This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.

We demonstrate that the electronic gap of a graphene bilayer can be controlled externally by applying a gate bias. From the magnetotransport data (Shubnikov–de Haas measurements of the cyclotron mass), and using a tight-binding model, we extract the value of the gap as a function of the electronic density. We show that the gap can be changed from zero to midinfrared energies by using fields of $\lesssim 1\mathrm{V}/\mathrm{nm}$, below the electric breakdown of ${\mathrm{SiO}}_2$. The opening of a gap is clearly seen in the quantum Hall regime.

Straining Graphene's Electronic States: The conduction electrons in graphene, single sheets of graphite, can have very high mobilities. Under the influence of an applied magnetic field, a series of energy steps, or Landau levels, can be observed that correspond to the conduction electrons traveling in cyclotron orbits. Recent theoretical work has indicated that if graphene layers are strained, the strain field creates a pseudomagnetic field that should also lead to observable Landau levels. Levy et al. (p. 544 ) used scanning tunneling microscopy to probe the energy levels of graphene grown on a platinum surface, which forms highly strained “nanobubbles.” The strain is equivalent to applying very high magnetic fields (in excess of 300 tesla). Thus, the electronic properties of graphene can indeed be modified using applied strain.

Two-dimensional carbon, or graphene, is a semimetal that presents unusual low-energy electronic excitations described in terms of Dirac fermions. We analyze in a self-consistent way the effects of localized (impurities or vacancies) and extended (edges or grain boundaries) defects on the electronic and transport properties of graphene. On the one hand, point defects induce a finite elastic lifetime at low energies with the enhancement of the electronic density of states close to the Fermi level. Localized disorder leads to a universal, disorder independent, electrical conductivity at low temperatures, of the order of the quantum of conductance. The static conductivity increases with temperature and shows oscillations in the presence of a magnetic field. The graphene magnetic susceptibility is temperature dependent (unlike an ordinary metal) and also increases with the amount of defects. Optical transport properties are also calculated in detail. On the other hand, extended defects induce localized states near the Fermi level. In the absence of electron-hole symmetry, these states lead to a transfer of charge between the defects and the bulk, the phenomenon we call self-doping. The role of electron-electron interactions in controlling self-doping is also analyzed. We also discuss the integer and fractional quantum Hall effect in graphene, the role played by the edge states induced by a magnetic field, and their relation to the almost field independent surface states induced at boundaries. The possibility of magnetism in graphene, in the presence of short-range electron-electron interactions and disorder is also analyzed.