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

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spheroid or sphere). In very small particles, the magnetostatic energy required for coherent reversal is less than the exchange energy required for curling, so that coherent reversal is preferred.

To what actual sizes do the critical diameter ratios D/D0 calculated above correspond? This question is difficult to answer precisely because of uncertainty in the value of the

An experimental problem that occurs in almost all small particle work is that it is extraordinarily difficult to produce a three-dimensional assembly of very small particles, each of which has exactly the same size and shape. Trying to work with single particles, instead of assemblies of particles, is at least equally difficult, both because of the small magnetic moments involved and because of the problems in isolating and mounting a single particle about 100 atoms in diameter.

We see from the preceding discussion that a magnetic particle may have all its atomic moments parallel in zero field, but that during the reversal process the magnetization may be nonuniform. Some workers in the field of micromagnetics have taken the position that a particle is truly a single domain only when its spins are always parallel, both before and during a reversal. Thus there can be two definitions of critical size:

1.Static. A particle is a single domain when all its spins are parallel in zero field. If it spontaneously breaks up into domains in zero field, then the critical size Ds has been exceeded. This is the sense of the term “single domain” as it has been used in this book. With this criterion we estimated, in Section 9.5, that the critical size for single-domain particles was generally several hundred angstroms, and that the critical size would be larger, the larger the crystal anisotropy K.

2.Dynamic. A particle is a single domain only when its spins are all parallel at all times, both in zero field and during reversal in an applied field. Thus, according to this micromagnetic definition, only particles which reverse coherently are single domain, and the critical size for single-domain behavior is to be found from the intersection of curves like those shown in Figs. 11.9 and 11.10. For spherical iron particles

˚

this critical size was estimated above as 170 A, which is certainly smaller than the size estimated according to static definition (1) above.

What experimental evidence is there for reversal by curling? The electro-deposited, elongated iron particles examined by Luborsky, and mentioned earlier, had c/a

¼ ˚

values larger than 10 and diameters D( 2a) ranging from 125 to 215 A, corresponding to D/D0 values of 1.04–1.79. Their coercivities Hci were all about 1800 Oe (hci ¼ 0.17), as shown in Fig. 11.9. This size independence of the coercivity suggests that these particles reverse by fanning rather than curling. This is not unexpected, because the peanut-like shape of these particles is not a good approximation of a cylinder.

In order to deal with particles more nearly resembling the idealized models of micromagnetics, F. E. Luborsky and C. R. Morelock [J. Appl. Phys., 35 (1964) p. 2055] examined the properties of iron and iron–cobalt alloy whiskers. These have straight, smooth sides and are structurally the most nearly perfect of all crystals. The coercivities were found to be very size dependent over the large range of whisker diameters investigated, from about 200 to

thicker whiskers deviated from the predictions of curling theory, so that some other reversal mechanism must be acting instead of, or in addition to, curling.

The discussion of incoherent reversal mechanisms in this section hardly does justice to the large amount of theoretical and experimental work done, by the authors cited and others, to examine and test these hypotheses. In particular, two lines of investigation in addition to those mentioned above have been pursued:

1.Angular variation of the coercivity. The way which hci varies with the angle a between the applied field and the axis of the specimen depends in a predictable way on the mode of reversal. Therefore, a measurement of this variation offers a

means of distinguishing between various possible modes. We have so far considered only the case of a ¼ 0.

2.Rotational hysteresis. (See also Section 13.3.) Suppose a specimen is rotated 3608 in an applied field H of constant magnitude. Then, if any irreversible processes

of spin rotation or domain wall motion occur during this rotation, there will be a rotational hysteresis loss Wr. The value of Wr, which is a function of H, can be found from a torque curve; Wr is equal to 2p times the average value of the torque during a full rotation of the sample. The rotational hysteresis integral R is defined by

another. Details are given by I. S. Jacobs and F. E. Luborsky [J. Appl. Phys., 28 (1957) p. 457].

Although the idea that small ferromagnetic particles should show single-domain behavior and have high coercivity has led to a great deal of important theoretical and experimental work, and also led directly to a new class of permanent magnet materials, it has gradually become clear that real permanent magnet materials do not represent a direct embodiment of the simple single-domain particle model. The Stoner–Wohlfarth model, and its various derivatives and developments, guided permanent magnet research for many years, and must be regarded as a success even if they fail to correspond exactly to the structure and behavior of the latest and best permanent magnets.

11.5MAGNETIZATION REVERSAL BY WALL MOTION

Relatively large particles can have quite substantial coercivities, if their magnetocrystalline anisotropy constant K is large. For example, the curves for barium ferrite (K ¼ 3.3 106 erg/cm3 ) in Fig. 11.1 show that a coercivity Hci of about 3000 Oe can be obtained in particles 1 mm in diameter. Yet we estimated, in Problem 9.4, that the critical size Ds for single

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˚ m ˚

domains in barium ferrite was 730 A. Therefore, 1 m (10,000 A) particles must surely be multidomain. Two problems then arise:

1.Why are the coercivities so high, if domain-wall motion is a relatively easy process, as is usually assumed?

2.Why does the coercivity decrease as the particle size increases?

We shall look for answers to these questions in this section. Note that the discussion is restricted to particles owing their coercivity to crystal, rather than shape, anisotropy.

The two questions posed above are made even more pointed by some micro-magnetics calculations made by Brown. Consider a perfect crystal in the form of a spheroid, with the spheroid axis and the easy crystal axis parallel to the z-axis. Let it be saturated in the þz direction and therefore a single domain. A reversing field is then applied in the 2z direction. Brown showed that the field required for reversal is given by

for a crystal of any size, whether larger or smaller than Ds. (Here Hci is a particular value of the true field H acting on the crystal, where H is the sum of the applied field Ha and the demagnetizing field Hd. Both of these are acting in the 2z direction, and negative signs are understood.) This value of Hci is just the value obtained in Section 9.12 for coherent rotation. For barium ferrite,

2K ¼ 2(3:3 106) ¼ 17 kOe ¼ 1:4 MA=m: Ms 380

It would not be surprising to find this value approached in single-domain particles smaller

˚

than Ds, for example, in 100 A particles. In fact, coercivities as large as 11 kOe have been observed in barium ferrite. But in particles larger than Ds the coercivities are much less than expected; in 1 mm particles Hci is only 3 kOe, and it decreases still further as the particle size increases (Fig. 11.1). This large discrepancy between theory and experiment for large particles is known as “Brown’s paradox.”

What is the mechanism of magnetization reversal in particles larger than Ds, i.e., in particles large enough to contain domain walls? It cannot be coherent rotation, because the observed coercivities are much less than 2i/Ms. The only other alternative is that one or more domain walls are nucleated in the particle; the wall then moves through the particle and reverses its magnetization. We may consider this wall nucleation process in some detail, with reference to the perfect spheroidal crystal mentioned earlier, now assumed to be larger than Ds. When saturated in the þz direction, it is a single domain with all spins parallel. When a reversing field H acts in the 2z direction, wall formation presumably begins at the surface, say at the point B of Fig. 11.11a. The spins along a line AB intersecting the surface are shown in Fig. 11.11b, with the spin on the surface atom rotated 458 out of the xz plane by the field. This rotation tends to rotate the spin on the next atom, because of exchange coupling, and Fig. 11.11c–e shows successive steps in the rotation. In Fig. 11.11e a complete wall, assumed for simplicity to be only four atoms thick, has been formed. In Fig. 11.11f the wall has moved inward, producing a new domain 2 magnetized antiparallel to the original domain 1. Because the crystal has been assumed perfect, there are no

Fig. 11.11 Domain wall nucleation at a crystal surface. Only the projections of the spins on the xz plane are shown.

hindrances to the motion of the newly formed wall; under pressure of the nucleating field, it flashes across the crystal in a single Barkhausen jump, and the magnetization reversal is completed. (The just-nucleated wall will move entirely across the crystal only if the applied field exceeds the demagnetizing field. See the discussion of Fig. 11.16.) The hysteresis loop is rectangular, just as it would be if all the spins reversed coherently. But this whole process of wall nucleation and wall motion has its beginnings in spin rotation, on a few atoms at the surface, against the crystal anisotropy forces, and this requires a field equal to 2K/Ms.

The essential point is that the process of nucleating a domain wall in a perfect crystal is just as difficult as coherent rotation. Either process requires spin rotation against the same anisotropy forces.

The assumption made in earlier parts of this book is that a saturated crystal larger than Ds will break up into domains when the saturating field is reduced to zero, in order to lower its magnetostatic energy. In a perfect crystal this may not be true, because the difficulty of wall nucleation is an energy barrier between the single-domain state and the lower-energy, multidomain state. The perfect crystal will spontaneously divide into domains only if

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where Nd is the demagnetizing factor along the axis of magnetization z. For a given material in the form of a spheroid, the process of subdivision into domains is easiest when Nd has a maximum value of 4p (cgs) or 1 (SI), in a thin oblate spheroid (disk) with the plane of the disk normal to z. For such a disk of barium ferrite,

Because this is larger than Ms (¼ 380 emu/cm3), a perfect crystal of barium ferrite in the form of a spheroid cannot spontaneously break up into domains, whatever its size or shape. The same is not true of a high-Ms, low-K material like iron. A perfect iron crystal in the shape of a sphere will form domains, but one in the form of a prolate spheroid of sufficiently high axial ratio will not.

The demagnetizing field of a thin disk of barium ferrite [Hd ¼ 4pMs ¼ 4800 Oe (cgs), Hd ¼ Ms ¼ 365 kA/m (SI)] amounts to a substantial fraction of the theoretical coercivity of 17,000 Oe (1.4 MA/m). Thus the applied field required to reverse the magnetization is 17,000 2 4800 ¼ 12,000 Oe (960 kA/m). Inasmuch as actual crystals of barium ferrite tend to be plate-shaped, with easy axis normal to the plate, the figure of 12,000 Oe is a more realistic value of the theoretical upper limit of coercivity for reversal by wall motion than 17,000 Oe, which applies to a crystal that has a negligible demagnetizing field.

Brown’s paradox is based on calculations assuming a perfect crystal in the form of a perfect spheroid. Real crystals contain imperfections and are not perfect spheroids. Interior imperfections include such defects as dislocations, solute atoms, interstitials, and vacancies; not only may the shape depart from spheroidal in various ways, but the surface may be marred by pits, bumps, steps, cracks, and scratches. Brown’s paradox is resolved by invoking the action of some of these imperfections in lowering the field required to nucleate a domain wall. The condition for wall nucleation in a single particle is that

The applied field required may be much less than expected if (a) Hd or Ms is larger than normal, or (b) K is smaller than normal. The maximum value of Hd is 4pMs (cgs) or Ms (SI) only in a perfect spheroid; near sharp corners, for example, Hd can become very much larger. Because the value of Ms is determined by the magnetic moment per atom and the exchange coupling between adjacent atoms, it may change locally, up or down, in the vicinity of vacancies, interstitials, or the core of dislocations where the strains are very large. Similarly, the local value of K, which is due to spin-orbit coupling, may be changed by imperfections or small-scale inhomogeneities in the chemical composition of the particle.

Of these several possibilities, local variations in the demagnetizing field Hd are usually assumed to be the most likely nucleators of domain walls. Among surface imperfections, pits are no help at all. As shown in Fig. 11.12a, the field at the base of the pit, due to the poles on the pit walls, acts to increase the magnetization at the base of the pit and to counteract the field Ha which is acting to reverse the magnetization. On the other hand, a surface bump, shown in Fig. 11.12b, produces local fields which act to demagnetize the particle and aid Ha.

graph paper with 1-mm squares give the scale. An even shorter reversing coil, 13 turns and 0.8 mm long, was wound on the center of the solenoid. The whisker is initially saturated in one direction by applying a 50-Oe pulse with the solenoid; it remains saturated in that direction after the field has returned to zero. A large pulse field is then applied in the opposite direction with the short central coil. If the region of the whisker inside this coil reverses its magnetization, an emf will be induced in the solenoid and can be detected with an oscilloscope. If no reversal is observed, the magnitude of the reverse pulse field is increased until one is observed. The capillary and whisker is then moved a short distance in the solenoid to place another portion of the whisker in the central coil. An example of such measurements along the length of an 8 mm whisker, which varies in thickness from 19 to 1 mm along its length, is shown in Fig. 11.14b. The coercivity is extremely variable along the length, ranging from less than 50 to a maximum of 504 Oe (4–40 kA/m) at a point where the whisker is 12.4 mm thick. Microscopic examination of the whisker showed that visible surface defects corresponded to the positions of the local minima in the nucleating field, but not all field minima had a matching visible surface defect.

The theoretical coercivity of iron is

2K ¼ 2(4:8 105) ¼ 560 Oe ¼ 45 kA=m: Ms 1714

The maximum observed value of 504 Oe is 90% of the theoretical maximum. These experiments therefore support the view that a perfect crystal does have a coercivity of 2K/Ms and that imperfections can drastically lower this value.

The experiments just described relate to the difficulty of nucleating, or unpinning, a domain wall and give no information on the field required to move a wall once it is free, or on its velocity. In other work, however, DeBlois [J. Appl. Phys., 29 (1959) p. 459] observed that a field as small as 0.008 Oe (0.6 A/m) could move a domain wall in an iron whisker. This field is about 1025 times that required to nucleate a wall in the more nearly perfect portions of a whisker. This result is not surprising; because whiskers are the most nearly perfect crystals known, they offer minimal resistance to wall motion. DeBlois also measured domain wall velocity and found it very high. The velocity depends on the applied field, and at 10 Oe (800 A/m) it was about 1 km/sec or 2200 mph.

Further evidence of the role of surface imperfections as nucleating points comes from measurements of coercivity before and after polishing. The coercivity of iron whiskers can be increased by several hundred oersteds by electropolishing and that of YCo5 particles by several thousand oersteds by chemical polishing. This means that there is not a single curve relating coercivity to particle size for a given material, as in Fig. 11.1, but a whole family of curves, each for a different degree of surface roughness.

A further effect was uncovered by Becker, who showed that the coercivity of particles increased, not only with increasing surface smoothness, but also with increasing magnitude of the field used to magnetize the particles [J. J. Becker, IEEE Trans. Mag., 5 (1969) p. 211]. Now a dependence of the coercive field, as defined in Section 1.8, on the maximum magnetizing field Hm is quite normal, because the coercive field is the reverse field required to reduce M or B to zero after the specimen has been partially magnetized in the forward direction. Thus in Fig. 11.15a, an increase in Hm from 1 to 2 increases the intrinsic coercive force Hci from 1 to 2. But that the coercivity, measured after saturation in the forward direction, should depend on Hm is quite unexpected; whether Hm has the

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Fig. 11.15 Dependence of coercivity on maximum value Hm of magnetizing field. (a) Schematic. (b) Experimental results on aligned particles of SmCo5. [J. J. Becker, J. Appl. Phys., 39 (1968) p. 1270.]

value 3 or 4, the coercivity should have the value 3. Actually, the coercivity is found to increase from 3 to 4 when Hm is increased from 3 to 4. The experimental data are shown in Fig. 11.15b for assemblies of particles of SmCo5 in various size ranges. The easy axes of these particles (the c-axes of their hexagonal unit cells) were aligned with the field, and saturation is essentially complete for fields of 10 kOe and above. Yet substantial increases in Hci are observed, even for changes in Hm beyond 20 kOe. If coercivity is defined as the limiting value of the coercive field as Hm is increased, then the quantity plotted in Fig. 11.15b is coercive field, and the fields Hm are not really saturating the material. The retentivity Mr also increases with Hm but the effect is relatively small. Thus, when Hci doubles in magnitude, the increase in Mr is only about 10%.

If the high coercivity of particles such as these is due to the difficulty of freeing domain walls that are pinned or trapped at imperfections, then the dependence of coercivity on Hm must mean that the “strength” of a pinning site depends on how hard the domain wall is driven into it by the magnetizing field in the forward direction. The mechanism of this behavior is not understood. (In terms of Fig. 9.36, where wall motion was discussed in terms of energy changes in the system, pinning sites must correspond to potential energy wells with almost vertical sides, and the sides must become more nearly vertical as Hm increases.) If a pinning site is an arrangement of closure domains near an imperfection, as suggested in Fig. 11.13, it is conceivable that the particular configuration of the closure domains, and therefore the strength of the pinning site, could depend on the