## 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 ThMn_{12}-type SmFe_{12} 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 SmFe_{11}Ti and Sm(Fe,Co)_{11}Ti magnets, and even to the state-of-the-art permanent magnet Nd_{2}Fe_{14}B. 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 4*f*-electron orbitals.

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## Introduction

ThMn_{12}-type SmFe_{12} 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 *u**n**i**a**x**i**a**l* magnetocrystalline anisotropy (MCA) *K*_{u}, obtaining a single-crystalline phase of the ThMn_{12} structure is extremely difficult^{9,10,11,12,13}. In particular, thermodynamically stable large (bulk)-scale production of SmFe_{12} single crystals is in high demand for industrial applications such as electric motors and generators. To stabilize the ThMn_{12} 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, SmFe_{11}Ti and SmFe_{11}V (SmFe_{10}V_{2}) exhibit *μ*_{0}*M*_{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 SmFe_{12}^{22,23}. In contrast, the replacement of 20 at.% Fe at the 8f site with Co enhances *μ*_{0}*M*_{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_{0.8}Co_{0.2})_{12} superior to the state-of-the-art permanent magnet Nd_{2}Fe_{14}B (*μ*_{0}*M*_{s} = 1.57 T, *K*_{u} = 4.5 MJ m^{−3}, and *T*_{c} = 585 K)^{25,26,27}. However, the ThMn_{12} structure of this ternary compound could only be synthesized with a limited film thickness of no more than 0.5 *μ*m^{22,23}. In addition to the structural instability, to maximize the permanent hard magnetic properties, the grain size of SmFe_{12}-based magnets must be prepared close to the single-domain (SD) size (~51−54 nm)^{28,29}. However, preparing nanometre-sized ThMn_{12}-type SmFe_{12} in a practical sample is quite difficult, which must be resolved to fully utilize SmFe_{12} as a practical high-performance permanent magnet. The development of thermodynamically stable bulk-scale SmFe_{12} 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 ThMn_{12}-type SmFe_{12} 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 3*d* or 3*p* metal substitutional atom). In contrast to conventional TM substitutional elements, simple metal (SM) Al and Ga substitutional atoms stabilize the ThMn_{12}-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 SmFe_{11}Ti and Sm(Fe,Co)_{11}Ti 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 Nd_{2}Fe_{14}B, 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 4*f*-electron shells.

## Results

### Structural stability

We first inspected the structural stability of ThMn_{12}-type SmFe_{12} structure upon M (M = Ti − Ga and Al) replacement. The formation enthalpy is defined as *H*_{f} = (*E*_{0} − ∑_{i}*μ*_{i}*N*_{i})*N*_{A}/*N*, where *E*_{0} 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 SmFe_{12} 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−x}M_{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 distribution^{24}, as indicated in Fig. 1a–c. The optimized lattice parameters of SmFe_{10}M_{2} with the preferred M site are listed in Table 2. In agreement with an experiment^{33}, ThMn_{12}-type SmFe_{12} 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 calculations^{34}. Nevertheless, the single-crystalline ThMn_{12} phase can be obtained in SmFe_{11}Ti 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}.

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−x}Ti_{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 Sm_{2}Fe_{17}, Sm_{3}Fe_{29}, Sm_{6}Fe_{23}, and SmFe_{2} 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), Fe_{2}Ti 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), Fe_{2}Ti + 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−x}Ti_{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 SmFe_{11}Ti, \({H}_{{{{\rm{f}}}}}^{{{{\rm{elm}}}}}=-2.6\) kJ mol^{−1} and \({H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}=0.2\) kJ mol^{−1}. Since the single-crystalline SmFe_{11}Ti phase is practically achieved in the bulk form^{15,16,17}, the latter value (0.2 kJ mol^{−1}) of SmFe_{11}Ti is referred to as a threshold of the ThMn_{12}-phase stability in the present study. The aforementioned experimental observations of the reduced *μ*_{0}*M*_{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.

Figure 3a presents the \({H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}\) values of SmFe_{12−x}M_{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 SmFe_{11}M, which, except for SmFe_{11}Ti, is not shown for simplification of the figure. Hereafter, we thus refer to the results corresponding to the practically acceptable stoichiometry SmFe_{10}M_{2}, unless specifically mentioned. These results are supported by the experiment in which Ti and V dopants occupied the 8i site^{39}. 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 SmFe_{12} 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).

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 ThMn_{12} 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 ThMn_{12} 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−x}Ti_{x} and SmFe_{12−x}V_{x} phases are synthesized with a nearly perfect ThMn_{12} structure when x ≥ 1^{14,15,16,17,20,21}. Furthermore, a very recent experiment reported that the stability of SmFe_{8.8}Co_{2.2}Ti was greatly improved by Ga addition^{33}. The authors also predicted theoretically that Ga is the most effective stabilizing elements with higher *μ*_{0}*M*_{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 ThMn_{12} 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 *μ*_{0}*M*_{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 SmFe_{12}, the experimental *μ*_{0}*M*_{s} of 1.94 T at 0 K determined by fitting with low-temperature measurements^{22} 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 *μ*_{0}*M*_{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 *μ*_{0}*M*_{s} are 1.78, 2.53, and 2.34 *μ*_{B}, respectively. The Sm 4*f* spin moment (−5.44 *μ*_{B}) is antiparallel to the Fe 3*d* spin moment, which is mediated by the Sm 5*d* 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 4*f* electrons of Sm (antiparallel to its spin moment according to the 3rd Hund’s rule, thus parallel to the Fe 3*d* spin moment) to *μ*_{0}*M*_{s} is substantial, whereas those from the Fe 3*d* orbitals are <0.1 *μ*_{B}.

In the Λ-shape curve (Slater − Pauling-like) shown in Fig. 3b, a peak occurs for M = Co; *μ*_{0}*M*_{s} increases from 1.64 T for SmFe_{12} to 1.68 T for SmFe_{10}Co_{2}. A similar trend was found in the room-temperature measurement^{22}, as indicated by the circle symbols in Fig. 3b; 1.78 T for SmFe_{9.6}Co_{2.4} and 1.64 T for SmFe_{12}. But this is not the case in the extrapolated zero-temperature measurement^{22}; *μ*_{0}*M*_{s} decreases from 1.94 T for SmFe_{12} to 1.88 T for SmFe_{9.6}Co_{2.4}. The difference could be attributable to the aforementioned shortcomings as well as to dissimilar Co concentrations in the present theory (i.e., SmFe_{10}Co_{2}) and experiment (i.e., SmFe_{9.6}Co_{2.4})^{22}. Both the theory and experiment reveal that the early TMs (Ti − Mn) greatly lower *μ*_{0}*M*_{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 SmFe_{11}Ti, the present *μ*_{0}*M*_{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 *μ*_{0}*M*_{s} (1.2−1.3 T). Overall, *μ*_{0}*M*_{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 SmFe_{10}M_{2} (M = Ti − Ga and Al). We recall from ref. ^{24} that the *K*_{u} values of SmFe_{12} 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 K^{23}. 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 K^{23}. According to Fig. 3c, substantial reductions obviously appear for the other TM substitutions, including the reference SmFe_{11}Ti system (*K*_{u} = 9.8 MJ m^{−3}). Similar reductions were also observed for SmFe_{11}Ti (7.2−3.9 MJ m^{−3} at 77−300 K)^{15,16,17} and SmFe_{10}V_{2} (1.58 MJ m^{−3} at 300 K) in the experiments^{21}. Cr and Mn even turn the magnetic easy axis from the *c* axis to the *a**b* plane (which is improper for permanent magnets). In contrast, the SM substitutes (Ga and Al), which stabilize the ThMn_{12} 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 ThMn_{12}-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.

### 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 *μ*_{0}*M*_{s}, while Ti, V, Ga, and Al improve the SmFe_{12} stability. To support this scenario, we extended our calculations to the quaternary stoichiometries SmFe_{10}CoM and SmFe_{9}Co_{2}M (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 SmFe_{10}CoAl and SmFe_{9}Co_{2}Al compounds obtained from our total energy minimization in Fig. 5a and b, respectively.

Figures 5c, d and e show the \({H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}\), *μ*_{0}*M*_{s}, and *K*_{u} values, respectively, of SmFe_{10}CoM and SmFe_{9}Co_{2}M for M = Ti, V, Ga, and Al. Here, the formation enthalpy \({H}_{{{{\rm{f}}}}}^{{{{\rm{bin}}}}}\) is calculated against Fe_{3}Co + Fe_{3}M + 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 SmFe_{11}Ti. The obtained *μ*_{0}*M*_{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 SmFe_{10}M_{2} (Fig. 3c) is roughly preserved for both SmFe_{10}CoM and SmFe_{9}Co_{2}M; *K*_{u} reaches the largest values of 12.5 MJ m^{−3} for SmFe_{9}Co_{2}Al and 14.1 MJ m^{−3} for SmFe_{10}CoAl. 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 SmFe_{12} (12.8 MJ m^{–3}) and SmFe_{11}Ti (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 Nd_{2}Fe_{14}B.

### 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} = *λ***L** ⋅ **S**, where *λ* is the atomic SOC parameter. We provide the atom-by-atom contributions to MCA energy (*E*_{MCA}) for the selected SmFe_{12}, SmFe_{10}Co_{2}, SmFe_{10}CoAl and SmFe_{9}Co_{2}Al 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 SmFe_{12} and SmFe_{10}Co_{2}, Sm dominates *E*_{MCA}, whereas the contributions from the Fe and Co 3*d* orbitals are 2 orders of magnitude smaller. The two equivalent Sm sites in the unit cells of SmFe_{12} and SmFe_{10}Co_{2} 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 Sm_{1} − Al_{1} (Sm_{1} − Al_{2}) and Sm_{2} − Al_{1} (Sm_{2} − Al_{2}) separations are 3.08 (4.79) and 4.88 (3.11) Å, respectively. We find that the Sm_{1} site predominantly contributes to the large *E*_{MCA} and that Sm_{2} undermines *E*_{MCA}. The same trend holds for the orbital magnetic anisotropy (Δ*μ*_{L}) in Fig. 5g, defined as \({{\Delta }}{\mu }_{{{{\rm{L}}}}}={({\mu }_{{{{\rm{L}}}}})}_{c}-{({\mu }_{{{{\rm{L}}}}})}_{a}\), consistent with the Bruno theory^{42}; Δ*μ*_{L} is larger by 1 − 2 orders of magnitude in the Sm 4*f* orbitals than in the 3*d* orbitals. The contribution from the Al 3*p* orbitals to *E*_{MCA} (Δ*μ*_{L}) is essentially zero.

### Thermodynamic phase stability

The Helmholtz free energy can be written as *F*(*T*, *V*) = *E*_{0}(*V*) + *F*_{el}(*T*, *V*) + *F*_{vib}(*T*, *V*) + *F*_{mag}(*T*, *V*), where *E*_{0}(*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}: Δ*F*(*T*, *V*) = *F*(*T*, *V*) − ∑_{i}*N*_{i}*F*_{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 SmFe_{12}, SmFe_{10}Co_{2}, SmFe_{10}CoAl, and SmFe_{9}Co_{2}Al. In Fig. 6a, the same for SmFe_{11}Ti 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.

Figure 6b shows the temperature-dependent free energy change Δ*F*(*T*, *V*) and its contributions, i.e., Δ*F*_{el}(*T*, *V*), Δ*F*_{vib}(*T*, *V*), and Δ*F*_{mag}(*T*, *V*), for SmFe_{12}, SmFe_{11}Ti, SmFe_{10}Co_{2}, SmFe_{10}CoAl, and SmFe_{9}Co_{2}Al. As a generic for all compounds, Δ*F*(*T*, *V*) decreases as temperature increases, which is almost entirely accounted for by the vibrational contribution. The other two contributions, Δ*F*_{el}(*T*, *V*) and Δ*F*_{mag}(*T*, *V*), are insignificant. For SmFe_{12}, Δ*F*(*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 Δ*F*(*T*, *V*) occurs much earlier around 200 K^{34}. 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 study^{34}. The Ti and Co substitutional atoms reduce the onset temperature significantly; around 200 K for SmFe_{11}Ti and 500 K for SmFe_{10}Co_{2}. This indicates that SmFe_{11}Ti is thermally stable as observed in the experiments^{15,16,17}. For SmFe_{10}CoAl and SmFe_{9}Co_{2}Al, Δ*F*(*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})/4*n*_{i}*z*_{ij}*S*_{i}*S*_{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,j}*z*_{ij}*J*_{ij}. In this approach, the spin in the *i*th sublattice is assumed to interact with the *j*th 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 SmFe_{12}, SmFe_{11}Ti, SmFe_{10}Co_{2}, SmFe_{10}CoAl, and SmFe_{9}Co_{2}Al. 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*^{″}-Fe_{16}N_{2}^{47}, and B2-FeCo^{46}. 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 SmFe_{12} and SmFe_{11}Ti, 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 SmFe_{12}, and the Fe−Al interactions are negligible (thus not shown).

### Temperature-dependent intrinsic magnetic properties

We plot the temperature-dependent *μ*_{0}*M*_{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 SmFe_{12}, SmFe_{11}Ti, and SmFe_{9.6}Co_{2.4}^{15,16,17,22,23}. For both SmFe_{12} and Sm(Fe,Co)_{12}, the present theory reproduces the experimental trends of *μ*_{0}*M*_{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 SmFe_{10}Co_{2} than in SmFe_{12} 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 SmFe_{10}CoAl and 8 MJ m^{−3} for SmFe_{9}Co_{2}Al (Fig. 7c).

Table 3 summarizes the present theoretical Curie temperature *T*_{c}, anisotropy field *μ*_{0}*H*_{a}( = 2*K*_{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 SmFe_{12}, SmFe_{11}Ti, SmFe_{10}Co_{2}, SmFe_{10}CoAl, and SmFe_{9}Co_{2}Al, respectively. The fairly good consistency of the predicted values of SmFe_{12} (554 K), SmFe_{11}Ti (568 K), and SmFe_{10}Co_{2} (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 *μ*_{0}*H*_{a} and *κ*, overall the agreement between the present theory and the experiment^{22} is satisfactory. In order to further justify a potential replacement of Sm(Fe,M)_{12}-based magnets for the currently best permanent magnet Nd_{2}Fe_{14}B, 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 Nd_{2}Fe_{14}B in Table 3. The obtained results of *T*_{c} and *μ*_{0}*H*_{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 Nd_{2}Fe_{14}B in the entire temperature range up to 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 *μ*_{0}*M*_{s} and *K*_{u} with temperature, as discussed regarding Fig. 7. For SmFe_{12}, 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_{8.8}Co_{2.2}Ti^{29}. For SmFe_{10}CoAl and SmFe_{9}Co_{2}Al, *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 (SmFe_{12} and SmFe_{10}Co_{2}) in the entire temperature range up to 500 K.

## 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 ThMn_{12}-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}, *μ*_{0}*H*_{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 SmFe_{11}Ti and Sm(Fe,Co)_{11}Ti 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 Nd_{2}Fe_{14}B. 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 ThMn_{12} structure, leading to practical realization of SmFe_{12}-based high-performance permanent magnets.

## Methods

We adopted the WIEN2k package^{54} with the generalized gradient approximation (GGA) for the exchange-correlation functional^{55}. 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})/*v**o**l**u**m**e*, 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 zone^{56}. 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. 4*s*^{1}3*d*^{7}, 4*s*^{1}3*d*^{8}, and 6*s*^{2}5*p*^{6}4*f*^{6} are treated as valence electrons for Fe, Co, and Sm, respectively. The strongly correlated 4*f* 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 bands^{24}. 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 code^{58} within the harmonic approximation in the DFPT^{59} 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}.

## Data availability

The data that support the findings of this study are available upon reasonable request to the corresponding authors.

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## Acknowledgements

This work is supported by the Future Materials Discovery Program (Grant No. 2016M3D1A1027831) and the Basic Research Program (Grant No. 2020R1F1A1067589) through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT and the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government (MOTIE) (Grant No. 20192010106850, Development of magnetic materials for IE4 class motor).

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S.C.H. and D.O. conceived the project and wrote the paper. T.O. performed the calculations. All authors reviewed the paper.

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Ochirkhuyag, T., Hong, S.C. & Odkhuu, D. Intrinsic hard magnetism and thermal stability of a ThMn_{12}-type permanent magnet.
*npj Comput Mater* **8**, 193 (2022). https://doi.org/10.1038/s41524-022-00821-8

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DOI: https://doi.org/10.1038/s41524-022-00821-8