Charge-transfer biexciton annihilation in a donor–acceptor co-crystal yields high-energy long-lived charge carriers

Organic donor–acceptor (D–A) co-crystals have attracted much interest due to their important optical and electronic properties. Co-crystals having ⋯DADA⋯ π-stacked morphologies are especially interesting because photoexcitation produces a charge-transfer (CT) exciton, D˙+–A˙−, between adjacent D–A molecules. Although several studies have reported on the steady-state optical properties of this type of CT exciton, very few have measured the dynamics of its formation and decay in a single D–A co-crystal. We have co-crystallized a peri-xanthenoxanthene (PXX) donor with a N,N-bis(3-pentyl)-2,5,8,11-tetraphenylperylene-3,4:9,10-bis(dicarboximide) (Ph4PDI) acceptor to give an orthorhombic PXX–Ph4PDI ⋯DADA⋯ π-stacked co-crystal with a CT transition dipole moment that is perpendicular to the transition moments for Sn ← S0 excitation of PXX and Ph4PDI. Using polarized, broadband, femtosecond pump–probe microscopy, we have determined that selective photoexcitation of Ph4PDI in the single co-crystal results in CT exciton formation within the 300 fs instrument response time. At early times (0.3 ≤ t ≤ 500 ps), the CT excitons decay with a t−1/2 dependence, which is attributed to CT biexciton annihilation within the one-dimensional ⋯DADA⋯ π-stacks producing high-energy, long-lived (>8 ns) electron–hole pairs in the crystal. These energetic charge carriers may prove useful in applications ranging from photovoltaics and opto-electronics to photocatalysis.

1. Single crystal X-ray structure data. PXX-Ph 4 PDI crystallized from chloroform and methanol. The structure was solved with XT intrinsic phasing solution package. The structure was refined with XL least squares minimization.
PXX was imported through the FRAGFEND command and solved isotropically. PXX lies on a symmetry operator, so the molecule was set in Part -1 and each position given 0.125 occupation.
The PXX moiety was constrained with EADP and FLAT commands and Squeeze was used to remove disordered toluene solvent molecules. Dfix 2.1 was applied to C9-C9A, and Dfix 2.4 was applied to C10-C10A. C13 experiences disorder out of the symmetry element it resides on, not modeled here. The structure was deposited in the Cambridge Crystallographic Data Center (CCDC 1991812). Figure S1. The needle-like crystals have two short axes of similar magnitude and one significantly longer axis. Indexing the crystal faces after determining the unit cell reveals that the small face at the end of the needle corresponds to the (100) plane, indicating the long axis along the macroscopic crystal corresponds to the 7 Å a axis of the unit cell, or the direction of π -π stacking.

Pump and probe spot sizes
The pump and probe beams were independently focused on a glass slide and imaged through the microscope setup on a CMOS camera (see Experimental section). The mode shape and size of each of the spots was extracted using a 2d fit to the Gaussian function Where are the center x and center y positions, the widths along the two principal , , 1 , 2 , axes and the angle of rotation around the z axis, respectively. The FWHM for each principal axis was calculated using   Where are the number of molecules per unit cell that can absorb a pump photon, , , , , excitation volume, unit cell volume, spot radius and crystal thickness.

Excitation density and fraction of molecules excited calculations
The excitation density is then = so that = 7 x 10 19 cm -3 . An upper bound on the fraction of donor-acceptor pairs excited in the cocrystal with each laser pulse, , can be calculated by assuming every molecule that can = / absorb a photon does so. The crystal thickness varies the most between crystals but is assumed to be comparable to the crystal width as imaged on the microscope. With these values we get < 1.05 × 10 9 , > 7.85 × 10 9 < 0.13

Calculation of the fraction of CT excitons adjacent to one another
The initial average fraction of occupied CT sites in the crystal is , and the distribution of the CT states in the crystal is Poissonian with an average spacing of between CT states. Since the 1/ lattice is discrete, the distance between each pair of CT excitons is a random variable, , which is distributed geometrically with probability parameter : We now examine an arbitrary site in this lattice. Setting as the event of having the previous site occupied (whether the current site is occupied or not), and as the event of having the examined site occupied, the probability of having two nearest neighbors CT excitons in the crystal is:

Calculation of reorganization energies and charge transfer rates
The internal nuclear reorganization energies, for PXX  PXX •+ ,  I = 0.08 eV, and Ph 4 PDI  Ph 4 PDI •- I = 0.13 eV, 1 were obtained by calculating the geometry-optimized ground state energies of the neutral species, E 0 , the ionic species, E, and the energy of the ions at the ground state geometries, E 0 . The calculations were performed with the B3LYP theory level functional using the 6-31G** basis set. All calculations were run in QChem 5.1. 2 The value of the environmental contribution to the reorganization energy,  S , was computed starting with eq 1 from the main text:

Model Hamiltonian for calculating polarization-dependent steady-state absorption spectra
The electronic Hamiltonian reads: Here, the summations run over donor ( ) and acceptor ( ) molecules, and are the donor and acceptor transition energies, respectively, is the nearest-neighbor CT energy respectively.
The transition dipole moment operator is The parameters used to calculate the polarized absorption spectra in Figure 3a in the main text are summarized in Table S3. The vibrational quantum and donor and acceptor HR factors were taken from the energy difference and relative intensities of the vibronic progression observed in the monomer linear absorption spectra (yielding roughly equal parameters for donor and acceptor).
The cationic and anionic HR factors were taken to be equal and taken so as to reproduce the CT band in the co-crystal absorption spectrum. Note that all vibrational parameters are in good S12 agreement with those adopted in detailed modeling of Ph 4 PDI reported previously, although the absence of spectral overlap and optical brightness of the CT band renders the present determination of these parameters less ambiguous. 7

A maximum number of vibrational quanta of 3 was taken,
where it should be noted that vibrational states were always expressed in their respective eigenbasis (corresponding to , , and CT electronic states).

1
Values for the donor and acceptor transition energies as well as the relative CT energy were 1 ← 0 obtained by aligning the associated bands appearing in the spectral simulations with those observed experimentally, and qualitatively match those obtained in our TD-DFT calculations ( Figure 6 in the main text). For simplicity, the HOMO-HOMO and LUMO-LUMO overlap factors were assumed to have equal magnitude, and their value was inferred from the modified vibronic progression observed in the linear absorption spectrum of the co-crystal. The (relative) signs of these overlap factors have very little impact on the spectral properties. Owing to symmetry considerations, they only affect the phase mixing of donor and acceptor singlet states, the degree of which is small as these states are separated by 400 meV. We therefore take the overlap factors to have equal signs. The dipole moments were aligned in accordance with our TD-DFT calculations, and their magnitudes were scaled to match the intensities of the three absorption bands observed experimentally. For the lineshape function, a Lorentzian was taken with a linewidth of 100 meV. Periodic boundary conditions were applied, in order to mimic extended cocrystal sizes. S13 Table S3. Parameters used to calculate the absorption spectra in Figure 3a in the main text. it is a CT transition. The calculated wavelength for this transition is 772 nm, which is close, but red-shifted relative to the position of the experimental CT band seen in Figure 3.
Examining the energy levels of the D-A pair, we see that there is another good candidate transition to generate a CT state, namely from the D-A pair HOMO, localized on the PXX, to the D-A pair LUMO+1, with contribution mainly from the LUMO+1 level of the isolated Ph 4 PDI.
Nevertheless, it was found in our calculations that this transition carries negligible oscillator strength and is therefore not shown in Figure S5.