Unraveling the charge transfer variation of tetrathiafulvalene-based organic crystals through fragment charge difference calculation

Tetrathiafulvalene (TTF) is a well-known electron donor. It contains 14 π electrons in two dithiolylidene rings of TTF.^1,2 TTF and its derivatives can be semiconducting, redox-active, and even superconducting (by forming organic salts). The crystalline forms of TTF-based compounds are often assembled by weak interactions, such as van der Waals force.³ These organic crystals inherit the above-mentioned properties and bridge to a range of cutting-edge applications,^3–6 such as organic electronics,^2,7,8 smart materials,^9,10 and superconductors.^11,12

Since the first report from Wudl et al.¹³ about TTF, the charge transfer characteristics of TTF-based compounds have been widely investigated both experimentally and theoretically.^1,3,14 Chemical modification is a common method to alter the charge transfer properties of TTF-based crystals.² For instance, hexamethylene-tetrathiafulvalene (HM-TTF) achieves the highest mobility among the measured TTF derivative crystals (11.2 cm² V⁻¹ s⁻¹ by Takahashi et al.¹⁵ and 3.6 cm² V⁻¹ s⁻¹ by Kanno et al.¹⁶), but bis(ethylenedithio)-tetrathiafulvalene (BEDT-TTF) exhibits low carrier mobility of ∼0.005 cm² V⁻¹ s⁻¹.¹⁷ Meanwhile, organic crystals with the same chemistry but different phases (a.k.a. polymorphs) are also known to exhibit various charge transfer properties.^18–21 In other words, the charge transfer properties of organic crystals depend not only on the delocalized electron distribution at molecules but also on the packing of molecules. Taking TTF polymorphic crystals as an example, the experimentally obtained mobility of α-TTF (denoted as 1) can be up to 1.2 cm² V⁻¹ s⁻¹. The highest mobility of β-TTF (denoted as 2) is, however, only about 0.23 cm² V⁻¹ s⁻¹.²² 

Weak interactions between molecules make electrons mainly localize at individual molecules in organic crystals. For investigating charge transfer properties of the weak interaction systems, the hopping transport theory and the semiclassical Marcus theory^3,14,23 are extensively employed. The estimated hopping rate of carriers, k, between adjacent molecules is shown as follows:

where V is the transfer integral (transfer integral of hole transport in this study), λ is the reorganization energy, ℏ is reduced Planck constant, and k_B and T are Boltzmann constant and temperature. Herein, the free energy difference between the initial and final states is zero by only considering the self-exchange reaction.²⁴ 

At a preliminary stage, the transfer integral can be estimated by an Energy-Splitting-in-Dimer (ESD) method,^25,26

where V_ESD is the transfer integral of a dimer, and E₁ and E₂ are the energies of the highest occupied molecular orbital (HOMO); HOMO-1 for holes transport [or the lowest unoccupied molecular orbital (LUMO); LUMO+1 for electrons transport]. The method can reasonably predict symmetric dimers by assuming that the excess charge is uniformly distributed over both sites. The ESD method can, however, lead to overestimation due to the nonzero site energy difference in asymmetric dimer cases.²⁷ As an improvement, site energy correction (SEC) methods have been widely employed to calculate the transfer integral based on dimers,^27–30 

where Ψ_A and Ψ_B are the orbital wave functions of quantum states A and B, and $Ĥ$ is a Hamiltonian operator. The expression of $Ĥ$ can be described as follows:

where $am+$ and a_n are creation and annihilation operators, and ϵ_m and V_mn are site energy and transfer integral. The predictions from the SEC methods are more reliable than those from the ESD methods due to the consideration of spatial asymmetry in the majority of dimers. Based on the SEC predictions, Li et al. suggested that adding more aromatic rings to the TTF skeleton could lower intramolecular reorganization energies, but there is no discernible improvement in the hole mobilities of the benzene-exposed TTF derivatives.²⁸ Nan and Li studied the phase dependence of hole mobilities in dibenzo-tetrathiafulvalene crystals (α phase and β phase).²⁹ Although these two phases have similarly predicted hole mobilities in three-dimensional space, their two-dimensional predictions are quite different. SEC method, in general, has its limitation, and it fails to contain many fundamental and crucial details of charge transfer in dimers, such as difference in energy, E₂ − E₁, in Eq. (2).

The fragment charge difference (FCD) method considers spatial asymmetry and includes key factors for charge transfer (e.g., E₂ − E₁), which can provide more details to evaluate the charge transfer variation of dimers. So far, the FCD methods have also been applied to investigate intermolecular charge transfer properties in recent years.^31–34 Here, we have adapted an FCD method to study a collection of TTF-based crystals given in Table I. We have analyzed systematically the effects of polymorph (α-TTF, 1 versus β-TTF, 2) and the crystals with homologous molecular packings (1 versus dithiophene-tetrathiafulvalene, DT-TTF, 3). To study the combinational effects, we have rationalized the high mobility in HM-TTF (4) and the low mobility in BEDT-TTF (5).

A crucial factor in determining the strength of electronic coupling is the transfer integral, V V, which is expected to have a high absolute value for a high charge hopping rate. Dimers extracted from crystals have been used as models for calculation. In practice, the transfer integral calculated by the SEC method is based on HOMO states of monomers of one dimer,²⁷ which can be described as follows:

where h₁₂ is the nonorthogonal transfer integral, h₁₁ and h₂₂ are the site energies, and S₁₂ is the overlap integral.^1,27

In this study, we have calculated the atomic orbital coefficients of dimers and monomers at B3LYP/6-311G*(d, p) level using the Gaussian 16 package.^39–41 The transfer integral was then calculated using the CATNIP program by considering the orthogonalization of the relevant quantum states.⁴² 

Similar to the generalized Mulliken–Hush (GMH) method,⁴³ the FCD methods^31,44 calculate the electronic coupling matrix elements between a donor and an acceptor. The transfer integral (V_FCD) can be described as follows:

where E₁ and E₂ are the energy of two adiabatic states Ψ₁ and Ψ₂ (HOMO and HOMO-1 energy levels of a dimer), Δq₁ and Δq₂ are the charge differences between donor and acceptor in these two adiabatic states, respectively, and Δq₁₂ is the corresponding off-diagonal term (like the transition dipole moment in GMH method). Herein, considering two electronic eigenstates (adiabatic states) for the transfer integral of dimers applies to most cases. Using only two adiabatic states for very localized states (the strength of electronic coupling in dimers is usually weak) can overestimate the transfer integral of dimers, which can lead to an overestimation of mobilities.^33,45

We defined the term H_s as an orbital distribution correction (ODC) factor with the range of 0.0–0.5. When H_s is close to 0.5, the charges on the donor and acceptor are nearly equal. The term ΔE represents the difference between E₂ and E₁. The orbital energy and the wave function of a dimer have been calculated by an FCD method using a standard SEC method as a comparison. The orbital composition analysis was completed by using the multiwfn 3.7 program.^46,47 The orbital contribution of monomers was conducted by the Hirshfeld method.⁴⁸ 

Another important factor to determine the hopping rate is reorganization energy (λ) including internal and external inputs. In organic crystals, the weak interactions make the external contributions negligible due to surrounding media. Therefore, the intramolecular contribution was mainly considered in this study.^3,24,28 The λ was described as follows:

where $E0*$ and $E+*$ are the energies of neutral species and cation species with the original geometries, and E₀ and E₊ are the energies of neutral species and cation species in the lowest-energy geometries.

After obtaining the hopping rate based on the above-mentioned calculation, carrier mobility can be predicted by the Einstein relation^3,49 without considering an external electric field,

where e, k_B, and T are the electron charge, Boltzmann constant, and temperature, respectively, and D is the diffusion coefficient.³ D can be estimated for a spatially isotropic organic crystal by

where n is the dimension of an organic crystal, r_i is the centroid distance of two monomers in a dimer, and k_i and p_i (⁠$pi=ki/∑iki$⁠) are the hopping rate and the relative probability for i th pathway, respectively.

In terms of a specific pathway in a given organic crystal, the carrier mobility^49–51 was described as follows:

Based on the aforementioned description, we could predict average mobilities and mobilities along a specific pathway of TTF-based crystals. In the next section (Sec. III), we will present multiple factors to analyze the charge transfer properties of TTF-based crystals, which can be used to quickly estimate whether TTF-based organic crystals have high mobility.

1 exhibits a herringbone crystal structure as shown in Fig. 1(a). All six extracted dimers from 1 were considered to calculate the mobilities of specific transfer pathways shown in Figs. 1(c) and S1. As described in the Computational Details section, the difference in energy of E₂ − E₁ (ΔE), ODC factor (H_s), and centroid distance (r_i) are important factors to obtain high mobilities in organic crystals. A comparison of three factors for the six dimers is shown in Fig. 1(b). In addition, the values of the three factors and the reorganization energy for the dimers 1.1–1.6 are given in Table S1.

The factors of dimer 1.1 with the highest mobility (1.63 cm² V⁻¹ s⁻¹) and dimer 1.5 with the lowest mobility (0.0053 cm² V⁻¹ s⁻¹) are highlighted in Fig. 1(b). It can be seen that the ΔE of dimer 1.1 (211.3 meV) is significantly higher than that of dimer 1.5 (5.8 meV). Both dimers have H_s values close to 0.5, indicating that the molecules in these dimers have an equal distribution of charge. r of dimer 1.1 is only half of that of dimer 1.5.

Figure 1(d) shows the values of the transfer integral of the six transfer pathways (dimers 1.1–1.6). The strong anisotropic electronic coupling strength in 1 is indicated by the differential values of the transfer integral. Figure 1(e) shows that the carrier mobility along the dimer 1.1 (1.63 cm² V⁻¹ s⁻¹) is close to the maximum experimental mobility value for 1 (1.20 cm² V⁻¹ s⁻¹).²² The average mobilities of 1 determined by the FCD and SEC methods fall within the experimentally measured range of hole mobilities. The findings demonstrate that the FCD method is not only a reliable method for calculating the transfer integral and mobilities of a dimer but also a facile tool to identify the critical parameters (ΔE and H_s).

Polymorphs 1 and 2, with significantly different crystal packings, are selected to analyze the effect of different packings on carrier mobility. The packing of 2 is relatively complicated as shown in Fig. S2. According to the crystal structure of 2, two charge transfer centers have been identified, and the dimers of 2 have been extracted based on the two centers, respectively (the detailed information is shown in Figs. S2(a) and S2(b). For instance, 2.1 and 2.1′ represent the transfer pathways based on the first and second charge transfer centers. As shown in Figs. 2(a) and S3(a), despite dimers 2.1 and 2.1′ having high transfer integrals (57 and 54 meV respectively), the values of transfer integral mainly fall in a range from 10 to 30 meV. The mobilities at the two centers show a fluctuation between about 0.1 and 0.4 cm² V⁻¹ s⁻¹. The calculated average mobilities of 2 are in the experimental hole mobility values range.

Since polymorphs are assumed to have the same reorganization energy, the other three factors (ΔE, H_s, and r_i) at the two transfer centers enable us to investigate the variation (Fig. S4 and Table S2). Both dimers 2.1 and 2.1′ possess the highest mobilities in the two transfer centers, respectively, and they also have similar values in the three factors. In contrast, the H_s value of dimer 2.4 (0.077) is only about one-third of dimer 2.1 (0.226). Although the H_s value of dimer 2.6′ is close to 0.5, the ΔE (0.7 meV) is very low.

Figures 2(c) and 2(d) exhibit the comparison of HOMO distribution between 1 and 2. Unlike dimers 1.1 and 1.5 with an even HOMO distribution, dimers 2.1 and 2.4 have an uneven orbital distribution. The ratio of charge difference of dimer 2.1 is Δq1: Δq2 = 94.6%: 5%. The ratio of charges difference in dimer 2.4 is Δq1: Δq2 = 99.5%: 0.5%. Figures 2(e) and 2(f) show the comparison of four factors, including H_s, ΔE, r_i, and λ. The dimers with the highest mobilities, dimers 1.1 and 2.1, and the lowest mobilities, dimers 1.5 and 2.4, are selected to compare the four factors, respectively. ΔE and r of dimer 1.1 (211 meV, 0.402 nm) are smaller than those of dimer 2.1 (251 meV, 0.550 nm) as shown in Fig. 2(e). The H_s value of 1.1 (0.500) is, however, more than twice that of dimer 2.1 (0.226), which causes the mobility of dimer 1.1 higher than dimer 2.1. As for dimers 1.5 and 2.4 in Fig. 2(f), their low mobilities are mainly attributed to the low ΔE (5.8 meV) of dimer 1.5 and low H_s value (0.077) of dimer 2.4, respectively.

3 has herringbone crystal packing similar to that of 1 (Figs. S5 and S6). Figure 3(a) shows that dimers 3.1–3.3 have higher values of transfer integral (35–55 meV) than dimers 3.4–3.6 (0.3–6 meV), exhibiting a considerable extent of anisotropic electronic coupling. Similarly, Fig. 3(b) shows dimers 3.1–3.3 with mobility values of 0.39–0.48 cm² V⁻¹ s⁻¹, which is significantly higher than dimers 3.4–3.6 (0.0002–0.07 cm² V⁻¹ s⁻¹). The three-factor comparison (ΔE, H_s, and r_i) for the dimers 3.1–3.6 is shown in Fig. S7 and Table S3. It should be noticed that the HOMO distributions of dimers 1.1, 3.1, 1.5, and 3.6 are all even [shown in Figs. 3(c) and 3(d)], indicating an H_s value close to 0.5.

As shown in Figs. 3(e) and 3(f), the four-factor comparisons demonstrate a clear effect of chemical modification. Under similar face-to-face packings, the ΔE in dimer 3.1 (110 meV) is only half of that in dimer 1.1 (211 meV), leading to the mobility (0.42 cm² V⁻¹ s⁻¹) of 3.1 is much smaller than 1.1 (1.63 cm² V⁻¹ s⁻¹). Meanwhile, dimers 1.5 and 3.6 have larger r values (0.839 and 1.493 nm) and smaller ΔE values (6 and 0.7 mV) than those for dimers 1.1 and 3.1, which could lead to the poor mobilities in dimers 1.5 and 3.6 (0.006 and 0.0003 cm² V⁻¹ s⁻¹).

Organic crystal assembled by HM-TTF building block, 4, has the highest hole mobility among the reported TTF-based crystals.^16,53,54 As shown in Table I and Fig. S8, HM-TTF molecules are arranged in an edge-by-edge fashion along the x–z plane. The positions of dimers (4.1-4.8) and the extracted dimers of 4 are displayed in Fig. S8. In terms of our multiple-factor analysis, the highest and the lowest mobilities of dimers 4.1 and 4.8 have similar trends to that of dimers 1.1 and 1.5 (Fig. S9 and Table S4).

Figure 4(a) shows the values of the transfer integral of dimers 4.1–4.8. The transfer integral of dimer 4.1 is higher than other transfer pathways similar to 1, indicating that 4 has anisotropic electronic coupling strengths. The mobility of dimer 4.1 (3.39 cm² V⁻¹ s⁻¹) in Fig. 4 is close to the measured mobility (2.20 cm² V⁻¹ s⁻¹).¹⁶ As shown in Figs. 4(c) and 4(d), dimers 4.1 and 4.8 also exhibit even HOMO distributions compared to dimers 1.1 and 1.5, which is consistent with their H_s values close to 0.5.

We have obtained a deeper insight into the high mobility of 4 through our four-factor analysis [Fig. 4(e)]. While the ΔE, λ, and H_s in dimer 4.1 are close to those in dimer 1.1, the r of dimer 4.1 (0.638 nm) has a larger value than dimer 1.1 (0.402 nm), indicating that longer intermolecular distance could contribute to the high mobility of 4.1. As for the comparison of dimers 1.5 and 4.8 in Fig. 4(f), too large r and too small ΔE are the main reasons for the low mobilities of these two dimers similar to the circumstance of dimers 1.5 and 3.6.

Taking TTF-based crystals as examples, we have analyzed the effects of packings and chemistries on the charge transfer properties of the molecular crystals. Our FCD predictions allow us to bridge structure/chemistry and transfer integral/mobility with the multiple-factor analysis. Based on our current investigation of the trends, it seems that a large ΔE, an H_s value close to 0.5, an appropriate r (neither too large nor too small), and a small λ value are the main features for high mobility along a given transfer pathway in a TTF-based organic crystal. However, it should be noted that the current multiple-factor analysis based on the FCD method has its limitations inevitably. One of them requires uniform intermolecular packing along a transfer pathway in a crystal. As shown in Figs. S10 and S11 and Table S5, although the dimer 5.1 in BEDT-TTF (5) has factors for high charge transfer integral and mobility predicted by our multiple-factor analysis, the intermolecular packing varies along the charge transfer pathways in 5 with low mobilities reported experimentally.¹⁷ 

Our current analysis has merely focused on a collection of TTF-based organic crystals. However, the idea and method can be applied in a range of molecular material systems assembled by weak interactions. The analysis based on dimers can also be adapted to molecular materials without crystallinity. Xie et al.⁵⁵ have recently reported an amorphous coordination polymer (Ni-based tetrathiafulvalene tetrathiolate) with high electronic conductivity where the theoretical analysis was conducted by a density functional theory method (a projector-augmented-wave pseudopotential with a Perdew–Burke–Ernzerhof functional calculation in Quantum espresso). Our multiple-factor analysis for dimers could provide further insights into the charge transfer properties at the intermolecular level.

In conclusion, we have adapted the FCD method to predict the charge transfer properties of TTF-based polymorphs and crystals with homologous molecular packings. We have verified the FCD predictions with the well-accepted SEC predictions. The FCD method renders us the ability to compare the differences in the energy of E₂ − E₁ (ΔE), ODC factor (H_s), centroid distance (r_i), and reorganization (λ) to rationalize the variation in charge transfer properties in TTF-based organic crystals. The results from our evaluation based on these factors can contribute to designing TTF-based organic crystals with more desired electrical properties. Furthermore, the FCD predictions combined with multiple-factor analysis on dimers can be applied to study the intermolecular charge transfer properties for a myriad of molecular material systems.

See the supplementary material for more details on dimer positions and charge transfer properties of TTF-based crystals.

This work was funded by the National Natural Science Foundation of China (Grant No. 52103224), the Science and Technology Commission of Shanghai Municipality (Shanghai Sailing Program, Grant No. 20YF1420400), and the start package grant from Shanghai Jiao Tong University to Dr. T.W. B.C. thanks the Hyogo Science and Technology Association (4019) for funding. Z.D. acknowledges funding from the Lee Kuan Yew Postdoctoral Fellowship (Grant No. 22-5930-A0001).

The authors have no conflicts to disclose.

Yakui Mu: Conceptualization (equal); Formal analysis (lead); Investigation (equal); Methodology (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (supporting). Tan Wang: Investigation (supporting); Writing – review & editing (supporting). Zeyu Deng: Investigation (supporting); Validation (supporting); Writing – review & editing (supporting). Bun Chan: Investigation (supporting); Methodology (supporting); Validation (lead); Writing – review & editing (supporting). Tiesheng Wang: Conceptualization (lead); Formal analysis (supporting); Funding acquisition (lead); Investigation (equal); Methodology (supporting); Supervision (lead); Writing – original draft (supporting); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.