# Triplet superconductivity in ferromagnets due to magnon exchange

## TL;DR

This paper explores how magnon exchange in ferromagnets can induce triplet superconductivity, revealing critical parameters and temperature-dependent behavior that differ from conventional BCS theory.

## Contribution

It introduces a microscopic Eliashberg and RPA-based framework to describe triplet pairing mediated by spin waves in ferromagnets, highlighting the role of magnon exchange.

## Key findings

- Superconductivity occurs only if the electron-spin wave coupling exceeds a critical value.
- Pairing exists within a specific temperature window, vanishing outside it.
- The results differ from traditional BCS theory, emphasizing dynamic magnon effects.

## Abstract

We consider the superconducting pairing induced by spin waves exchange in a ferromagnet with both conduction and localized electrons, the latter being described as spins. We use the microscopic Eliashberg theory to describe the pairing of conducting electrons and the RPA approach to treat the localized spins assuming an exchange coupling between the conducting electrons and spins. In the framework of non relativistic Hamiltonian twe found that he spin wave exchange results in equal spin electron pairing described by the two components of the order parameter, $\Delta^{\uparrow}$ (both spins up) and $\Delta^{\downarrow}$ (both spins down). Due to the conservation of total spin projection on the axis of the spontaneous ferromagnetic moment, the spin wave exchange at low temperatures includes an emission of magnons and an absorption of thermal magnons by the conduction electrons. The absorption and emission processes depend differently on the temperature, with the absorption being progressively suppressed as the temperature drops. As a result, the superconducting pairing exists only if the electron-spin wave exchange parameter $g$ exceeds some critical value $g_c$. At $g>g_c$ pairing vanishes if the temperature drops below the lowest point $T_{cl}$ or increases above the upper critical point $T_{ch} \approx T_m$ (the Curie temperature) where the spin waves cease to exist. This behavior inherent to the spin carrying glue is in an obvious disagreement with the results of conventional BCS approach which assumes that the effective electron-electron attraction is simply proportional to the static magnetic susceptibility.