Calculating the Efficiency of Exciton Dissociation at the Interface

Perspective pubs.acs.org/JPCL

Calculating the Efficiency of Exciton Dissociation at the Interface between a Conjugated Polymer and an Electron Acceptor S. D. Baranovskii,*,† M. Wiemer,† A. V. Nenashev,‡ F. Jansson,† and F. Gebhard† †

Department of Physics and Material Sciences Center, Philipps-Universität, D-35032 Marburg, Germany Institute of Semiconductor Physics and Novosibirsk State University, 630090 Novosibirsk, Russia

‡

ABSTRACT: The decisive feature of any material designed for photovoltaic applications is the dissociation efficiency of photogenerated excitons. This efficiency is essentially governed by the Coulomb attraction between electrons and holes. Because the dielectric constant in organic materials is rather low (εr ≈ 3), the exciton binding energy is much larger than the thermal energy at room temperature, so that thermally governed dissociation seems improbable. Experiments show that while the dissociation probability for electron−hole pairs is indeed very low in the bulk samples, it becomes close to unity at intrinsic interfaces between two organic materials, an electron donor (usually a conjugated polymer) and an electron acceptor (usually a fullerene derivative). The driving force for this dissociation is still a matter of controversy. This Perspective provides a theoretical analysis of possible mechanisms for the efficient dissociation of electron−hole pairs at internal organic interfaces despite the strong Coulomb attraction between the charges.

I

t is one of the main challenges in the research on organic materials to reveal the mechanism responsible for the efficient dissociation of electron−hole pairs (EHPs) created by light. The interest of researchers in the dissociation problem is caused mainly by perspectives of photovoltaic applications of organic semiconductors. Such materials usually exhibit very high light absorption coefficients, which, combined with lowcost processing, makes them promising for applications in solar cells.1,2 However, it is not the efficient light absorption itself but rather the combination of the efficient absorption with the efficient photocurrent generation that makes a material favorable for photovoltaic applications. The decisive step in photocurrent generation is the dissociation of EHPs created by light. In the dissociation process, the electron and hole must overcome their mutual Coulomb attraction, determined by the energy EC =

e2 4πεrε0r

One of the main challenges in research on organic materials is to reveal the mechanism responsible for the efficient dissociation of electron−hole pairs created by light, with photovoltaic applications in organic semiconductors. current report. Interested readers can find a comprehensive description of the research field in recent review articles.1,3 Here, we only briefly describe the state of the research field. The first organic solar cells tested in the 1970s used a single material, sandwiched between two electrodes. Because the thermal energy kT is much lower than the binding energy of excitons generated by light, most excitons recombined, avoiding their dissociation in the bulk material. Therefore, the power conversion of single-layer organic solar cells was far below 1%.5 The power conversion has been essentially improved by introducing a second organic semiconductor layer in a planar geometry. The first bilayer solar cell was reported in 1986.6 The working mechanism of such devices is the following.1 The light is usually absorbed in the so-called donor material, a holeconducting organic semiconductor. The optically generated excitons can diffuse within the donor material toward the planar interface with the second material, the electron acceptor, which

(1)

where e is the elementary charge, εr is the relative dielectric constant of the surrounding media, ε0 is the permittivity of vacuum, and r is the electron−hole separation. While in inorganic semiconductors with εr > 10 the binding energy of excitons is on the order of 10 meV, in organic semiconductors, in which εr is typically between 2 and 4,3 the binding energy of excitons is on the order of 1 eV.4 The puzzling question arises then regarding the mechanism that could provide an efficient dissociation of EHPs with such a huge binding energy as compared to the thermal energy at room temperature, kT ≃ 0.025 eV. Numerous experimental and theoretical studies have been dedicated to the dissociation problem of EHPs in organic semiconductors. There is no chance to review all of them in the © 2012 American Chemical Society

Received: January 31, 2012 Accepted: April 13, 2012 Published: April 13, 2012 1214

dx.doi.org/10.1021/jz300123k | J. Phys. Chem. Lett. 2012, 3, 1214−1221

The Journal of Physical Chemistry Letters

Perspective

is chosen to be strongly electronegative. At the interface, the electron moves to a state of much lower energy within the acceptor semiconductor, whereas the hole remains within the donor material so that the exciton dissociates. For an efficient dissociation, the energy gain of the electron by moving into the acceptor material should be larger than the exciton binding energy Eb. Such bilayer solar cells achieved a power conversion of about 1%.6 Bilayer solar cells had the following deficiency: The only way for excitons to dissociate was to reach the donor−acceptor (DA) interface. However, the exciton diffusion length in disordered polymers and small molecules used for photovoltaic applications is about 10 nm, much lower than the sample thickness of about 100 nm, which is necessary for an efficient light absorption. Therefore, only a small fraction of excitons created close to the DA interface can profit from the large energy gain, transferring their electrons into the acceptor material. In 1995, a novel concept was introduced accounting for the low exciton diffusion length in disordered organic semiconductors, the so-called bulk heterojunction solar cell.1,7 In such systems, donor and acceptor materials interpenetrate one another so that the internal DA interface is much more developed than that in planar DA systems. This DA interpenetration is usually achieved by spin coating a polymer−fullerene blend or by co-evaporation of conjugated small molecules.1 The most studied materials for the bulk heterojunction solar cells are the donor−acceptor combination poly(3-hexylthiophene) (P3HT)−[6,6]-phenyl-C61 butyric acid methyl ester (PCBM) and copper phthalocyanine-C60 for solution-processed and thermally evaporated solar cells, respectively.1 In such systems, excitons can dissociate at the developed DA interface over the whole extent of the solar cell active layer. Organic devices based on blends of conjugated polymers and fullerene derivatives are currently considered as the most promising candidates for photovoltaic applications.1,3,8,9 As described above, the photocurrent in such conjugated polymer−fullerene blends is dominated by the dissociation efficiency of bound EHPs at the DA interfaces.9 There is, however, no consensus among researchers on the very basic physics of the dissociation process. Some experiments indicate that excitons generated in the bulk of the polymer (electron donor) diffuse to the polymer−fullerene interface, where electrons are captured by the fullerene molecules (electron acceptor) while holes stay within the polymer material.10 Other experiments evidence an energy transfer from the polymer to the fullerene so that excitons generated in the polymer material are first transferred into the fullerene due to Förster processes, and only afterward do holes return into the polymer material via the polymer−fullerene interface.11 In any of these scenarios, once the exciton has been dissociated with the help of a suitable acceptor material, the electron and hole reside on the acceptor and donor components, respectively, but are still bound by the Coulomb force.1,3 Estimating the separation between the charges as r = 0.6 nm,12 an interaction energy of Eb ≈ 0.6 eV results from eq 1, taking εr ≈ 3. This binding energy represents a large energetic barrier to charge generation at the DA interface.1,3 Because the thermal energy kT at room temperature is less than Eb by at least an order of magnitude, it is hardly possible that the dissociation of EHPs can happen due to a thermal activation over such a huge energy barrier. Despite numerous experimental and theoretical studies, the mechanism responsible for the efficient dissociation of EHPs at donor−acceptor organic interfaces has not yet been revealed. Bisquert and Garcia-Belmonte13 suggested that quasi-free carriers present in the system can strongly shield the Coulomb interaction within EHPs, enhancing the dissociation efficiency.

Despite numerous experimental and theoretical studies, the mechanism responsible for the efficient dissociation of EHPs at donor−acceptor organic interfaces has not yet been revealed. Using Boltzmann statistics, they estimated the Debye screening length and found that at concentrations of quasi-free carriers above n ≈ 1017 cm−3, which is the carrier density that was found under steady-state illumination in several bulk heterojunction solar cells,13 the screening effects can become essential for separations within EHPs above 10 nm. Because these separations are often estimated to be smaller than 10 nm, for instance, ∼1 nm,3,12 the screening by quasi-free carriers needs to be studied in more detail before their effect on the dissociation efficiency is finally clarified. Recent calculations by Isaacs et al.14 in the framework of density functional theory showed a very promising way to come closer to understanding the dissociation problem in detail. Considering particular orientations between the fullerene molecule and the donor molecules, Isaacs et al. were able to estimate the interaction energy within the dissociating EHP. However, the physical mechanism governing the dissociation of EHPs has yet to be clarified. Clarke and Durrant3 emphasized the importance of the entropy factor for the dissociation process of EHPs, pointing at higher entropy for the independent electron and hole as compared to that of the bound EHP. The entropy term in the consideration of Clarke and Durrant diminishes essentially the activation barrier for dissociation of EHPs.3 Surely, this factor does play an essential role in the dissociation process. One should however keep in mind that this factor is inherently present in the Onsager theory. The latter has nevertheless been proven as unable to fit the results obtained by straightforward computer simulations.15,16 Therefore, more study is necessary to clarify the role of entropy terms for the dissociation problem. We discuss several possible solutions for the efficient dissociation without being able to finally resolve the problem. The simplest solution of the problem would be if the problem did not exist at all. So far, nobody has considered such an option. We should keep in mind, however, that the problem how to explain efficient dissociation at internal organic interfaces becomes so dramatic only if one takes a low value of the dielectric constant, εr ≈ 3 in eq 1, for granted. Although reported values of εr between 2 and 4 measured in organic semiconductors3 are indeed low, it is not necessarily true that such a low value for εr is valid in the vicinity of the internal interface between two organic materials. It has been shown in numerous experimental and theoretical studies that the dielectric constant in a mixture of two materials diverges in the vicinity of the percolation threshold.17−21 If the dielectric constant in the vicinity of the internal DA interface diverges, it could be the reason for the efficient screening of the Coulomb attraction within the dissociating EHP. If so, there would be no puzzling problem of a huge Coulomb energy barrier. One should, however, study to which extent the concept of the divergent dielectric constant can be applied to dissociation of close EHPs at DA organic interfaces. According to the concept of a divergent dielectric constant,17−21 two materials building the mixture should have essentially different conducting properties, which is probably true for organic DA blends. On the other 1215



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with respect to electrons captured in the fullerene (acceptor), thus reducing the Coulomb attraction within EHPs. This approach was studied by computer simulations assuming a homogeneous smearing of holes along polymer chains.16 An alternative approach was suggested by Arkhipov et al. based on the assumption that a dipolar layer can be formed in the dark at a polymer/acceptor interface.12 Below, we present theoretical calculations within the modified dipolar model and also within the model without dipoles, though beyond the assumption of the homogeneous smearing of charges. For both approaches, we study the dependencies of the dissociation probability on material parameters and on the electric field at the DA interface. Following Arkhipov et al.,12 we consider the polymer material as a set of one-dimensional chains parallel to the DA interface. We place the chains equidistantly with separation r between the neighbors, as shown in Figure 1. The chains are

hand, a divergent dielectric constant is essentially a macroscopic property of a system in the vicinity of the percolation threshold.17−21 One should study to what extent the DA organic blend fulfills the demand of closeness to the percolation threshold. Furthermore, even if the divergence of the dielectric constant in organic blends was confirmed, it should be further studied at which spatial scales one could use the high values of εr. The separation between the electron and hole within the dissociating pair is on the order of 1 nm.12 The dielectric constant εr could hardly diverge at such a microscopic scale. However, the value of εr in the vicinity of the organic DA interface could be essentially higher than εr ≈ 3. In such a case, the binding energy within the dissociating EHP would be much less than the estimated value Eb ≈ 1 eV,1,3,22 and thermal activation at room temperatures over a barrier with a small height would not be as puzzling. This route to the dissociation problem of EHPs at organic DA interfaces needs to be studied first before any further conclusions can be made. Because such studies are not yet available, we follow below a traditional approach1,3,22 of using small εr values and concomitantly large binding energies Eb within the dissociating EHPs and discuss possible mechanisms that could provide efficient dissociation under such unfavorable conditions. Onsager theory,23,24 is a commonly used approach for considering the dissociation of EHPs in organic disordered semiconductors. It has been applied to describe the dissociation of excitons as well as polaron pairs.1,3 Onsager calculated the probability that a pair of oppositely charged ions separated by some distance r0 in a weak electrolyte would escape recombination undergoing diffusive/drift motion under the combined influence of the mutual Coulomb attraction and external electric field F. The recombination event in the Onsager theory happens instantly as soon as the ions meet in space. This theory was extended in 1984 by Braun, who introduced the finite lifetime τ0 for the bound state of the two opposite charges and applied the theory to the dissociation of EHPs (called chargetransfer states).25 More recently, Wojcik and Tachiya26 further improved Braun’s approach with the finite lifetime for the recombination event, going beyond the assumption by Braun of dissociation and recombination following exponential kinetics. The Braun−Onsager theory is nowadays the most commonly applied approach for describing the EHP dissociation in organic DA blends.1,9,27 It has been shown, however, by straightforward computer simulations15,16 that the Onsager model fails to describe the dissociation of EHPs in disordered organic materials. One possible reason for the failure of the Onsager approach could be that the system of recombining EHPs is unable to achieve thermal equilibrium within the time scale of recombination kinetics in the energetically disordered environment typical for organic semiconductors.15 Deibel et al.16 suggested another reason for the failure of the Onsager description, namely, that the theory does not account for hopping transport and energetic or spatial disorder. Rubel et al. recently suggested a mathematically exact description for the dissociation of geminate EHPs taking into account hopping transport in systems with energetic and spatial disorder, though only in one-dimensional chains.28 Although the DA blends under consideration are three-dimensional systems, we will see below that essential models suggested so far to describe dissociation of EHPs in such systems allow a one-dimensional consideration. Being armed with this theoretical technique allows us turn to discussing the possible mechanisms of EHP dissociation. Deibel et al.16 suggested that delocalization of holes along polymer (donor) chains brings the holes to remote positions

Figure 1. Schematic view of the dissociation model and the corresponding energy diagram.

not necessarily parallel to each other. Holes are delocalized along the polymer chains.12,16 An exciton, optically generated in a polymer, can diffuse toward the interface and dissociate into a short geminate EHP. The electron is assumed to be immobile and tied to the acceptor separated by the distance r from the nearest polymer chain. The hole from the exciton can move via the system of polymer chains starting on the chain nearest to the interface.12,16 The motion of the hole is affected by the potential produced by the negative point charge −e of the electron and by a constant electric field F assumed directed perpendicular to the DA interface. We will numerate polymer chains by integer numbers starting from the chain nearest to the interface (chain number 1). The energy of the hole on the nth chain is determined by the one-dimensional Schrödinger equation29 −

2 ℏ2 d ψ (y) + Un(y)ψ (y) = Enψ (y) 2meff dy 2

(2)

Here, ψ(y) is the hole wave function, meff is the hole effective mass on the polymer chain, assumed equal for all chains, Un is the potential energy of the hole on the n th chain Un(y) = −

e2 4πεrε0 y 2 + xn2

− eFxn (3)

and xn is the distance from the electron to the nth chain. In our model 1216


The Journal of Physical Chemistry Letters xn = nr

(n = 1, 2, 3, ...)

Perspective

and kT = 0.025 eV were used for the calculations; the same values were used by Deibel et al.16 and Rubel et al.28 Figure 3

(4)

where r is the distance between neighboring chains, as well as the distance between the electron and the first chain. The hole energies on polymer chains along with the hole wave functions are exactly calculated by solving eqs 2−4 numerically. In contrast to the approach of Deibel et al.,16 the wave functions of holes are not assumed to be homogeneously smeared along the polymer chains, but they are rather calculated to be concentrated in a restricted space close to the electron, as schematically shown in Figure 2. It has been also shown that

Figure 3. Field dependence of the dissociation probability for EHPs at a DA interface for different hole effective masses meff given in the units of the free electron mass.29,31

shows that a dissociation efficiency close to unity, as found experimentally,32 can be obtained for realistic effective masses meff ≥ me only at high electric fields F > 107 V/m. However, experimental studies33,34 show that the electric field is not a determining parameter of the photogeneration flux in solar cells. Hence, one can conclude that the model considered above is not sufficient to explain the experimental data. Therefore, we turn below to the alternative model based on the assumption of a dipolar layer formed in the dark on the internal DA interface. Arkhipov et al.12 were the first who suggested a model for dissociation of EHPs at a DA interface based on the assumption that a dipolar layer can be formed at the interface. Recently, the existence of dipoles has been confirmed experimentally for the interfaces between fullerene and various polymers35 and theoretically for the interface between fullerene and pentacene.36 These results strongly support the model for dissociation of EHPs at the DA organic interfaces based on the existence of the dipolar layer. However, the theoretical study of the model by Arkhipov et al.12 led to the conclusion that exciton dissociation is energetically favorable only if the effective mass of a hole in the polymer material meff is smaller than ≃0.3 of the free electron mass me. Such a small meff value is unrealistic,37,38 causing general pessimism with respect to the model based on dipoles.3,37 Therefore, we consider below the model suggested by Arkhipov et al.12 in more detail. The model is shown schematically in Figure 4. It presumes that at least a partial charge separation occurs at the interface between a conjugated polymer and a fullerene molecule in the dark. This creates an array of discrete dipoles distributed regularly along the interface, so that the negative charges of the dipoles, −αe, are situated on the acceptor side of the interface and the positive charges of dipoles, +αe, are situated on the polymer chain nearest to the interface (chain 1). The magnitude α of the dipole strength is a model parameter. The potential energy V of the on-chain hole is determined by its interaction with the electron and with the partial charges of the dark dipoles. Although an on-chain hole is supposed to be in an extended state, it cannot move freely along the chain nearest to the interface, mainly because of the positive partial dipole charges residing on the same chain to the right and to the left of the hole.12 The dipoles are assumed to be immobile

Figure 2. Schematic view of the nonhomogeneous spread of the hole wave functions (shown in red) along the polymer chains.

these energies could be well estimated using a harmonic approximation.29 A hole placed initially on the chain closest to the DA interface can either perform a jump to the second-nearest chain with the hopping rate a1→2 or recombine with the electron captured on the fullerene acceptor within a characteristic lifetime τ0. If the hole performs a hop to the second chain, it can further continue the hopping motion via remote chains. The hopping rate depends on the energy levels of a hole on the corresponding chains. We imply the Miller−Abrahams expression30 for the hopping rate of a hole an→n±1 from the nth chain to chain n ± 1 ⎛ 2r En ± 1 − En + |En ± 1 − En| ⎞ an → n ± 1 = ν0 exp⎜ − − ⎟ ⎝ α ⎠ 2kT

(5)

Here, ν0 is the attempt-to-escape frequency, α is the localization length of the hole in the direction of the hopping transition, that is, perpendicular to the chains, k is the Boltzmann constant, and T is the temperature.30 The hole starts from the chain closest to the electron. We consider the EHP to be dissociated when the hole reaches the Nth chain, where N is a large enough number. In the calculations described below, N was taken to be so large that the results do not depend on N. The probability of dissociation p is calculated using the exact solution obtained by Rubel et al.28 τ0 p= E −E N − 1 −1 τ0 + ∑n = 1 an → n + 1 exp nkT 1 (6)

(

)

After substituting the exact hole energies into eqs 5 and 6, one obtains the dissociation probabilities p plotted as a function of electric field F for different values of the hole effective mass meff in Figure 3. The parameters τ0·ν0 exp[−2r/α] = 104 and εr = 3.5 1217



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chain to the DA interface). For calculations of the dissociation probability via eqs 5 and 6, the combination of parameters τ0·ν0 exp[−2r/α] = 104 and kT = 0.025 eV was taken to be equal to those of Deibel et al.16 and Rubel et al.28 As a result, one obtains the dissociation probabilities p plotted as a function of the electric field F for different values of the hole effective mass meff and for different values of the dipole strength α shown in Figure 5. It is clearly seen in Figure 5 that already at the dipole strength α = 0.2 and meff = me, the dissociation probability is close to unity in the whole range of considered electric fields, whereas it decreases fast with decreasing electric field without dark dipoles (α = 0) or without kinetic energy (meff = ∞). This result confirms the idea of Arkhipov et al.12 that the presence of dark dipoles at the DA interface decisively enhances the dissociation of EHPs. Furthermore, the data in Figure 5 evidence

Figure 4. Scheme of the model with discrete dipoles along the DA interface.

just as the optically generated electron after the latter is captured by the acceptor.12 This squeezing of the hole leads to quantization effects and supplies the hole with some kinetic energy Ek, which can be estimated using a harmonic approximation Ek ≅

ℏ2 ∂ 2V 2meff ∂y 2

(7)

where ℏ is Planck’s constant, meff is the hole effective mass along the polymer chain, and y is the coordinate along the polymer chain. Arkhipov et al.12 calculated in the harmonic approximation the energy E1 of the hole on the polymer chain nearest to the interface (chain 1) and the energy E2 of the hole on the secondnearest chain to the interface (chain 2). The condition E1 > E2 was considered as the dissociation criterion. In the calculations, only two dipoles adjacent to the dissociating EHP were considered. Furthermore, the distance from the photogenerated hole to the positive charges of the adjacent dark dipoles was considered equal to the interdipole distance a as if the photogenerated EHP is placed on the top of a dark dipole already existing at the DA interface.12 Our consideration differs from that of Arkhipov et al. in three respects.39 First, the photogenerated EHP is placed between the two already existing dipoles, which makes the distance between the hole and the adjacent positive charges of those dipoles equal to a/2. Second, in our calculation of the hole energies on various polymer chains, we take into account the Coulomb potentials from up to N = 100 dipoles to each side of the photogenerated EHP along the DA interface, not only from the two neighboring dipoles. Our calculations show that the results converge rapidly for N > 20. Third, we do not use the comparison between E1 and E2 as a dissociation criterion, but we rather calculate the hole energies on all chains and insert them into eqs 5 and 6 to obtain the dissociation probability p as a function of electric field F. Arkhipov et al.12 only compared the energy of the optically generated hole on the polymer chain nearest to the interface with its energy on the next-nearest chain and estimated the conditions under which it is energetically favorable for the hole to hop away. Following Arkhipov et al.,12 we used for our calculations of the hole energy parameters εr = 3 and a = r = 6 Å (distances between the dark dipoles are set equal to those between the adjacent chains and to that between the electron and the nearest

Figure 5. Field dependence of the dissociation probability for EHPs at a DA interface with a layer of discrete dipoles for different hole effective masses meff and for different dipole strengths α.31,39

that a high dissociation probability can be achieved in the theory assuming reasonable values of the hole effective mass in organic materials, meff ≈ me.

The presence of dark dipoles at the DA interface decisively enhances the dissociation of EHPs. The question arises on how important it is to consider dipoles as discrete species equidistantly distributed along the DA interface as suggested by Arkhipov et al.12 and considered above. In order to answer this question, we have investigated an alternative model in which the dipole charge is smeared homogeneously along the two chains, one in the donor material with a positive charge and one in the acceptor material with a negative charge. These chains are the ones adjacent to the DA interface.31 The chains are considered to be homogeneously charged cylinders, and therefore, a new model parameter, the cylinder radius R, has to appear in the final result, while the dipole strength parameter α from above can be kept as a measure for the charge density of cylinders. The model is shown schematically in Figure 6. It is worth noting that in the smeared-dipole model, no quantization effects similar to those arising in the case of discrete dipoles can appear. Therefore, the only reason for the high hole energy on the chain nearest to the interface, the 1218



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a high dissociation yield at reasonable electric fields of around 107 V/m can be reached only with hole effective masses meff ≤ 0.3me, which are too small to be assumed for organic materials due to the low electronic bandwidth in such systems.37−39 In our calculations, we considered the dissociation of EHPs using a one-dimensional analytic theory with the exact solution.28 The question might arise on how important was the one-dimensional structure of the dissociation path. As has been emphasized recently by Gregg,40 dissociation in two- or three-dimensional systems is always more efficient than that in one-dimensional systems. However, one should recognize that in our consideration, charge carriers are delocalized along the polymer chains within the conjugation length, that is, parallel to the DA interface, which provides a second dimension in addition to the dimension of the dissociation path perpendicular to the interface. This dimension in fact makes our consideration two-dimensional. Therefore, we do not expect much inaccuracy related to using the one-dimensional theory of Rubel et al.28 One can conclude that the results of the analytical theory given in this Perspectives show that delocalization of holes along polymer chains enhances the dissociation probability of EHPs at the interfaces between the conjugated polymer (electron donor) and a fullerene derivative (electron acceptors). This happens due to the quantization energy of holes on polymer chains. However, this effect by itself is not capable of explaining the very high dissociation efficiency at low electric fields seen in numerous experiments on polymer/fullerene blends. Additional effects such as, for instance, the formation of dark dipoles at the DA interfaces can play the decisive role for the dissociation process, as suggested by Arkhipov et al.12 Consequential calculations in the framework of Arkhipov’s model provide high values for the dissociation probability assuming a relatively low strength of dark dipoles, α ≈ 0.2, and realistic values of hole effective masses meff ≥ me for organic materials. Furthermore, the calculations show that while the model with discrete dipoles distributed equidistantly along the DA interface is capable of explaining the high efficiency of the dissociation processes, the model with continuous smearing of the dipole charge along the interface fails to account for high dissociation efficiency at reasonable material parameters. Finally, we highlight the following not yet resolved problems. The magnitude of the dielectric constant εr that governs the attraction between the electron and hole in the vicinity of the DA interface is unclear. Provided that the low value of εr is valid, as currently assumed by all researchers, the decisive problem is to clarify the properties of the dipolar layer along the DA interface because this layer provides the only reasonable yet suggested explanation for the efficient dissociation of EHPs at DA interfaces. Furthermore, one should critically analyze the theoretical tools presented in this Perspective. In particular, one should check to which extent the exact theory of dissociation in systems with energetic and spacial disorder developed so far only for one-dimensional systems28 is applicable to real threedimensional systems, even though the dissociation is treated as motion along the axis perpendicular to the DA interface. Moreover, the energies of the dissociating charges in the only promising model with discrete dipoles have been so far calculated only within the harmonic approximation.12,39 It is necessary to study the stability of these results with respect to removing the harmonic assumption. In any case, one needs more information about the morphology of the DA organic interfaces. As long as this information is missing, it remains

Figure 6. Scheme of the model with smeared dipoles along the DA interface.

condition favorable for dissociation of EHPs, can be a high potential energy due to repulsion of the positively charged hole by the positively charged polymer chain. Under such circumstances, the magnitude of the cylinder radius R plays the decisive role. The smaller the radius, the larger the potential energy of a hole on the polymer chain nearest to the DA interface and, concomitantly, the higher the dissociation probability for EHPs. In order to estimate the very possibility of obtaining considerable dissociation probabilities in the framework of this model, let us assume the smallest conceivable value for the cylinder radius, R = 1 Å. Calculations show that for such a small cylinder radius, the hole is almost homogeneously smeared along the polymer chains. This is in line with the assumption of Deibel et al.16 Calculating the hole energies on polymer chains as electrostatic energies caused by the homogeneously charged dipoles and also taking into account the interaction of the hole with the optically generated electron, eqs 5 and 6 yield the field dependencies of the dissociation probability shown in Figure 7 for a dipole strength

Figure 7. Field dependence of the dissociation probability for EHPs at a DA interface with a continuous layer of dark dipoles for different hole effective masses meff.31 Where not mentioned, the dipole strength is α = 0.1.

of α = 0.1. All material parameters were taken to be equal to those considered above for the model of discrete dipoles. Again, a considerable increase of the EHP dissociation probability is obtained as compared to the result without any interface dipoles and kinetic energy (α = 0, meff = ∞). However, 1219



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difficult to make reliable theoretical predictions with respect to the driving mechanisms for the efficient dissociation of EHPs at internal organic interfaces.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: baranovs@staff.uni-marburg.de. Notes

The authors declare no competing financial interest. Biographies Professor Sergei Baranovskii received his Ph.D. (1981) in Theoretical Physics from the Ioffe Physical-Technical Institute of the Russian Academy of Sciences in St. Petersburg, where he worked as a senior researcher until 1990. Since 1990, he has been working at the Philipps-Universität Marburg, Germany, where he got a Habilitation in Theoretical Physics in 1995. His research interests are devoted to charge transport and optical properties of organic and inorganic disordered solids. Martin Wiemer is a M.Sc. student at the Department of Physics at Philipps-Universität Marburg. He received his B.Sc. in 2010 with a study of the interface dipole-assisted exciton dissociation in organic solar cells. His current research interest is the scattering of charge carriers on a disorder potential in inorganic semiconductors. Dr. Alexey V. Nenashev is a senior researcher at Rzhanov Institute of Semiconductor Physics (Novosibirsk, Russia). He received his Ph.D. from the same institution in 2004. His research interests include electronic structure of semiconductor quantum dots, electronic transport in disordered systems such as quantum dot arrays, and organic materials. Dr. Fredrik Jansson is a postdoctoral researcher at the department of Physics at Philipps-Universität Marburg. He received the Ph.D. degree in 2011 from the Åbo Akademi University, Turku, for studying charge transport in disordered materials. Florian Gebhard has been professor of Physics at the Department of Physics of the Philipps-Universität Marburg since 1998. His research interests focus on phenomena caused by electron−electron interactions in solids such as magnetism of transition metals and Mott insulators. He also works on the microscopic theory of optical absorption and electronic transport in organic materials.

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ACKNOWLEDGMENTS The authors are indebted to Prof. V. Vinokur and Prof. T. Baturina for bringing their attention to the problem of divergent dielectric constant and to J. H. Grönqvist for providing the figures and illustrations. Financial support from the Deutsche Forschungsgemeinschaft and that of the Fonds der Chemischen Industrie is gratefully acknowledged.

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REFERENCES

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(29) Nenashev, A. V.; Baranovskii, S. D.; Wiemer, M.; Jansson, F.; Ö sterbacka, R.; Dvurechenskii, A. V.; Gebhard, F. Theory of Exciton Dissociation at the Interface Between a Conjugated Polymer and an Electron Acceptor. Phys. Rev. B 2011, 84, 035210. (30) Miller, A.; Abrahams, E. Impurity Conduction at Low Concentrations. Phys. Rev. 1960, 120, 745−755. (31) Nenashev, A. V.; Wiemer, M.; Jansson, F.; Baranovskii, S. D. Theory to Exciton Dissociation at the Interface Between a Conjugated Polymer and an Electron Acceptor. J. Non-Cryst. Solids 2012, DOI: 10.1016/j.jnoncrysol.2011.12.036. (32) Ö sterbacka, R.; Pivrikas, A.; Juška, G.; Poškus, A.; Aarnio, H.; Sliaužys, G.; Genevičius, K.; Arlauskas, K.; Sariciftci, N. S. Effect of 2-D Delocalization on Charge Transport and Recombination in BulkHeterojunction Solar Cells. IEEE J. Sel. Top. Quantum Electron. 2010, 16, 1738−1745. (33) Mauer, R.; Howard, I. A.; Laquai, F. Effect of Nongeminate Recombination on Fill Factor in Polythiophene/Methanofullerene Organic Solar Cells. J. Phys. Chem. Lett. 2010, 1, 3500−3505. (34) Jamieson, F. C.; Agostinelli, T.; Azimi, H.; Nelson, J.; Durrant, J. R. Field-Independent Charge Photogeneration in PCPDTBT/ PC70BM Solar Cells. J. Phys. Chem. Lett. 2010, 1, 3306−3310. (35) Osikowicz, W.; de Jong, M. P.; Salaneck, W. R. Formation of the Interfacial Dipole at Organic−Organic Interfaces: C60/Polymer Interfaces. Adv. Mater. 2007, 19, 4213−4217. (36) Verlaak, S.; Beljonne, D.; Cheyns, D.; Rolin, C.; Linares, M.; Castet, F.; Cornil, J.; Heremans, P. Electronic Structure and Geminate Pair Energetics at Organic−Organic Interfaces: The Case of Pentacene/C60 Heterojunctions. Adv. Funct. Mater. 2009, 19, 3809− 3814. (37) Peumans, P.; Forrest, S. R. Separation of Geminate Charge-Pairs at Donor−Acceptor Interfaces in Disordered Solids. Chem. Phys. Lett. 2004, 398, 27−31. (38) Meng, H. F.; Lai, C. M. Translationally Invariant Polarons in Conjugated Conducting Polymers. Phys. Rev. B 1997, 55, 13611− 13617. (39) Wiemer, M.; Nenashev, A. V.; Jansson, F.; Baranovskii, S. D. On the Efficiency of Exciton Dissociation at the Interface Between a Conjugated Polymer and an Electron Acceptor. Appl. Phys. Lett. 2011, 99, 013302. (40) Gregg, B. A. Entropy of Charge Separation in Organic Photovoltaic Cells: The Benefit of Higher Dimensionality. J. Phys. Chem. Lett. 2011, 2, 3013−3015.

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Calculating the Efficiency of Exciton Dissociation at the Interface

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