Finite differences and finite elements

Figure 3.11 depicts the hysteresis loops obtained from simulations of a nickel sphere (diameter $d$=200nm, $K_1$=0). The loops in the upper left, upper right and lower left of the figure show loops yielded with the OOMMF software, whilst the lower right loop was computed with magpar.

Figure: Hysteresis loops for nickel spheres of diameter $d$=200nm with no magnetocrystalline anisotropy with varying applied field offsets in $xy$ ($\theta$) and $xz$ ($\phi$). The top left hysteresis loop was performed with OOMMF with applied field offsets $\theta$=45$^o$, $\phi$=0$^o$. The top right loop (OOMMF) has $\theta$=10$^o$, $\phi$=0$^o$. The bottom left loop (OOMMF) has $\theta$=$30^o$, $\phi$=18$^o$. The bottom right hysteresis loop was performed with magpar using an applied field offset of $\theta$=4$^o$, $\phi$=6$^o$

For each simulation, the applied field direction was varied in $\theta$ and $\phi$, respectively the azimuth and polar angle of the applied field. These are relative to the orientation of the finite difference grid, aligned with the x-, y- and z-axes.

All the loops show a similar pattern just below saturation, with small openings in the hysteresis where the magnetisation moves from the single domain state into the vortex state.

It is interesting to note that in the loops computed with OOMMF there is an ``opening'' in the hysteresis loop around $B_x$=0, the size of which varies with the direction of the applied field, although it is never quite eliminated. The results from magpar do not display this characteristic.

These data show that the inner loop depends on the angles $\theta$ and $\phi$. For the spherically symmetric system that we wish to simulate, how the coordinate system of the simulation software is aligned relative to the direction of the applied field should be irrelevant. It is therefore likely that the inner openings in the hysteresis loops are an artefact of the finite difference simulation technique as these vary substantially as a function of this direction.

Initially, we assume the applied field is zero and the magnetisation forms a vortex with the core pointing in the $x$ direction as shown later in figure 3.15, and a very small field is applied in the opposite direction (i.e. $-x$) such that the vortex structure of the magnetisation is not significantly affected. Since the overall magnetic moment of the magnetisation is finite and points in the direction of the moments in the vortex core, it is therefore energetically favourable for the system to align the vortex core with the applied field.

In order for this to happen, the vortex core needs to turn around by 180 degrees (i.e. point in $-x$ rather than $+x$. If there is no magnetocrystalline anisotropy in the system, the spherically symmetric sphere should allow the core to rotate round in either $\theta$ or $\phi$, similar to a typical Stoner-Wohlfarth particle. The spherical symmetry should not allow the occurrence of the ``inner'' hysteresis loop indicated in figure 3.11; this is supported by the results presented in the lower right of figure 3.11 computed with the finite element code.

It is plausible therefore to assume that directions along the discretisation axis are either favoured or avoided by the system when a finite difference grid is introduced for symmetric geometries.