Flat cylinder

We compute a hysteresis loop as shown in figure 3.3 in the following way: initially, a magnetic field is applied in the simulation such that the magnetisation across the cylinder is homogeneous (see figure 3.4).

Figure 3.4: Overview of the cylindrical geometry (left) with the magnetisation (represented by the cones) in a high applied field (right). The rectangular surface on the right is a cut-plane through the sample, the colour of which reflects the scalar value of the in-plane $x-y$ angle.

Figure 3.5: Flat cylinder with homogeneous magnetisation (point $a$ in figure 3.3). The colour is representative of the in-plane $x$-$y$ angle. There is a small out-of-plane magnetisation shift due to demagnetising energy at the edges which is not present in a high applied field (figure 3.4)

Figure 3.6: The flower state and the onion state shown together in a cylinder of diameter 200nm and height 40nm in zero applied field. The flower state is shown by the coloured cut-plane ($xz$) and exists in the $x$-$z$ plane. The onion state is shown by monochrome streamlines which follow the magnetisation direction; this exists in the $x$-$y$ plane.

For $B_x =$ 0mT (point $a$ in figure 3.3) the magnetisation appears roughly homogeneous (figure 3.5) but closer inspection (figure 3.6) reveals a slight out-of-plane magnetisation shift at either end of the sample due to a small contribution from the dipolar interaction. This can also be seen in figure 3.5 (Bertram, 1994), and overall an onion state occurs, present to minimise the dipolar surface charges (Ha et al., 2003); this is shown in figure 3.6, and the homogeneous magnetisation state at this point can be seen with a normalised colour scale in figure 3.5 and corresponds to point $a$ in figure 3.3.

As the applied field is reduced further a vortex forms in the $x$-$y$ plane (figure 3.7). Note that this vortex (point $b$ in figure 3.3) does not appear until after the applied field has passed zero; the vortex appears here as there is an energy barrier which it must first overcome. This results in the vortex forming slightly off-centre, so that when the vortex is created the magnetisation jumps from a high positive value to approximately $-0.1M_S$, where $M_s$ is the saturation magnetisation (see figure 3.5). The position of the core of the vortex is a reflection of the amount of magnetisation which is following the applied field to minimise the Zeeman energy.

As the field is further reduced, the core of the vortex can be seen to pass through the cylinder in the negative $y$ direction until $M_x = -0.7M_s$ (figure 3.8), corresponding to point $c$ in figure 3.3; at this point the core of the vortex is close to the edge of the cylinder and disappears with another increase in the magnitude of the applied magnetic field, leaving the magnetisation once more homogeneous in the direction of the applied field, which is the opposite situation to that in figure 3.5.

Figure 3.7: Flat cylinder entering the vortex state (point $b$ in figure 3.3). The direction of the cones corresponds to the direction of the magnetisation; their colour shows their angle in the $x$-$y$ plane -- yellow is 0$^\circ$ from +$x$, blue is 180$^\circ$ from +$x$. The translucent cutplane in $y$-$z$ shows the magnetisation pattern is consistent throughout the height, and the grey isosurface outlines the core of the vortex.

Figure 3.8: Flat cylinder just before leaving the vortex state (point $c$ in figure 3.3).

Taking a cylinder of diameter 200nm, we study the microstructures of the magnetisation which form for different heights of a cylinder between 5nm and 100nm. Using the technique outlined in section 2.7.2 we assign a uniform magnetisation in the $x$ direction, allow the system to relax in zero applied field, and make observations.

Figure 3.9: Height dependence of the remanent state in cylinders - vertical cutplanes in the $xz$ plane of a cylinder of diameter 200nm in zero applied field are shown. The images are (left) a cylinder of height 35nm, (centre) $h=$75nm and (right) $h=$85nm. The colour variation represents the $z$ component of the magnetisation.

$h$In geometry, the height of an object, usually measured along the $z$ axis

Figure 3.9 shows the flower state, clearly visible in the shorter cylinders; however as the height is increased, the dipolar energy exerts a greater influence, as in figure 3.6. The dipolar energy continues to exert more and more pressure on the cylinder to abandon its flower state in favour of a vortex -- at a height of 75nm the top and bottom of the cylinder are completely dominated by the magnetostatic energy. Increasing the height a little further from here to 80nm causes this structure to collapse and fall into the vortex state at zero field.

Figure 3.10: Phase diagram of the remanent magnetisation state for nickel cylinders of varying height and diameter

Figure 3.10 shows the reversal behaviour of nickel cylinders dependent on height and diameter. As height is increased, the single domain behaviour disappears and vortex reversal behaviour occurs; the same is true as the diameter is increased.