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

Fig. 11.22 Copper–cobalt equilibrium diagram (ASM Handbook).

distribution of particles with uniaxial anisotropy. The initial susceptibility will therefore be given by Equation 3.15:

392 FINE PARTICLES AND THIN FILMS

This equation applies to materials which obey the Curie law (Equation 3.7). However, susceptibility vs temperature measurements showed that the Cu–Co alloy obeyed the Curie–Weiss law (Equation 3.8) with a small value of the constant u, about 5–10 K. There is, therefore, a small interaction between the precipitate particles. Because of the observed Curie–Weiss behavior, Equation 11.41 was modified to

The value of Ms in this equation was taken as the value for pure cobalt (1422 emu/cm3 or kA/m). From the particle volumes V given by Equation 11.43, the radius R of the particles, assumed spherical, could be calculated. These values are shown by the lower curve of Fig. 11.23, which shows how the particle size increases with aging time. These particles are very small. They are invisible with the light microscope, and could be seen only with great difficulty in the electron microscopes available at the time. At the shortest aging

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time of 3 min, their diameter is only about 25 A, or less than 10 unit cells. Probably no other experimental tool could have disclosed this information as readily as these rather simple magnetic measurements.

The observed value of the saturation magnetization Msa of the alloy disclosed the amount of the magnetic phase present, independent of its particle size. The upper curve of Fig. 11.23 therefore shows that precipitation of b is essentially complete in only 3 min at 6508C. Thereafter, the changes that occur in the alloy consist solely of an increase in particle size and a decrease in the number of particles, with a nearly constant total volume of precipitate. This process, called coarsening or ripening, occurs by the dissolving into solid solution of some particles and the growth of others, in order to decrease the total surface energy of the precipitate. This kind of behavior, initial rapid nucleation followed by a

Fig. 11.23 Effect of aging at 6508C on the saturation magnetization Msa of a Cu–Co alloy and on the effective radius R of the precipitate particles. [J. J. Becker, Trans. AIME, 209 (1957) p. 59.]

decrease in the nucleation rate to zero, is typical of precipitation from a supersaturated solid solution and is technically known as continuous precipitation. The cobalt-rich precipitate particles are believed to have the fcc crystal structure, rather than hcp, and to owe their uniaxial nature to shape anisotropy. If the particles are actually egg-shaped, then the radius R shown in Fig. 11.23 is an average radius.

After about 100 min at 6508C, the particle volume increased beyond the critical value Vp appropriate to the measuring temperature (room temperature), and hysteresis appeared. The time to grow particles larger than Vp can be shortened from 100 to about 10 min by raising the aging temperature from 650 to 7008C, as shown by Fig. 11.24. Hysteresis, as evidenced by a nonzero value of the coercivity, begins to appear after about 10 min. Figure 11.24 is interesting because it shows in one single curve, on a time base, all the changes that are schematically displayed in Fig. 11.2 on a particle-size base. Up to about 10 min, the particles behave superparamagnetically. At longer times the particles have grown to such a size that thermal energy alone can no longer rapidly overcome the energy barrier KV for reversal, and a nonzero coercive field Hci is needed in addition. With further growth, Hci increases to a maximum at about 800 min. The particles then become multidomain, and Hci decreases.

Fine magnetic particles dispersed in a nonmagnetic matrix also exist in many kinds of rocks. The magnetic phase is usually a ferrimagnetic oxide. Studies in rock magnetism often involve the problem of determining the direction and magnitude of the earth’s field in the remote past when the rock was formed. This brings up the question of the stability of the original magnetization. Has it been altered by subsequent changes in the earth’s field? One must then consider the possibility of magnetization relaxation, not over a period of the order of 100 sec, but over geological times.

In the copper–cobalt alloy cited in this section and in most rocks, the magnetic particles responsible for the superparamagnetic behavior of the material are discrete particles of a second phase. This condition is not essential, because a single-phase solid solution can also be superparamagnetic if it has local inhomogeneities of the right kind. For example, in a solid solution containing 90 mol% Zn ferrite and 10% Ni ferrite, there may be small clusters containing more than the average number of magnetic ions (Fe and Ni), surrounded

Fig. 11.24 Effect of aging at 7008C on the coercivity Hci of a Cu–Co alloy. [J. J. Becker, Trans. AIME, 209 (1957) p. 59.]

Fig. 11.26 Hysteresis loops measured at 4.2K of an alloy of Ni þ 26.5 atomic percent Mn after cooling with and without a field of 5 kOe in the positive direction. [J. S. Kouvel, C. D. Graham, and I. S. Jacobs, J. Phys. Rad., 20 (1959) p. 198.]

after cooling in a field in the positive direction, whether the previous saturating field was applied in the positive or negative direction. (Note that we are dealing here with shifted major loops, representing saturation in both directions. A shifted minor loop, in which saturation is not achieved in one or both directions, is not evidence for exchange anisotropy and can be obtained in any material.)

Both the Co–CoO particles and the Ni3Mn alloy display unidirectional, rather than uniaxial, anisotropy. The anisotropy energy is therefore proportional to the first power, rather than the square, of the cosine:

where K is the anisotropy constant and u is the angle between Ms and the direction of

magnet in a field (Equation 1.5).

The unusual properties of field-cooled Co–CoO particles are due to exchange coupling between the spins of ferromagnetic Co and antiferromagnetic CoO at the interface between

the particle to state (a), with a positive retentivity. This ideal behavior is achieved in the observed loop of Ni3Mn (Fig. 11.26) but not in the Co–CoO loop (Fig. 11.25).

There are evidently three requirements for the establishment of exchange anisotropy: (1) field-cooling through TN, (2) intimate contact between ferromagnetic and antiferromagnetic, so that exchange coupling can occur across the interface, and (3) strong crystal anisotropy in the antiferromagnetic. Actually, the role of field-cooling is only to give the specimen as a whole a single easy direction. If it is cooled in zero field, the exchange interaction occurs at all interfaces, leading, in the ideal case, to a random distribution of easy directions in space and zero retentivity, as shown in Fig. 11.26.

The exchange anisotropy exhibited by disordered Ni3Mn is believed to be due to composition fluctuations in the solid solution, leading to the formation of Mn-rich clusters. These clusters would be antiferromagnetic, because the exchange force between Mn–Mn nearest neighbors is negative, as in pure manganese. Just outside the cluster the solution would be richer in nickel than the average composition, and the preponderance of Ni–Ni and Ni–Mn nearest neighbors would cause ferromagnetism there. (Ordered Ni3Mn, in which all nearest neighbor pairs are Ni–Mn, is ferromagnetic.) We may therefore conclude that the establishment of exchange anisotropy does not require a two-phase system like Co–CoO; it can also occur in a single-phase solid solution having the right kind of inhomogeneity. The latter behavior resembles that described in the last paragraph of Section 11.7.

Exchange anisotropy remained a scientific curiosity for many years, but has found widespread use in the data-reading heads of hard disk drives. In the spin valve, considered in more detail in Chapter 15, a thin ferromagnetic layer is held with its magnetization in a fixed direction by being placed in exchange contact with an antiferromagnetic layer. In this application, the effect is sometimes known as spin bias. This application has led to considerable additional theoretical and experimental work on exchange coupling and exchange anisotropy. For further details, see R. M. O’Handley [Modern Magnetic Materials, Principles and Applications, Wiley (2000) p. 437ff].

11.9PREPARATION AND STRUCTURE OF THIN FILMS

We turn now to thin films. Much of the early research on magnetic thin films was done because of their prospective application as memory elements in digital computers. This application was never commercially successful, but was followed by the intense development of bubble domain memories, which also required the use of patterned magnetic thin films to control the motion of the bubbles. This development, too, eventually came to naught. But a rich variety of other uses, mainly related to various aspects of computer technology, have arisen. These include the original idea of magnetic memory, although in a somewhat different form from the earlier ideas. Some of these will be discussed in Chapter 15.

Here we will consider some of the basic behavior of magnetic thin films, dealing primarily with the widely used alloy called 80 permalloy (80 wt% Ni, 20% Fe), which has low anisotropy and low magnetostriction combined with reasonably high magnetization, and is easily made in thin-film form. It is the obvious choice when a soft magnetic thin film is required.

Thin films are made by depositing atoms onto a substrate, and are almost always measured and used while bonded to the substrate. Interactions between the substrate and

Fig. 11.29 Thin-film preparation by thermal evaporation.

The difference in thermal expansion coefficients between film and substrate leads to stresses in the film (and small stresses in the substrate), and the film itself has microstress due to various imperfections. The effect of these stresses is minimized by choosing a film composition with magnetostriction is as close as possible to zero. Stress anisotropy is therefore avoided because magneto-elastic effects are always proportional to the product of the magnetostriction l and the stress s.

11.10INDUCED ANISOTROPY IN FILMS

A uniaxial anisotropy can often be produced in a magnetic thin film if the deposition is carried out in the presence of an applied magnetic field as in Fig. 11.29. The induced anisotropy is described by the energy relation

where u is the angle between Ms and the easy axis, which is the direction of the applied field during deposition. The value of Ku is generally of the order of 1000–3000 ergs/cm3 or 100–300 J/m3 for 80 permalloy, which is similar to the value observed for magnetically annealed bulk alloys (Fig. 10.5).

400 FINE PARTICLES AND THIN FILMS

The measurement of Ku is difficult, because the specimen size is so very small. Two methods are available:

1. Measurement of the Torque Curve. This is done in the usual way, but the torque magnetometer has to be extremely sensitive. The volume of a circular film specimen

is proportional to the volume. A typical bulk specimen would have about the same diameter and be about 0.1 cm thick. Thus the film specimen has a volume of only 1024 that of the bulk specimen, and the torque magnetometer must have a sensitivity 104 times higher than usual.

2. Measurement of the Hysteresis Loop. An 80 permalloy film 1 cm in diameter and

measured with a vibrating-sample magnetometer. From measurements of the loop in the easy and hard directions, the anisotropy field HK can be determined. The value of Ku is then calculated from the relation HK ¼ 2Ku/Ms. (Details are given in Section 12.4). The origin of the induced anisotropy in magnetic thin films is presumably directional order, the same as in bulk alloys.

11.11DOMAIN WALLS IN FILMS

Except in special cases. the magnetization in thin films lies in the plane of the film, because a huge demagnetizing field would act normal to the plane of the film if Ms were turned in that direction. Domains in the film extend completely through the film thickness, and the walls between them are mainly of the 1808 kind, roughly parallel to the easy axis of the film.

However, two new kinds of domain walls can exist in thin films. The first is the Ne´el wall, first suggested on theoretical grounds by Ne´el (see Section 9.2.1). Ordinary walls such as those found in bulk materials can also exist in thin films; they are then specifically called Bloch walls to distinguish them from Ne´el walls.

Ne´el showed that the energy per unit area g of a Bloch wall is not a constant of the material but depends also on the thickness of the specimen, when the thickness is less than a few thousand angstroms. The magnetostatic energy of the wall them becomes appreciable, relative to the usual exchange and anisotropy energy. Free poles are formed where the wall intersects the surface, as indicated in Fig. 11.30a where only the central spin in the wall is shown. When the specimen thickness t is of the same order of magnitude as the wall thickness d, the field created by these poles constitutes an appreciable magnetostatic energy. To calculate this energy Ne´el approximated the actual wall, a nonuniformly magnetized rectangular block in which the spins continuously rotate from the direction þy to 2y, by a uniformly magnetized elliptic cylinder. This is sketched in Fig. 11.30b; its major axis c is infinite. When magnetized along the a-axis, its demagnetizing coefficient is