Effect of dynamic disorder on charge carrier dynamics in Ph4DP and Ph4DTP molecules

Abstract

Electronic structure calculations were used to study the charge transport and optical properties of 2,2′,6,6′-tetraphenyldipyranylidene (Ph4DP) and its sulfur analogue 2,2′,6,6′-tetraphenyldithiopyranylidene (Ph4DTP) based molecules. The dynamic disorder effect is included while calculating the charge transfer kinetic parameters such as rate coefficient, disorder drift time, hopping conductivity and mobility of charge carriers through the kinetic Monte Carlo simulations. The existence of degeneracy levels promotes the delocalization of charge carrier and charge transfer. Theoretical results show that if the orbital splitting rate is larger than the static charge transfer rate (O_R > k_static), the charge transfer is kinetically favored. If O_R < k_static, the charge carrier is potentially trapped in the localized site. In the case of O_R ∼ k_static, the charge carrier motion is not affected by the dynamic disorder. The calculated hole mobility in Ph4DP and Ph4DTP molecules is 0.04 and 0.03 cm² V⁻¹ s⁻¹ and is in agreement with the experimental results. It has been found that fluorine and chlorine substituted Ph4DP molecules have good ambipolar transporting character. The absorption and emission spectra were calculated using the time dependent density functional theory (TDDFT) method at the CAM-B3LYP/6-31G(d,p) and M062X/6-31G(d,p) levels of theory. The calculated absorption spectra are in agreement with the experimental results.

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1. Introduction

Studying charge transport behavior of organic materials is of great interest because of its relevance in the development of optoelectronic devices such as organic light emitting diodes,^1–4 thin film transistors,^5–8 organic photovoltaic cells and field effect transistors.^9–12 The advantage of organic materials rather than inorganic semiconductors is their relatively low cost, lower molecular weight and tunable electronic structure.^12,13 The optoelectronic properties of organic semiconductors are improved by the structural modification, substitution of suitable electron donating (ED), electron withdrawing (EW) groups, and active heterocyclic compounds.¹⁴ The substitution of different EW and ED groups alters the delocalization of electron density on the frontier molecular orbitals of the molecule. The substitution of heterocyclic groups influences the structure and optoelectronic properties such as molecular packing, conjugation length, bandwidth and ionic state properties.¹⁵ The self-aggregation and phase properties are controlled by the addition of appropriate side chains.^16–18

At high temperature (T > 150 K), the charge carrier mean free path is shorter than the intermolecular spacing, and the wave function of the charge carrier is localized on particular molecule due to weak coupling between the electronic states.^12,19,20 Therefore, the charge transport in these materials is due to thermally activated hopping mechanism rather than band like transport.^12,17,21,22 Understanding the charge carrier dynamics along the localized sites is difficult due to the interaction between nuclear and electronic degrees of freedom. It has been shown in earlier studies that the dynamic disorder along the charge transfer path decreases the effect of electron-phonon scattering on the charge carrier motion and provides the dynamic localization of charge carrier rather than the static localization.^19,23–25 Here, the dynamic disorder, such as nuclear degrees of freedom, perturbs the localized charge carrier and so the coefficients of charge carrier wave function are no longer zero at its boundaries.²⁵ The perturbed localization is called as dynamic localization which is responsible for charge flux which facilitates the charge transfer (CT) kinetics, and unperturbed localization is named as static localization (or Anderson localization).^25–27 In this dynamic fluctuation regime, the charge transport behaviour is termed as diffusion limited crossover from non-adiabatic localization to adiabatic delocalization.^{12,19,20,28,29} The structural fluctuations leads to breakdown of the Franck–Condon (FC) principle and makes significant impact on charge carrier motion.^30,31 Even in crystal packing, at room temperature, the molecules are fluctuating from their equilibrium position.³² Hence, for better understanding of charge transfer phenomena in organic crystals, the charge transport properties should be studied at molecular level.

Altan Bolag et al.³³ have synthesized 2,2′,6,6′-tetraphenyldipyranylidene (Ph4DP), its sulfur analogue 2,2′,6,6′-tetraphenyldithiopyranylidene (Ph4DTP) crystals and their derivatives. These molecules have quinoid structure and core is attached with tetrathiafulvalene (TTF), a well-known family of π-electron donor groups. The Ph4DTP possesses quasi-planar conformation with phenyl rings tilted by 12° relative to the core.³³ The Ph4DP and Ph4DTP are reported as new p-type semiconductors due to their hole mobility and on/off ratios, and they have the following advantages, first, they have an isoelectronic structure with TTF, their cation and dication states are stable. Second, they have the extended π-conjugated structure which is favorable for strong intermolecular interaction leading to high charge carrier mobility. Third, the preparation method is simple and substituents can easily be introduced. Finally, due to the presence of sulfur atom in Ph4DTP, the molecule exhibits with high polarizability nature which provides strong π–π interaction,³⁴ and the presence of oxygen atom in Ph4DP reduces the steric repulsion which is responsible for the planar structure and π-stacking aggregation in crystal packing. The chemical structure of Ph4DP and Ph4DTP molecules is shown in Fig. 1. The experimental results on these molecules motivated us to study the charge transport and optical properties of Ph4DP, Ph4DTP and their substituted analogues. The time-dependent density functional theory (TD-DFT) method is used to calculate the absorption and emission spectra of Ph4DP and Ph4DTP molecules. This study will provide information to tune the optoelectronic properties of organic semiconductors.

Fig. 1 The chemical structure of tetraphenyldipyranylidene derivatives.

2. Theoretical formalism

Based on tight binding Hamiltonian approach, the presence of excess charge in a π-stacked molecular system is expressed as^13,35where, a_i⁺ and a_i are creation and annihilation operators, ε_i(θ) is the site energy, energy of the charge when it is localized at i^th molecular site and J_i,j is the charge transfer integral or electronic coupling. By considering a two state model, the energy eigenvalue equation can be written as,

HC − ESC = 0 (2)

where, H, C and S are the Hamiltonian, orbital coefficient and spatial overlap matrix element of a two state system for which the Hamiltonian is written as,and the spatial overlap matrix,here, site energy ε₁ = 〈ψ₁|Ĥ|ψ₁〉 and J₁₂ = 〈ψ₁|Ĥ|ψ₂〉 are diagonal and off-diagonal matrix elements of the Hamiltonian.

The charge transfer rate between the localized sites is described by semi-classical theory of Marcus–Hush model, which coupled the density of states (DOS) and square of the effective charge transfer integral (J_eff),^22,36,37

The density of states (DOS) are weighted by the Franck–Condon factor, ρ_FCT and is calculated by using reorganization energy (λ),²²

where, k_B is Boltzmann constant and T is the temperature. Here, the two key parameters, the effective charge transfer integral (J_eff) and reorganization energy (λ) determines the charge transfer rate.

The effective charge transfer integral (J_eff) is calculated from charge transfer integral, spatial overlap integral and site energy as,^38,39

As described above the site energies, ε_i and ε_j are the energy of a charge when it is localized at i^th and j^th molecules, respectively, and J_ij represents the electronic coupling between HOMO (or LUMO for electron) of nearby molecules i and j. As described in previous studies,^13,31,39 the J, S and ε are calculated by using the fragment molecular orbital approach³⁹ as implemented in the Amsterdam density functional (ADF) program.^40,41 In ADF calculation, we have used the Becke–Perdew (BP) exchange correlation functional^42,43 with triple-zeta plus double polarization (TZ2P) basis set.

The reorganization energy (λ) is the energy change associated with relaxation of molecular geometry due to the presence of excess charge on a molecule. The reorganization energy is evaluated directly from the adiabatic potential energy surfaces of neutral, cation and anion geometries.^44,45 Within this approximation, the reorganization energy is defined as,

λ^± = [E^±(g^o) − E^±(g^±)] + [ E^o(g^±) − E^o(g^o)] (8)

where, E^±(g^o) is the total energy of a molecule with an excess (positive or negative) charge in the optimized neutral geometry, E^±(g^±) is the total energy of optimized ionic geometry, E^o(g^±) is the total energy of neutral molecule in ionic geometry and E^o(g^o) is the total energy of optimized neutral molecule. The neutral, cationic and anionic geometries were optimized using density functional theory method, B3LYP^46–48 in conjunction with the 6-31G(d,p)³² basis set using GAUSSIAN 09 program package⁴⁹ and these energy values are used to find the reorganization energy for hole (λ⁺) and electron (λ⁻) transport.

It has been shown in earlier studies that the structural fluctuation in the form of stacking angle change have significant impact on charge carrier mobility.^31,41,50 Thus, the effective charge transfer integral (J_eff) calculated at different stacking angle is used to study the rate of charge transfer between the localized sites. Based on the charge transfer rate calculated from Marcus eqn (5) kinetic Monte Carlo simulation is performed to calculate the mean squared displacement of the charge carrier in the π-stacked system. The motion of charge carrier in the disordered molecular system is described in the form of thermal diffusion process.¹³ The diffusion co-efficient (D) is calculated from the time evolution of mean squared displacement as,

Based on the Einstein diffusion model, the drift mobility of the charge carrier is calculated from the diffusion coefficient, D as,⁵¹

As described in our previous study the hopping conductivity is calculated on the basis of density flux model and is described as,⁵²

where, ε is electric permittivity of the medium and is the rate of transition probability (or charge transfer rate). At room temperature the structural fluctuation in the form of stacking angle change will affect the charge transport.^30,31 During the Monte Carlo simulation the stacking angle fluctuation up to 6° from the equilibrium position is allowed^41,53 and is assumed that the stacking angle fluctuation from the equilibrium value is harmonic and molecules are bounded within the unit cell.

In the present work, the charge transport calculations were performed with experimental crystal structure of fluorinated Ph4DP and unsubstituted Ph4DTP molecules. To find the crystal structure of unsubstituted Ph4DP and chlorine substituted Ph4DP the DFT calculations were performed by using Vienna Abinitio Simulation Package (VASP)^54–56 with projected augmented wave potential, force convergence of 0.02 eV Å⁻¹ and energy convergence of 0.001 eV. The optimized structure of Ph4DP has triclinic structure with P1̄ space group and the crystal structure of Ph4DTP has monoclinic structure with the space group of C2/c. In the unit cell, the fluorinated Ph4DP and unsubstituted Ph4DTP molecules are packed in co-axial manner along the a-axis and b-axis, respectively, and the corresponding intermolecular distance is 6.08 and 5.52 Å. In the optimized structure of unsubstituted Ph4DP and chlorine substituted Ph4DP, the molecules are arranged in parallel along the a-axis with the intermolecular distances of 4.13 and 5.92 Å, respectively.

To calculate the emission spectra, the excited state geometry of Ph4DP, F and Cl substituted Ph4DP and Ph4DTP molecules has been optimized in dichloromethane medium by using time-dependent density functional theory (TD-DFT) method at B3LYP/6-31G(d,p) level of theory. Based on the ground and excited states geometry, the absorption and emission spectra were calculated using TD-DFT method at CAM-B3LYP/6-31G(d,p) and M06-2X/6-31G(d,p) level of theories. Tomasi's⁵⁷ polarized continuum model (PCM) in self-consistent reaction field (SCRF) theory is used to include the solvent effect on the optical properties of the studied molecules. The SWizard program^58,59 was used to plot and analyse the absorption and emission spectra of the studied molecules. The spectra were generated using the following Gaussian function,

where, ε(ω) is the molar extinction coefficient in M⁻¹ cm⁻¹, ω is the energy of the allowed transition in cm⁻¹, f_I is the oscillator strength and Δ_1/2 is the half-bandwidth and is fixed as 3000 cm⁻¹.

3. Results and discussion

3.1. Frontier molecular orbitals

In the π-stacked organic molecules the excess positive charge migrates through the highest occupied molecular orbital (HOMO) and the excess negative charge migrates through the lowest unoccupied molecular orbital (LUMO) of nearby molecules. That is, the charge transport, optical absorption and emission properties of the π-stacked molecules strongly depends on the delocalization of electron density on the frontier molecular orbitals of the individual molecules. The density plot of the HOMO and LUMO of Ph4DP, F substituted Ph4DP, Cl substituted Ph4DP and Ph4DTP molecules are obtained at B3LYP/6-31G(d,p) level of theory and are shown in Fig. 2 and 3. It has been observed that HOMO is delocalized on the center of heptacyclic ring and LUMO is delocalized over the entire molecule. The distribution of HOMO and LUMO on the studied molecules exhibits π-orbital character. It has been observed that the substitution of F or Cl and O or S atoms on the core does not alter the delocalization of electron density on the frontier molecular orbitals. The energy of HOMO, LUMO and energy gap between HOMO and LUMO of the studied Ph4DP and Ph4DTP molecules are calculated at B3LYP/6-31G(d,p) level of theory and are summarized in Table 1. It has been observed that the HOMO and LUMO energies of Ph4DP molecule is −4.03 and −1.48 eV, respectively. The F and Cl substitution decreases the HOMO level by 0.2 and 0.4 eV and the LUMO level around 0.14 and 0.45 eV which is in agreement with the experimental results.³³

Fig. 2 Plot of highest occupied molecular orbital (HOMO) of the Ph4DP and Ph4DTP molecules.

Fig. 3 Plot of lowest unoccupied molecular orbitals (LUMO) of the Ph4DP and Ph4DTP molecules.

Table 1 Molecular orbital energies (E_HOMO, E_LUMO), energy gap (ΔE), ionization potential (IP), electron affinity (EA) and reorganization energy (λ) for Ph4DP, F and Cl substituted Ph4DP and PH4DTP molecules

Molecule	EH (in eV)	EL (in eV)	Energy gap EH–EL (in eV)		Ionization potential (in eV)		Electron affinity (in eV)		Reorganization energy (in eV)
			Theory	Exp.^a	Vertical	Adiabatic	Vertical	Adiabatic	Hole	Electron
a Values taken from ref. 33.
Ph4DP	−4.03	−1.48	2.55	2.02	5.25	5.10	0.41	0.58	0.28	0.31
F-Ph4DP	−4.23	−1.62	2.60	2.25	5.44	5.28	0.55	0.73	0.31	0.33
Cl-Ph4DP	−4.44	−1.93	2.50	2.18	5.59	5.45	0.92	1.08	0.29	0.30
Ph4DTP	−4.19	−1.75	2.43	1.98	5.38	5.23	0.67	0.93	0.30	0.51

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3.2. Reorganization energy and ionic state properties

The reorganization energy due to the presence of excess positive (λ⁺) and negative (λ⁻) charges on the studied molecules has been calculated using eqn (8) and are summarized in Table 1. The presence of excess charge on the molecules alters the structural parameters significantly. The selected geometrical parameters of the studied molecules in the neutral and ionic states are summarized in Table S1.†

It has been observed that the λ⁺ calculated for unsubstituted, and substituted Ph4DP and Ph4DTP molecules are almost same and is nearly 0.29 eV. As observed from Table S1,† the presence of excess negative charge on the Ph4DTP significantly alters the structural parameters, particularly the dihedral angles. As given in Table 1, the λ⁻ calculated for Ph4DP molecules is around 0.3 eV. Notably, the reorganization energy corresponding to presence of excess negative charge on Ph4DTP is 0.51 eV. In agreement with previous experimental results,³³ the above results show that the migration of positive charge on the studied molecules is more favorable than the migration of negative charge.

The ionization potential (IP) and electron affinity (EA) are the basic properties and determines the stability, injection barrier and charge polarity of a molecule. The adiabatic ionization potential (AIP), vertical ionization potential (VIP), adiabatic electron affinity (AEA) and vertical electron affinity (VEA) are calculated using total energy of neutral and ionic states and are summarized in Table 1. Among the studied Ph4DP molecules, the unsubstituted Ph4DP molecule has minimum ionization potential of 5.52 and 5.10 eV for vertical and adiabatic excitations. The IP of Ph4DTP is higher than that of Ph4DP by 0.13 eV. The substitution of F and Cl atoms on Ph4DP molecule increases the IP and EA by 0.2 to 0.5 eV. That is, the creation of hole on Ph4DP molecule is easier than on the other studied molecules. Among the studied molecules, the unsubstituted Ph4DP is having minimum EA of 0.41 and 0.58 eV for vertical and adiabatic excitations. The substitution of Cl atoms on Ph4DP increases the EA by 0.5 eV. Tabulated values clearly show that the AIP is smaller than the VIP, and the AEA is higher than the VEA. That is, the studied molecules relax more when there is an excess negative charge. This result is in agreement with the results obtained from reorganization energy values.

3.3. Effective charge transfer integral

The effective charge transfer integral, J_eff for hole and electron transport in studied molecules has been calculated by using eqn (7) and are summarized in Table 2. For F substituted Ph4DP and Ph4DTP the ADF calculations were performed for the dimer structure taken from crystal structure data and for other molecules the dimer structure is taken from the optimized structures as described in Section 2. It has been observed that the structure, substitution and intermolecular arrangement determines the J_eff. In the case of Ph4DP molecules, the F substituted Ph4DP molecule is having maximum J_eff of 0.1 eV for hole transport. In this molecule, the intermolecular distance is 6.08 Å and the arrangement of molecules is such that the HOMO of each molecule is interacting constructively. For electron transport, the Ph4DTP molecule has maximum J_eff of 0.1 eV. The chlorine substituted Ph4DP molecule has minimum J_eff of 0.03 for hole transport and 0.02 eV for electron transport which is due to unequal contribution of HOMO (or LUMO) of each monomer on the dimer HOMO (LUMO for electron transport). The earlier studies^{19,28,30,31,52,60} show that the structural fluctuation at room temperature provides the significant effect on charge transfer integral and mobility. In the present study, while calculating the charge transfer kinetic parameters the variation of J_eff due to structural fluctuation in the form of stacking angle is included.

Table 2 The effective charge transfer integral (J_eff) for hole and electron transport in Ph4DP, F and Cl substituted Ph4DP and Ph4DTP molecules

Molecules	Intermolecular distances (Å)	Effective charge transfer integral (Jeff in eV)
		Hole	Electron
Ph4DP	4.13	0.02	0.07
F-Ph4DP	6.08	0.10	0.03
Cl-Ph4DP	5.92	0.03	0.02
Ph4DTP	5.52	0.005	0.10

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3.4. Charge transfer kinetics

The computed effective charge transfer integral (J_eff) and reorganization energy (λ) are used to calculate the charge transfer rate using eqn (5) and (6). During the kinetic Monte Carlo (KMC) simulations, the excess charge is propagated based on the charge transfer rate calculated from semi-classical Marcus theory. The time evolution of mean squared displacement 〈X²(t)〉 is used to calculate the carrier mobility by using eqn (9) and (10). As shown in Fig. 4 and S1,† 〈X²(t)〉 is linearly increasing with time. The results show that the CT is the normal diffusion process in which the charge carrier does not reach the end of hopping site within the simulation time. As shown in Fig. 5, the survival probability for the charge carrier at particular site is decreasing exponentially with respect to time. The calculated rate coefficient from survival probability graph is used in eqn (11) to calculate the hopping conductivity.⁵²

Fig. 4 Mean squared displacement of electron in Ph4DP molecule with respect to time.

Fig. 5 Survival probability of negative charge in Ph4DP molecule with respect to time.

As described in previous studies,^30,31,52 the time evolution of CT kinetics is studied on the basis of rate coefficient (k), dispersive parameter (a), survival probability (P(t)), thermal disorder S(t) and disorder drift time (S_t). The results are shown in Fig. 4–7 and S2† for hole and electron transport in the studied molecules. Here, the disorder drift time S_t is the simulation time at which the dynamic disorder is maximum. On the basis of statistical relation, the disorder drift and the possible number of electronic states (Z) along the charge transfer path are related as,⁶¹

As described earlier, the dynamic disorder due to structural fluctuation promotes the carrier dynamics from static to dynamic localization.^{12,19,24,25,29} As shown in Fig. 7, from the disorder drift curve the rate of splitting of energy states (O_R) can be calculated as

where, Z_d and Z₀ are the number of electronic states at time t = S_t and at time t = 0, respectively. The energy distribution among the possible electronic states is studied by calculating the dispersed energy difference due to the dynamic disorder and is described aswhere, ΔE_S0 is the difference in energy distribution among the electronic levels in the absence of dynamic disorder. As shown in Fig. 8, the ratio of dispersed energy difference, is calculated by using the disorder drift, S(t).

Fig. 6 Time evolution of the rate coefficient for electron transport in Ph4DP molecule.

Fig. 7 Disorder drift with respect to time for electron transport in the Ph4DP molecule.

Fig. 8 Time evolution of dispersal energy difference ratio for electron transport in the Ph4DP molecule.

It has been observed that, the dispersive parameter for hole transport in Ph4DP and electron transport in Ph4DTP is nearly 1 which shows that the CT kinetics follows static non-Condon effect and the charge decay along the charge transfer path is coherence.³¹ For other cases, the dispersive parameter is in the range of 0.6 to 0.75 and the CT kinetics follows the intermediate regime between static and kinetic non-Condon effect. Here, the wave function of the charge carrier is slowly decaying with respect to time and the charge carrier takes the motion along the sequential hopping sites. In the present study, the existence of degeneracy levels per unit time is calculated from the timely varying disorder drift curve (see Fig. 7) and is associated with the orbital splitting rate, O_R. As given in Table 3, for hole transport in chlorine and fluorine substituted Ph4DP molecules, the orbital splitting rate is 5.4 × 10¹² and 2 × 10¹² s⁻¹, respectively, showing the coupling strength between the HOMO of each monomer and responsible for good hole mobility of 0.17 and 0.34 cm² V⁻¹ s⁻¹, and rate coefficient of 2.38 and 4.91 ps⁻¹, respectively. In these molecules the hole drift time by the dynamic fluctuation is 156 and 68 fs which is smaller than that of other studied molecules. Here, the dynamic disorder enhances the orbital splitting rate (see Table 3) which is responsible for the large bandwidth and the maximum hopping conductivity of 0.13 and 0.26 S cm⁻¹, respectively. The hole mobility in unsubstituted Ph4DP and Ph4DTP molecules is around 0.04 cm² V⁻¹ s⁻¹ which is in agreement with experimental values of 0.02 and 0.05 cm² V⁻¹ s⁻¹, respectively. For electron transport, unsubstituted Ph4DP, F and Cl substituted Ph4DP molecules are having hopping conductivity of 0.4, 0.12 and 0.21 S cm⁻¹ and their corresponding rate coefficient is 7.54, 2.19 and 4.02 ps⁻¹. While comparing unsubstituted Ph4DP and F substituted Ph4DP, it has been observed that the F substituted Ph4DP molecule is having slightly higher electron mobility, which is due to a small difference in intermolecular distance (see Table 4). As given in Table 4, in comparison with Ph4DTP, the Ph4DP molecules are having significant orbital splitting rate and electron transporting ability. The disorder drift time, S_t calculated for Ph4DTP molecule is 8.08 × 10⁴ fs which is higher than that of the Ph4DP molecules and clearly shows the poor electron transport in Ph4DTP molecule.

Table 3 Calculated charge transfer kinetic parameters, rate coefficient (k), hopping conductivity (σ_Hop), mobility (μ), disorder drift time (S_t), charge transfer time (τ_CT), dispersive parameter (a) and orbital splitting rate (Z_d − Z₀)/S_t for hole transport in Ph4DP and F and Cl substituted Ph4DP and Ph4DTP molecules

Molecule	Inter-molecular distance (Å)	k (ps^−1)	σHop (S cm^−1)	μ (cm^2 V^−1 s^−1)	St (fs)	τCT (fs)	a	(Zd − Z0)/St (ps^−1)
Ph4DP	4.13	0.85	0.04	0.04	1.62 × 10^3	1.39 × 10^3	0.97	0.20
F-Ph4DP	6.08	2.38	0.13	0.17	156	65	0.56	1.95
Cl-Ph4DP	5.92	4.91	0.26	0.34	68.2	506	0.60	5.41
Ph4DTP	5.52	0.52	0.03	0.03	1.73 × 10^3	2.31 × 10^4	0.51	0.16

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Table 4 Calculated charge transfer kinetic parameters rate coefficient (k), hopping conductivity (σ_Hop), mobility (μ), disorder drift time (S_t), charge transfer time (τ_CT), dispersive parameter (a) and orbital splitting rate (Z_d − Z₀)/S_t for electron transport in Ph4DP and F and Cl substituted Ph4DP and Ph4DTP molecules

Molecule	Inter-molecular distance (Å)	k (ps^−1)	σHop (S cm^−1)	μ (cm^2 V^−1 s^−1)	St (fs)	τCT (fs)	a	(Zd − Z0)/St (ps^−1)
Ph4DP	4.13	7.54	0.40	0.16	94	133	0.68	2.11
F-Ph4DP	6.08	2.19	0.12	0.21	820	754	0.72	0.46
Cl-Ph4DP	5.92	4.02	0.21	0.32	217	2.69 × 10^3	0.62	1.45
Ph4DTP	5.52	0.06	3.19 × 10^−3	0.06	8.08 × 10^4	681	0.91	0.01

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By comparing the disorder drift time (S_t) and charge transfer time (τ_CT) (inverse of the static CT rate, τ_CT = 1/k_static), in the static case the structural fluctuation effect on charge carrier dynamics is studied. The static CT rate (k_static) is directly calculated from the eqn (5) without invoking the effect of structural fluctuations. In chlorine substituted Ph4DP molecule, the disorder drift time, S_t (∼68 fs) is lesser than the hole transfer time, τ_CT (∼500 fs), and the dynamic hole transfer rate 4.9 × 10¹² s⁻¹ is higher than the static CT rate 2 × 10¹² s⁻¹. Whereas, in the case of fluorine substituted Ph4DP molecule, the S_t (156 fs) is greater than the τ_CT (65 fs), and the static CT rate (∼15.4 ps⁻¹) is relatively higher than the dynamic rate (∼2.4 ps⁻¹). As given in Table 3, for unsubstituted PH4DP molecule, the calculated hole drift time (∼1.6 ps) and static hole transfer time (∼1.4 ps) are comparable, and the static and dynamic hole transfer rates are also comparable (0.71 × 10¹² s⁻¹ and 0.85 × 10¹² s⁻¹). As observed from Table 4, in Ph4DTP molecule, the electron transfer time (τ_CT = 0.68 ps) is lesser than the disorder drift time (81 ps) and the static rate is higher than the dynamic rate. The calculated disorder drift time for electron transport in Ph4DTP molecule shows that the electron survives longer time on the localized electronic state and may be potentially trapped. The above results show that when the carrier drift time is lesser than the static CT time (S_t < τ_CT), the dynamic rate (inclusion of fluctuation) is higher than the static CT rate (absence of fluctuation), and if S_t > τ_CT, the dynamic CT rate is lesser than the static CT rate. When S_t and τ_CT are comparable, the static and dynamic CT rates are also comparable. The above observations are in agreement with the previous studies.^30,52

As given in Tables 3 and 4, if the orbital splitting rate is larger than the static charge transfer rate (O_R > k_static), then the charge transfer is kinetically favorable due to the formation of large bandwidth, and if O_R < k_static, the charge carrier is potentially trap on the localized site. In the latter case the energy of the carrier is not enough for drift from the trapped site. Based on eqn (15), the dispersed energy difference ratio is calculated and the plot of with respect to time is shown in Fig. 8 and S2.† As expected is minimum at time t = S_t. That is, at t = S_t the possible dispersed electronic states are closer to each other which enhances the delocalization of the charge carrier on the nearby molecules and rate of charge transfer. The above results show that the disorder drift time S_t is acting as the crossover point between the delocalized band transport and localized hopping transport. At this crossover point, the probability of the charge carrier is equally distributed on the nearby molecules and the diffusion process is limited.^19,24,28,52 Among the studied molecules, F and Cl substituted Ph4DP molecules are having good hole and electron transporting ability (see Table 3).

3.5. Absorption spectra

The absorption spectra of studied molecules are calculated using TD-DFT method at CAM-B3LYP/6-31G(d,p) and M06-2X/6-31G(d,p) level of theories in dichloromethane medium. The calculated absorption spectra, oscillator strength and corresponding orbital transitions are summarized in Table 5. To study the nature and the energy of the singlet–singlet electronic transition, the first three low lying electronic transitions energies are calculated. The absorption intensity is directly related with the dimensionless oscillator strength value and the dominant absorption bands are the transitions with higher oscillator strength. The experimental absorption wavelength is available for all the studied molecules and two absorption peaks around 450–470 nm and around 265–280 nm have been reported. As observed from Table 5 and Fig. 9, the calculated absorption spectra at CAM-B3LYP/6-31G(d,p) and M06-2X/6-31G(d,p) level of theories in dichloromethane medium are comparable and are in agreement with the experimental values.³³ The absorption spectra of the molecules at CAM-B3LYP/6-31G(d,p) method is discussed in detail. The calculated absorption spectra of the studied molecules exhibit two intense peaks. As shown in Fig. 9 for Ph4DP molecules, the first peak is observed around 440 nm and it corresponds to HOMO to LUMO and HOMO to LUMO+1 transitions. The second intense peak is observed at 260 nm and is due to HOMO−1 to LUMO transitions. Similarly, Ph4DTP molecule exhibits two intense peaks. The first peak observed at 470 nm corresponds to excitation of electron from HOMO to LUMO with the maximum oscillator strength value of 1.4. The second band is observed at 265 nm and is associated with the HOMO−1 → LUMO transition. The absorption pattern shows that the unsubstituted Ph4DP and F substituted Ph4DP have similar spectra. The chlorine substituted Ph4DP molecule has intense absorption spectra at 454 nm with the maximum oscillator strength of 1.76 and is agreement with the experimental value.³³ It has been observed that the introduction of Cl in Ph4DP enhances the charge transporting ability as well as intensity of absorption and emission spectra (see Fig. 9 and 10).

Table 5 Experimental absorption wavelength (λ_abs), calculated absorption wavelength, energy, orbital transition and oscillator strength (in a.u) of Ph4DP, F and Cl substituted Ph4DP and Ph4DTP molecules at CAM-B3LYP/6-31G(d,p) and M06-2X/6-31G(d,p) level of theories in dichloromethane medium

Molecule	Exp.^a	Orbital transitions^b	CAM-B3LYP/6-31G(d,p)			M06-2X/6-31G(d,p)
			λabs		f	λabs		f
			(nm)	(eV)		(nm)	(eV)
a Values taken from ref. 28.b H and L represent HOMO and LUMO, respectively.
Ph4DP	457	H → L+1	452	2.74	0.14	455	2.72	0.14
	273	H → L	441	2.81	1.65	441	2.82	1.60
		H−1 → L	260	4.77	0.84	260	4.78	0.84
F-Ph4DP	447	H → L+1	446	2.78	0.13	449	2.76	0.14
	265	H → L	436	2.84	1.67	436	2.85	1.62
		H−1 → L	261	4.75	0.84	260	4.76	0.83
Cl-Ph4DP	467	H → L+1	461	2.69	0.17	464	2.67	0.17
	280	H → L	454	2.73	1.76	454	2.73	1.71
		H−1 → L	266	4.66	1.08	266	4.66	1.07
Ph4DTP	465	H → L	471	2.63	1.42	472	2.63	1.37
	263	H → L+1	424	2.92	0.10	433	2.86	0.09
		H−1 → L	267	4.65	0.86	267	4.65	0.86

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Fig. 9 The absorption spectra of the Ph4DP and Ph4DTP molecules computed at CAM-B3LYP/6-31G(d,p) level of theory in dicholoromethane medium.

Fig. 10 The emission spectra of the Ph4DP and Ph4DTP molecules computed at CAM-B3LYP/6-31G(d,p) level of theory in dicholoromethane medium.

3.6. Emission spectra

The emission spectra of the studied molecules were calculated using time-dependent density functional theory method at CAM-B3LYP/6-31G(d,p) and M06-2X/6-31G(d,p) level of theories. The calculated emission energy and the corresponding oscillator strength of unsubstituted Ph4DP, F and Cl substituted Ph4DP and Ph4DTP molecules are summarized in Table 6. The computed emission spectra at CAM-B3LYP/6-31G(d,p) and M06-2X/6-31G(d,p) level of theories are similar. Using the SWizard program, the emission spectra calculated at CAM-B3LYP/6-31G(d,p) level of theory is plotted and is shown in Fig. 10. The unsubstituted Ph4DP molecule exhibits the intense emission peak at 527 nm due to HOMO → LUMO transition and the corresponding oscillator strength is around 1.6. As observed in Table 6, the substitution of F on Ph4DP molecule does not affect the emission spectra and the Cl substitution red shifts the emission spectra by 15 nm. Notably, the emission spectra of Ph4DTP is red-shifted by 50 nm with respect to Ph4DP, that is, the substitution of sulfur atom decreases the HOMO–LUMO energy gap and the emission energy.

Table 6 Calculated emission wavelength and energy and oscillator strengths (in a.u) for Ph4DP, F and Cl substituted Ph4DP and Ph4DTP molecules corresponding to electronic transition between HOMO and LUMO energy levels at CAM-B3LYP/6-31G(d,p) and M06-2X/6-31G(d,p) level of theories in dichloromethane medium

Molecule	Orbital transitions^a	CAM-B3LYP/6-31G(d,p)			M06-2X/6-31G(d,p)
		λemi		f	λemi		f
		(nm)	(eV)		(nm)	(eV)
a H and L represent HOMO and LUMO, respectively.
Ph4DP	H → L	527	2.35	1.57	526	2.36	1.53
F-Ph4DP	H → L	521	2.38	1.59	520	2.39	1.55
Cl-Ph4DP	H → L	541	2.29	1.68	541	2.29	1.63
Ph4DTP	H → L	577	2.15	1.35	575	2.16	1.30

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4. Conclusions

The effect of structural fluctuation on charge transfer in Ph4DP and Ph4DTP molecules has been studied by using kinetic charge transfer parameters such as rate coefficient, disorder drift time, orbital splitting rate, hopping conductivity and mobility. The calculated hole mobility of 0.04 and 0.03 cm² V⁻¹ s⁻¹ for unsubstituted Ph4DP and Ph4DTP molecules is in agreement with the experimental values. The F and Cl substituted Ph4DP molecules have hole mobility of 0.17 and 0.33 cm² V⁻¹ s⁻¹, and their corresponding hopping conductivity is 0.13 and 0.26 S cm⁻¹, respectively. Relatively larger orbital splitting rate in substituent Ph4DP enhances the bandwidth and delocalization of the charge carrier, which facilitate the charge transfer. It has been observed that the molecules with high orbital splitting rate and less disorder drift time possess good charge transport. Theoretical results show that if the orbital splitting rate is lesser than the static charge transfer rate, the charge carrier is potentially trapped on the localized site. The calculated absorption spectra of the studied Ph4DP and Ph4DTP molecules are in agreement with experimental results.

Acknowledgements

The authors thank the Department of Science and Technology (DST), India for awarding research project under Fast Track Scheme.

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Footnote

† Electronic supplementary information (ESI) available: The optimized structure of Ph4DP and Ph4DTP molecules are given in Fig. S1. Mean squared displacement, survival probability, time dependence of rate coefficient, disorder drift with in time scale of simulation and time evolution of dispersal energy difference ratio for hole and electron transport in studied molecules is given in Fig. S2. The calculated geometrical parameters of the studied molecules are summarized in Table S1. Calculated effective charge transfer integral (J_eff, in eV) for hole and electron transport in Ph4DP and Ph4DTP molecules is summarized in Table S3. See DOI: 10.1039/c4ra15779f

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