Figure 1 shows a representation of Py disk with 2 µm in diameter and thickness of 50 nm. In addition, stadium-like geometry with 1 µm of minor axes and 4 µm for the major (thickness of 50 nm). The figure also shows the coordinate axes directions and the nomenclature of parallel field when applied along the x axis and perpendicular field when applied along the z axis. The FMR simulations were performed with a field parallel (θ = 0o) to the structures plane as well for a perpendicular field (θ = 90o).
Figure 2 presents the circular geometry used during the simulation and its respective dimensions. Next, the same figure shows the dynamics of the magnetization component my (component perpendicular to the external field at x). This dynamic is displayed together with the corresponding FFT. From this result we can verify that for the 80 kA/m of external field, the dynamics is typical of a well-behaved slightly damped movement and from this we can verify in the FFT that the resonance occurs at 43.05 GHz.
Close to 33 GHz, an absorption occurs that can be associated with the formation of non-homogeneous magnetization structures. The papers of Saavedra et al. [10] and Vavasori et al. [15] are a reference to suggest that for circular structures the formation of vortex-like structures should occur, and for stadium-like geometries, the domain walls generate complex reversal modes. It is to be expected that this peak will occur because, according to the results of Silva et al. [16], there is a non-homogeneous moment structure, due to edge effects.
At this point we can make use of Kittel's equations for the FMR in an infinite plane, in order to verify the model in circular geometry with parallel and perpendicular field of equations (4) and (5). According to the model, the resonance frequencies f0 as a function of the external field B0 can be obtained through the following expressions:
$$2\pi {f}_{0}=\gamma {\left[{B}_{0}\left({B}_{0}+{\mu }_{0}M\right)\right]}^{1/2},externalfieldonxdirection$$
4
$$2\pi {f}_{0}=\gamma \left({B}_{0}-{\mu }_{0}M\right),externalfieldonzdirection\left(perpendicularfield\right)$$
5
At a first glance (without going into details on the exact values), the resulting peaks from the FFT to the disk comply with equations (4) and (5). As we mentioned at the beginning of this work, the main focus is to evaluate the effectiveness of the model in which an object with a thickness of 50 nm and micrometric lateral dimensions is considered a plane.
Figure 4 presents a stage-type geometry as another example of this situation. The sequence presents the used geometry and then the magnetization dynamics with its corresponding FFT for the field parallel to the x axis (see Fig. 1). As we can see the main absorption peak appears at 43.55 GHz, 0.5 GHz higher than in the case of the disk shown in Fig. 2. In the same sequence as in Fig. 4, the dynamics of magnetization for the field perpendicular to the plane of the structure are also presented, with an absorption peak at 4.70 GHz (0.25 GHz higher than in the disk). The differences in the values of the frequency of the peaks is an important point because qualitatively the two structures follow the behavior established by the Kittel model. On the other hand, there are changes in the peak value for parallel and perpendicular field (comparing the circle and the stadium).
Figure 5 shows the dynamics of each magnetization component for the two field situations. We can verify a phase change structure along the stadium plane. Compared with the result of Fig. 3, it is easy to verify that this is the main signature that leads to changes in the frequency value of the two geometries. From these results we can draw two important pieces of information. The first one is that to use the Kittel model it is very important to have the sample saturated and above all, it is essential that the moments oscillate in phase. The other important piece of information is that the condition for which this occurs strongly depends on the geometry under study. Works in the literature present situations in which the Kittel model is not valid in fields below the anisotropy field [17].