Ultrafast Charge Separation and Nongeminate Electron-Hole Recombination in Organic Photovoltaics
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J un Ultrafast Charge Separation and Nongeminate Electron-Hole Recom-bination in Organic Photovoltaics
Samuel L Smith ∗ a and Alex W Chin a Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XXFirst published on the web Xth XXXXXXXXXX 200X
DOI: 10.1039/b000000x
The mechanism of electron-hole separation in organic so-lar cells is currently hotly debated. Recent experimen-tal work suggests that these charges can separate on ex-tremely short timescales ( <
100 fs). This can be understoodin terms of delocalised transport within fullerene aggre-gates, which is thought to emerge on short timescales be-fore vibronic relaxation induces polaron formation. How-ever, in the optimal heterojunction morphology, electronsand holes will often re-encounter each other before reach-ing the electrodes. If such charges trap and cannot sepa-rate, then device efficiency will suffer. Here we extend thetheory of ultrafast charge separation to incorporate po-laron formation, and find that the same delocalised trans-port used to explain ultrafast charge separation can ac-count for the suppression of nongeminate recombinationin the best devices.
The best solution-processed organic photovoltaic cells (OPVs)now exhibit efficiencies exceeding 9% . Devices consistof a nanostructured ”heterojunction” morphology of inter-mixed electron donor and acceptor semiconductors . Usuallyfullerene-derivatives are used as the electron acceptor. Pho-tons are absorbed within the device generating tightly boundexcitons. These excitons diffuse to interfaces between donorand acceptor semiconductor, where electron and hole can sep-arate into free charges. However in order to separate, chargesmust overcome their mutual Coulomb attraction, which is anorder of magnitude greater than thermal energies at room tem-perature . Experimentally, charge separation has been ob-served on ultrafast timescales ( <
100 fs) . This observationis incompatible with conventional theories of charge transportin organic media, and new proposals have emerged . Ithas been proposed that ultrafast charge transfer is sustainedby spatially coherent delocalised states, which arise on shorttimescales before molecular vibrations can respond to thepresence of charges . It is assumed that more localisedpolarons will form on longer timescales, although
Bakulin etal. found that delocalised states could be repopulated at late a Theory of Condensed Matter, Cavendish Laboratory, University of Cam-bridge, UK. E-mail: [email protected] times by an infrared pulse . Electrons and holes remaining inclose proximity at long times are thought to trap into boundcharge transfer (CT) states at the interface, while separatedcharges are free to generate a photocurrent.Electron hole pairs generate a dipolar electric field as theyseparate, this field can be observed as a Stark shift in the op-tical spectra of neighboring molecules. Using this signature, G ´ elinas et al. directly observed the separation of charges onfemtosecond timescales . They found that electron and holeseparated by a few nanometres within just 40 fs, but that thiswas only observed in devices containing nanoscale aggregatesof the fullerene-derivative electron acceptor PC BM. Along-side this experiment, we presented a simple phenomenolog-ical model of ultrafast charge separation through delocalisedstates of small acceptor crystals. This model has since beensupported by more detailed modeling of PCBM crystallites .Our central proposition states that in order for ultrafast chargeseparation to occur, the effective bandwidth of the crystal-lite LUMO should be similar in magnitude to the electrostaticbinding energy of electron and hole across the interface.The current theoretical model is not complete, since it onlydescribes charge transport within a few hundred femtosecondsof exciton dissociation. Once charges are free, they will dif-fuse through the device. However, in the heterojunction mor-phology many electrons and holes will re-encounter each otherbefore reaching the electrodes. This may lead to re-trappingand nongeminate exciton recombination, lowering the deviceefficiency . In efficient devices, either re-trapping must besuppressed or the trapped CT states that form must themselvesbe able to separate into free charges. In a recent experiment, Rao et al. noted the absence of nongeminate triplet excitonsat open circuit in an efficient PIDT-PhanQ:PC BM device .Since three quarters of nongeminate CT states should havetriplet character, they concluded that such CT states were ableto separate long after exciton dissociation first occurs, thusavoiding the formation of triplet excitons. To explain theseobservations and build a complete description of charge sep-aration on both femtosecond and nanosecond timescales, weextend our description of ultrafast charge separation to takeaccount of vibronic relaxation. ||
Bakulin etal. found that delocalised states could be repopulated at late a Theory of Condensed Matter, Cavendish Laboratory, University of Cam-bridge, UK. E-mail: [email protected] times by an infrared pulse . Electrons and holes remaining inclose proximity at long times are thought to trap into boundcharge transfer (CT) states at the interface, while separatedcharges are free to generate a photocurrent.Electron hole pairs generate a dipolar electric field as theyseparate, this field can be observed as a Stark shift in the op-tical spectra of neighboring molecules. Using this signature, G ´ elinas et al. directly observed the separation of charges onfemtosecond timescales . They found that electron and holeseparated by a few nanometres within just 40 fs, but that thiswas only observed in devices containing nanoscale aggregatesof the fullerene-derivative electron acceptor PC BM. Along-side this experiment, we presented a simple phenomenolog-ical model of ultrafast charge separation through delocalisedstates of small acceptor crystals. This model has since beensupported by more detailed modeling of PCBM crystallites .Our central proposition states that in order for ultrafast chargeseparation to occur, the effective bandwidth of the crystal-lite LUMO should be similar in magnitude to the electrostaticbinding energy of electron and hole across the interface.The current theoretical model is not complete, since it onlydescribes charge transport within a few hundred femtosecondsof exciton dissociation. Once charges are free, they will dif-fuse through the device. However, in the heterojunction mor-phology many electrons and holes will re-encounter each otherbefore reaching the electrodes. This may lead to re-trappingand nongeminate exciton recombination, lowering the deviceefficiency . In efficient devices, either re-trapping must besuppressed or the trapped CT states that form must themselvesbe able to separate into free charges. In a recent experiment, Rao et al. noted the absence of nongeminate triplet excitonsat open circuit in an efficient PIDT-PhanQ:PC BM device .Since three quarters of nongeminate CT states should havetriplet character, they concluded that such CT states were ableto separate long after exciton dissociation first occurs, thusavoiding the formation of triplet excitons. To explain theseobservations and build a complete description of charge sep-aration on both femtosecond and nanosecond timescales, weextend our description of ultrafast charge separation to takeaccount of vibronic relaxation. || irst we briefly reiterate our model of ultrafast charge sepa-ration . This model is illustrated in figure 1a, in figure 1b weillustrate our parallel theory of trapped pair separation at latetimes (developed below). We assume that ultrafast charge sep-aration occurs when an exciton reaches an interface betweendonor molecules and a small acceptor crystallite. This crystal-lite is modeled by an FCC lattice of localised single electronenergy levels, which are coherently coupled to their nearestneighbours. We take a Gaussian distribution of site energiesto introduce disorder. The electronic eigenstates of this crys-tallite are delocalised standing waves, with bandwidth B. Thenwe introduce a Coulomb well surrounding a donor site adja-cent to one face of the crystal, this well models the hole leftbehind after an electron is injected into the crystallite. TheHamiltonian, H S = (cid:229) i E i | i ih i | − (cid:229) n . n . J | i ih j | . (1) E i = s i − e pe e r r i . s i represents the Gaussian disorder on eachsite, e r labels the dielectric constant, and r i labels the distancebetween the i th acceptor site and the hole. J labels the trans-fer integral between nearest neighbours, and the bandwidth B = J . The maximum depth of the Coulomb well within theacceptor lattice, W = e / pe e r r . If the bandwidth B is muchsmaller than W, then the electronic eigenstates at the inter-face will localise and ultrafast charge transport will not occur.However if B & W , then a set of band-states will survive, de-localised across the entire crystallite. Resonant coupling be-tween the incoming exciton and these delocalised states candrive ultrafast charge separation.A detailed DFT study of the popular electron acceptorPC BM was recently published, which supports this pic-ture . Despite the Coulomb well induced by the hole,nanoscale crystallites were found to exhibit delocalised statesnear the interface, able to support resonant coupling with in-coming excitons. The naive LUMO bandwidth is ∼ ∼ BM exhibits three closely spaced low ly-ing molecular orbitals; incorporating all three low lying bandsenhances the effective bandwidth to ∼ ∼ W Interface BulkCE CT Fig. 1
Left (a): Electronic eigenstates on ultrafast timescales. If W . B, then a set of delocalised states will survive with signicantweight near the interface. These states enable ultrafast chargeseparation. Right (b): On longer timescales vibronic relaxationlowers the energy of the CT state, while leaving higher lying statesunchanged. Thermal fluctuations promote the trapped electron intohigher lying states, excitation into the delocalised states above CEcan enable charge separation. each lattice site to an effective molecular vibration , H = H S + H B + H I , (2) H B = w (cid:229) i a † i a i , (3) H I = g (cid:229) i ( a i + a † i ) | i ih i | . (4)We express the electronic subsystem in terms of the generalwavefunction | y i = (cid:229) i C i | i i , and the vibrations in terms ofposition and momentum operators X i ( t ) = h a i + a † i i , P i ( t ) = h a i − a † i i . These obey the Heisenberg equation of motion ˙ O = i [ H , O ] . We assume that vibrations oscillate slowly comparedto electronic modes. This allows us to treat the vibrationssemi-classically, making the approximation H I → g (cid:229) i X i | i ih i | . Thus far we have neglected damping in the vibrationalmode, which is necessary to reach the relaxed lowest energyCT state. This can easily be included phenomenological intothe equations of motion,˙ X i = − i w P i , (5)˙ P i = − i ( w X i + g | C i | ) − g ˙ P i . (6)Damping will drive both P i and ˙ P i to zero. Thus in the relaxedCT state, X i = − g | C i | / w . Inserting this result into equa-tion 2, we obtain an effective non-linear Hamiltonian for theelectronic eigenstates of the relaxed, maximally trapped CTstates at long times , H eff = (cid:229) i ( E i − D | C i | ) | i ih i | + (cid:229) n . n . J | i ih j | . (7)The reorganization energy D = g / w ∗ . This equation can besimply understood. At long times the hole is assumed to lie ∗ In this simple model, the energy penalty arising from H B is precisely half thepolaronic stabalisation arising from H I . The relevant reorganization energyhere corresponds to the upper potential energy surface of Marcus theory; if H B is identical for occupied and unoccupied electronic states then D = D Marcus / | ext to an acceptor crystallite, while the electron lies in thelowest available eigenstate of the acceptor lattice. The vibra-tional reorganization on each electronic site is proportional tothe charge density on that site. This lowest energy eigenstate | y i = (cid:229) i C i | i i can be found by a simple iterative procedure.Since the vibrational modes are assumed slow compared toelectronic motion, the instantaneously accessible higher en-ergy eigenstates can be found by holding the vibration reor-ganization on each site fixed and solving H eff , which is nowlinear, for the higher lying states.Vibronic relaxation will tend to localise the CT eigenstatenear the interface. If the electron is localised on a single sitethen the corresponding site energy will be reduced by D . Thiswill provide a significant barrier to charge separation. How-ever if the couplings between neighboring lattice sites are suf-ficiently strong to preserve a delocalised CT , then the cor-responding reorganization energies of the relevant lattice siteswill be reduced; lowering the barrier to charge separation. Ad-ditionally, delocalisation will have decreased the exciton re-combination rate.We now investigate the properties of the relaxed CT statenumerically. We take a dielectric constant of 3, a donor-acceptor nearest neighbour separation r = = unit cells. Finally we fix the nearestneighbour coupling J =
25 meV (implying a bandwidth B = eigenstate as D is varied between 0 and 0.4 eV. For reference,and to illustrate the simulation error, we also include the de-localisation of the early time unrelaxed lowest eigenstate ofequation 1. Each data point is averaged over 10000 runs, andthe delocalisation is measured via the inverse participation ra-tio † . As discussed in our earlier paper, at early times beforevibrational relaxation can occur even the lowest eigenstate ofthe acceptor lattice is delocalised over many sites. This occursdespite the presence of a deep Coulomb well near the inter-face. By contrast, when D is large, the relaxed CT eigenstatewhich forms on long timescales is fully localised. In this limitnongeminate trapped electron hole pairs would be unlikely toseparate. As D is reduced below 0.3 eV however, the relaxedCT state begins to delocalise.To further explore this transition, in figure 2b we presentthe binding energy of the electron in the relaxed CT state aswell as the population fraction of this state which lies on theacceptor site neighboring the donor. The binding energy ismeasured from the bottom of the LUMO band of bulk accep-tor crystal, E B = E + J ‡ . The binding energy of the CT is † The inverse participation ratio, IPR = / ( (cid:229) i | C i | ) , where the sum runs overall lattice sites.‡ Note that FCC lattices show an unusual band structure. In the infinite bulk D e l o c a li s a t i on ( I n v . P a r t. R a t i o ) Reorganisation energy / meV
Unrelaxed CT Relaxed CT -500-400-300-200-100 0 0 100 200 300 400 0 0.2 0.4 0.6 0.8 1 B i nd i ng ene r g y / m e V P op . d . ne i ghbou r i ng dono r Reorganisation energy / meV
Binding energyPopulation density 0 10 20 30 40 50 60 0 10 20 30 40 50 60 D e l o c a li s a t i on ( a . r e l a x . ) Delocalisation (b. relax.) -200 0 200 400-200 0 200 400 B i nd i ng E ne r g y ( a . r e l a x . ) / m e V Binding Energy (b. relax.) / meV
Fig. 2
Top left (a): Delocalisation of the lowest electroniceigenstate with and without vibronic relaxation. At early times,before relaxation occurs, the state is delocalised. At long times thelocalisation depends on the reorganization energy. Top right (b):Binding energy of the charge pair, and population densityneighboring the donor. Bottom panel: Delocalisation and bindingenergy of all electronic eigenstates before and after relaxation.States are labelled by their energy, and then plotted against eachother. The CT state (red) is more localised and more stronglybound after relaxation occurs. Higher lying states (green), whenaveraged over disorder runs, are not significantly affected by therelaxation process. Consequently these states lie on the line y = x (blue). little affected until D >
100 meV, after which it rises steeply.Meanwhile as D is reduced the population density neighbor-ing the donor site falls, enhancing the CT state lifetime andgiving the charge pair more opportunity to separate. Typicalvalues for the reorganization energy of p -conjugated organicmolecules lie in the range 0.1-0.3 eV
21 § ; which suggests thateven the fully relaxed CT state may remain delocalised overseveral sites near the interface, if the bandwidth is sufficientlylarge.Thus far we have compared the lowest eigenstate of the ac- crystal without disorder, there are equal numbers of eigenstates above andbelow the isolated site energy E =
0. However the bandwidth, which extendsfrom − J to + J , is not symmetric.§ There is some uncertainty over the intramolecular reorganization energy offullerene derivatives. Kwiatkowski et al. computed a theoretical Marcus valueof 0.13 eV for C
60 22 , and
Cheung et al. computed a similar value of 0.14 eVfor PC BM . These results include both upper and lower potential surfaces,suggesting a reorganisation energy here of ∼
70 meV. However
Savoie et al. directly compute the reorganisation energy of the upper potential surface (theanion) of PC BM and obtain a much smaller value of just 15 meV . ||
Savoie et al. directly compute the reorganisation energy of the upper potential surface (theanion) of PC BM and obtain a much smaller value of just 15 meV . || eptor crystallite at early and late times. However, so long asthe molecular modes can be treated as slow, we may also com-pare the relaxed and unrelaxed higher lying electronic eigen-states. In the bottom panel of figure 2, we take D = ), it has much less effecton the higher lying states. This is easily understood, sincethe vibrational reorganization is determined by the CT statedensity. It explains why Bakulin et al. were able to opticallyre-excite the CT state to the delocalised band states typicallyobserved immediately after exciton dissociation .After vibrational relaxation has occurred, thermal fluctua-tions in the vibrational modes attached to each lattice site drivespontaneous transitions from the relaxed CT state to higherlying states. Applying time dependent perturbation theory andassuming that vibrational fluctuations are in thermal equilib-rium, the rate of such transitions is given by R → a = p J ( E a ) e ( E a − E ) / k B T − (cid:229) i | C ai C i | . (8)The spectral density of a single, over-damped vibrationalmode, J ( E ) = D E g / ( E + ¯ h g ) . In figure 3a we take adamping timescale 2 p / g = ¶ . Thisdenotes the typical time required for an electron to be ex-cited, from the maximally trapped CT state, to one of theunbound eigenstates with binding energy E B >
0. As demon-strated above, these are the same states which drive ultrafastcharge separation; each such event presents an opportunity forelectron and hole to separate.We focus first on the green curve at 300K. Since transitionsare driven by thermal fluctuations in the vibrational modes, theescape timescale diverges as D →
0. It exhibits a broad mini-mum at D ≈
70 meV and rises rapidly for D > ; therefore, so long as D < ≈ . eV ) may notonly support ultrafast charge separation, but can also assistthe thermal separation of electrons and holes on timescalessufficiently fast to suppress nongeminate recombination. Thisunified description of ultrafast charge separation and the dis-sociation of trapped pairs was summarised in figure 1, bothphenomena depend crucially on the formation of nanoscale ac-ceptor crystallites ( & nm ). Whereas ultrafast charge sep- ¶ The probability distribution of rates in the presence of static disorder is highlyskewed, due to the exponential factor in equation 8. Therefore we take theaverage over the binding energy and the overlap function, before calculatingthe rate using these mean values. E sc ape t i m e sc a l e / n s Reorganisation energy / meV D e l o c a li s a t i on ( I n v . P a r t. R a t i o ) Bandwidth / meV
Unrelaxed CT D = 150 meV D = 400 meV 0.001 0.01 0.1 1 10 100 1000 0 100 200 300 400 E sc ape t i m e sc a l e / n s Reorganisation energy / meV
200 meV 300 meV 400 meV 500 meV 600 meV
Fig. 3
Top left (a): The transition timescale from the CT state tohigher lying, unbound states, at a range of temperatures. Top right(b): The delocalisation of the CT state for D =
0, 150 and 400 meV,as a function of bandwidth. Bottom (c): The escape timescale at300K for a range of bandwidths. All points (a-c) averaged over 1000runs. aration is temperature independent , the curves at 200/400Kemphasize that the separation of trapped nongeminate pairs ishighly temperature dependent. These predictions agree wellwith the observations of Rao et al. .Thus far we have only considered an efficient device withrelatively large bandwidth, B = state as the bandwidth is variedfrom 0 to 0.8 eV, for three different values of the reorganiza-tion energy. Delocalisation rises rapidly once the bandwidthis increased beyond a particular threshold; for moderate reor-ganization ( D = = .Also problematic is the assumption of a single broad LUMOband; as discussed earlier, the CT eigenstates of PCBM crys-tallites, the most popular electron acceptor, are formed fromthe mixing of three closely spaced narrow bands . In fu-ture work we hope to investigate these two effects in more de- | ail, here we prefer to preserve a simple and intuitive pictureof the acceptor crystallite.We have not allowed any delocalisation of the hole. It islikely that in real systems the hole is partially delocalisedalong the donor polymer; this will lower the binding energybetween electron and hole, reducing the acceptor bandwidthrequired to enable ultrafast charge separation and suppress ex-citon recombination . Additionally, we have pessimisticallyassumed that the charge pair is able to relax into the maxi-mally trapped CT state described by equation 7. In reality,the thermal fluctuations in the vibrational modes obstruct therelaxation process. Consequently, at sufficiently high temper-atures a stable vibronically relaxed CT state will never form.In this instance the eigenstates of the acceptor crystallite willresemble their early time counterparts at all times. To quantifythis temperature, we note that the typical fluctuation scale ofmolecular vibrations on a lattice site is set by k B T ∼
25 meVat room temperature. Meanwhile the typical vibrational reor-ganization energy on a site is given by D / D , where D is thedelocalisation of the CT . Consequently, once D exceeds 5-10sites, thermal fluctuations are similar in scale to the molecularreorganization, and the formation of a polaronic CT state islikely to be inhibited.The discussion above has focused on the thermal seperationof nongeminate pairs, however the mechanism is equally ap-plicable to the seperation of trapped geminate pairs. As such,these results may rationalise the apparent contradiction be-tween ultrafast charge transport and the observations of Van-dewal et al. ; who found that sub-gap excitation of trappedCT states can efficiently generate free charges.In conclusion, there is significant experimental evidencethat charge pairs are able to separate on ultrafast timescalesat the interfaces between organic donor semiconductors andfullerene-derivative crystallites. This observation is best un-derstood by treating the electronic eigenstates within thesecrystallites as delocalised. We have shown that simple modelsof this phenomenon can be extended to incorporate vibronicrelaxation, which has previously been thought to localise thesestates. By contrast, here we show that the electronic eigen-states can remain delocalised and continue to assist the sepa-ration of trapped electron-hole pairs on long timescales. Thiswork underlines the importance of crystallinity and nanoscalemorphology for OPV performance.We acknowledge funding from the Winton Programme forthe Physics of Sustainability and we thank Richard H. Friend,Neil Greenham, Akshay Rao and Dan Credgington for helpfulcomments on the manuscript. References
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