# Theoretical investigations on the magnetic properties of the (Sc,Nb)Fe2 compounds

D. Benea, S. El Bidaoui, L.V.B. Diop, O. Isnard

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D. Benea, S. El Bidaoui, L.V.B. Diop, O. Isnard. Theoretical investigations on the magnetic properties of the (Sc,Nb)Fe2 compounds. Solid State Communications, 2025, 404, pp.116074. 10.1016/j.ssc.2025.116074. hal-05219787

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# Theoretical investigations on the magnetic properties of the (Sc,Nb) Fe₂ compounds

D. Benea<sup>a, *,</sup> , S. El Bidaoui<sup>b, c</sup> , L.V.B. Diop<sup>d</sup> , O. Isnard<sup>b, c</sup>

## ARTICLE INFO

Communicated by Luis Brey

### Keywords:

A. ab-initio calculations

B. transition metal intermetallics

C. Magnetic properties

## ABSTRACT

The theoretical investigations on the electronic and magnetic properties of the Sc₁ₓNbₓFe₂ (x = 0–0.6) Laves type intermetallic compounds with the C14 MgZn₂-type structure are presented. The magnetic properties are determined by Fe 6h atoms forming two Kagomé hexagonal sublattices, separated by Fe 2a sites and Nb/Sc atoms in the interstices. ScFe₂ is a ferromagnet, but Nb for Sc substitution reduces interatomic distances and promotes antiferromagnetic interaction between Fe spins. The present theoretical study may provide an insight into the influence of the chemical composition and interatomic spacing on the magnetic properties (magnetic moments, exchange-coupling parameters, and Curie temperatures) of the Sc₁ₓNbₓFe₂ series of compounds. The spin-polarized fully relativistic Korringa-Kohn-Rostoker (SPR-KKR) band structure method has been employed, using the coherent potential approximation (CPA) to deal with substitutional disorder. The available experimental data are used to give a more comprehensive description of the electronic and magnetic properties.

## 1. Introduction

The magnetic Laves phase RM₂ (R = 3d, 4d, 5d, or rare earth and M = 3d transition metal) is of great interest for research due to the extremely varied magnetic behavior of these materials. The magnetic behavior shown by Laves phases comprise (a) strong or weak ferromagnetism, if M = Fe and R is a rare earth, Sc or Hf, (b) Pauli paramagnetism (for RNi₂), (c) magnetism of itinerant electrons (in RCO₂) or (d) antiferromagnetism in RMn₂, NbFe₂ and TiFe₂ [1–6]. These compounds show often non-collinear magnetic structures, some of them being frustrated [2,3,7].

For the Laves phase with M = Fe, a competition between ferromagnetic and antiferromagnetic interactions has been reported, as the Fe magnetic moment is very sensitive to the local environment (coordination number, nearest neighbors, atomic distances and bonding type). For instance, in the Hf₁ₓTaₓFe₂ C14 Laves phases, a first-order ferromagnetic to antiferromagnetic transition is evidenced upon increasing temperature [8]. On the other hand, NbFe₂ is an itinerant system exhibiting a quantum critical point [9,10]. In particular, in the Nb₁ₓFe₂+y Laves phase, the magnetic ground state can be tuned by changing the Nb/Fe ratio, with weakly ferromagnetic behaviour at

Nb-rich and Fe-rich sides and, respectively, with antiferromagnetic or spin-density-wave (SDW) order close to stoichiometric composition.

ScFe₂ is a polymorphic compound that can crystallize in the three Laves phase structures (C14, C36, and C15) [11,12]. Earlier investigations on Sc₁ₓNbₓFe₂ (x = 0–1) compounds revealed that these compounds exhibit dimorphism as they can crystallize either in cubic C15 (space group Fd-3m) or hexagonal C14 (space group P6₃/mmc) symmetry [13,14]. The C14 Laves phase was found more stable by electronic band structure calculations, but the small energy difference to C15 Laves phase (ΔE/kg ~ 58 K) explains the dimorphism [13]. Nb doping promotes the C14 phase formation, which is obtained without additional heat treatments for x > 0.15 [13]. ScFe₂ exhibits a ferromagnetic ordering, but the Curie temperature drops by Nb for Sc substitution, according to the magnetic measurements of the Sc₁ₓNbₓFe₂ compounds with C14 single phases [14]. Low-temperature magnetic investigations reveal that ScFe₂ has a spontaneous magnetization of approximately 2.3 µB/f.u. Substituting Nb for Sc leads to a slight increase in magnetization for x < 0.25 and a sharp drop for x > 0.35. Still, the ferromagnetic order is absent even at 2K for the Nb-rich samples with x > 0.75 [14]. A transition from a short-range ferromagnet in ScFe₂ to systems with competing exchange interaction for Nb-rich compositions

<footnote><sup>a</sup> Faculty of Physics, Babeş-Bolyai University Cluj-Napoca, Kogălniceanu str 1, 400084, Cluj-Napoca, Romania</footnote>

<footnote><sup>b</sup> Université Grenoble Alpes, Institut Néel, Grenoble, F-38042, France</footnote>

<footnote><sup>c</sup> CNRS, Institut Néel, 25 rue des Martyrs, F-38042, Grenoble, France</footnote>

<footnote><sup>d</sup> Université de Lorraine, CNRS, IJL, F-54000, Nancy, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: diana.benea@ubbcluj.ro (D. Benea).</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** The C14 hexagonal MgZn₂-type of structure (space group P6₃/mmc) type of structure Sc₁ₓNbₓFe₂ (x = 0–1) compounds.</figcaption></figure>

has been recently evidenced [15] revealing the large sensitivity of Fe magnetism on the Nb for Sc substitution. These interesting results motivated us to further investigate the Sc₁ₓNbₓFe₂ system by electronic band structure calculations.

Theoretical electronic band structure investigations have been performed and are reported here for different magnetic ground states in Sc₁ₓNbₓFe₂ alloys with x ≤ 0.6. The effects of the chemical composition and atomic spacing on the electronic and magnetic properties of the Sc₁ₓNbₓFe₂ compounds are extensively studied in order to determine a possible explanation of the unusual magnetization behavior exhibiting a sharp decrease for Nb content x > 0.2.

## 2. Calculations details

Electronic band structure calculations have been carried out within the Density Functional Theory using the Munich SPRKKR band structure program package [16]. Multiple scattering theory is used in the KKR-Green's function formalism, upon which the theoretical computations are based [17]. The fully relativistic approach was used, meaning that all relativistic effects were taken into account, including spin-orbit coupling. The angular momentum expansion of the basis functions was taken up *l* = 2. The exchange and correlation effects by local spin density approximation (LSDA) which employs Vosko et al.'s parametrization was used within this study [18]. The k-space integration was carried out using the special points approach [19]. The substitutional disorder in the system has been treated using the Coherent Potential Approximation (CPA) theory [20]. The full potential approach, which implies the treatment of the potential and charge density without any shape approximations in the interstitial region and inside the muffin-tins, has been applied [16,17].

We used microscopic models for the magnetic interactions as complementary method to investigate the magnetic behaviour of solids. This widely used approach relies on the classical Heisenberg Hamiltonian, which can be represented by the following formula:

$$H_{\text{ex}} = - \sum_{ij} J_{ij} \hat{e}_i \cdot \hat{e}_j \qquad (1)$$

where the summation is performed on all lattice sites *i* and *j* and $\hat{e}_i/\hat{e}_j$ are the unit vectors of magnetic moments on sites *i* and *j*, respectively. The *J*<sub>ij</sub> exchange coupling parameters between the Fe spin moments have been calculated as a function of distance using the expression derived by Liechtenstein et al. from the magnetic force theorem within the atomic spheres approximation [21]. The Curie temperatures have been estimated by the mean-field approach using the following expression [21–23]:

<table><caption>**Table 1**
Lattice constants (from Ref. [14]) of the Sc₁ₓNbₓFe₂ (x = 0–1) compounds.</caption><thead><tr><th></th><th>a<sub>lat</sub> (Å)</th><th>c<sub>lat</sub> (Å)</th></tr></thead><tbody><tr><td>ScFe<sub>2</sub></td><td>4.974</td><td>8.135</td></tr><tr><td>Sc<sub>0.9</sub>Nb<sub>0.1</sub>Fe<sub>2</sub></td><td>4.964</td><td>8.115</td></tr><tr><td>Sc<sub>0.8</sub>Nb<sub>0.2</sub>Fe<sub>2</sub></td><td>4.949</td><td>8.089</td></tr><tr><td>Sc<sub>0.7</sub>Nb<sub>0.3</sub>Fe<sub>2</sub></td><td>4.930</td><td>8.056</td></tr><tr><td>Sc<sub>0.6</sub>Nb<sub>0.4</sub>Fe<sub>2</sub></td><td>4.916</td><td>8.035</td></tr><tr><td>Sc<sub>0.5</sub>Nb<sub>0.5</sub>Fe<sub>2</sub></td><td>4.897</td><td>8.004</td></tr><tr><td>Sc<sub>0.4</sub>Nb<sub>0.6</sub>Fe<sub>2</sub></td><td>4.885</td><td>7.983</td></tr></tbody></table>

$$T_{c}^{\text{rough-MFA}} = \frac{2}{3k_{B}} \sum_{i} J_{0i} \qquad (2)$$

The summation of $J_{ij}$ on all coordination shells up to 17 Å surrounding lattice site *i* yields the $J_{0i}$ exchange-coupling parameters in this expression, while the Curie temperature can be obtained by summing across lattice sites.

## 3. Results and discussions

### 3.1. Crystal structure

The Sc₁ₓNbₓFe₂ (x = 0–1) compounds crystallize in the C14 hexagonal MgZn₂-type crystal structure (space group symmetry P6₃/mmc) (Fig. 1). According to earlier reports, the C14 phase is more stable than the C15 Laves phase, the last one being stable at high temperature only and obtained by quenching [11,12]. Within this unit cell, two inequivalent Fe sites are occupied (2a and 6h), while the Sc/Nb atoms occupy one single Wyckoff position 4f. The Fe 6h atoms form.

Kagomé planes perpendicular to the c-axis. The Fe atoms form chains of tetrahedra along the c-axis, with the base built by the Fe 6h atoms and the Fe 2a atoms on top. The lattice constants decrease with Nb substitution for Sc since Nb has smaller metallic radii as Sc [24]. The theoretical calculations use the experimental lattice constants summarized in Table 1 [14].

### 3.2. Density of states

Band structure calculations using the SPR-KKR band structure method have been performed for ScFe₁ₓNbx (x = 1 - 0.6) in the ferromagnetic spin configuration. The total density of states (DOS) for ScFe₂ and ScFe₀.₄Nb₀.₆ are presented in Fig. 2. According to the DOS plots, the main contribution at the Fermi level is due to the 3d-electrons of Fe. The valence bandwidth is enhanced by Nb for Sc substitution. The exchange-splitting inScFe₀.₄Nb₀.₆ is reduced compared with ScFe₂ due to the extra electrons of Nb filling the spin-down bands which are shifted to lower energies.

The DOS at the Fermi level (E<sub>F</sub>) in ScFe₂ is situated in a dip of the spin-down channel. Nb doping shifts the spin-down band to lower energy, rising the Fermi level DOS in Sc₀.₄Nb₀.₆Fe₂ alloy. Also, a local peak in DOS at the Fermi level is observed in the spin-down channel of the Sc₀.₄Nb₀.₆Fe₂ alloy. It is important to note that in ScFe₂, the DOS of Fe at the Fermi level has similar magnitude for the 2a and 6h sites, suggesting that both should be magnetically active, as shown in previous reports [13,14,25].

### 3.3. Magnetic properties

The ScFe₂ compound with the C14 Laves structure is a ferromagnet with a Curie temperature of 542 K and the average magnetic moment on Fe of 1.37 µ<sub>B</sub> at 4.2 K [26]. Sankar et al. reported a saturation magnetization of 2.90 µ<sub>B</sub>/f.u. for ScFe₂ in C14 Laves structure [27], comparable with the corresponding values for C14 Laves HfFe₂ (3.20 µ<sub>B</sub>/f.u.) [28]. Smaller values of about 2.33 µ<sub>B</sub>/f.u. at 4 K have been measured recently
<figure><img src="image_4.png" /><figcaption>Fig. 2. The density of states (DOS) of the ScFe₂ and Sc₀.₄Nb₀.₆Fe₂ alloys. The Fermi level is the origin of the energy scale.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Table 2**</figcaption></figure>

The calculated magnetic moments (in Bohr magnetons) of the Sc₁ₓNbₓFe₂ alloys in ferromagnetic spin configuration.

<table><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="5">Ferromagnetic</th></tr><tr><th>Fe 2a</th><th>Fe 6h</th><th>Sc</th><th>Nb</th><th>tot./f.u.</th></tr></thead><tbody><tr><td rowspan="2">x = 0</td><td>m<sub>s</sub>(μ<sub>B</sub>)</td><td>1.47</td><td>1.65</td><td>-0.49</td><td>-</td><td>2.67</td></tr><tr><td>m<sub>1</sub>(μ<sub>B</sub>)</td><td>0.03</td><td>0.04</td><td>-</td><td>-</td><td>0.08</td></tr><tr><td rowspan="2">x = 0.1</td><td>m<sub>s</sub>(μ<sub>B</sub>)</td><td>1.49</td><td>1.68</td><td>-0.45</td><td>-0.62</td><td>2.80</td></tr><tr><td>m<sub>1</sub>(μ<sub>B</sub>)</td><td>0.03</td><td>0.04</td><td>-</td><td>-</td><td>0.08</td></tr><tr><td rowspan="2">x = 0.2</td><td>m<sub>s</sub>(μ<sub>B</sub>)</td><td>1.59</td><td>1.70</td><td>-0.42</td><td>-0.55</td><td>2.89</td></tr><tr><td>m<sub>1</sub>(μ<sub>B</sub>)</td><td>0.04</td><td>0.05</td><td>-</td><td>-</td><td>0.09</td></tr><tr><td rowspan="2">x = 0.3</td><td>m<sub>s</sub>(μ<sub>B</sub>)</td><td>1.58</td><td>1.68</td><td>-0.39</td><td>-0.50</td><td>2.89</td></tr><tr><td>m<sub>1</sub>(μ<sub>B</sub>)</td><td>0.05</td><td>0.05</td><td>-</td><td>-</td><td>0.10</td></tr><tr><td rowspan="2">x = 0.4</td><td>m<sub>s</sub>(μ<sub>B</sub>)</td><td>1.53</td><td>1.66</td><td>-0.36</td><td>-0.44</td><td>2.87</td></tr><tr><td>m<sub>1</sub>(μ<sub>B</sub>)</td><td>0.05</td><td>0.05</td><td>-</td><td>-</td><td>0.11</td></tr><tr><td rowspan="2">x = 0.5</td><td>m<sub>s</sub>(μ<sub>B</sub>)</td><td>1.42</td><td>1.54</td><td>-0.31</td><td>-0.40</td><td>2.66</td></tr><tr><td>m<sub>1</sub>(μ<sub>B</sub>)</td><td>0.07</td><td>0.06</td><td>-</td><td>-</td><td>0.14</td></tr><tr><td rowspan="2">x = 0.6</td><td>m<sub>s</sub>(μ<sub>B</sub>)</td><td>1.28</td><td>1.45</td><td>-0.28</td><td>-0.37</td><td>2.49</td></tr><tr><td>m<sub>1</sub>(μ<sub>B</sub>)</td><td>0.07</td><td>0.07</td><td>-</td><td>-</td><td>0.17</td></tr></tbody></table>

by El Bidaoui et al. [14]. A large anisotropy field of Hₐ = 4.5 T has been determined, with an easy c-axis at 300 K [27]. It has been shown that the ground magnetic state of ScFe₂ can be changed to antiferromagnetic by applying a magnetic field above 30 GPa [29].

Our theoretical calculations for ScFe₂ (Table 2) found on Fe 2a and 6h crystallographic sites spin magnetic moments of 1.47 µB and 1.65 µB, respectively, along with an induced negative moment for Sc of -0.49 µB. The theoretical magnetic moment along the c-axis (2.75 µB/f.u.) shows good agreement with the experimental values [14,27].

According to the magnetic measurements of El Bidaoui et al. at 5K [14], the magnetization increases by Nb for Sc substitution in Sc₁ₓNbₓFe₂ up to x ≈ 0.25, followed by a sharp drop for x > 0.35. On the other hand, Z. Jing-Ting et al. [13] found ferromagnetism for x < 0.15 and the coexistence of ferro- and antiferromagnetic phases for 0.15 < x < 0.5. Nevertheless, the structural analysis of their systems revealed a biphasic mixture of hexagonal C14 and the cubic C15 Laves phase, making the findings less conclusive. The theoretical calculations found similar values of magnetic moments in C14 and C15 Laves phases [13].

* Ferromagnetic spin configuration

Accounting for a ferromagnetic spin configuration, as reported in ScFe₂ compound, fully relativistic band structure calculations have been performed. The calculated magnetic moments vs. Nb content in the Sc₁ₓNbₓFe₂ alloys are shown in Table 2.

The Fe spin moments rise by Nb for Sc substitution, reaching the highest values (1.59 µB on Fe 2a and 1.70 µB on Fe 6h, respectively) at x = 0.2. As the Nb concentration exceeds x = 0.2, the Fe spin moments begin to decrease. Increased Nb content with x > 0.2 results in a significant drop in spin moment for the Fe 2a site, reaching the value of 1.28 µB at x = 0.6. The spin magnetic moment on the Fe 6h site shows a similar decrease, reaching a value of 1.45 µB at x = 0.6.

On the other hand, the negative sign of the observed magnetic moments on both Sc and Nb clearly indicates that the magnetic moment is induced by the polarization of the surrounding Fe atoms. The induced negative spin magnetic moments for Sc range between -0.49 µB (x = 0) and -0.28 µB (x = 0.6) while for Nb the values are between -0.62 (x = 0.1) µB and -0.37 (x = 0.6) µB. In a compound with both magnetic and nonmagnetic atoms, the nonmagnetic atoms can develop an induced magnetic moment due to the polarization caused by interactions with the magnetic atoms, often through exchange interactions or hybridization effects. If the spin moment of the magnetic atom decreases, the induced moment on the nonmagnetic atom typically decreases as well. As a consequence, the Sc and Nb spin moment decrease can be related to the decrease of Fe magnetization, as the exchange interactions between Fe and Sc/Nb are weakened, leading to a corresponding reduction in the polarization of the nonmagnetic Sc and Nb atoms.

It is noteworthy that the calculated orbital magnetic moments are non-zero only for Fe sites. The orbital magnetic moments of 0.03 µB (Fe 2a) and 0.04 µB (Fe 6h) amount to about 2 % of the total magnetization in ScFe₂. However, there is no significant difference in the orbital magnetic moment on the two inequivalent Fe atomic positions, their magnitude being very similar for the two sites whatever the Nb composition. This weakness of the orbital moment confirms the almost total quenching of the orbital moment by the high symmetry of the crystal electric field at the Fe positions in ScFe₂ crystal structure. However, the orbital magnetic moment increases remarkably upon Nb for Sc substitution reaching twice larger values of 0.07µB/Fe for x = 0.6 composition. This may result from the substitution induced distortion of the crystal structure.

As a result, the calculated total magnetic moments in Sc₁ₓNbₓFe₂ alloys have similarities to the experimental measurements at low temperatures [14]. Still, the theoretical calculations in ferromagnetic spin configuration do not explain the sharp decrease in magnetization for x > 0.35 observed in magnetic measurements [14].

* Ferrimagnetic spin configuration

The substitution of Nb for Sc leads to a complex magnetic diagram
<figure><img src="image_6.png" /><figcaption>Fig. 3. The ferromagnetic vs. ferrimagnetic spin configuration in the Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys. For simplicity, the Sc/Nb atoms are not shown.</figcaption></figure>

<table><caption>**Table 3**
The calculated spin magnetic moments (in Bohr magnetons) of the Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys in ferrimagnetic spin configuration.</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="5">Ferrimagnetic</th><th rowspan="2">E<sub>Ferri</sub>-E<sub>FM</sub> (meV/atom)</th></tr><tr><th>Fe 2a</th><th>Fe<sup>+</sup> 6h</th><th>Fe<sup>-</sup> 6h</th><th>Sc</th><th>Nb</th><th>tot./f.u.</th></tr></thead><tbody><tr><td>x = 0</td><td>m<sub>1</sub>(μ<sub>B</sub>)</td><td>1.27</td><td>1.60</td><td>-1.27</td><td>-0.13</td><td>-</td><td>0.88</td><td>26.95</td></tr><tr><td>x = 0.1</td><td>m<sub>2</sub>(μ<sub>B</sub>)</td><td>1.33</td><td>1.63</td><td>-1.23</td><td>-0.13</td><td>-0.19</td><td>0.83</td><td>23.17</td></tr><tr><td>x = 0.2</td><td>m<sub>3</sub>(μ<sub>B</sub>)</td><td>1.34</td><td>1.61</td><td>-1.22</td><td>-0.13</td><td>-0.18</td><td>0.82</td><td>17.35</td></tr><tr><td>x = 0.3</td><td>m<sub>4</sub>(μ<sub>B</sub>)</td><td>1.25</td><td>1.57</td><td>-1.26</td><td>-0.10</td><td>-0.15</td><td>0.70</td><td>11.48</td></tr><tr><td>x = 0.4</td><td>m<sub>5</sub>(μ<sub>B</sub>)</td><td>1.01</td><td>1.52</td><td>-1.27</td><td>-0.09</td><td>-0.12</td><td>0.59</td><td>5.05</td></tr><tr><td>x = 0.5</td><td>m<sub>6</sub>(μ<sub>B</sub>)</td><td>0.62</td><td>1.42</td><td>-1.27</td><td>-0.05</td><td>-0.06</td><td>0.30</td><td>-0.14</td></tr><tr><td>x = 0.6</td><td>m<sub>7</sub>(μ<sub>B</sub>)</td><td>0.35</td><td>1.35</td><td>-1.24</td><td>-0.03</td><td>-0.03</td><td>0.23</td><td>-20.68</td></tr></tbody></table>

with composition-dependent ferro- and antiferromagnetic phases in C14 Laves phases of the Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys [7,15]. The spin configuration is not explicitly defined in Mössbauer experiments for the studied alloys [7]. In the similar C14 Laves phase of TiFe<sub>2</sub>, the antiparallel aligned Fe 6h spins on neighboring planes result in an antiferromagnetic phase. Due to the symmetry (Fe 2a are inversion centres), the spin moments on Fe 2a sites vanish [30].

On the other hand, in the parent compound ScFe<sub>2</sub>, the magnetic

ground state is sensitive to the interatomic distances, and when pressure is applied, an antiferromagnetic spin configuration is formed [29]. The decrease of the lattice constants by Nb for Sc substitution may also promote an antiparallel spin arrangement in Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys.

In order to investigate the stability of a ferrimagnetic spin arrangement in Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys, the Fe spin structure presented in Fig. 3(b) has been constructed from the ferromagnetic structure of Fig. 3(a) by inverting the spins on adjacent Fe 6h Kagomé planes. In the present investigations, as the spin magnetic moment on Fe 2a site is supposed to be nonzero, the spin configuration in Fig. 3(b) is ferrimagnetic. However, it is important to note that in Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys, a noncollinear spin arrangement cannot be ruled out.

The calculated magnetic moments for the ferrimagnetic configuration are shown in Table 3. The spin moments on Fe 2a sites are slightly lower than the corresponding values in ferromagnetic configuration, but they follow a similar dependence on Nb concentration, showing a maximum at x = 0.2 (1.34 μ<sub>B</sub>). The Fe 6h spin moments have different values depending on the considered.

orientation. The Fe 6h spin moments oriented parallel to the c-axis (denoted Fe 6h↑) are comparable with the corresponding values in ferromagnetic configuration (maximum of 1.63 μ<sub>B</sub> at x = 0.1). In contrast, Fe 6h spin moments oriented antiparallel to the c-axis (Fe 6h ↓) vary between -1.21 μ<sub>B</sub> and -1.27 μ<sub>B</sub> rather unrelated to the Nb content (x). A ferrimagnetic spin configuration occurs with non-zero Fe 2a spin moments, for a Nb content of x > 0.4. In this collinear spin configuration, non-compensating magnetic moments on the Fe 6h sites are obtained.

Reduced Fe 2a spin magnetic moments and significantly lower average Fe 6h spin magnetic moments lead to decreased induced spin magnetic moments in Sc and Nb. The resulting total spin moments in this

<figure><img src="image_7.png" /><figcaption>Fig. 4. The calculated total spin magnetic moments in ferro- and ferrimagnetic spin configurations for Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys, together with the experimental data of El Bidaoui et al. [14]. The dotted line links the total spin magnetic moments of the most stable spin configuration.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 5. The total energy vs. volume dependence for ferromagnetic (FM) and ferrimagnetic ('Ferri') spin structures in Sc<sub>0.7</sub>Nb<sub>0.3</sub>Fe<sub>2</sub> alloy.</figcaption></figure>

ferrimagnetic configuration are much lower compared with the values calculated for the ferromagnetic spin configuration (Table 2). The calculated total magnetic moment dependence on the Nb content shows a low decrease for $x \le 0.2$, followed by a rather sharp decrease for $0.3 \le x \le 0.6$.

According to the total energy calculations for the Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys (Table 3), the ferromagnetic spin configuration is favored for low Nb content ($x < 0.4$), while a ferrimagnetic spin structure is more stable for $x \ge 0.5$. This finding is in line with experimental results recently published [15] where careful analysis of the magnetization curve has shown the evolution of the magnetism of Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys towards itinerant electron weak ferromagnetism and trend to antiparallel spin alignment.

* Ferromagnetic and ferrimagnetic spin configuration vs. experiment

The calculated values of total spin magnetic moments in ferro- and ferrimagnetic spin configuration are plotted in Fig. 4, together with the experimental data of El Bidaoui et al. [14]. The dotted line connects the total spin magnetic moments of the most stable spin structures in the Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys, allowing for straightforward comparison with experimental results. As can be seen in Fig. 4, the agreement between the theoretical calculations and experimental measurements is improved significantly by accounting for the most stable spin structure. Furthermore, the theoretical computations indicate that the transition from ferro-to ferrimagnetic spin configuration may explain the steep decrease in magnetization measurements at $x \approx 0.3$.

As Table 1 shows, the lattice constants decrease upon Nb for Sc substitution, that results in shorter Fe-Fe interatomic distances. Lower interatomic distances may promote antiparallel coupling of Fe 6h spins in adjacent planes, as in other Fe-based Laves phases [31]. The ferromagnetic to antiferromagnetic phase transition in ScFe<sub>2</sub> under pressure provides further evidence for this theory [29].

* Volume dependent spin structures

According to Asano et al. [32], the bandwidth of Fe in AFe<sub>2</sub> (where A is a transition metal element) compounds is enhanced as the number of valence electrons of the A atom increases. This enhancement is due to the reduced spacing, which promotes the hybridization between the d-bands of Fe and A atoms. The width of the Fe d-bands is increased by Nb for Sc substitution in Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys, as can be seen in Fig. 2. The bandwidth increase diminishes spin polarization energy gain, lowering the ferromagnetic order stability.

The spacing decrease can also be achieved by applying hydrostatic pressure. Experimental investigations show that about 30 GPa is required to shift the ferromagnetic spin ordering to antiferromagnetic in ScFe<sub>2</sub> [29].

The relationship between spin ordering and interatomic spacing for the Sc<sub>0.7</sub>Nb<sub>0.3</sub>Fe<sub>2</sub> alloy is discussed below. The total energy vs. unit cell volume is illustrated in Fig. 5 for both ferro- and ferrimagnetic spin configurations. The c/a ratio was kept constant in our study, at the experimental value. The ferrimagnetic configuration consisting of antiparallel aligning of Fe 6h spins exhibits higher stability in the Sc<sub>0.7</sub>Nb<sub>0.3</sub>Fe<sub>2</sub> alloy, as illustrated in Fig. 5, at a volume below the value obtained from the experiment.

The Fe spin magnetic moments dependence on volume in Sc<sub>0.7</sub>Nb<sub>0.3</sub>Fe<sub>2</sub> alloy is shown in Table 4. In the ferromagnetic spin arrangement (FM), the magnetic moments on both Fe sites drop monotonically as volume decreases, a result in agreement with volume dependence of Fe magnetization reported elsewhere for a wide range of iron containing intermetallics [33]. The spin magnetic moment on Fe 2a sites is more affected by the space shrinkage, particularly for $V < 180 \AA^3$. The spin moments at the Fe 6h sites exhibit a gradual decrease, changing from 1.73 $\mu_B$ to 0.80 $\mu_B$ at a volume shrinkage of 14 %. The spin magnetic moment at the Fe 2a sites diminishes from 1.72 $\mu_B$ to 0.13 $\mu_B$ over the same volume shrinkage, resulting in a substantial decrease of the overall magnetization from 3 $\mu_B$ to 1.02 $\mu_B$.

In ferrimagnetic spin configuration ('Ferri'), the Fe 2a spin moments are lower than the corresponding values in FM spin arrangement. The Fe<sup>1</sup> 6h spin moments decrease by volume shrinkage with a comparable rate as in the FM spin configuration. At $V \le 191 \AA^3$, the Fe 2a spins moments are zero, and the Fe<sup>1</sup> 6h and Fe<sup>1</sup> 6h spins compensate each other, resulting in a fully compensated antiferromagnetic state. This

<caption>**Table 4**</caption>

The volume-dependent magnetic moments in the Sc<sub>0.7</sub>Nb<sub>0.3</sub>Fe<sub>2</sub> alloy, calculated for ferromagnetic (FM) and ferrimagnetic spin configurations, respectively.

<table><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="3">FM</th><th colspan="4">'Ferri'</th></tr><tr><th>Fe 2a</th><th>Fe 6h</th><th>total/f.u.</th><th>Fe 2a</th><th>Fe<sup>1</sup> 6h</th><th>Fe<sup>1</sup> 6h</th><th>total/f.u.</th></tr></thead><tbody><tr><td>V(Å<sup>3</sup>)</td><td>a,c(Å)</td><td>m<sub>6</sub>(μ<sub>B</sub>)</td><td>m<sub>6</sub>(μ<sub>B</sub>)</td><td>m<sub>6</sub>(μ<sub>B</sub>)</td><td>m<sub>6</sub>(μ<sub>B</sub>)</td><td>m<sub>6</sub>(μ<sub>B</sub>)</td><td>m<sub>6</sub>(μ<sub>B</sub>)</td><td>m<sub>6</sub>(μ<sub>B</sub>)</td></tr><tr><td>173</td><td>4.7298<br>7.7291</td><td>0.13</td><td>0.80</td><td>1.02</td><td>0</td><td>0.97</td><td>-0.97</td><td>0</td></tr><tr><td>177</td><td>4.7698<br>7.7944</td><td>0.14</td><td>0.85</td><td>1.17</td><td>0</td><td>1.05</td><td>-1.05</td><td>0</td></tr><tr><td>182</td><td>4.8098<br>7.8598</td><td>1.03</td><td>1.28</td><td>2.13</td><td>0</td><td>1.14</td><td>-1.14</td><td>0</td></tr><tr><td>186</td><td>4.8498<br>7.9252</td><td>1.30</td><td>1.48</td><td>2.53</td><td>0</td><td>1.24</td><td>-1.24</td><td>0</td></tr><tr><td>191</td><td>4.8898<br>7.9905</td><td>1.45</td><td>1.59</td><td>2.73</td><td>0</td><td>1.32</td><td>-1.32</td><td>0</td></tr><tr><td>196 (exp.)</td><td>4.9298<br>8.0559</td><td>1.58</td><td>1.68</td><td>2.89</td><td>1.25</td><td>1.57</td><td>-1.26</td><td>0.74</td></tr><tr><td>201</td><td>4.9698<br>8.1213</td><td>1.72</td><td>1.73</td><td>3.00</td><td>1.39</td><td>1.67</td><td>-1.31</td><td>0.82</td></tr></tbody></table>
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 6.** The exchange-coupling parameters between Fe 2a - Fe 2a (a), Fe 2a-Fe 6h (b), and Fe 6h-Fe 6h (c) spin pairs, represented as a function of distance (in units of lattice constant *a*) for different cell volumes in the Sc<sub>0.7</sub>Nb<sub>0.3</sub>Fe<sub>2</sub> alloy.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 7.** The Curie temperatures $T_c$ calculated for Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys by mean field approximation (MFA) together with the experimental data of El Bidaoui et al. [14].</figcaption></figure>

fully compensated ferrimagnetic state is stable for about 5 % shrinkage of the experimental volume, as can be seen in Fig. 5. As a result, the magnetic behavior of the Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloy under pressure is similar to that experimentally observed in ScFe<sub>2</sub> [29]. It is interesting to point out that in such a magnetic configuration, the compensation of the magnetization of Fe<sup>1</sup> 6h and Fe<sup>1</sup> 6h leads to symmetry-induced cancelation of the molecular field experienced at the Fe 2a position, a result in excellent agreement with the calculated disappearance of the Fe 2a magnetic moment in this AF-type ordering [30].

The volume-dependent computed exchange-coupling parameters $J_{ij}$ for Fe spins in the Sc<sub>0.7</sub>Nb<sub>0.3</sub>Fe<sub>2</sub> alloy are displayed in Fig. 6. The exchange coupling between the central spin and the neighboring spins is shown as a function of distance (in units of lattice constant *a*). The central spins belong to Fe 2a in Fig. 6a and b, and Fe 6h in Fig. 6(c), respectively.

As shown in Fig. 6, the exchange interactions between Fe spins decay rapidly beyond the second coordination shells due to the localized nature of Fe's 3d states. The short-range exchange indicates that Fe magnetic moments are well localized, typical for 3d transition metals.

The highest exchange-coupling parameters occur between Fe 6h spins, while the lowest values occur between Fe 2a spins. Reminding that Fe 6h positions are three times more numerous than the 2a ones, this proves the dominant role of the Fe 6h on the overall exchange and molecular field in the compound. By decreasing the unit-cell volume, the Fe 2a-Fe 2a and Fe 2a-Fe 6h exchange-coupling parameters drop to zero. Nonetheless, non-zero values of Fe 6h-Fe 6h exchange-coupling parameters are obtained at a volume decrease of roughly 10 %. These findings demonstrate that applying pressure maintains magnetic order, despite the absence of spin moments on Fe 2a sites.

### • Curie temperatures

The experimental Curie temperature in the C14 phase of Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys is 530 K for ScFe<sub>2</sub>, and decreases with Nb content, reaching a value of 120 K for x = 0.6 [14]. Another experimental study indicates a slightly different Curie temperature values of 545 K for the C14 Laves phase of ScFe<sub>2</sub> [27]. A detailed study of the composition dependence revealed that the Curie temperature $T_C$ is relatively unaffected by Nb doping up to x = 0.1, but for Nb content x > 0.2, the magnetic ordering temperature $T_C$ decreases almost linearly with x [14]. Other $T_C$ measurements for a mixture of C14 and C15 Laves phases of Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys show a rather similar trend by Nb doping, but for a reduced measurement range (0 < x < 0.35) [13].

The theoretical Curie temperatures ($T_C$) of Sc<sub>1-x</sub>Nb<sub>x</sub>Fe<sub>2</sub> alloys
calculated by mean-field approximation (MFA) [21–23] for FM spin configuration are shown in Fig. 7, together with the experimental results [14]. As can be seen in Fig. 7, the MFA calculated $T_C$ increases slightly up to $x = 0.2$ ($T_C$ = 709K) before decreasing for $x > 0.2$. The theoretical calculations follow the experimental declining behaviour only for $x > 0.2$. When comparing the experimental results to theoretical calculations, it is important to note that MFA computations tend to overestimate Curie temperatures by about 20% [34]. Consequently, the calculated $T_c$ values enable a qualitative analysis rather than $T_c$ estimations. The 20% difference is the order of magnitude of the difference between pure ScFe₂ and Sc₀.₈Nb₀.₂Fe₂, when the spin configuration is purely ferromagnetic.

The qualitative agreement with the experimental results might be improved by accounting for the fact that the ferrimagnetic state is favored upon increasing Nb content, according to the theoretical calculations (Table 3). More accurate theoretical predictions of the Curie temperatures can be obtained using more reliable computations as the Monte Carlo approach or random phase approximation (RPA).

## 4. Conclusions

We present theoretical investigations on the electronic and magnetic properties of the Sc₁₋ₓNbₓFe₂ alloys ($x \le 0.6$). According to our calculations, band-filling effects in DOS due to additional electrons from Nb can be found. Calculations of the electronic band structure indicate that ScFe₂ displays ferromagnetic order, with a greater magnetic moment observed at the Fe 6h position compared to the Fe 2a position. The orbital moments in ScFe₂ are rather low; however, substituting Nb for Sc results in a significant rise in Fe orbital moments. Calculations of magnetic moments for both Sc and Nb atoms reveal spin polarization induced by Fe spin moments, which decreases with Nb content $x$. In ferromagnetic spin configuration, the magnetic moments on Fe 2a and Fe 6h increase in the concentration range $0 \le x \le 0.2$ before decreasing for $x > 0.3$, in qualitative agreement with the experiment. Similar behavior is obtained for the mean field approach (MFA) calculated Curie temperature. However, theoretical calculations taking into account the ferromagnetic spin arrangement are unable to reproduce the sharp drop of the experimental magnetization for $x > 0.3$. The antiparallel spin alignment of Fe 6h spins was considered, as found in Ti₂Fe or Sc₁₋ₓTiₓFe₂ (C14 Laves phase) [29,30]. The agreement between the theoretical calculations and magnetization measurements is improved by taking into account the ferrimagnetic spin configuration, which is more stable for $x > 0.5$. The evidence of the volume dependence of the spin ground state may add support to the change of the magnetic spin ordering by Nb for Sc substitution in the Sc₁₋ₓNbₓFe₂ alloys.

### CRediT authorship contribution statement

**D. Benea:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.
**S. El Bidaoui:** Visualization, Validation, Formal analysis, Data curation.
**L.V.B. Diop:** Writing – review & editing, Writing – original draft, Validation, Formal analysis, Data curation.
**O. Isnard:** Writing – review & editing, Writing – original draft, Validation, Supervision, Investigation, Formal analysis, Data curation.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

This work was supported by a grant from the Executive Agency for Higher Education, Research, Development and Innovation Funding (UEFISCDI), project number 21PCE/2025-PN-IV-P1-PCE-2023-0969, within PNCD IV Program 5.1 - Idei.

### Data availability

Data will be made available on request.

### References

[1] J.J.M. Franse, R.J. Radwanski, Handbook magn. Mater. 7 (1993) 307, https://doi.org/10.1016/S1567-2719(05)80046-0.

[2] R. Ballou, J. Deportes, R. Lemaire, B. Ouladdiaf, J. Appl. Phys. 63 (1988) 3487, https://doi.org/10.1063/1.340745.

[3] R. Ballou, C. Lacroix, M.D. Nunez Regueiro, Phys. Rev. Lett. 66 (1991) 1910, https://doi.org/10.1103/PhysRevLett.66.1910.

[4] Y. Yamada, H. Nakamura, Y. Kitaoka, K. Asayama, K. Koga, A. Sakata, T. Murakami, J. Phys. Soc. Jpn. 59 (1990) 2976, https://doi.org/10.1143/JPSJ.59.2976.

[5] T. Nakamichi, J. Phys. Soc. Jpn. 25 (1968) 1189, https://doi.org/10.1143/JPSJ.25.1189.

[6] P.J. Brown, J. Deportes, B. Ouladdiaf, J. Phys.: Condens. Mat. 4 (1992) 10015, https://doi.org/10.1088/0953-8984/4/49/028.

[7] Y. Nishihara, J. Magn. Magn. Mater. 70 (1987) 75, https://doi.org/10.1016/0304-8853(87)90367-2.

[8] L.V.B. Diop, D. Benea, S. Mankovsky, O. Isnard, J. Alloys Compd. 643 (2015) 239, https://doi.org/10.1016/j.jallcom.2015.04.134.

[9] S. Friedemann, W.J. Duncan, M. Hirschberger, T.W. Bauer, R. Küchler, A. Neubauer, M. Brando, C. Pfleiderer, F. Malte Grosche, Nat. Phys. 14 (2018) 62, https://doi.org/10.1038/nphys4242.

[10] A. Alam, D.D. Johnson, Phys. Rev. Lett. 107 (2011) 206401, https://doi.org/10.1103/PhysRevLett.107.206401.

[11] O.I. Bodak, B.Ya Kotur, I.S. Gavrilenko, V.Ya Markiv, V.G. Ivanchenko, Dopov. Akad. Nauk. Ukr. RSR (4) (1978) 365-370.

[12] H. Okamoto, J. Phase Equilib. Diff. 38 (2017) 829, https://doi.org/10.1007/s11669-017-0590-3.

[13] Z. Jing-Ting, H. Yibole, B. Narsu, Z.Q. Ou, O. Haschuluu, O. Tegus, F. Guillou, Intermetallics 136 (2021) 107252, https://doi.org/10.1016/j.intermet.2021.107252.

[14] S. El Bidaoui, L.V.B. Diop, F. Maccari, O. Isnard, J. Alloys Compd. 985 (2024) 174013, https://doi.org/10.1016/j.jallcom.2024.174013.

[15] S. El Bidaoui, L.V.B. Diop, O. Isnard, Phys. Rev. B 110 (2024) 224406, https://doi.org/10.1103/PhysRevB.110.224406.

[16] The Munich SPR-KKR package, version 7.7, H. Ebert et al. http://ebert.cup.uni-muenchen.de/SPRKKR.

[17] H. Ebert, D. Ködderitzsch, J. Minar, Rep. Prog. Phys. 74 (2011) 096501, https://doi.org/10.1088/0034-4885/74/9/096501.

[18] S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200, https://doi.org/10.1139/p80-159.

[19] H. Monkhorst, J. Pack, J. Phys. Rev. B 13 (1976) 5188, https://doi.org/10.1103/PhysRevB.20.1504.

[20] J.S. Faulkner, Prog. Mater. Sci. 27 (1982) 1, https://doi.org/10.1016/0079-6425(82)90005-6.

[21] A.I. Liechtenstein, M.I. Katsnelson, V.P. Antropov, V.A. Gubanov, J. Magn. Magn. Mater. 67 (1987) 65, https://doi.org/10.1016/0304-8853(87)90721-9.

[22] P.W. Anderson, Solid State Phys. 14 (1963) 99, https://doi.org/10.1016/S0081-1947(08)60260-X.

[23] P. Nieves, S. Arapan, J. Maudes-Raedo, R. Marticorena-Sánchez, N.L. Del Brío, A. Kovacs, C. Echevarria-Bonet, D. Salazar, J. Weischenberg, H. Zhang, O. Yu Vekilova, R. Serrano-López, J.M. Barandiaran, K. Skokov, O. Gutfleisch, O. Eriksson, H.C. Herper, T. Schrefl, S. Cuesta-López, Comput. Mat. Sci. 168 (2019) 188, https://doi.org/10.1016/j.commatsci.2019.06.007.

[24] S.S. Barsanov, J. Struct. Chem. 3 (1962) 590.

[25] K. Terao, H. Yamada, E. Hoffmann, P. Entel, J. Phys. Soc. Jpn. 69 (2000) 1430, https://doi.org/10.1143/JPSJ.69.1430.

[26] K. Ikeda, T. Nakamichi, T. Yamada, M. Yamamoto, J. Phys. Soc. Jpn. 36 (1974) 611, https://doi.org/10.1143/JPSJ.36.611.

[27] S.G. Sankar, W.E. Wallace, AIP Conf. Proc. 29 (1976) 334, https://doi.org/10.1063/1.30654.

[28] J. Beloševic-Cavor, V. Koteski, N. Novakovic, G. Concas, F. Congiu, G. Spano, Eur. Phys. J. B 50 (2006) 425, https://doi.org/10.1140/epjb/e2006-00160-7.

[29] K. Rupprecht, J. Lu, G. Wortmann, J. Przewoznik, J. Zukrowski, Acta Phys. Pol. 100 (2001) 789.

[30] H.R. Rechenberg, L. Morellon, P.A. Algarabel, M.R. Ibarra, Phys. Rev. B 71 (2005) 104412, https://doi.org/10.1103/PhysRevB.71.104412.

[31] S. Asano, S. Ishida, J. Phys. Cond. Mater. 1 (1989) 8501, https://doi.org/10.1088/0953-8984/1/44/020.
[32] S. Asano, S. Ishida, J. Phys. F: Met. Phys. 18 (1988) 501, https://doi.org/10.1088/0305-4608/18/3/020.

[33] O. Isnard, D. Fruchart, J. Alloy Comp. 205 (1994) 1, https://doi.org/10.1016/0925-8388(94)90757-9.

[34] K. Sato, L. Bergqvist, J. Kudrnovský, P.H. Dederichs, O. Eriksson, I. Turek, B. Sanyal, G. Bouzerar, H. Katayama-Yoshida, V.A. Dinh, T. Fukushima, H. Kizaki, R. Zeller, Rev. Mod. Phys. 82 (2010) 1633, https://doi.org/10.1103/RevModPhys.82.1633.