Semiclassical theory of a disordered two-dimensional SNS-junction
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- Institutt for fysikk 
Two-dimensional hybrid superconductor-semiconductor structures provide a promising platform for realising networks of Majorana fermions for fault-tolerant quantum computing, and, as such, are attracting a lot of attention.Recently, two-dimensional superconductor-normal material-superconductor (SNS) junctions created in such hybrid systems, exhibited dramatic asymmetries in their Fraunhofer-like patterns of the critical current as a function of applied magnetic field.It was proposed that, apart from spin-orbit interaction and Zeeman effects, disorder inside the junction can play an important role in the appearance of these asymmetries. This thesis investigates the role of disorder in two-dimensional SNS-junctions by developing a toy model in a semiclassical picture.This model assumes two distinct paths across the junction, connected by beamsplitters at the normal material-superconductor (NS) interfaces, enclosing a magnetic flux.By describing these paths as ballistic, one-dimensional nanowires, and using a scattering matrix approach to describe the beamsplitters, we develop a method for calculating the transmission and reflection coefficients of the junction as a whole.This allows us to control the coupling between the nanowires and the NS-interfaces, as well as to introduce an asymmetric probability injection into the two arms and to incorporate a difference in the two path lengths. The supercurrent through the junction is found from the energy of the Andreev bound states, which allows us to investigate the critical current as a function of the magnetic flux penetrating the surface enclosed by the two paths.We find that, in the absence of spin-dependent effects such as spin-orbit interaction and the Zeeman effect, none of the combinations of asymmetric probability injection, different path lengths, modifications of the chemical potential or NS coupling strength produces asymmetries in the critical current, such as observed in the experiment.We thus conclude that either our toy model is not sufficiently complex, e.g., one would need more than two interfering trajectories, or that disorder alone is not sufficient to produce asymmetric patterns of critical currents.Further research is required to in order to to determine which conclusion is correct.