Transport theory and finite element methods for mesoscopic superconducting devices

Doctoral thesis, 2022

At low temperatures, electrons in a superconductor exhibit pairing correlations that result in a macroscopic, phase-coherent ground state. This leads to peculiar electromagnetic properties such as the flow of dissipationless charge currents and expelling of external magnetic fields.
This thesis investigates superconductors that are brought out of equilibrium through injection of charge and heat from normal-metal reservoirs. In particular for unconventional superconductors, where the pairing correlations have a non-trivial orbital symmetry, the resulting non-equilibrium is thus far only partially explored. A better understanding is desirable both from a fundamental point of view as well as for applications in superconducting devices.
As a step in this direction, we study transport in mesoscopic superconducting hybrid structures with arbitrary mean free path using the quasiclassical theory of superconductivity. In order for fundamental conservation laws to be satisfied, a description of the non-equilibrium state requires a fully self-consistent solution of the underlying equations. We present strategies on how such a self-consistent solution can be achieved. Using these techniques, we investigate the non-linear steady-state response of both conventional and unconventional superconductors to an external voltage- or temperature-bias. Specifically, we study charge transport in a conventional s-wave and an unconventional d-wave superconductor under voltage bias, the thermoelectric effect due to elastic impurity scattering in both systems, and the influence of spectral rearrangements on a suggested sub-dominant s-wave order-parameter component in d-wave superconductors. Lastly, we introduce a finite element method for the quasiclassical theory. It can be used to study transport in two or more dimensions where geometric effects such as current focussing and dilution can occur. We present exemplary results based on this method for equilibrium transport in two dimensions.