Dynamical magnetic behavior of anisotropic spinel-structured ferrite for GHz technologies

We have fabricated a high quality magnetic Ni0.5Zn0.5Fe2O4 ferrite powder/polymer composite sheet consisting of common and environmentally friendly elements only. The sheet was then tested for its dynamic permeability by irradiating with electromagnetic waves with frequencies up to 50 GHz. Two different originally developed methods were used for the high-frequency permeability measurements, a short-circuited microstrip line method and a microstrip line-probe method. It is challenging to measure the dynamic permeability of magnetic thin films/sheets beyond 10 GHz because of the low response signal from these materials. However, the two methods produced essentially equivalent results. In the frequency dependent permeability profile, the maximum position of the profile, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu ^{\prime \prime }_{max}$$\end{document}μmax″, shifted towards higher frequencies upon increasing an applied (strong) static external magnetic field, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{dc}$$\end{document}Hdc. A linear relationship between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu^ {\prime \prime }_{max}$$\end{document}μmax″ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{dc}$$\end{document}Hdc for the entire range of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{dc}$$\end{document}Hdc was observed even at small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{dc}$$\end{document}Hdc. In general, the spinel-structured Ni-based ferrites exhibit low magnetic anisotropy, but the present sample showed a uniaxial-anisotropic behavior in the parallel direction of the sheet. Our Ni0.5Zn0.5Fe2O4 powder/polymer composite sheet thus exhibits high performance at GHz frequencies, and should be applicable e.g. as an anisotropic electromagnetic wave-interference material.

In this study, we evaluate the dynamic permeability behavior of ferrites under the irradiation of an electromagnetic wave of GHz frequency through two different originally developed measurement techniques, (i) shortcircuited MSL 10,15,19 , and (ii) MSL-probe 11,20 methods. Our aim is to provide a comprehensive understanding of the dynamical magnetic behaviors of ferrites in the GHz region. For both measurement methods, the same sample, i.e., Ni 0.5 Zn 0.5 Fe 2 O 4 powder/polymer composite sheet from the same sample batch, was used. We used polyvinyl alcohol (PVA) as a polymer source. It should be noted that the spinel-structured Ni 0.5 Zn 0.5 Fe 2 O 4 is widely commercialized because of its simpler crystal structure compared to the AFe 12 O 19 ferrites with the magnetoplumbite structure. Therefore, it is an optimal material for the present study.

Results and discussion
Verification of the quality of the Ni 0.5 Zn 0.5 Fe 2 O 4 powder. The high sample quality was an essential requirement for the present study. Hence we started the study by confirming the phase purity and the fundamental structural and physical properties of the synthesized Ni 0.5 Zn 0.5 Fe 2 O 4 ceramic powder. First of all, the XRD pattern shown in Fig. 1a confirms the cubic spinel structure (lattice parameter determined at 8.398 Å in excellent agreement with the value of 8.383 Å given in the JCPDS card) for the sample without any impurity phase peaks. Moreover, the half-width-at-half-maximum (HWHM) value of 0.05° for the strongest (311) peak indicates excellent crystallinity.
Further evidence of the high sample quality of our Ni 0.5 Zn 0.5 Fe 2 O 4 ceramic powder was obtained from the microstructural study. From the FE-SEM image (Fig. 1b) it could be seen that the powder consists of crystallites larger than 100 μm. The local-crystal structure observation by high-angle annular dark-field (HAADF) with an incident-electron beam from [011] revealed well-ordered arrangements of atoms (Fig. 1c). Moreover, we confirmed the Ni, Zn, Fe, and O signals from the TEM-EDS analysis. Finally, Fig. 1d displays an enlarged image of Fig. 1c, and its inset shows the simulated atomic positions of the spinel-structured ferrite. For this simulation, the atomic positions were taken from a literature 21 . In the simulated image, oxygen atoms are drawn in red, A-site cations in tetrahedra in green, and B-site cations in octahedra in blue. There are two distinct B-site cations in the simulated image, marked by blue circles and a blue arrow. It is noteworthy to point out that these two types of B-site cations can be recognized in Fig. 1d. The simulated atomic positions and the atomic configurations are consistent with the interpretation of Fig. 1d. In Fig. 1e we display a simulated diffraction pattern for a spinelstructured ferrite with a thickness of 4.2 nm. By comparing the observed (Fig. 1d) and simulated ( Fig. 1e) images, the high/low-color contrasts of both the images are consistent with the following interpretation: B-site cations marked by blue circles exhibit higher contrast, while the position exhibiting indistinct contrast is due to atoms indicated by a blue arrow. Furthermore, the A-site cation indicated by a green arrow exhibits low contrast in Fig. 1d, which agrees with the simulated result (Fig. 1e).
We also confirmed the magnetic property of our Ni 0.5 Zn 0.5 Fe 2 O 4 ceramic powder; from Fig. 1f it shows the typical soft magnetic behavior with a low coercivity ( H c ) value of 13 Oe, and saturation-magnetization ( M s ) of 78 emu/g, in agreement with the values reported by other groups 22,23 . Mössbauer spectroscopy enabled us to simultaneously determine the oxidation state, chemical surrounding and magnetic structure; we performed Mössbauer measurements on our Ni 0.5 Zn 0.5 Fe 2 O 4 ceramic powder at room temperature without applying an external magnetic field. The spectrum consisted of several magnetically broadened components and one paramagnetic component. The paramagnetic component was assigned to a Fe atom surrounded only by Zn atoms, thereby the local magnetic coupling of this Fe species was broken. The isomer shift values indicated that all the Fe atoms in the Ni 0.5 Zn 0.5 Fe 2 O 4 ceramic powder are trivalent ( Fe 3+ ) in a high-spin state without any traces of divalent Fe ( Fe 2+ ). This confirms that the present sample is fully oxygenated, indicating that a high resistivity is expected.
Static physical behavior of the Ni 0.5 Zn 0.5 Fe 2 O 4 powder/polymer composite sheet. Prior to the preparation of sheet-shaped Ni 0.5 Zn 0.5 Fe 2 O 4 powder/polymer composites, the Ni 0.5 Zn 0.5 Fe 2 O 4 ceramic powder was crushed using a grinding machine to achieve homogeneous-sized particles leading to homogeneous physical properties. Afterward, the powder was classified based on the sizes of particles by the use of 180, 120, and 90 μm sieves, successively. The distribution of the particle size of the ceramic Ni 0.5 Zn 0.5 Fe 2 O 4 powder after the classification is shown in Fig. 2a. The largest number of particles occurs at approximately 50 μm, while the median value of the particle size is 25 μm. By using homogeneous-sized Ni 0.5 Zn 0.5 Fe 2 O 4 ceramic powder, we prepared a sheet-shaped Ni 0.5 Zn 0.5 Fe 2 O 4 powder/polymer composite as described in the Method section. The appearance of the prepared sheet is shown in Fig. 2b. The average thickness of the sheet is 225 μm. The FE-SEM image shown in Fig. 2c reveals that Ni 0.5 Zn 0.5 Fe 2 O 4 particles with various shapes are embedded in the polymer matrix. Also seen is that the sheet exhibits some surface roughness. The distribution of constituent elements of the sheet analyzed by EDS is in Fig. 2d; in the EDS mapping red corresponds to Fe, whereas C is denoted as green. It is inferred that the location of Fe corresponds to the existence of ferrite particles. In contrast, the signal from C shows the distribution of the polymer. In the present sheet, C is selectively detected at the boundary of the Ni 0.5 Zn 0.5 Fe 2 O 4 particles, thereby each Ni 0.5 Zn 0.5 Fe 2 O 4 particle is isolated by the polymer matrix. The surface resistivity of the Ni 0.5 Zn 0.5 Fe 2 O 4 -powder/polymer composite sheet measured under an application voltage of 10 V at room temperature is 1.09 × 10 9 Ω, which is sufficiently high. Thus, the influence of eddy currents in the sheet can be ruled out when high frequency electromagnetic wave is irradiated on the sample. This is a great advantage for materials applied in high-frequency technologies.
The magnetic properties of the Ni 0.5 Zn 0.5 Fe 2 O 4 -powder/polymer composite sheet were measured by applying an external magnetic field parallel with (//) and perpendicular to (⊥) the surface of the sheet (Fig. 2e) Fig. 1f), and the sheet basically repeated this behavior. Indeed, H c values for the Ni 0.5 Zn 0.5 Fe 2 O 4 -powder/polymer composite sheet are only 12 Oe for parallel, and 17 Oe for perpendicular configurations. There is a configuration dependence for the saturation-magnetic field in Fig. 2e, which suggests a presence of magnetic anisotropy, i.e., that the magnetic easy axis is along the parallel direction in the sheet. The distribution of each Ni 0.5 Zn 0.5 Fe 2 O 4 particle is not artificially controlled inside the sheet (see Fig. 2c), so that the magnetic properties of the Ni 0.5 Zn 0.5 Fe 2 O 4 particles dispersed in the sheet should vary from particle to particle. The different magnetic behaviors seen in Fig. 2e between the parallel and perpendicular configurations could be induced by a shape anisotropy, which is generated during the sheet preparation by the Doctor-Blade method (see Method section). As a consequence, we infer that the present Ni 0.5 Zn 0.5 Fe 2 O 4 -powder/polymer composite sheet would be the parallel (in-plane) magnetic anisotropy.
Dynamic physical behavior of the Ni 0.5 Zn 0.5 Fe 2 O 4 powder/polymer composite sheet. The complex permeability of the Ni 0.5 Zn 0.5 Fe 2 O 4 -powder/polymer composite sheet measured by the short-circuited MSL method 10,15,19 is shown in Fig. 3a where the real ( µ ′ ) and imaginary ( µ ′′ ) parts of the relative permeability are  This confirms the reliability of the present evaluation method. The high-frequency electromagnetic wave loss phenomena should be addressed. The frequency dependence of the complex permittivity (Fig. 3b) and complex permeability (Fig. 3c) were measured using a network analyzer (ENA E5080A, Keysight Technologies Inc.) on the present sheet sample fabricated in a toroidal shape. We measured in the frequency range of 0.1 to 9 GHz in these experiments. From the obtained permittivity and permeability, the loss tangents, i.e., tan δ ε = ε ′′ ε′ (Fig. 3d) and tan δ µ = µ ′′ µ′ (Fig. 3e), respectively, were calculated. We inferred that the permittivity (Fig. 3b) and tan δ ε (Fig. 3d) can be categorized into three regions, I, II, and III. In the region I, the low frequency range of f ≤ 1 GHz , the dispersion of permittivity (Fig. 3b) and tan δ ε (Fig. 3d) rapidly decreased. According to Patil et al. 25 , the decrease in permittivity at low frequencies is attributed to the interfacial polarization between two different phases; Ni 0.5 Zn 0.5 Fe 2 O 4 and PVA in the present case. Differing permittivies and conductivities between Ni 0.5 Zn 0.5 Fe 2 O 4 and PVA induce space charges under the application of an electric field on the composite sheet. The space charges accumulate at the interface between Ni 0.5 Zn 0.5 Fe 2 O 4 and PVA, leading to interfacial polarization 25,26 .
The response time of the interfacial polarization is denoted as 25 , where ω max = 2πf max . In the low frequency region, which corresponds to region I in this case, τ is relatively long 25 according to Eq. (1). Consequently, the decrease in permittivity is prominent in region I. This is the reason for the relatively dull-negative slopes of permittivity (Fig. 3b) and tan δ ε (Fig. 3d) in the range of 1 < f ≤ 5 GHz (region II), compared with region I. On the other hand, region III is highly arguable. We observed an anomalous peak in ε ′ − f profile at ∼ 6.1 GHz (Fig. 3b). The plausible explanation for region III is electron hopping. We previously described that our sample exhibited a high surface resistivity value. However, it does not mean that electrons responsible for conduction of the sample do not exist. Namely, an electron hopping between Fe 2+ /Fe 3+ and between Ni 3+ /Ni 2+ could partially account for the conduction 25 , though the conductivity is quite low in our sample. Note that the Mössbauer measurements concluded that the present Ni 0.5 Zn 0.5 Fe 2 O 4 ceramic powder exhibited Fe 3+ only, but the detection limit of the Mössbauer measurements is approximately 2% of Fe. Thus, small amounts of Fe 2+ could exist in the present sample. When the sample is subjected to high-frequency electromagnetic waves, the electron hopping between Fe 2+ /Fe 3+ and between Ni 3+ /Ni 2+ cannot be neglected; the electron hopping between these cation pairs leads to a local displacement of the space charges, resulting in the interfacial polarization in accordance with the direction of the electric field 25,27,28 . When the frequency of irradiated-electromagnetic waves on the sample is increased in region III, the electron hopping would be activated and frequent. Consequently, the interfacial polarization of the sample tends to be increased. After passing through the "critical frequency" at ∼ 6.1 GHz , the electron hopping cannot follow the alternating electric field any more 26 , which leads to an abrupt deterioration of permittivity (Fig. 3b). This could be the reason for the anomalous peak 25 in Fig. 3b, while a minimum value was observed in Fig. 3d owing to an inverse relationship between ε ′ and tan δ ε . We speculate that the anomalous peak in ε ′ − f profile in category III (Fig. 3b) is due to the electron hopping.
We further measured frequency dependence of permeability (Fig. 3c) to discuss the magnetic loss, tanδ µ (Fig. 3e). We obtained the maximum tanδ µ value of approximately 1.0 at ∼ 3.6 GHz (Fig. 3e). As is generally recognized, tanδ µ of magnetic materials consists of the sum of eddy-current loss ( tanδ e ), hysteresis loss ( tanδ h ) and remanent loss ( tanδ r ). As previously mentioned, the influence of tanδ e is negligible for the present sheet. Thus, we consider only tanδ h and tanδ r .
To discuss tanδ h , we must consider the stress sensitivity of the samples. In general, stresses such as magnetostriction, compressive and tensile stresses of magnetic materials cause stress-induced deformation of the hysteresis loops, which give rise to a non-zero tanδ h 29,30 in the high-frequency band. For polycrystalline samples, the stress effects and hence also tanδ h are complicated because various types of stresses are possible with random directions 31 . When we prepared the Ni 0.5 Zn 0.5 Fe 2 O 4 -powder/polymer composite sheet, we ground the Ni 0.5 Zn 0.5 Fe 2 O 4 ceramic powder sufficiently to attain fine Ni 0.5 Zn 0.5 Fe 2 O 4 particles. It is possible that this procedure could generate some internal stresses on the Ni 0.5 Zn 0.5 Fe 2 O 4 particles. On the other hand, mechanical or electric stresses were not applied for the present sample, so that the present sheet should be free from external stresses. tanδ h consists of the sum of internal and external stresses, but the areas of the hysteresis loops of the sheet (Fig. 2e) are small. Hence, an influence of tanδ h is rather small for the present case.
For tanδ r , the resonance linewidths of the µ ′ − f and µ ′′ − f profiles are good indicators, because the linewidths determine the amount of tanδ r 31 . It has been reported that four independent factors determine the resonance linewidths, and hence tanδ r : (1) spin-lattice relaxation, (2) porosity and/or nonmagnetic inclusions, (3) magnetocrystalline anisotropy, and (4) surface roughness. (2) plays the role of local demagnetization centers to induce demagnetization effects owing to inhomogeneous internal magnetic fields. This results in broadening of the resonance linewidths 32,33 . We infer that (2) would be the most probable origin of tanδ r for the present composite material among these four factors.
As a conclusion, the tanδ µ value in the present Ni 0.5 Zn 0.5 Fe 2 O 4 -powder/polymer composite sheet (Fig. 3e) could be mainly due to tanδ r , which is caused by (2). In this study, we achieved dynamic-physical behavior for our Ni 0.5 Zn 0.5 Fe 2 O 4 -powder/polymer composite sheet comparable to that previously reported for a material with essentially similar composition 24 , except the tanδ µ value of ~ 1.0 in the vicinity of ∼ 3.6 GHz (Fig. 3e). To overcome large loss tangents, the formation of epitaxially-grown single-crystal Ni 0.5 Zn 0.5 Fe 2 O 4 films on the flexible underlayers could be a solution. We also evaluated the complex permeability by the MSL-probe method, (ii). Although the methods (i) and (ii) represent different approaches (see Method section), we emphasize that the dynamic µ ′ − f and µ ′′ − f behaviors were essentially consistent. In the MSL-probe method, we applied a strong static external magnetic field, H dc , on the sheet sample during the measurements. The strength of H dc was controlled by changing the strength of an applied DC current with/without using yokes. As a consequence, H dc was varied from 0 to ~ 7 kOe. In this case, H dc was applied parallel to a high frequency-alternating current (AC), I rf , which propagates along the MSL. As a result, the configuration of the directions among longitudinal direction of the rectangle-shaped sheet sample, H dc , and I rf are parallel. Figure 4a-f show the results of the µ ′ − f and µ ′′ − f behaviors upon applying various H dc fields obtained by method (ii). In contrast to Fig. 3a, these figures are plotted with a linear scale for the X-axis. The maximum frequency value of electromagnetic wave irradiated for these measurements is 50 GHz.
Note that the measurement condition of Fig. 3a is the same as that of Fig. 4a ( H dc = 0 ). Upon increasing H dc , the position of µ ′′ max (indicated by dashed arrows) shifted towards higher frequencies. Changes in f r as a function of H dc are summarized in Fig. 4g; we clarified the apparent shift of FMR through H dc . From Fig. 4g, the relationship between f r and H dc assuming a linear relation is:   Fig. 4g is reasonable from the perspective of the FMR theory reported by Kittel 34 . In the present case, the linearity is concluded for the entire range of H dc (Fig. 4g), such that Ni 0.5 Zn 0.5 Fe 2 O 4 -powder/polymer composite sheet could be applicable in a wide range of GHz-operational frequencies 35 even at small H dc . From Eq. (2), the value of f r(cal) is determined to be 6.25 GHz at H dc = 0 . However, we did not observe any apparent µ ′′ max in Fig. 4a ( H dc = 0 ), and hence the determination of f r is impossible from Fig. 4a. This is contradictory to the experiment (Fig. 4a) and the mathematical extrapolation [Eq. (2)]. We should also mention that the f r value is estimated to be 7.5 × 10 −1 GHz from Fig. 3a ( H dc = 0 ); this is the only difference between methods (i) and (ii), which reveals the difficulties of precise evaluations under GHz-frequency bands.
We studied a relationship between the Gilbert damping constant, α, and f r (Fig. 5). For this discussion, f r values were obtained from Fig. 4g, whereas α values were determined by the µ ′′ − f profiles obtained by method (ii). We utilized following equation 36 for calculating the damping constant, f is denoted as 1 2 × (full width at half maximum of the µ ′′ − f peak). From Fig. 5, we found that the damping constant drastically decreased as f r increases (increases in H dc ). The α max value is 3.40 × 10 −1 at f r = 7.01 GHz ( H dc = 3.20 × 10 −1 kOe, which corresponds to Fig. 4b), while the value of α min is 6.52 ×10 −2 at f r = 2.35 × 10 GHz ( H dc = 7.02 kOe corresponding to Fig. 4f). In epitaxiallygrown spinel ferrite films with the chemical formula of Ni 0.65 Zn 0.35 Al 0.8 Fe 1.2 O 4 possessing a similar composition as the present sample, low damping constant values have been reported; 2.6 × 10 −3 ≤ α ≤ 3.5 × 10 −337 . These values are one-to-two orders of magnitude lower than our sample.
A magnetic-based material exhibiting a high α value implies that the FMR phenomenon of the sample is complicated 38 . In the case of spinel-structured ferrite films, the constituent of cations 39 and high density of crystal defects generated by dislocations and/or antiphase boundaries 40−42 affect the damping. This results in a defect-mediated damping 37 . These factors are potentials for "extrinsic" damping, which leads to high α value 43−48 . However, from microstructural investigations in this study, we observed neither remarkable antiphase boundaries nor misfit dislocations (Fig. 1c-e). In the magnetic field dependent magnetization behaviors, the structural defects such as antiphase boundaries and so on could lead to a large H c value and "S-shaped" hysteresis curve 49 , but we obtained a rather small value for H c as well as "regular-shaped" hysteresis curves (Figs. 1f and 2e). By considering these experimental facts, we can eliminate the possibility of defect-mediated damping behind the high α values in the present case.
By taking into account the discussions above, we suggest that the damping of the present Ni 0.5 Zn 0.5 Fe 2 O 4 powder/polymer composite sheet could be largely related to the 3d electrons from Fe and Ni atoms. Although further detailed studies are necessary, intuitive considerations would be possible; the coupling between 3d electrons, 3d spins, and phonons is sufficiently strong to induce high α values 38 in the present Ni 0.5 Zn 0.5 Fe 2 O 4 powder/polymer composite sheet. In the epitaxially-grown spinel structured MgFe 2 O 4 thin film deposited on MgAl 2 O 4 (100) substrate, the α value was reported to be approximately 8 × 10 −249 . Our α min value ( 6.52 × 10 −2 at f r = 2.35 × 10 GHz with the application H dc of 7.02 kOe) is comparable to that value 49 , though the present sheet consists of polycrystalline Ni 0.5 Zn 0.5 Fe 2 O 4 particles.
Therefore, the significance of the present study is the high potential of the present Ni 0.5 Zn 0.5 Fe 2 O 4 powder/ polymer composite sheet composed of an environmentally friendly ferrite with cost-effective elements applicable (2) f r(cal) = 2.59H dc + 6.25,

Conclusions
In conclusion, we applied two different originally developed techniques, short-circuited MSL 10,15,19 and MSL-probe 11,20 methods, to a Ni 0.5 Zn 0.5 Fe 2 O 4 powder/polymer composite sheet prepared from high-quality Ni 0.5 Zn 0.5 Fe 2 O 4 ceramic powder. Although the two techniques are based on different experimental setups/systems, the dynamic permeability behaviors at GHz bands are essentially consistent.
A linear relationship between f r(cal) and H dc was obtained for the entire measured range, up to 50 GHz. The linearity between f r(cal) and H dc at sufficiently small required H dc suggests that the present sheet has potential applications to power-efficient devices 37 . Further study on the damping is necessary, but the damping process of the current Ni 0.5 Zn 0.5 Fe 2 O 4 powder/polymer composite sheet could be dominated by a strong coupling between 3d electrons, 3d spins, and phonons.
Although the present sheet is composed of polycrystalline Ni 0.5 Zn 0.5 Fe 2 O 4 , the sheet showed soft magnetism exhibiting parallel magnetic anisotropy, high surface resistivity, small required H dc for FMR, an intrinsic damping process, and comparable damping constant to the epitaxially-grown thin film 49 . These results indicate that highperformance devices that can be utilized in frequency bands up to several tens of GHz. For example, the present Ni 0.5 Zn 0.5 Fe 2 O 4 -powder/polymer composite sheet is magnetically anisotropy, such that "anisotropic suppression" of electromagnetic waves with GHz frequencies could be possible 50 ; the present sample can be expected to be useful as an anisotropic-EMI material in the GHz band, which is essential for high-frequency electronic devices. After drying the sheet under ambient conditions in air, the sheet was removed from PET film, and was heat-treated to dry completely at 105 °C for 1 h; this is a monolayered sheet with the thickness of 166 μm on the average. Afterward, we piled two monolayered sheets, and performed a heat treatment at 150 °C for 6 min by applying a mechanical pressure to attain higher density of the Ni 0.5 Zn 0.5 Fe 2 O 4 powder/polymer composite sheet. We focus only on the double-layered Ni 0.5 Zn 0.5 Fe 2 O 4 powder/polymer composite sheet instead of a monolayered one in this study.

Synthesis of impurity-free spinel-structured
Fundamental characterizations. The phase purity and crystallinity were evaluated by X-ray diffraction (XRD, SmartLab, Rigaku Corporation). The magnetic field dependent magnetization behaviors were measured in the range of ± 10 kOe at room temperature using a vibrating sample magnetometer (VSM, C7-10A, Toei Industry Co., Ltd.). The Ni 0.5 Zn 0.5 Fe 2 O 4 powder was embedded in a polymer matrix (Pattex 100% glue, Henkel) on a Fe-free Al foil, and room-temperature 57 Fe Mössbauer spectra were collected at 12 mm/s (FGX-222ST Mössbauer spectrometer, Topologic Systems Inc.) using a 50 mCi Cyclotron 57 Co/Rh source. The microscopic crystal structures of the sample were observed by field-emission scanning electron microscopy (FE-SEM, JSM-7000F, JEOL Ltd.) and field-emission transmission electron microscopy (FE-TEM, JEM-ARM200F, JEOL Ltd.), respectively.
Evaluations of dynamic permeability in the GHz-frequency region. For the dynamic permeability in the GHz-frequency region, (i) short-circuited MSL 10,15,19 and (ii) MSL-probe 11,20 methods were employed and their results were compared. In the case of (i), originally developed sample jigs were used to produce electromagnetic radiation of suitable frequencies to irradiate the sample for the measurements. The sheet sample was cut into a rectangle and placed at the short end of the MSL. In method (i), a sufficiently strong H dc was applied to saturate the Ni 0.5 Zn 0.5 Fe 2 O 4 powder/polymer composite sheet using a permanent magnet (Fig. 3a). I rf , AC with high frequency, was propagated along the MSL, such that an AC magnetic field, H rf , is generated perpendicular to the direction of the I rf . In this method, we regarded the overall experimental system as an "electric circuit model" including the sample and the jigs 10,15,19 . We attained one of the scattering matrixes of an electric circuit, S 11 as an output signal from the measurements in this method. Using S 11 , the admittance,Y 1 , of the electric circuit can be calculated. H dc was applied parallel to the longitudinal direction of the rectangle-shaped sheet sample. Therefore, the direction of H dc is perpendicular to I rf . Under the application of H dc , the electric circuit model of Scientific Reports | (2021) 11:614 | https://doi.org/10.1038/s41598-020-79768-z www.nature.com/scientificreports/ the experimental system changes, so that Y 1 is replaced by Y 2 10,15,19 . By the use of Y 1 and Y 2 , we can determine the relative values of complex permeability of the sample. The details of the method and calculation procedures are mentioned elsewhere 10,15,19 .
In method (ii), a "self-developed" probe composed of a MSL on a flexible substrate was used for the measurements 11,20 . The MSL probe was set on the surface of sheet sample with a gap between the sample and the probe of approximately 200 μm. Before the measurements, a calibration of the network analyzer had been carefully performed by applying H dc ≈ 8kOe without the sample. Non-magnetic signals were perfectly subtracted from coaxial cables and jigs throughout the calibration. Similar to method (i), we applied sufficiently strong H dc after the sheet sample was placed in the measurement device in the method (ii). This procedure enables us to saturate the specimen 20 . After that, the permeability measurements were carried out. In this method, the overall experimental system is also regarded as an electric circuit model. The output signal from the measurements is S 21 , which is one of the other scattering matrixes of an electric circuit. Using the obtained S 21 , we can determine the relative values of complex permeability of the sample. For the details of method (ii), refer to literatures 11,20 .

Data availability
All data generated or analyzed during this study are included in this published article.