Graphyne-Derived C₁₂-GR and C₁₆-GY as Potential Nanocapacitor Materials

Abstract

Graphyne (GY)-derived carbon allotropes such as C₁₂-GR and C₁₆-GY, recently predicted through mechanically induced reconstruction pathways [Kotoko, K. J.; Sodoga, K.; Shaidu, Y.; Seriani, N.; Borah, S.; Beltako, K. Uniaxial Tensile-Induced Phase Transition in Graphynes. J. Phys. Chem. C 2024, 128, 17058], constitute emerging two-dimensional carbon phases with unconventional bonding topologies. In this work, we investigate their structural, electronic, optical, and electrostatic energy-storage properties using first-principles calculations. Both monolayer and AA-stacked bilayer configurations exhibit semi-metallic behavior, accompanied by strong dielectric screening and broadband optical absorption extending from the infrared to the ultraviolet region. The presence of acetylenic linkages in C₁₆-GY enhances the electronic polarizability and increases the density of states near the Fermi level, leading to a stronger response under external electric fields. Under out-of-plane bias, the bilayer systems display reversible field-induced charge accumulation and significant electrostatic energy storage, yielding effective gravimetric capacitance values on the order of ∼100 F/g for C₁₂-GR and above ∼120 F/g for C₁₆-GY within the idealized bilayer capacitor model considered here. These values compare favorably with previously reported theoretical carbon-based nanocapacitor architectures. Overall, the results suggest that mechanically reconstructed GY derivatives may provide promising platforms for nanoscale electrostatic energy storage and nanoelectronic applications.

Introduction

The remarkable diversity of carbon hybridizations (sp, sp², and sp³) has enabled the emergence of a vast family of carbon allotropes, ranging from naturally occurring materials such as graphite and diamond, to engineered low-dimensional systems including fullerenes, carbon nanotubes, and graphene (GR). The latter, composed of a single layer of sp²-hybridized carbon atoms arranged in a honeycomb lattice, has revolutionized the landscape of two-dimensional (2D) materials due to its exceptional mechanical strength, ultrahigh electron mobility, thermal conductivity, and the presence of massless Dirac Fermions in its electronic band structure. − However, the absence of a bandgap in GR poses a fundamental limitation for its integration into semiconductor-based electronics, where on/off switching behavior is crucial.

This inherent shortcoming has catalyzed a widespread search for alternative 2D carbon materials that retain GR’s outstanding attributes while offering greater versatility in electronic properties. Among the most promising candidates are graphynes (GYs), , a class of 2D carbon allotropes composed of both sp- and sp²-hybridized atoms. These materials are designed by introducing acetylenic (–CC–) linkages into the GR framework, yielding a rich spectrum of geometric motifs and electronic structures. Depending on the type and density of these triple bonds, GY allotropes such as α-, β-, γ-, and extended forms like [14,14,18]-GY can display a broad spectrum of electronic behaviors, including semi-metallicity, direct or indirect bandgaps, and even Dirac-like dispersions. −

The intrinsic porosity, π-conjugation diversity, and structural tunability of GYs have attracted considerable interest for applications beyond electronics. Their unique architectures make them promising candidates for hydrogen storage, ion transport, photocatalysis, and energy storage systems, particularly in supercapacitors and batteries. − Unlike GR, which requires postsynthesis modifications to open a bandgap, GYs inherently offer structural degrees of freedom to engineer their electronic behavior via topological design, strain, doping, or stacking. ,−

In addition to their electronic flexibility, GYs are mechanically versatile. − Their atomically thin and porous nature allows them to respond dramatically to external perturbations such as strain, leading to possible topological transformations and phase transitions under mechanical deformation. These transformations can yield new, previously unreported phases with altered atomic frameworks and potentially distinct physical properties. Such mechanically induced reconfigurations are of particular interest for the rational design of 2D materials. They enable access to a broader range of functional properties without relying on chemical functionalization or external substrates.

In parallel with these efforts, the growing demand for compact and efficient energy storage systems has sparked considerable interest in nanoscale capacitors, which offer ultrafast charge/discharge dynamics and high power delivery capabilities. Unlike conventional batteries, these devices exhibit rapid energy storage and release, operate at high power densities, and maintain excellent cycling stability over extended use. − Such characteristics make them highly attractive for integration in next-generation electronics, power regulation circuits, and energy harvesting platforms. To enhance capacitive performance, various material systems have been explored, including metal oxides, conductive polymers, and carbon-based nanostructures. Among the latter, carbon allotropes such as GR, , carbon nanotubes, , and GYs have exhibited particularly promising performances, with specific capacitances typically reported in the ranges of 5.6–24.5 F/g and up to 20–40 F/g. These results are largely attributed to their intrinsically high surface area, excellent electrical conductivity, and outstanding chemical stability. Notably, GR-based heterostructures have shown improved charge storage capacities, particularly in bilayer configurations combining GR with materials such as silicene or hexagonal boron nitride. − These advances underscore the importance of tailoring the structural and electronic features of carbon nanostructures to achieve desirable capacitive behavior at the nanoscale.

In light of these developments, understanding the functional potential of emergent GY-based phases becomes a vital task. Yet, studies that integrate both optical and capacitive properties remain scarce, limiting a comprehensive assessment of their applicability in energy and nanoelectronic technologies. In our recent work, we predicted two novel carbon allotropes, C₁₂-GR and C₁₆-GY, through mechanical reconstruction of α-GY and extended GY frameworks, respectively. These structures exhibit distinctive topologies: C₁₆-GY incorporates acetylenic linkages, while C₁₂-GR contains no such bonding motifs, features that significantly affect their electronic delocalization and dielectric behavior. Importantly, their dynamical and mechanical stability was validated through extensive molecular dynamics (MD) simulations, which revealed that both allotropes maintain structural integrity under thermal agitation and mechanical strain. Furthermore, their elastic and failure properties surpass those of several widely studied 2D materials, including germanene, silicene, and stanene, positioning them as highly resilient candidates for future nanoscale devices.

A comprehensive understanding of these 2D phases is still lacking in the literature, particularly regarding their electronic structure, optical response, and capacitive behavior at the atomic scale. Such fundamental insights are essential to fully assess their potential and to determine their suitability for advanced technological applications. In this context, the present work provides a comprehensive first-principles investigation of C₁₂-GR and C₁₆-GY, aiming to elucidate their electronic band structures, work functions, optical responses, and energy storage capabilities. By employing density functional theory (DFT) along with advanced post-processing techniques, we seek to uncover the structure–property relationships that govern their behavior. This effort bridges fundamental design and practical application, positioning these GY-derived nanosheets as promising candidates for next-generation nanoelectronic devices and nanoscale energy storage technologies.

Computational Methods

DFT Calculations

All calculations were performed within the framework of DFT using the Quantum ESPRESSO package. For electronic properties and phonon calculations, the projector augmented-wave method is adopted in conjunction with the generalized gradient approximation in the Perdew–Burke–Ernzerhof (PBE) form. , While the PBE exchange-correlation functional provides a reliable description of ground-state properties for carbon-based systems, it is known to underestimate band gaps. Consequently, absolute band gap values and optical excitation energies should be interpreted with caution. Nevertheless, PBE has been shown to reliably capture qualitative trends in electronic structure, charge distribution, and relative stability, especially in sp²- and sp/sp²-bonded carbon systems, which are the focus of the present study. A plane-wave cutoff energy of 80 Ry is employed to ensure convergence of total energies and forces. Atomic positions were relaxed until the residual forces on each atom were smaller than 10^–5 Ry/Bohr, and the total energy variation was below 10^–8 Ry. The Brillouin zone is sampled using Monkhorst-Pack grids of 19 × 19 × 1 for the SCF step and 76 × 76 × 1 for the NSCF step in the case of the C₁₂-GR system. For the C₁₆-GY system, 19 × 13 × 1 and 76 × 52 × 1 grids are employed for the SCF and NSCF calculations, respectively. Optical properties were computed separately using norm-conserving Hamann pseudopotentials within the same PBE framework, in order to obtain accurate dielectric functions and absorption spectra. These calculations are performed on the dense NSCF meshes mentioned above, following SCF calculations to guarantee well-converged charge densities. Phonon spectra are evaluated within the finite-displacement supercell method as implemented in the Phonopy code. Supercells of 3 × 2 × 1 combined with Γ-centered 3 × 3 × 1 k-point meshes were employed for both monolayer and bilayer configurations to extract the harmonic interatomic force constants. For bilayer systems, long-range interlayer van der Waals interactions were explicitly included through the Grimme DFT-D3 dispersion correction. Finally, a vacuum spacing exceeding 20 Å are introduced along the out-of-plane direction to eliminate spurious interactions between periodic images and to properly describe the electrostatic potential profile relevant for work function and nanocapacitor analyses.

Theoretical Framework for Optical Properties

The optical absorption behavior as a function of frequency is characterized by the absorption coefficient α­(ω), which is derived from the complex dielectric response ε­(ω) = ε₁(ω) + iε₂(ω). Within the independent-particle framework, the imaginary part ε₂(ω), associated with photon-induced electronic transitions, is computed based on the interband momentum matrix elements: ,

$${\epsilon }_2(\omega )=\frac{2\pi e^2}{\mathrm{\Omega }{\epsilon }_0}\sum _{\mathbf{k},v,c}{|⟨{\mathrm{\Psi }}_{\mathbf{k}}^c|\mathbf{u}·\mathbf{r}|{\mathrm{\Psi }}_{\mathbf{k}}^v⟩|}^2\delta ({ϵ}_{\mathbf{k}}^c-{ϵ}_{\mathbf{k}}^v-\omega )$$ 1

where ω represents the incident photon energy, Ω is the unit cell volume, and ε₀ is the permittivity of vacuum. The summation includes all k-points and transitions from valence (v) to conduction (c) states. The operator u·r captures the interaction of the electric field polarization with the electronic states. The real part ε₁(ω) is determined from ε₂(ω) via the Kramers–Kronig transformation, , which ensures the analytical consistency of the dielectric function. From the computed dielectric components, the complex refractive index ñ(ω) = n(ω) + ik(ω) is obtained, where $n(\omega )=\sqrt{(|\epsilon (\omega )|+\mathrm{R}\mathrm{e}[\epsilon (\omega )])/2}$ is the refractive index and $k(\omega )=\sqrt{(|\epsilon (\omega )|-\mathrm{R}\mathrm{e}[\epsilon (\omega )])/2}$ is the extinction coefficient. Finally, the absorption coefficient is expressed in terms of the extinction coefficient as α­(ω) = 2ω k(ω)/c, where c is the speed of light in vacuum.

Results and Discussion

Structural Properties

The atomic configurations of the newly predicted C₁₂-GR and C₁₆-GY monolayers are presented in Figure . Panels (a) and (b) display top views of the two lattices, while panels (c) and (d) show side views of AA-stacked bilayers used for interlayer interaction analysis. In both structures, carbon atoms are arranged in a periodic 2D structure composed of rings with different sizes and symmetries. C₁₂-GR exhibits a compact and planar 2D architecture composed entirely of sp²-hybridized carbon atoms arranged in a periodic structure of non-hexagonal rings, notably 12-membered cycles, with no acetylenic linkages. Its primitive cell, delimited by the red dashed parallelogram in Figure a, is hexagonal with equal lattice parameters (a = b = 5.18 Å) and an angle of γ = 120°, corresponding to the P6/mmm space group. In contrast, C₁₆-GY is characterized by the integration of linear acetylenic (–CC–) linkages connecting sp² carbon nodes. This results in a more open and porous structure, visible in Figure b, with larger ring motifs and extended conjugation paths. The presence of triple bonds introduces directionality and symmetry breaking, leading to a monoclinic lattice with unequal lattice constants (a = 5.19 Å, b = 7.51 Å) and a nonorthogonal angle β = 110.23°, placing the structure in the Pm space group.

1.

Top and side views of (a,c) C₁₂-GR and (b,d) C₁₆-GY structures. Panels (a,b) show monolayer top views, highlighting the primitive unit cell (red dashed parallelogram) and the supercell (blue rectangle). Atoms are labeled (C₀, C₁, ...) for bond length analysis. Panels (c,d) depict AA-stacked bilayers used to determine the equilibrium interlayer spacing d _in.

The main lattice constants, crystallographic symmetries, and representative bond lengths of the two materials are summarized in Table , which highlights the purely sp² nature of C₁₂-GR in contrast to the mixed sp/sp² character of C₁₆-GY, along with the presence of triple bonds characteristic of acetylenic units.

1. Structural Parameters of C₁₂-GR and C₁₆-GY Monolayers .

property	C12-GR	C16-GY
crystal system	hexagonal	monoclinic
Bravais lattice	hP	mP
space group (no.)	P6/mmm (191)	Pm (6)
lattice constants [Å]	a = b = 5.182	a = 5.194, b = 7.505
lattice angles [deg]	α = β = 90, γ = 120	α = γ = 90, β = 110.23
atoms per unit cell	6	8
bonding type	pure sp2	mixed sp/sp2
Representative Bond Lengths [Å]
C0–C1 (sp/sp2–sp2)	1.357	1.336
C1–C3 (sp2–sp2)	1.419	1.421
C2–C3 (sp2–sp2)	1.419	1.413
C3–C4 (sp2–sp2)	1.354	1.354
C0–C5 (sp–sp)	-	1.248
interlayer distance d in [Å]	3.263	3.285

^aBond lengths are measured between labeled atoms in Figure .

To better understand the bonding environments, atoms are labeled (C₀, C₁, ..., C₅) in Figure for bond length evaluation. The C₀–C₁ bond in both materials corresponds to a short sp-sp² or sp²-sp² bond of approximately 1.34–1.35 Å, indicating strong covalent interaction. The C₁–C₃ and C₂–C₃ bonds, both around 1.41–1.42 Å, are consistent with standard C–C single bonds in aromatic frameworks. In C₁₆-GY, the C₀–C₅ bond is notably shorter (∼1.25 Å), characteristic of a triple bond between sp-hybridized carbon atoms, confirming the presence of acetylenic linkers.

Side views of the bilayer configurations [Figure c,d] show that both systems retain planar morphologies with relatively weak interlayer undulations. The equilibrium interlayer distance d _in, evaluated using DFT-D3 dispersion corrections, is indicated by red arrows in Figure c,d. These bilayers serve as reference systems to study out-of-plane properties such as interlayer binding energy and nanocapacitor performance. The total energy variations as a function of interlayer distance are shown in Figure for both materials. C₁₂-GR [panel (a)] exhibits a minimum energy at d _in = 3.263 Å, while C₁₆-GY [panel (b)] shows a slightly smaller optimal separation of ∼3.285 Å. Despite the enhanced porosity of C₁₆-GY, the similar equilibrium distances suggest that both structures experience comparable van der Waals attraction due to their flat π-conjugated layers. The structural differences outlined above, namely, the symmetry, porosity, and bonding topology are expected to have a direct impact on the electronic, optical, and transport properties of these materials. In the following sections, we systematically explore these aspects using first-principles calculations to assess their functional potential for nanocapacitor and electronic applications.

2.

Total energy as a function of interlayer separation d _in for (a) C₁₂-GR and (b) C₁₆-GY bilayers. The equilibrium spacing is highlighted with red dashed lines and reported in the legends. Calculations were performed using DFT-D3 dispersion corrections.

Energetic and Dynamical Stability

The structural stability of the proposed C₁₂-GR and C₁₆-GY phases was assessed through both energetic and vibrational analyses. The thermodynamic stability was first evaluated by computing the formation energy per atom using the expression

$$\mathrm{\Delta }H=\frac{E_{\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{e}\mathrm{t}}-n\times E_{\mathrm{C}}}{n}$$ 2

where E _sheet is the total energy of the mono or bilayer structure, E _C is the energy of an isolated carbon atom, and n is the number of carbon atoms in the system. The calculated cohesive energies show that monolayer C₁₂-GR has a value of −7.61 eV/atom, which slightly decreases to −7.55 eV/atom in the AA-stacked bilayer, indicating a weak but stabilizing interlayer interaction. A similar behavior is observed for C₁₆-GY, with energies of −7.64 eV/atom for the monolayer and −7.58 eV/atom for the bilayer. These values are considerably more negative than those reported for experimentally synthesized graphtetrayne (−7.11 eV/atom), γ-graphyne (−7.21 eV/atom), graphdiyne (−7.20 eV/atom), and graphene (−7.46 eV/atom), − confirming the thermodynamic stability of both phases and the intrinsic stability of their AA-stacked bilayer configurations.

To complement the energetic analysis and confirm the viability of these systems, their dynamical stability was assessed through phonon dispersion calculations, as depicted in Figure . For the mono and bilayer structures, the phonon spectra display no imaginary modes in the Brillouin zone, indicating the absence of lattice instabilities. This result reinforces the dynamic stability inferred from our earlier MD simulations. A notable feature of the C₁₆-GY spectra is the presence of high-frequency phonon modes around 2100–2200 cm^–1, clearly visible in both monolayer (Figure a) and bilayer forms (Figure b). These modes are characteristic of –CC– stretching vibrations and serve as spectroscopic fingerprints of the acetylenic linkages inherent to the sp-hybridized carbon atoms ,, in C₁₆-GY. By contrast, these modes are absent in C₁₂-GR (Figure a,b), whose vibrational spectrum is limited to a maximum frequency below 2000 cm^–1, consistent with a purely sp²-hybridized carbon network. ,

3.

Phonon dispersion curves along high-symmetry directions in the Brillouin zone: (a) monolayer and (b) bilayer configurations of C₁₂-GR, and (c) monolayer and (d) bilayer configurations of C₁₆-GY. The absence of imaginary frequencies confirms the dynamical stability of all systems. No high-frequency modes are observed in C₁₂-GR, consistent with the absence of triple bonds. In contrast, the prominent high-frequency modes above 2100 cm^–1 in C₁₆-GY correspond to vibrations of the –CC– triple bonds.

Moreover, the acoustic phonon branches in all structures exhibit the expected behavior for 2D crystals: the transverse and longitudinal acoustic modes increase linearly from the Γ point, while the out-of-plane flexural mode follows a quadratic dispersion. In bilayer systems, additional low-frequency flat modes emerge below 100 cm^–1, corresponding to interlayer shear and breathing vibrations, which provide insight into weak van der Waals interactions between the layers. Together, the combination of low formation energies, the absence of imaginary phonon modes, and the presence of characteristic high-frequency vibrations in C₁₆-GY confirms the stability of these two novel GY-derived carbon allotropes. These findings support their potential for integration in nanoelectronic, optoelectronic, and energy storage applications.

Electronic Properties

The electronic band structures and density of states (DOS) of C₁₂-GR and C₁₆-GY, in both monolayer and bilayer configurations, are depicted in Figure . These analyses reveal that both materials exhibit semi-metallic character, albeit with distinct orbital features and band dispersions stemming from their contrasting structural motifs.

4.

Electronic band structures and projected density of states (PDOS) for (a) mono and (b) bilayer C₁₂-GR, with insets highlighting the semi-metallic band overlap near the Fermi level; and (c) mono and (d) bilayer C₁₆-GY, where insets illustrate the effect of acetylenic linkages on band flatness and the DOS near E _F.

Figure a presents the electronic band structure of monolayer C₁₂-GR along the high-symmetry path M–K–Γ–M. The system exhibits a semi-metallic nature, with the valence band maximum (VBM) and conduction band minimum (CBM) exhibiting a slight overlap around the Γ point. The zoomed-in inset highlights the presence of a small indirect band crossing near the Fermi level E _F, confirming the absence of a true gap. This behavior resembles that of GR but with less linear dispersion and reduced band velocity, suggesting a deviation from ideal Dirac-like dispersion due to lattice reconstruction. The PDOS confirms that the states near E _F are dominated by carbon 2p orbitals, which are responsible for π and π* states in sp²-hybridized carbon frameworks. The minimal contribution from 2s orbitals reinforces the π-conjugated nature of electronic states, which typically enable high in-plane carrier mobility in graphitic systems. In the AA-stacked bilayer C₁₂-GR [Figure b], the overall band topology is preserved. The slight energy shift of bands near E _F results from weak interlayer van der Waals coupling, but no significant bandgap opening occurs. The degeneracy lifting is minimal, indicating negligible interlayer orbital hybridization. The semi-metallic character remains intact, suggesting that interlayer stacking does not significantly degrade or enhance the metallicity of the system. The DOS retains a low but finite value at E _F, which implies preserved metallic conductivity with a limited density of carriers.

Figure c illustrates the electronic structure of monolayer C₁₆-GY. The system is likewise semi-metallic, but its band structure is qualitatively different. Notably, the band dispersion is less symmetric, and flatter conduction bands appear near the Fermi level, particularly along the Γ–Y–C direction. These bands originate from localized states associated with sp-hybridized carbon atoms forming acetylenic (–CC–) linkages. Such bonding introduces stronger σ–π separation and reduces band dispersion due to electron localization in triple bonds. The inset shows a larger indirect overlap between VBM and CBM, compared to C₁₂-GR, with the VBM lying slightly above and the CBM slightly below E _F. This reinforces the semi-metallic character but with enhanced DOS around E _F, as confirmed in the right panel. For the bilayer C₁₆-GY [Figure d], stacking results in more pronounced modifications. The CBM and VBM become flatter and show increased overlap, especially along the Y–H path. This behavior suggests enhanced electron–electron interactions or partial interlayer state hybridization. Additionally, the enhanced DOS at E _F may signal increased screening and electronic compressibility, which could promote stronger electron–phonon coupling favorable for specific applications such as energy storage or catalysis. Overall, both phases are intrinsically semi-metallic. However, C₁₆-GY distinguishes itself through: (i) the presence of localized states associated with triple bonds, (ii) higher DOS near E _F, (iii) flatter bands that may suppress carrier velocities and reduce mobility but enhance electronic capacitance, and (iv) an asymmetric and a more complex electronic landscape that may influence transport properties.

The observed semi-metallicity and rich orbital character suggest promising electronic tunability through functionalization or doping. Furthermore, the combination of dispersive and localized states provides a favorable platform for coexisting fast transport channels (through delocalized π states) and charge trapping or storage zones (due to localized states).

Work Function

Planar-averaged electrostatic-potential profiles (Figure ) give work function values of 5.43 eV for monolayer C₁₂-GR and 5.40 eV for monolayer C₁₆-GY. When a second sheet is added in AA stacking, Φ decreases to 4.99 eV for C₁₂-GR and 4.96 eV for C₁₆-GY. These magnitudes are higher than those of pristine GR (4.5–5.05 eV) − and γ-GY (4.29 eV), and they rival the reported work function of graphdiyne (5.14 eV). The magnitude of Φ is closely linked to atomic topology. In C₁₂-GR, an entirely sp² carbon network without acetylenic linkages produces a high surface charge density; the accompanying steep electrostatic gradient pushes the Fermi level downward and elevates Φ. By contrast, the more open sp/sp² framework of C₁₆-GY reduces the surface charge slightly, explaining a work function drop of about 0.3 eV. In both lattices, the semi-metallic character pins E _F near mid-π band, discouraging surface-state pinning and conferring robustness on Φ. Adding a second sheet systematically lowers the work function by 0.44 eV, an effect attributable to surface-dipole reduction as the upper layer partially screens interfacial charge, bringing Fermi level E _F closer to vacuum level E _vac. Owing to their large work functions, C₁₂-GR and C₁₆-GY may be relevant for tunneling barriers in nanoscale electronic architectures, for contact engineering in 2D electronic devices, and for selective charge extraction interfaces in photovoltaic systems. Within metal–insulator–metal structures, a high work function generally increases the Schottky barrier height and suppresses leakage currents. High-work function materials such as MoO _x (with x < 3) and RuCl₃ have already attracted considerable interest for low-resistance contacts in two-dimensional devices. − In this context, the GY-based bilayers investigated here combine relatively large work functions with lightweight all-carbon architectures and tunable electronic properties induced by stacking configuration. Although explicit transport, catalytic, or sensing properties were not directly investigated in the present work, the calculated electronic characteristics suggest that these systems could constitute interesting candidates for future studies targeting nanoscale electronic and interfacial applications. Therefore, the present results primarily establish the intrinsic electronic and electrostatic properties of C₁₂-GR and C₁₆-GY bilayers, while prospective applications in optoelectronics or nanoelectronic devices remain to be explored through dedicated transport and interface investigations.

5.

Planar-averaged electrostatic potential profiles used to extract the work function of each structure. (a) Monolayer and (b) AA-stacked bilayer of C₁₂-GR; (c) monolayer and (d) AA-stacked bilayer of C₁₆-GY. In every panel, the solid blue curve is the electrostatic potential, the black dashed line marks the Fermi level, and the gray dotted line indicates the vacuum level; the red double-headed arrow gives the resulting work function Φ.

Optical Properties

Figures and present the full set of calculated optical responses for the monolayer and AA-stacked bilayer configurations of C₁₂-GR and C₁₆-GY. In all panels, results for both thicknesses are superimposed: solid lines represent the monolayer, while dashed lines correspond to the bilayer. Blue and magenta curves denote light polarization along the x- and y-directions, respectively.

6.

Calculated optical response of C₁₂-GR. (a) Real part ε₁(ω) and (b) imaginary part ε₂(ω) of the dielectric function; (c) absorption coefficient α­(ω) for both thicknesses, shown together with the AM1.5 solar spectrum irradiance (right-hand axis); (d) refractive index n(ω); (e) extinction coefficient k(ω); and (f) energy-loss function L(ω). In all panels, solid lines correspond to the monolayer and dashed lines to the AA-stacked bilayer. Blue and magenta curves represent light polarization along the x- and y-directions, respectively. The shaded background highlights the visible spectral range (1.55–3.25 eV).

7.

Calculated optical response of C₁₆-GY. (a) Real part ε₁(ω) and (b) imaginary part ε₂(ω) of the dielectric function; (c) absorption coefficient α­(ω) for both thicknesses, shown together with the AM1.5 solar spectrum irradiance (right-hand axis); (d) refractive index n(ω); (e) extinction coefficient k(ω); and (f) energy-loss function L(ω). In all panels, solid lines correspond to the monolayer and dashed lines to the AA-stacked bilayer. Blue and magenta curves represent light polarization along the x- and y-directions, respectively. The shaded background highlights the visible spectral range (1.55–3.25 eV).

Figures a and a display the real part ε₁(ω) of the dielectric function. We observe that, at low photon energies, the monolayers exhibit extremely large static dielectric constants, reflecting a pronounced electronic polarizability, an observation already noted for the classical GYs. − In the corresponding bilayers, these values drop sharply about 15 for C₁₂-GR [Figure a] and 17.81/12.45 for C₁₆-GY along the x/y directions [Figure a] because interlayer van der Waals coupling constrains the independent dipolar oscillations of the two sheets.

These large dielectric permittivities arise from the semi-metallic character of the materials, consistent with the well-established inverse relationship between the dielectric constant and the electronic band gap, expressed as ${ϵ}_1(0)\simeq 1+{(\hslash {\omega }_{\mathrm{p}}/E_{\mathrm{g}})}^2$ , where ω_p denotes the characteristic plasmon frequency. ,, This combination of metallic-like screening and low-density carbon backbones highlights their strong potential for high-energy-density capacitive storage, a behavior intrinsically linked to their pronounced electronic polarizability. Beyond the static limit, frequency intervals where ε₁ becomes negative 3.2–4.0 eV for the C₁₂-GR bilayer and 2.3–3.0 eV for C₁₆-GY (predominantly for y polarization) signal interband plasmonic resonances. − These resonances are highly promising for sub-wavelength light confinement and for the design of high-performance, spectrally selective absorbers.

Figures b and b show the imaginary part ε₂(ω) of the dielectric function for C₁₂-GR and C₁₆-GY, respectively. This quantity reflects the joint DOS and the strength of optically allowed interband transitions. For C₁₂-GR monolayer [Figure b], two dominant peaks at 2.2 and 3.2 eV, assigned to π → π* transitions, are followed by a UV resonance at 5.7 eV. AA stacking introduces a low-energy resonance at 1.1 eV and densifies the spectrum up to 5.5 eV, a direct consequence of electronic states delocalized over both sheets. Monoclinic C₁₆-GY is more anisotropic [Figure b]: x polarization shows a single maximum at 2.4 eV, whereas y polarization supports four features between 1.6 and 5.4 eV. In the bilayer, additional low-energy resonances emerge (1.1–2.3 eV in x; 1.1–5.4 eV in y), multiplying infrared absorption channels and suggesting suitability for polarization-selective filters and spectral modulators.

It is worth noting, however, that for the C₁₂-GR bilayer, the dielectric function remains relatively isotropic, with ε₂ values slightly enhanced compared to the monolayer, yet without significant alteration in spectral features. In contrast, the C₁₆-GY bilayer retains a pronounced optical anisotropy, particularly along the y direction. The intensity and multiplicity of the peaks in ε₂ remain substantial, although slightly red- or blue-shifted relative to the monolayer. These observations indicate that interlayer interactions in C₁₆-GY do not quench its intrinsic anisotropic optical response, in contrast to the behavior observed in C₁₂-GR.

Figures c and c show the absorption spectra of C₁₂-GR and C₁₆-GY, respectively. Despite their semi-metallic character, both materials absorb strongly in the visible range. C₁₂-GR monolayer [Figure c] displays an isotropic peak centered at 3.2 eV (∼6.2 × 10⁵ cm^–1) that grows to 7.0 × 10⁵ cm^–1 and broadens in the bilayer, indicating additional interband transitions. In C₁₆-GY [Figure c], the monolayer y polarization reaches 7.7 × 10⁵ cm^–1 at 2.5 eV, coinciding with the maximum of the solar spectrum and 1.15 × 10⁶ cm^–1 at 5.2 eV, whereas the x-polarized intensities remain lower. Stacking enhances all intensities, attaining a record 1.4 × 10⁶ cm^–1 and unveiling a discrete new peak at 4.9 eV, a profile well-suited to photothermal applications and broadband photodetectors.

The evolution of the refractive index further corroborates these trends, as shown for C₁₂-GR [C₁₆-GY] in Figure d [d]. Monolayer C₁₂-GR [Figure d] maintains n ∼ 3.3 in the visible, while the bilayer develops an infrared maximum accompanied by a visible attenuation, favorable to multi-band transparent components. In C₁₆-GY [Figure d], the index reaches 4.3 (near-IR, y polarization), falls to 1.6 around 2.6 eV, then rises again in the UV. Stacking lowers the static values (3.5 in x, 4.1 in y) and introduces a pronounced peak at 0.9 eV (5.1 in y). The remarkably low visible-range minima (1.1–1.2) open promising avenues for anisotropic metamaterials and nonlinear optics.

Figures e and e display the extinction coefficient k(ω) for C₁₂-GR and C₁₆-GY, further completing the optical response analysis. In C₁₂-GR [Figure e], an intense infrared peak signals semi-metallicity, followed by a plateau (k ∼ 1.4) and a maximum at 3.3 eV; in the bilayer, this plateau widens (1.2–3.3 eV) and rises to k ∼ 2.2, reflecting hybrid interlayer bands. For C₁₆-GY [Figure e], the monolayer y polarization shows an ultra-strong regime (k > 4 below 1 eV) and two visible peaks at 1.75 and 2.7 eV, whereas the x-polarized response remains more modest. Stacking broadens and blueshifts the x resonances and amplifies the y ones up to 5.4 eV, indicative of delocalized states conducive to controlled dissipation.

Finally, Figures f and f show the energy-loss function L(ω) = ε₂(ω)/|ε­(ω)|², which highlights the plasmonic potential of C₁₂-GR and C₁₆-GY. For C₁₂-GR [Figure f], L(ω) remains weakly dissipative (<0.1) up to 2.5 eV before exhibiting a resonance at 3.6 eV whose intensity doubles in the bilayer. In contrast, C₁₆-GY [Figure f] concentrates its plasmons in the visible: monolayer peaks at 2.85 eV (x) and 3.1 eV (y) are strongly enhanced in the bilayer (0.55 at 2.9 eV in x; 0.40 at 3.1 eV in y). Such directionally tuned activity within the 1.6–3.4 eV window is ideal for nanoscale surface plasmonics, selective absorption modulators, and high-sensitivity optical sensors.

In summary, C₁₂-GR couples infrared transparency to moderate UV plasmons, making it attractive for transparent electrodes, antireflection coatings, and near-UV plasmonics. C₁₆-GY combines pronounced spectral anisotropy with reinforced visible-range plasmonic activity, positioning it as a prime candidate for high-resolution polarized filters, broadband detectors, and directional metamaterials. The ability to tailor these responses through simple stacking thus provides an additional degree of freedom for engineering next-generation optoelectronic devices.

Capacitive Performance of Bilayer Structures

Building upon the promising characteristics of GY-based materials for energy storage applications, − we evaluate here the capacitive behavior of C₁₂-GR and C₁₆-GY bilayers subjected to an out-of-plane electric field, as illustrated in the inset of Figure b. The bilayer systems are modeled as atomically thin electrostatic nanocapacitors, in which the two carbon layers act as charge accumulation planes separated by the interlayer distance d _in. A uniform external electric field is applied perpendicular to the layers along the non-periodic z direction under periodic boundary conditions, while dipole correction is employed to suppress artificial electrostatic interactions between periodic replicas. To ensure proper isolation of the bilayer system, a sufficiently large vacuum region (>20 Å) is introduced along the out-of-plane direction. Under the applied electric field, opposite charges accumulate on the two layers, resulting in field-induced charge separation and electrostatic energy storage analogous to that of an ideal nanoscale parallel-plate capacitor.

8.

Capacitive performance of C₁₂-GR and C₁₆-GY bilayers under an external electric field. (a) Stored energy E _c, (b) induced charge magnitude |Q|, and (c) specific capacitance C as functions of the applied electric field $\mathrm{E⃗}$ . The inset in (b) illustrates the orientation of the external field relative to the bilayer geometry. C₁₆-GY consistently exhibits higher energy storage, charge accumulation, and capacitance compared to C₁₂-GR, highlighting its enhanced potential for nanocapacitor applications.

In periodic systems under a finite electric field, P reflects the degree of charge separation induced along the field direction and is obtained from the modern theory of polarization as implemented in DFT codes via Berry-phase formalism. , Physically, P corresponds to the net polarization per unit cell and is directly sensitive to atomic displacements and electronic rearrangements under the field. In this context, the induced surface charge per monolayer is estimated using the relation |Q| = |P|/d _in, where d _in denotes the interlayer spacing. The electrostatic energy stored in the system is computed as the difference in total energy between the field-biased and zero-field configurations, using the expression $E_{\mathrm{c}}=E_{\mathrm{t}\mathrm{o}\mathrm{t}}(\mathrm{E⃗})-E_{\mathrm{t}\mathrm{o}\mathrm{t}}(\mathrm{E⃗}=0)$ with both quantities obtained from self-consistent field (SCF) calculations performed on fully relaxed geometries. Finally, the specific capacitance per unit mass is calculated from the classical electrostatic relation C = Q ²/(2mE _c), where m = (N _C × M _C)/N _A represents the total mass of carbon atoms in the bilayer. Here, N _C, M _C, and N _A denote the number of carbon atoms in the structure, the molar mass of carbon, and Avogadro’s number, respectively. This formulation enables a direct and quantitative comparison of the capacitive performance of GY-derived bilayers under realistic applied electric field conditions.

The stored energy E _c in Figure a follows a parabolic dependence on the electric field, reflecting the expected electrostatic behavior where energy scales as Q ². In both materials, the energy storage increases symmetrically with the field magnitude, but C₁₆-GY consistently exhibits larger E _c values across the entire range. This superior storage capability is likely attributed to the enhanced polarizability of the C₁₆-GY lattice, promoted by the presence of acetylenic linkages (–CC– bonds) absent in C₁₂-GR. These chains enhance π-electron delocalization and strengthen dipole formation under an external electric field, thereby promoting more efficient field-induced charge separation.

Moreover, the semi-metallic character of both constituent monolayers, discussed previously in this work, facilitates electronic redistribution under an external electric field and promotes efficient charge accumulation on the two layers. This behavior further strengthens the electrostatic response and enhances the energy storage capability of the bilayer systems. As reported in Table , the maximum energy density reaches E _c ≈ 1.24 eV for C₁₆-GY, significantly higher than the E _c ≈ 0.83 eV obtained for C₁₂-GR.

2. Comparison of the Calculated Electrostatic Parameters and Gravimetric Capacitance of the Present Bilayer Systems with Previously Reported Carbon-Based Nanocapacitors and Supercapacitor Materials.

system	|Q| (×10–2 e)	E c (eV)	C (F/g)
Present Work
C12-GR bilayer	∼51.0	∼0.83	∼100
C16-GY bilayer	∼79.0	∼1.24	≥120
Previous DFT Studies
GR/(h-BN)n/GR	3.5–6.0	0.033–0.093	24.5–54.4
γ-GY homo/hetero-bilayers	30.0–40.0	0.5–0.7	20.0–40.0
Experimental Studies
graphdiyne nanostructures	-	-	∼71.4
nanoporous triazine-based GDY	-	-	∼250
graphene prepared from graphene oxide	-	-	141.0
nitrogen-doped graphene	-	-	225.2
amorphous mesoporous carbon cubes	-	-	205.0

Figure b further shows that the induced charge |Q| increases linearly and symmetrically with the applied field in both systems, confirming a fully reversible polarization response in the linear dielectric regime. Owing to its extended π-conjugation and larger unit cell, C₁₆-GY systematically accumulates more charge than C₁₂-GR across the entire field range, reaching peak values of |Q| ≈ 0.79 e compared to |Q| ≈ 0.51 e.

The evolution of the gravimetric capacitance, shown in Figure c, exhibits a characteristic non-monotonic behavior: C is minimal at weak fields, where polarization and stored energy remain small, and rises sharply as the field increases. Under strong field excitation, C₁₆-GY attains capacitance values exceeding 120 F/g, markedly higher than the ∼100 F/g observed for C₁₂-GR.

These values should be interpreted as effective electrostatic capacitances associated with field-induced polarization and charge accumulation within the idealized bilayer nanocapacitor configuration considered here. Accordingly, the present approach does not include electrolyte effects, ion adsorption, or explicit quantum capacitance contributions arising from the electronic DOS near the Fermi level. In realistic devices, the total capacitance would result from the series combination of electrostatic, quantum, and electrolyte–interface capacitances. − To place the present results into a broader context, Table compares the calculated electrostatic parameters and gravimetric capacitances of the investigated bilayers with representative carbon-based nanocapacitor and supercapacitor systems previously reported in the literature. A first important observation is that the capacitance values obtained for C₁₂-GR and especially C₁₆-GY are significantly larger than those predicted for several previously studied DFT nanocapacitor models, including GR/(h–BN) _n /GR structures and γ-GY homo/hetero-bilayers. In particular, the specific capacitance of C₁₆-GY exceeds the values reported for these reference systems by approximately a factor of 2 to 3, while simultaneously exhibiting substantially larger induced charge accumulation and electrostatic energy storage.

The enhanced performance of C₁₆-GY can be physically attributed to the combined effects of its acetylenic network, enlarged porous geometry and semi-metallic electronic character. The presence of extended –CC– linkages promotes stronger π-electron delocalization and facilitates field-induced charge redistribution across the bilayer. As discussed previously from the electronic band structures, the semi-metallic nature of both constituent monolayers enables efficient carrier redistribution under an external electric field without the energetic limitations typically encountered in semiconducting or insulating nanocapacitors. This behavior contributes to the strong polarization response, larger accumulated charge, and consequently enhanced electrostatic energy storage capability observed in C₁₆-GY.

Although several experimentally realized porous carbon and GDY-based electrodes exhibit even larger electrochemical capacitances, often exceeding 200 F/g, these systems generally involve electrolyte environments, ion adsorption processes, defect engineering, heteroatom doping, or pseudocapacitive mechanisms − that are not considered in the present idealized electrostatic model. In contrast, the capacitance values reported here are obtained solely from intrinsic field-induced charge separation within pristine bilayer systems. From this perspective, the calculated values remain fully competitive with a broad range of carbonaceous nanoscale energy-storage materials and highlight the intrinsic electrostatic potential of graphyne-derived bilayers.

In addition to the absolute performance metrics, the symmetry of the responses with respect to field inversion (i.e., $\mathrm{E⃗}\to -\mathrm{E⃗}$ ) indicates low hysteresis and excellent reversibility, critical features for reliable and sustainable energy storage operation. The sharp but symmetric dip in C near $\mathrm{E⃗}$ = 0 is physically meaningful and expected, as the charge displacement and thus stored energy are both minimal in the absence of an external bias.

Furthermore, the nearly symmetric evolution of |Q|, E _c, and C under positive and negative electric fields suggests a stable and reversible polarization mechanism dominated by electrostatic charge redistribution rather than irreversible structural or electronic transformations. Such behavior is particularly desirable for nanoscale capacitor architectures requiring high cycling stability and reproducible charge–discharge characteristics.

It is worth emphasizing that both bilayers are semi-metallic, as confirmed by band structure analysis discussed earlier. Their capacity to redistribute charge without energy barriers, unlike in semiconducting or insulating nanocapacitors, enables fast polarization dynamics and enhanced cyclability. Moreover, the absence of a bandgap may reduce dielectric breakdown risks under moderate electric fields.

Another notable aspect emerging from Table is that the present bilayer systems achieve these favorable capacitance values while preserving relatively simple all-carbon architectures, without requiring chemical functionalization, defect introduction, or composite formation. This intrinsic simplicity may constitute an advantage for future nanoscale device integration and theoretical modeling.

Overall, this analysis indicates that C₁₆-GY, owing to its acetylenic architecture, metallic conductivity, and strong dielectric response, exhibits enhanced energy storage, charge accumulation, and effective specific capacitance compared to C₁₂-GR. These findings suggest that both bilayer systems, particularly C₁₆-GY, may represent promising platforms for nanoscale electrostatic capacitive applications and motivate further investigations including electrolyte environments, interface effects, and explicit quantum capacitance contributions in realistic device conditions.

Conclusion

We have presented a comprehensive first-principles study of two mechanically reconstructed GY-derived carbon allotropes, C₁₂-GR and C₁₆-GY, focusing on their structural stability as well as their electronic, optical, and capacitive properties. Both materials are found to be energetically and dynamically stable in monolayer and AA-stacked bilayer forms, and they exhibit an intrinsic semi-metallic character dominated by π-derived electronic states. The optical response of these systems is characterized by strong dielectric screening, broadband absorption, and plasmonic features whose spectral positions and intensities are modulated by layer stacking. In particular, the mixed sp/sp² bonding network of C₁₆-GY induces enhanced electronic polarizability and pronounced optical resonances, distinguishing it from the purely sp²-bonded C₁₂-GR phase. When subjected to an external out-of-plane electric field, both bilayer structures display reversible polarization behavior and efficient charge redistribution, resulting in significant electrostatic energy-storage capability. The calculated specific capacitance values, reaching approximately 100 F/g for C₁₂-GR and exceeding 120 F/g for C₁₆-GY, are competitive with or superior to those reported for several representative carbon-based nanocapacitor systems. These performances are closely linked to the semi-metallic electronic structure and strong field-induced polarization of the materials.

From a synthesis perspective, C₁₂-GR and C₁₆-GY remain theoretically predicted graphyne-derived phases. Nevertheless, recent progress in the synthesis of graphdiyne and γ-GY materials suggests that increasingly complex 2D carbon networks may become experimentally accessible. In particular, C₁₂-GR, composed exclusively of sp² carbon atoms, may exhibit enhanced structural robustness, whereas C₁₆-GY combines sp and sp² hybridizations through acetylenic linkages similar to those already realized in graphdiyne-related systems. These structural similarities indicate that strain engineering, surface-assisted growth, or mechanochemical reconstruction approaches could represent plausible pathways for future experimental realization. Taken together, these findings provide a detailed structure–property relationship for mechanically reconstructed graphyne derivatives and suggest that C₁₆-GY, in particular, may represent a promising platform for nanoscale electrostatic capacitive applications. Further experimental validation and device-level investigations will be essential to assess the practical feasibility of these materials and to explore additional tuning strategies through stacking, functionalization, or heterostructure design.

The authors declare no competing financial interest.