Theoretical calculations of the high-pressure phases of SnO2

Computational Materials Science 72 (2013) 86–92

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Theoretical calculations of the high-pressure phases of SnO2 F. El Haj Hassan a,b,⇑, S. Moussawi a, W. Noun a, C. Salameh a, A.V. Postnikov c a

Université Libanaise, Faculté des sciences (I), Laboratoire de Physique et d’Electronique (LPE), Elhadath, Beirut, Lebanon Condensed Matter Section, the Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34014 Trieste, Italy c Laboratoire de Physique des Milieux Denses, Université de Lorraine, 1 Bd. Arago, 57078 Metz, France b

a r t i c l e

i n f o

Article history: Received 21 December 2012 Received in revised form 4 February 2013 Accepted 7 February 2013 Available online 8 March 2013 Keywords: FP-LAPW Phase transition Band structure Thermal expansion Debye temperature Heat capacity Grüneisen parameter

a b s t r a c t Total-energy calculations of tin dioxide, done at seven different phases under conditions of hydrostatic pressure in the rutile, CaCl2-type, a-PbO2, pyrite, ZrO2 orthorhombic, fluorite, and cotunnite structures using first principle full potential-linearized augmented plane wave (FP-LAPW) plus local orbitals method within the density functional theory (DFT). The structural properties at equilibrium as at high pressure are investigated by using generalized gradient approximations (GGAs) that are based on the optimization of total energy. For band structure calculations, both GGA and modified Becke–Johnson (mBJ) of the exchange–correlation energy and potential, respectively, are used. Pressure–temperature dependent thermodynamic properties including the bulk modulus, thermal expansion, Debye temperature, heat capacity and Grüneisen parameter, are calculated using model based on the quasi-harmonic approximation (QHA). Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction One of the interesting phenomena that may occur under applied pressure is a sudden change in the arrangement of the atoms, i.e., a structural phase transition. The Gibbs free energies of the different possible arrangements of atoms vary under compression, and at some stage it becomes favorable for the material to change the type of atomic arrangement. A phase transition is said to have occurred with a change in crystal symmetry. The properties of the high-pressure phases may be very different from those under normal conditions. Meanwhile, there has been considerable interest in the high-pressure behavior of metal dioxides. Tin dioxide (SnO2) also called cassiterite whish is a wide-band-gap semiconductor has attracted increasing interests owing to its outstanding electrical, optical, and electrochemical properties and is attractive for potential applications such as catalytic support material, transparent electrodes for flat panel displays, solar cells, gas sensors, varistors, and optoelectronic devices [1–4]. The behaviors of bulk SnO2, such as phase transition, electronic properties, lattice dynamics, and optical properties have attracted sustained investigations both in experiments and theories. SnO2 at ambient temperature and pressure crystallizes in the tetragonal rutile-type structure, belonging to the P42/mnm space group, since ⇑ Corresponding author at: Université Libanaise, Faculté des sciences (I), Laboratoire de Physique et d’Electronique (LPE), Elhadath, Beirut, Lebanon. Tel.: +961 5 460494; fax: +961 5 461496. E-mail address: [email protected] (F. El Haj Hassan). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.02.011

Jiang et al. [5] found that the tetragonal rutile-type structure could  at transform into the cubic pyrite phase with the space group pa3 a pressure of 18 GPa. Moreover, Shieh et al. [6] by means of X-ray diffraction analysis demonstrated the existence of four phase transitions. The observed sequence of phases for SnO2 is rutile-type  ? ZrO2 (P42/mnm) ? CaCl2-type (Pnnm) ? pyrite-type (pa3) orthorhombic phase I (Pbca) ? cotunnite-type (Pnam). While Haines and Léger [7] by using angle-dispersive X-ray diffraction analysis found three phase transitions. Rutile-type SnO2 underwent a second-order transition to an orthorhombic CaCl2-type phase at 11.8 GPa under hydrostatic conditions and to an orthorhombic a-PbO2-type (Pbcn) phase above 12 GPa under nonhydrostatic conditions. Both the a-PbO2-type and the CaCl2-type phases  transformed to a modified fluorite-type phase (Fm3m) above 21 GPa at ambient temperature. There are also many other experimental works [8–14]. In theoretical study, Gracia et al. [15] calculated the equations of states and the phase transitions of high-pressure polymorphs SnO2 using density functional theory at the B3LYP level. Mazzone [16] calculated the binding and fragmentation energies of SnO2 based on the extended Debye-Hükel approximation. Sevik and Bulutay [17] investigated the elastic, electronic, and lattice dynamical properties of some metal dioxides (SiO2, GeO2, and SnO2) in ‘‘i-phase’’ within the framework of density functional theory. Jacquemin et al. [18] made some calculations on the electronic and optical properties by the Kohn–Korringa–Rostoker method. We explored in our previous theoretical work [19] base on DFT with FP-LAPW method, two structural phase transitions with increasing pressure, from the rutile to the

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F. El Haj Hassan et al. / Computational Materials Science 72 (2013) 86–92

2. Computational details The calculations were performed using the scalar relativistic full potential linearized augmented plane wave FP-LAPW + lo method [21–23] within the framework of DFT [24,25] as implemented in WIEN2 K [26] code. For structural properties the exchange–correlation potential was calculated using the generalized gradient approximation (GGA) in the new form PBEsol proposed by Perdew et al. [27] which is an improved form of the most popular Perdew– Burke–Ernzerhof (PBE) GGA [28]. In addition, and for electronic properties, we also applied the modified Becke–Johnson (mBJ) [29] scheme, this new potential give a very accurate band gaps of semiconductors and insulators with an orbital independent exchange–correlation potential which depends solely on semilocal quantities. In the FP-LAPW + lo method, the wave function, charge density and potential were expanded by spherical harmonic functions inside non-overlapping spheres surrounding the atomic sites (muffin-tin spheres) and by plane waves basis set in the remaining space of the unit cell (interstitial region). An meshes of 30, 36, 36, 45, 32, 35 and 30 special k points for rutile-type, CaCl2-type, a-PbO2-type, pyrite-type, ZrO2-type, fluorite-type and cotunnitetype respectively, were taken in the irreducible wedge of Brillouin zone. The maximum l quantum number for the wave function expansion inside atomic spheres was confined to lmax = 10. The plane wave cutoff of Kmax = 8.0/RMT (RMT is the smallest muffin-tin radius in the unit cell) is chosen for the expansion of the wave functions in the interstitial region while the charge density is Fourier expanded up to GMAX = 14 (Ryd)1/2. Both the plane wave cutoff and the number of k-points were varied to ensure total energy convergence. In order to consider the relativistic effects in our calculation, the electronic states were classified into two categories, the core, and valence states. The core states that are completely confined inside the corresponding muffin-tin spheres were treated fully relativistic, while for the valence states that are rather non-localized we used the scalar relativistic approach that includes the mass velocity and Darwin s-shift but omits spin–orbit coupling.

(0 0 0; 1/2 1/2 1/2) and four oxygen atoms in the following positions ±(u u 0; u + 1/2 1/2-u 1/2). It is clearly seen that the oxygen atomic positions depend on the internal parameter u. These parameters can be optimized by calculating forces on the nuclei and using the damped Newton scheme [30] to find equilibrium atomic positions. The force on each atom after relaxation decreased to less than 1.0 mRyd au1 for all crystals considered in this work. The total energies were then calculated as a function of volume and the obtained data fitted to the Murnaghan equation of state [31]:

" # B0 V ðV 0 =VÞB’0 V 0 B0 EðVÞ ¼ 0 þ 1 þ E0  0 ; B0 B00  1 B0  1

ð1Þ

Where 2

B0 ¼ V

d E dV

2

ð2Þ

:

The c/a ratio has also been optimized at a constant volume. Our results showed that the most stable structure of rutile-type SnO2 occurs when the axial ratio c/a = 0.673 and u = 0.306. The same strategy of calculations followed for the other structures under pressure. The E–V curves of the rutile, CaCl2-type, a-PbO2-type, pyrite, ZrO2-type, fluorite and cotunnite SnO2 phases under the respective phase transition pressures are shown in Fig. 1. To determine the transition pressure at T = 0° K, the enthalpy, H = E0 + PV should be considered [32]. For a given pressure, a stable structure is one for which enthalpy has its lowest value as shown in Fig. 2. The structural parameters, bulk properties and the transition pressures for all the studied SnO2 phases are listed in Table 1. These values agree well with available experimental and other calculation values, indicating that the methods used in our calculations are reliable and reasonable. The rutile to CaCl2-type structural transformation under highpressure was observed at 11.6 GPa which is in reasonable agreement with the reported experimental value 11.8 [7]. Both shows similar E–V curves because the symmetry of the CaCl2-type structure is orthorhombic, having the Pnnm space group, which is only a slight distortion of the P42/mnm space group. This distortion induces the existence of two internal parameters. SnO2 undergoes transition at 16.8 GPa to the orthorhombic a-PbO2 structure, space groupe Pbcn, this transition will have a reconstructive character because the corresponding space groupe is not sub character of P42/mnm. In the a-PbO2 structure, four internal parameters have been optimized.

-12652.60

Total energy per molecule (Ry)

 phase. Recently, Li et al. [20] CaCl2-type phase and to the cubic pa3 studied the optical properties, including dielectric function, reflectivity, absorption, refractive index, and electron energy-loss spectrum, of the high-pressure phase SnO2 in the rutile, pyrite, fluorite, and cotunnite structures by using the DFT plane-wave pseudopotential method. From the descriptions above, we have found that the thermodynamic properties of SnO2 under pressure have sustained rare investigations especially in theoretical up to now. However, thermal properties of solid are closely correlated with various fundamental physical properties, such as specific heat, melting point, interatomic bonding, equation of states, Debye temperature, thermal expansion coefficient. Therefore, in this work, we investigate the structural and electronic properties of SnO2 at equilibrium volume as well as under high pressure up to 80 GPa, by using FP-LAPW method in frame of DFT. Finally, the quasi-harmonic Debye model was successfully applied to determine the thermal properties. In the subsequent text, the computational details are given in Section 2. The results are presented and discussed in Section 3. Section 4 is the conclusion.

-12652.65

-12652.70 Pnam Fm3m Pbca pa3

-12652.75

Pbcn

-12652.80

Pnnm

3. Results and discussions 3.1. Structural properties The rutile-type SnO2 belongs to space group P42/mnm. It is characterized by the two lattice parameters a and c and the internal parameter u. its unit cell contains two tin atoms are set at

P42 /mnm

-12652.85 160 170 180 190 200 210 220 230 240 250 260 270 280 290

Volume per molecule (a.u.3) Fig. 1. Calculated total energies as a function of primitive cell volume of the SnO2 polymorphs for rutile, CaCl2-type, a-PbO2, pyrite, ZrO2 orthorhombic, fluorite, and cotunnite structures.

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F. El Haj Hassan et al. / Computational Materials Science 72 (2013) 86–92

Our calculations shown that SnO2 undergoes at 20 GPa a new  its unit cell contains 8 transition to cubic phase, space group pa3, atoms with the Sn4+ and O2 ions located at (0 0 0) and (u u u) respectively. This phase transition agrees well with the experimental value of 21 GPa [7]. At 41 GPa our results show a reconstructive transition to the ZrO2-type orthorhombic phase I, space group Pbca. In this structure, seven oxygen anions are placed around a Sn4+ cation, and nine internal parameters have been optimized. We have studied a further transition to the fluorite-type structures; it was found at 61 GPa. The fluorite-type structure, space  is an 8-fold oxygen coordinated structure. The differgroup Fm3m,  and Fm3m  space groups arises from a modificaence between pa3 tion to the oxygen positions. In both structures the cations form a  fcc sublattice, and the anions lie on 8c sites (u = 0.25) in the Fm3m  SnO2 optimized structure u = 0.349. structure, whereas in the pa3 This difference yields a cation coordination number 6 + 2 and 8 for the pyrite and fluorite-type structures, respectively. Our calculations predict a transformation from rutile to orthorhombic II cotunnite-type structure at 68 GPa. This phase, space group Pnam, has nine oxygen anions placed around the Sn4+ cation.

Enthalpy per atom relative to rutile (Ry)

0.04

0.02

0.00

-0.02

P42 /mnm Pnnm Pbcn Pa-3 Pbca Fm-3m Pnam

-0.04

-0.06

-0.08 0

10

20

30

40

50

60

70

80

Pressure (GPa) Fig. 2. Static enthalpy per atom in different phases of the SnO2, relative to the rutile phase, as calculated with GGA.

Table 1 Structural properties of SnO2 under pressure, the lattice parameters a, b, and c, are in Å; zero-pressure bulk modulus values, B0, are in GPa; B00 and the pressure transition values, PT, is in GPa. Phase

Space group

a a

Rutile

P42/mnm

Exp. Other theo.b,c Ours

4.737 4.715 4.673 4.776

CaCl2-type

Pnnm

Exp.a Other theor.b Ours

4.653 4.708 4.808

a-PbO2-type

Pbcn

Exp.a Other theor.b

4.707

b

4.631 4.720 4.691 5.710

c

x

3.186 3.194 3.149 3.212

0.307 0.306 0.306

3.155 3.195 3.226

0.282 0.3059 0.309

4.737

5.746

 Pa3

Exp.a Other theor.b,c Ours

4.888 5.066 5.005 5.116

ZrO2-type

Pbca

Exp.e Other theorb

9.304 9.970

10.076

Fluorite-type

Cotunnite-type

 Fm3m

Pnam

Exp.d Other theor.b,c Ours

5.08 4.993 4.972 5.088

Exp.e Other theor.b

5.016 5.326

5.167

5.364 Sn O O

a b c d e

[7]. [15]. [20]. [5]. [6].

0.165 0.388

0.25 0.418

0.0 0.269

0.164 0.393

0.25 0.422

4.6

0.882 0.788 0.979

0.028 0.363 0.742

0.265 0.122 0.496

0.882 0.793 0.976

0.031 0.376 0.739

0.268 0.140 0.501

212

290 288 269 204

3.028 6.668 0.25 0.25 0.75

0.092 0.406 0.354

0.261 0.372 0.984

0.25 0.25 0.75

0.086 0.400 0.354

6.917

208

11.8 12 11.6

4.7

16.8

4.7

21 17 17 20 50–74 18

5.1

41

4.5

52.1 24 24 61

417 180 0.258 0.368 0.010

PT

12 17

259 285

0.25

3.437

4.8

328 281 271 216

0.25

5.904 3.379

205 221 245 192

201

5.076

Sn O O Ours

0.0 0.277

4.731 5.022

Sn O O

B00

204 231

0.349 4.893 5.113

B0

204 231 195

0.352 0.335

Sn O O Ours

0.264 0.3063 0.305

5.279

Sn O Pyrite-type

z

5.246

Sn O Ours

y

54–117 33

4.7

68

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F. El Haj Hassan et al. / Computational Materials Science 72 (2013) 86–92

20

20

Energy (eV)

16

16

SnO2 Fluorite-type

12

12

8

8

4

4

0

0

-4

-4

-8

-8

GGA

-12

mBJ

-12

-16

-16

-20

-20

Γ

W

L

Γ

X

W K

W

Γ

Γ

L

X

W

K

Fig. 3. Band structure of the SnO2 polymorphs calculated along high symmetry directions using GGA (left panel) and mBJ (right panel) at the pressure value corresponding to its presumed stability.

Our theoretical results show that the rutile, CaCl2-type and aPbO2 phases present good bulk modulus values of 192, 195 and 201 GPa which are close to the experimental values of 205, 204 and 204 GPa from Ref. [7]. Our calculated B0 of pyrite, ZrO2-type and fluorite phases are less agree with the experimental values. The calculated bulk modulus of cotunnite phase is 208 GPa, agrees with the previous 180 GPa [15] but also has obvious underestimation of the experimental values 417 GPa [6].

Table 2 The band gap calculated of the high-pressure polymorphs of SnO2 within different approximations and compared with the available experimental and theoretical values. Phase

Gap energy Eg (eV) Exp.

Ours

Other theo.

a

Rutile CaCl2-type a-PbO2-type Pyrite-type ZrO2-type Fluorite-type Cotunnitetype

3.2. Electronic properties The crucial role of the studied compound in a variety of technologically devices has necessitated precise knowledge of the fundamental energy gap as well as the alignment of the main conduction-band valleys. The self consistent scalar relativistic band structures of SnO2 were obtained at equilibrium volume as well as at high pressure within the GGA, and mBJ schemes [29]. The usual problem of the DFT gap underestimation encourages the researcher to find a suitable solution; mBJ is a modified version of the exchange potential proposed by Becke and Johnson [33]. The agreement with experiment is very good for solids and it is of the same order as the agreement obtained with the hybrid functionals or the GW methods. This semilocal exchange potential mimics very well the behavior of orbital-dependent potentials and leads to calculations which are barely more expensive than GGA calculations. Therefore, it can be applied to very large systems in an efficient way. As a prototype the results obtained for the band structures of fluorite-type within GGA and mBJ approximations are shown in Fig. 3. The overall behavior of the band structures calculated by these two approximations is very similar except for the value of their band gap, which is higher within mBJ as presented in Table 2. Direct band gap energy at the highly symmetric U-point has been observed for all the phases except the fluorite which is an indirect band gap semiconductor at W–U points. The total density of state (TDOS) and the partial density of state (PDOS) at equilibrium of all phases are calculated. The conduction band consists essentially of Sn 5s and Sn 5p orbits while O 2s states also have a little contribution. The lowest lying bands shown in Fig. 3 arise from the oxygen 2s states. The upper valence bands that lie above these bands are mainly due to Sn 5p and O 2p states.

Space group

a b c

P42/mnm Pnnm Pbcn  Pa3

2.9

Pbca  Fm3m Pnam

GGA

MBJ

DFTLDAb

DFTB3LYPc

0.832 0.889 1.162 0.849 0.838 0.143 0.522

2.760 3.021 3.222 2.536 2.015 2.001 1.936

1.38

3.50 3.58 3.80 3.55 3.44 3.01 2.84

1.90 1.32 1.64

[14]. [20]. [15].

3.3. Thermal properties Finally, to investigate same thermal properties, we used the quasi-harmonic Debye model [34] in which the non-equilibrium Gibbs function G (V; P, T) is written in the form:

G ðV; P; TÞ ¼ EðVÞ þ PV þ Av ib ½hðVÞ; T;

ð3Þ

where E(V) is the total energy per unit cell, PV corresponds to the constant hydrostatic pressure condition, h(V) is the Debye temperature and Avib is the vibrational term which can be written using the Debye model of the phonon density of states as:

Av ib ðh; TÞ ¼ nkT



  9h h þ 3 lnð1  eh=T Þ  D ; 8T T

ð4Þ

where n is the number of atoms per formula unit, D(h/T) represents the Debye integral and for an isotropic solid, h is expressed as:

hD ¼

rffiffiffiffiffiffiffi  h 2 1=2 i1=3 h BS 6p V n f ðrÞ : k M

ð5Þ

M being the molecular mass per unit cell, BS is the adiabatic bulk modulus, which is approximated given by the static compressibility:

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F. El Haj Hassan et al. / Computational Materials Science 72 (2013) 86–92 2

BS ffi BðVÞ ¼ V

d EðVÞ dV

2

580

ð6Þ

:

P42/mnm 575

Details on f(r) can be found elsewhere [35,36]. 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:

ð7Þ

570 565

TD (K)

   @G ðV; P; TÞ ¼ 0: @V P;T

By solving Eq. (7), one can obtain the thermal equation-of-state (EOS) V(P, T). The heat capacity CV and the thermal expansion coefficient a are given by [37]:

    h 3h=T  h=T ; C V ¼ 3nk 4D T e 1

560 555 550 545

ð8Þ

540 535

a¼

cC V BT V

50

100

150

200

250

300

350

400

450

500

T (K) Fig. 5. Relationships of Debye temperature h with temperatures of the SnO2 polymorphs for different structures.

d ln hðVÞ : d ln V

ð10Þ

Through the quasi-harmonic Debye model, one could calculate the thermodynamic quantities of any temperatures and pressures of compounds from the calculated EV data. The thermal properties are determined in temperature range from 0 to 500 K, where the quasi harmonic model remains fully valid. The temperature effects on the bulk modulus are shown in Fig. 4. From this figure, we can find that T < 100 K, B nearly keeps constant. When T > 100 K, the bulk modulus decreases with increasing temperature at given structure, but the rate of increase is moderate. The Debye temperature is an important fundamental parameter and closely related to many physical properties of solids, such as specific heat and melting temperature. It is well known, that below Debye temperature, quantum mechanical effects are very important in understanding the thermodynamic properties, while above Debye temperature quantum effects can be neglected. The calculated Debye temperature h are depicted in Fig. 5 Accordingly, the Debye temperature remains constant up to 100 K then it decreases with increasing temperature. The heat capacity of a substance not only provides essential insight into its vibrational properties, but is also mandatory for many applications. The isochoric (Cv) heat capacities of SnO2 were

220

calculated using Eq. (6) and are displayed in Fig. 6. It is seen that all the structures have the same behavior. At high temperature, the constant volume heat capacity Cv tends to the Dulong-Petit limit which is common for all solids. At sufficiently low temperatures, Cv is proportional to T3. Our calculated value of Cv at room temperature is about 60 J mol1 K1 for all the phases which is comparable with the experimental value of Cp in the rutile-type (55.3 J mol1 K1 [38]). Thermal expansion is the tendency of matter to change in volume in response to a change in temperature. The thermal expansion coefficient a is shown in Fig. 7. Our calculated value of a at room temperature in the rutile-type is 3.8  105 K1 which is much smaller than the experimental value of 1.17  105 K1 given by Ref. [39]. The Grüneisen parameter (c) describes the effect that changing the volume of a crystal lattice has on its vibrational properties, and, as a consequence, the effect that changing temperature has on the size or dynamics of the lattice, and it has been widely used to characterize and extrapolate the thermodynamic behavior of a material at high pressures and temperatures, such as the thermal expansion coefficient and the temperature dependence of phonon frequen-

75

P42 /mnm

Pnnm Pbcn Pa-3 Pbca Fm-3m Pnam

210 205 200

70 65 60 55 50

Cv (J/mol K)

215

B (GPa)

0

ð9Þ

:

where is the Grüneisen parameter which is defined as

c¼

Pnnm Pbcn Pa-3 Pbca Fm-3m Pnam

195 190

45

P4 2/mnm

40

Pnnm Pbcn Pa-3 Pbca Fm-3m Pnam

35 30 25 20

185

15 180

10 5

175

0 0

50

100

150

200

250

300

350

400

450

500

T (K) Fig. 4. The temperature dependence of bulk modulus of the SnO2 polymorphs for different structures.

0

50

100

150

200

250

300

350

400

450

500

T (K) Fig. 6. The temperature dependent heat capacity Cv of the SnO2 polymorphs for different structures.

F. El Haj Hassan et al. / Computational Materials Science 72 (2013) 86–92 5

(ii) The zero-pressure values of the band gap energy have been calculated by using the Becke–Johnson (mBJ) of the exchange–correlation potential for the first time: rutiletype, 2.76 eV; CaCl2-type, 3.021 eV; a-PbO2, 3.222 eV; pyrite-type, 2.536 eV; ZrO2 orthorhombic, 2.015 eV; fluorite, 2.001 eV; and cotunnite-type 1.936 eV. (iii) The thermodynamic properties of SnO2 have been predicted systematically using the density functional theory (DFT) and quasi-harmonic Debye model. It is found that the high temperature leads to a smaller bulk modulus, a large heat capacity, a smaller Debye temperature, a larger Grüneisen parameter and a larger thermal expansion coefficient for a given structure. The heat capacity and the thermal expansion coefficient tend to be constant value at high temperatures.

4

-5

α (10 /K)

3

2

P42/mnm

1

Pnnm Pbcn Pa-3 Pbca Fm-3m Pnam

0

0

50

100

150

200

250

300

350

400

450

91

500

T (K) Fig. 7. The temperature dependent coefficient of thermal expansion a of the SnO2 polymorphs for different structures.

The reported calculations provide new structural and electronic results from first principles for tin dioxide. Hence, this study is part of large theoretical effort to explore the properties of this compound. Acknowledgments

2.44

One of the authors (F. H. H.) would like to thank the Lebanese National council for the scientific research (CNRS) and the Doctoral School of Sciences and Technology in the Lebanese university (EDST) for their financial support during the realization of this work.

2.42 2.40

Grüneisen parameter (γ)

2.38 2.36 2.34 2.32

References

2.30

P42/mnm

2.28

Pnnm Pbcn Pa-3 Pbca Fm-3m Pnam

2.26 2.24 2.22 2.20 2.18 2.16 2.14

0

50

100

150

200

250

300

350

400

450

500

T (K) Fig. 8. Temperature dependence of Grüneisen parameter c of the SnO2 polymorphs for different structures.

cies. Grüneisen constant c as a function of temperature at different structures are shown in Fig. 8. Our calculated value of c equal to 2.4 for the rutil-type at room temperature which agree well with the experimental value of 2.58 [39]. c increases with increasing temperatures for a given structure. 4. Concluding remarks We have applied a FP-LAPW method to study the structural, electronic and thermal properties of SnO2 compound at equilibrium as well as at high pressure. The main results can be summarized as follows: (i) Starting from the rutile-type structure six phase transitions under high-pressure conditions have been found at 11.6, 16.8, 20, 41, 61 and 68 GPa, which corresponding to the CaCl2-type, a-PbO2, pyrite-type, ZrO2 orthorhombic, fluorite, and cotunnite structures, respectively.

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