Phonon-assisted ultrafast charge separation in the PCBM band structure

Organic solar cells separate strongly bound electron-hole pairs into free charges at interfaces between electron donor and acceptor organic semiconductors. Recently, electron-hole separation was observed on femtosecond time scales near crystallite phases of the fullerene derivative [6,6]-phenyl-${\mathrm{C}}_{71}$-butyric acid methyl ester (PCBM), which is incompatible with conventional Marcus theories of organic transport. Here we show that ultrafast charge transport in PCBM arises from its broad range of electronic eigenstates, provided by three closely spaced electronic bands near the lowest unoccupied molecular orbital. The highest band provides a charge transfer state resonant with delocalized states of the lower two bands away from the interface. This state acts as a bridge between the donor phase and the acceptor bulk, bypassing the trapped charge-transfer (CT) states below. Vibrational fluctuations enable rapid electronic transitions across this bridge, which can drive the electron more than 4 nm away from the interface within 100 fs. All this is demonstrated within a simple tight-binding Hamiltonian containing transfer integrals no larger than 8 meV.

Figure 1

For simplicity, we depict two bands only. Narrow bands localize near the interface when the Coulomb energy drop between neighboring sites approaches the bandwidth. We assume the donor injects electrons into the interfacial state of the higher band. This state lies at the same energy as the delocalized states of the lower band, and consequently the states weakly mix. Ultrafast charge separation occurs when the electron successfully tunnels into the band before relaxing down into the trapped interfacial state. This tunneling is enhanced by fluctuations of the site energies near the interface, while $\delta$ describes the typical scale of these fluctuations.

Figure 2

(a) The density of states of a ${\left(7\mathrm{nm}\right)}^3$ aggregate, neglecting the nearby hole. (b) The density of states, when the hole is introduced 1 nm away from the center of one face of the aggregate. We neglect disorder for best comparison with Ref. [12], and each eigenstate is convolved with a 5 meV Gaussian to generate a continuous distribution. (c) A plot of eigenstate energies and their electron-hole separations. Each point corresponds to a single eigenstate, and the eigenstates of 100 disorder realizations are shown. (d) Mean electron-hole separation for coherent propagation of an electron (averaged over 1000 simulations).

Figure 3

(a) Mean electron-hole separation under Redfield dynamics at 300 K, for damping time scales $\tau =10$, 100, and 1000 fs. Note the log scale. (b) Mean electron-hole separation under Redfield dynamics at 4 K. All simulations are averaged over 1000 disorder realizations.

Figure 4

(a) Electron dynamics at 300 K, with $\tau =100$ fs, for a range of aggregate sizes. (b) Electron dynamics for a range of band offsets ${\mathrm{\Delta }}_3$. (c) Electron-hole separation at 300 K under fluctuations in the off-diagonal couplings between states. (d) Electron-hole separation at 4 K under off-diagonal fluctuations. All simulations are averaged over 1000 disorder realizations.