A mechanically strong and ductile soft magnet with ultralow coercivity

Soft magnetic materials (SMMs) are indispensable components in electrified applications and sustainable energy supply, allowing permanent magnetic flux variations in response to high frequency changes of the applied magnetic field, at lowest possible energy loss 1 . The global trend towards electrification of transport, households and manufacturing leads to a massive increase in energy consumption due to hysteresis losses 2 . Therefore, minimizing coercivity, which scales the losses in SMMs, is crucial 3 . Yet, meeting this target alone is not enough: SMMs used for instance in vehicles and planes must withstand severe mechanical loads, i.e., the alloys need high strength and ductility 4 . This is a fundamental design challenge, as most methods that enhance strength introduce stress fields that can pin magnetic domains, thus increasing coercivity and hysteretic losses 5 . Here, we introduce a new approach to overcome this dilemma. We have designed a multicomponent alloy with ferromagnetic matrix and paramagnetic coherent nanoparticles of well-controlled size (~91 nm) and high volume fraction (55%). They impede dislocation motion, enhancing strength and ductility. Yet, their small size, low coherency stress and small magnetostatic energy create an interaction volume below the magnetic domain wall width, leading to minimal domain wall pinning, thus maintaining the material’s soft magnetic properties. The new material exhibits an excellent combination of mechanical and magnetic properties outperforming X-ray diffraction (XRD) measurements carried out in an X-ray system (Diffractometer Advance A25-X1) with Co K α radiation 35 40 mA). Electron backscatter diffraction (EBSD) characterizations were conducted in a Zeiss-Crossbeam focus ion beam scanning electron microscope at 15 kV. Electron channelling contrast imaging (ECCI) characterizations were performed using a Zeiss-Merlin high-resolution field emission electron microscope at 30 kV. Transmission electron microscopy (TEM) analysis including selected area electron diffraction (SAED) was conducted in a JEOL JEM 2100 at 200 kV. Scanning transmission electron microscopy (STEM) images were collected using a probe-corrected Titan Themis 60-300 (Thermo Fischer Scientific) microscope. To modify the Z-contrast characteristics of the imaging mode, high-angle annular dark-field (HAADF) micrographs with a convergence angle of 23.8 mrad were acquired at 300 kV. The resulting collection angle ranges from 73 mrad to 200 mrad. Further energy-dispersive X-ray spectroscopy (EDS) analysis was conducted using Thermo Fischer Scientific’s Super-X windowless EDS detector at an acceleration voltage of 300 kV. Atom probe tomography (APT) experiments were performed in a local electrode atom probe (LEAP 5000 XR) from Cameca Instruments Inc and analyzed with commercial AP Suite software (v 6.1). A pulse frequency of ~125 kHz, a pulse energy of ~40 pJ, and a temperature of ~60 K was used. The detection rate was kept at a frequency of 1 ion per 100 pulses.

Lowest possible coercivity is a primary goal for soft magnetic materials (SMMs), to reduce hysteresis-related energy losses, noise, heating and the associated material damage [1][2][3] . Also, new SMMs with higher strength and ductility are needed, to operate under mechanically demanding loading conditions for safety-critical parts in transportation and energy 4 . This multi-property profile creates a fundamental dilemma. Mechanical strength of metallic materials is produced by lattice defects and their elastic interactions with linear lattice faults that carry inelastic deformation, referred to as dislocations. However, the defects also interact with the magnetic domain walls and pin them. The loss in domain wall motion increases coercivity so that the materials lose their soft magnetic features. Therefore, current SMMs follow the design rule of avoiding lattice defects to minimize coercivity 5 . Vice versa, increasing an alloy's mechanical strength requires enhancing its internal stress level through defects such as dislocations, grain boundaries and precipitates 6 . This means that the task of making soft magnets mechanically strong is a trade-off between two mutually exclusive design strategies, namely, mechanical strength vs. unaffected domain wall motion.
The theory of the grain size dependence of coercivity 7 shows its proportionality to the 6 th power of the grain size for the case of nanocrystalline materials, a relation that can be also applied to particles 8 . Current design of SMMs has thus focused on using small particles (<15 nm) 8,9 and grain sizes (<100 nm) [10][11][12] . According to magnetic strain theory, the coercivity depends on the energy required to displace domain walls to overcome lattice barriers [13][14][15] . Here we introduce particles into multicomponent massive solid solution matrix and increase their size from the commonly used range of 5~15 nm to 90~100 nm. With that, the internal stress level and the overall elastic coherency misfit energy are reduced through the smaller specific surface area (total surface area per unit of volume) of the particles, caused by the coarsening. We then propose that the particle design must follow four main rules. First, minimal pinning of domain walls requires a well-tuned and -controlled particle size distribution with optimum balance between the decrease in specific surface area and the increase in magnetostatic energy during particle coarsening.
Second, the particle size must remain smaller than the domain wall width to prevent strong pinning, i.e. strong resistance against spin rotation 6 . Third, the particles' chemical composition and crystal structure determine their saturation magnetization; therefore, antiferromagnetic elements are usually excluded. Fourth, strengthening of the alloys is determined by the interaction between dislocations and particles and by the friction forces exerted on dislocations in the massive solid solution matrix. Thus, intrinsically strong intermetallic particles with minimal lattice misfit are employed. These require high forces for dislocation cutting (giving strength), but repeated cutting by ensuing dislocations emitted by the same source shear them with gradual ease along the remaining and gradually reducing particle cross sections (providing ductility).
These different mechanism considerations had next to be translated into a corresponding compositional alloy design concept. This is mainly guided by the requirement for (a) a ferromagnetic matrix with (b) high solid solution contribution, components that trigger the formation of (c) strong and stable intermetallic phases with (d) small lattice misfit relative to the matrix. These considerations have led us to the non-equiatomic iron-nickel-cobalt-tantalumaluminium (Co27.7Fe32.6Ni27.7Ta5.0Al7.0, at.%) multicomponent alloy (MCA). We synthesized the material in a vacuum induction melting furnace followed by conventional hot rolling and homogenization (details of the processing procedures and chemical compositions are provided in the Methods section). Through further isothermal heat treatments (1~100 h at 1173 K), we obtain a wide range of particle sizes (24~255 nm). The particles have L12 structure and a complex composition, as presented in detail below. Fig. 1 shows the structural characterization of the MCA with medium particle size (M-MCA, where "M" stands for "medium particle size") after annealing at 1173 K for 5 h. The M-MCA shows an average grain size of 85.3 ± 25.6 µm (including annealing twin boundaries) according to electron backscatter diffraction (EBSD) analysis shown in Fig. 1a. The electron channelling contrast imaging (ECCI) analysis shows that the L12 particles have a high number density (7.2×10 20 m -3 ) and a large volume fraction (~55%) in homogeneous distribution within the face-centered cubic (fcc) matrix (Fig. 1b). The lattice misfit (~0.48%) between the fcc matrix and the L12 particles has been calculated using their lattice parameters 16 acquired from the X-ray diffraction (XRD) patterns (Fig. 1c) by Rietveld simulation 17 . Such a small lattice mismatch reduces the capillary driving forces for further coarsening and the uniform dispersion prevents plastic localization at high strength 18,19 . The central beam dark-field (DF) transmission electron microscopy (TEM) analysis reveals that the average size of the L12 particles is 90.8 ± 35.8 nm, The elemental partitioning between the L12 precipitates and fcc solid solution matrix is characterized by atom probe tomography (APT). Fig. 1e shows the three-dimensional (3D) distribution of the volume investigated by APT, highlighted by a set of iso-surfaces delineating regions containing above 25 at.% Fe. Fig. 1f shows the one-dimensional (1D) compositional profiles acquired along the cylinder in Fig. 1e. The profiles show that Fe partitions into the fcc matrix (36 at.%), whereas the L12 particles are enriched in Ni (40 at.%), Ta (13 at.%), and Al (9 at.%). The compositions of the fcc and L12 phases were determined by averaging over 3 APT data sets (including 10 L12 particles) as Fe36Co28Ni26Al7Ta3 and Ni40Co26Ta13Fe12Al9 (at.%), respectively.
Besides these intragranular nanoparticles, we also observed two types of grain-boundary variants: (a) Coarse grain boundary particles (160.2 ± 55.3 nm) with the same crystal structure and composition (Extended Data Fig. 1a-c) as the ones inside the grains; (b) incoherent particles with a minor fraction (<0.3%) at triple-points of the grains with different structure (cubic Fd-3m, Fig.   1c) and composition (Ta40.0Co26.1Fe20.1Ni11.1Al2.7, at.%, Extended Data Fig. 1e). These two types of particles are both promoted by the high diffusion rate along the grain boundaries. The current strategy by tuning the particle size also enables overcoming the strengthductility trade-off, significant for advanced alloys with gigapascal-level strength. Fig. 2a displays the tensile stress-strain curves of the M-MCA at room-temperature (red curve). The yield strength (σ # ) is 904 ± 11 MPa with an ultimate tensile strength (σ $%& ) of 1336 ± 21 MPa, and an elongation at fracture (ε ( ) up to 53.6 ± 1.5%, averaged from four tests. Accordingly, the M-MCA has a high σ $%& ×ε ( value of ~71.6 GPa%. To reveal the improvement in strength and ductility achieved by the well-controlled particle size distribution, the mechanical response of a material variant with identical chemical composition ~Fe32Co28Ni28Ta5Al7 (at.%) but smaller particle size (S-MCA, annealed for 1 h, producing an average particle size of ~24 ± 15 nm), larger particle size (L-MCA, annealed for 100 h, average particle size of ~255 ± 49 nm) and the particle-free Fe35Co30Ni30Ta5 (at.%) alloy 9 , are also presented in Fig. 2a.
Compared with the single-phase Fe35Co30Ni30Ta5 (at.%) alloy with a relatively low value of σ # of ~501 MPa, the notable increase in yield strength (σ #~9 04 MPa) of the M-MCA can be attributed to the precipitation strengthening of the L12 particles with a high volume fraction (~55%).
More importantly, such improvement in the strength of the M-MCA is achieved at no expense of ductility, which is fundamentally different from the case of the S-MCA where a significant loss in ductility is observed with increasing strength. The good ductility is correlated with the high workhardening capability, as shown in Fig. 2b. Further increase of particle size at longer annealing time (100 h) in the L-MCA leads to the decrease of ductility (53%) and ultimate tensile strength (14%) as compared to those of the M-MCA. This is related to the mechanical weakness, strain localization 6 and embrittlement in the particle-free zone adjacent to the grain boundary caused by solute depletion (Extended Data Fig. 2a). This interfacial weakening has been confirmed by the associated fracture morphologies, with typical ductile fracture with fine dimples in the M-MCA material (see Fig. 2b inset) and intergranular fracture in the L-MCA material (Extended Data Fig.   2b).  Fig. 2c). A further refined and curved microband structure is observed at high local strains +,of ~80% (Extended Data Fig. 4c). The quantification of the evolution of the average microband spacing reveals a microband refinement process during straining (Extended Data Fig.   4d). The gradually reduced microband spacing causes a higher passing stress and thus enhanced strain hardening. This has been proposed to explain the good strength-ductility combinations in some high manganese steels 25-27 and MCAs 28 . Hence, the observed dynamic microband refinement and particle shearing are the prevalent deformation mechanisms in the current MCAs. No Orowan looping was observed, even when increasing the average particle size up to 255 nm for the L-MCA (Extended Data Fig. 2c), as the average particle spacing (Extended Data Table 1) remains far below the critical value (~3094 nm, see Extended Data for detailed calculation) for the activation of dislocation bowing around particles, a mechanism referred to as Orowan effect. Furthermore, the stress required for shearing particles in the M-MCA with a medium particle size (~91 ± 36 nm) with high volume fraction (~55%) is 2.2 times larger than that for the S-MCA with a smaller particle size (~24 ± 15 nm) with low volume fraction (43%) (see Extended Data for detailed calculation). Therefore, the high critical shear stress required for cutting the L12 particles and the dynamic microband refinement during plastic deformation lead to the strong strain hardening capacity of the M-MCA. The mechanism behind this transition is the change in the intrinsic spin alignment, which is related to the change in chemical composition (Extended Data Figure 6) and ordering during annealing. The overall increase in saturation magnetization of the MCAs as a function of particle coarsening is attributed to the change in fcc matrix composition due to elemental partitioning (Extended Data Fig. 6), i.e., specifically to the resultant higher concentration of (Fe+Co) in the fcc matrix. This effect enhances the total average magnetic moment per formula unit and leads to higher a Ms.
To gain further insight into the mechanism behind the magnetic response of the M-MCA, we investigated the magnetic domain structure in terms of magneto-optical Kerr effect (MOKE) microscopy ( Fig. 3c) under different applied magnetic field strengths. Starting from the alternating current (AC) demagnetized state to an applied field of 40 kA/m, the nucleation of magnetic domains is uniformly distributed within the grain. Further increasing the applied field (~155 kA/m) leads to domain wall movement and growth of the energetically favourable domains. The domains grow smoothly within the grains while the movement is hindered at the grain/twin boundaries (Extended Data Fig. 7). Fig. 3d summarized the statistically averaged particle size distribution with respect to the coercivity of all the MCA samples at different annealed states. The data is acquired by developing an automated processing protocol, as shown in Extended Data Fig. 8. The coercivity first decreases from 763 A/m (S-MCA, average particle size ~24 nm) to 78 A/m (M-MCA, average particle size ~91 nm) and then increases to 1745 A/m (L-MCA, average particle size ~255 nm).
Both the average particle size and the grain size increase monotonously with increasing annealing time (Extended Data Table 1). Since the grain size of the MCAs material is above the critical single-domain size, its coercivity decreases with grain coarsening, following the model for the grain size dependence of the coercivity as Hc ∝1/D (where D is the grain size) 11 . However, the magnitude of the decrease in coercivity due to grain coarsening according to the model is  Therefore, the significant drop in coercivity at the early particle coarsening stage (≤91 nm) is attributed to the gradual reduction of the overall coherency stresses between matrix and particles, due to their average size increase (see Extended Data for further calculation details). More specific, the values of the specific surface area×integrated lattice misfit decrease from 1. These considerations show that the nanoscale size distribution of the particles must be carefully controlled to minimize their pinning effect on domain wall movement, which determines the alloy's coercivity. This is achieved here by an optimal balance between the release of structural defects (e.g., interfacial elastic distortion, dislocation density) just down to a level required to maintain high mechanical strength and the rise of the pinning effect from the magnetostatic energy, while keeping the particle size below the domain wall width during particle coarsening.
To highlight the pronounced combination of mechanical and soft magnetic properties of the M-MCA with optimal particle size, we compare it with existing soft magnetic materials in an Ashby-plot showing the ultimate tensile strength (σ $%& )×elongation at fracture (ε ( ) against the intrinsic coercivity, Fig. 4a. This comparison shows that the σ $%& ×ε (   Here we have developed a new material which unifies so far mutually exclusive properties, namely, high mechanical strength (1336 MPa), high tensile ductility (54%), extremely low coercivity (78 A/m) and good saturation magnetization (100 Am 2 /kg). We realized this in a new class of bulk SMMs through a nanoparticle dispersion with well-controlled size (~91 nm), magnetic properties, coherency strain, strength and interface energy. The design strategy is opposite to that generally applied in conventional SMM design. Instead of using smallest microstructure features (particle size <15 nm) to avoid magnetic wall pinning as in conventional SMMs, we have chosen a relatively coarse particle dispersion with tuned particle/matrix interfacial coherency stresses and paramagnetic properties to minimize magnetic pinning of domain walls on the one hand (soft magnetism) and maximize interaction strength with dislocations on the other hand (strength and ductility). Such a particle design concept can be applied also to many other precipitation-containing soft magnetic alloys, suited to serve in a broad spectrum of mechanicallydemanding energy, automotive and aerospace applications. T

Materials preparation. The bulk MCA ingot with a predetermined nominal composition of
Co27.7Fe32.6Ni27.7Ta5.0Al7.0 (at%) was first cast in a vacuum induction furnace using pure metallic ingredients (purity higher than 99.8 wt.%) under high-purity argon (Ar) atmosphere. The as-cast ingot with a dimension of 40 mm×60 mm×20 mm (length×width×thickness) was then hot-rolled at 1473 K to an engineering thickness reduction of 50% (final thickness ~10 mm). After hot-rolling, The alloy sheets were then homogenized at 1473 K for 10 min in Ar atmosphere followed by water-quenching. To obtain a wide size distribution of the particles, further isothermal heat treatments were conducted at 1173 K, lasting from 1 h up to 100 h (1 h, 2 h, 5 h, 20 h, 50 h, 100 h) in Ar atmosphere and followed by water quenching. The exact chemical composition of the MCA measured by wet-chemical analysis is Co28.0Fe32.0Ni28.1Ta4.7Al7.2 (at.%), which is close to the pre-designed composition. In addition, the bulk ingots (~50 g) with the compositions identical to that of the fcc (Fe36Co28Ni26Al7Ta3, at%) matrix phase and the L12 particle (Ni40Co26Ta13Fe12Al9, at.%) in the M-MCA derived from the APT analysis were also cast, respectively, by arc-melting under Ar atmosphere. The ingots were remelted six times to achieve chemical homogeneity.

Estimation of interfacial coherency stress
The coherency stress at the L12/fcc interface is determined by integrating the lattice misfit across the interface as: where 34 5 /(77 is the L12/fcc interface area related to the average particle size ( ), the volume fraction of the L12 particles ( ) and the overall volume ( ) as follows: The estimated interfacial constrained misfit value is 1.09×10 6 , 4.08×10 5 and 1.96×10 5 for the S-MCA, M-MCA and L-MCA, respectively. Therefore, the significant decrease in the interfacial coherency stress is expected to play an essential role in releasing the pining effect on domain wall movement with particle coarsening for the MCAs with the particle size below the domain wall width.

Estimation of dislocation density
The dislocation density ( ) in the fcc matrix can be calculated through the Williamson-Smallman relationship as 57 : where b is microstrain, D is crystallite size acquired from the XRD profiles (Extended Data Table   1), and is the Burgers vector (for fcc structure, = √2 2 f × (77 ) 58 . The dislocation density in the fcc matrix is thus estimated to be 1.50×10 14 m -2 , 9.32×10 13 m -2 and 5.38×10 13 m -2 for the S-MCA, M-MCA and L-MCA, respectively. Based on the above estimation, the considerable improvement in the coercivity also derives from the decrease of dislocation density in the fcc matrix.

Estimation of particle shearing stress
Based on the experimental observation ( Fig. 2c and Extended Data Fig. 2c), particle shearing is the primary deformation mechanism in the investigated MCAs. The strengthening contribution of particle shearing (∆ ) is estimated according to 59 : where 2 is the mean spacing of the particles, 2 ≈ s c > , is the average radius of the particles ( = /2). is the volume fraction of the particles shown in Extended Data Table 1, is the force exerted on the particles. The shearing strength is expressed as: by using the relation ∝ @/c and introducing constant k. The effect of particle strengthening of the M-MCA is then estimated to be two times larger than that of the S-MCA ( ∆ wVwxy / When considering the volume fraction of the particles to be constant, the mean spacing of the particles increases with increasing the particle size. As a result, the force required for shearing particles increases until the Orowan mechanism is activated, i.e., dislocations bowing the particles becomes easier than shearing. The critical mean spacing of the particles is determined by 59 : G=84 GPa is the adopted shear modulus 60 . Consequently, the critical mean spacing of the particles is calculated as 3094.3 nm.

Estimation of magnetostatic energy
The magnetostatic energy ( b ) determines the coercive force which interacting between the paramagnetic particles (for M-MCA and L-MCA) and domain wall movement according to the formula 33 : where µ [ = 4π×10 −7 H/m is the permeability of vacuum, is the average particle radius, b is the saturation magnetization of the fcc matrix. For M-MCA and L-MCA, in which the L12 phase is paramagnetic (Extended Data Fig. 5), the b of the fcc matrix is considered as the overall b of the alloy. The values of b significantly increase with increasing the particle size, that is, it varies from 6.5×10 -24 (M-MCA) to 1.53×10 -22 (L-MCA). The notable increase in magnetostatic energy results in a strong magnetic pinning effect.

Estimation of domain wall width
Strong pinning arises and results in the deterioration of coercivity when the microstructure defects have a comparable dimension to the domain wall thickness ( 1 ). As a result, the estimation of the 1 helps understand the ultralow coercivity in the current work is given by 61,62 : constant,and [ are the Curie temperature and lattice parameter of the fcc matrix, respectively (Extended Data Fig. 5d and Extended Data Table 1

Author Information
Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Readers are welcome to comment on the online version of the paper. Correspondence and requests for materials should be addressed to D. Raabe (d.raabe@mpie.de) or Z. Li (lizhiming@csu.edu.cn).

Supplementary Files
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