Atomistic Approach To Simulate Processes Relevant for the

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Atomistic Approach To Simulate Processes Relevant for the Efficiencies of Organic Solar Cells as a Function of Molecular Properties. II. Kinetic Aspects Charlotte Brückner,† Frank Würthner,‡ Klaus Meerholz,§ and Bernd Engels*,† †

Institut für Theoretische Chemie, Universität Würzburg, Emil-Fischer-Straße 42, 97074 Würzburg, Germany Institut für Organische Chemie, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany § Department Chemie, Universität zu Köln, Luxemburgerstr. 116, 50939 Köln, Germany ‡

S Supporting Information *

ABSTRACT: The individual steps of the light-to-energy conversion process in the vicinity of the interfaces of organic solar cells are investigated with kinetic Monte Carlo simulations employing Marcus hopping rates obtained from quantum-chemical calculations. A chemically diverse set of ptype semiconducting molecules in heterojunction with fullerene C60 is used. Starting with exciton diffusion, exciton dissociation, charge generation, and charge separation are modeled on an atomistic level. Numerous aspects were already analyzed, but comprehensive simulations including all three processes in amorphous model interface systems and a comparison of various different molecular p-type semiconductors seem to be missing. Our investigation identifies several important kinetic effects that could limit device efficiencies, such as the strong reduction of charge transport rates in the vicinity of the interface due to Coulomb interactions between the charges, the importance of adjusting the relative rates of exciton transfer and dissociation, and the impact of morphology. Charge drift velocities and hole mobilities obtained from the simulations compare well with experimental values indicating that the main effects are covered by the simulations. A correlation between experimental short-circuit currents and simulated charge drift velocities suggests that slow charge-transfer processes could represent a major efficiency-limiting parameter in organic solar cells.

I

diminished. Finally the charges are recollected at the electrodes (Step 5: charge recollection).6 Intense research in the field has brought about a plethora of polymers7 and small organic molecules8,9 for OSCs, all with advantages and drawbacks. In order to further optimize device efficiencies, structure−property relationships would be extremely helpful to correlate molecular and aggregate properties10 with the efficiencies of the individual steps of the light-toenergy conversion process. Simulations of these processes are a field of intense research so that a complete review of previous investigations is beyond the scope of this work. Hence we can only focus on some examples. Further information can be taken from reviews on charge transport,11 on exciton transport,12 and on the charge-transfer and recombination processes.13 Exciton and charge transport in the disordered semiconducting layers are usually considered to be incoherent; i.e., they are viewed as successive individual hopping processes of excitons and charges between localized states.14−16 The

n recent years, organic solar cells (OSCs) have attracted much research interest, and promising device efficiencies were observed, especially for the bulk heterojunction (BHJ) cell architecture where p-type and n-type semiconducting layers are intermixed.1,2 Assuming the so-called “cold exciton breakup”,3 the light-to-energy conversion in these OSCs can be described as a five-step process.4,5 By light absorption, an exciton is created in one of the semiconducting layers (Step 1: light absorption) and diffuses within the respective bulk phases (Step 2: exciton diffusion/transport). If it reaches the interface between the p-type and the n-type semiconductor, it dissociates into a charge-transfer state across the interface (Step 3: photoinduced charge transfer). At first, depending on the exact energies, the charge-transfer state can still be bound due to significant Coulomb attraction between the geminately formed electron and hole. Despite this Coulomb attraction, the electron and the hole can overcome their mutual binding energy so that they migrate independently through the respective semiconducting layer (Step 4: hole/electron separation and transport). This step can be further subdivided into charge separation taking place near the interface and charge transport occurring as soon as the mutual attraction has © XXXX American Chemical Society

Received: November 13, 2016 Revised: December 13, 2016 Published: December 19, 2016 A

DOI: 10.1021/acs.jpcc.6b11340 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Table 1. Color Code Used To Indicate the Positions of the p-Type Semiconductors and the Fullerenes (Corresponding to the Grid of the KMC Simulations) and the Exciton Couplings and Ratesa

a

The same color code is used for corresponding charge couplings and rates.

associated transport properties such as hole and electron mobilities or exciton diffusion lengths in disordered organic semiconductors are usually calculated with kinetic Monte Carlo simulations (KMC)17 employing Marcus theory for the underlying rates.15,18−21 KMC approaches have also been widely used to simulate mobilities and their field and temperature dependence within the Bäs sler disorder model22−25 and other phenomenological approaches, which use additional experimental data as input parameters and are not based on Marcus rates.26,22 However, as pointed out by Andrienko et al., only ab initio calculations of all input parameters, i.e., couplings and reorganization energies, without using additional fitting parameters, provide clear-cut relationships between molecular parameters and resulting transport properties.11 KMC simulations combined with first-principle calculations of couplings were conducted by Li et al. for mobilities in disordered films of anthracene derivatives27 and by Nelson et al. for electron mobilities in fullerene C60-based transistors.28

Transport parameters were also obtained from simulations of discotic mesophases such as phthalocyanines29 and perylene derivatives.30 Koster et al. modified existing KMC approaches to efficiently apply them to disordered semiconductors where no periodicity can be exploited.31 Hole mobilities in tris(8hydroxyquinolinato)aluminum were investigated by various groups. Andrienko et al.32,33 and Nelson et al.34 employed KMC-based approaches to compute hole mobilities. Van Voorhis et al. highlighted the influence of disorder on these hole mobilities in tris(8-hydroxyquinolinato)aluminum using a combination of constrained DFT (density functional theory) calculations35 of charge transport couplings36 and QM/MM (quantum mechanics/molecular mechanics) simulations.37 The impact of disorder on charge transport properties was also tackled by Andrienko et al.38−40 and by Gennett et al.41 for several compounds. Exciton diffusion 42 was also simulated using KMC schemes.43−46 For instance, Köse et al. determined exciton diffusion lengths in poly(3-hexylthiophene) (P3HT) employing B


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Figure 1. Variety of p-type molecular semiconductors, which we employed as model systems for our calculations in heterojunction with fullerene.

pentacene:C60 interface and identified different influencing factors of the morphology on the charge-transfer and recombination rate, respectively.61 Another study on chargetransfer and recombination rates on dimers composed of an αsexithiophene donor and either C60 or PBI (perylene-3,4:9,10tetracarboxylic acid bisimide) as the acceptor rationalized that the lower device performances observed for OSCs based on PBI acceptors could be related to the faster charge recombination rate.62 The strong influence of the interfacial morphology5 on the charge-transfer and charge-separated state was demonstrated by Janssen et al.63 and by Pinto.64 Van Voorhis et al. studied photoinduced charge-transfer couplings in a heterodimer composed of a zinc phthalocyanine donor and a PTCBI (3,4,9,10-perylenetetracarboxylic-bisbenzimidazole) acceptor using constrained DFT.36 Troisi and Liu et al. emphasized the importance of the influence of conformational rearrangements at the interfaces on the respective rates.65 They computed absolute rates of charge recombination and charge separation for the P3HT:PCBM system as well and provided a detailed survey of influencing parameters.66 Voityuk et al. presented a benchmark study on the performance of DFT in the prediction of charge-transfer rates at heterojunctions to fullerene C60. A comprehensive comparison of excitations of several (polymeric) donor−PCBM pairs conducted by Geerlings et al. indicated that high experimental OSC performances are related to a low oscillator strength of the

KMC simulations based on calculated coupling parameters and energy landscapes extracted from experimental measurements.47,48 This system was also investigated by Lu et al.49 Beljonne et al. analyzed the impact of morphological order on exciton transport in organic oligomers and concluded that exciton couplings decrease while the spectral overlap increases with increasing temperature.50 It was also found that rates obtained from the spectral overlap approach reduce more strongly at higher temperature than Marcus rates using only one effective mode.50 Van Voorhis et al. combined quantummechanical calculations for Marcus transfer rates with KMC simulations to study triplet exciton diffusion in organic thin films.51 In the high-temperature regime, exciton diffusion lengths obtained from KMC simulations based on Marcus rates agree nicely with experimental values from organic crystals.52 Fewer studies exist on the photoinduced charge-transfer step at the interfaces of OSCs, which is crucial for high device efficiencies.53 Its adequate description is more difficult, especially since its mechanism is still under debate and presumably depends on the involved molecules, the incident radiation, and other factors.3,54−58 However, if the electron transfer is assumed to occur from a vibrationally relaxed S1 state of the donor to the ground state of the acceptor (cold exciton breakup),59 Marcus theory still applies.60 A number of systems were addressed by Brédas at al. They calculated charge-transfer and recombination rates for different heterodimers at the C


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Figure 2. Approach for an analysis of the interfacial kinetics using diverse molecular p-type semiconductors in heterojunction with fullerene C60.

charge-transfer state.67 Yi et al.68 and Beljonne et al.69 investigated the role of hot and/or delocalized charge-transfer states in the charge-transfer and recombination step. The previous paragraphs clearly illustrate the interest in the subject. However, to the best of our knowledge, no comprehensive atomistic simulation of the complete threestep process from exciton diffusion over charge transfer and recombination to charge separation has yet been described (for macroscopic simulations see ref 70). Furthermore, while many studies address the influence of different mutual crystallographic orientations of p- and n-type semiconductors at the interface,71,72 implications of different amorphous morphologies on optoelectronic processes have been critically analyzed only in some investigations. Very recently, Volpi et al. thoroughly analyzed the charge separation process in the anthracene:fullerene model system for different BHJ morphologies. Their investigation focuses especially on the influence of the strength and orientation on the electric field. Due to its potentially limited influence, they did not include exciton diffusion.73 In this context, we model also the exciton diffusion in the ptype semiconductor and calculate efficiencies for exciton dissociation at the interface. Additionally, our simulations comprise the subsequent charge separation up to the limit

where the interactions between the charges have considerably decreased. Recombination from the interfacial charge-transfer state (see Table 1), fluorescence, and known exciton trapping mechanisms were included as loss mechanisms in our simulations. Pure charge transport effects are not taken into account because they require much larger model devices which we could not handle due to computer limitations. Since motions of holes and electrons are independent for these processes, other simulation strategies could be used.74,75,16 Moreover, the choice of our approach of hopping between individual sites implies that we cannot account for exciton dissociation mechanisms involving exciton delocalization across oligomeric aggregates76,77 or vibrationally excited states.55 Table 1 summarizes all simulated processes and displays the symbols used in this paper. To emphasize the relationship between exciton transport and charge transport processes in the p-type semiconducting phase, both types of couplings/rates are indicated in red. To reveal possible structure−property relationships for ptype semiconductors, and in contrast to most investigations in the literature focusing on a single compound, we simulate the processes for a chemically diverse set of p-type semiconducting molecules (Figure 1). In all simulations, we use fullerene C60 as the n-type counterpart because it is commonly used in D


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The Journal of Physical Chemistry C experimental devices.78 As already mentioned, for a more profound understanding of the influence of the morphology, we conduct the simulations with different amorphous morphologies for each interface composed of p-type semiconducting molecules and fullerenes C60. Three different morphologies were taken from a previous complementary investigation79 that focused on the thermodynamic aspects of the involved processes. In this previous investigation, model devices of the organic donor:acceptor interfaces were generated in a multistep approach using molecular dynamics (MD) that mimics experimental layer deposition techniques. We started with a monolayer from the crystal structure of a given p-type semiconductor and created some vacancies to increase the flexibility of the system. In several consecutive MD steps, we simulated the deposition of additional layers on top of the first. To take adhesion forces into account, we applied a harmonic potential. We repeated the procedure for the three basic crystallographic orientations of the crystal structure of the underlying p-type semiconductor. This allowed for a more complete sampling of conformational space and yielded three different model systems per donor−acceptor combination. Using these model interfacial systems, we calculated in a second step in our previous work the energetics of the relevant excitonic, polaronic, and charge-transfer states to identify important energetic loss mechanisms and to understand how they depend on the morphology and on the electronic properties of the molecules. Subsequently, we analyzed which processes are in principle feasible; e.g., the possibility of cold exciton transfer was discussed. The dimer approach was employed; i.e., all relevant energies were computed with suitable quantum-chemical calculations on homo- or heterodimers. To take into account environmental effects, a continuum approach with local electric fields was used. For more information, we refer to our previous investigation.79 However, only the combination of thermodynamics and kinetics provides an adequate description of the processes at organic:organic interfaces. Therefore, in the present work, we concentrate entirely on the corresponding kinetic aspects. We conduct KMC simulations based on Marcus rates to investigate the efficiencies of the various processes in more detail. We consider only systems where charge separation is thermodynamically possible according to our previous investigation. Therefore, we do not employ any energetic restrictions in the KMC simulations. We had to introduce various approximations to keep the simulations computationally feasible. However, as we will show later, the computed KMC efficiencies agree well with available experimental external/internal quantum efficiencies (EQE/IQE). A good coincidence between computed and measured hole mobilities is also achieved. Both comparisons indicate that our simulations take into account the most important effects. Hence, the revealed structure−property relationships should be valid. Moreover, the main interest of this work is to provide insight into fundamental kinetic effects of the light-to-energy conversion process rather than to accurately predict transport parameters. The paper is organized as follows. In the section “Description of the Theoretical Approach”, we discuss the employed theoretical approaches to compute the rates of the various processes, the protocol of our KMC simulations, and the models to analyze the outcome. In the subsequent section “Results and Discussion”, we present and analyze the results for selected systems and compare them with available experimental data. Subsequently, we compare computed macroscopic

transport parameters with their experimental counterparts. This offers more insight but allows also for a critical evaluation of our model. Finally, we combine all data to elucidate trends and to derive structure−property relationships being of interest for the design of molecules and aggregates with improved properties.

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DESCRIPTION OF THE THEORETICAL APPROACH Our simulation protocol is outlined in Figure 2. We begin with a description of the calculation of all types of rates (Figure 2, red box, “rates”, gray boxes) using Marcus theory. Nonadiabatic Marcus transfer rates kif depend on the coupling Vif between the initial and the final diabatic state, the reorganization energy λ, and the driving force ΔG. k if =

Vif 2 ℏ

2 π ·e−(λ +ΔG) /4λkBT λkBT

(1)

ℏ is the Planck constant, T the (ambient) temperature, and kB the Boltzmann constant. In hopping processes between localized sites in organic thin films, the initial and final states correspond to two adjacent monomers. Couplings Vif (for exciton diffusion, charge transport, and charge transfer) were computed for next-neighbor dimers that were selected with a distance criterion (see Supporting Information for more details). No dielectric constant was used as no screening takes place between nearest neighbors.52 Exciton couplings Vex between two monomers are obtained as the adiabatic splitting80,81 of the first two excitations {E1, E2} of the dimer. We employed ZINDO82,83 that gave quite reliable excitation energies within a benchmark.84 Vex =

1 (E2 − E1) 2

(2)

Charge transport couplings between two monomers for hole (electron) transport are calculated as the adiabatic splitting of the HOMO and HOMO−1 (the LUMO and LUMO+1) of the neutral dimer.81,85 Vhole =

1 (εHOMO − εHOMO − 1) 2

Velectron =

1 (εLUMO + 1 − εLUMO) 2

(3)

(4)

86

The INDO method was shown to deliver coupling values which agree well with corresponding DFT values.15 The adiabatic splitting method is only justified as long as monomer excitation energies or orbitals are well-separated.81 This is the case for all excitation and orbital energies except for those of TBA and of the fullerene. Using an approximate scheme described in more detail in the Supporting Information, we could still retrieve coupling values for electron transport in the fullerene phase and for hole transport in the TBA layer. Since we are interested in trends rather than in absolute values, the resulting uncertainties represent a smaller problem for C60 that is used for all interfaces. Exciton couplings in simulations of large systems are often approximated as dipole−dipole couplings. However, it was shown that this provides reasonable values only for sufficiently large monomer separations.52,80 Similarly, charge transport couplings are commonly expressed as a simple exponential function of the intermolecular distance, like in Miller− Abrahams hopping rates.15 Indeed, the molecular orbital E


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oscillator strength f, the refractive index n, and the wavenumber ν̃.96,97,96

interactions and orbital overlap decrease exponentially with increasing intermolecular distance. As shown by Brédas and coworkers,15 this equally holds for the couplings. Yet, they similarly emphasize that charge transport couplings depend on the wave function overlap and not just the spatial overlap, as suggested by the simple exponential dependence. As a consequence, the computationally efficient approach to approximate couplings by an exponential function of the intermolecular distance totally ignores the strong variations of couplings with slight differences in intermolecular arrangements. This could result in erroneous couplings. Couplings for the photoinduced charge transfer and for the recombination (back-electron transfer) were calculated using the generalized Mulliken−Hush formalism.87,88 VCT =

k rad = n3ν 2̃ f

As TDDFT obeys the Thomas−Reiche−Kuhn sum-rule, ωB97X-D is used for the calculation of oscillator strengths and wavenumbers entering into the radiative decay rates. It should be noted that oscillator strengths and wavenumbers for absorption were used to approximate the radiative decay rates. Exciton self-trapping processes are known for various perylene-based compounds, e.g., for PBI100 aggregates and PTCDA101 crystals (perylene-tetracarboxylic dianhydride). In DIP single crystals, trapping does not take place due to steric hindrances. In amorphous thin films, however, DIP changes its orientation so that trapping becomes feasible. To take this effect into account, we approximated the trapping rate constant by 1012 s−1, which is based on the timescales found for the trapping in PBI (215 fs100) and PTCDA (∼400 fs101). For merocyanines, exciton trapping can occur due to photoinduced intramolecular torsional motions.102 A complete torsion is not possible due to steric strain in amorphous thin films, but feasible partial torsions are already sufficient to trap an exciton efficiently. The corresponding rate constant (5 × 1012 s−1) was approximated on the basis of laser-spectroscopic measurements (decay time: ∼100 fs103) assuming that the characteristic timescales are very similar for all merocyanines. Reliable simulations of the various processes require the inclusion of the influence of the environment on excitons and charges. First, this involves a realistic model of the structural disorder in amorphous thin films because structural disorder strongly influences the hopping probabilities of excitons and charges due to different distances between the hopping sites. Additionally, disorder influences the driving forces of all processes since in contrast to well-ordered crystals the structural disorder in thin films also induces varying site energies. These variations result from local electric fields and a site-specific polarization and delocalization. Finally, the polarizability of the environment also strongly influences the charge separation process72,104 because it leads to a screening of the Coulomb attraction between the electron and hole. Our approach to generate realistic disordered amorphous bulk phases and interfaces was already described in our previous work.79 From our previous work, we also adopt our procedure that used an effective epsilon to mimic the screening effects on the Coulomb forces.79 The variations in the site energies resulting from dimer arrangements and environmental polarization have to be taken into account as well. They are particularly decisive because resulting driving forces ΔG enter exponentially into the Marcus rate equation. Dimer-based calculations as performed in our previous work79 would be perfect because they include near-neighbor and delocalization effects. However, this approach turned out to be computationally problematic for the KMC simulations. Hence, in the spirit of the Bässler model,23,25,105 we computed for each compound the corresponding densities of states (DOSs) of the disordered energy landscapes of excitons and charges using our previously calculated dimer-state energies.79 Employing corresponding Gaussian disorder parameters σ in the KMC simulations,22 we obtained for all hopping steps of each compound a distribution of driving forces that includes site-energy variations statistically. The respective disorder parameters for excitons and charges are given in the Supporting Information. Please note that we use a constant disorder parameter for the whole p-type semi-

μ12 (E2 − E1) (μ2 − μ1)2 + 4μ12 2

(6) 98,99

(5)

{Ei} are the energies of the excited and the charge-transfer state (the charge-transfer state and the ground state); {μi} are their respective dipole moments; and μ12 is the transition dipole moment between the two states. As often found in the literature, the projection of the transition dipole moment along the dipole difference vector is used to exclude any small locally excited component of the transition dipole moment.89,90 ZINDO is again employed. The excited-state dipole moments μ2 could not be directly calculated. Therefore, the dipole moment difference μ2 − μ1 was assumed to be the dipole moment of an electron−hole pair located at the interface with the electron situated on the fullerene and the hole on the molecular p-type semiconductor. Semi-empirical methods inherently cannot describe overlap-dependent contributions to charge-transfer states correctly, but they provide reliable electrostatic contributions to the charge-transfer states and ensure the correct inclusion of the Coulomb interaction between the electron and the hole (in contrast to TDDFT91). Because both effects seem to be more important than the overlap as discussed by Nelson et al.,13 ZINDO couplings should be sufficiently reliable. It should be mentioned that more elaborate schemes are not feasible due to computer limitations. Due to the 3-fold degenerate LUMO of fullerene C60, three energetically close-lying charge-transfer states exist for all heterodimers. Hence three different coupling values and corresponding rates are calculated. As described by Brédas et al.,62 all reorganization energies as well as the driving forces for the excited-state charge transfer and the recombination are calculated in monomer calculations using ωB97X-D92/cc-pVDZ,93 which was shown to be equally reliable for neutral, charged, and excited states.84,94 Coulomb interactions involved in the charge-transfer states are included afterward in the KMC simulations. As pointed out by Troisi et al.66 and Voityuk et al.,95 computations of external reorganization energies strongly depend on the theoretical approach. However, they are usually approximately about one magnitude smaller than internal reorganization energies.72 Hence for trends, it seems to be justified to include their influence only implicitly via ΔG (see below). Two exciton loss mechanisms, i.e., fluorescence and exciton self-trapping, are taken into account in the KMC simulations (yellow boxes, Figure 2). Fluorescence rates were obtained from a simplified version of the Strickler−Berg relationship that can be used to relate the rate of radiative decay, krad, to the F


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Figure 3. Demonstration of additional parameters used to evaluate the KMC results. The approximated interfacial plane is the center plane between the centers of mass of the p-type semiconducting molecules and the fullerenes, respectively.

previously described,79 interfacial model systems for all donor− acceptor combinations were generated for three mutually orthogonal crystallographic orientations. All three crystallographic orientations are treated separately. Moreover, for each molecule and each crystallographic orientation, a second slightly larger interface model is used to check the consistency of our results. Additional parameters used in the KMC simulations (Figure 2, blue box) are further illustrated in Figure 3. The p-type semiconducting phase is on the right-hand side of Figure 3, the fullerene phase on the left. Due to the low extinction coefficient of fullerene C60,108 we assume that excitons are only created in the p-type semiconducting phase. Each trajectory should include exciton diffusion through the bulk p-type phase, exciton dissociation at the interfaces, and subsequent charge separation because we are mainly interested in the interplay of these processes. Hence, each trajectory starts by randomly exciting monomers located in the red box indicated in Figure 3. This ensures that each exciton diffuses across a certain minimal distance, which was chosen as a function of the given system size (see Supporting Information). The exciton generations are uniformly distributed in the red box; i.e., morphological dependencies and anisotropic light absorption109,110 are not taken into account. Consequently, we exclude events that do not include all processes, e.g., direct excitations of interfacial charge-transfer states54 or the dissociation of an exciton far from the interface.65,111 The overall efficiency of a simulation is given by the percentage of the generated excitons that successfully dissociate into charge-separated states,72 i.e., into charges that have

conductor phase. A potentially different degree of the disorder at the interface compared to the bulk phase as pointed out by van Voorhis59 is neglected. This approach could be too approximate for exact values, but it should be sufficiently reliable to describe trends in the processes resulting from the varying molecular and aggregate properties. It should be kept in mind that since we do not exclude energetic uphill processes in the simulations if they are kinetically feasible, slightly endothermic transitions in the disordered DOS are possible. Doing this also avoids double-counting of the effect’s disorder in the additionally included trapping. Dynamic disorder106 is neglected; i.e., variations of the coupling values are not taken into account. We implement the KMC in the spirit of Houili et al.107 (Figure 2, red center box). The generated disordered interfaces (described in our previous investigation79) are used to calculate the grid of the centers of mass (Figure 2, blue box). Hopping processes take place between these grid points. No periodicity is used; i.e., as soon as a hopping particle arrives at a boundary of the system, it is reflected. We conduct the KMC simulations with a fixed number of steps that is proportional to the size and dimensions of the system (see Supporting Information). To prove that the number of steps does not influence the outcome, we repeated the simulations with a 6-fold number of steps (see Supporting Information). The differences turned out to be insignificant. It should be noted that because of the fixed number of KMC steps, overall simulation times differ. At least 15 and 20 starting points are chosen in most cases for the KMC (see below). For each point, 100 trajectories are calculated, and results were averaged. It should be kept in mind that as G


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where zsep is the position of the charge for which the Coulomb attraction has dropped below the threshold discussed above. tch is the time needed to reach this point. It should be noted that the position of the hole is used to define final charge separation; i.e., the hole must have reached a certain distance from the interface (only one blue arrow in Figure 3). Effective velocities for excitons and charges obtained from an average over all MC trajectories can be compared with their experimental counterparts. As pointed out by Andrienko et al.,33 drift/diffusion velocities and resulting mobilities obtained from simulations on disordered finite systems are usually overestimated. In order to correct for finite-size effects115 of the simulations, a simple relationship between the mean displacement and the mobility was used. Details can be found in the Supporting Information. Troisi et al. pointed out that average velocities are only meaningful when all rates are approximately equal.65,116 The aspect was also experimentally addressed by Yan et al.117 In our simulations, exciton and charge velocities were similar, while only charge-transfer and recombination rates differed occasionally by several magnitudes. Hence averaging of exciton/charge velocities for several MC trajectories is physically grounded. In line with our previous investigation,79 all neutral and excited-state monomer geometries were optimized at the SCSCC2118−120/cc-pVDZ93 level of theory. The Turbomole program package was used.121 Charged geometries were obtained using ωB97X-D92/cc-pVDZ.93 All charge, exciton, charge-transfer, and recombination reorganization energies and the driving forces as well as the fluorescence rates were obtained with ωB97X-D92/cc-pVDZ.93 Please note that excited-state structures of the merocyanines MD353 and HB194 optimized at this level of theory are not twisted.102 An ultrafine grid was employed. Charge couplings were obtained using INDO86 if parameters were available. ZINDO82,83 was used for all other couplings and for the charge transport couplings between sulfur-containing molecules. When computing recombination and charge-transfer couplings, occupied orbitals of fullerene C60 were frozen. The Gaussian program package was used.122

escaped their mutual Coulomb attraction. As this is related to the experimental internal quantum efficiency (IQE), we designate this quantity as a “KMC quantum yield” (Figure 2, left purple box). We consider an electron−hole pair as separated as soon as the electron−hole distance has acquired a certain value so that the mutual Coulomb attraction has significantly dropped (see Supporting Information). It should be noted that geminate pairs at the interface should be considered as fluctuating charge pairs with variable intermolecular spacings that dissociate and can potentially recombine again.112 Since the latter aspect is ignored in our simulations, resulting charge generation efficiencies might be overestimated. Even more important than these overall efficiencies is an understanding of how given molecular and aggregate properties influence the individual processes, as this allows for the elucidation of the bottleneck of the light-to-energy conversion and for an identification of structure−property relationships. To obtain such information, we further analyzed the trajectories to determine the efficiencies of the individual one-step processes and the influence of loss mechanisms (Figure 2, left purple box). For example, the KMC quantum yield can be reduced due to exciton losses which, apart from the discussed loss mechanisms of fluorescence and self-trapping, also include the probability that an exciton trajectory simply does not reach the interface on a reasonable timescale (Figure 3, green ovals). To shed light on the influence of a given loss mechanism on the overall efficiency, we also compute the individual efficiencies. The same applies to the subsequent processes (exciton dissociation and charge separation). Also for these processes, various loss mechanisms exist for which the relative importance has been individually evaluated. Since we want to elucidate trends rather than absolute values, we did not evaluate the efficiencies of the electron transport in the fullerene phase because fullerene C60 is the n-type semiconductor in all our systems. Therefore, the role of the fullerene C60 is not addressed in detail, in contrast to investigations by Friend and Rao and co-workers113 and by Ratner et al.114 emphasizing the “unequal partnership” between donors and acceptors in terms of charge delocalization. The efficiency of a given process strongly depends on the efficiencies of the other processes, e.g., exciton loss mechanisms will be less important if the exciton moves very fast and in a direct way to the interface. In contrast, if the exciton moves mainly parallel to the interface, the probability of competing loss mechanisms increases significantly. We compute effective exciton and charge velocities for each successful trajectory. They provide insight into how efficiently excitons or charges move toward or away from the interface. The effective velocities for excitons |v|ex,eff are defined as |v|ex,eff =

|zabs − zCT| tex

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RESULTS AND DISCUSSION In the following, we present an analysis of the efficiencylimiting influence of various kinetically determined processes to establish relevant structure−property relationships. Subsequently, we discuss the importance of the computed effective velocities for the KMC efficiencies and correlate them with experimental data. We start with the three different DIP:fullerene systems to introduce the general features (see Figures S2−S4 in the Supporting Information for a comparison of the three model systems). This discussion allows us to identify three important kinetic effects potentially limiting device efficiencies which we then generalize to a broader range of molecules by discussing representative examples for all three kinetic effects. Subsequently, transport parameters extracted from the simulations are compared to available experimental mobilities and short-circuit currents. The achieved good agreement demonstrates that despite the introduced approximations, e.g., the intentional inclusion of slightly endothermic steps, our model covers the most important interactions. This is also indicated by the fact that the computed KMC efficiencies possess the same magnitude as experimentally available external/internal quantum efficiencies (IQE/EQE).123,124 It should be noted that the simulated quantity is related to the

(7)

zabs is the position where the exciton was generated, while zCT is the corresponding value of the respective charge-transfer state at the interface. Projections of the respective three-dimensional position vectors onto the direction perpendicular to the interface were employed. tex is the time between the generation and the dissociation of the exciton. The effective velocities for charges are defined accordingly |v|ch,eff =

|zCT − zsep| tch

(8) H


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Figure 4. KMC quantum yields (see above) for the KMC simulations on the three different DIP morphologies. For more information, see text. See Table 2 for a detailed description of the colors and the numbers. Chemical structure of DIP.

IQE. We only compare to EQE values if IQE data are not available. General Features of Interfacial Kinetics: DIP:Fullerene interface. In Figure 4 composite pie charts illustrate the efficiencies of the processes for the three different DIP:fullerene interfaces. All colors are described in more detail in Table 2. The underlying crystallographic orientations are used to designate the systems. The efficiencies are normalized to the number of generated excitons, i.e., the number of KMC trajectories. Values in brackets are referenced to the number of dissociated excitons (Table 2). The left charts summarize the relative efficiencies of the processes up to the exciton dissociation into a geminate electron−hole pair. For the a-bplane, 28% of the generated excitons undergo charge transfer at the interface (“dissociated excitons”, Figure 7). The remaining 72% of the excitons decay radiatively (“fluorescence”, 7%), are trapped (“trapping”, 14%), or continue migrating through the p-type semiconducting phase (“exciton migration”, 52%). The right pie charts correspond to the charge dissociation efficiency. For the a-b-system, only 36% of the dissociated excitons accomplish the charge separation process (“charge separation”), while the majority of charges remains bound (64%, “charge migration”); i.e., with respect to the generated excitons, only 10% would contribute to the photocurrent. For both other interfaces the trends are different. Both possess considerably higher exciton dissociation efficiencies (56% and 79%, respectively). They result due to considerably lower losses due to exciton migration (green). Simultaneously, loss mechanisms (fluorescence, trapping) and exciton migration become less important. This suggests that especially in the b-c-

system, excitons diffuse and dissociate sufficiently fast to outperform all competing loss processes. Another considerable difference between the morphologies becomes evident from the number of excitons that dissociate. For the (a-b)-DIP:fullerene interface, most dissociated excitons apparently remain bound (64%). In contrast, for the a-c- and b-c-morphologies, charge separation is considerably more successful. For the (a-c)DIP:fullerene interface, 83% of the dissociated excitons separate completely (73% for the (b-c)-DIP:fullerene interface). Nevertheless, while the variations between the a-b- and b-cmorphologies in the exciton dissociation lie between 28% (ab) and 79% (b-c), the corresponding differences between the successful charge separation of the dissociated excitons are only 36% and 78%. More insight into the differences is gained from the computed couplings and rates for the various DIP:fullerene interfaces which are summarized in Figures 5−7. Red squares are used to indicate the centers of mass of the molecular p-type semiconductors, and black circles correspond to the centers of mass of fullerenes. Two coordinates in a plane perpendicular to the interfacial plane are employed to indicate the positions of the centers of mass of the molecules; i.e., the molecules’ positions are described by their projections into a plane perpendicular to the interface. The upper left panels show the exciton couplings between the p-type semiconducting molecules (red arrows, see Table 1) and the photoinduced chargetransfer couplings (blue arrow, see Table 1). The corresponding charge transport couplings of the p-type semiconductor (red arrows) and fullerene (black arrow) are given in the upper right panel together with photoinduced charge-transfer and I


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The Journal of Physical Chemistry C Table 2. Color Code Used To Indicate the KMC Efficienciesa

a

The percentages (except for those in brackets) are obtained by referencing to the number of excitons formed in the excitation box (= to the number of KMC trajectories).

recombination couplings (blue arrows). The corresponding rates which, aside from the couplings, also include the influence of the energetics on the processes are shown in the lower panels. Both couplings and rates are given to clarify effects arising from the couplings (Vif eq 1) and from the driving forces (ΔG eq 1) which both enter the Marcus rate equation. The driving force varies due to the effects of disorder, positiondependent Coulomb forces, local electric fields, and screening effects of the environment. Please note that the Coulomb forces used for the visual representation of the rates in Figures 5−7 are calculated for a hypothetical symmetric charge pair at the interface; i.e., the positive charge in the donor phase interacts with a mirror charge in the acceptor phase. Although charge separation in general and also in the KMC simulations is not necessarily symmetric with respect to the interface, qualitative conclusions drawn from visualizations like Figures 5−7 can be generalized. The monomer reorganization energies (λ eq 1), which also enter into the Marcus equation, are constant for all

given DIP:fullerene systems. The size of the coupling strengths and the transfer rates are coded in the widths of the different arrows in Figures 5−7. A linear relationship is used to translate coupling strengths, while a logarithmic scale is used for the rates. The differences in the rates computed for the respective morphologies easily explain the trends of the KMC efficiencies depicted in Figure 4. Despite the disorder, for the (a-b)DIP:fullerene interface, most DIP molecules are situated in an edge-on orientation on top of the fullerenes. This morphology favors exciton and charge transport parallel to the interfacial plane (Figure 5, lower right panel) due to large exciton couplings within the standing DIP stack and small couplings between adjacent stacks. For both other morphologies, exciton transport toward the interface is considerably more efficient. This is mostly due to the fact that the DIP stacks adopt two nonequivalent lying orientations on top of the fullerene phase J


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Figure 5. Couplings for excitons (upper left panel) and charges (upper right panel) and the corresponding rates for exciton transport (lower left panel) and charge transport (lower right panel) for the (a-b)-DIP:fullerene interface (edge-on orientation of DIP molecules on fullerene C60).

so that maximal couplings (along the stack) are orientated perpendicular to the fullerene surface. However, for all morphologies, the hole transport rates significantly decrease in the immediate vicinity of the interface (Figures 5−7, lower right panels). We will refer to these regions as charge transport depletion zones in the following. They arise due to the significant Coulomb attraction between the geminate electron−hole pair, which is only insufficiently screened in organic semiconductors. The Coulomb attraction between different sites varies most in the direct vicinity of the interface, where the electron−hole separation is in average still small, and for hopping processes in a perpendicular direction to the interfacial plane. A large difference of the Coulomb attraction between the initial and the final hopping site leads to large and positive contributions to ΔG (assuming a hopping process away from the interface where the Coulomb attraction decreases) and hence to a reduction of the charge transport hopping rates. Large charge transport depletion zones in the interfacial region rule out the possibility of a “cold CT state” mediated charge separation process and suggest the participation of vibrationally/electronically excited CT states and delocalization.13 Please keep in mind that we focus on kinetics, while thermodynamic implications of the Coulomb attraction were already discussed in our previous work.79 Figures 5 and 6 predict that the charge depletion zone for the (a-b)-DIP:fullerene interface is smaller than that for its b-ccounterpart. This results because the Coulomb attractions are larger for the face-on orientations of DIP molecules on fullerene (b-c) than for the edge-on orientation (a-b) because

the distances are smaller. As a consequence, despite the weak charge transport couplings in the direction orthogonal to the interface in the a-b-systems, its relative charge transport is not as inefficient as could be expected. This also explains why the differences in the successful exciton dissociation events between the a-b- and b-c-systems vary more than the corresponding successful charge separation processes (Figure 4); i.e., the driving force reduces the differences. The electron transport in the fullerene phase (upper right panels, black arrows see Table 1) is considerably more isotropic than the hole transport in the three DIP morphologies. This results from the spherical shape of the fullerene (Figure 4, upper right panel, blue arrows) whose importance for the isotropic character of electron transport has also been addressed by Ratner et al.114 and discussed by Nelson et al.13 Strong differences between the DIP morphologies also appear in the rates of the photoinduced charge transfer process in which an exciton dissociates into a geminate electron−hole pair at the interface (blue arrows in the lower left panels of Figures 5−7), although the differences in the corresponding coupling constants are rather small (given as blue arrows in the upper left panels of Figures 5−7). Nevertheless, the couplings enter quadratically in the Marcus rate equation (eq 1). For the (a-b)-DIP:fullerene interface, the photoinduced charge-transfer rates are very small and attain only in one region of the interface considerable values (see Figure 5, lower left panel). A visual inspection of the morphology at this point reveals the existence of a slip-stacked conformation of DIP molecules on fullerene C60, inducing larger rates. The generally low chargeK


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Figure 6. Couplings for excitons (upper left panel) and charges (upper right panel) and the corresponding rates for exciton transport (lower left panel) and charge transport (lower right panel) for the (a-c)-DIP:fullerene interface (face-on orientation of DIP molecules on fullerene C60).

region). Thus, their impact as an efficiency-limiting loss mechanism might decrease. Smaller variations between the interfaces are also found for the recombination process in which an electron−hole pair at the interface recombines to the ground state. In all cases the corresponding rates vanish since all rates are smaller than 5 × 107 s−1. Among others, Troisi et al.65 and Castet et al.104 pointed out that recombination rates are usually inferior to photoinduced charge-transfer rates. In a description based on the Marcus rate equation, this stems from the fact that the recombination of an interfacial charge-transfer state to the neutral ground state is located in the Marcus-inverted region, which has also been experimentally measured (see for example ref 126 or 127). Our calculations suggest that reaction energies for the recombination of the charge-transfer state to the DIPfullerene ground state often exceed 1−1.5 eV, thus explaining the vanishing rates. As discussed above, disorder is included via a random normal distribution of site energies with a disorder parameter extracted from our previous investigation,79 including variations in the environment, delocalization, and intermolecular interactions. These effects of disorder are not included in the illustration of the rates as shown in Figure 5 to Figure 7. In the KMC simulations, however, an increasing degree of disorder leads to decreased exciton and charge dissociation efficiencies. Once the exciton or charge has arrived at an energetically low-lying site of the DOS, further hopping is kinetically hampered. 128 Furthermore, as pointed out by Gennett et al.,41 disorder affects intrinsically slow rates more than fast rates. The latter

transfer rates at the (a-b)-DIP:fullerene interface are in line with predictions of Heremans et al.125 for the pentacene:fullerene interface, where interactions and charge-transfer rates considerably decreased for an edge-on orientation of the pentacene molecules on fullerene C60. For both other morphologies, considerably more routes allowing for a fast photoinduced charge-transfer process exist (see the number of blue arrows in the lower left panel of Figure 6 and Figure 7; see Table 1) indicating another possible reason for the less efficient exciton splitting for the (a-b)-DIP:fullerene interface. Notably, the fastest rates for the charge-transfer process are not necessarily observed on those heterodimers with the largest charge-transfer couplings. This results from the existence of interfacial trap states for the a-c- and the b-c-orientation. The largest couplings are observed for those heterodimers with the closest contacts between the DIP molecule and the fullerene (leaving aside the exact conformation). However, the same heterodimers possess also the lowest-lying charge-transfer states due to the very high Coulomb binding energy arising from short electron−hole separations on the nearby monomers. Consequently, the driving force for the photoinduced chargetransfer process increases so that, according to our calculations, the charge transfer is in the Marcus-inverted region for very deep interfacial charge-transfer states. Resulting rates decrease, contrasting observed trends for the couplings. From a device perspective, this is important since as long as the weak coupling limit of Marcus theory applies,104 populating deep interfacial charge-transfer states is kinetically hindered (Marcus-inverted L


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Figure 7. Couplings for excitons (upper left panel) and charges (upper right panel) and the corresponding rates for exciton transport (lower left panel) and charge transport (lower right panel) for the (b-c)-DIP:fullerene interface (rather face-on orientation of DIP molecules on fullerene C60).

Figure 8. KMC quantum yields for the KMC simulations on the (a-b)-rubrene:fullerene morphology. For more information, see the text. See Table 2 for a detailed description of the colors and the numbers.

(2) A “charge transport depletion zone” was identified in the direct vicinity of the interface. It results from the significant Coulomb binding energy of an electron−hole pair whose changes enter into the Marcus rate equation. It is well-known that charge separation can be an energetic uphill process,6 but the consequences on the spatial distribution of rates have not yet been fully assessed. Moreover, for the DIP:fullerene system, the extent of this depletion zone was shown to be morphologydependent and may contrast with advantageous efficiencies of exciton and charge transport. It should be kept in mind that the depletion zone is not necessarily symmetric; it depends in each trajectory on the positions of both the electron and the hole of a geminate electron−hole pair. (3) Loss mechanisms like radiative decay or trapping usually only compete with exciton/charge transport and dissociation/ separation if rates of the latter processes are slow; i.e., the

are just reduced, while the former can become so small that other competing processes (i.e., fluorescence, trapping, etc.) take over. We will discuss this in more detail below. Before turning to other systems, the presented discussion of the kinetic aspects of the DIP:fullerene system with different morphologies led us to identify three important kinetic effects potentially limiting device efficiencies. (1) Random exciton and charge migration can impede the completion of exciton dissociation or charge separation. For the DIP:fullerene system, it was found that random exciton and charge migration can be favored by certain interface morphologies, when transport parameters direct exciton and charge hopping in “wrong” directions (i.e., parallel to the interfacial plane) and/or when the follow-up processes like exciton dissociation are slow. M


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Figure 9. Couplings for excitons (upper left panel) and charges (upper right panel) and the corresponding rates for exciton transport (lower left panel) and charge transport (lower right panel) for the (a-b)-rubrene:fullerene interface (face-on orientation of rubrene molecules on fullerene C60).

occurrence of fluorescence and trapping depends on the relative rates. Although radiative decay rates for organic substances are on the order of 109 s−197several magnitudes slower than hopping and charge transferthey become noticeable in the DIP:fullerene system because charge transfer under certain conditions is even slower. To generalize these aspects, we will discuss the above stated kinetic effects for representative examples of our test set of ptype semiconductors. Random Exciton Migration Caused by Fast Exciton/ Charge Transport Rates Combined with Slow ChargeTransfer Steps. We start with a comparison of the (a-b)- and the (b-c)-rubrene:fullerene system (see Figures S5−S7 for a comparison of the structures). The KMC simulations for the ab-system given in Figure 8 predict that all generated excitons successfully dissociate. This contrasts completely with the results of the b-c-system for which almost no exciton dissociation occurs; the quantum yields for exciton splitting and charge separation are only 0.3% and 0.0%, respectively (not shown). The fluorescence yield is 0% for both systems as well. To gain more insight, Figure 9 shows the distribution of couplings and rates for the a-b-system, while Figure 10 displays corresponding results for the b-c-system. As known from diverse experimental measurements,129,130 exciton couplings and transfer rates in rubrene are high compared to other processes. These high absolute rates are responsible for the inexistence of radiative decay in both systems, which cannot compete with fast exciton diffusion on the timescales of the simulation.

As for the different morphologies of the DIP:fullerene systems, the exciton diffusion and charge-transfer couplings differ for rubrene:fullerene, but in this case, the strong differences in efficiencies of the exciton dissociation mainly result from the different rates of the charge-transfer step. In the a-b-system, charge-transfer couplings in combination with the driving forces lead to large charge-transfer rates. This is not the case for the b-c-system. In this system, charge-transfer couplings exist, but due to unfavorable driving forces, the corresponding rates are smaller than for the a-b-system (no blue arrows in the left lower panel of Figure 10). The unfavorable driving forces result from large charge separation in the interfacial charge-transfer states in the b-c-system. It is worth emphasizing that absolute charge-transfer rates at the bc-interface are not exceptionally low, but they are very low compared to the exciton transport rates. Whenever an exciton can either dissociate or continue diffusing, the probability for further exciton diffusion is several orders of magnitude larger than the probability for exciton dissociation. The implications of unfavorable driving forces on the rates also seem to be responsible for the low charge separation efficiency in the a-b-system where only 9% of the dissociated excitons undergo charge separation. Figure 9 (lower right panel) indicates that despite charge transport couplings directed toward and perpendicular to the interface, the corresponding rates are very small and vanish in the vicinity of the interface due to inefficient screening of the Coulomb attraction between the charges by the apolar environment. This leads to a large charge depletion zone for the a-b-system so that N


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Figure 10. Couplings for excitons (upper left panel) and charges (upper right panel) and the corresponding rates for exciton transport (lower left panel) and charge transport (lower right panel) for the (b-c)-rubrene:fullerene interface (edge-on orientation of rubrene molecules on fullerene C60).

Figure 11. KMC quantum yields for the KMC simulations on the (a-c)-HB194:fullerene morphology. For more information, see the text. See Table 2 for a detailed description of the colors and the numbers.

charge separation is significantly hampered. The charge transport rates of the b-c-system are only parallel to the interface. This indicates that even if an exciton splitting had taken place, charge separation would be difficult. Influence of Molecular Characteristics on the Charge Transport Depletion Zone. For the (a-c)-HB194:fullerene system, our simulations predict a high exciton dissociation efficiency of 72%. The amount of exciton migration of 22% (Figure 11) is comparable to the (a-c)-DIP:fullerene system (26%) for which the exciton dissociation efficiency is only 56% because trapping and fluorescence loss processes are more significant. However, the most noticeable feature of the HB194:fullerene system is the very high charge separation efficiency as soon as an exciton dissociates. Charge dissociation efficiencies amount to 68% (Figure 12) or to 94% if referenced

to the number of dissociated excitons, in accordance with high experimental performances.131 This is even larger than for the (b-c)-DIP:fullerene system where 86% of the dissociated excitons successfully split into two separate charges. Again, the rates and couplings shown in Figure 12 provide more insight. Rates and couplings indeed only slightly favor migration toward the interface, in line with the existence of some exciton migration (Figure 12, left panels, red arrows; see Table 1). The high efficiencies result due to comparably high charge-transfer rates (blue arrows, lower left panel, Figure 11) that enable fast photoinduced charge transfer so that exciton dissociation efficiencies are high in spite of rapid exciton transport. In rubrene, large exciton transport rates hamper exciton dissociation because exciton hopping has a considerably higher probability than exciton dissociation. Nevertheless, the O


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Figure 12. Couplings for excitons (upper left panel) and charges (upper right panel) and the corresponding rates for exciton transport (lower left panel) and charge transport (lower right panel) for the (a-c)-HB194:fullerene interface.

Figure 13. KMC quantum yields for the KMC simulations on the (a-b)-squaraine:fullerene morphology. For more information, see the text. See Table 2 for a detailed description of the colors and the numbers.

formation of interfacial charge-transfer states but also has very beneficial effects on the spatial distribution of rates, efficiently promoting charge transport especially in the vicinity of the interface. This high efficiency of the charge separation process near the interface may be an important contribution to the high efficiencies of merocyanine-based OSCs. Thus, it countervails the disadvantages by dipolar disorder of polar materials in the bulk for charge carrier transport.25,105 Similar effects are observed for the (a-b)-squaraine:fullerene system whose results are shown in Figure 13 and Figure 14. High exciton dissociation efficiencies (86%, Figure 13) are computed that arise from strong exciton couplings, resulting in high exciton transfer rates and high charge-transfer rates (see blue arrows, lower left panel, Figure 14). Charge separation efficiencies are also very high, attaining 81% (Figure 13) or 94% referenced to the number of exciton dissociation events. Charge

main reason for the high charge separation efficiency of the HB194:fullerene system is the quasi-inexistence of a charge transport depletion zone (Figure 12, lower right panel). For the DIP and the rubrene systems, the Coulomb interactions between separating charges lead to endothermic driving forces thataside from thermodynamic implications disregarded in this investigationconsiderably reduce charge transport rates in comparison to the corresponding coupling constants. In contrast, in the HB194:fullerene system, the rates barely decline compared to the couplings (comparison of red arrows in right panels of Figure 12; see Table 1). This can be attributed to the efficient screening of the Coulomb interaction by the polar environment that reduces the changes in the electrostatic energy between hopping sites. Hence, in addition to the results of our previous investigation,79 the high effective epsilon of the polar merocyanine environment not only prevents the P


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Figure 14. Couplings for excitons (upper left panel) and charges (upper right panel) and the corresponding rates for exciton transport (lower left panel) and charge transport (lower right panel) for the (a-b)-squaraine:fullerene interface.

Figure 15. KMC quantum yields for the KMC simulations on the (a-b)-TAM:fullerene morphology. For more information, see the text. See Table 2 for a detailed description of the colors and the numbers.

Structure−Property Relationships for Further Loss Mechanisms Such as Fluorescence and Migration. To further explore the influence of loss mechanisms such as fluorescence and to study how their importance depends on the relative size of the rates of the other processes, we use TAM as an example.132 The efficiencies are displayed in Figure 15, while the distribution of couplings and rates is shown in Figure 16. As can be seen from Figure 15, the fluorescence quantum yield amounts to 23%, which considerably exceeds the yields of all previously investigated systems. This is surprising because the absolute radiative decay rate in TAM (see Supporting Information) is approximately equal to those of the other molecules. Finally, less than a third of the incident excitons undergo dissociation (30%, Figure 15), and almost 50%

transport rates in Figure 14 demonstrate again the almost inexistent charge transport depletion in the direct vicinity of the interface. Alike to the HB194:fullerene system, this can be in part ascribed to the high effective epsilon of the squaraine environment, screening Coulomb interactions. The large linear shape of the squaraine molecules also supports the charge separation process. As already pointed out in our previous investigation,79 initial electron−hole separations are larger on a squaraine−fullerene dimer than for example on a DIP− fullerene dimer. The Coulomb binding energy decreases along with its variations along potential transport pathways. Nevertheless, also rubrene has a favorable shape for charge separation but still very low efficiencies (9%, Figure 8). This underlines the importance of the screening ability of the environment. Q


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Figure 16. Couplings for excitons (upper left panel) and charges (upper right panel) and the corresponding rates for exciton transport (lower left panel) and charge transport (lower right panel) for the (a-b)-TAM:fullerene interface.

Figure 17. KMC quantum yields for the KMC simulations on the (a-b)-TAA:fullerene morphology. For more information, see the text. See Table 2 for a detailed description of the colors and the numbers.

transfer rates. Due to the long dwell time, the much slower fluorescence can occur. This is another example showing that disorder influences especially those rates that are intrinsically slow. In the case of the TAM system, this concerns the low interdimer exciton transfer rates which diminish with increasing disorder. The inexistence of fluorescence in the rubrene or the DIP bulk phase clearly demonstrated that fast exciton transfer rates effectively suppress the probability of radiative decay. Low exciton couplings can only lead to high exciton dissociation efficiencies if excitons are not confined to dimers and if additionally charge transfer is so fast that excitons arriving at the interface directly dissociate. An example for this situation is the TAA system shown in Figure 17 and Figure 18. It possesses an exciton dissociation efficiency of 82% in spite of low exciton diffusion rates (Figure 18, lower left panel). Charge dissociation occurs with a probability of almost 50%, an

continue migrating through the p-type semiconducting bulk phase. An investigation of the charge-transfer rates (Figure 16, lower left panel) clearly shows that the low exciton dissociation efficiency is not due to slow charge-transfer rates, which lie within the normal range. In fact, the significant losses due to fluorescence result from low exciton transfer rates. As already mentioned in our previous investigation,79 the TAM bulk phase is prone to the formation of charge-transfer complexes.133 Indeed, dimers exist where the composing monomers interact more strongly with each other than with all other neighboring molecules. This is obvious from the pairs of coupled p-type semiconductors given in Figure 16 (left panels). An exciton localized on such a dimer is trapped because it hops back and forth between the composing monomers, while exciton transfer to a neighboring dimer is highly improbable due to the lower R


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Figure 18. Couplings for excitons (upper left panel) and charges (upper right panel) and the corresponding rates for exciton transport (lower left panel) and charge transport (lower right panel) for the (a-b)-TAA:fullerene interface.

acceptable efficiency within our model systems (Figure 17). The promising results for this three-dimensional molecular semiconductor with accepting groups are well in line with experimental findings showing the utility of related compounds in organic optoelectronics.134,135 According to our calculations, the importance of the accepting groups results not only from the better overlap of the absorption with the solar spectrum136 but also from tuned transport levels137,138 resulting in only slightly exothermic charge transfers and correspondingly high charge-transfer rates (see eq 1, Figure 18). This coincides with experimentally observed increases in the external quantum efficiencies due to the introduction of accepting groups in investigations of Roncali et al.134 Comparison of Effective Charge and Exciton Velocities and Resulting Transport Parameters to Macroscopic Experimental Data. We have discussed kinetic consequences of distinct molecular and morphological properties for diverse interfaces in the previous sections and identified the importance of several kinetic loss channels. The discussion was entirely based on a molecular perspective. From a device perspective, it would be desirable to define single molecular parameters that directly influence the macroscopic device efficiency. To do so, molecular parameters obtained from our atomistic simulations are correlated with macroscopic device properties such as hole mobilities. This also allows for a critical evaluation of our model because computed and measured values can be compared. Hence, in addition to KMC quantum yields (see above), we computed velocities for charge transport

and exciton diffusion from our simulations (Table 3), which we discuss in the following. Table 3. Simulated Velocities for the Discussed Systems in Heterojunction with Fullereneq HB194 squaraine rubrene DIP MD353 DPP TAM TAA

exciton velocity [cm/s]

charge drift velocity [cm/s]

451.25 2501.07 98.05 38.65 95.15 712.47 1.57 28.10

95.55 107.41 1.01 8.20 56.39 118.84 0.90 8.72

q

Charge velocities refer only to positive charges in the p-type semiconducting layer.

Simulated hole velocities (Table 3, right column) are in the order of 1−100 cm/s. This is in good accordance with experimental values, which lie in this range and can reach maximum values of 106 cm/s as pointed out by Karl et al.139,140 In view of the approximations of the employed methods and models, the very good numerical agreement with the expected order of magnitude is surprising but indicates that our model includes the major effects correctly, which justifies our approach. To further support our approach, hole mobilities for blends were derived from the drift velocities using standard device parameters (see Supporting Information)141−144 and S


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100 cm/s (Table 3) and assuming an average exciton lifetime in the nanosecond regime (5 ns), our simulations predict approximate exciton diffusion lengths of 5 nm, which compare well with reasonable estimates for disordered organic thin films.150 The generally higher exciton transport velocities result from the isoenergetic character of exciton hopping, in contrast to charge transport, where the significant Coulomb interaction hampers fast hole transfer away from the interface into the bulk p-type layers. Please keep in mind that as stated above, energetic uphill processes are allowed if they are kinetically possible. The short-circuit current of a functional OSC device is directly related to the external quantum efficiency, determined by the quantum efficiencies of the individual processes; i.e., it should correlate with the least optimal process.97 According to our simulations, slow charge transport in the vicinity of the interface constitutes a major efficiency-limiting process. If this prediction is valid, our effective hole mobilities characterizing the charge transport process should correlate with experimental short-circuit currents. This is indeed the case as shown in Figure 20 using available experimental values (Table 5). The correlation between experimental short-circuit currents and the simulated drift velocities for charge carriers is very good (R2 = 88%) in spite of the chemical diversity of the underlying p-type semiconductors. This indicates the importance of high charge carrier mobilities for high short-circuit currents and OSC performances, which has also been investigated with macroscopic models, for example by Blom et al.,154 Dyakonov et al.,155 Riede et al.,156 Albrecht et al.,157 and Resendiz et al.158 Experimental evidence was provided for example by Nguyen et al.159 and by Samuel et al.160 However, the presented results are the first direct link between simulations with an atomistic resolution and macroscopic quantities; i.e., they provide a clearcut dependence between molecular charge drift velocities and experimental short-circuit currents.

correlated with available experimental values (Table 4). Mobility values obtained for blends were used if available; otherwise, bulk mobilities are employed in the correlation. Table 4. Experimental Values for Hole Mobilitiesa

a

hole mobility [104 cm2/(V s)]

source

squaraine rubrene DIP

1.90 0.08 0.30

DPP

2.50

145 146 139 large variations147 123

Amorphous/average values are used.

The correlation is given in Figure 19. The coincidence is good although simulated values are about 1 order of magnitude too high in line with findings of Andrienko et al.148 Nevertheless, trends are correctly reproduced, and the coefficient of correlation, the R2-value, amounts to almost 1. This again supports our model. It should be noted that the calculation of drift velocities and hole mobilities from the simulated, purely diffusive charge transport processes is somewhat inconsistent. Another possibility to estimate mobilities is via the diffusion coefficient obtained from the mean displacements in the kinetic Monte Carlo simulations and the Einstein relation. Nevertheless, it was shown for organic crystals that resulting charge transport mobilities are qualitatively correct but also overestimated.149 It was suggested that this results, at least for organic crystals, from the neglect of the nonlocal electron−phonon coupling.149 A good agreement with experimental data is also found for exciton diffusion. The simulated exciton diffusion velocities (first column, Table 3) are about 1 to 2 orders of magnitude higher (factor of 2−25) than the simulated charge drift velocities. Again, a comparison with experimental results indicates that the order of magnitude of exciton diffusion is physically plausible: taking our averaged exciton velocities of

Figure 19. Correlation of experimental and simulated hole mobilities. T


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Figure 20. Correlation of simulated charge drift velocities with experimental short-circuit currents.

able to screen these Coulomb interactions, our simulations predict quite high efficiencies. The molecular shape of the molecular p-type semiconductors can also be advantageous as shown for the squaraine. However, the comparison to rubrene, which also possesses an advantageous linear shape but still results in lower efficiencies, shows that the screening ability is more important. It is well-known that the charge separation can be an energetic uphill process,56,68,161 but its influence on the spatial distributions of the rates has not yet been fully assessed. Furthermore, the simulations also indicate that the ability to screen these interactions is another important reason for the efficiencies of dipolar or quadrupolar merocyanine-,141 diketopyrrolopyrrole-,123 and squaraine145-based organic solar cells. Up to now, only efficient exciton and charge transfer due to dimer formation in the bulk phase of merocyanines10 and the equivalent J-aggregation in squaraines162 were assumed to be the main reasons. (2) Random exciton and charge migration can also decrease quantum efficiencies. These effects strongly depend on the given morphology of the p-type semiconductor and on the anisotropy of the spatial distribution of rates. Such effects become more pronounced if they are accompanied by unfavorable rates of the follow-up processes, i.e., exciton dissociation or charge separation. (3) The interplay between the various rates is slightly less important. If exciton transport is slow, fluorescence becomes competitive. Similarly, fast exciton dissociation in systems with aligned energy levels can outcompete exciton diffusion, resulting in high exciton dissociation yields. Finally, to further evaluate our model, we use the computed microscopic parameters to calculate macroscopic transport quantities that can be compared to their experimental counterparts. We found a nice agreement of experimental and theoretical values which indicates that the revealed trends are indeed realistic.

Table 5. Experimental Values for OSC Short-Circuit Currentsa HB194 MD353 squaraine rubrene DIP DPP

short-circuit current [mA/cm2]

source

8.24 4.00 6.48 2.35 0.80 9.85

151 141 145 152 153 123

a

Either fullerene C60 or PCBM was used as acceptors in the experimental devices. It should be, however, noted that we inherently assumed low charge carrier concentrations in our model because independent trajectories for single charge carriers are calculated.

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CONCLUSION Using a kinetic Monte Carlo approach with rates obtained from the Marcus rate equation in conjunction with ab initio calculations of couplings and energies allowed us to model the individual steps of the light-to-energy conversion near the interfaces of organic solar cells on an atomistic scale. This shed light on several important kinetically controlled processes at the interfaces and highlighted the importance of adjusting relative rates for exciton transport, charge transfer, and charge transport to suppress the occurrence of energy losses and to prevent random charge and exciton migration. The soundness of our model is supported by the fact that computed KMC efficiencies agree quite nicely with their experimental counterparts, i.e., with internal quantum efficiencies (IQE). Our simulations reveal three important kinetic effects that potentially limit device efficiencies: (1) For most p-type semiconductors, charge migration in the vicinity of the interface represents the most important efficiency-limiting process. This leads to an interfacial charge transport depletion zone where charge transport processes are very slow. The underlying reasons are unfavorable driving forces which result mainly from the Coulomb interactions between the charges. Consequently, for compounds which are U


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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b11340. Distance criterion used for selection of dimers, approximate scheme for obtaining couplings between fullerene molecules, intramolecular photoisomerization for HB194 and MD353, effective epsilon values used for the calculation of the Coulomb attraction between geminate electron−hole pairs, derivation of Gaussian disorder parameters for energetic disorder, determination of number of MC steps per MC trajectory, threshold values for minimal exciton diffusion and charge separation, absolute radiative decay rates for all molecules, extrapolation scheme for larger systems, device parameters used for calculation of hole mobilities, visualization of the geometrical arrangements as a function of the crystallographic orientation for the first layer of the p-type semiconductor (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone number: (+49) 931-31-85394. Fax number: (+49) 931-31-85331. ORCID

Frank Würthner: 0000-0001-7245-0471 Bernd Engels: 0000-0003-3057-389X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the DFG for funding in the framework of the SPP1355, the FOR1809, and the GRK2112. CB thanks the Gaussian technical support for technical assistance and Dr. Vera Stehr and Dr. Christof Walter for fruitful discussions.

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REFERENCES

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