First-Principles Study on the Stability, Site Preference, Electronic Structure and Magnetism of Alloyed Fe3B with Ni3P-Type Structure

First-principles calculations were performed to investigate the site preference of alloying elements, and the effect of alloying elements on stability, electronic structure and magnetism of Ni3P-type Fe3B. The calculated energies suggested that all studied compounds are thermodynamically stable while it is relatively difficult to synthesize the (Fe2.875,Cu0.125)B, (Fe2.875,W0.125)B and (Fe2.875,Nb0.125)B. The (Fe2.875,W0.125)B is the most stable compound from the view of cohesive energy. Mn element prefers to occupy the Fe2 site, whereas the others are more likely to reside in the Fe1 site. It can be found from the electronic structures that the DOSs of both Fe3B and alloyed Fe3B are dominated by Fe-d states, and all the compounds mainly contain Fe-B covalent bond, Fe-Fe covalent bond and Fe-Fe metallic bond. Based on the magnetic moments (Ms) results, it can be known that the Fe3B, (Fe2.875,Mn0.125)B, (Fe2.875,Co0.125)B, (Fe2.875,Ni0.125)B and (Fe2.875,Cu0.125)B are ferromagnetic compounds, whereas the others are ferrimagnetic compounds. Only Mn and Co are able to enhance the magnetism of Fe3B. Moreover, Mn is the most favorable candidate for improving the magnetic properties of Fe3B among the alloying elements. These results can be used to guide the composition design and performance optimization of magnetic materials containing Fe3B with Ni3P-type structure.


Introduction
Magnetic materials play a major role in improving the performance of devices in the field of energy applications, data storage, refrigeration technologies, etc. [1]. Metastable Fe 3 B is an attractive magnetic compound due to its large saturation magnetization and reasonably strong magnetic anisotropy. In particular, it appears in various important magnetic materials, such as nanocomposite permanent magnets (e.g., Fe 3 B/Nd 2 Fe 14 B) [2][3][4][5] and Fe-B-based amorphous and nanocrystalline soft magnetic alloys [6][7][8][9]. For the nanocomposite permanent magnets, Fe 3 B, as the main phase and providing large saturation magnetization, interacts with hard magnetic phase (e.g., Nd 2 Fe 14 B) by exchange-coupling interaction, achieving large energy production. Whereas, in the latter case, nanocrystalline Fe 3 B precipitated from liquid phase or amorphous precursor is generally undesirable, mainly because it is magnetically harder than Fe-based solid solution such as α-(Fe, Si) and α-(Fe, Co, Ni). Further, to improve the magnetic properties, alloying has been widely investigated and used [4,7,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. It was found that three effects can be achieved by alloying elements to affect the magnetic properties of these materials, as follows: (1) partitioning into phases and directly modifying their magnetic properties; (2) changing phase stability further to affect crystallization behaviors, and consequently influencing the size, distribution, content, and type of phases and (3) accelerating nucleation and/or suppressing the growth of phases by segregation. Numerous experimental studies have been conducted. Results

Computational Details
The Ni 3 P-type Fe 3 B has a body-centered-tetragonal structure with a space group of I-4. Each unit cell contains 24 Fe atoms and 8 B atoms; that is, each cell consists of 8 Fe 3 B formula units. B atoms only have one Wyckoff site 8 g (0.288 0.046 0.491), whereas Fe atoms have three different Wyckoff sites, which are Fe1 (8 g (0.078 0.111 0.244)), Fe2 (8 g (0.364 0.031 0.988)) and Fe3 (8 g (0.166 0.220 0.751)), respectively, with equal quantities. The crystal structure of Ni 3 P-type Fe 3 B is shown in Figure 1. To obtain alloyed Fe 3 B, in the unit cell one Fe atom (Fe1 or Fe2 or Fe3) was replaced by alloying elements M (M = Ti, V, Cr, Mn, Co, Ni, Cu, Mo, W or Nb) to generate alloyed borides Fe 23 M 1 B 8 , namely, (Fe 2.875 , M 0.125 )B. All calculations were carried out by Vienna ab initio simulation package (VASP) based on density functional theory (DFT) [38]. The interactions between valence electrons and ionic cores were treated by projector-augmented wave (PAW) pseudopotential [39], where for B a standard version of PAW pseudopotential was used, and an extended M_pv version was applied for all alloying elements M. The generalized gradient approximation (GGA) method in the scheme of Perdew-Burke-Ernzerhof (PBE) was used to deal with exchangecorrelation potentials [40]. The Monkhorst-Pack method was used to generate a k-points mesh in the first irreducible Brillouin zone (BZ). After extensive convergence tests, a cutoff energy of 550 eV and a 5 × 5 × 11 k-points mesh were determined for all calculations. The convergence criteria were as follows: for electronic relaxation, the maximum energy change was set to 1 × 10 −7 eV/atom, and for ion relaxation, the maximum force acting on each atom was set to 0.001 eV/Å. In addition, spin polarization was considered for all the calculations.
Materials 2022, 15, x FOR PEER REVIEW 3 of 12 maximum energy change was set to 1 × 10 −7 eV/atom, and for ion relaxation, the maximum force acting on each atom was set to 0.001 eV/Å. In addition, spin polarization was considered for all the calculations.

Geometry Optimization
The equilibrium cell parameters of Ni3P-type Fe3B and its alloyed counterparts, together with some available experimental and calculated results, are presented in Table 1. The data of Fe3B show that the maximum deviations of lattice constants and cell volume between the calculated values and the previous results [28,41,42] are less than 2% and 4%, respectively, suggesting that our calculation method is credible. Note that small lattice distortion was triggered by alloying elements in all alloyed Fe3B, which is related to the differences of atomic radius and electronegativity between alloying elements and Fe. Figure 2 shows the atomic radius of alloying elements and the cell volume of all studied compounds. As can be seen, the variation trends of cell volume are almost identical to that of the atomic radius of alloying elements, implying that the difference of atomic radius dominates the lattice distortion. In addition, it can be found from Figure 2 that the cell volumes of the compounds are different from each other when the alloying elements occupy a different Fe site. This is mainly attributed to the different neighboring environments of different Fe sites. Previous research [29] reported that the Fe1, Fe2 and Fe3 have 12, 10 and 10 iron neighbors, respectively, and have 2, 4 and 3 boron neighbors, respectively. However, the present results demonstrate that the Fe3 has 11 iron neighbors and 2 boron neighbors. It is well known that Fe3B is a metastable phase, and thus it is relatively difficult to produce it, and further to obtain high-quality polycrystalline X-ray diffraction (XRD) data. The existing XRD data on Fe3B, PDF#39-1316, is marked with "?", implying its lower reliability. Therefore, we simulated the XRD data of Fe3B based on the optimized crystal structure, and the pattern is given in Figure 3a. It can be found that the pattern of PDF#39-1316 is roughly consistent with the simulated one, especially at lower angles. Figure 3b shows the three patterns of Fe2B from the experiment [43], high-quality PDF card data and the simulation based on optimized crystal structure. They are in good agreement with each other, indicating that the simulated result is reliable. Thus, it can be inferred that the simulated XRD data of Fe3B is probably more credible than that of PDF#39-1316.

Geometry Optimization
The equilibrium cell parameters of Ni 3 P-type Fe 3 B and its alloyed counterparts, together with some available experimental and calculated results, are presented in Table 1. The data of Fe 3 B show that the maximum deviations of lattice constants and cell volume between the calculated values and the previous results [28,41,42] are less than 2% and 4%, respectively, suggesting that our calculation method is credible. Note that small lattice distortion was triggered by alloying elements in all alloyed Fe 3 B, which is related to the differences of atomic radius and electronegativity between alloying elements and Fe. Figure 2 shows the atomic radius of alloying elements and the cell volume of all studied compounds. As can be seen, the variation trends of cell volume are almost identical to that of the atomic radius of alloying elements, implying that the difference of atomic radius dominates the lattice distortion. In addition, it can be found from Figure 2 that the cell volumes of the compounds are different from each other when the alloying elements occupy a different Fe site. This is mainly attributed to the different neighboring environments of different Fe sites. Previous research [29] reported that the Fe1, Fe2 and Fe3 have 12, 10 and 10 iron neighbors, respectively, and have 2, 4 and 3 boron neighbors, respectively. However, the present results demonstrate that the Fe3 has 11 iron neighbors and 2 boron neighbors. It is well known that Fe 3 B is a metastable phase, and thus it is relatively difficult to produce it, and further to obtain high-quality polycrystalline X-ray diffraction (XRD) data. The existing XRD data on Fe 3 B, PDF#39-1316, is marked with "?", implying its lower reliability. Therefore, we simulated the XRD data of Fe 3 B based on the optimized crystal structure, and the pattern is given in Figure 3a. It can be found that the pattern of PDF#39-1316 is roughly consistent with the simulated one, especially at lower angles. Figure 3b shows the three patterns of Fe 2 B from the experiment [43], high-quality PDF card data and the simulation based on optimized crystal structure. They are in good agreement with each other, indicating that the simulated result is reliable. Thus, it can be inferred that the simulated XRD data of Fe 3 B is probably more credible than that of PDF#39-1316.

Thermodynamic Stability and Site Preference of Alloying Elements
In order to analyze the thermodynamic stability of alloyed Fe3B and the site p ence of alloying elements, formation enthalpy (Efor) and cohesive energy (Ecoh) were lated according to the following equations: In the equations, n is the number of Fe2B per unit cell, and Etotal ((Fe1−x,Mx)3B denotes the total energy of (Fe1−x,Mx)3B per unit cell. Ecry (Fe), Ecry (B) and Ecry (M) a energy of one atom for simple substance bcc Fe, α-B and M (bcc (V, Cr, Mo, W an hcp (Co and Ti) and fcc (Ni and Cu) and α-Mn), respectively. Finally, Eiso (Fe), Eiso ( Eiso(M) are the energy of one isolated atom for Fe, B and M (Ti, V, Cr, Mn, Co, Ni, C W and Nb), respectively.
The calculated formation enthalpy and cohesive energy of Ni3P-type Fe3B and loyed counterparts are shown in Figure 4. The results suggest that all studied comp are thermodynamically stable, as their formation enthalpy and cohesive energy are tive. The Ec value of Fe3B is consistent with previously calculated results (−22.732 e [28] and −22.576 eV/f.u. [29]), and its Efor value is comparable with previously theo values (−0.864 eV/f.u. [28] and −0.880 eV/f.u. [44]). Although the formation enthalp (Fe2.875,Cu0.125)B, (Fe2.875,W0.125)B and (Fe2.875,Nb0.125)B are negative, Cu, W and Nb c easily enter the Fe3B lattice to replace the Fe atom since their formation enthalpi greater than that of Fe3B. Similarly, Ni is also unable to easily substitute the Fe atom Fe2 site. (Fe2.875,Mo0.125)B has the lowest formation enthalpy, which means that it is to fabricate than the others. From the view of cohesive energy, (Fe2.875,W0.125)B is the stable compound among those considered. The site preference of alloying elements judged by formation enthalpy and cohesive energy; namely, for the (Fe2.875,M0.1 which the M atom occupies the Fe1, Fe2 and Fe3 site, respectively, the lower ener notes that the M atom prefers to occupy this site. Hence, Figure 4 indicates that all al elements prefer to occupy Fe1, Fe3 and Fe2 sites in turn except for Mn, Mo and W thermore, Mn prefers to occupy Fe2, Fe3 and Fe1 sites in turn, while for Mo and order is Fe1, Fe2 and Fe3. It is necessary to emphasize that, apart from Mn, all the al elements prefer to enter the Fe1 site in the Ni3P-type Fe3B.  Table 1. Cell parameters of Ni 3 P-type Fe 3 B and its alloyed counterparts.

Model
Cell Parameters

Thermodynamic Stability and Site Preference of Alloying Elements
In order to analyze the thermodynamic stability of alloyed Fe 3 B and the site preference of alloying elements, formation enthalpy (E for ) and cohesive energy (E coh ) were calculated according to the following equations: In the equations, n is the number of Fe 2 B per unit cell, and E total ((Fe 1−x ,M x ) 3 B, cell) denotes the total energy of (Fe 1−x ,M x ) 3  The calculated formation enthalpy and cohesive energy of Ni 3 P-type Fe 3 B and its alloyed counterparts are shown in Figure 4. The results suggest that all studied compounds are thermodynamically stable, as their formation enthalpy and cohesive energy are negative. The E c value of Fe 3 B is consistent with previously calculated results (−22.732 eV/f.u. [28] and −22.576 eV/f.u. [29]), and its E for value is comparable with previously theoretical values (−0.864 eV/f.u. [28] and −0.880 eV/f.u. [44]). Although the formation enthalpies of (Fe 2.875 ,Cu 0.125 )B, (Fe 2.875 ,W 0.125 )B and (Fe 2.875 ,Nb 0.125 )B are negative, Cu, W and Nb cannot easily enter the Fe 3 B lattice to replace the Fe atom since their formation enthalpies are greater than that of Fe 3 B. Similarly, Ni is also unable to easily substitute the Fe atom of the Fe2 site. (Fe 2.875 ,Mo 0.125 )B has the lowest formation enthalpy, which means that it is easier to fabricate than the others. From the view of cohesive energy, (Fe 2.875 ,W 0.125 )B is the most stable compound among those considered. The site preference of alloying elements is also judged by formation enthalpy and cohesive energy; namely, for the (Fe 2.875 ,M 0.125 )B in which the M atom occupies the Fe1, Fe2 and Fe3 site, respectively, the lower energy denotes that the M atom prefers to occupy this site. Hence, Figure 4 indicates that all alloying elements prefer to occupy Fe1, Fe3 and Fe2 sites in turn except for Mn, Mo and W. Furthermore, Mn prefers to occupy Fe2, Fe3 and Fe1 sites in turn, while for Mo and W the order is Fe1, Fe2 and Fe3. It is necessary to emphasize that, apart from Mn, all the alloying elements prefer to enter the Fe1 site in the Ni 3 P-type Fe 3 B.

Electronic and Magnetic Properties
To understand the bonding characteristics and magnetic properties of Fe3B and alloyed Fe3B, their spin-polarized total densities of states (TDOSs) and partial densities of states (PDOSs) were calculated, and the results are exhibited in Figure 5. Since Mn preferentially occupies the Fe2 site, whereas the others are more likely to occupy the Fe1 site, the DOS calculations for all alloying elements except for Mn are based on the model that the M atom resides in the Fe1 site; similarly, for Mn the calculations are based on the Mn atom preferentially occupying the Fe2 site. Figure 5 shows that all DOSs have a similar shape, suggesting that the compounds have similar bonding and magnetic characteristics. The absence of an energy gap at Fermi level (Ef) indicates that a metallic bond should be

Electronic and Magnetic Properties
To understand the bonding characteristics and magnetic properties of Fe 3 B and alloyed Fe 3 B, their spin-polarized total densities of states (TDOSs) and partial densities of states (PDOSs) were calculated, and the results are exhibited in Figure 5. Since Mn preferentially occupies the Fe2 site, whereas the others are more likely to occupy the Fe1 site, the DOS calculations for all alloying elements except for Mn are based on the model that the M atom resides in the Fe1 site; similarly, for Mn the calculations are based on the Mn atom preferentially occupying the Fe2 site. Figure 5 shows that all DOSs have a similar shape, suggesting that the compounds have similar bonding and magnetic characteristics. The absence of an energy gap at Fermi level (E f ) indicates that a metallic bond should be contained in all compounds. In addition, it is easy to see that the TDOSs are greatly dominated by the Fe-d states. From the lower energy level to the higher energy level, all TDOSs and PDOSs of Fe and B are composed of the lower valence band, the upper valence band and unoccupied conduction band, in which there is an energy gap between the lower valence band and the upper valence band. Moreover, the lower valence band of TDOSs primarily consists of Fe-s, Fe-p and B-s states, whereas the upper valence band is mostly determined by the Fe-d states; additionally, a small number of Fe-p, Fe-s, B-p and M-d states can be observed. Due to the negligible contribution from M-s and M-p states, only the M-d states are presented in Figure 5. As can be seen, for 3d transition metals, more and more M-d states appear in the upper valence band with increasing valences, which is in good accord with the case of alloyed Fe 2 B [45]. The PDOSs reveal that there is a strong hybridization between the Fe-d states and the B-p states in the energy range of about −6 eV to −3 eV, which implies that all the compounds possess strong Fe-B covalent bonds. Moreover, the large overlaps between the Fe-d states and the Fe-s, Fe-p states suggest the existence of Fe-Fe covalent bonds.
To gain insight into bonding nature, we checked the nearest-neighbor table of atoms, which was calculated on the basis of the optimized Fe 3 B crystal structure. The type of bond and their bond length are summarized in Table 2, and the visualized Fe-B and Fe-Fe bonds with typical bond length are presented by charge-density plots in Figure 6. From Table 2, four Fe-B bond lengths can be found: 2.12 Å (Fe3-B bond), 2.15 Å (Fe2-B bond), 2.16 Å (Fe1-B bond and Fe2-B bond) and 2.18 Å (Fe2-B bond). Figure 6a gives a Fe2-B bond of 2.18 Å, clearly demonstrating that this bond is covalent due to the appearance of elongated contours between the Fe2 atom and the B atom. Thus, it can be concluded that all Fe-B bonds are covalent, and the Fe3-B bond has the largest bond energy because it has the shortest bond length. The Fe-Fe bonds have many more bond lengths because the quantity of Fe atoms is three times that of B atoms. Moreover, as can be seen from Figure 6  to the addition of alloying element M, M-Fe bonds and M-B bonds can also be observed in the alloyed Fe3B. Moreover, the bonding strength of the other bonds is affected by the M atoms, revealed by the nearest-neighbor table of atoms for the alloyed Fe3B. These facts mean that it is very difficult to obtain the rule of the effect of alloying elements on the bonding behaviors of Fe3B. Therefore, we did not carry out a further analysis about the bonding behaviors of alloyed Fe3B.   Fe1-B Fe2-B Fe3-B Fe1-Fe1 Fe2-Fe2 Fe3-Fe3 Fe1-Fe2 Fe1-Fe3 Fe2-Fe3 B Additionally, as can be seen from Figure 5, all the TDOSs are mainly composed of two large majority-spin and minority-spin DOS peaks. Moreover, it is interesting to note that one of the minority-spin DOS peaks is located at the unoccupied conduction band, indicating that the majority-spin electrons are greater in number than the minority-spin electrons. In other words, all these compounds are magnetic. Moreover, their magnetism is significantly affected by Fe-d states. Here, in order to quantitatively analyze the effects of alloying elements and their site preference on the magnetic properties of Fe3B, we calculated the magnetic moments (Ms) values of unit cells, and the Ms values of different elements and the interstices of unit cells. For the calculations, the default Wigner-Seitz radii were employed, and only spin magnetic moments were considered because orbital Ms is very small. The results are presented in Table 3. It can be found that for Fe3B the calculated Ms of Fe is 1.921 μB, which is consistent with previously obtained theoretical values 2.02 μB [28] and 1.99 μB [29]. Additionally, the Ms values of Fe1 (2.00 μB), Fe2 (1.80 μB) and Fe3 (1.97 μB) match well with the previous results (Fe1 (2.14 μB), Fe2 (1.89 μB) and Fe3 (1.95 μB)) [29]. It is necessary to point out that although the magnetic properties of some borides, such as Fe3B [23], (Fex,Cr1−x)3B [23], (Fex,Mn1−x)3B [12] and (Fex,Co1−x)3B [11], have been measured by experiments, the crystal structures of these borides were not carefully and clearly distinguished. Thus, these experimental data will be discussed in a following work about the alloyed Fe3B with Ti3P-type structure, since according to the calculated results of the present work and the next work, their crystal structures can probably be distinguished. Additionally, as can be seen from Figure 5, all the TDOSs are mainly composed of two large majority-spin and minority-spin DOS peaks. Moreover, it is interesting to note that one of the minority-spin DOS peaks is located at the unoccupied conduction band, indicating that the majority-spin electrons are greater in number than the minority-spin electrons. In other words, all these compounds are magnetic. Moreover, their magnetism is significantly affected by Fe-d states. Here, in order to quantitatively analyze the effects of alloying elements and their site preference on the magnetic properties of Fe 3 B, we calculated the magnetic moments (Ms) values of unit cells, and the Ms values of different elements and the interstices of unit cells. For the calculations, the default Wigner-Seitz radii were employed, and only spin magnetic moments were considered because orbital Ms is very small. The results are presented in Table 3. It can be found that for Fe 3 B the calculated Ms of Fe is 1.921 µ B , which is consistent with previously obtained theoretical values 2.02 µ B [28] and 1.99 µ B [29]. Additionally, the Ms values of Fe1 (2.00 µ B ), Fe2 (1.80 µ B ) and Fe3 (1.97 µ B ) match well with the previous results (Fe1 (2.14 µ B ), Fe2 (1.89 µ B ) and Fe3 (1.95 µ B )) [29]. It is necessary to point out that although the magnetic properties of some borides, such as Fe 3 B [23], (Fe x ,Cr 1−x ) 3 B [23], (Fe x ,Mn 1−x ) 3 B [12] and (Fe x ,Co 1−x ) 3 B [11], have been measured by experiments, the crystal structures of these borides were not carefully and clearly distinguished. Thus, these experimental data will be discussed in a following work about the alloyed Fe 3 B with Ti 3 P-type structure, since according to the calculated results of the present work and the next work, their crystal structures can probably be distinguished.  and the cost of production, it can be concluded that among all the studied alloying elements, Mn is the most favorable candidate for improving the magnetic properties of Fe3B.

Conclusions
We systematically investigated the effect of alloying elements on the stability, electronic structure and magnetism of Ni3P-type Fe3B, as well as the site preference of alloying

Conclusions
We systematically investigated the effect of alloying elements on the stability, electronic structure and magnetism of Ni 3 P-type Fe 3 B, as well as the site preference of alloying elements, via first-principles density functional calculations, resulting in the following conclusions.
(1) Negative formation enthalpy and cohesive energy suggest that all studied compounds are thermodynamically stable, while Cu, W and Nb do not easily enter the Fe 3 B lattice. The lowest formation enthalpy implies that the (Fe 2.875 ,Mo 0.125 )B can be produced more easily, whereas the lowest cohesive energy means that the (Fe 2.875 ,W 0.125 )B is the most stable compound.