Results

Figure 4.2: Phase diagram of the remanent magnetisation states for cones where the applied field was originally in the $+x$ direction. The symbols represent computed points, dashed lines are guides to the eye.

Figure 4.2 shows the phase diagram for the remanent magnetisation states in cones where $10\mathrm{nm} \leq d \leq 100\mathrm{nm}$ and $10\mathrm{nm} \leq h \leq 100\mathrm{nm}$ and the applied field was originally in the $+x$ direction (see figure 4.1). Where $h$ is less than 20nm, the remanent state is a single domain state with the magnetisation pointing in the $x$ direction (figure 4.1, left). If $h$ is above 50nm and the ratio $h/d$ is high (i.e. a tall, thin cone) then the single domain state in $z$ is preferable due to shape anisotropy (figure 4.1, second from left). We have observed two types of single domain states in $z$: one with the magnetisation pointing up towards the tip of the cone and the other with the magnetisation pointing down towards the base. The single domain states in $x$ and $z$ are in agreement with experimental data in Ross et al. (2001).

For larger $h$ and $d$ the demagnetising energy grows and the remanent state is the vortex state with the magnetisation in the core of the vortex pointing out of the $xy$ plane (figure 4.1, right).

For intermediate values of $h$ we find at large $d$ the buckle state (figure 4.1, centre), in which the overall magnetisation points in $x$ but around the centre of the $xy$ plane this bends slightly upwards and downwards in $z$. The buckling is an indication of the growing dipolar energy of the single-domain state.

At smaller $d$ for intermediate $h$ the remanent state is a C-shaped configuration (figure 4.1, second from right). The C state is related to the single domain state in $z$ by the shape anisotropy driving the magnetisation to point primarily in the -$z$ direction. In larger diameters the magnetisation will attempt to reduce the demagnetisation energy -- the bending of the magnetisation in the $+x$ direction close to the base of the cone shows the history of the system: prior to the field being reduced to zero the magnetisation was pointing in the $+x$ direction.

Figure: Hysteresis loop and reversal mechanism for a cone where $d=h=$100nm and the applied field is across the diameter. Points E, F, G and H are explained in figure 4.4.

Figure 4.3 shows the complete hysteresis loop for a cone with $d$=$h$=100nm, computed using magpar. When the applied field is reduced from saturating the magnetisation in the $+x$ direction, it forms an in-plane vortex (i.e. where the magnetisation circulates in $yz$ and the vortex core points in the $x$ direction) shown at point A. The same behaviour is observed for the magnetisation reversal of spheres (Eisenstein and Aharoni, 1975, Boardman et al., 2005b) of similar size. Note that this configuration is not observed in the absence of an applied field and therefore this state is not shown in figure 4.1.

The in-plane vortex is replaced by an out-of-plane vortex (i.e. where the magnetisation circulates in $xy$ and the vortex core points in the $z$ direction) shown at point B after overcoming an energy barrier. The core of this vortex is anchored at the tip of the cone and compensates for the applied field by tilting the core, allowing the majority of the magnetisation to align with the applied field, thus minimising Zeeman energy.

A further reduction of the field (point C) causes the core of the vortex to shift to the centre of the cone. Reducing the field below zero causes the vortex to bend in the opposite direction to point B. At $B_{\mathrm{x}}=0$mT the magnetisation is in the vortex state as shown in figure 4.1 (right). Another energy barrier needs to be overcome to destroy the out-of-plane vortex, leaving the magnetisation with an in-plane vortex (point D) with the core pointing in the opposite direction to the vortex at point A.

Once the field is sufficiently high the in-plane vortex aligns into a homogeneous saturated magnetisation in the -$x$ direction for $\vert B_{\mathrm{x}}\vert\geq250$mT.

Figure 4.4 shows specifically the magnetisation at points E, F, G and H from figure 4.3 to explain the subtle ``kinks'' in the hysteresis loop. The top row shows a magnetisation cross section along the height of the cone and the bottom row that along the base. The middle row is a schematic representation of the two vortices corresponding to the cross sections above and below.

Figure: Reversal mechanism detail for points E, F, G and H in figure 4.3. Top: $yz$ cutplanes of magnetisation shaded by the $x$ component of the magnetisation ($M_x$). Middle: schematics indicating vortex behaviour (solid arrows for the in-plane vortex shown the in top row, dashed arrows for the out-of-plane vortex shown in the bottom row). Bottom: $xy$ cutplanes of the magnetisation at the base of the cone.

Starting at point E, the system contains two vortices: the out-of-plane vortex with the core pointing in the +$z$ direction introduced at point B, and the formation of an in-plane vortex with the core in the -$x$ direction parallel to the applied field. Increasing the field in the -$x$ direction causes the in-plane vortex to become more dominant (point F). A further increase of the applied field in the -$x$ direction allows the in-plane vortex to become even more influential, moving the out-of-plane vortex to the edge of the sample (point G). There is a small energy barrier present to force the out-of-plane vortex from the system; once this has been overcome only the in-plane vortex remains (point H).