Wavenumber-dependent Gilbert damping in metallic ferromagnets

TL;DR

This paper demonstrates the existence of a wavenumber-dependent Gilbert damping term in metallic ferromagnets, revealing a new imaginary effective field related to spin transport, with results consistent with spin pumping theories.

Contribution

It reports the first experimental evidence of a $k^2$ dependent damping term in metallic ferromagnets, linking it to spin pumping and transverse spin relaxation lengths.

Findings

Presence of a $k^2$ damping term in NiFe, Co, and CoFeB.

Magnitude of the $k^2$ term is 0.07-0.1 nm$^2$.

Results align with transverse spin relaxation lengths from spin pumping.

Abstract

New terms to the dynamical equation of magnetization motion, associated with spin transport, have been reported over the past several years. Each newly identified term is thought to possess both a real and an imaginary effective field leading to fieldlike and dampinglike torques on magnetization. Here we show that three metallic ferromagnets possess an imaginary effective-field term which mirrors the well-known real effective-field term associated with exchange in spin waves. Using perpendicular standing spin wave resonance between 2-26 GHz, we evaluate the magnitude of the finite-wavenumber ($k$) dependent Gilbert damping $\alpha$ in three typical device ferromagnets, Ni$_{79}$Fe$_{21}$, Co, and Co$_{40}$Fe$_{40}$B$_{20}$, and demonstrate for the first time the presence of a $k^2$ term as $\Delta\alpha=\Delta\alpha_0+A_{k}\cdot k^2$ in all three metals. We interpret the new term as the continuum analog of spin pumping, predicted recently, and show that its magnitude, $A_{k}$=0.07-0.1 nm$^2$, is consistent with transverse spin relaxation lengths as measured by conventional (interlayer) spin pumping.