Analytical dispersion relations of three graphynes

By using tight-binding approximation, we analytically investigate three interesting two-dimensional carbon allotropes: α-graphyneα-graphyne, 6,6,12-graphyne and β-graphyneβ-graphyne, which contain anisotropic Dirac points at their fermi surfaces. We obtain concrete analytical expressions of the energy dispersion relations in the vicinity of the Diarc points of the three graphynes. Furthermore, we find that there exists an unusual phenomenon in the β-graphyneβ-graphyne, that is, the quantum Hall effect diminishes when its Dirac points are all located at the M points, which is different both from the common material and graphene.