A Theoretical Study on the Design, Structure, and Electronic

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A Theoretical Study on the Design, Structure and Electronic Properties of Novel Forms of Graphynes Naga Venkateswara Rao Nulakani, and Venkatesan Subramanian J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b03562 • Publication Date (Web): 27 Jun 2016 Downloaded from http://pubs.acs.org on July 2, 2016

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The Journal of Physical Chemistry

A Theoretical Study on the Design, Structure and Electronic Properties of Novel Forms of Graphynes Naga Venkateswara Rao Nulakani1, 2 and Venkatesan Subramanian*,1, 2 1

Chemical Laboratory, CSIR-Central Leather Research Institute, Adyar, Chennai - 600 020, India. 2 Academy of Scientific and Innovative Research (AcSIR), CSIR-CLRI Campus, Chennai, India.

Abstract α-, β-, γ- and 6,6,12-graphynes are well established one atom thick two-dimensional (2D) materials in the graphyne family. These 2D sheets have been mainly designed by incorporating an acetylenic linker (−C≡C−) in graphene with different ratios. The graphdiynes and their higher order 2D architectures have also been studied to elucidate the effect of length of linker (−C≡C−C≡C−) on the structure-property relationship. In the present investigation, we have modelled the three novel analogues of α-graphyne by increasing the acetylenic linkers and expanding the sp2 network. The structure, stability and electronic properties of novel forms of graphyne architectures were examined by using the computational methods within the frame work of Density Functional Theory (DFT). The molecular dynamics simulations show that only one system is thermodynamically stable and rule out the existence of other two newly designed systems. The electronic structure calculations reveal that, the stable 2D sheet exhibit semimetallic Dirac point features. Further, the semi-metallic carbon sheet has massless Dirac fermions (m* = 0.014 mo) akin to those of γ-graphyne and graphdiyne. The predicted Fermi velocity (vf( K→M) = 7.13 × 105 m/s) of the novel 2D sheet is higher than that of α-graphyne and close to that of graphene. Furthermore, the electronic properties of armchair and zigzag nanoribbons of stable 2D sheet have also been investigated. Interestingly, one of the zigzag nano ribbons shows linear band dispersion (Dirac point) in the proximity of the Fermi level and others exhibit semi-conducting to metallic properties. Corresponding Author *E–mail: [email protected]; [email protected] Tel.: +91 44 24411630. Fax: +91 44 24911589.

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1. INTRODUCTION Over a decade, the members on the atlas of two-dimensional (2D) materials have been continuously increasing1–7 due to their potential applications in various fields including chemistry, solid state physics and medicine. In 2004, Geim and his co-workers have successfully peeled a one atom thick two-dimensional monolayer8,9 (graphene) from the 3D van der Waals solid such as graphite using optical microscopy, scanning electron microscopy and atomic-force microscopy techniques. The graphene exhibits exceptional physical, chemical,10,11 electronic,12 transport13 and optical properties14 and owns outstanding applications15 in various fields which are far superior to those of graphite. In fact, graphene opens up the new directions in the fields of material science, optoelectronics, photovoltaic cells, energy storage and composite materials. Since then, enormous research attention has been devoted to design and development of novel 2D materials using both experimental as well as theoretical methods. For instance, group IV elemental (C,16–19 Si,20 Ge,21,22 Sn23,24) 2D materials, boron nitride (h-BN),25 MXenes,26–28 graphitic nitride (g-C3N4),29 mono-, di-, tri-chalcogenides (MX, MX2, MX3)30–36 and phosphorene37 are the different forms of 2D materials which are extensively studied in the literature. Further, these monolayer 2D materials are the basic building blocks for the constructions of layered materials or van der Waals solids. In addition to this, the 2D materials exhibit broad range of conducting properties.7 The multiway procedures in the experimental synthesis of carbon based materials impart the special preference to them over other 2D materials. Further, the profound diversity in the electronic structure of these materials also provokes the design and development of novel carbon allotropes. For example, numerous 2D carbon allotropes were reported in literature with distinct electronic properties including metallic,38,39 semi-metallic,16,17 semi-conducting40–43 and


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topologically insulating18 in nature. This is due to the fact that the metallic carbons are often associated with many interesting properties including superconductivity,44,45 negative differential resistance,46 and phonon–plasmon coupling.47 Further, the metallic carbons can be effective as catalysts due to the high electron density48 at the Fermi level and show magnetic behavior when Stoner-like criterion is satisfied.49,50 The semi-metallic (zero band gap and zero electron density) carbon allotropes exhibit completely different properties to those of metallic carbons. The semimetallic carbons show unique electronic features such as massless and massive Dirac fermions,51 half-integer quantum Hall effect,52 Klein tunneling,53 pseudo-diffusive conduction54 and conducting edge states in topological insulators.55,56 These properties are mainly originated from the linear dispersion relation in the electronic energy within the Brillouin zone. For instance, graphene with a non-zero band gap shows above mentioned electronic properties which are completely difficult to attain from the regular 2D or 3D semi-conductors. Even though the semimetallic carbon allotropes exhibit highly interesting electronic properties, they are not fully exploited in the field of semi-conductor based electronics due to semi-metallic nature which leads to the low standby power dissipation. Opening the band gap in these kinds of 2D Dirac materials or design of novel materials with similar characteristic bands becomes a challenging task in the field of materials science. For example, graphene is subjected to covalent or noncovalent functionalization57–59 with various materials,60–63 strain,64,65 electric fields,66 lattice mismatch67–69 and doping70,71 to create an energy gap in its electronic spectra.72,73 Hence, semiconductors, semi-metallic Dirac materials and metallic carbon phases are highly inevitable members in the field of material science for the development of next generation electronic materials.



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Among various classes of carbon based materials, graphynes74 are the competitive 2D materials75 to the graphene. For example, some of the graphynes are furnished with direction dependent electronic properties,75 self-doping capacity and topologically insulating properties18 which are superior to graphene. In fact, different forms of graphynes were created by insertion of sp hybridized C atoms76 in the graphene in different ratios. For instance, the α-graphyne77 has been created by incorporating an acetylenic linker (−C≡C−) between each pair of C atom in graphene. Further, the effect of the length78,79 of the linker (···−C≡C−C≡C−···) on the structure related properties of graphyne have been investigated extensively. Furthermore, the structural and electronic properties of other graphyne and graphdiyne flat carbon networks have also been scrutinized with the aid of theoretical methods.80–83 However, none of these studies have been concentrated on the addition of acetylenic linkers in the side by side manner. In the present study, we have created three α-graphyne analogues by replacing the some selected C−C bonds in graphene with acetylenic linkers. These three 2D sheets comprise of different number of acetylenic linkers which are placed adjacent to each other. Further, the structural stability of the newly designed three carbon allotropes has also been scrutinized. However, the electronic properties of the thermodynamically stable 2D carbon sheet show semimetallic properties with linear band dispersion relation. The armchair and zigzag nanoribbons of the same 2D sheet exhibit width depending electronic properties similar to those of graphene nanoribbons. 2. COMPUTATIONAL DETAILS All the density functional theory (DFT) based calculations were carried out using Vienna ab initio Simulation Package (VASP).84–86 The generalized gradient approximation (GGA)87 parameterized by Perdew-Burke-Ernzerhof (PBE) was adopted to calculate the exchange and


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correlation energy. The all-electron projector augmented wave (PAW)88 method was adopted with carbon 2s2 2p2 treated as valence electrons. The plane waves were extended with a kinetic energy cutoff of 520 eV. The energy minimization is performed using the conjugate gradient iterative technique with a Monkhorst-Pack89 of 8 × 8 × 1 k-mesh in the Brillouin zone until the energy between two consecutive steps is less than 10-8 eV and the Hellmann-Feynman force on each atom was less than 10-3 eV/atom. Comparatively a dense mesh of 24 × 24 × 1 k-points were used for the electronic structure calculations. The phonon band dispersion curves were calculated using the open source Phonopy package90 and the forces were obtained employing the VASP package. The dynamic stability of graphyne sheets was further examined by performing the firstprinciples finite temperature molecular dynamics (FPMD) simulation for 10 ps at 300 K with a time step of 1 fs using NVT ensemble. The nanoribbons were optimized with a Monkhorst-Pack of 1 × 1 × 8 k-mesh and electronic structure calculations were carried out with 1 × 1 × 24 k-mesh with the same thresholds for the energy and Hellmann-Feynman force. The van der Waals corrected density functional theory (vdW-DF2)91 was employed to explore the structural and electronic properties of carbon bilayers. In all cases, the periodic images were separated by a vacuum of 15 Å along the required directions. 3. RESULTS AND DISCUSSION 3.1 Design. The schematic representation of creation of novel 2D carbon allotropes from one of the well-studied carbon networks such as graphene is presented in Figure 1. It shows the optimized geometries of various graphyne networks. The α-graphyne is created by replacing each C−C bond of a graphene with an acetylenic (−C≡C−) linker. Similarly, the α-2, α-3 and α-4 graphynes are created by incorporating −C≡C− linkers in some selected C−C bonds of 2 × 2 × 1, 3 × 3 × 1 and 4 × 4 × 1 super cells of graphene. The α-graphyne contains two sp2- and six sp- hybridized



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carbon atoms (in the form of acetylenic linkers) with in the unit cell. Each acetylenic linker in αgraphyne has been doubled, tripled and quadrupled in the α-2, α-3 and α-4 graphynes as depicted in Figure 1. To elicit the dynamical stability of these novel 2D sheets, we have performed First Principles Molecular Dynamics (FPMD) simulations for 10 ps with a time step of 1 fs at 300 K. It can be seen from Figure 2a that the average value of the total energy of α-2 graphyne is constant throughout the simulation. Further, Figure 2b shows that the temperature is maintained at 300 K during the simulation. Furthermore, there are no obvious changes in the α-2 graphyne sheet after 10 ps MD simulation. These results assert that the α-2 graphyne is stable at the room temperature. On the other hand, the α-3 and α-4 graphynes are not stable and dismantled into other structures. Further, we have also tried the different conformations of α-2, α-3 and α-4 graphynes which are arrived by joining the C-C bonds in other directions as shown in Supporting Information ((SI) Figure S1). For example, α-2a is emerged from the α-2 graphyne by joining the two acetylenic linkers in order to form two five- and one four-membered rings. Similarly, α-2b and α-2c are obtained by the joining of 4 and 6 acetylenic linkers of α-2 graphyne, respectively. The same strategy is applied for α-3 and α-4 graphynes to arrive at the stable conformations. However, none of these structures obtained from α-2, α-3 and α-4 are stable at room temperature. Hence, further theoretical calculations on these 2D carbon architectures were not carried out in this study and periodic DFT calculations were performed only on α-2 graphyne. The phonon mode analysis was conducted on α-2 graphyne to gain insights into the dynamical stability. The phonon band dispersion is plotted in Figure 2c along the high symmetry k-points throughout first Brillouin zone. The absence of imaginary frequencies in the complete phonon band spectra authenticates the dynamical stability of α-2 graphyne.


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3.2 Structural properties. This section gives a detailed account on the complete structural parameters of α-2 graphyne. The α-2 graphyne exhibits the p6/mmm plane group, which is akin to that of graphene. The unit cell of α-2 graphyne comprises 20 C atoms in the plane spanned by the lattice constants a = b = 9.48 Å. However, α-2 graphyne contains three chemically inequivalent C atoms such as C1, C2 and C3 in the unit cell. These C1, C2 and C3 atoms successively occupy the Wyckoff positions at 2d (0.66667, 0.33333, 0.50000), 6m (0.48876, 0.24438, 0.50000) and 12q (0.39525, 0.32040, 0.50000) as shown in Figure 3. The α-2 graphyne exhibits three distinct bond lengths which include d1 (C1–C2) = 1.46 Å, d2 (C2–C3) = 1.39 Å and d3 (C3–C3) = 1.23 Å. It is evident from these results that the C3–C3 bond length is similar to that of C≡C (sp–sp) bond length whereas the C2–C3 and C1–C2 bond lengths are close to the sp–sp2 and sp2–sp2 bond lengths of αand β-graphynes.92 In general, the bond strength between two atoms in any molecule is inversely related to the bond length of those particular atoms. Hence, the C3–C3 bond exhibit comparatively higher bond strength fallowed by C2–C3 and C1–C2 bonds in α-2 graphyne. Further, the same sheet shows three different bond angles such as θ1 (C2–C1–C2) = 120.0°, θ2 (C3–C2–C3) = 113.18° and θ3 (C2–C3–C3) = 176.59°. The scrutiny of these results elicits that the bond lengths and bond angles slightly deviate from the regular sp and sp2 geometrical parameters. Hence, we have calculated the cohesive energy of α-2 graphyne to estimate the strength of C–C bonds in α-2 graphyne. The cohesive energy of α-2 graphyne (-7.09 eV/atom) is comparable to that of α- (-7.01 eV/atom), β- (-7.09 eV/atom) and γ-graphyne (-7.24 eV/atom).92 The areal density of the α-2 graphyne (0.26 atom/Å2) is lower than that of graphene (0.38 atom/Å2) and phagraphene (0.37 atom/Å2). This suggests that, the α-2 graphyne can be considered as a porous carbon phase and often propitious for the design of lightweight materials.



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3.3 Electronic Properties. In this section, we describe the electronic properties of α-2 graphyne. The results from electronic structure calculations of α-2 graphyne are presented in Figure 4. The band structure is plotted along the high symmetry k-points such as Γ(0.00, 0.00, 0.00) → M(0.00, 0.50, 0.00) → K(-0.33, 0.67, 0.00) → Γ(0.00, 0.00, 0.00). It can be seen from Figure 4a that, the energies of valence and conduction bands (VB and CB) are degenerate at a single point on the Fermi surface. This degenerate point is located at a high-symmetric k-point (K) with in the first Brillouin zone (see Figure 4b). Further, these bands are linearly dispersed with the k-vector in the proximity of the Fermi level. The valence and conduction bands meet the degenerate point with a slope (∂E/∂k) of ±26.3 eV Å along a line that passes through the high-symmetry points Γ and K. Further, the slopes remain same and the curvature becomes zero when approaching the degenerate point along a line which is perpendicular to the former Γ to K line. We have also computed the 3D band structure of α-2 graphyne and plotted on an energy scale of -0.06 eV to 0.06 eV, which is depicted in Figure 4c. It suggests that the energy of fermions (electrons and holes) is dispersed in the form a double cone nearer to the Fermi level with in the First Brillouin zone. These features substantiate that the degenerate point of VB and CB on the Fermi level can be considered as the Dirac point. The results are essentially same even if we consider the other high-symmetric k-point, K՛ instead of K. Hence, there are two Dirac points on the Fermi surface at the high-symmetric K and K՛ points in the hexagonal Brillouin zone of α-2 graphyne. The two Dirac points symmetrically related to one another by the rotation of the Brillouin zone with an angle of 60°. The effective mass (m*) of holes and electrons in α-2 graphyne (m* = 0.014 mo) is nearly same in both symmetric directions and lower than those of γ-graphyne (0.08 to 0.21mo)92 and graphdiyne (0.091 to 0.095 mo).41 The Fermi velocity (vf) of α-2 graphyne was computed using the E(q)/ħ|q| relation. The α-2 graphyne exhibits the band slopes (Fermi


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velocities, vf) of ±29.5 eV Å (vf( K→M) = 7.13 × 105 m/s) along K→M and ±26.3 eV Å (vf( K→Γ) = 6.36 × 105 m/s) along K→Γ directions. These Fermi velocities are comparable to those of other 2D carbon allotropes such as graphene (8.22 × 105 m/s), α-graphyne (6.77 × 105 m/s), βgraphyne (3.87 – 6.77 × 105 m/s),92 T-graphene (~ 106 m/s)93 and phagraphene (3.43 – 6.48 × 105 m/s).17 In order to understand the orbital contributions for the formation of linear band dispersion in the proximity of the Fermi level, we have plotted the total and partial density of states (TDOS and PDOS) in Figure 5. The TDOS of α-2 graphyne gradually decreases to zero around the Fermi level. This feature corroborates that the α-2 graphyne is semi-metallic in nature. It is evident from the PDOS that the pz orbitals of C atoms are mostly responsible for the formation of linearly dispersed VB and CB in the proximity of the Fermi level which is akin to that of graphene and graphynes. Further, the band decomposed charge densities (Figures 5b and c) of VB and CB also reinforce that the pz bonding and anti-bonding orbitals of C atoms in α-2 graphyne are predominantly responsible for the formation of linear band dispersion nearer to the Fermi level. 3.4 Structural and Electronic Properties of Nanoribbons. We have also investigated the structure-property relationship of aesthetically pleasing 1D nanoribbons (NRs) of α-2 graphyne. Similar to graphene94 and α-graphyne95 nanoribbons, we have created two regular types of nanoribbons such as armchair (Figure 6a) and zigzag (Figure 6b) by cleaving the α-2 graphyne along (110) and (100) zero directions, respectively. The unsaturated carbon atoms at the edges are passivated by hydrogen atoms. Following the conventional notation of graphene NRs, the armchair and zigzag NRs of α-2 graphyne are indicated by the width of the respective nanoribbon. For example, the notations NA-ANR and NZZNR represent the armchair and zigzag NRs with the widths NA and NZ, respectively. In this



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study, the width (NA) of α-2 graphyne ANRs has been varied from NA = 2 to NA = 10 while the width (NZ) of ZNRs has been changed from NZ = 1 to NZ = 5 to evaluate the effect of nanoconfinement on the electronic structure. Further, the widths of ANRs are classified into three families such as NA = 3p+2, 3p+1 or 3p (where p = 0, 1, 2, 3…..), similar to those of graphene nanoribbons. Based on this, the nanoribbons with NA = 2, 5, 8 are in the category of 3p+2 family and with NA = 4, 7, 10 fall in the category of 3p+1 family. The remaining NRs such as NA = 3, 6, 9 belong to the 3p family. However, the nanoconfinement of NRs questions the stability of the systems. Hence, to confirm the dynamical stability of armchair and zigzag NRs of α-2 graphyne, we have evaluated the phonon band dispersion relation of 2-ANR and 1-ZNR along the NR axis. It can be seen from Figures 7a and b that both armchair and zigzag NRs of α2 graphyne are dynamically stable. The electronic structure calculations unravel that, all the nine α-2 graphyne ANRs exhibit the direct band gap at the center of the Brillouin zone. The variation in the band gaps of the nine 1D ANRs of α-2 graphyne is shown in Figure 7c. Results reveal that, the band gaps of 3p family ANRs are higher than those of 3p+1 family. Further, the band gaps of 3p+2 ANRs are intermediate between the band gaps of 3p and 3p+1 family ANRs for the widths considered in the present study. It is obvious from the previous studies94,95 that the quantum confinement of nanoribbons plays a major role in deciding the band gap of the nanoribbons. Consequently, the band gaps of NRs which belong to a particular nanoconfinement or family (3p) are higher in nature than those of other nanoconfinement (3p+1 and 3p+2) NRs. Overall results show that the band gap of NA-ANRs of α-2 graphyne is inversely related to the width of the nanoribbon in a particular family. The representative band structures for each family of α-2 graphyne ANRs are depicted in Figure 8. It is important to mention here that the current level of density functional


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theory (PBE−GGA) underestimates the band gap of the ANRs. In fact, it is well known that the hybrid density functional methods are employed to explore the band gap of the NRs more accurately. However, the employment of more accurate hybrid methods are computationally more demanding in the present model systems as the unit cells (5- to 10-ANRs) are composed of more than 100 atoms. Hence, we have examined the band gap of smallest ANRs (2- to 4-ANR) from each family with hybrid density functional method (HSE06)96 and the results are listed in Table S1. It shows that 2-ANR exhibits the large band gap and 4-ANR shows the narrow band gap whereas the band gap of 3-ANR is intermediate between those of 2- and 4-ANRs. Overall, the band gap of each nanoribbon increases by an amount of 0.5 eV in each family. The same trend may be obtained for the remaining (5- to 10-ANRs) nanoribbons. It is important to mention here that the DFT methods are also effectively used by numerous research groups to explore the band gap of various nanomaterials. For example, Yue et. al., have probed the magnetic and electronic properties97 of α-graphyne nanoribbons. Further, Chen and his co-workers have investigated the electronic structure of Graphane/Fluorographene Bilayer.98 Furthermore, Zeng and his research group99 have predicted the band gaps of B2C nanoribbons and nanotubes using DFT methods. In fact, it is a mammoth task to enumerate all the literature information on prediction of band gaps of semi-conducting materials using DFT methods. These studies clearly authenticate the usefulness of DFT methods in predicting the band gaps of the nanomaterials. Based on the above mentioned reasons, we have restricted ourselves to use the PBE-GGA level of theory to predict the band gaps α-2 graphyne nanoribbons. On the other hand, all NZ-ZNRs considered in this study are metallic in nature except 1ZNR. It can be seen from the Figure 8 that, the band structure corresponding to NZ = 1 exhibits semi-metallic behavior similar to that of α-2 graphyne. The VB and CB of 1-ZNR are degenerate



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at a contact point (D) on the Fermi level. Further, the amplified band structure ensures that the VB and CBs nearer to the Fermi level of 1-ZNR exhibit the linear band dispersion around the contact point. The calculated band slopes of the VB and CB of 1-ZNR along D→Γ (D→Z) are 16.24 (-30.01) and -21.78 (24.71) eV Å, respectively. The corresponding Fermi velocities are around vf( D→Γ) = 3.95 × 105 m/s and vf( D→Z) = 7.26 × 105 m/s for holes and vf( D→Γ) = 5.27 × 105 m/s and vf( D→Z) = 5.97 × 105 m/s for electrons. These results suggest that the Fermi velocities of 1-ZNR are also close to those of α-2 graphyne sheet. The semi-metallic Dirac point feature of 1ZNR is transformed to metallic feature for the remaining NRs such as 2-, 3-, 4- and 5-ZNRs. The panel corresponding to NZ = 5 in Figure 8 presents electronic band structure for 5-ZNR. In fact, the band profiles of 2-, 3- and 4-ZNRs are similar to that of 5-ZNR. It indicates that the ZNRs of α-2 graphyne are metallic in nature and the metallicity is independent of the width of the ZNR (except 1-ZNR). It is important to mention here that, the graphene NRs exhibit width dependent magnetic properties. In order to investigate the magnetic properties of ANRs and ZNRs of α-2 graphyne, we have also performed spin polarized calculation. However, all the NRs of α-2 graphyne are stable in nonmagnetic ground state. 3.4 Structural and Electronic Properties of Bilayers. Finally, we have analyzed the probability of formation of layered structures using α-2 graphyne similar to graphite and h-BN. We have stacked the α-2 graphyne monolayers in twodifferent ways such as AA and AB stacking modes as shown in Figures 9a and b, respectively. In AA stacking, the chemically inequivalent C atoms of first layer are exactly placed on the same atoms of the second layer, whereas in AB stacking the first layer is shifted laterally to the centers of hexagons of second layer. Results indicate that the α-2 graphyne prefers to form AB stacking rather than AA stacking. We have calculated the interlayer binding energy for different layered structures by subtracting the sum of energies of individual layers from the energy of bilayer. It is


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evident from Figure 9c that the α-2 graphyne monolayers are strongly stacked together in AB stacking and exhibit a maximum binding energy of 37 meV/atom at an interlayer distance of 3.5 Å. Further, computed electronic properties of bilayer of α-2 graphyne show that, the conduction and valence bands joins at the K-point on the Fermi level similar to that of α-2 graphyne. The linear band dispersion of α-2 graphyne monolayer changes to parabolic dispersion in the case of bilayer around the K-point in the proximity of the Fermi level. Further, the hole and electronic energy states are doubled in bilayer of α-2 graphyne when compared to those of single layer α-2 graphyne sheet (see Figure 9d). Findings from electronic structure calculations of bilayer of α-2 graphyne are comparable to those of bilayer graphene. 4. CONCLUSIONS In this study, the structural and dynamical stabilities of a series of two-dimensional (2D) graphyne networks and their isomers have been examined with the aid of First-Principles Density Functional Theory based calculations. Results show that, there is another analogue of αgraphyne which exhibits semi-metallic Dirac point features. The novel 2D Dirac material is christened as α-2 graphyne. The α-2 graphyne shows different Fermi velocities (6.36 × 105 and 7.13 × 105 m/s) along different crystallographic high symmetric directions. The calculated effective mases of holes and electrons of α-2 graphyne confirm that the charge carriers are lighter than those of γ-graphyne and graphdiyne. Further studies have been carried out to evaluate the structural and electronic properties armchair and zigzag nanoribbons (NRs) of α-2 graphyne. The findings from electronic structure calculations show that all armchair NRs are semi-conducting while the zigzag NRs are metallic in nature except 1-ZNR. Interestingly, semi-metallic linear band dispersion is observed in the case of 1-ZNR similar to its 2D parent system. Further, the electronic structure of bilayer of α-2 graphyne is akin to that of bilayer graphene. These results



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substantiate that the α-2 graphyne and its nanoribbons can be used as potential alternatives to develop carbon based futuristic faster electronic materials.

ACKNOWLEDGMENTS: The authors acknowledge the Multi-Scale Simulation and Modelling (MSM) project (No. CSC0129) funded by CSIR and Design and Development of Two Dimensional van der Waals Solids and their Applications project (No. EMR/2015/000447) funded by DST for the financial support. One of the authors (NVR Nulakani) wish to thanks DST for SRF.

Supporting Information. The geometries of various graphyne networks and HSE06 band gaps are presented. This material is available free of charge via the Internet at http://pubs.acs.org.


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Figure 1. Design of different forms of α-graphynes from graphene. (a) Unit cell of graphene, (b) 2 × 2 × 1, (c) 3 × 3 × 1 and (d) 4 × 4 × 1 supercells of graphene networks. The unit cells of (e) αgraphyne, (f) α-2 graphyne, (g) α-3 graphyne and (h) α-4 graphynes were created by replacing each of the highlighted C−C bonds (yellow in color) with an acetylenic (−C≡C−) linker in the corresponding graphene network.

Figure 2. Structural stability of the α-2 graphyne sheet. (a) Fluctuation of total energy and (b) fluctuation of temperature with respect to simulation time and (c) phonon band dispersion along the high symmetric points in the first Brillouin zone.



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Figure 3. The three chemically inequivalent Wyckoff positions of C atoms in α-2 graphyne.

Figure 4. Electronic properties of the α-2 graphyne sheet. (a) Full band structure for a unit cell and the inset represents the first Brillouin zone with the high symmetry k-points along which the bands are plotted (b) amplification of bands around the Dirac point and (c) the three-dimensional band structure at the Dirac point (the Fermi level is set as 0 eV in all cases).


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Figure 5. Calculated electronic properties and band decomposed charge densities of α-2 graphyne sheet. (a) Total and partial density of states and partial charge density corresponding to (b) VB and (c) CB nearer to the Fermi level of α-2 graphyne (20 k-points are considered from the Dirac point (K) along K→Γ and K→M directions to plot the charge densities). The isosurfaces value is set to 0.001 e/bhor3. The Fermi level is set as 0 eV in the DOS panel.

Figure 6. Geometric structures of the (a) armchair and (b) zigzag α-2 graphyne nanoribbons. The NA and NZ indicate the width of armchair and zigzag the nanoribbons, respectively. The grey spheres are C atoms and white spheres are H atoms. The red dashed lines represent the unit cell of respective nanoribbon.



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Figure 7. Calculated phonon band structure of (a) 2-ANR (b) 1-ZNR and (c) variation of band gaps of armchair (NA) nanoribbons of α-2 graphyne for three different families.

Figure 8. Calculated electronic properties of the armchair (NA) and zigzag (NZ) nanoribbons of α-2 graphyne for different widths. The 1-ZNR shows linear band dispersion in the proximity of the Fermi level around the contact point D.


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Figure 9. Calculated structural and electronic properties of bilayer of α-2 graphyne. (a) AA stacking, (b) AB stacking, (c) variation of the binding energy as a function of interlayer distance in AB stacking and (d) electronic band structure of bilayer of α-2 graphyne (The C atoms are represented in blue and black colors in (a) and (b) for clarity).



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A Theoretical Study on the Design, Structure, and Electronic

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