Photoinduced Charge Transport in a BHJ Solar Cell Controlled by an External Electric Field

This study investigated theoretical photoinduced charge transport in a bulk heterojunction (BHJ) solar cell controlled by an external electric field. Our method for visualizing charge difference density identified the excited state properties of photoinduced charge transfer, and the charge transfer excited states were distinguished from local excited states during electronic transitions. Furthermore, the calculated rates for the charge transfer revealed that the charge transfer was strongly influenced by the external electric field. The external electric field accelerated the rate of charge transfer by up to one order when charge recombination was significantly restrained. Our research demonstrated that photoinduced charge transport controlled by an external electric field in a BHJ solar cell is efficient, and the exciton dissociation is not the limiting factor in organic solar cells.Our research should aid in the rational design of a novel conjugated system of organic solar cells.


Results
Properties of the excitation state. A C 60 molecule, also known as a buckyball, consists of 60 carbon atoms and has a good electron affinity. A BT molecule (see Fig. 1), with an absorption region ranging from 300 to 800 nm, mainly consists of methine (= CH-). Owing to its broad spectral response range, a BT molecule has a strong ability to capture light. In this study, we combined a BT molecule and a PC 61 BM molecule to investigate the properties of a BT-PC 61 BM molecular complex in an excited state.
To simulate the optical properties of a BT-PC 61 BM molecule, the excited state electronic transitions were calculated using the time-dependent density functional theory (TDDFT) method, and a long-range-corrected functional (CAM-B3LYP) was employed for the non-Coulombic part of the exchange functional. The relevant results for the calculated absorption spectroscopy of the BT-PC 61 BM molecule are shown in Fig. 2. The BT-PC 61 BM molecule had two clear absorption peaks at 605 and 353 nm. The main absorption band was in the range 300 to 800 nm, demonstrating strong absorption of visible light. The vertical lines indicate the oscillator strength of BT-PC 61 BM, whereas the blue lines express the intramolecular electric charge transfer in PC 61 BM and the red lines express the intermolecular electric charge transfer between the BT molecule and PC 61 BM molecule. Furthermore, the intermolecular electric charge transfer occurs in the range 300 to 400 nm, and an absorption peak approximately 600 nm arises from the action of the BT molecule. Table 1 lists the vertical excitation energies and oscillator strengths for the thirty excited states, as calculated using the TDDFT method based on the optimized  ground-state structure of the BT-PC 61 BM molecule, and these data also support our conclusion. The first four excited states of BT-PC 61 BM correspond to intramolecular electric charge transfer, whereas the fifth and sixth excitation states correspond to electric charge transfer between the BT molecule and the PC 61 BM molecule. In this study, we focused on the intermolecular electric charge transfer associated with the fifth and sixth excitation states. The 3D real-space analysis method has been used to analyse the charge transfer in conjugated polymers because the charge difference density (CDD) determines the orientation and results of the charge transfer in molecular and molecular-metal systems 13 . As shown in Fig. 3, two types of excited state exist in the absorption band of BT-PC 61 BM: one is the strong resonance charge transfer (CT) excited state, in which electron transfer occurs in the strong absorption region, and the other is weak resonance charge transfer. S 22 and S 23 are strong absorption excited states. S 23 and S 24 are pure intermolecular electric charge transfer excited states in which the electrons are mainly localized in the PC 61 BM molecule, and the holes are mainly localized in the main chain of the BT molecule, which demonstrates that electrons will be completely transferred from the BT molecule to the C 60 molecule. S 4 , S 5 , S 6 and S 7 are weak absorption excited states. In the S 4 excited state, the electric charge is redistributed in the PC 61 BM molecule. The electrons and holes are mainly located in C 60, which demonstrates that the S 4 excited state is a localized excited (LE) state. For the S 5 and S 6 excited states, electrons are mainly transferred from the main chain of the BT molecule to C 60 . However, states S 5 and S 6 are not pure intermolecular electric charge transfer excited states, as some of the holes are still located in C 60 . Nevertheless, we are certain that S 5 and S 6 are the lowest intermolecular electric charge transfer states. Figure 3 shows the qualitative visualization analysis with 3D charge difference density for charge transfer. To provide quantitative analysis of the charge transfer, Δ r is introduced to measure charge-transfer length 18 ,  where i and j traverse all of the occupied and virtual molecular orbitals, respectively, and ϕ is the orbital wave function. The larger Δ r is the stronger charge transfer. The calculated results are summarized in Table 1, in which the S 5 and S 6 states have larger charge-transfer lengths, 3.497 Å and 4.502 Å, respectively. These calculations, along with the qualitative visualization analysis with 3D charge difference density, justify the previous conclusion that the excited-states S 5 and S 6 are the lowest intermolecular electric charge transfer states. The excited state S7 (with S6) is degenerate and is also an intermolecular charge transfer excited state.
Charge difference density and Δ r can be used to qualitatively and quantitatively analyse the charge transfer properties of the complex molecule in this study, but the electron-hole coherence on electronic transition cannot be demonstrated. In the two-dimensional (2D) site representation, photoexcitation creates an electron-hole pair or exciton by moving an electron from an occupied orbital to an unoccupied orbital. Each element of the transition density matrix reflects the dynamics of an exciton projected on a pair of atomic orbitals, indicated by indices, and increases the probability of finding one charged particle on site x and the second one on site y. The number of charged particles reflects the strength of the coherence between the donor and acceptor, which is defined by different colours of the element 15 . Figure 4 visualizes the electron-hole coherences, spatial span and primary sites of electron transitions. For S 1 and S 2 , the electron-hole coherences are strong in the BT molecule and PC 61 BM, respectively. They are the π-π* transition of the inner donor and acceptor, as the electrons and holes are all localized in those units (see Fig. 4a,b). Therefore, S 1 and S 2 should be categorized as LE states. Furthermore, S 6 is an intermolecular charge transfer excited state owing to the electron and hole coherence between BT and PC 61 BM. In other words, the electrons strongly cohere with holes between the the inner region of the BT molecules and the outer left region of PC61BM inner BT and left outer PC 61 BM, but this coherence is weak between the the inner region of the BT molecule and the outer right region of PC61BM (see the 2D transition density matrix in Fig. 4c).  The charge transfer integral (electronic coupling matrix). The electronic coupling strength directly influences the electron transfer rate and determines the method of electron transfer. In this study, we used the generalized Mulliken-Hush (GMH) method to calculate the electronic coupling strength 19 .
The expression is written as where μ tr is the transition dipole moment, Δ μ is the difference in the dipole moments between the S 0 and S n states, and Δ E is the vertical excitation energy. Δ μ can be obtained by the finite electric field method 20,21 . The external electric field F ext is written as where E exc (0) = Δ E = E j -E i is the excitation energy at the zero field and Δ α is the polarizability. There is widespread consensus that hyperpolarizabilities can be simulated by using the quadratic response theory method. This approach was widely used for studying two-proton absorption (TPA) and to investigate the nonlinear optical (NLO) properties of nonlinear optical materials [22][23][24][25][26][27] . An experimental aspect, second harmonic generation (SHG), has been used to study photoinduced charge separation dynamics in organic semiconductor thin films using the time-resolved spectra technique. Some meaningful results have been obtained, including charge separation from a gradient in excitation density and differential electron/hole mobility in model systems of fullerene (C70) and semiconductor interfaces 28,29 . However, it is reasonable to believe that using quadratic response theory will not considerably affect our qualitative results for photoinduced charge transport efficiency controlled by external electric field. Therefore, the second harmonic generation (SHG) effect was not considered with the time-dependent density functional method. Figure 5 shows a nonlinear effect for S 5 and S 6 , which are the lowest intermolecular electric charge transfer excited states. The data were obtained with the excited state electronic transitions energies calculation for the BT-PC 61 BM molecular system using the TDDFT method, and the difference in the dipole moments was obtained by fitting the calculated results of the S 5 and S 6 excited states with Eq. (3). Our calculations show the lowest intermolecular electric charge transfer state changes when we increase or decrease the external electric field. When the external electric field is F ext = 1 × 10 −4 au, the lowest intermolecular electric charge transfer state is the S 5 or S 6 excited state. As the external electric field increases to 5 × 10 −4 au, the lowest intermolecular electric charge transfer excited state changes to the S 2 or S 3 excited state. As the external electric field decreases to − 5 × 10 −4 au, the lowest intermolecular electric charge transfer excited state changes to the S 8 or S 9 excited state. We first visualized and analysed the excited states under different external electric fields to determine the lowest intermolecular electric charge transfer states; then, we determined the fit for the lowest intermolecular electric charge transfer states (see Fig. 5(b)). According to Eq. (3), the differences in dipole moment (Δ μ) for CT 1   importance. The free energy (Δ G) of exciton separation is expressed as Δ G CT , and the free energy of electric charge recombination is expressed as Δ G CR [30][31][32][33][34][35][36] . Δ G CR is written as where E IP (D) is the ionization potential of electrons in a donor and E EA (A) is the ionization potential of holes in an acceptor. The calculated E IP (D) and E EA (A) are the energies of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of the donor and acceptor, respectively. These quantities are normally estimated from the geometry optimization of the isolated PC 61 BM adduct and BT by using the DFT method. The calculated Δ G CR is − 1.5060 eV. Δ G CT can be obtained by the Rehm-Weller equation: where Δ E 0-0 is the lowest excited state energy of a free radical donor and E b is the exciton binding energy. The exciton binding energy is taken as the difference between the electronic and optical band gap energies. The electronic band gap can be approximated as the energy difference of HOMO and LUMO. The calculated E b is 0.2685 eV and Δ G CT is − 0.7686 eV. The negative value of Δ G CT means that electron transfer is thermodynamically favourable.
We used the Marcus model to calculate the rate of exciton separation and charge recombination, respectively. Figure 6(a) shows the calculated exciton separation rate for BT-PC 61 BM. The exciton separation rate increases by one order of magnitude with the external electric field. According to the above calculations, the increase in exciton separation occurs due to the increase in Δ G CT and λ CT caused by the increase in the external electric field. From previous studies, we know that during the formation of the charge-separated state (D + A − ), the Coulomb attraction pulls the charges back together to undergo first-order geminate recombination 37,38 , that is the electron and hole may be trapped in each others' Coulomb well and forces the electric charge to recombine from the electric charge separation state at longer timescales. Also, this electric charge recombination process may occur again during exciton transport. As shown in Fig. 6(b), the change in the charge recombination rate calculated by the Marcus model with an external electric field is different from the exciton separation rate. The charge recombination rate decreases linearly with increases in the external electric field. By comparing Fig. 6(a,b), one can clear find that the initial CT dissociation rate is much larger than the recombination rate. And the relative slowness of the charge recombination rate originates from the vanishingly small electronic couplings and because recombination occurs deep in the inverted Marcus region (|Δ G CR | = 1.5060 eV ≫ λ CR = 0.4882 e V, at F ext = 0). Additionally, the charge recombination rate decreases by 6 orders when the external electric field changes from − 5 × 10 −4 au to 5 × 10 −4 au, which is larger than exciton separation rate. The calculated charge separation rate variations are quite small for typical electric field values for organic solar cells. This result agrees with recent charge separation studies using time-resolved electric-field induced second harmonic generation 39 . The researchers concluded that the influence of the external field on the initial charge separation was minor and that the external electric field mainly influenced the charge collection via high carrier mobility in polymers 40 .
Finally, the hot charge transfer exciton mechanism in organic photovoltaics may offer an alternative explanation for our computational results that the exciton separation rate is much higher than that of charge recombination 41 . The excess energy from charge separation results in a charge pair at an initial distance in the Coulomb potential. If this initial distance is longer than the critical escape distance, where the Coulomb binding energy is less than the thermal activation energy, charge separation occurs. Otherwise, the charge pair relaxes towards contact and probably eventual recombination 5 . For polymer/ fullerene interfaces, the excess energy of hot CT excitons may still play a role in assisting charge separation, even at sufficiently low temperatures 42 . This is mainly because hot CT excitons have relative weaker bonds from the Coulomb potential, and these excitons can be easier to dissociate. Electronic coupling from a donor exciton to a hot charge transfer exciton across the D/A interface can also be higher than the normal charge transfer expected from energy resonance. As a result, hole delocalization can further reduce the Coulomb attraction effect and accelerate the charge transfer process 43 . According to the results of our theoretical simulation on the BT-PC 61 BM complex, the calculated E b is 0.2685 eV and Δ G CT is − 0.7686 eV respectively, excitons can sufficiently separate into electrons and holes at F ext = 0 from the interfacial hot CT excitons perspective. Therefore, for this photoactive material in organic solar cells, the excitons separate into electrons and holes before they arrive at the donor-acceptor interface. But, according to our results, we must state that exciton dissociation is not the limiting factor in organic solar cells. Noted that, whether the interfacial energy gradient can overcome the Coulomb trap or not clearly should be system specific. These conclusions can greatly improve the photoelectric transformation efficiency of organic solar cells and decrease the loss of electrons in the transport process. Our findings are in accordance with a recent study revealing the enhanced probability of charge separation from highly delocalized hot interfacial charge transfer states 44 , and provide an important design principle for new organic photovoltaics materials.

Discussion
In this study, the excited states of BT and PC 61 BM molecules were examined with a quantum-mechanical method. The BT-PC 61 BM complex has a broader spectrum response region than regular response regions, and strong absorption peaks are located in the visible light range. The lowest electric charge transfer state changes as the external electric field changes. The BT-PC 61 BM complex molecule has a high electric charge separation rate (3.1334 × 10 13 s −1 ) and a low electric charge recombination rate (1.4221 × 10 5 s −1 ) according to the Marcus model at F ext = 0. After adding an external electric field, the charge recombination rate calculated with the Marcus model decreased linearly as the external electric field increased. The charge recombination rate decreased by 6 orders when the external electric field increased from − 5 × 10 −4 au to 5 × 10 −4 au. The exciton separation rate increased with increases in the external electric field. The exciton separation rate is increased by one order.

Methods
Charge transfer rate. For nonadiabatic and adiabatic reactions, the charge transfer is handled using different approximations. We chose the classic Marcus mode. In classic Marcus theory, Marcus proposed three assumptions about an adiabatic reaction: 1) The reactants have the same energy as the products, and the energy is conserved before and after the state-to-state transition; 2) The Frank-Condon principle is met; 3) The energy split is large, and the transition probability is 1 at the position of the transition state 45 .
Exciton dissociation and charge recombination relate to the electron transition reaction. In the semi-classical limit from Marcus theory, the charge transfer rate is expressed as where k B is the Boltzmann constant, h is Planck's constant, V da is the electronic coupling between the initial and final states, λ is the reorganization energy, and T is the temperature (T = 300 K).
Reorganization energy. The reorganization energy is an important parameter for characterizing the electron and energy transfer. According to the rate expression (Eq. (6)), the rate reaches its maximum value when the reorganization energy is at its minimum value [46][47][48] . In our research, we focused on the relationship between the inner reorganization energy and electron transfer. Before and after the electron transfer, the entire system will relax at a new steady state. The dissipative energy in the relaxation process is the reorganization energy. In other words, the reorganization energy is the relaxation energy. The reorganization energy is written as where λ i is the internal reorganization energy arising from the change in equilibrium geometry of the donor and acceptor sites upon electron transfer and λ s is the outer reorganization energy. The inner reorganization energy consists of two sections 49,50 : Scientific where E(A − ) and E(A) are the energies of the neutral acceptor A at the anion geometry and the optimal ground-state geometry, respectively, and E(D) and E(D + ) are the energies of the radical cation at the neutral geometry and optimal cation geometry. λ s in Eq. (7) is the outer reorganization energy, which is related to the change in electronic and nuclear polarizations in electron transfer. The outer reorganization energy can be expressed as where R D and R A are the radii of the donor and acceptor, respectively. q D and q A are the electric charge of the donor and acceptor, respectively. r DA is the donor-acceptor distance. ε 0 is the dielectric constant in a vacuum. ε op is the optical dielectric constant of the surrounding media and ε s is the relative dielectric constant of the molecule 51-54 .
Quantum chemical calculation method. BT and PC 61 BM were chosen as the photoactive materials for organic solar cells. All quantum chemistry calculations were performed by Gaussian 09 software 55 .
The ground-state geometries of the BT-PC61BM molecule, isolated PC 61 BM adduct and BT were optimized using density functional theory (DFT) 56 , the B3LYP functional 57,58 , and the 6-31G(d) basis set. The molecular structures of isolated BT and PC 61 BM are shown in Fig. 1. To calculate the reorganization energies of the charge transfer reaction in Marcus theory, the cationic ground state geometry of isolated BT and the anionic ground state geometries of the isolated PC 61 BM adduct were also optimized with DFT, the B3LYP functional, and the 6-31G(d) basis set. Then, the single point energies of these two neutral acceptors at the anionic geometry and optimal ground-state geometry, as well as the single point energies of the radical cation at the neutral geometry and optimal cation geometry, were calculated at the same level of theory. Although previous theoretical studies have revealed that the conventional hybrid B3LYP functional could be sufficiently accurate for calculating charge transfer excited-states in some systems 59,60 , the long-range-correction should be considered in quantum chemical calculations of large systems, such as the organic solar cell donor-acceptor heterojunction in this study 61 . Therefore, to simulate the optical absorption properties, a calculation of the excited state electronic transitions BT-PC 61 BM molecular system was performed using time-dependent density functional theory (TDDFT) 62 , the long-range-corrected functional (CAM-B3LYP) 63 , and the 6-31G(d) basis set. The long-range-corrected functional was employed for the non-Coulombic part of the exchange functional. Furthermore, geometry optimization of the lowest excited state of the isolated donor and the lowest excited state of the radical cation state was performed with TD-DFT, CAM-B3LYP, and the 6-31G (d) basis set. The Generalized Mulliken-Hush (GMH) model was employed to calculate the charge transfer integral (electronic coupling matrix). To investigate the effect of an external electric field on the excited-state properties of the molecules, the finite field method was employed, and the direction of the electric field is shown in Fig. 1. Fields ranging from − 5 × 10 −4 to 5 × 10 −4 au were used. This result 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. To visualize charge transfer during electronic transitions, two-dimensional (2D) site space analysis (transition density matrix) and three-dimensional (3D) real space analysis (charge difference density) were performed, which were described in detail in our previous article 13 .