States -- microstructures of magnetisation

At nanometre length scales in magnetic samples, particularly interesting states occur (see figure 2.14) as a result of the system attempting to reduce its overall energy.

The single-domain state, also called the monodomain state (see figure 2.14, top left), occurs when an infinitely large external field is applied to a magnetic material. In small particles, the single-domain state is often maintained as the field is reduced since the exchange energy is the most dominant term.

The C state (see figure 2.14, top centre) is known as such because the magnetisation direction roughly reflects the curve of the letter ``C'', tending to point along some direction in one part of the sample and gradually changing to the opposite direction in another part of the sample.

The S state (see figure 2.14, top right) is also named after the shape of the letter it reflects. The magnetisation undulates along the sample pointing initially in one direction, gradually turning towards another direction and then finally pointing back in the initial direction.

A cuboidal geometry of a certain size with a saturated magnetisation can fall into the flower state when an applied field is removed (see figure 2.14, bottom left). In this state the magnetic moments at the extremities point out of the sample along the overall magnetisation, and into the sample at the other side of the overall magnetisation. Further examples showing the C, S and flower states can be seen in Huang (2003).

Figure 2.14: Common metastable states of magnetisation microstructures. Top row: (left) single-domain state -- homogeneous magnetisation, (centre) C state and (right) S state. Bottom row: (left) flower state, (centre) vortex state and (right) onion state. The colour indicates the in-plane angle of magnetisation; the square samples are of size order $\approx$200nm, the circular samples of size order $\approx$500nm. Parameters for isotropic nickel ($A$ = 8.5$\times$10$^{-12}$ J/m, $M_s$ = 4.93$\times$10$^5$ A/m, $K_1$ = $K_2$ = 0 J/m$^3$) were used in these sample simulations.

At lower fields, or in larger sample sizes, the vortex state might occur (see figure 2.14, bottom centre). This is where the magnetisation in a sample curls in order to minimise its dipolar energy, except at the centre, or core, of the vortex, where a minimisation of exchange energy causes the magnetisation here to point in one particular direction; in this case out of the plane.

In ring samples the onion state (see figure 2.14, bottom right) is likely to occur as an applied field is reduced. This state often occurs prior to vortex nucleation. The majority of the magnetisation is homogeneous, however towards the edges the magnetisation tends to follow the shape of the sample.