Dirac Materials in a MatrixWay

Abstract

Recent years have been the platform of discovery of a wide range of materials, like d-wave superconductors, graphene, and topological insulators. These materials do indeed share a fundamental similarity in their low-energy spectra namely the fermionic excitations. There carriers behave as massless Dirac particles rather than conventional fermions that obey the usual Schrödinger Hamiltonian. A surprising aspect of most Dirac materials is that many of their physical properties measured in experiments can be understood at the non-interacting level. In spite of the large effective coupling constant in case of graphene, it has been observed that the interactions do not seem to play a major key role. Controlling the electrons at Dirac nodes in the first Brillouin zone needs the interplay of sublattice symmetry, inversion symmetry and the time-reversal symmetry. In this article, we have used explicit fundamental symmetry to understand the basic features of Dirac materials occurring in three diverse systems in a compact 2 × 2 matrix way. Furthermore, the robustness of the Dirac cones has also been explored from the scientific notion of topological physics. In addition, an elementary introduction on the three dimensional (3D) topological insulators and d wave superconductors will shed light in their respective fields. Furthermore, we have also discussed the way to evaluate the effective mass tensor of the carriers in the two dimensional (2D) Dirac materials. This methodology has also been critically extended to three dimensional (3D) topological insulators and d wave superconductors.