Tuning charge density waves and magnetic switching in carbon nanowires encased in boron nitride nanotubes†

Chi Ho Wong*^a, Zongliang Guo^d, King Cheong Lam^a, Chun Pong Chau^a, Wing Yu Chan^a, Chak-yin Tang^c, Yuen Hong Tsang^d, Leung Yuk Frank Lam^b and Xijun Hu^b
aDivision of Science, Engineering and Health Studies, School of Professional Education and Executive Development, The Hong Kong Polytechnic University, Hong Kong, China. E-mail: roy.wong@cpce-polyu.edu.hk
^bDepartment of Chemical and Biological Engineering, The Hong Kong University of Science and Technology, Hong Kong, China
^cDepartment of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, China
^dDepartment of Applied Physics, The Hong Kong Polytechnic University, Hong Kong, China

First published on 5th July 2025

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New concepts Our research introduces the groundbreaking concept that exotic charge density waves (CDW) are not limited to transition metal composites but can also emerge in carbon nanowires due to the proximity effect. This proposes reconsidering the traditional understanding of CDW formation, as our findings demonstrate that carbon in the cumulene phase can exhibit unconventional CDW patterns without the need for complex electron correlations, periodic lattice distortion or spin–orbit coupling. What sets this work apart is the coexistence of ferromagnetism or ferrimagnetism with CDW in carbon chains, a phenomenon previously thought to be exclusive to transition metals. This insight opens new avenues for materials design, as it suggests that carbon-based materials can exhibit complex exotic properties. Furthermore, our findings highlight that carbon nanowires can achieve a sharp insulator-to-conductor transition through CDW modulation under constant voltage excitation. This has profound implications for quantum electronics and marks our work as a significant contribution to topological science, sparking interest in future research and applications across organic and materials science.

1. Introduction

Charge density waves (CDW) and their interactions with ferromagnetism or ferrimagnetism are prominent areas of study in topological condensed matter research, particularly due to the applications of resistive switching and resonant oscillations for topological computing systems.^1–5 However, CDW often competes with non-zero net magnetization (e.g. ferromagnetism and ferrimagnetism), with only a few exceptional cases comprising transition elements documented.^1–4 CDW usually coexists with antiferromagnetism (AFM)⁶ such as in iron-based superconductors. The technical problem is that the combination of AFM and CDW poses challenges in field-effect applications, as tuning the CDW state with an external field is ineffective.⁷ Therefore, ferromagnetic or ferrimagnetic CDW materials can be crucial for these quantum applications.^8–12

However, many of these desirable exotic properties originate from heavy transition metals. The presence of spin–orbit coupling and p–d–f shell hybridization complicates the tuning of these quantum phenomena,¹³ as any adjustment necessitates a simultaneous readjustment of multiple parameters. This raises questions about whether these complex quantum phenomena can also be realized in systems that rely solely on p-shell electrons, which present a clearer framework for tuning. In this regard, a LCC, commonly known as carbyne,^14–16 represents a promising option for this framework. Although a linearly straight carbon chain is always non-magnetic, 400 K ferromagnetism has been experimentally observed in a parallel array of finite-length carbon chains with the help of periodic kink structures.¹⁷ Despite its high Curie temperature, the creation of CDW patterns along these ferromagnetic chains is still an open question.

To enable the coexistence of CDW in a magnetic carbon chain and meanwhile allow CDW to respond to external perturbations for quantum switching, we need to address at least seven huge challenges. First, periodic kink structures are required to induce magnetism in the chain.¹⁷ Since the internal carbon chain within the nanotube always remains straight,^17–19 we need to come up with another strategy to induce magnetism in the carbon chain without these kink structures. Second, it is essential to ensure that organic magnetism is not antiferromagnetic. Otherwise, it cannot react with the external magnetic field for tuning the CDW states. Third, ferromagnetic or ferrimagnetic exchange interactions should be maintained at room temperature, which is often challenging in organic materials. Fourth, the lattice spin must be carefully balanced: sufficiently large to enable couplings with CDW, but not so large that it overwhelms their competing effects.^1–4 Fifth, while CDW is always observed in heavy transition elements,^1–4 it is an uphill struggle to create CDW patterns from the light atom (carbon), without relying on strong electronic correlation or spin–orbit coupling. Sixth, we need to determine the cut-off length of the internal chain within the nanotube. If the chain is very short, we have to ensure that its finite length can still maintain the desired properties. Finally, we need to analyze whether the CDW state could be influenced by external perturbation (e.g. magnetic field or electric field) to assess the potential for quantum switching.

On the other hand, creating bulk carbynes remains a significant challenge despite extensive experimental efforts.^14–16 Current manufacturing techniques can only connect ∼6400 carbon atoms in a 1D structure while double-walled carbon nanotubes (DWCNT)^18,19 serve as the host. Predicting the cut-off length limit for carbon chains on other substrates or nanoreactors presents a considerable experimental challenge. Although the science of bulk carbon chains can be predicted using repeating units in ab initio software for valuable insights,¹⁴ uncertainty about the actual length presents a significant challenge in correcting for size effects because desired properties may vanish when the carbon chain becomes finite. To carefully consider the size effect, the carbon chain array model developed by C. H. Wong et al.²⁰ can estimate the likely range of chain lengths and lateral separations within array structures, allowing ab initio software to later assess whether the desired magnetic properties remain valid in a finite scale.¹⁷ In 2023, C. H. Wong et al. developed the carbyne@nanotube model,¹⁹ which examines how these nanotube structures influence the cut-off length of the internal carbon chain.¹⁸

To explore magnetic carbon chains within a nanotube structure, carbon nanotubes are not ideal hosts due to the involvement of magnetic transition elements during CNT fabrication,²¹ which may lead to false magnetic detection of carbon chains. Hence, we are seeking an alternative host, i.e. boron nitride nanotubes (BNT),²² to see if there is a chance to overcome the seven challenges for achieving the exotic applications.

Regarding the stability of the host, recent advancements in the fabrication of a substantial proportion of single walled BNT has been achieved using a continuous CO₂ laser ablation system.²³ However, research is ongoing to improve quality control. When the laser (1000 W, 10.6 μm) focused on the boron filaments, they melted into a molten boron ball. Nitrogen gas dissociated on the surface, transforming into nitrogen atoms, which then reacted with boron to form BNT.²³ Although both CNT and BNT possess hexagonal structures, the key differences in their bonding and charge distribution affect the stability of integrating carbon chains into BNT. In CNT, the purely covalent bonds exhibit symmetric charge distribution, facilitating the formation of internal chains. In contrast, the B–N bond is partially ionic, leading to asymmetric charge distribution, which further complicates in synthesizing the internal chain. However, if the mentioned seven challenges can be addressed for short internal chains, instead of lengthy chains within BNT, sample fabrication could become easy.

2. Computational details

We combine the ab initio and Monte Carlo methods to simulate the results. Unless otherwise stated, we employ the LDA–PWC functional within the CASTEP software to relax the LCC@BNT composites.²⁴ We use a SCF tolerance of 1 × 10⁻⁵ eV and allow up to 9000 SCF cycles to obtain the relaxed geometry of LCC@BNT. The magnetism and CDW are computed using Dmol3 software in BIOVIA Materials Studio. Then the Monte Carlo carbyne@nanotube model¹⁹ is employed to reduce the chain length in a finite scale. If the corrected chain length becomes very short, recalculating the physical properties in the form of a non-periodic supercell is necessary. The ab initio-calculated atomic coordinates are fed into the Hamiltonian of the carbyne@nanotube model in order to update the atomic coordinates of the internal carbyne at finite temperatures, where the steps have been listed in the ESI.†¹⁹

We revisit the Hamiltonian of the carbyne@nanotube model¹⁹ in an approximated form,

The adjacent bond distance of carbon is _n. The lateral distance between LCC and BNT is r. The type of covalent bond formed between adjacent carbon atoms is described by stochastic variables j and n, where j represents the bond energies¹⁹ (1 for single bond: E₁ = 348 kJ mol⁻¹; ^eq_n,1 = 154 pm, 2 for double bond: E₂ = 614 kJ mol⁻¹; ^eq_n,2 = 134 pm, and 3 for triple bond: E₃ = 839 kJ mol⁻¹; ^eq_n,3 = 120 pm)^19,20 for the nth carbon atom. For instance, E_n,j (n = 500, j = 2) refers to the 500th carbon atom (n = 500) forming a double bond (j = 2) with respect to the 499th carbon atom (n − 1 = 499). The temperature T_bj corresponding to bond dissociation = bond energy/Boltzmann constant k_B.^19,20 We initialize the structure of carbyne@nanotube under various chirality numbers (N_c,M_c). The initial bond type of the internal LCC is j = 2. To assess the stability of the carbyne within the BNT environment, we introduce a chain-stability factor,¹⁹ defined as . The van der Waals VDW term between BNT and LCC along the circular plane is . The σ and φ are the VDW constants,^19,20 which can be interpreted from isothermal compressibility and the sample length τ_s.^19,20 More specifically, the van der Waals constant φ and σ are calculated by solving two formulae, and at equilibrium,²⁰ where ζ is isothermal compressibility. The computed φ and σ are 8.2 × 10⁻²³ J and 1.2 × 10⁻¹⁰ m accordingly. Since the volume response to pressure changes at constant temperature and entropy is nearly identical in solids, the distinction between isothermal compressibility and isentropic compressibility becomes rare. Therefore the isothermal compressibility can be set as , where v_s is the speed of sound in solid with density ρ.²⁰

Based on the ab initio data of a kink structured carbon chain, the angular energy term is calibrated by fitting (cosθ + 1)² function, with the J_A of ∼600 kJ mol^−119,20 and the pivot angle θ along the carbon chain. The angular energy increases when the pivot angle is non-linear. As the pivot angle increases by θ, the angular energy (E_angular) of the carbon chain also rises. We then subtract the angular energy of a kink structured carbon chain from the linear energy (E_linear) of a straight carbon chain, defining the difference as the net angular energy = E_angular − E_linear = J_A(cosθ + 1)².

To simulate the dynamic behavior of the carbyne chain within the nanotube environment, we employ the Monte Carlo approach. During the iterative process, the atomic coordinates and the types of covalent bonds in the carbon chain are amended at finite temperatures. In each Monte Carlo step (MCS), the simulation follows these steps:¹⁹

• Randomly select an atom in the carbon chain.

• Calculate the initial Hamiltonian of the system.

• Propose a trial range of spatial fluctuations and a trial type of covalent bond (Fig. S1 in the ESI†) for the selected atom.

• Estimate the trial Hamiltonian based on the proposed changes.

• If the trial Hamiltonian is less positive (or more negative) than the initial Hamiltonian, the trial states are accepted. Otherwise, the system reverts to the previous states.

• The trial range of atomic fluctuations is also controlled by the Hooke's factor (ESI†).

Thermal excitation is another opportunity to accept or reject the trial states in parallel, which is governed by the comparison between a random number with the Boltzmann factor.^19,20 The Monte Carlo MCS process continues until the energy vs. MCS becomes flattened, typically after approximately 250000 steps based on our previous result.¹⁹ The data are then averaged over the last 20000 equilibrium data points. Furthermore, we analyze the cut-off length of the internal chains in BNT versus CNT. This approach allows us to compare the stability and length of LCC within distinct nanotube structures. After correcting the size effect using the carbyne@nanotube model, we estimate the exotic properties of the internal carbon chain again.

3. Results and discussion

3.1. Long LCC@BNT as a ferrimagnetically frustrated CDW conductor

Despite an isolated straight carbon chain being always non-magnetic,¹⁷ our search for magnetism in a straight carbon chain begins with introducing the proximity interactions from BNT. The carbon chain is inserted in zigzag nanotubes (N_c,M_c) with the diameters between ∼0.4 nm and ∼1.7 nm, where N_c is from 5 to 21 and M_c is 0. For N_c < 11, the internal carbon chain is symmetric along the circular plane of the BNT. However, when N_c ≥ 11, the internal carbon chain automatically breaks this symmetry and is displaced towards the BNT surface (inset of Fig. 1(a)). If the tube diameter is small, the internal chain benefits from a uniform lateral van der Waals force along the angular plane. However, when the tube diameter exceeds ∼0.9 nm (or N_c ≥ ∼11), the internal chain loses the uniform van der Waals protection from the nanotubes. We observe that the internal chain tends to partially regain van der Waals protection by displacing the internal carbon chain (inset of Fig. 1(a)) radially. Without this radial displacement, the carbon chain risks losing all its van der Waals protection for maintaining stability within the nanotube structure. By adjusting its geometric arrangement radially, the carbon chain can still interact with the surrounding nanotube walls, allowing it to harness some of the lateral van der Waals forces. This partial van der Waals interaction helps to provide a degree of stability and support, even in the absence of the full protective effect initially present at smaller diameters.

The pivot angles between the carbon chains range from 179.80 to 179.96 degrees for the tube diameters analyzed in Fig. 1(a), suggesting that the chains can be regarded as straight.¹⁷ Therefore, we cannot apply the periodic kink-structure strategy¹⁷ to induce magnetism in the carbon chains, which typically requires a threshold pivot angle sharper than 170 degrees.¹⁷ Our simulations indicate that only the (17,0)BNT and (19,0)BNT exhibit magnetism. The average magnitude of magnetic moment of the carbon chain in (17,0)BNT and (19,0)BNT are both ∼0.6μ_B, which is at least three times larger than the average magnetic moment of the kink-structured carbon chains.¹⁷ Moreover, the local lattice spin of the internal carbon chain in (19,0)BNT is slightly greater, measuring ∼0.9μ_B, compared to ∼0.7μ_B in (17,0)BNT, but no magnetic properties are observed in the B and N atoms. Consequently, we focus solely on the average spin DOS of the C atoms in Fig. 1(b), which illustrates their magnetic nature. Using a 9-atom-long carbon chain enclosed by BNT forms a repeated unit, rather than using a basic unit of 3-atom-long carbon chain, because it more clearly draws the oscillations along the chain axis. If a 2-atom-long (or 4-atom-long) carbon chain is used per repeated unit, the carbon atoms are too far apart (or too close together) to form effective trial covalent bonds. Consequently, those configurations fail to relax the system to its ground state in our ab initio simulations. If we use a unit cell of 3-atom-carbon-chain@BNT, it generates a pattern of -[1st C atom: −0.3μ_B; 2nd C atom: +0.8μ_B; 3rd C atom: −0.3μ_B], where the 1st and 3rd C atoms have the same magnetic moment. If we duplicate or extend this unit cell, the 4th C atom would also be assigned −0.3μ_B manually. But this manual extension may trigger a problem in observing science. For materials with well-known magnetism, such as bulk iron, using translational symmetry to assign the magnetic moment by extending the repeated unit is not problematic. However, to investigate the magnetic moment under unusual magnetism, like that seen in the internal carbon chain, we should check not only whether magnetic frustration occurs but also check if it impacts the magnetic moments of adjacent C sites at the frustrated regions. Therefore, we selected a longer unit cell, specifically a 9-atom carbon chain within BNT, which not only reveals oscillations but also indicates that the magnetic moments near the frustrated interface are weakened. Since the local magnetism of the carbon atoms exhibits a stronger magnetic moment inside (19,0)BNT, we analyze the distribution of magnetic moments of the chain within (19,0)BNT in Fig. 1(c), where we observe the two spin directions of carbon atoms compete, leading to the emergence of magnetic frustration, akin to a kagome lattice.²⁵

We observe a short-range CDW pattern in the carbon chain within the (19,0)BNT between the first and third carbon atoms in Fig. 1(d). The CDW structure should have restored a DOS (E_F) of 0.83 electron per eV and 0.14 electrons per eV for the 4th and 5th carbon atom, respectively. However, magnetic frustration between the sixth and seventh carbon atoms disrupts the long-range establishment of the CDW. It is because the differential electron (charge) density at the Fermi surface triggers an equivalent effect of local magnetism based on Maxwell's equations,^6,26–29 which is further influenced by the lattice spin distribution. As a result, the DOS (E_F) in the vicinity of the magnetically frustrated region is affected, where the “bridge” is the local magnetism. This also explains why the DOS (E_F) at the 5th carbon differs from that of the first and third carbon atoms. This CDW, despite its short range, distinguishes our findings from a conventional CDW pattern. A conventional CDW in a chain with two dissimilar bond lengths under periodic alignment refers to a phenomenon where the electron density in the chain becomes trivially modulated in a periodic manner, typically in response to lattice distortions. In other words, this includes the modulation of electron density, where the charge density varies periodically along the chain, resulting in regions of enhanced and reduced charge density conventionally. The periodic distortion can lower the overall energy of the system, stabilizing the conventional CDW state. This modulation, commonly called the Peierls transition, always leads to semiconducting or insulating behavior. Here, the carbon chain is in the cumulative phase (same bond length), thereby ruling out the possibility of having two dissimilar bond lengths under periodic alignment,³⁰ and therefore it can be defined as an unconventional CDW phenomenon. The long carbon chain in zigzag BNT prefers a cumulative phase over a polyyne phase because the creation of alternating single and triple bonds leads to conflicts in bond type and bond geometry under a translational symmetry. This is similar to how spin-up and spin-down configurations conflict with geometry. When these bond and location conflicts arise, a significant number of lone pair electrons are generated from the internal chain, undermining its stability.

The non-zero magnetic moment of carbon, despite its linear configuration, indicates that the proximity effect from the BNT is influential. This is particularly noteworthy since we do not observe a similar effect in CNT hosts of the same diameter. The nearest lateral distance illustrated in Fig. 1(a) indicates that the proximity interaction between the BNT and the carbon chain is maximized at a radial separation of ∼0.5 nm for both (17,0)BNT and (19,0)BNT. Both the straight carbon chain and the isolated boron–nitride nanotube are originally non-magnetic in their separate systems. To explore the proximity effect between these non-magnetic materials and how one of them (carbon chain) can become magnetic, we compare the cases of “LCC@(19,0)BNT vs. LCC@(19,0)CNT”, and “LCC@(17,0)BNT vs. LCC@(17,0)CNT”. To ensure that the atomic coordinates remain consistent in these composites, we replace the B and N atoms with C atoms without conducting another round of geometric optimization. This approach ensures that the proximity effect can be compared under the same atomic coordinates. In these comparative studies, the LCC@(19,0)CNT and LCC@(17,0)CNT can be considered as non-magnetic. In other words, the intrinsic electric field established by the Group III (boron) and Group V (nitrogen) elements could be responsible for creating a magnetic carbon chain, even in the absence of a periodic kink structure. In the previous study, we observed that the periodic E-field from dopants could also trigger magnetism along the carbon chain.³¹ Although no dopants are applied to the internal carbon chain periodically, the B and N atoms have created a periodic electric field, resembling the effects of periodic on-site electric potential modulation.¹⁷ The periodic kink-structured 400 K ferromagnetic carbon chains are laterally spaced by 0.5 nm in the experiment,¹⁷ which is consistent with the lateral distance observed in the two magnetic LCC@BNT composites (Fig. 1(a)).

3.2. Length limitation in LCC@BNT

Before we predict the finite size effects of magnetism and charge density waves, we have to determine how long the carbon chain can exist within the BNT. The carbyne@nanotube model¹⁹ can be considered as a reasonable tool to assess the stability factor of the internal carbon chain. By replacing CNT with BNT in this model, we investigate the stability of a finite-length LCC as a function of the BNT diameter. The carbon chain exhibits the strongest mechanical strength (i.e. very strong covalent bond), if not the absolute strongest.^14,15 However, its magnetic moment is only a fraction of the Bohr magneton, indicating that the magnetic energy is significantly weaker than the covalent bond between the carbon atoms. As a result, its weak magnetic interaction is understandably not included in the Hamiltonian. While the longest LCC achieved to date has been through encapsulation in a (6,4)CNT,¹⁸ our analysis begins by comparing the cut-off LCC length within (6,4)CNT versus (6,4)BNT to observe whether there are significantly distinct levels of protection and subsequently interprets the cut-off LCC length within (19,0)BNT. Going back to the carbyne@nanotube model of LCC@(6,4)CNT, a more rapid downturn in the chain stability factor of ∼0.7 is observed above ∼5750 carbon atoms,¹⁹ which almost matches the experimental cut-off LCC length of ∼6400 atoms.¹⁸ Applying the same criteria (chain stability factor ∼ 0.7) to judge the cut-off LCC length inside (6,4)BNT at 300 K, the estimated value drops to only ∼900-atom-long in Fig. 2(a). In other words, the use of BNT is unlikely to suit long carbon chain production under the same tube diameter and chirality. The stability factor of ∼0.7 is equivalent to an average bond length of 1.39 Å, close to the average single and triple carbon–carbon bond distance of (1.21 + 1.54)/2 = 1.37 Å. The lower stability factor of ∼0.55 corresponds to an average bond length of ∼1.50 Å, which is still less than the maximum permittable C–C length ∼1.73 Å.³² The finite polyyne chain with N = 900 carbon atoms exhibits increasing stability as the temperature is lowered, as shown in Fig. 2(b). This phenomenon can be explained by the reduction in thermal fluctuations at low temperatures. When the temperature is lowered, the thermal energy available to the atoms decreases. As a result, the carbon atoms within the polyyne chain vibrate less vigorously. This reduced atomic motion allows the carbon atoms to favor and occupy their ground state more steadily. At low temperatures, the minimization of thermal fluctuations leads to a higher degree of stability for the polyyne chain. The carbon atoms are less likely to be displaced from their optimal positions, and the overall structure of the chain becomes more stable.

According to Fig. 3(a), the formation of the polyyne phase ([C–C]) is energetically favorable¹⁸ in the 900-atom-long LCC inside (6,4)BNT, even up to room temperatures. The percentage of the polyyne phase remains above 95% even though we observe that the polyyne phase starts to pale under thermal excitations. Correspondingly, as shown in Fig. 3(b), the formation of the cumulene phase (CC bonds) is tiny. Our Monte Carlo simulator also detects rare signals, less than 0.1%, for the formation of [C–C] or [–C–C] phase. We further study and examine the relationship between the cut-off LCC length and the chirality numbers of BNT in Fig. 3(c). The results indicate that as the BNT radius increases from approximately 0.25 nm to 0.5 nm, the cut-off LCC length shows a relatively linear decrease. The curve then takes on an exponential-like shape, approaching around 10 atoms for (19,0)BNT. The abrupt change in the slope at N_c = 12 is due to the formation of an asymmetric shape of BNT (see the inset of Fig. 1(a)). No data are computed for BNT structures thinner than (5,0), as this may trigger the formation of covalent bonds between the carbon chain and the BNT surface which would violate the underlying assumption of the Hamiltonian, where only van der Waals interactions exist between the carbon chain and the BNT.

We further analyze the cut-off length of the LCC within CNT versus BNT in the presence of a Group VII dopant. Doping with a Group VII element into nanotube is anticipated to shorten the LCC length due to the increased number of lone pair electrons on the nanotube, which leads to enhanced electrostatic repulsion. Therefore, the test specimen, fluorine, is selected as the dopant for the nanotube. We arbitrarily select N_c = 10 and compare the cut-off LCC lengths in (10,0)CNT versus (10,0)BNT. In the case of (10,0)BNT, approximately 1% fluorine doping dramatically reduces the LCC length from 600 to less than 100 atoms. In contrast, the LCC length in (10,0)CNT only decreases from about 1000 to 800 with the same level of fluorine doping. These results suggest that impurity control is more critical for growing LCC within BNT compared to CNT hosts.

The choice of setting the maximum Monte Carlo steps (MCS) to 250000 has been justified,¹⁹ as this is large enough to allow the composite system to reach equilibrium. This is supported by evidence presented in Fig. 5 of ref. 19, which shows that the energy versus the MCS plot for a 15000-atom-long carbyne inside a carbon nanotube (CNT) has reached equilibrium for MCS values exceeding ∼180000.¹⁹ In the current study, the longest internal carbyne is only up to ∼1000 atoms. Since the required MCS duration is proportional to the chain length of carbon, the chosen MCS duration of 250000 steps is suitable for this investigation.

3.3. The short LCC@BNT as a ferrimagnetic CDW insulator

Based on the LCC@(19,0)BNT model, we anticipate that the internal carbon chain is connected by approximately 10 atoms. Hence, we constructed a non-periodic supercell with 9 carbon atoms surrounded by (19,0)BNT. After geometric optimization of the finite supercell, the proximity effect resulted in the nearest lateral distance being ∼0.55 nm, which falls within the expected range for magnetism, where it breaks the symmetry along the circular plane of BNT again. However, no magnetism is observed in this internal carbon chain surprisingly. We suspect that the disappearance of the internal finite-length LCC is due to the formation of free radical electrons at the opposite ends of the chain (4 free radical electrons are detected). Therefore, to activate the re-emergence of magnetism, we employ hydrogen termination, which is a common method for terminating chain growth.³³ Geometric optimization of finite-length H–C₉–H within (19,0)BNT is conducted again under the same ab initio setup, where the supercell is drawn in the inset of Fig. 4(a). After hydrogen termination, the carbon atoms reappear magnetism in the supercell structure. Ferrimagnetic state is observed in the finite carbon chain in (19,0)BNT without magnetic frustrations, as depicted in Fig. 4(a) and (b). The finite size effect causes a reduction in the highest local magnetism from 0.9μ_B to 0.6μ_B, where the strong exchange interaction is ∼24 meV (or ∼280 K). When compared to the infinitely long case, Fig. 4(c) shows a more pronounced long-range CDW pattern along the entire carbon chain, with a complete cut-off of DOS at the even-numbered carbon sites, which is a sign of unconventional insulators.^9,11 To enable the electron tunnel between odd-numbered carbon sites on the Fermi surface, the tunneling barrier of 2.3 eV needs to be overcome as shown in Fig. 4(d).

The H–C₉–H within (19,0)BNT contains over ∼300 atoms in the supercell which raises a high computational cost. Fortunately, in 2018, we analyzed the magnetic states of carbon chains and discovered that using either the LDA or GGA level provides a very accurate prediction of experimental ferromagnetism above room temperatures.¹⁷ Similarly, this magnetism focuses exclusively on carbon chain again, thereby avoiding the complexities associated with transition metals. Additionally, no spin–orbit coupling is observed in the internal carbon chains. Hence, our selected DFT functional is a sensible choice,²⁴ as it highlights that the coexistence of magnetism and CDW effects in the carbon chain does not involve complex electron–electron interactions. Dmol³ and CASTEP are accurate ab initio software for nanowires,^33–37 where their accuracies have been justified.³⁷ The average magnitude in magnetic moment of LCC@(19,0)BNT computed using LDA–PWC in Dmol3 software is approximately 0.6μ_B, while GGA–PW91 or GGA–PBE in CASTEP software yields around 0.5μ_B. The cut-off LCC length calculated using GGA–PW91, GGA–PBE, LDA–PWC, and LDA–VWN methods is consistent, showing only a 1–2% variation in the computed length only. The CDW patterns obtained using GGA–PW91, GGA–PBE, LDA–PWC, and LDA–VWN are also consistent, indicating that the computed results are not highly sensitive to the choice of DFT functional or software.

We summarize the approach to triggering magnetisms in carbon chains. Self-assembled kink structures,¹⁷ doping,^17,31 and BNT suggest a link to trigger p-orbital magnetism in carbon chains. The main distinction is in how the electric field modulation is applied, which could influence whether the system exhibits ferromagnetism, ferrimagnetism, magnetic frustration, charge density waves, or the emergence of a quantum tunneling gap. For a parallel array of carbon chains, when a self-assembly kink structure occurs, it creates an extra periodic modulation in atomic charge density due to the strong local electric field at these kink points. By comparing scenarios with and without kinks, we understand that this atomic charge density modulation along the chain can be a mechanism to trigger p-orbital magnetism experimentally.¹⁷ Later, we suggested a similar approach using doping to induce p-orbital magnetism.³¹ However, the concentration of dopants needs careful adjustment (around 10%) to avoid disrupting chain stability by electrostatic repulsion. These periodic dopants introduce additional charge, creating sharp kinks alongside free radical electrons, significantly altering their magnetism. Both the kink structure and doping serve to generate extra E-field modulation along the chain longitudinally. However, this modulation can be too strong in certain areas and too weak in others, depending on the location of kink or dopants. Thus, we sought a method to impose an electric field modulation without using kink structures or dopants.

To trigger magnetism in carbon chains without relying on kink structures, we propose using the proximity effect to impose an electric field modulation along the chain. In this work, we focus on using BNT to invoke magnetism in the internal carbon chain through this proximity effect, where the source is the periodic E-field between boron and nitrogen along the nanotube. In our previous studies,^17,31 we found that high concentrations of dopants on the carbon chain can eliminate magnetism. Since boron and nitrogen atoms are densely packed in the nanotube, a wider nanotube-to-chain separation is necessary. The key challenge remains meeting the threshold for triggering magnetism throughout the entire chain, which we can adjust by varying the diameter of the BNT.

3.4. Field tuning of the insulator-to-conductor transition and further scientific insights

Looking ahead, we foresee a future in which organic materials could play a role in the development of next-generation field-tuning CDW devices, leading to novel spin transport.^1–12 Comparing the distinct CDW patterns in Fig. 1 and 4, we observe that magnetic fluctuations can disrupt a long-range CDW pattern, where ordered spins promote ferrimagnetism and long-range CDW on a finite scale. The sharp gradients in DOS(E_F) within the CDW structure generate local magnetism equivalently, in line with Maxwell's equations.^6,26–29 Consequently, an exotic coupling arises between this local magnetism and the underlying magnetic background. Thus, one of the potential applications is that applying a magnetic field perpendicular to the easy axis of ferrimagnetic alignments induces spin fluctuations that anticipate CDW modulation, modifying the electronic properties.

To produce an abrupt insulator-to-conductor transition, should a magnetic field or an electric field be applied? Based on Fig. 1, if applying an external magnetic field could help adjust the CDW pattern associated with electronic properties by triggering subtle spin fluctuations, it may not be suitable to produce a rapid insulator-to-conductor transition. Although Fig. 1 involves an infinite long LCC@BNT, the underlying science regarding to the subtle spin fluctuations should align with the finite scale. Conversely, applying a constant electric excitation of 2.3 V along the chain axis may be possible for overcoming the tunneling barrier and achieving a rapid insulator-to-conductor transition, as we see it in the finite case (Fig. 4). Using a constant electric excitation to trigger the insulator-to-conductor transition may be more superior than using external magnetic field because the external magnetic field may introduce magnetic noise within the ferrimagnetic carbon chain.

This research leads to more exotic findings in organic science. Contrary to our initial expectation that a linearly straight carbon chain would not exhibit magnetism,¹⁷ we discovered that interactions resulting from the broken symmetry in non-magnetic BNT create magnetic proximity to the carbon chain. This is highly unusual, but it suggests that, in the field of organic science, there may be multiple pathways to trigger magnetism in organic substances. The periodic kink structures in a short carbon chain arranged in parallel have shown ferromagnetism experimentally. In contrast, a short carbon chain enclosed by BNT could generate ferrimagnetism, while increasing the length of the internal carbon chain within BNT emerges magnetic frustration.²⁵ By replacing the substrate and scaling the carbon chain, we could alter the type of magnetism in the carbon chain, highlighting its vast potential applications in spintronics. In addition, we have found that CDW and ferrimagnetism are compatible in this sample, whereas ferromagnetism and CDW are not compatible here. This indicates that the co-occurrence of ferromagnetism and CDW presents a much tougher challenge comparably.^1–4 The emergence of CDW patterns in a fully organic material is particularly noteworthy because these unusual properties are always found in transition elements only.^38–41 Our findings⁴² demonstrated that carbyne has the potential to enter the realm of topological science when the substrate is carefully selected. Notably, a Dirac gap at M point is observed in Ru-metalated carbyne under a ruthenium substrate.⁴² Interestingly, when a different substrate (BNT) is chosen, carbyne surprisingly exhibits a charge density wave (CDW) state. This creates a new way of thinking that a much more exotic science in organic materials could be uncovered through the exploration of other carbyne–substrate combinations.

Conventional CDW patterns typically require periodic lattice distortion, which leads to the formation of an ordered quantum fluid of electrons at the Fermi level.⁴ However, the cumulative phase does not exhibit periodic lattice distortion. Hence, our discovery of the CDW pattern without periodic lattice distortion in Section 3.1 marks a significant departure from conventional CDW patterns. Shortening the internal carbon chain to a finite length ultimately transforms the system into polyyne, where periodic lattice distortion occurs. The CDW pattern in Fig. 4(c) appears as a superposition of both conventional and unconventional cases. Upon examining the DOS values, the unconventional case predominates. The combined effect of both conventional and unconventional cases results in a more pronounced CDW effect, leading to an alternating complete vanishing of DOS(E_F). In Fig. 4(c), the presence of zero density of states at even-numbered sites supports the identification of ferrimagnetic insulators.^10,11 These materials could act as spin-filtering tunneling barriers for tunneling magnetoresistance.¹² While ferrimagnetic insulators could allow the transport of spin momentum without charge, this unique property has attracted considerable interest in dissipation-less electronic and spintronic devices, solid-state quantum computing, and magnetic tunneling junctions.^1–5

Magnetism in finite graphite sheets with edges has been extensively studied. In graphite ribbons with zigzag edges, unique ‘edge states’ emerge, which can lead to ferromagnetism.⁴⁴ In contrast, carbon chains can exhibit magnetic properties through various mechanisms, such as utilizing periodic kink and branch structures to impose a periodic electric field along the chain or adjusting the doping levels periodically for the same mechanism. Furthermore, straight configurations of carbon chains induced by the proximity effect of a periodic electric field in boron nitride nanotubes support magnetic behavior. This fundamental difference highlights the diverse ways magnetism can manifest in carbon-based materials. Furthermore, unlike transition metal nanowires, which rely on d-shell electrons with unpaired spins for magnetism, the carbon chain exhibits 100% p-orbital magnetism, demonstrating that their magnetic properties are fundamentally different in nature.

We acknowledge that creating a carbon chain inside single-wall BNT is a challenging task. However, this does not imply that it is without hope. In an experiment conducted by Ryo Nakanishi et al.,⁴³ the successful fabrication of single-wall BNT with a very small radius of 0.35 nm inside a CNT represents a crucial first step toward achieving our goal while our optimal BNT radius is much larger. Scientific advancements often come with difficulties, but the experimental support^18,43 they provide keeps hope alive. We believe that with continued effort, the creation of a carbon chain inside single-wall BNT should be possible. Each step forward in our research not only enriches organic science but also inspires optimism for future breakthroughs. The challenges we face are substantial, but they may not be insurmountable, as foundational frameworks (experimental fabrication of LCC@CNT and BNT@CNT) are already in place.^18,23,43 We remain confident that our efforts will lead us to new scientific horizons.

4. Conclusions

Our study offers valuable insights to reconsider the conventional belief that organic materials cannot be promising candidates for emerging exotic charge density waves (CDW). We demonstrate that the broken symmetry of a long carbon chain within boron nitride nanotubes can induce a short-range CDW alongside a ferrimagnetically frustrated state. After addressing the size effects of the carbon chain, we find that these exotic properties persist in finite-length carbon chains, revealing a transition to a pure ferrimagnetism with a long-range CDW state, characterized by a strong exchange interaction of ∼24 meV. The optimal LCC@BNT composite has shown a quantum tunnelling gap that may favor a rapid transition from insulator to conductor. This underscores the vast potential for quantum switching and spin computing applications.

Author contributions

Conceptualization: C. H. W.; methodology: C. H. W.; computation: C. H. W.; validation: C. H. W.; formal analysis: C. H. W., L. Y. F. L., X. H.; supervision: C. H. W.; writing manuscript: C. H. W.; editing manuscript: C. H. W., Z. G., C.-Y. T., Y. H. T.; software: C. H. W., C.-Y. T.; data curation: C. H. W., Z. G.; visualization, C. H. W., K. C. L., C. P. C., W. Y. C.; resource: C. H. W., C.-Y. T., Y. H. T, L. Y. F. L., X. H.

Conflicts of interest

The authors declare no conflict of interests.

Data availability

Data are available upon reasonable request.

Acknowledgements

The authors thank the Department of Industrial and Systems Engineering at The Hong Kong Polytechnic University for providing computational support.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5mh00664c

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