Negative differential resistance and transport regularity in aromatic cyclo[n]carbon-based (n = 4k + 2) molecular devices

Abstract

Cyclo[18]carbon (C₁₈), which has been successfully synthesized and extensively studied, exhibits unique electronic transport properties. Cyclo[n]carbons (C_n, n = 4k + 2), sharing a similar π-electron conjugation with C₁₈, potentially demonstrate comparable or even superior transport characteristics. In this study, three C_n devices with different types of electrodes have been investigated using density functional theory combined with the nonequilibrium Green's function. Our findings reveal that the devices with large-diameter carbon rings tend to exhibit the negative differential resistance (NDR) effect. The magnitude, position, and number of NDR peaks are remarkably affected by the number of carbon atoms and the type of electrode. However, the NDR effect does not appear in devices with odd-numbered rings. This study sheds light on the remarkable electronic transport properties of C_n molecules and offers valuable insights for the development of advanced carbon-based molecular devices.

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1. Introduction

Carbon allotropes, such as graphene,¹ carbon nanotubes,² and fullerenes,³ have fascinated scientists for decades due to their remarkable physicochemical⁴ and potential applications.^5–7 A recent milestone was the successful synthesis of C₁₈, which introduced a novel zero-dimensional carbon allotrope to this diverse family.⁸ Comprising two-coordinated sp-hybridized carbon atoms, C₁₈ exhibits remarkable electrical characteristics,^9,10 opening up numerous avenues for the development of carbon-based nanodevices.

The successful synthesis of C₁₈ has reignited a fresh wave of research into its electrical properties. For instance, the report by Stasyuk et al. illustrated the similar electron accepting properties and excellent electron transfer properties of C₁₈ to those of C₆₀, indicating the potential of C₁₈ as a promising candidate for further investigation in electronic transport applications.¹¹ Subsequently, Zhang et al. first constructed three different types of two-probe molecular devices based on C₁₈, and uncovered ohmic characteristic, quasi-Schottky feature, and current-limiting functions,¹² proving a fundamental prediction for the transport properties of novel C₁₈-based molecular devices. To realize the versatility and high performance of these novel carbon-based molecular devices, Hou et al. designed a device using C₁₈ and two graphene electrodes based on a common molecular device model, demonstrating negative differential resistance (NDR), rectification, and spin-filtering effects. This study expanded the application of C₁₈ in the field of spintronic devices and provided new insights into the construction of future single-molecule switching devices.¹³ Furthermore, Tang et al. proposed a cross-plane spin-filtering device using AA-stacked C₁₈ as a central molecule sandwiched between two zigzag-edged graphene nanoribbons (ZGNRs).¹⁴ This device showcased a magnetoresistance (MR) of up to 12 480% when vertically clamped between two ZGNRs, along with a rectification ratio of 2926 upon inserting a transition metal atom (V). This study further confirmed that C₁₈ exhibits marvellous electronic transport properties and may be used in functionally integrated molecular devices.

Therefore, C₁₈ is an excellent candidate for carbon-based electronic devices, with its outstanding properties, including NDR behavior and rectification effects, and can be viewed as a promising material for the design of various functional devices, such as molecular switches,^15,16 molecular rectifiers,^17,18 field-effect transistors,^19,20 memory devices^21,22 and so on.^23,24

Such superior physicochemical properties of C₁₈ have aroused the interest of scientists in exploring other molecules within the class of C_n. Indeed, as early as 10 years ago, Fowler et al. investigated the double aromaticity of C_n with n = 6–30, revealing that the aromatic properties of the two delocalized subsystems are independent and adhere to the Hückel rule of aromaticity.²⁵ Since the introduction of C₁₈, many researchers have embarked on studying a broader range of larger C_n. For example, Baryshnikov et al. studied a series of even-numbered C_n with carbon atom counts in the range of n = 6–100 and odd-numbered ring carbons in the range of n = 5–29 using more precise computational methods.^26,27 Their study found that in even-numbered C_n, electron delocalization and aromatic properties change with the size of the molecular rings, whereas odd-number C_n exhibit distinct electronic structure characteristics, influenced by an additional carbon atom (a carbene). However, most of these studies on C_n predominantly focused on their electronic structures and aromatic properties, employing theoretical calculation.^11,28,29 The electronic transport properties of another molecule of C_n remain largely unexplored.

In experimental endeavours, since Scriven et al.³⁰ improved the protocol for C₁₈ using dehalogenation of the bromocyclocarbon precursor, C₁₀ and C₁₄ have also been successfully produced by tip-induced dehalogenation on a NaCl surface.³¹ Meanwhile, recent experimental breakthroughs have demonstrated that scientists can now leverage diverse design frameworks and sophisticated fabrication and analysis techniques to realize single-molecule devices with specific functionalities.^32,33 For instance, Capozzi et al. achieved a high rectification ratio in a single-molecule diode through precise environmental control,³⁴ while Meng et al. developed a chemically gated, fully reversible, dual-mode single-molecule transistor based on a robust graphene–molecule–graphene junction.³⁵ These advancements not only suggest the feasibility of synthesizing additional C_n molecules in the future but also further validate the controllable fabrication and precise design of C_n-based molecular devices. Regarding electronic structure, C_n share a similar π-electron conjugate with C₁₈, implying comparable or even superior transport characteristics. Furthermore, as the number of carbon atoms increases, the transport properties of C_n may exhibit regularity. Therefore, it is both natural and timely to study the transport properties of this new form of carbon allotropes.

It is worth noting that the selection of electrodes for molecular devices also plays a crucial role in determining electronic properties.³⁶ Thus, we construct three types of devices with different metallic electrodes, each consisting of a molecule (Fig. 1a) coupled to two electrodes (Fig. 1b). We systematically investigate electronic transport properties of C_n and the transmission mechanism using first-principles methods based on the density functional theory (DFT)³⁷ and nonequilibrium Green's function (NEGF).³⁸ This research serves as a cornerstone for emphasizing the potential importance in the domain of carbon-based nanodevices.

Fig. 1 (a) Structures of C_n. (b) Device configuration of C_n-chain, C_n-ribbon and C_n-tube. (c) I–V curves of C_n-chain (n = 4k + 2). (d) I–V curves of C_n-ribbon (n = 4k + 2). (e) I–V curves of C_n-tube (n = 4k + 2).

2. Computational methods

To construct C_n-based devices, we considered three different types of metallic electrodes, including carbon chains, zigzag graphene nanoribbons (ZGNRs), and zigzag (3,0) carbon nanotubes (CNTs). All three electrode materials are carbon-based, ensuring strong interactions with the molecules. Carbon chains, as the simplest 1D electrodes, directly reflect the intrinsic transport properties of the C_n molecules. Carbon nanotubes, as an alternative 1D electrode configuration, allow us to investigate how different kinds of 1D materials affect the transport behavior. Graphene nanoribbons, functioning as quasi-2D electrodes, were selected to explore the effect of electrode dimensionality on transport characteristics. For the sake of discussion, they are named C_n-chain, C_n-ribbon, and C_n-tube, respectively. The configuration of these devices is illustrated in Fig. 1b. The C_n ranging from C₁₀ to C₃₄ are sandwiched between two semi-infinite electrodes. The molecular structures of C_n were optimized using the ωB97XD/def2-TZVP functional³⁹ in the Gaussian 16 program package, which aligns excellently with the experimental findings reported by Kaiser et al.⁸ Utilizing these optimized molecular structures, we construct a series of two-probe devices. In the device with ZGNR electrodes, which consist of 4-atom-wide nanoribbons, molecules are connected to the electrode using alligator clips. For the other two devices, they are directly coupled to the electrodes through a carbon atom.

Electron transport properties were investigated using a combination of density functional theory (DFT) and the nonequilibrium Green's function (NEGF) method, as implemented in the Atomistix Toolkit (ATK) package.^40,41 The exchange–correlation function is a generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE), while the double-zeta plus polarization (DZP) basis is chosen for all atoms. The k-point sampling set is 1 × 1 × 100, and the transport direction is along the C axis. A supercell with over 10 Å of vacuum space is used between neighboring cells to avoid interactions between periodic images. The cut-off energy for the density mesh and the electron temperature are set to 75 Ha and 300 K, accordingly. During the device structure optimization, the C_n molecules are fixed to preserve the integrity of their molecular geometry. Geometries are optimized until all residual forces on each atom are smaller than 0.01 eV Å⁻¹.

The electronic properties of the devices were probed by computing their currents, transmission spectra, eigenstates, and molecular projection self-consistent Hamiltonian (MPSH). The current through the device was calculated using the Landauer–Buttiker equation:⁴²

Here, f_1,2(E) denotes the Fermi function of the source and drain electrodes; e and h are the electron charge and Planck constant, respectively, and T(E) represents the transmission function, quantifying the probability of electron transmission from source to drain, which can be shown as follows:⁴³

T(E,V) = tr(Γ₁G^RΓ₂G^A)

where G^R and G^A are the retarded and advanced Green functions of the conductor part. Γ₁ and Γ₂ respectively represent the coupling matrices between the central region and the two electrodes. They are defined based on the self-energies and of the electrodes, with the formula:
G^R and G^A are the retarded and advanced Green's functions of the central region. They are defined as and (where δ is a positive infinitesimal approaching zero, used to ensure its causal property is different from that of G^R), where H is the Hamiltonian of the central region.

3. Results and discussion

3.1. Transport properties of C_n-chain devices

First, let us focus on the C_n-chain devices. To investigate the transport properties of C_n-chain, we calculate current–voltage (I–V) curves within a bias region of [0 V, 2 V], as shown in Fig. 1c. It is evident that all the devices exhibit linear currents at lower biases (0–0.4 V), following the Ohmic law.

The conductance of C_n-chain devices under zero bias is present in Fig. 2a. It follows that from C₁₀-chain to C₂₆-chain, the conductance shows a notable increase as the number of carbon atoms rises, but decreases once the number of carbon atoms exceeds 26. This variation in conductance under zero bias can be qualitatively understood from the electronic structure of C_n. Specifically, a smaller HOMO–LUMO gap usually indicates better conductivity, where HOMO and LUMO denote the highest occupied molecular orbital and the lowest unoccupied molecular orbital, respectively. Fig. 2b depicts the frontier molecular orbitals of C_n, where the n ranges from 10 to 34. It is crucial to note that the gaps are greatly underestimated when using the PBE function. However, we are only qualitatively concerned with the change in bandgap size as the number of carbon atoms increases. From Fig. 2b, we can see that the HOMO and HOMO−1 orbitals, as well as the LUMO and LUMO+1 orbitals, are degenerate in each molecule, and a clear pattern emerges—the HOMO–LUMO gap becomes smaller with an increasing number of carbon atoms. From C₁₀ to C₂₆, the gap shrinks quickly, but as the C_n becomes larger, it decreases slowly. This observation is consistent with previous literature, which reported a transition from a globally delocalized to a bond-localized topology between C₂₆ (delocalized) and C₃₀ (localized).²⁶ This transition obstructs the electronic transport and thus affects the I–V curves to some extent.

Fig. 2 (a) The conductance of C_n-chain at zero bias. (b) Frontier molecular orbitals energies of C_n. (c) The resonant tunneling model for transport through a C₁₀-chain junction.

When the bias voltage exceeds 0.4 V, the C_n-chain device with n ≥ 22 exhibits the NRD effect. Taking C₂₆-chain as an example, when the voltage continuously increases from 0.4 to 2.0 V, the current rises to an extreme value of 82.52 μA, and then drops to 80.10 μA at 1.8 V, leading to a NDR peak. Furthermore, as the carbon rings become larger, the bias voltage for the extreme current decreases. To be specific, the NDR peak is observed at 1.8 V for the C₂₂-chain, whereas it occurs at 1.2 V for the C₃₄-chain. Since the NDR effect is indispensable for several electronic components such as the amplifier, logic gate, Esaki and resonant tunnelling diodes,^44–46 we expect the C_n-chain device with large-diameter carbon rings to have a promising application in the NDR-based electronic components.

To comprehensively understand the underlying origins of the distinctive transport properties in the C_n-chain (n = 4k + 2) devices, we examine the transmission spectrum which reflects interiorly the current according to the Landauer–Buttiker equation. Fig. 3a presents the quantum transmission function of all C_n-chain devices in the equilibrium states (zero-bias). In general, under equilibrium states, the large transmission coefficient near the Fermi level mainly contributes to the transport current. It is evident that there are broad peaks around the Fermi level for all the C_n-chain transport systems, which can be ascribed to the strong coupling between the molecules and the carbon chain electrodes. When the molecular orbitals strongly couple with the electrodes, the broadening of the orbitals causes the molecular energy levels to approach the Fermi level of the electrodes, thus promoting resonant tunnelling, leading to an increase in the current magnitude and endowing the devices with excellent transport properties. Under zero bias voltage, the transmission coefficients of these devices exhibit the following hierarchy: C₂₆-chain > C₃₀-chain > C₃₄-chain > C₂₂-chain > C₁₈-chain > C₁₄-chain > C₁₀-chain, precisely consistent with the conductance at zero bias. Notably, the coefficient of C₂₆ exceeds 1, suggesting the presence of two transmission channels in devices and the highest currents. Upon careful observation, we can discern that the shapes of the transmission near the Fermi level transition from concave to convex with an increase in the number of carbon atoms. This transformation implies that the expansion of the bias window may lead to NDR effects.

Fig. 3 (a) Transmission spectra of devices C_n-chain (n = 4k + 2). The dotted line represents the Fermi energy level. (b) Transmission eigenstates of the devices under equilibrium states. (c) MPSH states of C_n (n = 4k + 2) molecules. The dominant molecular orbitals are circled by colored boxes.

To elucidate the physical underpinnings of the transport phenomena, the transmission eigenstates under equilibrium states are illustrated in Fig. 3b. These eigenstates unveil the delocalized electronic states spanning the entire devices, from the left electrodes to the central region and right electrodes. However, as the number of carbon atoms steadily increases, the bond-localized states in the central molecules of the devices gradually shift towards being delocalized. The bond-localized states are observed in the C₁₀-chain and C₁₄-chain devices. The electronic state of the C₁₀-chain is locally distributed along both sides of the carbon ring and is oriented perpendicular to the plane of the C₁₀-chain, due to the cumulene structure. Serving as the transition state between the cumulene structure and polyalkyne-like structure, the electronic state of the C₁₄-chain is localized on both sides of the carbon ring but along the in-plane orbitals of the C_n. Once the atom count surpasses 18, the polyynic structure causes the eigenstates to distribute along the in-plane orbitals and localize on one side of the C_n. The localization of electronic states in the upper half of C_n is primarily attributed to its intrinsic structural characteristics. The alternating short and long bonds in C_n induce spatially heterogeneous electron transport pathways when the devices are formed. The interfacial carbon atoms at electrode coupling sites exhibit closer proximity to the upper-half carbon atoms, creating geometric asymmetry that facilitates preferential electron propagation through the highly conjugated upper region (short-bond dominated).

Fig. 3c presents the MPSH eigenvalues around the Fermi level. The MPSH can be perceived as the renormalized molecular orbital of C_n that interacts with the electrode.⁴⁷ It can be seen that the LOMO orbitals of C_n with polyynic structures share similar features with the equilibrium transmission eigenstates of devices except C₂₆. In the case of C₂₆-chain, there are two primary channels, one from the LUMO and another from the HOMO. On the other hand, the eigenstates of C₁₀-chain and C₁₄-chain mainly originated from the LUMO frontier orbitals. The dominant orbitals can also be analyzed by the resonant tunneling model for transport through junctions. Take C₁₀ as an example. Fig. 2c illustrates that the energy of the LUMO orbital is nearly aligned with the electrode's energy, so electrons preferentially hop along the LUMO orbital. Other tunneling models of the remaining C_n-chain are provided in Fig. S1 (ESI†) to prove the main transport orbitals of the devices.

To shed light on the origin of NDR in the large carbon rings, we consider C₁₄-chain and C₃₀-chain as illustrative examples and compute their transmission spectra under typical biases of 1.2, 1.6, 1.8, and 2.0 V. As illustrated in Fig. 4a, the effective integral area of the transmission curve for C₁₄-chain exhibits an augmentation with the elevation of bias voltage, leading to a sustained increase in the current. In the case of the C₃₀-chain (Fig. 4d), the NDR effect occurs after 1.2 V, primarily due to a valley entering the integral area. Subsequently, with the emergence of a new peak within the bias window at 2.0 V, the current begins to rise once again. This correspondence aligns well with the I–V curve.

Fig. 4 (a) Transmission spectra of the C₁₄-chain device at typical biases. The insets show the transmission eigenstates of the Fermi level at corresponding biases. (b) MPSH states of the dominant orbitals for the C₁₄-chain at four typical biases. (c) Distance between the Fermi level and the dominant orbitals of C₁₄-chain under bias voltages. (d) Transmission spectra of the C₃₀-chain device at the typical bias. (e) MPSH states of the dominant orbitals for the C₃₀-chain at four typical biases. (f) Distance between the Fermi level and the dominant orbitals of C₃₀-chain under bias voltages.

When external bias voltages are applied, the molecular orbitals that serve as potential transmission channels may undergo changes.^48–50Fig. 4b and c show the changes in the HOMO state and orbital position of the C₁₄. Obviously, the dominating orbital for the C₁₄-chain is completely delocalized under various biases, resulting in a continuous increase in the current. Although the position of the HOMO moves away from the Fermi level in Fig. 4c, the transmission peak remains within the bias window without significantly affecting the transmission spectra. In contrast, the dominant orbital for the C₃₀-chain exhibits gradual localization with increasing voltage (Fig. 4e), leading to a decrease in the current. Such a change is primarily driven by the movement of the LUMO orbitals. The LUMO states quickly move away from the Fermi level when the bias voltage increases after 1.2 V. Although the HOMO–LUMO gaps (HLGs) are becoming smaller, the change cannot compete with the rapid movement of the dominant orbital (Fig. 4f). Consequently, the NDR effect occurs after 1.2 V. As the bias voltage continues to increase, the movement of the orbitals slows down, and the HLGs remain constant. This results in more electronic states transferring from the LUMO to the HOMO, and thereby increases the current value. All of these variations can be reflected in changes in the transmission eigenstates (Fig. 4a and d). We have revealed interesting transport properties of even-numbered aromatic C_n-based devices (n = 4k + 2) with different types of metallic electrodes. We show the NDR effect appearing in these devices composed of large diameter carbon rings. This phenomenon is attributed to the electron localization of dominant orbitals under bias voltage.

For comparison, it is interesting to investigate the device using an odd-numbered C_n, since the electron localization feature is closely related to the number of π-electrons. Thus, we constructed the C_n-chain with n = 4k + 3, as illustrated in Fig. 5a, where n ranges from 11 to 27. Despite some odd-numbered C_n differing by only one atom from even-numbered carbon rings, their transport properties exhibit significant differences compared to the even ones. From the I–V curve shown in Fig. 5b, all of these odd-numbered C_n-chains exhibit nearly Ohmic behavior at lower bias levels and generally manifest higher conductivity values than even C_n-chains. As a result, the transport properties of the C_n are sensitive to its number of carbon atoms and exhibit a clear even–odd behavior, similar to carbon atomic wires.^51–53 The Ohmic behaviors of odd-numbered C_n are primarily attributed to the relatively small HOMO–LUMO gaps, which are typically less than 0.5 eV (Fig. S2, ESI†). This characteristic makes them approach metallic properties. Therefore, odd-numbered C_n have a higher density of conductive electrons, making it easier to maintain a linear relationship between current and voltage.

Fig. 5 (a) Device configuration of C_n-chain (n = 4k + 3). (b) The I–V curves of C_n-chain (n = 4k + 3). (c) Transmission spectra of C₁₅-chain devices at typical biases. (d) Transmission eigenstates of the devices under equilibrium states.

Taking the C₁₅-chain device as an example, we plot its transmission spectra, transmission eigenstates, and MPSH in Fig. 5c and d. At zero bias, the transmission around the Fermi level for C₁₅-chain is larger than that of the C₁₄-chain, contributing to the large electronic conductivity. The primary channel originates from the HOMO orbitals, while the subsidiary channel is derived from the LUMO orbitals. These results are consistent with the energy alignments under the equilibrium states as shown in Fig. S3a (ESI†), because the HOMO and LUMO orbitals of the C₁₅ are close to the Fermi levels under zero bias voltage. Besides, Fig. 5d shows that the eigenvalues of the main channel become large and the eigenstate becomes delocalized, which can also be reflected in the completely delocalized HOMO across various bias voltages (Fig. S3b, ESI†). When the bias voltage is applied up to 2.0 V, a peak enters the right region of the bias window (Fig. 5c), and this peak originates from the LUMO, because it moves closer to the Fermi level of the electrodes (Fig. S3c, ESI†). The transmission spectra, transmission eigenstates, and MPSH of other C_n-chain (n = 4k + 3) are shown in Fig. S4 (ESI†).

3.2. Transport properties of C_n-ribbon and C_n-tube devices

The transport properties of the same molecule can exhibit significant variations when interfaced with different electrodes, owing to the molecular–electrode interfacial effects. Here, we employ ZGNRs and CNTs as electrodes to compare the transport properties of devices with different electrodes. The central regions of both devices consist of C_n ranging from C₁₀ to C₂₆.

We first focus on the C_n-ribbon devices. The I–V curves of the C_n-ribbon (n = 4k + 2) are presented in Fig. 1d. Distinct from the C_n-chain devices, the C_n-ribbon exhibit nonlinear characteristics, instead of Ohmic behaviour owing to the interaction between the electrode and C_n. A noteworthy observation is that all C_n-ribbon configurations manifest the NDR effect at 0.3 V. Furthermore, with an increase in the number of carbon atoms, the extreme current for NDR peaks decreases. Another phenomenon observed is that as the number of carbon atoms increases, the number of NDR peaks increases. When the number of carbon atoms is less than 22, only one bias voltage shows the NDR effect, whereas with 22 or more carbon atoms, the NDR peaks appear at two or more bias voltages, leading to the current oscillation. Such an oscillatory behaviour can be utilized in molecular devices for the desired switching functionalities.^54–56

For a more comprehensive exploration of the NDR effect and the regularity, we calculated the transmission spectra and corresponding eigenstates of C₁₄-ribbon and C₂₂-ribbon as representatives under the typical bias voltage, and the results are depicted in Fig. 6a and b. Different from the carbon chain electrodes, both C₁₄-ribbon and C₂₂-ribbon exhibit two distinct peaks (denoted as peak I and peak II) under equilibrium states. Other devices also have a similar phenomenon and can be found in Fig. S5a (ESI†). These isolated but salient resonant peaks are associated with the resonant electron transport through molecular energy levels and play a significant role in the transport process.^57,58 The analysis of transmission eigenstates reveals different transport paths for the C_n-ribbon with different electrodes. Through a comparative analysis of the transmission eigenstates and MPSH from Fig. S5b and c (ESI†), we can deduce that peak I of the devices predominantly originates from orbitals neighbouring the HOMO, and peak II stems from orbitals near the LUMO.

Fig. 6 (a) Transmission spectra of C₁₄-ribbon device at typical biases. (b) Transmission spectra of the C₂₂-ribbon device at the typical bias. The insets show the transmission eigenstates of the marked peaks at corresponding biases.

When the bias voltage ascends to 0.3 V, peak I of these two devices enters the bias window, resulting in a surge of current to a local maximum. Additionally, peak I of C₁₄-ribbon is larger than that of C₂₂-ribbon. With the bias voltage continuing to grow to 0.6 V, peak I gradually vanishes in both devices, leading to the first NDR effect. Peak II emerges into the bias window and dominates the I–V curves at 1.8 V. As a result, the current rises again. However, the peak II of C₂₂-ribbon declines at 1.9 V, triggering the second NDR effect. These multi-NDR behaviours can be applied to multiple-valued memory, analogy-to-digital converters, and multiple-valued logics.⁵⁹

The spatial distributions of the major transport channel are present in the insets of Fig. 6. Take C₂₂-ribbon as an example. The transmission eigenstate is delocalized at 0.3 V but becomes localized at 0.6 V. To further elucidate the localization of the transmission channels at the left electrode, we systematically investigated the positions of molecular orbitals and device density of states (DDOS) under bias voltages. As shown in Fig. S6b (ESI†), within the 0.6 V bias range, the dominant molecular orbital (HOMO−1) shifts away from the Fermi level with increasing bias. This results in a reduction of electrons transported through this molecular orbital. Meanwhile, the DDOS at 0.6 V also indicates that a large number of electrons are localized near the left-hand bias window and cannot spread uniformly across the entire device. Consequently, it becomes difficult for electrons to be injected from the left electrode into the right electrode, leading to the localization of electron states in the eigenchannels. This phenomenon elucidates the sharp decline in transmission and the subsequent decrease in current above 0.3 V. As the bias increases to 1.8 V, the eigenstate of C₂₂-ribbon becomes delocalized again, resulting in an upsurge in current. The eigenstates turn localized again at 1.9 V, which contributes to a subsequent reduction in current. In this manner, a repetitive occurrence of current oscillations takes place.

Another type of electrode we use to compare the electron transport properties of C_n (n = 4k + 2) is the carbon nanotube. Fig. 1e presents the I–V characteristics of the C_n-tube. Remarkably, the I–V curve continues to demonstrate a distinct degree of regularity. The NDR effect emerges when the number of carbon atoms exceeds 18. Specifically, the large-size carbon configurations exhibit NDR effects at lower bias voltages than small molecules. This trend suggests that the increased carbon ring size accelerates the onset of NDR effects.

Following a similar approach to the previous devices, we compute transmission spectra, corresponding eigenstates, and MPSH of the C_n-tube (n = 4k + 2). We take C₁₄-tube and C₂₂-tube as examples, as shown in Fig. 7a and b. One can see that the two resonant peaks on the opposite side of the Fermi level for the C₂₂-tube are closer than those in C₁₄-tube under the equilibrium state. For both devices, peaks on the right are much closer to the Fermi energy. Besides, as the number of central carbon atoms increases, we can see the same pattern in Fig. S7a (ESI†), where the peak of C₂₆-tube is the closest to the Fermi level. Notably, one of the dominant peak eigenstates is along the p_z-orbital, and the other is along the p_x-orbital (Fig. S7b, ESI†). The MPSH, as shown in Fig. S7c (ESI†), also provides insightful information that the transmission peaks I and II of the C_n-tube with a polyalkyne-like structure are more likely to be contributed by the LUMO and LUMO+2 orbitals.

Fig. 7 (a) Transmission spectra of the C₁₄-tube device at typical biases. (b) Transmission spectra of the C₂₂-tube device at the typical bias. The insets show the transmission eigenstates of the marked peaks at corresponding biases.

When voltage is applied, peak to the right of the Fermi energy level of the C₁₄-tube gradually comes into the bias window after 0.6 V, while peak to the left of the Fermi energy level appears within the bias window after 1.6 V, making the current gradually increase. For the C₂₂-tube, both dominant peaks enter the bias window at 0.6 V, thus the integral area of transmission spectra is the local maximum. Between 0.6 V and 1.0 V, the peaks decrease and the bias window increases, causing a plateau in the I–V curve. When the bias increases to 1.4 V, both peaks for the C₂₂-tube disappear, leading to a sudden drop in current. However, a new peak appears around the Fermi level, making the current rise again.

The eigenstates of the major transport channel for C₁₄-tube and C₂₂-tube devices are demonstrated in the insets of Fig. 7. The transmission eigenstates of the C₁₄-tube are localized when the bias voltage is 0.3 V, and become delocalized at 0.6 V, 1.0 V and 1.4 V. A new delocalized electronic state also appears at 1.6 V, leading to a steady increase in the integral area of transmission spectra and a rise in current. For the C₂₂-tube, when the bias voltage is set to 0.6 V and 1.0 V, both the dominant peaks in the bias window correspond to strong delocalized eigenstates. As the bias continuously increases to 1.4 V, the eigenstate of C₂₂-tube becomes localized again accompanying the disappearance of dominant transmission peaks.

Our systematic investigation of three types of C_n-based molecular devices reveals that large-diameter rings universally exhibit prominent NDR effects with notable electrode-independence. Compared to existing C₁₈-based devices and conventional biphenyl molecular junctions,^60–62 our designed devices demonstrate richer transport modalities: Ohmic behaviours near zero bias, NDR effects, and multiple NDR effects. Moreover, the electron-accepting properties of C_n and their dual π-electron delocalized system enable our devices to exhibit higher currents and larger NDR peak-to-valley ratios. These findings provide theoretical guidance for the development of multifunctional molecular devices and establish a new paradigm for designing carbon-based nanoelectronics. However, the development and fabrication of molecular devices face several challenges.^32,63,64 These challenges encompass precise control over electrode fabrication, adjustment of the molecule–electrode interface coupling, and the design of functional molecules with tailored properties. Additionally, the scalability, reproducibility, and device stability of molecular devices are also issues that need to be considered. To practically apply C_n-based molecular devices in future electronic technologies, it is crucial to address these challenges. Nevertheless, despite these obstacles, we believe that through the integration of advanced theoretical calculations and experimental innovations, breakthroughs will be achieved in this field. Although these aspects are beyond the scope of this paper, we earnestly hope that our research findings can serve as a wellspring of inspiration for future investigations, catalysing further progress in molecular electronics.

4. Conclusions

In summary, the transport properties of C_n (n = 4k + 2) were investigated using a combination of DFT and NEGF methods. The results indicate that all the devices with larger carbon rings exhibit NDR effects due to the gradual localization of electronic states. The larger the carbon rings, the lower voltage at which the NDR occurs. Notably, when carbon chains serve as electrodes, even-numbered C_n exhibited clearer NDR behavior compared to the ohmic behavior of odd-numbered C_n, highlighting an even–odd behavior in C_n devices. In the devices with ZGNRs as electrodes, electronic states of C_n are oriented perpendicular to the carbon atom plane, contrasting with the carbon chain and CNTs electrodes. Our findings provide insight into the NDR effect in C_n, which is valuable for tailoring nanoelectronics devices.

Data availability

The data supporting this article have been included as part of the ESI.†

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We would like to acknowledge the support from the National Natural Science Foundation of China (NNSFC) (Grant No. 52171038). This work is also supported by the Special Funding in the Project of the Taishan Scholar Construction Engineering and the program of Jinan Science and Technology Bureau (2020GXRC019) as well as new material demonstration platform construction project from Ministry of Industry and Information Technology (2020-370104-34-03-043952-01-11) and the Key Research and Development Plan of Shandong Province (2021SFGC1001).

Notes and references

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5tc00070j

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