Intrinsic hard magnetism and thermal stability of a ThMn₁₂-type permanent magnet

Abstract

Herein, we theoretically demonstrate that simple metal (Ga and Al) substitutional atoms, rather than the conventional transition metal substitutional elements, not only stabilize the ThMn₁₂-type SmFe₁₂ and Sm(Fe,Co)₁₂ phases thermodynamically but also further improve their intrinsic magnetic properties such that they are superior to those of the widely investigated SmFe₁₁Ti and Sm(Fe,Co)₁₁Ti magnets, and even to the state-of-the-art permanent magnet Nd₂Fe₁₄B. More specifically, the quaternary Sm(Fe,Co,Al)₁₂ phase has the highest uniaxial magnetocrystalline anisotropy (MCA) of about 8 MJ m⁻³, anisotropy field of 18.2 T, and hardness parameter of 2.8 at room temperature and a Curie temperature of 764 K. Simultaneously, the Al and Ga substitutional atoms improve the single-domain size of the Sm(Fe,Co)₁₂ grains by nearly a factor of two. Numerical results of MCA and MCA-driven hard magnetic properties can be described by the strong spin-orbit coupling and orbital angular momentum of the Sm 4f-electron orbitals.

Introduction

ThMn₁₂-type SmFe₁₂ alloy is known for its attractive intrinsic hard magnetic properties^{1,2,3,4,5,6,7,8}, which make it a potential high-performance permanent magnet. Despite its high saturation magnetization M_s and strong uniaxial magnetocrystalline anisotropy (MCA) K_u, obtaining a single-crystalline phase of the ThMn₁₂ structure is extremely difficult^{9,10,11,12,13}. In particular, thermodynamically stable large (bulk)-scale production of SmFe₁₂ single crystals is in high demand for industrial applications such as electric motors and generators. To stabilize the ThMn₁₂ structure, a third substitutional metal element, including Ti or V^{14,15,16,17,18,19,20,21}, is essential. However, inclusions of these early transition metal (TM) elements are severely detrimental to intrinsic magnetic properties. For example, SmFe₁₁Ti and SmFe₁₁V (SmFe₁₀V₂) exhibit μ₀M_s of 1.14−1.16 and 1.12 (0.81) T and K_u of 3.9−4.8 and 1.92 (1.58) MJ m⁻³ at room temperature^15,16,17,21, respectively, which are much lower than the corresponding values (1.64 T and 5.4−5.67 MJ m⁻³) of the parent SmFe₁₂^22,23. In contrast, the replacement of 20 at.% Fe at the 8f site with Co enhances μ₀M_s up to 1.78 T and K_u up to 6.2 MJ m⁻³ at room temperature^22,23,24. These findings, in addition to the enhanced Curie temperature (T_c = 859 K), make ternary Sm(Fe_0.8Co_0.2)₁₂ superior to the state-of-the-art permanent magnet Nd₂Fe₁₄B (μ₀M_s = 1.57 T, K_u = 4.5 MJ m⁻³, and T_c = 585 K)^25,26,27. However, the ThMn₁₂ structure of this ternary compound could only be synthesized with a limited film thickness of no more than 0.5 μm^22,23. In addition to the structural instability, to maximize the permanent hard magnetic properties, the grain size of SmFe₁₂-based magnets must be prepared close to the single-domain (SD) size (~51−54 nm)^28,29. However, preparing nanometre-sized ThMn₁₂-type SmFe₁₂ in a practical sample is quite difficult, which must be resolved to fully utilize SmFe₁₂ as a practical high-performance permanent magnet. The development of thermodynamically stable bulk-scale SmFe₁₂ single crystals with improved SD grain sizes and desirable intrinsic magnetic properties is thus essential and of timely importance.

In this paper, we propose a possible solution to realize an otherwise unstable ThMn₁₂-type SmFe₁₂ permanent magnet through systematic first-principles density functional theory (DFT), density functional perturbation theory (DFPT), and Monte Carlo (MC) simulations on ternary Sm(Fe,M)₁₂ and quaternary Sm(Fe,Co,M)₁₂ alloys (M is a 3d or 3p metal substitutional atom). In contrast to conventional TM substitutional elements, simple metal (SM) Al and Ga substitutional atoms stabilize the ThMn₁₂-type Sm(Fe,Co)₁₂ structure thermodynamically with an improved SD grain size and simultaneously enhanced K_u. We further predict that intrinsic hard magnetic properties at an elevated temperature of the proposed quaternary Sm(Fe,Co,Al)₁₂ and Sm(Fe,Co,Ga)₁₂ compounds in the present study are superior to those of the widely investigated SmFe₁₁Ti and Sm(Fe,Co)₁₁Ti compounds, and even to the present and reported values by the experimental^25,26,27 and previous theoretical studies^30,31 for the state-of-the-art permanent magnet Nd₂Fe₁₄B, including a higher T_c. We attribute the physical origin of the large MCA and MCA-driven hard magnetic properties to the strong spin-orbit coupling (SOC) and orbital angular momentum of the Sm 4f-electron shells.

Results

Structural stability

We first inspected the structural stability of ThMn₁₂-type SmFe₁₂ structure upon M (M = Ti − Ga and Al) replacement. The formation enthalpy is defined as H_f = (E₀ − ∑_iμ_iN_i)N_A/N, where E₀ is the total energy of the system and μ_i and N_i are the chemical potential and the number of decomposable component i, respectively. N_A is the Avogadro number and N is the total number of atoms in the computed unit cell. Our computed unit cell is composed of 2 Sm atoms at the 2a site and 24 Fe atoms at the inequivalent crystallographic 8f, 8i, and 8j sites (Wyckoff positions). Their optimized atomic coordinates along with the lattice parameters a, b, and c of SmFe₁₂ are given in Table 1. The present a = b and c values are 8.481 and 4.661 Å, respectively, in agreement with the experimental (8.44 and 4.81 Å)^5,22 and previous theoretical results (8.46−8.49 and 4.68−4.81 Å)^8,32. Correspondingly, the three different substitution sites of M atoms for SmFe_12−xM_x (x = 0.5, 1, and 2) were considered and denoted as M(8f), M(8i), and M(8j). For each configuration, from the total energy minimization, the M atoms were identified to prefer a uniform distribution²⁴, as indicated in Fig. 1a–c. The optimized lattice parameters of SmFe₁₀M₂ with the preferred M site are listed in Table 2. In agreement with an experiment³³, ThMn₁₂-type SmFe₁₂ is not a stable phase and decomposes into bulk-Sm and α-Fe phases, as expected from the calculated positive H_f values of 2.9 kJ mol⁻¹ from the standard DFT and 6.5 kJ mol⁻¹ from the DFT plus U (DFT + U) with U_eff = 6 eV. A slightly smaller value of H_f = 1.8 kJ mol⁻¹ was reported in the previous DFT calculations³⁴. Nevertheless, the single-crystalline ThMn₁₂ phase can be obtained in SmFe₁₁Ti form^15,16,17. For a ternary system, simply considering decomposition into the final constituent elements is not sufficient for predicting the structural stability^35,36,37,38. The most competitive binary decomposable phases for Sm−Fe−M are identified by constructing the convex hull phase diagram^35,36,37,38.

Table 1 Optimized lattice parameters a, b, and c (Å), and angles α, β, and γ (^∘), and internal atomic coordinates of ThMn₁₂-type SmFe₁₂.

Fig. 1: Atomic structures used for calculations.

ThMn₁₂-type atomic structures of SmFe_12−xM_x (when x = 2) for the a 8f, b 8i, and c 8j site M substitutes. Only the most stable configuration of the distribution patterns of the M atoms is shown for each substitution site.

Table 2 Optimized lattice parameters a, b, and c (Å), and spin magnetic moment μ_S (μ_B per atom) of the different atom types in SmFe₁₀M₂ for M = Ti-Ga and Al.

We show heat map of the calculated formation enthalpy of the ternary Sm−Fe−Ti system in Fig. 2a. The solid nodes on the phase diagram represent the present ternary SmFe_12−xTi_x phases (x = 0.5, 1, and 2), while the known binary and elemental phases are indicated by the open nodes. In the Sm−Fe−Ti diagram, since all Sm−Fe binary phases are unstable against their elemental bulk-Sm and α-Fe phases, we have chosen the bulk-Sm and α-Fe as decomposable phases rather than a Sm−Fe binary phase. We find that the H_f values of the known binary Sm₂Fe₁₇, Sm₃Fe₂₉, Sm₆Fe₂₃, and SmFe₂ phases are 8.4, 9.2, 4.9, and 0.9 kJ mol⁻¹, respectively. In the Ti-poor region (i.e., x = 0.5), Fe₂Ti is chosen as the decomposable binary phase for Sm−Fe−Ti, as indicated by the dotted lines in Fig. 2a. In the Ti-rich region (i.e., x = 1 and 2), Fe₂Ti + FeTi decomposition is taken into account (solid lines). The numerical values of the obtained ${H}_{{{{\rm{f}}}}}^{{{{\rm{elm}}}}}$ (against the elemental decomposition) and ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}$ (against the binary decomposition) of SmFe_12−xTi_x phases are shown in Fig. 2b for x = 0, 0.5, 1, and 2. Both ${H}_{{{{\rm{f}}}}}^{{{{\rm{elm}}}}}$ and ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}$ decrease as x increases and reach the negative values of −10.7 and −4.2 kJ mol⁻¹ at x = 2, respectively. For SmFe₁₁Ti, ${H}_{{{{\rm{f}}}}}^{{{{\rm{elm}}}}}=-2.6$ kJ mol⁻¹ and ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}=0.2$ kJ mol⁻¹. Since the single-crystalline SmFe₁₁Ti phase is practically achieved in the bulk form^15,16,17, the latter value (0.2 kJ mol⁻¹) of SmFe₁₁Ti is referred to as a threshold of the ThMn₁₂-phase stability in the present study. The aforementioned experimental observations of the reduced μ₀M_s and K_u upon the Ti substitution^15,16,17 have also been obtained in our calculations, as shown in Fig. 2c, which will be discussed more explicitly in the following paragraphs.

Fig. 2: Structural stability and intrinsic magnetic properties of Sm−Fe−Ti system.

a Heat map of the formation enthalpy of the ternary Sm−Fe−Ti system. Open nodes mark the known elemental and binary phases. Solid nodes represent the present ternary SmFe_12−xM_x (x = 0.5, 1, and 2) phases. Dotted and solid lines are the projections of the convex hull construction into compositional space. b Formation enthalpy H_f and c saturation magnetization μ₀M_s (left) and uniaxial magnetocrystalline anisotropy K_u (right) of SmFe_12−xM_x (x = 0, 0.5, 1, and 2). The corresponding experimental results for SmFe₁₂ measured at 0−300 K^22,23 and SmFe₁₁Ti at 4.2−300 K^15,16,17 are indicated by the open symbols, where the result at 0 K is determined by fitting with low-temperature measurements²².

Figure 3a presents the ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}$ values of SmFe_12−xM_x (M = Ti − Ga and Al; x = 0.5 and 2) for M(8f), M(8i), and M(8j). Early TMs, Ti to Mn, prefer the 8i site rather than the 8f and 8j sites for both x = 0.5 and 2. A similar trend is also observed for SmFe₁₁M, which, except for SmFe₁₁Ti, is not shown for simplification of the figure. Hereafter, we thus refer to the results corresponding to the practically acceptable stoichiometry SmFe₁₀M₂, unless specifically mentioned. These results are supported by the experiment in which Ti and V dopants occupied the 8i site³⁹. Late TMs, except for Co(8f), prefer the 8j site. The different site preferences can be interpreted in correlation with the atomic radii and electronegativity. In the SmFe₁₂ structure, the 8i site (3.06 and 4.98 Å) is farther from Sm than the 8j (3.04 and 4.82 Å) and 8f (3.21 and 3.21 Å) sites (Table 1). Ti − Mn with large atomic radii thus naturally prefer the 8i site. In contrast, Ni − Ga attract Sm more than Ti − Mn, as Ni − Ga (1.65−1.91) have higher electronegativities than Ti − Mn (1.54−1.66).

Fig. 3: Impact of single-element substitution on the stability and intrinsic magnetic properties at 0 K.

a Formation enthalpy ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}$ of SmFe_12−xM_x with x = 0.5 (open) and 2 (solid symbols) for M = Ti−Zn (transition metals), Ga and Al (simple metals) for the 8f, 8i, and 8j sites. Horizontal dashed line corresponds to the calculated ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}$ value of SmFe₁₁Ti, a threshold of the ThMn₁₂-phase stability. b Saturation magnetization μ₀M_s and c uniaxial magnetocrystalline anisotropy K_u of SmFe₁₀M₂ with the preferred M site. Solid and open circles indicate the experimental data for SmFe₁₁M and SmFe₁₀V₂/SmFe_9.6Co_2.4 measured at 0−300 K^{14,15,16,17,20,21,22,23}, respectively, where the result at 0 K is determined by fitting with low-temperature measurements²².

As predicted from the ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}$ values (Fig. 3a), the Ga (${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}=-1.5$ kJ mol⁻¹) and Al (−3.1 kJ mol⁻¹) substitutional atoms stabilize the ThMn₁₂ structure. These SMs improve the stability comparably to the conventional stabilizing early TM substitutional elements Ti (−4.2 kJ mol⁻¹) and V (0.2 kJ mol⁻¹). On the other hand, the dopant amount of x = 0.5 is not sufficient for stabilizing the ThMn₁₂ phase as the aforementioned basic requirement of the crystal formation (${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}} \,<\, 0.2$ kJ mol⁻¹; as indicated by the horizontal dashed line in Fig. 3a) is not fulfilled. Practically, SmFe_12−xTi_x and SmFe_12−xV_x phases are synthesized with a nearly perfect ThMn₁₂ structure when x ≥ 1^{14,15,16,17,20,21}. Furthermore, a very recent experiment reported that the stability of SmFe_8.8Co_2.2Ti was greatly improved by Ga addition³³. The authors also predicted theoretically that Ga is the most effective stabilizing elements with higher μ₀M_s over Ti, V, and Cr dopants. Our prediction in support with these experimental and theoretical studies suggests a possible synthesis of the single-crystalline ThMn₁₂ structure, including a series of Sm(Fe,Co)₁₂ and Sm(Fe,Co,Ti)₁₂ compounds, with Al and/or Ga, at least at a low temperature.

Intrinsic magnetic properties

The Slater − Pauling-like behavior of μ₀M_s is evident in Fig. 3b for the late TMs (Mn − Zn). For each M, only the result obtained by the full-potential calculations for the preferred substitution site is presented. For SmFe₁₂, the experimental μ₀M_s of 1.94 T at 0 K determined by fitting with low-temperature measurements²² is underestimated in the present theory (1.64 T). This discrepancy might be attributed to the existence of the α-Fe phase and surface/interface effect in a real thin-film sample, which are fully ignored in the present simulation. To obtain microscopic insight, we decompose μ₀M_s into the atomic-level spin magnetic moment (μ_S) in Table 2. The spin moment contributions of the Fe(8f), Fe(8i), and Fe(8j) atoms to μ₀M_s are 1.78, 2.53, and 2.34 μ_B, respectively. The Sm 4f spin moment (−5.44 μ_B) is antiparallel to the Fe 3d spin moment, which is mediated by the Sm 5d orbitals^40,41. The magnitude of μ_S adheres to the high-spin state. Nevertheless, the orbital moment contribution (μ_L = 2.25 μ_B) of the 4f electrons of Sm (antiparallel to its spin moment according to the 3rd Hund’s rule, thus parallel to the Fe 3d spin moment) to μ₀M_s is substantial, whereas those from the Fe 3d orbitals are <0.1 μ_B.

In the Λ-shape curve (Slater − Pauling-like) shown in Fig. 3b, a peak occurs for M = Co; μ₀M_s increases from 1.64 T for SmFe₁₂ to 1.68 T for SmFe₁₀Co₂. A similar trend was found in the room-temperature measurement²², as indicated by the circle symbols in Fig. 3b; 1.78 T for SmFe_9.6Co_2.4 and 1.64 T for SmFe₁₂. But this is not the case in the extrapolated zero-temperature measurement²²; μ₀M_s decreases from 1.94 T for SmFe₁₂ to 1.88 T for SmFe_9.6Co_2.4. The difference could be attributable to the aforementioned shortcomings as well as to dissimilar Co concentrations in the present theory (i.e., SmFe₁₀Co₂) and experiment (i.e., SmFe_9.6Co_2.4)²². Both the theory and experiment reveal that the early TMs (Ti − Mn) greatly lower μ₀M_s (0.75 − 0.95 T)^16,17,21 because of their antiparallel spin coupling to Fe (Table 2). Thus, the concentration of these early TMs should be kept at a minimal. For SmFe₁₁Ti, the present μ₀M_s is 1.31 T, which is comparable with 1.28−1.14 T at 4.2−300 K in the experiments^15,16,17. In contrast to the early TMs, the nonmagnetic late TMs and SMs are not as detrimental to μ₀M_s (1.2−1.3 T). Overall, μ₀M_s is mainly affected by the magnetic moments of the M atom and its neighboring Fe atoms, as addressed explicitly in Table 2.

Figure 3c shows the computed K_u of SmFe₁₀M₂ (M = Ti − Ga and Al). We recall from ref. ²⁴ that the K_u values of SmFe₁₂ are 12.8 MJ m⁻³ in the DFT calculation and 10.6 MJ m⁻³ in the DFT + U calculation (with U_eff = 6 eV). Both values are in reasonable agreement with the low-temperature experimental values of 10.76−11.1 MJ m⁻³ at 100−10 K²³. The K_u results thus refer to those from DFT calculations without the U_eff parameter to minimize computational complications. From a theoretical point of view, we here would like to remind that direct comparison with experiment, particularly on the precise magnitude of MCA, requires some caution, as an accurate treatment of f-electron systems by first-principles is quite challenging. Nevertheless, our theory for M = Co (K_u = 11.2 MJ m⁻³) further supports the low-temperature experiment in which the substitution of Co atoms for 20 at.% Fe at the 8f site reduced K_u to 9.73 MJ m⁻³ at 10 K²³. According to Fig. 3c, substantial reductions obviously appear for the other TM substitutions, including the reference SmFe₁₁Ti system (K_u = 9.8 MJ m⁻³). Similar reductions were also observed for SmFe₁₁Ti (7.2−3.9 MJ m⁻³ at 77−300 K)^15,16,17 and SmFe₁₀V₂ (1.58 MJ m⁻³ at 300 K) in the experiments²¹. Cr and Mn even turn the magnetic easy axis from the c axis to the ab plane (which is improper for permanent magnets). In contrast, the SM substitutes (Ga and Al), which stabilize the ThMn₁₂ structure, still preserve the MCA uniaxiality. In particular, K_u reaches as high as 7.6 MJ m⁻³ for M = Al, which is 6.3 MJ m⁻³ for M = Ga. Hence, in the discussion below we focus mainly on Al rather than Ga. Note that this uniaxial MCA makes Sm(Fe,M)₁₂-based magnets a potential high-performance permanent magnet beyond the other ThMn₁₂-type Nd(Fe,M)₁₂ magnets with a biaxial MCA^12,13,18,19.

Recalling the favorable formation (${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}} \,<\, 0$ in Fig. 3a), in a real sample, Al (which has the smallest atomic radius among the M) could occupy either the 8j or 8i site, or both. These two sites are energetically competitive and differ in ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}$ by only within 1.4 kJ mol⁻¹. For a given temperature, we estimate the occupation probability $\left\langle {N}_{\nu }\right\rangle$ of the Al substitution sites (ν = 8f, 8i, and 8j) by using Maxwell-Boltzmann statistics. As shown in Fig. 4, the occupation probability of the Al substitute at the 8j (8i) site decreases (increases) with temperature; $\langle {N}_{{{{\rm{8i}}}}}\rangle :\langle {N}_{{{{\rm{8j}}}}}\rangle =0.30:0.57$ at 300 K and 0.33:0.48 at 500 K. A certain amount of the 8f site could also be occupied by the Al atoms in a high-temperature sample. We further find that K_u is almost independent of the stable 8i (10.9 MJ m⁻³) and 8j (7.6 MJ m⁻³) sites. From a practical viewpoint, this substitution-site-independent uniaxial feature of the MCA is worth noting, in addition to the abundance of Al on earth.

Fig. 4: Temperature-dependent occupation probability.

Occupation probability $\left\langle {N}_{\nu }\right\rangle$ of the Al substitution sites (ν = 8f, 8i, and 8j) at an elevated temperature in SmFe₁₀Al₂.

Impact of multielement substitution

We next explored the effects of multielement substitution on the intrinsic permanent magnetic properties. According to Fig. 3, only Co substitution maximizes μ₀M_s, while Ti, V, Ga, and Al improve the SmFe₁₂ stability. To support this scenario, we extended our calculations to the quaternary stoichiometries SmFe₁₀CoM and SmFe₉Co₂M (M = Ti, V, Ga, and Al). For each M, we considered several different substitutional configurations of the M atoms while the Co atoms were kept fixed at their optimized 8f sites, as shown in Fig. 1a. As examples, we illustrate the most favorable structures of SmFe₁₀CoAl and SmFe₉Co₂Al compounds obtained from our total energy minimization in Fig. 5a and b, respectively.

Fig. 5: Impact of multielement substitution on the stability and intrinsic magnetic properties at 0 K.

Optimized atomic structures of a SmFe₁₀CoAl and b SmFe₉Co₂Al. Blue and orange spheres represent the Co and Al substitutional atoms, respectively. Other atomic symbols are the same as those used in Fig. 1. c Formation enthalpy ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}$, d saturation magnetization μ₀M_s, and e uniaxial magnetocrystalline anisotropy K_u of SmFe₁₀CoM and SmFe₉Co₂M for M = Ti, V, Ga, and Al. Horizontal dashed lines correspond to the calculated results of SmFe₁₁Ti for reference. f Atom-by-atom contribution to magnetocrystalline anisotropy energy E_MCA and g orbital magnetic anisotropy Δμ_L of SmFe₁₂ (denoted as 1:12), SmFe₁₀Co₂ (1:10:2), SmFe₉Co₂Al (1:9:2:1), and SmFe₁₀CoAl (1:10:1:1).

Figures 5c, d and e show the ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}$, μ₀M_s, and K_u values, respectively, of SmFe₁₀CoM and SmFe₉Co₂M for M = Ti, V, Ga, and Al. Here, the formation enthalpy ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}$ is calculated against Fe₃Co + Fe₃M + FeCo + FeM decomposition. All the quaternary phases considered here, except for M = V, are stable as their ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}$ values are <0.2 kJ mol⁻¹ for SmFe₁₁Ti. The obtained μ₀M_s ranges from 1.3 T (M = Ti and V) to 1.45 T (M = Ga and Al), which are sufficient values for practical permanent magnet applications. In our further prediction, as shown in Fig. 5e, the main trend of the K_u of SmFe₁₀M₂ (Fig. 3c) is roughly preserved for both SmFe₁₀CoM and SmFe₉Co₂M; K_u reaches the largest values of 12.5 MJ m⁻³ for SmFe₉Co₂Al and 14.1 MJ m⁻³ for SmFe₁₀CoAl. We emphasize that the latter value is the highest K_u achieved in the present study, which is 1.3−4.3 MJ m⁻³ larger than those of SmFe₁₂ (12.8 MJ m^–3) and SmFe₁₁Ti (9.8 MJ m⁻³). More remarkably, these K_u values (12.5−14.1 MJ m⁻³) are even superior to those obtained in the present (5.4 MJ m⁻³) and previous theoretical (4.31 MJ m⁻³)³¹ and experimental studies (4.5 MJ m⁻³)²⁷ for the state-of-the-art permanent magnet Nd₂Fe₁₄B.

Microscopic origin of magnetic anisotropy

The microscopic origin of MCA is the SOC interaction between the atomic orbital angular momentum (L) and the atomic spin angular momentum (S), given by H_SOC = λL ⋅ S, where λ is the atomic SOC parameter. We provide the atom-by-atom contributions to MCA energy (E_MCA) for the selected SmFe₁₂, SmFe₁₀Co₂, SmFe₁₀CoAl and SmFe₉Co₂Al compounds in Fig. 5f. Here, E_MCA is scaled down to the microscopic atomic level (meV atom⁻¹), rather than the macroscopic energy density (MJ m⁻³). For SmFe₁₂ and SmFe₁₀Co₂, Sm dominates E_MCA, whereas the contributions from the Fe and Co 3d orbitals are 2 orders of magnitude smaller. The two equivalent Sm sites in the unit cells of SmFe₁₂ and SmFe₁₀Co₂ are no longer symmetrically equivalent in the quaternary compounds due to the presence of the 8j-site substitutional atoms. In the optimized structure, the 2 Sm atoms are also not equally separated from the Al(8j) sites. As described in Fig. 5a, the Sm₁ − Al₁ (Sm₁ − Al₂) and Sm₂ − Al₁ (Sm₂ − Al₂) separations are 3.08 (4.79) and 4.88 (3.11) Å, respectively. We find that the Sm₁ site predominantly contributes to the large E_MCA and that Sm₂ undermines E_MCA. The same trend holds for the orbital magnetic anisotropy (Δμ_L) in Fig. 5g, defined as ${{\Delta }}{\mu }_{{{{\rm{L}}}}}={({\mu }_{{{{\rm{L}}}}})}_{c}-{({\mu }_{{{{\rm{L}}}}})}_{a}$, consistent with the Bruno theory⁴²; Δμ_L is larger by 1 − 2 orders of magnitude in the Sm 4f orbitals than in the 3d orbitals. The contribution from the Al 3p orbitals to E_MCA (Δμ_L) is essentially zero.

Thermodynamic phase stability

The Helmholtz free energy can be written as F(T, V) = E₀(V) + F_el(T, V) + F_vib(T, V) + F_mag(T, V), where E₀(V) is the zero-temperature total energy of the system, and F_el(T, V), F_vib(T, V), and F_mag(T, V) are the electronic, vibrational, and magnon contributions of the free energy, respectively. The phase stability against phase decomposition into the most competitive decomposable compounds can be described thermodynamically by the change in the free energy^43,44: ΔF(T, V) = F(T, V) − ∑_iN_iF_i(T, V), where N_i and F_i(T, V) are the number and free energy of decomposable compounds i, respectively. The temperature-dependent free energy contributions of F_el(T, V), F_vib(T, V), and F_mag(T, V) are shown in Fig. 6a for SmFe₁₂, SmFe₁₀Co₂, SmFe₁₀CoAl, and SmFe₉Co₂Al. In Fig. 6a, the same for SmFe₁₁Ti is also shown as a reference. All systems exhibit similar trends in the free energy against temperature; F_el(T, V) and F_vib(T, V) decrease as temperature increases, whereas F_mag(T, V) increases with temperature. Obviously, the former two contributions, particularly the vibrational, mainly determine the temperature-induced changes in the free energy.

Fig. 6: Thermodynamic phase stability.

Temperature-dependent a free energy contributions of the electronic F_el(T, V), vibrational F_vib(T, V), and magnon F_mag(T, V), and b free energy change ΔF(T, V) of SmFe₁₂, SmFe₁₁Ti, SmFe₁₀Co₂, SmFe₁₀CoAl, and SmFe₉Co₂Al. In b the contributions of the electronic ΔF_el(T, V), vibrational ΔF_vib(T, V), and magnon ΔF_mag(T, V) to ΔF(T, V) are also presented for each compound. Open symbols at 0 K represent the formation enthalpy ${H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}$.

Figure 6b shows the temperature-dependent free energy change ΔF(T, V) and its contributions, i.e., ΔF_el(T, V), ΔF_vib(T, V), and ΔF_mag(T, V), for SmFe₁₂, SmFe₁₁Ti, SmFe₁₀Co₂, SmFe₁₀CoAl, and SmFe₉Co₂Al. As a generic for all compounds, ΔF(T, V) decreases as temperature increases, which is almost entirely accounted for by the vibrational contribution. The other two contributions, ΔF_el(T, V) and ΔF_mag(T, V), are insignificant. For SmFe₁₂, ΔF(T, V) changes its sign from positive to negative at temperature around 1000 K. Similar results were reported in previous calculations although the sign change in ΔF(T, V) occurs much earlier around 200 K³⁴. The main cause for this discrepancy is the choice of the exchange-correlation in the computation, as mentioned early; DFT + U in the present study and DFT in the previous study³⁴. The Ti and Co substitutional atoms reduce the onset temperature significantly; around 200 K for SmFe₁₁Ti and 500 K for SmFe₁₀Co₂. This indicates that SmFe₁₁Ti is thermally stable as observed in the experiments^15,16,17. For SmFe₁₀CoAl and SmFe₉Co₂Al, ΔF(T, V) remains negative over the temperature range, which suggests an acceptable thermodynamic stability of Sm(Fe,Co,Al)₁₂ magnets under realistic conditions.

Exchange interaction

The exchange interaction in the Heisenberg model is calculated to determine the intrinsic hard magnetic properties at an elevated temperature. When an external magnetic field is not applied, the Heisenberg spin Hamiltonian is expressed as $H=-(1/2){\sum }_{i\ne j}{J}_{ij}{{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{j}-{K}_{{{{\rm{u}}}}}{\sum }_{i}{({{{{\bf{S}}}}}_{i}\cdot {{{\bf{e}}}})}^{2}$, where J_ij is the exchange coupling interaction between two spins S_i (at the i site) and S_j (at the j site), and e is the unit vector along the magnetic easy axis. The exchange interaction parameters were estimated by the constrained local moment approach in the DFT calculations through J_ij = (Δ_ij − Δ_i − Δ_j)/4n_iz_ijS_iS_j^45,46, where Δ_ij is the energy difference between the magnetic ground state and the excited state (i.e., the magnetic moments at sites i and j are inverted). Δ_i (Δ_j) refers to the inverted magnetic moment at the i (j) site. n_i is the number of atoms in sublattice i, and z_ij is the number of nearest neighbor sites in sublattice j relative to sublattice i. The exchange coupling integral J_ex is the sum of J_ij, i.e., J_ex = ∑_i,jz_ijJ_ij. In this approach, the spin in the ith sublattice is assumed to interact with the jth neighbor only while the other possible intrasublattice exchange interactions are disregarded. The present results of the exchange interaction parameters in the analyses below have to be thus double checked with a more precise state-of-the-art approach.

Figure 7a shows the calculated J_ij values for the different atomic couplings in SmFe₁₂, SmFe₁₁Ti, SmFe₁₀Co₂, SmFe₁₀CoAl, and SmFe₉Co₂Al. The positive (negative) exchange parameter reflects the preference for spin parallel (antiparallel) coupling between the two magnetic moments. To inspect the validity of our results, we first calculated the exchange interactions of the prototypical ferromagnetic systems, including α-Fe, α^″-Fe₁₆N₂⁴⁷, and B2-FeCo⁴⁶. For α-Fe, the nearest- and next-nearest-neighbor Fe−Fe interactions are 16.8 and 7.6 meV, respectively. These values are in good agreement with previous calculations^48,49. The third-nearest-neighbor interactions are found to be rather small, although J_ij is a long-range interaction. As shown in Fig. 7a, the Fe−Fe exchange parameters are more or less retained in SmFe₁₂ and SmFe₁₁Ti, while the Sm−Fe interactions prefer antiparallel coupling. The Fe(8i)−Fe(8i) interaction is the strongest among the Fe−Fe interactions since the Fe(8i) atom has the largest spin magnetic moment compared to the other two sites (Table 2). In the Sm−Fe−Co and Sm−Fe−Co−Al systems, the Co substitute enhances all the first-nearest-neighbor exchange parameters, which indeed occurs quite often in Fe−Co magnetic systems^46,49. The second-nearest-neighbor Fe−Fe and Sm−Fe interactions are not much altered from those of SmFe₁₂, and the Fe−Al interactions are negligible (thus not shown).

Fig. 7: Temperature-dependent intrinsic magnetic properties.

a Exchange interaction parameters J_ij as a function of the interatomic distance, b saturation magnetization μ₀M_s versus temperature, and c uniaxial magnetocrystalline anisotropy K_u versus temperature of SmFe₁₂, SmFe₁₁Ti, SmFe₁₀Co₂, SmFe₁₀CoAl, and SmFe₉Co₂Al. In a the nearest- and next-nearest-neighbor exchange interactions of the α-Fe structure are shown in the open circles. Open symbols in b and c denote the experimental data for SmFe₁₂, SmFe₁₁Ti, and SmFe_9.6Co_2.4, taken from refs. ^{15–17,22,23}.

Temperature-dependent intrinsic magnetic properties

We plot the temperature-dependent μ₀M_s and K_u obtained from the constrained MC simulation in Fig. 7b and c, respectively. In the MC simulation, we adopted an 8.47 nm × 8.47 nm × 4.64 nm cell with 26,000 atoms under periodic boundary conditions. One thousand equilibrations, a critical damping of 0.1, and a time step of 1 fs were imposed. The open symbols in Fig. 7b and c denote the available experimental data for SmFe₁₂, SmFe₁₁Ti, and SmFe_9.6Co_2.4^{15,16,17,22,23}. For both SmFe₁₂ and Sm(Fe,Co)₁₂, the present theory reproduces the experimental trends of μ₀M_s and K_u versus temperature, even though small discrepancies in the absolute values are detected. The main possible causes of such small discrepancies have already been addressed in previous paragraphs. Moreover, the K_u values from the present theory become larger in SmFe₁₀Co₂ than in SmFe₁₂ at T > 200 K, which occurs at a slightly higher temperature of ~300 K in the experiment^22,23. In our prediction, the room-temperature K_u values are 7.8 MJ m⁻³ for SmFe₁₀CoAl and 8 MJ m⁻³ for SmFe₉Co₂Al (Fig. 7c).

Table 3 summarizes the present theoretical Curie temperature T_c, anisotropy field μ₀H_a( = 2K_u/M_s), and hardness parameter $\kappa (={({K}_{{{{\rm{u}}}}}/{\mu }_{0}{M}_{{{{\rm{s}}}}}^{2})}^{1/2})$ at 0, 300, and 500 K in comparison with the available experimental data. We predict T_c to be about 554, 568, 885, 665, and 764 K for SmFe₁₂, SmFe₁₁Ti, SmFe₁₀Co₂, SmFe₁₀CoAl, and SmFe₉Co₂Al, respectively. The fairly good consistency of the predicted values of SmFe₁₂ (554 K), SmFe₁₁Ti (568 K), and SmFe₁₀Co₂ (885 K) with the experimental values (555, 597, and 859 K)^22,50,51 conforms the reliability of our calculations. Notably, the T_c (665−764 K) of Sm(Fe,Co,Al)₁₂ fulfils the basic requirement of high-performance permanent magnets (i.e., T_c ≥ 550 K), as suggested in ref. ⁹. For μ₀H_a and κ, overall the agreement between the present theory and the experiment²² is satisfactory. In order to further justify a potential replacement of Sm(Fe,M)₁₂-based magnets for the currently best permanent magnet Nd₂Fe₁₄B, we also compare our results for the present Sm(Fe,M)₁₂ systems to the results reported by the previous theoretical and experimental studies for Nd₂Fe₁₄B in Table 3. The obtained results of T_c and μ₀H_a as well as κ of the present quaternary Sm(Fe,Co,Al)₁₂ systems are superior to the previously reported theoretical (602 K and 2.4−7 T)^30,31 and experimental values (585 K, 4.5−8.8 T, and 1.5)^25,26,27 of Nd₂Fe₁₄B in the entire temperature range up to 500 K.

Table 3 Predicted Curie temperature T_c (K), anisotropy field μ₀H_a (T), and hardness parameter κ of SmFe₁₂, SmFe₁₁Ti, SmFe₁₀Co₂, SmFe₁₀CoAl, and SmFe₉Co₂Al at 0, 300, and 500 K.

Magnetic grain size

The magnetic grain size plays an important role in practice to maximally utilize the intrinsic magnetic properties of a permanent magnet. A permanent magnet exhibits maximal coercivity when the magnet domain grains reach the SD size. Here we estimate the minimal and maximal limits of the stable SD size. The SD size ranges from the minimal stable particle diameter ${D}_{{{{\rm{sp}}}}}={(60{k}_{{{{\rm{B}}}}}T/{K}_{{{{\rm{u}}}}})}^{1/3}$ to the domain threshold diameter ${D}_{{{{\rm{sd}}}}}=72{({A}_{{{{\rm{ex}}}}}{K}_{{{{\rm{u}}}}})}^{1/2}/{\mu }_{0}{M}_{{{{\rm{s}}}}}^{2}$ (A_ex is the exchange stiffness constant and $={J}_{{{{\rm{ex}}}}}\left\langle {S}^{2}\right\rangle /{V}^{1/3}$)^52,53. If grain sizes are smaller than D_sp, then the particle behaves like a superparamagnet, and if grain sizes are larger than D_sd, then the grains are energetically favorable for splitting into multiple domains. The estimated temperature-dependent D_sp and D_sd are reported in Fig. 8a and b, respectively. For all compounds, both D_sp and D_sd increase with increasing temperature up to T_c. The increase is primarily a reflection of the reduced μ₀M_s and K_u with temperature, as discussed regarding Fig. 7. For SmFe₁₂, we find D_sp = 3.5 nm and D_sd = 48.6 nm at room temperature. These values at room temperature do not change much upon Co substitution, in reasonable agreement with the experimental SD size of 51 nm in SmFe_8.8Co_2.2Ti²⁹. For SmFe₁₀CoAl and SmFe₉Co₂Al, D_sp = 3 and 4.5 nm and D_sd = 90.5 and 86.3 nm at room temperature, respectively. We note that their stable SD regimes are approximately two times wider than those of the other two compounds (SmFe₁₂ and SmFe₁₀Co₂) in the entire temperature range up to 500 K.

Fig. 8: Temperature-dependent magnetic grain size.

a Minimal stable grain size D_sp and b single-domain threshold size D_sd versus temperature of SmFe₁₂, SmFe₁₀Co₂, SmFe₁₀CoAl, and SmFe₉Co₂Al. Vertical dotted lines denote the Curie temperatures T_c. Open circle in b indicates an experimental single-domain size (51 nm) of SmFe_8.8Co_2.2Ti at room temperature, taken from ref. ²⁹.

Discussion

Using systematic DFT, DFPT, and MC simulations, we reveal that the substitutes of simple metal atoms (Ga and Al) for the Fe sites, rather than the conventional TM substitutional elements, stabilize the ThMn₁₂-type Sm(Fe,Co)₁₂ structure with desirable intrinsic magnetism. In particular, the optimal intrinsic magnetic properties of the quaternary Sm(Fe,Co,Al)₁₂ compounds, including the highest K_u of 8 MJ m⁻³, μ₀H_a of 18.2 T, κ of 2.8 at room temperature and a T_c of 764 K, are predicted to be superior to those of the widely investigated SmFe₁₁Ti and Sm(Fe,Co)₁₁Ti magnets, and even to the present (K_u = 5.4 MJ m⁻³) and previously reported theoretical (4.31 MJ m⁻³, 2.4 − 7.0 T, and 602 K)^30,31 and experimental results (4.5 MJ m⁻³, 8.8 T, and 1.5 at room temperature and 585 K)^25,26,27 and for the state-of-the-art magnet Nd₂Fe₁₄B. We further predict that the Ga and Al substitutional atoms also significantly improve the SD size of the Sm(Fe,Co)₁₂ grains. We hope that the present prediction may resolve the major problem of the structural and thermal instabilities of the ThMn₁₂ structure, leading to practical realization of SmFe₁₂-based high-performance permanent magnets.

Methods

We adopted the WIEN2k package⁵⁴ with the generalized gradient approximation (GGA) for the exchange-correlation functional⁵⁵. This method accurately deals with both the core and valence electrons and is suitable for f-electron magnetic systems. Herein, K_u is defined as K_u = (E_a − E_c)/volume, where E_a and E_c are the total energies with magnetization along the a and c directions, respectively. A total of 1271 k points (or a 11 × 11 × 21 k-point mesh) were used in the irreducible Brillouin zone⁵⁶. The convergence of K_u with respect to the number of k points was seriously checked. The atom-by-atom contribution to E_MCA was obtained by switching on/off the SOC of individual atom types in the WIEN2k calculations. The lattice and ionic coordinate relaxations were performed by the Vienna ab initio simulation package (VASP)⁵⁷ version 5.4.4. 4s¹3d⁷, 4s¹3d⁸, and 6s²5p⁶4f⁶ are treated as valence electrons for Fe, Co, and Sm, respectively. The strongly correlated 4f electrons were treated with the Hubbard model in the DFT + U scheme. The effective onsite U_eff parameter (U − J) was chosen as 6 eV, which is sufficient to split the f-orbital bands into lower and upper Hubbard bands²⁴. An energy cutoff of 600 eV, a 9 × 9 × 17 k-point mesh, and a high force convergence criterion of 10⁻³ eV Å⁻¹ were imposed for the structure optimization. The phonon dispersion and thermodynamic properties were carried out by using the PHONOPY code⁵⁸ within the harmonic approximation in the DFPT⁵⁹ implemented in the VASP. For the phonon calculations, we employed a 2 × 2 × 1 supercell and a 3 × 3 × 3 k-point mesh in the interpolation of the force constants matrices. Numerical calculations of temperature-dependent intrinsic magnetic properties and T_c were carried out using MC simulation in the VAMPIRE software package^60,61.