• disorder
• graphene
• many-body effects

Graphene is a two-dimensional (2D) system of Carbon atoms packed in a honeycomb lattice. The recent isolation of this one-atom thick crystal has attracted considerable attention, both for the new physics which it exhibits and because it may pave the way for carbon-based electronics. In the absence of doping, graphene is a gapless semiconductor in which the conduction and valence bands touch at two inequivalent points, called Dirac points, at the corners of the hexagonal Brillouin zone. Near either of these points the energy bands are conical and the electronic states are described by a massless Dirac equation, in which the spin degree-of-freedom is replaced by a sublattice degree-of-freedom, referred to in the literature as \emph{pseudospin}. Most importantly, this implies that the eigenstates are endowed with a definite chirality, i.e. a well defined projection of the pseudospin along the momentum direction.

Recent experiments have demonstrated that close to charge neutrality the system is highly inhomogeneous and breaks up into electron and hole puddles. One key challenge for graphene research has been the identification of the main source of scattering which induces these density modulations and limits the mobility of current samples. Even though the problem is not yet fully understood, it has been recognized that the approximately linear dependence of conductivity on carrier density suggests that charged impurities trapped close to the graphene sheet could play a very important role in limiting graphene's mobility, partly obscuring the intrinsic properties of graphene's massless Dirac fermions (MDFs).

In this Thesis we discuss two different issues related to disorder in current graphene samples. On the one hand, we propose to engineer artificially systems of MDFs in standard parabolic-band 2D electron gases by employing suitable periodic external potentials. This route to create "artificial graphene" offers potentially unprecedented opportunities to study fundamental interactions of MDFs in high-mobility semiconductor structures.
On the other hand, we provide a microscopic study (accounting for electron-electron interactions) of the impact of charged impurities on the density profiles and local density-of-states of ordinary exfoliated graphene sheets deposited on substrates such as Silicon dioxide.

In the first Chapter we review some fundamental aspects concerning the quantum theory of the electron liquid. Particular attention is given to linear response theory, Landau theory of normal Fermi liquids, and density functional theory.

Chapter 2 is devoted to the properties of graphene in the absence of disorder, i.e. to "homogeneous" graphene. We first review the electronic single-particle band structure of graphene and then comment on the impact of electron-electron interactions at the level of the random phase approximation. In the last section we present the main original result of this Chapter (Gibertini et al. arXiv:0904.4191v1 and Phys. Rev. B 79 (R), in press): we show that modulating a standard 2D electron gas with a long-wavelength external periodic potential with hexagonal symmetry can lead to the creation of isolated massless Dirac points with tunable Fermi velocity. We provide detailed theoretical estimates to realize such artificial graphene-like system and discuss an experimental realization in a modulation-doped GaAs quantum well.

In the last Chapter we consider the effects of disorder on standard exfoliated graphene samples on Silicon dioxide substrates. After a brief introduction to the main experimental results, we first review a Kohn-Sham-Dirac (KSD) density-functional-theory scheme [Polini et al., Phys. Rev. B 78, 115426 (2008)] that treats slowly-varying external potentials and electron-electron interactions on an equal footing. We then report on original continuum-model electronic structure calculations based on the KSD scheme for graphene sheets under the influence of local scatterers distributed in space in the same way as in the sample studied experimentally by Zhang et. al. (arXiv:0902.4793v1). We first verify the reliability of the spectroscopic method used by Zhang et. al. to measure density and then assess what can be learned about the nature of these scatterers based on the detailed comparison between theory and experiment (Gibertini et. al., in preparation).