We present a theoretical treatment of the quantum mechanical properties of electron transport, focusing on the mathematical and numerical developments that enable calculation of the electronic structure and transport characteristics in nanoscaled solid-state devices, in particular devices that exploit quantum effects to establish energy filtering effects.

First, we model band-to-band tunneling in heterostructure semiconductors. We develop a mathematically consistent heterostructure envelope function formalism, using only existing bulk k.p parameters. Invoking the quantum-transmitting boundary method (QTBM) we implement this formalism into a numerical solver that calculates the transmission coefficients in Zener diodes. Unlike for semi-classical models, we observe significant reflections at the material interface of broken-gap heterostructures, although no effective bandgap is present. Moreover, our heterostructure framework can also be tied up to model and optimize tunneling field-effect transistors.

Next, we develop an atomistic empirical pseudopotential solver to determine the electronic structure of large structures, using a hybrid approach that switches between reciprocal and real space for optimal computational efficiency. Combined with Bardeen's transfer Hamiltonian method, we calculate the electric current flowing between two crossed graphene nanoribbons, which exhibits resonance-like and plateau-like regions originating from the one-dimensional density-of-states of the individual ribbons.

Finally, we turn to the phase space description of quantum mechanics, using the time dependent Wigner distribution function that obeys the Wigner-Liouville equation. To alleviate the difficulties inherent to numerical manipulation of the latter, we recast it in terms of the spectrally decomposed force field. We show that this new form, noticeably resembling the Boltzmann equation, greatly facilitates physical interpretation. Based on these improvements, we develop a one-dimensional time marching solver, incorporating self-consistent coupling to Poisson's equation and adopting the relaxation time approximation. Calculating both the transient and the steady-state regimes of a resonant tunneling diode, we compare the steady-state Wigner functions, without any preliminary assumptions regarding the non-equilibrium quantum state, to those of the ballistic QTBM approximation, showing that the latter is only adequate when scattering can be neglected and no bound states are present. Eventually, we derive an envelope function expansion of the Wigner function to model multi-band phenomena.