Boosting thermoelectric performance of molecular junction by utilizing edge branches and weak coupling

Molecular junctions (MJs) with lengths between 1 and 3 nm have been widely studied in the field of molecular electronics because they can realize a series of electronic device functions, such as single-molecule field-effect transistors,^1,2 optoelectronic devices,^3,4 negative differential resistors,^5,6 molecular rectifiers,^7,8 and other functional devices. Along with the significant progress in thermoelectric materials, especially in nanostructure design,^9,10 low-dimensional application,^11,12 new alloy development,^13,14 and interface engineering optimization,^15,16 MJs have also seen developments in this field. Owing to their inherent low-dimensional characteristics, pronounced interfacial phenomena, and unique means of molecular-level regulation, MJs possess distinct advantages over traditional bulk thermoelectric materials in thermoelectric applications.^17–24 In fact, as early as 2007, Reddy et al.¹⁷ conducted the first measurement of the Seebeck coefficient (S) in single-molecule junctions (SMJs), finding that its magnitude and sign depend on the distance of the HOMO/LUMO energy levels from the Fermi level and predicted that these structures held great potential to serve as highly efficient thermoelectric components. First, the quasi-one-dimensional molecular junctions exhibit pronounced quantum confinement effects, with sharp and narrow peaks in the density of states, which facilitates the realization of a larger Seebeck coefficient. Second, the discreteness of molecular energy levels can lead to the breaking of the Wiedemann–Franz law, thereby enhancing the thermoelectric conversion efficiency. Additionally, due to the significant mismatch between the vibrational spectra of the bulk electrodes and the discrete molecules, the molecular junctions exhibit lower phonon thermal conductivity. Therefore, in recent years, scholars have conducted extensive and in-depth research on the thermoelectric properties of molecular junctions.

The performance of thermoelectric components is crucially determined by the dimensionless thermoelectric figure of merit ZT, which is defined as ZT = S² GT/κ, where S represents the Seebeck coefficient, G denotes the electric conductance, and κ signifies the total thermal conductance contributed by phonons and electrons, respectively. Within the Landauer framework, G in MJs is directly proportional to the transmission function [T_el(E_F)] for electrons traversing from one electrode to another electrode through a central molecule, whereas S is approximately proportional to the slope of −lnT_el(E) at the Fermi level.^20–22 In recent years, both in theoretical and experimental studies, common strategies for optimizing the thermoelectric characteristics of MJs primarily focus on improving power factor (S²G) through the implementation of mechanical strain,^25,26 tuning the molecule–electrode coupling,^27–30 manipulating the alignment between frontier molecular orbitals and electrode's Fermi level,^31–33 increasing individual molecules' length,^34–36 as well as modifying molecular conformation and orientation.^37–39 For example, Li et al.³⁶ shown that the thermopower in DNA-based MJs can be manipulated through length and sequence variations, with the effect of thermopower being minimal and length-insensitive in hopping regimes, and significant and length-sensitive in tunneling regimes, whereas the electric conductance decreases linearly as length increases. Rincón-García et al.²⁶ experimentally revealed that the magnitude and sign of the thermopower of Sc₃N@C₈₀-based MJs can be modulated by applying pressure or altering molecular orientation, which is attributed to the transmission resonance being close to the Fermi energy undergo external stimuli. In addition, the Seebeck coefficient can be significantly enhanced in MJs by achieving resonance effects and^40,41 quantum interference effects (QIs).^42–44 Although these strategies can effectively enhance the electric conductance or the Seebeck coefficient, the power factor of MJs still cannot reach the optimal value (approximately ∼10⁻² pW/K²) due to the significant coupling that exists between the two. Therefore, it is also crucial to optimize the thermal properties of MJs by suppressing the contribution of phononic thermal conductance (κ_ph) relative to electronic thermal conductance (κ_el).^45,46

In most strongly covalently bonded junctions, thermal conductance is predominantly governed by phonons, especially when carbon materials are used as electrode materials. As an ideal electrode material for constructing stable solid-state MJs, graphene has attracted much attention from researchers due to its low resistance, flexibility, excellent optical transparency, low roughness, and easy surface modification.^3,47–49 For example, Jia et al.³ successfully designed a reversible molecular optoelectronic switch by combining a single diarylethene molecule with graphene electrodes. While the graphene-based MJs have higher thermal conductance than metal-based MJs because the resonance match between the vibration spectrum of graphene and the vibration energy levels of molecular in graphene-based molecular electronic devices promotes the coupling effect between molecular vibrations and the graphene lattice, which limits on their potential for thermoelectric applications. Therefore, it is important to develop strategies that can effectively enhance both conductance and Seebeck coefficient while simultaneously reducing thermal conductance. At present, few have focused on how to effectively coordinate the electrical transport and thermal transport in such devices. Markussen et al.⁵⁰ obtained large thermal properties by introducing ultrathin nanotrees or alkane chain groups at the edge of silicon nanowires. This is primarily due to the fact that the local resonance of the edge branched structure weakens the backbone phonon transport, causing a lower thermal conductance; The local electronic states of the branched structure only exist in the deep of the backbone energy band, thus the influence on the electrical transport is relatively weak. In recent years, numerous researchers have begun to focus on the use of branched structures to regulate the thermal and thermal properties of large-scale materials.^51–53 Consequently, the use of molecules containing branched structures or the introduction of branched structures can synergistically modulate the thermoelectric properties of molecular junctions.

In this paper, we study the thermoelectric properties of naphthacene-based and rubrene-based graphene junctions using density functional theory combined with the non-equilibrium Green's function method. By increasing the length of naphthacene and rubrene chains, large figures of merit can be obtained at specific chemical potentials. It has been discovered that the phonon thermal conductance of rubrene molecular junctions is obviously weakened, and it is even half of the electron thermal conductance. In addition, the electrical transport properties of both remain essentially the same. Furthermore, the use of π-coupling contact methods greatly reduces the phonon transport and enhances the thermal properties of the molecular junctions.

In this work, the electron–phonon interaction and phonon–phonon interaction were ignored in the calculation process because the system size is smaller than the inelastic mean free path for the phonons. It means that the transport process of electron and phonon is ballistic, which has been confirmed in previous theories and experiments.⁶⁰ 

Figure 2 shows the calculated electron transport properties of molecular junctions as naphthacene and rubrene molecular with n (n = 1, 2, 3, and 4) units (denoted by nN or nR), respectively. Obviously, all molecular junctions have a large resonant transmission peak at the Fermi level, as shown in Fig. 2(a), and these peaks remain unchanged with an increase of the molecule length. From Fig. 2(b), it is found that the electronic structure of 4-ZGNR shows metallic properties and two bands are formed at the Fermi level due to the existence of a serrated edge structure.⁶¹ Additionally, because of the serrated edge structure of naphthacene and rubrene molecules, electrons transmitted from the left electrode to the right electrode are not scattered. The calculated transmission eigenstates of 1 N and 1R graphene junctions at the Fermi level are shown in Fig. 2(c). The results show that the wave function transfer path of the junctions is mainly along the zigzag edge, and the wave amplitude does not change when passing through the central molecule. Besides, the hybridization type between the phenyl groups substituted hydrogen atoms and edge carbon atoms is sp² hybridization. It means that the conductance (G = G₀T(E_F), electrical conductance quantum, G₀ = 2 e²/h ≈ 77.5 μS) is insensitive to the molecular length or the pendant branches at the edge of the naphthacene molecule.

In order to investigate the effect of molecular length and pendant branches on thermoelectric transport, we calculated the electrical conductance and Seebeck coefficient for the two series of naphthacene and rubrene-based molecular junctions. As shown in Fig. 3(a), the conductance curve exhibits the same phenomenon as the transmission spectrum, with a resonant conductance peak appearing near the Fermi level. In addition, it is obvious that the conductance in the vicinity of the Fermi level is not sensitive to the increase in molecular length. Figure 3(b) illustrates this with the conductance ln(G/G₀) on the naphthacene and rubrene molecular junctions differing in the number of molecule units. In strong contrast to length-dependent electrical conductance in previous studies,⁶² the data, as expected, document a nearly length-independent behavior, indicating the presence of resonant tunneling transport in molecular junctions. To optimize the numerator of ZT, we calculate the Seebeck coefficient (S) against chemical potential (μ) over a range of −0.5 < μ< 0.5, as shown in Fig. 3(c). It is clear that the value of S reaches its maximum value due to the high slope of the transport curve near the chemical potential of 0.05 eV, rather than the near resonant transmission peak. Comparison between naphthacene- and rubrene-based molecular junctions, it is evident that the Seebeck coefficient of both have the same trend, the maximum value of S is 59.6 μV/K for n = 1, 32.6 μV/K for n = 2, 68.5 μV/K for n = 3, and 34.8 μV/K for n = 4, which is similar to the order of magnitude of data reported in many experimental literature studies.^17,63,64 The calculate maximum value of power factor (S²G) is close to 0.35 pW/K² for rubrene-based molecular junction when n = 3, as shown in Fig. 3(d), which is conducive to achieve large thermoelectric figure of merit. The unfortunate reality is that the coupling relationship between G and S is not decoupled by the influence of molecular length and edge branching, which provides a significant obstacle for regulating the thermoelectric performance of molecular junctions in the electronic level. Therefore, it is necessary to discuss the influence of molecular length and edge branching on phonon transport.

Next, we explore the influence of molecular length and pendant branches on the thermal transport of naphthacene- and rubrene-based molecular junctions. In order to ensure the accuracy of the thermal conductance, we first calculate the partial phonon spectrum of 4-ZGNR electrode, as shown in Fig. 4(d). It is observed that no imaginary frequency modes (denoted by ν > 0 THz) in the phonon dispersion near the Γ point exist, which indicates that the structure is dynamically stable.^65,66 Figure 4(a) shows the thermal conductance (κ_ph) of naphthacene- and rubrene-based molecular junctions with different molecular length (denoted by κ_nN and κ_nR) as a function of temperature. It is found that κ_ph of rubrene-based molecular junctions is lower than that of the naphthacene-based molecular junctions. As the n increase from 1 to 4, the room-temperature κ_ph of naphthacene-based molecular junctions is maintained around 0.5 nW/K, while the room-temperature κ_ph of rubrene-based molecular junctions decrease from 0.34 to 0.29 nW/K, then to 0.27 nW/K, and finally to 0.26 nW/K. As can be seen in Fig. 4(b), the ratio χ presents the reduction magnitude of κ_nN relative to κ_nR vs temperature and exhibits an initial decline followed by an increase, eventually stabilizing. At high temperatures, all the phonon modes are excited and contribute to the phonon thermal conductance. At this time, χ tends to be a constant value. While at low temperatures, only the low frequency phonons are excited, it can be observed form Fig. 4(e) that the transmission coefficients of the phonon modes of Rubrene-based molecular junctions in the range of 0–10 THz are decreased comparable to those of naphthacene-based molecular junctions. Therefore, χ shows a downward trend before 50 K. It is clear that the maximum value of χ increase from 30% to 40%, then to 43%, and finally to 45% as the molecular length increase, which implies that the pendant branches in rubrene-based molecular junctions is conducive to suppress the phonon transport.

To illustrate the reason for a significant reduction in κ_nR, we calculated the phonon transmission spectra of molecular junctions as a function of frequency for different lengths, as depicted in Figs. 4(c) and 4(e). The results show that the transmission coefficients of rubrene-based molecular junctions in the entire frequency range is lower than that of naphthacene-based molecular junctions, particularly significant in the low frequency range of 0–10 THz. In addition, it is worth noting that there are valley peaks in the rubrene-based molecular junctions at different frequency. For example, at 1 THz, as molecular lengths increase (meaning an increase in the number of branches), the valley peak shift toward the zero axis. This means the occurrence of multiple scattering from the points of attachment of the branches to the trunk or Fano resonance scattering.⁶⁷ In addition, the phonon transport resonance become narrows with increasing length, which reduces thermal conductance.³⁴ The spatial distribution of the density of states provides an intuitive physical picture of the localized phonon resonance. As shown in Fig. 5, we compared the phonon local density of states (PLDOS) of 3 N and vdW-3R at 1.5, 6, 28 THz, respectively. It is clear that the PLDOS of 3R are strongly localized at these frequencies, which leads to the corresponding transmission coefficient approaching zero. These phenomena fully illustrate that the edge phenyl group of the Rubrene molecules have an inhibitory effect on phonon transport, thereby resulting in a corresponding decrease in phonon thermal conductance.

However, the thermal conductance κ consists of two parts: the electronic thermal conductance (κ_el) and the phonon thermal conductance (κ_ph). Figure 6 shows the electron thermal conductance and the ratio κ_ph/κ_el for different configurations as a function of chemical potential at room temperature. The results in Fig. 6(a) are present that κ_el of naphthacene-based molecular junctions are roughly the same as that of rubrene-based molecular junctions at the same length within the selected chemical potential range of −0.5 ∼ 0.5 eV, which are similar to the results of electrical conductance. Comparison between Figs. 6(a) and 6(b) reveals that κ_el is comparable with κ_ph for naphthacene-based molecular junctions in the vicinity of the Fermi level. However, the phonon contribution is lower than the electronic contribution for rubrene-based molecular junctions, with a maximum twice lower at a length of 4. In addition, the ratio κ_ph/κ_el is lower for the rubrene-based molecular junctions than for the naphthacene-based molecular junctions, which makes the former more attractive for thermoelectricity.

Figure 6(c) shows the resulting thermoelectric figure of merit ZT. We observed that the thermoelectric performance of rubrene-based molecular junctions are better than that of naphthacene-based molecular junctions at 300 K. The maximum value of ZT reaches 0.13 at n = 3 in the vicinity of Fermi energy due to the larger S and lower κ. In addition, as shown in Fig. 6(d), we calculate the variation of maximum ZT with molecular length at T = 50, 100, 200,300, 400, 500 K. It is obvious that the ZT tends to decrease with increasing temperature, and the maximum ZT of junctions with different lengths obtain at 200 or 300 K. However, the maximum ZT can only reach 0.13, which is much less than 1. This is mainly due to the resonant tunneling of electrons, which leads to the thermal conductance of electrons and phonons in an order of magnitude, or even greater than the thermal conductance of phonons. As a result, the decrease of thermal conductance is not obvious for the improvement of thermoelectric properties.

At present, π–π stacked molecular junction has been synthesized in experiments^68–70 and have received widespread attention in tuning the thermoelectric properties, as junctions with weak coupling contacts have lower thermal conductivity compared to covalent bond contacts.^71–74 Based on this theme, we construct the vertical rubrene-based molecular junction as a weak van der Waals (vdW) coupling between electrodes and molecular, schematically illustrated in the inset of Fig. 7(a). Figure 7 presents the calculated phonon transmission spectra T_ph, phonon thermal conductance κ_ph, electron transmission spectra, and figure of merit ZT of covalent and vertical junctions of rubrene for n = 3 units (named 3R and vdW-3R), respectively. Compared with the results of 3R, it is evident that the phonon transport of vdW-3R is further suppressed by the weak vdW interaction between electrodes and molecular. As shown in Fig. 7(a), the transmission coefficient significantly decreases in the full frequency range, and only a few cross-plane phonon modes participate in phonon transport.⁵⁰ In order to intuitively reflect the underlying physical principles, we compared the PLDOS of 3 N and vdW-3R at 2 and 20 THz, respectively. As shown in Fig. 8, clearly, after constructing a weak vdW coupling, only a small portion of phonons transmits through the ZGNR. This fully demonstrates that weak vdW coupling can further suppress phonon transport. Moreover, the phonon thermal conductance κ_ph of vdW-3R is significantly reduced, as shown in Fig. 7(b), and κ_ph is not sensitive to the applied temperatures. The room-temperature κ_ph of vdW-3R is predicted to be about 0.02 nW/K, which is approximately decreased by 91% than that of 3R, as shown in ratio χ of Fig. 7(b). The inset of Fig. 7(b) shows κ_ph/ κ_el, we discover that the phonon thermal conductance is significantly greater than the electron thermal conductance at −1.0 ∼ 1.0 eV, indicating that the phonon thermal conductance still plays a dominant role in this structure. However, the electron transport of vdW-3R is also affected by the weak coupling contact, as shown in Fig. 7(c), where the resonant transmission peak is suppressed at the Fermi level and accompanied by a large transmission gap. Although the conductance peak is reduced, according to the previous studies, the Seebeck coefficient will be an enhanced in the transmission gap, and the power factors of vdW-3R and 3R at the maximum ZT value are 0.09 and 0.18 pW/K², respectively. This means that the enhancement of the thermoelectric performance of the device is mainly attributed to the significant decrease in phonon thermal conductance of the van der Waals structure. In addition, the transmission peak contributed by HOMO and LUMO become narrow and sharpness due to the weak overlap of interlayer π-electron states, indicating the lower conductance and larger Seebeck coefficient. Interestingly, the maximum ZT value of vdW-3R is significantly increased, up to ∼1.1 at chemical potential of 0.25 eV, which is nearly an order of magnitude higher than that of 3R junction. Therefore, the thermoelectric performance of molecular junction can be further enhanced by introducing the edge branch and weak interaction.

In conclusion, we investigated the thermoelectric properties of graphene junctions based on naphthacence (C₁₈H₁₂) and rubrene (C₄₂H₂₈) molecules using density functional theory combined with a non-equilibrium Green's function method. The results indicate that the thermoelectric performance of rubrene-based graphene junctions is superior than that of the naphthacence-based graphene junctions. This is mainly due to the fact that the local vibration of the edge phenyl group of the rubrene molecules blocking the phonon transmission, and the phenyl group does not affect the electron transmission along the backbone. Furthermore, the thermal conductance of the rubrene junctions is reduced by nearly half compared to that of naphthalene junctions. Nevertheless, the room-temperature ZT of the rubrene junction is only close to 0.2 due to large thermal conductance. By constructed to weak coupling rubrene-based graphene junctions, its phonon thermal conductance is reduced by about 91% compared to a strongly coupled junction. The room-temperature ZT is increased to 1.1, nearly an order of magnitude higher than the covalent junction. It is highly desirable to minimize the phonon thermal conductance of self-assembled molecular films while maintaining their electrical properties, which will provide a new approach for optimizing the thermoelectric performance of molecular junctions in the future.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12204066 and 12074046).

The authors have no conflicts to disclose.

Bing-Yu Gan: Data curation (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Xin-Yi Liu: Formal analysis (equal); Methodology (equal). Wen-Si Tang: Conceptualization (equal). Xuan-Hao Cao: Conceptualization (equal); Formal analysis (equal). Zhi-Qiang Fan: Supervision (equal); Visualization (equal). Dan Wu: Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.