Quasiparticle scattering from vortices in d-wave superconductors. II. Berry phase contribution

TL;DR

This paper investigates the Berry phase's role in quasiparticle scattering in d-wave superconductors, focusing on a simplified model of a single branch cut between vortices, and calculates the resulting scattering cross section.

Contribution

It introduces a detailed analysis of Berry phase effects on quasiparticle scattering using elliptical coordinates and solves the Whittaker-Hill equations for this purpose.

Findings

Derived scattering cross sections for Berry phase effects

Solved the Dirac equation in elliptical coordinates for this problem

Compared Berry phase scattering with superflow effects qualitatively

Abstract

In the mixed state of a d-wave superconductor, Bogoliubov quasiparticles are scattered from magnetic vortices via a combination of two effects: Aharonov-Bohm scattering due to the Berry phase acquired by a quasiparticle upon circling a vortex, and effective potential scattering due to the superflow swirling about the vortices. In this paper, we consider the Berry phase contribution in the absence of superflow, which results in branch cuts between neighboring vortices across which the quasiparticle wave function changes sign. Here, the simplest problem that captures the physics is that of scattering from a single finite branch cut that stretches between two vortices. Elliptical coordinates are natural for this two-center problem, and we proceed by separating the massless Dirac equation in elliptical coordinates. The separated equations take the form of the Whittaker-Hill equations, which we solve to obtain radial and angular eigenfunctions. With these eigenfunctions in hand, we construct the scattering cross section via partial wave analysis. We discuss the scattering effect of Berry phase in the absence of superflow, having considered the superflow effect in the absence of Berry phase in a separate paper. We also provide qualitative comparison of transport cross sections for the Berry phase and the Superflow effects. The important issue of interference between the two effects is left to future work.