# The quasiclassical theory for interfaces with spin--orbit coupling

## TL;DR

This paper develops a new quasiclassical boundary condition framework that explicitly includes spin--orbit coupling, enabling the analysis of superconducting hybrid structures with heavy metal layers and ferromagnets, revealing long-range triplet superconductivity.

## Contribution

It introduces a novel set of boundary conditions for quasiclassical theory that incorporate spin--orbit coupling, advancing the understanding of superconducting interfaces in spintronics.

## Key findings

- Derived boundary conditions accounting for spin--orbit coupling.
- Demonstrated emergence of long-range triplet superconductivity.
- Applied theory to superconductor-ferromagnet systems.

## Abstract

In recent years, substantial progress has been made regarding the application of quasiclassical theory on superconducting hybrid structures. This theoretical framework is reliant on a proper set of boundary conditions in order to describe multilayered systems. With the advent of the field of superconducting spintronics, systems which combine heavy metal layers, in which there is large spin--orbit coupling, with ferromagnets have received a great deal of attention, due to their potential for generating long range triplet superconductivity. In contrast to interfaces of strongly spin polarized materials, which are well understood, a quasiclassical theory for interfaces in systems where there is significant spin--orbit coupling does not yet exist. After reviewing the quasiclassical theory for interfaces, we here solve this problem by deriving a new set of boundary conditions which take spin--orbit coupling explicitly into account. We then go on to apply these boundary conditions to a superconductor-ferromagnet (SF) bilayer and an SFS Josephson weak link, demonstrating the emergence of long range triplet superconductivity in these systems.