First-principles study of structural, mechanical, and thermodynamic properties of cubic Y2O3 under high pressure

Author’s Accepted Manuscript First-principles study of structural, mechanical, and thermodynamic properties of cubic Y2O3 under high pressure Xian Zhang, Wenhua Gui, Qingfeng Zeng www.elsevier.com/locate/ceri

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To appear in: Ceramics International Received date: 4 October 2016 Revised date: 15 November 2016 Accepted date: 24 November 2016 Cite this article as: Xian Zhang, Wenhua Gui and Qingfeng Zeng, First-principles study of structural, mechanical, and thermodynamic properties of cubic Y2O under high pressure, Ceramics International, http://dx.doi.org/10.1016/j.ceramint.2016.11.176 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

First-principles study of structural, mechanical, and thermodynamic properties of cubic Y2O3 under high pressure Xian Zhanga,*, Wenhua Guia, Qingfeng Zengb a

School of Advanced Materials and Nanotechnology, Xidian University, Xi’an 710071, China

b

International Center for Materials Descovery, School of Materials Science and Engineering,

Northwestern Polytechnical University, Xi’an 710072, Shaanxi, PR China *Corresponding author at: School of Advanced Materials and Nanotechnology, Xidian University, Xi’an 710071, China. Tel./fax: (0086) 182 2050 7206 E-mail address: [email protected] (X. Zhang)

Abstract The structural, mechanical, and thermodynamic properties of cubic Y2O3 crystals at different hydrostatic pressures and temperatures are systematically investigated based on density functional theory within the generalized gradient approximation. The calculated ground state properties, such as equilibrium lattice parameter a0, the bulk modulus B0, and its pressure derivative B0′ are in favorable agreement with the experimental and available theoretical values. The pressure dependence of a/a0 and V/V0 are also investigated. Furthermore, the elastic constants Cij, bulk modulus B, shear modulus G, Young’s modulus E, the ductile or brittle (B/G), Vickers hardness Hv, isotropic wave velocities and sound velocities are calculated in detail in a pressure range from 0 to 14 GPa. It was found that the Debye temperature decreases monotonically with an increase in pressure, the calculated elastic anisotropic factors indicate that Y2O3 has low anisotropy at zero pressure, and that its elastic anisotropy increases as the pressure increases. Finally, the thermodynamic properties of Y2O3, such as the dependence of the heat capacities CV and CP, the thermal expansion coefficient α, the isothermal bulk modulus, and the Grüneisen parameter γ on 1

temperature and pressure, are discussed from 0 to 2000 K and from 0 to 14 GPa, respectively, applying the non-empirical Debye model in the quasi-harmonic approximation. Keywords: Y2O3, first principles, mechanical properties, thermodynamic properties, quasi-harmonic Debye model

1. Introduction The binary sesquioxides R2O3 are promising materials in the science and technology of many applications such as high-temperature materials, electronics, catalysts, photonics, nuclear materials, chemicals, biomaterials, etc. Therefore, a better understanding of binary systems can help in the design of new oxide materials with desirable properties and would stimulate further advances in the field. In recent years, the most widely studied materials employed for optoelectronic devices that accomplish these high-pressure and high-temperature (HP and HT) properties are several sesquioxide like: Y2O3, In2O3, Sb2O3, Sc2O3, Bi2O3, and Ti2O3 [1-10]. Several cubic-bixbyite structure under HP-HT conditions showed a transformation to a very dense and unusual oxide structure of the α-Gd2S3-type with Pnma space group. Yttria (Y2O3), with a cubic bixbyite structure, is an important rare-earth sesquioxide. The cubic structure of Y2O3 with space group of Ia3-( Th7 ) (No. 206) contains a total of 16 formula units (80 atoms) in unit cell. The unit cell contains two nonequivalent yttrium Wyckoff sites, Y1 atom located at the 8b sites and Y2 at the 24d sites, and one type of O atom located at 4e Wyckoff sites, respectively. Y1 (Y2) is surrounded by six oxygen atoms in the form of a perfect (distorted) octahedron (namly, Y1O6 and Y2O6 polyhedral units), and O atom is surrounded by four Y atoms 2

in the form of a distorted tetrahedron. This compound exhibit three structural polymorphisms: cubic, hexagonal corundum-type (space group R3c, No. 167) and monoclinic Rh2O3-type (space group Pbca, No. 61) structures, commonly known as C-Y2O3, H-Y2O3, and M-Y2O3, respectively. The cubic form of Y2O3 being stable at room temperature and ambient pressure. The structural stability of yttria with pressure and temperature makes it useful in many industrial applications and technological areas. Due to its excellent chemical stability [11, 12], low emission and small absorption coefficient in the IR region at high temperature, outstanding refractoriness (T ≈ 2410℃), optical transparency over a broad spectral range (0.2-0.8 μm), large band gap (5.5~6.0 eV), high hardness, low phonon energy (430 cm–1), high refraction index (1.92-1.99), Y2O3 transparent ceramics are used as excellent IR-windows materials and used in phosphor matrixes, scintillators, missile domes, waveguides and anti-wear anti-reflection coatings for mid-infrared lenses, solid oxide fuel cells, high-temperature protective coatings [13–16]. Recently, Y2O3 has also received attention as a promising candidate for replacing silicon dioxide (SiO2) as a gate dielectric material in metal-oxide-semiconductor (MOS) transistors and put it down to its high dielectric contant (ε = 14-19) [17-20]. Moreover, Y2O3 is also an important oxide-based phosphor material and is used as a host material in rare-earth-doped lasers [21]. In recent years, the significance of yttria has attracted extensive experimental and theoretical studies on the electronic, mechanical and opticalproperties [22-25]. The measurement of the elastic moduli of yttria from 300 to 1473 K have been 3

reported by James and et al. [26], and found it to be stable up to about 2705 K. Ab-initio molecular dynamics [27] study of the structure of Y2O3 up to 5000 K indicated melting at 3150 K. Raman spectra of C-type bixbyite structured R2O3 (R = Yb, Sc, Er, Y, Ho, Gd, and Sm) are studied by Abrashev and et al. [28]. The stability of the crystalline phases of Y2O3 at high pressure has also been studied by calculating [29] enthalpy in various phases using the ab-initio method. Phase transitions and thermodynamic properties of yttria, Y2O3 has been studied by Bose and et al. [30]. Ramzan and et al. [31] have been extensively researched the electronic, mechanical and optical properties of Y2O3 with the GGA-PBE approximation the HSE06 hybrid-density functional. In 1990, Ching and et al. [32] repoted a self-consistent band structure and optical properties calculation of Y2O3, using the orthogonalized linear combination of atomic orbitals (OLCAO) method in the local-density approximation (LDA). Ahuja and et al. [33] carried out a study on the electronic and optical properties of ceramic Sc2O3 and Y2O3 using compton spectroscopy and first principles calculations. Badehian and et al. [34] studied the elastic, structural, electronic, thermodynamic, and optical properties of cubic yttria ceramic within the density functional theory, implemented in the WIEN2k (2011) code [35]. Zheng and et al. [36] investigated the native point defects in bulk yttria through the firstprinciples calculation using pseudopotential plane waves approach. Swamy and et al. [37] analyzed the thermodynamic properties and phase diagram of Y2O3. Ou and et al. [38] studied the vacancy formation energies and electronic structures of Y2O3 by first principles. Mueller and et al. [39] used the linear muffin-tin orbital (LMTO) method 4

within the atomic sphere approximation for the calculation of the electronic structure of yttria. Mudavakkat and et al. [40] evaluated the structure, morphology, and optical properties of nanocrystalline yttrium oxide. Manning and et al. [41] measured the elastic properties of polycrystalline yttrium oxide, holmium oxide, and erbium oxide at high temperature. Also, some studies have reported [42] that bulk modulus values deviate too much from experimental values (around 150 Gpa [26, 43]). While the bulk modulus and elastic moduli are valuable for designing such optical components. In the present study, we have investigated the effect of tstructural, mechanical, and thermodynamic properties of cubic Y2O3 at high pressures (0-14 GPa) and temperatures (0-2000 K), by using first-principles calculations combined with the quasi-harmonic Debye model. 2. Computational details and theoretical method In the present work, the Cambridge Serial Total Energy Package (CASTEP) [44] based on density functional theory (DFT) was used, under ambient pressure conditions. Furthermore, the norm-conserving pseudo-potential [45] was adopted to describe the interactions of valence electrons (O 2s22p4 and Y 4s24p65s24d1) with ion cores. The exchange-correlation energy was evaluated using GGA-PBE [46]. A quasi-Newton minimization algorithm using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) [47] scheme was used, which provides a very efficient and robust way to explore the optimum crystal structure with minimum energy. The k-point meshes for Brillouin zone sampling are constructed using the Monkhorst–Pack scheme [48]. A plane wave cut-off energy of 600 eV and a 4×4 ×4 k-point mesh are found to be 5

sufficient to converge the total energy to better than 1 meV/atom. This assured a high level of convergence with respect to all parameters: a self-consistent computational convergence precision of within 5.0×10–7 eV/atom, a total energy variation of 5.0×10–6 eV/atom, a maximum ionic Hellmann–Feynman force of within 0.01 eV/Å, a maximum stress of within 0.02 GPa, and a maximum ionic displacement of within 5.0 ×10–4 Å. The investigations on the thermodynamic properties of Y2O3 under high pressure and temperature are of great practical significance in engineering applications. In this paper, to investigate the thermodynamic properties of Y2O3, we have used the quasi-harmonic Debye model, as implemented in the Gibbs program [49]. The non-equilibrium Gibbs function G* (V ; P, T) can be written as follows:

G V ; P, T   E V   pV  AVib  V  ; T 

(1)

where E(V) represents the total energy per unit cell, and can be obtained by electronic structure calculations, P is the constant hydrostatic pressure condition, Θ (V) is the Debye temperature, and AVib is the vibrational Helmholtz free energy, which can be written using the quasi-harmonic approximation [50] and the Debye model of phonon density of states as [51],

9     AVib  ; T   nkT   3ln 1  e /T   D     T  8 T

(2)

where n is the number of atoms per formula unit, D (Θ/T) is the Debye integral. For an isotropic solid, Θ can be taken as [51], D y 

y

3 x3 dx y 3 0 e x  1

(3)

6



 6π V n  2

kB

12

1/3

f  

BS M

(4)

where the Poisson ratio σ is taken to be 0.25, f (σ) is given by Francisco and et al. [52], and BS denotes the adiabatic bulk modulus, which is approximated by the static compressibility and expressed by [52],  d 2 E V   BS  B V   V   2  dV 

(5) 13

  21   3 2 11   3 2  1        f    3  2        31     31        

(6)

Therefore, the non-equilibrium Gibbs function G* (V; P, T) as a function of V, P, and T can be minimized with respect to volume V by taking,  G  V ; P, T     0    V   P,T

(7)

Other thermodynamic properties can be obtained by solving Eq. (7) and the equation of state (EOS), and the isothermal bulk modulus BT, the heat capacity at constant volume CV, and the heat capacity at constant pressure CP, and the thermal expansion coefficient α are given by [53],  p  BT  p, T   V    V 

(8)

    3 T  CV  3nkB  4 D     /T  1  T  e

(9)

Cp  Cv 1   T 



(10)

 Cv

(11)

BTV

where γ is the Grüneisen parameter, which is defined as, 7

 

d ln  V  d ln V

(12)

3. Results and discussion 3.1. Electronic and structural properties As shown in Fig. 1, Y2O3 has a cubic bixbyite structure with the space group Ia3 (No. 206) at ambient temperature and pressure conditions, which is similar to In 2O3. The cubic unit cell contains 80 atoms (two primitive cells of Y2O3) with two nonequivalent yttrium cation sites, Y1 and Y2, occupying the 8b (1/4, 1/4, 1/4) and 24d (–0.029, 0, 0.25) crystallographic positions, respectively, and one type of O atom at the 48e (0.3910, 0.1510, 0.3804) site. Y1 (Y2) is surrounded by six oxygen atoms in the form of a perfect (distorted) octahedron (Y1O6 and Y2O6 polyhedral units), and the O atom is surrounded by four Y atoms in the form of a distorted tetrahedron. We have used the experimental data for initial input with a lattice constant of 10.604 Å, and optimized the ion positions. To determine the ground state structure of Y2O3, several different lattice parameter a are set to calculate the total energy E corresponding primitive cell volume V. The plot of calculated total energies versus volume for Y2O3 is given in Fig. 2. By fitting the calculated energy-volume (E-V) to the third-order Birch-Murnaghan equation of state (EOS) [54], in which the energy-volume relationship is expanded as, B ' -1 B0V0  1  V0  0 V B'  E V   E0  '  0  '0    ' V ( B0 - 1)  B0  ( B0 - 1)  V   

(13)

we can determine the equilibrium structure parameters, the bulk modulus B0 and its pressure derivative B0′. All the equilibrium parameters using LDA, GGA-PBE, and 8

GGA+U approximations are given in Table 1, together with other theoretical data and available experimental data for comparison. It may be noted that the LDA and GGA-PBE calculations are underestimated to experimental data (10.604Å) [55] by about 0.41% and 0.09%, and GGA+U calculations are overestimated by about 0.745% for lattice parameters a, respectively. Bulk modulus is a fundamental physical property of solids and it can also be used as a measure of the average bond strengths of atoms of the given crystals. Therefore, its value is an important indicator for the several purposes. It also can be seen that our result of bulk modulus B0 and its pressure derivative B0′, compressibility K are well consistent with the experimental data [26, 56–60] and theoretical data [30, 31, 40, 60]. As can be seen from Table 1, there are very a few differences between structural parameters with coulomb potential (U) and without U. Thus we only report calculation results that did not consider coulomb potential U. Agoston et al.[4] and Gomis et al. [9] measured the bulk modulus B0, its pressure derivative B0′ within density functional theory calculations and obtained value B0 = 178.87 GPa , B0′=5.15 for C-In2O3 and B0= 147 GPa , B0′=5 for C-Ti2O3, respectively. Form the results obtained for bulk modulus, it observed that our results (see Table 1) are in good agreement with theoretical date [4, 9]. This is because the C-Ti2O3, C-In2O3 and C-Ti2O3 are very similar in structure and properties. It is also indicated that the calculated method performed in our work is reasonable. In order to show how the structural parameters under pressure in this compound behave, the equilibrium ratios of normalized lattice parameters a/a0 and normalized volume V/V0 as functions of pressure are plotted in Fig. 3, where a0 and V0 are the 9

equilibrium structure parameters at T = 0 K and P = 0 GPa, respectively. By fitting the calculated data to a third-order polynomial, we obtained their relationships at the temperature T = 0 K, a/a0 = 1.00073 – 1.85×10–3P+4.659×10–5P2 – 1.81187×10–7P3

(14)

V/V0 = 1.00262 – 5.57×10–3P+1.1875×10–4P2 – 2.88194×10–7P3

(15)

We note that with the increase in pressure, the ratios of lattice parameter a/a0 decrease and the volume V/V0 shrinks constantly, indicating that the cell is being compressed. The reason of this change is that when pressure increases, atoms in the interlayers become closer, and their interactions become stronger. The effects of temperature and pressure on the cell are opposite. Generally, electronic properties of solid materials are studied by calculating the energy band structure and density of states. In our work, the energy band gap value has been calculated to be 4.296 eV, 5.748 eV, and 4.133 eV using GGA-PBE, GGA+U, and LDA, respectively (listed in Table 2, together with other theoretical results and available experimental data). The calculated GGA+U band gap is larger than that of obtained from GGA-PBE and LDA, and is in good agreement with the value 5.5–6eV obtained through experiment [20, 39, 61, 62]. The results show that the overall band profiles improve the band gap energy with respect to the other theatrical results [27, 30, 41] and they are in good agreement with the experimental results. Our calculated band gaps with LDA and GGA-PBE are underestimated in comparison with the experimental data. The reason of this underestimation is that the Kohn–Sham one particle equation does not provide the quasiparticle excitation energies. To our 10

knowledge, the modified GGA +U formalism better band gap than the GGA because it has orbital independent exchange-correlation potential which depends only on semilocal quantities. 3.2. Mechanical properties and anisotropy The elastic constants determine the response of a crystal to external force, and they have a significant role in our understanding of the mechanical behaviors of materials. These properties play an important part in providing valuable information about the binding characteristic between adjacent atomic planes, anisotropic character of binding, and structural stability. Hence, to study the stability of C-Y2O3, we have calculated the elastic constants at normal a pressure using CASTEP package. The elastic constants are defined by means of a Taylor expansion of the total energy E(V, δ), for the system with respect to a small strain δ of the lattice primitive cell volume V. The energy of a strained system is expressed as [63]   1 E V ,    E V0 , 0   V0   ii i   Cij  ii j  2 ij  i 

(16)

where E (V0, 0) is the energy of the unstrained system with the equilibrium volume V0, and δ is the small-strain corresponding volume, τi is the stress tensor, ξi is a Voigt index factor, and Cij is the elastic tensor. For cubic crystals, there are only three independent elastic constants, namely, C11, C12, and C44. And two shear moduli, c and c′, where c = C44, c′ = (C11 – C12)/2, corresponding to shear along the (100) and (110) planes, respectively, which can be used to estimate distortion in a certain plane of Y2O3 when it is expanded or compressed. The conditions for the mechanical stability of crystal structures are given by[64] 11

~

~

~

~

~

C  0, C11 | C12 |, C11 2 C12  0 ~

(17) ~

Where C  C  p(  1, 4) and C12  C12  p . Once three elastic constants C11, C12, and C44 have been calculated, bulk modulus B, and isotropic shear modulus G can be calculated simply using the subsequent clear expressions: GV  GR 2 B  BR B V 2

G

(18) (19)

As is known to all, the arithmetic average of the Voigt (BV, GV) and the Reuss (BR, GR) can be commonly used to estimate the bulk modulus B and shear modulus G, which can be evaluated by the Voigt–Reuss–Hill scheme[65] BV  BR 

C11  2C12 3

(20)

GV 

C11  C12  3C44 5

(21)

GR 

5C44 (C11  C12 ) 4C44  3(C11  C12 )

(22)

Here, GV is Voigt’s shear modulus corresponding to the upper bound of G values, and GR is Reuss’s shear modulus for cubic crystals corresponding to the lower bound values. Young’s modulus E, Poisson ratio v, and compressibility coefficient K are important fundamental parameter closely connected to many physical properties such as internal strain, thermoelastic stress, bonding forces, hardness, sound velocity, fracture toughness. The expressions for the Young’s modulus, Poisson’s ratio and compressibility coefficient are given by E 

9 BG 3B  G

(23) 12

3B  2G 23B  G  1 3 K  B C11  2C12 ν

(24) (25)

Our theoretical elastic constants C11, C12, C44, bulk modulus B (GPa), Young's modulus E (GPa), shear modulus G (GPa), and Poisson's ratios ν for Y2O3 at P = 0 GPa and T = 0 K are summarized in Table 3, together with the available theoretical [30, 31] or experiment [26] data. The obtained elastic constant, bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, and other important physical quantities at 0-14 GPa using GGA-PBE approroximation are listed in Table 4. When the pressure is less than 14 GPa, the calculated elastic constants listed in Table 4, satisfied the above stability certeria [see Eq. (17)], indicating that the crystal structure of Y2O3 remains elastically stable, i.e., it is still cubic. The variation in the elastic constants C11, C12, C44, modulus (B, G, E) and cauchy pressure CP of Y2O3 with hydrostatic pressure are plotted in Fig. 4. It is noted that C11 and C12 increase linearly with an increase in pressure, while the varying amplitude of C44 is moderate compared with C11 and C12 (Fig. 4 a). It is clear that the C11 and C12 are more sensitive to the change of pressure than the C44, at the same time, it also indicates that Y2O3 becomes increasingly difficult to compress as the pressure increases. The variation in the modulus (B, G, E) in Fig. 4 (b), Young’s modulus and the shear modulus exhibit a slight decrease as the pressure increases. Table 4 also shows that the hydrostatic pressure has a remarkable effect on the microhardness, which decreases with increasing hydrostatic pressure. The similar elastic properties of Y2O3 using PBE-GGA approximation at 0–7 GPa have also been calculated by 13

Badehian et al. [34], and obtained the elastic constants satisfy the cubic stability condition and showed that Y2O3 is elastically stable. It can be seen that the gareement between our results and previous theoretical date [34] is good. Pettifor [66] reported that the angular character of atomic bonding in metals and compounds is depicted through the Cauchy pressure, which is defined as CP=C12–C44. The negative (positive) values show the brittle (ductile) nature of the compound. Table 4 shows that the Cauchy pressure CP is 40.3 GPa at 0 GPa. According to above empirical laws, the Y2O3 behaves in a ductile manner under pressure. Furthermore, in this work, the effect of the hydrostatic pressure on the Cauchy pressure was also investigated, and the results are plotted in Fig 4. (b). Evidently, an increase in the hydrostatic pressure causes an increase in the Cauchy pressure, indicating an increase in the ductility of crystalline Y2O3. Elastic anisotropy or isotropy is an important physical property of materials, and has a vital role in technological and industrial applications. Ranganathan et al. [67] introduced the concept of the universal anisotropy index: for cubic Voigt and Reuss approximations, the anisotropy index can be simplified to AU = 5 (GV/GR – 1), and for an isotropic material, AU is equal to zero, and deviation of AU from zero indicates the presence of elastic anisotropy. The dependence of the calculated anisotropy factor AU on pressure for Y2O3 is shown Table 4. It is obvious that Y2O3 has low anisotropy at zero pressure, and the anisotropy of Y2O3 becomes stronger as the pressure increases. A three-dimensional (3D) surface representation of the elastic anisotropy of cubic Y2O3, expressed by the variation of Young’s modulus with crystal direction, is 14

further considered. The directional dependence of Young’s modulus can be described as [68],



1 1    S11  2 S11  S12  S 44  l12l22  l12l32  l22l32 E 2  



(26)

where l1 = sin θ cos φ, l2 = sin θ sin φ, and l3 = cos θ and represent the direction cosines with respect to the x, y and z directions of the lattice, and S11, S12, and S44 are the elastic compliance coefficients. The 3D directional dependence of an isotropic system exhibits a spherical shape, while the degree of deviation from spherical reflects the extent of anisotropy. The projected configurations of the directional Young’s moduli on the xy, yz, and xz planes revealed significant similarity, denoting that the strength of Y2O3 would not change with the crystallographic plane. As an example, the 3D representation and corresponding 2D projections of Young’s modulus for Y2O3 compounds are presented in Figs. 5a–5d at four different hydrostatic pressures (i.e., 0, 4, 8, and 14 GPa). It can be seen that the projected configuration of the directional Young’s modulus would vary from a square to quasi-square shape as the hydrostatic pressure rises from 0 to 14 GPa. This suggests that a Y2O3 single crystal tends to become slightly anisotropic at higher hydrostatic pressures. This result is consistent with that of the pressuredependent anisotropy index AU values listed in Table. 4. The angular character of atomic bonding is related to material characteristics such as brittleness or ductility. Ductility and brittleness are both crucial features in the manufacture of materials, and have a close relationship with B and G. According to the Pugh criterion [69], the critical threshold value for distinguishing the physical 15

properties of materials is around 1.75. If B/G > 1.75, a polycrystal possesses a ductile character, otherwise the material exhibits brittle behavior. The computed B/G ratio for Y2O3 was about 2.24, as listed in Table 4, indicating that crystalline Y2O3 is a ductile material. Poisson’s ratio is generally used to describe the stability of a crystal against shear and provides more information about the characteristics of the bonding forces than elastic constants [51]. The values of the Poisson ratio ν for covalent materials is small (ν = 0.1), whereas for ionic materials a typical value of ν is 0.25. In this study, the calculated poisson ratio ν is also given in Table 4, the value of ν for Y2O3 is about 0.306 with GGA-PBE method at P = 0 GPa and T = 0 K, so a considerable ionic contribution should be assumed for this compound. As shown in the Table 4, the result shows the Poisson ratio for cubic phase Y2O3 increases slightly with pressure, ranging from 0.306 to 0.366 under pressure between 0 and 14 GPa, The results indicates that the ionic contribution to interatomic bonding and the interatomic forces are mainly central forces for Y2O3. As a measure to resist penetration, deformation, abrasion, and wear, hardness plays an important role in industrial applications. Given to the above investigation, Y2O3 quite possibly possesses excellent hardness. Recently, many studies have shown that the elastic modulus G and B both influence Hv for many materials. In this work, we adopt the Tian’s formula to predict the hardness [35]: Hv = 0.92(G/B)1.137G0.708. The results of Vickers hardness Hv calculated above equation of Y2O3 under pressure are listed in Table 4. The calculated results are 7.705 GPa at P = 0 GPa. Obviously, 16

the Hv show a monotonic dencrease with pressure in the range of 0-14 GPa, indicating that the Hv of Y2O3 is significantly influenced by pressure. The Debye temperature, as an important fundamental parameter, is closely related to many physical properties of solids, such as elastic constants, melting temperature, and specific heat. Based on the Debye theory, the vibration of a solid can be considered as an elastic wave. The Debye temperature ΘD can be calculated from the elastic constant data, since ΘD is proportional to the averaged sound velocity VM, via [70], 13

h  3n  N ρ  ΘD    A  VM k  4π  M 

(27)

where h is Planck’s constant, k is Boltzmann’s constant, NA is Avogadro’s number, M is the molecular weight, ρ is the density, and n is the number of atoms per formula unit. In the present paper, for Y2O3, n = 5 and M = 225.8099. The averaged sound velocity can be computed from [71] VM

1 1 2    3  3  3  VL VS 

1 3

(28)

where VL and VS are the longitudinal and shear-wave velocities, respectively, VL  B  4 / 3G  /  1 2

(29)

VS  G /  

(30)

1

2

The velocity of sound for longitudinal and shear waves (VL and VS), Debye average velocity (VM), Debye temperature (ΘD) have been investigated using Eqs. (27)-(30), respectively, shown in Table 5. The sound velocities depend on elastic moduli through the bulk modulus (B) and shear modulus (G). Thus, the greater the 17

elastic modulus, the greater is the sound velocity. Moreover, we calculated the pressure dependence of mass density, melting temperature Tm listed in Table 5 and also the sound velocities versus pressure up to 14 GPa are depicted in Fig. 6 (a). It is observed that the sound velocities VS and VM, , and Debye temperature decrease monotonously as pressure increases, while longitudinal shear waves VL and mass density increases gradually with increasing pressure. It should be noted that our calculated Debye temperature at T = 0 K and P = 0 GPa is 556.113 K, which is less than the result of Ref. [34] (ΘD = 533.42 K). The values of the Debye temperature decrease with increasing pressure from 0 to 14 GPa in Table 5. Specifically, there is about 1.25%, 1.15%, 0.93%, 1.38%, 2.39%, 3.32%, and 3.76% decreases in the Debye temperature as hydrostatic pressure increase, indicating that hydrostatic pressure has less influence on the Debye temperature of Y2O3. The melting temperature Tm of cubic cystalline solids has been estimated using an empirical relation [72]: Tm = 553 + 5.91C11 ± 300 K. The melting temperature under pressure for Y2O3 is listed in Table 5. Quantitative analysis of Table 5 indicates that the melting temperature increase with increasing pressure. The single-crystal elastic wave velocities can be computed from the elastic constants by resolution of the Christoffel equation [73]: (cijklkjkk –ρɷ2)ul = 0, where cijkl is the elastic constant tensor element, ρ is the mass density, ω is the vibrational angular frequency, ul is the displacement amplitude, and kj is the wave vector component of the vibrational wave. The solutions of this equation are of two types: a longitudinal wave VLA with polarization parallel to the direction of propagation and 18

VTA1 and VTA2 donoted the shear-wave velocities polarized in the (110) and (001) planes along the propagation direction, respectively. The calculated elastic wave velocities along [100], [110], and [111] directions are depicted in Fig. 6b. It is found that Y2O3 has low anisotropy in both longitudinal and shear-wave velocities. Both the longitudinal velocities along the directions [100], [110], and [111] slowly increase with an increase in pressure, while the shear-wave velocities along the directions [1 00], [110], and [111] move slightly downward instead of upward. 3.3. Thermodynamics properties Study of thermodynamic properties of materials is of great importance in order to extend our knowledge about their specific behaviours when they are put under severe constraints such as high pressure and high temperature environment. In our work, the thermodynamic properties of cubic Y2O3 are determined in the temperature range 0–2000 K in GGA-PBE approximation, where the quasi-harmonic model remains fully valid. The effect of pressure is studied in a pressure range from 0 to 14 GPa. We began with a study of the isothermal bulk modulus. Fig. 7 presents in detail the variation of the isothermal bulk modulus with different temperature and different pressure. The variation of the bulk modulus with pressure at various temperatures is shown in Fig.7 (a). It is seen that the bulk modulus B rapidly increases almost linearly with pressure at different temperatures. The variation of bulk modulus with temperatures at P = 0, 2, 4, 6, 8, 10, 12, and 14 GPa is shown in Fig.7 (b). It is clear that when the temperatures is less than 100K, the isothermal bulk modulus nearly remains constant (≈157.9 GPa at zero temperature and zero pressure), but it 19

drops remarkably at temperatures higher than 100 K up to 2000 K. Correspondingly, when T < 100 K, the primitive cell volume of both compounds has a small change; when T > 100 K, the cell volume changes rapidly as T increases, and rapid volume variation makes the B rapidly decreases. The calculated zero-pressure bulk modulus of Y2O3 is 154.5 GPa and 142.8 GPa at T = 300 and 2100 K, respectively. An inspection of Figs. 7 (a) and 7 (b) reveals that the effect of increasing pressure on the isothermal bulk modulus is the same as that of decreasing temperature. It is obvious that the effect of pressure on the bulk modulus is more important than that of temperature. The heat capacity can be used to analyze the vibrational properties of solids, serving as a bridge between thermodynamics and microscopic structure. It provides an essential insight into vibrational properties, and has many applications. Fig. 8 illustrates the temperature dependence of the isochoric heat capacity CV and the isobaric heat capacity CP of Y2O3 at 0, 2, 4, 6, 8, 10, 12, and 14 GPa. A comparison of Fig. 8a with Fig. 8b reveals that CV and CP exhibit very similar behavior, with both increasing sharply with rising temperature and being proportional to T3 at low temperatures (T < 500 K). However, at high temperatures, the CV approaches a constant value, CP increases monotonously with increments of the temperature. The values follow the Debye model at low temperature (CV(T) ~T3) and the classical behavior (CV(T) ~ 3R for mono-atomic solids) is found at sufficient high temperatures, obeying Dulong and Petit’s Rule [74]. Our calculated values (≈ 123.74 Jmol−1 K−1) are close to the experimental and theoretical data obtained from various sources [75] 20

(≈ 120–140 Jmol−1 K−1) and the data given in the JANAF table [76] (≈116.74 Jmol−1 K−1). From Fig. 8, one can also see that the influences of the temperature on the heat capacity are much more significant than that of the pressure on it. The Grüneisen parameter γ is a very important physical parameter in relation to condensed matter. It can describe alterations in the vibration of a crystal lattice resulting from an increase or decrease in volume with temperature change. The study of γ is very important for thermodynamic properties (such as the temperature dependence of phonon frequencies), elasticity, and the non-syntony of matter. This parameter indicates the anharmonicity of atomic vibrations, and reveals the relationship between the expansion coefficient and other properties. In Fig. 9, we have plotted the Grüneisen parameter γ of Y2O3 at various pressure and temperature. It can be observed that the γ decreases dramatically with an increase in pressure at a given temperature. Meanwhile, at higher temperatures, the γ decreases more rapidly with the increasing pressure. As shown in Fig. 7(b), at higher temperatures (T > 500 K), γ increases monotonously with temperature at a given pressure. These results are due to the fact that the effect of temperature on the Grüneisen parameter γ is not as significant as that of pressure, and there will be a large thermal expansion at a low pressure. The indisputable fact is that the effect of increased pressure on the material is the same as decreased temperature of the material. The thermal expansion coefficient α is a significant thermodynamic parameter in theoretical and practical applications, it can be obtained from the temperature derivative of lattice constant. The variations of the thermal expansion a with 21

temperature at different pressures (P = 0, 2, 4, 6, 8, 10, 12, and 14 GPa) is presented in Fig. 10 for Y2O3 compound. In α–T graphs, it is noted that α increases exponentially (α T3) at lower temperatures (T < 600 K) and increases slowly (approaches a linear increases) at higher temperatures (T > 600 K), we have finally found that the thermal expansion coefficient α converges to a constant value at high temperatures. This is due to the inadequacy of the quasi-harmonic approximation at high temperatures and low pressures. The effects of pressure on the thermal expansion coefficient α are very small at low temperatures, the effects are increasingly obvious as the temperature increases. In this work, the thermal expansion coefficient has been calculated as 0.153×10−5(K−1) at T = 300 K at zero presssure and as 0.511×10−5 (K−1) at 2000 K, respectively. Our calculated value is very compatible with 0.203×10−5 (K−1) at T = 300 K which has been calculated by Chase [76]. 4. Conclusions In this work, the structural, elastic, and thermodynamic properties of cubic Y2O3 or α-Y2O3 under pressure have been predicted by first-principles calculations in combination with the quasi-harmonic Debye model. The value of the equilibrium lattice parameter of the ground state structure are found to be 10.560Å, 10.594Å, and 10.683 Å, using LDA, GGA-PBE, and GGA+U approximations, respectively, which is in good agreement with the experimental data (10.604 Å). The pressure dependence of a/a0 and V/V0 are also investigated, and both are found to decrease with increasing pressure. By fitting the third-order Birche–Munaghan EOS, the bulk modulus B0 and its pressure derivative B0′ are determined as 145.4 GPa, 165.2 GPa, 155.6 GPa and 22

5.010, 5.003, 4.849, respectively. The band gap of Y2O3 are calculated, and show that the GGA+U approximations is more accurate, with a value of 5.743 eV (experimental data, 5.5–6 eV). The predicted elastic constants and elastic modulus of Y2O3 do satisfy the mechanical stability criteria, implying that this cubic single crystal is a mechanically stable system at pressure less than 14 GPa. The compressibility coefficient K (6.333×10−3 GPa at 0 GPa), the ratio B/G (2.33 at 0 GPa), and Poisson’s ratios ν for Y2O3 are obtained under pressure, which indicates that Y2O3 is a potentially compressible and behaves in a ductile manner at pressures up to 14 GPa. Moreover, the Vickers hardness, the Debye temperature, the melting temperature, the isotropic wave velocities, and the sound velocities as well as the elastic anisotropy of Y2O3 are also investigated at various pressures. The Debye temperature decreases monotonically with the increase of pressure, and Y2O3 has low anisotropy for both longitudinal and shear-wave velocities. In addition, we carried out a study of the thermodynamic properties using the quasi-harmonic Debye model at pressures of 0–14 GPa and temperatures of 0–2000 K. The heat capacity (CV and CP), the thermal expansion coefficient α, and the Grüneisen parameter γ are systematically calculated. We predict that the thermal expansion coefficient α of Y2O3 is 0.153×10−5 K−1 at 0 GPa and ambient temperature. It can be found that the thermal expansion coefficient converges to a nearly constant value at high pressures and temperatures. The heat capacity CV is proportional to T3 at low temperatures, and it converges to the Dulong–Petit limit (≈ 123.74 Jmol−1 K−1) at high temperatures.We hope our theoretical study do some help to the experiment. 23

Acknowledgments This study is financially supported by the National Natural Science Foundation of China (Grants No.51372203, No.51332004 No. 51571166 and No.11304238)，We also thanks for the support for the computational resources by the High-performance Super Computing Center in Xidian University. References [1] J. Qi, J. F. Liu, Y. He, W. Chen, C. Wang, Compression behavior and phase transition of cubic In2O3 nanocrystals, J. Appl. Phy. 109 (2011) 063520-1-6. [2] T. de Boer, M. F. Bekheet, A. Gurlo, R. Riedel, A. Moewes, Band gap and electronic structure of cubic, rhombohedral, and orthorhombic In2O3 polymorphs: Experiment and theory, Phys. Rev. B. 93 (2016) 155205-1-7. [3] A. Gurlo, D. Dzivenko, P. Kroll, R. Riedel, High-pressure high-temperature synthesis of Rh2O3-II-type In2O3 polymorph, phys. stat. sol. 2 (2008) 269–271. [4] P. Agoston, K. Albe, Thermodynamic stability, stoichiometry, and electronic structure of bcc-In2O3 surfaces, Phys. Rev. B. 84 (2011) 045311-1-20. [5] B. G. Domene, J. A. Sans, F. J. Manjón, Sergey V. Ovsyannikov, L. S. Dubrovinsky, D. M. Garcia, O. Gomis, D. Errandonea, H. Moutaabbid, Y. L. Godec, H. M. Ortiz, A. Muñoz, P. R. Hernández, C. Popescu, Synthesis and High-Pressure Study of Corundum Type In2O3, J. Phys. Chem. C. 119 (2015) 29076−29087. [6] A. L. J. Pereira, L. Gracia, D. S. Perez, R. Vilaplana, F. J. Manjon, D. Errandonea, M. Nalin, A. Beltran, Structural and vibrational study of cubic Sb2O3 under high pressure, Phys. Rev. B. 85 (2012) 174108-1-11. [7] S. V. Ovsyannikov, E. Bykova, M. Bykov, M. D. Wenz, A. S. Pakhomova, K. Glazyrin, H. P. Liermann, L. Dubrovinsky, Structural and vibrational properties of single crystals of Scandia, Sc2O3 under high pressure, J. Appl. Phys. 118 (2015) 165901-1-8. [8] A. J. Pereira, D. Errandonea, A. B.Gracia, O. Gomis, J. A. Sans, B. G. Domene, A. 24

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Figures

Fig. 1. The structure of cubic Y2O3. (a). A conventional cell containing 80 atoms, (b) and (c). two different sites for Y atoms,donoted as Y1 and Y2, respectively. (d) the site of an O atom. The red spheres represent oxygen atoms and the gray spheres represent yttrium atoms.

-409.675

GGA BM EOS Fit

-409.700

Energy(Hartree)

-409.725 -409.750 -407.625

GGA+U BM EOS Fit

-407.640 -407.655 -407.670 -407.685 -407.700 -409.250 -409.275 -409.300 -409.325 -409.350 -409.375

LDA BM EOS Fit

560

570

580

590

600

610

620

630

640

650

660

3 Volume(bohr )

Fig. 2. Energy as a function of the primitive unit cell volume of cubic Y2O3 using the GGA-PBE, GGA+U, and LDA approximation.

31

1.00

a/a0

Structural parameter ratios

V/V0 0.98

0.96

0.94 0

2

4

6

8

10 12 14

Pressure (GPa)

Fig. 3. The normalized parameters a/a0 and V/V0 as functions of pressure.

320

(a) C11

C12

C44

280

(b) B G E Cauchy pressure

280 240

240

Modulus (GPa)

Elastic constants (GPa)

320

200

160

200 160 120

120

80

80

40 0

2

4

6

8 10 12 14

0

2

4

6

8 10 12 14

Pressure (GPa)

Pressure (GPa)

Fig. 4. (a) The elastic constant as a function of pressure of Y2O3 at T = 0 K. (b) Moduli versus pressure.

32

(a)

(b)

(c)

(d) 33

Fig. 5. Directional dependence of Young’s modulus for Y2O3 (GPa) and its projection onto the x–y, y–z, and x–z planes associated with four different hydrostatic pressures: (a) 0 GPa, (b) 4 GPa, (c) 8 GPa, and (ed) 14 GPa

(a)

(b)

VL

7

VS VM

6

5

4

VLA(110)

7

Sound velocity(kms-1)

Isotropic wave velocity (kms-1)

VLA(111)

8

8

VLA(100)

6

5

VTA(100) VTA2(110) 4

VTA(111) VTA1(110 )

3

3 0

2

4

6

2

8 10 12 14

0

Pressure (GPa)

2

4

6

8 10 12 14

Pressure (GPa)

Fig. 6. (a) Longitudinal wave velocity VL, shear-wave velocity VS, and average velocity VM versus pressure. (b) the calculated pressure dependences of the sound velocity in Y2O3.

34

(a)

240

0 K 100 K 300 K 500 K 1000 K 1500 K 2000 K

230 220

Bulk modulus (GPa)

(b)

210 200

14GPa 12GPa 10GPa 8GPa

190

6GPa

180

4GPa 2GPa

170

0GPa

160 150 140

0 2 4 6 8 10 12 14

0

500

Pressure (GPa)

1000 1500 2000

Temperature (K)

Fig. 7. Bulk modulus B of Y2O3 as a function of pressure and temperature. (a) Dulong--Petit limit

140

120

120

100

80 60 40

0 GPa 2 GPa 4 GPa 6 GPa 8 GPa 10 GPa 12 GPa 14 GPa

-1 -1

-1

-1

0 GPa 2 GPa 4 GPa 6 GPa 8 GPa 10 GPa 12 GPa 14 GPa

CP (Jmol K )

100

CV (Jmol K )

(b) Dulong--Petit limit

140

80 60 40 20

20

0

0 0

500

1000

1500

2000

Temperature (K)

0

500

1000

1500

2000

Temperature (K)

Fig. 8. Temperature and pressure dependence of the heat capacity for Y2O3.

35

(a)

(b)

2.68 2.64

Grüneisen parameter 

0 GPa 2 GPa 4 GPa 6 GPa 8 GPa 10 GPa 12 GPa 14 GPa

0 K 100 K 300 K 500 K 1000K 1500K 2000K

2.60 2.56 2.52 2.48 2.44 2.40 2.36 2.32 0

2

4

6

8 10 12 14

0

Pressure (GPa)

500

1000 1500 2000

Temperature (K)

Fig. 9. (a) Pressure and (b) temperature dependence of the Grüneisen parameter γ for cubic Y2O3.

0.6

Thermal expension (10-5K-1)

0.5

0.4

0 GPa 2 GPa 4 GPa 6 GPa 8 GPa 10 GPa 12 GPa 14 GPa

0.3

0.2

0.1

0.0 0

500

1000

1500

2000

Temperature (K)

Fig. 10. Temperature-dependent behavior of the expansion coefficient α for Y2O3 at P = 0, 2, 4, 6, 8, 10, 12, and 14 GPa.

36

Table Table 1. The equilibrium lattice constant (a0), equilibrium volume (V0), bulk modulus (B0), the pressure derivative of bulk modulus (B0′ ), compressibility coefficient K (10-3 GPa) for Y2O3 obtained in our work compared with other studies.

This work

Other studies

Method

a0 (Å)

V0 (Å)

B0 (GPa)

B0 ′

K (GPa)

GGA-PBE GGA+U LDA Potential model Ab-initio GGA-PBE HSE06 GGA-PBE GGA-WC LDA Brillouin spectroscopy

10.594 10.683 10.560 10.61 a 10.63 a 10.71 b 10.60 b 10.630 c 10.658 c 10.527 c 10.392 d

594.43 609.65 588.73 597.2 a 600.0 a 612.0 b 601.2 b

165.2 155.6 145.4 138 a 170 a 153.86 b 150.64 b 136.85 c 140.81 c 140.36 c 180 -183 d

5.003 4.849 5.010

6.05 6.43 6.88

4.4230 c 4.7526 c 4.5554 c 4.01d ,7.65 d 6.92 d

7.12 c 7.1 c 7.3 c 5.494 d

10.6018 e 10.604 f 10.603 g

14.904 e

Experiment

149.5±1.0 e 144 h , 146.2 i 148.9 j

6.6 e 6.9 h , 6.8 i 6.7 j

a: Ref.[30]; b: Ref.[31]; c: Ref.[34]; d: Ref.[42]; e: Ref.[26]; f: Ref.[55]; g: Ref.[56]; h: Ref.[57]; i: Ref.[58]; j: Ref.[59];

Table 2. Calculated energy band gap for cubic Y2O3 using various approximations. Present work

Band gap/eV

Other calculations

GGA-PBE

GGA+U

LDA

GGA-PBE

mBJ-GGA

LDA

LMTO

HSE06

Experiment

4.296

5.748

4.13

4.3 a

5.7 b

4.5 b

4.5 d

6.0 e

5.5 f , 6.0 g

4.8 b

4.5 c

a: Ref.[27]; b: Ref.[30]; c: Ref.[40]; d: Ref.[60]; e: Ref.[31]; f: Ref.[20]; g: Ref.[61]; h: Ref.[62];

37

5.8 h

Table 3. Values of the elastic constant Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), and Possion’s ratio ν for various approximations for cubic Y2O3. This work

GGA-PBE GGA+U LDA Theoretical [30] Theoretical [31] Experiment [26]

C11

C44

C12

B

GV

GR

GH

E

ν

243.3 233.1 212.3 206 a 233 b 219.3 223.7 ±0.6

85.9 79.5 88.0 59 a 72 b 72.79 74.6 ±0.7

126.2 116.8 115.9 103 a 139 b 120.68 112.4 ±1.1

165.2 155.6 148.0

74.9 70.9 72.1

72.4 69.3 66.2

73.7 70.2 69.2

192.4 182.8 179.4

0.306 0.304 0.298

62.43 66.3 ±0.8

165 173 ±2

0.32 0.307 ±0.003

149.5

GV, GR, and GH, represent the Voigt, Reuss, and average polycrystalline shear moduli, respectively. a. Potential model and b.ab initio.

Table 4. Values of the elastic constant Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), B/G, Young’s modulus E (GPa), universal elastic anisotropy index AU, Possion’s ratio ν, Cauchy pressure CP (GPa), compressibility coefficient K (10–3 GPa–1), and Vickers hardness Hv (GPa), for Y2O3 with GGA-PBE approroximation P (GPa) C11 (GPa) C44 (GPa) C12 (GPa) B (GPa) G (GPa) GV (GPa) GR (GPa) B/G E (GPa) U A

ν CP (GPa) Hv (GPa) K (10–3GPa–1)

0

2

4

6

8

10

12

14

243.3 85.9 126.2 165.2 73.7 74.9 72.4 2.24 192.4 0.179 0.306 40.3 7.705 6.052

249.1 84.6 137.0 174.4 71.7 73.2 70.3 2.43 189.2 0.206 0.319 52.5 6.898 5.734

258.7 85.5 147.0 184.2 72.0 73.6 70.5 2.56 191.2 0.221 0.327 61.6 6.536 5.428

269.2 85.6 156.2 193.8 72.5 74.0 71.0 2.67 193.4 0.210 0.334 70.5 6.24 5.159

278.3 86.4 169.1 205.5 71.9 73.7 70.1 2.86 193.1 0.258 0.343 82.7 5.748 4.866

285.7 85.0 179.2 214.7 70.5 72.3 68.6 3.05 190.6 0.268 0.352 94.2 5.272 4.657

292.2 84.4 189.4 223.6 69.2 71.2 67.2 3.23 188.2 0.300 0.36 105.0 4.866 4.471

298.2 83.5 198.0 231.4 68.0 70.1 65.9 3.40 185.8 0.320 0.366 114.5 4.536 4.321

38

Table 5. Calculated density ρ, shear-wave velocity VS, longitudinal velocity VL, average sound velocity VM, Debye temperature ΘD, and melting temperature Tm of Y2O3. –3

P (GPa)

ρ (g·cm )

VL (m/s)

VS (m/s)

VM (m/s)

ΘD (K)

Tm (K)

0 2 4 6 8 10 12 14

5.033 5.087 5.139 5.188 5.235 5.281 5.325 5.369

7.235 7.285 7.385 7.483 7.587 7.645 7.702 7.746

3.825 3.755 3.744 3.738 3.705 3.653 3.605 3.559

4.276 4.204 4.197 4.193 4.162 4.108 4.058 4.011

556.113 549.122 549.700 550.943 548.420 542.811 537.632 535.212

1530±300 1584±300 1642±300 1699±300 1768±300 1822±300 1875±300 1921±300

39