Stability of magnetic vortex with respect to displacement of its center in a nano-scale circular cylinder made of soft ferromagnetic material is studied theoretically. The mode of vortex displacement producing no magnetic charges on the cylinder side is proposed and the corresponding absolute single-domain radius of the cylinder is calculated as a function of its thickness and the exchange length of the material. In cylinders with the radii less than the single-domain radius the vortex state is unstable and is absolutely prohibited (except if pinned by material imperfections), so that the distribution of the magnetization vector in such cylinders in no applied magnetic field is uniform (or quasi-uniform). The phase diagram of nano-scale cylinders including the stability line and the metastability region obtained here is presented.The small magnetic nano-scale cylinders made of soft magnetic materials recently gained attention due to the progress in fabrication and observation techniques and also because of their possible applications in magnetic random access memory (MRAM) devices. Within such a cylinders of circular shape in a certain range of sizes the magnetic vortices are frequently observed (see e.g. [1,2,3,4]). For applications, which usually try to avoid vortex formation (such as MRAM cells, [5]), it is important to know the sizes of the cylinder where the vortices do not form.In thin ferromagnetic cylinders with the thickness L of the order of a few L E (where L E = C/M 2 S is the exchange length, C is the exchange constant of the material, M S is the saturation magnetization) distribution of the magnetization vector can be assumed uniform along the cylinder axis. Then, there are two characteristic sizes of the cylinder important for the presence of the vortex state versus the uniformly magnetized one in zero applied magnetic field. The first is the single-domain radius R EQ , which is defined as a radius of the cylinder (at a given thickness) in which the energies of the uniformly magnetized state and the state with the vortex are the same [6]. In cylinders with radii R < R EQ the uniformly magnetized state has a lower energy than the vortex state. However, the metastable vortices still may be present in cylinders with radii below R EQ . There is another characteristic radius, the absolute single domain radius R S of the cylinder, which is obtained from the requirement that the vortex is unstable (therefore is absolutely prohibited) in cylinders with R < R S .The rigorous calculation of R S requires evaluating the second variation of the energy functional including the long-range dipolar interactions, which is currently beyond possibilities of analytical methods. The other way to estimate the stability radius is to assume the precise * Electronic address: metlov@fzu.cz way (mode) the vortex loses its stability and then to calculate the stability radius with respect to that process. There is an infinite set of possible candidate modes. Provided calculation of the energies is rigorous, the result for a pa...

The energy of a displaced magnetic vortex in a cylindrical particle made of isotropic ferromagnetic material (magnetic dot) is calculated taking into account the magnetic dipolar and the exchange interactions. Under the simplifying assumption of small dot thickness the closed-form expressions for the dot energy is written in a non-perturbative way as a function of the coordinate of the vortex center. Then, the process of losing the stability of the vortex under the influence of the externally applied magnetic field is considered. The field destabilizing the vortex as well as the field when the vortex energy is equal to the energy of a uniformly magnetized state are calculated and presented as a function of dot geometry. The results (containing no adjustable parameters) are compared to the recent experiment and are in good agreement. 75.75.+a, 75.25.+z, 75.60.-d Introduction. The unusual magnetic properties of sub-micron cylindrical magnetic particles and their periodical two-dimensional arrays drawn much attention because of their possible potential as a magnetic storage as well as an interesting model system for studying of the magnetization reversal 1-4 . The applications of such patterned magnetic films for magnetic information storage (MRAM, for instance) are promising.In this work the isolated magnetic particles made of isotropic (soft) magnetic material shaped as circular cylinders (referred hereafter as "dots") are considered. The magnetic properties of the dots are governed by the magnetic dipolar and the exchange interactions. When the external magnetic field is absent, three parameters of the dimension of length define completely the magnetic structure of a dot. These parameters are: the dot radius R, the dot thickness L and the exchange length L E = C/M 2 S , where C is the exchange constant and M S is the saturation magnetization of the material. The phase diagram of small L/L E 4, R/L E 4 dots in no applied magnetic field showing the range of dot parameters where the magnetic vortex (curl) is the ground state was calculated by Usov and co-workers 5,6 . Vortex states were indeed observed in polycrystalline Co 7 and F eN i 4,8-11 cylindrical dots.So far there is no theory of magnetization reversal of a dot in a vortex state when the magnetic field h = H/4πM S applied in the direction perpendicular to the cylinder axis. This process taking place via nucleation, displacement and annihilation of a single magnetic vortex 11 is considered in this work. There are three characteristic fields describing this process: the "vortex" nu-

For magnetic materials containing many lattice imperfections (e.g., nanocrystalline magnets), the relativistic Dzyaloshinski-Moriya (DM) interaction may result in nonuniform spin textures due to the lack of inversion symmetry at interfaces. Within the framework of the continuum theory of micromagnetics, we explore the impact of the DM interaction on the elastic magnetic small-angle neutron scattering (SANS) cross section of bulk ferromagnets. It is shown that the DM interaction gives rise to a polarization-dependent asymmetric term in the spin-flip SANS cross section. Analysis of this feature may provide a means to determine the DM constant.

The assumption of a certain hierarchy of soft ferromagnet energy terms, realized in small enough flat nanoelements, allows us to obtain explicit expressions for their magnetization distributions. By minimizing the energy terms sequentially, from the most to the least important, magnetization distributions are expressed as solutions of the Riemann-Hilbert boundary value problem for a function of complex variable. A number of free parameters, corresponding to positions of vortices and antivortices, still remain in the expression. Thus, the presented approach is a factory of realistic Ritz functions for analytical (or numerical) micromagnetic calculations. Examples are given for multivortex magnetization distributions in a circular cylinder, and for two-dimensional domain walls in thin magnetic strips.