Polaron Structure and Transport in Fullerene Materials: Insights from

Article pubs.acs.org/JPCC

Polaron Structure and Transport in Fullerene Materials: Insights from First-Principles Calculations Kenley M. Pelzer,† Maria K. Y. Chan,‡ Stephen K. Gray,‡ and Seth B. Darling*,‡,§ †

Department of Chemistry, The James Franck Institute, and the Institute for Biophysical Dynamics, The University of Chicago, 929 East 57th Street, Chicago, Illinois 60637, United States ‡ Center for Nanoscale Materials, Argonne National Laboratory, 9700 South Cass Avenue, Building 440, Argonne, Illinois 60439, United States § Institute for Molecular Engineering, The University of Chicago, 5747 South Ellis Avenue, Chicago, Illinois 60637, United States S Supporting Information *

ABSTRACT: Organic semiconductors offer a low-cost alternative to inorganic semiconductors. However, their usefulness is limited by a relatively low mobility of polaron charge carriers. Past research indicates a positive correlation between charge density and charge mobility in organic semiconductors. This relationship is usually attributed to the phenomenon of excess charges filling traps. Here, we explore whether charge density may also affect mobility via influence on intermolecular couplings. Density functional theory (DFT) with a long-range corrected (LC-BLYP) functional is used to calculate charge densities and electronic couplings of negative charges on C70 fullerenes in the presence of nearby negative point charges, which provides an upper limit calculation of the influence of nearby polarons. We find that in C70 systems with relatively low couplings, the presence of additional charges has an effect of maximizing intermolecular couplings and hence transport. This effect drops off quickly with distance, suggesting that it is relevant only at extremely high charge densities that are an unlikely event in current C70 devices. The effect of charge density on couplings may be useful in understanding transport in very limited regions of C70 materials where the local charge density is high; however, it is unlikely to affect overall device performance.

1. INTRODUCTION Organic semiconductors have attracted much attention as alternatives to traditional silicon-based inorganic electronics, showing promise as active materials in a range of technologies including field-effect transistors (FETs), light-emitting diodes (LEDs), and organic photovoltaics (OPVs).1,2 The low cost, light weight, and flexibility of organic semiconductors3,4 make them particularly attractive. A major limiting factor, however, is their rate of charge transport. Charges in organic semiconductors are coupled to some degree to nuclear motions, forming polarons (charges dressed by phonons). These charge carriers in organic semiconductors generally have a mobility of 10−5−10−4 cm2 V−1 s−1, far lower than the charge mobility of 103 cm2 V−1 s−1 for typical inorganic crystalline semiconductors.5 One factor contributing to these low mobilities is the fact that the electronic couplings (which determine the probability of charge transfer between different molecules) are relatively weak due to their intermolecular character.6 Increasing the intermolecular couplings, and thus the speed of this slow journey to the electrode, has the potential to increase the efficiency and economic viability of organic semiconductor technology. One factor that must be considered in the modeling of charge transport is the influence of other charges. Much of the work modeling charge transport in organic semiconductors has neglected charge density effects,7 and the available literature on © 2014 American Chemical Society

the effects of charge density has reached conflicting conclusions on the nature of its impact. Ciuchi and Fratini model small polarons in organic FETs and argue that polaron−polaron interactions increase the thermal activation barrier to polaron hopping;8 similarly, in inorganic semiconductors, a positive correlation between charge density and activation energy was suggested to explain resistivity measurements for small polaron conduction in substituted magnetites.9 However, a significant body of literature argues that increased charge density usually has a positive effect on charge mobility in organic semiconductors. Pivrikas et al. experimentally measure mobility in fullerene FETs and find decreased activation energy and increased charge mobility with increased charge density.10 Theoretical literature on various organic semiconductor systems has argued that increases in charge carrier density lead to increases in charge mobility,11−14 and a positive correlation between charge density and charge mobility has been demonstrated experimentally for poly(3-hexylthiophene) (P3HT)/[6,6]-phenyl C61-butyric acid methyl ester (PCBM) solar cells.5 Experimental work in LEDs and FETs has found substantial evidence that charge density has a positive effect on hole mobility.15−18 It is not clear whether the relationship Received: May 19, 2014 Revised: August 28, 2014 Published: September 2, 2014 21785

dx.doi.org/10.1021/jp504923x | J. Phys. Chem. C 2014, 118, 21785−21797

The Journal of Physical Chemistry C

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process beyond the effects of traps and an altered chemical potential. The arrangement of charges that we study corresponds to an extremely high charge density and addresses the question of whether these effects become irrelevant at the charge densities that occur in current C70 devices. This work makes no assumptions about the nature of the charge transport process: our results are relevant to charge transport in C70 systems regardless of whether polarons travel via thermally activated hopping or via band transport. We will use the term “polaron” to refer to our charges because electron−phonon coupling in organic semiconductors is generally at least comparable to, if not larger than, electron−electron coupling.2 However, both smaller polarons traveling via thermally activated hopping and larger polarons traveling via band-like transport exhibit a mobility that is a function of couplings.1 The precise dependence on coupling differs between various models of charge mobility; however, we can qualitatively examine the relationship between charge density and mobility irrespective of the nature of the transport process. We examine only the effect of like charges: the effect of negative point charges on a single negative polaron. The idea that polarons can be regarded as point charges has been suggested with the argument that the localization of charges in organic semiconductors makes point charges a reasonable approximation.19,23 These point charges can represent other charge carriers or charged defects in the device. Because we fix the position of the point charges in all calculations, these charges most accurately represent trapped charge carriers or charged defects. Using mobile point charges to represent mobile charge carriers is problematic because of the gas-phase nature of our simulations (discussed below); due to Coulombic repulsion, the point charges would move away from the fullerene(s) with a freedom of movement that would not accurately represent the movement of charges in a solid, disordered system. Because fullerenes primarily serve as electron-acceptors rather than electron-donors, we do not consider positive polaron/positive point charge interactions. Examining the influence of unlike charges is complicated by the possibility of charge recombination, a consideration that is beyond the scope of this study. Also, although ambipolar transport has been reported for some compounds, many organic semiconductors contain a single type of charge carrier, making the examination of like charges sufficient for studying charge density effects in these devices. By using point charges rather than simulating a charge delocalized over an adjacent fullerene (which is computationally infeasible given current resources), we neglect delocalization and screening that would be present in the case of an adjacent polaron. Therefore, our results should be viewed as an upper limit of the effects of nearby polarons. Of the various materials that have been explored as possible electron acceptors in FETs, C60 fullerenes10,22,24−37 and C70 fullerenes27 and their derivatives have been used in a large number of studies and have shown some of the more promising electron mobilities.3 C70 has also been studied extensively as an electron-acceptor in OPVs,38−42 with C70-based acceptors offering increased photoabsorption over a large energy range relative to the traditional C60-based acceptors.43 Because of the usefulness of this increased photoabsorption, we treat C70 rather than C60 in this study of fullerenes. Although we treat only C70, the results are likely to be at least qualitatively relevant to charge transport between other fullerene molecules.

between density and mobility persists at all charge densities: some authors have suggested that below a certain critical concentration, the charge carriers are essentially independent and the effect on mobility disappears;12 yet in their study of P3HT/PCBM, Shuttle et al. find a dependence of mobility on charge density for concentrations as low as 2 × 1015 cm−3.5 In an experimental study of hole mobility in hole-only diodes and FETs, the authors find that mobility is density-independent for charge densities 1016 cm−3.16 Given the wide range of charge densities exhibited by various organic semiconductors and the conflicting evidence regarding the threshold for charge density dependence, inspecting charge density effects at the molecular level may provide useful insights into the importance of charge density in organic semiconductors. In particular, the question of charge density effects is relevant to FETs, which operate at fairly high charge densities of 1018−1019 cm−3.7 Hanwell et al. argue that Coulombic effects of other charges should be considered in modeling charge transport in FETs, and perform Monte Carlo simulations in which the interactions between charge carriers have a significant impact on the transport process.19 It is not as obvious that charge density effects are important in OPVs and organic LEDs, which tend to operate at somewhat lower charge densities of 1015−1017 cm−3.5,20,21 Predicting the importance of charge density effects is complicated by the fact that charge density in organic semiconductors is not necessarily uniform throughout the material; Coehoorn et al. point out that in a disordered system, there are likely to be small regions in which the current density is much larger than the average.12 Also, the presence of trapped charges in organic semiconducting devices may affect the properties of mobile charges. Thus, understanding how nearby charges affect the electronic structure and transport properties of charge carriers may provide useful insights into the transport process. Charge density can influence charge transport in several ways. If the steady-state distribution of charges in an organic semiconductor is nonuniform, an increased charge density in a limited region will increase transport simply due to an increased chemical potential and hence increased diffusion. Coulombic interactions between charges are also argued to play an important role, with like charges repelling and avoiding one another.19 The mechanism most frequently discussed, which is relevant irrespective of charge distribution, is the hypothesis that charge density affects charge transport by the filling of traps caused by structural or chemical defects. It is argued that when additional charges fill traps, the remaining charges are allowed to experience trap-free transport.2,5,13,14,18,22 Blom et al. present a slightly different argument to explain the density dependence in LEDs, arguing that the charges first fill the energetically lowest localized states of an organic semiconductor, with higher energy states filled only at increased charge density. The higher energy charges then have a lower activation energy for hopping to adjacent sites.17 Zhou et al. propose yet another effect of adjacent charges, in which nearby charge carriers alter the site energies of each transport site in a way that changes the energy barrier for hopping events.23 When barriers become high, charges are effectively trapped. In this study, we use density functional theory (DFT) to take a more detailed look at the effect of nearby charges on polarons in C70 fullerenes, examining whether perturbations by adjacent charges can affect the charge carrier electronic structure and electronic couplings in a way that influences the transport 21786



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point charges then correspond approximately to charges placed on the nearest-neighbor or next-nearest-neighbor C70 molecule. It is important to note that these distances between the fullerenes and the point charges are not intended to reflect the average charge densities in current fullerene semiconducting devices. Rather, the distances that we explore reflect situations with extremely high charge densities that, at most, would be present only in limited regions of these systems. If a charge were present on nearly every fullerene (the density that is implied when we use distances such as 4−8 Å), the overall charge density would be on the order of ∼1021−1022 cm−3. In contrast, FETs have charge densities of only 1018−1019 cm−3 (while OPV and LED densities are even lower). The presence of two charges on adjacent fullerenes relative to a particular polaron thus represents an extremely high charge density that is an unlikely event in current organic semiconducting devices. The case of point charges 4 Å from the edge of the fullerene could be interpreted as a case in which the effects of the surrounding environment distort the electron density of the nearby fullerenes, pushing their electron density toward the polaron of interest. This event is even more unlikely, and it is difficult to imagine a system in which this event would be sufficiently frequent to impact overall device efficiency. The examination of these very high charge densities (much higher than the average densities in any current device) serves the following purposes. First, if significant inhomogeneity is present in the distribution of charges, these results may be relevant to limited regions of current fullerene devices. Second, inspecting a very high charge density is a useful toy model for understanding the physics that may underlie density/mobility relationships. Third, the results may be relevant to fullerene materials or other organic semiconductors that will be designed in the future, in which extremely high charge densities may be a more frequent event than they are in current devices. Last, and most importantly, the examination of very high densities is an effective way to disprove the hypothesis that the effect of density on couplings mediates the density/mobility relationship. Results indicating that the effect of point charges on electronic structure is insignificant even at extremely high charge densities serves as powerful evidence that for current fullerene devices, with average densities that are much lower than those that we examine here, these effects are unlikely to affect overall efficiencies. Studying the presence of a single point charge would be somewhat more relevant to low-density systems; however, the effects of a single point charge are arguably trivial, because the dominant effect would likely be a Coulombic repulsion that pushes the charges away from one another. For the case of a charge carrier with additional charges on opposite sides, although Coulombic repulsion would still promote a movement of the charges away from one another, the presence of two charges would complicate this predictable movement, particularly if one or both of the charges was trapped rather than mobile. Regions with a large number of trapped charges would be particularly likely to restrict this movement, because the Coulombic repulsion of trapped charges in the surroundings would complicate the tendency of the charges to move away from one another. In this case, the effects on electronic structure and transport properties are not as obvious. These less obvious effects are the focus of this study. We apply the LC-BLYP hybrid functional,44−46 a long-range corrected (LRC) functional, in our DFT calculations. Use of an LRC functional with an appropriate value of the range-

First, we examine the effects of nearby charges on the structure of the charge density distribution of a polaron on a C70 molecule. Next, we apply DFT to calculate the effect of nearby charges on electronic couplings between two C70 molecules.

2. METHODS 2.1. DFT Methods. Adjacent charges are treated simply as point charges in the DFT calculations, and each calculation utilizing point charges places two charges near each fullerene molecule, on opposite sides, as pictured in Figure 1.

Figure 1. (a) Structure of C70 monomer with point charges placed 4 Å from the molecule’s edge. (b) Structure of C70 dimer with point charges placed 4 Å from the molecule’s edge. The point charges represent the effect of other negatively charged polarons in an organic semiconductor.

For the monomer calculations, point charges are placed at 4, 8, and 10 Å from the edge of the fullerene. For the dimer calculations, the point charges are placed at 4, 8, and 16 Å from the edge of the fullerene. For our dimer calculations, we assume an edge-to-edge distance between the two fullerenes of about 2.9 Å. If we think of the position of the point charge as representing the center of the fullerene molecule on which the charge resides, and assume that all fullerenes are packed with an edge-to-edge difference of about 2.9 Å, placing a charge at the center of an adjacent fullerene would correspond to placing the point charge ∼6.4−7 Å from the edge of the fullerene of interest (depending on the relative orientation of the molecules, and assuming that the long axes of adjacent molecules are either parallel or orthogonal; more complex packings would give similar distances). This suggests that representing the position of the point charge as 8 Å from the edge of the fullerene is more accurate than a position of 4 Å from the edge of the fullerene if we wish to understand a situation in which charges reside on adjacent fullerenes. The 21787



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qualitative purposes (visualization of the charge density isosurfaces) this is not a problem. The polaron charge density was computed for a case with no nearby point charges and also cases with two negative point charges placed on the long axis of the fullerene at varying distances (one point charge on each side). An image of a fullerene with point charges 4 Å away is pictured in Figure 1. Distance from the fullerene was defined as the distance from the edge of the fullerene. Point charges were placed at 4, 8, and 10 Å from the fullerene edge. The point charges were frozen during the optimizations while all atoms of the fullerene were allowed to move. Monomer calculations used D5h symmetry for the case of no point charges and C5v for the cases with point charges. Although the presence of point charges slightly breaks the D5h symmetry of the whole system leading to the C5v symmetry (because the charges were placed according to the edge of the fullerene, not according to the symmetry plane), the fullerene, an inherently D5h molecule, retains D5h symmetry even in the optimized structures. We visualize the polaron charge density isosurfaces using VESTA (Visualization for Electronic and Structural Analysis) software version 3.1.4.67 Details of the isosurface imaging are described in the Supporting Information. 2.2.2. Electronic Coupling. The electronic coupling VAB, which is indicative of the probability of charge transfer between different molecules, is the key quantity we study in this work. We calculate the electronic coupling between a pair of fullerenes according to the Koopman’s theorem “energy splitting in dimer” method, or KT-ESD, in which the coupling is calculated as

separation parameter (μ) can improve the overdelocalization problem of DFT,47 an advantage in the study of organic semiconductors, where the extent of delocalization of charges in a particular material is not at all obvious. Although electronic structure software generally offers a default value of μ, various studies have shown the importance of determining the most accurate μ value for a particular calculation.47−57 We describe our procedure of tuning the range-separation parameter in the Supporting Information. To explore the significance of tuning the range-separation parameter, all calculations presented in this Article were repeated for both the default value of μ and the tuned value of μ. The geometry of the neutral monomer was optimized with the B3LYP functional44,58−60 and a 6-311G* basis set.61 For the monomer, all other calculations were performed using LCBLYP44−46 with the basis set 6-31+G*.62 For the dimer, the ccpVDZ basis set63 was used. Details on the choice of basis set are described in the Supporting Information. All DFT calculations were performed with the NWChem software package, version 6.1.1.64 Maximal symmetry was used in all calculations to reduce computational costs. 2.2. Application to C70 Monomer and Dimer. We assume that for C70 materials, the relevant unit for the mobile charge is a single C70 monomer (rather than concerning ourselves with delocalization over several fullerenes). This is supported by electron paramagnetic resonance (EPR) studies that failed to observe spin-density delocalization between neighboring fullerenes in polymer/fullerene OPVs.65 The same study also noted identical EPR spectra for C60 fullerenes regardless of the polymer used in the fullerene/polymer blend, suggesting that the perturbation of the fullerene electronic structure by the polymer in fullerene/polymer blends is negligible. Therefore, our calculations simply treat a single isolated C70 monomer for inspection of the effect of point charges on charge density, and an isolated C70 dimer for calculations of interfullerene coupling. Calculations were performed in the gas phase due to the difficulty of defining an implicit solvent model that would be generalizable to all fullerene organic semiconductors, where the “solvent” would consist of other fullerenes or polymer molecules. Because the C70 molecules in our dimer calculation are quite close together, it is highly unlikely that any other molecule would penetrate between them, and molecules around the perimeter would be unlikely to have a major effect on coupling patterns. Therefore, including implicit (or explicit) solvent would be unlikely to drastically change our results. The fact that the static dielectric constant of C70 solid is low (4.066) also suggests that screening effects from the surrounding molecules will be limited and will not have a major effect on the electronic structure of a single C70 molecule. 2.2.1. Charge Density on C70 Monomer. In our treatment of an excess charge on a monomer, we allow the positions of the nuclei to optimize in response to the excess charge, to account for the electron−nuclear deformation coupling in a polaron. We define the polaron charge density as the charge density difference between the geometry-optimized anion and the geometry-optimized neutral monomer. The difference in the two geometries results in spurious charge density differences near the nuclei, but due to the rigidity of the fullerene, the change in nuclear positions between the neutral and anionic molecules is sufficiently small (with all atoms changing position by ≤0.11 Å for the cases we present here) that for our

VAB =

E − EL ΔE = L+1 2 2

where EL refers to the energy eigenvalue of the lowest unoccupied molecular orbital (LUMO) and EL+1 refers to the energy eigenvalue of the LUMO+1 of the neutral dimer.2,68,69 Because of its simplicity, this is a commonly used method for the estimation of transfer integrals in organic semiconductors,2,70 and the calculated ΔE compares reasonably well to the results of more advanced methods such as using Complete Active Space Self-Consistent Field (CASSCF)71 and CASSCFState Interaction (CASSI)72 to calculate ΔE as the difference between the ground and excited states. The KT-ESD method is only accurate when the two charge-localized structures A−−B and A−B− can be obtained from one another via a symmetry operation; thus, we are restricted to highly symmetric dimer configurations.2,68 For dimers, single point energy calculations were performed rather than geometry optimizations. Although geometry optimization of donor−acceptor complexes has been shown to be useful in the analysis of charge transfer,73 optimizing our very large systems would present problems with computational cost. Cost aside, it is unlikely that optimization would lead to substantial changes in the geometry of the fullerenes, considering that our geometry optimizations of the monomer showed changes in position of 100%) upon adding nearby point 21791



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charges. The other category (structures 2, 3, 4, 6, and 9) consists of structures with higher initial couplings (>0.05 eV) that are mostly unaffected by the presence of point charges. Structure 7 is a somewhat intermediate case, in which the initial −1 coupling is 0.036 eV (0.038 eV) for μ = 0.2 a−1 0 (μ = 0.33 a0 ) and the change in coupling upon adding charges 4 Å away is −1 +29.1% (+21.6%) for μ = 0.2 a−1 0 (μ = 0.33 a0 ). When couplings are already high, adjacent charges are insignificant. When couplings are low, adjacent charges have a substantial effect in enhancing transport. Examining the LUMO and LUMO+1 of these structures provides clues as to what distinguishes these two categories of dimer structures (other than the magnitude of coupling in the case with no charges). As noted by Tamura and Tsukada in their study of C60 dimer couplings,76 the LUMO in our system has a bonding nature and the LUMO+1 has an antibonding nature. However, the bonding/antibonding nature is not necessarily complete. For the simple case of coupling between two ethylene molecules, Brédas et al.74 point out that the bonding/antibonding nature of the LUMO/LUMO+1 is not complete and both bonding and antibonding interactions may occur within each molecular orbital. Visualization of the isosurfaces of our orbitals indicates that the extent to which the LUMO/LUMO+1 orbitals take on a bonding/antibonding character is influenced by the slight variations in orientation distinguishing the various dimer configurations. As an example, in Figure 5a, using μ = 0.2 a−1 0 , we show the LUMO/LUMO+1 for dimer structure 1, which has the lowest coupling of the nine structures in the absence of point charges. In Figure 5b, we show orbital isosurfaces of the LUMO/LUMO+1 for dimer structure 2, which has the highest coupling of the nine structures without point charges. We see that, although both dimers have significant bonding character in the LUMO, the LUMO isosurface for structure 2 suggests a slightly higher charge density between the two monomers and hence more bonding character. Considering the LUMO+1, the structure 2 isosurface suggests a lower charge density in the middle and hence more antibonding character than the LUMO+1 of structure 1. The extent of bonding/antibonding character is reflected in the LUMO/LUMO+1 energy gap and hence the couplings. One rather crude way to quantify the extent to which an orbital is “bonding” or “antibonding” is to measure the fraction of charge density of the orbital that is found in the region between the two molecules. Given that the closest atoms in each dimer pair are 2.92 Å apart, we defined the region “between the two molecules” as a 2.9 Å wide block in the center of the system, as illustrated in Figure 6. The density within this block is summed and then divided by the total density of the orbital (as measured by the sum of all density in a 40 Å × 40 Å × 40 Å box around the dimer) to give the fraction of the total charge density of the orbital that is contained in the region between the two molecules. This fraction of the total orbital density will be referred to as ni for each orbital i. If coupling is highest when the LUMO has the most bonding character and the LUMO+1 has the most antibonding character, we would expect coupling to be higher when nLUMO − nLUMO+1 is higher. This trend is plotted in Figure 7a and b, where we see an initial steep increase and then some saturation of the effect as nLUMO − nLUMO+1 increases. The same trend is observed for cases with no point charges and cases with charges 4, 8, and 16 Å away, all of which are shown together in Figure 7a and b. Given the trend between nLUMO − nLUMO+1 and couplings, one might

Figure 5. Isosurfaces of the LUMO and LUMO+1 orbitals for dimer structures 1 (a) and 2 (b).

Figure 6. Each LUMO/LUMO+1 orbital density was calculated for a 40 Å × 40 Å × 40 Å block surrounding the dimer. The 2.9 Å thick red slice in the center of the cube represents the region used in calculating ni, the fraction of the orbital’s density that is between the two molecules.

expect that the effect of adding charges on nLUMO − nLUMO+1 would determine the change in couplings. The effect on nLUMO − nLUMO+1 of adding charges 4 Å from the fullerenes will be referred to as Δ[nLUMO − nLUMO+1], and the change in coupling upon adding charges 4 Å away will be referred to as ΔVAB. If all nine dimer structures are considered and ΔVAB is plotted 21792



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Figure 7. ni reflects the fraction of the orbital’s density that is found between the two fullerene molecules as a quantitative approximation to “bonding” and “antibonding” character. In (a) (μ = 0.2) and (b) (μ = 0.33), we plot coupling VAB versus nLUMO − nLUMO+1 for all dimer calculations, including both dimer calculations without point charges and dimer calculations with point charges at varying positions. In (c) (μ = 0.2) and (d) (μ = 0.33), we show the percent change in coupling relative to Δ[nLUMO − nLUMO+1] for the case with charges 4 Å away from the fullerenes relative to the case with no point charges.

relative to Δ[nLUMO − nLUMO+1], a loose positive relationship appears, as shown in Figure 7c and d. The results in Tables 1 and 2 show that the effect on electronic coupling drops off fairly quickly as the charges are moved to a greater distance from the dimer, although the tuned value of the range-separation parameter (Table 1) suggests more significant effects at 8 Å (up to +55.4%) than the default value (Table 2), which shows a maximal change of +27.6%. When charges are 16 Å away, the effect is small regardless of the structure of the dimer or the value of the range-separation parameter. These results are consistent with the insensitivity to more distant charges that we observed when visualizing polaron charge densities in section 3.1, and it again appears that charge density will only have a significant effect on electronic structure if charges reside on adjacent fullerenes. As mentioned in the Introduction, the fact that screening effects are neglected suggests that our results probably represent an upper bound of the effect of nearby charges. As in the case of the monomer, with significant effects occurring only at distances of 4−8 Å, these effects on coupling will only occur in extremely charge-dense regions. In the case in which a charge resides on each fullerene (which, as discussed above, corresponds to an edge-to-point-charge distance of ∼6.4−7 Å, if the point charge represents the center of the adjacent fullerene), the charge density would be ∼1021 cm−3. Considering that FETs have charge densities of only 1018−1019 cm−3 (while OPV and LED densities are even lower), these effects probably do not occur over substantial ranges in any current semiconducting devices, although local effects in limited regions may still be present. Of nine possible dimer orientations, we observe four that experience a significant increase in electronic coupling upon

adding nearby point charges. The difficulty lies in assessing the likelihood of interfullerene orientations that experience this sensitivity in a particular solid-state fullerene material. Our sample of nine structures is much too small to make claims about the precise likelihood of such orientations, and a sample large enough to make such claims is at the present time prohibitively expensive. A proper assessment of the likelihood of this event would also require methods that allow for asymmetric dimer structures, which we do not treat here. However, we believe that these results are likely to be qualitatively useful in predicting the effect of adjacent charges on fullerene transport irrespective of molecular orientation. Ideally, calculations would reveal a link between dimer structure and the coupling’s sensitivity to nearby charges such that the sensitivity could be predicted by structural characteristics rather than requiring electronic structure calculations. However, we find no obvious pattern among the structures studied here. Our strongest predictive property is the magnitude of the coupling in the absence of point charges, with more weakly coupled dimers exhibiting greater sensitivity to charges. The other trends apparent in Figure 7, although informative regarding the electronic structure properties that influence the couplings, appear to only tell part of the story and are not strongly predictive in and of themselves. For example, on the basis of the positive relationship between nLUMO − nLUMO+1 and couplings and the fact that charges are influential only when initial couplings are low, one might expect that the influence of charges would be significant only when the initial nLUMO − nLUMO+1 is low. This trend exists but is weak. Thus, although the trends that we have discussed here are likely to have a significant effect in determining the sensitivity of couplings to nearby charges, other determining factors must be 21793



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charge densities of current C70 devices, these effects are probably only relevant to inhomogeneous systems that have limited regions of extraordinarily high charge density. When the effect of charges on electronic couplings in dimers is calculated, we find that the presence of nearby charges has an effect of maximizing coupling: while dimer orientations for which coupling is already relatively high are insensitive to charges, orientations with low initial couplings experience increases in electronic coupling of >100% when charges are placed 4 Å from the edges of the fullerenes. As in the case of effects on the polaron charge density of a monomer, the effect of nearby charges on electronic couplings drops off quickly as the distance between the charges and the fullerenes is increased. Again, the drop-off with distance suggests that this effect is insignificant in most regions of current C70 devices. Even given a very high charge density in a limited region, these effects still might be destroyed by the simple Coulombic repulsion of the negative charges, leading the charges to be pushed to a distance where the effects are no longer significant. This movement would not occur in all cases; the adjacent charges may be trapped, or may be constricted in their movement by other trapped or mobile charges. However, the consideration of Coulombic repulsion between the polaron and the adjacent charges makes the case in which these effects are important even more unlikely. Distances of 4, 8, and 16 Å all correspond to extremely high charge densities that exceed the average charge densities in current C70 devices. However, the 8 Å case has somewhat more relevance to actual systems than the 4 Å case, which corresponds to a charge not only being present on an adjacent fullerene but the electron density being somewhat shifted toward the polaron of interest. Although possible, we have no reason to assume that this event would be more frequent than other electron density distributions. For the 8 Å case, we see a significant effect of adjacent charges on couplings only in the case of a tuned range-separation parameter. Thus, our results are in agreement with the wealth of literature suggesting that the effect of tuning the range-separation parameter is significant in calculations employing an LRC functional. Our results suggest that the electronic structure effects discussed in this Article are relevant only to limited, very charge-dense regions of the system. Such charge-dense regions may occur due to the inhomogeneity of the charge distribution pointed out by Coehoorn et al.12 Particularly near the point of charge injection, charge densities may be higher than the average charge density in the device. However, the extreme inhomogeneity necessary to have very charge-dense regions should not be assumed; this inhomogeneity may or may not be present in current and future C70 devices. It is important to note that the trends we observe may or may not be relevant to positive polarons influenced by nearby positive charges. Although our results certainly suggest that the effect of charge density on electronic couplings should be considered in understanding the mobility of positive charges in C70 semiconductors, calculations treating positive polarons in electron-donating organic materials would be necessary to definitively extend our conclusions to positive charge carriers. In considering the applicability of these results to specific semiconducting devices, it may also be helpful to test whether particular functionalizations of C70 molecules influence the trends that we explore here. A limitation of this work is the fact that we treat isolated monomers and dimers, whereas organic semiconducting contain fullerenes that are packed closely with many other

elucidated to form a truly predictive model of the effect of charges on couplings. For the qualitative purposes of visualizing the polaron charge density on a C70 monomer, tuning the range-separation parameter does not affect our results on the influence of point charges. However, in calculating the effect of point charges on couplings, it appears that tuning the value of the range-separation parameter does significantly influence our results. Using the default value of μ leads to moderately decreased effects of charges, given that the more sensitive structures all show a greater increase in coupling with the tuned value of μ. This may be partly due to the fact that the initial couplings are slightly lower when using the tuned value of μ. We might speculate that the greater delocalization of the polaron that we observed with the default value of μ in section 3.1 contributes to the greater couplings in the absence of point charges, but this interpretation is difficult to confirm with the present dimer calculations (in which we treat a neutral system and do not have any rigorous information on the extent to which a polaron in the dimer would localize or delocalize). For the various structures and arrangements of point charges, the coupling calculated with μ = 0.2 a−1 0 is always fairly close to the value calculated with μ = 0.33 a−1 0 , with all differences having a magnitude of ≤10%. Whether a difference of 10% in couplings matters depends on the question being asked; for our purposes, these variations are relevant. Using the default value of μ leads to a lower estimate of the effects of point charges relative to the using the tuned value of μ, with these differences particularly important when charges are placed at 8 Å (which is probably the most relevant position of charges, given the (100) d-spacing of 9.7(1) Å in fullerene crystals43). We must be cautious in assuming that the tuned value of μ provides more accurate values of the couplings relative to the default value, because the tuning procedure was based on gap fitting according to Koopmans’ theorem and did not involve any benchmarking in terms of couplings. Previous work has indicated that tuning the value of μ according to gap fitting leads to improvements in accuracy in terms of excitation energies49 and simulated optical-absorption properties;47 therefore, it is reasonable to suspect that the tuned value of μ also leads to improvements in accuracy in terms of electronic coupling. However, this cannot be proven with the present data; all we can say conclusively is that the value of μ does affect electronic couplings and it is worthwhile to consider optimizing μ for a particular system when calculating couplings.

4. CONCLUSIONS Increased charge density has been found to increase charge mobility in organic semiconductors in a variety of experimental and theoretical studies, with this effect usually explained by the filling of traps permitting trap-free transport for the remaining charges. Here, we use first-principles calculations with point charges adjacent to fullerene molecules to explore whether, irrespective of trap effects, charge density may affect electronic structure in ways that promote transport. The fact that adjacent charges will to some extent repel the electron density of a polaron on a fullerene molecule is expected; however, by rigorously calculating the charge density of a polaron and visualizing its isosurface, we gain insight into the range of distances over which these effects are significant. We find that these effects drop off quickly as the distance of the point charges is increased, suggesting that such an effect would only be present in extremely charge dense regions. Given the 21794



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Computational Science Graduate Fellowship, under grant number DE-FG02-97ER25308.

fullerenes and/or polymers. Any effects that may result from large numbers of nearby fullerenes/polymers are neglected here. Unfortunately, performing the calculations that we have described on groups of a large number of fullerenes is intractable due to problems of computational cost (particularly considering that a wide range of fullerene orientations would need to be considered to represent amorphous regions of an organic semiconductor). However, we believe that at least qualitatively, studies of the isolated monomer and dimer can provide information on the properties of fullerenes in bulk materials. Our results indicate that the positive correlation between charge density and mobility that has been presented in a range of experimental studies may, in limited regions with high charge density, be partially mediated by effects on electronic structure and couplings. These results are not in conflict with the common hypothesis that charge density influences mobility by allowing trap-free transport; rather, both mechanisms may contribute to the observed trends. We also note that the effects on electronic structure noted here would be present regardless of whether the polaron is experiencing the effects of another mobile polaron or of a trapped charge. In fact, the effects of trapped charges may be more significant than the effects of mobile charges, because such charges would not be pushed away by Coulombic repulsion. A system with a large number of trapped charges presents a very charge-dense landscape to each mobile polaron, in which the close proximity of charges that is necessary to influence electronic structure is a likely event. We conclude that in most systems, increases in mobility due to density effects on electronic couplings are probably rare events that do not significantly impact overall device efficiency, and trap effects dominate. However, in limited regions with extremely high charge density, increased density may promote charge transport not only via trap-filling but also via the mechanism of altered electronic couplings. Whether this infrequent event influences overall mobility will depend on the importance of these limited regions in charge transport in a particular device.

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ASSOCIATED CONTENT

S Supporting Information *

(1) A discussion of how the basis set and functional were chosen, (2) a description of how the range-separation parameter was chosen for the long-range corrected functional, (3) a visualization of polaron isosurfaces using isosurface values different from those used in the main text, and (4) geometry files for fullerene dimers. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: (630) 252-4580. Fax: (630) 252-4646. E-mail: darling@ anl.gov. Notes

The authors declare no competing financial interest.

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REFERENCES

ACKNOWLEDGMENTS

Use of the Center for Nanoscale Materials was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under contract no. DE-AC0206CH11357. K.M.P. acknowledges the support of the DOE 21795



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