Dirac Cone Materials: Graphene and Beyond Graphene

The present chapter contains a detailed consideration of graphene and some other innovative Dirac materials, whose discovery without exaggeration has been one of the greatest advances in condensed matter physics in the twenty-first century. We provide a brief but rigorous theoretical review of the main aspects of graphene science and, specifically, its unique electronic behavior. We will go over several crucial materials beyond graphene and demonstrate that each new lattice of this family has a less symmetric band structure. We lay out a detailed review of the \(\alpha -\mathcal {T}_3\) model, a dice lattice, and Lieb and Kagome lattices, whose low-energy spectrum exhibits a purely flat or dispersionless band in addition to a regular Dirac cone. A finite-gap \(\alpha -\mathcal {T}_3\) model will be also addressed. Importantly, \(\alpha -\mathcal {T}_3\) also represents a rare class of materials in which the Berry phase enters their Hamiltonian directly.