GITAM, Department of Engineering Physics


 

Domain theory of Ferromagnetism

In order to explain the fact that ferromagnetic materials with spontaneous magnetisation could exist in the demagnetised state Weiss proposed the concept of magnetic domains. Weiss built on earlier work carried out by Ampère, Weber and Ewing suggesting their existence. The findings of this work revealed that within a domain large numbers of atomic moments are aligned  typically 1012-1018, over a much larger volume than was previously suspected. The magnetisation within the domain is saturated and will always lie in the easy direction of magnetisation when there is no externally applied field. The direction of the domain alignment across a large volume of material is more or less random and hence the magnetisation of a specimen can be zero.

Magnetic domains exist in order to reduce the energy of the system. A uniformly magnetised specimen as shown in figure 5(a) has a large magnetostatic energy associated with it. This is the result of the presence of magnetic free poles at the surface of the specimen generating a demagnetising field, Hd. From the convention adopted for the definition of the magnetic moment for a magnetic dipole the magnetisation within the specimen points from the south pole to the north pole, while the direction of the magnetic field points from north to south. Therefore, the demagnetising field is in opposition to the magnetisation of the specimen. The magnitude of Hd is dependent on the geometry and magnetisation of the specimen. In general if the sample has a high length to diameter ratio (and is magnetised in the long axis) then the demagnetising field and the magnetostatic energy will be low.

The break up of the magnetisation into two domains as illustrated in figure 5(b) reduces the magnetostatic energy by half. In fact if the magnet breaks down into N domains then the magnetostatic energy is reduced by a factor of 1/N, hence figure 5(c) has a quarter of the magnetostatic energy of figure 5(a). Figure 5(d) shows a closure domain structure where the magnetostatic energy is zero, however, this is only possible for materials that do not have a strong uniaxial anisotropy, and the neighbouring domains do not have to be at 180º to each other.

(a)

(b)

(c)

(d)

 

Figure 5: Schematic illustration of the break up of magnetisation into domains (a) single domain, (b) two domains,
(c) four domains and (d) closure domains.

The introduction of a domain raises the overall energy of the system, therefore the division into domains only continues while the reduction in magnetostatic energy is greater than the energy required to form the domain wall. The energy associated a domain wall is proportional to its area. The schematic representation of the domain wall, shown in figure 6, illustrates that the dipole moments of the atoms within the wall are not pointing in the easy direction of magnetisation and hence are in a higher energy state. In addition, the atomic dipoles within the wall are not at 180º to each other and so the exchange energy is also raised within the wall. Therefore, the domain wall energy is an intrinsic property of a material depending on the degree of magnetocrystalline anisotropy and the strength of the exchange interaction between neighbouring atoms. The thickness of the wall will also vary in relation to these parameters, as a strong magnetocrystalline anisotropy will favour a narrow wall, whereas a strong exchange interaction will favour a wider wall.

Figure 6: Schematic representation of a 180º domain wall.

A minimum energy can therefore be achieved with a specific number of domains within a specimen. This number of domains will depend on the size and shape of the sample (which will affect the magnetostatic energy) and the intrinsic magnetic properties of the material (which will affect the magnetostatic energy and the domain wall energy).

The microscopic ordering of electron spins characteristic of ferromagnetic materials leads to the formation of regions of magnetic alignment called domains.

The main implication of the domains is that there is already a high degree of magnetization in ferromagnetic materials within individual domains, but that in the absence of external magnetic fields those domains are randomly oriented. A modest applied magnetic field can cause a larger degree of alignment of the magnetic moments with the external field, giving a large multiplication of the applied field.

These illustrations of domains are conceptual only and not meant to give an accurate scale of the size or shape of domains. The microscopic evidence about magnetization indicates that the net magnetization of ferromagnetic materials in response to an external magnetic field may actually occur more by the growth of the domains parallel to the applied field at the expense of other domains rather than the reorientation of the domains themselves as implied in the sketch.

 

Some of the more direct evidence we have about domains comes from imaging of domains in single crystals of ferromagnetic materials. The sketches above are after Young and are adapted from magnified images of domain boundaries in single crystals of nickel. They suggest that the effect of external magnetic fields is to cause the domain boundaries to shift in favor of those domains which are parallel to the applied field. It is not clear how this applied to bulk magnetic materials which are polycrystalline. Keep in mind the fact that the internal magnetic fields which come from the long range ordering of the electron spins are much stronger, sometimes hundreds of times stronger, than the external magnetic fields required to produce these changes in domain alignment. The effective multiplication of the external field which can be achieved by the alignment of the domains is often expressed in terms of the relative permeability.

Domains may be made visible with the use of magnetic colloidal suspensions which concentrate along the domain boundaries. The domain boundaries can be imaged by polarized light, and also with the use of electron diffraction. Observation of domain boundary movement under the influence of applied magnetic fields has aided in the development of theoretical treatments. It has been demonstrated that the formation of domains minimizes the magnetic contribution to the free energy.