The structural phase transition and elastic properties of zirconia under high pressure from first-principles calculations

Solid State Sciences 13 (2011) 938e943

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The structural phase transition and elastic properties of zirconia under high pressure from first-principles calculations HaiSheng Ren a, Bo Zhu b, Jun Zhu a, b, **, YanJun Hao b, *, BaiRu Yu b, YanHong Li b a b

Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China College of Physical Science and Technology, Sichuan University, Chengdu 610064, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 September 2010 Received in revised form 22 December 2010 Accepted 26 February 2011 Available online 5 March 2011

The structural phase transition and elastic properties of monoclinic, orthoI and orthoII zirconium dioxide (ZrO2) are investigated by using pseudopotential plane-wave methods within the PerdeweBurke eErnzerhof (PBE) form of generalized gradient approximation (GGA). Our calculated equilibrium structural parameters of ZrO2 are in good agreement with the available experimental data. On the basis of enthalpy versus pressure data obtained from our theoretical calculations for high pressure, we find that phase transition pressure from monoclinic to orthoI and orthoI to orthoII are ca. 7.94 GPa and 11.58 GPa, respectively, which are in good agreement with the experimental observations. Especially, the elastic properties of orthoII ZrO2 under high pressure are studied for the first time. We note that the elastic constants, bulk moduli, shear moduli, compressional and shear wave velocities as well as Debye temperature of orthoII ZrO2 increase monotonically with increasing pressure. By analyzing G/B, the brittle-ductile behavior of ZrO2 is assessed. In addition, polycrystalline elastic properties are also obtained successfully for a complete description of elastic properties. Crown Copyright Ó 2011 Published by Elsevier Masson SAS. All rights reserved.

Keywords: Phase transition Elastic properties First-principles study ZrO2

1. Introduction As one of the groups IV-B oxides, ZrO2 is a highly attractive ceramic material because of its great potential for development of solid oxide fuel cell electrolytes and microelectronic gate dielectrics [1e3]. The compound exhibits many outstanding properties, such as high dielectric constants, corrosion-resistant and high refractive, which can be used as gate-dielectric materials in metal-oxidesemiconductor (MOS) devices, in metallurgy and as a thermal barrier coating in engines [4]. In addition, it has been reported to have potential superhard high-pressure phases and therefore hold potential as refractory materials [5,6]. Therefore, it is of great interest to physicists in both experimental [7e16] and theoretical investigations [17e20]. At ambient condition, zirconia stabilizes in the monoclinic phase (space group: P21/c, No. 14). At ambient pressure and temperature above 1400 K, the monoclinic phase transforms to a tetragonal phase (space group: P42/nmc, No. 137), and at

* Corresponding author. College of Physical Science and Technology, Sichuan University, Chengdu 610064, China. Tel.: þ86 02885990273; fax: þ86 02885412322. ** Corresponding author. Tel.: þ86 02885418566; fax: þ86 02885412323. E-mail addresses: [email protected] (J. Zhu), [email protected] (YanJun Hao).

a temperature of ca. 2570 K the tetragonal phase transforms into a cubic fluorite structure (space group: Fm3m, No. 255), while at ambient temperature and under high pressures the several orthorhombic phases are also observed. The first orthorhombic phase (space group: Pbca, No. 61, labeled as orthoI) exits in the range of 3.0e11.0 GPa [7e13] and the another orthorhombic phase (space group: Pnma, No. 62, labeled as orthoII) is in the range of 9.0e25.0 GPa [8e10,14]. Recently, Leger et al. [8] used angulardispersive x-ray in situ powder diffraction experiment found the third orthorhombic structure (labeled as orthoIII) when pressure increased above 42 GPa, but the correct space group of this phase in ZrO2 has not yet been determined. Very recently, Desgreniers et al. [15] observed the orthoII is stable up to roughly 70 GPa at room temperature using diamond anvil cell (DAC) techniques. In a subsequence report, Ohtaka et al. [16] using both the sample annealing and in situ laser-heating techniques found orthoII can be stable to a pressure of 100 GPa and a temperature of 2500 K and no post-orthoII phase was observed. Theoretically, Dewhurst et al. [17] predicted the transition monoclinic to orthoI and orthoI to orthoII at 4.10 GPa and 19.00 GPa, respectively, based on the local density approximation (LDA) functional of Ceperley and Alder, as parameterized by Perdew and Zunger. Terki et al. [18] employed the full-potential linearaugmented-plane wave (FP-LAPW) method to study the structural phase transformation of ZrO2 under pressure and discovered that

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HaiSheng Ren et al. / Solid State Sciences 13 (2011) 938e943

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the monoclinic to orthoI and orthoI to orthoII phase transition pressures are at 7.92 GPa and 12.15 GPa, respectively. John et al. [19] found the transition pressure from monoclinic to orthoI and orthoI to orthoII are at 6.64 Gpa and 9.20 GPa by using density functional theory (DFT) within the generalized gradient approximation (GGA). In another theoretical work of Ozturk et al. [20] also predicted the transition sequence monoclinic / orthoI / orthoII by using the pseudopotential plane-wave methods within density function theory and generalized gradient approximation. They found that monoclinic to orthoI and orthoI to orthoII phase transition pressures are 6.32 GPa and 11.26 GPa, respectively. From above experimental and theoretical work, we can see that the phase transition pressures given by experiments have a larger range and are disagreement between different theoretical calculations. On the other hand, elastic properties of a solid are important because they relate to various fundamental solid-state phenomena such as inter-atomic bonding, equations of state, and phonon spectra. Elastic properties are also linked thermodynamically with specific heat, thermal expansion, Debye temperature, and Grüneisen parameter. Most importantly, knowledge of elastic constants is essential for many practical applications related to the mechanical properties of a solid: load deflection, thermoelastic stress, internal strain, sound velocities, and fracture toughness [21,22]. To data, only a few theoretical methods have been applied to calculate the elastic constants of ZrO2 at zero temperature and zero pressure, such as Lewis and Catlow (LC), TroulliereMartins (TM), projector augmented wave (PAW) method [23] and Lattice Dynamics (LD) method [24]. However, there are no investigations on the elastic constants of orthoII ZrO2 under high pressure. Therefore, in the present work, we predict the structure phase transition and elastic properties of ZrO2 at high pressures using the plane-wave pseudopotential density functional theory (DFT) method. This paper is organized as follows: In Section 2, we make a brief review of the theoretical method. The calculated results with some discussions are presented in Section 3 and compared with the previous experimental and theoretical results. Conclusions are summarized in Section 4.

the maximum ionic displacement is less than 0.001 Å; and the maximum stress is less than 0.05 GPa. These parameters are carefully tested. And we can find that they are sufficient in leading to well converged total energy and elastic constants calculations.

2. Theoretical methods

3.1. Pressure-induced structural phase transition

2.1. Total energy electronic structure calculations

In order to obtain the total energy E and the corresponding volume V of ZrO2, a series of pressures from 5 GPa to 15 GPa interval 1 GPa are set for geometry optimization, in which atomic coordinates and cell parameters (angles and axes lengths) are allowed to relax at each pressure. The calculated energyevolume (EeV) points are fitted to the third-order BircheMurnaghan equation of state [31]. The obtained lattice parameters a, b, c and V0, the bulk modulus B0, and the first order pressure derivatives of bulk modulus B’0 at 0 K are listed in Table 1, along with the available experimental data [8,32e35]. Obviously, the agreement among them is good. As it is well known, the thermodynamic stable phase at some given pressure and temperature is the one with the lowest Gibbs free energy G

Total energy electronic structure calculations are performed based on the density functional theory (DFT), within the generalized gradient approximation (GGA) with the functional of Perdew, Burke, and Ernzerhof (PBE) [25] for the exchange-correlation potential, as implemented through the Cambridge Serial Total Energy Package (CASTEP) code [26]. In pseudopotential methods, the effect of core electrons and nuclei is replaced by an effective ionic potential, and only the valence electrons, which are directly involved in chemical bonding, are considered. The valence electrons for Zr are in the 4s24p64d25s2 and O are in the 2s22p4 configurations. In order to obtain accurate results, the cut-off energy of plane-wave basis set is 450 eV for monoclinic and orthoI, 500 eV for orthoII, which are tested to be fully converged with respect to total energy for many different volumes. The special points sampling integration over the Brillouin zone are carried out using the Monkhorst-Pack method with a 7  7  7, 5  7  7 and 8  10  7 special k-point mesh for monoclinic, orthoI and orthoII, respectively. The self-consistent is considered to be converged when the total energy is less than 1  106 eV/atom. The BrodydenFletcher-Goldfarb-Shanno (BFGS) minimization scheme [27] is used in geometry optimization. The tolerances are set as the difference in total energy being less than 1  105 eV/atom; the maximum ionic HellmanneFeynman force is less than 0.03 eV/Å;

2.2. Elastic constants In order to calculate the elastic constants under hydrostatic pressure, the strains to be non-volume conserving are used, which are appropriate for the calculation of the elastic wave velocities. The elastic constants Cijkl with respect to the finite strain variables is defined as [28,29]


Cij ¼

 vsij ðXÞ v[kl x

(1)

where sij and ekl are the applied stress and Eulerian strain tensors and X and x are the coordinates before and after the deformation. For the isotropic stress, we have [29,30]

Cijkl ¼ cijkl þ

cijkl ¼

 P 2dij dkl  dil djk  dik djl 2

1 v2 EðxÞ VðxÞ v[ij v[kl

(2)

! (3) X

where Cijkl denotes the second-order derivatives with respect to the infinitesimal strain (Eulerian). The fourth-rank tensor C has generally 21 independent components. However, this number is greatly reduced when taking into account the symmetry of the crystal. For a monoclinic crystal, they are reduced to thirteen independent components, i.e. C11, C22, C33, C44, C55, C66, C12, C13, C15, C23, C25, C35 and C46. For an orthorhombic crystal, they are reduced to nine independent components, i.e. C11, C22, C33, C44, C55, C66, C12, C13 and C23. 3. Results and discussion

G ¼ E þ PV  TS

(4)

where E, S, P, and V are the internal energy, vibrational entropy, pressure and volume, respectively. Our theoretical calculations are performed at T ¼ 0 K, therefore, the Gibbs free energy G is equivalent to the enthalpy, H ¼ E þ PV, where P ¼ ӘE/ӘV. In general, the equilibrium phase transition pressure is determined by calculating the total energy curves of the two random phases and finding the common tangent, which is difficult to calculate accurately. Therefore, we calculate the enthalpy of ZrO2 corresponding to the monoclinic, orthoI and orthoII structures. Due to the tiny difference

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HaiSheng Ren et al. / Solid State Sciences 13 (2011) 938e943

Table 1 Calculated and experimental lattice constants (Å), Angles b (◦), primitive cell volume three low temperature ZrO2 polymorph (Å3), bulk modulus (GPa), and its first pressure derivative of bulk modulus of ZrO2 at T ¼ 0 K and P ¼ 0 GPa (In bracket, we list zero temperature lattice parameters of orthoI and orthoII phases at P ¼ 6 GPa and P ¼ 20 GPa, respectively). a

b

c

b

V0

B0

B’0

Monoclinic

Present work Exp.

5.175 5.150a

5.251 5.210a

5.352 5.310a

99.49 99.23a

35.87 35.10a

168 95e189b

3.66 4e5b

OrthoI

Present work Exp.

10.137 (10.043) 10.086c

5.294 (5.255) 5.262c

5.133 (5.075) 5.091c

34.43 (33.44) 33.77c

203 128181c

4.29 4.2e5d

OrthoII

Present work Exp.

5.596 (5.487) 5.587e

3.370 (3.261) 3.329e

6.537 (6.391) 6.485e

30.85 (28.57) 30.12e

238 332e

4.78

Structure

a b c d e

Ref. Ref. The Ref. The

[32]. [8]. volume was estimated at T ¼ 873 K and P ¼ 6 GPa (Ref. [33]). [34]. volume was estimated at T ¼ 1023 K and P ¼ 20 GPa (Ref. [35]).

in enthalpy between two random structures, we plot the enthalpy difference as a function of pressure as shown in Fig. 1. From this figure, we can see that the phase transitions of monoclinic to orthoI and orthoI to orthoII structure occur at ca. 7.94 GPa and 11.58 GPa, respectively, which are in good agreement with the available experimental data [7e14] and theoretical calculation of Terki et al. [18].

3.2. Elastic properties In Table 2, we list the calculated elastic constants of ZrO2 at T ¼ 0 K and P ¼ 0 GPa, together with experimental values [36] and the previous theoretical calculations [23,24]. Obviously, our calculated results are in good agreement with other theoretical results [23,24], except for those of Giuseppe et al. [23] using the LC method, in which ionic coordinates were fixed at the experiment values. It can be found that both our calculated values and other theoretical results [23,24] on individual elastic constants in monoclinic ZrO2 are in somewhat disagreements with experimental data [36]. This might be due to monoclinic ZrO2 is prone to twinning, and is therefore difficult to grow large homogeneous crystals required for elasticity measurements as discussed by Léger et al. [8] and Chan et al. [36], but we also note that part of the discrepancy is probably due to the exchange-correlation functional

employed in the calculations. Since individual elastic constants differences are hard to quantify for Zirconia studied. Instead of considering individual elastic constants, we compute the aggregate moduli in the ReusseVoigt bounds [37] of monoclinic ZrO2, using the formulas given in Ref. [37]. The obtained bulk and shear moduli are B ¼ 179 GPa and G ¼ 87 GPa, respectively, which are in good agreement with experimental data [36] B ¼ 187 GPa and G ¼ 93 GPa. The calculated pressure dependences of the elastic constant of orthoII ZrO2 at zero temperature are shown in Fig. 2. It is found that the nine elastic constants increase monotonously with the applied pressure. C11, C22 and C33 increase quickly with the increasing pressure, and C44 has a moderate increase as well as C13, C12, C23, C55 and C66. For monoclinic and orthorhombic structure, the mechanical stability criteria can be obtained using the formulas given in Ref. [38]. The calculated theoretical Cij values satisfy these conditions for any pressure p, ensuring the mechanical stability of ZrO2 under pressure. The isotropic aggregate bulk modulus B and shear modulus G can be obtained from elastic constants according to the VoigteReusse Hill (VRH) average scheme [37]. For monoclinic and orthorhombic phase, the bulk modulus BV and BR and shear modulus GV and GR can be also obtained using the expressions given in Ref. [38] (The subscript V denotes the Voigt bound, R denotes the Reuss bound). The arithmetic average of Voigt and Reuss bounds is termed as the VoigteReusseHill approximations [39]

Enthalpy difference (meV/atom)

G ¼

monoclinic

0.5

B ¼

1 ðB þ BV Þ 2 R

(5)

Thus, the isotropically averaged aggregate velocities for compressional (yp) and shear waves (ys) are written as

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 ffi 4 yp ¼ Bþ G p 3

orthoII

0.0

1 ðG þ GV Þ 2 R

ys ¼

pffiffiffiffiffiffiffiffiffi G=p

(6)

11.58 GPa -0.5

orthoI

-1.0

7.94GPa -1.5

6

7

8

9

10

11

12

13

Pressure (GPa) Fig. 1. Enthalpy as a function of pressure for ZrO2 in the monoclinic, orthoI and orthoII structures at T ¼ 0 K.

with r the density. In Figs. 3 and 4, we present the pressure dependence of bulk modulus, shear modulus, and velocities for compressional (yp) and shear waves (ys). Obviously, the bulk modulus, shear modulus, and velocities for compressional (yp) and shear waves (ys) show a linear increase with increasing pressure. In order to predict the brittle and ductile behavior of solids, Pugh [40] introduced a simple relationship that the ratio of bulk to shear modulus (B/G) is associated with ductile or brittle characters of material. A high B/G value corresponds to ductility, whereas a low ratio is associated with brittleness. Pugh [40] gave a critical value for ductileebrittle transition. If B/G < 1.75, the material behaves in a brittle manner, otherwise, in a ductile manner. For orthoII ZrO2 at 0 K and 0 GPa, our calculated value of B/G is 2.55, which shows that

HaiSheng Ren et al. / Solid State Sciences 13 (2011) 938e943

941

Table 2 The elastic constants Cij in GPa of ZrO2 at T ¼ 0 K and P ¼ 0 GPa, together with the experimental data and other theoretical results. Parameter

C11

C22

C33

C44

C55

C66

C12

C13

C15

C23

C25

C35

C46

Monoclinic

This work LCa TMa PAWa LDb Expc

324 389 337 337 347 361

364 426 354 351 364 408

253 355 267 268 274 258

80 113 77 79 88 100

70 106 70 70 108 81

114 132 113 114 122 126

153 233 157 155 164 142

71 154 89 84 102 55

30 39 26 26 28 21

145 145 156 153 156 196

3.36 23.40 4.32 4.28 17.00 31.20

4.36 13.60 0.69 1.91 11.00 18.20

15.3 18.6 15.2 14.6 44.0 22.7

OrthoI

This work LCa TMa PAWa

392 417 349 349

375 484 393 397

334 424 355 352

84 130 86 87

85 125 84 84

111 156 116 115

160 222 152 150

134 188 124 125

110 164 121 120

OrthoII

This work LCa TMa PAWa LDb

422 578 426 422 463

299 340 293 293 400

351 223 335 327 429

46 85 57 52 31

74 45 71 69 113

118 114 118 117 126

142 146 147 145 165

173 154 181 178 193

128 51 118 114 149

a b c

From Lewis and Catlow (LC), Troullier-Martins (TM), and projector augmented wave (PAW) method (Ref. [23]). From Lattice Dynamics (LD) method (Ref. [22]). Ref. [36].

this material is ductile. This conclusion is in accordance with that of Caravaca et al. [41]. With increasing pressure from 15 GPa to 40 GPa, the B/G ratio changes in the vicinity of 2.52, i.e. orthoII ZrO2 maintains ductile properties. On the other hand, Frantsevich [42] proposed distinguishing brittleness and ductility by Poisson’s ratio

s¼

3B  2G 2ð3B þ GÞ

(7)

According to Frantsevich, the critical value of Poisson’s ratio of a material is 1/3. For ductility materials such as ceramics, the Poisson’s ratio is more than 1/3. Therefore, the obtained Poisson’s ratio for orthoII ZrO2 (Table 3) also indicates the ductility character of the materials in the same order as B/G. It is well known that anisotropy of elasticity is an important implication in engineering science and in crystal physics. So we turn to discuss the elastic anisotropy of orthoII ZrO2 at different pressures. A measure of the degree of anisotropy can be provided by the shear anisotropic factors in the bonding between atoms in different planes. The shear anisotropic factor for the {100} shear planes between the <011> and <010 > directions is [43]

A1 ¼

4c44 c11 þ c33  2c13

(8)

For the {010} shear planes between the <101> and <001> directions is

A2 ¼

4c55 c22 þ c33  2c23

(9)

For the {001} shear planes between the <110> and <010> directions is

A3 ¼

4c66 c11 þ c22  2c12

(10)

For isotropic crystals, the factors A1, A2, and A3 must be one, while any value smaller or greater than one is a measure of the degree of elastic anisotropy possessed by the crystal. The anisotropic factors A1, A2, and A3 obtained from our theoretical calculations at different pressures are presented in Fig. 5. It can be found that A1 is close to one-half of A2 and A3 with pressures, which corresponds to strains related to {100} shear plane where the

450 700

400 600

C22

500

C33 C44

400

C55 C66

300

C12

350

Modulus(GPa)

Elastic constants (GPa)

C11

300

B G

250 200

C13

200

C23

100

150 100

15

18

21

24

27

30

33

36

39

42

15

18

21

24

27

30

33

36

39

42

Pressure(GPa)

Pressure (GPa) Fig. 2. Calculated pressure dependence of Cij of orthoII ZrO2 at T ¼ 0 K.

Fig. 3. Pressure dependence of bulk modulus and shear modulus in orthoII ZrO2 at T ¼ 0 K

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HaiSheng Ren et al. / Solid State Sciences 13 (2011) 938e943

10

1.5

ABc

1.4

9

1.3

Anisotropic factors

Velocity (Km/s)

8 7

Vs Vp

6 5

ABb A3

1.2 1.1

A2

1.0 0.9 0.8

A1

0.7

4

0.6 3

15

20

25

30

35

15

40

18

21

Fig. 4. Predicted compressional and shear wave velocities of the orthoII ZrO2 as a function of pressure at T ¼ 0 K.

elastic constants are weak. Furthermore, for orthorhombic crystals, the shear anisotropic factors are not sufficient to describe the elastic anisotropy, so we go on investigating the anisotropy of the linear bulk modulus. The elastic anisotropy arises from the anisotropy of the linear bulk modulus in addition to the shear anisotropy. The anisotropy of bulk modulus along the a, b and c axes, i.e, Ba, Bb and Bc can be expressed as [43]

dP L Ba ¼ a ¼ da 1þaþb

(11)

dP Ba ¼ a db

(12)

dP Ba ¼ b db

(13)

Bc ¼ c

27

30

33

36

39

42

Fig. 5. Anisotropic factors of orthoII ZrO2 at T ¼ 0 K as a function of pressure.

is the most compressible and the c-axis the least compressible at any pressure. The anisotropy of the bulk modulus along the a- and c-axes with respect to the b-axes can be expressed as

Ba Bb Bc ¼ Bb

ABb ¼ ABc

(17)

when a value of one indicates elastic isotropy, while any departure from one represents elastic anisotropy. The anisotropies of the linear bulk modulus ABb and ABc with pressure are also presented in Fig. 5. we can see that ABb and ABc decrease by 21.2% and 11.1%, as the applied pressure changes from 15 GPa to 40 GPa. In addition, the percentage elastic anisotropies for bulk modulus AB and shear modulus AG in polycrystalline materials are defined as

BV  BR BV þ BR G  GR ¼ V GV þ GR

AB ¼

where L, a and b are defined as 2

L ¼ c11 þ 2c12 a þ c22 a2 þ 2c13 b þ c33 b þ 2c23 ab

(14)

a ¼

ðc11  c12 Þðc33  c13 Þ  ðc23  c13 Þðc11  c13 Þ ðc33  c13 Þðc22  c12 Þ  ðc13  c23 Þðc12  c23 Þ

(15)

b¼

ðc22  c12 Þðc11  c13 Þ  ðc11  c12 Þðc23  c12 Þ ðc22  c12 Þðc33  c13 Þ  ðc12  c23 Þðc13  c23 Þ

(16)

The obtained Ba, Bb and Bc of ZrO2 as a function of pressure are shown in Fig. 6. Obviously, Ba, Bb and Bc increase with the increasing pressure. It is noted that the bulk modulus along the c-axis is the largest and along the b-axis is the smallest, implying that the b-axis Table 3 Bulk modulus B (GPa), shear modulus G (GPa), Poisson’s ratio s, anisotropic factors ABb, ABc, AB, AG and Debye temperature Q (K) of orthoII ZrO2 at different pressure. P

B

G

s

ABb

ABc

AB

AG

Q

0 15 20 25 30 35 40

214 299 324 346 367 387 406

84 118 131 138 147 154 161

0.3264 0.3257 0.3218 0.3239 0.3233 0.3243 0.3287

2.2348 1.3652 1.2296 1.1504 1.0937 1.0751 1.0337

1.4195 1.5107 1.4560 1.4236 1.3854 1.3426 1.3202

0.01679 0.00442 0.00305 0.00259 0.00229 0.00192 0.00183

0.06172 0.02337 0.01397 0.01288 0.01186 0.01044 0.01023

546.03 638.15 672.99 689.22 709.61 724.79 739.59

AG

Linear bulk modulus Ba, Bb, and Bc (GPa)

Bb ¼ b

24

Pressure (GPa)

Pressure (GPa)

(18)

1500 1400 1300 1200 1100 1000 900

Ba Bb

800

Bc

700 15

18

21

24

27

30

33

36

39

Pressure (GPa) Fig. 6. Linear bulk modulus Ba, Bb, and Bc of orthoII ZrO2 at T ¼ 0 K as a function of pressure.

HaiSheng Ren et al. / Solid State Sciences 13 (2011) 938e943

943

where B and G denote the bulk and shear modulus, and the subscripts V and R represent the Voigt and Reuss approximations. A value AB ¼ 0 (BR ¼ BV) is associated to isotropic elastic constants, while AB ¼ 100% is associated to the largest possible anisotropy. It can be noted from the Table 3 that the bulk modulus anisotropy AB and shear modulus anisotropy AG decrease by 68.7% and 26.8%, respectively, indicating that orthoII ZrO2 is less anisotropic in compressibility than in shear with the increasing pressure.

properties, Poisson’s ratio, elastic anisotropy and Debye temperature have been also evaluated.

3.3. Debye temperature

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As is known to all, the Debye temperature is an important physical quantity and closely related to the elastic constants, specific heat, thermal coefficient, and melting temperature. At low temperatures, the vibrational excitations arise solely from acoustic vibrations, and the Debye temperature Q calculated from elastic constants should be the same as that determined from specific heat measurements. From the elastic constants, it can be obtained the elastic Debye temperature (Q), since the Debye temperature may be estimated from the averaged sound velocity ym by the following equation [44]

Q ¼

 
h 3n NA r 1=3 ym k 4p M

(19)

where h is the Planck’s constant, k the Boltzmann’s constant, NA the Avogadro’s number, n the number of atoms per formula unit, M the molecular mass per formula unit, r (r ¼ M/V) the density, and ym is obtained from

ym

!#1=3 " 1 2 1 ¼ þ 3 y3s y3p

(20)

where ys and yp are compressional and shear wave velocity, respectively, which can be obtained from Eq. (6). The computed Debye temperature of monoclinic ZrO2 is 567 K, which is in reasonable agreement with low-temperature specific-heat measurements which yields a Debye temperature of 575 K [45]. The results for orthoII ZrO2 at different pressure P are also summarized in Table 3. 4. Conclusions In summary, we have performed first-principles calculations to investigate the structural phase transitions and elastic properties of ZrO2 at T ¼ 0 K. By analyzing the enthalpy-pressure curves for monoclinic, orthoI and orthoII structure, we find transition pressures from monoclinic to orthoI and orthoI to orthoII are ca. 7.94 GPa and 11.58 GPa, which are in reasonable with the available experimental data and the theoretical calculation of Terli et al. The elastic constants of orthoII under high pressures are also predicted. From the elastic constants, the bulk modulus, shear modulus, compressional and shear wave velocities are obtained successfully. We find the bulk moduli, shear moduli, compressional and shear wave velocities increase monotonically with increasing pressure. By analyzing G/B, we find that orthoII ZrO2 maintains ductile properties under pressures. For a complete description of elastic

Acknowledgements We acknowledge the support for this work by the Fundamental Research Funds for the Central Universities (2009SCU11124). References