3. Particle size distribution

As discussed before, the high-frequency permeability can be greatly enhanced by controlling the shapes of particles and alignment of flakes (i.e., a thin film can be treated as a large quantity of well-aligned flakes in a plane). In this section, we will focus on the effect of size distribution of flakes on the dynamic permeability. We choose the composites containing Fe-Cu-Nb-Si-B alloy particles (a.k.a. "FINEMET" alloy) for this purpose. The reason to choose this Fe-based nanostructured material for its special nanoscale structure is the nano-grains are well dispersed in an amorphous matrix (it can be thought as a core-shell structure). The amorphous matrix has the larger resistivity which can reduce the detrimental effects of eddy current on high-frequency permeability. Besides, both amorphous phase and nano-grains are ferromagnetic, which avoids the effect non-magnetic coatings have on ferromagnetic particles (it will be discussed in the next section), and as a result both phases contribute to the permeability.

treated at 540°C are characterized by a transmission electron microscopy (TEM) and shown in Figure 4. Figure 4a is the bright-field TEM image, whereas Figure 4b shows the typical features of Fe-based nanostructured alloys, where the amorphous matrix is surrounding the nanocrystalline grains. The average grain size is about 14.38 nm. XRD results show the nanocrystalline grains to be α-Fe(Si). It should be noted that the effect of annealing and ball milling on phase transitions can be investigated using Mössbauer technique and XRD measurements. Especially, Mössbauer technique can detect fine distinctions in the transformations of crystal structures due to the high-energy ball milling [5]. Using the milling procedure described in the previous section, we can effectively transform the annealed ribbons into flakes, which are shown in Figure 5. These flakes are divided into two categories by a shaking sieve: the large flakes (Figure 5a) and the small flakes (Figure 5b). The typical thickness of flakes of both categories is also shown in Figure 5c and d. For large flakes, the width of flakes is about 23–111 μm, average width is estimated to be about 81.1 μm and average thickness was found to be 4.5 μm. For small flakes, the width of flakes is about 3–21 μm, average width is estimated about 9.4 μm and

High-Frequency Permeability of Fe-Based Nanostructured Materials

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

Frequency dependence of permeability of these two kinds of flakes is shown in Figure 6. At the beginning of measurement frequency (i.e., at 0.5 GHz), the real

<sup>s</sup> ) is about 1.3 for large flakes and about 0.9 for small flakes. Inter-

about 4.6 for large flakes and about 3.9 for small flakes. Initial imaginary part of

estingly, it should be pointed out that wide magnetic loss spectra (μ � f) can be found for both kinds of flakes, as indicated in Figure 6b. Wide magnetic loss peak is advantageous for electromagnetic noise attenuation composites with a wide working frequency band. The loss peak (μ″ is maximum) is found to be 3.0 GHz for large flakes and 5.5 GHz for small flakes, whereas <sup>μ</sup>″ (max) is 2.1 for the composite with

For materials with high resistivity, such as ferrites magnetic loss above 1 GHz is often ascribed to the natural resonance mechanism. The frequency of natural resonance is closely associated with the magnetocrystalline anisotropy as per Snoek's law. One of typical magnetic materials in this case is M-type hexagonal ferrites. In our case, however, the sources of the observed broad magnetic loss peaks are believed to be due to the distribution of localized magnetic anisotropy field, which is the resultant of distribution of shapes (shape anisotropy fields), and distribution of interaction fields among particles. Moreover, eddy current effect is another cause

TEM photographs of Fe-Cu-Nb-Si-B nanostructured alloys. (a) Image showing nano-grains. The inset showing the select area electron diffraction. (b) Nano-grains circled in the high-resolution image (copyright, 2014,

large flakes, and for composites with small flakes, <sup>μ</sup>″ (max) is about 1.6.

s

), is found to be

part of permeability, which can be named "initial permeability" (μ<sup>0</sup>

thickness is averaged to be 1.3 μm.

permeability (μ″

Figure 4.

IOP).

121

The phase transition temperatures of a "FINEMET" alloy can be identified by taking a differential scanning calorimeter (DSC) curve. As shown in Figure 3, two exothermal peaks are found in the DSC curve: one at 530°C (called Tx1) and the other at 672°C (called Tx2). Tx1 is known as the primary crystallization temperature, and Tx2 is the secondary crystallization temperature. In order to form nanoscale grains, as-prepared ribbons may be annealed above Tx1 (see zone I) or Tx2 (see zone II). It was previously reported that nanoscale Fe(Si) grains can be formed when the alloys are annealed above Tx1. Annealing at different temperatures and with different time will give rise to different volume fractions of ferromagnetic phases (amorphous phase and nanocrystallized phase), size of nano-grains and different kinds of magnetic phases (T > Tx2, zone II), such as Fe-B phases. More details on the phase transitions can be found in literature [4]. We believe that controlling the annealing (in other words, phase transition) will give us abundant ways to tailor the high frequency of Fe-based nanostructured materials to meet the specific requirement of an EM wave absorbing application. Here, we propose that without preparing materials with new compositions, just annealing the nanocrystalline soft magnetic alloys (such as FINEMET, NANOPERM or HITPERM) to tailor the high-frequency permeability is an economic approach from the perspective of mass production.

Preparation of FINEMENT amorphous ribbon, annealing treatments and milling processes can be found in our published paper [5]. Nanostructures of sample heat

Figure 3. DSC curve of FINEMET alloy.

3. Particle size distribution

Electromagnetic Materials and Devices

phases contribute to the permeability.

approach from the perspective of mass production.

Figure 3.

120

DSC curve of FINEMET alloy.

As discussed before, the high-frequency permeability can be greatly enhanced by controlling the shapes of particles and alignment of flakes (i.e., a thin film can be treated as a large quantity of well-aligned flakes in a plane). In this section, we will focus on the effect of size distribution of flakes on the dynamic permeability. We choose the composites containing Fe-Cu-Nb-Si-B alloy particles (a.k.a. "FINEMET" alloy) for this purpose. The reason to choose this Fe-based nanostructured material for its special nanoscale structure is the nano-grains are well dispersed in an amorphous matrix (it can be thought as a core-shell structure). The amorphous matrix has the larger resistivity which can reduce the detrimental effects of eddy current on high-frequency permeability. Besides, both amorphous phase and nano-grains are ferromagnetic, which avoids the effect non-magnetic coatings have on ferromagnetic particles (it will be discussed in the next section), and as a result both

The phase transition temperatures of a "FINEMET" alloy can be identified by taking a differential scanning calorimeter (DSC) curve. As shown in Figure 3, two exothermal peaks are found in the DSC curve: one at 530°C (called Tx1) and the other at 672°C (called Tx2). Tx1 is known as the primary crystallization temperature, and Tx2 is the secondary crystallization temperature. In order to form nanoscale grains, as-prepared ribbons may be annealed above Tx1 (see zone I) or Tx2 (see zone II). It was previously reported that nanoscale Fe(Si) grains can be formed when the alloys are annealed above Tx1. Annealing at different temperatures and with different time will give rise to different volume fractions of ferromagnetic phases (amorphous phase and nanocrystallized phase), size of nano-grains and different kinds of magnetic phases (T > Tx2, zone II), such as Fe-B phases. More details on the phase transitions can be found in literature [4]. We believe that controlling the annealing (in other words, phase transition) will give us abundant ways to tailor the high frequency of Fe-based nanostructured materials to meet the specific requirement of an EM wave absorbing application. Here, we propose that without preparing materials with new compositions, just annealing the nanocrystalline soft magnetic alloys (such as FINEMET, NANOPERM or HITPERM) to tailor the high-frequency permeability is an economic

Preparation of FINEMENT amorphous ribbon, annealing treatments and milling processes can be found in our published paper [5]. Nanostructures of sample heat

treated at 540°C are characterized by a transmission electron microscopy (TEM) and shown in Figure 4. Figure 4a is the bright-field TEM image, whereas Figure 4b shows the typical features of Fe-based nanostructured alloys, where the amorphous matrix is surrounding the nanocrystalline grains. The average grain size is about 14.38 nm. XRD results show the nanocrystalline grains to be α-Fe(Si). It should be noted that the effect of annealing and ball milling on phase transitions can be investigated using Mössbauer technique and XRD measurements. Especially, Mössbauer technique can detect fine distinctions in the transformations of crystal structures due to the high-energy ball milling [5]. Using the milling procedure described in the previous section, we can effectively transform the annealed ribbons into flakes, which are shown in Figure 5. These flakes are divided into two categories by a shaking sieve: the large flakes (Figure 5a) and the small flakes (Figure 5b). The typical thickness of flakes of both categories is also shown in Figure 5c and d. For large flakes, the width of flakes is about 23–111 μm, average width is estimated to be about 81.1 μm and average thickness was found to be 4.5 μm. For small flakes, the width of flakes is about 3–21 μm, average width is estimated about 9.4 μm and thickness is averaged to be 1.3 μm.

Frequency dependence of permeability of these two kinds of flakes is shown in Figure 6. At the beginning of measurement frequency (i.e., at 0.5 GHz), the real part of permeability, which can be named "initial permeability" (μ<sup>0</sup> s ), is found to be about 4.6 for large flakes and about 3.9 for small flakes. Initial imaginary part of permeability (μ″ <sup>s</sup> ) is about 1.3 for large flakes and about 0.9 for small flakes. Interestingly, it should be pointed out that wide magnetic loss spectra (μ � f) can be found for both kinds of flakes, as indicated in Figure 6b. Wide magnetic loss peak is advantageous for electromagnetic noise attenuation composites with a wide working frequency band. The loss peak (μ″ is maximum) is found to be 3.0 GHz for large flakes and 5.5 GHz for small flakes, whereas <sup>μ</sup>″ (max) is 2.1 for the composite with large flakes, and for composites with small flakes, <sup>μ</sup>″ (max) is about 1.6.

For materials with high resistivity, such as ferrites magnetic loss above 1 GHz is often ascribed to the natural resonance mechanism. The frequency of natural resonance is closely associated with the magnetocrystalline anisotropy as per Snoek's law. One of typical magnetic materials in this case is M-type hexagonal ferrites. In our case, however, the sources of the observed broad magnetic loss peaks are believed to be due to the distribution of localized magnetic anisotropy field, which is the resultant of distribution of shapes (shape anisotropy fields), and distribution of interaction fields among particles. Moreover, eddy current effect is another cause

#### Figure 4.

TEM photographs of Fe-Cu-Nb-Si-B nanostructured alloys. (a) Image showing nano-grains. The inset showing the select area electron diffraction. (b) Nano-grains circled in the high-resolution image (copyright, 2014, IOP).

<sup>N</sup><sup>⊥</sup> <sup>¼</sup> <sup>α</sup><sup>2</sup> r α2 <sup>r</sup> � 1

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

(μ<sup>0</sup>

equations as follows:

Table 1.

123

Demagnetization factors of two kinds of flakes.

� 1 �

High-Frequency Permeability of Fe-Based Nanostructured Materials

ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 α2 <sup>r</sup> � 1

Nh <sup>¼</sup> <sup>1</sup> � <sup>N</sup><sup>⊥</sup>

(z, the real partflakes drops more rapidly ty and the loss peak. Theower frequencies, with increasing frequency, eddy current becomes a more serious issue in

conducting materials. When they work above 1 GHz, the eddy current effect will be unavoidable. The electromagnetic wave will interact with the part of magnetic materials which are within the so-called "skin depth." As a result, high-frequency permeability spectrum depends on this eddy current effect. Stronger eddy current effect will give rise to the smaller volume fraction of magnetic materials interacting with the EM wave. Consequently, a faster dropping of μ<sup>0</sup> � f spectrum is observed. For electromagnetic wave absorbing application, the simplest example is that composites containing the flakes work as single layer on a perfectly conducting substrate (such as the surface of aircrafts). The absorbing properties of a normal incident EM wave can be assessed by the reflection loss (RL, in dB) based on the

the large flakes. As we pointed out, FINEMET alloys are metallic and well-

Zin ¼ Z<sup>0</sup>

ffiffiffi μ ε r

tanh j

� � � �

<sup>R</sup>:L: <sup>¼</sup> 20 log Zin � <sup>Z</sup><sup>0</sup>

where "d" is the thickness of composite layer, "c" is the velocity of light, Z0 is the impedance of free space and Zin is the characteristic impedance at the free

Small flakes 0.816 0.092 0.908 0.075 Large flakes 0.919 0.041 0.960 0.038

2πfd c

Zin þ Z<sup>0</sup>

ffiffiffiffiffi

� � � �

N<sup>⊥</sup> Nh (N<sup>⊥</sup> + Nh) N⊥Nh

με <sup>p</sup> � � (7)

(8)

where α<sup>r</sup> is the width/thickness ratio (often called "aspect ratio") of a flake. The demagnetization factor along the direction of thickness and width is N<sup>⊥</sup> and Nh, respectively. The saturation magnetization of material under studied is denoted as Ms. Magnetocrystalline anisotropy field is denoted as Hk. The total magnetic anisotropy field is given in the denominator of Eq. (3). For our samples, these two kinds of flakes are obtained under the same milling process and made from the same material. Therefore, large flakes and small flakes have same Ms and Hk values. The main differences among them are size distribution, aspect ratio and thickness. The aspect ratio is calculated as 7.23 and 18.02, respectively, for small flakes and large flakes according to the measured geometrical parameters. Subsequently, demagnetization factors (N<sup>⊥</sup> and Nh) have been calculated as per Eqs. (5) and (6) and are shown in Table 1. It can be seen that larger flakes having the larger initial permeability values are because they have smaller Nh values, as indicated by Eq. (3). Moreover, the finding that their magnetic loss peaks are found at lower frequencies can be understood according to Snoek's law: the inversely proportional relationship between the initial permeability and the loss peak. The μ<sup>0</sup> � f spectrum of large flakes drops more rapidly than small flakes. When f > 1.8 GHz, the real part

� arssin

!

ffiffiffiffiffiffiffiffiffiffiffiffiffi α2 <sup>r</sup> � <sup>1</sup> <sup>p</sup> αr

<sup>2</sup> (6)

(5)

s

#### Figure 5.

SEM images of two categories of flakes: (a) large flakes; (b) small flakes; (c) typical thickness of large flakes; (d) typical thickness of small flakes (copyright, 2015, AIP).

#### Figure 6.

High-frequency permeability of composites contained different sizes of Fe-Cu-Nb-Si-B flakes. (a) μ<sup>0</sup> � f spectra and (b) μ � f spectra (copyright, 2015, AIP).

for broadening the spectra of permeability. In order to interpret the observed dissimilarities in the high-frequency permeability of composites with these two categories of flakes, Snoek's law with shape factors included is employed and given as below:

$$\mu\_s = 1 + \frac{4\pi M\_s}{(H\_k + 4\pi M\_s N\_h)}\tag{3}$$

$$f\_r = \frac{\gamma}{2\pi} \sqrt{H\_k^2 + 4\pi M\_\circ H\_k (N\_\perp + N\_h) + (4\pi M\_\circ)^2 N\_\perp N\_h} \tag{4}$$

High-Frequency Permeability of Fe-Based Nanostructured Materials DOI: http://dx.doi.org/10.5772/intechopen.86403

$$N\_{\perp} = \frac{a\_r^2}{a\_r^2 - 1} \times \left(1 - \sqrt{\frac{1}{a\_r^2 - 1}} \times \arcsin\frac{\sqrt{a\_r^2 - 1}}{a\_r}\right) \tag{5}$$

$$N\_h = \frac{1 - N\_\perp}{2} \tag{6}$$

where α<sup>r</sup> is the width/thickness ratio (often called "aspect ratio") of a flake. The demagnetization factor along the direction of thickness and width is N<sup>⊥</sup> and Nh, respectively. The saturation magnetization of material under studied is denoted as Ms. Magnetocrystalline anisotropy field is denoted as Hk. The total magnetic anisotropy field is given in the denominator of Eq. (3). For our samples, these two kinds of flakes are obtained under the same milling process and made from the same material. Therefore, large flakes and small flakes have same Ms and Hk values. The main differences among them are size distribution, aspect ratio and thickness. The aspect ratio is calculated as 7.23 and 18.02, respectively, for small flakes and large flakes according to the measured geometrical parameters. Subsequently, demagnetization factors (N<sup>⊥</sup> and Nh) have been calculated as per Eqs. (5) and (6) and are shown in Table 1. It can be seen that larger flakes having the larger initial permeability values are because they have smaller Nh values, as indicated by Eq. (3). Moreover, the finding that their magnetic loss peaks are found at lower frequencies can be understood according to Snoek's law: the inversely proportional relationship between the initial permeability and the loss peak. The μ<sup>0</sup> � f spectrum of large flakes drops more rapidly than small flakes. When f > 1.8 GHz, the real part (μ<sup>0</sup> (z, the real partflakes drops more rapidly ty and the loss peak. Theower frequencies, with increasing frequency, eddy current becomes a more serious issue in the large flakes. As we pointed out, FINEMET alloys are metallic and wellconducting materials. When they work above 1 GHz, the eddy current effect will be unavoidable. The electromagnetic wave will interact with the part of magnetic materials which are within the so-called "skin depth." As a result, high-frequency permeability spectrum depends on this eddy current effect. Stronger eddy current effect will give rise to the smaller volume fraction of magnetic materials interacting with the EM wave. Consequently, a faster dropping of μ<sup>0</sup> � f spectrum is observed.

For electromagnetic wave absorbing application, the simplest example is that composites containing the flakes work as single layer on a perfectly conducting substrate (such as the surface of aircrafts). The absorbing properties of a normal incident EM wave can be assessed by the reflection loss (RL, in dB) based on the equations as follows:

$$Z\_{in} = Z\_0 \sqrt{\frac{\mu}{\varepsilon}} \tanh\left(j\frac{2\pi fd}{c}\sqrt{\mu\varepsilon}\right) \tag{7}$$

$$R.L. = 20\log\left|\frac{Z\_{\text{in}} - Z\_0}{Z\_{\text{in}} + Z\_0}\right|\tag{8}$$

where "d" is the thickness of composite layer, "c" is the velocity of light, Z0 is the impedance of free space and Zin is the characteristic impedance at the free


Table 1.

Demagnetization factors of two kinds of flakes.

for broadening the spectra of permeability. In order to interpret the observed dissimilarities in the high-frequency permeability of composites with these two categories of flakes, Snoek's law with shape factors included is employed and given

High-frequency permeability of composites contained different sizes of Fe-Cu-Nb-Si-B flakes. (a) μ<sup>0</sup> � f spectra

SEM images of two categories of flakes: (a) large flakes; (b) small flakes; (c) typical thickness of large flakes;

4πMs ð Þ Hk þ 4πMsNh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>k</sup> <sup>þ</sup> <sup>4</sup>πMsHkð Þþ <sup>N</sup><sup>⊥</sup> <sup>þ</sup> Nh ð Þ <sup>4</sup>πMs <sup>2</sup>

(3)

(4)

N⊥Nh

μ<sup>s</sup> ¼ 1 þ

H2

q

(d) typical thickness of small flakes (copyright, 2015, AIP).

Electromagnetic Materials and Devices

fr <sup>¼</sup> <sup>γ</sup> 2π

and (b) μ � f spectra (copyright, 2015, AIP).

as below:

122

Figure 6.

Figure 5.

as NiFe2O4 ferrite) so as to realize two objectives: to reduce the permittivity and to

The impacts of spinel ferrite coating layer on the high-frequency permittivity and permeability are shown in Figure 11. When the flakes are coated, ε<sup>0</sup> drops from 61.49 to 33.02 at 0.5 GHz, whereas <sup>ε</sup>″ also drops from 21.39 to 1.16 at 0.5 GHz. As shown, the complex permittivity values are significantly decreased within the lower frequency band and are believed to result from increased resistivity. The permeability values are also found to be a little reduced, due to the fact that Ms value of

SEM images of (a) uncoated flakes and (b) coated flakes. (c) TEM image of a coated flake. Inset in (a): typical

thicknesses of the sample. Inset in (b): surface roughness of a coated flake (copyright, 2015, IEEE).

NiFe2O4 layers were fabricated on the Fe-Cu-Nb-Si-B flakes using a lowtemperature chemical plating route. In a simplified description, Fe-Cu-Nb-Si-B nanocrystalline flakes were added into a flask containing a bath solution at 333 K. As for the bath solution, the well-designed molar ratios of NiCl2, FeCl2 and KOH solutions were prepared in the flask. Meanwhile, oxygen gas was introduced into the solution until the chemical reactions were completed. Subsequently, the flask was heated at 333 K for 40 min. When the chemical reaction was finished, the deionized flakes were collected and dried at 333 K for 12 h. Elaborative experimental descriptions can be found in our published paper [7]. The morphologies of coated flakes are presented in Figure 9a and b, respectively. TEM images in Figure 9c show that the thickness of coating layer is estimated to be about 17.73– 55.61 nm. The energy dispersion spectrum (EDS) and XRD measurements of uncoated and coated flakes confirm the formation of NiFe2O4 spinel ferrite. XRD patterns are given in Figure 10, which indicate the formation of spinel ferrite phase. The magnetic hysteresis loops of coated and uncoated flakes are given in Ref. [7] and show that the saturation magnetization of coated flakes drops from 129.33 to

absorb the EM wave by the high-resistivity layer.

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

High-Frequency Permeability of Fe-Based Nanostructured Materials

96.54 emu/g.

Figure 9.

125

Figure 7.

Contour maps showing the absorbing properties of single layer composites with different flakes: (a) large flakes and (b) small flakes (copyright, 2015, AIP).

#### Figure 8.

(a) Composites filled with smaller flakes and with different thickness and (b) composite filled with different flakes but with same thickness (4 mm) (copyright, 2015, AIP).

space/absorber interface. In this chapter, all "t" values are in mm unit. "μ" and "ε" are the measured relative complex permeability and permittivity, respectively. The measured permittivity values can be found and have been studied in our paper [6]. The potential absorbing performances of composites with different thickness values are illustrated in Figure 7. Clearly, composites containing smaller flakes will have excellent absorbing performances in terms of RL as well as reduced absorber thickness. The superior absorbing properties are also shown in Figure 8 by selecting a few thicknesses of single layer of composites filled with smaller flakes.

#### 4. Coating treatments

Since high-frequency permittivity of metallic flakes is much larger than their permeability, this difference will cause a serious mismatch of impedance (Zin >> Z0), which will deteriorate the absorbing properties of flakes. Common means of reducing impedance mismatch is to coat the metallic particles with a layer of oxides with high resistivity (such as SiO2, TiO2, Al2O3, etc.). Although these layers are effective in decreasing the permittivity, they are not ferro- (or ferri-) magnetic and cannot take part in the absorbing of EM wave via magnetic losses. Therefore, we propose to coat the FINEMET flakes with ferrimagnetic layer (such

### High-Frequency Permeability of Fe-Based Nanostructured Materials DOI: http://dx.doi.org/10.5772/intechopen.86403

as NiFe2O4 ferrite) so as to realize two objectives: to reduce the permittivity and to absorb the EM wave by the high-resistivity layer.

NiFe2O4 layers were fabricated on the Fe-Cu-Nb-Si-B flakes using a lowtemperature chemical plating route. In a simplified description, Fe-Cu-Nb-Si-B nanocrystalline flakes were added into a flask containing a bath solution at 333 K. As for the bath solution, the well-designed molar ratios of NiCl2, FeCl2 and KOH solutions were prepared in the flask. Meanwhile, oxygen gas was introduced into the solution until the chemical reactions were completed. Subsequently, the flask was heated at 333 K for 40 min. When the chemical reaction was finished, the deionized flakes were collected and dried at 333 K for 12 h. Elaborative experimental descriptions can be found in our published paper [7]. The morphologies of coated flakes are presented in Figure 9a and b, respectively. TEM images in Figure 9c show that the thickness of coating layer is estimated to be about 17.73– 55.61 nm. The energy dispersion spectrum (EDS) and XRD measurements of uncoated and coated flakes confirm the formation of NiFe2O4 spinel ferrite. XRD patterns are given in Figure 10, which indicate the formation of spinel ferrite phase. The magnetic hysteresis loops of coated and uncoated flakes are given in Ref. [7] and show that the saturation magnetization of coated flakes drops from 129.33 to 96.54 emu/g.

The impacts of spinel ferrite coating layer on the high-frequency permittivity and permeability are shown in Figure 11. When the flakes are coated, ε<sup>0</sup> drops from 61.49 to 33.02 at 0.5 GHz, whereas <sup>ε</sup>″ also drops from 21.39 to 1.16 at 0.5 GHz. As shown, the complex permittivity values are significantly decreased within the lower frequency band and are believed to result from increased resistivity. The permeability values are also found to be a little reduced, due to the fact that Ms value of

#### Figure 9.

space/absorber interface. In this chapter, all "t" values are in mm unit. "μ" and "ε" are the measured relative complex permeability and permittivity, respectively. The measured permittivity values can be found and have been studied in our paper [6]. The potential absorbing performances of composites with different thickness values are illustrated in Figure 7. Clearly, composites containing smaller flakes will have excellent absorbing performances in terms of RL as well as reduced absorber thickness. The superior absorbing properties are also shown in Figure 8 by selecting a

(a) Composites filled with smaller flakes and with different thickness and (b) composite filled with different

Contour maps showing the absorbing properties of single layer composites with different flakes: (a) large flakes

Since high-frequency permittivity of metallic flakes is much larger than their

few thicknesses of single layer of composites filled with smaller flakes.

permeability, this difference will cause a serious mismatch of impedance (Zin >> Z0), which will deteriorate the absorbing properties of flakes. Common means of reducing impedance mismatch is to coat the metallic particles with a layer of oxides with high resistivity (such as SiO2, TiO2, Al2O3, etc.). Although these layers are effective in decreasing the permittivity, they are not ferro- (or ferri-) magnetic and cannot take part in the absorbing of EM wave via magnetic losses. Therefore, we propose to coat the FINEMET flakes with ferrimagnetic layer (such

4. Coating treatments

Figure 7.

Figure 8.

124

and (b) small flakes (copyright, 2015, AIP).

Electromagnetic Materials and Devices

flakes but with same thickness (4 mm) (copyright, 2015, AIP).

SEM images of (a) uncoated flakes and (b) coated flakes. (c) TEM image of a coated flake. Inset in (a): typical thicknesses of the sample. Inset in (b): surface roughness of a coated flake (copyright, 2015, IEEE).

greatly suppressed in coated particles, which means that the high-resistivity coating

Contour maps showing the absorbing properties (in terms of RL) of composites with different thicknesses.

The impact of coating with a high-resistivity ferrite on EM absorbing performances is shown in Figure 12. We use a contour map to illustrate the reflection loss of absorbers filled with flakes which are coated and uncoated. For composites containing uncoated flakes, the complex ε values are much larger than the complex μ values. Due to this large difference in ε and μ, the impedance mismatch (Zin and Z0) according to Eq. 7 is significant, and consequently it results in the worsening of absorbing properties compared to absorbers filled with coated flakes. Obviously, the high-resistivity coating can effectively lessen the impedance mismatch and improve absorbing performance. Moreover, the thickness of absorbing composite is

ferrite effectively reduces the impact of eddy currents on μ values.

(a) Uncoated flakes and (b) coated flakes (copyright, 2015, IEEE).

High-Frequency Permeability of Fe-Based Nanostructured Materials

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

greatly decreased if coated flakes are used as absorbent composite.

section is 10 nm ("y" axis) 20 nm ("x" axis).

equation:

127

Figure 12.

5. Origins of multi-peaks in the intrinsic permeability spectra

As discussed before, broad permeability spectra are commonly observed in the composites filled with magnetic particles. The debate on the causes is intense. We believe that a broad intrinsic permeability spectrum results from superposition of many natural resonance peaks. Intrinsic permeability can be retrieved from measured permeability using one of the mixing rules [8]. However, origins of multipeaks in the intrinsic permeability spectra have not been well answered, and it is essential for the designing of electromagnetic devices or materials. In order to exclude impact of non-intrinsic factors (e.g., inhomogeneous microstructure, eddy current, size distribution and particle shape) on the understanding of the origin of multi-peaks, we design several "1 3" iron nanowire (Fe-NW) arrays. Each array has a different interwire spacing. Each nanowire in the array is identical in geometry. Each one has a cuboid shape: the length is 100 nm (set as the "z"axis). The cross

In this study, we will discuss the impact of interwire distance on the intrinsic

permeability for only two interwire distances: 2.5 and 60 nm, as depicted in Figure 13. The static magnetic properties and dynamic responses of Fe nanowire arrays are simulated using micromagnetic simulation. First-order-reversal-curve (FORC) technique is used to simulate and analyse the impact of interwire distance on the static magnetic properties. The dynamic response of magnetization under excitation of pulse magnetic field can be described by the Landau-Lifshitz-Gilbert

Figure 10. XRD patterns of uncoated and coated flakes (copyright, 2015, IEEE).

Figure 11.

The impact of ferrite coating on the permittivity and permeability of flakes. (a) ε<sup>0</sup> � f and (b) ε � f. (c) μ<sup>0</sup> � f and (d) μ � f (copyright, 2015, IEEE).

coated flakes is less than that of uncoated flakes; see Figure 11c and d. Since the spinel ferrite layer has a smaller Ms value than the FINEMET alloy, the reduced permeability can be understood as per Snoek's law. It is interesting to point out that μ � f spectra of coated flakes are not significantly fluctuated as much as the uncoated flakes. The previous section showed that eddy current effect in uncoated flakes is strong and results in a large reduction of μ values. Such a large reduction is High-Frequency Permeability of Fe-Based Nanostructured Materials DOI: http://dx.doi.org/10.5772/intechopen.86403

Figure 12.

Contour maps showing the absorbing properties (in terms of RL) of composites with different thicknesses. (a) Uncoated flakes and (b) coated flakes (copyright, 2015, IEEE).

greatly suppressed in coated particles, which means that the high-resistivity coating ferrite effectively reduces the impact of eddy currents on μ values.

The impact of coating with a high-resistivity ferrite on EM absorbing performances is shown in Figure 12. We use a contour map to illustrate the reflection loss of absorbers filled with flakes which are coated and uncoated. For composites containing uncoated flakes, the complex ε values are much larger than the complex μ values. Due to this large difference in ε and μ, the impedance mismatch (Zin and Z0) according to Eq. 7 is significant, and consequently it results in the worsening of absorbing properties compared to absorbers filled with coated flakes. Obviously, the high-resistivity coating can effectively lessen the impedance mismatch and improve absorbing performance. Moreover, the thickness of absorbing composite is greatly decreased if coated flakes are used as absorbent composite.

## 5. Origins of multi-peaks in the intrinsic permeability spectra

As discussed before, broad permeability spectra are commonly observed in the composites filled with magnetic particles. The debate on the causes is intense. We believe that a broad intrinsic permeability spectrum results from superposition of many natural resonance peaks. Intrinsic permeability can be retrieved from measured permeability using one of the mixing rules [8]. However, origins of multipeaks in the intrinsic permeability spectra have not been well answered, and it is essential for the designing of electromagnetic devices or materials. In order to exclude impact of non-intrinsic factors (e.g., inhomogeneous microstructure, eddy current, size distribution and particle shape) on the understanding of the origin of multi-peaks, we design several "1 3" iron nanowire (Fe-NW) arrays. Each array has a different interwire spacing. Each nanowire in the array is identical in geometry. Each one has a cuboid shape: the length is 100 nm (set as the "z"axis). The cross section is 10 nm ("y" axis) 20 nm ("x" axis).

In this study, we will discuss the impact of interwire distance on the intrinsic permeability for only two interwire distances: 2.5 and 60 nm, as depicted in Figure 13. The static magnetic properties and dynamic responses of Fe nanowire arrays are simulated using micromagnetic simulation. First-order-reversal-curve (FORC) technique is used to simulate and analyse the impact of interwire distance on the static magnetic properties. The dynamic response of magnetization under excitation of pulse magnetic field can be described by the Landau-Lifshitz-Gilbert equation:

coated flakes is less than that of uncoated flakes; see Figure 11c and d. Since the spinel ferrite layer has a smaller Ms value than the FINEMET alloy, the reduced permeability can be understood as per Snoek's law. It is interesting to point out that

The impact of ferrite coating on the permittivity and permeability of flakes. (a) ε<sup>0</sup> � f and (b) ε � f. (c) μ<sup>0</sup> � f

μ � f spectra of coated flakes are not significantly fluctuated as much as the uncoated flakes. The previous section showed that eddy current effect in uncoated flakes is strong and results in a large reduction of μ values. Such a large reduction is

Figure 10.

Figure 11.

126

and (d) μ � f (copyright, 2015, IEEE).

XRD patterns of uncoated and coated flakes (copyright, 2015, IEEE).

Electromagnetic Materials and Devices

#### Figure 13.

The equilibrium states of magnetization for two kinds of interwire distance: (a) D = 2.5 nm and (b) D = 60 nm (only partial sections of nanowire are presented for saving space).

$$\frac{\partial \overrightarrow{\boldsymbol{M}} \, \boldsymbol{\epsilon}(\boldsymbol{r}\_{i}, \boldsymbol{t})}{\partial \boldsymbol{t}} = -\boldsymbol{\chi} \, \overrightarrow{\boldsymbol{M}} \, (\boldsymbol{r}\_{i}, \boldsymbol{t}) \times \overrightarrow{\boldsymbol{H}}\_{\text{eff}}(\boldsymbol{r}\_{i}, \boldsymbol{t}) - \frac{\boldsymbol{\chi}\boldsymbol{a}}{\boldsymbol{M}\_{\text{s}}} \overrightarrow{\boldsymbol{M}} \, (\boldsymbol{r}\_{i}, \boldsymbol{t}) \times \left[\overrightarrow{\boldsymbol{M}} \, (\boldsymbol{r}\_{i}, \boldsymbol{t}) \times \overrightarrow{\boldsymbol{H}}\_{\text{eff}}(\boldsymbol{r}\_{i}, \boldsymbol{t})\right] \tag{9}$$

More details on simulation procedures and setting parameters can be found in our published paper [9]. FORC approach is based on the Preisach hysteresis theory and is helpful in investigating factors that determine local magnetic properties of materials. According to FORC measurement procedure, the array is at first saturated at a field value (Hs) in one direction; next, the magnetic field is decreased to a field (called "reversal field," Ha); and then the array is again saturated from Ha to Hs, which will trace out one partial magnetization curve (i.e., a FORC curve). Following the same procedure, a series of FORC curves can be obtained starting from different Ha values to the Hs value, which will fill the interior of the major hysteresis loop. Each data point of a FORC curve is denoted as M(Ha, Hb), where Hb is the applied field. According to the Preisach model, a major hysteresis loop is consisted to be a set of hysterons, and the probability density function ρ (Ha, Hb) of hysteron ensemble can be calculated by a mixed second derivative as follows, where ρ wha, Hb) is often called the FORC distribution, and is expressed as

$$\rho(H\_a, H\_b) = -\frac{1}{2} \frac{\partial^2 M}{\partial H\_a \partial H\_b} \tag{10}$$

Corresponding FORC distributions are shown in Figures 14a and 15a. Obviously as seen in Figure 14, two stronger resonance peaks at f = 5.75 and 19.5 GHz and several weak resonance peaks at f = 11.75, 15, 22.25, 24.25 and 26.5 GHz are found for the nanowire array with D as 2.5 nm. The previous studies of both ours and by others show that the resonance peaks found at lower frequency of 5.75 GHz are often named "edge mode" [10], which are resulting from precessions at the ends of nanowires. The frequencies of "edge mode" found by us are very close to those reported by others [10, 11]. The peak found at 19.5 GHz can be identified as "bulk mode." The eigenfrequency of bulk mode for an isolated nanowire can be calculated

FORC distribution and intrinsic permeability of nanowire array when D = 60 nm (copyright, 2018, IOP).

FORC distribution and intrinsic permeability of nanowire array when interwire distance (D) is 2.5 nm

High-Frequency Permeability of Fe-Based Nanostructured Materials

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f g H þ ð Þ Nx � Nz Ms � f g H þ ð Þ Ny � Nz Ms

where H (the external DC magnetic field) is along the easy axis, demagnetization factor is <sup>N</sup> and <sup>γ</sup><sup>0</sup> is the gyromagnetic ratio (for Fe: 2.8 � <sup>10</sup><sup>6</sup> Hz/Oe). For our nanowire arrays, Nx + Ny + Nz = 1, Nz is 0, Nx is 1/3 and Ny is 2/3. Then, the eigenfrequency frequency (natural resonance frequency) of bulk mode is calculated as 28.2 GHz. This calculated value is larger than those for bulk mode simulated in the arrays with different D values. This difference is a result from the fact that the calculated value is based on Eq. (11) which is for uniform precession without considering interactions among nanowires. However, the simulated values are obtained under the circumstance of interaction among NW. It was found that "the

(11)

by the following equation:

Figure 14.

Figure 15.

129

(copyright, 2018, IOP).

f ¼ γ<sup>0</sup>

q

Usually, the FORC distribution is illustrated in the diagram of (Hu vs. Hc), as shown in Figure 14a. Hu is the local interaction field and Hc the local coercivity. Relationship between (Ha, Hb) coordinate system and (Hu, Hc) coordinate system is given as Hc = (Hb � Ha)/2 and Hu = (Ha + Hb)/2.

Clearly, there are no domain walls existing in our simulations, as shown in Figure 13. Therefore, the impact of domain wall on the permeability can be excluded. In addition, coercivity is therefore only decided by localized effective magnetic field (no domain wall movements involved). Localized magnetic fields decide the precession of local magnetizations, which will precess around these effective local fields. Each precession has an eigenfrequency, which is also called "natural resonance frequency." Under the excitation of a pulse magnetic field ((h(t) = 100 exp(�10<sup>9</sup> t), t in second, hour in A/m) is) perpendicular to "z" axis, the simulated intrinsic permeability spectra are shown in Figures 14b and 15b.

High-Frequency Permeability of Fe-Based Nanostructured Materials DOI: http://dx.doi.org/10.5772/intechopen.86403

Figure 14.

∂ M ! ri ð Þ ; t <sup>∂</sup><sup>t</sup> ¼ �<sup>γ</sup> <sup>M</sup>

Figure 13.

!

Electromagnetic Materials and Devices

ri ð Þ� ; t H !

(only partial sections of nanowire are presented for saving space).

given as Hc = (Hb � Ha)/2 and Hu = (Ha + Hb)/2.

((h(t) = 100 exp(�10<sup>9</sup>

128

eff ri ð Þ� ; <sup>t</sup> γα

ρ wha, Hb) is often called the FORC distribution, and is expressed as

ρð Þ¼� Ha; Hb

1 2

Usually, the FORC distribution is illustrated in the diagram of (Hu vs. Hc), as shown in Figure 14a. Hu is the local interaction field and Hc the local coercivity. Relationship between (Ha, Hb) coordinate system and (Hu, Hc) coordinate system is

Clearly, there are no domain walls existing in our simulations, as shown in Figure 13. Therefore, the impact of domain wall on the permeability can be excluded. In addition, coercivity is therefore only decided by localized effective magnetic field (no domain wall movements involved). Localized magnetic fields decide the precession of local magnetizations, which will precess around these effective local fields. Each precession has an eigenfrequency, which is also called "natural resonance frequency." Under the excitation of a pulse magnetic field

simulated intrinsic permeability spectra are shown in Figures 14b and 15b.

∂2 M ∂Ha∂Hb

t), t in second, hour in A/m) is) perpendicular to "z" axis, the

Ms M !

The equilibrium states of magnetization for two kinds of interwire distance: (a) D = 2.5 nm and (b) D = 60 nm

More details on simulation procedures and setting parameters can be found in our published paper [9]. FORC approach is based on the Preisach hysteresis theory and is helpful in investigating factors that determine local magnetic properties of materials. According to FORC measurement procedure, the array is at first saturated at a field value (Hs) in one direction; next, the magnetic field is decreased to a field (called "reversal field," Ha); and then the array is again saturated from Ha to Hs, which will trace out one partial magnetization curve (i.e., a FORC curve). Following the same procedure, a series of FORC curves can be obtained starting from different Ha values to the Hs value, which will fill the interior of the major hysteresis loop. Each data point of a FORC curve is denoted as M(Ha, Hb), where Hb is the applied field. According to the Preisach model, a major hysteresis loop is consisted to be a set of hysterons, and the probability density function ρ (Ha, Hb) of hysteron ensemble can be calculated by a mixed second derivative as follows, where

ri ð Þ� ; t M

!

ri ð Þ� ; t H !

h i

eff ri ð Þ ; t

(9)

(10)

FORC distribution and intrinsic permeability of nanowire array when interwire distance (D) is 2.5 nm (copyright, 2018, IOP).

Figure 15.

FORC distribution and intrinsic permeability of nanowire array when D = 60 nm (copyright, 2018, IOP).

Corresponding FORC distributions are shown in Figures 14a and 15a. Obviously as seen in Figure 14, two stronger resonance peaks at f = 5.75 and 19.5 GHz and several weak resonance peaks at f = 11.75, 15, 22.25, 24.25 and 26.5 GHz are found for the nanowire array with D as 2.5 nm. The previous studies of both ours and by others show that the resonance peaks found at lower frequency of 5.75 GHz are often named "edge mode" [10], which are resulting from precessions at the ends of nanowires. The frequencies of "edge mode" found by us are very close to those reported by others [10, 11]. The peak found at 19.5 GHz can be identified as "bulk mode." The eigenfrequency of bulk mode for an isolated nanowire can be calculated by the following equation:

$$f = \mathbf{y}' \sqrt{\{H + (\mathbf{Nx} - \mathbf{Nz})\mathbf{M}\_t\} \times \{H + (\mathbf{Ny} - \mathbf{Nz})\mathbf{M}\_t\}}\tag{11}$$

where H (the external DC magnetic field) is along the easy axis, demagnetization factor is <sup>N</sup> and <sup>γ</sup><sup>0</sup> is the gyromagnetic ratio (for Fe: 2.8 � <sup>10</sup><sup>6</sup> Hz/Oe). For our nanowire arrays, Nx + Ny + Nz = 1, Nz is 0, Nx is 1/3 and Ny is 2/3. Then, the eigenfrequency frequency (natural resonance frequency) of bulk mode is calculated as 28.2 GHz. This calculated value is larger than those for bulk mode simulated in the arrays with different D values. This difference is a result from the fact that the calculated value is based on Eq. (11) which is for uniform precession without considering interactions among nanowires. However, the simulated values are obtained under the circumstance of interaction among NW. It was found that "the

calculated fr" and "the simulated fr" are in good agreement in an isolated nanowire [10]. Our simulations show that when the interwire distance increases, FORC diagrams and the intrinsic μ f spectra vary differently. In addition, the reversal process of nanowire array is found to be different. Please refer to our published paper for more details [9]. When interwire distance increases, magnetization reversal behaviour progressively changes from "the sequential mode" to "the independent mode." Moreover, the finding that the amount of weak resonance peaks decreases with increasing D is worth noting. With regard to the origin of weak resonance peaks in Figure 14b, we presume that it is the consequence of superposition of interaction field, magnetocrystalline field and shape anisotropy field. As depicted in FORC diagrams (see Figures 14 and 15), the interaction field can be either negative or positive. Effective field acting on some magnetization can be smaller (or larger) than the effective field related to "bulk mode" in an isolated nanowire. Accordingly, some resonance frequency is smaller (or larger) than that of bulk mode, which can be clearly observed in Figure 14. With increasing interwire distance, interaction among nanowires gradually disappears; therefore, these smaller peaks gradually vanish. From the stand point of electromagnetic (EM) attenuation application, it means that volume fraction of magnetic particles in a composite should not be diluted in order to have many resonance peaks for expanding the attenuation bandwidth. As shown in Figure 15, there are only two strong resonance peaks found when D is 60 nm, which may be explained using the same physical mechanism. The "calculated fr" (bulk mode) is about 28.2 GHz and is close to the "simulated fr" (about 25 GHz).

electrochemically deposited polycrystals and multimagnetic domains, which will

As stated before, high-frequency permeability of Fe-based conducting nanostructured materials can be tailored by shapes, phase transitions, coating and size distributions. Most importantly, all imaginary parts of permeability are positive (μ″ <sup>&</sup>gt; 0), which are accepted as an unalterable natural principle. This is correct when there are no other excitations except AC magnetic field based on the fact that positive <sup>μ</sup>″ manifests energy dissipation of magnetization precession. What happens to the imaginary parts of permeability if there is another excitation, which can compensate the energy loss during the precession? Furthermore, is it possible to tailor the high-frequency permeability by electric current? If possible, it will be possible for us to design intelligent electromagnetic devices. In this section, we propose for the first time an approach to accomplish such a goal via spin transfer torque (STT) effect. According to STT effect, when polarized electrons flow through zone of metallic ferromagnetic material with nonuniformly oriented magnetic moments (e.g., magnetic domain walls), the spins of electrons will exert torques on them, which will change the dynamic responses of precession of magnetic moments [13]. When STT is strong enough, it can even switch the direction of magnetic moments. Here we only demonstrate an example using micromagnetic

give rise to plentiful irreversible magnetization processes.

High-Frequency Permeability of Fe-Based Nanostructured Materials

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

6. Negative imaginary parts of permeability (μ″ <sup>&</sup>lt; 0)

simulation; our detailed results can be found in Ref. [12]. The object of

dM ! dt <sup>¼</sup> <sup>γ</sup><sup>H</sup> ! eff� M ! þ α Ms Ms ! � dM ! dt ! � <sup>u</sup>

Figure 16.

131

gyromagnetic ratio (2.21 � <sup>10</sup><sup>5</sup> mA�<sup>1</sup> <sup>S</sup>�<sup>1</sup>

micromagnetic simulations is an isolated Fe nanowire with a length of 400 nm (set as "x" axis) and the diameter of 10 nm, as shown in Figure 16. The dynamic response of magnetization (a group of magnetic moments in a direction, also can be called "macro spin") can be simulated under two external excitations: AC magnetic field and electrically polarized current. The AC magnetic field is applied along the "z" axis, and polarized electrons are flowing along the "x" axis. The software of micromagnetic simulation is a three-dimensional object-oriented micromagnetic framework (OOMMF). The dynamic responses of precession are simulated by solving the Landau-Lifshitz-Gilbert (LLG) equation as a function of time. When the STT effect is incorporated, the modified LLG equation is expressed as [13, 14]:

> ! � <sup>∇</sup> � � !

where α (damping constant) is set as 0.01 for the simulations; γ is the Gilbert

Inhomogeneous orientation of magnetic moments of an isolated nanowire (copyright, 2018, IOP).

M ! þ β Ms <sup>M</sup> !

� u ! � <sup>∇</sup> � � !

); M is the magnetization; Heff is the

M h i !

(12)

Impact of interwire distance on interaction among NWs is shown in Figures 14a and 15a. Obviously, with increasing interwire distance (D), data points approach together (results of other D values were shown in Ref. 9), and Hu becomes "zero" when D is equal or greater than 60 nm. Here, when reversal resistance comes only from localized effective field (Heff), then effective field can be approximated by coercivity field (i.e., Hc ≈ Heff). Hence, when D increases, localized Hc is always larger than zero. The "scattered" characteristic of Hc value vanishes when D is 60 nm, which means that reversal behaviour of M changes into the "independent mode," as shown in Ref. 9. In addition, peak magnitude of "edge mode" in an isolated "cylindrical" nanowire is usually smaller than "bulk mode" [10, 11]. However, all peak magnitudes of "edge mode" in our "cuboid" shape nanowires are comparable with bulk mode, which suggests that high-frequency magnetic loss due to the "edge mode" cannot be neglected in our case. Moreover, smaller fr value of "edge mode" means that localized effective magnetic field governing its precession is weaker. It means that reversal of magnetization commonly starts from the ends of nanowires, which has been observed in our study.

To better understand the changes of intrinsic μ f spectra, studies on difference in magnetic moment orientations in equilibrium states are necessary, as illustrated in Figure 13. Apparently, magnetic moment orientations around the ends of nanowires at equilibrium states are distinct. Such orientation pattern of magnetic moments strongly relies on the local effective magnetic field. The first-order- reversal-curve diagram is useful to know the distribution of local effective magnetic fields. Hence, we believe any factor affecting the equilibrium state of magnetic moments will change the intrinsic μ f spectra. If the equilibrium state is rebuilt, then the permeability spectra under same excitation can be recovered too (we name it as "memory effect"). Finally, we want to point out that such traits of "scattered" values of Hc and Hu in these simulations differ from the continuous distribution measured in most nanowires deposited in nanoporous templates (such as AAO templates). The reason is that continuous distribution of Hc and Hu in a FORC diagram results from inhomogeneity in nanopore sizes of template, nanoscale

calculated fr" and "the simulated fr" are in good agreement in an isolated nanowire [10]. Our simulations show that when the interwire distance increases, FORC diagrams and the intrinsic μ f spectra vary differently. In addition, the reversal process of nanowire array is found to be different. Please refer to our published paper for more details [9]. When interwire distance increases, magnetization reversal behaviour progressively changes from "the sequential mode" to "the independent mode." Moreover, the finding that the amount of weak resonance peaks decreases with increasing D is worth noting. With regard to the origin of weak resonance peaks in Figure 14b, we presume that it is the consequence of superposition of interaction field, magnetocrystalline field and shape anisotropy field. As depicted in FORC diagrams (see Figures 14 and 15), the interaction field can be either negative or positive. Effective field acting on some magnetization can be smaller (or larger) than the effective field related to "bulk mode" in an isolated nanowire. Accordingly, some resonance frequency is smaller (or larger) than that of bulk mode, which can be clearly observed in Figure 14. With increasing interwire distance, interaction among nanowires gradually disappears; therefore, these smaller peaks gradually vanish. From the stand point of electromagnetic (EM) attenuation application, it means that volume fraction of magnetic particles in a composite should not be diluted in order to have many resonance peaks for expanding the attenuation bandwidth. As shown in Figure 15, there are only two strong resonance peaks found when D is 60 nm, which may be explained using the same physical mechanism. The "calculated fr" (bulk mode) is about 28.2 GHz and is

Impact of interwire distance on interaction among NWs is shown in Figures 14a and 15a. Obviously, with increasing interwire distance (D), data points approach together (results of other D values were shown in Ref. 9), and Hu becomes "zero" when D is equal or greater than 60 nm. Here, when reversal resistance comes only from localized effective field (Heff), then effective field can be approximated by coercivity field (i.e., Hc ≈ Heff). Hence, when D increases, localized Hc is always larger than zero. The "scattered" characteristic of Hc value vanishes when D is 60 nm, which means that reversal behaviour of M changes into the "independent mode," as shown in Ref. 9. In addition, peak magnitude of "edge mode" in an isolated "cylindrical" nanowire is usually smaller than "bulk mode" [10, 11]. However, all peak magnitudes of "edge mode" in our "cuboid" shape nanowires are comparable with bulk mode, which suggests that high-frequency magnetic loss due to the "edge mode" cannot be neglected in our case. Moreover, smaller fr value of "edge mode" means that localized effective magnetic field governing its precession is weaker. It means that reversal of magnetization commonly starts from the ends of

To better understand the changes of intrinsic μ f spectra, studies on difference in magnetic moment orientations in equilibrium states are necessary, as illustrated in Figure 13. Apparently, magnetic moment orientations around the ends of nanowires at equilibrium states are distinct. Such orientation pattern of magnetic moments strongly relies on the local effective magnetic field. The first-order- reversal-curve diagram is useful to know the distribution of local effective magnetic fields. Hence, we believe any factor affecting the equilibrium state of magnetic moments will change the intrinsic μ f spectra. If the equilibrium state is rebuilt, then the permeability spectra under same excitation can be recovered too (we name it as "memory effect"). Finally, we want to point out that such traits of "scattered" values of Hc and Hu in these simulations differ from the continuous distribution measured in most nanowires deposited in nanoporous templates (such as AAO templates). The reason is that continuous distribution of Hc and Hu in a FORC diagram results from inhomogeneity in nanopore sizes of template, nanoscale

close to the "simulated fr" (about 25 GHz).

Electromagnetic Materials and Devices

nanowires, which has been observed in our study.

130

electrochemically deposited polycrystals and multimagnetic domains, which will give rise to plentiful irreversible magnetization processes.
