Ferromagneticity and Spinel

The spin of an electron combined with its orbital angular momentum results in a magnetic dipole moment and creates a magnetic field. But, the total dipole moment of all the electrons in many materials which have a filled electron shell is zero. Only atoms with partially filled shells can undergo a net magnetic moment in the absence of an external field. Ferromagnetic materials contain many atoms with unpaired spins. When the tiny magnetic dipoles are aligned in the same direction, they create a measurable macroscopic field. These magnetic dipoles tend to align in parallel to an external magnetic field, an effect called paramagnetism (see Fig. 1.2(a)). A related but much weaker effect is diamagnetism (see Fig. 1.2(b)), due to the orbital motion induced by an external field, resulting in a dipole moment opposite to the applied field. Ferromagnetism involves an additional phenomenon---the dipoles tend to align spontaneously even without any applied field (see Fig. 1.2(c)). This is a purely quantum-mechanical effect. According to the classical electromagnetism, two nearby magnetic dipoles will tend to align in opposite directions. However, they tend to align in the same direction because of the Pauli exclusive principle: two electrons with the same spin cannot sit at the same "position", which effectively

reduces the energy of their electrostatic interaction compared to electrons with opposite spin. This difference in energy is called the exchange energy.

The exchange interaction is primarily responsible for the ordering of atomic moments occurring in magnetic solids and for two other major magnetic ordering types, antiferromagnetism and ferrimagnetism. For instance, in iron (Fe) the exchange interaction between two atoms is about 1000 times stronger than that of classical interaction. There are a small number of "exotic" ferromagnets in which the exchange interactions are exceptionally weak, and then the classical dipole-dipole Figure 1.2: Properties of paramagnetism, diamagnetism, and ferromagnetism.

interaction may become the dominant ones. However, such systems become ferromagnetic only at very low temperature, usually below 1 K. But if the Curie point in a given material is higher than a few kelvins, then its ferromagnetism is surely produced by exchange interaction. In such systems the classical dipole-dipole interaction may only give rise to secondary effects.

For the long range, the advantage of exchange energy is overtaken by the classical tendency of dipoles to anti-align. This is why, in an equilibriated ferromagnetic material, the dipoles in the whole material are not aligned. Rather, they organize into magnetic domains (also known as Weiss domains) that are aligned at short range, but at long range adjacent domains are anti-aligned. The boundary between two domains, where the magnetization flips, is called a domain wall (i.e., a Bloch/Néel wall, depending upon whether the magnetization rotates parallel/perpendicular to the domain interface) and is a gradual transition on the atomic scale.

Thus, an ordinary piece of iron generally has little or no net magnetic moment.

However, if it is placed in a strong enough external magnetic field, the domains will re-orient in parallel with that field, and will remain re-oriented when the field is turned off, thus creating a "permanent" magnet. The domains don't go back to their original minimum energy configuration when the field is turned off because the

domain walls tend to become 'pinned' or 'snagged' on defects in the crystal lattice, preserving their parallel orientation. This is shown by the Barkhausen effect: as the magnetizing field is changed, the magnetization changes in thousands of tiny discontinuous jumps as the domain walls suddenly "snap" past defects. This magnetization as a function of the external field is described by a hysteresis curve.

Although this state of aligned domains is not a minimal-energy configuration, it is extremely stable and has been observed to persist for millions of years in seafloor magnetite aligned by the Earth's magnetic field. Alloys used for the strongest permanent magnets are "hard" alloys made with many defects in their crystal structure where the domain walls "catch" and stabilize. The net magnetization can be destroyed by heating and then annealing the material without an external field, however. The thermal motion allows the domain boundaries to move, releasing them from any defects to return to their low-energy unaligned state.

As the temperature increases, thermal motion and entropy competes with the ferromagnetic tendency for dipoles to align. When the temperature rises beyond a certain point, called the Curie point, there is a second-order phase transition and the system can no longer maintain a spontaneous magnetization, although it still responds paramagnetically to an external field. Below that temperature, there is a spontaneous symmetry breaking and random domains form. The Curie temperature itself is a

critical point, where the magnetic susceptibility (χ) is theoretically infinite and, although there is no net magnetization, domain-like spin correlations fluctuate at all length scales.

Spinel is an important class of mixed-metal oxides, which has the general chemical composition of AB2O4. Atom “A” is a divalent ion of radius between 80 and 110 pm, such as Fe, Mn and Cu; atom “B” is a trivalent atom of radius between 75 and 90 pm, such as Ti, Fe and Co. The majority of spinel compounds belong to the space group Fd3m (F _/ 32/m; O ; No. 227 in the International Tables). The structure consist of a cubic close-packed array of 32 oxygen ions, which forms 64 tetrahedral interstices and 32 octahedral interstices in one unit cell (containing eight formula units (AB2O4)8) [10]. In a normal spinel structure, e.g. MgAl2O4, all the

Figure 1.3: Arrangement of atoms within the MgAl2O4 unit cell.

trivalent cations (Al³⁺) are located in half the octahedral sites, while all the divalent cations (Mg²⁺) occupy 1/8 of the tetrahedral sites. Figure 1.2 shows a typical spinel structure. Notice the red oxygen atoms, the green “A” atoms are in the tetrahedral holes, and the grey “B” atoms in the octahedral holes.

Cobalt ferrite, CoFe2O4 (CFO), has an inverse spinel structure. The normal crystal structure of an AB2O4 spinel consists of the A²⁺ atoms occupying the tetrahedral coordination sites and the B³⁺ atoms occupying the octahedral sites [11].

An inverse spinel is an alternative arrangement where the divalent ions swap with half of the trivalent ions so that the Co²⁺ now occupy octahedral sites i.e. Fe(CoFe)O4. The Co cation occupies one half of the octahedral coordination sites and half the Fe³⁺

cations occupy the other half of the octahedral coordination sites as well as all of the tetrahedral coordination sites.