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Materials preparation

The bulk MCA ingot with a predetermined nominal composition of Fe32.6Co27.7Ni27.7Ta5.0Al7.0 (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 dimension 40 mm × 60 mm × 20 mm (length × width × thickness) was then hot rolled at 1,473 K to an engineering thickness reduction of 50% (final thickness 10 mm). After hot rolling, the alloy sheets were then homogenized at 1,473 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 1,173 K, lasting from 1 h up to 100 h (1 h, 2 h, 5 h, 20 h, 50 h and 100 h) in Ar atmosphere and followed by water quenching. The exact chemical composition of the MCA measured by wet-chemical analysis is Fe32.0Co28.0Ni28.1Ta4.7Al7.2 (at.%), which is close to the predesigned composition. In addition, the bulk ingots (50 g) with 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.

Analytical methods

XRD measurements were carried out in an X-ray system (Diffractometer D8 Advance A25-X1) with Co Kα radiation (λ = 1.78897 Å, 35 kV and 40 mA). EBSD characterizations were conducted in a ZEISS Crossbeam focused ion beam scanning electron microscope at 15 kV. ECCI characterizations were performed using a ZEISS MERLIN high-resolution field-emission electron microscope at 30 kV. TEM analysis including selected-area electron diffraction 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 Fisher Scientific) microscope. To modify the Z-contrast characteristics of the imaging mode, high-angle annular dark-field (HAADF) micrographswith 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 Fisher Scientific’s Super-X windowless EDS detector at an acceleration voltage of 300 kV. APT experiments were performed in a local electrode atom probe (LEAP 5000 XR) from Cameca Instruments Inc. and analysed with commercial AP Suite software (v6.1). 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.

Mechanical response measurements

Room-temperature uniaxial tensile tests were performed using flat tensile specimens at an initial strain rate of 1 × 10−3 s−1. The tensile specimens were machined along the rolling direction from the alloy sheets by electrical discharge machining. The specimens with a total length of 60 mm, a gauge length of 30 mm, a gauge width of 5 mm and a thickness of 2 mm were used to investigate the bulk tensile properties. Further, smaller tensile specimens with a total length of 20 mm, a gauge length of 10 mm, a gauge width of 2 mm and a thickness of 1 mm were used to measure the local strain evolution by the digital image correlation method. At least four specimens for each condition were tested to confirm reproducibility. Further, to clarify the relation between global strain–stress behaviour and microstructure evolution, we also conducted interrupted tensile tests on different global true strains (that is, 5%, 15% and 25%), and the microstructures in the middle part of the deformed regions were then characterized accordingly.

Magnetic response measurements

The magnetic response was evaluated using the Quantum Design Magnetic Property Measurement System (MPMS) equipped with a standard Vibrating Sample Magnetometry (VSM) option. Cuboid specimens of dimensions 3 mm × 3 mm × 1 mm (length × width × thickness) were used for the measurements. The hysteresis loops M(H) were performed in an external magnetic field of ±800 kA m−1 at a magnetic field-sweeping rate of 1 kA m−1 at 10 K, 300 K, 500 K and 800 K, respectively. The temperature dependence of magnetization M(T) analysis was carried out under an applied field of 40 kA m−1 from 10 K to 1,000 K with a temperature-sweeping rate of 10 K min−1.

The magnetic domain patterns were characterized by a MOKE ZEISS microscope (Axio Imager.D2m). The domain wall movement was captured under an applied magnetic field of ±155 kA m−1. Before the measurement, a background image was collected as a reference in the AC demagnetized state. The images acquired at different applied fields were enhanced by subtracting the background image using KerrLab software.

Physical response measurements

The electrical resistivity response was evaluated using the Quantum Design Physical Property Measurement System (PPMS) equipped with an Electrical Transport Option (ETO) option. Cuboid specimens of dimensions 6 mm × 2 mm × 1 mm (length × width × thickness) were used for the measurements. The resistivity ρ values are calculated by:

in which R is the reported resistance, A is the cross-sectional area through which the current is passed and l is the voltage lead separation. The resistance value of each measurement is obtained by averaging those from 100 times of current passing. At least three specimens for each condition were tested.

Thermodynamic calculations

The equilibrium compositions of the fcc matrix and L12 particles in the Fe32Co28Ni28Ta5All7 (at.%) alloy at 1,173 K were calculated using the Thermo-Calc software (v.2022a) equipped with the High Entropy Alloys database TCHEA v.4.2. The calculated equilibrium compositions for the fcc and L12 phases in the Fe32Co28Ni28Ta5Al7 (at.%) alloy are Fe36Co31Ni23Ta4Al6 and Ni63Ta13Fe6Co3Al15 (at.%), respectively.

Estimation of particle size (edge length)

The size distribution is statistically analysed by applying a batch image-processing protocol with several 2D-projected ECC images of all the MCA samples at different annealed states (Extended Data Fig. 7). The average particle size (edge length) of the L12 particles is estimated by:

$$d=\sqrt{\frac{\sum {S}_{i}}{i}}$$

in which d is the average particle size, Si is related to the area of each particle acquired from the 2D-projected ECC images by the batch image-processing protocol and i is the total particle number. The particle size of the M-MCA is also characterized by DF-TEM (Fig. 1d) and bright-field TEM (Extended Data Fig. 1). The TEM results fit well with the value acquired by ECC images.

Estimation of interfacial coherency stress

The coherency stress at the L12–fcc interface is determined by integrating the lattice misfit across the interface as:

$$\delta ={S}_{{\rm{L}}{1}_{2}/{\rm{fcc}}}\sum {\delta }_{x}$$

in which \({S}_{{\rm{L}}{1}_{2}/{\rm{fcc}}}\) is the L12–fcc interface area related to the average particle size (d), the volume fraction of the L12 particles (f) and the overall volume (V) as follows:

$${S}_{{\rm{L}}{1}_{2}/{\rm{fcc}}}=\frac{Vf}{{d}^{3}}\cdot 6{d}^{2}=\frac{6Vf}{d}$$

δx is the varying lattice misfit as a function of distance (x) from the L12–fcc interface determined by the following equation:

$${\delta }_{x}=2\times \left[\frac{{a}_{x}^{{\rm{L}}{1}_{2}}-{a}_{x}^{{\rm{fcc}}}}{{a}_{x}^{{\rm{L}}{1}_{2}}+{a}_{x}^{{\rm{fcc}}}}\right]$$

\({a}_{x}^{{\rm{L}}{1}_{2}}\) and \({a}_{x}^{{\rm{fcc}}}\) are the lattice parameters of the L12 and fcc phases at the interfacial region, respectively. Such values were calculated using the L12–fcc interfacial chemical compositions acquired from the APT datasets with Vegard’s relation51:

$${a}_{x}^{{\rm{L}}{1}_{2}}={a}_{0}^{{\rm{L}}{1}_{2}}+{\sum }_{i}{\varGamma }_{i}^{{\rm{L}}{1}_{2}}{x}_{i}^{{\rm{L}}{1}_{2}}$$

$${a}_{x}^{{\rm{fcc}}}={a}_{0}^{{\rm{fcc}}}+{\sum }_{i}{\varGamma }_{i}^{{\rm{fcc}}}{x}_{i}^{{\rm{fcc}}}$$

in which \({a}_{0}^{{\rm{L}}{1}_{2}}\) and \({a}_{0}^{{\rm{fcc}}}\) are the average lattice parameters for the L12 particles and the fcc matrix, respectively, derived from the Rietveld simulation based on the XRD measurements, as shown in Extended Data Table 1. \({\varGamma }_{i}^{{\rm{L}}{1}_{2}}\) and \({\varGamma }_{i}^{{\rm{fcc}}}\) are the Vegard coefficients for the L12 and fcc phases, respectively, obtained from the ordered Ni3Al phase and the disordered fcc phase in the Ni-base superalloys52, as shown in Extended Data Table 2. Note that the above-calculated lattice misfit δl represents the theoretical unconstrained state. This can be related to the constrained misfit (ε) by elasticity theory as below53:

$$\varepsilon =\frac{3}{2}{\delta }_{l}$$

The estimated interfacial constrained misfit value is 1.09 × 106, 4.08 × 105 and 1.96 × 105 for the S-MCA, M-MCA and L-MCA, respectively. Therefore, the marked decrease in the interfacial coherency stress is expected to play an essential role in releasing the pinning effect on domain wall movement with particle coarsening for the MCAs with particle size below the domain wall width.

Estimation of dislocation density

The dislocation density (ρ) in the fcc matrix can be calculated through the Williamson–Smallman relationship as54:

$$\rho \,=\,\frac{2\sqrt{3}{\left({\varepsilon }_{{\rm{s}}}^{2}\right)}^{1/2}}{Db}$$

in which εs is microstrain, D is crystallite size acquired from the XRD profiles (Extended Data Table 1) and b is the Burgers vector (for fcc structure, \(b=\sqrt{2}/2\times {a}_{{\rm{fcc}}}\))55. The dislocation density in the fcc matrix is thus estimated to be 1.50 × 1014 m−2, 9.32 × 1013 m−2 and 5.38 × 1013 m−2 for the S-MCA, M-MCA and L-MCA, respectively. On the basis of 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

On the basis of 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 (Δτ) is estimated according to56:

$$\Delta {\tau }_{{\rm{S}}{\rm{h}}{\rm{e}}{\rm{a}}{\rm{r}}{\rm{i}}{\rm{n}}{\rm{g}}}=\frac{F}{b\cdot 2\lambda }$$

in which 2λ is the mean spacing of the particles, \(2\lambda \approx \sqrt{\frac{2}{f}}\cdot d\), d is the average particle size, f is the volume fraction of the particles shown in Extended Data Table 1 and F is the force exerted on the particles. The shearing strength is expressed as:

$$\Delta {\tau }_{{\rm{S}}{\rm{h}}{\rm{e}}{\rm{a}}{\rm{r}}{\rm{i}}{\rm{n}}{\rm{g}}}=k\sqrt{fd}$$

by using the relation Fd3/2 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 (\(\Delta {\tau }_{{\rm{M}}-{\rm{M}}{\rm{C}}{\rm{A}}}/\Delta {\tau }_{{\rm{S}}-{\rm{M}}{\rm{C}}{\rm{A}}}=\frac{k\sqrt{{f}_{{\rm{M}}-{\rm{M}}{\rm{C}}{\rm{A}}}{\cdot r}_{{\rm{M}}-{\rm{M}}{\rm{C}}{\rm{A}}}}}{k\sqrt{{f}_{{\rm{S}}-{\rm{M}}{\rm{C}}{\rm{A}}\cdot }{r}_{{\rm{S}}-{\rm{M}}{\rm{C}}{\rm{A}}}}}\)).

When considering the volume fraction of the particles to be constant, the mean spacing of the particles increases with increasing particle size. As a result, the force required for shearing particles increases until the Orowan mechanism is activated, that is, dislocations bowing the particles becomes easier than shearing. The critical mean spacing of the particles is determined by56:

$$\Delta {\tau }_{{\rm{S}}{\rm{h}}{\rm{e}}{\rm{a}}{\rm{r}}{\rm{i}}{\rm{n}}{\rm{g}}}=\frac{F}{b\cdot 2\lambda }=\Delta {\tau }_{{\rm{O}}{\rm{r}}{\rm{o}}{\rm{w}}{\rm{a}}{\rm{n}}}=\frac{Gb}{2\lambda }$$

G = 84 GPa is the adopted shear modulus57. Consequently, the critical mean spacing of the particles is calculated as 3,094.3 nm. However, in the current MCAs, the volume fraction of the L12 phase is not constant even after annealing at 1,173 K for 100 h. This is because the alloys have not yet reached the thermodynamical equilibrium state, as indicated by both thermodynamic calculations and APT analysis (Extended Data Fig. 5).

Estimation of magnetostatic energy

The magnetostatic energy (Es) determines the coercive force that interacts between the paramagnetic particles (for M-MCA and L-MCA) and domain wall movement according to the formula38:

$${E}_{{\rm{s}}}=\frac{1}{2}{\mu }_{0}\frac{1}{3}{{M}_{{\rm{s}}}}^{2}{d}^{3}$$

in which μ0 = 4π × 10−7 H m−1 is the permeability of vacuum, d is the average particle size and Ms is the saturation magnetization of the fcc matrix. For the M-MCA and L-MCA, in which the L12 phase is paramagnetic (Extended Data Fig. 4), the Ms of the fcc matrix is considered as the overall Ms of the alloy. The values of Es markedly increase with increasing particle size, that is, it varies from 1.57 × 10−24 (M-MCA) to 3.65 × 10−23 (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 (δw). As a result, the estimation of the δw to help understand the extremely low coercivity in the current work is given by58,59:

$${\delta }_{{\rm{w}}}={\rm{\pi }}{({A}_{{\rm{ex}}}/{K}_{1})}^{1/2}$$

in which Aex = kBTc/2a0 is the exchange stiffness, kB = 1.380649 × 10−23 J K−1 is Boltzmann’s constant and Tc and a0 are the Curie temperature and lattice parameter of the fcc matrix, respectively (Extended Data Fig. 4d and Extended Data Table 1). K1 is the first magnetocrystalline anisotropy constant. The value of K1 (M-MCA) is taken from the Co–Fe system60,61based on the composition of the fcc matrix (Fig. 1f) as 10.4 kJ m−3 (Al and Ta are non-ferromagnetic elements that do not show any magnetic moment, the chemical composition of the fcc phase Fe36Co28Ni26Al7Ta3 (at.%), in the M-MCA is thus considered as Co31(Fe+Ni)69 (at.%)). The domain wall thickness of the M-MCA is therefore estimated to be 112 nm. Similarly, the domain wall thicknesses of the S-MCA and L-MCA are calculated as 103 nm and 117 nm, respectively.

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