Abstract
Simultaneously enhancing the uniaxial magnetic anisotropy (\(K_u\)) and thermal stability of \(\alpha ^{''}\)phase Fe\(_{16}\)N\(_{2}\) without inclusion of heavymetal or rareearth (RE) elements has been a challenge over the years. Herein, through firstprinciples calculations and rigidband analysis, significant enhancement of \(K_u\) is proposed to be achievable through excess valence electrons in the Fe\(_{16}\)N\(_{2}\) unit cell. We demonstrate a persistent increase in \(K_u\) up to 1.8 MJ m\(^{\text {}3}\), a value three times that of 0.6 MJ m\(^{\text {}3}\) in \(\alpha ^{''}\)Fe\(_{16}\)N\(_{2}\), by simply replacing Fe with metal elements with more valence electrons (Co to Ga in the periodic table). A similar rigidband argument is further adopted to reveal an extremely large \(K_u\) up to 2.4 MJ m\(^{\text {}3}\) in (Fe\(_{0.5}\)Co\(_{0.5}\))\(_{16}\)N\(_{2}\) obtained by replacing Co with Ni to Ga. Such a strong \(K_u\) can also be achieved with the replacement by Al, which is isoelectronic to Ga, with simultaneous improvement of the phase stability. These results provide an instructive guideline for simultaneous manipulation of \(K_u\) and the thermal stability in 3donly metals for REfree permanent magnet applications.
Similar content being viewed by others
Introduction
Alphaphase iron has been known for its extraordinary magnetic properties, including high saturation magnetization (\(\mu _0 M_s\)) and Curie temperature (\(T_c\)), in addition to its relatively simple fabrication and low price. These intriguing features make it a potential champion ever for highperformance permanent magnet applications^{1,2,3,4}. However, the main drawback of \(\alpha\)Fe is its negligible uniaxial magnetic anisotropy (\(K_u\)) on the order of \(\mu\)eV per atom^{1,2,3,4}. Two practical approaches to enhance \(K_u\) in \(\alpha\)Fe are (1) alloying with heavymetal (HM) or rareearth (RE) elements, with the most prominent examples being FePt^{5} and Nd\(_2\)Fe\(_{14}\)B^{6, 7}, and (2) reducing the crystal symmetry from the cubic (\(c/a=1\)) to tetragonal phase (\(c/a\ne 1\))^{8,9,10}. In (1), 5d or 4f electrons possess inherently large spinorbit coupling (SOC) and orbital angular momentum (L), but the inclusion of these HM and RE elements is not desirable in terms of price and is detrimental to \(\mu _0 M_s\) and \(T_c\). In (2), the energy levels of the 3d orbitals evolve in tetragonal symmetry, particularly around the Fermi level (\(E_F\)), which in turn enhances \(K_u\)^{11}.
The tetragonal phase of \(c/a\ne 1\) is now accessible in epitaxial Fe films with a diverse choice of latticemismatched substrates. Nevertheless, such tetragonal distortion is feasible only for limited film thicknesses of a few nanometers^{12, 13}. In contrast, a bulkscale tetragonal structure of the \(\alpha ^{''}\)phase (c/a \(=\) 1.1) is favored when 12.5 at.% N is embedded into the \(\alpha\)Fe structure with octahedral interstitial sites, forming a 16:2 (Fe\(_{16}\)N\(_2\)) stoichiometry^{14}. Since a surprisingly large magnetic moment of 2.6−3 \(\mu _B\) per Fe atom was reported^{15,16,17}, \(\alpha ^{''}\)phase Fe\(_{16}\)N\(_2\) has received enormous attention as a possible 3donly permanent magnet. However, the practical implementation of \(\alpha ^{''}\)Fe\(_{16}\)N\(_2\) in obtaining monophasic samples is quite difficult as \(\alpha ^{''}\)Fe\(_{16}\)N\(_2\) decomposes into \(\alpha\)Fe and \(\gamma ^\prime\)Fe\(_4\)N at a low temperature of approximately 500 K^{14}. Numerous efforts have been made to improve the thermal stability of \(\alpha ^{''}\)Fe\(_{16}\)N\(_2\); the most successful approach is Ti addition but the magnetic properties are greatly suppressed^{18, 19}. In addition to the weak thermal stability, another major obstacle that hampers practical applications is the still insufficient \(K_u\), which ranges from 0.4 to 1 MJ m\(^{\text {}3}\), depending on the sample preparation and film thickness^{20,21,22}. In the research community, search for enhancing \(K_u\) while improving the thermal stability of \(\alpha ^{''}\)Fe\(_{16}\)N\(_{2}\) in the bulk has been thus very intensive and remains unresolved.
In this article, we propose a possible mechanism of tuning the number of valence electrons to simultaneously enhance the thermal stability and \(K_u\) by a few times in Fe\(_{16}\)N\(_{2}\) and (Fe\(_{0.5}\)Co\(_{0.5}\))\(_{16}\)N\(_{2}\) apart from the aforementioned approaches (1) and (2), using firstprinciples calculations and rigidband model analysis. We predict a persistent increase in \(K_u\) up to 2.4 MJ m\(^{\text {}3}\), which is four times that (0.6 MJ m\(^{\text {}3}\)) of Fe\(_{16}\)N\(_{2}\), by replacing Fe with metal elements with more valence electrons (Co to Ga and Al in the periodic table). Such a supreme \(K_u\) is discussed in connection with the mutual mechanisms of the Jahn−Teller orbital splitting and excess electroninduced energy level changes in the electronic structure.
Results and discussion
Figure 1a displays the \(\alpha ^{''}\)phase structure of Fe\(_{16}\)N\(_{2}\). The optimized a and c/a are 5.69 Å and 1.1, respectively. The corresponding values in an experiment are 5.72 Å and 1.1^{14}. In the unit cell, 16 Fe atoms occupy 3 inequivalent sites at the Wyckoff positions of 4e, 8h, and 4d, while 2 N are at the octahedral interstices with the 4 Fe(8h) coordination^{14}. These 4e, 8h, and 4d sites differ in magnetic moment (Table 1): 2.14 (2.33), 2.36 (2.45), and 2.82 (3.05) \(\mu _B\) in the present theory (experiment^{17}), respectively. The mechanism is associated with the nonidentical Fe–N bond lengths: Fe(4e)–N: 1.83, Fe(8h)–N: 1.95, and Fe(4d)–N: 3.24 Å. Furthermore, the energy levels of their 3d orbitals evolve near \(E_F\), particularly in the spindown channel (Fig. 1b).
Only one Fe atom in the 16 Fe unit cell is replaced by a 3dmetal atom (M \(=\) Co−Ga and Al), which corresponds to approximately 5.5 at.% doping. For each M, we have considered three different substitution sites, i.e., M(4e), M(8h), and M(4d). The \(\alpha ^{''}\)phase stability upon M replacement can be inspected by the enthalpy of formation: \(H_f = (H  \sum _i \mu _i N_i)N_A/N\), where \(\mu _i\) and \(N_i\) are the chemical potential and number of decomposable components i, respectively. \(N_A\) and N are the Avogadro constant and the number of atoms in the unit cell. The obtained \(H_f\) values of Fe\(_{16}\)N\(_{2}\) are 3.14 kJmol\(^{\text {}1}\) against (\(\alpha\)Fe)+N\(_2\) and 0.80 kJ mol\(^{\text {}1}\) against (\(\alpha\)Fe)+(\(\gamma ^\prime\)Fe\(_4\)N) decomposition. The small negative value of the latter implies that the \(\alpha ^{''}\)Fe\(_{16}\)N\(_{2}\) phase is stable at a low temperature but most likely decomposes into the \(\alpha\)Fe and \(\gamma ^\prime\)Fe\(_4\)N phases at an elevated temperature, as observed experimentally^{14}. From the ternary Fe−M−N phase diagram, as an example for M \(=\) Al in Fig. 1c, Fe\(_4\)N and Fe\(_3\)M phases are identified as the most competitive binary decomposable phases to Fe\(_{15}\)M\(_1\)N\(_2\). For M \(=\) Ni (Cu and Zn), Fe\(_4\)N+FeNi(Cu/Zn)+Fe decomposition has been considered since Fe\(_3\)Ni (Fe\(_3\)Cu/Zn and FeCu/Zn) is unstable.
Figure 1d presents the calculated \(H_f\) of Fe\(_{15}\)M\(_1\)N\(_2\) (M \(=\) Co−Ga and Al) for M(4e), M(8h), and M(4d). All the M elements prefer 4dsite replacement, which in turn splits the neighboring Fe(4d) sites into Fe(4d)\(_c\) along the c axis and Fe(4d)\(_{ab}\) on the ab plane with dissimilar magnetic properties, as addressed in the following paragraphs. The \(\alpha ^{''}\)phase becomes unstable upon Co to Cu replacements. In contrast, the replacement of Zn, Ga, and Al improves the \(\alpha ^{''}\)phase stability with \(H_f\) enhanced by 0.04−0.8 kJ mol\(^{\text {}1}\) in magnitude, as their nitrides (ZnN, GaN, and AlN) have higher standard enthalpies of formation, in the range of 100 to 320 kJ mol\(^{\text {}1}\)^{23,24,25}, than FeN (47 kJ mol\(^{\text {}1}\))^{25}. The completely filled dorbitals of the Zn and Ga elements provide extra stability to the system, as these elements have a symmetrical distribution of electrons and larger exchange energies than Fe^{26}.
We further investigate the structural stability at an elevated temperature using ab initio molecular dynamic (AIMD) simulation. Figure 2a,b present the fluctuations of the total free energy of the selected Fe\(_{16}\)N\(_{2}\) and Fe\(_{15}\)Al\(_1\)N\(_2\) phases for given temperatures 300, 500, and 600 K, respectively. The total energy of Fe\(_{16}\)N\(_{2}\) decreases immediately within a few fs by 1.3 (at 300 K)−2.2 eV/f.u. (at 600 K), which is associated with the thermal motion and relocation of atomic coordinates. Furthermore, the energy variation during the AIMD simulation increases with temperature and reaches 3.2 eV/f.u. at 600 K, where the \(\alpha ^{''}\)phase structure is largely distorted, as indicated in the insets in Fig. 2a. On the other hand, the energy fluctuation of Fe\(_{15}\)Al\(_1\)N\(_2\) phase is rather small within 1.5 eV/f.u. even at 600 K (Fig. 2b). In particular, the \(\alpha ^{''}\) phase tends to maintain up to 600 K (insets in Fig. 2b), although marginal phonon vibrations and atomic coordinate distortions occur during the AIMD simulation.
In Fig. 3a, an even more notable finding is the persistent increase in \(K_u\) as M changes from Ni to Ga, reaching the largest value of 1.85 MJ m\(^{\text {}3}\) for Ga replacement. This value is more than 3 times the enhancement attained for \(\alpha ^{''}\)Fe\(_{16}\)N\(_{2}\). The present \(K_u\) of Fe\(_{16}\)N\(_{2}\) is 0.6 MJ m\(^{\text {}3}\), which is within the range of experimental values of 0.4−1 MJ m\(^{\text {}3}\)^{20,21,22}. The enhanced \(K_u\) of the \(\alpha ^{''}\)phase is associated with tetragonal distortion. The Jahn−Tellerlike dorbital level splitting when \(\alpha \rightarrow \alpha ^{''}\) offers more electronic energy level degrees of freedom^{10}. More specifically, the tetragonal distortion splits the cubic \(e_g\) and \(t_{2g}\) levels around \(E_F\): singlets \(a_1\) (\(d_{x^2\text {}y^2}\)) and \(b_1\) (\(d_{3r^2\text {}z^2}\)), and singlet \(b_2\) (\(d_{xy}\)) and doublet e (\(d_{yz,xz}\)), respectively. Evidently, the energy levels near \(E_F\) of the spindown electrons, especially the \(a_1\) and \(b_2\) states, differ at the 4e, 8h, and 4d sites (Fig. 1b). Further analyses indicate that the difference comes from their dissimilar hybridization with N2p orbitals.
Our analysis of atom resolved magnetocrystalline anisotropy energy (MAE) in Fig. 3c indicates that MAE in this \(\alpha ^{''}\)phase distributes unequally over the unit cell: 0.14, 0.18, and 0.03 meV at the 4e, 8h, and 4d sites, respectively. Here, MAE is scaled down to the microscopic atomic level (meV/atom), rather than the macroscopic energy density (MJ m\(^{\text {}3}\)). From the kresolved (minorityspin) eigenvalue analysis in Fig. 1b, featured bands with conelike shapes occur: the minimum of \(a_1\) (parabolic) and maximum of \(b_2\) (reverse parabolic) dispersions touch at the \(\Gamma\) point. In particular, for Fe(8h), such a conical \(a_1b_2\) pair appears right at \(E_F\). In association with their reduced eigenvalue difference across \(E_F\), the SOC matrix term in the Hamiltonian can thus increase the positive contribution to MAE, according to the perturbation theory^{11}.
According to Fig. 3d, the Fe(4d)\(_c\) site plays a major role in the M(Cu to Ga)induced enhancement in MAE (\(\Delta \text {MAE}\)) rather than the Fe(4d)\(_{ab}\) site. The contributions to MAE from the other 4e and 8h sites cannot be ignored although minor. Meanwhile, the conical \(a_1b_2\) pair of the Fe(4d)\(_c\) site moves gradually toward \(E_F\) with the Cu to Ga replacement (Fig. 4a), which reflects the rigidband model. A similar phenomenon is not present for the Fe(4d)\(_{ab}\) site because of its longer separation (4 Å) from M than Fe(4d)\(_c\) (3.1 Å). We therefore attribute MAE in Fe\(_{15}\)M\(_1\)N\(_2\) to the joint effects of the Jahn−Teller level splitting and the suppliedelectroninduced level changes of the dorbitals.
In the rigidband picture, the shift of the electronic states is related to the change in the energy of the Bloch state with M (\(\Delta \varepsilon _k\)), as \(\rho (\varepsilon ) = \rho _0(\varepsilon )  [ \partial \rho _0(\varepsilon _k) /\partial \varepsilon _k ] \Delta \varepsilon _k\)^{27}, where \(\rho (\varepsilon )\) and \(\rho _0(\varepsilon )\) are the density of states (DOS) of Fe(4d)\(_c\) in Fe\(_{15}\)M\(_1\)N\(_{2}\) and Fe\(_{16}\)N\(_{2}\), respectively. For a small amount of M, \(\Delta \varepsilon _k\) is also small and thus independent of k, where the shape of the band structure remains the same but displaced by \(\Delta \varepsilon _k\). Eventually, for the Ga replacement, the conical \(a_1b_2\) pair of the Fe(4d)\(_c\) site shifts down and appears near \(E_F\), which in turn enhances MAE. Here, the Jahn−Teller argument is not applicable, as the c/a (1.1) of \(\alpha ^{''}\)Fe\(_{16}\)N\(_{2}\) remains almost the same upon M replacement (Table 1).
In accordance with the \(a_1b_2\) shift (Fig. 4a), the replacement element that can maximize MAE is Ga, in line with the obtained \(K_u\) in Fig. 3a. To support this scenario, we explore the replacement element Al, which is isoelectronic to Ga. Remarkably, we find \(K_u\) value of 1.80 MJ m\(^{\text {}3}\) for the Al replacement (Fig. 3a), similar to that (1.85 MJ m\(^{\text {}3}\)) for the Ga replacement. Accordingly, similar electronic features of the \(a_1b_2\) bands and \(\Delta \varepsilon _k\) at the Fe(4d)\(_c\) site are identified in Fig. 4a for the same group elements, Al and Ga. Furthermore, as mentioned early in Fig. 1d, the inclusion of Al in Fe\(_{16}\)N\(_2\) greatly improves the \(\alpha ^{''}\)phase stability beyond the other Mreplacements. From a practical viewpoint, the Ga and Al replacements (particularly, Al) for Fe are desirable for REfree permanent magnets because of their abundances on earth.
In line with the argument we outlined thus far, an even larger \(K_u\) may be achieved if more Fe in Fe\(_{16}\)N\(_{2}\) are replaced with metal elements with more valence electrons. To test this scenario, we replace half of the Fe in Fe\(_{16}\)N\(_{2}\) with Co. From the total energy minimization, all the 4d and 4e sites are occupied by Co, forming the B2phase, while 2 N prefer the 4 Fe(8h) coordinated octahedral interstices on the same ab plane. As Co has 1 more electron and stronger SOC than Fe, we find that the \(K_u\) in (Fe\(_{0.5}\)Co\(_{0.5}\))\(_{16}\)N\(_{2}\) is 1.65 MJ m\(^{\text {}3}\). This value is more than double that (0.6 MJ m\(^{\text {}3}\)) of \(\alpha ^{''}\)Fe\(_{16}\)N\(_{2}\). A similar argument can be applied for other replacements such as Ni and Zn (not shown here). Furthermore, the enhanced c/a (1.17) of (Fe\(_{0.5}\)Co\(_{0.5}\))\(_{16}\)N\(_{2}\), compared with 1.1 in Fe\(_{16}\)N\(_{2}\), is clearly an additional cause of the large \(K_u\)^{8}.
Remarkably, the Mreplaced (Fe\(_{0.5}\)Co\(_{0.5}\))\(_{16}\)N\(_{2}\) compounds exhibit a trend similar to, but with notably enhanced numerical values, that in Fe\(_{16}\)N\(_{2}\): a nearly linear increase in \(K_u\) from Cu to Ga (Fig. 3b). Eventually, Ga replacement leads to a \(K_u\) as high as 2.44 MJ m\(^{\text {}3}\). Such supreme value of \(K_u\) can also be achieved for Al (2.41 MJ m\(^{\text {}3}\)). These values are more than 4 times that of Fe\(_{16}\)N\(_{2}\) and more than half the value of 4.5 MJ m\(^{\text {}3}\) of the typical REmagnet Nd\(_2\)Fe\(_{14}\)B^{28}. Similar to in Fe\(_{16}\)N\(_{2}\), Co next to M on the c axis (denoted Co\(_c\)) produces the largest \(\Delta \text {MAE}\) compared with other sites (Fig. 3f), although MAE is larger for Fe than for Co (Fig. 3e). At this Co\(_c\) site, it is manifested that the main mechanism of enhancing \(K_u\) is the displacement of the (unoccupied) \(a_1\) band toward \(E_F\) as M changes from Cu to Ga and Al in Fig. 4b.
We believe that the present argument is rather general and can be applied to other magnetic materials. To better justify the excesselectroninduced enhancement in \(K_u\), we forcibly increase the number of valence electrons in Fe\(_{16}\)N\(_{2}\) and (Fe\(_{0.5}\)Co\(_{0.5}\))\(_{16}\)N\(_{2}\). This approach reflects an excess electron that is uniformly accumulated over all Fe rather than at a specific site neighboring the M replacement. For both Fe\(_{16}\)N\(_{2}\) and (Fe\(_{0.5}\)Co\(_{0.5}\))\(_{16}\)N\(_{2}\), \(K_u\) increases linearly as the number of excess electrons (\(\Delta \Omega\)) increases (Fig. 5a). Nearly the same values of \(K_u\) of 1.3 MJ m\(^{\text {}3}\) in Znreplaced Fe\(_{16}\)N\(_{2}\) and 2.4 MJ m\(^{\text {}3}\) in Gareplaced (Fe\(_{0.5}\)Co\(_{0.5}\))\(_{16}\)N\(_{2}\) are reproduced at \(\Delta \Omega\) \(=\) 0.2 e/atom. From the simplified DOS analyses in Fig. 5b for Fe\(_{16}\)N\(_{2}\), the unoccupied \(a_1\) bands of all the Fe sites displace toward \(E_F\) upon an increase in \(\Delta \Omega\), while the occupied \(b_2\) state is rather insensitive. This result again reveals that the dorbital level change induced by supplied electrons is the main mechanism of the \(K_u\) enhancement.
We now would like to highlight the intrinsic hard magnetic properties, including maximum theoretical energy product (BH)\(_{\text {max}}\), anisotropic field \(\mu _0 H_a\), and hardness parameter \(\kappa\), of the present compounds. The sufficiently large \(\mu _0 M_s\) and thus (BH)\(_{\text {max}}\), defined as (BH)\(_{\text {max}}\) \(=\) (1/4)\(\mu _0 M_s^2\)^{29}, are worth noting. In Table 1, Fe\(_{15}\)M\(_1\)N\(_{2}\) exhibits \(\mu _0 M_s\) of 2.24−1.98 T, while the (Fe\(_{0.5}\)Co\(_{0.5}\))\(_{16}\)N\(_{2}\)based compounds have a slightly lower magnetization of 1.96−1.80 T. These values are far beyond those of the best highperformance permanent magnets, for example, 1.61 T for Nd\(_2\)Fe\(_{14}\)B^{28}. The predicted \(\mu _0 H_a\) (\(=2K_u/M_s\))^{29} increases from 0.65 T for Fe\(_{16}\)N\(_{2}\) to more than 2 (3) T for the Ga/Alreplaced Fe\(_{16}\)N\(_{2}\) ((Fe\(_{0.5}\)Co\(_{0.5}\))\(_{16}\)N\(_{2}\)), as shown in Table 1. Additionally, typical permanent magnets possess a hardness parameter \(\kappa\) (\(=\) (\(K_u/\mu _0 M_s^2\))\(^{1/2}\)) close to or greater than 1^{29}. The calculated \(\kappa\) of the Mreplaced (Fe\(_{0.5}\)Co\(_{0.5}\))\(_{16}\)N\(_{2}\) (Mreplaced Fe\(_{16}\)N\(_{2}\)) ranges within 0.75−1.10 (0.41−0.78), values which are 2−3 times that (0.38) of Fe\(_{16}\)N\(_{2}\). We finally explore the magnetization dynamics of \(\alpha\)Fe and Fe\(_{15}\)M\(_1\)N\(_{2}\) for the selected M \(=\) Fe, Zn, Ga, and Al. From the temperature dependent magnetization in Fig. 6, the absolute value of \(T_c\) can be estimated by fitting the magnetization data to the function M(T) \(=\) (1\(T\)/\(T_c\))\(^{\beta }\)^{30}. We find that the calculated \(T_c\) values of \(\alpha\)Fe and \(\alpha ^{''}\)Fe\(_{16}\)N\(_{2}\) are 1026 and 925 K, respectively, which are in reasonable agreement with the experimental ones (1044 and 813 K)^{31, 32}. Furthermore, as listed in Table 1, the obtained \(T_c\) values (804−855 K) of the stable compounds (M \(=\) Zn, Ga, and Al) are sufficient to fulfill the basic requirement (no less than 550 K) of permanent magnets^{33}.
Conclusion
In summary, we show, using firstprinciples calculations and rigidband model analysis of \(\alpha ^{''}\)phase Fe\(_{16}\)N\(_{2}\), that \(K_u\) can be scaled up by a few times upon the substitution of metal elements with more valence electrons than Fe, from Co to Ga, without the inclusion of RE and HM elements. More remarkably, the replacement by simple metals (Al and Ga) has potential for simultaneous enhancements of \(K_u\) and the thermal stability, which would make \(\alpha ^{''}\)Fe\(_{16}\)N\(_{2}\) a possibly REfree permanent magnet, along with its high Curie temperature and low materials price. Furthermore, we demonstrate that a similar argument, as a general rule, is applicable to suitable systems to achieve enhanced intrinsic hard magnetic properties and improved thermal stability. We hope that our results can be used as a guideline for subsequent experimental investigations of REfree highperformance permanent magnetic materials.
Methods
The densityfunctional theory (DFT) calculations were performed using the projector augmented wave (PAW) method^{34}, as implemented in the Vienna ab initio simulation package (VASP)^{35}. The exchangecorrelation interactions are treated with the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE)^{36}. We used an energy cutoff of 500 eV and a \(11 \times 11\times\)11 Brillouin zone (BZ) kpoint mesh to relax the lattice parameters and atomic coordinates until the largest force decreased to below 10\(^{\text {}2}\) eV/Å. The total energy method is applied to obtain \(K_u\), which is expressed as \(K_u = (E_a  E_c)/volume\), where \(E_a\) and \(E_c\) are the total energies with magnetization along the a and c axes, respectively. To obtain wellconverged \(K_u\), we impose a denser kpoint mesh of \(15 \times 15\times\)15 with a smaller smearing of 0.05 in the Gaussian method, where the convergence of \(K_u\) with respect to the number of k points and smearing parameter is ensured. In tetragonal symmetry, \(K_u\) is expressed as \(K_u \approx K_1 \text {sin}^2 \theta + K_2 \text {sin}^4 \theta\), where \(K_1\) and \(K_2\) are the magnetic anisotropy constants and \(\theta\) is the polar angle between the magnetization vector and the easy axis (c axis in the present system). For \(\theta = \pi /2\), \(K_u = K_1 + K_2\). It is a formidable task to ensure numerical results of \(K_u\) with all electron methods, if we start from scratch. To this end, we have also performed fullpotential calculations using the WIEN2K package^{37}, adopting the optimized lattice constants and ionic positions obtained from the VASP calculations. The two methods produce consistent results. In the AIMD simulation, we adopted the Noséthermostat algorithm to model a canonical ensemble^{38}. A time step of 1 fs and 10000 ionic steps were used for the total simulation time of 10 ps with the \(\Gamma\)point BZ integration, where the lattice parameters and atomic coordinates are allowed to relax at constant volume. The numerical calculations for magnetization dynamics and \(T_c\) were carried out using Monte Carlo simulation based on the Heisenberg model in the VAMPIRE package^{30}. Here, the Heisenberg spin Hamiltonian is defined by
where \(J_{ij}\) is the exchange interaction between two spins \(\mathbf{S} _i\) at the i site and \(\mathbf{S} _j\) at the j site. The exchange interaction parameters, from the first to the third nearest neighbor atoms, were estimated by the constrained local moment approach in the VASP calculations. More detailed methodology is provided in Ref.^{9}.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Chikazumi, S. Physics of Magnetism (Wiley, New York, 1964).
McCurrie, R. A. Ferromagnetic Materials. Structure and Properties (Academic Press, London, 1994).
Skomski, R. & Coey, J. M. D. Permanent Magnetism (Institute of Physics, Bristol, 1999).
Coey, J. M. D. Magnetism and Magnetic Materials (Cambridge University Press, Cambridge, 2010).
Sun, S., Murray, C. B., Weller, D., Folks, L. & Moser, A. Monodisperse FePt nanoparticles and ferromagnetic FePt nanocrystal superlattices. Science 287, 1989 (2000).
Sagawa, M., Fujimura, S., Togawa, N., Yamamoto, H. & Matsuura, Y. New material for permanent magnets on a base of Nd and Fe. J. Appl. Phys. 55, 2083 (1984).
Yamamoto, H., Matsuura, Y., Fujimura, S. & Sagawa, M. Magnetocrystalline anisotropy of R\(_2\)Fe\(_{14}\)B tetragonal compounds. Appl. Phys. Lett. 45, 1141 (1984).
Burkert, T., Nordstrom, L., Eriksson, O. & Heinonen, O. Giant magnetic anisotropy in tetragonal FeCo alloys. Phys. Rev. Lett. 93, 027203 (2004).
Odkhuu, D. & Hong, S. C. Firstprinciples prediction of possible rareearth free permanent magnet of tetragonal FeCo with enhanced magnetic anisotropy and energy product through interstitial nitrogen. Phys. Rev. Appl. 11, 054085 (2019).
Odkhuu, D., Rhim, S. H., Park, N., Nakamura, K. & Hong, S. C. Jahn–Teller driven perpendicular magnetocrystalline anisotropy in metastable ruthenium. Phys. Rev. B 91, 014437 (2015).
Wang, D. S., Wu, R. Q. & Freeman, A. J. Firstprinciples theory of surface magnetocrystalline anisotropy and the diatomicpair model. Phys. Rev. B 47, 14932 (1993).
Andersson, G. et al. Perpendicular magnetocrystalline anisotropy in tetragonally distorted FeCo alloys. Phys. Rev. Lett. 96, 037205 (2006).
Winkelmann, A., Przybylski, M., Luo, F., Shi, Y. & Barthel, J. Perpendicular magnetic anisotropy induced by tetragonal distortion of FeCo alloy films grown on Pd(001). Phys. Rev. Lett. 96, 257205 (2006).
Jack, K. H. The occurence and the crystal structure of \(\alpha\)iron nitride; a new type of interstitial alloy formed during the tempering of nitrogenmartensite. Proc. R. Soc. A 208, 216 (1951).
Kim, T. K. & Takahashi, M. New magnetic material having ultrahigh magnetic moment. Appl. Phys. Lett. 20, 492 (1972).
Mitsuoka, K., Miyajima, H., Ino, H. & Chikazumi, B. Induced magnetic moment in ferromagnetic Fe alloys by tetragonally elongated lattice expansion. J. Phys. Sot. Jpn. 53, 2381 (1984).
Coey, J. M. D. The magnetization of bulk \(\alpha ^{^{\prime \prime }}\)Fe\(_{16}\)N\(_{2}\). J. App. Phys. 76, 6632 (1994).
Wang, H. Y. et al. The effect of Ti addition on the thermal stability of \(\alpha ^{^{\prime \prime }}\)Fe\(_{16}\)N\(_{2}\). J. Phys. D 30, 2932 (1997).
Wang, H. Y., Jiang, E. Y. & Wu, P. Enhancement of the thermal stability of Fe\(_{16}\)N\(_2\) by Ti addition. J. Magn. Magn. Mater. 177, 1285 (1998).
Kita, E., Shibata, K., Yanagihara, H., Sasaki, Y. & Kishimoto, M. Magnetic anisotropy in spherical Fe\(_{16}\)N\(_{2}\) coreshell nanoparticles determined by torque measurements. J. Magn. Magn. Mater. 310, 2411 (2007).
Ji, N. et al. Perpendicular magnetic anisotropy and high spinpolarization ratio in epitaxial FeN thin films. Phys. Rev. B 84, 245310 (2011).
Ogawa, T. et al. Challenge to the synthesis of \(\alpha ^{^{\prime \prime }}\)Fe\(_{16}\)N\(_{2}\) compound nanoparticle with high saturation magnetization for rare earth free new permanent magnetic material. Appl. Phys. Express 6, 073007 (2013).
Mellor, J. W. A Comprehensive Treatise on Inorganic and Theoretical Chemistry (Wiley, New York, 1964).
Ranade, M. R. et al. Enthalpy of formation of gallium nitride. J. Phys. Chem. B. 104, 4060 (2000).
Gupta, R. et al. Formation of iron nitride thin films with Al and Ti additives. J. Appl. Phys. 111, 130520 (2012).
Verma, N.K., Khanna, S.K. & Kapila, B. Comprehensive Chemistry XII (Laxmi Publications LTD, 2019).
Stern, E. A. Rigidband model of alloys. Phys. Rev. 157, 544 (1967).
Sagawa, M., Fujimura, S., Yamamoto, H., Matsuura, Y. & Hirosawa, S. Magnetic properties of rareearthironboron permanent magnet materials. J. Appl. Phys. 57, 4094 (1985).
Skomski, R. & Coey, J. M. D. Magnetic anisotropy: How much is enough for a permanent magnet. Scr. Mater. 112, 3 (2016).
Asselin, P. et al. Constrained Monte Carlo method and calculation of the temperature dependence of magnetic anisotropy. Phys. Rev. B 82, 054415 (2010).
Leger, J. M., LoriersSusse, C. & Vodar, B. Pressure effect on the curie temperatures of transition metals and alloys. Phys. Rev. B 6, 4250 (1972).
Sugita, Y. et al. Giant magnetic moment and other magnetic properties of epitaxial grown Fe\(_{16}\)N\(_2\) singlecrystal films. J. Appl. Phys. 70, 5977 (1991).
Coey, J. M. D. Permanent magnets: Plugging the gap. Scr. Mater. 67, 524 (2012).
Blochl, P. E. Projector augmentedwave method. Phys. Rev. B 50, 17953 (1994).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169 (1996).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Blaha, P., Schwarz, K., Madsen, G.K.H., Kvasnicka, D. & Luitz, J. WIEN2K, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties, Technische Universitat Wien, Vienna (2001).
Bylander, D. M. & Kleinman, L. Energy fluctuations induced by the Nose thermostat. Phys. Rev. B 46, 13756 (1992).
Acknowledgements
This research was supported by Future Materials Discovery Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (Grant No. 2016M3D1A1027831), the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korean government (MOTIE) (Grant No. 20192010106850, development of magnetic materials for IE4 class motor), and the Incheon National University Research Grant in 20180438.
Author information
Authors and Affiliations
Contributions
D.O. and S.C.H conceived the study and wrote the manuscript. D.O. performed the calculations.
Corresponding authors
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Odkhuu, D., Hong, S.C. Simultaneous tuning of the magnetic anisotropy and thermal stability of \(\alpha ^{''}\)phase Fe\(_{16}\)N\(_{2}\). Sci Rep 11, 7823 (2021). https://doi.org/10.1038/s41598021870772
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598021870772
This article is cited by

Intrinsic hard magnetism and thermal stability of a ThMn12type permanent magnet
npj Computational Materials (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.