Parameters

We use the material parameters for Ni$_{80}$Fe$_{20}$ permalloy ($J_s$ = 1.0$\;$T, $A$ = 1.3 $\times10^{-11}$ J/m, $K_1$ = 0.0$\;$J/m$^3$) (Skomski and Coey, 1999) and a damping constant $\alpha$ of 0.25 to improve convergence.

We perform simulations on cones where the overall diameter $d$ of the base ranges from 10nm to 100nm, the height $h$ between the base and the tip ranges from 10nm to 100nm and the external magnetic field is applied along the $x$ direction of the cone (see figure 4.1).

The mesh for the magpar hybrid FE/BE simulation is created using NETGEN (Schöberl, 2003). For the cone where $d=100$nm and $h=100$nm, the mesh contains 210825 tetrahedra, with the largest element having an edge length smaller than 4nm, which is below (Donahue and McMichael, 1997) the calculated exchange length $\lambda_{\mathrm{ex}} = \sqrt{2A/ \mu_0M_s^2}$ (Kronmüller and Fähnle, 2003) of 5.71nm.

The remanent magnetisation configurations using OOMMF and magpar are in agreement providing the shape is sufficiently well resolved in both cases. To ensure this is the case with OOMMF, the cones were discretised into 8000 cells, in the largest case the cell edge length is 5nm with a maximum adjacent cell spin angle of 0.5 radians.

Starting from an initially uniform magnetisation state pointing in the $+x$ direction, we apply a magnetic field of 500mT capable of maintaining a nearly homogeneous magnetisation and reduce this field in steps of 1mT until the magnetisation is reversed to compute the hysteresis loop. When the applied field is reduced to zero, we classify the magnetisation pattern to create the remanence phase diagram. Figure 4.1 shows schematic plots of the observed remanent states.

Figure 4.1: Representation of remanent magnetisation states; the solid arrows indicate the magnetisation direction, the dotted arrow shows the vortex core direction. From left to right: single domain state in $x$, single domain state in $+z$ (filled arrow) and $-z$ (hollow arrow), buckle state (Koltsov et al., 2000), C state and out-of-plane vortex state.