Open access peer-reviewed chapter

High-Frequency Permeability of Fe-Based Nanostructured Materials

Written By

Mangui Han

Submitted: 14 July 2018 Reviewed: 18 April 2019 Published: 16 May 2019

DOI: 10.5772/intechopen.86403

From the Edited Volume

Electromagnetic Materials and Devices

Edited by Man-Gui Han

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Frequency dependence of permeability of magnetic materials is crucial for their various electromagnetic applications. In this chapter, the approaches to tailor the high-frequency permeability of Fe-based nanostructured materials are discussed: shape controlling, particle size distribution, coating treatments, phase transitions and external excitations. Special attention is paid on the electromagnetic wave absorbing using these Fe-based magnetic materials (Fe-Si-Al alloys, Fe-Cu-Nb-Si-B nanocrystalline alloys and Fe nanowires). Micromagnetic simulations also have been presented to discuss the intrinsic high-frequency permeability. Negative imaginary part of permeability (μ″ < 0) is proved possible under the spin transfer torque effect.


  • complex permeability
  • electromagnetic wave absorbing
  • negative imaginary part of permeability
  • micromagnetic simulation
  • natural resonance

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′ > 0) and lower imaginary parts of permeability (μ″ > 0 and μ″ → 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 μ″ 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.

Figure 1.

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

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′ 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′ 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 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]

Figure 2.

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

μ s 1 f r 2 = γ 2 π 2 × 4 π M s × H k + 4 π M s D z E1

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

μ s 1 f r = 2 3 γ 4 π M s E2

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


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.

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.

Figure 3.

DSC curve of FINEMET alloy.

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.

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).

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).

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 s ), is found to be about 4.6 for large flakes and about 3.9 for small flakes. Initial imaginary part of permeability ( μ s ) 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 μ″ (max) is 2.1 for the composite with large flakes, and for composites with small flakes, μ″ (max) is about 1.6.

Figure 6.

High-frequency permeability of composites contained different sizes of Fe-Cu-Nb-Si-B flakes. (a) μ′ ∼ f spectra and (b) μ″ ∼ f spectra (copyright, 2015, AIP).

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 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:

μ s = 1 + 4 π M s H k + 4 π M s N h E3
f r = γ 2 π H k 2 + 4 π M s H k N + N h + 4 π M s 2 N N h E4
N = α r 2 α r 2 1 × 1 1 α r 2 1 × ars sin α r 2 1 α r E5
N h = 1 N 2 E6

where αr is the width/thickness ratio (often called “aspect ratio”) of a flake. The demagnetization factor along the direction of thickness and width is N 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 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′ ∼ f spectrum of large flakes drops more rapidly than small flakes. When f > 1.8 GHz, the real part (μ′(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 well-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′ ∼ f spectrum is observed.

N Nh (N + Nh) NNh
Small flakes 0.816 0.092 0.908 0.075
Large flakes 0.919 0.041 0.960 0.038

Table 1.

Demagnetization factors of two kinds of flakes.

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 μ ε tanh j 2 πfd c με E7
R . L . = 20 log Z in Z 0 Z in + Z 0 E8

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 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.

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).


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 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 low-temperature 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.

Figure 9.

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).

Figure 10.

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

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′ drops from 61.49 to 33.02 at 0.5 GHz, whereas ε″ 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 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 greatly suppressed in coated particles, which means that the high-resistivity coating ferrite effectively reduces the impact of eddy currents on μ values.

Figure 11.

The impact of ferrite coating on the permittivity and permeability of flakes. (a) ε′ ∼ f and (b) ε″ ∼ f. (c) μ′ ∼ f and (d) μ″ ∼ f (copyright, 2015, IEEE).

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.

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).


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 multi-peaks 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:

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).

M r i t t = γ M r i t × H eff r i t γα M s M r i t × M r i t × H eff r i t E9

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

ρ H a H b = 1 2 2 M H a H b E10

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.

Figure 14.

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

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(−109t), t in second, hour in A/m) is) perpendicular to “z” axis, the simulated intrinsic permeability spectra are shown in Figures 14b and 15b. 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:

Figure 15.

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

f = γ H + Nx Nz M s × H + Ny Nz M s E11

where H (the external DC magnetic field) is along the easy axis, demagnetization factor is N and γ′ is the gyromagnetic ratio (for Fe: 2.8 × 106 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).

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


6. Negative imaginary parts of permeability (μ″ < 0)

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

Figure 16.

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

d M dt = γ H eff × M + α Ms M s × d M dt u M + β Ms M × u M E12

where α (damping constant) is set as 0.01 for the simulations; γ is the Gilbert gyromagnetic ratio (2.21 × 105 mA−1 S−1); M is the magnetization; Heff is the effective magnetic field which consists of demagnetization field, exchange interaction, anisotropic field and external applied field; and finally, β (nonadiabatic spin transfer parameter) is set as 0.02 and is used to consider the impact of temperature on the dynamics of precession. The vector u is defined as

u = g μ B P 2 eM s J E13

where g is the Landé factor, e is the electron charge, μB is the Bohr magneton, J is the current density and P is the polarization ratio of current and set as 0.5. The simulation parameters for iron (Fe) nanowire are Ms = 17 × 105 A/m, magnetocrystalline anisotropy constant, K1 = 4.8 × 104 J/m3, exchange stiffness constant and A = 21 × 10−12 J/m. Also, the nanowire is discretized into many tetrahedron cells (cell size: 5 × 1.25 × 1.25 nm). The cell is smaller than the critical exchange length. To obtain high-frequency permeability spectra, these steps are followed: Firstly, the remanent magnetization state should be acquired after the external field is removed. Secondly, a pulse magnetic field is applied perpendicular to the long axis (x-axis) of the wire. The pulse magnetic field has the form h(t) = 1000 exp(−109t) (t in second, hour in A/m). A polarized current (density J is 3.0 × 1012 A/m2) is flowing along x-axis. Thirdly, the dynamic responses of magnetization in time-domain are recorded under these two excitations. Using the Fast Fourier Transform (FFT) technique, the dynamic responses in frequency domain are obtained. The high-frequency permeability spectra are calculated based on the definition of permeability:

μ f = 1 + m f h f = 1 + χ f " f E14

where m(f) is magnetization under both excitations and the pulse magnetic field (h) works as the perturbing field. The following relations exist: μ′ = 1 + χ′(f), μ″ = χ″(f). The differences in the high-frequency permeability under different excitations have been investigated.

When single nanowire is under the excitation of only an AC magnetic field (h), two Lorentzian-type resonance peaks are found in the permeability spectrum, as shown in Figure 17a. One is located approximately at 18 GHz, and the other is located approximately at 31.5 GHz. According to our previous studies, the major resonance is called “bulk mode,” which has larger magnetic loss and is manifested by the larger μ″ value, which is ascribed to the precession of perfectly aligned magnetic moments within the nanowire body excluding the misaligned magnetic moments at both ends of nanowire. The minor resonance peak located at 18 GHz is often called “edge mode,” whose smaller magnitudes of real and imaginary parts are due to smaller volume fractions of magnetic moments at the ends of nanowire. The edge mode is due to the precession and resonance of misaligned magnetic moments. Physically, if external field is not applied, the magnetizations are aligned along the local effective field (Heff). Once the perturbation field (h) is applied to excite the precession, M will move away from their equilibrium positions with small angles, and M will then precess around Heff. When precession goes on, the y-component of M (My) will have nonzero values, as shown in Figure 17b. During the period of precession, the energy absorbed from the AC magnetic field will be gradually dissipated via damping mechanisms. M will restore to its equilibrium positions, and My will be zero; see Figure 17b. The energy loss is manifested by a positive imaginary part of permeability (μ″ > 0). When the nanowire is excited under both the pulse magnetic field and polarized current, the high-frequency permeability spectrum is naturally found to be different and is illustrated in Figure 17c. The previous minor resonance becomes the major resonance with negative μ″ values. The “My” component does not vanish gradually; see Figure 17d. According to the STT effect, a spin-polarized current flows through the nanowire; STTs only act on the magnetic moments at the ends of nanowire. These STTs will counteract the torques due to effective magnetic field, which will then bring the magnetizations back to their equilibrium positions. When the STT is strong enough, it can maintain the precession angle (μ″ = 0) and even enlarge the precession angle (μ″ < 0) or switch the direction of magnetization. Additionally, negative μ″ values are found only for “edge mode” at lower frequency. This is because the STT effect only acts on magnetic moments with a spatially nonuniform orientation, which are only found at both ends of nanowires. The cause for minor peak becoming major peak under STT effect is due to the fact that the spin transfer torques are consistently acting on the magnetic moments; therefore, the precession angle is enlarging, and as a result, My component is also increased. From the perspective of total energy (Etotal) of magnetic moment ensemble, when no STT effect is involved, Etotal first increases due to the excitation of pulse magnetic field and then gradually attenuates due to energy loss via damping mechanisms. However, when STT effect exists, Etotal is not attenuated to a constant; see Figure 18.

Figure 17.

Permeability spectra and responses of My component without and with STT excitation (J ≠ 0) (copyright, 2018, Elsevier).

Figure 18.

The variation of total energy without and with STT excitation (copyright, 2018, Elsevier).

Finally, it should be pointed out that although others have also reported negative μ″ in other materials, there are no convincing physical mechanisms that have been provided to support their results (μ″ < 0), and negative μ″ is mostly likely due to measurement errors.


7. Conclusions

In this chapter, we have shown several approaches to tailor the high-frequency permeability of ferromagnetic materials: shapes, particle size distribution, heat treatments, coating and spin transfer torque effect. Micromagnetic simulations are used to explain the origins of multi-peaks in permeability spectra of nanowire array. In addition, just like the simultaneous excitation method (pulse H field + STT effect) shown here, we want to propose the other possibilities (pulse H field + the other excitation) to tailor μ ∼ f spectra and to realize negative imaginary parts of permeability (μ″ < 0), such as femtosecond laser pulses or photomagnetic pulses.



This work is supported by the National Science Foundation of China (Grant No. 61271039) and the International Scientific Collaboration of Sichuan Province (Grant No. 2015HH0016).


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Written By

Mangui Han

Submitted: 14 July 2018 Reviewed: 18 April 2019 Published: 16 May 2019