Elastic, electronic and optical properties of cotunnite

Physica B 407 (2012) 4495–4501

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Elastic, electronic and optical properties of cotunnite TiO2 from first principles calculations Tariq Mahmood a,b, Chuanbao Cao a,n, Faheem K. Butt a, Haibo Jin a, Zahid Usman a, Waheed S. Khan a, Zulfiqar Ali a, Muhammad Tahir a, Faryal Idrees a, Maqsood Ahmed b a b

Research Center of Materials Science, School of Material Science and Engineering, Beijing Institute of Technology, Beijing 100081, People’s Republic of China Centre for High Energy Physics, University of the Punjab, Lahore 54590, Pakistan

a r t i c l e i n f o

abstract

Article history: Received 14 March 2012 Received in revised form 2 August 2012 Accepted 6 August 2012 Available online 10 August 2012

The ultrasoft pseudopotential technique is used to explore the elastic, electronic and optical properties of cotunnite TiO2 using LDA and GGA proposed by Perdew Wang (PW91), Perdew–Burke–Ernzerhof (PBE) functional as defined by Wu and Cohen (PBEWC) and PBE functional for solids (PBESOL). The calculated elastic constants bulk modulus, shear modulus and Young’s modulus are in agreement with the previous theoretical reports. From our investigated shear anisotropy factors (A1, A2, and A3), we infer that cotunnite TiO2 is strong anisotropy in case of A1 and A2 and less anisotropy in case of A3. The value of mean sound speed and Debye temperature are calculated using the obtained values of elastic moduli. The calculated structural parameters are in accord with the reported experiment and theoretical results. Our obtained values of direct bandgaps show an improvement over the other previous theoretical reports. The values of the dielectric constant (e1(o)) of cotunnite TiO2 calculated within LDA and GGA approximations are 7.655 (LDA (CA-PZ)), 7.578 (GGA (PW91)), 7.685 (GGA (WC)) and 7.655 (GGA (PBESOL)), which are slightly higher than the experimental values of rutile (6.69) and anatase (6.55) polymorphs. The obtained values of the refractive index are consistent with rutile TiO2 and higher than anatase phase. The investigated imaginary part of dielectric constant and absorption spectrum reflect that the cotunnite TiO2 is a weak photocatalytic material as compared to anatase and similar to rutile phases. & 2012 Elsevier B.V. All rights reserved.

Keywords: DFT Bandgap Elastic constants Anisotropy factor Debye temperature Dielectric constant

1. Introduction Recently, researchers are immensely interested to find the hardest material to substitute diamond. The identification of such super-hard materials is based on largest bulk modulus (B) and shear modulus (G) [1–4]. Experimentally and theoretically, it is easier to work out the material’s bulk modulus (B) than shear modulus (G). In this domain, metal oxides have secured immense attention as they owe low reactivity with atmosphere. Before the emergence of cotunnite TiO2, the stishovite (SiO2) rutile structure was supposed to be the hardest material among metal oxides [3,5–7]. Dubrovinsky et al. discovered in 2001 that the cotunnite phase of TiO2 is the hardest material at high temperature ( 41000 K) and pressure ( 460 GPa) [8] , and increase the research interest in metal oxides. In 2004, Mattesini et al. synthesized cotunnite and cubic phases of TiO2 by heating (1900– 2100 K) anatase sample at low pressure (48 GPa), using a laser heated diamond anvil cell [9]. Later on, more sophisticated

n

Corresponding author. Tel.: þ86 10 6891 3792; fax: þ 86 10 6891 2001. E-mail address: [email protected] (C. Cao).

0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.08.006

experimental techniques along with density functional theory (DFT) have been employed to explain the cotunnite TiO2 structure. In theoretical solid state physics, the density functional theory (DFT) is one of the best tools to investigate the physical and chemical properties of the solid materials. We can predict properties (mechanical and magnetic, etc.) of condense matter physics, which are difficult to get experimentally. Many computational studies have been presented that explain the structural, mechanical and electronic properties of cotunnite TiO2 [10–16]. Daisuke Nishio-Hamane et al. [17] and Yahya Al-Khatatbeh et al. [18] have identified the cotunnite TiO2 at high temperature both experimentally and theoretically. The phonon and dielectric properties of orthorhombic structures (cotunnite and brookite) of TiO2 have been investigated within the framework of density functional perturbation theory (DFPT) by Shojaee et al. [10] and Haruhiko et al. have discovered a post-cotunnite phase of TiO2 at 161 GPa and 0 K with 210 GPa bulk modulus in both density functional theory (DFT) and high pressure experiment [11]. They have found that the cotunnite TiO2 is the hardest oxide among the other phases of TiO2 showing its stability at room temperature. The structural, electronic and elastic properties of cotunnite have been studied by Ming-Yu Kuo

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T. Mahmood et al. / Physica B 407 (2012) 4495–4501

et al. [19] and Carvaca et al. [14,15]. Despite all other phases of TiO2, there are few studies on the electronic and elastic properties of the cotunnite phase. The optical properties of cotunnite TiO2 are not fully explored to date. Due to the hardness of cotunnite TiO2, it motivates to investigate the detailed electronic, elastic and optical properties, so that its suitable future applications can be predicted. In our work, we investigate the elastic, electronic and optical properties of cotunnite TiO2 structure using local density approximation (LDA) and generalized gradient approximation (GGA) as exchange correlation functions within the framework of density functional theory (DFT). We considered the ultrasoft pseudopotential plane-wave (PP-PW) method in order to investigate the subjected properties. This paper is divided as follows: computational details are presented in Section 2. The elastic, electronic and optical properties are addressed in Section 3. Additionally, using the elastic constants (Cij), Debye temperature and sound wave speeds are also calculated and are presented in Section 3. We have summarized our work in Section 4.

2. Computational method The Cambridge Serial Total Energy Package (CASTEP) is used to calculate the total energy, electronic structure, elastic constants and optical spectra. This package uses ultrasoft pseudopotentials to describe the ion–electron interaction and makes use of a plane wave basis set to expand the wave functions for the total energy minimization. Ultrasoft pseudopotentials proposed by Vanderbilt [20] facilitate to perform calculations with the least cutoff energy. To attain the precise exchange and correlation energies of a specified ionic configuration, we use the LDA functional proposed by Ceperley and Alder [21] as reformulated by Perdew and Zunger [22] and GGA functional proposed by Perdew Wang (PW91) [23], Perdew–Burke–Ernzerhof functional (PBE) as parameterized by Wu and Cohen (PBEWC) [24] and PBE functional for solids (PBESOL) [25]. Pseudopotential calculations are performed for titanium (3s, 3p, 4s, and 3d) and oxygen (2s and 2p) electrons. The cutoff energy 800 eV is choosed to expand the electronic wave functions. The k-points 4n4n4 and 7n7n7 were selected to perform the energy and optical calculation for brillouin zone (BZ) integration respectively. The Broyden–Flecher–Goldfarb–Shanno algorithm is used to perform the geometry optimization for cotunnite TiO2. The convergence of the energy is set to 2.0 10 6 eV/atom ˚ All calculations are and displacement is set less than 0.002 A. considered as spin polarized.

3. Results and discussion 3.1. Elastic properties The calculated equation of state’s parameters of cotunnite TiO2 with LDA and GGA based methods are summarized in Table 1. The

third order of Birch Murnaghan equation of state is used to fit the volume data for cotunnite polymorphs of TiO2 [26]: 5=3 " 2=3 # V V PðVÞ ¼ 1:5B0 1 V0 V0 ( n

10:75ð4B00 Þ

"

V V0

#)

2=3

1

ð1Þ

where P(V) is the pressure as a function of volume, V0 is the equilibrium volume, B0 is the bulk modulus, and B00 is the pressure derivative of the bulk modulus. The calculated values of equilibrium volume by using prescribed methods are in good agreement with the experiment [8,17] and theoretical results [16]. The present value of bulk modulus (393.5 GPa) by using LDA (CA-PZ) method is consistent with the experiment value (431 GPa) reported by Dubrovinsky et al. [8]. On the other hand, the obtained values of bulk modulus by using GGA methods are in good agreement with the experiment (306 GPa) [17] and theoretical values (301 and 306 GPa) [10,16] but lower than Dubrovinsky et al. [8] and higher than the theoretical values (220 and 235 GPa) [12]. The pressure derivative of the bulk modulus should be close to 4.0 [27] for hard materials which is apparent from the current investigated values (3.6 (LDA), 4.7 (GGA-WC) and 4.4 (GGA-PBESOL)) of B00 , which are in good agreement with the experiment [17] and theoretical values (3.9 (LCAO-HF), 4.0) [8,12]. The generalized Hook’s law defines the relation of stress and strain tensors elements such as si ¼Cijej (si ¼ stress tensor element, ej ¼strain tensor element, and Cij ¼elastic constant matrix) [28]. The elastic constant matrix contains nine independent elastic constants of an orthorhombic system and is written as [29]: 2 3 0 0 0 C 11 C 12 C 13 6 7 0 0 0 7 C 22 C 23 6 6 7 6 0 0 0 7 C 33 6 7 C ij ¼ 6 ð2Þ 7 0 0 7 C 44 6 6 7 7 6 0 5 C 55 4 C 66

The comparison of our calculated elastic constants, bulk modulus, shear modulus, Young’s modulus and shear anisotropic factors in all LDA (CA-PZ) and GGA (PW91, WC, and PBESOL) with other theoretical results [10,12,13,19] are presented in Table 2. The present studies elastic constants are very low compared to the reported results by Koci et al. at zero pressure [30]. From Table 2 it can be seen that the calculated values of elastic constants in the LDA (CA-PZ) are noticeably larger than all GGA (PW91, WC, and PBESOL) calculations. Our calculated elastic constants, bulk modulus and shear modulus in the LDA are consistent with the theoretical results [10,19] using LDA exchange correlation functions but lower than the values reported by Zhao Jian-Zhi et al. [13] and Ming-Yu Kuo et al. (NFP-LMTO) [19]. The obtained values of elastic constants in GGA (PW91) are in good agreement with theoretical results [10,12].

Table 1 Calculated equation of state parameters for cotunnite TiO2. Cotunnite TiO2

Present work LDA (CA -PZ)

Ref. [10]

Ref. [12]

Experiment

LAPW

PAW

Ref. [8]

220 4.0

235 4.0

431 1.35

GGA PW91

V0 (A˚ 3) B0 (GPa) B00

Ref. [16]

WC

PBESOL

96

105.7

100.5

100.3

393.5 3.6

285.3 3.1

318.7 4.7

263 4.4

LDA

PW91

LDA (CA-PZ)

301

187

306 4.57

104.56

105.064

Ref. [17] 90.9 306 4.0


Following relations are used to determine the values of Poisson ratio (v, in Table 3) and Young’s modulus (Y in Table 2) [31]: Y¼

9BG 3B þ G

and

v¼

3B2G 2ð3B þ GÞ

ð3Þ

Additionally, the shear anisotropy factors (A1, A2, and A3) were calculated to understand the mechanism of elastic properties. The shear anisotropy factors for the [1 0 0], [0 1 0] and [0 0 1] planes between the /0 1 1S and /0 1 0S, /1 0 1S and /0 0 1S, /1 1 0S and /0 1 0S directions are formulated as [32]: A1 ¼

4C 44 , C 11 þ C 33 2C 13

A2 ¼

4C 55 , C 22 þ C 33 2C 23

and

A3 ¼

4C 66 C 11 þ C 22 2C 12

ð4Þ The value of these shear anisotropy factors is one, indicates an elastic isotropy, and if the values fluctuate around one (41o), it shows a certain degree of elastic anisotropy. For cotunnite TiO2, our calculations (Table 2) infer that the prominent anisotropy

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coefficients are found to be A1 and A2, which are less then one and less anisotropic in case of A3 (41). While, Caravaca et al. [14] reported that cotunnite TiO2 is anisotropic in case of A1 (o1) and less anisotropic in case of A2 ( 1) and A3 ( 1). An important parameter in thermoelastic properties of solids is Debye temperature, which is related to sound speeds and describes an atomic vibration’s temperature. The Debye temperature (YD), mean sound speed (vm), transverse speed (vt) and longitudinal speed (vl) are calculated from the following relations: " !#1=3 1=3 _ 6p2 N 1 2 1 YD ¼ vm , vm ¼ þ 3 , kB V 0 3 v3t vl sffiffiffiffi G B0 þ ð4=3ÞG 1=2 , vl ¼ ð5Þ vt ¼

r

r

where kB is Boltzmann’s constants, _ ¼ h=2p, h is Planck’s constant, N is the number of ions in the unit cell and V0 is the equilibrium volume. Predicted Debye temperature values together with sound

Table 2 Calculated elastic constants (GPa), bulk modulus (GPa), shear modulus (GPa), young’s modulus (GPa) and shear anisotropic factor for cotunnite TiO2. Elastic moduli

C11 C12 C13 C22 C23 C33 C44 C55 C66 B G Y A1 A2 A3

LDA (CA-PZ)

547 206 221 397 183 430 72 88 156 281 105 280 0.538 0.763 1.173

GGA

Ref. [10]

Ref. [12]

Ref. [13]

Ref. [30]

PW91

WC

PBESOL

LDA

PW91

GGA (PAW)

LDA

LDA

NFP-LMTO

454 182 185 198 68 173 59 13 121 260 98 289 0.955 0.505 1.680

505 195 202 339 145 362 71 54 144 241 86 230 0.613 0.525 1.268

506 194 207 327 130 340 70 57 142 228 85 227 0.648 0.560 1.276

555 234 237 408 196 450 89 109 159 301 119

474 190 202 258 90 224 62 13 127 187 55

478 165 191 291 85 279 58 38 129 200 77

563 330 254 645 343 782 231 203 254 348 203 525

688 258 240 510 253 649 129 133 204 370 162

646 250 229 475 283 632 148 203 246 362 163

Shear anisotropy factors

Table 3 Calculated Poisson coefficient, transverse, longitudinal, mean sound speeds (ms-1), Debye temperature (K) and density (g/cm3) for cotunnite TiO2. Physical quantities

Present work

Ref. [14]

LDA (CA–PZ)

v vt vl vm

GGA

0.333 4377 8763 4910 927 5.48

YD

r

PW91

WC

PBESOL

SIESTA

NFP-LMTO

0.444 7197 8900 7619 1438 5.02

0.340 4032 8199 4527.5 854.5 5.29

0.334 4005 8025 4493 848 5.3

0.31 5436 10327 6078 904

0.304 5350 10078 5976 899

Table 4 Calculated lattice constants and energy bandgaps for cotunnite TiO2. Cotunnite TiO2

Present work LDA (CA-PZ)

Ref. [10]

Ref. [16]

Ref. [19]

Experiment

LDA (CA-PZ)

LDA

Ref. [8]

GGA PW91

WC

PBESOL

LDA

PW91

Ref. [17]

˚ a (A)

5.188

5.178

5.218

5.215

5.181

5.214

5.259

5.204

5.163

5.119

˚ b (A) ˚ c (A)

3.063

3.199

3.111

3.110

3.043

3.150

3.145

3.046

2.989

2.982

6.104

6.385

6.191

6.185

6.069

6.256

6.322

6.067

5.966

5.957

Eg (eV)

1.773

2.038

1.822

1.826

1.5

1.6

1.72

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velocities, Poisson ratio and density are given in Table 3. The obtained values of Debye temperature and Poisson coefficient in LDA (CA-PZ) and GGA (PW91, WC, and PBESOL) are in good agreement with the reported theoretical results [14]. Likewise, the present values of sound speeds in GGA (PW91) are consistent and in case of LDA (CA-PZ), GGA (WC, PBESOL) are lower but comparable to with Cravaca et al. [13]. There is no experimental report for comparison. 3.2. Electronic properties The calculated structural parameters including lattice constants and the electronic bandgaps of cotunnite TiO2 with LDA and GGA based methods are summarized in Table 4. The obtained ˚ b¼3.063 A, ˚ and c¼6.104 A˚ by using lattice constants a ¼5.188 A, LDA (CA-PZ) method are in consistent with the experimental results [8,17] compared to the previous studies with LDA [16,19]. ˚ b ¼3.199 A, ˚ While, the obtained lattice constants (a ¼5.178 A, ˚ (a ¼5.218 A, ˚ b¼3.111 A, ˚ and c¼6.191 A) ˚ and and c ¼6.385 A), ˚ b ¼3.110 A, ˚ and c¼6.185 A) ˚ by using GGA (PW91), (a¼5.215 A, GGA(WC) and GGA (PBESOL) methods are slightly higher but comparable to the experimental and theoretical results. The calculated bandgaps (1.773 eV, 2.038 eV, 1.822 eV, and 1.826 eV) of cotunnite TiO2 by employing LDA (CA-PZ), GGA (PW91), GGA (WC) and GGA (PBESOL) are as good as the other theoretical results [10,19]. We did not find any experimental bandgap energy result of cotunnite TiO2 to compare with the present results. The graphical view of band structures of cotunnite TiO2 with respect to listed methods are presented in Fig. 1(a–c). The bottom of the conduction bands and top of the valance bands are occurred at G symmetry points which confirm nature of the bandgap as direct bandgap. The Fermi level is considered at 0 eV (top of the valance band) in the bandgaps 1.773 eV (Fig. 1(a)), 2.038 eV (Fig. 1(b)), 1.822 eV (Fig. 1(c)) and 1.826 eV (Fig. 1(d)), defining cotunnite TiO2 polymorph as a semiconductor material. Fig. 2 displays the projected and total density of states (PDOS and DOS) of cotunnite structure. In Figs. 1 and 2, the band structure Ti 3d and O 2p states are dominant to construct the conduction and valance bands respectively. The lower states far from the fermi level are

made up of O 2s states (not displayed here). The calculated valance bands (Fig. 1(a–c)) have bandwidth of 6.88 eV (LDA (CA-PZ)), 6.12 eV (GGA (PW91)), 6.6 eV (GGA (WC) and GGA (PBESOL)) have excellent agreement with the theoretical value of 6.94 [19]. Our calculated conduction bands have bandwidth of 2.967 eV (LDA (CA-PZ)), 2.452 eV (GGA (PW91)), 2.743 eV (GGA (WC)) and 2.774 eV (GGA (PBESOL)). It can be seen from Fig. 2 that in the conduction band Ti 3d states are dominant and O 2p states are predominant, while in the valance band O 2p states are dominant and Ti 3d states are predominant. The character of the bandgap for different applied methods is same with maximum energy bandgap of 2.038 eV as obtained in GGA (PW91). The major contribution of O 2p and Ti 3D states in valance and conduction bands and transitions across the bandgaps are also involved in bonding.

Fig. 2. Total and partial density of states of cotunnite polymorphs TiO2.

Fig. 1. Predicted band structures of cotunnite TiO2 obtained by: (a) LDA (CA-PZ), (b) GGA (PW91), (c) GGA (WC), and (d) GGA (PBESOL) methods.


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Fig. 3. Predicted optical properties of cotunnite TiO2: (a) imaginary part of dielectric constants, (b) real part of dielectric constants, (c) refractive Index, (d) reflectivity, (e) absorption, and (f) energy loss function.

3.3. Optical properties A range of interband and intraband optical transitions are related to the electronic structure of materials which can be identified with the help of tensor components of complex dielectric function. However, in case of semiconductor materials, the direct and indirect interband transitions are considered instead of intraband transitions. The direct interband transitions have a significant contribution to the dielectric function. Therefore, the direct interband transitions are taken into account,

whereas indirect interband transitions are neglected due to their negligible contribution to the dielectric function [33]. We know that the optical properties such as dielectric constants, refractive index, absorption and energy loss function are the main features to characterize the materials. These optical properties are linked with complex dielectric function which can be defined by the following mathematical relation:

eðoÞ ¼ e1 ðoÞ þ ie2 ðoÞ

ð6Þ

4500


where e1(o) ¼real part and e2(o) ¼imaginary part. The imaginary part of the dielectric function is defined as [34]:

e2 ðq-Ou^ ,hoÞ ¼

2e2 p X

Oe0

2 cck 9u:r9cvk dðEck Evk EÞ

ð7Þ 4. Conclusion

k,v,c v

c

where the symbols e, u, ck , and ck represent electronic charge, incident electric field, the valence band and conduction band wave functions at k point respectively. The Kramers–Kronig relation [35] is used to obtain the real part (e1(o)) of the dielectric function from the imaginary part (e2(o)). The energy loss spectrum (L(o)), absorption coefficient (I(o)), refractive index (N(o)) and reflectivity (R(o)) are related with the dielectric function (e(o)) as follows: 1 e2 ðoÞ ¼ 2 ð8Þ LðoÞ ¼ Im eðoÞ e1 ðoÞ þ e22 ðoÞ IðoÞ ¼

2ko c

ð9Þ

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eðoÞ þ e1 ðoÞ NðoÞ ¼ 2

ð10Þ

2

RðoÞ ¼

ðn1Þ2 þ k 2

2

ðn þ 1Þ þk

cotunnite phase has weak photocatalytic property for visible light.

ð11Þ

Fig. 3 demonstrates the calculated imaginary part (e2(o)), real part (e1(o)) of the dielectric function, refractive index (N(o)), reflectivity (R(o)), absorption spectra (I(o)) and energy loss function (L(o)). The imaginary part (e2(o)) of the dielectric function starts from 1.0 eV and major peaks (Fig. 3(a)) are identified at 4.2 eV (GGA-PW91), 4.7 eV (GGA (WC and PBESOL)) and 4.9 eV (LDA (CA-PZ)) in the respective GGA and LDA functions which belong to R–R direct transition due to Ti 3d and O 2p direct interband transitions as shown in Fig. 2. Our obtained values of dielectric constants (Fig. 3(b)) are 7.655, 7.578, 7.685 and 7.655 in LDA (CA-PZ), GGA (PW91), GGA (WC) and GGA (PBESOL). p Furtherffiffiffiffiffiffiffiffiffiffiffiffi more, our computed values of refractive index (N¼ e1 ðoÞ), where e(o)¼0 are 2.76, 2.75, 2.77 and 2.76 with employing LDA (CA-PZ), GGA (PW91), GGA (WC) and GGA (PBESOL) respectively as shown in Fig. 3(c). The energy loss spectrum (L(o)) is a vital characteristic to approximate the consumption of fast moving electrons within the material’s medium as shown in Fig. 3(f). The prominent peaks of energy loss spectrum (L(o)) are positioned in the range of 38.3 eV–38.5 eV where possibly the electrons are unbound and start plasma oscillation. The first major peak of I(o) lies from 6.0 eV to 9.5 eV and second major peaks occurred at 37 eV as displayed in Fig. 3(e). The absorption peak at 37 eV is due to d-band transitions from the valence band to the conduction band. From Fig. 3(d), the prominent peaks of reflection spectra are occurred in the same energy limits as absorption spectrum. Furthermore, it can be observed that the major peaks of reflection spectra occurred where the peaks of refractive index, imaginary and real parts of the dielectric function, start decreasing and become zero. The obtained values of dielectric constants (Fig. 3(b)) and refractive index (Fig. 3(c)) are approximately equal to rutile (6.693 [36] and 2.71 [37]) TiO2 and higher than anatase (6.55 [38] and 2.56 [39]) phase of TiO2. There are no experimental and theoretical reports on the cotunnite optical properties for comparison except E. Shojaee et al. [10] who studied the electronic and static dielectric permittivity tensors of different phases of TiO2 including cotunnite. Our investigated results with LDA and GGA approach are presenting detailed optical properties of the cotunnite TiO2 for the first time. Present studies suggest that the

The elastic, electronic and optical properties of the least studied cotunnite structure of TiO2 are investigated by first principles in the LDA (CA-PZ) and GGA (PW91, WC and PBESOL) exchange correlation functions. Elastic properties of cotunnite structure are reported in details and a comparison is given with the previous theoretical results. To understand the mechanism of elastic properties, the shear anisotropy factors (A1, A2, and A3) for the [1 0 0], [0 1 0] and [0 0 1] planes between the /0 1 1S and /0 1 0S, /1 0 1S and /0 0 1S, /1 1 0S and /0 1 0S directions are also investigated respectively. The prominent anisotropy coefficients for cotunnite TiO2 are A1 and A2. The Debye temperature, Poisson coefficient, density, mean sound speed (vm) along with transverse (vt) and longitudinal sound speeds (vl) are calculated with the help of the elastic moduli values. Our calculated elastic properties reveal that cotunnite TiO2 is one of the hard polymorphs of TiO2. The structural parameters and energy bandgaps for cotunnite structure are calculated and compared with the experimental and theoretical values. Our obtained values of equilibrium structural parameters are in good agreement with the previous reported experimental and theoretical results. The obtained energy bandgaps are consistent with the theoretical results; there is no experimental report on the energy bandgaps of this structure. The nature of our predicted energy bandgaps is directed at symmetry point G. The resultant values of the valance bands 6.88 eV (LDA (CA-PZ)), 6.12 eV (GGA (PW91)), 6.6 eV (GGA (WC) and GGA (PBESOL)) are in excellent agreement with the theoretical value (6.94) calculated by Vanderbilt et al. The dielectric constants, refractive index, absorption spectra, energy loss spectra and reflection spectra for this polymorph are reported. Our obtained values of dielectric constants (e1(o)) ((Fig. 3(b)) are 7.655, 7.578, 7.685 and 7.655 in LDA (CA-PZ), GGA (PW91), GGA (WC) and GGA (PBESOL)) are different from rutile (6.69) and anatase (6.55) polymorphs. On the other hand the obtained values of the refractive index (2.76, 2.75, 2.77, and 2.76 by employing LDA (CA-PZ), GGA (PW91), GGA (WC) and GGA (PBESOL)) are in consistent with rutile (2.71) and differ with to anatase (2.56) phase. The investigated imaginary part of dielectric constant and absorption spectrum reflect that the cotunnite polymorph of TiO2 is a weak photocatalytic material as compared to anatase and same as rutile phases. To the best of our knowledge, there is no experimental report on the optical properties of cotunnite polymorph, due to the complexity in its preparation.

Acknowledgment This work was supported by National Natural Science Foundation of China (20471007, 50972017) and the Research Fund for the Doctoral Program of Higher Education of China (20101101110026). References [1] [2] [3] [4] [5] [6]

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