Locally Broken Crystal Symmetry Facilitates Singlet Exciton Fission

Letter pubs.acs.org/JPCL

Locally Broken Crystal Symmetry Facilitates Singlet Exciton Fission Piotr Petelenz* and Mateusz Snamina The K. Gumiński Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, Ingardena 3, 30-060 Kraków, Poland ABSTRACT: Photovoltaic yield is normally limited to at most two charge carriers per photon. In solid pentacene this limit may be potentially bypassed owing to singlet exciton fission into a pair of triplets. The process occurs via a superexchange mechanism mediated by charge-transfer (CT) configurations and is sensitive to their energies. As demonstrated recently, these strongly depend on the local environment of the two molecules on which the charges reside. Using a multiscale model, here we show that in the crystal bulk approximate local symmetry affects CT state energetics in a way unfavorable for fission, so that at the places where this symmetry is broken the fission probability is enhanced by up to an order of magnitude. These fission-favorable locations entail the vicinity of vacancies, specific impurities, and interfaces, such as crystallite boundaries. Hence, photovoltaic yield might be substantially increased by using nanoscopically disordered pentacene rather than highly ordered specimens.

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The above observation is rooted in the dimer model,4−8,10 parametrized by ab initio calculations of Zeng et al.5 The minimum basis needed to describe it consists of two Frenkel state diabats, each representing the lowest intramolecular singlet excitation residing at one of the moieties (eg and ge), two CT configurations of opposite polarities (ac and ca), and the singlet component of the triplet pair state (tt); the notation follows ref 5. The energies of the diabatic configurations involved and the off-diagonal matrix elements of the Hamiltonian that govern their mixing are schematically depicted in Figure 1. Only the Frenkel configurations carry substantial transition moment from the ground state, which (or the ensuing population) may be transmitted to the eigenstate of tt parentage. Direct coupling between the pertinent diabats being negligible,5,8−10 this is mediated by the two virtual CT states of opposite polarity and is governed by the effective superexchange coupling constant introduced in ref 8. The crucial point is that the superexchange terms contributed by these two channels are governed by charge-transfer integrals of opposite signs (−90 and 64 meV,5 as visualized), so they interfere destructively.8 The leading contribution comes from the lower CT state, owing to its larger coupling constants and smaller energy separation from the neutral configurations. The large splitting (up to 0.8 eV) between the two CT diabats of the isolated model dimer is due primarily to intermoiety charge-quadrupole (CQ) interactions.5 When the dimer is embedded in the perfectly ordered pentacene crystal matrix, these are counterbalanced by their counterparts that engage the surrounding molecules.11 The compensation, strictly regimented by lattice symmetry, is almost complete, diminishing the energy separation between the two CT

olid pentacene is a promising material for photovoltaic applications. This is due to the proclivity of its lowest excited singlet state to undergo fission into a pair of triplets, potentially leading to an increased yield of charge carriers per photon. The process is 100% effective; it occurs on the femtosecond scale1 and preserves excitation-induced vibrational coherence.2,3 The obvious model to study fission is a dimer representing the nearest-neighbor pair of molecules extracted from the pentacene crystal.4 The calculations of its electronic eigenstates, carried out independently within the extended multiconfigurational quasi-degenerate perturbation theory (XMCQDPT)5 and the active space decomposition strategy,6 provide a reliable set of parameters to use in multiscale models. The latter approach has the additional advantage of dealing with the pertinent dimer embedded in its crystalline environment (disregarding, however, the change of dimer-environment polarization energy upon excitation). Although parametrized in a less sophisticated method, the papers by Berkelbach et al.7−9 represent the most complete methodology to date, including the effects of vibronic coupling and of the coupling to the surrounding thermal bath. In the most recent paper,9 the crystal was simulated by a large cluster, including the interaction of the pertinent dimer with the surrounding crystal matrix. Our present Letter is concerned with some peculiar consequences of this interaction. According to the present consensus,5−10 fission in pentacene occurs via superexchange mechanism mediated by chargetransfer (CT) configurations, and for this reason it is sensitive to their energies. These strongly depend on the local environment.11,12 In contrast to the common situation where crystal field splits the energy levels of the crystal contituents, in this case it almost entirely obliterates the energy gap between the CT states of opposite polarities. This is unfavorable for fission. © XXXX American Chemical Society

Received: April 5, 2016 Accepted: May 6, 2016

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DOI: 10.1021/acs.jpclett.6b00746 J. Phys. Chem. Lett. 2016, 7, 1913−1916

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The Journal of Physical Chemistry Letters

other positions. In the CT state diagonal energies the interaction with the surrounding crystal molecules was evaluated by means of the SCCF method11 with extrapolation to the infinite crystal limit based on the results of Tsiper and Soos,14 as in ref 11. Vibronic interactions were implicitly included in the strong coupling approximation by consistently scaling all CT integrals by the approximate vibrational overlap factor of exp(−1), as had been previously done by Yamagata et al.13 and Zeng et al.5 In view of complex vibronic activity in acenes,15 this approach, rooted in single-mode approximation, would be insufficient to describe the dynamics but should suffice for energy considerations. The lowest eigenstate of the Hamiltonian described above is dominated by the spectroscopically mute tt configuration, but the Frenkel admixture it contains opens it to direct generation by light absorption, which yields the fission product without populating the eigenstate of Frenkel provenance. Following Zeng et al., for the descriptor of this (direct, D) fission channel’s efficiency (tentative figure of merit, DFM) we take the global Frenkel contribution to the (singlet component of) the triplet pair state (tt). Its value calculated for the triplet pair located in the bulk of the cluster with no vacancy is decreed to be its reference unit. Alternatively, the tt-derived eigenstate may emerge by radiationless conversion from the eigenstate of Frenkel provenance. The rate constant for this (indirect, I) fission channel is proportional to the (squared) superexchange matrix element (cf. ref 8), which (summed over the two contributing Frenkel configurations) will be our other figure of merit (IFM). The values of this matrix element calculated for the two Frenkel configurations of the dimer in cluster bulk (with no vacancy) are 3 and 9 meV (in fair agreement with ref 8); the sum of their squares serves as the IFM unit. Our present calculations reveal that for some triplet pairs located in the direct neighborhood of the vacancy both figures of merit are increased, with the amplification factors (depending on the specific position) in the range between 5 and 7 for DFM and between 5 and 8 for IFM. Conceivably, the amplification effect might be strengthened by combining the influence of several vacancies, e.g., at cluster boundary. Our test example, representing the model dimer embedded in the (1,1,0) face of the crystal (mimicked by a 32molecule test cluster), is displayed in Figure 2 (without the faded part). In this case, the calculations yield the amplification factors DFM = 14 and IFM = 17, i.e., an order-of-magnitude fission enhancement with respect to the bulk.

Figure 1. Schematic diagram of energetic relations in the model dimer. The broken horizontal lines represent the diagonal energies of the CT configurations in the isolated dimer; their solid counterparts correspond to the CT states of the embedded dimer. The straight arrows depict the off-diagonal couplings; the absolute values of the latter are coded by line widths. The color of line fillings (blue or red) shows the sign of the coupling constant (negative or positive, respectively).

configurations to about 0.02 eV. This tends to equalize the contributions from the two destructively interfering channels, thereby reducing the superexchange matrix element and consequently the fission efficiency. Apparently, this effect could be alleviated by removing some of the surrounding molecules, which would break the local symmetry and partly restore the dominance of the prevailing superexchange channel. Self-consistent charge field (SCCF) calculations11 we performed demonstrate that when a molecule is removed from the vicinity of the target dimer, a part of the charge-quadrupole compensating terms is eliminated. Consequently, the superexchange coupling should be enhanced, leading to an increased triplet yield. This idea has been tested by our present calculations for model pentacene clusters. The clusters, each with a centrally located vacancy, contained 43 molecules, and the treatment was based on the recently proposed multiscale approach of ref 12 (where details of the method and parametrization may be found). As previously reported, the Hamiltonian was constructed in the basis of diabatic functions representing the Frenkel (one per molecule) and the CT excitations. In this latter class, we included all CT configurations with the electron transferred from the molecule at the (m, n, 0) position (where the hole is left) to the molecules located, alternatively, at the (m ± 1/2, n ∓ 1/2, 0), (m ± 1/2, n ± 1/2, 0), (m ± 1, n, 0) or (m, n ± 1, 0) positions, with (m, n) running over all molecules of the cluster. As in previous work,12 the above set was complemented by a single tt diabat, residing at a pair of molecules in the (±1/2, ∓1/2, 0) relative position. These two molecules represent the unit cell of the crystal and correspond to the archetypal model dimer.4−8,10 To assess the dependence of fission descriptors on the location of the triplet pair to be generated, the location of the tt diabat was varied, sampling various positions around the vacancy. We have continued our previous strategy of using the Hamiltonian matrix elements from ref 5 for the pentacene dimer corresponding to the (±1/2, ∓1/2, 0) relative position of the molecules and those calculated by Yamagata et al.13 for

Figure 2. Model dimer (magenta) embedded in the test cluster (cyan) extracted from the (1,1,0) face of a pentacene crystallite (dark blue). The faded part (light blue) represents another, differently oriented, crystallite. 1914

DOI: 10.1021/acs.jpclett.6b00746 J. Phys. Chem. Lett. 2016, 7, 1913−1916

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The Journal of Physical Chemistry Letters

sample consisting of adjoining crystallites seems to promise further increase of our descriptors, probably up to DFM ≅ IFM ≅ 50. The fact that a differently oriented or shifted pentacene crystallite may serve as the stabilizing dielectric is rationalized by observing that, in contrast to the polarization contribution (which is negative-definite), the charge-quadrupole interaction energy (which may have either sign) is very sensitive to orientation. In consequence, owing to the displacement and/or rotation of the adjoining crystallite, the pattern of its molecular quadrupoles is no longer related by symmetry to that on the cluster side, so that their interaction with the charges in the model dimer now fails to compensate the corresponding intracluster terms (which guarantees large CT state splitting), and yet the crystal polarization is substantial (providing the expected CT state stabilization). This favors fission at crystallite boundaries. The smaller the crystallites, the larger the share of the regions where lattice symmetry in the ab plane is locally perturbed as described. This implies that photovoltaic yield could be substantially increased by using nanoscopically disordered pentacene instead of a highquality crystal. The optimum size of the crystallites (in the ab plane, their thickness in the c′ direction being practically irrelevant) should be a compromise between the advantage of local symmetry breaking and the disadvantage of lowered sample density leading to reduced polarization energy. It is our conjecture that this size is rather small, probably on the order of a few nanometers. Although we believe our present model to be sound, the descriptors we have introduced are crude and are expected to be only qualitatively valid; other factors may outbalance their influence. Vibrational coherence, preserved in fission,2,3 seems to be a crucial ingredient of the process. As revealed by studies on other systems,16 it is extremely sensitive to the details of the energy spectrum. However, even the most accurate part of our input, extracted from the results of advanced ab initio calculations,5 differs from that generated by another, equally sophisticated, method.6 Having no claim to that level of precision, we consider these slight differences irrelevant, but they might be crucial for coherence and ultimately for the experimental result. Our main objective here is to expose a potentially useful aspect of fission-related energetics, treating pentacene as a model case. To our knowledge, no experimental study of the fission efficiency dependence on pentacene texture has been published. The thin films used in the prime reference on the topic1 are characterized as “polycrystalline”. Lacking information on crystallites size, it is impossible to surmise how far from the single-crystal limit the measured fission time (about 80 fs) may be. About 2-fold enhancement was reported17 between the tetracene single-crystal and polycrystalline samples. Profound analogies between the two hydrocarbons suggest that our present conclusions might be applicable there. However, as in tetracene the tt state lies higher than the lowest singlet exciton, and fission is thermally activated which makes the process slower by orders of magnitude, it would be premature to venture definite conclusions concerning the underlying mechanism. Nonetheless, implicit expectation that fission should be the fastest in a single crystal10 appears equally risky. Growing good quality crystals is a difficult art and a tedious procedure. However, in most contexts, high crystal quality is crucial for success. It is encouraging to find out that for harnessing singlet fission to improve the yield of photovoltaic

According to further calculations, the model dimers embedded in other crystallographic faces exhibit figures of merit that are either slightly lower than unity or substantially larger than unity, although lower than the above values. This suggests that the overall triplet yield for a random set of dispersed crystallites is expected to be larger than that for an equivalent amount of pentacene in the form of a single crystal. We speculate that, statistically, further fission enhancement might be achieved if the crystallites could be shaped in a controlled way to expose primarily their (1, ±1, 0) face. Admittedly, for the time being we see no feasible way of doing that, but we hope that in the future it may become attainable owing to further progress in nanotechnology. Once some charge-quadrupole contribution to the CT state splitting is reinstated by breaking the local symmetry (as at the cluster boundary), further FM increase could be attainable by lowering the energies of both intradimer CT configurations, which would strengthen their superexchange-mediating coupling both to Frenkel excitations and to the triplet pair state. This can be done by replacing the vacuum extending outward from the cluster boundary with another dielectric; its polarization by the charges of the model cluster should provide the desired stabilization. As an example, we represented the “dielectric” on the former “vacuum side” (faded area in Figure 2) by another pentacene crystallite, but shifted and/or rotated with respect to the model cluster. To reduce the computational effort without sacrificing reasonable accuracy, out of this other crystallite we explicitly included the three pentacene molecules that were the closest to the model dimer (magenta in Figure 2). The part of the model dimer’s CT state stabilization energies, contributed by the rest of the crystallite (extrapolated to its infinite size), was extracted from the results of Tsiper and Soos,14 as in ref 11. We checked that the outcome of this latter procedure was practically the same as if the rest of the crystallite were treated as infinite polarizable continuum and that, compared to the interaction with the explicitly represented molecules, it contributed only a minor correction. The three admolecules were assumed to form a linear array extracted from the pentacene (1,1,0) face. The array, always located in the direct neighborhood of the model dimer (within the limits set by van der Waals radii), was as a whole displaced along the interface with the main cluster by a fraction of the lattice spacing and/or rotated by some angle around the c′ axis (like the faded part of Figure 2). Then the following procedure was repeated for several values of the shift and of the angle. Using the SCCF method11,12 we evaluated the CT state stabilization energies for the model dimer, still embedded in its original cluster, but now with the three extra molecules added in close neighborhood on the “vacuum” side. These energies, corrected by including the contribution from more distant parts of the crystallite (vide supra), were employed in constructing the Hamiltonian of the original 32-molecule cluster, which was subsequently diagonalized. The resultant eigenvectors (for DFM) and SCCF CT state energies (for IFM) yielded the sought-after figures of merit. For various orientations of the simulated crystallite the obtained FM values ranged from about 6 to about 100 for both descriptors. When some deficit with respect to the vacuum reference occurred, it was usually minor, while in the surplus cases the figures of merit exceeded the vacuum value several times. On average, comparing to a set of separated crystallites, a 1915

DOI: 10.1021/acs.jpclett.6b00746 J. Phys. Chem. Lett. 2016, 7, 1913−1916

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The Journal of Physical Chemistry Letters

(10) Beljonne, D.; Yamagata, H.; Bre̋das, J. L.; Spano, F. C.; Olivier, Y. Charge-Transfer Excitations Steer the Davydov Splitting and Mediate Singlet Exciton Fission in Pentacene. Phys. Rev. Lett. 2013, 110, 226402. (11) Petelenz, P.; Snamina, M.; Mazur, G. J. Charge-Transfer States in Pentacene: Dimer versus Crystal. J. Phys. Chem. C 2015, 119, 14338−14342. (12) Petelenz, P.; Snamina, M. Charge-Transfer Coupling of an Embedded Pentacene Dimer with the Surrounding Crystal Matrix. J. Phys. Chem. C 2015, 119, 28570−28576. (13) Yamagata, H.; Norton, J.; Hontz, E.; Olivier, Y.; Beljonne, D.; Brédas, J. L.; Silbey, R. J.; Spano, F. C. The Nature of Singlet Excitons in Oligoacene Molecular Crystals. J. Chem. Phys. 2011, 134, 204703. (14) Tsiper, E. V.; Soos, Z. G. Electronic Polarization in Pentacene Crystals and Thin Films. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 68, 085301. (15) Ito, S.; Nagami, T.; Nakano, M. Density Analysis of Intra- and Intermolecular Vibronic Couplings toward Bath Engineering for Singlet Fission. J. Phys. Chem. Lett. 2015, 6, 4972−4977. (16) Novelli, F.; Nazir, A.; Richards, G. H.; Roozbeh, A.; Wilk, K. E.; Curmi, P. M. G.; Davis, J. A. Vibronic Resonances Facilitate ExcitedState Coherence in Light-Harvesting Proteins at Room Temperature. J. Phys. Chem. Lett. 2015, 6, 4573−4580. (17) Piland, G. B.; Bardeen, C. J. How Morphology Affects Singlet Fission in Crystalline Tetracene. J. Phys. Chem. Lett. 2015, 6, 1841− 1846.

devices just the opposite trend is expected. In the future, this may save some unnecessary effort. Besides, our present arguments should apply to a wide class of analogous crystals, where deliberately created nanoscale texture could lead to increased efficiency. The insight the above results offer indicates that fission is favored irrespective of the factor that causes the lowering of local symmetry. Qualitatively the same effect as that operative at crystallite boundaries is expected to occur also at other interfaces, e.g., in heterojunctions, especially the bulk ones, where the interface area is large. The interface with fullerene seems to be especially promising, because the fullerene molecule, having no quadrupole moment, would introduce no risk of reducing the splitting between the two CT configurations, resulting from charge-quadrupole interactions with the pentacene side of the junction. The methodology we have presented here may be applied also in that case; it allows one to estimate the expected fission amplification at the interface between any molecular crystallites.

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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 This research was performed with equipment purchased with the financial support of the European Regional Development Fund in the framework of the Polish Innovation Economy Operational Program (Contract no. POIG.02.01.00-12-023/ 08).

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REFERENCES

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DOI: 10.1021/acs.jpclett.6b00746 J. Phys. Chem. Lett. 2016, 7, 1913−1916