On the Inapplicability of Electron-Hopping Models for the Organic

Mar 8, 2013 - Thus, we decided to delete the solvent molecules in the tc and mc structures to allow for a more consistent comparison with the model st...
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On the Inapplicability of Electron-Hopping Models for the Organic Semiconductor Phenyl-C61-butyric Acid Methyl Ester (PCBM) Fruzsina Gajdos,† Harald Oberhofer,‡ Michel Dupuis,§ and Jochen Blumberger*,† †

University College London, Department of Physics and Astronomy, London WC1E 6BT, United Kingdom Technical University Munich, Theoretical Chemistry, Lichtenbergstr. 4, D-85747 Garching, Germany § Pacific Northwest National Laboratory, Richland, Washington, United States ‡

S Supporting Information *

ABSTRACT: Phenyl-C61-butyric acid methyl ester (PCBM) is one of the most popular semiconductors in organic photovoltaic cells, but the electron-transport mechanism in the microcrystalline domains of this material as well as its preferred packing structure remain unclear. Here we use density functional theory to calculate electronic-coupling matrix elements, reorganization energies, and activation energies for available experimental and model crystal structures. We find that the picture of an excess electron hopping from one fullerene to another does not apply for any of the crystalline phases, rendering traditional rate equations inappropriate. We also find that the cohesive energy increases in the order body-centered-cubic < hexagonal < simple cubic < monoclinic < triclinic, independently of the type of dispersion correction used. Our results indicate that the coupled electron-ion dynamics needs to be solved explicitly to obtain a realistic description of charge transfer in this material. SECTION: Energy Conversion and Storage; Energy and Charge Transport

O

solving the coupled electron-ion dynamics in fullerene materials from first-principles. Our current theoretical understanding of charge transport in OSCs is still far from complete, especially at ambient and high temperatures. Historically, two descriptions have been advocated, one based on a band-like (delocalized) conduction mechanism similar to the case of inorganic semiconductors7,8 and the other based on polaronic (localized) charge carriers hopping between molecules or molecular fragments.9,10 None of these limiting cases seem to provide a fully satisfactory description. The band-like mechanism relies on “Bloch” states and breaks down for higher temperatures where the mean free path becomes comparable to the intermolecular lattice spacing.11 The hopping model may be used to describe thermally activated transport in crystals with defects that act as traps12 or in disordered films where the mobility was shown to increase with temperature.13,14 However, for ultrapure single crystals of C60,15,16 pentacene,17 and rubrene,18 charge transport was found to decrease with increasing temperature, even at room temperature, indicating that activated hopping models are inappropriate for these systems. More recently, it has been suggested that charge carriers could diffuse through OSCs as localized waves, with localization not being induced by local lattice distortions and the associated reorganization energy λ (also referred to as “local electron−

rganic semiconductors (OSC) are envisaged as one of the most promising materials for alternative solar cell technologies.1−3 They are light and flexible and relatively cheap and easy to produce from renewable sources, in contrast with most inorganic semiconducting materials. They have also found first applications in organic light-emitting diodes and plastic electronics, and they are envisioned in many other applications as well such as e-paper and radio-frequency identification tags. However, a major disadvantage of OSCs is their modest charge conduction, which ultimately limits the performance of these devices. Thus, much effort is currently being invested both experimentally and theoretically to synthesize and process OSCs with high charge mobilities4 and to understand charge transport and structure−property relationships in these materials.5,6 Here we focus on the intrinsic electron-transport mechanism in crystalline phenyl-C61-butyric acid vinyl ester (PCBM), which is arguably the most common electron-accepting material in bulk-heterojunction OSCs. Carrying out dispersioncorrected density functional theory calculations, we find that the excess electron (polaron) is not localized on a single PCBM molecule in any of the experimentally determined crystal structures. This result is a consequence of the inherent electron-transfer parameters describing ET between two PCBM molecules and is expected to remain valid if thermal fluctuations are taken into account. Consequently, traditional charge-hopping models, which have been used extensively by the community, are inappropriate for the description of charge transfer in this system. Instead, our results point to the need for © 2013 American Chemical Society

Received: January 30, 2013 Accepted: March 8, 2013 Published: March 8, 2013 1012

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phonon coupling”) but by thermal fluctuations of the electronic coupling matrix element Hab between neighboring molecules (also referred to as “nonlocal electron−phonon coupling”).8,19−21 Yet another recent theoretical investigation proposed the simultaneous existence of band-like and dynamically localized states with the former dominating below room temperature and the latter dominating for higher temperature.22 In previous work, we have investigated charge transport in crystalline C6023 at room temperature and in the limit of a very low excess electron concentration (to be distinguished from fullerenes with a stoichiometric number of excess electrons such as alkali-doped C60 salts24). We have found that for a small but significant fraction of thermally accessible configurations in the fcc-crystal lattice a small polaron cannot be formed. We investigate if this initial finding can be generalized to a wider class of fullerene materials, specifically to PCBM. It is not obvious whether our findings for C60 also apply to PCBM because the presence of the butyric-vinyl ester side chain in the latter material gives rise to profound differences in the ionic and electronic structure and in the lattice dynamics. Whereas unsubstituted C60 forms only the fcc-phase above 255 K25 PCBM was reported to form triclinic (tc),26 monoclinic (mc),26 and hexagonal (hex)27,28 crystals depending on the solvent and crystallization conditions. (See Figure 1.) The different packing

experimental value at 255 K (∼14.16 Å25) by mc > sc > hex > bcc (numerical values summarized in Table S1 in the SI). The energetic order for the model structures is in agreement with previous LDA-based calculations, which gave sc > hex > bcc.32 For each of the optimized crystal structures we determined the unique directions for electron transfer between pairs of PCBM molecules including the nearest and next-nearest neighbors. (The coupling for more distant pairs becomes negligibly small.) The corresponding electronic coupling matrix elements (Hab) were calculated according to eq 1 using the fragment orbital density functional theory (FODFT) approach of ref 23 Hab = ⟨ϕDN + 1|hbKS|ϕAN + 1⟩

(1)

N+1 where hKS b is the one-particle Kohn−Sham Hamiltonian, ϕD is the singly occupied molecular orbital (SOMO) of the negatively charged donor molecule PCBM−, and ϕN+1 is the A SOMO orbital of the negatively charged acceptor molecule PCBM−. The orthogonalization of the two sets of Kohn−Sham orbitals for donor and acceptor molecules was carried out according to Lowdin. The assumptions underlying eq 1 were extensively discussed in our previous work.23,33 The electronic level structure of PCBM is described in detail in the SI and illustrated in Figure S1. The electronic couplings obtained are summarized in Tables 1 and 2 for tc- and mc-PCBM and in Table S2 in the SI for hex-, sc-, and bcc-PCBM. The notation [xyz]nm denotes ET along the crystallographic direction xyz from the electron donating PCBM molecule n to the accepting PCBM molecule m, as defined in the deposited cif file.26 The couplings for all crystal structures and directions are shown in Figure 2 as a function of the distance (R) between the centers of mass of the two PCBM molecules. The tc and hex structures exhibit the largest values, followed by the mc and sc structures and the bcc

Figure 1. Experimental crystal structures of PCBM (A,26 B26) and model structures obtained in the present work from DFT energy minimization (D−F). An enlarged view of a PCBM dimer is shown in panel C.

motifs will have a strong impact on electronic coupling, which will no longer be isotropic as in the case for C60. Moreover, the presence of the dipolar side chain is expected to lead to an increase in the ionic response to charge transfer compared with C60. The computations that we report in the following are detailed in the Supporting Information (SI). Two dispersioncorrected (DC) PBE functionals are used: one based on dispersion-corrected atom-centered pseudopotentials (PBEDCACP)29 and the other based on the correction according to Grimme (PBE-D).30 We have investigated their accuracy by optimizing the lattice constant of fcc-C60 and obtain a value of 14.25 ± 0.05 Å for both functionals, overestimating the 1013

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Table 1. Electron-Transfer Parameter for Triclinic PCBM in the Minimum Energy Structure ET direction

R (Å)

Hab (meV)

λ (meV)

ΔE‡ad (meV)

[11̅1]24 [010]12 [010̅ ]34 [010]13 [000]13 [1̅10]12 [110̅ ]34 [000]41 [000]32 [011̅]42 [000]21 [000]43 [1̅1̅1̅]41 [001̅]32 [100]21 [100]34 [100]14 [100]32 [1̅10]42 [100]11 [100]22 [100]33 [100]44 [001̅]31 [001̅]42 [110̅ ]31

9.87 9.88 9.88 9.91 9.92 9.95 9.95 10.00 10.00 10.08 10.11 10.11 10.18 10.18 13.20 13.20 13.42 13.42 13.68 13.83 13.83 13.83 13.83 14.19 14.51 14.53

17.31 44.64 37.26 9.74 41.78 14.87 25.24 99.39 151.34 82.73 12.96 39.86 3.09 8.27 0.02 0.35 0.52 0.48 3.28 0.46 0.10 0.83 0.72 3.72 17.92 0.10

137 138 138 138 138 138 138 138 138 138 138 138 138 138 146 146 146 146 147 147 147 147 147 148 148 148

19 3 7 25 4 21 14 −13 −20 −9 23 5 32 27 36 36 36 36 33 36 37 36 36 33 21 37

kET (Hz) 6.03 2.06 1.72 2.10 1.93 4.63 1.07 3.90 5.03 3.42 3.59 1.83 1.97 1.50 8.40 2.13 4.64 4.02 2.01 3.66 1.68 1.19 8.98 2.57 5.68 1.80

× × × × × × × × × × × × × × × × × × × × × × × × × ×

1011 1012 1012 1011 1012 1011 1012 1012 1012 1012 1011 1012 1010 1011 105 108 108 108 1010 108 107 109 108 1010 1011 107

Table 2. Electron-Transfer Parameter for Monoclinic PCBM in the Minimum Energy Structure ET direction

R (Å)

Hab (meV)

λ (meV)

ΔE‡ad (meV)

[010]41 [010]32 [010]31 [011]21 [000]12 [001̅]42 [100]23 [000]14 [001]̅ 43 [010]43 [010]22 [010]44

9.99 9.99 10.10 10.18 10.18 10.22 12.97 12.97 13.15 13.15 13.76 13.76

2.85 2.92 20.48 28.28 20.38 49.21 25.26 1.42 1.06 1.09 0.15 0.08

138 138 138 139 139 139 145 145 146 146 147 147

32 32 17 12 17 1 15 35 35 35 37 37

kET (Hz) 1.68 1.77 7.83 1.23 7.75 2.24 9.90 3.62 1.97 2.10 3.96 1.03

× × × × × × × × × × × ×

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

schematically illustrated to the right of Figure 3. In the orientation shown in panel A the orbital of the donor (red/blue for positive/negative isosurfaces) overlaps constructively with the orbital of the acceptor (orange/silver for positive/negative isosurfaces). Hence, Hab is large. In panel B, the separation distance remains the same but the lobes of the interacting orbitals are rotated by 90° with respect to one another. Consequently, the overlap is destructive and Hab drops by a factor of ∼40. Finally, in panel C the overlap is constructive again but the distance is increased by 4.5 Å, resulting in a coupling that is in between the values for A and B. It appears that up to the second nearest neighbor distance of 14.5 Å orientation is even more important than distance for maintaining high electronic couplings. The second essential parameter determining the ET mechanism is reorganization energy. It was divided into an inner-sphere contribution, λi, obtained from DFT calculations

structure. The latter exhibits by far the smallest coupling values. Overall, the data for Hab span five orders of magnitude, ranging from 151 meV for the [000]32 direction in the tc structure (R = 10 Å) to 0.01 meV for the [1̅10] direction in the sc structure (R = 14.5 Å). A striking feature is the nonexponential decrease in Hab with distance R. We can explain this behavior by considering three different donor−acceptor configurations in the tc crystal (Figure 3). In the configurations shown in panels A and B, Hab takes values of 151 and 4 meV although the donor− acceptor distance is the same, R = 10.0 Å. Conversely, Hab takes rather similar values for donor−acceptor distances that differ by several angstroms; compare panels B and C. Clearly, in addition to distance it is the orientation of the PCBM molecules that determines Hab. The latter is proportional to the overlap of the electron-donating and -accepting orbitals, which in turn depends on the orientation. The overlap of the orbitals is 1014

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reorganization energy is small due to the rigidity of the fullerene C60 cages, which was shown to scale as 1/N for conjugated organic molecules, where N is the number of π electrons.9 Our value for PCBM is only slightly larger than that for pure C60 (93 meV), indicating that the reorganization contribution from the side chain is small but should not be neglected. Compared with previous estimates we find that the PBE functional gives a somewhat smaller λi than hybrid functionals (136 meV for B3LYP/6-311G*,34 0.14 eV for B3LYP/6-31G*35). The small outer-sphere contribution is due to the low static dielectric constant of fullerenes. The value for λo may be slightly underestimated because we have used the static dielectric constant of pure C60, which may in fact be somewhat larger for PCBM due to the dipolar side chain. Considering these uncertainties, a total reorganization energy of 0.14 to 0.20 eV seems to be an appropriate range for crystalline PCBM. Inserting the computed electronic couplings and reorganization energies into eq 2, we obtain the adiabatic activation free energy for electron transfer, ΔE‡ad

Figure 2. Electronic coupling matrix element (Hab) as a function of the distance (R) between the centers of mass of two PCBM molecules. Each data point corresponds to a unique electron-transfer direction in the tc, mc, sc, bcc, and hex crystals. The reorganization energy (λ) is shown in black lines and the coupling value Hab = (3/8)λ, for which the activation energy for electron transfer vanishes (ΔE‡ad = 0), in red lines.

‡ ΔEad =

−

⎡ (λ + ΔE)2 λ + ΔE − ⎢|Hab| + 4λ 2 ⎣⎢

⎤ (λ + ΔE)2 + |Hab|2 ⎥ ⎥⎦ 4

(2)

The values are compiled in Tables 1 and 2 and Table S2 in the SI for zero electron-transfer energy (ΔE = 0), which was found to be negligibly small for all structures. The activation energies are, in general, very small due to the low reorganization energy, with values not exceeding 1.5kBT at 300 K. Importantly, we find that in the tc and hex structures there is at least one direction where the activation energy is negative. Similarly, also in the mc and sc structures there exist directions where the activation energy almost vanishes. The criterion for the activation energy to vanish, ΔE‡ad = 0, is fulfilled when Hab = (3/8)λ at ΔE = 0. This limit is indicated in Figure 2 in red lines. Remarkably, all structures exhibit coupling values that are beyond or very close to this limit except in case of bcc-PCBM, where nearest neighbors are relatively distant. The physical picture that emerges from our calculations is illustrated in Figure 4, where we show the Marcus parabola for

Figure 3. Interpretation of electronic coupling values by consideration of orbital symmetry. The isosurfaces for the SOMO of the PCBM− donor molecule are shown in red/blue and the ones for the acceptor molecule are shown in orange/silver. The symmetry of the lobes of the orbitals where the interaction is the greatest, that is, between donor and acceptor, is indicated in the panel to the right. The distance between the centers of mass of the PCBM donor and acceptor molecules and the corresponding electronic coupling values are indicated for each pair.

Figure 4. Localization of excess charge in an electron donor−acceptor (D A) system. Diabatic and adiabatic energy curves for electron transfer are drawn in dashed (black) and solid lines (red), respectively. In panel A, electronic coupling is much smaller than reorganization energy, Hab ≪ λ, so that the excess electron is fully localized on the donor. In panel B, electronic coupling is larger but still relatively small, Hab < (3/8)λ. Consequently, only a small fraction of the excess electron is delocalized over the electron acceptor. In panel C, electronic coupling is so strong, Hab ≥ (3/8)λ, that the transition state in panels A and B turns into a minimum. As a consequence, the excess electron is delocalized over donor and acceptor.

and an outer-sphere contribution, λo, obtained from continuum theory. See the SI for computational details. Both contributions are rather small, λi = 112 meV and λo = 25−36 meV, giving a total reorganization energy of λ = 137−148 meV depending on the separation distance of the C60 cages. The inner-sphere 1015

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three different ET regimes. In the panel to the left Hab ≪ λ, in the middle panel Hab < λ, and in the right panel Hab > (3/8)λ. The former two cases can be regarded as nonadiabatic and adiabatic ET (see ref 23 for a more precise definition) between two localized states. In the third case, the electronic coupling is so strong that charge-localized states do not exist. At this point, the transition state for ET turns into a minimum and the excess charge is delocalized over donor and acceptor. This is the case for electron transfer between two PCBM molecules in the experimental tc structure and in the modeled hex structure. If one takes into account thermal effects and zero-point vibrational motion (ℏω = 62 meV for the C60 breathing mode), then the electron transfer is also activationless for mc-, sc-, and bcc-PCBM. Thus, the picture of an excess electron hopping from one fullerene to another does not apply to crystalline PCBM. If one does insert the computed electron transfer parameters for PCBM into the semiclassical rate equation ‡ kET = κelνn exp( −β ΔEad )

approximations, the electronic coupling matrix elements they report are on the same order of magnitude as the couplings we obtain here from a full DFT fragment-orbital approach using a plane-wave basis set. This is a technical but important detail because ultimately the transport mechanism will depend on how the magnitude of the off-diagonal elements compares with the differences in site energies. In addition to the monoclinic structure that Cheung and Troisi investigated, we considered here a number of additional crystal structures (albeit without including dynamics), especially the triclinic structure that according to our calculations exhibits the highest cohesive energy, thus generalizing the results to essentially all crystal structures of PCBM known to date. In conclusion, the nonexistence of a finite barrier for certain ET directions makes the application of standard ET rate theories inapplicable. We have previously shown that this is the case for crystalline C6023 and we have shown here that this key result also holds for crystalline PCBM and is very likely to hold for many other ordered fullerene materials unless chemical substituents give rise to large reorganization energies. Alternative transport models such as Landauer theory can be used if the electrons are faster than the nuclei, that is, in the opposite limit for which hopping is valid. However, Landauer theory breaks down if the electron-nuclei coupling is large, as is the case for OSCs. Clearly, approaches that integrate over the electronic or ionic degrees of freedom at the outset of the calculation should be avoided, suggesting that nonadiabatic methods such as Ehrenfest or surface-hopping molecular dynamics be more appropriate. The current calculations of electron-transfer parameter will be used for further developments along these lines.

(3)

where κel is the electronic transmission coefficient (dependent on Hab, see ref 23 for explicit expression) and νn is the effective nuclear frequency (C60 breathing mode), then one obtains rates in excess of picoseconds. (See Tables 1 and 2.) This is faster then the nuclear relaxation times. Hence, a crucial assumption of ET rate theory is violated. Nonetheless, electron mobilities based on similar rate equations seem to give reasonable estimates for C6023,36 and fullerene derivatives.34 Thus, it seems that the frequency factor κelνn in eq 1 is, perhaps fortuitously, a useful effective descriptor for the electron dynamics of the system, even though the underlying charge-transport mechanism bears no similarity to the hopping model for which the frequency factor was derived.37 We would like to note that the factors that make the hopping model inapplicable are not tied to the fact that we have studied only static crystal structures. Even if we allow for thermal fluctuations, there will be at any instant in time a significant fraction of donor−acceptor pairs where the coupling is so large that the hopping model breaks down. This is because the crystal structure represents the potential energy minimum and retains a large Boltzmann weight at room temperature. We also expect that the presence of solvent molecules will not change this picture. It seems unlikely that the nearest neighbor couplings at van der Waals separation will be significantly affected by solvent molecules in lateral directions. Of course, it is possible that second nearest neighbor couplings are affected, in particular, when solvent molecules are in between donor and acceptor. For these configurations, the couplings are expected to be higher than the current estimates, which assume vacuum between donor and acceptor. However, this will even further strengthen our conclusion that hopping models are inappropriate. Our findings are in agreement with a previous analysis of charge transfer in PCBM by Cheung and Troisi, even though these authors used rather different methods to arrive at this result. Classical molecular dynamics simulation of monoclinic PCBM was carried out and electronic couplings were sampled along the trajectory. To make the calculations feasible, electronic couplings between two PCBM molecules were approximated by the overlap of the lowest unoccupied molecular orbitals of methyl-C60 times a proportionality factor, and the molecular orbitals were approximated by the semiempirical ZINDO method. Despite these rather drastic

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ASSOCIATED CONTENT

S Supporting Information *

Supporting results (computational methods, electronic level structure of C60 and PCBM, Figure S1, and Tables S1−S2). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS F.G. is supported by an IMPACT Ph.D. studentship cosponsored by University College London and Pacific Northwest National Laboratory (PNNL), H.O. is supported by a Humboldt Society Research Fellowship, and J.B. is supported by a Royal Society University Research Fellowship. M.D. was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences (BES), Division of Chemical Sciences, Geosciences and Biosciences. PNNL is a multiprogram national laboratory operated for DOE by Battelle. We acknowledge that the results in this paper have been achieved using the PRACE Research Infrastructure resource JUGENE based in Germany at Juelich and the United Kingdom’s High Performance Computing Materials Chemistry Consortium (funded by Engineering and Physical Sciences Research Council, EP/F067496) for access to the high-performance computing facility HECToR. 1016

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