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Direct Optical Generation of Long-Range Charge-Transfer States in Organic Photovoltaics Haibo Ma* and Alessandro Troisi* One of the most debated points in the emerging field of organic photovoltaics is the mechanism of formation of free charges at the interface between a donor (D) and an acceptor (A) material from the initial exciton, which would typically be localized in the donor.[1–9] Very useful insights into the mechanism of generation of free charges can be obtained by direct optical generation of the lowest charge-transfer (CT) state (CT1) at the donor/ acceptor (D/A) interface, as illustrated remarkably by Vandewal et al.,[9] who observed that the internal quantum efficiency (IQE) is essentially independent of whether excited states with an energy higher than that of CT1 are excited or not. The direct excitation of CT states from the ground state (GS) was previously reported by several research groups.[10–13] It is generally assumed that the CT state optically formed is the one with the closest possible hole–electron (h–e) distance, i.e., a short-range CT state, and that the direct optical generation of long-range CT states is unfeasible owing to the vanishing oscillator strengths. The accurate assignment of such CT absorptions is complex due to their very low oscillator strengths, but obviously crucial for understanding the relevant optoelectronic mechanisms in organic devices. The aim of this work is therefore to find out whether long-range CT states can be generated by direct optical excitation like their short-range CT counterparts. The critical role of energetically higher CT states (or electronically “hot” CT excitons) in the photocurrent generation process has been proposed as a result of many experiments and theoretical calculations in recent years.[4–7,14–19] In such states hole and electron are usually more spatially separated and, accordingly, considered to be able to avoid the possible trapping in Coulomb well of the bound CT states during charge separations (a view not easily reconcilable with that given by Vandewal et al.[9]). Very recent ab initio modeling of excitonic and CT states in the poly(thieno[3,4-b]-thiophene benzodithiophene) (PTB1)/ [6,6]-phenyl-C61-butyric acid methyl ester (PCBM) mixture revealed that the CT states are located below the bright excitonic state and thus should be directly accessible via internal

Dr. H. Ma Key Laboratory of Mesoscopic Chemistry of MOE School of Chemistry and Chemical Engineering Institute of Theoretical and Computational Chemistry Nanjing University Nanjing 210093, China E-mail: [email protected] Prof. A. Troisi Department of Chemistry and Centre of Scientific Computing University of Warwick CV4 7AL, Coventry, United Kingdom E-mail: [email protected]

DOI: 10.1002/adma.201402294

Adv. Mater. 2014, DOI: 10.1002/adma.201402294

conversion processes.[20] On the other hand, direct optical generation of long-range CT states will have much higher quantum efficiency than the internal conversion process if it is feasible. It is widely believed that the oscillator strengths for the direct generation of long-range CT states are vanishing because of the vanishing overlap between hole and electron wavefunctions. For the same reason low-lying short-range CT states have already a much smaller oscillator strength than that of the dipoleallowed excitations to Frenkel exciton (FE) states in the donor material. Nevertheless, the total absorption intensity in a given energy range is governed not only by the oscillator strength of a single transition but also by the density of states (DOS) of the excited states. Although the oscillator strength of a single transition from the GS to an individual higher long-range CT state is much lower than that of the transition from the GS to the lower short-range CT state, the DOS of higher energy long-range CT states is conversely much larger than that of the lower bound CT states.[4,18,19] Therefore, the direct excitation of long-range CT states may be still possible but a quantitative analysis is needed. A detailed quantum chemical investigation of the electronic structures of various CT excited states at the D/A interface may help in assigning the optical absorption spectra and clarifying the possible role of long-range CT states. This clarification will be crucial to correctly interpret the experimental spectra and to construct a consistent model for the photophysical processes in organic solar cells. Herein we aim to determine whether direct optical generation of long-range CT states is possible at an organic D/A interface by using first-principles quantum chemical calculations of the excited states of several mediumto large-sized model heterojunction systems in conjunction with model Hamiltonians. We initially performed first-principles time-dependent density functional theory (TDDFT)[21,22] calculations for some medium-sized realistic molecular clusters of D/A heterojunction, before performing a study of larger systems on a simplified model. Here we chose the widely used tetracene and perylene-3,4,9,10-tetracarboxylic dianhydride (PTCDA) as the D and A molecules in the heterojunction. The structural model was built according to the experimental crystal structure for both tetracene[23] and PTCDA[24] and the intercomponent distance was chosen as 4 Å, as shown in Figure 1. Unfortunately, conventional first-principle calculations of a large number of CT states for molecular aggregates containing more than hundreds of atoms are beyond current computational capabilities due to the huge computational costs. Although some approximate approaches, including constrained density functional theory (CDFT)[25] and classical micro-electrostatic (ME) techniques,[26] were successfully applied for the energetic calculations for the D/A heterojunctions in recent years, such methods cannot be used to evaluate the absorption intensities of various

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Figure 1. a) Schematics of various CT states for D(m × n)/A(m × n) heterojunction model: localized bound CT state (1), localized long-range CT state (2), and delocalized long-range CT state (3). b) Part of the experimental crystal structure of the tetracene/PTCDA system for the TDDFT calculation.

and the 6–31G(d) Pople polarized valence double-zeta basis set. Figure 2a and Figure 2b show the h–e separation and the oscillator strength (f ) for CT states of the tetracene(4 × 3)/ PTCDA(2 × 3) system. They show that the lowest-lying CT states are short-range separated while the most of the other higher CT states are long-range separated with large h–e distances which can be even more than 20 Å. At the same time, f decreases rapidly with increasing h–e separation, i.e., values of f for long-range CT states with h–e distance larger than 15 Å can be two or more orders of magnitude smaller than those for the short-range ones, while the latter ones are already very small, usually less than 0.01; this is the reason why the direct excitation of long-range CT states was generally regarded as impossible. However, although the longrange CT states have much smaller oscillator strengths than their short-range CT counterparts, they have much higher DOS (see Figure 2c) due to the much larger number of the possible linear combinations of many different long-range charge-separated configurations. The number of long-range h–e pairs is (m × n)2 for D(m ×n)/A(m × n), which is much larger than that of bound h–e pairs that only scales as m. As a consequence, the total absorption intensity for these longrange CT states can be competitive to that for lowest-bound CT state given that the D/A heterojunction is not too small, as shown in Figure 2d, although the value of f of a single transition from the GS to a higher long-range CT state is much lower than that of the transition from the GS to the lower short-range CT state. The absorption intensities for both the short-range CT states and the long-range counterparts increase simultaneously when one enlarges the length m of D/A interface Figure 2. a) Hole-electron (h–e) separation distance versus excitation energy and b) oscillator [D(2 × 2)/A(2 × 2)→D(4 × 2)/A(2 × 2)]. This strength (f ) versus h–e separation distance for CT states of tetracene(4 × 3)/ PTCDA(2 × 3). result can be understood because increasing c) Density of states (DOS) for CT states of the realistic tetracene/PTCDA heterojunction aggrethe interface length can produce more bound gates with different sizes. d) Optical absorption for CT states of tetracene/PTCDA heterojunch–e pairing possibilities and also more tion aggregates with different sizes. For each D(m1 × n1)/A(m2 × n2) system, the lowest m1 × n1 × m2 × n2 CT excited states were calculated by using REM-TDDFT and used for the analysis. long-range CT configurations. However,

CT states due to their neglect of the couplings between different CT configurations and the consequent charge delocalization. To include such coupling terms in the electronic structure calculations, here we adopted our newly developed renormalized exciton method (REM),[27–29] which is based on the assumption that the low-lying delocalized electronic state can be approximately described by a linear combination of many local excitations or ionizations through effective couplings between them[30] and which is capable of calculating various delocalized electronic states including excitations and ionizations and incorporating correctly the screening effect with high accuracy for large molecular aggregates. Here REM calculations were implemented at the TDDFT level with the longrange-corrected exchange-correlation functional CAM-B3LYP[31]

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H = ∑ E ex | ij 〉〈ij | + ij

∑V

FE

| ij 〉〈i ′ j ′ |

same unit transition dipole moment, so ∑ ij c 2ij (the weight of the FE configurations) is used to estimate the oscillator strength of a given adiabatic state ψn. We took the general parameter values of VFE = 0.07 eV, Eex = 0.25 eV, C = 0.96 eV, and VDA = 0.08 eV for organic photovoltaics.[18] To reduce the number of parameters it is convenient to consider the case in which VD =VA =V. The parameter V, which determines the bandwidth of the valence band of the donor and the conduction band of the acceptor, is the most important parameter affecting the charge delocalization of CT excited states and their absorption, as illustrated in Figure 3a, where we compare optical absorptions of the D/A heterojunction D(8 × 8)/A(8 × 8) with different V values. Increasing values of V (upon aggregation) has the double effect of decreasing the energy of the edge of the lowest energy CT transition and increasing the intensity of higher energy CT transition. Such redshift behavior for a low-energy short-range CT band agrees well with the finding from recent experiments[32–35] that the donor or acceptor aggregation from a more amorphous system can decrease the lowest absorption energy of CT states since the value of V increases upon aggregation. When such V values are large enough as compared to VDA, like 0.06 and 0.08 eV, the absorption intensity of long-range CT states can become larger than that of low-lying short-range CT states (see Figure 3a), in accordance with the previous REM-TDDFT calculations for realistic tetracene/PTCDA system. Therefore, a long-range CT band may be detected experimentally provided that the electronic couplings between organic molecules are strong enough in highly ordered heterojunction aggregates. Indeed, the values of V computed with our REM-TDDFT calculations for realistic tetracene(4 × 3)/PTCDA(2 × 3) are 0.06 ± 0.02 eV, i.e., in the regime where long-range CT excitations are more intense. The results of the calculation of the optical absorption of the D/A heterojunction aggregates with different system sizes are

{ ij , i ′ j ′ }

C | ij , kl 〉〈ij , kl | + ∑ ∑ VA | ij , kl 〉〈ij , k ′l ′ | r ij ,kl ij ,kl ij {kl ,k ′ l ′ }

−∑ −∑

∑V

D

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the number of long-range CT configurations increases more rapidly with m and so the relative intensity of the long-range absorption increases with a growing value of m. When we enlarge the thickness n of D/A interface instead of the length m (D(4 × 2)/A(2 × 2) →D(4 × 3)/A(2 × 3)), we find that the absorption intensity of long-range CT states can be further increased but that for the bound ones is decreased. This effect is because expanding the thickness of the D/A interface without increasing the length [D(4 × 2)/A(2 × 2) →D(4 × 3)/A(2 × 3)] will only produce more long-range CT configurations but no more bound h–e pairs and the couplings between the short- and more new long-range CT configurations will decrease the DOS for low-lying bound CT states (see Figure 2c). The excitation energy for the dipoleallowed FE state at a tetracene molecule is 2.58 eV and, accordingly, most of the long-range CT excited states are still lower than these “optical-allowed” FE excitations. This result implies that these long-range CT states can be also energetically accessible via internal conversion process from the bright FE state, which is in agreement with the recent finding for the PTB1/ PCBM system by Borges et al.[20] From the above analysis of our REM-TDDFT results in Figure 2 for medium-sized tetracene/PTCDA model aggregates, the absorption intensity of long-range CT excited states can increase with the size of the model aggregated system and can be competitive with or even larger than that of bound CT excited states, given that the model D/A heterojunction system is not too small. To broaden the validity of our finding for medium-sized D/A heterjunctions, and determine the parameters affecting the intensity of long-range CT absorption, we considered a simplified model[4,18,19] for larger D/A interface clusters:

| ij , kl 〉〈i j , kl |

i

with Vξ, ξ = (FE, D, A, DA), denoting the couplings between different diabatic electronic configurations [ ij for FE localized in site (i,j) and ij , kl for a CT one with an electron transferred from a neutral donor molecule on site (i,j) to the acceptor molecule on site (k,l)]. For the interactions between Frenkel and CT excitons, i.e., the last term in the model Hamiltonian, we only considered the dominant couplings between FEs i1 at the interface and their nearest local CT configurations i1, i1 . In addition, Eex is the on-site energy for the FE 2 configuration ij , C = 4 πεe R0 is the Coulomb energy between an electron and a hole at nearest-neighboring sites, and rij ,kl is the distance between site (i,j) and (k,l) in the lattice spacing unit R0. The symbol {ij , i′ j ′} or {kl , k ′l ′} indicates that the summation is limited to the nearest-neighboring pairs of sites. After the diagonalization of Hˆ one can get the adiabatic solution Ψ n = ∑ ij c ij ij + ∑ ijkl c ijkl ij , kl where cij and cij,kl correspond to the configuration coefficients. We assume all FEs ij have the

Adv. Mater. 2014, DOI: 10.1002/adma.201402294

Absorption

kl { ij , i ′ j ′ }

+ ∑VDA | i1〉〈i1|

(a)

V=0.02 eV V=0.04 eV V=0.06 eV V=0.08 eV

′ ′

D(4× 4)/A(4 × 4) D(6 × 6)/A(6 × 6) D(7 × 7)/A(7 × 7)

(b)

D(8 × 8)/A(8× 8) D(9 × 9)/A(9× 9)

(c)

Eex=0.15 eV Eex=0.25 eV Eex=0.35 eV

-1.5

-1.0

-0.5

Energy/eV

0.0

0.5

Figure 3. a) Optical absorption of model D/A heterojunction D(8 × 8)/ A(8 × 8) by model Hamiltonian calculations with different V values. b) Optical absorption of model D/A heterojunction aggregates with different sizes by model Hamiltonian calculations with V = 0.06 eV. c) Optical absorption of model D/A heterojunction D(8 × 8)/A(8 × 8) by model Hamiltonian calculations with different Eex values. In this model, zero energy corresponds to the dissociated charges separated at infinite distance.

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reported in Figure 3b. The absorption intensity here has a very similar trend to that shown in the realistic tetracene/PTCDA system in Figure 2d: there are two main peaks, of which the low energy one corresponds to the short-range CT band and the high energy one to the long-range CT band. The intensity increases slightly for the low-lying bound CT states but increases rapidly for the long-range CT states when the model system is enlarged from D(4 × 4)/A(4 × 4) to D(8 × 8)/A(8 × 8). This result can be expected as increasing the interface width m produces more bound h–e pairing possibilities (scaling as m) but many more long-range CT configurations [scaling as (m × n)2]. The oscillator strength f decreases exponentially for long-range separations as illustrated in Figure 2, so the absorption intensity will saturate by increasing the D/A interface thickness n. When the absorption intensity is saturated with large values of n, it only depends on m and it is proportional to m, i.e., the interface length. For this reason, in Figure 3b, the absorption intensity for long-range CT states does not increase too rapidly when the model system is further enlarged from D(8 × 8)/A(8 × 8) to D(9 × 9)/A(9 × 9). To understand why the oscillator strengths of long-range CT states are non-vanishing, we further examined the energy range and the charge delocalization of these electronic states. From Figure 2d and Figure 3b such states have an excitation energy very close to local FEs, which implies that the increase of their oscillator strengths may be ascribed to the resonance with the dipole-allowed local excitations. To see how the local FE excitation energy affects the optical generation of longrange CT states, in Figure 3c we illustrated the absorption of a D(8 × 8)/A(8 × 8) model with different Eex values. Both the intensity and peak position of long-range CT band are greatly dependent on the FE excitation energy value, which verifies that the nonvanishing oscillator strengths of long-range CT states are mainly caused by the resonance with those dipoleallowed FE states. However, near-degeneracy in energy is not the only prerequisite condition for the effective resonance. In Figure 4, we illustrate the population of three representative CT states, CT3397, CT3398, and CT3469 (n in CTn denotes their rank when ordered from low to high energy) with largest oscillator strengths, of D(8 × 8)/A(8 × 8) of various h–e

Figure 4. Weight of h–e configurations with different separation distances in three representative CT states of D(8 × 8)/A(8 × 8).

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configurations with different separation distances. Another state CT3399 which has a very close excitation energy to the others but vanishing oscillator strength is also shown in Figure 4 as a comparison. All the four states are not dominated by only one single h–e paring configuration and the contributions by the short-range (rh–e less than four spacing units) ones for the three ones with large oscillator strengths are noticeable, although the largest contributions still come from longrange CT configurations. This observation means the charges for all these states are quite delocalized and many long-range CT states have tails that touch the D/A interface, as shown in the schematic in Figure 1a. Such tails, of which the intensity is determined by the coupling V, can promote the couplings of long-range CT states with dipole-allowed FE states and, accordingly, their oscillator strengths are non-vanishing. The same V parameter that controls the charge delocalization and the optical absorption of long-range CT states also has a crucial role in facilitating charge separation from a bright exciton and charge transport. [2–4,7,18,36,37] In contrast, the states like CT3399, which have nearly the same energy but no tails at the D/A interface, cannot have resonance with FE states and accordingly their oscillator strengths are still vanishing. This verifies that charge delocalization with tails touching the D/A interface is also a prerequisite for producing non-vanishing oscillator strengths for long-range CT states. Although we have shown that the absorption intensity of a long-range CT band is much stronger than that of a lower energy short-range CT band for large molecular aggregates, from the oscillator strength of CT3397, CT3398 and CT3469 in Figure 4 the absorption intensity of long-range CT band is still much lower than that of dipoleallowed FE states which are localized in donor molecules and assumed to have unit transition dipole moment. Herein we focused only on a crystalline D/A heterojunction, although such a perfect morphology might be not the case in actual devices. Obviously the disorders in realistic systems may influence the electronic structure picture significantly. Recent calculations of exciton dissociation rates in model organic photovoltaic interfaces revealed that charge-dissociation rate decreases with an increasing amount of disorder, but the exciton dissociates into partially separated hole and electrons regardless of the disorder, which suggests that the essential physics of the electronic structure is unaffected by the chemical details of the interface.[19] Therefore, our analysis in this work should be insightful for interpreting the experimental absorptions and further study of the disorder effect will be also important for more accurately quantitative predictions for actual devices. Direct optical generation of long-range CT states is feasible but this does not imply a priori that free charges will be produced directly from the dissociation of these states. It is still possible to generate electrons and holes from the dissociation of low-energy short-range CT states because the relaxation of the optical excited system may result in the population of these states. In summary, a simple but new picture for the optical generation of long-range CT states emerges from the above calculations based on both first principle theories and model Hamiltonians. The oscillator strength of a single transition from GS to one long-range CT state is very low, but non-vanishing due

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Acknowledgements The authors acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 21373109), the National Basic Research Program of China (Grant No. 2011CB808604), and the European Research Council (Grant No. 615834). Received: May 21, 2014 Revised: June 13, 2014 Published online:

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