Multiple vortex states

As the diameter of a part-spherical geometry is increased, further characteristics are observed which resemble those of the full sphere.

Figure 5.13 shows the hysteresis loop for an isotropic nickel half-sphere of diameter 350nm. This hysteresis loop shares some similarity with that of a smaller half-spherical system, such as that in figure 5.10 -- it has zero remanence and there are openings in the hysteresis loop. The characteristic difference between the two loops is there are two energy barriers: the first occurs when $B_x$ is around 100mT, followed by another when $B_x$ is approximately 70mT, corresponding to the kink in the curve at 60mT in figure 5.10.

Figure 5.14 reveals the magnetic microstructures in this model. The image on the left shows the magnetisation when $B_x$ is 80mT. An in-plane vortex has formed and the magnetisation circulates in the $yz$ plane, with the core of the vortex pointing in the $x$ direction with the applied field. The image on the right shows the magnetisation after $B_x$ has been reduced to 65mT. The in-plane vortex has been replaced by a perpendicular vortex: the magnetisation now circulates in the $xy$ plane with the core of the vortex pointing in the $z$ direction. The core of the vortex is off-centre to allow the majority of the magnetisation to point in the direction of the applied field.

Figure 5.13: Hysteresis loop for an isotropic nickel half-sphere of diameter 350nm. Two energy barriers can be seen: one at $\approx$100mT and another at $\approx$70mT. The first indicates the transition from the single-domain state to the in-plane vortex state, the second the transition from the in-plane vortex state to the perpendicular vortex state.

Figure 5.14: Two vortex states in an isotropic nickel half-sphere of diameter 350nm. The images show vector cut-planes of magnetisation in the $y$-$z$ direction -- applied field was in the $x$ direction. The in-plane vortex (left) is clearly visible at $B_x\approx$80mT. The perpendicular vortex (right) is in a field of $\approx$65mT, and is off-centre to compensate for the applied field.

Figure 5.15 shows the hysteresis loop when the diameter of the nickel half-sphere is increased to 750nm. Qualitatively, the same behaviour is observed as when the half-sphere diameter is 350nm. Point A shows the fully-saturated magnetisation at a high applied field in the $+x$ direction. When this is reduced to overcome the first energy barrier (point B) then the in-plane vortex forms, with the core of the vortex pointing in the direction of the applied field. As the field is reduced to immediately prior to the second energy barrier at around 75mT (point C), the magnetisation around the core of the vortex deviates further away from the $x$ direction to minimise dipolar energy.

Point D shows the magnetisation immediately after the second energy barrier has been overcome. The in-plane vortex has disappeared to be replaced by a perpendicular vortex, with the core pointing in the $z$ direction. The majority of the magnetisation remains in the direction of the applied field. Reducing the field to zero (point E) sees the vortex core move across the sample in the $y$ direction, causing the average $M_x$ to be zero. Increasing the applied field in the opposite direction ($-x$) causes the vortex core to move further along the $y$ direction (point F) allowing the majority of the magnetisation to point in the $-x$ direction, similar to point D.

Increasing the field further in the $-x$ direction overcomes another energy barrier present around -125mT (point G), causing the perpendicular vortex to disappear and an in-plane vortex with the core pointing in $-x$ to form, which disappears when the applied field is increased to -140mT (point H).

Figure 5.15: Hysteresis loop for isotropic nickel half-sphere of diameter 750nm simulated with magpar. The coloured sections of the smaller images represent the magnetisation in the $x$ direction

Figure 5.16: Vortex ``pinning'' in a part-sphere of height $0.75d$. After entering an in-plane vortex state (left), the upper part of the vortex core is situated about the centre of the $xy$ plane while the lower part is towards $+y$. When the field is reduced to zero (centre), the upper and lower parts of the vortex core are around the centre of the $xy$ plane. If a negative field is then applied (right), then the lower part of the vortex core moves towards $-y$, but the upper part remains at the centre of $xy$.