Comparative study of methodologies to compute the intrinsic Gilbert damping: interrelations, validity and physical consequences

TL;DR

This paper compares various methods for calculating intrinsic Gilbert damping in ferromagnets, analyzing their validity, interrelations, and physical implications, especially in different material dimensions and conditions.

Contribution

It provides a unified theoretical framework to compare damping calculation methods, clarifies their assumptions, and highlights their applicability and limitations across different magnetic systems.

Findings

Most methods agree for bulk Fe, Co, Ni due to weak spin-orbit coupling.

Damping from electronic broadening is additive and can be isolated.

Damping diverges in clean bulk magnets but remains finite in monolayers.

Abstract

Relaxation effects are of primary importance in the description of magnetic excitations, leading to a myriad of methods addressing the phenomenological damping parameters. In this work, we consider several well-established forms of calculating the intrinsic Gilbert damping within a unified theoretical framework, mapping out their connections and the approximations required to derive each formula. This scheme enables a direct comparison of the different methods on the same footing and a consistent evaluation of their range of validity. Most methods lead to very similar results for the bulk ferromagnets Fe, Co and Ni, due to the low spin-orbit interaction strength and the absence of the spin pumping mechanism. The effects of inhomogeneities, temperature and other sources of finite electronic lifetime are often accounted for by an empirical broadening of the electronic energy levels. We show that the contribution to the damping introduced by this broadening is additive, and so can be extracted by comparing the results of the calculations performed with and without spin-orbit interaction. Starting from simulated ferromagnetic resonance spectra based on the underlying electronic structure, we unambiguously demonstrate that the damping parameter obtained within the constant broadening approximation diverges for three-dimensional bulk magnets in the clean limit, while it remains finite for monolayers. Our work puts into perspective the several methods available to describe and compute the Gilbert damping, building a solid foundation for future investigations of magnetic relaxation effects in any kind of material.