Nonlinear transport behaviors in anti-aromatic cyclo[n]carbon-based (n = 4k) molecular devices

Abstract

Recently, a doubly anti-aromatic cyclo[16]carbon (C₁₆) has been successfully synthesized. Similar to doubly aromatic C_n (n = 4k + 2), anti-aromatic C_n (n = 4k), including C₁₆, feature vertically discrete π-electron conjugate systems, endowing them with unique electronic properties. However, the exploration of the transport properties of anti-aromatic C_n-based molecular devices is still in a rudimentary stage. In this study, we report on the electron transport properties of anti-aromatic C_n-based devices with three different types of electrodes, using density functional theory (DFT) combined with the nonequilibrium Green's function (NEGF) method. Our findings reveal that all devices exhibit nonlinear transport behavior regardless of the electrode used, including current-limiting functions, multiple-negative differential resistance (multi-NDR) effect, and current oscillation behaviors. Furthermore, as the number of carbon atoms increases, each type of device shows a discernible pattern in its transport properties. The results reveal the regularity of transport in C_n-based molecular devices and offer theoretical guidance for the development of next generation carbon based molecular devices.

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1. Introduction

All-carbon materials,¹ including graphene,² fullerenes,³ and carbon nanotubes,^4,5 characterized by sp² hybridization, stand as next-generation materials primed to catalyse disruptive advancements.^6–8 Another noteworthy category is represented by cyclo[n]carbons,^9,10 molecular rings composed of n carbon atoms showcasing rich electrical properties,^11,12 opening numerous additional new avenues for the development of carbon-based nanodevices.^13–16

Synthesizing carbon rings poses a challenge due to their chemical instability. Nevertheless, researchers have achieved the synthesis of C₁₈ by eliminating carbon monoxide from an epoxide molecule (C₂₄O₆), paving the way for the creation of other elusive carbon-rich molecules through atomic manipulation.¹⁷ Subsequently, the synthetic pathway was improved for C₁₈ by increasing the yield from 13% to 64% through the dehalogenation of a bromocyclo hydrocarbon precursor (C₁₈Br₆).¹⁸ This enhancement suggests that dehalogenation of cycloalkane precursors holds promise and enables the exploration of another C_n through surface chemistry. More recently, Gao et al. synthesized and characterized C₁₆ (an anti-aromatic carbon isomer) for the first time using tip-induced surface chemistry with C₁₆(CO)₄Br₂ as a precursor.¹⁹ Meanwhile, Sun et al. also prepared C₁₀ and C₁₄ through tip-induced dehalogenation of fully chlorinated naphthalene (C₁₀Cl₈) and anthracene (C₁₄Cl₁₀) molecules, followed by ring-opening via the anti-Bergman reaction.²⁰ Recently, they further extended the on-surface retro-Bergman ring-opening reaction and successfully produced C₁₂ and C₂₀.²¹ These successive syntheses of these C_n represent a significant milestone in carbon ring structural chemistry, instilling confidence and motivating further exploration in this field.²²

C_n molecules can be categorized into aromatic and anti-aromatic molecules based on their aromaticity.^9,23–27 For instance, C_n molecules with n = 4k + 2 (where k is an integer) are termed as aromatic C_n molecules, such as C₁₀, C₁₄, and C₁₈. Aromatic C_n molecules possess a 4n + 2 π electron system, resulting in a closed-shell electronic configuration that enhances stability. Conversely, when n = 4k, they are referred to as anti-aromatic molecules, characterized by a 4n π electron system, exhibiting inferior stability compared to aromatic counterparts. Several studies have focused on the electron transport properties of aromatic C_n molecules. For instance, diverse transport behaviors have been observed in C₁₈ molecular devices, encompassing ohmic, quasi-Schottky, and current-limiting functions, providing fundamental insights into the transport properties of novel C₁₈-based molecular devices.²⁸ Hou et al. engineered a novel multifunctional all-carbon-based molecular integrated device of C₁₈ which demonstrated a negative differential resistance (NDR) effect, spin-filtering effect, and giant switching effect concurrently, offering a versatile approach for manipulating the spin current.²⁹ These investigations on aromatic C_n serve as valuable references for the design of electronic spin devices and low-dimensional nano-integrated circuits. However, due to the high reactivity and low stability of anti-aromatic C_n, research on the electron transport properties of anti-aromatic C_n and its family is notably lacking, with only sporadic studies focusing on the electronic structure and transport properties of C₁₆.^30,31 Nevertheless, three anti-aromatic C_n (C₁₆, C₁₂, and C₂₀) have been successfully synthesized, and methods for synthesizing C_n have been improved, offering prospects for synthesizing other C_n in the future.¹⁹

Anti-aromatic C_n share similarities with their aromatic C_n counterparts, featuring a double π-electron off-domain system, suggesting comparable transport properties. Besides, studies on other anti-aromatic materials also suggest that these molecules tend to exhibit smaller highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) gaps and more diverse charge transport behaviors compared to aromatic molecules.^32–34 These properties make anti-aromatic electron acceptors promising candidates for applications in nano-electronic devices. Moreover, the transport characteristics of anti-aromatic C_n may demonstrate regularity with increasing numbers of carbon atoms. Therefore, we constructed three types of devices with different electrodes possessing metallic properties and systematically investigated their electron transport properties and mechanisms using a first-principles approach combining DFT³⁵ and NEGF.³⁶ Our findings aim to offer insights for the design of devices leveraging anti-aromatic C_n molecules.

2. Computational methods

The C_n molecules (Fig. 1a) are initially optimized using the ωB97XD/def2-TZVP function,^37,38 renowned for its significant non-local Hartree–Fock exchange energy (>40%).^25,39 This optimization method ensures the accurate reproduction of C_n structures observed in experimental settings. After optimization, each molecule exhibits a distinct pattern of alternating short and long bonds. They are sandwiched between carbon chain electrodes, graphene nanoribbon electrodes and carbon nanotube electrodes to form distinct devices. All three electrode materials are carbon-based materials, ensuring strong interaction with the molecules. Carbon chains, as the simplest 1D electrodes, can directly reflect the intrinsic transport properties of the C_n, carbon nanotubes provide an alternative 1D electrode configuration, and graphene nanoribbons, as quasi-2D electrodes, are chosen to explore the impact of electrode dimensionality on transport characteristics. Specifically, the carbon chain electrode comprises metallic carbon chains, the graphene nanoribbon electrode features a width of 4 atoms, and the carbon nanotubes adopt the (3,0)-type zigzag configuration. These three devices, illustrated in Fig. 1b, c, and d, are designated as the C_n-chain, C_n-ribbon and C_n-tube, respectively, with n representing the number of carbon atoms.

Fig. 1 (a) Structures of C_n (n from 12 to 32). Device configurations of (b) C_n-chains, (c) C_n-ribbons, and (d) C_n-tubes. I–V curves of (e) C_n-chains, (f) C_n-ribbons, and (g) C_n-tubes.

To investigate the electron transport properties, a combination of DFT and NEGF methods was employed, utilizing the ATK software package.^40,41 A vacuum layer exceeding 10 Å was introduced between adjacent cells to minimize periodic interactions. The Perdew–Burke–Ernzerhof (PBE)⁴² generalized gradient approximation (GGA)⁴³ was utilized for the exchange–correlation function. A double zeta polarization (DZP) basis set was adopted for all atoms to enhance computational accuracy. Sampling of k-points was conducted at a density of 1 × 1 × 100, with the transport direction aligned along the C-axis. The cut-off energy and electron temperature of the density grid were set to 75 Ha and 300 K, respectively. Prior to the computation of electron transport properties, all devices underwent optimization using the quasi-Newton method, ensuring that residual stresses on each atom remained below 0.01 eV Å⁻¹.

The transport coefficient is defined as:⁴⁴

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 are the coupling functions related to the left and right electrodes.

The current through the device was calculated using the Landauer–Buttiker equation:⁴⁵

where 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. T(E) represents the transmission function.

3. Results and discussion

3.1. Transport properties of C_n-chain devices

Before investigating its transport properties, we calculated the electron localization function (ELF) of molecules for equilibrium states as ELF is a function that reflects the localization of electrons in different regions of a molecule.⁴⁶ As shown in Fig. S1 (ESI†), the ELF isosurfaces for both types of C–C bonds in C_n exhibit a toroidal shape, indicating that both short and long C–C bonds possess a double π interaction character. Notably, the wider ELF isosurface observed for the short C–C bonds suggests a stronger π interaction compared to that observed for the long C–C bonds, which is consistent with the previous ELF study of C₁₈.⁴⁷

In order to study the transport characteristics of the anti-aromatic C_n devices under carbon chain electrodes, we calculated their I–V characteristic curves within the bias region [0.0 V, 2.0 V]. The large currents are observed in the C_n-chain device configurations, which can be attributed to stronger coupling between the molecules and the carbon chain electrodes. When molecular orbitals couple strongly with the electrodes, orbital broadening brings the molecular levels closer to the Fermi level of the electrodes. This facilitates resonant tunnelling, resulting in larger currents. The current–voltage curves reveal that all the antiaromatic C_n devices exhibit nonlinear transport behaviors (Fig. 1e), contrasting with aromatic C_n, where nonlinear transport behaviors typically occur only in large C_n (Fig. S2c, ESI†). Notably, as the number of carbon atoms increases, the transport characteristics gradually shift from a current-limiting effect (atoms < 16) to a NDR effect. Since the NDR effect is indispensable for several electronic components such as the amplifier, logic gate, Esaki and resonant tunnelling diodes,^48–50 we expect the C_n-chain device with large-diameter carbon rings to have a promising application in the NDR-based electronic components. More interestingly, when the bias voltage exceeds 1.6 V, the current at the same bias voltage decreases with the increase in the number of carbon atoms, following the trend: C₁₂-chain > C₁₆-chain > C₂₀-chain > C₂₄-chain > C₂₈-chain > C₃₂-chain. Our research findings are consistent with Baryshnikov et al.'s study on the magnetically induced current density of cyclo[n]carbons.²⁶ Their research elucidated a trend of decreasing electron delocalization with increasing molecular ring size, which is correlated with transport properties. Hence, this correlation substantiates the phenomena observed in our study. At the same time, this trend is also different from aromatic C_n devices, where the current at the same bias increases with the number of carbon atoms (atoms < 26) under low biases. In order to take into account the changes in transport properties due to the different electrode–molecule connection configurations, we also calculated the I–V curves for devices with alternative connection structures. As shown in Fig. S3 (ESI†), although the current in the I–V curves varies, the same transport trend remains for the C_n molecules: the current decreases as the size of the C_n increases, under the same bias voltage.

The band gaps of antiaromatic C_n molecules are all quite similar (Fig. S4a, ESI†). When they form devices, the molecular band gaps show considerable fluctuations due to the instability of antiaromatic C_n rings (Fig. S4b, ESI†), so they do not show a regular pattern at low bias voltages. Besides, the band gap of C₁₆ is the largest, which is due to the fact that the bond length alternation (BLA) of C₁₆ is the strongest among these molecules, and the BLA, in turn, is related to the off-domain nature of the electrons, which affects the electron transport of C₁₆.²⁶ It is worth noting that in this study, we ignore the band gap error caused by the GGA functional and only qualitatively compare the size of the band gaps. As the voltage increases, the HOMO and LUMO gradually return to their normal positions, both within the energy transport range, and both participate in the transport processes. The number of electron excitations increases, making electron transport mainly dependent on the number of carbon atoms passing through the central region, and the greater the number of atoms passing through, the greater the resistance and the greater the electron consumption, so the current decreases as the number of carbon atoms increases.

We calculated the electron transmission spectra of each device in their equilibrium states. As depicted in Fig. 2a, the electron transmission spectra near the Fermi energy level for all devices share a similar shape, characterized by two distinct and independent sharp peaks. Notably, the values of transmission peaks I and II decrease with an increase in the number of carbon atoms, aligning consistently with the observed current trends. This observation suggests that electron transport in anti-aromatic C_n devices is suppressed compared to their aromatic counterparts. Moreover, the presence of valleys within the peaks indicates that as the bias window widens, the current diminishes, leading to either a current-limiting effect or a NDR effect.

Fig. 2 (a) Transmission spectra of the device C_n-chain. The dotted line represents the Fermi energy level. (b) Maximum transmission eigenvalues and transmission eigenstates of peaks I and II at equilibrium states.

Fig. 2b illustrates the spatial distribution of dominant orbital eigenvalues and eigenstates of transmission peaks I and II at equilibrium states. In each device, transmission peaks I and II consistently consist of multiple transmission channels. The characteristic states of dominant orbitals reveal the electronic states of C_n devices from the left electrode, through the centre region, to the right electrode in a delocalized manner. The electronic states associated with transmission peaks I are predominantly distributed along the in-plane direction of the C_n. Furthermore, these states gradually localize as the number of carbon atoms increases, consistent with the variation observed in the transmission spectrum. In contrast, the electronic states corresponding to transmission peaks II are perpendicular to the plane of the C_n and remain largely unaffected by changes in the number of carbon atoms.

To comprehensively understand the transmission mechanism, we selected the C₁₆-chain as an example and conducted calculations on the eigenstates of four molecular projected self-consistent Hamiltonian (MPSH) quantities⁵¹ around the Fermi energy level of the device and transmission pathways in its equilibrium states. From Fig. S5a (ESI†), we can see that the dominant peak I of the C₁₆-chain is predominantly contributed by the HOMO, whereas the dominant peak II is mainly contributed by the LUMO, other devices show the similar feature as well except for the smallest C₁₂-chain and the largest C₃₂-chain (Fig. S6, ESI†). For the C₁₂-chain and C₃₂-chain, the dominant molecular orbitals are the orbitals close to the HOMO and LUMO. Furthermore, Fig. S5b (ESI†) shows that the local asymmetry of the transmission pathway is similar to that of the aromatic C₁₈-cased device studied by Zhang et al.,²⁸ but what differs is electrons primarily injected into the carbon ring along a long bond and subsequently transmitted to the right electrode along a long bond again. However, weaker reflection paths exist primarily along short bonds, which partially neutralize electron propagation.

According to the Landauer–Buttiker formula, it is known that the transmission spectrum under non-zero bias provides valuable data for analysing the current–voltage curves.⁵² Taking the C₁₆-chain and C₂₄-chain devices as examples, we further compute the electron transmission spectra under typical bias voltages. The electron transmission spectra of the C₁₆-chain device at 0.4 V, 0.8 V, 1.2 V, 1.6 V, and 2.0 V are given in Fig. 3a. When the bias voltage is below 0.8 V, the integral area within the bias window is predominantly contributed by the dominant peak I. The electronic states of the dominant peaks are mainly along the in-plane direction of C₁₆, which are the P_z-orbitals of the C₁₆, and the P_z-orbitals are more affected by the BLA, which affects the uniformity of electron transport, so the current of C₁₆ at low bias is smaller in comparison. However, beyond 0.8 V, transmission peak II gradually emerges within the bias window, co-dominating transport with peak I, resulting in a continued increase in current. As the bias voltage increases from 1.2 V to 1.6 V, both transmission peaks I and II begin to appear within the bias window, leading to a further increase in the integration area and subsequent current rise. However, beyond 1.6 V, all of transmission peak I has entered the bias window, and the appearance of a valley slows down or even halts the increase in integration area within the bias window, resulting in a current-limiting effect. It's important to note that the molecular orbitals in the molecular device providing transmission channels for electrons may alter after the application of external bias. Hence, Fig. 3b displays the spatial distribution of orbital states under the bias corresponding to transmission peaks I and II. It's evident that the main transmission eigenstates gradually localize with increasing bias, aligning with the changes observed in transmission peaks I and II.

Fig. 3 (a) Transmission spectra of the device C₁₆-chain at typical biases. (b) Maximum transmission eigenvalues and transmission eigenstates of peaks at corresponding biases.

The electron transmission spectra and eigenstates under non-zero bias for the C₂₄-chain device are illustrated in Fig. S7 (ESI†). Similar to the C₁₆-chain device, from 0 V to 1.6 V, further increases in the integration area led to an increase in current. However, beyond 1.6 V, both dominant transmission peaks have fully entered the bias window, and the emergence of new peaks is insufficient to counteract the appearance of the valley, resulting in a NDR effect. The spatial distribution of orbital states under bias indicates that as the values of transmission peaks I and II gradually decrease, the main transmission eigenstates gradually localize. Therefore, it does not draw as much current as the C₁₆-chain at the same bias voltage.

3.2. Transport properties of C_n-ribbon devices

When graphene nanoribbons are used as electrodes, the IV curves of all devices exhibit NDR effects, as shown in Fig. 1f. For devices with fewer than 16 carbon atoms, the current stabilizes within a certain range after the NDR effect appears within the bias range. However, for devices with more than 16 carbon atoms, the current rises again with the increase in bias after the NDR effect appears. Moreover, for devices with more than 28 carbon atoms, the NDR effect appears multiple times within the bias range, similar to the large carbon ring of aromatic C_n devices (Fig. S2d, ESI†). It is noteworthy that at low bias (<0.4 V), the current follows the pattern C₁₂-ribbon > C₁₆-ribbon > C₂₀-ribbon > C₂₄-ribbon > C₂₈-ribbon > C₃₂-ribbon. This observation is consistent with that of devices with carbon chain electrodes at high bias voltage. Additionally, when the current rises again, the corresponding bias voltage also follows this pattern.

Similar to the study of devices with carbon chain electrodes, we calculated the electron transmission spectrum of each device at equilibrium states. As shown in Fig. 4a, the electron transmission peaks at the Fermi energy level at zero bias indicate that the transmission probability decreases with the increase in the number of carbon atoms, consistent with the current law at low bias. When the number of carbon atoms is less than 20, the device has only one strong transmission peak in the energy range. However, when the number of carbon atoms exceeds 20, multiple transmission peaks appear in the energy range, suggesting the possibility of multiple occurrences of the NDR effect with the increase of the bias window. Additionally, when the number of carbon atoms exceeds 20, the distance between two adjacent transmission peaks (highlighted by coloured dashed arrows) gradually decreases, indicating the sequence of current recovery.

Fig. 4 (a) Transmission spectra of the device C_n-ribbon. The dotted line represents the Fermi energy level. (b) Transmission eigenvalues and transmission eigenstates of peaks at equilibrium states.

Fig. 4b depicts the eigenstates of the transmission peaks at equilibrium states, where each transmission peak consists of only one transmission channel. The transmission eigenstates reveal the delocalized electronic states of the devices. However, unlike the carbon chain electrodes, the distribution of the electronic states of the spikes on the right side of the Fermi energy level is mainly along the direction perpendicular to the plane of the C_n, instead of the in-plane direction to the plane of the C_n.

Taking the C₁₆-ribbon device as an example, the molecular MPSHs and the transmission pathways are calculated (Fig. S8, ESI†). As shown in Fig. S8a (ESI†), the eigenstates of the dominant peak in the C₁₆-ribbon are mainly contributed by the LUMO. Dominant orbitals for other C_n-ribbon devices are shown in Fig. S9 (ESI†). All of them have similar electron cloud distribution, all along the out-of-plane direction of the carbon rings. Simultaneously, the transmission pathways are similar to those in the C₁₆-chain. However, there are more reflective paths compared to the C₁₆-chain configuration, resulting in weaker currents (Fig. S8c, ESI†).

Next, we further recorded the electron transmission spectra of the C₁₆-ribbon device and the C₂₄-ribbon device at typical bias voltages. As depicted in Fig. 5a, when the bias voltage applied to the C₁₆-ribbon device is below 0.8 V, the transmission peak on the right side of the Fermi energy level gradually enters the bias window, while the transmission peak on the left side decreases, reaching its maximum integrated area at 0.8 V, coinciding with the peak current of the C₁₆-ribbon device. After 0.8 V, the transmission peak on the right side gradually diminishes within the bias window, almost disappearing by 1.2 V, leading to a sudden drop in the integrated area and the emergence of a NDR effect. However, with further increases in bias voltage, the second transmission peak on the right side gradually appears within the bias window, resulting in a slight increase in current. Fig. 5b illustrates the spatial distribution of the orbital states of the transmission peaks of the device at the corresponding bias voltages. We see that the transmission is mainly along the P_x-orbital, which is less affected by the BLA. However, the HOMO–LUMO gap is also small, so the transport is mainly affected by the molecular size at low bias, which further explains why the devices show transport regularity at low bias. It is also evident that the change in the transmission eigenstates of the device aligns with the variations in its transmission peaks, with the probability of electron transmission being highest at 0.8 V and lowest at 1.6 V.

Fig. 5 (a) Transmission spectra of the device C₁₆-ribbon at typical biases. (b) Transmission eigenvalues and transmission eigenstates of peaks at corresponding biases.

The electron transmission spectrum, transmission eigenvalues and transmission eigenstates of the C₂₄-ribbon device under non-zero bias voltages are shown in Fig. S10 (ESI†). At the initial stage (0–1.2 V), the transmission peaks demonstrate similar patterns to those of the C₁₆-ribbon (Fig. S10a, ESI†). However, beyond 1.2 V, a secondary peak on the right side gradually emerges within the bias window, leading to a subsequent increase in current. As observed in Fig. S10b (ESI†), at 1.2 V, the electronic states associated with the transmission peaks predominantly localize on the left electrode and within the device. This localization impedes electron transmission towards the right electrode, thereby yielding the lowest current. Subsequently, beyond 1.2 V, the electronic states sporadically distribute towards the right electrode, facilitating a rebound in the current.

3.3. Transport properties of C_n-tube devices

When employing carbon nanotubes as electrodes, all devices still exhibit nonlinear transport behavior compared to aromatic C_n-based devices (Fig. S2e, ESI†). As depicted in Fig. 1g, the devices with fewer than 16 carbon atoms exhibit a NDR effect at only one bias point, while those with more than 20 carbon atoms display this effect at two or more bias points, accompanied by current oscillations. Similar to devices with carbon chain electrodes, after applying a bias voltage exceeding 1.6 V, the current at the same bias voltage decreases with an increase in the number of carbon atoms in the central region, following the trend: C₁₂-tube > C₁₆-tube > C₂₀-tube > C₂₄-tube > C₂₈-tube > C₃₂-tube. The reason for this phenomenon is the same as for devices with carbon chain electrodes.

The electron transmission spectra of each device at equilibrium are depicted in Fig. 6a. At zero bias, the peak values of the transmission peaks near the Fermi energy level for all the devices exceed 1, indicating their strong transport capability. In addition, the electron transmission spectra at the Fermi energy evolve from a single transmission peak in the C₁₂-tube device to two peaks in the C₂₄-tube device and eventually to three peaks in the C₃₂-tube device. The reason is that as the number of carbon atoms increases, the LUMO+1 gradually approaches the Fermi level of the electrodes (from Fig. S12 (ESI†), we can also deduce that peak III originates from the resonance of the LUMO+1 with the electrode). Consequently, multiple molecular orbitals resonate with the Fermi level of the electrodes, leading to the emergence of multiple resonance peaks. These transmission peaks suggest that as the bias window widens, the base of the peak enters the bias window, leading to a reduction in the effective integration area and the occurrence of NDR effects multiple times, resulting in current oscillation. The spatial distribution of eigenstates for the transmission peaks I, II and III at equilibrium state is presented in Fig. 6b. Each transmission peak comprises multiple transmission channels, with the largest transmission channel corresponding to a higher transmission eigenvalue. The eigenstates of transmission peaks I and II indicate that their electronic states are mainly distributed along the in-plane direction of the C_n, while the eigenstate of transmission peak III indicates that its electronic states are mainly distributed along the in-plane direction perpendicular to the C_n, and the off-domain nature of all three transmission peaks is notably prominent.

Fig. 6 (a) Transmission spectra of the device C_n-tube. The dotted line represents the Fermi energy level. (b) Maximum transmission eigenvalues and transmission eigenstates of peaks I, II, and III at equilibrium states.

As shown in Fig. S11a (ESI†) (other molecule orbitals are depicted in Fig. S12, ESI†), the dominant peak I and dominant peak II in the equilibrium state of the C₁₆-tube device are contributed by the LUMO and LUMO+1, respectively. Analysis of the transmission paths reveals two primary transmission routes between the electrode and the C₁₆ (Fig. S11c, ESI†). One main path involves electron transmission along the coupling atom to the carbon atom along the long bond, then to the right electrode via the upper part of the ring carbon. The other path entails electron transfer directly from the carbon nanotube to the carbon atom without passing through the coupling atom, subsequently transmitted to the right electrode. Numerous reflection paths exist in the transmission process, increasing the probability of NDR effects.

Similar to the previous calculations, we further explore the electronic transmission spectra and transmission eigenstates corresponding to the transmission spectra under typical biases, using the C₁₆-tube and C₂₄-tube devices as examples. As depicted in Fig. 7a, during the initial stage (0–0.4 V), the transmission spectrum within the integration area of the C₁₆-tube device is primarily contributed by the dominant peak I. At 0.8 V, transmission peak II and the neighbouring peaks move closer to the Fermi energy level, resulting in an increased integration area and current. By 1.0 V, a weaker peak and a stronger peak emerge within the bias window after the merging of three peaks, leading to a reduction in current and the onset of a NDR effect. Subsequently, beyond 1.2 V, the integration area expands, accompanied by an increase in current within the bias window. The spatial distribution of the orbital states of the transmission peaks at the corresponding bias voltages, illustrated in Fig. 7b, shows that peaks I and II consist of multiple transmission channels, all along the in-plane direction of the C₁₆. In contrast, the peak on the right side of peak II consists of only a single transmission channel and is along the out-of-plane direction of the C₁₆. Notably, the variations in transmission eigenstates under bias align consistently with the changes in transmission peaks.

Fig. 7 (a) Transmission spectra of the device C₁₆-tube at typical biases. (b) Maximum transmission eigenvalues and transmission eigenstates of peaks at corresponding biases.

The electron transmission spectrum of the C₂₄-tube device under non-zero bias is shown in Fig. S13a (ESI†). As the bias voltage exceeds 0.4 V, peak I shifts towards higher energies, causing its bottom to enter the bias window, and resulting in a decrease in the integration area and current. By 0.8 V, the peaks on the right side of the Fermi energy level coalesce into a single peak, progressively entering the bias window with its expansion, and thereby increasing the current. At 1.2 V bias, all peaks enter the bias window, culminating in a local maximum of the current. Beyond 1.2 V, the valleys gradually encroach upon the bias window with its further expansion, leading to a decline in current. After 1.4 V, multiple transmission peaks gradually infiltrate the bias window, resulting in a resurgence of current. The electronic states corresponding to the transmission peaks exhibit increased off-domain characteristics, consistent with their alterations (Fig. S13b, ESI†).

In our studies on these three types of devices, we find that all devices exhibit nonlinear transport characteristics. Compared to the existing C_n molecular devices, our anti-aromatic C_n molecular devices exhibit more diversified transport behaviors, such as current oscillation effects, NDR effects, and multi-NDR effects, demonstrating the versatility of our molecular devices.⁵³ In comparison to other molecular devices, the electron acceptor properties of anti-aromatic C_n, along with their dual π-electron off-domain systems, show higher current and a larger NDR peak-to-valley ratio^54–57 (Table S1, ESI†). These outstanding properties of our designed molecular devices are fundamental to the application of molecular-scale devices and play a critical role in the design of nanoscale electronic components, including high-frequency oscillators, mixers, multipliers, logic circuits, and analog-to-digital converters.^58–60 Moreover, experimental studies on two-dimensional molecular devices have demonstrated nonlinear transport behavior, thereby providing a robust experimental validation of the theoretical predictions.^61–63 We are confident that our DFT results contribute valuable theoretical insights and predictive guidance for understanding the nonlinear transport properties of molecular devices.

3.4. Main challenges and opportunities in the fabrication and integration of molecular devices

Molecular electronics, as a promising alternative to silicon-based microelectronics, has made significant strides over the past few decades in both theoretical and experimental research. In this study, the nonlinear transport behaviors of anti-aromatic C_n molecules have been discovered for the first time using theoretical studies, which provide theoretical guidance and pave the way for their application to electronic nanocircuits. Nevertheless, the path from theoretical designs of single-molecule devices to experimental fabrication and large-scale integration remains a significant challenge.

From an experimental perspective, key factors such as the fabrication of electrodes, control of molecule–electrode interface coupling, and the design of functional molecules are critical in determining the performance and stability of molecular devices. C_n molecules are chemically active, but their synthesis efficiency has been improving in recent years (as mentioned in the Introduction section), providing a viable basis for their application to practical devices. Besides, the carbon-based electrodes utilized in this study address many issues associated with metallic electrodes, such as incompatibility and unintended mobility. However, the performance of these electrodes is highly dependent on factors such as the precise diameter and chirality of carbon nanotubes, as well as the edge configuration of graphene.^64–66 Consequently, achieving atomic-scale precision in electrode fabrication, precisely controlling molecular conformations within graphene gaps, and optimizing contact configurations remain formidable technical hurdles. Moreover, the contact chemistry and coupling strength at the molecule–electrode interface play a pivotal role in determining the electrical characteristics and long-term stability of molecular devices. While commonly employed strategies, such as self-assembled monolayers (SAMs) and Langmuir–Blodgett (L–B) films, facilitate molecular connections, these methods often result in unstable surface bonds, leading to electronic instability and stochastic behavior.⁶⁷ Furthermore, as the number of connecting molecules is reduced to the single-molecule scale, the device becomes increasingly sensitive to the atomic structure and local environmental factors, exacerbating variability.⁶⁸ For example, molecular devices are highly susceptible to external influences, including temperature, humidity, light, and electromagnetic fields, with even minor fluctuations destabilizing their performance.^69–71 Robust and reliable methodologies are therefore urgently required to address these challenges, including fabrication of electrodes, the intramolecular connections, and external environments, to ensure precise control over molecular device behavior.

The ultimate goal for molecular electronics is to fabricate commercial devices and apply them to daily life.^72,73 Hence, integrating molecular electronic components into modern microelectronics for easier information readout is an inevitable choice. Unfortunately, efficient integration strategies for bridging hard electronics with the vast store of the soft-molecular world are still lacking. Challenges such as maintaining a clean environment during integration, applying mild contact forces, addressing incompatibilities between silicon-based chips and molecular devices, and ensuring the reversibility of integration processes remain complex and require meticulous control. However, advances such as polymer interlayers, graphene films, and specialized metal layers have significantly improved device yields, enabling the development of parallel integrated junction arrays fabricated with specific functionality.^71,74–76 Developing the actual device is still much more complex and would certainly introduce a range of disruptive and destructive effects, and these issues are beyond the scope of the manuscript discussion.

Despite these obstacles, molecular electronics holds immense potential to complement silicon-based technologies, enabling ultra-miniaturization and providing functionalities that surpass conventional limits. It is evident that the convergence of advanced theoretical calculation and experimental innovation will unlock the transformative potential of molecular-scale electronics. This synergy will bridge the divide between rigid electronic systems and soft molecular architectures, paving the way for ground-breaking advancements and a promising future in the field.

4. Conclusions

In this study, we explored three novel molecular devices featuring anti-aromatic C_n molecules as the central unit, sandwiched between different types of electrodes-carbon chains, ZGNRs, and CNTs. Our investigation revealed that all devices exhibit nonlinear transport properties, including current-limiting functions, negative differential resistance, and current oscillation behaviors, distinguishing them from aromatic C_n. Additionally, we observed that for devices with carbon chains and CNTs electrodes, at high bias voltages, the current at the same bias voltage decreases with the increase in the number of carbon atoms. Conversely, in devices with ZGNR electrodes, this trend manifests with the application of small bias voltages. These findings offer fundamental insights into the transport properties of novel doubly anti-aromatic cyclo[n]carbon-based molecular devices, unveiling underlying regularities, and providing theoretical guidance for future applications in molecular 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 was 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 the new material demonstration platform construction project from the Ministry of Industry and Information Technology (2020-370104-34-03-043952-01-11) and the Key Research and Development Plan of Shandong Province (2021SFGC1001).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4tc04033c

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