Density Functional Theory Study on the Complete Substitutions of Nd and Fe by Other Rare-earth and Transition-metal Elements in Nd2Fe14B Compound①

RAO Shuang LIN Chen-Sheng HE Zhang-Zhen CHAI Guo-Liang②

a (College of Chemistry, Fuzhou University, Fuzhou 350108, China) b (State Key Laboratory of Structural Chemistry, Fujian Institute of Research on the Structure of Matter, Chinese Academy of Sciences (CAS), Fuzhou 350002, China)

ABSTRACT To search for proper alternatives to improve the magnetic properties of Nd2Fe14B, using first-principles density functional theory calculations we have systematically studied the R2M14B (R = lanthanides from La to Lu; M = Mn, Fe, Co, and Ni) compounds with the isomorphic structure of Nd2Fe14B. The results show that for rare-earth elements, Pr is the most suitable choice for considering as an alternative of Nd. As for the substitution of Fe in Nd2Fe14B by other transition-metal elements, Co is much more suitable than Mn and Ni because the latter two result in too significant reduction of the magnetic moment.

Keywords: permanent magnet materials, Nd2Fe14B, density functional theory calculations;

1 INTRODUCTION

Nd2Fe14B, the third-generation of rare-earth permanent magnet material, is the most widely used one in contemporary magnet[1,2]. As a key component, Nd2Fe14B frequently appears in the control system of inverter air conditioner, wireless communication system and other fields[3]. Global demand for high-performance Nd2Fe14B has been growing rapidly[4]. Although rare-earth resources are abundant in China, it is hard to mine high-performance Nd2Fe14B and ensure the utilization and recovery[4,5]. Considering the atomic radius and similar chemical properties of adjacent chemical elements, many efforts have been done in order to search for a substitute for Nd-Fe-B by completely replacing Nd with other rare-earth elements or Fe with other metal elements or non-metal elements.

The substitution of Fe in Nd2Fe14B and Nd2Fe14C by other transition metal elements such as Mn, Co, and Ni has been extensively studied. For instance, in the study of Nd2Fe14-xMxB it is found that the partial substitution of Fe by Ni (Mn and Co) results in an increase (decrease) of lattice constant c and that Co (Mn) substitution in Nd2Fe14-xMxB results in an increase (decrease) of magnetization[6-8]. Similarly, in Nd2Fe14-xMxC system the Co and Ni (Mn) substitution for Fe can lead to an increase (decrease) of Curie temperature[9]. There are six distinct sites of Fe and two different sites of Nd in Nd2Fe14B system, which are Fe(16k1), Fe(16k2), Fe(8j1), Fe(8j2), Fe(4c), Fe(4e), Nd(4g) and Nd(4f)[10]. The preference of substitution site has also been studied in Nd2Fe14-xMxB. In the case of Si, Ge and Sn considered for M, they prefer to occupy the Fe(4c) site. Meanwhile, they lead to a reduction of volume and little change of magnetic moments of Fe atoms[11]. However, in the case of Ti or Nb considered for M, the most energetically favorable substitution site is the Fe(8j2) site[12,13]. The replacement of Nd in Nd2Fe14B by other rare earth elements has also been studied recently. It is found that Ce prefers to the Nd-4g site irrespective of the doping concentration[10]. La prefers to the Nd-4f (4g) site in low (high) doping concentration, while Pr behaves just in the opposite way to La[10,14]. The systematical study on the substitution of Fe and Nd in Nd2Fe14B was rarely performed in literature, which is of great importance to screen the proper substitution elements for improving the magnetic properties (i.e., magnetization, Curie temperature, and coercivity) of Nd2Fe14B.

In order to search for possible substitution elements for Nd and Fe in Nd2Fe14B, we have performed the first-principles calculations within generalized gradient approximation (GGA) to systematically study the R2M14B (R = lanthanides from La to Lu; M = Mn, Fe, Co, and Ni) compounds with the isomorphic structure of Nd2Fe14B. In this work, we con- centrated on the effect of R and M substitution on the lattice parameters and magnetic moments of R2M14B. Meanwhile we also checked the stability of R2M14B by analyzing the calculated cohesive energy.

2 COMPUTATIONAL DETAILS

To understand the influence of rare-earth and transition metal elements substitution on the magnetic properties of Nd2Fe14B, we have performed the density functional theory (DFT) calculations on a series of compounds R2M14B (R represents lanthanides from La to Lu, and M = Mn, Fe, Co, and Ni). The DFT calculations on R2M14B were carried out using Vienna ab-initio simulation package (VASP) code[15,16]. The ion-electron interaction was described by the projector- augmented wave (PAW) method[17]. The recommended standard PAW potentials in the VASP code were chosen for the lanthanides, transition metal, and boron atoms. The open-core (OC) PAW potentials were particularly used for Ce-Lu, where 4f electrons were treated as spin-polarized core electrons. The electron wavefunction was expanded by the plane wave basis set with an energy cutoff set to 400 eV after we investigated the literature and tested some values[18]. The exchange-correlation interaction was treated within the spin-polarized generalized gradient approximation (GGA) and the Perdew-Becke-Ernzerhof (PBE) exchange correlation functional was used[19]. The Brillouin zone integration was approximated by the Monkhorst-Pack k-point sampling method and a k-grid of 5 × 5 × 4 was employed[20]. Convergence criteria for the structure relaxation were set to 10-6eV and 0.01 eV/Å for the energies and forces, respec- tively. The experimental lattice constants of Nd2Fe14B are a = b = 8.812 Å and c = 12.215 Å[21]. Based on the experimental data, the lattice parameters and atomic positions were entirely relaxed by conjugate gradient algorithm.

Here we first obtained the spin moments of valence electrons of R2M14B compounds from the self-consistent field (SCF) collinear spin-polarized calculations without spin-orbit coupling (SOC) for their relaxed crystal structures. Because the previous DFT calculations with SOC on Nd2Fe14B showed that the orbital moments of Fe atoms in Nd2Fe14B are the order of magnitude of 0.04 µB/atom[10,14], we ignored the orbital moments of transition metal atoms in R2M14B compounds. Based on the ionic model[22-24], the orbital moments of 4f electrons of each rare-earth atom in R2M14B compounds were approximated as gJJ, where gJis the Land´e g-factor and J is the total angular moment of 4f electrons of rare-earth elements, as given by Hund’s rule. This is because the average magnetic moment per R atom in R2Fe14B (R = Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, and Yb) estimated from the 4 K magnetization data is very close to the magnetic moment gJJ of the corresponding trivalent R ions[24]. For the less (more) than half-filled 4f electron shell of rare-earth elements, its orbital moment aligns parallel (antiparallel) to the spin moment of 3d electrons of transition metal atoms in R2M14B[23,25]. The total magnetic moments of R2M14B compounds were estimated from the spin moments of valence electrons and the approximated orbital moments of rear-earth 4f electrons. We should point out that this approximation does not take into account the contribution from the orbital moments of transition metal atoms and the SOC. Such an approximation has been employed in literature[26-28]to estimate the magnetic moments of NdFe11Ti, Nd2Fe14B, and RFe12. The obtained magnetic moment of Nd2Fe14B is in good agreement with the experiment value[27]. The reliability of the open-core treatment for the rear-earth atoms in Nd2Fe14B has been discussed in literature, e.g., in Ref.[27], and it was found that the magnetic moment of Nd2Fe14B calculated with GGA+OC and GGA+U shows the same tendency.

3 RESULTS AND DISCUSSION

3. 1 Crystal structure

The neutron powder diffraction analysis and single-crystal X-ray diffraction experiments showed that Nd2Fe14B crystalli- zes in the tetragonal CaCu5-type structure with space group P42/mnm. Each crystallographic unit cell contains four chemical formula units, namely 68 atoms. The Nd2Fe14B crystallographic unit cell is depicted in Fig. 1, where the symmetry-inequivalent sites of atoms (i.e., Wyckoff positions) are labelled. The Nd atoms occupy two different sites, 4f and 4g. The Fe atoms contain six distinct sites of 4c, 4e, 8j1, 8j2, 16k1, and 16k2. The nonmetal element boron (B) lies in 4g site. The total magnetic moment of Nd2Fe14B reported in experiments is 37.1 µB/formula unit (f.u.)[29]. The lattice constants of Nd2Fe14B obtained from the X-ray diffraction data are a = b = 8.812 Å and c = 12.215 Å[21]. The calculated values of lattice constants (a, c) of Nd2Fe14B are presented in Table 1: a = 8.7518 Å and c = 12.1082 Å. We can see that the difference between the theoretically predicted and experimentally measured values for the lattice constants of Nd2Fe14B is less than 1%. Therefore, the calculated results are in good agreement with the available experimental values, indicating that the present calculation setup is good enough to study the structure and magnetic properties of R2M14B.

Fig. 1. Schematic drawing of crystal structure of Nd2Fe14B. Nd atoms occupy two distinct symmetry sites: 4g (x, -x, 0) and 4f (x, x, 0). Fe atoms occupy six different symmetry sites: 4c (0, 1/2, 0), 4e (0, 0, z), 8j1 (x, x, z), 8j2, 16k1 (x, y, z), and 16k2. B atoms occupy one symmetry site: 4g

Table 1. Calculated Lattice Constants (a and c, in Å) of R2Fe14B Compounds, where R Represents the Lanthanide Elements from La to Lu

Table 1 lists the calculated lattice constants of R2Fe14B along a comparison with the available experiment results. Table S1 of Supplementary Information lists the calculated lattice constants a and c of R2Mn14B, R2Co14B and R2Ni14B. To make the comparison more transparent, the change of lattice constants of R2M14B with respect to the rare-earth element R is illustrated in Fig. 2. For the lanthanides (R) from La to Lu, both lattice constants a and c of R2M14B (M = Mn, Fe, Co, and Ni) have a tendency toward decrease. When the R changes from La to Lu, the reduction of lattice constant a of R2M14B is less than that of c of R2M14B. For a given rare-earth element (R) in R2M14B, the change of c as the transition metal element (M) is also more significant than that of a. Therefore, we can anticipate that the (partial) substitu- tion of La, Ce, and Pr for Nd in Nd2Fe14B would lead to a volume expansion for Nd2Fe14B, while the (partial) substitution of other lanthanides (namely from Pm to Lu) would cause a volume shrinkage. The (partial) substitution of Mn, Co, and Ni for Fe in Nd2Fe14B would give rise to a volume shrinkage. Fig. 3 presents the calculated unit cell volumes (Ω) of R2M14B compounds, which are also listed in Table S2 of Supplementary Information accordingly. As discussed above, the volume of R2M14B increases from to La to Ce and then decreases from Ce to Lu in each case of M = Mn, Fe, Co and Ni. For each rare-earth element compound, the general trend is Ω(R2Fe14B) > Ω(R2Mn14B) > Ω(R2Ni14B) ≈ Ω(R2Co14B).

Fig. 2. Lattice constants (a and c, in Å) of R2M14B

Fig. 3. Lattice volume (Ω, in Å3/cell) of R2M14B

Lanthanide contraction refers to the shrinkage of atomic radius of lanthanide elements from La to Lu due to the shielding effect from 5s5p electrons on 4f electron. As the filled 4f electrons increase, more 4f electrons gather around their nucleus unable to escape, making the atomic radius smaller[30]. When lanthanide element reacts with other elements and turns into trivalent ion, the trend of monotonic decreasing of ionic radii still exists from La3+to Lu3+. As a result, the monotonic decrease of lattice constant (a, c) and volume (Ω) in Figs. 2 and 3 can be attributed to the lanthanide contraction. In the case of Ce2M14B compounds, Ce may not take the completely trivalent state, but it is somewhere between trivalent and tetravalent. Some 4f electrons of Ce in its own compound are localized and cannot get bonding, while others react with transition metals M and B[31,32].

3. 2 Magnetic moments

As already mentioned in the above section of computa- tional method, the following approximations were taken to calculate the total magnetic moments of R2M14B. First, the magnetic moments of R and M atoms are collinear. Second, the total magnetic moment of R includes both the spin and orbital magnetic moments. The orbital moment of M atom is ignored because the orbital moments of Fe atoms in Nd2Fe14B are small, at the order of magnitude of 0.04 µB/atom, so that the magnetic moment of M atom includes only the contribution of valence electrons. Third, the orbital moment of 4f electrons of R atom in R2M14B was approximated to the magnetic moment gJJ of the corresponding trivalent R ion (R3+), as shown in Table 2.

Table 2. Total Angular Momentum J, the Land´e gJ Factor, and Magnetic Moments gJJ for the Trivalent Lanthanide Ions (R3+)[24, 25, 41]. The Average Magnetic Moment mR Per R Atom in R2Fe14B Compound was Estimated from the 4 K Magnetization Data[24]

To check the reliability of the present method used to study the magnetic moments of R2M14B, we first compare the obtained local magnetic moments of Fe atoms in Nd2Fe14B and total magnetic moment of Nd2Fe14B with the previous calculation results and the available experiment data, as shown in Table 3. Our calculation results are in good agreement with those obtained by Tatesu et al.[27]using the similar GGA+OC approach. The slight difference in the local magnetic moment of Fe atoms may be caused by the different radius size used in the estimation of local magnetic moments. The Fe-8j2(-4e) sites have the largest (smallest) local magnetic moments, which are consistent with the trend in the previous LDA/GGA+U+SOC calculation[10,14]and experiment[33]results. The local magnetic moments of Fe atoms at the 16k1, 16k2 and 4c sites are underestimated by ～0.3 µB/atom with respect to the experiment results[33], which is caused partially by the ignored SOC between the Nd 4f and Fe 3d electrons and also by the ambiguity of radius size used in estimating the local magnetic moments. The obtained total magnetic moment of Nd2Fe14B is about 37.63 µB/f.u. and in good agreement with the experimental value (37.1 µB/f.u.)[33]. Therefore, the present method is accurate enough to study the magnetic moments of R2M14B.

In Nd2Fe14B, we completely replace Nd with other 14 lanthanide elements to obtain R2Fe14B series of compounds. At first, we pay attention to the total spin magnetic moments (denoted as µS) of R2Fe14B, which was obtained directly from the collinear spin-polarized calculations. The orbital magnetic moments of the 4f electrons of rare-earth atoms in R2Fe14B were approximated to the magnetic moment gJJ of the corresponding trivalent rare-earth ions based on the ionic model[22-24], as listed in Table 2. Among the 15 rare-earth elements of lanthanide, the La and Lu have none orbital moments. The orbital magnetic moment of Eu is quenched completely to zero. When the number of electrons in the 4f electron shell is less than half for a rare-earth element, the total magnetic moment (denoted as µt) of R2M14B is µt= µS+ gJJ(R); and more than half of the corresponding is µt= |µS- gJJ(R)|. The same way was done for the cases of R2Mn14B, R2Co14B and R2Ni14B. The calculated results of total magnetic moments (µt) of R2M14B are displayed in Fig. 4 and listed in Table S3 of Supplementary Information.

Table 3. Local Magnetic Moments of Fe Atoms (mFe, in µB) and Total Magnetic Moments (µt, in µB/f.u.) of Nd2Fe14B Obtained by the Distinct Computational Methods. The Experimental Results are Listed for Comparison

Fig. 4. Calculated spin magnetic moment (µS, in µB/f.u.) and total magnetic moment (µt, in µB/f.u.) of R2M14B

Now we look at the dependence of total magnetic moments of R2Fe14B on the rare-earth elements. As the total spin magnetic moment of R2Fe14B is almost around 31.0 µB, irrespective of the different rare-earth elements, the change in total magnetic moments R2Fe14B follows the trend in the orbital moments of the 4f electrons of R atoms. For the same transition metal, the magnetic moments of Pr2M14B and Nd2M14B are almost the same. Compared with the Fe and Co series (R2Fe14B and R2Co14B), the spin magnetic moments of the Mn and Ni series (R2Mn14B and R2Ni14B) are too small. For the light rare-earth elements with less than a half-filled 4f electron shell, the order of total magnetic moments is µt(R2Fe14B) > µt(R2Co14B) > µt(R2Mn14B) > µt(R2Ni14B). For the heavy rare-earth elements with more than a half-filled 4f electron shell, their 4f orbital magnetic moments are antiparallel to the spin magnetic moments of transitional metal 3d electrons, and thus the corresponding total magnetic moments of R2M14B are significantly reduced with respect to that of Nd2Fe14B. From this aspect, the substitution of Nd by other light rare-earth elements is more favorable to maintain the magnetic properties of Nd2Fe14B. In the case of light rare-earth elements, the magnetic moment of R2Co14B series is slightly smaller than those of R2Fe14B systems so that it would be worthwhile to consider the incorporation of Co into R2Fe14B.

3. 3 Cohesive energy

The cohesive energy Ecohof a compound is defined as the energy difference between its total energy and the sum of the total energies of its constituent atoms. For R2M14B, its Ecohis calculated according to the following equation:

where ER, EM, and EBare the total energies of isolated rare-earth, transition metal, and boron atoms, respectively. ER2M14Bis the total energy of R2M14B. The more negative value of Ecohindicates that R2M14B is more stable with respect to the corresponding constituent atoms.

Fig. 5. Cohesive energy (Ecoh, in eV/f.u.) of R2M14B

We present the calculated cohesive energies of R2M14B in Fig. 5 and list them in Table S4 of Supplementary Information. For all lanthanide elements studied here, the general trend is that the cohesive energies of R2M14B take an order of R2Mn14B > R2Ni14B > R2Co14B > R2Fe14B. It is remarkably noted that the cohesive energies of R2Mn14B are much larger than those of R2Ni14B, between which the energy differences are around 19.5 eV/formula unit. However, the cohesive energies of R2Co14B are very close to those of R2Fe14B, between which the energy differences are around 1.0 eV/formula unit. Therefore, the formation of R2Mn14B would be thermodynamically unstable as compared with R2Ni14B, R2Co14B, and R2Fe14B. It also suggests that the incorporation of Mn into R2Fe14B would be more energetically costly as compared with the incorporation of Ni and Co into R2Fe14B. On the other hand, the Co substitution for Fe in R2Fe14B would be more thermodynamically favorable than the Ni and Mn substitution. Now let us see the dependence of cohesive energies of R2Fe14B on the rare-earth elements. If taking the cohesive energy of Nd2Fe14B as a reference, the relative energy differences in the cohesive energies of other R2Fe14B (R ≠ Nd) are in a -0.90～0.28 eV/formula unit range. Therefore, the substitution of Nd by other lanthanide elements would be thermodynamically favorable. Indeed, many compounds of R2Fe14B serials have been synthesized in experiment, as listed in Table 1.

4 CONCLUSION

To search for an alternative for Nd2Fe14B, we have studied the substitution of other rare-earth elements for Nd along with transition elements Mn, Co and Ni for Fe, which implies sixty compounds. Within the framework of density functional theory, we optimized the crystal structures and calculated the magnetic moments and cohesion energies of these sixty compounds, finding that the substitution leads to small changes in both lattice constants and volumes of these candidates with respect to those of Nd2Fe14B. For the rare-earth elements, Pr is the most possible nominee to Nd. As for transition-metal elements, Co substitution manages with an effort to keep the magnetic moments, while Mn and Ni substitutions make the magnetic moments of Nd2Fe14B too small to be considered. For transition-metal substitutions, the most suitable choice for Fe is Co, followed by Ni, while the Mn substitution is thermodynamically unfavorable with a serious disadvantage, namely a significant reduction of magnetic moments. Therefore, the incorporation of Mn into Nd2Fe14B shall not be considered. In this proposition of complete replacing for Nd2Fe14B, it might be better to consider Pr2Fe14B. In addition, although Pr2Fe14B and Nd2Fe14B have similar amount of total magnetic moments, further comparisons need to be made from mining cost and magneto-crystalline anisotropy, which will be studied in our future work. Furthermore, the complete substitution may be thermodynamically unfavorable in many cases for Nd2Fe14B, so we will consider the partial substitution as well as doping some other elements in the next step.

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