Introduction

Electromagnetic (EM) interference has emerged as a crucial problem with the rapid development of high-frequency communication techniques and micro-miniaturization of electronic information devices, which has aroused the rapid development of EM wave absorption materials1,2,3,4,5,6. Among all these achievements, soft magnetic materials occupy an important position in designing materials because of their excellent magnetic loss abilities induced by high saturation magnetization (Ms)5,7,8. However, their ferromagnetic resonance behaviors could be impeded by the contradictory relationship between loss ability and frequency, known as the Snoek limit ((μi − 1)fr = 4γMs/3)9. Consequently, dielectric materials such as carbon, MXene, and ceramic have been introduced to obtain satisfactory EM wave absorption at gigahertz6,10,11. However, an approach to cross the Snoek limit by manipulating the ferromagnetic materials themselves is still lacking, thus limiting the development of high-frequency magnetic loss materials and the optimization of composited materials.

Shape anisotropy, which is constructed through controllable preparation processes, has been recognized as a promising pathway to address this shortcoming12. The flake geometry could induce an effective in-plane anisotropy in soft magnetic materials to break the Snoek limit, which could optimize the magnetic loss ability at gigahertz13,14,15,16,17. Under these guidelines, various soft magnetic materials, such as flake-like FeNi318, ZnCo2O4 crystalline nanosheets19, flake-like Sm1.5Y0.5Fe17−xSix20, Ce2Fe17N microflakes13, and carbonyl iron microflakes21, have been investigated, in which their ferromagnetic resonance frequencies have been extended to gigahertz regions. Nevertheless, it should be noted that their loss abilities, reflected by the imaginary part of the complex permeability (μ“), still exhibit a decreasing tendency as the frequency increases to the high-frequency region. For example, the μ“ values of Sm1.5Y0.5Fe17Si, FeSiAl/MnO2, and FeSiAl/ferrite are 0.6, 1.2, and 1.05 at 5.0, 5.0, and 6.0 GHz, respectively. The values of Sm1.5Y0.5Fe15.5Si1.5, barium ferrite, and FeSiAl/Al2O3 ceramics were reduced to 0.38, 0.31 and 0.48 at 9.5, 9.1 and 9.4 GHz20,22,23,24,25. Therefore, a soft magnetic material with high ferromagnetic resonance frequency and magnetic loss capacity is urgently needed to meet the requirements of high-frequency applications.

Cementite (Fe3C) with high saturation magnetization can be an ideal candidate to achieve the above requirements because of its high saturation. However, there remains a great lack of controllable synthesis of morphology through traditional chemistry methods because the structures of Fe3C are difficult to restrict by coating agents. Here, we propose a pathway via solid-state phase transformation to obtain the large-scale preparation of Fe3C microflakes with crystal orientation. The morphology can be adjusted by solid-state phase transformation, and the material can be obtained by electrochemical dealloying at a constant voltage (CV) of −0.4 V. The electromagnetic response performances and theoretical simulations indicate that the ferromagnetic resonance frequency relies on the increased effective in-plane anisotropy induced by the flake geometry, which can be regulated by adjusting the isothermal quenching temperatures of eutectoid steels. As a result, the 2D Fe3C microflakes treated at 550 °C could achieve the highest anisotropy, which can achieve an excellent magnetic loss ability (μ“) of 0.9 at a frequency of 11.56 GHz. Thus, our results clarify the relationship between anisotropy and high-frequency magnetic loss ability, proposing a new route for designing high-performance microwave absorption materials.

Results

According to the Fe–C binary phase diagram and the isothermal quenching curves of eutectoid steel (0.77 wt%[C]) (Fig. 1a), the microstructures of pearlite can be changed by the isothermal quenching process within the temperature regions of 550 °C to the temperature of eutectoid transformation (marked by A1)26,27,28. The average interlamellar distance S0 of different pearlite microstructures, so-called pearlite (650 °C to A1), sorbite (600–650 °C), and troostite (550–600 °C), can be evaluated by the empirical formula of S0 (nm) = 8.02  ×   103/ΔT (ΔT refers to the difference between the heat treatment temperature (800 °C) and the isothermal quenching temperature)27,28, which indicates that the thicknesses of cementite can be adjusted by designing the isothermal temperatures. In this context, the original eutectoid steels were first cut into a cuboid with a size of 20  ×   10  ×   5 mm3 and annealed at 800 °C for 3 h to completely austenitize. Then, the eutectoid steel precursors were placed into the molten salt to realize the isothermal quenching process, during which the austenite transformed into pearlite, sorbite, or troostite. Finally, the precursors were set as the electrode and connected to the electrochemical workstation in CV mode (−0.4 V). The 2D Fe3C microflakes, denoted as Fe3C−x (x stands for quenching temperature), were prepared by electrochemical dealloying of the ferrite textures within a solution of KCl and C6H5Na3O729, as illustrated in Fig. 1b. The corresponding current curves of the electrochemical dealloying process with decreasing tendencies can be recognized, as shown in Supplementary Fig. S1.

Fig. 1: Synthesis process (design route of solid-state phase transformation) of Fe3C microflakes.
figure 1

a Fe–C binary phase diagram and isothermal quenching curve of eutectoid steel and schematic illustrations of pearlite, sorbite, and troostite, and the white and blue stripes represent ferrite and Fe3C, respectively. (A: austenite, B: bainite, F: ferrite, L: liquid, Ld/Ld’: ledeburite, M: martensite, P: pearlite, Mf: martensite finish, MS: martensite start, S0: average interlamellar distance) (b) Schematic illustrations of the electrochemical dealloying process and the morphology of Fe3C microflakes, and the Fe3C microflake (right) is derived from the blue cementite band (middle) after the dealloying process.

Full size image

As shown in Fig. 2a, a typical pearlite structure can be observed in all the samples using scanning electron microscopy (SEM), in which the bright and dark stripes are the cementite and ferrite textures, respectively. The corresponding thicknesses of cementite were collected and summarized in Fig. 2e and Supplementary Fig. 2. The thicknesses decreased from 22.0 to 15.7 nm as the quenching temperatures decreased from 700 to 500 °C, implying that the Fe3C structures can be manipulated by the isothermal quenching process. The microstructures of the eutectoid steel precursors after the dealloying process (Fe3C-700) were characterized by SEM and shown in Supplementary Fig. 3. It can be seen that the ferrite has been etched, and the precursor was constructed by the flake structural cementite. Subsequently, the microstructures of the 2D Fe3C microflakes were characterized by SEM and transmission electron microscopy (TEM), as shown in Fig. 2b, c. Here, an irregular flake structure could be observed in all the samples. It can be noticed that the thickness of Fe3C is nanoscale, but the width and length are microscale, which results in a high aspect ratio for reinforcing the shape anisotropy to enhance natural resonance performance. The elemental distributions and crystalline structures of Fe3C were detected by energy-dispersive X-ray spectroscopy (EDS) and X-ray diffraction (XRD), respectively (Supplementary Figs. S4 and S5). The schematic diagram of pearlite has been exhibited in Fig. 2d and the corresponding definition of thickness has been displayed. Two composed elements, carbon, and iron, were uniformly distributed in the Fe3C microflakes, and their phase structures were highly identical to the standard phase card PDF#85-131730,31, indicating that the 2D Fe3C microflakes were successfully obtained through the electrochemical dealloying process.

Fig. 2: Microstructure characterizations.
figure 2

a SEM images of eutectoid steel isothermal quenching under different temperatures, scale bar: 2 μm. b High-resolution SEM images of the Fe3C-700, Fe3C-625, and Fe3C-550 micro-flakes, scale bar: 10 μm. c High-resolution TEM image, atomic EDS maps, and SAED image of Fe3C-700 microflakes, scale bar: 1 μm. d Schematic diagram of pearlite, electrochemical dealloying process, and analysis directions. e Summarization of the thickness of Fe3C microflakes (Fitting line: y = 2.024–0.00003x, R2 = 0.698). The number refers to the isothermal quenching temperature, and source data are provided as a Source Data file.

Full size image

Using spherical aberration-corrected TEM, we further investigated the microstructures of these 2D Fe3C microflakes, in which high-resolution TEM (HR-TEM) images of Fe3C-700, Fe3C-625, and Fe3C-550 were obtained and shown in Supplementary Fig. 6. As the atomic EDS mapping demonstrated, the iron and carbon atoms are uniformly posited in the 2D microflakes. The enlarged view of HR-TEM images and the corresponding inverse fast Fourier transform (IFFT) images (Supplementary Fig. S6B) further disclose the lattice structures of Fe3C-700, in which the red spots represent iron atoms and the blue spots are carbon. It can be found that the zone axis of all the Fe3C is [010], which has been identified by the selected area electron diffraction (SAED) images shown in Supplementary Figs. S6A and S732,33. It is concluded that the crystal structure of 2D Fe3C microflakes is orthorhombic with the space group of Pnma. The magnetocrystalline anisotropy can be inferred, where the easy axis of [001] is located in the plane of 2D Fe3C microflakes and the second easy axis of [010] is located out-of-plane34. Thus, the intrinsic in-plane anisotropy introduced by the solid-state phase transformation assisted by electrochemical dealloying can effectively improve the magnetic loss ability in the high-frequency region.

The magnetic performances of all the samples were analyzed through the hysteresis loops, as exhibited in Supplementary Figs. S8 and S9. The saturation magnetizations (Ms) are 97.7, 103.5, 114.6, 119.9, 109.2, 128.3, and 125.8 emu/g, respectively, and the coercivities (Hc) are 279.6, 271.8, 194.4, 114.4, 106.5, 104.2, and 69.6 Oe. The Ms increased with the enhanced thicknesses of Fe3C microflakes, while the Hc exhibited an inverse phenomenon. The Fe3C-550 microflakes presented the highest Hc among all the samples. Because the coercivity is proportional to the anisotropy constant35, it can be inferred that the increase in coercivity is the consequence of the increased anisotropies of the 2D Fe3C microflakes. Thus, the shape anisotropy can be efficiently manipulated by adjusting the isothermal quenching conditions36.

The microwave response performances of all the Fe3C microflakes were measured by a vector network analyzer (VNA) in the frequency region of 2–18 GHz, as demonstrated in Supplementary Figs. S10 and S11. The measurements of each sample were repeated more than 5 times. In general, the dielectric loss ability should not be observed in the single magnetic loss material. In all the Fe3C microflakes, it can be seen that the complex permittivity values of the real parts (ε‘) are approximately centered at 20 and the imaginary parts (ε“) are centered at 0. However, some resonance peaks can be noticed in the complex permittivity patterns, both in real and imaginary parts. To clearly demonstrate the position of these resonance peaks, the superimposed figure of complex permittivity and complex permeability has been employed and shown in Supplementary Fig. 12. It can be noticed that the values can be found in two areas: the high-frequency turning point of the real part, and both ends of the resonance peak of the imaginary part. Such a phenomenon is mainly associated with the transformation between the permittivity and permeability, which is realized by the connection between the Fe3C microflakes because of their high anisotropy microstructures5. According to studies such as close-packed Ni nanoparticles, multi-walled carbon nanotubes, and sub-nanometer clusters in nanocages, the aggregation of conductive medium would construct the conductive network, resulting in a reinforced polarization behavior for increasing the dielectric loss ability5,37,38. Therefore, the irregular shape of Fe3C microflakes would form a conductive network in the EM testing sample, which can induce polarization around the natural resonance and thus result in the vibration of complex permittivity.

From the results of the complex permeability, typical ferromagnetic resonances can be found in all the samples, and their frequencies are within the regions of 9.47–11.56 GHz. The complex permeability of all the samples was fitted through the Landau–Lifshitz–Gilbert (LLG) equation based on five different experimental results, and the details of the fitting process have been demonstrated in the method section. Here, an increased resonance peak (fr) can be observed along with the decreased isothermal temperature (Fig. 3a, b). It can be noticed that the fr of Fe3C-700 microflakes, containing the lowest magnetic anisotropy of all the samples, can even reach 9.73 GHz, which is far higher than that of previously reported soft magnetic materials such as c-axis oriented hcp-(CoIr) thin films (4.5 GHz)39, Fe20Ni80 and Co20Fe60B20 material-modulated stripe-patterned thin films (5.8 GHz)40, and Sm1.5Y0.5Fe17-xSix and their composites (5.0 GHz)20. The results indicate that the magnetic loss abilities can be significantly optimized in the 2D Fe3C microflakes by adjusting the isothermal temperature, ascribed to the manipulation of shape anisotropies. In the optimized Fe3C-550 microflakes, which present the highest anisotropy, fr can be enhanced to 11.56 GHz (Fig. 3c). For the magnetic loss ability (imaginary part of the complex permeability, μ“)6, the values of all the Fe3C microflakes reached 1.05, 1.27, 1.43, 0.86, 0.82, 1.00, and 0.90 at 9.58, 10.07, 10.24, 10.33, 10.87, 11.00, and 11.56 GHz, respectively. The magnetic loss abilities of Fe3C microflakes can be effectively improved in high-frequency regions compared with the Sm1.5Y0.5Fe15.5Si1.5 (0.38, 9.1 GHz)20, ferrites (0.31, 9.1 GHz)22, and FeSiAl (0.48, 9.4 GHz)24. Furthermore, to visually illustrate the optimized magnetic loss abilities, the pattern of fr versus μ“ of Fe3C microflakes was compared to the other reported magnetic loss materials in Fig. 3d (Supplementary Refs. 1–26), in which the Fe3C samples with increased anisotropy could possess satisfactory magnetic loss performances at gigahertz among the related reports. Furthermore, the magnetic loss angles (tan δM)41 were calculated by the complex permeability to further imply the loss performances of Fe3C microflakes (Fig. 3e). Similar to the complex permeability, the loss abilities were improved along with the increased anisotropy.

Fig. 3: Magnetic loss abilities.
figure 3

a Real part (μ‘) and b imaginary part (μ“) of the complex permeability at the frequency regions of 6–14 GHz, in which the colored fitted lines are fitted from the experimental lines (gray lines) through the Landau–Lifshitz–Gilbert (LLG) equation. c Summarization of the natural resonance frequencies (Fitting line: y = 9.348 + 0.299x, R2 = 0.956), effective absorption bandwidth, reflection loss and the corresponding thickness of Fe3C. d Comparison of fr versus μ“ of reported soft magnetic materials and Fe3C. e The tan δM of Fe3C microflakes in the frequency region of 6–14 GHz. Source data are provided as a Source Data file.

Full size image

The microwave absorption properties of the 2D Fe3C microflakes were evaluated by transmit-line theory42,43,44, as shown in Fig. 3c and Supplementary Fig. 13. The minimum reflection loss (RLmin) values of Fe3C are −52.09, −44.39, −42.02, −52.82, −45.73, −43.29, and −56.99 dB, respectively, in which −10 and −20 dB indicate that 90% and 99% incident EM energy can be attenuated42,43,45. For the effective absorption bandwidth (EAB≤−10 dB), the values of 2.55, 2.60, 2.75, 3.05, 2.60, 3.75, and 2.69 GHz with thicknesses of 1.20, 2.60, 1.25, 1.25, 1.25, 1.10, and 1.35 mm can be observed. The 2D Fe3C microflakes exhibit excellent microwave absorption performance only through their magnetic loss abilities. For the impedance-matching performances of Fe3C microflakes, the values of |Zin/Z0|6 were collected and are shown in Supplementary Fig. 14. Commonly, the values in the regions of 0.8 and 1.2 should be considered as excellent impedance-matching performances6,46. Here, the 2D Fe3C microflakes could obtain satisfactory matching performances in the entire frequency region (2–18 GHz), indicating that the manipulation of shape anisotropy could simultaneously improve the impedance matching and microwave absorption performances in high-frequency regions.

Therefore, it is concluded that a high-frequency magnetic loss ability can be achieved in Fe3C microflakes with enhanced shape anisotropy. To further identify the details of such performances, Lorentz transmission electron microscopy (L-TEM) was employed, and the corresponding in-focus (Fig. 4a, b) and overfocus (Fig. 4c, d) images are shown, in which strip domain structures can be observed in the Fe3C-700 microflakes. The nucleation of strip domains can result from the appearance of intrinsic out-of-plane anisotropy, which is similar to the manipulation of magnetic domains in sputtered (or evaporated) magnetic thin films by adjusting the perpendicular anisotropy47,48. The strip domain structures under a static magnetic field (Hext = 0 mT) can be reconstructed by micromagnetic simulations with mumax349 by considering the shape anisotropy as an effective in-plane anisotropy (Ki), as shown in Fig. 4e. The color denotes the x-component of magnetization. The simulated results are consistent with the calculated distribution of magnetization from contrast L-TEM images by the classic transport of intensity equation (TIE) method50 (see Fig. 4f). Thus, the ferromagnetic resonance behaviors of Fe3C microflakes under different temperatures can also be obtained from the static magnetic domains by solving the Landau–Lifhiz–Gilbert equation numerically.

Fig. 4: L-TEM characterizations of Fe3C-700 microflakes.
figure 4

a In-focus L-TEM images and b enlarged view of the red dashed square, scale bar: 500 nm. c Overfocus L-TEM images (scale bar: 2 μm) and d enlarged view of the red dashed square (scale bar: 500 nm). e Simulated distribution of magnetization by micromagnetic simulations, scale bar: 2 μm. f Calculated distribution of magnetization from contrast L-TEM images by the classic transport of intensity equation (TIE) method, and the color denotes the x-component of magnetization, scale bar: 500 nm. Source data are provided as a Source Data file.

Full size image

Figure 5a shows the ferromagnetic resonance spectra of the Fe3C microflakes under different temperatures after obtaining the static magnetic structures with different values of Ki, which are induced by the shape anisotropy of Fe3C microflakes with different thicknesses. The resonant frequencies of the simulated Fe3C microflakes increase from 15.1 GHz to 17.8 GHz with the increase in Ki from 5 × 104 J/m3 to 7  ×   104 J/m3 with an interval of 0.5 × 104 J/m3, in which the chosen values of Ki are considered for fitting the change in coercivity (Supplementary Figs. S8 and S9) and the change in frequency is consistent with the experimental results. The difference between the simulation and experimental values of frequency could result from the defect in a real material system, where the ideal material model is considered in micromagnetic simulations. Thus, it is inferred that the increased resonant frequencies can be ascribed to the increase in effective in-plane anisotropy induced by flake geometries, which can be manipulated by adjusting the isothermal quenching temperatures. Furthermore, the amplitude distribution of resonance at 15.1 GHz for the Fe3C-700 microflakes is shown in Fig. 5b, c, which is focused on the position of domain walls and named localized spin wave modes. Therefore, the absorption of EM for Fe3C microflakes at high frequency could originate from the appearance of large numbers of domain walls in stripe domains. The corresponding phase distribution of the Fe3C-700 microflakes is also shown in Fig. 5d, e. The phase is almost uniform in a single domain wall, which is similar to the propagation of spin waves vertically along the domain wall.

Fig. 5: Simulated ferromagnetic resonance behaviors of Fe3C microflakes.
figure 5

a Simulated results of the imaginary part of the complex permeability for Fe3C microflakes under different temperatures. b Amplitude distribution of resonance at 15.1 GHz for the Fe3C-700 microflakes and c enlarged view of the region in the white dashed square. d Phase distribution of resonance at 15.1 GHz for the Fe3C-700 microflakes and e enlarged view of the region in the black dashed square. Source data are provided as a Source Data file.

Full size image

Discussion

In summary, 2D anisotropic Fe3C microflakes with crystal orientations were prepared from eutectoid steel precursors through solid-state phase transformation assisted by electrochemical dealloying. It can be recognized that the high-frequency ferromagnetic resonance behavior is strictly correlated to the shape anisotropy, which can be manipulated by adjusting the isothermal quenching temperatures. The stripe domain structures are observed directly without a biased magnetic field using in situ L-TEM, which can be reconstructed by micromagnetic simulations with in-plane and out-of-plane anisotropies. Consequently, the natural resonance frequencies of Fe3C microflakes have been increased to the regions of 9.47–11.56 GHz, while the mostly optimized Fe3C-550 microflakes can reach 11.56 GHz with a μ“ value of 0.9. Moreover, the RLmin and EAB≤−10 dB of Fe3C-550 microflakes could reach −52.09 dB (15.85 GHz, 2.90 mm) and 2.55 GHz (1.20 mm), respectively, exhibiting an optimized microwave absorption property. The present study provides an intrinsic insight into the role of anisotropy in high-frequency magnetic loss ability and provides a new route to design high-performance microwave absorption materials. Furthermore, this method can stimulate research interests in obtaining functional materials from other traditional structural materials. We believe that this work also provides some preliminary research basis for the realization of structural–functional integrated materials.

Methods

Electrochemical dealloying for synthesizing Fe3C micro-flakes

The eutectoid steel (0.77 wt%[C]) was firstly cut into the blocks with the size of 20 × 10 × 5 mm3 and annealed at 800 °C for 3 h in the ambient atmosphere. Then, the blocks were placed into the molten salt for 30 min to suffer the isothermal quenching process. Here, the isothermal temperature was achieved by two different salt groups (wt%): (i) 30% KCl + 20% NaCl + 50% BaCl2 for 700 °C and 675 °C, (ii) 30% NaCl + 32% BaCl2 + 48% CaCl2 for 550 °C to 650 °C. After that, the blocks were fixed as an electrode in the solution of KCl and C6H5Na3O7 to engage the electrochemical dealloying process for 48 h. The electrochemical dealloying process was obtained by an electrochemical workstation working in constant voltage (CV) mode with a voltage of −0.4 V. The as-made Fe3C micro-flakes were washed with the mixture of HCl (4%) and ethanol (96%) and the deionized water for several times and then dried in the vacuum drying oven for 12 h.

Structural characterizations

The scanning electron microscope (SEM) was characterized by the JEOL JSM-IT500HR/LA with an accelerating voltage of 20 kV. The microstructure and morphology of the micro-flakes were characterized by the field-emission transmission electron microscope (FE-TEM) (Thermo Scientific Talos F200S), and the energy-dispersive X-ray spectroscopy (EDS) was obtained by Bruker Super Lite X2. The high-resolution TEM images were obtained by the spherical aberration-corrected transmission electron microscope (JEOL ARM-200). The statistic of the micro-flake thickness is calculated via the SEM images using the open-access ImageJ software. The crystalline structure of Fe3C micro-flakes was calculated by the open-access VESTA software. The magnetic properties were measured via the vibrating sample magnetometer (VSM, ADE EV9) with a maximum applied field of 15 kOe. The Lorentz transmission electron microscope (L-TEM) images under a static magnetic field were analyzed by Thermo Scientific Talos F200S. X-ray diffraction (XRD) was performed using a SmartLab9kW (Rigaku) with a scan step of 0.04°.

Electromagnetic response performances

The electromagnetic response performance was obtained by the vector network analyzer (VNA, N5222A, Keysight Co., Ltd.) equipped with a Type-N 50 Ω coaxial airline (2–18 GHz, Ceyear CETC). The VNA was calibrated by the 2-port short-open-load-thru (SOLT) calibration standard (85054B), and the samples were uniformly mixed with the paraffin at a weight percentage of 50 wt% and then processed to a toroidal shape of 7.00 mm outer diameter and 3.04 mm inner diameter. The reflection loss (RL) and efficient absorption bandwidth (EAB) were calculated via the transmit-line theory using the following equations (S3 and S4)6,43.

$${R}\left({{{{{\rm{dB}}}}}}\right)=20{{{{\mathrm{lg}}}}}\left|\frac{{{Z}}_{{{{{{\rm{in}}}}}}}-1}{{{Z}}_{{{{{{\rm{in}}}}}}}+1}\right|$$
(1)
$${Z}_{{{{{\rm{in}}}}}}=\sqrt{\frac{{\mu }_{r}}{{\varepsilon }_{r}}}\tanh i\frac{2\pi f}{c}\sqrt{{\mu }_{r}{\varepsilon }_{r}}d$$
(2)

In with Zin is normalized input impedance, c is the light velocity in the free space, f is the frequency, and d is the thickness of the absorber.

Fitting process of complex permeability

The fitting of the complex permeability of all the samples has been processed by OriginPro 2021 (9.8.0.200, OriginLab Corporation). All the fitting line is fitted by five different experimental lines, and the frequency region was set at 6–14 GHz. The complex permeability can be fitted based on the classical LLG as follows51,52:

$$\mu ^{\prime}\,=\,1+{\chi }_{0}\frac{1+({\alpha }^{2}-1){\left(\frac{f}{{f}_{r}}\right)}^{2}}{{\left[1-\left(1+{\alpha }^{2}\right){\left(\frac{f}{{f}_{r}}\right)}^{2}\right]}^{2}+{\left(2\alpha \frac{f}{{f}_{r}}\right)}^{2}}$$
(3)
$$\mu ^{\prime \prime }={\chi }_{0}\frac{\alpha (\frac{f}{{f}_{r}})\left[1+\left(1+{\alpha }^{2}\right){\left(\frac{f}{{f}_{r}}\right)}^{2}\right]}{{\left[1-\left(1+{\alpha }^{2}\right){\left(\frac{f}{{f}_{r}}\right)}^{2}\right]}^{2}+{(2\alpha \frac{f}{{f}_{r}})}^{2}}$$
(4)

Where χ0 = μi-1is the initial susceptibility, α is the damping constant and f is the operating frequency. During the fitting process, the χ0, α, and fr have been set as a, b, and c, and the variates including μ“, μ‘, and f have been employed to fit the constant a, b, and c. In addition, the natural resonance frequency fr’ can be obtained through the equation as follows13:

$${f}_{r}^{{\prime} }=\frac{{f}_{r}}{\sqrt{1+{\alpha }^{2}}}$$
(5)

Where the fr and α are obtained from the above fitting process.

Theoretical calculations

The micromagnetic simulations were conducted using the mumax349. The material parameters are the following: The saturation magnetization is Ms = 8 × 105 A/m. The exchange constant A = 6 × 10−12 J/m. The anisotropy constant of out-of-plane (Kz) and in-plane (Ki) are Kz = 1 × 105 J/m3 and Ki = 5 × 104 J/m3, respectively. The thickness is set as 20 nm. The damping constant is α = 0.1, which is obtained by averaging the fitted damping constant in Fig. 3b, as shown in Supplementary Table 1. It should be noted that the α in the simulation usually affects the intensity of amplitude in the position of resonant peaks, which does not affect the position of resonant peaks (see Supplementary Fig. 15)53. Thus, in micromagnetic simulations, the influence of anisotropy on ferromagnetic resonances is studied by fixing α at 0.1. The mesh size is 10 × 10 × 2 nm3. The static magnetic structures were obtained by minimizing the total energy (Et) of the magnetic system, including Heisenberg exchange energy (Eex), anisotropy energy (Eani), demagnetization energy (Edeg), and Zeeman energy (Ezeem)54. The magnetization dynamical behaviors in the magnetic system were obtained by solving the Landau–Lifshitz–Gilbert (LLG) equation:

$$\frac{d{{{{{\boldsymbol{M}}}}}}}{dt}=-|\bar{\gamma }|{{{{{\boldsymbol{M}}}}}}\times {{{{{{\boldsymbol{H}}}}}}}_{{{{{\rm{eff}}}}}}-\frac{|\bar{\gamma }|\alpha }{{M}_{s}}{{{{{\boldsymbol{M}}}}}}\times ({{{{{\boldsymbol{M}}}}}}\times {{{{{{\boldsymbol{H}}}}}}}_{{{{{\rm{eff}}}}}})$$
(6)

where \(\bar{\gamma }\) is the gyromagnetic constant, M = (Mx, My, Mz) denotes the magnetization vector, Heff is the effective field and can be calculated by \({{{{{{\rm{H}}}}}}}_{{{{{\rm{eff}}}}}}=\frac{\partial {{E}}_{t}}{\partial {{{{{\boldsymbol{M}}}}}}}\). A perpendicular sinc field hz was applied to excite the dynamics of the magnetic system and was given by the following:

$${h}_{z}(t)={h}_{a}\,\sin [2\pi {f}_{c}(t-{t}_{0})]/[2\pi {f}_{c}(t-{t}_{0})]$$
(7)

where t0 = 10 ps and fc = 50 GHz. The field amplitude ha is fixed at 5 mT. The fast Fourier transform (FFT) of Mz/Ms was used to obtain the ferromagnetic resonance spectra. The amplitude and phase distributions were obtained by performing the FFT of every mesh in the magnetic system, where the intensity at 15.1 GHz for the Fe3C-700 microflakes was plotted in the x–y plane.