Vortex mechanics in planar nano-magnets

A collective-variable approach for the study of non-linear dynamics of magnetic textures in planar nano-magnets is proposed. The variables are just arbitrary parameters (complex or real) in the specified analytical function of a complex variable, describing the texture in motion. Starting with such a function, a formal procedure is outlined, allowing a (non-linear) system of differential equations of motion to be obtained for the variables. The resulting equations are equivalent to Landau-Lifshitz-Gilbert dynamics as far as the definition of collective variables allows it. Apart from the collective-variable specification, the procedure does not involve any additional assumptions (such as translational invariance or steady-state motion). As an example, the equations of weakly non-linear motion of a magnetic vortex are derived and solved analytically. A simple formula for the dependence of the vortex precession frequency on its amplitude is derived. The results are verified against special cases from the literature and agree quantitatively with experiments and simulations.Comment: 7 pages, 1 figure, published versio

Two-dimensional topological solitons in small exchange-dominated cylindrical ferromagnetic particles

A general approach allowing to construct the magnetization distributions of arbitrary topological charge in small exchange-dominated cylindrical ferromagnetic particles is presented. The exchange energy functional is minimized by these distributions exactly. The magnetostatic energy is accounted partially, so that it facilitates a choice between the topologically equivalent exchange-only solutions. The resulting magnetization distributions can be easily generalized to a variety of non-circular cylindrical shapes by means of a conformal transformation. As an example a magnetic structures of a thin circular ferromagnetic cylinder both with centered and displaced magnetic vortex and of a finite Bloch domain wall in an elongated particle is given.Comment: 5 pages, 2 figures, RevTe

Two-dimensional topological solitons in soft ferromagnetic cylinders

A simple approach allowing to construct closed-form analytical zero-field magnetization distributions in cylindrical particles of a small thickness and an arbitrary shape (not necessarily circular) is presented. The approach is based on reduction of the non-linear Euler equations for magnetization vector field to the classical linear Riemann-Hilbert problem. The result contains all the distributions minimizing the exchange energy functional and the surface magnetostatic contribution exactly, except for the neighbourhood of topological singularities on the cylinder faces where the result is approximate. The completeness of the analysis permitted to find a new type of a topological soliton in the case of circular cylinder. Also, an example of magnetic vortex in a triangular cylinder is given to investigate the role of the particle corners.Comment: 3 pages, 2 figures, RevTeX 3 pages, 2 figures, RevTex (evaluated the integral (4) and added the equivalent formula (5), minor corrections to improve English, new Ref. [2]

Vortex precession frequency and its amplitude-dependent shift in cylindrical nanomagnets

Frequency of free magnetic vortex precession in circular soft ferromagnetic nano-cylinders (magnetic dots) of various sizes is an important parameter, used in design of spintronic devices (such as spin-torque microwave nano-oscillators) and characterization of magnetic nanostructures. Here, using a recently developed collective-variable approach to non-linear dynamics of magnetic textures in planar nano-magnets, this frequency and its amplitude-dependent shift are computed analytically and plotted for the full range of cylinder geometries. The frequency shift is positive in large planar dots, but becomes negative in smaller and more elongated ones. At certain dot dimensions a zero frequency shift is realized, which can be important for enhancing frequency stability of magnetic nano-oscillators.Comment: 6 pages, 2 figure

Large vortex state in ferromagnetic disks

Magnetic vortices in soft ferromagnetic nano-disks have been extensively studied for at least several decades both for their fundamental (as a "live" macroscopic realization of a field theory model of an elementary particle) as well as applied value for high-speed high-density power-independent information storage. Here it is shown that there is another stable vortex state with large thickness-dependent core profile in nano-scale ferromagnetic disks of several exchange lengths in size. Its energy is computed numerically (in the framework of Magnetism@home distributed computing project) and its stability is studied analytically, which allows to plot it on magnetic phase diagram. In cylinders of certain geometries large vortices exist on par with classical ones, while being separated by an energy barrier, controllable by tuning the geometry and material of ferromagnetic disk. They can be an excellent candidate for magnetic information storage not only because the resulting disk sizes are among the smallest, able to support magnetic vortices, but also because it is the closest to the classical vortex state of all other known metastable states of magnetic nano-cylinder. It means that memory, based on switching between these two types of magnetic vortices, may, potentially, achieve the highest possible rate of switching.Comment: 5 pages, 1 figure, slightly shortened version with no substantial change

Magnetic neutron scattering by magnetic vortices in thin submicron-sized soft ferromagnetic cylinders

Using analytical expressions for the magnetization textures of thin submicron-sized magnetic cylinders in vortex state, we derive closed-form algebraic expressions for the ensuing small-angle neutron scattering (SANS) cross sections. Specifically, for the perpendicular and parallel scattering geometries, we have computed the magnetic SANS cross sections for the case of small vortex-center displacements without formation of magnetic charges on the side faces of the cylinder. The results represent a significant qualitative and quantitative step forward in SANS-data analysis on isolated magnetic nanoparticle systems, which are commonly assumed to be homogeneously or stepwise-homogeneously magnetized. We suggest a way to extract the fine details of the magnetic vortex structure during the magnetization process from the SANS measurements in order to help resolving the long standing question of the magnetic vortex displacement mode.Comment: 8 pages, 5 figure

Metastable states of sub-micron scale ferromagnetic periodic antidot arrays

The magnetic textures on nanoscale possess topological features due to the continuity of the magnetization vector field and its boundary conditions. In thin planar nanoelements, where the dependence of the magnetization across the thickness is inessential, the textures can be represented as a soup of 2-d topological solitons, corresponding to magnetic vortices and antivortices, which are the solutions of Skyrme's model. Topology of the element (of the boundary conditions) then imposes the restrictions on properties and locations of these objects. Periodic arrays of magnetic antidots have topology with infinite connectivity. In this work we classify and build an approximate analytical representation of metastable magnetization textures in such arrays and prove the conservation of their topological charge.Comment: 4 pages, 2 figure

A simple analytical description for the cross-tie domain wall structure

A closed form analytical expression for the magnetization vector distribution within the cross-tie domain wall in an isotropic ferromagnetic thin film is given. The expression minimizes the exchange energy functional exactly, and the magnetostatic energy by means of an adjustable parameter (wall width). The equilibrium value of the wall width and the film thickness corresponding to the transition between the N{\'e}el and the cross-tie walls are calculated. The results are compared with the recent experiments and are in good qualitative agreement.Comment: 4 pages, 2 figures, RevTex 4, to appear in Appl. Phys. Lett. in Oct 200

Map of metastable states for thin circular magnetic nano-cylinders

Nano-magnetic systems of artificially shaped ferromagnetic islands, recently became a popular subject due to their current and potential applications in spintronics, magneto-photonics and superconductivity. When the island size is close to the exchange length of magnetic material (around 15 nm), its magnetic structure becomes markedly different. It determines both static and dynamic magnetic properties of elements, but strongly depends on their shape and size. Here we map this dependence for circular cylindrical islands of a few exchange lengths in size. We outline the region of metastability of "C"-type magnetic states, proving that they are indeed genuine and not a result of pinning on particle imperfections. A way to create the smallest particles with guaranteed magnetic vortex state at zero field becomes evident. It is expected that the map will help focus the efforts in planning of experiments and devices.Comment: 3 pages, 1 figur

Analytical approximations to the core radius and energy of magnetic vortex in thin ferromagnetic disks

The energy of magnetic vortex core and its equilibrium radius in thin circular cylinder were first presented by N.A. Usov and S.E. Peschany in 1994. Yet, the magnetostatic function, entering the energy expression, is hard to evaluate and approximate. In this communication precise and explicit analytical approximations to this function (as well as equilibrium vortex core radius and energy) are derived in terms of elementary functions. Also, several simplifying approximations to the magnetic Hamiltonian and their impact on theoretical stability of magnetic vortex state are discussed.Comment: 5 pages, 1 figur