Cullity B.D. Introduction to Magnetic Materials. Second Edition (2008)

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equivalent to Lc of Chapter 9.) Particles of size Ds and smaller change their magnetization by spin rotation, but more than one mechanism of rotation can be involved (Section 11.4).

3.As the particle size decreases below Ds the coercivity decreases, because of thermal effects, according to

where g and h are constants. This relation is well understood (Section 11.6).

4.Below a critical diameter Dp the coercivity is zero, again because of thermal effects, which are now strong enough to spontaneously demagnetize a previously saturated assembly of particles. Such particles are called superparamagnetic (Section 11.6).

The magnetic hardness of most fine particles is due to the influence of shape and/or magnetocrystalline anisotropy. In order to study the effect of either of these alone, one can try to eliminate shape anisotropy by making spherical particles, or make elongated particles of a material having low or zero crystal anisotropy. In order to form a practical magnet, the fine magnetic particles must be compacted, with or without a nonmagnetic binder, into a rigid assembly. An important variable is then the packing fraction p, defined as the volume fraction of magnetic particles in the assembly. The variation of coercivity with p may depend on the kind of anisotropy present.

When shape anisotropy prevails, the coercivity Hci decreases as p increases, because of particle interactions. The following relation has been proposed:

where Hci(0) is the coercivity of an isolated particle ( p ¼ 0). However, the theoretical basis for this expression is unclear. Some materials obey this relation but many do not. Figure 11.3 shows the behavior of elongated Fe–Co alloy particles having an axial ratio

˚

greater than 10 and an average diameter of 305 A or 30.5 nm.

The nature of the particle-interaction problem is indicated in Fig. 11.4, for elongated single-domain particles. In (a), part of the external field of particle A is sketched, and this field is seen to act in the þz direction on particle B below it, but in the 2z direction on particle C beside it. Thus the “interaction field” depends not only on the separation of the two particles but also on their positions relative to the magnetization direction of the particle considered as the source of the field. Suppose the Ms vectors of these particles had all been turned upward by a strong field in the þz direction. This field is then reduced to zero and increased in the 2z direction. The field of A at C now aids the applied field, and C would reverse its magnetization at a lower applied field than if A were absent; the coercivity would therefore be lowered. The opposite conclusion would be reached if we considered only the pair of A and B particles. In either case we have a relatively easy two-body problem to deal with. A quantitative study of the interactions of three particles is very difficult, and the exact solution of the many-body problem represented by Fig. 11.4b, a cylindrical compact of a large number of particles, is virtually impossible. If a reversing field Ha is applied in the 2z direction, what is the true field acting on the shaded particle in the interior of the compact? Note that it is not enough to

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individual particles no longer exist. They become grains or crystallites in a polycrystalline material, and their magnetic behavior will depend the coupling between them. This topic will be considered further in Chapter 14.

11.4MAGNETIZATION REVERSAL BY SPIN ROTATION

In Section 9.12 we examined the hysteresis loops of uniaxial single-domain particles reversing their magnetization by rotation, as calculated by Stoner and Wohlfarth. The assumption made there was that the spins of all the atoms in the particle remained parallel to one another during the rotation. This mode of reversal is called coherent rotation, or the

Stoner–Wohlfarth mode.

For spherical single-domain particles of iron, with their easy axes aligned with the field, the intrinsic coercivity Hci due to crystal anisotropy is equal to

2K ¼ 2(4:8 105) ¼ 560 Oe ¼ 45 kA=m, from Equation 9:51: Ms 1714

Aligned elongated particles, reversing by coherent rotation, have a coercivity due to shape anisotropy given by Equation 9.46, namely,

where Na is the demagnetizing factor along the short axis and Nc along the long axis. The

For aligned iron particles we calculated, in Table 9.2, that Hci should be 10,100 Oe (800 kA/m) for an axial ratio c/a of 10, and 10,800 Oe (865 kA m) for infinite elongation. About 1955 a method was developed for making very thin, elongated, iron particles with axial ratios from about 1 to well over 10 by electrodeposition on a mercury cathode. (Details of the method are given in Section 14.10.) Measurements on aligned, dilute compacts of these particles showed that Hci increased with c/a but did not exceed about 1800 Oe for c/a larger than 10 [F. E. Luborsky, J. Appl. Phys., 32 (1961) p. 171S]. These results showed that the observed coercivity was larger than could be explained by crystal anisotropy, but not as large as expected for shape anisotropy. In fact, it was less than 20% of the theoretical value. This discrepancy forced theoreticians to question the assumption on which Equation 11.4 is based, namely, coherent rotation of the spins. They therefore examined possible modes of incoherent rotation, in which all spins do not remain parallel, to see if such modes would give coercivities in better accord with experiment. The two most important of these incoherent modes are magnetization fanning and curling.

11.4.1Fanning

This mode was suggested by the shape of the electrodeposited iron particles observed with the electron microscope. Figure 11.5a shows that these particles have a kind of “peanut shape,” characterized by periodic bulges rather than smooth sides. This suggests that they can be approximated by the “chain of spheres” model shown at A and B of Fig. 11.5b for two-sphere chains. I. S. Jacobs and C. P. Bean [Phys. Rev., 100 (1955)

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dipoles therefore causes the pair to have a uniaxial anisotropy with an easy axis along the line joining the dipoles. This has been called interaction anisotropy.

In the fanning mode, the spins in one sphere are not parallel to those in the adjacent sphere at the point of contact. Some exchange energy is therefore introduced. But exchange energy is essentially short range, which means that the spins contributing to this energy in the fanning mode form only a small fraction of the total. Thus the total exchange energy is considered to be small, and it can be made still smaller by imagining the spheres to be slightly separated. It is neglected in these calculations.

To find the coercivity of the two-sphere chain in a field H parallel to the chain axis, we note that the potential energy Ep in the field is 2mH cos u, when H is antiparallel to m. The total energy is then

This equation is of the same form as Equation 9.37 for a uniaxial particle reversing coherently. Therefore, a fanning reversal is also characterized by a rectangular hysteresis loop, and the coercivity is the field at which the moments will flip from u ¼ 0 to u ¼ 1808. To find it we set d2E/du2 equal to zero and proceed in exactly the same way as in Section 9.12 The result, for the intrinsic coercivity, is

or three times as large as Hci for fanning. The easier reversal for the fanning mode is due to the fact that the field must overcome an energy barrier for reversal only one-third as high for fanning as for coherent rotation (Problem 11.1). The physical reason is suggested in Fig. 11.6. Fanning brings north and south poles closer together, thus reducing the spatial extent of the external fields of the spheres and hence the total magnetostatic energy.

Jacobs and Bean also computed the coercivity for linear chains longer than n ¼ 2, where n is the number of spheres in the chain, both for chains aligned with the field and for chains randomly oriented. Similar calculations were made for reversal mechanism C, where n is now the axial ratio of the prolate spheroid, by means of Equation 11.4. The results, for chains aligned with the field, are shown in Fig. 11.7 both in terms of the intrinsic coercivity

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Hci of iron particles and in terms of the reduced intrinsic coercivity, defined as

which gives Hci as a fraction of the maximum attainable coercivity in a prolate spheroid undergoing coherent rotation. The predictions of the fanning theory are seen to be in rather close accord with the experimental results on highly elongated iron particles, suggesting that the reversal mechanism is incoherent rotation. This agreement is somewhat surprising, since fanning mechanism A assumes negligible exchange interaction between adjacent spheres because of their minimal contact. But if the contact area between spheres is large, the exchange forces would favor coherent-rotation mechanism B. The actual shape of these particles, seen in Fig. 11.5a, is that of a cylinder with periodic bulges, which is closer to a chain of squashed-together spheres than to a chain of spheres in point contact. Nevertheless, there must be something in the structure of these particles that forces them to reverse incoherently rather than coherently.

11.4.2Curling

This mode of incoherent reversal was investigated theoretically by the methods of micromagnetics. The calculations are too intricate to reproduce here, and only the main results will be given. Consider a single-domain particle in the shape of a prolate spheroid, initially magnetized in the þz direction parallel to its long axis. Suppose a field is then applied in the 2z direction, causing each spin to rotate about the radius, parallel to the xy plane, on which it is located. This curling mode is illustrated in perspective in Fig. 11.8a, where the reversal is shown as about one-fourth completed. When the reversal has gone half way, the spins are all parallel to the xy plane and form closed circles of flux in all cross sections, as shown in Fig. 11.8b. If the axial ratio of the spheroid approaches infinity, so that it approximates an infinite cylinder, the spins are always parallel to the surface during a curling reversal, so that no free poles are formed and no magnetostatic energy is involved. The energy barrier to a curling reversal is then entirely exchange energy, because the spins are not all parallel to one another during the reversal. In contrast, the coherent rotation shown in Fig. 11.8c and d produces free poles on the surface, and therefore magnetostatic energy, but no exchange energy. We can also conclude that, if highly elongated particles in an assembly reverse by curling, the coercivity should be independent of the packing fraction, because no magnetostatic energy is involved, again in sharp contrast to coherent rotation.

As the axial ratio c/a of the spheroid (c ¼ semi-major axis, a ¼ semi-minor axis) changes from infinity (for the very long cylinder) to unity (for the sphere), some magnetostatic energy will be generated during a curling reversal, because the spins are no longer always parallel to the particle surface. The energy barrier then includes exchange and magnetostatic energy, and the importance of the latter increases as c/a decreases. For values of c/a less than infinite the coercivity will therefore depend on the packing fraction.

The calculations show that the coercivity in the curling mode is markedly sizedependent. A convenient and fundamental unit of length with which to measure size is

Fig. 11.8 (a) and (c) are curling and coherent-rotation modes. (b) and (d) are cross sections normal to the z axis after a 908 rotation from the þz direction.

where A is the exchange stiffness or exchange constant. It is a measure of the force tending to keep adjacent spins parallel to one another, i.e., of the torsional stiffness of the spin–spin coupling; it is related to the exchange integral Jex of Equation 4.31 by

where S is the spin, a the lattice parameter, and n is the number of atoms in the unit cell (2 for bcc, 4 for fcc).

The micromagnetics calculation of shape anisotropy begins with the assumption that the particle is a single domain with all spins initially parallel to the þz direction in zero field. Crystal anisotropy is ignored and so are thermal effects. The coercivity is then found by calculating the field in the 2z direction that is just enough to supply the energy required for a reversal by curling. The result is found to depend both on particle size and shape, as follows:

1. Infinite Cylinder. If the diameter of the cylinder is D, then the reduced coercivity is