Theoretical chemistry of $\alpha$-graphyne: functionalization, symmetry   breaking, and generation of Dirac-fermion mass
  We investigate the electronic structure and lattice stability of pristine and functionalized (with either hydrogen or oxygen) $\alpha$-graphyne systems. We identify lattice instabilities due to soft-phonon modes, and describe two mechanisms leading to gap opening in the Dirac-fermion electronic spectrum of these systems: symmetry breaking, connected with the lattice instabilities, and partial incorporation of an $sp^3$-hybrid character in the covalent-bonding network of a buckled hydrogenated $\alpha$-graphyne lattice that retains the symmetries of the parent pristine $\alpha$-graphyne. In the case of an oxygen-functionalized $\alpha$-graphyne structure, each O atom binds asymmetrically to two twofold-coordinated C atoms, breaking inversion and mirror symmetries, and leading to the opening of a sizeable gap of 0.22 eV at the Dirac point. Generally, mirror symmetries are found to suffice for the occurrence of gapless Dirac cones in these $\alpha$-graphyne systems, even in the absence of inversion symmetry centers. Moreover, we analyze the gapless and gapped Dirac cones of pristine and functionalized $\alpha$-graphynes from the perspective of the dispersion relations for massless and massive free Dirac fermions. We find that mirror-symmetry breaking mimics a Dirac-fermion mass-generation mechanism in the oxygen-functionalized $\alpha$-graphyne, leading to gap opening and to isotropic electronic dispersions with a rather small electron-hole asymmetry. In the hydrogen-functionalized case, we find that carriers show a remarkable anisotropy, behaving as massless fermions along the M-K line in the Brillouin zone and as massive fermions along the $\Gamma$-K line. 
R. Longuinhos, E. A. Moujaes, S. S. Alexandre, and R. W. Nunes
cond-mat.mtrl-sci cond-mat.mes-hall