Josephson effects in topological superconductors

In the last decades major progress in experimental solid state physics was achieved, which kindled increasing interest in research and possible applications. Topological superconductors present a platform to observe Majorana fermions and offer various interesting applications, one of them being quantum computers. We introduce the concept of topological superconductors, this includes superconductivity, topology and their creation. Additionally a formalism to analyze various junctions is introduced. We are going to use this formalism for various junctions and models for topological superconductors. We start with a junction of a topological and non topological superconductor, a quantum dot is put in between to allow a supercurrent through the junction. For the topological superconductor a model, that allows analytical insights, is used. Afterwards we use a model for the topological superconductor, that is closer to experiments, but needs numerical calculations. Before we check the changes to the junction with this model, we discuss some general properties of the model and analyze a simpler junction first. The next junction we discuss, has a different focus. Instead of the supercurrent we are going to focus on the states of the boundary of the wires, configurations with even and odd parity are connected through the topological superconductor and give us an interesting platform to study bound states. Afterwards we dive deeper into models closer to experiments for topological superconductors. We look at our approach to analyze the junctions and summarize interesting and useful properties. Using these properties we try to get some analytical understanding beyond the numerical calculations. We apply this method on different models and discuss some junctions including these models afterwards.