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Site preference of dopant elements in Nd2Fe14B

Ab initio calculations of formation energy and mixing energy for (Nd,R)2(Fe,Co)14B [R = rare-earth elements other than Nd] are presented to address the site preference of dopants and the corresponding magnetic properties. Contrasting trends between magnetic calculations and non-magnetic calculations are discussed in conjunction with the nature of localized moments in the metallic ferromagnetism at high temperatures. Implications on the optimal heat treatment temperature to maximize the intrinsic properties are discussed referring to the experimental Curie temperature.


I. INTRODUCTION
Rare-earth permanent magnets are one of the key materials for sustainable solutions of the energy problem. It takes strong magnetization, accordingly strong coercive force that intrinsically originate in the uni-axial magnetic anisotropy, high Curie temperature, and the structure stability for a magnetic material to be useful enough in practical applications of permanent magnets. Unfortunately we often encounter a trade-off between those prerequisites. The champion magnet fabricated on the basis of Nd 2 Fe 14 B 1,2 has been a remarkably successful case where strong magnetization and anisotropy comes with robust structure, but a drawback has been its low Curie temperature that is only half of elemental Fe with the body-centered-cubic structure. Most notably, recent developments along the line of 1:12 materials including NdFe 12 N 4-6 and Sm(Fe,Co) 12 7 fabricated on a special substrate to go beyond Nd 2 Fe 14 B have shown a promising route to stronger magnetization and anisotropy than Nd 2 Fe 14 B in the typical temperature range 300 K ≤ T ≤ 450 K for practical usage with the reasonably elevated Curie temperature up to 800 K, but at the sacrifice of bulk structure stability.
In order to resolve such trade-off in ferromagnetic compounds for permanent magnets, we look into the issue of structure stability monitoring the trend of ferromagnetism in doped Nd 2 Fe 14 B to inspect the possibility for gaining both. One of the most popular dopant elements is Co following the celebrated Slater-Pauling curve as the promising route toward gaining both of magnetization and Curie temperature with a small amount of Co. Unfortunately it is known that Co degrades the intrinsic uni-axial magneto-crystalline anisotropy (MCA) in the particular crystal structure of Nd 2 Fe 14 B 3 and the possible supplement can come from doped rare-earth elements. Indeed Dy-doping had been the solution to sustain hightemperature MCA for the coercivity. Since heavy rare earth (HRE) elements are not so abundant, alternative solutions to minimize or even eliminate the usage of extra HRE have been in high demand recently 8 . It is hoped that the present study would help from the solid state physics point of view in finding a route toward the mini-mum usage of extra rare earth elements by inspecting the energetics of doping rare earth elements in conjunction with simultaneous doping of transition metals. Possible relevance of the heat treatment temperature in sample preparation for realizing a particular site selective doping is discussed referring to the intrinsic Curie temperature and the robustness of localized magnetic moments in the target materials.
In the next section we describe our ab initio methods combining two codes, namely, Korringa-Kohn-Rostoker (KKR) Green's function method on the basis of coherent potential approximation (CPA) as implemented in AkaiKKR 9 and ab initio structure optimization as implemented in another open-source package OpenMX 10 . Main results on doped Nd 2 Fe 14 B are given in Sec. III and their implications on the nature of localized magnetic moments and heat treatment temperature are discussed in Sec. IV. Final section is devoted for conclusions and outlook.

II. METHODS AND TARGET MATERIALS
A. Ab initio calculations for formation energy We start with looking at the structure stability of R 2 T 14 B where R=Nd, Sm, Dy, and Y and T=Fe or Co with calculating the formation energy ∆E f which is defined as follows: Here E tot [M] is the calculated energy for a given material M yielded from ab initio structure optimization utilizing OpenMX within the standard pseudopotential data provided therein 11 . Results from magnetically polarized and non-magnetic calculations are used for discussions without taking into account the relativistic effects. Even though spin-orbit interaction is crucial in getting MCA, the non-relativistic set up should be good enough to address the structure stability and ferromagnetism considering their dominating energy scales. We take GGA-PBE exchange correlation to describe the electronic structure dominated by ferromagnetic 3d-electrons most correctly. For rare-earth elements we use pseudopotentials with open-core approximation. The basis set we took was Nd8.0_OC-s2p2d2f1, Sm8.0_OC-s2p2d2f1, Dy8.0_OC-s2p2d2f1, Y8.0-s3p2d2f1, Fe6.0S-s2p2d1, Co6.0S-s2p2d2f1, and B7.0-s3p3d2. There is some subtlety with the partial core correction (PCC) in the opencore approximation which is set by the following entry in the input to OpenMX <scf.pcc.opencore Nd 1 Fe 0 B 0 scf.pcc.opencore> in case of Nd 2 Fe 14 B. Analogous set up is basically recycled for all calculations involving rare-earth elements in the present work. However, a problem in convergence can happen with Sm 2 Co 14 B with this particular setup which can be worked around if the calculation is done without PCC. We do both of (a) open-core approximation with PCC and (b) without PCC. Results from the setup in a) is basically taken whenever available.
For the calculations of R 2 T 14 B the energy cutoff is set to be 500 Ry and the number of k points was 64. Number of k points was sometimes increased up to 216 to ensure that calculated results with k points being 64 are already converging. The above choice of the computational setup seems to give a reasonable precision in an acceptable computational cost after monitoring the convergence with respect to the richness of the basis set, energy cutoff, and the number of k points. The experimental structure for Nd 2 Fe 14 B found in Ref. 3 is taken as the starting structure and ab initio structure optimization has been done for R 2 T 14 B.
Energy of the reference elements is calculated in the analogous way, taking the experimental structure of dhcp-Nd, trigonal Sm, dhcp-Dy, hcp-Y bcc-Fe, hcp-Co and α-B as the starting structure 12 . The number of k points for these reference systems is taken to be 512 and the rest of the set up is the same as is done for R 2 T 14 B. Unfortunately the structure of the elemental boron comes with polymorphism that is not yet well controlled from first principles 14 . The ground-state energy of Boron may not have been well fixed. Thus the absolute values of the calculated formation energy for R 2 T 14 B need be taken with some reservation while the relative trends between materials should be able to be robustly addressed. Thus calculated trend of the formation energy is shown in Fig. 1. The details are described in Sec. III A.

B. Ab initio calculations for mixing energy
Calculated formation energy fixes the structure stability at the stoichiometric limits and the structure stability of off-stoichiometric compounds ( 14 B is further addressed via mixing energy as defined in the following ways The trend of calculated mixing energy can point to the relative site preference of dopants. Whenever needed, the formation energy of (Nd,R) 2 (Fe,Co) 14 B can be restored with the following relation out. The advantage of OpenMX results is that formation energy of the stoichiometric compounds and discretely substituted compounds can both be directly addressed while fractional substitution can be more easily addressed via AkaiKKR. We combine these two approaches to cover relevant dopants for possible improvement of magnetic properties of Nd 2 Fe 14 B. Namely, all magnetic rare-earth elements which reasonably converged on the 2:14:1 structure (Pr, Sm, Gd, Tb, Dy, Ho, Er, Tm, Yb 3+ ) to partially replace Nd and Co to partially replace Fe. Convergence with a fictitious Eu 2 Fe 14 B seems to involve some problems and the task of answering them is split for other study.
Effects of non-magnetic light rare earth elements, La and Ce 4+ , explicitly involve 4f -electrons in the valence state. This is qualitatively different from what we deal with in the present work where 4f electrons are so well localized that energetics can be safely addressed only with 3d-electrons from Fe, 5d-electrons from Nd, and the respective substitutes. Energetics issue involving 4felectrons in (La,Ce)-based compounds is addressed in a separate work 16 .

III. RESULTS
Calculated formation energy of R 2 T 14 B where R=Nd, Sm, Dy, and Y and T=Fe or Co employing OpenMX is inspected in Sec. III A. In Sec. III B we present results for the mixing energy in Nd 2 (Fe,Co) 14 B obtained with OpenMX to elucidate the 3d-electron magnetism and energetics. Contrasting outcome from magnetic and non-magnetic calculations is discussed referring to past experimental claims. Then in Sec III C we discuss 5delectron physics in (Nd,R) 2 Fe 14 B where R is magnetic and trivalent rare earth elements other than Nd based on both of OpenMX and AkaiKKR results for the mixing energy. Finally in Sec. III D calculated formation energy of (Nd,Dy) 2 (Fe,Co) 14 B as obtained with OpenMX is presented.

A. Formation energy of stoichiometric compounds
Calculated energy of the reference elemental materials and the target compounds R 2 T 14 B [R=Nd, Sm, Dy, Y and T=Fe,Co] on the basis of OpenMX pseudopotentials and the choice of basis as described in Sec. II is summarized in Tables I and II, respectively. The label (a) and (b) for materials involving rare-earth elements denotes explicit incorporation of partial charge correction in open-core approximation has been included (a) or not included (b), respectively, as described in Sec. II A.
Calculated trend of the formation energy is tabulated at the rightmost column in Table II and plotted in Fig. 1 with the results basically from setup (a). Only for Sm 2 Co 14 B the results with setup (b) is plotted. Even though the structure stability of Nd 2 Fe 14 B is well es-   Fig. 2. We note that the ambiguity in the absolute value of calculated formation energy is cancelled out in the mixing energy. Discrete substitution of Fe by Co is done in OpenMX and one Co atom is put in one of the 56 possible Fe atomic sites in the unit cell. Magnetic calculations show that Fe(8j 1 ) site is most favorable for Co to substitute and Fe(8j 2 ) site is most unfavorable, which is in remarkable agreement with the past experimental claim measured by Mössbauer effect 18 . The same Mössbauer measurement 18 claimed less population in Fe(16k 1 ) and some preference in Fe(16k 2 ) which is again in good agreement with the trend in calculated mixing energy on the basis of ferromagnetic ground states while the expected preference of Fe(4c) and avoidance of Fe(4e) seems to be at variance with the experimentally observed slight avoidance on Fe(4c) and preference for Fe(4e) 18 .
Corresponding trends of magnetization is shown in Fig. 3. On top of open-core approximation results, the contribution of two Nd atoms in the formula unit has been restored with g J J(J + 1) for one Nd atom, where J = 9/2 and g J = 8/11. It is noted that all possible ways to dope Co into Nd 2 Fe 14 B lead to inferior magnetization.  This is not actually recovered by the volume effects even if the lattice shrinks upon doping of Co. While some doping of Co lead to slightly enhanced structure stability in Fig. 2, ferromagnetism is sacrificed here.
The calculated trend in the magnetization of Co-doped Nd 2 Fe 14 B is at variance with the Slater-Pauling curve. It may well be the case that Co-induced elevation of Curie temperature can render the room temperature magnetization as a function of Co concentration resembling the Slater-Pauling curve as was reported in some of the past experimental measurements 19 . In the ground state, we conclude that no Slater-Pauling curve would be observed in Nd 2 (Fe,Co) 14 B with the observed trend in Fig. 3 and also based on CPA data obtained with AkaiKKR. More systematic data for the appearance or the disappearance of Slater-Pauling curve in 4f -3d intermetallics will be presented elsewhere 20 .

C. (Nd,R)2Fe14B [R =Nd]
Calculated mixing energy for (Nd,R) 2 Fe 14 B with R=Sm, Dy, and Y with OpenMX is shown in Fig. 4. Discrete replacement is done in such a way that just one dopant rare earth atom replaces one of the eight Nd atoms in the host Nd 2 Fe 14 B and ab initio structure optimization has been done with OpenMX to yield the energy on the basis of OpenMX pseudopotentials with the setup a) and the choice of basis as described in Sec. II. Preference of dopant Dy for Nd(4f ) and calculated mixing energy is in agreement with the results in the previous work 17 . The same thing is done on the basis of continuous replacements following KKR-CPA for which the results obtained with AkaiKKR are shown in Fig. 5 which is a plain extension of what was done for Dy in Ref. 17 toward other rare-earth elements. In the latter 10% of the host Nd atoms are replaced either selectively Nd(4f )/Nd(4g) sublattice or uniformly on a fixed lattice of Nd 2 Fe 14 B that is known from past experiments 3 . The results for R=Sm and Dy look reasonably consistent between OpenMX and AkaiKKR considering the similar replacement ratio 1/8 in the former and 10% in the latter, and also the expected minor effects of lattice variation in the small concentration range of dopants. With AkaiKKR we have basically covered all relevant rareearth elements and see that except for Pr all dopant rare earth prefers Nd(4f ) site. Here we have verified that the difference between magnetic and non-magnetic calculations is minor. magnetic calculations as presented in Fig. 2 is included here. The overall trend in Co-position dependence of the calculated formation energy in (Nd,Dy) 2 (Fe,Co) 14 B is basically just a shift from what is observed for Nd 2 (Fe,Co) 14 B.
We note that presence of Co would affect the site distribution ratio of Dy over Nd(4f ) and Nd(4g). Because the energetics in the site preference of Co dominate over the energetics of Dy by a factor of two or three, the seemingly unfavorable situation in Fig. 4 where Dy resides in Nd(4g) site can be less unlikely when Co sufficiently populates Fe(8j 1 ) site.

A. Localized magnetic moments at high temperatures
We have seen that the results from magnetic and nonmagnetic calculations for Co-doped Nd 2 Fe 14 B show contrasting trends in Fig. 2. Even stronger contrast is actually seen for mixing energy calculations for Fe-doped SmCo 5 with the simpler crystal structure. Calculated mixing energy for Sm(Co,Fe) 5 is shown in Fig. 7. SmCo 5 makes the cell boundary phase in Sm-Co based magnet which was the champion magnet in 1970's and is still of practical importance for special-purpose permanent magnet used under extreme conditions. The data is collected with OpenMX where we discretely replaced one Co atom by a dopant Fe atom. There are two sublattices, Co(2c) and Co(3g), and site preference of dopant Fe points to opposite site, namely, magnetic calculations show Fe prefers Co(2c) while according to non-magnetic calculations Fe prefers Co(3g) site. Result of this non-magnetic calculation is consistent with the past calculation 21 . Considering the high Curie temperature of SmCo 5 at 1000 K that can even be higher than temperatures in sample fabrication processes, it looks likely for SmCo 5 to sustain the magnetic nature through the heat treatment processes. On the other hand the low Curie temperature of Nd 2 Fe 14 B could mean that a high-temperature sample fabrication involving Co would be more consistent with the results of non-magnetic calculations in Fig. 2.
The contrast is seen here when some experiments on Nd 2 (Fe,Co) 14 B 18 show remarkable agreement with our magnetic calculations for the preference of Co among the Fe sublattices that dominate the magnetization. Even with the low Curie temperature, high-temperature processes for the sample fabrication preserves the message of magnetic electronic state presumably due to the robustness of localized magnetic moments in Nd 2 Fe 14 B. On the other hand, experimentally found trends for the site preference of Fe and Co in 1:5 materials seem to point to the possibility that Fe prefers Co(3g) site 18 , being consistent rather with the above data from non-magnetic calculations. This presumably reflects the fragility of localized magnetic moments at high temperatures in Co-based 1:5 materials. Low (high) Curie temperature comes from relatively weak (strong) exchange couplings among the robust (fragile) localized magnetic moments in Nd 2 Fe 14 B (SmCo 5 ) at high temperatures, respectively. It seems that robustness of localized magnetic moments and strength of exchange couplings are traded off. Ab initio quantification of this scenario on the basis of local moment disorder (LMD) picture for finite-temperature magnetism 23-27 is desired.

B. Implications on the heat treatment temperature
The messages of magnetic calculations and nonmagnetic calculations for the site preference correspond to low-temperature limit and high-temperature limit of the sample fabrication processes, respectively. It is reasonable to see in the literature some experimental claims which point to various cases of the site preference of Co in Nd 2 (Fe,Co) 14 B 3 considering the relatively low Curie temperature of the host material Nd 2 Fe 14 B. It is expected that the electronic state of Nd 2 Fe 14 B may not be quite ferromagnetic in the middle of very high temperature process when the sample is prepared. The precise boundary on the temperature axis between the non-magnetic region and the paramagnetic region with the localized magnetic moments fluctuating in the heat bath is yet to be explored with LMD. When the experimental processing temperature is only slightly above the Curie temperature at 585 K, LMD calculations would be most realistic. Closeness of ferromagnetic results or nonmagnetic results to the experimental reality depends on the processing temperature. If the sample is fabricated at very high temperatures and quenched, the site preference found in the resultant sample might well be consistent with the results from non-magnetic calculations. On the other hand, site preference in well annealed sample at relatively low temperature should reflect the results from ferromagnetic calculations. Heat treatment protocols for Sm-Co magnets has been extensively studied and good chemical composition in the cell boundary phase for the permanent-magnet utility has been nailed down 29 and the corresponding ab initio inspection of the intrinsic magnetism at finite temperatures has been carried out 30 . For the 1:5 compounds in the cell boundary phase of Sm-Co magnets, such effects of annealing temperature were discussed 28 . We expect in more widely applicable context that those sample fabrication processes which are mostly off-equilibrium can in principle be improved or even optimized referring to these intrinsic properties at thermal equilibrium to better control the chemical composition and site preference.

V. CONCLUSIONS AND OUTLOOK
We have presented ab initio results of the formation energy and mixing energy for doped Nd 2 Fe 14 B. Revealed trends in the site preference compared to experimental observation point to the validity of localized magnetic moment picture for Nd 2 Fe 14 B at high temperatures while fragile nature in the localized moments in SmCo 5 at high temperatures seems to have been indicated. This may be in contrast to the trends in the robustness of localized magnetic moments in the ground state 22 and extra care must be taken in describing the finite-temperature magnetism. Various trade-off situations have been elucidated between structure stability and magnetization, or between high Curie temperature and robustness in localized magnetic moments. At the moment the best we can do is to identify a good compromise and we have provided ab initio data that can be of potential use for that.
Referring to the outcome of magnetic and nonmagnetic calculations, the heat-treatment temperature in the sample fabrication process compared to the intrinsic Curie temperature can tell which site preference would be realized at least energetically. Two extreme cases, so to say high-temperature limit and low-temperature limit, have been inspected in the present study while the truth for finite-temperature magnetism and structure stability lies somewhere in between -ab initio studies based on the LMD picture for finite-temperature magnetism of