Article pubs.acs.org/JPCC
Modeling Singlet Fission in Rylene and Diketopyrrolopyrrole Derivatives: The Role of the Charge Transfer State in Superexchange and Excimer Formation Claire E. Miller, Michael R. Wasielewski,* and George C. Schatz* Department of Chemistry and Argonne-Northwestern Solar Energy Research (ANSER) Center, Northwestern University, Evanston, Illinois 60208-3113, United States S Supporting Information *
ABSTRACT: Singlet fission (SF) is being explored as a way to improve the efficiency of organic photovoltaics beyond the Shockley-Queisser limit; however, many aspects of the SF mechanism remain unresolved. The generally accepted mechanisms provide simplified models of SF that equivocate over whether a charge transfer (CT) state is involved in SF. A one-step superexchange model allows the CT state to act as a virtual state, reducing the effect of large Gibbs free energy values from SF rate calculations. Also, extending superexchange to an excimer-mediated process allows for further refinement of the triplet formation model. Application of the superexchange and excimer-mediated models to a variety of rylene and diketopyrrolopyrrole derivatives provides new insights into the role of the CT and excimer states, providing a semiquantitative description of SF that is dictated by the CT state energy.
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INTRODUCTION
To rectify this contradiction, Berkelbach et al. proposed that SF proceeds through a superexchange mechanism.6 Superexchange is a phenomenon often discussed in the context of electron transfer in photosynthetic reaction center proteins and electron donor−acceptor molecules;7−9 however, applied to SF it allows the CT state to be used as a virtual state along the path to form two triplets. Coupling the CT and 1(S1S0) states allows for an overall coupling matrix element that is a function of both the one- and two-electron matrix elements, resulting in a coupling value large enough for SF to occur on an ultrafast time scale (Figure 1, blue pathway). Applications to pentacene have suggested that the CT mechanism plays an important role.6 In related but distinct work, several studies have observed excimer-like states prior to triplet formation in the SF process.10−15 The exact nature of the excimer is often debated as having CT character, 1(S1S0) character, 1(T1T1) character, or
Singlet fission (SF) is the process by which a singlet exciton is energetically down-converted to produce two triplet excitons provided the energy of the singlet state is greater than or equal to two times that of the triplet state, E(S1) ≥ 2E(T1). Originally discovered in 1965,1 SF is now being studied as a method for improving the efficiency of organic photovoltaics (OPVs). In single-junction OPVs, the Shockley-Queisser limit states that the theoretical maximum efficiency is 33%.2 However, assuming SF proceeds with 100% efficiency, the maximum theoretical efficiency of an OPV can be as high as 45%.3 However, being able to reach this maximum efficiency requires a deeper understanding of how SF works through mechanistic studies. In past work it has been proposed that there are two mechanisms for SF:4 (1) a “direct” mechanism where two electrons move simultaneously between the singlet exciton and an adjacent ground state chromophore, 1(S1S0), to yield the correlated triplet pair state, 1(T1T1), and (2) a “mediated” mechanism where a charge transfer (CT) intermediate state precedes triplet formation allowing one electron to move at a time. Analysis of CT energies and electronic coupling matrix elements provide conflicting results as to which mechanism is more likely.5 For SF to proceed efficiently by the mediated mechanism, the CT state energy must lie between that of the 1 (S1S0) and 1(T1T1) states; however, this is rarely the case, as CT energies generally lie well above the S1 energy resulting in large positive values of the Gibbs free energy (ΔG). Alternatively, the coupling matrix elements for a one-electron process are much larger than that of a two-electron process, often by up to 2 orders of magnitude,5 and it is unlikely that the coupling for the direct two-electron process is sufficient to produce the ultrafast SF rates observed.6 © 2017 American Chemical Society
Figure 1. Schematic of the SF superexchange mechanism involving the CT state (blue pathway) and with the addition of fast excimer formation (red and purple pathway). Received: March 21, 2017 Revised: April 26, 2017 Published: April 26, 2017 10345
DOI: 10.1021/acs.jpcc.7b02697 J. Phys. Chem. C 2017, 121, 10345−10350
Article
The Journal of Physical Chemistry C a mixture of the three. The excimer state is defined here as |Exc⟩ = α|S1S0⟩ + β|S0S1⟩ + γ|C0A1⟩ + δ|C1A0⟩, where C = cation and A = anion and the contribution from the last two charge transfer terms is small, while the CT state is defined as |CT⟩ = γ|C0A1⟩ + δ|C1A0⟩. In the excimer-mediated process, SF occurs in two steps, first, a fast transition from the 1(S1S0) state to the excimer intermediate occurs, which is followed by a second slower transition from the excimer intermediate to 1 (T1T1) using the CT state in a superexchange interaction (Figure 1, purple and red pathway). In the relatively low dielectric environment of organic solids, the CT state energy should be higher than that of the excimer as indicated in Figure 1. Since the energies of these states are not known explicitly, the influence of these states on the SF rates are treated phenomenologically as described in detail below.
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RESULTS AND DISCUSSION In this work, we apply the superexchange and excimer-mediated models to the series of rylene and diketopyrrolopyrrole (DPP) chromophores shown in Figure 2 that have been the subject of Figure 3. Electron movement and coupling elements of the four pathways of SF. Electrons are color coded to more easily follow movement.
Berkelbach et al.6 showed that the coupling matrix element for the superexchange model of SF is as follows: ̂ 1 1(1)⟩ = ⟨S1S(0) ̂ 1 1(0)⟩ ⟨S1S(1) 0 |Vel|TT 0 |Hel|TT 1
−2
⟨ CA|Ĥel|S1S0⟩⟨1TT|Ĥel|1CA⟩ + ⟨1AC|Ĥ el|S1S0⟩⟨1TT|Ĥ el|1AC⟩ [E(CT) − E(TT)] + [E(CT) − E(S1)]
(1)
This equation combines the two pathways that start from the S1S0 configuration (pathways 1 and 2), and a similar equation can be written for the two pathways that start from the S0S1 configuration (pathways 3 and 4). The first term, the direct two-electron coupling matrix element, is small compared to the four 1e coupling matrix elements and will be ignored. Equation 1 can therefore be rewritten in terms of the HOMO (H) and LUMO (L) couplings as follows:
Figure 2. Structures of all molecules in this work.
recent theory16,17 and experiments.13−15,18−22 The series includes several chromophores observed to proceed through an excimer intermediate and several with direct triplet formation. DFT calculations were performed using the Amsterdam Density Functional (ADF) package23 at the B3LYP/TZ2P level on each chromophore replacing substituents or tails with a methyl group or hydrogen atom. By comparing the predicted time constants to experimental results, we can gain insight into the role of CT energies in determining why one mechanism of SF may prevail over the other, and how accurate is the extended theory. Importantly, since there are a large number of molecules available for comparisons between theory and experiment, we are able to show that application of the extended theory to SF provides a semiquantitative description of the variation of rates with molecular properties. Superexchange, as applied to SF, assumes a virtual CT state is used on the path to two triplets. There are four possible pathways from a singlet state to two triplet states, which stem from two different starting configurations, S1S0 and S0S1, and two different CT configurations, CA and AC, where C represents a cation and A represents an anion. Figure 3 shows the four pathways and coupling elements that represent each step of the process.
VSE(S1S0) = 2
VLLVLH − VHHVHL [E(CT) − E(TT)] + [E(CT) − E(S1)] (2a)
VSE(S0S1) = 2
VLLVHL − VHHVLH [E(CT) − E(TT)] + [E(CT) − E(S1)] (2b)
There are two ways in the literature to evaluate Vif: (1) using Fock matrix elements,24 Jif, or (2) using the effective Fock matrix elements,25 Jeff,if, where the overlap integral, Sif, is included as a correction factor, 1
Jeff,if =
Jif − 2 Sif (e i + ef ) 1 − Sif2
(3)
There is a normalization factor of
2 3
for the off-diagonal
elements in both methods of evaluating Vif. In some cases, the contribution of the overlap integral is negligible compared to that of the Fock matrix elements. Fock matrix elements and overlap integrals were calculated by placing two monomers at the slip-stacking distance taken from the experimental crystal structures13−15,18,21,22 and using the fragment orbital approach 10346
DOI: 10.1021/acs.jpcc.7b02697 J. Phys. Chem. C 2017, 121, 10345−10350
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The Journal of Physical Chemistry C Table 1. Regular and Effective Fock Matrix Elements for Selected Molecules in meV (meV)
JHH
JHL
JLH
JLL
Jeff,HH
Jeff,HL
Jeff,LH
Jeff,LL
Me-TDPP Ph4-PDI Ph4-TDI tBu2-TER
15.7 −249 −158 241
−161 310 −273 −57.0
176 310 157 57.0
122 352 103 −20.4
13.6 −115 −77.1 133
−101 174 −160 −37.2
111 174 89.5 37.2
81.6 195 51.9 −18.1
The superexchange model provides many improvements over the two originally proposed models, but does not describe the entire picture seen experimentally. Several chromophores appear to undergo SF by going through an excimer intermediate before reaching the final triplet state. Krylov and co-workers have published results27,28 dealing with excimermediated SF and have developed a model to determine the rates of both steps of the process, 2 2π 2 1 k1 = e α(λ +ΔG1) /4λkBT |V | ℏ 4πλkBT (5a)
as implemented in ADF. Table 1 shows the differences between Jif and Jeff,if for a sample of the chromophores in this study, while Tables S2−S4 give the input parameters used in these calculations. The magnitude of Jeff,if is about 50−60% of Jif, meaning the overlap integrals should not be neglected in this case. Moving forward, Jeff,if will be used for all calculations, and the final expression for the rate is determined from the couplings just defined via the Marcus theory formula: 2 2π 2 1 k= e−(λ +ΔG) /4λkBT |V | ℏ 4πλkBT (4) In this expression, the CT state energy ΔG and the reorganization energy λ are calculated using the Direct Reaction Field (DRF) Force Field method as demonstrated by Mirjani et al.26 This requires crystal structures of the chromophores to be known, which is the case for the molecules in this study. Tails and substituents on the molecules have been removed and replaced with either a methyl group or a hydrogen atom to make the structure similar to that used in the DFT calculations. All other aspects of the crystal structures were preserved. One of the drawbacks of standard TDDFT methods is that they do not describe triplet states well and often underestimate triplet energies. For example, the singlet and triplet energies of 3,6-bis(thiophen-2-yl)-DPP (TDPP) were calculated to be 2.1 and 0.94 eV, respectively, resulting in ΔG = −0.22 eV. However, the experimentally determined singlet and triplet energies of TDPP are 2.2 and 1.1 eV, respectively, resulting in ΔG = 0 eV. To observe the effect of ΔG on the rate, the curves in Figure 4 were created by holding the singlet energy constant
k2 =
2π 2 |V | ℏ
2 1 e−α(λ +ΔG2) /4λkBT 4πλkBT
(5b)
The new parameter, α, is a value between 0 and 1 that describes the stabilization of the excimer state relative to the initial and final states, with α = 1 corresponding to the excimer being at the S1S0 state and smaller α values corresponding to larger admixtures of CT states. Krylov and co-workers used a value of 0.5 for their analysis and focused on other parameters, such as excimer energy, in their work. In our studies we assume that the value of α will vary based on the individual coupling elements of each derivative. A qualitative comparison of the TDPP coupling elements and the step of the process to which they correspond was used to define α. Assuming the excimer state exhibits some CT character, the coupling elements can be assigned to the two steps of the process as was done in Figure 1. Table 2 shows the Table 2. Effective Fock Matrix Elements in meV Corresponding to the Two Steps of SF Pathway 1 and Experimental Time Constants in ps for Both Steps of the Excimer-Mediated SF Process in TDPP for Qualitative Comparison Me C6 EH
Jeff,LL (meV)
Jeff,LH (meV)
τ1 (ps)
τ2 (ps)
81.6 129 71.1
111 47.0 15.6
2.7 0.9 16.0
22.1 336 1600
coupling elements for pathway 1 and the experimental time constants13 for both steps of the excimer-mediated SF process. For the Me derivative, both coupling elements are large and both steps of the SF process occur rapidly in τ = 2.7 and 22.1 ps. The C6 and EH derivatives both have larger coupling matrix elements for the excimer formation step which is consistent with the fact that both form the excimer quickly, but form triplet more slowly on the 100−1000 ps time scale. From this analysis, α is chosen as follows: J1 α= J1 + J2 (6)
Figure 4. SF time constants for each of the three TDPP derivatives as a function of ΔG. The dashed lines represent the experimental time constants and the ΔG to which they correspond.
at the TDDFT result and varying the triplet energy for all three derivatives of TDPP: methyl (Me), n-hexyl (C6), and 2ethylhexyl (EH). The time constants for both the Me and the EH derivatives match experiment around 0 eV. Although the C6 derivative does not match exactly at 0 eV, the experimental ΔG value provides closer agreement to the experimental rate than that determined through TDDFT. For the purpose of testing the accuracy of the model, experimentally determined singlet and triplet energies will be used.
where J1 and J2 are the effective Fock matrix elements that correspond to the first and second step of the relevant SF 10347
DOI: 10.1021/acs.jpcc.7b02697 J. Phys. Chem. C 2017, 121, 10345−10350
Article
The Journal of Physical Chemistry C Table 3. Experimental and Predicted Time Constants (in ps) with Both Models for All Molecules Studieda experiment Me-TDPP C6-TDPP EH-TDPP PhDPP PhTDPP Ph4-PDI TDI Ph4-TDI tBu2-TER tBu4-TER a
superexchange
excimer-mediated
τ1 (ps)
τ2 (ps)
τ (ps)
τ1 (ps)
τ2 (ps)
2.7 0.9 16 42
22.9 336 1600
8.13 14.9 956 18 300 88.0 132 000 18.6 1.10 15 958 11 140
0.013 0.005 0.15 16.1 0.06 0.000 11 14400 2.73 0.005 0.003
0.96 10.1 1590 5590 5.88 20.7 4.7e7 121 240 187
