Mirror symmetry origin of Dirac cone formation in rectangular two-dimensional materials

The Dirac cone (DC) band structure of graphene was thought to be unique to the hexagonal symmetry of its honeycomb lattice. However, two-dimensional (2D) materials possessing rectangular unit cells, e.g. unitary 6,6,12-graphyne and binary t1/t2-SiC, were also found to have DC band features. In this work, a “mirror symmetry parity coupling (MSPC)” mechanism is proposed to elaborate on the DC formation process of 6,6,12-graphyne with the tight-binding method combined with density functional calculations. First, atoms of unit cells are divided into two groups, each of which possesses its own mirror symmetry. Second, wave atom functions within each group are combined into two sets of normalized orthogonal wave functions with an odd and even parity symmetry, respectively, followed by couplings among intragroups and intergroups. The MSPC mechanism, in general, can explain the origins of the DC band structures of a category of 2D materials possessing mirror symmetry and rectangular or hexagonal unit cells. The important role of symmetry analysis, especially mirror symmetry, in understanding DC formation is demonstrated, which may serve as a critical design criterion for novel DC materials.