Theoretical Considerations for Experiments to Create and Detect Localised Majorana Modes in Electronic Systems

This thesis presents our work on building a bridge between the theoretical proposals for the condensed matter realisation of peculiar localised excitations, known as Majorana modes, and experiments to search for them. The main focus in the first two sections is on charge sensing of localised Majorana modes in two distinct systems. First, we address the properties of charged quasiparticles in the $\nu=5/2$ fractional quantum Hall regime. In particular, we focus on the case where these particles are trapped by disorder, often in close proximity to one another. Next, we consider one-dimensional semi-conducting wires with strong spin-orbit coupling and proximity-induced superconductivity. The Majorana modes in this system are predicted to be charge-neutral. We show, however, that when the wire is short enough, there is a uniform charge distribution along the wire, and we show how the presence of this charge depends on system parameters. A third portion is related to HgTe quantum wells, another system predicted to host Majorana modes when coupled to a superconductor. Here we consider a HgTe well in the metallic regime, coupled to two superconducting strips. We compute the Josephson coupling in the presence of spin-orbit interactions and in-plane external magnetic fields.