Introduction

Soft magnetic materials with stable permeability (μ) produce alternating magnetic flux under magnetic fields driven by current, and serve as the fundamental materials for basic electronic components such as inductor and transformer1. In the newest more-than-Moore integrated circuit packaging technologies, PCB-embedded inductors above 100 MHz are designed2, which require soft magnetic materials with stable permeability at such high frequency. Various Fe-based magnetic alloys with good static magnetic properties, i.e. large saturation magnetization (Ms > 200 Am2/kg) and high initial permeability (μi > 10000) have been developed3. However, soft magnetic materials are subject to working under alternative field, and the most fundamental requirement is a suitable permeability at working frequency from engineering perspective. Traditional Fe-based magnetic alloys suffer disastrous permeability “collapse” above kilohertz due to low electrical resistivity (ρ < 102 μΩ·cm) and domain wall resonance, leading to the malfunction of circuits containing the inductor. Spinel and hexagonal ferrites possess high resistivity (ρ > 1012 μΩ·cm), and thus can serve applications up to gigahertz4. Nevertheless, the low Ms (~50 Am2/kg) is not ideal for miniaturized inductors, and their low Curie temperatures (~200 °C) induces magnetic deterioration in integrated circuits. It remains challenging to prepare magnetic materials with simultaneous frequency-stable permeability and large saturation magnetization.

Soft magnetic composites, cold-pressed from insulated metallic magnetic particles, have shown great potential in high-frequency inductors due to their simultaneously high Ms and high ρ5. However, current soft magnetic composites use metallic magnetic particles with a multidomain structure6,7,8,9,10,11, which still leads to permeability decline above megahertz frequency as a result of domain wall resonance12. The coercivity also deteriorates (~800 A/m) compared with bulk materials (\( < \) 80 A/m)9,13,14, due to the internal stress introduced during cold-pressing process. Moreover, traditional soft magnetic composites crack easily due to the weak interlock capacity of spherical particles and bonding strength of the metal/coating interface. The cold pressing process also inevitably damages the integrity of insulating coatings and decreases electrical resistivity of soft magnetic composites. Evidently, new magnetic topology of particles and consolidation technique should be exploited to address the drawbacks in present soft magnetic composites.

A magnetic vortex is a basic magnetic topology in micro or nano magnetic materials. It contains a curling spin structure around a central region where magnetic moments are pointing out of plane to avoid creating a singularity15. Magnetic vortexes have been extensively investigated by virtue of their zero magnetostatic energy, topological protection and high resonance frequency15. However, previous research primarily focuses on two-dimensional magnetic vortexes and their potential employment in spintronics16. There are few reports on three-dimensional magnetic vortex materials and their application in traditional magnetic devices, which is mostly due to the difficulties in industrial scale production and effective consolidation of ultrafine magnetic particles. Cold sintering technique, a recently reported low-temperature low-pressure consolidation process, provides the possibility of consolidating micro or nano particles17,18. The basic process involves the uniform wetting of powders by aqueous solution, partially dissolution of solid surfaces in the hydrothermal environment, and the precipitation of mass at particle–particle interfaces19. Although cold sintering technique is commonly used for fabricating advanced ceramic materials, the dissolution-precipitation process also provides the possibility of one-step insulation and consolidation of ultrafine metallic particles with moderate pressure and temperature.

Here, we report on the effective cold sintering of soft magnetic composite from ultrafine FeSiAl particles. This composite features isolated magnetic vortexes rather than domain walls, as is the case in traditional multidomain materials from magnetic structure perspective. FeSiAl (known as Sendust alloys) presents zero magneto-crystalline anisotropy and magnetostriction constant simultaneously, which meets the intrinsic requirements of magnetic vortex formation. Gas atomization and airflow classification are used to produce micro-spherical FeSiAl particles with vortex structure in tens of kilograms. During the cold sintering process, the Al2SiO5/SiO2/Fe2(MoO4)3 multilayered heterostructure eventually forms, which magnetically isolates neighboring vortexes and covalently bonds these ultrafine FeSiAl particles. This unique structure endows the composite with frequency-stable permeability, relatively large magnetization saturation, low coercivity and high ultimate compressive strength.

Results and discussion

Microstructure characterization

The cold sintering process is shown in Supplementary Fig. 1. Ultrafine FeSiAl spherical particles (d50 = 2.6 μm, Supplementary Fig. 2) are mixed homogeneously with water and (NH4)6Mo7O24·4H2O (AMT). Afterwards, the slurry is cold sintered under 400 MPa and 250 °C for 1 h. The corresponding composite is designated as CS-AMT&H2O. For comparison, cold-sintered samples from FeSiAl powders and H2O (termed as CS-H2O), pure FeSiAl powders (termed as CS-powder-only), as well as traditional cold-pressed samples from phosphoric acid passivation with silicon resin binding (termed as CP-PA&SR), and with polyvinyl alcohol binding (termed as CP-PVA) are also prepared. Meanwhile, ultrafine (d50 = 2.0 μm) carbonyl iron and FeSiCrB amorphous particles are also mixed with water and AMT, and cold-sintered under the same condition to validate the universality of vortex-based composites. The corresponding composites are designated as CS-AMT-Fe&H2O and CS-AMT-Am&H2O, respectively.

The sectional view of CS-AMT&H2O composite by scanning electron microscopy (SEM) is shown in Supplementary Fig. 3, 4, which reveal highly compacted structure with low porosity. The energy dispersive spectroscopy (EDS) elemental mappings in Supplementary Fig. 3b and c indicate that Fe, Si and Al remain in the particle matrix, while Mo, Fe and O fill the inter-particle space. The microstructure of CS-AMT&H2O composite is investigated by transmission electron microscopy (TEM) in Fig. 1. For a FeSiAl particle with diameter of 2.0 μm in Fig. 1a, magnetic vortex structure is observed in Fig. 1b by differential phase contrast (DPC), which is entirely distinct from the multidomain structure in traditional soft magnetic material. Figure 1c presents the high-resolution TEM (HRTEM) image of the particle, in which (022) and (0\(\bar{2}\)2) planes of face-centered cubic (FCC) FeSiAl with d-spacing of 2.01 Å can be identified. The selected area electron diffraction (SAED) pattern in Fig. 1g confirms the monocrystal structure of the particle viewed along [100] direction. The characterizations of other particles in Supplementary Fig. 5 also demonstrate the ubiquity of monocrystal and vortex structure in the particles we adopted. Thus, the effect of grainboundary on magnetic properties can be excluded in later discussion.

Fig. 1: Microstructure and domain structure characterization of CS-AMT&H2O composite.
figure 1

a, b TEM and DPC image of the composite. c HRTEM image of FeSiAl particle. d White square region in (a). e STEM image of the dash rectangular region in (d). f HRTEM of sublayer III. g SAED pattern of FeSiAl matrix. h–j FFT patterns of sublayer I, II, III. k–p High angle angular dark field (HAADF) images and EDS elemental mapping of white square region in (d).

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Meanwhile, a distinct coating can be observed on the surface of FeSiAl particle, as indicated by white arrow in Fig. 1a. HRTEM image of the white square zone in Fig. 1d reveals that this coating layer is composed of three disparate sublayers. Enrichment of Al/Si in sublayer I, Si in sublayer II, and Mo/Fe in sublayer III are observed from the EDS mappings in Fig. 1k–p. Sublayer I is attributed to be Al2SiO5 viewed along axis of [100], which is confirmed by the diffraction spots of (022) and (040) planes in fast Fourier transform (FFT) in Fig. 1h. The interface between FeSiAl matrix and sublayer I is further investigated with spherical aberrated scanning transmission electron microscopy (STEM) in Fig. 1e, which clearly reveals the epitaxial growth of Al2SiO5 (022) plane on the FeSiAl (110) plane. Sublayer II presents no diffraction signal in the FFT image (Fig. 1i), indicating the formation of amorphous SiO2. The HRTEM image in Fig. 1f shows the nanocrystalline structure of sublayer III, in which (222) and (200) planes of Fe2(MoO4)3 with d-spacings of 2.95 Å and 4.70 Å are identified. The polycrystalline structure of sublayer III is also confirmed by FFT pattern with diffraction rings of Fe2(MoO4)3 indexed in Fig. 1j. The observation of Al2SiO5/SiO2/Fe2(MoO4)3 multilayered heterostructure is in consistent with the X-ray photoelectron spectroscopy (XPS) results in Supplementary Fig. 3e, f, which prove that metallic FeSiAl particles are partially oxidized during cold sintering.

Static and high-frequency properties of composites

We begin illustrating the static properties of vortex-based composite by exploring a single FeSiAl particle. Because it is still challenging to observe 3D magnetic structure directly, the projected DPC image for a FeSiAl particle with size of 150 nm is taken to illustrate the vortex structure (Supplementary Fig. 6). We also use micromagnetic simulation to investigate the magnetic feature of FeSiAl particle with different diameters (Fig. 2a). Theoretically, magnetic structure and coercivity of a particle depend on its geometry, intrinsic anisotropies and inter-particle interaction. For spherical FeSiAl particle, the effects of shape anisotropy, intrinsic magnetocrystalline anisotropy, as well as magnetostriction (stress anisotropy) can be excluded. It is found that for particles smaller than 50 nm, magnetic monodomain state is the most stable configuration. However, its coercivity is higher than 100 kA/m, which is not desired for soft magnetic composites. Although superparamagnetic behavior with small coercivity can be induced in monodomain nanoparticles, their magnetization will also be deteriorated due to thermal perturbation20. The magnetic structure transforms to vortex state from 50 nm to ~3.0 μm (Fig. 2b), and evolves to multidomain state with larger diameters. Meanwhile, the particle coercivity continues decreasing with increased diameters. We take the particle size distribution into consideration and develop a method to estimate the theoretical coercivity of the composite (Supplementary Table 1), with a value of 91 A/m obtained. As is well known, micromagnetic simulation represents the behaviour at 0 K, and thermal perturbation could decrease coercivity. Thus, the simulated value is higher than the room-temperature value (48 A/m) of CS-AMT&H2O composite (shown as red star point in Fig. 2a).

Fig. 2: Domain structure and static magnetic properties of FeSiAl particles.
figure 2

a Simulated domain structure and coercivity of FeSiAl particle with different diameters, the star point shows the experimental coercivity of CS-AMT&H2O. b Reconstructed semi sphere and disk vortex of FeSiAl particle with a diameter of 2.6 μm. c Simulated near-zero hysteresis loops for isolated and contacted four-vortex assemblies, the insets show the residual magnetization. d Measured near-zero hysteresis loops for CS-AMT&H2O and CS-H2O. e, f SEM fracture surfaces for CS-AMT&H2O and CS-H2O. g DPC image of FIB-thinned CS-AMT&H2O with 6 vortexes.

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Theoretically, magnetic exchange coupling takes place if the distance between two particles is smaller than exchange length (~10 nm). This inter-particle interaction may remarkably affect the static properties of vortex assembly. Figure 2c simulates the hysteresis loops for isolated and contacted four-particle vortex assemblies (simulation parameters are provided in method section), with coercivity of 64 A/m and 230 A/m obtained respectively. The remanence-state magnetization configuration (insets of Fig. 2c) for contacted vortex assembly illustrates that interfacial magnetic moment are exchange-coupled and aligned parallelly, which induce an effective anisotropic field and lead to larger Hc. On the contrary, isolated vortex facilitates magnetization reversal process and is beneficial for smaller Hc. It is interesting that isolated vortex assembly presents smaller slope of hysteresis loop than contacted assembly, which indicates a lower permeability. This can be explained from its higher Zeeman energy from the simulation results (Supplementary Fig. 7). Simulations of assemblies with more particles show similar results in Supplementary Fig. 8. In experimental demonstration, we compare the hysteresis loops for CS-AMT&H2O and CS-H2O composites in Fig. 2d, from which coercivity of 48 A/m and 270 A/m are obtained, respectively. For the CS-H2O composite, only H2O is applied during the cold sintering process. FeSiAl particles are supposed to contact with each other after the evaporation of H2O. The experimental values are in good consistence with simulation, signifying that the inter-particle interaction in CS-AMT&H2O can be neglected, and most FeSiAl particles act as independent magnetic vortexes. From the fracture surface shown in Fig. 2e, f, it is clear that FeSiAl particles in CS-AMT&H2O are well separated while particles in CS-H2O contact with each other (as indicated by the white arrow Fig. 2f). In the DPC image of FIB-thinned CS-AMT&H2O sample with 6 vortexes (Fig. 2g), we observed that most FeSiAl particles are magnetically separated by Al2SiO5/SiO2/Fe2(MoO4)3 multilayered heterostructure. Meanwhile, two contacted particles are also observed (Supplementary Fig. 9). At the contact point, neighboring magnetic moments are aligned parallelly to reduce the exchange energy at interface. At this circumstance, multidomain structure has lower energy (Supplementary Fig. 10), and is more stable than isolated vortex, in which neighboring magnetic moments along the domain wall are aligned antiparallelly. Electron holography characterization is also applied to confirm the vortex and multidomain structure of FeSiAl particles in Fig. 2f (Supplementary Fig. 11). Above results confirm the role of continuous Al2SiO5/SiO2/Fe2(MoO4)3 multilayered heterostructure as magnetic isolator in realizing single-magnetic-vortex composite with reduced coercivity.

Figure 3a, b plot the permeability spectra of CS-AMT&H2O composite and contrast samples. CS-AMT&H2O presents an initial permeability (μi, defined as permeability at 1 MHz) of 13 and remains almost constant up to 1 GHz. Meanwhile, CS-AMT-Fe&H2O and CS-AMT-Am&H2O composites present similar permeability stability. However, CS-H2O and CP-PA&SR composites present drastic permeability decline around 100 MHz. In the imaginary permeability, only one resonance peak above 1 GHz can be found for CS-AMT&H2O, CS-AMT-Fe&H2O and CS-AMT-Am&H2O composites, while additional resonance peak at 100 MHz is observed for CS-H2O and CP-PA&SR counterparts, which may correspond to the domain wall resonance. In Fig. 3c, CS-AMT&H2O also presents the highest qualify factor. For traditional CP-PA&SR composite, the high molding pressure (2 GPa) destroys the integrity of insulation coating (Supplementary Fig. 12), resulting in a similar frequency dependence of permeability to CS-H2O. Above results confirm the crucial role of magnetic isolation on permeability stability. Figure 3d, e compare the Ms, μi and maximal stable frequency (defined as frequency of 90% μi) of traditional magnetic composites, ferrites and our composites. We succeed in obtaining simultaneous high maximal stable frequency (up to 1 GHz), large Ms (105 ~ 176 Am2/kg, Supplementary Fig. 13) and high μi in the vortex-based composites made from ultrafine FeSiAl, Fe and amorphous FeSiCrB particles. Above excellent magnetic properties enable the vortex-based soft magnetic composites to be perfect magnetic core materials for high-frequency inductors. Figure 3f shows the performance of PCB-embedded inductor (Supplementary Fig. 14) made from CS-AMT&H2O. One can find that the inductor presents stable inductance below 1 GHz and much higher qualify factor at 100 MHz than literatures21,22,23,24,25,26, which is of great application potential in integrated circuits.

Fig. 3: High-frequency performance comparison.
figure 3

a–c Frequency dependence of real permeability (\({\mu }^{{\prime} }\)), imaginary permeability (\({\mu }^{{{{\hbox{'}}}{{\hbox{'}}}}}\)) and quality factor (Q) of different composites. d, e Comparison of Ms, μi, and maximal stable frequency between vortex-based soft magnetic composites in this work and literature results6,7,8,9,10,13,14,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56. Ferrites materials are colored by grey because of their high sintering temperature and low Curie temperature. f Performance of PCB-embedded inductor made from CS-AMT&H2O composite.

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The frequency-stable permeabilities in CS-AMT&H2O, CS-AMT-Fe&H2O and CS-AMT-Am&H2O composites originate from the vortexes that are magnetically isolated. As predicted by Snoek27, the permeability of magnetic material could remain stable until nature resonance frequency, which describes the uniform Larmor procession of all spins in the material. The nature resonance frequency (fr) is always at GHz, and limited by the product of Ms and electron gyromagnetic ratio (γ). The validation of Snoek’s predication is based on idealized assumptions of perfect monocrystals, rotational magnetization process etc., but most real magnetic materials deviate from ideality, such as polycrystalline and multidomain structure. The actual permeability declines drastically at frequencies that are much below those predicted ones. A magnetically isolated vortex, where spins are aligned spirally, facilitates the magnetization rotation process. This magnetic configuration matches well with Snoek’s assumption and only a nature resonance peak above 1 GHz is observed. On the contrary, the magnetic vortexes in CP-PA&SR and CS-H2O composites contact with each other, which induces multidomain structure and destabilizes the permeability against increasing frequency.

Cold sintering mechanism and mechanical strength

In practical application, high-frequency inductors are always packaged in printed circuit board by epoxy resin sealant at high pressure. Thus, adequate mechanical strength should be instilled in the inductor materials. Figure 4a compares the compression curve and mechanical strength of cold-sintered composites with those of traditional cold-pressed composites. It is observed that CS-AMT&H2O composite presents an ultimate compressive strength of 337.1 MPa, much higher than other samples. This result demonstrates that CS-AMT&H2O composite is consolidated efficaciously even though the pressure is much lower than that applied in cold-pressed samples (400 MPa vs. 2 GPa). The effective consolidation of ultrafine particles has been a bottleneck in powder metallurgy. To unveil the cold sintering mechanism of CS-AMT&H2O composite, the shrinkage curve of CS-AMT&H2O, CS-H2O and CS-powder-only composites during cold sintering process are recorded in Fig. 4b. In the case of CS-powder-only composite (indicated by green line), no obvious shrinkage is observed until 200 °C, after which creep deformation occurs, resulting in a total shrinkage of 1.19% at the end of heat preservation stage. For CS-H2O composite (indicated by blue line), the volume remains almost unchanged below the boiling point of water (100 °C). The water evaporation continues until the temperature reaches 200 °C, leading to a shrinkage of 0.35%. At temperature higher than 200 °C and in the heat preservation stage, a shrinkage rate similar to CS-powder-only composite and total shrinkage of 1.04% are observed, which correspond to the creep deformation of FeSiAl particles. However, in the case of CS-AMT&H2O composite (indicated by red line), the shrinkage begins with the onset of pressure, reaches the first stage at 200 °C and continues in the heat preservation stage, leading to a final shrinkage of 10.67%.

Fig. 4: Mechanical properties and sintering behavior of different composites.
figure 4

a Compression curve comparison between cold-sintered and cold-pressed composites. b Shrinkage curves of cold-sintered composites CS-AMT&H2O, CS-H2O and CS-powder-only. The inset shows the magnified curve for CS-H2O and CS-powder-only samples in the heating stage.

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High mechanical strength is the premise of engineering application. The crucial role of Al2SiO5 transition layer in enhancing the mechanical strength of CS-AMT&H2O composite is discussed by first principle calculation. The formation energy (γ) and interaction energy (Eint) of \({(110)}_{{{\mbox{FeSiAl}}}}/{(022)}_{{{{\mbox{Al}}}}_{2}{{\mbox{Si}}}{{{\mbox{O}}}}_{5}}\) interface (Fig. 5a) are calculated to be −1.45 eV and −15.04 eV. The negative γ indicates a favorable formation regime of interface, while the negative Eint reveals that the slabs are strongly bonded at the interface. Figure 5b shows the three-dimensional charge density difference in CS-AMT&H2O composite, where charge accumulation and charge reduction are coloured as yellow region and cyan region respectively. The charge density differences are mainly observed around the \({(110)}_{{{\mbox{FeSiAl}}}}/{(022)}_{{{{\mbox{Al}}}}_{2}{{\mbox{Si}}}{{{\mbox{O}}}}_{5}}\) interface. New Fe-O and Al-O bonds are found at the interface with lengths of 1.89 Å and 1.83 Å, respectively (Supplementary Fig. 15). The partial density of states (PDOS) in Fig. 5c and elements involved at the interface (Supplementary Fig. 15) reveal zero-band gap of the interface, suggesting an interfacial covalent bonding. The two-dimensional charge density differences in Fig. 5d illustrates that the interfacial interaction originate from the covalent nature of Fe-O and Al-O bonds. Due to the covalently bonded interface, the epitaxial Al2SiO5 sublayer adheres strongly to the metallic FeSiAl matrix. Moreover, compared with damaged and discontinuous FePO4 coating layer in traditional CP-PA&SR composite, the interparticle space in CS-AMT&H2O composite is filled with Fe2(MoO4)3 nanocrystals. The Al2SiO5 layer also acts as transition layer between metallic FeSiAl matrix and interparticle oxides, provides continuous binding of neighboring particles, and endows CS-AMT&H2O composite with good mechanical strength.

Fig. 5: First principle investigation of the interface.
figure 5

a STEM image and crystal structure of \({(110)}_{{{\mbox{FeSiAl}}}}/{(022)}_{{{{\mbox{Al}}}}_{2}{{\mbox{Si}}}{{{\mbox{O}}}}_{5}}\) interface. b Three-dimensional charge density difference of interface. c PDOS of FeSiAl, Al2SiO5 and interface. d Two-dimensional charge density difference of interface.

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Since the natural oxidization layer at FeSiAl surface is amorphous, as demonstrated in Supplementary Fig. 16, we conclude that this multilayered Al2SiO5/SiO2/Fe2(MoO4)3 heterostructure is formed in the hydrothermal environment of cold sintering process. As revealed by the sintering shrinkage curve in Fig. 4b, AMT aqueous solution can exist at temperature up to 200 °C, providing a hydrothermal environment in which FeSiAl surface is oxidized and dissolved. Al2SiO5 has more negative formation energy than Al2O3 and SiO2 mixture, while Fe2(MoO4)3 has more negative formation energy than Fe2O3 and MoO3 mixture (Supplementary Table 2), providing the thermodynamic principle for the formation of Al2SiO5/SiO2/Fe2(MoO4)3 multilayered heterostructure. In the solvent evaporation process, Al3+ and Si4+ with less solubility precipitate epitaxially as Al2SiO5 transition layer on the fresh surface of FeSiAl matrix. After that, excessive Si4+ precipitates as amorphous SiO2 sublayer outside Al2SiO5. Fe3+ and (MoO4)2+ with lager solubility in the solution finally crystalize as Fe2(MoO4)3 outer layer in the anaphasis of cold sintering.

Perspectives

Soft magnetic materials are subjected to working under alternative field. The most important and fundamental requirement in practical application is a suitable permeability at given frequency. In the past, permeability spectrum is phenomenologically related to domain wall resonance and domain rotation in multidomain/monodomain materials. Our results demonstrate that magnetic vortex, as the transitional state from monodomain to multidomain, can present frequency-stable permeability up to gigahertz frequencies when magnetically isolated. By applying ultrafine metallic magnetic particles with vortex structure, we can obtain large saturation magnetization simultaneously. The universality of frequency-stable permeability in isolated magnetic vortex can be validated by the cold-sintered composites from ultrafine Fe and amorphous FeSiCrB particles despite of their low mechanical strength. However, we believe that vortex-based SMC could also be achieved by FeSi, FeNi, FeNiMo or any other ultrafine soft magnetic particles. Meanwhile, ascribed to its high resonance frequency, we convinced that vortex-based composite also has great potentials in microwave applications such as radio frequency (RF) oscillators and spintronic devices.

In summary, we have proposed vortex-based FeSiAl soft magnetic composite with frequency-stable permeability, relatively large saturation magnetization, low coercivity and high mechanical strength. The magnetic vortexes are magnetically isolated and covalently bonded by the Al2SiO5/SiO2/Fe2(MoO4)3 multilayered heterostructure that is formed during cold sintering. The composite’s permeability maintains stable up to 1 GHz with a value of 13. Meanwhile, Ms of 105 Am2/kg, low Hc of 48 A/m and excellent ultimate compressive strength of 337.1 MPa can be achieved. Our study reveals the high frequency characteristics of magnetic vortex and sheds light on developing high-frequency magnetic devices.

Methods

Cold sintering of CS-AMT&H2O composite

Ultrafine FeSiAl particles with nominal composition of Fe85Si9.5Al5.5 (wt%) were prepared through gas atomization and airflow classification by Hunan Hualiu New Materials Co., Ltd, China. In gas atomization process, the FeSiAl ingot is melted at 1580 °C for 20 min in nitrogen atmosphere and atomized by a pressure of 5.5 MPa. 100 g FeSiAl particles and 8 g ammonium molybdate tetrahydrate ((NH4)6Mo7O24·4H2O) were mixed with water and dried to prepare the precursor. The precursor was mixed with 20 wt% water, homogenized with a pestle and mortar, and subsequently placed in a cylinder die (diameter of 12.7 mm) and cold-sintered at 250 °C in air for 1 h under a uniaxial pressure of 400 MPa. The cold-sintered samples were machined into toroid and strip for dynamic magnetic measurements.

Preparation of contrast samples

For CS-AMT-Fe&H2O and CS-AMT-Am&H2O composites, ultrafine (d50 = 2.0 μm) carbonyl iron and FeSiCrB amorphous particles are also mixed homogeneously with water and AMT, and cold-sintered under the same condition to CS-AMT&H2O. For CS-H2O composite, pure FeSiAl particles were mixed with 20 wt% water and conducted the above cold sintering process. For CS-powder-only composite, only FeSiAl particles were subjected to the cold sintering process. For CP-PA&SR composite, FeSiAl particles were passivated by 0.6 wt% phosphoric acid and mixed with 1.5 wt% silicon resin (SH-9602, provided by LSSH New Materials Co., Ltd, China), then pressed in a toroidal die under a uniaxial pressure of 2 GPa. For CP-PVA composite, FeSiAl particles were mixed with 8% polyvinyl alcohol and pressed in a toroidal die under a uniaxial pressure of 2 GPa.

Characterization

The crushing strength were measured by universal testing machine (WDW-200M, ZLC) with a loading rate of 0.2 mm/min. The magnetic properties were measured by a superconducting quantum interference device (SQUID, MPMS-XL-5, Quantum Design). The permeability spectra were measured by an impedance analyzer (E4991A, Agilent). The sample was machined to 3.0 × 1.5 × 0.5 mm to evaluate the performance of micro-inductors with three-turn windings. The valence states were investigated by X-ray photoelectron spectroscopy (XPS, Nexsa, ThermoFisher). The microstructure was observed by scanning electron microscopy (SEM, JSM-1T500HR, JEOL) and transmission electron microscope (TEM, JEM-ARM200F, JEOL) with a probe aberration corrector. The magnetic domain structure was observed using a TEM (Talos F200S, FEI) with differential phase contrast (DPC) mode. The elemental distribution was characterized by high-angle annular dark-field detector (HAADF) and energy dispersive X-ray detector (EDX). Before TEM observation, the composites were polished by a focused ion beam (FIB, Strata 400 S, FEI).

First principle calculation

Calculations were done within the framework of spin-polarized density functional theory (DFT) as implemented in the Vienna ab-initio simulation package (VASP). The Perdew–Burke–Ernzerhof (PBE) version of generalized-gradient approximation (GGA) was adopted to describe the exchange–correlation interaction among electrons. Hubbard U correction was included with effective U values of 4.3 eV for Fe-3d orbitals and 2.0 eV for Mo-4d orbitals. DFT-D2 method was used to calculate the van der Waals interactions. The plane wave cut-off was set to 420 eV. The Brillouin zone was sampled with a size-dependent G-centered k-point mesh, i.e., 7 × 7 × 7 and 5 × 5 × 7 for the primitive cells of FeSiAl and Al2SiO5, 7 × 5 × 1 and 5 × 4 × 1 for FeSiAl/Al2SiO5 simulations. The lattice parameters and atomic positions were fully relaxed until the variation of total energy was within 10−5 eV and the final force on each atom was less than 0.01 eV/Å. To simulate the interfaces, we adopted a periodic model that includes a four-layer slab of FeSiAl (110) and an adequate thickness of Al2SiO5 (022) slab respectively, with the lattice constants of 6.71 Å, 8.82 Å and 6.67 Å, 8.82 Å along a and b axes, respectively, and 15 Å thick vacuum in between slabs. The formation energy (γ) and interaction energy (Eint) of interfacial structure are calculated by formula (1) and (2):

$${\gamma }=\frac{1}{A}({{E}_{{surf}}^{{inter}}-\,E}_{{surf}}^{A}-{E}_{{surf}}^{B})$$
(1)
$${E}_{{{{{\mathrm{int}}}}}}=({{E}_{{surf}}^{{inter}}-E}_{{surf}}^{C}-{E}_{{surf}}^{D})$$
(2)

Here, \({E}_{{surf}}^{{inter}}\) is the interface energy, \({E}_{{surf}}^{A}\) and \({E}_{{surf}}^{B}\) correspond to the energies of two fully relaxed surfaces, \({E}_{{surf}}^{C}\) and \({E}_{{surf}}^{D}\) represent energy of surfaces separated from the interface. In addition, we calculated the difference between the charge densities of CS-AMT&H2O composite. The charge densities are calculated using the following equation: Δρ = (\({\rho }_{{surf}}^{{inter}}\) – \({\rho }_{{surf}}^{A}\) – \({\rho }_{{surf}}^{B}\)), here, \({\rho }_{{surf}}^{{inter}}\), \({\rho }_{{surf}}^{A}\) and \({\rho }_{{surf}}^{B}\) are the charge densities of interface and corresponding individual surface slabs, respectively.

Micromagnetic simulation

The micromagnetic simulation was carried out by mumax software. The cell size was varied from 0.1 × 0.1 × 0.1 nm3 to 50 × 50 × 50 nm3 depending on the simulation scale. The material parameters of FeSiAl, i.e., the saturated magnetization (Ms), the magnetocrystalline anisotropy constant (K1), and the exchange stiffness (Aex) were set to be 1.0 T, 0 J·m−3, and 13 × 10−12 J·m−1. In the simulation of Fig. 2b, the cell size and diameter for bigger particle are 20 nm and 2560 nm respectively. In the isolated configuration, the smaller particle has a diameter of 1000 nm, and the distance between each particle is 100 nm. In the contacted configuration, the smaller particle has a diameter of 1240 nm, and it overlaps with neighboring particles by 20 nm.