**2. Ferrites**

paramagnetic resonance (EPR) has been known for ~50 years, so in some way, this is a situation both classic and new. Ferrimagnetic and superparamagnetic NPs are more complex, and their

We first describe the general basis of the classic resonance technique, where a microwave signal of constant frequency is applied on the sample, which is simultaneously subjected to a sweeping magnetic field, in order to achieve the resonance conditions. We tried to establish the main differences between the response of classic and metallic paramagnetic phases and

We then briefly review the novel properties associated with the nanometric size, followed by a short account of crystal structure and magnetic interactions in ferrites. EPR is briefly described, illustrated with some recent results. A basic description of the superparamagnetic phase is then given, with a review of some of the theoretical models proposed, as well as some of the most representative experimental results. The ferrimagnetic phase is then described with an accent on the differences originated by the exchange interactions, magnetocrystalline anisotropy, demagnetization fields, etc., and their effects on the response signal. The general response of magnetic nanoparticles in the ordered state is discussed. In the conclusion, finally, an attempt is made to establish a correlation between these phases and their resonance signals.

In addition to the changes related to the decrease in scale down to the nanometric range, magnetic materials provide another source of novel properties. Many of the critical parameters in magnetism are found in the 1 to 200-nm scale of length, see Table 1 [1]. Two of the most interesting changes in magnetic structure as a consequence of the reduction in size are the

Bulk samples of ferro- and ferrimagnetic materials are divided into magnetic domains (separated by domain walls) in order to decrease the magnetostatic energy, i.e., the energy associated with the presence of magnetic flux just outside the sample surface. Inside the magnetic domains exchange energy is at a minimum (all spins are parallel coupled) as well as magnetocrystalline energy (all spins are oriented into easy axes), and the magnetization of domains is oriented to provide a continuous magnetic flux inside the sample, thus avoiding any external flux. This is why most magnetic materials do not manifest any attraction or repulsion force (in the absence of an applied magnetic field). Magnetostatic energy is therefore eliminated, except for a small contribution from domain walls, where spins rotate from the orientation of a domain toward the orientation of the neighboring domain. There is also a small contribution to both exchange and anisotropy energy due to domain walls, as spins cannot be strictly parallel, neither oriented into easy directions within the domain wall. These contribu‐ tions, however, are small as domain wall thickness is in the 10- to 100-nm range. Domain walls represent a very sensitive equilibrium in ferromagnetic materials. They can be displaced by very small applied, fields and their dynamics of propagation though defects (pinning sites)

change from multidomain to single domain and superparamagnetism.

have a fundamental importance for soft magnetic materials [2].

the shapes of magnetically ordered (ferro- or ferrimagnetic) phases in the bulk state.

resonance response is currently an active research subject.

**1.1. Nanomagnetism**

210 Advanced Electromagnetic Waves

Spinel ferrites are a large family of materials, with the structure of the natural spinel mineral, MgAl2O4, first determined by Bragg [3]. The spinel is a very stable crystal structure; it is almost enough to satisfy the conditions of neutral electric charge, relatively small cation radii, and a ¾ cation-to-anion ratio [2]. These conditions allow several cation combinations such as 2,3 (as in Ni2+Fe3+2O4), 2,4 (as in Co2GeO4), 1,3,4 (as in LiFeTiO4), 1,3 (as in Li0.5Fe2.5O4), 1,2,5 (as in LiNiVO4), and 1,6 (as in Na2WO4). Most of ferrites with significant magnetic properties are of the 2,3 type and contain Fe3+. An important ferrite is magnetite Fe2+Fe3+2O4, also referred as Fe3O4, which is the oldest known magnetic material. The name "magnetic" is derived from Magnesia, which is the region where magnetite was first discovered in Greece in the 6th century BC. It is interesting that magnetite, the magnetic material first reported in history, is currently an extremely active field in basic and applied research. This is certainly due to its fascinating magnetic and electric properties, associated with the coexistence of ferrous and ferric cations in equivalent crystal sites. A related magnetic material also particularly interest‐ ing is maghemite (or γ-Fe2O3), which possesses only Fe3+ and the same spinel structure often obtained by oxidation of magnetite. Since the cation to anion ratio is not ¾, it contains vacancies in a fraction of cation sites; by using the spinel formula, it can be represented as □1/3Fe3+8/3O4. Here, □ stands for vacancies.

The crystal symmetry can be understood by considering a face centered cubic (fcc) lattice of oxygen, leading to two kinds of cation sites: 64 tetrahedral sites and 32 octahedral sites for a unit cell, which is formed by 8 times the basic formula D2+T3+2O4 (D2+ stands for a divalent cation, and T3+ for a trivalent cation). Only one-eighth of tetrahedral sites is occupied and half of octahedral sites as well. The space group is Fd3m. This structure is shown in Figure 1.

In the spinel mineral MgAl2O4, Mg2+ cations occupy tetrahedral sites and Al3+ cations are found in octahedral sites, which appears as a stable arrangement, as far as divalent cations are surrounded by less anions (four anions in the tetrahedral site) than trivalent cations with higher electric charge and enclosed by 6 anions. This structure is known as the *normal* spinel, and is indicated by using the system (D2+) [T3+2]O4. Parenthesis indicates occupancy of tetrahedral sites, also known as "A" sites, and square brackets show cations on octahedral sites or "B" sites. A different cations distribution is (T3+)[T3+D2+)O4, where the divalent cation goes to an octahedral site and trivalent cations are found both on tetra and octahedral sites. This is the *inverse* spinel. An intermediate cation distribution has also been observed for some ferrites, (D1–δ Tδ)[DδT2-δ]O4, where δ is the degree of inversion.

The cation distribution in spinels was a problem for some time, but it is now well understood. The involved energies are the *elastic* energy, associated with the lattice deformation produced by cation radii differences. The *electrostatic* energy, also known as the Madelung energy, which depends on the overall electric charge distribution; divalent cations on small sites and trivalent cations on larger sites should stabilize the spinel. The *crystal field* energy has also a large influence on cation site "preference," and it is related mainly to the geometry of *d*-orbitals (or electronic orbitals for nontransition cations). *d*<sup>5</sup> orbitals, for instance, can occupy both types of sites as these orbitals (in high-spin state) have spherical geometry (see Table 2).


**Table 2.** Cation preferences for spinel sites. Adapted from [2].

Magnesia, which is the region where magnetite was first discovered in Greece in the 6th century BC. It is interesting that magnetite, the magnetic material first reported in history, is currently an extremely active field in basic and applied research. This is certainly due to its fascinating magnetic and electric properties, associated with the coexistence of ferrous and ferric cations in equivalent crystal sites. A related magnetic material also particularly interest‐ ing is maghemite (or γ-Fe2O3), which possesses only Fe3+ and the same spinel structure often obtained by oxidation of magnetite. Since the cation to anion ratio is not ¾, it contains vacancies in a fraction of cation sites; by using the spinel formula, it can be represented as □1/3Fe3+8/3O4.

The crystal symmetry can be understood by considering a face centered cubic (fcc) lattice of oxygen, leading to two kinds of cation sites: 64 tetrahedral sites and 32 octahedral sites for a unit cell, which is formed by 8 times the basic formula D2+T3+2O4 (D2+ stands for a divalent cation, and T3+ for a trivalent cation). Only one-eighth of tetrahedral sites is occupied and half of octahedral sites as well. The space group is Fd3m. This structure is shown in Figure 1.

In the spinel mineral MgAl2O4, Mg2+ cations occupy tetrahedral sites and Al3+ cations are found in octahedral sites, which appears as a stable arrangement, as far as divalent cations are surrounded by less anions (four anions in the tetrahedral site) than trivalent cations with higher electric charge and enclosed by 6 anions. This structure is known as the *normal* spinel, and is indicated by using the system (D2+) [T3+2]O4. Parenthesis indicates occupancy of tetrahedral sites, also known as "A" sites, and square brackets show cations on octahedral sites or "B" sites. A different cations distribution is (T3+)[T3+D2+)O4, where the divalent cation goes to an octahedral site and trivalent cations are found both on tetra and octahedral sites. This is the *inverse* spinel. An intermediate cation distribution has also been observed for some ferrites,

The cation distribution in spinels was a problem for some time, but it is now well understood. The involved energies are the *elastic* energy, associated with the lattice deformation produced by cation radii differences. The *electrostatic* energy, also known as the Madelung energy, which depends on the overall electric charge distribution; divalent cations on small sites and trivalent cations on larger sites should stabilize the spinel. The *crystal field* energy has also a large influence on cation site "preference," and it is related mainly to the geometry of *d*-orbitals (or electronic orbitals for nontransition cations). *d*<sup>5</sup> orbitals, for instance, can occupy both types of

> **Tetrahedral Octahedral Undistinguished** Cd2+ Co2+ Fe3+ Zn2+ Ni2+ Mn2+

> > Li1+

Cu2+ Mg2+ Fe2+ Mo2+

sites as these orbitals (in high-spin state) have spherical geometry (see Table 2).

Here, □ stands for vacancies.

212 Advanced Electromagnetic Waves

(D1–δ Tδ)[DδT2-δ]O4, where δ is the degree of inversion.

**Table 2.** Cation preferences for spinel sites. Adapted from [2].

**Figure 1.** Representation of the spinel structure. The unit cell can be divided into octants (above). The detailed struc‐ ture of two octants is illustrated below. Large spheres are oxygen, small black spheres indicate cations in tetrahedral sites, and small white spheres represent cations in octahedral sites [2].

An outstanding characteristic of the spinel structure is that it admits an extremely large variety of *total solid solutions*. The divalent cations in the spinel formula can be formed by a combination of two (or more) cations; Fe3+ can also by partially or totally substituted by another trivalent cation, always maintaining the spinel crystal structure. A very well studied system (or "family") is Zn-Ni ferrites with the general formula Zn*x*Ni1-*x*Fe2O4, with 0 ≤ *x* ≤ 1. The end compositions are nickel ferrite, NiFe2O4 (for *x* = 0), and zinc ferrite, ZnFe2O4 (*x* = 1). Nickel ferrite is an inverse spinel, while zinc ferrite is a normal one; due to the features of magnetic interactions in spinel ferrites, nickel ferrite is ferrimagnetic with a Curie point about 858 K, while zinc ferrite is antiferromagnetic with a Néel temperature about 9 K. The properties of this family of compounds can be "tailored" between these two extreme behaviors just by varying the chemical composition.

The magnetic structure and interactions in ferrites can be understood on the basis of superex‐ change interactions between two transition cations separated by an oxygen, as shown in Figure 2. The electron spin up in the *p*-orbital of the oxygen can occupy for a short period of time the empty half of the *d*-orbital in the transition cation at right, if the occupied state in this cation is spin down. As the *p*-orbital of oxygen exhibits, for the same given period of time, an available site for an electron with spin up, the electron of the transition cation on the left side of the oxygen can occupy this place, if its spin is pointing up. This mechanism leads therefore to an antiparallel arrangement of spins on the transition cations. In spinels, there are two sublattices, a tetrahedral one (with one cation per formula) and an octahedral one (with two cations per formula). In most cases, a resultant appears and most ferrites possess a net magnetization. The Zn-Ni ferrite family is a very good example of the richness of magnetic properties and structure of ferrites.

**Figure 2.** Superexchange interactions in ferrites.

The magnetic interactions in spinel ferrites can be easily understood by considering two B sites and one A site about an oxygen, see Figure 3. The center and right side of this figure, where the large circle represents the oxygen ion, the small circle above it stands for the cation on the A site, and the two small circles below represent B sites. Note that this arrangement is 3D, but for this schematic illustration, it has been simplified on a plane. Nickel ferrite is an inverse spinel, Ni2+ goes to B sites, one ferric ion occupies the other B site, and the other Fe3+ is located on the A site. All magnetic interactions between cations are antiparallel (superexchange).

**Figure 3.** Schematic spin arrangements in spinel ferrites. Left: a section of the crystal structure showing the cation occu‐ pancy about an oxygen. Center: the ferrimagnetic spin orientation of the crystal sites for nickel ferrite. Right: the anti‐ ferromagnetic structure for the zinc ferrite.

There are A–O–B interactions between the cation on A site and the cations on B sites, and an B–O–B interaction between the cations on the two B sites. However, due to the axial symmetry of oxygen *p*-orbitals, superexchange interaction becomes more efficient when cations and oxygen are in an axial arrangement. This makes a strong difference between interactions; A– O–B interaction is far more efficient than B–O–B interactions since the latter has a 90° angle, while the former is closer to 180°. As a result, both spins on B sites are antiparallel to the A spin. The magnetic moment of Fe3+ on A site cancels with the magnetic moment of Fe3+ on B site, and the resulting magnetic moment is the spin value for nickel Ni2+ (2.3 Bohr magnetons), as shown in Figure 3 center. This is a strong interaction; the Curie transition is about 858 K.

In the case of zinc ferrite, iron ions occupy the B sites and zinc is located on the A site. Since Zn2+ is a 3*d*<sup>10</sup> ion, it has no magnetic moment, and the only interaction is B–O–B, leading to an antiparallel arrangement of spins (Figure, 3 right). The spins are identical, and the material becomes antiferromagnetic. The Néel temperature is about 9 K, and this clearly illustrates the weakness of this superexchange interaction, as compared with AOB of the nickel ferrite case. Based on transition temperatures, A–O–B interaction is roughly 95 times stronger than B–O– B interaction.
