## 4 Properties of GR
### Band Properties  
GR possesses a nonzero band gap that is not present in G [55; 56; 132]. By utilizing LDA and GGA approaches (Figure 16a), the band gap of graphyne-\(n\) was calculated to be about 0.5 eV, and it shows a small dependence on acetylenic linkage [55; 56; 122]. The HSE or GW approaches that have self-interaction correction were utilized to calculate the band gap of GR with the comparison of silicon, and the calculated value was \(\sim\)1 eV [56; 122; 132; 133]. It has been observed that GR allows a high radiative recombination rate over Si and a light absorption rate over Si because of the direct band gap. From DFT results, the location of the conduction band minimum and valence band maximum was calculated, and it has been located at the M point of the Brillouin Zone of the hexagonal structure. To understand the projected band structure of graphyne, it can be divided into two parts, i.e., sp\({}^{2}\) and sp [133]. It can be seen (Figure 16) that at the band edges the wide \(P_{z}\pi\)_to \(\pi^{*}\), deep-lying \(\sigma\)_to \(\sigma^{*}\)_bands are situated, and the inside \(P_{z}\pi\)_to \(\pi^{*}\)_consists of another band \(P_{x}-P_{y}\pi\)_to \(\pi^{*}\)[56]. In sp-sp interaction, it originates the \(P_{x}-P_{y}\pi\)_to \(\pi^{*}\)_interaction that is not present in G. This leads to anisotropic optical properties in GR [56]. In the range of band gap energy from 0 to 8 eV, this structure adsorbs more energy when in-plane polarized light falls on the material with respect to out-of-plane polarized light [56]. According to the results of first-principles calculations [65], the Dirac cone is also present in \(\alpha\), \(\beta\), and \(\gamma\)-graphyne Figure 16e. In \(\alpha\) GR the Dirac cone is located at higher symmetry points K and K', while in \(\beta\)- and \(\gamma\)-graphyne, the Dirac cone is situated at lower symmetry points. It has been observed that in 6,6,12-graphyne the Dirac cone is located below and above the Fermi level, which means that it has the property of self-doping [65; 135]. This shows that the Dirac cone can be found at any point, and it is not associated with a hexagonal structure, and also it is not limited to G.  
Various studies have explored the presence/absence of Dirac cones, and many studies have explored the importance of binding \(P_{z}\pi\)_to \(\pi^{*}\)_[133; 135; 136; 137; 138]. This idea proposed that the impact of acetylenic linkage -C\(\equiv\)C- is rival to the renormalized direct hopping term among vertex atoms. Because of this bonding, a tight-binding model can be applied to all types of GR, whereas GR has a hexagonal structure like G. With the existence of Dirac cones, the magnitude and combination of renormalized hopping terms can be determined. By utilizing the above-mentioned model, with the presence of Dirac cones, the new GR-based structures are also predicted. Graphyne 14, 14, 14 has a rhombic symmetry, and it acts the same as deformed G in the direction of the armchair direction (Figure 17). The 14,14,18-graphyne is another that has a rectangular symmetry, and the slope band dispersion shows 5.0 eV/A on the X-axis and 4.1 eV/A on the Y-axis with anisotropy linear band dispersion  
Figure 16: (**a**–**c**) Electronic band structures of \(\alpha\)-, \(\beta\)-, and 6,6,12-graphyne. The figure is adapted from reference [113] @ 2018 MDPI. (**d**) Band structure of \(\gamma\)-graphyne calculated using HSE and GGA methods showing direct band gap nature. The figure is adapted from reference [134]. (**e**) Projected electronic states of sp and sp\({}^{2}\) carbon atoms The figure is reprinted with permission from reference [49] @ 2013 Elsevier. (Figure 18) [133]. A complicated tight-binding model was proposed to check the effect of spin-orbit coupling (C-O-S) on Dirac cones [139]. This model also includes the \(\sigma\) bonding and its effects, which makes it more complicated. By utilizing intrinsic SOC, the nontrivial gap can be opened in \(\alpha\)-, \(\beta\)-graphyne.