Evaluating the appropriateness of γ-graphyne derivatives as electrode materials for supercapacitors

DFT calculations were used to study the quantum capacitance of pure, B/Al/Si/N/P-doped, and defective γ-graphyne. Due to the direct relationship between capacitance and electronic states around the Fermi level, structures' electronic properties were evaluated by DOS plots. The results of integrated specific quantum capacitance in the range of water stability potential show an improvement of capacity in each p and n-type doping. The calculated cohesive energies of doped structures reflect the stability enhancement. Also, the stability/capacitance of single and double vacancies in two distinct positions (sp and sp2) were examined. The results illustrate stability retention and quantum capacitance improvement of these defective structures. Among the doped structures, the maximum quantum capacitance is 2251.10 F/gr belonging to the aluminum doped structure (in the sp position). For the defective structures, the maximum quantum capacitance is 4221.69 F/gr belonging to removing two sp carbon atoms. These quantum capacitances significantly improved compared to the pristine structure (1216.87 F/gr) and many other structures. These stunning results can contribute to the design of appropriate structures as electrode materials for high-efficiency supercapacitors.


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 studies [38][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) package 42 .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 stability 2,38  www.nature.com/scientificreports/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.
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.
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 follows 5,47 where dσ is the change of charge density and dϕ 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 (µ) is rigidly shifted by eϕ g , the excess charge density (σ ) can be obtained by 48,49 where D(E) is the density of states, f (E) is the Fermi-Dirac distribution function, E is the energy concerning E F , e is the elementary charge.Therefore the C Q is given by 10,27,50 : where the thermal broadening function F T (E) obtains from the derivative of the Fermi-Dirac distribution func- tion relative to energy and expressed as 27 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 follows 51,52

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 sp 2 -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 (1) The optimized geometrical structure of pristine γ-graphyne.In the left image, S 1 and S 2 represent the sp and sp 2 atoms, respectively.In the right image, the optimized bond lengths and angles are shown.
bonds, (2) aromatic double bonds, and (3) the connections between the benzene ring and the acetylene groups (a single bond).
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 computations 53,54 .Furthermore, generally, the length of a single σ bond, double (σ + π), and triple (σ + 2π) bonds are ∼1.21Å, ∼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.
Furthermore, the results reveal that the contribution of sp 2 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 sp 2 carbons.All the more explicitly, the states near the Fermi level are mainly contributed by the carbon's p z orbitals.In contrast, the p x and p y 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 S 1 (sp) and S 2 (sp 2 ), 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.
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 S 1 and S 2 -doped graphyne (after the optimization) are listed in Tables 1 and 2, respectively.
The results revealed that the structure retained its planar shape after B-doping in both S 1 and S 2 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 sp 2 .For the S 2 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 S 2 site is 0.021 eV, more negative than the other in the S 1, which indicates a little more stability of S 2 than S 1 .
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 S 1 site increases from 1.221 Å in pristine graphyne ( C ≡ C ) to 1.845 Å in Al-doped structure ( Al ≡ C ), and the bond of the aluminum-benzene ring increases from 1.407 Å to 1.920 Å.In S 2 , 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 S 1 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 S 2, a 31.03%increment was observed in the Al=C bond length.To compare the stability of structures, the cohesive energy was obtained for S 1 -AlGY, − 8.383 eV, and  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 S 2 site, is negligible, and a slight change is observed in the geometric structure.When the nitrogen atom is doped in the S 1 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 S 2 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 S 1 site and − 8.53 eV for the S 2 site.It indicates that substitution of nitrogen atom instead of an sp carbon atom gave the structure a little more stability than replacement with sp 2 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 S 1 site was obtained, − 8.459 eV, and at the S 2 , − 8.347 eV, so S 1 -PGY is more stable than S 2 -PGY.S 2 doped state preserves its planar geometry, but doping in the S 1 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 S 1 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 S 1 site, and the boron atom itself plays a significant role in producing new states.Furthermore, the contribution of the adjacent sp 2 carbon atom at the illustrated interval is lower, and the effect of farther atoms is negligible.In the S 2 case, the boron atom and the neighboring sp 2 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 S 1 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 sp 2 carbon atoms.In the S 2 -AlGY structure, the contribution of one adjacent sp 2 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 S 1 position.In contrast, the Si atom has a more influential performance than other atoms in the S 2 site.
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 S 1 position, the sharp peak of states comes from the sp carbon atoms adjacent to the phosphorus, and adjacent sp 2 carbon atoms have a much lower contribution in forming new states.In the S 2 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 (sig- nified by C Q ) of pristine and doped γ-graphyne appear for S 1 -doped graphyne (Fig. 6) and S 2 -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.
The results reveal that boron atoms have improved the capacitance in both S 1 and S 2 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 S 2 site relative to the S 1 site The maximum quantum capacitance observed for the S 1 case is 1105.82F/gr at 0.052 V, and the S 2 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 S 1 -AlGY quantum capacitance is 2251.10F/gr at 0.022 V, and for S 2 -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 ≈ 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 (S 1 and S 2 ) 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 S 1 position and 1372.84F/gr at − 0.20 V for the S 2 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 S 1 -PGY has a symmetrical quantum capacitance with a maximum value of 3454.16F/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.S 2 -PGY also has a maximum quantum capacitance of 1192.77F/gr at − 0.29 V.
Vol:.( 1234567890 8, all doped atoms except Si improved the quantum capacitance near zero potential.Among the S 1 cases, nitrogen-doped as the cathode, and boron and aluminum-doped can be used as anode in asymmetric supercapacitors.Phosphorus-doped graphyne in the S 1 site with symmetrical behavior is suitable for symmetric supercapacitors electrode.According to Fig. 9, in the S 2 position, nitrogen and phosphorus-doped graphyne   perform better at negative potentials and are recommended as cathodes in asymmetric supercapacitors.In contrast, S 2 -BGY performs better at positive bias and is more suited as an anode.In symmetric supercapacitors, S 2 -AlGY can also be used as an electrode. 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 S 1 and S 2 sites to create defective graphyne.Our findings demonstrated that an in-plane rearrangement occurs after optimizing the defective graphyne by removing a sp or sp 2 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 sp 2 carbon atom of the ring.Also, when a sp 2 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 S 2 position causes identical changes in bond lengths and angles similar to the optimized structure of the defective graphyne in the S 1 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 V 1 -sp and V 1 -sp 2 structures before and after the optimization processes.The point defect is represented with V 1 .
Following these two cases, the defect was examined by removing two carbon atoms (see Fig. 11, S 1 ), which are indicated in the plots with V 2 .For this case, no in-plane rearrangement was seen after optimizing for the S 1 -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.In the other case, in the absence of two carbon atoms at the S 2 position, the sp carbon atoms adjacent to the defect site approach the near sp 2 atoms and form new bonds (see Fig. 11, S 2 ).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 S 1 one.The new bonds have a 1.454 Å bond length, were created after optimization, and the angle between sp 2 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 V 2 defects is shown in Fig. 10.
For V 1 defects, as represented in Fig. 12, removing each of the sp or sp 2 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 (V 1 -sp and V 1 -sp 2 in Fig. 12).Due to removing two sp carbon atoms (V 2 -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 sp 2 carbon atoms from the graphyne structure at the Fermi level (V 2 -sp 2 in Fig. 12).We expect a quantum capacitance diagram similar to the pure graphyne.
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 sp 2 ) 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 V 1 -spGY was equal to 1037.23 F/gr at − 0.04 V, and the sp 2 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  Removing two sp 2 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 V 2 -sp structure can be a suitable material as an anode electrode in asymmetric supercapacitors.While The structures with a single atom defect (V 1 ) in both cases are suggested for the symmetric supercapacitor electrode.

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 sp 2 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.S 1 -and S 2 -NGY are recommended as the cathode, and S 1 -and S 2 -BGY as the anode among studied structures.Nevertheless, S 2 -PGY shows a symmetrical behavior.So, it is suitable for symmetric supercapacitors.In contrast, S 1 -PGY is recommended as the cathode in asymmetric supercapacitors.In contrast, aluminum in the S 1 position makes the structure more suitable for the anode, but S 2 -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 V 2 -sp structure seems desirable regarding electronic properties and quantum capacitance.However, the structure of V 2 -sp 2 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 2251.10F/gr and 4221.69F/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 far 4,10,18,56-58 .

Figure 3 .Figure 4 .
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.

)Figure 5 .
Figure 5. Projected density of states of doped graphyne.S 1 and S 2 panels are sp and sp 2 atoms doping positions, respectively.The Fermi energy has been rescaled at zero.In the state of Si-doped in the S 2 position, the total PDOS is in the out-of-defined axes range.So, it is not seen in the plot.

Figure 6 .
Figure 6.Integrated specific quantum capacitance of undoped and S 1 -doped graphyne.S 1 is the position of the sp atom.

Figure 7 .
Figure 7. Integrated specific quantum capacitance of undoped and S 2 -doped graphyne.S 2 is the position of the sp 2 atom.

Figure 10 .
Figure10.Top view of structures corresponds to V 1 -sp and V 1 -sp 2 before and after optimization processes.

Figure 11 .
Figure 11.The top view of structures corresponds to V 2 -sp (removing two sp carbon atoms) and V 2 -sp 2 (two sp 2 carbon atoms) before and after optimization processes.

Figure 12 .
Figure 12.The density of states of V 2 -sp (removing two sp carbon atoms) and V 2 -sp 2 (two sp 2 carbon atoms) defective graphynes.The Fermi energy has been rescaled at zero.

Table 1 .
Electronic character, optimized bond lengths, and cohesive energy of pristine and S 1 -doped graphyne.

Table 2 .
Electronic character, optimized bond lengths, and cohesive energy of pristine and S 2 -doped graphyne.theangle inside the ring decreases from 120° to 113.989° in the doping position.Cohesive energy for S 1 -SiGY (silicon-doped graphyne), − 8.485 eV, and S 2 -SiGY, − 8.449 eV, was obtained, reflecting the same stability of S 1 and S 2 states.