Comparative study on structural, electronic, optical and mechanical properties of normal and high pressure phases titanium dioxide using DFT

In this paper, a Self-consistent Orthogonalized linear combination of atomic orbitals (OLCAO) technique with a generalized gradient approximation such as Perdew–Burke–Ernzerhof Solid (GGA-PBE SOL) has been used to scrutinize the structural, optical, electronic and mechanical properties of normal pressure phase (Anatase and Rutile) and high pressure phase i.e., cubic (Fluorite and Pyrite) TiO2. Electronic and optical properties of normal pressure phases of TiO2 are also investigated using (Meta) MGGA-Tran and Blaha (TB09) and obtained results are a close approximation of experimental data. It is seen that the virtually synthesized structural parameter for cubic and tetragonal phases of TiO2 are consistent with experimental and theoretical data. From the effective mass of charge carriers (m*), it can be observed that pyrite TiO2 is having lower effective mass than the fluorite and hence shows higher photocatalytic activity than fluorite. Furthermore, it is seen that fluorite is more dense than anatase, rutile and pyrite TiO2. From the theoretical calculations on the optical properties, it can be concluded that optical absorption occursin the near UV region for high and normal pressue phases of TiO2. Again from the reflectivity characteristics R(ω), it can be concluded that TiO2 can be used as a coating material. Elastic constants, elastic compliance constants, mechanical properties are obtained for anatase, rutile, fluorite and pyrite TiO2. A comparison of the results with previously reported theoretical and experimental data shows that the calculated properties are in better agreement with the previously reported experimental and theoretical results.

There exist several approaches to theoretically investigate the properties of this material such as local density approximation (LDA) and generalized gradient approximation (GGA), MGGA under the framework of DFT along with various exchange-correlation functional. Gong sai et al [26], used TB-mBJ (Tran-Blaha modified Becke-Johnson) potentialand obtained properties were in better agreement with the experimental results and considered to be much better compared to LDA and GGA approach. Zhi-Gang Mei et al [27], used LDA and GGA along with various exchange correlations for calculating structural, mechanical and phonon properties of rutile and anatase TiO 2 and found that GGA-PBES provide accurate structural and mechanical properties for both the phases. Samat et al [14], calculated structural, optical and electronic properties of brookite TiO 2 using GGA with various exchange-correlation and found that structural properties calculated using GGA-WC are in good agreement with experimental values than the remaining exchange-correlation. Shatendra Sharma et al [28], calculated the electronic and optical properties for Strontium Sulphide (SrS) using LDA, GGA, and MGGA and observed that MGGA gives the bandgap value more close to experimental value while LDA and GGA gives underestimated results. Dash et al [25,29], used OLCAO-LDA-Perdew and Zunger (PZ) (1981) method to investigate various properties of anatase and cubic TiO 2 and found an improvement in mechanical properties. Coronado et al [30], also used OLCAO-GGA method to verify the experimental data with theoretical data.
From the literature, it reveals that all the properties of TiO 2 are sensitive to exchange correlation used. OLCAO is able to give effective improvement in terms of different elastic and mechanical properties rather than plane wave and other theoretical results. GGA-PBESOL provides better structural and mechanical properties than other exchange correlations. Hence, the first objective of this work is to carry outa detail analysis of the structural, electronic, optical and mechanical properties of high pressure and normal pressure phases of TiO 2 using OLCAO-GGA-PBES. Second objective is to provide a detail comparision of the obtained results of high pressure and normal pressure phases of TiO 2 with each other and with previously available experimental and theoretical data. However, GGA also causes overestimation of lattice constants and underestimation of bandgap value. Third objective is to get results consistent with experimental data.As experimental data is available for normal pressure phases of TiO 2 , MGGA-TB09 [46] is used here for analyzing its electronic and optical properties. The last objective is to calculate the m * of charge carriers for the cubic phase of TiO 2 to find its possible application in photo-catalytic activity and compared with other phases of TiO 2 .
The rest of the paper is arranged as follows. Section 2 explains the adopted computational details for the analysis of all properties of TiO 2 . Section 3 details the results and discussion on obtained properties. Finally, sections 4 and 5 gives the conclusion and future scope of the work.

Materials and methodology
Here, computations are carried out for anatase, rutile, fluorite and pyrite TiO 2 . Anatase and rutile belogs to tetragonal crystal system whereas fluorite and pyrite belongs to cubic crystal system. Anatase [29], rutile [39] and fluorite [11] TiO 2 structures are created using experimental lattice parameters and wyckoff positions whereas for pyrite the lattice constants considered are a=b=c=4.844 Å [25]. Structure has been optimized using maximized force of 0.005eV/Å and maximum step length of 0.5 Å. Zero constraints are considered during optimization. The Limited Memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) [47][48][49][50] is adopted for the optimization of all structures because of its effectivity for estimation of parameter in machine learning. It is based on the approximate Hessian matrix. At every iteration, it updates the approximated Hessian matrix by using products of vector-vector. With minimum iterations, it obtains its local minimum without sticking at the time of calculation. Furthermore, energy minimization is also carried out by varying lattice constants for all the structure under consideration to obtain the characteristic plotfortotal energy versus total volume, which are shown in figures 2(a) and (b) for high and normal pressure TiO 2 , 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. OLCAO method [51] is applied here in the framework of Density Functional Theory (DFT), which is an all-electron technique applied for calculating 3p 6 4s 2 3d 2 and 2s 2 2p 4 states as valence electrons for Titanium and Oxygen atom respectively. The optimized lattice structures of fluorite, pyrite, anatase and rutileTiO 2 are shown in figures 1(a), (b), (c) and (d), respectively. The prediction of the molecular orbitals by this method is accurate due to the orthogonal simulation pattern. The LCAO method initially assumes that the total number of atomic orbitals is equal to the total number of molecular orbitals included in the linear expansion. We have used GGA with PBE SOL [52] as exchange-correlation functional. Also used MGGA-TBO 9 exchange-correlation functional to analyzed electronic and optical properties of normal pressure phases of TiO 2 . Other parameters like van der Waals corrections, spin-orbit coupling and Hubbard Uare disabled. The Density mesh cut off is taken as 140 Hartree for all the structures. Sampling is done using the Monkhorst -Pack scheme [53] and set at 6×6×7 for anatase and rutile, 12×12×12 for fluorite and 7×7×7 for pyrite. State of art norm-conserving pseudopotentials have been used [54] for both Titanium and oxygen atoms.
The biggest benefit of approximate Linear Combinations of Atomic Orbitals (LCAO) method is its correctness in observable properties of molecules over other methods. The approximations used in LCAO method are superior than that of self-consistency field calculations. In addition, the approximation used in LCAO method is benificial because they provide a relation between an orbital description and chemical intuition. GGA-PBE is most widely used approximation. PBE is the improved to PBESOL by making changes in two parameters. PBESOL provides improvement in equilibrium properties of bulk. By restoring gradient expansion for exchange, PBESOL provide lattice parameters lower than PBE and cohesive energies with less accuracy. Electronic properties such as bandgap value using GGA are underestimated because a single exchangecorrelation potential is not continuous across the gap. To enhance electronic properties MGGA family of functional extend the GGA approximation by additionally depending on the Laplacian of the density and kinetic energy density. Hence, MGGA gained substantial achievement in the improvement of electronic properties.

Results and discussion
3.1. Structural parameter Structural optimization of anatase, rutile, fluorite and pyrite TiO 2 is carried out by varying lattice constant and finding out the lowest energy point. For simulation, a totally relaxed Wyckoff position and experimental cell volume [11,29,39] is considered. The lattice constant as obtained from the figures 2(a) and (b) are a=4.804 Å for fluorite and a=4.869 Å for pyriteand a = 3.796 Å, c = 9.617 Å for anatase and a = 4.616 Å, c = 2.961 Å, for rutile. Table 1 represents a comparison of the calculated density, structural parameters, volume and bandgap (Eg) with previously reported theoretical and experimental values.

Electronic properties
The energy band structure is an important property of any material which describes the optical and electronic properties. The energy bandgap is defined as the minimum energy required to create an electron and hole pair in the semiconductor. Whereas, the optical bandgap is the excitation energy which determines the onset of vertical interband transitions. Two properties namely, the band structure and total density of states are studied here and analyzed in detail. Figure 3(a) represents the band structure and 3 (b) represents the total density of states (TDOS) for fluorite TiO 2 . From the band diagram of fluorite TiO 2 , it can be observed that Fermi energy is more   Table 1. Density, the lattice constant, volume, and bandgap of fluorite and pyrite TiO 2 in comparison with previous data where D represents direct bandgap nature and ID represents indirect bandgap nature.

Phase
Method  From the band diagram of anatase TiO 2 , it can be observed that Fermi energy is more close to the valence band. Valence band maxima and conduction band minima lie at different points X and G in the energy band implying anatase TiO 2 to be of indirect bandgap having a value of 2 eV. Bandgap value obtained using MGGA-TBO9 is 3.3 eV which is 3% higher than experimental value [30].

Effective mass of electron and hole (m * )
It is the mass of charge carriers (e−/h+) when they respond to any type of interaction in a crystal lattice. The effective mass highly depends on the crystal structure or electronic band structure in solids [33]. It is usually stated in the unit of the rest mass of an electron, m e (9.11×10 −31 kg). Equation (1) represents, the transfer rate of photogenerated electrons and holes which is inversely proportional to effective mass.
Where, m * representsan effective mass of (e−/h+), k is the wave vector,ħ is Planck constant, v is the transfer rate of photogenerated electrons and holes. Thus, a smaller effective mass is desired to get higher photocatalytic activity. Effective mass isusually calculated using equation (2) and is represented as Where, E is the energy of an electron at wavevector k in that band. Table 2 gives a comparison of the calculated effective mass of cubic TiO 2 with an effective mass of other phases of TiO 2 . From table 2, m * of an electron in pyrite is smaller than fluorite and brookite but it is higher than anatase and rutile. The transfer rate of hole and electrons in pyrite is faster than fluorite. This leads to higher photocatalytic activity in pyrite than fluorite. Also, pyrite is an indirect bandgap semiconductor, therefore conduction band minima aad valence band maxima lies at different k points. Thus, a lifetime of photogenerated electron and hole increases in pyrite compared to fluorite. This is in agreement with [33], that indirect bandgap semiconductor has better photocatalytic activity than direct bandgap semiconductor.

Optical properties
The optical properties are calculated by considering energy level up to the phonon energy, (40 eV), for cubic phases and (12 eV) for tetragonal phases using GGA-PBE SOL. MGGA-TB09 is used to calculate optical properties of normal pressure phases of TiO 2 . For calculating the optical properties, equations given in ref [33] are used. As the dielectric function is a complex quantity and so it contains real and imaginary parts.   [33] 0.0948(average of m * from G to Z and G to M) 0.1995(average of m * from B to G and B to M) Rutile [33] 0.0949(average of m * from G to Z and G to M) 0.5620(average of m * from G to Z and G to M) Brookite [33] 1.4610(m * from G to Z) 0.4345(m * from G to Z ) good absorption value which ranges from 0 to 15 eV of photon energy for fluorite and 0 to 14 eV for pyrite TiO 2 . Therefore, cubic TiO 2 can provide good electrical conductivity. Table 3 represents the comparison of calculated ε 1 (0), η(0) and R(0) with previously reported data.From reflectivity spectra, it is found that TiO 2 can provide good electrical conductivity and very much suitable as a coating material.

Mechanical properties
Mechanical properties give an idea about the nature of forces acting in solids, phonon spectra, and interatomic potential and thereby specifying its hardness, stability, etc. Before doing all the calculations we checked positive definiteness of stiffness matrix [43,61] using equation (3) for cubic phase, 0, 0, 2 0 3 11 12 11 44 11 12 Tetragonal phase will be mechanically stable, if it satisfies the Born-Huang criteria [62] If equation (3) is fulfilled by cubic material then it is considered as mechanically stable and if equation (4) is fulfilled by tetragonal material then it is considered as mechanically stable. Table 4 gives a comparison of calculated elastic properties with previously reported data. Elastic compliance s ij are calculated using equations given in [63]. Table 5 shows the comparision of Elastic compliance sij for fluorite, pyrite, anatse and rutile TiO 2 with available theoretical data. From table 4, it is apperant that cubic and tetragonal phase TiO 2 satisfies respective equations (3) and (4) and hance they are mechanically stable.
Again, Bulk and shear moduli are used to measure the hardness of the material. Two different theories are there to calculate the bulk and shear modulus namely, Reuss theory [67] and Voigt theory [68]. According to Reuss Theory B R and G R are given by Where, s 11 , s 12, and s 44 are the compliance matrix elements of cubic TiO 2 . According to Voigt Theory B V , G V is given by Table 3. Comparison of calculated ε 1 (0), η(0) and R(0) of fluorite , pyrite, anatase and rutile TiO 2 with previously available data.
Phase Method (ε xx (0)) (ε zz (0)) (η(0)) (R(0)) References According to the Hill [69] approximation, the bulk modulus B Hill and shear modulus G Hill is given by While Vicker's hardness [70] is also used to check the hardness of the material and it is given by Obtained Vicker's hardness show that rutile is harder than other three polymorphs of TiO 2 where, k=G/ B=Pugh's modulus ratio. Young's modulus is the measure of the stiffness of the material and it is given by Poisons ratio (v) is used to classify material and to check the ductile and brittle property of the material. v=0.25 [71,72] for ionic material v=small approximately equal to 0.1 [69,73] for covalent materials. For brittle material,v<0.33 and for ductile materialv>0.33.Poisons ratio (v) [69,73] is given by  From the calculated v, we cay say that all the four structures are ductile in nature. If A=1, the material is isotropic else the material is anisotropic. Anisotropy (A) of material is calculated using equation (11) for cubic phase and equation (12) for tetragonal phase From calculated A, it is clear that all the structures are anisotropic in nature. Lame constant (μ, λ) are calculated using the following equation Table 6 gives a comparison of calculated mechanical properties with previously reported data.

Conclusion
This paper presents a comparative study on the optical, mechanical, structural and electronic properties of normal and high pressure phases of TiO 2 using OLCAO-GGA-PBE-SOL for the first time. MGGA-TBO9 reults on electronic and optical properties of normal pressure phases of TiO 2 are also analyzed. The computed results are then compared with the previously reported experimental and theoretical data. From the comparision, we can find that lattice constant for anatase and rutile vary only by 0.4% from experimental data [39] whereas it shows 1.3% variation with experimental data for fluorite [11]. However results are much better than the theoretical data [6,25,33,34]. Bandgap values calculated using OLCAO-GGA-PBE-SOL for all the phases of TiO 2 are consistent with other theoretical data but underestimated compared to experimental data [30].
Bandgap values obtained using MGGA-TBO9 is the approximation of experimental data [30]. Effective mass analysis of cubic TiO 2 shows that transfer rate is faster in pyrite and hence exhibits higher photocatalytic activity than fluorite. Dielectric constant value calculated using OLCAO-GGA-PBE-SOL for all the structure are higher than other theoretical data [6,23,26,32,38]. Dielectric constant value for anatase and rutile using GGA-PBES are close to the experimental value [57,58]. Dielectric constant and refractive index calculated using MGGA-TBO9 for anatase and rutile are very much close to experimental data [40,41,59,60] and better than [26,37,38]. Elastic constat of rutile TiO 2 are in excellent agreement with experimental data [65]. Obtained bulk modulus value for anatase, rutile and fluorite vary by 2%,4.7% and 33% from experimental data [2,44,74] and better than [27,29,31,35]. As experimental data of pyrite TiO 2 is not available, obtained properties are compared withother theoretical data and results are found to be consistent with previously reported data. OLCAO-GGA-PBE-SOL provides better results for structural and mechanical properties of normal pressure phases compared to high pressure phases. MGGA-TB09 reults on optical and electronic properties are the approximation of experimental data. This comparative analysis of TiO 2 using OLCAO-GGA-PBE-SOL and MGGA-TB09 will be helpful for future theoretical as well as experimental investigations.This analysis can be also helpful to study various properties of other material using the two methods described earlier.

Future scope
Among many candidates for photo-catalysts, TiO 2 is the only material suitable for industrial use because of its efficient photoactivity, the highest stability and lowest cost. Also it is suitable for solar cell applications. Both these applications required small bandgap material. But TiO 2 exibits large bandgap value. So, to make it suitable for above application one need to narrow the the bandgap using suitable metal and non-metal doping. So, one can verify the properties of doped and undoped TiO 2 using two methods mentioned in this paper. This will help to find its future applications.