Cite this Article
Dash Debashish, Pandey Chandan K, Chaudhury Saurabh, Tripathy Susanta K. Structural, electronic, and mechanical properties of cubic TiO2: A first-principles study. Chinese Physics B, 2018, 27(1): 017102  
Permissions
Structural, electronic, and mechanical properties of cubic TiO2: A first-principles study
Dash Debashish1, †, Pandey Chandan K1, Chaudhury Saurabh1, Tripathy Susanta K2
Department of Electrical Engineering, National Institute of Technology, Silchar, Assam 788010, India
Department of Electronics and Communication Engineering, National Institute of Technology, Silchar, Assam 788010, India

 

† Corresponding author. E-mail: debashishdashnits@gmail.com

Abstract

We present an analysis of structural, electronic, and mechanical properties of cubic titanium dioxide (TiO2) using an all electron orthogonalzed linear combinations of atomic orbitals (OLCAO) basis set under the framework of density functional theory (DFT). The structural property, especially the lattice constant a, and the electronic properties such as the band diagram and density of states (DOS) are studied and analyzed. The mechanical properties such as bulk moduli, shear moduli, Young’s Moduli, and Poison’s ratio are also investigated thoroughly. The calculations are carried out on shear moduli and anisotropy factor for cubic TiO2. The Vickers hardness is also tested for fluorite and pyrite cubic-structured TiO2. Furthermore, the results are compared with the previous theoretical and experimental results. It is found that DFTbased simulation produces results which are approximation to experimental results, whereas the calculated elastic constants are better than the previous theoretical and experimental values.

PACS: 71.15.Mb;77.84.Bw
Keyword:density functional theory;cubic TiO2;structural;electronic;and mechanical properties
1. Introduction

TiO2 is found in nature in many different structures and crystalline forms such as monoclinic, orthorhombic, simple tetragonal, and cubic.[1] It provides many different utilizations when it is grown in different polymorphic forms. In nature, titanium dioxide forms in three different ways, namely, rutile, anatase, and brookite.[2] Since the last decade, different types of advanced materials have been invented by applying high pressure on the crystal structure for various electronic applications. Yugui et al. synthesized platinum nitride as a noble nitride material under high pressure and high temperature in its crystalline form.[3] They found out the electronic properties using Raman scattering and the mechanical properties like bulk modulus and shear modulus using Voight, Reuss, and Hill (VRH) theory. Similarly, Wuming et al. described different mechanical properties of the earth and analyzed how the high pressure variations change the earth’s mantle.[4] Li et al. used laser light excitation to investigate the Raman and photoluminescence spectra of Y2O3/Eu3+ and Y2O3/Eu3+/Mg2+ nanorods under high pressure and noticed that a structural transition from cubic to amorphous takes place when the pressure increases above 24 GPa.[5] They also found that when the pressure is 8 GPa, distorted phases of Raman and photoluminescence spectra become visible.[5] Moreover, Wang et al. investigated disordered materials using the synchrotron radiation-based technique and the pressure-induced phase transformation of disordered alloys. They reported the mechanical behavior and how Shanghai synchrotron radiation facility (SSRF) helps to investigate various structural properties of different disordered materials.[6,7] Thus, it is clear that high pressure plays as an important role of catalyst in converting a material from one form to another for different applications. In the last decade, titanium dioxide has been extensively studied both theoretically and experimentally due to its numerous advantages like non-toxicity, low cost, long-term stability and also due to its superb catalytic feature.[8] By applying high pressures, titania is grown in many different forms such as orthorhombic, columbite, cotunnite, monoclinic baddeleyite, and cubical.[9] When a high pressure is applied to rutile/anatase type polymorphs (eg., SnO2, PbO2, HfO2, and RuO2), they get converted to the isostructural type of fluorite (CaF2) or pyrite (FeS2) structure. Investigations are still in the infant stage to explore different applications of titanium dioxide polymorphs. Some distinguished industrial and commercial applications of TiO2 such as self-cleaning tiles, self-cleaning paints, windows panes, transparent windows and glasses, high efficiency solar cells, dynamic random access modules, and creating super hard materials are highly encouraged in the present era.[10] However, considering future conservation of energy and cleanliness of environment, the solar cell is the most important application for which cubic TiO2 is highly preferred.[11] In general, the cubic TiO2 is synthesized from anatase TiO2 by heating at a high temperature of 1900–2100 K in a diamond-anvil cell under a pressure of 48 GPa.[12–14] Investigations on both phases of cubic TiO2 have been carried out by many researchers both experimentally and theoretically. Initially, Swamy and Muddle[15] concluded that the theoretical values of pyrite TiO2 are closer to the experimental values, except a lower bulk modulus. However, Liang et al.[13] found little difference between both the phases of cubic TiO2, which contradicts the facts given by Swamy and Muddle.[15] According to Mattesini et al.,[16,17] the only difference between the two cubic phases of TiO2 is the positions of oxygen atoms located. In pyrite TiO2, the oxygen atoms are located at ±(0.34,0.34,0.34); whereas in fluorite structure, these are at ±(0.25,0.25,0.25). Later on, Kim et al. [18] performed a number of experiments and concluded that pyrite TiO2 is unstable because of the imaginary frequencies in the phonon spectra during the entire pressure range, however fluorite TiO2 is stable because of the absence of such imaginary frequencies. Miloua et al.[8] studied both cubical TiO2 in detail using the full potential linearized augmented plane wave method. Similarly, Zhao et al. investigated various material parameters such as diffusion and interfacial energy using atomic simulations.[19] In the same way, Hu et al. carried out a deep study on the electronic and other properties using density functional theory (DFT) and other ultra-soft pseudopotentials.[20]

From the literature, it is clear that numerous investigations have been carried out for natural TiO2. Additionally, a few investigations on the structural, electronic, and elastic properties for a catalytic material are also available. However, exploration of material properties under high pressure is quite neglected. Detailed investigations on cubic TiO2 in a systematic way are rare due to ambiguity of experimental and simulated results. Hence, detailed investigations of structural, electronic, and elastic properties of cubic TiO2 under high pressure are motivated for different applications. In this paper, a systematic ab-initio investigation of the structural, electronic, and mechanical properties of cubic TiO2 under high pressure is carried out using the orthogonalized linear combinations of atomic orbitals (OLCAO) method. The rest of the paper is organized as follows. Section 2 describes various computational methods adopted in this study of TiO2. The results and discussion are present in Section 3. Finally, conclusions are drawn in Section 4.

2. Computational methods

In this paper, computations are carried out only for fluorite and pyrite type TiO2. Fluorite is known as the conventional cubic cell, having a space group of , whereas pyrite TiO2 is known to be isostructural to the modified fluorite structure, having a space group of .[21] The lattice parameters considered here are a = b = c = 4.787 Å for fluorite and a = b = c = 4.844 Å for pyrite TiO2, and the structure has been optimized using maximized force. Zero constraints are assumed for optimization and the limited memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) algorithm[22–25] is utilized as the optimization method because of its popularity for parameter estimation in machine learning. The BFGS algorithm is a quasi-Newton method, which converges as early as possible compared to other algorithms like the conjugate gradient (CG) optimization technique. The LBFGS algorithm works on the approximate Hessian matrix. In each iteration, it updates the approximated Hessian matrix by using vector–vector products. With fewer iterations, it reaches into its local minimum without sticking at the time of calculation. But for intermediate calculations, it needs a huge amount of memory for storing the temporary approximated Hessian matrix; and if more intermediate results are stored, then there is a chance of losing convergence. To avoid this problem, it is preferable to use the limited memory BFGS. Furthermore, energy minimization by varying lattice constants is also carried out. During the simulation, the default lattice parameters of both crystals are varied within ±15% and then we obtain the characteristic curves between the total energy in Hartree and the total volume in Bohr3, which are shown in Figs. 1(a) and 1(b) for fluorite and pyrite TiO2, respectively. The lowest energy lattice constants are considered for simulation because minimizing the total energy of the crystal determines an appropriate set of linear combination of coefficients. For simulation, the orthogonalized linear combinations of atomic orbitals method[26] is adopted here, which is an all-electron technique adopted for calculating 3p64s23d2 and 2s22p4 states as valence electrons for Ti and O, respectively. The optimized lattice structures of fluorite and pyrite TiO2 are shown in Figs. 3(a) and 3(b), respectively. This method correctly predicts the molecular orbitals due to the orthogonal simulation pattern. The LCAO method initially assumes that the total number of molecular orbitals is equal to the total number of atomic orbitals included in the linear expansion. In other words, n atomic orbitals combine to form n molecular orbitals. The OLCAO is implemented in the framework of density functional theory having the exchange correlation type of local density approximation (LDA). For the simulation of various parameters, Dirac–Bloch exchange with Perdew–Zunger (PZ) correlation[27] is considered. Other parameters like Hubbard U, spin–orbit coupling, and van der Waals corrections are disabled. The grid mesh cut-off is taken as 1633 eV (60 Hartree). The Monkhorst–Pack scheme[28] is used for the k-point sampling, which is 12 × 12 × 12 for fluorite TiO2 with a total of 864 symmetric points in the first Brillouin zone and 4 × 4 × 4 for pyrite TiO2 with 32 symmetric points in the first Brillouin zone.[29] Double zeta polarized basis sets are used for calculation. For the oxygen atoms, the Fritz-Haber-Institute (FHI) [Z = 6] LDA PZ pseudopotential has been used, whereas for the titanium atoms, the FHI [Z = 4] LDA PZ pseudopotential has been used.

Fig. 1. (color online) Total energy vs. total volume for (a) fluorite-structured TiO2 and (b) pyrite-structured TiO2.
Fig. 3. (color online) (a) Band diagram for fluorite TiO2 and (b) energy band gap between VBM and CBM.
3. Results and discussion

With the framework presented in the previous section, the simulation results and analysis on the structural, electronic, and mechanical properties of fluorite and pyrite TiO2 are presented in this section in detail. The simulations are carried out in a HP workstation with 12 GB RAM (Z380) using Virtual Nano Lab software from Quantumwise Atomistix Tool Kit (ATK).[30–32]

3.1. Structural properties

First of all, a systematic structured optimization has been carried out for both phases of TiO2 over a wide range of lattice parameters. As all the physical properties are related with the total energy, so those lattice constants which give a minimum total energy are considered as the equilibrium lattice constants. If the total energy is obtained, then any of the physical properties related to the total energy can be easily calculated.

Both the fluorite and pyrite titanium dioxide structures are cubic in nature and are considered for simulation. The optimized lattice constants for fluorite and pyrite structures are shown in Fig. 2. The geometrical configuration of the pyrite phase is given with the distance between Ti and O atoms dTi − O = 1.97 Å, the distance between two O ions dO − O = 2.57 Å, and the distance between two titanium atoms dTi − Ti = 3.42 Å. Similarly, for the fluorite phase, dTi − O = 2.07 Å, dO − O = 2.39 Å, and dTi − Ti = 3.38 Å.

Fig. 2. (color online) (a) Optimized fluorite TiO2 with distance between Ti’s, O’s, and Ti–O atoms. (b) Optimized pyrite TiO2 with distance between Ti’s, O’s, and Ti–O atoms. White balls represent Ti atoms and red balls represent O atoms.

The equilibrium lattice constants, the bulk modulus, and other properties have been simulated using Birch–Murnaghan’s equation of state (EOS)[33,34] in LCAO-based calculations. The totally relaxed Wyckoff positions and cell volume are considered during simulation. The results are compared with the theoretical and experimental values reported earlier. The lattice parameter obtained for fluorite-structured TiO2 is 1.7% smaller compared to the experimental values.[35] Here, for all the calculations, the optimized minimum energy lattice parameters computed earlier are used. The bulk moduli of fluorite and pyrite TiO2 using OLCAO are found to be 296.2 and 270.83, respectively. The bulk modulus of fluorite TiO2 is higher (about 3%–4%) compared to the theoretical results given in Refs. [8], [9], [12], and [35]. It can be seen that the calculated value is very close to the other exchange correlation-based values given in Refs. [10] and [13]. This small difference may be due to better optimization algorithm and the use of different density-functional-based electronic structure. Similarly, for the pyrite structure, the calculated bulk modulus is much closer (about 0.7% to 3.17% undervalued) to the other calculated values given in Refs. [9], [12], and [13].

3.2. Elastic properties

Elastic property is directly related to various solid state properties and creates a link between the mechanical and dynamical behaviors of crystals. It also provides information regarding the interatomic potential, phonon spectra, and more importantly the nature of the forces operating in solids. Much information regarding material properties can be obtained from the elasticity matrix. Generally, it provides the bonding character between adjacent atomic planes, the bonding anisotropy, the stability of the structure, and the stiffness of the material. So their ab-initio calculation requires precise ways of computation. Basically, the forces and elastic constants are functions of the first-order and the second-order derivatives of the potentials.[36–39]

Here, to compute the independent elastic constants Cij, assumptions of small lattice distortions are made in order to remain within the elastic domain of the crystal. As both crystals are cubic in nature, there are only three independent elastic constants C11, C12, and C44.[9] With the help of these three elastic independent constants, the mechanical stability can be determined. Initially, the positive definiteness of the stiffness matrix should be checked before proceeding to further calculations. The conditions are as follows:[40,41]

If all the above conditions are satisfied, then the material is mechanically stable.

The bulk modulus (B) and shear modulus (G) are calculated using two different theories, such as Reuss theory and Voigt theory. BR and GR are used here to represent the bulk and the shear moduli obtained using the Reuss theory,[42] whereas BV and GV represents the bulk and the shear moduli obtained using the Voigt theory.[43] Under the Reuss approximation, BR and GR are expressed as

where s11, s12, and s44 are the compliance matrix elements of both phases of TiO2. By using the Voigt theory, BV and GV are expressed as

Hill[44] suggested that the actual effective elastic moduli of anisotropic crystalline materials should be approximated by the arithmetic mean of bulk and shear moduli. As per the Hill approximation, the bulk modulus BHill and shear modulus GHill are given as

It is well known that the bulk and shear moduli are measures of hardness of any crystalline solid. The bulk modulus is a measure of resistance to volume change by the applied pressure. The shear modulus is a measure of resistance to reversible deformations upon shear stress. Thus, the shear modulus is a better predictor of hardness than the bulk modulus. Generally, the shear modulus mainly depends on C44. The larger C44 is, the larger the shear modulus will be. C44 of pyrite TiO2 is larger because the deformity of oxygen atoms from cubic to rhombohedral largely enhances the rigidity against the shape deformations at some directions. If the shear modulus is larger, the material is harder. Thus, the shear modulus is a measure of resistance to the shape change and is more pertinent to hardness. Here, for the fluorite-structured TiO2, the bulk modulus is found to be 296.2, whereas the shear modulus is 68.126. For pyrite TiO2, the bulk modulus is 270.83 but the shear modulus is about 100 which is quite large compared to that of fluorite TiO2. So pyrite TiO2 is much harder than fluorite TiO2. Later, by using Vickers hardness, it is also proved that pyrite TiO2 is harder than fluorite TiO2. All the calculated values are listed in Tables 1 and 2.

Table 1.

Structural, electronic, and elastic properties of cubic TiO2 using the OLCAO, single-crystal (SC), Voight–Reuss–Hill (VRH), and Hashing–Shtrikman (HS) schemes and other exchange correlation methods used with LDA.

.
Table 2.

Elastic constants Cij, calculated anisotropy A, shear modulus G, Young’s modulus Y, Poisson’s ratio υ, and Vickers hardness Hv.

.

There is another averaging scheme known as Hashing–Shtrikman (HS). The shear modulus using HS scheme[8,45] is expressed as By averaging both, the shear modulus in the HS scheme can be given as Whereas the shear modulus for single crystal is calculated as By comparing all the shear moduli, it is observed that GVRH < GHS < GSC for both structures of cubic TiO2.

Young’s modulus can be defined as the ratio of stress and strain, which basically is used to measure the stiffness of the solid. If the Young’s modulus is high, then the material is stiffer. The Young’s modulus has an important impact on the ductility. When it increases, the covalent nature of the material increases. Direct calculation of Young’s modulus from the bulk and shear moduli is avoided in the simulation due to the existence of many different alternative ways of calculation. Here the Young’s modulus is calculated from compliance matrix directly, as it is not influenced by any conventions. Thus it is given as

where Si is the inverse of the elastic constants matrix which is also known as the elastic compliance matrix. The calculated Young’s modulus is presented in Table 2. The Poisson’s ratio on the other hand is indicative of the degree of directionality of the covalent bonds. For covalent materials, the Poisson’s ratio is small (υ = 0.1), whereas for ionic materials, a typical value of υ is 0.25.[46,47] The interatomic bonding forces can be known from Poisson’s ratio. For a brittle material, the Poisson’s ratio is very small, whereas for a ductile material, it is larger than 0.33.[48] Poisson’s ratio (υ) can be expressed as[49] For fluorite type TiO2, υ is 0.393, which means that the material is ductile and fluorite has better interatomic bonding forces. Whereas for pyrite-type TiO2, υ is 0.335. So it can be seen that the pyrite material is also ductile but there exists a weak interatomic bonding force compared to fluorite-type TiO2. All the calculated values are given in Table 2. For proving the ductility of the crystal, Pugh provided a parameter B/G. If B/G > 1.75, then the material is ductile else brittle. For fluorite TiO2, B/G is 4.34, whereas for pyrite TiO2, it is 2.71. Both the crystal phases satisfy the Pugh’s criteria,[50] both can be treated as a ductile material.

Many high symmetry crystals exhibit a low degree of elastic anisotropy and vice versa.[51] The anisotropy is dependent on the orientation of the elastic modulus. Anisotropic behavior is an important indicator in solid state physics. As such, the zener anisotropic factor is defined as[52] If any crystal has an anisotropic value of unity, then that material exhibits isotropic properties; whereas if the value is less or more than unity, then it represents a varying degree of anisotropy. Fluorite-type TiO2 has high symmetry points, so it provides a very low anisotropy of 0.19. However, due to the low symmetry nature of pyrite type TiO2, it provides a high anisotropy of 0.73. Another way of measurement of elastic anisotropy is the percentage of anisotropy in compression and shear[44,45,52,53] For any crystals, these two values can range from 0 (isotropic) to 100% (anisotropy). Here, for the two types of titanium dioxide, the compression anisotropy is very small whereas the shear anisotropy is large.

Again, testing of hardness of the material can be done for polycrystalline material based on the squared Pugh’s modulus ratio (k = G/B) and the shear modulus (G) as[54] From the calculated Vickers hardness, it can be seen that pyrite-structured TiO2 is harder than fluorite-structured TiO2.

3.3. Electronic properties

To illustrate the electronic properties, we focus primarily on two major parameters, namely, the band structure and the total density of state (TDOS). The band structure usually gives a detailed idea about the electronic and optical properties. The energy band structures of cubic fluorite and pyrite single crystals have been calculated along the high symmetry directions in the first Brillouin zone using OLCAO–Perdew–Zunger (PZ) method. The band structures of both crystals are calculated along the special lines connecting the high-symmetry points, Γ(0, 0, 0), X(1/2, 0, 0), M(1/2, 1/2, 0), Γ(0.005, 0.005, 0), R(1/2, 1/2, 1/2) in the k-space.

The energy band diagrams calculated using LDA for both fluorite TiO2 and pyrite TiO2 are shown in Fig. 3. It can be seen from Fig. 3 that the Fermi level lies at the lower edge of the conduction band in the fluorite-structured TiO2 and has a direct band gap with a simulated value of band-gap energy of 0.89 eV, which are the characteristics of a narrow band semiconductor material. It means that the fluorite-structured TiO2 can absorb light with longer wavelength, which is mainly applied to split water into oxygen and hydrogen. The top of the valence band (valence band maximum, VBM) and the bottom of the conduction band (conduction band minimum, CBM) lie near to the Γ point of the optimized maximum peak of the crystal. The estimated band gap energy is found to be very close to that in Ref. [12], while compared with some other work,[9] the result is contradictory too. As seen from Figs. 3 and 5(a), the electronic structure of fluorite type TiO2 is mainly influenced by Ti 3d and O 2p states, as the edges of O 2s states are far below the Fermi level, nearly at an edge distance of 8 eV. From Fig. 5(a), it is observed that the unoccupied regions (0.5 eV to 1.5 eV and 2.5 eV to 5 eV) are positioned mainly by Ti 3d states. There are 14 bands which are below the Fermi level and have a width of approximately 8.5 eV. This is primarily due to O 2p states, but has a notable Ti 3d contribution. If compared with other phases of TiO2, it can be seen that the TDOS of fluorite TiO2 has some divergent characteristic (which may be due to alternation of the local symmetry throughout the Ti atoms). Except for the splitting of Ti 3d states (of the empty orbitals), the remaining occupied states between O 2p and Ti 3d (from −8 eV to 0 eV) are however hybridized, which of course provides outcomes, more degree of co-valent bonding between Ti–O of fluorite TiO2. The major peaks appear on the top of the valence band and the bottom of the conduction band of fluorite-structured TiO2 near to −5.5 eV, −2.5 eV, 1.5 eV, 2.5 eV, 4 eV, and 4.9 eV respectively, which are due to hybridization of Ti 3d and O 2p orbitals.

Fig. 5. Total density of states for (a) fluorite TiO2 and (b) pyrite TiO2.

The bandgaps given in Refs. [9] and [29] are indirect in nature and found to be 1.061 eV and 1.16 eV, respectively. From the simulation, the band gap found for fluorite TiO2 varies from 0.89 eV to 1.79 eV. The energy band gap found in this study is ~ 20% less than DFT-GGA-based simulation[29] and ~ 16% less than that in Ref. [12]. LDA mostly provides the direct band gap, whereas GGA provides an indirect band gap in most of the simulation studies.[9,12,29] Thus it can be inferred that the simulated energy band gap in this study is quite close to other established approaches.

The simulated energy band diagram for pyrite-type TiO2 is shown in Fig. 4. It can be seen from the figure that the energy band gap has quite similar behavior to fluorite-type TiO2 and exhibits an indirect band gap with an energy of 1.18 eV. The VBM lies near to the Γ point and the CBM lies very close to the R point of BZ. As seen from Figs. 4 and 5(b), the electronic structure of pyrite-type TiO2 is mainly influenced by Ti 3d and O 2p states, as the edges of O 2s states are far below the Fermi level nearly at an edge distance of 9.5 eV. From Fig. 5(b) of total DOS, it is observed that the unoccupied regions (0.5 eV to 3 eV and 4 eV to 6.25 eV) are positioned primarily by Ti 3d states. There are 16 bands which are below the Fermi level and have a width of approximately 6.75 eV. They are primarily due to O 2p states, but have a notable Ti 3d contribution. If compared with other phases of TiO2, it is seen that the TDOS of pyrite TiO2 has some divergent characteristic, which is probably due to alternation of the local symmetry throughout the Ti atoms. Except for this splitting of Ti 3d states in empty orbitals, the remaining occupied states between O 2p and Ti 3d (from −6.7 eV to 0 eV) are however hybridized, which means more degree of covalent bonding between Ti–O of pyrite TiO2. The major peaks appear in the VBM and CBM of pyrite-structured TiO2 nearly at −5.8 eV, −5.25 eV, 1.8 eV, 2.9 eV, 4.85 eV, and 5.9 eV respectively, which are generated by hybridization of Ti 3d and O 2p orbitals. According to Refs. [9], [12], and [29], the indirect band gap varies from 1.438 eV to 1.451 eV. Here, the simulated band gap is 17.94%–20.27% lower than other simulated results. This is an inherent drawback of DFT and associated difference in crystal structure.

Fig. 4. (color online) (a) Band diagram for pyrite TiO2 and (b) energy band gap between VBM and CBM.

Another most important material property is the DOS, which also describes the electronic property of a material. In general, it shows the energy representation for describing molecular dynamics and spectroscopy. Mathematically, it is expressed as g(E) which describes the number of quantum states available within an energy range from E to E + dE. In other words, the DOS is a measure of number of electron (or hole) states per unit volume at a given energy. The total densities of state of fluorite TiO2 and pyrite TiO2 are illustrated in Figs. 5(a) and 5(b). From Fig. 5, it can be seen that the lowest valence band occurs at −20 eV for fluorite TiO2 whereas for pyrite TiO2, it starts well before −20 eV. The valence band between −20 eV and −15 eV in fluorite TiO2 is mainly contributed by O-s state, whereas in the energy range between −7.5 eV and 0 eV, it is mainly contributed by O-p state. For pyrite TiO2, the valence band between −20 eV and −13 eV is mainly contributed by O-s state, whereas from −10 eV to 0 eV, it is contributed by O-p state. The highest occupied valence bands for fluorite are essentially dominated by O-1s and Ti-2p states. The Ti-1s state also contributes to the valence band but the density of this state is very small compared to O-1s and Ti-2p states. In the case of pyrite, the highest occupied valence bands are dominated by O-1s and Ti-1s states. A little contribution is also seen from Ti-2p and Ti-3d orbitals towards valence band formation. The lowest occupied conduction band just above the Fermi energy level is mainly contributed by Ti-3d and O-2p states for both crystals. Other orbital atoms also contribute, but their contributions are negligibly small compared to Ti-3d and O-2p orbital atoms.

4. Conclusion

A simulation-based study is presented here on the structural, mechanical, and electronic properties of high pressurized fluorite and pyrite TiO2 using orthogonal LCAO basis sets under the framework of DFT. All the results are compared with the previously calculated LDA ones and found an improvement over the elastic properties such as bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, and so on. Ductility of both structures of TiO2 is addressed from their calculated values. It is ensured that all types of ductility conditions are satisfied in both structures of cubic TiO2. Further, the compound is found to be mechanically stable, elastically anisotropic (tested through various ways) and exhibits ductile property. As there is no perfect experimental data available so far, this work can be considered as very accurate and very much close to work carried out in Refs. [8]–[10], [12], [13], and [21]. The calculated bulk moduli are also very accurate and may be useful for the future study. The calculated lattice parameters of this binary compound are in good agreement with the experimental data with a deviation of less than 1%. Moreover, the band structure with LDA is found to be very much closer to the experimental value. But an underestimated energy bandgap is obtained because a single exchange–correlation potential is not continuous across the gap. The gap can be enhanced in two ways. The first one is to implement Green’s function instead of density functional theory. The Green’s function can help to study self-energy of quasi particles in a many-particle system. The second way can be applying self-interaction correction (SIC). Here, self-interaction in the Hartree term is removed by an orbital-by-orbital correction to the exchange–correlation potential. The detailed investigation on DOS is carried out to understand the fluorite and pyrite TiO2 structures.

Reference
[1] Zhao J Z Wang G T Liang Y C 2008 Chin. Phys. Lett. 25 4356
[2] Landmann M Rauls E Schmidt W G 2012 J. Phys.: Condens. Matter 24 1
[3] Fan L B Zhang P Has S Wang Y X Liu P D Wang W Yao Y G 2007 Physics 36
[4] Xie H Hou W Bai W 2005 Physics 34
[5] Wang J H Li Z P Liu B Liu B B 2017 Chin. Phys. 26 026101
[6] Wang X Shi Z Jiang J 2009 Physics 38
[7] Wan L Cao L Zhang W H Han Y Y Chen T X Liu L Y Guo P P Feng J Y Xu F Q 2012 Acta Phys. Sin. 61 186801 in Chinese
[8] Miloua R Kebbab Z Benramdane N Khadraoui M Chiker F 2011 Computational Material Science 50 2142
[9] Jun L Q Zhang N C Liu F S Liu Z T 2014 Phys. Scr. 89 1
[10] Zhou X F Dong X Qian G R Zhang L Tian Y Wang H T 2010 Phys. Rev. B 82 1
[11] Zhang J Zhou P Liu J Yu J 2014 Phys. Chem. Chem. Phys. 16 20382
[12] Mahmood T Cao C Tahir M Idrees F Ahmed M Tanveer M Aslam I Usman Z Ali Z Hussain S 2013 Physica 420 74
[13] Liang Y Zhang B Zhao J 2008 Phys. Rev. B 77 1
[14] Kong X G Yu Y Gao T 2010 Eur. Phys. J. B. 76 365
[15] Swamy V Muddle B C 2007 Phys. Rev. Lett. 98 035502
[16] Mattesini M Almeida J S de Dubrovinsky L Dubrovinskaia N Johansson B Ahuja R 2004 Phys. Rev. 70 115101
[17] Mattesini M Almeida J S de Dubrovinsky L Dubrovinskaia N Johansson B Ahuja R 2004 Phys. Rev. 70 212101
[18] Kim D Y Almeida J S de Koci L Ahuja R 2007 Appl. Phys. Lett. 90 171903
[19] Wang Y Y Ding J H Liu W B Huang S S Ke X Q Wang Y Z Zhang C Zhao J J 2017 Chin. Phys. 26 026102
[20] Hu X L Zhao R X Luo Y Song Q G 2017 Chin. Phys. 26 023101
[21] Muscat J Swamy V Harrison N M 2002 Phys. Rev. 65 224112
[22] Broyden C G 1970 J. Inst. Maths Applics 6 76
[23] Fletcher R 1970 Computer Journal 13 317
[24] Goldfarb D 1970 Mathematics of Computation 24 23
[25] Shanno D F 1970 Mathematics of Computation 24 657
[26] Ching W Y Xu Y N French R H 1996 Phys. Rev. 54 13546
[27] Perdew J P Zunger A 1981 Phys. Rev. 23 5048
[28] Monkhorst H J Pack J D 1976 Phys. Rev. 13 5188
[29] Zhu T Gao S P 2014 J. Phys. Chem. 118 11385
[30] Atomistix ToolKit version 2014.3 QuantumWise A/S (www.quantumwise.com)
[31] Brandbyge M Mozos J L Ordejón P Taylor J Stokbro K 2002 Phys. Rev. 65 165401
[32] Soler J M Artacho E Gale J D García A Junquera J Ordejón P Sánchez-Portal D 2002 J. Phys.: Condens. Matter 14 2745
[33] Birch F 1947 Phys. Rev. 71 809
[34] Murnaghan F D 1944 Proc. Natl. Acad. Sci. USA 30 244
[35] Lu W Wang H Hu Y Huang H Gu H 2009 Physica 404 79
[36] Delogoz E Ozisik H Colakoglu K Surucu G Ciftci Y O 2011 J. Alloy Compd. 509 1711
[37] Ozisik H Deligoz E Colakoglu K Ciftci Y O 2011 Com. Mat. Sci. 50 1057
[38] Ozisik H Deligoz E Colakoglu K Surucu G 2010 Phys. Status Solidi 4 347
[39] Ateser E Ozisik H Colakoglu K Deligoz E 2011 Com. Mat. Sci. 50 3208
[40] Born M Huang K 1982 Dynamical Theory and Experiment Springer 1
[41] Wu Z J Zhao E J Xiang H P Hao X F Liu X J Meng J 2007 Phys. Rev. B 76 1
[42] Reuss Von A 1929 Z. Angew Math. Mech. 9 149
[43] Voight W 1928 Lehrbuch der Kristallphysik: Verlag und Druck, Von B. G. Teubner in Leipzig und Berlin 9 62
[44] Hill R 1952 Proc. Phys. Soc. 65 349
[45] Simmons G Wang H 1971 Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook Cambridge MIT Press
[46] Bannikov V V Shein I R Ivanovskii A L 2007 Phys. Stat. Sol. 3 89
[47] Fu H Li D Peng F Gao T Cheng X 2008 Comp. Mat. Sci. 44 774
[48] Clark Stewart J Segall Matthew D Pickard Chris J Hasnip Phil J Probert Matt I J Refson K Payne M C 2005 Z. Kristallogr 220 567
[49] Caravaca M A Mino J C Perez V J Casali R A Ponce C A 2009 J. Phys.: Condens. Matter 21 1
[50] Pugh S F 1954 The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 45 823
[51] Tvergaard V Hutchinson J W 1988 J. Am. Chem. Soc. 71 157
[52] Tripathy S K Kumar V 2014 Material Science and Engineering 182 52
[53] Chung D H Buessem W R Vahldiek F W Mersol S A 1968 New York Plenum Press
[54] Chen X Q Niu H Li D Li Y 2011 Intermettalics 19 1275