Mathematical Modelling of Charge Transport in Nanoscale Structures

This thesis presents an original generalization of the single-state Schrödinger equation for the description of charge transport in semiconductors operating in the ballistic regime, based on a manipulation of the single-electron dispersion relation. Specifically, a Taylor expansion of the Kane dispersion relation is performed, leading to additional terms in the Hamiltonian operator. Truncating this expansion at first order with respect to the non-parabolicity factor α yields a fourth-order equation, at second order a sixth-order equation, and so on, thus generating a hierarchy of increasingly higher-order models. Focusing on the fourth-order one, the problem is formulated on a finite spatial domain and transparent boundary conditions are derived to simulate transport in a quantum coupler, where the active region represents an electronic device and the contacts act as charge reservoirs. The well-posedness of the problem is studied, a generalized expression for the current is derived, and a novel numerical scheme is introduced for solving the fourth-order Schrödinger equation. Numerical simulations reveal interference effects arising from the richer wave structure generated by the higher-order terms. A further original contribution is the extension of the formulation of both the boundary conditions and the current to arbitrary orders in the hierarchy. In particular, a comparative analysis is carried out between the current computed using second- and fourth-order models, as well as between the probability density and current density obtained from the full Kane dispersion and from the parabolic band approximation. The numerical analysis highlights significant challenges related to computational efficiency, which are addressed by adapting the WKB method to the fourth-order Schrödinger equation. Comparisons with traditional finite-difference schemes are presented, leading to the introduction of an efficient numerical approach for solving the equation which was not previously investigated. Beyond this main line of research, a novel study is presented on a macroscopic transport model derived from the Wigner–Boltzmann equation for GaAs, which also incorporates the Kane dispersion relation. The thesis further includes three additional original contributions: a perturbative stochastic model based on Random Ordinary Differential Equations (RODEs) for extracting more informative inferences from diabetic patient data; a theoretical investigation of the relationship between the Hilbert space of random variables and the n-dimensional case space; and research results developed within the framework of the European project AMBEATion (“Analog/Mixed Signal Back End Design Automation based on Machine Learning and Artificial Intelligence Techniques”, H2020-MSCARISE-2020). All the works presented in this thesis are entirely original and provide independent and innovative contributions to their respective fields.