Enhanced Phase Stability of Sm₂(Fe, Al)₁₇C_x

Abstract

Aluminum doping can improve the phase stability of metastable compound Sm₂Fe₁₇C_x with a high carbon content (x > 1.5). We investigated the preferential site substitution of Al, chemical bonding, and structural stability in Sm₂(Fe,Al)₁₇C₃ using first-principle calculations. Our results reveal a strong correlation between the preferential substitution of Fe by Al and the atomic site chemical environment, which affects the overall phase stability. Specifically, Al preferentially occupies the 9d site in Sm₂(Fe,Al)₁₇C₃. At the same time, Al prefers the site 6c in its parent phase Sm₂(Fe,Al)₁₇. Partial replacement of Fe with Al leads to a more negative formation energy, indicating enhanced thermodynamic stability. Crystal Orbital Hamilton Population (COHP) and Crystal Orbital Bond Index (COBI) analysis suggest that insertion of carbon weakens the bonding strength of Sm-Fe (18f) and Sm-Fe (18h), resulting in metastability of Sm₂Fe₁₇C_x. Doping Al strengthens Al-Fe, Al-Sm, Sm-Fe (18f, 18h) and Fe–C bonding in Sm₂(Fe,Al)₁₇C₃, as revealed by calculated COHP and COBI. These effects contribute to improved phase stability in the Al-doped 2:17 interstitial compound.

1. Introduction

High-performance rare-earth (RE) permanent magnets, particularly Nd-Fe-B, are widely applied in electric vehicles (EVs), robotics, consumer electronics, etc. [1,2]. The growth in the demand for the Nd-Fe-B magnet has resulted in a supply bottleneck for critical RE metals such as Pr, Nd, Dy, and Tb. The development of RE-Fe-based permanent magnets with more earth-abundant RE such as La, Ce, and Sm has gained renewed interest. The promising candidate compositions include Ce₂Fe₁₄B, Sm₂Fe₁₇N_x, Sm(Fe,Ti)₁₂, etc. [3,4,5].

The interstitial compound Sm₂Fe₁₇N₃ has a rhombohedral Th₂Zn₁₇-type structure (2:17) and displays excellent intrinsic magnetic properties such as a large saturation magnetization of Js = 15.4 kG, a strong magnetocrystalline anisotropy field of H_K = 140 kOe, and a theoretical maximum energy product (BH)_max of 59.7 MGOe at room temperature [6]. However, the synthesis of Sm₂Fe₁₇N₃ is achieved through a gas–solid reaction of Sm₂Fe₁₇ powder with N₂ or NH₃, typically in the temperature range of 450–500 °C. Sm₂Fe₁₇N₃ is metastable and decomposes into SmN, α-Fe and N₂ at temperatures as low as <650 °C [6,7], which prevents densification into bulk magnets via high-temperature sintering and hot pressing. Hence, high-performance Sm₂Fe₁₇N₃ magnetic powder is typically used as feedstock powder for bonded magnets [8,9]. However, there are intensive research efforts targeting full densification of Sm-Fe-N powder into bulk magnets [10,11,12,13,14].

The isostructural Sm₂Fe₁₇C_2.5 displays excellent intrinsic magnetic properties of a J_S of 13.6 KG, a H_K of 150 kOe, and a theoretical (BH)_m of 46 MGOe [15,16,17]. One advantage of Sm₂Fe₁₇C_x is that it can be prepared using an arc- or induction-melting process, which provides an opportunity for making bulk magnets using high-temperature processes such as sintering and/or hot-pressing. However, the arc-melted Sm₂Fe₁₇C_x alloy with a 2:17 single phase has a carbon content, x, limited to 1.0 [17]. It was discovered that partial replacement of Fe by Ga, Si, or Al can stabilize the 2:17 phase with carbon content, x, up to 3.0 [18,19,20,21]. The arc-melted Sm₂Fe₁₅Al₂C_1.5 alloys with a 2:17-type single-phase structure have typical magnetic properties of J_s = 110 emu/g, T_c = 576 K, H_A = 110 kOe [20]. Melt-spun ribbon of Sm₂Fe₁₅Al₂C_1.5 shows a coercivity of 9.4 kOe [22]. The partial substitution of Fe by Co also enhances the formation of the 2:17 phase in arc-melted Sm₂(Fe,Co)₁₇C_x alloy [23]. The rational design of the chemical composition of Sm₂(Fe, M)₁₇C_x (M = Al, Si, Ga, Co, etc.) can enhance phase stability and enable the production of sintered or hot-press magnets.

The structure has been experimentally and theoretically investigated to understand the role of doped elements, such as Al, Si, Ga, and Co, in the 2:17 phase. Neutron diffraction measurements show that Al, Si, Co, and Ga tend to substitute Fe with a strong preference to 18h sites in Nd₂Fe_17−xM_x, isostructural to Sm₂Fe_17−xM_x [24]. With increasing M content, some alloying elements also occupy other sites, such as 18f and 9d. X-ray diffraction also indicates that Al, Si, and Ga substitute for Fe with a strong preference to 18h sites in Sm₂Fe_17−xM_x compounds with M = Al, Si and Ga [21]. The preferential Fe substitution has been qualitatively explained using the Miedema model [24,25]. Despite extensive research on Sm₂(Fe,M)₁₇C_x interstitial compounds, a systematic investigation into the preferential site occupancy of dopant elements, chemical bonding, and their effect on phase stability remains lacking. The electron structure of Sm₂Fe₁₇ and Sm₂Fe₁₇C_x has been reported with a focus on magnetic properties [26,27,28]. Understanding the stabilization mechanisms of the 2:17 structure in Sm₂(Fe,M)₁₇C_x is crucial for accelerating the development of high-performance Sm₂(Fe,M)₁₇C_x permanent magnets.

Phase stability can generally be evaluated by calculating formation energy and cohesive energy, as well as chemical bonding analysis. The formation energy and cohesive energy can be routinely calculated using first-principle total energy calculations, such as the density functional theory (DFT) approach. To evaluate the chemical bonding in crystal solids, the Crystal Orbital Hamilton Population (COHP) scheme is an efficient and reliable tool for extracting chemical-bonding information based on electronic-structure calculations [29,30,31]. COHP partitions the band-structure energy into orbital-pair interactions. It is a “bond-weighted” density of states (DOS) between a pair of adjacent atoms. A COHP plot indicates bonding and anti-bonding contributions to the band-structure energy. The energy integral of the COHP (ICOHP) shows the contribution of a specific “chemical bond” to the band energy, or the bond strength. The COHP has been implemented in several DFT-related software such as Stuttgart TB-LMTO-ASA, SIESTA, CRYSTAL, and LOBSTER [30,32,33,34]. Another intuitive method for quantifying covalent bonding in solids is the crystal orbital bond index (COBI) [35]. COBI’s qualitative interpretation resembles COHP, but it is directly related to the classic chemical bond order. The energy-integrated value of COBI (ICOBI) reflects the bonding characteristics. The large (>0.9) and small (<0.1) values correspond to a covalent and ionic bonding, respectively. The moderate value of ICOBI means a mixing bonding state.

In this study, we employed first-principle electronic structure calculations and chemical bonding analysis to examine the phase stability of Sm₂(Fe,Al)₁₇C_x. To understand the role of interstitial atoms, carbon, in phase stability, we have also investigated the Fe substitution by Al in its parent phase Sm₂(Fe,Al)₁₇. We have discussed the effect of Al doping on interatomic chemical bonding, such as Al-Fe, Al-Sm, Sm-Fe, Sm-C and Fe-C. Finally, we examined the interrelation among Al site preferential substitution, formation energy, chemical bonding, and the overall phase stability of Sm₂(Fe,Al)₁₇C_x.

2. Results and Discussion

2.1. Formation Energy and Site Preference of Al in Sm₂(Fe,Al)₁₇ and Sm₂(Fe,Al)₁₇C₃

The Th₂Zn₁₇-type structure of Sm₂Fe₁₇ can be derived from the CaCu₅-type structure, where Fe–Fe atom pairs systematically replace one-third of the Sm atoms [16]. These Fe–Fe pairs occupy the 6c crystallographic sites, are aligned along the hexagonal c-axis, and exhibit short interatomic distances. The resulting structure is densely packed and layered, consisting of alternating Sm–Fe and Fe–Fe layers. Within the unit cell, Sm atoms occupy the 6c sites, while Fe atoms are distributed across four distinct sites: 6c (Fe₁), 9d (Fe₂), 18f (Fe₃), and 18h (Fe₄). In the interstitial compound Sm₂Fe₁₇C₃ (Figure 1), carbon atoms reside at the 9e sites. Each Sm atom is coordinated by three nearest-neighbor carbon atoms. This local environment significantly alters the effective crystal field at the Sm site, inducing a strong easy-axis magnetocrystalline anisotropy (MCA) in Sm₂Fe₁₇C₃ [16].

To evaluate the effect of the Fe substitution by Al in Sm₂(Fe,Al)₁₇C₃ and Sm₂(Fe,Al)₁₇, we take the crystallographic data of Sm₂Fe₁₇ and Sm₂Fe₁₇C₃ as a starting point to construct the supercell of Sm₂(Fe, Al)₁₇ and Sm₂(Fe, Al)₁₇C₃ [36,37]. One Al atom is doped at a specific Fe site in the 2:17 unit cell. It has a nominal composition of Sm₂Fe₁₆Al₁C₃ (Sm₂Fe₁₇Al) or 5.8% of Fe replaced by Al in the supercell. Since 2:17 has four crystallographically inequivalent Fe sites in the unit cell, there are four geometrically different patterns for doping Al into the unit cell. In the DFT calculation, the structures are fully relaxed. As shown in

Table 1. Formation energy (E_f) of Sm₂Fe₁₆Al and Sm₂Fe₁₆AlC₃ with Al at different crystallographic sites. The unit of E_f is electron-volt per formula unit (eV/f.u.).

	Al@6c	Al@9d	Al@18f	Al@18h
Sm2Fe16Al	−0.0996	−0.0752	−0.0903	−0.0961
Sm2Fe16AlC3	−0.1904	−0.1966	−0.1632	−0.1736

, the Al at the sites 9d and 6c has a lower formation energy (E_f) in Sm₂(Fe,Al)₁₇C₃, which implies that the Al atoms prefer the sites 9d and 6c. In contrast, the value of E_f is more negative for Al at the sites of 6c and 18h in Sm₂(Fe, Al)₁₇ (Table 1). The results reveal that the carbon interstitial atoms modified the chemical environment of the Fe atoms at different sites and affected the occupancy preference of Al.

The derived substitution energies (E_sub) based on Formula (2) are also shown in Figure 2. All the values of E_sub are negative, indicating that doping a small amount of Al lowers the formation energy and enhances the 2:17 structure stability. The trend agrees with the formation energy calculations, i.e., the Al atoms at the 9d- and 6c sites have the most negative E_sub values in Sm₂Fe₁₆AlC₃ and Sm₂Fe₁₆Al, respectively.

The site occupancy preferences of Al also depend on temperature. Figure 3 displays the temperature dependence of the site occupancy of Al in Sm₂(Fe, Al)₁₇ and Sm₂(Fe, Al)₁₇C₃ over the four Fe (Al) sites, respectively. In Sm₂(Fe, Al)₁₇, the occupancy probability p_i of Al at the sites 6c and 18h decreases while the p_i values of Al at the sites 18f and 9d increase with increasing temperature. Elevated temperatures facilitate a random distribution of Al atoms across different lattice sites. The reason is that the contribution from configuration entropy to total free energy increases linearly with increasing temperature, which significantly lowers the total free energy. Similarly, with increasing temperature, the p_i values of Al at the 9d and 6c sites decrease while those at the sites 18f and 18h increase in Sm₂(Fe,Al)₁₇C₃. Sm₂(Fe, Al)₁₇ and Sm₂(Fe, Al)₁₇C₃ alloys are typically prepared using an arc- or induction-melting process plus a high-temperature heat treatment (e.g., 1200 K) and quenching. The Al occupancy scheme at high temperature is expected to be frozen in the final state. The actual Al distribution at different sites depends on the processing conditions, such as temperature and cooling approaches. The surviving Al occupancy scheme in the final state determines the phase stability and magnetic properties of Sm₂(Fe, Al)₁₇C₃. In other words, the phase stability and properties depend on the alloy processing history.

The normalized p_i values of Al at different crystallographic sites have been derived using Formula (4) and displayed in Figure 4. In Sm₂(Fe, Al)₁₇, the normalized p_i values of Al at the 6c and 18h sites are higher than 1.0 while those at 18f and 9d sites are less than 1.0 (Figure 4a). Above 1000 K, Al has a strong preference for the 6c site, a moderate preference for the 18h site, and almost randomly occupies at the 18f sites (the normalized p_i value is nearly 1.0) while it tends to avoid the 9d site. In contrast, Al shows a strong preference for the 9d site and a moderate preference for the 6c site while avoiding the 18f and 18h sites in Sm₂(Fe,Al)₁₇C₃ (Figure 4b). The results agree with the calculated formation energy and substitution energy. Although high temperatures weaken the preferential site occupancy of Al, the preferential trends remain unchanged, i.e., Al prefers 6c and 9d in Sm₂(Fe, Al)₁₇ and Sm₂(Fe, Al)₁₇C₃, respectively.

To understand the drastic difference in preferential site occupancy in Sm₂(Fe,Al)₁₇ and Sm₂(Fe,Al)₁₇C₃, we analyze the crystallographic and chemical environments of the different sites from Wigner–Seitz calculation using DIDO95 [38].

Table 2. The nearest neighbors (NN) and the Wigner–Seitz volume (WSV, ų) of atomic sites in Sm₂Fe₁₇ and Sm₂Fe₁₇C₃. The NN atoms are listed in order of increasing interatomic distance.

Site	Sm2Fe17		Sm2Fe17C3
	NN	WSV	NN	WSV
Sm (6c)	6Fe3, 1Fe1, 9Fe4, 3Fe2	31.7	3C, 3Fe4, 1Fe1, 6Fe3, 6Fe4, 3Fe2	32.8
Fe1 (6c)	1Fe1, 1Sm, 3Fe2, 3Fe4, 6Fe3	12.2	1Fe1, 1Sm, 3Fe4, 3Fe2, 6Fe3	12.8
Fe2 (9d)	4Fe3, 4Fe4, 2Fe1, 2Sm	11.1	4Fe3, 4Fe4, 2Fe1, 2Sm	11.9
Fe3 (18f)	2Fe2, 2Sm, 2Fe3, 4Fe4, 2Fe1	11.5	1C, 2Fe2, 2Fe3, 2Fe4, 2Sm, 2Fe4, 2Fe1	11.9
Fe4 (18h)	2Fe2, 2Fe4, 3Sm, 2Fe3, 1Fe1, 2Fe3	11.7	1C, 2Fe2, 2Fe3, 3Sm, 2Fe4, 1Fe1, 2Fe3	12.5
C (9e)			4Fe3, 4Fe4, 2Sm	2.6

lists the nearest neighbors (NN) and each crystallographic site’s Wigner–Seitz volume (WSV) in Sm₂Fe₁₇ and Sm₂Fe₁₇C₃. As shown in Table 2, the Fe₁ site has the largest site volume (12.2 ų), while the Fe₄ site has the second largest site volume (11.7 ų) and the largest number of Sm NNs (three Sm atoms) in Sm₂Fe₁₇. The large-sized Al atoms (atomic radius 1.43 Å) have a strong preference for the Fe₁ site and a moderate preference for the Fe₄ site. The size effect in Sm₂Fe₁₇ mainly dominates the site preference of Al. In addition to the size effect, Al prefers to form more Sm-Al bonds. The predicted Al occupancy agrees with the experiments [21].

In Sm₂Fe₁₇C₃, the interstitial carbon atoms (9e) change the site volume of the neighboring Fe sites and modify their chemical environments. The carbon is coordinated by two Sm atoms, two Fe₃ atoms and two Fe₄ atoms in Sm₂Fe₁₇C₃ (Figure 5). Al avoids the 18f and 18h sites (Figure 2, Figure 3 and Figure 4), which are neighboring to the carbon atoms, implying the Al-C may have weak bonding or even an anti-bonding interaction. Although the Fe₁ site has a larger site volume than the Fe₂ site, while Fe₂ has more Sm NNs (Table 2), Al exhibits a strong preference for the Fe₂ site and a moderate preference for the Fe₁ site, respectively. The results indicate that the site preference depends on the site volume and chemical environment. In Sm₂Fe₁₇C₃, Al preferential site occupancy is dominated by the site chemical environment, i.e., it favors Sm-Al bonding but avoids Al-C bonding.

2.2. Chemical Bond in Sm₂Fe₁₇C₃ and Sm₂(Fe,Al)₁₇C₃

As shown in Figure 5, the interstitial carbon atom resides at the 9e site, at the center of an octahedral cage formed by two Sm and four Fe atoms. The insertion of carbon atoms displaces neighboring Sm and Fe atoms, resulting in local structural distortions that compromise overall structural stability. To gain deeper insight into the enhanced phase stability of Sm₂(Fe, Al)₁₇C₃, we have calculated the COHP and COBI for different atomic pairs in Sm₂Fe₁₇, Sm₂Fe₁₇C₃ and the model compound Sm₂Fe₁₄Al₃C₃, in which Al replaces all Fe₂ atoms.

As illustrated in Figure 6, the -pCOHP curve for the Sm–C interaction (red) remains positive up to the Fermi level in both Sm₂Fe₁₇C₃ (Figure 6a) and Sm₂Fe₁₄Al₃C₃ (Figure 6b), indicating strong bonding character. In contrast, the COHP curves for Fe₃–C (green) and Fe₄–C (blue) exhibit small negative values near the Fermi level, suggesting the presence of anti-bonding interactions in both compounds, which contribute to the metastability of the interstitial 2:17 phase.

Figure 7 displays the calculated ICOHP and ICOBI of different atomic pairs in Sm₂Fe₁₇, Sm₂Fe₁₇C₃ and Sm₂Fe₁₄Al₃C₃. The Fe-C bonding is much stronger than Sm-C ones in Sm₂Fe₁₇C₃, as indicated by much larger ICOHP (−4.95 eV to −5.25 eV) of Fe-C than that of Sm-C (−2.95eV). A similar situation is observed in Sm₂Fe₁₄Al₃C₃ (Figure 7). The calculated ICOBI also confirms that the Fe-C bonding (~0.7) is much stronger than Sm-C bonding (~0.35) in these two interstitial phases. This imbalance in bonding strength is responsible for the local structural distortion of the carbon-centered octahedra cage (Figure 5) and contributes to the metastability of the interstitial 2:17 phase. Insertion of carbon atoms also weakens the bonding strength of Sm-Fe₃ and Sm-Fe₄. Calculated ICOHP of Sm-Fe₃ and Sm-Fe₄ are −2.34 eV and −2.00 eV in Sm₂Fe₁₇, respectively. Carbon insertion decreases the ICOHP values to −1.65 eV for Sm-Fe₃ and −1.95 eV for Sm-Fe₄ in Sm₂Fe₁₇C₃, respectively. The weak bonding of Sm-Fe₃ and Sm-Fe₄ is partially responsible for the local structural distortion of the carbon-centered octahedral cage, which in turn lowers the overall structural stability of the 2:17 interstitial carbide.

Doping Al partially restores the bonding strength of Sm-Fe₃ and Sm-Fe₄, as indicated by increased ICOHP and ICOBI values (Figure 7). ICOHP values change from −1.65 eV to −1.70 eV for Sm-Fe₃ and from −1.95 eV to −2.04 eV for Sm-Fe₄ upon partial replacement of Fe by Al. Furthermore, partial replacement of Fe by Al also slightly enhances the chemical bonding of Fe-C (Figure 7), which may contribute to the enhanced structural stability of the 2:17 interstitial phase.

Substitution of Fe by Al at the 9d site significantly alters the bonding behavior between Al (Fe₂ in parent phase Sm₂Fe₁₇C₃) and its neighboring atoms. As shown in Figure 8, the -pCOHP curves for Al–Fe₁, Al–Fe₃, and Al–Fe₄ atomic pairs remain positive up to the Fermi level, indicating strong bonding interactions. Compared to Fe₂–Fe pairs, the Al–Fe bonding exhibits higher -pCOHP values at the Fermi level, although their peak COHP values (around –2.8 eV) are slightly lower. The results indicate stronger Al-Fe bonding interaction near the Fermi level. Additionally, the -pCOHP curve for Al–Sm is positive at the Fermi level whereas the Fe₂–Sm interaction is nearly zero (Figure 8). The results suggest that Al substitution transforms the Fe₂–Sm pair from a non-bonding to a bonding state near the Fermi level—an effect that contributes to enhanced phase stability of Sm₂(Fe, Al)₁₇C₃.

Basically, doping Al enhances chemical bonding strength in Sm₂(Fe,Al)₁₇C₃ structure. Beyond strengthening Al–Fe and Al–Sm bonds, Al doping also influences Sm-Fe and Fe–C interactions, collectively contributing to improved structural and phase stability in the 2:17 interstitial compound.

3. Computational Method and Details

The electronic structure calculations were performed in the framework of DFT with the projector augmented wave (PAW) method, implemented in the QUANTUM ESPRESSO package (version 7.4) [39]. The exchange correlation function was approximated using a generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof for solids (PBEsol) [40]. The PAW atomic potential was taken from the PSlibrary 1.0 generated by A. Dal Corso [41]. The relativistic effects were included by solving a scalar relativistic wave equation. Due to the strong on-site correlation effects, the Sm 4f electrons were localized as atomic-like states [42]. The 4f electrons of Sm were treated as an open-core state in the calculation. The wave functions were expanded in plane-wave basis sets truncated at a cutoff energy of 50 Ry and the charge densities were truncated at 400 Ry. The structures (lattice parameters and atomic positions) were fully optimized, keeping the unit cell shape fixed. Brillouin zone integrations were performed on an 11× 11× 11 k-point grid, and the Marzari–Vanderbilt broadening [43] was applied with a smearing width of 5 mRy. The formation energy (E_f) of Sm₂Fe₁₆Al and Sm₂Fe₁₆AlC₃ was derived from DFT total energy calculations to evaluate the phase stability.

$$E_f\left({Sm}_2{Fe}_{17}\right)=E_{tot}-2E_{tot\left(Sm\right)}{-17E}_{tot\left(Fe\right)}$$

(1a)

$$E_f\left({Sm}_2{Fe}_{17}{C}_3\right)=E_{tot}-2E_{tot\left(Sm\right)}{-17E}_{tot\left(Fe\right)}-{3E}_{tot\left(C\right)}$$

(1b)

$$E_f\left({Sm}_2{Fe}_{16}Al\right)=E_{tot}-2E_{tot\left(Sm\right)}{-16E}_{tot\left(Fe\right)}-E_{tot\left(Al\right)}$$

(1c)

$$E_f\left({Sm}_2{Fe}_{16}Al{C}_3\right)=E_{tot}-2E_{tot\left(Sm\right)}{-16E}_{tot\left(Fe\right)}-E_{tot\left(Al\right)}-{3E}_{tot\left(C\right)}$$

(1d)

Here, E_tot, E_tot(Sm), E_tot(Fe), E_tot(Al), E_tot(C) are the total energy of the compounds, hexagonal Sm, bcc Fe, fcc Al and carbon, respectively.

The substitution energy (E_sub) of the Al atom in Sm₂Fe₁₆Al (Sm₂(Fe,Al)₁₇C₃) is calculated as the change in the formation energy of Sm₂Fe₁₆Al (Sm₂Fe₁₆AlC₃) relative to Sm₂Fe₁₇ (Sm₂Fe₁₇C₃). The calculation details of E_sub in rare-earth iron intermetallic phases have been previously reported [44].

$$E_{sub}\left({Sm}_2{Fe}_{16}Al\right)=E_f\left({Sm}_2{Fe}_{16}Al\right)-E_f\left({Sm}_2{Fe}_{17}\right)$$

(2a)

$$E_{sub}\left({Sm}_2{Fe}_{16}Al{C}_3\right)=E_f\left({Sm}_2{Fe}_{16}Al{C}_3\right)-E_f\left({Sm}_2{Fe}_{17}{C}_3\right)$$

(2b)

To understand the effect of temperature on the Al occupancy scheme, the occupancy probability of Al at each Fe site in Sm₂(Fe, Al)₁₇C₃ has been described using the Maxwell–Boltzmann distribution [45]. The formula can be expressed as

$$p_i={\displaystyle \frac{g_i\mathrm{e}\mathrm{x}\mathrm{p}(-\raisebox{1ex}{$∆E_i$}\!\left/ \!\raisebox{-1ex}{$k_{B}T$}\right.)}{\sum g_{i}exp(-\raisebox{1ex}{$∆E_i$}\!\left/ \!\raisebox{-1ex}{$k_{B}T$}\right.)}}$$

(3)

where g_i, ΔE_i, T, and k_B are the multiplicity of the crystallographic site i, internal energy change (i.e., the substation energy E_sub), temperature, and Boltzmann’s constant, respectively.

To highlight the site occupancy preferences, we normalize the site occupancy probability to that expected for a random distribution [46], which can be expressed as

$$Norm. p_i={\displaystyle \frac{51exp(-\raisebox{1ex}{$∆E_i$}\!\left/ \!\raisebox{-1ex}{$k_{B}T$}\right.)}{\sum g_{i}exp(-\raisebox{1ex}{$∆E_i$}\!\left/ \!\raisebox{-1ex}{$k_{B}T$}\right.)}}$$

(4)

The difference between Formulas (3) and (4) is the normalization factor g_i/51. Within this framework, the normalized occupancy p_i will be 1 for a random distribution of Al at a specific site. If the value of normalized p_i is much higher or lower than 1, Al has a strong preference or avoidance of the crystallographic site.

Furthermore, the PAW functions were projected to a Slater-type orbital (STO) with the Local Orbital Basis Suite Towards Electronic-Structure Reconstruction (LOBSTER version 5.1.1) software [29,30]. LOBSTER quantifies the interatomic interactions and estimates the quantum chemical bonding characteristics using COHP and COBI [30,35]. The energy-integrated values (up to the Fermi level) of these quantities, namely ICOHP and ICOBI, can be interpreted as a measure of covalent bond strength.

4. Conclusions

In summary, a strong correlation exists between the preferential substitution of Fe by Al, the resulting chemical bonding, and the phase stability of the interstitial compound Sm₂(Fe,Al)₁₇C₃. Insertion of carbon weakens the bonding strength of Sm-Fe₃ and Sm-Fe₄, contributing to the metastability of Sm₂Fe₁₇C₃ as indicated by calculated ICOHP and ICOBI. Partial replacement of Fe with Al lowers the formation energy, indicating enhanced thermodynamic stability, and leads to preferential occupancy of Al at the 9d site in Sm₂(Fe,Al)₁₇C₃. COHP and COBI analysis reveal that doping Al strengthens Al-Fe, Al-Sm, Sm-Fe₃, Sm-Fe₄ and Fe–C bonding in Sm₂(Fe,Al)₁₇C₃. These effects contribute to improved phase stability in the Al-doped 2:17 interstitial compounds.