Thesis presented July 07, 2023

Abstract:
Since the discovery of the quantum Hall (QH) effect, physicists have realized that the distinction between an insulator and a conductor is not as simple as the band theory of solids would suggest. Indeed, a normal metal in the QH regime becomes insulating in the bulk and exhibits conducting edge states. Such a system with an insulating bulk and a conducting boundary is called a topological insulator, which is the origin of the modern research field of the topological phases of matter. In this thesis, we study the consequences of inducing superconducting correlations in different types of boundary modes through three projects. We first consider a QH region in contact with a superconductor (SC), i.e., a QH-SC junction. Due to successive Andreev reflections, the QH-SC interface hosts hybridized electron and hole edge states called chiral Andreev edge states (CAES). We theoretically study the energy spectrum and the transport properties of these CAES by using microscopic, tight-binding and effective approaches. Interestingly, we find that their transport properties strongly depend on the contact geometry and the value of the filling factor.
The second project is an extension of the first one in which we study the coupling between counter- propagating pairs of CAES in QH-SC-QH junctions. The presence of the second QH region allows for the non-local scattering processes of elastic co-tunneling and crossed Andreev reflection while normal and Andreev reflections are still allowed. We study the energy spectrum of the counter-propagating pairs of CAES by using a two-dimensional microscopic model and we develop a one-dimensional effective model to investigate the transport properties the junction.
​ In the last project, we consider the helical modes of a higher order topological insulator (HOTI). A HOTI generalizes the concept of topological insulator so that the boundary modes appear at corners or hinges. Here we investigate the effects of Zeeman and superconducting couplings on helical hinge modes. The Zeeman coupling spatially splits the helical pair into two chiral states enclosing a quantum anomalous Hall region, while the superconducting coupling divides the helical modes into two helical Majorana modes. The combination of both Zeeman and superconducting couplings leads to different splitting scenarios depending on the ratio between the two couplings. We derive the corresponding wave functions and analyze the different splitting scenarios by performing tight-binding simulations.
Keywords:
Quantum Hall Effect, Superconductivity, Andreev Reflection, Boundary Modes, Topological Insulators, Majorana Modes

On-line thesis.