• integer quantum Hall effect
• graphene
• envelope functions
• shot noise

This thesis presents the results obtained in the field of the numerical simulation of conduction in low-dimensional carrier systems on different materials.
Motivated by the significant interest aroused by the potential of graphene for the development of an alternative technology with margin for improvement larger than the current CMOS, we have taken into account some important issues related to transport in monolayer and bilayer graphene.
In the case of monolayer graphene, an analysis of transport in large area samples has been carried out within the envelope function approximation. An efficient numerical method has been developed for the simulation of conduction in the presence of a quite general electrostatic potential. The method provides an alternative to the atomistic approaches, that in the case of the considered do-
main would require an enormous computational burden. Some peculiarities of transport in monolayer graphene have been studied in detail, by inscribing them into the framework of the recent theory of non Hermitian Hamiltonians and the of the spontaneous breaking of the PT symmetry.
The simulations of transport on bilayer graphene have been oriented to the explanation of scanning gate spectroscopy measurements performed in the presence of the integer quantum Hall effect. We started form a semiclassical model, initially proposed for ordinary two-dimensional systems, and we have developed a numerical method able to reproduce with good accuracy the experimental results and to explain them.
The last subject we deal with is the suppression of the shot noise power spectral density in mesoscopic semiconductor devices. We simulated transport in quantum wires defined in GaAs/AlGaAs hetherostructures in the presence of one-dimensional and two-dimensional disorder, investigating the possibility that a completely diffusive transport regime be established in real samples, and thus a 1/3 suppression of the shot noise power spectral density measured. The obtained results clearly indicate that it is unlikely to measure such a suppression and therefore contribute to explain the exisisting disagreement between theory and experiments.