First-Principles Calculations of the Pressure Stability and Elasticity of

Apr 7, 2014 - Department of Materials Engineering, Monash University, Clayton Campus, Clayton, Victoria 3800, Australia. §. CSIRO Process Science and...
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First-Principles Calculations of the Pressure Stability and Elasticity of Dense TiO2 Phases Using the B3LYP Hybrid Functional Varghese Swamy*,†,‡ and Nicholas C. Wilson§ †

Advanced Engineering Program and School of Engineering, Monash University Malaysia, Jalan Lagoon Selatan, Bandar Sunway, 46150 Selangor Darul Ehsan, Malaysia ‡ Department of Materials Engineering, Monash University, Clayton Campus, Clayton, Victoria 3800, Australia § CSIRO Process Science and Engineering, Bayview Avenue, Clayton, Victoria 3169, Australia ABSTRACT: The crystal structures, pressure−volume equations of state, and pressure stability up to 100 GPa of the experimentally observed and theoretically proposed dense TiO2 phases (rutile, columbite, baddeleyite, OI, cotunnite, fluorite, pyrite, and Pca21) were calculated using the hybrid B3LYP exchange-correlation functional and two independent Gaussian-type basis functions. Overall, the B3LYP results are in good agreement with the experimental data on the structures and elastic properties. The B3LYP functional also shows superior performance relative to projector-augmented wave/pseudopotential-based planewave density functionals in predicting the elastic behaviors for most of the TiO2 phases with the exception of fluorite-TiO2. The latter structure shows significant sensitivity to the choice of basis functions. The order of phase stability with increasing pressure predicted by B3LYP is rutile → columbite → baddeleyite → OI → cotunnite, in agreement with available experimental results. In the pressure range of 20−50 GPa, the B3LYP total energy for the recently proposed Pca21 structure is very close to that for one or more of the accepted stable phases (columbite, baddeleyite, OI, and cotunnite), suggesting potential stabilization of the Pca21 structure at high temperatures. The B3LYP electron density-ofstates projections suggest large band gaps for the high-pressure phases OI and Pca21.

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and cubic-TiO2.15,17,18,21,25 The purpose of this paper is to critically review the recent published work and present new first-principles computational results in order to resolve existing discrepancies relating to pressure-dependent stability and compressive behaviors of the dense TiO2 phases at 0 K and to a pressure of 100 GPa. The rich polymorphism of TiO2 has been reviewed earlier.3,4 Of the low-pressure TiO2 forms, anatase, rutile, brookite, and TiO2(B) are known to exist in nature; ramsdellite-TiO2 (TiO2(R)) and hollandite TiO2 (TiO2(H)) are synthetic forms. At elevated pressures, the observed and proposed TiO2 phases include rutile, columbite (Pbcn, α-PbO2 structure, TiO2-II), baddeleyite (monoclinic ZrO2, P21/c), OI, cotunnite, fluorite, pyrite (Pa3̅), and Pca21. The pressure-dependent structural phase transition sequence of baddeleyite → OI → OII is by now well established for the three closely related group IVB metal dioxides TiO2, ZrO2, and HfO2. Fluoritestructured ZrO2 and HfO2 are also well established; stable fluorite-TiO2, however, has been controversial. Mattesini et al.9 proposed fluorite structure for the new TiO2 phase they observed at 48 GPa and 1900−2100 K in their high-pressure diamond-anvil cell synthesis starting with anatase. The presence of strong and overlapping X-ray diffraction (XRD) peaks due to

INTRODUCTION The phase stability and physical properties of compact TiO2 crystal structures have been the subject of a number of recent experimental and computational investigations.1−31 The highpressure synthesis of cotunnite-TiO2 (Pnma, PbCl2-structure, also referred to as orthorhombic-II or OII) at ∼60 GPa and 1100 K,1 orthorhombic-I TiO2 (Pbca, OI) at ∼28 GPa and 1300−1500 K,2 and “cubic-TiO2” with a nominal fluorite (Fm3m) structure at 48 GPa and 1900−2100 K9 triggered much of this interest. Apart from the enormous research efforts directed at the octahedrally coordinated anatase (I41/amd) and rutile (P42/mnm) forms of TiO2 principally from the perspective of photocatalyst and photovoltaic applications (for example, see refs 32 and 33), the interest in the highcoordinated denser (high-pressure) TiO2 structures is due in part to certain attractive properties attributed to them. The first combined experimental and computational investigation identified cotunnite-TiO2 as an ultrahard and ultrastiff oxide.1,5 First-principles calculations of fluorite-TiO2 suggested it to be a potential solar energy material with optical absorptive transitions in the visible region that are 3−4 orders of magnitude larger than the conventional state-of-the-art solar cell with anatase,8,12 a high static dielectric constant (κ) material with potential semiconductor applications,11 and a potential ultrahard/ultrastiff material.13 More recent computational and experimental investigations have, however, returned more modest values for the mechanical properties of cotunnite© 2014 American Chemical Society

Received: November 19, 2013 Revised: March 29, 2014 Published: April 7, 2014 8617

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reflection in the original experimental XRD pattern pointed to the fluorite structure. However, the closeness of the experimental K0 value to that computed for pyrite indicated that at least to moderate pressures pyrite is a candidate TiO2 structure, with fluorite as the preferred structure at high pressures. Kim et al.12 investigated the phonon density of states for both pyrite- and fluorite-TiO2 to 100 GPa using firstprinciples calculations and found that pyrite was dynamically unstable at all pressures whereas fluorite becomes dynamically stable at pressures beyond 55 GPa. They used this result to further assign fluorite structure to the “cubic-TiO2”. For a fluorite-based, disordered (Ti0.50Zr0.26Mg0.14Cr0.10)O1.81 solid solution phase, referred to as TZMC, a K0 = 140 ± 4 GPa with a large pressure derivative of K′ = 11 ± 1 was first reported34 and later revised to K0 = 164 ± 4 GPa and K′ = 4.3 ± 73.5 Keeping in view the calculated significant K0 reduction when going from ideal fluorite to distorted fluorite (i.e., pyrite) TiO2, Swamy et al.36 (see p. 2333) suggested that any deviation from the ideal fluorite structure via distortion or disorder may lead to smaller K0 values in relation to the ideal structure and called for further investigations to reconcile the experiment− theory dichotomy in the K0 values for “cubic-TiO2”. Crystalchemical arguments invoking cation/anion radius ratio, R(Ti4+)/RO2− or R(Zr4+)/RO2−, in the ideal fluorite structure have been used to explain the smaller experimental K0 values observed for both TZMC and the so-called cubic-TiO2.35 First-principles calculations of cubic-TiO2 (fluorite or pyrite) published subsequently reported larger K0 values compared to the experiment,14,15,17,25,28 but not as large as the values reported by Swamy and Muddle13 for fluorite-TiO2. To illustrate, the planewave, pseudopotential GGA and local density approximation (LDA) pressure−volume equations of state (P−V EoS) for fluorite- and pyrite-TiO2 reported by Liang et al.15 are identical: GGA K0 = 272 GPa for both pyrite and fluorite and LDA K0 = 320−324 GPa. Using their computational results, Liang et al.15 supported the identification of the “cubic-TiO2” as having fluorite structure. They further suggested that the fluorite K′ values reported by Swamy and Muddle13 are too low, i.e., K′ < 4. Although the requirement for a K′ ∼ 4 is based on the Eulerian finite-strain theory (K′ = 4 for the second-order Birch−Murnaghan EoS), in reality experimental P−V data have been fitted with a range of K′ values (see ref 37). In fact, the recommended K′ value for rutile is ∼7,38,39 but values as large as 10.6 have been reported [ref 40, p. 397]. Furthermore, Liang et al.15 suggested that the different computational results obtained by Swamy and Muddle13 could be an “artifact” due to insufficient convergence, even though the computations reported by the latter authors were carried out under very tight computational constraints, as detailed in their Supporting Information. In a DFT (GGA) calculation by Zhou et al.,23 the starting columbite showed abrupt crystal structural changes at 43 GPa. Symmetry analysis and XRD pattern simulation by the authors suggested Pca21 space group for the new TiO2 phase at 43 GPa where “cubic-TiO2” was expected to be stable.9 The computed P−V EoS parameters for Pca21 TiO2 were: V0 = 115.46 Å3, K0 = 207 GPa, and K′ = 4.24. It may be noted that the Pca21 structure results from large displacements of O atoms and relatively smaller displacements of Ti atoms from their respective positions in the fluorite lattice. In other words, Pca21 is a distorted fluorite with larger compressibility compared to fluorite and pyrite, in conformity with the previous suggestion.36

Figure 1. Crystal structures of the dense TiO2 phases considered in this study.

OII and gold (the latter used as an internal pressure standard) did not allow them a definitive identification of the structure either as fluorite or a closely related distorted fluorite structure such as pyrite. The relatively low experimental zero-pressure bulk modulus (K0) value of 202 ± 5 GPa with a pressure derivative K′ = 1.3 ± 1 reported for the putative fluorite-TiO29 in comparison to the known K0 values for TiO2 structures with Ti−O octahedral coordinations prompted Swamy and Muddle13 to carry out first-principles calculations of the structure and elastic properties of both fluorite- and pyrite-TiO2 using gradient-corrected (generalized gradient approximation, GGA) density functional theory (DFT) and hybrid density functional theory−Hartree− Fock (HF) formulations. The calculations yielded a smaller zero-pressure unit cell volume (V0 = 112.13 ± 0.06−112.75 ± 0.06 Å3) and remarkably larger K0 (∼390 GPa) for fluorite compared to the corresponding values for pyrite (V0 = 117.26 ± 0.04−118.62 ± 0.12 Å3 and K0 = 220−258 GPa). It may be noted that structurally the two cubic phases are quite similar: in fluorite each Ti atom at (0,0,0) is 8-fold (cubic) coordinated by O atoms at ±(0.25,0.25,0.25), whereas in pyrite, each Ti atom at (0,0,0) is coordinated to 6 + 2 O atoms at ±(x,x,x), with x ∼ 0.34, in a distorted polyhedron. Swamy and Muddle13 proposed a correlation between the Ti−O coordination number of the crystal structures and K0 for the various TiO2 phases and suggested that fluorite-TiO2 is potentially ultrahard. Without arguing for either fluorite or pyrite structure, they noted that the absence of the 210 (hkl) 8618

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Table 1. Lattice Parameters and Atomic Coordinates for Selected TiO2 Structures phase/space gr. (#) OI Pbca (61)

method exp

atom (pos.)

a

Ti O1 O2 LMTOb Ti O1 O2 B3LYPc Ti O1 O2

cotunnite Pnma (62)

expd Ti O1 O2 B3LYPe Ti O1 O2

Pca21 (29)

B3LYPf Ti O1 O2 GGAg Ti O1 O2

a (Å)

b (Å)

c (Å)

x

y

z

9.046(2) 0.886(1) 0.794(2) 0.971(2) 9.052 0.885 0.805 0.945 9.138 0.886 0.791 0.972 5.163(2) 0.264(1) 0.346(1) 0.012(2) 5.1246 0.2463 0.3580 0.0247 4.8177 0.5405 0.2455 0.3896 4.84 0.5428 0.2477 0.3893

4.834(1) 0.034(2) 0.379(2) 0.738(1) 4.836 0.049 0.401 0.690 4.853 0.041 0.381 0.735 2.989(1) 0.25 0.25 0.75 2.9144 0.25 0.25 0.75 4.5212 0.7273 0.5578 0.0918 4.51 0.7265 0.5629 0.0933

4.621(1) 0.272(1) 0.156(1) 0.505(2) 4.617 0.256 0.135 0.464 4.671 0.257 0.145 0.511 5.966(2) 0.110(1) 0.422(1) 0.325(1) 5.9312 0.1151 0.4250 0.3375 4.5573 0.2123 0.4643 0.2944 4.55 0.2112 0.4608 0.2989

a Experimental data from ref 2. bLinear muffin tin orbital (LMTO) calculations from ref 1. cThis study at 28 GPa, using the TVAE basis set. dAt 61 ± 2 GPa from ref 1. eThis study, at 60 GPa, using the TVAE basis set. fThis study, at 43 GPa, using the TVAE basis set. gAt 43 GPa from ref 23.

In this paper, we present first-principles computational results on the 0 K crystal structure, P−V EoS, and phase stability for the dense TiO2 phases (Figure 1) to 100 GPa obtained using the popular B3LYP hybrid HF-DFT formalism. Electronic density of state projections are also presented for selected phases. The present computations were carried out with even more stringent constraints and better optimization methods compared to the previous work.13 We have also carried out additional computations using the most recently published Peintinger−Oliveira−Bredow triple-zeta valence with polarization (pob-TZVP) Ti and O basis functions41 to verify the influence of basis sets on the structure, elastic properties, and electronic density of states. We compare our results with available experimental data as well as with recent first-principles calculations employing the GGA-planewave approach. This work is also relevant to the recent discussions regarding performances of various exchange-correlation functionals.42−50 We demonstrate that our results obtained with all-electron localized basis functions and the hybrid functional are for the most part in much better agreement with experiments than the published planewave-GGA DFT results.

used hybrid Hartree−Fock/Kohn−Sham Becke (exchange)/ Lee−Yang−Parr (correlation) B3LYP functional,53,54 as in our previous work.13 In the AE-LCAO approach, the Bloch orbitals of the crystal are constructed from a linear combination of Gaussian-type orbitals with s, p, or d symmetry. We employed two independent Ti and O basis sets in our calculations: the triple valence all-electron (TVAE) (ref 4) and the Peintinger− Oliveira−Bredow triple-zeta valence with polarization (pobTZVP) basis sets.41 The reciprocal space integration was carried out by sampling the Brillouin zone using an 8 × 8 × 8 Monkhorst−Pack grid. The accuracy of the results with local basis functions in CRYSTAL is determined by five truncation criteria for bielectronic integrals (Coulomb and exchange series) [ref 52]. These Gaussian overlap criteria are labeled ITOL1 (overlap threshold for Coulomb integrals), ITOL2 (penetration threshold for Coulomb integrals), ITOL3 (overlap threshold for HF exchange integrals), ITOL4 (pseudo-overlap−HF exchange series), and ITOL5 (pseudo-overlap−HF exchange series). Compared to our previous work13 in the present calculations we used much tighter Gaussian overlap criteria: ITOL1, ITOL2, ITOL3, ITOL4, and ITOL5 values of 10−9, 10−9, 10−9, 10−9, and 10−18, respectively. Structure optimizations were carried out under very strict convergence criteria: energy thresholds (in au) for SCF convergence of 10−9−10−10 and for structure optimization −10−9. Again, unlike our previous calculations using numerical gradients for energy minimization, the present calculations were performed using

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COMPUTATIONAL METHODS We performed 0 K total energy calculations of the TiO2 structures (Figure 1) by employing the all-electron periodic linear combination of atomic orbitals (AE-LCAO) approach implemented in the CRYSTAL package.51,52 The exchangecorrelation energies were evaluated using the most commonly 8619

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the CRYSTAL09 version52 that uses analytical gradients with respect to unit cell parameters to calculate stress tensors.55 All the external and internal atomic coordinates of the crystal structures were optimized simultaneously to obtain minimum energy configurations. The energy minimizations started with published experimental crystal structure where available (rutile, columbite, baddeleyite, OI, and cotunnite). For pyrite and fluorite TiO2, theoretical start values were used, whereas for Pca21 we used the results from Zhou et al.23 as start values. At specified hydrostatic pressures, the enthalpy was calculated from the relation H = E + PV where H is the enthalpy, E the internal energy, and V the unit cell volume. The Birch−Murnaghan EoS56 is commonly used for fitting experimental P−V data.37 We used the following third-order Birch−Murnaghan EoS for analyzing the P−V relationships of the individual phases P = 3/2K 0[(V0/V )7/3 − (V0/V )5/3 ] {1 + 0.75(K ′ − 4)[(V0/V )2/3 − 1]}

(1)

The crystal structures of the TiO2 phases presented in Figure 1 were plotted using the VESTA package.57

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RESULTS AND DISCUSSION Structures and Equations of State. The crystal structures computed in our work using the B3LYP functional are in very good agreement with room-temperature experimental data. We include selected structures with large degrees of freedom to illustrate the performance of the B3LYP functional in Table 1. Among the TiO2 phases considered here, the elastic properties of rutile are the most investigated, both experimentally and computationally. In Figure 2, experimental data18,58,59 on the pressure dependence of rutile lattice parameters are compared with B3LYP results using both TVAE and pob-TZVP basis sets. Excellent prediction of the anisotropic compressive behavior of rutile is obtained to the highest measured pressure. The P−V EoS data compiled in Figure 2 and Table 2 for the dense TiO2 phases suggest significant variations not only between experimental and theoretical determinations but also within experimental and theoretical determinations themselves. Most of the experimental determinations of V0, K0, and K′ were carried out using in situ XRD volume−pressure data obtained in static compression and fitted to an EoS, the Birch− Murnaghan EoS in most cases. While P values can be arbitrarily chosen in first-principles computations, including negative values, the P range available to experiments is determined by the stability/metastability of the structure under investigation and the quality of the data obtained. For some high-P structures that are not stable down to ambient pressure, this would mean considerable extrapolations of the structural parameters from high pressures to ambient. Cotunnite, OI, baddeleyite, and “cubic-TiO2”, with limited high-pressure stability ranges, are normally not pressure-quenchable: they revert to the columbite structure upon decompression typically around 4−8 GPa.1,2,9,18 Similarly, in our calculations we have found limited P stability ranges for baddeleyite (10−50 GPa) and Pca21 (6−44 GPa) as well as significant abrupt changes in the compression curves for OI (around 45 and 60 GPa) and cotunnite (∼40 GPa). While we did not attempt detailed characterizations of the structures obtained at pressures beyond the mentioned stability fields for the above phases, it has been possible to easily relate some of the structural transitions. For example, Pca21 TiO2

Figure 2. Comparison of calculated (unbroken curves, TVAE basis sets; dashed curves, pob-TZVP basis sets) and experimental pressuredependent variation of unit cell parameters. Experimental data: rutile− ref 18 (diamonds and triangles), ref 58 (crosses), and ref 59 (circles); columbite−ref 18 (green diamonds), ref 21 (blue diamonds), ref 60 (plus sign), and ref 61 (squares); baddeleyite−ref 7 (circles), ref 18 (green diamonds and triangles), and ref 21 (blue diamonds); OI−ref 2 (circles), ref 18 (green diamonds and triangles), and ref 21 (blue diamonds); cotunnite−ref 18 (green diamonds and triangles) and ref 21 (blue triangle); and cubic TiO2−fluorite (green curves), pyrite (blue curves), Pca21 (red curves), and open circles (experimental data from ref 9).

relaxes into the columbite structure below 6 GPa, in excellent agreement with the experimental behavior of high-pressure TiO2 phases upon decompression. At 0 GPa, the computed (TVAE basis sets) unit cell parameters for columbite are a = 4.5836 Å, b = 5.5471 Å, c = 4.9197 Å, and V = 125.084 Å3. The columbite structure obtained at 0 GPa starting from Pca21 has almost identical unit cell parameters with a = 4.5851 Å, b = 5.5457 Å, c = 4.9197 Å, and V = 125.096 Å3. Similarly, the highpressure transition of Pca21 at P > 44 GPa can be characterized as approaching the OI symmetry. At 50 GPa, the computed unit cell starting with Pca21 has the following unit cell parameters: a = 4.8347 Å, b = 4.4463 Å, c = 4.5086 Å, and V = 96.9185 Å3. The unit cell parameters of OI at 50 GPa are a = 8.8870 Å, b = 4.810 Å, c = 4.5698 Å, and V = 195.351 Å3. Note the following correspondence: aPca21 = bOI; bPca21 = 1/2aOI; cPca21 = cOI; and V = 97.6753 Å3 for the equivalent Pca21 unit cell. When analyzing the P−V EoS for the above phases, it is important to remember that even though the commonly used EoS such as Murnaghan, Birch−Murnaghan, and Vinet equations are based on sound theories of elastic/cohesive 8620

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Table 2. Comparison of Theoretical and Experimental P−V EoS Parameters for TiO2 Phases phase rutile P42/mnm

columbite Pbcn

baddeleyite P21/c

orthorhombic-I Pbca

cotunnite Pnma

Pca21 fluorite Fm3m pyrite Pa3̅ “cubic-TiO2”

method

V0 (Å3)

K0 (GPa)

K′

reference

B3LYP (TVAE) GGA (PAW) GGA (PAW) experiment experiment B3LYP (TVAE) GGA (PAW) GGA (PAW) experiment experiment experiment experiment B3LYP (TVAE) GGA (PAW) GGA (PAW) experiment experiment experiment experiment experiment B3LYP (TVAE) GGA (PAW) GGA (PAW) experiment experiment experiment B3LYP (TVAE) GGA (PAW) GGA (PAW) experiment experiment experiment B3LYP (TVAE) GGA-PAW B3LYP (TVAE) GGA-PAW GGA-Psp B3LYP (TVAE) GGA-PAW GGA-Psp experiment

63.45(0.004) 64.8 64.36 62.44 62.4(0.02) 125.1(0.03) 127.2 127(0.2) 122.19(5) 122.37 122.12(0.4) 122.2(7) (fixed) 117.32(0.32) 121.2 119.84(0.16) 112.24 112.25 110.48(5) 112.24(0.6) 115.31 226.98(0.14) 229.6 229.68(0.32) 218.10d 218.16(0.5) 223.57(0.6) 103.18(0.6) 105.2 103.88(0.4) 105.09d 101.12(0.4) 100.56 116.05(0.17) 115.46 112.55(0.04) 112.70 112.11 117.59(0.05) 117.73 116.65 115.50(2)

226(7) 200 216(2) 211(1) 235(10) 244(2) 208 188 ± 4 253(4) 258(8) 253(12) 206(4) 223(13) 145 157(1) 290(10) 304(6) 303(5) 298(21) 175(5) 265(5) 220 209(2) 318(3) 314(1) 222(14) 296(20) 235 261(7) 431(10) 312(34) 306(9) 199(8) 207 392(4) 277 272 258(2) 258 272 202(5)

5.42(0.7) 5.6 4.0 (fixed) 6.76 4.0 (fixed) 3.38(0.07) 4.0 4.0 (fixed) 4.0 (fixed) 4.1(0.3) 4.0 (fixed) 4.0 (fixed) 3.5(0.4) 4.0 4.0 (fixed) 4.0 (fixed) 3.9(2) 3.9(2) 4.0 (fixed) 4.0 (fixed) 3.19(0.19) 4.1 4.0 (fixed) 4.0 (fixed) 4.0 (fixed) 4.0 (fixed) 3.90(0.23) 4.0 4.0 (fixed) 1.35(0.1) 4.0 (fixed) 4.0 (fixed) 5.44(0.43) 4.24 2.06(0.04) 4.07 4.66 4.05(0.05) 4.27 4.58 1.3(1)

this study ref 22 ref 18 ref 38 ref 18 this study ref 22 ref 18 ref 60 ref 61 ref 18 ref 21 this study ref 22 ref 18 ref 61 ref 1 ref 7 ref 18 ref 21 this study ref 22 ref 18 ref 2 ref 18 ref 21 this study ref 22 ref 18 ref 1 ref 18 ref 21 this study ref 23 this study ref 23 ref 15 this study ref 23 ref 15 ref 9

Table 3. P−V EoS Parameters for Selected Dense TiO2 Phases Computed Using the pob-TZVP Basis Set phase

space group

V0 (Å3)

K0 (GPa)

K′

columbite baddeleyite orthorhombic-I cotunnite Pca21 fluorite pyrite

Pbcn P21/c Pbca Pnma Pca21 Fm3m Pa3̅

124.78(0.07) 117.08(0.26) 226.27(0.09) 103.35(0.18) 115.83(0.59) 112.87(0.09) 116.59(0.03)

237(3) 208(6) 252(2) 255(7) 196(17) 256(4) 281(2)

3.26(0.13) 3.54(0.12) 3.70(0.07) 4.65(0.15) 4.50(0.54) 4.34(0.09) 3.75(0.05)

as the exchange-correlation functional, completeness/transferability of the basis sets, “hardness” of the pseudopotentials, and computational approaches and strategies, including the control parameters. The most recent experimental determinations of the EoS by Nishio-Hamane et al.21 reported consistently lower K0 values for columbite, baddeleyite, OI, and cotunnite compared to most previous experimental determinations (Table 2 and Figure 2). Our B3LYP calculations using the TVAE basis sets predict P−V data for these phases in much closer agreement with the majority of the experiments than those obtained with GGA (Figure 2 and Tables 2 and 3). In fact, planewave GGA calculations as well as Nishio-Hamane et al.21 experimental data severely underestimate the P−V EoS for most of these phases (Table 2). With the exception of fluorite and baddeleyite, the P−V relations for the remaining TiO2 structures computed by both the AE-LCAO basis sets are quite similar. The pob-TZVP basis sets predict fluorite P−V relation in good agreement with

properties,56,62 the extrapolation of the P−V data over large P ranges to ambient conditions can lead to uncertainties in the derived V0, K0, and K′ values. For instance, if we included computed volume data for cotunnite-TiO2 to pressures up to 200 GPa, a very large K0 value is obtained, much closer to the original experimental value than that listed in Table 1. Again, the computational results will depend critically on factors such 8621

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On the other hand, the larger pob-TZVP basis sets, optimized for the DFT method,41 are transferable and yield fluorite EoS parameters similar to the DFT data listed in Table 2. The B3LYP V0 and K0 values for Pca21 are similar to the GGA predictions and close to the experimental data for “cubic TiO2” (ref 9). Taking into account the present results as well as other experimental and computational reports, the K0 value for cotunnite TiO2 reported by Dubrovinsky et al.1 appears to be too large, potentially resulting from a number of factors including structure extrapolation from very high pressures to ambient, limited P range for the experimental P−V data (30− 60 GPa) and large stress gradients within the diamond-anvil cell, as also suggested previously.18,21 Phase Stability. The phase stability of “cubic TiO2” (fluorite, pyrite, or Pca21) has been the most controversial in the literature. In the case of pyrite, apart from the unit cell constant a, the oxygen internal coordinate x is the other variable (the structure reduces to fluorite when x = 0.25) that influences the structure, stability, and mechanical properties. As seen in Figure 3, at 0 GPa pyrite with x ∼ 0.34 is the lowestenergy configuration and has a larger volume. However, previous work12 has shown that pyrite TiO2 is dynamically unstable at all pressures up to 100 GPa. It can be seen in the enthalpy−pressure plot (Figure 4) that fluorite becomes relatively more stable at high pressures, but large energy differences do exist between the experimentally established high-pressure phases (columbite, baddeleyite, OI, and cotunnite) and the cubic structures. It may be noted here that Nishio-Hamane et al.21 considered the cubic TiO2 observed in the experiment by Mattesini et al.9 as metastable, resulting from high nonhydrostatic conditions. The case of Pca21, however, is markedly different: the relative enthalpy differences between this phase and the stable highpressure phases baddeleyite and OI are relatively small in the pressure range of 20−50 GPa, suggesting that temperature can stabilize the former in this pressure range. As discussed by Zhou et al.,23 Pca21 satisfies the experimental XRD and bulk modulus data quite well. Although one may suspect the two unidentified XRD peaks (with d-values of ∼2.74 and 1.9 Å) in

Figure 3. Variation of total energy (a) and unit parameter (b) for pyrite TiO2 as a function of internal oxygen coordinate calculated using the TVAE basis sets.

GGA results, and the baddeleyite P−V EoS computed using these latter basis sets is closer to the Nishio-Hamane et al.21 data (Figure 2). It is now obvious that the discrepant bulk modulus value obtained for fluorite in the previous AE-LCAO calculations13 resulted from the use of the TVAE basis sets. It may be noted, however, that there has not been any evidence for nontransferability of these and related basis functions as they have been successfully used for a variety of Ti-containing systems (see, for example, refs 4 and 63−67). Clearly, with the smaller TVAE basis sets inferior results are obtained for fluorite.

Figure 4. Enthalpy versus pressure plot for the various TiO2 phases to 100 GPa calculated with the TVAE basis sets. The transition pressures for the stable phases are indicated. 8622

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Figure 5. Electronic density of states for selected TiO2 polymorphs calculated using the TVAE basis sets (a) and pob-TZVP basis sets (b).

the 58 GPa pattern of Figure 2 in ref 18 originated from Pca21, we are unsure of this due to the instability of the Pca21 phase at this pressure. Rutile is widely believed to be the thermodynamically stable phase at pressures close to ambient. However, as with previous GGA calculations (for example, refs 4, 22, and 68), B3LYP also predicts very close energies for rutile and columbite at 0 GPa (Figure 4). Indeed, rutile versus columbite relative stability seen in Figure 4 could be fortuitous: our earlier calculations using the CRYSTAL06 version69 and the TVAE basis sets with various levels of detail (TVAE* and TVAE**) suggested slightly enhanced stability for columbite at 0 GPa. The extreme sensitivity of the TiO2 phase stability relations to the choice of basis sets and computational parameters has already been discussed.4 The 0 K phase stability sequence with increasing pressure to 100 GPa predicted by B3LYP in the present work is rutile → columbite → baddeleyite → OI → cotunnite with approximate transition pressures of 2, 17, 29.5, and 48.5 GPa, respectively. This stability sequence is in agreement with the experimental data.1,2,21,70−72 The published projector-augmented wave (PAW)/pseudopotential planewave GGA calculations23,29 of Pca21 TiO2 have not carried out a comprehensive phase stability analysis as has been done here, and therefore, we are unable to compare the relative performances of GGA and B3LYP with regard to phase stability. Electronic Structure. The computed total electronic density of states (DOS) for OI, Pca21, and cotunnite are plotted, along with rutile as a reference, in Figure 5. The DOS for rutile was calculated at 0 GPa, while for OI, Pca21, and cotunnite the DOS were calculated at pressures of 28, 44, and 60 GPa, respectively. In addition to the total DOS, Mulliken projected DOS of the Ti and O contributions are also given. For the four polymorphs plotted, the valence band is predominantly O 2p in character, with a small contribution from the Ti 3d states. The DOS predicts all four polymorphs to be insulators, with the conduction band composed mainly of Ti 3d states mixed with a small contribution from the O 2p. In the case of rutile, the crystal field splitting from the octahedral coordination can be seen as the splitting of t2g and eg orbitals in the conduction band.

Owing to the incorporation of HF exchange, the B3LYP functional is known to give bandgap results that are a significant improvement over the LDA and GGA approaches, the latter well-known for considerably underestimating the bandgap.73 The bandgap of rutile is predicted here to be 3.40 eV, comparable to the experimental values74,75 of 3.00−3.16 eV. As anticipated, the B3LYP bandgap is closer to experimental data than GGA predictions.22,24 The present rutile bandgap value is similar to the HSE06 hybrid functional prediction24 of 3.2 eV but much better than the 4.50 eV value obtained with the standard hybrid PBE0 functional.68 The bandgap is predicted to widen for Pca21 and OI to 4.28 and 4.20 eV, respectively, whereas for cotunnite the bandgap remains about the same. The bandgap obtained for Pca21 is rather different from that predicted for “cubic TiO2” earlier.8 Widening of the bandgap for the Pbca and OI structures was reported using the GGA,22 although the predicted band gap values are much smaller.

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CONCLUSIONS

Following a comprehensive analysis of the elastic data (P−V EoS) and pressure-dependent 0 K stability of experimentally observed and theoretically proposed dense TiO2 phases (rutile, columbite, baddeleyite, OI, cotunnite, fluorite, pyrite, and Pca21), the crystal structures, P−V EoS, and pressure stability up to 100 GPa of the phases were calculated using the hybrid B3LYP exchange-correlation functional and two independent Gaussian-type basis sets. The B3LYP results are in good agreement, overall, with experimental data on the structures and elastic properties. The B3LYP results are also superior to the results obtained with projector-augmented wave-/pseudopotential-based planewave GGA functionals in regard to the elastic properties of most TiO2 phases with the exception of fluorite. Our comparative analysis using the two independent AE-LCAO Gaussian-type basis sets shows significant sensitivity of fluorite structure to the choice of Ti and O basis functions. The present results allow resolution of the discrepant elastic data reported earlier for fluorite TiO2.13 In view of the present results and other published data, it can now be concluded that the bulk modulus value reported originally1 for cotunnite TiO2 is too large. The phase stability sequence with increasing 8623

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pressure predicted by B3LYP is rutile → columbite → baddeleyite → OI → cotunnite, in absolute agreement with available experimental data. The B3LYP total energy for the recently proposed Pca21 structure is very close to one or more of columbite, baddeleyite, OI, and cotunnite in the pressure range of 20−50 GPa, suggesting potential stabilization of the Pca21 structure at elevated temperatures. As anticipated, B3LYP, in combination with either set of basis function, provides highly satisfactory predictions of the TiO2 electron density of states and bandgaps.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Computations were carried out at the National Computational Infrastructure − National Facility of Australia (http://nf.nci. org.au/). V.S. is grateful to the Advanced Engineering Program, Monash University Malaysia, for partial financial support.

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REFERENCES

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