1. Introduction

Ferrimagnetism is similar to ferromagnetism in many ways [1–4]. They all have hysteresis curves as the applied magnetic field changes, resulting in the saturation magnetization (4πMs), the coercive field (Hc), and the remnant polarization (Br). A ferri- or ferromagnetic material can be used to fabricate a hard or a soft magnet depending on its coercivity (Hc). Hard magnets are characterized by a high Hc, indicating that their magnetization is difficult to change and will retain their magnetization in the absence of an applied field as shown in Figure 1(a). On the contrary, the soft magnets have low Hc values and, normally, have very weak, remnant magnetic field (low Br) as in Figure 1(b). Hard magnetism has been extensively used to make the permanent magnets and provides a strong DC magnetic field, while the soft magnetism can be used for the AC systems.

Soft magnetic materials include electrical steels and soft ferrites [3, 4]. Unlike the ferromagnetic metals which are conductors, soft ferrites have low electric conductivity, i.e., they are dielectric materials. The electrical steels have extensive applications in low-frequency systems, such as generators, motors, and transformers, while the soft ferrites are suitable for the high-frequency applications, such as circulators, isolators, phase shifters, and high-speed switches.

This chapter will focus on the properties of the soft ferrites and their applications at high-frequency systems. The ferrites are crystals having small electric conductivity compared to ferromagnetic materials. Thus they are useful in highfrequency devices because of the absence of significant eddy current losses. Ferrites

#### Figure 1.

The hysteresis (B-H) curves for (a) hard ferrimagnetism and (b) soft ferrimagnetism. The remnant polarization (Br) and the coercive field (Hc) for the hard ferrites should be as large as possible. On the other hand, for the soft ferrites, the remnant polarization (Br) and the coercive field (Hc) are very small or even close to zero.

are ceramic-like materials with specific resistivities that may be as much as 10<sup>14</sup> greater than that of metals and with dielectric constants around 10 to 16 or greater. Ferrites are made by sintering a mixture of metal oxides and have the general chemical composition MOFe2O3, where M is a divalent metal such as Mn, Mg, Fe, Zn, Ni, Cd, etc. Relative permeabilities of several thousands are common [5, 6]. The magnetic properties of ferrites arise mainly from the magnetic dipole moment associated with the electron spin [2].

The magnetic dipole moment precesses around the applied DC magnetic field by treating the spinning electron as a gyroscopic top, which is a classical picture of the magnetization process. This picture also explains the anisotropic magnetic properties of ferrites, where the permeability of the ferrite is not a single scalar quantity, but instead is a generally a second-rank tensor or can be represented as a matrix. The left and right circularly polarized waves have different propagation constant along the direction of the external magnetic field, resulting in the nonreciprocity of a propagating wave. Since the permeability should be treated as a tensor (matrix), not a scalar permeability, it is generally much difficult to understand and to have intuition, even for the researchers.

right-hand circularly polarized (RHCP) wave, the fields rotate clockwise at a given position from the source looking in the direction of propagation. The magnetic dipole moment m processes around the H0 field vector, like a top spinning precess around the z-axis at the Larmor frequency ω0. The spinning property depends on the applied DC bias magnetic field. Figures 2(a) and (b) shows the RHCP and LHCP waves with the gyrating frequency of ω. When the RHCP wave is propagating along the direction of the DC bias field, it corotates with the precession of the magnetic dipole moments. On the other hand, the left-hand circularly polarized (LHCP) wave will counter-rotate with the precession of the dipole moments. A linearly polarized incident wave can be decomposed into RHCP and LHCP waves of equal amplitude. The orientation of the linearly polarized wave changes after the wave propagates a certain distance because of the distinct propagation constants. The phenomenon is the famous Faraday's rotation [5, 6]. This unique property has various applications, such as phase shifters, isolators, and circulators. However, it is difficult to follow for students and even researchers in that the

Larmor precession of a magnetic moment m around the applied DC bias field H<sup>0</sup> (¼ H0z^) with (a) a righthand circularly polarized (RHCP) wave and (b) a left-hand circularly polarized (LHCP) wave. The frequency of the Larmor precession in both cases are the same, i.e., the Larmor frequency ω<sup>0</sup> ¼ μ<sup>0</sup> ð Þ γH<sup>0</sup> . H<sup>þ</sup>

<sup>t</sup> are the transverse components of the incident waves which rotate clockwise (RHCP) and counterclockwise (LHCP) from the source viewpoint looking in the direction of propagation. The thumb points the direction of the wave propagation, and the fingers give the rotation of the transverse components [7].

t

permeability is a tensor, not just a simple proportional constant.

Figure 2.

Ferrite Materials and Applications

DOI: http://dx.doi.org/10.5772/intechopen.84623

and H�

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the ferrite magnetic resonance (FMR) or gyromagnetic resonance [7].

Here we consider the simplest case for the pedagogic purpose—a circularly polarized plane wave is normally incident upon a semi-infinite medium. The wave characteristics such as the propagation constant k and the wave impedance Z are associated with the permeability μ, which is a tensor for the ferrite medium [5]. By finding the preferred eigenvalues, it will be shown that the properties of μ depend on the DC bias field H0, the saturated magnetization Ms, and the operating frequency ω. By adjusting the frequency of the incident wave, i.e., ω, the permeability μ changes, especially close to the Larmor frequency (ω0). Such an effect is called
