The absence of intraband scattering in a consistent theory of Gilbert damping in metallic ferromagnets

TL;DR

This paper clarifies the conditions under which intraband scattering contributes to Gilbert damping in metallic ferromagnets, showing it is absent beyond second order in spin-orbit coupling and emphasizing the importance of Coulomb interactions.

Contribution

It provides an analytical resolution to the inconsistency in damping calculations, demonstrating the limits of Kambersky's formula and highlighting the role of Coulomb interactions.

Findings

Intraband scattering terms are absent beyond second order in spin-orbit coupling.

Kambersky's formula is valid only to second order in the spin-orbit parameter.

Including Coulomb interactions is crucial for accurate damping calculations at large spin-orbit coupling.

Abstract

Damping of magnetization dynamics in a ferromagnetic metal is usually characterized by the Gilbert parameter alpha. Recent calculations of this quantity, using a formula due to Kambersky, find that it is infinite for a perfect crystal owing to an intraband scattering term which is of third order in the spin-orbit parameter xi This surprising result conflicts with recent work by Costa and Muniz who study damping numerically by direct calculation of the dynamical transverse spin susceptibility in the presence of spin-orbit coupling. We resolve this inconsistency by following the Costa-Muniz approach for a slightly simplified model where it is possible to calculate alpha analytically. We show that to second order in the spin-orbit parameter xi one retrieves the Kambersky result for alpha, but to higher order one does not obtain any divergent intraband terms. The present work goes beyond that of Costa and Muniz by pointing out the necessity of including the effect of long-range Coulomb interaction in calculating damping for large xi. A direct derivation of the Kambersky formula is given which shows clearly the restriction of its validity to second order in xi so that no intraband scattering terms appear. This restriction has an important effect on the damping over a substantial range of impurity content and temperature. The experimental situation is discussed.