Micromagnetic modeling of anisotropic damping in magnetic nanoelements

Abstract

We report a numerical implementation of the Landau-Lifshitz-Baryakhtar theory that dictates that the micromagnetic relaxation term obeys the symmetry of the magnetic crystal, i.e., replacing the single intrinsic damping constant with a tensor of corresponding symmetry. The effect of anisotropic relaxation is studied in a thin saturated ferromagnetic disk and an ellipse with and without uniaxial magnetocrystalline anisotropy. We investigate the angular dependence of the linewidth of magnonic resonances with respect to the given structure of the relaxation tensor. The simulations suggest that the anisotropy of the magnonic linewidth is determined by two factors: the projection of the relaxation tensor onto the plane of precession and the ellipticity of the latter. 4 These terms are now widely used for the description of magnetic relaxations in magnetic thin films 5,6 and patterned magnetic media. 7 The microscopic mechanism behind the magnetic losses in metals has also been suggested, e.g., in Gilmore et al. 8 However, recent experimental data urge the development of new micromagnetic approaches to the description of magnetic losses, i.e., by introducing higher-order terms within the Gilbert approach, 9 inert relaxation, 10 and by generalizing the magnetization dynamics and relaxation within the framework of Onsager's kinetic equations.11 The latter approach shows that the relaxation part of the equation of precession should obey the crystallographic symmetry of the media, thereby replacing the single intrinsic damping constant with a tensor. The reason behind anisotropic relaxation in magnetic media is the symmetry of the spin-orbit and s-d interaction, 12 which couples the spins to the other subsystems (degrees of freedom), i.e., the lattice and the free electrons, respectively. It is worth noting that an angular dependence of magnetic losses (or effects associated with the same physics) has already been reported experimentally.13-18 However, analytical and numerical approaches still have to be developed, especially for nanoscale structures where the magnetic resonances are strongly confined, and so, their spectra are discrete.In this paper, we report on a numerical implementation of Baryakhtar's theory 11 within the mumax2 micromagnetic framework.19 Furthermore, we systematically investigate the influence of anisotropic relaxation on the angular dependence of ferromagnetic resonance (FMR) linewidths in a nanoscale magnetic disk and an ellipse.We start from the general Baryakhtar equation (LLBar),where M, γ LL , H are the magnetization vector, the positively defined gyromagnetic ratio, and the effective internal field, respectively. The first term in the equation defines the torque, while the second and third describe the local and nonlocal 20 relaxations, respectively.λ(M) andλ (e) (M) are the relaxation tensors of relativistic and exchange nature, respectively, and in general are functions of the magnetization vector. These tensors are in fact operators that describe how crystallographic and magnetic sym...

The classical two-sublattice Landau–Lifshitz–Bloch equation for all temperatures

“…One should also note an alternative macroscopic approach suggested by V. Bar'yakhtar [15], based on the symmetry principle which was recently also used to model ultra-fast dynamics [16]. Also the M3TM model [17], proposed by Koopmans as well as the self-consistent Bloch (SCB) equation [18] were used for this purpose.…”

Gate-controlled magnon-assisted switching of magnetization in ferroelectric/ferromagnetic junctions

“…We start from the Landau-Lifshitz-Baryakhtar equation (LLBar) [37][38][39] for the magnetiza-tion dynamics at the FM interface,…”

Size dependence of the relaxation rate for non-equilibrium redistributions of the magnetization in Ni–Fe heterestuctures: Exchange vs. relativistic damping scenarios

“…The explanation [11] of this unusual dynamics in the Ni-Ru-Fe trilayer has been provided on the basis of the Landau-Lifshitz equation with the relaxation term proposed by Bar'yakhtar [20][21][22], called also LLBar equations [23]. LLBar equations were derived on the basis of general symmetry arguments and Onsager relations, and employ the concept of both the relativistic (local) and exchange (nonlocal) spin relaxations.…”