First principles lattice dynamics study of SnO2 polymorphs

Journal of Alloys and Compounds 633 (2015) 272–279

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First principles lattice dynamics study of SnO2 polymorphs I. Erdem a,⇑, H.H. Kart a, T. Cagin b a b

Department of Physics, Pamukkale University, Kinikli Campus, 20020 Denizli, Turkey Department of Material Science and Engineering, Texas A&M University, College Station, TX 77843-3003, USA

a r t i c l e

i n f o

Article history: Received 27 October 2014 Received in revised form 23 January 2015 Accepted 27 January 2015 Available online 13 February 2015 Keywords: SnO2 Density functional theory Structural properties Lattice dynamics calculations Phonon dispersions

a b s t r a c t The structural properties of SnO2 polymorphs in the sequential order of observed phases in experiments are determined by the density functional theory (DFT) calculations based on local density approximation (LDA) of ultra soft pseudo potentials (US-PPs). Phonon dispersion relations are calculated by the lattice dynamics calculations. Shifts in the infrared (IR) active optical modes due to polarization (LO/TO splitting) are also calculated. Moreover, softening of B1g mode at the rutile-CaCl2 second-order ferroelastic phase transition is confirmed. Thermal properties, such as temperature behavior of bulk modulus and thermal expansion in the rutile phase are obtained by employing quasiharmonic approximation (QHA). They are in good agreement with the available experimental results. Dynamic stabilities of SnO2 polymorphs except for the rutile phase are checked for the first time by using phonon dispersions. The rutile, CaCl2, pyrite, ZrO2 and cotunnite type structures have shown thermodynamical stability. The cause of a-PbO2 phase showing nearly stability is discussed in the light of experimental studies. However, the fluorite type structure is definitely instable even at different pressures. It may not be one of SnO2 polymorphs. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction The cassiterite, SnO2, ranks among the most fruitful raw material of state of the art technologies. High capability of lithium storage makes it irreplaceable in lithium-ion batteries [1,2]. Core–shell nano structures of SnO2 are applied frequently in gas sensors, dyesensitized solar cells and high performance supercapacitors [3–6]. Its enhanced photocatalytic property with TiO2 enables to clean water, soil and air in health and environmental sciences [7]. Scientific researches on the material studied in this work keep their popularity at present and the polymorphs of SnO2 are investigated by both experimentally and theoretically as analogues of group IVB Ti, Zr, Hf dioxides, group IVA Si, Ge, Pb dioxides or rutile-structured (isomorphic) Ru, Ir, Nb, V, Mn dioxides [8–15]. Structural phase transitions can be driven by applying pressure, and heating or both of them to the material. Two most important results of the experimental study reported by Prakapenka et al. [10] are that the high pressure polymorphs of SiO2 and GeO2 and the sequence of phases depend on the starting phase of material and pressure/temperature history of the system under investigation. Ohtaka et al. [11] have stated in their experimental study of ZrO2 that a large amount of deviatoric stresses deteriorating the volume of the sample is

⇑ Corresponding author. Tel.: +90 536 229 64 34; fax: +90 258 296 36 31. E-mail address: [email protected] (I. Erdem). http://dx.doi.org/10.1016/j.jallcom.2015.01.235 0925-8388/Ó 2015 Elsevier B.V. All rights reserved.

produced at compression processes. Hence, releasing of these stresses and overcoming kinetic barriers of phase transitions can be provided by laser annealing [11]. Haines et al. [12] have reported that the cation size of dioxides has also considerable importance for phase transition. Neutron scattering and X-ray experiments of structures of SnO2 polymorphs as well as theoretical studies using different types of potentials and exchange–correlation functionals have been performed [16–24]. Experimentally observed and theoretically predicted phase sequence of SnO2 polymorphs is rutile-type, CaCl2type, a-PbO2-type, pyrite-type, ZrO2-type and cotunnite-type structures. Ono et al. [17] have reported that reliable phase transformations of SnO2 can be obtained at temperatures higher than 1000 K and the CaCl2 phase has not been detected in the X-ray experiment up to 30 GPa. In the X-ray experiment up to 117 GPa performed by Shieh et al., the a-PbO2 phase has not been detected [19]. A new phase called fluorite (space group No. 225) which is predicted between the ZrO2 and the cotunnite structures in the phase sequence by Hassan et al., Gracia et al. and Li et al. in their DFT studies based on PBEsol form of GGA, B3LYP and LDA potentials, respectively [22–24]. However, the fluorite phase existing before cotunnite type structure has not been proposed in our previous DFT study using Perdew–Burke–Ernezrhof (PBE) parameterized GGA potential [20]. On the other hand, some theoretical researchers have predicted a direct phase transition from rutile

I. Erdem et al. / Journal of Alloys and Compounds 633 (2015) 272–279

phase to pyrite phase disregarding CaCl2 and a-PbO2 phases [21,24]. It is possible to encounter discrepancies in both experimental and theoretical literature of polymorphs of SnO2. It can be beneficial to examine these polymorphic phase transitions from some other perspective such as thermodynamical stability. While lattice dynamics properties have been measured by Raman, infrared and Brillouin spectrum experiments [25–28], they have been calculated by DFT method, rigid ion or shell model theoretically as well [13,29–34]. The aim of this study is to investigate polymorphs of SnO2 by first-principles structural and lattice dynamics calculations which have taken its place in literature as a powerful method. In addition, temperature dependencies of some physical properties such as bulk modulus, Gibbs free energy and thermal expansion coefficient are calculated by means of a successful collaboration of DFT and quasi harmonic approximation. The rest of the paper is organized as follows: Computational method is defined in Section 2, results and discussions are presented in Section 3. Finally, conclusion arising from this work is given in Section 4.

273

tensor of rutile phase are obtained by density functional perturbation theory (DFPT) property as implemented in VASP code. Phonon dispersions of polymorphs of SnO2 of which the rutile one includes non-analytic term corrections are evaluated. Thermal properties of rutile phase are obtained via quasi harmonic approximation. 3. Results and discussion 3.1. Structural properties Lattice parameters and other structural properties such as bulk modulus and its pressure derivative are determined through total energy calculations as a function of cell volume per formula unit. In total energy calculations, data are fitted to a third-order Birch–Murnaghan equation of states (EOS) which is given as follows [48]: 9 8" #3 "  #2 "  2=3  2=3 #= 2=3 9V 0 B0 < V 0 V0 V0 0 EðVÞ ¼ E0 þ  1 B0 þ 1 64 : ; 16 : V V V

2. Computational method

ð1Þ

Total energy calculations have been carried out via DFT method as implemented in Vienna Ab-initio Simulation Package (VASP) code [35–38]. Vanderblit ultra soft pseudo potentials (US-PPs) including exchange–correlation part generated by Ceperley and Alder within local density approximation (LDA) are employed for interactions [39–41]. While US-PP of oxygen atom has 2s22p4 valence states, two different types of US-PPs are used for Sn atoms having valence states 5s25p2 and 4d105S25p2 which are referred in the text as Sn and Sn(d) PPs, respectively. Sn (d) PP may reflect some physical properties such as electronic band gap more accurately than Sn PP, since the former takes into account the contribution of 4d10 electrons in the chemical bonding of SnO2 markedly [42]. The kinetic cut-off energy value of 500 eV and a criterion of 106 eV per formula unit for energy convergence are taken in structural calculations. K-point meshes for Brillouin zone integrations are converged at 8  8  10 for rutile phase and CaCl2 phase, 6  6  6 for phases of a-PbO2 and pyrite, 6  10  10 for ZrO2 phase and 6  10  6 for cotunnite phase in the Monkhorst–Pack scheme [43]. The structural information and Wyckoff sites of SnO2 polymorphs are listed in Table 1. A fine relaxation with a force criterion of 108 eV/Å between ions is performed at the first step of lattice dynamics calculations. Then, super cells of 2  2  3 for rutile phase, 1  2  2 for ZrO2 phase and 2  2  2 for CaCl2, a-PbO2, pyrite and cotunnite phases are produced according to small displacement (direct) method as implemented in PHONOPY code [44–47]. 2  2  2 meshes of k-points are used for force calculations performed by VASP code. Born effective charges and the macroscopic dielectric

Here, E(V) is total energy, E0 and V0 are energy and volume at equilibrium, respectively. B0 is the bulk modulus at the pressure of 0 GPa and B00 is the first derivative of bulk modulus with respect to pressure. Gibbs free energy G = E0  TS + PV turns into enthalpy H = E0 + PV, since calculations in DFT are carried out at the temperature of T = 0 K. Enthalpies of two phases at a phase transition from a phase to another one become equal which enables to determine transition pressure PT. Total energy calculations are carried out by using US-PPs 5s25p2 valence electrons for Sn and 2s22p4 valance electrons for O. Structural parameters calculated for the pressure value at which they are obtained are given in Table 2 to make comparison with the corresponding experimental results. Phase transition pressures are also given in Table 2. We have determined a phase sequence of rutile ? pyrite ? ZrO2 ? cotunnite at the pressures of 12.3 GPa, 23.6 GPa and 36.0 GPa, respectively. While Zhu et al. [21] have reported the same phase sequence, Li et al. [24] have suggested a sequence of rutile ? pyrite ? fluorite ? cotunnite in their theoretical studies with LDA. Although allowing ionic relaxations in DFT calculations reflects a very small amount of thermal effects, these relaxations are insufficient to compensate and overcome thermal barrier totally in a phase transition. Hence, our LDA calculations show that a phase transition from rutile phase to pyrite phase takes place directly. Haines and Léger have observed the same direct phase transition (disregarding the CaCl2 and a-PbO2 phases) in their experimental study when rutile structured SnO2 is not heated within the pressure range of stability of these phases [18]. On the other hand, there is not any experimental study where

Table 1 Structural information about the polymorphs of SnO2. Name of phase, space group, space group number (S. G. No), number of formula unit per unit cell (Z), code of Wyckoff sites and coordinates of basis atoms are indicated, sequentially. Phase

Space group

S.G. No. Z Wyckoff Positions sites O Sn O

Rutile P42/mnm 136 CaCl2 Pnnm 58 a-PbO2 Pbcn 60

2 2a 2 2a 4 8d

3 Pa

205

4 8c

61 62

8 8c 4 4c

Pyrite

Pbca ZrO2 Cotunn. Pnma

4f ±(u, u ,0), ±(½ + u, ½  u, 0.5) 4 g ±(u, v, 0), (½  u, ½ + v, 0.5), (½ + u, ½  v, 0.5) 4c ±(u, v, w), ±(½  u, ½  v, w + ½),  ±(u  , v, ½  w) ±(u + ½, ½  v, w), 4a ±(w, w, w), ±(w + ½, ½  w, w), ±(w, w + ½, ½  w), ±(½  w, w, w + ½)  ;u  , v + ½, ½  w; ½  u, v  , w + ½) 8c ±(u, v, w; u + ½, ½  v, w 4c ±(u, 1/4, v; u + ½, 1/4, ½  v)

Sn (0, 0, 0), (1/2, 1/2, 1/2) (0, 0, 0), (1/2, 1/2, 1/2) ±(0, v, 1/4), ±(1/2, v + ½, 1/4) (0, 0, 0), (0, 1/2, 1/2), (1/2, 0, 1/2), (1/2, 1/2, 0)  u  , v + ½, ½  w; ½  u, v  , w + ½) ±(u, v, w; u + ½, ½  v, w; ±(u, 1/4, v; u + ½, 1/4, ½  v)

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Table 2 Comparison of calculated structural data of SnO2 polymorphs with available experimental results. Lattice parameters a(Å), b(Å), c(Å) and internal parameters u, v, w are given with pressure values (GPa) at which they are calculated. Bulk modulus B0 (GPa) and its pressure derivative B00 of phases are at zero pressure. Transition pressures are given in GPa [18,19]. This study LDA (Sn) Rutile-type Pressure (GPa) a (Å) c (Å) u B0 (GPa) B00 PT (GPa)

Exp.a

LDA (Sn(d))

0 4.726 3.185 0.306 200 4 12.3

CaCl2-type Pressure (GPa) a (Å) b (Å) c (Å) u v B0 (GPa) B00 PT (GPa)

Exp.b

0 4.711 3.184 0.306 192 4 7.2

Ambient 4.737 3.186 0.307 205 7 11.8

Ambient 4.746 3.189

12.6 4.607 4.601 3.149 0.306 0.304 192 4 9.6

12.6 4.653 4.631 3.155 0.330 0.282 204 8 12.6

13.6 4.678 4.536 3.144

0 4.693 5.686 5.199

Ambient 4.744 5.707 5.209

13.6

28.8

a-PbO2-type Pressure (GPa) a (Å) b (Å) c Sn 0.000 0.167 0.250

u v w B0 (GPa) B00 PT (GPa)

a

186 4 14.2

208 4 21.0 25.0 4.936 0.353 261 7

Pyrite-type Pressure (GPa) a (Å) w B0 (GPa) B00 PT (GPa)

27.3 4.901 0.344 226 3 23.6

28.6 4.886 0.344 217 4 35.6

ZrO2-type Pressure (GPa) a (Å) b (Å) c (Å)

60.9 9.195 4.877 4.683

62.8 9.177 4.872 4.675 O(1) 0.803 0.396 0.176 206 4 43.5

Sn 0.884 0.034 0.269

u v w B0 (GPa) B00 PT (GPa)

211 3 36.0

Cotunnite-type Pressure (GPa) a (Å) b (Å) c (Å)

115.1 5.037 3.023 5.889

u v w B0 (GPa) B00 b

O 0.270 0.394 0.422

Sn 0.251 0.250 0.118 120 4

116.6 5.020 3.027 5.902 O(1) 0.354 0.250 0.430 117 4

28.8 4.905

64

O(2) 0.970 0.734 0.495

O(2) 0.023 0.750 0.332

Sn 0.891 0.025 0.242

Sn 0.255 0.250 0.114

64 9.304 4.893 4.731 O(1) 0.810 0.400 0.143 259 4 74.0 117.0 5.016 3.028 5.904 O(1) 0.348 0.250 0.390 417 4

O(2) 0.984 0.738 0.469

O(2) 0.043 0.750 0.333

From Ref. [18]. From Ref. [19].

fluorite phase of SnO2 has been observed, but the CaCl2 and a-PbO2 phases are observed and their lattice parameters are refined in experiments [17–19]. Furthermore, it is a well-known fact that SnO2 crystallizes in tetragonal symmetry called as rutile at

ambient conditions, its lattice dynamics calculations performed by the pseudo potential with 5s25p2 valence states for Sn in the rutile structure show a small amount of thermodynamic instability as well.

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rutile-type CaCl2-type -PbO2-type pyrite-type ZrO2-type cotunnite-type fluorite-type

Total Energy (eV)

-19.5

-20.0

-20.5

-21.0

-21.5

30

31

32

33

34

35

36

37

V(A3 ) Fig. 1. Total energy versus volume (per formula unit) with LDA ultra soft PPs Sn(d) and O for SnO2 polymorphs which are rutile (tetragonal), CaCl2 (orthorhombic), aPbO2 (orthorhombic), pyrite (cubic), ZrO2 (orthorhombic), cotunnite (orthorhombic), fluorite (cubic) phases.

Upon including 4d electrons to the valance states in the US-PP of Sn (4d105S25p2), the phonon dispersions of rutile phase have improved the description of thermodynamic stability and the whole phase sequence including CaCl2 and a-PbO2 phases is achieved. Inclusion of 4d10 states into valence makes the PP harder. Haman has reported that hard potentials are useful in sensitive structural separations due to the capability of accurate bandcharge shapes [49]. Structural calculations are repeated with this new hard potential. Total energy-volume curves fitted to Birch– Murnaghan EOS for tin dioxides polymorphs are shown in Fig. 1. The lattice parameters calculated at about the pressures where they are refined in the experimental studies are given in Table 2. They are in excellent agreement with the experimental results as seen in Table 2. Transition pressure PT listed in this table indicates the pressure at which the phase transition occurs from parent phase to the following phase, such as phase transition from pyrite to ZrO2 takes place at 35.6 GPa. Bulk moduli of SnO2 polymorphs are in good agreement with the available experimental results, except for high pressure polymorphs of pyrite, ZrO2 and cotunnite which are calculated as 217 GPa, 206 GPa, 117 GPa, respectively. They deviate from their experimental values as 17%, 20% and 72%, respectively. These discrepancies between theoretical and experimental values may be explained as the following: Experiments are performed by compressing the sample to a critical pressure value then decompressing to ambient pressure. If a phase disappears at ambient pressure during decompression process, an extrapolation is made then fitting the relevant data to an EOS gives bulk modulus. However, tradition in DFT method not only compression to a critical pressure and decompression to ambient pressure are taken into account, but also decompression is extended to negative pressures. The system under investigation is relaxed around the equilibrium volume from 0.05 to 0.05 percent by the steps of 0.01%. To exemplify this idea the cotunnite phase which is observed above 54 GPa pressure in the experiment is reexamined [19]. EOS fit of volume-energy data calculated at pressure ranges greater than 53 GPa results in more realistic value of 369 GPa for bulk modulus. When decompression to negative values is included bulk modulus is calculated as 117 GPa instead of 369 GPa. The same reason may also underlie underestimation of phase transition pressures.

Fig. 2. Phonon dispersions of rutile SnO2 in the main symmetry directions by taking LO/TO splitting of IR active modes into account.

Table 3 Phonon dispersions at the zone center of rutile SnO2 evaluated by taking the LO/TO splitting of IR modes into account are compared with the data of experiments [25–27] and other available DFT calculations with LDA pseudo potential [29–32]. Frequencies of modes are in units of THz. Mod (THz) This study Exp.a Exp.b Exp.c Theor.d Theor.e Theor.f Theor.g A1g A2g B1g B2g Eg

18.11 10.19 2.94 22.27 14.24 4.41

ð1Þ

B1u

ð2Þ

B1u A2u (TO) Euð1Þ (TO)

a

c d e f g

18.98 10.79 3.12 22.93 14.03 4.41

18.51 9.60 2.48 22.00 13.86 4.15

19.36 8.94 4.32 22.23 14.59 3.93

14.24

15.14

17.54

16.91

16.57

17.47

13.82 6.75

14.30 7.31

13.82 7.24

13.67 6.69

13.71 6.00

14.23 5.34

Euð2Þ (TO)

8.42

8.78

8.57

8.54

8.11

8.16

Euð3Þ (TO) A2u (LO)

16.97

18.53

18.43

18.38

17.51

18.24

19.10 8.00

21.13 8.27

19.70 8.36

20.09 8.06

19.44 7.55

14.22 8.16

Euð1Þ (LO)

b

19.13 19.00 19.07 19.13 11.93 10.98 3.00 3.69 3.14 23.44 23.26 23.29 22.84 14.27 14.24 14.21 14.08 4.20 4.40

Euð2Þ (LO)

10.61

10.97

12.18

10.04

9.20

8.16

Euð3Þ (LO)

21.59

23.08

21.09

22.33

21.34

18.24

From From From From From From From

Ref. Ref. Ref. Ref. Ref. Ref. Ref.

[25]. [27]. [26]. [31]. [29]. [32]. [30].

3.2. Lattice dynamics properties First principles lattice dynamics calculations have been performed by using Sn(d) and O US-PPs at the super cell of 2  2  3 for rutile SnO2 within direct (frozen phonon) method. The macroscopic dielectric tensor and Born effective charges have been calculated by DFPT implemented in VASP code in order to take account of shift (LO/TO splitting) of infrared (IR) active modes due to polarization. Macroscopic static dielectric tensor  and Born effective charge tensors Z⁄ are given as follows:

0

4:187

¼B @ 0:000

0:000

0:000

0:000

1

C 4:175 0:000 A; 0:000 4:507

ð2Þ

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and

0

1 3:945 0:378 0:000 B C Z ðSnÞ ¼ @ 0:378 3:945 0:000 A; and 0:000 0:000 4:275 0 1 1:972 0:598 0:000 B C Z  ðOÞ ¼ @ 0:598 1:972 0:000 A; 

0:000

Fig. 3. Pressure behavior of mode frequencies of rutile SnO2 at C point up to the pressure of 10 GPa including LO/TO splitting of IR active modes.

a

c

0:000

ð3Þ

2:136

respectively. The phonon dispersions calculated in the main symmetry directions of Brillouin zone by adding non-analytic term corrections to the dynamical matrix are displayed in Fig. 2. Mode frequencies at the zone center (C) involving LO/TO splitting are calculated by taking arithmetic mean of the values at the two different paths (A–C and C–M). The evaluated mode frequencies at C point are compared with available experimental and other LDA theoreti-

b

d

Fig. 4. (a) Helmholtz free energies and corresponding volumes. Temperature dependencies of (b) Gibbs free energy, (c) bulk modulus and (d) thermal expansivity from 0 K up to 990 K, respectively.

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cal results in Table 3. The overall agreement with experimental results of the zone center phonon modes is obtained. The first three acoustic mode values of A1, A2 and A3 have been overlapping at the frequency of 0 THz. The Raman active modes of A1g, B1g, B2g and Eg , IR active modes of A2u, and 3Eu and silent modes of 2B1u and A2g deviate from the corresponding experimental values as in the range

of 0.–5.3%, 3.3–9.6% and 5–14.6%, respectively. Some experimental and theoretical scientists have reported that softening of B1g mode takes place at the pressure dependence of mode frequency, which indicates that a second-order ferroelastic phase transition to the orthorhombic CaCl2 phase occurs [27,30,31]. Pressure behaviors of mode frequencies of rutile SnO2 are depicted up to 10 GPa pressure

a

b

c

d

e

f

Fig. 5. The dispersions without LO/TO splitting of SnO2 polymorphs in the structures of (a) rutile (ambient), (b) CaCl2 (7.07 GPa), (c) a-PbO2 (9.75 GPa), (d) pyrite (15 GPa), (e) ZrO2 (33.96 GPa) and (f) cotunnite (43.62 GPa), respectively.

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in Fig. 3. All optical mode frequencies increase with pressure except for E(1) u (TO) and B1g. The softening of B1g mode continues as the pressure increases, which reflects the phase transition. Our prediction of 7.13 GPa for this transition pressure is compatible to the value of 7 GPa calculated by Parlinsky and Kawazoe [31]. However, Gupta et al. [30] have reported that LDA calculation underestimate the value of transition pressure, while GGA calculations propose a transition between 12 and 14.6 GPa which is closer to experimental values of around 12 GPa than LDA results. The lattice dynamics calculations have been performed under harmonic approximation up to this point. By the assumption of mode frequency wj = wj (V,T) as a function of volume and temperature, anharmonic (concerning temperature) contributions causing shifts to the mode frequencies of Raman active modes are formulated as [27]:

      @ ln wj b @ ln wj @ ln wj ¼ þ ; @T j @P T @T P V

ð4Þ

where j is the mode index of phonon, b is the thermal expansion coefficient and j is the isothermal compressibility. Peercy and Morosin have reported that pure-volume contribution (first term in the right-hand side of Eq. (4)) dominates pure-temperature contribution (the second term) through the temperature range of 0–500 K. Moreover, anharmonic contributions to the self-energies are the order of 1% or less of the harmonic energies [27]. Lan et al. [29] have investigated nonharmonic contributions by making comparison between anharmonic (pure temperature) and quasiharmonic (volumetric) contributions in the temperature range of 83–873 K. Under the guidance of these experimental results, applying the quasi harmonic approximation to rutile structure of SnO2 may be efficient to obtain important realistic results of some thermal properties. Vibrational (phonon) contributions to the Helmholtz free energy F in the quasi harmonic approximation (QHA) can be expressed by adding thermal contribution to the zero-point term [50]:

F vib ¼

X 1X  wðqsÞ þ kB T ln ½1  expðhwðqsÞ=kB T ; h 2 qs qs

ð5Þ

where q, s, ⁄, w, kB and T are the wave vector, band index, Planck’s constant, mode frequency, Boltzmann’s constant, and temperature, respectively. Total energy and lattice dynamics calculations have been performed for eleven different volumes around the equilibrium volume. Gibbs free energy in QHA is given as [46]:

GðT; pÞ ¼ ½EðVÞ þ F vib ðT; VÞ þ pV;

ð6Þ

where V is volume, E(V) is structural electronic total energy, T is temperature and p is pressure. Thermal properties can be calculated by means of PHONOPY program which employs related thermodynamic functions on the principle of minimization of Gibbs free energy at finite temperature [44–47]. Helmholtz free energies as a function of volume at different temperatures between 0 K and 990 K, and the variation of Gibbs free energy with temperature are depicted in Fig. 4a and b, respectively. Temperature dependencies of bulk modulus and thermal expansivity are shown in Fig. 4c and d, respectively. Gibbs free energy and bulk modulus are decreasing with temperature, while thermal expansivity increasing. Our bulk modulus calculated by QHA at 300 K is 182.6 GPa and thermal expansivity at 350 K is 11.2  (106/K). They are in acceptable agreement with the experimental results of 222 GPa and 11.7  (106/K) [27]. We have shown in our previous DFT study by using PBE parameterized GGA potential that experimentally observed and refined SnO2 polymorphs show mechanical stability [20]. We are interested in dynamical stabilities of these polymorphs in this study. Their dispersion relations are obtained around the transition

pressures instead of zero pressure, since most of them can only be retained at elevated pressures. The dispersions without LO/TO splitting of rutile phase (zero pressure), CaCl2 phase (7.07 GPa), a-PbO2 phase (9.75 GPa), pyrite phase (15.00 GPa), ZrO2 phase (33.96 GPa) and cotunnite phase (43.62 GPa) are calculated for the first time except for rutile. They are presented in Fig. 5a–f, respectively. They have shown entire dynamical stability except for a-PbO2 phase which is nearly stable. The reason for this nearly stable behavior needs to be investigated by experimentalists. Shieh et al. [19] have reported that not observing the a-PbO2 phase in their experiment may be caused by over-stepping its stability field and requirement of thermal activation or high shear stresses in the formation of this phase. X-ray diffraction study under room temperature and non-hydrostatic conditions in the range of 12–21 GPa pressure has shown that upon applying compression to material small amount of a-PbO2 is formed, and the phases of CaCl2 and a-PbO2 have undergone a phase transition to pyrite structure simultaneously, without finishing the formation of a-PbO2 phase entirely. Haines and Léger have achieved the structural refinement of a-PbO2 phase which constitutes half of the sample after recovering to ambient conditions by decompression [18]. Hence, although phonon dispersions of a-PbO2 phase show dynamical instability, it can be considered as a hard phase, but not metastable or instable. Production of a-PbO2 phase under special conditions (only decompression for instance) needs to be investigated by experimentalists. Our lattice dynamics calculations have shown that in the fluorite structured SnO2 presents dynamical instability definitely. So, it is impossible to propose it as one of the SnO2 polymorphs according to our calculations. 4. Conclusions The observed phase sequence of SnO2 polymorphs is as the following: rutile, CaCl2, a-PbO2, pyrite, ZrO2, cotunnite. Ab-initio calculations of Sn with LDA type US-PP (5s25p2 as valence) neglect the phases of CaCl2 and a-PbO2. Therefore, upon substituting it the one with 4d105s25p2 (as valence), it has become possible to make structural optimization of all phases. The structural results are in excellent agreement with experiments, but transition pressures are underestimated. Phonon dispersions of rutile phase involving LO/TO splitting of IR active modes are in good accord with the experimental values. Softening of B1g mode in rutile phase is demonstrated through the pressure dependencies of mode frequencies up to 10 GPa pressure. Thermal properties are evaluated by applying QHA and the results are compatible to experimental results. To the best of our knowledge, phonon dispersions of polymorphs of SnO2 are calculated for the first time except for rutile phase in this study. Although phonon dispersions of a-PbO2 phase show small amount of instability, it can be considered as one of the polymorphs of SnO2 since it is evaluated and refined experimentally, but fluorite phase showing throughout instability may not be proposed as a candidate to be a polymorph of SnO2. Consequently, first principles structural and lattice dynamics calculations based on DFT are reliable and successful tools in the investigation of group IVA, IVB and rutile structured dioxides. Acknowledgements Authors would like to thank the following institutions for carrying out this work. Calculations are performed in the Department of Physics by using the opportunities of the project supported by Pamukkale University (BAP Project No: 2012FBE002). Some of the calculations are performed on TUBITAK-ULAKBIM clusters. Corresponding author also thanks to Science Fellowships and

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