A nodal flexible-surface three-dimensional carbon network with potential applications as a lithium-ion battery anode material

Abstract

Topological semimetals (TSMs) with nodal flexible-surfaces are currently garnering paramount importance due to their exceptional electronic properties. However, nodal flat-surfaces and nodal spheres have only been observed in a few quantum models due to strict symmetry constraints. In this study, we introduce a cyclooctatetraene (COT)-based novel three-dimensional (3D) carbon network using density functional methods. Computational results show that the novel 3D COT network is dynamically stable and exhibits a linear dispersion relation near the Fermi level. Remarkably, the 3D band spectrum of the COT network exhibits a rare crossbar-like topological nodal flexible-surface. We identify key factors behind the emergence of a nodal flexible-surface using the analyses of orbital-projected band structures, tight-binding electronic bands, atom/orbital-projected density of states and band-decomposed charge densities. The π-bonding and anti-bonding orbitals of C atoms predominantly contribute to the origin of topological semimetallic properties. Fermi velocities of holes and electrons (∼2.01–8.70 × 10⁶ m s⁻¹) in the 3D COT network are found to be higher than that of graphene along specific crystallographic directions. Furthermore, we have investigated the application of the 3D COT network as an anode material for Li-ion batteries. Our findings indicate that the 3D COT network shows promise as the anode in the Li-ion batteries due to its low energy diffusion barriers (0.008–0.68 eV), enabling rapid charge–discharge rates. Additionally, it exhibits a higher theoretical specific capacity (627.5 mA⁻¹ h g⁻¹) compared to graphite and an optimal average open circuit voltage of 0.85 V.

---

1. Introduction

The discovery of topological phenomena in condensed matter systems, such as quantum Hall effects and topological insulators, has sparked significant scholarly interest in the field of topological semimetals (TSMs).^1–4 TSMs are identified primarily by the presence of a linear band dispersion in three-dimensional (3D) momentum space at the Fermi level. The linear band dispersion gives the TSMs unique characteristics. For example, it is primarily responsible for the formation of Dirac and Weyl fermions in condensed matter systems. The Dirac and Weyl fermions exhibit massless nature and behave in a fundamentally different manner from that of conventional Schrödinger fermions.^5–8 Furthermore, it is responsible for the realization of specific properties in TSMs such as Fermi arcs, drumhead surface states, chiral anomaly and extreme magnetoresistance.^9–13 As a result, TSMs have become a focus of scientific investigation in the fields of solid-state chemistry, condensed matter physics and materials science.^14–22

The classification of TSMs is primarily based on the dimensional arrangement of their linear band crossing points (Fig. 1). For instance, quantum materials that exhibit linear band crossings at isolated k-points are classified as zero-dimensional (0D) nodal-point TSMs, as depicted in Fig. 1(a). Noteworthy examples of nodal-point TSMs include Dirac materials, Weyl materials and materials with multiple band crossing points. Conversely, when these isolated linear band crossing points align coherently along a continuous 1D line or 2D loop (see Fig. 1(b) and (c)), they give rise to nodal-line or nodal-loop TSMs.^17,23 Similarly, materials with band crossing points congregated within a two-dimensional (2D) plane can be regarded as nodal flat-surface TSMs.²⁴ It is important to note that these 2D flat nodal surfaces can exhibit curvature, which results in various curved nodal surfaces. For example, Huang and colleagues have proposed a higher-dimensional nodal sphere TSM, as presented in Fig. 1(d). They demonstrated that the band crossing points in the Si₃N₂ material approximate the geometry of a sphere.²⁵

Fig. 1 Arrangement of nodal points (linear band crossing points) in 0-, 1-, 2- and 3-dimensional space in different topological semi metals. (a) Nodal point, (b) nodal line, (c) nodal loop and (d) nodal sphere.

In a recent report, Chen and co-workers generated novel topological phases within a family of 3D carbon nanotube architectures using tight-binding methodology.²⁶ These phases encompass nodal tubes, nodal crossbars, nodal surfaces and nodal spheres. Each of these topological phases is distinguished by distinctive curved nodal surfaces. Quantum materials furnished with these intriguing topological states have been christened as nodal flexible-surface TSMs.²⁶ Higher-dimensional nodal surface TSMs, which include 2D nodal flat-surface, nodal sphere and nodal flexible-surface TSMs, exhibit exceptional electronic properties.²⁶ For instance, 2D nodal flat-surface TSMs show intrinsic pseudospin degrees of freedom and demonstrate unique plasmonic behaviours within excited coarse-grained quasiparticles.^27,28 Conversely, nodal sphere TSMs display strong quantum oscillations and peculiar plasmon excitations.²⁵ However, the nodal flexible-surfaces are relatively rare in condensed matter systems compared to other topological phases. This prompts an exploration of alternative materials that may exhibit higher-dimensional topological phases within the realm of topological semimetals.

On the contrary, rechargeable energy storage devices are key components in the design and development of portable to advanced energy storage solutions. They hold significant importance not only in reducing reliance on the dependence on carbon-intensive fuels but also in meeting the escalating energy demands of our modern era.^29–32 For this purpose, lithium-ion batteries (LIBs) stand out among other energy storage and conversion technologies due to their promising qualities such as high energy density, longer lifetime and flexible and lightweight design.^33–35 However, commonly used anode materials in the Li-ion batteries, such as graphite and silicon, have limitations in terms of storage capacity and cyclability. For example, graphite has a maximum theoretical specific capacity of 372 mA h g⁻¹ and silicon has a charge capacity of almost 10 times higher than that of graphite.^36–38 The low theoretical storage capacity of graphite limits the performance of LIBs. On the contrary, the extremely high charge capacity of silicon leads to the fracture of the anode materials due to the massive expansion, resulting in a very short lifetime. Thus, the anode materials with good storage capacity and extended life span are of utmost importance.

Overall, materials that exhibit an ample number of adsorption sites, nano-porosity, mechanical stability and low diffusion barriers exhibit promising potential as superior anode materials for LIBs. Previous studies^{17,19,20,39,40} have demonstrated that these desirable characteristics are inherently present in carbon allotropes featuring π-conjugation. The π-conjugation in the carbon allotropes not only facilitates robust adsorption of Li atoms but also enables seamless electronic charge transfer from the Li atoms to the carbon allotrope. Consequently, it facilitates the formation of lithium ions, thereby effectively mitigating dendrite formation, a common issue found in lithium-ion batteries. Moreover, the extended π-conjugation typically reduces the energy gap, rendering the materials either semiconductors or metals. These combined attributes significantly augment the performance of LIBs. The π-bonding and anti-bonding states are accountable for inducing topological features such as Dirac nodal points, Dirac nodal lines, Dirac nodal loops and nodal surfaces in many carbon allotropes.^{17,19,20,39,40} Evidently, the carbon allotropes featuring topological semi-metallic properties show enhanced lithium-ion battery performance. Therefore, the exploration of the carbon allotropes with π-states not only facilitates novel topological phases but also holds considerable promise for the advancement of lithium-ion battery technology.

In this study, we present a comprehensive investigation into the geometrical properties, structural stabilities, electronic configuration and Li-ion battery anode material applications of a cyclooctatetraene (COT)-based three-dimensional carbon network using density functional theory (DFT)-based first-principles methodologies. The results of this study reveal that the 3D COT network is energetically viable and structurally stable up to a maximum temperature of 1500 K. Electronic structure analysis suggests that the 3D COT network is a nodal flexible-surface semimetal. The results obtained here are the first of their kind and have never been reported before. In addition, high storage capacity and low diffusion energy barriers for Li migration suggest that the 3D COT network could serve as a potential anode material for the Li-ion batteries. Overall, this study contributes to the exploration of novel Li-ion battery anode materials and higher-dimensional nodal flexible-surface phenomena using a carbon-based 3D network, which is relatively challenging to obtain in 3D quantum models.

2. Computational details

A comprehensive investigation of the geometrical and electronic properties of a 3D carbon network obtained from the cyclooctatetraene (COT) unit was conducted using the Vienna ab initio simulation package^41,42 coupled with periodic density functional theory, employing the projector augmented wave (PAW) method.⁴³ The exchange–correlation potential was defined within the framework of the generalized gradient approximation (GGA) as proposed by Perdew–Burke–Ernzerhof (PBE).⁴⁴ Previous studies show that the PBE-GGA functional is adequate to accurately describe most properties of solid-state systems, specifically carbon allotropes.^{15,18,21,45,46} For geometry optimization and electronic structure calculations, the Monkhorst–Pack scheme was incorporated to span the 3D Brillouin zone, with the k-point grids of 3 × 3 × 12 and 7 × 7 × 14, respectively.⁴⁷ The tight convergence criteria for the conjugate-gradient algorithm-based self-consistent electronic iterations were set below 10⁻⁸ eV for energy and 10⁻³ eV Å⁻¹ for Hellmann–Feynman residual forces on individual atoms compared to previous studies.^{15,18,21,45,46} To assess the structural stability of the 3D COT network, phonon mode analysis was conducted using the supercell approach facilitated by the Phonopy package.⁴⁸ Additionally, ab initio molecular dynamics (AIMD) simulations were performed for a duration of 10 picoseconds (ps) using the canonical (NVT) ensemble on the 3D COT network to explore its thermal stability. The long-range van der Waals (vdW) interactions were accounted for using Grimme's DFT-D3 method.⁴⁹ The climbing-image nudged elastic band (CI-NEB) approach is used to compute minimum energy diffusion paths and corresponding energy barriers for the metal atom.⁵⁰ Furthermore, other details of the CI-NEB approach were described in the ESI.†

3. Results and discussion

3.1. Structural details

It is evident from previous studies that several novel 3D carbon allotropes were predicted by joining or substituting different motifs in the existing carbon allotropes. For example, Su and co-workers have designed one of the renowned carbon allotropes, T-carbon, by substituting the carbon tetragon at each carbon atom of diamond.⁴⁵ Yang et al. have proposed four novel carbon architectures (L-carbon, C-c-C₄, H-c-C₄, and H-2c-C₄) by optionally substituting carbon atoms in diamond with carbon tetrahedrons.⁵¹ Zhong and co-workers have also designed two new carbon allotropes, namely cubic-graphite and T5-carbon, by altering the carbon atoms in diamond with different units.⁵² Similarly, the cyclooctatetraene (COT)-based three-dimensional (3D) carbon network can be derived by substituting the tetrahedral carbon atoms with a tetrahedral-like COT ring and C–C single bonds with double acetylenic linkers as depicted in Fig. 2(a). The ground state geometry of the 3D COT network along different projections is shown in Fig. 2(b) and (c). The 3D COT network crystalizes in the I4_1/amd space group (space group number: 141). The fundamental building block of the 3D COT network, i.e., the unit cell, comprises a total of 64 carbon atoms within the volume spanned by the three mutually perpendicular lattice vectors of magnitude a = b = 14.44 Å and c = 4.26 Å. Out of the 64 carbon atoms, half of the C atoms exhibit the sp² hybridized state (hereafter denoted as C1) and are used to form the four cyclooctatetraene rings, while the other half belong to the sp hybridized state (hereafter denoted as C2) and are utilized to form the sixteen acetylenic linkers as depicted in Fig. 2(c). Moreover, four cyclooctatetraene rings are joined to one another through the sixteen acetylenic linkers in the unit cell. The 3D COT network exhibits four distinct covalent bond lengths, namely d₁–d₄, between the C atoms of the unit cell. Here, d₁ and d₂ represent bond lengths between the C1 type C atoms of the COT ring, while d₃ is the bond length between the C1–C2 type C atoms, and d₄ is the bond length of the two C2 type C atoms. The results show that the equilibrium atomic bond lengths, d₁–d₄, are around 1.42, 1.47, 1.41 and 1.23 Å, which are following the sp²–sp² > sp²–sp > sp–sp hybridized carbon bond lengths. We have also presented the complete details of the geometrical parameters of the 3D COT network in ESI,† Fig. S1. Overall, the unit cell of the 3D COT network entirely consists of two inequivalent carbon atoms, namely C1 and C2, occupying atomic positions at 2p (0.44911, 0.88092, and 1.21597) and 2x (0.41349, 0.79146, and 1.15941), respectively. We have calculated the equilibrium atomic density (ρ) of different carbon allotropes at the same level of theory, and the results are presented in Table S1 (ESI†). The results show that the 3D COT network has a lower atomic density (1.44 g cm⁻³) than the rest of the 3D carbon allotropes, including diamond, graphite, bct-C₄⁵³ and T-carbon.⁴⁵ The 3D COT network exhibits nano-porosity along certain crystallographic directions similar to tetragon-based carbon architectures, including T-carbon, L-carbon, C-c-C₄, H-c-C₄, H-2c-C₄, sp²-diamond, cubic-graphite and T5-carbon. The 3D COT network may have potential applications in hydrogen storage and alkali-ion battery anode material devices. (see Fig. 2(d)).

Fig. 2 (a) Schematic representation of the creation of a 3D COT network from diamond. (b) and (c) Different projections of the unit cell of the 3D COT network. (d) Crystalline structure. Different colour codes are used in (c) to represent the different types of carbon atoms.

3.2. Structural stability

3.2.1. Energetics. The structural stability of the 3D COT network has been verified by conducting total energy, phonon band dispersion, ab initio molecular dynamics simulation and elastic constant analysis. Fig. 3(a) shows that the 3D COT network is metastable against bct-C₁₆,⁵³ CNT (4, 4), diamond and graphite with a higher energy of 0.21, 0.62, 0.76 and 0.89 eV per atom, respectively. However, we noticed that the 3D COT network may be synthesized using experimental methods as it is energetically preferable not only over some theoretically proposed 3D carbon phases (penta-graphene⁵⁴ and T-carbon⁴⁵) but also a few experimentally synthesized allotropes of carbon, such as C₂₀ fullerene⁵⁵ and the smallest armchair carbon nanotubes, CNT (2, 2).^56,57 We have computed cohesive energies (E_c) of different carbon allotropes at the same level of theory as that of the 3D COT network. The 3D COT network exhibits a higher cohesive energy (E_c) than the T-carbon, C₂₀, penta-graphene and CNT (2, 2) carbon allotropes as shown in Table 1. This indicates the stronger bonding strength between the atomic constituents of the 3D COT network. However, the E_c of the 3D COT network is slightly lower than that of graphite, diamond, bct-C₁₆ and CNT(4,4), indicating a comparative bonding strength. The E_c of the 3D COT network is also close to that of tetragon-based carbon allotropes derived from diamond.^51,52 Overall, lattice constants, E_c values, electronic structure and atomic density values of the different carbon allotropes included in this study closely match with those of previous studies.^45,58 This clearly validates our functional and methodology.

Fig. 3 Structural stability of the 3D COT network. (a) Total energy per atom of the different carbon allotropes, (b) the phonon band spectrum and (c) the total energy fluctuation of the 3D COT network during AIMD simulations at different temperatures.

Table 1 Cohesive energy (E_c) of different carbon allotropes. All the E_c values are calculated at the same level of theory

S. no.	C allotrope	E c (eV per atom)
1	3D COT network	−7.09
2	T-carbon	−6.68
3	bct-C16	−7.30
4	Diamond	−7.85
5	Graphite	−7.98
6	Penta-graphene	−7.08
7	CNT (2, 2)	−7.05
8	CNT (4, 4)	−7.71
9	C20	−6.84

---

3.2.2. Dynamical stability. The phonon band spectrum of the 3D COT network is illustrated in Fig. 3(b). It exhibits positive phonon frequencies across the entire first tetragonal Brillouin zone, except for a tiny imaginary frequency (3i cm⁻¹) around the Γ-point (see Fig. S2, ESI†). It is evident from previous studies that negative frequencies, which are less than 10 cm⁻¹, can be ignored.^59–63 Furthermore, the highest phonon mode of the 3D COT network is found to be around 2161 cm⁻¹ at the centre (Γ) of the tetragonal Brillouin zone, which greatly exceeds the highest phonon mode frequency of graphite (≈1600 cm⁻¹). These results clearly affirm the dynamical stability of the 3D COT network. The substantial phonon gap (560 cm⁻¹) of the COT network starting from 1550 cm⁻¹ at the tetragonal Brillouin zone centre (Γ) may be helpful in identifying the 3D COT network in the experimental synthesis.

3.2.3. Thermal stability. To explore the thermal stability of the 3D COT network, ab initio molecular dynamics (AIMD) simulation studies were conducted with a time step of 1 fs for a duration of 10 ps using the canonical ensemble (NVT) at different temperatures including 300, 500, 1000, 1500 and 2000 K. Previous studies show that the AIMD simulation for a duration of 10 ps is adequate to capture the thermal stability of the different carbon allotropes.^15,17,18,64 It is evident from Fig. S3 (ESI†) that temperatures were maintained well during the all-AIMD simulations. The variation in the total energy is confined to a narrow region throughout the AIMD simulation performed at 300, 500, 1000 and 1500 K, as depicted in Fig. 3(c). However, it can be observed that the total energy dropped to a lower value after 1500 fs at 2000 K. Snapshots of the 3D COT network at the end of each AIMD simulation are presented in Fig. S4 (ESI†). The geometrical analysis also revealed that the 3D COT network maintained its structural integrity until 1500 K and transformed to a new phase at 2000 K by forming the tetra-rings using the acetylenic linkers. These results ascertain that the 3D COT network is thermally stable at room temperature, and it can maintain its structural integrity up to 1500 K once realized using experimental techniques.

3.2.4. Mechanical stability. To assess the mechanical stability, we examined the independent elastic stress tensors of the 3D COT network through an analysis of the stress–strain relationship. With its tetragonal crystal symmetry, the mechanical stability of the 3D COT network can be defined by six distinct elastic constants, namely C₁₁, C₃₃, C₄₄, C₆₆, C₁₂ and C₁₃. Our computed results revealed that different elastic constant values are around C₁₁ = 339, C₃₃ = 5, C₄₄ = 4.0, C₆₆ = 32, C₁₂ = 209 and C₁₃ = 19 GPa. Evidently, the elastic constants of the 3D COT network satisfy the Born criteria, such as C₁₁, C₃₃, C₄₄, C₆₆ > 0, C₁₁–C₁₂ > 0, C₁₁–C₃₃–2C₁₃ > 0 and 2C₁₁–C₃₃ + 2C₁₂ + 4C₁₃ > 0 for a tetragonal system. Furthermore, we determined the bulk (B) and shear (G) modulus of the 3D COT network using the Voigt–Reuss–Hill approximation. B and G values are found to be around 67 GPa and 21 GPa, respectively. The B/G ratio of the 3D COT network suggests that the 3D COT network is ductile nature. Collectively, these results provide strong evidence for the mechanical stability of the 3D COT network.

3.3. Electronic structure

In this section, we provide a detailed exploration of the electronic band spectrum of the 3D COT network. Initially, we investigated the 2D electronic band structure of the 3D COT network within the first 3D tetragonal Brillouin zone, as depicted in Fig. 4(a). The conduction band (CB) and valence band (VB) of the 3D COT network exhibit a linear dispersion in the proximity of the Fermi level, as shown in Fig. 4(b). Furthermore, these linearly dispersed bands intersect each other along the high-symmetry line segments Z → A, M → Γ, Γ → Z, and X → Γ at four different off-symmetry k-points, which include D1 (0.2815, 0.2815, 0.5000), D2 (0.4000, 0.4000, 0.0000), D3 (0.0000, 0.0000, 0.1345) and D4 (0.0000, 0.3235, 0.0000) in the proximity of the Fermi level. The linear dispersion observed in the VB and CB closely resembles the Dirac linear dispersion characteristics of topological semimetals. Hence, these four band crossing points, namely D1 to D4, located in the vicinity of the Fermi level, can aptly be regarded as Dirac nodal points. Interestingly, the Dirac point at D3 coincides precisely with the Fermi level, whereas those at D1, D4, and D2 are situated slightly below and slightly above the Fermi level, respectively. These Dirac points manifest n-type at D1 and p-type at D2 and D4 self-doping behaviours. Furthermore, we have computed the Fermi velocities of holes and electrons along different crystallographic directions using v_f = E(q)/ℏ|q|, at the four Dirac points. The estimated Fermi velocities of electrons and holes from the Dirac points D1–D4 to different crystallographic directions are in the range of 2.01–8.70 × 10⁶ m s⁻¹. It is evident from Table S2 (ESI†) that the Fermi velocities are isotropic and maximum around Dirac point D2, while minimum and anisotropic around Dirac point D3. The anisotropy in the Fermi velocities at the Dirac points D1 and D4 is intermediate between that of D2 and D3.

Fig. 4 Electronic structure of the 3D COT network. (a) Electronic bands are computed along the irreducible portion of the tetragonal Brillouin zone and (b) electronic band diagram of the 3D COT network along specific high-symmetry points. The coloured dots represent the location of Dirac points (D1, D2, D3 and D4) in the irreducible Brillouin zone.

Subsequently, we performed electronic structure calculations of the 3D COT network at various k_z-planes to gain deeper insight into the linear dispersion of the VB and CB, and the interconnectivity between the four Dirac nodal points (D1–D4). We have computed the 3D band structure of the VB and CB around the Dirac points, D2 and D4, for which k-point coordinates are zero along the z-axis (i.e., k_z = 0). We have spanned the 2D k_xy-plane with finely gridded k-points and computed the energy dispersion of the valence and conduction bands in the entire tetragonal Brillouin zone as presented in Fig. S5 (ESI†). The 3D band structure of the VB and CB in the entire tetragonal Brillouin zone is presented in Fig. 5(a). We have depicted the location of the Dirac points, D2 and D4, with green and blue dots in the 3D band structure, at which the VB and CB intersect each other with linear dispersion. Furthermore, the 3D band structure shows that the VB and CB not only cross each other at the Dirac points D2 and D4 but also intersect each other at a series of points in the tetragonal Brillouin zone. These band crossing points form a four-fold X-shaped closed loop as presented in the lower panel of Fig. 5(a). Furthermore, the VB and CB exhibit linear dispersion around all these band crossing points. Hence, the series of X-shaped closed loops of intersecting points are similar to the previous Dirac points, D2 and D4. Thus, contact points inherently present in the X-shaped closed loop can be referred to as Dirac nodal points similar to previous studies. Overall, the 3D COT network possesses a series of X-shaped new Dirac points besides D2 and D4 at k_z = 0.

Fig. 5 3D electronic band structure of the valence and conduction bands of the COT network at different k_z planes, (a) k_z = 0.0, (b) k_z = 0.1, (c) k_z = 0.2, (d) k_z = 0.3, (e) k_z = 0.4 and (f) k_z = 0.5. Red dots in lower panels represent the alignment of the Dirac nodal points of the corresponding 3D band structure at the respective k_z plane.

Similarly, we have computed the 3D band structures of the VB and CB at various k_xy-planes by varying k_z values from 0.0 to 0.5, in steps of 0.01. The representative 3D band diagrams at k_z = 0.1, 0.2, 0.3, 0.4, and 0.5 along with the alignment of the Dirac nodal points are presented Fig. 5(b)–(f). The alignment of the Dirac nodal points is almost similar at k_z = 0.0 and k_z = 0.1. However, the single four-fold X-shaped closed loop of the Dirac nodal points splits into four individual closed loops as one moving from k_z = 0.2 to k_z = 0.4. Furthermore, the size of individual Dirac nodal loops decreases as k_z increases from 0.2 to 0.4. Finally, these four individual Dirac nodal loops turn to four curved lines at k_z = 0.5. We have also shown the location of the Dirac point D1, which lies on the Z → A high-symmetry line segment at the k-point coordinates, 0.282, 0.282, 0.500.

Finally, we have plotted the Dirac nodal points present in the various closed loops of the VB and CB at different k_z values in Fig. 6. The Dirac nodal points formed from a four-fold X-shaped closed loop of the Dirac nodal points in the k_z range 0.0 to 0.2, four individual closed loops of Dirac nodal points in the range of the k_z values from 0.2 to 0.4, and a curved line of the Dirac nodal points at k_z = 0.5. The varying colour represents varying k_z values in Fig. 6. Overall, the Dirac nodal points at different k_z-planes form a closed nodal surface. This closed topological nodal surface of the 3D COT network is similar to that of higher dimensional topological nodal surfaces as observed in the tight-binding models of square carbon nanotube networks and can be called a nodal flexible-surface.²⁶

Fig. 6 3D nodal flexible-surface of the 3D COT network. The varying colours indicate the different k_z values.

We conducted the analysis of the orbital and atom-projected electronic band structure of the 3D COT network to elucidate specific atomic orbitals contributing to the emergence of linearly dispersed electron and hole bands in close proximity to the Fermi level. As depicted in Fig. 7(a), our findings indicate that p_z orbitals associated with the carbon atoms of the C1 and C2 type play a predominant role in forming these linearly dispersed bands situated nearly close to the Fermi level. Furthermore, our investigation of the atom and orbital-projected density of states (DOS) provides additional evidence that the π bonding and anti-bonding orbitals of the C atoms within the 3D COT network are the primary driving factors behind the origin of the linearly dispersed valence and conduction bands as represented in Fig. 7(b). The decomposed charge densities corresponding to the VB and CB around the Dirac point, D1, are depicted in Fig. 7(c) and (d). This clearly substantiates that the hole and electronic bands in the proximity of the Fermi level are primarily obtained from the p_z bonding and antibonding orbitals of C atoms.

Fig. 7 Orbital and atom projected (a) electronic band structure and (b) density of states. Partial charge densities associated with the (c) VB and (d) CB. The isosurface value is set to 0.4 × 10⁻⁴ e bohr⁻³.

Furthermore, to explain the emergence of a nodal flexible-surface, in the first-principles band structure of the 3D COT network, we proposed a realistic 3D tight-binding (TB) model given as follows,

In the above eq., parameters , t_ij, and represent the onsite potential of the ith site, hopping integral between the jth and ith site and creation (annihilation) operator for electrons with spin σ defined for the lattice site i(j), respectively. The summation 〈i,j〉 runs over all neighbouring sites with different bond lengths. The orbital projected electronic band structure and density of states (PDOS) reveals that the p_z orbital of the C1 and C2-type C atoms predominantly contributes near the Fermi level. Therefore, the Hamiltonian with single orbital per site (p_z) basis is a reasonable consideration for exploring the necessary low-energy information of the system. The tight-binding band structure can be obtained by solving the eigenvalue problem for the Fourier transformed tight-binding Hamiltonian. However, we reduced the degree of difficulty of the problem by downfolding the Hamiltonian by means of a real space decimation approach as reported in previous studies.^65,66 In particular, we aimed to encode the information of all the carbon atoms taking part in acetylenic linkages into the remaining sites. For a lattice system, Schrödinger's equation can be written down as a set of difference equations as given below.

Here, , t_nm, E, and φ_m are the onsite potential, hopping parameter (as also mentioned above), eigen energy and probability amplitude of site m. This decimation process can be well understood for a piece of simple lattice chain depicted in Fig. 8(a), where the acetylenic linkage is between C and D. The difference equations can be written down for the system as

Fig. 8 Tight binding parameters of the 3D COT network. (a) TB model for the COT network with hopping parameters and (b) band structure of the COT network computed using the TB model.

Using above equations, we obtained a low-energy renormalized hopping parameter between A and B sites, . However, the onsite potentials of remaining decimated sites remain invariant under the above transformation. Our approach essentially maps the Hamiltonian to a matrix with a reduced dimensionality (to half) without losing any electronic information near the Fermi level. This equivalence has been established as the decimated Hamiltonian (32 × 32) exhibits a band structure that precisely mimics the first-principles results near the Fermi level, as presented in Fig. 8(b). Here, we have considered a uniform onsite potential , ∀i for all the carbon atoms that also serves as a reference zero energy or Fermi level of the system. Besides, the hopping parameters are expressed in terms of the hopping integral of sp² hybridized graphene γ = −2.7 eV as follows t = γ, t₁ = 0.98γ, and r = −0.68γ. These results strongly substantiate that the p_z orbitals of the C atoms are mainly responsible for shaping the linearly dispersed fermion bands near the Fermi level.

Overall, all the carbon allotropes that are derived from the Diamond by the substitution of the carbon-tetragon^45,51,52 are mainly semiconductors with energy gaps ranging from 1.5 to 5.5 eV, whereas the 3D COT network exhibits a nodal-flexible surface. The C–C single bonds in the carbon tetragons of the tetragon-based carbon allotropes make them semiconductors, while the presence of π-conjugation imparts unique nodal-flexible surface properties to the 3D COT network.

3.4. Applications of the 3D COT network as an anode material for the Li-ion batteries

It is evident from previous studies that π-conjugated topological carbon networks are found to be promising anode materials.^{17,19,20,39,40} Furthermore, the unique nanoporous structure, semi-metallic properties and mechanical stability inherent in the 3D COT network have spurred us to explore its potential applications as an anode material for the Li-ion batteries. To achieve this goal, we comprehensively explored the four different parameters, which include binding energies, diffusion barriers, specific capacity and open circuit voltage. We employed a 1 × 1 × 2 super cell of the 3D COT network to scrutinize the binding energies (E_b) of the Li atoms at three symmetrically non-equivalent binding sites including S₁, S₂, and S₃, as illustrated in Fig. 9(a). Specifically, S₁ is positioned at the centre of the acetylenic linkers, S₂ at the centre of cyclooctagon rings, and S₃ at the centre of the nanoporous channel along the z-axis. We estimated the binding energies of the Li atoms at these adsorption sites using,where E(Li@COT) and E(COT) represent the energies of the 3D COT network with and without a Li atom, and n is the number, and μ(Li) is the chemical potential of the Li atoms. The chemical potential of the Li atoms is determined based on the per-atom energy of the bcc bulk Li crystal. Our findings indicate that the Li atom exhibits minimum and maximum binding energies at the adsorption sites, S₂ (−0.87 eV) and S₃ (−2.14 eV), respectively. The binding energy of a Li atom at S₁ (−1.54 eV) is found to be intermediate between that at S₂ and S₃. Furthermore, we have also examined the adsorption feasibility of Li atoms at the other binding sites; for example, on top of the C–C bonds and at hollow positions of the COT network.

However, we have found that the Li atoms landed on either of S₁, S₂ or S₃ adsorption sites from the other fragments of the COT network during the atomic relaxation. To get intrinsic details about the adsorption mechanism of the Li atoms, we conducted an analysis of Bader charges on different atomic constituents of the Li-adsorbed COT network, along with examining the charge densities. The results revealed that the Li atom imparted net charges of 0.90, 0.92 and 0.91 e to the adjacent C atoms of the COT network at the adsorption sites, S₁, S₂ and S₃, respectively. Furthermore, the difference in the charge densities (Δρ) of the Li atom, COT network and Li adsorbed COT network is calculated using,

Δρ = ρ_COT+Li − (ρ_COT + ρ_Li), (5)

where ρ_COT, ρ_COT+Li and ρ_Li represent the charge densities of the COT network before Li adsorption, after Li adsorption and for the Li atom within the complex geometry, respectively. The depletion of the charge density around Li at the all-adsorption sites substantiates that Li exhibits cationic character upon adsorption into the COT network (see Fig. 9(b)–(d)). Evidently, the strong ionic interactions between the COT network and Li atoms result in favourable binding energies and that can endure the formation of dendrites or clustering of the Li atoms, which often leads to performance issues and safety concerns in the Li-ion batteries.

Fig. 9 (a) Distinct adsorption sites of the 3D COT network for the Li atoms, together with four distinct pathways for diffusion. Charge density difference plots depicting the variation in charge density resulting from the adsorption of Li at positions (b) S₁, (c) S₂, and (d) S₃. The isosurface threshold is set to 0.001 e bohr⁻³ for all charge density maps.

In addition, the diffusion kinetics of Li ions adsorbed into the 3D COT network have a great impact on the rapid charge and discharge processes of the anode materials. Hence, we have investigated the facile movement of the Li ions between the most favourable adsorption sites (i.e., S₁, S₂, and S₃) of the COT network, employing the climbing-image nudged elastic band (CI-NEB) method.⁵⁰ Additional details of the CI-NEB approach are described in the ESI.† We have examined the diffusion energy barriers of the Li ions along four possible paths, namely path 1 (S₁ → S₂ → S₁), path 2 (S₃ → S₁ → S₃), path 3 (S₂ → S₃ → S₂) and path 4 (S₃ → S₃) as indicated by black, blue, red and purple arrows in Fig. 9(a). It is evident from Fig. 10(a) that the Li ion can diffuse form one adsorption site to another adsorption site with an energy barrier of 0.68 eV along path 1, 1.27 eV along path 2, 0.66 eV along path 3 and 8 meV along path 4. Furthermore, path 4 is almost barrier-less and energetically a viable path for the movement of the Li ions during the charge–discharge process (see Fig. 10(b)).

Fig. 10 Diffusion energy barriers of Li in the 3D COT network along (a) path 1, path 2, path 3 and (b) path 4. (c) Equilibrium geometries of the lithiated 3D COT phases at different concentrations, (d) formation energies of intermediate lithiated COT phases and (e) open circuit voltage.

The open circuit voltage (OCV) profile is another essential parameter to evaluate the performance of electrode materials. Hence, we have computed the OCV profile by systematically increasing the number of the Li atoms in the COT network. The atoms were strategically introduced at energetically preferred S₃ sites, followed by S₁ and S₂ sites. We have varied the concentration of the Li ions (x) from 0 to 1 in increments of 0.1 by considering the formula unit of the lithiated COT network as Li_xC_3.2. The equilibrium geometries of lithiated COT phases at different concentrations are presented in Fig. 10(c). At each Li concentration, the binding energy per Li atom has been computed and plotted in Fig. S6 (ESI†). It shows that the average binding energy decreases with the Li concentration due to the interatomic repulsion between the Li atoms. Furthermore, we have estimated the formation energy of the lithiated COT phases at different Li concentrations using,

E_f(Li_xC_3.2) = E(Li_xC_3.2) − xE(LiC_3.2) − (1 − x)E(C_3.2), (6)

where E(Li_xC_3.2), E(LiC_3.2) and E(C_3.2) denote the total energies of the Li_xC_3.2 phase and fully lithiated and non lithiated COT networks, respectively. Fig. 10(d) illustrates that formation energies at each Li concentration clearly form a convex hull, except at x = 0.8, indicating the stability of these intermediate phases. To assess the stability of the intermediate phases lying on the convex hull, we conducted optimization of the COT network by removing the Li atoms. The results demonstrated that the COT network regains its original lattice and geometrical parameters after the removal of the Li atoms, emphasizing the recyclability of the 3D COT network. Finally, we determined the open circuit voltage by assuming that the charge–discharge process at the COT network can be defined by a following typical half-cell reaction versus Li/Li⁺,

Li_x2C_3.2 + (x₁ − x₂)Li⁺ + (x₁ − x₂)e⁻ ↔ Li_x1C_3.2 (7)

By neglecting minimal contributions from the entropy and enthalpy to the change in the total energy, the OCV profile can be obtained at different Li concentrations (x₂ < x₁) using the following approximation,

Here, E(Li_x1C_3.2) and E(Li_x2C_3.2) indicate the total energies of the lithiated COT network at two distinct Li concentrations, denoted as x₁ and x₂, μ(Li) signifying the chemical potential of the Li atom and corresponding to the energy of the Li atom in the bcc bulk phase. We have computed the OCV profile using the energies of the intermediate phases, which lies on the convex hull. It is evident from Fig. 10(e) that the OCV indicates a positive voltage until x = 0.9, after which it decreases to −0.67 V when x reaches 1. This drop suggests that further addition of the Li atoms to the COT network is unfeasible. Consequently, we have estimated the maximum storage capacity (C_M) of the COT network using C_M = zxF/M, where z is the valence number, x is the concentration of Li, F is the Faraday constant (26.8 A h mol⁻¹), and M is the mass of the electrode material in the formula unit. These computational findings corroborate that the COT network exhibits promising potential for applications in Li-ion battery anode materials, boasting a maximum theoretical storage capacity of 627.5 mA h g⁻¹. We have also listed the diffusion energy barriers, maximum storage capacity and OCV of 3D COT along with those of other 3D carbon frameworks available in the literature in Table 2. This shows that the storage capacity and OCV of the 3D COT network are higher than those of the rest of the carbon allotropes (except bco-C₂₀) included in Table 1. Furthermore, the diffusion energy barrier (8 meV) along the nanoporous channel (along the z-axis) is lower than that of other 3D carbon allotropes. These results proclaim that the 3D COT network could show great potential to serve as a promising anode for the Li-ion batteries.

Table 2 Theoretical specific storage capacity, diffusion energy barriers, open circuit voltage (OCV) and electronic conductivity of different 3D carbon allotropes

Carbon allotrope	Storage capacity (mA h g^−1)	Diffusion energy barrier (eV)	OCV (V)	Conductivity
3D COT network	628	0.008–0.68	0.85	Topological semimetal
Graphite^37	372	0.4	0.11	Dirac semimetal
bco-C20^22	893	0.02–0.12	0.41	Dirac semimetal
m-C16^20	558	0.25	0.56	Topological semimetal
bco-C16^19	558	0.019	0.23	Topological semimetal
bct-C40^67	893	0.01–0.06	0.42	Topological semimetal
IGN^68	298	0.004	0.66	Topological semimetal
HZGM-42^21	425 (638)	0.02–0.18	0.50 (0.19)	Topological semimetal
IGYN^69	496	0.3–0.45	0.67	Semiconductor
3D-BPC2^70	407	0.29	0.48	Metal
D14^71	319	0.41	—	Metal
Carbon honeycomb^72	319	0.41	—	Metal
Hex-C18^73	496	0.02–0.14	—	Metal

---

Finally, we have provided a possible synthetic route for the experimental synthesis of a 3D COT network based on earlier reports on similar carbon allotropic materials. Exploring feasible synthetic approaches for the experimental realization of carbon networks holds profound significance to unlock their practical applications. In general, sp and sp² hybridized carbon networks, such as graphyne,⁷⁴ graphdiyne⁷⁵ and graphtetrayne,⁷⁶ 2D materials featuring one, two and four acetylene linkages between benzene moieties, are synthesized using cross-coupling reactions between halobenzene and appropriate alkyne compounds. For example, Cui and co-workers have successfully fabricated γ-graphyne using hexabromobenzene (PhBr₆) and calcium carbide (CaC₂) as precursors employing the ball-milling method to facilitate the cross-coupling reaction as shown in Fig. S7 (ESI†).⁷⁴ Similarly, graphdiyne and graphtetrayne 2D materials were synthesized by substituting calcium carbide with higher-order acetylenic (viz., silyl protected ethynyl and butadienyl) scaffolds using the Sonogashira cross-coupling reaction. In a parallel fashion, the synthesis of the 3D COT network may be envisaged employing the mechanochemistry and cross-coupling reaction methods between octabromocyclotetraene (C₈Br₈) and calcium carbide or other appropriate acetylenic compounds, as illustrated in Fig. 11. This synthetic methodology may serve as a viable route for the experimental realization of the 3D COT network.

Fig. 11 Proposed synthetic procedure for the synthesis of the 3D COT network. (a) Different projections of C₈Br₈ and (b) the 3D COT network.

Overall, the 3D COT network exhibits unique electronic features, such as nodal flexible-surface states, which are relatively rare to observe in the 3D quantum models. The condensed matter system furnished with such higher dimensional topological phases are of paramount importance as they unlock novel electronic properties; for example, strong quantum oscillations and unique exciton and plasmon behaviours. Obviously, the 3D COT network emerges as a potential platform for realizing such intriguing electronic phenomena and thus opens new avenues for exploring the application potential in materials science and technology. Furthermore, the low atomic density combined with nano-porous nature, energetic favourability and optimal adsorption energies for Li, low diffusion energy barriers, and high storage capacity may position the 3D COT network as a potential candidate for Li-ion battery energy storage devices.

4. Conclusions

In this work, ab initio density functional theory was employed to predict a unique three-dimensional carbon network based on a cyclooctatetraene (COT) unit. We thoroughly investigated the structural stability of this 3D COT network by examining various factors, including total energy, phonon band dispersion, thermal fluctuations at different temperatures and independent elastic constants. Our computational findings suggest that the COT network, while energetically metastable compared to diamond and graphite, demonstrates dynamic and mechanical stability. Furthermore, through molecular dynamics simulations, we established that the COT network remains stable up to a maximum temperature of 1500 K. Electronic structure calculations have unveiled that the COT network features intriguing linear band dispersion patterns near the Fermi level and adorned with nodal crossbar-like topological features. Orbital/atom projected band structures, density of states and tight binding models have been included to unravel atomic and orbital factors contributing to the formation of nodal flexible-surfaces. Carbon allotropes featuring such distinctive band characteristics hold promise for the exploration of phenomena like high-temperature superconductivity and strong-correlation effects, owing to their substantial electronic density of states at the Fermi level. Open circuit voltage (0.85 V), maximum storage capacity (627.5 mA⁻¹ h g⁻¹) and low diffusion energy barriers (0.008–0.68 eV) corroborate the propensity of the 3D COT network as a potential anode material for Li-ion batteries.

Author contributions

Naga Venkateswara Rao Nulakani: conceptualization, investigation, data curation, methodology, validation, visualization and writing the original draft. Arka Bandyopadhyay: tight binding methodology and the band structure. Mohamad Akbar Ali: conceptualization, supervision, writing, review and editing.

Conflicts of interest

There are no conflicts of interest to disclose.

Acknowledgements

NVRN and MAA thank the supercomputer facility at Khalifa University of Science and Technology, at Abu Dhabi, UAE. MAA thanks Khalifa University of Science and Technology for Faculty Start-up grant #8474000461.

References

1. M. Z. Hasan and C. L. Kane, Rev. Mod. Phys., 2010, 82, 3045–3067.
2. L. Fu, C. L. Kane and E. J. Mele, Phys. Rev. Lett., 2007, 98, 106803.
3. B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett., 2006, 96, 106802.
4. C. L. Kane and E. J. Mele, Phys. Rev. Lett., 2005, 95, 146802.
5. S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S. Jia, A. Bansil, H. Lin and M. Z. Hasan, Nat. Commun., 2015, 6, 7373.
6. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Nature, 2005, 438, 197–200.
7. S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele and A. M. Rappe, Phys. Rev. Lett., 2012, 108, 140405.
8. L. Meng, J. Wu, J. Zhong and R. A. Römer, Nanoscale, 2019, 11, 18358–18366.
9. S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia and M. Z. Hasan, Science, 2015, 349, 613–617.
10. I. Belopolski, K. Manna, D. S. Sanchez, G. Chang, B. Ernst, J. Yin, S. S. Zhang, T. Cochran, N. Shumiya, H. Zheng, B. Singh, G. Bian, D. Multer, M. Litskevich, X. Zhou, S.-M. Huang, B. Wang, T.-R. Chang, S.-Y. Xu, A. Bansil, C. Felser, H. Lin and M. Z. Hasan, Science, 2019, 365, 1278–1281.
11. J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan, M. Hirschberger, W. Wang, R. J. Cava and N. P. Ong, Science, 2015, 350, 413–416.
12. E. Zhang, R. Chen, C. Huang, J. Yu, K. Zhang, W. Wang, S. Liu, J. Ling, X. Wan, H.-Z. Lu and F. Xiu, Nano Lett., 2017, 17, 878–885.
13. T. Yang, S. Ding, Y. Liu, Z. Wu and G. Zhang, Phys. Chem. Chem. Phys., 2022, 24, 8208–8216.
14. Z. Zhao, Z. Zhang and W. Guo, J. Mater. Chem. C, 2020, 8, 1548–1555.
15. D. Ni, Y. Shen, W. Sun and Q. Wang, J. Mater. Chem. A, 2022, 10, 7754–7763.
16. C. Zhong, Y. Chen, Y. Xie, S. A. Yang, M. L. Cohen and S. B. Zhang, Nanoscale, 2016, 8, 7232–7239.
17. H. Chen, S. Zhang, W. Jiang, C. Zhang, H. Guo, Z. Liu, Z. Wang, F. Liu and X. Niu, J. Mater. Chem. A, 2018, 6, 11252–11259.
18. Y. Qie, J. Liu, S. Wang, Q. Sun and P. Jena, J. Mater. Chem. A, 2019, 7, 5733–5739.
19. J. Liu, S. Wang and Q. Sun, Proc. Natl. Acad. Sci. U. S. A., 2017, 114, 651–656.
20. H. Xie, Y. Qie, M. Imran and Q. Sun, J. Mater. Chem. A, 2019, 7, 14253–14259.
21. J. Liu, X. Li, Q. Wang, Y. Kawazoe and P. Jena, J. Mater. Chem. A, 2018, 6, 13816–13824.
22. S. Wang, Z. Peng, D. Fang and S. Chen, Nanoscale, 2020, 12, 12985–12992.
23. N. V. R. Nulakani and V. Subramanian, J. Mater. Chem. C, 2018, 6, 7626–7634.
24. Y. Qie, J. Liu, S. Wang, Q. Sun and P. Jena, J. Mater. Chem. A, 2019, 7, 5733–5739.
25. J. Wang, Y. Liu, K.-H. Jin, X. Sui, L. Zhang, W. Duan, F. Liu and B. Huang, Phys. Rev. B, 2018, 98, 201112.
26. S.-Z. Chen, S. Li, Y. Chen and W. Duan, Nano Lett., 2020, 20, 5400–5407.
27. J. Wang, X. Sui, S. Gao, W. Duan, F. Liu and B. Huang, Phys. Rev. Lett., 2019, 123, 206402.
28. W. Wu, Y. Liu, S. Li, C. Zhong, Z.-M. Yu, X.-L. Sheng, Y. X. Zhao and S. A. Yang, Phys. Rev. B, 2018, 97, 115125.
29. M. Armand and J.-M. Tarascon, Nature, 2008, 451, 652–657.
30. D. D. Sarma and A. K. Shukla, ACS Energy Lett., 2018, 3, 2841–2845.
31. G. Xu, B. Ding, J. Pan, P. Nie, L. Shen and X. Zhang, J. Mater. Chem. A, 2014, 2, 12662–12676.
32. N. Mahmood, C. Zhang, H. Yin and Y. Hou, J. Mater. Chem. A, 2014, 2, 15–32.
33. B. Dunn, H. Kamath and J.-M. Tarascon, Science, 2011, 334, 928–935.
34. S. Zhao, W. Kang and J. Xue, J. Mater. Chem. A, 2014, 2, 19046–19052.
35. Y. Zhu, X. He and Y. Mo, J. Mater. Chem. A, 2016, 4, 3253–3266.
36. V. B. Shenoy, P. Johari and Y. Qi, J. Power Sources, 2010, 195, 6825–6830.
37. E. M. Erickson, C. Ghanty and D. Aurbach, J. Phys. Chem. Lett., 2014, 5, 3313–3324.
38. J.-M. Tarascon and M. Armand, Nature, 2001, 414, 359–367.
39. J. Liu, S. Wang, Y. Qie, C. Zhang and Q. Sun, Phys. Rev. Mater., 2018, 2, 25403.
40. S. Wang, Z. Chen, B. Yang, H. Chen and E. Ruckenstein, J. Colloid Interface Sci., 2019, 555, 431–437.
41. G. Kresse and J. Furthmüller, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169–11186.
42. G. Kresse and J. Furthmüller, Comput. Mater. Sci., 1996, 6, 15–50.
43. P. E. Blöchl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 17953–17979.
44. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865–3868.
45. X.-L. Sheng, Q.-B. Yan, F. Ye, Q.-R. Zheng and G. Su, Phys. Rev. Lett., 2011, 106, 155703.
46. W.-J. Yin, Y.-E. Xie, L.-M. Liu, R.-Z. Wang, X.-L. Wei, L. Lau, J.-X. Zhong and Y.-P. Chen, J. Mater. Chem. A, 2013, 1, 5341.
47. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Solid State, 1976, 13, 5188–5192.
48. A. Togo, F. Oba and I. Tanaka, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 134106.
49. S. Grimme, J. Antony, S. Ehrlich and H. Krieg, J. Chem. Phys., 2010, 132, 154104.
50. G. Henkelman, B. P. Uberuaga and H. Jónsson, J. Chem. Phys., 2000, 113, 9901–9904.
51. L. Yang, H. Y. He and B. C. Pan, J. Chem. Phys., 2013, 138, 024502.
52. C. He, L. Sun, C. Zhang and J. Zhong, Phys. Chem. Chem. Phys., 2013, 15, 680–684.
53. Y. Cheng, X. Feng, X. Cao, B. Wen, Q. Wang, Y. Kawazoe and P. Jena, Small, 2017, 13, 1602894.
54. S. Zhang, J. Zhou, Q. Wang, X. Chen, Y. Kawazoe and P. Jena, Proc. Natl. Acad. Sci. U. S. A., 2015, 112, 2372–2377.
55. H. Prinzbach, A. Weiler, P. Landenberger, F. Wahl, J. Wörth, L. T. Scott, M. Gelmont, D. Olevano and B. v Issendorff, Nature, 2000, 407, 60–63.
56. Y. L. Mao, X. H. Yan, Y. Xiao, J. Xiang, Y. R. Yang and H. L. Yu, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 71, 033404.
57. X. Zhao, Y. Liu, S. Inoue, T. Suzuki, R. O. Jones and Y. Ando, Phys. Rev. Lett., 2004, 92, 125502.
58. K. Umemoto, R. M. Wentzcovitch, S. Saito and T. Miyake, Phys. Rev. Lett., 2010, 104, 125504.
59. A. Mahata, P. Garg, K. S. Rawat, P. Bhauriyal and B. Pathak, J. Mater. Chem. A, 2017, 5, 5303–5313.
60. J. Zhou, J. Huang, B. G. Sumpter, P. R. C. Kent, Y. Xie, H. Terrones and S. C. Smith, J. Phys. Chem. C, 2014, 118, 16236–16245.
61. Z. Zhang, X. Liu, B. I. Yakobson and W. Guo, J. Am. Chem. Soc., 2012, 134, 19326–19329.
62. N. Duhan and T. J. Dhilip Kumar, Phys. Chem. Chem. Phys., 2024, 26, 11140–11149.
63. D. D. Nematov, A. S. Burhonzoda, K. T. Kholmurodov, A. I. Lyubchyk and S. I. Lyubchyk, Nanomaterials, 2023, 13, 2657.
64. D.-C. Yang, R. I. Eglitis, Z.-J. Yi, C.-S. Liu and R. Jia, J. Mater. Chem. C, 2022, 10, 10843–10852.
65. A. Bandyopadhyay, S. Datta, D. Jana, S. Nath and Md. M. Uddin, Sci. Rep., 2020, 10, 2502.
66. A. Bandyopadhyay and D. Jana, Rep. Prog. Phys., 2020, 83, 056501.
67. S. Wang, Z. Chen, B. Yang, H. Chen and E. Ruckenstein, J. Colloid Interface Sci., 2019, 555, 431–437.
68. J. Liu, S. Wang, Y. Qie, C. Zhang and Q. Sun, Phys. Rev. Mater., 2018, 2, 025403.
69. C. Zhong, W. Zhang, G. Ding and J. He, Carbon, 2019, 154, 478–484.
70. U. Younis, I. Muhammad, F. Qayyum, Y. Kawazoe and Q. Sun, J. Mater. Chem. A, 2020, 8, 25824–25830.
71. D. Fan, A. A. Golov, A. A. Kabanov, C. Chen, S. Lu, X. Li, M. Jiang and X. Hu, J. Phys. Chem. C, 2019, 123, 15412–15418.
72. J. Hu and X. Zhang, Eur. Phys. J. B, 2018, 91, 76.
73. J. Liu, T. Zhao, S. Zhang and Q. Wang, Nano Energy, 2017, 38, 263–270.
74. Q. Li, Y. Li, Y. Chen, L. Wu, C. Yang and X. Cui, Carbon, 2018, 136, 248–254.
75. G. Li, Y. Li, H. Liu, Y. Guo, Y. Li and D. Zhu, Chem. Commun., 2010, 46, 3256–3259.
76. J. Gao, J. Li, Y. Chen, Z. Zuo, Y. Li, H. Liu and Y. Li, Nano Energy, 2018, 43, 192–199.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4tc00019f

---