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Theoretical chemistry of $α$-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) $α$-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 $α$-graphyne lattice that retains the symmetries of the parent pristine $α$-graphyne. In the case of an oxygen-functionalized $α$-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 $α$-graphyne systems, even in the absence of inversion symmetry centers. Moreover, we analyze the gapless and gapped Dirac cones of pristine and functionalized $α$-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 $α$-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 $Γ$-K line.