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

magnitude Hm of the field that originally drove the main wall into the vicinity of the imperfection.

Becker has suggested [J. Appl. Phys., 39 (1969) p. 1270] that a single particle can have a number of pinning sites, each characterized by the field Hn necessary to nucleate, or unpin, a domain wall, and that the particular site that operates depends on the value of the previously applied field Hm. The predicted behavior of a single particle is shown in Fig. 11.16a where M is plotted against the applied field Ha. If walls are present and free to move, the hysteresis loop will have a low coercivity Hw, the wall-motion coercivity. If the interior defect concentration is so low that there is negligible resistance to wall motion, then the loop will have a negligible width and a slope of 1/Nd, where Nd is the demagnetizing factor of the particle. Saturation is achieved at Ha ¼ Hd. If the field is

Fig. 11.16 Hysteresis loops of a single particle. (a) Theoretical. (b) Observed for a particle of SmCo5. [J. J. Becker, IEEE Trans. Mag., 5 (1969) p. 211.]

382 FINE PARTICLES AND THIN FILMS

increased to Hm, walls are pinned so strongly that nucleating fields Hn, which depend on Hm, are needed to cause reverse wall motion. If a wall nucleates at Hn1, it would immediately move until M drops to the upper branch of the wall-motion hysteresis loop, which would then describe the magnetization as Ha was further decreased. If a wall did not nucleate until Hn2, this field would also be the coercivity. If the wall nucleated at Hn3, the loop would be square.

With a vibrating-sample magnetometer, Becker was able to measure the hysteresis loop of a single particle of SmCo5, about 0.2 by 0.5 mm in size, and the result is shown in Fig. 11.16b. A field of 18 kOe was applied before drawing the curves shown. The narrow portion of the loop is due to wall motion, and it is curved because of the irregular shape of the sample. The four closely spaced vertical lines are four magnetization jumps that took place after four successive magnetizations to a field of þ18 kOe. If Hm was less than 13 kOe, no such jumps were observed. When Hm was between 17 and 28 kOe, jumps occurred at Ha equal to about zero, as shown. When Hm was increased to 30 kOe, the jump did not take place until Ha was 22000 Oe. (These values refer to the upper branch of the loop; the values for the lower loop were similar but not exactly the same; in fact, for the loop shown, no jumps are evident for Hm ¼ 219 kOe.) These observations demonstrate that the strong dependence of Hci on Hm for assemblies of particles (Fig. 11.15b) is an inherent property of the individual particles, caused by a variation in the strength of pinning sites with Hm. The fact that a crystal can have more than one wall-nucleating field, for a constant value of Hm, has been demonstrated for barium ferrite by C. Kooy and U. Enz [Philips Res. Rep., 15 (1960) p. 7].

As mentioned in Section 11.3, the coercivity of an assembly of multidomain particles varies inversely with the diameter D of the particles, in accordance with Equation 11.1. This variation can be understood, at least qualitatively, with the following assumptions:

1.The particles are uniaxial, with their easy axes all parallel to the reversing field.

2.When a free 1808 wall is formed by a reverse field Hn, either by nucleation or unpinning, it moves completely through the particle and reverses its magnetization, which means that Hn must exceed Hd.

3.Walls are nucleated only at surface defects.

4.The “strength” of these defects varies in such a way that all values of Hn from zero up to an upper limit of 2K/Ms are equally probable. (A defect is strong if a small field can nucleate or unpin a wall at the defect.)

5.The probability of a defect existing, per unit surface area of particle, is constant.

If the particles are large, the probability that a particular particle has a strong defect is high, because of the large surface area per particle. That particle will therefore reverse at a low Hn. The particle can be expected to have strong defects with a range of strengths, but the strongest defect present determines the reversing field Hn. Moreover, the reversal of a large particle will cause a relatively large change in the magnetization M of the assembly of particles. For example, if there are only eight particles, the reversal of one of them would cause M to decrease from Ms to 3/4Ms, as in Fig. 11.17a. The coercivity Hci of the assembly (the field required to reverse half the particles) is expected to be low because of the high probability that many particles will contain strong defects. Small particles, on the other hand, each have a small surface area and therefore fewer defects. The range of probable Hn values per particle is thus more restricted; some particles will

Fig. 11.17 Demagnetization curves of assemblies of particles.

contain only weak defects and some only strong ones. The demagnetization curve is therefore expected to resemble Fig. 11.17b, with a large coercivity.

11.6SUPERPARAMAGNETISM IN FINE PARTICLES

Consider an assembly of uniaxial, single-domain particles, each with an anisotropy energy density E ¼ K sin2u, where K is the anisotropy constant and u the angle between Ms and the easy axis. If the volume of each particle is V, then the energy barrier DE that must be overcome before a particle can reverse its magnetization is KV. Now in any material, fluctuations of thermal energy are continually occurring on a microscopic scale. In 1949 Ne´el pointed out that if single-domain particles became small enough, KV would become so small that energy fluctuations could overcome the anisotropy forces and spontaneously reverse the magnetization of a particle from one easy direction to the other, even in the absence of an applied field. Each particle has a magnetic moment m ¼ MsV and, if a field is applied, the field will tend to align the moments of the particles, whereas thermal energy will tend to disalign them. This is just like the behavior of a normal paramagnetic, with one notable exception. The magnetic moment per atom or ion in a normal para-

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magnetic is only a few Bohr magnetons. But a spherical particle of iron 50 A in diameter contains 5560 atoms and has the relatively enormous moment of (5560)(2.2) ¼ 12,000 mB. As a result, Bean coined the very apt term superparamagnetism to describe the magnetic behavior of such particles. This subject has been reviewed by C. P. Bean and J. M. Livingston [J. Appl. Phys., 30 (1959) p. 120S].

If K ¼ 0, so that each particle in the assembly has no anisotropy, then the moment of each particle can point in any direction, and the classical theory of paramagnetism will apply. Then the magnetization curve of the assembly, consisting of magnetic particles in a nonmagnetic matrix, will be given by Equation 3.13, or

Here M is the magnetization of the assembly, n the number of particles per unit volume of the assembly, m(¼MsV ) the magnetic moment per particle, a ¼ mH/kT, and. nm ¼ Msa is the saturation magnetization of the assembly. As a consequence of the large value of m, the variable a ¼ mH/kT can assume large values at ordinary fields and temperatures. Thus the full magnetization curve, up to saturation, of superparamagnetic particles can be easily observed, whereas very high fields and low temperatures are required for ordinary

384 FINE PARTICLES AND THIN FILMS

paramagnetic materials, as we saw in Chapter 3. At the other extreme, if K is nonzero and the particles are aligned with their easy axes parallel to one another and to the field, then the moment directions are severely quantized, either parallel or antiparallel to the field, and quantum theory will apply. Then, in accordance with Equation 3.36,

where the hyperbolic tangent is just a special case of the Brillouin function. In the intermediate case of nonaligned particles of nonzero K, neither Equation 11.23 nor Equation 11.24 will apply. Nor will these equations be obeyed if, as is usually true, all particles in the assembly are not of the same size, because then the moment per particle is not constant. Nevertheless, two aspects of superparamagnetic behavior are always true:

1.Magnetization curves measured at different temperatures superimpose when M is plotted as a function of H/T.

2.There is no hysteresis, so that both the retentivity and coercivity are zero. We are therefore dealing with particles having diameters smaller than the critical value Dp of Fig. 11.2.

Both of these features are illustrated by the measurements on fine iron particles dispersed in solid mercury shown in Fig. 11.18. The curves for 77 and 200K in Fig. 11.18a show typical superparamagnetic behavior, and they superimpose when plotted as a function of H/T, as shown in Fig. 11.18b. But at 4.2K, the particles do not have enough thermal energy to come to complete thermal equilibrium with the applied field during the time required for the measurement, and hysteresis appears. (The 4.2K curve is only half of the complete hysteresis loop.)

Hysteresis will appear and superparamagnetism will disappear when particles of a certain size are cooled to a particular temperature, or when the particle size, at constant temperature, increases beyond a particular diameter Dp. To determine these critical values of temperature or size, we must consider the rate at which thermal equilibrium is approached. Consider first the case of zero applied field. Suppose an assembly of uniaxial particles has been brought to some initial state of magnetization Mi by an applied field, and the field is then turned off at time t ¼ 0. Some particles in the assembly will immediately reverse their magnetization, because their thermal energy is larger than average, and the magnetization of the assembly will begin to decrease. The rate of decrease at any time will be proportional to the magnetization existing at that time and to the Boltzmann factor e2KV/kT, because this exponential gives the probability that a particle has enough thermal energy to surmount the energy gap DE ¼ KV required for reversal. Therefore,

Here the proportionality constant f0 is called the frequency factor and has a value of about 109 sec21. This value is only slightly field dependent and can be taken as constant. The constant t is called the relaxation time. (For spherical particles with cubic anisotropy the energy barrier is not KV but 14 KV if K is positive with k100l easy directions, or 121 KV if K is negative with k111l easy directions.) To find how the remanence Mr decreases

Fig. 11.18 Magnetization curves of iron particles 44 A in diameter. [I. S. Jacobs and C. P. Bean, Magnetism, Volume 3, G. T. Rado and H. Suhl, eds., Academic Press (1966).]

with time we rearrange the terms of Equation 11.24 and integrate:

386 FINE PARTICLES AND THIN FILMS

The value of t is the time for Mr to decrease to 1/e or 37% of its initial value. (If the initial state is the saturated state, then Mi ¼ Ms.) From Equation 11.24 we have

Because the particle volume V and the temperature T are both in the exponent, the value of t

Equation 11.28, of only 0.1 sec. An assembly of such particles would therefore reach thermal equilibrium (Mr ¼ 0) almost instantaneously; Mr would appear to be zero in any normal measurement and the assembly would be superparamagnetic. On the other hand,

or 100 years. An assembly of such particles would be very stable, with Mr essentially fixed at its initial value.

Because t varies so very rapidly with V, it follows that relatively small changes in t do not produce much change in the corresponding value of V. Thus it becomes possible to specify rather closely an upper limit Vf for superparamagnetic behavior by letting t ¼ 100 sec define the transition to stable behavior. The value of 100 sec is chosen because it is roughly the time required to measure the remanence of a specimen. If this time were increased to 1000 sec, the corresponding value of Vf would be only slightly increased.

whence KVp/kT equals 25. Therefore, the transition to stable behavior occurs when the energy barrier becomes equal to 25 kT. For uniaxial particles,

and the corresponding diameter Dp can be calculated for any given particle shape. It is this value which in Fig. 11.2 marks the upper limit of superparamagnetism, and the terms “stable” and “unstable” in that diagram refer to particles which have relaxation times longer and shorter, respectively, than 100 sec. The value of Dp for spherical cobalt

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particles is 76 A or 7.6 nm at room temperature.

For particles of constant size there will be a temperature TB, called the blocking temperature, below which the magnetization will be stable. For uniaxial particles and the

The iron particles of Fig. 11.18 must have a blocking temperature between 77 and 4.2K, because they ceased to behave superparamagnetically on cooling through this range. The results of Problem 11.5 show that these particles must have been slightly elongated, rather than spherical.

Figure 11.19 summarizes some of these relations for spherical cobalt particles. The vari-

8 ˚ ation of Dp with temperature shows that 20 C is the blocking temperature for particles 76 A

t ˚

Fig. 11.19 Temperature dependence of the relaxation time for spherical cobalt particles 76 A in diameter and of the critical diameter Dp of spherical cobalt particles.

in diameter; above this temperature particles of this size have enough thermal energy to be superparamagnetic, while below this temperature they are stable and show hysteresis.

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The other curve shows how the relaxation time of 76 A particles varies with temperature; at 208C t is 100 sec.

The next point to consider is the effect of an applied field on the approach to equilibrium. Assume an assembly of uniaxial particles with their easy axes parallel to the z axis. Let the assembly be initially saturated in the þ z direction. A field H is then applied in the 2z direction, so that Ms in each particle makes an angle u with þz. The total energy per

which is the same as Equation 9.37 with a ¼ 1808. The energy barrier for reversal is the difference between the maximum and minimum values of E, and it is left to the reader to show that this barrier is

The barrier is therefore reduced by the field, as shown by the curves of Fig. 9.37. Particles larger than Dp are stable in zero field and will not thermally reverse in 100 seconds. But

388 FINE PARTICLES AND THIN FILMS

when a field is applied, the energy barrier can be reduced to 25 kT, which will permit thermally activated reversal in 100 sec. This field will be the coercivity Hci, given by

When V becomes very large or T approaches zero, Hci approaches 2K/Ms as it should, because 2K/Ms is the coercivity when the field is unaided by thermal energy. If we put this limiting value equal to Hci,0 and substitute Equation 11.30 into Equation 11.35, we obtain the reduced coercivity

The coercivity therefore increases as the particle diameter D increases beyond Dp, as indicated qualitatively in Fig. 11.2. A quantitative comparison of Equation 11.36 with experiment appears in the lower part of Fig. 11.20 for slightly elongated, randomly oriented particles of a 60 Co–40 Fe alloy dispersed in mercury. The agreement is very good. (Measurements were made at three different temperatures, as indicated. The curve is independent of temperature, however, because both coordinates are normalized.) The experimental curve tails off at D/Dp values less than 1; this is due to the particles having a range of sizes, instead of a single size, in each specimen. For these alloy particles Ds equals about 5Dp; larger particles are therefore multidomain, and the coercivity decreases as the size increases.

Equation 11.35 also predicts the variation of coercivity with temperature, for particles of constant size. Particles of the critical size Vp have zero coercivity at their blocking temperature TB and above, where TB is given by Equation 11.32. Therefore

This relation is plotted in Fig. 11.21. Similar curves have been found experimentally. However, too much generality should not be attached to Equation 11.37. The value of TB is so extremely large for the particles in ordinary permanent magnets that any observed variation of Hci with temperature is due, not to the thermal effects discussed here, but to other causes, such as the variation of K and/or Ms with temperature.

Particles larger than Vp have a nonzero retentivity, because thermal energy cannot reverse their magnetization in 100 sec. To find the relation between retentivity and size we combine Equations 11.27 and 11.28 to obtain

Fig. 11.20 Particle size dependence of the coercivity (lower curve) and retentivity (upper curve) of a Co–Fe alloy. Solid lines are calculated. [E. F. Kneller and F. E. Luborsky, J. Appl. Phys., 34 (1963) p. 656.]

Substituting Equation 11.31 into this and putting t ¼ 100 sec, we obtain

The variation of Mr/Mi with V/Vp, or D/Dp, predicted by this equation, is extremely rapid, as illustrated by Problem 11.7. It is plotted as the upper curve of Fig. 11.20, and its slope is so large that it appears vertical. The experimental retentivities do not agree with theory as well as the coercivities, because the retentivities, which vary so rapidly with size, are more sensitive to the presence of a distribution of sizes. When particles of size Vp are cooled below TB, the retentivity rises very rapidly to its maximum value.

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Fig. 11.21 Theoretical temperature dependence of the coercivity of single-domain fine particles. TB ¼ blocking temperature.

The marked dependence of the magnetic properties of very fine particles on their size means that particle sizes can be measured magnetically. An example is given in the next section.

11.7SUPERPARAMAGNETISM IN ALLOYS

To produce specimens suitable for magnetic studies, fine particles of magnetic metals, alloys, and nonmetals may be prepared by a variety of methods and then dispersed in a nonmagnetic medium to avoid particle interactions,. Alternatively, some alloys can be heat treated so that fine particles of a magnetic phase are precipitated out of a nonmagnetic matrix. Magnetic measurements on such a material can furnish valuable information about the very early stages of precipitation, before the precipitate is visible in the optical microscope.

An alloy of copper with 2 wt% cobalt has received the most attention. The Cu–Co equilibrium diagram is shown in Fig. 11.22. The copper-rich phase is nonmagnetic, and the Co phase, which contains about 90% Co, is magnetic. In the experiments of J. J. Becker [Trans. AIME, 209 (1957) p. 59] the alloy was heated to 10108C, where the solubility of cobalt in copper exceeds 2%. The homogeneous a solid solution was then quenched to room temperature. Subsequent aging treatments at 650 and 7008C allowed the magnetic phase to precipitate in reasonable time periods. The magnetic properties of the alloy were measured after each treatment.

Magnetization curves measured after aging up to 100 min at 6508C had the typical superparamagnetic shape, with zero coercivity. The size of the precipitate particles was determined from the initial susceptibility x of the alloy and its saturation magnetization Msa. The Langevin law will apply either for particles with no anisotropy or for a random