The magnetocrystalline anisotropy (MCA) energy of the giant saturation magnetization candidate material α″–Fe16N2 was investigated using first-principles electronic-structure calculations. The plane-wave density-functional theory (DFT) code Quantum ESPRESSO was employed to study the effect of different DFT approaches on the system, particularly the influence of exchange-correlation functionals and pseudopotential methods. The MCA energies obtained this way are within the range of previous theoretical and experimental results, while exhibiting significant variation between the different approaches. The role and limitations of these approaches in the view of Fe16N2 band structure are discussed in detail.

One of the most promising candidates for an environment-friendly rare-earth-free magnetic material is the ordered iron nitride phase α″–Fe16N2.1 First discovered by Jack in 1951,2 reports of “giant” saturation magnetization Ms (well above Fe-Co alloy) by Kim and Takahashi in 19723 and Sugita et al. in 19914 raised interest in its use in magnetic recording applications. However, subsequent experimental evidence yielded significantly lower values and results by the mid-1990s were deemed inconclusive.5 A decade later, long-running work by Wang’s group once again obtained samples with giant saturation magnetization,6–8 leading to a resurgence of interest in this material.

Meanwhile, attempts to theoretically replicate such high saturation magnetization by means of the density-functional theory (DFT) have largely remained inconclusive, with most approaches yielding comparatively lower magnetic moments. In particular, more conventional methods such as the local-density approximation (LDA) and generalized gradient approximation (GGA) have particular difficulty explaining a higher magnetization.5 Several more advanced paradigms have been employed to bridge this gap and give insight into a potential mechanism for higher magnetization, including the addition of a Hubbard correction term U,9 hybrid functionals incorporating exact exchange terms such as the Heyd–Scuseria–Ernzerhof (HSE) functional,10 or the Green’s function–based GW method.10,11 Despite these efforts, the precise mechanism is still under debate, though recent Mössbauer spectra12 and polarized neutron diffraction experiment13 has provided support for the DFT calculation based on the “cluster+atom” physical model put forth by Ji et al. nearly a decade ago.14 

For use as a permanent (hard) magnet, not only is a high saturation magnetization desirable, but also a high coercivity, corresponding to a resistance to demagnetization.1 At the microscopic level, this is associated with the magneto-crystalline anisotropy (MCA), i.e., the energetical preference for one direction of magnetization over another. The higher the energy difference between magnetization along this easy axis and elsewhere (the MCA energy), the more stable is its direction of magnetization. Fe16N2 exhibits uniaxial anisotropy, with previous experiments giving a range of anisotropy constants in the range of Ku ∼ 4.4 × 106 – 1.9 × 107 emu/cc (which includes both thin-film and nanoparticle measurements15–18 — see Table I). This is lower than typical strong permanent magnets, e.g., Nd2Fe14B, FePt, CoPt, or SmCo5 with Ku ranging from ∼ 5×107 – 2×108 emu/cc.19 

TABLE I.

Summary of selected previous observations of MCA energy in Fe16N2, both experimental and theoretical. Lattice constants used in quoted theoretical results are included for context; some results omitted for brevity.

Lattice const. (Å)
PaperRef.acKu (105 erg/cc)Notes
(Exp.) Takahashi 1999 15  … … 200 Ku taken as K1 + K2 
Kita 2007 16  … … 44 Nanoparticles 
Ji 2011b 17  … … 100  
Li 2019 18  … … 190  
(Theo.) Ke 2013 11  5.72 6.29 144/131 LDA/GGA 
Zhou 2014 22  5.68 6.23 100/140 GGA/GGA+U 
Zhao 2016 20  5.68 6.22 64 GGA (pure Fe16N2
Han 2019 21  5.69 6.22 86 GGA (pure Fe16N2
Li 2019 18  5.72 6.29 50/160 GGA/GGA+U (U = 4 eV) 
Lattice const. (Å)
PaperRef.acKu (105 erg/cc)Notes
(Exp.) Takahashi 1999 15  … … 200 Ku taken as K1 + K2 
Kita 2007 16  … … 44 Nanoparticles 
Ji 2011b 17  … … 100  
Li 2019 18  … … 190  
(Theo.) Ke 2013 11  5.72 6.29 144/131 LDA/GGA 
Zhou 2014 22  5.68 6.23 100/140 GGA/GGA+U 
Zhao 2016 20  5.68 6.22 64 GGA (pure Fe16N2
Han 2019 21  5.69 6.22 86 GGA (pure Fe16N2
Li 2019 18  5.72 6.29 50/160 GGA/GGA+U (U = 4 eV) 

Unlike the question of magnetic moment, theoretical calculations of MCA energy have put less emphasis on the choice of DFT approach, for the most part employing mainly the GGA,20,21 though some have considered LDA11 or GGA+U.18,22 (Compare also Table I.) Given their effect on magnetic moment, it would be reasonable to likewise consider more systematically the influence of the choice of DFT approach on the MCA energies. In this paper, the MCA energy is thus calculated using a variety of DFT approaches, including variant exchange-correlation (XC) functionals, choice of pseudopotential (PP), and choice of applied Hubbard U, and the results compared both to previous values and amongst each other.

Electronic structure calculations were performed using the plane-wave “pwscf” code from the Quantum ESPRESSO package.23,24 Two different XC functionals were considered: the LDA in the form proposed by Perdew and Zunger,25 and the GGA in the form proposed by Perdew, Burke, and Ernzerhof.26 The GGA+U method was also considered; due to technical limitations, two variants were employed: a simplified formulation with effective Hubbard correction UeffUJ,27 and a more explicit formulation using both U and J terms due to Liechtenstein et al.28 

Under GGA XC, two different pseudopotentials were compared: Projector-Augmented Wave (PAW) pseudopotentials from the QE PSlibrary,29 and the SG15 Optimized Norm-Conserving Vanderbilt (ONCV) pseudopotentials (here labeled NC).30,31 The LDA calculations’ PAW PP were also taken from the PSlibrary. Both scalar-relativistic and fully-relativistic variants were used as described below. The following combinations were considered: (1) LDA XC with PAW PP; (2) GGA XC with PAW PP; (3) GGA XC with NC PP; and (4) GGA+U with PAW PP. For GGA+U, two choices for U and J were considered, motivated by Ji et al. (2010):14 (4a) U = 1.0 eV for all Fe (the “baseline” case), and (4b) U = 4.0 eV on the Fe adjacent to N, 1.0 eV elsewhere (the “high-moment’ case); in both cases, J = U/10 was chosen throughout, and in the simplified method, Ueff was taken equal to UJ.

Crystal structures were fully relaxed using each of the above combinations; for GGA+U, the simplified Ueff formulation was used for relaxation. The unit cell volume, shape, and atomic positions were allowed to relax within a scalar-relativistic approach i.e., neglecting terms involving spin–orbit coupling (SOC), until the forces acting on each atom were smaller than 0.025 eV/Å and all stress tensor elements smaller than 1 kbar (0.1 GPa). The calculations were performed with a kinetic energy cutoff of 150 Ry (∼2000 eV) and the Brillouin zone (BZ) was sampled by using the Monkhorst–Pack scheme with 8 × 8 × 8 zone-centered grid for structure optimization, with Marzari–Vanderbilt cold smearing of width 0.14 eV.

The MCA energy calculations were performed using the respective relaxed structures derived above for each XC–PP combination. For these calculations, the k-point mesh density was increased to 24 × 24 × 24 with smearing narrowed to 0.07 eV. The SOC terms (and fully-relativistic PP) were included; for GGA+U, the explicit U+J method was employed. The MCA energy was calculated as the difference in total energy between states with magnetization along the easy (001) and hard (100) directions, i.e., ΔEMCA = E(100)E(001).

Density of states (DOS) and band structures were calculated in the ground state (with M along the easy axis) without SOC, with the finer k-mesh and employing the tetrahedron method due to Blöchl.32 For GGA+U, the explicit U+J method was used.

The experimental crystal structure of α″–Fe16N2 is body-centered tetragonal, with lattice constants a = 5.72 Å and c = 6.29 Å, and Fe atoms in three different Wyckoff positions: 4e and 8h (adjacent to the interstitial N atoms), and 4d.2 See Figure 1 (produced using Mathematica, Version 11.3.0, by Wolfram Research, Inc.33)

FIG. 1.

Crystal Structure of α″–Fe16N2, indicating tetragonal axes, unit cell, and specific Fe Wyckoff positions.

FIG. 1.

Crystal Structure of α″–Fe16N2, indicating tetragonal axes, unit cell, and specific Fe Wyckoff positions.

Close modal

Since MCA can be strongly affected by strain,11,18 the crystal structures were relaxed for each combination of XC and PP (compare Table II). In all but the high-M GGA+U case, the relaxed unit cell volume is somewhat lower than experiment, particularly along the c direction. Additionally, the LDA structure has notably lower a compared to both GGA and experiment, while NC and PAW PP yield fairly similar structures. (The agreement of the latter in particular is consistent with previous research on consistency between DFT methods.34) The addition of Hubbard U expectedly leads to increased unit cell volume, particularly for the high-M case: here, the relaxed volume is now larger than experiment.

TABLE II.

Summary of results – relaxed lattice structure, magnetization, and MCA energies – under various combinations of XC and PP. Magnetization and anisotropy are shown both over the conventional unit cell, and in equivalent macroscopic units. (Compare also Table I.)

MethodLattice const. (Å)Magnetization
XCPPacMB/u.c.)Equiv. Ms (emu/cc)ΔEMCA (µeV/u.c.)Equiv. Ku (105 erg/cc)
LDA PAW 5.3750 6.2084 28.94 1496 1144 102 
GGA PAW 5.6874 6.2196 38.30 1765 909 71 
GGA NC 5.7054 6.2298 39.46 1818 510 40 
GGA+U+J (baseline) PAW 5.7240 6.2245 39.70 1805 632 50 
GGA+U+J (high-M) PAW 5.9486 6.3048 43.94 1826 3983 286 
MethodLattice const. (Å)Magnetization
XCPPacMB/u.c.)Equiv. Ms (emu/cc)ΔEMCA (µeV/u.c.)Equiv. Ku (105 erg/cc)
LDA PAW 5.3750 6.2084 28.94 1496 1144 102 
GGA PAW 5.6874 6.2196 38.30 1765 909 71 
GGA NC 5.7054 6.2298 39.46 1818 510 40 
GGA+U+J (baseline) PAW 5.7240 6.2245 39.70 1805 632 50 
GGA+U+J (high-M) PAW 5.9486 6.3048 43.94 1826 3983 286 

For reference, we briefly consider also the total magnetization: whereas most results are fairly well in line with previous LDA or GGA results (typically around 38.5–39.5 µB/unit cell),5,10,11 note that the LDA result is abnormally low (below 30 µB/unit cell), while the high-moment case exhibits a higher magnetization as expected. (For comparison, the lower range of experimental values are equivalent to around 36–42 µB/unit cell;5 with high-moment measurements as high as 51 µB/unit cell.7)

Notably, in most cases the MCA energy is rather low in comparison to experimental observations (compare Tables I and II). With most measurements of Ku around 1–2 × 107 erg/cc, the equivalent anisotropy constant only reaches 0.7 × 107 erg/cc under GGA with PAW PP, and decreases further to 0.5 × 107 erg/cc with NC PP. Among non-U methods, only LDA exceeds 107 erg/cc, but with significantly shorter lattice parameter a and quite lower magnetization. Interestingly, adding a uniform Hubbard term U = 1 eV also decreases equivalent Ku to 0.5 × 107 erg/cc, whereas the asymmetric high-M variant leads to greatly increased Ku ∼ 2.8 × 107 erg/cc (though here the lattice is also greatly expanded).

Spin-resolved and site-projected density-of-state plots for each case are displayed in Fig. 2. The general shape remains similar, with significant spin polarization. Going from LDA to GGA to GGA+U, the density of bands thins out as the majority-spin states increasingly shift below the Fermi level. At this point, Fe atoms approach strong ferromagnetism, with nearly-full spin-up polarized d-band. Apparently, the increased magnetic moments result from a strong increase of the exchange splitting. Notably, the distributions for GGA/PAW, GGA/NC, and GGA+U baseline are fairly similar, despite the clear discrepancies in MCA energy among these three cases. The high-moment case exhibits particularly striking shifts in the partial DOS of Fe4d in particular, and significantly lower occupation of the minority spin states.

FIG. 2.

Spin-resolved, site-projected DOS plots for each combination of XC/PP: (a) LDA/PAW, (b) GGA/PAW, (c) GGA/NC, (d) GGA+U baseline, (e) GGA+U high moment.

FIG. 2.

Spin-resolved, site-projected DOS plots for each combination of XC/PP: (a) LDA/PAW, (b) GGA/PAW, (c) GGA/NC, (d) GGA+U baseline, (e) GGA+U high moment.

Close modal

This can also be observed in the spin-resolved band structure, of which we include excerpts close to the Fermi energy for select cases in Fig. 3. While in all studied cases the system is undoubtedly metallic, the band structures are quite complex, with many bands crossing the Fermi level, creating a complex topology of Fermi surface sheets.

FIG. 3.

Selected band structure plots for the following cases: (a) LDA/PAW, (b) GGA/PAW, (c) GGA+U high moment.

FIG. 3.

Selected band structure plots for the following cases: (a) LDA/PAW, (b) GGA/PAW, (c) GGA+U high moment.

Close modal

In summary, different DFT approaches from the LDA up to GGA+U were used to calculate the MCA energy in Fe16N2, and their results were compared and discussed accordingly. In addition, the effect of relaxed crystal structure, DOS, and band structure were considered for additional context. Notably, only LDA and GGA+U (high moment) produced anisotropy as high as typical measured anisotropy constants, though in both cases the relaxed structure deviates significantly from experimental values, and the latter rather overestimates MCA. The failure to predict the correct lattice parameter by GGA+U at high U is not unexpected, as the exchange portion of GGA+U is much stronger than that of simple LDA/GGA. On the other hand, neglecting strong correlation in LDA approaches may lead to the failing DFT in this case for a proper description of mentioned properties.

Due to the fine precision involved, even small changes that minimally affect crystal structure can have a noticeable effect on calculated MCA energy – in particular, despite small differences elsewhere, GGA/PAW and GGA/NC values differ by almost a factor of 2.

Obvious targets for further study include a more systematic study and/or first-principles calculation of U, J values; considering additional more advanced XC; as well as cross-referencing the interaction of alloying and/or strain with the above with respect to the MCA energy.

Authors deeply appreciate the Minnesota Supercomputing Institute (MSI) for providing computational resources used for the DFT calculations (see http://www.msi.umn.edu), as well as the Robert F. Hartmann Endowed Chair.

Dr. Jian-Ping Wang has equity and royalty interests in and serves on the Board of Directors for Niron Magnetics, Inc., a company involved in the commercialization of Fe16N2 magnet. The University of Minnesota also has equity and royalty interests in Niron Magnetics, Inc. These interests have been reviewed and managed by the University of Minnesota in accordance with its Conflict of Interest policies.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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