Point Singularities in Micromagnetism

Abstract

Solutions of the micromagnetic equations for the equilibrium distribution of magnetization may have singular points, at which no spontaneous magnetization exists, because the limit of the magnetization vector for an approach to this point on different rays has different values. The boundary conditions at these points are considered. The exchange-energy density goes to infinity at such a point, but the condition that its integral should be a minimum leads to no definite boundary condition. For a special class of such singularities, namely those at which every direction of the magnetization is accepted once on a small sphere around it, the conditions for a minimum of the field energy are derived.

New Type of Stable Particlelike States in Chiral Magnets

“…In the limiting case H = 0 and D → 0, the expression in Eq. (A1.8) shows a linear dependence on R and converges to the energy dependence derived earlier for singularity in pure ferromagnets [42]. Thus, the energy contribution to the total energy related to the small volume around the singularity resulting from the Dzyaloshinskii-Moriya interaction is proportional to R 3 .…”

Rotating Vortex Dipoles in Ferromagnets

“…This is a somewhat simplified realization of the BP studied in Ref. [20]. Its energy is approximately 4π times the length of the vortex or the antivortex line.…”

Exchange Explosions: Magnetization Dynamics during Vortex-Antivortex Annihilation

“…In the continuum theory of micromagnetism, a BP can be defined as a region inside a ferromagnet, where the magnetization collapses to zero [24]. On a shell that encloses the BP, any magnetization direction can be found [23]. In some cases, a BP is therefore similar to a magnetic hedgehog structure.…”

“…Usually, the exchange length is considered to be the smallest relevant length scale in structurally homogeneous ferromagnetic materials. However, there are exceptional cases where large deviations of the magnetization can occur in a ferromagnet even on atomic distances [23]. The approach of the vortex and the antivortex proceeds until their cores meet in one point, leading to a dramatic and sudden annihilation process.…”

“…The approach of the vortex and the antivortex proceeds until their cores meet in one point, leading to a dramatic and sudden annihilation process. The annihilation is accomplished by a micromagnetic singularity [23] (Bloch point) that propagates vertically through the sample. The propagating Bloch point (BP) dissolves simultaneously both the vortex and the antivortex structure.…”