The phenomenological theory of magnetization relaxation (Review Article)

Abstract

The review is devoted to systematic description of results on relaxation of magnetization in magnetically ordered crystals previously obtained by the authors. The ideas of the phenomenological theory of magnetism formulated by Landau and Lifshitz are analyzed. A general method of constructing the dissipation function for both magnetically ordered systems and paramagnets is described. In the case of magnetically ordered systems the dissipation of exchange and relativistic nature is considered. It is found that for constructing the dissipation function it is necessary to take into account not only the symmetry of a crystal, but also the laws of conservation of magnetization. It is shown that in the case of a ferromagnet, the ground state is characterized by a continuous degeneracy parameter; the Landau-Lifshitz relaxation term gives qualitatively incorrect results (abnormally large attenuation of spin waves). According to the proposed method the spectra of spin waves and their attenuation were calculated and analyzed for ferromagnets with uniaxial, tetragonal and cubic symmetry as well as for two-sublattice uniaxial ferrites. It was found that the relaxation of the magnetization vector has a two-step character in ferromagnets and a multistep character in ferrites. In ferrites, the fastest process is the relaxation of length of the antiferromagnetic vector. It is shown that this relaxation is caused by the exchange interaction between sublattices of a ferrite and is enhanced by the exchange interactions within the sublattices. The relaxation of the total magnetization of a ferrite is much slower and, as in the case of a simple ferromagnet, is described by non-uniform exchange interactions and relativistic interactions. The results obtained are in a good agreement with recent experimental data.

Relativistic theory of magnetic inertia in ultrafast spin dynamics

“…A different kind of spin relaxation due to the exchange field has been introduced by Bar'yakhtar et al [19]. In the Landau-LifshitzBar'yakhtar equation spin dissipations originate from the spatial dispersion of exchange effects through the second order space derivative of the effective field [20,21]. A further recent work predicts the existence of extension terms that contain spatial as well temporal derivatives of the local magnetization [22].…”

Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz-Gilbert equation of spin dynamics

“…Brataas et al proposed a scattering theory formulation [15] of the Gilbert damping which is equivalent to a Kubo linear-response formulation. A different form of the relaxation term caused by spatial dispersion of the exchange interaction-this in contrast to the isotropic medium assumption made in the LL equation-was proposed by Bar'yakhtar and co-workers [10,20,21].…”

Graded-index magnonics

“…Within the second direction, researchers will face the challenge of developing a completely new theoretical formalism that will fully account for the rich and exciting complexities inherent to the Landau-Lifshitz equation. In addition to the already discussed anisotropic dispersion of magnetostatic and dipole-exchange spin waves, the challenges include the non-linearity of the Landau-Lifshitz equation [75,76] and exotic contributions to the magnetic energy, such as the magneto-elastic coupling [77,78] and (nonreciprocal) Dzyaloshinskii-Moriya [51,52] interaction, and magnetic dissipative function [79,80].…”