The Stability, Electronic Structure, and Optical Property of TiO2

Article pubs.acs.org/JPCC

The Stability, Electronic Structure, and Optical Property of TiO2 Polymorphs Tong Zhu and Shang-Peng Gao* Department of Materials Science, Fudan University, Shanghai 200433, P. R. China ABSTRACT: Enthalpies of nine TiO2 polymorphs under different pressures are presented to study the relative stability of the TiO2 polymorphs. It is important to include the dispersion correction for van der Waals interaction and the Hubbard U term for the Ti 3d orbital in the DFT-GGA calculation to correctly reproduce the relative stability of rutile, brookite, and anatase. Band structures for the TiO 2 polymorphs are calculated by density functional theory with generalized gradient approximation, and the band energies at high symmetry k-points are corrected using the GW method to accurately determine the band gap. The differences between direct band gap energies and indirect band gap energies are very small for rutile and columbite-structured TiO2, indicating a quasi-direct band gap character. For optical response calculations, two-particle effects have been included by solving the Bethe− Salpeter equation for Coulomb correlated electron−hole pairs. TiO2 with baddeleyite, pyrite, and fluorite structures has optical transitions in the violet light region. structured metal oxides at 0 GPa can be transformed to fluoritestructured phases at high pressure and Rietveld refinement of X-ray diffraction data from three rutile-structured oxides (SnO2, PbO2, and RuO2) revealed that the high-pressure phase in these systems actually adopts a pyrite structure,29 it has been postulated that this cubic phase of TiO2 adopts a fluorite or pyrite structure. Whether or not there are other possible structures is still an open question. For the electronic and optical properties of TiO 2 polymorphs, the experimental studies are mainly about the mineral phases. The electronic band structure of rutile has reported values of 3.3 ± 0.5 eV30 (photoemission spectroscopy (PES) and inverse photoemission spectroscopy (IPES)) and 3.6 ± 0.2 eV31 (PES and IPES for rutile (110) surface). Hardman et al. reported the measurement of the valence-band structure of rutile along the Γ−Δ−X and Γ−Σ−M directions by PES.32 For anatase and brookite, there are no reported measurements of the electronic band gap from combined PES and IPES measurements. The optical band gap or direct exciton energy have been measured by absorption, photoluminescence, and Raman-scattering techniques.33−44 The electronic and optical properties of doped TiO2 and the influence of dopant on photocatalytic properties have been investigated by DFT.45−47 It has been generally observed for semiconductors and insulators that the band gap is underestimated in the density functional theory (DFT) calculations with local density approximation (LDA) or generalized gradient approximation

1. INTRODUCTION Even after half a century of research,1,2 investigation of the fundamental properties of TiO2 crystal phases remains very important due to their important role in effectively utilizing solar energy, for instance, photocatalytic splitting of water into H2 and O2,3 photovoltaic generation of electricity,4 degradation of environmentally hazardous materials,5,6 and reduction of CO2 into hydrocarbon fuels.7 The gap between valence and conduction bands and the optical absorption property are vital to all of these applications. TiO2 exists in many polymorphs. Among them, rutile, anatase, and brookite8 are well-known minerals in nature. Furthermore, TiO2 has a rich phase diagram at elevated pressure. High pressure X-ray diffraction9−14 and Raman spectroscopy15−20 studies have proved that rutile and anatase can be transformed to a columbite (α-PbO2) phase at high pressure. Recent X-ray diffraction studies reported that the columbite phase is only formed at about 7 GPa during decompression from a higher pressure phase,12−14 while the transformation of rutile and anatase directly to columbitestructured TiO2 has been observed at 5 GPa15−20 in Raman studies. Columbite-structured TiO2 was also discovered in the suevite from the Ries crater in Germany.21 It has been found that columbite-structured TiO2 is transformed to a baddeleyite structure between 12 and 17 GPa in X-ray diffraction12,22 and Raman17,19 studies. However, calculations indicated the transition pressure to be 2623 or 31 GPa.24 A TiO2 polymorph with cotunnite structure was observed at pressures higher than 37.4 GPa.25−27 A number of observations suggested a transformation to a cubic phase at about 60 GPa but without sufficient data available to fully determine the structure.19,28 However, because it has been known that several rutile© 2014 American Chemical Society

Received: December 19, 2013 Revised: April 18, 2014 Published: May 7, 2014 11385

dx.doi.org/10.1021/jp412462m | J. Phys. Chem. C 2014, 118, 11385−11396

The Journal of Physical Chemistry C

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(GGA) for the exchange correlation functional.48,49 Ab initio many-body perturbation theory with GW approximation is regarded as an accurate way to predict the band structure.48,50−52 Available band gap data calculated by the GW method in the literature for TiO2 polymorphs are listed in Table 1.8,53−59 There are still no theoretical band structures for

2. COMPUTATIONAL METHODS The planewave pseudopotential method is employed for structure optimization and ground state electronic structure calculation. The Perdew−Burke−Ernzerhof GGA functional is used to describe the exchange-correlation potential.63 In the determination of structure and enthalpy under different pressures, two different approaches are employed: one is the standard ab initio DFT-GGA and the other adopts the Grimme scheme for semiempirical dispersive interactions64−66 and an effective Hubbard U term of 2.5 eV for the Ti 3d orbital (referred to as DFT-GGA+D+U in the following). Following the work of Albuquerque et al.,65,66 a van der Waals (vdW) radius of 1.4214 Å is chosen for Ti and a default vdW radius of 1.3420 Å is chosen for oxygen. Structural optimization is carried out using the Broyden−Fletcher−Goldfarb−Shanno minimization (BFGS).67 All pseudopotentials in this work are generated using the OPIUM package68 in the Troullier−Martins scheme.69 For the pseudopotential of the Ti, it has been reported that dividing the n = 3 shell of Ti into frozen core (3s, 3p) and valence (3d) contributions introduces a significant error into the band gap energy.54 Therefore, 3s, 3p, 3d, 4s, and 4p are treated as valence states for Ti. The plane wave cutoff energy is set to 1633 eV (60 hartree) in the DFT calculation. The Monkhorst−Pack scheme is used for the k-point sampling in the first Brillouin zone. A 6 × 6 × 8 k-point grid is chosen for rutile. A 6 × 6 × 3 k-point grid is chosen for anatase. A 4 × 4 × 4 k-point grid is chosen for TiO2 polymorphs with brookite, columbite, baddeleyite, and pyrite structures. A 4 × 6 × 4 k-point grid is chosen for cotunnite-structured TiO2. A 12 × 12 × 12 k-point grid is chosen for fluorite-structured TiO2 with a primitive cell adopted in the calculation. A 6 × 6 × 4 k-point grid is chosen for the tridymite phase. Band energies at high symmetry k-points are corrected using the standard one-shot G0W0 method.48,51,52 Following the standard approach, Kohn−Sham eigenvalues and eigenfunctions will be first obtained by DFT-GGA calculation and then used as a starting point to do the GW correction.48,51,52 In order to include all the high symmetry k-points and consider the calculation efficiency, 4 × 4 × 4 and 6 × 6 × 2 k-points are both chosen for tridymite-structured TiO2. The screening in the GW calculation is treated with the plasmon pole approximation.48,52 The dielectric matrix is evaluated at an imaginary frequency of 13.6 eV for all TiO2 polymorphs. The polarization function, which is necessary to evaluate the screened interaction, is calculated within the random phase approximation. The numbers of bands used to calculate the screening and the self-energy in the GW method are chosen to be 344 for rutile, 384 for anatase and columbite-structured TiO2, 512 for brookite and pyrite-structured TiO2, 560 for baddeleyite-structured TiO2 and cotunnite-structured TiO2, 332 for fluorite-structured TiO2, and 320 for tridymitestructured TiO2. The cutoff energy of the planewave is set to 490 eV (18 hartree) to represent the independent particle susceptibility and the dielectric matrix and to generate the exchange part of the self-energy operator. In order to get the excitation properties, such as the absorption of light, the interaction between the excited electron and the hole is included by solving the Bethe−Salpeter equation.70−73 The Tamm−Dancoff approximation that neglects the two off-diagonal coupling parts in the two-particle Hamiltonian is applied. The Haydock iterative technique is

Table 1. Theoretical Electronic Band Gaps Calculated by the GW Method Reported in the Literaturea EgapGW

TiO2 structure rutile anatase brookite columbite baddeleyite cotunnite pyrite fluorite

b

3.34(I)/3.38(D), 3.59(D),c 3.40(D),d 3.46(D),e 3.73f 3.56,b 3.83,c 3.70,d 3.73,e 4.05,f 3.79g 3.45,e 3.68f not given not given not given not given 2.367(2.369)h

For rutile, “D” in the bracket means direct band gap energy and (I) in the bracket means indirect band gap energy. b(LDA+GW).54 c(PBE +GW).55 d(DFT+U+GW).56 e(PBE+GW). 57 f(HSE06+GW).57 g (PBE+GW).58 h(PBE+GW).59 a

columbite and baddeleyite phases of TiO2. Mattesini et al. demonstrated the possible optical transition in the visible light region for pyrite-structured and fluorite-structured TiO2 from the DFT calculation.60 The GW method59 and quantum Monte Carlo61 studies show that fluorite TiO2 has a band gap energy lying in the visible light region, though the GW calculation for pyrite is still absent. There are still no experiments that reported the visible light absorption for these two materials up to now. The optical absorption process accompanies the generation of electron−hole pairs. An accurate way to treat the electron− hole interaction is solving the Bethe−Salpeter equation (BSE) for Coulomb correlated electron−hole pairs. Kang and Hybertson54 as well as Chiodo et al.55 investigated the optical excitation energies of rutile and anatase TiO2. Lawler et al. investigated the birefringence of rutile and anatase phase TiO2.62 Landmann et al. have calculated the three natural occurring TiO2 polymorphs rutile, anatase, and brookite.57 MBPT calculations of band structure and optical property for newly proposed structures, such as columbite, baddeleyite, cotunnite, and pyrite structures, are still lacking (Table 1). In this paper, a new possible TiO2 polymorph with tridymite structure is proposed. For the convenience of discussion, we classified TiO2 polymorphs in three categories: phases that have been found in minerals: the well-known rutile, anatase, and brookite, and the less well-known columbite-structured TiO2 which has been found in shocked garnet gnersses, high pressure phases that have been reported in the literature (baddeleyite, cotunnite, pyrite, and fluorite), and a new tridymite-structured TiO2 phase we proposed in this paper. The structure optimization and electronic band structure calculations are carried out for these nine TiO2 polymorphs. The GW method is adopted to calculate the band energies at high-symmetry kpoints in the first Brillouin zone. The BSE method is used to calculate the optical absorption spectrum. The GW band gap energies and the optical absorption spectrum based on the BSE method would be helpful for future research work on electrical, optical, and transport properties of these TiO2 polymorphs. 11386



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Figure 1. Polyhedra structures for the TiO2 polymorphs: (a) rutile, (b) anatase, (c) brookite, (d) columbite, (e) baddeleyite, (f) cotunnite, (g) pyrite, (h) fluorite, and (i) tridymite. Ti and O atoms are represented by big blue and small red spheres, respectively.

3. RESULTS AND DISCUSSION 3.1. Structure and Relative Stability of TiO2 Polymorphs. The polyhedral structure of TiO2 polymorphs investigated in this work is given in Figure 1. By searching possible structures of AB2 types, we proposed a new TiO2 polymorph with tridymite (a high-temperature polymorph of quartz) structure. Phonon DOS of tridymite-structured TiO2 are displayed in Figure 2 and all the phonon modes have

used to obtain the macroscopic dielectric function by iterative applications of the Hamiltonian on a set of vectors in the electron−hole space. The Coulomb term of the BS Hamiltonian is evaluated using the truly nonlocal screening function W. In the BSE calculation, we use the Kohn−Sham eigenvalues and wave functions to construct the transition space. To consider the effect of the self-energy correction, the Kohn−Sham energies are corrected by a scissors operator with its energy given by the difference between the direct band gap calculated with the DFT-GGA and GW methods at the k-points where the valence band maximum (VBM) locates. This permits us to avoid a cumbersome GW calculation for each state included in the transition space. A complex shift of 0.15 eV is used to avoid divergences in the expression for the macroscopic dielectric function. ABINIT52,74,75 code is employed for band structure and optical property calculation. Because the DFT+U has not been implemented for the norm-conserving pseudopotential in the ABINIT code, Quantum Espresso76 code is used to calculate the structure and enthalpy. We have used the same set of pseudopotentials in both ABINIT and Quantum Espresso for consistency. The phonon density of states (DOS) of tridymitestructured TiO2 are calculated by DFT-GGA using the CASTEP code77 within Materials Studio 6.1 Package. The norm-conserving pseudopotential is required for linear response calculations or finite displacement calculations that require LO−TO splitting correction in the CASTEP code.

Figure 2. Phonon density of states for tridymite-structured TiO2. The inset shows the enlarged figure from −3.0 to 3.0 THz.

positive (real) frequencies, indicating that tridymite-structured TiO2 can be mechanically stable. The tridymite-structured TiO2 belongs to space group P63/mmc. Each Ti ion is tetrahedrally coordinated to four O ions. Geometry optimized structures at zero pressure for the nine TiO2 polymorphs are listed in Table 2. The structures of the 11387


a

11388

a = b = 5.900,b 5.922c; c = 9.620,b 9.655c

tridymite P63/mmc (194)

Available experimental data are also listed for comparison. bDFT-GGA. cDFT-GGA+D+U.

a = b = c = 4.787,b 4.779c (conventional cubic cell)

Ti 4c (x, 1/4, z) x = 0.2577c; z = 0.0960c O1 4c (x, 1/4, z) x = 0.3653c; z = 0.4124c O2 4c (x, 1/4, z) x = 0.4937c; z = 0.8494c Ti 4a (0, 0, 0) O 8c (x, x, x) x = 0.3411,b 0.3427c Ti 4a (0, 0, 0) O 8c (1/4, 1/4, 1/4) Ti 4f (1/3, 2/3, z) z = 0.4377,b 0.4377c O1 2c (1/3, 2/3, 1/4) O2 6g (1/2, 1/2, 0)

Ti 4e (x, y, z) x = 0.2755,b 0.2780c; y = 0.0584,b 0.0493c; z = 0.2177,b 0.2089c O1 4e (x, y, z) x = 0.0607,b 0.0714c; y = 0.3180,b 0.3369c; z = 0.3571,b 0.3393c O2 4e (x, y, z) x = 0.4499,b 0.4417c; y = 0.7587,b 0.7599c; z = 0.4550,b 0.4758c

Wyckoff positions Ti 2a (0, 0, 0) O 4f (x, x, 0) x = 0.3052,b 0.3045c, 0.30572 Ti 4a (0, 0, 0) O 8e (0, 0, z) z = 0.2068,b 0.2087,c 0.208174 Ti 8c (x, y, z) x = 0.1290b, 0.1293c; y = 0.0906,b 0.1007c; z = 0.8623,b 0.8626c O1 8c (x, y, z) x = 0.0100,b 0.0099c; y = 0.1483,b 0.1497c; z = 0.1833,b 0.1833c O2 8c (x, y, z) x = 0.2298,b 0.2316c; y = 0.1081,b 0.1125c; z = 0.5365,b 0.5369c Ti 4c (0, y, 1/4) y = 0.1779,b 0.1707c O 8d (x, y, z) x = 0.2708,b 0.2696c; y = 0.3803,b 0.3840c; z = 0.4191,b 0.4196c

structure parameters a = b = 4.618,b 4.581,c 4.587 (15 K),72 4.593 (295 K),72 4.594 (298 K)73 c = 2.948,b 2.979,c 2.954 (15 K),72 2.959 (295 K),72 2.958 (298 K)73 a = b = 3.780,b 3.801,c 3.782 (15 K),72 3.785 (295 K),72 3.7842 (298 K)74 c = 9.628,b 9.515,c 9.502 (15 K),72 9.512 (295 K),73 9.5146 (298 K)74 a = 9.207,b 9.173,c 9.18475 b = 5.477,b 5.453,c 5.44775 c = 5.140,b 5.159,c 5.14575 a = 4.553,b 4.528,c 4.541,12 4.53521 b = 5.546,b 5.503,c 5.493,12 5.49921 c = 4.897,b 4.913,c 4.906,12 4.90021 a = 4.827,b 4.787c b = 4.876,b 4.890c c = 5.081,b 4.994c β = 100.192,b 99.471c a = 5.166c b = 3.170c c = 6.301c a = b = c = 4.844,b 4.834c

fluorite Fm3̅m (225)

pyrite Pa3̅ (205)

cotunnite Pnma (62)

baddeleyite P21/c (14)

columbite Pbcn (60)

brookite Pbca (61)

anatase I41/amd (141)

rutile P42/mnm (136)

TiO2 structure and space group

Table 2. Space Group, Lattice Constants (in Å) and Angles (in deg), Multiplicities, and Wyckoff Letters of Nonequivalent Atoms and the Corresponding Relative Atomic Coordinates for TiO2 Polymorphsa

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Figure 3. Enthalpies (in eV, for one TiO2 formula unit) of TiO2 polymorphs under different hydrostatic pressures calculated by DFT-GGA (a) and DFT-GGA+D+U (b). The inset shows the enthalpy between 0 and 11 GPa.

pressure, our calculated enthalpy order using the standard DFT-GGA method, the same as DFT-GGA calculations published previously by other people,82,83 is anatase < brookite < rutile (Figure 3a), which is in disagreement with experimental results.84,85 After adding the semiempirical dispersion correction term for the van der Waals interaction and the Hubbard U term for the Ti 3d orbital, an enthalpy order agreeing with the experiment, rutile < brookite < anatase, is obtained (Figure 3b). Relative to rutile, brookite is 1.65 kJ/mol and anatase is 4.41 kJ/mol higher in enthalpy from the DFT-GGA+D+U calculation, that is, close to the enthalpy of formation reported in the CRC Handbook of Chemistry and Physics:84 antase is 4.85 kJ/mol (1.16 kcal/mol) higher than rutile at 0 K and 5.02 kJ/mol (1.2 kcal/mol) higher than rutile at 298.15 K, brookite is 2.09 kJ/mol (0.5 kcal/mol) higher than rutile at 298.15 K but larger than the 0.71 kJ/mol for brookite and 2.61 kJ/mol for anatase measured by high temperature oxide melt drop solution calorimetry.85 The enthalpy (energy) of tridymite-structured TiO2 at zero pressure is lower than rutile from standard DFT-GGA calculation but is between fluorite-structured TiO2 and pyritestructured TiO2 from the DFT-GGA+D+U calculation. At elevated pressure, tridymite-structured TiO2 becomes less stable. This indicates that the newly proposed tridymitestructured TiO2 cannot be obtained from high pressure phase transformation from the nature minerals. From the DFT-GGA (DFT-GGA+D+U) calculations, the transformation from the columbite structure to the baddeleyite structure occurs at about 11.3 GPa (12.8 GPa), that is, in reasonable agreement with experimental observations that the columbite structure transforms to the baddeleyite structure between 12 and 17 GPa.12,17,19,22 The phase transformation from the baddeleyite structure to the cotunnite structure occurs at about 40.9 GPa (39.8 GPa), agreeing with the experiment

rutile and brookite structures calculated by DFT-GGA are in reasonable agreement with the available experimental data.78−80 However, the value of c for the anatase (9.628 Å) is about 1.3% larger than the experimental one (9.502 Å).78 This relative larger discrepancy for the value of c is also found in other calculations.81,82 Recent DFT calculations performed with inclusion of semiempirical dispersion correction of the Grimme scheme on TiO2 by Albuquerque et al. solved this problem and gave an anatase structure in good agreement with the experiment.65,66 Our calculation after adding the semidispersion correction term for the van der Waals interaction and the Hubbard U term for the Ti 3d orbital also gives a lattice parameter (c = 9.515 Å) agreeing well with the experiment data (c = 9.502 Å). For consistency, theoretical lattice parameters and atomic positions calculated by DFT-GGA+D+U (Table 2) are used in the band structure and optical property calculation in this paper. Experimental lattice parameters at high pressures for baddeleyite-structured TiO2 and cotunnite-structured TiO2 have been measured.22,25,26 Our calculated lattice parameters are very close to those reported by Nishio-Hamane et al.25 at a series pressure. For example, at 22.5 GPa, the discrepancy between DFT-GGA+D+U calculated lattice parameters (a = 4.579 Å, b = 4.854 Å, c = 4.703 Å, β = 97.73°) for baddeleyitestructured TiO2 and experimental data (a = 4.587 Å, b = 4.843 Å, c = 4.720 Å, β = 97.89°)25 is less than 0.4%. At 59.6 GPa, DFT-GGA+D+U calculated parameters for cotunnite-structured TiO2 (a = 5.007 Å, b = 2.938 Å, c = 5.864 Å) also agree very well with experimental data (a = 5.028 Å, b = 2.930 Å, c = 5.889 Å).25 The discrepancies between DFT-GGA and DFTGGA+D+U calculated lattice parameters are trivial in both cases. Enthalpies of the TiO2 polymorphs under hydrostatic pressure from 0 to 70 GPa are shown in Figure 3. At zero 11389



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Table 3. Band Energies (in eV) at Special k-points in the First Brillouin Zones of TiO2 Polymorphs Calculated by the DFTGGA and GW Methodsa rutile

anatase

DFT-GGA Z A M Γ R X

GW

Ev

Ec

Ev

Ec

−1.44 −0.69 −1.09 0 −1.05 −0.75

2.95 2.57 1.89 1.85 1.89 2.50

−1.39 −0.54 −0.97 0.24 −0.95 −0.62

4.61 4.18 3.57 3.54 3.47 4.15

Z Γ Δ X P N

GW

Ec

Ev

Ec

−0.16 −0.48 0 −0.05 −0.24 −0.66

2.42 2.07 3.19 3.29 3.51 3.17

−0.06 −0.43 0.11 0.06 −0.12 −0.63

4.11 3.68 5.12 5.34 4.92

Ev Γ Z T Y S X U R

Ec

0 −0.18 −0.27 −0.16 −0.47 −0.56 −0.66 −0.51

baddeleyite GW

DFT-GGA

Ev

Ec

Ev

Ec

0 −0.28 −0.50 −0.23 −0.33 −0.11 −0.40 −0.78

2.63 2.59 2.93 3.08 2.96 2.72 2.84 2.99

0.31 −0.02 −0.26 0.09 −0.05 0.19 −0.14 −0.54

4.46 4.40 4.84 4.95 4.89 4.60 4.70 4.85

Z Γ Y A B D E C

GW

Ev

Ec

Ev

Ec

2.58 2.38 2.81 2.68 2.15 2.30 2.57 2.59

−0.16 0.27 −0.20 −0.56 0.15 −0.34 −0.77 −0.54

4.33 4.01 4.52 4.42 3.78 3.98 4.22 4.33

Ev Γ Z T Y S X U R H

DFT-GGA

GW

DFT-GGA

Ec

Ev

Ec

−0.27 −1.38 −0.91 0

2.25 1.48 2.15 1.78

0.06 −1.21 −0.60 0.35

3.93 3.04 3.74 3.38

W L Γ X K

GW

Ec

Ev

Ec

−0.24 −2.56 −0.88 0 −0.45

1.91 1.51 1.16 1.86 1.90

−0.02 −2.57 −0.75 0.25 −0.19

3.48 3.07 2.64 3.25 3.40

Γ A H K M L

4.21 4.55 4.63 4.33 4.79 4.67 4.78 4.83 GW

Ev

Ec

2.02 0.19 2.38 −0.59 2.45 0.01 2.31 0.19 2.83 −0.73 2.48 −0.02 2.40 −0.17 2.80 −0.62 2.19 0.30 tridymite

DFT-GGA

Ev

Ec

2.38 0.35 2.64 0.15 2.70 0.04 2.46 0.20 2.81 −0.17 2.70 −0.21 2.85 −0.37 2.89 −0.20 cotunnite

Ec

−0.14 −0.88 −0.30 −0.10 −0.96 −0.32 −0.44 −0.90 0

fluorite

Ev

GW Ev

DFT-GGA

−0.41 0 −0.46 −0.77 −0.11 −0.57 −0.96 −0.78

pyrite

X R M Γ

DFT-GGA

Ev

columbite DFT-GGA Γ Z T Y S X U R

brookite

DFT-GGA

3.52 3.91 4.13 3.85 4.46 4.09 3.98 4.40

GW

Ev

Ec

Ev

Ec

0 0 −0.46 −0.14 0 −0.25

3.22 3.48 4.07 3.65 3.63 3.76

−0.02 −0.02 −0.61 −0.20 −0.04 −0.34

5.65 5.90 6.47 6.06 6.04 6.15

a

The energies of the valence band maximum and the conduction band minimum are indicated by bold font. The valence band maximum from DFTGGA calculation is set to 0 eV.

that the cotunnite phase appeared at 37.4 GPa.25 The cotunnite phase is the most stable phase above 40.9 GPa (39.8 GPa) to at least 70 GPa, in agreement with the experimental observation.25 At about 29 GPa (23 GPa), the enthalpy−pressure diagram of brookite implies a phase transition, accompanied with an abrupt change of lattice constant and internal atomic positions. Structure data shows that the structure still belongs to space group Pbca (no. 61) after the transition. At about 46 GPa (55 GPa), the enthalpy−pressure diagram of columbite also implies a phase transition. The enthalpy−pressure diagram labeled as “columbite” in Figure 3 above 46 GPa (55 GPa) actually corresponds to a new phase with space group P42/nmc (no. 137) with Ti occupying the 2a and O occupying the 4c Wyckoff positions. 3.2. Electronic Band Structure and Density of States. The band energies at high symmetry k-points from both DFTGGA and GW calculations are listed in Table 3. For easy comparison, the band gaps calculated by the DFT-GGA and GW methods are listed in Table 4. First, we look at the electronic band structures and DOS for the four mineral phases (Figure 4). The projection-sphere radii in the analysis of local DOS and its angular-momentum projections are 1.2 Å (2.30 bohr) for Ti and 0.75 Å (1.41 bohr) for O. The DFT-GGA

Table 4. Electronic Band Gaps (in eV) for TiO2 Polymorphs with Rutile, Anatase, Brookite, Columbite, Baddeleyite, Pyrite, Fluorite, Cotunnite, and Tridymite Structures Calculated by the DFT-GGA and GW Methods EgapDFT rutile anatase brookite columbite baddeleyite cotunnite pyrite fluorite tridymite

1.85 2.07 2.38 2.59 2.15 2.02 1.48 1.16 3.22

(D) (I) (D) (I) (I) (I) (I) (I) (D)

EgapGW 3.23 3.57 3.86 4.09 3.51 3.22 2.69 2.39 5.67

(I) (I) (D) (I) (I) (I) (I) (I) (D)

band structure of rutile (Figure 4a) indicates that it has a direct band gap of 1.85 eV at the Γ point. However, with GW correction, rutile has an indirect band gap of 3.23 eV with the VBM at Γ and the conduction band minimum (CBM) at R. The lowest conduction band energy at R is slightly lower than that at Γ by about 0.07 eV. Kang and Hybertsen54 have found the same phenomenon and reported an indirect GW band gap 11390



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Figure 4. Band structure and the corresponding DOS of (a) rutile, (b) anatase, (c) brookite, and (d) columbite calculated by DFT-GGA. Yellow points indicate the values obtained with the GW method. The valence band maximum from the DFT-GGA calculation is set to 0 eV.

Figure 5. Band structure and the corresponding DOS calculated by DFT-GGA for four TiO2 polymorphs that have been proposed in high-pressure studies: (a) baddeleyite, (b) cotunnite, (c) pyrite, and (d) fluorite. Yellow points indicate the values obtained with the GW method. The valence band maximum is set to 0 eV.

of 3.34 eV and a direct band gap of 3.38 eV at Γ. The direct GW band gap of 3.30 eV at Γ from our calculation agrees well with the experimental result of 3.3 ± 0.5 eV measured by PES and IPES.30 Note that it is the primitive cell employed in the band structure calculation for anatase, so the high-symmetry k-point labels are similar to that used by Labat et al.53 and Catatayud et al.81 but are different from the high symmetry k-point labels corresponding to the conventional unit cell.8,57 DFT-GGA calculation shows that anatase has an indirect band gap with the VBM at Δ (0, 0, 0.44) between Γ and X (0, 0, 0.5) and the CBM at Γ (Figure 4b). The indirect GW band gap from X to Γ is 3.62 eV. A minimum band gap of 3.57 eV can be deduced approximately from GW band energies at X and the DFT band dispersion from X to Δ. Brookite has a direct band gap at Γ (Figure 4c), and its GW band gap (3.86 eV) is larger than that of anatase (3.57 eV) and rutile (3.30 eV). Our calculation shows that columbite-structured TiO2 has an indirect band gap with the VBM at Γ and CBM at Z (Figure 5a). From the GW calculation, the indirect band gap for columbite is 4.09 eV and

the minimum direct band gap at Γ is 4.15 eV. The direct band gap is only 60 meV larger than the indirect band gap. There are still no electronic band structure data reported for columbitestructured TiO2 in the literature. TiO2 polymorphs with baddeleyite, cotunnite, pyrite, and fluorite structures have been studied in the literature as high pressure phases. GW calculation indicates that the minimum band gap of the baddeleyite TiO2 is indirect with the VBM at Γ and the CBM at B (Table 2). The GW band gap of the baddeleyite TiO2 is 3.51 eV. The VBM of cotunnite-structured TiO2 is at a point H (0.35, 0, 0) between Y and Γ. The GW energy at the H point is inferred from the DFT-GGA energy at H and the GW corrections at Y and Γ. The minimum GW band gap of cotunnite-structured TiO2 from H to Γ is 3.22 eV. Parts c and d of Figure 5 show the band structure of two TiO2 polymorphs with a cubic lattice: pyrite-structured TiO2 and fluorite-structured TiO2. Pyrite-structured TiO2 has an indirect band gap with VBM at Γ and CBM at R, and its DFTGGA band gap of 1.48 eV is close to the DFT-LDA band gap of 1.44 eV reported by Mattesini et al.60 The band gap calculated 11391



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close to those at A and M points. The band gap calculated by the GW method is 5.67 eV, larger than the other TiO2 polymorphs studied in this paper. Tridymite-structured TiO2 could not have any potential application in visible light photocatalysis, as its band gap is too large. Because it has a direct band gap, tridymite-structured TiO2 might have some potential application in optoelectric devices that require an ultraviolet wavelength corresponding to its band gap. 3.3. Optical-Absorption Properties. Real and imaginary parts of the frequency-dependent macroscopic dielectric function ε(ω) calculated by the Bethe−Salpeter equation for four mineral TiO2 phases (rutile, anatase, brookite, and columbite-structured TiO2) are presented in Figure 7a-d. The dielectric functions for rutile and anatase, which are tetragonal structure, are resolved into two componentsthe in-plane component E⊥c and the out-of-plane component E∥cto study their optical anisotropy. Three dielectric components parallel to the a, b, and c axes are resolved for the brookite and columbite-structured TiO2 that have an orthorhombic crystal lattice. Compared with experimental spectra obtained by means of spectroscopic elipsometry for the imaginary part of the dielectric function of rutile,87 the peaks below 6 eV from the calculated results agree well with the experimental data both for the location and for the amplitude, whereas the peaks above 6 eV red shift slightly (0.2−0.4 eV) and the amplitude is higher comparing to the experimental data. For the real part of the dielectric function, the value of Re ε(ω = 0) is very close to the experimental data. This justifies the reliability of the macroscopic static dielectric constants in Table 5 obtained from Re ε(ω = 0). For anatase, the main features in the experimental spectra88 are also reproduced. For columbite-structured TiO2, a very sharp peak appears on the absorption threshold of the imaginary dielectric function for E∥a. As many optical experiments actually probe the absorption coefficients, the normal-incidence absorption coefficients are also calculated. The experimental absorption coefficients for rutile and anatase are derived from the experimental dielectric functions87,88 (Figure 8). The agreement near the absorption threshold is poor due to a constant value (0.15 eV) giving the

by the GW method is 2.69 eV for pyrite-structured TiO2. DFTGGA calculation shows fluorite-structured TiO2 has an indirect band gap of 1.16 eV with the VBM at X and CBM at Γ (Figure 5d), comparable with the DFT-LDA value of 1.44 eV60 and DFT-GGA value of 1.04 eV.59 The indirect band gap from X to Γ for fluorite-structured TiO2 from GW calculations is 2.39 eV, about 0.3 eV smaller than the pyrite-structured TiO2. Among the nine TiO2 polymorphs studied, the band gap of TiO2 with pyrite and fluorite structures meets the requirement of a photocatalyst candidate with visible light catalytic activity: 2− 3.1 eV.86 The tridymite-structured TiO2 has a direct band gap at Γ (Figure 6a). The maximum valence band energy at Γ is very

Figure 6. Electronic states and optical absorption property of a newly proposed tridymite-structured TiO2: (a) the band structure and the corresponding DOS calculated by DFT-GGA and the values obtained with the GW method are indicated by the yellow points; (b) imaginary part of the complex dielectric function calculated by solving the Bethe−Salpeter equation.

Figure 7. Polarization-dependent real and imaginary parts of the complex dielectric function calculated by solving the Bethe−Salpeter equation for mineral TiO2 polymorphs: (a) rutile, (b) anatase, (c) brookite, and (d) columbite. The experimental dielectric functions (ref 87 for rutile and ref 88 for anatase) are shown for comparison. 11392



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Table 5. Calculated Macroscopic Static Dielectric Constants of TiO2 Polymorphs rutile

anatase

brookite

columbite

baddeleyite

cotunnite

pyrite

fluorite

tridymite

5.71 (E⊥c)

5.12 (E⊥c)

7.00 (E∥a) 7.51 (E∥b) 6.53 (E∥c)

8.14 (E∥a) 7.42 (E∥b) 6.71 (E∥c)

9.12

2.39 (E⊥c)

4.98 (E∥c)

6.59 (E∥a) 4.79 (E∥b) 6.03 (E∥c)

7.78

7.33 (E∥c)

5.31 (E∥a) 4.28 (E∥b) 4.40 (E∥c)

2.40 (E∥c)

Figure 8. Absorption coefficients for (a) rutile, (b) anatase, and (c) brookite. The absorption coefficients derived from experimental dielectric functions (ref 87 for rutile and ref 88 for anatase) are shown for comparison.

baddeleyite, the absorption threshold energy of the E∥b component is lower than that of the E∥a and E∥c components. The first absorption peak A locates at 3.36 eV, that is, smaller than the band gap of 3.51 eV, implying an excitonic nature. Although the band gap energy of the baddeleytie-structured TiO2 is higher than the visible light range, there is a considerable optical transition probability at 3.1 eV due to the attractive interaction of electron and hole. It can be found that pyrite-structured TiO2 and fluorite-structured TiO2 show optical absorption in the visible light range. Both pyritestructured TiO2 and fluorite-structured TiO2 have a first absorption peak at about 3.3 eV (peak A). The prominent peak A of fluorite-structured TiO2 is separated from leading features B and C. Although the visible light absorption of pyritestructured TiO2 and fluorite-structured TiO2 can be deduced from band structure, the absorption spectra are still essential because these can be directly compared with experimental light absorption spectra and this gives valuable information on optical transition probability corresponding to specific light wavelength. The imaginary parts of dielectric functions (absorption spectra) for tridymite-structured TiO2 are shown in Figure 6b. The optical absorption spectra can serve as a good reference for structure characterization. The absorption threshold energies for tridymite-structured TiO2 are higher than those of the other TiO2 polymorphs shown in Figures 7 and 9, and the spectra feature differences can be easily recognized. The tridymite structure has a hexagonal lattice, and anisotropic optical absorption can be identified from the in-plane (E⊥c) and out-of-plane (E∥c) components.

complex shift to avoid divergences in the continued fraction in the iterative Haydock technique to calculate the macroscopic dielectric function, and this value mimics the experimental broadening of the absorption peaks. The calculated imaginary parts of dielectric function (absorption spectrum) for TiO2 polymorphs with baddeleyite, cotunnite, pyrite, and fluorite structures are shown in Figure 9. There are still no relevant experimental spectra available for these phases found in high pressure studies. The baddeleyite structure has a monoclinic unit cell, and the cotunnite structure has an orthorhombic unit cell. Three components along the a, b, and c cell vector directions are given separately. For

4. CONCLUSIONS Standard DFT-GGA calculation cannot reproduce the correct order of stability between rutile, brookite, and anatase. By adding the semiempirical dispersion correction term for the van der Waals interaction and the Hubbard U term for the Ti 3d orbital, an enthalpy order agreeing with experiments can be obtained and the enthalpy differences are comparable with experimental results. The GW band gaps for nine TiO2 polymorphs are 3.23 eV for rutile, 3.86 eV for brookite, 3.57 eV for anatase, 4.09 eV for

Figure 9. Imaginary parts of the complex dielectric function calculated by solving the Bethe−Salpeter equation for TiO2 polymorphs with baddeleyite, cotunnite, pyrite, and fluorite structures. A tick at 3.1 eV is specially added to discern the optical transitions in the visible light range. 11393



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columbite-structured TiO2, 3.51 eV for baddeleyite-structured TiO2, 3.22 eV for cotunnite-structured TiO2, 2.69 eV for pyritestructured TiO2, 2.39 eV for fluorite-structured TiO2, and 5.67 eV for tridymite-structured TiO2. Among the nine TiO2 polymorphs, brookite and tridymite-structured TiO2 have a direct band gap. For rutile and columbite-structured TiO2, the direct band gap energies at Γ are very close to the indirect band gap energies, indicating a quasi-direct band gap character. Optical absorption properties are analyzed on the basis of the macroscopic dielectric function calculation by solving the twoparticle function BSE. We have shown that TiO2 polymorphs with baddeleyite, pyrite, and fluorite structures have optical transitions in the region of the violet visible light. The GW band gap energies and optical absorption spectra present in this article can serve as a guide in the promising applications for the TiO2 polymorphs.

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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 This work is supported by the State Key Development Program of Basic Research of China (Grant No. 2011CB606406). The computational resources utilized in this research are provided by Shanghai Supercomputer Center, and we would like to thank Dr. Tao Wang for his support in using the supercomputer facility.

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The Stability, Electronic Structure, and Optical Property of TiO2

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