Dirac cones in two-dimensional systems: from hexagonal to square lattices.

The influence of lattice symmetry on the existence of Dirac cones was investigated for two distinct systems: a general two-dimensional (2D) atomic crystal containing two atoms in each unit cell and a 2D electron gas (2DEG) under a periodic muffin-tin potential. A criterion was derived under a tight-binding approximation for the existence of Dirac cones in the atomic crystal. When the transfer hoppings are assumed to be single functions of the distance between atoms, it was shown that the probability of observing Dirac cones in the atomic crystal gradually decreases before being reduced to zero when the lattice changes from hexagonal to square. For a 2DEG with full square symmetry, a Dirac point exists at the Brillouin zone corners, where the energy dispersion is parabolic not linear. These results suggest that conventional Dirac fermions (such as those in graphene) are difficult to achieve in a square lattice with full symmetry (wallpaper group p4mm).