Charge Carrier Mobilities and Singlet Fission Dynamics in

Article pubs.acs.org/JPCC

Cite This: J. Phys. Chem. C XXXX, XXX, XXX-XXX

Charge Carrier Mobilities and Singlet Fission Dynamics in Thienoquinoidal Compounds ZhiYe Zhu,† Hang Zang,† Yi Zhao,*,† and WanZhen Liang*,‡ †

Department of Chemical Physics, University of Science and Technology of China, Hefei 230026, China State Key Laboratory of Physical Chemistry of Solid Surfaces, Collaborative Innovation Center of Chemistry for Energy Materials, and Department of Chemistry, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, China

‡

S Supporting Information *

ABSTRACT: The charge carrier mobilities and singlet fission (SF) dynamics in newly synthesized three thienoquinoidal compounds (ThBF, TThBF, and BThBF) are theoretically characterized by combining a time-dependent wavepacket diffusion method and electronic structure calculations. It is found that all three compounds have quite large electron and hole mobilities and that ThBF possesses the property of p-type semiconductor, whereas TThBF and BThBF show the behavior of n-type semiconductor. For the SF efficiency, TThBF should be the best candidate among the three molecular crystals due to a strong electronic coupling. The obvious correlation between charge carrier mobilities and the SF dynamics is not found; however, the expansion of exciton wave function in the aggregates of these compounds can accelerate the SF process, consistent with other experimental and theoretical observations.

1. INTRODUCTION The ever increasing energy demand and deleterious environmental effects of nonrenewable fossil fuels has led to a search for alternative energy sources. Solar energy is regarded as a viable alternative choice because of the abundance of solar radiation reaching the Earth’s surface. Organic photovoltaic (OPV) devices using sunlight have been a hot topic in trials during the few last decades, endowed with good properties such as large area, low cost, lightweight, and mechanical flexibility.1−8 Despite these benefits, the power conversion efficiency (PCE) of OPV devices needs to be further improved for practical applications. Through the synthesis of low-bandgap polymers with high charge-carrier mobilities and the posttreatment for the accurate modification of active-layer morphology, the PCE has greatly risen.9,10 However, in single-junction solar cells the Shockley−Queisser limit11 places a maximal efficiency because of unavoidable losses,12 such as incomplete absorption, thermodynamic loss, and radiative recombination. Several ways of reducing losses have been proposed.13−16 One of them is singlet fission (SF), which converts one singlet exciton into two triplet excitons through an intermediate state of two entangled triplets on two neighboring molecules to produce two electron−hole pairs that are both capable of charge separation. SF was first invoked in 1965 to explain observations made on crystalline anthracene17 and then was firmly established in crystalline tetracene.18−20 Although SF was later found in carotenoids21 and conjugated polymers,22 interest gradually abated until Nozik et al.23 pointed out its potential utility for increasing the efficiency of solar cells. The authoritative reviews effectively summarize the state of the field up to 2013.15,16 © XXXX American Chemical Society

Since then, more and more monomeric and polymeric SF materials based on polycylic aromatic compounds have come into vision, and some nonpolycylic aromatic SF materials based on biradicaloid character have been reported, too.24−30 Meanwhile, many theoretical papers appeared to provide a deep understanding of SF process.31−36 To exhibit SF character, two conditions are required for the molecules. One is the energy condition, E(S1), E(T2) ≥ 2E(T1); i.e., both the lowest energy of excited singlet state (S1) and the second excited triplet state (T2) have to be equal to or higher than twice the energy of the lowest triplet state (T1). The other is the morphology condition,37−41 which requires a sufficiently appropriate distance and packing style between the neighboring molecules or chromophores. In order to apply the SF molecules to the practical OPV devices, some additional conditions should be satisfied, i.e., broad and intense light absorption, high charge carrier mobility, larger SF rate than other relaxation rates, appropriate energy levels of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), and good chemical stability.42−45 Recently, three nonpolycyclic aromatic thienoquinoidal compounds (ThQs), 2,5-bis(fluorene-9-ylidene)-2,5-dihydrothiophene (ThBF), 2,5-bis(fluorene-9-ylidene)-2,5dihydrothieno[3,2-b]-thiophene (TThBF), and 5,5-bis(fluorene-9-ylidene)-5,5′-dihydro-2,2′-bithiophene (BThBF), have been synthesized as the potential SF candidates.46 These compounds are based on a thienoquirnoid-biradical resonance, Received: April 15, 2017 Revised: October 1, 2017 Published: October 2, 2017 A

DOI: 10.1021/acs.jpcc.7b03573 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

Figure 1. Scheme of states and electronic couplings involved in SF process in a dimer AB model. A*B and AB* are initial exciton states; A+B− and A−B+ are charge transfer (CT) states; and ATBT is the triplet−triplet TT state.

thus the values of E(T1) can be lowered to satisfy SF condition. Furthermore, the ThQs are able to form thermally stable films for the design of OPV devices. In spite of these benefits, their PCEs are quite low. To unveil the insights, the detailed structure−function relationships, especially carrier mobilities and SF dynamics, should be known. Here, we theoretically investigated the ThQs by combining a time-dependent wavepacket diffusion (TDWPD) method47−49 for the description of carrier dynamics and quantum-chemistry methods for the construction of an effective model Hamiltonian. We will show that ThQs have high electron and hole mobilities, manifesting good organic semiconductors, and that the TThBF should be the best candidate for a high SF yield among three structures. The possible mechanism is further discussed. The paper is organized as follows. Section 2 outlines the theoretical approaches and computational details. Section 3 shows the calculated results for the construction of model Hamiltonian and for the carrier mobilities and SF dynamics, respectively. A concluding remark is given in Section 4.

for the organic aggregate with N monomers, where e ii represents the energy of monomer i and eij,i≠j is the electronic coupling between monomers i and j. To correctly describe the SF dynamics, however, more electronic states should be involved. It is generally accepted that there is a direct pathway from singlet exciton (SE) states to triplet−triplet (TT) states and a superexchange pathway via charge transfer (CT) states.52−54 For instance, in a dimer AB model, these states include SE states A*B and AB*, CT states A+B− and A−B+, and a TT state ATBT. The schematic graph of these states with orbital distributions as well as the electronic couplings among them are shown in Figure 1. For N monomers, He involves N SE states, 2(N − 1) CT states, and (N − 1) TT states, and it has a form He = H SE + H CT + HTT + H int SE

where H , H , and H N

H SE =

2. THEORETICAL AND COMPUTATIONAL METHODS It is known that charge carrier dynamics plays an important role in the determination of OPV device efficiency. Commonly, the hopping model is used to describe the carrier dynamics in organic semiconductors, and the coherent band-like model is adopted in conventional inorganic crystals. In perfect organic crystals, however, neither a single band-like model nor the hopping type model alone can correctly predict charge carrier mobility.50,51 Recently, the TDWPD method47−49 has been proposed to describe the carrier dynamics in organic materials. It unifies these two models with the incorporation of the coherence and tunneling effects and is suitable for the present study. To describe carrier dynamics, we first have to construct the Hamiltonian which can be explicitly written as H = He + Hph + He − ph (1)

(3)

read

∑ EnSEBn†Bn + ∑ (JnSE,n+ 1Bn†Bn+ 1 + h. c. ) n=1

(4)

N−1

H CT =

∑ (EnCT,n+ 1Cn†,n+ 1Cn,n+ 1 + EnCT+ 1,nCn†+ 1,nCn+ 1,n) n=1 N−1

+

∑ (JnCT,n+ 1Cn†,n+ 1Cn,n+ 1 + h. c. ) n=1 N−2

+

∑ (JnCT+ 1,nCn†+ 1,nCn+ 1,n+ 2 + h. c. ) n=1

N TT

H

=

∑

(5)

N−2 † EnTT , n + 1Dn , n + 1Dn , n + 1

n=1

+ h. c. )

+

∑ (JnTT,n+ 1Dn†,n+ 1Dn+ 1,n+ 2 n=1

(6)

Here, the creation operator B†n generates a SE state at the n-th monomer; the creation operator C†n,n+1 creates an electron and a hole residing at the n-th and (n + 1)-th monomer, respectively; the creation operator D†n,n+1 spans two triplet states over the nth and (n + 1)-th monomer; and Bn, Cn,n+1, and Dn,n+1 are the corresponding annihilation operators. In the above model, only the exciton coupling between two adjacent molecules is considered, which is typical in organic aggregates. Generally, the CT−CT coupling JCT is negligible because it requires the exchange of two particles. Compared to SE−SE couplings JSE,

N

∑ |i⟩eij⟨j| i,j

TT

N−1

n=1

where He is the pure electronic Hamiltonian, and Hph and He−ph describe vibrational or phonon motions and carrier−phonon interactions, respectively. For carrier mobility calculations, the expression of He is quite simple, and it is written as He =

CT

(2) B


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Figure 2. Optimized molecular structures and the electronic distributions of HOMO and LUMO orbitals for (a) ThBF, (b) TThBF, and (c) BThBF monomers. Isovalue = 0.02.

TT−TT couplings JTT are quite small because multistep processes are involved even in the nearest-neighboring TT energy transfer. The Hamiltonian Hint represents the interactions among different-type exciton states

⟨q2(t )⟩ =

H

=

∑

+

† V nSE+−1,CT n Bn + 1Cn + 1, n

+ h. c. )

∑ (VnTT, n+−1CTDn†, n+ 1Cn, n+ 1 + VnTT+ 1,−nCTDn†, n+ 1Cn+ 1, n + h. c. ) n=1 N−1

+

† SE − TT † ∑ (V nSE, n−+TT 1 Bn Dn , n + 1 + V n + 1, n Bn + 1Dn , n + 1 + h. c. ) n=1

3. RESULTS AND DISCUSSION 3.1. Molecular Geometries and Electronic-State Energies. The initial geometries of ThBF, TThBF, and BThBF are taken from their crystal structures,46 and they are optimized at the theoretical level of B3LYP/6-311G** without symmetric constraints. The vibrational frequency calculations are further performed to ensure the absence of vibrational instabilities in optimized structures. Figure 2 displays the optimized structures for these three compounds. Compared with the unoptimized structures, the optimized ones do not change much, and they still exhibit planar skeleton structures. Generally, the molecule with a planar structure easily possesses a π-conjugate property, which is actually verified by the charge distribution of HOMOs and LUMOs shown in Figure 2. The HOMO and LUMO phases of TThBF are obviously different from those of the other two, which may result from the parity of double bond number in the bridge fragment. TThBF has 4 (even) double bonds, whereas ThBF and BThBF have 3 and 5 (odd) double bonds. At the optimized geometries, we calculate the vertical excitation energies of first singlet excited states (S1) and show the results in Table 1. S1 mainly comes from electronic transition of HOMO → LUMO, and the vertical excitation

⎛ p2 ⎞ 1 2 nj 0 2⎟ ⎜ + ω ( x − | n ⟩ x ⟨ n | ) ∑∑⎜ nj nj nj ⎟ 2 2 n=1 j=1 ⎝ ⎠ N

(9)

In realistic applications, electronic structure calculations are performed to construct the effective model Hamiltonian. The locally modified Q-Chem program package55 is adopted to calculate the energies of diabatic electronic states and the electronic couplings between these states.

(7)

where VSE‑TT n,n+1 represents direct interactions between the SE and TT states and VSE−CT and VTT−CT are the indirect supern,n+1 n,n+1 exchange interactions between SE and CT states and interactions between TT and CT states, respectively. The Hamiltonians Hph and He−ph for vibrational motions and carrier−phonon can be obviously written as Hph + He − ph =

n

where r is the distance between two adjacent sites. The carrier mobility can be estimated by the Einstein relationship e μ= D kBT (10)

n=1 N−1

+

∑ ⟨n|n2ρ(t )|n⟩r 2 = ∑ ρnn (t )n2r 2 n

† (V nSE, n−+CT 1 Bn Cn , n + 1

where d is the dimension of the system and

⟨q (t)⟩ is the mean squared displacement of carrier 2

N−1 int

⟨q2(t )⟩ , t →∞ 2dt

D = lim

Nph

(8)

where Nph is the number of phonons; ωnj is the frequency of jth mode on site n; and x0nj is the corresponding displacement from equilibrium geometry and \is related to the coefficient cnj ⎛ π ⎞ cnj2 cnj of spectral density Jn (ω)⎜ = 2 ∑j ω δ(ω − ωnj)⎟ by xnj0 = 2 . ωnj ⎝ ⎠ nj The corresponding reorganization energy is given by 1 λn = ∑j 2 ωnj2(xnj0 )2 . Once the effective Hamiltonian is constructed, the carrier dynamics can be obtained by solving the time-dependent stochastic Schrödinger equation in the TDWPD method (the corresponding formulation is outlined in the Supporting Information (SI)). In the site representation, the carrier wave function can be written as |ψ(t)⟩ = ∑Nn cn(t)|n⟩. The population dynamics ρnn(t) is thus straightforwardly predicted by ρnn(t) = ⟨c*n (t)cn(t)⟩, and the diffusion coefficient D is calculated from C


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in Figure 3(a) dominantly contribute to the superexchange channel in the SF process (3 different geometries in fact due to the symmetry). We thus calculate the energies of CT states constructed from these 6 dimers, and the results are listed in Table 2. In CDFT calculations, the dimer’s geometries are directly taken from molecular crystals without geometric optimization. For TThBF and BThBF monomers, their ground-state dipole moments are zero, while for ThBF, it is not. This difference leads to the quite different CT-state energies of A+B− and A−B+ for dimer 1,2, indicating that CTstate energies of ThBF are more easily affected by the polarizable environment. Interestingly, those CT-state energies are not far from the corresponding adiabatic excitation energies of localized excited states. It is predicated that the SF processes of ThQs are significantly mediated by CT states. 3.2. Electronic Couplings. The electronic couplings between these diabatic states are directly calculated as ⟨Φi| H|Φj⟩,59,60 where Φi are the wanvefunctions of diabatic states. In the construction of diabatic states of the dimer, we employ the four-electron four-orbital model,15,16 which only allows four electrons in two HOMOs of two monomers to transit to their LUMOs, and all other core electrons are kept freezed. At first, the dimer is partitioned into the fragments, each of which represents a single molecule. Then a constrained locally projected SCF method is applied to get the MOs which are totally projected on a single molecule.61 Finally, with respect to these MOs, we build the diabatic states and calculate their couplings. We have implemented this scheme into a locally modified Q-chem software. Table 3 lists the calculated coupling values based on the four-electron, four-orbital model at the HF/6-311G** level in the gas phase. In all dimer arrangements, it is found that dimer 1,2 possess the much stronger direct couplings and indirect SF couplings via CT states than the other dimers, suggesting that SF may mainly occur along this πstacking direction. To further compare the direct and indirect pathways for the SF process, we calculate the effective couplings of indirect pathway of dimers 1 and 2 according to the superexchange mechanism,34 and the obtained values are 6.35, 26.57, and 27.14 meV for ThBF, TThBF, and BThBF, respectively. These couplings are obviously larger than the direct couplings, manifesting that the indirect pathway may be more important than the direct one. To demonstrate the computational accuracy of electronic couplings, we calculate VSE−CT and VCT−TT based on the oneelectron orbital approximation approach,62−66 which only considers HOMO−HOMO and LUMO−LUMO couplings. The electronic couplings VSE−SE between the localized excited states have also been calculated as the Coulomb interaction between the transition densities of interacting moieties by

Table 1. Vertical and Adiabatic Excitation Energies (in eV) of S1 and T1 for the Monomers of ThQs by (TD-)B3LYP/ PCMa

a

monomer

E(S1)ver

E(S1)adi

E(T1)adi

ThBF TThBF BThBF

2.2182(1.39) 2.0152(1.93) 1.7488(2.02)

2.0026 1.8311 1.5848

0.7808 0.5728 0.4183

The data in parentheses are the corresponding oscillator strengths.

energies follow the order: ThBF > TThBF > BThBF. Without exception, the excitation energies are very much dependent on the bridge’s conjugate lengths, and they decrease with the increase of conjugate lengths, agreeing well with an experimentally observed absorption behavior.46 To correctly describe SF dynamics, it is more appropriate to adopt the adiabatic excitation energies of related electronic states. We thus optimize the geometries of S1 and T1 for ThQs. The optimized geometries do not have dramatic changes and still keep planar skeletons like the S0 geometries. The calculated adiabatic excitation energies, listed in Table 1, follow the same order as the vertical excitation energies. The energies of triplet states are smaller than those of the first singlet excited states; especially the adiabatic energies satisfy the SF condition, i.e., E(S1)adi ≥ 2E(T1)adi. The energies of singlet and triplet excited states are approximated as ES1S0 = ES0S1 ≃ ES1 + δ1 and ETT ≃ 2ET1 + δ2, respectively, where ES1 and ET1 are the calculated adiabatic excitation energies of singlet and triplet states of an isolated monomer at the theoretical level of (TD-)B3LYP/6-311G**, where δi covers the interaction between two monomers and the polarization effect of the surrounding molecules. Here the latter effect is treated by the polarizable continuum model (PCM), integral-equation formalism PCM (IEFPCM).56,57 The dielectric and optical dielectric constants are set to be 3 and 2. The energies of CT states for the dimers are calculated by the constrained DFT (CDFT) approach.58 The polarization effect of the surrounding molecules on CT states is covered by the nonequilibrium PCM model. Table 2 lists the calculated Table 2. Energies (in eV) of CT States of Different Dimers of ThQs in the Gas Phase and Aggregates in aggregate molecule ThBF

TThBF

BThBF

in gas phase

dimer

EA+B−

EA−B+

EA+B−

EA−B+

1,2 3,4 5,6 1,2 3,4 5,6 1,2 3,4 5,6

2.7922 3.5044 3.5312 1.8510 2.6886 2.6885 1.8704 2.7082 2.7083

2.2821 2.2858 2.2659 1.8510 1.8615 1.8616 1.8704 1.8842 1.8841

2.4110 4.3707 3.9161 2.3594 3.9201 3.9111 2.2640 3.3578 3.3695

2.3071 2.6513 2.5899 2.4453 2.1027 2.0712 1.9368 1.9910 2.0137

∫ d r ⃗ ∫ d r ′⃗ ρAT *( r ⃗)( |r −1 r ′| + gXC( r ⃗ , r ′⃗ ))ρBT ( r ′⃗ ), where gXC is

T indicates the the exchange-correlation kernel and ρA/B transition density for the excited state of monomer A/B.67 With those single-electron approximation and transition density-based methods, a series of DFT XC functionals are adopted. Table S1 lists the electronic couplings of ThBF dimers calculated from different approaches, and it shows that all the results are comparable although HF slightly overestimates the coupling of VSE−SE compared with DFT results. Therefore, it is expected that the electronic couplings listed in Table 3 can be safely used to simulate the SF dynamics. It is noted that although the orbital phases of TThBF are different from those of ThBF and BThBF the magnitudes of

energies of CT states for the dimers in the gas phase and molecular aggregates. The energies of CT states are quite different for the face and edge stacked pairs, indicating that ECT is sensitive to the dimer arrangement. Figure 3(a) and (b) displays the arrangement of BThBF dimers (the other ThQs have very similar arrangement with BThBF). There are 12 dimer geometries around a monomer. However, 6 among them D


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Figure 3. Crystal structures and the selected charge hopping pathways for BThBF: (a) from view along x axis and (b) between different layers.

Table 3. Calculated Couplings of Different ThBF, TThBF, and BThBF Dimers at the HF/6-311G** Level Based on a FourElectron, Four-Orbital Model in the Gas Phasea molecule ThBF

TThBF

BThBF

a

dimer

VS0S1−CA

VS0S1−AC

VS1S0−CA

VS1S0−AC

VAC−TT

VCA−TT

VSE−TT

VSE−SE

1,2 3,4 5,6 7 1,2 3,4 5,6 1,2 3,4 5,6

−61.69 11.15 16.68 0.42 36.64 18.04 18.52 −96.59 −8.58 −8.93

59.53 7.05 6.89 −0.90 144.40 −6.72 −6.63 −161.71 −6.01 −5.98

49.79 8.77 5.62 −0.90 144.47 −6.63 −6.70 −162.14 −5.97 −6.00

−61.24 9.29 17.14 0.42 36.73 18.51 18.04 −97.05 −8.97 −8.56

−67.52 18.40 −7.75 0.74 112.71 −10.19 −12.51 80.61 6.14 −8.15

55.11 −1.22 16.70 0.74 113.12 −12.56 −10.21 81.80 −8.30 6.23

−0.83 −0.01 −0.003 0.001 −2.05 0.001 0.001 2.60 0.001 −0.002

157.49 −101.56 −91.78 22.59 −310.88 140.21 140.25 −230.91 −95.51 −95.49

CA and AC are short for CT states A+B− or A−B+, respectively. Units: meV.

couplings do not show the phase dependence except the signs. Indeed, the model of four-electron and four-orbital predicts that the phase change of orbital wave function only affects the sign of the coupling but not the absolute value. This can be confirmed by values in Table 3. Renaud demonstrates that intermolecular vibrational modes can speed up SF in perylenediimide crystals with some energy and coupling constraints.68 We find that the couplings remarkably depend on the arrangement of two monomers as shown in Figure S2 for TThBF. Around the crystal stable geometry, however, the electronic couplings show very weak variations, and the oneelectron couplings stay relatively large. In addition to these conditions, the large energy difference between the singlet state and the double triplet state manifests that the Condon approximation should be suitable in dynamic simulations in the present work. Similar to SF, charge transfer has 12 different pathways as shown in Figure 3(a) and (b). We calculate the couplings along those pathways for electron transfer (Ve) and for hole transfer (Vh) by using the one-electron approximation model, and the results are listed in Table 4. We also show the intermolecular mass−center distances in this table for the mobility calculations. For all ThQs, the most prominent pathways for charge transfer are still routes 1 and 2, which are through a displaced π-stacking configurations. This indicates that 1 and 2 directions might be the dominant conducting channels. 3.3. Reorganization Energies. The reorganization energy contains the inner part induced by intramolecular vibrations and the outer part caused by the polarization of the surrounding medium. For organic solids, the contribution from electronic polarization of surrounding molecules is quite small (typically lower than 0.01 eV). It can be safely

Table 4. Intermolecular Mass−Center Distances (d) and Electronic Couplings (V) of Hole and Electron Transfer for All Pathways of ThQsa molecule ThBF

TThBF

BThBF

a

pathway

d (Å)

Ve

Vh

1,2 3,6 4,5 7 8 9 10 11 12 1,2 3,4,5,6 7,8 9,10 11,12 1,2 3,4,5,6 7,8 9,10 11,12

5.412 10.810 11.001 12.791 13.937 9.243 19.738 15.726 10.514 5.405 11.600 9.334 14.282 10.786 5.420 12.362 9.909 12.528 9.909

34.54 5.67 5.85 2.52 0.15 3.02 0.21 0.90 2.47 100.40 7.19 6.37 2.84 1.54 103.01 5.43 2.77 1.88 1.51

30.37 8.45 13.43 0.70 0.09 0.83 0.08 0.42 0.54 30.42 13.62 3.39 2.41 0.76 50.42 7.85 2.33 0.85 0.44

Units: meV.

neglected,69 so the calculations are performed in the gas phase. For the calculation of inner reorganization energy, we adopt the vertical derivative method on the basis of normalmode analysis (NMA).70 For instance, to calculate the reorganization energy of the electron transfer process, we first optimize a neutral monomer A and get its vibrational E


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λ e{h} are fitted to have a Lorentzian line shape 34 λiωγ 1 J(ω) = π ∑i with a uniform broadening parameter (ω − ω )2 + γ 2

frequencies ωj on the ground state S0 and then calculate the gradient of A− at the optimized geometry of A, and finally project out the normal-mode coordinate shifts x0j in the displaced harmonic oscillator model. The reorganization energy on the monomer A for electron transfer is then given by 1 λ = ∑ λi = ∑ 2 ωi2(xi0)2 . Table 5 lists the reorganization energies of monomers from neutral S0 to cation state S+0 for hole transfer, to anion state S−0

i

γ = 40 meV to smoothly generate fluctuation energies. Since the couplings along the direction of dimers 1 and 2 are obviously larger than those along the other directions, the onedimensional model should be enough to predict mobilities. In order to incorporate the band-like effect, we use 10 monomers and find that convergent results can be obtained. Figure 5

Table 5. Reorganization Energies of Monomersa molecule

S0 → S+0

S0 → S−0

S0 → S1

ThBF TThBF BThBF

0.1002 0.1038 0.0906

0.1803 0.1797 0.1730

0.1836 0.1841 0.1276

Cation and anion structure are labeled as S+0 and S−0 respectively. Units: eV.

a

for electron transfer, and to neutral S1 for exciton transfer, respectively. The reorganization energies of ThQs for electron and exciton transfer are very similar but obviously larger than those for hole transfer. The reorganization energies become smaller for the monomer with more conjugation units except for the TThBF molecule. To further reveal detailed information, we exhibit the mode-specific reorganization energies of ThQs for electron, hole, and exciton transfer in Figure 4. The main contribution to the reorganization energy comes from the mode with 1484 cm−1 (BThBF) and 1543 cm−1 (ThBF), which belong to the stretching motions of backbone. However, for TThBF, a low-frequency mode of 45 cm−1 makes a considerable contribution to the reorganization of λS0 → S1 and λe, making them larger than those of ThBF and BThBF. This mode shows a large vibrational magnitude of the central part of the molecule along the molecule plane. A further analysis shows that the electron density difference71 between the cation state or S1 and the ground state is localized at the central part of the molecule (see Figure S1 in SI), consistent with the behavior of the reorganization energy. Therefore, one may change the motion of stretching mode of backbone or the vibration of the molecule plane to control the magnitudes of reorganization energies. 3.4. Electron and Hole Mobilities. Charge carrier mobility in materials is one of the important characters for the description of the functional property. With the calculated structural parameters, it is easy to use the TDWPD method to get the mobilities. In the concrete performance, we use the elements eii = 0, eij = Ve{h}(i ≠ j) in the pure electronic Hamiltonian. The spectral densities of reorganization energies

Figure 5. ⟨q2(t)⟩ relative to time (t) of pathways 1,2 in the Y−Z plane for ThQs at a temperature of 300 K by the TDWPD method. Mobilities can be calculated from eq 10.

displays ⟨q2(t)⟩ for electron and hole transfer, respectively, along the dimer 1 pathway at a temperature of 300 K. It is observed that the ⟨q2(t)⟩ shows a good linear relationship with time, and diffusion coefficients D are thus well-defined and easily obtained from this linear dependence. Table 6 lists the mobilities of electron and hole transfer obtained from D at 300 and 1000 K. For all ThQs, the Table 6. Simulated Drift Mobility Values at the Temperature of 300/1000 K for ThQ Compounds Based on Equation 10a μh

a

μe

molecule

pathway

300 K

1000 K

300 K

1000 K

ThBF TThBF BThBF

1,2 1,2 1,2

13.25 12.51 40.37

3.17 3.01 11.62

9.03 56.25 69.48

2.11 16.30 18.05

Units: cm2 ·V−1·S−1.

mobilities decrease with increase of temperature, opposite to the behavior from the hopping model. The possible reason may be from both the carrier coherent motion and the quantum effect of vibrational-mode motions with high frequencies.50,72,73 Interestingly, the electron mobilities are proportional to the conjugate length of ThQs, whereas the hole mobilities do not

Figure 4. Contribution of the vibrational modes to the reorganization energy λh, λe, and λS0 → S1 of ThQs. F


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Figure 6. Singlet fission processes in dimers (a) and aggregates (b) along path 1 direction for ThQs.

In the simulations, we use two models, a dimer model and a one-dimensional chain model with 10 monomers, to investigate detailed SF dynamics and the effect of aggregation. In the aggregate model, the SE population is initially at the central monomer. The results show that the convergent population dynamics can be obtained in the chain model because most of the exciton wave function is located within the aggregate chain during the SF process. Figure 6 exhibits the SF population evolution for ThBF, TThBF, and BThBF, respectively, with the dimer and aggregate models. For all ThQs, the SF processes are quite similar except the SF rates. Initially, the SE population rapidly decays; the transient CT population is generated during the SF process; and finally, all the population is distributed on TT states. Due to the populated CT states, the CT states cannot be only explained as virtual states, and the mechanism of step by step via CT states should also play a rule. It has been discussed36 that the increase of TT state population should be exponential once the CT states are occupied. Indeed, the population of TT states can be fitted by an exponential function P(t) = P0 + Ae−t/τ, where the inverse of τ naturally corresponds to the SF rate, i.e., k = 1/τ. Table 7 lists the SF rates of ThQs from both the dimer and aggregate models. It is seen that the SF rates from the aggregate model are about 5 times larger than the corresponding rates from the dimer model, manifesting that the expansion of exciton wave function helps the SF process in the three ThQ compounds. A similar phenomenon has been observed from other organic aggregates.75−79 In ThQs, the SF rate in the TThBF crystal is larger than those in the ThBF and BThBF crystals especially in aggregates.

have an obvious dependence. This property leads to very different behaviors of materials: the ThBF belongs to a p-type semiconductor, whereas TThBF and BThBF are n-type semiconductors because of larger electron mobilities. It is also noted that the magnitudes of hole and electron mobilities for these compounds are obviously larger than those of conventional organic semiconductors. As the static disorders of electronic-state energies are incorporated in the realistic crystals, these mobilities should become small. The present calculations thus present the upper limits of mobilities. 3.5. Singlet Fission. The SF dynamics is more complicated than the electron and hole transfer because more electronic states are involved. To catch the dominant property of the SF process, we still use the one-dimensional model along the dimer 1 direction, as adopted in the description of charge mobility. The adiabatic energies of SE, CT, and TT states, listed in Tables 1 and 2, and the electronic coupling values shown in Table 3 are employed to form the effective model Hamiltonian. To account for the polarization effect of environmental molecules on the electronic couplings, in the simulation of SF dynamics, we scale all the calculated electronic coupling values for the dimers in the gas phase as shown in Table 3 by a screening factor f = 0.65, which was introduced to cover the environmental screening effect on Frenkel exciton− exciton coupling by Megow et al.74 Here all the calculated electronic couplings are scaled by this factor. The spectral density with Lorentzian line shape from the reorganization of the exciton state34 is used to generate fluctuation energies for all electronic states. T = 300 K is set. G


Article

The Journal of Physical Chemistry C Notes

Table 7. Simulated SF Rate k for ThQs in the Dimer and Aggregate Modela ThBF

a

TThBF

The authors declare no competing financial interest.

■

BThBF

model

dimer

aggregate

dimer

aggregate

dimer

aggregate

k

0.45

3.70

1.29

6.25

0.11

0.91

ACKNOWLEDGMENTS The author thanks So Kawata and Yong-Jin Pu for offering the crystal structures. This work is partially supported by NSFC (Grants 21290193, 21373163, 21573177).

Units: ps−1.

■

We notice that there are no obvious parameter differences among ThQs except the exciton−exciton couplings and modespecific reorganization energies. As we use the reorganization energies from CT states in numerical simulations, the qualitative results of SF dynamics do not change for ThQ compounds. It is thus expected that the maximum SF rate in the TThBF may come from the strongest exciton−exciton coupling which can accelerate the expansion of exciton wave function to increase the SF rate.36 To confirm the mechanism, we set the exciton−exciton coupling in TThBF to −200 meV, close to the value in BThBF, and we obtain a similar SF rate to that in BThBF.

4. CONCLUDING REMARKS By combining the time-dependent wavepacket diffusion method for carrier dynamics and electronic structure calculations for the construction of the effective Hamiltonian, we have investigated charge carrier mobilities and singlet fission dynamics in ThBF, TThBF, and BThBF compounds, newly synthesized organic crystals. The calculated results demonstrate that ThBF is a p-type semiconductor, whereas TThBF and BThBF are n-type semiconductors, although these compounds have similar structures. Furthermore, TThBF should be the best potential candidate for singlet fission in those compounds, originated in the large exciton−exciton coupling to accelerate the exciton expansion and SF process. On the basis of these detailed calculations, a conclusion can be drawn that thienoquinoidal-based compounds may have wide application prospects as promising organic semiconductor materials and as potential singlet fission materials in organic photovoltaic devices.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b03573. Detailed formula for the TDWPD method and expressions for electronic couplings. Electron density difference between the cation state, the first singlet excited state, and the ground state. Electronic couplings of ThBF dimers calculated by different approaches (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Phone: +86-592-2184300. *E-mail: [email protected]. Phone: +86-592-2189197. ORCID

ZhiYe Zhu: 0000-0001-9479-9498 Hang Zang: 0000-0002-1797-6857 Yi Zhao: 0000-0003-1711-4250 WanZhen Liang: 0000-0002-5931-2901 H


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