Magnetic dipolar interaction between correlated triplets created by singlet fission in tetracene crystals

Singlet fission can potentially break the Shockley–Queisser efficiency limit in single-junction solar cells by splitting one photoexcited singlet exciton (S1) into two triplets (2T1) in organic semiconductors. A dark multiexciton state has been proposed as the intermediate connecting S1 to 2T1. However, the exact nature of this multiexciton state, especially how the doubly excited triplets interact, remains elusive. Here we report a quantitative study on the magnetic dipolar interaction between singlet-fission-induced correlated triplets in tetracene crystals by monitoring quantum beats relevant to the multiexciton sublevels at room temperature. The resonances of multiexciton sublevels approached by tuning an external magnetic field are observed to be avoided, which agrees well with the theoretical predictions considering a magnetic dipolar interaction of ∼0.008 GHz. Our work quantifies the magnetic dipolar interaction in certain organic materials and marks an important step towards understanding the underlying physics of the multiexciton state in singlet fission.


Hyperfine coupling < 5×10 -4 GHz
Supplementary Note 1. Theoretical modelling the quantum beats involving the sublevels of ME state.
In crystalline tetracene, the spin-orbital interaction is negligibly weak. Singlet fission (SF) process is induced by the Hamiltonian operator el sp The isolated triplet dipole (e.g., α or β, as represented by "Tri") can be described by the Hamiltonian 1, 2 Here, the first item represents the Zeeman shift due to the applied magnetic field, where S , B , B  , and g are the spin operator, the external magnetic field, the Bohr magneton, and the Lande g -factor, respectively. The latter two terms give zero-field Hamiltonian with parameters of * D and * E characterized by ESR experiments 3 . The interaction Hamiltonian between the two paired triplets can be written as 4 where the interaction strength follows the inverse cubic dependence on the separation Here, the spin operators are replaced by Pauli matrices for a system with spin s=1, i.e., x 0 1 0 The three eigenstates and their eigenvalues for an isolated dipole are obtained by diagonalizing this Hamiltonian (6). We denote the three states (energies) as |x> (E x ), |y> (E y ), and |z> (E z ) for the isolated triplet dipole in analogue to the literature. When the mutual interaction is not considered, the nine ME sublevels of the correlated triplet dipoles can be expressed as a simple combination of states in the form of Here, the first term ( α+β zero H ) is the zero-field Hamiltonian for two isolated triplets, the second term ( α+β Zeeman H ) is the field-dependent Hamiltonian for two isolated triplets, and the last term includes the interaction described in (4). The first term can be diagonalized with the interaction-free solution given above, i.e., 9 α+β The total spin-dependent Hamiltonian is then given in the form of The energies of the nine ME sublevels ( i TP |   ) for two coupled triplets can then be calculated by diagonalizing the Hamiltonian (11). For clarity, the sublevels The singlet state (s=0) can be described as where | , |0 , and |  are the eigenfunctions of Pauli matrices. Only the ME sublevels with non-zero mappings to the singlet state (i.e., The quantum beats can be simulated with kinetic equations that describe the population of singlet state relevant to the exciton fusion from the ME sublevels. Here, we extend the method derived by Burdett & Bardeen. 1 In principle, to fully describe the evolution involving the singlet state and nine ME sublevels, one should consider the Hamiltonian for the 10-level system as, 10 10 n n1 1n n 2 n 2 | n n | (| n 1| |1 n |) where the energy |1 is the initial singlet state with 0 energy, and 1n M represents the transition matrix elements that couple state 1 to the ME sublevels. The dynamical behaviors of the density-matrix elements are governed by the quantum Liouville equation, It is very complicated to fully simulate the dynamics of this 10-level system. To simplify the mathematical procedure, only the ME sublevels involved in the exciton fission/fusion processes will be considered. The mapping to the singlet state (i.e. When no external magnetic field is applied, the eigenvalues for the spin-dependent Hamiltonian of an isolated triplet dipole can be directly calculated with the well-characterized parameters of * D and * E 3 , i.e., When the interaction is not considered, only three ME sublevels (|xx>, |yy>, and |zz>) have non-zero mappings to the singlet state at zero field case. The energy separations between these levels can well explain the three beat frequencies observed at zero magnetic field 1, 5 .
When an external magnetic field is applied along x axis, the term of Zeeman shift in Hamiltonian will lead to significant changes of the ME sublevels. For the interaction-free case, the energies for ME sublevels can be directly calculated as combinations of eigenvalues of Hamiltonian (6). The values are plotted in Supplementary Fig. 1a, where some ME sublevels are degenerate (i.e., |xy>/|yx>, |xz>/|zx>, and |yz>/|zy>). We calculate the mappings of ME sublevels to the singlet state as shown in Supplementary Fig. 2a. The mappings of ME sublevels (|xy>, |yx>, |xz>, and |zx>) to the singlet state are zero, indicating that these sublevels are not involved in the exciton fusion/fission processes. The quantum beats can then be simulated by considering evolutions of the rest five sublevels. The degenerate sublevels of |yz> and |zy> are taken as a single level to simplify the mathematical procedure. The simulated amplitudes of quantum beating oscillations are plotted as functions of the beat frequency and magnetic field magnitude in Supplementary Fig. 3.
The theoretically calculated results can well explain the measured data in the high frequency range (> 0.1 GHz) (Fig. 2c), but the calculation fails to account for the unusual low-frequency beat signal near 420 Gauss.
When the magnetic dipolar interaction is considered, the ME sublevels cannot be regarded as linear combinations of eigenstates for isolated triplets. To calculate the energies of these ME sublevels, we exactly diagonalize the Hamiltonian (11). The displacement vector between the two correlated triplet dipoles is assumed to be along the orientation of the nearest neighbor as suggested in literature 6,7 . Supplementary  Fig. 1b. Since the ME sublevels with non-zero singlet mappings contribute to the exciton fusion/fission processes, the beats relevant to the 3-4 sublevels or 6-7-9 sublevels contribute to the measured low-frequency beat signal as highlighted in Supplementary Fig. 1d. Considering the mappings of ME sublevels to the singlet state, the beat signal related to the 3-4 sublevels is dominant as discussed in the manuscript.
In the above calculation, the eigenstates (eigenenergies) of the basis of nine states

The matrix form ( I ) of the interaction Hamiltonian
i nt H is given by:

Supplementary Note 2. Angle-dependent measurements near the level crossing resonance.
In time-resolved fluorescence (TRFL) traces, the intensity of delayed fluorescence decays in the time scale of nanoseconds. The damping of the oscillatory components makes a poor visibility for the low-frequency beat signal. As described in the manuscript, we introduce a perturbation by tilting the magnetic field in the xy plane (Fig. 2a) to enhance the visibility as depicted in Supplementary Fig. 4. The interaction between the correlated triplets opens a gap ( Supplementary Fig. 4a,b), which is further increased when the field tilting is introduced (Supplementary Fig. 4c).
When the tilting angle is small ( 5    ), the perturbation-induced gap size is linearly dependent on  ( Supplementary Fig. 4d), which can be used to interpolate the gap size for zero value of  ( Supplementary Fig. 4e). We note here that the effect of a slight tipping of magnetic field with respect to the xy plane can be neglected.
Besides the above discussed avoided level crossings at ~ 420 Gauss, the calculated energy alignments of ME sublevels in Supplementary Fig. 1b Fig. 5b). These results explain the absence of low-frequency beat in the field range near 104 Gauss (Fig. 2c).

Supplementary Note 3. Control experiments with magnetic field applied along z axis
We have also calculated the ME sublevels when a magnetic field is applied along z axis (Supplementary Fig. 6). The obtained energies are plotted in Supplementary Fig.   6, and no level-crossing resonance exists. The energy alignments of ME sublevels are similar for calculation with and without the magnetic dipolar interaction ( Supplementary Fig. 6a,c). Without interaction, there are five sublevels including two degenerated ones with non-zero mappings to the singlet state ( Supplementary Fig. 6b); When the interaction is included, the avoided degeneracy between two ME sublevels (i.e., |xy> & |yx> for interaction-free case, Supplementary Fig. 6b) results in one sublevel (# 6) having their total mappings to the singlet state ( Supplementary Fig. 6d).
These facts lead to very similar simulated results of quantum beat for both interaction-free and interaction-involved models with the procedure described above.
Considering these sublevels, we simulated the beating amplitudes as functions of the beat frequency and field magnitude, which can well reproduce the experimental data ( Supplementary Fig. 7). This control experiment verifies that the avoided level-crossing resonance is a prerequisite for observing abnormal low-frequency beat in the TRFL traces.

Supplementary Note 4. Measurement at the strong field limit.
When the external magnetic field is sufficiently strong, the first term (i.e., the Zeeman shift) in Hamiltonian (3) is predominant in comparison to the other two terms.
In this strong field limit, two ME sublevels dominate the mappings to singlet state (Supplementary Fig. 2b) 8 , which exhibits a single-frequency beating ( Supplementary   Fig. 3 and 7b). The strong-field limit had been solved with a perturbative approach 8 .
Here, in order to extract the weak strength of magnetic dipolar interaction, we performed the exact diagonalization of the Hamiltonian (11) for accurate solutions.
We considered the case with field applied in the xy plane and changed the field direction with angle Φ in respect to x axis. The interaction will introduce a gap when Φ is tuned to approach the degeneracy of the two sublevels. The strength can be then extracted by analyzing the Φ-dependent beat frequency as discussed in the manuscript.

Supplementary Note 5. The separation distance between the correlated triplet excitons.
For tetracene, highly efficient SF has been observed in crystalline tetracene but not in tetracene solution, suggesting that the intermolecular interactions are essential for the SF process. It is generally believed that SF process creates two triplets in two nearest neighboring molecules when one photo-excited molecule shares its energy with its neighboring molecule 6,7 . Supplementary Fig. 8 schematically shows the herringbone structure of molecule alignments in tetracene crystals. There are four possible configurations for two nearest neighbors with different directions as labeled in Supplementary Fig. 8. The parameters for these configurations are listed in Supplementary Table 1. Two molecules aligned with r 1 configuration have the shortest separation distance, which has been frequently employed in modelling SF in crystalline tetracene 6,7 . Here, the four possible configurations of two neighboring molecules and the strengths of magnetic dipolar interactions for these configurations are considered (Supplementary Table 1).
The effect of magnetic dipolar interaction described by Hamiltonian (4) is sensitive to the interaction strength (X) and the displacement vector ( 0 / R R ) between the two correlated triplets. We fit the experimental data recorded at the cases of level-crossing resonance (~ 420 Gauss, Fig. 3b) and strong-magnetic-field limit with the displacement vectors for all four configurations (Supplementary Table 1). The best fitted parameters of interaction strengths and equivalent distances are listed in Supplementary  Table 2) is over one-order magnitude weaker than that calculated for two nearest neighboring molecules (Supplementary Table 1). Basically, the equivalent separation distance is 2~4 times larger than the intermolecular distance, which can be explained by the effect of exciton delocalization 9-12 as discussed in the manuscript.
These results indicate the exciton size may play an important role for achieving highly-efficient SF in organic materials, which might be the reason of much less inefficient SF (~ 3%) in bis(tetracene) molecules 13  Magnetic field alignment. For the magnetic-field-dependent experiments, we employed a pre-calibrated magnetic coil driven by a DC power supply. The field direction could be rotated 360˚ by a step motor. The field magnitude was tuned by the applied current with accuracy better than 0.02 % (equivalent to 0.2 Gauss at the field magnitude of 1000 Gauss).
To align the sample at a desirable direction relative to the magnetic field, the sample was mounted on a multi-axis stage with the rotation/tilt accuracy better than 1 arc minute. Tetracene crystals have molecules aligned in a herringbone pattern in the ab plane ( Supplementary Fig. 9e). The magnetic axes of the crystal are different from those of single molecules (Inset, Supplementary Fig. 9e). As suggested in previous literature, the magnetic axes (x, y, z) of the crystal sample can be determined with the pre-determined crystallographic axes by a transformation matrix as established in the electron spin resonance experiments 3 . Briefly, the new axes of a' and c' are defined, respectively, as the in-plane directions normal to b axis and the direction normal to ab plane as depicted in Supplementary Fig. 9f we kept the excitation flux at a low level of ~ 4 nJ cm -2 (photon flux ~ 8 × 10 9 cm -2 ).

Data analysis.
The quantum beats manifest themselves as oscillations which are entangled with multi-exponential decay components in TRFL traces 1,5 . To extract the amplitudes and frequencies of quantum beating signals, the multi-exponential decay components were subtracted ( Supplementary Fig. 10a). The resultant damped oscillatory components ( Supplementary Fig. 10b)  More frequencies of quantum beats appear when increasing the field up to 90 Gauss.
At 420 Gauss, in addition to the above GHz beating components, an anomalous low-frequency beat signal emerges. At 1000 Gauss, a single frequency beat becomes dominant similar to the case of strong-field limit 8 . For the low-frequency beating signal, the frequency has also been evaluated by directly measuring the oscillation period, which gives similar results as that obtained with the method of Fourier transform. The error in estimating the beat frequency is less than 1×10 -3 GHz.
Since the magnetic dipolar interaction is of small magnitude, we have evaluated the measurement errors and some other possible mechanisms which could affect the accuracy of our measurements. As listed in Supplementary Table 3, we have considered the error bounds relevant to the uncertainties in measuring the beat frequency, the magnitude and angle of magnetic field, as well as the spatial heterogeneity of magnetic field. Moreover, we also estimated the possible effect of the hyperfine interaction. In an isolated tetracene molecule, the hyperfine coupling constants for C-H radicals have been reported to be 1.15, 1.55, and 4.25 Gauss at three different nonequivalent sites of the molecule, respectively 19,20 . Following the procedures in literature 21,22 , the estimated magnitude of hyperfine coupling for a triplet exciton is less than 2.8 Gauss, corresponding to a deviation of ~ 5×10 -4 GHz in beating frequency at the high-field limit. In the crystalline sample employed in our study, the effect of hyperfine coupling for triplet exciton is much less efficient since the hyperfine interaction for triplet exciton in organic crystals should be "washed out" as well-established in early literatures 23,24 .