Dissipation function of magnetic media

Abstract

A general method of constructing a dissipation function is developed for disordered magnetic media and for magnetically ordered systems. As an example it is shown for a ferromagnet that not only the invariance with respect to uniform rotations of the body but also the law of conservation of magnetization must be taken into account in order to construct a dissipation function. It is found that in ferromagnets the dissipation term in the equations of motion for the magnetization is a sum of Bloch and Landau-Lifshitz-Gilbert relaxation terms. The region of applicability of the relaxation term in the Landau-Lifshitz form is determined. The damping of spin waves in a ferromagnet with tetragonal symmetry is calculated. A procedure is formulated for transitioning from a ferromagnet with lower symmetry to a ferromagnet with a continuous degeneracy parameter. In this case the relaxation process can be systematically described by means of the dissipation function described in this article. It is shown how the relaxation term of a general form for ferromagnets becomes the Bloch relaxation term for paramagnets. It is shown that the magnetization vector relaxes in two stages. First the magnetic moment relaxes in magnitude quite rapidly as a result of exchange enhancement and then the magnetization relaxes slowly to its equilibrium direction. The second stage qualitatively corresponds to the relaxation picture described by the Landau-Lifshitz model.

Thermodynamically self-consistent non-stochastic micromagnetic model for the ferromagnetic state

“…In contrast to the Landau-Lifshitz and Landau-Lifshitz-Bloch models, the relativistic relaxation tensor obeys the crystallographic and magnetic symmetries of the system. So let us expand it into powers of M around the highest symmetry magnetic state M = 0 [19], i. e.…”

Micromagnetic modeling of anisotropic damping in magnetic nanoelements

“…Equation (3) is employed in the present study with an internal field combining the contributions from the Zeeman, magnetocrystalline, magnetodipolar, and exchange energies. The latter also includes a contribution from the phenomenological field that accounts for the exchange energy arising from changes of the magnetization vector length, 11,22,[26][27][28] i.e., the exchange energy that is stored in thermal magnons, and cannot be explicitly accounted for on the micromagnetic scale. This field pushes the magnetization vector length back to its equilibrium value at the given temperature.…”

“…22 However, where the anisotropy constant is varied, the corresponding components of the damping tensor are kept constant to draw a straight comparison between LL and LLBar models. A more rigorous treatment, where relations between the components of the relaxation tensor and magnetocrystalline anistoropy constants are taken into account, is expected to change our results quantitatively, but not qualitatively.…”

“…The magnetic parameters are close to that of cobalt, i.e., a saturation magnetization of M s = 1440 × 10 3 A/m, exchange stiffness constant of A ex = 2.1 × 10 −11 J/m, and uniaxial anisotropy strength of |K u | = 5.2 × 10 5 J/m 3 (varied in some of the simulations). Cobalt has a hexagonal lattice for which the relaxation tensor is diagonalα = αν, 11,22 whereν is a diagonal tensor which characterizes the relative degree of anisotropic damping. Cobalt is also characterized by a relatively strong magnetocrystalline anisotropy, making it an ideal candidate for our case study and for possible experiments.…”

“…The coefficients of the expansion take into account crystallographic symmetry, while the combinations of M represent symmetry invariants of the magnetic ground state. The expansion is taken in the vicinity of the absolutely symmetric state M = 0, 22 …”