Shallow distance-dependent triplet energy migration mediated by endothermic charge-transfer

Conventional wisdom posits that spin-triplet energy transfer (TET) is only operative over short distances because Dexter-type electronic coupling for TET rapidly decreases with increasing donor acceptor separation. While coherent mechanisms such as super-exchange can enhance the magnitude of electronic coupling, they are equally attenuated with distance. Here, we report endothermic charge-transfer-mediated TET as an alternative mechanism featuring shallow distance-dependence and experimentally demonstrated it using a linked nanocrystal-polyacene donor acceptor pair. Donor-acceptor electronic coupling is quantitatively controlled through wavefunction leakage out of the core/shell semiconductor nanocrystals, while the charge/energy transfer driving force is conserved. Attenuation of the TET rate as a function of shell thickness clearly follows the trend of hole probability density on nanocrystal surfaces rather than the product of electron and hole densities, consistent with endothermic hole-transfer-mediated TET. The shallow distance-dependence afforded by this mechanism enables efficient TET across distances well beyond the nominal range of Dexter or super-exchange paradigms.

is the two-electron operator. In the initial state, the donor (D) is in the spin-triplet excited state and the acceptor (A) is in its ground state; in the final state, the donor is in its ground state and the acceptor is in its triplet excited state.

Harcourt et al.'s formula
Harcourt et al. have reformulated the electronic coupling matrix element for excitation energy transfer by including through-configuration terms. 2 They find that the coupling matrix element, in the case of triplet energy transfer, can be approximated as: where the two-electron exchange term K is the same as defined above. For the QD-molecule donor-acceptor systems studied herein, because of the difficulty to compute the absolute values for the various terms defined in eq. S10, it remains unclear how the through-configuration terms compare with K. Nonetheless, because T ET and T HT should scale with Ψ and Ψ , respectively, the scaling relationship in eq. S9 should still hold.

Supplementary Note 2. Electronic coupling matrix for charge-transfer mediated triplet energy transfer
Charge-transfer (CT) mediated TET involves sequential transfer of an electron ad a hole from the donor to the acceptor. In this case, using the above procedures, it is easy to show that the rates of electron ( ) and hole transfer ( ) processes scale as : When the donor is a QD, these scaling relationships can be further simplified: .
In order to derive how the overall, apparent TET rate scales with QD wavefunctions, we consider the following coupled kinetic reaction: * − 1 ( −1 ) where QD * -A and QD-A * are the initial and final states, respectively, CT is intermediate charge-transfer state, k 1 and k 2 are electron and hole transfer rates, and k -1 is the backward reaction rate of the first step. As we are dealing with the situation that CT is an endothermic state with higher energy than QD * -A, the formation of CT via k 1 should be much slower compared to its decay via k -1 and k 2 . Thus, the concentration of CT should always be low (undetectable on TA spectra in our experiment), allowing us to apply the steady-state approximation to solve the kinetic equations: Therefore, and the apparent rate for the formation of acceptor triplets is: Under the situation that k 2 are much faster than k 1 and k -1 (as is in our case), the overall TET rate is reduced to k 1 and it should simply scale as |Ψ | 2 or |Ψ | 2 depending on whether the first step is an electron or hole transfer process.

Supplementary Note 3. Calculations of carrier wavefunctions in QDs
The probability densities (squared wavefunctions) of the band edge electron and hole of CdSe/ZnS QDs were simulated using a single-band effective mass approximation model (EMA). The Coulomb interaction between the electron and the hole was also ignored in our simulation but can be readily added using first-order perturbation when calculating optical energy gaps.
A model of a particle in 3-region spherical potentials is employed. The three regions from the center to the outside of the sphere correspond to the CdSe core, ZnS shell and oleate ligand layer, and their radius or thickness are defined as the radius of CdSe core size, ZnS shell thickness (0.27 nm per monolayer multiplied by the layer numbers) and 1.5 nm, respectively. Thus, the potentials were set as: in which stands for the radius CdSe core QDs in the unit of nm (which is 1. The effective mass of the electron (hole) in the above-mentioned three regions was defined as: where 0 is the free electron mass and The oxidation potential energy of ACA triplet excited state ( where is the triplet energy of ACA which is 1.83 eV.
The oxidation potential energy of ACA singlet excited state ( where , is exciton binding energy in ACA which is reported to be ~0.8 eV. 8 Note that all the Coulombic energy terms are defined as positive values here.

Redox potential energies of QDs
The oxidation potential energy of ground state CdSe/1.2ZnS QDs ( + ⁄ ) is revealed as an anodic peak in Supplementary Figure . Unlike molecules, this peak is irreversible for QDs. 9 Therefore the anodic peak of ACA + /ACA pair is used as a new reference:

TET and CT driving forces
Using the state potential energies determined above, rather than commonly-adopted single-particle potential energies (such as HOMO and LUMO for molecules and electron and hole levels for QDs), allows us to facilely calculate the free energy changes (i.e., driving forces) from reactant states to product states, because the electron correlation and exchange energy terms are already included in the state potential energies. An additional term that needs to be added, when applicable, is the Coulomb binding energy of the charge separated states.
For direct TET from photoexcited QDs to ACA, the free energy change can be written as: Here is the stabilization energy resulting from the above-mentioned charge separated state (QD --ACA + ) binding energy, which can be estimated as ~50 meV for CdSe/1.2ZnS QDs using the Gauss's theorem. Therefore, this hole transfer process is energetically uphill by ~0.03 eV. This electron transfer process is strongly disallowed.
There is another electron transfer process that is involved in endothermic hole-transfer-mediated triplet migration, electron transfer from QDto ACA + to form 3 ACA * . The free energy change of this process can be written as: We note that all the calculations above are made for CdSe/1.2ZnS QDs for which the CV measurement was performed. We assume that QDs of other shell thicknesses have very similar TET or CT driving forces for the following reasons. First, the changes in exciton binding energy and QD charging energy with shell thickness almost mutually cancels out, as evidenced by the negligible exciton absorption peak shift from the thinnest shell CdSe/0.5ZnS QDs to the thickest shell CdSe/3.9ZnS QDs.
Second, according to our calculation, the term changes by less than 30 meV from the thinnest to thickest shell QDs. Thus, TET and CT driving forces can be treated as shell-thickness-independent in our experiments. ( 34). 12 The average lifetime was calculated as:
Triplet energy transfer quantum yield is defined as: The average transfer energy transfer rate from QD to each anchored ACA molecule is calculated as: in which n ACA stands for the average number of anchored ACA molecules per QD.
The kinetics fitting parameters and calculated energy transfer yields and rates are listed in Supplementary

Supplementary Note 6. Kinetic simulation
We simulated the temporal evolution of QD * -ACA, QD --ACA + (i.e. CT state) and QD-3 ACA * species using the coupled rate equations of eqs. S39-41, where k 1 and k 2 are hole and electron transfer rates, respectively, and k -1 is the backward hole transfer rate. The value of k 1 was from the experimental data, whereas k 2 and k -1 were estimated according the Marcus equation by assuming that the reorganization energies were the same for these CT processes. We