Magnetic order of Dy 2 Fe 14 B/Fe interface

1. Introduction

Various nowadays technologies require application of modern hard-magnetic materials without or with reduced amount of rare earth elements [1–6]. In this field, very promising are nanostructured alloys composed of different crystalline phases for which, magnetic interactions can lead to an appearing of new and unique properties. Recently we have reported ultra-high coercivity of Fe-Nb-B-RE (6 at.% of Nb, RE = Tb, Dy) bulk nanocrystalline alloys produced by the vacuum suction casting technique, i.e. more than 7 T at the room temperature and after some field annealing [7,8]. It should be underlined that in the field of bulk materials this is a unique feature, giving new opportunities to designing new magnetic materials. It was shown that the examined alloys contain some ultra-hard magnetic objects (Tb₂/Dy₂Fe₁₄B) that do not directly contribute to the magnetization processes but, via direct interactions, can influence magnetic characteristics of the whole alloy. This type of materials has a potential to be a base for a new type of spring-exchange composites with magnetic properties better than conventional Nd-based permanent magnets.

The origin of the ultra-high coercivity of Fe-Nb-B-RE (6 at.% of Nb, RE  = Tb, Dy) is related to the presence of the specific irregular dendrite-like particles of hard magnetic RE₂Fe₁₄B phase [7,8]. Furthermore, it seems that the hard magnetic properties of the investigated compounds can be significantly improved by an introduction of some magnetically soft phases that are directly coupled with RE₂Fe₁₄B. In such composite, the hard and soft components can be a source of high anisotropy and high remanence, respectively.

The aim of this work is to study, by the first-principle investigations, magnetic coupling between Dy₂Fe₁₄B hard and Fe soft magnetic phases for different interface directions. Such a combination can be found in real magnetic composites, and therefore, interactions between them are of great importance regarding designing of new high-efficient hard magnetic materials. Moreover, the kind magnetic order and interactions parameters, determined by the DFT calculations, should be useful for further high-level simulations.

2. Calculation methods and procedures

The properties of magnetic coupling between Dy₂Fe₁₄B and Fe-bcc were calculated by spin-polarized calculations offered by OpenMX software [9]. The many-body interaction between electrons and ions was modeled by norm-conserving pseudopotentials within the frame of density functional theory (DFT). In order to reduce consumption of computational resources, we used an open-core pseudopotential for Dy atoms that locates 4f state in the set of core states while 5s, 5p, 5d and 6s states are considered as a valence ones.

The valence states of the pseudopotentials selected for Fe and B consists of 3p, 3d, 4s and 2s, 2p states, respectively. All pseudopotentials were selected from the OpenMX database (ver. 2013). The Kohn-Sham Bloch functions are defined as linear combination of pseudo-atomic basis functions centered on each atomic site. Thus, a basis set was composed of s2p2, s2p2d1 and s2p2d1f1 functions for B, Fe and Dy atoms, respectively. In order to achieve well convergent results, the 500 Ry kinetic energy cutoff and 5x5x5 k-points Monkhosrt-Pack [10] grid in the reciprocal space were applied. Moreover, the Perdew-Wang [11] generalized-gradient approximation of the exchange-correlation functional was employed in our simulations.

Firstly, the assumed crystallographic structure of Dy₂Fe₁₄B was relaxed in order to obtain energy minimum, regarding interatomic distances. Next, the calculations were carried out for the two interface directions: (i) [001] Dy₂Fe₁₄B/Fe-bcc (117 atoms) and (ii) [100] Dy₂Fe₁₄B/Fe-bcc (132 atoms). In the both cases, a distance distances D between Dy₂Fe₁₄B and Fe-bcc was changed from 1.5 Å to 5 Å. Finally, the supercell used in our simulations was composed of Dy₂Fe₁₄B unit cell and thin layer of body-centered Fe-bcc structure, separated by distance D, as shown in Figs 1 and 2.

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Fig. 1.

The supercell of the [100] interface direction.

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Fig. 2.

The supercell of the [001] interface direction.

Here we provide results of first-principle investigations of magnetic coupling between Dy₂Fe₁₄B hard and Fe soft magnetic phases for the two mentioned interface directions.

The crystallographic parameters of Dy₂Fe₁₄B compound was measured by Herbst et al. [12]. Results of their neutron powder diffraction experiments show that Dy₂Fe₁₄B compound exhibit P4₂/mnm space group (No. 136) at 77 K and 293 K. An increase of temperature causes a slight increase of both tetragonal lattice parameters (a, c) and magnetic moment per formula unit (from ∼18 to ∼21 𝜇 _B ). It was also reported that magnetic moments of Dy and Fe are collinear along c-axis and antiparallel (ferrimagnetic). The crystallographic data provided by Herbst et al. [12] were used as starting point for structural relaxation of Dy₂Fe₁₄B unit cell. The final values of lattice parameters and coordinates of atomic positions are in good agreement with the experimental data [12] which validates the selected pseudopotentials and methodology. The structural parameters of the Dy₂Fe₁₄B unit cell are as follows: a = b = 8.776 Å, c = 11.945 Å, Dy(4f) = (0.268, 0.268, 0), Dy(4g) = (0.146, −0.146, 0), Fe(16k ₁) = (0.224, −0.435, 0.126), Fe(16k₂) = (0.037, 0.359, 0.175), Fe(8j ₁) = (0.098, 0.098, 0.201), Fe(8j ₂) = (0.317, 0.317, 0.246), Fe(4e) = (0.5, 0.5, 0.384), Fe(4c) = (0.0, 0.5, 0.0), B(4g) = (0.374, −0.374, 0).

The main results of the performed calculations are dependencies of the system energy as a function of the distance D between the Dy₂Fe₁₄B and Fe-bcc phases, which, for the [100] direction, is shown in Fig. 3.

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Fig. 3.

Energy of the system as a function of the interface distance D in the [100] direction.

The energy was calculated for the two cases, i.e. parallel (the black line, FM) and aniparallel (the red line, AFM) alignment of Fe (in Dy₂Fe₁₄B) and Fe-bcc magnetic moments. One can see that the energy minimum occurs for D = 1.5 Å and for the AFM magnetic order. Interesting is the fact that for D < 1.3 Å the FM order is energetically preferred.

In the second case (the [001] direction), the performed simulations reveal only ferromagnetic alignment of the Dy₂Fe₁₄B and Fe-bcc magnetic moments, see Fig. 4. Moreover the minimum of energy is observed for D = 2 Å.

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Fig. 4.

Energy of the system as a function of the interface distance D in the [001] direction.

Finally, for the analyzed system, one can determined the diagram presented in Fig. 5. For the [001] direction the Fe atomic magnetic moments of the both Dy₂Fe₁₄B and Fe-bcc phases are aligned parallel. As was expected, the Dy magnetic moments are coupled antiferromagnetically to Fe. More complicated is the case of the [100] direction, for which, the change of magnetic order was revealed. In fact, for D < 1.3 Å the ferromagnetic coupling is preferred, while for higher interface distances the antiferromagnetic order of Fe (Dy₂Fe₁₄B) – Fe-bcc is observed.

In order to compare energies calculated for two considered interfaces we have divided them by the number of atoms. The value of −84.4 Hartree was obtained for the minimum energy (AFM order) of the [100] interface, while the [001] interface has minimum energy (FM order) at the level of −80.9 Hartree. This means that coupling of the iron atoms to [100] plane of the Dy₂Fe₁₄B unit cell is more likely.

In real magnetic composites, containing magnetically hard and soft phases, one can expect fluctuations of the interface distances as well as directions that can be caused by, for example, internal stresses or some kind of crystal disorder usually present in the inter-grain boundaries. This means that the both FM and AFM magnetic order can be found, and therefore, some magnetic frustrations can be expected. In the case of the so-called spring-exchange magnets, such frustrations can be an additional source of magnetic anisotropy which in the field of permanent magnets is desired.

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Fig. 5.

The diagram of the interface magnetic order for the two directions of [001] and [100].

Based on the performed DFT calculations, the conclusions can be summarized as follows:

For the [001] Dy₂Fe₁₄B/Fe interface, the ferromagnetic coupling is preferred in various range of the distance D.