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Abstract

Charge transport rate is one of the key parameters determining the performance of organic electronic devices. Based on density functional theory, exchange-correlation functionals which adequately account for non-covalent interactions, such as M06-2X and wB97XD, should significantly improve the accuracy of charge transport rate calculations for large systems with non-covalent interactions. In this work, the B3LYP hybrid functional, the variant hybrid functional M06-2X, and the long-range-corrected wB97XD functional were used to perform geometry optimizations and charge transport rate calculations on 11 variants of tetrabenzo[a,d,j,m]coronene, including tetrabenzo[a,d,j,m]coronene itself and its tetra-substituted and octa-substituted derivatives. Our results indicate that the molecular geometries of these benzocoronene semiconductors are large quasi-planar conjugated π systems, and the incorporation of different substituents significantly affects their frontier molecular orbitals. The hole carrier mobility (µ₊) and electron carrier mobility (µ₋) of the methoxy-substituted derivatives (TBC(OCH₃)₄ and TBC(OCH₃)₈) were relatively low. The results of the tetrabenzo[a,d,j,m]coronene molecules studied were consistent with using the aforementioned M06-2X, wB97XD, and B3LYP methods. We found that the octa-substituted derivatives (TBCF₈, TBCCl₈, TBC(CH₃)₈, and TBC(CN)₈) could be used as p-type organic semiconductor materials.

Keywords

charge carrier mobility charge transport properties coronene derivatives density functional theory organic semiconductors

The charge transport rate of tetrabenzo[a,d,j,m]coronene derivatives was calculated using density functional theory method with three different functionals, respectively.

Introduction

Semiconductors are light, flexible, easy to chemically modify, and able to form large films at low temperatures. Thus, research on the use of cheap organic semiconductors on microelectronic and optoelectronic devices has recently received increasing levels of attention.¹ Since carrier transport is an extremely important physical process in semiconductors, charge transport rate is one of the key parameters determining the performance of organic electronic devices. In the aggregated state, the structure, molecular orientation, and condensed phase formed by molecular aggregation of a semiconductor are important factors affecting its charge transport properties. To design novel organic semiconductors that will improve the performance of modern devices, it is necessary to understand the relationship between an organic semiconductor’s charge transport properties and its structure. At present, highly reliable and computationally cheap quantum chemical methods, such as density functional theory (DFT), are being widely used in the field of organic optoelectronics to validate the results of experimental studies and to predict the performance of novel organic semiconductor molecules. Based on DFT quantum chemistry, researchers have calculated and predicted the intrinsic electron and hole transport rates of organic molecular systems with various molecular structures, substituents, or hybridizations, including molecular derivatives such as thiophene,² bithiazole,³ tetracene,⁴ anthracene,^5,6 pentacene,^7–9 heptacenes,¹⁰ diindole–diimide,¹¹ perylene,^12,13 triphenylene,^14,15 truxene,^16,17 coronene,¹⁸ and phthalocyanine.^19,20 These theoretical studies on the relationship between the molecular structures of different systems and their charge transport properties are extremely useful for guiding experimental studies.

Aromatic compounds with large conjugated π-electron systems are an important class of materials for organic semiconductors. In particular, discotic liquid crystals (DLCs), composed of a rigid core surrounded by flexible alkyl chains, are a novel type of aromatic organic semiconductor.^21,22 DLCs have excellent solubility in common solvents and can self-assemble in a temperature-dependent manner to form highly ordered columnar phases. These molecules are also capable of self-healing, and their charge carriers can migrate between the aromatic cores of the columns. DLCs are therefore a typical one-dimensional semiconductor material. Benzocoronene derivatives are a class of DLCs with high intrinsic carrier mobility. In particular, hexa-peri-hexabenzocoronenes (HBCs) are typical DLCs, and their hole carrier mobility can reach 1.0 cm² V⁻¹ s⁻¹.¹ Recently, other topological structures such as tetrabenzo[a,d,j,m]coronenes (TBCs),²³ octabenzocircumbiphenyl (c-OBCBs),^24,25 hexathienocoronenes (HTCs),²⁶ and tetrapyrene-fused coronenes (TPCs)²⁷ have been reported in quick succession. Furthermore, B-N hybridized benzocoronenes have also been synthesized.^28,29 The topological differences and hybridization of polycyclic aromatic molecules can significantly affect their optoelectronic properties.³⁰ n-type semiconductor materials can be obtained by reducing the energy level of the lowest unoccupied molecular orbital (LUMO) of polycyclic aromatic molecules, either through hybridization of the rigid core using negatively charged heteroatoms or through the incorporation of strong electrophilic groups around the molecule (e.g. halogens or cyano groups).^4,12

Electron energy band models and discontinuous hopping models are often used to simulate charge transport between adjacent semiconductor molecules. Energy band models are usually suitable for inorganic semiconductors with periodic structures at low temperatures.^31,32 Hopping models, on the contrary, provide reasonable predictions for the charge mobility trends of organic semiconductors that exhibit significant temperature dependencies at room temperature.^33–35 A number of studies have used discontinuous hopping models to simulate and calculate carrier mobility during charge transport processes. In theory, if the exact exchange-correlation function is known, DFT can produce extremely accurate quantum chemistry calculations. Significant efforts are being made to develop accurate exchange-correlation functionals for property calculations using a range of DFT functionals, from local density functionals to doubly hybridized functionals.^30,36–38 In the literature, the B3LYP hybridized functional, which contains 20% Hartree–Fock (HF) exchange, is frequently used to calculate the carrier mobility of charge transport processes.³⁹ The aromatic compounds of the conjugated π-electron system aggregated into a one-dimensional semiconductor via non-covalent interactions, such as π–π interactions, Van der Waals forces, hydrogen bonds, halogen bonds, or CH–π interactions. DFT methods such as M06-2X, CAM-B3LYP, LC-ωPBE, and wB97XD, which adequately account for non-covalent interactions, should be better suited, compared with B3LYP, for calculating the charge mobility of large conjugated π-electron systems that aggregate via non-covalent interactions. Our group has previously tested the variant M06-2X hybrid functional for carrying out quantum calculations on the charge mobility of benzocoronene semiconductor molecules.¹⁸ Furthermore, we have evaluated the use of B3LYP, M06-2X, and long-range-corrected CAM-B3LYP and wB97XD functionals for calculating the charge mobility of hybrid phthalocyanine–tetrabenzoporphyrin macrocycles.¹⁹

In this work, the B3LYP, M06-2X, and wB97XD functionals were used with the 6-31+G(d,p) basis set to perform geometry optimizations and charge transport rate calculations on 11 tetra- and octa-substituted TBC derivatives. In our work,

We studied how different numbers and types of substituents in the rigid aromatic core of tetrabenzocoronenes affected their charge transport properties. These substituents include –F, –Cl, –CH₃, –OCH₃, and –CN.

We evaluated the strengths and weaknesses of density functionals when calculating the carrier mobility of large conjugated π-electron systems, including functionals such as M06-2X, wB97XD, and B3LYP.

Theoretical methodology

All calculations were performed using the Gaussian 09 software package⁴⁰ and the B3LYP hybrid functional;⁴¹ the variant hybrid functional M06-2X,⁴² HF exchange integral component of which is 56%; and long range correction functional wB97XD^43,44 of DFT,⁴⁵ HF exchange integral component of which are short-range (α) 22.2%, long-range (β) 100%, ω = 0.2. The different functions and 6-31+G(d,p) basis set were first used to optimize the geometries of the ground state for neutral molecules and charged counterparts. The frequency calculations were performed at the same level based on the optimized geometries. All the frequencies are real, showing that all the geometries obtained are energy minima on the potential energy surfaces. Subsequently, we had calculated the energy of the highest occupied molecular orbital (HOMO) and the LUMO, the HOMO–LUMO energy gap of the single-molecule optimized structure, the internal reorganization energies for hole and electron transfer (λ₊, and λ₋, respectively), and the charge transfer matrix element for hole and electron transfer (t₊, and t₋, respectively) of the dimer.

The mobility of the charge carrier μ can be obtained using the Einstein equation, which is expressed as

= \frac{e D}{k_{B} T}

(1)

where e is the electron charge (1.60 × 10⁻¹⁹ C), k_B is the Boltzmann constant (1.38 × 10⁻²³ J/K), T is the absolute temperature, and D is the average diffusion coefficient of the charge starting from a molecule and toward all directions in three-dimensional space, as given by equation (2). The diffusion coefficient is expressed as

D = \frac{1}{6} \underset{i}{Σ} r_{i}^{2} k_{i} p_{i}

(2)

where r_i is the distance between adjacent molecules, k_i is the charge transfer rate constant between adjacent molecules, and p_i is the probability, denoted by

p_{i} = \frac{k_{i}}{\sum_{i} k_{i}}

of charge transfer to molecule i.

The π-conjugated organic semiconductor materials have the properties of one-dimensional charge carrier migration. The average diffusion coefficient can be simplified as

D = \frac{1}{2} r^{2} k

where r is the disk spacing between adjacent discotic molecules, and k is the charge transfer rate constant between adjacent molecules. The carrier mobility is obtained by substituting into equation (1)

μ = k \frac{r^{2}}{2 k_{B} T / e}

(3)

According to the semi-classical model of Marcus charge transfer,^46–50 the constant of charge transfer rate between adjacent molecules is expressed as follows

k = (\frac{4 π^{2}}{h}) t^{2} {(4 π λ k_{B} T)}^{- 0.5} \exp [\frac{- λ}{({4 k}_{B} T)}]

(4)

where h is the Planck constant (6.626 × 10⁻³⁴ J s), t is the charge transfer matrix element, λ is the charge reorganization energy, and T is the absolute temperature. Under a certain temperature, λ and t are the main parameters affecting the charge transport rate constant. To achieve a larger charge transport rate constant, the molecule should have smaller reorganization energy λ and larger intermolecular charge transport matrix element t.

The reorganization energy λ was directly calculated using the insulation potential surfaces, that is, the recombination energy λ₊ of the transmission hole and the recombination energy λ₋ of the transmission electrons are calculated as follows

\begin{array}{l} λ_{+} = E ({Ar}^{+} / Ar) - E ({Ar}^{+} / {Ar}^{+}) + \\ E (Ar / {Ar}^{+}) - E (Ar / Ar) \end{array}

(5)

\begin{array}{l} λ_{-} = E ({Ar}^{-} / Ar) - E ({Ar}^{-} / {Ar}^{-}) + \\ E (Ar / {Ar}^{-}) - E (Ar / Ar) \end{array}

(6)

where E(Ar⁺/Ar) represents the total energy of the cationic single point energy based on the Ar neutral molecular configuration optimization, and E(Ar⁺/Ar⁺) represents the total energy when the Ar⁺ ion is optimized configuration.

The charge transfer matrix element characterizes the coupling strength of electron–electron interaction, and several methods have been proposed to evaluate the transfer integral within a molecular dimer. The simplest way is the frontier orbital energy level splitting method.^51,52 That is, the closed shell system was formed by adding one electron to the molecular/molecular cation system. The energy level splitting of HOMO and HOMO – 1 at the transition state was calculated. One half is the hole transfer matrix element t₊. The closed shell system is formed by removing one electron from the molecular/molecular anion system. The energy level splitting of LUMO and LUMO + 1 at the transition state is calculated. One half is the negative charge transfer matrix element t₋. The transfer matrix elements for holes and electrons can be estimated by halving these values. The mean squares of the charge transfer matrix elements of the coronene derivatives were calculated using equation (7) according to the Boltzmann distribution using the energies of the dimers at different dihedral angles (E_i) and the corresponding matrix elements (t_i) at a temperature of 298.15 K. The root of <t²> is the charge transfer matrix element t.⁵³ The carrier transport rates, µ, were also calculated for the nine molecules using equations (3) and (4)

\begin{array}{l} n_{i} = \frac{exp(- E_{i} / k T)}{Σ_{j} \exp (- E_{j} / k T)} \\ < t^{2} > = \sum_{i} n_{i} {t_{i}}^{2} = \frac{\sum_{i} {t_{i}}^{2} \exp (- E_{i} / k T)}{\sum_{j} \exp (- E_{j} / k T)} \end{array}

(7)

Results and discussion

Molecule design and geometry optimization

Geometry optimizations and frequency calculations were performed on the 11 molecules shown in Figure 1 (TBC and its tetra- and octa-substituted derivatives) using the Gaussian 09 E.01 program, with M06-2X/6-31+G(d,p), B3LYP/6-31+G(d,p), and wB97XD/6-31+G(d,p) levels of theory; therefore, their stable geometries were obtained. All the resulting geometries had C_2h symmetry, and the structural parameters of the left and right sides were identical. The structure-optimized dihedral angles between the central benzocoronene ring and its four surrounding benzenoid rings at the B3LYP/6-31+G(d,p) level are listed in Table 1. Figure 2 shows the optimized geometries of TBC, tetra-substituted TBC(CH₃)₄, and octa-substituted TBC(CH₃)₈ using the B3LYP/6-31+G(d,p) method. The structural optimization results of all coronene derivatives using the wB97XD/6-31+G(d,p) method are shown in Supplemental Figure S1. The coronene ring in all 11 molecular structures was planar, with some degree of distortion between the central coronene ring and its four surrounding benzenoid rings. The dihedral angle between two adjacent benzenoid rings of the central ring varied between 22.3° and 23.8°, and it was found that the type and number of substituents had minimal impact on molecular structure. In addition, the molecular optimization structures obtained using three different functionals are very similar.

Figure 1.

Molecular structures of tetrabenzo[a,d,j,m]coronene and its derivatives.

Table 1.

Structure-optimized dihedral angles (θ in degree)^a for coronene derivatives using different methods.

Methods	Compound/dihedral angles	θ ₁	θ ₂	θ ₃	θ ₄
B3LYP/6-31+G(d,p)	TBC	−156.8	−156.6	23.3	23.3
	TBCF₄	−156.5	−156.7	23.4	23.4
	TBCF₈	−157.5	−157.7	22.7	22.7
	TBCCl₄	−156.6	−156.8	23.2	23.2
	TBCCl₈	−156.9	−157.2	22.9	22.9
	TBC(CH₃)₄	−156.8	−157.1	23.1	23.1
	TBC(CH₃)₈	−156.8	−157.1	23.0	23.0
	TBC(OCH₃)₄	−157.1	−156.9	22.9	22.9
	TBC(OCH₃)₈	−157.2	−156.9	22.9	22.9
	TBC(CN)₄	−156.3	−156.6	23.5	23.5
	TBC(CN)₈	−156.2	−156.4	23.7	23.7
M06-2X/6-31+G(d,p)	TBC	−157.0	−156.7	23.1	23.1
	TBCF₄	−156.9	−157.2	22.9	22.9
	TBCF₈	−157.5	−157.7	22.4	22.4
	TBCCl₄	−156.9	−157.2	22.9	22.9
	TBCCl₈	−157.4	−157.7	22.5	22.4
	TBC(CH₃)₄	−157.1	−157.3	22.8	22.8
	TBC(CH₃)₈	−157.3	−157.1	22.8	22.8
	TBC(OCH₃)₄	−157.2	−156.9	22.9	22.9
	TBC(OCH₃)₈	−157.1	−156.8	23.1	23.1
	TBC(CN)₄	−156.9	−156.7	23.2	23.2
	TBC(CN)₈	−156.3	−156.5	23.6	23.6
wB97XD/6-31+G(d,p)	TBC	−157.6	−156.8	23.3	23.3
	TBCF₄	−156.5	−156.7	23.4	23.4
	TBCF₈	−157.1	−157.4	22.7	22.7
	TBCCl₄	−156.6	−156.8	23.3	23.3
	TBCCl₈	−156.9	−157.2	22.9	22.9
	TBC(CH₃)₄	−156.8	−157.1	23.1	23.1
	TBC(CH₃)₈	−157.1	−156.9	23.0	23.0
	TBC(OCH₃)₄	−157.2	−156.9	22.9	22.9
	TBC(OCH₃)₈	−157.2	−156.9	22.9	22.9
	TBC(CN)₄	−156.6	−156.4	23.5	23.5
	TBC(CN)₈	−156.2	−156.4	23.7	23.7

TBC: tetrabenzo[a,d,j,m]coronene.

θ₁ = ∠1-2-3-4, θ₂ = ∠1-2-5-6, θ₃ = ∠4-3-2-5 and θ₄ = ∠3-2-5-6.

The angles footnote references Figure 2 where the atoms are defined.

Figure 2.

Optimized structures of coronene derivatives TBC, TBC(CH₃)₄, TBC(CH₃)₈ using the B3LYP/6-31+G(d,p) method.

It is worth noting that single-crystal experiments have shown that tetramethyl-substituted derivative TBC(CH₃)₄ can also have another conformational stereoisomer with D₂ symmetry, which are the two benzenoid rings in the opposite corner pointing in the opposite direction. The single crystal grown in solution is dominated by the C_2h configuration crystals, and the two configurations can be converted at room temperature.^23,54 So for simplicity of discussion, we use C_2h configuration molecules for further calculations.

Frontier molecular orbitals

Frontier molecular orbitals are an important factor for determining the electrical properties of materials. The energy levels of a material’s HOMO and LUMO directly affect the effective injection capability between a pair of electrodes in practical applications. The intrinsic frontier orbital energies of the 11 molecules were calculated at the B3LYP/6-31+G(d,p) level of theory. The energy gap of organic semiconductors ranges roughly between 1.4 and 4.2 eV (135.08–405.24 kJ/mol). In Figure 3, it is shown that the energy gap of TBC and its 10 derivatives ranged between 3.02 and 3.20 eV. Hence, all 11 coronene derivatives are organic semiconductors. It is noted that HOMO eigenvalues lie well above the negative of the ionization energy when generalized gradient approximation (GGA) functionals are used. However, comparing a set of compounds that are related a specific problem with eigenvalues will cancel out.⁵⁵

Figure 3.

HOMO energies (E_H), LUMO energies (E_L), and HOMO–LUMO gaps (E_g) of tetrabenzo[a,d,j,m]coronene and its derivatives at the B3LYP/6-31+G(d,p) level.

Figure 3 illustrates the HOMO and LUMO frontier orbitals and the energy gap diagrams of TBC and its 10 derivatives. Detailed data and graphics are shown in Supplemental Table S1 and Figure S2. In these figures, it is shown that tetra-substituted and octa-substituted TBC derivatives have similar HOMO and LUMO electron cloud density distributions, and that these electron clouds are uniformly distributed on the coronene ring and the four surrounding benzenoid rings. Compared with TBC and its methane-substituted derivatives, which are TBC(CH₃)₄ and TBC(CH₃)₈, the eight remaining TBC variants exhibited differences in the delocalization of their electron clouds due to the heteroatoms’ (F, Cl, O, and CN) n-electron contributions toward the HOMO and LUMO orbitals. This caused significant changes in the energy levels of these orbitals. Based on comparisons with the HOMO and LUMO of TBC, it was found that methane and methoxy substitution increased HOMO and LUMO energies, and that the change in the energy gap (E_g) was insignificant. Tetrahalogen substitution caused the HOMO and LUMO to decrease in energy, but the change in E_g was relatively small. The HOMO and LUMO of the dicyano derivatives were both lower in energy than those of TBC, and the E_g was also smaller. In comparisons between the five octa-substituted compounds and their corresponding tetra-substituted compounds, the effects of substitution were quite significant; in particular, the frontier orbitals of octacyano-substituted TBC(CN)₈ were lower in energy than those of tetracyano-substituted TBC(CN)₄. This is consistent with previous reports in the literature, where strongly electrophilic substituents (e.g. –F and –CN) were found to significantly reduce frontier orbital energies.^7,12

Reorganization energy of the molecules

The hole and electron transport reorganization energies of the 11 aforementioned molecules, λ₊ and λ₋, were calculated using (a) the energy of the optimized neutral and cationic structures, (b) the energy of the cation species with the geometry of the neutral species, and (c) the energy of the neutral species with the geometry of the cation species, as shown in Table 2. These calculations were performed at the B3LYP/6-31+G(d,p), M06-2X/6-31+G(d,p), and wB97XD/6-31+G(d,p) levels of theory.

Table 2.

Organization energies λ (meV/mol) of 11 coronene derivatives at the different density functional methods.

Compound	λ	B3LYP	M06-2X	wB97XD	Compound	λ	B3LYP	M06-2X	wB97XD
TBC	λ ₊	129(123)₂₂	198	222
	λ ₋	173	243	273
TBCF₄	λ ₊	171(169)₂₂	246	266	TBCF₈	λ ₊	178	244	249
TBCF₄	λ ₋	191	256	275		λ ₋	216	283	311
TBCCl₄	λ ₊	152(154)₂₂	223	224	TBCCl₈	λ ₊	154	222	207
	λ ₋	177	247	274		λ ₋	189	259	289
TBC(CH₃)₄	λ ₊	134(136)₂₂	204	230	TBC(CH₃)₈	λ ₊	134	206	231
	λ ₋	171	236	267		λ ₋	173	242	270
TBC(OCH₃)₄	λ ₊	199	271	301	TBC(OCH₃)₈	λ ₊	211	271	294
	λ ₋	199	294	276		λ ₋	231	294	318
TBC(CN)₄	λ ₊	114	180	206	TBC(CN)₈	λ ₊	116	182	209
	λ ₋	168	246	276		λ ₋	163	242	271

TBC: tetrabenzo[a,d,j,m]coronene.

When the B3LYP method is used to calculate the study molecule or ion-related energy, the delocalization effect will be overestimated, resulting in the calculation that the resulting hole and electron transport recombination energies are too small. Based on the data shown in Table 2, the B3LYP method gave the lowest reorganization energies for hole and electron transport, whereas the long-range-corrected wB97XD method gave the highest energies. All three methods produced the same trends in terms of the relative hole and electron transport reorganization energies for each molecule. The hole transport reorganization energies (λ₊) obtained for TBC, TBCF₄, TBCCl₄, and TBC(CH₃)₄ were consistent with those reported by Pola et al.²³ From equation (4), it can be inferred that the charge transport rate constant decreases with increasing reorganization energy; hence, increases in reorganization energy are detrimental for charge transport. Compared with TBC, the –CH₃-substituted TBC(CH₃)₄ and TBC(CH₃)₈ derivatives had similar reorganization energies, while the –F-substituted TBCF₄ and TBCF₈ derivatives exhibited a slight increase in reorganization energy. The –OCH₃-substituted TBC(OCH₃)₄ and TBC(OCH₃)₈ derivatives had significantly higher reorganization energies, whereas the –CN-substituted TBC(CN)₄ and TBC(CN)₈ derivatives had significantly lower reorganization energies.

Charge transport matrix elements and rate constants

When studying the carrier mobility of molecules, it is necessary to calculate the energies corresponding to different rotational angles between the neutral and ionic species of a dimer, E_i, and also the corresponding energy gap between HOMO and HOMO – 1, and LUMO and LUMO + 1. These calculations will yield the matrix elements t of hole and electron transport. In order to simplify the model, the dimers were not further structurally optimized. We carried out these calculations with angles ranging from 0° to 180°, in 20° intervals. The charge transport matrix elements characterize the coupling strength of electron–electron interactions, which is a major factor for carrier mobility and charge transport rates. The value of the matrix element increases as coupling strength increases, and hence so does charge mobility; high coupling strengths are therefore beneficial for charge transport. To facilitate this analysis, Figure 4 illustrates the frontier diagram corresponding to different rotational angles between the neutral and ionic species of a dimer that were obtained by stacking stable TBC(CH₃)₈ neutral and ionic species geometries. These calculations were performed at the B3LYP/6-31+G(d,p) level of theory. For the TBC(CH₃)₈ neutral and ionic species dimer, the electron clouds of the cationic and neutral species overlapped to an extent, but no overlap was found between the electron clouds of the anionic and neutral species, as shown in Figure 4. Therefore, the coupling strength between cationic and neutral species was greater than that between anionic and neutral species, and the matrix element for hole transport was also greater than that of electron transport. We then predicted that the hole transport rate constant and the hole mobility of TBC(CH₃)₈ are greater than its electron transport rate constant and electron mobility, respectively.

Figure 4.

The frontier molecular orbital of molecule TBC(CH₃)₈ and its molecular ion formed dimer at different angles at the B3LYP/6-31+G(d,p) level. The HOMO of molecule TBC(CH₃)₈ and its positive ion formed dimer (A–C). The LUMO of molecule c and its anion formed dimer (D–F).

Since the 11 benzocoronene molecules included in this study are large quasi-planar conjugated π systems, an excessively small centroid distance between the monomers of a dimer would lead to overlaps between the side groups of the neutral and ionic species, whereas an excessively large centroid distance would significantly weaken their interactions and reduce carrier mobility values. Table 3 shows the charge transport matrix elements and rate constants of hole and electron transport in a dimer with centroid distances ranging from 0.45 to 0.50 nm, calculated at the B3LYP/6-31+G(d,p) level of theory. The data in Table 3 indicate that the charge transport matrix elements and rate constants of all 11 molecules generally increased with decreasing centroid distance. The differences between the results corresponding to centroid distances of 0.50 and 0.45 nm were significant, whereas most of the calculated results with centroid distances of 0.47 and 0.45 nm were relatively similar. To obtain a suitable method for calculating the carrier mobility of TBC-like molecules and thus reduce computational workloads, we used M06-2X and wB97XD functionals with dimer centroid distances of 0.47 and 0.45 nm. The hole and electron transport matrix elements and charge transport rates calculated for the 11 TBC variants are shown in Table 4. From the comparison of data in Tables 3 and 4, it can be seen that the wB97XD method uses the DFT-D2 dispersion correction and long-range correction to calculate the weak interaction between dimers. The hole and electron transport matrix obtained from the calculation of the center-to-center distance of the same dimer is significantly larger than that obtained by the B3LYP method, therefore impairing the effect of the increase in recombination energy on the decrease in the molecular carrier transport rate. The result of the synergistic effect makes the hole/electron transport matrix elements and rate constants calculated using wB97XD and B3LYP methods relatively similar, whereas the results of the calculations of the M06-2X method deviate toward lower values. For the 11 TBC variants studied in this work, all three methods resulted in identical qualitative trends.

Table 3.

Charge transport matrix element t (meV/mol) and charge transport rate constant k (s⁻¹) of 11 molecules at the B3LYP/6-31+G(d,p) level.

Compound	Distance	0.50 nm	0.47 nm	0.45 nm	Compound	Distance	0.50 nm	0.47 nm	0.45 nm
	t					k
TBC	t ₊	16.1	34.6	39.6(41)₂₂	TBC	k ₊	3.5 × 10¹²	7.8 × 10¹²	2.1 × 10¹³
	t ₋	14.3	15.2	25.9		k ₋	1.5 × 10¹²	2.3 × 10¹²	5.1 × 10¹²
TBCF₄	t ₊	11.8	15.7	19.4	TBCF₄	k ₊	1.1 × 10¹²	1.9 × 10¹²	2.9 × 10¹²
	t ₋	20.8	19.1	18.7		k ₋	2.6 × 10¹²	2.2 × 10¹²	2.1 × 10¹²
TBCF₈	t ₊	13.6	28.4	52.4	TBCF₈	k ₊	1.3 × 10¹²	5.8 × 10¹²	2.0 × 10¹³
	t ₋	16.0	20.1	32.3		k ₋	3.0 × 10¹¹	4.7 × 10¹¹	1.2 × 10¹²
TBCCl₄	t ₊	14.4	24.3	43.7	TBCCl₄	k ₊	2.0 × 10¹²	5.9 × 10¹²	2.0 × 10¹³
	t ₋	17.5	21.1	32.2		k ₋	2.2 × 10¹²	3.2 × 10¹²	5.1 × 10¹²
TBCCl₈	t ₊	22.6	38.4	45.2	TBCCl₈	k ₊	4.9 × 10¹²	1.4 × 10¹³	2.0 × 10¹³
	t ₋	14.6	24.7	29.0		k ₋	1.3 × 10¹²	3.8 × 10¹²	5.1 × 10¹²
TBC(CH₃)₄	t ₊	16.4	27.2	44.3	TBC(CH₃)₄	k ₊	3.4 × 10¹²	9.2 × 10¹²	2.5 × 10¹³
	t ₋	11.9	16.1	23.9		k ₋	1.1 × 10¹²	2.0 × 10¹²	4.5 × 10¹²
TBC(CH₃)₈	t ₊	25.0	40.8	42.2	TBC(CH₃)₈	k ₊	7.9 × 10¹²	2.1 × 10¹³	2.2 × 10¹³
	t ₋	20.1	26.0	34.8		k ₋	3.1 × 10¹²	5.1 × 10¹²	9.2 × 10¹²
TBC(OCH₃)₄	t ₊	14.8	14.0	23.7	TBC(OCH₃)₄	k ₊	1.2 × 10¹²	1.1 × 10¹²	3.1 × 10¹²
	t ₋	26.0	31.2	40.2		k ₋	3.7 × 10¹²	5.3 × 10¹²	8.8 × 10¹²
TBC(OCH₃)₈	t ₊	5.8	14.5	20.4	TBC(OCH₃)₈	k ₊	1.6 × 10¹¹	9.9 × 10¹¹	2.0 × 10¹²
	t ₋	24.6	22.7	35.6		k ₋	2.2 × 10¹²	1.2 × 10¹²	4.8 × 10¹²
TBC(CN)₄	t ₊	15.3	27.3	35.2	TBC(CN)₄	k ₊	3.9 × 10¹²	1.2 × 10¹³	2.0 × 10¹³
	t ₋	23.0	36.1	44.2		k ₋	4.3 × 10¹³	1.1 × 10¹³	1.6 × 10¹³
TBC(CN)₈	t ₊	11.0	18.5	28.4	TBC(CN)₈	k ₊	1.9 × 10¹²	5.5 × 10¹²	1.3 × 10¹³
	t ₋	10.7	13.7	19.3		k ₋	9.9 × 10¹¹	1.6 × 10¹²	3.2 × 10¹²

TBC: tetrabenzo[a,d,j,m]coronene.

Table 4.

Charge transport matrix element t (meV/mol), charge transport rate constant k (s⁻¹), and transport rate of 11 molecules at the different density functional methods.

Compound	Distance	0.47 nm		0.45 nm			0.47 nm		0.45 nm
	t	M06-2X	wB97XD	M06-2X	wB97XD	k	M06-2X	wB97XD	M06-2X	wB97XD
TBC	t ₊	30.1	43.8	38.8	54.0	k ₊	5.0 × 10¹²	8.2 × 10¹²	8.40 × 10¹²	1.2 × 10¹³
	t ₋	21.5	25.0	28.0	34.2	k ₋	1.5 × 10¹²	1.5 × 10¹²	2.5 × 10¹²	2.7 × 10¹²
TBCF₄	t ₊	19.6	21.2	26.0	42.9	k ₊	1.2 × 10¹²	1.0 × 10¹²	2.1 × 10¹²	4.6 × 10¹²
	t ₋	26.9	19.8	31.4	29.2	k ₋	2.0 × 10¹²	9.1 × 10¹¹	2.8 × 10¹²	1.9 × 10¹²
TBCF₈	t ₊	28.6	40.2	43.2	60.0	k ₊	2.6 × 10¹²	5.0 × 10¹²	6.0 × 10¹²	1.1 × 10¹³
	t ₋	19.2	26.8	25.5	38.6	k ₋	7.6 × 10¹¹	1.1 × 10¹²	1.3 × 10¹²	2.2 × 10¹²
TBCCl₄	t ₊	10.9	20.8	16.0	50.2	k ₊	4.9 × 10¹¹	1.5 × 10¹²	1.0 × 10¹²	8.1 × 10¹²
	t ₋	24.4	28.6	28.4	37.3	k ₋	1.8 × 10¹²	1.9 × 10¹²	2.5 × 10¹²	3.2 × 10¹²
TBCCl₈	t ₊	38.1	42.7	44.9	51.2	k ₊	6.0 × 10¹²	8.2 × 10¹²	8.4 × 10¹²	1.2 × 10¹³
	t ₋	26.1	29.0	23.3	34.1	k ₋	1.8 × 10¹²	1.6 × 10¹²	1.5 × 10¹²	2.2 × 10¹²
TBC(CH₃)₄	t ₊	15.1	24.7	26.9	36.4	k ₊	1.2 × 10¹²	7.4 × 10¹²	3.7 × 10¹²	1.5 × 10¹³
	t ₋	25.1	26.0	26.2	39.4	k _—	2.2 × 10¹²	1.7 × 10¹²	2.4 × 10¹²	3.8 × 10¹²
TBC(CH₃)₈	t ₊	20.4	37.8	42.3	62.7	k ₊	2.1 × 10¹²	7.1 × 10¹²	9.0 × 10¹²	1.5 × 10¹³
	t ₋	15.9	24.6	21.3	39.8	k ₋	8.3 × 10¹¹	1.4 × 10¹²	1.5 × 10¹²	3.8 × 10¹²
TBC(OCH₃)₄	t ₊	16.4	17.0	22.6	39.5	k ₊	5.9 × 10¹¹	4.9 × 10¹¹	1.1 × 10¹²	2.6 × 10¹²
	t ₋	29.8	18.8	32.7	23.8	k ₋	2.3 × 10¹²	8.0 × 10¹¹	2.8 × 10¹²	1.3 × 10¹²
TBC(OCH₃)₈	t ₊	10.8	9.0	21.6	20.8	k ₊	2.7 × 10¹¹	1.4 × 10¹¹	1.1 × 10¹²	7.9 × 10¹¹
	t ₋	24.8	24.8	34.6	24.5	k ₋	1.1 × 10¹²	8.4 × 10¹¹	2.1 × 10¹²	8.3 × 10¹¹
TBC(CN)₄	t ₊	17.8	34.7	23.5	32.0	k ₊	2.2 × 10¹²	6.3 × 10¹²	3.8 × 10¹²	5.2 × × 10¹²
	t ₋	19.7	20.3	26.5	26.0	k ₋	1.2 × 10¹²	8.4 × 10¹¹	2.2 × 10¹²	1.5 × 10¹²
TBC(CN)₈	t ₊	22.3	22.1	29.5	31.4	k ₊	3.4 × 10¹²	2.6 × 10¹²	5.9 × 10¹²	4.8 × 10¹²
	t ₋	22.6	22.3	31.3	31.6	k ₋	1.7 × 10¹²	1.1 × 10¹²	3.2 × 10¹²	2.3 × 10¹²

TBC: tetrabenzo[a,d,j,m]coronene.

Carrier mobility

The B3LYP method overestimates the delocalization effect. Compared with the B3LYP method, the variant hybrid functional M06-2X method takes into account the weak intermolecular interactions between dimers. The long-range-corrected functional wB97XD method calculating the weak interaction between dimers is considered both the DFT-D2 dispersion correction and the long-range correction, which is more conducive to improving the accuracy of the weak interaction between dimeric molecules. From Table 2, we can see that the B3LYP method calculates the reorganization energy data obtained, which are significantly greater than the results obtained by the M06-2X or wB97XD method. From the comparison between Tables 3 and 4, we know that the B3LYP method calculates the resulting charge transfer matrix, which is much smaller than the calculated value by the M06-2X or wB97XD method. The result of the above synergistic effect makes the carrier mobility obtained by the calculation of M06-2X and wB97XD very similar.

The carrier mobilities calculated at the B3LYP/6-31+G(d,p), M06-2X/6-31+G(d,p), and wB97XD/6-31+G(d,p) levels of theory using different centroid distances are shown in Table 5. In this table, it is shown that the hole transport rates of TBC, TBCCl₄, TBCF₄, and TBC(CH₃)₄ that were calculated using a centroid distance of 0.45 nm at M06-2X/6-31+G(d,p) and wB97XD/6-31+G(d,p) levels of theory are consistent with the results of the single-crystal field-effect transistors test reported by Tao et al.²³ We have found that the M06-2X and wB97XD functionals were superior to B3LYP, as the former methods were able to adequately account for non-covalent interactions. wB97XD was especially effective in accounting for the non-covalent interactions of dimers and is thus the most suitable method for these calculations.

Table 5.

Transport rate μ (cm² V⁻¹ s⁻¹) of 11 molecules using B3LYP/6-31+G(d,p), M06-2X/6-31+G(d,p), and B97XD/6-31+G(d,p) levels.

Compound	Distance	0.50 nm	0.47 nm			0.45 nm
	μ	B3LYP	B3LYP	M06-2X	wB97XD	B3LYP	M06-2X	wB97XD
TBC	μ ₊	0.167	0.331	0.215	0.352	0.820	0.328	0.474
	μ ₋	0.074	0.075	0.064	0.064	0.198	0.100	0.105
TBCF₄	μ ₊	0.053	0.082	0.051	0.045	0.114	0.083	0.179
	μ ₋	0.126	0.094	0.086	0.038	0.083	0.108	0.074
TBCF₈	μ ₊	0.0636	0.246	0.112	0.212	0.767	0.235	0.427
	μ ₋	0.056	0.077	0.032	0.048	0.183	0.052	0.086
TBCCl₄	μ ₊	0.100	0.251	0.021	0.023	0.739	0.055	0.316
	μ ₋	0.106	0.136	0.078	0.064	0.292	0.098	0.124
TBCCl₈	μ ₊	0.237	0.604	0.258	0.396	0.803	0.258	0.534
	μ ₋	0.064	0.160	0.078	0.068	0.210	0.078	0.090
TBC(CH₃)₄	μ ₊	0.163	0.394	0.050	0.104	0.961	0.146	0.195
	μ ₋	0.053	0.086	0.095	0.074	0.175	0.094	0.150
TBC(CH₃)₈	μ ₊	0.380	0.895	0.128	0.292	0.876	0.353	0.574
	μ ₋	0.148	0.219	0.036	0.059	0.360	0.058	0.148
TBC(OCH₃)₄	μ ₊	0.046	0.058	0.025	0.021	0.120	0.043	0.102
	μ ₋	0.179	0.228	0.099	0.034	0.346	0.109	0.049
TBC(OCH₃)₈	μ ₊	0.008	0.043	0.012	0.006	0.077	0.043	0.031
	μ ₋	0.110	0.082	0.047	0.036	0.186	0.084	0.032
TBC(CN)₄	μ ₊	0.187	0.526	0.093	0.071	0.799	0.149	0.202
	μ ₋	0.207	0.450	0.052	0.036	0.619	0.086	0.058
TBC(CN)₈	μ ₊	0.094	0.234	0.143	0.111	0.505	0.230	0.187
	μ ₋	0.048	0.069	0.071	0.047	0.125	0.126	0.091

TBC: tetrabenzo[a,d,j,m]coronene.

All three methods exhibited the same trends: for–F-, –Cl-, –CH₃-, and –CN-substituted TBCs, octa-substituted derivatives had better charge transport properties than their tetra-substituted counterparts. The CH₃-substituted derivatives (both tetra- and octa-substituted) consistently exhibited low carrier mobilities. From comparisons between TBC and its 10 derivatives, we found that the reorganization energies associated with hole and electron transport (λ₊ and λ₋) were higher in the four –F and –OCH₃ derivatives, and the increase in reorganization energies was highest for the TBC derivative with eight –OCH₃ substituents (TBC(OCH₃)₈). –Cl and –CN substitution reduced reorganization energies, whereas –CH₃ tetra-substitution and octa-substitution did not have a significant impact on reorganization energies. The octa-substituted –F, –Cl, –CH₃, and –CN derivatives of TBC had hole transport matrix elements that were 1.1–1.6 times larger than their electron transport matrix elements, and their μ₊ values were consequently 2.1–5.0 times that of their μ₋ values. Therefore, these molecules can be used as high-performance hole transport materials (p-type semiconductor materials). Methoxy-substituted TBC(OCH₃)₄ and TBC(OCH₃)₈ molecules have large reorganization energies and low charge transport matrix elements; hence, their μ₊ and μ₋ values were both very small, and they had poor charge transport properties. The cyano-substituted TBC(CN)₄ and TBC(CN)₈ molecules have highly electrophilic cyano groups, which expand their conjugation surfaces. In theory, this should greatly improve electron transport performance. However, as these molecules have large electron transport reorganization energies and small electron transport matrix elements, the synergistic effect of these properties could be greater than the effects of the expanded conjugation surface. Consequently, the μ₊ values of TBC(CN)₄ and TBC(CN)₈ greatly exceeded their μ₋ values.

Conclusion

In this work, we used DFT functionals such as B3LYP, M06-2X, and wB97XD with the 6-31+G(d,p) basis set to perform geometry optimizations and charge transport rate calculations on 11 variants of TBC, including TBC itself and its tetra- and octa-substituted derivatives. We found that these benzocoronenes are organic semiconductors. The molecular geometries of these molecules are all large quasi-planar conjugated π systems, and the incorporation of different substituents had a significant impact on the energies of their frontier orbitals. Compared with TBC, the methane- and methoxy-substituted derivatives of TBC had higher HOMO and LUMO energies, but their differences in energy gap (E_g) were insignificant. All halogen-substituted derivatives exhibited lower HOMO and LUMO energies and a slight decrease in E_g. The two cyano-substituted derivatives had significantly reduced HOMO and LUMO energies and a narrowed E_g. Our carrier mobility calculations indicated that octa-substituted TBCF₈, TBCCl₈, TBC(CH₃)₈, and TBC(CN)₈ molecules could be used as hole-type (p-type) organic semiconductor materials. The hole and electron mobilities (μ₊ and μ₋) of methane-substituted TBC(OCH₃)₄ and TBC(OCH₃)₈ molecules were both quite small, whereas the cyano-substituted TBC(CN)₄ and TBC(CN)₈ molecules had much greater μ₊ values than μ₋ values. Based on the findings of this work, we conclude that the M06-2X and wB97XD functionals, which are able to adequately account for non-covalent interactions, are superior to the B3LYP functional for calculating the carrier mobilities of organic semiconductors made from large one-dimensional conjugated π-electron systems.

Supplemental Material

SI2018 – Supplemental material for Theoretical investigations on charge transport properties of tetrabenzo[a,d,j,m]coronene derivatives using different density functional theory functionals (B3LYP, M06-2X, and wB97XD)

Supplemental material, SI2018 for Theoretical investigations on charge transport properties of tetrabenzo[a,d,j,m]coronene derivatives using different density functional theory functionals (B3LYP, M06-2X, and wB97XD) by Ziran Chen, Yuan Li, Zhanrong He, Youhui Xu and Wenhao Yu in Journal of Chemical Research

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Science and Technology Department of Sichuan Province (grant number 2015GZ0343) and the Project of Sichuan Provincial Department of Education (grant numbers 16ZB0059, 17ZA0346).

ORCID iD

Wenhao Yu

Supplemental material

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Theoretical investigations on charge transport properties of tetrabenzo[ a,d,j,m ]coronene derivatives using different density functional theory functionals (B3LYP,M06-2X,and wB97XD)

Abstract

Keywords

Introduction

Theoretical methodology

Results and discussion

Molecule design and geometry optimization

Frontier molecular orbitals

Reorganization energy of the molecules

Charge transport matrix elements and rate constants

Carrier mobility

Conclusion

Supplemental Material

SI2018 – Supplemental material for Theoretical investigations on charge transport properties of tetrabenzo[a,d,j,m]coronene derivatives using different density functional theory functionals (B3LYP, M06-2X, and wB97XD)

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Supplemental material

References

Supplementary Material