Rational Design of Diketopyrrolopyrrole-Based Small Moleculesas Donating Materials for Organic Solar Cells

Abstract

A series of diketopyrrolopyrrole-based small molecules have been designed to explore their optical, electronic, and charge transport properties as organic solar cell (OSCs) materials. The calculation results showed that the designed molecules can lower the band gap and extend the absorption spectrum towards longer wavelengths. The designed molecules own the large longest wavelength of absorption spectra, the oscillator strength, and absorption region values. The optical, electronic, and charge transport properties of the designed molecules are affected by the introduction of different π-bridges and end groups. We have also predicted the mobility of the designed molecule with the lowest total energies. Our results reveal that the designed molecules are expected to be promising candidates for OSC materials. Additionally, the designed molecules are expected to be promising candidates for electron and/or hole transport materials. On the basis of our results, we suggest that molecules under investigation are suitable donors for [6,6]-phenyl-C₆₁-butyric acid methyl ester (PCBM) and its derivatives as acceptors of OSCs.

1. Introduction

Solar cells have been widely considered as a next-generation clean and renewable energy resources to relieve the global energy crisis. Accordingly, various solar cells have been developed. Among them, organic solar cells (OSCs) have attracted considerable research interest because of their outstanding advantages such as low cost, light weight, easy fabrication, and flexible features [1,2,3,4,5]. Since the pioneering work of Tang on donor-acceptor thin film OSCs [6], intensive interdisciplinary efforts have been focused on improving the power conversion efficiencies (PCEs) of OSCs by using novel materials and the different device structures [7,8]. It has been demonstrated that the PCEs of the device for solution-processed polymer bulk heterojunction (BHJ) solar cells has exceeded 10% [9,10]. Unfortunately, polymeric materials suffer from batch-to-batch variation, polydispersity, indefinite molecular weight, and impurity [11]. On the other hand, solution-processed small-molecule OSCs have also attracted increasing attentions because they can deliver respectable PCEs, which are comparable to those of polymer solar cells [12,13]. These trigger the momentum for the rational design of small molecules with novel structural features used for solar cells. Small molecule OSC materials are more suitable than those of polymers because of their advantages in terms of well-defined molecular structure, accurate molecular weight, high purity, no end group contaminants, easy mass-scale production, and better batch-to-batch reproducibility [14,15,16]. Furthermore, small molecule organic materials could exhibit higher charge carrier mobility, as well as tuning more easily their band structure to absorb sunlight efficiently [17]. These advantages make small molecule OSCs strong competitors to polymer solar cells. Small molecule organic materials are well on their way to outperform polymers in OSC applications [18,19]. Therefore, it is believed that small-molecule OSCs are promising to realize their commercial application in the future. However, the PCEs of small-molecule OSCs still lagged behind that of the polymer counterparts [20,21,22]. Therefore, it is still a significant challenge for the research community to develop novel, high-performance, and desirable donor materials to improve the PCEs of small molecule OSCs. In order to maximize small molecule OSC performance, the energy of the frontier molecular orbitals (FMO) including the HOMO and LUMO of the designed and synthesized donor material should be suitable to those of [6,6]-phenyl-C₆₁-butyric acid methyl ester (PCBM) and its derivatives [23,24], an excellent acceptor of OSCs [25]. Furthermore, donating material should exhibit a broad absorption region, high and balanced charge transfer properties, and high ambient stability. A number of studies demonstrated some guidelines to experiments, where useful insights have been provided to help understand the nature of the molecules [26,27,28,29].

It is well known that there are three major parameters that determine the PCEs of OSCs: Open circuit voltage (V_oc), short circuit current (J_sc), and fill factor (FF) [30]. In order to realize highly-efficient small molecule-based OSCs, an ideal donor material should have a low energy gap and deep highest-occupied molecular orbital energy level. A low energy gap is beneficial for effective optical absorption, providing a larger J_sc, while a deep highest-occupied molecular orbital energy level can increase the V_oc. In addition, the high hole mobility is also crucial for the carrier transport to improve the J_sc and FF [31]. In OSC research, the push-pull organic compounds, containing π-center with electron donors and acceptors on the terminal sites of the conjugated backbone, are regarded as one of the most promising molecular materials [12,13,18]. This type of molecular structure can enhance intramolecular charge transfer (ICT), yielding higher molar absorptivity. At the same time, they can lower the material band gap and extend the absorption spectrum towards longer wavelengths. Additionally, the electronic energy levels and band gaps can be tuned effectively through adjusting the acceptor and donor units, and π-bridge units or length [32,33]. Among the various push-pull organic compounds investigated, diketopyrrolopyrrole (DPP)-based molecules have been regarded as a promising core building block for small molecule OSC materials due to their strong light absorption, good photochemical stability, excellent charge carrier mobility, and easy synthesis [34,35,36,37]. Recently, some DPP-based small molecule OSC materials have been reported [38]. It was found that these molecules have good charge transport, film morphology, and optical properties.

With the above considerations, in this work, we report the investigation of both optical and charge transporting properties from a theoretical point of view for DPP-based small molecules. We designed a series of DPP-based small molecules with DPP derivative 2,5-bis(2-ethylhexyl)pyrrolo[3,4-c]pyrrole-1,4(2H,5H)-dione (BEDPP) as core, thiophene or furan as conjugate π-bridges (CB), and aromatic derivatives (Ar) as end groups for OSCs applications (Scheme 1). The purpose of this molecular architecture was to investigate the relationship between topologic structure and optical, as well as electronic, properties to provide a demonstration for the rational design of a novel candidate for small molecule OSC materials. We have also predicted the mobility of the designed molecules.

Scheme 1. Molecular structures of the investigated molecules.

Scheme 1. Molecular structures of the investigated molecules.

2. Results and Discussion

2.1. Frontier Molecular Orbitals

To characterize the optical and electronic properties, it is useful to examine the FMOs of the compounds under investigation. The qualitative molecular orbital representations of the HOMOs and LUMOs of the compounds under investigation are plotted in Figure 1. The total and partial densities of states (TDOS and PDOS) on each fragment of the investigated molecules around the HOMO-LUMO gaps were calculated based on the current level of theory. The HOMOs and LUMOs contributions of individual fragments (in%) to the FMOs of the investigated molecules are given in Table 1. The FMOs of the compounds under investigation show π characteristics as visualized in Figure 1. The distribution patterns of FMOs are spread over the whole molecules for the investigated molecules. It implies that the spatial overlap between the HOMOs and LUMOs are strong, which may result in stronger optical absorption corresponding to the transition from HOMOs to LUMOs. The HOMOs of the investigated molecules are mainly localized on the BEDPP and CB fragments with only minor contributions from Ar fragments. The sum contributions of BEDPP and CB fragments of HOMOs are larger than 80%, while the corresponding contributions of Ar fragments are within 20%, respectively. Similar phenomena are found for the LUMOs of 1a, 1b, 1e, 2a, 2b, and 2e. The LUMOs mainly reside at the BEDPP and CB fragments with only minor contributions from Ar fragments. The sum contributions of BEDPP and CB fragments of LUMOs are larger than 84.6%, while the corresponding contributions of Ar fragments are within 15.4%, respectively. However, for the LUMOs of 1c, 1d, 2c, and 2d, the sum contributions of BEDPP and CB fragments are almost equal to those of Ar fragments. These results reveal that the different π-bridge units and end groups have obvious effects on the distribution of FMOs for the compounds under investigation. The distribution patterns of the FMOs of the compounds under investigation provide a remarkable signature for the intramolecular charge transfer (ICT) character of the vertical S₀ → S₁ transition. Analysis of the FMOs indicates that the excitation of the electron from the HOMOs to LUMOs leads the electronic density to flow mainly from the BDDPP fragments to CB and Ar fragments for 1a, 1b, and 2b. The percentages of charge transfer are the differences between the contributions of fragments for LUMOs and the corresponding contributions for HOMOs in the compounds under investigation. The percentages of charge transfer from BEDPP fragments to CB and Ar fragments for 1a, 1b, and 2b are 8.5%, 12.8%, and 5.2%, respectively. For 1c, 1d, 2c, and 2d, the excitation of the electron from the HOMOs to LUMOs leads the electronic density to flow mainly from BEDPP and CB fragments to Ar fragments, while the corresponding electronic density flow mainly from BEDPP and Ar fragments to CB fragments for 2a. However, for 1e and 2e, the excitation of the electron from the HOMOs to LUMOs leads the electronic density to flow mainly from Ar fragments to BEDPP and CB fragments, the percentages of charge transfer from Ar fragments to BEDPP and CB fragments are about 12.1% and 13.6%, respectively. The results displayed in Table 1 reveal that the BEDPP fragments serve as donors and CB and Ar fragments serve as acceptors for 1a, 1b, and 2b, while the BEDPP and CB fragments serve as donors and Ar fragments serve as acceptors for 1c, 1d, 2c, and 2d. The BEDPP and Ar fragments serve as donors and CB fragments serve as acceptor for 2a. However, the Ar fragments serve as donors and BEDPP and CB fragments serve as acceptors for 1e and 2e.

Figure 1. The electronic density contours of the frontier orbitals for the compounds under investigation at the B3LYP/6-31G(d,p) level.

Figure 1. The electronic density contours of the frontier orbitals for the compounds under investigation at the B3LYP/6-31G(d,p) level.

Table 1. The HOMOs and LUMOs contributions of individual fragments (in%) to the FMOs of the investigated molecules at the B3LYP/6-31G(d,p) level.

Species	HOMOs			LUMOs
	BEDPP a	CB b	Ar c	BEDPP a	CB b	Ar c
1a	55.3	30.2	14.5	46.8	37.8	15.4
1b	61.1	29.2	9.7	48.3	37.5	14.2
1c	58.7	29.0	12.4	28.0	25.3	46.8
1d	52.1	28.5	19.4	23.4	23.3	53.3
1e	55.2	28.2	16.6	55.3	40.2	4.50
2a	54.4	30.7	14.9	52.6	33.1	14.3
2b	57.9	30.2	11.9	52.7	32.7	14.6
2c	56.6	29.8	13.6	31.0	21.1	47.9
2d	54.6	27.7	17.7	26.4	19.3	54.4
2e	51.4	29.2	19.4	60.0	34.2	5.80

^a BEDPP: 2,5-bis(2-ethylhexyl)pyrrolo[3,4-c]pyrrole-1,4(2H,5H)-dione moieties; ^b CB: conjugate bridge moieties; ^c Ar: aromatic moieties.

Table 1. The HOMOs and LUMOs contributions of individual fragments (in%) to the FMOs of the investigated molecules at the B3LYP/6-31G(d,p) level.

Species	HOMOs			LUMOs
	BEDPP a	CB b	Ar c	BEDPP a	CB b	Ar c
1a	55.3	30.2	14.5	46.8	37.8	15.4
1b	61.1	29.2	9.7	48.3	37.5	14.2
1c	58.7	29.0	12.4	28.0	25.3	46.8
1d	52.1	28.5	19.4	23.4	23.3	53.3
1e	55.2	28.2	16.6	55.3	40.2	4.50
2a	54.4	30.7	14.9	52.6	33.1	14.3
2b	57.9	30.2	11.9	52.7	32.7	14.6
2c	56.6	29.8	13.6	31.0	21.1	47.9
2d	54.6	27.7	17.7	26.4	19.3	54.4
2e	51.4	29.2	19.4	60.0	34.2	5.80

^a BEDPP: 2,5-bis(2-ethylhexyl)pyrrolo[3,4-c]pyrrole-1,4(2H,5H)-dione moieties; ^b CB: conjugate bridge moieties; ^c Ar: aromatic moieties.

Another way to understand the influence of the optical and electronic properties is to analyze the E_HOMO, E_LUMO, and E_g. The E_HOMO, E_LUMO, and E_g of the designed molecules, PCBM and its derivatives bisPCBM and PC70BM were calculated and the results are given in Figure 2. As shown in Figure 2, for molecules with thiophene π-bridge (1a–1e), the E_HOMO values are in the order of 1d > 1a > 1b > 1e > 1c and the sequence of E_LUMO values is 1e > 1a > 1b > 1d > 1c. Therefore, the E_g values are in the order of 1e > 1b > 1a > 1c > 1d. This shows that molecules with benzo[d]thiazole (BT), benzo[c][1,2,5]thiadiazole (BTD), and 9H-carbazole (CZ) end groups possess lower E_HOMO, while a molecule with a thieno[3,4-b]pyrazine (TP) end group has higher E_HOMO compared with molecules with thiophene end groups. For E_LUMO, molecules with BT, TP, and BTD end groups possess lower E_LUMO, while molecules with CZ end groups has higher E_LUMO in comparison with molecules with thiophene end groups. The E_g value of molecules with BT and CZ end groups are larger, while the corresponding values of molecules with TP and BTD end groups are smaller than that of molecules with thiophene end groups. For molecules with furan π-bridge (2a–2e), the E_HOMO values are in the order of 2a > 2d > 2e > 2b > 2c and the sequence of E_LUMO values is 2a > 2e > 2b > 2d > 2c. Thus, the E_g values are in the order of 2e > 2b > 2a > 2c > 2d. This indicates that molecules with BT, BTD, TP, and CZ end groups possess lower both E_HOMO and E_LUMO values compared with molecules with thiophene end groups. The BT and CZ end groups increase, while the TP and BTD end groups decrease the E_g values compared with that of molecules with thiophene end groups. Furthermore, compared the E_HOMO, E_LUMO, and E_g of molecules with thiophene π-bridges with those with furan π-bridges, one can find that the E_HOMO, E_LUMO, and E_g values of with furan π-bridges are larger than those of molecules with thiophene π-bridges. These results suggest that the different π-bridges and end groups have effects on the E_HOMO, E_LUMO, and E_g for the compounds under investigation.

It is well-known that PCBM, bisPCBM, and PC70BM are excellent acceptors for organic solar cells [39,40,41]. Therefore, we choose these three fullerene derivatives as acceptors in our work. As shown in Figure 2, the E_LUMO values of the designed molecules are higher than those of PCBM, bisPCBM, and PC70BM, respectively. The differences between the E_HOMO of the designed molecules and the E_LUMO of PCBM are 1.563~1.836 eV, while the corresponding values of bisPCBM and PC70BM are larger than 1.649 and 1.642 eV, respectively. These results imply that the designed molecules can provide better matches of FMOs to PCBM, bisPCBM, and PC70BM. Therefore, different π-bridges and aromatic end groups can tune the FMOs of derivatives more suitable to PCBM, bisPCBM, and PC70BM.

Figure 2. Evaluation of calculated FMO energies for investigated molecules as well as FMO energies for PCBM, bisPCBM, and PC70BM at the B3LYP/6-31G(d,p) level.

Figure 2. Evaluation of calculated FMO energies for investigated molecules as well as FMO energies for PCBM, bisPCBM, and PC70BM at the B3LYP/6-31G(d,p) level.

2.2. Absorption Spectra

Table 2 presents the absorption region R, the longest wavelength (λ_abs) of absorption spectrum, the oscillator strength (f), and main configurations of the designed molecules. The absorption wavelengths λ_abs and the oscillator strength f of the first fifteen excited states for the compounds under investigation are listed in Tables S1 and S2 in the Supporting Information. The λ_abs value of 1a is in agreement with the experimental result [38], the deviation is 9 nm. This reveals that the level of theory we selected is reasonable for this type of system. The absorptions of the compounds under investigation are assigned to the S₀ → S₁ electronic transitions and HOMOs → LUMOs excitations play a dominant role. From Table 2, one can find that the λ_abs of 1c, 1d, 2c, and 2d have bathochromic shifts, while the corresponding λ_abs values of 1b, 1e, 2a, 2b, and 2e have hypsochromic shifts compared with that of the parent compound 1a. The λ_abs values are in the order of nd > nc > na > nb > ne (n = 1 and 2), which is in excellent agreement with the corresponding reverse order of E_g values displayed in Figure 2. It indicates that the introduction of BT and CZ end groups decrease, while the introduction of TP and BTD end groups increase the λ_abs values compared with molecules with thiophene end groups for designed molecules. For 1a–1e, the sequence of their R values is 1d > 1c > 1a > 1b > 1e. It suggests that the introduction of BT and CZ end groups results in smaller R values, while the introduction of TP and BTD end groups lead to the increase of R values compared with molecules with thiophene end groups. However, for 2a–2e, their R values are in the order of 2d > 2c > 2b > 2e > 2a. It indicates that the introduction of BT, CZ, TP, and BTD end groups leads to the increase of R values compared with the thiophene end group. The oscillator strength for an electronic transition is proportional to the transition moment [42]. In general, larger oscillator strength corresponds to larger experimental absorption coefficient. The order of the predicted f values are in the decreasing order of nb > na > ne > nc > nd (n = 1 and 2). This indicates that the introduction of BT end group increases, while the introduction of BTD, TP, and CZ end groups slightly decreases the f values compared with thiophene end groups for the designed molecules. Furthermore, a careful inspection of the results displayed in Table 2 reveals clearly that the λ_abs and R values of molecules with BTD and TP end groups are larger than those of other molecules. It suggests that molecules with BTD and TP end groups can lower the material band gap and extend the absorption spectrum towards longer wavelengths. The λ_abs and R values of molecules with BT and CZ end groups are smaller slightly than those of parent molecule 1a. Therefore, molecules under investigation own the large λ_abs, f, and R values. The designed molecules could be used as solar cell material with intense, broad absorption spectra.

Table 2. Predicted absorption region R, the longest wavelength of absorption, corresponding oscillator strength f, and main configurations of the compounds under investigation at the TD-B3LYP/6-31G(d,p)//B3LYP/6-31G(d,p) level.

Species	λabs (nm)	f	Main configurations	R (nm) a
1a	599	0.97	H → L (0.71)	295
1b	588	0.98	H → L (0.71)	281
1c	684	0.82	H → L (0.70)	340
1d	746	0.80	H → L (0.70)	387
1e	575	0.87	H → L (0.70)	261
2a	583	0.74	H → L (0.71)	209
2b	581	0.81	H → L (0.71)	288
2c	669	0.61	H → L (0.70)	354
2d	716	0.56	H → L (0.70)	357
2e	574	0.73	H → L (0.70)	271
Exp b	590

^a R denotes for the difference of the longest and shortest wavelength values with oscillator strength larger than 0.01 considering the first fifteen excited states; ^b Experimental data for 1a were taken from Ref. [38].

Table 2. Predicted absorption region R, the longest wavelength of absorption, corresponding oscillator strength f, and main configurations of the compounds under investigation at the TD-B3LYP/6-31G(d,p)//B3LYP/6-31G(d,p) level.

Species	λabs (nm)	f	Main configurations	R (nm) a
1a	599	0.97	H → L (0.71)	295
1b	588	0.98	H → L (0.71)	281
1c	684	0.82	H → L (0.70)	340
1d	746	0.80	H → L (0.70)	387
1e	575	0.87	H → L (0.70)	261
2a	583	0.74	H → L (0.71)	209
2b	581	0.81	H → L (0.71)	288
2c	669	0.61	H → L (0.70)	354
2d	716	0.56	H → L (0.70)	357
2e	574	0.73	H → L (0.70)	271
Exp b	590

^a R denotes for the difference of the longest and shortest wavelength values with oscillator strength larger than 0.01 considering the first fifteen excited states; ^b Experimental data for 1a were taken from Ref. [38].

2.3. Reorganization Energies

Understanding the relationship between molecular structure and charge transport property of the material is a key factor for designing good candidates for solar cell devices. It is well-known that the lower the reorganization energy values, the higher the charge transfer rate [43,44]. The calculated reorganization energies for hole and electron are listed in Table 3. The results displayed in Table 3 show that the calculated λ_h values of 1c, 1d, 2c, and 2d are smaller than that of N,N′-diphenyl-N,N′-bis(3-methlphenyl)-(1,1′-biphenyl)-4,4′-diamine (TPD), which is a typical hole transport material (λ_h = 0.290 eV) [45]. The calculated λ_h values of 1a, 1b, 2a, and 2b are slightly larger than that of TPD. This implies that the hole transfer rates of 1c, 1d, 2c, and 2d might be higher than that of TPD, while the hole transfer rates of 1a, 1b, 2a, and 2b are almost equal to that of TPD. The λ_h values of 1e and 2e are larger than that of TPD. It suggests that their hole transfer rates might be lower than that of TPD. The λ_e values of the designed molecules except for 1e are smaller than that of tris(8-hydroxyquinolinato)aluminum(III) (Alq3), which is a typical electron transport material (λ_e = 0.276 eV) [46], indicating that the electron transfer rates of the designed molecules except for 1e might be higher than that of Alq3. For molecules with thiophene π-bridges (1a–1e), both the λ_h and λ_e value of 1d and 1c are smaller, while the corresponding values of 1b and 1e are larger than those of 1a. This suggests that the introduction of TP and BTD end groups increases, while the introduction of BT and CZ end groups decreases the electron and hole transfer rates compared with molecules with thiophene end groups. For molecules with furan π-bridges (2a–2e), the λ_h values of 2b, 2d, and 2c are smaller, while the corresponding value of 2e is larger than that of 2a. On the contrary, the λ_e values of 2b, 2d, and 2e are larger, while the corresponding value of 2c is smaller than that of 2a. This indicates that the introduction of BT, BTD, and TP end groups increases, while the introduction of CZ end group decreases the hole transfer rates compared with thiophene end groups. However, BT, TP, and CZ end groups decrease, while BTD end groups increase the electron transfer rates compared with thiophene end groups. From Table 3, one can find that ne (n = 1 and 2) have the largest λ_h and λ_e values, while nc (n = 1 and 2) own the smallest λ_h and λ_e values, respectively. Inspection of the results displayed in Table 3 reveals that the designed molecules can be used as promising hole transport materials except for molecules with CZ end groups. The designed molecules can be used as promising electron transport materials from the stand point of the smaller reorganization energy except for molecules with thiophene π-bridges and CZ end groups.

Table 3. Calculated λ_e and λ_h (both in eV) of the compounds under investigation at the B3LYP/6-31G(d,p) level.

Species	λh	λe
1a	0.293	0.160
1b	0.295	0.199
1c	0.266	0.154
1d	0.258	0.171
1e	0.489	0.285
2a	0.301	0.153
2b	0.291	0.159
2c	0.273	0.143
2d	0.287	0.191
2e	0.426	0.210

Table 3. Calculated λ_e and λ_h (both in eV) of the compounds under investigation at the B3LYP/6-31G(d,p) level.

Species	λh	λe
1a	0.293	0.160
1b	0.295	0.199
1c	0.266	0.154
1d	0.258	0.171
1e	0.489	0.285
2a	0.301	0.153
2b	0.291	0.159
2c	0.273	0.143
2d	0.287	0.191
2e	0.426	0.210

2.4. Calculated Crystal Structure and Transport Properties

We calculated the mobility of the designed molecules to study their charge transport property. The total energies of the predicted crystal structures for the designed molecules in different space groups are summarized in Tables S3 and S4 in the Supporting Information. The lattice constants of the designed molecules with the lowest total energies are listed in Table S5 in the Supporting Information. The transmission paths are selected according to the optimized crystal structures. We arbitrarily choose one molecule in the crystal as the carrier donor and take all its neighboring molecules as paired elements. Each pair is defined as a transmission path. Then the charge transfer integral can be calculated according to the transmission path. The mobility can be estimated from the Einstein relation. We predict the mobility of the designed molecules with the lowest total energies. In order to understand the crystal structures effect on the charge transfer integral and mobility, we select 1c as representative of the system under investigation. The predicted crystal structures of 1c with two lowest total energies belong to space groups Pī and Pbca and shown in Figure 3. We predict the charge transfer integral and mobility of 1c in these two space groups. The most important pathways (dimers) in space groups Pī and Pbca are shown in Figure 4. Then the charge transfer integral can be calculated according to the transmission path, and the mobility can be estimated from the Einstein relation. The calculated transfer integrals of 1c for holes and electrons in space groups Pī and Pbca are listed in Table 4. The calculated transfer integrals of others designed molecules with the lowest total energies for holes and electrons are listed in Table S6 in the Supporting Information. The data in Table 4 demonstrate that the electronic coupling is determined by the relative distance and orientations of the interacting molecules [47]. Furthermore, 1c possesses the largest absolute electron and hole coupling values in pathways 1, 3, and 5 for space group Pī and in pathways 3, 4, and 5 for space group Pbca. It reveals that the orientation of the interacting molecules is the key factor of hole or electron coupling for 1c, because the co-facial stacking structure is expected to provide more efficient orbital overlap leading to the most efficient charge transfer route [47]. The calculated electron and hole mobility of the compounds under investigation are listed in the Table 5. The values of hole mobility of 1c for Pī and Pbca space group (2.00 × 10⁻³ and 2.14 × 10⁻³ cm²/Vs) are larger than that of TPD (1.0 × 10⁻³ cm²/Vs) [48], respectively. The values of electron mobility of 1c for Pī and Pbca space group are 8.97 × 10⁻² and 5.11 × 10⁻² cm²/Vs, respectively. The values of electron mobility for Pī and Pbca space group are larger than that of hole mobility, respectively. It shows that different space groups lead to different mobility values; that is to say, the stacking structure is the most important factor for the molecular mobility property. The theoretical prediction shows that 1c can be made as hole and electron transfer materials used for solar cells. Moreover, it also has balanced charge transport characteristics, which agrees with the result of reorganization energy.

The results displayed in Table 5 show that the hole mobility values of 1a, 1c, 1d, 2a, and 2b are larger, while the corresponding values of 1b, 1e, and 2c–2e are smaller than that of TPD. Furthermore, from Table 5, one can find that the hole mobility values are in the orders of 1d > 1a > 1c > 1b > 1e and 2b > 2a > 2d > 2e > 2c for 1a–1e and 2a–2e, respectively. It suggests that the introduction of the TP end group can increase, while the introduction of BT, BTD, and CZ end groups leads to the decrease for the hole mobility values compared with molecule with thiophene end groups for 1a–1e. The introduction of the BT end group increases, while the introduction of BTD, TP, and CZ end groups decreases the hole mobility values compared with molecule with thiophene end groups for 2a–2e. The sequences of electron mobility are 1d > 1c > 1a > 1b> 1e and 2a > 2c > 2b > 2d> 2e for 1a–1e and 2a–2e, respectively. This clearly shows that the electron mobility values can be decreased by introduction of BT and CZ end groups, while introduction of BTD and TP end groups increases the electron mobility values compared with molecule with thiophene end groups for 1a–1e. The introduction of BT, BTD, TP, and CZ groups leads to the decrease for the electron mobility values compared with molecule with thiophene end groups for 2a–2e. Inspection of the results displayed in Table 5 reveals clearly that the electron mobility values of 1a–1d, 2a, 2c–2e are larger than those of hole mobility values, respectively. However, the electron mobility value of 2b is smaller than that of hole mobility value. The electron mobility value of 1e is almost equal to that of hole mobility value. Considering the reorganization energy for designed molecules above, the compounds under investigation can be made as electron transfer materials except for 1e, particularly for 1a and 1c. The 1a, 1c, 1d, 2a, and 2b can be made as hole transfer materials. This suggests that the compounds under investigation can be made as hole and/or electron transfer materials used for solar cells.

Figure 3. Herringbone structures of 1c in different space groups.

Figure 3. Herringbone structures of 1c in different space groups.

Figure 4. Crystal structures and hopping routes of 1c in different space groups.

Figure 4. Crystal structures and hopping routes of 1c in different space groups.

Table 4. The center-center distance and the corresponding hole and electron coupling between the dimer in all of the nearest neighbor pathways for 1c in different space groups.

Space Groups	Pathway	Distance (Å)	Electron Coupling (eV)	Hole Coupling (eV)
Pī	1	13.450	−9.00 × 10−3	−3.22 × 10−4
	2	13.450	−9.00 × 10−3	−3.22× 10−4
	3	16.936	−3.28 × 10−7	1.25 × 10−6
	4	16.936	−3.28 × 10−7	1.25 × 10−6
	5	11.223	1.50 × 10−3	−3.20 × 10−3
	6	13.509	1.50 × 10−8	1.04 × 10−8
Pbca	1	13.373	−5.86 × 10−6	8.19 × 10−6
	2	13.373	−5.86 × 10−6	8.19 × 10−6
	3	14.824	−5.80 × 10−3	2.50 × 10−3
	4	14.824	−5.80 × 10−3	2.50 × 10−3
	5	14.824	−5.80 × 10−3	2.50 × 10−3
	6	15.591	−6.50 × 10−3	5.79 × 10−4

Table 4. The center-center distance and the corresponding hole and electron coupling between the dimer in all of the nearest neighbor pathways for 1c in different space groups.

Space Groups	Pathway	Distance (Å)	Electron Coupling (eV)	Hole Coupling (eV)
Pī	1	13.450	−9.00 × 10−3	−3.22 × 10−4
	2	13.450	−9.00 × 10−3	−3.22× 10−4
	3	16.936	−3.28 × 10−7	1.25 × 10−6
	4	16.936	−3.28 × 10−7	1.25 × 10−6
	5	11.223	1.50 × 10−3	−3.20 × 10−3
	6	13.509	1.50 × 10−8	1.04 × 10−8
Pbca	1	13.373	−5.86 × 10−6	8.19 × 10−6
	2	13.373	−5.86 × 10−6	8.19 × 10−6
	3	14.824	−5.80 × 10−3	2.50 × 10−3
	4	14.824	−5.80 × 10−3	2.50 × 10−3
	5	14.824	−5.80 × 10−3	2.50 × 10−3
	6	15.591	−6.50 × 10−3	5.79 × 10−4

Table 5. The electron and hole mobility of the designed molecules. [T = 298 K, in cm²/Vs].

Species	Space Groups	Electron Mobility	Hole Mobility
1a	P212121	1.05 × 10−2	8.18 × 10−3
1b	Pna21	2.21 × 10−4	1.18 × 10−4
1c	Pī	8.97 × 10−2	2.00 × 10−3
1c	Pbca	5.11 × 10−2	2.14 × 10−3
1d	C2/c	0.136	9.57 × 10−3
1e	P21/c	2.45 × 10−6	7.75 × 10−6
2a	P21	7.561 × 10−3	1.77 × 10−3
2b	P21	4.22 × 10−5	7.13 × 10−2
2c	Cc	8.18 × 10−4	1.31 × 10−5
2d	P212121	3.14 × 10−5	2.56 × 10−5
2e	Pī	2.55 × 10−5	1.78 × 10−5

Table 5. The electron and hole mobility of the designed molecules. [T = 298 K, in cm²/Vs].

Species	Space Groups	Electron Mobility	Hole Mobility
1a	P212121	1.05 × 10−2	8.18 × 10−3
1b	Pna21	2.21 × 10−4	1.18 × 10−4
1c	Pī	8.97 × 10−2	2.00 × 10−3
1c	Pbca	5.11 × 10−2	2.14 × 10−3
1d	C2/c	0.136	9.57 × 10−3
1e	P21/c	2.45 × 10−6	7.75 × 10−6
2a	P21	7.561 × 10−3	1.77 × 10−3
2b	P21	4.22 × 10−5	7.13 × 10−2
2c	Cc	8.18 × 10−4	1.31 × 10−5
2d	P212121	3.14 × 10−5	2.56 × 10−5
2e	Pī	2.55 × 10−5	1.78 × 10−5

3. Computational Methods

All calculations have been carried out using Gaussian 09 package [49]. The equilibrium structures of the compounds under investigation, including neutral, cationic, and anionic molecules, were optimized using the B3LYP functional with the 6-31G(d,p) basis set. The harmonic vibrational frequency calculations using the same methods as for the geometry optimizations were used to ascertain the p3resence of a local minimum. To compare the energies of the FMOs for donors and acceptors, the electronic properties of PCBM and its derivatives were calculated at the B3LYP/6-31G(d,p) level based on the optimized structures at the B3LYP/6-31G(d) level. The absorption spectra of the compounds under investigation were predicted using the TD-B3LYP/6-31G(d,p) method based on the optimized geometries.

According to Marcus theory [43,44], the charge transport can be considered as a hopping process in the organic solid. The charge transfer rate can be represented by means of the following equation:

$$K=(\raisebox{1ex}{$V^2$}\!\left/ \!\raisebox{-1ex}{$\hslash $}\right.){\left(\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$\mathrm{\lambda }{k}_{\mathrm{B}}T$}\right.\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\exp \left(-\raisebox{1ex}{$\mathrm{\lambda }$}\!\left/ \!\raisebox{-1ex}{$4k_{\mathrm{B}}T$}\right.\right)$$

(1)

where T represents the absolute temperature, k_B is the Boltzmann constant, λ and V correspond to the reorganization energy and transfer integral, respectively. Inspection of Equation (1) clearly reveals that the reorganization energy and transfer integral play a dominant role in determining the charge transfer rate. The minimal reorganization energy and the maximal transfer integral can lead to the increase of charger transfer rates. For the reorganization energies λ, they consist of external reorganization energy (λ_ext) and internal reorganization energy (λ_int). λ_ext represents the effect of polarized medium on charge transfer. The internal reorganization is due to the change between ionic and neutral states. The predicted λ_ext values in pure organic condensed phases are much smaller than their λ_int counterparts and can be neglected [50,51,52]. Moreover, there is a clear correlation between λ_int and charge transfer rate in literature [53]. Therefore, we focus on the λ_int of the isolated active organic π-conjugated systems exclusively. The reorganization energies for electron (λ_e) and hole (λ_h) can be calculated by equations [54]:

$$\mathrm{\lambda }\mathrm{e}=\left({{\displaystyle E}}_0^{-}-{{\displaystyle E}}_{-}^{-}\right)+\left({{\displaystyle E}}_{-}^0-{{\displaystyle E}}_0^0\right)$$

(2)

$$\mathrm{\lambda }\mathrm{h}=\left({{\displaystyle E}}_0^{+}-{{\displaystyle E}}_{+}^{+}\right)+\left({{\displaystyle E}}_{+}^0-{{\displaystyle E}}_0^0\right)$$

(3)

where ${{\displaystyle E}}_0^{+}$(${{\displaystyle E}}_0^{-}$) is the energy of the cation (anion) calculated with the optimized structure of the neutral molecule. Similarly, ${{\displaystyle E}}_{+}^{+}$(${{\displaystyle E}}_{-}^{-}$) is the energy of the cation (anion) calculated with the optimized cation (anion) structure, ${{\displaystyle E}}_{+}^0$(${{\displaystyle E}}_{-}^0$) is the energy of the neutral molecule calculated at the cationic (anionic) state, and $E_0^0$ is the energy of the neutral molecule in ground state. For comparing with the interested results reported previously [45,46], the λ_e and λ_h of the molecules were calculated at the B3LYP/6-31G(d,p) level on the basis of the single point energy.

For the charge transfer integral of hole (electron) transfer, they can be predicted using equation [55,56]:

$$V_{i\mathrm{j}}=\langle {\mathrm{\phi }}_1^0|{\widehat{F}}^0|{\mathrm{\phi }}_2^0\rangle$$

(4)

V_ij is the transfer integral. ${\mathrm{\phi }}_1^0$ and ${\mathrm{\phi }}_2^0$ represent the unperturbed frontier orbital of molecules 1 and 2 in the dimer, respectively. ${\widehat{F}}^0$ is the Kohn-Sham-Fock operator of the dimer obtained with the unperturbed density matrix, which can be evaluated by the molecular orbitals and density matrix of the two individual molecules. The pw91pw91/6-31G(d) method is employed to calculate the transfer integral. This method gave a reasonable description for intermolecular coupling term previously [57]. The molecular crystal structure is predicted by the module Polymorph of the software package Materials Studio [58]. The geometry of the cluster models used in the present study was taken from the B3LYP/6-31G(d,p) level.

The drift mobility of hopping μ, can be evaluated from the Einstein equation:

$$\mathrm{\mu }=\frac{e}{k_{\mathrm{B}}T}D$$

(5)

where e is the electronic charge, D is the diffusion coefficient, which can be evaluated as [59]:

$$D=\underset{t\to \infty }{\lim }\frac{1}{2n}\frac{\langle x{(t)}^2\rangle }{t}\approx \frac{1}{2n}{\displaystyle \sum _i{d}_{\mathrm{i}}^2}{k}_{\mathrm{i}}{p}_{\mathrm{i}}=\frac{1}{2n}\frac{{\displaystyle {\sum }_i{d}_{\mathrm{i}}^2{k}_{\mathrm{i}}^2}}{{\displaystyle {\sum }_i{k}_{\mathrm{i}}}}$$

(6)

Here d_i is the center mass distance to neighbor i, n is the spatial dimension of the crystal, and k_i is the hopping rate due to charge transfer to ith neighbor. P_i represents the relative probability for charge transfer to a particular ith neighbor.

$$p_{\mathrm{i}}=\frac{k_{\mathrm{i}}}{{\displaystyle {\sum }_j{k}_{\mathrm{i}}}}$$

(7)

4. Conclusions

In the present work, we investigated a series of DPP-based small molecules with DPP derivative 2,5-bis(2-ethylhexyl)pyrrolo[3,4-c]pyrrole-1,4(2H,5H)-dione (BEDPP) as core, thiophene or furan as conjugate π-bridges (CB), and aromatic derivatives (Ar) as end groups for OSC applications. The FMO analyses have turned out that the molecules can lower the material band gap and extend the absorption spectrum towards longer wavelengths. Our results reveal that the molecules under investigation own the longest wavelength of absorption spectra, oscillator strength, and absorption region values. They could be used as solar cell material with intense broad absorption spectra. Our results suggest that the optical, electronic, and charge transport properties are affected by the introduction of different end groups. The mobility values of the designed molecules are also investigated. Our results show that the designed molecules are expected to be promising candidates for electron and/or hole transport materials for solar cells. On the basis of the investigated results, we suggest that molecules under investigation are suitable donors for PCBM, bisPCBM, and PC70BM as acceptors of solar cells.