High-entropy spinel ferrites MFe2O4 (M = Mg, Mn, Fe, Co, Ni, Cu, Zn) with tunable electromagnetic properties and strong microwave absorption

Abstract: Ferrites are the most widely used microwave absorbing materials to deal with the threat of electromagnetic (EM) pollution. However, the lack of sufficient dielectric loss capacity is the main challenge that limits their applications. To cope with this challenge, three high-entropy (HE) spinel-type ferrite ceramics including (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄, (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄, and (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄ were designed and successfully prepared through solid state synthesis. The results show that all three HE MFe₂O₄ samples exhibit synergetic dielectric loss and magnetic loss. The good magnetic loss ability is due to the presence of magnetic components; while the enhanced dielectric properties are attributed to nano-domain, hopping mechanism of resonance effect and HE effect. Among three HE spinels, (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ shows the best EM wave absorption performance, e.g., its minimum reflection loss (RL_min) reaches −35.10 dB at 6.78 GHz with a thickness of 3.5 mm, and the optimized effective absorption bandwidth (EAB) is 7.48 GHz from 8.48 to 15.96 GHz at the thickness of 2.4 mm. Due to the easy preparation and strong EM dissipation ability, HE MFe₂O₄ are promising as a new type of EM absorption materials.

Keywords: high-entropy ceramics (HECs); spinel-type ferrite; electromagnetic (EM) wave absorption; dielectric loss; magnetic loss

1 Introduction

With the rapid development of modern science and technology, more and more electromagnetic (EM) devices appear in people’s lives. However, every coin has double sides, namely, it may have devastating effects to human health and the environment to a certain extent along with convenience brought to the daily life ^[1–3]. Therefore, preventing and controlling of EM pollution have already become a crucial task ^[4,5]. Microwave absorbing materials are a kind of materials which can absorb the energy of EM waves projected on their surfaces, and convert them into other forms of energy (mainly thermal energy) through the dielectric and magnetic losses with almost zero reflection ^[6–10]. Hence, microwave absorbing materials are expected to regard as a sharp facility to address EM pollution issues ^[11–13]. In general, EM absorbing materials need to meet the following requirements: (1) thin thickness; (2) light weight and excellent chemical stability; (3) wide effective absorption bandwidth (EAB) and EM impedance match; (4) strong absorbing capability ^[14,15]. 

Ferrites (such as FeO·Fe₂O₃ and BaO·6Fe₂O₃) have the advantages of high permeability, high resistivity, and good impedance matching performance, which mainly dissipate EM waves through self-polarization, hysteresis loss, domain wall resonance, and natural resonance ^[16–20]. Moreover, compared with other microwave absorbing materials, ferrites are the earliest and most widely used microwave absorbing materials because of their facile preparation and low cost ^[6,21]. However, ferrites as microwave absorbing materials still have many shortcomings, such as insufficient dielectric loss capacity, which results in the narrow EAB ^[22]. To improve the dielectric properties, an effective strategy is to fabricate nano-ferrite; however, it needs complex manufacturing process and precise composition control ^[21]. These problems hinder the wide spread application of ferrites. 

“High-entropy (HE)” is a new material design paradigm, which provides vast space for composition design and property tuning ^[5,23–29]. High-entropy ceramics (HECs) are solid solutions formed by multi-principal elements in equal or near equal atomic ratios. HECs have broad prospective applications due to their good stability and unique properties, such as adjustable thermal expansion coefficient, high hardness, good stability, and corrosion resistance ^[23]. In particular, HECs show adjustable EM characteristics. For oxides, disorder cations in HECs cause valence compensation, which increases the defect concentration, narrows the band gap, and enhances the ionic conductivity ^[30]. Meanwhile, the sublattices of HECs are randomly occupied by a variety of components. Through composition design, the defect concentration, disorder degree, and energy band structure can be controlled, such that the dielectric properties can be tuned ^[31]. In addition, by adjusting the proportion of magnetic ions, the exchange coupling at the ferromagnetic interface can be optimized, and the resonance intensity can be adjusted ^[32]. Besides, recent studies have shown that HE ceramics, such as HE rare earth hexaborides, HE carbides, and HE rare earth silicide ceramics present enhanced EM properties, and exhibit excellent microwave absorbing properties ^[5,33–38]. Therefore, introducing HE state into ferrites is expected to solve the problem of insufficient dielectric properties of ferrite-based EM absorbing materials.

Since the combination of dielectric and magnetic losses can be realized using the HE design paradigm ^[34,36], this study aims to use the design method of HE to enhance the EM properties of ferrites so as to realize the purpose of simple, high-efficiency, and large-scale preparation of ferrites with high absorbing performance. To achieve such a goal, three HE spinel-type ferrites (HE MFe₂O₄, M = Mg, Mn, Fe, Co, Ni, Cu, Zn) ceramics including (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2) Fe₂O₄ , (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2 )Fe 2O4 , and (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄ were designed and prepared by a one-step solid-state reaction method. The specific design criteria of HE MFe₂O₄ are as follows: (1) The difference in ion radius (Table 1) is within 15% to ensure the easy formation of solid solution; (2) each oxide can react with ferrum oxide to form a single phase spinel ferrite; (3) metal ions should be magnetic ions or exhibit magnetic structure in spinel ferrite. To demonstrate the composition–microstructure–property relationship, the phase composition, microstructure, magnetic properties, and EM wave loss abilities of HE MFe₂O₄ were investigated systematically. 

Table 1 Cation radii for the design of HE MFe₂O₄ (M = Mg, Mn, Fe, Co, Ni, Cu, Zn); the data are obtained from the list of the revised effective ionic radii ^[39] 

2 Experimental 

2. 1 Synthesis of HE MFe₂O₄

Three HE MFe₂O₄ ceramics including (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄, (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄, and (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄ were prepared by a solid state reaction method. MgO, MnO, FeO, CoO, NiO, CuO, and ZnO powders (99.9% purity, 0.5 μm, HWRK Chem Co., Ltd., Beijing, China) were used as start materials. The solid-state reaction for the synthesis of HE MFe₂O₄ can be described by Reaction (1): 

4MO + 8FeO + 2O₂ = 4MFe₂O₄ (1) 
(M = Mg, Mn, Fe, Co, Ni, Cu, Zn)

Firstly, the metal oxide powders were blended with ethanol and ball-milled with agate grinding balls for 4 h. And then the mixture was dried at 60 ℃ for 12 h. After that, the mixed powders were screened through a 200-mesh screen and dry pressed into a cylinder green body using a 30 mm diameter stainless-steel die. Eventually, the green bodies were heated to 1150 ℃ for 5 h in air to obtain HE MFe₂O₄.

2. 2 Characterization 

The crystal structure and phase composition of the HE MFe₂O₄ were analyzed by X-ray diffraction (XRD) using an X-ray diffractometer (D8 Advance, Bruker, Germany) with a step size of 0.02° at a scan rate of 2 (°)/min. The lattice constants and crystallite sizes were obtained by the Total Pattern Solution Software (TOPAS, Bruker Corp., Karlsruhe, Germany) using whole pattern fitting (WPF) XRD refinement method. To investigate the chemical state, the X-ray photoelectron spectroscopy (XPS) was obtained using Thermo Scientific K-Alpha (Thermo, USA). The microstructure of HE MFe₂O₄ was characterized by the scanning electron microscope (SEM, Apollo 300, CamScan, UK) coupled with the energy-dispersive spectroscope (EDS Inca X MAX 80T, Oxford, UK). Transmission electron microscopy (TEM) and EDS were conducted on the FEI Talos F200S instrument (FEI Talos F200S, Hillsboro, USA) at an accelerating voltage of 200 kV. The room temperature magnetic properties were investigated by the vibrating sample magnetometer (VSM, Lakeshore 7400, Columbus, USA). 

The dielectric and magnetic parameters were measured by using an Agilent N5224A vector network analyzer (VNA) at the range of 1–18 GHz. By measuring the scattering parameters (S parameters: S11 symbolizes the tested reflection coefficient data, and S21 stands for the transmission data) of the samples, the data were automatically converted into the relative complex permittivity (εr = ε′ + jε″, where ε′ and ε″ are the real and imaginary parts of permittivity, respectively) and permeability (μr = μ′ + jμ″, where μ′ and μ″ are the real and imaginary parts of permeability, respectively). The EM absorption properties of HE MFe₂O₄ were evaluated with the reflection loss (RL), which was determined by Eqs. (2) and (3) based on the transmission line theory and the metal backplane model ^[40]: 

where Z₀ is the impedance of free space, Z_in is the input impedance, μ_r is the relative complex permeability, ε_r is the relative complex permittivity, f is the frequency of the EM wave, d is the absorber thickness, and c is the light velocity (3 × 10⁸ m/s). It can be seen from Eqs. (2) and (3) that the EM wave absorption properties are related to EM parameters. When the RL is less than −10 dB, more than 90% of the EM wave is absorbed, and the frequency range with the RL below −10 dB is defined as the EAB ^[7]. 

The samples for the measurement of EM parameters were prepared as follows. After grinding, the HE MFe₂O₄ powders were mixed with paraffin as a binder at the mass ratio of 7 : 3. The composites were miscible at about 70 ℃, and then the molten mixtures were filled into the coaxial ring mold, and subsequently compacted into toroidal-shaped samples (external diameter = 7.00 mm; inner diameter = 3.04 mm). 

3 Results and discussion 

3. 1 Phase identification

Figure 1 shows the XRD patterns of the three HE MFe₂O₄ and those of MgFe₂O₄, MnFe₂O₄, Fe₃O₄ (FeFe₂O₄), CoFe₂O₄, NiFe₂O₄, CuFe₂O₄, and ZnFe₂O₄ obtained from ICDD/JCPDS cards. It can be clearly seen that all HE MFe₂O₄ samples present a single-phase MgAl₂O₄ type spinel structure with a Fd3^▔m (227) space group. The positions of the diffraction peaks of the three HE MFe₂O₄ vary with the composition. Compared with the other two HE MFe₂O₄, the diffraction peaks of (Mg_0.2Mn_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄ shift slightly to lower angle directions, which implies that (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ has larger interplanar spacings and lattice constant. The lattice constants and theoretical densities of HE MFe₂O₄ and several typical spinel ferrites are listed in Table 2. It can be noticed that the lattice constants of the three HE MFe₂O₄ samples are only larger than that of NiFe₂O₄ but smaller than those of other spinels ^[23,25]. 

Fig. 1 XRD patterns of the three HE MFe₂O₄ together with those of MgFe₂O₄, MnFe₂O₄, Fe₃O₄, CoFe₂O₄, NiFe₂O₄, CuFe₂O₄, and ZnFe₂O₄ obtained from ICDD/JCPDS cards.

Table 2 Lattice constants and theoretical densities of HE MFe₂O₄ and those of several typical spinel ferrites

As presented in Fig. 2, the chemical state and bonding of HE MFe₂O₄ samples have been explored by XPS. XPS shows the coexistence of characteristic peaks of C 1s, O 1s, Mg 1s, Mn 2p, Fe 2p, Co 2p, Ni 2p, Cu 2p, and Zn 2p (Fig. 2(a)). In the spinel structure, it is a common phenomenon that some Fe3+ that should be at the [FeO₆] octahedron will occupy the tetrahedral site, and M²⁺ will occupy the vacant octahedral site. This phenomenon is the intrinsic defect of cation disorder in spinel, and the highly disordered cation distribution has a great influence on the dielectric properties of spinel ^[41]. This defect can be described by Reactions (4) and (5): 

M^X_V + V^'''_Fe → M^'_Fe + V^''_M (4) 
Fe^X_V  + V^''_M→ Fe^•_M + V^'''_Fe (5) 

The XPS results of Fe element in Fig. 2(b) confirm the existence of this phenomenon in HE MFe₂O₄ samples. The only peak related to Fe²⁺ is located at 710.6 eV, which reveals that all of the Fe²⁺ ions are in the octahedron site. The Fe³⁺ ions in the octahedral site and tetrahedral site correspond to the distinguished peaks at 712.1 and 714.6 eV, respectively ^[42]. The area ratios of n(Fe³⁺(tet)) : n(Fe³⁺(oct)) of (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄, (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄, and (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄ are 1 : 2.06, 1 : 1.90, and 1 : 2.34, respectively, being different from the normal value of the spinel structured ferrites, which will enhance the magnetism of the HE materials ^[43,44]. Meanwhile, the XPS results reveal that the ratio of Fe³⁺ : M²⁺ in HE MFe₂O₄ is less than the stoichiometric ratio of spinel, which indicates that there are more defects in the HE crystal. Reaction (6) written below takes into account charge-compensating defects of Schottky anion vacancies: 

3MO → 2M'_Fe +M^X_M +3O^X_O + V_O^‥  (6)

The conductivity of ferrites is attributed to the hopping of electrons from M²⁺ and Fe³⁺ ions ^[45]. The increased defects and oxygen vacancies in HE MFe₂O₄ improve the carrier mobility, thereby enhancing the ionic conductivity and dielectric loss capacity. 

Fig. 2 (a) XPS survey and (b) Fe 2p XPS results of HE MFe₂O₄.

Figure 3 shows the crystal structure and computer-simulated equilibrium morphology of HE MFe₂O₄. Based on Fig. 4, it can be found that the (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄, (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄, and (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄ all present octahedral morphology with slightly different orientation. Their average grain sizes are 2.91, 1.91, and 2.07 μm, respectively. The triangle grain on the octahedral is the growth step of {111} face of cubic spinel. The stability energy of crystal growth unit (U) greatly influences the grain morphology, which can be calculated by Eq. (7) ^[46]: 

where N is the Avogadro constant, e is the electron charge, n is the Born exponent, m is the total number of the ions in 1 mol growth units, Z_i and Z_j represent the electrovalence of ions i and j, respectively, and R_ij is the distance between ions i and j. In spinel, different U will affect the stacking mode of growth units and make it present different morphologies. When the growth rate of {200} faces is the fastest, the {111} faces will determine the crystallite shape, which is octahedral four-sided pyramids. Meanwhile, if the preferred orientation is , the relative growth rates on {200} and {111} faces (V₍₂₀₀₎/V₍₁₁₁₎) will decrease, and the grain shape changes from octahedron to truncated pyramid, i.e., its morphology appears as a triangle plane ^[47]. This explains why the grains of (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ are mostly octahedral, and those of (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄ are mostly plane triangles. 

Fig. 3 (a) Crystal structure with marked (111) plane and (b–e) computer-simulated morphologies of spinel ferrites grown with different growth rates on {111} and {200} faces viewed in the [100] direction. 

Fig. 4 SEM images and average grain sizes of (a, d) (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄, (b, e) (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄,and (c, f) (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄. Ideally, in a spinel structure, Fe³⁺ forms [FeO₆] octahedron with the nearest six O²⁻, and M²⁺ forms [MO₄] tetrahedron with the nearest four O²⁻. Since the [FeO₆] octahedron and [MO₄] tetrahedron overlap orderly in [111] direction, the crystallites usually present three-dimensional equal length morphology, such as regular octahedron ^[48]. 

The microstructure of (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ was further investigated by means of TEM. The high-resolution transmission electron microscopy (HRTEM) image shown in Fig. 5(b) reveals the existence of nano-domains within the (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ grain (Fig. 5(a)), which proves that the grain is composed of multiple nano-domains. The existence of nano-domains will greatly enhance the interfacial polarization effect under irradiating by EM fields. The EDS mappings in Fig. 5(c) reveal that the elements distribute homogeneously in the (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ grain. 

Fig. 5 (a) TEM image, (b) HRTEM image, and (c) scanning transmission electron microscopy (STEM) image and EDS mappings of (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄. 

3. 2 Magnetic properties

The hysteresis loops of the HE MFe₂O₄ are shown in Fig. 6. It can be clearly seen that all HE MFe₂O₄ samples are soft magnetic materials. The saturation magnetization and coercivity can be determined from the hysteresis loops, which are shown in Table 3. (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ has the highest saturation magnetization, whereas (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄ has the strongest coercivity. Both the coercivity and saturation magnetization of (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2) Fe₂O₄ are the lowest. 

Fig. 6 Hysteresis loops of HE MFe₂O₄ samples. 

Table 3 Saturation magnetization and coercivity of HE MFe₂O₄ and several typical spinel ferrites ^[49–53] 

The magnetic properties of HE MFe₂O₄ are originated from the super exchange between tetrahedral site and octahedron site of Fe group ions. According to the Hund’s first rule, the 3d electrons in the ions of Mn²⁺, Fe²⁺, Fe³⁺, Co²⁺, Ni²⁺, and other Fe group elements will be filled into different orbits as much as possible, which makes them have larger electron spin magnetic moment, and therefore showing stronger magnetism ^[54]. Because (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ contains most of these ions, it shows the strongest saturation magnetization. The magnetic properties of spinel ferrites are also related to the size distribution of Fe group ions. The ferromagnetism of ferrites is mainly due to the super-exchange interaction of Fe group elements between tetrahedral site and octahedron site ions. Because the 3d electrons of Mn²⁺, Fe²⁺, Fe³⁺, Co²⁺, and Ni²⁺ are more than 5 and less than 10, and have negative exchange integral with O²⁻, HE MFe₂O₄ samples show ferrimagnetism ^[55,56]. Although Mg²⁺ has no 3d electrons, it tends to occupy the tetrahedral site, which makes Fe³⁺ exist at both tetrahedral site and octahedron site, making them also exhibit ferrimagnetism ^[57]. Besides, the double-exchange interaction between Fe cations with different valence states also has a positive effect on the ferromagnetism of ferrites, but it is weaker than the above two factors ^[58–60]. 

Coercivity is the magnetic field intensity that must be applied to reduce the induced magnetic intensity of the magnetized ferromagnet to zero. Materials with higher coercivity are ideal for high density magnetic recording ^[61,62]. Generally speaking, the higher the magnetic saturation strength of the same type of ferrite, the greater the coercivity is. However, (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄ with the middle saturation magnetization has the strongest coercivity. This is because the coercivity is strongly affected by the change of grain size. The relationship between coercivity (H_c) and average grain size (D) is as follows ^[63]: 

H_c = C/D  (8) 

where C is a constant related to the material. As can be seen in Fig. 4, (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄ has the smallest grain size, which explains why it has the strongest coercivity. 

3. 3 EM wave absorption properties 

The real (ε′, μ′) and imaginary (ε″, μ″) parts of permittivity and permeability for HE MFe₂O₄ are illustrated in Fig. 7. According to Eqs. (2) and (3), the microwave absorbing properties are related to the complex permittivity, permeability, and thickness of a material. The real parts of permittivity (ε′) and permeability (μ′) symbolize the EM storage capacity, while the imaginary parts of permittivity (ε″) and permeability (μ″) present the dielectric and magnetic energy dissipation ability of the material, respectively. As far as HE MFe₂O₄ is concerned, a highly similar monotonically descending trend of the real part of dielectric permittivity (ε′) versus frequency is evident. This trend at low frequencies (1–10 GHz) indicates the dispersion phenomenon; while at high frequencies (10–18 GHz), it is associated with orientation polarization of intrinsic electric dipoles, space charge polarization, interfacial polarization, and the corresponding dielectric relaxation effects ^[64]. The real part of permittivity (ε′) of (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ is higher than that of the other two HE MFe₂O₄ samples in the frequency range from 1 to 18 GHz, which is attributed to the effect of Cu²⁺ and Zn²⁺ ions on the hopping mechanism of ferrites ^[65–67]. The imaginary part of permittivity (ε″) of HE MFe₂O₄ is displayed in Fig. 7(c), and the enhanced imaginary part is attributed to the enhancement of ionic conductivity as we have mentioned in Section 3.1. The dielectric loss reduces significantly for three HE MFe₂O₄ samples at 16–18 GHz. This phenomenon is related to the resonance frequency of electrons and external EM field. When the jumping frequency of electrons between tetrahedral site and octahedron site ions of ferrites becomes approximately equal to the frequency of the applied field, this phenomenon can be observed ^[68]. This situation has been found in many previous works in the low frequency region, and it is attributed to the substitution of Fe group elements by other ions (such as Mg²⁺) at octahedron site changing the resonance frequency of ferrites ^[69–73]. Due to the existence of a large number of defects in HE ceramics and the interaction between the primary components, the dielectric constant of HE ceramics changes abnormally, which explains the above phenomenon that appears in the high frequency region ^[31]. 

Figures 7(b) and 7(d) suggest that all HE MFe₂O₄ samples show similar trend of permeability (μ′ and μ″), which is due to the natural resonance and natural exchange resonance of ferrites ^{[16,19,21,74]}. The performance of natural resonance is that the real part of permeability decreases rapidly, while the imaginary part of permeability has an obvious oscillation peak ^[75,76]. The different magnetic saturation strength and coercivity of HE MFe₂O₄ with various compositions are helpful to understand the composition-dependent permeability among samples. From Fig. 7(d), it can be seen that the natural resonance peak of (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ with the strongest saturation magnetization appears at 4–12 GHz, whilst that of (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄ with the weakest coercivity is located at 6–14 GHz, and the imaginary part of permeability of the former is stronger. Although there is no obvious peak in the 1–18 GHz band for (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄, the values of μ″ increase slightly in the frequency band of 10–16 GHz, which indicates that the natural resonance phenomenon is located in this band and is not prominent. 

Fig. 7 EM parameters of HE MFe2O4: (a) real part of permittivity (ε′), (b) real part of permeability (μ′), (c) imaginary part of permittivity (ε″), and (d) imaginary part of permeability (μ″). 

The dielectric tangent loss (ε″/ε′) and the magnetic tangent loss (μ″/μ′) represent the EM loss ability of a material. These values of HE MFe₂O₄ are shown in Figs. 8(a) and 8(c). Interestingly, the enhanced dielectric loss tangent by the HE design decreases rapidly at 16–18 GHz, and the value of magnetic loss tangent increases rapidly at this band, which suggests that they can complement each other. The different trend in the change of dielectric and magnetic losses provides good whole-band matching for the HE MFe₂O₄ samples, indicating that the HE design strategy is effective to realize broadband absorption. 

Fig. 8 (a) Frequency dependences of dielectric tangent loss (ε″/ε′), (b) Cole–Cole semicircles, (c) magnetic tangent loss (μ″/μ′), and (d) C0–f curves of HE MFe₂O₄. 

Through the above analysis, it can be concluded that the dielectric loss of HE MFe₂O₄ is mainly caused by the interfacial polarization of nano-domains and the resonance absorption between tetrahedral site and octahedron site ions. The polarization mechanism mainly includes dipole orientation polarization and interfacial polarization, which can be explained by the Debye theory as follows ^[77]: 

where ε_s is the static dielectric constant and ε_∞ is the dielectric constant at infinite frequency. Therefore, when the relationship between ε′ and ε″ presents a semicircular ring, it confirms the existence of Debye relaxation process ^[78]. Intriguingly, all three HE MFe₂O₄ samples show the obvious Cole–Cole semicircle in Fig. 8(b), which reflects the polarization relaxations of HE MFe₂O₄. 

The magnetic tangent loss is similar to the imaginary part of permeability (μ′), which reveals the natural resonance of the materials. In addition, another possible reason that may significantly affect the magnetic loss in GHz range is the eddy current loss, which can be given by Eq. (10) ^[79]: 

where μ₀ is the permeability in vacuum and σ is the electrical conductivity. If the C₀ curve is a horizontal line regardless of frequency, the eddy current effect is the main loss mechanism ^[77]. Apparently, the C₀ curve in Fig. 8(d) is frequency dependent in 1–4.4 GHz, but the variation range of C₀ in high frequency region is very small with frequency. Therefore, it can be concluded that both the eddy current loss and the natural resonance loss contribute to the magnetic loss. 

Due to the presence of multiple loss factors, it is necessary to use an index to measure the comprehensive loss ability. Attenuation constant (α) is an important parameter to measure the comprehensive ability of EM loss of a material. This parameter can be evaluated by Eq. (11) ^[80]: 

The complex permittivity and permeability have great influences on the total loss ability of a material. Figure 9(a) exhibits the attenuation constant (α) of HE MFe₂O₄ samples. The α curves of all HE MFe₂O₄ samples remain steadily in the range of 1–10 GHz, which shows that under the synergistic effect of stable dielectric loss, the violent oscillation of magnetic loss does not have a negative impact on the loss capacity of the samples. Meanwhile, the attenuation constant peaks of (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ and (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄ coincide with the position of natural resonance peak, indicating that the magnetic loss caused by natural resonance has an important contribution to the loss capacity of these two HE MFe₂O₄ at this frequency. 

Fig. 9 (a) Comparison of attenuation constant (α) and (b) impedance match (Z_in/Z₀) of HE MFe₂O₄ samples at 2.2 mm. 

The better impedance matching can ensure that the microwave can enter into the microwave absorber. To evaluate the impedance matching, the characteristic impedance values (Z = Z_in/Z₀) of HE MFe₂O₄ samples are calculated and shown in Fig. 9(b). It can be found that the Z_in/Z₀ for all HE MFe₂O₄ samples are close to 1 in the frequency range of 7.8–18 GHz, showing the best input impedance matching. 

Based on the foregoing analysis and discussion, the complementary dielectric and magnetic loss abilities will render the materials with good absorbing properties in the whole band. The RL is considered as an index to assess the EM wave absorption properties,i.e., when the RL value is less than −10.0 dB, the corresponding frequency range is defined as the EAB. As shown in Fig. 10(a), the minimum reflection loss(RL_min) of (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ reaches −35.10 dB at 6.78 GHz with a thickness of 3.5 mm, and the optimized EAB is 7.48 GHz from 8.48 to 15.96 GHz with a thickness of 2.4 mm. Figure 10(b) shows that the largest EAB of (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄ reaches 6.12 GHz at the thickness of 2.2 mm from 10.86 to 16.98 GHz, and the RL_min = −26.11 dB at 14.26 GHz when the thickness is 2.1 mm. Figure 10(c) shows that the RLmin of (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄ is −27.4 dB at 11.88 GHz with the thickness of 2.5 mm, and the maximum EAB is 6.46 GHz at the thickness of 2.2 mm from 10.86 to 17.32 GHz. Besides, when the thickness of (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ is 2.2 mm, its EAB can still reach 6.8 GHz from 10.18 to 16.98 GHz, which suggests that this HE MFe₂O₄ has the best EM absorbing properties. It should be noted that the RL_min values of all three HE MFe₂O₄ samples are located at their natural resonance peaks, which validates that magnetic loss is the dominant factor of EM loss. In addition, the RL_min and EAB shift to higher frequencies when the thickness decreases, which is ascribed to the destructive interference caused by the quarter-wavelength attenuation as explained by Eq. (12) ^[81]: 

where f_m is the peak frequency of the RL_min. 

Fig. 10 3D representations of RL values and 2D contour plots of (a) (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄, (b) (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄, and (c) (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄. 

Figure 11 summarizes the EM wave absorption properties of ferrite-based ^{[9,16,20,82–89]}, oxide-based ^[3,15,90], carbide-based [5,91], and silicide-based materials ^[92], and some HE ceramics ^[5,33,36,38]. The materials in the upper left region have a wide EAB while maintaining a thin thickness, which can be considered as thin thickness, light weight, wide bandwidth, and strong absorption materials for modern electronic and communication devices. It is worth noting that HE MFe₂O₄ samples synthesized in this work are located in the upper left region, which demonstrates that the microwave absorbing properties of single-phase HE MFe₂O₄ synthesized in this work are better than those of nano-ferrites and ferrite-based composites.

Fig. 11 EAB and corresponding thickness of HE MFe₂O₄ samples and various kinds of reported EM absorbing materials: SnO₂@Ni ^[3], HfC ^[5], (Ti_0.2Zr_0.2Hf_0.2Nb_0.2Ta_0.2)C ^[5], ZnO/Co [15], Graphite/CoZnFe2O4 [16], CF@ CoFe₂O₄@MnO ^[20], (Tm_0.2Y_0.2Pr_0.2Gd_0.2Tb_0.2)₃Si₂C₂/ (Tm_0.2Y_0.2Pr_0.2Gd_0.2Tb_0.2)₂O₃ ^[33], (Y_0.2Nd_0.2Eu_0.2E_r0.2Yb_0.2) B₆/(Y_0.2Nd_0.2Eu_0.2Er_0.2Yb_0.2)B₄ ^[36], (MnNiCuZn)_xCo_1-xFe₂O₄/Graphene ^[38], Ni_0.5Zn_0.5Nb_0.04Fe_1.96O₄ ^[82], Sr_0.9Ba_0.1Fe₁₂O₁₉ ^[83], Graphene/Fe₃O₄ ^[84], NiFe₂O₄ ^[85], Ni_0.5Zn_0.5Fe₂O₄ ^[86], Li_0.4Mg_0.6Fe₂O₄/TiO₂ ^[87], Graphene@Fe₃O₄@WO₃@PAN ^[88], Fe₃O₄ ^[89], MnO₂ ^[90], SiC foam ^[91], Gd₅Si₄ ^[92]. 

4 Conclusions 

In summary, three HE MFe2O4 ceramics including (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄, (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄, and (Mg_0.2Fe_0.2Co_0.2Ni_0.2Zn_0.2)Fe₂O₄ were successfully synthesized by a solid-state reaction method at 1150 ℃ with a holding time of 5 h. The single-phase HE spinel-type ferrites were formed for all three compositions based on XRD results. XPS results show that the HE design improves the ionic conductivity of the HE MFe₂O₄ by increasing the defect and cation disorder. These three HE MFe₂O₄ samples show octahedron morphology with average grain sizes of 2.91, 1.91, and 2.07 μm. In addition, nano-domains are observed within the HE MFe₂O₄ grain. The HE MFe₂O₄ ceramics not only show improved dielectric properties, but also maintain good magnetic properties. Magnetic property measurement shows that (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ has the strongest saturation magnetization of 56.10 emu/g, while (Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄ with the smallest grain size has the strongest coercivity of 204.52 Oe. Excellent EM absorbing properties depend on the coordination between dielectric loss and magnetic loss. The RL_min value of (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ reaches –35.10 dB at 6.78 GHz at the thickness of 3.5 mm, and the optimized EAB is 7.48 GHz from 8.48 to 15.96 GHz at the thickness of 2.4 mm. With the thickness of (Mg_0.2Mn_0.2Fe_0.2Co_0.2Ni_0.2)Fe₂O₄ from 1.5 to 5.5 mm, the effective absorbing coverage frequency ranges from 3.48 to 18 GHz. Considering the convenient manufacture of HE MFe₂O₄ ceramics as well as their low density, low cost, and adjustable EM properties, the current research suggests that the HE design paradigm is powerful for the design of thin and wide-band EM wave absorbing materials.

References: Omitted