4.3 Magnetization reversal in ideal Fe elongated particles

Fig. 4.5 presents the results obtained from our simulation for the hysteresis cycles of ribbons having different widths and bulk biaxial magnetocrystalline anisotropy exclusively. The magnetic moment configuration (see Figs. 4.6 and 4.8) occurring during these processes allowed us to identify two different reversal modes associated to the particles having large and small aspect ratio, respectively.

An initial result, observed at all the applied field values and in all the simulated particles, was the fact that the orientation of the magnetic moments did not depend on the position along the system thickness. That result could be ``a priori'' expected from the particle thickness value which was below the Fe exchange correlation length, $l_{ex,Fe}$. Thus, in the following we will discuss all the moment configurations present in the modeled systems in terms of 2D moment distributions corresponding to planes parallel to the two largest dimensions of the particles.

Figure 4.5: Hysteresis loops for different particle sizes, varying their width.

Figure: X-Y view of the magnetic moments configuration for $400\times 40\times 4 nm^3$ particle: (a) Remanent state, (b) Applied field $H = 925 Oe$ and (c) Applied field $H = 930 Oe$. Every arrow represents $10\times 6$ cells.

In the case of the particles having the largest aspect ratios (see the complete reversal sequence in Fig. 4.6, where data corresponding to the particle having $400 nm \times 40 nm \times 4 nm$ are presented) the remanent state corresponded to the well known ``S-type'' configuration, which only differs from the single domain one in the quasi-parallel-to-the-surface moment structures present at the particle ends. Those structures are a result of the minimization of the magnetostatic energy and their presence results in the spreading of the magnetic poles that should be localized at the particle surfaces in the case of the uniform magnetization configuration. The ``S-type'' configuration evolves with the increase of the demagnetizing field to the formation of closure-like domains at both ends of the particle and finally to a collective irreversible rotation which leads to a ``S-type'' state symmetric to that corresponding to the magnetizing field remanence. The hysteresis loops presented in Fig. 4.5 evidence that the coercive force decreases with the decrease of the aspect ratio value see (Fig. 4.7).

Figure 4.7: Comparison of the experimental and the simulational data obtained by exclusively considering bulk biaxial magnetocrystalline anisotropy. The line corresponds to the value predicted for the Stoner-Wohlfarth model with cubic anisotropy.

The field at which the reversal occurs can be compared to the classical shape anisotropy field associated to the particle dimensions. For that purpose we can consider the demagnetizing field model for thin films discussed in [O'Handley 00, Pages 42-43]. If the dimensions of the system are $w$ (width) $>h$ (height) $> t$ (thickness), then the demagnetizing factors, $N_w$ and $N_h$, associated to the two larger dimensions are in SI units:

$$N_w\thickapprox \frac{2t}{\pi w},N_h\thickapprox \frac{2t}{\pi h}$$ (4.1)

and the corresponding anisotropy field, $H_s$

$$H_s=M_s(N_w-N_h).$$ (4.2)

In the case of the particle having the larger aspect ratio, $w = 400 nm$, $h = 30 nm$, and $t = 4 nm$, this results in a shape anisotropy field of $H_s = 0.172 T$, which is significantly larger than our simulational result for the coercive force ( $H_c= 0.1265 T$). This fact is related to the moment configurations obtained from our micromagnetic simulations at the remanence and, more concretely, to the inhomogeneous moment structures present at the ends of the particles.

In the case of the particles having smaller aspect ratios, both the reversal mode and the associated moment configurations are more complex than those previously discussed. At the saturation remanence (see Fig. 4.8(a)), the moments at a layer close to the surface of the particles point essentially parallel to that surface, whereas those at the inner core of the particles are oriented along the direction of one of the magnetocrystalline (100) easy axes. The transition from the ``S-type'' configuration characteristic of the elongated particles to that present in the small aspect ratio ones is illustrated in Fig. 4.9 in which we have plotted the evolution with the aspect ratio of the orientation of the moment present at the center of the system. These data allow to know that only for particle widths above $200 nm$ the inner moments point parallel to the magnetocrystalline easy axis.

Figure: X-Y view of the magnetic moments configuration for $400\times 175\times 4 nm^3$ particle: (a) Remanent state, (b) Applied field $H = 150 Oe$, (c) Applied field $H = 155 Oe$ and (d) Applied field $H = 460 Oe$. Every arrow represents $10\times 6$ cells.

Figure 4.9: Angular deviation at the remanence from the applied direction at the center of the particle.

The next steps of the magnetization reversal of these small aspect ratio particles are shown in Fig. 4.8. From this figure it is possible to see how the first partially reversed region is the particle core which moments rotate irreversibly from their direction at the remanence to that of the closest easy axis forming the smallest angle with the applied demagnetizing field. For a field slightly below that corresponding to this rotation the particle exhibits uniform magnetization domains separated by charged 45º walls. Since for the particles having small aspect ratio most of the volume of the particle corresponds to the core moments their reversal governs the coercive force value (evaluated from the $M_x = 0$ condition).

It is interesting to point out that the field corresponding to the particle core reversal approaches for sufficiently small aspect ratios the value corresponding to the Stoner-Wohlfarth (SW) model discussed in Section 5.3.2 (see Fig. 4.7 where we show the width dependence of the core reversal field and the corresponding SW field). Thus, for these conditions the surface layer minimizes the stray fields linked to the finite size of the particles and, very interestingly, the system behaves globally as an infinite one, reversing through a coherent rotation mechanism.

Once the particle core experiences the low field irreversible rotation their moments rotate reversibly towards the field direction. In the large width particles and for a field just above that corresponding to the rotation, the wall-like structure allowing the transition from the core moments direction to the orientation of the moments in the long axis uniform magnetization surface layer, sweeps an angle of ca. $135^\circ$. Further reorientation of the core moments towards the applied field reduces both the swept angle and the width of this wall. The reversal process is completed by the annihilation of the core-surface wall. This process occurs at a field approximately independent from the sample size and through the irreversible rotation of the surface moments. Immediately after that annihilation the system moment configuration is of the ``S-type''.

The efficiency of the surface structures to minimize the magnetostatic energy contribution is also relevant from the point of view of the analysis of the interparticle dipolar interactions that should here be largely reduced in comparison to a case in which the surface moment structures were absent. However, it is not clear that the pole avoidance is the best way to minimize the total energy of the many particles. An efficient compensation of charges in neighboring particles can be more advantageous than their avoidance.

Finally, in Fig. 4.7 is plotted, for comparison, the simulational and experimental coercivity. Whereas for reduced widths the simulational results are up to two times larger than the experimental ones (which is in principle plausible due to the absence in the model of different types of defects, i.e.: morphological, associated to reduced crystallinity regions, etc., that could help to reduce the coercivity) for widths larger than ca. $70 nm$ the experimental data are slightly larger than the simulational ones, clearly suggesting that mechanisms different from the presence of the mentioned defects should account for the differences between the experimental and the simulational data. Note also that in experimental situations the irreversible jumps of the hysteresis cycles are normally smoothed due to slight distribution of geometrical and magnetic parameters between different objects.