Spatially inhomogeneous states of charge carriers in graphene

The interaction of two-dimensional quasiparticles characterized by a linear dispersion E = ±u|p| (graphene) with impurity potentials is studied. It is shown that discrete levels corresponding to localized states are present in a one-dimensional potential well (quantum wire), whereas such states are absent in a two-dimensional well (quantum dot). The cross section for the scattering of electrons (holes) of graphene by an axially symmetric potential well is determined. It is shown that the cross section tends to a constant value in the limit of infinite particle energy. The effective Hamiltonian is derived for a curved quantum wire of graphene.