Demagnetization factors of two kinds of flakes.
Abstract
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.
Keywords
- 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 (
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.
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
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
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.
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” (
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:
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
N⊥ | Nh | (N⊥ + Nh) | N⊥Nh | |
---|---|---|---|---|
Small flakes | 0.816 | 0.092 | 0.908 | 0.075 |
Large flakes | 0.919 | 0.041 | 0.960 | 0.038 |
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:
where “
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.
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
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
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:
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 (
Usually, the FORC distribution is illustrated in the diagram of (
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 “
where
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.,
To better understand the changes of intrinsic
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 (
where
where
where
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
Finally, it should be pointed out that although others have also reported negative
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
Acknowledgments
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).
References
- 1.
Snoek JL. Dispersion and absorption in magnetic ferrites at frequencies above one mc/s. Physica. 1948; 14 :207. DOI: 10.1016/0031-8914(48)90038-X - 2.
Han M, Deng L. Mössbauer studies on the shape effect of Fe84.94Si9.68Al5.38 particles on their microwave permeability. Chinese Physics B. 2013; 22 (8):083303. DOI: 10.1088/1674-1056/22/8/083303 - 3.
Acher O, Adenot AL. Bounds on the dynamic properties of magnetic materials. Physical Review B. 2000; 62 :11324. DOI: 10.1103/PhysRevB.62.11324 - 4.
McHenry ME, Willard MA, Laughlin DE. Amorphous and nanocrystalline materials for applications as soft magnets. Progress in Materials Science. 1999; 44 :291. DOI: 10.1016/S0079-6425(99)00002-X - 5.
Han M, Guo W, Wu Y, et al. Electromagnetic wave absorbing properties and hyperfine interactions of Fe–Cu–Nb–Si–B nanocomposites. Chinese Physics B. 2014; 23 (8):083301. DOI: 10.1088/1674-1056/23/8/083301 - 6.
Wu Y, Han M, Liu T, Deng L. Studies on the microwave permittivity and electromagnetic wave absorption properties of Fe-based nano-composite flakes in different sizes. Journal of Applied Physics. 2015; 118 :023902. DOI: 10.1063/1.4926553 - 7.
Wu Y, Han M, Deng L. Enhancing the microwave absorption properties of Fe-Cu-Nb-Si-B nanocomposite flakes by coating with spinel ferrite NiFe2O4. IEEE Transactions on Magnetics. 2015; 51 (11):2802204. DOI: 10.1109/TMAG.2015.2444656 - 8.
Lagarkov AN, Rozanov KN. High-frequency behavior of magnetic composites. Journal of Magnetism and Magnetic Materials. 2009; 321 :2082. DOI: 10.1016/j.jmmm.2008.08.099 - 9.
Deng D, Han M. Discussing the high frequency intrinsic permeability of nanostructures using first order reversal curves. Nanotechnology. 2018; 29 :445705. DOI: 10.1088/1361-6528/aad9c2 - 10.
Gerardin O, Le Gall H, Donahue MJ, Vukadinovic N. Micromagnetic calculation of the high frequency dynamics of nano-size rectangular ferromagnetic stripes. Journal of Applied Physics. 2001; 89 :7012. DOI: 10.1063/1.1360390 - 11.
Han XH, Liu RL, Liu QF, Wang JB, Wang T, Li FS. Micromagnetic simulation of the magnetic spectrum of two magnetostatic coupled ferromagnetic stripes. Physica B. 2010; 405 :1172-1175. DOI: 10.1016/j.physb.2009.11.030 - 12.
Han M, Zhou W. Unusual negative permeability of single magnetic nanowire excited by the spin transfer torque effect. Journal of Magnetism and Magnetic Materials. 2018; 457 :52-56. DOI: 10.1016/j.jmmm.2018.02.031 - 13.
Thiaville A, Nakatani Y, Miltat J, Suzuki Y. Micromagnetic understanding of current-driven domain wall motion in patterned nanowires. Europhysics Letters. 2005; 69 :990. DOI: 10.1209/epl/i2004-10452-6 - 14.
Yan M, Kakay A, Gliga S, Hertel R. Beating the walker limit with massless domain walls in cylindrical nanowires. Physical Review Letters. 2010; 104 :057201. DOI: 10.1103/PhysRevLett.104.057201