Domain theory

The micromagnetic approach is capable, in principle, of predicting the equilibrium magnetic configurations of any system where the exchange stiffness A(r)

Figure 1: (a) A Bloch wall, and (b) a N´eel wall.

and anisotropy Ea(r) can be specified throughout. Hysteresis may be deduced, knowing the magnetic history in an applied field H'(t ). No account is taken of temperature. It is impractical to implement micromagnetic theory in any but idealized situations. The problem is mathematically complex, and real materials contain local defects and disorder which cannot be specified precisely, but which nonetheless tend to dominate the magnetization process.
Domain theory is an attempt to reduce this complexity to manageable proportions. It postulates the existence of large regions of uniform magnetization in a macroscopic sample, which are separated by planar regions – the domain walls – where the magnetization rotates from one easy direction to another. Domain observations support the model. If domains exist, so must domain walls. An applied field changes the net magnetization of the sample, either by causing the walls to move or by making the magnetization in the domains rotate towards the applied field direction. The magnetostatic energy depends on the wall positions and the domain orientations.
Domain theory breaks down in very soft magnetic materials, especially in thin film elements where the demagnetizing field is small. There, instead of domains, states with continuous rotation of magnetization tend to form.
Now we look more carefully into the structure of the domain walls. A flip of magnetization from one plane of atoms to the next would be prohibitively expensive, costing or about 0.1 Jm⁻². Magnetization rotates continuously over many interatomic distances under the combined influence of exchange and anisotropy. Dimensional analysis gives the wall width ,which is of order 10–100 nm, and thewall energy ,which is of order 1 mJ m⁻². The association of energy with the wall area means that the domain wall behaves like an elastic membrane or soap film. Two common types of domain wall are illustrated in Fig. 1.