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SORDER, AND PHASE TRANSITION IN CONDENSED SYSTEM

Nonlinear Oscillations of Magnetization for Ferromagnetic Particles in the Vortex State and Their Ordered Arrays A. Yu. Galkina, b and B. A. Ivanova, c* a

Institute of Magnetism, National Academy of Sciences of Ukraine, Ministry of Education and Science, Bulvar Akademika Vernadskogo 36
b, Kiev, 03142 Ukraine b G. V. Kurdyumov Institute for Metal Physics, National Academy of Sciences of Ukraine, Bulvar Akademika Vernadskogo 36, Kiev, 03142 Ukraine c Taras Shevchenko National University of Kyiv, Vladimirskaya ul. 64, Kyiv, 03127 Ukraine *e
mail: [email protected] Received January 26, 2009

Abstract—The dynamics of magnetization oscillations with a considerable amplitude and a radial symmetry in small ferromagnetic particles in the form of a thin disk with a magnetic vortex has been investigated. The collective variables that describe radially symmetric oscillations of the magnetization dynamics for particles in the vortex state are introduced, and the dependence of the particle energy is studied as a function of these variables. The analytical expressions describing the frequency of magnetization oscillations with a radial sym
metry, including nonlinear oscillations, are derived using the collective variables. It is shown that the presence of a magnetic field oriented perpendicular to the particle plane reduces the oscillation frequency and can lead to hybridization of this mode with other modes of spin oscillations, including the mode of translational oscil
lations of the vortex core. The soliton solutions describing the propagation of collective oscillations along the chain of magnetic particles are found. PACS numbers: 75.50.Ee, 75.45.+j, 75.50.Xx, 75.50.Tt DOI: 10.1134/S1063776109070103

tion do not produce a demagnetizing field. The state with a single vortex in a cylindrical particle is the sim
plest state of the magnetization with a closed magnetic flux [16], and with an increase in the particle size, it arises immediately after the quasi
uniform single
domain state. In other words, the vortex state for micron and submicron particles (from 0.1 µm to sev
eral microns in diameter [17] and 20–300 nm in thick
ness) is an alternative to the conventional domain structure. The vortex state has been experimentally observed in cylindrical particles with a diameter 2R = 200–800 nm, and the magnetization distribution in such particles with a thickness h = 20–40 nm varies only slightly over the particle thickness. It is clear that the presence of a nonuniform magnetization state can strongly change the dynamic properties of the particle as compared to the properties of a uniformly magne
tized disk [18–23].

1. INTRODUCTION Vortices (topologically nontrivial distributions of magnetization) arise in submicron mesoscopic sam
ples of soft magnetic materials that have an approxi
mately spherical shape (they have been frequently referred to as the magnetic dots). At present, interest expressed by researchers in vortex states observed in magnetic dots [1–3] and magnetic ri