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

Since Equations 4.15 and 4.26 are identical, it follows from molecular-field theory that Tc, the temperature at which the spontaneous magnetization becomes zero, and u, the temperature at which the susceptibility becomes infinite, are identical. It is left as an exercise to show that the Curie constant per gram given by Equation 4.25 is the same as that given by Equation 3.46.

At temperatures well below the Curie temperature, for example at room temperature in most ferromagnets, even a very strong applied field produces only a small increase in the spontaneous magnetization ss or Ms already produced by the molecular field. Thus, for T ¼ 0.5 Tc, line 2 in Fig. 4.8 shifts to position 20 when a field is applied, but the increase in relative magnetization, from P to P0, is very slight, because the magnetization curve is so nearly flat in this region. To increase the magnetization of a ferromagnetic specimen from zero to ss or Ms usually requires, at room temperature, applied fields less than 1000 oersted or 80 kA/m. In this chapter “applied field” means the true field acting inside the specimen. It is the difference between the actual applied field and any demagnetizing field that may be present. The magnetization of the specimen is then the same as that of each domain in the demagnetized state, and the specimen is sometimes said to be in a state of “technical saturation” or simply “saturation.” To produce any appreciable increase in magnetization beyond this point requires fields 100 times stronger, and such an increase is called forced magnetization. It represents an increase in the magnetization of the domain itself. At any temperature above 0K an infinite field is required to produce absolute saturation, for which ss ¼ s0 or Ms ¼ M0. At 0K this condition can be reached with fields of 102 –103 Oe, or 8–80 kA/m. (At 0K the obstacles to absolute saturation no longer include thermal agitation, because ss equals s0 in each domain of the demagnetized state. The obstacles are then only the “ordinary” ones inherent in domain wall motion and domain rotation; these processes are discussed in Chapters 7–9.)

Forced magnetization beyond the value of the spontaneous magnetization is called the para-process by Russian authors. It is represented, at temperature T1, by points lying along the line AB of Fig. 4.9. The increase in magnetization beyond ss caused by a given increase in field is larger the closer T1 is to the Curie temperature; this and other phenomena occurring near Tc have been described in detail by K. P. Belov [Magnetic Transitions, Consultants Bureau (1961), translated from Russian], L. F. Bates [Modern Magnetis, 4th ed., Cambridge University Press (1961)], and R. M. Bozorth [Ferromagnetism, Van Nostrand (1951); reprinted by IEEE Press (1993)].

Figure 4.9 summarizes the results of molecular-field theory. The temperature u at which the susceptibility x becomes infinite, and 1/x becomes zero, is the same as the temperature Tc at which the spontaneous magnetization appears. Careful measurements have shown that the situation near the Curie point is not that simple. Two deviations from the theory, illustrated in Fig. 4.10, are observed:

1.The curve of 1/x vs T is a straight line at high temperatures but becomes concave upward near the Curie point. The extrapolation of the straight-line portion cuts the

blurred. The fuzziness of the transition is attributed to spin clusters. These are small groups of atoms in which the spins remain parallel to one another over a small temperature range above up; as such they constitute a kind of magnetic short-range order. These clusters of local spin order exist within a matrix of the spin disorder which constitutes a true paramagnetic material, and they gradually disappear as the temperature is raised. Conversely, below uf there is a long-range order of spins even in the absence of an applied field; this is precisely what spontaneous magnetization means.

There is an extensive literature on the behavior of ferromagnets in the temperature region near the Curie point, which is generally categorized under the topic critical indices. The book by Chikazumi (see Table 4.1) treats this subject in some detail.

Another and perhaps more serious disagreement between molecular-field theory and experiment involves the magnitudes of the magnetic moment per atom below and above the Curie point. At absolute zero, where complete saturation is attained, the specific magnetization s0 is given by the maximum magnetic moment per atom in the direction of the field multiplied by the number of atoms per gram, or

where N is Avogadro’s number and A the atomic weight. For iron, substituting the value of s0 from Appendix 2, we have

Similar calculations for cobalt and nickel yield the values listed in the second column of Table 4.2. According to molecular-field theory, these elements should exhibit the same atomic moments above the Curie point, but this is not found to be true. From the observed Curie constants in the paramagnetic region we can calculate the effective moments per atom; these appear in the third column. Values of mH can then be calculated from observed values of meff for any assumed value of J; two such values of mH are shown for each metal in

We will conclude this section on the molecular field by calculating its magnitude. From Equation 4.26 for J ¼ 12, we have for the molecular field coefficient:

Therefore, the molecular field in iron at room temperature is

Hm ¼ (gr)ss ¼ (3:15 104)(218:0) ¼ 6:9 106 Oe ¼ 550 106 A=m

The corresponding values for cobalt and nickel are 11.9 and 14.7 106 oersteds (950 and 1200 106 A/m), respectively. These fields are very much larger than any continuous field yet produced in the laboratory, and some 104 times larger than the fields normally needed to achieve technical saturation. It is therefore not surprising to find that the application of even quite large fields produces only a slight increase in the spontaneous magnetization already produced by the molecular field. Finally, it should be stressed again that the molecular field is not a real field, but rather a force or torque tending to make adjacent atomic moments parallel to one another; it is called a field, and measured in field units, because it has the same kind of effect as a real field.

4.3EXCHANGE FORCES

The Weiss theory of the molecular field says nothing about the physical origin of this field. However, the hypothesis that Hm is proportional to the existing magnetization implies that the phenomenon involved is a cooperative one. Thus, the greater the degree of spin alignment in a particular region of a crystal, the greater is the force tending to align any one spin in that region. The cooperative nature of the phenomenon is clearly shown by the way in which ss decreases with increasing temperature (Fig. 4.7). Near absolute zero the decrease is slight, but, as the temperature is raised, thermal energy is able to reverse more and more spins, thus reducing the aligning force on those spins which are still aligned. The result is a more and more rapid breakdown in alignment, culminating in almost complete disorder at the Curie point.

In seeking for a physical origin of the molecular field, we might at first wonder if it could be entirely magnetic. Figure 4.12a, for example, shows a set of atoms, represented by small circles, each having a net magnetic moment. The directions of these moments, considered as classically free to point in any direction, are indicated by the arrows. Each atom, considered as a magnetic dipole, produces an external field like that of Fig. 1.6, and the sum of the fields of all the dipoles at the point P would be a field from left to right, tending to increase the alignment of the moment on an atom placed at that point. The calculation of this field at P would involve a summation over all the dipoles in the specimen. The calculation becomes easier if we replace the set of dipoles with a continuous medium of average magnetization M, as in Fig. 4.12b and ask: What is the field at P in the center of

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Fig. 4.12 Field at an interior point due to surrounding magnetized material.

a spherical hole in the material? This field, called the Lorentz field, is due to the north and south poles produced on the sides of the hole and is exactly equal in magnitude to the demagnetizing field of a solid magnetized sphere in empty space. We have already calculated this field in Section 2.6 and found it to be 4p/3M (cgs) or 1/3M (SI). For saturated iron at room temperature this is equal to 4p/3 (1714) or about 7200 Oe (cgs) or 575 kA/m (SI). So purely magnetostatic forces are too small by about a factor of 1000 to account for the molecular field.

The physical origin of the molecular field was not understood until 1928, when Heisenberg showed that it was caused by quantum-mechanical exchange forces. About a year earlier the new wave mechanics had been applied to the problem of the hydrogen molecule, i.e., the problem of explaining why two hydrogen atoms come together to form a stable molecule. Each of these atoms consists of a single electron moving about the simplest kind of nucleus, a single proton. For a particular pair of atoms, situated at a certain distance apart, there are certain electrostatic attractive forces (between the electrons and protons) and repulsive forces (between the two electrons and between the two protons) which can be calculated by Coulomb’s law. But there is still another force, entirely nonclassical, which depends on the relative orientation of the spins of the two electrons. This is the exchange force. If the spins are antiparallel, the sum of all the forces is attractive and a stable molecule is formed; the total energy of the atoms is then less for a particular distance of separation than it is for smaller or larger distances. If the spins are parallel, the two atoms repel one another. The exchange force is a consequence of the Pauli exclusion principle, applied to the two atoms as a whole. This principle states that two electrons can have the same energy only if they have opposite spins. Thus two hydrogen atoms can come so close together that their two electrons can have the same velocity and occupy very nearly the same small region of space, i.e., have the same energy, provided these electrons have opposite spin. If their spins are parallel, the two electrons will tend to stay far apart. The ordinary (Coulomb) electrostatic energy is therefore modified by the spin orientations, which means that the exchange force is fundamentally electrostatic in origin.

The term “exchange” arises in the following way. When the two atoms are adjacent, we can consider electron 1 moving about proton 1, and electron 2 moving about proton 2. But electrons are indistinguishable, and we must also consider the possibility that the two electrons exchange places, so that electron 1 moves about proton 2 and electron 2 about proton 1. This consideration introduces an additional term, the exchange energy, into the expression for the total energy of the two atoms. This interchange of electrons takes place at a very high frequency, about 1018 times per second in the hydrogen molecule.

The exchange energy forms an important part of the total energy of many molecules and of the covalent bond in many solids. Heisenberg showed that it also plays a decisive role in ferromagnetism. If two atoms i and j have spin angular momentum Sih=2p and Sjh=2p, respectively, then the exchange energy between them is given by

where Jex is a particular integral, called the exchange integral, which occurs in the calculation of the exchange effect, and f is the angle between the spins. If Jex is positive, Eex is a minimum when the spins are parallel (cos f ¼ 1) and a maximum when they are antiparallel (cos f ¼21). If Jex is negative, the lowest energy state results from antiparallel spins. As we have already seen, ferromagnetism is due to the alignment of spin moments on adjacent atoms. A positive value of the exchange integral is therefore a necessary condition for ferromagnetism to occur. This is also a rare condition, because Jex is commonly negative, as in the hydrogen molecule.

According to the Weiss theory, ferromagnetism is caused by a powerful “molecular field” which aligns the atomic moments. In modern language we say that “exchange forces” cause the spins to be parallel. However, it would be unrealistic to conclude from this change in terminology that all the mystery has been removed from ferromagnetism. The step from a hydrogen molecule to a crystal of iron is a giant one, and the problem of calculating the exchange energy of iron is so formidable that it has not yet been solved. Expressions like Equation 4.29, which is itself something of a simplification and which applies only to two atoms, have to be summed over all the atom pairs in the crystal. Exchange forces decrease rapidly with distance, so that some simplification is possible by restricting the summation to nearest-neighbor pairs. But even this added simplification does not lead to an exact solution of the problem. In the present state of knowledge, it is impossible to predict from first principles that iron is ferromagnetic, i.e., merely from the knowledge that it is element 26 in the periodic table.

Nevertheless, knowledge that exchange forces are responsible for ferromagnetism and, as we shall see later, for antiferroand ferrimagnetism, has led to many conclusions of great value. For example, it allows us to rationalize the appearance of ferromagnetism in some metals and not in others. The curve of Fig. 4.13, usually called the Bethe–Slater curve, shows the postulated variation of the exchange integral with the ratio ra/r3d, where ra is the radius of an atom and r3d the radius of its 3d shell of electrons. (It is the spin alignment of some of the 3d electrons which is the immediate cause of ferromagnetism in Fe, Co, and Ni.) The atom diameter is 2ra and this is also the distance apart of the atom centers, since the atoms of a solid are regarded as being in contact with one another. If two atoms of the same kind are brought closer and closer together but without any change in the radius r3d of their 3d shells, the ratio ra/r3d will decrease. When this ratio is large, Jex is small and positive. As the ratio decreases and the 3d electrons approach one another

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Fig. 4.13 Bethe–Slater curve (schematic).

more closely, the positive exchange interaction, favoring parallel spins, becomes stronger and then decreases to zero. A further decrease in the interatomic distance brings the 3d electrons so close together that their spins must become antiparallel (negative Jex). This condition is called antiferromagnetism.

The curve of Fig. 4.13 can be applied to a series of different elements if we compute ra/r3d from their known atom diameters and shell radii. The points so found lie on the curve as shown, and the curve correctly separates Fe, Co, and Ni from Mn and the next lighter elements in the first transition series. (Mn is antiferromagnetic below 95K, and Cr, the next lighter element, is antiferromagnetic below 378C; above these temperatures they are both paramagnetic.) When Jex is positive, its magnitude is proportional to the Curie temperature (see below), because spins which are held parallel to each other by strong exchange forces can be disordered only by large amounts of thermal energy. The positions of Fe, Co, and Ni on the curve agree with the fact that Co has the highest, and Ni the lowest, Curie temperature of the three.

Although the theory behind the Bethe–Slater curve has received much criticism, the curve does suggest an explanation of some otherwise puzzling facts. Thus ferromagnetic alloys can be made of elements which are not in themselves ferromagnetic; examples of these are MnBi and the Heusler alloys, which have approximate compositions Cu2 MnSn and Cu2MnAl. Because the manganese atoms are farther apart in these alloys than in pure manganese, ra/r3d becomes large enough to make the exchange interaction positive.

Inasmuch as the molecular field and the exchange interaction are equivalent, there must be a relation between them. We can find an approximate form of this relation as follows. Let z be the coordination number of the crystal structure involved, i.e., let each atom have z nearest neighbors, and assume that the exchange forces are effective only between nearest neighbors. Then, if all atoms have the same spin S, the exchange energy between one atom and all the surrounding atoms is

Eex ¼ z( 2JexS2),

when all the spins are parallel. But this is equivalent to the potential energy of the atom considered in the molecular field Hm. If the atom has a magnetic moment of mH in the direction of the field, this energy is

Epot ¼ mHHm: :

But the molecular field coefficient (gr) is related to the Curie temperature u by Equation 4.26. When J ¼ S (pure spin), this substitution gives

which shows that the exchange integral is proportional to the Curie temperature, as mentioned above. For a body-centered cubic structure like that of iron, for which z ¼ 8, and for S ¼ 12, Equation 4.31 gives Jex ¼ 0:25 ku. A more rigorous calculation gives Jex ¼ 0:34 ku for this case.

Exchange forces depend mainly on interatomic distances and not on any geometrical regularity of atom position. Crystallinity is therefore not a requirement for ferromagnetism. The first discovery of an amorphous ferromagnet was reported in 1965 by S. Mader and A. S. Nowick [Appl. Phys. Lett., 7 (1965) p. 57]. They made amorphous thin films of cobalt-gold alloys by co-depositing the two metals from the vapor on a substrate maintained, not at room or elevated temperatures, but at 77K. These alloys were both ferromagnetic and amorphous, as judged from electron-diffraction photographs, and they retained their amorphous condition when heated to room temperature. Many ferromagnetic amorphous alloys have since been made, and several are produced commercially. See Chapter 13.

4.4BAND THEORY

The band theory is a broad theory of the electronic structure of solids. It is applicable not only to metals, but also to semiconductors and insulators. It leads to conclusions about a variety of physical properties, e.g. cohesive, elastic, thermal, electrical, and magnetic. When the band theory is applied specifically to magnetic problems, it is sometimes called the collective-electron theory. This application of band theory was first made in 1933–1936 by E. C. Stoner and N. F. Mott in the United Kingdom and by J. C. Slater in the United States. Our task in this section is to apply it to Fe, Co, and Ni in an attempt to explain the mH values of these metals at 0K, namely, 2.22, 1.72, and 0.60 Bohr magnetons per atom, respectively (Table 4.2). These are important numbers, and any satisfactory theory of magnetism has to account for them.

We will begin by reviewing the electronic structure of free atoms, i.e., atoms located at large distances from one another, as in a monatomic gas. The electrons in such atoms occupy sharply defined energy levels in accordance with the Pauli exclusion principle. This principle states that no two electrons in the atom can have the same set of four quantum numbers. Three of these numbers define the level (“shell”) or sublevel involved, while the fourth defines the spin state of the electron (spin up or spin down). The Pauli principle can therefore be alternatively stated: Each energy level in an atom can contain a maximum of two electrons, and they must have opposite spin. Table 4.3 lists the various energy levels and the number of electrons each can hold, in terms of X-ray notation