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J. Phys. Chem. C 2010, 114, 6808–6813
Unified Theory of Geminate and Bulk Electron-Hole Recombination in Organic Solar Cells Maria Hilczer*,† and M. Tachiya*,‡ Institute of Applied Radiation Chemistry, Technical UniVersity of Lodz, Wroblewskiego 15, 93-590 Lodz, Poland, and National Institute of AdVanced Industrial Science and Technology (AIST), AIST Tsukuba Research Center, Central 5, Tsukuba, Ibaraki 305-8565, Japan ReceiVed: December 30, 2009; ReVised Manuscript ReceiVed: February 28, 2010
Geminate and bulk charge recombination in organic solar cells are extensively studied because they are major loss processes for cell efficiency. It was recently found that the observed bulk charge recombination rate constants in organic solar cells are more than 3 orders of magnitude smaller than those predicted from Langevin theory. In order to resolve this discrepancy, we develop a unified theory of geminate and bulk charge recombination by taking into account that recombination between an electron and a hole occurs at a nonzero separation with a finite intrinsic recombination rate. On the basis of this theory we successfully explain the observed results on bulk recombination rate constants. We also analyze how recombination between electrons and holes in geminate and bulk recombination phases is affected by various physical parameters, such as the intrinsic recombination rate, the diffusion coefficients of electrons and holes, the Onsager radius, and an external electric field, and show how they are optimized to suppress recombination loss. I. Introduction Organic solar cells are highly expected to be part of the next generation of solar cells and have recently been studied extensively. The processes occurring in organic solar cells are summarized as follows.1 Absorption of photons by the working mediumgeneratesexcitons.Theexcitonsdiffusetoadonor-acceptor interface. Geminate recombination phase starts when electron transfer from an exciton to an acceptor at the interface generates an electron and a hole which are separated by a certain distance. They recombine with a finite rate at this distance or get separated by Brownian motion under the influence of mutual Coulomb potential and an external electric field. Some fraction of them comes back to the original distance after separation to certain distances. Among that fraction some fraction recombines and the other gets separated again. This is repeated many times. When some fraction already recombines and the other ultimately gets separated to sufficiently long distances, geminate recombination phase ends and bulk recombination phase starts. In bulk recombination no electron and hole are paired. Each electron can recombine with any hole indiscriminately, regardless of whether they come from the same pair or different pairs. In the meantime the electrons and holes move to the respective electrodes. The bulk recombination phase ends when the electrons and holes reach the respective electrodes. Recombination between electrons and holes is a loss process for the efficiency of organic solar cells, whether it occurs in geminate recombination phase or in bulk recombination phase. So it is very important to suppress geminate and bulk charge recombination in organic solar cells. Geminate charge recombination is analyzed on the basis of Onsager theory,2 while bulk charge recombination is analyzed on the basis of Langevin theory.3 It was recently found4–9 that observed bulk recombination constants are more than 3 orders of magnitude smaller than those predicted from Langevin theory. Deibel et al.9 tried to * To whom correspondence should be addressed,
[email protected] and
[email protected]. † Technical University of Lodz. ‡ AIST.
explain this discrepancy by taking into account inhomogeneous and different concentration distributions of electrons and holes between two electrodes. Jusˇka et al.7 invoked two-dimensional Langevin theory to explain the discrepancy. Groves and Greenham10 considered the effects of the constraint of electrons and holes to their respective transport phases and the energetic disorder of the phases. However, the reason for this enormous discrepancy between theory and experiment still remains unresolved.11–13 In geminate charge recombination only one electron and one hole are involved in recombination, while in bulk charge recombination a practically infinite number of electrons and holes are involved. However, the way in which an electron and a hole recombine after they approach each other to a certain distance is essentially the same, whether it is geminate recombination or bulk recombination. Concerning this point, both Onsager and Langevin theories assume that recombination between an electron and a hole occurs when their separation is zero. For geminate recombination it is known for many years that the observed escape probabilities are much higher than those predicted from Onsager theory. This discrepancy is due to the inappropriate assumption in Onsager theory that recombination between an electron and a hole occurs when their separation is zero. Now geminate recombination is analyzed on the basis of the extended Onsager theory which was originally developed by Sano and Tachiya14 and assumes that recombination between an electron and a hole occurs at a nonzero separation with a finite intrinsic recombination rate. This suggests that the discrepancy between theory and experiment on bulk recombination rate constants may also be due to the inappropriate assumption in Langevin theory that recombination between an electron and a hole occurs when their separation is zero. In this paper we present a unified theory of geminate and bulk charge recombination and extend Langevin theory in the same way as Onsager theory was extended. By use of the extended Langevin theory, we resolve the enormous discrepancy between theory and experiment on bulk recombination rate constants. On the basis of the unified theory we also analyze
10.1021/jp912262h 2010 American Chemical Society Published on Web 03/23/2010
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how recombination between electrons and holes in geminate and bulk recombination phases is affected by various physical parameters such as the intrinsic recombination rate, the diffusion coefficients of electrons and holes, the dielectric constant of the medium, the temperature, and the strength of an external electric field and show how they are optimized to suppress recombination loss in organic solar cells. In section II a unified theory of geminate and bulk charge recombination is presented, and Langevin theory of bulk charge recombination is extended to the case in which recombination between an electron and a hole occurs at a nonzero separation with a finite intrinsic recombination rate. In section III the discrepancy between Langevin theory and observed bulk recombination rate constants is resolved by use of the extended Langevin theory. We also analyze how recombination between electrons and holes in geminate and bulk recombination phases is affected by various physical parameters and show how they are optimized to suppress recombination loss. The paper concludes with section IV. II. Unified Theory A. Geminate Charge Recombination. In geminate charge recombination an electron and a hole, which are initially separated by r, perform Brownian motion and recombine, if they meet. One important quantity in geminate recombination is the probability φ(r) that the pair will not recombine for ever. This probability is called the escape probability. In 1938 Onsager2 studied geminate charge recombination in the presence of an external electric field and calculated the escape probability by assuming that recombination between an electron and a hole occurs when their separation is zero. This assumption leads to a very low escape probability. So, if Onsager theory is applied to explain the observed escape probability, it fails. To reconcile Onsager theory with the observed results, it is occasionally speculated that a charge pair is initially formed with an excess energy which separates the pair to a long distance before geminate recombination starts. However, it is now established15 that in most organic photoconductors the quantum yield for carrier generation is independent of excitation wavelength. The reason Onsager theory fails to explain the observed results is due to its unrealistic assumption that recombination occurs when the distance between an electron and a hole is zero. In organic solar cells it is believed that an electron and a hole recombine at a nonzero separation with a finite intrinsic recombination rate. Braun16 tried to extend Onsager theory to the case in which an electron and a hole recombine at a nonzero separation with a finite intrinsic recombination rate. However, the exact extension of Onsager theory to this case is a very difficult problem, especially in the presence of an external electric field. So Braun extended it in a very empirical way. He considered the escape probability for an electron and a hole which are initially at the contact distance. He assumed that recombination occurs only at the contact distance R. He further assumed that the escape probability φ(R) is given by the ratio of the separation rate constant ks(R) to the sum of the separation rate constant and the recombination rate constant kr(R)
can be defined and the escape probability is given by eq 1. However, this is not the case. Braun theory assumes that an electron and a hole perform Brownian motion instead of ballistic motion,17 as seen clearly from the fact that the sum of diffusion coefficients of electron and hole appears in his theory. If they perform Brownian motion, it is now established14 that the separation kinetics follows t-1/2 law at long times. In other words, in geminate recombination the first-order separation rate constant cannot be defined. So Braun used for ks(R) an equation Onsager18 derived under the condition that practically infinite numbers of electrons and holes, instead of just one electron and one hole, are present in the system and that the separation rate and the recombination rate are balanced. Although that equation itself is a correct equation under this condition, it does not hold under the condition that just one electron and one hole are present in the system. In the latter condition a steady state is never established. Several years before Braun’s work Sano and Tachiya14 correctly extended Onsager theory to the case in which an electron and a hole recombine at a nonzero separation R with a finite back electron transfer rate kBET. For the purpose of the present paper, it is more convenient to consider the recombination probability κ(r) instead of the escape probability, which is defined by κ(r) ) 1 - φ(r). They have shown that the recombination probability satisfies the following differential equation and the associated boundary conditions
(
D
(1)
If the separation rate and the recombination rate both follow first-order kinetics, the first-order rate constants ks(R) and kr(R)
(2)
|
) -p[1 - κ(R)]
(3)
r)R
lim κ(r) ) 0
(4)
rf∞
where D is the sum of the diffusion coefficients of the electron and the hole, V(r) is the sum of the Coulomb potential and the potential due to the external electric field E. The intrinsic reactivity parameter p is related to the back electron transfer rate kBET between an electron and a hole separate by R through p ) kBETR. At low external electric fields the angularly averaged recombination probability is calculated from eqs 2-4 as
(
κ(r, E) ) κ(r, E ) 0) 1 + a1
(
-rc/R
1-e
1-
(
-e-rc/r + e-rc/R 1 -rc/r
(1 - e
eErc 2kBT
1 - e-rc/r
κ(r, E ) 0) )
a1 ) ks(R) φ(R) ) ks(R) + kr(R)
∂κ ∂r
)
1 ∇ν∇κ ) 0 kBT
D ∇2κ -
[
-rc/R
)1-e
Drc pR2
Drc
)
) )
pR2 Drc 1pR2
(
(5)
(6)
)]
(7)
Here rc is the Onsager radius and given by rc ) e2/(4πε0εkBT) with ε as the dielectric constant of the medium. If we put R ) 0 in eqs 5-7, we recover the Onsager result2
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(
κO(r, E) ) (1 - e-rc/r) 1 -
e-rc/r eErc 1 - e-rc/r 2kBT
Hilczer and Tachiya
)
(8)
Figure 1a shows the recombination probability κ(R,E) as a function of the external electric field E at low electric fields for different values of Drc/pR2 in the case of ε ) 4, R ) 10 Å, and T ) 300 K. The intrinsic reactivity parameter p can be changed by orders of magnitude by changing the combination of donor and acceptor or by changing the temperature. According to Figure 1a, the recombination probability changes by orders of magnitude by changing the value of Drc/pR2. The change of the recombination probability with the external electric field is not so great. Wojcik and Tachiya19 have shown that the escape probability calculated by using the Braun model is quite different from the exact one calculated by using eqs 2-4. They have also shown that by modifying the Braun model it is possible to obtain the result which agrees with the exact one except at very high electric fields. Their modified equation should be very useful because it is very simple but gives accurate results except at very high electric fields. B. Bulk Charge Recombination. In bulk recombination practically infinite numbers of electrons and holes which are present in the system with concentrations ce and ch perform Brownian motion and recombine, if they meet. In bulk recombination the recombination rate is described by the following equation
dce dch ) ) -kcech dt dt
an external electric field. The theoretical foundation of the SCK approach is established in ref 22. In the SCK approach the bulk rate constant k is calculated from ref 23
(
D∇ ∇w +
(
D
∂w w ∂ν + ∂r kBT ∂r
Figure 1. The recombination probability κ(R, E) (a) and the ratio of the bulk rate constant k(E) to the Langevin rate constant kL (b) as functions of the external electric field E at low electric fields for different values of Drc/pR2 in the case of ε ) 4, R ) 10 Å, and T ) 300 K.
(10)
)|
) pw(R)
(11)
r)R
lim w(r) ) 1
(12)
rf∞
k ) 2πDR2
∫0π
(
∂w w ∂ν + ∂r kBT ∂r
)|
sin θ dθ
(13)
r)R
where w(r) is the concentration distribution of electrons around a hole put at the origin. At low external electric fields the bulk rate constant k(E) is calculated as23
(
k(E) ) k(E ) 0) 1 + b1
(
-rc/R
1-e
(
1-
b1 )
eErc 2kBT
)
4πDrc
k(E ) 0) )
(9)
where k is the bulk rate constant. Geminate recombination is characterized by the recombination probability, while bulk recombination is characterized by the bulk rate constant. In 1903 Langevin3 studied bulk charge recombination and calculated the bulk rate constant by assuming that recombination between an electron and a hole occurs when their separation is zero. Following the Smoluchowski-Collins-Kimball (SCK) approach,20,21 we can extend Langevin theory to the case in which an electron and a hole recombine at a nonzero separation R with a finite intrinsic recombination rate in the presence of
)
w ∇ν ) 0 kBT
Drc pR2
-rc/R
1-e
(
)
1-
Drc pR2
(14)
)
(15)
e-rc/R
1-
Drc pR2
)
(16)
In the absence of an external electric field, the bulk rate constant is given by eq 15. If we put R ) 0 in eq 15, we recover the Langevin result3
kL ) 4πDrc
(17)
According to eq 16, if Drc/pR2 < 1, b1 is positive, so the bulk rate constant increases with increasing E, while if Drc/pR2 > 1, the bulk rate constant decreases with increasing E. Figure 1b shows the ratio of the bulk rate constant k(E) to the Langevin rate constant kL as a function of the external electric field E at low electric fields for different values of Drc/pR2 in the case of ε ) 4, R ) 10 Å, and T ) 300 K. On comparison of Figure 1a and Figure 1b, the dependences of κ(R, E) and k(E)/kL on the external electric field E look very similar. However, on comparison of eqs 5-7 and eqs 14-16 precisely, the exact dependences of κ(R, E) and k(E)/kL on E are different. When Drc/pR2 ) 0, according to eqs 5-7, κ(R, E) decreases with increasing E, while, according to eqs 14-16, k(E)/kL increases with increasing E. However, this difference is not discernible from Figures 1a and 1b because the decrease of κ(R, E) and the increase of k(E)/kL with increasing E are both extremely small in the case of Drc/pR2 ) 0. Onsager stated in ref 18 that the bulk rate constant is not affected by an external electric field. This conclusion is different from ours. The reason Onsager obtained this conclusion is because he assumed that recombination between
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J. Phys. Chem. C, Vol. 114, No. 14, 2010 6811
an electron and a hole occurs when their separation is zero. In fact, if we put R ) 0 in eqs 14-16, we have the same conclusion as Onsager’s. C. Relation between Geminate and Bulk Charge Recombination. As we already mentioned, the way in which an electron and a hole recombine after they approach each other to a certain distance is essentially the same, whether it is geminate recombination or bulk recombination. Therefore the recombination probability and the bulk rate constant should be related. In the absence of an external electric field this relation can be derived in the following way. Let us introduce the rate constant kfe with which an electron and a hole approach each other to a distance R for the first time. The subscript fe stands for first encounter. The bulk rate constant k is obtained by multiplying this rate constant by the probability that an electron and a hole which are initially separated by R will ultimately recombine, namely, the recombination probability κ(R)
k ) kfeκ(R)
(18)
kfe can be calculated in the following way. Assume that recombination occurs at a separation R with an infinite intrinsic recombination rate (p ) ∞). In this case, once an electron and a hole approach each other to the distance R for the first time, they immediately recombine. Therefore kfe is equal to the bulk rate constant k for p ) ∞, so we have from eq 15
kfe )
4πDrc
(19)
1 - e-rc/R
Substitution of eq 19 in eq 18 yields the following relation between k and κ(R)
4πDrc k ) κ(R) 1 - e-rc/R
(20)
III. Applications of the Unified Theory A. Analysis of Observed Bulk Recombination Rate Constants. Here we explain, on the basis of the theory developed in section II, why observed bulk rate constants in organic solar cells are more than 3 orders of magnitude smaller than those predicted from Langevin theory. According to eq 15, the bulk rate constant in the absence of an external electric field increases with increasing p and has the following maximum value at p ) ∞
kp)∞(E ) 0) )
4πDrc 1 - e-rc/R
In organic solar cells the dielectric constant ε of the medium is about 4, so the Onsager radius rc is about 150 Å. If the reaction radius R is about 10 Å, e-rc/R in eq 23 is negligible. Therefore the maximum value given by eq 23 is practically equal to the Langevin rate constant kL. This indicates that observed bulk rate constants should be in general smaller than the Langevin rate constant. The fact that the observed bulk rate constants are much smaller than the Langevin rate constant indicates that in those systems p is very small. In order for recombination to occur, an electron and a hole have to first approach each other by diffusion. This approach takes some time. After they approach, they recombine. This recombination also takes some time. When p is very small, recombination is very slow. In this case it takes more time for recombination than for approach by diffusion. In other words, the overall recombination is controlled not by diffusion but by recombination. Jusˇka et al.6 and Deibel et al.8 found that the ratio of the observed bulk rate constant to the Langevin rate constant increases with decreasing temperature. This result may be explained in the following way. Using eqs 15 and 17, the ratio of the observed bulk rate constant k to the Langevin rate constant kL in the absence of an external electric field is given by
k ) kL
1 -rc/R
1-e
Drc
+
pR2
24
Koster et al. noticed that the bulk rate constant and the escape probability should be related. They claimed that the bulk rate constant k is expressed in terms of the escape probability φ(R) as
k ) [1 - φ(R)]kL
(21)
where kL is the Langevin rate constant. If we use eq 17 and the relation κ(r) ) 1 - φ(r), we find that their equation is different from the exact one by a factor of 1/(1 - e-rc/R). In the presence of an external electric field a simple relation between k(E) and κ(R, E) such as eq 20 does not hold. In this case the flux D[∂w/∂r + (w/kBT)∂V/∂r]r)R in eq 13 and the recombination probability κ(R, E) are both angular dependent, and the recombination rate constant k(E) is expressed as
k(E) ) 2πDR2
(
w ∂ν + ∫0π κ(R, E) ∂w ∂r kBT ∂r
)|
sin θ dθ r)R
(22)
where w is the solution of eqs 10-13 for p ) ∞ and is angular dependent.
(23)
(24) -rc/R
e
If the temperature dependences of the diffusion coefficient D and the intrinsic reactivity parameter p are given by
( ) ∆ED kBT
(25)
( )
(26)
D ) D∞ exp -
p ) p∞ exp -
∆Ep kBT
respectively, the temperature dependence of D/p is given by
( )
D ∆E ) A exp p kBT
(27)
where A ) D∞/p∞ and ∆E ) ∆ED - ∆Ep. We have calculated the temperature dependence of k/kL on the basis of eqs 24 and 27 by taking A and ∆E as adjustable parameters. In Figure 2 the theoretical results are compared with the experimental data by Jusˇka et al.6 (a) and by Deibel et al.8 (b). The values of the
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Figure 2. Temperature dependence of the ratio of the observed bulk rate constant k to the Langevin rate constant kL obtained by Jusˇka et al.6 (a) and Deibel et al.8 (b). The theoretical curve given by eqs 24 and 27 was fitted to the experimental data by taking A and ∆E as adjustable parameters. The values of the other parameters used in the calculation are ε ) 4 and R ) 10 Å.
parameters obtained are A ) 8.76 × 104 Å and ∆E ) -0.243 eV for the data by Jusˇka et al. and A ) 2. 42 × 104 Å and ∆E ) -0.239 eV for those by Deibel et al. It is interesting to note that the values of ∆E obtained for two different data sets are almost equal and that the values of A have the same order of magnitude. The negative values of ∆E indicate that the activation energy of the intrinsic reactivity parameter p is larger than that of the diffusion coefficient D. If the activation energy of the diffusion coefficient is about 0.11 eV,25 the activation energy of p is estimated as 0.35 eV. As already mentioned, p/R stands for the back electron transfer rate kBET between an electron and a hole separated by R. According to the Marcus theory, the activation energy of the electron transfer rate depends on the free energy change of reaction and the reorganization energy. The activation energy of 0.35 eV is not unusual for the electron transfer rate. If the mobility of charges is 10-8 m2/(V s) at room temperature, the back electron transfer rate is estimated as 10 s-1. This rate is significantly slower compared with the lifetime of the contact charge transfer state observed by Veldman et al.26 However, it should be pointed out that the contact charge transfer state decays not only by back electron transfer but also by hopping of the electron or the hole to a next acceptor or a donor molecule. If the latter process occurs on the nanosecond time scale, the observed lifetime of the contact charge transfer state should be in the nanosecond range. We should also add that in the present analysis of experimental data the effect of hetrojunction is not taken into account. In both panels of Figure 2 systematic deviation between theory and experimental data can be observed. A part of this deviation may be due to the effect pointed out in ref 9. B. Dependences of Charge Recombination Loss in Organic Solar Cells on Various Physical Parameters. The recombination probability and the bulk rate constant are affected by various physical parameters such as the back electron transfer rate, the diffusion coefficients of electrons and holes, the dielectric constant of the medium, and the temperature. Now let us consider on the basis of eqs 5-7 and eqs 14-16 how
Hilczer and Tachiya these parameters are optimized in the absence of an external electric field to suppress recombination loss in organic solar cells. In organic solar cells the recombination probability for an electron and a hole which are initially at the contact distance R is important, so here we consider only that. When the intrinsic reactivity parameter p which is given by p ) kBETR in terms of the back electron transfer rate kBET and the contact distance R is increased from zero to infinity, the recombination probability monotonically increases from 0 to 1, and the bulk rate constant also monotonically increases from 0 to 4πDrc/(1 - e-rc/R). Therefore the recombination loss is suppressed by reducing p. When the diffusion coefficient D is increased from zero to infinity, the recombination probability monotonically decreases from 1 to 0, while the bulk rate constant monotonically increases from 0 to 4πpR2erc/R. Therefore, we have a trade-off concerning the value of D. When the Onsager radius rc which is given by rc ) e2/(4πε0εkBT) in terms of the dielectric constant ε is increased from zero to infinity, the recombination probability monotonically increases from 1/(1 + D/pR) to 1, and the bulk rate constant also monotonically increases from 4πDR/(1 + D/pR) to infinity. Therefore, the recombination loss is suppressed by reducing rc. Summarizing the above results, in the absence of an external electric field the recombination loss is suppressed by reducing the intrinsic reactivity parameter p and the Onsager radius rc, and by taking an optimal value for the diffusion coefficient D. In our theory the effect of the dielectric constant of the medium is included in the Onsager radius, while the effect of the temperature is included in the Onsager radius, the intrinsic reactivity parameter and the diffusion coefficient. Now let us consider how the external electric field is optimized to suppress recombination loss. The effect of an external electric field E on the recombination probability is described by the second term in the parentheses on the righthand side of eq 5. Since a1 in eq 7 is always negative, the recombination probability decreases with increasing E. However, according to Figure 1a, this decrease is negligible in magnitude, if Drc/pR2 e 103. Concerning the bulk rate constant, we have already pointed out that if Drc/pR2 < 1, the bulk rate constant increases with increasing E, while if Drc/pR2 > 1, it decreases with increasing E. However, according to Figure 1b, the dependence of the bulk rate constant on E is practically negligible in magnitude if Drc/pR2 e 103. Summarizing the above results, the effects of an external electric field on the recombination probability and the bulk rate constant are both negligible, if Drc/pR2 e 103, while if Drc/pR2 > 103, the external electric field decreases both the recombination probability and the bulk rate constant. Therefore, the recombination loss is suppressed by increasing the field strength E, if Drc/pR2 > 103, but practically not affected by E, otherwise. IV. Concluding Remarks In this paper we have developed a unified theory of geminate and bulk charge recombination. On the basis of this, we have analyzed recent experimental results on bulk charge recombination rate constants in organic solar cells and explained why the observed recombination rate constants are more than 3 orders of magnitude smaller than those predicted from Langevin theory and why the ratio of the former to the latter increases with decreasing temperature. By using the unified theory we have also analyzed how recombination between electrons and holes in geminate and bulk recombination phases is affected by various physical parameters such as the back electron transfer rate, the diffusion coefficients of electrons and holes, the dielectric constant of the medium, the temperature, and the
Bulk Electron-Hole Recombination strength of an external electric field. We have shown that in the absence of an external electric field the recombination loss is suppressed by reducing the intrinsic reactivity parameter p and the Onsager radius rc, and by taking an optimal value for the diffusion coefficient D. In our theory the effect of the dielectric constant of the medium is included in the Onsager radius, while the effect of the temperature is included in the Onsager radius, the intrinsic reactivity parameter, and the diffusion coefficient. In the presence of an external electric field, the recombination loss is suppressed by increasing the field strength E, if Drc/pR2 > 103, but practically not affected by E, otherwise. The theory developed in this paper should be useful to analyze charge recombination processes in organic solar cells and to optimize various physical parameters in order to suppress recombination loss. Charge recombination processes in organic solar cells are also affected by the constraint of electrons and holes to their respective transport phases10,12,27 and the energetic disorder of the phases.10 Theoretical work along this line is in progress. References and Notes (1) For review see, for example, papers in Acc. Chem. Res. 2009, 42 (11). (2) Onsager, L. Phys. ReV. 1938, 54, 554. (3) Langevin, P. Ann. Chim. Phys. 1903, 28, 433. (4) Pivrikas, A.; Jusˇka, G.; Mozer, A. J.; Scharber, M.; Arlauskas, K.; Sariciftci, N. S.; Stubb, H.; Osterbacka, R. Phys. ReV. Lett. 2005, 94, 176806. (5) Shuttle, C. G.; O’Regan, B.; Ballantyne, A. M.; Nelson, J.; Bradley, D. D. C.; Mello, J. D.; Durrant, J. R. Appl. Phys. Lett. 2008, 92, 093311.
J. Phys. Chem. C, Vol. 114, No. 14, 2010 6813 (6) Jusˇka, G.; Genevicius, K.; Nekrasas, N.; Sliauzys, G.; Dennler, G. Appl. Phys. Lett. 2008, 93, 143303. (7) Jusˇka, G.; Genevicius, K.; Nekrasas, N.; Sliauzys, G.; Osterbacka, R. Appl. Phys. Lett. 2009, 95, 013303. (8) Deibel, C.; Baumann, A.; Dyakonov, V. Appl. Phys. Lett. 2008, 93, 163303. (9) Deibel, C.; Wagenpfahl, A.; Dyakonov, V. Phys. ReV. B 2009, 80, 075203. (10) Groves, C.; Greenham, N. C. Phys. ReV. B 2008, 78, 155205. (11) Street, R. A. Appl. Phys. Lett. 2008, 93, 133308. (12) Maturova, K.; van Bavel, S. S.; Wienk, M. M.; J. Janssen, R. A. J.; Kemerink, M. Nano Lett. 2009, 9, 3032. (13) Pensack, R. D.; Banyas, K. M.; Asbury, J. B. J. Phys. Chem. C, DOI: 10.1021/jp905061y. (14) Sano, H.; Tachiya, M. J. Chem. Phys. 1979, 71, 1276. (15) Weiss, D. S.; Abkowitz, M. Chem. ReV. 2010, 110, 479. (16) Braun, C. L. J. Chem. Phys. 1984, 80, 4157. (17) For the theory of geminate recombination in the case when an electron and a hole perform ballistic motion, see: (a) Tachiya, M. Radiat. Phys. Chem. 1988, 32, 37. (b) Tachiya, M. J. Chem. Phys. 1988, 89, 6929. (c) Tachiya, M.; Schmidt, W. F. J. Chem. Phys. 1989, 90, 1471. For the theory of bulk recombination in this case, see:(d) Tachiya, M. J. Chem. Phys. 1987, 87, 4108. (18) Onsager, L. J. Chem. Phys. 1934, 2, 599. (19) Wojcik, M.; Tachiya, M. J. Chem. Phys. 2009, 130, 104107. (20) von Smolucowski, M. Z. Phys. Chem. 1917, 92, 129. (21) Collins, F. C.; Kimball, G. E. J. Colloid Sci. 1949, 4, 425. (22) Tachiya, M. Radiat. Phys. Chem. 1983, 21, 167. (23) Isoda, K.; Kouchi, N.; Hatano, Y.; Tachiya, M. J. Chem. Phys. 1994, 100, 5874. (24) Koster, L. J. A.; Smits, E. C. P.; Mihailetchi, V. D.; Blom, P. W. M. Phys. ReV. B 2005, 72, 085205. (25) Deibel, C. Private communication. (26) Veldman, D.; Ipek, O.; Meskers, S. C. J.; Sweelssen, J.; Koetse, M. M.; Veenstra, S. C.; Kroon, J. M.; van Bavel, S. S.; Loos, J.; Janssen, R. A. J. J. Am. Chem. Soc. 2008, 130, 7721. (27) Peumans, P.; Forrest, S. R. Chem. Phys. Lett. 2004, 398, 27.
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