Physics of Magnetism

156 CHAPTER 14. SOFT-MAGNETIC MATERIALS

grains is quite substantially reduced by exchange interaction (Herzer, 1989,
1996). The critical scale where the exchange energy starts to balance the anisotropy energy
is given by the ferromagnetic-exchange length

where A represents the average exchange energy as already introduced in Chapter 12. For

the value of the exchange length is about The quantity

is a measure of the minimum length scale over which the direction of the magnetic

moments can vary appreciably. For example, it determines the extent of the domain-wall

width, as was discussed in Chapter 12. However, the magnetization will not follow the

than the exchange length

randomly oriented easy axes of the individual grains if the grain size, D, becomes smaller

Instead, the exchange interaction will force the magne­

tization of the individual grains to align parallel. The result of this is that the effective

anisotropy of the material is an average over several grains and, hence, will strongly reduce

in magnitude. In fact, this averaging of the local anisotropies is the main difference with

large-grain materials where the magnetization follows the randomly oriented easy axes of

the individual grains and where the magnetization process is controlled by the full magne­

tocrystalline anisotropy of the grains. A more detailed description by means of which one

can quantitatively describe this dramatic reduction in anisotropy will be presented for the

interested reader in the next section.

14.3. THE RANDOM-ANISOTROPY MODEL

The random-anisotropy model has originally been developed by Alben et al. (1978) to

describe the soft-magnetic properties of amorphous ferromagnets. The advent of nanocrys­

talline magnetic materials has shown, however, that the model is of substantial technical

relevance and more generally applicable than considered by Alben. The random-anisotropy

model has been applied to nanocrystalline soft-magnetic materials by Herzer (1989,1996)

and the simplified version of the model presented in the review by Herzer (1996) will be

followed here.

A schematic diagram representing an assembly of exchange-coupled grains of size D

is given in Fig. 14.3.1. The volume fraction of the grains is and their easy magnetization

directions are statistically distributed over all directions. The effective anisotropy constant,

, relevant to the magnetization process of the whole material, can be obtained by aver­

aging the individual grain anisotropies over the grains contained

within the ferromagnetic-correlation volume determined by the exchange length

For a finite number N of grains contained within the exchange volume, there will

always be some easiest direction determined by statistical fluctuations. Thus, the averaged

anisotropy-energy density is determined by the mean fluctuation amplitude of the anisotropy

energy of the N grains, that is,