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Do “Hot” Charge-Transfer Excitons Promote Free Carrier Generation in Organic Photovoltaics? Guangjun Nan, Xu Zhang, and Gang Lu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b04652 • Publication Date (Web): 15 Jun 2015 Downloaded from http://pubs.acs.org on June 24, 2015

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The Journal of Physical Chemistry

Do “Hot” Charge-Transfer Excitons Promote Free Carrier Generation in Organic Photovoltaics? Guangjun Nan, Xu Zhang, and Gang Lu*

Department of Physics and Astronomy, California State University Northridge, Northridge, California 91330-8268, United States

Corresponding Author *Phone: 1-818-677-2021. E-mail: [email protected]

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ABSTRACT. In organic photovoltaics, the role of hot charge-transfer (CT) excitons on free carrier generation is currently under intense debate. In this paper, we carry out first-principles time-dependent density functional theory calculations to examine hot CT dissociation in polymer/fullerene heterojunctions. We reveal that whether or not hot CT states promote charge separation depends on excitation spectral range and crystallinity of the donor and acceptor phases. We find that while the crystallinity of the donor phase underlies the energy dependence of CT exciton dissociation, the crystallinity of the acceptor determines charge separation efficiency. We propose a theory that can reconcile contradictory experimental observations and provide insight into hot CT dissociation. Crucially, the timescale of hot CT dissociation is found to be comparable to the timescale of its relaxation to the lowest-lying CT state, which is localized in all interfacial models considered here.

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I. INTRODUCTION Fundamental understanding of charge separation at donor/acceptor (D/A) interfaces in organic photovoltaics (OPVs) is crucial for rational design of high performance OPV devices.1,2 Central to this understanding is the role of “hot” charge-transfer (CT) excitons on free carrier generation.3-5 Hot CT excitons refer to nascent excited states generated by above-gap excitations with electron and hole straddled across the D/A interface. Whether or not the excess energy of hot CT excitons could be harvested to promote free carrier generation is an open question and a subject of intense debate and confusion.6,7 On one hand, numerous experiments have suggested that hot CT excitons can facilitate long-range charge separation.8-12 For example, Dimitrov et al.8 observed that an increase of photon energy of 0.2 eV above the optical band gap could double both quantum yield of free charges and internal quantum efficiency (IQE) of low-band gap polymer/fullerene blends. Similarly, Grancini et al.9 reported that the IQE of polymer/fullerene PCPDTBT/PC60BM blends increased monotonically with excitation energy. Their observations were rationalized according to the hypothesis that hot CT states are more delocalized in nature and thus more prone to ultrafast charge separation. Other groups, on the other hand, have reported opposite results.13-15 Vandewal et al.13 measured IQE for a wide range of OPVs based on polymer/fullerene, small-molecule/C60 and polymer/polymer blends, and found that IQE was essentially independent of excitation energy. Armin et al.14 have arrived at the same conclusion based on polymer-fullerene PCPDTBT/PC60BM devices. Moreover, Lee et al.16 reported an oscillatory IQE spectrum for polymer-fullerene blends, suggesting a negligible impact of hot excitons on charge separation. To understand charge separation mechanisms at D/A interfaces in OPVs, numerous theoretical simulations have also been carried out; some of which have inspired and formed the basis of the present work. For example, it has been proposed that long-range charge separation at D/A heterojunctions is

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facilitated by the formation of delocalized CT states,17 and charge separation in efficient OPVs occurs through hot CT state delocalization rather than energy-gradient-driven intermolecular hopping.18 In our paper, we also take the view that the delocalized CT states are the gateway to long-range charge separation. It has been shown that the density of states (DOS) of hot CT excitons is much higher than that of lower-energy and shorter-range counterparts.19 As a result the long-range CT states could be generated directly by the above-gap optical excitations. In our paper, we also find that high DOS of hot CT states plays a crucial role in understanding the conflicting experimental observations. The dependence of CT states on molecular packing, system size, and intermolecular interactions has been examined for pentacene-C60 model interfaces.20 In our paper, we follow the same strategy but with a different focus. Lastly, the importance of acceptor orbitals above the lowest-unoccupied-molecularorbital (LUMO), particularly LUMO+1 in exciton dissociation, has been emphasized.21 In the present paper, we corroborate the importance of these LUMO+ orbitals in long-range charge separation. The previous theoretical and experimental studies12,17,18 have shown that a more delocalized CT exciton would have a higher propensity to dissociate into free carriers. Hence, in this paper, we focus on the electron-hole (e-h) distance of CT excitons, which characterizes their spatial delocalization. We evaluate the e-h distance as a function of excitation energy and molecular stacking at D/A heterojunctions based on time-dependent density functional theory (TDDFT) calculations.22-26 The aim of this work is to provide a general theory that can reconcile the contradictory experimental observations and shed light into hot exciton dissociation in OPVs. The paper is organized as follows. Section II describes the computational models and outlines the theoretical method for computing the energy and charge density of CT states. A theoretical model to estimate the timescale of free carrier generation due to hot CT states is proposed. Section III shows the dependence of e-h distance on excitation energy and

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crystallinity of the donor and acceptor phases. A theoretical framework is proposed to reconcile the contradictory experimental observations. The conclusion is provided in Section IV.

II. COMPUTATIONAL MODELS AND THEORETICAL METHODS

Figure 1. (a) The molecular structure of T4 and T6. The grey, white and yellow circles represent C, H and S atoms, respectively. (b) The atomic structure of D/A heterojunctions studied in this work. (c) A schematic diagram of an optimized D/A interfacial structure. T4 and T6 are used to represent the donor in heterojunctions (I)-(VI) and (VII)-(VIII), respectively.

D/A Interfacial Models. In this work, eight computational models are adopted to represent D/A heterojunctions in poly3-hexylthiophene (P3HT)/fullerene blends. As shown in Fig. 1, we model the donor, P3HT, by oligothiophenes with varying conjugation lengths and the acceptor, fullerene, by C60

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molecule - a common practice in quantum mechanical simulations.17,20,27 We simulate the D/A heterojunctions of different levels of “crystallinity” by varying four following length-scales: (1) the conjugation length of oligothiophenes, from four thiophene rings (T4) to six thiophene rings (T6); (2) the π-π stacking width of the donor, from one layer of oligothiophene to four layers; (3) the stacking thickness of the acceptor in the normal direction of the D/A interface, from two C60 layers to eight C60 layers; (4) the stacking width of the acceptor parallel to the D/A interface, from one C60 layer to three C60 layers. The schematic pictures of the computational models are summarized in Fig. 1(b). Except for the model (VI) in which C60 molecules are arranged in a face-centered-cubic (fcc) lattice, all other acceptor models take a two-dimensional lattice. The experimental fcc lattice parameter28 is used in the simulations. For all the D/A interfaces studied in Fig. 1(b), atomic relaxation is performed to determine the equilibrium interfacial structure. An example of the optimized interfacial structure is shown in Fig. 1(c). The intermolecular distance between C60 molecules and the adjacent π-π stacking distance of the donor molecules are similar to the corresponding experimental values.28,29 Born-Oppenheimer molecular dynamics (BOMD) has also been carried out for these interfacial models and they are thermodynamically stable. Clearly, the computational models represent an idealization of the realistic D/A interfacial structures, which remain largely unknown. However, it is hoped that by comparisons between the various models, one can gain physical insight that would be relevant to the realistic interfaces. In this paper, “crystallinity” is loosely used to represent different levels of structural order that are amenable to quantum mechanical simulations. Computational Methods and Parameters. In this work, the ground state calculations are based on projector-augmented-wave (PAW) pseudopotentials30 and Perdew-Burke-Ernzerhof (PBE) functional31 as implemented in the Vienna Ab-initio Simulation Package.32. The semi-empirical DFT-D2 method33 is used to account for the van der Waals corrections in determining the interfacial atomic structures. The

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Gamma k-point is sampled with an energy cut-off of 400 eV in all calculations. BOMD is performed at 300 K for 600 fs to examine the thermodynamic stability of the interfacial structures. In this paper, we focus on singlet CT excitons, which dominate free carrier generation. To calculate the energy and charge density of singlet CT states, we use TDDFT method with a range-separated hybrid functional.24,25 Similar functionals have also been used by others for CT excitons.17,20 The TDDFT calculations are performed with the PAW pseudopotentials30 and PBE functional31 for the shortrange exchange-correlation interaction. The Brillouin zone is sampled at the Gamma point and the plane-wave energy cutoff is 400 eV. To determine the range-separation parameter µ from firstprinciples, we follow the procedure proposed by Stein et al.34 More specifically, µ is obtained by minimizing the following objective function:

J (µ) =

∑

i = D , A−

µ ,i ε HOMO + IPiµ ( Ni )

(1)

µ ,i where IPiµ ( N i ) = Egsi ( N i − 1; µ ) − E gsi ( N i ; µ ) , i = D, A− . Here, ε HOMO is the highest-occupied-molecular-

orbital (HOMO) energy per a specific choice of µ ; the ionization potential IPiµ ( N i ) results from the energy difference between the ground state energies of the N-1 and the N electron systems per the same

µ . The total energies and eigenvalues are calculated using the range-separated PBE functional and 631g(d,p) basis set which are available in Gaussian 09 program package.35 In this work, µ is fixed at 0.2 bohr-1 determined based on an interface between a T4 donor molecule and a C60 acceptor molecule. This value is similar to that used by Tamura et al.17 for modeling charge separation at P3HT/PCBM interfaces. More detailed description of method on charge density calculations can be found in the Appendix and references.24,25 For a given CT state with charge density ρ ( x, y, z ) where x-y plane is parallel to the D/A interface and z is normal to the interface, the e-h distance in z-direction can be

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computed as the first moment of the charge density,

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∫∫∫ ρ ( x, y, z ) zdxdydz . Note that the hole and the

electron have opposite signs of charge density. In this paper, the e-h distance is always defined in zdirection.

Figure 2. Schematic diagram for charge separation rate calculation. The solid curve represents the e-h binding energy as a function of e-h distance.

Estimate of Charge Separation Timescale. To estimate charge separation timescale due to hot CT states, we adopt the following model. We assume that charge separation takes place with the electron hopping from one acceptor molecule to the next and overcoming the e-h binding energy barrier, as shown schematically in Fig. 2. In this model, the e-h binding energy is taken as the Coulomb interaction scaled by the effective dielectric constant of 4. The charge-transfer (or hopping) rate of the electron can be estimated by the semi-classical Marcus equation:36 12  ( ∆G + λ ) 2  V2  π  k= .   exp  −  h  λ k BT  4 k T λ B  

(2)

Here, h = h 2π with h being the Planck constant; kB is the Boltzmann constant; T is the temperature, and λ is the reorganization energy of C60. ∆G and V represents the e-h binding energy difference, and

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the transfer integral (electronic coupling) between the initial and the final state for each hop, respectively. The charge separation process is complete when the e-h binding energy is comparable to the thermal n

energy at 300 K. Thus the charge separation time τ can be estimated as τ = ∑ i =1

1 where n is total ki

number of hops towards the charge separation. The reorganization energy, λ , has two contributions:37 (1) the energy difference for a neutral C60 between its equilibrium geometry and the relaxed geometry of the anion state; (2) the energy difference for the C60 anion between its equilibrium geometry and the relaxed geometry in its neutral state. We have calculated λ as 140 meV from the adiabatic potential energy surfaces of the neutral and anionic species38 using B3LYP functional and the 6-31G(d,p) basis set. The reorganization energy λ is fixed in Eq. (2) for each hop. To determine the electronic coupling V, we use the site-energy-correction method.39 A number of calculations have been carried out with one C60 molecule fixed and the other rotated around its center (the intermolecular distance remains the same). We find that the electronic coupling V ranges from 5 meV to 60 meV for different rotations. Here we select V to be 30 meV in Eq. (2) for each hop. All these quantum chemical calculations are performed with Gaussian 09 program package.35

III. RESULTS AND DISCUSSION A CT state is a superposition of intermolecular excitations from the occupied orbitals of the donor to the unoccupied orbitals of the acceptor. As it becomes apparently later, the entire CT manifold falls into different bands or groups, each specified by a pair of non-negative integers, according to energies. For example, a CT band formed by intermolecular excitations from HOMO-m band of the donor to LUMO+n band of the acceptor is denoted by (m,n). The contribution of higher energy bands, such as LUMO+1 band, of the acceptor to charge separation is thus considered.

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We have calculated the e-h distance of CT states as a function of excitation energy for various heterojunctions in Fig. 1. First, we focus on the heterojunction (III). As shown in Fig. 3(a), we find that within each CT band, the e-h distance increases monotonically in excitation energy. However, the e-h distance is not a monotonic function of excitation energy as far as the entire CT manifold is concerned. Once the excitation energy exceeds the CT band gap (EG), the e-h distance exhibits an oscillatory behavior. In Fig. 3(b), we plot the charge density distribution of four CT states as labeled in Fig. 3(a).

Figure 3. (a) The e-h distance as a function of CT energy for CT band (0,0) and (0,1) in heterojunction (III). (b) The charge density for the four CT states labeled in (a). The electron (hole) density is shown in blue and red (yellow). The value of the iso-surface is 10-4 e/Å3. (c) The e-h distance as a function of CT energy for the heterojunction (III) when the thermal fluctuation is considered. (d) A snapshot of the heterojunction (III) during BOMD stimulations which is used to calculation the e-h distance of CT states in (c).

The states i and ii are at the bottom and top of the CT band (0,0), while iii and iv are at the bottom and top of the CT band (0,1), respectively. As shown in the figure, within the same CT band, the higher energy state has a larger e-h distance; but across the bands, the higher energy state iii actually has a

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smaller e-h distance than the lower energy state ii. This is the first important result of the paper, which suggests that whether or not a hot CT state promotes long-range charge separation depends on its energy in a complicated manner. There is no simple monotonic relationship between the excess energy of a CT state and its propensity for charge separation. One cannot assume that a higher-energy CT state is always more delocalized than a lower-energy counterpart. To examine whether the results hold true when structural and energetic disorder is present, we performed BOMD simulations for the heterojunction (III) and selected a number of snapshots along the MD trajectory. For each snapshot, we examine the e-h distance as a function of CT energy and find the same dependence as discussed above. In Fig. 3(c), we show such dependence for a particular snapshot whose structure is displayed in Fig. 3(d). It is evident that the e-h distance shows the same behavior as that in Fig. 3(a) within the same CT band and across the bands. Therefore, the monotonic increase of the e-h distance within the same CT band and the precipitous drop of the e-h distance across the CT bands, as a function of the excitation energy, hold true even if the structural and energetic disorder is present. The oscillatory e-h distance of hot CT states can be understood from the following consideration of CT energy: A D ECT = ELUMO − EHOMO + ∆ ( m, n ) − 1 ( ε r ) .

(3)

A D Here ELUMO represents the center of the acceptor’s LUMO band, and EHOMO represents the center of the

A D donor’s HOMO band. Clearly, ELUMO and EHOMO are independent of the e-h distance r.

∆ ( m, n ) = ∆ Dm + ∆ An where ∆Dm denotes the energy gap between donor’s HOMO-m and HOMO band, and ∆An is the energy gap between acceptor’s LUMO+n and LUMO band. ε is the effective dielectric

constant of the interface. For the simplest case of m = n = 0 , Eq. (3) recovers the expression commonly found in literature.40 For a given CT band, ∆ ( m, n ) takes a single value, and to a good approximation is

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independent of r. Hence, ECT is proportional to − 1 r as shown in Fig. 3(a). However, if multiple CT bands are considered, ∆ ( m, n ) would take a distinct value for each band, hence the CT energy would exhibit a “band structure” as a function of e-h distance; in other words, the e-h distance would appear oscillatory as a function of CT energy.

Figure 4. The CT energy as a function of e-h distance for CT band (0,0) in blue and CT band (0,1) in red. (a)-(d) correspond to the heterojunction (I)-(IV), respectively.

It has been a long sought-after goal to establish a relationship between charge separation and interfacial morphology, in particular the crystallinity of donor and acceptor phases.41-43 In the following, we attempt to establish such correlations by examining the dependence of e-h distance of CT states on the levels of structural order in the donor and acceptor phases. In Fig. 4, we display the e-h distance as a function of CT energy for heterojunctions (I)-(IV) with the identical donor structure but increasing acceptor size in the normal direction of the D/A interface. Here, the crystallite size of the donor is represented by the dimensions of the computational model. As the crystallite (or model) size of the

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acceptor increases, the largest e-h distance rises from 1.8 nm to 7.8 nm while the average e-h distance goes up from 1.2 nm to 4.2 nm. Clearly, the acceptor size normal to the D/A interface is crucial for charge separation. As the crystallite size of the acceptor increases along the D/A interface, e.g., from the heterojunction (II) to (VI), the e-h distance of CT states also rises. Moreover, the e-h distance of the lowest-lying CT (CT1) state barely changes. In Fig. 5 we display the charge density of CT1 state as well as a hot CT state with an excess energy of 0.5 eV at the heterojunction (II) and (VI). For both heterojunctions, the electron in CT1 state is localized at the interface. The electron of the hot CT state is only slightly more removed from the interface than CT1 state for both heterojunctions.

Figure 5. Charge density of CT1 state (a) and a hot CT state (b) with an excess energy of 0.5 eV for heterojunction (II), respectively. Charge density of CT1 state (c) and the hot CT state (d) for heterojunction (VI), respectively. The electron (hole) density is shown in blue (yellow). The value of the iso-surface is 10-4 e/Å3. The nearest-neighbor distance between molecules is the same for the upper and lower panels.

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Figure 6. The e-h distance as a function of excitation energy for different CT bands. (a), (b), (c), and (d) correspond to heterojunction (III), (VII), (V) and (VIII), respectively. (e) The charge density of CT1 state and a hot CT state with an excess energy of 0.66 eV labeled in (c) for the heterojunction (V). The electron (hole) density is shown in blue (yellow). The value of the iso-surface is 10-4 e/Å3.

The dependence of e-h distance of CT states on the crystallinity of the donor phase is shown in Fig. 6 where we display the e-h distance as a function of CT energy for four interfaces (III), (V), (VII), (VIII) with the identical acceptor structure. As the conjugation length of oligothiophene increases from T4 to T6, the energy gap between (0,0) and (1,0) bands (blue curves) and that between (0,1) and (1,1) bands (red curves) decreases, as compared between Fig. 6(a) and (b).The same trend is also evident in comparison of Fig. 6(c) and (d). As the π-π stacking width of the donor changes from one layer to four layers, the CT bandwidth EW rises significantly from 0.1 eV to 0.5 eV as shown in Fig. 6(a) and (c). The same trend is also observed between Fig. 6(b) and (d). Therefore, as the crystallinity of the donor phase increases, the CT band gap decreases while the CT bandwidth increases, leading to overlaps and intercalations among multiple CT bands. As a result, at a given excitation energy, there are multiple CT states of different e-h

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distances. In realistic D/A heterojunctions of relatively high crystallinity, the conjugation length and the π-π stacking width of the donor are significantly larger than those modeled here, hence the number of CT states with the same energy (or the density of states of hot CT excitons) would be much higher than what is shown in Fig. 6(c) and (d). Since all these states of the same energy can be accessed by the same excitation, their contribution to charge separation has to be averaged, scaled by their respective oscillator strength. As the oscillator strength of a CT state depends exponentially on its e-h distance,19 the CT states with lower e-h distances would contribute more to the average e-h distance. Crucially, as shown in Fig. 6(d), the distribution of e-h distances shows little dependence on excitation energy. Apparently, the energy-dependent characteristic of individual CT bands is now “smeared” out by a high density of CT states. This is another important result suggesting that in heterojunctions with relatively higher crystallinity of the donor phase, CT exciton dissociation is nearly energy independent, thus the hot CT states would not promote free carrier generation. Equally important, we find that the delocalization of electron (or hole) is determined primarily by the crystallite size of the fullerene (or P3HT) phase. Owing to strong π-π stacking interaction between adjacent P3HT chains, the hole is often delocalized or spread over the donor layers. In contrast, the van der Waals interaction between the fullerenes is weaker, hence the electron can be localized on some fullerene molecules. This contrast can be seen in Figure 6(e) where the charge density of CT1 state and a hot CT state in heterojunction (V) is shown. While the electron is localized in CT1 state and spreads out in the hot CT state, the hole is always delocalized over the donor layers, even for CT1 state. Therefore, given the same crystallite size of the donor and acceptor phase, the electron tends to be more removed from the interface than the hole does for a hot CT state. Hence, for the pair of materials studied here, the crystallinity of the donor phase determines the energy dependence of long-range charge separation, while the crystallinity of the acceptor is crucial for the efficiency of long-range charge separation. It

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should also be noted that the charge-nuclear feedback or polaronic effect may lead to a slight localization of the hole. However, since the hole is relatively closer to the interface, the e-h distance will not be changed substantially by the polaronic effect. Finally, we try to reconcile the contradictory experimental observations with the insight gleaned from the simulations. Grancini et al.9 and Dimitrov et al.8 have observed monotonic increase of IQEs in the spectral range of 1.5 - 2.6 eV, and 1.4 - 2.1 eV, respectively. Both observations are in line with the simulation results as displayed in Fig. 6(c) where the e-h distance rises monotonically with the CT energy from 0.8 eV to 2.0 eV. Since the experimental spectral ranges are 1.1 eV and 0.7 eV, narrower than the CT band gap of 1.2 eV, the e-h distance should retain the single-band behavior, i.e., a monotonic increase in energy. Once the spectral range exceeds the band gap as in Lee et al.,16 the IQE would exhibit an oscillatory behavior as predicted by the simulations. In this case, the experimental spectral range is 1.6 eV (from 1.2 to 2.8 eV), beyond the CT band gap of 1.2 eV. Finally, there are reports of flat and energy-independent IQEs,13,14 which echo our results in Fig. 6(d). We thus conclude that in OPVs with relatively higher crystallinity in the donor phase, the IQE would display a flat and energy-independent behavior. The OPVs with relatively lower crystallinity in the donor phase, on the other hand, would show energy-dependent IQE. In both cases, the actual e-h distance is primarily determined by the crystallinity of the acceptor, as observed in experiments.41 A competing theory has been proposed to account for the energy-independent IQEs.13 The theory hypothesizes that the relaxation of hot CT states to CT1 state is much faster than long-range charge separation via these hot CT states. Thus the overall IQE does not depend on energy, but rather on the nature of CT1 state. If CT1 state is sufficiently delocalized, the IQE would be higher, and conversely, the IQE would be lower. To verify this hypothesis, we have estimated the rate of hot CT dissociation. More specifically, a hot CT state with an intermediate e-h distance of 4 nm and an excess energy of 0.27 eV is

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considered here as an example. Based on the semi-classical Marcus theory,36 we find that in ten hops the charge separation is completed, and the corresponding charge separation time is ~1 ps at 300 K, similar to the experimental and simulation results reported by others.10,17 Critically, this timescale is comparable to the relaxation timescale (~1 ps) of hot CT states to CT1 state as observed experimentally.4,9 Hence our calculations do not support the hypothesis that the relaxation timescale is much faster than that of charge separation of hot CT states. Furthermore, for all the interfacial models studied here, CT1 state is found to be localized, which was also observed by others.19,44 Although we cannot rule out the possibility that in some heterojunctions CT1 state could be sufficiently delocalized and hot CT states could relax much faster than their dissociation, we believe that our theory is more plausible, independent of the unsubstantiated hypothesis.

IV. CONCLUSIONS To summarize, we have studied interfacial charge separation of hot CT states in a number of model oligothiophene/C60 heterojunctions based on first-principles TDDFT calculations. We find that for OPVs with relatively lower crystallinity of the donor phase, the long-range charge separation depends on excitation energy. When the spectral coverage of the excitation is narrower than the relevant CT band gap, the e-h distance of the hot CT states increases monotonically in energy. Otherwise, the e-h distance would exhibit an oscillatory dependence on energy. For OPVs with relatively higher crystallinity of the donor phase, the e-h distance of the hot CT states is averaged over a large number of CT bands, which renders it nearly energy-independent. While the crystallinity of the donor phase underlies the energy dependence of long-range charge separation, the crystallinity of the acceptor is responsible for charge separation efficiency. We propose a theory that can resolve the conflicting experimental observations on

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hot CT dissociation. Finally, we find that the timescale for hot CT exciton dissociation is comparable to their relaxation to CT1 state, which is localized in general.

APPENDIX Range-Separated TDDFT Method To properly describe long-range CT excitations, the Coulomb repulsion operator 1 r is divided into a short-range and a long-range part by using the error function:26 1 erfc ( µ r ) erf ( µ r ) = + , r r r

(4)

where r is the distance between electrons and µ is a range-separation parameter. The range-separated exchange-correlation (XC) potential is given by ' VˆXC = VˆXC , DFT − VˆXLR, DFT + VˆXLR, HF ,

(5)

where VˆXC , DFT is the PBE XC potential,31 VˆXLR, DFT is the long-range part of the PBE exchange potential, and VˆXLR, HF is the long-range part of the Hartree-Fock exchange potential. To account for the long-range Coulomb screening effect, VˆXLR, HF is divided by the dielectric constant ε which is set to 4 in this work. In the TDDFT calculation, a self-consistent PBE calculation is first carried out to determine the KohnSham (KS) eigenvalues {ε i } and eigenfunctions {ϕi } , which are then used to construct the rangeseparated KS (RS-KS) Hamiltonian matrix elements as24 occ  erf ( µ r ) *  ϕ k ϕi  , Hˆ ij' = ε iδ ij − ∫ ϕ *j ( r ) VXLR, DFT  ρ ( r )  ϕi ( r ) dr − ∑ ϕ *jϕ k εr k   

18


(6)

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f ( r1 ) g ( r2 )  1  where the Coulomb inner product is defined as  f g  ≡ ∫ dr1dr2 . The RS-KS eigenvalues r1 − r2  r 

{ε } ' i

and eigenfunctions {ϕi' } can thus be computed by a direct diagonalization of the RS-KS

Hamiltonian. The excited energies and states can be subsequently determined by solving the non-Hermitian eigenvalue equations of Casida:23 A  * B

B   XI   1 0   XI   = ωI    , *  A   YI   0 -1   YI 

(7)

where ωI is the Ith excitonic energy, and the elements of matrices A and B are given by

Aijσ ,klτ = δ i ,k δ j ,lδ σ ,τ ( ε 'jσ − ε i'σ ) + K ijσ ,klτ ,

(8)

Bijσ ,klτ = Kijσ ,lkτ .

(9)

Here, i, k and j, l indicate the occupied and virtual orbitals, respectively; σ and τ are spin indices. The coupling matrix elements Kijσ ,klτ are given by

K ijσ ,klτ

 1  =  nij' σ nkl'*τ  − δστ r  

δ 2 ( E XC , DFT − E XLR, DFT ) '*  ' erf ( µ r ) '*  ' n jlτ  + ∫ nijσ nklτ dr1dr2 ,  nikσ εr δρσ(0) ( r1 ) δρτ( 0) ( r2 )  

(10)

where nij' σ ≡ ϕi'*σ ϕ 'jσ . According to the assignment ansatz of Casida,23 the many-body wavefunction of the excited state I can be written as

Φ I ≈ ∑ zI ,ij aˆ †jσ aîσ Φ 0 ,

(11)

ijσ

where zI ,ij = ( X I ,ij + YI ,ij )

ωI ; Φ 0 is the ground state many-body wavefunction taken to be the single

Slater determinant of the occupied KS orbitals; aˆ †jσ and aîσ are the creation and annihilation operator, respectively, thus aˆ †jσ aîσ excites an electron from the ith occupied KS orbital to the jth virtual KS orbital

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with spin σ . With the density operator ρˆ ( r ) = ∑ k =1 δ ( r − rk ) of an N-electron system, the charge N

density of the Ith excitonic states can be written as

ρ I (r) = Φ I ρˆ ( r ) Φ I = ρ ground ( r ) + ∑ zI*,ij zI ,ij 'φ *j ( r ) φ j ' ( r ) − ∑ z*I ,ij zI ,i ' jφi*' ( r ) φi ( r ) . i , jj '

(12)

ii ', j

Here, ρ ground is the charge density of the ground state, and the second and the third terms on the righthand-side of the above equation represent the charge density of the quasi-electron and quasi-hole, respectively. Based on the charge density, we can identify the CT states with the electron and the hole straddled across the interface.

Corresponding Author

*Phone: 1-818-677-2021. E-mail: [email protected] Notes

The authors declare no competing financial interest. Acknowledgments

This work was supported by the NSF PREM Program (DMR-1205734) and the Army Research Office (W911NF-13-1-0147).

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