3. Results and Discussion
In order to show and prove the proposed approach, systems of ferromagnetic sphere, regular box (a = b = c) and tetragonal box (a = b, c = 2a) were simulated with the use of different values of the scaling factor n. The chosen calculation procedure was the same as the one presented in [22,24,25]. The system parameters were as follows: number of system nodes 40 × 40 × 40 = 64,000, r = 0.28 nm, Jij = 1 × 10−2 eV, Ki = 0 (perfect soft magnet), Si = 1, kBT = 1 × 10−5 eV, D = 2.18 eVnm3 and n = 1, 10, 50 and 100. The other parameters appearing in the presented algorithm are Nrelax = Navr = 400, Pcl = 0.001 and θ = π/100. The initial 3D systems (for n = 1) and their magnetic states (H = 0) for n equal to 10, 50 and 100 are depicted in Figure 3, Figure 4 and Figure 5, respectively. The arrows represent magnetic moments assigned to the nodes, and color depends on the arrow direction. For all cases with n = 1, the magnetic moment alignment is ferromagnetic because the contribution of the dipolar energy is marginal. With the development of the system size, the appearance of different magnetic structures dependent on the n parameter and the shape of the objects can be observed. In the case of n = 50 and for all analyzed objects, the depicted configurations of magnetic moments have pure vortex (spheres) and vortex-like (boxes) characters. For the higher value of n, more complex magnetic structures were detected. Such magnetic structures are expected and experimentally observed in, for example, magnetically soft amorphous and nanocrystalline iron-based alloys, such as the so-called fingerprint domains.
The second demonstration of the proposed scaling rules was a simulation of magnetization processes for the tetragonal box with magnetic anisotropy (K = 5 × 10−5 eV) directed along the z-axis. Magnetic field, also applied in the z-axis, ranged from 1.2 to −1.2 T in order to obtain a full hysteresis loop. The n parameter was equal to 100 and the remaining parameters of the system were the same as for the previous example (see Figure 5). In this configuration, a quadratic hysteresis loop was expected and obtained using the above-described procedure. Figure 6e shows a normalized magnetization (z-component of average magnetic moment) in a function of the applied external magnetic field, as shown in the graph in the center. Additionally, magnetic moment configurations in fields around the magnetization jump are presented as a background. It can be seen that the reverse magnetization process started from a “seed” nucleated on the top surface (Figure 6a) and then the domain expanded, reaching new saturation according to the direction of the external magnetic field. The Figure 6c depicts the growing domain responsible for the rapid magnetization change. This scenario was expected due to the fact that the magnetic moments on the top (or bottom) surface are more weakly coupled in comparison to the magnetic moments in the volume. Moreover, for this irregular shape and value of n, the dipolar interactions tend to reverse the directions of the magnetic moments, especially near the top and bottom surfaces (see also Figure 5).