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

350 INDUCED MAGNETIC ANISOTROPY

Fig. 10.12 Uniaxial anisotropy constant Ku of rolled polycrystalline Ni–Fe alloys. (a) Randomly oriented grains rolled to 33% reduction in thickness. (b) Cube-texture material rolled to 55% reduction. [After G. W. Rathenau and J. L. Snoek, Physica, 8 (1941) p. 555.]

Oxford University Press (1997)], who has done much of the basic work in this field. This directional order is created not by diffusion, as in magnetic annealing, but by slip.

How this can occur is shown in Fig. 10.13 for a perfectly ordered superlattice like FeNi3.

¯

Slip by unit distance on the (111) plane in the [011] direction, resulting from the passage of a single dislocation, has created like-atom pairs across the slip plane where none existed before; the double lines indicate BB pairs, with axes in the [011] direction. This direction therefore becomes a local easy axis. If a second dislocation passes, the crystal becomes perfectly ordered again, with no like-atom pairs. Thus directional order is created only if an odd number of dislocations pass along the slip plane. When a face-centered cubic crystal is deformed by rolling, at least two slip systems (combination of a slip plane of the form f111g and a slip direction of the form k110l) must operate to accomplish the observed change in shape, which is a lengthening in the rolling direction, a reduction in thickness, and little or no increase in width. There are thus at least two orientations of the local easy axes, corresponding to the two orientations of active slip planes.

A further complication is that an ordered crystal is never completely ordered throughout its volume, but is made up of a number of “antiphase” regions. In order–disorder language, these are called domains, but the word “region” will be substituted here to avoid confusion with magnetic domains. The crystal structure is out of step at the boundaries of these regions, and only within them is the order perfect. When the slip distance is large enough to bring a substantial portion of one antiphase region into contact with another across the slip plane, new orientations of like-atom pairs become possible. In this way slip on the (111) plane, for example, can create BB pairs not only in the [011] direction shown in Fig. 10.13 but also in [110] and [101]. The same result is expected when only

Fig. 10.13 Formation of AA and BB pairs by slip in an A3B superlattice. [S. Chikazumi, Physics of Magnetism, Oxford University Press (1997).]

short-range order exists. The various orientations of like-atom-pair axes in the deformed crystal then combine to give a single easy axis for the whole specimen.

The theory of Chikazumi et al. takes into account the orientations of the active slip systems when a crystal of a particular orientation is rolled, together with the orientations of like-atom pairs created by slip on these systems, and it yields a prediction of the easy axis of the rolled crystal. Chikazumi and his co-workers tested such predictions by rolling single crystals of the FeNi3 composition and measuring the resultant anisotropy; the agreement with theory was very good.

If we leave aside the complex details of the theory, we can still understand, in a qualitative way, why rolling an alloy single crystal should induce in it a magnetic anisotropy. Given the facts (1) that directional order of like-atom pairs creates an easy axis in some alloys, (2) that slip can create like-atom pairs where none existed before, (3) that the axes of such pairs must be oriented in particular ways with respect to the slip planes, and, finally, (4) that the slip planes which operate have particular orientations in the crystal, it follows that the local easy axes due to like-atom pairs must have particular

352 INDUCED MAGNETIC ANISOTROPY

orientations throughout the crystal. These local easy axes then combine to give a net overall anisotropy to the crystal as a whole. This general argument cannot, of course, predict the orientation of the resultant easy axis, or, for that matter, even show that the resultant anisotropy should be uniaxial rather than, say, biaxial.

When an Ni–Fe polycrystal with randomly oriented grains is rolled, the RD becomes an easy direction (Fig. 10.12). This result is not understood. Bitter-pattern examination of the rolled material shows that the direction of the easy axis, as judged by the orientation of 1808 walls, varies from one grain to another; the observed anisotropy is thus an average over all the grains in the specimen. Theoretical analysis is forbiddingly complex. When a polycrystal is deformed, as many as five slip systems can operate within a single grain, because the change in shape of that grain must conform to the changes in shape of all the neighboring grains, a requirement not imposed on a rolled single crystal.

There is little doubt that slip-induced anisotropy is due to directional order, because the magnitude of Ku varies with composition, and thus with the likelihood of having like-atom pairs, in much the same way after rolling (Fig. 10.12) as after magnetic annealing (Fig. 10.5). However, it is not clear why the two curves in Fig. 10.12 should peak at different compositions. The magnitude of Ku after rolling is some 50 times larger than after magnetic annealing, which shows that slip can produce a far larger concentration of like-atom pairs.

Roll magnetic anisotropy has also been observed in Fe–Al, Ni–Co, and Ni–Mn alloys. Chin et al. [G. Y. Chin, J. Appl. Phys., 36 (1965) p. 2915; G. Y. Chin, E. A. Nesbitt, J. H. Wernick, and L. L. Vanskike, ibid., 38 (1967) p. 2623] have extended the theory and experiments to deformation by wire drawing, roll flattening (rolling of wire), flat drawing (drawing through a rectangular die), and plane strain compression. The last of these simulates rolling, but the compression is carried out in a die which prevents the lateral spreading which occurs when single crystals of certain orientations are rolled.

10.6PLASTIC DEFORMATION (PURE METALS)

Krause and Cullity [R. F. Krause and B. D. Cullity, J. Appl. Phys., 39 (1968) p. 5532] cut disk specimens from a flat strip of annealed, polycrystalline nickel and measured torque curves in an applied field of 9 kOe or 72 kA/m. The result is shown in Fig. 10.14 (curve labeled “Annealed”), where u is the angle between the applied field and the long axis of the strip, which is the direction in which the strip was rolled before being annealed. There is a very weak, biaxial anisotropy, like that described by Equation 7.8, due to a small amount of preferred orientation; the easy axes are at 0 and 908 to the strip axis.

If the strip is now stretched a few percent, the torque curve is radically changed. The maxima are now some ten times higher and occur at different values of u. A fairly strong uniaxial anisotropy has been created, with the easy axis parallel to the direction of elongation. Compression has the reverse effect; the anisotropy is again uniaxial, but the easy axis is at 908 to the direction of compression. This anisotropy is also governed by Equation 10.1, where u is the angle between the magnetization and the deformation axis (tensile or compressive). Fourier analysis of the torque curves for deformed specimens permits the separation of the uniaxial anisotropy caused by deformation from the weak biaxial anisotropy caused by texture. The value of Ku was found to increase rapidly

Fig. 10.14 Torque curves for annealed nickel and for nickel after plastic tension and compression of 2.5%. [After R. F. Krause and B. D. Cullity, J. Appl. Phys., 39 (1968) p. 5532.]

Examination of the deformed specimens by X-ray diffraction revealed the presence of residual stress, because the X-ray lines were both shifted and broadened. The line shift, as mentioned in Section 9.9, indicates a stress which is more or less constant over most of the specimen volume, and the broadening reflects stress variations about the mean in this volume. The X-ray observations are consistent with the assumed stress distribution of Fig. 9.35, for a specimen previously deformed in tension. The material can then be viewed as consisting of regions in longitudinal compression (C regions) and regions in tension (T regions). The C regions comprise most of the specimen volume and are responsible for the X-ray effects, while the smaller T regions add nothing observable to the X-ray pattern. The C regions have been tentatively identified with the subgrains which form within each plastically deformed grain, and the T regions with the subgrain boundaries. After plastic compression, the stress distribution of Fig. 9.35 is inverted; most of the specimen is then in a state of tensile stress in the direction of previous compression. These conditions are not peculiar to nickel but have been observed in a number of metals and alloys. Actually, the stress state at the surface of the deformed nickel was found to be one of biaxial compression after plastic tension and biaxial tension after plastic compression. This complication does not invalidate the argument given below and will be ignored in what follows.

The observed magnetic anisotropy after plastic deformation is attributed to the residual stress described above. Because residual stresses must form a balanced-force system, a

354 INDUCED MAGNETIC ANISOTROPY

simple argument leads to the conclusion that they could not cause any net anisotropy. To see this, we take Equation 8.23, which governs stress anisotropy, and write the magnetoelastic energy as

Suppose that a residual tensile stress sT exists, in certain portions of the specimen, parallel to a particular axis from which u is measured, and a residual compressive stress sC exists in other portions, parallel to the same axis. Let fT and fC be the volume fractions of the specimen in residual tension and compression, respectively. Then the magnetoelastic energy per unit volume of such a specimen becomes

If relative cross-sectional areas of the regions under tension and compression are assumed equal to their relative volumes, a balance of forces requires that

where sT is taken as positive and sC as negative. The magnetoelastic energy is therefore zero, provided that the specimen is completely saturated at all values of u. Then Ms in the tensile regions, Ms in the compressive regions, and the applied field are all parallel to one another, and it is then justifiable to write Equation 10.3 in terms of a single angle u between Ms and the stress axis. Then, whatever the nature of the residual stress state, no magnetic anisotropy will result. Physically, for any value of u, the torque on Ms due to the regions in tension is exactly balanced by an opposing torque on Ms due to the regions in compression.

Some portions of the specimen, however, are not fully saturated at all angles u, and this condition causes the observed anisotropy. After plastic tension, for example, most of the specimen volume is in compression parallel to the deformation axis; these regions (the C regions) are therefore easy to magnetize along this axis, because lsi is negative for nickel. The average stress in the T regions is necessarily larger than that in the C regions, because the cross section of the T regions is smaller. Therefore, the Ms vectors in the T regions, initially at right angles to the deformation axis, strongly resist rotation by a field applied along that axis. The T regions are thus difficult to saturate, but they comprise such a small fraction of the total volume, estimated at about one-tenth, that the specimen as a whole acts as though it had an easy axis parallel to the deformation axis. The same argument in reverse will account for the easy axis at right angles to the axis of previous compression.

Presumably plastic tension or compression would produce the same kind of anisotropy in other polycrystalline ferromagnetic metals, or in alloys. In alloys this anisotropy would be superimposed on any anisotropy that might result from deformation-induced directional order.

10.7MAGNETIC IRRADIATION

The physical and mechanical properties of any material are usually changed when it is bombarded with neutrons, ions, electrons, or gamma rays. These property changes are due to the atomic rearrangements, called radiation damage, brought about by the radiation, and the kind of rearrangement depends on the kind of radiation. For example, neutrons, being uncharged particles, are highly penetrating. When incident on a solid, they can travel

The anisotropy here is not due to directional order in a solid solution, but to three superimposed crystal anisotropies, with one stronger, per unit volume of specimen, than the other two. These results of Ne´el et al. are remarkable, not only for the large anisotropy produced, but also for the fact that long-range ordering had never before been observed in this system at the composition FeNi.

10.8SUMMARY OF ANISOTROPIES

The magnitudes j K j of the anisotropy constants at room temperature are compared in Fig. 10.16 for the various anisotropies we have considered:

1.Grouped in the upper part of the figure are the magnetocrystalline anisotropies of Fe, Ni, and Co, of some ferrites, and of YCo5. The last is typical of the huge anisotropy found in the RCo5 compounds, where R is yttrium or certain of the rare earths.

2.Next are some anisotropies due to shape, for prolate ellipsoids. Shape anisotropy not

only depends on the length/diameter ratio l/d, but also is proportional to M2s . The values shown were calculated for iron (Ms ¼ 1714 emu/cm3 or 1.717 106 A/m).

3.The stress anisotropy constant is proportional to the product of stress and the magnetostriction lsi. The anisotropies shown were calculated for lsi¼ 20 1026, taken as typical of many materials.

4.The two examples of anisotropy due to directional order are (a) polycrystalline, magnetically annealed 60 Ni–40 Fe (Fig. 10.5), and (b) polycrystalline 50 Ni–50 Fe with a cube texture, rolled 55% (Fig. 10.12).

5.“Stretched Ni” refers to the uniaxial anisotropy found in polycrystalline nickel plastically elongated 2.5% (Section 10.6).

6.The final entry in the figure shows the anisotropy developed in a single crystal of 50 Ni–50 Fe by the treatment described in Section 10.7.

360 FINE PARTICLES AND THIN FILMS

11.2SINGLE-DOMAIN VS MULTI-DOMAIN BEHAVIOR

We have already examined, in Section 9.5, the theoretical reasons for believing that a single crystal will become a single domain when its size is reduced below a critical value Lc of a few hundred angstroms. There is ample experimental evidence of various kinds that single-domain particles do exist, and can have high values of coercivity.

An originally multi-domain particle can be kept in a saturated state only by a field larger than the demagnetizing field; once this field is removed, the magnetostatic energy associated with the saturated state breaks the particle into domains and reduces M, and the demagnetizing field, to some lower value. (But see Section 11.5 for some possible exceptions.) A single-domain particle is by definition always saturated in the sense of being spontaneously magnetized in one direction throughout its volume, but

small field. A well-pivoted compass needle is an exact analog to the Ms vector of a lowanisotropy single-domain particle. The needle is a permanent magnet with a demagnetizing field of more than 50 Oe, but the Earth’s field, of a few tenths of an oersted, can easily rotate it.

Another way of viewing this experiment is to focus on the energies involved. A saturated sphere of magnetic material, whatever its size, must always have a magnetostatic energy Ems of (1/2)(4p/3)(M2s ) per unit volume (cgs units), according to Equation 7.59. If the sphere is originally multidomain, this energy must be supplied by the applied field; if it is a single domain, this energy is always present, at all field strengths including zero.

11.3COERCIVITY OF FINE PARTICLES

In magnetic studies on fine particles the single property of most interest is the coercivity, for two reasons: (1) it must be high, at least exceeding a few hundred oersteds, to be of any value for permanent-magnet applications, and (2) it is a quantity which comes quite naturally out of theoretical calculations of the hysteresis loop.

The coercivity of fine particles has a striking dependence on their size. As the particle size is reduced, it is typically found that the coercivity increases, goes through a maximum, and then tends toward zero. This is clearly shown for three different materials in Fig. 11.1; for the other three the maximum in coercivity has not yet been reached. The very large range of the variables should be noted: the coercivities vary over three orders of magnitude and the particle sizes over five. The smallest particles are less than 10 unit cells across, while the largest are 0.1 mm in diameter.

The mechanism by which the magnetization of a particle reverses is different in different parts of the curves of Fig. 11.1. Much of this chapter is devoted to an examination of these different mechanisms. In anticipation of the results of later sections, Fig. 11.2 shows schematically how the size range is divided, in relation to the variation of coercivity with particle diameter D. Beginning at large sizes, we can distinguish the following regions:

1.Multidomain. Magnetization changes by domain wall motion (Section 11.5). For some, but not all, materials the size dependence of the coercivity is experimentally