Introduction

The crystal structure is a crucial factor for determining the physical and chemical properties because the atomic arrangement dominates the atomic-orbital hybridization1,2,3,4,5. In a multicomponent alloy system, even with a specific atomic ratio, a number of geometrically available crystal structures can be devised6,7. Although this geometrical possibility means that there is still room to investigate crystal-structure control to realize a fine or drastic tuning of material properties, only thermodynamically stable phases at low and high temperatures have been commonly formed which follow the phase diagrams8, except for the recently reported cases under extreme conditions, such as size9, pressure10, and lattice strain11. For instance, in the case of the FePd3 alloy system targeted in this research, only the thermodynamically stable L12 (Cu3Au-type) phase has been experimentally obtained5,7,8. Therefore, a synthetic strategy of stabilizing a crystal structure is deserved.

To create an ordered alloy with such an atomic arrangement as the layered structures shown in Supplementary Fig. 1, we paid attention to the inter-element miscibility—the miscibility between two elements at an atomic level—in a binary-alloy phase diagram8. Recently, the transformation from A1- to L10-FeNi structures was achieved by introducing and withdrawing interstitial N atoms12. The Fe–Ni–N compound formed during this process possessed a crystal structure with a unit cell composed of one Fe–N layer sandwiched by two Ni layers in such a way as to reflect the better binding affinity of Fe with N, such as N2 dissociation by Fe catalyst13. This structural transformation caused by the interelement affinity of an interstitial N with Fe and Ni inspired us to analogically consider that the transformation from well-known to unexplored structures of a binary alloy could be achieved by the difference in the miscibility of a substitutional third element with the two kinds of elements in binary alloys.

To demonstrate this idea, we focused on the binary alloy of Pd and Fe, because many third elements can be miscible with Pd and immiscible with Fe in the binary phase diagram8. The element In was selected as a substitutional third element. Between the well-known L12-FePd3 and L10-FePd phases in the Fe–Pd system, we chose the L12-FePd3 as a target crystal phase because of its likeliness to change crystal structure from an isotropic L12 structure to a highly anisotropic structure when selectively locating the In atoms at specific sites. Such a structural change would, in turn, alter the physical properties. However, according to previous reports, Fe–Pd–In ternary alloys have been rarely reported except for the In-rich ternary alloys (In/(Fe+Pd+In) >20 atomic percent (at.%))14,15 and the In-poor L12-type FePd3 ternary alloy phases (In/(Fe+Pd+In) <1.2 at.%)16. This indicates the difficulty in producing other Fe–Pd–In ternary alloys with a small amount of In composition (e.g., 1.2 < In/(Fe+Pd+In) < 20 at.%).

In this work, we create the Fe–Pd–In ternary alloy phase (11 < In/(Fe+Pd+In)< 14 at.%) with a Z3-based structure by using nanoparticulate precursor powders, because the uniformly mixing of Fe, Pd, and In at the nanoscale is crucial for the formation of Z3-type structure. Experimental results and first-principles calculation show that the stabilization of Z3-Fe(Pd,Ind)3 is induced by not a nanosize or a kinetic effect but a specific inter-element miscibility of In, which is miscible with Pd but immiscible with Fe. This idea that the inter-element miscibility of the third element works as a stabilizer of an ordered structure gives us an opportunity for the experimental discovery of a variety of novel ordered structures.

Results

Creation of layer-structured Fe–Pd–In alloy nanoparticles

Fe–Pd alloy nanoparticles (NPs) containing a small amount of In atoms were synthesized by a step-by-step chemical synthesis (Supplementary Table 1); (i) synthesis of 23-nm Pd NPs (Fig. 1a)17, (ii) alloying of In with Pd NPs to form A1-PdInx NPs (Fig. 1b), (iii) growth of FeOy shells on A1-PdInx NPs (Fig. 1c)17,18 and (iv) diffusion of Fe atoms into A1-PdInx NPs through the reductive annealing of FeOy shells at 600 °C or 800 °C for 3 h under a 4% H2 gas flow (Ar balance) (Fig. 1d). First, we analysed the NPs with the Pd/In/Fe atomic ratio of 63/14/23 (Pd/In = 82/18 at.% and Pd/Fe = 73/27 at.%), which was confirmed by energy-dispersive X-ray spectrometry (EDX). Figure 1e–g show Rietveld refinement for the powder X-ray diffraction (XRD) patterns and the high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) image of the final NPs. These NPs could not be assigned to any structures in the Fe−Pd and Pd−In systems8, but it was found that they had the P4/mmm structure with the atomic coordinates of Fe1a (0, 0, 0), Fe1c (0.5, 0.5, 0), Pd4i (0.5, 0, 0.23), Pd1b (0, 0, 0.5), and Pd1d (0.5, 0.5, 0.5) in the unit cell, where the superscripts refer to the multiplicities and Wyckoff letters. Therefore, the NPs had a superlattice structure with an alternate stacking of a single Fe-atomic layer and three Pd-atomic layers along the c axis (Fig. 1e–g), which is called the Z3-FePd36,7,19. However, it was quite difficult to detect the In atoms from the intensity ratio of the XRD peaks and the Z contrast of the HAADF-STEM image, because the electron density of In is very close to that of Pd. To identify the In sites in the Z3 framework, we performed elemental mapping by EDX at the atomic resolution. As a result, In atoms were found to be selectively excluded from both the Fe and In atoms and preferentially occupied the 1d or 1b site, which corresponded to the middle Pd layer among the three Pd layers (Fig. 2a–e). Consequently, the resulting Z3 structure with the In atoms at the 1d site was precisely denoted to be Z3-Fe(Pd,Ind)3.

Dependence of Fe–Pd–In alloy phase on In quantity

Next, to investigate the In composition inducing the formation of the Z3-type structure, the Pd–In@FeOy core@shell NPs (Pd/Fe ≈ 70/30 at.%) were annealed at 800 °C for 3 h (Supplementary Table 1). Powder XRD patterns verified that the reductive annealing for Pd–In@FeOy core@shell NPs with the Pd/In of 89/11 at.%, 85/15 and 83/17 at.%, and 73/27 at.% gave the L12-type phase, the Z3-type phases and a mixture of all three phases (A1-Fe60Pd40, Pnma-PdIn2, and I4/mmm phases), respectively (Supplementary Fig. 2). This indicated that the Z3-type structure can be formed in a narrow range of In compositions.

To further confirm the In sites in not only the Z3-type but also the L12-type structures, we performed the X-ray absorption fine structure (XAFS) measurement for these structures at the Fe-, Pd- and In-K edges. The In environment determined from the extended XAFS (EXAFS) regions at the In-K edge revealed that the In atoms in the L12-type and Z3-type structures were surrounded only by approximately 12 Pd atoms, that is, L12-(Fe,In)Pd3 and Z3-Fe(Pd,Ind)3 were formed by partially replacing Fe and Pd (1d site) far from the Fe with In, respectively (Supplementary Fig. 3 and Table 2). This tendency strongly implied that the inter-element miscibility of In with other elements, that is, immiscible with Fe and miscible with Pd, dominated the In sites.

Investigation of driving force for stabilizing Z3-type structure

Although we found that the formation of Z3-type structure depends on the In composition, the driving force for stabilizing Z3-type structure is yet to be clarified. Besides the thermodynamic effect, the kinetic and/or the nano-size effects9 should be considered as the candidates of the driving force for forming Z3-type structure. First, to investigate whether Z3-type structure was a kinetically or a thermodynamically stable phase below 800 °C, we conducted in situ XRD measurement for A1-PdInx@FeOy@SiO2 nanoparticles under the reductive annealing at 800 °C (SiO2 was coated on A1-PdInx@FeOy nanoparticles to avoid the interparticle fusion). As a result, only Z3-type phase was formed by the reductive annealing at 800 °C for 4 min and further annealing at 800 °C for 25 h did not change the crystal phase (Supplementary Fig. 4). Therefore, we concluded that the Z3-type structure was a thermodynamically stable phase below 800 °C.

Next, to investigate whether a particle size contributes to stabilizing the Z3-type structure, we synthesized micrometer-scale Fe–Pd–In ternary alloy particles via the interparticle sintering of PdIn@FeOy nanoparticles by the reductive annealing at 800 °C for 3 h, which gave the Z3-type structure (Supplementary Fig. 5). On the other hand, when the reductive annealing was conducted for a mixture of Pd@FeOy NPs (Pd/Fe = 74/26 at.%) and In powder (Pd/In = 82/18 at.%), in which In was inhomogeneously distributed, at 800 °C for 3 h, the Z3-type structure did not form (Supplementary Fig. 6). These results strongly support that the nanoparticulate precursor powders composed of homogeneously mixed Fe, Pd, and In at the nanoscale are crucial for the formation of Z3-type structure even if the Z3-type structure is not stable specifically at the nanoscale9.

To determine why the Z3-Fe(Pd,Ind)3 structure was preferentially formed, we undertook a theoretical approach using first-principles calculations. First, we calculated the formation energies (Eform) of various FePd3 phases in L12 and Z3 structures before and after adding In by using the equation below;

$${E}_{{{{{{\rm{form}}}}}}}=E[{{{{{{\rm{Fe}}}}}}}_{x}{{{{{{\rm{Pd}}}}}}}_{y}{{{{{{\rm{In}}}}}}}_{z}]-(x\times \mu [{{{{{\rm{Fe}}}}}}]+y\times \mu [{{{{{\rm{Pd}}}}}}]+z\times \mu [{{{{{\rm{In}}}}}}])$$
(1)

where E[FexPdyInz] represents the total energies of Z3- or L12-FexPdyInz ((x, y, z) = (2, 6, 0), (2, 5, 1), or (1, 6, 1)) and μ[Fe], μ[Pd] and μ[In] are the chemical potentials of Fe, Pd and In, respectively, which are equivalent to their total energies of the ground states. As shown in Fig. 3a, the calculation results showed that the L12-(Fe1,In1)Pd6 and Z3-Fe2(Pd5,In1d) structures were most stable in each structural type, which was in good agreement with the In sites in L12-type and Z3-type structures estimated from XAFS analysis. Furthermore, the differences in Eform values between L12-(Fe2-x,Inx)Pd6 and Z3-Fe2(Pd6-x,Inxd) (0 < x < 1) (see Methods) showed that the Z3-type structure became more stable than the L12-type structure from x > 0.48, or In/(In+Pd) > 8 at.% (Fig. 3b). The calculation results also agreed with the experimental tendency in terms of the phase transition and the critical In/Pd at.% from the L12-type to Z3-type phases when increasing the In composition. Therefore, although the L12-FePd3 structure is thermodynamically stable for FePd3 systems5,7,8, introducing a small amount of In atoms into FePd3 systems makes the Z3-type structure more stable than the L12-type structure.

The In sites shown in the experimental and theoretical data were likely to reflect the inter-element miscibility of In with Fe and Pd. To investigate whether the inter-element miscibility of In contributed to the stabilization of Z3-type phase, we conducted the same first-principles calculations for the nine 12–14 groups elements (M = Zn, Cd, Hg, Ga, In, Tl, Ge, Sn, and Pb) in the periodic table as that used for In. Very interestingly, the Z3-type structure was more stable than the L12-type structure when introducing Cd, Hg, In, Tl and Pb which possessed the same miscibility as In with Fe and Pd8,20. On the other hand, the Z3-type structure remained unstable when adding Zn, Ga, Ge, and Sn which are miscible with Fe and Pd8 (Fig. 3C). Since these calculations do not consider other candidates except for isotropic L12-type and anisotropic Z3-type phases, one should note the differences in these formation energies at this composition, however, in the Fe–Pd–Pb system, the formation of Z3-Fe(Pd2.5,Pb0.5d) structure was confirmed by the similar step-by-step chemical synthesis as the synthetic process inducing Z3-Fe(Pd,Ind)3 structure (see Methods and Supplementary Fig. 7). These results strongly support our claim that a third element selected by specific inter-element miscibility possess the potential to stabilize a binary alloy like the Z3 structure.

Physical properties of Z3-Fe(Pd,Ind)3 structure

Finally, we investigated the novel physical properties characteristic of the Z3-type structure. First, the Z3-type structure was expected to show high coercivity because of having a L10-type PdFePd trilayer17,21,22. Magnetization–magnetic field (MH) curves were measured for the L12-(Fe,In)Pd3 and Z3-Fe(Pd,Ind)3 NPs (Pd/Fe ≈ 70/30 at.%) synthesized by annealing at 600 °C for 3 h with the Pd/In at.% of 89/11 and 83/17, respectively, at room temperature by means of a vibrating sample magnetometer (VSM). Both the L12-type and Z3-type NPs showed a ferromagnetic feature with almost the same saturation magnetization, while the Z3-type NPs possessed a coercivity that was 15 times higher than the L12-type NPs and behaved as a magnetically hard phase similar to the L10 NPs21,22 (Fig. 4). This coercivity enhancement could be explained by the drastic increase in the magnetic anisotropy energy23 induced by the crystal structure change from L12 to Z3 frameworks (−1.38 μeV atom–1 for L12-type structure and −0.213 meV atom–1 for Z3-type structure (see Methods)).

As another structure-specific physical property, the Z3-type structure is also likely to possess a hydrogen-storage ability, owing to one Pd–In ordered alloy layer sandwiched by two Pd layers similar to the Pd-rich alloy structure24. However, it was found from the pressure-composition isotherms at 30 °C and 60 °C that this Z3-type structure had no hydrogen-storage ability (Supplementary Fig. 8). This hydrogen-storage ability can be understood by the small amount of 4d hole of Pd in the Z3-type structure shown by the density of states (DOS), which are obtained from first-principles calculations (Supplementary Fig. 9)25. The DOS feature also showed no significant difference in the DOS near the Fermi level and the d-bandwidths and centers of Fe and Pd between the Z3-Fe2(Pd5,In1d) and Z3-Fe2Pd6 structures. No significant change in DOS of Z3-Fe2(Pd5,In1d) and Z3-Fe2Pd6 structures indicates that the physical properties characteristic of the Z3-FePd3 structure are still observed even if a small amount of In atoms are introduced into the Z3-FePd3 framework.

Discussion

We proposed the concept that the inter-element miscibility stabilizes a binary alloy with an ordered structure. As an example, we demonstrated that introducing a small amount of In atoms enabled the formation of Z3-Fe(Pd,Ind)3 NPs with an almost identical DOS as the Z3-FePd3 structure, where the synthetic procedure using the nanoparticulate precursor powders such as A1-PdInx@FeOy NPs were crucial for creating the Z3-type structure. These results indicate that ordered alloy structures can be experimentally discovered according to the binary-phase diagrams8,20 rather than searching for synthesizable structures using first-principles calculations26,27. We expect that a variety of ordered alloy nanostructures will be discovered that exhibit unexpected and excellent properties.

Methods

Materials

All reagents and solvents were commercially available. Sodium tetrachloropalladate(II) (Na2PdCl4, 98%), polyvinylpyrrolidone (PVP, Mw ≈ 55,000), ascorbic acid (AA, 99 + %), potassium bromide (KBr, ≥ 99%), oleylamine (OAm, 80–90%), 1-octadecene (ODE, 90%), indium(III) chloride (InCl3, 98%), indium powder (In, >99.99%), eicosane (99%), ammonium hydrogen solution (NH3aq, 28–30%), tetraethyl orthosilicate (TEOS, 98%) and polyoxyethylene nonylphenylether (IGEPAL@CO-520, Mw = 441) were purchased from Sigma-Aldrich. Iron pentacarbonyl (Fe(CO)5, 95%) was purchased from Kanto Chemical Corporation. Oleic acid (OA, > 85%) was purchased from Tokyo Chemical Industry. Lead (II) acetate trihydrate (Pb(OAc)2·3H2O), ethanol, acetone, n-hexane, chloroform, and cyclohexane were purchased from Wako Pure Chemical Industries. Boron nitride (BN, > 99.5%) powder was purchased from Nacalai Tesque. Chemical reagents were used as received without further purification except for the synthetic process of Pd–Pb@FeOy NPs.

Synthesis of Pd NPs

Pd NPs were synthesized by the previously reported procedure17. After the rapid injection of an aqueous solution (30 mL) of Na2PdCl4 (1.5 g) and KBr (8.0 g) into an aqueous solution (270 mL) of PVP (2.8 g) and AA (0.66 g), the reaction solution was kept at 80 °C for 3 h and then cooled to room temperature. The above black solution of Pd seeds (15 mL) was injected into an aqueous solution (250 mL) of Na2PdCl4 (1.5 g) and KBr (3.0 g) at 40 °C and then an aqueous solution (25 mL) of PVP (2.8 g) and an aqueous solution (25 mL) of AA (1.8 g) were injected every 30 min at 40 °C. This solution was kept at 40 °C for 48 h, heated from 40 °C to 60 °C and kept at 60 °C for 48 h. Finally, the reaction solution was cooled to room temperature and the Pd NPs with an average edge length of 23 nm were collected by centrifugation with acetone and then purified with an ethanol/acetone (1/4 vol.%) mixed solvent.

Synthesis of Pd–In alloy NPs

To react the Pd NPs with the In precursor in an organic solution, the surface ligand passivating the Pd NPs was exchanged from PVP to OAm by heating a mixture of OAm (10 mL), chloroform (20 mL) and Pd NPs (0.5 mmol) at 50 °C for 30 min. The solution was cooled to room temperature and the OAm-protected Pd NPs were collected by centrifugation with ethanol and purified twice with a chloroform/ethanol (2/3 vol.%) mixed solvent. After the ligand exchange, the OAm-protected Pd NPs were stirred with the mixture of InCl3 (nIn mmol), OAm (32 mL) and OA (3.3 mL) at T °C for 3 h (Supplementary Table 1). Then, the organic solution was cooled to room temperature and the Pd–In alloy NPs were collected by centrifugation with ethanol and purified twice with a chloroform/ethanol (3/5 vol.%) mixed solvent.

Synthesis of Pd–In@FeOy NPs

The FeOy shells were grown on the Pd–In alloy NPs by the almost same procedure as the previous report17, namely, injecting Fe(CO)5 (nFe mmol) into the ODE solution (63 mL) containing the Pd–In alloy NPs (0.50 mmolPd), OAm (2.0 mL) and OA (3.2 mL) with stirring at 80 °C for 1 h under an N2 atmosphere, and the reaction temperature was increased from 80 °C to 180 °C at the heating rate of 3 °C/min (Table S1), where the amount of OAm was sensitive for the decomposition of Fe(CO)5 decomposition18. Finally, after the solution was kept at 180 °C for t h and cooled to room temperature, the Pd–In@FeOy NPs were collected by centrifugation with ethanol, purified twice with a n-hexane/ethanol (3/1 vol.%) mixed solvent containing OAm (0.50 mL) and OA (0.50 mL), washed 3 times with ethanol and dried in a desiccator overnight.

Synthesis of Pd–Pb alloy NPs

After the surface ligands on Pd seeds were exchanged as the same process as Fe–Pd–In system, the OAm-protected Pd seeds were stirred with the mixture of Pb(OAc)2·3H2O (0.3 mmol), distilled OAm (36 mL), and OA (4 mL) at 160 °C for 3 h after degassing by freeze-pump-thaw cycle (3 times). After the reaction, the organic solvent was cooled to room temperature and the Pd–Pb alloy NPs were collected by centrifugation with ethanol and purified twice with a chloroform/ethanol (3/5 vol.%) mixed solvent.

Synthesis of Pd–Pb@FeOy NPs

The FeOy shells were covered on the Pd–Pb alloy NPs by injecting Fe(CO)5 (7.4 mmol) at 80 °C for 1 h under an N2 atmosphere into the mixture solution of ODE (32 mL), distilled-OAm (1.0 mL) and the Pd–Pb alloy NPs (0.25 mmolPd) after degassing by freeze-pump-thaw cycle (3 times) and increasing from 80 °C to 140 °C at the heating rate of 3 °C/min. After the solution reached at 140 °C and was cooled to room temperature, the Pd–Pb@FeOy NPs were collected by centrifugation with ethanol, purified twice with a n-hexane/ethanol (5/3 vol.%) mixed solvent containing OAm (0.50 mL) and OA (0.50 mL), washed one time with ethanol and dried in a desiccator overnight.

Reduction-diffusion process for Pd–In@FeOy and Pd–Pb@FeOy NPs

The Pd–In@FeOy-NPs powders were heated at the rate of 10 °C/min, annealed at 600 °C or 800 °C for 3 h and cooled at the rate of 10 °C/min under an Ar+4% H2 gas flowing at 0.5 L/min. The Pd–Pb@FeOy-NPs powders were also annealed at 600 °C for 3 h via the same heating and cooling process under the same atmosphere.

Growth of SiO2 shell on A1-PdInx@FeOy NPs

In a typical method28, the SiO2 shell was coated on A1-PdInx@FeOy NPs by adding TEOS (200 μL) in the cyclohexane solution (68 mL) containing A1-PdInx@FeOy NPs (0.25 mmolPd), IGEPAL@CO-520 (9.6 mL) and NH3aq (1.6 mL) with stirring at room temperature for 16 h. After the reaction, the A1-PdInx@FeOy@SiO2 NPs were collected by centrifugation with MeOH, purified with MeOH and twice with EtOH and dried in a desiccator overnight.

XAFS measurement

Fe K-edge, Pd K-edge, and In K-edge EXAFS measurements were performed by using the BL01B1 beamline of SPring-8 in Japan, where the incident X-ray beam was monochromated by a Si(311) double crystal monochromator. The sample powder was mixed with BN powder and pressed into a pellet. XAFS spectra of the L12-(Fe,In)Pd3 pellet were measured at 300 K, while XAFS spectra of the Z3-Fe(Pd,Ind)3 pellet were measured at 10 K in a Cu folder attached on a cryostat. The EXAFS analysis was carried out by using the REX2000 Ver. 2.5 program (Rigaku Co.)29. In the curve-fitting analysis of the EXAFS oscillation and Fourier transform (FT) of EXAFS, the phase shift and the back-scattering amplitude function of the Fe–Fe, Fe–Pd, Pd–Pd, Pd–In and Fe–In were estimated from α-Fe (PDF#00-006-0696), L10-FePd (PDF#03-065-9971), Pd (PDF#00-046-1043), B2-PdIn (PDF#03-065-4804) and the crystal information file obtained from Rietveld refinement for Z3-Fe(Pd,Ind)3, respectively, by using the FEFF8 program30.

The Pd NPs, the Pd–In alloy NPs, and the Pd–In@FeOy NPs before and after the reduction-diffusion process were dispersed in chloroform and dropped on amorphous carbon-coated copper grids (JEOL). The TEM samples of the Fe–Pd–In ternary alloy NPs were prepared by annealing the Pd–In@FeOy NPs on TEM grids. TEM images were recorded on a JEM1011 (JEOL) at an acceleration voltage of 100 kV. HAADF-STEM images and EDX maps were obtained by a JEM-ARM200CF at an acceleration voltage of 120 kV for the samples prepared by dropping Fe–Pd–In(Pb) ternary alloy NPs on TEM grids.

Powder XRD measurement

The XRD patterns were recorded on a PANalytical X’Pert Pro MPD diffractometer with Cu Kα radiation (λ = 1.542 Å) at 45 kV and 40 mA. Rietveld refinements were performed by using the computer program RIETAN-2000 (ref. 31).

EDX measurement and SEM observations

The EDX measurements were performed using EDAX APOLLO XF attached by scanning electron microscope (SEM, HITACH S-4800) at 20 kV, which detected Fe-K, Pd-L, In-L, and Pb-M peaks. The SEM images were observed at 30 kV.

M–H hysteresis loop

The magnetization–magnetic field (MH) curves of L12-(Fe,In)Pd3 and Z3-Fe(Pd,Ind)3 structures were obtained by a VSM (TOEI VSM-5) under a magnetic field (H) of –20 to 20 kOe at room temperature (~25 °C). To avoid rotating the grains during the magnetic measurements, the samples were fixed with eicosane (melting point of 36.7 °C).

Hydrogen-storage capacity

The pressure–composition isotherms were recorded on a BELSORP-HP and measured at 30 °C and 60 °C for the sample with the Pd/Fe and Pd/In molar ratio of 65/35 at.% and 84/16 at.%, respectively (Supplementary Figs. 8 and 10).

First-principles calculations

We performed first-principles calculations for the Z3- and L12-type Fe–Pd–M systems within density functional theory. We used the computational code OpenMX which was based on optimized pseudopotentials and pseudo-atomic-orbital basis functions32. The generalized gradient approximation of Perdew-Burke-Ernzerhof was selected for an exchange-correlation functional33. In each calculation, we replaced one Fe or Pd atom with one M (M = Zn, Cd, Hg, Ga, In, Tl, Ge, Sn, and Pb) atom to determine the stability of the systems (Supplementary Tables 37). There are two different Wyckoff positions of Pd labeled by 4i and 1d that are similar to 1b in Z3-type FePd3. For simplicity, we noted them as Z3-Fe2(Pd5,M1i) and Z3-Fe2(Pd5,M1d). For the basis sets, we chose s2p2d2 configurations for Fe and s2p2d2f1 for Pd and M. The cut-off radius for Fe was 6.0 atomic units (a.u.) and 7.0 a.u. for Pd and In. The cut-off energy was 500 rydberg, and the k-point mesh was 30 × 30 × 16 and 21 × 21 × 30 for the Z3- and L12-type Fe–Pd–M systems, respectively. Conventional cells were chosen as 1 × 1 × 1 and √2 × √2 × 1 for the Z3- and L12-type Fe–Pd–M systems, respectively. These conventional cells had the same number and kinds of atoms in a unit cell, which allowed us to compare the energy difference between these systems easily. Obtained calculation results were at 0 K and we did not consider the reaction paths of how the In-doped FePd3 structure was fabricated because of the limitation of the first-principles calculations. The formation energies of L12-(Fe2-x,Inx)Pd6 and Z3-Fe2(Pd6-x,Inxd) (EL12 and EZ3, respectively) were estimated by comparing EL12 and EZ3 under the same number and kinds of atoms as follows (0 < x < 1);

$${E}_{L12}=x\times E[L{1}_{2}-({{{{{{\rm{Fe}}}}}}}_{1},{{{{{{\rm{In}}}}}}}_{1}){{{{{{\rm{Pd}}}}}}}_{6}]+(1-x)\times E[L{1}_{2}-{{{{{{\rm{Fe}}}}}}}_{2}{{{{{{\rm{Pd}}}}}}}_{6}]+x\times \mu [{{{{{\rm{Fe}}}}}}]$$
(2)

and

$${E}_{Z3}=x\times E[Z3-{{{{{{\rm{Fe}}}}}}}_{2}({{{{{{\rm{Pd}}}}}}}_{5},{{{{{{{\rm{In}}}}}}}_{1}}^{d})]+(1-x)\times E[Z3-{{{{{{\rm{Fe}}}}}}}_{2}{{{{{{\rm{Pd}}}}}}}_{6}]+x\times \mu [{{{{{\rm{Pd}}}}}}]$$
(3)

where E[phase] is the Eform of each phase. Furthermore, in order to analyse the magnetic anisotropy of Z3- and L12-type Fe–Pd–In in Fig. 4, we compared the total energies obtained by non-collinear calculations in which spin directions on each atom were fixed within 0< θ(°) <90 and 0< φ(°) <45 (i.e., from [001] to [100] at φ = 0 and from [100] to [110] at θ = 0). Conventional cells in these non-collinear calculations were expanded to 2 × 2 × 1 and 2 × 2 × 2 for Z3-Fe8(Pd20,In4d) and L12-(Fe7,In1)Pd24 systems, respectively (Supplementary Table 8).