Examination of the relativistic Dirac equation and its implications for 2D hexagonal materials
Abstract With the advent of the metal–insulator transition (MIT) as a tool for adjusting the properties of lower dimensions materials, with nanoscale dimensions in one or more of the coordinate directions, it is worthwhile to derive analytical formulas for the electronic energy in k-space, for a generic two-dimensional hexagonal lattice as a prelude for their use in nanoscopic transport devices (using the MIT) or other solid state devices. This is done in the tight binding (TB) approximation, since it allows a tractable analytical development, with some refinements possible. This lattice structure is applicable to many 2D materials like graphene, BN, MoS2, and many others. This chapter treats the relativistic energy expression (Section 9.1 ), the Dirac conditions obtained from linear energy representation and its Hamiltonian (Section 9.2 ), the allowable α i and β Dirac matrices (Section 9.3 ), choosing specific α i and β sets and their satisfaction of the Dirac conditions (Section 9.4 ), the non-uniqueness of the Dirac matrices and their transformations (Section 9.5 ), the Dirac equation (Section 9.6 ), the plane wave form of the Dirac equation (Section 9.7 ), eigenvalues of the plane wave Dirac equation (Section 9.8 ), eigenvectors of the plane wave Dirac equation and comparison to graphene eigenvectors (Section 9.9 ), spinor eigenvectors with transverse momentum plane wave Dirac equation (Section 9.10 ), transforming from one Dirac matrix set to another (Section 9.11 ), and the transformed plane wave Dirac equation for transverse momentum (Section 9.12 ).