Spin Statistics for Triplet–Triplet Annihilation Upconversion: Exchange Coupling, Intermolecular Orientation, and Reverse Intersystem Crossing

Triplet–triplet annihilation upconversion (TTA-UC) has great potential to significantly improve the light harvesting capabilities of photovoltaic cells and is also sought after for biomedical applications. Many factors combine to influence the overall efficiency of TTA-UC, the most fundamental of which is the spin statistical factor, η, that gives the probability that a bright singlet state is formed from a pair of annihilating triplet states. The value of η is also critical in determining the contribution of TTA to the overall efficiency of organic light-emitting diodes. Using solid rubrene as a model system, we reiterate why experimentally measured magnetic field effects prove that annihilating triplets first form weakly exchange-coupled triplet-pair states. This is contrary to conventional discussions of TTA-UC that implicitly assume strong exchange coupling, and we show that it has profound implications for the spin statistical factor η. For example, variations in intermolecular orientation tune η from to through spin mixing of the triplet-pair wave functions. Because the fate of spin-1 triplet-pair states is particularly crucial in determining η, we investigate it in rubrene using pump–push–probe spectroscopy and find additional evidence for the recently reported high-level reverse intersystem crossing channel. We incorporate all of these factors into an updated model framework with which to understand the spin statistics of TTA-UC and use it to rationalize the differences in reported values of η among different common annihilator systems. We suggest that harnessing high-level reverse intersystem crossing channels in new annihilator molecules may be a highly promising strategy to exceed any spin statistical limit.

We confirm our assignments (to singlet or triplet states) of spectral features in our transient absorption data by comparing the dynamics. Fig. S1a shows singlet and triplet PIA spectra in the visible spectral region extracted using multivariate curve resolution alternating least squares (MCR-ALS) 1,2 . Reference spectra used as the starting point of the deconvolution were obtained by time-averaging the data from 0.5-1 ps and 3-7 ns. The concentrations were constrained to be non-negative. The spectra resulting from the MCR-ALS procedure agree well with reported singlet 3 and triplet 4 absorption spectra for rubrene in solution.
The singlet and triplet dynamics extracted using MCR-ALS (Fig. S1b) are characteristic of singlet fission 5 . The triplet population rises with a time constant of approximately 3 ps, accompanied by a 50% reduction in the singlet population with a similar time constant. This suggests that singlet fission (S 1 → 1 (TT)) and triplet-pair fusion ( 1 (TT) → S 1 ) occur simultaneously with a time constant of 2 × 3 ps = 6 ps ∼ 10 ps.
In Fig. S1c, we show that the PIA bands at 680 nm and 1170 nm match the singlet dynamics, although an additional fast component is present at 1170 nm. Fig. S1d shows the dynamics at 850 nm and 960 nm, each with the dynamics at 1170 nm, weighted by the absorbance difference between the two wavelengths, subtracted. This removes the singlet component, yielding the residual triplet dynamics, which match those extracted in the visible region by MCR-ALS. This confirms that the peaks at 850 nm and 960 nm arise from triplet excited states.

Alternative explanations for the pump-push-probe data
We find that the effect of our 800 nm push pulses is to enhance the T 1 to T 3 photo-induced absorption (PIA) when the 400 nm pump is present. In this section we investigate the predicted change in triplet PIA for three possible scenarios and show that only the proposed HL-RISC can give rise to an enhancement. Let the triplet PIA signal X induced by the 400 nm pump have magnitude A at some arbitrary delay time. Then we have X 0,0 = 0 (S1) where X i,j denotes the signal with the pump on (i = 1), pump off (i = 0), push on (j = 1) or push off (j = 0).
Then, for example, the pump-probe signal is given by:

The push acts as a second pump
The first case we consider is that the push pulse acts as a pump from S 0 to S N (for example through two-photon absorption) or even from S 0 to T 1 . Let the triplet PIA signal induced by the push alone have magnitude B at the same arbitrary delay time: When the push is preceded by the 400 nm pump pulse, some of the ground state has already been depleted by the first pump pulse. As a result, the magnitude of the push-induced triplet PIA will be less than B, by an amount δ 1 , giving The pump-push-probe signal in this case is and so even if the push pulse acts as a second pump, the effect is to reduce the pump-push-probe signal rather than enhance it.

Internal conversion from T 2 to T 1
The second case to consider is that the push pulse excites the T 1 to T 2 transition but that T 2 undergoes internal conversion to T 1 . In this case, the push pulse has no effect in the absence of the pump: but causes a reduction δ 2 in the pump-induced PIA due to depletion of the T 1 state: and therefore again resulting in a reduced pump-push-probe signal where the magnitude δ 2 of the signal reduction would decrease to zero as the T 1 state is repopulated by internal conversion from T 2 .

HL-RISC from T 2 to S 1
In this case, the push pulse excites the T 1 to T 2 transition and T 2 undergoes rapid HL-RISC to form S 1 . S 1 then undergoes singlet fission, forming T 1 +T 1 . In the absence of the pump, the push pulse has no effect: When the pump pulse is present, T 1 is again depleted by amount δ 3 . However, since each S 1 state formed by the subsequent HL-RISC produces two T 1 states through singlet fission, the triplet PIA is enhanced by an amount 2δ 3 , giving and hence a pump-push-probe signal of Thus, HL-RISC is predicted to produce an enhancement of the triplet PIA, and the dynamics of the enhancement should match the pump-probe singlet fission dynamics.

Quantitative analysis of the pump-push-probe signal magnitude
Having demonstrated that HL-RISC is the most likely explanation for the push-induced enhancement of the triplet PIA, we can estimate the expected magnitude of the enhancement from known and estimated triplet absorption cross sections and measured pulse intensities. First, we estimate the density of triplet excitons responsible for a given ∆T /T signal at 510 nm. We can write the triplet PIA absorbance at 510 nm as where α and σ are the triplet absorption coefficient and cross section, respectively, at 510 nm, d is the film thickness and n T is the number density of triplet excitons. Transforming into the measurement units of ∆T /T gives the following expression for the number density of triplet excitons: The triplet absorption cross section in the vicinity of 510 nm has been measured as 1.2 × 10 −16 cm 2 for rubrene in solution 3 . Using this value, and our film thickness of 125 nm, gives n T = 4.5 × 10 18 cm −3 for our maximum ∆T /T signal of −0.0067 and n T = 3.9 × 10 18 cm −3 for ∆T /T = −0.0058, the value just before the arrival of the push pulse (see Fig. 7 in the main text). We can cross check these triplet exciton densities against the singlet exciton density calculated from the measured pump pulse intensity and the absorption of the film at 400 nm. In general, we have where F A and F S are the fraction of incident λ = 400 nm light absorbed and scattered/reflected by the film, respectively, and P is the pump pulse intensity in units of J cm −2 . By fitting a scattering background to the absorption spectrum of the film shown in Fig. S2a, we estimate F A ∼ 0.1 and F S ∼ 0.3 (from absorbances of 0.05 and 0.15). We measured P to be 0.2 mJ cm −2 (see Experimental Section). Using these values in Equation S15 results in an initial photoexcited singlet exciton density of n S ∼ 2.3 × 10 18 cm −3 . Assuming that singlet fission forms triplet excitons with a yield of 200%, we would expect a maximum triplet exciton density of n T ∼ 4.6 × 10 18 cm −3 , which agrees very well with the values calculated above from the solution cross section and the measured ∆T /T signal.
We are now in a position to estimate the number density of triplet excitons that are re-excited by the push pulse, n T , and hence obtain estimates for the push-induced ∆T /T signal from Equation S14.
The number density of triplets re-excited by the push can be evaluated from Equation S15, where P now represents the λ = 800 nm push pulse intensity which we measured to be 1.2 mJ cm −2 . F S is now the fraction of 800 nm light that is scattered by the film which we again obtain from the absorption spectrum, finding F S ∼ 0.09. The unknown parameter is F A , the fraction of 800 nm light absorbed by the T 1 → T 2 transition in the film. We have where the triplet-triplet absorbance at 800 nm is given by: A 800 = n T σ d log 10 (e) = n T σ σ σ d log 10 (e). (S17) Here σ and σ are the triplet absorption cross sections at 510 nm and 800 nm respectively and n T = 3.9 × 10 18 cm −3 (see above) is the triplet exciton density just before the arrival of the push pulse. The ratio of triplet-triplet absorption cross sections at 510 nm and 800 nm can be estimated from measured triplet PIA spectra for rubrene. We find significant variation between different measurements. For example the triplet PIA spectrum reported by Miyata et al. 6 for rubrene single crystals gives σ /σ ∼ 0.3. We suggest that this is an upper bound, given the significant spectral overlap between singlet and triplet PIA bands in the vicinity of 800 nm in their data. We can obtain a lower bound from our own transient absorption data in Fig. 5, main text. Although we cannot use the data at 800 nm due to residual fundamental in that probe region, we can take the ratio of the PIA signal at 510 nm and 850 nm (for t > 1000 ps) as a lower bound, giving σ /σ ∼ 0.08.
These upper and lower bounds for the triplet-triplet cross section ratio result in a range of 1.6 × 10 17 cm −3 < n T < 6.2 × 10 17 cm −3 for the push-induced triplet density, in other words the T 2 exciton density caused by the push pulse. In the HL-RISC picture, pushing from T 1 to T 2 results in the net gain of one T 1 . This is because the push removes one T 1 state but HL-RISC from T 2 to S 1 , followed by singlet fission from S 1 to T 1 +T 1 , adds back two T 1 states.
Thus we can simply convert n T into an expected push-induced ∆T /T signal at 510 nm by rearranging Equation S14. Assuming that all T 2 states populated by the push undergo HL-RISC, we predict that the magnitude of the push-induced ∆T /T signal should lie between −2.4 × 10 −4 and −9.3 × 10 −4 . Our measured value of −3.5 × 10 −4 (see Fig. 7b, main text) lies within this range, providing additional justification for our interpretation of the pump-push-probe data in terms of HL-RISC.
The rate equations for model 2 (Fig. 8 in the main text) can be written as follows: Since we assumed an effective linear annihilation rate constant k T T A , the equations were solved numerically using linear algebra. Definitions of rate constants and overlap factors are given in the main text. Table S1 gives the values of the rate constants used in the simulations and indicates the source of each one. Fig. S4 shows the sensitivity of the model to different rate constants. k SF 100 Approximated from TA data (Fig. S1) Approximated from TA data (Fig. S1) k T S 10 Lower bound on triplet hopping rate constant, estimated from the TTA rate constant in Ref. 9, larger values have no effect (Fig. S4) Approximate, estimated from onset of bimolecular TTA in Fig. 4 k IC1 17 Energy gap law (Fig. 6) k IC2 12 Energy gap law (Fig. 6) k IC21 8 Energy gap law (Fig. 6) k RISC 5000 Approximate instrument response of pump-push-probe Figure S4. Sensitivity of model 2 to different rate constants. Each rate constant was varied by ±3 orders of magnitude about its original value and the effect on η monitored. In particular, we highlight that the value of η does not depend on k SF or k T T A and that increasing the triplet hopping rate constants further also has little impact. Note that k IC is the sum of K IC1 and k IC2 .
Supplementary Tables S2 and S3 give non-exhaustive experimental literature values for the spin statistical factor of diphenylanthracene (DPA) and rubrene and the T 2 energy in rubrene. These values were used to indicate the spread of reported experimental values shown on Fig. 9d of the main text.   25 and pentacene-bridge-pentacene dimers 26 . Singlet and quintet triplet-pair states are mixed under weak exchange coupling. As such, various mechanisms have been proposed for the formation of 5 (TT) from 1 (TT), all based on fluctuating inter-triplet exchange interactions 25,27 . For instance, the hopping of one triplet in the pair onto a neighbouring molecule necessarily weakens the exchange coupling [28][29][30] . Alternative suggestions include polaronic (exciton-lattice) distortion, molecular relaxation or hopping to sites with different J 25 . The latter mechanism has been used to explain the changes in fluorescence of DPH crystals under strong magnetic fields 31 and is supported by the identification of several distinct triplet-pair sites in TIPS-tetracene with differing exchange interactions 30,32 .
Several recent trEPR studies have claimed that 1 (TT) and 5 (TT) can interconvert [33][34][35][36] . However, we agree with Atkins and Evans 37 that a direct interconversion is suppressed under strong exchange coupling and that instead the conversion from 1 (TT) to 5 (TT) and vice versa is mediated by weakly exchange-coupled singlet-quintet mixtures, i.e. 1/5 (T...T). This kind of mediated interconversion is explicitly included in model 2 (Fig. 8, main text) in which we clearly distinguish between weakly and strongly exchange coupled triplet-pair states.
EPR experiments are sensitive only to pure, non-zero spin states. As such, whilst 1/5 (T...T) states do not show up in these experiments that does not mean that they do not play a role in the dynamics.