Phase dynamics of oscillating magnetizations coupled via spin pumping

TL;DR

This paper develops a theoretical framework to analyze the phase dynamics of coupled ferromagnetic magnetizations influenced by spin pumping, revealing synchronization behaviors depending on oscillation modes and current regions.

Contribution

It introduces a novel formalism combining magnetization equations with spin transport, providing new insights into phase synchronization phenomena in spintronic systems.

Findings

Spin pumping induces in-phase synchronization around the easy axis.

Out-of-plane oscillations show in-phase or antiphase synchronization depending on current.

Phase difference depends on oscillation direction, clockwise or counterclockwise.

Abstract

A theoretical formalism is developed to simultaneously solve equation of motion of the magnetizations in two ferromagnets and the spin-pumping induced spin transport equation. Based on the formalism, a coupled motion of the magnetizations in a self-oscillation state is studied. The spin pumping is found to induce an in-phase synchronization of the magnetizations for the oscillation around the easy axis. For an out-of-plane self-oscillation around the hard axis, on the other hand, the spin pumping leads to an in-phase synchronization in a small current region, whereas an antiphase synchronization is excited in a large current region. An analytical theory based on the phase equation reveals that the phase difference between the magnetizations in a steady state depends on the oscillation direction, clockwise or counterclockwise, of the magnetizations.