Magnetic field effects and the role of spin states in singlet fission

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Chemical Physics Letters 585 (2013) 1–10

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FRONTIERS ARTICLE

Magnetic ﬁeld effects and the role of spin states in singlet ﬁssion Jonathan J. Burdett 1, Geoffrey B. Piland 1, Christopher J. Bardeen ⇑ Department of Chemistry, University of California – Riverside, Riverside, CA 92506, United States

a r t i c l e

i n f o

Article history: Available online 23 August 2013

a b s t r a c t Singlet ﬁssion is a photophysical process that has promise for increasing the efﬁciency of solar cells. The dynamics depend on triplet spin states and can be inﬂuenced by external magnetic ﬁelds. In 4-electron systems, ﬁssion takes an initial singlet state into a superposition of triplet pair states. Direct evidence for this superposition state is provided by quantum beats in the delayed ﬂuorescence of tetracene crystals. The beat frequencies depend on crystal orientation with respect to the magnetic ﬁeld, consistent with predictions based on solving the full spin Hamiltonian. Magnetic ﬁeld effects on the kinetics are analyzed in terms of a hybrid quantum-kinetic model. The magnetic ﬁeld has no effect on the initial ﬂuorescence decay rate but affects the decay after the triplet pair states begin to equilibrate with the singlets. The long-time behavior of the ﬂuorescence decay reﬂects association and separation of triplet pairs and relaxation into different spin states. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction The ability to harvest solar photons and convert their energy into electrical energy in a cost-effective and efﬁcient manner would have broad implications for human society. First-generation photovoltaic cells based on bulk semiconductor crystals have been followed by second generation thin ﬁlm technologies that promise similar efﬁciencies but lower costs. So-called ‘third generation’ photovoltaic technologies are projected to both lower costs and raise power conversion efﬁciencies by taking advantage of new types of physical phenomena [1]. An example of a physical process that could be harnessed to increase the overall efﬁciency of the solar energy conversion process is multiple exciton generation (MEG) [2]. When Shockley and Queisser calculated the thermodynamic limit for a single junction solar cell [3], they assumed that the absorption of a single photon could only create a single electron hole pair. When a photon has an energy above the bandgap, it is absorbed but quickly relaxes to the bottom of the band, in a process analogous to Kasha’s Rule for molecules. The excess energy of the photon is dissipated as heat and cannot contribute to useful energy production. This limitation is a major factor as to why the maximum efﬁciency of such a cell, under the solar spectrum, is only 32%. In principle, if the photon energy is twice the bandgap, it can produce two electron–hole pairs at the bandgap energy. A material in which this process was highly efﬁcient could boost the efﬁciency of the solar cell by up to 30%, or raise the theoretical limit of energy conversion from 32% to 43% [4,5]. The possible efﬁ⇑ Corresponding author. 1

E-mail address: [email protected] (C.J. Bardeen). These authors contributed equally to this work.

0009-2614/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2013.08.036

ciency gain in solar cells resulting from such a 1 ? 2 conversion of excitons has driven a resurgence of interest in multi-exciton processes in both inorganic and organic semiconductors. In a traditional inorganic semiconductor, Wannier excitons can undergo MEG through several processes that may be enhanced in nanostructured materials [6–8]. But MEG in these materials usually has a threshold energy greater than 2 the bandgap, and the overall efﬁciency of this process is still under debate, since MEG must compete with rapid relaxation processes that occur as the exciton relaxes to the bottom of the band [9–11]. Organic semiconductors, wherein optical excitation typically produces neutral Frenkel excitons, provide a qualitatively different physical situation. In these systems, the neutral exciton states are analogous to molecular excited states and exchange interactions result in energetically distinct spin states, namely singlets and triplets. In a sense, molecular semiconductors have two bandgaps: the optically allowed S0 ? S1 singlet bandgap, and the optically forbidden S0 ? T1 triplet bandgap. If the energy of the S1 state E(S1) is twice the energy of the T1 state E(T1), i.e. E(S1) P 2 E(T1), then it is possible for the optically excited singlet exciton to spontaneously split into a pair of triplet excitons through a process known as singlet ﬁssion (SF) [12]. The ability to use different spin excitons that also have different energies provides some potential advantages when compared to MEG in inorganic systems. First, the fact that the singlet and triplet states cannot easily interconvert means that SF can take place from the fully relaxed singlet state and does not have to compete with intraband relaxation. Second, SF is a bulk property of molecular systems and does not require quantum conﬁnement to enhance its efﬁciency. Lastly, the triplet excitons produced tend to have low radiative decay rates and long lifetimes, which should make it easier to harvest them for charge production. All of these

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characteristics have led to a revival of research in SF, which was ﬁrst observed in 1965 [13] and was the subject of considerable research interest in the 1970’s [14]. SF has been experimentally observed in several different classes of molecular systems, and there exist extensive reviews that nicely summarize both the ﬁrst generation of research and more recent advances in the ﬁeld [14,15]. Recent experiments have shown that the triplets produced via SF can produce appreciable photocurrent in photovoltaic devices [16–19]. There are two requirements that must be fulﬁlled in order to generate efﬁcient SF in an organic semiconductor: energy and spin conservation. Theory and experiment have identiﬁed several classes of molecules that fulﬁll the energetic condition E(S1) P 2 E(T1) [20], including polyacene derivatives [21–25], carotenoids [26], isobenzofurans [27], and polydiacetylene [28]. In this Letter, we use crystalline tetracene [29] as a model system and focus on the role of spin conservation. A quantitative model that describes how the spin states affect the SF dynamics is necessary both to understand the spectroscopic kinetics and to make contact with older work in the ﬁeld, wherein magnetic ﬁeld effects were used almost exclusively to characterize SF in molecular crystal systems [14,30,31]. In this Letter, we will begin with a general discussion of spin states in 4-electron systems, showing how SF can occur in these systems with conservation of the total spin angular momentum. We then show how a hybrid quantum-kinetic model, basically an extension of that proposed by Merriﬁeld more than 40 years ago [32], can describe both the electronic structure of the triplet pair states and their changing dynamics in a magnetic ﬁeld. The good agreement between our calculations and the experimental ﬂuorescence dynamics in crystalline tetracene provides evidence that this model can provide a good description of the SF process without the need to invoke additional intermediate states or coherences. To aid the reader who is not familiar with the ﬁeld, this paper brieﬂy reviews some of our previous results in order to place the new results in their appropriate context. 2. Spin states in multielectron systems Closed shell molecules have two electrons paired together in their HOMO in a singlet spin state. When one of those electrons is promoted to the LUMO, e.g. by photon absorption, there are four possible spin states. These states are eigenfunctions of the total 2 spin angular momentum operator ^ S2 ¼ ð^ S1 þ ^ S2 Þ and can be found by diagonalizing the matrix of this operator in the single spin product basis, i.e. |aai, |abi, |bai, |bbi where |ai, |bi are the usual eigenS1z . In this basis, we states of the one-electron spin operators ^ S21 and ^ ﬁnd one singlet (eigenvalue S = 0) and three triplet (eigenvalue S = 1) solutions [33]:

1 jSð2Þ i ¼ pﬃﬃﬃ ðjabi jbaiÞ 2 ð2Þ

ð1Þ

jT 1 i ¼ ji ¼ jbbi

ð2aÞ

1 ð2Þ jT 0 i ¼ j0i ¼ pﬃﬃﬃ ðjabi þ jbaiÞ 2

ð2bÞ

jT þ1 i ¼ jþi ¼ jaai

ð2cÞ

ð2Þ

Here |ai and |bi are the usual single electron spin functions, and the superscript refers to the number of electron spins. The jT ð2Þ n i triplet states are often referred to as the ‘high-ﬁeld’ basis since they are also eigenstates of the ^ Sz operator and thus describe the eigenstates when a strong magnetic ﬁeld Bz is applied to the system, for example in an electron spin resonance experiment. In the literature they are also deﬁned as the | i, |0i, | + i states, as shown in Eqs.

(2a)–(2c). In the absence of a magnetic ﬁeld, there exists an effective dipole–dipole interaction between the two electrons that leads to the ‘zero-ﬁeld’ Hamiltonian (given in detail in Section IV below) with three different eigenstates [14,34]:

1 jxi ¼ pﬃﬃﬃ ðjbbi jaaiÞ 2

ð3aÞ

i jyi ¼ pﬃﬃﬃ ðjbbi þ jaaiÞ 2

ð3bÞ

1 jzi ¼ pﬃﬃﬃ ðjabi þ jbaiÞ 2

ð3cÞ

Note that either of the triplet basis sets in Eqs. (2a)–(2c) or Eqs. (3a)–(3c) can be used to provide a complete description of the 2electron triplet subspace. When we consider two molecules A and B, we now have a 42 electron system with a total ^ S2 ¼ ð^ S1 þ ^ S2 þ ^ S3 þ ^ S4 Þ . We can diagonalize this spin operator in the 4-electron product basis, i.e. |aaaai, |aaabi, etc. When we diagonalize the 16 16 matrix, we ﬁnd two solutions with eigenvalues S = 0 (singlet states), nine solutions where S = 1 (triplet states), and ﬁve where S = 2 (quintet states). We are concerned with the two singlet solutions. The ﬁrst one can be factored as the product of 2-electron single states:

1 1 ð4Þ jS1 i ¼ pﬃﬃﬃ ðjabiA jbaiA Þ pﬃﬃﬃ ðjabiB jbaiB Þ 2 2

ð4Þ

All this solution tells us is that two molecules, each in a singlet state, combine to make an overall singlet state. Examples include the S0S0, S0S1 and S1S1 state combinations. The second solution is more interesting:

1 ð4Þ jS2 i ¼ pﬃﬃﬃ ðjaaiA jbbiB þ jbbiA jaaiB Þ 3 1 þ pﬃﬃﬃ ðjabiA þ jbaiA ÞðjabiB þ jbaiB Þ 2 3

ð5Þ

This wavefunction cannot be reduced to a product of singlets, but it can be written as a superposition of product pairs of 2-electron triplet states:

1 ð4Þ jS2 i ¼ pﬃﬃﬃ ðjxiA jxiB þ jyiA jyiB þ jziA zB Þ 3 1 ¼ pﬃﬃﬃ ðj0iA j0iB j þ iA j iB j iA j þ iB Þ 3 ð4Þ

ð6Þ

ð4Þ

A transition from jS1 i to jS2 i can be accomplished without ﬂipping a spin, in contrast to intersystem crossing. Thus a transition from the singlet to the triplet pair manifold is a spin-allowed process, and this is the key for understanding why SF can be such a rapid excited state relaxation pathway. S0 + S1 ? 1(T1T1) is an allowed sequence of events, as long as we remember that the triplets must be produced in a superposition state with overall singlet character, ð4Þ where jS2 i = 1(T1T1). Note that Eq. (4) implies that SF into a pair of singlets is also a spin-allowed process. The 1 ? 2 production of low-energy emissive singlet states from a high energy singlet state could have practical applications as a way to downconvert photons. But such a process has to compete with 1 ? 1 singlet-to-singlet energy transfer processes (e.g. Forster transfer or internal conversion) and is difﬁcult to observe in molecular systems [35]. 3. Electronic spin structure of triplet pair states In the previous section, we saw how an initially excited molecð4Þ ular singlet state jS1 i could transition to a triplet pair superposið4Þ tion state jS2 i in a spin-conserving process. Direct evidence for the creation of this superposition state by SF was ﬁrst obtained

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Figure 1. Schematic diagram illustrating ﬁssion and fusion between the initially excited singlet state and the 3 triplet pair states with singlet character in the zeroﬁeld basis.

by Chabr et al. and later in more detail by our group through the observation of quantum beats in the delayed ﬂuorescence of crystalline tetracene [36–38]. The basic idea is outlined in Figure 1, where SF projects the initially excited singlet population onto the |xxi, |yyi, and |zzi triplet pair states. The resulting superposition evolves for a period of time, and then couples back to the luminescent singlet state through triplet fusion. While the wavefunction is in the triplet pair superposition state, it acquires a phase and this phase leads to constructive and destructive interference in the triplet fusion pathways, which in turn leads to observable quantum beats in the singlet ﬂuorescence. Analysis of the oscillations in Figure 2 can provide valuable information about the SF process, and details can be found in reference [38]. In that paper, we analyzed our results using a density matrix model that assumed the existence of four states: one singlet plus three triplet pair states, wherein long-lived coherences between the triplet pair states were allowed. This model permitted us to quantitatively simulate our ﬂuorescence data (the simulation is overlaid with the data in Figure 2). The modeling allowed us to address two questions about the SF process. The ﬁrst was whether the initially excited singlet decays directly to the triplet pair, or whether an indirect pathway takes the singlet through an intermediate state. One can think of the SF reaction as being similar to a laser pulse that creates a coherent superposition of triplet sublevels if its bandwidth is greater than the level spacing. The triplet pair formation time is analogous to the pulse duration. A slow overall SF rate cannot impulsively excite any quantum beats, while a very rapid SF rate will excite all three of them equally. Both the magnitude and the relative amplitudes of the oscillations were

3

well-reproduced for the case where the formation time of the triplet pairs mirrored the decay time of the singlet, as expected for the one-step ‘direct’ mechanism for SF [15]. Our results did not necessarily rule out the ‘indirect’ two-step mechanism for SF, but they do suggest that lifetime of any intermediate state is so short at room temperature that it is indistinguishable from the single-step mechanism. The second question involved the structure of the triplet pair. If two triplets, each with a magnetic dipole, were trapped next to each other, the dipole–dipole interaction would be expected to shift the triplet energies and change the beat frequencies by a noticeable amount. To within the experimental error, we did not observe such a shift. We hypothesized that either (1) if the triplets are stationary, they must be spaced apart farther than the nearest neighbor distance in the crystal, or (2) the triplets are moving so rapidly that the dipole–dipole interaction averages to zero. It may seem surprising that spin coherence can be maintained while the excitons diffuse, but other experiments have suggested that spin coherence in organics can be surprisingly robust [39]. The spin coherence can be regarded as a fortunate side product of the SF reaction that allowed us to probe the properties of the triplet pairs without having to detect them directly. 4. Magnetic ﬁeld effects on ﬁssion and fusion The data in Figure 2 shows that in the zero-ﬁeld case, the triplets produced by SF behave as independent excitons whose spin states are the standard zero-ﬁeld wavefunctions. If the sample is placed in a magnetic ﬁeld, both the energies and character of the spin states will change in predictable ways. We should be able to predict how these changes affect the experimental observable, in our case the time-resolved ﬂuorescence signal. In reference [40] we outlined a version of the Johnson–Merriﬁeld theory to describe the kinetic effects that result when a magnetic ﬁeld is applied to a disordered sample, namely amorphous rubrene. In this Letter, we take a similar approach to describe how both the triplet pair energies and the kinetics change as a magnetic ﬁeld is applied to an ordered system, namely crystalline tetracene. 4.1. Magnetic ﬁeld effects on energy levels and quantum beats We ﬁrst consider the quantum mechanical part of the problem, starting with the Hamiltonian for two molecules A and B (4 electron spins) in an external magnetic ﬁeld B. The orientation of A and B are ﬁxed by the crystal packing, while the orientation of the magnetic ﬁeld is ﬁxed by the position of the crystal between the poles of the magnet. In the case of tetracene, the molecular x and z-axes are located in the ab plane of the crystal, and if we de-

Figure 2. (a) Time resolved ﬂuorescence of a solution grown single crystal of tetracene (black) along with simulated data convolved with an instrument response (red). Details of the simulation can be found in reference [38]. (b) Normalized Fourier transforms of the extracted frequencies from solution grown single crystal of tetracene (black) along with simulated data convolved with an instrument response (red) using the same parameters as (a).

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^ AB is a much simpliﬁed version of the expression that describes of H the full dipole–dipole interaction between two triplets [41]. But for the purposes of this paper, this term serves mainly to break the degeneracy of triplet pair states at high B-ﬁeld and the results reported below do not depend on its exact form. The total Hamiltonian can be solved using either the ‘zero-ﬁeld’ basis or the ‘high ﬁeld’ basis. In this Letter we will use the ‘zeroﬁeld’ basis. The new eigenvectors {|/li} with l = 1–9, are linear combinations of the simple product basis functions given in Eqs. (2a)–(2c) or (3a)–(3c). We can project these new wavefunctions onto the singlet wavefunction to obtain their singlet character C lS , given by Figure 3. (a) Tetracene labled with the molecular axes. (b) A c-axis projection of the crystal lattice detailing the orientation of the magnetic ﬁeld in the crystal axes (a and b) and the crystal ﬁne structure tensor axes(x⁄ and z⁄). h indicates the angle from the b-axis in the ab plane. (c) An illustration of the crystal axes and the crystal ﬁne structure tensor axes superimposed on each other with the magnetic ﬁeld in the ab plane.

ﬁne the angle of rotation of the ab-plane in the magnetic ﬁeld as shown in Figure 3, the overall Hamiltonian in terms of the rotation angle h consists of three parts:

^ magnetic ðhÞ þ H ^ zero-field ðhÞ þ H ^ AB ^ total ðhÞ ¼ H H

ð7Þ

where

^ magnetic ¼ gb½hA ^SA þ hA ^SA þ hA ^SA þ gb½hB ^SB þ hB ^SB þ hB ^SB H x x y y z z x x y y z z

ð8Þ

¼ D½ð^SAz Þ þ 13 ð^SA Þ þ ð^SBz Þ þ 13 ð^SB Þ 2 2 2 2 þE½ð^SA Þ ð^SA Þ þ ð^SB Þ ð^SB Þ

ð9Þ

^ zero-field H

2

x

2

y

2

x

2

y

^ AB ¼ X ^SA ^SB H

ð10Þ

^ magnetic describes the interaction of an external magnetic ﬁeld H with the electron spin system, where ^ Six ; ^ Siy ; ^ Siz are the x⁄, y⁄ and z⁄ components of the two-electron spin operators for molecule i in the crystal, g is the gyromagnetic ratio, and b is the magnetic ﬁeld coupling constant. To ﬁnd the B-ﬁeld projections on the molecular crystal axes, we ﬁrst calculate the projections of the magnetic ﬁeld vector onto the crystal unit cell axes: i

i

ha ¼ BSinðhi Þ

hb ¼ BCosðhi Þ

ð11Þ

where B is the strength of the magnetic ﬁeld and the angle h is shown in Figure 3. Once projected onto the crystal axes, the vectors are then projected onto the molecular axes using the following i i i directional cosine table [14] to obtain the values for hx ; hy ; hz :

0

x

y

z

1

B C a B 0:9634 0:2634 0:0372 C B C B b B 0:0269 0:2463 0:9714 C C B C c @ 0:2663 0:9330 0:2390 A

ð12Þ

1 ð4Þ hS2 j/i i ¼ pﬃﬃﬃ ðhxxj þ hyyj þ hzzjÞj/i i ¼ C lS 3

ð13Þ

The assumption underlying all theoretical models for SF is that only states with singlet character C lS –0 will participate in SF. If B = 0, there are only three zero-ﬁeld triplet pair states that have equal values for C lS , i.e. the |xxi, jyyi, and jzzi states (each with a dif^ AB , there are again ferent energy). At high ﬁeld, in the absence of H three states with equal singlet character: the |00i, | + i, and | + i states. The | + i, and | + i states are degenerate, while the energy difference between the |00i and | + i states depends on the angle h. Previous workers used ﬁrst order perturbation ^ zero is the perturbation) to estimate this energy theory (where H splitting as

1 DE00=þ ¼ 3D cos2 c þ 3E ðcos2 a cos2 b Þ 3 ^ zero j00i hþ jH ^ zero j þ i ¼ h00jH

ð14Þ

where a⁄, b⁄, and c⁄ are the angles between the magnetic ﬁeld and the ﬁne structure tensor axes x⁄, y⁄, and z⁄ respectively. Given h and the cosine table in Eq. (12), one can ﬁnd hx and hy and then extract the angles a⁄, b⁄, and c⁄. In other words, h uniquely determines these angles, although writing the full h dependence of Eq. (14) would result in a very cumbersome expression. For tetracene, we expect the three distinct quantum beat frequencies observed at zero-ﬁeld to collapse to one quantum beat frequency whose dependence on angle h can be deduced from Eq. (14). Indeed, this is what is observed experimentally for a subset of angles h = 0°–60°, as shown in Figure 4. The experimental oscillatory component of the delayed ﬂuorescence signal, over a 15 ns interval, is shown for seven different crystal orientations in the B-ﬁeld. The Fourier transforms of these signals are shown in Figure 4b. For all crystal orientations in the high ﬁeld, there is only a single dominant Fourier component, rather than the three components seen in Figure 2b. In Figure 5, we plot the oscillation frequency as a function of crystal angle, along with the theoretical prediction obtained by both ^ total ðhÞ and by using the perturbation numerical diagonalization of H expression given in Eq. (14). In the limit of high magnetic ﬁeld strength (B > 103 Gauss), the frequencies predicted by the exact solution and by Eq. (14) agree quantitatively. 4.2. Merriﬁeld kinetic model

Note that since the x⁄z⁄ plane coincides with the ab crystal plane, the angle of y⁄ with the B-ﬁeld does not change. Thus we do not have to consider the full three-dimensional projection of the B-ﬁeld onto the molecular axes, which considerably simpliﬁes the treat^ zero-field describes the effects of interactions between ment. Next, H electrons on a single molecule which breaks the degeneracy of the three triplet states even in the absence of a magnetic ﬁeld. These interactions lead to the zero-ﬁeld splitting, as parameterized by the constants D⁄ and E⁄. Lastly, the interaction between electrons on ^ AB in Eq. (10) where the different molecules A and B is given by H X term gives the strength of the spin–spin coupling. This version

The effect of the magnetic ﬁeld on the quantum beats can be modeled using the quantum mechanical model developed in the previous section. The same calculations can also be used to generate inputs for a kinetic model that predicts the overall rates of SF and TF. It should be noted that the kinetic schemes we will present in Sections 4.2 and 4.3 only take into account populations terms and thus cannot describe the coherence terms that give rise to the quantum beats in the previous section. In this section, we give an overview of the Merriﬁeld kinetic model [32,42], since it is the simplest model that can explain the salient features of the

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Figure 4. (a) Oscillatory component of the experimental time resolved delayed ﬂuorescence data taken with various B-ﬁeld angles h. (b) Fourier transform power spectra of the oscillations in part (a).

9 9 X X dNS1 ¼ k2 jC ls j2 NS1 þ k2 jC lS j2 NðT1T1Þl dt l¼1 l¼1

ð17aÞ

dNðT1T1Þl ¼ k2 jC lS j2 NS1 ðk2 jC lS j2 þ k1 ÞNðT1T1Þl þ k1 N2T1 dt

ð17bÞ

9 X dNT1 ¼ 2k1 NðT1T1Þl k1 N2T1 dt l¼1

ð17cÞ

Previous workers, not being able to directly observe the N(T1T1)l and NT species indicated in Eq. (16), realized that they could simply remove the intermediate N(T1T1)l population from the problem by dN 0. They could then explicassuming steady state conditions ðT1T1Þl dt itly solve for the time-dependence of the S1 and T1 populations, ﬁnding

Figure 5. Plot of oscillation frequency of the quantum beats from Figure 4a versus the angle of the B-ﬁeld relative to the crystal b-axis. Also shown are the calculated frequencies from perturbation theory (Eq. (14)) and an exact calculation obtained by diagonailzing the full spin Hamiltonian in Eq. (7).

dNS1 ¼ cS NS1 þ cT N2T1 dt

ð18aÞ

dNT1 ¼ 2cS NS1 cT N2T1 dt

ð18bÞ

where

cS ¼

9 X k2 jC lS j2 l¼1

ﬂuorescence dynamics. In section 4.3, we will present an expanded model that will be used to analyze our own data. In both models, we assume that the transition matrix element coupling the singlet to a speciﬁc triplet pair state |/li is proportional to C lS (l = 1–9). The rate coupling the singlet to a speciﬁc triplet pair state l will be proportional to jC lS j2

Rate of Fission ðsinglet ! j/i iÞ ¼ k2 jC lS j2

ð15aÞ

Rate of Fusion ðj/i i ! singletÞ ¼ k2 jC lS j2

ð15bÞ

Note that k2 – k2 in general. The use of the 2/2 subscripts is to be consistent with the notation of earlier workers [12,14], who reserved k1/k1 to describe triplet association processes that are important for fusion of free triplets created by S0 ? T1 excitation. The kinetic scheme for the populations of the singlet state NS1, the 9 possible triplet pair states N(TT)l, and the ‘free’ triplets NT, is

k1 k2 S1 ðT 1 T 1 Þ T 1 þ T 1 k2

k1

and is described by the following equations:

ð16Þ

1 9

1 þ ejC lS j2

cT ¼ k1

9 X

ejC lS j2 l 2 l¼1 1 þ ejC S j

ð19aÞ

ð19bÞ

and e ¼ kk12 . These are the ‘classical’ results for SF, but it should be emphasized that they are derived under somewhat restrictive conditions and cannot describe the full time evolution of NS1, in particular the early-time decay that takes place before the N(T1T1)l population is established. Eqs. (18a), (18b) show how the C lS singlet overlap coefﬁcients are the key quantities for understanding magnetic ﬁeld effects on SF kinetics. Note that if the intermediate 1 (T1T1) states did not exist, the rates of NS1 and NT1 decays would not depend on C lS and there would be no magnetic ﬁeld dependence. P This is true as long as 9l¼1 jC lS j2 ¼ 1. This condition is equivalent to saying that the norm of the singlet vector is conserved under unitary transformations of the spin Hamiltonian, which is typically the case. The Merriﬁeld model provides a way to predict the kinetics of SF and TF, subject to some approximations. As described in the previous section and shown in Eq. (6), in the high-ﬁeld there are three triplet pair states with equal singlet character. Thus one would naively expect to have the exact same situation as in the zero-ﬁeld

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case: three triplet pair states, each with jC lS j2 ¼ 13. But Merriﬁeld ^ AB term can realized that if X – 0 in Eq. (10), the presence of the H break the degeneracy of the | + i and | + i states, generating two new superposition states

1 jT i ¼ pﬃﬃﬃ ðj þ i j þiÞ 2

ð20Þ

Only |T+i has C lS –0 in this new basis, and thus only |00i and |T+i have singlet character in a high magnetic ﬁeld, given the geometry in Figure 3. Fewer triplet states with singlet character leads to decreased ﬁssion and fusion rates, with a concomitant increase in the prompt ﬂuorescence (cS is smaller) and a decrease in the delayed ﬂuorescence (cT is smaller). It should be emphasized, however, that different molecular arrangements can yield qualitatively different magnetic ﬁeld effects. In reference [40] we showed that random molecular orientations allow the magnetic ﬁeld to distribute singlet character over all nine triplet pair states on average, leading to the opposite effect on ﬁssion and fusion rates when compared to the ordered case. Eqs. (18a), (18b) were sufﬁcient to analyze early experiments that measured the magnetic ﬁeld effects on the ‘prompt’ and ‘delayed’ ﬂuorescence signals, but those early experiments had some important limitations. Although the designations ‘prompt’ and ‘delayed’ suggest that their temporal behavior was measured, in reality ‘prompt’ ﬂuorescence referred to the total luminescence after direct excitation of the S0 ? S1 transition (proportional to 1/cS), while ‘delayed’ ﬂuorescence was generated by excitation of the S0 ? T1 transition in the near infrared (proportional to cT) [43,44]. In practice, the calculated dependence of cS and cT on magnetic ﬁeld should mirror the observed dependence of the timeintegrated ﬂuorescence signals. The good correspondence between the calculated cS/cT dependence on B-ﬁeld strength and orientation and the ﬂuorescence signal observed after excitation of either the S0 ? S1 or S0 ? T1 transitions provided conclusive evidence for SF in tetracene [30,31,45]. But with the advent of modern time-resolved spectroscopic techniques, we are now in a position to examine the full time-dependence of the SF process and directly probe the formation and relaxation of the 1(T1T1) intermediate. The signature of this intermediate may have been detected indirectly via optically detected magnetic resonance experiments on tetracene [46], but without determining its kinetic properties. 4.3. Expanded kinetic model Given the new information available from picosecond time-resolved ﬂuorescence measurements, one goal of our work has been to extend the Merriﬁeld model to describe how the ﬂuorescence signal changes on all timescales. Suna provided a more detailed model of TF based on the density matrix [47–49], but in this Letter we will continue with the kinetic approach. In reference [40] we derived a set of kinetic equations to describe the ﬂuorescence dynamics in amorphous rubrene. In that work, we found that we did not have to consider the transition from geminate pair to ‘free’ triplets that interact with triplets generated by other excitation events. This limit can be attained through a combination of slow exciton diffusion and low excitation densities, as was the case for our experiments in amorphous rubrene. That kinetic model was sufﬁcient to describe the qualitative magnetic ﬁeld dependence of the decays but did not quantitatively match the experimental data. We attributed the discrepancy to the fact that the model did not take into account spatial diffusion of triplet excitons between ﬁssion and fusion events. In this section, we extend our earlier model in order to take exciton diffusion into account, albeit in a crude way. The kinetic scheme is similar to one proposed by Bouchriha and coworkers

Figure 6. Schematic representation of the kinetic model. (TT)l refers to a associated triplet pair state while (T T) refers to a spatially separated triplet pair state. This diagram only shows transfer rates to the lth triplet, but transfer can occur between the singlet state and all of the triplet pair states. krad represents the radiative decay from the singlet states, k2 and k2 determine the ﬁssion and fusion rates, respectively, ktrip is the relaxation rate out of the triplet manifold and krelax transfers population between the separated triplet pair states.

[50] and is outlined in Figure 6. We consider two types of triplet pairs: associated and separated. We also consider a spin–lattice relaxation process, parameterized by krelax, which only occurs in the separated pairs and serves to redistribute the population across the 9 possible spin states. This kinetic model assumes that diffusion of the triplet excitons occurs on a timescale more rapid than spin relaxation. The kinetic equations are given below 9 X dNS1 ¼ ðkrad þ k2 ÞNS1 þ k2 jC ls j2 NðTTÞl dt l¼1

ð21aÞ

dNðTTÞl ¼ k2 jC ls j2 NS1 ðk2 jC ls j2 þ k1 ÞNðTTÞl þ k1 NðT TÞl dt

ð21bÞ

dNðT TÞl ¼ k1 NðTTÞl ðk1 þ ktrip þ krelax ÞNðT TÞl dt X1 krelax NðT TÞj þ 8 j–l

ð21cÞ

The rate constants are deﬁned as follows: krad is the radiative rate, k2 refers to the rate of ﬁssion, k2 is the rate of fusion, k1 is the rate at which triplets become spatially separated, k1 is the rate at which triplets move into a distance at which fusion is possible, ktrip is the relaxation rate out of the triplet manifold, and krelax is the rate at which a triplet pair state relaxes into other triplet pair states. Note that krelax is only operative for the separated triplet pairs. This is equivalent to saying that the same processes that underly triplet exciton hopping also enable spin–lattice relaxation. The main difference between the model in Eq. (21) and the standard Merriﬁeld model in Eqs. (17a), (17b), (17c) is that the k1/1 processes in our model do not lead to ‘free’ triplets that can recombine with any other triplet (which would give rise to a kinetic term proportional to N 2T1 ) but instead describe the spatial separation of the geminate pair. These equations will be valid under low laser intensity conditions, where our experiments are typically performed. We will use this kinetic approach to model the magnetic ﬁeld effect on the SF process in polycrystalline tetracene thin ﬁlms. We choose this system rather than tetracene single crystals because the quantum beats arising from spin state coherences are largely suppressed in the polycrystalline ﬁlm [38], possibly due to rapid spin relaxation at the surfaces of the crystallites [51]. This allows us to use a kinetic theory that considers only populations, rather than a more complicated density matrix that is required

J.J. Burdett et al. / Chemical Physics Letters 585 (2013) 1–10

to take quantum coherences into account. We should also note that the initial SF rate is also faster in polycrystalline ﬁlms by roughly a factor of 2 [52,53]. Our ﬁrst task is to determine the C lS coefﬁcients that are needed to ﬁx the relative transition rates between singlet and triplet pair states. The sample is assumed to consist of a mosaic of small crystallites, all of which have their ab planes oriented parallel to the plane of the substrate, but randomly rotated with respect to the magnetic ﬁeld. The C lS coefﬁcients are calculated by solving the Hamiltonian in Eq. (7) for different crystal orientations. Once the nine jC lS j2 values have been calculated for a speciﬁc orientation of the crystal in the magnetic ﬁeld, the angle is rotated and another set of jC lS j2 values is calculated for the new orientation. To illustrate how jC lS j2 changes with crystal angle, in Figure 7 we show jC lS j2 values for all nine triplet pair states as the magnetic ﬁeld is increased from 0 to 800 Gauss for three different orientations of the tetracene ab plane with respect to the B-ﬁeld. We chose the angles 18.8°, 23.8°, and 28.8° to illustrate how the number of triplet pair states with singlet character changes as the crystal angle varies. At 18.8o, the expected 3 ? 2 change in the number of triplet pairs with singlet character is observed. Note that state 4 has a jC lS j2 value twice as large as that of state 6 at high ﬁeld, as expected based on Eq. (20). At 28.8°, we again have the 3 ? 2 evolution, but the relative amplitudes of states 4 and 6 have been reversed. This reversal occurs at 23.8°, where the |00i and |T+i states become degenerate. At this point, there is only one state with singlet character at high ﬁeld. With only one state, rather than two, participating in the SF/TF processes, we would expect to see a signiﬁcant effect on the ﬂuorescence kinetics, as discussed previously. If we were studying a single crystal, we would expect to see a narrow peak in the ﬂuorescence intensity at this angle, as many previous workers have [14,30,31,45]. However, here we are concerned with the polycrystalline ﬁlm, where we average over all angles and the Bﬁeld induced changes in the ﬂuorescence dynamics are dominated by the 3 ? 2 change in states with single character. The C lS coefﬁcients are used as inputs to scale the transition rates between the singlet and triplet pair states. The rate constants

7

k2 and k2 are ﬁxed, as are the other kinetic parameters. In order to use the jC lS j2 values as inputs for the kinetic calculations, we have two choices. In reference [40], we simply summed over the energy-ordered jC lS j2 values for all orientations in order to get a set of nine averaged jC lS j2 values, which were used as inputs for a single kinetic calculation. A more rigorous calculation would involve performing individual kinetic calculations for each orientation and then summing up all these time-dependent contributions to obtain a total NS1(t) signal. It turns out that both approaches result in very similar dynamics for both amorphous rubrene and polycrystalline tetracene. We use the latter approach for the simulations in this Letter, since it is physically more reasonable to assume that the triplet dynamics occur within single crystal domains, and that the total ﬂuorescence is given by a sum over signals emanating from all these domains. In Figure 8a, we show the experimental ﬂuorescence decay in a 1 ns window, with and without an applied B-ﬁeld. In order to simulate the observed ﬂuorescence decays, we solved Eqs. (21a)–(21c) using the parameters summarized in Table 1 and plot the results in Figures 8b (krelax = 0.3 ns1) and c (krelax = 0.3 ns1). Both the experimental and simulated signals indicate that the initial decay of the ﬂuorescence, over the ﬁrst natural log within the ﬁrst 200 ps or so, is not sensitive to the magnetic ﬁeld. This is because the initial decay of the singlet, before any triplet pair states are signiﬁcantly populated, is given by

! n X dNS1 l 2 ﬃ krad þ k2 jC S j NS1 ¼ ðkrad þ k2 ÞNS1 dt i¼0

ð22Þ

where the last equality arises from the conservation of the norm of the singlet state under unitary transformation, as discussed earlier. Additionally, this simple theory predicts that if there is no population transfer from the triplet states back into the singlet state (k2 = 0), then Eq. (22) becomes exact and no magnetic ﬁeld effect on the singlet decay is expected. Of course, the triplet pair states may equilibrate with other types of singlet states, for example charge-transfer states that eventually give rise to free carriers,

Figure 7. Histograms of singlet character (vertical axis) for each of the 9 triplet pair states as a function of magnetic ﬁeld strength (long horizontal axis). The triplet pair states are ordered from lowest to highest energy. The three h values correspond to the angle of the crystal b-axis with respect to the B-ﬁeld.

8

J.J. Burdett et al. / Chemical Physics Letters 585 (2013) 1–10

Figure 8. (a) Experimental ﬂuorescence decay in the 1 ns window without magnetic ﬁeld (black) and with a 8 kG magnetic ﬁeld (red). (b) Simulated ﬂuorescence decay displaying the magnetic ﬁeld effect using the model given Figure 6 with krelax = 0.3 ns1 (c) Simulated ﬂuorescence decay displaying the magnetic ﬁeld effect using the model given Figure 6 with krelax = 0.

Table 1 Kinetic parameters used in Eq. (17) to model the data in Figures 8 and 9. Kinetic parameters (ns1) krad

k2

k2

k1

k1

krelax

ktrip

0.18

9.3

1.0

0.2

0.1

0.3

0.004

Energetic parameters (cm1) D⁄

E⁄

Xint

0.0062

0.0248

0.001

and the B-ﬁeld may affect observables that are sensitive to such states. The important point is that in order to see magnetic ﬁeld effects, some form of back-and-forth population transfer between the triplet pair states and another state with singlet character is required, and this two-way transfer requires some time to become established. At longer times, after 200 ps, the singlet and triplet pair states are both populated and one begins to see an enhanced ‘prompt’ ﬂuorescence signal under the applied B-ﬁeld. In the 1 ns time window, the signal is most sensitive to k2 and k2. k2 = 9.3 ns1 is determined by the initial decay rate of the ﬂuorescence and is the same as that reported in previous work on polycrystalline tetracene thin ﬁlms [53–57]. The fusion rate k2 = 1 ns1 is an order of magnitude slower than k2, consistent with the results of our experiments on single crystal tetracene delayed ﬂuorescence [38]. These values give a reasonably good representation of the observed signal shape, where the initial exponential decay crosses over to a slower decay due to the replenishment of the singlet population via TF. The enhancement of the ‘prompt’ ﬂuorescence in this time range reﬂects the fact that at high ﬁeld there are only two triplet pair states for the population to partition into, and thus an average of 13 of the population resides in the emissive singlet state, rather than 14 as in the zero-ﬁeld case. The triplet pair feedback into the singlet state in this time regime leads to an enhanced

ﬂuorescence signal that partially cancels out the rapid decay that would be observed if only SF were operative. The simulated signals for both krelax = 0.3 ns1 and krelax = 0 are very similar in this time window, which is not surprising since the timescale of separation and spin–lattice relaxation are both longer than 1 ns. The longer time ﬂuorescence decay dynamics are more sensitive to the TF and exciton diffusion rates. Figure 9a shows the experimental ﬂuorescence decays, while Figure 9b and c show the simulated curves using the parameters in Table 1. The k1/k1 rates determine the level of the delayed ﬂuorescence signal, since they limit the ability of the separated triplets to recombine. In Table 1, the separation rate k1 = 0.2 ns1 is a factor of 2 greater than the association rate, k1 = 0.1 ns1, reﬂecting the fact that association is less likely as the excitons diffuse apart. These two values are chosen to make the level of the simulated delayed ﬂuorescence comparable to the experimental level. The value of krelax does not have a strong effect on the absolute magnitude of the delayed ﬂuorescence signal. It does, however, inﬂuence the crossing of the high-ﬁeld and zero-ﬁeld decay curves, seen experimentally at around 20 ns delay, where the high-ﬁeld delayed ﬂuorescence signal dips below the level of the zero-ﬁeld delayed ﬂuorescence. In Figure 9b, the simulated signal for krelax = 0.3 ns1 shows the convergence of the two curves on this timescale, whereas in Figure 9c the curves for krelax = 0 remain well-separated. We were unable to reproduce the more pronounced crossing seen in the experimental data, however, suggesting that this model needs further reﬁnement. From a physical standpoint, the suppression of the delayed ﬂuorescence signal in the high-ﬁeld case is related to the fact that the triplet pairs can randomize their spin populations on longer timescales. Once the triplets become distributed across the nine possible spin states, at high-ﬁeld there are only two channels back to the singlet (|00i, |T+i) rather than the three available at zero-ﬁeld (|xxi, |yyi, |zzi). Note that if krelax = 0, and the triplet pairs are not allowed to randomize their spin states, then the separated pairs are always trapped in states with singlet character and it does not matter whether there are two or three such states. In this case, changing the number of singlet gateway states using the magnetic

Figure 9. (a) Experimental ﬂuorescence decay in the 100 ns window without magnetic ﬁeld (black) and with a 8 kG magnetic ﬁeld (red). (b) Simulated ﬂuorescence decay displaying the magnetic ﬁeld effect using the model given Figure 6 with krelax = 0.3 ns1. (c) Simulated ﬂuorescence decay displaying the magnetic ﬁeld effect using the model given Figure 6 with krelax = 0.

J.J. Burdett et al. / Chemical Physics Letters 585 (2013) 1–10

ﬁeld has no effect on the TF rate and no crossing of the high- and zero-ﬁeld decay curves is observed in Figure 9c. If, on the other hand, we turn on krelax and allow population to redistribute across all triplet pair states, then the decreased number of gateway states at high B-ﬁeld will suppress TF and decrease the amount of delayed ﬂuorescence as seen in Figure 9b. In a previous paper [40], we only considered associated triplet pairs, and in that case the combination of rapid fusion and radiative loss from the singlet state could also cause a curve crossing, even if krelax = 0. The problem with that treatment is that we had trouble describing the highly nonexponential character of the decay curves. In this Letter, by taking diffusion into account using associated and separated pairs, we postulate a different mechanism where spatial separation of the triplets is accompanied by spin relaxation. This model gives a more accurate representation of the data, although the match is still not exact. Discerning the precise role of spin relaxation using ﬂuorescence measurements alone is probably asking too much of a single technique, and in the future we hope to utilize other types of experimental methods to gain a fuller picture of the triplet exciton dynamics after the SF event.

5. Conclusions In this Letter, we have reviewed the basic properties of triplet electron spin states as they relate to SF. We have tried to extend the work of pioneering researchers like Merriﬁeld, Suna, Pope and Swenberg to look at both the mechanism of SF and how it is affected by the presence of a magnetic ﬁeld. Direct evidence that SF produces the theoretically predicted triplet superposition state is provided by the observation of quantum beating in the delayed ﬂuorescence of tetracene crystals. The dependence of both the quantum beat frequencies and the ﬂuorescence decay rates on magnetic ﬁeld and crystal orientation are consistent with the predictions of a quantum mechanical model that takes the detailed spin Hamiltonian into account. The magnetic ﬁeld also affects the kinetics of the SF/TF processes, which we can understand in terms of an expanded Merriﬁeld model. This model allows us to make several qualitative observations. First, when the full time-dependence of the singlet population is considered, we ﬁnd that the magnetic ﬁeld has no effect on the very initial ﬂuorescence decay rate, but only becomes observable at later times when the triplet pair states begin to equilibrate with the singlets. This implies that the existence of a magnetic ﬁeld effect on the singlet decay depends on the exchange of population back and forth between the singlet and triplet pair states. If this exchange is not possible, e.g. due to energetic mismatches, then the absence of a magnetic ﬁeld effect does not necessarily rule out the presence of SF. Second, the longer time behavior of the ﬂuorescence decay reﬂects association and separation of the triplet pairs, along with the relaxation of the triplets into different spin states. In particular, our model suggests that the crossing point of the delayed ﬂuorescence curves at high and zero-ﬁeld is sensitive to the spin relaxation dynamics of the triplets. While our approach is general, it should be emphasized that our experimental results are speciﬁc to the tetracene crystal system. How a magnetic ﬁeld affects the spin states and thus the SF/TF rates depends on multiple variables, including molecular packing, level energetics, the magnetic ﬁeld strength and its orientation with respect to the sample. All these factors will affect the spin Hamiltonian and thus the distribution of singlet character C lS across the triplet pair states. It is conceivable that in a different system, the application of a magnetic ﬁeld could lead to no observable effect, or one in the opposite direction. Magnetic ﬁeld effects can provide valuable information on the SF process but cannot be used a simple way to determine its presence or absence. A last point is

9

that the question of what aspects of the electronic structure Hamiltonian lead to rapid SF ﬁssion rates is still an open question. The dependence of the SF rate on molecular separation and orientation was originally analyzed by considering the interaction between singlet and triplet exciton manifolds [58,59], but now an increasing number of theoretical studies point to the role of charge-transfer states in SF [15,60–64]. In the ﬁnal analysis, SF dynamics must be considered in the context of an individual sample’s characteristics. In addition to molecular-level structure and packing, our different results on polycrystalline and single crystal tetracene suggest that sample morphology (grain boundaries, surface effects) can affect the overall singlet/triplet interconversion rates, for example by affecting spin relaxation. Acknowledgements This work was supported by the National Science Foundation under grant CHE-1152677. G.B.P. was supported by a Department of Education Graduate Assistance in Areas of National Need (GAANN). References [1] Workshop, B. E. S. Basic research needs for solar energy utilization; D.O.E.: Washington, DC, 2005. [2] A.J. Nozik, Chem. Phys. Lett. 457 (2008) 3. [3] W. Shockley, H.J. Queisser, J. Appl. Phys. 32 (1961) 510. [4] M.C. Hanna, A.J. Nozik, J. Appl. Phys. 100 (2006) 074510/1–074510/8. [5] H. Shpaisman, O. Niitsoo, I. Lubomirsky, D. Cahen, Sol. Energy Mater. Sol. Cells 92 (2008) 1541. [6] A.J. Nozik, Physica E 14 (2002) 115. [7] R.D. Schaller, V.I. Klimov, Phys. Rev. Lett. 92 (2004) 186601/1–186601/4. [8] O.E. Semonin, J.M. Luther, S. Choi, H.Y. Chen, J. Gao, A.J. Nozik, M.C. Beard, Science 334 (2011) 1530. [9] A.J. Nozik, M.C. Beard, J.M. Luther, M. Law, R.J. Ellingson, J.C. Johnson, Chem. Rev. 110 (2010) 6873. [10] M.C. Beard, J. Phys. Chem. Lett. 2 (2011) 1282. [11] L.A. Padilha, J.T. Stewart, R.L. Sandberg, W.K. Gai, W.K. Koh, J.M. Pietryga, V.I. Klimov, Acc. Chem. Res. (2013). ASAP. [12] M. Pope, C.E. Swenberg, Electronic Processes in Organic Crystals and Polymers, Oxford University Press, New York, 1999. [13] S. Singh, W.J. Jones, W. Siebrand, B.P. Stoicheff, W.G. Schneider, J. Chem. Phys. 42 (1965) 330. [14] C.E. Swenberg, N.E. Geacintov, Excitonic interactions in organic solids, in: J.B. Birks (Ed.), Organic Molecular Photophysics, vol. 1, Wiley & Sons, Bristol, 1973, pp. 489–564. [15] M.B. Smith, J. Michl, Chem. Rev. 110 (2010) 6891. [16] P.J. Jadhav, A. Mohanty, J. Sussman, J. Lee, M.A. Baldo, Nano Lett. 11 (2011) 1495. [17] B. Ehrler, M.W.B. Wilson, A. Rao, R.H. Friend, N.H. Greenham, Nano Lett. 12 (2012) 1053. [18] P.J. Jadhav et al., Adv. Mater. 24 (2012) 6169. [19] D.N. Congreve et al., Science 340 (2013) 334. [20] I. Paci et al., J. Am. Chem. Soc. 128 (2006) 16546. [21] A. Rao, M.W.B. Wilson, J.M. Hodgkiss, S. Albert-Seifried, H. Bassler, R.H. Friend, J. Am. Chem. Soc. 132 (2010) 12698. [22] M.W.B. Wilson, A. Rao, J. Clark, R.S.S. Kumar, D. Brida, G. Cerullo, R.H. Friend, J. Am. Chem. Soc. 133 (2011) 11830. [23] W.L. Chan, M. Ligges, A. Jailaubekov, L. Kaake, L. Miaja-Avila, X.Y. Zhu, Science 334 (2011) 1541. [24] W.L. Chan, M. Ligges, X.Y. Zhu, Nat. Chem. 4 (2012) 840. [25] C. Ramanan, A.L. Smeigh, J.E. Anthony, T.J. Marks, M.R. Wasielewski, J. Am. Chem. Soc. 134 (2012) 386. [26] C. Wang, M.J. Tauber, J. Am. Chem. Soc. 132 (2010) 13988. [27] J.C. Johnson, A.J. Nozik, J. Michl, J. Am. Chem. Soc. 132 (2010) 16302. [28] G. Lanzani, G. Cerullo, M. Zavelani-Rossi, S.D. Silvestri, Phys. Rev. Lett. 87 (2001) 187402/1–187402/4. [29] C.E. Swenberg, W.T. Stacy, Chem. Phys. Lett. 2 (1968) 327. [30] N. Geacintov, M. Pope, F. Vogel, Phys. Rev. Lett. 22 (1969) 593. [31] R.E. Merriﬁeld, P. Avakian, R.P. Groff, Chem. Phys. Lett. 3 (1969) 155. [32] R.E. Merriﬁeld, J. Chem. Phys. 48 (1968) 4318. [33] I.N. Levine, Quantum Chemistry, Prentice Hall, New Jersey, USA, 2000. [34] S.P. McGlynn, Molecular Spectroscopy of the Triplet State, Prentice-Hall, Englewood Cliffs, NJ, 1969. [35] G. Klein, R. Voltz, Int. J. Radiat. Phys. Chem. 7 (1974) 155. [36] M. Chabr, U.P. Wild, J. Funfschilling, I. Zschokke-Granacher, Chem. Phys. 57 (1981) 425. [37] J. Funfschilling, I. Zschokke-Granacher, S. Canonica, U.P. Wild, Helv. Phys. Acta 58 (1985) 347.

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Dr. Jonathan Burdett received his Ph.D. in physical chemistry from the University of California at Riverside in 2012. Since 2012, he has been working with Professor Wickramasinghe at the CaSTL center at the University of California at Irvine as a postdoctoral researcher in order to study the application of atomic force microscopy to obtaining time resolved single molecule spectroscopy. His research interests include ultrafast spectroscopy of nanoscale materials.

Geoffrey B. Piland was born in San Diego, CA, in 1987. He received his B.S. in chemistry from California State University, San Marcos in 2011. Currently, He is a graduate student at University of California, Riverside, where his research includes the photophysics of singlet ﬁssion in organic semiconductors such as rubrene and tetracene.

Professor Bardeen received his B.S. in chemistry from Yale University in 1989, and a Ph. D. in chemistry from U. C. Berkeley in 1995 under the guidance of Prof. Charles V. Shank. After a post-doc with Kent Wilson at U. C. San Diego, he became an assistant professor at U. Illinois, Urbana-Champaign in 1998. In 2005 he moved to the University of California, Riverside. His research interests include the experimental study of exciton dynamics in molecular crystals and organic photovoltaic materials, as well as the photomechanical response of crystalline nanostructures composed of photoreactive molecules.