# Fullerene-Based Photoactive Layers for Heterojunction Solar Cells: Structure, Absorption Spectra and Charge Transfer Process

^{1}

^{2}

^{*}

## Abstract

**:**

_{3}, [70]PCBM/APFO

_{3}and [60]PCBM/APFO

_{3}, were studied with density functional theory (DFT), and the vertical excitation energies were calculated within the framework of the time-dependent DFT (TD-DFT). Visualized charge difference density analysis can be used to label the charge density redistribution for individual fullerene and fullerene/polymer complexes. The results of current work indicate that there is a difference between [60]PCBM and [70]PCBM, and a new charge transfer process is observed. Meanwhile, for the fullerene/polymer complex, all calculations of the twenty excited states were analyzed to reveal all possible charge transfer processes in depth. We also estimated the electronic coupling matrix, reorganization and Gibbs free energy to further calculate the rates of the charge transfer and the recombination. Our results give a clear picture of the structure, absorption spectra, charge transfer (CT) process and its influencing factors, and provide a theoretical guideline for designing further photoactive layers of solar cells.

## 1. Introduction

_{3}([6,6]phenyl-C71-butyric acid-methyl ester) and [60]PCBM/APFO

_{3}on the basis of experimental report [21]; the name APFO

_{3}is the abbreviation of APFO

_{3}(poly[2,7-(9,9-dioctylfluorene)-alt-5,5-(4,7′-di-2-thienyl-2′,1′,-3-benzothiadiazole)]. The parameters affecting charge transfer and charge recombination, were estimated and compared. Moreover, the developed 3D real-space analysis was used to investigate the excited states feature and charge transfer properties of the binary system.

## 2. Methods

_{3}, [70]PCBM/APFO

_{3}and [60]PCBM/APFO

_{3}can be seen from Figure 1. The side chains of APFO

_{3}were replaced by hydrogen atoms in order to save computational cost, on consideration that they merely aid in improving solubility and have negligible influence on optical properties [23,24]. Although the omission of the side chains is a common decision in this field, it should be done with caution because the side chains can affect conformational torsion of the backbone of some oligomers [25]. The ground state geometries were optimized with density functional theory (DFT) [26], using B3LYP functional [27,28,29] and 6-31G (D) basis set. For the calculations of inner reorganization energies, the cationic ground state geometry of APFO

_{3}, and anionic ground state geometries of [70]PCBM and [60]PCBM were optimized, using the DFT//B3LYP/6-31G(D). Then the energies of neutral acceptors at the anionic geometry and the optimal ground-state geometry were calculated by using the DFT//B3LYP/6-31G(D), respectively; and the energies of the radical cation at the neutral geometry and optimal cation geometry were calculated on the same functional and basis set. Based on the optimized neutral structures, the time-dependent DFT (TD-DFT) method [30] with long-range corrected functional Cam-B3LYP [31] and basis set 6-31G (D) was used to obtain the optical absorption properties. To calculate the charge transfer integral (electronic coupling matrix), the Generalized Mulliken-Hush (GMH) model and the finite field method on the excitation energy of the donor-acceptor heterojunction were employed (which will be discussed below).

**Figure 1.**Structures of APFO

_{3}(poly[2,7-(9,9-dioctylfluorene)-alt-5,5-(4,7′-di-2-thienyl- 2′,1′,-3-benzothiadiazole)] (n = 1 and n = 2) and C70-fullerene based acceptor.

_{i}(r) and the unoccupied $\varphi $

_{a}(r) orbital [32,33]; in this equation the first and the second terms stand for hole and electron, respectively.

## 3. Results and Discussion

#### 3.1. Energy Levels and Band Gap

_{3}. The LUMO of C70P is slightly higher than that of C60P. While, the LUMOs of the binary system are closed to that of fullerenes, their HOMOs verge on HOMOs of APFO

_{3}, which leads to charge transfer controlling by transition from HOMO to LUMO and can take place from APFO

_{3}to fullerenes. Compared to the isolated donor or acceptor, the donor-acceptor complex has a decreased trend of HOMO-LUMO band gap.

**Figure 2.**Energy levels of polymers and fullerene, where (APFO

_{3})

_{n=1}, (APFO

_{3})

_{n=2}, [C70]PCBM, [C60]PCBM, [C60]PCBM/(APFO

_{3})

_{n=1}and [C70]PCBM/(APFO

_{3})

_{n=1}are abbreviated as A(n = 1), A(n = 2), C70P, C60P, C60P-A and C70P-A, respectively.

#### 3.2. Optical Absorption of Donor, Acceptor and the Donor-Acceptor Complex

_{3}, vertical excitation energies and oscillator strengths for the five excited states were calculated, which are listed in Table 1. For n = 1 and n = 2, the absorption spectra cover the UV-visible region, and have one common property, i.e., their first excited state (S

_{1}) has high oscillator strength, compared to the other energetically low lying states. Transition density in Figure 3 shows the strength and orientation of the transition moment for calculated excited states. For n = 1, red electrons are mainly located on the left unit and green holes reside on the right unit, and thus the transition moment is singlet direction. In comparison, the orientation of the transition moment for n = 2 is unchanged, and the electron and hole are distributed over two monomers, which results in the increased strength of the transition moment. Due to the proportional relationship between oscillator strength with the transition energy (E

_{ge}) and transition moment (μ

_{ge}), $f=(8{\text{\pi}}^{2}{m}_{e}/3{\text{e}}^{2}h){E}_{ge}{\text{\mu}}_{ge}^{2}$ [36,37], APFO

_{3}(n = 2) displays a larger oscillator strength than APFO

_{3}(n = 1) under the condition of similar transition energy (Table 1). The week absorption of S

_{2}can be explained by TD analysis, and Figure 3 shows there are the two sub-transition dipole moments with the “tail to tail” character since more holes are mainly localized on both sides of APFO

_{3}, which results to a large extent in the weakness of the total transition dipole moment. So the total transition dipole moment of S

_{2}state is smaller than that of S

_{1}state. Turning to the charge transfer character of APFO

_{3}, the redistribution of electron density during photo-excitation was visualized with charge difference density (see Figure 3). It was found that S

_{1}and S

_{2}have some intramolecular CT character, where electron transfer is transferred from two-sided fluorene and thiophene units to the middle unit; while the S

_{3}state at 3.78 eV is essentially an $\text{\pi}-\text{\pi}*$ excited state.

**Table 1.**Calculated transition energies (eV, nm) and oscillator strengths (f) for polymer (n = 1 and n = 2).

States | n = 1 | n = 2 | Experiment | ||
---|---|---|---|---|---|

eV (nm) | f | eV (nm) | f | nm | |

S_{1} | 2.48(500.84) | 1.3006 | 2.40(515.68) | 2.8379 | 540 |

S_{2} | 3.51(353.23) | 0.0299 | 2.52(491.41) | 0.0625 | – |

S_{3} | 3.78(328.04) | 1.3606 | 3.40(364.33) | 0.1901 | 384 |

S_{4} | 4.09(302.68) | 0.0862 | 3.49(355.25) | 0.0006 | – |

S_{5} | 4.31(287.55) | 0.0023 | 3.67(337.96) | 1.7567 | – |

**Figure 5.**Charge difference density (CDD) of [C70]PCBM, where the green and red stand for the hole and electron, respectively.

_{3}and [C70]PCBM/APFO

_{3}are shown in Figure 6, and transition energies and oscillator strengths are listed in Table 2. For [C60]PCBM/APFO

_{3}, its excited states are classed as three kinds of excitation, in which S1 and S3 states represent two typical locally excited states. Table 2 shows that the strongest absorption peak of [C60]PCBM/APFO

_{3}corresponding to S3 state with f = 1.1259, and electron-hole pairs is located on APFO

_{3}(for S3). This state is a local-excited state; however, intramolecular charge transfer takes places on the molecular skeleton of APFO

_{3}, which displays the same character as the CT states of APFO

_{3}monomer. The S1, S2, S4–S9, S11, S13, S14, S15, S17–S20 states are local-excited states by exciting C60 (See Figure S2). Additionally the lowest intermolecular charge transfer excited state is the S10 state, peaking at 433 nm (Figure 6); this state can be expected to undergo a direct electron transfer from donor to acceptor, resulting in the charge separation. Similar CT excited states are found to be S12 and S16 states (See Figure S2).

**Table 2.**Calculated transition energies (eV, nm) and oscillator strengths (f) for [C60]PCBM/APFO

_{3}and [C70]PCBM/APFO

_{3}, respectively.

States | [C60]PCBM & APFO_{3} | [C70]PCBM& APFO_{3} | ||
---|---|---|---|---|

eV (nm) | f | eV (nm) | f | |

S_{1} | 2.42(511.43) | 0.0017 | 2.27(545.29) | 0.0014 |

S_{2} | 2.46(504.84) | 0.0026 | 2.45(506.59) | 0.1925 |

S_{3} | 2.48(500.52) | 1.1259 | 2.48(500.88) | 0.9239 |

S_{4} | 2.53(490.32) | 0.0004 | 2.61(474.33) | 0.0127 |

S_{5} | 2.55(486.25) | 0.0000 | 2.66(466.35) | 0.0159 |

S_{6} | 2.68(463.09) | 0.0001 | 2.71(457.64) | 0.0006 |

S_{7} | 2.73(454.88) | 0.0004 | 2.72(456.05) | 0.0432 |

S_{8} | 2.78(445.36) | 0.0000 | 2.74(452.86) | 0.0663 |

S_{9} | 2.84(437.07) | 0.0003 | 2.79(443.97) | 0.0020 |

S_{10} | 2.86(433.12) | 0.0053 | 2.79(443.84) | 0.0024 |

S_{11} | 2.87(431.69) | 0.0008 | 2.81(442.00) | 0.0023 |

S_{12} | 2.94(421.83) | 0.0005 | 2.85(435.34) | 0.0000 |

S_{13} | 2.95(419.91) | 0.0013 | 2.89(428.27) | 0.0045 |

S_{14} | 2.99(415.21) | 0.0018 | 2.96(419.22) | 0.0006 |

S_{15} | 3.00(412.93) | 0.0001 | 2.97(416.85) | 0.0009 |

S_{16} | 3.09(401.58) | 0.0005 | 3.01(411.92) | 0.0022 |

S_{17} | 3.10(400.60) | 0.0002 | 3.02(410.38) | 0.0020 |

S_{18} | 3.14(394.37) | 0.0010 | 3.05(405.93) | 0.0050 |

S_{19} | 3.18(389.81) | 0.0153 | 3.08(402.16) | 0.0000 |

S_{20} | 3.46(358.09) | 0.0029 | 3.10(399.78) | 0.0000 |

**Figure 6.**Charge difference density (CDD) of [C60]PCBM/APFO

_{3}and [C70]PCBM/APFO

_{3}, where the green and red stand for the hole and electron, respectively.

_{3}, the charge difference density in Figure 6 reveals that there are also three kinds of excited state: (a) local-excited state of C70 (S1, S4, S5–S8, S11, S12, S15–S20, see Figure S3); (b) an entire intermolecular CT state (S9, S10, S13, S14) and (c) an intramolecular CT state of APFO

_{3}coupled with local-excited states of [C70]PCBM (S2 and S3); the lowest intermolecular charge transfer excited state is the state S9, peaking at 444 nm.

#### 3.3. Rate of Charge Transfer in the Marcus Theory

_{DA}is the electronic coupling (charge-transfer integral) between donor and acceptor, ΔG is the free energy change for the electron transfer reaction, k

_{B}is the Boltzmann constant, h is Planck’s constant, and T is the temperature (we set T = 300K in our calculations). Firstly, the Generalized Mulliken-Hush (GMH) model was used to estimate the charge transfer integral (electronic coupling matrix) [39]. In terms of the two states (S

_{0}and S

_{n}states) the formulation, electronic coupling matrix can be written as:

_{tr}as well as the corresponding dipole moment difference Δμ between the initial and final electronic states. The Δμ in the above equation was calculated using the Hellmanne Feynman theorem, as the analytical derivative of the excited-state energy with respect to an applied electric field. For the dimer system of fullerene/polymer, the first charge transfer state for [70]PCBM/APFO

_{3}and [C60]PCBM/APFO

_{3}corresponding to the pure intermolecular charge transfer excited state identified as the fully charged separation state, pointed to the final state in order to obtain the electronic coupling. The transition energy dependent on the static electric field F can be expressed as [40]:

_{3}and [C70]PCBM/APFO

_{3}(13.39286 a.u. and 10.41667 a.u.), respectively. According to Equation (3), the electronic coupling strengths (V

_{DA}) are calculated to be 329.2 cm

^{−1}(0.04081 eV) and 260.2 cm

^{−1}(0.03226 eV), respectively.

Complex | States | $\Delta $U (a.u.) | U (a.u.) | V_{DA} (cm^{−1}) |
---|---|---|---|---|

[C60]PCBM/APFO_{3} | S_{10} | 13.39286 | 0.1910 | 329.2 |

[C70]PCBM/APFO_{3} | S_{9} | 10.41667 | 0.1204 | 260.2 |

_{CR}are −1.81 eV for [C60]PCBM/APFO

_{3}and −1.837 eV for [C70]PCBM/APFO

_{3}, as can be seen from Table 4, and negative values signify the process of electron recovery is spontaneous thermodynamically for these two systems. ΔG

_{CT}can be estimated by using the Rehm-Weller equation, $\Delta {G}_{CT}=-\Delta {G}_{CR}-\Delta {E}_{0-0}$, where $\Delta {E}_{0-0}$ is the energy of the lowest excited state of free-base donor. The calculated Gibbs Free energy differences ΔG

_{CT}, are all negative values (see Table 4), which means that electron transfer is thermodynamically favorable for these two systems. There is a directly competitive process between intermoleular charge transfer and charge recombination, and thus it is expected to maximize intermoleular charge transfer and minimize charge recombination for designing high-efficiency solar cells.

Complex | Δ
G_{CR} | λ | Δ
G_{CT} | V_{DA} | K_{CT} (×10^{13}) | K_{CR} (×10^{7}) |
---|---|---|---|---|---|---|

[C60]PCBM/APFO_{3} | −1.810 | 0.7 | −0.6655 | 0.04082 | 3.2811 | 0.13517 |

[C70]PCBM/APFO_{3} | −1.837 | 0.7 | −0.6400 | 0.03265 | 2.0304 | 0.036515 |

_{3}and fullerene derivatives, since the inner reorganization energy arises from the change in equilibrium geometry of the donor (D) and acceptor (A) sites consecutive to the gain or loss of electronic charge upon electron transfer. For the outer reorganization energy, it originates from the electronic and nuclear polarization/relaxation of the surrounding medium, which is not easy to estimate quantitatively in the solid state. So, the total reorganization energy in the calculations is adopted from experimental results. The energies of the neutral acceptor (A) at the anionic geometry and optimal ground-state geometry ($E({A}^{-})$ and $E(A)$, and subsequently the energies of the cation donor at the neutral geometry and optimal cation geometry ($E(D)$ and $E({D}^{+})$) were calculated only individually.

_{CT}) and recombination rates (K

_{CR}) can be simulated from these parameters using Equation (2). When comparing APFO

_{3}/[C60]PCBM with APFO

_{3}/[C70]PCBM, it was found that the introduction of [C70]PCBM did not obviously increase the value of K

_{CT}(K

_{CT}= 3.2811×10

^{13}s

^{−1}for [C60]PCBM/APFO

_{3}and 2.0304 × 10

^{13}s

^{−1}for [C70]PCBM/APFO

_{3}) owing to the similar values of λ, ΔG

_{CT}and V

_{DA}. However, it obviously reduces the rate of charge recombination, and the value of K

_{CR}is calculated to be 0.13517 × 10

^{7}s

^{−1}(for [C60]PCBM/APFO

_{3}) and 0.036515 × 10

^{7}s

^{−1}(for [C70]PCBM/APFO

_{3}), respectively.

#### 3.4. Effect of Electronic Field on CT Rate

_{DA}and ΔG. When considering this kind of perturbation, the electronic field has influence on the free energy by means of additional change energy ΔμF (where μ and F represent the dipole moment of a radical pair and the strength of the external electronic field), and thus under the external electronic field, $\Delta G(F\ne 0)=\Delta G(F)-\Delta uF$ (where F ≠ 0). The external electronic field dependent ${V}_{DA}(F\ne 0)$ can be induced by extending the GMH model: ${V}_{DA}(F\ne 0)=\frac{{\text{\mu}}_{tr}(F)\Delta E(F)}{\sqrt{{\left(\Delta {\text{\mu}}_{F}\right)}^{2}+4{\left({\text{\mu}}_{tr}(F)\right)}^{2}}}$. Inserting term of $\Delta G(F\ne 0)$ and ${V}_{DA}(F\ne 0)$ into Equation (2), we can rewrite the Marcus theory as:

_{F}, then get the value of V

_{DA}for the S

_{9}excited state of [C70]PCBM/APFO

_{3}because it is an (intermolecular charge transfer) ICT state. Figure 7 shows the relationship between electronic field and rate of charge transfer. For [C70]PCBM/APFO

_{3}, it was found that the rate of charge transfer is increased along with the external electronic field as a whole. In addition, we also discussed the effect of the individual values of V

_{DA}and ΔG on the rate of charge transfer. When only considering the effect of Δ $\text{\mu}F$, the rate is almost unchanged with the external electronic field (see blue line in Figure 7), i.e., when F = 4 × 10

^{−5}a.u., K

_{CT}= 2.0733 × 10

^{13}S

^{−1}, and F = 12 × 10

^{−5}a.u., K

_{CT}= 2.1141 × 10

^{13}S

^{−1}). When only ${V}_{DA}(F\ne 0)$ is considered, the CT rate generally grows in response to the increase of the external electronic field. While for F = 4 × 10

^{−5}and 8 × 10

^{−5}, the rate is approximately equal, and in a purely computational way the reason can be explained by the fact that the subequal values of transition energies and transition moments result in the very closed ${V}_{DA}(F\ne 0)$. Noted that, along with the increasing electronic field, obviously the CT rate increases, that is, when F = 0, K

_{CT}= 2.0304 × 10

^{13}S

^{−1}and F = 12 × 10

^{−5}, K

_{CT}= 6.2186 × 10

^{13}S

^{−1}). When the combination of ${V}_{DA}(F\ne 0)$ and $\Delta G(F\ne 0)$, it was found that the strength and shape by simultaneously considering the two factors are similar with those under the condition of only ${V}_{DA}(F\ne 0)$, which means that the influence of the electronic coupling matrix on the rate exerts a leading position.

**Figure 7.**Calculated rates of CT under different electronic fields (a.u.), where blue, red and blank lines are $\Delta G(F\ne 0)$, ${V}_{DA}(F\ne 0)$ and combination of two factors, respectively.

## 4. Conclusions

_{3}, fullerene, [C60]PCBM/APFO

_{3}and [C70]PCBM/APFO

_{3}. Molecular orbital energies show that the LUMO of [C70]PCBM is slightly higher than that of [C60]PCBM, and the LUMO of the binary system is closed to that of fullerenes. Additionally the HOMOs verge on HOMOs of APFO

_{3}, which leads to the fact that charge transfer controlled by transition from HOMO to LUMO can take place from APFO

_{3}to fullerenes. For the C70 derivative, absorption spectra and charge difference density show that the absorption peak comes from the local excitation of C70 monomer, and there are three kinds of CT originating from intramolecular CT between C70 and the benzene ring and internal composition. Moreover, the excited states of [C60]PCBM/APFO

_{3}and [C70]PCBM/APFO

_{3}were studied, and locally excited states and charge transfer states were found with CDD analysis. Based on Marcus theory, the calculated rate of charge transfer is of a certain magnitude for [C60]PCBM/APFO

_{3}and [C70]PCBM/APFO

_{3}, while the calculated recombination rate demonstrated the process of charge recombination is more likely to happen for the [C60]PCBM/APFO3 than the [C70]PCBM/APFO

_{3}. Upon introducing increasing electronic field, the free energy and electronic coupling matrix show a variety of different changes; however, it was found that the changed electronic coupling matrix under increasing electronic field may have even key impacts on the CT rate.

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Li, Y.; Qi, D.; Song, P.; Ma, F.
Fullerene-Based Photoactive Layers for Heterojunction Solar Cells: Structure, Absorption Spectra and Charge Transfer Process. *Materials* **2015**, *8*, 42-56.
https://doi.org/10.3390/ma8010042

**AMA Style**

Li Y, Qi D, Song P, Ma F.
Fullerene-Based Photoactive Layers for Heterojunction Solar Cells: Structure, Absorption Spectra and Charge Transfer Process. *Materials*. 2015; 8(1):42-56.
https://doi.org/10.3390/ma8010042

**Chicago/Turabian Style**

Li, Yuanzuo, Dawei Qi, Peng Song, and Fengcai Ma.
2015. "Fullerene-Based Photoactive Layers for Heterojunction Solar Cells: Structure, Absorption Spectra and Charge Transfer Process" *Materials* 8, no. 1: 42-56.
https://doi.org/10.3390/ma8010042