The effect of ternary additions on the structural and mechanical properties of B2 phase FeCo magnetic alloy

FeCo alloy constitutes an important class of soft-magnetic materials with a wide range of technological applications. This class of materials offers exceptional magnetic properties due to their rare combination of good properties like high mechanical strength, and high saturation magnetization. Fortunately, this alloy has good magnetic properties, but shows low levels of ductility at room temperature. This study intends to use density functional theory (DFT), to understand how ternary alloying with Pd and Mn can improve the ductility of B2 FeCo system. We probe the relationship between stability and composition of the ternary Fe50Co50−xMx (M = Pd, Mn) with different M content. Furthermore, the study also investigates how ternary alloying affects the stability of the B2 FeCo alloy. There is a significant enhancement of ductility with Mn addition than Pd ternary addition. It is noteworthy that small composition of Pd (< 10 at.%) does not compromise the valuable properties of FeCo. The findings suggest that Fe50Co50−xMx alloys can be used for the future development of magnetic components with good strength. The effect of ternary additions on the structural and mechanical properties of B2 phase FeCo magnetic alloy.


Introduction
The FeCo alloy is widely known as a soft-magnetic material (SMM) which represent an important group of engineering materials [1]. This material can be magnetized and demagnetized with ease, even in the existence of very little excitation field [2], resulting in high permeability. It has high reliability, low cost, and suitability for mass production, just like other commercial products [3]. Transformers, motors, inductors, and generators all use soft-magnetic materials in their magnetic cores [4]. The B2 FeCo binary alloy at lower temperatures than 1000 K is thermodynamically stable in a long-range ordered structure, which results in poor ductility of the alloy. At room temperature this alloy show levels of ductility. Attempts were made to enhance the mechanical properties of FeCo alloy mainly using heat treatment and deformation approaches [1,[5][6][7].
This study aims to explore ways of enhancing the ductility of B2 FeCo at room temperature through ternary alloying, within a concentration range of 0 to 50%. Moreover, we will be determining further the impact of the ternary additions on the magnetic, mechanical, thermodynamic, and structural properties of the alloy. We intend to enhance the ductility and workability of B2 FeCo alloys by ternary alloying using Pd and Mn, and further by determining the influence of these elements on the magnetic valuables of the alloy. These alloying elements were chosen based on their ability to improve the ductility of the material and because these elements have large atomic radii compared to Co [8]. It was reported that the slip systems are larger in B2 FeCo alloys [9,10], which contribute to the ductility of the material, hence the use of ternary elements with larger atomic radii will help reduce the effect caused by slip systems.
Although Co is good at enhancing mechanical properties, Fe has higher magnetic strength and is not expensive, easily accessible and has a slightly higher melting temperature (1 811 K [11]) than Co (1 768 K [11]). Furthermore, considering the application of Fe-Co, the substitution of ternary elements will be done on the Co-site of the alloy to avoid compromising the magnetic properties if Fe is to be reduced.

Methodology
The calculations in this study are purely computational. The DFT [12,13] was used employing the Vienna ab initio simulation package (VASP) code [14] along with the projector augmented wave (PAW) pseudopotential [15]. Calculations were performed considering the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional [16] because it gave better results compared to other functionals. Proper convergence was deduced before any of the properties could be calculated and an energy cut-off of 500 eV was used, to achieve a good convergence of the parameters. The k-spacing of 0.2 1/Å (12 × 12 × 12) for B2 FeCo was used according to Monkhorst and Pack [17]. On the 2 × 2 × 2 supercell structure, substitutional search tool was used to substitute Co with Pd and Mn at different atomic compositions (0 ≤ x ≤ 50) and the Pm-3 m symmetry was used for all the structures (as shown in appendix). The generated structures were subjected to full geometry optimization. All calculations were subjected to spin polarization to take account of the magnetization. The strain value of 0.005% was used for calculating the elastic properties.

Structural and thermodynamic properties
The equilibrium lattice constants were determined from relaxed structures, where the volume and unit cell were allowed to change. This was done for all the B2 Fe 50 Co 50−x M x (M = Pd, Mn) structures. Fig. 1(a) demonstrates the equilibrium lattice constants for the binary Fe 50 Co 50 and ternary Fe 50 Co 50−x M x (0 ≤ x ≤ 50) alloys. It was found that the lattice constant of Fe 50 Co 50 is 2.844 Å (2.843 Å [18]) which compares well with the experimental value in parenthesis to within 0.035%. The lattice constant increases with an increase in Pd content, this is attributed to the large atomic radius of Pd (1.69 Å [19]) compared to Co (1.52 Å [19]). At 6.25 at.% Pd, the lattice constant was found to be 2.870 Å (2.872 Å [20]) which is in good agreement with the theoretical value in the parenthesis to within 0.070%. Furthermore, a minimal reduction is seen with an increase in Mn, this is understood since the atomic radius of Mn (1.61 Å [19]) is smaller than that of Pd (1.69 Å [19]). Now, the thermodynamic stability of the binary Fe 50 Co 50 and ternary Fe 50 Co 50−x M x (0 ≤ x ≤ 50) alloys was determined from heats of formation (ΔH f ). The ∆H f can be estimated by the formula: where E c is the calculated total energy of the compound and Ei is the calculated total energy of each element in the compound. For a structure to be stable, the heat of formation must have a negative value (∆H f < 0) otherwise a positive value implies instability. The results of heats of formation are shown in Fig. 1 (b). Any alloy with very low heats of formation is taken to be stable, whereas the more positive it is the less stable.
The heats of formations for the binary B2 Fe 50 Co 50 was found to be stable with the value of -0.057 eV/atom (-0.065 eV/atom [21]) which is in good agreement with the theoretical value in parenthesis to within 3%. In the case of the B2 Fe 50 Co 50−x M x , the heats of formation increase with an increase in Pd and Mn content. This suggests that thermodynamic stability is not enhanced with Pd and Mn addition. (1)

Measurement of ductility
To investigate the ductility and brittleness behaviour of the B2 Fe 50 Co 50-x M x alloys, we have determined the two quantities: Poisson and the Pugh's ratio (B/G) at different compositions (0 ≤ x ≤ 50). Note that the structure is considered ductile when Poisson's ratio (σ) is greater than 0.26 and otherwise brittle [22]. The σ for B2 Fe-Co was found to be 0.286 (0.290 [18]) which agrees with the theoretical value to within 1%, implying that the material is ductile. The effect of Pd and Mn ternary additions on the alloy is shown in Fig. 2. As the content of Pd is increased, the σ values were found to increase and be greater than 0.26 in the entire concentration which is a condition of ductility. Similar was observed with the addition of Mn, although Pd has a greater effect compared to Mn.
The B/G ratio was also calculated and evaluated to determine the ductility and brittleness of the structures. Pugh [23] indicated that the ductile phase has a higher B/G ratio (> 1.75) while a brittle phase has a smaller B/G ratio (< 1.75). Fig. 3 (a, b) show the behaviour of the B/G ratios as functions of at.% Pd and Mn, respectively. It is seen that the B/G ratio for the binary Fe 50 Co 50 was found to be 1.998 (B/G > 1.75) condition of ductility. Furthermore, it was observed that the B/G ratio is greater than the unit at 6.25 at.% Pd indicating the ductility of the structure. At above 18.75, the structure reveals a brittle manner as the ratio is less than the unit as shown in Fig. 3 (a). In the case of Fe 50 Co 50-x Mn x , the B/G ratio increases gradually with increasing concentrations of Mn. This suggests that the B2 FeCo alloy showed good ductility in nature and the ductility became stronger with high concentrations of Mn and with a small concentration of Pd (< 10 at.%).

Magnetic properties
The total magnetic moments were calculated to check the magnetic strength of both binary Fe 50 Co 50 and ternary Fe 50 Co 50-x M x (M = Pd, Mn) systems. A positive value of magnetic moment indicates good magnetic strength. Fig. 4 shows the magnetic moments of Fe 50 Co 50-x M x alloys as a function of M atomic composition. The magnetic moment of the binary B2 FeCo alloy was found to be 4.530 μB (4.479 μB [18]) which is in good agreement with the theoretical value in the parenthesis to within 2%. In the case of Fe 50 Co 50-x M x , the magnetic moments slightly decrease with an increase in M addition.
This behaviour suggests that the magnetic strength is not enhanced, however, it is also not compromised. At 50 at.% Pd (0.00 μB), the structure transition from ferromagnetic to diamagnetic, similar observations were discussed elsewhere [24].

Conclusion
The study employed the ab initio density functional theory (DFT) to explore the ductility and stability of binary Fe 50 Co 50 and ternary Fe 50 Co 50-x M x (M = Pd, Mn) alloys. The additions of the dopants on Fe 50 Co 50 alloy were performed using the supercell approach and different compositions (6.25, 18.75, 25, 31.25, 43.75 and 50) were evaluated. This study revealed that Pd and Mn have the potential to improve the ductility of the B2 FeCo alloy. The findings suggest that Fe 50 Co 50-x M x alloys can be used for the future development of magnetic components with good strength.