Majorana zero modes in smooth 1d junctions and cylindrical nanowires

A numerical method to obtain the spectrum of smooth normal-superconductor junctions in
nanowires, able to host Majorana zero modes, is presented (Chapter 2). Softness in the potential and superconductor interfaces yields opposite effects on the protection of Majorana modes. While a soft potential is a hindrance for protection, a soft superconductor gap transition greatly favors it. Our method also points out the possibility of extended Majorana states when propagating modes are active far from the junction, although this requires equal incident fluxes in all open channels.
The one dimensional infinite model is very useful in order to study in a simple and qualitative way the behavior of Majorana zero modes in thin and long quantum nanowires. Nevertheless, a more realistic approach (Chapter 3) should pay attention to the role of the transverse additional dimensions as well as the finite size of the nanowire.