Magnetic properties and magnetocrystalline anisotropy of Nd2Fe17, Nd2Fe17X3, and related compounds

The electronic and magnetic properties of Nd2Fe17 and Nd2Fe17X3 (X = C or N) compounds have been calculated using the first-principles density functional calculations. Among these, the nitrogen and carbon interstitial compounds exhibit all of the required properties such as a saturation moment of 1.6 T, Curie temperature of 700–750 K, however easy magnetic axis lies in the planar direction making them less attractive for permanent magnet applications. The calculated magnetocrystalline anisotropy energy is found to be −2.7 MJ/m3 for Nd2Fe17C3 and −4.7 MJ/m3 for Nd2Fe17N3. We further explored the possibility of changing the easy axis direction through La/Ce alloying at Nd site. Although the MAE is found to be smaller in magnitude for all the La/Ce alloys it still maintains planar direction.

Nd spin moment opposite to that of the Fe and energy difference between ground state and non magnetic was found to be as 360, 460, and 510 meV on per Fe atom basis. The calculated magnetic moments are summarized in Table 1. The variation in Nd orbital and spin magnetic moments for Nd 2 Fe 17 N 3 and Nd 2 Fe 17 C 3 is presented in Fig. 1(a) and (b) respectively. As can be seen from Fig. 1(a) the orbital magnetic moments (M L ) is positive for rare earth atom, indicating that the direction of the orbital magnetic moment is opposite to the spin magnetic moment. This is consistent with the Hund's rule for half filled rare earth ions. The calculated orbital moment of Nd atoms without U is 1.49 μ B for both Nd 2 Fe 17 N 3 and Nd 2 Fe 17 C 3 . On varying U in GGA + SOC + U calculations, although the orbital moment of Nd atoms increases, overall the orbital moment displays a weak dependence on U. The rate of increase of orbital moments with U is higher for carbon interstitial compound than in nitrogen interstitial compound. The variation of Nd spin moments with U parameter is also shown in the Fig. 1(b). The Nd spin moments are almost independent of U values used.
The calculated Fe orbital moments for Nd 2 Fe 17 lie between 0.04 to 0.05 μ B . On introducing interstitial nitrogen/carbon atom the Fe orbital moments are increased, and the enhancement is higher for Nd 2 Fe 17 C 3 . The total magnetic moments for the crystallographically nonequivalent Fe sites are plotted in Fig. 2. The calculated total magnetic moments agree reasonably with experiments. Furthermore, on introducing nitrogen/carbon interstitial atoms, while the spin moment of Fe-18f and Fe-18h sites, which are close to N/C atoms decreases, the moments on the distant Fe-9d site are enhanced as shown in Fig. 2 and Table 1. This trend of Fe moments on nitrogenation/carbonation is in good qualitative agreement with previous studies and has been attributed to hybridization between N(C) and Fe atoms 21 . Similar variation in magnetic moments upon nitrogenation and carbonization has also been observed for Sm 2 Fe 17 . Regardless of this slight variation the average Fe-spin moment of these systems remains in the range of 2.3-2.5 μ B , which is significantly higher than the value for BCC Fe 2.2 μ B . The experimental values of Fe moments for Nd 2 Fe 17 N 3 by neutron powder diffraction measurements show large variation  19 , indicating the sensitivity of magnetic properties on nitrogen stoichiometry. This large variation is mainly due to the difference in Fe-Fe distance depending upon on nitrogen stoichiometry. As explained above due to the sensitivity of magnetic properties on nitrogen/ carbon content, a good quantitative agreement between experiments and theory is unlikely. Nonetheless our calculated total magnetic moment of 36.3 μ B for Nd 2 Fe 17 N 3 lies well within the range proposed by experimental measurements as shown in Table 1. The calculated Nd total magnetic moments are also listed in Table 1. We see that due to the presence of 4f electrons, Nd atom has a large (~1.47 μ B ) orbital moment. For both carbon and nitrogen interstitial compounds the orbital moment of Nd is slightly increased from 1.47 μ B to 1.50 μ B . The magnetic moment for La/Ce substituted compounds are also listed in Table 1. For the substituted compounds the magnetic moment on Fe and Nd sites mostly remain unchanged. On the other hand the magnetic moment on the substituted RE atomic sites (La/Ce) is greatly reduced as La has no f electrons and Ce has only one outermost f electron. Next, we calculate the magnetocrystalline anisotropy energy (MAE) constant K 1 , which is shown in Table 1. MAE is an important parameter for generating high coercivity in a permanent magnets. For accurate estimate of MAE values a proper treatment of strongly correlated f electrons is essential by applying a onsite Hubbard U parameter. Here a U value of 3 and 5 eV is used for Ce, and Nd, respectively. The convergence of MAE with respect to U parameter is shown in Fig. 3. For Nd 2 Fe 17 N 3 where the experimental data of MAE is available, our calculated MAE with U Nd as 5 eV is in reasonable agreement with the experimental data. We found the parent compound Nd 2 Fe 17 to be planar with K 1 value as −2.1 MJ/m 3 , which is in fair agreement with the experimentally measured value of −3.7 MJ/m 312 . Unlike Sm 2 Fe 17 where carbon and nitrogen interstitial atoms switch the direction of MAE from planar to axial, Nd 2 Fe 17 N 3 and Nd 2 Fe 17 C 3 does not exhibit this change in MAE sign and still maintain planar anisotropy. This planar anisotropy is less desirable for permanent magnet applications. However the magnitude of K 1 is quite significant and a value of −2.7 MJ/m 3 and −4.7 MJ/m 3 was obtained for Nd 2 Fe 17 C 3 and Nd 2 Fe 17 N 3 , respectively. The measured anisotropy values for Nd 2 Fe 17 N 3 are also listed in Table 1, which is in fair agreement with our calculated value. Given the high magnetization along with large magnitude of K 1 one might obtain a coercivity higher than 1.5 Tesla in these compounds if the MAE was uniaxial. To explore the possibility of switching the sign of MAE and reducing the rare earth content in these magnets next we study the effect of La/Ce substitution on MAE. As the MAE is sensitive to the nature of electronic structure around the Fermi energy, it can be controlled by tuning the band structure around the Fermi energy. In practice this can be done, for example, by doping/alloying. Theoretical calculations were done in order to find a possible alloy based on NdLaFe 17 X 3 and NdCeFe 17 X 3 (X = C or N) with uniaxial anisotropy. We thus began with substituting one out of two Nd atoms in the primitive cell by La/Ce. Results of these calculations are summarized the in Table 1. As seen, for all the La/Ce substituted systems the MAE still remains planar. While for Ce substituted compounds the MAE is reduced by 25-31% (on going from nitrogen interstitial to carbon interstitial), a higher 40-60% reduction was observed for La substituted compounds. Figure 4 presents the density of states (DOS) for Nd 2 Fe 17 N 3 . The total DOS illustrates behavior of a ferromagnetic system. In the Fig. 4(b) and (c) the DOS for RE-4f and Fe-3d states are shown. On comparing the total DOS with partial DOS in the lower panels, we can see that the states in the vicinity of the Fermi level are mainly composed of Fe-3d states. The RE-4f states are not occupied in the spin-up channel and are partially occupied in the spin-down channel, confirming that the Nd spin-moments align in the opposite direction compared to the Fe moments. Due to Hubbard U parameter the Nd spin down band split into lower and upper Hubbard bands separated by 6 eV (marked by red arrow in Fig. 4(b)). To understand the effect of La/Ce substitution at the Nd site on the magnetic properties of these compounds, we next analyze their DOS. The total and partial DOS for the these compounds are shown in Figs The only significant change is in the RE-f states, which are shifted to higher energies. As shown in Fig. 4(d) upon La/Ce substitution the Nd DOS around Fermi level is reduced. This reduction in RE DOS is responsible for the reduction in magnitude of MAE on La/Ce substitution ( Table 1).
The important insights into the effect of SOC on magnetic properties can be obtained from perturbation theory [25][26][27] . According to this theory the MAE can be described by the electronic structure near the Fermi energy, in-terms of coupling between occupied and unoccupied levels. To obtain information about which regions are particularly important to the MAE, the band structures after applying SOC with magnetization along either in plane or out of plane directions are plotted in Fig. 5(a) and (b)). From these bands the MAE contribution per k-point can be evaluated using the magnetic force theorem [28][29][30] , by taking the difference of the sum over occupied energy eigenvalues for different magnetization directions, which is also plotted (blue line, right y-axis) in     Fig. 6(a) in close vicinity of Fermi level (E F ), the majority spin DOS is similar for the two compounds. However, as Sm is exchanged for Nd more electrons are added into the system and the minority spin states become occupied, whereby these 4f levels are  The MAE of a permanent magnet is dependent on SOC, and can be further quantified by considering its dependence on orbital moments. As shown by Bruno et al. 26 by using perturbation theory that if deformations of the Fermi surface can be neglected and the MAE is dominated by spin-diagonal coupling, the MAE and orbital magnetic-moment anisotropy are proportional. The relation between MAE and orbital magnetic anisotropy is presented in Fig. 7. As can be seen from Fig. 7 the MAE is directly proportional to the orbital magnetic anisotropy. For Sm 2 Fe 17 C 3 , and Sm 2 Fe 17 N 3 where a uniaxial anisotropy is obtained the orbital magnetic moments is also maximum along easy axis. On the other for Nd 2 Fe 17 C 3 , and Nd 2 Fe 17 N 3 where the anisotropy is in hexagonal plane, orbital magnetic moment is maximum along the hard magnetization axis.

Discussions
We present a computational study of intrinsic magnetic properties such as saturation magnetization and MAE for Nd 2 Fe 17 , Nd 2 Fe 17 X 3 and related compounds. For all compounds the rare earth and Fe atoms spin moments are anti-aligned in the ground state. The calculated magnetic properties of the base compounds agree well with available experimental data. The treatment of the Nd-f electrons in our calculations is different from the previously reported first principles calculations. For example the calculation of Lai et al. 15 which were performed within orthogonalized linear combination of atomic orbitals (OLCAO) method, reported a decrease in total magnetic moments from 37.3 μ B /unit-cell (for to Nd 2 Fe 17 ) to 36.3 μ B /unit-cell (for Nd 2 Fe 17 N 3 ) which is contradictory to the experimental observation 19 , where on nitrogenation an increase in total magnetic moment has been reported. As shown in Table 1 the observed enhancement of magnetic moments in Nd 2 Fe 17 upon nitrogenation (for Nd 2 Fe 17 N 3 ) is nicely captured by our GGA + SOC + U calculations. Similarity Drebov et al. 20 used open-core method for treatment of f-electrons by assuming RE 3+ configuration. Such treatment of f electrons cannot reproduce some of the observed magnetic properties. For instance within open core method a moment of 2.5 μ B has been reported on Nd site, which is overestimated on comparing with the available experimental data 19 (1.6 μ B ). On the other hand, within our calculations the f-electrons are treated as valence electrons and, as a result we could correctly reproduce the reported moment for Nd site. Overall our all-electron calculations can reproduce a number of experimentally observed properties. For example, experimentally 19 for Nd 2 Fe 17 Fe-9d site has the lowest magnetic moment of 1.6 μ B . Upon introducing the interstitial nitrogen the moment of Fe-9d increases significantly and for Nd 2 Fe 17 N 3 the Fe-9d site has the maximum moment. This is nicely produced by our calculations as shown in Fig. 2. Furthermore the calculated MAE for Nd 2 Fe 17 and Nd 2 Fe 17 N 3 is in fair agreement with the available experimental data as shown in Table 1.
Although Nd 2 Fe 17 X 3 compounds has many of the desirable properties for a permanent magnet such as high Curie point, large saturation magnetization, the easy axis of magnetization lies in the hexagonal plane hindering it practical application. To investigate the possibility of switching the MAE direction from hexagonal plane to c-axis we also investigated the effect of La/Ce substitution at Nd site. These calculations indicate that although the MAE on La/Ce substitution is reduced by 30-40%, its still lies in the hexagonal plane, which is not very promising for technological applications. The MAE has been also analyzed in terms of the electronic structure and by using the magnetic force theorem to compute k-point resolved contributions to the MAE. For Nd 2

Methods
Density functional theory (DFT) calculations in the generalized gradient approximation 31 (GGA) were performed with the full-potential linearized augmented plane waves (FP-LAPW) method as implemented in WIEN2k [32][33][34] . The sphere radii were set to 2.50, 1.88, 1.61, and 1.61 Bohr for Nd, Fe, N and C. All calculations were performed with the experimental lattice parameters 8,35,36 . The structure relaxations were performed within spin-polarized calculations without spin orbit coupling and all internal coordinates were relaxed until internal forces on atoms were less than 1 mRyd/Bohr. For structural relaxation 500 k-points were in the full Brillouin zone. SOC was included in the MAE calculations within a second variational approach 37 . For all the calculations a RK max = 7 was used. RK max is typically defined as the product of the smallest muffin-tin sphere and the largest reciprocal lattice vector, and describes the basis set size for a calculation. For La/Ce substitution at Nd site, one out of two Nd atoms in the primitive cell was replace by La/Ce. These substituted structure were subsequently relaxed to their ground state by minimizing the forces on all the atoms. All the calculations are performed in the collinear spin alignment. The magnetic anisotropy energy (MAE) is obtained by calculating the total energies of the system with spin obit coupling (SOC) as K = E a − E c , where E a and E c are the total energies for the magnetization oriented along the a and c directions, respectively. Positive (negative) K corresponds to uniaxial (planar) anisotropy. For MAE calculations the convergence with respect to K-points was carefully checked all the MAE results reported in this paper correspond to 2000 reducible K-points in full Brillouin zone. To correctly treat the strong interactions between the Nd/Ce-f electrons, the Hubbard "U" correction was applied with U Nd = 5.0 eV, and U Ce = 3.0 eV with the Hund's coupling parameter J as zero. For DFT + U calculations, the standard self interaction correction (SIC) method 38,39 was used where onsite Coulomb interaction for localized orbitals is parametrized by U effective = U − J. The values of U parameter were obtained by optimized various magnetic properties with respect to available experimental data, and lies in the typical range that has been used successfully to describe various properties of Nd/Ce compounds before 22,[40][41][42][43] .