Three-dimensional model

A cubic arrangement of antispheres of radius 150nm cut from a sample of nickel of dimensions 600nm $\times$ 600nm $\times$ 150nm was prepared and the remanent magnetisation computed for this system with a finite difference cell edge length of 5nm. These particular dimensions were chosen as they represent the largest system which can be simulated in a timely fashion.

The magnetic microstructures which this system creates in zero applied field after being relaxed from an initially homogeneous state in the $x$ direction can be seen in figure 6.5.

Figure 6.6: Magnetisation of a cobalt hexagonal antidot array in zero field; direction of initial applied field $\theta=30^{o}$, $d$ = 300nm, $r \over R$ = 0.4, $h$ = 0.05$d$

Unfortunately it would appear that the effect of the edges in this system overwhelmingly influences the microstructure, so such a small sample does not provide a particularly useful insight into an effectively infinite array of antispheres. In particular, the relatively thin physical walls in the $z$ direction which exist in this arrangement apparently act as pinning centres of sorts; since it is difficult for the system to influence the middle of these walls, the system must have a significantly large external magnetic field applied to it for these spins to reverse their direction of magnetisation.

For more useful results to be obtained, a much larger system is required. Instead of $3 \times 3$ antisphere centres being present in the system, we estimate $5 \times 5$ or more centres would be necessary to reduce the effects of the sample edges enough to accurately reflect the microstructures which form within the experimental sample.

Figure 6.7: Hysteresis loop for a 2d array of hexagonally arranged permalloy antidots ($d=100$nm, hole radius to spacing period radius ratio $r/R=0.4$) in an in-plane applied field offset 30 degrees from the $x$ direction

Zhukov et al. (2003) perform experimental measurements on large hexagonal arrays of isotropic Ni$_{50}$Fe$_{50}$ permalloy antispheres ($M_s$ = 1.1$\times$10$^{6}$ A/m, $A$ = 5.85$\times$10$^{-12}$ J/m, $K_1$ = 0 J/m$^3$); this permalloy variant has a particularly small exchange length ( $\lambda_{\mathrm{ex}}$ = 2.76 nm) and as such requires a finer mesh. When this fine mesh is combined with the need for a large number of antisphere centres, it becomes unrealistic to perform even one simulation of the resulting structure.