The polarity of magnetic vortex core can be switched by current or magnetic field through a vortex-antivortex pair creation and annihilation process, in which the significant change of the exchange energy during the switching takes an important role. To further unveil the energetic origin of magnetic vortex switching, we investigate the evolution of the maximum exchange energy density of the sample by using micromagnetic finite-element simulations based on the Landau-Lifshitz-Gilbert equation including the adiabatic and the nonadiabatic spin torque terms. Our micromagnetic calculations indicate that maximum exchange energy density for the considered sample must exceed a critical value of ~3.0106 J/m3 in order to achieve the magnetic vortex switching. The threshold value corresponds to the maximum exchange energy density at the time of creation of new vortex-antivortex pair. Following the nucleation of antivortex, the maximum exchange energy density increases rapidly with the antivortex approaching the original vortex. The maximum exchange energy density can become large at the time of annihilation of two vortexes. To explain well the critical value of the local maximum exchange energy density, we use the rigid vortex model(in which the spin distribution is unchangeable while vortex is displaced) to develop an analytical model. For a magnetic vortex confined in a thin ferromagnetic nanodisk, the magnetization distribution is unchanged along the thickness and can be seen as a two-dimensional model when the thickness is less than or on the order of the exchange length. The components of vortex magnetization vector in a ferromagnetic dot can be expressed by using a complex function w(,). Corresponding to the trivortex state appearing in vortex core reversal process, the local exchange energy density Wex around the vortexes cores is obtained. Simultaneously, we obtain the maximum exchange energy density:Wex2.3106 J/m3. In a realistic system, the shape of vortexes will deform during the vortex core reversal, which leads to the analytical result lower than the simulation value. Based on this reason, the analytical result matches well with our simulation value.
In comparison with uniformly magnetized states, vortex structures demonstrate a rich frequency spectrum of spin-wave (SW) excitations. However, a detailed theoretical description of the magnetic modes is generally still a challenge due to the difficulty of analytic calculation, except for the well-defined symmetric circular states. In contrast, the method of micromagnetic simulations combined with Fourier analysis is shown to be very powerful for gaining insight into the nature of magnetic excitation modes. Vortex excitation modes have been reported to be directly influenced by the geometric symmetry of the elements and/or the nature of the initial perturbation of pulse field. In order to understand how the reduced symmetry affects the vortex SW modes, we perform the micromagnetic simulations on vortex modes excited in a submicron-sized thin ellipse. In order to excite the spin-wave modes, a short in-plane Gaussian field pulse is applied along the short axis direction. After the pulse, the off-centered vortex core moves following an elliptical trajectory around its equilibrium position. Simulations provide the time evolution of the local magnetizations (at each discretization point) and dynamics of the spatially averaged magnetization. To determine the mode frequencies, the spectrum is obtained from the average magnetization through Fourier transformation from time domain the frequency domain. By means of Fourier analysis, a variety of azimuthal SW modes can be observed in the excitation spectrum. The ellipse in single vortex state has a twofold rotational symmetry with a rotation of πup around the z-axis (out-of plane) and can be described by the C2 group. The observed azimuthal modes can be divided into two categories according to their symmetry. Two modes occur alternately with increasing azimuthal number, indicating that the magnetic excitation modes remain to keep the symmetry of the ellipse structure. Their frequencies are found to increase linearly with the azimuthal index number. An increase of the SW frequency with increasing number of nodal planes is rather well known, which results from the competition between exchange and dipolar energy terms. According to the temporal evolution of the ellipse's spatially averaged energy densities, our micromagnetic simulation shows that the average exchange energy is significantly higher than the magnetostatic energy, suggesting that the exchange interaction plays a more important role in the excitation modes. The exchange energy density is mainly focused on the core origin while the largest contribution of the magnetostatic energy is distributed near the long axis. Thus, we can conclude that the exchange interaction provides the principal contribution to the vortex energy in such small ellipses with a single vortex state, resulting in the increasing frequency versus the azimuthal number, that is observed.
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