Low-Energy Charge-Transfer Excitons in Organic Solids from First

Letter pubs.acs.org/JPCL

Low-Energy Charge-Transfer Excitons in Organic Solids from FirstPrinciples: The Case of Pentacene Sahar Sharifzadeh,*,† Pierre Darancet,‡,§ Leeor Kronik,‡ and Jeffrey B. Neaton*,† †

Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States Department of Materials and Interfaces, Weizmann Institute of Science, Rehovoth 76100, Israel § Department of Applied Physics and Applied Mathematics, Columbia University, New York 10027, United States ‡

S Supporting Information *

ABSTRACT: The nature of low energy optical excitations, or excitons, in organic solids is of central relevance to many optoelectronic applications, including solar energy conversion. Excitons in solid pentacene, a prototypical organic semiconductor, have been the subject of many experimental and theoretical studies, with differing conclusions as to the degree of their charge-transfer character. Using first-principles calculations based on density functional theory and many-body perturbation theory, we compute the average electron−hole distance and quantify the degree of charge-transfer character within optical excitations in solid-state pentacene. We show that several low-energy singlet excitations are characterized by a weak overlap between electron and hole and an average electron−hole distance greater than 6 Å. Additionally, we show that the character of the lowest-lying singlet and triplet excitons is well-described with a simple analytic envelope function of the electron−hole distance. SECTION: Spectroscopy, Photochemistry, and Excited States

T

symmetry, such as pentacene, there is no net dipole moment in the ground or excited state, and thus CT excitons will not exhibit linear Stark shifts. Such an uncertainty can also exist when interpreting electronic structure calculations. Since solid pentacene takes up a centrosymmetric structure, the excitedstate dipole moment must vanish, necessitating a definition of CT more general than an induced dipole. Although more general definitions of CT have been developed for donor− acceptor complexes,23 no such definition exists for excitations in the solid state. Semiempirical model Hamiltonian studies, which define CT character as electron and hole residing on nearest-neighbor molecules, predict that the lowest energy exciton in pentacene is a mixture of CT and Frenkel character.24 First-principles time-dependent density functional theory (TDDFT)-computed transition orbital densities for pentacene clusters suggest that the lowest energy exciton is delocalized, and that the electron and hole occupy the same space (no CT),13 while visual representations of the exciton wave function from prior calculations25−27 support the conclusion that the lowest energy exciton in pentacene is CT-like. In this Letter, we introduce a new general analysis of twoparticle electron−hole wave functions aimed at understanding the nature of the exciton and providing a more general definition of CT, addressing the uncertainties in previous

here has been much interest in understanding the nature of low-energy excitations in organic solids due to their many applications in optoelectronics, such as photovoltaics, where they are central to efficiency and function.1−4 In particular, the nature of low-energy excitons in the pentacene crystal has been a topic of extensive study. In addition to being a prototypical crystalline organic semiconductor with high carrier mobility,5 pentacene has garnered interest for its propensity for singlet fission (see, e.g.,6−16). Here, a photoinduced singlet excited state downconverts to two triplets, resulting in long-lived excitons and potential solar conversion quantum efficiencies exceeding 100%. The efficacy of singlet fission depends sensitively on the nature of the singlet wave function and its dynamics; for example, excitations of charge-transfer (CT) nature, in the sense that electron and hole reside on different molecules, can accelerate singlet fission.2 While it has been traditionally assumed that low-energy excitations in organic semiconductors are of single-molecule character (Frenkel type) due to weak solid-state screening and intermolecular interactions,17 there is experimental and computational evidence for CT excitons in organic solids. For pentacene, the degree of CT character continues to be debated. Experimentally, momentum-dependent electron-loss spectroscopy studies suggest some CT-character,18−20 in the sense that a Frenkel description does not properly describe measurement, while electric field modulation studies are inconclusive.21,22 The difficulty in interpreting electric field modulation data arises because, for a system with inversion © XXXX American Chemical Society

Received: May 23, 2013 Accepted: June 18, 2013

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Figure 1. (a) Computed optical absorption spectrum of the pentacene crystal, compared with measurements from ref 32. Because the measurements of ref 32 were performed with unpolarized light, we average our computed spectrum for incident polarizations along all three unit cell directions. The computed energy of the first singlet and triplet is indicated by dashed vertical lines. (b,c) Two-dimensional electron−hole correlation function, -̅ S(r), in the (ab) plane (averaged over the c axis) and the (ac) plane (averaged over b axis), respectively, for the singlet (top) and triplet (bottom). Panel d shows the bulk crystal structure (ref 33), and panels e and f show the projection of the atomic structure onto the (ab) and (ac) planes.

0.1 eV of the experimental optical gap determined by optical absorption or ellipsometry for pentacene and 3,4,9,10-perylene tetracarboxylic dianhydride (PTCDA). Our new analysis centers on the normalized electron−hole wave function, ψS(re, rh), a solution of the BSE for excited state S, expressed as a linear combination of tensorial products of single-excitation determinants. We define the electron−hole correlation function as

studies. Using the exciton wave function, and its corresponding electron−hole correlation function, we quantify the extent and directionality of the solid-state exciton, and its degree of CT character. We apply our approach to solid pentacene, and find that for the lowest energy excitation at 1.75 eV, there is a 94% probability that electron and hole reside on different molecules. We demonstrate that this state exhibits unambiguous CT character, with an average electron−hole separation of 8 Å. This CT character persists for the next three low-lying states. Furthermore, we show that the spatial distribution and CT character of the lowest-energy exciton is well-approximated by the product of the noninteracting excitonic wave function and a smooth envelope function. We calculate the nature and energy of optical excitations within the pentacene molecule and bulk crystal (see Figure 1 for crystal structure) using first-principles DFT and manybody perturbation theory. We employ the BerkeleyGW package,28 with computational details following ref 26 (see Supporting Information (SI)). Quasiparticle excitation energies are computed via the GW approximation, as a firstorder correction to a DFT starting point, in this case the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE).29 We take a “one-shot” G0W0@PBE approach, i.e., the quasiparticle wave functions are assumed to be well-approximated by the DFT-PBE Kohn−Sham states, and their energies are corrected perturbatively. Subsequently, optical excitations are computed by explicit inclusion of electron−hole interactions, via a Bethe−Salpeter equation (BSE) approach. The accuracy of this GW/BSE approach for predicting excitation energies of prototypical organic crystals was carefully examined in previous work.26 In particular, the computed lowest vertical singlet excitation energy was within

-S(r) =

∫Ω d3rh |ψS(re = rh + r, rh)|2

(1)

where - S describes the probability of finding electron and hole separated by the vector r = re − rh, and Ω is the volume of a primitive cell. Previous attempts to visualize exciton delocalization in periodic structures using semiempirical methods30 or time-dependent DFT31 have relied on the equivalent quantities in these theories. The average distance between electron and hole is determined from - S as ⟨r⟩ = ∫ d|r|- S(|r|)|r|, where the integral runs over all |r| in Ω. We compute the integral in eq 1 as a discrete sum over rh, with the hole sampled at 88 high hole probability positions on the two molecules in the primitive cell. These positions are ∼0.5 Å above and below the plane of the molecule at each carbon atom. For each hole position, - S(r) is computed on an evenly spaced, dense grid with volume element 0.077 Å3, spanning an 8 × 8 × 4 supercell for the two lowest energy excited states and a 10 × 10 × 6 supercell for higher energy states. These large real-space supercells are necessary for good convergence of the exciton wave functions. We verify that the electronic and hole sampling are sufficient by ensuring that application of the same procedure to the noninteracting 2198

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The computed - S(r) indicates that the singlet exciton is delocalized over many molecules in the (ab) planethe direction of significant π-stacking and high dispersionand localized within one plane of molecule along the c axis. Specifically, there is significant probability at 4−7 Å away from the origin within the (ab) plane. These high probability locations are along the crystal’s ± a-axis and the ± (a−b) direction, consistent with electron transfer from one molecule to its four neighbors in those directions. This finding indicates considerable CT character for the singlet, in the sense that electron and hole, on average, reside on different molecules, confirming previous conclusions based on a visual representation of the GW/BSE exciton for a single hole position.25−27

electron−hole wave function retains the periodicity of the lattice (see SI for details). With a sufficiently dense sample of electronic and hole coordinates throughout the supercell, - S(r) will integrate to unity. Because we sample a nonuniform subset of hole coordinates, in practice we normalize - S(r) manually, by dividing the function by its cumulative sum, i.e., -S(r) ≃

∑h |ψS(re = rh + r, rh)|2 ∑e , h |ψS(re, rh)|2

(2)

With eq 1, we can now quantify the degree of CT character within the excited state as the probability that electron and hole do not reside on the same molecule. We compute this value as the partial integration of - S(r), - CT S = 1 −

∫r∈Ω

d3r-S(r) Mol

(3)

where ΩMol corresponds to the volume/molecule in the crystal and - S is determined via eq 2. For the case of complete CT, - CT S will integrate to unity and for the case of a pure Frenkel exciton, it will vanish. For convenience, we carry out the integration on a uniform mesh over a cylindrical volume element, which is a useful symmetry in the case of the pentacene molecule. This cylinder has the same volume as that occupied by one molecule in the crystal, with a radius of 2.72 Å and length 14.88 Å. In Figure 1, we compare the computed optical absorption spectrum for solid pentacene with experiment.32 Because of the solid-state environment, each peak in the absorption spectrum consists of a series of excitations. The onset of absorption is at 1.74 eV, within 0.1 eV of the experiment and in good agreement with previous GW/BSE calculations.25−27,34 For the gas-phase molecule, the computed lowest excitation energy is 2.2 eV, also within 0.1 eV of experiment.26 The first peak maximum is at 1.86 eV, in good agreement with experiment, and corresponds to the fourth excitation. Up to an energy of ≃2.4 eV, the computed and measured spectrum agree well. Above this energy range, due in part to incomplete convergence, the agreement is merely qualitative. For comparison, we also study the lowest energy triplet excitation, which is reduced in energy relative to the singlet in pentacene because of the significant exchange interaction between electron and hole in the singlet state.35 We predict a triplet excitation energy of 0.9 and 1.2 eV for the molecule and solid, respectively, in good agreement with previous calculations25,27 and with the measured value of 0.86 eV from pump−probe fluorescence experiments of pentacene-doped tetracene.36 Figure 1b,c shows a two-dimensional representation of - S(r) for the first excited singlet and triplet states, where the third axis is integrated out. For the triplet state (bottom panel), - S(r) projected on the (ab) plane exhibits an oval shape near the origin. This shape is equivalent in spatial extent to the size of one molecule in the (ab) plane, indicating a high probability of electron and hole residing at the same location. - S(r) drops significantly at electron−hole separations of ≳2 Å. In the (ac) plane, the triplet exhibits significant correlation up to ≃15 Å, a length scale corresponding to the molecule’s long axis (see Figure 1c). For the singlet state, in contrast to the triplet, there is very little probability and electron−hole overlap at short distances.

Figure 2. Electron−hole radial distribution function, - S(|r|), (top) and cumulative distribution function (bottom) for (a) the lowest lying singlet, and (b) the lowest-lying triplet. The black line shows the cumulative distribution function when electron and hole are restricted to the same molecule (see eq 3). This restricted cumulative sum indicates the degree of CT character; specifically, the limit of the cumulative sum at large distance, subtracted from 1, provides a measure for CT.

Figure 2 shows the radial distribution of the electron−hole distribution function, - S(|r|), throughout the supercell. The radial distribution function for the triplet shows a narrow peak near the origin, followed by a smaller peak at ≃2.5 Å, with vanishing probability beyond 5 Å. By contrast, for the singlet, - S(|r|) is negligible at separations smaller than 3 Å but shows a broad peak at ≃5.0 Å. For the singlet, the average distance between electron and hole is ≃8 Å (compared with 2 Å in the gas-phase). For the lowest triplet solid-state exciton, the average electron−hole distance is less than ≃1 Å. Figure 2 also shows the cumulative distribution function of - S(|r|) (integral over all |r|) and the partial integral, with |r| restricted to electron and hole residing on one molecule. The restricted integral captures only a fraction of the cumulative sum for the singlet state (6%), whereas it captures the triplet state in almost its entirety. As mentioned previously, the pentacene crystal structure is centrosymmetric and thus there is no dipole moment in the excited state. Though individual molecules may exhibit a local dipole moment, as computed in some well-defined manner (e.g., ref 37), the net dipole must vanish. Nonetheless, as 2199

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smaller than the average electron−hole distance) and has a full width half-maximum (fwhm) of 6.7 Å, while for the triplet, it peaks at 0 Å and has a fwhm of 5.3 Å. The singlet envelope function resembles a p-type state, with a node at the origin. The fact that the lowest energy excited state has a node is inconsistent with a simple Bohr model of the exciton. Unlike the triplet state, the singlet does not exactly follow this model because of strong repulsive shortrange exchange interactions; the CT character of singlet excitons can then be understood simply by the destabilization of the nodeless state by short-range exchange interactions. To conclude, we used first-principles calculations to characterize the low-energy optical excitations of crystalline pentacene. We showed that, while the low-energy singlets possess no net dipole, they exhibit a large degree of charge transfer character, in agreement with previous interpretations.25−27 Moreover, through definition of the electron−hole correlation function, we provided a rigorous measure of the electron−hole distance and charge-transfer character in the excited state. Our analysis is general and can be applied to quantifying charge transfer in complex organic photovoltaic materials. Finally, we showed that the complex, structured electron−hole correlation function for the low lying states can be understood as a product of the noninteracting RPA function and a simple analytical envelope function, paving the way to efficient semiempirical modeling of excitations in organic assemblies.

shown in Figure 2, the solid-state singlet is characterized by little overlap between electron and hole and 94% of CT character as defined in eq 3. This analysis results in a lower bound on the CT probability, because electron and hole may be located on different molecules with the distance between them being less than the size of one molecule (e.g., electron on the edge of one molecule and hole on the nearest edge of the neighboring molecule). That the first excited singlet state in pentacene has significant CT character agrees qualitatively with semiempirical Hamiltonian studies that predict that the lowest energy exciton consists of 48% CT character, defined as electron and hole residing on nearest-neighbor molecules.24 The CT character of the exciton persists for the next three singlet states (see SI); the average electron−hole distances in the second, third, and fourth singlets are respectively 7, 9, and 6 Å, with CT character of 80, 82, and 81%. It is interesting that, although there is a clear CT character in the fourth state, this state has significant oscillator strength (i.e., is a bright state). Figure 1 and Figure 2 show corrugated fine-structure due the influence of atomic-scale features on the electron−hole distribution probability. However, the CT nature and degree of delocalization are unchanged when the atomic features of - S(r) are suppressed. This can be seen from the electron− hole correlation envelope function, defined as

- SEnv(r) =

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-S(r) - RPA(r)

(4)

ASSOCIATED CONTENT

S Supporting Information *

RPA

where is the electron−hole correlation function associated with the noninteracting wave function, ψRPA(re, rh), obtained from the random phase approximation (RPA) as a simple product of the highest energy valence and lowest energy conduction Kohn−Sham wave functions, integrated over the Brillouin zone. - Env S (|r|), shown in Figure 3, is a smooth function because the complex structure associated with the atomic-scale charge density variations is divided out. The distance-dependent envelope function can be fit very well to a Gaussian function. For the singlet state, this function peaks at 6.1 Å (slightly

Computational details and the electron−hole correlation function associated with the RPA wave function and high energy excited states. 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] (S.S.); [email protected] (J.B.N.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Work at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC0205CH11231. P.D. was funded by the Helios Solar Energy Research Center. Partial support for this work was also provided through Scientific Discovery through Advanced Computing (SciDAC) Partnership program funded by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research and Basic Energy Sciences. Work at the Weizmann Institute of Science was additionally supported by the Israel Science Foundation and the Lise Meitner Minerva Center for Computational Chemistry. We also acknowledge funding from the United States-Israel Binational Science Foundation (BSF). We thank the National Energy Research Scientific Computing (NERSC) center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC0205CH11231, for computational resources.

Figure 3. The two-dimensional envelope electron−hole correlation function, -̅ Env S (r), in the crystal’s (ab) plane (averaged over the c axis) for (a) the singlet state and (b) the triplet state. Panels c and d show the radial distributions - Env S (|r|) for the singlet and triplet, respectively. 2200

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