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

268 MAGNETOSTRICTION AND THE EFFECTS OF STRESS

Equation 8.28, ls becomes equal to lsi, as it should. On the other hand, demagnetized nickel under high compressive stress has all domain vectors parallel to the specimen axis. The value of kcos2ul1 is 1, and the observed ls is zero, in accordance with Fig. 8.23. Under high tensile stress all vectors are at right angles to the axis, kcos2 ul1 ¼ 0, and the observed ls is 3/2 of the normal, stress-free value (lsi), again in accordance with experiment.

In connection with effect of stress on magnetostriction, it is of interest to enquire into the shape of l, M curves. First, consider an extreme example, nickel under sufficient tension so that all Ms vectors are initially at right angles to the axis. Then Equation 8.27 becomes

where u is the angle between Ms vectors and the axis at any particular field strength. But

The linear relation between l and M2 predicted by this equation has been verified by experiment. Now, in measurements on “ordinary” polycrystalline specimens, it is sometimes impossible to reach magnetic saturation because the field source is not strong enough. Then a plot of l vs M2 may be a reasonably straight line in the high-field region, say for values of M greater than about 0.7 or 0.8 Ms. This line can be extrapolated to saturation to obtain a value of ls. The reason for this behavior is that the magnetization of most specimens changes almost entirely by rotation in the high-field region, just as the magnetization of the stressed nickel specimen changes by rotation over the whole range of M, from 0 to Ms.

8.7APPLICATIONS OF MAGNETOSTRICTION

When a material is subjected to an alternating magnetic field, the variation of B (or M ) with H traces out a hysteresis loop. At the same time, the variation of l with H traces out another loop. The latter is actually a double loop, sometimes called a butterfly loop, as illustrated for nickel in Fig. 8.25, because the magnetostrictive strain does not change sign when the field is reversed. The material therefore vibrates at twice the frequency of the field to which it is exposed. This magnetostrictive vibration is one source of the humming sound emitted by transformers. (A transformer contains a “core” of magnetic material subjected to the alternating magnetic field generated by the alternating current in the primary winding.) This sound, sometimes called “60-cycle hum,” actually has a fundamental frequency of twice 60, or 120 Hz. In Europe and other countries where the ac power frequency is 50 Hz, the hum is at 100 Hz.

Conversely, if a partially magnetized body is mechanically vibrated, its magnetization will vary in magnitude about some mean value because of the inverse magnetostrictive effect, and this alternating magnetization will induce an alternating emf in a coil wound around the body.

Fig. 8.25 Hysteresis in the magnetization and magnetostriction of nickel. [L. W. McKeehan, J. Franklin Inst., 202 (1926) p. 737.]

These two effects are exploited in the magnetostrictive transducer. It is one of a class of electromechanical transducers which can convert electrical energy into mechanical energy, and vice versa. The shape and size of a magnetostrictive transducer depends on the nature of the application. In early applications, nickel was commonly used as the magnetostrictive material, but this has been replaced by a rare earth-iron compound of the approximate composition Tb3Dy7Fe19, known as Terfenol-D, prepared by directional solidification to have a strong crystallographic texture. This compound has a saturation magnetostriction greater than 1023, about 15 times larger than nickel, and can be driven to saturation in an applied field of about 1000 Oe or 80 kA/m. In this case the direct correlation between anisotropy and magnetostriction does not exist; the Tb/Dy ratio is chosen so that positive and negative anisotropy contributions cancel, but the magnetostriction retains the large value characteristic of rare earth elements and compounds.

Magnetostrictive transducers using metallic elements are limited to fairly low frequencies, in the kHz range, by eddy-current effects. They have several applications:

1.Underwater Sound. The detection of an underwater object, such as a submarine or a school of fish, is accomplished by a sonar (sound navigation and ranging) system. An “active” sonar generates a sound signal with a transmitting transducer and listens for the sound reflected from the submerged object with a receiving transducer, called a hydrophone. A “passive” sonar listens for sound, such as engine noise, generated by the submerged object. An echo sounder is an active sonar designed to measure the depth of the ocean bottom.

270MAGNETOSTRICTION AND THE EFFECTS OF STRESS

2.Ultrasonic Sound Generators. These are also sound transducers, but here the emphasis is not on sound as a signal but on sound as a mechanical disturbance, usually in some liquid medium. The most important application of this kind is ultrasonic cleaning of metal and other parts, both in the laboratory and in manufacturing operations. The parts to be cleaned are immersed in a solvent which is agitated by an ultrasonic generator. Removal of dirt and grease is much faster and more complete by this method than by simple immersion or scrubbing, and dirt can be dislodged from crevices that are difficult to reach in other ways.

3.Large Force-Small Displacement Positioners. Terfenol-type transducers are useful when well-controlled small displacements requiring large forces are required, as in making small changes in the shape of very large mirrors in telescopes, or in changing the shape of a mechanical part under load so that it acts as if it had infinite stiffness. In comparison with piezoelectric transducers, which change shape under an applied voltage, magnetostrictive transducers provide much larger forces, but require much greater operating power.

8.8DE EFFECT

Another consequence of magnetostriction is a dependence of Young’s modulus E of a magnetic material on its state of magnetization. When an originally demagnetized specimen is saturated, its modulus increases by an amount DE. The value of DE/E depends greatly on the way in which it is measured, as will be explained below.

When a stress is applied to a demagnetized specimen, two kinds of strain are produced:

1.Elastic 1el, such as occurs in any substance, magnetic or not.

2.Magnetoelastic 1me, due to the reorientation of domain vectors by the applied stress. This strain is zero in the saturated state, because no domain reorientation can occur.

For an applied tensile stress, 1me is always positive, whatever the sign of lsi. (If a rod of positive lsi is stressed in tension, Ms vectors will rotate toward the axis, and the rod will lengthen. If lsi is negative, Ms vectors will rotate away from the axis, and the rod will lengthen.)

As a result of these two kinds of strain, the modulus in the demagnetized state is

We have seen in Section 8.6 that 1me depends on the magnitude of the applied stress and on the strength of whatever other anisotropy, such as crystal anisotropy, is present. Figure 8.26

Fig. 8.26 Elastic and magnetoelastic contributions to the stress–strain curve.

shows three kinds of stress–strain curves, with the differences between them greatly exaggerated. One is for a saturated specimen, and the other two are for demagnetized specimens:

(a) one with some strong other anisotropy; and (b) one with weak anisotropy. In (b) the maximum magnetoelastic strain is soon developed, and the curve rapidly becomes parallel to the curve for the saturated specimen. In (a) this occurs only at a higher stress, because of the stronger opposing anisotropy. We see therefore that the DE effect is actually rather complicated, in that E depends, not only on the degree of magnetization, but also on the stress (or strain) and the strength of the other anisotropy present. If DE is measured from a conventional stress–strain curve, the stress level will be reasonably high, and the measured DE/E will be small, typically a few percent. But if DE is measured from the resonant frequency of a small-amplitude vibration, the stress level will be very low, and the value of DE/E may be very large, up to several hundred percent.

The DE effect is a special case of what is more generally called the modulus defect. Whenever any mechanism is present which can contribute an extra strain (inelastic strain) in addition to the elastic strain, the modulus will be smaller than normal. Magnetoelastic strain is just one example of such an extra strain; other examples are the strain contributed by dislocation motion and the strain due to carbon atoms in iron moving into preferred positions in the lattice.

8.9MAGNETORESISTANCE

The magnetoresistance effect is a change in the electrical resistance R of a substance when it is subjected to a magnetic field. The value of DR/R is extremely small for most substances, even at high fields, but is relatively large (a few percent) for strongly magnetic substances. The resistance of nickel increases about 2%, and that of iron about 0.3%, when it passes from the demagnetized to the saturated state. The magnetoresistance effect is mentioned here because of its close similarity to magnetostriction, and because a magnetoresistance measurement, like a magnetostriction measurement, can disclose the presence or absence of preferred domain orientations in the demagnetized state.

As in the case of magnetostriction, we will be interested only in the change in R between the demagnetized and saturated states (the region of domain wall motion and domain rotation), and we will ignore the small change in R that occurs during forced magnetization beyond saturation. A magnetoresistance measurement is usually made on a rod or wire

272 MAGNETOSTRICTION AND THE EFFECTS OF STRESS

specimen with the measuring current i and the applied field H both parallel to the rod axis. Then, in most ferromagnetics, the observed effect is an increase in R of any domain as the angle u between the current i and the Ms vector of that domain decreases. The physical origin of the magnetoresistance effect lies in spin-orbit coupling: as Ms rotates, the electron cloud about each nucleus deforms slightly, as shown by the existence of magnetostriction, and this deformation changes the amount of scattering undergone by the conduction electrons in their passage through the lattice.

The relation between the saturation magnetoresistance change (DR/R)s and the initial domain arrangement is exactly analogous to Equation 8.28 for magnetostriction, namely,

where (DR/R)si is the change observed when a specimen is brought from the ideal demagnetized state to saturation.

Two ways of greatly increasing magnetoresistance have been developed. The resistance of a very thin layer of a normal (nonmagnetic) metal sandwiched between two layers of magnetic materials shows a substantial difference ( 10%) in resisitivity depending on whether the magnetizations in the magnetic layers are parallel or antiparallel. This gives what is known as Giant MagnetoResistance, or GMR; such a structure can be used to read the magnetic fields above a track of stored data in a hard disk drive. And certain perovskite oxides containing manganese and rare earths show an even larger magnetoresistance (100% or more), leading them to be called Colossal MagnetoResistive materials (CMR). They are also candidates for magnetic field detectors in computer drives. In this application, it is not necessary to determine the magnitude of the field above the recorded surface; it is simply necessary to detect the presence (or absence) of a field of a specified magnitude. These effects will be considered in more detail in Chapter 15.

Stress alone can change the resistance, an effect called elastoresistance, because stress alters the orientations of the Ms vectors. Stress can therefore change the observed magnetoresistance, just as it can change magnetostriction.

PROBLEMS

8.1Find the saturation magnetostriction of a cubic crystal in the ,110. direction, in terms of l100 and l111.

8.2A single-crystal disk of a cubic material is cut with the plane of the disk parallel to

¯

(110). A strain gage is cemented on one face of the disk, oriented to measure strain in the [001] direction, and another gage is cemented on the opposite face, oriented to measure strain in the [110] direction. The disk is placed in a saturating magnetic field aligned parallel to the [001] direction, and then rotated about an axis normal to its plane. Let u be the angle between the [001] direction and the field direction, measured so that positive u rotates the field direction from [001] to [110]. Find the equation that describes the strain measured by each gage as a function of u.

8.3Compute lsi for a cobalt single crystal when the measured strain is parallel to the applied field, and the applied field is at (a) 608 and (b) 908 to the easy c-axis.

Compare with Fig. 8.9.

PROBLEMS 273

8.4Consider a cubic material, with l100, l111, and K1 positive, and K2 ¼ 0. The magnetization Ms in a single domain lies in the [010] direction, and a tensile stress is applied in the [100] direction. How does the direction of Ms change as the stress increases from zero to some large value?

8.5If u is the angle between the local magnetization direction and some arbitrary direction in space, show that the average value of cos2 u is 1/3 if the magnetization

directions are random.

Fig. 9.2 Structure of a 1808 domain wall.

units of J/m or erg/cm, and may be regarded as a macroscopic equivalent of the exchange energy J. The quantity df/dx represents the rate at which the direction of local magnetization rotates with position in the wall. The series expansion of cos f is

Dropping the terms in f4 and higher powers because f is assumed to be small, and substituting into Equation 9.1b, we have

The first term is independent of angle and can therefore be dropped. The extra exchange energy existing within the wall is given by the second term

The anisotropy energy is given in the general case by

278 DOMAINS AND THE MAGNETIZATION PROCESS

where f is measured from the easy axis. For uniaxial anisotropy, g(f) ¼ Ku sin2 f, and for cubic anisotropy with magnetization confined to a f100g plane, g(f) ¼ K1 sin2 f cos2 f.

The wall energy is given by the sum of the exchange and anisotropy energies, integrated over the thickness of the wall:

Note that the symbol s is used here to denote the surface energy of a domain wall. Some authors use g. Greek s has previously been used in this book to denote magnetization per unit mass, and also stress.

We consider the case of a 1808 domain wall lying in the yz plane, with the direction of magnetization rotating about the x-axis while remaining parallel to the yz plane. Then x denotes position in the wall, and f denotes the direction of the local magnetization. Nature will choose the magnetization pattern [x ¼ f(f) or f ¼ f(x)] that minimizes the total wall energy s.

The mathematical treatment of the domain wall is usually handled as a problem in the calculus of variations. It seems conceptually and mathematically easier to think in terms of the torque acting on the local magnetization. The torque resulting from the exchange energy is

Physically, we can see that if the angle between neighboring spins is constant (df/dx ¼ const.), the exchange torques acting on each spin due to its neighbors will be equal and opposite, canceling to zero. So only if df/dx is not constant, and d2f/dx2 is not zero, is there a net exchange torque.

The torque resulting from the anisotropy energy is

At equilibrium, these torques must be equal and opposite, giving zero net torque, at each point in the domain wall. Therefore,

everywhere in the wall.

Multiplying by @f/@x and integrating over x, the first term in Equation 9.8 becomes