Assessment of magnetism and magnetocrystalline anisotropy of (Mn$_x$Fe$_{1-x}$)$_2$P$_{1-y}$Si$_y$ (x,y = 0 or 1/3) compounds for permanent magnets

Abstract

In order to calculate the prospects for new transition-metal rich and rare-earth free permanent magnets, we have investigated the electronic and magnetic properties of Fe$_{2}$P and related Mn- or/and Si-substituted compounds using full potential linearized augmented plane wave (FPLAPW) method under generalized gradient approximation (GGA) exchange-correlation (XC) functional. In agreement with the experiment, our calculated total magnetic moment and magnetocrystalline anisotropy (MCA) for Fe$_{2}$P are found to be 3.03 $\mu$ $_B$/f.u. and 538 $\mu$eV/f.u. (2.58 MJ/m$^{3}$), respectively. We find that the substitution of Mn or/and Si in Fe$_{2}$P not only enhances the magnetization but also boosts the maximum energy product by preserving the hard magnetic behavior. Our theoretical predictions provide useful insights to enhance magnetization with tolerable loss of MCA for Mn or/and Si substituted Fe$_{2}$P. Thus, we propose rare-earth free prospective candidates for permanent magnets through Fe$_{2}$P based compounds.

Introduction

The maximum energy product ((BH)$_{max}$), which describes magnetic energy storage, is a crucial parameter for advanced permanent magnets (PMs) to convert mechanical energy to electrical energy (and vice versa) in many electronic and industrial applications^1,2,3,4. The value of (BH)$_{max}$ can be enhanced significantly by combining high saturation magnetization of the soft magnetic phases and high anisotropy of hard magnetic phases. Presently, rare-earth (RE) elements based on Nd-Fe-B and Sm-Co are the strongest PMs due to their large saturation magnetization (M$_s$) and high magnetocrystalline anisotropy (MCA) energy, originating from a unique combination of d/f orbitals of transition metal (TM) / RE elements. However, recent socioeconomic pressures associated with global RE supplies have made it necessary to explore routes for the development of more economically viable RE free alternative permanent magnetic materials⁵. This quest has motivated many researchers towards RE free magnets. The most essential requirement for viable RE free magnets^6,7,8 is to have a magnetic response lying between that of alnico and Nd-Fe-B. To ensure high M$_s$ and high Curie temperature (T$_c$), it is necessary to use a large fraction of suitable TMs, but then the realization of having a valuable MCA is likely to be a challenge.

The presence of heavy TMs such as Pd and Pt in the compound improves their MCA and makes them suitable for PMs^9,10; however, these are not economical. On the other hand, 3d TM based compounds seem to be possible alternatives to RE materials for tuning M$_s$ and MCA, suitable for PMs. Among such materials, iron phosphide (Fe$_{2}$P) stands out as a particularly exciting candidate, due to its large MCA and considerable M$_{s}$. In addition, Fe$_{2}$P is composed of inexpensive and widely available elements^11,12,13. It is of widespread interest in (i) solid state chemistry because of its special hexagonal crystal structure with two non-equivalent iron sites (the tetrahedral Fe$_{3f}$ and the pyramidal Fe$_{3g}$)¹⁴, (ii) material science due to its diverse magnetic properties under different conditions of temperature, pressure, composition, and magnetic field^15,16,17 and (iii) earth science, due to its abundance in many meteorites and terrestrial garnet peridotites with implications for the Earth$^{'}$s core^18,19,20.

In the last half-century, many experimental and theoretical researchers have investigated on hexagonal Fe$_{2}$P^21,22,23,24. It is a ferromagnet having T$_{c}$ as 209 K accompanied by a first-order transition and a large MCA (2.32 MJ/m$^3$) along with magnetic moments of 0.90 $\mu$ $_B$ and 1.70 $\mu$ $_B$ on Fe$_\textit{3f}$ and Fe$_\textit{3g}$sites, respectively²⁵. The magnetism of Fe$_{2}$P is extremely sensitive to pressure and also to small concentrations of chemical dopants such as Mn, Co, Ni, Si and B^26,27,28. Japel et al.²⁹ reported that Fe$_{2}$P can undergo a phase transition from mP26 to a new, possibly monoclinic, structure between 30 and 50 GPa. Koumina et al.³⁰, using neutron diffraction, showed first-order ferro-paramagnetic phase transition for Fe$_{2}$P at T$_{c}$ = 217 K, associated with a discontinuous change in the lattice parameters of the hexagonal unit cell. Hudl et al.³¹ studied FeMnP$_{0.5}$Si$_{0.5}$ by Mossbauer spectroscopy along with ab-initio calculations and found an unexpectedly high magnetic hyperfine field for Fe atoms located at the tetrahedral 3f sites: the saturation moment being 4.4 $\mu$ $_B$/f.u. at low temperatures.

Costa et al.³² investigated the large magnetic anisotropy of Fe$_{2}$P using density functional theory (DFT) based full potential linear muffin-tin orbital (FPLMTO) method. They obtained a uniaxial MCA of 664 $\mu$eV/f.u., in agreement with experimental observations. Wiendlocha et al.³³ studied electronic structure of Fe$_{2-x}$T$_{x}$P (T = Ru and Rh) systems using Korringa-Kohn-Rostoker method. They found that the magnetic moment appearing on Fe$_{3f}$ in the ferromagnetic state completely vanishes in the disordered local moment (DLM) state, while the moment on Fe$_{3g}$ is only slightly changed. A systematic theoretical study of the electronic and magnetic properties of Fe$_{2}$P$_{1-x}$Si$_{x}$ (x = 0.5), performed by Severin et al.³⁴, showed that the magnetic moments vary substantially from 0.7 $\mu$ $_B$ to 2.6 $\mu$ $_B$ at different sites. The ab-initio electronic structure calculations by Czirjak et al.³⁵ to study the properties of Fe$_{2}$P$_{1-x}$Si$_{x}$ in ferromagnetic and paramagnetic states produced the lattice parameters, atomic positions, and magnetic properties in good agreement with experiment and other theoretical results.

Ou et al.³⁶ reported the structural and magnetic properties of 3g-site disordered Mn$_{0.66}$Fe$_{1.29}$P$_{1-x}$Si$_x$ melt-spun ribbons with a high Si content of 0.34 $\le$ x $\le$ 0.42 to maintain the structural stability of the hexagonal crystal. They found a large magnetic moment of 4.57 $\mu$ $_B$ for x = 0.34. Fujii et al.²⁵ prepared single crystal of hexagonal Fe$_{2}$P by solid vapor reaction and performed measurements of magnetization, susceptibility, thermal expansion, and electrical resistivity. They found the anisotropic constant (K$_{1}$) for the studied sample as 2.32 KJ/m$^{3}$ at 4.2 K, which is noticeably higher than that of 3d-transition compounds.

The literature reveals the potential of introducing a foreign element at 3g and 2c/1b sites of Fe and P, respectively, in enhancing the T$_{c}$, MCA, M$_s$ and (BH)$_{max}$. Further, a reliable first-principles investigation of potential materials for permanent magnets is possible with ease due to tremendous increase in computational resources now-a-days^37,38. In this work, we plan to check the effect of Mn-substitution at 3g-site of Fe$_{2}$P using same first-principles approach. Experimentally, it has been shown that the stability of a hexagonal crystal structure of Fe$_{2}$P with Mn-substitution can be preserved via a fractional replacement of P (0.24 $\le$x < 0.50) by Si³⁹. Thus, we aim to consider P-replacement by Si at 2c/1b sites additionally. We expect interesting and authentic results due to the employment of full potential approach in the present study. Furthermore, we also plan to explain the trends expected in MCA on the basis of spin-orbit coupling (SOC) and local crystal environment in Fe$_2$P after Mn- and Si-substitutions additionally.

Results and discussion

Fe$_{2}$P exhibits a hexagonal crystal structure with two non-equivalent iron sites (tetrahedral Fe$_{3f}$ and pyramidal Fe$_{3g}$) as depicted in Fig. 1. The present pristine and substituted compounds can collectively be represented by compositional formula (Mn$_x$Fe$_{1-x}$)$_2$P$_{1-y}$Si$_y$ (x,y as 0 or 1/3). Among (Mn$_x$Fe$_{1-x}$)$_2$P$_{1-y}$Si$_y$, (i) Fe$_2$P$_{2/3}$Si$_{1/3}$ can be designed for x = 0, y = 1/3 via substitution of P$_\textit{1b}$ atom by Si, (ii) (Mn$_{1/3}$Fe$_{2/3}$)$_2$P for x = 1/3, y = 0 via substitution of Fe$_\textit{3g}$ atom by Mn, and (iii) (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$ for x = 1/3, y =1/3 via substitution of the Fe$_\textit{3g}$ and P$_\textit{1b}$ atoms by Mn and Si, respectively, in the unit cell of Fe$_2$P. It is noted that for x = 0, y = 0 in compositional formula, we get pristine Fe$_2$P compound. The top views of all four compounds are shown in Fig. 2

Fig. 1

Hexagonal unit cell of Fe$_2$P (space group 189: P$\overline{6}$2m). Fe$_{3f}$/Fe$_{3g}$ represents tetrahedral/pyramidal site.

Fig. 2

Top view of hexagonal unit cell of (Mn$_x$Fe$_{1-x}$)$_2$P$_{1-y}$Si$_y$ (x,y = 0 or 1/3) compounds.

The optimized lattice parameters are listed in Table 1. In pristine Fe$_{2}$P, various bond lengths, Fe$_{3f}$-P$_{1b}$, Fe$_{3g}$-P$_{1b}$, Fe$_{3f}$-P$_{2c}$ and Fe$_{3g}$-P$_{2c}$are found to be 2.268 Å, 2.379 Å, 2.198 Å, and 2.450 Å, respectively, which are in agreement with the previous reports²². The substitution of Fe or/and P at respective sites in Fe$_2$P affects the local environment and as a result, the bond lengths relax to new values. The impact on the lattice parameters is significant, however, the change in volume of resultant compounds is only within 2–4$\%$ (Table 1). This, therefore, indicates considerable distortion in the tetrahedral and pyramidal shapes within the unit cell.

Table 1 Optimized lattice parameters (a,c) in Å, position coordinates (x$_1$, x$_2$) in Å, and relaxed bond lengths (l) in Å, for all compounds. Atomic positions: 3g (x$_1$, 0, 1/2); 3f (x$_2$, 0, 0); 2c (1/3, 2/3, 0), and 1b (0, 0, 1/2).

A fruitful insight about the origin of magnetic moment and metallicity is provided by the computed total density of states (DOS). The spin polarized total DOS of studied compounds (Fig. 3) indicates the metallic behavior for all the cases. Further, for (Mn$_{1/3}$Fe$_{2/3}$)$_2$P and (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$, a lower value of occupied minority DOS is clearly visible in the vicinity of the Fermi level (E$_F$). Since the difference of occupied majority and minority DOS is proportional to the final spin moment appearing on the system^40,41, thus, Mn or/and Si substitution leads to an enhanced moment compared to the case of pristine compound, Fe$_{2}$P.

Fig. 3

Calculated spin-resolved total DOS per unit cell of hexagonal (Mn$_x$Fe$_{1-x}$)$_2$P$_{1-y}$Si$_y$ (x,y = 0 or 1/3) compounds. E$_F$ is shifted to 0 eV.

As evident from spin resolved total and site-specific DOS of Fe$_{2}$P and Fe$_2$P$_{2/3}$Si$_{1/3}$ (Fig. 4), the total DOS is dominated by 3d states of both kind of Fe atoms, whereas in (Mn$_{1/3}$Fe$_{2/3}$)$_2$P and (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$, the Mn-3d DOS also contributes prominently along with the Fe-3d DOS. The 3d bands of both the TMs hybridize to generate large spin moment on Fe and Mn atoms. The 3d-orbitals of Fe and Mn atoms in majority spin channel (MAC) can be easily traced in the occupied region whereas in the minority spin channel (MIC), these are partially occupied, resulting in strong ferromagnetism for all the compounds. Further, in (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$, Mn$_{3g}$-3d and Fe$_{3g}$-3d states prominently hybridize with Fe$_{3g}$-3d states (d-dhybridization)⁴², resulting in reduction of DOS of MIC states in the occupied region. Since the difference in DOS of MAC and MIC decides magnetic moment^40,41 and with introduction of Mn+Si substitutions, the DOS of MIC decreases. Thus, the difference in DOS of MAC and MIC increases which leads to increase in net magnetic moment of resultant compound after substitution as evident from Table 2. A similar reason may be quoted for Fe$_2$P$_{2/3}$Si$_{1/3}$. The sp-element P/Si, present in the compounds, also plays an important role in deciding the final value of the magnetic moment of the compounds by aligning P/Si-p states antiferromagnetically with TM-3d states. This contribution is although small, but is essential for the stability of the compound⁴³.

Fig. 4

Calculated total and site specific (at Mn/Fe-3g and Fe-3f) spin-resolved DOS of (Mn$_x$Fe$_{1-x}$)$_2$P$_{1-y}$Si$_y$ (x,y = 0 or 1/3) compounds.

Table 2 summarizes the spin and orbital magnetic moments of the studied Fe$_{2}$P based compounds along with the available theoretical and experimental data^44,45,46,47. For Fe$_2$P, the calculated spin magnetic moments are 2.265 $\mu$ $_B$ and 0.859 $\mu$ $_B$ for Fe$_{3g}$ and Fe$_{3f}$, respectively. The total spin magnetic moment (3.008 $\mu$ $_B$/f.u.) also has a tiny contribution from the moment induced on P atom. The smaller spin moment of Fe$_{3f}$ is evident from a smaller exchange splitting of DOS for Fe$_{3f}$ than that for Fe$_{3g}$ (Fig. 4). On Mn-substitution, not only the moment at the substitutional site gets enhanced but also due to interaction of Mn with Fe, some extra moment is induced on both inequivalent Fe atoms. This extra moment induced on both Fe-3g and -3f sites is a consequence of d-d hybridization. Since Fe atom has four unpaired electrons whereas Mn has five, thus, on Mn-substitution, d-d hybridization becomes stronger and some extra moments get induced on both inequivalent Fe atoms. Moreover, Si-substitution further enhances the moment at both 3g and 3f sites. The maximum local moment (2.916 $\mu$ $_B$) is obtained for Mn at 3g site in (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$. In all three substitutional compounds, the total moments are found to be considerably larger than that of pristine Fe$_{2}$P, suggesting these as stronger magnetic materials. The orbital magnetic moments at various sites in pristine and substituted compounds are found to be feeble (Table 2) as expected due to large crystal field effect in TMs. However, they play a vital role in the emergence of MCA, as seen ahead.

Table 2 Calculated spin ($\mu$$_s$) and orbital ($\mu$$_l$) moments at individual sites, and the total magnetic moment/f.u. ($\mu$$_t$) in Fe$_2$P and its derivatives.

An enormous value of magnetic moment in a material is not the only criterion to approve it as a permanent magnet. To authenticate the compound in this category, a subtle quantity is the uniaxial magnetic anisotropy, defined as the energy required to align moments from easy axis to hard axis of magnetization. There are two fundamental contributions to the magnetic anisotropy. The first one is the dipole-dipole interaction, which leads to the magnetic shape anisotropy. The other contribution is MCA due to the SOC of electrons in magnetic materials.

To calculate the MCA, the difference between equilibrium energy for magnetization along the hard axis <100> and easy axis <001> has been considered, which is listed in Table 3. Our calculated MCA constant (K) for Fe$_{2}$P is 538 $\mu$eV/f.u. (2.58 MJ/m$^{3}$). This is very close to the experimental value²⁵, 500 $\mu$eV/f.u. (2.32 MJ/m$^{3}$), compared to the previously calculated value (664 $\mu$eV/f.u.)³². Keeping in mind the extremely delicate nature of the MCA and the typical low energy differences ($\mu$eV) associated with the calculations, our results are in acceptable agreement. However, on Mn or/and Si substitution in Fe$_2$P, the value of K gets reduced to 454, 389 and 316 $\mu$eV/f.u., for Fe$_2$P$_{2/3}$Si$_{1/3}$, (Mn$_{1/3}$Fe$_{2/3}$)$_2$P and (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$, respectively. This reduction can be explained by second order perturbation theory^48,49,50 in terms of energy arising from SOC between the states below and above the E$_F$. This energy is inversely proportional to the energy difference between occupied and unoccupied k-states i.e.

$$\begin{aligned} \Delta E_{SO}^{(2)}=-\sum _{n,\sigma }^{occ.}\sum _{n^{'},\sigma ^{'}}^{unocc.}\frac{\left| \left\langle n,\sigma \left| H_{SO} \right| n^{'},\sigma ^{'} \right\rangle \right| ^{2}}{E_{n^{'},\sigma ^{'}}^{0}-E_{n,\sigma }^{0}} \end{aligned}$$

(1)

On Mn or/and Si substitutions, DOS in occupied region near E$_F$ gets reduced (Fig. 3). Therefore, fewer states contribute to MCA, resulting in its reduction compared to the pristine case. To obtain a better understanding of MCA, a careful analysis of electronic band structure near the E$_F$ has been made. To extract qualitative ideas about the trends in MCA, we have restricted ourselves to analyze the bandstructure for the majority spin only. Fig. 5(a) and (b) show the bandstructures for Fe$_{2}$P and Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$, respectively, along high symmetry directions in the Brillouin zone for majority spin only along <100> and <001> directions of magnetization. Since the SOC constant of 3d TMs is very low (50–100meV)⁵¹ and only the states in the vicinity of E$_F$ can play an important role in deciding various properties of the materials, we have examined mainly the bandstructure near E$_F$, which clearly shows a change in splitting of energy bands when the magnetization direction changes from <001> to <100>.

Fig. 5

Majority spin bandstructure of (a) Fe$_{2}$P and (b) (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$ including SOC for magnetization axis along the crystallographic axes <100> (solid lines) and <001> (dotted lines). E$_F$ is shifted to 0 eV for both cases.

The k-points resolved MCA is plotted along high symmetry directions in Fig. 6 as a difference of the changes in band energies for magnetization direction <100> and <001>. For Fe$_{2}$P, the overall difference comes out to be positive. In (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$, the substitutions of Mn and Si atoms at their respective sites lead to different consequences that results in the significant change in the band energies at all the symmetry points in Brillouin zone which is clearly visible in Fig. 5(b). Moreover, the changes in band energies for magnetization to align at <100> from <001> direction is almost inverted in the stretch between symmetry point $\Gamma$ and A and between H and L (Fig. 6). This brings a reduction in its MCA as compared to pristine Fe$_{2}$P. Therefore, in (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$, the overall difference, though positive, has a smaller magnitude. But, still this MCA lies within the limit of permanent magnetic character.

Fig. 6

k-points resolved MCA of Fe$_{2}$P and Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$ along high symmetry directions. E$_F$ is shifted to 0 eV.

As stated earlier, the magnitude of MCA is a suitable indicator for the use of Fe$_{2}$P based compounds as the materials for PMs, however, the confirmation as a genuinely suitable material for PMs depends also on other factors like saturation magnetization (M$_s$), anisotropic field (H$_c$), hardness parameter ($\kappa$) and maximum energy product ((BH)$_{max}$). The two important parameters for assessing permanent magnetic behavior are H$_c$ and $\kappa$. The equations for ${H}_c$, as anticipated by empirical Kronmüller equation after ignoring local demagnetizing factor^52,53, and $\kappa$⁵⁴ are given by:

$$\begin{aligned} H_{c} = \frac{2K}{\mu _{0}M_{s}}\alpha \end{aligned}$$

(2)

and

$$\begin{aligned} \kappa = \sqrt{\frac{K}{\mu _{0}{M_{s}^2}}} \end{aligned}$$

(3)

where $\alpha$ is the microstructure constant. In real systems, its value is less than one and depends on the shape and size of the grain, temperature, and other extrinsic quantities but in present work, H$_c$ was estimated by setting $\alpha$ as 1.

The anisotropic field is a field which is required to remove the MCA. On neglecting factors like local demagnetizing and microstructure constant, it is equal to coercive field. Moreover, we can say that it provides an upper limit to the coercive field. In permanent magnets, best coercivity which can be achieved in a practical magnet rarely exceeds 20%−30% of the anisotropy field associated with the MCA⁵⁵.

In PMs, the figure of merit that defines their magnetic quality is the (BH)$_{max}$. It is the maximum energy density that can be stored in the magnet. The (BH)$_{max}$ is the result of a combination of large enough M$_s$ and MCA; the latter bestows the magnet with coercivity. An upper limit for the (BH)$_{max}$⁵⁶ can be expressed in terms of the M$_s$, an intrinsic material parameter, as:

$$\begin{aligned} (BH)_{max} = {\left\{ \begin{array}{ll}\frac{\mu _{0}M_s^2}{4} & H_c> M_s/2\\ \frac{\mu _{0}M_sH_c}{2} & H_c < M_s/2\end{array}\right. } \end{aligned}$$

(4)

Eq. (4) shows that a large M$_s$ is desirable for a strong permanent magnet. The typically hard magnetic materials have $\kappa$ values often more significant than 1 ($\kappa$ > 1). The calculated M$_{s}$, H$_{c}$, $\kappa$ and (BH)$_{max}$ compiled in Table 3, present a complete picture for the studied compounds. The highest M$_{s}$ among all studied compounds is 1.38 T for (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$. Alnico⁵⁷ with M$_{s}$ as 1.40 T is a famous RE free permanent magnet. Thus, the obtained value of M$_{s}$ of (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$ is quite good for permanent magnets. The M$_s$ values of other famous magnetic materials; SmCo$_{5}$⁵⁸, Nd$_{2}$Fe$_{14}$B⁵⁹, NdFe$_{16}$⁶⁰, $\alpha ^{\prime \prime }$ - Fe$_{16}$N$_{2}$⁵⁷, SmFe$_{12}$⁶¹, Sm$_2$Fe$_{17}$N$_3$⁶², SrFe$_{12}$O$_{19}$⁶³, CeFe$_{12}$O$_{19}$⁶³ and W-type SrFe$_{18}$O$_{27}$(SrW)⁶⁴ are 1.14 T, 1.86 T, 2.20 T, 2.68 T, 1.64 T, 1.52 T, 0.60 T, 0.59 T and 0.65 T, respectively. Due to comparable value of M$_s$ with some of these famous magnetic materials, (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$ holds potential for permanent magnets.

We note that the (BH)$_{max}$ and M$_s$ are enhanced consistently and significantly on Mn or/and Si substitution (Table 3). The predicted (BH)$_{max}$ values of all the substituted compounds are good enough to qualify them as PMs, since larger the (BH)$_{max}$, more is the energy stored in the PMs. In (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$, although the MCA reduces to some extent, yet its $\kappa$ is > 1. Therefore, its hard magnetic behavior remains preserved and its (BH)$_{max}$ is almost 70% higher as compared to that for pristine compound Fe$_{2}$P. The prediction of magnetic parameters of this compound is of unique importance as it has been already synthesized by Höglin et al.³⁹ in stable Fe$_2$P like hexagonal structure. In totality, a significant value of magnetic moment and a considerably decent MCA along with a larger (BH)$_{max}$, make the resultant compound; (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$ probably an even better candidate for RE free permanent magnets^65,66,67,68 than Fe$_2$P.

Table 3 Calculated MCA constant (K) (in units of $\mu$eV/f.u. and MJ/m$^3$), nature of MCA, saturation magnetization (M$_s$), maximum energy product ((BH)$_{max}$), anisotropic field (H$_{c}$) and magnetic hardness parameter ($\kappa$) of all the considered systems.

Conclusion

In a quest to achieve enhanced energy product ((BH)$_{max}$) and saturation moment (M$_{s}$), Mn or/and Si-substitution in Fe$_{2}$P was considered. Well converged electronic structure calculations by utilizing the FPLAPW method were performed to determine the M$_{s}$, MCA constant (K), H$_c$, (BH)$_{max}$ and $\kappa$ for pristine and all substitutionally derived Fe$_{2}$P compounds. The K value of an optimized Fe$_{2}$P structure was obtained to be 538 $\mu$eV/f.u., which is in very close agreement with the experiment. All the studied compounds show impressive values of M$_{s}$ and (BH)$_{max}$ while maintaining decent MCA and $\kappa$. The maximum magnetic moment among all cases was found to be 4.15 $\mu$ $_B$ for (Mn$_{1/3}$Fe$_{2/3}$)$_2$P$_{2/3}$Si$_{1/3}$, which also maintains $\kappa$ > 1. Therefore, our first-principles calculations show that Mn or/and Si substituted Fe$_{2}$P compounds are very promising for permanent magnet applications. In summary, the selective substitution of Mn atom at the 3g site along with Si-impurity in Fe$_{2}$P offers a new alternative to RE permanent magnetic materials. We hope this work will motivate the researchers to realize the proposed compounds experimentally also as the suitable materials for permanent magnets.

Method

To obtain the optimized lattice parameters for all (Mn$_x$Fe$_{1-x}$)$_2$P$_{1-y}$Si$_y$(x,y as 0 or 1/3) compounds, starting with the experimental lattice parameters²⁵, the relaxations of resultant unit cells were first performed using DFT⁶⁹ based projector augmented wave (PAW) method as implemented in VASP^70,71. In order to avoid residual pressure and fully optimizing these structures, the force convergence of 10$^{-7}$ eV/${\text{\AA }}$ and energy convergence of 10$^{-8}$eV were used. The generalized gradient approximation (GGA) under Perdew-Burke-Ernzerhof (PBE) parameterization⁷² was selected to incorporate the exchange-correlation (XC) potentials. After relaxation, we observed that placing Si on 1b site is energetically more favorable than on 2c site. Similarly, placing Mn on 3g site is energetically more favorable than on 3f site. Thus, we selected only the substitution of Fe by Mn at 3g site and P by Si at1b site for finding the magnetic response. All the crystal structures were drawn by using the Visualization for Electronic and Structural Analysis (VESTA) program⁷³.

After obtaining the relaxed unit cells, the final ground-state self-consistent calculations were performed using the more sophisticated full potential linearized augmented plane wave (FPLAPW)⁷⁴ method as implemented in WIEN2k⁷⁵ under spin-polarized GGA-PBE XC potentials. The core states were treated fully relativistically, whereas, for the valence states, a scalar relativistic approach was used. Additionally, valence electronic wavefunctions inside the muffin-tin sphere were expanded up to l$_{max}$ = 10. The radii of the muffin-tin sphere (R$_{MT}$) for various atoms were taken in such a way as to ensure nearly touching spheres and thus avoiding the possibility of charge leakage. The plane wave cut-off parameters were decided by R$_{MT}$k$_{max}$ = 7 (where k$_{max}$ is the largest wave vector of the basis set) and G$_{max}$ = 12 a.u.$^{-1}$ was used for Fourier expansion of potential in the interstitial region. The energy threshold was set at −6.0 Ry between the core and valence state electrons.

The k-space integration was carried out using the modified tetrahedron method⁷⁶ with a dense k-mesh of $13\times 13\times 20$ for high resolution. Strict convergence criteria, 10$^{-6}$ e for charge and 10$^{-6}$ Ry for energy difference, were employed to obtain an accurate value of MCA with $\mu$eV precision in the final calculations. Firstly, the scalar relativistic calculations were performed to obtain self-consistent potentials. After fixing the potential for each case, relativistic effects were included with the second variational treatment of SOC, keeping reduced symmetry for each magnetization direction⁷⁷.

Generally, the MCA constant (K) is calculated as MCA energy per unit volume, and its positive/negative indicates uniaxial/in-plane MCA. The MCA energy values were calculated by using the magnetic force theorem⁷⁸ as under:

$$\begin{aligned} E= \sum _{j,k}^{occu}\epsilon _{j}(\hat{n}_{2},k) - \sum _{j,k}^{occu}\epsilon _{j}(\hat{n}_{1},k) \end{aligned}$$

(5)

where $\epsilon _{j}$ is the Kohn-Sham eigenvalue evaluated for each magnetization orientation. n̂, j labels the occupied states and k is a particular k-point in the Brillouin zone. $\hat{\textrm{n}}_{1}$= $<001>$ is the direction along c-axis and $\hat{\textrm{n}}_{2}$ represents a perpendicular direction of magnetization.