Graphene is one of the most remarkable materials ever found. Also one of the most studied. It presents a number of unique features that have attracted the attention of both theoreticians and experimentalists. Investigation of its properties has led to breakthroughs, not only from the perspective of fundamental research but also from the point of view of applied science. Theoretically conjectured long ago, it was concretely obtained in 2004 by Geim and Novoselov [243]. Graphene properties include an outstanding mechanical robustness, being orders of magnitude stronger than steel; high electric and thermal conductivities, despite the absence of a Fermi surface; finite resistivity even without impurities, despite the absence of a gap; and relativistic dispersion relation for the active electrons, implying their kinematics are described by the Dirac equation and not by the Schrödinger equation, among others. This last property makes of graphene a concrete realization of the Dirac sea, a concept that in spite of not manifesting itself in nature, in the absence of matter has enabled Dirac to predict the existence of antimatter. The observation of antimatter in vacuo and the subsequent concrete realization of the Dirac sea in a material system such as graphene is an outstanding example of the great unity that exists in physics. Other properties of Dirac particles such as the Klein tunneling have been observed as well in graphene. As a consequence of charge conjugation symmetry, both electrons and holes possess the same mobility in graphene, a feature that is not found, for instance, in regular (Si or Ge based) semiconductors. Further properties of this extraordinary material include, for instance, the occurrence of the integer and fractional quantum Hall effects, in the presence of an external perpendicular magnetic field and the Zitterbewegung.

Crystal Structure and Tight-Binding Approach

Graphene is a one-atom-wide sheet of carbon with a sp2 hybridization, assembled in a honeycomb crystal structure, consisting of a Bravais triangular lattice with spacing a and a base of two atoms, respectively placed at (0, 0) and (0, h), with respect to the Bravais lattice sites.