Critical state-induced emergence of superior magnetic performances in an iron-based amorphous soft magnetic composite

Soft magnetic composites (SMCs) play a pivotal role in the development of high-frequency, miniaturization and complex forming of modern electronics. However, they usually suffer from a trade-off between high magnetization and good magnetic softness (high permeability and low core loss). In this work, utilizing the order modulation strategy, a critical state in a FeSiBCCr amorphous soft magnetic composite (ASMC), consisting of massive crystal-like orders (CLOs, ∼1 nm in size) with the feature of α-Fe, is designed. This critical-state structure endows the amorphous powder with the enhanced ferromagnetic exchange interactions and the optimized magnetic domains with uniform orientation and fewer micro-vortex dots. Superior comprehensive soft magnetic properties at high frequency emerge in the ASMC, such as a high saturation magnetization (M s) of 170 emu g−1 and effective permeability (μ e) of 65 combined with a core loss (P cv) as low as 70 mW cm−3 (0.01 T, 1 MHz). This study provides a new strategy for the development of high-frequency ASMCs, possessing suitable comprehensive soft magnetic performance to match the requirements of the modern magnetic devices used in the third-generation semiconductors and new energy fields.

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Introduction
Soft magnetic materials and related devices including transformers, inductors, electric motors, and electronics play crucial roles in energy storage, conversion, and transmission, which have revolutionized our lives [1,2].With the development of new energy vehicles and 5G technology, greater challenges are posed to the high-frequency performances of electronic power devices [3,4].Soft magnetic composites (SMCs) consisting of metallic powders coated by an electrical insulating layer [5,6], possess abundant distributed air gaps and interfaces.As a result, SMCs exhibit several advantages over traditional laminated steel cores, such as extremely low eddy current loss, three-dimensional (3D) isotropic magnetic behavior and flexible machine design and assembly, thus standing out from other soft magnetic materials for high-frequency applications [5].Compared with the conventional SMCs such as Fe, Fe-Si, Fe-Ni and Fe-Si-Al [7,8], amorphous soft magnetic composites (ASMCs) possess higher resistivity, lower coercivity (H c ) and much lower core losses (P cv ), thereby considered as the preferential choice for medium and highfrequency applications [9,10].
Several representative ASMC systems, including atomized powders of FeSiBC, FeSiBCCr and FeSiBP, have been developed in the past decade [11][12][13].However, since the cooling rate of atomization, especially the gas atomization, is well below that of conventional melt spinning method, the content of ferromagnetic elements in the ASMCs is limited to a lower level than that of the amorphous ribbons [14].Consequently, the saturation magnetizations (M s ) of the ASMCs are far below those of the typical amorphous ribbons such as METGLASS [15] and letting alone Fe-Si crystalline SMCs [16], which has become a main drawback restricting the miniaturization of modern electronics.Apart from the M s , effective permeability (µ e ) and P cv are also two important soft magnetic properties of SMCs.To improve the comprehensive properties, enduring efforts mainly focused on process modification have been devoted.On one hand, numerous insulation layers including organic adhesives such as epoxy resins (EPs), phenol resins and silicone resins, as well as inorganic substances such as silica (SiO 2 ), magnesia (MgO) and titania (TiO 2 ) have been modified to increase the resistivity of ASMCs, thus reducing the eddy current loss (P e ) [12,17,18].However, the reduction of P cv is generally at the expense of M s and µ e .On the other hand, the direct way to increase M s and µ e is to increase the density of ASMCs through high-pressure compaction, particle size matching and sintering [10], yet these strategies usually introduce large internal stresses or destroy the insulation layer, resulting in the compromise of P cv .Therefore, breaking the trade-off between high M s /µ e and low P cv is a tricky problem for the broad applications of ASMCs.
The soft magnetic properties of ASMCs not only depend on the molding process-related parameters including the insulation layer, compaction density, air gaps and microcracks but also the intrinsic properties of the amorphous powders.Improving the intrinsic soft magnetic properties of amorphous powders provides another way to achieve the synergy of magnetization and magnetic softness.Recently, a novel strategy to construct an amorphous nanocrystalline transitional microstructure between amorphous and traditional nanocrystalline alloys was proposed to overcome the trade-off between saturation magnetic induction (B s ) and H c [19].However, because of the absence of nucleation element (Cu) and large atoms (Nb, Mo and V, etc.) to suppress diffusion, it is hard to achieve the controlled precipitation of fine nanocrystals (<20 nm) for the amorphous alloy systems such as FeSiBC(Cr) [20].It should be emphasized that one of the important features of amorphous alloys is the existence of abundant orders [21].Therefore, it may be feasible to construct a distinctive amorphous structure in which specific orders rather than nanocrystals are embedded for the classical amorphous systems.
In this work, a distinct amorphous structure with a critical state was developed based on an order modulation strategy to fabricate ASMCs with prominent comprehensive soft magnetic properties.Annealing amorphous alloys under a transverse or longitudinal magnetic field is beneficial to the formation of medium-range order (MRO) [22,23].Compared with the unidirectional magnetic field annealing, the rotating magnetic field annealing (RFA) not only promotes the formation of ordered structures, but also can reduce the unexpected field-induced magnetic anisotropy that has a detrimental effect on magnetic softness [24].Inspired by this, a critical-state amorphous structure (CSAS), manifested as 1 nm-sized crystal-like orders (CLOs) sparsely dispersed in the amorphous matrix, was obtained through an ingenious annealing treatment assisted by a rotating magnetic field.This enabled the ASMCs to possess a feature of 'Two High and One Low' in soft magnetic property, i.e. high M s of 170 emu g −1 , high µ e (100 kHz) of 65 and low P cv (0.1 T, 100 kHz) of 560 mW cm −3 .In particular, the critical-state ASMCs possessed superior high-frequency characteristics with a constant µ e up to 10 MHz.The microstructure and magnetic domains were investigated systematically to elucidate the magnetic softness-magnetization synergy of the critical-state ASMCs.

Structural design strategy
The modification of soft magnetic properties of the Fe-based alloys can be achieved through a universal strategy of order modulation as illustrated in figure 1(a).When alloys possess a certain order, a specific property emerges.For example, the Fe-based amorphous alloys with long-range disordered structures show low H c , while their crystalline counterparts such as pure iron and silicon steel with long-range ordered atomic arrangements possess high B s .A common strategy to achieve the combination of high magnetization and magnetic softness in the Fe-based amorphous alloys is nanocrystallization to construct a confined ordered structure [25,26].The key of this method is the precipitation of fine (10-20 nm) and uniform nanocrystals with a volume percentage larger than 70% [27].A typical alloy system is FINEMET-type amorphous alloy containing moderate Fe element, appropriate metalloids (B and Si, to increase GFA), and a small number of Cu (to promote nucleation) and Nb (V, Mo, etc. to suppress growth) elements [28].However, there are lots of Fe-based amorphous compositions without the nucleation-promoting (Cu) and diffusion-inhibiting (Nb, V, Mo, etc.) elements, possessing only narrow or undetectable crystallization window [25,29,30].For these alloy systems, such as the typical atomized powders mentioned above, the classical nanocrystallization method cannot be applied to.In this scenario, the improvement of soft magnetic properties and achieving the magnetization and magnetic softness synergy is a major challenge.The order modulation between amorphous and nanocrystalline structures, benefiting from the multiscale orders in amorphous alloys, open a new door to the solution of this dilemma.Herein, we provide an innovative structural design strategy, i.e.CSAS consisting of considerable CLOs (∼1 nm) rather than nanocrystals dispersed in an amorphous matrix, which led to the emergence of superior comprehensive soft magnetic properties.On one hand, the avoidance of nanocrystals overcame the fatal disadvantage of harsh nucleation-growth processes in unfavorable amorphous systems [30].On the other hand, the introduction of massive CLOs with α-Fe features improved the soft magnetic properties [19].

Materials:
Commercial FeSiBCCr amorphous powders with D 50 = 18 µm (Advanced Technology (Bazhou) Special Powder Co., ltd) were used as the raw materials.Phosphoric acid, EP, zinc stearate, and acetone used in this study were purchased from the Macklin Reagent company.All chemicals were of analytical grade and used without any further purification.

Sample preparation:
Toroidal magnetic cores were prepared by the following steps.(1) Passivation: The FeSiBCCr amorphous powder was placed in acetone containing phosphoric acid (1 wt.% of the powder) and passivated for 10 min to form a suitable inorganic oxide layer.(2) Organic coating: EP (2 wt.% of the powder) was fully dissolved in acetone.The passivated powders were, then, put into this solution with continuous stirring until the complete evaporation of acetone.The powder was dried in a vacuum at 50 • C. (3) Cold compaction: The coated powder (∼1.5 g) was compressed into a ring with an outer diameter of 13 mm and an inner diameter of 8 mm under 2 GPa pressure.(4) Heat treatment: heat treatment was performed in a compound magnetic field heat treatment furnace to release the internal stress and modify the microstructure of the amorphous powder.In this study, the rotating magnetic field was achieved through a unidirectional magnetic field parallel to the upper surface of the magnetic core, combined with a rotating sample stage.The schematic diagram is shown in figure 1(c).

Thermal and structural characterizations:
The Curie temperature (T C ) and crystallization temperature (T x ) of the atomized amorphous powder were measured by differential scanning calorimeter (DSC, Netzsch 404 F3).Phase structures of the samples were analyzed by x-ray diffraction (XRD, Bruker D8 ADVANCE), Cu Kα.The morphology and elemental composition of the powder, the coating layer, and the cross section of cores were studied by a scanning electron microscopy (SEM, Verios 5UC) coupled with energy dispersive x-ray spectroscopy (EDS).Nanoscale structural features of the samples were characterized by a high-resolution transmission electron microscope (HRTEM, Talos F200X G2).The specimens for the TEM analysis were prepared using a Focused ion beam/scanning electron microscope (FIB, Helios 5UX).X-ray photoelectron spectrometer (XPS, ESCALAB XI+) and Fourier transform infrared spectrometer (FTIR, INVENIO-R) were used to investigate the composition and chemical state of the insulation layer.

Magnetic tests:
Magnetic properties inclusive of M s , µ e and frequency dependence of P cv were measured by a magnetic property measurement system (MPMS, Quantum Design), an impedance analyzer (Agilent 4294A), and an AC B-H analyzer (IWATSU, SY-8218), respectively.The magnetic domains analyses were performed by magnetic force microscopy (MFM, Bruker Dimension Icon) combined with Lorentz TEM (Talos F200X G2).

Molecular dynamics simulation:
The coupled molecular dynamics and spin dynamics simulations were performed to account for the structural relaxation dynamics from the magnetic and lattice interaction [31,32].The selected composition was Fe 95 Co 5 , with the spin-lattice Hamiltonian of the system written as: where ⃗ r i , ⃗ p i , ⃗ s i , and ⃗ m i designate the atomic position, atomic momentum, the normalized magnetic moment, and the mass of atom i, respectively.U (⃗ r ij ) is the interatomic potential energy, described by the embedded-atom method (EAM).The force-filed parameters of the EAM potential were taken from [33].H mag (⃗ r,⃗ s) is the interatomic magnetic energy: where, µ i , ⃗ H ext , and J (r ij ) stand for the Landé factor, the atomic magnetic moment, the external magnetic field, and the exchange parameter, respectively.J (r ij ) is the magnetic exchange term, defined as: where Θ (R c − r ij ) is the Heaviside step function and R c is the distance cutoff.The parameters, a, b, and d, are the fitting coefficients for the exchange interaction.H N ′ eel (⃗ r,⃗ s) is the Néel interaction, used to describe the interaction between pairs of magnetic spins.The Néel interaction that only considers the magnetic dipole interaction can be expressed as: where ⃗ e ij =⃗ r ij /r ij , g (r ij ) can be written in the form of the Bethe-Slater function as the exchange term: where a, b, and d are the three coefficients fitted to the magnetoelastic interaction.These interactions have been implemented in the SPIN package of LAMMPS [32,34].The required force-field parameters include the EAM potential parameters and the magnetic parameters for the exchange term and the Néel term.The magnetic parameters are taken from the works of P Nieves for Fe [35] and D Beaujouan for Co [36].The crossinteraction parameters between Fe and Co atoms utilize the value averaging over Fe and Co for a, b, and d.The mechanical forces have been offset by the exchange interaction to counter the implicitly comprised magnetic interaction in the EAM potential.
The simulation contains 10 000 atoms with periodic boundary conditions applied in all dimensions.Samples were prepared at 2000 K, then cooled down to 750 K with a cooling rate about 1.0 × 10 14 K s −1 , and the simulation timestep 1 fs.The external magnetic field was applied on the samples at 750 K, with the direction of the magnetic field rotating in x-y plane in a period about 7.2 × 10 −11 s.To compare the nucleation incubation time, two magnetic intensities (i.e.H = 0 and 0.1 T) were simulated.The final crystallization time was calculated by averaging 20 samples with different random seeds in initial configurations.The fine powders exhibit high sphericity and smooth surface except a small number of irregular particles.The amorphous features of the atomized powders were confirmed through XRD and DSC as shown in figures 1(d) and S1(b).Broad diffusion humps at ∼45 • and ∼80 • were detected in the XRD pattern and a sharp exothermic peak corresponding to crystallization was observed in the DSC curve.The T x was determined to be 570 • C. To achieve good high-frequency properties, passivation with phosphoric acid followed by the application of insulating EP was applied as shown in figure 1 The chemical states of the FeSiBCCr amorphous powders before and after the coating process were analyzed by FTIR and XPS.Broad bands at ∼1634 cm −1 , corresponding to the infrared absorption of -OH groups, can be detected for all the samples, and its intensity enhances following passivation and EP coating (figure S2(f)).Additionally, the presence of the Fe-O vibrational peak and the enhanced C-O-C peak indicate the formation of phosphates during passivation.Following the insulation coating, some specific peaks including C-H, C-C, and C=O, etc, originating from EP can be observed.On the other hand, the XPS spectra show the appearance of P 2p after passivation, while the disappearance of Fe 2p after EP coating (figure S2(g)).Figures S2(h)-(k) display the C 1s and O 1s XPS spectra.Compared with the untreated powder, the absence of Fe-O, Cr-O, and Si-O species as well as the presence of C-O and C=O groups in the coated powder can be observed.The results, thus, indicate that the surface of the particle is covered by the passivation film and EP insulation layer, completely.

FeSiBCCr ASMC with critical state
A series of RFA treatments under 0.1 T were performed to construct the CSAS of the FeSiBCCr powder.It is worth noting that a faint endothermic peak appears at 375 • C before crystallization (figure S1(b)), which is attributed to the magnetic transition of the amorphous phase from ferromagnetic to paramagnetic state.The RFA treatments were, therefore, conducted near T x to slightly lower than the Curie temperature (T C ), and the corresponding XRD patterns of the annealed powders are displayed in figure 1(d).At 550 • C (close to T x ), global crystallization, manifested as multiple sharp crystalline peaks, was detected.With the annealing temperatures below 500 • C, no distinction can be made between the XRD patterns of the annealed and the atomized samples, indicating a completely amorphous structure.However, an inconspicuous crystalline peak appears at ∼45 • for the sample annealed at 500 • C, which may correspond to the initial stage of crystallization.Therefore, the contiguous state formed by annealing the sample at temperatures below 500 • C via RFA (in short, 500-RFA), such as the 475-RFA, can be referred to a critical state of the amorphous alloy.
TEM characterization was performed to identify the critical-state amorphous alloy.Figure 2(a) shows the HRTEM image of the atomized FeSiBCCr powder.The disordered structure and halo-like selected area electron diffraction (SAED) pattern further verify the fully amorphous state of the raw powder.Some coarsen crystals including α-Fe(Si) and FeB phases with diameters larger than 500 nm precipitate in the amorphous matrix of the 500-RFA powder (figure 2(b)).However, only miserly nanocrystals (Region A) are observed in the 475-RFA sample (figure 2(c)), which signifies that it corresponds to a critical state between the amorphous and nanocrystalline alloys.The fine nanocrystals are also ascertained to be α-Fe(Si) from the SAED pattern (figure 2(d)) and HRTEM image (figure 2(e)).It is noteworthy that a pair of diffraction spots within the first ring, weaker in intensity than sharp spots from bcc spots, can also be observed, which is consistent with the result of [19].This observation suggests the existence of some MROs in the amorphous matrix [37], which generally ranges from 0.5-2 nm for amorphous alloys [38].Typical MROs in the Fe-based amorphous alloys are reported as a kind of CLOs, playing an important role in tuning mechanical and soft magnetic properties [19,39,40].
The auto-correlation function (ACF) process was used to analyze the local features of CSAS in detail.Figures 2(f) and (g) display the HRTEM of the amorphous region and corresponding inverse Fourier transform (IFT)-filtered HRTEM images of the 475-RFA powder.Although the sample exhibits a completely disordered structure at first glance, there are some local regions with incomplete fringe-like patterns such as the Regions D and E in figure 2(g).Remarkable lattices can be observed in their ACF images (figures 2(j2) and (k2)), and these regions belong to perfect CLOs (PCLOs).The translational symmetry in ACF image is analogous to that of α-Fe as depicted in figure S3, which was also reported in other Fe-based amorphous alloys [39,40].Differently, no translational symmetry can be observed in both the TEM and ACF images of the fully disordered regions (figures 2(h1) and (h2)).Furthermore, an intermediate structure transiting from the disordered structure to PCLO, exhibiting weaker lattice feature in the ACF image, can also be screened, which is marked as imperfect CLO (ICLO) as shown in figures 2(i1) and (i2).To quantify the CLOs, two-dimensional (2D) ACF mappings consisting of 576 segments with the size of 1.01 × 1.01 nm 2 of the 475-RFA sample were obtained as shown in figure 2(l).
The red boxes are the PCLOs and the green regions correspond to the ICLOs.The areal fraction of the CLOs was calculated to be 21.2%.Moreover, these CLOs distribute uniformly in the amorphous matrix.Although some CLOs already exist in the atomized powder (figure S4), their areal fraction is only 11.3%, which is much less than that of the critical-state amorphous alloy.It means that an appropriate RFA can promote the formation of CLOs without obvious precipitation of nanocrystals.Based on the above results, it is concluded that a CSAS with ∼1 nm-sized CLOs dispersed in an amorphous matrix was obtained in the FeSiBCCr amorphous powder.To further elucidate the effect of RFA on the order modulation of Fe-based amorphous alloys, molecular dynamic simulation was performed, and the crystallization behaviors of the samples under different magnetic fields were obtained (figure 2(m1)).It can be seen that the incubation period of crystallization under the rotating magnetic field of 0.1 T decreases clearly, compared with the original sample (figure 2(m2)).The averaged incubation period of 20 independent samples decreases from 3.512 to 2.337 µs (figure 2(m3)), signifying acceleration of the nucleation process.Therefore, it can be concluded that RFA promotes the ordering of Fe-based amorphous alloys, leading to the formation of CSAS.

Improved comprehensive soft magnetic properties
Figure 3(a) shows the hysteresis loops of the amorphous powders annealed at different conditions, and all samples are magnetic softness manifesting as S-shape curves.The M s first increases rapidly from 155 emu g −1 to a higher value of 168 emu g −1 annealing at 350 • C, and then further increases to 178 emu g −1 as the critical-state is achieved at 475 • C. When the annealing temperature elevates to 550 • C, crystallization is completed and the M s increases to a value as high as 180 emu g −1 .In general, the best suitable temperature range for magnetic field annealing treatment to improve M s of amorphous alloys occurs at a temperature near T C because of an enhanced ferromagnetic exchange coupling [41].In the present case, the optimal annealing temperature is appreciably higher than T C and approaching T x .Compared with the raw powder, the M s of the 475-RFA sample enhances from 155 to 178 emu g −1 , increased by 14.8%.It means that RFA at a temperature far higher than T C to construct a CSAS can significantly enhance the M s of amorphous alloys.
Although the coating layer is deleterious to M s to a certain extent as shown in the insert of figure 3(a), it endows the magnetic core with improved high-frequency performances.Figure 3(b) shows the frequency dependence of µ e for magnetic cores RFA at different conditions.All samples possess considerable frequency stability maintaining a constant µ e within the frequency range up to 10 MHz.Compared with the original ASMC, the value of µ e increases from 19 to 65 when the ASMC is annealed at 475 • C corresponding to the formation of CSAS, much larger than that of the sample annealed near T C of 350 • C [39].Nevertheless, once the critical state is exceeded, such as RFA at 500 • C, the µ e reduces sharply because of the precipitation of coarsened crystals in the amorphous matrix (figure 2(b)).As the amorphous powder crystallizes completely at 550-RFA, the magnetic softness dissipates, showing a quite low µ e .In general, the µ e of ASMC depends on three aspects: the compact density of ASMCs, the release of internal stress and the intrinsic structure of the amorphous powder [6].Because of the same fabrication procedure of the original ASMCs, the compact density of different magnetic cores can be regarded as fixed.The maximum µ e obtained at the CSAS is attributed to the complete elimination of the internal stress and the unique structure.On one hand, the higher annealing temperature is beneficial to release the internal stress, eliminating the unfavorable pinning effect of the magnetic domains, hence, improving µ e .On the other hand, the formation of considerable CLOs with α-Fe(Si) feature results in the suppression of magneto-crystalline anisotropy and the reduction of saturation magnetostriction, which is also responsible for the increase of µ e [42,43].
The quality factor Q is another important parameter for high-frequency applications, reflecting the energy storage and loss performance of soft magnetic materials during alternating magnetization [5]. Figure S6 displays the Q value versus frequency of the ASMCs annealed under different conditions.For all the samples, with the elevation of frequency, the Q value first increases and then decreases symmetrically, exhibiting a peak near 1 MHz.The full width at half maximum peak of each curve starts at ∼100 kHz and ends at ∼5 MHz, which signifies the optimal application range of the studied ASMCs.In brief, the Q values are improved to a certain extent for higher annealing temperature, however, the variation is not monotonous.The maximum Q value exceeds 60, implying a low core loss.Figure 3(c) shows the P cv of the ASMCs measured at a series of frequencies under a field of 0.1 T. It is clear that the P cv increases drastically with the frequency increment.At the same frequency, initially, the P cv declines significantly (to 560 mW cm −3 at 100 kHz) as the annealing temperature elevates to 475 • C, and then increases when the temperature exceeds the critical value.Due to the uncontrolled crystallization occurring at 550-RFA, the P cv increases sharply and the values over 25 kHz cannot be detected.In general, the total core losses P cv can be divided into three parts including the hysteresis loss (P h ), eddy current loss (P e ), and residual loss (P r ) [44][45][46].The P r results from the relaxation and resonance losses [5], and can be avoided in this study (figure S7(a)).Figure S7(b) shows the frequency dependence of the separated losses.The P h and P e exhibit the same variation trend with the change of frequency and annealing condition.The minimum values of both P h and P e were obtained for the 475-RFA ASMC with CSAS.As a representative, the P cv of 475-RFA was measured up to megahertz to further elucidate the development of the loss at higher frequencies (figure S8).It can be seen that the P h is predominant within 200 kHz, while P e increases drastically after 300 kHz and is much larger than the P h at megahertz.That is why the ASMCs with intrinsic high resistivity and insulation layer are preferred at high frequencies.
To highlight the prominent comprehensive soft magnetic properties of the ASMC with CSAS, previously reported data including M s , µ e (100 kHz) and P cv (0.1 T, 100 kHz) of the ASMCs prepared by the atomized powders were summarized [11,12,17,18,[47][48][49][50][51], and compared with the current study in figures 3(d) and (e).It can be seen that µ e of most ASMCs is less than 60 combined with a limited M s lower than 150 emu g −1 which is not favorably comparable to some of the traditional metallic magnetic powders such as the FeSi powder [16].To further enhance the M s of the ASMCs, crystalline FeCo powder with extremely high M s is often added as another magnetic phase [47,50].Through this strategy, although the M s of SMCs can be enhanced to a high value even near 200 emu g −1 , the µ e decreases remarkably, to less than 20 [50].However, the ASMC developed by order modulation strategy not only has a prominent µ e of 65 that surpasses most of the ASMCs, but also possesses a high M s of 170 emu g −1 comparable to that of SMC containing 50 wt.%FeCo powder, but, at a cost well below it.Apart from the 'double high' of M s and µ e , the ASMC with CSAS also possesses low P cv .As shown in figure 3(e), the increment of µ e is usually accompanied by the compromise of P cv , presenting a trade-off relationship.Herein, the ASMC with CSAS achieve the combination of a high µ e of 65 and a significantly low P cv of 560 mW cm −3 (0.1 T, 100 kHz), and especially 70 mW cm −3 (0.01 T, 1 MHz), overcoming the trade-off dilemma of µ e -P cv .

Magnetic structure origin of the magnetic softness
Results of this study indicate that designing amorphous powder with critical-state structure is an effective strategy to obtain superior comprehensive soft magnetic properties and of great promise to break the µ e -P cv trade-off relationship of ASMCs.Subsequently, we attempt to unveil the magnetic structure origin of the enhanced comprehensive soft magnetic properties.Figures 4(a)-(d) show the 3D MFM images of the ASMCs.It should be noted that the samples used for MFM observation were polished parallel to the direction of the magnetic field applied during the annealing treatment, and each image was obtained in an individual particle to eliminate the influence of interface and non-magnetic substances.The average domain width was also determined and plotted in figure 4(e).The original ASMC exhibits a typical maze-like domain, which indicates the coexistence of complex anisotropies resulting from the inner field and large internal stress [41].Hence, a high applied magnetic field is needed to drive the movement of the domain walls along with their multipath rotation towards the field direction during magnetization, leading to a poor magnetic softness, i.e. low µ e and high P cv .Under the condition of 350-RFA, the domains become more ordered and wider, with the domain width increasing from 0.49 to 1 µm, forming a branch-like structure (figure 4(b)), associated with a low-stress state.Nevertheless, the rugged edges and irregular arrangement of the domains cause a strong pinning effect [52], resulting in a certain degree of improvement in the soft magnetic properties but not reaching the optimal level.As the temperature increases to 475 • C, regular stripe-like domains with a larger width of 1.75 µm and smooth edge were obtained (figure 4(c)).No obvious pinning site can be observed, which facilitates the unobstructed movement of the domains and leads to the improved magnetic softness including high µ e and low P cv .On the contrary, after the full crystallization of the amorphous powder, the mazelike domains reappear (figure 4(d)), yet become more delicate (compared to the original sample) with the domain width sharply reducing to 0.47 µm, resulting in the rapid deterioration of soft magnetic properties.Furthermore, dynamic analyses of the magnetic domain wall were conducted by Lorentz TEM (LTEM).As shown in figure 4(f), the domain walls of the original ASMC are dendritic and some micro-vortex dots can be seen (circled by yellow rings).In contrast, the 475-RFA sample possesses a smooth domain wall with fewer micro-vortex dots (figure 4(h)), leading to a low H c [53].Under the same applied magnetic field, the domain walls in the 475-RFA sample move more easily and further than that in the original ASMC (figures 4(g) and (i)), consistent with the results of MFM analyses.Therefore, the regular domains and weak pinning effect of the domain walls contribute to the superior magnetic softness of the FeSiBCCr ASMC with critical-state structure.
The CSAS is a transitional state between the amorphous and classical dual-phase nanocrystalline alloys as shown in figure 1(a).Different from the reported results [19], due to the absence of Cu and V elements in amorphous alloy system, the controlled precipitation of fine nanocrystals is difficult to realize in the FeSiBCCr amorphous powder.However, the CSAS, containing only miserly crystal nuclei accompanied with the remarkable increment of CLOs (∼1 nm), is a desired structure.Both structures can alleviate the trade-off between magnetization and magnetic softness, indicating the promising prospect of order modulation based on amorphous features to modify soft magnetic properties.The commonality in obtaining the two transitional structures is to anneal the amorphous alloy under complex fields, which is designed as a temperature field assisted by a rotating magnetic field in this study.In addition, the appropriate application of stress or electric field may also be feasible to construct transitional or criticalstate amorphous alloy because of their analogous effects on magnetic properties [54,55], which sheds new light on the design of high-performance soft magnetic amorphous alloys.

Conclusions
On base of the order modulation strategy, a CSAS with multitudinous CLOs dispersed in the amorphous matrix was obtained through an elaborate annealing treatment under a rotating magnetic field.Arising from the unique microstructure, the critical-state ASMC of FeSiBCCr possesses high M s up to 170 emu g −1 , constant µ e of 65, stability up to 10 MHz, and an extra low P cv (0.01 T, 1 MHz) of 70 mW cm −3 , thus realizing the combination of superior magnetization and softness.The high M s benefits from the enhanced exchange interactions caused by the abundant formation of 1 nm-sized CLOs with the feature of α-Fe(Si).The remarkable magnetic softness is attributed to the regular magnetic domains together with the weak pinning effect of the domain walls induced by the CSAS.Such a strategy for constructing a critical-state amorphous alloy and domain structure can allow us to develop novel ASMCs to break the trade-off relation between magnetization and magnetic softness, and promote the development of modern electronics, especially in high-frequency fields.

Future perspectives
ASMCs, combining the features of conventional metallic soft magnetic materials and soft ferrites, are ideal high-frequency soft magnetic materials with promising comprehensive properties.Typical fabricating processes include powder preparation, insulation coating, compaction molding, and heat treatment/structural modulation, with each step playing a crucial role in the final properties of ASMCs.The complex process causes some tricky problems to reconcile.For example, the introduction of non-magnetic coating layers reduces the eddy current loss, yet at the expense of magnetization, and large-stress compaction is conductive to high magnetization, while the inevitable internal stress is deleterious to magnetic softness.Nevertheless, the flexible process also provides us with more possibilities to develop high-performance soft magnetic materials.In the further, on the one hand, the synergism of high permeability, low core loss, high magnetization and high application frequency can be achieved through process optimization, such as using novel coating layers (magnetic and insulated) and new techniques in compaction.On the other hand, developing new powder compositions and modifying the intrinsic microstructure such as the order modulation strategy and nanocomposites engineering allow us to break the trade-off between high magnetization and superior magnetic softness.It is certainly possible to develop highperformance ASMCs and entirely soft magnetic materials, and the scientific community has already launched some activities that will promote the transform of the power electronics fields, especially the third-generation semiconductor related devices.

Figure 1 .
Figure 1.Structural design strategy and fabrication of the FeSiBCCr ASMC with critical-state amorphous structure (CSAS).(a) Schematic diagram showing the structural design strategy to optimize the soft magnetic properties of the Fe-based alloys through order modulation.(b) and (c) schematic illustration of the preparation process and rotating magnetic field annealing (RFA) treatment of the amorphous soft magnetic composite (ASMC).(d) XRD patterns of the amorphous powders annealed under different conditions.

4. 1 .
Figure S1(a) shows the SEM morphology of the FeSiBCCr amorphous powder prepared by the multistage atomization.The fine powders exhibit high sphericity and smooth surface except a small number of irregular particles.The amorphous features of the atomized powders were confirmed through XRD and DSC as shown in figures 1(d) and S1(b).Broad diffusion humps at ∼45 • and ∼80 • were detected in the XRD pattern and a sharp exothermic peak corresponding to crystallization was observed in the DSC curve.The T x was determined to be 570 • C. To achieve good high-frequency properties, passivation with phosphoric acid followed by the application of insulating EP was applied as shown in figure1(b).The morphologies of the passivated and coated powders are shown in figures S2(a) and (b), respectively.Compared with the untreated powder (figure S1(a)), the surfaces of the coated powders are rougher.The total thickness of the coating layer is approximately 110 nm (figure S2(c1)), composed of the passivation compounds and EP (figures S2(c2)-(c6)).Figure S2(d) shows the morphology of the polished cross-section of the annealed ASMC.The corresponding EDS mappings are shown in figures S2(e2)-(e6).It can be seen that Fe, Si, and Cr elements distribute uniformly in the amorphous powders, while O, C, and P elements concentrate in the gaps between particles.The chemical states of the FeSiBCCr amorphous powders before and after the coating process were analyzed by FTIR and XPS.Broad bands at ∼1634 cm −1 , corresponding to the infrared absorption of -OH groups, can be detected for all the samples, and its intensity enhances following passivation and EP coating (figureS2(f)).Additionally, the presence of the Fe-O vibrational peak and the enhanced C-O-C peak indicate the formation of phosphates during passivation.Following the insulation coating, some specific peaks including C-H, C-C, and C=O, etc, originating from EP can be observed.On the other hand, the XPS spectra show the appearance of P 2p after passivation, while the disappearance of Fe 2p after EP coating (figure S2(g)).FiguresS2(h)-(k) display the C 1s and O 1s XPS spectra.Compared with the untreated powder, the absence of Fe-O, Cr-O, and Si-O species as well as the presence of C-O and C=O groups in the coated powder can be observed.The Figure S1(a) shows the SEM morphology of the FeSiBCCr amorphous powder prepared by the multistage atomization.The fine powders exhibit high sphericity and smooth surface except a small number of irregular particles.The amorphous features of the atomized powders were confirmed through XRD and DSC as shown in figures 1(d) and S1(b).Broad diffusion humps at ∼45 • and ∼80 • were detected in the XRD pattern and a sharp exothermic peak corresponding to crystallization was observed in the DSC curve.The T x was determined to be 570 • C. To achieve good high-frequency properties, passivation with phosphoric acid followed by the application of insulating EP was applied as shown in figure1(b).The morphologies of the passivated and coated powders are shown in figures S2(a) and (b), respectively.Compared with the untreated powder (figure S1(a)), the surfaces of the coated powders are rougher.The total thickness of the coating layer is approximately 110 nm (figure S2(c1)), composed of the passivation compounds and EP (figures S2(c2)-(c6)).Figure S2(d) shows the morphology of the polished cross-section of the annealed ASMC.The corresponding EDS mappings are shown in figures S2(e2)-(e6).It can be seen that Fe, Si, and Cr elements distribute uniformly in the amorphous powders, while O, C, and P elements concentrate in the gaps between particles.The chemical states of the FeSiBCCr amorphous powders before and after the coating process were analyzed by FTIR and XPS.Broad bands at ∼1634 cm −1 , corresponding to the infrared absorption of -OH groups, can be detected for all the samples, and its intensity enhances following passivation and EP coating (figureS2(f)).Additionally, the presence of the Fe-O vibrational peak and the enhanced C-O-C peak indicate the formation of phosphates during passivation.Following the insulation coating, some specific peaks including C-H, C-C, and C=O, etc, originating from EP can be observed.On the other hand, the XPS spectra show the appearance of P 2p after passivation, while the disappearance of Fe 2p after EP coating (figure S2(g)).FiguresS2(h)-(k) display the C 1s and O 1s XPS spectra.Compared with the untreated powder, the absence of Fe-O, Cr-O, and Si-O species as well as the presence of C-O and C=O groups in the coated powder can be observed.The

Figure 2 .
Figure 2. Microstructure characterizations and identification of the CSAS for the FeSiBCCr powders.TEM images of (a) the atomized amorphous powder, (b) the annealed powder at 500 • C for 30 min under an RFA of 0.1 T, and (c) the critical amorphous powder with 30 min of 0.1 T RFA heat treatment at 475 • C. (d) and (e) SAED pattern and HRTEM image of the crystal nucleus-contained region A in (b).Identification of the critical structure of the 475-RFA amorphous powder: (f) HRTEM images of the fully amorphous region; (g) IFT-filtered HRTEM image of (f); (h1)-(h3), (i1)-(i3), (j1)-(j3), and (k1)-(k3), the IFT-filtered HRTEM images and corresponding ACF and FT images of the B (amorphous structure, AS), C (imperfect crystal-like order, ICLO), D (perfect crystal-like order, PCLO), and E (PCLO) regions in (g), respectively; (l) 2D-ACF mappings of the HRTEM image in (f).(m1) Atomic model of the Fe 95 Co 5 amorphous alloy used for the molecular dynamics simulation and achievement of the RFA; (m2) Representative crystallization dynamics curves of the amorphous alloys under the rotating magnetic field of 0 and 0.1 T; (m3) The averaged incubation period and slope of the crystallization process of 20 independent samples.

Figure 3 .
Figure 3. Comprehensive soft magnetic properties of the FeSiBCCr powders and ASMCs.(a) M-H curves of the powders treated under different annealing conditions and the insets showing the magnetization curves of the CSAS powders before and after coating.Frequency dependence of (b) µe and (c) Pcv for the raw and annealed ASMCs.Summary of the soft magnetic properties of typical ASMCs.(d) µ e/100 kHz versus Ms and (e) µ e/100 kHz versus Pcv (0.1 T, 100 kHz) of the reported ASMCs.

Figure 4 .
Figure 4. Magnetic domain structures of the FeSiBCCr ASMCs under different annealing conditions using MFM and LTEM.3D-MFM images of the (a) raw, (b) 350-RFA, (c) 475-RFA, and (d) 550-RFA SMCs.(e) The variation of domain width and domain features of ASMCs under different heat treatments.LTEM images of domain wall states under different applied fields of the (f), (g) raw and (h), (i) CSAS powders.