Quasiclassical calculation of spontaneous current in restricted geometries

TL;DR

This paper develops a modified quasiclassical approach using the Schophol-Maki transformation for calculating spontaneous currents and magnetic moments in finite-sized inhomogeneous superconductors, relevant to unconventional superconductivity and quantum computing.

Contribution

It introduces a stable numerical method for 2D quasiclassical calculations in finite geometries, enabling analysis of spontaneous currents in complex superconducting structures.

Findings

Demonstrates the method on d-wave superconductor islands with subdominant order parameters.

Analyzes grain boundary junctions between arbitrarily oriented d-wave superconductors.

Reveals conditions for time-reversal symmetry breaking in these systems.

Abstract

Calculation of current and order parameter distribution in inhomogeneous superconductors is often based on a self-consistent solution of Eilenberger equations for quasiclassical Green's functions. Compared to the original Gorkov equations, the problem is much simplified due to the fact that the values of Green's functions at a given point are connected to the bulk ones at infinity (boundary values) by ``dragging'' along the classical trajectories of quasiparticles. In finite size systems, where classical trajectories undergo multiple reflections from surfaces and interfaces, the usefulness of the approach is no longer obvious, since there is no simple criterion to determine what boundary value a trajectory corresponds to, and whether it reaches infinity at all. Here, we demonstrate the modification of the approach based on the Schophol-Maki transformation, which provides the basis for stable numerical calculations in 2D. We apply it to two examples: generation of spontaneous currents and magnetic moments in isolated islands of d-wave superconductor with subdominant order-parameters s and d_{xy}, and in a grain boundary junction between two arbitrarily oriented d-wave superconductors. Both examples are relevant to the discussion of time-reversal symmetry breaking in unconventional superconductors, as well as for application in quantum computing.