An introduction to the physics of graphene layers

In this chapter we revise the basic physics of a single layer and a double layer of graphene. In both cases, and starting from a tight-binding description, we show how to construct the effective continuum models. Also for the single layer and for the bilayer, both with zigzag edges, we discuss the existence of a zero energy band made of surface states localized near the edges of the sample. The spectrum of the single layer in the presence of a magnetic field is studied and its relation with the half-odd integer quantum Hall effect is discussed. For the bilayer the electronic bulk properties are studied and the unconventional quantum Hall effect is addressed. The concept of a biased bilayer is introduced together with its energy spectrum.