In this thesis, four different numerical techniques are implemented for the purpose of simulating resonant tunnelling diodes (RTDs). The chosen methods were: piecewise constant transfer matrix (TMM-C), piecewise linear transfer matrix (TMM-L), quantum transmitting boundary method (QTBM), and the Crank-Nicolson method (CN). The numerical methods converged compared with the known analytic simulation for plane waves tunnelling through a single barrier. To better represent the semiconductor-based RTDs, the effective mass approximation was adopted with accompanying modifications to the Hamiltonian operators to ensure the continuity of the wave function and its derivative. Using the Tsu-Esaki formula, the current density was calculated as a function of bias voltage for two different RTD devices. The numerically obtained current density was of the same order of magnitude as referenced experimental values but differed significantly enough to require better models if engineering applications are decided. The models in this thesis were able to display resonant tunnelling and a negative differential resistance (NDR), giving them plausible educational value.

Experimental studies presented in this thesis have shown the first realisation of resonant tunnelling transport through two impurities in a vertical double-barrier tunnelling diode; have proved the chiral nature of charge carriers in graphene by studying ballistic transport through graphene $p$-$n$ junctions; have demonstrated significant differences of $1/f$ noise in graphene compared with conventional two-dimensional systems. Magnetic field parallel to the current has been used to investigate resonant tunnelling through a double impurity in a vertical double-barrier resonant tunnelling diode, by measuring the current-voltage and differential conductance-voltage characteristics of the structure. It is shown that such experiments allow one to obtain the energy levels, the effective electron mass and spatial positions of the impurities. The chiral nature of the carriers in graphene has been demonstrated by comparing measurements of the conductance of a graphene $p$-$n$-$p$ structure with the predictions of diffusive models. This allowed us to find, unambiguously, the contribution of ballistic resistance of graphene $p$-$n$ junctions to the total resistance of the $p$-$n$-$p$ structure. In order to do this, the band profile of the $p$-$n$-$p$ structure has been calculated using the realistic density of states in graphene. It has been shown that the developed models of diffusive transport can be applied to explain the main features of the magnetoresistance of $p$-$n$-$p$ structures. It was shown that $1/f$ noise in graphene has much more complicated concentration and temperature dependences near the Dirac point than in usual metallic systems, possibly due to the existence of the electron-hole puddles in the electro-neutrality region. In the regions of high carrier concentration where no inhomogeneity is expected, the noise has an inverse square root dependence on the concentration, which is also in contradiction with the Hooge relation.