3.2 Saturation magnetization

(Ms or 4πMs), and the resonance linewidth (ΔH). The behavior of the spin wave linewidth should be considered when the field strength of an electromagnetic wave exceeds a threshold value, that is, the high-power condition. For the general pur-

Ferrites are ceramic-like materials with relative dielectric constants around 10 to 16 or greater. The resistivities of ferrites may be as high as 1014 greater than that of metals. Since ferrites are dielectric materials. The dielectric properties (ε<sup>r</sup> þ iεi) always play an important role with or without the influence of the magnetic field. The perturbation method is the most commonly employed resonant technique [8, 9]. The perturbation method is very good for small-size and low-dielectric samples. When measuring the high-dielectric samples, however, the fields and the resonant frequency change drastically. The perturbation technique may lead to a reduced accuracy. Recently, the field enhancement method was proposed [10, 11]. The field enhancement method operates at a condition opposite to the perturbation method. The resonant frequency and quality factor alter significantly and depend on not only the geometry of the cavity but the sample's size and complex permittivity as well. Luckily, both the perturbation method and the field enhancement method agree well for samples with the dielectric constant below 50, which is

Figure 3 shows the ideal of the field enhancement method. Figure 3(a) shows the resonant frequency as functions of the dielectric constant (εr) using a simulation setup as in Figure 3(b). The ingot-shaped sample has a diameter of 16.00 mm and thickness of 5.00 mm. The solid blue line depicts the simulation result for the field enhancement method. From the measured resonant frequency, we can derive the corresponding dielectric constant. On the other hand, the dashed black line is obtained using a sample with the same diameter but much thinner in thickness

perturbation method. The imaginary part of the permittivity (εi) or the loss tangent ( tan δ ¼ εi=εr) can then be determined from the measured resonant frequency and the quality factor. The field enhancement method has very wide measuring range from unity to high-κ dielectrics and from lossless to lossy materials [10, 11].

(a) Resonant frequency versus dielectric constant based on full-wave simulations. The solid curve can be divided into three regions: low, transition, and ultrahigh. The dashed line is simulated with a much thinner sample of 1.00 mm in thickness, which exhibits the properties similar to those of perturbation. (b) Schematic diagram of the field enhancement method. It consists of a cylindrical resonant cavity and a metal rod. The sample is placed on the top of the metal rod. The metal rod focuses and enhances the electric field significantly. An SubMiniature version A (SMA) 3.5-mm adapter couples the wave from the top of the cavity [11].

of 1.0 mm. The response of the 1-mm-thick sample quite resembles the

pose, only the first three properties will be used in the ferrite simulation.

3.1 Dielectric properties

Electromagnetic Materials and Devices

suitable for most of the ferrites.

Figure 3.

142

Ferrites have a strong response to the applied magnetic field. The magnetic properties of ferrites arise mainly from the magnetic dipole moment associated with the electron spin. Relative permeabilities of several thousands are common. The saturation magnetization (Ms or 4πMs) of a ferrite plays a key role as shown in Section 2. Researchers or engineers use the saturation magnetization as a design parameter that enters into the initial selection of a ferrimagnetic material for microwave device applications. Typical ferrimagnets exhibit values of 4πMs between 300 gauss (G) and 5000 G. Static or low-frequency methods are generally used to measure 4πMs [12]. From the measured hysteresis loop as shown in Figure 4, one can determine the saturation magnetization Ms.

Note that the saturation magnetization is denoted as Ms in the SI unit, but since the values are generally displayed in Gaussian unit (gauss, G), 4πMs is commonly used. Also, the internal bias H<sup>0</sup> is different from the applied H-field (Ha). Demagnetization factor should be considered [5, 6]. The demagnetization factor allows us to calculate the H-field inside the sample denoted as H0. In all, measurement of the saturation magnetization from the dynamic hysteresis loop characteristics can be used for the design and simulation of ferrite devices.

### 3.3 Resonance linewidth

The loss of ferrite material is related to the linewidth, ΔH, of the susceptibility curve near resonance. Consider the imaginary part of the susceptibility χ″ xx versus the bias field H0. The linewidth ΔH is defined as the width of the curve of χ″ xx versus H0, where χ″ xx has decreased to half of its peak value. For a fixed microwave frequency ω, resonance occurs when ω<sup>0</sup> ¼ μ0γHr, such that ω ¼ ω<sup>0</sup> ¼ μ<sup>0</sup> ð Þ γHr . The linewidth, ΔH, is defined as the width of the curve of χ″ xx versus H0, where χ″ xx has decreased to half its peak value. This is the idea that is introduced in [5]. However, obtaining the relation of χ″ xx versus H<sup>0</sup> is not easy. Here we adopt another commonly used technique [9, 12].

#### Figure 4.

The hysteresis curve regarding the magnetization M and the internal bias H0. When the applied internal magnetic field H<sup>0</sup> is large enough, the magnetization will be saturated, denoted as (Ms or 4πMs). When H<sup>0</sup> decreases to zero, the remnant polarization is denoted as Mr. The polarization will change sign (from positive to negative) when H<sup>0</sup> is greater than �Hc which is called the coercive field.

The idea is normally implemented using a TE10n (n even) cavity in the X-band region [9, 12]. The test sample is placed at the H-field maximum. The sample is spherical with a diameter of approximately 0.040 inches, which is much easier to estimate than the internal bias field H0. A cross-guide coupler is used with the coupling iris. The loaded Q (QL) of the empty cavity should be 2000 or greater. The sample, mounted on a fused silica or equivalent rod, is positioned away from the cavity wall at a point of minimum microwave electric field and maximum microwave magnetic field. A power meter can be used to read off the half-power points by adjusting the DC magnetic field and measuring the difference in H0-field directly [12]. Figure 5 shows how the resonant linewidth is determined.

The three key parameters are obtained in three experimental setups under different sizes and geometries of the samples. If the samples' properties are slightly different or the machining error is not negligible, the error will be large or even unacceptable. The ultimate goal is to integrate the measurements and to extract the parameters using one experimental setup. The three key parameters will be used in the design of the microwave ferrite devices in the next session.
