Six-fold symmetry origin of Dirac cone formation in two-dimensional materials

Dirac materials possess many excellent electrical properties, resulting that the search and design of Dirac materials have become a hot research area. Revealing the formation conditions of Dirac cone (DC) can provide theoretical guidance for the search and design of Dirac materials. To obtain the necessary conditions for the formation of DC of two-dimensional (2D) materials with six-fold symmetry (SFS), the DC formation mechanism was analyzed by the 'divide-and-couple' approach in the framework of tight-binding theory, confirmed by the subsequent density functional theory calculations. The simple '6n + 2' rule was proposed to determine whether the 2D materials with SFS have DCs, i.e. when the number of atoms in a unit cell is 6n + 2, the systems would possess DCs at the vertex of Brillouin zone for the 2D materials composed of the elements of the IV main group. Moreover, the '3n + 1' rule was derived as the condition for the DC formation in graphene-like silagraphene with SFS and used to design a silagraphene Si6C8 with DCs. Understanding the DC formation mechanism of 2D materials with SFS not only provides theoretical guidance for designing novel Dirac materials but also sheds light on the symmetry origin of the formation mechanism of DC.