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Generally speaking, when the applied field is reduced from the saturation value, the competitive effects of magnetic (exchange and magnetostatic) interactions and the random magnetocrystalline anisotropy lead to magnetization ripple when the magnetization deviates locally from its saturation value [112, 128, 129]. Figure 14 supposes to simulate a hysteresis cycle of a permalloy-based thin film and a standard Kerr-microscopy image. The ripple formation and its changes through the hysteresis cycle are clearly appreciated. The remanence value decreases monotonically with an increase of the ripple angleā€”the greater the misalignment of grains is due to the anisotropy randomness, the less is the remanence of the corresponding polycrystalline film [130].

In general, two competitive mechanisms influence the development of the coercivity: the nucleation and the pinning. Nucleation occurs normally at the places where some local conditions (for example, reduced internal field at the sharp corners of a magnetic element) favors such a process. Modeling of the demagnetization process, for example, in Co bars [112] with the grain structure described by Voronoi polyhedra, shows [128, 129] that as the field increases toward the negative directions, the ripple becomes more pronounced, with the consequent formation of vortices at the ends of the bars. The magnetization vortex, together with a domain wall, is the most elementary structure resulting from nucleation and magnetization reversal. Expansion of reversed regions during hysteresis is achieved through their motion and annihilation in the magnetic element.

Furthermore, in nanostructured samples (for example, simulations of FePt materials [130]), multiple nucleation centers due to "magnetic charges" inside the grains with large deviations of easy axes take place. This drastically reduces the nucleation field in comparison to a textured sample.

Once the magnetization reversal nucleus is formed, it propagates in the form of some collective microstructure (domain wall or a vortex). The pinning effects the propagation. The same way as the grains with a large misalignment of easy axes could induce the nucleation, they also can serve as pinning centers [130]. The greater the misalignments between grain easy axes are, the more difficult it is for a domain wall to cross the boundary and thus the greater is

Figure 14. Simulation example of a hysteresis cycle in a Permalloy thin film with (D) = 40 nm average grain diameter, random anisotropy, and an additional easy-axis anisotropy. The ripple structure is clearly visible and becomes more pronounced with the increase of an external field. (Courtesy of D. Berkov).

Figure 14. Simulation example of a hysteresis cycle in a Permalloy thin film with (D) = 40 nm average grain diameter, random anisotropy, and an additional easy-axis anisotropy. The ripple structure is clearly visible and becomes more pronounced with the increase of an external field. (Courtesy of D. Berkov).

the external field that is necessary to push the domain wall through it. If the texture is small, there are fewer nucleation centers. However, once a domain of a reversed direction is nucleated, it sweeps the entire film quickly, producing the complete reversal. Consequently, the squareness of the hysteresis loop is high, and the coercive field is large. Figure 15 illustrates the variation of the coercive field as a function of the system texture in a one-dimensional system of grains. More complete micromagnetic simulations in 3D FePt films with 3D easy axes orientation [130] also have shown that the coercivity decreases for more randomness of the easy axes distribution so that it is maximum for a textured sample. However, intermediate texture and a random easy axes sample have revealed almost the same coercivity. This is explained by the competition between nucleation of domains and their motion.

Depending on the particular microstructure realization [128], different reversal mechanisms could occur. This results in dispersion of coercive fields, which may be especially evidenced in small-size magnetic nanoelements [128].

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