Electron Theory on Grain-Boundary Structures and Local Magnetic Properties of Neodymium Magnets

Abstract

A fundamental understanding of microstructures is indispensable in improving neodymium-magnets performance at high temperatures. Thus, it is of significant importance to clarify atomic structures and local magnetic properties of interphase interfaces in microstructures, based on electron theory. We studied interfaces between the main phase of neodymium magnets, Nd₂Fe₁₄B, and a subphase NdO_x using massively parallel first-principles electronic-structure calculations with the K computer. As well as the known Cu-addition effect on wettability improvement in metallic Nd subphase, we recognized that some of the added Cu atoms at the (001) interface improve the local magnetic anisotropy of Nd at the interface. Furthermore, we found that the substitution of Fe in the (001)-surface of main-phase grains with Zn can also improve the stability of magnetic anisotropy.

 

This Paper was Originally Published in Japanese in J. Japan Inst. Met. Mater. 81 (2017) 26–31. In order to more precisely explain the background, the methods, and the results, some parts of the contents were revised. The unit of the energy used in Figs. 2, 3, 4, 5, 7, and 8 was changed, and the Refs. 14) and 23) were added.

1. Introduction

There exists a demand for improved Nd-Fe-B magnets performance, in particular, their application in turbines for wind power generation, as well as their utilization at high temperatures in hybrid cars. A fundamental understanding of microstructures is essential to realize such performance improvement in Nd-Fe-B magnets. In Nd-Fe-B magnets, the single-crystal particles of the main phase, the Nd₂Fe₁₄B intermetallic compound, bond with each other through subphases known as grain boundary phases¹⁾. As there are virtually no factors that hinder magnetic domain-wall propagation within the main phase, it is necessary to optimize the microstructures of Nd-Fe-B magnets in order to prevent the magnetization reversal due to the magnetic domain-wall propagation into the main phase for improving the coercivity²⁾. Therefore, determining the condition of the domain-wall penetration into the main phase is a key to increasing the performance of Nd-Fe-B magnets. That is, the information on the atomic structures and local magnetic states of the interfaces between different phases in the material should be investigated carefully. It has gradually become possible to obtain such information in the atomic scale owing to the progress in experimental techniques in recent years. For example, the interfaces between the main phase and a subphase of Nd-Fe-B magnets can be observed in the atomic scale using scanning transmission electron microscopy³⁾. However, determining the detailed position of atoms at the interfaces of Nd-Fe-B magnets is difficult via experiments alone. Remarkable progress in electron theory has been made in recent years: First-principles electronic-structure calculations, based on density functional theory, can be applied to analyze atomic positions and electronic states theoretically from the basic principles of quantum mechanics and statistical mechanics by considering only fundamental physical constants and numerical parameters. These numerical parameters do not influence on results when sufficient preliminary calculations are made. Therefore, we can use first-principles calculations to obtain complementary knowledge to the position of the atoms and the magnetic states at the interface of Nd-Fe-B magnets where it is not possible to gain sufficient information from experiments alone.

Annealing processes are important when optimizing the microstructure of Nd-Fe-B magnets, where one valid approach known to improve coercivity is Cu addition^4–13). In particular, as the melting point of a Nd-rich phase, which is one of the subphases, becomes lower when it is alloyed with Cu, it has been clarified that a better quality of the microstructure around grain boundaries can be realized by improving wettability through the Cu diffusion process^4,5). Furthermore, during the Cu diffusion process, it has been concluded from the three-dimensional atom probe that Cu atoms are distributed in the subphase so as to surround the crystal grains of the main phase Nd₂Fe₁₄B completely⁴⁾. However, the position of those Cu atoms is not clear in the atomic scale, and we lack the sufficient knowledge regarding the direct impact of the added Cu on magnetism.

Generally, the subphases of Nd-Fe-B magnets can be classified into two types: the two-particle-grain-boundary phase and the grain-boundary-multipoint phase. The grain-boundary-multipoint phase can easily function as generating the cores of magnetization reversal owing to stray magnetic fields. Therefore, it is important to understand the details about the domain-wall penetration into the main phase at the interface between the main phase and the grain-boundary-multipoint phase. Clarifying the atomic position and local magnetic state of the interface between the main phase and the grain-boundary multipoint is expected to be useful for gaining a microscopic understanding.

Therefore, in this study, we performed large-scale first-principles calculations using the K computer¹⁴⁾ for the interface between the main phase Nd₂Fe₁₄B and the NdO_x phase which is a grain-boundary-multipoint phase known to exist adjacent to the main phase. In this paper, we discuss our findings regarding the effect of adding Cu into Nd-Fe-B magnets¹⁵⁾. First, we introduce the first-principles method and our models for interface atomic structures. Then, we provide the results of our large-scale calculations. We see that Cu easily substitutes itself for the Fe atoms in the first layer of the main-phase interface. Furthermore, we show that Cu can exist stably in the interstitial sites, instead of substitutional sites, between the main phase and the subphase. In particular, on substituting the interfacial Fe atoms, the in-plane magnetic anisotropy of the interface Nd in the main phase was reduced to approximately 40%. Therefore, the interface anisotropy tends to recover the uniaxial anisotropy of the main phase. In addition, we performed a theoretical analysis on the effect of Fe substitution with the different types of atoms other than Cu.

2. Computational Method

First-principles calculations were performed on the basis on density functional theory within the generalized gradient approximation¹⁶⁾. The OpenMX code¹⁷⁾ was adopted utilizing the pseudo-atomic localized basis sets, which is suitable for efficient parallel calculations on the K computer. We used the open-core pseudopotentials for Nd that consider the spin polarization of the open-shell 4f electrons in Nd as core states. This is justified by the fact that the Nd 4f electrons can be considered as the same as those in an isolated Nd atom, as far as the magnetic anisotropy of Nd is concerned. In first-principles calculations, we considered the spin collinear structure, where all spins are parallel or antiparallel with each other, without spin–orbit couplings for valence electrons. The magnetic anisotropy constant of Nd was calculated as a post-process of first-principles calculations using crystal-field analysis¹⁸⁾, where the orbital angular momenta of 4f electrons are taken into account. Furthermore, the magnetic anisotropy of Fe atoms was calculated with perturbation theory for spin–orbit couplings¹⁹⁾. All atomic positions were optimized so that the force exerted to all the atoms becomes numerically zero.

The crystal structure of the main phase is already well known²⁰⁾. Moreover, the Nd sublattice in the NdO_x phase has the fcc structure^21,22). In this study, oxygen atoms were allocated randomly at the tetrahedral sites in NdO_x with oxygen density x = 0.25. Since the total energy of the system with oxygen atoms at tetrahedral sites is lower than that of the system with oxygen at octahedral sites as identified for this composition determined by first-principles calculations²³⁾. Experimental studies reported that an interface structure is flat in the atomic scale, if an interface is formed on the main phase (001) surface³⁾. We performed calculations under the Bloch periodic boundary conditions for the interface systems containing more than 200 atoms per unit cell so that the lattice mismatch between the main phase and the subphase is sufficiently small at ～1.2%. To be consistent with experimental results showing a low Cu density around the grain boundary of Nd-Fe-B magnets⁴⁾, one Cu atom was placed at a variety of positions. In order to examine where Cu should exist around the interface, the formation energies for the Cu addition were calculated by considering the total energies of the interface structures together with the chemical potentials of fcc Cu, dhcp Nd, and bcc Fe.

3. Atomic Structure and Magnetic State of the Interface

Figure 1 shows the five types of the interface structures used in this study. The atomic positions in each structure were optimized. Structure I does not contain Cu, whereas in structure II, Cu is located in the interstitial space between the main phase and subphase. In structure III, Cu substitutes itself for the Fe atoms in the first layer of the main-phase side of the interface. In structure IV, Cu substitutes for the Nd atom in the first layer of the main-phase interface. In structure V, the Nd atom in the first layer of the subphase interface is substituted with Cu. The formation energy attributed to Cu in these structures is shown in Fig. 2, where 1 MJ/mol for 1 mol of Cu equals to 10.4 eV for single Cu atom. In structures II and III, adding Cu into these systems is energetically stable. However, in structures IV and V, Cu substitution at Nd sites is unstable, because the formation energies are positive.

Fig. 1

Atomic structures of Nd₂Fe₁₄B-NdO_x interfaces (a) without Cu, (b) with Cu at an interstitial position, (c) with an interface Cu_Fe, (d) with an interface Cu_Nd in the main phase, and (e) with an interface Cu_Nd in the subphase.

Fig. 2

Formation energies of interface Cu and the averaged change in the single-ion magnetic anisotropy constant of interface Nd. Note that 1 MJ/mol for 1 mol of Cu equals to 10.4 eV for single Cu atom.

The changes of the averaged magnetic anisotropy of Nd in the first layer of the main-phase interface are shown in Fig. 2. The magnetic anisotropy improvement toward a uniaxial direction in structure III is the best among the five structures, which can be seen by comparing the values of −530 J/mol (−5.5 meV) in structure I with that of −320 J/mol (−3.3 meV) in structure III. In other words, Cu substitution weakens the in-plane magnetic anisotropy of Nd in the first layer of the main-phase interface in structure III by approximately 40%. Thus, some Cu atoms that are dispersed within the subphase after the grain-boundary diffusion process may be placed as in structure III, and then improve the magnetic properties of Nd-Fe-B magnets directly. This comes from the fact that structure III can exist stably from the perspective of the formation energy. Furthermore, a decrease in the number of magnetic atoms, i.e., Fe, at the interface due to the Cu substitution results in weakened magnetic couplings between the main phase and the subphase, which also contributes to improved magnetic properties.

Figure 3 shows the magnetic anisotropies of the Nd atoms in the main phase for each layer. Nd₂Fe₁₄B has a tetragonal structure with (001) planes, namely the Nd-Fe-B plane, in every 0.6 nm as seen in Fig. 1. Considering the main phase [001] direction as the z direction, the z = 0 plane in Fig. 1 is the Nd-Fe-B plane at the interface that is the nearest neighbor to the added Cu. For the bulk single crystal of Nd₂Fe₁₄B, the magneto-crystalline anisotropy constants of Nd are positive; therefore, the magnetic anisotropy is uniaxial along the [001] direction. However, the magnetic anisotropy of Nd in the first layer of the interface, where Cu is not present, is negative, resulting in the in-plane magnetic anisotropy. In contrast, the magnetic anisotropy of Nd at z = −0.6 nm is uniaxial, which is practically equivalent to that of Nd in a single crystal. This comes from the fact that the electronic states of the inner region of Nd₂Fe₁₄B are not affected by the existence of the interface because of the screening effect owing to metallic electron states. In structure III, where Cu is present at the interface, the in-plane anisotropy at z = 0 is weakened due to Cu. On the contrary, the in-plane anisotropy at z = −1.2 nm is equivalent to an interface where Cu is not present, the changes in the magnetic anisotropy are extremely small.

Fig. 3

Single-ion magnetic anisotropy constants of Nd atoms for Nd₂Fe₁₄B bulk single crystals and the structures I and III, which are averaged in each (001) layer of Nd₂Fe₁₄B. Positive values represent the uniaxial magnetic anisotropy, while negative ones correspond to the in-plane anisotropy.

The major part of the magnetic moment of Nd-Fe-B magnets originates from the spin magnetic moment of Fe atoms. Figure 4 (a) shows the spin magnetic moment of each Fe atom divided by the Bohr magneton. The Bohr magneton is the spin magnetic moment of one electron, and Fig. 4 (a) represents the spin polarization at each Fe site. We can see that the Fe magnetic moment increases near the interface. Figure 4 (b) shows the magnetic anisotropy of Fe calculated with the perturbation theory for the spin–orbit couplings¹⁹⁾. The magnetic anisotropy of Fe in the Nd-Fe-B plane for bulk single crystals is in-plane. However, it can be perpendicular or in-plane at the interface. In particular, the in-plane anisotropy tends to weaken recovering the uniaxial anisotropy due to the Cu addition.

Fig. 4

(a) Spin moments per Bohr magneton for 3d electrons of Fe atomic sites in Nd₂Fe₁₄B bulk single crystals and the structures I and III, which are averaged for atoms with the same z coordinates. (b) The magnetic anisotropy energy coming from Fe 3d electrons.

We examined the case in which Cu is distributed to a location further away from the interface in order to investigate more in details. First, Cu atoms present deep inside of the NdO_x phase do not directly affect the magnetic properties of the Nd-Fe-B magnets, because the NdO_x phase is nonmagnetic at room temperature. Conversely, if Cu atoms are dispersed within the main phase, they have an extremely significant effect on the magnetic properties. Therefore, we investigated these effects in the following. Figure 5 (a) shows the calculated formation energy of Cu substitution for Fe in the bulk and the (001) surface of the main phase. For the surface system, the vacuum region is set over z = 0. The surface is terminated by the (001) plane, i.e., the Nd-Fe-B plane containing Nd, Fe, and B. This figure clarifies that Cu substitution is unstable inside the main phase while it occurs at the surface. In the case where one Fe atom in the main-phase surface is substituted with Cu, the formation energy and the change in the averaged magnetic anisotropy of Nd at z = 0 are shown in Fig. 5 (b) for the bulk and the (001) surface. The figure shows that the Nd substitution with Cu atoms is unstable. Furthermore, we can see that the anisotropy tends to be uniaxial by substituting Fe atoms with Cu as in the case of the interface.

Fig. 5

(a) Formation energies of the Cu substitution at Fe sites for bulk single crystals and (001) surfaces. The surface layer is located at z = 0, where positive values of z correspond to the vacuum region for the surface. (b) Formation energies of the Cu substitution at the Fe and Nd sites at z = 0 for bulk single crystals and (001) surfaces together with averaged changes in the magnetic anisotropy constants of Nd atoms at z = 0.

The enhancement of the uniaxial anisotropy of Nd is attributed to the Fe substitution with Cu; the interaction between Fe 3d electrons and Nd 5d electrons becomes weaker. As a result, the distribution of 5d electrons becomes more in the z direction. When the 5d electrons are distributed in the z direction, the 4f electrons with large orbital angular momentum are distributed within the (001) plane, thereby strengthening the uniaxial anisotropy of Nd. Figure 6 shows the region with increased electron density by the Cu substitution for the bulk single crystal, where one Fe atom at the 4c site in the Nd-Fe-B plane is substituted with Cu. This figure does not include the 4f electron distribution. The electron density surrounding the Cu atom is reduced, indicating the fact that the interaction between the 3d electrons and 5d electrons is weakened. Furthermore, we can see that the electron density near the Nd site is elongated in the z direction, and the uniaxial anisotropy of Nd is enhanced as described above.

Fig. 6

The difference in the electron-density distribution between the Cu-added Nd₂Fe₁₄B bulk single crystal and the pristine one for the region close to the (001) plane including Nd, Fe, and B. Only the positive change is shown for clarity. Due to the open-core pseudopotential used in the first-principles calculations, the distribution of 4f electrons is not included in the figure.

4. Substitution Effects of Various Elements for the Nd₂Fe₁₄B(001) Surface

In the previous section, we discussed the theoretical analysis on the effect of Cu addition. Here, we further investigate the effects of the Fe substitution with other atoms other than Cu from a perspective of the possibility of the addition of different types of atoms. Figure 7 shows the values for the formation energy for the substitution of the Fe site at z = 0 in the main phase with various atoms, together with the changes in the averaged magnetic anisotropy of Nd, in the same manner with Fig. 5 (b). The substitution of Fe with Cu, Zn, and Ga has low formation energies and improves the magnetic anisotropy of the Nd atoms significantly. Of course, the substitution will not occur unless these elements can diffuse within the subphases. However, if the added elements can reach the main-phase surface, the addition of Cu, Zn, and Ga can be considered feasible. In fact, the validity of the addition of not only Cu^4–13) but also Ga²⁴⁾ has been confirmed in experiments.

Fig. 7

Formation energies of substitution for Fe at z = 0 by various elements at the (001) surface of the main phase together with averaged changes in the magnetic anisotropy constants of Nd atoms at z = 0.

The most significant improvement of the magnetic anisotropy of Nd by the substitution of Fe with Zn is mainly attributed to the following reasons. i) In the case of Cu, in which the 3d-electron spin polarization is 0.06, and the elements to the left of Cu in the periodic table, those 3d states are partially occupied. However, in Zn, it is completely occupied, and the interaction with Nd 5d electrons and Zn 3d electrons is extremely weak. ii) The chemical-bonding states of Ga and Ge differ considerably from those of the transition metals such as Fe. However, in Zn, the number of s and p electrons contributing to the chemical bonding is 1.89, and this does not differ significantly from the value of 1.30 for Fe. Therefore, the electron density distribution of Nd 5d electrons tends to be in the out-of-plane direction resulting in the enhancement of the uniaxial anisotropy of the Nd 4f electrons.

Considering the main-phase crystals from a crystallographic perspective, there exist two types of Nd sites—the 4f site and 4g site²⁰⁾. Their definitions are somewhat ambiguous, but in this paper, the 4f site is defined as these that is separated from surrounding sites by a larger distance. Figure 8 shows the changes in the Nd magnetic anisotropy when the Fe site in the main-phase (001) surface is substituted with Ni, Cu, Zn, and Ga. Regardless of the types of the added element, Nd at the 4g site has significantly improved-magnetic anisotropy as compared to that of Nd at the 4f site. This is because the distance of 317 pm between the substitution site, and the 4g site is smaller than that of 337 pm for the case of the 4f site: The interaction between the 3d electrons of transition metals and 5d electrons of Nd at the 4g site is weaker than that in the case of Nd at the 4f site.

Fig. 8

Changes in the magnetic anisotropy constants of Nd atoms at z = 0 by substitution of Ni, Cu, Zn, or Ga for Fe at z = 0 for the (001) surface of the main phase.

5. Summary

In this paper, we focused on the microstructure interfaces of Nd-Fe-B magnets examined in the atomic scale with electron theory, because changes in the magnetic anisotropy of only the single atomic layer at surfaces or interfaces can have a major impact on coercivity owing to the domain-wall motion²⁵⁾. We used the K computer to perform large-scale first-principles calculations for the interface between the main phase, Nd₂Fe₁₄B, and one of the subphases, NdO_x. With clarifying stable Cu positions around the interface, we have found that the magnetic anisotropy of the interface Nd improves for the case where the interface Fe atoms are substituted by Cu. The mechanism of this improvement is considered as re-distribution of the Nd 5d electrons coming from the change in the interaction between the 3d electrons in the transition metals and the 5d electrons in Nd. Furthermore, we have clarified that, in addition to Cu and Ga, Zn is also effective for the substitution of Fe at the surface of the main phase.

It is effective to combine large-scale first-principles calculations of the interface between different phases, as introduced in this paper, and spin models to evaluate the coercivity of Nd-Fe-B magnets theoretically. Furthermore, the frontiers of science for magnets are tackled through a combination of large-scale experiments and first-principles calculations²⁶⁾. The research introduced in this paper is on materials science of permanent magnets approached with the viewpoint of condensed matter physics. It is of great pleasure if this article stimulates interests for such interdisciplinary research attempts.

A part of this research was supported by ESICMM, the element strategy initiative center for magnetic materials outsourced by MEXT, and the CDMSI, “Creation of new function devices/high performance materials to support the next generation.” Some of the large-scale first-principles calculations were performed on the K computer (HPCI project Nos. hp120086, hp140150, hp150014, and hp160227), TSUBAME at the Tokyo Institute of Technology, and the supercomputers at the Institute for Solid State Physics, The University of Tokyo. We are grateful to S. Hirosawa, K. Hono, T. Ohkubo, T. Nakamura, T. Ozaki, T. Miyake, H. Tsuchiura, A. Sakuma, S. Miyashita, and H. Akai for fruitful discussions.

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