#
Magnetic Properties of the Fe_{2}B Alloy Doped with Transition Metal Elements

^{*}

## Abstract

**:**

_{1−x}M

_{x})

_{2}B alloys have been calculated using the spin-polarized relativistic Korringa–Kohn–Rostoker (SPR-KKR) band structure method. The transition metal elements M (M = Co, Ni, Mo, Ta, W and Re) considered in the present study are reported to form stable M

_{2}B or FeMB alloys with a tetragonal Cu

_{2}Al structure type. The experimental studies show that the Fe

_{2}B alloy has a large magnetization (173 Am

^{2}/kg), a large Curie temperature (1017 K) and a relatively large anisotropy constant K

_{1}(−0.80 MJ/m

^{3}), but the alloy is inappropriate for permanent magnet applications due to in-plane easy magnetization axis (EMD). The present investigations show the magnetocrystalline anisotropy behaviour by doping with selected d-elements aiming to find an appropriate dopant which is able to switch the EMD from planar to axial and to enhance the magnetocrystalline anisotropy energy (MAE) value without a major decrease of magnetization and Curie temperature.

## 1. Introduction

_{2}B compounds occur as precipitates in amorphous alloys, but they can also be obtained from the high-temperature synthesis route [2]. Studies on the iron borides also reveal their potential use for H storage and fuel cell applications [3,4] or in the magnetically induced hyperthermia [5].

_{2}B [6] show their excellent magnetic properties characterized by a large saturation magnetization (173 Am

^{2}/kg) and a relatively large magnetic moment per Fe (about 1.9 µ

_{B}). A Curie temperature of 1017 K has been measured for Fe

_{2}B polycrystals [6]. Previous investigations on the magnetic anisotropy of the Fe

_{2}B alloy [7,8] showed a peculiar behaviour with increasing temperature. At a low temperature, the Fe

_{2}B alloy is characterized by an in-plane easy magnetization direction (EMD) which changes the sign, becoming axial with increasing temperature (at 524 K) and reaching a maximum value of ~200 kJ/m

^{3}at ~750 K [8].

_{2}B, a practical route to obtain a rare-earth-free material with an easy-axis anisotropy is to substitute Fe with heavier transition elements, such as 4d or 5d, which have stronger spin-orbit coupling and could support the enhanced easy-axis anisotropy needed for a permanent magnet material [9,10].

_{2}B-based alloys, as the rules to increase the anisotropy are difficult to predict [11]. For example, an axial anisotropy has been found experimentally in (Fe

_{1−}

_{x}Co

_{x})

_{2}B alloys for 0.2 ≤ x ≤ 0.35 [7].

_{1−x}M

_{x})

_{2}B alloys (M = Co, Ni, Mo, W, Ta and Re) are presented in the following. Our theoretical results are compared with the previous experimental and theoretical data [6,7,9,12] in order to show the reliability of the present study. As suggested by Erdström et al. [12], the coherent potential approximation (CPA) for the substitutional disorder as well as the full potential (FP) approach for the potential symmetry are expected to allow us to give an accurate description for the MAE of the (Fe

_{1−x}M

_{x})

_{2}B alloys. The possibility of obtaining a semihard magnetic alloy with appropriate intrinsic magnetic properties to be integrated in a nanocomposite magnet is discussed.

## 2. Materials and Methods

_{1−x}M

_{x})

_{2}B alloys, the calculations were performed in the relativistic mode, accounting for all the relativistic effects, including spin-orbit coupling. The angular expansion of the basis set was taken up to l = 2 for 3d metal Fe and l = 3 for 4d/5d metals M. The generalized gradient approximation within the parametrization of Perdew et al. (GGA-PBE) has been used to account for the exchange and correlation effects [15]. The special points method of Monkhorst et al. was used for the k-space integration [16]. Self-consistent band structure calculations have been performed using the full potential approach and the total energy convergence criterion of 10

^{−6}Ry.

_{ij}exchange-coupling parameters as a function of distance for all magnetic atoms were performed using the expression derived by Liechtenstein [19] based on the magnetic force theorem.

_{ij}as [13,19]

_{ij}exchange-coupling parameters within the mean field approach [19,20] have the following expression:

## 3. Results and Discussions

_{2}B alloy crystallizes in the tetragonal Cu

_{2}Al structure type (I4/mcm space group) with Fe atoms sitting on 8h and B atoms on 4a crystallographic sites. The structure can be seen as alternate layers of Fe and B. The Fe atoms are in a tetragonal surrounding, with two of the Fe–B distances being shorter and the other two being longer than in bcc Fe. The crystal structure of the Fe

_{2}B alloy created using the XCrySDen visualization program [21] is presented in Figure 1.

_{2}B alloys show a linear lattice constant dependence by the occupation of the 8h crystallographic site by a binary combination of transitional metal [7,22]. Based on this evidence, the lattice constants used in the theoretical calculations were obtained using the linear dependence of the reported experimental lattice constants [1,7,23,24,25,26]. In addition, the experimental free parameter of the 8h crystallographic site for Fe

_{2}B has been considered [1].

#### 3.1. Density of States Calculations

_{1−x}M

_{x})

_{2}B alloys (M = Co, Ni, Mo, Ta, W and Re) has been investigated using density of states (DOS) calculations (Figure 2). The exchange splitting of total DOS is visible for all (Fe

_{1−x}M

_{x})

_{2}B alloys, suggesting their ferromagnetic order. In order to see the evolution of the DOS by doping with magnetic as well as non-magnetic atoms, the DOS of doped alloys (with M content fixed at x = 0.20) is represented along with that of the undoped Fe

_{2}B alloy. One can notice a different behaviour when the (Fe

_{1−x}M

_{x})

_{2}B alloys are doped with 3d elements (M = Co, Ni) and with 4d/5d-elements (M = Mo, Ta, W, Re). In the 3d doping, the spin-up bands are shifted to the higher energy, diminishing in this way the exchange splitting. On the other hand, the 4d/5d-doping induces a broadening of the electronic bands, which increase from Ta to Re, through an increase of the valence electron numbers. Due to the DOS broadening, the exchange splitting as well as the total magnetic moments of the doped alloys decrease.

#### 3.2. Magnetic Moments

_{2}B found a spin magnetic moment of m

_{s}= 3.84 µ

_{B}/f.u. and an orbital moment of 0.09 µ

_{B}/f.u. (Table 1). The spontaneous magnetization µ

_{0}M

_{s}calculated value is 1.64 T, slightly higher than the value of 1.52 T measured by Coene et al. [27] at room temperature. The calculated mass magnetization is 177.8 Am

^{2}/kg, in good agreement with the value determined by Wang et al. (173 Am

^{2}/kg) [6]. The spin magnetic moment for Fe atoms is 1.99 µ

_{B}, close to the values obtained by Takacs et al. using Mössbauer spectroscopy at 20 K (1.908 µ

_{B}) [28].

_{s}) and orbital (m

_{l}) moments in (Fe

_{1−x}M

_{x})

_{2}B alloys vs. the doping amount x of M elements are represented in Figure 3a,b, respectively. The m

_{s}of Fe ranges between 1.97 and 2.03 µ

_{B}in the investigated doping range (0–0.3). For low doping (x = 0.04), all doping elements would induce an increase of the spin magnetic moment of Fe, but this behaviour changes by increasing the doping content. For large doping (0.12 < x < 0.30), one can distinguish between the doping elements which reduce the Fe spin moment (Re and Ni) and doping elements which enhance the m

_{s}of Fe (M = Co, W and Mo). The Fe orbital moment (m

_{l}) dependence is almost linear with doping.

_{l}is obtained by doping with 4d/5d elements, which is more pronounced for 5d elements (Ta, W and Re).

_{1−x}M

_{x})

_{2}B alloys are represented in Figure 3c vs. the doping amount x. The Co spin moment (~1.28 µ

_{B}) is almost independent of the doping content, whilst the m

_{s}of Ni ranges between 0.42 and 0.38 µ

_{B}, decreasing with x. The non-magnetic 4d/5d doping elements (Figure 3d) have small induced negative spin moments between −0.1 and −0.25 µ

_{B}(M = Ta, W and Mo) or small positive induced magnetic moments up to 0.04 µ

_{B}(M = Re), slightly dependent on the doping content. The boron spin moment vs. doping content x (Figure 3e) shows its negative polarization, which is reduced by an increase of doping amount x and is more pronounced for 5d elements (Re and W) and less pronounced for Co and Ni doping. The M and B orbital moments doping dependencies are negligible and not shown.

_{1−x}M

_{x})

_{2}B alloys (M = Co, Ni, Mo, W, Ta and Re) have been used to calculate the magnetization µ

_{0}M

_{s}, shown in Figure 4. As expected, the 3d magnetic elements (Co and Ni) have less impact on the magnetization of (Fe

_{1−x}M

_{x})

_{2}B alloys, but Co is more effective in keeping the large magnetization needed for a permanent magnet. On the other hand, doping with non-magnetic 4d and 5d elements induces a stronger decrease in magnetization. Between the 4d and 5d elements, doping with Ta produces a much stronger magnetization decrease compared with Re due to different M-Fe hybridization.

#### 3.3. The Magnetocrystalline Anisotropy Energy (MAE)

_{2}B calculated using the torque method is shown in Table 1. As can be seen in Table 1, MAE (K

_{1}) shows an in-plane easy magnetization direction (EMD) with a value of K

_{1}= −1.09 MJ/m

^{3}, in agreement with a previous experiment (−0.80 MJ/m

^{3}[8]).

_{1}in kJ/m

^{3}) for the (Fe

_{1−x}M

_{x})

_{2}B alloys (M = Co, Ni, Mo, W, Ta and Re) are shown in Figure 5. As can be seen in Figure 5, the (Fe

_{1−x}M

_{x})

_{2}B alloys have positive anisotropy in a specific doping range only for M = Co, Re and Ta. The case of (Fe

_{1−x}Co

_{x})

_{2}B has been considered extensively by Edström et al. [12]. The theoretical calculations of Edström et al. [12] by Wien2k and FPLO codes using general gradient approximation (GGA) for exchange-correlation potential reproduced the measurements at low temperatures of K

_{1}(x) in (Fe

_{1−x}Co

_{x})

_{2}B very well. The use of coherent potential approximation (CPA) instead of virtual crystal approximation (VCA) removes the MAE overestimation for the Fe-rich alloys. More accurate numerical values of MAE have been obtained using the full potential approach (FP) instead of atomic spheres approximation (ASA) [12].

_{1−x}Co

_{x})

_{2}B alloys obtained in our calculations, with the same interval for the Co amount with positive MAE. The maximum value of K

_{1}of 0.315 MJ/m

^{3}for x = 0.24 is slightly lower than the anisotropy constant K

_{1}at room temperature which was obtained for x = 0.25 (K

_{1}= 450 kJ/m

^{3}) [7]. On the other hand, the theoretical calculations of Edström et al. [12] found K

_{1}= 0.77 MJ/m

^{3}for x = 0.3, whilst the experimental K

_{1}value in the low temperature limit is 0.51 MJ/m

^{3}for x = 0.3 [9]. The numerical differences can be attributed to the different methods used for obtaining the lattice parameters, based on experimental parameters in the present work and by theoretical lattice relaxation [12]. On the other hand, the structural relaxation effects were found to be less effective in the case of Fe ferromagnetic alloys with only ~0.5% reduction of MAE [29].

_{1−x}M

_{x})

_{2}B alloys is by M = Re with K

_{1}exceeding 1.2 MJ/m

^{3}for x ≥ 0.16. The largest value of K

_{1}has been obtained for the (Fe

_{0.76}Re

_{0.24})

_{2}B alloy (1.5 MJ/m

^{3}). A small positive K

_{1}is obtained for Ta at x ≥ 0.28, but the Ta doping is less effective for the purpose of the present study, due to an already reduced magnetization value.

_{1}were found for the other 5d and 4d elements considered in the present study (M = Mo, Ta and W). One should mention the (Fe

_{1−x}Ni

_{x})

_{2}B alloys which have an increased K

_{1}close to that for M = W, despite the much lower spin-orbit coupling of M = Ni. Other influences on magnetocrystalline anisotropy except spin-orbit coupling might be effective in the case of (Fe

_{1−x}Ni

_{x})

_{2}B alloys, for example, d-band filling or crystal field altering [11].

_{2}B alloy as well as in a Re-doped alloy (x = 0.12) with a positive value of K

_{1}have been investigated. The calculations of K

_{1}performed for Fe

_{2}B considered a mixing amount δ between the Fe atoms sitting on 8h and B atoms on 4a crystallographic sites, according to the formula ${\left({\mathrm{F}\mathrm{e}}_{1-\mathsf{\delta}}{\mathrm{B}}_{\mathsf{\delta}}\right)}_{2}^{8h}{\left({\mathrm{B}}_{1-2\mathsf{\delta}}{\mathrm{F}\mathrm{e}}_{2\mathsf{\delta}}\right)}^{4a}.$ The K

_{1}dependence on the intermixing of Fe/B atoms in Fe

_{2}B is shown in Figure 6a. As can be seen in Figure 6a, the disorder effects on the anisotropy constant K

_{1}of Fe

_{2}B are significant, as an amount of mixing of δ = 0.05 would decrease the absolute value of anisotropy from −0.19 meV/f.u. to −0.12 meV/f.u. A similar investigation has been performed for the (Fe

_{0.88}Re

_{0.12})

_{2}B system, considering the intermixing of Re/B atoms (Figure 6b) described by the formula ${\left({\mathrm{F}\mathrm{e}}_{0.88}{\mathrm{R}\mathrm{e}}_{0.12-\mathsf{\delta}}{\mathrm{B}}_{\mathsf{\delta}}\right)}_{2}^{8h}{\left({\mathrm{B}}_{1-2\mathsf{\delta}}{\mathrm{R}\mathrm{e}}_{2\mathsf{\delta}}\right)}^{4a}.$ The anisotropy constant K

_{1}of (Fe

_{0.88}Re

_{0.12})

_{2}B (0.105 meV/f.u. for δ = 0) decreased at zero for δ~0.04, becoming negative for δ > 0.04 with a rate of decrease of ~0.02 meV/f.u. per δ mixing amount, which is detrimental for the purpose of reaching a large axial anisotropy in (Fe

_{1−x}Re

_{x})

_{2}B alloys. The mixing of Fe/B atoms between the 8h and 4a crystallographic sites as well as the mixing of Re/B atoms between the same crystallographic sites are not energetically favourable, as was found by the total energy calculations. Substitutional disorder could occur in the experiment during the preparation (by rapid quenching) or by thermic activation. In the case of the Fe

_{2}B alloy, disorder effects could explain the differences in the K

_{1}values obtained in the experimental investigations.

#### 3.4. The Curie Temperatures

_{1−x}M

_{x})

_{2}B alloys (M = Co, Ni, Mo, W, Ta and Re) are shown in Figure 7. The Curie temperature calculated using the mean field approach (MFA) for Fe

_{2}B gives a value of 1284 K, which is about 20% higher than the experimental value of 1017 K [6]. Knowing this deficiency of the MFA [30], we used also a more realistic estimated Curie temperature, calculated as ${T}_{c}^{estim}=0.8{T}_{c},$ to discuss the T

_{c}behaviour of the alloys with different M doping.

_{c}were obtained for the 3d magnetic elements; M = Co, with estimated Curie temperatures over 950 K over the whole doping range x ≤ 0.3. For Ni, ${T}_{c}^{estim}$ ≥ 650 K for x ≤ 0.3. The Curie temperature for M = Re, despite its steep decrease with x content of Re, keeps a ${T}_{c}^{estim}$≥ 530 K for x = 0.28. The alloy with the highest anisotropy constant K

_{1}(1.5 MJ/m

^{3}) is (Fe

_{0.76}Re

_{0.24})

_{2}B, which has a magnetization µ

_{0}M

_{s}= 1.22 T and an estimated Curie temperature ${T}_{c}^{estim}$ = 595 K, compatible with operating at temperatures under ~250 °C. The calculated hardness parameter for the (Fe

_{0.76}Re

_{0.24})

_{2}B alloy is κ = 1.1, slightly beyond the limit for hard magnets [10]. A higher µ

_{0}M

_{s}and Curie temperature but lower anisotropy constant K

_{1}were obtained for the (Fe

_{0.8}Re

_{0.2})

_{2}B alloy (µ

_{0}M

_{s}= 1.3 T, MAE of 1.35 MJ/m

^{3}, hardness parameter κ = 1.01, ${T}_{c}^{estim}$ = 660 K), which would make it suitable for permanent magnet applications. Unfortunately, the relatively large amount of expensive Re is prohibiting the production of such magnetic material on a large scale.

## 4. Conclusions

_{2}B through doping with selected transition elements M = Co, Ni, Mo, W, Ta and Re show a decrease in the total magnetic moment and Curie temperatures of the alloys by increasing the doping amount, which is more pronounced for non-magnetic 4d/5d elements. The evolution of the anisotropy constant K

_{1}from planar for the Fe

_{2}B alloy to axial was found only for M = Re, Co and Ta in the investigated doping range (x ≤ 0.28). The largest theoretical value of the anisotropy constant within the investigated alloys was obtained for (Fe

_{0.76}Re

_{0.24})

_{2}B with K

_{1}= 1.5 MJ/m

^{3}, which is characterized by a magnetization µ

_{0}M

_{s}= 1.22 T and an estimated Curie temperature ${T}_{c}^{estim}$ = 595 K. The intrinsic properties of (Fe

_{1−x}Re

_{x})

_{2}B alloys with x = 0.2–0.24 are compatible with those of semihard alloys; these are recommended for further experimental investigations. On the other hand, disorder effects have a strong influence on the MAE of the Fe

_{2}B-based alloys, which must be accounted for by experimental investigations.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The tetragonal Cu

_{2}Al structure type (I4/mcm space group) of the Fe

_{2}B alloy, with Fe atoms (red spheres) and B atoms (black spheres).

**Figure 2.**(

**a**) The density of states (DOS) of 3d-doped (Fe

_{0.8}M

_{0.2})

_{2}B alloys (M = Co, Ni) and (

**b**) the density of states (DOS) of 4d- and 5d-doped (Fe

_{0.8}M

_{0.2})

_{2}B alloys (M = Mo, Ta, W and Re).

**Figure 3.**The dependence of the spin and orbital moments of Fe (

**a**,

**b**), M (

**c**,

**d**) and B (

**e**) vs. doping content x in the (Fe

_{1−x}M

_{x})

_{2}B alloys.

**Figure 4.**Composition dependence of magnetization µ

_{0}M

_{s}in (Fe

_{1−x}M

_{x})

_{2}B alloys (M = Co, Ni, Mo, W, Ta and Re).

**Figure 5.**Anisotropy constants K

_{1}for (Fe

_{1−x}M

_{x})

_{2}B alloys (M = Co, Ni, Mo, W, Ta and Re).

**Figure 6.**(

**a**) Disorder effects on anisotropy constant K

_{1}of the Fe

_{2}B alloy. (

**b**) Disorder effects on anisotropy constant K

_{1}of (Fe

_{0.88}Re

_{0.12})

_{2}B alloys. Mixing of δ atoms of Fe/B and Re/B has been considered between 8h and 4a crystallographic sites, respectively.

**Figure 7.**(

**a**) The mean field approach (MFA) calculated Curie temperatures for the (Fe

_{1−x}M

_{x})

_{2}B alloys (M = Co, Ni, Mo, W, Ta and Re) and (

**b**) the estimated Curie temperatures as ${T}_{c}^{estim}=0.8{T}_{c}$.

**Table 1.**Spin and orbital magnetic moments together with the magnetocrystalline anisotropy constant K

_{1}for the Fe

_{2}B alloy.

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**MDPI and ACS Style**

Benea, D.; Pop, V.
Magnetic Properties of the Fe_{2}B Alloy Doped with Transition Metal Elements. *Magnetochemistry* **2023**, *9*, 109.
https://doi.org/10.3390/magnetochemistry9040109

**AMA Style**

Benea D, Pop V.
Magnetic Properties of the Fe_{2}B Alloy Doped with Transition Metal Elements. *Magnetochemistry*. 2023; 9(4):109.
https://doi.org/10.3390/magnetochemistry9040109

**Chicago/Turabian Style**

Benea, Diana, and Viorel Pop.
2023. "Magnetic Properties of the Fe_{2}B Alloy Doped with Transition Metal Elements" *Magnetochemistry* 9, no. 4: 109.
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