Quasiclassical theory for the superconducting proximity effect in Dirac materials

TL;DR

This paper develops a quasiclassical theoretical framework for analyzing the superconducting proximity effect in Dirac materials with spin-momentum locking, providing new insights into the influence of exchange fields and transport direction.

Contribution

It derives the first-order quasiclassical equations for Dirac materials and applies them to study the proximity effect, including phase-shifts and conductance in S/N and S/F structures.

Findings

Exchange field effects depend on its orientation relative to transport direction.

Normal components of exchange field attenuate Cooper pairs, while parallel components induce phase shifts.

Conductance varies with region length and exchange field strength.

Abstract

We derive the quasiclassical non-equilibrium Eilenberger and Usadel equations to first order in quantities small compared to the Fermi energy, valid for Dirac edge and surface electrons with spin-momentum locking, as relevant for topological insulators. We discuss in detail several of the key technical points and assumptions of the derivation, and provide a Riccati-parametrization of the equations. Solving first the equilibrium equations for S/N and S/F bilayers and Josephson junctions, we study the superconducting proximity effect in Dirac materials. Similarly to related works, we find that the effect of an exchange field depends strongly on the direction of the field. Only components normal to the transport direction lead to attenuation of the Cooper pair wavefunction inside the F. Fields parallel to the transport direction lead to phase-shifts in the dependence on the superconducting phase difference for both the charge current and density of states in an S/F/S-junction. Moreover, we compute the differential conductance in S/N and S/F bilayers with an applied voltage bias, and determine the dependence on the length of the N and F regions and the exchange field.