Elastic constants of Dy 2 Fe 14 B crystal by ab-initio calculations

Abstract

This paper concerns the elastic properties of Dy₂Fe₁₄B crystal investigated by ab-initio calculations. We determined the elastic constants of the considered crystal: C ₁₁ = 297, C ₁₂ = 92, C ₁₃ = 99, C ₃₃ = 287, C ₄₄ = 82, C ₆₆ = 80 GPa. Moreover, estimated from elastic constants physical properties (acoustic wave velocities, elastic moduli, Debye and melting temperature) are consistent with the reference data obtained for other (RE)₂Fe₁₄B crystals.

Keywords

Permanent magnet atomistic modelling elastic properties

1. Introduction

Modern nanostructured composites containing different magnetic phases are very required in nowadays technologies. Recently, we have reported an ultra-high coercivity of the Fe-Nb-B-RE nanocrystalline alloys produced by the vacuum suction casting technique. The magnetic coercivity is more than 7 T at the room temperature which is unique feature among nanocrystalline materials. As it was shown, magnetic characteristics of the alloys are strongly related to their dendrite-like microstructure with significant contribution of structural and magnetic disorder. The phase analysis reveals a coexistence of magnetically hard (RE)₂Fe₁₄B irregular grains embedded in relatively soft or non-magnetic surroundings. It seems that appearance of internal stresses is an additional factor causing magneto-elastic anisotropy and, in a consequence, may be one of a source of the observed magnetic hardening effect [1–3].

Taking into consideration the above, determination of the elastic properties of (RE)₂Fe₁₄B ternary compounds is of great importance since they can give a possibility for further analysis of magneto-elastic coupling, interactions between dislocations or residual stresses. Unfortunately, extremely few articles were devoted to this issue as the main scientific impact was focused on investigations of various magnetic properties of (RE)₂Fe₁₄B compounds. For example, Turilli et al. [4] reported the elastic constants of (RE)₂Fe₁₄B (Re = Y, Nd, and Pr) compounds which were measured at room temperature using acoustic wave propagation method.

The aim of the present work is to determine all elastic constants of tetragonal Dy₂Fe₁₄B crystal within the frame of ab-initio quantum calculations based on density functional theory (DFT). The resulting data, reported for the first time, are of importance for various applications and provide valuable information on the nature of the materials of interest.

2. Method

The elastic properties of Dy₂Fe₁₄B crystal were estimated from spin-polarized calculations offered by OpenMX software [5] based on density functional theory (DFT). The many-body interaction between electrons and ions was modeled by norm-conserving pseudopotentials. 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 wave functions were defined as a linear combination of pseudo-atomic basis functions centered on each atomic site. The basis set was composed of s2p2, s2p2d1 and s2p2d1f1 orbitals for B, Fe and Dy atoms, respectively (for more details refer to OpenMX website). The cutoff radius of the pseudo-atomic wave functions was 8.0 a.u., 6.0 a.u. and 7.0 a.u. for Dy, Fe and B atom, respectively. In order to achieve a well convergent results, the kinetic energy cutoff of 500 Ry and 5 × 5 × 5 k-point Monkhosrt-Pack [6] grid in the reciprocal space were applied. Moreover, the Perdew-Wang [7] generalized-gradient approximation of the exchange-correlation functional was employed in our simulations.

3. Results and discussion

The crystallographic parameters of Dy₂Fe₁₄B crystal was measured by Herbst et al. [8]. Results of their neutron diffraction experiments show that Dy₂Fe₁₄B crystal exhibit P42∕mnm space group (No. 136) at 77 K as well as at 293 K. An increase of the temperature causes a slight increase of the tetragonal lattice constants as well as the magnetic moment per formula unit. It was also reported that the magnetic moments correspondig to Dy and Fe atoms are collinear along the c-axis and antiparallel. In the present work, the crystallographic data provided by Herbst et al. [8] were used as a starting point for the structural relaxation of Dy₂Fe₁₄B unit cell. This task was conducted until the stress tensor components achieved the values lower than 10⁻⁷ Hartree∕a.u.³. The obtained lattice constants, the coordinates of atomic position and the crystal density (see Table 1) are in good agreement with experimental data [8]. This conclusion positively verifies the method selected for our simulations.

The open-core treatment of 4f orbitals assumes their location in core states and Russel-Sanders coupling to estimate magnetic moment of Dy₂Fe₁₄B compound. Therefore, the calculated spin magnetic moment of the Dy2Fe14B unit cell was corrected by subtraction of 4f shell magnetic moment (∼10 μB). In a consequence, the total magnetic moment of Dy₂Fe₁₄B takes the value of 11.31 μB/f.u. which is close to 12.75 μB/f.u. theoretically obtained by Kitagawa [9].

Fig. 1.

Atomic arrangement of Dy₂Fe₁₄B crystal (Dy – blue, Fe – gold, B – green).

The tetragonal lattice of Dy₂Fe₁₄B compound is characterized by the following elastic constants: C ₁₁, C ₁₂, C ₁₃, C ₃₃, C ₄₄, C ₆₆. Their calculation requires application of six distortions to the unstrained crystal. Among many possible sets of unit cell deformation we have selected the method proposed by Reshak et al. [10] due to its simplicity and accuracy. Table 2 shows the strain tensor components represented by Voigt notation as well as corresponding expansions of the elastic energy. The elastic tensor C _ij can be obtained from the second-order terms (∼δ²) and were calculated from the second derivatives of the DFT total energy with respect to the applied strains tuned by the five values of δ parameter: 0, ±0.01, ±0.02. It is worth noting that the calculated elastic constants (see Table 2) fulfills stability conditions derived for tetragonal crystals [12]. Indeed, one can easily verify that: $C_{33}(C_{11}+C_{12})-2C_{13}^{2}>∼0$ , C ₄₄ > 0, C ₆₆ > 0 and, C ₁₁ − C ₁₂ > 0. To the best of our knowledge, this is the first work that reports elastic constants of Dy₂Fe₁₄B crystal. Calculated by us elastic constants can be compared with the values obtained theoretically (Table 2) for much more investigated Nd₂Fe₁₄B compound. One can conclude that, the biggest difference ∼60 GPa is observed for C ₁₁ and C ₃₃ however, the difference C ₁₁ C ₃₃ is less than 10 GPa for both cases. The lack of data concerning mechanical properties for Dy₂Fe₁₄B suggests necessity of further theoretical as well as experimental investigations in this important area.

Table 1

Lattice constants (a, c), crystal density 𝜌 as well as Wyckoff coordinates (x, y, z) and spin moment (𝜇_S) of atomic sites for Dy₂Fe₁₄B

	Present work				Ref. [8]
	a = 8.776 Å, c = 11.945 Å				a = 8.776 Å, c = 11.945 Å
	𝜌 = 8.071				𝜌= 8.07 [11]
	x	y	z	𝜇_S[𝜇_B]	x	y	z	𝜇_S [𝜇_B]
Dy(4f)	0.268	0.268	0.0	−11.2	0.2666	0.2666	0.0	−8.9
Dy(4g)	0.146	−0.146	0.0	−11.2	0.1424	−0.1424	0.0	−9.2
Fe(16k₁)	0.224	−0.435	0.126	2.3	0.2232	−0.4339	0.1271	2.6
Fe(16k₂)	0.037	0.359	0.175	2.4	0.0367	0.3582	0.1744	2.5
Fe(8j₁)	0.098	0.098	0.201	2.4	0.0968	0.0968	0.1995	2.5
Fe(8j₂)	0.317	0.317	0.246	2.8	0.3169	0.3169	0.2457	3.0
Fe(4e)	0.5	0.5	0.384	2.2	0.5	0.5	0.3846	2.4
Fe(4c)	0.0	0.5	0.0	2.5	0.0	0.5	0.0	2.5
B(4g)	0.374	−0.374	0.0	−0.4	0.3720	−0.3720	0.0	0.0

Table 2

The unit cell distortions applied to Dy₂Fe₁₄B crystal and corresponding expansions of elastic energy as well as calculated elastic constants C _ij (GPa) and compliance parameters S = C ⁻¹ [10⁻³ (1/GPa). V ₀ refers to the volume of the unstrained unit cell. Strain tensor component (Voigt notation) that were not mentioned are equal to zero

Strain			Elastic energy
ϵ₁ = δ, ϵ₂ = δ			E (δ) = E (0) + V ₀ [(C ₁₁ + C ₁₂)δ² + O (δ³)]
ϵ₁ = δ, ϵ₂ = δ, , ϵ₃ = (1 + δ)⁻² −1			E (δ) = E (0) + V ₀ [(C ₁₁ + C ₁₂ + 2C ₃₃ − 4C ₁₃)δ² + O (δ³)]
ϵ₃ = δ			E (δ) = E (0) + V ₀ [(C ₃₃ ∕2)δ² + O (δ³)]
ϵ₁ = [(1 + δ)∕(1 −δ)]^1∕2, ϵ₂ = [(1 −δ)∕(1 + δ)]^1∕2			E (δ) = E (0) + V ₀ [(C ₁₁ − C ₁₂)δ² + O (δ³)]
ϵ₃ = δ², ϵ₄ = δ, ϵ₅ = δ			E (δ) = E (0) + V ₀ [4C ₄₄ δ ² + O (δ³)]
ϵ₁ = (1 + δ²)^1∕2 − 1, ϵ₂ = (1 + δ²)^1∕2 − 1, ϵ₆ = δ			E (δ) = E (0) + V ₀ [2C ₆₆ δ ² + O (δ³)]
Dy₂Fe₁₄B	C ₁₁ = 297	C ₁₂ = 92	C ₁₃ = 99	C ₃₃ = 287	C ₄₄ = 82	C ₆₆ = 80
	S ₁₁ = 3.998	S ₁₂ = −0.880	S ₁₃ = −1.076	S ₃₃ = 4.226	S ₄₄ = 12.195	S ₆₆ = 12.500
Nd₂Fe₁₄B [13]	C ₁₁ = 234	C ₁₂ = 92	C ₁₃ = 88	C ₃₃ = 228	C ₄₄ = 64	C ₆₆ = 54

Table 3

Selected mechanical properties (elastic moduli in GPa) of Dy₂Fe₁₄B crystal calculated from elastic constants

Property	Formula	Value
Voigt bulk modulus	$B_{V}=\frac{2C_{11}+∼2C_{12}+∼4C_{13}+C_{33}}{9}$	162
Voigt shear modulus	$G_{V}=\frac{2C_{11}-C_{12}+∼4C_{13}+C_{33}+∼6C_{44}+∼3C_{66}}{15}$	75
Reuss bulk modulus	$B_{V}=\frac{1}{2S_{11}+∼2C_{12}+∼4S_{13}+S_{33}}$	162
Reuss shear modulus	$G_{R}=\frac{15}{8S_{11}-∼4S_{12}-∼8S_{13}+∼4S_{33}+∼6S_{44}+∼3S_{66}}$	87
Averaged bulk modulus	$B=\frac{B_{V}+B_{R}}{2}$	162
Averaged shear modulus	$G=\frac{G_{V}+G_{R}}{2}$	81
Young’s modulus	$E=\frac{9BG}{3B+G}$	209
Poisson ratio	${\nu}=\frac{1}{2}\frac{3B-2G}{3B+G}$	0.29

Because we have no data to compare with and to check reasonableness of our results, we have calculated the values of other physical quantities associated with the elastic constants (see Table 3). One of them are longitudinal (v _l) and transverse (v _t) elastic wave velocities for isotropic materials. They were calculated using well known formula: $v_{l}=\sqrt{(3B+∼4G)/3{\rho}}$ , $v_{l}=\sqrt{G/{\rho}}$ , where 𝜌 is material’s density while B and G stand for average bulk and shear modulus, respectively. As a result we received the following values: (v _l, v _t) = (5790 m∕s, 3172 m∕s), which are similar to velocities (4650, 3150), (4520, 2870), (4640, 3060) obtained experimentally for Y₂Fe₁₄B, Pr₂Fe₁₄B and Nd₂Fe₁₄B compounds [4], respectively.

Another important physical parameter is the Debye temperature 𝛩_D which is a widely used to describe solid-state phenomena that are related to the lattice vibration. Here we considered two approaches to calculate the Debye temperature. First one, proposed by Anderson [14], assumes that at low temperatures the acoustic vibrational modes are more significant than optical ones. In a consequence, the Debye temperature can be expressed by the equation $\begin{eqnarray}\displaystyle {\Theta}_{D}=\displaystyle \frac{h}{k_{B}}\sqrt[3]{\frac{3n}{4{\pi}V_{0}}}\sqrt[3]{\displaystyle \frac{3}{v_{l}^{-3}+2v_{t}^{-3}}} & & \displaystyle\end{eqnarray}$ (1) where h, k _B, V ₀, n = 68 are Planck and Boltzmann constants, volume of unstrained unit cell and number of atoms in unit cell, respectively. Using elastic constants for Dy₂Fe₁₄B we obtained 𝛩_D = 442 K which is very close to the values 429 K, 390 K, 414 K calculated for Y₂Fe₁₄B, Pr₂Fe₁₄B and Nd₂Fe₁₄B crystals, respectively.

The second approach to estimation of the Debye temperature was detailed described by Siethoff [15]. This model assumes excitation of low-energetic and long-wave phonon modes. Certainly, for tetragonal crystals the following formula is hold $\begin{eqnarray}\displaystyle {\Theta}_{D}=C_{t}n^{-1/6}\sqrt{(a^{2}c)^{1/3}\displaystyle \frac{G_{t}}{M}} & & \displaystyle\end{eqnarray}$ (2) where C _t = 25. 68 ± 0. 42 K√m kg N⁻¹, $Gt=\sqrt[3]{C_{44}C_{66}(C_{11}-C_{12})/2}$ and M = (2m _Dy + 14m _Fe + m _B)∕17 = 65.74 ×10⁻³ kg is the averaged molecular weight of the Dy₂Fe₁₄B compound. The calculated value of the Debye temperature 𝛩_D = 458 K coincides with the value of 442 K obtained from Anderson’s model.

Finally, we have employed the empirical formula by Fine et al. [16] which relates the melting temperature with the elastic constants of tetragonal metallic crystals $\begin{eqnarray}\displaystyle T_{M}=354+\displaystyle \frac{4.5(2C_{11}+C_{33})}{3}. & & \displaystyle\end{eqnarray}$ (3) Substitution of the calculated elastic constants into the above equation allowed to find the melting temperature for Dy₂Fe₁₄B crystal of 1676 K. The obtained melting point is somewhat higher than the temperature of approximately 1533 K reported by Schneider et al. [17]. This discrepancy is likely caused by approximate character of Fine’s approach.

4. Conclusion

In sum, we calculated elastic constants of Dy₂Fe₁₄B crystal. The obtained values reflect the trend of the data measured or calculated for other (RE)₂Fe₁₄B crystals. The results presented in this article can be useful for future development of composite materials build of hard (Dy₂Fe₁₄B) and soft (e.g., Fe) magnetic phases. This work suggests further theoretical as well as experimental investigations of a structural and mechanical properties of interfaces composed of hard and soft magnetic phases.

Footnotes

Acknowledgements

The authors gratefully admit the sponsorship of the the National Science Center of Poland under Grants 2015/19/B/ST8/02636 and 2016/21/B/ST8/02737.

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