The response of charge transfer properties to electric fields in organic semiconductors: a comprehensive theoretical investigation†

Hengyue Zhang‡ ^abcd, Jie Zhu‡ ^ab, Xinxin Niu ^ab, Qian Zhang ^ab, Yajing Sun *^ab and Weigang Zhu *^a
aKey Laboratory of Organic Integrated Circuits, Ministry of Education, Tianjin Key Laboratory of Molecular Optoelectronic Sciences, Department of Chemistry, School of Science, Tianjin University, Tianjin, 300072, PR China. E-mail: syj19@tju.edu.cn
^bHaihe Lab of ITAI, Tianjin 300051, PR China
^cSchool of Chemical Engineering and Technology, Tianjin University, Tianjin, 300354, PR China
^dCollege of Chemistry, Nankai University, Tianjin, 300071, PR China

First published on 25th July 2024

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Weigang Zhu

Weigang Zhu received his B.Eng. degree in Applied Chemistry with the highest Honors from the University of Electronic Science and Technology of China (2011) and PhD in Physical Chemistry from the Institute of Chemistry Chinese Academy of Sciences with Prof. Wenping Hu (2016). He then carried out postdoctoral training at Northwestern University under Prof. Tobin J. Marks (2016–2019), and thereafter has been serving as the tenured Associate Professor in Chemistry and Materials Science at Tianjin University since 2020. His research encompasses time-resolved optical spectroscopy, electron paramagnetic resonance spectroscopy, device operation technology, nonlinear optics, and metal oxides.

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Introduction

Organic semiconductors have become a hot topic in the fields of materials science and electronic engineering due to their low cost, lightweight, flexibility, non-toxicity, and biodegradability.^1–4 Additionally, these materials are increasingly significant in the field of energy conversion and sustainability, particularly those with high power conversion efficiencies (PCEs), which are used as new organic photovoltaic (OPV) materials.^5–8 As the core of organic electronic devices, the charge transport properties of organic semiconductors are crucial. Therefore, enhancing their mobility has been a focus of research. For example, in OPV devices, high mobility effectively facilitates the separation of electrons and holes;⁹ in organic thermoelectric devices, high conductivity is key to improving the thermoelectric conversion efficiency;¹⁰ furthermore, in organic field-effect transistors (OFETs), which are fundamental components of wearable flexible devices, charge transport characteristics critically influence their application potential. In addition, OFETs, the fundamental units of various wearable flexible devices, and their charge transport properties determine their application potential. However, due to their sensitivity, the performance of organic functional materials can be severely affected by various working environmental factors such as molecular stacking modes,¹¹ interface situations,¹² external strain,^13–17 pressure,¹⁸ temperature,^19,20 and electric fields.^20–26 Electric fields, in particular, as an inevitable factor in the working environment of OFETs, should receive focused attention regarding their impact on mobility. Despite this, current research on this aspect is still relatively limited.

Currently, a substantial amount of experimental evidence has demonstrated that electric fields significantly affect the charge transport properties. For instance, a study in 2004 found that increasing the gate-source voltage in OFETs could significantly enhance mobility.²⁰ Additionally, researchers have observed that electric fields affect the orientation and position of organic semiconductor molecules,²² as well as the growth process of single crystals.²³ At the same time, some theoretical studies are also underway. Sancho-García et al.²⁴ found that the impact of the electric field along the molecular long axis on the internal reorganization energies and interchain transfer integrals in oligomers can be negligible. Afterwards, Förster et al.²⁵ performed similar investigation on polymers and discovered that the response to electric fields along different directions is distinct, and the electric field may cause reduction in reorganization energies and increase in transfer integrals. Zhan et al.²⁶ have also studied the response of reorganization energy in oligomers to electric fields in different directions. Lu et al.²¹ provided the carrier mobility that was influenced by the electric field along the molecular long axis and thoroughly analysed the reasons, indicting a linear relationship between the mobility and electric field. More comprehensive research work on the direct impact of an external electric field on the charge transport properties in organic semiconductors is still highly welcomed.

In this study, we selected 2,6-diphenylanthracene (DPA) as our model material, which demonstrates high carrier mobility.²⁷ Widely used in the organic semiconductor layers of organic field-effect transistors (OFETs),^27–29 DPA's superior performance underpins our choice and aims to enhance the practical and extendable aspects of our research. Additionally, we employed multiscale theoretical computation methods to investigate the effects of electric fields in various orientations on its geometric structure, stacking behaviour, and charge transport characteristics. By utilizing periodic boundary conditions (PBCs) and density functional theory (DFT), we have conclusively demonstrated that an external electric field significantly alters the intra- and intermolecular geometric structures, thereby significantly impacting charge transport characteristics. Additionally, we use a device-level simulation to determine the actual carrier mobility within OFETs. These simulation results further confirm that by adjusting the electric field, the carrier mobility can be effectively controlled. Based on these findings, we believe this research will provide key insights that will drive the development of the organic electronics industry.

Methods

Geometry optimization

DPA is a semiconductor molecule featuring an anthracene core with phenyl groups at the 2 and 6 positions, and it demonstrates satisfactory charge transport mobility. The molecular structure of DPA is characterized by its aromaticity, which imparts rigidity and results in a uniform orbital distribution across the anthracene, as shown in Fig. 1. When molecules aggregate into solid states, the DPA single-crystal has layered characteristics with in-plane herringbone packing mode. Each molecule in this arrangement interacts with six nearest neighbours, forming molecular dimers. The single crystal belongs to the P2(1)/c space group. The initial crystal structure of DPA is obtained from experimental measurements.²⁷ The structural optimization is conducted under external electric fields with PBCs, and the external electric field is along the x, y, and z axes. These computational studies are performed using the first-principles toolkit CP2K,³⁰ employing the Perdew–Burke–Ernzerhof (PBE)³¹ functional and the 6-31G(d) basis set.

Charge transport properties

We adopted the hopping model to depict the charge transport process, which is extensively utilized in the study of organic semiconductors to effectively characterize charge transport dynamics.³² In this model, charge transfer occurs between two adjacent molecules. Here, the carrier hop rate k within a molecular dimer can be written as,³³

	(1)

where ℏ is the reduced Planck constant, n_j the population of the jth normal mode, and ω_j represents the frequency of the ith vibration mode. S_j is the Huang–Rhys factor, a dimensionless factor used to characterize the strength of electron–phonon coupling. V is the transfer integral, partly indicating the strength of the electronic coupling between adjacent molecules. The coupling strength between m and n states of two adjacent molecules can be written as V_mn,^34,35

	(2)

V_mn is the transfer integral within a molecular dimer, e_m = 〈Φ_m|H|Φ_m〉, V⁰_mn = 〈Φ_m|H|Φ_n〉, O_mn = 〈Φ_m|O|Φ_n〉. Φ_m(n) is the frontier molecular orbital of each single molecule in a dimer. And in this work we ignored the site energy difference for there is only one DPA molecular in the crystal structure.

By combining the frequencies and Huang–Rhys factors of all phonon modes, we could obtain the reorganization, λ, which could reflect the overall electron–phonon coupling strength during the charge transport process,^36–38

	(3)

After determining the charge transfer rates using eqn (1), the charge transport process is modelled by the Pauli master equation (PME) under a Markov assumption. This equation is solved using an iterative numerical scheme to derive the charge carrier mobility, denoted as μ. The calculations for reorganization energy and the transfer integral are performed at B3LYP/6-31G(d) and PBE0/6-31G(d) levels, respectively. The frontier orbitals are calculated at the B3LYP/6-31G(d) level. The charge transfer property calculations are facilitated by the MOMAP software.^33,39–41 The determination of the geometry and electron structure is conducted using the Gaussian 09 computational package,⁴² with external electric fields applied. Furthermore, considering the dielectric properties of the DPA molecules, we simulated the electric field distribution in actual OFET devices. Detailed settings for these simulations are provided in the ESI.†

Results & discussion

Molecular and packing structure

This study first explores the impact of electric fields on the molecular structure and stacking modes of DPA crystals. We found that electric fields primarily affect the bond lengths, bond angles, and dihedral angles within the molecule. Particularly, due to the rigid conjugated structure of the aromatic rings, changes in bond lengths are more pronounced than changes in bond angles, as shown in Fig. S2.† Additionally, the electric field along the molecular long axis (i.e., the z-axis) has a notably greater impact on bond lengths. Representative bond lengths and dihedral angles are displayed in Fig. 2a and b, with corresponding atomic labels indicated in Fig. 1a. From Fig. 2a, it is evident that the lengths of C–C bonds in the aromatic ring, such as C3–C4 and C7–C8, change monotonically with the external electric field. When an external electric field is applied along the z-axis, the C3–C4 bond length decreases, while the C7–C8 bond length increases as the shielding effect of the electron clouds weakens. This opposite trend in electric field response is due to the shift in charge distribution: the electric field along the z-axis causes the electron cloud to shift in the opposite direction and accumulate at the C7–C8 end of the anthracene, thus strengthening the bond between atoms at this end and leading to the compression of relevant bonds. Conversely, at the C3–C4 end, the reduction in charge density results in an increase in bond lengths. Unlike the bonds within the aromatic ring, the effect of the electric field on the ring-connecting bonds, such as C6–C11, is non-directional. The high symmetry of the C6–C11 bond results in a symmetrical impact from the electric field. Additionally, due to weaker interactions, the electric field's influence on the C6–C11 bond is more significant, approximately twice that within the rings. The external electric field also significantly affects the dihedral angle between the benzene and anthracene planes, as seen in Fig. 2b. When an electric field of −0.005 a.u. along the z-axis is applied, the dihedral angle increases by 0.5°. This increase indicates an improvement in π-conjugation due to electron gathering. Moreover, the dihedral angle is also affected by electric fields along the x and y axes, as these directions are related to the dihedral twisting vibration.

Using periodic boundary conditions (PBCs), we investigated the effects of external electric fields on the stacking structure of DPA solids. This effect is primarily manifested through changes in the distances between the nearest molecular dimers. During the optimization of DPA single crystals, we discovered that in-plane electric fields have a more significant impact on molecular stacking compared to vertical electric fields, which is distinctly different from the response of intramolecular chemical bonds and dihedral angles. As shown in Fig. 2c and d, the response of molecular stacking distances to electric fields is closely related to the relative positions of molecular dimers and the direction of the electric field. The DPA crystal belongs to the P2(1)/c space group, and in the absence of electric fields, the crystal structure includes a centre of symmetry, a screw axis, and a slip plane. These three symmetry elements ensure that dimers 1–4 (D1–4) have the same molecular distances, as do D5/D6, which are 4.823 Å and 6.245 Å, respectively. When an electric field is applied, the symmetry of the crystal is partially disrupted. For example, an electric field along the x-direction destroys the screw axis and the centre of symmetry, breaking the degeneracy of D1/D2 and D3/D4. Similarly, an electric field along the y-direction destroys the slip plane and the centre of symmetry, breaking the degeneracy of D1/D4 and D2/D3. Specifically, an electric field of 0.005 a.u. along the y-axis increases the distance between dimers D1 and D4 by about 6 × 10⁻² Å, and decreases the distance between dimers D2 and D3 by about 6 × 10⁻² Å. This significant response of the molecular stacking structure to the applied electric field will further affect the charge transport properties.

Charge transport properties

After fully considering the impact of an external electric field on the geometric structures, we calculated the charge transport performance of DPA based on its most stable structure under various electric field conditions. Since DPA crystals exhibit a layered structure and the charge transfer rates between layers are extremely low (negligible), as shown in Fig. S3,† we primarily focused on the carrier mobilities within the layers. The in-plane charge transport in DPA demonstrates significant anisotropy, and we have studied the mobilities along the x and y directions, denoted as μ_x and μ_y, respectively.

First, as shown in Fig. 3a and b, the carrier mobility always exhibits a negative response to the electric field. Particularly, when an electric field of 0.005 a.u. is applied along the z-direction, the mobility decreases significantly by about 90%, indicating that the effect of the electric field is considerable. More importantly, compared to the in-plane electric field, the out-of-plane electric field, E_z, has a more significant inhibitory effect on mobility, at least three times greater than that of the in-plane fields. This is mainly because the additional electrostatic interactions introduced by E_z rearrange the electronic structure of DPA and disrupt the original intramolecular delocalized charge distribution (see Fig. 3c). This more localized charge distribution significantly enhances electron–phonon coupling, which can be measured by the reorganization energy, λ_h. The larger λ_h indicates stronger electron–phonon interactions, which hinder charge transport in organic materials. As can be seen from Fig. 3d, when E_z reaches 0.005 a.u., the reorganization energy is twice that of the absence of an electric field. Meanwhile, the in-plane electric fields (E_x and E_y) also cause a slight increase in reorganization energy.

Fig. 3e illustrates the changes in the contributions of each atom to the improvement of reorganization energy under an applied electric field. The red parts indicate contributions to an increase, while the blue parts indicate contributions to a decrease. Under the electric fields of E_x and E_y, each atom of DPA contributes slightly but uniformly to the increase in reorganization energy. In contrast, under E_z, due to the rearrangement effect on charge distribution, there is a sharp increase in the reorganization energy at the charge aggregation end, resulting in enhanced electron–phonon interactions and suppressed in-plane mobility. By analysing various phonon modes, we observed that the in-plane electric field enhances the Huang–Rhys factors for vibrational modes aligned with the direction of the electric field. Specifically, E_y leads to an increase in the Huang–Rhys factors of phonon modes at frequencies of 21.98 cm⁻¹ and 21.45 cm⁻¹, from 0 to 1.08 and 1.16, respectively. These frequencies correspond to the normal modes that oscillate along the y-axis, as shown in Fig. S5.† According to eqn (2), an external electric field will induce the interaction of vibration modes along the direction of the electric field with charge carriers, thereby restricting the reorganization of molecular structures and suppressing charge transport.

Additionally, by comparing Fig. 3a and b, we can observe that electric fields along different in-plane directions produce clearly anisotropic effects on charge mobility. For example, the electric field along the x-axis, E_x, significantly suppresses mobility in the y-direction, about three times more than in the x-direction (30% vs. 10%). The suppressive effect of the in-plane electric field on mobility can be understood by analysing the transfer rates of various hopping paths. The electric field in the x-direction disrupts the crystal's symmetry centre and slide plane, thereby eliminating the degeneracy between D1 and D3/D4. In D3 and D4, the increase in intermolecular distance leads to a reduction in transfer integrals. Moreover, this increased distance enlarges the angle between the charge transfer direction and the electric field direction, reducing the projection value, as shown in Fig. 4c. The dashed line represents the relative position when applying external electric fields, while the solid line represents the position without electric fields. Due to these changes, the reduction in hop rates has a more pronounced impact on the y-direction. Conversely, as the distances between D1 and D2 decrease, the transfer integrals between them slightly increase. The reduction in molecular distance decreases the angle between the transport direction and the x-direction, thereby increasing the ratio of projection along the x-axis than y-axes. This to some extent mitigates the suppression effect on D3/D4. Thus, E_x has a more subtle impact on mobility in the x-direction compared to the y-direction. Similarly, when an electric field is applied in the y-direction, the disruption of symmetry leads to a decrease in the intermolecular distances within D1/D4, and an increase in D2/D3, as shown in Fig. 4d. Following a pattern similar to that of E_x, the angle between the direction of D2 or D3 and the x-direction decreases, thereby enhancing the inhibitory effect of the electric field on mobility in the x-direction. This effect is consistent with previous mechanisms; when E_y is applied, the angle between the directions of D2/D3 and the x-direction decreases, which in turn makes the inhibitory effect of E_y on charge transport more pronounced in the x-direction.

In summary, by calculating the mobility of DPA under electric fields of various magnitudes and directions, we have discovered that electric fields significantly influence the in-plane mobility of organic semiconductors. Particularly, the influence of the electric field along the molecular long axis is the most pronounced. This is primarily due to the electrostatic effect induced by the electric field, which causes a redistribution of the electron structure within the molecules, shifting the charge distribution from a delocalized to a localized state. This localization of charge not only enhances electron–phonon coupling but also weakens the intermolecular electron coupling, as shown in Fig. S6.† Additionally, the impact of the electric field in the in-plane direction also exhibits anisotropic characteristics. This is mainly because the in-plane electric field alters the molecular stacking structure, thereby anisotropically modulating the charge transport properties.

Mobility in OFETs

To further investigate the effects of source, drain, and gate electrodes on the charge transport characteristics in real electronic devices, we fabricated an OFET device using Silvaco TCAD Atlas. We constructed a bottom-gate top-contact OFET device and set the electrode voltages according to commonly used experimental parameters. Gold electrodes were employed for the source and drain, with voltages set at 0 V and −20 V, respectively. The gate electrode was made of silicon, with a voltage set at −40 V, and silicon dioxide was selected as the dielectric layer between the gate and the organic functional layer. We simulated the electric field distribution within the semiconductor layer under specific operating voltages, as shown in Fig. 5a. To visually represent the electric field distribution within the semiconductor layer, we plotted the electric field variation along the charge transport direction, both in-plane (x/y direction) and out-of-plane (z direction). As depicted in Fig. 5b, the electric field significantly decreases with increasing distance from the source to the drain electrodes. Over a span of 300 nm, the electric field strength can reach approximately 10⁷ V m⁻¹.

Based on the theoretical mobility under discrete electric fields, we performed a numerical fitting to establish the relationship between mobility and the electric field vector. Under the influence of unavoidable external electric fields, the mobilities in the x and y directions decreased by 2.5% and 0.5%, respectively. Therefore, our study confirms that when organic semiconductors operate within a device, the inherent charge transport properties are modified due to the external electric fields induced by the working electrodes.

Conclusions

In this work, we have discovered that the electric field would have serious and direction-dependent influence on the charge transport properties in single-crystal organic semiconductors. This effect originates from the unsymmetric rearrangement of molecular structure and packing modes by external electric fields. The electric fields along the out-of-plane (z-) direction bring the greatest reduction in carrier mobility. At such electric field of 0.005 a.u., the carrier mobility could reduce by more than 80%. These drastic reductions could be ascribed to the increase of reorganization energies and reduction of transfer integrals. The reorganization energy improved by 73% under 0.005 a.u. electric field along the z-direction, which is mainly due to the charge localization induced by E_z, leading to a stronger electron–phonon interaction. On the other hand, due to the increase in intermolecular distance, such electric fields could also lead to a decrease in the transfer integrals. Besides, it is found that the in-plane electric field has an anisotropic suppression effect on mobility. This is primarily due to the diverse breaking of crystal symmetry. By constructing an OFET, the mobility under the operating voltage has also been explored. It was found that the operating voltage of the device can also lead to an intrinsic mobility reduction of more than 2%.

In summary, we have combined first-principles calculations and PBCs, revealing the mechanism of the external electric field effects on charge transport properties in organic semiconductors. It is found that an external electric field can inhibit the charge mobility by changing the geometry, electron structure, and stacking patterns. By constructing an organic electronic device, OFET, we have discovered that the operating voltages would have a non-negligible impact on device performance. Therefore, device properties can be regulated by optimizing the operating voltage. However, we still anticipate more studies to further comprehend the electric field response of the conductivity of organic semiconductors. For example, understanding the structure–effect relationship between the material conductivity and electric field response, recognizing the impact of site energy differences under an applied electric field on the conductivity of organic semiconductors, and exploring how to fabricate efficient organic electronic devices through electric field regulation are all extremely valuable research directions. We believe that our work can deepen the understanding of the charge transport mechanisms in organic semiconductors and it will contribute to the development of high-performance organic electronic devices, energy technology, and photocatalysts.

Data availability

The data supporting this article have been included as part of the ESI.†

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This research was supported by Haihe Laboratory in Tianjin (22HHXCJC00007, A Multi-Scale and High-Efficiency Computing Platform for Advanced Functional Materials), and the National Natural Science Foundation of China (U21A6002, Research on Key Materials for Organic Short-Wave Infrared Detection). The calculations were performed on the National Supercomputer Center in Tianjin (Tianhe 3F) and the Scientific Computing Center of CIC, Tianjin University. The authors gratefully acknowledge HZWTECH for providing computation facilities.

Notes and references

1. H. Dong, X. Fu, J. Liu, Z. Wang and W. Hu, Adv. Mater., 2013, 25, 6158–6183 CrossRef CAS PubMed .
2. H. Sirringhaus, Adv. Mater., 2014, 26, 1319–1335 CrossRef CAS PubMed .
3. T. Sekitani, U. Zschieschang, H. Klauk and T. Someya, Nat. Mater., 2010, 9, 1015–1022 CrossRef CAS PubMed .
4. Y. Sun, H. Geng, Q. Peng and Z. Shuai, ChemPhysChem, 2020, 21, 952–957 CrossRef CAS PubMed .
5. Y. Chen, X. Wan and G. Long, Acc. Chem. Res., 2013, 46, 2645–2655 CrossRef CAS PubMed .
6. S. A. Gevorgyan, N. Espinosa, L. Ciammaruchi, B. Roth, F. Livi, S. Tsopanidis, S. Züfle, S. Queirós, A. Gregori, G. A. D. R. Benatto, M. Corazza, M. V. Madsen, M. Hösel, M. J. Beliatis, T. T. Larsen-Olsen, F. Pastorelli, A. Castro, A. Mingorance, V. Lenzi, D. Fluhr, R. Roesch, M. M. D. Ramos, A. Savva, H. Hoppe, L. S. A. Marques, I. Burgués, E. Georgiou, L. Serrano-Luján and F. C. Krebs, Adv. Energy Mater., 2016, 6, 1600910 CrossRef .
7. K. Fukuda, K. Yu and T. Someya, Adv. Energy Mater., 2020, 10, 2000765 CrossRef CAS .
8. S. Hong, H. Kang, G. Kim, S. Lee, S. Kim, J.-H. Lee, J. Lee, M. Yi, J. Kim, H. Back, J.-R. Kim and K. Lee, Nat. Commun., 2016, 7, 10279 CrossRef CAS PubMed .
9. O. Inganäs, Adv. Mater., 2018, 30, 1800388 CrossRef PubMed .
10. Q. Zhang, Y. Sun, W. Xu and D. Zhu, Adv. Mater., 2014, 26, 6829–6851 CrossRef CAS PubMed .
11. A. T. John, D. George and M. Hariharan, J. Phys. Chem. C, 2023, 127, 3389–3397 CrossRef CAS .
12. H. Dong, L. Jiang and W. Hu, Phys. Chem. Chem. Phys., 2012, 14, 14165 RSC .
13. G. Giri, E. Verploegen, S. C. B. Mannsfeld, S. Atahan-Evrenk, D. H. Kim, S. Y. Lee, H. A. Becerril, A. Aspuru-Guzik, M. F. Toney and Z. Bao, Nature, 2011, 480, 504–508 CrossRef CAS PubMed .
14. X. Zheng, H. Geng, Y. Yi, Q. Li, Y. Jiang, D. Wang and Z. Shuai, Adv. Funct. Mater., 2014, 24, 5531–5540 CrossRef CAS .
15. X. Lu, Y. Sun, Z. Zhang, Z. Shuai and W. Hu, Chin. Chem. Lett., 2021, 32, 1233–1236 CrossRef CAS .
16. T. Kubo, R. Häusermann, J. Tsurumi, J. Soeda, Y. Okada, Y. Yamashita, N. Akamatsu, A. Shishido, C. Mitsui, T. Okamoto, S. Yanagisawa, H. Matsui and J. Takeya, Nat. Commun., 2016, 7, 11156 CrossRef CAS PubMed .
17. S. Sun, J. Zhu, Z. Wang, Y. Huang, Y. Hu, X. Chen, Y. Sun, L. Li and W. Hu, Adv. Mater., 2023, 35, 2306975 CrossRef CAS PubMed .
18. K. Sakai, Y. Okada, S. Kitaoka, J. Tsurumi, Y. Ohishi, A. Fujiwara, K. Takimiya and J. Takeya, Phys. Rev. Lett., 2013, 110, 096603 CrossRef CAS PubMed .
19. L. Lin, J. Fan, S. Jiang, Z. Wang and C.-K. Wang, Chem. Phys. Lett., 2017, 688, 19–25 CrossRef CAS .
20. M. Zhu, G. Liang, T. Cui and K. Varahramyan, Solid-State Electron., 2005, 49, 884–888 CrossRef CAS .
21. X. Lu, Y. Sun and W. Hu, J. Mater. Chem. A, 2021, 9, 21044–21050 RSC .
22. K. Kotsuki, S. Obata and K. Saiki, Langmuir, 2014, 30, 14286–14291 CrossRef CAS PubMed .
23. K. Kotsuki, S. Obata and K. Saiki, Langmuir, 2016, 32, 644–649 CrossRef CAS PubMed .
24. J. C. Sancho-García, G. Horowitz, J. L. Brédas and J. Cornil, J. Chem. Phys., 2003, 119, 12563–12568 CrossRef .
25. A. Förster, F. Günther, S. Gemming and G. Seifert, J. Phys. Chem. C, 2017, 121, 3714–3723 CrossRef .
26. X. Zhan, H. Shi, H. Liu and J. Y. Lee, J. Comput. Chem., 2017, 38, 304–311 CrossRef CAS PubMed .
27. J. Liu, H. Zhang, H. Dong, L. Meng, L. Jiang, L. Jiang, Y. Wang, J. Yu, Y. Sun, W. Hu and A. J. Heeger, Nat. Commun., 2015, 6, 10032 CrossRef CAS PubMed .
28. X. Gao, S. Duan, J. Li, D. Khan, Y. Zou, L. Zheng, J. Liu, X. Ren and W. Hu, J. Mater. Chem. C, 2018, 6, 12498–12502 RSC .
29. F. Yang, L. Sun, J. Han, B. Li, X. Yu, X. Zhang, X. Ren and W. Hu, ACS Appl. Mater. Interfaces, 2018, 10, 25871–25877 CrossRef CAS PubMed .
30. T. D. Kühne, M. Iannuzzi, M. Del Ben, V. V. Rybkin, P. Seewald, F. Stein, T. Laino, R. Z. Khaliullin, O. Schütt, F. Schiffmann, D. Golze, J. Wilhelm, S. Chulkov, M. H. Bani-Hashemian, V. Weber, U. Borštnik, M. Taillefumier, A. S. Jakobovits, A. Lazzaro, H. Pabst, T. Müller, R. Schade, M. Guidon, S. Andermatt, N. Holmberg, G. K. Schenter, A. Hehn, A. Bussy, F. Belleflamme, G. Tabacchi, A. Glöβ, M. Lass, I. Bethune, C. J. Mundy, C. Plessl, M. Watkins, J. VandeVondele, M. Krack and J. Hutter, J. Chem. Phys., 2020, 152, 194103 CrossRef PubMed .
31. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865–3868 CrossRef CAS PubMed .
32. J. L. Brédas, J. P. Calbert, D. A. Da Silva Filho and J. Cornil, Proc. Natl. Acad. Sci. U. S. A., 2002, 99, 5804–5809 CrossRef PubMed .
33. G. Nan, X. Yang, L. Wang, Z. Shuai and Y. Zhao, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 79, 115203 CrossRef .
34. E. F. Valeev, V. Coropceanu, D. A. Da Silva Filho, S. Salman and J.-L. Brédas, J. Am. Chem. Soc., 2006, 128, 9882–9886 CrossRef CAS PubMed .
35. H. Geng, Q. Peng, L. Wang, H. Li, Y. Liao, Z. Ma and Z. Shuai, Adv. Mater., 2012, 24, 3568–3572 CrossRef CAS PubMed .
36. H. Geng, Y. Niu, Q. Peng, Z. Shuai, V. Coropceanu and J.-L. Brédas, J. Chem. Phys., 2011, 135, 104703 CrossRef PubMed .
37. J. R. Reimers, J. Chem. Phys., 2001, 115, 9103–9109 CrossRef CAS .
38. X. Niu, Y. Dang, Y. Sun and W. Hu, J. Energy Chem., 2023, 81, 143–148 CrossRef CAS .
39. Y. Niu, W. Li, Q. Peng, H. Geng, Y. Yi, L. Wang, G. Nan, D. Wang and Z. Shuai, Mol. Phys., 2018, 116, 1078–1090 CrossRef CAS .
40. Z. Shuai, D. Wang, Q. Peng and H. Geng, Acc. Chem. Res., 2014, 47, 3301–3309 CrossRef CAS PubMed .
41. Z. Shuai, H. Geng, W. Xu, Y. Liao and J.-M. André, Chem. Soc. Rev., 2014, 43, 2662 RSC .
42. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery Jr, J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski and D. J. Fox, Gaussian 09 Revision E.01, 2016 Search PubMed .

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4ta03043e

‡ These authors contributed equally.

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