Thermodynamics and Magnetism of SmFe₁₂ Compound Doped with Co and Ni: An Ab Initio Study

Abstract

Ni-doped Sm(Fe_1−xCo_x)₁₂ alloys are investigated for their magnetic properties. The Sm(Fe,Co)₁₁M₁ compound (M acts as a stabilizer) with the smallest (7.7 at.%) rare-earth-metal content has been recognized as a possible contender for highly efficient permanent magnets thanks to its significant anisotropy field and Curie temperature. The early transition metals (Ti-Mn) as well as Al, Si, and Ga stabilize the SmFe₁₂ compound but significantly decrease its saturation magnetization. To keep the saturation magnetization in the range of 1.4–1.6 T, we suggest replacing a certain amount of Fe and Co in the Sm(Fe_1−xCo_x)₁₂ alloys with Ni. Ni plays the role of a thermodynamic stabilizer, and contrary to the above-listed elements, has the spin moment aligned parallel to the spin moment of the SmFe₁₂ compound, thereby boosting its saturation magnetization without affecting the anisotropy field or Curie temperature.

1. Introduction

Three main material parameters determine the intrinsic properties of hard magnetic materials: (i) the saturation magnetization, M_s; (ii) the Curie temperature, T_c; (iii) the anisotropy field, H_a [1]. These three parameters all need to be of a significant value for an economically compelling permanent magnet, i.e., μ₀M_s ~≥ 1.25 T, T_c ~≥ 550 K, and μ₀H_a ~≥ 3.75 T (μ₀ is the vacuum permeability). By mixing transition-metal (TM) with rare-earth-metal (RE) atoms in various intermetallic compounds [1,2], one can produce a material with these important magnetic properties.

The advancement of Nd-Fe-B sintered magnets in the 1980′s by Sagawa et al. [3,4,5,6] led to the introduction of iron-containing compounds that were of interest due to their considerable magnetic moments and anisotropies, which are necessary for industrially functional permanent magnetic materials [7,8,9,10,11,12]. Currently, the Nd₂Fe₁₄B₁-based sintered magnets (or so-called Neomax magnets) are considered the technological standard due to their robust performance as a result of their considerable saturation magnetization (µ₀M_S = 1.61 T, anisotropy field, µ₀H_a = 7.6 T, sufficient coercivity, µ₀H_c ≥ 0.8 T) at room temperature [1,3,4,5,6,7,8,9,10,11,12,13] and enormous maximum energy production (|BH|_max~515 kJ/m³) [14]. A deficiency of the Nd₂Fe₁₄B₁ magnets is their low Curie temperature, T_c = 588 K, and the notable deterioration of the anisotropy field with temperature [12]. Additionally, as was mentioned by Goll et al. [15], the corrosion resistance of the Nd₂Fe₁₄B₁ magnets is limited, which may lead to their faster aging. Nevertheless, the permanent magnet market is currently largely monopolized by RE-based alloys, namely Nd-Fe-B magnets and Sm-Co magnets for high-temperature operation [16,17,18].

However, all RE elements are considered critical materials, and their use in wind generators, electric vehicles, and other applications that require powerful magnets is often minimized. Because some RE elements are not naturally abundant, numerous studies have been committed to the creation of high-performance permanent magnets with a diminished content of the critical RE materials. The Nd content of the Nd₂Fe₁₄B₁ compound is 11.8 at.%. Aside from the Nd₂Fe₁₄B₁ compound, the RE(Fe,Co)₁₁M₁ compound, which crystalizes with the ThMn₁₂-type structure, has been recognized as a possible contender for high-performance permanent magnets where the content of 3d metals produces very large µ₀M_S and T_c, and a considerable a/c ratio results in a significant magnetocrystalline anisotropy energy (MAE) [16,17,18,19,20]. RE(Fe,Co)₁₁M₁ alloys have a decreased RE content (7.7 at.%) as compared with Nd₂Fe₁₄B₁ magnets. However, the SmFe₁₂ compound is thermodynamically unstable as a bulk material. In order to stabilize the REFe₁₂ phase in bulk form, the replacement of Fe with a stabilizer M, where M = Ti [15,16,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92], V [27,29,30,32,37,39,40,41,44,49,82,85,90,93,94,95,96,97,98,99,100,101,102,103,104], Al [32,49,105,106], Si [27,30,32,35,49,107,108,109,110,111,112,113,114,115], Ga [40,82,85], Cr [27,29,32,37,49,92], Mn [37,92], Mo [27,29,32,37,39,49,72,106,116,117,118,119,120], Nb [121], W [27,29,32,49], Re [49], H [19,56,57,58,59,60,61,62,67,77], N [11,43,59,61,65,68,72,77,92,93,101,122,123,124], and C [59], has been studied. The composition range x for stabilizing the REFe_12−xM_x phase differs depending on M [125].

So far, Ti is known to be the favorite stabilizing element for REFe_12−xM_x alloys with x ~ 1 [49]. The samarium-based compound SmFe₁₁Ti₁ has the following magnetic properties: µ₀H_a = 8.3–10.5 T, T_c = 584 K, and μ₀M_s = 1.14–1.22 T [23,34], and is smaller than that of Nd₂Fe₁₄B₁ (μ₀M_s = 1.61 T) [4,12,23,34]. The iron-rich REFe₁₂ compounds with the ThMn₁₂-type structure earned attention for prospective permanent magnet utilization with regard to maximizing μ₀M_s while minimizing Ti inclusion [20,31,35,38,46,47,52,65,68,69,71,72,73,78,84,87,88,89,90,91,96,108,109,110,111,113,115,122,126,127,128,129].

Commonly, large transition metals, such as TM = Ti, V, Cr, Nb, Mn, Mo, W, and Re, preferentially fill the 8i sites to stabilize the SmFe₁₁TM structure [14,49,76]. Al, Si, and Ga also stabilize the structure by filling the 8f sites [49,59,76]. According to Kobayashi et al. [71], stabilization of RE(Fe,Ti)₁₂ structures, when the Ti atoms substitute part of the smaller Fe in the 8i sites, can be considered as a reduction in the local mismatches in interatomic distances limiting the orbital hybridization. The average Fe-Fe interatomic distances for Fe (8i), Fe (8j), and Fe (8f) sites are 0.271, 0.259, and 0.251 nm, respectively, where the interatomic distance on the 8i site is longer than the Fe-Fe interatomic distance in the bcc structure (r_Fe = 0.126 nm, 2r_Fe = 0.252 nm) [71]. The replacement of some Fe atoms with atoms such as Ti (r_Ti = 0.147 nm) stabilizes the structure while diminishing the magnetization of Ti-free compounds [49]. Therefore, it is essential to keep the concentration of stabilizers as low as possible. The prejudiced occupancy of Fe and Ti atoms at the 8i site was also evident from a neutron diffraction experiment for the YFe₁₁Ti₁ compound [33].

According to Tozman et al. [75], the amount of magnetization of SmFe₁₁M₁ can be increased using two methods. The first method to improve the magnetization is Co replacement on the Fe site. Magnetic and structural properties of Sm(Fe_1−xCo_x)₁₁Ti₁, Sm(Fe_1−xCo_x)₁₀Ti₂, and Sm(Fe_1−xCo_x)₁₀Si₂ alloys have been explored by Cheng et al. [31] and Solzi et al. [35]. It was found that the easy magnetization direction (EMD) in these alloys occurs to change from axial, SmFe₁₁Ti₁, SmFe₁₀Si₂, SmFe_10.8Ti_1.2, to conical, and finally to planar with elevated Co content. The planar MAE of the SmCo₁₁Ti₁ compound was also discovered by Ohashi et al. [42] and Wang et al. [55]. A diagram representing the change in EMD with temperature for each REFe₁₁Ti₁ compound (RE = Y, Pr, Nd, Sm, Gd, Th, Dy, Ho, Er, and Tm) was presented by Kou et al. [50]. Tereshina et al. [52] presented the substitution effects of Fe by Co on magnetic properties of RE(TM,Ti)₁₂ (RE = Y, Sm; TM = Fe, Co) single crystals. They showed that REFe_1−xCo_xTi₁ alloys have axial EMD for x ~ ≤ 0.5 μ₀M_s with a maximum at x ≈ 0.3 [52]. Co has a strong influence on the MAE of the Sm-sublattice; the first anisotropy constant, K₁, displays a maximum at x ≈ 0.1 and drastically decreases at x > 0.1. Finally, the Curie temperature grows monotonically with an increase in Co content [31,35,52].

The second method to improve the magnetization is to diminish the Ti content at 8i sites. Phase stability can be reached by Zr or Y replacement for Sm, which decreases the resident substructure encompassing the Fe (8i) sites. Suzuki et al. [65] utilized Zr to develop a (Nd,Zr)(Fe,Co)_11.5Ti_0.5N magnet. In order to diminish the lattice mismatch in interatomic distances quoted above, the part of the RE element in the 2a sites can be substituted with the smaller Zr atoms (the atomic radius of Zr is ~12% less than that of Nd). Thus, a combined alloying with both Zr and TM (for example, Ti) could make possible to achieve stability with the ThMn₁₂-type magnet with less than one TM (for example Ti) atom per formula unit. Kuno et al. [69] mentioned that the hard magnetic characteristics of strip-cast (Sm_0.8Zr_0.2)(Fe_0.75Co_0.25)_11.5Ti_0.5 alloy are μ₀M_s = 1.58 T, μ₀H_a = 7.41 T at low temperature (T = 20 °C), which are similar to those of the Nd₂Fe₁₄B₁ magnet (µ₀M_S = 1.61 T, µ₀H_a = 7.6 T). However, the Curie temperature (T_c = 880 K) significantly outperforms the Nd₂Fe₁₄B₁ magnet (T_c = 584 K).

Hagiwara et al. [73,78] synthesized via arc melting (and annealed) the (Sm_0.8Y_0.2)(Fe_0.8Co_0.2)_11.5Ti_0.5 magnet with µ₀M_S = 1.50 T, µ₀H_a = 11 T at room temperature and T_c = 820 K. Both the μ₀H_a and T_c significantly outperform those of the Nd₂Fe₁₄B magnet.

Sakuma et al. 2016 [123] discovered that Zr replacement stabilizes the (Nd_1−αZr_α)(Fe_0.75Co_0.25)_11.25Ti_0.75N_1.2−1.4 (α = 0–0.3) magnets and displays a higher μ₀M_S = 1.62 T at α = 0.1 than the Nd₂Fe₁₄B₁ magnet. According to Tozman et al. [75], for the Sm(Fe_1−xCo_x)₁₁Ti₁ alloys, the largest μ₀M_s = 1.43 T and µ₀H_a = 10.9 T at room temperature are accomplished for the Sm_0.94(Fe_0.81Co_0.19)₁₁Ti_1.08 magnet (x = 0.19), with a corresponding Curie temperature of 800 K. The μ₀M_s could be improved by decreasing the Ti replacement for Fe from Ti₁ to Ti_0.5 and with partial replacement of Sm by Zr. According to Tozman et al. [75], for the (Sm_0.77Zr_0.24)(Fe_0.80Co_0.19)_11.5Ti_0.65 magnet, μ₀M_s = 1.53 T and µ₀H_a = 8.4 T, with T_c = 830 K.

The REFe₁₂ compounds are unstable as bulk materials, although they have been seen in thin films [11,13,75,80,82,93,124,126,127,128,129,130,131]. Hirayama et al. [126] synthesized epitaxial Sm(Fe_1−xCo_x)₁₂ films (x = 0, 0.1 and 0.2) (without stabilizing elements) on W- and V-buffered single-crystalline MgO substrates. The intrinsic hard magnetic properties of the Sm(Fe_0.8Co_0.2)₁₂ thin films are higher than those of Nd₂Fe₁₄B₁ at room temperature (μ₀M_s = 1.78 T, μ₀H_A = 12 T, (BH)_max = 630 kJ/m³), while the Curie temperature is also improved (T_C = 859 K). Additionally, Tozman et al. [80,84] reported that μ₀M_s increases via partial replacement of Sm in Sm(Fe_0.8Co_0.2)₁₂ with Zr. For the (Sm_0.74Zr_0.26)(Fe_0.8Co_0.2)₁₂ magnet, μ₀M_s = 1.86 T, µ₀H_a = 9.8 T, and T_c = 675 K [80,84]. Ogawa et al. [129] evaluated the magnetic hardness factor, κ = (K₁/μ₀M_s²)^1/2 of the Sm(Fe_1−xCo_x)₁₂ magnets at 300 K and found that κ is equal to 1.38, 1.28, and 1.24 for x = 0.0, 0.07, and 0.2, respectively. Because κ should be larger than one for a compound to be regarded as a permanent magnet material [18], the Sm(Fe_1−xCo_x)₁₂ alloys have a prospective use as the main phase of the high-performance permanent magnet material if the ThMn₁₂-type phase can be bulk-stabilized.

Table 1. Saturation magnetization, anisotropy field, Curie temperature, and maximum energy product values of 1:12 magnets.

Material	${\mathit{\mu }}_0{\mathit{M}}_{\mathit{s}}\left(\mathit{T}\right)$	${\mathit{\mu }}_0{\mathit{H}}_{\mathit{a}}\left(\mathit{T}\right)$	${\mathit{T}}_{\mathit{c}}\left(\mathit{K}\right)$	${\left|\mathit{B}\mathit{H}\right|}_{\mathit{m}\mathit{a}\mathit{x}}\left(\frac{\mathit{k}\mathit{J}}{{\mathit{m}}^3}\right)$	References
Nd2Fe14B1	1.61	7.6	588	515	[7,8,9,10,11,12,13,14]
SmFe11Ti1	1.14–1.22	8.3–10.5	584	258	[23,34]
(Sm0.8Zr0.2)(Fe0.75Co0.25)11.5Ti0.5	1.58	7.4	880	495	[69]
(Sm0.8Y0.2)(Fe0.80Co0.20)11.5Ti0.5	1.50	11.0	820	447	[73,78]
Sm0.94(Fe0.81Co0.19)11Ti1.08	1.43	10.9	800	406	[75]
(Sm0.77Zr0.24)(Fe0.80Co0.19)11.5Ti0.65	1,53	8.4	830	465	[75]
(Sm0.92Zr0.08)(Fe0.75Co0.25)11.35Ti0.65	1.47	>9.0	843	429	[76]
Sm(Fe0.8Co0.2)12	1.78	12.0	859	630	[126]
(Sm0.74Zr0.26)(Fe0.8Co0.2)12	1.86	9.8	875	688	[80,84]

summarizes the intrinsic properties of some ThMn₁₂-type structure magnets together with properties for Neomax (Nd₂Fe₁₄B₁). Note that the Sm(Fe_0.8Co_0.2)₁₂ and (Sm_0.74Zr_0.26)(Fe_0.8Co_0.2)₁₂ magnets are epitaxially grown thin films and are unstable in bulk.

Multiple theoretical approaches have investigated RE(Fe,Co)₁₁TM₁ magnets. These include augmented spherical waves within the local spin density-functional (ASW-LSDF), molecular dynamics (MD) and Monte Carlo (MC), embedded atom method (EAM), full-potential linearized muffin orbital (FPLMTO), tight-binding linear-muffin orbital atomic sphere approximation (TB-LMTO-ASA) in conjunction with computational high-throughput screening (HTS), hybrid-exchange density functional theory (HE-DFT), linear combination of atomic orbitals (LCAO), pseudopotential Vienna ab initio (VASP) package, full-potential linearized augmented plane wave (FLAPW) within WIEN2K package, Hubbard I method (LDA + U), Quantum Materials Simulator and projected augmented-wave (QMAS-PAW), Korringa–Kohn–Rostoker coherent-potential approximation (AkaiKKR-CPA), mean field approximation (MFA), and Bayesian optimization (BO) approaches [11,85,92,125,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148].

Coehoorn [125] performed ASW-LSDF calculations for the YFe₁₂ and YFe₈TM₄ (TM = Ti, V, Cr, Mn, Mo, and W) compounds with the ThMn₁₂-type structure, in which the TM atoms occupy the (8i) sites. These calculations showed that the YFe₁₂ compound is thermodynamically unstable. The calculated reduction in saturation magnetization of the YFe_12−xM_x (1 ≤ x ≤ 3) compounds after substituting Fe(8i) atoms by an M atom is in a good accordance with the experimental data [125]. Chen et al. [132] used classical MD to show that each TM = (Cr, V, Mo, and Ti) substantially reduces the cohesive energy of the Sm(Fe,TM)₁₂ compound and stabilizes the ThMn₁₂-type structure. The sequence of site preference occupation is 8i > 8j > 8f, with the 8i occupation resulting in extreme energy reduction.

Ke and Johnson [133] used the FPLMTO method to study the intrinsic properties of the RE(Fe_1−xCo_x)₁₁TiZ alloys, where RE = Y and Ce, interstitially dopped by Z = H, C, and N. The calculated μ₀M_s, μ₀H_a, and T_c values of the YFe₁₁Ti₁, YCo₁₁Ti₁, CeFe₁₁Ti₁, and CeCo₁₁Ti₁ compounds agree with the experimental data, although the Ti atom should have a strong preference to occupy the 8i sites, as observed by Moze et al. [33]. The site inclination of Co atoms in the Y(Fe_1−xCo_x)₁₁Ti₁ compound with x < 0.4 is: 8j > 8f >8i concurs with the neutron scattering experiments by Liang et al. [131]. According to Ke and Johnson [133], the maximum magnetization of the Y(Fe_1−xCo_x)₁₁Ti₁ and Ce(Fe_1−xCo_x)₁₁Ti₁ alloys occurs at x = 0.2, while in experiments it takes place at x = 0.3 [54] and x = 0.1–0.15 [15] for the Y(Fe_1−xCo_x)₁₁Ti₁ and Ce(Fe_1−xCo_x)₁₁Ti₁ alloys, respectively. In addition, the calculated T_c of the RECo₁₁Ti₁ compound is significantly higher than the calculated T_c of the REFe₁₁Ti₁ compound [133], which agrees well with the experimental data [64].

Körner et al. [134] performed theoretical investigations (HTS-TB-LMTO-ASA) of the 1280 compounds derived from the ThMn₁₂-type crystal structure in order to estimate their μ₀H_a and (BH)_max. HTS has led to several auspicious phases such as NdFe₁₂X or NdFe₁₁TiX (X = B, C, N) with (BH)_max~600 kJ/m³ and µ₀H_a~10 T. Butcher et al. [135] used the HTS-TB-LMTO-ASA methodology to study the influence of the substituted atom site preference on μ₀M_s and μ₀H_a of the REFe₁₁A-type compound, where A = Al, Si, Ti, V, Cr, Co, Ni, and Cu. Körner et al. [136] used the HTS-TB-LMTO-ASA methodology to screen the magnetic properties of over 2000 different RE-TM-X compounds with 1-11 and 1-11-X (the YN_i9In₂-type) structures as prospective candidates for new hard-magnetic phases.

Skelland et al. [137] used the combined MD, Morse potentials, and modified EAM (MEAM) to investigate the probability distributions for Ti substitutions in RETM_12−xTi_x (RE = Nd and Sm; TM = Fe and Co) alloys over the atom positions within the structure. The authors concluded that Ti substitution, while stabilizing the structure, also results in a considerable reduction in the spin magnetic moment due to Ti aligning anti-parallel to the net internal magnetization and reducing the magnetic moment of the neighboring iron atoms. The 8i position has the best probability for substitution followed by the 8j and 8f positions, and this conclusion agrees with the results by Miyake et al. [138].

Martinez-Casado et al. [79] performed an ab initio study of the electronic structure of CeFe₁₁Ti₁. The authors used the HE-DFT methodology, which incorporates a combined approach of atom-centered LCAO, PW-VASP, and FPLMTO. The calculations confirmed that Ti stabilizes the CeFe₁₁Ti structure and Ce⁺³ causes the large MAE. Odkhuu et al. [139] performed ab initio (Wein2k, FP-LDA + U, VASP) calculations of μ₀M_s and μ₀H_a of the SmFe₁₂ compound dopped by B, C, and N, indicating a possible way to improve these properties through interstitial doping of the SmFe₁₂ compound by 2p electron elements.

Hirayama et al. and Miyake et al. [11,138] (QMAS-PAW) found that Ti substitution in the NdFe₁₂ compound diminishes μ₀M_s substantially, whereas it scarcely alters MAE. They also showed that interstitial nitrogenation substantially raises μ₀M_s and the uniaxial MAE of the NdFe₁₁Ti compound. Using the same QMAS-PAW methodology, Harashima et al. [140] proposed that in the NdFe₁₁TiX compound, where X = B, C, N, O, and F, nitrogen is the most suitable dopant for permanent magnets in terms of μ₀M_s and MAE. Harashima et al. [92] performed first-principles calculations (QMAS-PAW) of the magnetic properties of the NdFe₁₁M compound, where M = Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn. They discovered that the calculated formation energy of the NdFe₁₁Co compound is negative compared to the Nd₂Fe₁₇ compound (the Th₂Zn₁₇-type structure), bcc-Fe, hcp-Co, and the CoFe compound (CsCl-type structure), indicating that Co could stabilize the ThMn₁₂ structure in bulk form. Contrary to Ti replacement, Co replacement does not diminish μ₀M_s meaningfully. Harashima et al. [141] studied (QMAS-PAW) the effects of interstitial nitrogenation on the magnetic properties of the ThMn₁₂-type compounds REFe_11−xTi_x (x = 0, 1) with RE = Nd, Sm, Y. They found that the evolution in MAE by nitrogenation is self-sufficient of Ti replacement and does not seem to have much dependence on the RE element [141]. Matsumoto et al. [142] performed finite temperature ab initio (QMAS-PAW, AkaiKKR) MC calculations of μ₀M_s and MAE to find improvements in the indirect 4f(Nd)-3d(Fe) exchange coupling to enhance the high-temperature properties of the NdFe₁₂N-type compounds.

Using AkaiKKR-CPA formalism, Fukuzawa et al. [143] assessed intersite magnetic exchange couplings in the NdFe₁₂ and NdFe₁₂X (X = B, C, N, O, and F) compounds with the ThMn₁₂ structure. They found that that MAE deteriorated rapidly with increasing temperature in the X = N system, albeit nitrogenation was dominant compared to other dopants in terms of improving the low-temperature magnetic properties. The MFA, with exchange couplings as input parameters, was used to calculate T_c. The introduction of X (X = B, C, N, O, and F) causes substantial variations in the magnetic couplings of the NdFe₁₂ compound and has a momentous consequence in terms of T_c. Fukazawa et al. [144] presented a first-principles study of REFe_12−xCr_x (RE = Y, Nd, Sm) compounds with the ThMn₁₂ structure using the QMAS -PAW package to calculate the lattice parameters of the REFe_12−xM_x (M = Cr, Co) system. In this study, Fe and the dopants (Co or Cr) were treated within the KKR-CPA (AkaiKKR method), and T_c was calculated within MFA. The calculations indicated an improvement in T_c over the addition of Cr, x ≤ 0.5 in the REFe₁₂ REFe_12−xCr_x compound. On the contrary, above x > 0.5, T_c diminishes as the Cr concentration increases.

Harashima et al. [145] performed an ab initio investigation (QMAS-PAW) to study the effects of RE(M) site replacement and pressure on the structural stability of the REFe₁₂ compounds, where RE = La, Pr, Nd, Sm, Gd, Dy, Ho, Er, Tm, Lu, Y, or Sc, and the group-IV element M = Zr or Hf. They found that the formation energy has a firm resemblance with the atomic radius of RE metal, in particular the formation energy with respect to simple substances diminishes as the atomic radius diminishes, except for RE = Sc and M = Hf. The calculated formation energies with respect to the RE₂Fe₁₇ compound (in both Th₂Zn₁₇ and Th₂Ni₁₇-type structures) and bcc Fe were also reported. These calculations show that the formation energy is positive, which means that one cannot make the REFe₁₂ phase more stable than the RE₂Fe₁₇ phase, and partial replacement for Fe is needed to stabilize the ThMn₁₂ structure.

Fukazawa et al. [146] used BO to optimize the composition of the REFe₁₂-type compound; specifically, the composition of (RE_1−α,Z_α)(Fe_1−βCo_β)_12−γTi_γ (RE= Nd, Sm, Y; Z = Zr, Dy) was improved in terms of the formation energetics, μ₀M_s, μ₀M_a and T_c. This scheme is based on the ab initio calculations (QMAS-PAW) and implementation of the KKR-CPA (AkaiKKR-CPA) technique to treat the compositional disorder. The authors identified 10 ThMn₁₂-type systems with the highest value of μ₀M_s, although only Nd(Fe_0.7Co_0.3)₁₂ with μ₀M_s = 1.89 T and T_c = 1143 K and (Sm_0.7Zr_0.3)(Fe_0.9Co_0.1)₁₂ with μ₀M_s = 1.77 T and T_c = 1002 K have the negative formation energy with respect to the unary system. The highest T_c = 1310 K was projected for the Sm(Fe_0.2Co_0.8) alloy, which has a negative formation energy in respect to the unary system and a significant μ₀M_s = 1.47 T.

Dirba et al. [85], both experimentally and theoretically, studied how the particle size and alloy composition influence the thermal stability of the ThMn₁₂-type phase. A definite stabilizing effect is accomplished by replacing Fe with elements such as Ti, V, or Ga. However, aligning to ab initio calculations (QMAS-PAW), the SmFe₁₁M compound, where M = Ti and V, shows lower energy in respect to the combination of α-Fe and SmFe₂ (resistance to decomposition), with preferable occupation of 8i sites for M. The SmFe₁₁Ga compound (M = Ga) also exhibits lower energy with respect to the combination of α-Fe and SmFe₂ with preferential occupation of 8j sites for Ga, but is not as stabilized as for M = Ti and V. However, the Ga dopant shows a smaller reduction in μ₀M_s in comparison with Ti or V dopants. Therefore, Ga is the most successful stabilizing element when μ₀M_s is considered.

Matsumoto et al. [147] used combined ab initio (AkaiKKR-CPA) methodology and experimental data from neutron diffraction to inspect the optimal chemical composition of the SmFe₁₂ compound doped by Zr, Ti, and Co. They found that in the Zr-substituted SmFe₁₂ magnet, Zr occupies both Sm(2a) and Fe(8i) roughly equally in terms of energetics. While using Ti as the stabilizing element cannot be avoided, it was found that the lower bound is around 0.5 Ti atoms, which is close to what has been attained experimentally [65,68,69,75]. The Zr dopant also improves the structural stability and μ₀M_s of the SmFe₁₂ magnet, which is in perfect agreement with the experimental data [80]. Dasmahapatra et al. [148] performed an ab initio study (LCAO-CRYSTAL and PW-CASP) on the effects of doping the CeFe₁₁Ti compound with Co and Ni. The authors of [149] showed that the addition of Co (Ni) slightly improves (diminishes) μ₀M_s and μ₀H_a.

In our previous papers [149,150,151], we suggested increasing μ₀M_s and (BH)_max in the well-known SmCo₅ and YCo₅ magnets by substituting Co with Fe and using Ni as a thermodynamic stabilizer.

In most studies, the REFe₁₂ (ThMn₁₂-type structure) magnets have been stabilized in the form RE(Fe,Co)₁₁M₁, where M₁ = Ti, V, Al, Si, Ga, Cr, Mn, Mo, Nb, Ta, W, Re, H, and N. A handful of papers have been published where the REFe₁₂ magnet dopped with Ni was studied. Yang et al. [91] studied the crystallographic and magnetic properties of the replaced Y(Fe_1−xM_x)₁₁Ti₁ alloys, where M = Co, Ni, and Si. The thermomagnetic analysis indicated that μ₀M_s has a maximum at x = 0.07 for the Y(Fe_1−xNi_x)₁₁Ti₁ alloys. T_c increases when Fe is substituted by Ni. According to Yang et al. [91], the increases in μ₀M_s and T_c can be assumed to be preferential site occupation by Ni atoms.

According to Harashima et al. [92], Ni substitution (NdFe₁₁Ni₁) yields negative formation energy values relative to the NdFe₁₂ compound, bcc Fe, and fcc Ni, and the most stable site for Ni is 8j. Ni substitution does not diminish the magnetization significantly, as in the case of Ti, V, Cr, and Mn dopants.

Suski et al. [152] studied the structure and magnetic properties of the SmMn_12–xNi_x alloys, which are stable in the ThMn₁₂-type structure for the Ni concentration, 4 ≤ x ≤ 8. Suski et al. [152] asserted that the SmMn_12–xNi_x alloys are magnetic at the low Néel temperature T_N ≈ 60 K, due to antiferromagnetic ordering in the Sm sublattice, whereas the Mn and Ni sublattices are nonmagnetic. Suski et al. [153] studied the structure and magnetic properties of the NdMn_12–xNi_x alloys, which are stable in the ThMn₁₂-type structure for the Ni concentration of 3.6 ≤ x ≤ 6. The absence of any maximum in the temperature reliance of the magnetic susceptibility indicates the investigated alloys are nonmagnetic.

The main intent of the present research is to investigate the effects of nickel on the phase stability of the Sm(Fe_1–xCo_x)₁₂ alloys and to evaluate their magnetic properties. We utilize ab initio calculations using the following techniques: (i) the fully relativistic exact muffin-tin orbital method (FREMTO) and (ii) the full-potential linear muffin-tin orbital method (FPLMTO). Both methods include all relativistic effects, such as spin–orbit coupling (SOC). The application of these two methods allows us to establish accurate results that are independent of technical implementation and relies on the individual strength and durability of each method. Related details of the ab initio computational methods are outlined in Section 2. The results of the density-functional theory (DFT) calculations of the ground-state properties of the Sm(Fe-Co-Ni)₁₂ alloys are outlined in Section 3. We demonstrate the results of the DFT calculations of the magnetic properties of the Sm(Fe-Co-Ni)₁₂ alloys in Section 4. Lastly, a discussion and concluding remarks are outlined in Section 5.

2. Computational Methods

The fundamental framework for the present modeling is based on density-functional theory (DFT). In this approach, there is an unavoidable approximation for the electron exchange and correlation interactions. Here, we apply the generalized gradient approximation for this purpose. This exchange and correlation functional works extremely well for many elements and compounds in the periodic table but cannot determine the localized nature of the electrons. This is no issue for the 3d transition elements Fe, Co, and Ni that we study, while for the rare-earth element Sm, the 4f electrons are localized and require special treatment.

The 3d transition metals Fe, Co, and Ni are generally very well described within DFT, but for very good orbital moments one can include a simple parameter-free scheme that couples the orbital moments, i.e., an orbital polarization (OP) mechanism, which is otherwise not included in conventional DFT. This recipe results in excellent spin and orbital moments for Fe, Co, and Ni [154].

Samarium metal poses a more challenging problem for DFT. Here, the 4f electrons are known to be localized (atomic-like and essentially non-bonding) with magnetic properties closer to the free Sm³⁺ atom but with a magnetic moment largely quenched by crystal-field effects. DFT-GGA-type calculations are not able to describe the nature of these 4f states but exaggerate their itineracy or band-like character. For energetics such as the structural stability, the DFT-GGA approximation works better than one would expect, and the formation energies are reproduced via experiments for the SmCo₅ (doped with Fe and Ni) very nicely [149,150]. For spectroscopic or magnetic properties, however, one must go beyond DFT-GGA to achieve an accurate theory. In the literature, DFT in connection with large intra-atomic Coulomb repulsion and a Hubbard U parameter (DFT + U) has been applied for Sm systems [155]. The more advanced dynamical mean-field theory (DMFT) has also been utilized for SmCo₅ [156]. A more practical approach, which allows for accurate MAE for SmCo₅, is the application of the “standard rare-earth model” (SRM), where the 4f electrons are treated as localized by being constrained to occupy core states that do not hybridize with the band states. This technique is very similar to DMFT in the Hubbard I approximation [157]. In the SRM, the magnetic-moment contribution from the Sm 4f electrons is constrained to zero, while a small induced magnetic moment still forms on the Sm atom (~0.3 μ_B) from other valence electrons. This is reasonable because for Sm metals the net magnetic moment is estimated to be very small due to spin and orbital moment cancelation (0.1 μ_B) [158].

The SRM approach, coupled with an orbital polarization extension of DFT (see the next section), proved to be an excellent model for the magnetic anisotropy energies in SmCo₅ and related systems [149,150], and we adopt it here for all MAE calculations.

Here, we employ two methods (described below)—one is very well suited for alloy calculations, and one is appropriate for MAE calculations within the standard rare-earth model. The calculations we refer to as EMTO are carried out using Green’s function based on the improved screened Korringa–Kohn–Rostoker formalism, where the one-electron potential is described by the optimized overlapping muffin-tin (OOMT) potential spheres [159,160]. Inside the potential spheres, the potential is spherically symmetric, while it is constant between the spheres. The radius of the potential spheres, the spherical potential inside these spheres, and the constant value in the interstitial region are determined by minimizing (i) the difference between the exact and overlapping potentials and (ii) the inaccuracy that is caused by the overlap between the spheres. Within the EMTO formalism, the one-electron states are resolved exactly for the OOMT potentials. As a product of the EMTO calculation, one can solve the self-consistent Green’s function of the system and the complete, non-spherically symmetric charge density. Finally, the total energy is obtained from the full charge density formalism [161]. We regard 6s, 5p, 5d, and 4f states for Sm and 4s and 3d states for Co and Fe as valence states. The Kohn–Sham orbitals are expanded in terms of spdf exact muffin-tin orbitals. The completeness of the muffin-tin basis was analyzed thoroughly in [159]. The generalized gradient approximation (GGA-PBE) [162] is chosen for the electron exchange and the correlation energy functional. Integration over the Brillouin zone assumes a 25 × 25 × 25 grid of k-points applying the Monkhorst–Pack technique [163]. The moments of the density of states, which are needed for the kinetic energy and valence charge density, are calculated by integrating Green’s function over a complex energy contour with a 2.2 Ry diameter while using a Gaussian integration technique with 40 points on a semi-circle enclosing the occupied states. When included, SOC is carried out through the four-component Dirac equation [164].

Accounting for compositional disorders, the EMTO method is linked with the coherent potential approximation (CPA) [165,166]. The ground-state properties of the Sm(Fe_1–x–yCo_xNi_y)₁₂ alloys are acquired from EMTO-CPA calculations. The paramagnetic state of the Sm(Fe_1–x–yCo_xNi_y)₁₂ alloys is modeled within the disordered local moment (DLM) approximation [167,168]. The equilibrium atomic density of the alloy is attained from a Murnaghan fit to the total energy versus the atomic volume curve [169].

For the magnetic properties, we adopt the highest level of theory (least approximations) that is implemented in a full-potential scheme, i.e., no structural approximations, which includes spin–orbit interactions, as explained [170] for early and late lanthanides [171,172]. The full-potential linear muffin-tin orbital (FPLMTO) accomplishes this in ways that are detailed in [173]. For the Sm 4f electrons, we apply two fundamentally different approaches. First, as with the EMTO method, the Sm 4f electrons are treated as band electrons. In terms of magnetic approximations, we perform scalar-relativistic approximations, i.e., no spin–orbit coupling and no orbital moment (4f-band-scalar), the same approximations but including spin–orbit coupling (4f-band-SO), and finally also approximations with the orbital polarization added (4f-band-SO + OP). Of these three approaches, the 4f-band-SO is most like the EMTO method outlined above. Second, we assume the standard rare-earth model (SRM), i.e., we constrain the 4f electrons to occupy non-hybridizing core states, while all d states (on Sm and the transition metal component) are orbitally polarized (SRM-SO + OP). For the valence states, we use two energy parameters coupled with each basis function, and these parameters have different values for pseudocore states (4s and 4p) and valence states (5s, 5p, 4d, and 5f). For the transition metals, analogous set ups are used. The spin–orbit interaction and the orbital polarization operate on the d and f orbitals for best accuracy of the orbital magnetism. OP is introduced in FPLMTO in a self-consistent fashion, so there are no fixed parameters. Because of the way it is constructed, the orbital polarization often enhances spin–orbit coupling, leading to better orbital moments [173]. Both the generalized gradient and the local-density approximations (GGA-PBE and LDA) [162,174] are applied.

In computing the MAE, i.e., the difference in total energy between spins pointing in plane (100) or perpendicular to the plane along the z axis (001), only small differences are observed. To resolve these small numbers, one must be very careful in calculating the total energy for the two cases in as consistent a manner as possible. Here, we decide to remove all crystal symmetry so that the calculations are identical from a crystal symmetry standpoint. This approach is less efficient but in practice more accurate due to errors associated with differing k points and Brillouin zones, and other numerical approximations are minimized. Brillouin zone integrations around 3000 k points are carried out using Fermi–Dirac broadening corresponding to room temperature (300 K). Finally, the alloy compositions between iron and cobalt are modeled within the virtual-crystal approximation (VCA). This simply means that an Fe_0.9Co_0.1 alloy is treated as a one-atom species with an atomic charge of 26.1.

3. Thermodynamic Properties of the Sm(FeCo-Ni)₁₂ Alloys

The SmFe₁₂ compound crystallizes in the body centered orthogonal to the ThMn₁₂-type structure (space group I4/mmm, no. 139; see Figure 1). The structure contains Sm atom in the 2a Wycoff position and 12 Fe atoms that occupy three inequivalent Wyckoff positions, 8f, 8i, and 8j, respectively. According to the Pearson symbol (tI26), the usual SmFe₁₂ supercell contains 26 atoms but can be represented by the so-called reduced supercell with 13 atoms (1 Sm and 12 Fe) used in the present calculations.

EMTO calculations revealed that the SmFe₁₂ compound has an equilibrium atomic volume Ω₀ equal to 13.19 ų, which is in fair accordance with previous calculations, 13.02 ų [141], but smaller than the experimental value for the SmFe₁₂ films, Ω₀ = 13.64 A³ [176]. The calculated formation energy of the SmFe₁₂ compound with respect to the unary systems α-Sm and α-Fe is H = +6.13 mRy/atom, which confirms that the SmFe₁₂ compound is thermodynamically unstable.

EMTO calculations revealed that the SmCo₁₂ compound has an equilibrium atomic volume Ω₀ = 12.42 ų. The calculated formation energy of the SmCo₁₂ compound in respect to the unary systems, α-Sm and α-Co, is H = +1.06 mRy/atom, which confirms that the SmCo₁₂ compound is thermodynamically unstable.

Figure 2 shows the formation energy of Sm(Fe_1−xCo_x)₁₂ alloys calculated with respect to the unary systems α-Sm, α-Fe, and α-Co. We assume that Sm atoms solely occupies the 2a Wyckoff position and Fe and Co atoms are randomly distributed on 8f, 8i, and 8j Wyckoff positions (see Figure 1). We could have expected the formation energy to decrease linearly as Co atoms gradually replace Fe atoms on their sites. However, this Figure shows that 20% of the Co solute decreases the formation energy of the Sm(Fe_0.8Co_0.2)₁₂ compound by about ~36% relative to the undoped SmFe₁₂ compound. In the compositional range between Sm(Fe.₅Co.₅)₁₂ and SmCo₁₂, the formation energy changes very slowly from ~+2 mRy/atom to ~+1 mRy/atom.

The calculated (EMTO) value of the formation energy of three configurations (a) SmNi(8f)₄Co(8i)₄Co(8j)₄, (b) SmNi(8i)₄Co(8f)₄Co(8j)₄, and (c) SmNi(8j)₄Co(8f)₄Co(8i)₄, with respect to the unary systems α-Sm, α-Co, and α-Ni, are +0.047 mRy/atom, −0.031 mRy/atom, and −1.65 mRy/atom, respectively. Thus, assuming that 1/3 of Co atoms are replaced by Ni atoms, one can obtain the energetically favorable configuration when 4 Ni atoms occupy all 8j sites and the remaining 8 Co atoms occupy 8i and 8f sites, resulting in a stable SmNi₄Co₈ compound with a formation energy H = −1.65 mRy/atom. This result agrees with the conclusion of Harashima et al. [92], who found that the Ni dopant prefers to occupy 8j sites when stabilizing the NdFe₁₁Ni₁ compound.

In order to explore the stabilizing effects of the addition of nickel metal to the Sm(Co_1–xFe_x)₁₂ alloys, we performed EMTO-CPA calculations of the formation energy of the quasi-binary SmNi₄(Fe_1−xCo_x)₈ alloys relative to the unary systems α-Sm, α-Fe, α-Co, and α-Ni, where 4 Ni atoms occupy the 8j sublattice and occupation of the 8i and 8f sublattices gradually changes from pure Fe to pure Co metals. As can be seen from Figure 3, the formation energies of the quasi-binary SmNi₄(Fe_1−xCo_x)₈ alloys are negative (stable alloys) within the whole compositional range. Even if the calculated heat of formation of the unstable SmFe₁₂ compound (+7.38 mRy/atom) is approximately seven times as large as the heat of formation of the also unstable SmCo₁₂ compound (+1.06 mRy/atom), the heats of the formation of the end points, SmNi(8j)₄Co(8i,8f)₈ and SmNi(8j)₄Fe(8i,8f)₈ compounds, are almost equal at −1.65 mRy/atom and −1.71 mRy/atom, respectively. Consequently, the occupation of the 8j sublattice by Ni atoms plays a critical role in defining the sign and magnitude of the heat of formation of the Sm (Fe,Co,Ni)₁₂ compound. EMTO calculations revealed that the equilibrium atomic volumes of the SmNi(8j)₄Fe(8i,8f)₈ and SmNi(8j)₄Co(8i,8f)₈ compounds are Ω₀ = 12.98 ų and 12.40 ų, respectively.

As we mentioned in the Introduction, Harashima et al. 2018 [145] found that the calculated formation energy of the REFe₁₂ compound in respect to the RE₂Fe₁₇ compound and α-Fe is positive, indicating that the REFe₁₂ compound is unstable toward the decomposition of the RE₂Fe₁₇ compound and α-Fe. The possible reactions of the REFe₁₂ compound have been suggested in [85,92,104,145,177]. In this paper, we studied the reactions suggested by Schönhöbel et al. [104], assuming the decomposition of the Sm(Fe_1−xV_x)₁₂ alloys to the Sm₂Fe₁₇ compound, α-V, and α-Fe:

$$∆H_f\left(SmF{e}_{12-x}{V}_x\right)=\frac{1}{26}\left[2E\left(SmF{e}_{12-x}{V}_x\right)-E\left(S{m}_{2}F{e}_{17}\right)-2xE\left(\alpha -V\right)-\left(7-2x\right)E\left(\alpha -Fe\right)\right]$$

(1)

Schönhöbel et al. [104] performed calculations for 26 atoms (doubled) of ThMn₁₂-type supercells. Taking into consideration that in the EMTO method the total energy is calculated in the [Ry/atom] unit, x = 4 for the SmNi₄Fe₈ compound, while Equation (1) transforms (in the EMTO formalism) to:

$$∆H_f\left(SmF{e}_{12}\right) =\frac{1}{26}\left[26E\left(SmF{e}_{12}\right)-19E\left(S{m}_{2}F{e}_{17}\right)-7 E\left(\alpha -Fe\right)\right],$$

(2)

$$∆H_f\left(SmF{e}_{8}N{i}_4\right) =\frac{1}{26}\left[26E\left(SmF{e}_{8}N{i}_4\right)-19E\left(S{m}_{2}F{e}_{17}\right)-8E\left(\alpha -Ni\right)+E\left(\alpha -Fe\right)\right]$$

(3)

$$∆H_f\left(SmC{o}_{12}\right) =\frac{1}{26}\left[26E\left(SmC{o}_{12}\right)-19E\left(S{m}_{2}C{o}_{17}\right)-7 E\left(\alpha -Co\right)\right],$$

(4)

$$∆H_f\left(SmC{o}_{8}N{i}_4\right) =\frac{1}{26}\left[26E\left(SmC{o}_{8}N{i}_4\right)-19E\left(S{m}_{2}C{o}_{17}\right)-8E\left(\alpha -Ni\right)+E\left(\alpha -Co\right)\right].$$

(5)

Table 2. The decomposition energy of the reaction RE(Fe,Co,Ni)₁₂ → RE₂(Fe, Co)₁₇ + TMs.

Material	∆E (mRy/atom)
SmFe12	+7.79
SmCo12	+2.13
SmNi4Fe8	−1.29
SmNi4Co8	−0.59

presents the decomposition energy of the reaction RE(Fe,Co,Ni)₁₂ → RE₂(Fe, Co)₁₇ + TMs. As previously mentioned, the SmFe₁₂ and SmCo₁₂ compounds are unstable (positive heat of formation reported in Figure 2). Moreover, each of theThMn₁₂-type compounds without Ni is unstable relative to decomposition into the Th₂Zn₁₇-type compound and TMs. On the contrary, the SmNi₄Fe₈ and SmNi₄Fe₈ compounds have negative formation energies (see Figure 3) and are also stable toward decomposition into the Th₂Zn₁₇-type compounds and TMs. These results highlight the promising phase stability behavior for the SmNi₄Fe₈- and SmNi₄Fe₈-based magnets.

4. Magnetic Properties of the Sm(FeCo-Ni)₁₂ Alloys

The total moment of the SmFe₁₂ compound calculated in the present study using the EMTO method (m^total ≈ 25.7211 μ_B/f.u.) at the equilibrium volume (Ω₀ = 13.19 ų) is smaller compared to the one calculated using VASP (m^total = 26.56 μ_B/f.u.) [141]. Taking into consideration the calculated total moment per atom ($m_{at.}^{tot}=1.9785 {\mu }_B$) and the calculated density of the SmFe₁₂ compound (ρ = 7.69 g/cm³), one can estimate M_s = $m_{at.}^{tot} {\mu }_B\frac{\rho N_A }{M_{SmF{e}_{12}}}=$1.39 MA/m and μ₀M_s = 1.75 T, where μ_B = 9.274 × 10⁻²⁴ Am², N_A = 6.0221 × 10²³ atoms/mole, and $M_{SmF{e}_{12}}$= 61.12 g/mol (i.e., the average atomic weight per atom of the SmFe₁₂ compound). Thus, the maximum energy product of the SmFe₁₂ compound is ${\left|BH\right|}_{max}$ = $\frac{1}{4}{\mu }_0{M}_s^2$ = 607.98 kJ/m³, where μ₀ = 4π × 10⁻⁷ $\frac{kg•m}{se{c}^2{A}^2}$ is the permeability of free space. These predictions are in good agreement with both calculated (VASP results, μ₀M_s = 1.83 T) [141] and experimental (μ₀M_s = 1.64 T for 0.5 μm thick SmFe₁₂ films) results [126].

Regarding the SmCo₁₂ compound, the calculated (EMTO) total moment is equal to 17.4368 μ_B/f.u. at the equilibrium volume (Ω₀ = 12.42 ų). Thus, one can estimate M_s = 1.00 MA/m, μ₀M_s = 1.26 T, and |BH|_max = 315.19 kJ/m³ based on the calculated total moment per atom ($m_{at.}^{tot}$ = 1.3413 μ_B) and the calculated density (ρ = 8.20 g/cm³) of the SmCo₁₂ compound.

Considering substitution of Fe by Ni in the SmFe₁₂ compound to form the SmNi₄Fe₈ compound, where 4 Ni atoms occupy the 8j sublattice and 8 Fe atoms are equally colligated on the 8i and 8f sublattices, the resulting site-projected spin (m^(s)) and orbital magnetic moments (m^(o)) and the total moment (m^total ≈ 19.9349 μ_B/f.u.) calculated (EMTO) at the equilibrium volume (Ω₀ = 12.98 ų) are presented in

Table 3. Site-projected spin (m^(s)) and orbital (m^(o)) magnetic moments for the SmNi₄Fe₈, SmNi₄(Fe_.9Co_.1)₈, SmNi₄(Fe_.8Co_.2)₈, and SmNi₄Co₈ compounds, where 4 Ni atoms occupy the 8j sublattice and 8 (Fe,Co) atoms are equally distributed on the 8i and 8f sublattices: m^total= 19.9349, 19.9778, 19.5601, and 13.5833 μ_B/f.u, respectively.

	Sm1(2a)	Fe1(8f)/Co1(8f)	Fe2(8i)/Co2(8i)	Ni3 (8j)
SmNi4Fe8
m(s) (μB)	+3.7182	−2.1482/-	−2.1763/-	−0.6144
m(o) (μB)	−3.1649	−0.0652/-	−0.0658/-	−0.0522
SmNi4(Fe.9Co.1)8
m(s) (μB)	+3.6956	−2.2330/−1.4267	−2.2568/−1.3874	−0.5977
m(o) (μB)	−3.2281	−0.0678/−0.0861	−0.0693/−0.0800	−0.0514
SmNi4(Fe.8Co.2)8
m(s) (μB)	+3.7452	−2.2866/−1.4492	−2.2898/−1.3992	−0.5936
m(o) (μB)	−3.2376	−0.0686/−0.0869	−0.0698/−0.0809	−0.0481
SmNi4Co8
m(s) (μB)	+4.1589	-/−1.5299	-/−1.4692	−0.5420
m(o) (μB)	−2.8154	-/−0.0899	-/−0.0678	−0.0329

. Using the calculated total moment per atom ($m_{at.}^{tot}$ =1.5345 μ_B) and the calculated density of the SmNi₄Fe₈ compound (ρ = 8.19 g/cm³), M_s = 1.10 $MA/m,$ μ₀M_s = 1.38 T, and |BH|_max = 377.55 kJ/m³ can be estimated.

By progressively substituting Fe with Co from SmNi₄Fe₈ to SmNi₄Co₈, the calculated (EMTO) site-projected spin (m^(s)) and orbital moments (m^(o)), as well as the total moments (m^total) of the SmNi₄(Fe_0.9Co_0.1)₈, SmNi₄(Fe_0.8Co_0.2)₈, and SmNi₄Co₈ compounds, where 4 Ni atoms occupy the 8j sublattice and 8 (Fe_1−xCo_x) atoms are equally distributed on the 8i and 8f sublattices, are presented in Table 3. In more detail (summarized in

Table 4. Atomic volume (Ω₀), density (ρ), total moment (m^total), saturation magnetization (M_s and μ₀M_s), and maximum energy product (|BH|_max) as calculated by the EMTO method and using the FPLMTO method with different approaches: 4f-band-scalar, 4f-band-SO, 4f-band-SO + OP, and SRM + OP (see main text for explanation of these models).

Material	Theory	Ω0 (Å3)	Ρ  $\left(\frac{\mathit{g}}{\mathit{c}{\mathit{m}}^3}\right)$	${\mathit{m}}^{\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}}$  $\left(\frac{{\mathit{\mu }}_{\mathit{B}}}{\mathit{f}.\mathit{u}.}\right)$	${\mathit{M}}_{\mathit{s}}$  $\left(\frac{\mathit{M}\mathit{A}}{\mathit{m}}\right)$	${\mathit{\mu }}_0{\mathit{M}}_{\mathit{s}}$  $\left(\mathit{T}\right)$	${\left|\mathit{B}\mathit{H}\right|}_{\mathit{m}\mathit{a}\mathit{x}}$  $\left(\frac{\mathit{k}\mathit{J}}{{\mathit{m}}^3}\right)$
SmFe12	EMTO	13.19	7.69	25.7	1.39	1.75	608.0
SmCo12	EMTO	12.42	8.82	17.4	1.00	1.26	315.2
SmNi4Fe8	EMTO	12.98	8.19	19.9	1.10	1.38	377.6
	4f-band-scalar	12.84	8.28	16.2	0.90	1.14	254.4
	4f-band-SO	12.84	8.28	18.7	1.04	1.31	339.4
	4f-band-SO + OP	12.90	8.24	20.1	1.12	1.41	388.6
	SRM + OP	12.98	8.19	22.5	1.24	1.57	481.3
SmNi4(Fe0.9Co0.1)8	EMTO	12.92	8.25	19.9	1.10	1.39	382.3
	4f-band-scalar	12.79	8.33	15.6	0.87	1.09	237.5
	4f-band-SO	12.79	8.33	18.1	1.01	1.27	319.7
	4f-band-SO + OP	12.85	8.29	19.6	1.08	1.37	371.2
	SRM + OP	12.93	8.24	22.0	1.21	1.53	462.5
SmNi4(Fe0.8Co0.2)8	EMTO	12.89	8.30	19.6	1.08	1.36	368.2
	4f-band-scalar	12.74	8.40	15.0	0.84	1.06	221.7
	4f-band-SO	12.74	8.40	17.4	0.97	1.23	298.2
	4f-band-SO + OP	12.80	8.36	19.0	1.06	1.33	352.6
	SRM + OP	12.87	8.31	21.4	1.18	1.42	441.7
SmNi4Co8	EMTO	12.47	8.77	13.6	0.78	0.98	189.7
	4f-band-scalar	12.33	8.87	9.60	0.56	0.70	97.08
	4f-band-SO	12.33	8.87	11.8	0.69	0.86	146.3
	4f-band-SO + OP	12.37	8.84	13.6	0.79	0.99	193.1
	SRM + OP	12.49	8.76	16.1	0.92	1.16	265.6

), for the SmNi₄(Fe_0.9Co_0.1)₈ compound at the equilibrium volume (Ω₀ = 12.92 ų), the calculated total magnetic moment is m^total ≈ 19.9778 μ_B/f.u. Taking into consideration the calculated total moment per atom ($m_{at.}^{tot}$= 1.5367 μ_B) and the calculated density of the SmNi₄(Fe_0.9Co_0.1)₈ compound (ρ = 8.25 g/cm³), one can estimate M_s = 1.10 MA/m, μ₀M_s = 1.39 T, and |BH|_max = 382.27 kJ/m³. For the SmNi₄(Fe_0.8Co_0.2)₈ compound at the equilibrium volume (Ω₀ = 12.89 ų), the calculated total magnetic moment is m^total ≈ 19.5601 μ_B/f.u. Taking into consideration the calculated total moment per atom ($m_{at.}^{tot}$ = 1.5046 μ_B) and the calculated density of the SmNi₄(Fe_0.8Co_0.2)₈ compound (ρ = 8.30 g/cm³), one can estimate M_s = 1.08 MA/m, μ₀M_s = 1.36 T, and |BH|_max = 368.16 kJ/m³. For the SmNi₄Co₈ compound at the equilibrium volume (Ω₀ = 12.47 ų), the calculated total magnetic moment is m^total ≈ 13.5833 μ_B/f.u. Taking into consideration the calculated total moment per atom ($m_{at.}^{tot}=$ 1.0449 μ_B) and the calculated density of the SmNi₄Co₈ compound (ρ = 8.77 g/cm³), one can estimate M_s = $0.78 MA/m,$ μ₀M_s = 0.98 T, and |BH|_max = 189.70 kJ/m³.

The compositional dependences of μ₀M_s and |BH|_max of the SmNi₄(Fe_1−xCo_x)₈ alloys for x = 0.0, 0.1, 0.2. and 1.0 are shown in Figure 4 and Figure 5, respectively. μ₀M_s initially increases and reaches a maximum at x = 0.1. This result is in an agreement with the experimental results for the SmTi₁Fe_11−xCo_x and SmTi₂Fe_10−xCo_x alloys [31,35,52], where μ₀M_s reaches its maximum at x ≈ 0.2 [31,35] or x ≈ 0.3 [52]. A further Co concentration increase causes a sharp decrease in μ₀M_s. The calculated trend for |BH|_max as a function of the Co substitution is similar to μ₀M_s.

Regarding the Curie temperature (T_c), a mean-field treatment can be formulated as [178,179]:

$$T_c=\frac{2}{3}\times \frac{E_{tot}^{DLM}-E_{tot}^{AF} }{k_B}$$

where $E_{tot}^{DLM} and E_{tot}^{AF}$ are the ground-state total energies of the DLM and the AF (antiferromagnetic) states, respectively, and k_B is the Boltzmann constant. Thus, an estimation of T_c can be accomplished from the total energy difference between the ferromagnetic (or antiferromagnetic) and paramagnetic (DLM) states. Nevertheless, in line with [179], the diversity between the total energies can be replaced by the diversity between the effective single-particle (one atomic specie) energies, which are instantly consociated with AF and DLM states (the so-called MFA treatment). In the present work, $E_{tot}^{DLM}\mathrm{and} E_{tot}^{AF}$ are calculated at the equilibrium volumes for DLM and AF states, respectively.

Figure 6 shows the calculated (within the EMTO-CPA technique) T_c values of the pseudo-binary SmNi₄(Fe_1−xCo_x)₈ alloys, where 4 Ni atoms occupy the 8j sublattice and 8 (Fe_1−xCo_x) atoms are shared on the 8i and 8f sublattices. The calculated T_c values are equal to 853 K, 928 K, 995 K, and 1120 K for the SmNi₄Fe₈, SmNi₄(Fe_0.9Co_0.1)₈, SmNi₄(Fe_0.8Co_0.2)₈, and SmNi₄Co₈ alloys, respectively. These values are substantially higher than the Curie temperature of the generally used Neomax (Nd₂Fe₁₄B₁) magnet (588 K) [12].

In Table 4, our FPLMTO-calculated magnetic properties for the SmNi₄(Fe_1−xCo_x)₈ alloys are shown. The obtained properties depend on the approximation of the problematic 4f electrons on samarium. The simplest approach is a 4f-band model that ignores spin–orbit coupling. Better models include spin–orbit (SO) and orbital polarization (OP). The best approach is to confine the 4f electrons in core states in the standard rare-earth model with OP (SRM + OP) for the d states. The largest total moments are obtained from SRM + OP because here the Sm moment (anti-parallel to the total unit-cell moment) is the smallest.

We can compare our FPLMTO magnetic moments with those of the fully relativistic EMTO method (see Table 3). The reason these models give different total magnetic moments is because they predict vastly different samarium magnetic moment. In Figure 7, we show the different results for the models.

Figure 7 shows that the SRM + OP model has the smallest Sm moment and the greatest total magnetic moment (Table 4). The 4f-band-EMTO approach is second. We believe there are technical differences in the EMTO and FPLMTO codes that give rise to the very different Sm moments, because relativistic DFT values (with spin–orbit coupling but no orbital polarization), i.e., EMTO and 4f-band-SO (FPLMTO), should in principle be close but they are not. For the FPLMTO calculations, the Sm magnetic moment depends very weakly on composition parameter x, while there is some dependence in the EMTO. We should point out that for the 3d (Fe, Co, and Ni) metals, the magnetic moments are almost the same for all models.

Next, we present our MAE results. Here, we apply the SRM + OP model for the MAE because the other approaches (4f-band) produce unrealistically large MAE due to the inappropriate treatment of the 4f electrons (not shown). Before calculating the MAE, i.e., the energy difference between the magnetic spin moment aligned in the x-y plane and along the z axis, we carefully optimize the crystal structure. In other words, both the lattice parameters a and c are optimized to give the lowest total energy of the tetragonal crystal. The differences between a and c for the iron-rich compound SmNi₄(Fe_1−xCo_x)₈ (here x = 0, 0.1, and 0.2) are actually very small, while for x = 1 the total atomic volume decreases by about 4%. We present the calculated equilibrium atomic volume (Ω₀) and μ₀H_a (with H_a = 2K₁/M_s) [17]) in

Table 5. Calculated atomic volume, magnetic anisotropy energy, first anisotropy constant, anisotropy field, and magnetic hardness parameter for the SmNi₄(Fe_1−xCo_x)₈ alloys, where x = 0, 0.1, 0.2, and 1.

Compound	Ω0	$\mathit{M}\mathit{A}\mathit{E} \left(\frac{\mathit{m}\mathit{e}\mathit{V}}{\mathit{f}.\mathit{u}.}\right)$	${\mathit{K}}_1\left(\frac{\mathit{M}\mathit{J}}{{\mathit{m}}^3}\right)$	μ0Ha (T)	κ
SmNi4Fe8	12.98	3.98	3.78	6.09	1.40
SmNi4(Fe0.9Co0.1)8	12.93	5.09	4.85	8.02	1.62
SmNi4(Fe0.8Co0.2)8	12.87	6.49	6.22	10.54	1.88
SmNi4Co8	12.49	9.53	9.41	20.13	2.98

.

As was mentioned in [17,180], the relation (interchange) between the magnetic anisotropy and saturation magnetization is the key criterion for a magnet to be fabricated in any possible shape with a good chance of resisting self-demagnetization. This criterion may be formulated in terms of the magnetic hardness parameter of the materials, $\kappa =\surd \left(\frac{K_1}{{\mu }_0{M}_s^2}\right)$, where κ > 1 is an empirically required rule to fabricate a good permanent magnet. As reported in Table 5, all calculated magnetic hardness parameters of the SmNi₄(Fe_1−xCo_x)₈ magnets meet the manufacturability criteria (i.e., κ > 1).

Finally, the compositional dependence of the anisotropy field (μ₀H_a) of the SmNi₄(Fe_1−xCo_x)₈ alloys is shown in Figure 8. We find that on the iron-rich side of the compound, adding a small amount of cobalt increases μ₀H_a in a linear fashion.

5. Discussion and Conclusions

The present calculations show that the SmNi₄(Fe_1−xCo_x)₈ alloys could be stable; fabricated in bulk form within the whole compositional range; and have significant μ₀M_s values of 1.38–1.57 T, 1.39–1.53 T, and 1.36–1.42 T (depending on the model); T_c values of 853 K, 928 K, and 995 K; μ₀H_a values of 6.09 T, 8.02 T, and 10.54 T; and κ values of 1.40, 1.62, and 1.88 for x = 0.0, 0.1, and 0.2, respectively. As mentioned in the Introduction, the economic requirement for a permanent magnet is fulfilled if μ₀M_s ~≥ 1.25 T, T_c ~≥ 550 K, μ₀H_a ~≥ 3.75 T, and κ > 1 [1,2,17], which means the SmNi₄(Fe_1−xCo_x)₈ magnets could be suitable for industrial applications. The most promising SmNi₄(Fe_0.9Co_0.1)₈ magnet outperforms most of the intrinsic properties of the widely used Neomax magnets (T_c = 588 K and μ₀H_a = 7.60 T), except for μ₀M_s = 1.61 T (Neomax), resulting in |BH|_max = 382.27–462.5 kJ/m³, which is 74.2–89.8% of |BH|_max~515 kJ/m³ (Neomax). Stabilization of the bulk SmFe₁₂ magnet is achieved by doping it with Ti, Co, Zr, and Y. According to [138,142,181], for early transition metals (TM = Ti-Mn), the stabilization of the SmFe₁₁TM₁ magnet is accompanied by substantial reduction in the magnetic moment. In contrast, the late transition metals (TM = Co, Ni, Cu, and Zn), when serving as stabilizers, increase the magnetic moment of the SmFe₁₂ magnet. As was mentioned in the Introduction, Harashima et al. [92] found that Ni substitution yields a negative formation energy of the NdFe₁₁Ni₁ compound relative to the NdFe₁₂ compound, bcc Fe, and fcc Ni. Our calculations confirm the observation made by Harashima et al. [92] that the most stable site for Ni is 8j, in contrast to Ti (the most stable site for Ti is 8i), for which the spin magnetic moment aligns anti-parallel to the internal magnetization of the SmFe₁₂ compound and the spin moment of Ni aligns parallel to the internal net magnetization of the SmFe₁₂ compound, thereby boosting its μ₀M_s.

In accordance with our previous calculations [149,150], substituting most of Co with Fe in the SmCo₅ magnet (the CaCu₅-type structure) and using Ni as a thermodynamic stabilizer engenders the formation of the SmFe₃CoNi magnet, which has exceptional magnetic properties, such as a high T_c = 1103 K, powerful magnetic anisotropy (K₁ = 9.2 MJ/m³, about twice that of the Nd₂Fe₁₄B₁ magnet at K₁ = 4.9 MJ/m³), and |BH|_max = 362 kJ/m³, which is about 70% of |BH|_max~515 kJ/m³ (Neomax). Ni has also been used as the stabilizing element in the YCo₅ magnet (the CaCu₅-type structure) in order to provide the maximum amount of Fe to favor high magnetization of the YFe₃CoNi magnet |BH|_max~309 kJ/m³ [151].

When considering why Ni acts as a stabilizer, it is important to realize that the favored structure depends rather sensitively on the number of d electrons in the compound, i.e., the filling (occupation) of electrons in the d-band. Figure 9 shows the calculated formation energies (EMTO) of the SmFe₁₂, SmCo₁₂, and SmNi₁₂ compounds in respect to the unary systems α-Sm, α-Fe, α-Co, and α-Ni as a function of the number of 3d electrons per TM atom.

Increasing the 3d electron count (more Ni) greatly stabilizes the SmTM₁₂ compound. An obvious consequence of this behavior is that one should be able to recover the crystal stability of a Sm(Fe,Co)₁₂ compound by doping it with 3d electrons from nickel. From Figure 9, one can conclude that an Sm(Fe_1−x−yCo_xNi_y)₁₂ alloy with at least 7.76 3d electrons per TM atom is stable. This result is consistent with the previously discovered stabilization of the hexagonal SmTM₅ (the CaCu₅-type structure) compound [149]. According to Söderlind et al. [149], the efforts to enhance the magnetism of the stable SmCo₅ magnets through the substitution of some amount of Co for Fe were limited because the SmFe₅ compound does not exist in the CaCu₅-type structure. To compensate for the reduction in d electrons due to the Fe/Co substitution (Fe has about one 3d electron less than Co), we simply added nickel, which has about one more 3d electron than cobalt, so that the structural stability was ensured. Nickel acts as a thermodynamic stabilizer in the Sm-Fe-Co-Ni alloys, and in present case, adding a certain amount of Ni helps to stabilize the SmNi₄(Fe_1−xCo_x)₈ magnets in the ThMn₁₂-type structure.

Our calculations revealed that the SmNi₄Co₈ magnet exhibits a significant axial MAE with K₁ = 9.41 MJ/m³. As is noted in the Introduction, the planar MAE of the SmCo₁₁Ti compound was discovered by Ohashi et al. [42] and Wang et al. [55]. Tereshina et al. [52] concluded that Co has a strong influence on the MAE of Sm-sublattice in the case of the REFe_1−xCo_xTi alloys, causing changes in MAE from axial to planar when x ~≥ 0.5. Our calculations did not reproduce this behavior.

In conclusion, we showed that the Sm(Fe_1−xCo_x)₁₂ alloys, including the end points SmFe₁₂ and SmCo₁₂, could be stabilized by replacing a certain portion of Fe or Co atoms with Ni atoms and could perhaps be synthesized. These contemporary permanent magnets are assumed to possess outstanding magnetic characteristics, namely a significant energy product and anisotropy field, as well as a high Curie temperature that significantly outperform Neomax magnets. Nevertheless, it is essential to accentuate that although these intrinsic properties are required for profitable permanent magnets, this is not a plentiful condition. It is also crucial to advance the anisotropic polycrystalline microstructure of a magnet to provide both high coercivity and remanence, e.g., to develop an applicable grain boundary phase that can grow in equilibrium with the main phase, in order to optimize the extrinsic magnetic properties.