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Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

Automatic High-Throughput Screening Scheme for Organic Photovoltaics: Estimating the Orbital Energies of Polymers from Oligomers and Evaluating the Photovoltaic Characteristics Yutaka Imamura,*,† Motomichi Tashiro,‡ Michio Katouda,§ and Masahiko Hada† †

Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, 1-1 minami-Osawa, Hachioji, Tokyo 192-0397, Japan ‡ Department of Applied Chemistry, Toyo University, Kujirai 2100, Kawagoe, Saitama 350-8585, Japan § Advanced Institute for Computational Science, RIKEN, 7-1-26, Minatojima-minami-machi, Chuo-ku, Kobe, Hyogo 650-0047, Japan S Supporting Information *

ABSTRACT: A theoretical search for organic photovoltaic materials is immensely helpful for easily identifying candidate materials and ultimately achieving high power conversion efficiencies for solar cells. In this study, an automatic scheme for screening organic photovoltaic (OPV) materials has been developed. This scheme mainly includes three steps, namely, the automatic generation of thiophene-based semiconducting polymers composed of donor and acceptor units, estimation of orbital levels by Hückel-based models, and an evaluation of photovoltaic characteristics. A numerical assessment confirmed that the screening tool could be applied to any calculations with a basis set that includes diffuse functions. An examination of 380 donor−acceptor-type polymers demonstrated that the geometric effects such as effective conjugation length and distortion in the polymers affected the orbital levels and were important to consider in the scheme for screening an ideal OPV material. In addition, the photovoltaic characteristics were evaluated and promising acceptor units for photovoltaic materials were obtained. Thus, the proposed methodology was suitable for high-throughput screening of promising donor/acceptor units.

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INTRODUCTION Organic photovoltaics (OPVs) have attracted significant attention as next-generation solar cells because of their flexibility and low-cost solution processing.1,2 The tunable physical properties such as absorption spectra and molecular orbital levels permit the design of OPV materials with desired open-circuit voltage (VOC), short-circuit current density (JSC), and fill factor (FF).3 The power conversion efficiencies (PCEs) of the single-junction devices have increased from a few percent (in the early 2000s) to more than 10% (by 2014). Such rapid progress can be attributed to an intensive research effort on OPV materials.4−10 OPV cells are composed of electron donor materials, usually semiconducting polymers, and electron acceptor materials, typically fullerene derivatives. Recently, non-fullerene electron acceptors have been developed and it was found that the PCEs of devices based on these acceptors increased up to 13%.11−13 Therefore, the non-fullerene electron acceptors are considered promising for their high PCEs and cost-effectiveness. For a long time, the ultimate goal of the OPV research community has been to find novel materials that exhibit a PCE of 15% or more. Virtually, an infinite number of candidate materials can be generated because of the intrinsic degrees of © XXXX American Chemical Society

freedom in organic molecules. However, they cannot be experimentally examined because of time and financial constraints. Therefore, the theoretical screening of candidate materials is important, especially for organic photovoltaics. In the recent past, several theoretical approaches have been reported for this purpose. In 2006, Scharber et al. proposed a very simple but powerful model that could predict the PCE on the basis of only a few parameters such as the acceptor and donor orbital levels.3 This model has been widely used as a guide to predict the OPV PCEs.2 On the basis of the predictions by this model, OPV materials with more than 8% PCEs have been reported, although several OPV materials with a small energy loss during the electron transfer process also exhibited a high PCE.14,15 Aspuru-Guzik et al. worked with OPV materials in a simulation environment under the Harvard Clean Energy Project.16−19 They generated candidate materials by combinatorial syntheses, calculated them by quantum chemical methodologies, and analyzed them with experimental data using cheminformatics methodologies. However, their Received: August 23, 2017 Revised: November 30, 2017 Published: December 1, 2017 A

DOI: 10.1021/acs.jpcc.7b08446 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C Scheme 1. Schematic Illustration of Material Screening

(1) Automatic generation of polymers. The polymers were automatically generated by our program with the help of several other programs.25−27 The following steps are also illustrated in Scheme 1. (a) The basic units of donor−acceptor type polymers were first generated in a simplified molecular-input line-entry system (SMILES) by combining the acceptor and donor units from the libraries. (b) Next, hydrogen atoms were added into the donor− acceptor type polymers by RDKit.27 (c) The monomer/dimer inputs were then generated by RDKit.27 (d) The geometries of the monomer and dimer were optimized by semiempirical methods using MOPAC.26 (e) The monomer and dimer geometries were also optimized by the first-principles method using Gaussian.25 (f) Finally, the orbital levels of the monomer/dimer were obtained and the HOMO−LUMO energy gaps of the polymers were estimated by using the Hückel-based method. All of the above processes were automatic and controlled by our program written in Python. (2) Analytical model based on the Hückel theory for estimating the orbital energies. The original analytical model based on the Hückel theory,28 which was proposed in our previous study,21 was first reviewed. In this model, to avoid the polymer calculations, we had incorporated the physical properties of the polymers using the Hückel model based on calculations of the monomer and the dimer. The energy of the jth orbital, εj, in a one-dimensional chain model with N sites can be analytically obtained as

approach was based on the physical properties of the oligomers and not polymers. Larsen et al. and our group have worked on estimating the orbital energies of polymers based on tightbinding models using calculations of the oligomers.20,21 This simple estimation method of orbital energies can be a powerful tool for screening OPV materials. There have also been other studies on OPV materials.22−24 To date, the search for OPV materials by combinatorial synthesis has been mostly limited to oligomers.16−19 As shown in our previous study, the physical properties of polymers cannot be directly linked to those of monomers.21 Further calibration of the search scheme is required in order to theoretically develop a new OPV material. The search scheme should satisfy three conditions toward establishing a high-throughput screening, namely, (1) the scheme should be based on the physical properties of the polymers (not oligomers) so as to be reliable and efficient, (2) the scheme should be carried out at a reasonable computational cost that enables the examination of a large number of candidate organic materials, and (3) the scheme should be automatic (from material generation to evaluation of photovoltaic characteristics) for practical reasons. On the basis of the above requirements, this study proposes an automatic material search scheme that includes polymer generation, estimation of orbital energies at low cost, and evaluation of photovoltaic characteristics. In the following discussion, we have first explained the detailed procedures of the proposed scheme. Next, we have examined the scheme with a wide variety of basis sets in terms of the calculation time. The Hückel-based models have then been assessed numerically by evaluating the highest occupied molecular orbital (HOMO)− lowest unoccupied molecular orbital (LUMO) energy gaps, and compared with accurate one-dimensional periodic boundary condition (PBC) calculations. Finally, the photovoltaic characteristics such as JSC, VOC, and PCE of the 380 automatically generated polymers have been evaluated by using their frontier orbital levels and the accuracy of the scheme and promising units for OPV materials have also been discussed.

⎛ j ⎞ εj = α + 2β cos⎜ π⎟ ⎝N + 1 ⎠

(1)

where α and β are fixed parameters for any value of N. By assuming that the HOMO and LUMO for the polymers are considered as sites in eq 1 and the polymer is solely composed of the HOMO and LUMO of the basic unit as shown in Scheme 2, the energies for the HOMO and LUMO bands based on the Hückel theory can be expressed as eqs 2a and 2b.

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METHODOLOGY In this study, a scheme consisting of (1) automatic polymer generation, (2) estimation of orbital energies, and (3) evaluation of photovoltaic characteristics for a high-throughput screening of OPV materials was developed, as described below.

⎛ j ⎞ εHOMO,j = αHOMO + 2βHOMO cos⎜ π⎟ ⎝N + 1 ⎠ B

(2a)


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The Journal of Physical Chemistry C Scheme 2. Monomer and Polymer Model of the N-mera

Polymer Monomer Monomer εGap = εLUMO − εHOMO + 2(βLUMO − βHOMO)

(7)

The original Hückel model (eqs 4 and 7) is hereafter referred to as Hückel model 1. In this study, a modified Hückel model was proposed on the basis of the previous model. In principle, eqs 3a and 3b assume a planar geometry of the polymer. Although a geometrical distortion occurs between a monomer and a dimer, the distortion effect is implicitly considered in the determination of parameters such as βHOMO and βLUMO. However, the distortion from the monomer to the infinite polymer may be nonlinear and cannot be easily predicted. In other words, the distortion effect from the monomer to the dimer is not maintained from the monomer to polymer. In fact, a preliminary numerical assessment showed that nonlinear distortion occurred for the polymers. In order to examine the distortion effect in more detail, it is necessary to modify eqs 4a and 4b. Therefore, we defined an index of the distortion as follows. (1) The equation ax + by + cy + d = 0 was fit by using the x, y, z Cartesian coordinates of all atoms on each ring by the least-squares method, and a normal vector ν⃗1 = (a, b, c) was obtained from the fitting function. (2) The inner product between ν⃗1 and ν⃗2 for the two rings was calculated, and the angle between the rings was estimated by cos θ = v1v2/|v1||v2|. By using the above distortion index, the following modified versions of eqs 4a, 4b, and 7 were proposed

Here, α and β represent the orbital energies of a monomer and the resonance integral, respectively.21

a

⎛ j ⎞ εLUMO,j = αLUMO + 2βLUMO cos⎜ π⎟ ⎝N + 1 ⎠

(2b)

The HOMO and LUMO energies of the N-mer oligomer, i.e., the top of the HOMO energy band and the bottom of the LUMO energy band, are given as ⎛ N ⎞ εHOMO,N = αHOMO + 2βHOMO cos⎜ π⎟ ⎝N + 1 ⎠

(3a)

⎛ 1 ⎞ εLUMO,1 = αLUMO + 2βLUMO cos⎜ π⎟ ⎝N + 1 ⎠

(3b)

εHOMO,∞ = αHOMO − 2βHOMO cos(θm − θp)/cos(θm − θd) (8a)

εLUMO,∞ = αLUMO + 2βLUMO cos(θm − θp)/cos(θm − θd)

As N becomes positive infinite, eqs 4a and 4b become

(8b)

εHOMO,∞ = αHOMO − 2βHOMO

(4a)

εLUMO,∞ = αLUMO + 2βLUMO

(4b)

Polymer Monomer Monomer εGap = εLUMO − εHOMO + 2(βLUMO − βHOMO)

cos(θm − θp)/cos(θm − θd)

where θm, θd, and θp are the distortion angles for the monomer, dimer, and polymer, respectively. The smallest inner product was chosen for calculating cos θ for the monomer, dimer, and polymer in the case of multiple rings. The physical meaning of the term cos(θm − θp)/cos(θm − θd) was used to remove the distortion effect from the monomer to dimer by dividing with cos(θm − θd), and in order to incorporate the distortion effect from the monomer to polymer, the expression was multiplied with cos(θm − θp). This Hückel model (eqs 8 and 9) is hereafter referred to as Hückel model 2. It is noted that eqs 8 and 9 require a priori polymer calculations because the distortion in the polymer must also be included. In this study, we proposed a model for investigating how the geometric distortion affects the estimation of orbital energies. A previous study has revealed that the HOMO−LUMO energy gap cannot be extrapolated with respect to chain lengths.29 For overcoming this difficulty, the following model was proposed in order to consider an effective conjugation length29

and correspond to the HOMO and LUMO energies for the infinite one-dimensional polymer. In order to obtain εHOMO,N and εLUMO,1, four parameters, namely, αHOMO, αLUMO, βHOMO, and βLUMO, should be determined from the calculations of the oligomer. As a simple and straightforward approach, the four parameters are determined by applying eqs 3a and 3b to the monomer and dimer calculations. This type of determination was termed as the two-point model in the previous study.21 Through elementary algebra, the four parameters are determined as follows (1) Monomer αHOMO = εHOMO

(5a)

(1) Monomer αLUMO = εLUMO

(5b)

Dimer (2) Monomer βHOMO = εHOMO − εHOMO

(6a)

(2) Dimer Monomer βLUMO = εLUMO − εLUMO

(6b)

εMonomer/Dimer HOMO

(9)

εMonomer/Dimer LUMO

N ‐ mer Polymer Monomer Polymer −d(N − 1) εGap = εGap + (εGap − εGap )e

where and are the orbital energies of the HOMO and LUMO, respectively, for the monomer/dimer. The HOMO−LUMO gap for the infinite one-dimensional polymer from eqs 4a and 4b is given as

(10)

where d is a fitting parameter. The model was originally proposed by Meier et al. for explaining absorption spectra and C


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The Journal of Physical Chemistry C

enough peaks and is not realistic. Therefore, we used the simple model described by eqs 15 and 16 in this study. The VOC was estimated simply as an energy difference between the LUMO of an acceptor and the HOMO of a polymer with an energy loss of 0.3 eV, as used in the Scharber model:3

later used by the Brédas group for describing the dihedral barrier heights.30 In the case of the dimer, eq 10 can be rewritten as Dimer Polymer Monomer Polymer −d εGap = εGap + (εGap − εGap )e

(11)

Combining eq 11 and eqs 8, 9 is difficult because the former equation is formulated under the assumption that the polymer limit is known. Therefore, the exponential function was expanded up to the first order and the polymer gap was described by a simpler expression Polymer Monomer Dimer Monomer εGap = εGap + (1/d)(εGap − εGap )

Acceptor Polymer eVOC = εLUMO − εHOMO − 0.3 eV

The energy loss of 0.3 eV is considered to be small, which indicates that the OPV devices are well formed. Although the fill factor, FF, depends on the manufacturing process of the photovoltaic devices, a widely accepted FF value of 0.7 was adopted in this study. Finally, using VOC, JSC, and FF, the PCE of the device could be estimated as

(12)

which can be extended to eqs 8 and 9. By introducing a parameter c (=1/d) into eqs 8 and 9, the following models were obtained:

PCE =

εHOMO,∞ = αHOMO − 2cβHOMO cos(θm − θp)/cos(θm − θd) (13a)

(13b) Polymer Monomer Monomer εGap = εLUMO − εHOMO + 2c(βLUMO − βHOMO)

Jsc =

∫ EQE(ω)ΦAirMass1.5(ω) dω

Polymer Polymer Polymer εGap = εLUMO − εHOMO

Pin

(19)

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(14)

COMPUTATIONAL DETAILS The automatic generation scheme was supported by RDKit.27 The preliminary geometry optimization was done by using PM7 in MOPAC as a semiempirical method.26 The geometry optimization for the oligomer calculations from the firstprinciples was carried out using density functional theory with the Perdew−Burke−Ernzerhof-one-Perdew−Burke−Ernzerhof (PBE1PBE) functional,36 and the basis set was 6-31G(d).37 The geometry optimization for the one-dimensional PBC at the PBE1PBE/6-31G(d) level was also performed for comparison purposes. For the OPV materials, the hybrid functionals including 20−25% Hartree−Fock exchange were adopted, and were reported to reproduce the appropriate orbital levels.21,22 All density functional theory calculations for oligomer and PBC calculations were carried out by the Gaussian 09 program.25

The determination of c needs a certain training set of polymers, but the value of c can be applied to other systems if c does not depend on the systems chosen. The model described by eqs 13 and 14 has been hereafter referred to as Hückel model 3. (3) Estimation of photovoltaic characteristics. Photovoltaic characteristics such as VOC and JSC were evaluated according to the previous study by using the frontier orbital levels estimated by the Hückel-based models.3,22,31 On the basis of the assumption that optical absorption occurs perfectly at energies above εPolymer , JSC could be evaluated by numerically integrating Gap in the energy range from εPolymer to the edge of short Gap wavelength light (280 nm) as follows Polymer EQE(ω) = 0.65Θ(ℏω − εGap ) dω

VOCJSC FF

In real OPV devices, the PCE depends on the morphology of the interfaces for planar and bulk heterojunction geometries. Equation 19 does not consider the details about the interfaces but examines the basic potential of OPV materials to choose promising donor/acceptor units. In order to estimate the PCE more accurately, a more detailed examination is required, which includes the interface model.

εLUMO,∞ = αLUMO + 2cβLUMO cos(θm − θp)/cos(θm − θd)

cos(θm − θp)/cos(θm − θd)

(18)

(15)

■

(16)

RESULTS AND DISCUSSION Efficiency of the Proposed Scheme for Material Screening. First, we discuss the efficiency of the proposed scheme for material screening. This scheme mainly consists of three processes: (1) automatic polymer generation, (2) estimation of orbital energies, and (3) evaluation of photovoltaic characteristics. Since the geometry optimization in the second process required high computation power for calculations based on the first-principles, which is essentially a rate-determining step, the following discussion focuses on this issue. It is known that geometry optimization by semiempirical calculations is not computationally negligible yet not ratedetermining. The other procedures such as estimating the orbital levels by the Hückel models and evaluating the photovoltaic characteristics are in fact computationally negligible. Figure S1 shows a target system, which was a typical planar polythiophene. The initial geometries for the monomer, dimer, and polymer were obtained via optimization by PM7 using MOPAC. Table S1 lists the total CPU times of the geometry

(17)

where Θ is the Heaviside step function and ΦAirMass1.5 is the Air Mass 1.5 solar spectrum.32 The value of the parameter external quantum efficiency (EQE) was set to 0.65, similar to the previous studies.3,22,31 Although this approximation utilized the DFT-calculated HOMO−LUMO energy gap instead of the optical gap and did not consider the oscillator strengths explicitly,33,34 JSC was appropriately estimated in the previous studies using this method.3,22 Numerical assessment also reveals that HOMO−LUMO gaps are close to experimental gaps of polymers with an appropriate Hartree−Fock exchange of 20− 30%.35 There are two alternative approaches for the current methodology, where the absorption spectra are calculated by considering either an imaginary dielectric constant or the excited states of the oligomers by time-dependent density functional theory. The first approach is basically available for only pure DFT and not hybrid DFT. Therefore, the problem of underestimation arises in the absorption spectra. The latter approach requires extensive calculations of the oligomer for D


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Figure 1. Libraries of the (a) acceptor and (b) donor units used for the automatic generation of OPV materials.

in SCF calculations, which lead to a high computational cost.38 For the aug-cc-pVTZ and 6-311++G basis sets including the significantly diffuse functions, the proposed scheme could complete the geometry optimizations and estimate the orbital energies of the polymers, even though the SCFs for the onedimensional PBC calculations could not be converged. However, the poor SCF convergence could be overcome by the current scheme. Although the current assessment was carried out on a specific machine and the timing data might be different on a different machine, the clear advantage of the proposed scheme is that it can estimate the orbital energy levels of the polymer with diffuse functions without any poor SCF convergence. Finally, we discuss the computational scaling of SCF calculations. For PBC calculations, the effective computational scaling of SCF calculations for PBC calculations with the periodic fast multipole method (FMM) is controlled by O(kn3) for diagonalization or O(n2N) for the formation of the Fock matrix by Schwarz screening, where n and N represent the number of basis functions in a unit cell and the number of reference unit cells and k is the number of k points, respectively. On the other hand, the effective computational scaling of the formation of the Fock matrix for oligomers with FMM is O(n3) for the diagonalization. Therefore, as a unit cell is small and the number of basis functions is small, PBC calculations can be faster. However, as n becomes larger, oligomer calculations are more efficient. HOMO, LUMO, and HOMO−LUMO Gap of the Polymers Predicted by Calculations of the Monomer and the Dimer. We first numerically estimated the orbital

optimizations for the monomer and dimer using PBE1PBE exchange-correlation functionals with a wide variety of basis sets: 6-31G(d), 6311G(d,p), 6311G(2dp,2p), cc-pVDZ, and ccpVTZ. For comparison, the result of the one-dimensional PBC calculations of the polymer is shown in Table S1. If the sum of the total CPU times of the monomer and dimer geometry optimizations is lower than the total CPU time of the polymer, the current scheme may be considered as more efficient than the polymer calculation on this type of machine. For relatively small basis sets such as 6-31G(d), 6311G(d,p), and cc-pVDZ, the sum of the total CPU times of the monomer and dimer optimizations was comparable to the total CPU time of the polymer geometry optimizations. For example, the sum of the total CPU times of the monomer and dimer optimizations was lower than the total CPU time of polymer geometry optimization by cc-pVDZ, while the trend was reversed for 6-31G(d) and 6-311G(d,p). A different situation arises when a large basis set is adopted, especially when the diffuse functions are included. For example, the sum of the total CPU times of the monomer and dimer were 5451 and 8625 min for 6-311G(2dp,2p) and cc-pVTZ, respectively, while those of polymers were 6490 and 10139 min. It is worth noting that the polymer calculations for ccpVTZ and 6-311G(2df,2p) needed adjustment of a few options (NCellK and CellRange) in addition to the default setting for self-consistent field (SCF) convergence. These explicit options are given in Table S1 in the Supporting Information. This peculiar difference between the computational times using compact and large basis sets can be attributed to the numerical instability induced by the diffuse functions in the large basis sets E


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Table 1. Standard Deviations (SDs), Mean Errors (MEs), and Mean Absolute Errors (MAEs) of the Monomer, Dimer, and Three Hückel Models for HOMO, LUMO, and HOMO−LUMO Energy Gaps in eV monomer HOMO LUMO HOMO−LUMO gap

dimer

Hückel model 1

Hückel model 2

Hückel model 3

SD

ME

MAE

SD

ME

MAE

SD

ME

MAE

SD

ME

MAE

SD

ME

MAE

0.22 0.23 0.38

−0.28 0.39 0.67

0.29 0.40 0.68

0.16 0.14 0.27

−0.07 0.13 0.20

0.13 0.15 0.27

0.18 0.15 0.29

0.13 −0.14 −0.28

0.17 0.15 0.30

0.15 0.11 0.22

0.11 −0.11 −0.22

0.15 0.12 0.24

0.14 0.10 0.21

0.03 −0.01 −0.04

0.10 0.07 0.15

For Hückel model 1 based on eqs 4 and 7, as shown in Table 1, the mean absolute error and standard deviation for the LUMO were ∼0.15 eV, similar to those for dimer calculations. On the other hand, the mean absolute error and the standard deviations for HOMO were 0.17 and 0.18 eV, which are larger than those for the dimer calculations (0.13 and 0.16 eV). As shown in Figure 2g and h, behaviors of the linear regression analyses are similar to the dimer calculations. The HOMO− LUMO gaps were tolerably reproduced with a mean absolute error of 0.30 eV. The slope and intercept for the linear regression analysis in Figure 2i were 0.932 and 0.442, respectively, with R2 = 0.705, and were comparable to those determined in the dimer calculations. Therefore, the Hückel models performed reasonably but did not show a significant improvement over the dimer calculations. We also examined the Hückel models based on eqs 8 and 9, which incorporated the distortion effect. Before discussing the HOMO and LUMO energies and the HOMO−LUMO energy gaps, we numerically assessed whether the distortion could be appropriately evaluated by the expression cos θ = ν1ν2(|ν1||ν2|)−1. Table S2 shows the values of the inner products for a donor−acceptor type molecule, as shown in Figure S2. Since the donor and acceptor units are essentially planar, the inner products of the rings of each donor or acceptor unit are close to unity. In contrast, the inner products of the rings from the donor and acceptor units are close to 0.7, which correspond to an angle of ∼45°. Therefore, the distortion of the index was evaluated as expected. Hückel model 2 (based on eqs 8 and 9) reduces the mean absolute errors for the HOMO and LUMO to 0.15 and 0.12 eV, respectively, which varies by more than 0.02 eV when compared with Hückel model 1. The standard deviations are reduced to 0.15 and 0.11 eV for HOMO and LUMO, respectively, which leads to a better linear regression analysis; the R2 values of 0.820 and 0.938 for the HOMO and LUMO were larger than those determined in the monomer, dimer, and Hückel model 1 calculations. Regarding the HOMO−LUMO gap, Hückel model 2 gave a mean absolute error of 0.24 eV with a small standard deviation of 0.22 eV, even though the tendency of underestimation remained. The results from linear regression analysis were also acceptable in this case. The slope and the intercept determined by such analysis (shown in Figure 2l) were 1.001 and 0.229 with R2 = 0.823. These results indicate that the distortion effect is of importance for an accurate estimation of the orbital levels. Lastly, Hückel model 3 (eqs 13 and 14) was evaluated. This model included the geometric effect of an effective conjugation length and distortion in the calculations. In this study, the value of c was determined to be 0.80, so that the mean error was close to zero for the HOMO, the LUMO, and their energy gap. It was found that Hückel model 3 showed the best performance for mean absolute errors and standard deviations among all models. The mean absolute errors were reduced to 0.10 and 0.07 eV for HOMO and LUMO energies, which led to a better

energies of the HOMO, LUMO, and HOMO−LUMO gaps for 380 polymers, which were automatically generated from the 20 acceptor and 19 donor units, as shown in Figure 1. According to the various reviews on OPV devices,39−42 where the measured PCEs of the experimentally synthesized polythiophene derivatives are reported, the libraries of the representative acceptor and donor units were chosen. In this study, the alkyl group of R was replaced with hydrogen in Figure 1. Table 1 shows the standard deviations, mean errors, and mean absolute errors in the energies of the HOMO and LUMO and the HOMO−LUMO gaps of the polymers as calculated by the three Hückel-based models from the accurate values, which were estimated by the one-dimensional PBC calculations at the PBE1PBE/6-31G(d) level. As the current target of 380 polymers is relatively small in size and the basis set of 631G(d) was one that did not include the diffuse functions, it was possible to perform one-dimensional PBC calculations by Gaussian. However, the large target size resulted in high computational cost and poor SCF convergence, making it difficult to perform PBC calculations. The results of the monomer and dimer calculations are also given for comparison. The numerical data are illustrated in Figure 2 with results of the linear regression analyses. For the monomer calculations, as shown in Figure 2a and b, and the mean errors in Table 1, the HOMO and LUMO energies were under- and overestimated, respectively. As shown in Table 1, the mean absolute errors for the HOMO and LUMO energies were 0.29 and 0.40 eV, and the standard deviations were 0.22−0.23 eV for both the HOMO and LUMO. By performing linear regression analysis, the slope and intercept were (0.806, −0.804) for the HOMO and (0.819, −0.846) for the LUMO, with the coefficients of determination (R2) as 0.633 and 0.740, respectively. These results showed that the linear regression analysis did not work well. The HOMO− LUMO gaps were severely overestimated by more than 0.65 eV, with standard deviations of 0.38 eV. Since the mean errors for the HOMO and LUMO were largely negative and positive, the HOMO−LUMO gaps were consequently overestimated. In contrast with the monomer calculations, the dimer calculations reduced the mean absolute errors to 0.13 and 0.15 eV for the HOMO and LUMO, with standard deviations of 0.16 and 0.14 eV (Table 1). It can be seen in Figure 2d and e that the mean absolute errors decreased. The improvement over the monomer calculations was also confirmed by linear regression analysis; the slope and intercept were 0.917 and −0.373 for the HOMO and 1.037 and −0.024 for the LUMO, with R2 values of 0.787 and 0.884, respectively. The HOMO− LUMO gap was reasonably reproduced with a mean absolute error of 0.27 eV. In this case, the slope and intercept determined from the linear regression analysis were 0.995 and −0.195 with R2 = 0.740, which was significantly better than the corresponding values for the monomer calculations, i.e., 0.743 and 0.110 with R2 = 0.564, as seen in Figure 2c and f. F


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Figure 2. HOMOs, LUMOs, and HOMO−LUMO gaps of the monomer, dimer, and polymer models for 380 polymers.

linear regression analysis; the R2 values of 0.836 and 0.949 for the HOMO and LUMO, respectively, were the largest among

all models. The HOMO−LUMO gap was also wellreproduced; the mean absolute error was 0.15 eV with the G


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Figure 3. JSC, VOC, and PCE values for the 380 polymers in the case of PBC calculations and the three Hückel models.

Table 2. Standard Deviations (SDs), Mean Errors (MEs), and Mean Absolute Errors (MAEs) of the Three Hückel Models for JSC, VOC, and PCE Hückel model 1 JSC VOC PCE

Hückel model 2

Hückel model 3

SD

ME

MAE

SD

ME

MAE

SD

ME

MAE

3.13 0.18 0.90

2.88 −0.13 0.32

3.16 0.17 0.69

2.31 0.15 0.66

2.14 −0.11 0.26

2.43 0.15 0.54

2.15 0.14 0.60

0.02 −0.03 −0.03

1.45 0.10 0.38

by using the orbital levels obtained by the three Hückel models and PBC calculations. Short-Circuit Current Density. The JSC values for the 380 polymers were estimated by eq 16 with HOMO and LUMO energies of one-dimensional PBC calculations at the PBE1PBE/6-31G(d) level and are illustrated in Figure 3a. On the horizontal axis, the polymers are shown, such that each acceptor unit shown in Figure 1a is fixed, and the donor units in Figure 1b are varied. The JSC values calculated by Hückel model 1 (eq 7 with eqs 4a and 4b), Hückel model 2 (eq 9 with eqs 8a and 8b), and Hückel model 3 (eq 14 with eqs 13a and 13b) have been illustrated in Figure 3d, g, and j. Table 2 shows the

small standard deviation of 0.21 eV. The slope and the intercept from the linear regression analysis shown in Figure 2o were 1.047 and −0.069, respectively, with R2 = 0.842. It is worth noting that the Hückel model including only the parameter c (eqs S1−S3) and no cosine term was also effective (as shown in Figures S3 and S4 and Tables S3 and S4 in the Supporting Information). The above results indicate that an effective conjugation length and the distortion effect are both important for an accurate estimation of the orbital levels. Photovoltaic Characteristics of the Polymers. The photovoltaic characteristics of the 380 polymers were evaluated H


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The Journal of Physical Chemistry C standard deviations, mean errors, and mean absolute errors of JSC from the accurate values, which were estimated by the onedimensional PBC calculations at the PBE1PBE/6-31G(d) level. As shown in Figure 3d and g, the JSC values determined by using Hückel models 1 and 2 were slightly overestimated, which can be confirmed by the mean errors of 2.88 and 2.14 mA cm−2, respectively. For Hückel models 1 and 2, the mean absolute errors were 3.16 and 2.43 mA cm−2. On the other hand, Hückel model 3 outperformed the other models; the mean absolute error in the JSC value was 1.45 mA cm−2. However, the overall trend in Figure 3a was reasonably similar to those in Figure 3d, g, and j. Open-Circuit Voltage. In order to estimate the VOC values, the [6,6]-phenyl-C61-butyric acid methyl ester (PCBM) fullerene derivative was adopted as an electron acceptor material. The LUMO level of PCBM was set to −4.3 eV for all polymers on the basis of the original Scharber model,3 although the different LUMO levels have also been theoretically estimated.22,31 It is well-known that the LUMO levels should be estimated by density functional theory with special care.43−46 The values of VOC for the 380 polymers, as estimated by eq 18 with the one-dimensional PBC calculations at the PBE1PBE/6-31G(d) level, and by eqs 4a and 8a, are illustrated in Figure 3b, e, h, and k. Table 2 lists the statistics of VOC, such as standard deviations, mean errors, and mean absolute errors. Hückel models 1−3 slightly underestimate the value of VOC; the mean errors were −0.13, −0.11, and −0.03 V, respectively, as shown in Figure 3e, h, and k. However, the overall trend in Figure 3b is well-reproduced in Figure 3e, h, and k. For the three Hückel models, the mean absolute errors were 0.17, 0.15, and 0.10 V, respectively. Power Conversion Efficiency. The PCEs of the 380 polymers, as estimated by eq 19 with one-dimensional PBC calculations at the PBE1PBE/6-31G(d) level, are illustrated in Figure 3c. For comparison, the PCEs of the Hückel models were calculated by multiplying VOC and JSC, which were estimated by eqs 16 and 18 with HOMO and LUMO orbital levels, as shown in Figure 3f, i, and l. The statistical data are listed in Table 2. The PCEs in Figure 3c were essentially reproduced in Figure 3f, i, and l, as shown by the mean absolute errors of 0.69, 0.54, and 0.38% for Hückel models 1−3, which is reasonable, since VOC and JSC were also reasonably reproduced. Therefore, the PCEs of the three Hückel models can be used as a screening tool in order to select the candidate materials with specific units having the desired quantities in a certain range, although a more accurate estimation is required to shortlist and choose the final candidates. Donor−Acceptor Type Polymers. The polymers have been discussed in terms of their JSC, VOC, and PCE values. The average photovoltaic characteristics for each acceptor or donor unit are summarized in Tables 3 and 4. Considering the JSC for each acceptor unit given in Table 3, it is evident that the polymer group including the diazine-containing unit donated as (10a) acceptor unit in Figure 1a exhibits the highest JSC value (close to 40 mA cm−2) and the average value is 24.19 mA cm−2. This group of polymers (circled by a solid line) was visually confirmed in Figure 3a, d, g, and j. The conspicuously high JSC value can be explained by the small HOMO−LUMO gap of ∼0.6 eV. However, the diazine family may not be suitable for organic photovoltaic materials because of the expected small VOC. The two groups, denoted as (3a) and (17a) acceptor units

Table 3. Photovoltaic Characteristics with Standard Deviations (SDs) for Each Acceptor Unit (Figure 1a) acceptor

JSC (mA cm−2)

VOC (V)

PCE (%)

(1a) (2a) (3a) (4a) (5a) (6a) (7a) (8a) (9a) (10a) (11a) (12a) (13a) (14a) (15a) (16a) (17a) (18a) (19a) (20a) SD

7.79 7.11 15.59 5.63 7.57 7.02 9.15 9.10 6.60 24.19 7.14 10.98 7.10 6.03 3.48 10.81 14.97 9.29 9.37 10.07 4.41

0.68 0.36 0.45 0.74 0.21 0.55 0.44 0.53 0.69 0.23 0.75 0.58 0.27 0.62 0.82 0.35 0.15 1.06 0.59 0.88 0.24

2.41 1.06 4.55 2.54 0.89 1.94 1.84 2.35 2.73 2.42 3.25 3.32 0.83 2.35 1.83 2.34 1.43 5.95 3.06 5.46 1.34

Table 4. Photovoltaic Characteristics with Standard Deviations (SDs) for Each Donor Unit (Figure 1b) donor

JSC (mA cm−2)

VOC (V)

PCE (%)

(1d) (2d) (3d) (4d) (5d) (6d) (7d) (8d) (9d) (10d) (11d) (12d) (13d) (14d) (15d) (16d) (17d) (18d) (19d) SD

3.52 12.10 3.51 16.08 8.57 11.63 11.69 11.72 13.01 10.00 3.30 3.46 7.10 9.31 14.44 14.80 9.66 3.65 11.97 4.12

0.94 0.56 0.78 0.15 0.75 0.50 0.34 0.41 0.48 0.65 0.87 0.75 0.73 0.60 0.40 0.16 0.66 0.60 0.06 0.24

1.89 3.86 1.64 1.78 3.87 2.96 2.32 2.49 3.65 4.21 1.80 1.58 3.18 3.46 3.81 1.71 3.53 1.47 0.71 1.02

in Figure 2a and confirmed by the dashed and dotted lines, respectively, in Figure 3a, d, g, and j, also have a high JSC value. The average of the JSC values for the two groups is ∼15 mA cm−2. The (3a) acceptor unit is the widely used diketopyrrolopyrrole (DPP) unit.47 The highest JSC obtained for DPP was close to 20 mA cm−2, which is deemed appropriate for the OPV materials. On the other hand, the (4d) cyclopentadithiophene donor unit had the highest average values of JSC (16.08 mA cm−2). The JSC values of the (15d) and (16d) donor units were also more than 14 mA cm−2. The (4d) and (16d) donor units had a common skeleton consisting of two thiophene and fivemembered rings, which seems to be the key for a small HOMO−LUMO gap. The standard deviation of the acceptor I


Article


especially effective when the basis set includes the diffuse functions. Using the libraries of the donor and acceptor units, 380 polymers were automatically generated and investigated in terms of the energy levels of the frontier orbitals and the HOMO−LUMO gaps of the polymers, as estimated from the monomer and dimer calculations. The numerical assessment demonstrated that the Hückel models were able to reasonably reproduce the HOMO and LUMO levels, except for the special cases where distortion and effective conjugation length should be considered. For example, when the polymers were nonplanar, it was necessary to incorporate the distortion effect in the evaluation of the orbital levels, especially when the polymers were unexpectedly distorted from the dimer to the polymer. Although it is difficult to incorporate the distortion effect in polymers without a priori information or polymer calculations, a model with an effective parameter that describes an effective conjugation length or distortion may improve the overestimation of the conjugation. Using the orbital levels, the photovoltaic characteristics such as JSC, VOC, and PCEs were estimated. On comparison with the PBC calculations, the models based on Hückel theory could predict promising polymers with specific groups in terms of their photovoltaic characteristics, which indicated that the photovoltaic characteristics were relatively insensitive to small deviations in the orbital levels, and we could predict promising units by using a model without including the distortion effect. The polymers with acceptor units, such as NDI, NOz, and DPP, which were predicted to be promising in our scheme, experimentally showed relatively high PCEs. As a polymer, the combinations of DPP and NOZ acceptor units with the thienothiophene donor unit seemed to be the most promising. The above results proved that the current scheme was quite effective in helping to choose the promising groups of polymers, although it may be necessary to consider more detailed processes such as charge transfer and charge separation on the interface in the models to further narrow the choice of candidate OPV materials. Further studies on this topic are in progress. Although calculations with only local functions (Gauss functions) have been performed in this work, one-dimensional PBC calculations with plane wave functions may also be performed with high efficiency. The combination of plane-wave based and local-function based methodologies is interesting and currently being investigated in our group. In this study, the scheme was restricted to 380 polymers. However, more polymers ought to be explored for identifying the most promising OPV materials. This can be achieved by generating larger libraries of donor and acceptor units, as well as the alkyl groups and spacer units. The alkyl groups affect the polymer geometry because of the steric effect, and the spacer units are essential for estimating the photovoltaic characteristics because of the extension of the conjugation length, in addition to reducing the steric effects between the donor and acceptor units. For instance, by considering 10 types of spacer and alkyl units in addition to the 19 donor and 20 acceptor units investigated in this work, the number of polymers would increase to 38,000, and would thus require a high-throughput screening scheme. The application of the proposed scheme in a donor−spacer−acceptor polymer with alkyl groups is under consideration in our group.

units was determined to be larger than that of the donor units, which is consistent with the fact that the choice of acceptor unit leads to a high JSC, as confirmed for the diazine family. The group of polymers with the (18a) and (20a) acceptor units, which include the naphthalene diimide (NDI) and naphthobisoxadiazole (NOz) moieties, as indicated by the solid and dashed lines in Figure 3b, e, h, and k, exhibit a high VOC. The high VOC values indicate that the HOMO levels are relatively deep in comparison with other acceptor units. As shown in Table 3, the average VOC values were calculated as 1.06 and 0.88 V for (18a)48−50 and (20a),14,51 respectively. These acceptor units have attracted interest because of their high reported PCEs. The (15a) acceptor unit also exhibits a high VOC. On the other hand, it can be seen from Table 4 that the (1d) and (11d) donor units have a high VOC. However, the corresponding average values of JSC are among the lowest values of the 19 donor units, i.e., 3.52 and 3.30 mA cm−2, respectively, which demonstrates that a high VOC and high JSC cannot be easily obtained owing to their intrinsic trade-off nature. Only three polymer groups, namely, (18a), (20a), and (3a), which include the NDI, NOz, and DPP acceptor units, as indicated by the solid, dashed, and dotted lines in Figure 3c, f, i, and l, respectively, exhibited more than 4% average PCE. The NDI and NOz groups showed a high PCE because of the multiplication between a medium JSC and high VOC; the average values of JSC and VOC were 9.29 mA cm−2 at 1.06 V and 10.07 mA cm−2 at 0.88 V for the NDI- and NOz-containing groups, respectively. On the other hand, the DPP-containing polymer group showed a high PCE because of the multiplication between a high JSC and a medium VOC; the average values of JSC and VOC were 15.59 mA cm−2 and 0.45 V, respectively. Although the acceptor units of (10a) and (17a) demonstrated high JSC values, it was not possible to have a high PCE because of the low VOC. Thus, a delicate balance between the JSC and Voc values is needed for a high PCE because of the trade-off relationship between them. As shown in Table 4, the PCE achieved in the case of the donor unit of (10d) was more than 4%. The (10d) unit has moderate values of both VOC and JSC; a high PCE is obtained. The (1d), (4d), and (11d) donor units could not exhibit a high PCE because either VOC or JSC were very low. Among all polymers, the highest PCEs were 8.38 and 8.24% for the polymers composed of DPP (3a) and NOz (20a) acceptor units and the thienothiophene (2d) donor unit. The standard deviations of the PCE values for the acceptor and donor units were 1.34 and 1.02%, which indicates that the choice of acceptor units is more important than that of the donor units. Experimentally, the polymers including the NDI (18a),48−50 NOz (20a),14,51 and DPP (3a)15,47 units are combined with PCBM and often used for OPV materials, which exhibit a high PCE. The above-mentioned results indicate that the proposed scheme succeeded in predicting promising OPV materials with specific acceptor units.

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CONCLUSION In this work, we have proposed a scheme to automatically generate thiophene-based semiconducting polymers composed of donor and acceptor units, estimate the orbital levels of the polymers generated from the oligomers by performing calculations at a reasonable cost, and evaluate the photovoltaic characteristics, and discuss the results in comparison with the experiments. First, we examined the automatic scheme for material screening and confirmed that the proposed scheme is J


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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b08446. Additional figures and tables (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Yutaka Imamura: 0000-0002-9527-6813 Motomichi Tashiro: 0000-0001-6039-4118 Michio Katouda: 0000-0001-7980-5386 Masahiko Hada: 0000-0003-2752-2442 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Prof. Akinori Saeki and Prof. Itaru Osaka for valuable discussions on the libraries of the donor and acceptor units. The calculations were performed at the Research Center for Computational Science, Okazaki, Japan, and the supercomputer systems of the Research Institute for Information Technology at Kyushu University and the Information Technology Center at Nagoya University. This study was supported in part by a Grant-in-Aid for Scientific Research on Innovative Areas “Coordination Asymmetry” from the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number JP17H05380; a Grant-in-Aid for Scientific Research on Innovative Areas “π-System Figuration: Control of Electron and Structural Dynamism for Innovative Functions” from JSPS KAKENHI Grant Number JP15H01006 and JP17H05169; and a Grant-in-aid for Young Scientists B: JSPS KAKENHI Grant Number JP15K17816, from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. We would like to thank Editage (www.editage.jp) for English language editing.

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