rate of exciton dissociation, we found it necessary to consider a much larger model because the delocalization of at least one of the charge carriers is an essential feature of the exciton dissociation process.1,5,17 Such a model is presented and justified herein.

elements that explain why some donor−acceptor pairs are more efficient than others: if CT2 is energetically accessible from the FE state, then it not only can enhance the charge-transfer rate but also can provide multiple exciton dissociation channels and then accordingly improve the charge separation efficiency as illustrated in Figure 1. (Herein, CT1, CT2, and other CT states

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ENERGETICS OF CT2 IN DONOR−ACCEPTOR PAIRS Frontier MOs of Representative Molecules. A low LUMO + 1 energy of an acceptor molecule or a high HOMO − 1 energy of a donor molecule may lead to a low-energy CT2 because low-energy transitions (HOMO (D) → LUMO + 1 (A) or HOMO − 1(D) → LUMO (A)) can be expected under these circumstances. Figure 2b reports the energies of the frontier MOs of selected representative electron donor and acceptor molecules listed in Figure 2a. As seen in Figure 2b, the acceptor molecules investigated in this work do not have LUMO/LUMO + 1 degeneracy, and their energy differences are quite variable, ranging from 0.14 eV for pyrene-tetraone (PyT) to 1.91 eV for 3,4,9,10-perylenetetracarboxylic dianhydride (PTCDA). Similarly, the donor molecules investigated in this work also lack HOMO − 1/HOMO degeneracy, and the energy difference is also quite variable, ranging from 0.72 eV for oligothiophene-6 (OT-6) to 2.31 eV for phthalocyanine (H2PC). Another important observation is that in many nonfullerene electron acceptor molecules both LUMO and LUMO + 1 are energetically lower than the LUMO of the corresponding donor molecules. Similarly, in many donor molecules, both HOMO and HOMO − 1 are energetically higher than the HOMO of the corresponding acceptor molecules. These facts imply that CT2 may be located below the FE state in many D/A heterojunctions. Frequency of CT2 Being Located Energetically below the FE State in D/A Pairs. To estimate how frequently CT2 can be located energetically below the FE state in D/A pairs, we considered all of the 16 possible D/A dimer complexes that could be constructed from different combinations of 4 widely used electron donor molecules (pentacene, tetracene, H2PC, and OT-6) and 4 popular nonfullerene electron acceptor molecules (PyT, pentacene-tetraone (PT), coronenehexaone (COHON), and dicyanomethyl-trinitrofluorenone (TNFCN)). The energy differences (between CT2 and the FE state, ΔCT2−FE) for each D/A pair are listed in Table 1. Half of the dimers (eight pairs) were found to have a negative ΔCT2−FE value. Not every D/A pair has CT2 that is energetically lower than the pair’s FE state, but this does happen quite often in many general D/A pairs, suggesting that the presence of a lowlying CT2 is one of the interface characteristics to be taken into account when different D/A pairs are considered. The energy gaps between LUMO and LUMO + 1 (ΔLUMOs) for the four acceptor molecules PyT, PT, COHON, and TNFCN are 0.14, 0.95, 0.71, and 0.82 eV, respectively, and the energy gaps between HOMO − 1 and HOMO (ΔHOMOs) for the four donor molecules pentacene, tetracene, H2PC, and OT-

Figure 1. Schematics of the possible new channel for photoinduced electron migration, red line.

are specified for a two-molecule system, i.e., a D/A pair. They do not have the usual meanings as the electronic excited states in large D/A heterojunction aggregates found elsewhere in the literature.) The aim of this work is therefore to answer the following two questions by using quantum chemical calculations in conjugation with phenomenological simulations: (i) Can more than one CT state be energetically lower than the FE state for general nonfullerene D/A pairs? (ii) Can the exciton dissociation efficiency in OPVs be modulated by tuning the energy difference between CT1 and CT2? The outline of this study is as follows: First, computational methods are briefly introduced. Second, we study the electronic structures of several widely used nonfullerene electron donor and acceptor molecules (Figure 2a) as well as their D/A dimers by quantum chemical calculations, and we find that a CT2 located below the dipole-allowed local exciton state can exist not only in nearly degenerate fullerene-based systems but also, maybe surprisingly, in some commonly used D/A pairs. Third, we investigate the influence of the CT2 position on exciton dissociation through a phenomenological model and different limits for the expression of the exciton dissociation rate. Fourth, we give the conclusions reached from our study.

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METHODS We calculated the electronic structures of selected representative nonfullerene electron donor and acceptor molecules by density functional theory (DFT) applied to their optimized neutral ground geometries. To avoid a possible underestimation of the fundamental gap and the energy of CT excited states by conventional DFT functionals, we adopted the range-separated ωB97X-D functional34 with polarized Pople basis set 6-31G(d). For each D/A pair selected, we optimized the neutral ground state (GS) geometry by ωB97X-D/631G(d) and then performed excited-state electronic structure calculation of this GS geometry with time-dependent density functional theory (TDDFT).35,36 The nature of the excited state (local excitation or charge-transfer character) was investigated by natural transition orbital37 analysis (Supporting Information, Figure S1). Comparisons with results obtained by other hybrid or range-separated density functionals as well as experimental observations of two representative molecules 27273

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Figure 2. (a) Selected electron donor and acceptor molecules studied in this work: pyrene-tetraone (PyT), 3,4,9,10-perylenetetracarboxylic dianhydride (PTCDA), pentacene-tetraone (PT), coronene-hexaone (COHON), pentacene-dion (PD), dicyanomethyl-trinitrofluorenone (TNFCN), phthalocyanine (H2PC), oligothiophene-4 (OT-4), and oligothiophene-6 (OT-6). (b) Frontier molecular orbital (MO) energy for selected electron donor and acceptor molecules.

Table 1. Energy Difference between CT2 and CT1 or between CT2 and the Dipole-Allowed FE State in eV for Various D/A Dimers in Their Ground-State-Optimized Geometriesa pentacene tetracene H2PC OT-6 a

PyT

PT

COHON

TNFCN

0.31/−0.36 0.16/−0.53 0.22/0.47 0.17/−0.56

0.98/0.75 0.80/0.20 1.12/1.26 0.76/0.02

0.50/0.12 0.69/−0.31 0.71/0.49 0.47/−0.61

0.80/−0.19 0.92/−0.39 0.76/0.33 0.87/−0.55

The number before each slash corresponds to ΔCT2−CT1, and the number after each slash corresponds to ΔCT2−FE.

between ΔLUMOs and ΔCT2−CT1 implies that the relative position

6 are 1.53, 1.65, 2.31, and 0.72 eV, respectively. Therefore, in most D/A pairs, ΔLUMOs has a much smaller value than ΔHOMOs (also illustrated in Figure 2b); accordingly, a low-energy CT2 may in many cases be mainly dominated by the excitation from HOMO (D) to LUMO + 1 (A). Figure 3 illustrates the relationship between ΔLUMOs for the acceptor molecule and ΔCT2−CT1 for the 16 D/A pairs. The quasi-linear relationship

of the two CT states is mainly determined by the energy levels of the acceptor molecule; thus, ΔLUMOs can be used as a rough estimate of ΔCT2−CT1. Effect of Intermolecular Distance on the Excited-State Energy-Level Alignment. It is well known that the energy of 27274

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change much with variations in intermolecular distance, indicating again that the relative position of the two CT states is mainly determined by the energy levels of the acceptor molecule and is not very dependent on the intermolecular couplings, both of which are in agreement with what is shown in Figure 3. Effect of Excited-State Geometry Relaxation on the Excited-State Energy-Level Alignment. Organic electron donor or acceptor molecules may undergo geometric reorganization within a specified new electronic state after an optical nonradiative transition has taken place.39,40 The reevaluation of the energy-level alignment under relaxed geometries of different electronic states is therefore highly desirable. Herein, we performed geometric optimizations for all three electronic states (GS, CT1, and CT2) of two dimers and also illustrated the energy-level alignment of these states at different optimized geometries (Figure 5). We found that for both dimers, in both CT1 and CT2, the intermolecular distance is slightly shorter than that of GS. Such slightly reduced intermolecular distances, as in CT1 and CT2, can decrease the e−h separation (for which the energies of CT1 and CT2 can be lowered) and can increase the electronic couplings between neighboring molecules (for which the energy of the FE state can also be reduced). More importantly, it is evident that both CT1 and CT2 are always energetically lower than the FE state in all three optimized geometries for pentacene/PyT and tetracene/COHON dimers. This verifies that more than one CT excited state can be energetically accessible from the FE state for a D/A pair during the realistic photoinduced electrontransfer process, even with a reorganization of the molecular geometry.

Figure 3. Relationship between ΔLUMOs of the acceptor molecule and the energy difference between CT2 and CT1 (ΔCT2−CT1) for 16 donor/ acceptor pairs.

CT states depends on the e−h distance, i.e., the intermolecular distance (R) between the donor and acceptor molecules. Therefore, we further examined the R dependence of the excitation energy of the CT states and the local FE state for two D/A pairs (pentacene/PyT and tetracene/COHON), and the results are presented in Figure 4. The data illustrate that the energy of the FE state slowly decreases with increasing intermolecular distance (R), whereas the energies of CT1 and CT2 increase with R because of the weakening of e−h binding. For both dimers, CT1 and CT2 lie below the FE state when R is less than 4.5 or 5 Å. It should be noted that the equilibrium intermolecular distance (R) for the ground state (GS) is around 3.5 Å for both dimers. Therefore, the two CT excited states are energetically accessible from the dipole-allowed local FE state in some D/A dimer systems, e.g., pentacene/PyT and tetracene/COHON, around their equilibrium GS geometries (Table 1). If the dielectric screening effect of the bulk materials is further considered (given that the relative permissivity of organic donor and acceptor molecules is usually around 2−4), then one can expect an additional stabilization of the CT states with respect to the FE state, increasing both the likelihood of CT2 being lower than the FE state and the range of the donor− acceptor distance where this happens. It can also be seen in Figure 3 that the energy gap between CT1 and CT2 does not

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ROLE OF CT2 IN EXCITON DISSOCIATIONS For the purpose of evaluating the role of more CT states on the exciton dissociation efficiency in organic photovolataics, we studied both the phenomenological model illustrated in Figure 6, which should capture the essential physics of the exciton dissociation process, and a variant discussed below. We started by considering a single donor molecule, coupled to a 9 × 9 square aggregate of acceptor molecules with lattice unit L (results are similar for larger aggregates). The Frenkel exciton is initially localized on the donor, and we assumed that there are two available LUMOs on each of the acceptors, separated by the energy gap ΔLUMOs. The dissociation of the Frenkel exciton generates states that are a linear combination of charge-transfer

Figure 4. Dependence of the CT and local FE excited states on the center-to-center intermolecular distance for a pentacene/PyT dimer (a) and a tetracene/COHON dimer (b). 27275

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Figure 5. Energy-level alignments for CT1, CT2, and the FE state of the pentacene/PyT dimer (a) and tetracene/COHON dimer (b) at optimized geometries for different electronic states.

The density of states (DOS) function that is generated by eq 1 is depicted in Figure 7. It displays, expectedly, two wellseparated CT bands for large values of ΔLUMOs that approach each other until they merge when ΔLUMOs is reduced to ∼1 eV. The lowest energy of an isolated hole and electron is the zero of the energy scale. The model Hamiltonian (eq 1) does not include the effect of disorder. However, it was noted15 that plausible values of disorder do not appreciably affect the shape of the DOS function, which is dominated by the magnitude of the transfer integral and the Coulombic term (the latter typically being one order of magnitude larger than the disorder). Disorder does affect the exciton localization and dynamics (irrelevant for this model, where we consider a donor at a fixed position) as well as hole and electron dynamics following the charge dissociation. We assume that the Frenkel state is only coupled to the CT configurations via a matrix element (VDA) by transferring an electron to the closest acceptor in the 9 × 9 aggregate (either LUMO or LUMO + 1), i.e., to either |15, 1⟩ or |15, 2⟩. The perturbation that couples to the Frenkel state with the CT manifold can be represented as

Figure 6. Schematics of the model system used to study the rate of exciton dissociation.

states |kl, α⟩, where the indices k and l indicate the position of the acceptor molecule (Figure 6) and the index α indicates transfer to the LUMO (α = 1) or the LUMO + 1 (α = 2). The Hamiltonian can be represented as Ĥ CT =

∑− kl , α

+

C rkl ,DA

|kl , α⟩⟨kl , α| +

∑ ∑ αα′ {kl , k ′ l ′}

̂ = VDA(|ΨFE⟩⟨15, 1| + |ΨFE⟩⟨15, 2|) + h. c . VDA

(2)

This allows a simple evaluation of the coupling (Vm,FE) between any CT state (ΨmCT) and the Frenkel state to be represented as

∑ ΔLUMOs|kl , 2⟩⟨kl , 2|

m m Vm ,FE = VDA(⟨15, 1|ΨCT ⟩ + ⟨15, 2|ΨCT ⟩)

kl

V |kl , α⟩⟨k′l′, α′|

We set the energy of the Frenkel state at EFE = 0.1 eV, i.e., slightly above the energy of an isolated hole and electron, and the energy of the CT states as EmCT. The coupling value (Vm,FE) is larger for CT states where the electron is closer to the hole (donor). To analyze the dependence of this coupling on the energy, we can introduce the quantity W(E), represented as

(1)

The first term represents the Coulombic hole−electron attraction, where rkl,DA is the distance between the donor and acceptor molecules in units of L. The constant C is set to 0.96 eV, corresponding to a lattice distance of L = 5 Å and a relative dielectric constant of 3. The second term indicates the additional energy needed to transfer the electron to the LUMO + 1 orbital instead of the LUMO. The last term represents the coupling between the CT states. The summation {kl, k′l′} is limited to nearest-neighbor sites. We present here the results obtained with a unique coupling parameter between CT states (V = 0.1 eV) because more general models, where this coupling is between different types of CT states, yield similar results.

W (E ) =

m − E)/|VDA|2 ∑m |Vm,FE|2 δ(ECT

(3)

which is essentially the density of the CT states weighted by the strength of the coupling with the Frenkel state. This function, illustrated in Figure 7, exhibits a clear maximum at the lowerenergy edge of the two CT bands, i.e., the energy region where the CT states are more strongly bound by a shorter hole− electron distance. 27276

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where IPRm is the inverse participation ratio of the CT states14 (a measure of delocalization) and λ0 is the reorganization energy for a system formed by a single donor and a single acceptor, here set to 0.25 eV.14 The computed rates, evaluated at kBT = 0.025 eV, are reported in Figure 7b as a function of the parameter ΔLUMOs. When ΔLUMOs is large, its value becomes irrelevant, and the charge dissociation rate is similar to that of a system with only one accessible LUMO per molecule. For ΔLUMOs values