# Nonuniform superconductivity and Josephson effect in conical ferromagnet

## TL;DR

This paper provides an exact theoretical analysis of superconductivity in a one-dimensional conical ferromagnet, revealing a persistent nonuniform phase, a current-carrying uniform state, and the realization of an anomalous Josephson junction with a spontaneous phase difference.

## Contribution

It offers an exact solution for superconductivity in a conical ferromagnet, demonstrating the nonuniform phase, current-carrying uniform state, and realization of an anomalous Josephson junction without approximations.

## Key findings

- Superconducting transition always occurs into a nonuniform phase.
- Uniform superconducting state carries a current and is not the ground state.
- Realization of an anomalous $$ Josephson junction with a spontaneous phase difference.

## Abstract

Using the Gorkov equations, we provide an exact solution for a one-dimensional model of superconductivity in the presence of a conical helicoidal exchange field. Due to the special type of symmetry of the system, the superconducting transition always occurs into a nonuniform superconducting phase (in contrast with the Fulde-Ferrell-Larkin-Ovchinnikov state, which appears only at low temperatures). We directly demonstrate that the uniform superconducting state in our model carries a current and thus does not correspond to the ground state. We study in the framework of the Bogoliubov-de Gennes approach the properties of the Josephson junction with a conical ferromagnet as a weak link. In our numerical calculations, we do not use any approximations (such as, e.g., a quasiclassical approach), and we show a realization of an anomalous $\phi_{0}$ junction (with a spontaneous phase difference $\phi_{0}$ in the ground state). The spontaneous phase difference $\phi_{0}$ strongly increases at high values of the exchange field near the borderline with a half-metal, and it exists also in the half-metal regime.