Inverse Problems in Disordered Systems

Abstract

It is a simple Quantum Mechanics problem to find the electronic conductance of a device by directly solving the Schrodinger Equation. Obviously, this is the case if and only if the underlying Hamiltonian is known, i.e., if all scattering sources are fully specified. Trying to perform the same task in reverse is significantly more challenging. For example, assuming that the device conductance is known, how can one infer about the Hamiltonian components from that information alone? To make matters worse, what if the device is made of a heavily disordered material? Questions of this type are generally labelled as Inverse Problems (IP) and are classified as those which attempt to obtain from a set of observations the causal factors that generated them in the first place. IP are integral parts of classical visualization tools but far less common in the quantum realm. Materials Science is ideal for applications of inverse problem since it involves studying structures for which the underlying Hamiltonians are rarely known. Identifying the precise Hamiltonian that generates a specific observable is a difficult process. In general, it consists of solving the Schrodinger equation with a Hamiltonian containing one (or more) parameter(s) that must be changed until the solution closely matches the original observation. Because of the gigantic number of possibilities, finding the exact configuration is computationally very demanding. Efficient codes and powerful computers alone are not sufficient to make this approach feasible and alternative ways of probing the phase space of possibilities are needed. In this thesis, I introduce a simple mathematically-transparent inversion technique capable of extracting structural and compositional information from a disordered quantum device and nanowire/nanosheet networks by looking at their electronic signatures. We put forward an efficient way that can quantify the overall concentration of scatterers in a device. In addition, a sudoku style technique is also presented which enables us not only to specify the total number of scatterers but also to determine how they are spatially distributed.