Applications of the gyrocoupling vector and dissipation dyadic in the dynamics of magnetic domains

Abstract

This paper extends the theory of magnetic domains with emphasis on recent developments in ``hard bubbles''. A spin configuration of a planar Bloch wall containing periodic Bloch lines is presented which minimizes the magnetostatic energy to first order in the parameter 2πMs2/Ku for arbitrary period. The form of this solution is found to suggest the form of the dynamic breakdown of this spin configuration. The remainder of the paper consists of applications of the gyrocoupling force and vector, fg = g × v and g = − (Ms/|γ|)sinθ(▿θ) × (▿φ), respectively, and the dissipation force and dyadic, fa = d · v and d = − α(Ms/|γ|)[(▿θ)(▿θ) + sin2θ(▿φ)(▿φ)]. The use of fg and fa produces results with fewer assumptions and with less calculation than with previous methods. The magnitude of g is found to be an invariant local measure of the ``hardness'' of the domains. Integrating fg and fα produces a general planar wall response function from which the hard bubble dynamic equation is obtained. It is found that the difference between the hard bubble and normal bubble damping parameters can be accounted for by examination of the hard bubble spin-wave spectrum. An estimate of the velocity required for the production of horizontal Bloch lines is made using fg. This velocity is a substantial fraction of the Walker velocity. The vector g is used as an aid in the visualization of the mechanism by which ion implantation suppresses hard bubbles. From the point of view of both mobility and hard bubble suppression, materials having a large in-plane anisotropy are found to be desirable.

Dynamic transformations of the internal structure of a moving domain wall in magnetic nanostripes

“…Similar oscillatory DW dynamics have been explained within the framework of the 1D model in bulk materials, 12 where the 1D Bloch wall periodically changes its polarization ͑the average magnetization in the transverse direction͒ via the transformation to the Néel wall and back. 26 Also, the numerical calculations of DW dynamics in 2D nanostripes 20,27,28 show that the DW motion is also periodic above H W but that the internal states in the course of DW oscillations are absolutely different from the prediction of Ref. 12.…”

“…Note that during the gyrotropic motion of AV or V after their nucleation, the total energy E tot and E ex do not change much; that is, the sum E Zeem + E dip is almost constant. The motion with almost constant E tot corresponds to the V/AV motion along equipotential lines ͑the damping term is small for Py and can be neglected for the gyromotion͒ under the influence of the gyroforce, which is perpendicular to the V/AV velocity v. 3,26,30 Related to the corresponding VW and AVW gyrotropic motions with either up-or down-core orientation, there are obvious correlations of the dynamic transformations between the TW and VW ͑or AVW͒ types. The V ͑⌳͒-shaped M configuration of the internal structure of TWs always leads to the upward ͑downward͒-core orientation of the VWs or AVWs to be transformed from the TWs themselves due to the rotation sense of the V and AV gyrotropic motions, which is determined by p only.…”

“…The AVW and VW types contain a single AV and a single V, respectively, inside the corresponding DWs. The AV and V states bear topological charges ͑vorticities͒ 30 and have nonzero gyrovectors 26,31 due to the existence of the vortex ͑antivortex͒ cores with M perpendicular to the nanostripe plane. Namely, the nonzero gyrovectors result in their gyrotropic ͑translation͒ motion in potential profiles influenced by the total magnetic energy E tot ͑X͒, which contains individual E ex , E dip , and E Zeem terms ͓Fig.…”

The influence of the edge effect on the skyrmion generation in a magnetic nanotrack

“…For small d 1 , i.e., 5 nm ≤ d 1 ≤ 17 nm, the magnetization forms a domain wall (DW) after an injection of 3 ns long pulse. It should be noted that there is a defect-like point 28 indicated by a yellow arrow within the DW, the configuration of which is similar to the two successive Bloch lines, 47 been observed in the generation of an isolated skyrmion in a nanodisk, which is annihilated once the skyrmion is formed in the presence of the spin transfer torque. 28 When d 1 ≥ 53 nm, the defect-like point will not occur, and a stabilized skyrmion (Sk) comes into being.…”

A general formalism for the determination of the effective mass of the nanoscale structural inhomogeneities of the domain wall in uniaxial ferromagnets

“…Note that the viscosity of the system is taken into account by introducing into the balance of forces a dissipative linear vector function [16], which, to simplify the consideration of the problem, we do not consider in our article. Its effect on the stationary motion of the BL and BP results in finite values of the mobility of the DW structural inhomogeneities, causing negligibly small additions to their effective masses [1,6,7].…”