Phase Stability and Mechanical Properties of the Monoclinic, Monoclinic-Prime and Tetragonal REMO₄ (M = Ta, Nb) from First-Principles Calculations

Abstract

YTaO₄ and the relevant modification are considered to be a promising new thermal barrier coating. In this article, phase stability and mechanical properties of the monoclinic (M), monoclinic-prime (M′), and tetragonal (T) REMO₄ (M = Ta, Nb) are systematically investigated from first-principles calculations method based on density functional theory (DFT). Our calculations show that M′-RETaO₄ is the thermodynamically stable phase at low temperatures, but the stable phase is a monoclinic structure for RENbO₄. Moreover, the calculated relative energies between M (or M′) and T phases are inversely proportional to the ionic radius of rare earth elements. It means that the phase transformation temperature of M′→T or M→T could decrease along with the increasing ionic radius of RE³⁺, which is consistent with the experimental results. Besides, our calculations exhibit that adding Nb into the M′-RETaO₄ phase could induce phase transformation temperature of M′→M. Elastic coefficient is attained by means of the strain-energy method. According to the Voigt–Reuss–Hill approximation method, bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio of T, M, and M’ phases are obtained. The B/G criterion proposed by Pugh theory exhibits that T, M, and M’ phases are all ductile. The hardness of REMO₄ (M = Ta, Nb) phases are predicted based on semi-empirical equations, which is consistent with the experimental data. Finally, the anisotropic mechanical properties of the REMO₄ materials have been analyzed. The emerging understanding provides theoretical guidance for the related materials development.

1. Introduction

The rare-earth tantalate and niobates with the formula REMO₄ (M = Ta, Nb) have attracted increasing attention due to their wide application, such as biomedicine, military technology, aerospace, remote sensing, and laser [1]. Moreover, YTaO₄ and the relevant modification are extensively investigated and supposed to be promising thermal barrier coatings (TBCs) [2,3,4] due to high phase stability, good mechanical properties, and thermal conductivity. Because of a ferroelastic toughening mechanism similar to the familiar ZrO₂-8 mol%YO_1.5 (8YSZ) materials, the high-temperature fracture toughness of YTaO₄ is very well [5]. It is well known that YTaO₄ has three different crystalline structures, such as monoclinic phase (M, space group I2/a), tetragonal phase (T, space group I41/a), and monoclinic-prime phase (M′, space group P2/a). The high-temperature phase transition is a second-order and displacive transformation when the equilibrium tetragonal (T) transited to the monoclinic (M) YTaO₄ phase [6]. Although yttrium tantalate has more superior advantages than YSZ, it still has some shortcomings as a new thermal barrier coating. To improve the properties of the yttrium tantalate, doping and modifying are important.

In the periodic table of elements, yttrium and lanthanides belong to the same group of elements and have similar outermost electronic structures, so YTaO₄ can be doped with lanthanides to change their properties. Therefore, it is of great significance to systematically study the influence of doping of lanthanide on the mechanical and thermal properties of YTaO₄. Up to now, a lot of experimental researches on RETaO₄ (RE = La, Nd, Gd, Dy, Yb) have been conducted. The mechanical properties of the M phase are found to be better than M’ phase, so it is necessary to stabilize the yttrium tantalate as the M phase below the phase transition temperature. It is studied that the YTaO₄ would be stabilized as an M phase when doping 15–30 mol % Nb into YTaO₄ materials at 1473 K [7]. It is discovered that the dopant of rare earth elements (Nd, Gd, Dy, Eu, Er, Lu, and Yb) can reduce the thermal conductivity of yttrium tantalate materials [8]. Besides, the mechanical properties and plasticity of RETaO₄ (RE = Nd, La, Sm, Gd, Eu, Dy) materials are found to change regularly and become worse and worse with the decrease of atomic radius [9]. In general, yttrium tantalate materials modified by rare earth elements have many advantages, such as great mechanical properties, better thermal stability, and a larger thermal expansion coefficient [10]. Therefore, understanding the doping effects of rare earth elements and Nb on YTaO₄ and its phase stability and mechanical properties are significant.

The main purpose of the present work is to systematically investigate the phase stability and mechanical properties of M-, T-, and M’-REMO₄ (RE = La, Nd, Gd, Dy, Y; M = Ta, Nb) phases by the first-principles calculation method. Phase stabilities of T-, M-, and M’-RETaO₄ or RENbO₄ along with the various rare earth elements are studied by comparing their calculated free energies using density functional theory (DFT), and then doping effects of rare earth elements or Nb on YTaO₄ are discussed. Elastic stiffness coefficient and elastic flexibility coefficient are attained by means of the strain-energy method. Bulk modulus, Young’s modulus, shear modulus, Poisson’s ratio, and hardness of T-, M-, and M’-REMO₄ (RE = La, Nd, Gd, Dy, Y; M = Ta, Nb) are obtained according to the Voigt–Reuss–Hill approximation method. The B/G criterion proposed by Pugh theory is used to analyze the ductility and brittleness of REMO₄ phases. Finally, the anisotropic mechanical properties of the REMO₄ materials have been analyzed. It is hoped that the regularity of the YTaO₄ materials doped by rare-earth elements or Nb can be determined through first-principles calculations and provide theoretical guidance for the related technological applications.

2. Methods

To theoretically investigate the effect of dopants on the relative stability and mechanical properties of REMO₄ (M = Ta, Nb) phases, the first-principles calculations based on density functional theory (DFT) were carried out as implemented in the Vienna Ab-Initio Simulation Package (VASP) [11,12]. The electron-ion interactions were described through projector augmented wave (PAW) [13] and the exchange-correlation functional was constructed by the generalized gradient approximation (GGA) proposed by Perdew–Burke–Ernzerhof (PBE) [14]. The energy cut-off is 550 eV and the converge total energy is less than 1.0 meV/atom. The conjugate gradient method was chosen to relax the structure of atomic positions, cell volumes, and cell shapes. When the residual forces are less than 0.02 eV/Å, the structural relaxations cease. The tetrahedron smearing method with Blochl corrections was used to perform the final self-consistent static calculations [15], and then obtain more accurate energy. For the computation of doping effects of Nb, supercells including 96 atomic sites are used for all structures. A 2 × 1 × 2 supercell is used for the M phase, a 2 × 2 × 2 supercell is used for the M’ phase, and a 2 × 2 × 1 supercell is used for the T phase. To model the doping concentration of 25%, 50%, 75%, we choose 4, 8, and 12 Nb atoms and replace the same amount of Ta atoms in supercells, respectively. Taking YTaO₄ as an example, the structures of M, M’, and T phases are listed in Figure 1.

Elastic Constants

In this work, first-principles calculations are used to calculate the mechanical properties of T-RETa(Nb)O₄, M′-RETa(Nb)O₄, and M-RETa(Nb)O₄ phases). When a very small strain was imposed on the equilibrium crystal, it would exhibit elastic deformation. The strain and stress can be expressed as:

$${\mathsf{\sigma }}_{\mathrm{ij}}=\frac{1}{{\mathrm{V}}_0}{\left[\partial \mathrm{E}\left(\mathrm{V},\mathsf{\epsilon }\right)/\partial {\mathsf{\epsilon }}_{\mathrm{ij}}\right]}_{\mathsf{\epsilon }=0}$$

(1)

According to the generalized Hooke’s law, the elastic constants can be defined as the second derivative of the total energy E (V, ε) to strain,

$${\mathrm{C}}_{\mathrm{ijkl}}=\frac{\partial {\mathsf{\sigma }}_{\mathrm{ij}}}{\partial {\mathsf{\epsilon }}_{\mathrm{kl}}}=\frac{1}{{\mathrm{V}}_0}{\left[\frac{{\partial }^2\mathrm{E}\left(\mathrm{V},\mathsf{\epsilon }\right)}{\partial {\mathsf{\epsilon }}_{\mathrm{ij}}\mathbf{\partial }{\mathsf{\epsilon }}_{\mathrm{kl}}}\right]}_{\mathsf{\epsilon }=0}$$

(2)

The total energy of a crystal could be expanded using the following Taylor form:

$$\mathrm{E}\left(\mathrm{V},{\mathsf{\epsilon }}_{\mathrm{ij}}\right)=\mathrm{E}\left({\mathrm{V}}_0,0\right)+{\mathrm{V}}_0{\displaystyle \sum }_{\mathrm{ij}}{\mathsf{\sigma }}_{\mathrm{ij}}{\mathsf{\epsilon }}_{\mathrm{ij}}+\frac{{\mathrm{V}}_0}{2}{\displaystyle \sum }_{\mathrm{ijkl}}{\mathrm{C}}_{\mathrm{ijkl}}{\mathsf{\epsilon }}_{\mathrm{ij}}{\mathsf{\epsilon }}_{\mathrm{kl}}{+}_{\cdots }$$

(3)

where E (V₀, 0) is the total energy and V₀ is the volume of the unstrained system. As shown in the above formula, the strain tensors subscripts (ij, kl) are explained in the Voigt notation scheme (11 = 1, 22 = 2, 33 = 3, 23 = 4, 31 = 5, and 12 = 6) [16]. Before and after the different strains, the total energy variation can be fit by using a multinomial formula. Then we can obtain a secondary coefficient. In this work, 8 distortions to the lattice cell are applied to the lattice cell, and the relaxation in all the strained unit cells was finished when the total energy was converged to less than 1.0 meV/atom.

The Young’s modulus (Y), shear modulus (G or μ), bulk modulus (B) and Poisson’s ratio (ν) of the polycrystalline crystal were acquired from independent single-crystal elastic constants. In general, the polycrystalline modulus can be approximately assessed by two methods (the Voigt method and the Reuss method) [17,18], and they are expressed as:

$$9{\mathrm{B}}_{\mathrm{V}}={\mathrm{C}}_{11}+{\mathrm{C}}_{22}+{\mathrm{C}}_{33}+2\left({\mathrm{C}}_{12}+{\mathrm{C}}_{13}+{\mathrm{C}}_{23}\right)$$

(4)

$$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\mathrm{B}}_{\mathrm{R}}$}\right.=\left({\mathrm{S}}_{11}+{\mathrm{S}}_{22}+{\mathrm{S}}_{33}\right)+2\left({\mathrm{S}}_{12}+{\mathrm{S}}_{13}+{\mathrm{S}}_{23}\right)$$

(5)

$$15{\mathrm{G}}_{\mathrm{V}}=\left({\mathrm{C}}_{11}+{\mathrm{C}}_{22}+{\mathrm{C}}_{33}\right)-\left({\mathrm{C}}_{12}+{\mathrm{C}}_{13}+{\mathrm{C}}_{23}\right)+3\left({\mathrm{C}}_{44}+{\mathrm{C}}_{55}+{\mathrm{C}}_{66}\right)$$

(6)

$$15/{\mathrm{G}}_{\mathrm{R}}=4\left({\mathrm{S}}_{11}+{\mathrm{S}}_{22}+{\mathrm{S}}_{33}\right)-4\left({\mathrm{S}}_{12}+{\mathrm{S}}_{13}+{\mathrm{S}}_{23}\right)+3\left({\mathrm{S}}_{44}+{\mathrm{S}}_{55}+{\mathrm{S}}_{66}\right)$$

(7)

where the subscripts R and V represent the Reuss and Voigt. The elastic compliance matrices were described as {S_ij}, which is obtained by the inverse matrix of the elastic constant {C_ij}⁻¹. The Voigt–Reuss–Hill approximation [19], which was obtained by the average of Voigt and Reuss bounds, was considered as the best estimation of the polycrystalline elastic modulus. It was indicated as:

$${\mathrm{B}}_{\mathrm{H}}=\left({\mathrm{B}}_{\mathrm{V}}+{\mathrm{B}}_{\mathrm{R}}\right)/2\hspace{1em}{\mathrm{G}}_{\mathrm{H}}=\left({\mathrm{G}}_{\mathrm{V}}+{\mathrm{G}}_{\mathrm{R}}\right)/2$$

(8)

In addition, the Poisson’s ratio and the polycrystalline elastic modulus can be obtained using the following relationship:

$${\mathrm{Y}}_{\mathrm{H}}=\frac{9{\mathrm{B}}_{\mathrm{H}}{\mathrm{G}}_{\mathrm{H}}}{3{\mathrm{B}}_{\mathrm{H}}+{\mathrm{G}}_{\mathrm{H}}}\hspace{1em}{\mathsf{\nu }}_{\mathrm{H}}=\frac{3{\mathrm{B}}_{\mathrm{H}}-2{\mathrm{G}}_{\mathrm{H}}}{2\left(3{\mathrm{B}}_{\mathrm{H}}+{\mathrm{G}}_{\mathrm{H}}\right)}$$

(9)

3. Results

3.1. Structural Properties and Thermodynamic Properties

In the present work, structure relaxations of T, M, and M′-REMO₄ (RE = Y, Dy, Gd, Nd, La; M = Ta, Nb) phases were performed. The crystal structures of M-, M’-RENbO₄, and RETaO₄ phases both belong to the monoclinic crystal structure, and the T phase is the tetragonal crystal.

Table 1. Calculated lattice parameters (Å) of M-, M′-, and T-RETaO₄ phases along with the experimental data.

Phase	Abbr.	Group	a	b	c	β	Remark
YTaO4	T	I41/a	5.23	5.23	11.06		cal
	M′	P2/a	5.15	5.53	5.34	96.40°	cal
			5.26	5.43	5.08	96.08°	exp [20]
	M	I2/a	5.362	11.071	5.093	95.58°	cal
			5.24	10.89	5.06	95.31°	exp [21]
DyTaO4	T	I41/a	5.24	5.24	11.06		cal
	M′	P2/a	5.34	5.52	5.15	96.58°	cal
		P2/a	5.32	5.48	5.14	96.52°	exp [20]
	M	I2/a	5.36	11.07	5.10	95.51°	cal
		I2/a	5.35	10.97	5.06	95.6°	exp [21]
GdTaO4	T	I41/a	5.27	5.27	11.17		cal
	M′	P2/a	5.38	5.55	5.19	96.75°	cal
		P2/a	5.36	5.52	5.17	96.66°	exp [20]
	M	I2/a	5.41	11.15	5.11	95.53°	cal
		I2/a	5.41	11.07	5.08	95.6°	exp [21]
NdTaO4	T	I41/a	5.37	5.37	11.48		cal
	M′	P2/a	5.47	5.66	5.28	96.83°	cal
		P2/a	5.43	5.60	5.24	96.77°	exp [20]
	M	I2/a	5.55	11.40	5.17	95.47°	cal
		I2/a	5.51	11.23	5.11	95.7°	exp [21]
LaTaO4	T	I41/a	5.44	5.44	11.69		cal
	M′	P2/a	5.52	5.77	5.34	96.75°	cal
	M	I2/a	5.65	11.57	5.19	95.63°	cal
	nM	P21/c	7.77	5.59	7.86	101.13°	cal
		P21/c	7.76	5.58	7.81	101.53°	exp [20]

and

Table 2. Calculated lattice parameters (Å) of M-, M′-, and T-RENbO₄ phases along with the experimental data.

Phase	Abbr.	Group	a	b	c	β	Remark
YNbO4	T	I41/a	5.25	5.25	11.08		cal
	M′	P2/a	5.11	5.45	5.29	96.44°	cal
	M	I2/a	5.31	10.97	5.07	94.42°	cal
		I2/a	5.29	10.94	5.07	94.32°	exp [21]
DyNbO4	T	I41/a	5.25	5.25	11.11		cal
	M′	P2/a	5.15	5.49	5.37	95.77°	cal
	M	I2/a	5.34	11.11	5.15	93.89°	cal
		I2/a	5.32	11.00	5.07	94.34°	exp [21]
GdNbO4	T	I41/a	5.29	5.29	11.22		cal
	M′	P2/a	5.18	5.53	5.41	95.85°	cal
	M	I2/a	5.38	11.21	5.18	93.74°	cal
		I2/a	5.37	11.09	5.11	94.37°	exp [21]
NdNbO4	T	I41/a	5.38	5.38	11.53		cal
	M′	P2/a	5.27	5.66	5.50	95.98°	cal
	M	I2/a	5.47	11.51	5.29	92.42°	cal
		I2/a	5.47	11.28	5.14	94.32°	exp [21]
LaNbO4	T	I41/a	5.45	5.45	11.74		cal
	M′	P2/a	5.33	5.76	5.55	95.86°	cal
	M	I2/a	5.57	11.53	5.20	94.40°	exp [21]

list the calculated information of the crystal lattice at 0 K and the experimental data [20,21]. Our calculated results are consistently consistent with the experimental values. Both the calculations and experiments show that the small rare earth atom in RETaO₄ or RENbO₄ phases have small volumes. Besides, the β angle of the M phase and M’ phase also gradually decrease with the decrease of the atomic radius of RE³⁺.

Figure 2a-j show the total energies for T, M, and M′-REMO₄ (RE = Y, Dy, Gd, Nd, La; M = Ta, Nb) phases, which are changed with a function of volume at 0 K. The equation of state (EOS) is used to fit the energy-volume. As we know, there are three crystalline structures in RETaO₄ materials, and they are monoclinic phase (M, space group I2/a), tetragonal phase (T, space group I41/a), and monoclinic-prime phase (M′, space group P2/a). At the high temperature, the stable phase is the T-RETaO₄, and it can transform to the M phase through a displacive transformation of T→M. However, the true equilibrium phase at low temperature is the M’ phase, which only can be obtained by means of synthesizing below the temperature of T→M transformation. Therefore, the M′ phase is the low-temperature phase of the RETaO₄ materials. As shown in Figure 2a–j, our calculated results show that the M phase and T phase are both less stable than the M’ phase. It implies that the M phase is metastable and the M’ phase is stable at low temperatures. This is consistent with the experimental results [22]. For RENbO₄ phases, only low-temperature M and high-temperature T phases are existent in the literature. As for comparisons, the M′-RENbO₄ structures are also calculated in this work. In Figure 2a–j, our calculations exhibit that the total energy of the M-RENbO₄ phase is larger than that of the M′ phase, so the M phase is a true equilibrium phase at low temperature. This is consistent with the experimental results [22] that M′-RENbO₄ crystalline structures do not exist in the RENbO₄ phases.

Figure 3 presents the relative energies (ΔE_{M′→T or M→T}) of M′- or M-REMO₄ with respect to that of T-REMO₄ for different rare-earth atoms of Y, Dy, Gd, Nd, and La. It is obvious that the relative energies are inversely proportional to the ionic radius of RE³⁺. The Gibbs free energy difference of M′→T or M→T phase transformation ΔG_{M′→T or M→T} can be expressed as ΔH_{M′→T or M→T} -ΔS_{M′→T or M→T}*T. If the differences in the enthalpy ΔH (≈ΔE) and entropy ΔS are assumed to be substantially unchanged 4 [23] and ΔS is supposed to be similar for the different rare-earth dopants, phase transformation temperature of M′→T or M→T may increase with the decreasing ionic radius of RE³⁺ for Y, Dy, Gd, Nd, and La. This is consistent with the measured results using a high-temperature X-ray diffractometer by Stubičan [24]. Figure 3 presents the comparison of the experimental transformation temperature and our calculated relative energies, which indicates that the relative energies and transformation temperature decrease with the increase of the rare earth ionic radius.

As M′ and M are the stable phases at the low temperature for the RETaO₄ and RENbO₄ structures, respectively, adding the Nb element into the M′-RETaO₄ phase should induce phase transformation of M′→M at the appropriate compositions. Based on the first-principles calculations, phase transformation of M′→M induced by the dopant of Nb is studied in this work. Figure 4 presents our calculated relative energies (ΔE_M′→M) of M′-RETa_xNb_1−xO₄ (x = 0.25, 0.5, 0.75) with respect to that of M-RETa_xNb_1−xO₄ (x = 0.25, 0.5, 0.75) for different rare-earth atoms of Y, Dy, Gd, Nd, and La. When ΔE_M′→M > 0, this means that M′ is thermodynamically stable. Conversely, when ΔE_M′→M < 0, it implies that M is thermodynamically more stable than M′. In Figure 4, the M′-RETa_xNb_1−xO₄ phase will transform into the M-RETa_xNb_1−xO₄ phase when the composition of the dopant Nb is about 0.5. It is worth mentioning that our calculated results are related to the phase transformation at 0 K, and transformation composition may decrease at the high temperature.

3.2. Mechanical Properties

The calculated elastic constants are listed in

Table 3. Elastic constants C_ij for the M phases of RETaO₄ and RENbO₄. All quantities are in GPa.

M-RETaO4	C11	C22	C33	C44	C55	C66	C12	C13	C23	C16	C26	C36	C45
La	217.07	168.49	230.98	43.28	55.86	56.26	61.27	106.50	90.21	25.28	6.27	−13.88	−1.14
Nd	229.36	193.08	255.53	50.63	62.02	67.21	63.53	116.64	98.38	21.92	1.88	−19.63	−2.09
Gd	249.50	212.45	270.47	58.12	61.57	83.45	70.90	131.87	95.84	13.11	−2.83	−20.28	−4.73
Dy	255.11	220.50	275.98	57.91	60.10	88.91	74.01	136.94	94.82	10.69	−4.04	−19.74	−4.96
Y	257.02	221.67	273.37	56.72	57.56	87.77	74.78	135.87	93.11	8.91	−4.74	−19.08	−5.23
M-RENbO4	C11	C22	C33	C44	C55	C66	C12	C13	C23	C16	C26	C36	C45
La	171.39	154.27	195.03	30.25	34.82	33.65	63.13	100.72	75.39	34.69	4.56	−23.70	3.09
Nd	157.05	169.74	224.90	29.37	43.43	47.48	49.18	117.27	84.96	45.25	8.44	−23.51	8.18
Gd	195.53	194.15	245.42	42.11	49.75	69.42	64.69	134.61	83.74	29.95	0.79	−22.30	2.39
Dy	205.94	201.75	249.32	45.85	49.62	75.96	68.74	140.48	81.48	25.77	−1.20	−21.93	0.68
Y	202.58	192.94	248.02	44.99	47.80	75.01	65.47	135.69	76.74	25.19	−1.17	−20.13	0.41

,

Table 4. Elastic constants C_ij for the M’ phases of RETaO₄ and RENbO₄. All quantities are in GPa.

M’-RETaO4	C11	C22	C33	C44	C55	C66	C12	C13	C23	C16	C26	C36	C45
La	238.64	162.73	253.65	52.01	60.31	59.34	86.70	113.51	87.66	16.34	−13.07	9.78	−10.43
Nd	266.93	178.92	271.78	62.93	62.67	66.14	91.90	118.95	90.78	15.35	−14.16	8.47	−9.67
Gd	288.41	176.27	296.06	66.70	59.66	71.33	88.50	119.39	90.73	14.69	−15.78	11.01	−7.09
Dy	296.31	173.88	304.01	67.04	56.06	71.97	84.00	118.53	89.13	14.15	−17.72	11.54	−6.91
Y	290.40	141.65	298.91	64.01	53.08	71.22	74.32	113.46	75.89	11.68	−20.59	10.52	−8.10
M’-RENbO4	C11	C22	C33	C44	C55	C66	C12	C13	C23	C16	C26	C36	C45
La	235.54	155.44	220.45	46.37	53.15	47.89	84.43	111.03	84.45	16.31	−6.62	−7.09	−10.34
Nd	258.48	181.92	240.56	53.68	57.23	50.10	91.42	113.48	88.80	17.39	−7.71	−7.47	−9.32
Gd	280.10	189.47	259.83	57.03	57.10	53.72	88.91	110.44	92.54	18.43	−2.49	−3.91	−4.55
Dy	292.68	188.65	271.23	57.82	55.63	55.05	88.54	112.08	96.52	18.88	−0.28	−1.63	−2.45
Y	288.33	181.77	271.43	56.05	53.58	54.05	84.43	110.17	94.51	17.74	−1.31	−1.75	−2.21

and

Table 5. Elastic constants C_ij for the T phases of RETaO₄ and RENbO₄. All quantities are in GPa.

T-RETaO4	C12	C11	C33	C44	C66	C13	C16
La	124.89	163.70	151.93	29.66	27.52	71.67	113.03
Nd	122.49	194.92	172.31	31.22	28.68	73.87	147.44
Gd	121.73	228.82	192.05	29.00	13.41	74.49	93.62
Dy	120.51	241.46	201.29	27.93	11.56	74.64	148.69
Y	118.20	237.39	198.37	26.25	12.87	71.61	69.08
T-RENbO4	C12	C11	C33	C44	C66	C13	C16
La	112.79	169.36	151.35	31.98	34.65	72.74	128.76
Nd	114.62	196.62	170.36	34.74	29.99	75.69	138.65
Gd	115.24	226.54	190.83	33.89	14.65	77.27	159.76
Dy	116.10	239.29	198.93	33.28	9.67	78.08	164.16
Y	111.40	233.13	197.32	31.69	9.17	75.04	146.88

. Because the T-YTaO₄ phase is tetragonal, C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, and C₆₆ can be determined through six deformation modes [25]. As M′-YTaO₄ and M-YTaO₄ phases are both monoclinic, the thirteen independent elastic constants can be obtained by applying thirteen distortions [26]. The total energies are varied before and after a set of different strains (±1%, ±2%, ±3%, and ±4%), and the elastic constants were calculated by the quadratic coefficients. For stable structures, the elastic constants need to meet the mechanical stability criterion [27]. The monoclinic system criteria are C₁₁ > 0, C₂₂ > 0, C₃₃ > 0, C₄₄ > 0, C₅₅ > 0, C₆₆ > 0, C₄₄C₆₆ − 2C₄₆ > 0, C₁₁ + C₂₂ + C₃₃ + 2(C₁₂ + C₁₃ + C₂₃) > 0, C₂₂ + C₃₃ − 2C₂₃ > 0. The tetragonal system criteria are C₁₁ > 0, C₃₃ > 0, C₄₄ > 0, C₆₆ > 0, C₁₁-C₁₂ > 0, C₁₁ + C₃₃ − 2C₁₃ > 0, 2C₁₁+C₃₃+2C₁₂+4C₁₃ > 0. All RETaO₄ (RE = La, Nd, Gd, Dy) and RENbO₄ (RE = La, Nd, Gd, Dy) materials meet the criterion of mechanical stability, and the structures are stable.

‘The polycrystalline elastic mechanical properties, such as shear modulus (G or μ), bulk modulus (B), Young’s modulus (Y), and Poisson’s ratio (ν) could be obtained through the Voigt and Reuss methods according to the calculated elastic constants. Using energy considerations, Hill [20] certificated the elastic moduli of the Voigt and Reuss methods are the upper and lower limits of polycrystalline constants. The practical elastic modulus can be estimated by the arithmetic means of these extremes. Generally, the bulk modulus is a measure of resistance to volume change by applied pressure. As seen from Figure 5 and

Table 6. Bulk modulus (GPa), shear modulus (GPa), B/G, Young modulus (GPa), Poisson ratio, and Vickers-hardness (Kg·N) for T-REMO₄ (M = Ta,Nb) phases.

T-RETaO4	B	G	B/G	E	ν	Hv	Remake
La	111.33	29.97	3.71	82.50	0.38	278.11	cal
Nd	120.97	36.28	3.33	98.95	0.36	319.77	cal
Gd	130.55	33.95	3.84	93.74	0.38	270.37	cal
Dy	134.21	33.87	3.96	93.71	0.38	261.17	cal
Y	131.14	33.86	3.87	93.52	0.38	268.04	cal
	128.9	52.7	2.45	139.1	0.32	-	exp [3]
T-RENbO4	B	G	B/G	E	ν	Hv	Remark
La	110.57	34.52	3.20	93.80	0.36	333.25	cal
Nd	120.38	38.68	3.11	104.82	0.35	347.78	cal
Gd	130.04	36.34	3.58	99.72	0.37	295.08	cal
Dy	134.24	34.04	3.94	96.69	0.38	262.75	cal
Y	130.41	33.26	3.92	91.97	0.38	263.96	cal

,

Table 7. Bulk modulus (GPa), shear modulus (GPa), B/G, Young modulus (GPa), Poisson ratio, and Vickers-hardness (Kg·N) for M’-REMO₄ (M = Ta,Nb) phases.

M’-RETaO4	B	G	B/G	E	ν	Hv	Remark
La	132.73	56.96	2.33	149.48	0.31	481.9	cal
Nd	142.66	64.43	2.21	168.01	0.30	515.73	cal
Gd	144.97	68.23	2.12	176.93	0.30	542.89	cal
Dy	144.00	68.59	2.10	177.04	0.30	550.51	cal
Y	128.67	65.20	2.19	167.34	0.28	588.38	cal
	132.7	66.1	2.01	170.2	0.29	-	exp [3]
M’-RENbO4	B	G	B/G	E	ν	Hv
La	126.58	50.02	2.53	132.60	0.33	434.37	cal
Nd	138.32	56.44	2.45	149.05	0.32	454.79	cal
Gd	143.03	61.34	2.33	161.00	0.31	484.34	cal
Dy	145.98	62.52	2.33	164.14	0.31	484.391	cal
Y	142.59	61.29	2.33	160.75	0.31	485.32	cal

and

Table 8. Bulk modulus (GPa), shear modulus (GPa), B/G, Young modulus (GPa), Poisson ratio, and Vickers-hardness (Kg·N) for M-REMO₄ (M = Ta,Nb) phases.

M-RETaO4	B	G	B/G	E	ν	Hv	Remark
La	122.08	52.80	2.31	138.45	0.32	483.34	cal
Nd	134.00	60.47	2.22	157.69	0.31	512.68	cal
	-	-	-	-	-	641	exp [8]
Gd	145.03	67.75	2.14	175.87	0.31	537.98	cal
	-	-	-	-	-	610	exp [8]
Dy	148.88	69.15	2.15	179.61	0.31	535.20	cal
	-	-	-	-	-	534	exp [8]
Y	148.68	68.32	2.18	177.74	0.30	528.35	cal
	183.7	63.2	2.91	170.1	0.34	378	exp [3]
M-RENbO4	B	G	B/G	E	ν	Hv	Remark
La	106.00	30.05	3.53	82.38	0.37	286.18	cal
Nd	107.85	39.70	2.72	106.08	0.331	393.53	cal
Gd	129.36	50.99	2.54	135.21	0.33	433.19	cal
Dy	134.53	54.51	2.47	144.08	0.32	449.5	cal
Y	129.95	54.12	2.4	142.57	0.32	463.94	cal

, the calculated shear modulus and bulk modulus of M-, M′-, T-RETaO₄, and RENbO₄ (RE = Y, Dy, Gd, Nd, La) is decreased with the increase of the rare-earth atoms, which indicate that the resistance to volume change through applied pressure is eventually lowered. Moreover, the calculated bulk modulus of rare-earth tantalate is regularly larger than and rare-earth niobates. The calculated shear modulus shows a similar trend, which means that the resistance to reversible deformations upon shear stress for RETaO₄ and RENbO₄ (RE = Y, Dy, Gd, Nd, La) is decreased with the increase of the rare-earth atoms. The ratio between bulk modulus and shear modulus, proposed by Pugh theory [28], can be used to empirically predict the brittleness and ductility of materials. A low B/G ratio is associated with brittleness, and a high value indicates its ductile nature. The empirically critical value which distinguishes ductile and brittle materials is around 1.75. In the present work, the calculated B/G of REMO₄ in Figure 6 is larger than 1.75, which means that all REMO₄ materials are ductile.

Young’s modulus E can be used to estimate the stiffness of materials. The calculations of M-, M′-, T-RETaO₄ or RENbO₄ in Figure 7 and Table 6, Table 7 and Table 8 suggest that Young’s modulus is decreased with an increase of the rare-earth atoms, which means that M-, M′-, and T-YTaO₄ are the stiffest, and then followed by DyTaO₄, GdTaO₄, NdTaO₄, and LaTaO₄. Moreover, our calculated results indicate that RETaO₄ is stiffer than RENbO₄. Poisson’s ratio (ν) is also related to the brittleness and ductility of materials. A compound is considered brittle if the ν is <0.26 [29]. The higher value of Poisson’s ratio is, the more ductile the material is. Thus, Table 6, Table 7 and Table 8 show that all REMO₄ materials are ductile. They are in good agreement with the results estimated by the B/G ratio. Besides, the value of Poisson’s ratio suggests that the ductility is inversely proportional to the rare earth atom of REMO₄ materials, and RENbO₄ is more ductile than RETaO₄.

Hardness is a very important mechanical property in applications. Hardness is defined as the resistance of a material to deformation and may be predicted using macroscopic and microscopic models. In this work, we use the semi-empirical equations of hardness proposed by Chen et al. [30] and Tian et al. [31] were used to study the hardness of the REMO₄ phases. The equations of these two models are defined as follow:

$${ \mathrm{H}}_{\mathrm{V}}=2{({\mathrm{k}}^2\mathrm{G})}^{0.585}-3$$

(10)

$${\mathrm{H}}_{\mathrm{V}}=0.92{\mathrm{k}}^{1.137}{\mathrm{G}}^{0.708}$$

(11)

where k = G/B, G and B are the shear modulus and the bulk, respectively. The obtained hardness of REMO₄ phases are shown in Figure 8 and Table 6, Table 7 and Table 8 presents a comparison between the calculated and experimental results, which exhibits a good consistency. Besides, our calculations suggest that the Vickers hardness of the T phase decreases, and the single-phase gradually increase with the decrease of the atomic radius.

The anisotropic mechanical properties of the compounds are very important in applications. Based on the G and B values from Reuss and Voigt, Ranganathan et al. [32] proposed a universal elastic anisotropy index A^U for crystal with any symmetry as shown below:

$${ \mathrm{A}}^{\mathrm{U}}=\frac{5{\mathrm{G}}_{\mathrm{Voigt} }}{{\mathrm{G}}_{\mathrm{Reuss} }}+\frac{{\mathrm{B}}_{\mathrm{Voigt}}}{{\mathrm{B}}_{\mathrm{Reuss}}}-6\ge 0$$

(12)

A^U is equal to zero when the single crystals are locally isotropic. The extent of single-crystal anisotropy can be expressed by the departure from zero indicates. The highly mechanical anisotropic properties exhibit large discrepancies from zero. The calculated elastic anisotropy is shown in Figure 9. Most values of A^U are lower than 1. The larger the value of A^U is, the stronger the anisotropy of the phase. The M-, M′- and T-REMO₄ are anisotropic, and the elastic anisotropy of M and M′ phases is larger than the tetragonal (T) phase.

4. Conclusions

In the present work, phase stability and mechanical properties of REMO₄ (RE = La, Nd, Gd, Dy, Y; M = Ta, Nb) are investigated by first-principles calculations. Some conclusions can be found. For RETaO₄, the M’ phase is more stable than the M phase at low temperature, and the T phase is only stable at high temperature. For RENbO₄, only the M phase is stable at low temperatures. This is consistent with the experimental results. Our calculated relative energies (ΔE_{M′→T or M→T}) of M′- or M-REMO₄ with respect to that of T-REMO₄ for different rare-earth atoms are inversely proportional to the ionic radius of RE³⁺. This implies that the phase transformation temperature of M′→T or M→T is decreased with the increase of the rare-earth atoms, which is consistent with the experimental data. Moreover, our calculations exhibit that adding Nb into M′-RETaO₄ can induce phase transformation of M′→M, and the doping concentration is about 50%. Besides, the elastic coefficient is attained by means of the strain-energy method. Bulk modulus, Young’s modulus, shear modulus, and Poisson’s ratio of T-, M-, and M’ phases are obtained according to Voigt-Reuss-Hill approximation, and anisotropic mechanical properties of the REMO₄ materials have been calculated. Finally, our calculated B/G exhibits that T-, M-, and M’ phases are all ductile, and the hardness of REMO₄ phases are predicted based on semi-empirical equations, which is consistent with the experimental data.