Calculation from First-Principles of Golden Rule Rate Constants for

Article pubs.acs.org/JPCC

Calculation from First-Principles of Golden Rule Rate Constants for Photoinduced Subphthalocyanine/Fullerene Interfacial Charge Transfer and Recombination in Organic Photovoltaic Cells Myeong H. Lee,†,‡ Eitan Geva,*,† and Barry D. Dunietz*,‡ †

Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States Department of Chemistry, Kent State University, Kent, Ohio 44242, United States

‡

ABSTRACT: The rates of interfacial charge transfer and recombination between the donor and acceptor layers play a key role in determining the performance of organic photovoltaic cells. The time scale and mechanism of these processes are expected to be impacted by the structure of the interface. In this paper we model the kinetics of those processes within the framework of a subphthalocyanine/fullerene donor/acceptor dimer model. Two likely configurations (on-top and hollow) in which the interfacial charge transfer and recombination may occur are studied. The corresponding rate constants are calculated within the fully quantum-mechanical framework of Fermi’s golden rule. All the input parameters (excitation energies, electronic coupling coefficients, normal-mode frequencies and coordinates, and Huang−Rhys factors) are obtained from density functional theory calculations with density functionals designed to yield accurate results in the case of noncovalently bound systems and charge transfer states. Multiple π−π* and charge-transfer excited states are identified and assigned. The kinetics of photoinduced charge transfer is obtained by solving a master equation using Fermi’s golden rule rate constants for the electronic transitions between the various excited states. Our results suggest that the hollow configuration may be superior to the on-top configuration and that maximizing its prevalence may improve the performance of subphthalocyanine/fullerene-based photovoltaic cells. commonly used as electron acceptors in OPV devices.13,14 SubPC is composed of three N-fused diiminoisoindole units linked around a boron atom forming cone-shaped structure whose photophysical and photochemical properties can be widely tuned by varying the peripheral atoms.15−17 SubPC molecules were shown to function well as light-harvesting antenna units with a relatively low excitation energy of ∼2.0 eV. 15,16,18 It has also been reported that subPC/C 60 heterojunctions can exhibit an enhanced open-circuit voltage and energy conversion efficiency depending on the fabrication scheme of the devices, which presumably impacts the structure of the interface.8−10,12 The molecular resolution of the subPC/C60 interface in practical devices has not been achieved. However, two likely donor−acceptor configurations, the on-top and hollow illustrated in Figure 1, can be identified by considering the molecular geometries. A schematic of the possible relative orientations of the two molecules is provided in Figure 1a, where a molecular perspective of the two principle orientations using dimer models is provided in Figure 1b. We use these

1. INTRODUCTION Organic photovoltaic (OPV) cells have recently attracted considerable attention due to their plasticity, synthetic tunability, and low manufacturing cost.1−7 In such cells, absorption of incident light by a molecule in the donor layer creates a localized, typically π−π*, electron−hole pair (or exciton): D + hν → D*. Excitons that migrate to the donor/ acceptor (D/A) interface may transform into interfacial electrostatically bound electron−hole pairs (or the interfacial charge-transfer (CT) state): D*A → D+A−. This CT state can then undergo either charge recombination (CR), via electronic relaxation to the ground state, D+A− → DA, or evolve into a charge-separated state that consists of f ree electron and hole charge carriers, D+A− → D+ + A−. In most cases, the donor and acceptor are noncovalently bound. However, the structure of the interface is believed to play an important role in determining CT and CR rates and thereby device performance.8,9 For example, open-circuit voltage10 and short-circuit current of OPV cells have been shown to depend strongly on the deposition protocol, which is hypothesized to affect the structure of the interface.11 In this paper, we focus our attention on subphthalocyanine/ fullerene-based solar cells.8−10,12 In this case, subphthalocyanine (SubPC) plays the role of donor and fullerene (C60) plays the role of the acceptor. Fullerenes and fullerene derivatives are © 2014 American Chemical Society

Received: February 3, 2014 Revised: April 1, 2014 Published: April 10, 2014 9780

dx.doi.org/10.1021/jp501199u | J. Phys. Chem. C 2014, 118, 9780−9789

The Journal of Physical Chemistry C

Article

2. THEORY AND COMPUTATIONAL DETAILS 2.1. Electron Transfer Rate Constant via Fermi’s Golden Rule. Let a and b be the final and initial electronic states, respectively. For example, a and b can stand for an excited CT and an excited π−π* state, respectively, in the case of a photoinduced CT process. Alternatively, a and b can stand for the ground and excited CT state, respectively, in the case of CR. The electronic transition between those two states can be described by the following two-state Hamiltonian (see Figure 2) Ĥ = |a⟩Hâ ⟨a| + |b⟩(ℏωba + Ĥb)⟨b| + |a⟩Vab⟨b| + |b⟩Vba⟨a| (1)

Figure 1. (a) Schematic view of the possible subPC/C60 interfacial configurations. SubPC and C60 are represented schematically by an inverted-umbrella-like shape (black) and a ball-like shape (red), respectively. Depending on relative orientation, two different configurations are possible: (1) the on-top configuration, where one subPC molecule caps one C60 and (2) the hollow configuration where the umbrella handle part of the subPC is inserted in the space between adjacent C60 molecules. (b) Optimized geometries of subPC/C60 dimers in the hollow and on-top configurations that are used to define the molecular models.

Figure 2. Schematic diagram of the potential energy surfaces for the ground, π−π*, and CT states. Here, CT occurs in the inverted region.

Here, ωba is the energy of state b, relative to that of state a, when each is in the corresponding equilibrium geometry; Vba = V*ab is the electronic coupling coefficient (assumed to be constant under the Condon approximation), and {Ĥ a, Ĥ b} are the nuclear Hamiltonians that correspond to each of the electronic states in the diabatic representation. The initial state is assumed to be described by a density operator of the system in state b with the nuclear degree of freedom (DOF) at thermal equilibrium

models as discussed below to investigate rates of photoinduced CT through on-top and hollow interfacial orientations. In the hollow configuration, subPC is anchored between several neighboring C60 molecules, so that it is in contact with several acceptor molecules. In contrast, in the on-top configuration, subPC sits on top of a single C60 molecule, so that the three N-fused diiminoisoindole units are positioned symmetrically around it (i.e., a single subPC molecule is “capping” a neighboring C60 molecule). Assuming that these two configurations dominate the interface, we hypothesize that fabrication schemes differ with respect to the relative subpopulation of each configuration, so that the difference in performance reflects the different subpopulation of these two configurations. In a recent paper, we successfully reproduced experimentally measured CT rate constants in covalently bound molecular D/A systems, by calculating them within the framework of Fermi’s golden rule (FGR).19 Importantly, key input parameters necessary for calculating the rate constants were obtained from first-principle calculations. Our goal in the present paper is to extend this approach to the case of the noncovalently bound subPC/C60 dimer in the above two key configurations. The structure of the remainder of this paper is as follows. The theory and computational details are presented in Section 2. The results are presented and discussed in Section 3. Concluding remarks and outlook are provided in Section 4.

ρ̂(0) = |b⟩

exp( −Ĥb/kBT ) exp( −Ĥb/kBT ) ⟨b| ≡ |b⟩ ⟨b| Zb Tr[exp(−Ĥb/kBT )] (2)

We also assume that the electronic coupling, |a⟩Vab⟨b| + |b⟩Vba⟨a|, can be treated as a small perturbation, so that the transition rate constant from state b to state a can be described by FGR20 ka ← b =

|Vba|2 2

ℏ

∞

∫−∞ dteiω tF(t ) ba

(3)

Here F(t ) = Zb−1Tr {exp(−Ĥb/kBT )exp(iĤbt /ℏ)exp( −iHâ t /ℏ)} (4)

Calculating F(t) fully quantum-mechanically for a general anharmonic many-body system is not feasible. One way for bypassing this obstacle is by employing semiclassical and mixed quantum-classical methodologies.20−22 However, here we choose to pursue another strategy, which is based on assuming that the nuclear Hamiltonians are harmonic and that the normal-mode frequencies and coordinates are independent of the electronic state 9781


The Journal of Physical Chemistry C ⎡ p2̂

N

Hâ =

∑⎢ α=1

α

⎢⎣ 2

⎡ p2̂

N

Ĥb =

+

∑⎢ α=1

α

⎢⎣ 2

Article

⎤ ωα2 (xα̂ − xαa ,eq)2 ⎥ , 2 ⎥⎦ +

⎤ ωα2 (xα̂ − xαb ,eq)2 ⎥ 2 ⎥⎦

2.2. Computational Details. All the electronic structure calculations reported in this paper were performed via Q-Chem 4.037 unless otherwise noted. Density functional theory (DFT) was employed for the ground-state, time-dependent density functional theory (TD-DFT) for excited states and chargeconstrained density functional theory (C-DFT)38 for determining the CT state geometry and energy as affected by the electrostatic environment (see below). All the calculations were performed with the 6-31G* basis set. Geometry optimization of the ground and the CT states was based on the ωB97X-D dispersion-corrected functional, designed for accurately describing noncovalent interactions.39,40 The optimal geometries of the π−π* excited state were assumed to be the same as the ground state optimal geometry. TD-DFT gas-phase calculations were based on the range-separated hybrid functional of Baer− Neuhauser−Livshits (BNL),41,42 designed to accurately describe CT states by employing a tunable range-separation parameter γ.41−47 We employed the J2(γ) range-separation parameter tuning scheme.48 Finally, the electronic coupling coefficient between electronic states, Vba, was obtained via the fragment-charge difference (FCD) method.49 The effect of the solid-state environment in the OPV device was modeled by treating it as a dielectric medium when performing the geometry optimization of the ground and the CT states. The value of the dielectric constant used was 4.2, which is close to the value of this parameter in subPC (3.9)50 and C60 (4−4.5).9,51−53 To this end, we employed the conductor-like polarizable continuum model (CPCM)54 with the switching/Gaussian (SWIG) method55−57 for surface discretization. The interaction with the dielectric continuum plays a particularly important role in determining the energy of the CT states. More specifically, the energy of the solvated CT state b in its equilibrium geometry is given by

(5)

Here, {x̂α}, {p̂α}, and {ωα} are the mass-weighted coordinates, momenta, and frequencies of the corresponding modes, and {xaα,eq} and {xbα,eq} are the equilibrium geometries of the corresponding electronic states (capped symbols, Â , represent quantum mechnaical operators). It should be noted that in this case ℏωba = ΔE = Eb − Ea corresponds to the energy gap between the electronic states when the system is in the corresponding equilibrium geometries. Both Marcus theory for CT23 and the Brownian oscillator model for electronic spectroscopy24 rely on assuming that the underlying Hamiltonian is of this form. Importantly, making this assumption also makes the calculation of the f ully quantum-mechanical F(t) tractable, so that it can be obtained analytically in closed form20,21,23,25−29 N

F(t ) = exp[ ∑ {−Sα(2nα + 1) + Sα[(nα + 1)e−iωαt α=1

+ nαeiωαt ]}]

(6) −1

Here, nα = ([e − 1] ) is the normal-mode occupancy at thermal equilibrium, and ω Sα = α (xαb ,eq − xαa ,eq)2 (7) 2ℏ (ℏωα/kBT)

is the Huang−Rhys factor30 (HRF). The HRFs describe the electron−vibration coupling strength in terms of the projection along the normal modes of the displacement between the equilibrium geometries of the initial and final states. The contribution of each normal mode to F(t) is therefore weighted by the corresponding HRF. The fully quantum-mechanical FGR approach employed here is distinctly different from the commonly used classical Marcus theory.31−33 In fact, the classical Marcus theory corresponds to the high-temperature and short-time limits of the FGR expression (eq 3 with F(t) from eq 6)19 kaM← b

|V |2 = ba ℏ

⎡ E ⎤ π exp⎢ − A ⎥ kBTEr ⎣ kBT ⎦

EbSol = EbGas − EbEnv − EbRel

Here, EGas b is the gas-phase (TD-DFT) energy of state b at the ground-state geometry, and EEnv > 0 represents the stabilization b when the dimer is placed in a dielectric continuum PCM EbEnv = EbGas ,Em − Eb ,Em Gas PCM PCM ≡ [EbGas ,CT − Eb ,Ground ] − [Eb ,CT − Eb ,Ground ]

EGas b,Em

(ℏωba − Er)2 4Er

(9)

where Er is the reorganization energy, which can be given in terms of the HRFs Er =

1 2

N

N

∑ ωα2(xαb ,eq − xαa,eq)2 = ∑ ℏωαSα α=1

α=1 34−36

(12)

EPCM b,Em

where the emission energies and are defined as the difference between the CT state and the ground-state energies evaluated at the equilibrium geometry of the state b in the gasphase and in the PCM environment, respectively, and ERel b represents the additional stabilization due to the relaxation of the equilibrium geometry from that of the ground state to that of the b state, obtained via C-DFT/PCM.19 The displacement in the equilibrium geometries between the initial and final electronic states was projected onto the normalmode coordinates of the ground state using the Dushin program to obtain the HRFs.58 The normal-mode coordinates and frequencies were calculated using Gaussian 09 (Revision A.02)59 with the ωB97X-D functional and 6-31G* basis set at the equilibrium geometry of the ground state as obtained via Gaussian.

(8)

In this limit, the electronic transition corresponds to an activated process, and the activation energy, EA, is given by EA =

(11)

(10) 19

As noted by others and demonstrated recently by us, the validity of these approximations can become questionable, in particular in the inverted region, Er < ℏωba, where the enhanced overlap between nuclear wave functions can turn nuclear tunneling, as opposed to barrier crossing, into the dominant mechanism for electronic transition.

3. RESULTS AND DISCUSSION 3.1. Excited States and Huang−Rhys Factors. In this paper, we consider electronic states with excitation energies 9782



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below 3.8 eV, to match the spectral range of OPV devices. The gas-phase electronic excitation energies, their oscillator strength, and corresponding charge separation between the donor and acceptor, ΔQ = |QD − QA|, for the hollow and ontop configurations of subPC/C60 are shown in Figure 3. These

Figure 4. Detachment and attachment of charge densities for (a) hollow and (b) on-top configurations. (The molecular subPC/C60 models are the same as introduced in Figure 1b.)

It should also be noted that the gas-phase energies of the CT states (CT1, CT2, and CT3) are higher than those of the π−π* states (EX4 and EX5). However, the order is reversed when the interaction with the solid-state environment (within PCM) is accounted for (see Table 1). In the case of the on-top configuration, we find six relevant excited states, which can be classified as belonging to one of three categories: (1) bright with weak CT character; (2) dark with strong CT character from donor to acceptor; and (3) bright with strong CT character from donor to acceptor. More specifically: 1. The excited states denoted EX5 and EX6 belong to group (1). These π−π* states are characterized by relatively large oscillator strengths (0.08−0.09) and small ΔQ (0.10e). However, unlike the bright π−π* states of the hollow configuration, these excitations are seen to be delocalized, i.e., to be spread across donor and acceptor. It should also be noted that as a result these states are also significantly less bright than the π−π* states of the hollow configuration. 2. The excited states denoted dCT1 and dCT2 belong to group (2). These states are characterized by low oscillator strength (0.00−0.01) and large ΔQ (1.19e, 0.87e) and are therefore identified as dark CT states (dCT). However, it should be noted that these states have ∼50% weaker CT character than the corresponding dark CT states in the hollow configuration, which can also be traced back to the more delocalized nature of these excitations.

Figure 3. Oscillator strength and charge separation ΔQ for (a) hollow and (b) on-top.

properties are also listed in Table 1. We characterize the different states by considering their detachment−attachment electron densities that are illustrated in Figure 4 following the procedure by Head-Gordon et al.60 In the case of the hollow configuration, we find five relevant excited states, which can be classified as either (1) bright with negligible CT character and localized on the donor or (2) dark with strong CT character from donor to acceptor. More specifically: 1. The excited states denoted EX4 and EX5 are seen to be localized on the donor and to possess large oscillator strengths (0.24) and small ΔQ (0.11e, 0.03e) and can be identified as donor π−π* excitations. 2. The excited states denoted CT1, CT2, and CT3 are characterized by small oscillator strength (0.00−0.01) and large ΔQ (1.86−1.93e), essentially corresponding to transferring one electron from the donor to the acceptor, and can therefore be identified as dark CT states.

Table 1. Oscillator Strength (OS), Charge Separation (ΔQ) in Units of e, and Excitation Energy (EGas) in eV Obtained from TD-DFT Calculation in the Gas Phasea hollow OS ΔQ EGas ESol

on-top

CT1

CT2

CT3

EX4

EX5

dCT1

dCT2

bCT3

bCT4

EX5

EX6

0.01 1.86 2.97 1.87

0.01 1.87 2.99 1.89

0.00 1.93 3.11 2.00

0.24 0.11 2.55 (2.55)

0.24 0.03 2.62 (2.62)

0.00 1.19 2.40 2.07

0.01 0.87 2.32 2.11

0.08 0.83 2.51 2.29

0.09 0.93 2.64 2.40

0.09 0.10 2.55 (2.55)

0.08 0.10 2.61 (2.61)

The stabilized CT state energy in the presence of a dielectric medium at its equilibrium geometry (ESol) is also shown in eV. (For the π−π* states with negligible ΔQ the gas-phase vertical excitation energy EGas is shown in parentheses instead.) a

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a few low-frequency (below 50 cm−1 for the hollow configuration and below 77 cm−1 for the on-top configuration) normal modes with distinctively large HRFs. As expected, these normal modes correspond to collective motions of the donor relative to the acceptor and reflect deformations due to charge separation. The reorganization energy, Er, obtained from HRFs using eq 10 is shown in Table 2 for each electronic transition. As expected, higher HRFs and thereby Er correlate with larger values of ΔQ. As a result, the HRFs and reorganization energies in the hollow configuration are larger than those in the on-top configuration. 3.2. Charge Transfer Rate Constants. We consider a scenario where photoexcitation to bright excited states (i.e., states with large oscillator strength) is followed by nonradiative transitions to lower-lying dark excited states. Bright states with CT character (e.g., states bCT3 and bCT4 in the on-top configuration) correspond to charge separation that occurs instantaneously upon photon absorption. Otherwise, the CT process follows radiationless transitions from bright excited states with small ΔQ to dark excited states with larger ΔQ on time scales described by the calculated rate constants. The FGR and classical Marcus rate constants for the bright to dark state transitions are given in Table 2. Also shown in this table are key parameters that impact these rate constants, namely, the electronic coupling coefficients Vba, reorganization energies Er, transition energies ΔE, and activation energies EA. It should be noted that the rate constants for several nonradiative electronic transitions ((bCT3 → dCT2), (bCT4 → dCT2), (bCT3 → bCT4), and (bCT4 → bCT3)) in the on-top configuration are negligibly slow and therefore are not included in Table 2. For these transitions the small rate constants are due to the similar value of ΔQ in the initial and final states and therefore the vanishing structural displacement as reflected by very small HRFs and reorganization energies. It should also be noted that Er < ΔE, except for the EX5 → bCT4 transition in the on-top configuration. This implies that CT takes place in the inverted region.34−36 In the case of the EX5 → bCT4 transition in the on-top configuration, Er is only slightly larger than ΔE. As a result, the electronic transition occurs in the normal region with an extremely small activation energy. While the average FGR rate constants in both configurations are similar, they are more broadly distributed in the hollow configuration (1.8 × 1010−2.7 × 1013 s−1) than in the on-top configuration (9.6 × 1010−2.7 × 1012 s−1). The fact that the classical Marcus rate constants for most transitions deviate from the FGR rate constants by no more than 1 order of magnitude implies that the assumptions underlying the classical Marcus limit are valid for these transitions, where a relatively small energetic barrier (EA) persists between the two electronic states. The only exception is observed for the (bCT4 → dCT1) transition in the on-top configuration, where the FGR rate constant is observed to be 5 orders of magnitude faster than the corresponding classical Marcus limit. This observation can be traced back to the fact that the amount of charge transferred in this case is rather small (ΔQ is 0.93 and 1.19 for the two states, respectively). As a result, Er ≪ ΔE for this transition and EA ≈ 15kBT, i.e., larger in comparison to other transitions. Under these conditions, nuclear tunneling can become more favorable than classical barrier crossing as the dominant transition mechanism, thereby leading to a FGR rate constant which is

3. The excited states denoted bCT3 and bCT4 belong to group (3). These states are characterized by oscillator strengths which are comparable to those of group (1) states (0.08−0.09) but also values of ΔQ which are comparable to those of group (2) states (0.83e, 0.93e). Those states can therefore be characterized as bright CT (bCT) states. However, it should be noted that like group (1) states in the on-top configuration they are not as bright as group (1) bright states in the hollow configuration and that like group (2) states in the on-top configuration they also have weaker CT character in comparison to group (2) dark CT states in the hollow configuration. Both trends can be traced back to the more delocalized nature of the excitations in the on-top configuration. It should also be noted that the ordering of the gas-phase energies of the three groups in the on-top configuration is (1) > (3) ≃ (2), which implies that the dCT states are more stable than the bright π−π* (EX) states. This trend is opposite to that observed in the hollow configuration where the gas-phase energies of the dark CT states were observed to be higher than the bright π−π* states. This difference can be traced back to a larger electron−hole binding energy in the on-top configuration, due to the proximity between the electron and the hole. As expected, this trend becomes even stronger when the interaction with the solid-state environment (within PCM) is accounted for (see Table 1). In Figure 5, we show the HRFs that correspond to the transition between the EX states and the CT state in the hollow configuration (ΔQ = 1.9e) and the dCT1 state in the on-top configuration (ΔQ = 1.2e). The HRF spectra are dominated by

Figure 5. Huang−Rhys factors for (a) hollow and (b) on-top configurations. The displacement is obtained between the ground state and the CT state geometries with charge constraint ΔQ = 1.9 for hollow and ΔQ = 1.2 for on-top configurations. 9784



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Table 2. Electronic Coupling (Vba), Reorganization Energy (Er), Transition Energy (ΔE), Activation Energy (EA) in eV, and Rate Constants in s−1 by Fermi’s Golden Rule (kFGR) and Classical Marcus theory (kM) for the (b → a) Electronic Transitions for Hollow and on-Top Configuration of SubPC/C60 b→a

|Vba|

EX4 → CT1 EX4 → CT2 EX4 → CT3 EX5 → CT1 EX5 → CT2 EX5 → CT3 EX5 → dCT1 EX5 → dCT2 EX6 → dCT1 EX6 → dCT2 EX5 → bCT3 EX5 → bCT4 EX6 → bCT3 EX6 → bCT4 bCT3 → dCT1 bCT4 → dCT1

hollow

on-top

0.054 0.076 0.032 0.023 0.003 0.009 0.011 0.034 0.020 0.015 0.005 0.005 0.012 0.004 0.012 0.050

ΔE

Er 0.352 0.352 0.352 0.352 0.352 0.352 0.205 0.146 0.205 0.146 0.146 0.161 0.146 0.161 0.098 0.051

0.687 0.669 0.551 0.756 0.738 0.620 0.479 0.446 0.531 0.498 0.259 0.150 0.311 0.202 0.220 0.329

Table 3. Electronic Coupling (Vb→a), Reorganization Energy (Er), Transition Energy (ΔE), Activation Energy (EA) in eV, and Rate Constants in s−1 by Fermi’s Golden Rule (kFGR) and Classical Marcus Theory (kM) for the Charge Recombination Reaction from the CT State to Ground State for Hollow Configurationa b

|Vba|

Er

ΔE

EA

kFGR

kM

0.047 0.148 0.008

0.352 0.352 0.352

1.867 1.885 2.003

1.632 1.670 1.938

2.4 × 107 1.9 × 108 1.2 × 105

5.6 × 10−15 1.2 × 10−14 9.3 × 10−22

kFGR

0.080 0.072 0.028 0.116 0.106 0.051 0.092 0.155 0.130 0.213 0.022 0.0002 0.047 0.003 0.038 0.376

× × × × × × × × × × × × × × × ×

1.2 2.7 9.5 1.4 1.8 5.3 3.8 1.3 7.5 9.6 4.4 7.9 1.2 5.0 2.1 2.7

kM 13

10 1013 1012 1012 1010 1011 1011 1012 1011 1010 1011 1011 1012 1011 1012 1012

3.6 9.7 9.4 1.6 2.7 3.1 1.1 1.1 9.0 2.1 5.0 8.5 1.0 6.1 1.7 6.5

× × × × × × × × × × × × × × × ×

1012 1012 1012 1011 109 1011 1011 1011 1010 109 1011 1011 1012 1011 1012 107

traced back to the larger value of ΔE for the CR compared to the CT process, which leads to slower CR rate constants in the inverted region, while the Vba is comparable for the CT and CR processes. Much slower CR rate constants for the on-top compared to the hollow configuration are mostly associated with the smaller HRFs for the on-top configuration, which leads to more rapid decrease of the rate constant as the ΔE increases. 3.4. Charge Transfer Kinetics. The overall CT kinetics involve transitions between multiple bright and dark states. Assuming that each of these transitions can be described by an FGR rate constant, as described above, one can describe the overall kinetics in terms of a master equation of the form

significantly larger than the corresponding classical Marcus rate constant. 3.3. Recombination Rate Constants. In an OPV device, the electrostatically bound electron−hole pair that constitutes the interfacial CT state can either dissociate into the desirable free electron and hole charge carriers or recombine by electronic relaxation to the ground state. The latter is a loss mechanism, whose time scale can play an important role in determining device performance. The FGR and classical Marcus CR rate constants for the hollow configuration of subPC/C60 are shown in Table 3. Also

CT1 CT2 CT3

EA

Pi(̇ t ) =

∑ [−kjiPi(t ) + kijPj(t )] (13)

j≠i

Here Pi(t) is the occupancy of excited state i at time t, such that ∑iPi(t) = 1, and kji is the FGR rate constant for the electronic transition from state i to state j (see Table 2). Since the CR is much slower than CT in this case, we do not include transitions from the excited states to the ground state in the master equation. We also assume that the initial state corresponds to photoexcitation at time t = 0 from the ground electronic state and is therefore given by

a

The on-top configuration CR rate constants are found to be less than 1 s−1 (not shown).

Pi(t = 0) =

shown in this table are key parameters that play a role in determining these rate constants, namely, the electronic coupling coefficients Vba, reorganization energies Er, transition energies ΔE, and activation energies EA. The classical Marcus CR rate constants are seen to be vanishingly small in comparison to the FGR CR rate constants, which can be traced back to the fact that CR occurs in the deep inverted region (ΔE ≫ Er), where nuclear tunneling is favored over barrier crossing. The FGR rate constants in this case suggest the existence of a rather wide 10 ns−10 μs time window for the interfacial CT state to dissociate into free charge carriers before CR takes place. Similar calculations in the case of the on-top configuration yielded significantly slower FGR CR rate constants (kFGR < 1 s−1) than the FGR CT rate constants. This behavior can be

|OSi | ∑j |OSj |

(14)

where OSi is the oscillator strength of excited state i. Namely, we impose direct photoexcitation of the interfacial dimer, which guarantees that the initial state is dominated by bright states. Figure 6a shows occupancies of the individual excited states as a function of time in the case of the hollow configuration. As expected, the initial state is a ∼50/50 mixture of the bright EX4 and EX5 π−π* states. The fact that kj,EX4 = 9.5 × 1012−2.7 × 1013 s−1 ≫ kj,EX5 = 1.8 × 1010−1.4 × 1012 s−1 implies that the occupancy of state EX4 is depleted much faster than that of state EX5. Similarly, transitions to different CT states occur selectively from different π−π* states and on different time scales. For example, while the π−π* state EX4 is coupled most strongly to the CT2 state, the π−π* state EX5 is coupled most 9785



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Figure 6. (a) Occupancy of individual excited state as a function of time for hollow configuration. (b) Total occupancy for the π−π* states and the CT states for hollow configuration. Inset shows the shorter time scale.

Figure 7. (a) Occupancy of individual excited states for on-top configuration. (b) Total occupancy for the π−π* states (EX5 and EX6) and for the CT states (dCT1, dCT2, bCT3, and bCT4). Inset shows the total occupancy for the bright (EX5, EX6, bCT3, and bCT4) and dark (dCT1 and dCT2) states.

initial total π−π* occupancy in this case is less than one, and the initial CT occupancy is larger than zero. The inset in Figure 7b shows the total occupancies of bright (EX5, EX6, bCT3, and bCT4) and dark (dCT1 and dCT2) states. The overall time scale on which the occupancy of bright states depletes and that on which the occupancy of CT states increases is very similar to that observed in the hollow configuration, with 90% of the initial total bright state occupancy depleted after 1.15 ps and 90% of the initial total π−π* occupancy after ∼0.80 ps. The total occupancies of the π−π* and CT states in both configurations are compared in Figure 8. The inset shows the total occupancies of the bright and dark states. The overall temporal behavior is rather similar despite the differences in details of the underlying mechanisms.

weakly to the CT2 state. As a result, CT2 state occupancy rises quickly after photoexcitation but remains essentially constant after that at a value that corresponds to the branching ratio, PEX4(0)kCT2,EX4/(kCT1,EX4 + kCT2,EX4 + kCT3,EX4). The latter should not be confused with the equilibrium occupancy of the CT2 state, which will only be obtained on a much slower time scale. The total occupancies of the π−π* states and of the CT states are plotted in Figure 6b. While nonexponential, CT is observed to occur on the subpicosecond time scale, with 90% of the charge transferred within 0.8 ps. Figure 7a shows occupancies of the individual excited states as a function of time in the case of the on-top configuration. The initial state is a mixture of the bright states, namely, states EX5, EX6, bCT3, and bCT4. Unlike the hollow configuration, the initial states already have some CT character due to the nonzero occupancy of the states bCT3 and bCT4. The two dCT states start out with zero occupancy and reach their steady-state occupancies after ∼1−2 ps. The steady-state occupancy of state dCT1 is much larger than that of the dCT2 state, as there are more pathways leading to state dCT1 than there are to state dCT2 (see Table 2). The occupancies of states bCT3 and bCT4 exhibit a slower initial depletion in comparison to states EX5 and EX6. This can be traced back to the CT character of states bCT3 and bCT4, which give rise to smaller HRFs when coupled with the dCT states. The overall π−π* (i.e., the sum of occupancies of EX5 and EX6) and CT (i.e., the sum of occupancies of dCT1, dCT2, bCT3, and bCT4) state occupancies in the on-top configuration are shown in Figure 7b. Since some of the CT states (bCT3, bCT4) have both large oscillator strength and large ΔQ, the

4. CONCLUSIONS Photoinduced CT and CR kinetics in a noncovalently bounded subPC/C60 dimer (embedded in a dielectric continuum to capture the effect of the solid-state environment) was modeled from first-principles within the framework of FGR for two representative interfacial configurations (on-top and hollow). All the input parameters (electronic coupling coefficients, normal-mode frequencies, HRFs, and transition energies) were calculated via DFT, TD-DFT, and C-DFT using state-of-the-art density functionals. Multiple accessible excited states were identified and assigned as π−π*, bright CT, and dark CT states. A multistate master equation with FGR rate constants was used to describe the overall CT kinetics. The detailed molecular picture underlying photoinduced CT and CR was found to be rather sensitive to the interfacial 9786



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that in our model result with initial charge transfer upon excitation. However, this effect is unlikely to play a significant role in real devices where the photoexcitations take place away from the interface. Thus, we conclude that the hollow configuration is likely to be more conducive for producing free charge carriers in a real device. Finally, we find that CR occurs on a much slower time scale than CT, which results from the CR following the deep inverted region with large ΔE. The molecular process in the farinverted regime depends on nuclear tunneling that can be addressed by the FGR scheme but is effectively prohibited in the classical Marcus theory. We also observe that CR in the hollow configuration is significantly faster than in the on-top configuration. This suggests that the on-top configuration has an advantage over the hollow configuration by providing a wider time window for the bound electron−hole pair to dissociate into free charge carriers. However, resolving accurately the time scale and mechanism of the dissociation process of the electron−hole pair into free charge carriers61−63 would be necessary for the complete molecular picture underlying the functionality of organic OPV cells and to establish the effect of the indicated differences on CR rates. Work on this direction is underway and will be reported in a future paper.

Figure 8. Total occupancy of the π−π* and CT states for hollow (H) and on-top (T) configurations. For on-top configuration the total occupancy for the bright (EX5, EX6, bCT3, and bCT4) and dark (dCT1 and dCT2) states is also shown in the inset.

configuration. While the well-defined bright donor-localized π−π* and dark CT states were found in the case of the hollow configuration, a more complex molecular picture was revealed in the case of the on-top configuration, where both π−π* and CT states were found to be more delocalized across the donor and the acceptor. As a result, the distinction between CT and non-CT states and between bright and dark states becomes less well-defined in the on-top configuration with π−π* states that are not as bright and CT states which are not as dark and which feature weaker CT character than in the hollow configuration. The FGR rate constants between the various electronic states were also found to be sensitive to the interfacial configuration. The time scale for CR was found to be significantly smaller in the hollow configuration. At the same time, the overall time scale for photoinduced CT between the donor and acceptor was found to occur on the ∼1 ps time scale for both interfacial configurations. The differences between the two configurations can be rationalized by considering the larger contact area between subPC and C60 in the on-top configuration. A larger contact area can explain the more delocalized nature of the excited states in the on-top configuration as well as its weaker CT character. The smaller HRFs can be explained by the smaller partial charges as well as by the fact that the tight fit limits the amount of deformation afforded by CT. The range of FGR rate constants is also dependent on the configuration (1.8 × 1010−1.4 × 1013 s−1 for the hollow configuration and 9.6 × 1010−2.7 × 1012 s−1 for the on-top configuration). For most of the electronic transitions the activation energy was small enough for the classical Marcus theory to be a reasonable approximation for the FGR CT rate constants. Interestingly, although one might expect that the ontop configuration with the larger contact area would feature faster rate constants, we find that the electronic coupling is comparable for both configurations (largest coupling was 0.076 and 0.050 eV for hollow and on-top, respectively) and that the rate constants for the hollow configuration are actually somewhat faster. The amount of CT is also larger for the hollow configuration (0.97e compared to 0.60e). Another advantage of the hollow configuration has to do with the fact that one subPC molecule is in contact with several neighboring C60 molecules as opposed to just one C60 molecule in the ontop configuration. The advantage offered by the on-top configuration stems from the presence of bright CT states

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Phone: +1 734 7638012. Fax: +1 734 6474865. *E-mail: [email protected]. Phone: +1 330 6722032. Fax: +1 330 6723816. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is pursued as part of the Center for Solar and Thermal Energy Conversion, an Energy Frontier Research Center funded by the US Department of Energy Office of Science, Office of Basic Energy Sciences under 390 Award No. DE-SC0000957. We are grateful to Prof. Jeffrey Reimers for sharing his Dushin code. We thank Dr. Shaohui Zheng for fruitful discussions and thank Dr. Hossein Hashemi and Prof. John Kieffer for sharing unpublished simulation results regarding the interface structure of subPC/C60. This work was supported in part by an allocation of computing time from the Ohio Supercomputer Center.

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Calculation from First-Principles of Golden Rule Rate Constants for

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