Theory of transport in nanoscale superconducting devices: charge imbalance and thermoelectric effects*

We present computational strategies for steady-state transport simulations of nanoscale superconductors coupled to normal-metal reservoirs. The Eilenberger-Larkin-Ovchinnikov nonequilibrium quasiclassical transport equations are solved self-consistently with self-energies to ensure charge-current conservation. Within the framework, different order parameter symmetries are studied for arbitrary impurity concentrations ranging from clean to dirty limits. In addition to the traditional quasi-1D treatment, we introduce a finite-element method and study real two-dimensional geometries.

For a voltage-biased s-wave nanowire coupled to two reservoirs [1], we find a critical voltage where superconductivity breaks down for currents below the critical current of the superconductor. We report the dependence of this critical voltage on mean free path and interface transparency. For unconventional d-wave superconductors, we find very long decay-lengths of charge imbalance and different nonequilibrium distributions compared to the s-wave case. In a temperature-biased setup, we find that d-wave superconductors exhibit a large thermopower for scattering phase-shifts between the Born and unitary limits.