ARTICLE pubs.acs.org/JPCC
Evaluation of Excitonic Coupling and Charge Transport Integrals in P3HT Nanocrystal Muhammet E. K€ose* Department of Chemistry and Biochemistry, North Dakota State University, Fargo, North Dakota, 58108 United States ABSTRACT: An attempt has been made to predict the crystallographic packing parameters of poly(3-hexylthiophene) (P3HT) nanocrystal with the help of both X-ray diffraction and computational methods. Excitonic coupling integrals were calculated using various methods such as point-dipole approximation (PDA), line-dipole approximation (LDA), ZINDO method, and transition density cube (TDC) method in crystallographic directions. ZINDO method leads to reasonably accurate integrals when compared to TDC results. PDA and LDA approaches fail badly in predicting the excitonic coupling integrals, though latter yields improved results. The calculations one more time show that PDA should be avoided in cases where interchromophoric distance is comparable to chromophore size. In general, the magnitude of both excitonic and charge transfer integrals are larger in stacking direction of chains than those in other crystallographic directions, suggesting preferential diffusion of both excitons and charge carriers between the chains. The hole transfer integrals were also found to be larger than those for electrons, confirming P3HT as p-type semiconducting material.
1. INTRODUCTION Organic photovoltaic devices usually consist of a donor polymer as light absorbing material and a fullerene derivative as electron acceptor in a bulk heterojunction device structure. Latest developments in device optimization as well as application of novel donor polymers led to power conversion efficiencies (PCEs) around 7%.14 These novel polymers usually have lower band gaps than well-known poly(3-hexylthiophene), P3HT, which has shown PCE more than 4%.5 The rapid development and applied nature of this field prevented the development of much needed understanding of exciton dynamics, charge transport, and charge dissociation mechanisms at donor/acceptor interfaces.6,7 Yet, P3HT is one of the most studied conjugated polymers in the field of organic photovoltaics.5,815 Apart from relatively high device efficiencies observed in bulk heterojunction devices of P3HT/PCBM blends, the semicrystalline behavior of P3HT films has important implications in charge transport and blend morphology.10,13 The effect of film preparing conditions, solvent choice, postprocessing film techniques has significant impact on the devices made from P3HT as well as the on the morphology of polymer film.5,12,1618 Regioregular P3HT films have ordered domains of crystalline phases at nanoscale.14,18 These nanoscale ordered domains of P3HT has been found to be parallel to the substrate surface and experiments show that plane of thiophene rings is normal to the substrate.19,20 The existence of such orientation is confirmed by the presence of multiple reflections (0 0 l), l = 13 in the X-ray diffraction (XRD) spectra. The cause of such preferred alignment is not well understood, though it is believed that the interactions between the hydrophobic P3HT side-chains and the hydrophilic SiO2 substrate cause such orientation of P3HT chains.21 r 2011 American Chemical Society
Although XRD patterns reveal important crystal geometry parameters of chain packing, it is not possible to predict the exact location of P3HT chains with respect to each other in the solid state. The packing geometry of the chains is vital in evaluation of accurate electronic coupling integrals for both understanding and simulation of energy transfer and charge transport dynamics in P3HT films. Even though P3HT is not a fully crystalline system, the semiordering of P3HT chains in the films are believed to influence the important device parameters such as exciton diffusion length, carrier mobility, and nanoscale blend morphology when mixed with fullerene derivatives.14,18,22 The goal of this work is to delineate crystal packing structure of ordered P3HT domains with the help of both experimental and theoretical methods and calculate electronic coupling parameters for both energy transfer and charge transport. To this end, we have utilized various theoretical approaches to calculate excitonic coupling integrals in various dimer configurations obtained within joint evaluation of theoretical and experimental results. In particular, we have compared and analyzed suitability of point-dipole approximation (PDA), line-dipole approximation (LDA), ZINDO method, and transition density cubed (TDC) methods for excitonexciton coupling interactions. Electronic splitting in dimer (ESD) approach was used in estimation of hole and electron transfer integrals.
2. METHODS X-ray Diffraction Measurements. Poly(3-hexylthiophene)2,5-diyls was purchased from Rieke Metals, Inc. and was used as Received: April 14, 2011 Revised: May 17, 2011 Published: June 03, 2011 13076
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received. X-ray diffraction spectrometer (Philips X’Pert MPD) was used for structural analysis of P3HT powder. The X-ray beam was generated by copper (KR) target, using a tube voltage of 45 kV at electron beam current of 45 mA. The scanning angle was every 0.05 step over the range of 240. Theoretical Approaches. The density functional theory (DFT) optimized geometries of all oligomers were obtained with B3LYP hybrid functional and 6-31G(d) basis set as implemented in Gaussian 03 suites of program.23 We have used plane symmetry for the conjugated backbone in optimization of geometries to save computational time due to the presence of long hexyl chains. Intermolecular interaction energies of TH10 dimers were calculated using ECEPP-05 force field, which has been parametrized for organic molecules with fixed bond lengths and angles. The details of the force field and the parameters can be found elsewhere.24 The intermolecular interaction energy is evaluated as follows Bij 332qi qj 6 EIIE ¼ Aij rij þ exp ð Cij rij Þ þ el k14 ijðj > iÞ ijðj > iÞ kij εrij
∑
∑
One has to utilize transition density and based on the distribution of the density coefficients on repeat units, the fractional dipoles should be calculated and then coupled with the other fractional dipoles on the interacting chromophore. The electronic coupling for pairwise dipoledipole coupling is given by ! kij μiD μjA ð4Þ VLDA ¼ Rij3 i, j 4πεo
∑
Similar to PDA, μiD and μiA are fractional dipoles on donor and acceptor chromophores and Rij is the distance between the dipoles on subunits of coupled chromophores. Equation 4 must satisfy following relations μD ¼
where μD and μA are the transition dipole moments of donor and acceptor chromophore, respectively. RDA is the interchromophore distance and k is the orientation factor of the dipoles f f f f k¼μ D μ A 3ðμ D n BDA Þðμ A n BDA Þ
ð3Þ
Here μBD and μBA are unit vectors in the direction of transition dipoles and B n DA is the unit vector connecting the centers of donor and acceptor transition dipoles. As mentioned above, PDA is not suitable for short interchromophore distances and is expected to break down for polymeric materials, especially in solid state. PDA does not take into account chromophore size and shape and therefore an improved model, namely, LDA, is proposed for interacting chromophores.26 LDA is an improved F€orster model that partially takes into account chemical structure and topology. Because polymers are composed of repetitive units, it is therefore reasonable to partition the dipole into small dipoles localized on monomeric moieties or repeat units.
ð5Þ
The most accurate electronic coupling calculation in energy transfer processes is given by a product of excited-state donor and ground-state acceptor with that of excited-state acceptor and ground-state donor25,27
ð1Þ Here, the calculated interaction energy (EIIE) is in the units of kcal 3 mole1. Aij, Bij, and Cij are nonbonded parameters for the atoms i and j separated by a distance of rij. The atomic charges, qi and qj, were obtained after a fit to reproduce the molecular electrostatic potential calculated at HF/6-31G(d) level on DFToptimized geometry. The scaling factors k14 and kijel are taken as unity since the interaction energy between two molecules is computed only. We have employed several theoretical methods to calculate the excitonic coupling integrals between two oligothiophenes. The excitonic coupling is most easily evaluated with PDA to describe the energy transfer based on F€orster mechanism.25 Although PDA fails badly when interchromophore distances are smaller than the size of the chromophores involved in energy transfer process, it is still widely used in the literature. The reason stems from the fact that it is easy to apply real systems and relatively meaningful results could be obtained in weak coupling regime. The electronic coupling term in PDA is obtained by using the following relation k μD μA ð2Þ VPDA ¼ 3 4πεo RDA
∑i μiD and μA ¼ ∑j μjA
V exc ¼ ÆψD ψjHDA jψD ψA æ
ð6Þ
where HDA represents the perturbation between two interacting chromophores. According Harcourt model,28 eq 6 can be expanded into terms that define Coulombic, exchange and some wave function overlap dependent interactions. In our analysis of P3HT nanocrystal packing configurations, we neglect all the exchange and wave function-overlap dependent terms as they will be much less significant in magnitude compared to the Coulombic coupling interactions. One can then reduce eq 6 further to the following form VTDC ¼
1 4πεo
D A
∑p ∑q
FD ðpÞFA ðqÞ rpq
ð7Þ
Here, FD and R FA are transition densities obtained using the equation; ∼ o,u Co,ujo(r)ju(r), with Co,u being CI expansion coefficients for the excited state whereas jo and ju are occupied and unoccupied molecular orbitals involved in the electronic transition, respectively. It is important to note that eq 7 is composed of two-electron integrals and directly related to the F€orster type energy transfer mechanisms.27,29 The transition dipole moment for the relevant excited state can be estimated by the following expression in x-coordinate μx ¼
∑i FðiÞxi
ð8Þ
Transition dipole moments can be estimated in the other axes in a similar fashion. We have checked the magnitude of each dipole moment for the chromophores used in this study. The slight deviations from what was calculated from transition density volumetric data and reported values in the Gaussian output were corrected by using appropriate scaling factors. The integrated transition density data over the space did not result in zero charge density in most cases. Therefore, we have added small amount of charge to each point in the transition density to minimize the errors that could be caused first order electrostatic interactions. However, even without these corrections, the calculated couplings have only 12% discrepancy. Quantum mechanical calculations on dimers of donor and acceptor chromophores can also be useful to estimate the excitonic interactions provided that the energy difference between 13077
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Figure 1. X-ray diffraction patterns of P3HT powder.
the lowest excited state and the higher lying state is much larger than the magnitude of electronic coupling.30 The half splitting in the excited state homodimer is equal to two-electron integral between the excited state of donor monomer and that of acceptor chromophore VZINDO ¼ ÆSD j
1 jSA æ jrpq j
ð9Þ
Here, ÆSD| and ÆSA| are the excited states of donor and acceptor, respectively. The excitonic coupling integral has been calculated using ZINDO method. ZINDO method is appealing due to low computational cost and good degree of accuracy compared to ab initio calculations.31,32 At this stage, we have utilized DFToptimized geometries for ZINDO calculations to evaluate the excitonic coupling integrals. The electronic coupling elements for hole and electron transfer were estimated as the half splitting of the HOMO and LUMO levels of the neutral state dimer oligothiophenes, respectively.33 The interchain transfer integrals are computed at the semiempirical INDO level for the stacks of oligothiophenes built along main crystallographic directions. ESD is known to yield erroneous results for nonsymmetric dimers,34 however it is not believed to be an issue for the symmetric dimer geometries studied here.
3. RESULTS AND DISCUSSION XRD of P3HT Powder. P3HT films exhibit a natural tendency to self-organize into crystalline lamellae. XRD spectrum of powder sample is shown in Figure 1. There are two major peaks observed at 5.5 and 23. The plane separations of these angles correspond to 16.6 and 3.9 Å, respectively. The latter is related to the lattice constant in the stacking direction of P3HT chains. The lattice constant in a-axis is then 16.7 Å. There are two more additional peaks in the XRD spectrum at 11 and 16. They are believed to be reflections of (0 0 2) and (0 0 3) planes, and also observed by others in the analysis of P3HT films and powders.10,19 P3HT Nanocrystal Packing Structure. We used a short oligomer (consists of 10 thiophene rings) of P3HT to determine the crystal packing geometry. Initially, optimizations with ECEPP-05 force field were carried out on rigid geometries to locate the stacking distance and relative orientation of P3HT chains. The stacking distance was estimated at 3.89 Å, which is in
excellent agreement with the experimental value. Cofacial stacking resulted in lateral displacements of chains along their short and long molecular axes. We have found two minima during a scan of relative displacements in longitudinal and traverse directions (Figure 2). The perfectly cofacial configuration is not the lowest energy conformation for the stack rather a displacement in the chain direction with l = 1.6 Å is estimated to be the most stable configuration for chain stacking.10 Yet, our calculations of intermolecular energies hint another possible configuration with t = 1.5 Å and l = 0.8 Å for P3HT chain packing, which is slightly higher in energy than the lowest energy configuration discussed above. The a-axis distance between the oligomers was also estimated with ECEPP-05 force field. However, the calculated a-axis (19.5 Å) is by far longer than the experimental result (16.7 Å) (Figure 3). Interdigitation should not occur since there is a large amount of steric repulsion when chains are forced to adopt an interdigitated structure. Packing geometries without interdigitation of alkyl groups have also been observed in other poly(3alkylthiophenes).35 Comparison of experimental and theoretical data indicates that P3HT chains should have nonplanar geometries in order to have such close a-axis separation. We then allowed chains to possess torsional degrees of freedom and later on estimated the packing geometry along a-axis. The predicted separation in a-axis has been found to be 16.8 Å and is in very good agreement with the experimental value (Figure 4). In such a conformation, the hexyl groups are twisted ∼110 with respect to the plane of thiophene backbone. It is important to note that the packing geometry in b-axis does not change with the nonplanar conjugated backbone. Excitonic Coupling. The magnitude of exciton diffusion length is an important factor for optimizing PCEs in organic solar cells. The extent of exciton diffusion is closely related to the efficient electronic coupling between interacting chromophores. The excitonic coupling integral has been evaluated with various theoretical methods as described in Methods. The goal here is to assess the applicability of various methods in estimating excitonic integrals as well as evaluate the relative strength of coupling integrals in crystallographic directions. The calculated excitonic coupling integrals are given in Table 1 for P3HT nanocrystal. The most accurate coupling integrals are estimated with TDC method.29 The coupling in stacking direction is approximately 6-fold higher than that in a-axis. We have also calculated the coupling integrals for the dimers brought together in chain direction. The relative strength of coupling in chain direction is similar to the coupling value in b-axis. The excitonic coupling integrals calculated with semiempirical ZINDO approach provide very similar results to TDC method. Although in a previous study ZINDO method has been found to underestimate excitonic coupling between thiophene oligomers,30 our calculations indicate that the coupling integrals match very well with those of renowned TDC method. The results from PDA largely overestimate the excitonic coupling integral in stacking direction and in contrast underestimate them in other directions. PDA predicts 8-fold larger electronic coupling integral in cofacial stacking than that obtained with TDC method. It is widely known that the F€orster coupling at short distances where intermolecular separation is comparable to chromophore sizes is problematic in estimating the electronic coupling between the interacting units.25 These calculations once more emphasize the inapplicability of PDA in evaluation excitonic coupling integrals and PDA should be 13078
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Figure 2. Scan of dimer interaction energies at various longitudinal and transverse directions at a stacking distance of 3.9 Å. The stacking motif in (a) has l = 0.8 Å and t = 1.5 Å offsets whereas those values are l = 1.6 Å and t = 0.0 Å in (b). Note that the molecular packing pictured in (b) stabilized by 0.03 kcal 3 mole1 per ring in comparison to packing in (a).
Figure 3. Dimer geometries used in predicting the excitonic and charge transport coupling parameters. Transition density of the thiophene decamer used in exciton coupling calculations is also shown. Purple color is for negative density while orange represents positive density.
avoided almost in all cases when dealing with conjugated materials in solid state.36 LDA provides better estimations than PDA but still lacks the required accuracy for exciton transport dynamics at variance with the conclusions of Barford who suggested justification of LDA between conjugated polymers for a chain of dimers.37 The fact that one should perform quantum mechanical calculations for generation of transition densities and utilize the transition density matrix to determine local dipoles makes LDA an unattractive method. Nonetheless, LDA still serves as an improved model over PDA.
We have considered a chromophore of 10-thiophene long above for excitonexciton coupling interactions. However, this is hardly the case in real polymeric materials where multiple chromophores (inhomogeneous broadening) contribute to the observed photophysical properties.25 Here we assumed chromophores spanning three to eight thiophene units and performed calculations for the excitons in crystal geometries. We first consider coupling along the chain direction where excitons sit next to each other. Table 2 summarizes the results of pairwise coupling of different excitons with TDC and PDA approach. 13079
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Figure 4. Predicted packing geometry of P3HT decamer chains in a-axis.
Table 1. Comparison of Coulombic Coupling Integrals Estimated with Various Methods in the Crystal Geometries Shown in Figure 3 decamer dimer
point dipole approximation (meV)
line dipole approximation (meV)
ZINDO (meV)
transition density cube method (meV) 5.4
x-axis (chain dir.)
0.9
2.6
7.1
a-axis
4.4
3.2
6.0
5.0
256.9
179.3
32.1
32.0
b-axis (stacking dir.)
Table 2. Calculated Excitonic Coupling Integrals with TDC Method for Various DonorAcceptor Chromophores along the Chain Directiona excitonic coupling in chain direction (in meV) TH8 TH7
TH8 11.8 (2.6)
TH7
TH6
TH5
TH4
TH3
12.4 (2.9)
16.0 (3.2)
15.4 (3.4)
15.2 (3.6)
14.8 (3.5)
13.2 (3.3)
16.4 (3.6)
16.4 (4.0)
17.0 (4.2)
17.0 (4.2)
20.0 (4.1)
18.4 (4.6) 22.2 (5.2)
19.2 (5.0) 22.0 (5.9)
19.0 (5.2) 22.2 (6.3)
TH6 TH5
27.2 (6.9)
TH4
26.4 (7.7) 30.6 (9.3)
TH3
a
Electronic coupling integrals obtained using point dipole approximation are given in the parentheses. As an example, transition density plots for TH8TH5 donoracceptor electronic coupling is shown along with their transition dipole moments (green arrows). Atoms are not illustrated in the transition density plots.
PDA results underestimate the TDC values by approximately 4 times, but still capture the trend exhibited with the TDC method. The excitonic coupling integrals decrease among THxTHx series as x increases. Although the strength of transition dipole moments decreases as the oligomer size decreases, the shrinking separation distance between the centers of transition dipoles causes stronger coupling between the excitons. In each column in Table 2, the THxTHy coupling integrals increase with the decreasing size of interacting excitons as well. This trend shows relatively larger electronic coupling values can be
obtained with the excitons localized on short thiophene oligomers. The excitonic interactions in between the donor and the acceptor chromophores of the same size are listed in Table 3 along a- and b-axes. There is a marked difference between PDA results and those of TDC method for stacked chromophores. As the strength of transition dipole moments decrease with the size of the chromophores, PDA results exhibit monotonically declining trend of electronic coupling integrals; in contrast, TDC predicts slight variation of excitonic coupling integrals. Interestingly, 13080
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Table 3. Excitonic Coupling Integrals with TDC Method Are Given for Various DonorAcceptor Chromophores in Stacking Direction with Molecular Offsets, l = 1.6 Å and t = 0.0 Åa donor/acceptor
b-axis (stacking dir.) (meV)
a-axis (meV)
Table 4. Charge Transfer Integrals for the Crystal Geometries Shown in Figure 3 decamer dimer
thole (meV)
telectron (meV)
25.9
21.0
x-axis (chain dir) a-axis
TH8/TH8
39.2 (314.3)
6.0 (5.4)
TH7/TH7
41.2 (259.8)
5.4 (4.5)
TH6/TH6
48.8 (204.1)
5.0 (3.5)
TH5/TH5
43.2 (151.9)
4.0 (2.6)
TH4/TH4
41.0 (102.0)
2.6 (1.8)
TH3/TH3
33.6 (58.2)
1.6 (1.0)
a
Electronic coupling integrals obtained using point dipole approximation are also given in the parentheses.
the largest interchromophore interactions are estimated with excitons that span six thiophene units according to TDC method. Similar calculations performed by Gierschner et al. resulted in a peak electronic integral value for a perfect cofacial TH4 dimer.36 However, they also reported the shift of maximum excitonic transfer integral to TH57 dimers at large separations (812 Å). This is expected since the perfect cofacial geometry adopted in their calculations does not represent the real case and its effect is diminished at large chromophore separations. PDA values are quite large in comparison to those of TDC method. These results again confirm that PDA should be avoided in all types of Coulombic coupling calculations as it can lead to severe errors in estimation energy transfer rates. PDA integrals have similar magnitudes with those of TDC in a-axis, the accuracy of PDA results are, however, still not good enough despite the similar trend observed by both approaches. Comparison of the magnitude of typical coupling values between cofacial dimer and adjacent chromophores on the same chain demonstrate that intrachain coupling is in general weaker than interchain coupling between the chromophores. For instance, TH8TH8 coupling integral in stacking direction is 39.3 meV whereas the coupling of similar size excitons along the chain is only 11.8 meV. Yet for smaller exciton such as TH3, the coupling integrals are very close to each other (33.6 meV vs 30.6 meV). Within this context, it is appropriate to discuss whether intramolecular or intermolecular exciton transport is faster for conjugated polymers in solid state. For P3HT nanocrystal, the excitonic coupling integrals in the stacking direction are expected to be larger than those along the chain direction. This is due to relatively short interchromophore separation distance between the ordered chain segments; however, this may not be true for amorphous polymers where average separation distance between the chains are longer than the ideal π-stacking distance observed in P3HT nanocrystal. Charge Transfer Integrals. The estimated hole and electron transport integrals with INDO method is given in Table 4. The magnitude of charge transfer integrals along b-axis is much larger than those in other crystallographic directions. Though, better comparison of charge transfer integrals along the chain direction should be performed for chromophores adjacent to each other as has been done above for excitonic interactions. Chromophore size should be varied to obtain in depth analysis of magnitude of charge transfer integrals and that will be the focus of another study. The hole transport coupling integral in cofacial configuration is roughly 2.5 times larger than electron transport integral, suggesting faster mobility of p-type carriers. Indeed, hole mobility
b-axis (stacking dir)
1.0
2.1
120.1
48.0
of P3HT is twice as large as its electron mobility measured with time-of-flight carrier mobility measurement technique.38 The other potential stacking geometry as shown in Figure 2A also has hole coupling integral much larger (66 vs 14 meV) than that for electron transfer, implying holes again as faster carriers in P3HT nanocrystal. The coupling integrals in a-axis, however, are a lot smaller than those in other axes. That means that the dominant pathways for carriers are along the stacking direction and in chain axis. Indeed, several research groups reported an increase in the mobility of carriers when the crystallinity of P3HT film increases.39,40 Others, however, underscored the importance of intergrain transport and reduced bridging of crystalline domains as the limiting situation for the macroscopic charge transport.41,42 Nanocrystalline domains of P3HT are perpendicular to the substrate surface and the plane of conjugated backbone is normal to the surface.13,19,43 Therefore, the weak coupling in a-axis should lead to poor mobility out of the P3HT nanocrystal when sandwiched between two electrodes, which could have significant impacts for device applications in organic electronics. One salient feature distinguishing carrier hopping from exciton migration is more evident preferential diffusion of holes and electrons in the medium. Thus, orientation of the interacting units plays a critical role in particle dynamics. The strength of excitonic coupling is usually determined by the distance, size, and relative orientation of chromophores. On the other hand, small lateral displacements in cofacial stacks for instance will have little impact on the magnitude of excitonic coupling integral. Yet, similar lateral displacement might have a huge impact on transfer integrals. The relative strength of excitonic coupling and charge transfer integrals in a-axis vs b-axis vary greatly for the latter, suggesting more preferential diffusion of charge carriers in P3HT nanocrystals.
4. CONCLUSIONS The combination of several theoretical approaches allowed us to estimate exciton transfer integrals and charge transfer integrals in crystallographic directions in ordered P3HT domains. PDA and LDA methods perform poorly in estimating the excitonic coupling between chromophores. On the other hand, ZINDO method yields reasonable estimates to the true coupling integral as verified by TDC approach. Both excitonexciton interactions and charge transfer integrals are largest in magnitude along the chain stacking direction. The relatively larger hole transfer integrals are in agreement with the experimental results for P3HT being a p-type semiconductor. Future studies from our group will focus on utilizing estimated coupling integrals to simulate exciton and charge transport dynamics with Monte Carlo methods for P3HT polymer. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. 13081
dx.doi.org/10.1021/jp203497e |J. Phys. Chem. C 2011, 115, 13076–13082
The Journal of Physical Chemistry C
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dx.doi.org/10.1021/jp203497e |J. Phys. Chem. C 2011, 115, 13076–13082
