# Quantum Mechanical Considerations (Magnetic Properties of Materials)

We have seen in the previous topic that the classical electromagnetic theory is quite capable of explaining the essentials of the magnetic properties of materials. Some discrepancies between theory and experiment have come to light, however, which need to be explained. Therefore, we now refine and deepen our understanding by considering the contributions which quantum mechanics provides to magnetism. We will see in the following that quantum mechanics yields answers to some basic questions. We will discuss why certain metals that we expect to be paramagnetic are in reality diamagnetic; why the paramagnetic susceptibility is relatively small for most metals; and why most metals do not obey the Curie-Weiss law. We will also see that ferromagnetism can be better understood by applying elements of quantum mechanics.

## Paramagnetism and Diamagnetism

We mentioned at the beginning of the previous topic that, for most solids, the dominant contribution to paramagnetism stems from the magnetic moment of the spinning electrons. We recall from topic 6 that each electron state may be occupied by a maximum of two electrons, one having positive spin and the other having negative spin (called spin up and spin down). To visualize the distribution of spins, we consider an electron band to be divided into two halves, each of which is thought to be occupied under normal conditions by an identical amount of electrons of opposite spin, as shown in Fig. 16.1(a). Now, if we apply an external magnetic field to a free electron solid, some of the electrons having unfavorably oriented spins tend to change their field direction. This can only be achieved, however, when the affected electrons assume an energy which is higher than the Fermi energy, EF, since all lower electron states of opposite spin direction are already occupied (Fig. 16.1(b)).

Figure 16.1. Schematic representation of the effect of an external magnetic field on the electron distribution in a partially filled electron band, (a) without magnetic field, (b) and (c) with magnetic field.

Thus, theoretically, the transfer of electrons from one half-band into the other would cause two individual Fermi energies (EF0 and Ef") to occur. Of course, this is not possible. In reality the two band halves shift relative to each other until equilibrium, i.e., a common Fermi energy, is reached (Fig. 16.1(c)).

Now, we recall from topic 6 that the electron distribution within a band is not uniform. We rather observe a parabolic distribution of energy states, as shown in Fig. 6.4. Thus, we refine our treatment by replacing Fig. 16.1(c) with Fig. 16.2, which depicts the density of states of the two half-bands. We observe a relatively large Z(E) near EF. Thus, a small change in energy (provided by the external magnetic field) may cause a large number of electrons to switch to the opposite spin direction.

We calculate now the susceptibility from this change in energy, DE. It is evident that DE is larger, the larger the external magnetic field strength |H|, and the larger the magnetic moment of the spinning electrons

Figure 16.2. Schematic representation of the density of states Z(E) in two half-bands. The shift of the two half-bands occurs as a result of an external magnetic field.

As mentioned already, the number of electrons. AN, transferred from the spin down into the spin up direction depends on the density of states at the Fermi energy, Z(EF), and the energy difference, AE (Fig. 16.2), i.e.,

The magnetization IMI of a solid, caused by an external magnetic field is, according to (14.8),

The magnetization is, of course, larger, the more electrons are transferred from spin down into spin up states. We thus obtain, for the present case,

which yields for the susceptibility

The spin magnetic moment of one electron equals one Bohr magneton, mB (see below). Thus, (16.5) finally becomes

The susceptibilities for paramagnetic metals calculated with this equation agree fairly well with those listed in Table 14.1 (see Problem 1). Thus, (16.6) substantiates, in essence, that only the electrons close to the Fermi energy are capable of realigning in the magnetic field direction. If we postulate instead that all valence electrons contribute to wpara we would wrongfully calculate a susceptibility which is two or even three orders of magnitude larger than that obtained by (16.6).

It is important to realize that the ever-present diamagnetism makes a sizable contribution to the overall susceptibility, so that w for metals might be positive or negative depending on which of the two components predominates. This will be elucidated now in a few examples.

To begin with, we discuss beryllium, which is a bivalent metal having a filled 2s-shell in its atomic state (see topic 3). However, in the crystalline state, we observe band overlapping (see topic 6), which causes some of the 2s-electrons to spill over into the 2p-band. They populate the very bottom of this band (see Fig. 16.3). Thus, the density of states at the Fermi level, and consequently, wpara, is very small. In effect, the diamagnetic susceptibility predominates, which makes Be diamagnetic.

In order to understand why copper is diamagnetic, we need to remember that for this metal the Fermi energy is close to the band edge (Fig. 5.22).

Figure 16.3. Overlapping of 2s- and 2p-bands in Be and the density of states curve for the 2p-band.

Thus, the density of states near EF and the paramagnetic susceptibility (16.6) are relatively small. Furthermore, we have to recall that the diamagnetic susceptibility (15.17),

is proportional to the square of an electron orbit radius, r, and proportional to the total number of electrons, Z, in that orbit. Copper has about ten 3d-electrons, which makes Z ^ 10. Further, the radius of d-shells is fairly large. Thus, for copper, wdia is large because of two contributions. The diamagnetic contribution predominates over the paramagnetic one. As a result, copper is diamagnetic. The same is true for silver and gold and the elements which follow copper in the Periodic Table, such as zinc and gallium.

Intrinsic semiconductors, which have filled valence bands and whose density of states at the top of the valence band is zero (Fig. 6.6) have, according to (16.6), no paramagnetic susceptibility and are therefore diamagnetic. However, a small paramagnetic contribution might be expected for highly doped extrinsic semiconductors, which have, at high enough temperatures, a considerable number of electrons in the conduction band (see topic 8).

We turn now to the temperature dependence of the susceptibility of metals. The relevant terms in both (16.6) as well as (16.7) do not vary much with temperature. Thus, it is conceivable that the susceptibility of diamagnetic metals is not temperature-dependent, and that the susceptibility of paramagnetic metals often does not obey the Curie-Weiss law. In fact, the temperature dependence of the susceptibility for different paramagnetic metals has been observed to decrease, to increase, or to remain essentially constant (Fig. 16.4). However, nickel (above TC) and rare earth metals obey the Curie-Weiss law reasonably well.

At the end of this section we remind the reader that in dilute gases (and also in rare earth metals and their salts) a second component contributes to paramagnetism.

Figure 16.4. Temperature dependence of the paramagnetic susceptibility for vanadium, chromium, and aluminum in arbitrary units.

It stems from a magnetic moment which is caused by the angular momentum of the orbiting electrons (Section 15.3). We mentioned already in Section 15.1 that this contribution is said to be "quenched" (nonexistent) in most solids.

Finally, we want to find a numerical value for the magnetic moment of an orbiting electron from a quantum-mechanical point of view. We recall from (15.5):

Now, quantum theory postulates that the angular momentum, mvr, of an electron is not continuously variable but that it rather changes in discrete amounts of integer multiples of \ only, i.e.,

If one combines (16.8) with (16.9) one obtains

Using n = 1 for the first electron orbit (ground state) yields, for the magnetic moment of an orbiting electron,

It was found experimentally and theoretically that the magnetic moment of an electron due to orbital motion as well as the magnetic moment of the spinning electron are identical. This smallest unit of the magnetic moment is given by (16.11) and is called the Bohr magneton,

which we already introduced without further explanation in (15.3).

## Ferromagnetism and Antiferromagnetism

The ferromagnetic metals iron, cobalt, and nickel are characterized by unfilled d-bands (see topic 3). These d-bands overlap the next higher s-band in a similar manner as shown in the band structure of Fig. 5.22. The density of states for a d-band is relatively large because of its potential to accommodate up to ten electrons. This is schematically shown in Fig. 16.5, along with the Fermi energies for iron, cobalt, nickel, and copper. Since the density of states for, say, nickel is comparatively large at the Fermi energy, one needs only a relatively small amount of energy to transfer a considerable number of electrons from spin down into spin up configurations, i.e., from one half-band into the other. We have already discussed in the previous section this transfer of electrons under the influence of an external magnetic field (Fig. 16.1). Now, there is an important difference between para-magnetics and ferromagnetics. In the former case, an external energy (i.e., the magnetic field) is needed to accomplish the flip in spin alignment, whereas for ferromagnetic materials the parallel alignment of spins occurs spontaneously in small domains of about 1-100 mm diameter. Any theory of ferromagnetism must be capable of satisfactorily explaining the origin of this energy which transfers electrons into a higher energy state.

Figure 16.5. Schematic representation of the density of states for 4s- and 3d-bands and the Fermi energies for iron, cobalt, nickel, and copper. The population of the bands by the ten nickel (3d + 4s)-electrons is indicated by the shaded area.

The energy in question was found to be the exchange energy. It is "set free" when equal atomic systems are closely coupled, and in this way exchange their energy. This needs some further explanation.

We digress for a moment and compare two ferromagnetic atoms with two identical pendula that are interconnected by a spring. (The spring represents the interactions of the electrical and magnetic fields.) If one of the pendula is deflected, its amplitude slowly decreases until all energy has been transferred to the second pendulum, which then in turn transfers its energy back to the first one and so on. Thus, the amplitudes decrease and increase periodically with time, as shown in Fig. 16.6. The resulting vibrational pattern is similar to that of two violin strings tuned at almost equal pitch. A mathematical expression for this pattern is obtained by adding the equations for two oscillators having similar frequencies, o1 and o2,

which yields

Equation (16.15) provides two frequencies, (o1 — o2)/2 and (o1 + o2)/2, which can be identified in Fig. 16.6. The difference between the resulting frequencies is larger, the stronger the coupling. If the two pendula vibrate in a parallel fashion, the "pull" on the spring, i.e., the restoring force, kx, is small. As a consequence, the frequency

(see topic 1) is likewise small and is smaller than for independent vibrations. (On the other hand, antisymmetric vibrations cause large values of k and n0.) This classical example demonstrates that two coupled and symmetrically vibrating systems may have a lower energy than two individually vibrating systems would have.

Figure 16.6. Amplitude modulation resulting from the coupling of two pendula. The vibrational pattern shows beats, similarly as known for two oscillators that have almost identical pitch.

Quantum mechanics treats ferromagnetism in a similar way. The exact calculation involving many atoms is, however, not a trivial task. Thus, one simplifies the problem by solving the appropriate Schrodinger equation for two atoms only. The potential energy in the Schrodinger equation then contains the exchange forces between the nuclei a and b, the forces between two electrons 1 and 2, and the interactions between the nuclei and their neighboring electrons. This simplification seems to be justified, because the exchange forces decrease rapidly with distance.

The calculation, first performed by Slater and Bethe, leads to an exchange integral,

A positive value for Iex means that parallel spins are energetically more favorable than antiparallel spins (and vice versa). We see immediately from (16.17) that Iex becomes positive for a small distance r12 between the electrons, i.e., a small radius of the d-orbit, rd. Similarly, Iex becomes positive for a large distance between the nuclei and neighboring electrons ra2 and rb1.

Iex is plotted in Fig. 16.7 versus the ratio rab/rd. The curve correctly separates the ferromagnetics from manganese, which is not ferromagnetic. Figure 16.7 suggests that if the interatomic distance rab in manganese is increased (e.g., by inserting nitrogen atoms into the manganese lattice), the crystal thus obtained should become ferromagnetic. This is indeed observed. The ferromagnetic alloys named after Heusler, such as Cu2MnAl or Cu2MnSn, are particularly interesting in this context because they contain constituents which are not ferromagnetic, but all contain manganese.

The Bethe-Slater curve (Fig. 16.7) suggests that cobalt should have the highest, and nickel (and the rare earth elements) the lowest, Curie temperature among the ferromagnetics because of the magnitude of their /ex values. This is indeed observed (Table 15.1). Overall, quantum theory is capable of explaining some ferromagnetic properties that cannot be understood with classical electromagnetic theory.

Figure 16.7. Exchange integral, Iex, versus the ratio of interatomic distance, rab, and the radius of an unfilled d-shell. The position of the rare earth elements (which have unfilled /-shells) are also shown for completeness.

We turn now to a discussion on the number of Bohr magnetons in ferromagnetic metals as listed in Table 16.1. Let us consider nickel as an example and reinspect, in this context, Fig. 16.5. We notice that because of band overlapping the combined ten (3d + 4s)-electrons occupy the lower s-band and fill, almost completely, the 3d-band. It thus comes as no surprise that nickel behaves experimentally as if the 3d-band is filled by 9.4 electrons. To estimate mB we need to apply Hund’s rule (Fig. 15.4), which states that the electrons in a solid occupy the available electron states in a manner which maximizes the imbalance of spin moments. For the present case, this rule would suggest five electrons with, say, spin up, and an average of 4.4 electrons with spin down, i.e., we obtain a spin imbalance of 0.6 spin moments or 0.6 Bohr magnetons per atom (see Table 16.1). The average number of Bohr magnetons may also be calculated from experimental values of the saturation magnetization, Ms0 (see Table 15.1). Similar considerations can be made for the remaining ferromagnetics as listed in Table 16.1.

We now proceed one step further and discuss the magnetic behavior of certain nickel-based alloys. We use nickel-copper alloys as an example. Copper has one valence electron more than nickel. If copper is alloyed to nickel, the extra copper electrons progressively fill the d-band and therefore compensate some of the unsaturated spins of nickel. Thus, the magnetic moment per atom of this alloy (and also its Curie temperature) is reduced. Nickel lacks about 0.6 electrons per atom for complete spin saturation, because the 3d-band of nickel is filled by only 9.4 electrons (see above). Thus, about 60% copper atoms are needed until the magnetic moment (and mB) of nickel has reached a zero value (Fig. 16.8). Nickel- copper alloys, having a copper concentration of more than about 60% are consequently no longer ferromagnetic; one would expect them to be diamagnetic. (In reality, however, they are strongly paramagnetic, probably owing to small traces of undissolved nickel.)

Zinc contributes about two extra valence electrons to the electron gas when alloyed to nickel. Thus, we expect a zero magnetic moment at about 30 at.% Zn, etc.

Table 16.1. Magnetic Moment,, at 0 K for Ferromagnetic Metals.

 Metal Fe Co Ni Gd

Figure 16.8. Magnetic moment per nickel atom as a function of solute concentration.

Palladium, on the other hand, has the same number of valence electrons as nickel and thus does not change the magnetic moment of the nickel atoms when alloyed to nickel. The total magnetization (14.8) of the alloy is, of course, diluted by the nonferromagnetic palladium. The same is also true for the other alloys.

We conclude our discussion by adding a few interesting details. The rare earth elements are weakly ferromagnetic. They are characterized by unfilled /-shells. Thus, their electronic structure and their density of states have several features in common with iron, cobalt, and nickel. They have a positive /ex (see Fig. 16.7).

Copper has one more valence electron than nickel, which locates its Fermi energy slightly above the d-band (Fig. 16.5). Thus, the condition for ferromagnetism, i.e., an unfilled d- or /-band is not fulfilled for copper. The same is true for the following elements such as zinc or gallium.

We noted already that manganese is characterized by a negative value of the exchange integral. The distance between the manganese atoms is so small that their electron spins assume an antiparallel alignment. Thus, manganese and many manganese compounds are antiferromagnetic (see Fig. 15.10). Chromium has also a negative /ex and thus is likewise antifer-romagnetic (see Table 15.2).