Table of Contents
5. Magnetic Materials
Exploring the Building Blocks of Magnetism
The three main classes of permanent magnet materials are: (i) hexagonal ferrimagnetic oxides, (ii) 3d - based ferromagnetic alloys and (iii) 3d - 4f intermetallics. The oxide ceramics are cheap to produce and offer modest magnetic properties with chemical stability at low cost. The 3d metallic alloys include steels which have a high magnetization, but little anisotropy. High - performance permanent magnets are iron - or cobalt - rich rare - earth intermetallics. The first of these, dating from the 1960s, was SmCo5, which has a high Curie temperature and excellent anisotropy but a mediocre spontaneous magnetization. Sm2Co17 - based magnets offer a greater magnetization, but cobalt is expensive. Since the magnetic moment of iron is larger than that of cobalt and the metal is two to three orders of magnitude cheaper, there was a need to develop iron - based compounds such as Sm2Fe17 have no suitable rare - earth - iron binaries. Iron - rich ternary phase Nd2Fe14B, which exhibits the best energy product and is now the basis of most high - performance magnets, was discovered in 1983. Since then, other ternary rare - earth iron phases with interstitial carbon or nitrogen such as Sm2Fe17N3 have been discovered which are suitable for permanent - magnet development.
Oxides: Essential Components of Magnetic Materials
Ferrites are iron oxides1 which include a high proportion of ferric (Fe3 + ) cations in their structure. The principal magnetic interaction is antiferromagnetic superexchange coupling of the spins of pairs of ferric cations via a bridging oxygen anion. A net magnetization arises when the crystal structure includes cation sublattices with unequal populations, so that the overall magnetic structure is ferrimagnetic. Most important from our point of view are the hexagonal ferrites, whose production tonnage far exceeds that of any other type of permanent magnet. They have a complex structure, so it is convenient to discuss
1Not to be confused with the iron - rich \(\alpha\) - phase in the Fe - C phase diagram.Table 5.1. Ionic radii of ions in oxides. Low - spin - state values are shown in brackets. The number of 3d electrons is shown in italics. \(R(O^{2 - })\)= 140 pm (1.4 Å).
| Fourfold | pm | Sixfold | pm | Sixfold | pm | 12 - fold | pm |
|---|---|---|---|---|---|---|---|
| \(Mg^{2 + }\) | 53 | \(Cr^{4 + }2\) | 55 | \(Ti^{3 + }1\) | 67 | \(Ca^{2 + }\) | 134 |
| \(Zn^{2 + }\) | 60 | \(Mn^{4 + }3\) | 53 | \(V^{3 + }2\) | 64 | \(Sr^{2 + }\) | 144 |
| \(Al^{3 + }\) | 42 | \(Cr^{3 + }3\) | 62 | \(Ba^{2 + }\) | 161 | ||
| \(Fe^{3 + }\) | 52 | \(Mn^{2 + }5\) | 83 | \(Mn^{3 + }4\) | 65 | \(Pb^{3 + }\) | 149 |
| \(Fe^{2 + }6\) | 78 (61) | \(Fe^{3 + }5\) | 64 | \(Y^{3 + }\) | 119 | ||
| \(Co^{2 + }7\) | 75 (65) | \(Co^{3 + }6\) | 61 (56) | \(La^{3 + }\) | 136 | ||
| \(Ni^{2 + }8\) | 69 | \(Ni^{2 + }7\) | 60 | \(Gd^{3 + }\) | 122 |
first simpler oxides which illustrate the principles involved before tackling the hexagonal ferrites themselves.
Most oxide structures are based on dense - packed oxygen arrays which contain both octahedral and tetrahedral interstices. The usual dense - packed structures exhibit ABCABC... or ABABAB... stacking sequences of hexagonal layers. The structures are respectively face - centred cubic (fcc) and hexagonal close - packed (hcp). Taking \(R(\text{O}^{2 - })=140\ \text{pm}\), the radii of the cations which will just fit in octahedral (sixfold) and tetrahedral (fourfold) sites are \((\sqrt{2}-1)R(\text{O}^{2 - })=58\ \text{pm}\) and \((\sqrt{3/2}-1)R(\text{O}^{2 - })=32\ \text{pm}\) respectively. Comparing these numbers to the ionic radii of \(\text{Fe}^{3 + }\) in octahedral and tetrahedral coordination, 64 and 52 pm, respectively (table 5.1), we see that the dense - packed oxygen arrays must distort to accommodate the interstitial ferric ions. Densities of oxides are typically about \(5000\ \text{kg}\ \text{m}^{-3}\). The different stacking possibilities and the presence of two types of interstice lead to a great variety of magnetic oxides. Important groups, with examples, are monoxides (MnO), dioxides (CrO2), sesquioxides (\(\alpha\) - Fe2O3), spinel ferrites (MnFe2O4), garnets (Y3Fe5O12), perovskites (GdFeO3) and hexagonal ferrites (\(\text{BaFe}_{12}\text{O}_{19}\)). Common iron oxides are \(\text{Fe}_{3}\text{O}_{4}\) (magnetite, the principal constituent of lodestone), \(\alpha\) - Fe2O3 (haematite), the metastable compound \(\gamma\) - Fe2O3 (maghemite), FeO (wüstite) and the hydroxide \(\alpha\) - FeOOH (goethite). All these oxides are black or brown, except Y3Fe5O12 which is green. All are insulators except Fe3O4 and CrO2.
The colours and electronic structure of transition - metal oxides are largely determined by the 3d cations. Normally, the orbital moment is quenched by the crystal field produced by the surrounding \(\text{O}^{2 - }\) ions, so the cation moment in Bohr magnetons is equal to the number of unpaired electrons. The \(\text{Fe}^{3 + }\) ion has a 3d5 configuration with five unpaired electrons which gives the largest possible 3d spin - only moment of \(5\ \mu_{\text{B}}\) per ion. However, because of the large volume of the oxygen anions and the tendency towards antiparallel sublattice coupling, this large cation moment does not translate into a particularly impressive
Figure 5.1. Some simple oxide structures.
Table 5.2. Antiferromagnetic monoxides.
| \(a_{0}\) (pm) | \(S\) | \(T_{\text{N}}\) (K) | \(\theta_{\text{p}}\) (K) | \(\mathcal{J}_{1}\) (K) | \(\mathcal{J}_{2}\) (K) | ||
|---|---|---|---|---|---|---|---|
| MnO | 445 R | \(3\text{d}^{5}\) | 5/2 | 117 | −610 | −7.2 | −3.5 |
| FeO | 431 R | \(3\text{d}^{6}\) | 2 | 198 | −570 | −7.8 | −8.2 |
| CoO | 426 T | \(3\text{d}^{7}\) | 3/2 | 291 | −330 | −6.9 | −21 |
| NiO | 418 R | \(3\text{d}^{8}\) | 1 | 525 | −1310 | −50 | −85 |
aligned {111} planes. Below \(T_{\text{N}}\), there is a tiny rhombohedral (R) or tetragonal (T) distortion of the cubic structure which is of magnetostrictive origin.
Another monoxide of interest as a model ferromagnet is EuO. It also has the NaCl structure (B1), with a Curie temperature of 78 K. Stoichiometric EuO is an insulator, with indirect exchange of the Eu2 + 4f7 ion cores via the unoccupied Eu 5d/6s orbitals.
Trivalent cations such as Ti3 + , V3 + , Cr3 + and Fe3 + form sesquioxides \(T_{2}\text{O}_{3}\) (table 5.3). They crystallize in the same basic structure of corundum or sapphire, which is based on an hcp oxygen array with cations in octahedral sites (figure 5.1). There are no 180° T - O - T bonds, but strong antiferromagnetic bonds with an angle of about 135° between a high Neél temperature of 950 K for \(\alpha\) - Fe2O3, the ferric sesquioxide, also known by its mineral name of haematite (figure 5.193).
Table 5.3. Antiferromagnetic sesquioxides and perovskites.
| a (pm) | c (pm) | \(T_{\text{N}}\) (K) | a (pm) | b (pm) | c (pm) | \(T_{\text{N}}\) (K) | ||
|---|---|---|---|---|---|---|---|---|
| \(\text{Ti}_{2}\text{O}_{3}\) | 514 | 1366 | 470 | \(\text{LaCrO}_{3}\) | 548 | 552 | 776 | 300 |
| \(\text{V}_{2}\text{O}_{3}\) | 503 | 1362 | 150 | \(\text{LaMnO}_{3}\) | 554 | 574 | 769 | 130 |
| \(\text{Cr}_{2}\text{O}_{3}\) | 496 | 1359 | 300 | \(\text{LaFeO}_{3}\) | 555 | 557 | 786 | 740 |
| \(\text{Mn}_{2}\text{O}_{3}\) | 504 | 1412 | 80 | \(\text{LaCoO}_{3}\) | 542 | 1307 | 80 | |
| \(\text{Fe}_{2}\text{O}_{3}\) | 504 | 1375 | 950 | \(\text{LaNiO}_{3}\) | 545 | 656 | — |
Figure 5.2. Magnetization curves of haematite (a) above and (b) below the Morin transition (\(T_{\text{M}} = 260\ K\)).
Haematite is the archetypical weak ferromagnet2. Uniaxial compounds sometimes adopt an antiferromagnetic structure where the two sublattices are inclined at a tiny angle, leading to a weak resultant magnetization. Phenomenologically, this Dzyaloshinskii - Moriya interaction arises from a small \(-\boldsymbol{D}\cdot(\boldsymbol{S}_{1}\times\boldsymbol{S}_{2})\) term in the Hamiltonian, where \(\boldsymbol{D}\) must lie along a high - symmetry axis. Atomically, this term is due to spin - orbit interaction. When \(\boldsymbol{D}\) is along the crystallographic \(c\) - axis of hexagonal, rhombohedral and tetragonal crystals, spin canting occurs if \(\boldsymbol{S}_{1}\) and \(\boldsymbol{S}_{2}\) lie in the basal plane. This is the case for haematite above the Morin temperature \(T_{\text{M}} = 260\ K\), where \(M\approx2\ kA\ m^{-1}\). A spin reorientation transition occurs at \(T_{\text{M}}\), and at lower temperatures magnetocrystalline anisotropy causes the sublattice magnetizations to lie along the \(c\) - axis direction, so that \(-\boldsymbol{D}\cdot(\boldsymbol{S}_{1}\times\boldsymbol{S}_{2}) = 0\) and the weak moment disappears (figure 5.2).
Another common antiferromagnetic structure type is perovskite (\(\text{CaTiO}_{3}\)). Here the large \(\text{Ca}^{2 + }\) and \(\text{O}^{2 - }\) ions together form an fcc array, with the \(\text{Ti}^{4 + }\) ions occupying oxygen - coordinated octahedral sites. Ca may be replaced
2This term has a different sense in metals and in antiferromagnetic ionic compounds.by a rare - earth ion \(R^{3 + }\) and \(\text{Ti}^{4 + }\) by a trivalent 3d cation \(T^{3 + }\). There is considerable structural flexibility and the lattice is generally distorted from the cubic ideal (figure 5.1). The rare - earth orthoferrites \(R\text{FeO}_{3}\), for example, have an orthorhombic structure with unit - cell parameters \((\sqrt{2}a_{0},\sqrt{2}a_{0},a_{0})\). As in MnO, superexchange bonds with six neighbours provide the dominant distorted magnetic interaction, and Néel temperatures for this series are around 650 K. The bond angles are reduced to about 160° by buckling of the octahedral network caused by the smaller ionic radius of \(R^{3 + }\) than \(\text{O}^{2 - }\). The rare - earth ions here are decoupled from the iron ions because they have equal numbers of iron neighbours on each antiferromagnetic sublattice. Note that orthoferrites are also weak ferromagnets with a canting angle of \(0.5^{\circ}\), compared to \(0.04^{\circ}\) in haematite. Other perovskites, such as \((\text{La}_{0.7}\text{Sr}_{0.3})\text{MnO}_{3}\) are ferromagnetic and metallic, both properties arising from the mixed valence of manganese imposed by the presence of \(\text{La}^{3 + }\) and \(\text{Sr}^{2 + }\). Electrons in a band formed from \(e_{\text{g}}\) orbitals (section 3.1.4.1) hop freely among the \(\text{Cr}^{3 + }\) Mn ion cores (Coey et al 1999). The itinerant electrons hop with spin memory and are strongly coupled to the ion cores by on - site Hund's rule exchange, which gives rise to a ferromagnetic interaction among the manganese known as double exchange.
An interesting dioxide is \(\text{CrO}_{2}\), which is a ferromagnet, crystallizing in the tetragonal rutile (\(\text{TiO}_{2}\)) structure (figure 5.1). Lattice parameters are \(a = 442\ \text{pm}\) and \(c = 292\ \text{pm}\). The \(\text{Cr}^{4 + }(3\text{d}^{2})\) is in a distorted octahedral site with a spin - only moment of \(2.0\ \mu_{\text{B}}\). Unlike \(\text{Cr}_{2}\text{O}_{3}\) which is an insulator, \(\text{CrO}_{2}\) is a metal with a partly filled sub - band. The ferromagnetic exchange follows from the second Goodenough - Kanamori rule. \(\text{CrO}_{2}\) has a Curie temperature of 396 K and the magnetization at room temperature is \(0.40\ \text{MA}\ m^{-1}(\mu_{0}M_{\text{s}} = 0.50\ \text{T})\). The magnetocrystalline anisotropy is small, \(K_{1}=20\ \text{kJ}\ m^{-3}\). Acicular (needle - like) \(\text{CrO}_{2}\) powders have \(D_{\text{e}}\approx0\) and therefore show shape anisotropy \(K_{1}^{\text{shape}}\) of up to \((1/4)\mu_{0}M_{\text{s}}^{2}=50\ \text{kJ}\ m^{-3}\) (equation (3.83)). As a rule, the dimensionless hardness parameter \((1.24,3.73)\)
\(\kappa=\sqrt{\frac{K_{1}}{\mu_{0}M_{\text{s}}^{2}}}\) (1.24)
should be greater than 1 for a permanent magnet. It cannot exceed 0.5 if the anisotropy is due only to shape. \(\text{CrO}_{2}\) particles, typically 300 nm long with an aspect ratio of 10:1, are used as tape magnets. Their coercivity is about 50 kA m-1.
Apart from \(\text{CrO}_{2}\) and \((\text{La}_{0.7}\text{Sr}_{0.3})\text{MnO}_{3}\), which are half - metallic ferromagnets, all the oxides considered in this section are basically antiferromagnetic. There is no demagnetizing field and therefore no shape anisotropy; the easy magnetic direction is not influenced by sample geometry. Antiferromagnets have a strong tendency to form magnetic domains. The easy magnetic direction is fixed by the balance between magnetocrystalline anisotropy
of crystal - field origin and magnetostatic dipole interactions. The dipole field obtained by summing equation (1.3) over all neighbours on a lattice is non - zero except in cubic lattices. The dipole interaction varies with temperature as the square of the atomic moment \(m\). Dipole fields and magnetocrystalline anisotropy fields, defined in an antiferromagnet as \(K_{1}/M_{\text{A}}\) where \(M_{\text{A}}\) is the sublattice magnetization, can be as high as 1 T in ferric oxides and small differences in the temperature dependence of the anisotropic contributions of opposite sign may lead to spin reorientation transitions, such as the Morin transition in haematite.
At first sight it is surprising that S - state ions such as \(Fe^{3 + }\) and \(Mn^{2 + }\) exhibit any crystal - field anisotropy, since the ground state is an orbital singlet \(^5S_{5/2}\) with a spherical charge distribution. The anisotropy is due to mixing of crystal - field excited states in a non - cubic environment. The relevant lowest - order crystal - field parameter \(A_2^0\) may be deduced, for instance, by measurements of the electric field gradient \(V_{zz}\) at the nucleus by \(^{57}\text{Fe}\) Mössbauer spectroscopy. In the absence of spin - orbit coupling, the energy splittings \(E_{an}\) are given by the crystal - field interaction, but a treatment of the spin - orbit coupling with second - order perturbation theory gives anisotropy contributions of the type \(A_2^0\) (section 3.3.6). A detailed analysis of the problem along the lines sketched in section 3.2
(Fuchikami 1965) reveals that strong uniaxial crystal fields yield an admixture of excited states that depends on the relative orientation of the crystal field and the ionic moment. In \(\alpha\) - \(\text{Fe}_{2}\text{O}_{3}\), the crystal - field anisotropy \(K_{1}\) is positive, thus favouring the easy axis, but the dipole contribution \(K_{1}^{\text{dip}}\) is negative and favours the basal plane. As a consequence, spin reorientation occurs when \(K_{1}^{\text{tot}}=K_{1}+K_{1}^{\text{dip}} = 0\). Powders of \(\alpha\) - \(\text{Fe}_{2}\text{O}_{3}\) exhibit hysteresis at room temperature, despite the smallness of \(K_{1}^{\text{tot}}\) (figure 5.2). This is because the weak moment \(M_{\text{s}}\) is also very small (\(|\boldsymbol{M}_{\text{s}}|\) of the order of 1 kA m-1) so that the macroscopic anisotropy field \(\mu_{0}H_{\text{a}} = 2K_{1}^{\text{tot}}/M_{\text{s}}\) is of the order of 5 T. Here the hardness parameter \(\kappa\approx10\), which shows that permanent magnetism is most easily achieved in materials where the magnetization is small. This is exploited in some magneto - optic recording media, which operate near their compensation temperature (section 5.1.2.2). The order of magnitude of both \(K_{1}\) and \(K_{1}^{\text{dip}}\) in ferric oxides with non - cubic sites and uniaxial lattice symmetry is 0.01 to 1 MJ m-3. The upper end of the range is sufficient for permanent magnetism to be envisaged. For a good permanent magnet we would like a ferrimagnetic compound where both contributions are positive or else a compound with a site - ion combination which leads to a very large value of \(K_{1}\).Cubic Ferrites: Insights into Their Structure and Magnetic Behavior
Spinels
A group of ferrimagnets which includes materials widely used for magnetic recording as well as soft magnets suitable for high - frequency applications are
Figure 5.3. Crystal structures of cubic ferrites: spinel structure (\(\text{Fe}_{3}\text{O}_{4}\)) and garnet structure (\(\text{Y}_{3}\text{Fe}_{5}\text{O}_{12}\)).
cubic ferrites with the spinel (\(\text{MgAl}_{2}\text{O}_{4}\)) structure (Brabers 1997). Spinels have general formula \([\text{A}][\text{B}_{2}]\text{O}_{4}\), where \([\ldots]\) and \(\{\ldots\}\) refer to tetrahedral and octahedral sites, respectively, and \(X\) is an S. The structure (figure 5.3) is based on an fcc \(X\) array which contains eight formula units where half the octahedral interstices (16d or B sites) and one eighth of the tetrahedral ones (8a or A sites) are filled by cations (Hill et al 1979).
Spinel ferrites have typical composition \(\text{MFe}_{2}\text{O}_{4}\) where \(M\) is a di - or trivalent cation. An important feature is that the \(M^{z + }\) ions do not necessarily occupy the tetrahedral [A] sites. Two extreme cation distributions are the normal one, \([\text{M}][\text{Fe}_{2}]\text{O}_{4}\), and the inverse one, \([\text{Fe}][\text{MFe}]\text{O}_{4}\). Of course, intermediate distributions are possible and the distribution found in a particular sample will depend both on its chemical composition and heat treatment. Usually,
inverse distributions are more common than normal ones, if only because the comparatively small \(\text{Fe}^{3 + }\) ions fit better into the small tetrahedral sites than most divalent cations (table 5.1). There are no 180° \(\text{Fe}_{\text{B}}\text{-O-Fe}_{\text{B}}\) paths in this structure, because of the incomplete occupancy of octahedral interstices. The principal exchange interaction in inverse spinel ferrites is superexchange via an \(\text{Fe}_{\text{A}}\text{-O-Fe}_{\text{B}}\) bond angle of about 125°. The bond angle depends on the special position parameter \(u\) of the 32 oxygen sites which are at \(u, u, u\); for an ideal fcc lattice, \(u = 1/4\sqrt{3}\). As a result of the superexchange, the structure is collinear ferrimagnetic with antiparallel alignment of A and B sublattices.
Magnetite (\(\text{Fe}_{3}\text{O}_{4}\)) is a special case of an inverse spinel where the two cations are iron ions in different valence states, \(\text{Fe}^{2 + }\) and \(\text{Fe}^{3 + }\). The tetrahedral sites are occupied by \(\text{Fe}^{3 + }\) and the octahedral states are occupied by an equal mixture of both ions, the overall cubic symmetry being maintained by rapid hopping of the sixth \(\downarrow\) electron of \(\text{Fe}^{2 + }\) among the \(\uparrow3\text{d}^{5}\) iron cores. As a result of this electron hopping, magnetite is a moderately good electrical conductor. Hopping ceases below the Verwey transition at \(T_{\text{V}} = 119\ K\), where the ferrous ions order in pairs and the symmetry of the crystal is reduced to triclinic. The magnetic moment per formula unit of \(\text{Fe}_{3}\text{O}_{4}\), \(m\approx4\ \mu_{\text{B}}\), equals that of \(\text{Fe}^{2 + }\), since the moments of the two \(\text{Fe}^{3 + }\) ions cancel because they are located on antiparallel sublattices.
Magnetite can be oxidized while retaining the spinel structure. The extreme is maghemite or \(\gamma - Fe_2O_3\) which includes B - site vacancies that may or may not be ordered in a tetragonal superstructure with \(c = 3a\); the formula is usually
The intrinsic anisotropy of cubic spinel ferrites is usually quite small: in magnetite, for example, the anisotropy field \(\mu_{0}H_{\text{a}}=-4K_{1}/3M_{\text{s}} = 0.031\ \text{T}\). An exception is \(\text{CoFe}_{2}\text{O}_{4}\), where the \(\text{Co}^{2 + }\) ion may develop an orbital moment of about \(1\ \mu_{\text{B}}\) giving a rather high anisotropy of \(0.18\ \text{MJ}\ m^{-3}\). Co surface - treated \(\gamma\) - \(\text{Fe}_{2}\text{O}_{3}\) has a coercivity of about 50 kA m-1 is a widely used particulate medium for video tapes. Good high - frequency materials should be insulators with the least possible anisotropy and magnetostriction to reduce hysteresis losses. The properties of the spinel ferrites are summarized in table 5.4. A continuous range of properties may be achieved by blending these basic compositions. Most cubic spinels \(\text{MFe}_{2}\text{O}_{4}\) have \(K_{1}<0\), indicating weak nickel - type (111) anisotropy, whereas Co ferrite exhibits quite strong iron - type (001) anisotropy (\(K_{1}>0\)). Likewise, there are spinels where the magnetostriction \(\lambda_{\text{s}}\) may be positive or negative.
Table 5.4. Magnetic properties of spinel ferrites at room temperature. N, normal; I, inverse.
| \(a_{0}\) (pm) | \(T_{\text{C}}\) (K) | \(M_{\text{s}}\) (MA m-1) | \(K_{1}\) (kJ m-3) | \(\lambda_{\text{s}}\) (10-6) | \(\rho_{\text{d}}\) (\(\Omega\) m) | ||
|---|---|---|---|---|---|---|---|
| \(\text{MgFe}_{2}\text{O}_{4}\) | I | 836 | 700 | 0.18 | −3 | −6 | 105 |
| \(\text{ZnFe}_{2}\text{O}_{4}\) | N | 844 | \(T_{\text{N}} = 9\) | 1 | |||
| \(\text{MnFe}_{2}\text{O}_{4}\) | I | 852 | 575 | 0.40 | −3 | −5 | 102 |
| \(\text{Fe}_{3}\text{O}_{4}\) | I | 840 | 860 | 0.50 | −12 | 40 | 10-1 |
| \(\text{CoFe}_{2}\text{O}_{4}\) | I | 839 | 790 | 0.45 | 180 | −110 | 105 |
| \(\text{NiFe}_{2}\text{O}_{4}\) | I | 834 | 865 | 0.33 | −7 | −17 | 102 |
| \(\text{Li}_{0.5}\text{Fe}_{2.5}\text{O}_{4}\) | 829 | 943 | 0.33 | −8 | −8 | 1 | |
| \(\gamma\) - \(\text{Fe}_{2}\text{O}_{3}\) | 834 | 1020a | 0.43 | ||||
| aEstimate; \(\gamma\) - \(\text{Fe}_{2}\text{O}_{3}\) converts to \(\alpha\) - \(\text{Fe}_{2}\text{O}_{3}\) at about 800 K. | |||||||
Table 5.5. Magnetic properties of selected iron garnets \(R_{3}\text{Fe}_{5}\text{O}_{12}\).
| \(R\) | \(a_{0}\) (pm) | \(T_{\text{C}}\) (K) | \(T_{\text{comp}}\) (K) | \(m\) (\(\mu_{\text{B}}\) f.u.-1) at \(T = 0\) | \(M_{\text{s}}\) (kA m-1) at RT | \(K_{1}\) (kJ m-3) at RT |
|---|---|---|---|---|---|---|
| Y | 1238 | 560 | — | 5.0 | 140 | −0.5 |
| Gd | 1247 | 564 | 290 | 16.0 | 10 | −0.6 |
| Dy | 1241 | 563 | 220 | 16.9 | 34 | −0.5 |
Lodestones, made of impure magnetite, were the original natural permanent magnets, which—magnetized by lightning—could have an energy product comparable to that of martensitic steel (about \(1\ \text{kJ}\ m^{-3}\)). Furthermore, single - domain magnetite particles—magnetized by cooling in the Earth's field through their superparamagnetic blocking temperature—are the source of the natural remanence of rocks such as basalt.
An interesting application of ferrites is ferrofluids. These magnetic liquids consist of small magnetic particles, typically magnetite or maghemite with a particle size of about 50 nm, covered by a surfactant such as oleic acid and dispersed in a carrier liquid such as kerosene. Due to the dilution of the magnetic phase, the saturation polarization of ferrofluids is of the order of 0.05 T. They are superparamagnetic.
Garnets
The other important family of cubic ferrites is the garnets (Gilleo 1980). Their structure is that of the mineral \((\text{Ca}_{3})_{3}[\text{Si}_{2}]\{\text{Al}_{2}\}\text{O}_{12}\), where calcium occupies large
24c dodecahedral sites, silicon occupies the tetrahedral 24d sites and aluminium occupies the octahedral 16a sites (figure 5.3). Garnets of general formula \(R_{3}\text{Fe}_{5}\text{O}_{12}\) are of particular interest. All the cations may be trivalent when \(R\) is a trivalent rare - earth or yttrium. Yttrium iron garnet \(Y_{3}\text{Fe}_{5}\text{O}_{12}\) (YIG) is a ferrimagnet with strong antiparallel superexchange coupling of the 24d and 16a sublattices. There is a net moment of about \(5\ \mu_{\text{B}}\) per formula unit at room temperature, corresponding to \(\mu_{0}M_{0}=0.25\ T\) and the Curie point is at 560 K (table 5.5). When \(R\) is a magnetic rare - earth these ions form a third magnetic sublattice which is coupled antiparallel to the 16a - site spins by a weak exchange interaction. This leads to a steep temperature dependence of the 24c sublattice moment. Some rare - earth iron garnets show a compensation temperature \(T_{\text{comp}}\) where the net magnetization of the three sublattices is exactly zero. The highest compensation point is for \(\text{Gd}_{3}\text{Fe}_{5}\text{O}_{12}\) (GIG). The low anisotropy, of the order of \(1\ \text{kJ}\ m^{-3}\) and the low magnetostriction make yttrium iron garnet particularly suited for microwave applications. In thin films, however, some anisotropy can be induced by strain when the film grows epitaxially on a substrate with a different lattice parameter. Hysteresis then appears near \(T_{\text{comp}}\) where \(M_{\text{s}}\approx0\) and the macroscopic anisotropy field \(\mu_{0}H_{\text{a}} = 2K_{1}/M_{\text{s}}\) can be large.
Hexagonal Ferrites: Unique Characteristics and Technological Uses
The ferrites most useful as permanent magnets have the hexagonal magnetoplumbite \([\text{Pb}(\text{Fe},\text{Mn},\text{Al},\text{Ti})_{12}\text{O}_{19}]\) or M - type structure shown in figure 5.4. Their properties have been reviewed by Kojima (1980). Lead may be completely replaced by divalent barium or strontium, and partially replaced by other cations such as calcium. The smaller cation sites may be entirely occupied by iron, as in \(\text{BaFe}_{12}\text{O}_{19}\) or \(\text{SrFe}_{12}\text{O}_{19}\). These oxides are good electrical insulators (\(\rho>10^{8}\ \Omega\ m\)) which helps reduce eddy - current losses in rotating electrical machines. The structure is based on spinel - like S - slabs, where a [111] direction becomes the hexagonal \(c\) - axis, interspersed with R - slabs containing \(M^{2 + }\) cations in an oxygen position. The S - slabs contain iron in the \(2a\) and half of the \(12k\) octahedral sites and in the \(4f_{1}\) tetrahedral sites. The R - slabs contain iron in the \(4f_{2}\) octahedral sites and the rest of the \(12k\) octahedral sites and in the \(2b\) trigonal bipyramidal site, indicated in figure 5.4. The latter site, which may be regarded as two tetrahedral sites back - to - back, has \(6m\) point symmetry and the crystal field there has a strongly uniaxial character, unrelated to any underlying cubic (octahedral or tetrahedral) symmetry. There are therefore five iron sublattices in all and superexchange via bridging oxygen leads to a collinear ferromagnetic structure, with iron on the \(12k\), \(2a\) and \(2b\) sites having their spins antiparallel to the \(4f_{1}\) and \(4f_{2}\) site iron. There are two formula units per unit cell, so that the anticipated low - temperature magnetization is \([(16 - 8)/2]\times5\ \mu_{\text{B}}=20\ \mu_{\text{B}}\) per formula unit, which corresponds to \(\mu_{0}M_{0}=0.66\ T\). Room - temperature values are considerably less (table 5.6).
Figure 5.4. The crystal structure of M - type hexagonal ferrites. The principal exchange paths and superexchange bond angles are indicated, together with values of the exchange constants. Coordination of the 2b site is emphasized. The R* - and S* - slabs are obtained by rotating the R - and S - slabs by \(\pi\) about \(c\).
Table 5.6. Intrinsic magnetic properties of hexagonal ferrites at room temperature.
| a (pm) | c (pm) | \(T_{\text{C}}\) (K) | \(m\) (\(\mu_{\text{B}}\) f.u.-1) at \(T = 0\) | \(M_{\text{s}}\) (MA m-1) | \(K_{1}\) (MJ m-3) | \(\mu_{0}H_{0}\) (T) | \(\kappa\) | |
|---|---|---|---|---|---|---|---|---|
| BaM | 589 | 2320 | 742 | 19.9 | 0.38 | 0.33 | 1.7 | 1.3 |
| SrM | 589 | 2304 | 746 | 20.2 | 0.38 | 0.35 | 1.8 | 1.4 |
| PbM | 590 | 2309 | 725 | 19.6 | 0.33 | 0.25 | 1.5 | 1.4 |
| BaW | 588 | 3250 | 728 | 27.6 | 0.41 | 0.30 | 1.5 | 1.2 |
| BaX | 588 | 5570 | 735 | 47.5 | 0.28 | 0.30 | 1.6 | 1.3 |
A family of hexagonal phases form between \(\text{BaO}\), \(\text{Fe}_{2}\text{O}_{3}\) and another divalent oxide (Sugimoto 1980). One other that contains the trigonal bipyramidal
site that helps to develop strong uniaxial anisotropy in ferric oxides is \(\text{BaFe}_{2}^{2 + }\text{Fe}_{16}^{3 + }\text{O}_{27}\) (W - ferrite). It is formed from a sequence of R - slabs and double S - slabs. The compound \(\text{Ba}_{3}\text{Fe}_{2}^{2 + }\text{Fe}_{24}^{3 + }\text{O}_{46}\) (X - ferrite) is obtained by stacking blocks of M and W. The M - ferrites have the highest Curie temperature, as well as the best room - temperature coercivity, so we focus our discussion on these materials.
Intrinsic Magnetic Properties
The principal superexchange paths are indicated in figure 5.4, together with values of the exchange constants. Note that the strongest coupling tends to coincide with the straightest bond angles. The exchange field and therefore the temperature dependence of the sublattice magnetization will be different at each site. The \(12k\uparrow\) sublattice in particular sees only a weak exchange field because of the near - cancellation of its coupling with \(4f_{2}\downarrow\) and \(2b\downarrow\) sublattices (figure 5.4). The temperature dependence of the \(^{57}\text{Fe}\) Mössbauer spectra (figure 4.8) gives an impression of the relative variation of the five sublattice magnetizations. Despite the high Curie temperature, the temperature dependence of the net magnetization at room temperature is quite large, \(d(\ln M)/dT=-0.0012/K\), because of the important contribution of the \(12k\) sites to the net magnetization and the strong temperature dependence of that sublattice magnetization. Magnetic properties of some hexagonal ferrites are summarized in table 5.6.
Much of the anisotropy of M - ferrites is due to the effect of the crystal field at the \(2b\) bipyramidal sites. There are also contributions from other sites, and the temperature dependence reflects the sum of all these crystal - field contributions and the magnetic dipole anisotropy. The principal term in the crystal - field interaction is \(B_{2}^{0}\)\(\tilde{O}_{2}^{0}\) (3.35) The splitting of the one - electron energy levels was shown in figure 3.6. Mixing of the \(^{4}F\) and other orbital excited states into the \(^{6}S\) ground state by the crystal - field interaction leads to an effective spin Hamiltonian for the \(2b\) sites of \(-D\hat{S}_{z}^{2}\) (section 3.1.4). The crystal - field parameters \(D(=B_{2}^{0}/3)\) at this and other \(\text{Fe}^{3 + }\) sites can be deduced from the electric field gradient at the nucleus, as follows. There is a significant nuclear quadrupole interaction only at the \(2b\) and \(12k\) sites, with the same sign of \(V_{zz}\), but with magnitudes in the ratio of approximately 4:1; the field gradient at the other sites is small by comparison. It follows that \(D_{2b}:D_{12k}=4:1\), so the contributions to the anisotropy from the two sites at zero temperature are in the ratio 2:3. At room temperature, the two contributions are of comparable magnitude and of the order of 1 T. There is also a microscopic dipole contribution to the anisotropy field of opposite sign, \(-0.6\ T\) at zero temperature, which changes sign just below \(T_{\text{C}}\) due to the rapid decline of the \(12k\) sublattice magnetization and positive contributions from some of the other sublattices. A comparison of the three main \(K_{1}\) contributions, \(K_{dip}\), \(K_{2b}\) and \(K_{12k}\) is shown in figure 5.5. The relation between \(K_{1i}\) and \(D_{i}\) for \(\text{Fe}^{3 + }\) is \(K_{1i}=-5N_{i}D_{i}\)
Figure 5.5. Some magnetization curves measured on a \(\text{BaFe}_{12}\text{O}_{19}\) crystal with the field applied perpendicular to the hexagonal \(c\) - axis (a), the temperature dependence of the magnetization and anisotropy field (b) and the total anisotropy constant (dashed curve) (c). The different contributions to \(K_{1}\) are indicated. (After Casimir et al 1959.)
where \(N_{i}\) is the number of ions per unit volume. The corresponding values of \(D_{2b}/k_{\text{B}}\) and \(D_{12k}/k_{\text{B}}\) are \(-0.6\ K\) and \(-0.15\ K\).
Many chemical substitutions can be made into the magnetoplumbite structure, but very few of them offer any substantial improvement of the intrinsic properties of \(M\), \(T_{\text{C}}\) and \(H_{\text{c}}\) relevant to hard magnetization beyond those already available in \(\text{BaFe}_{12}\text{O}_{19}\) and \(\text{SrFe}_{12}\text{O}_{19}\). The magnetization and Curie temperature are decreased by trivalent substitutions such as \(\text{Al}^{3 + }\), \(\text{Ga}^{3 + }\) or \(\text{Cr}^{3 + }\). Cobalt substitution is an interesting one. Divalent cations can be introduced into W - or Z - ferrites (\(\text{Ba}_{3}\text{Fe}_{2}^{2 + }\text{Fe}_{24}^{3 + }\text{O}_{46}\)) in place of \(\text{Fe}^{2 + }\) or into M - ferrites by introducing a quadrivalent ion at the same time, for example \(2\text{Fe}^{3 + }\to\text{Co}^{2 + }+\text{Ti}^{4 + }\). The anisotropy constant decreases rapidly with cobalt additions, changing sign at the approximate composition \(\text{Ba}(\text{Fe}_{10}\text{Co}\text{Ti})\text{O}_{19}\). Further increase of the cobalt content yields easy - plane anisotropy at low temperatures and a sequence of spin reorientations with increasing temperature. Physically, these transitions
Figure 5.6. The temperature dependence of the coercivity of \(\text{BaFe}_{12}\text{O}_{19}\) powder.
reflect the fact that the magnetizations of the \(\text{Co}^{2 + }(12k)\) and iron sublattices exhibit a different temperature dependence. As mentioned in section 3.1.1, magnetocrystalline anisotropy is realized via intersublattice exchange field, whose strength is proportional to the sublattice magnetization. A crude but visual way of interpreting the anisotropy contributions of \(\text{Fe}^{3 + }\) and \(\text{Co}^{2 + }\) ions is to consider spherical \((3\text{d}^{5})\) and oblate \((3\text{d}^{7})\) ions, respectively, in an environment where \(A_{2}^{0}<0\). The crystal - field - induced deformation of the \(3\text{d}^{5}\) shell makes it prolate, so that \(\text{Fe}^{3 + }\) and \(\text{Co}^{2 + }\) yield opposite anisotropy contributions.
An alternative way of achieving charge compensation is on the 2d site, for example \(\text{Ba}^{2 + }+\text{Fe}^{3 + }\to\text{La}^{3 + }+\text{Co}^{2 + }\). This produces a small increase of remanence, to \(\mu_{0}M_{\text{r}} = 0.45\ T\), which may be attributed to cobalt substituting on \(4f_{2}\) sites, and significantly - improved coercivity.
Coercivity
Although modest by comparison with the rare - earth intermetallics discussed later in this chapter, the anisotropy field for \(\text{BaFe}_{12}\text{O}_{19}\) and \(\text{SrFe}_{12}\text{O}_{19}\) is nonetheless a factor of three or four larger than the magnetization (table 5.6), which makes it feasible to develop permanent magnets from these materials. The hardness parameter \(\kappa\) is greater than 1. The coercivity of M - ferrites has an unusual temperature dependence, passing through a maximum at about 500 K (figure 5.6), which reflects the temperature dependence of the anisotropy field (figure 5.5(b)). Ferrite magnets rely on a dense structure of very fine isotropic or oriented grains to develop coercivity. The powders are generally composed of plate - like crys - tallites, as a consequence of the large \(c/a\) ratio. The critical single - domain particle size (section 3.2.2.3) for barium or strontium ferrite \(2R_{\text{sd}} = 72\kappa l_{\text{ex}}\) is \(0.9\ \mu\text{m}\). This means that powder of the type used for practical ceramic sin - tered and bonded magnets, which has a particle size of a few micrometres, will generally be multidomain in the virgin state. The variation of coercivity as a function of particle size in figure 5.7 shows that the smallest particles, where \(2R\) is much smaller than 100 nm, are superparamagnetic (section 3.4). Then there
Figure 5.7. The variation of coercivity of \(\text{BaFe}_{12}\text{O}_{19}\) with particle size.
is a region from \(0.1\) to \(1\ \mu\text{m}\) where the coercivity is relatively flat before falling off again as \(1/R\) for larger particles. Even in the plateau region, where superparamagnetic effects are negligible, the coercivity \(\mu_{0}H_{\text{c}}\approx0.4\ T\) is only a fraction of the Stoner - Wohlfarth and Brown predictions \(\mu_{0}H_{\text{c}} = 2K_{1}/M_{\text{s}}\) and \(\mu_{0}H_{\text{c}}>2K_{1}/M_{\text{s}}-\Delta M_{\text{s}}\), respectively. As discussed in section 3.5, this failure is related to the assumption of homogeneous ellipsoids on which both predic - tions rely. Apart from the virgin - curve behaviour, the angular dependence of the coercivity provides a means of distinguishing magnetization reversal mecha - nisms. If reversal involves coherent rotation of the magnetization, then \(H_{\text{c}}\) obeys the Stoner - Wohlfarth behaviour (section 3.3). On the other hand, in the case of localized nucleation only the field component in the original magnetization direction contributes to the magnetic reversal, so that \(H_{\text{c}}(\theta)=H_{0}/\cos\theta\), where \(\theta\) is the angle between the field and easy - axis directions. Industrial powder samples follow the \(1/\cos\theta\) law rather than the initial decrease predicted by the Stoner - Wohlfarth theory. Note, however, that the \(1/\cos\theta\) curve cannot cross the Stoner - Wohlfarth curve, so that rotation always dominates sufficiently close to \(\theta=\pi/2\). (Highly perfect fine powder made by chemical precipitation can have a coercivity as high as \(0.6\ T\) and the coercivity then tends to zero as \(\theta\) tends to \(\pi/2\).)
Ferrite Magnet Processing
Barium and strontium ferrites are manufactured as dense ceramics in quantities of the order of half a million tons per year (about 100 g for every person on Earth) and represent just over half of the total permanent - magnet market. Starting materials are usually powders of \(\text{BaCO}_{3}\) or \(\text{SrCO}_{3}\) and \(\alpha\) - \(\text{Fe}_{2}\text{O}_{3}\), together with small quantities of additives which help to control reaction kinetics, shrinkage and grain growth. Common sources of iron oxide are high - grade haematite ore and oxide produced by spray roasting of pickling solution from steel works.
Figure 5.9. Micrographs of sintered \(\text{BaFe}_{12}\text{O}_{19}\) magnets showing an oriented sample cut parallel (a) and perpendicular (b) to the alignment direction, and an isotropic magnet (c).
The main steps of the hard sintered ferrite production process (Stäblein 1980) are indicated schematically in figure 5.8.
The nucleation centres that destroy coercivity in hexagonal ferrites may be located at the particle surface or in the bulk. It is, therefore, necessary to develop a microstructure which surrounds the metastable, magnetically saturated state with energy barriers to prevent nucleation and propagation of reversed domains. For any given material this process may take years to optimize, and many small improvements being made along the way.
First the raw materials are thoroughly ground together, and the mixture prepared in granules a few millimetres in size for further processing. The density of the powder in the granules, greater than 50 vol.%, is sufficient to permit reaction of the constituents by solid - state diffusion during the first firing in air at 1150 °C. The carbonate decomposes, forming intermediate phases such as \(\text{BaFe}_{2}\text{O}_{4}\), which react with the remaining \(\alpha\) - \(\text{Fe}_{2}\text{O}_{3}\) to give \(\text{BaFe}_{12}\text{O}_{19}\). The coarse material produced by this reaction sintering process is milled to a fine powder in a vibrator or attritor mill. To make sintered magnets a wet or dry compact is die - pressed, after aligning in an applied field of 0.5 - 1.0 T if oriented magnets are required. The second firing lasts for several hours at 1200 - 1250 °C and this turns the 'green compact' into a dense (about 95 vol.%) sintered magnet body, with the microstructure illustrated in figure 5.9. Shrinkage parallel to the alignment direction is about 1.6 times as great as in the perpendicular direction, on account of the platelet growth of the M - ferrite crystallites. After finishing
Table 5.7. Properties of sintered and bonded \(\text{SrFe}_{12}\text{O}_{19}\) magnets.
| \(M_{\text{r}}\) (MA m-1) | \(H_{\text{c}}\) (kA m-1) | \((BH)_{\text{max}}\) (kJ m-3) | |
|---|---|---|---|
| Intrinsic \(\text{SrFe}_{12}\text{O}_{19}\) | 0.38 | — | 45a |
| Oriented sintered | 0.33 | 270 | 34 |
| Isotropic sintered | 0.18 | 310 | 9 |
| Oriented bondedb | 0.24 | 245 | 16 |
| Isotropic bondedc | 0.10 | 180 | 5 |
| aTheoretical maximum. | |||
| bInjection moulded. | |||
| cRubber bonded. | |||
to the required dimensions by slicing or grinding, the ceramic is magnetized, typically in a pulsed field of 1.0 to 1.5 T—about three times greater than the coercivity.
A different route is taken for polymer - bonded magnets. The advantages of the ease of processing and fabricating complex components to a near net shape from a magnetic/organic composite can outweigh the mediocre magnetic properties compared to dense anisotropic sintered magnets made of the same magnetic material. The coercivity does not change greatly when the magnetic powder is incorporated into an organic matrix, so coercive powder is a prerequisite for any bonded magnet. The ferrite powder is milled to a size of about 1 \(\mu\text{m}\) and then compounded with a thermoplastic, thermosetting or rubber agent. The mixture is injection moulded, die - pressed, extruded or rolled to give the required shape. Ferrite fill factors, \(f_{\text{m}}\), of 80% by volume can be achieved by die - pressing, whereas values of 60% are typical for injection moulding. Density and alignment are two key factors affecting the magnetic properties of any sintered or bonded magnet, since the energy product scales as \(M^{2}\). The magnetization is \(f_{\text{m}}(\cos\Theta)M_{\text{s}}\), where \(\Theta\) is the angle between the crystallite \(c\) - axis and the alignment direction. At \(f_{\text{m}} = 70\%\), \((BH)_{\text{max}}\) is reduced by half compared to a fully dense magnet. A more drastic loss of performance arises when the magnetic material is isotropic, composed of weakly interacting crystallites with randomly oriented \(c\) - axes. In this case \(M_{\text{r}}\leq M_{\text{s}}/2\), and even if the hysteresis loop is ideally rectangular with \(H_{\text{c}} = f_{\text{m}}M_{\text{s}}/2\), a loss of energy product by a further factor of four compared to a fully aligned magnet of the same density is inevitable (table 5.7). An approximate expression for the energy product of a bonded magnet when \(H_{\text{c}}>f_{\text{m}}M_{\text{s}}/2\) is \(\mu_{0}M^{2}f_{\text{m}}^{2}/4\).
Second - quadrant hysteresis loops for oriented and isotropic ferrite magnets are shown in figure 5.10. The magnetization and energy product are compared in table 5.7. A summary of bonded magnet processing is given in figure 5.11.
Figure 5.10. Demagnetizing curves of oriented (o). isotropic (i), sintered (s) and bonded (b) ferrite magnets (left) and temperature-dependent B:H curves (right ). Contours of constant energy product are shown as dashed curves.
Figure 5.11. Summary of materials and processes in bonded magnet manufacture.
