External Electric Field-Dependent Photoinduced Charge Transfer in a

Article pubs.acs.org/JPCC

External Electric Field-Dependent Photoinduced Charge Transfer in a Donor−Acceptor System for an Organic Solar Cell Peng Song,†,‡ Yuanzuo Li,‡,§ Fengcai Ma,*,† Tõnu Pullerits,*,∥ and Mengtao Sun*,‡ †

Department of Physics, Liaoning University, Shenyang 110036, P. R. China Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Science, P.O. Box 603-146, Beijing, 100190, P. R. China § College of Science, Northeast Forestry University, Harbin 150040, P. R. China ∥ Department of Chemical Physics, Lund University, Box 124, Lund 22100, Sweden ‡

ABSTRACT: External electric field is incorporated into the generalized Mulliken−Hush model and Marcus theory. With this new development, we have investigated the field-dependent electronic structure and rate of photoinduced charge transfer in organic donor−acceptor dyad using timedependent density functional theory and extensive multidimensional visualization techniques. The model is used to evaluate the influence of the external electric field on the electronic coupling between donor and acceptor. The reorganization energy and the free energy change of the electron transfer were calculated. It was found that the major effects in the external electric field dependent rate of the charge transfer originate from changes in the electronic coupling. The new theoretical approach not only promotes a deeper understanding of the connection between the external electric field, chemical structure, and optical and electronic properties of the donor−acceptor system, but also can be used for rational design of novel donor−acceptor system for organic solar cells.

1. INTRODUCTION Organic materials are becoming attractive alternatives over traditional inorganic materials in optoelectronics due to their numerous advantages, for example low cost, flexibility, largearea capability, and easy processing.1 They can be used for fabrication of transistors, photodiodes, solar cells, and (bio)chemical sensors.2−7 According to the current understanding, the photon-to-charge conversion in organic photovoltaic (OPV) devices can be described as a sequence of basic steps,8−10 in which charge transfer is one of the most important processes that facilitates the OPV device converting light energy. The key aspect we investigate here is the dissociation of the photogenerated excitons into separate charges. According to the Marcus theory,11 the electron transfer rate can be calculated as k=

⎛ (ΔG + λ)2 ⎞ 4π 3 2 | | exp V ⎜− ⎟ da 4λkBT ⎠ h2λkBT ⎝

are expected to be different for forward and backward reactions. We assign λ = λCT or λCR, and ΔG = ΔGCT or GCR, for the exciton dissociation or charge recombination, respectively. In the case of weak interaction, following Fermi’s Golden Rule, the rate of electron transfer is predicted to be proportional to | Vda|.2,13,14 A broad array of modern electronic structure methods has been applied to calculate Vda, and calculation protocols were established for evaluating electronic coupling matrix elements (Vda) for describing electron transfer process in D−A systems.15,16 Recently, Cave and Newton have introduced the generalized Mulliken−Hush (GMH) method,17 which allows accurate calculation of Vda in a wide variety of systems, independent of symmetry, geometrical constraints, or the number of interacting states. The advantage of GMH method is that only adiabatic state parameters such as the vertical excitation energy, stationary dipole moments, and transition dipole moment are needed. It should be noted that the nature of the electron−hole pairs created shortly after photon absorption and initial charge separation in organic solar cell systems may differ from that of a free well-separated electron−hole pairs, because of the Coulombic interaction, which has been proved to be one of the key parameters that govern the physics of many optoelectronic organic devices.8,18 The charges remain electro-

(1)

where λ is the reorganization energy, Vda is the electronic coupling matrix element between donor (D) and acceptor (A), ΔG is the free energy change for the electron transfer reaction. When taking ΔG as negative, the square dependence in (ΔG + λ) of the exponential term implies that the transfer rate displays a peak profile as a function of λ (which has a positive value) and reaches a maximum when |ΔG | is equal to λ.12 The reorganization energy and the free energy change, however, © XXXX American Chemical Society

Received: February 26, 2013 Revised: July 12, 2013

A

dx.doi.org/10.1021/jp401990z | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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energy, μi − μj = Δμ is the difference of stationary dipole moments, and μa − μd is the difference between stationary dipole moments of diabatic donor and acceptor states. All the parameters depend on the external electric field. Performing time-dependent density functional theory (TDDFT) calculations for a system in static uniform external electric field enables one to analyze the electrostatic properties of molecules in their excited states.23 Using a finite field method on the excitation energy, the transition energy dependence on the static electric field Fext can be expressed as24

statically attracted across the D−A interface; they must overcome the Coulombic attraction to achieve long-range charge separation. This process is facilitated through the formation of the delocalized excited states, which act as the gateway for charge separation. Such delocalization can play a critical role in overriding the Coulombic binding energy.10 It is important to recognize that in solar cells the delocalized excited state properties can be different due to the external electric field generated by the electrodes. However, how does the external electric field enable efficient long-range charge separation in organic bulk heterojunction? This fundamental scientific question remains unanswered. Because the binding energy of the lowest singlet exciton (i.e., the strength of the Coulombic attraction between the electron and hole) is significant in conjugated materials, the excitons are rather stable species. Consequently, the efficiency of the chargegeneration process is very low when a single compound forms the organic layer. In this paper, we investigate the external electric field-dependent photoinduced charge transfer in organic heterojunction mimicking the solar cells. The heterojunction is made from combination of an electrondonor component (a conjugated polymer) and an electronacceptor component (a molecule, for instance C60). As a model, we calculate the field-dependent photoinduced intramolecular electron transfer dynamics in an oligothiophene−fullerene dyad system. Such system shows efficient electron transfer from the oligomer to the fullerene;19−21 and moreover, organic photovoltaic cells have been fabricated using these materials.22 In such organic blends, the charge separation and further transport occurs under the influence of the external electric field generated via the difference of the work-functions of the electrodes. The structure of the article is as follows: in section 2, we present a framework of a method that can describe the external electric field dependent rate of electron transfer. We calculate separately the external field Fext dependence of the charge transfer integral Vda(Fext), change of free energy ΔG(Fext), and reorganization energy λ(Fext). Then, the external electric field dependent excited state properties are visualized with the twodimensional (2D) site representation and three-dimensional (3D) cube representation. In section 3 we present numerical results and data analysis. Concluding remarks are offered in the final section.

ΔEexc(Fext ) = ΔEexc(0) − ΔμFext −

= ΔEexc(Fext )

(3)

where Eexc(0) = ΔE = Ej − Ei is the excitation energy at zero field, Δμ is the change in the permanent dipole moment upon electronic transition (see eq 2), and Δα is the polarizability change. We can express the electronic coupling matrix element as Vda(Fext ) = γVda(0)

(4)

γ provides a convenient diagnostic indicator for influence of the external electric field on the electronic coupling matrix element Vda. We can distinguish three limiting cases, which we will investigate in detail. Case a: when |γ| = 1, then Vda(Fext) = Vda(0); the donor−acceptor coupling is independent of the external electric field. In this case, the “two-state” GMH model25 can be used for calculating Vda in the external electric field with arbitrary strength. Case b: when |γ| > 1, the electronic coupling between donor and acceptor is strengthened by the external electric field. Case c: when |γ| < 1, the electronic coupling between donor and acceptor is weakened by the external electric field. In cases b and c, the charge transfer integral is amplified or decreased by the external electric field. Therefore the external electric field effect should be taken into account, and eq 2 needs to be used to calculate Vda. We will also use 2D site representation8,26−28 and 3D cube representation8,27,28 to visualize the external electric field dependent charge transfer properties of electronic transition and of the charge transfer integral. For the exciton dissociation and charge recombination, ΔG = ΔGCT and ΔGCR, respectively,29 which can be estimated in different ways.12,30−33 Usually the two potential energy surfaces representing the reactants and the products do not have the same curvature. This can be well understood in terms of potential energy curves, as shown in Figure 1. Then ΔG has been estimated as the energy difference of the constituents in their final and initial states, accounting for the Coulombic attraction energy (ΔEcoul) between the two charges in the charge-separated state. Thus, for exciton dissociation, the external electric field-dependent ΔGCT (Fext) can be evaluated as12,31

2. THEORETICAL METHODS 2.1. The External Electric Field-Dependent Rate of Charge Transfer. In organic solar cells, the charge separation takes place under the influence of an external electric field. One of the key points is evaluating the external field effect on the electronic coupling Vda. For a given pair of diabatic states corresponding to the donor and acceptor, the external electric field dependent electronic coupling matrix element can be expressed using the GMH model: Vda(Fext ) = ΔEexc(Fext )

1 Δα Fext 2 2

ΔGCT(Fext ) = ED +(Fext ) + E A −(Fext ) − ED *(Fext )

μij (Fext )

− E A (Fext ) + ΔEcoul

μa (Fext ) − μd (Fext )

(5)

with

μij (Fext ) 2

ΔEcoul =

2

(μi (Fext ) − μj (Fext )) + 4μij (Fext )

∑∑ D+

A−

qD+q A− 4πε0εsr D+A−

−

∑∑ D*

A

qD *qA 4πε0εsrD * A

where ED*, ED+, EA and EA− represent the total energies of the isolated donor in the equilibrium geometries of the lowest excited state and of the cationic state and the total energies of

(2)

where, μij is the transition dipole moment between adiabatic electronic states i and j, ΔE = Ej − Ei is the vertical excitation B



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The exciton binding energy Eb mainly originates from the Coulombic attraction energy between the electron and hole and is usually taken as the difference between the electronic and optical band gap energy.34 The electronic band gap can be approximated as energy difference of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), while the optical gap is taken as the first singlet excitation energy. The charge carrier becomes free from the Coulombic attraction of an opposite charge when Eb ≤ kBT, which amounts to 25 meV at room temperature. Here, we set ΔEcoul ∼ Eb; the latter is adjusted by external electric field. After the electron transfer, an additional dipole moment is generated by the separated charges. This dipole moment interacts with the external electric field. Consequently, the reaction free energy of a radical pair with the total dipole moment μ in the charge separated state of D+A− is perturbed by external electric field and obtains an additional component. When the change in the free energy caused by this field-dipole interaction is considered, an additional expression for evaluating the field dependent free energy gap for the charge transfer reaction can be obtained as

Figure 1. Potential energy curves of the donor/acceptor pair in the ground state (DA), the lowest intramolecular excited state (D*A), and the lowest charge transfer state (D+A−).

the isolated acceptor in the equilibrium geometries of the ground state and of the anionic state, respectively. The ΔGCR has been estimated from expressions similar to eq 5, but involving the charge-separated state and the ground state: D

A

ΔGCR (Fext ) = E (Fext ) + E (Fext ) − E

ΔGCT(Fext ) − ΔGCT(0) = −ΔμCT Fext

D+

(Fext )

A−

− E (Fext ) + ΔEcoul

Thereby the influence of the external electric field on ΔGCT can be estimated by two different methods: by using eq 5 and eq 7, respectively. As we will see, eq 7 turns out to be a very efficient way to obtain correct ΔGCT(Fext). The reorganization energy includes two contributions: (i) the inner part λi, which describes the changes in the geometry of the donor and acceptor moieties upon charge transfer and

(6)

with ΔEcoul =

∑∑ D+

A−

qDqA 4πε0εsrDA

−

∑∑ D*

A

(7)

qD+q A− 4πε0εsr D+A−

Figure 2. The chemical structure of FQT dyad (a), fullerene adduct (b), and quarter-thiophene (c). Electric field is oriented along the x coordinate. C



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(ii) the outer part λs related to the change in electronic and nuclear polarizations that arise as a result of the whole (heterojunction and solvent molecule) geometric relaxation in the charge transfer process.31 The changes in geometry and charge distribution, induced by the application of the external electric field generated by the electrodes, are expected to modify mainly the inner part of the reorganization energy, on which we focus below.35 The inner reorganization energy λi arises from the change in equilibrium geometry of the donor and acceptor sites upon electron transfer and can be theoretically estimated with different approaches.8,12,31−33,35 Due to generally different potential energy curvature of the reactants and the products, here, two conditions are considered: in the first condition, λi1 corresponds to the difference between the energy of the reactants in the geometry characteristic of the products and that in their equilibrium geometry; in the second condition, λi2 corresponds to the difference between the energy of the products in the geometry characteristic of the reactants and that in their equilibrium geometry. The reorganization energy is calculated as the average of λi1 and λi2.12 In the case of exciton dissociation we obtain31 λ i = (λ i1 + λ i2)/2

dependent DFT (TD-DFT),42 long-range-corrected functional (CAM-B3LYP)43 and 6-31G(D) basis set, respectively. The long-range-corrected functional was employed for the nonCoulombic part of exchange functional. To investigate the effect of external electric field on the properties of excited-state of the molecules, the finite field method was employed, and the direction of the electric field is shown in Figure 2. The fields, ranging from −5 × 10−5 to 2 × 10−4 au, were used. This can be compared to the realistic strength of the electric field in the solar cell devices of up to 4 × 10−5 au (∼2 × 107 V/m). The field in a solar cell can be oriented in all possible directions. The real-space representation and site representation were described in our previous article.8

3. RESULTS AND DISCUSSION 3.1. Transition Energy and Character of the Excited States. We start with investigating the excited state properties without external electric fields. Results are summarized in Table 1. The electron−hole coherence (2D contour plot) of three kinds of electronic transitions can be seen in Figure 3, and the charge transfer is visualized by charge difference densities (CDD) presented in Figure 4. All the calculated excited states were carefully examined with these two representations, and classified as the intramolecular charge transfer (ICT) or the

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λ i1 = [ED *(Q P) + E A (Q P)] − [ED *(Q R ) + E A (Q R )]

Table 1. Selected Electronic Transition Energies (eV) and the Corresponding Oscillator Strengths (f), Main Compositions and CI Coefficients of FQT Dyad

(9)

λ i2 = [ED +(Q R ) + E A −(Q R )] − [ED +(Q P) + E A −(Q P)] (10)

states

where QR and QP refer to the equilibrium geometries of the reactants and products, respectively. We have evaluated all these terms at the DFT/TDDFT level. Here, all the parameters in eqs 8−10 are expected to depend on the strength of the applied external electric field, when the λi is Fext dependent. The treatment of the outer part of the reorganization energy λs will be discussed in the next section. 2.2. Quantum Chemical Calculations. All quantum chemical calculations in this work were done with Gaussian 09 software.36 The ground-state equilibrium geometries of the fullerene-linked quarter-thiophene (FQT) dyad, isolated fullerene adduct (F) and quarter-thiophene (QT) were optimized with density functional theory (DFT),37 B3LYP functional,38 631G(D) basis set. The molecular structures of FQT dyad, isolated fullerene adduct, and quarter-thiophene can be seen in Figure 2a−c, respectively. To study the reorganization energy and Gibbs free energy of charge transfer reaction in Marcus theory, the positively charged equilibrium geometry of isolated quarter-thiophene and the negatively charged equilibrium geometry of isolated fullerene adduct were also optimized with DFT, B3LYP functional, and 6-31G(D) basis set, and then the energy of the neutral/charged fullerene adduct, based on the optimized negative-charged/neutral equilibrium geometry were obtained from single point energy calculations at the same level of theory. Although previous theoretical studies have revealed that conventional hybrid B3LYP functional could be a sufficiently accurate way for calculating charge transfer excitedstates in some systems,39,40 the long-range-correction should be considered in quantum chemical calculations of large systems as the organic solar cell donor−acceptor dyad here.41 Therefore the electronic transitions of FQT and the geometry optimization of the lowest excited state of the isolated donor and the radical cation state were performed with time-

S1

transition energy (eV)a

fb

CIc

excited-state propertyd

2.3967 (517 nm) 2.4767 (501 nm) 2.5188 (492 nm) 2.5280 (490 nm) 2.7286 (454 nm) 2.7823 (446 nm) 2.7985 (443 nm) 2.8542 (434 nm) 2.9337 (423 nm) 3.0074 (412 nm) 3.0318 (409 nm) 3.0425 (408 nm) 3.1733 (391 nm)

0.0032

0.67319(H-1→L)

LE(F)

0.0000

0.59016(H-1→L+1)

LE(F)

0.0000

0.65049(H-2→L)

LE(F)

0.0000

0.67282(H-3→L)

LE(F)

0.0003

0.59078(H-4→L+1)

LE(F)

0.0002

0.53986(H-3→L+1)

LE(F)

0.0001

0.50999(H-4→L)

LE(F)

0.0024

0.54853(H-2→L+1)

LE(F)

0.0025

0.52015(H-2→L+2)

LE(F)

0.0160

0.56776(H-1→L+2)

LE(F)

0.0001

0.54112(H-3→L+2)

LE(F)

0.0005

0.43654(H-4→L+2)

LE(F)

0.7492

S14

3.2024 (387 nm)

0.7212

S15

3.2081 (386 nm)

0.1244

0.42743(H→L) 0.43815(H→L+3) 0.23058(H→L) 0.45865(H→L+3) 0.34851(H-4→L+2) 0.34455(H-2→L+2)

ICT LE(QT) ICT LE(QT) LE(F) LE(F)

S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13

a

The numbers in parentheses are the transition energy in wavelength. Oscillator strength. cCI coefficients are in absolute values. H stands for HOMO and L stands for LUMO. dF and QT in parentheses present that the density are localized on the fullerene adduct and quarter-thiophenes, respectively. b

D



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respectively. The transition from the ground state to the first excited state (S0→S1) can be clearly identified as LE(F). In order to investigate the effects of the external electric field on the transition energy, transition dipole moment, and other excited-state properties of the FQT dyad, the external electric field dependent excited-state characteristics were calculated by using the TDDFT method. The most relevant results for the evaluation of electron transfer are summarized in Table 2 and Figure 5. For all three selected excited-states, the transition energies decrease with increase of Fext (see Figure 5). The effect of Fext on S0→S13 transition energy is very pronounced. As can be seen from Table 2 and Figure 4, electronic character of the transitions (especially for S13 and S14) are sensitive to the external field Fext. The ICT character becomes more and more obvious for S13 excited state with the increase of the strength of the external electric field. When Fext is large enough, the S13 can be classified as an ICT state. The S14 behaves oppositely. It starts as mainly a LE(QT) transition with a considerable ICT character. The later diminishes while the field increases. In summary, the character of the S13 transition changes LE(QT)→ ICT and S14 character stays as LE(QT) with the minor ICT character decreasing while external electric field is increasing. The molecular FQT dyad, considered in this work, exhibits mainly an increase in permanent dipole moment upon excitation. Such dependence is demonstrated by calculating the excitation energy at zero field and at different field strengths ranging from 1 × 10−5 to 2 × 10−4 au and fitting a parabolic function eq 3 to the results (see Figure 5). As the S13 state is the lowest intermolecular charge transfer excited state, we only study the nonlinear effect for this state. The fit resulted in dipole moment change Δμ ∼ 6.7 D. The excess dipole moment was found to be in the direction of the long axis of the molecules studied. 3.2. Charge Transfer Integral. Equation 2 attributes the influence of the external electric field on the Vda to the field dependence of the various dipole moments and the transition energy. The relationship between Δμ and μij(Fext) at different strength of Fext plays a key role here. The fitting results of Δμ from eq 3 and the direct quantum chemical calculations of μij(Fext) for selected excited-states of FQT are plotted as a function of the amplitude of the external electric field in Figure 6. Two limiting cases, (Δμ)2 ≪ 4(μij(Fext))2 and (Δμ)2 ≫ 4(μij(Fext))2, can be considered. The external electric fielddependent electronic coupling Vda in eq 4 can be then simplified as

Figure 3. The 2D contour plots of the selected transition density matrix of the fullerene linked quarter-thiophenes dyad with different excited-state properties. The color bar is shown (absolute values of matrix elements, scaled to a maximum value of 1.0), and the electron− hole coherence increases with the increase of absolute values of matrix elements.

local excited (LE) transitions (see Table1). From the 2D contour plots of transition density matrix for ICT transitions, we can see the electron−hole coherence between quarterthiophene and fullerene of FQT dyad. Furthermore, from CDD, we can clearly identify electron transfer from thiophene to fullerene. 2D site and 3D cube representations also reveal localized excited states LE(F) and LE(QT), where electron and hole are both residing on fullerene and thiophenes, respectively. The strongest oscillator strength (f = 0.7492 and 0.7212) was found for the S0→S13 and S0→S14 transitions at about 400 nm, which is just around the strong absorption in the electronic absorption spectra of quarter-thiophene dyad.22 Without introducing external electric field, both these two transitions are found to be mixtures of LE(QT) and ICT with different CI coefficients. The proportion of LE(QT) and ICT components is for S 0 →S 13 : 0.44/0.43 and for S 0 →S 14 : 0.46/0.23,

Figure 4. Selected CDDs of FQT dyads at different external electric fields (Fext in 10−5 au). Green and red color represents the hole and electron, respectively. E


S14

S13

S1

TE f μij CI EP TE f μij CI EP CI EP TE f μij CI EP CI EP

2.3967 0.0032 0.2332 0.67319 LE(F) 3.1733 0.7492 3.1043 0.42743 ICT 0.43815 LE(QT) 3.2024 0.7212 3.0319 0.23058 ICT 0.45865 LE(QT)

Fext = 0


Fext = 1 2.3965 0.0032 0.2331 0.67326 LE(F) 3.1697 0.6355 2.8607 0.46146 ICT 0.40074 LE(QT) 3.2013 0.8605 3.3122 0.22692 ICT 0.49839 LE(QT)










Fext = 15


Fext = 20

Table 2. Selected Electronic Transition Energies (TE in eV) and Corresponding Oscillator Strengths (f), the Absolute Value of Transition Dipole Moment between Electronic States i and j (μij in au), the Dominating CI Coefficients and Corresponding Excited-State Properties (EP) of FQT Dyad under Different External Electric Field (Fext in 10−5 au)

The Journal of Physical Chemistry C Article

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Figure 7. The value of γ plotted vs the external electric field (in au). The S1, S13 and S14 represent the 1st, 13th, and 14th excited-states, respectively.

Figure 5. Excitation energy plotted vs electric field strength for the selected singlet excitation in FQT dyad. The S1, S13 and S14 represent the 1st, 13th and 14th excited-states, respectively.

the vector properties of Fext should be considered when the effect of external electric field is taken into account. 3.3. Reorganization Energy. For exciton dissociation to separate charges we start from initially excited QT. The reorganization energy has contributions from both electron donor and acceptor. The excited QT equilibrium geometry has to relax to the corresponding equilibrium in radical cation state. The corresponding values for fullerene are needed from the ground-state geometry to the radical anion geometry. The sum of these contributions adds up to λi2. λi1 corresponds to the backward process. If the reactant and product have identical symmetric potentials these two reorganization energies are equal. In general, they are not and commonly their average is used to calculate the reorganization energy.12 Thereby we obtain λiCT = 0.372 eV at Fext = 0. A detailed analysis of the calculation reveals that the most significant contribution to the reorganization energy of charge separation originates from pronounced geometric distortions of the conjugated thiophene molecule (QT) due to the D*→D+ transition. In the case of charge recombination, we count the energy required to promote QT from the geometry characteristic of the radical cation state to the ground-state geometry. The fullerene contribution is the same as for exciton dissociation. As a result we obtain λiCR = 0.240 eV at Fext = 0. The small value is rationalized by the fact that the geometry of QT is relative similar in the D+ and D states. Analogous to the previous studies,35 dependence of the inner reorganization energy, as a function of the external electric field is calculated and shown in Figure 8. The values of the inner reorganization energy of FQT dyad are found to decrease from 0.372 to 0.365 eV and from 0.240 to 0.234 for the exciton dissociation and charge recombination, respectively, following the increase of Fext at the range of −0.5−2.0 × 10−4 au. And the λiCT(Fext) and λiCR(Fext), at all the strengths of the selected Fext, are found to be mainly determined by the geometric distortions of QT. This is most probably due to the structural rigidity of fullerene molecule. The outer reorganization energy λs accounts for a substantial fraction of the total reorganization energy but it is difficult to obtain reliable quantitative theoretical estimate of it. It is possible to use a QM/MM44 or a polarizable force field45 to perform a direct evaluation of λs in the condensed phase. In this case the classical dielectric continuum models46 have to be resorted to.12 As a result, the substantial uncertainty of λs even for smaller molecules will be caused by the fact that the continuum model is intrinsically a crude approximation and the

Figure 6. The value of dipole moment plotted versus the external electric field (in au) The S1, S13, and S14 represent the 1st, 13th, and 14th excited-states, respectively. μ01, μ013, μ014 are the transition dipole moment from the ground state to the adiabatic electronic states S1, S13, S14, respectively.

⎡ ⎤ Δμ LE Vda (Fext ) = Vda(0)⎢1 − 2 Fext ⎥ = γVda(0) ΔE(Fext = 0) ⎦ ⎣

⎛ ⎞ (Δμ)2 ICT Vda (Fext ) = Vda(0)⎜⎜1 − Fext ⎟⎟ ΔE(Fext = 0)μij (Fext ) ⎠ ⎝ = γVda(0)

(11)

The first expression is valid for LE ((Δμ)2 ≪ 4(μij(Fext))2) and the second for the ICT state ((Δμ)2 ≫ 4(μij(Fext))2), the later at higher field strengths. In order to further investigate the influence of Fext on the electronic coupling matrix element Vda, we present the calculated γ values of selected excited states of FQT dyad using TDDFT data at different external electric fields. Note that the polymer bulk heterojunction is structurally disordered without any preferred orientation of the field in respect of the junction. Therefore we carry out the calculations for both positive and negative values of the field. As can be seen from Figure 7, the changes of Vda with the field (visualized via γ) for LE(F) and LE(QT) are rather small. At the same time for the ICT state the coupling strength between donor and acceptor increases substantially following the increasing of Fext. The values of γ < 1 are found when the electric field is in the opposite direction to the current as shown in Figure 2. Clearly, G



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Figure 8. Inner reorganization energy plotted versus the external electric field.

uncertainty in the dielectric constant caused by the different experimental conditions.33 In most cases the outer contribution to the reorganization energy is of the same order of magnitude as the inner part, and it is expected to be less sensitive to the chemical structure of the constituents.31 Recently, several authors leave λs as an adjustable parameter or use an order of magnitude estimate.47 In large systems like porphyrin-fullerene dyad, the outer reorganization energy of electron transfer reaction is 0.23 + 0.11 eV,48 in the photosynthetic reaction center ∼0.2 eV,13,49 and in the case of corrole-fullerene dyads in nonpolar solvent, the overall reorganization energy is 0.5 eV.50 On the basis of these results we choose λs to be equal to the inner reorganization energy. Thus, we take the overall reorganization as twice the inner reorganization energy at all strengths of Fext. The total reorganization energies λCT(Fext) and λCR(Fext) calculated in this way remain nearly unaffected by the application of the external electric field, as can be seen from Figure 8. It should be noted that we consider λs as the most ill-determined parameter in our study. 3.4. Free Energy Change. The external electric field dependent driving force for exciton dissociation and charge recombination is reported in Figure 9a. In the panel (a) ΔGCT(Fext) for exciton dissociation upon donor photoexcitation and ΔGCR(Fext) for charge recombination are calculated using eq 5 and 6. ΔGCT(Fext) becomes increasingly negative when the external electric field is increased (going from −0.64 eV at Fext = −0.5 to −0.79 eV at Fext = 2 × 10−4 au), and the decline reflects the size of electrostatic interaction between the separated charges and the electric field. At the same time, as expected, ΔGCR(Fext) becomes less negative following the increase of the electric field. The sum of ΔGCT(Fext) and ΔGCT(Fext) is consistent with the energy of the lowest excited states of QT at its corresponding strength of the electric field. This can be well understood in terms of potential energy curves shown in Figure 1. Also the external electric field dependent ΔG = ΔGCT(Fext) using eq 7 were calculated. The calculated ΔGCT(Fext) from eq 5 and 7 are plotted as a function of the external electric field in Figure 9b. The two methods provide almost identical field dependence. Clearly eq 7 provides a reliable method to evaluate the external electric field dependent free energy. When using the eq 7, optimization of the structure of the ground-state donor−acceptor system at different external electric field is not used. This will speed up calculations considerably. By

Figure 9. Gibbs free energy of the exciton dissociation and charge recombination reaction (ΔGCT and ΔGCR) plotted versus the external electric field: (a) calculated by using eq 5 and eq 6; (b) calculated by using eq 7.

investigating the validity of the eq 7 for describing the external electric field dependent ΔGCT(Fext), we conclude that Fext leads to orientation-dependent photoinduced charge-separation kinetics of the FQT complex. The external electric field can decrease the charge transfer efficiency in randomly oriented system, whereas the oriented charge-separation along the direction of Fext is enhanced. This makes ΔμCT (the change in the permanent dipole moment upon electronic transition) a key parameter to determine the effect of Fext on the free energy gap, as shown in eq 7. 3.5. External Electric Field Dependent Rate of Charge Transfer and Charge Recombination. We are now in a position to evaluate the external electric field dependent rates of exciton dissociation and charge recombination by inserting all the calculated parameters into eq 1. The rates for S13 (the lowest ICT excited states of FQT) can be seen from Figure 10a. The rate of exciton dissociation in this excited-state increases by a factor of 9 when varying the electric field from Fext = −5 × 10−5 to 2 × 10−4 au. A detailed analysis reveals that the charge transfer process occurs in the normal Marcus region when the field is small, since |ΔGCT | < λCT (0.69 versus 0.75 eV at Fext = 3 × 10−5 au, which is a typical realistic fields in solar cell devices). The rate gets faster when the strength of the external electric field is increased mainly because of the increase in electronic coupling but even due to the changes of ΔGCT and λCT. The transfer rate increases significantly when Fext > 7 × 10−5 au, which is partially because the |ΔGCT| and λCT converge toward a similar value in this region. It should be noted that the calculated charge separation rates (>1013 s−1) correspond to the limit where Marcus theory is not valid any more. The weak interaction condition between donor and acceptor molecules is not satisfied. Consequently the rate is so high that memory H



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This result is in good agreement with recent charge separation studies using time-resolved electric field induced second harmonic generation.55 The work concluded that the influence of the external field on the initial charge separation is minor, and the external electric field mainly influences the charge collection via high carrier mobility in polymers.56 We plot in Figure 10b also the evolution of the rate for exciton dissociation in the case of ΔGCT obtained from eq 7. We observe that, the charge transfer rates are almost unchanged as compared to that calculated from eq 1 with all the parameters depended on the electric field. It again testifies the validity of eq 7. Finally, we stress that the changes in the electron transfer rates rate with external electric field are mainly due to the charge transfer integral. The value of reorganization energy (Δλ ∼ 0.014 eV) and free energy (ΔGCT∼0.15 eV) are not very sensitive to the external electric field (see Figure 8 and 9). Most importantly, our results provide a new framework to understand charge transfer of organic systems under the external electric field, and the connection between the external electric field, chemical structure, and optical and electronic properties of the donor−acceptor system outline the basis for the design of improved organic photovoltaic (OPV) cells.

4. CONCLUSION We have analyzed the external electric field dependent rate of photoinduced charge transfer for donor−acceptor organic solar cells. It is found that the field mainly influences the electronic transition energies and charge transfer integral for the charge transfer excited states, but there is little influence on the electronic transition energy for the local excited states, inner reorganization energy, and free energy changes. We thoroughly analyze external field dependent charge transfer integral (Vda), reorganization energy (λ), free energy (ΔG), and the rate of charge transfer (kCT).

Figure 10. Calculated rate of exciton dissociation and charge recombination of the 13th excited-state of FQT dyad at different external electric field: (a) calculated by using eq 1; (b) calculated when ΔGCT is obtained from eq 7

effects of the interaction with the environment need to be considered. However, we expect that the qualitative trends of the calculated rates what we present will be valid.8 All calculated parameters (electronic couplings, reorganization energies, free energy of the reaction) remain unaffected by these considerations. We found that the rate of charge recombination is significantly smaller than the corresponding rate of charge separation (see Figure 10a). The extreme slowness of the recombination process originates from the vanishingly small electronic couplings and from the fact that recombination occurs deep into the inverted Marcus region (|ΔGCR | = 1.86 eV ≫ λCR = 0.48 eV, at Fext = 0).12 Simultaneously, with the formation of the charge-separated state (D+A−), the Coulombic attraction pulls the charges back together to undergo first order geminate recombination.51,52 Our analysis reveals that, exciton dissociation of the FQT dyad is thermodynamically favored, and the driving force is sufficiently strong to overcome the Coulombic attraction between the charges even at Fext = 0. As a result, the photogenerated charges can easily escape from their mutual attraction and yield free extractable charges. It has been also proposed that long-range exciton dissociation can occur even before the excitons reach the donor−acceptor interface of the organic solar cell.53 The hole delocalization can further reduce the Coulomb attraction effect and accelerate the charge transfer process.10 These results are in accord with the recent study revealing enhanced probability of charge separation from highly delocalized hot interfacial charge transfer states.54 For the typical electric field values for the organic solar cells the calculated charge separation rate variations are quite small.

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AUTHOR INFORMATION

Corresponding Author

*(M.T.S.) Tel: +86 10 82640779; Fax: +86-10-82649007; Email: [email protected]. (T.P.) E-mail: tonu.pullerits@ chemphys.lu.se. (F.C.M.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation of China (Grant No. 11274149), the Program of Shenyang Key Laboratory of Optoelectronic Materials and Technology (Grant No. F12-254-1-00), the Natural Science Foundation of Liaoning Province (Grant No. 20111035), the Fundamental Research Funds for the Central Universities (Grant No. DL12BB19) and the Research Project of Education Department of Heilongjiang Province (Grant No. 12533008). T.P. acknowledges financial support from STEM and the KAW Foundation.

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External Electric Field-Dependent Photoinduced Charge Transfer in a

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