steps in charge separation (see supplementary
materials for details).
Based on the above results, we propose a
simple phenomenological model of the heterojunction interface. Films without aggregated
fullerene domains do not exhibit a substantial
EA response, which implies that the mobile component of the charge pair is the electron at early
times. We model the electron motion on a nanoscale face-centered cubic lattice of acceptor sites,
consisting of localized single-electron energy levels that are coherently coupled to their nearest
neighbors (a tight binding model), giving rise
to a band of delocalized eigenstates with bandwidth B. We include a Gaussian distribution of
acceptor-site energies to introduce disorder. As expected, we find that when the static disorder of the
on-site energies is within the bandwidth, disorder
does not localize the states. The initial excitation
is a single exciton on an adjacent donor site.
We include a Coulomb well, of depth W, sur-
rounding the donor to model the presence of
the hole, which we assume does not move dur-
ing the first 200 fs of charge separation. The well
lowers the energies of electron states in the local
vicinity of the donor (Fig. 4A), introducing a
set of bound states and narrowing the energetic
width of delocalized states to ≈B-W. Typical struc-
tures of the emergent eigenstates are illustrated
in Fig. 4A (ϕ1-2), with corresponding examples of
the actual states given in fig. S11. We emphasize
that our model is only valid for a few hundred
femtoseconds after exciton dissociation when
delocalized states are briefly accessed; after this,
polaron formation will localize the electronic states
and holes and electrons will move via incoherent
hopping with comparable mobilities.
If the energy of the electron on the donor site
lies within the spread of fullerene eigenstates at
the interface, the electron can undergo resonant
transfer. When this energy lies below the conduction edge (CE, Fig. 4A), the initial electron
wave function can only mix with bound states
localized near the hole (ϕ1), forming a bound
charge-transfer state. However, if the initial electron energy lies within the conduction channel
above this edge, then the initial state is mixed
with states that are fully delocalized across the
fullerene lattice (ϕ2), enabling long-range charge
separation to occur.
The model described above assumes spatial
coherence between acceptor sites (
delocalization). To describe the dynamics of the system,
we consider two cases: (i) incoherent transitions
between a localized donor site and the delocalized acceptor eigenstates, or (ii) a fully coherent
case where the initial electron is described as
phase-coherent superposition of these delocalized eigenstates. We perform simulations for both
these models on a 5.25 nm by 5.25 nm by 5.25 nm
lattice of acceptor sites and for a range of couplings and disorder strengths between 30 and
50 meVand 0.1 and 0.2 eV, respectively (31, 32);
all parameters are described in full in the supplementary materials.
Incoherent transitions arise from a perturba-
tive treatment of the D-A coupling [case (i), see
supplementary materials]. This generates a set
of Fermi golden rule transition rates from the
localized donor site into the delocalized states
described earlier. Results are shown in Fig. 4B,
yellow curves. We find that charge separation
occurs via a single hop of ~4 nm. For case (ii),
we evolve the coherent superposition under the
Schrödinger equation, which generates a wave
packet that rapidly propagates across the Cou-
lomb well and into the acceptor crystallite (see
supplementary materials for details). Results
are shown in Fig. 4B, red curves. Again, for all
parameter values considered, electron and hole
separate by 3 to 5 nm within 300 fs. For both cases,
the separation distance is determined primarily by
the size of the acceptor aggregate in our model.
Currently, our experiments cannot differentiate be-
tween these two separation mechanisms.
We also consider charge dynamics under fully
incoherent localized dynamics (Marcus theory)—
i.e., in the absence of delocalization—for the same
parameters (see supplementary materials for details). This leads to rapid exciton dissociation,
with hole and electron on nearest-neighbor sites
(Fig. 4B, blue curve), to form tightly bound CT
states that are not expected to dissociate rapidly.
Our demonstration and model of short-time
charge delocalization and coherent motion is
very different from models for exciton coherence
and delocalization that have been suggested as
being key to efficient OPV performance (33).
Within our study, we find no need to invoke such
excitonic processes (see supplementary materials
Our results rationalize the apparent asymmetry
between efficient electron-hole capture in organic
light-emitting diodes (34) and near-unity photo-conversion quantum efficiencies in OPVs (25) by
revealing that ultrafast charge separation through
delocalized band-like states in fullerene aggregates
is key to efficient charge separation. Moreover,
the fast time scale for this process indicates that
efficient charge separation requires no excess energy beyond that needed to overcome the Coulomb
interaction. This is in contrast to Onsager-like
models that require excess energy in hot states.
Our results suggest that the real energy loss during
charge separation lies elsewhere—for instance, in
later energetic relaxation of charges through polaron
formation or the presence of defect-mediated gap
states—and that such energy losses are not
fundamental for efficient charge separation.
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Fig. 4. Model of initially accessible electronic states in fullerene derivatives and calculated
electron-hole separation distance. (A) Excited states before and immediately after charge transfer.
When excitons (S1) dissociate at interfaces with aggregated PC61/71BM, the isoenergetic charge transfer
places the electrons in delocalized band states, where they undergo wave-like propagation within the
PC61/71BM aggregate. In this model, the hole at the interface induces a well of depth W, reducing the
width of the band from its bulk value B to ~B-W. This system can sustain two typical electron wave
functions represented as ϕ1,2. The electron is either trapped at the interface (ϕ1) or propagating through
the band (ϕ2). (B) Calculation of electron-hole separation dynamics per charge pair for (i) injection of a
fully coherent electron wave packet; (ii) tunneling of the electron into delocalized states (Fermi golden
rule, FGR); and (iii) tunneling of the electron into localized states (i.e., Marcus-type electron transfer). The
multiple lines represent different values of disorder and couplings, spanning 100 to 200 meV and 30 to
50 meV, respectively (31, 32). r0 is the separation distance of the initial next-neighbor charge-transfer
state [typically 1.5 nm (9)], and l is the length of the PC61/71BM aggregate over which the wave function is
delocalized (5.25 nm for the calculation shown).