Results

Figure 5.9 shows a phase diagram of remanent magnetisation states for systems simulated with magpar where the height $h$ increases from $1/8d$ to $d$ in $1/8d$ steps and $d$ varies between 12.5nm and 125nm.

Figure 5.9: Phase diagram of reversal mechanisms for Ni$_{50}$Fe$_{50}$ permalloy part-spheres. The dotted and dashed lines are guides to the eye indicating reversal mechanism boundaries

We observe three distinct reversal mechanisms. Taking $d$=50nm, for $h/d\leq0.375$ the reversal is coherent; all the magnetic moments remain aligned and rotate homogeneously. Between $h/d=0.5$ and $h/d=0.875$ an out-of-plane vortex forms with a core perpendicular to the applied field after some initial energy barrier is overcome and this can freely move around the inside of the part-sphere with the applied field. This is similar to the behaviour seen in cylindrical particles shown in figures 3.3 to 3.8 and in other works (Boardman et al., 2004, Ha et al., 2003, Cowburn et al., 1999b, Li et al., 2002). We will now discuss the reversal mechanism in more detail.

Figure 5.10 shows the perpendicular vortex reversal behaviour. Point A shows the homogeneously aligned state at high applied field, though there is a small C-state-like shift in the $x$-$z$ direction at the extremities in order to minimise dipolar energy. At point B there is a shift into an S-like state in the $x$-$y$ direction, where the magnetic moments at the edges of the half-sphere persist in the applied field direction while the moments towards the centre are aligned a few degrees away from the $x$ direction into the $y$ direction. Reducing the field further overcomes an energy barrier and a perpendicular (i.e. the core of the vortex points in the $z$ direction) vortex is formed. Point C shows the remanent state of the half-sphere with this vortex in the centre; the net magnetisation in the $x$ direction is now zero. Point D shows the effects of a continued field reduction; the vortex has shifted further into the $y$ direction appropriate for allowing the majority of the magnetic moments to point in the negative $x$ direction. Finally, at point E the external field is now sufficiently low to remove the vortex from the system, and a homogeneously aligned state remains.

Figure: Reversal mechanism for $d$=50nm, $h$=0.5$d$

Figure 5.11 shows the reversal mechanism with an in-plane vortex for a sphere (i.e. $h/d=1.0$). Although the size and material differ from the sphere in section 3.5, there is a qualitative similarity we will review. Point A shows a homogeneous alignment of the magnetic moments in the $x$ direction, which persists until point B, where the field has been lowered enough to overcome the energy barrier and allow an in-plane (i.e. where the core points in the $x$ direction) vortex to form; this also allows the majority of the magnetisation to continue pointing in the $x$ direction.

As the field is further reduced, the $x$ component of the magnetisation outside the vortex core continues to follow the applied field; however the core remains pointing wholly in the direction of the initial applied field. At point C, after the field is reduced below zero the core of the vortex flips over, which is responsible for the ``minor'' hysteresis loop around $B_{\mathrm{x}}=0$. The vortex can exit the system when the field is further reduced and the magnetisation becomes homogeneous (point D).

Figure: Reversal mechanism for $d$=100nm, $h$=$d$

Figure 5.12: Hysteresis loops for a $d$=50nm half-sphere obtained with (solid line) OOMMF (finite difference method) and (dashed line) magpar (finite element/boundary element method)

Our simulation results agree with the computation of the critical radius
(O'Handley, 1999, p305) of single-domain to vortex state transition (equation 3.1) for Ni$_{50}$Fe$_{50}$ permalloy in spheres of radius 12.4nm ($d$=24.8nm): a single-domain remanent state is observed in our simulations of spheres of diameter 24nm and below where the exchange energy is dominant, whilst an in-plane vortex is in the remanent state when the diameter is 25nm or greater as the dipolar energy becomes preponderant.