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

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4.Anisotropy induced by

a.Magnetic annealing (Chapter 10).

b.Plastic deformation (Chapter 10).

c.Irradiation (Chapter 10).

5.Exchange anisotropy (Section 11.8).

Of these, only crystal anisotropy is intrinsic to the material. Strictly, then, all the others are extrinsic or “induced.” However, it is customary to limit the term “induced” to the anisotropies listed under item 4 above. All the anisotropies from 1 to 5 (except 4c) are important in practice, and any one may become predominant in special circumstances. In this chapter we will consider only crystal and shape anisotropy.

7.2ANISOTROPY IN CUBIC CRYSTALS

Suppose a single crystal with cubic structure is cut in the form of a disk parallel to a plane1 of the form f110g. This specimen will then have directions of the form k100l, k110l, and

k l ¯

111 as diameters, as shown in Fig. 7.1 for the plane (110). Measurements of magnetization curves along these diameters, in the plane of the disk, will then give information about three important crystallographic directions. The results for iron, which has a body-centered cubic structure, are shown in Fig. 7.2a, and those for nickel (face-centered cubic), in Fig. 7.2b.

For iron these measurements show that saturation can be achieved with quite low fields, of the order of a few tens of oersteds at most, in the k100l direction, which is accordingly called the “easy direction” of magnetization. This tells us something about domains in iron in the demagnetized state. As will become clear later, a domain wall separating two domains in a crystal can be moved by a small applied field. If we assume that domains in demagnetized iron are spontaneously magnetized to saturation in directions of the form k100l, then a possible domain structure for a demagnetized crystal disk cut parallel

¯

Fig. 7.1 The three principal crystallographic directions in the (110) plane of a cubic material.

¯

and (001). By convention, minus signs are placed above the index number. The indices of any one, enclosed in braces f100g, stand for the whole set. The indices of particular directions are enclosed in square brackets, such

¯ ¯ ¯

as the six cube-edge directions: [100], [010], [001], [100], [010], and [001]. These are directions of a form, and the whole set is designated by the indices of any one, enclosed in angular brackets k100l.

Fig. 7.2 Magnetization curves for single crystals of iron (a) and nickel (b).

to (001) would be that shown in Fig. 7.3a. It has four kinds of domains, magnetized parallel

¯ ¯

to four of the six possible easy directions, namely, [010], [100], [010], and [100]. Actually, an iron crystal disk of diameter, say, 1 cm, would contain tens or hundreds of domains, rather than the four shown in Fig. 7.3. However, it would still be true that all these domains would be of only four kinds, namely those with Ms vectors in the [010], [100],

¯ ¯

[010], and [100] directions. If a field H is now applied in the [010] direction, the [010] domain will grow in volume by the mechanism of domain-wall motion, as indicated in Fig. 7.3b. It does so because the magnetic potential energy of the crystal is thereby

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Fig. 7.3 Domain structures in a single-crystal disk of iron (schematic). The field H is applied in the [010] direction.

lowered; Equation 1.5 shows that the energy of a [010] domain in the field is 2MsH per unit

¯ þ ¯

volume, that of a [010] domain is MsH, and that of a [100] or [100] domain is zero. Continued application of the field eliminates all but the favored domain, and the crystal is now saturated (Fig. 7.3c). This has been accomplished simply by applying the low field required for domain wall motion. Since experiment shows that only a low field is needed to saturate iron in a k100l direction, we conclude that our postulated domain structure is basically correct and, more generally, that the direction of easy magnetization of a crystal is the direction of spontaneous domain magnetization in the demagnetized state. In nickel, Fig. 7.2b shows that the direction of easy magnetization is of the form k111l, the body diagonal of the unit cell. The direction k111l is also the direction of easy magnetization in all the cubic ferrites, except cobalt ferrite or mixed ferrites containing a large amount of cobalt. The latter have k100l as an easy direction.

Note that, on the scale of a few domains, as in Fig. 7.3b, a partially magnetized crystal is never uniformly magnetized, in the sense of M being everywhere equal in magnitude and direction, whether or not the crystal is ellipsoidal in shape. The notion of uniform magnetization predates the domain hypothesis. It has validity, for a crystal containing domains, only when applied either to a volume less than that of one domain, or to a volume so large that it contains many domains and has a net magnetization M equal to that of the whole crystal.

Figure 7.2a shows that fairly high fields, of the order of several hundred oersteds or tens of kiloamps per meter, are needed to saturate iron in a k110l direction. For this orientation of the field, the domain structure changes as in Fig. 7.4. Domain wall motion, in a low field, occurs until there are only two domains left (Fig. 7.4c), each with the same potential energy. The only way in which the magnetization can increase further is by rotation of the Ms vector of each domain until it is parallel with the applied field. This process is called domain rotation. The domain itself, which is a group of atoms, does not rotate. It is the net magnetic moment of each atom which rotates. Domain rotation occurs only in fairly high fields, because the field is then acting against the force of crystal anisotropy,

Fig. 7.4 Domain structures in a single crystal of iron (schematic). The field H is applied in the [110] direction.

which is usually fairly strong. Crystal anisotropy may therefore be regarded as a force which tends to hold the magnetization in certain equivalent crystallographic directions in a crystal. When the rotation process is complete (Fig. 7.4d), the domain wall in Fig. 7.4c disappears, and the crystal is saturated.

Because the applied field must do work against the anisotropy force to turn the magnetization vector away from an easy direction, there must be energy stored in any crystal in which Ms points in a noneasy direction. This is called the crystal anisotropy energy E. The Russian physicist Akulov showed in 1929 that E can be expressed in terms of a series expansion of the direction cosines of Ms relative to the crystal axes. In a cubic crystal, let Ms make angles a, b, c with the crystal axes, and let a1, a2, a3 be the cosines of these angles, which are called direction cosines. Then

where K0, K1, K2, . . . are constants for a particular material at a particular temperature and are expressed in erg/cm3 (cgs) or J/m3 (SI). Higher powers are generally not needed, and sometimes K2 is so small that the term involving it can be neglected. The first term, K0, is independent of angle and is usually ignored, because normally we are interested only in the change in the energy E when the Ms vector rotates from one direction to another. Table 7.1 gives the value of E when the Ms vector lies in a particular direction [u v w].

TABLE 7.1 Crystal Anisotropy Energies for Various Directions in a Cubic Crystal

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Fig. 7.7 Easy directions and planes in hexagonal crystals for all possible values of K1 and K2.

7.4PHYSICAL ORIGIN OF CRYSTAL ANISOTROPY

Crystal anisotropy is due mainly to spin-orbit coupling. By coupling is meant a kind of interaction. Thus we can speak of the exchange interaction between two neighboring spins as a spin–spin coupling. This coupling can be very strong, and acts to keep neighboring spins parallel or antiparallel to one another. But the associated exchange energy is isotropic; it depends only on the angle between adjacent spins, as stated by Equation 4.29, and not at all on the direction of the spin axis relative to the crystal lattice. The spin–spin coupling therefore cannot contribute to the crystal anisotropy.

The orbit-lattice coupling is also strong. This follows from the fact that orbital magnetic moments are almost entirely quenched, as discussed in Section 3.7 This means, in effect, that the orientations of the orbits are fixed very strongly to the lattice, because even large fields cannot change them.

There is also a coupling between the spin and the orbital motion of each electron. When an external field tries to reorient the spin of an electron, the orbit of that electron also tends to be reoriented. But the orbit is strongly coupled to the lattice and therefore resists the attempt to rotate the spin axis. The energy required to rotate the spin system of a domain away from the easy direction, which we call the anisotropy energy, is just the energy required to overcome the spin–orbit coupling. This coupling is relatively weak, because fields of a few hundred oersteds or a few tens of kilamps per meter are usually strong enough to rotate the spins. Inasmuch as the “lattice” consists of a number of atomic nuclei arranged in space, each with its surrounding cloud of orbital electrons, we can

Fig. 7.8 Spin–lattice–orbit interactions.

also speak of a spin–lattice coupling and conclude that it too is weak. These several relationships are summarized in Fig. 7.8.

The strength of the anisotropy in any particular crystal is measured by the magnitude of the anisotropy constants K1, K2, etc. Although there seems to no doubt that crystal anisotropy is due primarily to spin–orbit coupling, the details are not clear, and it is generally not possible to calculate the values of the anisotropy constants in a particular material from first principles.

Nor is there any simple relationship between the easy, or hard, direction of magnetization and the way atoms are arranged in the crystal structure. Thus in iron, which is body-centered cubic, the direction of greatest atomic density, i.e., the direction in which the atoms are most closely packed, is k111l, and this is the hard axis. But in nickel (face-centered cubic) the direction of greatest atomic density is k110l, which is an axis of medium hard magnetization. And when iron is added to nickel to form a series of face-centered cubic solid solutions, the easy axis changes from k111l to k100l at about 25% iron, although there is no change in crystal structure.

The magnitude of the crystal anisotropy generally decreases with temperature more rapidly than the magnetization, and vanishes at the Curie point. Since the anisotropy contributes strongly to the coercive field, the coercive field generally goes to zero together with the anisotropy. The combination of vanishing anisotropy and coercive field and nonvanishing magnetization leads to a maximum in permeability, especially the low-field or initial permeability. A maximum in permeability at or near the Curie point was noted by Hopkinson long before there was any theory to account for it, and is known as the Hopkinson effect; it can be used as a simple method to determine an approximate value of the Curie point.

7.5ANISOTROPY MEASUREMENT

The anisotropy constants of a crystal may be measured by the following methods:

1.Torque curves.

2.Torsion pendulum.

3.Magnetization curves (Section 7.6).

4.Magnetic resonance (Section 12.7).

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The first method is generally the most reliable and will be described, along with the closely related torsion-pendulum method, in this section. The other methods are left to later sections, or a later chapter.

7.5.1Torque Curves

A torque curve is a plot of the torque required to rotate the saturation magnetization away from an easy direction as a function of the angle of rotation. Consider first a uniaxial crystal, such as a hexagonal crystal, with an easy axis parallel to the c-axis. It is cut in the form of a thin disk with the c-axis in the plane of the disk, placed in a saturating magnetic field (usually provided by an electromagnet) directed in the plane of the disk, as in Fig. 7.9. The disk is rotated about an axis through its center, and the torque acting on the disk is measured as a function of the angle of rotation. Details of how the torque can be measured will be discussed later. If the field is strong enough, the magnetization Ms will be parallel to H and the angle between c and Ms, which we can call u, will be the same as the angle between c and H.

From Equation 7.3 the u-dependent part of the anisotropy energy, if K2 is negligible, is given by

When the energy of a system depends on an angle, the derivative of the energy with respect to the angle is a torque. Thus dE/du is the torque exerted by the crystal on Ms, and 2dE/du is the torque exerted on the crystal by Ms. (Clockwise torques are taken as positive, and the positive direction of u is measured from Ms to c.) Then the torque on the crystal per unit volume is

The torque L is in dyne-cm/cm3 if E is in erg/cm3, or in N m/m3 if E is in J/m3. Figure 7.10 shows how E and L vary with angle. For positive K1, the 0 and 1808 positions

Fig. 7.9 Uniaxial disk sample in a saturating magnetic field produced by an electromagnet. c ¼ easy axis.