1. Introduction

The high-frequency permeability values are crucial for many applications of electromagnetic (EM) materials. At low frequency region, a typical application is magnetic cores for transformers or inductors; these devices demand the larger real parts of permeability (μ<sup>0</sup> <sup>&</sup>gt; 0) and lower imaginary parts of permeability (μ″ <sup>&</sup>gt; 0 and <sup>μ</sup>″ ! 0) for magnetic materials. With the working frequency increases, many electronic devices work above gigahertz (GHz). The unwanted electromagnetic wave pollution is problematic, which can be overcome by using the electromagnetic wave attenuation composites with the absorbing frequency band falling into the GHz zone. Therefore, the electromagnetic wave absorbing materials require their working frequencies to be above 1 GHz and larger <sup>μ</sup>″ values for efficient absorption via the typical magnetic loss mechanism (natural resonance). Due to the small magnetocrystalline anisotropy constants of spinel ferrites, the magnetic loss peaks of spinel ferrites are below 1 GHz. Although the natural resonance frequencies of hexagonal ferrites can be a few GHz, the shortcoming of both ferrites (spinel and hexagonal) is the poor temperature stability; in other words, the permeability decreases faster with increasing environmental temperature, making these ferrites not suitable candidates for EM attenuation applications. The physical law governing the relationship of initial permeability, working frequency and magnetization is the well-known "Snoek's law" [1]. This law tells us that if we want to increase the

working frequency of a specific ferrite (Ms is constant), we have to sacrifice their permeability and vice versa. People often tailor the μ � f spectrum of a ferrite by cation substitution or microstructures by changing the sintering conditions. We think this technique is time-consuming and inefficient for mass production. In this chapter, we propose Fe-based nanostructured materials (Fe-Cu-Nb-Si-B alloys, also called "FINEMET") as ideal EM attenuation materials, which offer us greater freedom to tailor their high-frequency permeability spectra and possess good temperature stability. Firstly, we will discuss how to increase the permeability by controlling the shape of particles, and then we move to discuss the impacts of size distribution, phase transitions and coating on the permeability. Next, we will discuss the cause of frequently observed broad permeability spectra using the micromagnetic simulations. Finally, we will prove how we can obtain negative imaginary parts of permeability via spin transfer torque effect. Our discussions mainly focus on the electromagnetic wave absorbing (attenuation) applications, which require large imaginary parts of permeability and working frequency above 1 GHz.

#### 2. Shape controlling

Soft magnetic materials can give large initial permeability, but their working frequency is far below 1 GHz due to their small magnetocrystalline anisotropic constants. As per Snoek's law, if the working frequency is shifted to GHz range, the permeability has to be compromised. To increase the working frequency, we have to look at other anisotropy fields, such as shape anisotropy and stress (or external field)-induced anisotropy. Shape anisotropy is most commonly employed. Materials in forms of thin films, microwires, nanowires and flakes can possess a strong shape anisotropy. Here, we present how to enhance the permeability of a soft ferromagnetic material (Fe-Si-Al alloys) above 1 GHz via shape controlling. The Fe-Si-Al alloy ingot (composition: Fe84.94Si9.68Al5.38) was prepared using the hydrogen reduction method in a furnace with the starting materials (Si, Al and Fe with high purity). Subsequently, the Fe-Si-Al ingot was first pulverized into particles with irregular shapes; later these particles were further milled into flakes under different milling times (10, 20 and 30 hours, respectively). Our traditional ball milling process description can be found in our published paper [2]. The scanning electron microscopy images of Fe-Si-Al particles are shown in Figure 1. Clearly, the preliminarily pulverized particles have irregular shapes. The traditional ball milling process can transform them into flakes after 10, 20 or 30 hours of milling. The typical milling result is illustrated in Figure 1b showing flaky particles milled for 30 hours. The high-frequency complex permeability values were measured within 0.5–10 GHz using a network analyzer (Agilent 8720ET). The measured samples were prepared by mixing the Fe-Si-Al particles and wax homogeneously (weight ratio: alloy particles/wax = 4:1). The measured sample has an annular shape with inner and outer diameters of 3 and 7 mm, respectively.

values within the whole frequency range studied than those of particles with irregular shapes, as shown in Figure 2b. Our explanations for the enhanced permeability are as follows: Fe-Si-Al alloy is a metallic alloy with a conducting feature. The eddy current effect is stronger for irregular shape but can be greatly suppressed by milling the irregular particles into flakes that have lower thickness. Besides, Snoek's law

SEM pictures: (a) preliminarily pulverized particles and (b) particles with flaky shapes prepared by 30 hours

governing the relationship between the permeability and shape factor is given as [3]

where Dz is a factor related to the shape, also known as the demagnetization factor for the normal direction of the particle plane. Dz is about 4π/3 for a sphere and is about 4π for a flake. Accordingly, the enhanced permeability value can be observed by increasing Dz from (4π/3) to (4π) by controlling particle shapes. That is also the reason why people often fabricate ferromagnetic thin films to obtain enhanced high-frequency permeability. Thin films can be viewed as an extreme case of "all flakes well aligned," which therefore are found to have much larger permeability values resulting from the "flakes" and in-plane induced uniaxial anisotropy. Furthermore, the well-known Snoek's law for bulk materials describes

ð Þ <sup>μ</sup><sup>s</sup> � <sup>1</sup> fr <sup>¼</sup> <sup>2</sup>

No shape-related demagnetization factor is found in this equation.

3 γ0

� 4πM<sup>s</sup> � ð Þ Hk þ 4πMsDz (1)

4πM<sup>s</sup> (2)

ð Þ μ<sup>s</sup> � 1 f

Figure 1.

Figure 2.

119

of ball milling.

2 <sup>r</sup> <sup>¼</sup> <sup>γ</sup> 2π <sup>2</sup>

Dependence of high-frequency permeability on particle shapes (copyright, 2013, IOP).

High-Frequency Permeability of Fe-Based Nanostructured Materials

DOI: http://dx.doi.org/10.5772/intechopen.86403

The dependence of high-frequency complex permeability on particle shape is shown in Figure 2. Obviously, flaky particles have much larger values in both real and imaginary parts of permeability compared to the irregularly shaped particles. This finding is named "enhanced permeability." The enhanced permeability is more apparent at the lower frequency range. Besides, with increasing the milling hours, the enhanced permeability is stronger. For example, the μ<sup>0</sup> value of flakes after being milled for 30 hours is found to be 4.4 at 0.5 GHz. However, it is only 1.3 for the irregular shaped particle. Within 0.5–7 GHz, the μ<sup>0</sup> values of flakes after being milled for 30 hours are evidently larger than those of particles with irregular shapes. With regard to the imaginary part values (μ″), the flakes of Fe-Si-Al exhibit larger

#### Figure 1.

working frequency of a specific ferrite (Ms is constant), we have to sacrifice their permeability and vice versa. People often tailor the μ � f spectrum of a ferrite by cation substitution or microstructures by changing the sintering conditions. We think this technique is time-consuming and inefficient for mass production. In this chapter, we propose Fe-based nanostructured materials (Fe-Cu-Nb-Si-B alloys, also called "FINEMET") as ideal EM attenuation materials, which offer us greater freedom to tailor their high-frequency permeability spectra and possess good temperature stability. Firstly, we will discuss how to increase the permeability by controlling the shape of particles, and then we move to discuss the impacts of size distribution, phase transitions and coating on the permeability. Next, we will discuss the cause of frequently observed broad permeability spectra using the micromagnetic simulations. Finally, we will prove how we can obtain negative imaginary parts of permeability via spin transfer torque effect. Our discussions mainly focus on the electromagnetic wave absorbing (attenuation) applications, which require large imaginary parts of perme-

Soft magnetic materials can give large initial permeability, but their working frequency is far below 1 GHz due to their small magnetocrystalline anisotropic constants. As per Snoek's law, if the working frequency is shifted to GHz range, the permeability has to be compromised. To increase the working frequency, we have to look at other anisotropy fields, such as shape anisotropy and stress (or external field)-induced anisotropy. Shape anisotropy is most commonly employed. Materials in forms of thin films, microwires, nanowires and flakes can possess a strong shape anisotropy. Here, we present how to enhance the permeability of a soft ferromagnetic material (Fe-Si-Al alloys) above 1 GHz via shape controlling. The Fe-Si-Al alloy ingot (composition: Fe84.94Si9.68Al5.38) was prepared using the hydrogen reduction method in a furnace with the starting materials (Si, Al and Fe with high purity). Subsequently, the Fe-Si-Al ingot was first pulverized into particles with irregular shapes; later these particles were further milled into flakes under different milling times (10, 20 and 30 hours, respectively). Our traditional ball milling process description can be found in our published paper [2]. The scanning electron microscopy images of Fe-Si-Al particles are shown in Figure 1. Clearly, the

preliminarily pulverized particles have irregular shapes. The traditional ball milling process can transform them into flakes after 10, 20 or 30 hours of milling. The typical milling result is illustrated in Figure 1b showing flaky particles milled for 30 hours. The high-frequency complex permeability values were measured within 0.5–10 GHz using a network analyzer (Agilent 8720ET). The measured samples were prepared by mixing the Fe-Si-Al particles and wax homogeneously (weight ratio: alloy particles/wax = 4:1). The measured sample has an annular shape with

The dependence of high-frequency complex permeability on particle shape is shown in Figure 2. Obviously, flaky particles have much larger values in both real and imaginary parts of permeability compared to the irregularly shaped particles. This finding is named "enhanced permeability." The enhanced permeability is more apparent at the lower frequency range. Besides, with increasing the milling hours, the enhanced permeability is stronger. For example, the μ<sup>0</sup> value of flakes after being milled for 30 hours is found to be 4.4 at 0.5 GHz. However, it is only 1.3 for the irregular shaped particle. Within 0.5–7 GHz, the μ<sup>0</sup> values of flakes after being milled for 30 hours are evidently larger than those of particles with irregular shapes. With regard to the imaginary part values (μ″), the flakes of Fe-Si-Al exhibit larger

inner and outer diameters of 3 and 7 mm, respectively.

ability and working frequency above 1 GHz.

Electromagnetic Materials and Devices

2. Shape controlling

118

SEM pictures: (a) preliminarily pulverized particles and (b) particles with flaky shapes prepared by 30 hours of ball milling.

#### Figure 2.

Dependence of high-frequency permeability on particle shapes (copyright, 2013, IOP).

values within the whole frequency range studied than those of particles with irregular shapes, as shown in Figure 2b. Our explanations for the enhanced permeability are as follows: Fe-Si-Al alloy is a metallic alloy with a conducting feature. The eddy current effect is stronger for irregular shape but can be greatly suppressed by milling the irregular particles into flakes that have lower thickness. Besides, Snoek's law governing the relationship between the permeability and shape factor is given as [3]

$$\left(\left(\mu\_{\rm s} - 1\right)f\right)\_{r}^{2} = \left(\frac{\chi}{2\pi}\right)^{2} \times 4\pi \mathbf{M}\_{\rm s} \times \left(H\_{\rm k} + 4\pi \mathbf{M}\_{\rm s} D\_{\rm z}\right) \tag{1}$$

where Dz is a factor related to the shape, also known as the demagnetization factor for the normal direction of the particle plane. Dz is about 4π/3 for a sphere and is about 4π for a flake. Accordingly, the enhanced permeability value can be observed by increasing Dz from (4π/3) to (4π) by controlling particle shapes. That is also the reason why people often fabricate ferromagnetic thin films to obtain enhanced high-frequency permeability. Thin films can be viewed as an extreme case of "all flakes well aligned," which therefore are found to have much larger permeability values resulting from the "flakes" and in-plane induced uniaxial anisotropy. Furthermore, the well-known Snoek's law for bulk materials describes

$$(\mu\_s - 1)f\_r = \frac{2}{3}\gamma' 4\pi \mathbf{M}\_s \tag{2}$$

No shape-related demagnetization factor is found in this equation.
