Introduction

Due to the changing global landscape, fossil fuel depletion and expanding costs, air contamination, global warming, and increasing demand for portable systems and hybrid electric vehicles, researchers are encouraged to design more efficient energy storage devices1,2. Thus, developing electrochemical energy conversion and storage devices with high performance, low cost, and environmentally friendly for powering an increasingly diverse range of applications is one of the most critical challenges of today's dynamic society3,4,5,6.

Batteries, fuel cells, and supercapacitors are effective and practical electrochemical energy conversion and storage technologies7,8. Among these, supercapacitors are significant because of their high power density, not having memory effect, long life cycle, fast charging/discharging rate, good stability, and wide operating temperature range9,10,11.

Recently, carbon-based materials, such as activated carbons (AC)12, carbon nanotubes (CNTs)13, porous carbons (PCs)14,15, graphene, and its derivatives16,17, for example, graphyne (GY)18 and graphdiyne (GDY)19 have been investigated and utilized as electrode materials for supercapacitors. These non-poisonous materials have a high specific surface area, high electrical conductivity, a simple creation process, and proper resistance20. They have an adjustable energy gap and can maintain stability over wide temperature ranges. These features make them appropriate for use as the electrode of supercapacitors, but they suffer from low quantum capacitance with all these advantages21,22. In addition, they have a limited specific capacitance that outcomes from small quantum capacitance and, in fact, a lack of states near the Fermi level.

With some chemical modifications, Stoller et al.23 exploited graphite to graphene oxide (GO) for use as electrodes for a supercapacitor and expressed that their chemically modified graphene has good electrical conductivity, high surface area, and could be a hopeful candidate for electrochemical double layer capacitors (EDLCs)24. Also, Graphene papers have attracted much attention for electrodes of flexible supercapacitors due to their adjustable thickness, flexibility, and required electrical properties25,26. Mousavi et al. achieved significant quantum capacitances by doping and co-doping some adatoms on graphene and functionalization it for use as an electrode in asymmetric and symmetric supercapacitors4,27.

Graphyne is a two-dimensional (2D) structure proposed theoretically in 1987 by Bagman et al.28 with one atom thickness and a framework of sp and sp2 carbon atoms. In 2018, this structure was synthesized by Qiaodan et al.29. Their results confirmed the theoretical calculations by Bagman et al., such as semiconductor character and lattice parameter of 0.69 nm. The different allotropes of graphyne can be acquired by embedding acetylenic linkages into pure graphene30. As per this, α-, β-, δ, and γ-graphyne contain 100%, 66.67%, 41.67%, and 33.33%, of acetylenic groups in their structure, respectively. Acetylenic linkages make graphyne a material with fascinating structural, mechanical, thermal, optical, and particularly electrical properties. For instance, compared to graphene, the pore size of graphynes increases because of the acetylene bond, which decreases the average coordination number. Along these lines, graphyne pore size is more suitable for ion adsorption in electrolytes than that graphene.

On the other hand, the graphyne pores would make it an ideal candidate material for water desalination31. Graphyne has plentiful positively charged sites, increasing the oxygen reduction reaction and improving fuel cells' efficiency32. Among four types of graphynes, γ‐graphyne has been successfully synthesized, which has high stability, semiconductor characteristic, and widespread application in catalysis and energy‐related fields29,33. The theoretical analysis of γ-graphyne has been expanding in recent years. For example, Srinivasu et al.19 determined the electronic structure and bandgap energy of γ-graphyne by density functional theory (DFT). Liang et al.34 and Kang et al.35 investigated the influence of the single-atom defect on γ graphyne by DFT. They noticed an incredible distinction in the electronic and magnetic properties of two forms of γ-graphyne. Yang et al.36 developed a one-pot green method for fabricating nitrogen-doped γ-graphyne (NGY) through the ball-milling and calcination processes using NH4HCO3 as a nitrogen resource. Recently, Henriquede Araujo Chagas et al. used the all-atom molecular dynamics simulations to evaluate the properties of an aqueous mixture of two ionic liquids as an electrolyte for γ-graphyne supercapacitor. They showed that γ-graphyne can be an applicable electrode for use in supercapacitors37.

Furthermore, Chen et al.18 studied N-, P- and O- doped γ-graphyne as the electrode in supercapacitors. To utilize γ-graphyne in electronics and as an electrode of supercapacitors, tailoring the electronic structure of γ-graphyne by doping or creating defects is critical. In this work, we investigate the γ‐graphyne and its derivatives characteristics as electrode materials for supercapacitors by DFT method. We show how the doped atom (p-type and n-type), its position (sp or sp2), and different types of defects directly affect the properties and capacitance of structures. To this end, doped and defective structures' electronic and structural properties, quantum capacitance, and net charge are investigated. The density of states (DOS) and projected/partial density of states (PDOS) plots use to illustrate the electronic properties. The integrated specific quantum capacitance applies to the evaluation of structures' capacitance.

Computational methods

Initially, we selected a unit cell comprising 12 carbon atoms, as depicted in Fig. 1a. To investigate the impact of doping or defects on the quantum capacitance and cohesive energy, we conducted calculations using 2 × 2 supercells (Fig. 1b). The lattice parameter of pristine graphyne unit cell is optimized at 6.885 Å (or 13.769 Å for its supercell), which matches findings from other studies38,39,40. To examine the electronic properties and optimized geometrical structures of pristine and doped-graphyne sheets, we employed the density functional theory (DFT) framework with the plane wave basis sets and self-consistent field theory (PWSCF)41. The calculations were carried out in Quantum Espresso (QE) package42. The generalized gradient approximation (GGA) functional type developed by Perdew–Burke–Ernzerhof (PBE)43,44 approximates the electronic exchange–correlation energy. We employed the projector augmented wave (PAW)45 method to describe the interaction between core and valence electrons. Kohn–sham wave functions have been restricted with an energy cutoff of 100 Rydberg. The Brillouin zone was integrated using a 6 × 6 × 1 Monkhorst– Pack (M–P)46 of k-point mesh for geometry optimization and a 12 × 12 × 1 M–P of k-point mesh to obtain the density of state (DOS). The cohesive energy (\({E}_{coh}\)) was calculated using Eq. (1) to compare the studied structures' relative stability2,38. The cohesive energy refers to the required energy to separate the constituent atoms and bring them to an assembly of neutral-free atoms. The more stable structure demonstrates the more negative value of cohesive energy.

$${E}_{coh}=\frac{\left({E}_{complex}-(m{E}_{x}+n{E}_{y}+\dots )\right)}{\left(m+n+\dots \right)}$$
(1)

where \({E}_{coh}\) refer to cohesive energy, \({E}_{complex}\) is the energy per unit cell of each structure, \({E}_{x}\) and \({E}_{y}\) are the energy of single constituent atoms (the carbon and dopant atoms), and n and m correspond to the number of atoms per unit cell.

Figure 1
figure 1

(a) unit cell of pristine graphyne, (b) 2 × 2 supercell of pristine graphyne.

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Quantum capacitance is significant for non-metal structures, such as some carbon-based electrodes. This parameter refers to the capacitance of an electrode with quantized electronic levels. In principle, quantum capacitance indicates the simplicity of filling quantized levels with charge carriers (i.e., electrons or holes). For two-dimensional crystals, it is defined as follows5,47

$${C}_{Q }=\frac{d\sigma }{{ d\varphi }_{{\varvec{g}}}}$$
(2)

where \(d\sigma\) is the change of charge density and \({d\varphi }_{{\varvec{g}}}\) is the changes of graphyne surface potential. Assuming that the Fermi surface is displaced by applying potential and DOS is not affected by the electrode charge, and electrochemical potential \((\mu )\) is rigidly shifted by \({e\varphi }_{{\varvec{g}}}\), the excess charge density \((\sigma )\) can be obtained by48,49

$$\sigma =e\underset{-\infty }{\overset{+\infty }{\int }}D\left(E\right) \left(f\left(E\right)-f(E-\mu )\right) dE=e{\int }_{-\infty }^{+\infty }D\left(E\right) \left(f\left(E\right)-f(E-{ e\varphi }_{{\varvec{g}}})\right) dE$$
(3)

where \(D\left(E\right)\) is the density of states, \(f\left(E\right)\) is the Fermi–Dirac distribution function, \(E\) is the energy concerning EF, \(e\) is the elementary charge. Therefore the CQ is given by10,27,50:

$${{C}_{Q}}^{diff}=\frac{d\sigma }{{ d\varphi }_{{\varvec{g}}}}={e}^{2} {\int }_{-\infty }^{+\infty }D\left(E\right){F}_{T}\left(E-{ e\varphi }_{{\varvec{g}}}\right)dE$$
(4)

where the thermal broadening function \({F}_{T}\left(E\right)\) obtains from the derivative of the Fermi–Dirac distribution function relative to energy and expressed as27

$${F}_{T}\left(E\right)=-\frac{df}{dE}={\left(4KT\right)}^{-1} {\mathit{sec}h}^{2}\left(\frac{E}{2KT}\right)$$
(5)

In this equation, K is the Boltzmann constant, and T is the temperature, considered 300 K. In practical applications, studying the total stored energy is essential for assessing a supercapacitor's efficiency. This quantity is not based on differential capacitance but an integrated and complete charge–discharge cycle. In this way, to anticipate capacitance, we used the integrated quantum capacitance, which can be defined as a function of potential as follows51,52

$${C}_{Q}^{int}=\frac{Q}{V}=\frac{1}{Ve}{\int }_{0}^{V} {{C}_{Q}}^{diff}\left(V{\prime}\right)d{V}{\prime}$$
(6)

Results and discussion

Structural and electronic properties of pristine and doped γ-graphyne

Figure 2 shows the optimized structure of pure γ-graphyne. After complete relaxation, it is shown that the system maintains its planar form. As claimed, the γ-graphyne network includes two non-equivalent sorts of carbon atoms. The sp2-hybridized atoms form the hexagonal rings, and the sp-hybridized atoms form the acetylenic chains and create the pseudo-triangular pore. Three types of bonds can be considered for this structure: (1) triple acetylenic bonds, (2) aromatic double bonds, and (3) the connections between the benzene ring and the acetylene groups (a single bond).

Figure 2
figure 2

The optimized geometrical structure of pristine γ-graphyne. In the left image, S1 and S2 represent the sp and sp2 atoms, respectively. In the right image, the optimized bond lengths and angles are shown.

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The results demonstrated that the triple, aromatic, and single bond lengths are 1.221 Å, 1.424 Å, and 1.4070 Å, respectively, which concur with the previous computations53,54. Furthermore, generally, the length of a single σ bond, double (σ + π), and triple (σ + 2π) bonds are ∼1.21 Å, ∼1.38 Å, and ∼1.47 Å, respectively. Figure 3 indicates the projected density of states (PDOS) and partial density of states (PDOS) plots of pristine γ-graphyne. As shown, the density of states at the Fermi level is slight, and states near the Fermi level are mainly due to carbon atoms' p orbitals. The contribution of s orbitals is negligible compared to p orbitals and is not shown here.

Figure 3
figure 3

Calculated (a) projected density of states and (b) partial density of states of pristine γ-graphyne. The Fermi energy has been rescaled at zero.

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Furthermore, the results reveal that the contribution of sp2 carbon atoms in the conduction band is slightly more significant than sp atoms (see Fig. 3a). Therefore, the electronic character of the structure is more affected by sp2 carbons. All the more explicitly, the states near the Fermi level are mainly contributed by the carbon's pz orbitals. In contrast, the px and py orbitals begin adding states in the valence and conduction bands far from the Fermi energy.

The lack of DOS at the Fermi level demonstrates semiconductivity of γ-graphyne. Numerous theoretical and experimental studies show that doping with atoms can change carbon nanomaterials' properties. Depending on the coexistence of two types of C atoms in graphyne structure, we can consider two distinct situations for doping a heteroatom, signified by S1(sp) and S2 (sp2), which appear in Fig. 2. We have investigated the change of electronic properties and quantum capacitance induced by doping with Al, B, Si, N, and P in these two circumstances, as shown in Fig. 4.

Figure 4
figure 4

Top view of optimized structures corresponding to substitutional doping with B, Al, Si, N, and P in S1 (sp site) and S2 (sp2 site) of pristine graphyne (The black, pink, gray, green, blue, and orange atoms represent C, B, Al, Si, N, and P atoms, respectively).

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Among these five atoms, B and Al are p-type dopants. The p-type dopant has one electron less than carbon, leading to the electron-free cavity in the valance band of the closed-shell structure. Since this impurity is willing to accept an electron, it is known as an acceptor. On the other hand, N and P atoms are n-type dopants. The four outer electrons link with the carbon atom, while the fifth electron is in free movement and acts as a charge carrier. This free electron requires considerably less energy to move from the valence band to the conduction band. These impurities that enhance the carrier density by contributing extra electrons to the conduction band are called donors. The silicon atom is in the same group as the carbon atom and only has a larger atomic radius relative to carbon (Si 1.11 Å and C 0.77 Å). The bond lengths, cohesive energies, and electronic properties of S1 and S2- doped graphyne (after the optimization) are listed in Tables 1 and 2, respectively.

Table 1 Electronic character, optimized bond lengths, and cohesive energy of pristine and S1-doped graphyne.
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Table 2 Electronic character, optimized bond lengths, and cohesive energy of pristine and S2-doped graphyne.
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The results revealed that the structure retained its planar shape after B-doping in both S1 and S2 cases. Nevertheless, the bond lengths increased at the doping position compared to the pristine graphyne. It appears that the C-B bond length has expanded about 0.119 Å and 0.075 Å compared to the C–C, respectively, for sp and sp2. For the S2 one, the angle between the atoms in the benzene ring at the doped site decreases from 120° to 117.590° and between the adjacent atoms to 119.171°. Subsequently, the entry of boron dopant extends the lattice and lessens in-plane stiffness. As a result, the cohesive energy for the B-doped graphyne (BGY) in the S2 site is 0.021 eV, more negative than the other in the S1, which indicates a little more stability of S2 than S1.

Because of its larger atomic radius, aluminum influences the structure geometry and causes bond length increments at the doped site. When Al substitutes the C atom in γ-graphyne, it brings local strain around the doping site and folds the Al-doped γ-graphyne (AlGY) plane. As a result, the bond length at the S1 site increases from 1.221 Å in pristine graphyne (\(\mathrm{C}\equiv \mathrm{C}\)) to 1.845 Å in Al-doped structure (\(\mathrm{Al}\equiv \mathrm{C}\)), and the bond of the aluminum-benzene ring increases from 1.407 Å to 1.920 Å. In S2, bond lengths into the aromatic rings increase from 1.424 Å in graphyne to 1.868 Å in the Al-doped structure, but no protuberance was seen from the surface. In the quasi-triangular cavity, the triple bond length is slightly increased (up to 0.01 Å), which causes slight changes in the geometric structure of this area. According to the results, for the case of aluminum-doped graphyne in the S1 site, we observed a 51.17% increase in Al≡C bond length compared to the same position in pure graphyne, which makes the Al atom be over the surface. For the S2, a 31.03% increment was observed in the Al=C bond length. To compare the stability of structures, the cohesive energy was obtained for S1-AlGY, − 8.383 eV, and S2-AlGY, − 8.393 eV. Thus, the stability of the S1-AlGY is slightly less than that of the S2-AlGY, and both are less than that of pristine graphyne. When the C atom supplants with Si, the graphyne's geometrical structure is transformed, especially in the S1 case because of the larger radius of Si. Replacing an sp carbon atom (S1) with a Si atom changes the triple Si≡C bond length from 1.221 Å to 1.661 Å and changes the angle of acetylenic atoms from 180° to 126.876°. This change in bond length and angle has been attributed to the hybridization change of the Si atom from sp to sp355.

Entering this dopant atom reduces the length of the C–C bond attached to the sp carbons (1.375 Å) and increases the bond of Si-sp2 carbon (1.802 Å). In the S2 case, the length of the carbon-silicon bond (compared to 1.424 Å, the length of the aromatic carbon–carbon bond in pristine graphyne) changes to 1.756 Å. Likewise, the angle inside the ring decreases from 120° to 113.989° in the doping position. Cohesive energy for S1-SiGY (silicon-doped graphyne), − 8.485 eV, and S2-SiGY, − 8.449 eV, was obtained, reflecting the same stability of S1 and S2 states.

Nitrogen has the closest atomic radius to carbon and is suitable for doping in carbonic structures. By N-doping, the change of bond lengths and angles, particularly in the S2 site, is negligible, and a slight change is observed in the geometric structure. When the nitrogen atom is doped in the S1 site, the length of the nitrogen-carbon bond at the triple bond changes from 1.221 Å to 1.358 Å, and the nitrogen bond with the benzene ring decreases to 1.183 Å. At the S2 site, nitrogen entering causes the bond length to be slightly reduced from 1.424 Å in pristine graphyne to 1.417 Å in a doped structure. The cohesive energy of N-doped graphyne (NGY) was acquired at − 8.55 eV for the S1 site and − 8.53 eV for the S2 site. It indicates that substitution of nitrogen atom instead of an sp carbon atom gave the structure a little more stability than replacement with sp2 carbon. The presence of nitrogen impurities causes the carbon lattice to shrink and increment the in-plane stiffness.

The phosphorus atom has a larger atomic radius than carbon and nitrogen atoms. So, graphyne geometry are affected by doping with P atom. The cohesive energy for P-doped graphyne (PGY) at the S1 site was obtained, − 8.459 eV, and at the S2, − 8.347 eV, so S1-PGY is more stable than S2-PGY. S2 doped state preserves its planar geometry, but doping in the S1 position destroys the planar structure. Replacement of the phosphorus atom instead of carbon in the benzene ring increases the length of the P–C bond to 1.711 Å and decreases the angle between the phosphorus and the two side carbons from 120° to 114.928°. At the S1 site, the bond length of the phosphorus and adjacent sp carbon atom changes from 1.221 Å to 1.618 Å and the connection between phosphorus and benzene ring changes from 1.407 Å to 1.357 Å. In this case, the phosphorus atom has the highest height among all atoms in the cell and is located at 1.376 Å from the structure's surface.

Figure 5 shows the PDOS plots of the mentioned structures. As shown, all the structures got metallic except the silicon-containing system. In B-doped graphyne, the sp carbon atom adjacent to the boron atom in the S1 site, and the boron atom itself plays a significant role in producing new states. Furthermore, the contribution of the adjacent sp2 carbon atom at the illustrated interval is lower, and the effect of farther atoms is negligible. In the S2 case, the boron atom and the neighboring sp2 carbon contribute approximately equally to forming the states near the Fermi level. The PDOS plot of aluminum-doped graphynes shows that the doping with an aluminum atom in the S1 site makes a sharp peak at about 0.3 eV. Herein, unlike B-doping, the Al atom leads to creating the new states in the Fermi level for both adjacent sp and sp2 carbon atoms. In the S2-AlGY structure, the contribution of one adjacent sp2 carbon atom in the specified energy range is also higher than the Al atom, and Al is just slightly more effective at about 1.154 eV. For Si-doped graphyne, we found that the silicon atom creates electronic states similar to the adjacent sp carbon when placed in the S1 position. In contrast, the Si atom has a more influential performance than other atoms in the S2 site.

Figure 5
figure 5

Projected density of states of doped graphyne. S1 and S2 panels are sp and sp2 atoms doping positions, respectively. The Fermi energy has been rescaled at zero. In the state of Si-doped in the S2 position, the total PDOS is in the out-of-defined axes range. So, it is not seen in the plot.

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Nevertheless, it can be seen that Si does not change the electronic character of the structure because of a closed-shell electron configuration. For P-doped graphyne, in the S1 position, the sharp peak of states comes from the sp carbon atoms adjacent to the phosphorus, and adjacent sp2 carbon atoms have a much lower contribution in forming new states. In the S2 position, the phosphorus atom creates more new states at and around the Fermi surface than neighboring carbon atoms.

Quantum capacitance of pristine and doped γ-graphyne

Given what has been said, since quantum capacitance is straightforwardly related to the electronic density of states (Eq. 3), we anticipate that the amount of \({C}_{Q}\) should increment after doping. So, we have determined the specific integrated quantum capacitances (signified by \({C}_{Q}\)) of pristine and doped γ-graphyne appear for S1-doped graphyne (Fig. 6) and S2-doped graphyne (Fig. 7). According to the integrated quantum capacitance diagram in these figures, except Si, all doped atoms improved \({C}_{Q}\) around zero potential.

Figure 6
figure 6

Integrated specific quantum capacitance of undoped and S1-doped graphyne. S1 is the position of the sp atom.

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Figure 7
figure 7

Integrated specific quantum capacitance of undoped and S2-doped graphyne. S2 is the position of the sp2 atom.

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The results reveal that boron atoms have improved the capacitance in both S1 and S2 positions. The capacitance difference is slight at the beginning and end of the approximate stability potential of water. Nevertheless, a better quantum capacitance has been obtained by substituting the boron atom in the S2 site relative to the S1 site The maximum quantum capacitance observed for the S1 case is 1105.82 F/gr at 0.052 V, and the S2 is 1508.65 F/gr at 0.177 V. Since the maximum quantum capacitances are seen for both cases at positive bias, BGY can be a good choice for a positive electrode in asymmetric supercapacitors. Doping with aluminum improves the quantum capacitance at zero potential significantly. The maximum S1-AlGY quantum capacitance is 2251.10 F/gr at 0.022 V, and for S2-AlGY is 1401/64 F/gr at 0.005 V, which is positive and very close to zero for both structures. As indicated in Figs. 6 and 7, the Al-doped structure has a more symmetrical diagram, which can be said that this structure can function well as a symmetric supercapacitor electrode. SiGY has \({C}_{Q }\approx\) 0 at negative bias near the zero potential, similar to graphyne. Although the substitution of the Si atom has not improved quantum capacitance around zero potential and the maximum \({C}_{Q}\) for both cases (S1 and S2) is less than pristine graphyne. A slight improvement in the negative potentials can be seen, which is less than required.

The maximum \({C}_{Q}\) of NGY, in the approximate range of water stability potential, was obtained at 1146.48 F/gr at − 0.27 V for the S1 position and 1372.84 F/gr at − 0.20 V for the S2 position. In addition to that, the \({C}_{Q}\) of NGY in the two different sites at different potentials were nearly identical, and both performed better at negative potentials. NGY is proposed as a negative electrode in asymmetric supercapacitors based on these results.

Figure 6 shows that S1-PGY has a symmetrical quantum capacitance with a maximum value of 3454.16 F/gr at 0.02 V. However, this high capacitance value drops sharply in positive and negative potentials, and this structure only in a small range of potential shows high quantum capacitance. As a result, the \({C}_{Q}\) at the beginning and end of the water stability potential range is significantly lower. S2-PGY also has a maximum quantum capacitance of 1192.77 F/gr at − 0.29 V.

Figures 8 and 9 represent the calculated net charge on doped graphyne systems as supercapacitor electrodes. According to Fig. 8, all doped atoms except Si improved the quantum capacitance near zero potential. Among the S1 cases, nitrogen-doped as the cathode, and boron and aluminum-doped can be used as anode in asymmetric supercapacitors. Phosphorus-doped graphyne in the S1 site with symmetrical behavior is suitable for symmetric supercapacitors electrode. According to Fig. 9, in the S2 position, nitrogen and phosphorus-doped graphyne perform better at negative potentials and are recommended as cathodes in asymmetric supercapacitors. In contrast, S2-BGY performs better at positive bias and is more suited as an anode. In symmetric supercapacitors, S2-AlGY can also be used as an electrode.

Figure 8
figure 8

Calculated net charge on undoped and S1- doped graphyne.

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Figure 9
figure 9

Calculated net charge on undoped and S2-doped graphyne.

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Structural and electronic properties of defective γ-graphynes

The creation of defects is inescapable during material production. These defects significantly affect structures' electronic, optical, thermal, and mechanical properties (graphyne here). Therefore, we have considered removing one and two carbon atoms from both S1 and S2 sites to create defective graphyne. Our findings demonstrated that an in-plane rearrangement occurs after optimizing the defective graphyne by removing a sp or sp2 carbon atom (see Fig. 10). As shown, by removing one sp atom, the neighboring carbons stretch in the defect position and form a new bond with the sp2 carbon atom of the ring. Also, when a sp2 carbon atom is removed, its adjacent atoms move to the vacant position and form a new covalent bond. Therefore, the acetylenic bond, with a bond length of 1.222 Å, stretches and increases to 1.427 Å in the defective structure near the defect site after optimization steps. The aromatic bonds of the two rings adjacent to the defect region also changed from 1.424 Å to 1.484 Å. However, there were no bumps or indentations on the structure's surface, and the system stayed planar. The defect in the S2 position causes identical changes in bond lengths and angles similar to the optimized structure of the defective graphyne in the S1 site. Both systems had the same cohesive energy and stability, equaling − 8.478 eV, indicating that none is superior. However, after creating defects and optimization, the symmetry of graphyne has altered, diminishing its stability compared to the pure structure Fig. 9 shows the V1-sp and V1-sp2 structures before and after the optimization processes. The point defect is represented with V1.

Figure 10
figure 10

Top view of structures corresponds to V1-sp and V1-sp2 before and after optimization processes.

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Following these two cases, the defect was examined by removing two carbon atoms (see Fig. 11, S1), which are indicated in the plots with V2. For this case, no in-plane rearrangement was seen after optimizing for the S1-defective structure. The bond length in the aromatic ring in the two benzene rings closest to the defect area was altered (from 1.424 Å to 1.385 Å). The changes in bond lengths of triple bonds around the defect position were negligible. The calculated cohesive energy of the optimized planar structure was − 8.457 eV, demonstrating that it is less stable than pristine graphyne.

Figure 11
figure 11

The top view of structures corresponds to V2-sp (removing two sp carbon atoms) and V2-sp2 (two sp2 carbon atoms) before and after optimization processes.

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In the other case, in the absence of two carbon atoms at the S2 position, the sp carbon atoms adjacent to the defect site approach the near sp2 atoms and form new bonds (see Fig. 11, S2). As indicated by the outcomes, the bond lengths of the majority of atoms change after optimization. Despite these changes, the structure retains its planar geometry. The cohesive energy of this structure was achieved at − 8.464 eV, which shows that this type of defect is somewhat more stable than the S1 one. The new bonds have a 1.454 Å bond length, were created after optimization, and the angle between sp2 atoms in opened benzene 142.121° was obtained. Acetylenic bonds near the defect site, which were 1.221 Å before making the defect, also changed to 1.248 Å. The angle between them also decreased from 180° to 157.104°. The optimized structure for both V2 defects is shown in Fig. 10.

For V1 defects, as represented in Fig. 12, removing each of the sp or sp2 atoms from the graphyne structure can create new states at the Fermi level. Since the optimized structure for both single-atom vacancy graphynes is the same, the density of states diagram also shows the same electron properties for both structures, so we anticipate that they should have a similar quantum capacitance (V1-sp and V1-sp2 in Fig. 12). Due to removing two sp carbon atoms (V2-sp in Fig. 12), two significant peaks appear, one at the Fermi level and the other around − 0.5 eV. Nevertheless, no peaks are observed by separating the two sp2 carbon atoms from the graphyne structure at the Fermi level (V2-sp2 in Fig. 12). We expect a quantum capacitance diagram similar to the pure graphyne.

Figure 12
figure 12

The density of states of V2-sp (removing two sp carbon atoms) and V2-sp2 (two sp2 carbon atoms) defective graphynes. The Fermi energy has been rescaled at zero.

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Quantum capacitance of defective γ-graphynes

According to Fig. 13, as we guessed from the DOS diagram, removing one carbon atom (point defect) from the structure (sp or sp2) gives almost the same quantum capacitance result regardless of the initial defect location. In this case, the capacitance behavior is relatively favorable, and in the stability range of water, the structure has maintained its capacitance. The maximum quantum capacitance for the V1-spGY was equal to 1037.23 F/gr at − 0.04 V, and the sp2 one was equal to 1030.44 F/gr at − 0.44 V. This difference is probably related to the calculation errors. This difference is probably related to the calculation errors. Removing two sp carbon atoms improved the quantum capacitance at zero and positive potentials. The maximum quantum capacitance for this structure is 4221.69 F/gr, which is observed at − 0.03 V. Removing two sp2 carbon atoms leads to a defect that does not exhibit favorable quantum capacitance and thus may not be appropriate for use in water-electrolyte supercapacitors. So, as per Fig. 13, the V2-sp structure can be a suitable material as an anode electrode in asymmetric supercapacitors. While The structures with a single atom defect (V1) in both cases are suggested for the symmetric supercapacitor electrode.

Figure 13
figure 13

Quantum capacitance of different types of defective graphynes.

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Conclusion

We have investigated the doping and defect effects on γ-graphyne by density functional theory calculations. The results reflect that substituting all p-type and n-type dopants (B, Al, N, and P) instead of one of sp or sp2 carbon atoms in γ-graphyne positively affects the stability, electronic properties, and quantum capacitance. In this manner, such changes can obtain various quantum capacitance and properties from graphyne. All investigated dopants except Si improved quantum capacitance near zero potential. S1- and S2-NGY are recommended as the cathode, and S1- and S2- BGY as the anode among studied structures. Nevertheless, S2-PGY shows a symmetrical behavior. So, it is suitable for symmetric supercapacitors. In contrast, S1-PGY is recommended as the cathode in asymmetric supercapacitors. In contrast, aluminum in the S1 position makes the structure more suitable for the anode, but S2-AlGY can perform as an electrode in symmetric supercapacitors. Despite less stability related to other defective cases, evaluations of defective cases have shown that the V2-sp structure seems desirable regarding electronic properties and quantum capacitance. However, the structure of V2-sp2 does not have favorable electronic properties as a supercapacitor electrode. Creating a point defect in the structure of graphyne can likewise give good outcomes for its utilization as supercapacitor electrodes. Finally, the maximum quantum capacitances obtained for the doped and the defected structures are 2251.10 F/gr and 4221.69 F/gr (related to AlGY doped in the sp position and structure with removing two sp carbon atoms, respectively), which are the values more than most of the structures proposed so far4,10,18,56,57,58.