Marco Institucional de la Educación en Chile

2.8.1 The hysteresis loop

The hallmark of a magnetic system is thehysteresis loop. This is traditionally repre- sented graphically as the overall magnetisation of the sample against some applied magnetic field. The value of the applied field where the loop crosses zero mag- netisation is known as thecoercivefieldHcorBc, and this therefore represents the

amount of applied field required to reverse the magnetisation direction of the mag- net. Theremanent magnetisation Mr is the magnetisation which remains when the

applied field is reduced to zero.

Comparing the hysteresis loops, such as those in figure 2.10, of a soft and a hard magnet, one can make the observation that the softer magnet will have a nar- row hysteresis loop,i.e. the applied field necessary to reverse the magnetisation is relatively low, and the hard magnet will possess a comparatively wide hysteresis loop.

The point at which the overall magnetisation of a sample can no longer be in- creased (as all the magnetisation is pointing utterly in a single direction) — the saturationpoint orMs — is identified as a plateau at the extremes of applied field

in a hysteresis loop.

Also one should note that the area underneath the hysteresis loop is equivalent to the energy which, when the field is reversed, is converted into heat.

For the long-term storage of data, it is desirable to have a material with a wide hysteresis loop, and therefore a large coercive field, as this makes it more difficult for the said material to lose its magnetisation state. A narrow hysteresis loop is a characteristic beneficial for applications such as recording heads, as in these tem- porary magnetisation promotes easy switching between magnetisation states. The ideal hysteresis loops for applications in magnetic media can be seen in figure 2.11.

2.8.2 Domains

Figure 2.12 shows a relatively large (i.e. a size order of10−6 metres) ferromagnet

which contains domains. Domains can be thought of as the magnetic structures which form at small scales within magnets in particular circumstances (Hubert and Sch¨afer, 1998, 2000). Within these domains the magnetisation is parallel, though the overall magnetisation of any given domain is not in a particular direction. This gives rise to a mean magnetisation of approximately zero across a sample in zero field. Figure 2.13 illustrates an example of domains formed in a sample with a simple closed flux.

At high applied fields — what defines a high field is dependent on the type, size and shape of the magnet; it must be enough to fully saturate the magnetisation

-1 -0.5 0 0.5 1 applied field H / M_s -1 -0.5 0 0.5 1 magnetisation / Ms -1 -0.5 0 0.5 1 applied field H / M_s -1 -0.5 0 0.5 1

Figure 2.10:Two typical hysteresis loops — the left loop shows some permanently magnetic material, the right loop a softer magnet. The solid blue line indicates reducing field, the dashed red line indicates increasing field

Applied field -1 -0.5 0 0.5 1 Magnetisation medium head

Figure 2.11:Magnetic recording ideals. A square loop with a high coercivity is good for the long- term storage of data; an infinitely narrow loop with diagonal characteristics is desirable for the field switching required of read heads in magnetic media applications

Figure 2.12:A typical ferromagnet in zero field (left) and in an applied field (right)

1000nm

Figure 2.13:Flux closure (left), and (right) a larger sample attempting to close its flux through do- mains.

— no individual domains will form as the overall magnetisation in the sample is homogeneous at these fields; this can be considered to be asingle domain. However, when these fields are reduced, other domains can form in order to minimise the overall magnetisation, which often remains at zero field.

Smaller ferromagnets exhibit the property of magnetisation alignment with an applied magnetic field, though below a certain critical size they will not form do- mains but may form states (see section 2.8.3).

2.8.3 States — microstructures of magnetisation

At nanometre length scales in magnetic samples, particularly interesting states oc- cur (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.

TheCstate (see figure 2.14, top centre) is known as such because the magneti- sation 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.

TheSstate (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 sam- ple along the overall magnetisation, and into the sample at the other side of the

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≈200nm, the

circular samples of size order≈500nm. Parameters for isotropic nickel (A= 8.5×10−12

J/m,Ms= 4.93×105A/m,K1=K2= 0 J/m3) were used in these sample simulations. overall magnetisation. Further examples showing theC,Sandflowerstates can be seen inHuang(2003).

At lower fields, or in larger sample sizes, thevortexstate might occur (see fig- ure 2.14, bottom centre). This is where the magnetisation in a sample curls in order to minimise its dipolar energy, except at the centre, orcore, 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 theonion 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.