Spatially inhomogeneous states of charge carriers in graphene

We study an interaction of 2D quasiparticles with linear dispersion E = ±u|p| (graphene) with impurity potentials. It is shown that in 1D potential well (quantum wire) there are discrete levels, corresponding to localized states, whereas in 2D well (quantum dot) there are no such states. Scattering cross-section of electrons (holes) of graphene by an axially symmetric potential well is found and it is shown that for infinetily large energy of incoming particles the cross-section tends to a constant. The effective Hamiltonian for a curved quantum wire of graphene is derived and it is shown that the corresponding geometric potential cannot form 1D bound states.