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

Abstract

Alloys that are Ni-doped, such as the (Sm_1−yZr_y)(Fe_1−xCo_x)₁₂ and (Ce_0.5Sm_0.5)Fe₁₀Co₂ systems, are studied because of their magnetic properties. The (Sm_1−yZr_y)(Fe_1−xCo_x)_11−zTi_z and (Ce._1−xSm_x)Fe₉Co₂Ti alloys are considered contenders for vastly effective permanent magnets because of their anisotropy field and Curie temperature. Ti can act as a stabilizer for the SmFe₁₂ compound but substantially suppresses saturation magnetization. To maintain the saturation magnetization in the scope of 1.3–1.5 T, we propose substituting a particular quantity of Fe and Co in the (Sm_1−yZr_y)(Fe_1−xCo_x)₁₂ and (Ce_0.5Sm_0.5)Fe₁₀Co₂ alloys with Ni. By performing ab initio calculations, we show that Ni incorporation results in increased thermodynamic stability and, in contrast to Ti, has a parallel spin moment aligned to the moment of the SmFe₁₂ compound and improves its saturation magnetization without affecting the anisotropy field or Curie temperature.

1. Introduction

Rare earth-based magnets that exhibit the ThMn₁₂-type structure have garnered interest as hard magnetic materials. Specifically, the tetragonal REFe₁₂-based compound, which RE describes as a rare-earth metal, is studied because of its significant saturation magnetization (μ₀M_s), significant anisotropy field (μ₀H_a), and significant Curie temperature (T_c) [1,2,3,4,5,6,7,8]. An REFe₁₂ magnet has a lower RE concentration (7.7 at.%) in contrast to the extensively used Nd₂Fe₁₄B₁ magnet, or so-called Neomax, (11.8 at.%). However, the SmFe₁₂ compound is not considered to be stable from a thermodynamic standpoint in the bulk, but its favorable intrinsic properties, μ₀M_s = 1.64 T, μ₀H_a = 12 T, and T_c = 550 K [1], resemble epitaxially grown thin films. To sustain the REFe₁₂ phase in the bulk material, substituting Fe with a stabilizing metal M, where M = Ti, Nb, V, Mo, Cr, Mn, W, Re, Al, Ga, Si, H, and C, has been studied. The composition area x for stabilizing the REFe_12−xM_x phase is conditional on M [9].

It is known that Ti is favored to stabilize REFe_12−xM_x alloys with x~1 [2]. The samarium-based compound SmFe₁₁Ti₁ has a saturation magnetization μ₀M_s = 1.14–1.22 T that is smaller than the saturation magnetization of Nd₂Fe₁₄B₁ (μ₀M_s = 1.61 T) [10,11]. The stabilization of the SmFe₁₂M magnet is supported by considerable contraction of the magnetic moment because the spin addition to the magnetic moment of the stabilizing element aligns itself anti-parallel in direction to the internal magnetization of the SmFe₁₂ intermetallic compound. Therefore, it is critical to keep the concentration of stabilizer as small as possible.

In accordance with the works of Tozman et al. [12], the above-mentioned reduction of the saturated magnetization of the SmFe₁₁Ti₁ compound can be recovered by partial replacement of Fe, e.g., for the Sm(Fe_0.8Co_0.2)₁₁Ti magnet μ₀M_s = 1.43 T [12]. Another practice to enhance the saturation magnetization of the SmFe_12−xTi_x alloys is to reduce the Ti content, where instead the phase stability is assured by Zr or Y partially substituting on the Sm site [13,14,15]. Kuno et al. [13] presumed that an alloy that includes both Zr and M (for instance, Ti) could stabilize the ThMn₁₂-type magnet with less than a single M atom per formula unit: the strip-cast (Sm_0.8Zr_0.2)(Fe_0.75Co_0.25)_11.5Ti_0.5 magnet reaches a saturation magnetization μ₀M_s = 1.58 T and anisotropy field μ₀H_a = 7.41 T that are identical to those of Neomax (µ₀M_s = 1.61 T, µ₀H_a = 7.60 T) [16], although the Curie temperature of the (Sm_0.8Zr_0.2)(Fe_0.75Co_0.25)_11.5Ti_0.5 magnet, T_c = 880 K, substantially surpasses Neomax, T_c = 584 K [16]. Further improvement in saturation magnetization is possible for the Sm(Fe_0.8Co_0.2)₁₁Ti compound (μ₀M_s = 1.43 T), which can be accomplished by reducing the Ti content from Ti₁ to Ti_0.5 and with partial substitution of Sm by Zr as reported by Tozman et al. [12], where the magnetic properties of the (Sm_0.80Zr_0.20)(Fe_0.80Co_0.20)_11.5Ti_0.5 magnet are: μ₀M_s = 1.53 T and µ₀H_a = 8.4 T, with T_c = 830 K. Further improvement of Zr substituting for Sm allowed Tozman et al. [17,18] to produce a magnet with a large saturation magnetization, where μ₀M_s = 1.90 T, for the (Sm_0.82Zr_0.18)(Fe_0.8Co_0.2)₁₂ magnet, epitaxially grown thin films with the anisotropy field, µ₀H_a = 9.8 T, and Curie temperature, T_c = 671 K.

An excess of cerium metal exists, while its uses do not exceed production, making the mining economics of other less abundant and more technologically important rare earth metals very expensive. Thus, Ce-based magnets with the ThMn₁₂-type intermetallic compound would be perfect candidates for new-rare earth permanent magnets [19,20,21,22]. Theoretical analyses have shown that Ce might possibly stabilize the ThMn₁₂-type intermetallic compound [23,24,25,26,27]. Goll et al. [28] found that a CeTiFe_1−xCo_x arc melted (and quenched) magnet shows the maximum value of the saturation magnetization, μ₀M_s = 1.27 T, magnetic anisotropy energy (MAE), K₁ = 2.15 MJ/m³, and the maximum energy product, |BH|_max = 282 kJ/m³, at x ≈ 1.95, are considerably reduced compared to Neomax (µ₀M_s = 1.61 T, µ₀H_a = 7.60 T, and |BH|_max ~ 515 kJ/m³) [16]. In order to increase MAE and thus the coercivity of the CeFe₉Co₂Ti magnets, Gabay et al. [29], Martin-Cid et al. [30], Wuest et al. [31], Martin-Cid et al. [32], and Martin-Cid [33] proposed sectional replacement of Ce with Sm, resulting in the (Ce_1−xSm_x)Ti₁Co₂Fe₉ magnet films synthesized via melt-spinning [27,28,29,30,31,32,33]. According to [32,33], for the (Ce_0.5Sm_0.5)Ti₁Co₂Fe₉ magnet: saturation magnetization, μ₀M_s = 1.15 T, anisotropy field, µ₀H_a = 5.6 T, Curie temperature, T_c = 726 K, and maximum energy product, |BH|_max = 261.28 kJ/m³. Recently, Saito [34] discovered that the compelling increment of the CeFe₁₁Ti melt-spun ribbon coercivity can be reached by partial replacement of Sm with Ce.

In our past works [35,36,37,38], we suggest enhancing μ₀M_s and (BH)_max in the widely studied SmCo₅, YCo₅, and SmFe₁₂ magnets by replacing Co with Fe and using Ni as a stabilizer. As mentioned above, the stabilization of the SmFe₁₁M magnet is accompanied by a considerable decrease in the magnetic moment because of the anti-parallel spin alignment of the magnetic moment of a given stabilizing metal, M, to the internal magnetization of the SmFe₁₂ intermetallic compound. On the contrary, the spin moment of Ni exhibits parallel alignment with respect to the internal net magnetization of the SmFe₁₂ compound, thereby increasing its saturation magnetization. Recent calculations [38] demonstrate that the SmNi₄(Fe_1−xCo_x)₈ alloys of the ThMn₁₂ type structure could be stable and possibly manufactured in bulk form across the entire compositional range. They have compelling magnetic properties, such as: μ₀M_s values of 1.38–1.57 T, 1.39–1.53 T, and 1.36–1.42 T (model dependent); T_c values of 853 K, 928 K, and 995 K, and μ₀H_a values that are 6.09 T, 8.02 T, and 10.54 T, respectively. However, similar to the case of the undoped SmFe₁₂ magnet, the RE (Sm) content is 7.7 at.%. for the SmNi₄(Fe_1−xCo_x)₈ alloys. The partial substitution of Sm by Zr or Ce will decrease the content of critical RE.

The primary purpose of the current study is to study the influence of zirconium and cerium on the phase stability of the (Sm,Zr,Ce)(Fe-Co-Ni)₁₂ alloys and to assess their magnetic properties. We perform ab initio calculations using the following formalisms: (i) the fully relativistic exact muffin-tin orbital method (FREMTO) in conjunction with the coherent potential approximation (CPA) and (ii) the full-potential linear muffin-tin orbital method (FPLMTO), see [35,36,37,38] for details. The methods account for all relativistic effects, such as spin–orbit coupling (SOC). These two techniques provide accurate results that are independent of technical implementation and rely on the particular strength and durability of each technique. The results of the density-functional theory (DFT) calculations of the ground-state properties of the (Sm,Zr,Ce)(Fe-Co-Ni)₁₂ alloys are presented in Section 2. We discuss the results of the DFT calculations and the magnetic characteristics of the (Sm,Zr,Ce)(Fe-Co-Ni)₁₂ alloys in Section 3. Finally, an analysis and summary are presented in Section 4.

2. Thermodynamic Properties of the (Sm,R)Ni₄(Fe,Co)₈ Alloys: R = Zr, Ce

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

As mentioned in Ref. [38], the Sm(Fe_1−xCo_x)₁₂ alloys could be stabilized by substituting a particular portion of Fe or Co atoms with Ni atoms. The optimal configuration of these alloys has a stoichiometry of SmNi₄(Fe_1−xCo_x)₈, where a single Sm atom occupies the 2a Wyckoff position, 4 Ni atoms occupy the 8j Wyckoff position, and 8 (Fe_1−xCo_x) atoms are randomly distributed on the 8i and 8f Wyckoff positions (see Figure 1).

To analyze the stabilizing effects of the nickel addition to the (Sm_0.8Zr_0.2)(Fe_1−xCo_x)₁₂ compounds, we carried out EMTO-CPA calculations for the formation energy of the (Sm_0.8Zr_0.2)Ni₄(Fe_1−xCo_x)₈ compound relative to the reference states for unary systems α-Sm, α-Zr, α-Fe, α-Co, and α-Ni, where Sm and Zr atoms occupy the 2a sublattice, 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 2, the formation energies of the (Sm_0.8Zr_0.2)Ni₄(Fe_1−xCo_x)₈ alloys are negative within the whole compositional interval.

To avoid using the titanium metal as a stabilizer, e.g., CeFe₉Co₂Ti [30,31,32], we instead used the nickel metal, whose spin moments align parallel to the internal net magnetization of the SmFe₁₂ compound, thus enhancing its saturation magnetization. Although Martin-Cid et al. [32] discussed the probabilities of Co atom site occupation for the (Ce_1−xSm_x)Ti₁Co₂Fe₉ magnets, x = 0.00, 0.25, 0.50, 0.75, and 1.00, assuming that a single Ti atom occupies an 8i Wyckoff position, we estimated the lowest energy configuration for the CeNi₄Co₂Fe₆ magnet assuming that 4 Ni atoms occupy 8j Wyckoff positions in analogy with the calculated energetically favorable configuration for the SmNi₄(Fe_1−xCo_x)₈ alloys [38]. The EMTO-CPA calculations for the CeNi₄Co₂Fe₆ magnet revealed (see

Table 1. The relative energies of different CeNi₄Co₂Fe₆ magnet atomic configurations. The energy of the CeNi₄Co₂Fe₆ magnet, where 2 Fe and 2 Co atoms occupy the 8f Wyckoff positions, 4 Fe atoms occupy the 8i Wyckoff positions, and 4 Ni atoms occupy the 8j Wyckoff positions, is taken to be zero.

Configuration	ΔE (mRy/Atom)
Ce(Fe2Co2)8f(Fe4)8i(Ni4)8j	0.000
Ce(Fe0.75Co0.25)8f8j(Ni4)8j	0.470
Ce(Fe3Co1)8f(Fe3Co1)8i(Ni4)8j	0.642
Ce(Fe4)8f(Fe2Co2)8i(Ni4)8j	1.068

) that the configuration, where 2 Fe and 2 Co atoms occupy the 8f Wyckoff positions, 4 Fe atoms occupy the 8i Wyckoff positions, and 4 Ni atoms occupy the 8j Wyckoff positions, is the energetically favorable configuration.

To investigate the stabilizing effects of nickel additions to the (Ce_1−xSm_x)Fe₁₀Co₂ alloys, we carried out EMTO-CPA calculations of the formation energy of the (Ce_1−xSm_x)Ni₄Fe₆Co₂ alloys in respect to the reference states for unary systems α-Sm, α-Ce, α-Fe, α-Co, and α-Ni, where Ce and Sm atoms are randomly distributed on the 2a Wyckoff positions, 2 Fe and 2 Co atoms occupy the 8f Wyckoff positions, 4 Fe atoms occupy the 8i Wyckoff positions, and 4 Ni atoms occupy the 8j Wyckoff positions, within the whole compositional interval (0 ≤ x ≤1). The results of these calculations are shown in Figure 3. The calculations show the negative heat of formation of the (Ce_1−xSm_x)Ni₄Fe₆Co₂ alloys across the whole compositional interval, suggesting the possibility of the creation of bulk magnets. Notice that the experiments [28,29,30,31,32,33,34] resulted in the synthesizing of the melt-spun ribbons.

3. Magnetic Properties of the (Sm,R)Ni₄(Fe,Co)₈ Alloys: R = Zr, Ce

The total moment of the (Sm_0.8Zr_0.2)Ni₄Fe₈ compound, where Sm and Zr atoms occupy the 2a sublattice, 4 Ni atoms occupy the 8j sublattice, and 8 Fe atoms are equally distributed on the 8i and 8f sublattices, calculated in this study using the FREMTO-CPA method, is equal to (m^total ≈ 21.5431 μ_B/f.u.) at the equilibrium volume (Ω₀ = 13.499 ų). Taking into consideration the calculated total moment per atom ($m_{at.}^{tot}=1.6572 {\mu }_B$) and the calculated density of the (Sm_0.8Zr_0.2)Ni₄Fe₈ compound (ρ = 7.760 g/cm³), one can evaluate M_s = $m_{at.}^{tot} \left[{\mu }_B\right]\frac{\rho N_A }{\left(M_{\left(S{m}_{0.8}Z{r}_{0.2}\right)N{i}_{4}F{e}_8}\right)}=$ 1.1385 MA/m and μ₀M_s = 1.4307 T, where μ_B = 9.274 × 10⁻²⁴ Am², [μ_B] is the dimension for μ_B, N_A = 6.0221 × 10²³ atoms/mole, and $M_{\left(S{m}_{0.8}Z{r}_{0.2}\right)N{i}_{4}F{e}_8}$= 63.0819 g/mol (the average atomic weight per atom of the (Sm_0.8Zr_0.2)Ni₄Fe₈ compound). Thus, the maximum energy product of the (Sm_0.8Zr_0.2)Ni₄Fe₈ compound is ${\left|BH\right|}_{max}$= $\frac{1}{4}{\mu }_0{M}_s^2=$ 407.207 kJ/m³, where μ₀ = 4π × 10⁻⁷ $\frac{\mathrm{kg}·\mathrm{m}}{{\sec }^2{\mathrm{A}}^2}$ is the permeability of free space.

By sequentially replacing Fe with Co from (Sm_0.8Zr_0.2)Ni₄Fe₈ to (Sm_0.8Zr_0.2)Ni₄Co₈, the calculated (FREMTO-CPA) site-projected spin (m^(s)) and orbital moments (m^(o)), as well as the total moments (m^tot.) of the (Sm_0.8Zr_0.2)Ni₄(Fe_0.9Co_0.1)₈, (Sm_0.8Zr_0.2)Ni₄(Fe_0.8Co_0.2)₈, and (Sm_0.8Zr_0.2)Ni₄Co₈ compounds, where 4 Ni atoms occupy the 8j sublattice and 8 (Fe_1−xCo_x) atoms are equitably distributed on the 8i and 8f sublattices, are presented in

Table 2. Site-projected spin (m^(s)) and orbital (m^(o)) magnetic moments for the (Sm_0.8Zr_0.2)Ni₄Fe₈, (Sm_0.8Zr_0.2)Ni₄(Fe_0.9Co_0.1)₈, (Sm_0.8Zr_0.2)Ni₄(Fe_0.8Co_0.2)₈, and (Sm_0.8Zr_0.2)Ni₄Co₈ compounds, where Sm and Zr atoms occupy the 2a sublattice, 4 Ni atoms occupy the 8j sublattice, and 8 (Co,Fe) atoms are randomly distributed on the 8i and 8f sublattices. m^tot. = 21.5431, 21.5747, 20.5341, and 14.0884 μ_B/f.u, respectively.

Header	Sm1(2a)/Zr1(2a)	Fe1(8f)/Co1(8f)	Fe2(8i)/Co2(8i)	Ni3(8j)
(Sm0.8Zr0.2)Ni4Fe8
m(s) (μB)	+3.6760/+0.4193	−2.3803/-	−2.3076/-	−0.6086
m(o) (μB)	−3.2160/+0.0522	−0.0670/-	−0.0789/-	−0.0591
(Sm0.8Zr0.2)Ni4(Fe0.9Co0.1)8
m(s) (μB)	+3.6430/+0.4290	−2.4416/−1.5617	−2.3364/−1.4195	−0.6089
m(o) (μB)	−3.2713/+0.0411	−0.06558/−0.0986	−0.0740/−0.0876	−0.0572
(Sm0.8Zr0.2)Ni4(Fe0.8Co0.2)8
m(s) (μB)	+3.7200/+0.4205	−2.24451/−1.5511	−2.3554/−1.4329	−0.6047
m(o) (μB)	−3.2913/+0.0364	−0.0657/−0.0970	−0.0714/−0.0870	−0.0537
(Sm0.8Zr0.2)Ni4Co8
m(s) (μB)	+4.2200/+0.3425	-/−1.5670	-/−1.5001	−0.5530
m(o) (μB)	−2.8153/+0.0186	-/−0.0910	-/−0.0734	−0.0366

.

The FPLMTO (SRM + OP) calculated magnetic properties of the (Sm_0.75Zr_0.25)Ni_4-(Fe_1−xCo_x)₈ alloys are also presented in

Table 3. Atomic volume (Ω₀), density (ρ), total moment (m^tot.), saturation magnetization (M_s and μ₀M_s), and maximum energy product (|BH|_max) of the (Sm_1−yZr_y)Ni₄(Fe_1−xCo_x)₈ magnets as calculated by the FREMTO-CPA method and using the FPLMTO (SRM + OP) method.

Material	Theory	Ω0 (Å3)	ρ$(\frac{\mathbf{g}}{\mathbf{c}{\mathbf{m}}^3}$)	${\mathit{m}}^{\mathit{tot}.} \left(\frac{{\mathit{\mu }}_{\mathit{B}}}{\mathit{f}.\mathit{u}.}\right)$	${\mathit{M}}_{\mathit{s}}\left(\frac{\mathbf{MA}}{\mathbf{m}}\right)$	${\mathit{\mu }}_0{\mathit{M}}_{\mathit{s}} \left(\mathbf{T}\right)$	${\left|\mathit{B}\mathit{H}\right|}_{\mathit{m}\mathit{a}\mathit{x}}\left(\frac{\mathbf{kJ}}{{\mathbf{m}}^3}\right)$
(Sm0.8Zr0.2)Ni4Fe8	FREMTO-CPA	13.50	7.760	21.543	1.139	1.431	407.207
(Sm0.75Zr0.25)Ni4Fe8	SRM + OP	12.81	8.148	21.190	1.180	1.483	437.435
(Sm0.8Zr0.2)Ni4(Fe0.9Co0.1)8	FREMTO-CPA	13.49	7.783	21.575	1.140	1.433	408.401
(Sm0.75Zr0.25)Ni4(Fe0.9Co0.1)8	SRM + OP	12.81	8.196	20.800	1.157	1.454	420.549
(Sm0.8Zr0.2)Ni4(Fe0.8Co0.2)8	FREMTO-CPA	13.36	7.888	20.534	1.097	1.378	377.691
(Sm0.75Zr0.25)Ni4(Fe0.8Co0.2)8	SRM + OP	12.81	8.227	20.410	1137	1.429	406.135
(Sm0.8Zr0.2)Ni4Co8	FREMTO-CPA	12.59	8.573	14.088	0.799	1.003	200.268
(Sm0.75Zr0.25)Ni4Co8	SRM + OP	12.75	8.463	15.210	0.851	1.069	227.515

. Abbreviation SRM + OP stays for the standard rare-earth model + orbital polarization (see Ref. [38] for details).

The compositional dependence of μ₀M_s and |BH|_max of the (Sm_0.8Zr_0.2)Ni₄(Fe_1−xCo_x)₈ alloys for x = 0.0, 0.1, 0.2., and 1.0 is presented in Figure 4 and Figure 5, respectively. μ₀M_s first increases and approaches the maximum at x = 0.1. A further Co concentration increase prompts an acute decrease in μ₀M_s. The calculated tendency for |BH|_max as a function of the Co replacement is analogous to μ₀M_s.

With regards to the Curie temperature (T_c), a mean-field approximation (MFA) can be expressed as [41,42]:

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

(1)

where $E_{tot}^{DLM} \mathrm{and} E_{tot}^{FiM}$ are the ground-state total energies of the DLM (disordered local moment, see Ref. [38] for details) and FiM (ferrimagnetic) states, respectively, and k_B is the Boltzmann constant. Therefore, an evaluation of T_c can be based on the total energy difference between the ferrimagnetic and paramagnetic (DLM) states. Nonetheless, in line with [42], the diversity between the total energies can be substituted by the diversity between the effective single-particle (one atomic specie) energies, which are directly associated with DLM and FiM states (the so-called MFA treatment). In this paper, $E_{tot}^{DLM} \mathrm{and} E_{tot}^{FiM}$ are calculated at the equilibrium volumes for DLM and FiM states, correspondingly.

Figure 6 shows the calculated (within the FREMTO-CPA formalism) T_c values of the pseudo-binary (Sm_0.8Zr_0.2)Ni₄(Fe_1−xCo_x)₈ alloys, where 4 Ni atoms occupy the 8j sublattice and 8 (Fe_1−xCo_x) atoms are distributed on the 8i and 8f sublattices. The calculated T_c values are equal to 730 K, 792 K, 828 K, and 955 K for the (Sm_0.8Zr_0.2)Ni₄Fe₈, (Sm_0.8Zr_0.2)Ni₄(Fe_0.9Co_0.1)₈, (Sm_0.8Zr_0.2)Ni₄(Fe_0.8Co_0.2)₈, and (Sm_0.8Zr_0.2)Ni₄Co₈ alloys, respectively. These values are significantly higher than the Curie temperature of the Neomax (Nd₂Fe₁₄B₁) magnet (588 K) [16].

Calculated Curie temperature of the suggested (Sm_0.8Zr_0.2)Ni₄(Fe_0.8Co_0.2)₈ magnet, T_c = 828 K, which is of the same magnitude as the Curie temperature of the zirconium-doped (Sm_0.77Zr_0.24)(Fe_0.80Co_0.19)_11.5Ti_0.65 magnet, T_c = 830 K Tozman et al. [12], (Sm_0.92Zr_0.08)-(Fe_0.75Co_0.25)_11.35Ti_0.65 magnet, T_c = 843 K Gabay et al. [3], and (Sm_0.8Zr_0.2)(Fe_0.75Co_0.25)_11.5Ti_0.5 magnet, T_c = 880 K Kuno et al. [13], or yttrium-doped (Sm_0.8Y_0.2)(Fe_0.8Co_0.2)_11.5Ti_0.5 magnet, T_c = 820 K Hagiwara et al. [14]. All these synthesized magnets contain titanium, which decreases saturation magnetization. The suggested (Sm_0.8Zr_0.2)Ni₄(Fe_0.8Co_0.2)₈ magnets do not have this deficiency due to a lack of titanium and a credible alignment of nickel magnetic moments.

The calculated (FREMTO-CPA) site-projected spin (m^(s)) and orbital moments (m^(o)), as well as the total moments (m^total) of the (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ magnet where Ce and Sm atoms equally occupy 2a Wyckoff positions, 2 Fe and 2 Co atoms occupy the 8f Wyckoff positions, 4 Fe atoms occupy the 8i Wyckoff positions, and 4 Ni atoms occupy the 8j Wyckoff positions, are presented in

Table 4. Site-projected spin (m^(s)) and orbital (m^(o)) magnetic moments for the (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ magnet, where Ce and Sm atoms equally occupy 2a Wyckoff positions, 2 Fe and 2 Co atoms occupy the 8f Wyckoff positions, 4 Fe atoms occupy the 8i Wyckoff positions, and 4 Ni atoms occupy the 8j Wyckoff positions. M^tot. = 20.8942 μ_B/f.u.

(Ce0.5Sm0.5)Ni4Fe6Co2	Ce1(2a)	Sm1(2a)	Fe1(8f)	Co1(8f)	Fe2(8i)	Ni3(8j)
m(s) (μB)	+0.1564	+2.2586	−2.3247	−1.5709	−2.3645	−0.6108
m(o) (μB)	+0.0159	−3.1577	−0.0658	−0.0930	−0.0699	−0. 0608

.

According to our FREMTO-CPA calculations, the total magnetic moment of the (Ce_0.5Sm_0.5)Ni_4-Fe₆Co₂ compound is equal to m^total ≈ 20.8910 μ_B/f.u. at an equilibrium atomic volume Ω₀ ≈ 13.4473 ų. Taking into consideration the calculated total moment per atom, $m_{at.}^{tot.}$= 1.6073 μ_B, and the calculated density of the (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ compound, ρ ≈ 7.9120 g/cm³, one can evaluate M_s ≈ 1.1085 MA/m, μ₀M_s = 1.3929 T, and |BH|_max = 385.992 kJ/m³. The FREMTO-CPA and FPLMTO (SRM + OP) calculated magnetic properties of the (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ compound are shown in

Table 5. Atomic volume (Ω₀), density (ρ), total moment (m^total), saturation magnetization (M_s and μ₀M_s), and maximum energy product (|BH|_max) of the (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ magnet as calculated by the FREMTO-CPA method and using the FPLMTO method and SRM + OP.

Material	Theory	Ω0 (Å3)	ρ$(\frac{\mathbf{g}}{\mathbf{c}{\mathbf{m}}^3}$)	${\mathit{m}}^{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}\left(\frac{{\mathit{\mu }}_{\mathit{B}}}{\mathit{f}.\mathit{u}.}\right)$	${\mathit{M}}_{\mathit{s}}\left(\frac{\mathbf{M}\mathbf{A}}{\mathbf{m}}\right)$	${\mathit{\mu }}_0{\mathit{M}}_{\mathit{s}}\left(\mathbf{T}\right)$	${\left|\mathit{B}\mathit{H}\right|}_{\mathit{m}\mathit{a}\mathit{x}}\left(\frac{\mathbf{kJ}}{{\mathbf{m}}^3}\right)$
(Ce0.5Sm0.5)Ni4Fe6Co2	FREMTO-CPA	13.45	7.912	20.894	1.109	1.393	385.992
	SRM + OP	13.71	7.756	20.020	1.042	1.311	341.757

.

The mentioned saturation magnetization, M_s = $m_{at.}^{tot.} \left[{\mu }_B\right]\frac{\rho N_A }{M}=$ $m_{at.}^{tot.}\left[{\mu }_B\right]\frac{N_A }{V}$ = $\frac{m_{at.}^{tot.}}{{\Omega }_o}\left[{\mu }_B\right]$, where V = Ω₀N_A is the molar volume, represents the saturation magnetization calculated per atomic volume Ω₀. The expression M_s = $m_{at.}^{tot.} \left[{\mu }_B\right]\frac{N_A }{M}$ represents the saturation magnetization per unit mass. According to the present FREMTO-CPA calculations, the saturation magnetization per unit mass for the (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ magnet, M_s ≈ $\approx$140.096 $\frac{{\mathrm{Am}}^2}{\mathrm{kg}}$. According to Refs. [32,33], the saturation magnetization of the (Ce_0.5Sm_0.5)Ti₁Fe₉Co₂ magnet is equal to M_s = 117 $\frac{{\mathrm{Am}}^2}{\mathrm{kg}}$. Thus, the saturation magnetization and maximum energy product of the suggested (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ magnet are 1.197 and 1.434 times larger than the saturation magnetization and maximum energy product of the (Ce_0.5Sm_0.5)Ti₁Fe₉Co₂ magnet, respectively [30,32,33].

Calculated in a mean-field approximation (FREMTO-CPA), the Curie temperature of the (Ce_0.5Sm_0.5)Ni_4-Fe₆Co₂ magnet is T_c = 731.89 K.

One crucial quantity for an efficient and realistic magnet is how robust the direction of its magnetic field is. This property is measured or calculated in terms of magnetic anisotropy energy, i.e., the energy difference between the easy (higher energy) and hard (lower energy) directions. Naturally, these energies are very small relative to the total electronic energy of the compound, and to resolve the difference, the energies must be converged to at least 12 digits.

Here, we apply the SRM + OP model for the MAE because the other approximations (4f-band) produce an unreal large MAE due to the improper handling of the 4f electrons. Because the MAE is sensitive to the details of the crystal structure, we optimize the parameters, including the atomic volume, to produce the lowest total energy (structural relaxation). Namely, both the lattice parameters a and c are optimized to give the lowest total energy of the tetragonal crystal. These parameters are presented in

Table 6. Calculated crystal-structure parameters. Lattice constant a is in units of Å. x₁ and x₂ are atomic position parameters for the 3d-metal components.

Compound	a	$\mathit{c}/\mathit{a}$	x1	x2
(Sm0.75Zr0.25)Ni4Fe8	8.4249	0.557	0.359	0.277
(Sm0.75Zr0.25)Ni4(Fe0.9Co0.1)8	8.4249	0.557	0.359	0.277
(Sm0.75Zr0.25)Ni4(Fe0.8Co0.2)8	8.4249	0.557	0.359	0.277
(Sm0.75Zr0.25)Ni4Co8	8.3107	0.560	0.358	0.277
(Ce0.5Sm0.5)Ni4Fe6Co2	8.6443	0.552	0.361	0.277

. The differences between a and c for the iron-rich compound (Sm_0.75Zr_0.25)Ni₄(Fe_1−xCo_x)₈ (here x = 0, 0.1, and 0.2) are indeed very small, while for x = 1, the total atomic volume diminishes by about 4%.

There have been numerous attempts to calculate MAE accurately in rare-earth TM (TM-transition metal) systems. We found an efficient yet accurate procedure to do this in our previous investigation of the SmCo₅-type permanent magnets [35]. Specifically, treat the correlated 4f electrons on the rare-earth atom within the standard rare-earth model while including orbital polarization on the d electrons for the other atoms to ensure better orbital magnetic moments. The entire procedure is free of any parameters.

Calculating (FPLMTO) the total energy for the [001] and [100] spin directions of the 52-atom cell, we obtain the MAE that is listed in

Table 7. Calculated (FPLMTO: assuming the SRM + OP model) atomic volume, magnetic anisotropy energy, first anisotropy constant, anisotropy field, and magnetic hardness parameter for the (Sm_0.75Zr_0.25)Ni₄(Fe_1−xCo_x)₈ alloys, where x = 0, 0.1, 0.2, and 1, and (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ compound.

Compound	Ω0 (Å3)	$\mathit{M}\mathit{A}\mathit{E} \left(\frac{\mathbf{meV}}{\mathit{f}.\mathit{u}.}\right)$	${\mathit{K}}_1\left(\frac{\mathbf{MJ}}{{\mathbf{m}}^3}\right)$	μ0Ha (T)	κ
(Sm0.75Zr0.25)Ni4Fe8	12.81	0.592	0.570	0.966	0.571
(Sm0.75Zr0.25)Ni4(Fe0.9Co0.1)8	12.81	1.520	1.464	2.531	0.933
(Sm0.75Zr0.25)Ni4(Fe0.8Co0.2)8	12.81	2.529	2.435	4.283	1.214
(Sm0.75Zr0.25)Ni4Co8	12.75	2.816	2.724	6.402	1.730
(Sm0.5Ce0.5)Ni4Fe6Co2	13.71	6.695	6.027	11.568	2.101

. We also present the calculated anisotropy field, μ₀H_a = 2 K₁/M_s [43] and magnetic hardness parameter of the materials, $\kappa =\sqrt{\left(\frac{K_1}{{\mu }_0{M}_s^2}\right)}=\sqrt{\frac{{\mu }_0{H}_a}{2{\mu }_0{M}_s}}$ [43]. As is expected, the MAE increases with increasing Co content, and the reason is simply that Co has a larger orbital moment than Fe.

According to [43,44], the interchange between magnetic anisotropy and saturation magnetization defines resistance to the self-demagnetization of a magnet fabricated in any possible shape. The empirical rule requires κ ≥ 1 for good permanent magnet fabrication. As can be seen from Table 7, the (Sm_0.75Zr_0.25)Ni₄(Fe_0.8Co_0.2)₈, (Sm_0.75Zr_0.25)Ni₄Co₈, and (Sm_0.5Ce_0.5)Ni₄Fe₆Co₂ magnets meet the manufacturability standards (i.e., κ ≥ 1).

In

Table 8. Calculated (FPLMTO) saturated magnetization, anisotropy field, Curie temperature, maximum energy product, and magnetic hardness parameter for the (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ compound compared to the experimental data for the (Ce_0.5Sm_0.5)Fe₉Co₂Ti melt-spun magnetic ribbon [32,33].

Material	μ0Ms (T)	μ0Ha (T)	Tc (K)	${\left|\mathit{B}\mathit{H}\right|}_{\mathit{m}\mathit{a}\mathit{x}}$(kJ/m3)	κ
(Sm0.5Ce0.5)Ni4Fe6Co2	1.311	11.57	731.89	341.575	2.101
(Ce0.5Sm0.5)Fe9Co2Ti	1.150	5.60	726.00	261.380	1.560

, we compare the results of our calculations for the (Sm_0.5Ce_0.5)Ni₄Fe₆Co₂ magnet with the experimental data for the melt-spun magnetic ribbons (Ce_0.5Sm_0.5)Fe₉Co₂Ti, [32,33]. Both magnets have almost identical Curie temperatures; however, the suggested (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ magnet has a much higher anisotropy field and a higher maximum energy product. In both cases, k > 1, which satisfies an empirically required rule to manufacture a strong permanent magnet.

4. Discussion and Conclusions

The material of interest should satisfy the following magnetic property conditions to be considered a viable hard permanent magnet: μ₀M_s ~ ≥1.25 T, T_c ~ ≥550 K, μ₀H_a ~ ≥3.75 T, and κ > 1 [1,43]. The present calculations show that the (Sm_0.8Zr_0.2)Ni₄(Fe_1−xCo_x)₈ alloys are stable across the entire compositional range and have large μ₀M_s values between 1.43–1.48 T (depending on the model); T_c values of 730 K, 792 K, 828 K, and 955 K; μ₀H_a values of 0.966 T, 2.531 T, 4.283 T, and 6.421T; and κ values of 0.571, 0.933, 1.214, 1.730, for x = 0.0, 0.1, 0.2, and 1.0, respectively. For the (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ magnet, the present calculations indicate that it could be fabricated in bulk based on its thermodynamic stability and exhibits values for μ₀M_s = 1.31–1.39 T (depending on the model); and has an excellent value for μ₀H_a equal to 11.568 T; a high T_c value of 731.89 K; and κ = 2.101 > 1.

Table 9. Saturation magnetization, anisotropy field, Curie temperature, and maximum energy product values of 1:12 magnets and Nd₂Fe₁₄B₁.

Material	${\mathit{\mu }}_0{\mathit{M}}_{\mathit{s}} \left(\mathbf{T}\right)$	${\mathit{\mu }}_0{\mathit{H}}_{\mathit{a}} \left(\mathbf{T}\right)$	${\mathit{T}}_{\mathit{c} }\left(\mathbf{K}\right)$	${\left|\mathit{B}\mathit{H}\right|}_{\mathit{m}\mathit{a}\mathit{x}} \left(\frac{\mathbf{kJ}}{{\mathbf{m}}^3}\right)$	References
Nd2Fe14B1	1.61	7.6	588	515	[4,16,43]
(Sm0.8Zr0.2)(Fe0.75Co0.25)11.5Ti0.5	1.58	7.41	880	495	[13]
(Sm0.8Y0.2)(Fe0.80Co0.20)11.5Ti0.5	1.50	11.0	820	447	[14,15]
Sm0.94(Fe0.81Co0.19)11Ti1.08	1.43	10.9	800	406	[12]
(Sm0.77Zr0.24)(Fe0.80Co0.19)11.5Ti0.65	1.53	8.4	830	465	[12]
(Sm0.92Zr0.08)(Fe0.75Co0.25)11.35Ti0.65	1.47	>9.0	843	429	[3]
(Ce0.5Sm0.5)Ti1Co2Fe9	1.15	5.60	726	261	[32]
(Sm0.75Zr0.25)Ni4(Fe0.8Co0.2)8	1.43	4.28	828	406	Present
(Ce0.5Sm0.5)Ni4Fe6Co2	1.31	11.57	731	342	Present

summarizes the intrinsic properties of some reported [3,12,13,14,15,32] ThMn₁₂-type structure magnets that are compared to the characteristics for Neomax [4,16,43] along with our present results of calculations for the (Sm_0.8Zr_0.2)Ni₄(Fe_0.8Co_0.2)₈ and (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ magnets. The maximum energy product of the (Sm_0.8Zr_0.2)Ni₄(Fe_0.8Co_0.2)₈ and (Ce_0.5Sm_0.5)Ni_4-Fe₆Co₂ magnets is ~79% and 66% of the maximum energy product of Neomax, respectively; the anisotropy field of the (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ magnet is the largest among the listed magnets with ThMn₁₂-type structure, and the Curie temperature exceeds that of Neomax by 340 K and 143 K for the (Sm_0.8Zr_0.2)Ni₄(Fe_0.8Co_0.2)₈ and (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ magnets, respectively. Comparing the maximum energy products of our earlier suggested magnets with ThMn₁₂-type structure [34,35,36,37,38], the novel (Sm_0.8Zr_0.2)Ni₄(Fe_0.8Co_0.2)₈ magnet (|BH|_max = 406 kJ/m³) outperforms the SmNi₄(Fe_0.9Co_0.1)₈, SmCoNiFe₃, and YFe₃(Ni_0.3Co_0.7)₂ magnets with |BH|_max = 382 kJ/m³, 361 kJ/m³, and 351 kJ/m³, respectively.

According to our calculations, the maximum energy product of the (Sm_0.8Zr_0.2)Ni₄Fe₈ magnet, |BH|_max = 407.21 kJ/m³, (see Table 3) is larger than the undoped SmNi₄Fe₈ magnet, which has a |BH|_max = 377.6 kJ/m³ [38]. As was discovered by Tozman et al. [17], the addition of Zr to the SmFe₁₂ magnet not only stabilized it in the ThMn₁₂-type structure, but in addition, it increased the saturation magnetization of the Sm(Fe_0.8Co_0.2)₁₂ magnet in the thin film form. A similar phenomenon occurs in the case of the SmNi₄Fe₈ compound. It is well known that the spins of the samarium and TM (iron, cobalt, and nickel) atoms adjust in an antiparallel direction; however, the total spin moment of the SmFe₁₂ compound aligns parallel to the spin moments of TM. Zr substitution for Sm decreases the magnetic moment on the 2a site and, thus, increases the total magnetic moment of the (Sm_1−xZr_x)Fe₁₂ magnets. The reasons are straightforward, since the 4f electrons on samarium spin polarize and thus produce a spin moment. Replacing a fraction of Sm with Zr, that has no occupied 4f levels thus reduces the amount of 4f electrons and the spin moment. A charge transfer of d electrons from Zr to Fe increases the spin moment on iron. Therefore, if one can replace Sm with Zr, it reduces or eliminates the need for the expensive rare-earth metal and also improves the total magnetic moment of the compound. The same arguments are true for the (Sm_1−xZr_x)Ni₄Fe₈ magnets studied in the present work.

Recently, Kobayashi et al. [45] performed X-ray absorption fine structure (XAFS) and scanning transmission electron microscopy (STEM) to understand the magnetization increase associated with Zr replacement in the (Sm_1−xZr_x)(Fe_0.8Co_0.2)₁₂ alloys. They confirmed that the magnetic moment of Sm was two orders of magnitude smaller than the magnetic moment of Fe and Co, and thus the contribution to the total magnetization from 2a sites, occupied by Sm and Zr, is negligible. In addition, they found that the charge transfer from Zr (2a sites) atoms to the Fe (8f sites) atoms drives a magnetization increase in the (Sm_1−xZr_x)(Fe_0.8Co_0.2)₁₂ single crystalline films. By performing ab initio calculations, Matsumoto et al. [46] found that Zr(2a)-induced an increase in magnetization due to rearrangement (charge transfer) of the 4d-electon states from Zr to Fe sites, reducing the overlap of the majority-spin states on the Fermi level for Fe (8f). This is identical to the Slater-Pauling curve in the Fe_1−xCo_x alloys, where cobalt sums up one electron on the top of the 3d-electron band of iron; in the case of the (Sm,Zr)Fe₁₂ compound, the delocalized 4d-electrons of zirconium sum up one electron on the top of the 3d-electron band of iron.

In the case when Ce substitutes for Sm, the calculated maximum energy product also increases from |BH|_max = 377.6 kJ/m³ for the SmNi₄Fe₈ magnet [38] to 386.0 kJ/m³ for the (Ce_0.5Sm_0.5)Ni₄Fe₆Co₂ magnet. In this case, the increase in saturation magnetization is due to the decrease in the magnetic moment of the 2a sites that are equally occupied by Ce and Sm atoms. Analogous to the discussion above, cerium substitution for samarium decreases the number of spin-polarized 4f electrons because cerium has only about 1 4f electron while samarium has about five. The result is a smaller spin moment on the (2a) site and a larger total magnetic moment for the compound.

Numerous theoretical studies have been performed to understand the stability of the REFe₁₂ compound, which decomposes into the RE₂Fe₁₇ compound and α-Fe [27,46,47,48,49,50,51,52,53]. According to Fukazawa et al. [53], the stability of (RE_1−yZr_y)(Fe_1−xCo_x)₁₂ alloys exhibits general tendencies that are similar for RE = Y, Nd, and Sm: the stability of the 1:12 phase relative to the 2:17 (Th₂Zn₁₇-type structure) phase increases as Zr concentration increases and Co concentration decreases.

As was shown in Ref. [38], the SmNi₄(Fe_1−xCo_x)₈ alloys have a negative formation energy relative to the unary systems α-Sm, α-Fe, α-Co, and α-Ni, the whole composition range, with a pronounced minimum of −2.85 mRy/atom at x ≈ 0.4. According to the present calculations (Figure 2), the heat of formation of the (Sm_0.8Zr_0.2)Ni₄(Fe_1−xCo_x)₈ alloys, relative to the unary systems α-Sm, α-Zr, α-Fe, α-Co, and α-Ni, is also negative, reaching a minimum value of −3.26 mRy/atom at x ≈ 0.45. Both Ni and Zr play the role of stabilizers for the (Sm_0.8Zr_0.2)Ni₄(Fe_1−xCo_x)₈ alloys, and a decrease in the maximum value of the formation energy from −2.85 mRy/atom (SmNi₄(Fe_0.6Co_0.4)₈) to −3.26 mRy/atom ((Sm_0.8Zr_0.2)Ni₄ (Fe_0.55Co_0.45)₈) implies the (Sm_0.8Zr_0.2)Ni₄(Fe_1−xCo_x)₈ alloys will be stable against decomposition to the Th₂Zn₁₇-type structure compound and the respective TMs.

The (Ce_0.5Sm_0.5)Ni₄Co₂Fe₆ alloy is fairly stable with respect to the elements α-Sm, α-Ce, α-Fe, α-Co, and α-Ni and exhibits a significantly negative calculated formation energy of −5.735 mRy/atom. The (Ce_0.5Sm_0.5)Ni₄Co₂Fe₆ magnet is also predicted to be stable against decomposition into the Th₂Zn₁₇-type structure compound and its respective TMs.

In conclusion, we showed that the (Sm_0.8Zr_0.2)(Fe_1−xCo_x)₁₂ alloys could be stabilized by substituting a significant quantity of Fe or Co atoms with Ni atoms. These modern permanent magnets are predicted to have excellent magnetic characteristics, specifically a significant gain energy product ((Sm_0.8Zr_0.2)Ni₄(Fe_0.9Co_0.1)₈) and anisotropy field ((Sm_0.5Ce_0.5)Ni₄Fe₆Co₂), as well as a high Curie temperature that substantially surpasses Neomax magnets. However, it is imperative to mention that, though these intrinsic properties are necessary for permanent magnets, this alone is not a sufficient requirement. It is also necessary for the anisotropic microstructure of a magnet to exhibit both significant values for coercivity and remanence [6,7,8,25,34,45,53,54,55,56,57,58,59,60,61,62,63,64,65]. It is therefore necessary to establish a pertinent grain boundary phase that can grow in equilibrium with the matrix phase in order to improve the extrinsic magnetic properties.