Electric Field-Assisted Dissociation Yield of Bound Charge Pairs in

Feb 6, 2017 - (11) The solid lines were obtained from eq 26 and the dashed lines represent results from Braun's equation, as given by eq 30. The diffe...
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Electric Field-Assisted Dissociation Yield of Bound Charge Pairs in Low Permittivity Materials Kazuhiko Seki*,† and Mariusz Wojcik*,‡ †

Nanomaterials Research Institute (NMRI), National Institute of Advanced Industrial Science and Technology (AIST), Higashi 1-1-1, Tsukuba, Ibaraki 305-8568, Japan ‡ Institute of Applied Radiation Chemistry, Lodz University of Technology, Wroblewskiego 15, 93-590 Lodz, Poland ABSTRACT: The intrinsic recombination rate in low permittivity materials must be low to produce an appreciable dissociation yield. Previously, the electric field-assisted dissociation of the charge-transfer (CT) complex was studied using Braun’s approximate equation. We analyze this process using a mathematically strict theory and derive a more accurate analytical expression that reduces to Braun’s equation at low field strength. Our result is based on the assumption that, at a low intrinsic recombination rate, the charge distribution near the reaction radius is close to equilibrium. We also study other effects important for description of the electric fieldassisted dissociation yield, such as anisotropy of the initial separation rate and the effect of a large initial separation distance. We use our theoretical results to analyze experimental CT emission data.

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INTRODUCTION The photoelectric conversion efficiency of organic photovoltaic cells is lower than that of inorganic photovoltaic cells, although the former have several advantages, including softness, light weight, and flexibility.1 The photoelectric conversion efficiency of organic solar cells is limited by several factors, including the initial charge separation efficiency, the bound charge-pair dissociation efficiency, and the charge collection efficiency of the electrode.2 Among these factors, an increase in the dissociation efficiency is crucial to enhancement of the photoelectric conversion efficiency.1,2 In organic materials, the dissociation of the photogenerated charge pairs occurs against a strong Coulomb binding energy that originates from the low relative dielectric constants. Interestingly, efficient free carrier generation has been observed in organic photovoltaic cell materials, contrary to the behavior expected from theoretical models.1 This has aroused fundamental interest in the geminate charge separation mechanism in organic solids.1 This charge separation mechanism can be studied by applying external electric fields and measuring the field and temperature dependence of the dissociation yield.3−5 The fundamental theoretical models that have been used to study the electric field dependence of the dissociation yield and the dissociation constant date back to the works of Onsager. In 1934, Onsager solved the problem of Brownian motion under influence of an external electric field along with the Coulomb potential to determine the electric field dependence of the dissociation constant.6 The boundary conditions used were the equilibrium occupation probability in the vicinity of the origin at which one of the pair of charges is located and the zero population when the distance from the origin is infinite. This solution was used to study the geminate recombination of a pair of charge carriers for given initial separation and © 2017 American Chemical Society

orientation conditions with respect to the applied electric field.7 More precisely, the solution was transformed to satisfy the boundary conditions that represent perfect absorption at the origin and ultimate escape from recombination at r → ∞, where r denotes the distance from the origin. The extension of Onsager’s work to the case of absorbing boundary conditions at r = R > 0 is complex if the charge carrier distribution isotropy is broken by the external electric field, although a rigorous method is known for isotropic systems.8 In many organic solids, charge separation occurs in the charge-transfer (CT) complex that is formed after photoexcitation.9,10 The recombination radius can be associated with the size of the CT complex, and should thus be nonzero. Additionally, the recombination rate can be comparable to or even smaller than the rate of charge transfer from the CT complex to a surrounding molecule. On the basis of consideration of all these effects occurring in organic materials, the most appropriate boundary condition to represent the recombination and dissociation processes may be a partially reflecting boundary condition at a nonzero distance. Several theoretical approaches have been developed in attempts to extend Onsager’s works based on generalization of the boundary conditions.11−13 Among these works, the simplest and most widely used expression was obtained by Braun.12 Braun studied the dissociation of the CT complex, in which the charge separation distance is equal to the recombination radius and recombination occurs with a finite intrinsic rate. He considered the external field dependence of the dissociation yield, i.e., the Received: December 11, 2016 Revised: January 17, 2017 Published: February 6, 2017 3632

DOI: 10.1021/acs.jpcc.6b12470 J. Phys. Chem. C 2017, 121, 3632−3641

Article

The Journal of Physical Chemistry C

periods of random motion, the charge carriers lose their memory of the initial distribution. Using the formalism developed to analyze Braun’s theory, we derive a convenient analytical expression which better describes the field dependence of escape probability than Braun’s equation, and reduces to it in the weak field limit. We also study the dependence of escape probability on the initial separation distance of bound charge pairs (thermalization distance), and describe this dependence by a useful analytical formula. Finally, we apply our theoretical results to discuss the experimental CT emission data under an applied electric field.

escape probability of geminate pairs from recombination via diffusion under an external electric field. When Braun derived the geminate charge dissociation yield (or, in other words, escape probability) under the influence of an external electric field, both recombination and dissociation were described using their respective first-order rate constants.12 In principle, pair dissociation by diffusion is governed by nonexponential kinetics and the dissociation rate is timedependent. The assumption of exponential kinetics of the recombination and dissociation processes ignores the diffusive migration of carriers before ultimate dissociation. Recently, it was found from Monte Carlo simulations 14 that the recombination kinetics in the absence of an external field can be approximated at short times by an exponential kinetics. However, the algebraic slow decay was seen to persist for long periods because of the diffusion.8,15−17 Although Braun’s equation is simple and convenient, its implicit assumption of the exponential recombination kinetics is hardly justified. A rigorous analytical treatment of the geminate charge dissociation process was developed by Noolandi and Hong.11 However, their solution is represented by infinite summations of terms which involve coefficients that must be determined by solving infinite linear sets of equations.11,13,18 Because of its practical usability, Braun’s result has been widely applied in the molecular electronics field,2,3,19−21 although the exact expression that was obtained by Noolandi and Hong can be also used in the analysis of experimental data in certain cases.11,22 A detailed comparison between Braun’s approach and that of Hong and Noolandi was made by Wojcik and Tachiya.13 They indicated that the geometric effects of the intrinsic reaction rate should be modified in the Braun theory to make its results comparable with those of Hong and Noolandi.13,23 In the present paper, we refer to the version of Braun’s equation that includes the required modification. On the basis of an isotropic initial distribution, Wojcik and Tachiya simplified the solution of Hong and Noolandi. Then, using this simplified solution, they showed that Braun’s result deviates from that of Hong and Noolandi by increasing both the field strength and the intrinsic recombination rate. In this study, we first aim to show an alternative derivation of Braun’s result without the assumption of first order recombination kinetics. We analyze the recombination process by considering the role of anisotropy of the initial charge separation rate resulting from the presence of an external field. We show that Braun’s result follows from a specific form of the angular dependence of the initial charge distribution. In real experiments, the angular distribution of bound charge pairs formed as a result of the initial charge separation is usually unknown. However, this distribution can be realistically modeled using the Marcus equation to describe the charge separation rate. We obtain exact numerical results of the escape probability using the Marcus equation, and compare these results with those determined for the isotropic initial charge distribution, as well as those corresponding to Braun’s theory. We also study the conditions required for insensitivity of the escape probability to the assumed form of angular distribution of initially separated charge pairs, and find that this insensitivity appears when the escape probability is higher than ∼10−3 under zero field. We explain this effect by noting that a high escape probability requires a low intrinsic recombination rate, which in turn forces the charge carriers to execute random walks for long periods of time before recombination on average. During these

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CHARGE-PAIR DISSOCIATION FROM BOUND STATE We study the escape probability of an electron and a hole that are initially separated by a distance R and become free charge carriers when they get separated by an infinite distance. We assume that recombination occurs, with a finite intrinsic rate, when the electron and the hole are separated by R. In organic solids, the relative dielectric constant is approximately ∼4, so the Onsager radius defined by rc =

e2 4π ϵϵ0kBT

(1)

is ≈14 nm at room temperature, and is much larger than the recombination radius. Here, ϵ, ϵ0, e, kB and T denote the relative dielectric constant, the vacuum permittivity, the elementary charge, Boltzmann’s constant and the temperature, respectively. The Onsager radius is defined as the distance at which the Coulomb binding energy e2/(4πϵϵ0r) is equal to the thermal energy kBT. The pair probability distribution function for an electron and a hole separated by a distance r satisfies ⎡ ∇⃗V ⎤ ⎥=0 ∇⃗ ·D⎢∇⃗fs + fs kBT ⎦ ⎣

(2)

where D denotes the diffusion constant, and V(r, θ) represents the interaction potential that includes both the Coulomb potential and the external potential, V (r , θ ) = −

e2 − eEr cos θ 4π ϵϵ0r

(3)

Here, E and θ represent the external electric field strength and the polar angle between the applied electric field and the dipole vector of the charge pair, respectively. The equilibrium distribution is given by ⎛ V ⎞ feq ∝ exp⎜ − ⎟ ⎝ kBT ⎠

(4)

apart from a multiplication constant. eq 2 implicitly assumes an incoherent charge transfer mechanism and is applicable to the media where the charge-carrier scattering length is at least oneorder of magnitude smaller than the Onsager radius.24 The radial component of the probabilistic current can be defined as j(r , θ ) = Dfeq (r , θ )

∂ f (r , θ )/feq (r , θ ) ∂r s

(5)

At equilibrium, we have j(r, θ) = 0, which can be shown by substituting feq(r, θ) for fs(r, θ) in eq 5. 3633

DOI: 10.1021/acs.jpcc.6b12470 J. Phys. Chem. C 2017, 121, 3632−3641

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The Journal of Physical Chemistry C

the limit where r → 0, the steady-state distribution approaches the equilibrium distribution. This implies that the steady-state distribution given by eq 12 is close to the equilibrium distribution around the recombination radius. If the steadystate solution that is expressed by eq 12 is substituted into eq 5, then the radial component of the probabilistic current can be expressed as follows:

When electrons and holes are generated at a constant rate, they can either recombine or separate to an infinite distance at a specific rate. In the steady state, the escape probability is given by the output flow at an infinite distance divided by the input flow due to electron−hole generation. We denote the steady state input flow at R and an angle θ by Jin (θ). We then calculate the steady state output flow for the given input flow. We set the boundary condition to represent the flow balance between the input flow given by Jin (θ), the diffusional flow to the recombination radius, and the recombination flow at the recombination radius. This boundary condition can be expressed as 4πR2Jin (θ ) = −4πR2j(R , θ ) + k int(R , θ )fs (R , θ )

⎛ 1 ∂δf (r , θ ) jO (r , θ ) = D⎜ − + δf (r , θ )(b(r ) cos θ ∂r ̅ ⎝ rc ⎞ − rc /r 2)⎟ ⎠

(6)

It is convenient to express eq 13 using the series form

where the diffusional flow is given by eq 5 and kint denotes the intrinsic recombination rate at R. We first consider the case of isotropic reactivity, which is expressed as kint (R, θ) = ki. The other boundary condition assumed represents perfect absorption at an infinite distance to guarantee steady state outward flow toward infinity. This boundary condition has the form fs (r , θ ) → 0 as r → ∞

⎛ b(r ) ⎞ (1 − cos θ )⎟ × δf (r , θ ) = exp⎜ − ⎠ ⎝ 2 ∞

∫0

π

n=0

By substituting eq 15 into eq 14, we then obtain ⎡⎛ r ⎞ b(r ) (1 + cos θ ) − c2 ⎟δf (r , θ ) jO (r , θ ) = D⎢⎜ ⎢⎣⎝ 2r r ⎠

(8)

−

where r can be set at an arbitrary value that is larger than R if recombination occurs only at r = R. The steady state escape probability is then given by φ(R , E) = Jout /2π

∫0

π

sin θ dθ R2Jin (θ )

(10)

⎝ rc ⎠

(16)

(17)

⎞ ⎛ b(r ) δf (r , θ ) = exp⎜ − (1 − cos θ )⎟ ⎠ ⎝ 2

(11)

I0( 2b(rc)(1 + cos θ ) )

(18)

We next considered the case where the steady-state distribution is given by eq 12, as obtained by Onsager. The probability current was obtained from the steady-state distribution using eq 17 and is denoted by jO(r, θ). At the recombination radius, the probability current should be balanced with the separation rate and the recombination rate. By substituting jO (r, θ) for j (r, θ) in eq 6, we find that the steady-state current balance leads to

(12)

apart from the normalization constant, where feq (r, θ) is given by eq 11. δf (r, θ) represents the deviation of the steady state distribution from the equilibrium distribution that is caused by the dissociative current flow and is given by ∞

∫1/ r dy

I0( 2b(r )y(1 + cos θ ) ) exp(−y)

n=1

where δf (R, θ), as given by eq 15, is given approximately by

Based on Onsager’s paper,6 the steady state probability distribution of a dissociated charge pair at a distance r with azimuthal angle θ can be obtained as

⎛ b(r )(1 − cos θ ) 1⎞ δf (r , θ ) = exp⎜ − + ⎟ ⎝ 2 r̅ ⎠

⎛ ⎞n − 1

jO (r , θ ) = −Drcδf (r , θ )/r 2

Apart from the normalization constant, the equilibrium distribution that is expressed by eq 4 can be represented in terms of these dimensionless quantities as

fs (r , θ ) = feq (r , θ ) − δf (r , θ )

∞

∑ n⎜ r ⎟

Within the limits of r ≪ rc and eEr/(kBT) ≪ 2rc/r, eq 16 can be expressed as,

We also introduce two dimensionless quantities

feq (r , θ ) = exp(b(r ) cos θ + 1/ r ̅ )

⎛ ⎞ b(r ) 1 (1 − cos θ )⎟⎟ × exp⎜⎜ − rc ⎝ 2 ⎠

⎤ ⎛ b(rc) ⎞n /2 (1 + cos θ )⎟ In( 2b(rc)(1 + cos θ ) )⎥ ⎜ ⎥⎦ ⎝ 2 ⎠

(9)

r ̅ = r /rc and b(r ) = eEr /(kBT )

⎞n /2 (1 + cos θ )⎟ In( 2b(rc)(1 + cos θ ) ) ⎝ 2 ⎠ (15)

(7)

sin θ dθ r 2j(r , θ )|r > R

⎛ b(rc)

∑ r ̅ n⎜

We calculate the output probability flow, which expresses the dissociation rate of a charge pair, as Jout = −2π

(14) 7

4πR2Jin (θ ) = k ifs (R , θ ) + 4πDrcδf (R , θ )

̅

(19)

We show the resulting θ-dependence of Jin(θ) later in the paper. Onsager’s solution implies that the distribution is close to equilibrium within the vicinity of the origin at which one of the charges of a pair is located, in addition to the other condition that the distribution approaches zero at an infinite distance when the absorbing boundary is set such that it has a steady-state flow to the infinite distance.

(13)

where In(z) for n = 0, 1, 2, ..., denotes a modified Bessel function of the first kind.25 Dissociative current flow is induced by the perfectly absorbing boundary condition at an infinite distance from the origin. It should be noted that the same boundary condition is set in our problem, as shown in eq 7. In 3634

DOI: 10.1021/acs.jpcc.6b12470 J. Phys. Chem. C 2017, 121, 3632−3641

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The Journal of Physical Chemistry C In low-permittivity materials, the intrinsic reaction rate must be small to obtain an appreciable dissociation yield. The steadystate distribution that is given by Onsager’s solution may be appropriate when the intrinsic reaction rate is low, so that the distortion of the equilibrium distribution is also low in the vicinity of the origin. The escape probability can be calculated using eq 9. We begin by calculating the numerator of eq 9 using eq 8 and jO(r, θ). We note that the probability current remains continuous throughout the entire space except at the recombination radius, because charge separation is assumed to occur at the recombination radius only. In this case, it is convenient to evaluate Jout at r such that it satisfies r ≪ rc and eEr/(kBT) ≪ 2rc/r using jO(r, θ) as expressed by eq 17. We then substitute eq 17 into Jout = −2π

∫0

π

sin θ dθ r 2jO (r , θ )|r > R

J1(2 − 2βE ) ⎛ r ⎞ b(R ) kd(E) = 4πDrc exp⎜ − c ⎟ ⎝ R ⎠ sinh(b(R )) − 2β E

and b(R) = eER/(kBT). By further approximating b(R) ≪ 1, we obtain a result that is equivalent to Braun’s expression, apart from the R-dependence, as follows:

1+

(21)

Second, we calculate the denominator of eq 9 using eq 19 with jO (r, θ) as expressed by eq 17. Using eq 19, and evaluating the integral using π

sin θ dθ exp(b(r ) cos θ ) = 2

sinh(b(r )) b(r )

(22)

we then obtain 2π

∫0

π

sin θ dθ R2Jin (θ ) = k ̅int + 4πDrc

I1(2 b(rc) ) b(rc) (23)

where k̅int is defined as ⎡ I1(2 b(rc) ) ⎤ sinh(b(R )) ⎥ exp(rc /R ) − k ̅int = k i⎢ ⎢⎣ b(R ) b(rc) ⎥⎦

(24)

The escape probability is then obtained by substituting eqs 21 and 23 into eq 9 to give φ (R , E ) =

1 ⎛ k i sinh(b(R )) ⎜exp 1 + 4πDr b(R ) ⎝ c

( Rr ) I (2 c

1

b(rc) b(rc) )

1+

( Rr ) I (2

exp

c

1

b(rc) b(rc) )

(26)

when the assumption R/rc ≪ 1 is used. eq 26 above is one of the main results of this paper. The result can be rewritten using Bessel functions of the first kind by applying the identity25 J ( −z ) I1( z ) = 1 z −z

(27)

and βE = b(rc)/2 = e E/[8πϵϵ0(kBT) ] to give 3

kd(E) φ (R , E ) = k i + kd(E)

c

1

b(rc) b(rc) )

(30)

EFFECT OF ANISOTROPY OF THE INITIAL CHARGE DISTRIBUTION ON THE ESCAPE PROBABILITY As already mentioned, our analytical result of the field dependence of escape probability given by eq 26, which reduces to Braun’s expression [eq 30] at low fields, implies a specific form of angular dependence of the initial charge separation rate given by eq 19. In this section, we study the escape probability obtained for a realistic initial charge distribution calculated using the Marcus equation and compare the results with those calculated for the isotropic initial charge separation rate, as well as that given by eq 19. In the calculations, we use the Marcus equation in the following form

1 k i sinh(b(R )) 4πDrc b(R )

( Rr ) I (2

exp

■

⎞ − 1⎟ ⎠ (25)

≃

ki 4πDrc

The difference between eq 26 and eq 30 is given by the factor sinh(b(R))/b(R). This factor was mentioned as a correction factor for the dissociation constant in ref 6 of Onsager’s original paper.6 In this derivation, Braun’s expression was obtained from the steady-state flux without considering the kinetics. The use of exponential kinetics, which is implicitly assumed in the original derivation, is not required to derive Braun’s expression. In Braun’s original derivation of the expression, the dissociation rate constant is obtained from a combination of the dissociation constant and the bulk recombination rate constant that was given by Langevin.12 The bulk recombination rate is weakly dependent on the external field strength.27,28 The field dependence in his derivation therefore originates from the dissociation constant. We calculated the outward dissociative flux directly and did not need to use either the bulk recombination rate constant or the dissociation constant. A restriction inherent in our derivation is that we consider only the case where recombination occurs at a fixed distance R. If recombination occurs over a range of distances, the dissociation yield of charge pairs in the absence of external electric fields can be determined analytically using Padé approximants.17 In principle, this methodology can also be applied when an external electric field is present but its accuracy should then be tested since the Padé approximants reproduce the exact results of local reactivity only in the absence of external electric fields. Accurate results of the dissociation yield in the presence of an external electric field can be obtained numerically.29 Unfortunately, it is difficult to extend the theoretical approach developed in the present study to nonlocal reactions.

The solution is obtained as (see Appendix A for the full derivation)26

∫0

1

φ (R , E ) =

(20)

Jout = 4πDrcI1(2 b(rc) )/ b(rc)

(29)

2

⎡ (ΔG + λ)2 ⎤ cs ⎥ kcs(R , θ ) = νcs exp⎢ − 4λkBT ⎦ ⎣

(28)

(31)

where νcs is the frequency factor and ΔGcs can be written as

where kd(E) is defined by 3635

DOI: 10.1021/acs.jpcc.6b12470 J. Phys. Chem. C 2017, 121, 3632−3641

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The Journal of Physical Chemistry C ΔGcs(R , cos θ ) = ΔELL −

e2 1 − eRE cos θ 4π ϵϵ0 R

The conventional results of Braun’s model, given by eq 30, reproduce those obtained using eq 26 quite well at low electric fields. As seen from Figure 1, the former start to deviate from the latter when F = 5−10, which corresponds to E ∼ 107 V/m at room temperature and R = 1 nm. The deviation increases with increasing field strength. In Figure 2, the solid lines represent the θ-dependence of the initial charge separation probability given by

(32)

where ΔELL denotes the energy offset resulting from the difference between the lowest unoccupied molecular orbital (LUMO) level of the donor and that of the acceptor. The escape probability, which is averaged over the initial charge separation distribution, is obtained from π 1 φM(R , E) = π sin θ dθ ∫ sin θ dθ kcs(R , θ) 0

∫

0

φ(R , θ , E)kcs(R , θ )

(33)

where φ(R, θ, E) is calculated from the exact solution using the exact analytical method developed by Noolandi and Hong.11 The results of calculations in which we assume typical parameter values: ϵ = 4, R = 1 nm, λ = 0.3 eV, and ΔELL = −0.5 eV, are shown as plus signs in Figure 1. We also show in this

Figure 2. Initial charge separation probability as a function of θ for κ = kirc/(4πDR2) = 1, when ϵ = 4 and R = 1 nm. The red solid lines indicate the results from the Marcus rate equation, where λ = 0.3 eV, and ΔELL = −0.5 eV. The dashed lines are obtained using eq 19. The dash-dotted line indicates the isotropic distribution. Thick lines and thin lines correspond to results for b(rc) = 10 and b(rc) = 1, respectively. Figure 1. Escape probability as a function of the reduced electric field, F = b(rc), where ϵ = 4 and R = 1 nm. Red circles indicate results obtained using the isotropic charge separation probability as calculated from the exact solution.13 Blue pluses indicate results obtained using the Marcus charge separation rate in combination with the exact solution, λ = 0.3 eV, and ΔELL = −0.5 eV.11 The solid lines were obtained from eq 26 and the dashed lines represent results from Braun’s equation, as given by eq 30. The different lines correspond to three different values of κ = kirc/(4πDR2); these values of κ, from top to bottom, are 0.01, 0.1, and 1.

φiCS(R , E) =

kcs(R , θ ) π

∫0 sin θ dθ kcs(R , θ)

(34)

where kcs (R, θ) is obtained from the Marcus eq 31. In the same figure, the dash-dotted line indicates the result obtained when kcs (R, θ) is substituted by 1, which corresponds to the isotropic charge separation probability. One can see that φiCS (R, E) is higher when the direction of initial charge separation is opposite to the field direction. This is because the energy change associated with the initial charge separation satisfies then the condition for the Marcus inverted region, where the electron transfer rate decreases with increasing driving force of the transfer. The observed effect is rather weak at F = 1, but at F = 10 the Marcus separation rate is strongly anisotropic. The initial charge separation rate implicitly built into the analytical model standing behind Braun’s theory is represented in Figure 2 by the dashed lines. These results, obtained from eq 19, show an anisotropy of similar size as that of the Marcus model, but of the opposite angular characteristics. Interestingly, the observed anisotropies do not greatly affect the final values of the escape probability in the cases shown in Figure 1, which means that the angular dependence of the escape probability is very weak. This can be explained as follows. When the condition R ≪ rc is satisfied, the initially separated charges are

figure (as circles) the escape probability for the isotropic initial charge distribution, calculated using the exact method described in ref 13, and the analytical results given by eqs 26 and 30, represented by the solid and dashed lines, respectively. One can see that the escape probability obtained using the Marcus initial charge separation rate is very close to that calculated for the isotropic distribution. Noticeable differences appear only at high electric fields, and only in the case of a rather high intrinsic recombination rate which gives the zerofield escape probability below 10−3. Equation 26 very well approximates the exact results of escape probability shown in Figure 1. A small discrepancy appears only at high fields in the case of κ = 1, and is similar in size to the discrepancy between the results obtained using the Marcus equation and the isotropic distribution. 3636

DOI: 10.1021/acs.jpcc.6b12470 J. Phys. Chem. C 2017, 121, 3632−3641

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The Journal of Physical Chemistry C deep inside the sphere of the Onsager radius, where the Coulomb binding energy is much higher than the thermal energy, and also higher than the energy associated with an applied electric field. At a low intrinsic reactivity, which is required to achieve a sizable escape probability, it takes a long time on average for the charges to ultimately separate from each other as a result of the thermally activated process. During that time, the memory of the angular dependence is lost because of the stochastic thermal motion. For the above reasons, it may be expected that the angularly averaged escape probability is very weakly affected by an anisotropy of the initial charge separation rate.

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EFFECT OF INITIAL SEPARATION DISTANCE In this section, we consider a situation in which the initial separation distance of a geminate charge pair Rth is larger than the recombination radius R. Such a situation is theoretically possible when a charge pair undergoing the initial separation process has an excess energy which needs to be dissipated to the medium before the particles become thermalized.5,30−32 We denote the recombination probability of the charge pair initially separated by Rth as κr(Rth, R, E), and the corresponding escape probability as φ(Rth, R, E) = 1 − κr(Rth, R, E). In the absence of an external field, κr(Rth, R, 0) can be described by an exact analytical expression in the form8,33,34

Figure 3. Escape probability as a function of reduced electric field, F = b(rc), when ϵ = 4 and R = 1 nm. κ = kirc/(4πDR2) = 0.01. Pluses denote the exact results obtained using the isotropic charge separation probability when Rth = R. Circles denote the exact results obtained using the isotropic charge separation probability when Rth = 2 nm, Rth = 3 nm, and Rth = 4 nm, from bottom to top. The red lines show the results for eq 41, where κr,1(Rth) and φ(R, E) are given by eq 38 and eq 26, respectively.

1 − exp( −rc /R ) 4πDrc 1 = + κr(R th , R , 0) 1 − exp( −rc /R th) ki exp( −rc /R ) 1 − exp( −rc /R th)

∞ ⎛ r ⎞n − 1 ⎡ rc ⎤ κr ,1(r , E) = 1 − exp⎢ −b(r ) − ⎥ ∑ ⎜ ⎟ ⎣ r ⎦ n = 1 ⎝ rc ⎠

b(rc)(n /2) − 1In(2 b(rc) )

(35)

It should be noted that even in the limit where R → 0, the recombination probability is nonzero because of the divergent character of the Coulomb binding energy. The second term in eq 37, represents the correction required to take into account a nonzero recombination radius,18

If both R and Rth are much shorter than rc, eq 35 can be reduced to 4πDrc 1 ≈1+ exp( −rc /R ) κr(R th , R , 0) ki

(36)

which shows that the recombination probability is approximately independent of Rth. It is straightforward to show that eq 36 is equivalent to Braun’s expression given by eq 30 at E = 0 using limx→0 I1(2x)/√x = 1. The range of validity of eq 36 can be assessed by analyzing the exact results of escape probability plotted in Figure 3. At the electric fields close to zero, the escape probability calculated for Rth = 2 nm differs very little from that obtained for Rth = R = 1 nm. A significant effect of Rth can be seen only at larger values of this parameter. Interestingly, this form of the dependence of φ on Rth is observed not only at fields close to zero, but over a wide range of F. This means that eq 26 derived in Charge-Pair Dissociation from Bound State section, which well describes the electric field dependence of escape probability for Rth = R = 1 nm, is also applicable for larger values of Rth, at least up to Rth = 2 nm. We now derive an analytical expression that approximately describes the field dependence of escape probability over a wider range of Rth. In this derivation, we use the analytical results obtained for the case of perfectly absorbing boundary condition set at the recombination radius r = R, where R ≪ rc. Following Hong and Noolandi,18 the recombination probability in this case can be decomposed into two terms: κr , ∞(R th , R , E) = κr ,1(R th , E) + κr ,2(R th , R , E)

(38)

κr ,2(R th , R , E) =

⎤ b(rc) ⎡ ⎛ rc r⎞ exp⎢ −⎜ + c ⎟ + b(R th)⎥ ⎥⎦ ⎢⎣ ⎝ R th 8 R⎠ H(R th , E)

(39)

where H(r, E) can be expressed, in the limit of R ≪ rc as

18

H (r , E ) =

∫2

∞

dv1

∫0

2

dv2

∫0

2

v + v2 ⎤ ⎡ du exp⎢ −b(r ) 1 ⎣ 2 ⎥⎦

I0( b(rc)uv1 )I0( b(rc)uv2 )

(40)

It can be shown that when R ≪ rc, κr,2(Rth, R, E) is negligible in comparison to κr,∞(Rth, R, E). In the case of finite intrinsic reactivity, the recombination probability of a pair initially separated by the distance Rth can be expressed as the product of the probability of reaching the recombination radius for the first time, given approximately by κr,1(Rth, E), and the recombination probability of a pair initially separated by R, equal to 1 − φ(R, E). The overall escape probability of a pair initially separated by Rth is therefore approximately expressed as φ(R th , R , E) ≈ 1 − κr ,1(R th , E)(1 − φ(R , E))

(37)

(41)

The results of eq 41, where κr,1(Rth, E) and φ(R, E) are obtained using eqs 38 and 26, respectively, are shown in Figure 3 as red lines. We see that eq 41 quite well reproduces the exact

where κr,1(r, E) is equal to the recombination probability obtained by Onsager in the limit of R → 0 given by7 3637

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irradiation at the time zero. If we denote the radiative decay rate of this state by kf and the nonradiative decay rate by kn, the rate equation for the occupation probability pCT can be expressed as36

results shown by circles. A significant deviation appears only at high fields in the case of Rth = 4 nm, which originates from ignoring the contribution associated with κr,2(Rth) [cf. eq 37].

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EMISSION YIELD The charge dissociation yield in low permittivity materials is an important factor that gives us insight into the charge separation processes that occur in organic solar cells. The geminate-pair dissociation yield can be studied using the photocurrent or the photoemission of recombined pairs. Recently, an investigation of the charge separation process in organic solar cells has been performed using the photoemission from the CT state by appropriate selection of the donor and acceptor materials.21 In this section, we provide a theoretical study of the relationship between the emission from the CT state and the dissociation yield of the geminate pairs. To study photoemission of recombining charge pairs, one can use various theoretical models. In the simplest case, the rate of deactivation of an excited state responsible for the emission can be directly related to the intrinsic recombination rate ki, as shown schematically in Figure 4a.

∂ p (t ) = −(k f + kn + kd)pCT (t ) + ∂t CT

∫0

t

dt1

kdpCT (t1)κ r(R , E , t − t1)

(42)

where κr(R, E, t) represents the recombination rate of a geminate pair that dissociates from the CT state at time 0 and forms a CT state again at time t. The time-integrated quantity given by κ(R, E) = ∫ ∞ 0 dt1 κr(R, E, t1) is then equal to the recombination probability, 1 − φ (R, E). The quantum yield of the CT emission and the dissociation t1 ∞ yield can be expressed as ∫ ∞ 0 dt kf pCT(t) and ∫ 0 dt1 ∫ 0 dt2 φ(t1 − t2)kdpCT(t2), respectively. The quantum yield of the CT emission is obtained as I f (E ) = =

kf k f + kn + kd[1 − κ(R , E)] kf k f + kn + kdφ(R , E)

(43)

where the reversible dissociation and recombination properties of the CT state are taken into account by φ(R, E), which is given by eq 26. ki in φ(R, E) indicates the rate of recombination from the continuum of states to the CT state. Therefore, when the emission originates from the CT state, the recombination rate cannot be identified with the fluorescence rate. Even when ki/D in φ(R, E) can be estimated, the absolute value of D cannot be estimated for this dissociation scheme because the recombination rate differs from the emission rate. The dependence of the dissociation rate on the external field is incorporated into φ(R, E). Moreover, eq 43 indicates that 1/ If(E) is proportional to φ(R, E). Therefore, the escape probability φ(R, E) can be studied by analyzing 1/If(E) or other quantities related to it. In a similar manner to eq 43, which describes the quantum yield of CT emission, the dissociation yield can be expressed as kdφ(R, E)/[kf + kn + kdφ(R, E)]. Recently, the emissions from the CT states have been measured in polymer/fullerene blends.21,35 The experimental results were analyzed on the basis of the CT emission quenching defined by 1 − If(E)/If(0). When the recombination rate cannot be identified as the fluorescence rate, as considered in the model illustrated in Figure 4 (b), the quantum yield of the CT emission is given by eq 43. In this case, the CT emission quenching can be expressed as

Figure 4. Schematic representation of charge separation and recombination of geminate pair. ki represents the intrinsic recombination rate. kf and kn represent the radiative decay rate and the nonradiative decay rate, respectively. kd represents the charge separation rate from the CT state. (a) Charge separation of the mobile geminate pair state occurs directly from the excited state. Recombination occurs when the mobile geminate pair is at its shortest separation distance. (b) Charge separation occurs when the chargetransfer (CT) state is formed. (c) Charge separation of the mobile geminate pair state occurs before formation of the thermally relaxed CT state.

In a more complicated situation, when a charge transfer state involved in the photoemission is bound by a quantum resonance, so that its binding energy includes other components in addition to the Coulomb energy, this CT state should be excluded from the continuum of other states of separated charges, as shown schematically in Figure 4b. The dissociation of this CT state into the continuum of other states needs to be represented by an additional dissociation rate, which we denote as kd. In the theoretical model given below, we assume that the charge separation/recombination process occurs only through the relaxed CT state.35 We consider the transient occupation probability of the CT state after pulsed

1 − I f (E)/I f (0) =

kd[φ(R , E) − φ(R , 0)] k f + kn + kdφ(R , E)

(44)

The field-dependent escape probability that is given by φ(R, E) appears in both the numerator and the denominator of eq 44. Therefore, the quantity 1 − If(E)/If(0) is not very suitable to study the field dependence of the escape probability φ(R, E). It is much more convenient to study this field dependence using I f (0)/I f (E) − 1 = 3638

kd[φ(R , E) − φ(R , 0)] k f + kn + kdφ(R , 0)

(45)

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The Journal of Physical Chemistry C which is proportional to φ(R, E) − φ(R, 0). As shown in Figure 4b, recombination into the CT state occurs at the recombination radius of the mobile electron−hole pair. The recombination radius is the shortest distance between the mobile electron−hole pair and can be larger than the size of the CT state. As an example, we analyze the dissociation of a CT complex in a 1:1 blend of poly[2-methoxy-5-(3′,7′-dimethyloctyloxy)1,4-phenylenevinylene] (MDMO-PPV) and [6,6]-phenyl-C61 butyric acid methylester (PCBM).35 The experimental data were originally analyzed on the basis of the scheme shown in Figure 4 (b), using the quantity 1 − If(E)/If(0) to describe the CT quenching. These results were used in ref 35 to estimate the lower limit of the binding energy of the CT complex. Here, we analyze these experimental data to estimate the dissociation rate of the CT state relative to its deactivation rate. In Figure 5

state is ignored. In reality, direct dissociation of excitons can hardly be distinguished from the dissociation that occurs through a hot excited CT state.5,32 Using the model illustrated in Figure 4 (c), the emission yield is given by11 I f (E ) =

k f [1 − φ(R th , R , E)] k f + kn

(46)

When Rth = R, we can insert φ(Rth, R, E) = φ(R, E) into the above equation. In this case, the fluorescence emission quenching can be expressed as11 1 − I f (E)/I f (0) =

φ(R , E) − φ(R , 0) 1 − φ(R , 0)

(47)

and the emission quenching is thus proportional to φ(R, E) − φ(R, 0). In Figure 6, we compare the emission quenching described by 1 − If(E)/If(0), as a function of the electric field, for the

Figure 5. If(0)/If(E) − 1 as a function of electric field strength E (V/ m). Circles indicate experimental data for PPV−PCBM bulk heterojunctions.35 The solid line represents the fitting result when using eq 45 and eq 26. The dashed line represents the results for eq 45 and Braun’s equation as given by eq 30 when using the same parameter values that were obtained via the above fitting.

we fit eq 45 to the experimental data using eq 26 to describe φ(R, E). We assume ϵ = 3.4, 297 K, and R = 1 nm (the Onsager length is calculated as rc = 16.6 nm). As shown in Figure 5, we obtain a reasonable fit, from which we determine the ratios (kf + kn)/kd = 0.09 and ki/(4πDrc) = 0.0019. These results indicate a fast dissociation rate kd when compared with the deactivation rate. In the original paper, a value of kn = 0 was proposed.35 If this assumption is correct, then the dissociation rate is approximately ten times higher than the radiative rate. The dashed line in the figure shows the results obtained from eq 45 and Braun’s equation as given by eq 30 when using the same parameter values. The values of 1 − If(E)/If(0) are overestimated. We also performed a fitting in which the recombination radius R was used as a free parameter in addition to (kf + kn)/kd and ki/(4πDrc). From this fitting we obtained R = 0.99 nm, (kf + kn)/kd = 0.09 and ki/(4πDrc) = 0.0025 with the fitted line practically overlapped with the solid line. This confirms that the value of R is in fact close to 1 nm and that the description of the CT emission by eq 44 is reasonable. In addition to the charge separation/recombination models illustrated in Figure 4, parts a and b, one can consider another situation, shown schematically in Figure 4c, in which the initial charge separation occurs directly into the continuum of states and the CT state is formed later by recombination of the separated charges. More precisely, the CT state indicates here the relaxed CT state and any further dissociation of this CT

Figure 6. Emission quenching defined by 1 − If(E)/If(0) as a function of the reduced electric field, F = b(rc), when ϵ = 4 and R = 1 nm. kn/kf = 1 and κ = 0.001 in ki/(4πDrc) = κ(R/rc)2. The thick solid line indicates the case of charge separation without CT state formation that was calculated from eq 47. The thin solid line, dashed line, and dashdotted line indicate the results for charge separation via the CT state calculated from eq 44 for kd/kf = 100, 10 and 1, respectively.

models shown in Figure 4, parts b and c. The thick solid line indicates the results obtained using eq 47, which corresponds to the case when the charge separation occurs without formation of the CT state, i.e., the scenario shown in Figure 4c. The other lines indicate the emission quenching behavior when charge separation occurs through the formation of the CT state, i.e., the scenario shown in Figure 4b. The different lines represent the different values of kd/kf in the latter case. If the charge separation occurs via the CT state, the emission quenching is then dependent on the dissociation rate of the CT state. From Figure 6 we see that when kd < kf + kn the slope of the dependence 1 − If(E)/If(0) vs F is almost the same as that of direct initial charge separation. However, if the dissociation rate kd is higher than kf + kn, the slope decreases with increasing dissociation rate if formation of the CT state during the charge separation is assumed. 3639

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CONCLUSION Braun’s result was originally derived under the assumption of first order reaction kinetics. In this paper, we have derived an analytical expression [eq 26] that describes the dissociation yield and reduces to Braun’s equation when the external electric field is low, as characterized by b(R) = eER/(kBT) < 1. In low permittivity materials, a low recombination rate is required to obtain an appreciable dissociation yield. If the recombination rate at the recombination radius is sufficiently small, as defined by ki/(4πDrc) ≤ 1, the charge carrier then executes random walks and visits the recombination radius multiple times before finally separating to a large distance from its counterpart. We have shown that both Braun’s results and our results follow from the assumption that the charge distribution is close to equilibrium in the vicinity of the origin at which one charge of a pair is located. In our analytical derivation, we have also shown that to satisfy the flux balance at the recombination radius between the charge separation, recombination, and dissociative flows, a specific form of the charge separation rate is implicitly assumed that is isotropic in the zero-field limit and becomes anisotropic under an external field. We also studied sensitivity of the escape probability to the assumed form of angular dependence of the charge separation rate and compared the results with those obtained using a realistic initial angular dependence calculated by the Marcus equation. Using the exact numerical results, we have shown that this sensitivity is very limited in low-permittivity materials for both cases. In experiments performed on materials for organic photovoltaic applications, the geminate charge-dissociation mechanism was studied by application of strong external fields, which can be more than E ≥ 107 V/m.1,4,9,10,21 We have shown that Braun’s results begin to deviate from the exact ones when b(R) ≥ 1 and the intrinsic recombination rate exceeds a specific threshold value. It should be realized that in a polymer film of 100 nm thickness, the field strength 107 V/m is induced by applying a voltage of only 1 V. Equation 26 much better describes the charge-dissociation yields in this case. If the formation of mobile electron−hole pairs from the excited state involves an excess energy, then the initial separation distance of the thermalized electron−hole pair can be larger than the recombination radius. We studied the effects of the initial separation distance on the dissociation yield and found that the dissociation yield is weakly dependent on the initial separation distance. We also obtained an approximate analytical expression for the dissociation yield that included its dependence on the initial separation distance. The escape probability can be experimentally obtained from the photocurrent or from the emission from the CT state.21 We studied the case where charge separation occurs from the CT state. When the CT state is stabilized by a binding energy, and when the emission originates from the CT state, the recombination rate cannot then be identified as the emission rate. We analyzed the experimental data from the CT emission and estimated the dissociation rate from the CT state relative to the deactivation rate using our approximate analytical expression for the dissociation yield. We also found that the estimated CT emission differs considerably if Braun’s result is used to describe the dissociation yield. In the studies described in this paper, which concentrated on the effect of strong external fields on charge-carrier dissociation, we did not take into account the morphology of organic photovoltaic systems. In electron donor−acceptor blends,

where a heterojunction is formed between the donor and acceptor phases, even the equilibrium carrier distribution that occurs around the origin at which the counter charge is located can differ from that which occurs in homogeneous systems. Charge separation is known to be enhanced in the presence of heterojunctions and the electric field effect can be different from that which occurs in homogeneous systems, depending on the morphology, the carrier mobilities, and the energetic disorder.19,20,30,37−39

■

APPENDIX: DETAILED DERIVATION OF EQUATION 21 When eq 17 is substituted into Jout = −2πr 2

∫0

π

sin θ dθ jO (r , θ )|r > R

(48)

the integral can then be calculated using

∫0

π

⎛ b (r ) ⎞ sin θ dθ exp⎜ − (1 − cos θ )⎟ ⎝ 2 ⎠

I0( 2b(rc)(1 + cos θ ) ) =

exp[− b(r )] b(rc)

∫0

2 b(rc)

⎛ rz 2 ⎞ I1(2 b(rc) ) exp⎜ ⎟zI0(z) = 2 ⎝ 4rc ⎠ b(rc)

dz

(49)

where z = 2b(rc)(1 + cos θ ) is introduced and integration by parts together with r ≪ rc is then used to obtain the final equality. Integration by parts was performed using the relation d(zI1(z))/(dz) = zI0(z).25 We then obtain

■

Jout = 4πDrcI1(2 b(rc) )/ b(rc)

(50)

AUTHOR INFORMATION

Corresponding Authors

*(K.S.) E-mail: [email protected]. *(M.W.) E-mail: [email protected]. ORCID

Kazuhiko Seki: 0000-0001-9858-2552 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI under Grant No. 15K05406. M.W. would also like to acknowledge the support from the National Science Center of Poland under Grant No. DEC-2013/09/B/ST4/02956.

■

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