Question

Why (and in which regime) can the electronic properties of graphene be described by a Dirac-like Hamiltonian?

Answer 1

The remarkable properties and its strong potential to provide the material basis for a new generation of electronic devices,graphene has been the subject of massive research throughout the world since 2004.Graphene is a single atom thick two dimensional planar layer of carbon atoms in a hexagonal honey combed array composed of two superposed triangular sublattices .The band structure of graphene involves two nodal zero gap (Dirac) points (K,K) in the first Brillouin zone at which the conduction and valence bands touch.In the vicinity of these points ,the low energy dispersion relation of massless relativistic electrons ,so the electrons of graphene are described as Dirac fermions having no mass

The fundamental low-energy graphene electron/hole dispersion relation proportional to momentum, , which likens graphene carriers to massless relativistic Dirac fermions, is embodied in the Hamiltonian written in ‘pseudo-spin’ notation ( are Pauli matrices), which distinguishes the two triangular sub-lattices of the honeycomb lattice on which a graphene quasi-particle can be located, where the two zero-gap ‘Dirac’ points correspond to   

and ? is given in terms of graphene band-structure parameters as ( is the hopping parameter in the tight-binding approximation and ? is the lattice spacing): ? plays the role of a constant Fermi velocity independent of density. This Hamiltonian is responsible for features in graphene that are analogous to relativistic phenomena such as Klein tunnelling, ‘zitterbewegung’ and others. As in the study of massless relativistic neutrino fermions, pseudo-helicity, the component of pseudo-spin in the momentum direction, commutes with h?1 and its eigenvectors can be used as a basis in which is diagonal. Introducing the transformation from pseudo-spin basis to pseudo-helicity basis,.