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The Structure and Dynamics of Molecular Excitons Annu. Rev. Phys. Chem. 2014.65. Downloaded from www.annualreviews.org by Michigan State University Library on 01/15/14. For personal use only.
Christopher J. Bardeen Department of Chemistry, University of California, Riverside, California 92521; email: [email protected]
Annu. Rev. Phys. Chem. 2014. 65:127–48
Keywords
The Annual Review of Physical Chemistry is online at physchem.annualreviews.org
organic, semiconductor, solid state, spectroscopy, diffusion, fission
This article’s doi: 10.1146/annurev-physchem-040513-103654
Abstract
c 2014 by Annual Reviews. Copyright All rights reserved
The photophysical behavior of organic semiconductors is governed by their excitonic states. In this review, I classify the three different exciton types (Frenkel singlet, Frenkel triplet, and charge transfer) typically encountered in organic semiconductors. Experimental challenges that arise in the study of solid-state organic systems are discussed. The steady-state spectroscopy of intermolecular delocalized Frenkel excitons is described, using crystalline tetracene as an example. I consider the problem of a localized exciton diffusing in a disordered matrix in detail, and experimental results on conjugated polymers and model systems suggest that energetic disorder leads to subdiffusive motion. Multiexciton processes such as singlet fission and triplet fusion are described, emphasizing the role of spin state coherence and magnetic fields in studying singlet ↔ triplet pair interconversion. Singlet fission provides an example of how all three types of excitons (triplet, singlet, and charge transfer) may interact to produce useful phenomena for applications such as solar energy conversion.
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1. INTRODUCTION
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Many technological applications envisioned for organic electronic materials (e.g., photovoltaics, sensors, light-emitting diodes) involve the absorption and emission of light. The photophysical behavior of these materials is governed by the structure and dynamics of their low-lying electronic states. In solid-state materials, these excited electronic states can be thought of as mobile quasiparticles that carry energy, collectively termed excitons (1). This area of research lies at the interface of chemistry and physics, and progress depends on the coordinated efforts of experiment and theory. In Section 2 of this review, I begin with a general description of excitons in terms of a site Hamiltonian, with each site having two molecular orbitals. The aim of this section is not to develop a new theory of excitons but rather to provide a consistent framework that classifies the different types of excitons encountered in organic semiconductors. Section 3 briefly discusses experimental complications that arise in the study of organic semiconductors and how they can be mitigated. Section 4 discusses how the electronic characteristics of some of the exciton types described in Section 2 manifest themselves in the steady-state spectroscopy of these materials, with a particular emphasis on the photoluminescence properties of intermolecular Frenkel excitons. Sections 5 and 6 describe two aspects of exciton dynamics that are relevant to solar energy conversion: the spatial diffusion of excitons and fission/fusion reactions in which excitons can split and recombine. Space limitations preclude a comprehensive review, and it is impossible to include all references to relevant work from the many research groups working in this area. I apologize for any unintentional omissions.
2. OVERVIEW OF ELECTRONIC STATES IN MOLECULAR CRYSTALS The term exciton can encompass a wide variety of states with very different structures and behaviors. There exist several books and review articles on electronic states and excitons in organic systems (2–9). For the purposes of classifying different types of excitons in this review, I consider how the electronic states of molecules change as they assemble to form an aggregate or crystal. An isolated molecule in its ground state has two spin-paired electrons residing in its highest occupied molecular orbital (HOMO). The promotion of an electron into the lowest unoccupied molecular orbital (LUMO) leaves a hole in the HOMO, resulting in an excited singlet or triplet state, depending on the electron’s spin state. Figure 1a illustrates possible orbital configurations. The electron can be promoted into a LUMO on the same site as the hole or onto a neighboring site. Furthermore, the electron (hole) can travel from site to site via intersite electronic coupling. The most general electronic Hamiltonian for this system can be written in the position representation as + c n+ c n d n+ d n − U(r)c n+ c n d n+r d n+r + [te c n+ c n+1 + th d n+ d n+1 + H.C.] Hˆ el = −U(0) n r≥1 n n (1) + [V c n+ d n+ d n+1 c n+1 + H.C.], n
c n+
(c n ) are the electron creation (annihilation) operators for site n, d n+ (d n ) are the hole where creation (annihilation) operators for site n, and H.C. denotes the Hermitian conjugate of the preceding term. U(0) is the Coulomb energy of an electron-hole pair localized on a single site, and U(r) is the energy when the electron and hole reside on different sites. The transfer matrix element for nearest-neighbor electron (hole) transfer is given by te (th ), whereas the dipole-dipole interaction term V transfers an electron-hole pair from site n + 1 to site n. This site-basis Hamiltonian (10) is quite general and can be used to describe excitons in both the Frenkel and Wannier limits (11–13). For the purposes of this review, we consider only r = 1 terms, which limits us to the consideration 128
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a LUMO
b
....
....
ϕ–1 ϕ+2
ϕ+1 ϕ–2
1 — (ϕ+1 ϕ–2 – ϕ–1 ϕ+2) √2
ϕ1 ϕ*2
1 — (ϕ+1 ϕ–2 + ϕ–1 ϕ+2) √2 1 — (ϕ1 ϕ*2 – ϕ*1 ϕ2) √2
HOMO ϕ+n
ϕn–1
ϕ–n+1
ϕn+2 ϕ*1 ϕ2
Φ nCT = ϕ1 ϕ2…ϕ–n ϕ–n+1…ϕN
LUMO
(k)
....
.... ϕ*n
ϕn–1
ϕn+1
ϕ1 ϕ2
ϕ1 ϕ2
ϕ1 ϕ2
1
2
Dimer N = 2
∑n Bn(k)…ϕ+n ϕ–n+1…
ΨFrenkel =
1 — (ϕ1 ϕ*2 + ϕ*1 ϕ2) √2
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(k)
ΨCT =
∑n A(k)n …ϕ*n … D
2V
Aggregate N > 2
ϕn+2
Φ nFrenkel = ϕ1ϕ2…ϕ*n…ϕN Figure 1 (a) Schematic illustration of product basis states used to evaluate the Hamiltonian in Equation 1. The Frenkel (neutral) states involve an electron promoted to a lowest unoccupied molecular orbital (LUMO) on the same site, whereas the charge transfer (CT) states involve an electron promoted to a LUMO on a neighboring site. (b) At large separations, there is no interaction between molecules 1 and 2. The total wave function of an excited state is the simple product state. As the molecules approach each other, intermolecular interactions mix the neutral excited states to form superposition states. Similarly, the ionic states form delocalized charge resonance states. As an aggregate is formed, the neutral states grow into a Frenkel exciton band with a bandwidth of 2V and a solvatochromic shift D. Similarly, the ionic states give rise to the CT exciton band.
of so-called charge transfer (CT) excitons, where the charges separate only to nearest neighbors (14–17). This is usually justified by invoking the relatively low dielectric constant of organic solids as compared to inorganic semiconductors. For the U(0) term, we have a neutral basis consisting of states of the form Frenkel = φ1 φ2 . . . φn∗ . . . φ N .
(2a)
For the U(1) term, we have an ionic basis consisting of product wave functions of the form − . . . φN . CT = φ1 φ2 . . . φn+ φn+1
(2b)
If the interaction terms V/te /th did not exist, these product wave functions would be the N-fold degenerate eigenstates of the systems, separated by the energy difference U(1) − U(0), corresponding to the exciton binding energy. In many organic systems, this energy difference is much larger than the interaction terms, and we can consider the Frenkel and CT exciton problems separately. In this limit, the te /th terms couple CT states to each other, while the dipolar V interaction couples Frenkel states. The Frenkel part of Hˆ el can then be written as Bn+ Bn n + [V Bn+ Bn+1 + H.C.], (3) Hˆ Frenkel = U(0) n
n
where the Frenkel exciton creation/annihilation operators are Bn+ = c n+ d n+ and Bn = c n d n . For the CT part, we have + U(1)c n+ c n d n+1 d n+1 + [te c n+ c n+1 + th d n+ d n+1 + H.C.]. (4) Hˆ CT = n
n
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When Hˆ Frenkel or Hˆ CT is solved, we obtain a set of N eigenfunctions, enumerated by the index k, which are linear combinations of the basis states given in Equations 2a,b, i.e., (k) ∗ Frenkel = A(k) (5a) n ϕ1 ϕ2 . . . ϕ n . . . ϕ N , n (k)
CT =
− Bn(k) ϕ1 ϕ2 . . . ϕn+ ϕn+1 . . . ϕN .
(5b)
n
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In the momentum representation, k is the momentum quantum number that indexes the state. The delocalized exciton states Frenkel and CT can have new energies and different oscillator strengths as compared to their constituent localized states, and Figure 1b illustrates their evolution. The CT exciton states are cousins of the large-radius Wannier excitons often encountered in inorganic semiconductors. Note that I have not included coupling to the nuclear degrees of freedom in this treatment. I also assume that the sites 1 − N correspond to individual molecules, but the Hamiltonian in Equation 1 can also be applied to covalent polymer chains, where the sites correspond to repeat units (18, 19). Even given the simple picture in Figure 1, we can make some important points about exciton states in molecular crystals. First, the common perception that Frenkel excitons are more localized than CT excitons is not quite precise. It is the degree of charge separation within the basis functions that is different. Frenkel excitons comprise neutral states, whereas CT excitons comprise ionic states. Both types of states can have wave functions that are delocalized over N sites. Second, with regard to the value of N, for a perfect crystal, ideally N → ∞ and the new states form a continuous band. In practice, N is usually of the order of 1–100 in organic molecular systems, and we have aggregate states that exist within a larger solid or crystal. Many factors can conspire to limit the value of N, including structural defects and intermolecular vibrations that lead to time-dependent fluctuations in the te/h /V terms in the Hamiltonians given in Equations 3 and 4. The important point is that the finite value of N encountered in most organic systems tends to make the position-space Hamiltonian more useful than its momentum-space representation. Third, electron spin states play an important role in determining the energy of Frenkel exciton states but not CT excitons. In Frenkel exciton states, two electrons are localized on the same site, which means that the exchange interaction between them can be quite large (14, 20). This exchange interaction lowers the energy of the triplet states by up to 1 eV in some cases, effectively separating the Frenkel exciton into two different bands: singlet and triplet. These bands contain different numbers of states, as there are three possible triplet wave functions for every singlet. Because the ground state is singlet in character, spin selection rules make the triplet states effectively dark, with very weak absorption and phosphorescence. But these states are always present at energies below that of the singlet, and it is the 3:1 triplet:singlet ratio that leads to the 25% upper limit on the efficiency of organic light-emitting diodes based on fluorescence (21, 22). Accessing triplet states via intersystem crossing from the singlet requires a spin flip and is typically quite slow. But as shown below, multiple exciton interactions can provide spin-allowed ways to connect the singlet and triplet Frenkel exciton states. In deriving the Frenkel and CT Hamiltonians separately, we assume that the Frenkel and CT configurations do not interact. The total Hamiltonian is more accurately written as Hˆ tot = Hˆ Frenkel + Hˆ CT + Hˆ Frenkel−CT ,
(6)
where examples of the Hˆ Frenkel−CT term can be found in the work of Petelenz and colleagues (23, 24) and Spano and colleagues (25). This term gives rise to Frenkel exciton dissociation into a CT exciton, among other phenomena. Importantly, the division of electronic states into Frenkel versus CT excitons is an approximation that can break down when real physical systems are considered. 130
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A good example of this is recent work showing how the Frenkel-CT interaction leads to large changes in the Davydov splitting as polyacenes increase in size (25). Additionally, we assume that the Frenkel exciton band lies below the CT band. In fact, the positions of the two types of states depend on how the molecules pack in the crystal. The α- (face-to-face packing) and β- (edgeto-face herringbone packing) polymorphs of perylene provide examples of how different packing motifs for the same molecule can lead to low-lying CT and Frenkel states, respectively (26). Furthermore, small changes in molecular position can lower the CT states relative to the neutral state. The excitation of a Frenkel exciton followed by nuclear relaxation and CT is commonly called excimer formation in the chemistry literature but is also sometimes referred to as selftrapping (27, 28). Again, the important point is that a clean separation of the Frenkel and CT states is rarely found in nature. In summary, the simple idea of a single band gap between valence and conduction bands, commonly used to describe the situation in inorganic semiconductors, fails to capture the complexity of electronic states in organic semiconductors. In reality, these materials can be thought of as having three separate band gaps, corresponding to triplet Frenkel excitons, singlet Frenkel excitons, and CT excitons, respectively. Each band has its own energy, bandwidth, and photophysical properties.
3. EXPERIMENTAL CONSIDERATIONS IN THE STUDY OF SOLID-STATE ORGANIC MATERIALS This section discusses some experimental challenges in studying exciton dynamics in organic materials. The first issue revolves around the chemical composition of the sample. The presence of chemical impurities is much more of a concern for solid-state studies than for solution studies. In a solid sample, the effect of an impurity can be amplified owing to its ability to interact with neighboring molecules via energy and CT. The type and concentration of the chemical impurities depend on the system studied. Conjugated polymers often possess a distribution of chain lengths, conformations, and chemical defects. Small molecules can be purified to a high level using standard techniques such as sublimation and zone refining, providing a way to systematically address concerns about chemical purity. It is important to keep in mind that chemical contaminants can be introduced after the solid sample is prepared, with O2 and H2 O being the most common culprits. Even if the molecular composition of the sample is well controlled, one must pay careful attention to sample preparation to obtain consistent and reproducible results. For polymers, different spincoating conditions or solvents can lead to different degrees of aggregation and disorder, changing the spectroscopic properties considerably (29). For oligomers, crystal growth is still somewhat of an art, as the presence of different crystal polymorphs and mixed amorphous/crystalline regions can depend sensitively on sample growth conditions. One important point is that the required sample purity depends on the quantity being measured. For example, the fluorescence lifetime of a singlet exciton is in the nanosecond regime, which does not leave the exciton enough time to explore a large region of the sample before it decays back to the ground state. A contaminant present at the parts per million level will typically have little effect on this observable. Conductivity measurements, conversely, probe the dynamics of longlived charge carriers that diffuse throughout the sample and can encounter distant defects and contaminants. Reliable measurements of charge mobilities thus require highly purified samples with very low impurity concentrations (30, 31). A second issue is the need to distinguish between effects that arise from intrinsic exciton properties and those that arise from the optical properties of the macroscopic sample. For example, most single crystals are optically thick, absorbing so much light that simple transmission www.annualreviews.org • Structure and Dynamics of Molecular Excitons
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measurements become impossible, and one must use reflection measurements, in conjunction with a Kramers-Kronig analysis, to extract absorption spectra (32, 33). For fluorescence measurements, one must account for possible complications from the self-absorption of the high-energy portion of the luminescence, waveguiding from total internal reflection in a high-index crystal, and depolarization due to crystal birefringence. Such effects distort the steady-state spectra and also time-resolved signals. Many optical artifacts can be reduced or eliminated by using optically thin (90% in the single crystal (121). In a polycrystalline film, conversely, the singlet decay rate has been reported to be 40–100 ps (120, 121, 129). This is considerably faster than the single crystal data described above, suggesting that more disordered samples may have sites that can accelerate SF. Magnetic field effects on triplet kinetics provided some of the earliest evidence for SF in the polyacenes (130, 131). More recently, magnetic field–dependent measurements of SF in amorphous rubrene provided evidence of the importance of molecular orientation in SF (132). The triplet spin states, and thus the triplet pair states, change in the presence of a magnetic field. Although the detailed theory of how these states change in a magnetic field is somewhat complicated, the basic idea is relatively simple (133–137). In the absence of a magnetic field, only three out of nine possible triplet pair states (|xx , |y y , and |zz ) have singlet character, as per Equation 9. As the magnetic field is turned on, the triplet eigenstates change, and the distribution of singlet character within those states changes as well. For a pair of parallel oriented molecules, the number of triplet pair states with singlet character decreases from three (zero field) to two (high field). Alternatively, if the fissioning molecules have random orientations, then the number of states with singlet character increases from three to nine. Figure 7 illustrates the evolution of singlet character for the nine triplet pair states for different molecular arrangements. Figure 8a shows the fluorescence decay of amorphous rubrene films with and without an applied magnetic field, along with calculated signals for the case in which SF can occur only in parallel oriented molecular pairs (Figure 8b) and in randomly oriented molecular pairs (Figure 8c). The increased prompt fluorescence in the experimental data shows that even in amorphous films, SF occurs preferentially in aligned pairs of molecules.
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θ A = ϕA = 0 θ B = ϕB = 0
a Parallel 0.6 0.5 0.4 0.3 0.2 0.1 0 0
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= 45˚
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ss B-field/Gau
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ss B-field/Gau
Figure 7 Singlet character, defined as the magnitude squared of the overlap of the triplet pair and singlet spin states, of each triplet pair state as a function of magnetic field strength for different molecular orientations. (a) For parallel molecules, the number of states with singlet character decreases from three to two. (b) For a 45◦ angle between molecules, the number of states with singlet character increases from three to eight. (c) For a collection of randomly oriented molecular pairs, the number of states with singlet character increases from three to nine. Figure taken from Reference 132.
I emphasize that the photophysical mechanism of SF remains the subject of active study. The importance of relative chromophore orientations, now well established through experimental results, is being clarified by theory. Although originally thought to involve only the singlet and triplet exciton manifolds (138, 139), recent theoretical studies strongly implicate the role of CT states in accelerating the SF reaction (140–146). Thus SF may provide an example of how the interaction of all three types of excitons described in Section 2 can lead to novel phenomena that may be of practical utility. www.annualreviews.org • Structure and Dynamics of Molecular Excitons
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Figure 8 (a) Experimental fluorescence decays of amorphous rubrene in a 20-ns time window. (b) Simulated fluorescence decays, showing the enhancement in the prompt fluorescence when the molecules are assumed to be parallel. The application of a magnetic field decreases the number of triplet pair states with singlet coupling. This simulation agrees qualitatively with the experimental data. (c) Simulated fluorescence decay showing suppressed prompt fluorescence at high magnetic field when randomly oriented molecules are assumed. Figure taken from Reference 132.
7. CONCLUSION The electronic structure of conjugated organic solids can be complicated, and the existence of three different exciton types (Frenkel singlets, Frenkel triplets, and CT excitons) must be kept in mind when studying these materials. Furthermore, one must take considerable care to avoid experimental artifacts due to impurities, high optical densities, and annihilation effects. But the properties of these excitons, including delocalization, diffusion, and fission/fusion reactions, make them well worth studying. At the present time, a convergence of new experimental methods and theoretical advances has resulted in new insights. These studies will likely provide the framework for a predictive structure-function understanding of excitons in organic solids. It is hoped that this review helps facilitate the experimental advances needed to generate improved materials and usher in the era of organic electronics.
DISCLOSURE STATEMENT The author is not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.
ACKNOWLEDGMENTS This work was supported by the National Science Foundation under grant CHE-1152677.
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136. Klein G. 1978. Kinematics of triplet pairs in anthracene and tetracene crystals. Chem. Phys. Lett. 57:202–6 137. Hachani L, Benfredj A, Romdhane S, Mejatty M, Monge JL, Bouchriha H. 2008. Fluorescence detected magnetic resonance in organic systems: a pair-density matrix formalism approach. Phys. Rev. B 77:035212 138. Zimmerman PM, Zhang Z, Musgrave CB. 2010. Singlet fission in pentacene through multi-exciton quantum states. Nat. Chem. 2:648–52 139. Zimmerman PM, Bell F, Casanova D, Head-Gordon M. 2011. Mechanism for singlet fission in pentacene and tetracene: from singlet exciton to two triplets. J. Am. Chem. Soc. 133:19944–52 140. Greyson EC, Stepp BR, Chen X, Schwerin AF, Paci I, et al. 2010. Singlet exciton fission for solar cell applications: energy aspects of interchromophore coupling. J. Phys. Chem. B 114:14223–32 141. Greyson EC, Vura-Weis J, Michl J, Ratner MA. 2010. Maximizing singlet fission in organic dimers: theoretical investigations of triplet yield in the regime of localized excitation and fast coherent electron transfer. J. Phys. Chem. B 114:14168–77 142. Smith MB, Michl J. 2013. Recent advances in singlet fission. Annu. Rev. Phys. Chem. 64:361–86 143. Beljonne D, Yamagata H, Bredas JL, Spano FC, Olivier Y. 2013. Charge-transfer excitations steer the Davydov splitting and mediate singlet exciton fission in pentacene. Phys. Rev. Lett. 110:226402 144. Teichen PE, Eaves JD. 2012. A microscopic model of singlet fission. J. Phys. Chem. B 116:11473–81 145. Berkelbach TC, Hybertsen MS, Reichman DR. 2013. Microscopic theory of singlet exciton fission. I. General formulation. J. Chem. Phys. 138:114102 146. Berkelbach TC, Hybertsen MS, Reichman DR. 2013. Microscopic theory of singlet exciton fission. II. Application to pentacene dimers and the role of superexchange. J. Chem. Phys. 138:114103
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Review in Advance first posted online on December 2, 2013. (Changes may still occur before final publication online and in print.)
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The Structure and Dynamics of Molecular Excitons Annu. Rev. Phys. Chem. 2014.65. Downloaded from www.annualreviews.org by Michigan State University Library on 01/15/14. For personal use only.
Christopher J. Bardeen Department of Chemistry, University of California, Riverside, California 92521; email: [email protected]
Annu. Rev. Phys. Chem. 2014. 65:127–48
Keywords
The Annual Review of Physical Chemistry is online at physchem.annualreviews.org
organic, semiconductor, solid state, spectroscopy, diffusion, fission
This article’s doi: 10.1146/annurev-physchem-040513-103654
Abstract
c 2014 by Annual Reviews. Copyright All rights reserved
The photophysical behavior of organic semiconductors is governed by their excitonic states. In this review, I classify the three different exciton types (Frenkel singlet, Frenkel triplet, and charge transfer) typically encountered in organic semiconductors. Experimental challenges that arise in the study of solid-state organic systems are discussed. The steady-state spectroscopy of intermolecular delocalized Frenkel excitons is described, using crystalline tetracene as an example. I consider the problem of a localized exciton diffusing in a disordered matrix in detail, and experimental results on conjugated polymers and model systems suggest that energetic disorder leads to subdiffusive motion. Multiexciton processes such as singlet fission and triplet fusion are described, emphasizing the role of spin state coherence and magnetic fields in studying singlet ↔ triplet pair interconversion. Singlet fission provides an example of how all three types of excitons (triplet, singlet, and charge transfer) may interact to produce useful phenomena for applications such as solar energy conversion.
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1. INTRODUCTION
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Many technological applications envisioned for organic electronic materials (e.g., photovoltaics, sensors, light-emitting diodes) involve the absorption and emission of light. The photophysical behavior of these materials is governed by the structure and dynamics of their low-lying electronic states. In solid-state materials, these excited electronic states can be thought of as mobile quasiparticles that carry energy, collectively termed excitons (1). This area of research lies at the interface of chemistry and physics, and progress depends on the coordinated efforts of experiment and theory. In Section 2 of this review, I begin with a general description of excitons in terms of a site Hamiltonian, with each site having two molecular orbitals. The aim of this section is not to develop a new theory of excitons but rather to provide a consistent framework that classifies the different types of excitons encountered in organic semiconductors. Section 3 briefly discusses experimental complications that arise in the study of organic semiconductors and how they can be mitigated. Section 4 discusses how the electronic characteristics of some of the exciton types described in Section 2 manifest themselves in the steady-state spectroscopy of these materials, with a particular emphasis on the photoluminescence properties of intermolecular Frenkel excitons. Sections 5 and 6 describe two aspects of exciton dynamics that are relevant to solar energy conversion: the spatial diffusion of excitons and fission/fusion reactions in which excitons can split and recombine. Space limitations preclude a comprehensive review, and it is impossible to include all references to relevant work from the many research groups working in this area. I apologize for any unintentional omissions.
2. OVERVIEW OF ELECTRONIC STATES IN MOLECULAR CRYSTALS The term exciton can encompass a wide variety of states with very different structures and behaviors. There exist several books and review articles on electronic states and excitons in organic systems (2–9). For the purposes of classifying different types of excitons in this review, I consider how the electronic states of molecules change as they assemble to form an aggregate or crystal. An isolated molecule in its ground state has two spin-paired electrons residing in its highest occupied molecular orbital (HOMO). The promotion of an electron into the lowest unoccupied molecular orbital (LUMO) leaves a hole in the HOMO, resulting in an excited singlet or triplet state, depending on the electron’s spin state. Figure 1a illustrates possible orbital configurations. The electron can be promoted into a LUMO on the same site as the hole or onto a neighboring site. Furthermore, the electron (hole) can travel from site to site via intersite electronic coupling. The most general electronic Hamiltonian for this system can be written in the position representation as + c n+ c n d n+ d n − U(r)c n+ c n d n+r d n+r + [te c n+ c n+1 + th d n+ d n+1 + H.C.] Hˆ el = −U(0) n r≥1 n n (1) + [V c n+ d n+ d n+1 c n+1 + H.C.], n
c n+
(c n ) are the electron creation (annihilation) operators for site n, d n+ (d n ) are the hole where creation (annihilation) operators for site n, and H.C. denotes the Hermitian conjugate of the preceding term. U(0) is the Coulomb energy of an electron-hole pair localized on a single site, and U(r) is the energy when the electron and hole reside on different sites. The transfer matrix element for nearest-neighbor electron (hole) transfer is given by te (th ), whereas the dipole-dipole interaction term V transfers an electron-hole pair from site n + 1 to site n. This site-basis Hamiltonian (10) is quite general and can be used to describe excitons in both the Frenkel and Wannier limits (11–13). For the purposes of this review, we consider only r = 1 terms, which limits us to the consideration 128
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a LUMO
b
....
....
ϕ–1 ϕ+2
ϕ+1 ϕ–2
1 — (ϕ+1 ϕ–2 – ϕ–1 ϕ+2) √2
ϕ1 ϕ*2
1 — (ϕ+1 ϕ–2 + ϕ–1 ϕ+2) √2 1 — (ϕ1 ϕ*2 – ϕ*1 ϕ2) √2
HOMO ϕ+n
ϕn–1
ϕ–n+1
ϕn+2 ϕ*1 ϕ2
Φ nCT = ϕ1 ϕ2…ϕ–n ϕ–n+1…ϕN
LUMO
(k)
....
.... ϕ*n
ϕn–1
ϕn+1
ϕ1 ϕ2
ϕ1 ϕ2
ϕ1 ϕ2
1
2
Dimer N = 2
∑n Bn(k)…ϕ+n ϕ–n+1…
ΨFrenkel =
1 — (ϕ1 ϕ*2 + ϕ*1 ϕ2) √2
HOMO Annu. Rev. Phys. Chem. 2014.65. Downloaded from www.annualreviews.org by Michigan State University Library on 01/15/14. For personal use only.
(k)
ΨCT =
∑n A(k)n …ϕ*n … D
2V
Aggregate N > 2
ϕn+2
Φ nFrenkel = ϕ1ϕ2…ϕ*n…ϕN Figure 1 (a) Schematic illustration of product basis states used to evaluate the Hamiltonian in Equation 1. The Frenkel (neutral) states involve an electron promoted to a lowest unoccupied molecular orbital (LUMO) on the same site, whereas the charge transfer (CT) states involve an electron promoted to a LUMO on a neighboring site. (b) At large separations, there is no interaction between molecules 1 and 2. The total wave function of an excited state is the simple product state. As the molecules approach each other, intermolecular interactions mix the neutral excited states to form superposition states. Similarly, the ionic states form delocalized charge resonance states. As an aggregate is formed, the neutral states grow into a Frenkel exciton band with a bandwidth of 2V and a solvatochromic shift D. Similarly, the ionic states give rise to the CT exciton band.
of so-called charge transfer (CT) excitons, where the charges separate only to nearest neighbors (14–17). This is usually justified by invoking the relatively low dielectric constant of organic solids as compared to inorganic semiconductors. For the U(0) term, we have a neutral basis consisting of states of the form Frenkel = φ1 φ2 . . . φn∗ . . . φ N .
(2a)
For the U(1) term, we have an ionic basis consisting of product wave functions of the form − . . . φN . CT = φ1 φ2 . . . φn+ φn+1
(2b)
If the interaction terms V/te /th did not exist, these product wave functions would be the N-fold degenerate eigenstates of the systems, separated by the energy difference U(1) − U(0), corresponding to the exciton binding energy. In many organic systems, this energy difference is much larger than the interaction terms, and we can consider the Frenkel and CT exciton problems separately. In this limit, the te /th terms couple CT states to each other, while the dipolar V interaction couples Frenkel states. The Frenkel part of Hˆ el can then be written as Bn+ Bn n + [V Bn+ Bn+1 + H.C.], (3) Hˆ Frenkel = U(0) n
n
where the Frenkel exciton creation/annihilation operators are Bn+ = c n+ d n+ and Bn = c n d n . For the CT part, we have + U(1)c n+ c n d n+1 d n+1 + [te c n+ c n+1 + th d n+ d n+1 + H.C.]. (4) Hˆ CT = n
n
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When Hˆ Frenkel or Hˆ CT is solved, we obtain a set of N eigenfunctions, enumerated by the index k, which are linear combinations of the basis states given in Equations 2a,b, i.e., (k) ∗ Frenkel = A(k) (5a) n ϕ1 ϕ2 . . . ϕ n . . . ϕ N , n (k)
CT =
− Bn(k) ϕ1 ϕ2 . . . ϕn+ ϕn+1 . . . ϕN .
(5b)
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In the momentum representation, k is the momentum quantum number that indexes the state. The delocalized exciton states Frenkel and CT can have new energies and different oscillator strengths as compared to their constituent localized states, and Figure 1b illustrates their evolution. The CT exciton states are cousins of the large-radius Wannier excitons often encountered in inorganic semiconductors. Note that I have not included coupling to the nuclear degrees of freedom in this treatment. I also assume that the sites 1 − N correspond to individual molecules, but the Hamiltonian in Equation 1 can also be applied to covalent polymer chains, where the sites correspond to repeat units (18, 19). Even given the simple picture in Figure 1, we can make some important points about exciton states in molecular crystals. First, the common perception that Frenkel excitons are more localized than CT excitons is not quite precise. It is the degree of charge separation within the basis functions that is different. Frenkel excitons comprise neutral states, whereas CT excitons comprise ionic states. Both types of states can have wave functions that are delocalized over N sites. Second, with regard to the value of N, for a perfect crystal, ideally N → ∞ and the new states form a continuous band. In practice, N is usually of the order of 1–100 in organic molecular systems, and we have aggregate states that exist within a larger solid or crystal. Many factors can conspire to limit the value of N, including structural defects and intermolecular vibrations that lead to time-dependent fluctuations in the te/h /V terms in the Hamiltonians given in Equations 3 and 4. The important point is that the finite value of N encountered in most organic systems tends to make the position-space Hamiltonian more useful than its momentum-space representation. Third, electron spin states play an important role in determining the energy of Frenkel exciton states but not CT excitons. In Frenkel exciton states, two electrons are localized on the same site, which means that the exchange interaction between them can be quite large (14, 20). This exchange interaction lowers the energy of the triplet states by up to 1 eV in some cases, effectively separating the Frenkel exciton into two different bands: singlet and triplet. These bands contain different numbers of states, as there are three possible triplet wave functions for every singlet. Because the ground state is singlet in character, spin selection rules make the triplet states effectively dark, with very weak absorption and phosphorescence. But these states are always present at energies below that of the singlet, and it is the 3:1 triplet:singlet ratio that leads to the 25% upper limit on the efficiency of organic light-emitting diodes based on fluorescence (21, 22). Accessing triplet states via intersystem crossing from the singlet requires a spin flip and is typically quite slow. But as shown below, multiple exciton interactions can provide spin-allowed ways to connect the singlet and triplet Frenkel exciton states. In deriving the Frenkel and CT Hamiltonians separately, we assume that the Frenkel and CT configurations do not interact. The total Hamiltonian is more accurately written as Hˆ tot = Hˆ Frenkel + Hˆ CT + Hˆ Frenkel−CT ,
(6)
where examples of the Hˆ Frenkel−CT term can be found in the work of Petelenz and colleagues (23, 24) and Spano and colleagues (25). This term gives rise to Frenkel exciton dissociation into a CT exciton, among other phenomena. Importantly, the division of electronic states into Frenkel versus CT excitons is an approximation that can break down when real physical systems are considered. 130
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A good example of this is recent work showing how the Frenkel-CT interaction leads to large changes in the Davydov splitting as polyacenes increase in size (25). Additionally, we assume that the Frenkel exciton band lies below the CT band. In fact, the positions of the two types of states depend on how the molecules pack in the crystal. The α- (face-to-face packing) and β- (edgeto-face herringbone packing) polymorphs of perylene provide examples of how different packing motifs for the same molecule can lead to low-lying CT and Frenkel states, respectively (26). Furthermore, small changes in molecular position can lower the CT states relative to the neutral state. The excitation of a Frenkel exciton followed by nuclear relaxation and CT is commonly called excimer formation in the chemistry literature but is also sometimes referred to as selftrapping (27, 28). Again, the important point is that a clean separation of the Frenkel and CT states is rarely found in nature. In summary, the simple idea of a single band gap between valence and conduction bands, commonly used to describe the situation in inorganic semiconductors, fails to capture the complexity of electronic states in organic semiconductors. In reality, these materials can be thought of as having three separate band gaps, corresponding to triplet Frenkel excitons, singlet Frenkel excitons, and CT excitons, respectively. Each band has its own energy, bandwidth, and photophysical properties.
3. EXPERIMENTAL CONSIDERATIONS IN THE STUDY OF SOLID-STATE ORGANIC MATERIALS This section discusses some experimental challenges in studying exciton dynamics in organic materials. The first issue revolves around the chemical composition of the sample. The presence of chemical impurities is much more of a concern for solid-state studies than for solution studies. In a solid sample, the effect of an impurity can be amplified owing to its ability to interact with neighboring molecules via energy and CT. The type and concentration of the chemical impurities depend on the system studied. Conjugated polymers often possess a distribution of chain lengths, conformations, and chemical defects. Small molecules can be purified to a high level using standard techniques such as sublimation and zone refining, providing a way to systematically address concerns about chemical purity. It is important to keep in mind that chemical contaminants can be introduced after the solid sample is prepared, with O2 and H2 O being the most common culprits. Even if the molecular composition of the sample is well controlled, one must pay careful attention to sample preparation to obtain consistent and reproducible results. For polymers, different spincoating conditions or solvents can lead to different degrees of aggregation and disorder, changing the spectroscopic properties considerably (29). For oligomers, crystal growth is still somewhat of an art, as the presence of different crystal polymorphs and mixed amorphous/crystalline regions can depend sensitively on sample growth conditions. One important point is that the required sample purity depends on the quantity being measured. For example, the fluorescence lifetime of a singlet exciton is in the nanosecond regime, which does not leave the exciton enough time to explore a large region of the sample before it decays back to the ground state. A contaminant present at the parts per million level will typically have little effect on this observable. Conductivity measurements, conversely, probe the dynamics of longlived charge carriers that diffuse throughout the sample and can encounter distant defects and contaminants. Reliable measurements of charge mobilities thus require highly purified samples with very low impurity concentrations (30, 31). A second issue is the need to distinguish between effects that arise from intrinsic exciton properties and those that arise from the optical properties of the macroscopic sample. For example, most single crystals are optically thick, absorbing so much light that simple transmission www.annualreviews.org • Structure and Dynamics of Molecular Excitons
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measurements become impossible, and one must use reflection measurements, in conjunction with a Kramers-Kronig analysis, to extract absorption spectra (32, 33). For fluorescence measurements, one must account for possible complications from the self-absorption of the high-energy portion of the luminescence, waveguiding from total internal reflection in a high-index crystal, and depolarization due to crystal birefringence. Such effects distort the steady-state spectra and also time-resolved signals. Many optical artifacts can be reduced or eliminated by using optically thin (90% in the single crystal (121). In a polycrystalline film, conversely, the singlet decay rate has been reported to be 40–100 ps (120, 121, 129). This is considerably faster than the single crystal data described above, suggesting that more disordered samples may have sites that can accelerate SF. Magnetic field effects on triplet kinetics provided some of the earliest evidence for SF in the polyacenes (130, 131). More recently, magnetic field–dependent measurements of SF in amorphous rubrene provided evidence of the importance of molecular orientation in SF (132). The triplet spin states, and thus the triplet pair states, change in the presence of a magnetic field. Although the detailed theory of how these states change in a magnetic field is somewhat complicated, the basic idea is relatively simple (133–137). In the absence of a magnetic field, only three out of nine possible triplet pair states (|xx , |y y , and |zz ) have singlet character, as per Equation 9. As the magnetic field is turned on, the triplet eigenstates change, and the distribution of singlet character within those states changes as well. For a pair of parallel oriented molecules, the number of triplet pair states with singlet character decreases from three (zero field) to two (high field). Alternatively, if the fissioning molecules have random orientations, then the number of states with singlet character increases from three to nine. Figure 7 illustrates the evolution of singlet character for the nine triplet pair states for different molecular arrangements. Figure 8a shows the fluorescence decay of amorphous rubrene films with and without an applied magnetic field, along with calculated signals for the case in which SF can occur only in parallel oriented molecular pairs (Figure 8b) and in randomly oriented molecular pairs (Figure 8c). The increased prompt fluorescence in the experimental data shows that even in amorphous films, SF occurs preferentially in aligned pairs of molecules.
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θ A = ϕA = 0 θ B = ϕB = 0
a Parallel 0.6 0.5 0.4 0.3 0.2 0.1 0 0
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Figure 7 Singlet character, defined as the magnitude squared of the overlap of the triplet pair and singlet spin states, of each triplet pair state as a function of magnetic field strength for different molecular orientations. (a) For parallel molecules, the number of states with singlet character decreases from three to two. (b) For a 45◦ angle between molecules, the number of states with singlet character increases from three to eight. (c) For a collection of randomly oriented molecular pairs, the number of states with singlet character increases from three to nine. Figure taken from Reference 132.
I emphasize that the photophysical mechanism of SF remains the subject of active study. The importance of relative chromophore orientations, now well established through experimental results, is being clarified by theory. Although originally thought to involve only the singlet and triplet exciton manifolds (138, 139), recent theoretical studies strongly implicate the role of CT states in accelerating the SF reaction (140–146). Thus SF may provide an example of how the interaction of all three types of excitons described in Section 2 can lead to novel phenomena that may be of practical utility. www.annualreviews.org • Structure and Dynamics of Molecular Excitons
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Figure 8 (a) Experimental fluorescence decays of amorphous rubrene in a 20-ns time window. (b) Simulated fluorescence decays, showing the enhancement in the prompt fluorescence when the molecules are assumed to be parallel. The application of a magnetic field decreases the number of triplet pair states with singlet coupling. This simulation agrees qualitatively with the experimental data. (c) Simulated fluorescence decay showing suppressed prompt fluorescence at high magnetic field when randomly oriented molecules are assumed. Figure taken from Reference 132.
7. CONCLUSION The electronic structure of conjugated organic solids can be complicated, and the existence of three different exciton types (Frenkel singlets, Frenkel triplets, and CT excitons) must be kept in mind when studying these materials. Furthermore, one must take considerable care to avoid experimental artifacts due to impurities, high optical densities, and annihilation effects. But the properties of these excitons, including delocalization, diffusion, and fission/fusion reactions, make them well worth studying. At the present time, a convergence of new experimental methods and theoretical advances has resulted in new insights. These studies will likely provide the framework for a predictive structure-function understanding of excitons in organic solids. It is hoped that this review helps facilitate the experimental advances needed to generate improved materials and usher in the era of organic electronics.
DISCLOSURE STATEMENT The author is not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.
ACKNOWLEDGMENTS This work was supported by the National Science Foundation under grant CHE-1152677.
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