First-principles study on a new 2D carbon allotrope with an intrinsic wide bandgap: A comparison with α-graphyne

The versatile hybridization of carbon elements, i.e., sp, sp², and sp³, not only forms the basis of life and organic materials but also leads to a number of carbon allotropes with different properties that arise from specific carbon frameworks.¹ Besides the existing carbon allotropes in the natural world, e.g., diamond, graphite, and carbon black, many novel artificial carbon allotropes, e.g., zero-dimensional (0D) fullerene,² one-dimensional (1D) carbon nanotube,³ two-dimensional (2D) graphene,⁴ and three-dimensional (3D) fcc-carbon and diamond derivatives,^5–8 have undoubtedly sparked tremendous research interest in both the scientific community and industry. Specifically for the miracle material graphene, carbon atoms are arranged in a honeycomb lattice forming an ideal one-atom-thick layer, which is able to exhibit fascinating mechanical, electrical, and thermal transport properties coupled with tremendous inherent advantages of 2D materials.^9,10 The most remarkable feature of graphene is the presence of Dirac cones in its electronic structure, where the dispersions of electron and hole meet linearly at Dirac points.¹¹ However, the resulting semimetallic behavior of graphene with zero bandgap severely restricts its further applications in electronic devices.¹² As a result, various techniques and methods have been proposed to functionalize graphene with the aim to open a bandgap while retaining major benefits of graphene, e.g., via surface decoration, oxygenation, covalent modification, porous defects, etc.^13–15 However, these approaches usually involve a narrow bandgap but with a compromise on the crystalline quality, which often degrades electrical performance. On the other hand, benefiting from versatile hybridization forms of carbon atoms, the design and preparation of other 2D carbon allotropes with unique sp²-, sp²–sp³-, or sp²–sp-hybridized carbon networks have also become a promising area in the field of nanocarbon materials, exhibiting great potentials beyond graphene.

To date, a variety of 2D new carbon allotropes with different geometrical topologies and novel carbon network-dependent properties have been reported, such as T-Graphene,¹⁶ phagraphene,¹⁷ graphene+,¹⁸ twin graphene,¹⁹ diamane,²⁰ graphyne and graphdiyne,^21,22 and so on. Among these new 2D carbon allotropes, the graphyne and its derivative graphdiyne with mixed sp²–sp hybridization have been the focus of such research, inspired by the synthesis of γ-graphyne and graphdiyne nanoscale films,^23,24 whose 2D lattice can be visualized by inserting acetylene linkages (sp hybridization) into purely sp²-hybridized graphene.^21,22 Consequently, a variety of carbon networks can be constructed by changing the ratio and arrangement of acetylenic linkages, and a large material family has been developed, including α-, α2-, β-, γ-, δ-, 6,6,12-, 8,16,4-, T_4,4,4-, Pal-, Ph-graphyne, and so on.^25–33 In addition, our previous work indicates that the vertically stacked graphyne is able to achieve novel 2D carbon allotropes resulting from covalent bonding between adjacent layers.³⁴ 

However, semiconducting behavior remains rare in these carbon-based 2D materials. Correspondingly, for the graphyne family, the Dirac-cone structures with semimetallic characteristics have been observed in α-, α2-, β-, δ-, 6,6,12-, 8,16,4-, Pal-graphyne, and Ph-graphyne sheets, while only γ- and T_4,4,4-graphyne exhibit direct semiconducting features, regardless of a narrow bandgap, i.e., 0.45²⁵ and 0.34 eV,³¹ respectively. For other reported 2D carbon allotropes, buckled T-graphene, Phagraphene, and graphene+ have Dirac fermion characters, while twin graphene and its analogs possess intrinsic semiconducting natures with relatively wider bandgaps, e.g., 0.96, 1.30, and 1.79 eV for the twin graphene,¹⁹ twin 4–8 graphene,³⁵ and twin T-graphene,³⁶ respectively. Note that a semiconductor with a wide bandgap could have more advantages, especially in nanoelectronic devices and light-emitting applications. Herein, a new 2D carbon allotrope with structural similarity to α-graphyne has been proposed theoretically, which exhibits an intrinsic direct bandgap with a wide energy gap significantly larger than that reported for other 2D carbon allotropes. The structural and electronic properties of the new 2D carbon allotrope have been systematically investigated through a comparative study with α-graphyne by using first-principles approaches. Moreover, the vertical stacking and in-plane biaxial effects on the structural and electronic properties of the new 2D carbon allotrope have also been clarified in this work, providing robust approaches for the effective modulation of electronic properties in the new 2D carbon allotrope.

In this work, all first-principles calculations were performed by using the Quantum ESPRESSO (QE) package³⁷ on the basis of density functional theory (DFT). The Perdew–Burke–Ernzerhof (PBE) formed generalized gradient approximation (GGA) was used to deal with the exchange and correlation energy.³⁸ To well account for van der Waals interactions in the considered carbon networks, the empirical correction method (DFT-D3)³⁹ was incorporated into the calculations. The projected augmented wave (PAW) pseudopotentials⁴⁰ were used to represent the interactions between ionic cores and valence electrons. For 2D simulations with periodic boundary conditions, a vacuum space of at least 25 Å was inserted in the out-of-plane (perpendicular) direction of all 2D sheet models (a vertical lattice parameter of 30 Å) to avoid interaction between neighboring images. The cutoff for the plane-wave basis in the QE package was chosen to be 52 Ry with an energy precision of 10⁻⁸ eV employed in the self-consistent iterative calculations. All the atoms in the considered carbon networks, as well as the in-plane lattice parameters, were fully relaxed until the maximum force on each atom was below 10⁻³ eV/Å. The Brillouin zone (BZ) was integrated by a 12 × 12 × 1 Monkhorst-Pack k-point mesh.⁴¹ Phonon spectra were calculated by using the Phonopy package⁴² within the finite displacement supercell approach. Accordingly, the harmonic interatomic force constants (IFCs) were extracted by constructing 4 × 4 × 1 supercells for the considered 2D sheets.

The new 2D carbon allotrope proposed in this work exhibits a lattice similar to that of α-graphyne.²⁵ Thus, both the top and side views of the two carbon allotropes are shown in Fig. 1 for comparison. The α-graphyne can be visualized by uniformly inserting sp-hybridized acetylene linkages into each carbon bond of the pristine graphene, as illustrated by the top view of the α-graphyne sheet in Fig. 1(a). Consequently, the enlarged hexagonal honeycomb lattice possesses an optimized cell parameter of 6.959 Å. It can be observed in Fig. 1(a) that the one-atom-thick layer of α-graphyne is composed of two types of carbon atoms, i.e., the sp²- and sp-hybridized carbon atoms labeled as C1 and C2, respectively. The corresponding interatomic distances, i.e., C1–C1 (d_C1–C1), C1–C2 (d_C1–C2), and C2–C2 (d_C2–C2), are also tabulated in Table I, all of which are in good agreement with the previous work.²⁵ 

The optimized geometrical structure of the new 2D carbon allotrope is displayed in Fig. 1(b), where the structural similarity to α-graphyne can be confirmed by the similar top views of the two sheets. However, the structural deviations can be easily distinguished from the side views in Fig. 1. Specifically, in contrast to the one-atomic-thick feature of the α-graphyne, the new 2D carbon allotrope proposed in this work can be visualized as a honeycomb arrangement of carbon nanoparticles consisting of five carbon atoms, labeled as C₅ and circled in Fig. 1(b). Thus, for simplicity, the new 2D carbon allotrope is named C₅-6-ring, indicating the six-membered rings composed of C₅ nanoparticles in this 2D lattice. Accordingly, three types of carbon atoms are distinguished in this 2D carbon sheet, marked as C1, C2, and C3 in Fig. 1(b), respectively. It is interesting to note that the involvement of buckled structures in this new carbon sheet, i.e., the vertically stacked C1 and C3 atoms, leads to the absence of acetylene bonds between C2 atoms compared to that in the α-graphyne sheet, implying a purely sp²-hybridized carbon network. The corresponding atomic coordinates of the new 2D carbon allotrope are also attached in the supplementary material.

The optimized cell parameter of the new carbon allotrope, i.e., the C₅-6-ring sheet in Fig. 1(b), is 6.593 Å, which is relatively smaller than that of the α-graphyne sheet (6.959 Å). In addition, the representative interatomic distances in the C₅-6-ring sheet are also listed in Table I. Compared to the α-graphyne counterparts, it can be observed that the bond length between C2 atoms in the C₅-6-ring sheet is relatively stretched, which can be attributed to the absence of acetylene linkages. Thus, the reduced cell parameter of the C₅-6-ring sheet can be attributed to the buckled structure with respect to the C1/C3 atoms.

In addition, the dynamical stability of the 2D carbon sheets can be further verified by their phonon dispersion spectra, and the results are displayed in Figs. 2(a) and 2(b) for α-graphyne and C₅-6-ring sheets, respectively. Since there are 8/10 carbon atoms existing in the unit cell of the α-graphyne/C₅-6-ring sheet, 24/30 phonon branches can be detected in the dispersion spectrum in Figs. 2(a) and 2(b). Correspondingly, three acoustic phonon branches can be easily distinguished in Fig. 2 due to their phonon frequencies approaching zero at the Γ point in the BZ. Moreover, the maximum optical phonon frequency of the α-graphyne sheet (2262 cm⁻¹) is significantly higher than that of the C₅-6-ring sheet (1859 cm⁻¹), which can be attributed to the greater mechanical strength of acetylene linkages in the α-graphyne single layer. Specifically, the phonon branch with the highest frequencies corresponds to the acetylene bond stretching mode as confirmed in Fig. S1 in the supplementary material. In contrast, since the vertically stacked C1 and C3 atoms in the C₅-6-ring sheet lead to the absence of acetylene bonds between C2 atoms as shown in Fig. 1(b), the sp² hybridization, as well as the buckled structures, eventually result in the softening of the corresponding optical phonon branches as confirmed in Fig. 2(b) and Fig. S1 in the supplementary material. More importantly, no imaginary phonon frequencies can be detected in the whole spectrum of the C₅-6-ring sheet as shown in Fig. 2(b), further demonstrating the dynamical stability of the new 2D carbon allotrope.

The electronic structures of α-graphyne and C₅-6-ring sheets, including the electronic dispersion curves and the total/projected density of state (TDOS/PDOS), are displayed in Figs. 3(a) and 3(b), respectively. Obviously, there are pronounced deviations in the electronic structures of the two sheets. Specifically, for the single-layer α-graphyne in Fig. 3(a), the Dirac-cone characteristics similar to graphene can be observed around the Fermi level located at K points in the BZ, which agrees well with the previous report.²⁵ More interestingly, the C₅-6-ring sheet turns out to be a semiconductor and possesses an intrinsic direct bandgap (E_gap) with an amazing gap size of 2.888 eV, which is obviously larger than that of the reported 2D carbon allotropes, e.g., T_4,4,4-graphyne (0.34 eV³¹), γ-graphyne (0.45 eV²⁵), twin graphene (0.96 eV¹⁹), twin 4–8 graphene (1.30 eV³⁵), and twin T-graphene (1.79 eV³⁶) within the similar GGA approach. Simultaneously, the wide direct bandgap observed in this new 2D carbon allotrope also exceeds that of GaAs⁴³ and 2D MoS₂,⁴⁴ implying great potential and advantages in electronic devices and light-emitting applications. Correspondingly, the conduction band minimum (CBM) and the valence band maximum (VBM) are located at the Γ point as confirmed in Fig. 3(b).

Furthermore, based on the DOS/PDOS results in Fig. 3(a), it can be concluded that the electronic properties of the single-layer α-graphyne are gradually dominated by the contribution of C2 atoms (acetylene linkages) approaching the Fermi level. In contrast, for the C₅-6-ring counterpart as shown in Fig. 3(b), the CBM states are mainly contributed by C1/C3 atoms, while the VBM states are dominated by C2 atoms. Thus, regardless of the similar geometrical features of the two carbon sheets, i.e., similarly enlarged honeycomb lattices as shown in Fig. 1, our results demonstrate that the absence of acetylene bonds can lead to dramatic changes in the electronic structure of graphyne.

The effect of vertical stacking on the structural and electronic properties of the new 2D carbon allotrope has also been revealed by involving AA-stacked bilayer configurations. The initial AA-stacked bilayer configurations can be constructed by a simple duplication of the single layer followed by a vertical displacement. Interestingly, in addition to the typical vdW bilayer stack based on the C₅-6-ring single layer as shown in Fig. 4(a), a new bilayer configuration can be obtained due to the special buckled structure of the single layer, as shown in Fig. 4(b), based on an initial configuration with smaller interlayer distances, e.g., 1 Å (a value smaller than the typical sp²-hybridized bond length). The most pronounced difference between the two bilayer configurations is the different interlayer distance (h), which can be easily detected from their side views as shown in Figs. 4(a) and 4(b). Compared to the vdW stacking counterpart in Fig. 4(a), the reduced interlayer distance (h) of the new bilayer configuration in Fig. 4(b) results from the vertical covalent bonding between the C3 atoms, which eventually leads to the sp³-hybridized C3 atoms, implying the formation of a mixed sp²–sp³ carbon network. Therefore, the two optimized bilayer configurations in Figs. 4(a) and 4(b) are named vdW bilayer and sp²–sp³ bilayer, respectively. All the representative interatomic distances of the two bilayers are also summarized in Table I for comparison. It can be seen that the geometrical deviations between the vdW bilayer and the single-layer case are subtle, which can be attributed to the weak vdW interlayer interaction. In contrast, for the sp²–sp³ bilayer sheet, the reduced interlayer distance (h) leads to noticeable structural changes in each sublayer, i.e., the C₅ nanoparticles are vertically stretched with an increased d_C1–C3.

Furthermore, the thermodynamical stability of the two bilayer configurations can also be demonstrated by the calculated E_form values as listed in Table I. Both the E_form values of the two bilayer sheets are close to the single-layer counterpart, indicating their thermodynamical stability. Specifically, the vdW bilayer is relatively more energetically stable than its single-layer case, which has also been claimed in some vdW stacks based on other 2D materials,^45,46 while the sp²–sp³ bilayer is relatively less stable than the vdW stack although the deviation is subtle. To illustrate the dynamic stability of the two configurations, the phonon spectrum of the sp²–sp³ bilayer is displayed in Fig. 4(c). The doubled phonon branches, including three acoustic branches and 57 optical branches, can be observed in the complicated phonon spectrum due to the presence of two layers in the stack. Similar to the single-layer counterpart in Fig. 2(b), no imaginary phonon frequencies can be detected in the whole spectrum, suggesting the dynamical stability of the bilayer lattice. However, although the vdW bilayer possesses a minimum E_form as confirmed in Table I, imaginary phonon frequencies cannot be avoided in the phonon spectrum calculation for the vdW bilayer, indicating the dynamical instability of the AA-vdW stacking.

The electronic structures of the two bilayer configurations, i.e., the vdW bilayer and the sp²–sp³ bilayer, are plotted in Figs. 5(a) and 5(b), respectively. It can be observed that both sheets retain the direct semiconducting behavior but with different electronic bandgaps. Specifically, for the vdW stack, the electronic bandgap is reduced to a value of 2.552 eV compared to the single-layer counterpart (2.888 eV). Despite the subtle structural deviations in each sublayer as confirmed in Table I, the results in Fig. 5(a) indicate that vertical stacking is able to further modulate the electronic structure, which has also been demonstrated in other 2D materials.^44,47 Moreover, besides the reduced bandgap, the changes in DOS and PDOS of the vdW bilayer are relatively small with respect to the single-layer case in Fig. 3(b), e.g., the VBM states are still dominated by C2 atoms. For the sp²–sp³ bilayer in Fig. 5(b), a dramatically reduced bandgap appears in the electronic structure with a reduced E_gap value of 0.607 eV, suggesting a practical approach for the effective modulation of E_gap in the new 2D carbon allotrope via vertical stacking. In contrast to the vdW bilayer case, the DOS and PDOS results in Fig. 5(b) show obvious differences compared to the single-layer counterpart. Correspondingly, the VBM states in the sp²–sp³ bilayer are mainly contributed by C1 and C3 atoms. Simultaneously, two valence bands dominated by C1 atoms appear above the Fermi level, leading to a significantly reduced bandgap. Moreover, the deviations between C1 and C3 contributions in the PDOS of the sp²–sp³ bilayer can also be detected, which can be attributed to the different hybridization of C3 atoms as shown in Fig. 5(b).

As shown in Fig. 6(a), the biaxial tensile/compressive strains lead to an increase in the formation energy of the C₅-6-ring sheet, further demonstrating the accuracy of the optimized cell parameter (6.593 Å) for the unstrained 2D sheet. Simultaneously, no phase transition can be detected over the whole strain interval with the maximum strain level up to 10%, implying the superior mechanical performance of the new 2D carbon allotrope. Correspondingly, the specific geometrical changes in the strained 2D carbon network can be further revealed by the strain dependence of various bond lengths, i.e., d_C1–C2, d_C2–C2, and h_C1–C3 referred to Fig. 1(b). The results are displayed in Fig. 6(b), where monotonic strain-dependent features can be detected for all bond lengths. Specifically, the in-plane bond lengths, i.e., d_C1–C2 and d_C2–C2, stretch or contract monotonically with the increased biaxial tensile or compressive strains, while the vertical buckled height, i.e., h_C1–C3, exhibits the opposite behavior. More importantly, our result indicates that the intrinsic electronic bandgap of the C₅-6-ring sheet can be further modulated by the biaxial strains as confirmed in Fig. 6(c). Accordingly, the direct bandgap can be further enlarged by the biaxial tensile strains with a maximum E_gap value of 2.994 eV obtained at the 4% strain level, while a monotonic decrease in E_gap can be observed as the compressive strain level increases. Therefore, an effective modulation of the electronic bandgap ranging from 1.030 to 2.994 eV can eventually be realized in the new 2D carbon allotrope via strain engineering.

In summary, a new 2D carbon allotrope with structural similarities to α-graphyne, named C₅-6-ring, has been theoretically proposed in this work. The structural and electronic properties of the new 2D carbon allotrope, as well as the lattice stability, have been systematically investigated by first-principles calculations through a comparative study with α-graphyne. The new 2D carbon allotrope possesses an enlarged hexagonal honeycomb lattice, which can be visualized as the six-membered rings formed by carbon nanoparticles consisting of five carbon atoms. Interestingly, despite the structural similarity between the α-graphyne and the C₅-6-ring sheets, pronounced differences can be observed in their electronic structures. In contrast to the Dirac-cone features in the electronic structure of α-graphyne, the new 2D carbon allotrope exhibits intrinsic semiconducting behavior with a promising direct bandgap of 2.888 eV, which is obviously larger than that of the reported 2D carbon allotropes. Moreover, the structural stability of the new 2D carbon allotrope can be further demonstrated by the calculated formation energy close to the α-graphyne counterpart and the phonon spectrum without any imaginary frequencies. Moreover, the vertical stacking effect on the structural and electronic properties of the new 2D carbon allotrope has also been revealed in this work by involving the AA-stacked bilayer configurations. In addition to the typical vdW bilayer stack with a relatively reduced bandgap of 2.552 eV, a new bilayer configuration with a mixed sp²–sp³-hybridized carbon network can be obtained from the structural relaxation process, named sp²–sp³ bilayer, which possesses a dramatically reduced bandgap of 0.607 eV. More importantly, the electronic structures of the new 2D carbon allotrope can also be effectively modulated via strain engineering with a tunable bandgap ranging from 1.030 to 2.994 eV. Thus, the new 2D carbon allotrope proposed in this work exhibits great advantages and potential in electronic devices and light-emitting applications beyond other reported 2D carbon allotropes.

See the supplementary material for details of atomic coordinates in the optimized geometrical structure of the new 2D carbon allotrope and atomic vibrational modes corresponding to the phonon branch with the highest phonon frequency.

This work was supported by the Scientific Research Plan Projects of Shaanxi Education Department (No. 21JK0539).

The authors have no conflict to disclose.

Wentao Li: Conceptualization (lead); Data curation (lead); Validation (equal); Writing – original draft (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.