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

Fig. 9.21 Wall motion in a Ni–Co alloy platelet 165 mm across. Bitter patterns, dark field illumination.

snaps irreversibly to Fig. 9.21c. The reason that the 1808 walls are so thin, or invisible, in these photographs is that the crystal anisotropy is low in this alloy; the walls are accordingly very broad, produce only a weak field gradient above the surface, and attract little or no colloid. The 908 walls are thinner and attract more colloid.

The domain structure observed on the surface of a polycrystalline specimen is usually quite complex, because (a) the grain surfaces are rarely even approximately parallel to crystal planes of low indices like f100g or f111g, except in certain specimens having a high degree of preferred orientation, and (b) the grain boundaries tend to interrupt the continuity of magnetization from grain to grain. Because the easy directions of magnetization have different orientations in two adjoining grains, the grain boundary between them is also a domain boundary, although not of the usual kind since it cannot move or adjust its orientation. The normal component of Ms is rarely continuous across a grain boundary, which therefore becomes a source of free poles and magnetostatic energy. Although domains are not continuous across a grain boundary, domain walls often are, and Fig. 9.22 shows an example of this. Spike domains often form at grain boundaries to reduce the free pole density there, just as they do at the surface of some single crystals (Fig. 9.17b).

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Fig. 9.22 Domains in a polycrystalline sample of Fe þ 3% Si.

9.5SINGLE-DOMAIN PARTICLES

There are good reasons to believe that a sufficiently small magnetic particle will not contain domain walls and so will consist of a single domain. One simple argument is that, if the particle is smaller than the thickness of a domain wall, it cannot consist of two domains separated by a domain wall. Another way to approach the problem is to note that the total magnetostatic energy of a single domain particle of diameter D will vary as the particle volume, which is proportional to D3, while the domain wall area and energy will vary as the cross-sectional area of the particle, which is proportional to D2. If these are the only energies to consider, there must be a critical diameter Dc below which the single-domain state is favored.

The calculation of the critical size for single-domain behavior in specific cases turns out to be tricky and uncertain. The magnetostatic energy will depend on the saturation magnetization of the material and on the geometry of the particle. The domain wall energy will depend on the values of the anisotropy constant(s) and the exchange stiffness, and perhaps also the magnetostriction and elastic constants, as well as on the geometry. And the exact magnetization configuration in the tiny particle may not be as simple as a pure ferromagnetic state or two purely ferromagnetic regions separated by a domain wall whose structure and energy were calculated for an infinite solid.

Robert C. O’Handley [Modern Magnetic Materials, Principles and Applications, WileyInterscience (2000)] gives formulas for various cases. The solution generally ends up as a function of s/M2s , where s is the domain wall energy and Ms is the saturation magnetization, and the critical size for a single-domain particle is similar to the value of the domain wall thickness. So in general single-domain particles are expected to be less than 1000 atoms in diameter.

Experimental determination of a critical size for single-domain behavior is also uncertain. Particles in this size range are hard to prepare with controlled size and shape, and very difficult to isolate and measure. The particles in an assembly are unlikely to all

have exactly the same size and shape, and will interact magnetically unless the individual particles are sufficiently separated. Some theoretical and experimental approaches to this general problem will be discussed in Chapter 14.

9.6MICROMAGNETICS

The two preceding sections have been devoted to understanding the configuration of the local magnetization vector Ms in a specimen subjected to no external field. We reached two conclusions:

1.If the specimen exceeds a certain critical size, it will divide itself into domains, in each of which Ms is everywhere parallel, separated by domain walls, in which the direction of Ms varies with position.

2.If the specimen size is less than a critical value, Ms is everywhere parallel.

As far as large specimens are concerned, the problem is one of finding the domain arrangement of lowest total energy. For example, in considering the cubic crystal of Fig. 9.18, we found that the crystal would not consist solely of slab-like domains magnetized parallel and antiparallel to [010], because the magnetostatic energy of such an arrangement can be eliminated by putting in closure domains at the ends. But these domains contribute magnetoelastic energy. It is then necessary to make the closure domains smaller without, however, adding too much wall energy. Continuing in this manner, we finally arrive at an equilibrium width D of the main domains, which in turn defines the sizes of the other domains and the total amount of wall area. This whole process of devising a domain configuration of minimum energy has been criticized by W. F. Brown Jr [Magnetostatic Principles in Ferromagnetism, North-Holland (1962)] as being insufficiently rigorous. He points out that “the particular configuration devised is dependent on the ingenuity of the theorist who devised it; conceivably a more ingenious theorist could devise one with even lower free energy.”

Brown instead advocates a rigorous approach, called micromagnetics. Here we forget about domains and domain walls. But we allow the Ms vector, of constant magnitude, to have a direction which is a continuous function of its position x, y, z in the crystal. We then express the various energies (exchange, anisotropy, magnetostatic, etc.) in terms of these directions throughout the crystal. The resulting equations are then to be solved for the spin directions at all points. If the crystal is a large one, then the existence of domains (regions of parallel spin) and the positions of domain walls (regions of rapid change of spin direction) should come quite naturally out of the solution. If the crystal is very small, then the solution should indicate that the spins are everywhere parallel, making it a single domain. In each case we must begin with a complete physical description of the crystal; this would include such things as its crystal anisotropy, magnetostrictive behavior, size, shape, and the presence or absence of imperfections.

While there is no doubt that micromagnetics is more rigorous and intellectually satisfying than domain theory, the mathematics involved in the micromagnetics approach is of formidable complexity. As a result, only certain rather limited kinds of problems have been attempted. In passing, we note that the problem of determining the spin structure of a domain wall, treated in Section 9.2, is a simple problem in micromagnetics, because there we allowed the spin direction to be a function of position, subject to certain constraints.

now known to be due mainly to domain walls making sudden jumps from one position to another. Either way, the Barkhausen effect is evidence for the existence of domains, and it was the first evidence in support of Weiss’ hypothesis of 13 years earlier.

In a classic paper published in 1949, H. J. Williams and W. Shockley [Phys. Rev. 75 (1949) p. 178] reported direct visual evidence of jerky domain wall motion. They made a single crystal of Fe þ 3.8% Si in the form of a hollow rectangle (picture frame) with each side parallel to an easy direction of the form k100l and all faces parallel to f100g. The overall dimensions were 19 13 mm, and each side was 1 mm wide and 0.7 mm thick. They examined the whole surface of the polished crystal by the Bitter technique and found the particularly simple domain structure shown in Fig. 9.24a. This shows the demagnetized state, and the crystal contains only eight domains. The domain walls, shown as dashed lines, were found, by observation of the other side, to pass straight through the crystal normal to the plane of the drawing. A magnetizing coil was wound on one leg and a flux-measuring coil on the opposite leg of the sample, marked in the figure as the H and B coils, respectively. When the applied field H was clockwise, the domain wall in each leg moved outward until it reached the edge of the specimen at saturation; each leg of the crystal was then a single domain. While the wall AB, for example, was moving, it could be observed with a microscope focused on the top leg, as indicated in Fig. 9.24a. The wall motion was generally fairly smooth, but now and then jerky when the wall encountered an inclusion. The nature of this interaction will be described in the next section.

However, the Williams–Shockley experiment has an importance much more fundamental than its clarification of the Barkhausen effect. When the applied field was changed from its maximum clockwise value to its maximum counterclockwise value and back again, the crystal went through its hysteresis cycle, and the recorded loop is shown in Fig. 9.24b. At the same time photographs of the wall position in one leg were made through the microscope. It was found (Fig. 9.24c) that there was a direct linear relation between wall position measured on the film and the magnetization M of the specimen, as would be expected from the previous observation that the walls went straight through the crystal. This was the first experimental demonstration of a relation between domain wall motion and change in magnetization. It showed that observation of a domain wall at its intersection with a surface could be correlated with a real volume effect, as measured with the B coil.

Two other points of interest in this experiment are:

1. The hysteresis loop is “square,” where this word refers to the relative squareness of the loop corners rather than to the shape of the whole loop. The coercive field of less than 0.02 Oe (1.6 A/m) is a measure of the extreme magnetic softness of this specimen. And when the magnetization begins to reverse, a field change of less than 0.005 Oe (0.4 A/m) is enough to effect essentially complete reversal. Materials with square hysteresis loops are important in many applications, and we shall return to this subject later.

2.When a magnetized rod-shaped crystal is heated above the Curie point and cooled again, it becomes demagnetized, in order to reduce its demagnetizing field and the associated magnetostatic energy. But when this picture-frame crystal was cooled from the Curie point, it was observed to be saturated, clockwise or counterclockwise, each leg a single domain. Because it forms a closed magnetic circuit, this crystal can have no demagnetizing field. Therefore, its state of minimum energy is one of minimum

304 DOMAINS AND THE MAGNETIZATION PROCESS

Fig. 9.24 The picture-frame experiment. [H. J. Willliams and W. Shockley, Phys. Rev., 75 (1949) p. 178.]

domain wall area. This is the saturated state, with only four short 908 domain walls at the corners. Since the crystal anisotropy energy is zero, the only other source of energy is some magnetoelastic energy due to domain misfit at the corners.

In previous chapters we noted that magnetization can change as a result both of domain wall motion and domain rotation. The question then arises: in a typical polycrystalline specimen, what proportion of the total change in M is due to wall motion and what rotation?

Fig. 9.25 Magnetization processes.

This question has no precise answer, but a rough division is indicated in Fig. 9.25. Wall motion is the main process up to about the “knee” of the magnetization curve. From there to saturation, rotation predominates; in this region work must be done against the anisotropy forces, and a rather large increase in H is required to produce a relatively small increase in M. This division of the magnetization curve is rather arbitrary, because wall motion and rotation are not sharply divisible processes. In fact, at any one level of M, wall motion may be occurring in one portion of a specimen and rotation in another. And in certain orientations of a single-crystal specimen relative to the applied field, wall motion and rotation can occur simultaneously in the same part of the specimen. When magnetization occurs entirely by domain rotation, we expect the process to be reversible, with the same B vs H curve followed in both increasing and decreasing fields. Domain wall motion in real materials is irreversible, leading to different curves for increasing and decreasing fields.

9.8HINDRANCES TO WALL MOTION (INCLUSIONS)

Even in the very special case of the picture-frame crystal, an applied field greater than zero is required to move domain walls through the material. In other specimens, substantial wall motion may require fields of tens or hundreds of oersteds. Evidently real materials contain crystal imperfections of one sort or another which hinder the easy motion of domain walls. These hindrances are generally of two kinds: inclusions and residual microstress.

Inclusions may take many forms. They may be particles of a second phase in an alloy, present because the solubility limit has been exceeded. They may be oxide or sulfide particles and the like, existing as impurities in a metal or alloy. They may be simply holes or cracks. From a magnetic point of view, an “inclusion” in a domain is a region which has a different spontaneous magnetization from the surrounding material, or no magnetization at all. We will regard an inclusion simply as a nonmagnetic region.

One reason that an inclusion might impede wall motion is that the wall might tend to cling to the inclusion in order to decrease the area, and hence the energy, of the wall. When a wall arrives at a position bisecting an inclusion, as from Fig. 9.26a to Fig. 9.26b, the wall area decreases by pr2 (for a spherical inclusion of radius r) and the wall energy decreases by pr2s. But in 1944 Ne´el pointed out that free poles on an inclusion

308 DOMAINS AND THE MAGNETIZATION PROCESS

Fig. 9.29 Passage of a domain wall through an inclusion.

when the main wall suddenly snaps off the tubular domains irreversibly and jumps a distance to the right, leaving two spike domains attached to the inclusion, as in Fig. 9.29d. This is a Barkhausen jump. If the field is now reduced to zero and reversed, these changes occur in reverse. If the reversed field is strong enough to drive the wall well to the left of the inclusion, the inclusion will be left with spike domains pointing to the left.

The magnetostatic energy associated with the “naked” inclusion of Fig. 9.27a is proportional to the volume of the inclusion. The results of Problem 9.2 suggest that, in a material like iron, inclusions with a diameter of about 1 mm or larger will have spike domains attached to then in order to reduce this energy. Smaller inclusions will remain bare, because their magnetostatic energy is small. When the inclusion size is of the order

m ¼ ˚

of 0.01 m ( 100 A), it is smaller than the usual domain wall thickness. When such an inclusion is within a wall, it reduces the wall energy and hence tends to anchor the wall. Thus both large and small inclusions tend to impede wall motion, large ones because they have subsidiary domains which tend to stick to walls, small ones because they reduce the energy of any walls that contain them. Inclusions are most effective, per unit volume of inclusion, when the inclusion diameter is about equal to the wall thickness.

9.8.1Surface Roughness

If a sample is thin enough so that a single domain wall runs completely through the thickness, extending from the upper to the lower surface, then the roughness of the surfaces can affect domain wall motion. The wall will always seek to minimize its total area, and so will stick in valleys where the sample is thinnest. The fact that domain walls tend to form as plane surfaces will act to average out this effect if the surface roughness is random, but not if there are more or less continuous hills and valleys running parallel to the primary domain walls.

9.9RESIDUAL STRESS

The second kind of hindrance to domain wall motion is residual microstress. Before examining the magnetic effects of such a stress, we will digress in this section to consider residual stress in general, in order to get a clearer notion of what is meant by microstress.