Computational study on thermally activated delayed fluorescence of donor–linker–acceptor network molecules

Abstract

Using Density Functional Theory (DFT) and time-dependent DFT calculations, we have investigated the structure–property relationships of organic molecules with donor–linker–acceptor (DLA) frameworks which can be used as precursors of OLED materials. Two types of linkers, thiophene (π-conjugated) and ethylene (non-π-conjugated), with carbazole as the electron donor and cyano-substituted benzene as the electron acceptor in the DLA framework were chosen. The donor–linker–acceptor network allows the HOMO and LUMO orbitals to be spatially separated, reducing the overlap of the frontier orbitals and decreasing the exchange energy (J); this results in a smaller ΔE_ST (energy gap between the excited singlet and triplet states), thus allowing us to realize metal-free molecules suitable for OLED applications. Incorporation of donor and acceptor groups in the same moiety helps reduce the number of layers conventionally used in OLEDs, reducing the cost and simplifying the fabrication of the device. By enhancing the electron donating nature of the donor group and increasing the electron withdrawing nature of the acceptor group, we observed a decrease in exchange energy, which was further decreased by the non-conjugated linker. Important properties such as the changes in dipole moments, absorption and emission energies and their corresponding oscillator strengths, transport energy gaps, electron affinities, ionization potentials (vertical and adiabatic), reorganization energies, and exciton binding energies of various substituted molecules were investigated and their roles in determining the efficiencies are discussed. In order to account for the effects of solvents and their role in altering various properties, the studies were carried out for polar and non-polar solvent phases in addition to the gas phase calculations.

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Introduction

Ever-increasing energy demands and the depletion of non-renewable resources are creating a great necessity to devise more efficient technologies which harness power from renewable sources, consume less power and conserve the environment. Light emitting diodes (LEDs) have been widely investigated and have attained commercial-scale usage. Current research is focused on organic light emitting diodes (OLEDs), especially small molecule organic semiconductors, as they offer many advantages over their inorganic counterparts,¹ such as fabrication of devices with high uniformity and large surface areas. However, lower efficiencies and shorter lifetimes are the main drawbacks associated with these devices (OLEDs), and many new materials are being synthesized and tested every year.^1–11 In OLEDs, electrically injected charge carriers recombine to form singlet and triplet excitons in a 1 : 3 ratio;² as the transition between the triplet to singlet states is spin forbidden, it imposes a theoretical quantum efficiency of 25%. In order to overcome this problem, phosphorescent metal–organic complexes of heavy transition metals, such as Os, Pt, Ru, Re and Ir, are employed.^3–6 The heavy atom induces a spin–orbit coupling effect which can partially decrease the spin–orbit forbidden nature of the T₁ → S₀ transition. However, the usage of heavy metals imposes a challenge of high cost in addition to many environmental concerns.

Another way to increase the efficiency of OLEDs is by efficient spin upconversion from non-radiative triplet states (which are formed at a rate of 75%) to radiative singlet states.⁴ This can be achieved by minimizing the energy gap between S₁ and T₁. Thermally activated delayed fluorescence (TADF), or E-type delayed fluorescence, is directly proportional to the energy gap between the triplet and singlet states (ΔE_ST⁷), which is dependent on the electron exchange energy. Intramolecular charge transfer states (ICT), where the electrons and holes are decoupled on different orbitals, have zero to first order exchange energy, giving rise to E-type delayed fluorescence.⁸ Recently, Adachi and co-workers⁹ have synthesized metal-free, environmentally benign organic molecules which harness both singlet and triplet excitons for light emission through fluorescence decay channels with very high efficiencies.

OLEDs have a three-layered structure that consists of a hole transporting layer (HTL), an emitting layer (EL) and an electron transporting layer (ETL). Hole transporting materials should contain electron donating moieties which can form stable cation radicals, whereas electron-transporting materials should have electron withdrawing groups which can form stable anion radicals.¹⁰ A single layer structure is possible if the emitting material has balanced electron and hole mobilities. Such a structure would have a lower overall cost and simplified device fabrication.¹¹ Recently, Huang et al.¹¹ synthesized dipolar dibenzothiophene S,S-dioxide derivatives for use as materials for single layer OLED devices. However, such studies are rare, and there is great scope to model and synthesize these molecules.

In the current work, we have considered some important donor–linker–acceptor framework structures and investigated their electronic properties. This type of framework helps reduce the cost of OLEDs in two ways: first, the electron donating and withdrawing groups are separated, which allows the HOMO and LUMO to separate; this minimizes the overlap of the frontier orbitals and reduces the exchange energy (J), resulting in a small ΔE_ST and thus allowing the realization of metal free molecules suitable for OLED applications. Moreover, the incorporation of donor and acceptor groups in the same moiety helps reduce the number of layers. Excited state angular materials have stronger charge transfer character than the excited state of their linear analogues;¹² this enhances the charge transfer character of the lowest singlet excited state, favouring higher intersystem crossing yields.¹³ We have chosen carbazole as the electron donor and different cyano-substituted benzene groups as the acceptors. Thiophene and ethylene molecules were chosen as linkers. This type of framework allows the compound to adopt an angular geometry where the donor and acceptor moieties are separated by the linkers.¹⁴

Computational methodology

Ground state optimizations were carried out employing DFT with the hybrid functional B3LYP and the valence triple zeta polarized 6-311G(d) basis set. Excited state properties were determined by employing TD-DFT, using the same functional and basis set. B3LYP combines the Becke generalized gradient approximation (GGA) and exchange potential based upon Hartree Fock (HF) exchange along with the Lee–Yang–Parr correlation functional.^15,16 The hybrid functional B3LYP is found to give better results in calculating HOMO and LUMO energies, ionization potentials and electron affinities in comparison with regular local density and generalized gradient approximations.¹⁷ It is found to provide accurate ionization potentials, electron affinities and cohesive energies for a range of finite molecules.^17,18 B3LYP typically employs 20% of orbital exchange, which is accurate enough to calculate excitonic properties for organic molecules.¹⁹ Basis set effects in calculating excitation energies¹⁹ and other ground state properties^17–19 are found to be negligible beyond the 6-31G(d) basis set. Recent theoretical investigations by Gierschner et al.¹⁴ and a few other groups^20,21 have suggested that B3LYP gives a very good description of the singlet and triplet states of medium-sized molecules. Hence, we expected that various properties calculated employing the B3LYP functional and 6-311G(d) basis set would predict the properties of medium-sized organic molecules with reasonable accuracy. We employed the polarizable continuum model (PCM)^22–24 to model the bulk solvent effects. This model accounts for the solvent effect by considering the solute inside a cavity that is surrounded by structureless media which has the macroscopic characteristics of the solvent. To compute transition energies within the PCM framework, we employed the Linear-Response (LR) approach, which is computationally efficient and has well-defined transition properties.²⁴ All the calculations are performed employing Gaussian 09 software.²⁵

We investigated the excited singlet–triplet gaps (ΔE_ST), changes in dipole moments, absorption and emission energies and their corresponding oscillator strengths, HOMO–LUMO energies, electron affinities, ionization potentials, reorganization energies, and exciton binding energies for the chosen molecules to understand their properties for application in OLEDs. All the calculations were carried out in the gas phase as well as in solvent phases to understand how solvent effects alter the properties of the molecules under consideration. To account for the effects of polarity, we chose two different solvents, namely cyclohexane (non-polar, ε = 2.0165) and methanol (polar, ε = 32.613).

The preliminary strategy adopted to design molecules which exhibit low singlet–triplet energy gaps is as follows. The molecular energies of the lowest singlet energy (E_S) and the triplet energy (E_T) are given by

E_S = E_S + K + J (1)

E_T = E_T + K − J (2)

ΔE_ST = E_S − E_T = 2J (3)

which are dependent on the orbital energies (E), electron repulsion energy (K) and exchange energy (J). Since the unpaired electrons are distributed mainly on the HOMO and LUMO, the exchange energy (J) of these two electrons at the HOMO and LUMO are given by²⁶ψ_H and ψ_L represent the wavefunctions of the HOMO and LUMO, and “e” represents the charge of electron. From eqn (3), it is clear that E_ST increases as the exchange integral (J) value increases. In general, the wavefunctions of electrons in the HOMO and LUMO overlap significantly, resulting in large exchange energies of the order of 0.7 to 1.0 eV.^27,28 Thus, ΔE_ST can be reduced by decreasing the overlap between the HOMO and LUMO. By choosing a donor–linker–acceptor architecture, the HOMO and LUMO can be separated by a linker, as the HOMO localizes on the electron donating group and the LUMO on the electron withdrawing group. Carbazole was chosen as the electron donating group and cyano substituted benzene groups were chosen as electron withdrawing groups. To enhance the withdrawing effect, the number of cyano groups substituted on benzene was increased (shown in Fig. 1). In the same fashion, to increase the electron-donating nature of carbazole, it was substituted with methoxy groups, as shown in Fig. 1.

Fig. 1 Donor–linker–acceptor (DLA) molecules with thiophene (1a–e) and ethylene (2a–e) as linkers (linkers are shown in brackets).

The advantage of considering cyano groups is that they act as strong electron-withdrawing groups, enhancing the electron affinities of the resulting cyano-substituted aromatophors;^29,30 incorporation of alkyl or alkoxy groups enhances the solubility, which facilitates processing.¹⁰ Two types of linkers, a π-conjugated group, thiophene (C₄H₄S), and a non-conjugated group, ethylene (–CH₂–CH₂–), were considered.

Results and discussion

As a first step, we calculated ΔE_ST, which is crucial in determining the efficiencies of OLED materials, since the energy gap affects the rate of intersystem crossing. The emission quantum yield (Ø) is given by eqn (5), where k_r and k_nr correspond to the radiative and non-radiative rate constants given by eqn (6) and (7).^31–33

k_nr ≈ α exp(−βE_T) (6)

here, α, β, γ are constants and μ_S1 corresponds to the transition dipole moment (S₀ → S₁). From eqn (5) and (7), we observe that the rate of the radiative rate constants increases with lower ΔE_ST energies, enhancing the emission quantum yields. The first excited singlet and triplet geometries were optimized, and the energy values were calculated by employing TD-DFT. The energy levels are shown in Fig. 2, and the ΔE_ST gaps (in the gas and solvent phases) are shown in Table 1.

Fig. 2 Excited singlet and triplet energies for the molecules in different phases.

Table 1 Excited singlet and triplet gaps ΔE_ST (in eV) in the gas phase and solvent phases (cyclohexane and methanol) for the studied molecules

Molecules	Gas phase	Cyclohexane	Methanol
1a	1.05	1.20	1.14
1b	0.72	0.77	0.88
1c	0.29	0.39	0.66
1d	0.01	−0.14	0.06
1e	0.00	−0.10	0.04
2a	0.65	0.68	0.71
2b	0.01	0.01	0.05
2c	−0.12	−0.08	0.01
2d	0.01	0.01	0.01
2e	0.01	0.01	0.01

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From Table 1, it is evident that with the increase in substitution of electron-withdrawing and donating groups, the energy gap between the excited state singlet and triplets decreases. The energy gaps (ΔE_ST) of molecules with thiophene as a linker are larger (1a–c) than those of the molecules having ethylene as a linker (2a–c).

As the number of cyano groups attached to the benzene ring increased (1a to c), we observed a decrease in the ΔE_ST values. In the case of the thiophene linker molecules (TLMs), we noticed a steep decrease in ΔE_ST values with the substitution of electron donating methoxy groups (1d and e). Unlike the TLMs, the ΔE_ST values of the ethylene linker molecules (ELMs) were found to be very sensitive with respect to substitution. We observed a steep decrease in ΔE_ST from 0.65 eV to 0.01 eV (in the gas phase from 2a to b), and a similar trend is observed in the solvent phases, as shown in Table 1. The overall trend of decreasing ΔE_ST energies in the gas phase was found to be similar with respect to the solvent phases (cyclohexane and methanol). However, we noticed that in most cases, the ΔE_ST energies are higher (or equal) in polar media (methanol) in comparison with cyclohexane, which in turn are higher than the gas phase energies. The excited singlet and triplet energy states in the different phases are shown in Fig. 2.

To understand this effect of decreasing ΔE_ST energies, we plotted the HOMOs and LUMOs (gas phase) for the TLMs and ELMs, as shown in Fig. 3. From Fig. 3, it is clear that in the molecules where thiophene is the linker (TLMs), the HOMO and LUMO orbitals are slightly overlapped at the linker. On the other hand, in the case of the molecules where ethylene is the linker, the overlap is minimal or negligible. For example, in the case of 1a, the contribution of the donor, linker and acceptor towards the HOMO were found to be 89%, 9% and 2%, and those for the LUMO were 2%, 35% and 63%. For 2a, the contributions of the donor, linker and acceptor towards the HOMO were found to be 95%, 4% and 1%, and for the LUMO, they were 1%, 4% and 95%.

Fig. 3 HOMO and LUMO orbitals of molecules with thiophene linkers (1a–e) and ethylene linkers (2a–e). Grey, white, blue, red and yellow indicate carbon, hydrogen, nitrogen, oxygen and sulphur atoms.

This can be understood by the π-conjugation in thiophene, which allows electronic information to be shared between the HOMO and LUMO with the π electron network, whereas in the ELMs, the electronic information is blocked by the non-conjugated linker ethylene. Hence, the energy gaps of the excited state singlet and triplet are lower for the ELMs in comparison with the conjugated TLMs. We observed that with increasing strength of the electron donating and withdrawing substituents, there was a decrease in the ΔE_ST energies in the compounds with donor–linker–acceptor frameworks. The contributions of the donor, linker and acceptor towards the HOMOs and LUMOs for all the molecules in the gas phase are given in the ESI.†

From eqn (6) and (7), we can observe that for efficient OLED materials, higher energy T₁ states are required. With the increasing substitution of cyano groups on the benzene moiety for the TLMs, we observed an increase in the level of the T₁ states, as shown in Fig. 2 (1a–c); however, no such trend is observed for the ELMs (2a–c). With the substitution of electron donating methoxy groups on the carbazole moiety, we observed a decrease in both the S₁ and T₁ states (1d, e, 2d and e).

We calculated the dipole moments corresponding to the optimized ground (S₀) and excited states (S₁) in order to understand the structural changes which the molecules underwent upon excitation. The differences in the dipole moments from the ground state to the excited singlet state (S₀–S₁) are shown in Fig. 4. Negative dipole moments suggest larger dipole moments in the excited state, and positive values suggest higher dipole moments in the ground state. From Fig. 4, it is evident that the polarity of the solvent plays a significant role in altering the dipole moments of the molecules from the ground state to the excited state. The gas phase and cyclohexane solvent phase (ε = 2.0165) calculation results were found to be closer; on the other hand, in the solvent phase calculations with methanol, the changes in the dipole moments are significantly different from the gas phase. In the case of the TLMs, with increasing substitution, we observed an increase in the change in the dipole moment in the gas phase. The cyclohexane phase results were closer to the gas phase calculations; however, the magnitudes of the dipoles are slightly different. We did not observe any particular trend in the case of methanol medium for TLMs. The dipole moments of TLMs in the excited state are significantly higher compared to the ground state in methanol medium (negative dipole moment, as shown in Fig. 4). If the solute dipole moment in the excited state is greater than in the ground state, then the excited state is better stabilized compared to the ground state.³⁴ Hence, it can be concluded that the excited states of TLMs are more stabilized in polar media such as methanol, offering longer lifetimes for the corresponding singlet excitons. In contrast to the TLMs, the ELMs exhibit a different trend where the gas phase and non-polar solvent (cyclohexane) stabilize the excited state (2c–e) when compared to methanol medium. Another important conclusion which can be drawn based on the change in dipole moments is the geometrical relaxation experienced by the molecule upon excitation. The larger the difference in the dipole moments (between the ground state and the excited state), the greater the structural change the molecule undergoes. From Fig. 4, we observed that the TLMs undergo greater structural changes in comparison with the ELMs; this effect is greatly influenced by the solvent medium.

Fig. 4 Differences in dipole moments (in Debye) from S₀ → S₁ for thiophene and ethylene linker molecules (optimized ground and excited state dipole moments are considered).

We calculated the absorption and fluorescence energies and corresponding oscillator strengths (f), which give us information about the molecule aggregation, intensities and rates of transition dipole moments (in the solvent phases). From Table 2, it is clear that the absorption energies decrease with substitution due to the increasing conjugation, resulting in a bathochromic shift of the absorption maxima. The absorption energies of the TLMs decreased from 3.16 to 2.08 eV in non-polar media and from 3.24 to 2.24 eV in polar media. A similar trend was observed in the case of the ELMs. The increase in the intensity of the absorption indicates that the solute is most soluble in the solvent and that there is the lowest degree of aggregation.³⁵ The oscillator strength suggests the intensity of absorption or emission. Compared to the ELMs, the oscillator strengths of the TLMs were found to be higher, indicating that the aggregation of the TLMs is lower than that of the ELMs. The oscillator strengths of the TLMs are mostly higher (except 1a) in non-polar media than in polar media, whereas the ELMs exhibit higher oscillator strengths in polar media than in non-polar media (gas phase calculation results are given in the ESI†).

Table 2 Absorption energies in eV (A), oscillator strengths (f), in non-polar (NP) cyclohexane and polar (P) methanol

Molecule	(A) NP	(f) NP	(A) P	(f) P
1a	3.16	0.0572	3.24	0.0631
1b	2.81	0.2178	2.93	0.2069
1c	2.50	0.2117	2.71	0.1524
1d	2.23	0.2186	2.42	0.1342
1e	2.08	0.1783	2.24	0.1790
2a	3.60	0.0870	3.67	0.0885
2b	2.87	0.0551	3.10	0.0564
2c	2.42	0.0010	2.77	0.0593
2d	2.10	0.0017	2.47	0.0512
2e	1.89	0.0007	2.27	0.0105

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Another important factor which enhances the radiative rate constants is the transition dipole moment (eqn (7)). μ_S1 corresponds to the transition dipole moment (S₀ → S₁), which is correlated with the oscillator strengths of absorption. Quantum mechanical oscillator strengths (f) and transition dipole moments are related as³⁶

where m_e is the mass of the electron, [small variant theta, Greek, macron] is the energy of the transition, “h” is Planck's constant and μ_i² corresponds to the square of the transition dipole moment to the i^th state (here, it is to the S₁ state from the ground state). From eqn (8), we noted that higher oscillator strengths and larger energies of excitation favour larger transition dipole moments, thus enhancing the radiative rate constants. From the oscillator strengths corresponding to the absorption shown in Table 2, we observed that the TLMs exhibited higher oscillator strengths compared to the ELMs in both polar and non-polar media.

Absorption and fluorescence intensities are generally similar if there are negligible structural changes associated with the excitation. However, from Fig. 4, we observed significant changes in the dipole moments corresponding to the ground state and excited state, which in turn alters the intensity of fluorescence (oscillator strength) differently from the absorption, as shown in Table 3. The fluorescence intensity was higher in polar solvent compared to non-polar solvent, as is evident from the corresponding oscillator strengths shown in Table 3. TLMs exhibit higher oscillator strengths in comparison with ELMs. With increasing substitution, we observed a decrease in fluorescence intensity.

Table 3 Fluorescence energies in eV (F) and oscillator strengths (f) in non-polar (NP) cyclohexane and polar (P) methanol

Molecule	(F) NP	(f) NP	(F) P	(f) P
1a	2.67	0.0046	2.63	0.9914
1b	2.31	0.0003	2.44	0.6013
1c	2.03	0.0001	2.29	0.4003
1d	1.49	0.0004	1.68	0.0008
1e	1.39	0.0005	1.60	0.0005
2a	3.18	0.0630	3.24	0.1024
2b	2.36	0.0246	2.67	0.0493
2c	1.92	0.0005	2.33	0.0258
2d	1.39	0.0033	1.83	0.0189
2e	1.28	0.0003	1.73	0.0192

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The HOMO and LUMO energies are crucial because they are related to the hole and electron injection abilities,³⁷ and the values of the energy levels must be determined for choosing anodes and cathodes in the OLED set-up.³⁸ We calculated the HOMO and LUMO energies (or transport energy levels) in different media, as depicted in Fig. 5 (the energy values are provided in the ESI†). The transport energy gaps were found to decrease with the substitution of both electron withdrawing and electron donating groups in all the phases. In the case of the TLMs, the highest transport energy gap was found to be 3.64 eV (1a, gas phase) with a significant decrease in energy gap upon substitution from 1a to e; the lowest energy gap was 2.43 eV (1e, gas phase). Similar findings were observed in the ELMs with the highest transport energy gap of 4.02 eV (2a, gas phase) and the lowest energy gap of 2.12 eV (2e, gas phase).

Fig. 5 HOMO and LUMO energies of the molecules chosen for the study.

The solvent effects on the transport energy gaps were found to be profound, as is evident from Fig. 5, where we observed that the LUMO energy levels in solvent media are higher than in the gas phase calculations, resulting in larger energy gaps. The transport energy gaps in various media are given in Table 4. As the polarity of the solvent increases, the transport gap increases. For example, in the case of the ELMs with an increase in substitution, the difference in the transport gaps of the gas phase and the methanol phase was found to increase from 0.08 eV (2a) to 0.58 eV (2e). In the case of the TLMs, we observed that the difference between the transport energy gaps of the methanol and gas phases increased from 0.12 to 0.27 eV (from 1a to c) and slightly decreased to 0.24 and 0.21 eV for 1d and e. In the case of cyclohexane media, the transport energy gaps of all the molecules studied were found to be intermediate between the gas phase and methanol media.

Table 4 Transport energy gaps (HOMO–LUMO gaps) in the gas phase and in solvent phases (cyclohexane and methanol) of the studied molecules

Molecules	Gas phase	Cyclohexane	Methanol
1a	3.64	3.68	3.76
1b	3.23	3.27	3.39
1c	2.87	2.93	3.14
1d	2.58	2.62	2.82
1e	2.43	2.46	2.64
2a	4.02	4.04	4.10
2b	3.21	3.33	3.57
2c	2.69	2.88	3.22
2d	2.34	2.53	2.90
2e	2.12	2.32	2.70

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Ionization potentials and electron affinities give us information about the hole injection and electron injection capabilities of molecules. The ionization potentials (IP) and electron affinities (EA) (only adiabatic IP and EA are discussed here; vertical and adiabatic IP and EA values are provided in the ESI†) were calculated in order to understand the performance of the proposed molecules. Smaller ionization potentials favour easier hole injection, and larger electron affinities facilitate electron injection.³⁷ Fig. 6 depicts the trends of the IPs and EAs (adiabatic) of the TLMs and ELMs. The IP values depend on the HOMO energy levels, and the EA values depend on the LUMO levels. The electron withdrawing groups decrease the LUMO energies, and the electron donating groups increase the HOMO levels by donating electrons (Fig. 5).

Fig. 6 Adiabatic ionization potentials (I.P.) and electron affinities (E.A.).

The ionization potential energies (shown in Fig. 6) were found to be slightly altered with the substitution of electron withdrawing groups; however, we observed a significant reduction in IP values from 7.274 to 6.708 eV (1c to e, gas phase) with the substitution of electron donating groups, as they increase the HOMO energy levels, thus favouring the easy removal of electrons. The ionization potential increases from 7.274 to 7.328 eV (1a to c) and decreases to 6.708 eV (1e) for the TLMs; the ELMs also exhibit the same trend. From the histogram in Fig. 6, it is evident that the IP values are highest in the gas phase and lowest in the polar medium methanol, as the cations are stabilized by the polar media. A significant difference in the magnitude of the IP energy is observed between the gas phase and the polar medium methanol; they were found to be in the range of 1.4 to 1.7 eV. From Fig. 6, we observed that the magnitude of the electron affinities increased with increasing substitution of cyano groups (1a to c and 2a to c) from −0.829 to −1.714 eV (1a to c, gas phase) and a slight decrease upon substitution with electron donating groups from −1.687 to −1.657 (1d to e, gas phase). The electron affinities of the ELMs were found to be lower in comparison with the TLMs; however, the trend was found to be similar to that of the TLMs. Compared to the IP energies, a contrasting trend was observed, where the EAs were higher in energy in the polar solvent methanol compared to the gas phase. The cyclohexane IPs and EAs were found to be intermediate between the gas phase and methanol.

One of the important factors which determine the efficiencies of OLED materials is the reorganization energy (λ). “λ” is related to the rate of intermolecular charge transfer (k_et) from Marcus–Hush theory,³⁹ given by the formula

where “A” is a pre-factor related to the electronic coupling between adjacent molecules. From eqn (9), it is clear that lower reorganization energies are required for efficient charge transport processes. We calculated the reorganization energies for electrons (λ_electron) and holes (λ_hole) employing an adiabatic process, using the formulae given by eqn (10) and (11).

λ_e = (E⁻₀ − E⁻₋) + (E⁰₋ − E⁰₀) (10)

λ_h = (E⁺₀ − E⁺₊) + (E⁰₊ − E⁰₀) (11)

where E⁻₀, E⁺₀ represent anionic and cationic energies in neutral geometry; E⁻₋ and E⁺₊ represent anionic and cationic state energies in their anionic and cationic optimized geometries; and E⁰₋ and E⁰₊ are energies of neutral molecules in the anionic and cationic geometries, respectively. The reorganization energies calculated employing the above formula neglect the environmental relaxation and changes in electronic polarization between the isolated system and the solid state system; however, the calculated λ can act as the upper bound of the charge transport energies.⁴⁰ The reorganization energies of electrons and holes for the studied molecules are shown in Fig. 7.

Fig. 7 Reorganization energies corresponding to electrons and holes in different media.

From Fig. 7, we observe that λ_hole is found to be lower in comparison with λ_electron (except 1a), indicating that the hole transporting ability of these molecules is superior to that of electron transport. Compared to TLMs, ELMs exhibit lower reorganization energies (both λ_hole and λ_electron). From Fig. 7, we observed a significant difference in the magnitude of reorganization energies corresponding to the electrons and holes in different media; however, the trend was found to be similar in all the phases.

Lower ΔE values (ΔE corresponds to the modulus of energy difference between the reorganization of electrons and holes) are required for efficient balance of electrons and holes in the emitting layer.³⁷ The ΔE values of the studied molecules in different phases are given in Table 5.

Table 5 Energy differences between the reorganization of electrons and holes (ΔE) in different phases

Molecules	Gas phase	Cyclohexane	Methanol
1a	0.011	0.068	0.092
1b	0.189	0.190	0.159
1c	0.261	0.237	0.217
1d	0.184	0.148	0.117
1e	0.235	0.208	0.103
2a	0.036	0.154	0.271
2b	0.165	0.184	0.213
2c	0.182	0.203	0.214
2d	0.017	0.033	0.030
2e	0.099	0.092	0.104

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With the increase in substitution of cyano groups in the TLMs, we observed increases in the ΔE values from 0.011 to 0.261 eV (in the gas phase). With the substitution of electron donating methoxy groups, the ΔE value slightly decreases to 0.184 (1d, gas phase) and increases to 0.235 (1e, gas phase) with the substitution of an additional methoxy group. ELMs also followed a similar trend to that of the TLMs. The ΔE values in the solvent phases mostly followed a similar trend to that of the gas phase; however, they slightly differed in magnitude, as shown in Table 5.

Larger exciton binding energies are required in OLED materials to increase the probability of electron hole pair recombination.⁴¹ From eqn (12), we observed that the exciton binding energy is dependent on the transport gap and the optical energy. E_T is called the transport gap, which is the energy difference between the HOMO and LUMO; E_opt corresponds to the optical gap (S₀ to S₁). Exciton binding energy helps us understand the attraction–repulsion interplay between electrons and holes by comparing the variation of S₁ energy with respect to the HOMO–LUMO energy gap.⁴² The exciton binding energy is calculated by the formula given in eqn (12).

E.B.E. = E_T − E_opt (12)

In larger chains (and solids), the electrons and holes are well separated; their localized repulsions are negligible, and the HOMO–LUMO gap coincides with the optical gap.^43,44 However, in small and medium-sized molecules, we observe appreciable exciton binding energies due to the strong attraction between electrons and holes. From Fig. 8, we found that the HOMO–LUMO energy gaps are greater than the optical gaps due to the effective electron–hole attraction which stabilizes the HOMO–LUMO gaps. In solar cells, the excitation is optical (the transition is from S₀ to S₁); however, in the case of OLEDs, the injection is electrical, which results in the formation of triplet and singlet excitons (75% triplet and 25% singlet). Hence, we calculated the exciton binding energies corresponding to the triplet states (EBE_T) along with the singlet states (EBE_S), as shown in Fig. 8.

Fig. 8 Exciton binding energies corresponding to singlets (EBE_S) and triplets (EBE_T).

In general, excited triplet states are found to be lower compared to the corresponding singlets; hence, we observed (in Fig. 8) higher exciton binding energies for the triplets than for their corresponding singlets. In the case of the TLMs, the EBE_S were found to be higher in the solvent phase than in the gas phase, and we observed a decrease in the EBE_S from 1a to e (0.5 to 0.35 eV – gas phase; 0.52 to 0.38 eV – methanol medium). On the other hand, for the ELMs, the EBE_S were insensitive to both solvent media and substitution (2a to e); they remained close to ∼0.4 eV. In the case of the triplet exciton binding energies, there is a gradual decrease in the exciton binding energies for the TLMs (from 1a to e) from 0.83 to 0.61 eV in the gas phase and from 1.18 to 0.60 eV in methanol media. The ELMs exhibited lower EBE_T values compared to the TLMs, ranging from 0.85 to 0.44 eV in the gas phase and 0.91 to 0.48 eV in methanol.

Conclusions

We have considered donor–linker–acceptor framework molecules to study the properties of OLEDs. Various properties which effect the efficiencies of emission quantum yields (Ø), such as the energy gaps between the excited singlet and triplet states (ΔE_ST), dipole moments, HOMO and LUMO energies, transport energy gaps, ionization potentials, electron affinities, reorganization energies and exciton binding energies, were investigated. Calculations were carried out with a regular gas phase along with non-polar cyclohexane and polar methanol solvent media in order to understand the effects of solvent on these properties. The singlet–triplet energy gap, which is dependent on the exchange energy, can be reduced by separating the HOMO and LUMO with a linker. By enhancing the donating ability of the electron donors and the withdrawing capability of the electron acceptors, the energy gaps of the S₁ and T₁ states can be reduced to almost zero eV. A non-conjugated linker (ethylene) was found to be more efficient in decreasing the energy gap (ΔE_ST) compared to the conjugated linker thiophene, since the non-conjugated linker blocks the electronic information between the frontier orbitals more efficiently than the conjugated linker thiophene. Although the gas phase calculations in most of the cases (except the dipole moment changes from S₀ to S₁) predicted the trend of variation in properties with respect to substitution, the magnitude of the difference is appreciable in many of the cases. The solvent effects were found to be profound due to the polarity difference, and it was found that the transport energy gaps (HOMO–LUMO gaps) increased with increasing solvent polarity. Significant differences in the magnitudes of IP and EA in various media were found, as the frontier orbital energy levels varied in the different media due to the changes associated with polarity. The reorganization energies were calculated; we observed that λ_hole was lower in comparison with λ_electron (except in 1a), indicating that the hole transporting abilities of these molecules are superior to their electron transporting abilities. Compared to the TLMs, the ELMs exhibit lower reorganization energies (both λ_hole and λ_electron). Exciton binding energies (EBE) corresponding to the first excited singlet and triplet states were calculated. The EBE_T values of the triplets were higher than their corresponding singlets due to the lower energies of the triplets compared to their respective singlets. In the case of the TLMs, we observed a gradual decrease in the exciton binding energies with respect to substitution corresponding to both singlets and triplets. The ELMs exhibited a different trend compared to the TLMs, where we do not observe any significant changes corresponding to the EBE_S upon substitution; on the other hand, there is a gradual decrease in the exciton binding energy corresponding to the triplets.

Acknowledgements

K. Sumithra gratefully acknowledges the Council of Scientific and Industrial Research (CSIR), Government of India, New Delhi for financial support in the form of a project grant (01 (2748)/13/EMR-II), T. Vikramaditya thanks the University Grants Commission, New Delhi for providing a senior research fellowship and M. Saisudhakar acknowledges support from the same CSIR project for funding in the form of a senior research fellowship.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra00053c

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