# Generalized quasiclassical theory of the long-range proximity effect and spontaneous currents in superconducting heterostructures with strong ferromagnets

## TL;DR

This paper develops a generalized quasiclassical theory for long-range proximity effects and spontaneous currents in superconductor-strong ferromagnet heterostructures, highlighting the role of spin-dependent gauge fields from spin-orbit coupling or magnetic textures.

## Contribution

It introduces a unified framework incorporating spin-dependent gauge fields into quasiclassical equations, explaining long-range triplet correlations and spontaneous currents in strong ferromagnetic heterostructures.

## Key findings

- Derivation of Usadel equations for strong ferromagnets
- Identification of spin-dependent gauge fields as key to long-range effects
- Examples of Josephson systems with spontaneous currents

## Abstract

We present the generalized quasiclassical theory of the long-range superconducting proximity effect in heterostructures with strong ferromagnets, where the exchange splitting is of the order of Fermi energy. In the ferromagnet the propagation of spin-triplet Cooper pairs residing on the spin-split Fermi surfaces is shown to be governed by the spin-dependent Abelian gauge field which results either from the spin-orbital coupling or from the magnetic texture. The additional gauge field enters into the quasiclassical equations in superposition with the usual electromagnetic vector potential and results in the generation of spontaneous superconducting currents and phase shifts in various geometries which provide the sources of long-range spin-triplet correlations. We derive the Usadel equations and boundary conditions for the strong ferromagnet and consider several generic examples of the Josephson systems supporting spontaneous currents.